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+{"text":"\\section{Lower Bound}\n\nTo prove the lower bound on the round complexity of approximating (weighted) diameter in the quantum CONGEST model, we combine the reduction in~\\cite{Elkin06,SarmaHKKNPPW12,ElkinKNP14} and the graph gadget in~\\cite{AbboudCK16}.\n\n\n\n\\subsection{Reduction from Server Model}\n\\label{sec:reduction}\n\nWe briefly outline the reduction introduced by Elkin et al.~\\cite{Elkin06,SarmaHKKNPPW12,ElkinKNP14} from the Server model to prove the hardness of certain graph problems such as diameter and radius.\nWe will introduce a distributed network $G=(V,E)$ and embed a certain two-argument function $F:\\{0,1\\}^k\\times\\{0,1\\}^k\\to\\{0,1\\}$ into the network by showing that if the instance on the network $G$ has a low round-complexity protocol in the quantum CONGEST model, then there exists a low communication complexity protocol for $F$ in the quantum Server model.\nThus, the hardness of diameters and radius in the quantum CONGEST model is reduced to proving the lower bounds the communication complexity in the quantum Server model.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\linewidth]{figures\/example.pdf}\n\\caption{An example of constructed graph $G$.}\n\\label{fig:example}\n\\end{figure}\n\nThe network $G=(V,E)$ is depicted by Figure~\\ref{fig:example} where $V=V_S\\uplus V_A\\uplus V_B$ and $E=E_S\\uplus E_A\\uplus E_B\\uplus E'$.\nWe use $G[U]$ to denote the subgraph induced by vertex set $U\\subseteq V$, then $E_S,E_A,E_B$ are the edges in $G[V_S],G[V_A],G[V_B]$ respectively.\nAnd $E'$ denotes the edges between $V_S$ and $V_A\\uplus V_B$.\n\n$G[V_S]$ includes a full binary tree of height $h$ and $m$ disjoint paths of length $2^h-1$.\nEach of the $2^h$ leaves of the binary tree is connected to the nodes on the paths as depicted in Figure~\\ref{fig:example}.\nSuppose nodes of depth $i$ on the tree are $t_{i,1},\\cdots,t_{i,2^i}$ and nodes on the $i$-th path are $p_{i,1},\\ldots,p_{i,2^h}$ from left to right.\nThen $t_{h,1},\\ldots, t_{h,2^h}$ are the leaves of the binary tree in $G[V_S]$.\nFor each $i\\in[1,m]$ and $j\\in[1,2^h]$, there is an edge between $t_{h,j}$ and $p_{i,j}$.\nThus,\n$$\n\\begin{aligned}\nV_S & =\\left\\{t_{i,j}:i\\in[0,h],j\\in[1,2^i]\\right\\} \\\\\n& \\uplus\\left\\{p_{i,j}:i\\in[1,m],j\\in[1,2^i]\\right\\}, \\\\\nE_S & =\\left\\{\\{t_{i,j},t_{i-1,\\lceil j\/2\\rceil}\\}:i\\in[1,h],j\\in[1,2^i]\\right\\} \\\\\n& \\uplus\\left\\{\\{p_{i,j},p_{i,j-1}\\}:i\\in[1,m],j\\in[2,2^m]\\right\\} \\\\\n& \\uplus\\left\\{\\{t_{h,j},p_{i,j}\\}:i\\in[1,m],j\\in[1,2^m]\\right\\}.\n\\end{aligned}\n$$\n$V_A$ contains at least $m$ nodes, each of which is connected to $p_{i,1}$ for $1\\leq i\\leq m$.\n$V_B$ contains at least $m$ nodes, each of which is connected to $p_{i,2^h}$ for $1\\leq i\\leq m$.\nThose $2m$ edges are contained in $E'$.\nThe subgraphs $G[V_A]$ and $G[V_B]$ are decided by Alice's input and Bob's input, respectively.\n\n\\bigskip\n\nThe following lemma gives an efficient simulation of algorithms on network $G$ by the protocols in the quantum Server model.\n\n\\begin{lemma}[Quantum Simulation Lemma]\nSuppose Alice and Bob are given $(V_A,E_A)$ and $(V_B,E_B)$, respectively.\nFor any $T$-round ($T<2^h\/2$) distributed algorithm on network $G$ described above, there exists a communication protocol for Alice and Bob in the quantum Server model to simulate the algorithm with communication complexity $O(T\\cdot h\\cdot B)$,  where $B$ denotes the bandwidth in the CONGEST model.\n\\label{lem:simulation}\n\\end{lemma}\n\n\\begin{proof}\nThe proof of Lemma~\\ref{lem:simulation} follows closely with the proof in \\cite[Proof of Theorem 3.5]{ElkinKNP14}.\nThe protocol we will construct simulates the distributed algorithm round by round.\nThus, it also has $T<2^h\/2$ rounds of communication.\nIn the beginning, the server simulates all the nodes in $V_S$ which are independent of Alice and Bob's inputs.\nAnd in the end of the $r$-th round, the server simulates $p_{i,1+r},\\cdots,p_{i,2^h-r}$ on the $i$-th path and nodes $t_{h,1+r},\\cdots,t_{h,2^h-r}$ along with their ancestors on the binary tree, while Alice simulates the nodes on the left side and Bob simulates on the right side.\nMore formally, in the end of the $r$-th round, the server simulates\n$$\\left\\{p_{i,j}:i\\in[1,m],j\\in[1+r,2^h-r]\\right\\}\\cup\\left\\{t_{i,j}:i\\in[0,h],j\\in\\left[\\left\\lceil(1+r)\/2^{h-i}\\right\\rceil,\\left\\lceil(2^h-r)\/2^{h-i}\\right\\rceil\\right]\\right\\};$$\nAlice simulates\n$$V_A\\cup\\left\\{p_{i,j}:i\\in[1,m],j\\in[1,1+r)\\right\\}\\cup\\left\\{t_{i,j}:i\\in[0,h],j\\in\\left[1,\\left\\lceil(1+r)\/2^{h-i}\\right\\rceil\\right)\\right\\};$$\nBob simulates\n$$V_B\\cup\\left\\{p_{i,j}:i\\in[1,m],j\\in(2^h-r,2^h]\\right\\}\\cup\\left\\{t_{i,j}:i\\in[0,h],j\\in\\left(\\left\\lceil(2^h-r)\/2^{h-i}\\right\\rceil,2^i\\right]\\right\\}.$$\n\nWe describe the simulation of the computation and communication of a processor $v$ in the $r$-th round, and count the total communication complexity.\n\\begin{itemize}\n\\item If $v$ is owned by Alice or the server in the $(r-1)$-th round and will be owned by Alice in the $r$-th round, Alice needs the local information of $v$ in the $(r-1)$-th round and messages from $\\Gamma(v)$ (neighbours of $v$) to $v$ in the $r$-th round, which can be obtained by local computation and communication from the server to Alice since $T<2^h\/2$, which implies that each of $v$ and nodes in $\\Gamma(v)$ is owned by either Alice or the server in the $(r-1)$-th round for $r\\leq T$.\nSo in this case, we only need communication from the server to Alice in the Server model.\nThis part will not be counted to complexity by definition.\n\\item If $v$ is owned by Bob or the server in the $(r-1)$-th round and will be owned by Bob in the $r$-th round, no communication will be counted to complexity by the same argument as mentioned above.\n\\item If $v$ is owned by the server in both the $(r-1)$-th round and the $r$-th round,\nthe server needs the messages from $\\Gamma(v)$ to $v$.\nFor each node $u\\in\\Gamma(v)$ owned by Alice or Bob in the $(r-1)$-th round, Alice or Bob will simulates the local computation of $u$ in the $r$-th round, and send the message to the server.\n\\begin{itemize}\n\\item If $v$ is on the paths on $V_S$, none of $\\Gamma(v)$ is owned by Alice and Bob in the $(r-1)$-th round.\n\\item If $v$ is on the binary tree, node $u\\in\\Gamma(v)$ is owned by Alice in the $(r-1)$-th round only if all nodes of the same depth with $v$, meanwhile on the left side of $v$, are not owned by the server in the $r$-th round, and $u$ is the left-child of $v$.\nSimilarly, node $u\\in\\Gamma(v)$ is owned by Bob in the $(r-1)$-th round only if all nodes of the same depth with $v$, meanwhile on the right side of $v$, are not owned by the server in the $r$-th round, and $u$ is the right-child of $v$.\nIn the $r$-th round, there are at most $2h$ such $(u,v)$ in total.\n\\end{itemize}\n\\end{itemize}\nHence, a total of $O(T\\cdot h)$ messages, each of size $O(B)$, are sent from Alice or Bob to the server.\n\\end{proof}\n\n\n\n\\subsection{Hardness of Approximating Diameter}\n\nwe will use $G$ constructed above as a gadget to prove a lower bound on round complexity of approximating weighted diameter in the quantum CONGEST model.\nThe specific graph depicted in Figure~\\ref{fig:diameter} will contain $n=\\left(2^{h+1}-1\\right)+\\left(2s+\\ell\\right)\\left(2^h+2\\right)+2\\cdot2^s$ nodes, where parameters $h,s,\\ell$ are chosen as follows throughout this section.\n\\begin{equation}\n\\label{eqn:paramters}\nh\\text{ is some even number},s=3h\/2,\\ell=2^{s-h}.\n\\end{equation}\nThis choice makes $2^h=\\widetilde\\Theta(n^{2\/3}),2^s=\\widetilde\\Theta(n)$ and $\\ell=\\widetilde\\Theta(n^{1\/3})$.\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/Diameter.pdf}\n\\caption{\nGraph $G$ for approximating diameter.\nThe black edges are of weight $1$; the blue edges are of weight $\\alpha$; and weights of red edges are determined by inputs $x,y$, i.e, $w(\\{a_i,a^\\star_j\\})=\\alpha$ if $x_{i,j}=1$ and $w(\\{a_i,a^\\star_j\\})=\\beta$ if $x_{i,j}=0$, and $w(\\{b_i,b^\\star_j\\})=\\alpha$ if $y_{i,j}=1$ and $w(\\{b_i,b^\\star_j\\})=\\beta$ if $y_{i,j}=0$, for $i\\in[1,2^s]$ and $j\\in[1,\\ell]$.\n}\n\\label{fig:diameter}\n\\end{figure*}\n\n\\begin{theorem}[Restated]\nFor any constant $\\varepsilon\\in(0,\\frac12]$, any algorithm, with probability at least $\\frac{11}{12}$, computing a $(\\frac32-\\varepsilon)$-approximation of the weighted diameter in the quantum CONGEST model requires $\\Omega\\left(\\frac{n^{2\/3}}{\\log^2n}\\right)$ rounds, even when the unweighted diameter is $\\Theta(\\log n)$, where $n$ denotes the number of nodes.\n\\label{thm:diameter}\n\\end{theorem}\n\nOn network $G=(V,E)$ described in Section~\\ref{sec:reduction}, we specify $G[V_A]$ and $G[V_B]$.\nLet\n\\begin{equation*}\n\\begin{aligned}\nV_A & =\\left\\{a_1,\\cdots,a_{2^s}\\right\\}\\uplus\\left\\{a^0_1,a^1_1,\\cdots,a^0_s,a^1_s\\right\\}\\uplus\\left\\{a^\\star_1,\\cdots,a^\\star_\\ell\\right\\}, \\\\%\\text{ and} \\\\\nV_B & =\\left\\{b_1,\\cdots,b_{2^s}\\right\\}\\uplus\\left\\{b^0_1,b^1_1,\\cdots,b^0_s,b^1_s\\right\\}\\uplus\\left\\{b^\\star_1,\\cdots,b^\\star_\\ell\\right\\}.\n\\end{aligned}\n\\end{equation*}\nThe edges $E_A, E_B$ and $E'$ are specified as follows.\n$$\n\\begin{aligned}\nE_A & =\\left\\{\\{a_i,a^{\\text{bin}(i,j)}_j\\}:i\\in[1,2^s],j\\in[1,s]\\right\\} \\\\\n& \\uplus\\left\\{\\{a_i,a^\\star_j\\}:i\\in[1,2^s],j\\in[1,\\ell]\\right\\} \\\\\n& \\uplus\\left\\{\\{a_i,a_j\\}:i,j\\in[1,2^s],i\\ne j\\right\\}, \\\\\nE_B & =\\left\\{\\{b_i,b^{\\text{bin}(i,j)}_j\\}:i\\in[1,2^s],j\\in[1,s]\\right\\} \\\\\n& \\uplus\\left\\{\\{b_i,b^\\star_j\\}:i\\in[1,2^s],j\\in[1,\\ell]\\right\\} \\\\\n& \\uplus\\left\\{\\{b_i,b_j\\}:i,j\\in[1,2^s],i\\ne j\\right\\}, \\\\\nE' & =\\left\\{\\{a^0_i,p_{2i-1,1}\\},\\{b^1_i,p_{2i-1,2^h}\\}:i\\in[1,s]\\right\\} \\\\\n& \\uplus\\left\\{\\{a^1_i,p_{2i,1}\\},\\{b^0_i,p_{2i,2^h}\\}:i\\in[1,s]\\right\\} \\\\\n& \\uplus\\left\\{\\{a^\\star_i,p_{2s+i,1}\\},\\{b^\\star_i,p_{2s+i,2^h}\\}:i\\in[1,\\ell]\\right\\},\n\\end{aligned}\n$$\nwhere  $\\text{bin}(i,j)$ denote the $j$-th bit in binary expression of integer $i-1$.\n\nThe node pairs $(a^0_i, p_{2i-1,1})$, $(a^1_i,p_{2i,1})$, $(b^0_i,p_{2i,2^h})$, $(b^1_i,p_{2i-1,2^h})$ for $1\\le i\\le s$, and $(a^\\star_j,p_{2s+j,1})$, $(b^\\star_j,p_{2s+j,2^h})$ for $1\\le j\\le\\ell$ are connected.\nFor each $i\\in[1,2^s]$, $a_i$ is connected to $a^{\\text{bin}(i,j)}_j$ for each $j\\in[1,s]$, and $a_i$ is connected to $a^\\star_j$ for each $j\\in[1,\\ell]$.\nMoreover, $G[\\{a_1,\\cdots,a_{2^s}\\}]$ is a clique.\nThe edges in $G[V_B]$ are linked in the same way as the edges in $G[V_A]$.\n\nThe weights of the edges are specified as follows, which are also depicted in Figure~\\ref{fig:diameter}.\n\\begin{itemize}\n\\item The edges on the binary tree and the edges on the $2s+\\ell$ paths (including the endpoints in $V_A$ and $V_B$) are of weight $1$ (the black edges in Figure~\\ref{fig:diameter}).\n\\item Recall that Alice and Bob receive inputs $x,y\\in\\{0,1\\}^{2^s\\cdot\\ell}$ respectively.\n$x$ and $y$ are indexed by $x_{i,j}$ and $y_{i,j}$ for $i\\in[1,2^s],j\\in[1,\\ell]$ where $s$ and $\\ell$ are given in Eq.~\\eqref{eqn:paramters}.\nFor each $i\\in[1,2^s],j\\in[1,\\ell]$, $w(\\{a_i,a^\\star_j\\})=\\alpha$ if $x_{i,j}=1$ and $w(\\{a_i,a^\\star_j\\})=\\beta$ if $x_{i,j}=0$ ($\\alpha<\\beta$); weights of edges between $\\{b_1,\\cdots,b_{2^s}\\}$ and $\\{b^\\star_1,\\cdots,b^\\star_\\ell\\}$ are assigned according to $y$ in the same way (the red edges in Figure~\\ref{fig:diameter}).\n\\item The edges between the binary tree and the $2s+\\ell$ paths, those between $\\{a_1,\\cdots,a_{2^s}\\}$ and $\\{a^0_1,a^1_1,\\cdots,a^0_s,a^1_s\\}$, and those between $\\{b_1,\\cdots,b_{2^s}\\}$ and $\\{b^0_1,b^1_1,\\cdots,b^0_s,b^1_s\\}$ are of weight $\\alpha$; weights of edges inside $G[\\{a_1,\\cdots,a_{2^s}\\}]$ and $G[\\{b_1,\\cdots,b_{2^s}\\}]$ are also $\\alpha$ (the blue edges in Figure~\\ref{fig:diameter}).\n\\end{itemize}\n\nIt is sufficient to analyze the diameter of graph after contracting all edges of weight $1$ due to the following lemma.\nAn edge is contracted if the two endpoints are merged to one node, and the adjacent edges of the two endpoints are incident to it.\nIf there are parallel edges after contraction,  we only keep the one with the lowest weight.\n\n\\begin{lemma}\nGiven a weighted graph $(G,w)$ where $G=(V,E)$ and $w:E\\to\\mathbb N^+$.\nLet $G'$ be the graph after contracting all edges of weight $1$.\nWe have $D_{G',w}\\le D_{G,w}\\le D_{G',w}+n$ and $R_{G',w}\\le R_{G,w}\\le R_{G',w}+n$, where $n=|V|$.\n\\label{lem:contraction}\n\\end{lemma}\n\\begin{proof}\nFor any path $P$ in $G$, let $P'$ be the path in $G'$ obtained from $P$ after contraction.\nThen\n$$\\text{length}(P')\\leq\\text{length}(P)\\leq\\text{length}(P')+n$$\nas there are at most $n-1$ $1$-weight edges.\nThus we conclude the result.\n\\end{proof}\n\nFor inputs $x,y\\in\\{0,1\\}^{2^s\\cdot\\ell}$ received by Alice and Bob, define\n$$F(x,y)=\\bigwedge_{i\\in[1,2^s]}\\left(\\bigvee_{j\\in[1,\\ell]}\\left(x_{i,j}\\wedge y_{i,j}\\right)\\right),$$\ni.e., $F=\\text{AND}_{2^s}\\circ(\\text{OR}_\\ell\\circ\\text{AND}^\\ell_2)^{2^s}$.\nWe have the following lemma.\n\n\\begin{lemma}\n$D_{G,w}\\le\\max\\{2\\alpha,\\beta\\}+n$ if $F(x,y)=1$, and $D_{G,w}\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}$ otherwise.\n\\label{lem:reduction}\n\\end{lemma}\n\n\\begin{proof}\nThe graph $G'$ after contraction is given in Figure~\\ref{fig:contraction}.\nThe binary tree is contracted to node $t$.\nThe $2s+\\ell$ paths are contracted to nodes $a^0_1,a^1_1,\\cdots,a^0_s,a^1_s$ and $a^\\star_1,\\cdots,a^\\star_\\ell$ respectively.\nNote that $b_i$ is connected to $a^{\\text{bin}(i,j)\\oplus1}_j$ for $i\\in[1,2^s],j\\in[1,s]$.\nwe list upper bounds of the distances between any two nodes $u$ and $v$ in $G'$ on Table~\\ref{tab:distance} with the corresponding paths, except for the distance between $a_i$ and $b_i$ with $i\\in[1,2^s]$.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\linewidth]{figures\/Contraction.pdf}\n\\caption{\nGraph $G'$ after contraction.\nThe distance between any pair of nodes, except $a_i$ and $b_i$ for $i\\in[1,2^s]$, is at most $\\max\\{2\\alpha,\\beta\\}$; and the distance between $a_i$ and $b_i$ is at most $2\\alpha$ if there exists $j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$, otherwise it is at least $\\min\\{\\alpha+\\beta,3\\alpha\\}$.\nTherefore, the diameter is at most $\\max\\{2\\alpha,\\beta\\}$ if, for any $i\\in[1,2^s]$, there exists $j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$, otherwise it is at least $\\min\\{\\alpha+\\beta,3\\alpha\\}$.\n}\n\\label{fig:contraction}\n\\end{figure}\n\n\\begin{table*}[t]\n\\centering\n\\caption{\nDistance between nodes in $G'$.\nLet {\\rm router} be any node in $\\{a^0_1,a^1_1,\\cdots,a^0_s,a^1_s,a^\\star_1,\\cdots,a^\\star_\\ell\\}$.\n${\\rm adj}(i,j)$ denotes the integer after changing the $j$-th bit in binary expression of integer $i-1$, and ${\\rm ind}(i,j)$ is the smallest $z\\in[1,s]$ satisfying ${\\rm bin}(i,z)\\ne{\\rm bin}(j,z)$.\n}\n\\label{tab:distance}\n\\begin{tabular}{cccc}\n\\toprule[1.5pt]\n$u$                                         & $v$                                               & $d_{G',w}(u,v)$   & Path \\\\\n\\midrule[1.5pt]\n\\multirow{3.6}{*}{$t$} \t\t\t\t\t\t& router                                            & $\\le\\alpha$       & $\\left(t\\to v\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $a_i$ ($i\\in[1,2^s]$)                             & $\\le2\\alpha$      & $\\left(t\\to a^{\\text{bin}(i,0)}_0\\to a_i\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $b_i$ ($i\\in[1,2^s]$)                             & $\\le2\\alpha$      & $\\left(t\\to a^{\\text{bin}(i,0)\\oplus1}_0\\to b_i\\right)$ \\\\\n\\midrule\n\\multirow{6.4}{*}{$a_i$ ($i\\in[1,2^s]$)}    & $a_j$ ($j\\ne i,j\\in[1,2^s]$)                      & $\\le\\alpha$       & $\\left(a_i\\to a_j\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $a^{\\text{bin}(i,j)}_j$ ($j\\in[1,s]$)             & $\\le\\alpha$       & $\\left(a_i\\to a^{\\text{bin}(i,j)}_j\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $a^{\\text{bin}(i,j)\\oplus1}_j$ ($j\\in[1,s]$)      & $\\le2\\alpha$      & $\\left(a_i\\to a_{\\text{adj}(i,j)}\\to a^{\\text{bin}(i,j)\\oplus1}_j\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $b_j$ ($j\\ne i,j\\in[1,2^s]$)                      & $\\le2\\alpha$      & $\\left(a_i\\to a^{\\text{bin}(i,\\text{ind}(i,j))}_{\\text{ind}(i,j)}\\to b_j\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $a^\\star_j$ ($j\\in[1,\\ell]$)                      & $\\le\\beta$        & $\\left(a_i\\to a^\\star_j\\right)$ \\\\\n\\midrule\n\\multirow{5.2}{*}{$b_i$ ($i\\in[1,2^s]$)}    & $b_j$ ($j\\ne i,j\\in[1,2^s]$)                      & $\\le\\alpha$       & $\\left(b_i\\to b_j\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $a^{\\text{bin}(i,j)\\oplus1}_j$ ($j\\in[1,s]$)      & $\\le\\alpha$       & $\\left(b_i\\to a^{\\text{bin}(i,j)\\oplus1}_j\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $a^{\\text{bin}(i,j)}_j$ ($j\\in[1,s]$)             & $\\le2\\alpha$      & $\\left(b_i\\to b_{\\text{adj}(i,j)}\\to a^{\\text{bin}(i,j)}_j\\right)$ \\\\\n\t\t\t\t\t\t\t\t\t\t\t& $a^\\star_j$ ($j\\in[1,\\ell]$)                      & $\\le\\beta$        & $\\left(b_i\\to a^\\star_j\\right)$ \\\\\n\\midrule\nrouter \t\t\t\t\t\t\t\t\t\t& router                                            & $\\le2\\alpha$      & $\\left(u\\to t\\to v\\right)$ \\\\\n\\bottomrule[1.5pt]\n\\end{tabular}\n\\end{table*}\n\nRegarding the distance between $a_i$ and $b_i$ for $i\\in[1,2^s]$, if there exists $j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$, then $w(\\{a_i,a^\\star_j\\})=w(\\{b_i,a^\\star_j\\})=\\alpha$ and $d_{G',w}(a_i,b_i)\\le2\\alpha$ because of the path $(a_i\\to a^\\star_j\\to b_i)$ in $G'$.\nIf there is no $j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$, we claim that  $d_{G',w}(a_i,b_i)\\geq\\min\\{\\alpha+\\beta, 3\\alpha\\}$.\nFor any path between $a_i$ and $b_i$, if it contains exactly two edges, it is of the form $(a_i\\to a^\\star_j\\to b_i)$ for some $j\\in[1,\\ell]$ by the construction of $G'$, and it is of length at least $\\alpha+\\beta$ by the assumption.\nIf it contains at least three edges, it is of length at least $3\\alpha$.\n\nIf $F(x,y)=1$, then for any $i\\in[1,2^s]$, there exists $j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$.\nHence,\n$$\n\\begin{aligned}\n& d_{G',w}(a_i,b_i)\\le2\\alpha,\\forall i\\in[1,2^s], \\\\\n& D_{G',w}=\\max_{u,v}d_{G',w}(u,v)\\le\\max\\{2\\alpha,\\beta\\}.\n\\end{aligned}\n$$\nTherefore, $D_{G,w}\\le D_{G',w}+n\\le\\max\\{2\\alpha,\\beta\\}+n$ by Lemma~\\ref{lem:contraction}.\n\nIf $F(x,y)=0$, then there exists $i\\in[1,2^s]$ such that $x_{i,j}=0$ or $y_{i,j}=0$ for any $j\\in[1,\\ell]$.\nHence,\n$$\n\\begin{aligned}\n& d_{G',w}(a_i,b_i)=\\min_{\\text{path }P\\text{ from }a_i\\text{ to }b_i}\\text{length}(P)\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}, \\\\\n& D_{G',w}=\\max_{u,v}d_{G',w}(u,v)\\ge d_{G',w}(a_i,b_i)\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}.\n\\end{aligned}\n$$\nTherefore, $D_{G,w}\\ge D_{G',w}\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}$ by Lemma~\\ref{lem:contraction}.\n\\end{proof}\n\nCombining Lemma~\\ref{lem:simulation} and Lemma~\\ref{lem:reduction}, we have a reduction from computing $F$ in the Server model to approximating diameter in the quantum CONGEST model.\nTo prove the communication complexity of $F$ in the Server model, we adopt the following lemma.\n\n\\begin{lemma}[Lemma B.4 in \\cite{ElkinKNP14}, arXiv version]\nFunction ${\\rm VER}:\\{0,1,2,3\\}\\times\\{0,1,2,3\\}\\to\\{0,1\\}$ is defined by ${\\rm VER}(x,y)=1$ if and only if $x+y$ is equivalent to $0$ or $1$ modulo $4$, where $x,y\\in\\{0,1,2,3\\}$.\nLet $f:\\{0,1\\}^k\\to\\{0,1\\}$ be an arbitrary function.\nThen\n$$Q^{sv}_\\varepsilon(f\\circ{\\rm VER}^k)\\ge\\frac12\\text{deg}_{4\\varepsilon}(f)-O(1)$$\nfor any $0<\\varepsilon<1\/4$.\n\\label{lem:lifting}\n\\end{lemma}\n\nA read-once formula, which consists of AND gates, OR gates, and NOT gates, is a formula in which each variable appears exactly once.\nWe will need the following conclusion for approximate degree of read-once formulas.\n\n\\begin{lemma}[Theorem 6 in \\cite{AaronsonBKRT21}]\nFor any read-once formula $f:\\{0,1\\}^k\\to\\{0,1\\}$, $\\text{deg}_{1\/3}(f)=\\Theta\\left(\\sqrt k\\right)$.\n\\label{lem:read_once}\n\\end{lemma}\n\n\\begin{lemma}\nGiven $s,\\ell$ defined in Eq.~\\eqref{eqn:paramters} where $\\ell$ is a multiple of $4$, $F={\\rm AND}_{2^s}\\circ({\\rm OR}_\\ell\\circ{\\rm AND}^\\ell_2)^{2^s}$ with inputs $x,y\\in\\{0,1\\}^{2^s\\cdot\\ell}$, set\n\\[F(x,y)=\\bigwedge_{i\\in[1,2^s]}\\left(\\bigvee_{j\\in[1,\\ell]}\\left(x_{i,j}\\wedge y_{i,j}\\right)\\right).\\]\nIt holds that\n$$Q^{sv}_{1\/12}(F)=\\Omega\\left(\\sqrt{2^s\\cdot\\ell}\\right).$$\n\\label{lem:and_or_and}\n\\end{lemma}\n\\begin{proof}\nThe function $F$ can be rewritten as $F=f\\circ \\text{GDT}^{2^s\\cdot\\ell\/4}$, where\n$f=\\text{AND}_{2^s}\\circ\\text{OR}^{2^s}_{\\ell\/4}$ and $\\text{GDT}=\\text{OR}_4\\circ\\text{AND}^4_2$.\nObviously the function $f$ is a read-once formula.\nIt can be seen that the function VER is actually a {\\it promise version} of the function GDT where inputs $x,y\\in\\{0,1\\}^4$ satisfy\n$$x\\in\\{0011,1001,1100,0110\\},y\\in\\{0001,0010,0100,1000\\}.$$\nThus, the lower bound for $f\\circ\\text{VER}^{2^s\\cdot\\ell\/4}$ clearly implies the lower bound for $f\\circ\\text{GDT}^{2^s\\cdot\\ell\/4}$.\nTherefore,\n$$Q^{sv}_{1\/12}(f\\circ\\text{GDT}^{2^s\\cdot\\ell\/4})\\ge Q^{sv}_{1\/12}(f\\circ\\text{VER}^{2^s\\cdot\\ell\/4})\\ge\\frac12\\text{deg}_{1\/3}(f)-O(1)=\\Omega\\left(\\sqrt{2^s\\cdot\\ell}\\right).$$\nThe second inequality is due to Lemma~\\ref{lem:lifting} and the last inequality is due to Lemma~\\ref{lem:read_once}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:diameter}]\nLet $\\mathcal A$ be a $T$-round algorithm ($T<2^h\/2$) in the quantum CONGEST model which, for any weighted graph $(G,w)$, computes a $(\\frac32-\\varepsilon)$-approximation of $D_{G,w}$ (constant $\\varepsilon\\in(0,1\/2]$) with probability at least $11\/12$.\nAlice and Bob, who receive $x,y\\in\\{0,1\\}^{2^s\\cdot\\ell}$, respectively, construct the network $G$ as described above with parameters $h,s,\\ell$ given in Eq.~\\eqref{eqn:paramters}.\nThe number of nodes is\n$$n=\\left(2^{h+1}-1\\right)+\\left(2s+\\ell\\right)\\left(2^h+2\\right)+2\\cdot2^s=\\Theta\\left(2^{3h\/2}\\right).$$\nAnd the unweighted diameter is $D_G=\\Theta(h)=\\Theta(\\log n)$.\nLet $w$ be the weight function.\nDue to Lemma~\\ref{lem:simulation}, they can simulate $\\mathcal A$ on $(G,w)$ in the quantum Server model with communication complexity $O(T\\cdot h\\cdot B)$ where $B$ denotes the bandwidth.\nWith probability at least $\\frac{11}{12}$, Alice and Bob output an approximation $\\widetilde D_{G,w}$ satisfying $D_{G,w}\\le\\widetilde D_{G,w}\\le(\\frac32-\\varepsilon)D_{G,w}$.\nWe set $\\alpha=n^2$ and $\\beta=2n^2$.\nBy Lemma~\\ref{lem:reduction},\n$$\n\\begin{aligned}\n\\text{if }F(x,y)=1,\\widetilde D_{G,w} & \\le\\left(\\frac32-\\varepsilon\\right)D_{G,w}\\le\\left(\\frac32-\\varepsilon\\right)\\left(\\max\\{2\\alpha,\\beta\\}+n\\right) \\\\\n& =3n^2-\\left(2\\varepsilon n^2-\\left(\\frac32-\\varepsilon\\right)n\\right); \\\\\n\\text{if }F(x,y)=0,\\widetilde D_{G,w} & \\ge D_{G,w}\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}=3n^2. \\\\\n\\end{aligned}\n$$\nFor large enough $n$, Alice and Bob can distinguish whether $F(x,y)=1$ or not with probability at least $\\frac{11}{12}$ in the Server model, and thus $Q^{sv}_{1\/12}(F)=O(T\\cdot h\\cdot B)$.\nDue to Lemma~\\ref{lem:and_or_and},\n$$T=\\Omega\\left(\\frac{\\sqrt{2^s\\cdot\\ell}}{h\\cdot B}\\right)=\\Omega\\left(\\frac{2^h}{h\\cdot B}\\right)=\\Omega\\left(\\frac{n^{2\/3}}{ \\log^2 n}\\right),$$\nwhere the last equality is by the choice of $h$ and the the bandwidth $B=\\Theta(\\log n)$.\nTherefore, the round complexity of approximating diameter is $\\Omega\\left(\\min\\left\\{2^h\/2,\\frac{n^{2\/3}}{\\log^2n}\\right\\}\\right)=\\Omega\\left(\\frac{n^{2\/3}}{\\log^2n}\\right)$.\n\\end{proof}\n\n\n\n\\subsection{Hardness of Approximating Radius}\n\nWe choose the same set of parameters $h,s,\\ell$ given in Eq.~\\eqref{eqn:paramters}.\nThe argument is very close to the one for diameter.\n\n\\begin{theorem}[Restated]\nFor any constant $\\varepsilon\\in(0,\\frac12]$, any algorithm, with probability at least $\\frac{11}{12}$, computing a $(\\frac32-\\varepsilon)$-approximation of radius in the quantum CONGEST model requires $\\Omega\\left(\\frac{n^{2\/3}}{\\log^2n}\\right)$ rounds, even when the unweighted diameter is $\\Theta(\\log n)$, where $n$ denotes the number of nodes.\n\\label{thm:radius}\n\\end{theorem}\n\nThe weighted graph $(G,w)$ that we construct for showing hardness of approximating radius is almost the same except that we add a node $a_0$ in $V_A$ along with edges $\\{a_0,a_1\\},\\cdots,\\{a_0,a_{2^s}\\}$ of weight $2\\alpha$.\nHere we only show in Figure~\\ref{fig:radius} the graph $G'$ after contracting all edges of weight $1$ (the green edges are the new-added edges).\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.6\\linewidth]{figures\/Radius.pdf}\n\\caption{\nGraph $G'$ (after contraction) for approximating radius.\nThe additional green edges are of weight $2\\alpha$.\nThe eccentricity of any node, except $a_i$ for $i\\in[1,2^s]$, is at least $3\\alpha$; and the eccentricity of $a_i$ is at most $\\max\\{2\\alpha,\\beta\\}$ if there exists $j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$, otherwise it is at least $\\min\\{\\alpha+\\beta,3\\alpha\\}$.\nTherefore, the radius is at most $\\max\\{2\\alpha,\\beta\\}$ if there exist $i\\in[1,2^s],j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$, otherwise it is at least $\\min\\{\\alpha+\\beta,3\\alpha\\}$.\n}\n\\label{fig:radius}\n\\end{figure}\n\nFor inputs $x,y\\in\\{0,1\\}^{2^s\\cdot\\ell}$ define\n$$F'(x,y)=\\bigvee_{i\\in[1,2^s],j\\in[1,\\ell]}(x_{i,j}\\wedge y_{i,j}).$$\nWe have the following lemma.\n\n\\begin{lemma}\n$R_{G,w}\\le\\max\\{2\\alpha,\\beta\\}+n$ if $F'(x,y)=1$, and $R_{G,w}\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}$ otherwise.\n\\label{lem:reduction_radius}\n\\end{lemma}\n\\begin{proof}\nIt suffices to estimate the radius of $(G',w)$ by Lemma~\\ref{lem:contraction}.\nFor any node $v\\notin\\{a_0,a_1,\\cdots,a_{2^s}\\}$, $d_{G',w}(a_0,v)\\ge3\\alpha$.\nThis is because that any path from $a_0$ to $v$ is of the form $(a_0\\to a_i\\leadsto v)$ for some $i\\in[1,2^s]$, where $w(\\{a_0,a_i\\})=2\\alpha$, and the remaining edges on the path have total weight at least $\\alpha$.\nTherefore, $e_{G',w}(v)\\ge3\\alpha$ for any $v\\notin\\{a_1,\\cdots,a_{2^s}\\}$.\nTo estimate the eccentricity of $a_i$ for $i\\in[1,2^s]$, we have $d(a_i,v)\\le\\max\\{2\\alpha,\\beta\\}$ for any $v\\ne b_i$ as shown on Table~\\ref{tab:distance}, and $d_{G',w}(a_i,b_i)\\le2\\alpha$ if there exists $j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$, and $d_{G',w}(a_i,b_i)\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}$ otherwise.\n\nIf $F'(x,y)=1$, then there are $i\\in[1,2^s]$ and $j\\in[1,\\ell]$ such that $x_{i,j}=y_{i,j}=1$, and thus\n$$\n\\begin{aligned}\n& d_{G',w}(a_i,b_i)\\le2\\alpha,\\\\\n& e_{G',w}(a_i)=\\max_v d_{G',w}(a_i,v)\\le\\max\\{2\\alpha,\\beta\\}, \\\\\n& R_{G',w}=\\min_u e_{G',w}(u)\\le e_{G',w}(a_i)\\le\\max\\{2\\alpha,\\beta\\}.\n\\end{aligned}\n$$\nTherefore, $R_{G,w}\\le R_{G',w}+n\\le\\max\\{2\\alpha,\\beta\\}+n$ by Lemma~\\ref{lem:contraction}.\n\nIf $F'(x,y)=0$, then for any $i\\in[1,2^s]$ and $j\\in[1,\\ell]$, $x_{i,j}=0$ or $y_{i,j}=0$, and thus\n$$\n\\begin{aligned}\n& d_{G',w}(a_i,b_i)\\ge\\min\\{\\alpha+\\beta,3\\alpha\\},\\forall i\\in[1,2^s], \\\\\n& e_{G',w}(a_i)=\\max_v d_{G',w}(a_i,v)\\ge d_{G',w}(a_i,b_i) \\\\\n& \\qquad\\ge\\min\\{\\alpha+\\beta,3\\alpha\\},\\forall i\\in[1,2^s], \\\\\n& R_{G',w}=\\min_u e_{G',w}(u)\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}.\n\\end{aligned}\n$$\nTherefore, $R_{G,w}\\ge R_{G',w}\\ge\\min\\{\\alpha+\\beta,3\\alpha\\}$ by Lemma~\\ref{lem:contraction}.\n\\end{proof}\n\nSimilar to Lemma~\\ref{lem:and_or_and}, one can prove a lower bound on communication complexity of $F'$ in the quantum Server model.\n\n\\begin{lemma}\nGiven $s,\\ell$ defined in Eq.~\\eqref{eqn:paramters} where $2^s\\cdot\\ell$ is a multiple of $4$,\n$F'={\\rm OR}_{2^s\\cdot\\ell}\\circ{\\rm AND}^{2^s\\cdot\\ell}_2$ with inputs $x,y\\in\\{0,1\\}^{2^s\\cdot\\ell}$,\nset\n$$F'(x,y)=\\bigvee_{i\\in[1,2^s],j\\in[1,\\ell]}(x_{i,j}\\wedge y_{i,j}).$$\nIt holds that\n$$Q^{sv}_{1\/12}(F')=\\Omega\\left(\\sqrt{2^s\\cdot\\ell}\\right).$$\n\\label{lem:or_and}\n\\end{lemma}\n\\begin{proof}\nThe function $F'$ can be rewritten as $F'=f'\\circ \\text{GDT}^{2^s\\cdot\\ell\/4}$,\nwhere $f'=\\text{OR}_{2^s\\cdot\\ell\/4}$.\nNote that $f'$ is still a read-once formula.\nThus the rest of proof is the same as the one in Lemma~\\ref{lem:and_or_and}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:radius}]\nLet $\\mathcal A$ be a $T$-round algorithm ($T<2^h\/2$) in the quantum CONGEST model which, for any weighted graph $(G,w)$, computes a $(\\frac32-\\varepsilon)$-approximation of $R_{G,w}$ (constant $\\varepsilon\\in(0,\\frac12]$) with probability at least $\\frac{11}{12}$.\nAlice and Bob, who receive $x,y\\in\\{0,1\\}^{2^s\\cdot\\ell}$ as input, construct the weighted graph $(G,w)$ described above with the number of node $n=\\Theta(2^{3h\/2})$. The unweighted diameter $D_G=\\Theta(\\log n)$.\nDue to Lemma~\\ref{lem:simulation}, Alice and Bob can simulate $\\mathcal A$ on $(G,w)$ in the quantum Server model with communication complexity $O(T\\cdot h\\cdot B)$.\nThen with probability at least $\\frac{11}{12}$, Alice and Bob compute $\\widetilde R_{G,w}$ satisfying $R_{G,w}\\le\\widetilde R_{G,w}\\le(\\frac32-\\varepsilon)R_{G,w}$.\nWe set $\\alpha=n^2$ and $\\beta=2n^2$.\nDue to Lemma~\\ref{lem:reduction_radius},\n$$\n\\begin{aligned}\n& \\text{if }F'(x,y)=1,\\widetilde R_{G,w}\\le3n^2-\\left(2\\varepsilon n^2-\\left(\\frac32-\\varepsilon\\right)n\\right); \\\\\n& \\text{if }F'(x,y)=0,\\widetilde R_{G,w}\\ge3n^2. \\\\\n\\end{aligned}\n$$\nFor large enough $n$, Alice and Bob can compute $F'$ with probability at least $\\frac{11}{12}$ in the Server model, and thus $Q^{sv}_{1\/12}(F')=O(T\\cdot h\\cdot B)$.\nDue to Lemma~\\ref{lem:or_and}, $T=\\Omega\\left(\\frac{n^{2\/3}}{\\log^2n}\\right)$.\nTherefore, the round complexity of approximating radius is $\\Omega\\left(\\min\\left\\{2^h\/2,\\frac{n^{2\/3}}{\\log^2n}\\right\\}\\right)=\\Omega\\left(\\frac{n^{2\/3}}{\\log^2n}\\right)$.\n\\end{proof}\n\\section{Introduction}\n\nQuantum distributed computing has received great attention in the past decade~\\cite{Ben-OrH05,TaniKM12,ElkinKNP14,GallM18,GallNR19,IzumiG19,IzumiGM20,Censor-HillelFG22,MagniezN20,GavoilleKM09}.\nA large body of work has been devoted to investigating the quantum advantages in distributed computing.\nIn this paper, we are concerned with the CONGEST networks, which are one of the most fundamental models in distributed computing.\nIn a classical CONGEST network, the nodes synchronously exchange classical messages, and each channel has $O(\\log n)$-bit bandwidth, where $n$ is the number of nodes in the network.\nQuantum CONGEST networks were first introduced by  Elkin, Klauck, Nanongkai,\nand Pandurangan~\\cite{ElkinKNP14}, where the only difference is that the nodes exchange quantum messages and the bandwidth of each channel is $O(\\log n)$ qubits. The round complexity of diameter and radius  of unweighted graphs in classical CONGEST networks has been extensively studied~\\cite{AbboudCK16,AnconaCDEW20,HolzerW12,PelegRT12,FrischknechtHW12,HolzerPRW14}.\nLe Gall and Magniez~\\cite{GallM18} proved that quantum communication may save the round complexity in CONGEST networks if the graph has a low diameter.\n\nIn this paper, we further investigate the round complexity of computing the diameters and radius of weighted graphs in quantum CONGEST networks. We prove that quantum communication may also save the round complexity for both problems.\n\n\n\n\\subsection{Our Results}\n\\label{sec:results}\n\nThe following is one of our main results which asserts that quantum communication may save the round complexity for computing the weighted diameter and radius of a graph given that the graph has a low unweighted diameter.\n\n\\begin{theorem}\nThere exists a $\\widetilde O\\left(\\min\\left\\{n^{9\/10}D^{3\/10},n\\right\\}\\right)$-round distributed algorithm computing a $(1+o(1))$-approximation of the weighted diameter\/radius with probability at least $1-1\/\\text{poly}(n)$, in the quantum CONGEST model, where $D$ denotes the unweighted diameter.\n\\label{thm:upper_bound}\n\\end{theorem}\n\nHolzer and Pinsker in~\\cite{HolzerP15} and Abboud, Censor-Hillel and Khoury in~\\cite{AbboudCK16} proved that $(2-o(1))$-approximating the diameter and $(3\/2-\\varepsilon)$-approximating the (even unweighted) radius in the classical CONGEST network require $\\widetilde\\Omega(n)$ rounds, even when $D$ is constant.\nTherefore, Theorem~\\ref{thm:upper_bound} exhibits the advantages of quantum communication over classical communication in approximating the weighted diameter\/radius when $D=o(n^{1\/3})$.\n\nWe prove Theorem~\\ref{thm:upper_bound} by applying the framework of distributed quantum optimization introduced by Le Gall and Magniez in~\\cite{GallM18}.\nNote that the diameter and radius are the maximum and minimum of eccentricities respectively.\nIt will not give a sublinear-time algorithm if we simply apply a quantum search algorithm, because evaluating the eccentricity of one node takes $\\widetilde\\Theta\\left(\\sqrt n\\right)$ rounds (the lower bound is due to \\cite{ElkinKNP14}), and the searching process should require another $\\widetilde\\Theta\\left(\\sqrt n\\right)$ times of evaluation (the number of nodes with maximum\/minimum eccentricity maybe $O(1)$).\nThus the number of rounds in total will be $\\widetilde{\\Theta}(n)$.\\footnote{\n$\\widetilde O(\\cdot)$, $\\widetilde\\Omega(\\cdot)$, and $\\widetilde\\Theta(\\cdot)$ hide polylogarithmic factors.}\nOur algorithm is inspired by Nanongkai's algorithm~\\cite{Nanongkai14STOC} for approximating the weighted shortest paths in a classical network.\nThe algorithm constructs several small vertex sets and searches the node achieving the maximum\/minimum eccentricity within those sets, which turns out to be a good approximation of diameter\/radius.\nOur algorithm quantizes Nanongkai's algorithm using the standard technique \\cite{Bennett89} and further combines with the framework of distributed quantum optimization in~\\cite{GallM18}.\n\nWe also prove lower bounds for approximating weighted diameter and radius.\n\n\\begin{theorem}\nAny algorithm computing a $(3\/2-o(1))$-approximation of the weighted diameter\/radius requires $\\widetilde\\Omega(n^{2\/3})$ rounds, in the quantum CONGEST model, even when $D=\\Theta(\\log n)$.\n\\label{thm:lower_bound_raw}\n\\end{theorem}\n\nThe hardness of both problems is proved via the communication complexity of quantum Server models.\nThe Server model is a variant of two-party communication complexity models introduced in \\cite{ElkinKNP14}.\nCombining with the graph gadget in \\cite{AbboudCK16}, we get a reduction from the communication complexity of certain read-once functions to the round complexity of approximating the weighted diameter and radius.\nWe further apply a lifting theorem of quantum communication complexity~\\cite{ElkinKNP14} to obtain the desired lower bounds.\n\nCompared with Le Gall and Magniez's algorithm~~\\cite{GallM18} for unweighted diameter\/radius with $\\widetilde O\\left(\\sqrt{nD}\\right)$ rounds,  Theorem~\\ref{thm:lower_bound_raw} says that computing weighted diameter\/radius is strictly harder than unweighted diameter\/radius, when $D$ is small.\nWhile in the classical setting, computing weighted and unweighted diameter\/radius have the same round complexity $\\Theta(n)$~\\cite{AbboudCK16,BernsteinN19}\\footnote{The lower bound is proved in~\\cite{AbboudCK16}. The upper bound is followed from the $\\widetilde O(n)$-round algorithm for the exact weighted All-Pairs Shortest Paths (APSP) in~\\cite{BernsteinN19}.}.\n\n\n\n\\subsection{Related Works}\n\nA series of works started with the distance computation in the classical CONGEST network.\nEarlier, Frischknecht, Holzer, and Wattenhofer~\\cite{FrischknechtHW12} showed that computing the diameter of an unweighted graph with constant diameter requires $\\widetilde\\Omega(n)$ rounds, which is tight up to logarithmic factors since even computing All-Pairs Shortest Paths (ASAP) on an unweighted graph can be resolved in $O(n)$ rounds \\cite{HolzerW12,PelegRT12}.\nAbboud, Censor-Hillelet, and Khoury~\\cite{AbboudCK16} later gave the same lower bound of $\\widetilde\\Omega(n)$ for $(3\/2-\\varepsilon)$-approximating the diameter\/radius in sparse networks.\nBernstein and Nanongkai~\\cite{BernsteinN19} provided a $\\widetilde O(n)$-round algorithm computing the exact APSP on any weighted graph.\nAs a result, computing unweighted diameter\/radius and weighted diameter\/radius (exactly or with a small approximation ratio) have an almost tight complexity of $\\widetilde\\Theta(n)$ in the classical CONGEST network.\nIf a larger approximation ratio is allowed, there are $\\widetilde O(\\sqrt n+D)$-round algorithms for $3\/2$-approximating the diameter\/radius on any unweighted graph \\cite{HolzerPRW14,AnconaCDEW20}.\nBesides, Chechik and Mukhtar~\\cite{ChechikM20} showed a $\\widetilde O(\\sqrt nD^{1\/4}+D)$-round algorithm computing Single-Source Shortest Paths (SSSP) exactly on any weighted graph, which also gives a $2$-approximation of the diameter\/radius.\n\nAs for the quantum setting, while quantum computation offers advantages over classical computation in various settings such as query complexity and two-party communication complexity, the power of quantum computation in distributed computing has not been fully explored.\nIn the quantum CONGEST network, Elkin et al.~\\cite{ElkinKNP14} gave negative results for several problems such as minimum spanning tree, minimum cut, and SSSP, i.e., quantum communication does not speed up distributed algorithms for these problems.\nLe~Gall and Magniez~\\cite{GallM18} presented a $\\widetilde O\\left(\\sqrt{nD}\\right)$-round algorithm computing the diameter\/radius on any unweighted graph, along with a $\\widetilde O\\left(\\sqrt[3]{nD}+D\\right)$-round algorithm $3\/2$-approximating the diameter.\nThey also proved a $\\widetilde\\Omega(\\sqrt n+D)$-lower bound for computing the unweighted diameter, which was later improved to $\\widetilde\\Omega\\left(\\sqrt[3]{nD^2}+\\sqrt n\\right)$ by Magniez and Nayak~\\cite{MagniezN20}.\nThe above results are listed on Table~\\ref{tab:complexity}.\n\n\\begin{table}[t]\n\\centering\n\\scriptsize\n\\renewcommand\\arraystretch{1.5}\n\\caption{Complexity of computing diameter and radius in the CONGEST model.}\n\\label{tab:complexity}\n\\begin{tabular}{cccccccc}\n\\toprule[1.5pt]\n\\multirow{2}{*}{Problem} & \\multirow{2}{*}{Variant} & \\multirow{2}{*}{Approx.} & \\multicolumn{2}{c}{Upper bound $\\widetilde O(\\cdot)$} & ~ & \\multicolumn{2}{c}{Lower bound $\\widetilde\\Omega(\\cdot)$} \\\\\n\\cline{4-5} \\cline{7-8}\n~ & ~ & ~ & Classical & Quantum & ~ & Classical & Quantum \\\\\n\\midrule[1.5pt]\n\\multirow{7}{*}{diameter} & ~ & exact & $n$~\\cite{HolzerW12,PelegRT12} & $\\sqrt{nD}$~\\cite{GallM18} & ~ & $n$~\\cite{FrischknechtHW12} & $\\sqrt[3]{nD^2}+\\sqrt n$~\\cite{MagniezN20} \\\\\n~ & ~ & $3\/2-\\varepsilon$ & $n$ & $\\sqrt{nD}$ & ~ & $n$~\\cite{AbboudCK16} & $\\sqrt n+D$~\\cite{GallM18} \\\\\n~ & ~ & $3\/2$ & $\\sqrt n+D$~\\cite{HolzerPRW14,AnconaCDEW20} & $\\sqrt[3]{nD}+D$~\\cite{GallM18} & ~ & open & open \\\\\n~ & weighted & exact & $n$~\\cite{BernsteinN19} & $n$ & ~ & $n$ & $n^{2\/3}$ \\\\\n~ & weighted & $(1,3\/2)$ & $n$ & \\textcolor{red}{$\\min\\left\\{n^{9\/10}D^{3\/10},n\\right\\}$ (This work)} & ~ & $n$ & \\textcolor{red}{$n^{2\/3}$ (This work)} \\\\\n~ & weighted & $2-\\varepsilon$ & $n$ & $\\min\\left\\{n^{9\/10}D^{3\/10},n\\right\\}$ & ~ & $n$~\\cite{HolzerP15} & $\\sqrt n+D$ \\\\\n~ & weighted & $2$ & $\\sqrt nD^{1\/4}+D$~\\cite{ChechikM20} & $\\sqrt nD^{1\/4}+D$ & ~ & open & open \\\\\n\\midrule\n\\multirow{6}{*}{radius} & ~ & exact & $n$~\\cite{HolzerW12,PelegRT12} & $\\sqrt{nD}$ & ~ & $n$ & $\\sqrt[3]{nD^2}+\\sqrt n$ \\\\\n~ & ~ & $3\/2-\\varepsilon$ & $n$ & $\\sqrt{nD}$ & ~ & $n$~\\cite{AbboudCK16} & $\\sqrt n+D$ \\\\\n~ & ~ & $3\/2$ & $\\sqrt n+D$~\\cite{AnconaCDEW20} & $\\sqrt n+D$ & ~ & open & open \\\\\n~ & weighted & exact & $n$~\\cite{BernsteinN19} & $n$ & ~ & $n$ & $n^{2\/3}$ \\\\\n~ & weighted & $(1,3\/2)$ & $n$ & \\textcolor{red}{$\\min\\left\\{n^{9\/10}D^{3\/10},n\\right\\}$ (This work)} & ~ & $n$ & \\textcolor{red}{$n^{2\/3}$ (This work)} \\\\\n~ & weighted & $2$ & $\\sqrt nD^{1\/4}+D$~\\cite{ChechikM20} & $\\sqrt nD^{1\/4}+D$ & ~ & open & open \\\\\n\\bottomrule[1.5pt]\n\\end{tabular}\n\\end{table} \n\\section{Preliminaries}\n\n\\subsection{Graph Notations}\n\\label{sec:graph_notations}\n\nGiven a weighted graph $(G,w)$ where $G=(V,E)$ and $w:E\\to\\mathbb N^+$.\nThe length of a path is defined to be the sum of weights of edges on it, and the distance between nodes $u$ and $v$, denoted by $d_{G,w}(u,v)$, is the least length over all paths between them.\nThe eccentricity of a node $u$ is denoted by $e_{G,w}(u)=\\max_{v\\in V}d_{G,w}(u,v)$.\nThe radius of weighted graph $(G,w)$, denoted by $R_{G,w}$, is the minimum eccentricity over all nodes, i.e., $R_{G,w}=\\min_{u\\in V}e_{G,w}(u)$, while the diameter of $(G,w)$, denoted by $D_{G,w}$, is the maximum eccentricity of nodes, or equally, the maximum distance between any two nodes, i.e., $D_{G,w}=\\max_{u\\in V}e_{G,w}(u)=\\max_{u,v\\in V}d_{G,w}(u,v)$.\nThe unweighted diameter of graph $G$ is denoted by $D_G=D_{G,w^\\star}$ where $w^\\star(e)=1$ for all $e\\in E$, which is an essential parameter when $G$ represents the underlying graph of a distributed network.\n\n\\subsection{CONGEST Model}\n\nIn the classical CONGEST model, the communication network is a graph $G=(V,E)$ with $n$ nodes, and every node is assigned with a unique identifier.\nEach node represents a processor with unlimited computational power, i.e., the consumption of any local computation in a single processor is ignored.\nEach edge connecting two nodes represents a communication channel with $B=O(\\log n)$ bits of bandwidth.\nIn this article, we further consider the weighted graph $(G,w)$ as underlying network, where the weight of each edge is initially known to both of its endpoints.\n\nFor quantum version of the CONGEST model defined in \\cite{ElkinKNP14}, adjacent nodes are allowed to exchange \\textit{qubits} (quantum bits), i.e., the classical channels are now quantum channels with the same bandwidth $B=O(\\log n)$.\nEach node can locally do some quantum computation, and distinct nodes may own qubits with entanglement.\nIn this paper, we assume that initially all nodes do not share any entanglement, but the nodes can, for example, locally create a pair of entangled qubits, and send one to others.\n\nFor both classical and quantum CONGEST models, the algorithm is implemented round by round in a synchronous manner.\nIn each round, each node sends\/receives a message of $O(\\log n)$ (qu)bits to\/from each neighbor, and then does local computation according to local knowledge.\nThe algorithm halts when all nodes halt, and at the end of the algorithm, each node has its own output. We say an algorithm computes the diameter\/radius if all nodes output the correct answer.\nThe round complexity of an algorithm in this model is defined to be the number of communication rounds needed.\nAnd the round complexity of a distributed problem is the least round complexity of any algorithm solving it.\nOur focus here is the distance problems, mainly the computation of diameter and radius mentioned in Section~\\ref{sec:graph_notations}.\n\n\\subsection{Server Model}\n\nThe Sever model  is a variant of the two-party communication model, which was introduced by Elkin et al.~\\cite{ElkinKNP14} to prove lower bounds in the CONGEST model.\nThere are three players in the Server model: Alice, Bob, and the server.\nAlice and Bob receive the inputs $x$ and $y$ respectively, and want to compute $F(x,y)$ for some function $F$.\nThe server receives no input.\nAlice and Bob can exchange messages with the server.\nThe catch here is that the server can send messages for free.\nThus, the communication complexity counts only messages sent by Alice and Bob.\nNote that Alice and Bob can talk to each other by considering the server as a communication channel, so any protocol in the traditional two-party communication model can be implemented in the Server model with the same complexity.\n\nFor a two-argument function $F$ and $0\\le\\varepsilon<1$, we let $Q^{sv}_\\varepsilon(F)$ denote the communication complexity (in the quantum setting) of computing $F$ where for any inputs $x,y$, the algorithm must output $F(x,y)$ with probability at least $1-\\varepsilon$.\nFor Boolean function $f:\\{0,1\\}^n\\rightarrow\\mathbb{R}$ and $0\\le\\varepsilon<1$, the $\\varepsilon$-approximate degree of $f$, denoted by $\\text{deg}_\\varepsilon(f)$, is the smallest degree of any polynomial $p$ that $\\varepsilon$-represents $f$, i.e., $|p(x)-f(x)|\\le\\varepsilon$ for any input $x\\in\\{0,1\\}^n$.\n\\section{Algorithm}\n\nWe first introduce the framework of distributed quantum optimization in \\cite{GallM18}.\nGiven function  $f:X\\to\\mathbb Z$, where $X$ is a finite set, let $G=(V,E)$ be a network with a pre-defined node $\\text{leader}\\in V$.\nWe write $\\ket\\phi_v$ to denote a state in the memory space of node $v$.\nA specific register $\\ket\\cdot_I$ called \\textit{internal} and the control of the algorithm are centralized by the node leader.\nAssume that the following three quantum procedures are given as black boxes.\n\\begin{itemize}\n\\item {\\bf Initialization:} Prepare an initial state $\\ket0_I\\ket{\\text{init}}$ with some pre-computed information $\\ket{\\text{init}}$.\n\\item {\\bf Setup:} Produce a superposition from the initial state:\n$$\\ket0_I\\ket{\\text{init}}\\mapsto\\sum_{x\\in X}\\alpha_x\\ket x_I\\ket{\\text{data}(x)}\\ket{\\text{init}},$$\nwhere the $\\alpha_x$'s are arbitrary amplitudes and $\\text{data}(x)$ are information depending on $x$.\n\\item {\\bf Evaluation:} Perform the transformation\n$$\\ket{x,0}_I\\ket{\\text{data}(x)}\\ket{\\text{init}}\\mapsto\\ket{x,f(x)}_I\\ket{\\text{data}(x)}\\ket{\\text{init}}.$$\n\\end{itemize}\n\nThe following lemma provides an algorithm to search $x\\in X$ with high value $f(x)$ given the three procedures above.\n\n\\begin{lemma}[Theorem 2.4 in \\cite{GallM18}\\footnote{Although Le~Gall and Magniez write a slightly weaker statement, the lemma we claim here can be proven by the same argument in \\cite{GallM18}.}]\nAssume that {\\rm Initialization} can be implemented within $T_0$ rounds in the quantum CONGEST model, and that unitary operators {\\rm Setup} and {\\rm Evaluation} and their inverses can be implemented within $T$ rounds.\nLet $\\rho>0$ be such that $\\sum_{x\\in X:f(x)\\ge M}|\\alpha_x|^2\\ge\\rho$ where $M$ is unknown to all nodes.\nThen, for any $\\delta>0$, the node {\\rm leader} can find, with probability at least $1-\\delta$, some element $x$ such that $f(x)\\ge M$, in $T_0+O\\left(\\sqrt{\\log(1\/\\delta)\/\\rho}\\right)\\times T$ rounds.\n\\label{lem:optimization}\n\\end{lemma}\n\nThe three procedures will be described as deterministic or randomized procedures that combine the subroutines provided by Nanongkai~\\cite{Nanongkai14STOC} (also presented in Appendix~\\ref{sec:toolkits}).\nThey can be quantized using the standard technique \\cite{Bennett89}, with potentially additional garbage whose size is of the same order as the initial memory space.\n\nGiven a weighted graph $(G,w)$ where $G=(V,E)$ is a network and $w:E\\to\\mathbb N^+$, we show a quantum algorithm approximating $D_{G,w}$ and $R_{G,w}$ by proving Theorem~\\ref{thm:upper_bound}.\nWe only show the algorithm approximating the diameter.\nThe proof for radius is basically the same except that it finds the minimum (approximate) eccentricity instead of the maximum one.\n\nWe choose the parameters throughout this section.\n\\begin{equation}\n\\label{eqn:alg_parameters}\n\\varepsilon=1\/\\log n,r=n^{2\/5}D_G^{-1\/5},\\ell=n\\log n\/r,k=\\sqrt{D_G}.\n\\end{equation}\nAs mentioned in Section~\\ref{sec:results}, finding a node with maximum eccentricity among all nodes by directly applying a quantum search algorithm can hardly be done in $o(n)$ rounds.\nWe instead try to find a vertex set containing a node with maximum approximate eccentricity among $n$ vertex sets $S_1,\\cdots,S_n$, and then search such a node in this vertex set.\nEach set $S_i$ for $i\\in[1,n]$ is sampled by having each node $v\\in V$ join it independely with probability $r\/n$.\nFor such a random set and a node $s$ in it, Nanongkai showed in \\cite{Nanongkai14STOC} an efficient classical procedure to approximate its eccentricity (actually every node $v\\in V$ can know an approximation of the distance from $s$ to $v$).\n\n\n\n\\subsection{Computation of Approximate Eccentricity}\n\\label{sec:approx_ecc}\n\nFor convenience, we need to introduce several graph notations.\nGiven a weighted graph $(G,w)$, the hop distance between nodes $u$ and $v$, denoted by $h_{G,w}(u,v)$, is the minimum number of edges over all shortest paths between them.\nThe hop diameter of the weighted graph, denoted by $H_{G,w}$, is the maximum hop distance between any two nodes.\nFor $\\ell>0$, the $\\ell$-hop distance between $u$ and $v$, denoted by $d^\\ell_{G,w}(u,v)$, is the least length over all paths between them containing at most $\\ell$ edges.\nNote that $d^\\ell_{G,w}(u,v)=d_{G,w}(u,v)$ when $h_{G,w}(u,v)\\le\\ell$.\n\nIn general, Nanongkai~\\cite{Nanongkai14STOC} would approximate the bounded-hop distance, and sample a random set of key nodes as skeleton.\nThen it could approximate the distance from any key node $s$ to any node $v$ since, with high probability, any shortest path from $s$ to $v$ can be partitioned into bounded-hop shortest paths between key nodes, along with a tail path from some key node to $v$, as long as the number of key nodes is sufficiently large.\n\nHere we only list the necessary definitions of approximate bounded-hop distance, approximate distance, and approximate eccentricity.\nWe claim that these are good approximations.\nThe algorithms evaluating these quantities are presented in Appendix~\\ref{sec:toolkits}, and the detailed proof should be found in \\cite[arXiv version]{Nanongkai14STOC}.\nNote that we are given a weighted graph $(G,w)$ where $G=(V,E)$ and $w:E\\to\\mathbb N^+$.\n\n\\begin{lemma}[Theorem 3.3 in \\cite{Nanongkai14STOC}]\nGiven an integer $\\ell>0$.\nFor integer $i\\ge0$, define $w_i:E\\to\\mathbb N^+$ where $w_i(e)=\\left\\lceil\\frac{2\\ell w(e)}{\\varepsilon\\cdot2^i}\\right\\rceil$ for $e\\in E$.\nFor any $u,v\\in V$, the approximate bounded-hop distance is defined as\n$$\\widetilde d^\\ell_{G,w}(u,v)=\\min_i\\left\\{d_{G,w_i}(u,v)\\cdot\\frac{\\varepsilon\\cdot2^i}{2\\ell}:d_{G,w_i}(u,v)\\le\\left(1+\\frac2\\varepsilon\\right)\\ell\\right\\}.$$\nThen $d_{G,w}(u,v)\\le\\widetilde d^\\ell_{G,w}(u,v)\\le(1+\\varepsilon)d^\\ell_{G,w}(u,v)$.\n\\label{lem:bounded_hop_distance}\n\\end{lemma}\n\n\\begin{lemma}[Theorem 4.2 in \\cite{Nanongkai14STOC}]\nGiven a vertex set $S\\subseteq V$.\nLet the weighted complete graph $\\left(G'_S,w'_S\\right)$ be such that\n$$\n\\begin{aligned}\n& G'_S=\\left(S,\\tbinom{S}{2}\\right),w'_S:\\tbinom{S}{2}\\to\\mathbb N^+, \\\\\n& w'_S(\\{u,v\\})=\\widetilde d^\\ell_{G,w}(u,v),\\forall\\{u,v\\}\\in \\tbinom{S}{2}.\n\\end{aligned}\n$$\nFor node $v\\in S$, let $N^k_S(v)$ be the set of the $k$ nodes with the least distance from $v$ on $(G'_S,w'_S)$.\nAnd let the weighted complete graph $\\left(G''_S,w''_S\\right)$ be such that\n$$\n\\begin{aligned}\n& G''_S=\\left(S,\\tbinom{S}{2}\\right),w''_S:\\tbinom{S}{2}\\to\\mathbb N^+, \\\\\n& w''_S(\\{u,v\\})=\n\\begin{cases}\nd_{G'_S,w'_S}(u,v),\t& u\\in N^k_S(v)\\text{ or }v\\in N^k_S(u) \\\\\nw'_S(\\{u,v\\}), \t& \\text{otherwise}\n\\end{cases},\\forall\\{u,v\\}\\in \\tbinom{S}{2}.\n\\end{aligned}\n$$\nFor any $s\\in S$ and $v\\in V$, the approximate distance is defined as\n$$\\widetilde d_{G,w,S}(s,v)=\\min_{u\\in S}\\left\\{\\widetilde d^{4|S|\/k}_{G''_S,w''_S}(s,u)+\\widetilde d^\\ell_{G,w}(u,v)\\right\\}.$$\nIf $\\ell=n\\log n\/r$ and $S$ is sampled by having each node $v\\in V$ join it independetly with probability $r\/n$, then $d_{G,w}(s,v)\\le\\widetilde d_{G,w,S}(s,v)\\le(1+\\varepsilon)^2d_{G,w}(s,v)$ for all $s\\in S$ and $v\\in V$, with probability at least $1-2^{-cn}$, for some constant $c>0$ and sufficiently large $n$.\n\\label{lem:approx_distance}\n\\end{lemma}\n\n\\noindent\n{\\bf Remark.} We briefly explain why $\\widetilde d_{G,w,S}(\\cdot)$ is a good approximation.\nBy the choice of $\\ell$ and $S$, Lemma 4.3 in \\cite{Nanongkai14STOC} says that, with high probability, any $s\\in S,v\\in V$ and shortest path $\\left(s\\leadsto v\\right)$ on $(G,w)$ is of the form $\\left(s=s_1\\leadsto\\cdots\\leadsto s_m=u\\leadsto v\\right)$ such that $s_i\\in S$ for $i\\in[1,m]$, $h_{G,w}(s_{i-1},s_i)\\le\\ell$ for $i\\in[2,m]$, and $h_{G,w}(u,v)\\le\\ell$.\nApparently $\\widetilde d_{G,w,S}\\ge d_{G,w}(s,v)$.\nOn the other side,\n\\begin{equation*}\n\\begin{split}\n\\widetilde d_{G,w,S}(s,v) & =\\widetilde d^{4|S|\/k}_{G''_S,w''_S}(s,u)+\\widetilde d^\\ell_{G,w}(u,v) \\\\\n& \\le(1+\\varepsilon)d^{4|S|\/k}_{G''_S,w''_S}(s,u)+\\widetilde d^\\ell_{G,w}(u,v) \\\\\n& =(1+\\varepsilon)d_{G''_S,w''_S}(s,u)+\\widetilde d^\\ell_{G,w}(u,v) \\\\\n& \\le(1+\\varepsilon)\\sum_{i=2}^mw'_S(\\{s_{i-1},s_i\\})+\\widetilde d^\\ell_{G,w}(u,v) \\\\\n& \\le(1+\\varepsilon)\\sum_{i=2}^m\\widetilde d^\\ell_{G,w}(s_{i-1},s_i)+\\widetilde d^\\ell_{G,w}(u,v) \\\\\n& \\le(1+\\varepsilon)^2\\left(\\sum_{i=2}^md^\\ell_{G,w}(s_{i-1},s_i)+d^\\ell_{G,w}(u,v)\\right) \\\\\n& =(1+\\varepsilon)^2\\left(\\sum_{i=2}^md_{G,w}(s_{i-1},s_i)+d_{G,w}(u,v)\\right) \\\\\n& =(1+\\varepsilon)^2d_{G,w}(s,v).\n\\end{split}\n\\end{equation*}\nThe second and sixth lines are due to Lemma~\\ref{lem:bounded_hop_distance}.\nThe third line is due to Theorem 3.10 in \\cite{Nanongkai14STOC}, which says that $H_{G''_S,w''_S}<4|S|\/k$ since $(G''_S,w''_S)$ is the {\\it $k$-shortcut graph} of $(G'_S,w'_S)$.\n\n\\bigskip\n\nFor $i\\in[1,n]$, we rewrite $G''_{S_i},w''_{S_i},\\widetilde d_{G,w,S_i}(\\cdot)$ as $G''_i,w''_i,\\widetilde d_{G,w,i}(\\cdot)$ for short.\nFor any $s\\in S_i$, the approximate eccentricity is defined as $\\widetilde e_{G,w,i}(s)=\\max_{v\\in V}\\widetilde d_{G,w,i}(s,v)$.\nDefine two good events:\n\\begin{itemize}\n\\item {\\bf Good-Scale:} For all $i\\in[1,n]$, $|S_i|=\\Theta(r)$.\nBesides, let $v^\\star\\in V$ be a node with maximum eccentricity, i.e., $e_{G,w}(v^\\star)=D_{G,w}$, then $v^\\star$ joins $\\beta=\\Theta(r)$ sets $S_{i_1},\\cdots,S_{i_\\beta}$.\n\\item {\\bf Good-Approximation:} For all $i\\in[1,n]$ and $s\\in S_i,v\\in V$, $d_{G,w}(s,v)\\le\\widetilde d_{G,w,i}(s,v)\\le(1+\\varepsilon)^2d_{G,w}(s,v)$, thus $e_{G,w}(s)\\le\\widetilde e_{G,w,i}(s)\\le(1+\\varepsilon)^2e_{G,w}(s)$.\n\\end{itemize}\nBy Chernoff inequality and a union bound, the event Good-Scale occurs with probability at least $1-1\/\\text{poly}(n)$.\nBy Lemma~\\ref{lem:approx_distance}\nand a union bound, the event Good-Approximation occurs with probability at least $1-1\/\\text{poly}(n)$.\nTherefore, we can assume that the two events all happen in the following context.\n\n\n\n\\section{Quantization}\n\nFor each $i\\in[1,n]$, we define $f_i:S_i\\to\\mathbb Z$ where $f_i(s)=\\widetilde e_{G,w,i}(s)$ for $s\\in S_i$, and  $f:[1,n]\\to\\mathbb Z$ where $f(i)=\\max_{s\\in S_i}f_i(s)$ for $i\\in[1,n]$.\n\n\\begin{lemma}\nThe number of $i\\in[1,n]$ satisfying $f(i)\\ge D_{G,w}$ is $\\Theta(r)$.\nMoreover, $f(i)\\le(1+\\varepsilon)^2D_{G,w}$  for all $i\\in[1,n]$.\n\\label{lem:approximation}\n\\end{lemma}\n\n\\begin{proof}\n$$\n\\begin{aligned}\n& f(i_j)=\\max_{s\\in S_{i_j}}\\widetilde e_{G,w,i_j}(s)\\ge\\max_{s\\in S_{i_j}}e_{G,w}(s)\\ge e_{G,w}(v^\\star)=D_{G,w}, & \\forall j\\in[1,\\beta]; \\\\\n& f(i)=\\max_{s\\in S_i}\\widetilde e_{G,w,i}(s)\\le\\max_{s\\in S_i}(1+\\varepsilon)^2e_{G,w}(s)\\le(1+\\varepsilon)^2D_{G,w}, & \\forall i\\in[1,n].\n\\end{aligned}\n$$\n\\end{proof}\n\n\\begin{lemma}\nGiven $i\\in[1,n]$, there exists a quantum procedure performing the transformation\n$$\\bigotimes_{v\\in V}\\ket i_v\\ket0_{{\\rm leader}}\\mapsto\\bigotimes_{v\\in V}\\ket i_v\\ket{f(i)}_{{\\rm leader}}$$\nin the quantum CONGEST model, and taking $\\widetilde O\\left(D_G+\\frac n{\\varepsilon\\cdot r}+rk+\\sqrt r\\left(\\frac r{\\varepsilon\\cdot k}\\cdot D_G+r\\right)\\right)$ rounds, with probability at least $1-1\/{\\rm poly}(n)$.\n\\label{lem:evaluation}\n\\end{lemma}\n\n\\begin{proof}\nWe give the quantum procedure maximizing $f_i$ (thus evaluating $f(i)$) by following the framework of distributed quantum optimization:\n\\begin{itemize}\n\\item {\\bf $\\text{Initialization}_i$:} Perform the transformation\n$$\n\\bigotimes_{v\\in V}\\ket i_v\\mapsto\\bigotimes_{v\\in V}\\ket i_v\\ket0_I\\ket{\\text{init}_i},$$\nwhere\n$$\\ket{\\text{init}_i}=\\bigotimes_{v\\in V,u\\in S_i}\\left|\\widetilde d^\\ell_{G,w}(u,v)\\right\\rangle_v\\ket{G''_i,w''_i},$$\nand $d^\\ell_{G,w}(u,v)$ is given in Lemma~\\ref{lem:bounded_hop_distance}, $G''_i$ and $w''_i$ are given in lemma~\\ref{lem:approx_distance}.\n\n\\item {\\bf $\\text{Setup}_i$:} Perform the transformation\n$$\\bigotimes_{v\\in V}\\ket i_v\\ket0_I\\ket{\\text{init}_i}\\mapsto\\bigotimes_{v\\in V}\\ket i_v\\left(\\sum_{s\\in S_i}\\frac1{|S_i|}\\ket s_I\\ket{\\text{data}_i(s)}\\right)\\ket{\\text{init}_i},$$\nwhere\n$\\ket{\\text{data}_i(s)}=\\bigotimes_{v\\in V}\\ket s_v\\bigotimes_{v\\in V,u\\in S_i}\\left|\\widetilde d_{G''_i,w''_i}(s,u)\\right\\rangle_v$.\n\n\\item {\\bf $\\text{Evaluation}_i$:} Perform the transformation\n$$\\bigotimes_{v\\in V}\\ket i_v\\left(\\ket{s,0}_I\\ket{\\text{data}_i(s)}\\right)\\ket{\\text{init}_i}\\mapsto\\bigotimes_{v\\in V}\\ket i_v\\left(\\ket{s,f_i(s)}_I\\ket{\\text{data}_i(s)}\\right)\\ket{\\text{init}_i}.$$\n\\end{itemize}\n\nWe now analyze the round complexity:\n\\begin{itemize}\n\\item In $\\widetilde O\\left(D_G+\\frac n{\\varepsilon\\cdot r}+r\\right)$ rounds, each $v\\in V$ can know $\\widetilde d^\\ell_{G,w}(u,v)$ for each $u\\in S_i$, with high probability, due to Lemma~\\ref{lem:bounded_hop_mssp}.\nAfter that, the overlay network $\\left(G''_i,w''_i\\right)$ can be embedded in $O(D_G+rk)$ rounds due to Lemma~\\ref{lem:embedding} (we say that the network $G=(V,E)$ embeds an overlay network $G'=(V',E')$ with a weight function $w':E'\\to\\mathbb N^+$ if $V'\\subseteq V$ and for each $v\\in V'$, it stores each $e\\in E'$ incident to $v$ along with $w'(e)$ in the local memory).\nTherefore, the procedure $\\text{Initialization}_i$ can be implemented in $T_0=\\widetilde O\\left(D_G+\\frac n{\\varepsilon\\cdot r}+rk\\right)$ rounds.\n\\item The node leader can collect $S_i$ in $O(D_G+r)$ rounds.\nIt then prepares the quantum state $\\sum_{s\\in S_i}\\frac1{|S_i|}\\ket s_I$ and broadcasts to all nodes using CNOT copies, in $O(D_G)$ rounds.\nThus, the transformation\n$$\\bigotimes_{v\\in V}\\ket i_v\\ket0_I\\ket{\\text{init}_i}\\mapsto\\bigotimes_{v\\in V}\\ket i_v\\left(\\sum_{s\\in S_i}\\frac1{|S_i|}\\ket s_I\\bigotimes_{v\\in V}\\ket s_v\\right)\\ket{\\text{init}_i}$$\ncan be implemented in $O(D_G+r)$ rounds.\nBesides, the transformation\n$$\\bigotimes_{v\\in V}\\ket i_v\\bigotimes_{v\\in V}\\ket s_v\\ket{\\text{init}_i}\\mapsto\\bigotimes_{v\\in V}\\ket i_v\\bigotimes_{v\\in V}\\ket s_v\\bigotimes_{v\\in V,u\\in S_i}\\left|\\widetilde d^{4|S_i|\/k}_{G''_i,w''_i}(s,u)\\right\\rangle_v\\ket{\\text{init}_i}$$\ncan be implemented in $T_1=\\widetilde O\\left(\\frac r{\\varepsilon\\cdot k}\\cdot D_G+r\\right)$ rounds since Lemma~\\ref{lem:sssp_on_overlay} implies that, after the overlay network $\\left(G''_i,w''_i\\right)$ is embedded, each $v\\in V$ can know $\\widetilde d^{4|S_i|\/k}_{G''_i,w''_i}(s,u)$ for each $u\\in S_i$ within $T_1$ rounds.\nTherefore, the procedure $\\text{Setup}_i$ can be implemented in $T_1=\\widetilde O\\left(\\frac r{\\varepsilon\\cdot k}\\cdot D_G+r\\right)$ rounds.\n\\item For the procedure $\\text{Evaluation}_i$, recall that $f_i(s)=\\max_{v\\in V}\\widetilde d_{G,w,i}(s,v)$ where\n$$\\widetilde d_{G,w,i}(s,v)=\\min_{u\\in S_i}\\left\\{\\widetilde d^{4|S_i|\/k}_{G''_i,w''_i}(s,u)+\\widetilde d^\\ell_{G,w}(u,v)\\right\\}.$$\nBy definition, for any $v\\in V$ and $u\\in S_i$, $\\widetilde d^{4|S_i|\/k}_{G''_i,w''_i}(s,u)$ and $\\widetilde d^\\ell_{G,w}(u,v)$ have been stored in the local memory of $v$, i.e., $\\ket\\cdot_v$.\nThus, each $v\\in V$ can locally compute $\\widetilde d_{G,w,S_i}(s,v)$, and the node leader can compute the maximum by converge-casting in $O(D_G)$ rounds.\nSo the procedure $\\text{Evaluation}_i$ can be implemented in $T_2=O(D_G)$ rounds.\n\\end{itemize}\nBy Lemma~\\ref{lem:optimization}, there exists a quantum procedure maximizing $f_i$ in $\\widetilde O(T_0+\\sqrt r(T_1+T_2))$ rounds with high probability.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:upper_bound}]\nWe give a quantum procedure maximizing $f$ also by following the framework of distributed quantum optimization:\n\\begin{itemize}\n\\item \\textbf{Initialization} is a classical procedure which samples vertex sets $S_1,\\cdots,S_n$, and $\\ket{\\text{init}}$ represents the corresponding classical information.\n\\item \\textbf{Setup:} Perform the transformation\n$$\\ket0_I\\ket{\\text{init}}\\mapsto\\sum_{i=1}^n\\frac1n\\ket i_I\\ket{\\text{data}(i)}\\ket{\\text{init}},$$\nwhere $\\ket{\\text{data}(i)}=\\bigotimes_{v\\in V}\\ket i_v$.\n\\item \\textbf{Evaluation:} Perform the transformation\n$$\\ket{i,0}_I\\ket{\\text{data}(i)}\\ket{\\text{init}}\\mapsto\\ket{i,f(i)}_I\\ket{\\text{data}(i)}\\ket{\\text{init}}.$$\n\\end{itemize}\n\nWe now analyze the round complexity:\n\\begin{itemize}\n\\item $S_1,\\cdots,S_n$ are sampled locally in parallel, and the procedure Initialization is free, i.e., $T_0=0$.\n\\item The node leader prepares the quantum state $\\sum_{i=1}^n\\frac1n\\ket i_I$ and broadcast using CNOT copies to all nodes.\nTherefore, the procedure Setup can be implemented in $T_1=O(D_G)$ rounds.\n\\item The procedure Evaluation can be of $T_2=\\widetilde O\\left(D_G+\\frac n{\\varepsilon\\cdot r}+rk+\\sqrt r\\left(\\frac r{\\varepsilon\\cdot k}\\cdot D_G+r\\right)\\right)$ rounds by Lemma~\\ref{lem:evaluation}.\n\\end{itemize}\n\nBy Lemma~\\ref{lem:optimization} and Lemma~\\ref{lem:approximation}, there exists a quantum procedure that find, with high probability, some $i\\in[1,n]$ such that $D_{G,w}\\le f(i)\\le(1+\\varepsilon)^2D_{G,w}$, in\n$$\\widetilde O(T_0+\\sqrt{n\/r}(T_1+T_2))=\\widetilde O\\left(\\sqrt{n\/r}\\left(D_G+\\frac n{\\varepsilon\\cdot r}+rk+\\sqrt r\\left(\\frac r{\\varepsilon\\cdot k}\\cdot D_G+r\\right)\\right)\\right)$$\nrounds.\nBy the choice of the parameters in Eq.~\\eqref{eqn:alg_parameters}, Theorem~\\ref{thm:upper_bound} follows.\n\\end{proof}\n\\section{Toolkits in Nanongkai's Algorithm}\n\\label{sec:toolkits}\n\nLet $G=(V,E)$ be a distributed network with a weight function $w:E\\to\\mathbb N^+$ and a pre-defined node $\\text{leader}\\in V$.\nWe assume that each node initially knows $n=|V|$ and $W=\\max_{e\\in E}w(e)$.\nThe parameters $\\varepsilon,r,\\ell,k$ are chosen the same as in Eq.~\\eqref{eqn:alg_parameters}.\nWe follow the background of Section~\\ref{sec:approx_ecc}.\nGiven a vertex set $S\\subseteq V$, let $\\widetilde d^\\ell_{G,w}(\\cdot)$, $(G'_S,w'_S)$, $N^k_S(\\cdot)$, $(G''_S,w''_S)$, $\\widetilde d_{G,w,S}(\\cdot)$ be as defined in Lemma~\\ref{lem:bounded_hop_distance} and Lemma~\\ref{lem:approx_distance}.\nThe following lemmas and algorithms are summarized from \\cite[arXiv version]{Nanongkai14STOC}.\n\n\\begin{lemma}[Theorem 3.2 in \\cite{Nanongkai14STOC}]\nFor $s\\in V$ known to all nodes, there exists an algorithm (Algorithm~\\ref{alg:bounded_hop_sssp}) such that in $\\widetilde O(\\ell\/\\varepsilon)$ rounds, each $v\\in V$ knows $\\widetilde d^\\ell_{G,w}(s,v)$, and during the whole computation, each node broadcasts $O(\\log n)$ messages of size $O(\\log n)$ to its neighbors.\n\\label{lem:bounded_hop_sssp}\n\\end{lemma}\n\n\\begin{algorithm}\n\\caption{Bounded-Hop SSSP $(G,w,s,\\ell,\\varepsilon)$}\n\\label{alg:bounded_hop_sssp}\n\\begin{algorithmic}[1]\n\\Require Network $(G,w)$, source node $s$ and parameters $\\ell,\\varepsilon>0$.\n\\Ensure Each node $v$ knows $\\widetilde d^\\ell_{G,w}(s,v)$.\n\\State Initially, $\\widetilde d^\\ell_{G,w}(s,v)\\leftarrow\\infty$ for each $v\\in V$.\n\\For{$i=0$ to $\\log\\frac{2nW}\\varepsilon$}\n\\State Run bounded-distance SSSP with parameters $(G,w_i,s,(1+2\/\\varepsilon)\\ell)$ using Algorithm~\\ref{alg:bounded_distance_sssp}.\n\\For{each $v\\in V$} in parallel\n\\If{$d_{G,w_i}(s,v)\\le(1+2\/\\varepsilon)\\ell$}\n\\State $\\widetilde d^\\ell_{G,w}(s,v)\\leftarrow\\min\\left\\{\\widetilde d^\\ell_{G,w}(s,v),d_{G,w_i}(s,v)\\right\\}$.\n\\EndIf\n\\EndFor\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}\n\\caption{Bounded-Distance SSSP $(G,w,s,L)$}\n\\label{alg:bounded_distance_sssp}\n\\begin{algorithmic}[1]\n\\Require Network $(G,w)$, source node $s$ and parameter $L>0$.\n\\Ensure Each node $v$ knows whether $d_{G,w}(s,v)\\le L$, and if so, it further knows $d_{G,w}(s,v)$.\n\\State Initially, $d_{G,w}(s,s)\\leftarrow0$ and $d_{G,w}(s,v)\\leftarrow\\infty$ for each $v\\ne s$.\n\\State Let $t$ be the time this algorithm starts.\n\\For{round $r=t$ to $t+L$}\n\\For{each $v\\in V$} in parallel\n\\For{each message $(u,d_{G,w}(s,u))$ received in the previous round}\n\\If{$d_{G,w}(s,u)+w(\\{u,v\\})\\le L$}\n\\State $d_{G,w}(s,v)\\leftarrow\\min\\left\\{d_{G,w}(s,v),d_{G,w}(s,u)+w(\\{u,v\\})\\right\\}$.\n\\EndIf\n\\EndFor\n\\If{$d_{G,w}(s,v)=r-t$}\n\\State $v$ broadcasts message $(v,d_{G,w}(s,v))$ to all neighbors.\n\\EndIf\n\\EndFor\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{lemma}[Theorem 3.6 and Lemma 3.7 in \\cite{Nanongkai14STOC}]\nThere exist an algorithm (Algorithm~\\ref{alg:bounded_hop_mssp}) such that in $\\widetilde O(D_G+\\ell\/\\varepsilon+|S|)$ rounds, each node $v\\in V$ knows $\\widetilde d^\\ell_{G,w}(s,v)$ for each $s\\in S$, with probability of failure at most $n^{-c}$, for any constant $c>0$ and sufficiently large $n$.\n\\label{lem:bounded_hop_mssp}\n\\end{lemma}\n\n\\begin{algorithm}[!h]\n\\caption{Bounded-Hop Multi-Source Shortest Paths $(G,w,S,\\ell,\\varepsilon)$}\n\\label{alg:bounded_hop_mssp}\n\\begin{algorithmic}[1]\n\\Require Network $(G,w)$, set of source nodes $S$ and parameters $\\ell,\\varepsilon>0$.\n\\Ensure With high probability, each node $v$ knows $\\widetilde d^\\ell_{G,w}(s,v)$ for each $s\\in S$.\n\\State Assume that $S=\\{s_1,\\cdots,s_b\\}$.\nLet $\\mathcal A_i$ be the Algorithm~\\ref{alg:bounded_hop_sssp} with parameters $(G,w,s_i,\\ell,\\varepsilon)$ for each $i\\in[1,k]$ (each $\\mathcal A_i$ is of $T=\\widetilde O(\\ell\/\\varepsilon)$ rounds, and during the whole computation of $\\mathcal A_i$, each node broadcasts $O(\\log n)$ messages to its neighbors due to Lemma~\\ref{lem:bounded_hop_sssp}).\n\\State The node leader samples $\\Delta_1,\\cdots,\\Delta_b\\in[0,b\\log n]$ independently and uniformly at random for delaying algorithms $\\mathcal A_1,\\cdots,\\mathcal A_k$, and broadcasts them by pipelining in $O(D_G+b)$ rounds.\n\\For{$r=1$ to $T+b\\log n$}\n\\For{each $v\\in V$} in parallel\n\\State Let $a=|\\ \\{i\\in[1,b]:v\\text{ broadcasts a message in the }(r-\\Delta_i)\\text{-th round of }\\mathcal A_i\\}\\ |$.\n\\If{$a\\le\\lceil\\log n\\rceil$}\n\\State $v$ broadcasts these $a$ messages in the next $\\lceil\\log n\\rceil$ rounds.\n\\Else\n\\State The algorithm fails.\n\\EndIf\n\\EndFor\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\n\\newpage\n\n\\begin{lemma}[Theorem 4.5 in \\cite{Nanongkai14STOC}]\nAfter the overlay network $(G'_S,w'_S)$ is embedded, there exists an algorithm (Algorithm~\\ref{alg:embedding}) which further embeds the overlay network $(G''_S,w''_S)$ in $\\widetilde O(D_G+|S|k)$\nrounds.\n\\label{lem:embedding}\n\\end{lemma}\n\n\\begin{algorithm}\n\\caption{Embedding Overlay Network $(G,w,S,G'_S,w'_S,k)$}\n\\label{alg:embedding}\n\\begin{algorithmic}[1]\n\\Require Network $(G,w)$, set of source nodes $S$, overlay network $(G'_S,w'_S)$ and parameter $k>0$.\n\\Ensure It embeds the overlay network $(G''_S,w''_S)$.\n\\State Each node $s\\in S$ broadcasts the $k$ shortest edges incident to it on $(G'_S,w'_S)$ (this can be done in $O(D_G+|S|k)$ rounds).\n\\For{each $s\\in S$} locally\n\\State $s$ computes $N^k_S(s)$, along with the weight $w''_S(\\{s,v\\})=d_{G'_S,w'_S}(s,v)$ for each $v\\in N^k_S(s)$ (this can be done due to Observation 3.12 in \\cite{Nanongkai14STOC}).\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{lemma}[Lemma 4.6 in \\cite{Nanongkai14STOC}]\nFor node $s\\in S$ known to all nodes, after the overlay network $(G''_S,w''_S)$ is embedded, there exists an algorithm (Algorithm~\\ref{alg:sssp_on_overlay}) such that in $\\widetilde O\\left(\\frac{|S|}{\\varepsilon\\cdot k}\\cdot D_G+|S|\\right)$ rounds, each node $v\\in V$ knows for each $u\\in S$ the value of $\\widetilde d^{4|S|\/k}_{G''_S,w''_S}(s,u)$.\n\\label{lem:sssp_on_overlay}\n\\end{lemma}\n\n\\begin{algorithm}\n\\caption{SSSP on Overlay Network $(G,w,S,\\varepsilon,k,G''_S,w''_S,s)$}\n\\label{alg:sssp_on_overlay}\n\\begin{algorithmic}[1]\n\\Require Network $(G,w)$, set of source nodes $S$, parameters $\\varepsilon,k>0$, overlay network $(G''_S,w''_S)$ and source node $s\\in S$.\n\\Ensure Each node $v$ knows $\\widetilde d^{4|S|\/k}_{G''_S,w''_S}(s,u)$ for each $u\\in S$.\n\\State Let $\\mathcal A$ be the Algorithm~\\ref{alg:bounded_hop_sssp} with parameters $(G''_S,w''_S,s,4|S|\/k,\\varepsilon)$ ($\\mathcal A$ is of $T=\\widetilde O\\left(\\frac{|S|}{\\varepsilon\\cdot k}\\right)$ rounds, and during the whole computation of $\\mathcal A$, each node broadcasts $O(\\log n)$ messages to its neighbors due to Lemma~\\ref{lem:bounded_hop_sssp}).\n\\For{$r=1$ to $T$}\n\\State Let $a$ be the number of nodes in $G''_S$ that want to broadcast a message to its neighbors in $G''_S$ in the $r$-th round of $\\mathcal A$.\nCount $a$ and make every nodes in $G$ knows $a$ in $O(D_G)$ rounds.\n\\State Each node in $G''_S$, which wants to send a message to each of its neighbors in $G''_S$, broadcasts such message to all nodes in $G$ (this takes $O(D_G+a)$ rounds).\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\\section*{Acknowledgements}\nWe thank the anonymous reviewers' feedback.\nThis work was supported in part by, the National Key R\\&D Program of China 2018YFB1003202, National Natural Science Foundation of China (Grant No. 61972191), the Program for Innovative Talents and Entrepreneur in Jiangsu, and Anhui Initiative in Quantum Information Technologies (Grant No. AHY150100).\n\n\\bibliographystyle{acm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and main results}\nThe study of real-valued topological cocycles and real skew product extensions has been initiated by Besicovitch, Gottschalk, and Hedlund.\nBesicovitch \\cite{Be} proved the existence of point transitive real skew product extensions of an irrational rotation on the one-dimensional torus.\nFurthermore, he proved that none of them is minimal, i.e. there are always non-transitive points for a point transitive real skew product extension.\nThe main result in Chapter 14 of \\cite{G-H} can be rephrased to the dichotomy that a topologically conservative real skew product extension of a minimal rotation on a torus (finite or infinite dimensional) is either point transitive or it is defined by a topological coboundary and almost periodic.\nThis result and a generalisation to skew product extensions of a Kronecker transformation (cf. \\cite{LM}) exploit the isometric behaviour of a minimal rotation.\nA corresponding result apart from isometries is based on homotopy conditions for the class of distal minimal homeomorphisms usually called Furstenberg transformations (cf. \\cite{Gr}).\nHowever, in general this dichotomy is not valid, and counterexamples can be provided by the Rokhlin skew products of the so-called topological type $III_0$.\nThis motivates the study of topologically conservative real skew product extensions of compact flows apart from isometries and toral extensions, which is carried out in this note for \\emph{distal} minimal flows with Abelian compactly generated acting groups.\n\nThroughout this note we shall denote by $T$ a \\emph{compactly generated Abelian} Hausdorff topological group acting continuously on a compact metric phase space $(X,d)$ so that $(X,T)$ is a \\emph{compact metric flow}.\nIn the monograph \\cite{G-H} such an acting group $T$ is called \\emph{generative}, and notions of recurrence are provided for such Abelian acting groups apart from $\\mathbb Z$ and $\\mathbb R$.\nFor a $\\mathbb Z$-action on $X$ we let $T$ be the self-homeomorphism of $X$ generating the action by $(n,x)\\mapsto T^n x$, while in the case of a real flow we shall use the notation $\\{\\phi^t:t\\in\\mathbb R\\}$ for the acting group.\nWe call a flow \\emph{minimal} if the whole phase space is the only non-empty invariant closed subset, and then for every $x\\in X$ the \\emph{orbit closure} $\\bar{\\mathcal O}_T(x)=\\overline{\\{\\tau x:\\tau\\in T\\}}$ is all of $X$.\nA flow $(X,T)$ is \\emph{topologically transitive} if for arbitrary open neighbourhoods $\\mathcal U,\\mathcal V\\subset X$ there exists some $\\tau\\in T$ with $\\tau\\mathcal U\\cap\\mathcal V\\neq\\emptyset$, and it is \\emph{weakly mixing} if the flow $(X\\times X,T)$ with the diagonal action is topologically transitive.\nFor a topologically transitive flow $(X,T)$ with complete separable metric phase space there exists a dense $G_\\delta$-set of \\emph{transitive points} $x$ with $\\bar{\\mathcal O}_T(x)=X$, and a flow with transitive points is \\emph{point transitive}.\nIf $(X,T)$ and $(Y,T)$ are flows with the same acting group $T$ and $\\pi:X\\longrightarrow Y$ is a continuous \\emph{onto} mapping with $\\pi(\\tau x)=\\tau\\pi(x)$ for every $\\tau\\in T$ and $x\\in X$, then $(Y,T)=\\pi(X,T)$ is called a factor of $(X,T)$ and $(X,T)$ is called an extension of $(Y,T)$.\nSuch a mapping $\\pi$ is called a \\emph{homomorphism} of the flows $(X,T)$ and $(Y,T)$.\nThe set of bicontinuous bijective homomorphisms of a flow $(X,T)$ onto itself is the topological group $\\textup{Aut}(X,T)$ of \\emph{automorphisms} of $(X,T)$ with the topology of uniform convergence.\nTwo points $x,y\\in X$ are called \\emph{distal} if\n\\begin{equation*}\n\\inf_{\\tau\\in T}d(\\tau x,\\tau y)> 0 ,\n\\end{equation*}\notherwise they are called \\emph{proximal}.\nFor a general compact Hausdorff flow $(X,T)$ distality of two points $x,y\\in X$ is defined by the absence of any nets $\\{\\tau_n\\}_{n\\in I}\\subset T$ with $\\lim \\tau_n x=\\lim \\tau_n y$.\nA flow is called distal if any two distinct points are distal, and an extension of flows is called distal if any two distinct points in the same fibre are distal.\nAn important property of distal compact flows is the \\emph{partitioning} of the phase space into invariant closed minimal subsets, even if the flow is not minimal.\n\nSuppose that $\\mathbb A$ is an Abelian locally compact second countable (Abelian l.c.s.) group with zero element $\\mathbf 0_\\mathbb A$, and let $\\mathbb A_\\infty$ denote its one point compactification with the convention that $g+\\infty=\\infty+g=\\infty$ for every $g\\in \\mathbb A$.\nA cocycle of a compact metric flow $(X,T)$ is a continuous mapping $f:T\\times X\\longrightarrow \\mathbb A$ with the identity\n\\begin{equation*}\nf(\\tau,\\tau' x)+f(\\tau',x)=f(\\tau\\tau',x)\n\\end{equation*}\nfor all $\\tau,\\tau'\\in T$ and $x\\in X$.\nGiven a compact metric $\\mathbb Z$-flow $(X,T)$ and a continuous function $f:X\\longrightarrow \\mathbb A$, we can define a cocycle $f:\\mathbb{Z}\\times X \\longrightarrow \\mathbb A$ with $f(1,\\cdot)\\equiv f$ by\n\\begin{equation*}\nf(n,x)=\n\\begin{cases}\n\\sum_{k=0}^{n-1}f(T^k x) & \\textup{if}\\enspace n\\geq 1 ,\n\\\\\n\\mathbf 0_\\mathbb A & \\textup{if}\\enspace n=0 ,\n\\\\\n-f(-n,T^n x) & \\textup{if}\\enspace n < 0.\n\\end{cases}\n\\end{equation*}\nMoreover, there is a natural occurrence of cocycles of $\\mathbb R$-flows as solutions to ODE's.\nSuppose that $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ is a smooth flow on a compact manifold $M$ and $A:M\\longrightarrow\\mathbb R$ is a smooth function.\nThen a continuous real valued cocycle $f(t,m)$ of the flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ is given by the fundamental solution to the ODE\n\\begin{equation*}\n\\frac{d\\, f(t,m)}{d\\, t}=A(\\phi^t(m))\n\\end{equation*}\nwith the initial condition $f(0,m)=0$.\nThe \\emph{skew product extension} of the flow $(X,T)$ by a cocycle $f:T\\times X\\longrightarrow \\mathbb A$ is defined by the homeomorphisms\n\\begin{equation*}\n\\widetilde\\tau_f(x,a)=(\\tau x, f(\\tau,x)+ a)\n\\end{equation*}\nof $X\\times \\mathbb A$ for all $\\tau\\in T$, which provide a continuous action $(\\tau,x,a)\\mapsto\\widetilde\\tau_f(x,a)$ of $T$ on $X\\times \\mathbb A$ by the cocycle identity.\nFor a $\\mathbb Z$-flow $(X,T)$ this action is generated by\n\\begin{equation*}\n\\widetilde T_f(x,a)=(T x, f(x)+ a) .\n\\end{equation*}\nThe essential property of a skew product is that the $T$-action on $X\\times\\mathbb A$ commutes with the right translation action of the group $\\mathbb A$ on $X\\times\\mathbb A$, which is defined by $$R_b(x,a)=(x,a-b)$$\n for every $b\\in\\mathbb A$.\nThe orbit closure of $(x,a)\\in X\\times\\mathbb A$ under $\\widetilde\\tau_f$ will be denoted by $\\bar{\\mathcal O}_{T,f}(x,a)=\\overline{\\{\\widetilde\\tau_f(x,a):\\tau\\in T\\}}$.\n\nThe \\emph{prolongation} $\\mathcal D_T(x)$ of $x\\in X$ under the group action of $T$ is defined by\n\\begin{equation*}\n\\mathcal D_T(x)=\\bigcap\\{\\bar{\\mathcal O}_T(\\mathcal U):\\mathcal U\\enspace\\textup{is an open neighbourhood of}\\enspace x\\},\n\\end{equation*}\nand we shall use the notation $\\mathcal D_{T,f}(x,a)$ for the prolongation of a point $(x,a)\\in X\\times \\mathbb A$ under the skew product action $\\widetilde\\tau_f$.\n\nWhile the inclusion of the orbit closure in the prolongation is obvious, the coincidence of these sets is generic by a result from the paper \\cite{Gl3}.\nThis result, one of our main tools, is usually referred to as ``topological ergodic decomposition''.\n\n\\begin{fact}\\label{fact:o_p}\nFor every compact \\emph{metric} flow $(X,T)$ there exists a $T$-invariant dense $G_\\delta$ set $\\mathcal F\\subset X$ so that for every $x\\in\\mathcal F$ holds\n\\begin{equation*}\n\\bar{\\mathcal O}_{T}(x)=\\mathcal D_{T}(x) .\n\\end{equation*}\nFor a skew product extension $\\widetilde\\tau_f$ of $(X,T)$ by a cocycle $f:T\\times X\\longrightarrow \\mathbb A$ there exists a $T$-invariant dense $G_\\delta$ set $\\mathcal F\\subset X$ so that for every $x\\in\\mathcal F$ and \\emph{every} $a\\in \\mathbb A$ holds\n\\begin{equation*}\n\\bar{\\mathcal O}_{T,f}(x,a)=\\mathcal D_{T,f}(x,a) .\n\\end{equation*}\nThis assertion holds as well for the extension of $\\widetilde\\tau_f$ to $X\\times \\mathbb A_\\infty$ which is defined by $(x,\\infty)\\mapsto (\\tau x,\\infty)$ for every $x\\in X$, and given an $\\mathbb R^2$-valued topological cocycle $g=(g_1,g_2):T\\times X\\longrightarrow\\mathbb R^2$ for the extension of $\\widetilde\\tau_g$ to $X\\times(\\mathbb R_\\infty)^2$ which is defined by $(x,s,\\infty)\\mapsto (\\tau x,s+g_1(x),\\infty)$, $(x,\\infty,t)\\mapsto (\\tau x,\\infty,t+g_2(x))$, and $(x,\\infty,\\infty)\\mapsto (\\tau x,\\infty,\\infty)$, for every $x\\in X$ and $s,t\\in\\mathbb R$.\n\\end{fact}\n\n\\begin{proof}\nThe statement for a compact metric phase space and a general acting group is according to Theorem 1 of \\cite{AkGl}.\nThe other statements can be verified by means of the extension of $\\widetilde\\tau_f$ onto the compactification of $X\\times\\mathbb A$.\nThe coincidence of $\\bar{\\mathcal O}_{T,f}(x,a)$ and $\\mathcal D_{T,f}(x,a)$ for some $(x,a)\\in X\\times \\mathbb A$ implies this coincidence for all $(x,a')\\in\\{x\\}\\times \\mathbb A_\\infty$, since the extension of $\\widetilde\\tau_f$ to $X\\times\\mathbb A_\\infty$ commutes with the right translation on $X\\times \\mathbb A_\\infty$.\n\\end{proof}\n\n\\begin{remark}\\label{rem:o_p}\nIf $y\\in\\bar{\\mathcal O}_{T}(x)$ and $z\\in\\bar{\\mathcal O}_{T}(y)$, then $z\\in\\bar{\\mathcal O}_{T}(x)$ follows by a diagonalisation argument.\nA corresponding statement for prolongations is not valid, however follows from $x\\in\\bar{\\mathcal O}_{T}(y)$ and $z\\in\\mathcal D_{T}(y)$ that $z\\in\\mathcal D_{T}(x)$.\n\\end{remark}\n\nWe shall consider more general Abelian acting groups than $\\mathbb Z$ and $\\mathbb R$, hence the definition of recurrence requires the notions of a replete semigroup and an extensive subset of the Abelian compactly generated group $T$ (cf. \\cite{G-H}).\nWe recall that a semigroup $P\\subset T$ is replete if for every compact subset $K\\subset T$ there exists a $\\tau\\in T$ with $\\tau K\\subset P$, and a subset $E\\subset T$ is extensive if it intersects every replete semigroup.\nTherefore, a subset $E$ of $T=\\mathbb Z$ or $T=\\mathbb R$ is extensive if and only if $E$ contains arbitrarily large positive \\emph{and} arbitrarily large negative elements.\n\n\\begin{defi}\nWe call a cocycle $f(\\tau,x)$ of a minimal compact metric flow $(X,T)$ \\emph{topologically recurrent} if for arbitrary neighbourhoods $\\mathcal U\\subset X$ and $U(\\mathbf 0_\\mathbb A)\\subset \\mathbb A$ of $\\mathbf 0_\\mathbb A$ there exists an extensive set of elements $\\tau\\in T$ with\n\\begin{equation*}\n\\mathcal U\\cap\\tau^{-1}(\\mathcal U)\\cap\\{x\\in X: f(\\tau,x)\\in U(\\mathbf 0_\\mathbb A)\\}\\neq\\emptyset .\n\\end{equation*}\nSince $\\widetilde\\tau_f$ and the right translation on $X\\times\\mathbb A$ commute, this is equivalent to the \\emph{regional recurrence} of the skew product action $\\widetilde\\tau_f$ on $X\\times\\mathbb A$, i.e. for every open neighbourhood $U\\subset X\\times \\mathbb A$ there exists an extensive set of elements $\\tau\\in T$ with $\\widetilde\\tau_f(U)\\cap U\\neq\\emptyset$.\nA non-recurrent cocycle is called \\emph{transient}.\n\nA point $(x,a)\\in X\\times \\mathbb A$ is $\\widetilde\\tau_f$-recurrent if for every neighbourhood $U\\subset X\\times \\mathbb A$ of $(x,a)$ the set of $\\tau\\in T$ with $\\widetilde\\tau_f(x,a)\\in U$ is extensive.\nMoreover, a point $(x,a)\\in X\\times\\mathbb A$ is \\emph{regionally} $\\widetilde\\tau_f$-recurrent if for every neighbourhood $U$ of $(x,a)$ the set of $\\tau\\in T$ with $\\widetilde\\tau_f(U)\\cap U\\neq\\emptyset$ is extensive.\n\\end{defi}\n\n\\begin{remarks}\\label{rems:rec}\nIf $f(\\tau,x)$ is recurrent, then by Theorems 7.15 and 7.16 in \\cite{G-H} there exists a dense $G_\\delta$ set of $\\widetilde\\tau_f$-recurrent points in $X\\times\\mathbb R$.\n\nGiven a regionally $\\widetilde\\tau_f$-recurrent point $(x,a)\\in X\\times\\mathbb A$, every point in $\\{x\\}\\times\\mathbb A$ is regionally $\\widetilde\\tau_f$-recurrent.\nThe minimality of $(X,T)$ and Theorem 7.13 in \\cite{G-H} imply that every point in $X\\times \\mathbb A$ is regionally $\\widetilde\\tau_f$-recurrent, hence $f(\\tau,x)$ is recurrent.\n\nA cocycle $f(n,x)$ of a $\\mathbb Z$-flow is topologically recurrent if and only if $\\widetilde T_f$ is topologically conservative, i.e. for every open neighbourhood $U\\subset X\\times\\mathbb A$ there exists an integer $n\\neq 0$ so that $\\widetilde T_f^n(U)\\cap U\\neq\\emptyset$.\n\\end{remarks}\n\nOne of the most important concepts in the study of cocycles is the essential range, originally introduced in the measure theoretic category by Schmidt \\cite{Sch}.\n\n\\begin{defi}\\label{def:er}\nLet $f(\\tau,x)$ be a cocycle of a minimal compact metric flow $(X,T)$.\nAn element $a\\in \\mathbb A$ is in the set $E(f)$ of \\emph{topological essential values} if for arbitrary neighbourhoods $\\mathcal U\\subset X$ and $U(a)\\subset \\mathbb A$ of $a$ there exists an element $\\tau\\in T$ so that\n\\begin{equation*}\n\\mathcal U\\cap\\tau^{-1}(\\mathcal U)\\cap\\{x\\in X: f(\\tau,x)\\in U(a)\\}\n\\end{equation*}\nis non-empty.\nThe set $E(f)$ is also called the \\emph{topological essential range}.\nThe cocycle identity implies that $f(\\mathbf 1_T,x)=\\mathbf 0_\\mathbb A$ for all $x\\in X$ and hence $\\mathbf 0_\\mathbb A\\in E(f)$.\nMoreover, the essential range is always a closed \\emph{subgroup} of $\\mathbb A$ (cf. \\cite{LM}, Proposition 3.1, which carries over from the case of a minimal $\\mathbb Z$-action to a general Abelian group acting minimally).\n\\end{defi}\n\n\\begin{fact}\\label{fact:er}\nIf $f(\\tau,x)$ is a cocycle with full topological essential range $E(f)=\\mathbb A$, then $\\mathcal D_{T,f}(x,a)\\subset\\{x\\}\\times\\mathbb A$ holds for every $(x,a)\\in X\\times\\mathbb A$.\nBy Fact \\ref{fact:o_p} there exists a $T$-invariant dense $G_\\delta$ set $\\mathcal F\\subset X$ with $\\{x\\}\\times \\mathbb A\\subset\\bar{\\mathcal O}_{T,f}(x,a)$ for every $(x,a)\\in\\mathcal F\\times \\mathbb A$.\nFor every $\\tau\\in T$ follows that $\\{\\tau x\\}\\times \\mathbb A\\subset\\bar{\\mathcal O}_{T,f}(x,g)$, and by the minimality of the flow $(X,T)$ every $(x,a)\\in\\mathcal F\\times \\mathbb A$ is a transitive point for $\\widetilde\\tau_f$.\n\\end{fact}\n\nThroughout this note we shall use a notion of ``relative'' triviality of cocycles.\n\n\\begin{defi}\\label{def:rnt}\nLet $f_1(\\tau,x)$ and $f_2(\\tau,x)$ be $\\mathbb R$-valued cocycles of a minimal compact metric flow $(X,T)$.\nWe shall call the cocycle $f_2(\\tau,x)$ \\emph{relatively trivial} with respect to $f_1(\\tau,x)$, if for every sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ with $d(x_k, \\tau_k x_k)\\to 0$ and $f_1(\\tau_k,x_k)\\to 0$ it holds also that $f_2(\\tau_k,x_k)\\to 0$ as $k\\to\\infty$.\nFor a sequence $\\{\\tau_k\\}_{k\\geq 1}\\subset T$ and a point $\\bar x\\in X$ so that $\\tau_k\\bar x$ and $f_1(\\tau_k,\\bar x)$ are convergent, this implies that also $f_2(\\tau_k,\\bar x)$ is convergent.\n\\end{defi}\n\nBy the following lemma it suffices to verify an essential value condition ``locally''.\n\n\\begin{lemma}\\label{lem:trans}\nLet $(X,T)$ be a minimal compact metric flow with an \\emph{Abelian} group $T$ acting, and let $f(\\tau,x)$ be a cocycle of $(X,T)$ with values in an Abelian l.c.s. group $\\mathbb A$.\nIf there exists a sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ with $d(x_k, \\tau_k x_k)\\to 0$ and $f(\\tau_k,x_k)\\to a\\in \\mathbb A_\\infty$ ($\\mathbb R_\\infty\\times\\mathbb R_\\infty$ for $\\mathbb A=\\mathbb R^2$, respectively) as $k\\to\\infty$, then for every $x\\in X$ it holds that $(x,a)\\in\\mathcal D_{T,f}(x,\\mathbf 0_\\mathbb A)$.\nHence if $a\\in \\mathbb A$ is finite, then $a\\in E(f)$.\n\nNow let $g=(g_1,g_2):T\\times X\\longrightarrow\\mathbb R^2$ be a cocycle of the flow $(X,T)$ with a sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ so that $d(x_k,\\tau_k x_k)\\to 0$, $g_1(\\tau_k,x_k)\\to 0$, and $g_2(\\tau_k,x_k)\\nrightarrow 0$ as $k\\to\\infty$.\nThen there exist a point $\\bar x\\in X$ and a sequence $\\{\\bar\\tau_k\\}_{k\\geq 1}\\subset T$ so that $d(\\bar x,\\bar\\tau_k \\bar x)\\to 0$ and $g(\\bar\\tau_k,\\bar x)\\to(0,\\infty)$ as $k\\to\\infty$.\nMoreover, for an extension $(Y,T)$ of $(X,T)=\\pi(Y,T)$ there exists a sequence $\\{(\\tilde\\tau_k,y_k)\\}_{k\\geq 1}\\subset T\\times Y$ with $d_Y(y_k,\\tilde\\tau_k y_k)\\to 0$ and $(g\\circ\\pi)(\\tilde\\tau_k,y_k)\\to(0,\\infty)$ as $k\\to\\infty$.\n\\end{lemma}\n\n\\begin{proof}\nWe let $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ be a sequence with the properties above, and we may assume that $x_k\\to x'\\in X$ as $k\\to\\infty$.\nFor arbitrary neighbourhoods $\\mathcal U\\subset X$ and $U(a)$ of $a\\in \\mathbb A_\\infty$ we can fix an element $\\tau\\in T$ with $\\tau x'\\in\\mathcal U$, and since the group $T$ is Abelian it holds that $\\tau x_k\\to\\tau x'$ and $\\tau_k\\tau x_k=\\tau\\tau_k x_k\\to \\tau x'$ as $k\\to\\infty$.\nFrom the cocycle identity and the continuity of $f(\\tau,\\cdot)$ follows\n\\begin{eqnarray*}\nf(\\tau_k,\\tau x_k) & = & f(\\tau,\\tau_k x_k) + f(\\tau_k,x_k)+f(\\tau^{-1},\\tau x_k)\\\\\n& = & f(\\tau,\\tau_k x_k)+ f(\\tau_k,x_k)-f(\\tau, x_k) \\to a\n\\end{eqnarray*}\nas $k\\to\\infty$, and for all $k$ large enough it holds that $\\tau x_k$, $\\tau_k\\tau x_k\\in\\mathcal U$ and $f(\\tau_k,\\tau x_k)\\in U(a)$.\nSince the neighbourhoods $\\mathcal U$ and $U(a)$ were arbitrary, we have $(x,a)\\in\\mathcal D_{T,f}(x,\\mathbf 0_\\mathbb A)$ for every $x\\in X$ and $a\\in E(f)$ if $a\\neq\\infty$.\n\nIf $g(\\tau_k,x_k)\\nrightarrow(0,\\infty)$ as $k\\to\\infty$, then $E(g)$ has an element $(0,c)$ with $c\\in\\mathbb R\\setminus\\{0\\}$.\nSince $E(g)$ is a closed subspace of $\\mathbb R^2$, we can start over with a sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ so that $d(x_k,\\tau_k x_k)\\to 0$, $g_1(\\tau_k,x_k)\\to 0$, and $|g_2(\\tau_k,x_k)|\\to\\infty$ as $k\\to\\infty$.\nThe statement above implies that $(x,0,\\infty)\\in\\mathcal D_{T,g}(x,0,0)$ for every $x\\in X$, and by Fact \\ref{fact:o_p} we can select $\\bar x\\in X$ and a sequence $\\{\\bar\\tau_k\\}_{k\\geq 1}\\subset T$ so that $\\bar\\tau_k\\bar x\\to\\bar x$ and $g(\\bar\\tau_k,\\bar x)\\to(0,\\infty)$.\nFor an arbitrary point $\\bar y\\in\\pi^{-1}(\\bar x)$ we can select an increasing sequence of positive integers $\\{k_l\\}_{l\\geq 1}$ with $d_Y(\\bar\\tau_{k_{l+1}} \\bar y,\\bar\\tau_{k_l}\\bar y)\\to 0$ and $(g\\circ\\sigma)(\\bar\\tau_{k_{l+1}}(\\bar\\tau_{k_l})^{-1},\\bar\\tau_{k_l}\\bar y)\\to(0,\\infty)$ and put $\\{(\\tilde\\tau_l,y_l)=(\\bar\\tau_{k_{l+1}}(\\bar\\tau_{k_l})^{-1},\\bar\\tau_{k_l}\\bar y)\\}_{l\\geq 1}$.\n\\end{proof}\n\n\\begin{defi}\nLet $f(\\tau,x)$ be a cocycle of a minimal compact metric flow $(X,T)$ with values in an Abelian l.c.s. group $\\mathbb A$, and let $b:X\\longrightarrow \\mathbb A$ be a continuous function.\nAnother cocycle of the flow $(X,T)$ can be defined by the $\\mathbb A$-valued function\n\\begin{equation*}\ng(\\tau,x)=f(\\tau,x)+b(\\tau x)-b(x).\n\\end{equation*}\nThe cocycle $g(\\tau,x)$ is called \\emph{topologically cohomologous} to the cocycle $f(\\tau,x)$ with the \\emph{transfer function} $b(x)$.\nA cocycle $g(\\tau,x)=b(\\tau x)-b(x)$ topologically cohomologous to zero is bounded on $T\\times X$ and called a \\emph{topological coboundary}.\n\\end{defi}\n\nThe Gottschalk-Hedlund theorem (\\cite{G-H}, Theorem 14.11) characterises topological coboundaries of a minimal $\\mathbb Z$-action as cocycles bounded on at least one semi-orbit.\nThe generalisation to an Abelian group $T$ acting minimally is natural.\n\n\\begin{fact}\\label{fact:GH}\nA real valued topological cocycle $f(\\tau,x)$ of a minimal compact metric flow $(X,T)$ with an Abelian acting group $T$ is a coboundary if and only if there exists a point $\\bar x\\in X$ so that the function $\\tau\\mapsto f(\\tau,\\bar x)$ is bounded on $T$.\nFor the groups $T=\\mathbb Z$ and $T=\\mathbb R$ acting, the boundedness on a semi-orbit is sufficient.\n\nA real valued cocycle $f(\\tau,x)$ is also a topological coboundary if for every sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ with $d(x_k,\\tau_k x_k)\\to 0$ the set $\\{f(\\tau_k,x_k)\\}_{k\\geq 1}\\subset\\mathbb R$ is bounded.\n\\end{fact}\n\n\\begin{proof}\nSuppose that $\\tau\\mapsto f(\\tau,\\bar x)$ is bounded on $T$.\nBy the cocycle identity holds\n\\begin{equation*}\nf(\\tau,\\tau'\\bar x)=f(\\tau\\tau',\\bar x)-f(\\tau',\\bar x)\n\\end{equation*}\nfor all $\\tau,\\tau'\\in T$, and by the density of the $T$-orbit of $\\bar x$ follows the boundedness of $f(\\tau,x)$ on $T\\times X$ and thus the triviality of the subgroup $E(f)=\\{0\\}$.\nBy the density of the $T$-orbit of $\\bar x$ and the boundedness of $\\tau\\mapsto f(\\tau,\\bar x)$, the intersection $\\{x\\}\\times\\mathbb R\\cap\\bar{\\mathcal O}_{T,f}(\\bar x,0)$ is non-empty for every $x\\in X$.\nFor every $x\\in X$ this intersection is a singleton, since otherwise Lemma \\ref{lem:trans} proves a non-zero element in $E(f)$.\nHence the compact set $\\bar{\\mathcal O}_{T,f}(\\bar x,0)$ is the graph of a continuous function $b:X\\longrightarrow\\mathbb R$ with $f(\\tau,\\bar x)=b(\\tau\\bar x)$, and thus $f(\\tau,x)=b(\\tau x)-b(x)$ holds for every $(\\tau,x)\\in T\\times X$.\nFor $T=\\mathbb Z$ and $T=\\mathbb R$ the set of \\emph{limit points} of a semi-orbit is a $T$-invariant closed subset of $X$, which is non-empty by compactness and equal to $X$ by minimality.\nWe can conclude the proof as above, but using the semi-orbit.\n\nNow suppose that $f(\\tau,x)$ is not a topological coboundary and let $\\bar x\\in X$ be arbitrary.\nThen there exists a sequence $\\{\\tau'_l\\}_{l\\geq 1}\\subset T$ with $|f(\\tau'_l,\\bar x)|\\to\\infty$, and we may assume that $\\tau'_l \\bar x\\to x'$ as $l\\to\\infty$.\nSince $(X,T)$ is minimal, there exists sequence $\\{\\tau''_k\\}_{k\\geq 1}\\subset T$ with $\\tau''_k x'\\to\\bar x$ as $k\\to\\infty$.\nA diagonalisation with a sufficiently increasing sequence of positive integers $\\{l_k\\}_{l\\geq 1}$ yields for $\\tau_k=\\tau_k''\\tau_{l_k}'$ that $\\tau_k\\bar x\\to\\bar x$ and $|f(\\tau_k,\\bar x)|=|f(\\tau_k'',\\tau_{l_k}'\\bar x)+f(\\tau_{l_k}',\\bar x)|\\to\\infty$ as $k\\to\\infty$.\n\\end{proof}\n\nThe following lemma appeared originally in the paper \\cite{A} in a setting for $\\mathbb R^d$-valued cocycles of a minimal rotation on a torus.\n\n\\begin{lemma}\\label{lem:at}\nLet $f(\\tau,x)$ be a real valued topological cocycle of a minimal compact metric flow $(X,T)$ with an Abelian acting group $T$.\nIf the skew product action $\\widetilde\\tau_f$ is \\emph{not} point transitive on $X\\times\\mathbb R$, then for every neighbourhood $U\\subset\\mathbb R$ of $0$ there exist a compact symmetric neighbourhood $K\\subset U$ of $0$ and an $\\varepsilon>0$ so that for every $\\tau\\in T$ holds\n\\begin{equation}\\label{eq:s_l}\n\\{x\\in X:d(x,\\tau x)<\\varepsilon\\enspace\\textup{and}\\enspace f(\\tau,x)\\in 2K\\setminus K^0\\}=\\emptyset .\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $f(\\tau,x)$ is real valued and $\\widetilde\\tau_f$ is not point transitive.\nBy Fact \\ref{fact:er} the essential range $E(f)$ is a proper closed subgroup of $\\mathbb R$, and thus there exists a compact symmetric neighbourhood $K\\subset U$ of $0$ with $(2K\\setminus K^0)\\cap E(f)=\\emptyset$.\nIf the assertion is false for the neighbourhood $K$, then there exists a sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ with $d(x_k, \\tau_k x_k)\\to 0$ and $f(\\tau_k,x_k)\\to t\\in 2K\\setminus K^0$.\nNow Lemma \\ref{lem:trans} implies $t\\in E(f)\\cap 2K\\setminus K^0$, in contradiction to the choice of $K$.\n\\end{proof}\n\nWe shall commence the study of cocycles of distal minimal flows by the generalisation of the results for minimal rotations in \\cite{G-H} and \\cite{LM}.\n\n\\begin{proposition}\\label{prop:isom}\nLet $(X,T)$ be a minimal compact \\emph{isometric} flow with a compactly generated Abelian acting group $T$, and let $f(\\tau,x)$ be a topologically recurrent real valued cocycle of $(X,T)$.\nThen the cocycle $f(\\tau,x)$ is either a coboundary or its skew product extension $\\widetilde \\tau_f$ is point transitive on $X\\times\\mathbb R$.\n\\end{proposition}\n\n\\begin{proof}\nSuppose that the cocycle $f(\\tau,x)$ is not a coboundary and $\\widetilde \\tau_f$ is not point transitive.\nThen by Lemma \\ref{lem:at} there exist a compact symmetric neighbourhood $K$ of $0$ and an $\\varepsilon>0$ so that equality (\\ref{eq:s_l}) holds for every $\\tau\\in T$.\nFurthermore, if $L\\subset T$ is a compact generative subset, then $\\varepsilon>0$ can be chosen small enough so that for all $\\tau'\\in L$ and $x,x'\\in X$ with $d(x,x')<\\varepsilon$ it holds that\n\\begin{equation*}\nf(\\tau',x)-f(\\tau',x')\\in K^0 .\n\\end{equation*}\nBy Fact \\ref{fact:GH} we can fix a pair $(\\bar\\tau,\\bar x)\\in T\\times X$ with $d(\\bar x,\\bar\\tau\\bar x)<\\varepsilon$ and $f(\\bar\\tau,\\bar x)\\notin 2K$, since $f(\\tau,x)$ is not a coboundary.\nThe Abelian group $T$ acts on $X$ isometrically, and thus $d(\\bar x,\\bar\\tau\\bar x)<\\varepsilon$ implies that $d(\\tau'\\bar x,\\bar\\tau\\tau' \\bar x)=d(\\tau'\\bar x,\\tau'\\bar\\tau\\bar x)<\\varepsilon$.\nTogether with equality (\\ref{eq:s_l}) we can conclude for every $\\tau'\\in L$ that\n\\begin{equation*}\nf(\\bar\\tau,\\tau'\\bar x)=f(\\bar\\tau,\\bar x)-f(\\tau',\\bar x)+f(\\tau',\\bar\\tau\\bar x)\\notin 2K ,\n\\end{equation*}\nand hence both of the real numbers $f(\\bar\\tau,\\bar x)$ and $f(\\bar\\tau,\\tau'\\bar x)$ are elements of the one and the same of the disjoint sets $\\mathbb R^+\\setminus 2K$ and $\\mathbb R^-\\setminus 2K$.\nSince the set $L$ is generative in the Abelian group $T$ acting minimally on $X$, it follows by induction that $f(\\bar\\tau, x)$ is in the closure of one of the sets $\\mathbb R^+\\setminus 2K$ and $\\mathbb R^-\\setminus 2K$ for every $x\\in X$.\nThus we have a constant $c>0$ with $|f(\\bar\\tau^k,x)|> |k| c$ for every integer $k$, and we define a subset $P\\subset T$ by\n\\begin{equation*}\nP=\\cup_{k\\geq 1}\\bar\\tau^k\\cdot\\{\\tau\\in T:f(\\tau,\\cdot)<|k| c\/2\\} .\n\\end{equation*}\nGiven two integers $k,k'\\geq 1$ and $\\bar\\tau^k\\tau$, $\\bar\\tau^{k'}\\tau'\\in P$ with $f(\\tau,\\cdot)<|k| c\/2$ and $f(\\tau',\\cdot)<|k'| c\/2$, we can conclude that $\\bar\\tau^k\\tau\\bar\\tau^{k'}\\tau'=\\bar\\tau^{k+k'}(\\tau\\tau')$ with $f(\\tau\\tau',\\cdot)<|k+k'| c\/2$, hence $P$ is a semigroup.\nMoreover, the semigroup $P$ contains a translate of every compact set $L\\subset T$, since for large enough $k\\geq 1$ the inequality $f(\\tau,x)<|k| c\/2$ holds for every $\\tau\\in L$ and every $x\\in X$.\nTherefore $P$ is a replete semigroup in $T$ so that $|f(\\tau,x)|>c\/2$ holds for every $(\\tau,x)\\in P\\times X$, which contradicts the existence of a dense $G_\\delta$ set of $\\widetilde\\tau_f$-recurrent points (cf. Remarks \\ref{rems:rec}).\n\\end{proof}\n\nThe \\emph{Rokhlin extensions} and the \\emph{Rokhlin skew products} have been studied in the measure theoretic setting in \\cite{LL} and \\cite{LP}.\nWe shall introduce the notion of a \\emph{perturbed Rokhlin skew product}, which will be inevitable in our main result.\n\n\\begin{defi}\nSuppose that $(X,T)$ is a distal minimal compact metric flow and $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ is a distal minimal compact metric $\\mathbb R$-flow.\nLet $f:T\\times X\\longrightarrow\\mathbb R$ be a cocycle of $(X,T)$ with a point transitive skew product $\\widetilde\\tau_f$ on $X\\times\\mathbb R$.\nWe define the \\emph{Rokhlin extension} $\\tau_{\\phi,f}$ on $X\\times M$ by\n\\begin{equation*}\n\\tau_{\\phi,f}(x,m)=(\\tau x,\\phi^{f(\\tau,x)}(m)) ,\n\\end{equation*}\nwhich is an action of the group $T$ on $X\\times M$ due to the cocycle identity for $f(\\tau,x)$.\nIf $(\\bar x,0)$ is a transitive point for $\\widetilde\\tau_f$, then by the minimality of $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ every point $(\\bar x,m)$ with $m\\in M$ is a transitive point for $(X\\times M,T)$.\nSince the flow $(X\\times M,T)$ is distal by the distality of its components, it is even minimal.\n\nThe skew product extension of $(X\\times M,T)$ by the cocycle $(\\tau,x,m)\\mapsto f(\\tau,x)$ is the \\emph{Rokhlin skew product} $\\widetilde\\tau_{\\phi,f}$ on $X\\times M\\times \\mathbb R$ with\n\\begin{equation*}\n\\widetilde\\tau_{\\phi,f}(x,m,t)=(\\tau x,\\phi^{f(\\tau,x)}(m),t+f(\\tau,x)) .\n\\end{equation*}\nLet $g:\\mathbb R\\times M\\longrightarrow\\mathbb R$ be a cocycle of the flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$.\nThe $\\mathbb R$-valued map\n\\begin{equation*}\n(\\tau,x,m)\\mapsto f(\\tau,x)+g(f(\\tau,x),m)\n\\end{equation*}\ndefined on $T\\times X\\times M$ turns out to be a cocycle of the flow $(X\\times M,T)$ due to the cocycle identity for $g(t,m)$.\nThe skew product extension of this cocycle with\n\\begin{equation*}\n\\widetilde\\tau_{\\phi,f,g}(x,m,t)=(\\tau x,\\phi^{f(\\tau,x)}(m),t+f(\\tau,x)+g(f(\\tau,x),m)) .\n\\end{equation*}\nis called a \\emph{perturbed Rokhlin skew product} $\\widetilde\\tau_{\\phi,f,g}$ on $X\\times M\\times\\mathbb R$.\n\\end{defi}\n\nWe shall present at first the basic example of a topological Rokhlin skew product of topological type $III_0$, i.e. recurrent with a trivial topological essential range but not a topological coboundary.\n\n\\begin{example}\\label{ex:zi}\nLet $f:\\mathbb T\\longrightarrow\\mathbb R$ be a continuous function with a point transitive skew product extension $\\widetilde T_f$ of the irrational rotation $T$ by $\\alpha$ on the torus, and let $\\beta\\in(0,1)$ be irrational so that the $\\mathbb R$-flow $(\\mathbb T^2,\\{\\phi^t:t\\in\\mathbb R\\})$ defined by $\\phi^t(y,z)=(y+t,z+\\beta t)$ is minimal and distal.\nThe minimal and distal Rokhlin extension $T_{\\phi,f}$ on $\\mathbb T^3$ is\n\\begin{equation*}\nT_{\\phi,f}(x,y,z)=(x+\\alpha,y+f(x),z+\\beta f(x)),\n\\end{equation*}\nand putting $h(x,y,z)=f(x)$ for all $(x,y,z)\\in\\mathbb T^3$ gives a topological type $III_0$ cocycle $h(n,(x,y,z))$ of the homeomorphism $T_{\\phi,f}$ with the skew product extension $\\widetilde T_{\\phi,f}$.\nIndeed, since $\\widetilde T_f$ is point transitive, the cocycle $h(n,(x,y,z))$ is recurrent, but it is not bounded and therefore no topological coboundary.\nFurthermore, a sequence $\\{t_n\\}_{n\\geq 1}\\subset\\mathbb R$ with $t_n\\mod 1\\to 0$ and $(\\beta t_n)\\mod 1\\to 0$ cannot have a finite cluster point apart from zero, and hence $E(h)=\\{0\\}$.\nFor a point $\\bar x\\in\\mathbb T$ so that $(\\bar x,0)\\in\\mathbb T\\times\\mathbb R$ is transitive under $\\widetilde T_f$ and arbitrary $y,z\\in\\mathbb T$ the orbit closure of $(\\bar x,y,z,0)$ under the skew product extension of $T_{\\phi,f}$ by $h$ is of the form\n\\begin{equation*}\n\\bar{\\mathcal O}_{\\widetilde T_{\\phi,f}}((\\bar x,y,z),0)=\\bar{\\mathcal O}_{T_{\\phi,f},h}((\\bar x,y,z),0)=\\mathbb T\\times\\{(\\phi^t(y,z),t)\\in\\mathbb T^2\\times\\mathbb R:t\\in\\mathbb R\\} .\n\\end{equation*}\nThe collection of these sets is a partition of $\\mathbb T^3\\times\\mathbb R$ into $\\widetilde T_{\\phi,f}$-orbit closures.\n\\end{example}\n\nThe next example makes clear that the perturbation of a Rokhlin skew product by a cocycle is an essential component, which in general cannot be eliminated by continuous cohomology.\n\n\\begin{example}\\label{ex:pe}\nLet $T$, $f$, $h$, and $\\{\\phi^t:t\\in\\mathbb R\\}$ be defined as in Example \\ref{ex:zi}, and suppose that $g(t,(y,z))$ is a point transitive $\\mathbb R$-valued cocycle of the flow $\\{\\phi^t:t\\in\\mathbb R\\}$.\nFrom the unique ergodicity of the flow $\\{\\phi^t:t\\in\\mathbb R\\}$ follows $\\int_{\\mathbb T^2} g(t,(y,z)) d\\lambda(y,z)=0$ for every $t\\in\\mathbb R$, and after rescaling $g$ we can assume that $|g(t,(y,z))|<|t|\/2$ for every $t\\in\\mathbb R$ and $(y,z)\\in\\mathbb T^2$.\nWe define a function\n\\begin{equation*}\n\\bar h(x,y,z)= f(x)+g(f(x),(y,z))\n\\end{equation*}\nso that the cocycle of $T_{\\phi,f}$ is $\\bar h(n,(x,y,z))= f(n,x)+g(f(n,x),(y,z))$ for every $n$.\nSince the perturbation $g(f(n,x),(y,z))$ is unbounded, there cannot be a continuous transfer function defined on $\\mathbb T^3$ so that $\\bar h$ and $h$ are cohomologous.\nHowever, due to the condition $|g(t,(y,z))|<|t|\/2$ the set\n\\begin{equation*}\n\\mathbb T\\times\\{(\\phi^t(y,z),t+g(t,(y,z)))\\in\\mathbb T^2\\times\\mathbb R:t\\in\\mathbb R\\}\n\\end{equation*}\nis closed, and it coincides with $\\bar{\\mathcal O}_{\\widetilde T_{\\phi,f,g}}((\\bar x,y,z),0)$ if the point $(\\bar x,0)\\in\\mathbb T\\times\\mathbb R$ is transitive under $\\widetilde T_f$.\nThus the structure of the orbit closures is preserved as well as these sets provide a partition of $\\mathbb T^3\\times\\mathbb R$.\n\\end{example}\n\n\\begin{remark}\nThe structure of Example \\ref{ex:zi} can be revealed from the toral extensions of $T_{\\phi,f}$ by the function $(\\gamma h) \\mod 1$ for all $\\gamma\\in\\mathbb R$.\nThis distal homeomorphism of $\\mathbb T^4$ is transitive and hence minimal for rationally independent $1$, $\\beta$, and $\\gamma$.\nHowever, for $\\gamma=1$ and $\\gamma=\\beta$ the orbit closures collapse to graphs representing the dependence of $h$ and the action on the coordinates of the torus.\nThe same approach will not be successful with respect to Example \\ref{ex:pe}.\nIt can be verified that for every $\\gamma\\in\\mathbb R$ the toral extension of $T_{\\phi,f}$ by the function $(\\gamma\\bar h) \\mod 1$ is minimal on $\\mathbb T^4$.\n\\end{remark}\n\nThe main result of this note puts these examples into a structure theorem.\n\n\\begin{strth*}\nSuppose that $(X,T)$ is a distal minimal compact metric flow with a compactly generated Abelian acting group $T$ and that $f:T\\times X\\longrightarrow\\mathbb R$ is a topologically recurrent cocycle which is not a coboundary.\nThen there exist a factor $(X_\\alpha,T)=\\pi_\\alpha(X,T)$, a topological cocycle $f_\\alpha:T\\times X_\\alpha\\longrightarrow\\mathbb R$ of $(X_\\alpha,T)$, and a distal minimal compact metric $\\mathbb R$-flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$, so that the Rokhlin extension $(X_\\alpha\\times M,T)$ with the action $\\tau_{\\phi,f_\\alpha}$ is a factor $(Y,T)=\\pi_Y(X,T)$ of $(X,T)$.\nThe cocycle $f(\\tau,x)$ is topologically cohomologous to $(f_Y\\circ\\pi_Y)(\\tau,x)=f(\\tau,x)+b(\\tau x)-b(x)$ with a topological cocycle $f_Y:T\\times Y\\longrightarrow\\mathbb R$ of the flow $(Y,T)$ so that\n\\begin{equation}\\label{eq:fpy}\n\\mathcal D_{T, f_Y\\circ\\pi_Y}(x,0)\\cap(\\pi_\\alpha^{-1}(\\pi_\\alpha(x))\\times\\{0\\})=\\pi_Y^{-1}(\\pi_Y(x))\\times\\{0\\}\n\\end{equation}\nholds for all $x\\in X$.\nMoreover, there exists a cocycle $g:\\mathbb R\\times M\\longrightarrow\\mathbb R$ of the $\\mathbb R$-flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ so that the cocycle $(\\mathbbm 1+g)(t,m)= t+g(t,m)$ is topologically \\emph{transient} and\n\\begin{equation}\\label{eq:f_Y2}\nf_Y(\\tau,(x,m))=f_\\alpha(\\tau,x)+g(f_\\alpha(\\tau,x),m)=(\\mathbbm 1+g)(f_\\alpha(\\tau,x),m)\n\\end{equation}\nholds for every $\\tau\\in T$ and $(x,m)\\in Y=X_\\alpha\\times M$.\nThus the skew product $\\widetilde\\tau_{f_Y}$ on $Y\\times\\mathbb R$ is the perturbed Rokhlin skew product $\\widetilde \\tau_{\\phi,f_\\alpha, g}$.\n\\end{strth*}\n\n\\noindent We shall conclude the proof of this theorem in the next section of this note.\n\\medskip\n\nThe application of the structure theorem for a topological ergodic decomposition requires a suitable topology on the hyperspace of the non-compact space $X\\times\\mathbb R$.\nWe shall use the Fell topology on the hyperspace of \\emph{non-empty} closed subsets of a locally compact separable metric space.\nGiven finitely many open neighbourhoods $\\mathcal U_1,\\dots,\\mathcal U_k$ and a compact set $K$, an element of the Fell topology base consists of all non-empty closed subsets which intersect each of the open neighbourhoods $\\mathcal U_1,\\dots,\\mathcal U_k$ while being disjoint from $K$.\nThis topology is separable, metrisable, and $\\sigma$-compact (cf. \\cite{HLP}).\nThe Fell topology was introduced in \\cite{Fe} as a compact topology on the hyperspace of all closed subsets, with the empty set as infinity.\n\n\\begin{decth*}\nSuppose that $f:T\\times X\\longrightarrow\\mathbb R$ is a topologically recurrent cocycle of a distal minimal compact metric flow $(X,T)$ with a compactly generated Abelian acting group $T$.\nThe prolongations $\\mathcal D_{T,f}(x,s)\\subset X\\times\\mathbb R$ of the skew product action $\\widetilde\\tau_{f}$ with $(x,s)\\in X\\times\\mathbb R$ define a \\emph{partition} of $X\\times\\mathbb R$.\nThe mapping $(x,s)\\mapsto\\mathcal D_{T,f}(x,s)$ is continuous with respect to the Fell topology on the hyperspace of non-empty closed subsets of $X\\times\\mathbb R$, and the right translation on $X\\times\\mathbb R$ is a minimal continuous $\\mathbb R$-action on the set of prolongations.\nIf the cocycle $f(\\tau,x)$ is not a topological coboundary, then the set of all prolongations in the skew product is Fell compact.\n\\end{decth*}\n\n\\begin{tmath*}\nA recurrent cocycle $f(\\tau,x)$ apart from a coboundary has a minimal compact metric flow as a topological version of the \\emph{Mackey action}.\nIts phase space is the set of prolongations in the skew product with the Fell topology, with the right translation of $\\mathbb R$ acting on the prolongations.\nThis flow is a distal extension (possibly the trivial extension) of a weakly mixing compact metric flow (possibly the trivial flow).\nThe Mackey action is distal if and only if the perturbation cocycle $g(t,m)$ in the structure theorem is a topological coboundary.\n\\end{tmath*}\n\nWhile most of the properties of the topological Mackey action are part of the decomposition theorem, its structure as a distal extension of a weakly mixing flow will be verified in the next section of this note.\nThe proof of the decomposition theorem depends on the following general lemma on transient cocycles of minimal $\\mathbb R$-flows, which might be of independent interest.\n\n\\begin{lemma}\\label{lem:tr_coc}\nLet $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ be a minimal compact metric $\\mathbb R$-flow and let $h(t,m):\\mathbb R\\times M\\longrightarrow\\mathbb R$ be a \\emph{transient} cocycle of $(M,\\{\\phi^t:t\\in\\mathbb R\\})$.\nThen for every point $(m,s)\\in M\\times\\mathbb R$ the orbit $\\mathcal O_{\\phi,h}(m,s)$, the orbit closure $\\bar{\\mathcal O}_{\\phi,h}(m,s)$, and the prolongation $\\mathcal D_{\\phi,h}(m,s)$ under the skew product extension $\\widetilde{\\phi^t}_h$ coincide.\nThe mapping from points to their orbits in $M\\times\\mathbb R$ is continuous with a compact range with respect to the Fell topology, and the right translation on $M\\times\\mathbb R$ provides a minimal continuous $\\mathbb R$-action on the set of orbits.\nMoreover, for every $m\\in M$ the mapping $t\\mapsto h(t,m)$ maps $\\mathbb R$ \\emph{onto} $\\mathbb R$.\n\\end{lemma}\n\n\\begin{proof}\nSince prolongations are closed sets, it suffices to verify that for every $(m,s)\\in M\\times\\mathbb R$ the orbit and the prolongation coincide.\nOtherwise, there exist two points $(m,s), (m',s')\\in M\\times\\mathbb R$ so that $(m',s')$ is not in the $\\widetilde{\\phi^t}_h$-orbit of $(m,s)$, however there exists a sequence $\\{(t_k,m_k)\\}_{k\\geq 1}\\subset\\mathbb R\\times M$ so that $(t_k,m_k)\\to (+\\infty,m)$ and\n\\begin{equation*}\n\\widetilde{\\phi^{t_k}}_h(m_k,s)=(\\phi^{t_k}(m_k),s+h(t_k,m_k))\\to (m',s') .\n\\end{equation*}\nIf there exists a compact set $L\\subset\\mathbb R$ with $h([0,t_k],m_k)\\subset L$ for all $k\\geq 1$, then $h([0,\\infty),m)\\subset L$ since $m_k\\to m$, and by Fact \\ref{fact:GH} the cocycle $h(t,m)$ is a coboundary in contradiction to its transience.\nTherefore we have an increasing sequence of integers $\\{k_l\\}_{l\\geq 1}$, a sequence $\\{t'_l\\}_{l\\geq 1}\\subset\\mathbb R$ with $t'_l\\in[0,t_{k_l}]$, and $S\\in\\{+1,-1\\}$ so that\n\\begin{equation*}\nS\\cdot h(t'_l,m_{k_l})=\\max_{t\\in[0,t_{k_l}]}S\\cdot h(t,m_{k_l})\\to +\\infty\n\\end{equation*}\nas $l\\to\\infty$.\nFor every limit point $\\bar m$ of the sequence $\\{\\phi^{t'_l}(m_{k_l})\\}_{l\\geq 1}$ it holds that $S\\cdot h(t,\\bar m)\\leq 0$ for all $t\\in\\mathbb R$, and the mapping $t\\mapsto h(t,\\bar m)$ maps each of the sets $\\mathbb R^+$ and $\\mathbb R^-$ onto $S\\cdot \\mathbb R^-$.\nHence for every $t\\in\\mathbb R^+$ there exists a $t'\\in\\mathbb R^-$ with $h(t,\\bar m)=h(t',\\bar m)$, and by the density of the semi-orbit $\\{\\phi^t(\\bar m):t\\in\\mathbb R^+\\}$ (cf. the proof of Fact \\ref{fact:GH}) and the cocycle identity the open set\n\\begin{equation*}\nM_k=\\{m\\in M:|h(t,m)|<2^{-k}\\enspace\\textup{for some}\\enspace t<-k\\}\n\\end{equation*}\nis dense for every integer $k\\geq 1$.\nFor a point $m_k$ in the dense $G_\\delta$ set $\\bigcap_{t\\in\\mathbb Q}\\phi^t(M_k)$, we can find rational numbers $t_1,\\dots,t_k<-k$ so that $\\phi^{t_1+\\cdots+t_l}(m_k)\\in M_k$ and $|h(t_1+\\dots+t_l,m_k)|<k 2^{-k}$ for all $1\\leq l\\leq k$.\nSince $M$ is compact, there exists a point $\\tilde m\\in M$ each neighbourhood of which contains at least two distinct points out of the finite sequence $\\phi^{t_1}(m_k),\\dots,\\phi^{t_1+\\cdots+t_k}(m_k)\\in M_k$ for an infinite set of integers $k\\geq 1$.\nThus $(\\tilde m,0)$ is a regionally recurrent point for the skew product $\\widetilde{\\phi^t}_h$, in contradiction to the Remarks \\ref{rems:rec} and the transience of the cocycle $h(t,m)$.\n\nSuppose for some $m\\in M$ the mapping $t\\mapsto h(t,m)$ is not onto $\\mathbb R$, then either there exists a compact set $L\\subset\\mathbb R$ so that one of the inclusions $h([0,\\infty),m)\\subset L$, $h((-\\infty,0],m)\\subset L$ holds, or there exists a point $\\bar m\\in M$ as above.\nIn each of these cases, we can obtain a contradiction to the transience of $h(t,m)$ as above.\n\nLet $(m,s)\\in M\\times\\mathbb R$ be arbitrary, and let a Fell neighbourhood of its \\emph{closed} $\\widetilde{\\phi^t}_h$-orbit be defined by the open neighbourhoods $\\mathcal U_1,\\dots,\\mathcal U_k$ and a compact set $K$.\nObviously the $\\widetilde{\\phi^t}_h$-orbit of every point in a suitable neighbourhood of $(m,s)$ intersects each of the neighbourhoods $\\mathcal U_1,\\dots,\\mathcal U_k$.\nMoreover, the $\\widetilde{\\phi^t}_h$-orbit is disjoint from $K$ for every point in a suitable neighbourhood of $(m,s)$, since otherwise the $\\widetilde{\\phi^t}_h$-prolongation of $(m,s)$ intersects the compact set $K$.\nThis contradicts the coincidence of orbits and prolongations of $\\widetilde{\\phi^t}_h$, and hence the mapping of a point $(m,s)\\in M\\times\\mathbb R$ to its $\\widetilde{\\phi^t}_h$-orbit is Fell continuous.\nBy the surjectivity of the mapping $t\\mapsto h(t,m)$, every $\\widetilde{\\phi^t}_h$-orbit intersects the set $\\{(m,0):m\\in M\\}$.\nThe Fell compactness of the set of $\\widetilde{\\phi^t}_h$-orbits in $M\\times\\mathbb R$ follows, and the right translation on $M\\times\\mathbb R$ is a Fell continuous $\\mathbb R$-action by the definition of the Fell neighbourhoods.\nSince the points $(\\phi^t(m),0)$ and $(m,-h(m,t))$ are within the same $\\widetilde{\\phi^t}_h$-orbit, the minimality of $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ implies the minimality of the right translation action.\n\\end{proof}\n\n\\begin{proof}[Proof of the decomposition theorem]\nFor a topological coboundary $f(\\tau,x)$ the assertions are trivial, since the prolongations are just the right translates of the graph of the transfer function.\nIf the cocycle $f(\\tau,x)$ is not a coboundary, then we apply the structure theorem.\nFor a point $\\bar x_\\alpha\\in X_\\alpha$ with $\\bar{\\mathcal O}_{T,f_\\alpha}(\\bar x_\\alpha,0)=X_\\alpha\\times\\mathbb R$ and every $m\\in M$ follows from equality (\\ref{eq:f_Y2}) that $X_\\alpha\\times\\mathcal O_{\\phi,\\mathbbm 1+g}(m,0)\\subset\\bar{\\mathcal O}_{T,f_Y}(\\bar x_\\alpha,m,0)\\subset X_\\alpha\\times\\bar{\\mathcal O}_{\\phi,\\mathbbm 1+g}(m,0)$, and  by Lemma \\ref{lem:tr_coc} holds $\\mathcal O_{\\phi,\\mathbbm 1+g}(m,0)=\\bar{\\mathcal O}_{\\phi,\\mathbbm 1+g}(m,0)=\\mathcal D_{\\phi,\\mathbbm 1+g}(m,0)$.\nThe set of points $\\bar x_\\alpha\\in X_\\alpha$ with $\\bar{\\mathcal O}_{T,f_\\alpha}(\\bar x_\\alpha,0)=X_\\alpha\\times\\mathbb R$ is a dense $G_\\delta$ set, and for arbitrary $(x_\\alpha,m)\\in X_\\alpha\\times M$ and $(x_\\alpha',m',t')\\in\\mathcal D_{T,f_Y}(x_\\alpha,m,0)$ we can find a sequence $\\{(\\tau_k,\\bar x_k,m_k)\\}\\subset T\\times X_\\alpha\\times M$ with $(\\bar x_k,m_k)\\to (x_\\alpha,m)$, $\\tau_k(\\bar x_k,m_k)\\to (x_\\alpha',m')$, $f_Y(\\tau_k,(\\bar x_k, m_k))\\to t'$, and $X_\\alpha\\times\\mathcal O_{\\phi,\\mathbbm 1+g}(m_k,0)=\\bar{\\mathcal O}_{T,f_Y}(\\bar x_k,m_k,0)$.\nTherefore $(m',t')\\in\\mathcal D_{\\phi,\\mathbbm 1+g}(m,0)$ so that $\\mathcal D_{T,f_Y}(x_\\alpha,m,0)\\subset X_\\alpha\\times\\mathcal D_{\\phi,\\mathbbm 1+g}(m,0)$.\nWe conclude for every $(x_\\alpha,m,s)\\in X_\\alpha\\times M\\times\\mathbb R$ that\n\\begin{equation*}\n\\mathcal D_{T,f_Y}(x_\\alpha,m,s)=X_\\alpha\\times\\{(\\phi^t(m),s+t+g(t,m)):t\\in\\mathbb R\\} ,\n\\end{equation*}\nand these sets define a partition of $X_\\alpha\\times M\\times\\mathbb R$.\nThe Fell continuity of the mapping $(x_\\alpha,m,s)\\mapsto\\mathcal D_{T,f_Y}(x_\\alpha,m,s)$, the compactness of its range, and the minimality of the right translation follow directly from Lemma \\ref{lem:tr_coc}.\n\nSuppose that $(x,s)\\in X\\times\\mathbb R$ and $(x_\\alpha',m',s')\\in\\mathcal D_{T,f_Y}(\\pi_Y(x),s)$, and let $\\{\\tau_k\\}_{k\\geq 1}\\subset T$ and $\\{y_k\\}_{k\\geq 1}\\subset X_\\alpha\\times M$ be sequences with $y_k\\to\\pi_Y(x)$, $\\tau_k y_k\\to (x_\\alpha',m')$, and $f_Y(\\tau_k,y_k)\\to s'-s$.\nSince $\\pi_Y$ is a distal homomorphism, it is an open onto mapping, and the mapping $y\\mapsto\\pi_Y^{-1}(y)$ is continuous with respect to the Hausdorff metric $d_H$ (cf. \\cite{K}, p. 68, Theorem 1, and p. 47).\nTherefore we can define a sequence $\\{x_k\\in\\pi_Y^{-1}(y_k)\\}_{k\\geq 1}\\subset X$ so that $x_k\\to x$, $\\tau_k x_k\\to x'\\in\\pi_Y^{-1}(x'_\\alpha,m')$, and $(f_Y\\circ\\pi_Y)(\\tau_k,x_k)\\to s'-s$.\nBy equality (\\ref{eq:fpy}), for every $x''\\in\\pi_Y^{-1}(\\pi_Y(x'))$ there exists a sequence $\\{(\\bar\\tau_k,\\bar x_k)\\}_{k\\geq 1}$ with $\\bar x_k$ sufficiently close to $x_k$ so that $\\bar x_k\\to x$, $\\tau_k\\bar x_k\\to x'$, $(f_Y\\circ\\pi_Y)(\\tau_k,\\bar x_k)\\to s'-s$, $d(\\bar\\tau_k \\tau_k\\bar x_k,x'')<d_H(\\pi_Y^{-1}(\\pi_Y(\\tau_k x_k)),\\pi_Y^{-1}(\\pi_Y(x')))+2^{-k}$, and $(f_Y\\circ\\pi_Y)(\\bar\\tau_k,\\tau_k\\bar x_k)<2^{-k}$.\nFrom $d_H(\\pi_Y^{-1}(\\pi_Y(\\tau_k x_k)),\\pi_Y^{-1}(\\pi_Y(x')))\\to 0$ follows that $\\bar\\tau_k\\tau_k\\bar x_k\\to x''$ and $(f_Y\\circ\\pi_Y)(\\bar\\tau_k\\tau_k,\\bar x_k)\\to s'-s$, hence $(x'',s')\\in\\mathcal D_{T,f_Y\\circ\\pi_Y}(x,s)$.\nThus the $\\widetilde\\tau_{f_Y\\circ\\pi_Y}$-prolongations in $X\\times\\mathbb R$ are exactly the pre-images of the $\\widetilde\\tau_{f_Y}$-prolongations under the mapping $\\pi_Y\\times\\textup{id}_\\mathbb R$.\nIf a Fell neighbourhood of $\\mathcal D_{T,f_Y\\circ\\pi_Y}(x,s)$ is defined by the open neighbourhoods $\\mathcal U_1,\\dots,\\mathcal U_k$ and a compact set $K$ in $X\\times\\mathbb R$, then by the openness of $\\pi_Y$ the $\\pi_Y\\times\\textup{id}_\\mathbb R$-images of these sets define a Fell neighbourhood of $\\mathcal D_{T,f_Y}(\\pi_Y(x),s)$.\nThe Fell continuity of $(x,s)\\mapsto \\mathcal D_{T,f_Y\\circ\\pi_Y}(x,s)$, the compactness of the range, and the minimality of the right translation follow.\n\nLet $b:X\\longrightarrow\\mathbb R$ be the transfer function in the structure theorem.\nThe homeomorphism $(x,s)\\mapsto (x,s+b(x))$ on $X\\times\\mathbb R$ defines a homeomorphism of the hyperspace with the Fell topology commuting with the right translation.\nSince it maps $\\mathcal D_{T,f}(x,s)$ onto $\\mathcal D_{T,f_Y\\circ\\pi_Y}(x,s+b(x))$, it is a topological isomorphism of the right translation actions and all properties carry over.\n\\end{proof}\n\n\\begin{remarks}\nThough the compact metric flow $(X,T)$ is not necessarily itself a Rokhlin extension, the existence of a real-valued recurrent non-coboundary cocycle $f(\\tau,x)$ with a non-transitive skew product extension $\\widetilde\\tau_f$ implies the existence of a Rokhlin extension factor $(Y,T)=\\pi_Y(X,T)$ with a non-trivial flow $\\{\\phi^t:t\\in\\mathbb R\\}$ and a cocycle $f_Y(\\tau,y)$ of the flow $(Y,T)$ with $f_Y\\circ\\pi_Y$ cohomologous to $f$.\n\nThe surjectivity of the mapping $t\\mapsto (\\mathbbm 1+g)(t,m)$ for every $m\\in M$ implies that $\\{f(\\tau,x):\\tau\\in T\\}$ is dense in $\\mathbb R$ for all $x$ in a dense $G_\\delta$ subset of $X$.\n\nFor other well-known topologies on the hyperspace like the Vietoris topology and the Hausdorff topology with respect to the product metric on $X\\times\\mathbb R$, the continuity of the mapping $(x,s)\\mapsto\\mathcal D_{T,f}(x,s)$ can be disproved by Example \\ref{ex:pe}.\n\\end{remarks}\n\n\\section{Proof of the structure theorem}\n\nFurstenberg's structure theorem for distal minimal flows shall be our main tool for studying the structure of cocycles.\n\n\\begin{defi}\nLet $X$ and $Y$ be compact metric spaces, let $\\pi$ be a continuous mapping from of $X$ onto $Y$, and let $M$ be a compact homogeneous metric space.\nSuppose that $\\rho(x_1,x_2)$ is a continuous real valued function defined on the set\n\\begin{equation*}\nR_\\pi=\\{(x,x')\\in X^2: \\pi(x)=\\pi(x')\\}\n\\end{equation*}\nso that for every $y\\in Y$ the function $\\rho$ is a metric on the fibre $\\pi^{-1}(y)$ with an isometric mapping between $\\pi^{-1}(y)$ and $M$.\nThen $X$ is called an $M$-\\emph{bundle} over $Y$.\n\nNow let $(X,T)$ and $(Y,T)=\\pi(X,T)$ be compact metric flows with $X$ an $M$-bundle over $Y$.\nIf the function $\\rho$ satisfies $\\rho(x,x')=\\rho(\\tau x,\\tau x')$ for all $(x,x')\\in R_\\pi$ and $\\tau\\in T$, then $(X,T)$ is called an \\emph{isometric extension} of $(Y,T)$.\n\\end{defi}\n\n\\begin{fact}[Furstenberg's structure theorem \\cite{Fu}]\nLet $(X,T)$ be a distal minimal compact metric flow.\nThen there exist a countable ordinal $\\eta$ and factors $(X_\\xi,T)=\\pi_\\xi(X,T)$ for each ordinal $0\\leq\\xi\\leq\\eta$ with the following properties:\n\\begin{enumerate}\n\\item $(X_\\eta,T)=(X,T)$ and $(X_0,T)$ is the trivial flow.\n\\item $(X_\\xi,T)=\\pi_\\xi^{\\zeta}(X_{\\zeta},T)$ is a factor of $(X_{\\zeta},T)$ for all ordinals $0\\leq\\xi<\\zeta\\leq\\eta$.\n\\item For every ordinal $0\\leq\\xi<\\eta$ the flow $(X_{\\xi+1},T)$ is an isometric extension of $(X_\\xi,T_\\xi)$.\n\\item For a limit ordinal $0<\\xi\\leq\\eta$ the flow $(X_\\xi,T)$ is the inverse limit of the flows $\\{(X_{\\zeta},T):0\\leq\\zeta<\\xi\\}$.\n\\end{enumerate}\nA system $\\{(X_\\xi,T):0\\leq\\xi\\leq\\eta\\}$ with the properties above is called a quasi-isometric system or \\emph{I}-system.\n\\end{fact}\n\n\\begin{defi}\nAn \\emph{I}-system $\\{(X_\\xi,T):0\\leq\\xi\\leq\\eta\\}$ is called \\emph{normal} if for each ordinal $0\\leq\\xi<\\eta$ the flow $(X_{\\xi+1},T)$ is the maximal isometric extension of $(X_{\\xi},T)$ in $(X_{\\eta},T)$.\nThis \\emph{I}-system gives the minimal ordinal $\\eta$ to represent the compact metric flow $(X,T)=(X_\\eta,T)$ (cf. \\cite{Fu}, Proposition 13.1, Definitions 13.2 and 13.3).\n\\end{defi}\n\nIt will be essential that the fibres of all the isometric extensions are connected, with a possible exception of extension from the trivial flow to the minimal isometric flow $(X_1,T)$.\nFor a \\emph{normal} \\emph{I}-system this property will be ensured by results of the paper \\cite{MMWu}.\nThese results require the acting group to be generated by every open neighbourhood of a compact subset, and the group $T$ is even compactly generated.\n\n\\begin{proposition}\nLet $\\{(X_\\xi,T):0\\leq\\xi\\leq\\eta\\}$ be a normal \\emph{I}-system with a compactly generated group $T$ acting minimally and distally.\nThen for all ordinals $1\\leq\\xi<\\zeta\\leq\\eta$ the extension from $(X_\\xi,T)$ to $(X_\\zeta,T)$ has connected fibres.\n\\end{proposition}\n\n\\begin{proof}\nIn the following arguments we shall refer to terminology and results provided in the paper \\cite{MMWu}.\nAt first suppose that $1\\leq\\xi<\\eta$ is not a limit ordinal.\nLet $S(\\pi_{\\xi-1})\\subset X^2$ denote the relativised equicontinuous structure relation of the homomorphism $\\pi_{\\xi-1}:(X,T)\\longrightarrow(X_{\\xi-1},T)$, hence the flow $(X,T)\/S(\\pi_{\\xi-1})$ is the maximal isometric extension $(X_\\xi,T)$ of the flow $(X_{\\xi-1},T)$ in $(X,T)$.\nBy Theorem 3.7 of \\cite{MMWu} the homomorphism $\\pi_\\xi:(X,T)\\longrightarrow (X_\\xi,T)$ has connected fibres.\nFor a limit ordinal $1<\\xi<\\eta$ and an ordinal $0\\leq\\zeta<\\xi$, the same argument shows the connectedness of the fibres of $\\pi_{\\zeta+1}:(X,T)\\longrightarrow(X_{\\zeta+1},T)$.\nSince\n\\begin{equation*}\n(\\pi_\\xi)^{-1}(x_\\xi)=\\bigcap_{0\\leq\\zeta<\\xi}(\\pi_{\\zeta+1})^{-1}(\\pi_{\\zeta+1}^\\xi(x_\\xi))\n\\end{equation*}\nholds for every $x_\\xi\\in X_\\xi$, the fibre $(\\pi_\\xi)^{-1}(x_\\xi)$ is connected as the limit of a sequence of connected sets in a compact metric space (cf. \\cite{K}, p. 170, Theorem 14).\nThe hypothesis follows, since for all ordinals $1\\leq\\xi<\\zeta\\leq\\eta$ the fibres of the homomorphism $\\pi_\\xi^\\zeta$ are the images under $\\pi_\\zeta$ of the connected fibres of $\\pi_\\xi$.\n\\end{proof}\n\nWe shall henceforth assume that $\\{(X_\\xi,T):0\\leq\\xi\\leq\\eta\\}$ is a normal \\emph{I}-system with $(X_\\eta,T)=(X,T)$.\nFor every ordinal $1\\leq\\xi<\\eta$ we shall define a projection of the cocycles of $(X,T)$ to the cocycles of $(X_\\xi,T)$ by families of probability measures, using the fact that every distal extension of compact metric flows is a so-called \\emph{RIM}-extension (relatively invariant measure, cf. \\cite{Gl}).\nFor an isometric extension this \\emph{RIM} is unique (cf. \\cite{Gl}), and within the \\emph{I}-system the \\emph{RIM's} obey to an integral decomposition formula.\n\n\\begin{fact}\nFor every ordinal $0\\leq\\xi\\leq\\eta$ there exists a family of probability measures $\\{\\mu_{\\xi,x_\\xi}:x_\\xi\\in X_\\xi\\}$ on $X$ so that for every $x_\\xi\\in X_\\xi$ and $\\tau\\in T$ holds\n\\begin{equation*}\n\\mu_{\\xi,x_\\xi}(\\pi_\\xi^{-1}(x_\\xi))=1\\enspace\\textup{and}\\enspace\\mu_{\\xi,x_\\xi}\\circ \\tau^{-1}=\\mu_{\\xi, \\tau x_\\xi}.\n\\end{equation*}\nThe mapping $x_\\xi\\mapsto\\mu_{\\xi,x_\\xi}$ is continuous with respect to the weak-* topology on $C(X)^*$.\nFor a continuous function $\\varphi\\in C(X)$ and ordinals $1\\leq\\xi<\\zeta\\leq\\eta$ we have for all $x_\\xi\\in X_\\xi$ the equality that\n\\begin{equation}\n\\mu_{\\xi,x_\\xi}(\\varphi)=\\int_{X_\\zeta}\\mu_{\\zeta,y_\\zeta}(\\varphi)\\, d(\\mu_{\\xi,x_\\xi}\\circ\\pi_\\zeta^{-1})(y_\\zeta) .\n\\end{equation}\n\\end{fact}\n\n\\begin{proof}\nThe proof follows the inductive construction of an invariant measure for $(X,T)$ out of the unique \\emph{RIM's} of the extensions $(X_\\xi,T)=\\pi_\\xi^{\\xi+1}(X_{\\xi+1},T)$ with $0\\leq\\xi<\\eta$ (cf. Chapter 12 of \\cite{Fu}).\nThe generalisation to a relatively invariant measure is provided in \\cite{dVr}, p. 494, where the induction process is initiated with the family of point measures on $X_\\xi$ instead of the point measure on the trivial space $X_0$.\nSince the \\emph{I}-system and the \\emph{RIM's} for the isometric extensions remain fixed, the decomposition formula follows from the inductive construction.\n\\end{proof}\n\nGiven a cocycle $f(\\tau,x)$ of the flow $(X,T)$ and an ordinal $1\\leq\\xi<\\eta$, the \\emph{RIM} $\\{\\mu_{\\xi,x_\\xi}:x_\\xi\\in X_\\xi\\}$ defines a continuous function $f_\\xi:T\\times X_\\xi\\longrightarrow\\mathbb R$ by\n\\begin{equation*}\nf_\\xi(\\tau,x_\\xi)=\\mu_{\\xi,x_\\xi}(f(\\tau,\\cdot))=\\int_X f(\\tau,x)\\,d\\mu_{\\xi,x_\\xi}(x) .\n\\end{equation*}\nThe properties of the \\emph{RIM} imply for all $\\tau,\\tau'\\in T$ and $x_\\xi\\in X_\\xi$ that\n\\begin{eqnarray*}\nf_\\xi(\\tau,\\tau'x_\\xi)+f_\\xi(\\tau',x_\\xi)=\\mu_{\\xi,\\tau'x_\\xi}(f(\\tau,\\cdot))+\\mu_{\\xi,x_\\xi}(f(\\tau',\\cdot))=\\\\\n=\\mu_{\\xi,x_\\xi}(f(\\tau,\\cdot)\\circ\\tau')+\\mu_{\\xi,x_\\xi}(f(\\tau',\\cdot))=f_\\xi(\\tau\\tau',x_\\xi),\n\\end{eqnarray*}\ntherefore $f_\\xi$ is a cocycle of the flow $(X_\\xi,T)$.\nFurthermore, for ordinals $\\xi$, $\\zeta$ with $1\\leq\\xi<\\zeta\\leq\\eta$ and every $x_\\xi\\in X_\\xi$, the integral of the cocycle $(f_\\zeta-f_\\xi\\circ\\pi_\\xi^\\zeta)(\\tau,x_\\zeta)$ by the measure $\\mu_{\\xi,x_\\xi}\\circ\\pi_\\zeta^{-1}$ on $X_\\zeta$ vanishes for every $\\tau\\in T$:\n\\begin{eqnarray*}\n\\int_{X_\\xi}(f_\\zeta-f_\\xi\\circ\\pi_\\xi^\\zeta)(\\tau,x_\\zeta)\\, d(\\mu_{\\xi,x_\\xi}\\circ\\pi_\\zeta^{-1})(x_\\zeta) = \\hspace{3cm}\\\\\n= \\int_{X_\\xi}\\left( \\mu_{\\zeta,x_\\zeta}(f(\\tau,\\cdot))\\right)\\, d(\\mu_{\\xi,x_\\xi}\\circ\\pi_\\zeta^{-1})(x_\\zeta) - \\mu_{\\xi,\\pi_\\xi^\\zeta(x_\\zeta)} (f(\\tau,\\cdot)) = 0\n\\end{eqnarray*}\nSince the measure $\\mu_{\\xi,x_\\xi}\\circ\\pi_\\zeta^{-1}$ is supported by the connected fibre $(\\pi_\\xi^\\zeta)^{-1}(x_\\xi)$ in $X_\\zeta$, for every $\\tau\\in T$ and every $x_\\xi\\in X_\\xi$ the function $x_\\zeta\\mapsto (f_\\zeta-f_\\xi\\circ\\pi_\\xi^\\zeta)(\\tau,x_\\zeta)$ assumes zero on the fibre $(\\pi_\\xi^\\zeta)^{-1}(x_\\xi)$.\nThis property will be essential, as well as the representation of extensions of distal flows by so-called regular extensions.\n\n\\begin{fact}\\label{fact:reg_ex}\nLet $(X,T)$ be a distal minimal compact metric flow with a factor $(Y,T)=\\sigma(X,T)$.\nThen there exist a distal minimal compact Hausdorff flow $(\\tilde X,T)$ with $(X,T)=\\pi(\\tilde X,T)$ as a factor and a Hausdorff topological group $G\\subset\\textup{Aut}(\\tilde X,T)$ acting freely on $\\tilde X$ (i.e. $g(\\tilde x)=\\tilde x$ for some $\\tilde x\\in\\tilde X$ implies $g=\\textbf 1_G$).\nThe group $G$ acts strictly transitive on the fibres $\\tilde\\sigma^{-1}(\\tilde\\sigma(\\tilde x))=\\{g(\\tilde x):g\\in G\\}$ of the homomorphism $\\tilde\\sigma=\\sigma\\circ\\pi$ for every $\\tilde x\\in\\tilde X$, and $(X,T)$ is the orbit space of a subgroup $H$ of $G$ in $\\tilde X$ so that $\\pi$ is the mapping of a point in $\\tilde X$ to its $H$-orbit (cf. \\cite{El} 12.12, 12.13, and 14.26, with a direct proof in \\cite{MMWu}, Proposition 1.1).\n\nFor an isometric extension $(X,T)$ of $(Y,T)$, the flow $(\\tilde X,T)$ is metric and an isometric extension of $(Y,T)$, with a \\emph{compact metric} group $G$ and a compact subgroup $H$, hence called a compact metric group extension.\n\\end{fact}\n\n\\begin{remark}\\label{rem:d_n_m}\nThe construction above is also called the regulariser of an extension.\nIn \\cite{Gl2} it is verified that a compact Hausdorff flow $(\\tilde X,T)$ with these properties is metrisable if and only if the extension from $(Y,T)$ to $(X,T)$ is isometric.\n\\end{remark}\n\nStudying the skew product extensions $\\widetilde\\tau_{f_\\xi}:X_\\xi\\times\\mathbb R\\longrightarrow X_\\xi\\times\\mathbb R$ for the ordinals $0\\leq\\xi\\leq\\eta$ will require the following technical lemma.\n\n\\begin{lemma}\\label{lem:sub}\nLet the minimal compact metric flow $(Z,T)$ be an extension of the flow $(Y,T)=\\sigma(Z,T)$ and let $g(\\tau,y)$ be a real-valued cocycle of $(Y,T)$.\nLet $h(\\tau,z)$ be a real-valued cocycle of $(Z,T)$ so that for every $\\tau\\in T$ and $z'\\in Z$ the image of $\\sigma^{-1}(\\sigma(z'))$ under the function $z\\mapsto h(\\tau,z)$ is connected and includes zero.\nSuppose that there exist a compact symmetric neighbourhood $K\\subset\\mathbb R$ of $0$ and $\\varepsilon>0$ so that for all $z\\in Z$ and $\\tau\\in T$ with $d_Z(z,\\tau z)<\\varepsilon$ holds $(g\\circ\\sigma+h)(\\tau,z)\\notin 2K\\setminus K^0$.\nSuppose that there exists a $\\delta>0$ so that for all $\\tau\\in T$ and $z\\in Z$ with $d_Z(z,\\tau z)<\\delta$ holds $d_Z(z',\\tau z')<\\varepsilon$ for every $z'\\in\\sigma^{-1}(\\sigma(z))$.\nGiven $\\bar z\\in Z$ and a sequence $\\{\\bar\\tau_k\\}\\subset T$ so that $\\bar\\tau_k \\bar z$ converges and $(g\\circ\\sigma)(\\bar\\tau_k,\\bar z)\\to 0$ as $k\\to\\infty$, the sequence $\\{(g\\circ\\sigma+h)(\\bar\\tau_k,\\bar z)\\}_{k\\geq 1}$ is bounded.\nSimilarly, for a sequence $\\{\\bar\\tau_k\\}\\subset T$ so that $\\bar\\tau_k\\bar z$ converges and $(g\\circ\\sigma+h)(\\bar\\tau_k,\\bar z)\\to 0$, the sequence $\\{(g\\circ\\sigma)(\\bar\\tau_k,\\bar z)\\}_{k\\geq 1}$ is bounded.\n\\end{lemma}\n\n\\begin{proof}\nThere exists a $k_0\\geq 1$ so that for all $k,k'\\geq k_0$ holds $d_Z(\\bar\\tau_k\\bar z,\\bar\\tau_{k'}\\bar z)<\\delta$ and\n\\begin{equation*}\n(g\\circ\\sigma)(\\bar\\tau_{k'},\\bar z)-(g\\circ\\sigma)(\\bar\\tau_k,\\bar z)=(g\\circ\\sigma)(\\bar\\tau_{k'}\\bar\\tau_{k}^{-1},\\bar\\tau_k \\bar z)\\in K^0 .\n\\end{equation*}\nBy the choice of $K$, $\\varepsilon$, and $\\delta$ follows that $(g\\circ\\sigma+h)(\\bar\\tau_{k'}\\bar\\tau_k^{-1},z)\\notin 2K\\setminus K^0$ for all $z\\in\\sigma^{-1}(\\sigma(\\bar\\tau_k\\bar z))$.\nSince the range of $(g\\circ\\sigma+h)(\\bar\\tau_{k'}\\bar\\tau_k^{-1},z)$ on the fibre $\\sigma^{-1}(\\sigma(\\bar\\tau_k\\bar z))$ is connected and intersects $K^0$, we can conclude that $(g\\circ\\sigma+h)(\\bar\\tau_{k'}\\bar\\tau_k^{-1},\\bar\\tau_k \\bar z)\\in K^0$ for all $k,k'\\geq k_0$.\nTherefore the sequence $\\{(g\\circ\\sigma+h)(\\bar\\tau_k,\\bar z)\\}_{k\\geq 1}$ is bounded.\n\nProvided a sequence $\\{\\bar\\tau_k\\}\\subset T$ with convergent $\\bar\\tau_k\\bar z$ and $(g\\circ\\sigma+h)(\\bar\\tau_k,\\bar z)\\to 0$, there exists an integer $k_0\\geq 1$ so that for all $k,k'\\geq k_0$ holds $d_Z(\\bar\\tau_k\\bar z,\\bar\\tau_{k'}\\bar z)<\\delta$ and $(g\\circ\\sigma+h)(\\bar\\tau_{k'}\\bar\\tau_{k}^{-1},\\bar\\tau_k \\bar z)\\in K^0$.\nWe conclude as above that $(g\\circ\\sigma+h)(\\bar\\tau_{k'}\\bar\\tau_k^{-1},z)\\in K^0$ for all $k,k'\\geq k_0$ and $z\\in\\sigma^{-1}(\\sigma(\\bar\\tau_k\\bar z))$.\nSince $h(\\bar\\tau_{k'}\\bar\\tau_k^{-1},z)=0$ for some $z\\in\\sigma^{-1}(\\sigma(\\bar\\tau_k\\bar z))$, the sequence $\\{(g\\circ\\sigma)(\\bar\\tau_k,\\bar z)\\}_{k\\geq 1}$ is bounded.\n\\end{proof}\n\nAt first the step from an ordinal to its successor by an isometric extension shall be considered.\nThe ``local'' behaviour within the fibres of a compact group extension is similar to a skew product extension by a compact metric group, even if the global structure might be different since it does not necessarily split into a product.\n\n\\begin{lemma}\\label{lem:c_t}\nLet $\\gamma$ be an ordinal with $1\\leq\\gamma<\\eta$.\nIf there exists a sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X_{\\gamma+1}$ with $d_{\\gamma+1} (x_k,\\tau_k x_k)\\to 0$ so that $(f_\\gamma\\circ\\pi_\\gamma^{\\gamma+1})(\\tau_k,x_k)\\to 0$ and $f_{\\gamma+1}(\\tau_k,x_k)\\nrightarrow 0$ as $k\\to\\infty$ (or equivalently $(f_{\\gamma+1}\\circ\\pi_\\gamma^{\\gamma+1})(\\tau_k,x_k)\\nrightarrow 0$ and $f_\\gamma(\\tau_k,x_k)\\to 0$), then the skew product $\\widetilde\\tau_{f_{\\gamma+1}}$ is necessarily point transitive.\nTherefore, if $f_\\gamma(\\tau,x_\\gamma)$ is transient, then $f_{\\gamma+1}(\\tau,x_{\\gamma+1})$ is either transient or the skew product $\\widetilde\\tau_{f_{\\gamma+1}}$ is point transitive.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $\\widetilde\\tau_{f_{\\gamma+1}}$ is not point transitive and let $G\\subset\\textup{Aut}(Z,T)$ define a compact metric group extension of $(X_\\gamma, T)$ with $(X_{\\gamma+1},T)=\\pi(Z,T)$.\nThen the skew product extension $\\widetilde\\tau_{f_{\\gamma+1}\\circ\\pi}$ of the flow $(Z,T)$ is also not point transitive, and Lemma \\ref{lem:at} provides $K\\subset\\mathbb R$ and $\\varepsilon>0$.\nSince $G$ acts uniformly equicontinuous, there exists a $\\delta>0$ so that for all $z\\in Z$ and $\\tau\\in T$ with $d_Z (z,\\tau z)<\\delta$ follows $d_Z(k(z),k(\\tau z))=d_Z(k(z),\\tau k(z))<\\varepsilon$ for all $k\\in K$.\nFor every $z\\in Z$ the $G$-orbit of $z$ is all of the fibre $(\\pi_\\gamma^{\\gamma+1}\\circ\\pi)^{-1}((\\pi_\\gamma^{\\gamma+1}\\circ\\pi)(z))$.\nSince the $\\pi^{\\gamma+1}_\\gamma$-fibres are connected, for every $\\tau\\in T$ and $z'\\in Z$ the range of $(f_{\\gamma+1}-f_\\gamma\\circ\\pi^{\\gamma+1}_\\gamma)(\\tau,\\pi(z))$ on the fibre $(\\pi_\\gamma^{\\gamma+1}\\circ\\pi)^{-1}((\\pi_\\gamma^{\\gamma+1}\\circ\\pi)(z'))$ is connected and contains zero.\nHence Lemma \\ref{lem:sub} applies with $(Y,T)=(X_\\gamma,T)$, $\\sigma=\\pi_\\gamma^{\\gamma+1}\\circ\\pi$, $g=f_\\gamma$, and $h(\\tau,z)=(f_{\\gamma+1}-f_\\gamma\\circ\\pi^{\\gamma+1}_\\gamma)(\\tau,\\pi(z))$.\n\nHowever, given the sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X_{\\gamma+1}$ in the hypothesis, Lemma \\ref{lem:trans} provides a point $\\bar x\\in X_{\\gamma+1}$ and a sequence $\\{\\bar\\tau_k\\}\\subset T$ so that $(f_\\gamma\\circ\\pi_\\gamma^{\\gamma+1})(\\bar\\tau_k,\\bar x)\\to 0$ and $f_{\\gamma+1}(\\bar\\tau_k,\\bar x)\\to\\infty$ (or $(f_\\gamma\\circ\\pi_\\gamma^{\\gamma+1})(\\bar\\tau_k,\\bar x)\\to\\infty$ and $f_{\\gamma+1}(\\bar\\tau_k,\\bar x)\\to 0$).\nBy choosing a point $\\bar z\\in Z$ with $\\pi(\\bar z)=\\bar x$ and changing to a subsequence of $\\{\\bar\\tau_k\\}\\subset T$ with $\\bar\\tau_k\\bar z$ convergent, this contradicts to Lemma \\ref{lem:sub}.\n\nNow suppose that $f_\\gamma(\\tau,x_\\gamma)$ is transient and $f_{\\gamma+1}(\\tau,x_{\\gamma+1})$ is recurrent.\nLet $x'\\in X_{\\gamma+1}$ be so that $(x',0)$ is $\\widetilde\\tau_{f_{\\gamma+1}}$-recurrent (cf. Remarks \\ref{rems:rec}).\nSince $(x',0)$ is cannot be $\\widetilde\\tau_{f_\\gamma\\circ\\pi_\\gamma^{\\gamma+1}}$-recurrent, there exist a neighbourhood $V\\subset X_{\\gamma+1}\\times\\mathbb R$ of $(x',0)$ and a replete semigroup $P\\subset T$ so that $\\widetilde\\tau_{f_\\gamma\\circ\\pi_\\gamma^{\\gamma+1}}(x',0)\\notin V$ for every $\\tau\\in P$.\nGiven an arbitrary compact set $C\\subset T$, by Theorem 6.32 in \\cite{G-H} there exists a replete semigroup $Q\\subset P\\setminus C$.\nSince $(x',0)$ is $\\widetilde\\tau_{f_{\\gamma+1}}$-recurrent, we can inductively construct a sequence $\\{\\tau_k\\}_{k\\geq 1}\\subset P$ with $\\tau_k x'\\to x'$, $f_{\\gamma+1}(\\tau_k,x')\\to0$, and $(f_\\gamma\\circ\\pi^{\\gamma+1}_\\gamma)(\\tau_k,x')\\nrightarrow 0$.\nThe point transitivity of $\\widetilde\\tau_{f_{\\gamma+1}}$ follows by the preceding statement.\n\\end{proof}\n\nFurthermore, we shall study the case of transfinite induction to a limit ordinal.\nThe arguments are quite similar, however with an approximation of a limit ordinal instead of an isometric group extension.\n\n\\begin{lemma}\\label{lem:lim}\nSuppose that $\\gamma$ is a limit ordinal with $1<\\gamma\\leq\\eta$.\n\\begin{enumerate}\n\\item If for every ordinal $1<\\alpha<\\gamma$ there exists an ordinal $\\alpha\\leq\\xi<\\gamma$ so that $f_\\xi(\\tau,x_\\xi)$ has a point transitive skew product extension, then $f_\\gamma(\\tau,x_\\gamma)$ has a point transitive skew product extension.\n\\item If there exist an ordinal $1\\leq\\alpha<\\gamma$ and a sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X_\\gamma$ with $d_\\gamma (x_k,\\tau_k x_k)\\to 0$  so that $(f_\\xi\\circ\\pi_\\xi^\\gamma)(\\tau_k,x_k)\\to 0$ for every $\\alpha\\leq\\xi<\\gamma$ and $f_\\gamma(\\tau_k,x_k)\\nrightarrow 0$ as $k\\to\\infty$ (or equivalently $(f_\\xi\\circ\\pi_\\xi^\\gamma)(\\tau_k,x_k)\\nrightarrow 0$ for every $\\alpha\\leq\\xi<\\gamma$ and $f_\\gamma(\\tau_k,x_k)\\to 0$), then $\\widetilde\\tau_{f_\\gamma}$ is necessarily point transitive.\n\\item If there exists an ordinal $1\\leq\\alpha<\\gamma$ so that for all $\\alpha\\leq\\xi<\\gamma$ the cocycle $f_\\xi(\\tau,x_\\xi)$ is transient, then $f_\\gamma(\\tau,x_\\gamma)$ is either transient or its skew product extension is point transitive.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nSuppose that the skew product of $\\tau_{f_\\gamma}$ on $X_\\gamma\\times\\mathbb R$ is not point transitive, and let $K\\subset\\mathbb R$ and $\\varepsilon>0$ be provided by Lemma \\ref{lem:at}.\nSince $\\gamma$ is a limit ordinal and $(X_\\gamma,T)$ is the inverse limit of the flows $\\{(X_\\xi,T):0\\leq\\xi<\\gamma\\}$, we can choose an ordinal $\\zeta<\\gamma$ so that for all $x,x'\\in X_\\gamma$ with $\\pi_\\zeta^\\gamma (x)=\\pi_\\zeta^\\gamma (x')$ holds $d_\\gamma (x,x')<\\varepsilon\/3$.\nIf we put $\\delta=\\varepsilon\/3$, then $d_\\gamma(x',\\tau x')<\\delta$ for $x'\\in X_\\gamma$ and $\\tau\\in T$ implies $d_\\gamma (x,\\tau x)<\\varepsilon$ for all $x\\in(\\pi^\\gamma_\\zeta)^{-1}(\\pi^\\gamma_\\zeta(x'))$.\nThese conditions remain valid even if the ordinal $\\zeta$ will be increased later.\nSince the $\\pi_\\zeta^{\\gamma}$-fibres are connected, for every $\\tau\\in T$ and $x'_\\gamma\\in X_\\gamma$ the range of $(f_{\\gamma}-f_\\zeta\\circ\\pi^{\\gamma}_\\zeta)(\\tau,x_\\gamma)$ on the $\\pi_\\zeta^{\\gamma}$-fibre of $x'_\\gamma$ is connected and contains $0$.\n\nUnder the hypothesis (i), we can choose $(\\tau,x_\\zeta)\\in T\\times X_\\zeta$ so that $f_\\zeta(\\tau,x_\\zeta)\\in 2K\\setminus K^0$ and $d_\\gamma(x'_\\gamma,\\tau x'_\\gamma)<\\delta$ for some $x_\\gamma'\\in(\\pi_\\zeta^{\\gamma})^{-1}(x_\\zeta)$.\nThus $d_\\gamma(x_\\gamma,\\tau x_\\gamma)<\\varepsilon$ holds for all $x_\\gamma\\in(\\pi_\\zeta^{\\gamma})^{-1}(x_\\zeta)$, and for $x_\\gamma$ with $(f_{\\gamma}-f_\\zeta\\circ\\pi^{\\gamma}_\\zeta)(\\tau,x_\\gamma)=0$ this contradicts $f_\\gamma(\\tau,x_\\gamma)\\notin 2K\\setminus K^0$.\nThus assertion (i) is verified.\n\nWe apply then Lemma \\ref{lem:sub} with $(Z,T)=(X_\\gamma,T)$, $(Y,T)=(X_\\zeta,T)$, $\\sigma=\\pi_\\zeta^{\\gamma}$, $h=(f_{\\gamma}-f_\\zeta\\circ\\pi^{\\gamma}_\\zeta)$, and $g=f_\\zeta$.\nHowever, given the sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X_\\gamma$ in hypothesis (ii), Lemma \\ref{lem:trans} provides a point $\\bar x\\in X_\\gamma$ and a sequence $\\{\\bar\\tau_k\\}\\subset T$ so that $(f_\\zeta\\circ\\pi_\\zeta^{\\gamma},f_{\\gamma}-f_\\zeta\\circ\\pi^{\\gamma}_\\zeta)(\\bar\\tau_k,\\bar x)=(g\\circ\\sigma,h)(\\bar\\tau_k,\\bar x)\\to(0,\\infty)$ (or $(\\infty,0)$) and $\\bar\\tau_k\\bar x\\to\\bar z$ as $k\\to\\infty$.\nThis is a contradiction to Lemma \\ref{lem:sub} and verifies (ii).\n\nNow suppose that $f_\\zeta(\\tau,x_\\zeta)$ is transient and $f_{\\gamma}(\\tau,x_{\\gamma})$ is recurrent, and choose $x'\\in X_\\gamma$ so that $(x',0)$ is $\\widetilde\\tau_{f_{\\gamma}}$-recurrent.\nSince $(x',0)$ is cannot be $\\widetilde\\tau_{f_\\zeta\\circ\\pi_\\zeta^{\\gamma}}$-recurrent, there exist a neighbourhood $V\\subset X_\\gamma\\times\\mathbb R$ of $(x',0)$ and a replete semigroup $P\\subset T$ with $\\widetilde\\tau_{f_\\zeta\\circ\\pi_\\zeta^{\\gamma}}(x',0)\\notin V$ for every $\\tau\\in P$.\nBy induction exists a sequence $\\{\\tau_k\\}_{k\\geq 1}\\subset P$ with $\\widetilde{(\\tau_k)}_{f_{\\gamma}}(x',0)\\to(x',0)$ as $k\\to\\infty$, and by Lemma \\ref{lem:trans} there exist a point $\\bar x\\in X_\\gamma$ and sequence $\\{\\bar\\tau_k\\}\\subset T$ so that $(f_\\gamma,f_\\zeta\\circ\\pi^{\\gamma}_\\zeta)(\\bar\\tau_k,\\bar x)=(g\\circ\\sigma+h,g\\circ\\sigma)(\\bar\\tau_k,\\bar x)\\to(0,\\infty)$ and $\\bar\\tau_k\\bar x\\to\\bar x$.\nThis contradiction to Lemma \\ref{lem:sub} verifies the statement (iii).\n\\end{proof}\n\n\\begin{proposition}\\label{prop:max}\nIf the real-valued cocycle $f(\\tau,x)$ is topologically recurrent apart from a coboundary, then there exists a maximal ordinal $1\\leq\\alpha\\leq\\eta$ so that the skew product extension $\\widetilde\\tau_{f_\\alpha}$ is point transitive on $X_\\alpha\\times\\mathbb R$.\nThe cocycle $(f-f_\\alpha\\circ\\pi_\\alpha) (\\tau,x)$ is relatively trivial with respect to $(f_\\alpha\\circ\\pi_\\alpha)(\\tau,x)$.\n\\end{proposition}\n\n\\begin{proof}\nLet us first suppose that the cocycle $f_\\xi(\\tau,x_\\xi)$ is recurrent for every ordinal $1\\leq\\xi<\\eta$, and let $\\mathcal O=\\{1\\leq\\xi\\leq\\eta:{f_\\xi}(\\tau,x_\\xi)\\enspace\\textup{is \\emph{not} a coboundary}\\}$.\nThis set is non-empty since $f_\\eta(\\tau,x)$ is not a coboundary, and let $\\beta$ be its minimal element.\nIf $\\beta=1$, then by Proposition \\ref{prop:isom} the recurrent skew product $\\widetilde{\\tau}_{f_1}$ of the isometric flow $(X_1,T)$ is point transitive.\nIf $\\beta>1$, then Fact \\ref{fact:GH} provides a sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X_\\beta$ with $d_\\beta(x_k,\\tau_k x_k)\\to 0$ and $f_\\beta(\\tau_k, x_k)\\to\\infty$.\nFor all $1\\leq\\zeta<\\beta$ holds $(f_\\beta\\circ\\pi_\\zeta^\\beta)(\\tau_k, x_k)\\to 0$, and by the Lemmas \\ref{lem:c_t} and \\ref{lem:lim} (ii) $\\widetilde{\\tau}_{f_\\beta}$ is point transitive.\n\nIf $f_\\xi(\\tau,x_\\xi)$ is transient for an ordinal $1\\leq\\xi<\\eta$, then let $\\beta$ be the minimal element of the set $\\mathcal O=\\{\\xi<\\zeta\\leq\\eta:f_\\zeta(\\tau,x_\\zeta)\\enspace\\textup{is topologically recurrent}\\}$.\nThis set is non-empty since $f_\\eta(\\tau,x_\\eta)$ is topologically recurrent, and it follows from the Lemmas \\ref{lem:c_t} and \\ref{lem:lim} (iii) that $\\widetilde\\tau_{f_\\beta}$ is even point transitive.\n\nNow let $\\mathcal O=\\{1\\leq\\xi\\leq\\eta:\\widetilde\\tau_{f_\\zeta}\\enspace\\textup{is \\emph{not} point transitive for all}\\enspace\\xi\\leq\\zeta\\leq\\eta\\}$.\nIf $\\mathcal O$ is empty, then $\\widetilde\\tau_{f_\\eta}$ is point transitive and $\\alpha=\\eta$.\nOtherwise, the set $\\mathcal O$ has a minimal element $\\gamma>1$ since $\\widetilde\\tau_{f_\\beta}$ is point transitive for some $1\\leq\\beta\\leq\\eta$.\nSince $\\gamma$ cannot be a limit ordinal by Lemma \\ref{lem:lim} (i), there exists a maximal ordinal $\\alpha\\geq 1$ with point transitive $\\widetilde\\tau_{f_\\alpha}$.\nIf $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ is a sequence with $d (x_k,\\tau_k x_k)\\to 0$ and $(f_\\alpha\\circ\\pi_\\alpha)(\\tau_k,x_k)\\to 0$, then transfinite induction using the maximality of $\\alpha$ and Lemmas \\ref{lem:c_t}, \\ref{lem:lim} (ii) verifies that $(f_\\xi\\circ\\pi_\\xi)(\\tau_k,x_k)\\to 0$ for every $\\alpha\\leq\\xi\\leq\\eta$.\n\\end{proof}\n\nAfter the flow $(X_\\alpha,T)$ with a point transitive skew product extension $\\widetilde\\tau_{f_\\alpha}$ has been identified, we shall study the extension from $(X_\\alpha,T)$ to $(X,T)$.\nThere might be infinitely many isometric extensions in between, and therefore this extension is in general a distal extension.\nSince our construction will use the regulariser of this extension, it is necessary to leave the category of compact metric flows for the category of compact Hausdorff flows during the following construction (cf. Remark \\ref{rem:d_n_m}).\nHowever, the flow which will be constructed by means of the regulariser will be metric as a factor of the compact metric flow $(X,T)$.\n\n\\begin{proposition}\\label{prop:flow}\nThere exists a factor $(Y,T)=(X_\\alpha\\times M,T)=\\pi_Y(X,T)$ which is a Rokhlin extension of $(X_\\alpha,T)=\\rho_\\alpha(Y,T)$ by a distal minimal compact metric $\\mathbb R$-flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ and the cocycle $f_\\alpha(\\tau,x_\\alpha)$ so that for every $x\\in X$ holds\n\\begin{equation}\\label{eq:p_X}\n\\pi_Y^{-1}(\\pi_Y(x))\\times\\{0\\}=\\mathcal D_{T, f_\\alpha\\circ\\pi_\\alpha}(x,0)\\cap(\\pi_\\alpha^{-1}(\\pi_\\alpha(x))\\times\\{0\\}) .\n\\end{equation}\nThe $\\mathbb R$-flow $\\{\\psi^t:t\\in\\mathbb R\\}\\subset\\textup{Aut}(Y,T)$ defined by $\\psi^t(x_\\alpha,m)=(x_\\alpha,\\phi^t(m))$ for $(x_\\alpha,m)\\in Y=X_\\alpha\\times M$ fulfils for every $y\\in Y$ and every $t\\in\\mathbb R$ that\n\\begin{equation}\\label{eq:o_flow}\n\\bar{\\mathcal O}_{T, f_\\alpha\\circ\\rho_\\alpha}(y,0)\\cap(\\rho_\\alpha^{-1}(\\rho_\\alpha(y))\\times\\{t\\})\\subset\\{(\\psi^t(y),t)\\} ,\n\\end{equation}\nwith coincidence of these sets if $(\\rho_\\alpha(y),0)\\in X_\\alpha\\times\\mathbb R$ is transitive for $\\widetilde\\tau_{f_\\alpha}$.\n\\end{proposition}\n\n\\begin{proof}\nWe shall construct a factor $(Y,T)$ of $(X,T)$ and a flow $\\{\\varphi^t:t\\in\\mathbb R\\}\\subset\\textup{Aut}(Y,T)$, and then we shall represent $(Y,T)$ as a Rokhlin extension of $(X_\\alpha,T)$.\nLet $(\\tilde X,T)$ be a distal minimal compact Hausdorff flow with $(X,T)=\\pi(\\tilde X,T)$ and a Hausdorff topological group $G\\subset\\textup{Aut}(\\tilde X,T)$ acting freely on the fibres of $\\pi_\\alpha\\circ\\pi$ so that $(X,T)$ is the $H$-orbit space of a subgroup $H\\subset G$ (cf. Fact \\ref{fact:reg_ex}).\nFor an arbitrary point $\\tilde z\\in \\tilde X$ and $t\\in\\mathbb R$ we define a closed subset of $G$ by\n\\begin{equation}\\label{eq:G}\nG_{\\tilde z,t}=\\{g\\in G: (\\pi(g(\\tilde z)),t)\\in\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(\\pi(\\tilde z),0)\\} .\n\\end{equation}\nThe mapping $\\pi$ is open as a homomorphism of distal minimal compact flows, and hence for every $g\\in G_{\\tilde z,t}$ there exist nets $\\{\\tilde z_i\\}_{i\\in I}\\subset \\tilde X$ and $\\{\\tau_i\\}_{i\\in I}\\subset T$ with $\\tilde z_i\\to \\tilde z$, $\\tau_i\\pi(\\tilde z_i)\\to\\pi(g(\\tilde z))$, and $f_\\alpha(\\tau_i,\\pi_\\alpha\\circ\\pi(\\tilde z_i))\\to t$.\nSince the cocycle $(f_\\alpha\\circ\\pi_\\alpha)(\\tau,x_\\alpha)$ is constant on the fibres of $\\pi_\\alpha$ and $T$ is Abelian, it follows for every fixed $\\tau\\in T$ that\n\\begin{equation*}\n\\tau_i\\pi(\\tau \\tilde z_i)=\\tau_i\\tau\\pi(\\tilde z_i)=\\tau\\tau_i\\pi(\\tilde z_i)\\to\\tau\\pi(g(\\tilde z))=\\pi(\\tau g(\\tilde z))=\\pi(g(\\tau \\tilde z))\n\\end{equation*}\nand by the cocycle identity\n\\begin{eqnarray*}\nf_\\alpha(\\tau_i,\\pi_\\alpha\\circ\\pi(\\tau \\tilde z_i)) & = & f_\\alpha(\\tau_i,\\pi_\\alpha\\circ\\pi(\\tilde z_i))-f_\\alpha(\\tau,\\pi_\\alpha\\circ\\pi(\\tilde z_i))\\\\\n& & +f_\\alpha(\\tau,\\pi_\\alpha\\circ\\pi(\\tau_i\\tilde z_i))\\to t .\n\\end{eqnarray*}\nBy the density of the $T$-orbit of $\\tilde z$ and a diagonalisation of nets there exist for every $\\tilde x\\in \\tilde X$ nets $\\{\\tilde x_i\\}_{i\\in I}\\subset \\tilde X$ and $\\{\\tau'_i\\}_{i\\in I}\\subset T$ with $\\tilde x_i\\to \\tilde x$, $\\tau'_i\\pi(\\tilde x_i)\\to\\pi(g(\\tilde x))$, and $f_\\alpha(\\tau_i,\\pi_\\alpha\\circ\\pi(\\tilde x_i))\\to t$.\nTherefore\n\\begin{equation*}\n(\\pi(g(\\tilde x)),t)\\in\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(\\pi(\\tilde x),0)\n\\end{equation*}\nso that $g\\in G_{\\tilde x,t}=G_{\\tilde z,t}=G_t$.\nBy symmetry follows now that $G_{-t}=(G_t)^{-1}$.\n\nThen we fix a point $x'\\in X$ with $\\bar{\\mathcal O}_{T,f_\\alpha}(\\pi_\\alpha(x'),0)=X_\\alpha\\times\\mathbb R$ and $\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(x',0)=\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\pi_\\alpha}(x',0)$ (cf. Fact \\ref{fact:o_p}).\nThe set $G_t$ is non-empty for every $t\\in\\mathbb R$, since $\\bar{\\mathcal O}_{T, f_\\alpha}(\\pi_\\alpha(x'),0)=X_\\alpha\\times\\mathbb R$ and the compactness of $X$ ensure that\n\\begin{equation*}\n\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\pi_\\alpha}(x',0)\\cap\\pi_\\alpha^{-1}(\\pi_\\alpha(x'))\\times\\{t\\}\\neq\\emptyset .\n\\end{equation*}\nFor arbitrary $t,t'\\in\\mathbb R$ and $g\\in G_t$, $g'\\in G_{t'}$, we select $\\tilde x, \\tilde z\\in \\tilde X$ so that $\\pi(\\tilde x)=x'$ and $\\tilde x=g'(\\tilde z)$.\nThen we have\n\\begin{equation*}\n(x',t')=(\\pi(g'(\\tilde z)),t')\\in\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(\\pi(\\tilde z),0) ,\n\\end{equation*}\nand for $\\tilde y=g(\\tilde x)=gg'(\\tilde z)$ it holds that $(\\pi(\\tilde y),t)\\in\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\pi_\\alpha}(x',0)=\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(x',0)$.\nBy Remark \\ref{rem:o_p} follows $(\\pi(\\tilde y),t+t')\\in\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(\\pi(\\tilde z),0)$ so that $gg'\\in G_{t+t'}$.\nHence $G_t G_{t'}\\subset G_{t+t'}$ holds for all $t,t'\\in\\mathbb R$, and from $G_{-t}=(G_{t})^{-1}$ follows $(G_{t})^{-1} G_{t+t'}=G_{-t} G_{t+t'}\\subset G_{t'}$ so that $G_{t} G_{t'}= G_{t+t'}$.\nThus the Hausdorff topological group\n\\begin{equation*}\n\\tilde G=\\cup_{t\\in\\mathbb R}G_t\n\\end{equation*}\nhas the closed set $G_0\\supset H$ as a normal subgroup so that $G_t$ is a $G_0$-coset in $\\tilde G$ for every $t\\in\\mathbb R$.\nMoreover, the mapping $t\\mapsto G_t$ is a group homomorphism from $\\mathbb R$ into $\\tilde G\/G_0$.\nThe group $G_0$ is not necessarily compact, however its orbit space on $\\tilde X$ defines a partition into sets invariant under $H\\subset G_0$.\nHence this is also a partition of $X$, and the equivalence relation $R_Y$ of this partition of $X$ is $T$-invariant since $G_0\\subset\\textup{Aut}(\\tilde X,T)$.\nMoreover, $R_Y$ is closed in $X^2$, since definition (\\ref{eq:G}) implies that $(x,x')\\in R_Y$ if and only if $(x',0)\\in\\mathcal D_{T, f_\\alpha\\circ\\pi_\\alpha}(x,0)\\cap(\\pi_\\alpha^{-1}(\\pi_\\alpha(x))\\times\\{0\\})$.\nThe factor $(Y,T)=\\pi_Y(X,T)$ defined by the $T$-invariant closed equivalence relation $R_Y$ is an extension of $(X_\\alpha,T)=\\rho_\\alpha(Y,T)$, and equality (\\ref{eq:p_X}) follows.\nThe $\\mathbb R$-action $\\{\\varphi^t:t\\in\\mathbb R\\}\\subset\\textup{Aut}(Y,T)$ is well defined for every $y\\in Y$ and $t\\in\\mathbb R$ by\n\\begin{equation*}\n\\varphi^t(y)=G_t ((\\pi_Y\\circ\\pi)^{-1}(y))=G_t (\\{\\tilde x\\in \\tilde X:G_0(\\tilde x)=y\\}) .\n\\end{equation*}\nLet $\\{(t_k,y_k)\\}_{k\\geq 1}\\subset\\mathbb R\\times Y$ be a sequence with $(t_k,y_k)\\to (t,y)$, then $\\varphi^{t_k}(y_k)=G_0 g_k(\\tilde x_k)$ for a sequence $\\{\\tilde x_k\\}_{k\\geq 1}\\subset\\tilde X$ with $\\pi_Y\\circ\\pi(\\tilde x_k)=y_k$ and $g_k\\in G_{t_k}$.\nWe can assume that $\\tilde x_k\\to\\tilde x$ and $g_k(\\tilde x_k)\\to\\tilde z$ so that $(\\pi(\\tilde z),t)\\in\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(\\pi(\\tilde x),0)$ and $\\tilde z=g_t(\\tilde x)$ for some $g_t\\in G_t$.\nFrom $\\pi_Y\\circ\\pi(\\tilde x)=y$ and $\\varphi^{t_k}(y_k)=\\pi_Y\\circ\\pi(g_k(\\tilde x_k))\\to\\pi_Y\\circ\\pi(\\tilde z)=\\varphi^t(y)$ follows the continuity of the action $\\{\\varphi^t:t\\in\\mathbb R\\}$ on $Y$.\n\nWe turn to the inclusion (\\ref{eq:o_flow}).\nSuppose that $(y_i,t)\\in\\bar{\\mathcal O}_{T, f_\\alpha\\circ\\rho_\\alpha}(y,0)\\cap\\rho_\\alpha^{-1}(x_\\alpha)\\times\\{t\\}$ for some $x_\\alpha\\in X_\\alpha$ and $i\\in\\{1,2\\}$, and select $x\\in\\pi_Y^{-1}(y)$.\nBy the compactness of $X$ there exist points $x_i\\in\\pi_Y^{-1}(y_i)\\subset\\pi_\\alpha^{-1}(x_\\alpha)$ so that $(x_i,t)\\in\\bar{\\mathcal O}_{T, f_\\alpha\\circ\\pi_\\alpha}(x,0)$, and therefore $(x_2,0)\\in\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(x_1,0)$.\nThe equality (\\ref{eq:p_X}) implies that $y_1=\\pi_Y(x_1)=\\pi_Y(x_2)=y_2$, and thus for every $y\\in Y$ and $t\\in\\mathbb R$ holds\n\\begin{equation}\\label{eq:card_1}\n\\textup{card}\\{\\bar{\\mathcal O}_{T, f_\\alpha\\circ\\rho_\\alpha}(y,0)\\cap\\rho_\\alpha^{-1}(\\rho_\\alpha(y))\\times\\{t\\}\\}\\leq 1 .\n\\end{equation}\nMoreover, for $x_\\alpha=\\rho_\\alpha(y)$ follows $x_1=\\pi(g_t(\\tilde x))$ with $g_t\\in G_t$ and $\\tilde x\\in\\pi^{-1}(x)\\subset\\tilde X$.\nHence $y_1=\\pi_Y(x_1)=\\varphi^t(y)$ and inclusion (\\ref{eq:o_flow}) is verified.\nFor $\\widetilde\\tau_{f_\\alpha}$-transitive $(\\rho_\\alpha(y),0)$ the cardinality in (\\ref{eq:card_1}) is equal to $1$ for every $y\\in Y$ and $t\\in\\mathbb R$, and for $y'\\in\\rho_\\alpha^{-1}(\\rho_\\alpha(y))$ and $x_\\alpha\\in X_\\alpha$ holds $\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\cap\\rho_\\alpha^{-1}(x_\\alpha)\\times\\{0\\}=\\{(y_1,0)\\}$.\nWe fix a point $\\bar x\\in X_\\alpha$ with $\\widetilde\\tau_{f_\\alpha}$-transitive $(\\bar x,0)$.\nIf $(y_2,0)\\in\\mathcal D_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\cap\\rho_\\alpha^{-1}(x_\\alpha)\\times\\{0\\}$, then Remark \\ref{rem:o_p} implies that $(y_2,0)\\in\\mathcal D_{T,f_\\alpha\\circ\\rho_\\alpha}(y_1,0)$, and as above follows $y_1=y_2$.\nHence\n\\begin{equation}\\label{eq:homeo_Y}\n\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\cap\\rho_\\alpha^{-1}(x_\\alpha)\\times\\{0\\}=\\mathcal D_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\cap\\rho_\\alpha^{-1}(x_\\alpha)\\times\\{0\\}\n\\end{equation}\nholds for every $y'\\in\\rho_\\alpha^{-1}(\\bar x)$ and $x_\\alpha\\in X_\\alpha$.\nFor distinct $y',y''\\in\\rho_\\alpha^{-1}(\\bar x)$ follows\n\\begin{equation*}\n\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\cap\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\rho_\\alpha}(y'',0)\\cap\\rho_\\alpha^{-1}(x_\\alpha)\\times\\{0\\}=\\emptyset .\n\\end{equation*}\nIndeed, given a point $\\bar y$ in this intersection, for every sequence $\\{\\tau_k\\}_{k\\geq 1}\\subset T$ with $\\tau_k\\bar x\\to\\rho_\\alpha(\\bar y)$ and $f_\\alpha(\\tau_k,\\bar x)\\to 0$ follows by equality (\\ref{eq:card_1}) that $d_Y(\\tau_k y',\\tau_k y'')\\to 0$, in contradiction to the distality of $(Y,T)$.\nHence the mapping $\\iota:X_\\alpha\\times\\rho_\\alpha^{-1}(\\bar x)\\longrightarrow Y$\n\\begin{equation*}\n(x_\\alpha,y')\\mapsto\\rho_\\alpha^{-1}(x_\\alpha)\\cap\\{y\\in Y:(y,0)\\in\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\}\n\\end{equation*}\nis well-defined, one-to-one, and by equality (\\ref{eq:homeo_Y}) also continuous.\nFor a dense set of points $\\bar y\\in Y$ holds the $\\widetilde\\tau_{f_\\alpha}$-transitivity of $(\\rho_\\alpha(\\bar y),0)$, since $\\rho_\\alpha$ is open.\nWe can conclude for every $y\\in Y$ that $\\mathcal D_{T,f_\\alpha\\circ\\rho_\\alpha}(y,0)\\cap\\rho_\\alpha^{-1}(\\bar x)\\times\\{0\\}\\neq\\emptyset$, and thus $\\mathcal D_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\cap\\{y\\}\\times\\{0\\}=\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\cap\\{y\\}\\times\\{0\\}\\neq\\emptyset$ for some $y'\\in\\rho_\\alpha^{-1}(\\bar x)$.\nHence $\\iota$ is onto and by compactness $Y$ and $X_\\alpha\\times\\rho_\\alpha^{-1}(\\bar x)$ are homeomorphic.\n\nLet $\\{\\phi^t:t\\in\\mathbb R\\}$ be the restriction of $\\{\\varphi^t:t\\in\\mathbb R\\}$ to the $\\{\\varphi^t:t\\in\\mathbb R\\}$-invariant compact metric space $M=\\rho_\\alpha^{-1}(\\bar x)$.\nFor every $y'\\in M$ and $\\tau\\in T$ holds $\\bar{\\mathcal O}_{T, f_\\alpha\\circ\\rho_\\alpha}(y',0)\\cap\\rho_\\alpha^{-1}(\\bar x)\\times\\{-f_\\alpha(\\tau,\\bar x) \\}=\\{(\\phi^{-f_\\alpha(\\tau,\\bar x)}(y'),-f_\\alpha(\\tau,\\bar x))\\}$ and $\\widetilde{\\tau}_{f_\\alpha\\circ\\pi_Y}(\\phi^{-f_\\alpha(\\tau,\\bar x)}(y'),-f_\\alpha(\\tau,\\bar x))\\in\\rho_\\alpha^{-1}(\\tau\\bar x)\\cap\\{y\\in Y:(y,0))\\in\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\rho_\\alpha}(y',0)\\}$.\nTherefore $\\tau\\phi^{-f_\\alpha(\\tau,\\bar x)}(y')=\\iota(\\tau\\bar x,y')$ and $\\tau y=\\iota(\\tau_{\\phi,f_\\alpha}(\\bar x,y))$ for every $y\\in M$ and $\\tau\\in T$.\nThe minimality of $(Y,T)$ implies that $(X_\\alpha\\times M,T)$ and $(Y,T)$ are topologically isomorphic via $\\iota$.\nMoreover, for the mapping $\\psi^t(x_\\alpha,m)=(x_\\alpha,\\phi^t(m))$ with $\\psi\\in\\textup{Aut}(X_\\alpha\\times M,T)$ and every $m\\in M=\\rho_\\alpha^{-1}(\\bar x)$ and $t\\in\\mathbb R$ holds $\\varphi^t(\\iota(\\tau_{\\phi,f_\\alpha}(\\bar x,m)))=\\varphi^t(\\tau m)=\\tau\\varphi^t(m)=\\iota(\\tau_{\\phi,f_\\alpha}(\\bar x,\\phi^t(m)))=\\iota(\\psi^t(\\tau_{\\phi,f_\\alpha}(\\bar x,m)))$.\nBy the minimality of $(X_\\alpha\\times M,T)$ follows $\\psi^t=\\iota^{-1}\\circ\\varphi^t\\circ\\iota$ for every $t\\in\\mathbb R$.\nThe flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ is minimal and distal, since a non-transitive point $m'\\in M$ and a proximal pair $(m',m'')\\in M^2$, respectively, give rise a non-transitive point $(x_\\alpha,m')\\in Y$ and a proximal pair $((x_\\alpha,m'),(x_\\alpha,m''))\\in Y^2$, respectively.\n\\end{proof}\n\nIt should be mentioned that an ordinal $\\xi\\leq\\eta$ with $(Y,T)=(X_\\xi,T)$ does not necessarily exist.\nTherefore we shall define a cocycle $f_Y:T\\times Y\\longrightarrow\\mathbb R$ independently of the cocycles $f_\\xi(\\tau,x_\\xi)$, and it will turn out that $(f_Y\\circ\\pi_Y)(\\tau,x)$ can be chosen topologically cohomologous to $f$.\n\n\\begin{proposition}\\label{prop:res}\nThere exists a topological cocycle $f_Y(\\tau,y)$ of the flow $(Y,T)$ so that $(f_Y\\circ\\pi_Y)(\\tau,x)$ is topologically cohomologous to $f(\\tau,x)$ and $f_Y(\\tau,y)$ is relatively trivial with respect to $(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)$.\n\\end{proposition}\n\nWe shall prove another technical lemma first.\n\n\\begin{lemma}\\label{lem:tilde_coc}\nLet $(Z,T)$ be a distal minimal compact metric flow which extends $(X_\\alpha,T)=\\sigma_\\alpha(Z,T)$, and let $G\\subset\\textup{Aut}(Z,T)$ be a Hausdorff topological group preserving the fibres of $\\sigma_\\alpha$.\nSuppose that there exists a continuous group homomorphism $\\varphi:G\\longrightarrow\\mathbb R$ so that for every $g\\in G$ and every $z\\in Z$ it holds that\n\\begin{equation*}\n(g(z),\\varphi(g))\\in\\mathcal D_{T,f_\\alpha\\circ\\sigma_\\alpha}(z,0) .\n\\end{equation*}\nFurthermore, suppose that $h(\\tau,z)$ is a real-valued cocycle of $(Z,T)$ which is relatively trivial with respect to $(f_\\alpha\\circ\\sigma_\\alpha)(\\tau,z)$.\nThen there exists a continuous cocycle $\\bar h((\\tau,g),z)$ of the flow $(Z,T\\times G)$ with the action $\\{g\\circ\\tau:(\\tau,g)\\in T\\times G\\}$ so that $h(\\tau,z)=\\bar h((\\tau,\\mathbf 1_G),z)$ holds for every $(\\tau,z)\\in T\\times Z$ and the mapping $h\\mapsto\\bar h$ is linear.\nFor $z\\in Z$, $g\\in G$, and a sequence $\\{(\\tau_k,z_k)\\}_{k\\geq 1}\\subset T\\times Z$ with $z_k\\to z$, $\\tau_k z_k\\to g(z)$, and $(f_\\alpha\\circ\\rho_\\alpha)(\\tau_k,z_k)\\to\\varphi(g)$ holds\n\\begin{equation}\\label{eq:h_uni}\nh(\\tau_k,z_k)\\to\\bar h((\\mathbf 1_T,g),z)\\quad\\textup{as}\\enspace k\\to\\infty.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nWe put $F=(f_\\alpha\\circ\\sigma_\\alpha,h):T\\times Z\\longrightarrow\\mathbb R^2$ and fix a point $\\bar z\\in Z$ so that $\\bar{\\mathcal O}_{T,F} (\\bar z,0,0)$ and $\\mathcal D_{T,F} (\\bar z,0,0)$ coincide in $Z\\times(\\mathbb R_\\infty)^2$ (cf. Fact \\ref{fact:o_p}).\nFor every $g\\in G$ we fix a sequence $\\{\\tau_k^g\\}_{k\\geq 1}\\subset T$ with $\\widetilde{(\\tau_k^g)}_{f_\\alpha\\circ\\sigma_\\alpha}(\\bar z,0)\\to(g(\\bar z),\\varphi(g))$ as $k\\to\\infty$, with $\\{\\tau_k^{\\mathbf 1_G}=\\mathbf 1_T\\}_{k\\geq 1}$.\nSince $g\\in\\textup{Aut}(Z,T)$ and $f_\\alpha\\circ\\sigma_\\alpha\\circ g=f_\\alpha\\circ\\sigma_\\alpha$, we can conclude for every $t\\in T$ that $\\tau_k^gt\\bar z=t\\tau_k^g\\bar z\\to t g(\\bar z)=g(t\\bar z)$ as $k\\to\\infty$ as well as\n\\begin{equation}\\label{eq:transf}\n(f_\\alpha\\circ\\sigma_\\alpha)(\\tau_k^g,t\\bar z)=(f_\\alpha\\circ\\sigma_\\alpha)(t, \\tau_k^g\\bar z)+(f_\\alpha\\circ\\sigma_\\alpha)(\\tau_k^g,\\bar z)-(f_\\alpha\\circ\\sigma_\\alpha)(t,\\bar z)\\to\\varphi(g).\n\\end{equation}\nBy the relative triviality of $h(\\tau,z)$ with respect to $(f_\\alpha\\circ\\sigma_\\alpha)(\\tau,z)$, the sequence $\\{h(\\tau\\tau_k^g,t\\bar z)\\}_{k\\geq 1}$ converges for all $\\tau,t\\in T$.\nThus we can put\n\\begin{equation}\\label{eq:def_g}\n\\bar h((\\tau,g),t\\bar z)=\\lim_{k\\to\\infty} h(\\tau\\tau_k^g,t\\bar z)=h(\\tau,g(t\\bar z))+\\lim_{k\\to\\infty} h(\\tau_k^g,t\\bar z)\n\\end{equation}\nfor every $(\\tau, g,t\\bar z)\\in T\\times G\\times Z$.\nSuppose that there exist sequences $\\{(\\tau_k^{i},z_k^i)\\}_{k\\geq 1}\\subset T\\times Z$ for $i=1,2$ so that $z_k^i\\to z$, $\\tau_k^i z_k^i\\to g(z)=\\lim_{k\\to\\infty}g(z_k^i)$, and\n\\begin{equation*}\n(f_\\alpha\\circ\\sigma_\\alpha)(\\tau_k^{i},z_k^i)\\to\\varphi(g) \\quad\\textup{as}\\enspace k\\to\\infty,\n\\end{equation*}\nwhile for $i=1,2$ the limit points $\\bar h_i=\\lim_{k\\to\\infty} h(\\tau_k^{i},z_k^i)\\in\\mathbb R_\\infty$ are either distinct or both equal to $\\infty$.\nThen $(g(z),\\varphi(g),\\bar h_i)\\in\\mathcal D_{T,F} (z,0,0)$ for $i=1,2$, and for every $\\tau'\\in T$ follows from $g\\in\\textup{Aut}(Z,T)$ and the cocycle identity that\n\\begin{eqnarray*}\n(g(\\tau' z),\\varphi(g)+(f_\\alpha\\circ\\sigma_\\alpha)(\\tau',g(z))-(f_\\alpha\\circ\\sigma_\\alpha)(\\tau',z),h(\\tau',g(z))+\\bar h_i-h(\\tau',z))=\\hspace{-5mm}\\\\\n(g(\\tau' z),\\varphi(g),h(\\tau',g(z))+\\bar h_i-h(\\tau',z))\\in\\mathcal D_{T,F}(\\tau' z,0,0).\n\\end{eqnarray*}\nSince $\\bar{\\mathcal O}_T(z)=Z$, either there are distinct points $a_1,a_2\\in\\mathbb R_\\infty$ with $(g(\\bar z),\\varphi(g),a_i)\\in\\mathcal D_{T,F} (\\bar z,0,0)$ or it holds that $(g(\\bar z),\\varphi(g),\\infty)\\in\\mathcal D_{T,F} (\\bar z,0,0)$.\nIn either case, since $\\bar{\\mathcal O}_{T,F} (\\bar z,0,0)=\\mathcal D_{T,F} (\\bar z,0,0)$ in $Z\\times(\\mathbb R_\\infty)^2$, this contradicts to the relative triviality of $h(\\tau,z)$ with respect to $(f_\\alpha\\circ\\sigma_\\alpha)(\\tau,z)$.\nTherefore equality (\\ref{eq:h_uni}) holds true, and the definition (\\ref{eq:def_g}) extends uniquely from the $T$-orbit of $\\bar z$ to a continuous mapping $\\bar h:T\\times G\\times Z\\longrightarrow\\mathbb R$ since the action of $T\\times G$ on $X$ and $\\varphi$ are continuous.\n\nFor the cocycle identity let $(\\tau_1,g_1),(\\tau_2,g_2)\\in T\\times G$ be arbitrary with sequences $\\{\\tau_k^{g_1}\\}_{k\\geq 1},\\{\\tau_k^{g_2}\\}_{k\\geq 1}\\subset T$.\nBy equality (\\ref{eq:transf}) we select a sequence $\\{k_l\\}_{l\\geq 1}\\subset\\mathbb N$ with\n\\begin{equation*}\n\\tau_{k_l}^{g_2}\\tau_l^{g_1}\\bar z=\\tau_l^{g_1}\\tau_{k_l}^{g_2}\\bar z\\to g_2(g_1(\\bar z))\\enspace\\textup{and}\\enspace (f_\\alpha\\circ\\sigma_\\alpha)(\\tau_l^{g_1}\\tau_{k_l}^{g_2},\\bar z)\\to\\varphi(g_2)+\\varphi(g_1)=\\varphi(g_2 g_1)\n\\end{equation*}\nas $l\\to\\infty$.\nThus we can put $\\{\\tau_l^{g_2 g_1}\\}_{l\\geq 1}=\\{\\tau_l^{g_1}\\tau_{k_l}^{g_2}\\}_{l\\geq 1}$, and for every $t\\in T$ the equality (\\ref{eq:transf}) implies that $\\tau_{k_l}^{g_2}\\tau_2 t\\bar z\\to g_2(\\tau_2 t\\bar z)$, $\\tau_l^{g_1}\\tau_{k_l}^{g_2}\\tau_2 t\\bar z\\to(g_2 g_1)(\\tau_2 t\\bar z)$, and\n\\begin{eqnarray*}\n(f_\\alpha\\circ\\sigma_\\alpha)(\\tau_1\\tau_l^{g_1},\\tau_{k_l}^{g_2}\\tau_2 t\\bar z)=\\hspace{7.5cm}\\\\\n(f_\\alpha\\circ\\sigma_\\alpha)(\\tau_1,\\tau_l^{g_1}\\tau_{k_l}^{g_2}\\tau_2 t\\bar z)+(f_\\alpha\\circ\\sigma_\\alpha)(\\tau_l^{g_1}\\tau_{k_l}^{g_2},\\tau_2 t\\bar z)-(f_\\alpha\\circ\\sigma_\\alpha)(\\tau_{k_l}^{g_2},\\tau_2 t\\bar z)\\\\\n\\to (f_\\alpha\\circ\\sigma_\\alpha)(\\tau_1,g_2(\\tau_2 t\\bar z))+\\varphi(g_2g_1)-\\varphi(g_2) \\quad\\textup{as}\\enspace l\\to\\infty.\n\\end{eqnarray*}\nThe uniqueness according to equality (\\ref{eq:h_uni}) verifies that $\\bar h((\\tau_1,g_2 g_1 g_2^{-1}),g_2(\\tau_2 t\\bar z))=\\lim_{l\\to\\infty}h(\\tau_1\\tau_l^{g_1},\\tau_2\\tau_{k_l}^{g_2}t\\bar z)$, and therefore\n\\begin{eqnarray*}\n\\bar h((\\tau_1,g_2 g_1 g_2^{-1}),g_2(\\tau_2t\\bar z))+\\bar h((\\tau_2,g_2),t\\bar z)=\\hspace{4.5cm}\\\\\n=\\lim_{l\\to\\infty}h(\\tau_1\\tau_l^{g_1},\\tau_2\\tau_{k_l}^{g_2}t\\bar z)+\\lim_{l\\to\\infty}h(\\tau_2\\tau_{k_l}^{g_2},t\\bar z)=\\\\\n=\\lim_{l\\to\\infty}h(\\tau_1\\tau_l^{g_1}\\tau_2\\tau_{k_l}^{g_2},t\\bar z)=\\bar h((\\tau_1\\tau_2,g_2g_1),t\\bar z) .\n\\end{eqnarray*}\nWe substitute $g_2^{-1} g_1 g_2$ for $g_1$ and obtain from $\\bar{\\mathcal O}_T(\\bar z)=Z$ and the continuity of $\\bar h$ that cocycle identity is valid.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{prop:res}]\nLet $(Y_c,T)=\\pi_c(X,T)$ be the flow defined by the connected components of the fibres of $\\pi_Y$ (cf. \\cite{MMWu}, Definition 2.2), and let $\\rho$ be the homomorphism from $(Y_c,T)$ onto $(Y,T)=\\rho(Y_c,T)$.\nWith a \\emph{RIM} $\\{\\mu_{c,y}:y\\in Y_c\\}$ for the distal extension $(Y_c,T)=\\pi_c(X,T)$ we define a cocycle $f_c(\\tau,y)=\\mu_{c,y}(f(\\tau,\\cdot))$ for every $(\\tau,y)\\in T\\times Y_c$.\nWe fix a point $\\bar x\\in X$ with $\\mathcal D_{T,f_\\alpha\\circ\\pi_\\alpha}(\\tau\\bar x,0)=\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\pi_\\alpha}(\\tau\\bar x,0)$ for all $\\tau\\in T$.\nBy equality (\\ref{eq:p_X}) and $\\pi_c^{-1}(\\pi_c(\\tau\\bar x))\\subset\\pi_Y^{-1}(\\pi_Y(\\tau\\bar x))$ holds for all $\\tau\\in T$\n\\begin{equation*}\n\\bar{\\mathcal O}_{T,f_\\alpha\\circ\\pi_\\alpha}(\\bar x,0)\\cap(\\pi_c^{-1}(\\pi_c(\\tau\\bar x))\\times\\mathbb R)=\\pi_c^{-1}(\\pi_c(\\tau\\bar x))\\times\\{(f_\\alpha\\circ\\pi_\\alpha)(\\tau,\\bar x)\\} .\n\\end{equation*}\nLet $F(\\tau,x)$ be the $\\mathbb R^2$-valued cocycle $(f_\\alpha\\circ\\pi_\\alpha,f)$.\nWe shall verify that\n\\begin{eqnarray}\\label{eq:f_c}\n\\bar{\\mathcal O}_{T,F} (\\bar x,0,0)\\cap(\\pi_c^{-1}(\\pi_c(\\tau\\bar x))\\times\\{(f_\\alpha\\circ\\pi_\\alpha)(\\tau,\\bar x)\\}\\times\\mathbb R)=\\nonumber\\\\\n\\{(x,(f_\\alpha\\circ\\pi_\\alpha)(\\tau,\\bar x),b_\\tau(x)):x\\in\\pi_c^{-1}(\\pi_c(\\tau\\bar x))\\}\n\\end{eqnarray}\nfor every $\\tau\\in T$, in which $b_\\tau:\\pi_c^{-1}(\\pi_c(\\tau\\bar x))\\longrightarrow\\mathbb R$ is a continuous function.\nIndeed, for a sequence $\\{\\tau_k\\}_{k\\geq 1}\\subset T$ with $\\tau_k\\tau\\bar x\\to x\\in\\pi_c^{-1}(\\pi_c(\\tau\\bar x))$ and $(f_\\alpha\\circ\\pi_\\alpha)(\\tau_k,\\tau\\bar x)\\to 0$ follows by the relative triviality of $(f-f_\\alpha\\circ\\pi_\\alpha)(\\tau,x)$ the existence and uniqueness of the limit $b_\\tau(x)$ of $f(\\tau_k,\\tau\\bar x)$.\nIt also follows that for every $\\varepsilon>0$ there exists a $\\delta>0$ so that for all $\\tau\\in T$ and $x,x'\\in\\pi_c^{-1}(\\pi_c(\\tau\\bar x))$ with $d(x,x')<\\delta$ holds $|b_\\tau(x)-b_\\tau(x')|<\\varepsilon$.\nSince the fibres of $\\pi_c$ are connected, a covering of $X$ by $\\delta$-neighbourhoods provides a constant $D>0$ with $|b_\\tau(x)-b_\\tau(x')|<D$ for all $\\tau\\in T$ and $x,x'\\in\\pi_c^{-1}(\\pi_c(\\tau\\bar x))$.\nEquality (\\ref{eq:f_c}) shows for $x\\in\\pi_c^{-1}(\\pi_c(\\bar x))$ and $\\tau\\in T$ that $b_{\\mathbf 1_T}(x)+f(\\tau,x)=b_{\\tau}(\\tau x)$ and hence $|f(\\tau,\\bar x)-f(\\tau,x)|\\leq 2D$.\nSince $(f-f_c\\circ\\pi_c)(\\tau,x)$ assumes zero on each $\\pi_c$-fibre, for all $\\tau\\in T$ holds $|(f-f_c\\circ\\pi_c)(\\tau,\\bar x)|< 2D$ so that this is a coboundary.\n\nDue to Theorem 3.7 in \\cite{MMWu} the extension from $(Y,T)$ to $(Y_c,T)$ is an isometric extension, and by Fact \\ref{fact:reg_ex} there exists a compact group extension $(\\tilde Y,T)$ of $(Y,T)=\\rho(Y_c,T)$ by $G\\subset\\textup{Aut}(\\tilde Y,T)$ so that $(Y_c,T)=\\sigma(\\tilde Y,T)$ is the orbit space of a compact subgroup $H\\subset G$.\nFor every sequence $\\{(\\tau_k,\\tilde y_k)\\}_{k\\geq 1}\\subset T\\times\\tilde Y$ with $d_{\\tilde Y} (\\tilde y_k,\\tau_k\\tilde y_k)\\to 0$ and $(f_\\alpha\\circ\\rho_\\alpha\\circ\\rho\\circ\\sigma)(\\tau_k,\\tilde y_k)\\to 0$ holds $(f_c\\circ\\sigma)(\\tau_k,\\tilde y_k)\\to 0$.\nOtherwise, by Lemma \\ref{lem:trans} there exists a sequence $\\{(\\tilde\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ so that $d(x_k,\\tilde\\tau_k x_k)\\to 0$, $(f_\\alpha\\circ\\pi_\\alpha)(\\tilde\\tau_k,x_k)\\to 0$, and $(f_c\\circ\\pi_c)(\\tilde\\tau_k,x_k)\\to\\infty$, which contradicts to Proposition \\ref{prop:max} and the boundedness of the transfer function between $f$ and $f_c\\circ\\pi_c$.\nWe can apply Lemma \\ref{lem:tilde_coc} for the flow $(\\tilde Y,T)$, the cocycle $h=f_c\\circ\\sigma$, the group $G\\subset\\textup{Aut}(\\tilde Y,T)$, and the group homomorphism $\\varphi\\equiv 0$, and we obtain a real valued cocycle $\\bar h((\\tau,h),\\tilde y)$ with $\\bar h((\\tau,\\mathbf 1_G),\\tilde y)=(f_c\\circ\\sigma)(\\tau,\\tilde y)$ for every $(\\tau,\\tilde y)\\in T\\times\\tilde Y$.\nWe define a topological cocycle of $(Y,T)$ by $f_Y(\\tau,y)=\\mu_{Y,y}((f_c\\circ\\sigma)(\\tau,\\cdot))$, where $\\{\\mu_{Y,y}:y\\in Y\\}$ is the \\emph{RIM} for the extension $(Y,T)=\\rho\\circ\\sigma(\\tilde Y,T)$.\nFrom the cocycle identity $\\bar h((\\tau,g),\\tilde y)=(f_c\\circ\\sigma)(\\tau,g(\\tilde y))+\\bar h((\\mathbf 1_T,g),\\tilde y)=\\bar h((\\mathbf 1_T,g),\\tau\\tilde y)+(f_c\\circ\\sigma)(\\tau,\\tilde y)$ and the boundedness of $\\bar h((\\mathbf 1_T,g),\\tilde y)$ for $(g,\\tilde y)\\in G\\times\\tilde Y$ we can conclude that $(f_c\\circ\\sigma-f_Y\\circ\\rho\\circ\\sigma)(\\tau,\\tilde y)$ is uniformly bounded and a coboundary.\nHence also $(f_c-f_Y\\circ\\rho)(\\tau,y_c)$ is a coboundary, and the relative triviality of $f_Y(\\tau,y)$ with respect to $(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)$ can be verified as above for the cocycle $(f_c\\circ\\sigma)(\\tau,\\tilde y)$.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:inc}\nThe cocycle $(f_Y-f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)$ of the flow $(Y,T)$ can be extended to a cocycle $\\bar f((\\tau,t),y)$ of the $T\\times\\mathbb R$-flow $(Y,\\{\\psi^t\\circ\\tau:(\\tau,t)\\in T\\times\\mathbb R\\})$ so that\n\\begin{equation*}\n(f_Y-f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)=\\bar f((\\tau,0),y)\n\\end{equation*}\nfor every $(\\tau,y)\\in T\\times Y$.\nWe put $L_\\tau (x_\\alpha,m)=(\\tau x_\\alpha,m)=\\psi^{-f_\\alpha(\\tau,x_\\alpha)}(\\tau (x_\\alpha,m))$ for $(x_\\alpha,m)\\in X_\\alpha\\times M=Y$.\nThen for arbitrary $\\bar x\\in X_\\alpha$ there exists a continuous function $\\bar b:Y\\longrightarrow\\mathbb R$ with $\\bar b(\\rho_\\alpha^{-1}(\\bar x))=\\{0\\}$ so that for every $(\\tau,y)\\in T\\times Y$ holds\n\\begin{equation}\\label{eq:tf}\n\\bar f((\\tau,-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)),y)=\\bar b(\\psi^{-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)} (\\tau y))-\\bar b(y) =\\bar b(L_\\tau y)-\\bar b(y) ,\n\\end{equation}\nand $(\\tau,y)\\mapsto\\bar f((\\tau,-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)),y)$ is a topological coboundary of the distal flow $(Y,\\{L_\\tau:\\tau\\in T\\})$ with transfer function $\\bar b$.\nFor every $(x_\\alpha,m)\\in Y$ and $t\\in\\mathbb R$ holds\n\\begin{equation}\\label{eq:tF}\n\\bar f((\\mathbf 1_T,t),(x_\\alpha,m))= \\bar f((\\mathbf 1_T,t),(\\bar x,m))+\\bar b(x_\\alpha,\\phi^t(m))-\\bar b(x_\\alpha,m) .\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nSince Lemma \\ref{lem:tilde_coc} can be applied to the cocycle $h=f_Y-f_\\alpha\\circ\\rho_\\alpha$, the group $G=\\{\\psi^t:t\\in\\mathbb R\\}\\subset\\textup{Aut}(Y,T)$, and the group homomorphism $\\varphi=\\textup{id}_\\mathbb R$, it provides the cocycle $\\bar f((\\tau,t),y)$.\nThe mapping $f'((\\tau,t),y)=\\bar f((\\tau,t-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)),y)$ fulfils\n\\begin{eqnarray*}\nf'((\\tau,t),\\psi^{t'}(L_{\\tau'} y))+f'((\\tau',t'),y)=\\hspace{55mm}\\\\\n=\\bar f((\\tau,t-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,\\psi^{t'}(L_{\\tau'} y))),\\psi^{t'}(L_{\\tau'} y))+\\bar f((\\tau',t'-(f_\\alpha\\circ\\rho_\\alpha)(\\tau',y)),y)=\\hspace{-7mm}\\\\\n=\\bar f((\\tau\\tau',t+t'-(f_\\alpha\\circ\\rho_\\alpha)(\\tau\\tau',y)),y)=f'((\\tau\\tau',t+t'),y)\n\\end{eqnarray*}\nand is thus a cocycle of the \\emph{minimal} flow $(Y,\\{\\psi^t\\circ L_\\tau:(\\tau,t)\\in T\\times\\mathbb R\\})$.\nNow let $(\\tau,y)\\in T\\times Y$ be arbitrary.\nBy equality (\\ref{eq:o_flow}) and the density of $\\widetilde{\\tau}_{f_\\alpha}$-transitive points in $X_\\alpha$, there exists a sequence $\\{(\\tau_k,y_k)\\}_{k\\geq 1}\\subset T\\times Y$ so that $y_k\\to\\tau y$, $\\tau_k y_k\\to\\psi^{-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)}(\\tau y)$, and $(f_\\alpha\\circ\\rho_\\alpha)(\\tau_k,y_k)\\to -(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)$ as $k\\to\\infty$, and by equality (\\ref{eq:h_uni}) holds $(f_Y-f_\\alpha\\circ\\rho_\\alpha)(\\tau_k,y_k)\\to\\bar f((\\mathbf 1_T,-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)),\\tau y)$.\nThus\n\\begin{equation*}\n(f_Y-f_\\alpha\\circ\\rho_\\alpha)(\\tau_k\\tau,\\tau^{-1} y_k)\\to\\bar f((\\mathbf 1_T,-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)),\\tau y)+(f_Y-f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)\n\\end{equation*}\nas $k\\to\\infty$, and this limit coincides with $\\bar f((\\tau,-(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)),y)=f'((\\tau,0),y)$ due to the cocycle identity for $\\bar f((\\tau,t),y)$.\nSince $f_Y(\\tau,y)$ is relatively trivial with respect to $(f_\\alpha\\circ\\rho_\\alpha)(\\tau,y)$, for every $\\varepsilon>0$ there exists a $\\delta>0$ so that for every $(\\tau',y')\\in T\\times Y$ with $d_Y(y',\\tau' y')<\\delta$ and $|(f_\\alpha\\circ\\rho_\\alpha)(\\tau',y')|<\\delta$ holds $|(f_Y-f_\\alpha\\circ\\rho_\\alpha)(\\tau',y')|<\\varepsilon$.\nFrom $\\tau^{-1} y_k\\to y$, $\\tau_k y_k\\to L_\\tau y$, and $(f_\\alpha\\circ\\rho_\\alpha)(\\tau_k\\tau,\\tau^{-1} y_k)\\to 0$ follows for every $(\\tau,y)\\in T\\times Y$ with $d_Y(y,L_\\tau(y))<\\delta$ that $f'((\\tau,0),y)<\\varepsilon$.\nFact \\ref{fact:GH} implies that the cocycle $(\\tau,y)\\mapsto f'((\\tau,0),y)$ of the distal flow $(Y,\\{L_\\tau:\\tau\\in T\\})$ is a coboundary on the $\\{L_\\tau:\\tau\\in T\\}$-orbit closure $X_\\alpha\\times\\{m\\}$ with transfer function $b_m:X_\\alpha\\longrightarrow\\mathbb R$ for every $m\\in M$.\nSince $\\delta>0$ is valid for all $\\{L_\\tau:\\tau\\in T\\}$-orbit closures, the transfer functions $\\{b_m:m\\in M\\}$ are uniformly equicontinuous.\nWe fix a point $\\bar x\\in X_\\alpha$ and obtain from the cocycle identity for all $(\\tau,t)\\in T\\times\\mathbb R$ and $(x_\\alpha,m)\\in Y$ that\n\\begin{eqnarray*}\nf_{\\bar x}((\\tau,t),(x_\\alpha,m))=f'((\\tau,t),(x_\\alpha,m))-f'((\\mathbf 1_T,t),(\\bar x,m))=\\\\\n=b_{\\phi^t(m)}(\\tau x_\\alpha)-b_{\\phi^t(m)}(\\bar x)-b_{m}(\\tau x_\\alpha)+b_m(\\bar x) .\n\\end{eqnarray*}\nThe function $f_{\\bar x}((\\tau,t),(x_\\alpha,m))$ is also a cocycle of $(Y,\\{\\psi^t\\circ L_\\tau:(\\tau,t)\\in T\\times\\mathbb R\\})$ and bounded on $T\\times\\mathbb R\\times Y$, hence a coboundary with a transfer function $\\bar b:Y\\longrightarrow\\mathbb R$ so that $\\bar b(\\rho_\\alpha^{-1}(\\bar x))=\\{0\\}$.\nNow equality (\\ref{eq:tf}) follows, and equality (\\ref{eq:tF}) follows from $f'((\\mathbf 1_T,t),(x_\\alpha,m))=\\bar f((\\mathbf 1_T,t),(x_\\alpha,m))$ for all $t\\in\\mathbb R$ and $(x_\\alpha,m)\\in Y$.\n\\end{proof}\n\nWith these prerequisites we can conclude the proof of our main result.\n\n\\begin{proof}[Proof of the structure theorem]\nWe let all  elements of the theorem and the flow $\\{\\psi^t:t\\in\\mathbb R\\}\\subset\\textup{Aut}(Y,\\{L_\\tau:\\tau\\in T\\})\\cap\\textup{Aut}(Y,T)$ be defined according to the Propositions \\ref{prop:max}, \\ref{prop:flow}, \\ref{prop:res}, and \\ref{prop:inc}.\nWe fix a point $\\bar x\\in X_\\alpha$ so that $(\\bar x,0)$ is transitive for $\\widetilde\\tau_{f_\\alpha}$ and $\\bar b(\\rho_\\alpha^{-1}(\\bar x))=\\{0\\}$.\nThen we define a cocycle $g(t,m)$ of the distal minimal flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ by\n\\begin{equation*}\ng(t,m)=\\bar f((\\mathbf 1_T,t),(\\bar x,m)) \\quad\\textup{for all}\\enspace (t,m)\\in\\mathbb R\\times M.\n\\end{equation*}\nFrom equalities (\\ref{eq:tf}) and (\\ref{eq:tF}) follows for all $\\tau\\in T$ and $(x_\\alpha,m)\\in Y$ that\n\\begin{eqnarray*}\n(f_Y-f_\\alpha\\circ\\rho_\\alpha)(\\tau,(x_\\alpha,m))=\\bar f((\\tau,0),(x_\\alpha,m))=\\hspace{3cm}\\\\\n=\\bar f((\\mathbf 1_T,f_\\alpha(\\tau,x_\\alpha)),L_\\tau (x_\\alpha,m))+\\bar b(L_\\tau (x_\\alpha,m))-\\bar b(x_\\alpha,m)=\\\\\n=g(f_\\alpha(\\tau,x_\\alpha),m)+\\bar b\\circ \\tau(x_\\alpha,m)-\\bar b(x_\\alpha,m) .\n\\end{eqnarray*}\nHence equality (\\ref{eq:f_Y2}) holds for the cocycle $f_Y(\\tau,y)-\\bar b(\\tau y)+\\bar b(y)$ cohomologous to $f_Y(\\tau,y)$, and this cocycle will be substituted for $f_Y(\\tau,y)$ henceforth.\nFor every sequence $\\{(\\tau_k,x_k)\\}_{k\\geq 1}\\subset T\\times X$ with $(f_\\alpha\\circ\\pi_\\alpha)(\\tau_k,x_k)\\to 0$ it holds also that $(f_Y\\circ\\pi_Y)(\\tau_k,x_k)\\to 0$ as $k\\to\\infty$, and thus identity (\\ref{eq:p_X}) implies identity (\\ref{eq:fpy}).\n\nFor every ordinal $\\xi$ with $\\alpha\\leq\\xi\\leq\\eta$ we can apply the Propositions \\ref{prop:flow}, \\ref{prop:res}, and \\ref{prop:inc} to the distal minimal flow $(X_\\xi,T)$ and the cocycle $f_\\xi(\\tau,x_\\xi)$.\nWe obtain a factor $(Y_\\xi,T)=\\pi_{Y_\\xi}(X_\\xi,T)$ with $(X_\\alpha,T)=\\rho_\\alpha^\\xi(Y_\\xi,T)$, an $\\mathbb R$-flow $\\{\\psi_\\xi^t:t\\in\\mathbb R\\}\\subset\\textup{Aut}(Y_\\xi,T)$, a cocycle $f_{Y_\\xi}(\\tau,y_\\xi)$ of $(X_\\xi,T)$, and a cocycle $\\bar f_\\xi((\\tau,t),y_\\xi)$ of the flow $(Y_\\xi,T\\times\\mathbb R)$ extending the cocycle $(f_{Y_\\xi}-f_\\alpha\\circ\\rho^\\xi_\\alpha)(\\tau,y_\\xi)$.\nStriving for a contradiction to the maximality of the ordinal $\\alpha$ (cf. Proposition \\ref{prop:max}), we assume that the cocycle $(\\mathbbm 1+g)(t,m)$ of the minimal flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ is recurrent so that the cocycle $\\bar f_\\eta((\\mathbf 1_T,t),y_\\eta)+t$ of the minimal flow $((\\rho^\\eta_\\alpha)^{-1}(\\bar x),\\{\\psi_\\eta^t:t\\in\\mathbb R\\})$ is also recurrent.\nWe let $\\beta$ be the minimal element of the non-empty set of ordinals\n\\begin{equation*}\n\\{\\alpha\\leq\\xi\\leq\\eta: \\bar f_\\xi((\\mathbf 1_T,t),y_\\xi)+t\\enspace\\textup{is a recurrent cocycle of}\\enspace ((\\rho_\\alpha^\\xi)^{-1}(\\bar x),\\{\\psi_\\xi^t:t\\in\\mathbb R\\})\\} ,\n\\end{equation*}\nwith $\\beta>\\alpha$ since $f_{Y_\\alpha}=f_\\alpha$ and $\\bar f_\\alpha((\\mathbf 1_T,t),y_\\alpha)\\equiv 0$.\nWe fix a point $\\bar x_\\beta\\in(\\pi_\\alpha^\\beta)^{-1}(\\bar x)$ so that $(\\pi_{Y_\\beta}(\\bar x_\\beta),0)$ is a recurrent point for the skew product extension of the flow $((\\rho_\\alpha^\\beta)^{-1}(\\bar x),\\{\\psi_\\beta^t:t\\in\\mathbb R\\})$ by the cocycle $\\bar f_\\beta((\\mathbf 1_T,t),y_\\beta)+t$.\nThen there exists a sequence $\\{\\bar\\tau_k\\}_{k\\geq 1}\\subset T$ with $f_\\alpha(\\bar\\tau_k,\\pi_\\alpha^\\beta(\\bar x_\\beta))\\to\\infty$ so that\n\\begin{equation*}\n\\bar f_\\beta((\\mathbf 1_T,f_\\alpha(\\bar\\tau_k,\\pi_\\alpha^\\beta(\\bar x_\\beta))),\\pi_{Y_\\beta}(\\bar x_\\beta))+f_\\alpha(\\bar\\tau_k,\\pi_\\alpha^\\beta(\\bar x_\\beta))=f_{Y_\\beta}(\\bar\\tau_k,\\pi_{Y_\\beta}(\\bar x_\\beta))\\to 0\n\\end{equation*}\nfor $k\\to\\infty$, and by the cohomology of the cocycles $f_\\beta(\\tau,x_\\beta)$ and $(f_{Y_\\beta}\\circ\\pi_{Y_\\beta})(\\tau,x_\\beta)$ the sequence $f_\\beta(\\bar\\tau_k,\\bar x_\\beta)$ is bounded.\nHence there exists a sequence $\\{k_l\\}_{l\\geq 1}\\subset\\mathbb N$ so that $f_\\alpha(\\bar\\tau_{k_{l+1}},\\pi_\\alpha^\\beta(\\bar\\tau_{k_l}\\bar x_\\beta))\\to\\infty$, $d_\\beta(\\bar\\tau_{k_{l+1}} \\bar x_\\beta,\\bar\\tau_{k_l}\\bar x_\\beta)\\to 0$, and $f_\\beta(\\bar\\tau_{k_l},\\bar x_\\beta)$ is convergent.\nThen the sequence $\\{(\\tau_l,x_l)=(\\bar\\tau_{k_{l+1}}(\\bar\\tau_{k_l})^{-1},\\bar\\tau_{k_l}\\bar x_\\beta)\\}_{l\\geq 1}\\subset T\\times X_\\beta$ fulfils $f_\\alpha(\\tau_l,\\pi_\\alpha^\\beta(x_l))\\to\\infty$, $d_\\beta(x_l,\\tau_l x_l)\\to 0$, and $f_\\beta(\\tau_l,x_l)\\to 0$ for $l\\to\\infty$.\nHowever, for every $\\alpha\\leq\\xi<\\beta$ holds\n\\begin{equation*}\n\\bar f_\\xi((\\mathbf 1_T,f_\\alpha(\\tau_l,\\pi_\\alpha^\\beta(x_l))),\\pi_{Y_\\xi}\\circ\\pi_\\xi^\\beta(x_l))+f_\\alpha(\\tau_l,\\pi_\\alpha^\\beta(x_l))=f_{Y_\\xi}(\\tau_l,\\pi_{Y_\\xi}\\circ\\pi_\\xi^\\beta(x_l))\\to\\infty\n\\end{equation*}\nfor $l\\to\\infty$.\nOtherwise, since $|\\bar f_\\xi((\\mathbf 1_T,t),(x_\\alpha,m_\\xi))-\\bar f_\\xi((\\mathbf 1_T,t),(\\bar x,m_\\xi))|$ is uniformly bounded for all $t\\in\\mathbb R$, $x_\\alpha\\in X_\\alpha$, and $m_\\xi\\in M_\\xi$ (cf. identity (\\ref{eq:tF})), there exists a non-trivial prolongation in the skew product of the minimal flow $((\\rho_\\alpha^\\xi)^{-1}(\\bar x),\\{\\psi_\\xi^t:t\\in\\mathbb R\\})$ and its cocycle $\\bar f_\\xi((\\mathbf 1_T,t),y_\\xi)+t$, which sufficient for its recurrence (cf. Lemma \\ref{lem:tr_coc}).\nTherefore also $f_\\xi(\\tau_l,\\pi_\\xi^\\beta(x_l))\\to\\infty$ as $l\\to\\infty$, and depending on the type of the ordinal $\\beta$ follows either from Lemma \\ref{lem:c_t} or Lemma \\ref{lem:lim} that $\\widetilde\\tau_{f_\\beta}$ is point transitive, in contradiction to the maximality of $\\alpha$.\n\\end{proof}\n\n\\begin{proof}[Proof of the structure of the topological Mackey action]\nIn the proof of the decomposition theorem it is verified that the topological Mackey actions for the cocycle $f(\\tau,x)$ of $(X,T)$ and the transient cocycle $(\\mathbbm 1+g)(t,m)$ of $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ are topologically isomorphic.\nLet $\\{(M_\\xi,\\{\\phi_\\xi^t:t\\in\\mathbb R\\}):0\\leq\\xi\\leq\\theta\\}$ be the normal \\emph{I}-system for the distal minimal compact metric flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$ with the homomorphisms $\\sigma_\\xi:M\\longrightarrow M_\\xi$.\nFor every ordinal $0\\leq\\xi\\leq\\theta$ a cocycle $g_\\xi(t,m_\\xi)$ of $(M_\\xi,\\{\\phi_\\xi^t:t\\in\\mathbb R\\})$ is defined by a \\emph {RIM}.\nLet $\\beta$ be the minimal element of the set\n\\begin{equation*}\n\\theta\\in\\{0\\leq\\xi\\leq\\theta:(g-g_\\xi\\circ\\sigma_\\xi)(t,m)\\enspace\\textup{is a coboundary of}\\enspace(M,\\{\\phi^t:t\\in\\mathbb R\\})\\} .\n\\end{equation*}\nThe cocycle $(\\mathbbm 1+g_\\beta)(t,m_\\beta)$ is transient, since the cocycle $(\\mathbbm 1+g)(t,m)$ cohomologous to $(\\mathbbm 1+g_\\beta\\circ\\sigma_\\beta)(t,m)$ is transient.\nBy Lemma \\ref{lem:tr_coc} the right translation action $\\{R_b:b\\in\\mathbb R\\}$ acts minimally on the Fell compact space $D$ of orbits in $M_\\beta\\times\\mathbb R$.\nThe mapping $\\chi:M_\\beta\\longrightarrow D$ defined by $m_\\beta\\mapsto\\mathcal O_{\\phi_\\beta,(\\mathbbm 1+g_\\beta)}(m_\\beta,0)$ is Fell continuous, and for every $t\\in\\mathbb R$ holds $\\chi\\circ\\phi_\\beta^t(m_\\beta)=R_{(\\mathbbm 1+g_\\beta)(t,m_\\beta)}\\circ\\chi(m_\\beta)$.\nFor $\\beta=0$ the flow $(D,\\{R_b:b\\in\\mathbb R\\})$ is trivial and thus weakly mixing.\nIf $\\beta\\geq 1$, then $(D,\\{R_b:b\\in\\mathbb R\\})$ is a non-trivial minimal compact metric flow.\nIf it is not weakly mixing, then there exists a non-trivial equicontinuous factor $(D_1,\\{\\varphi^t:t\\in\\mathbb R\\})=\\nu(D,\\{R_b:b\\in\\mathbb R\\})$ with homomorphism $\\nu$ (cf. \\cite{KeRo}).\nWe shall use a generalised and relativised version of Theorem 1 in \\cite{Eg} to obtain a contradiction to the minimality of $\\beta$.\nSince $(D_1,\\{\\varphi^t:t\\in\\mathbb R\\})$ is a minimal and non-trivial flow, for each small enough $\\varepsilon>0$ holds $\\varphi^\\varepsilon(d_1)\\neq d_1$ for all $d_1\\in D_1$.\nWe shall verify as a sub-lemma that there are no sequences $\\{t_k\\}_{k\\geq 1}\\subset\\mathbb R$, $\\{m_k\\}_{k\\geq 1},\\{m'_k\\}_{k\\geq 1}\\subset M_\\beta$ so that $m_k\\to\\bar m$, $m'_k\\to\\bar m$, $\\phi_\\beta^{t_k}(m_k)\\to\\bar m'$, $\\phi_\\beta^{t_k}(m'_k)\\to\\bar m'$, and $g_\\beta(t_k,m'_k)-g_\\beta(t_k,m_k)\\to\\varepsilon$.\nIndeed,\n\\begin{gather*}\n\\nu\\circ\\chi\\circ\\phi_\\beta^{t_k}(m'_k)=\\varphi^{(\\mathbbm 1+g_\\beta)(t_k,m'_k)}\\circ\\nu\\circ\\chi(m'_k)\\to\\varphi^{\\varepsilon}(\\lim \\varphi^{(\\mathbbm 1+g_\\beta)(t_k,m_k)}\\circ\\nu\\circ\\chi(m'_k))\n\\end{gather*}\nand $\\lim\\varphi^{(\\mathbbm 1+g_\\beta)(t_k,m_k)}\\circ\\nu\\circ\\chi(m'_k)=\\lim\\varphi^{(\\mathbbm 1+g_\\beta)(t_k,m_k)}\\circ\\nu\\circ\\chi(m_k)=\\nu\\circ\\chi(\\bar m')$, by the equicontinuity of $(D_1,\\{\\varphi^t:t\\in\\mathbb R\\})$, imply that $\\nu\\circ\\chi(\\bar m')=\\varphi^{\\varepsilon}\\circ\\nu\\circ\\chi(\\bar m')$, which contradicts to the choice of $\\varepsilon$.\n\nIf $\\beta=\\gamma+1$ for some ordinal $0\\leq\\gamma<\\theta$, then the sub-lemma implies the uniform equicontinuity of the mapping $m_\\beta\\mapsto g_\\beta(t,m_\\beta)$ restricted on the $\\sigma_\\gamma^\\beta$-fibres and for all $t\\in\\mathbb R$.\nIndeed, otherwise we can find sequences as above with $\\sigma_\\gamma^\\beta(m_k)=\\sigma_\\gamma^\\beta(m'_k)$ for all $k\\geq 1$, and the condition on sufficiently small $\\varepsilon$ can be fulfilled by the connectedness of the $\\sigma_\\gamma^\\beta$-fibres.\nThus for all $t\\in\\mathbb R$ the cocycle $(g_\\beta-g_\\gamma\\circ\\sigma_\\gamma^\\beta)(t,m_\\beta)$ is uniformly equicontinuous and assumes zero on every connected $\\sigma_\\gamma^\\beta$-fibre.\nBy Fact \\ref{fact:GH} this is then a coboundary, in contradiction to the minimality of $\\beta$.\n\nIf $\\beta$ is a limit ordinal, then the sub-lemma applies to sequences so that for every ordinal $0\\leq\\xi<\\beta$ there exists an integer $k_\\xi\\geq 1$ with $\\sigma_\\xi^\\beta(m_k)=\\sigma_\\xi^\\beta(m'_k)$ for all $k\\geq k_\\xi$.\nHence there exists an ordinal $0\\leq\\zeta<\\beta$ so that $|g_\\beta(t,m_\\beta)-g_\\beta(t,m'_\\beta)|<\\varepsilon$ for all $m_\\beta,m'_\\beta\\in M_\\beta$ with $\\sigma_\\zeta^\\beta(m_\\beta)=\\sigma_\\zeta^\\beta(m'_\\beta)$ and for all $t\\in\\mathbb R$.\nIt follows that $(g_\\beta-g_\\zeta\\circ\\sigma_\\zeta^\\beta)(t,m_\\beta)$ is a coboundary, in contradiction to the minimality of $\\beta$.\n\nThe topological Mackey action of the transient cocycle $(\\mathbbm 1+g)(t,m)$ is topologically isomorphic to the topological Mackey action of the cohomologous cocycle $(\\mathbbm 1+g_\\beta\\circ\\sigma_\\beta)(t,m)$ (cf. the proof of  the decomposition theorem).\nThe weakly mixing flow $(D,\\{R_b:b\\in\\mathbb R\\})$ is a factor of the topological Mackey action of the cocycle $(\\mathbbm 1+g_\\beta\\circ\\sigma_\\beta)(t,m)$, since for every $(m,s)\\in M\\times\\mathbb R$ the mapping $\\sigma_\\beta\\times\\textup{id}_\\mathbb R$ maps the orbit $\\mathcal O_{\\phi,(\\mathbbm 1+g_\\beta\\circ\\sigma_\\beta)}(m,s)$ in $M\\times\\mathbb R$ to the orbit $\\mathcal O_{\\phi_\\beta,(\\mathbbm 1+g_\\beta)}(\\sigma_\\beta(m),s)$ in $M_\\beta\\times\\mathbb R$ continuously with respect to the Fell topologies.\nSuppose that there exists two distinct orbits $\\mathcal O, \\mathcal O'$ in $M\\times\\mathbb R$ within the same $\\sigma_\\beta\\times\\textup{id}_\\mathbb R$-fibre and $\\{t_k\\}_{k\\geq 1}\\subset\\mathbb R$ is a sequence with $R_{t_k}\\mathcal O\\to \\mathcal O''$ and $R_{t_k}\\mathcal O'\\to \\mathcal O''$.\nSince the mapping $t\\mapsto (\\mathbbm 1+g_\\beta)(t,m_\\beta)$ is onto $\\mathbb R$ for every $m_\\beta\\in M_\\beta$ (cf. Lemma \\ref{lem:tr_coc}), there exists a point $\\bar m\\in M_\\beta$ and distinct $m,m'\\in\\sigma_\\beta^{-1}(\\bar m)$ so that $(m,0)\\in\\mathcal O$ and $(m',0)\\in\\mathcal O'$.\nMoreover, for every integer $k\\geq 1$ we can select a real number $t'_k$ so that $(\\mathbbm 1+g_\\beta)(t'_k,\\bar m)=t_k$, and therefore $(\\phi^{t'_k}(m),0)\\in R_{t_k}\\mathcal O$ as well as $(\\phi^{t'_k}(m'),0)\\in R_{t_k}\\mathcal O'$.\nBy changing to a subsequence we can suppose that $\\phi^{t'_k}(m)\\to m_1$ and $\\phi^{t'_k}(m')\\to m_2$ as $k\\to\\infty$ with $m_1\\neq m_2$, by the distality of the flow $(M,\\{\\phi^t:t\\in\\mathbb R\\})$.\nHowever, since $(m_1,0),(m_2,0)\\in\\mathcal O''$, the point $(\\bar m,0)$ is a periodic point in $(D,\\{R_b:b\\in\\mathbb R\\})$ in contradiction to the transience of the cocycle $(\\mathbbm 1+g_\\beta)(t,m_\\beta)$.\nWe can conclude that $\\sigma_\\beta\\times\\textup{id}_\\mathbb R$ is a distal homomorphism of the topological Mackey action of the cocycle $(\\mathbbm 1+g_\\beta\\circ\\sigma_\\beta)(t,m)$ onto the weakly mixing flow $(D,\\{R_b:b\\in\\mathbb R\\})$.\n\\end{proof}\n\n\\textbf{Acknowledgement}: The author would like to thank Professor Jon Aaronson and Professor Eli Glasner for useful discussions and encouragement.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nGender classification has an important role in modern society for surveillance or smart adaptation systems. It would be advantageous if a computer system or a machine could correctly classify an individual's gender. For example, a surveillance camera system of mall shoppers could be beneficial to know the gender of the customers to create a proper strategy, or a sale-man robot \\cite{ref0} could use an appropriate and smart approach to communicate with customers based on their gender.\n\n\nHuman gender, an active and promising area of research, can be classified using either a recorded voice \\cite{ref1}\\cite{ref2}\\cite{ref3} or face image \\cite{ref4}\\cite{ref5}\\cite{ref6}\\cite{ref7}\\cite{ref8}\\cite{ref9}\\cite{ref10}\\cite{ref10a}. SexNet \\cite{ref4} is an early system for gender classification using face images. The system uses the back-propagation algorithm of a neural network to train the gender classifier and obtains an error rate of 8.1\\%. Based on this encouraging result, the system demonstrates that automatic gender classification by computers is feasible. However, the use of voice and face features for gender classification has limitations when the objects are distant from the sensor because it is difficult to obtain a high-quality recorded voice or face image from a distance.\n\nMany psychological and medical experiments \\cite{ref11}\\cite{ref12} have indicated that humans and their gender can be recognized using their gait features. Therefore, gait features appear as alternative cues for resolving the recognition problem that occurs at long distances. Compared with other biometric features, gait information has particular advantages:\n\\begin{enumerate}\n\\item Easily obtainable from public areas and from a distance: Even when the subject is distant from the camera, we remain able to capture their gait information with an acceptable level of quality for specific tasks such as gait recognition and gender classification.\n\\item Uses simple instruments: Capturing the gait features requires only a simple conventional camera that can be placed anywhere in public areas such as banks, parking lots, and airports.\n\\item Does not require collaboration with the subjects: Gait features can be captured easily, even without the subject's permission. Although this is an advantage, it raises the issue of the right to privacy.\n\\item It is difficult to forge or falsify gait features: Gait features indicate the walking manner of a human, characterizing their physical capability. Mimicking the gait of other people is difficult.\n\\end{enumerate}\n\nHowever, gait-based systems such as gender classification and gait recognition share the same challenges as indicated in Fig.~\\ref{fig:1}. These challenges arise from the environment including the viewpoint of camera changes or the subject's physical characteristics such as carrying a backpack, wearing a heavy coat, or displaying signs of an injury. These factors change the subject's appearance leading to a significant effect on their gait information as they move \\cite{ref13}\\cite{ref14}.\n\n\n\\begin{figure}[!b]\n  \\centering\n  \\includegraphics[scale=0.35]{Fig1}\n\\caption{Challenges in gait analysis of humans: change in viewpoint (top row), carrying an item (middle row), wearing a coat (bottom row)}\n\\label{fig:1}      \n\\end{figure}\n\n\nThis paper proposes a novel gender classification method for use with an arbitrary viewpoint. To improve the performance, we present a method to remove areas with an attachment, such as a heavy coat or backpack. A general flowchart of the proposed method is presented in Fig. ~\\ref{fig:2}, which includes two major phases: training and testing. During the training phase, after the preprocessing step, the distance signal (DS) model, viewpoint (VP) model, and view-dependent gender classifier are built. Before building the VP model, the average gait image (AGI) and the lower portion of the average gait image (LAGI) are generated. During the testing phase, after the human detection and preprocessing step, the viewpoint of the current object is estimated using the current silhouette image and the VP model from the training phase. The attachment-area removal module is then used to eliminate unwanted areas such as backpacks or bags to obtain an attachment-free silhouette. Based on the estimated viewpoint of the current object, the corresponding classifier of that viewpoint (built during the training phase) is applied to the attachment-free silhouette to classify the object gender. The contributions of this paper are as follows:\n\\begin{itemize}\n\\item Building a VP model for viewpoint estimation, allowing the proposed method to estimate the viewing direction automatically.\n\\item Building a DS model for attachment-area removal to eliminate the noise generated from carried objects, which significantly degrades the performance of the system.\n\\item Building a viewpoint-dependent gender classifier using an SVM \\cite{ref15} that allows the algorithm to function from any viewpoint.\n\\end{itemize}\n\n\\begin{figure}[!b]\n\\centering\n\\includegraphics[scale=0.25]{Fig2}\n\\caption{Flowchart of proposed gender-classification process}\n\\label{fig:2}\n\\end{figure}\n\nThis paper is organized as follows. Section 2 discusses related works. Section 3 introduces the proposed method including the training and testing phases. The details of the key modules, such as VP modeling and estimation, DS modeling, viewpoint-dependent classifier building, and attachment-area removal are discussed in this section. A pseudo-code of the overall flowchart is also presented in this section. Section 4 presents the experimental results on public datasets. Finally, Section 5 concludes this paper and provides some areas for future work.\n\n\\section{Related works}\nGait-based recognition techniques can be divided into two categories, marker-based and markerless methods. Using markers, in an early work, Kozlowski and Cutting \\cite{ref16}\\cite{ref17} attempted to attach a point-light display (marker-based method) to a human body to extract the gait information. With this system, a human observer can determine a subject's gender based on the signals obtained with an acceptable level of accuracy (63\\%). However, to capture the gait information, the subject is required to wear a swimsuit and special devices, which is inconvenient, unfriendly, and impractical in real circumstances.\n\nToday, owing to technical innovations in camera and sensor development, the human gait can be easily obtained without a point-light display, leading to the development of markerless methods. The markerless-based methods for gait recognition can be classified into the model and appearance-based approaches. Such categorization can also be used for gender classification.\n\nIn the model-based methods \\cite{ref18}\\cite{ref19}\\cite{ref20}, the human body is divided into various parts, the structures of which are then fitted using primitive shapes such as ellipses, rectangles, and cylinders. Then, the gait feature is encoded using the parameters of the primitive shapes to measure the time-varying motion of the subject. In \\cite{ref18}, L. Lee et al. divide a human silhouette into seven different parts corresponding to the head and shoulder region, the front of the torso, back of the torso, front thigh, back thigh, front calf and foot, and back calf and foot. They then use ellipses to fit the model and capture the parameters of the ellipses such as the mean, standard deviation, orientation, and magnitude of the major components as feature vectors for classification. Although such methods are robust to noise and occlusions, they typically require a relatively high computational cost.\n\nAppearance-based methods \\cite{ref21}\\cite{ref22}\\cite{ref23}\\cite{ref24}\\cite{ref25}\\cite{ref26} analyze the spatio-temporal shape and dynamic motion characteristics of the silhouette in a gait sequence without using a human body model. A gait energy image (GEI) \\cite{ref21} is frequently used to encode the gait features because it includes both static (body shape) and dynamic information (arm swings and leg movements). The GEI feature is defined as the average frame of the subject in the gait cycle. Compared to the model-based methods, the appearance-based methods are considerably faster. In \\cite{ref23}, instead of modeling the silhouette, Shiqi Yu et al. calculate the GEI and use it to create the seven-part model defined in \\cite{ref17}. Because the contribution of each part to the gender classification varies, the authors assign different weights to the parts based on their experiments. Such methods obtain highly accurate classification rates (approximately 95\\%). However, they were developed to the only function on a side view, making it inappropriate to apply in real applications.\n\nIn \\cite{ref13}\\cite{ref14}\\cite{ref27}\\cite{ref28}\\cite{ref29}\\cite{ref30}\\cite{ref31}\\cite{ref32}\\cite{ref47}, the researchers attempted to resolve gender classification from multiple viewpoints. In \\cite{ref31}\\cite{ref32}, De Zhang et al. build an invariant classifier by combining the GEIs of different viewpoints into a single third-order tensor. They then use multiple linear principal component analysis (PCA) to reduce the dimensions and apply a support vector machine (SVM) to create a discriminative gender classifier. From another perspective, Kale et al. \\cite{ref33} use complicated equations from the structure of motion \\cite{ref34} to eliminate the viewpoint effect by synthesizing the side view from other viewpoints. A final recognition task is conducted on the synthesized data. Issac et al. \\cite{ref47} propose a method to delineate the gait instance as a sequence of poses or frames based on the fact that humans tend to assume certain poses at each part of a gait cycle. The gender of each frame is predicted, and the gender decision of a sequence is then made using majority voting. However, none of the previous works considers solving the problem of a subject carrying an item or wearing a heavy coat, which are common situations in real applications that can significantly degrade the classification rate.\n\n\\section{Proposed method}\n\\label{sec:3}\n\n\\subsection{Dataset and preprocessing step}\n\\label{sec:31}\nThis paper proposes a method for gender classification from an arbitrary viewpoint, and therefore, a dataset with multiple camera views is required. For this purpose, the CASIA gait Dataset B \\cite{ref13}\\cite{ref14} was utilized throughout this study for illustrative and experimental purposes.\n\nA person is first detected using a histogram of oriented gradient (HOG) \\cite{ref35}. During the preprocessing step, a classic background subtraction \\cite{ref36} is then applied to obtain a person's silhouette. Because a person's size changes from frame to frame, it is necessary to normalize the human bounding box before the training process. Assume in frame $I_t$ at time $t$, that a human is detected with a bounding box $B_t$; denote their silhouette obtained from the background subtraction as $S_t$. To register the silhouette image, the center of the silhouette $P_t$ (reference point) at time t is computed as:\n\\begin{equation}\n\\label{eq1}\nP_t(x_0, y_0) = \\left(\\frac{M_{10}}{M_{00}}, \\frac{M_{01}}{M_{00}}\\right)\n\\end{equation}\nwhere $M_{ij}$ are the raw moments of the binary silhouette image defined by $M_{ij}=\\sum_{x}\\sum_{y}x^iy^jS_{t}(x,y)$.\n\nThe silhouette image $S$ is resized to the fixed height $h$ to maintain the human ratio scale. The resized image is then zero padded or cropped on both sides (left and right sides) to ensure that the silhouette image has the predefined width $w$.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\textwidth]{Fig3}\n\\caption{Example AGIs: models from different viewpoints}\n\\label{fig:3}\n\\end{figure*}\n\n\n\\subsection{Viewpoint modeling and estimation}\n\\label{sec:32}\n\\subsubsection{Viewpoint modeling}\n\\label{sec: 321}\nA person can move in an arbitrary direction under real circumstances. Therefore, in this paper, rather than using GEI \\cite{ref21} as a representation of a gait feature for classification, AGI is defined, as shown in Fig.~\\ref{fig:3}. The main difference between GEI and AGI is that the gait cycle information, which must be calculated into a GEI, is not required in an AGI. Furthermore, applying the gait cycle during the feature extraction step makes the entire algorithm inflexible because the gait cycle can be calculated accurately only when the person is captured from a side view, which is impractical under real circumstances. With the proposed method, all models are trained based on each viewpoint independently. Subsequently, we describe the details required for training for viewpoint $\\alpha$ using the data on that viewpoint. The AGI is defined as:\n\\begin{equation}\n\\label{eq2}\nAGI^{\\alpha}(x, y)=\\frac{1}{T}\\sum_{t=1}^{T}S_t^\\alpha(x,y)\n\\end{equation}\nwhere $T$, gait period, is defined adaptively using the video frame rate $f$ and approximate gait cycle time $\\mu$ as $T = \\mu*f$. According to \\cite{ref37}\\cite{ref38}, when the frame rate $f$ is 25 frames\/s, the value of the gait cycle time $\\mu$ must be 0.6 seconds to capture the most informative gait features; thus, $T = 0.6*f$ is used in the proposed method; $S_t^\\alpha$ is the silhouette image at time $t$ with the viewpoint $\\alpha$.\n\nDefining $\\gamma_k^\\alpha$ as $\\gamma_k^\\alpha=\\left\\{{AGI}_1^\\alpha, {AGI}_2^\\alpha, ..., {AGI}_n^\\alpha\\right\\}_k$, is the feature vector of subject $k$ in a viewpoint $\\alpha$, $\\alpha=\\overline{1, \\nu}$ and $k=\\overline{1,N}$, where $\\nu$ and $N$ are number of viewpoints and number of subjects (training samples), respectively. In fact, for the training step, a greater number of training samples $N$ is preferable. The corresponding label of $\\gamma_k^\\alpha$, denoted as set $L_k^\\alpha=\\{y_1^\\alpha, y_2^\\alpha, ..., y_n^\\alpha\\}_k, k=\\overline{1, N}; \\alpha=\\overline{1,\\nu}; y_i^\\alpha\\in\\{-1,1\\}$, is used to indicate the gender (\"-1\" for female and \"1\" for male).\n\nTo estimate the viewpoint for the input during the testing phase, we construct a viewpoint model $D$. This viewpoint model includes the viewpoint templates of an individual view. The viewpoint template is calculated as the average silhouette of all sequences from the $\\alpha$-th viewpoint. As observed, the viewpoint is clearly distinguished in the lower part of the silhouette and therefore, the $\\alpha$-th viewpoint template, denoted as ${LAGI}^\\alpha$, is extracted as the lower part of the average silhouette denoted as $LPS^\\alpha$ with a height of 0.715$h$ to $h$, as suggested by \\cite{ref39}, where $h$ is the height of the silhouette:\n\n\\begin{equation}\n\\label{eq3}\n{LPS}^\\alpha(x,y)=S^\\alpha(x,y), x=\\overline{0.715h, h}, y=\\overline{1, w}\n\\end{equation}\n\\begin{equation}\n\\label{eq4}\n{LAGI}^\\alpha(x,y) = \\frac{1}{N}\\sum_{t=1}^N{LPS_t^\\alpha(x,y)}\n\\end{equation}\n\nThe viewpoint model is then denoted as $D_{Low}=D=\\{{LAGI}^0,{LAGI}^1, \u2026,{LAGI}^\\nu \\}$, where $\\nu$ is the number of viewpoints. This viewpoint model $D_{Low}$ is used to estimate the viewpoint of a person walking during the testing phase.\n\n\\subsubsection{Viewpoint estimation}\nWith this method, the attachment-area removal module and gender classifier are dependent on the viewpoint; thus, the viewpoint is first estimated. During the testing phase, given the sequences of the silhouettes, the average gait image of the current walking subject, ${AGI}^c$, is calculated using Eq.~\\ref{eq2}. Then, ${LAGI}^c$  is extracted from the lower part of ${AGI}^c$  based on the size given in Eq.~\\ref{eq3}, rather than recalculating $LAGI$, as during the training phase.\n\nTo obtain viewpoint $\\alpha$ of the current walking subject, ${LAGI}^c$  is matched with each viewpoint template in viewpoint model $D_{Low}$ using the Euclidean distance. The least distance is then selected for the viewpoint estimation, as indicated in Fig. ~\\ref{fig:4}.\n\\begin{equation}\n\\label{eq5}\n\\alpha=\\min_j||{LAGI}^c-{LAGI}^j||_2, j=\\overline{0,\\nu}\n\\end{equation}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\textwidth]{Fig4}\n\\caption{Viewpoint model of 11 viewpoint templates where ${LAGI}^c$ is matched with the first template}\n\\label{fig:4}\n\\end{figure*}\n\n\n\n\\subsection{Distance signal modeling and attachment removal}\n\\subsubsection{Distance modeling}\nIn real applications, it is common to view people moving with attached objects such as bags or backpacks; similarly, their appearance can be significantly changed when wearing a heavy coat. The added area resulting from a held item or worn coat contributes nothing to the result of the gender classification. In actuality, these factors negatively influence the results of the classification. In this section, a distance signal (DS) model of humans under normal walking conditions (not holding anything and wearing thin clothes) from different viewpoints is proposed for removing these redundant attachments.\n\nGiven a set of silhouettes in movement direction $\\alpha$ , for each silhouette, a distance signal is built. Considering the mass reference point calculated by Eq. ~\\ref{eq1}, each point $P_i$  on the silhouette boundary is represented in polar coordinates by two parameters, $d_i$ and $\\theta_i$, which indicate the distance from the point $P_i$ to the reference point $P$, and the angle formed by the line connecting the point to the reference point ${PP}_i$ with the horizon ${PP}_h$, respectively. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\textwidth]{Fig5}\n\\caption{Example of distance signal (a) and distance signal model for different viewpoints (b)}\n\\label{fig:5}\n\\end{figure*}\n\n\n\\begin{equation}\n\\label{eq6}\nd_i=||P-P_i||_2\n\\end{equation}\n\n\\begin{equation}\n\\label{eq7}\n\\theta_i=\\arccos\\left(\\frac{\\overrightarrow{PP_i}*\\overrightarrow{PP_h}}{|\\overrightarrow{PP_i}||\\overrightarrow{PP_h}|}\\right)\n\\end{equation}\nwhere * is the dot product between two vectors and the value of angle $\\theta$ varies from $0^o$ to $360^o$ computed counterclockwise; $P_i$ and $P_h$ are depicted in Fig. \\ref{fig:5}a (left). The DS signal is then constructed by continuously concatenating these parameters from $P_h$ counterclockwise to define the signal presented in Fig. \\ref{fig:5}a (right). After building these DS signals for viewpoint $\\alpha$, denoted by ${DS}^\\alpha=\\{{DS}_1^\\alpha,\u2026,{DS}_n^\\alpha\\}$, the DS model for viewpoint $\\alpha$ is constructed using two curves, $MaDS^\\alpha$ and $MiDS^\\alpha$, which are defined as:\n\n\\begin{equation}\n\\label{eq8}\nMaDS^\\alpha=\\{d_i^{max}, \\theta_i\\}, \\text{where } d_i^{max} = \\max\\limits_{k=\\overline{1,N}}\\{d_k|\\theta_k\\}\n\\end{equation}\n\n\\begin{equation}\n\\label{eq9}\nMiDS^\\alpha=\\{d_i^{min}, \\theta_i\\}, \\text{where } d_i^{min}=\\min\\limits_{k=\\overline{1,N}}\\{d_k|\\theta_k\\}\n\\end{equation}\n\nThe distance signals are smoothed using the moving average technique with the number $n_{avg}=3$ before calculating the DS model. Fig. \\ref{fig:5}b illustrates the DS model for 11 viewpoints in our experiments.\n\n\n\\subsubsection{Attachment-area removal}\nDuring the testing phase, the DS signal of the silhouette of the current subject $DS^c$ is calculated in the manner described in the section 3.3.1. Given the viewpoint estimated using the viewpoint estimation module, the current $DS^c$ is projected to the corresponding viewpoint DS model. The current $DS^c$  is modified using the following rule to eliminate any attachments, if they exist:\n\n\\begin{equation}\n\\label{eq10}\nDS^c=\\{d_i^c, \\theta_i\\} \\text{where } d_i^c= \\begin{cases} d_i^c & \\text{if }  d_i^c \\leq d_i^{max} \\\\\nd_i^{min} & \\text{ otherwise}\n\\end{cases} \n\\end{equation}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\textwidth]{Fig6}\n\\caption{Example of original silhouette, DS of the silhouette, corrected DS of the silhouette, and silhouette reconstruction of the current subject from a side view}\n\\label{fig:6}\n\\end{figure*}\n\n\nFig. \\ref{fig:6} illustrates this process. In the figure, an example of a side-view silhouette input with an attachment is provided because in this view the attachment can be seen most clearly. Our goal is to remove the attachment from the human silhouette, therefore, the human silhouette is divided into three parts, as shown in Fig. \\ref{fig:6}a. The first part consists of head and shoulder with a height of 0 to 0.17$h$, as suggested in \\cite{ref45}\\cite{ref46}, where $h$ is the height of human silhouette. The second part includes the human torso and thigh with a height of 0.17$h$ to 0.715$h$. Finally, the last part is human calf with a height of 0.715$h$ to $h$, as suggested in \\cite{ref39}. The red vertical lines (Fig. 6b, c, d, e ) are drawn to separate these three parts from the human silhouette. Firstly, the input silhouette is converted into a distance signal $DS^c$, presented as violet curve in Fig. \\ref{fig:6}b. The white curves are the maximum distance signal $MaDS^\\alpha$ (upper) and minimum distance signal $MiDS^\\alpha$ (lower) from the DS model for a specific viewpoint ($\\alpha=90^o$). \n\n\n\nWhen human carrying a backpack or bag, the appearance of torso thigh part is\nchanged due to the attachment, thus only this part is taken into account for\ncorrection. As we observed from the experiments, a $DS^c$ with a value less than\n$MiDS^\\alpha$ is the noise from an imperfect background subtraction. A $DS^c$\nwith a value greater than $MaDS^\\alpha$ is considered as the attachment area\nfrom a subject carrying an item while walking. Using the DS model from\nSection 3.3.1, the attachment area can therefore be removed. The corrected\nversion of $DS^c$ is obtained by replacing the violated signal with the \ncorresponding values of the $MiDS^\\alpha$ curve at Point A, as shown in Fig. \n\\ref{fig:6}c. To avoid the problem of strict change in the resulted signal, we \ncontinue to look from the point A to point B in Fig. \\ref{fig:6}c to find the \npoint that has the smallest vertical distance between $MiDS^\\alpha$ and the \nresulted signal (point C). The segment of the resulted signal from A to C is again \nreplaced by $MiDS^\\alpha$, as presented in Fig. \\ref{fig:6}d. The signal in Fig. \n\\ref{fig:6}e is obtained by smoothing using the average filter $\\frac{1}{5}[1 1 1 1 1]$. The same process is applied to all segments of torso\/thigh part of the $DS^c$ curve. Finally, the corrected version of the $DS^c$ is used to reconstruct the silhouette of attachment-free, as shown in Fig. \\ref{fig:6}f. The updated version of the silhouette is then used to recalculate the AGI for gender classification. \n\n\n\\subsection{Gender classifier building}\nSVM \\cite{ref15} is a superior tool for a binary classification problem regarding minimizing the classification error and maximizing the margin between the two classes. Because gender classification is a binary classification task, a standard SVM with a linear kernel was selected to train the view-dependent classifiers. For solving a constrained quadratic optimization problem, we set the maximum number of iteration to 100.\n\nTo create the viewpoint-dependent classifier, the feature sets $\\gamma^\\alpha=\\{\\gamma_1^\\alpha,\\gamma_2^\\alpha, \u2026,\\gamma_N^\\alpha\\}$ and its corresponding labels $L^\\alpha=\\{L_1^\\alpha,L_2^\\alpha, \u2026,L_N^\\alpha\\}$ are used as inputs for the linear SVM. The $\\alpha$-th viewpoint classifier, obtained by using the SVM, is denoted as  $C_{gen}^\\alpha$. The multiple-view classifier is a collection of different viewpoint-dependent classifiers, which is denoted as $C_{gen}={C_{gen}^\\alpha}$ and $\\alpha=\\overline{1, \\nu}$. \n\nIn the testing phase, the viewpoint is estimated as discussed in section 3.2.2. Based on the estimated viewpoint $\\alpha$, the corresponding classifier $C_{gen}^\\alpha$ is automatically selected from  $C_{gen}$ to predict the person's gender in a current frame.\n\nThe algorithms 1 and 2 give a more detail description of the proposed method using pseudo-code in which all notations described above are used.\n\n\\begin{algorithm*}\n\\caption{The training phase}\n\\begin{algorithmic}[1]\n\\STATE\\textbf{Input}: $video\\texttt{\\_}sequences$ in the $training\\texttt{\\_}samples$ of male and female in all viewpoints \n\\STATE \\textbf{Output}: VP model, DS model, and $C_{gen}$ classifier\n\n\\FORALL {$\\alpha$ \\textbf{in} $views$}\n\t\\FORALL {$video\\texttt{\\_}sequence\\; k$ \\textbf{in} $training\\texttt{\\_}samples$ of viewpoint $\\alpha$ }\n\t\t\\FORALL {frame $t$ \\textbf{in} $video\\texttt{\\_}sequence$}\n\t\t\t\\STATE Human detection\n\t\t\t\\IF {no human detected}\n\t\t\t\t\\STATE Skip to the next frame\n\t\t\t\\ENDIF\n\t\t\t\\STATE Preprocessing to get the normalized silhouette $S^\\alpha$\n\t\t\t\\STATE Extract low part of the silhouette $S^\\alpha$ by Eq.~\\ref{eq3} and accumulate to $LPS_{k,t}^\\alpha$\n\t\t\t\\STATE Detect contour of a normalized silhouette $S^\\alpha$\n\t\t\t\\STATE Calculate  $d_i$ and $\\theta_i$ by Eq.~\\ref{eq6}, Eq.~\\ref{eq7}\n\t\t\t\\STATE Calculate $AGI^\\alpha$ and assign its label $y^\\alpha$ by Eq.~\\ref{eq2}\n\t\t\t\\STATE Append $AGI^\\alpha$ to vector $\\gamma_k^\\alpha$\n\t\t\t\\STATE Append $y^\\alpha$ to vector $L_k^\\alpha$\n\t\t\\ENDFOR\n\t\\ENDFOR\n\t\\STATE Calculate the $LAGI^\\alpha$ using $LPS_{t,f}^\\alpha$ by Eq.~\\ref{eq4} for VP model\n\t\\STATE Calculate $MaDS^\\alpha$ and $MiDS^\\alpha$ using $d_i$ and $\\theta_i$ by Eq.~\\ref{eq8}, Eq.~\\ref{eq9} for DS model\n\t\\STATE Train view-dependent classifier $C_{gen}^\\alpha$ using $\\gamma^\\alpha$ and $L^\\alpha$ as inputs of SVM\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm*}\n\n\n\\begin{algorithm*}\n\\caption{The testing phase}\n\\begin{algorithmic}[1]\n\\STATE \\textbf{Input}: $video\\texttt{\\_}sequence$ of a person in an unknown viewing angle\n\\STATE \\textbf{Output}: Gender information\n\\STATE Initialize $counter = 0$\n\\STATE Initialize empty vector $v$\n\\FORALL {frame $t$ \\textbf{in} $video\\texttt{\\_}sequence$}\n\t\\STATE Human detection\n\t\\IF {human detected}\n\t\t\\STATE Increase $counter$ by 1\n\t\\ELSE\n\t\t\\STATE $counter=0$\n\t\t\\STATE Skip to the next frame\n\t\\ENDIF\n\t\\STATE Preprocessing to get the normalized silhouette $S$\n\t\\STATE Append $S$ to the end of vector $v$\n\t\\IF {$counter\\geq15$}\n\t\t\\STATE Calculate the $AGI$ using a vector of silhouette $v$\n\t\t\\STATE Extract low part of average gait image of the current frame $LAGI^c$ from $AGI$\n\t\t\\STATE Estimate the viewpoint $\\alpha$ using $LAGI^c$ and VP model by Eq.~\\ref{eq5}\n\t\t\\STATE Remove the attachment area using estimated viewpoint $\\alpha$ and DS model by Eq.~\\ref{eq10}\n\t\t\\STATE Reconstruct silhouette $S\\rightarrow S$' and update $AGI\\rightarrow AGI$'\n\t\t\\STATE Predict the gender in current frame using the updated $AGI$' and the estimated viewpoint $\\alpha$\n\t\t\\STATE Remove the first element of $v$\n\t\\ENDIF\n\\ENDFOR\n\\end{algorithmic}\n\\end{algorithm*}\n\n\\section{Experimental results}\n\\subsection{Experimental dataset}\nThe CASIA Dataset B \\cite{ref13}\\cite{ref14} addresses our requirements of multiple camera views because it includes sequences of various people from 11 viewpoints (from $0^o$ to $180^o$) under different walking conditions such as walking normally, carrying a backpack, and wearing a coat. The CASIA Dataset B captures sequences of 124 individual people (31 females and 93 males). Each person is captured ten times to create ten different sequences including six sequences under normal walking conditions, two backpack-carrying sequences, and two coat-wearing sequences. Table~\\ref{tab:1} summarizes the information of the CASIA Dataset B.\n\n\n\n\n\\begin{table}[!b]\n\\centering\n\\caption{Summary of CASIA Dataset B}\n\\label{tab:1}      \n\\begin{tabular}{lll}\n\\hline\\noalign{\\smallskip}\n\\text{Walking condition} & \\text{\\#subjects} & \\text{\\#sequences}  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\text{Normal walking} & 6 & $6\\times124\\times11$ \\\\\n\\text{Carrying a bag} & 2 & $2\\times124\\times11$ \\\\\n\\text{Wearing a coat} & 2 & $2\\times124\\times11$ \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table}\n\nThe CASIA Dataset B includes background subtraction and thus, in the proposed system we are only required to resize and center the silhouette to the same size (144$\\times$144). For the AGI calculation, we must accumulate fifteen frames; it requires approximately 0.6 seconds to obtain the first gender-classification result when the frame rate is 25 fps, which can be considered a system delay.\n\nWe used the CASIA Dataset B for both training and testing using the same protocol as in \\cite{ref32}, which uses n-fold cross-validation. With this protocol, all 31 females were selected; 31 males were selected randomly from the CASIA Dataset B owing to a bias in the number of males in the dataset. The 31 females and 31 males were then grouped into 31 disjoint sets consisting of one female and one male. To create viewpoint-dependent classifiers, we use 30 sets for training. The remaining sets were used to test the system accuracy. The training and testing phases were repeated 31 times; the averages of the correct classification rate are listed for all experiments.\n\\subsection{Viewpoint-dependent classifiers test}\nThis test was used to validate the performance of only viewpoint-dependent classifiers under the assumption that the viewpoint was given. We conducted the test for both correct and incorrect viewpoint classifiers with respect to a specific viewpoint to observe the effect of viewpoint changes on the gender classification. Table \\ref{tab:2} displays the correct classification rates ($CCRs$) when using the corresponding classifier and a non-corresponding classifier (the viewpoint is given). The $CCR$ is defined as:\n\\begin{equation}\n\\label{eq11}\nCCR=\\frac{TP+TN}{N}\n\\end{equation}\nwhere $TP$ is the true positive referring to the cases in which the system correctly classifies positive samples (male to male), $TN$ is the true negative referring to the cases in which the system correctly classifies negative samples (female to female), and $N$ is the total number of samples. In these experiments, the male samples are labeled as 1 (positive samples) and the female samples are labeled as -1 (negative samples). As indicated in Table \\ref{tab:2}, applying a proper classifier for a specific viewpoint provides higher $CCRs$ (97.6\\% $\\pm$ 0.881) (for further description see Table \\ref{tab:2}).\n\n\\begin{table*}\n\\centering\n\\caption{CCRs (\\%) of viewpoint-dependent classifiers for specific viewing angle under normal walking conditions}\n\\label{tab:2}      \n\\begin{tabular}{llllllllllll}\n\\hline\\noalign{\\smallskip}\n\\multirow{2}{*}{\\text{Ground-truth viewpoint}} & \\multicolumn{11}{c}{Classifier} \\\\\n& $0^o$ & $18^o$ & $36^o$ & $54^o$ & $72^o$ & $90^o$ & $108^o$ & $126^o$ & $144^o$ & $162^o$ & $180^o$  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n$0^o$ &97.6&96.5&94.1&89.6&89.9&82.3&87.2&93.8&94.6&94.5&95.8 \\\\\n$18^o$&97.5&98.6&96.8&93.4&89.1&85.4&88.6&92.4&93.7&95.2&96.4  \\\\\n$36^o$&95.3&96.6&97.4&95.8&94.2&93.4&93.9&94.5&95.7&95.7&94.2  \\\\\n$54^o$&92.2&95.3&96.1&96.6&95.8&94.6&93.4&95.0&95.5&94.6&93.1 \\\\\n$72^o$&92.3&93.2&94.1&95.4&96.1&94.8&94.5&95.2&93.0&92.8&92.4 \\\\\n$90^o$&90.0&93.1&95.3&95.5&95.7&98.8&96.9&95.7&94.1&92.5&91.1 \\\\\n$108^o$&91.4&92.7&95.4&96.1&96.5&97.0&97.3&96.2&95.5&92.4&93.2 \\\\\n$126^o$&92.5&94.3&96.7&95.7&95.3&94.1&93.4&96.8&94.5&94.7&94.5 \\\\\n$144^o$&95.7&96.8&97.8&95.8&93.2&93.4&93.9&94.8&97.5&95.5&94.5 \\\\\n$162^o$&95.5&96.3&95.8&93.4&91.3&90.4&92.6&94.4&95.4&98.3&96.6 \\\\\n$180^o$&96.8&97.6&95.8&92.8&91.5&91.4&92.6&92.3&93.5&95.3&98.5 \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\begin{table*}\n\\centering\n\\caption{CCRs (\\%) of viewpoint-dependent classifiers of specific viewing angle when carrying a bag}\n\\label{tab:3}      \n\\begin{tabular}{llllllllllll}\n\\hline\\noalign{\\smallskip}\n\\multirow{2}{*}{\\text{Ground-truth viewpoint}} & \\multicolumn{11}{c}{Classifier} \\\\\n & $0^o$ & $18^o$ & $36^o$ & $54^o$ & $72^o$ & $90^o$ & $108^o$ & $126^o$ & $144^o$ & $162^o$ & $180^o$  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n$0^o$&94.3&91.6&89.7&84.8&81.5&79.6&80.3&83.9&85.2&89.9&92.6 \\\\\n$18^o$&92.8&94.4&93.5&89.9&83.1&80.1&82.8&86.4&85.4&92.2&91.7  \\\\\n$36^o$&89.8&92.1&93.7&90.7&85.2&82.0&88.8&90.2&86.4&90.5&89.1  \\\\\n$54^o$&88.0&88.3&90.1&91.2&89.1&84.5&89.1&90.6&87.9&88.3&86.4 \\\\\n$72^o$&88.1&89.4&89.5&90.4&90.6&86.2&89.4&89.1&89.3&88.8&84.4 \\\\\n$90^o$&82.3&82.8&83.1&85.5&86.4&87.4&87.1&85.3&85.1&83.6&82.1 \\\\\n$108^o$&82.2&82.4&83.9&84.2&85.5&86.7&89.8&87.4&87.0&85.4&83.7 \\\\\n$126^o$&87.4&87.3&88.3&90.7&83.3&85.5&88.1&91.2&90.5&89.7&88.5 \\\\\n$144^o$&88.7&89.3&90.6&90.1&82.2&83.4&87.3&90.2&91.4&89.8&88.4 \\\\\n$162^o$&91.4&91.2&90.8&89.4&81.1&82.1&85.6&89.6&90.7&93.1&92.2 \\\\\n$180^o$&93.2&92.5&91.0&90.7&80.5&80.6&84.1&87.1&89.1&91.4&94.8 \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}\n\\centering\n\\caption{CCRs (\\%) of viewpoint-dependent classifiers of specific viewing angle when wearing a coat }\n\\label{tab:4}      \n\\begin{tabular}{llllllllllll}\n\\hline\\noalign{\\smallskip}\n\\multirow{2}{*}{\\text{Ground-truth viewpoint}}& \\multicolumn{11}{c}{Classifier} \\\\\n& $0^o$ & $18^o$ & $36^o$ & $54^o$ & $72^o$ & $90^o$ & $108^o$ & $126^o$ & $144^o$ & $162^o$ & $180^o$  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n$0^o$&92.2&91.5&89.2&87.4&86.1&85.5&87.2&88.8&89.3&90.6&91.7 \\\\\n$18^o$&91.3&93.5&90.9&89.5&88.2&84.6&86.2&87.1&89.8&89.4&91.8  \\\\\n$36^o$&88.5&90.4&94.1&91.7&89.1&88.4&91.0&92.5&93.8&90.1&89.4  \\\\\n$54^o$&87.5&88.3&89.7&91.8&90.3&87.6&90.6&91.1&90.4&89.2&87.1 \\\\\n$72^o$&85.6&88.5&90.4&91.6&92.4&91.3&89.1&87.3&87.0&86.3&84.4 \\\\\n$90^o$&83.0&84.7&85.3&88.6&90.8&93.7&91.7&89.6&87.4&86.2&85.1 \\\\\n$108^o$&82.4&85.6&85.4&88.1&89.5&89.4&90.0&89.8&87.2&85.3&83.2 \\\\\n$126^o$&84.6&86.1&88.6&90.3&89.3&88.0&89.8&91.9&87.5&85.7&83.4 \\\\\n$144^o$&84.7&85.0&89.2&89.8&88.2&84.4&86.9&87.8&89.9&86.5&84.5 \\\\\n$162^o$&90.5&91.2&88.8&87.2&84.6&83.4&85.1&87.4&89.7&91.5&90.2 \\\\\n$180^o$&92.0&91.6&87.8&86.8&83.5&82.4&82.9&85.1&86.5&90.9&92.4 \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table*}\n\nWe also conducted experiments under more challenging conditions such as a person\ncarrying an item or wearing a coat because the CASIA Dataset B includes sequences\nof such conditions, which were not used in previous studies \\cite{ref22}\\cite{ref23}\\cite{ref30}\\cite{ref31}\\cite{ref40}\\cite{ref41}. As indicated in Tables \\ref{tab:3} and \\ref{tab:4}, the $CCR$ of the gender prediction was\nsignificantly decreased under the challenging conditions of a side view or\nnearside view, even when the proper classifier was applied for the specific \nviewpoint. This problem is understandable because our viewpoint-dependent \nclassifiers are built upon sequences under normal walking conditions.\n\n \nMoreover, for a side view or nearside view, the appearance of the person is clearly changed, both when carrying an item and when wearing a coat. The mean $\\pm$ std of both $CCRs$ while carrying a bag and wearing a coat were 92.0\\% $\\pm$ 2.3 and 92.1\\% $\\pm$ 1.4, respectively. \n\nSome examples of silhouette from the same persons in front view (top row) and side view (bottom row) are shown in Fig. \\ref{fig:7}, respectively. It is interesting to notice from the figure that even in different views, the head part of the silhouette also contains the classifiable gender information (head and hair style). This is to explain that even the $90^o$ classifier is used to test the silhouette in $0^o$, the accuracy is not too low as seen in Tables 2-3-4. However, when the correct view classifier is applied, many traits for gender classification such as head and hair style, chest and back, waist and buttocks, legs  \\cite{ref23} are taken into account to increase the performance.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.5]{Fig7}\n\\caption{Silhouette example of human in front view (first row) and side view (second row)}\n\\label{fig:7}\n\\end{figure}\n\n\n\n\\begin{table*}\n\\centering\n\\caption{Results of viewpoint estimation in terms of percentage for arbitrary viewpoint under normal walking conditions }\n\\label{tab:5}      \n\\begin{tabular}{llllllllllll}\n\\hline\\noalign{\\smallskip}\n\\multirow{2}{*}{\\text{Ground-truth viewpoint}}& \\multicolumn{11}{c}{Estimated viewpoint} \\\\\n& $0^o$ & $18^o$ & $36^o$ & $54^o$ & $72^o$ & $90^o$ & $108^o$ & $126^o$ & $144^o$ & $162^o$ & $180^o$  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n$0^o$&90.8&8.1&0&0&0&0&0&0&0&0&1.1 \\\\\n$18^o$&3.4&89.5&0&0&0&0&0&0&0&8.1&0  \\\\\n$36^o$&0&5.3&88.3&6.4&0&0&0&0&0&0&0  \\\\\n$54^o$&0&0&4.3&82.1&11.6&0&0&0&0&0&0 \\\\\n$72^o$&0&0&0&3.0&93.7&3.3&0&0&0&0&0 \\\\\n$90^o$&0&0&0&0&4.4&93.2&2.4&0&0&0&0 \\\\\n$108^o$&0&0&0&0&0&7.4&87.1&5.5&0&0&0 \\\\\n$126^o$&0&0&0&0&0&0&1.1&88.2&10.7&0&0 \\\\\n$144^o$&0&0&0&0&0&0&0&6.2&84.3&9.5&0 \\\\\n$162^o$&0&0&0&0&0&0&0&0&6.4&84.5&9.1 \\\\\n$180^o$&2.6&0&0&0&0&0&0&0&0&9.3&88.1 \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\begin{table*}\n\\centering\n\\caption{CCRs (\\%) of the viewpoint-dependent classifier of unknown viewing angle under different walking conditions without attachment-area removal module }\n\\label{tab:6}      \n\\begin{tabular}{llllllllllllll}\n\\hline\\noalign{\\smallskip}\n\\multirow{2}{*}{\\text{Walking condition}}& \\multicolumn{11}{c}{Classifier} \\\\\n& $0^o$ & $18^o$ & $36^o$ & $54^o$ & $72^o$ & $90^o$ & $108^o$ & $126^o$ & $144^o$ & $162^o$ & $180^o$ & \\text{Avg} & \\text{Unified}  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\text{Normal walking}&98.5&99.1&98.7&97.8&99.3&99.8&98.7&98.4&98.3&98.9&99.2&98.8 &90.3\\\\\n\\text{Carrying backpack}&94.6&94.4&93.7&92.6&91.3&87.5&90.1&91.7&92.5&93.7&94.9&92.5 &85.7\\\\\n\\text{Wearing coat}&92.1&93.7&94.4&92.6&93.2&94.1&91.3&92.6&90.1&92.3&93.4&92.7 &86.1\\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}\n\\centering\n\\caption{CCRs (\\%) of the viewpoint-dependent classifier of unknown viewing angle when including attachment-area removal module }\n\\label{tab:7}      \n\\begin{tabular}{lllllllllllll}\n\\hline\\noalign{\\smallskip}\n\\multirow{2}{*}{\\text{Walking condition}}& \\multicolumn{11}{c}{Classifier} \\\\\n& $0^o$ & $18^o$ & $36^o$ & $54^o$ & $72^o$ & $90^o$ & $108^o$ & $126^o$ & $144^o$ & $162^o$ & $180^o$ & \\text{Avg}  \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\text{Normal walking}&98.5&99.1&98.7&97.8&99.3&99.8&98.7&98.4&98.3&98.9&99.2&98.8 \\\\\n\\text{Carrying backpack}&94.8&94.6&94.5&94.8&93.5&95.1&94.4&94.3&93.4&93.9&94.9&94.4 \\\\\n\\text{Wearing coat}&93.1&93.8&94.6&93.4&93.7&94.5&93.1&93.5&92.3&92.6&93.7&93.5 \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\subsection{Viewpoint estimation test}\nViewpoint estimation is an important step in this work because it determines the DS model and classifier to be used for gender prediction. To test the accuracy of the viewpoint estimation module, we randomly selected ten sequences from a specific viewpoint under normal walking conditions from the CASIA Dataset B. This procedure was conducted as discussed in Section 3.2.2. The average percentages (for the ten sequences) of the viewpoint estimation are displayed in Table \\ref{tab:5}. As can be seen, given a sequence with a specific viewpoint from the CASIA Dataset B, the viewpoint estimated from the program did not match the given viewpoint (for the given $0^o$ degree sequence, the estimated viewpoints are $0^o$, $18^o$, and $180^o$ with probabilities of 90.8, 8.1, and 1.1, respectively). This is understandable because people change their gait features while walking. Moreover, the person's appearance from the front and rear views are similar, which results in a classification step, i.e., 90.8\\% for a $0^o$ classifier, 8.1\\% for an $18^o$  classifier, and 1.1\\% for a $180^o$ classifier were used to obtain the gender of the individual.\n\nAfter obtaining the viewing angle, the corresponding classifier was selected to conduct the gender prediction. Table \\ref{tab:6} displays the $CCRs$ for an unknown viewpoint under different walking conditions, i.e., normal walking, carrying an item, and wearing a coat. The CCRs under normal walking conditions (Table \\ref{tab:6}, r ow 1) are improved because the viewpoints were automatically calculated and the proper classifier was selected for the gender prediction. \n\n\nFor the given unknown $0^o$ viewpoint, $90.8\\%$ of the image sequences are estimated at $0^o$ viewing angle, $8.1\\%$ at $18^o$, and $1.1\\%$ at $180^o$, respectively, as shown in Table \\ref{tab:5}. Those image sequences are then sent to $0^o$, $18^o$, $180^o$ classifiers, respectively to calculate the $CCR$ at $0^o$ viewpoint. The performance using the corresponding classifiers under normal walking conditions increases the $CCRs$ from ($97.6\\% \\pm 0.881$) to ($98.8\\% \\pm 0.550$). The $CCRs$, while carrying a bag or wearing a coat (Table \\ref{tab:6}, rows 2 and 3), were not significantly improved because we used classifiers trained under normal walking conditions to predict the gender.\n\n\n\nFor the aim of proving the superiority of the view-dependent design, a unified classifier of all viewpoints is trained without considering the viewing angle using the same configuration of the SVM. The experimental result is reported in the last column of Table \\ref{tab:6}. As seen in the table, the average performance of view-dependent (penultimate column) increases significantly in each experimental scenario comparing to the unified classifier.\n\n\n\\begin{table*}\n\\centering\n\\caption{Comparison with related methods}\n\\label{tab:8}      \n\\begin{tabular}{llllll}\n\\hline\\noalign{\\smallskip}\n\\text{Compared methods} & \\text{\\#subjects} & \\text{Viewpoints} & \\text{Walking condition} & \\text{Reported} & \\text{Proposed method} \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\n\\text{Lee et al. \\cite{ref42}} & \\text{14 males, 10 females} & $90^o$ & \\text{Normal} & $84.5\\%$ & $100\\%$ \\\\\n\n\\text{Li et al. \\cite{ref44}} & \\text{31 males, 31 females} & $90^o$ & \\text{Normal} & $93.2\\%$ & $99.8\\%$\\\\\n\n\\text{Yu et al. \\cite{ref23}} & \\text{31 males, 31 females} & $90^o$ & \\text{Normal} & $95.9\\%$ & $99.8\\%$\\\\\n\n\\text{Huang et al. \\cite{ref43}} & \\text{30 males, 30 females} & $0^o, 90^o, 180^o$  & \\text{Normal} & $89.5\\%$ & $99.2\\%$\\\\\n\n\\text{Zhang De \\cite{ref32}} & \\text{31 males, 31 females} & $0^o, 18^o,..., 180^o$  & \\text{Normal} & $98.1\\%$ & $98.8\\%$\\\\\n\n\\text{NA}& \\text{31 males, 31 females} & $0^o, 18^o,..., 180^o$  & \\text{Bag-carrying} & \\text{NA} & $94.4\\%$\\\\\n\n\\text{NA}& \\text{31 males, 31 females} & $0^o, 18^o,..., 180^o$  & \\text{Coat-wearing} & \\text{NA} & $93.5\\%$\\\\\n\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Attachment removal test}\nThe attachment area and noise can be removed using the procedure discussed in \nSection 3.3.2. During the testing phase, the silhouette was corrected and updated\nusing the attachment-area removal module. The updated version of the AGI was \ncalculated based on the new version of the silhouette. As indicated in Table \\ref{tab:6} and \nTable \\ref{tab:7}, the $CCRs$ of a human carrying a backpack in the side view ($90^o$) is \nsignificantly improved with the attachment removal module (95.1\\%), without the \nattachment removal module (87.5\\%). More improvement in the $CCRs$ of 94.4\\% $\\pm$\n0.564 and 93.5\\% $\\pm$ 0.704, compared to the cases of no attachment removal, is indicated in Table \\ref{tab:7} because of the attachment-area removal module. Moreover, a significant improvement for the side view or nearside view is presented in Table \\ref{tab:7}, row 2 (carrying a bag). The $CCR$ values in Table \\ref{tab:7}, row 1 were not changed because in this case, the person was walking with a thin coat and not carrying any objects. The attachment-area removal module did not remove anything in this case since the distance signal was within the range of $MaDS$ and $MiDS$.\n\\subsection{Comparisons}\nThe dataset used in \\cite{ref42} consists of twenty-four subjects, 14 males and 10 females, walking in normal speed and stride. The camera was placed perpendicular to their walking path. In \\cite{ref23}\\cite{ref44}experiments, only side-view sequences of 31 males and 31 females were collected from CASIA Dataset B for gender classification evaluation. Huang et al. in \\cite{ref43} extracted only 30 males and 30 females from CASIA Dataset B in three viewing angles including $0^o$,$90^o$, and $180^o$.\nTable \\ref{tab:8} presents a $CCR$ comparison of the proposed method with other related works. In the side-view dataset, the proposed method attained $CCRs$ of 100\\% and 99.8\\% compared with 84.5\\% reported in \\cite{ref42} and 95.9\\% reported in \\cite{ref23} on a small dataset and the CASIA Dataset B under the same conditions, respectively. The proposed method was also tested on normal walking conditions in three viewing angles ($0^o,90^o$, and $180^o$) and achieved greater accuracy (99.2\\% on average) compared with 89.5\\% as in \\cite{ref43}.\n\nTo demonstrate the effectiveness of the proposed method for gender classification with multiple-viewing angles, we conducted a test on multiple-views of the CASIA Dataset B (11 viewing angles) and obtained CCRs of 98.8\\%, which is also greater than the state-of-the-art method, 98.1\\% reported in \\cite{ref32}.\n\nAs described in Section 4.1, this CASIA Dataset B included three categories. The first category contained videos of humans walking in a normal condition without any attachments. The two remaining categories were more challenging, containing videos of humans carrying a backpack and humans wearing a coat. Because of the attachments, the silhouettes were highly deformed, leading to significant degradation on the classification results (Table \\ref{tab:6} and Table \\ref{tab:7}). To the best of our knowledge, there are no experimental results reported for these two remaining datasets.\n\nApplying the proposed module to remove the attachments, we performed experiments on these two datasets in multiple viewpoints (11 viewing angles) in the same scenario as the first category. The CCRs on the challenging dataset images indicated promising results of 94.4\\% and 93.5\\% for the bag-carrying and coat-wearing images, respectively, as indicated in Table \\ref{tab:8}.\n\nFurther, because the proposed method uses simple operations for gender classification such as 2-dimensional signals (distance signal), and linear SVM it requires only 48 ms (20.8 frames per second) to process a frame after skipping the first 15 frames for the AGI calculation. This means that the algorithm can be applied to a surveillance application in real time.\n\n\n\n\\section{Conclusions and future works}\nGender information can be effectively obtained from a video surveillance system based on the gait feature of the subject. Instead of using a GEI, this paper employed an AGI, which is easier to calculate for a real application. To accurately predict the human gender in real applications, we created viewpoint-dependent classifiers, i.e., a VP model and a DS model. The VP model is used to estimate the viewing angle during the testing phase; any attachment area is then removed using the DS model. Finally, the gender information is provided through the use of the viewpoint-dependent classifier. A comparison with other state-of-the-art methods \\cite{ref23} confirmed that the proposed method achieved a high accuracy of 98.8\\% and can be applied to a real-world system. However, the results of this method depend mainly on the quality of the silhouette obtained during the background subtraction step, as shown in Fig. \\ref{fig:8}. In the figure, the first row shows the samples that the method correctly classifies the gender whereas the second row shows the ones that are wrongly classified due to bad quality. As future work, we will attempt to apply the raw RGB image of a person rather than a silhouette image using deep-learning techniques because color information is an important factor for gender classification accurately predict.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[scale=0.5]{Fig8}\n\\caption{Good and bad silhouettes obtained from the background subtraction process.}\n\\label{fig:8}      \n\\end{figure}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n{\\bf Introduction.}\nThe most general form of the neutrino electromagnetic vertex function \\cite{Giunti:2014ixa} is given by\n$\\Lambda_{\\mu}^{ij}(q) =  \\left( \\gamma_{\\mu} - {q}_{\\mu}\n\\slashed{q}\/q^{2} \\right) \\left[ f_{Q}^{ij}(q^{2}) + f_{A}^{ij}(q^{2})\nq^{2} \\gamma_{5} \\right] \\nonumber\n - i \\sigma_{\\mu\\nu} q^{\\nu} \\left[ f_{M}^{ij}(q^{2}) +\ni f_{E}^{ij}(q^{2}) \\gamma_{5} \\right]$ ,\nwhere $\\Lambda_{\\mu}(q)$ and form factors $f_{Q,A,M,E}(q^2)$ are $3\\times 3$ matrices in  the space of massive neutrinos.  In the case of coupling with a real photon ($q^2=0$) form factors provide four sets of neutrino electromagnetic characteristics: 1) the dipole magnetic moments $\\mu_{ij}=f_{M}^{ij}(0)$,\n2) the dipole electric moments $\\epsilon_{ij}=f_{E}^{ij}(0)$, 3) the millicharges $q_{ij}=f_{Q}^{ij}(0)$ and\n4) the anapole moments $a_{ij}=f_{A}^{ij}(0)$.\n\n\n\n\n{\\bf Neutrino dipole magnetic moments.} The most well understood and studied among neutrino electromagnetic characteristics are the neutrino magnetic moments.\nIn the Standard Model with massless neutrinos magnetic moments  of neutrinos are zero. Therefore, it is believed that the studies of neutrino electromagnetic properties open a window to {\\it new physics} \\cite{Giunti:2014ixa,Aprile:2020tmw,Studenikin:2008bd,Studenikin:2018vnp}. In a minimal extension of the Standard Model  the diagonal magnetic  moment of a Dirac neutrino is given \\cite{Fujikawa:1980yx} by\n$\\mu^{D}_{ii}\n  = \\frac{3e G_F m_{i}}{8\\sqrt {2} \\pi ^2}\\approx 3.2\\times 10^{-19}\n  \\Big(\\frac{m_i}{1 \\ \\mathrm{eV} }\\Big) \\mu_{B}$,\n$\\mu_B$ is the Bohr magneton. The Majorana neutrinos can have  only transition\n(off-diagonal) magnetic\nmoments  $\\mu^{M}_{i\\neq j}$. The same is valid also for the flavour neutrinos in the case of the Majorana mass states.\n\nThe most stringent constraints on the neutrino\nmagnetic moments are obtained with the reactor antineutrinos\n(GEMMA Collaboration \\cite{GEMMA:2012}):\n$\\mu_{\\nu_e} < 2.9 \\times 10^{-11} \\mu_{B}$,\nand solar neutrinos (Borexino Collaboration \\cite{Borexino:2017fbd}):\n${\\mu}_\\nu ^{eff}< 2.8 \\times\n10^{-11} \\mu _B$. The last limit can be translated to the upper limits for flavour neutrinos: $(\\mu_{\\nu_e}, \\mu_{\\nu_{\\mu, \\tau}}) \\sim (4, 6 ) \\times 10^{-11} \\mu _B$.\n\nNote that in general in the scattering experiments the neutrino\nis created at some distance from the detector as a flavor neutrino, which is a\nsuperposition of massive neutrinos. Therefore, the magnetic\nand electric moments that are measured in these experiments are not that of a\nmassive neutrino, but there are effective moments that take into account the neutrino mixing and oscillations\nduring the propagation between the\nsource and detector \\cite{Grimus:1997aa, Beacom:1999wx}.\nFor the recent and detailed study of the neutrino electromagnetic characteristics\ndependence on neutrino mixing see \\cite{Kouzakov:2017hbc}.\n\nA new phase of the GEMMA project for measuring the neutrino magnetic moment is now underway at the Kalinin Power Plant in Russia. The discussed next experiment \\cite{Belov:2015ufh} called GEMMA-3$ \/ \\nu  $GEN  is aimed at the further increase in sensitivity to the neutrino magnetic moment and will reach the level of\n$\\mu_{\\nu_e} \\sim (5{-}9) \\times 10^{-12}\\mu _B $.\nTo reach the  claimed limit on the neutrino magnetic moment  the $\\nu$GEN experiment setup reasonably improves characteristics in respect to those of the previous editions of the GEMMA project. The most important are the following \\cite{Lubashevskiy:2020}: 1) a factor of 2 increase in the total neutrino flux at the detector because of much closer location of the detector to the reactor core, 2) a factor of 3.7 increase in the total mass of the detector, 3) the energy threshold would be improved from $2.8 \\ keV$  to $ 200 \\ eV$. Furthermore, the $\\nu$GEN experimental setup is located in the new room at the Kalinin Power Plat with much better (by an order of magnitude) gamma-background conditions and on a moveable platform. The later gives an opportunity to vary online the neutrino flux and thus suppress systematic errors.\n\n\n\n\nThe observation of coherent elastic neutrino-nucleus scattering reported for the first time \\cite{Akimov:2017ade} by the\nCOHERENT experiment at the Spallation Neutron Source can be also used for constraining neutrino electromagnetic properties. For the case of neutrino magnetic moments, however, as it was shown in \\cite{Kosmas:2017tsq}\nand then confirmed in recent studies (see, for instance, \\cite{Miranda:2020tif}  ) the bounds for the flavour neutrino magnetic moments are of the order $\\mu_e , \\mu_\\mu \\sim 10^{-8} \\mu _{B}$.\n\n\n\n\nIn the recent studies \\cite{Miranda:2020kwy} it is shown that the puzzling results of the XENON1T collaboration \\cite{Aprile:2020tmw} at few keV electronic recoils  could be due to the scattering of solar neutrinos endowed with finite Majorana transition magnetic moments of the strengths lie within the limits set by the Borexino experiment  with solar neutrinos \\cite{Borexino:2017fbd}. The comprehensive analysis of the existing and new extended mechanisms for enhancing neutrino transition magnetic moments to the level appropriate for the interpretation of the XENON1T data  and leaving neutrino masses within acceptable values is provided in \\cite{Babu:2020ivd}.\n\nIn the most recent paper \\cite{Cadeddu:2019qmv} we have proposed an experimental setup to observe coherent elastic neutrino-atom scattering using electron antineutrinos from tritium decay and a liquid helium target. In this scattering process with the whole atom, that has not beeen observed so far, the electrons tend to screen the weak charge of the nucleus as seen by the electron antineutrino probe.\n Finally, we study the sensitivity of this apparatus to a possible electron\n neutrino magnetic moment and we find that it is possible\n to set an upper limit of about\n$\\mu_{\\nu} < 7 \\times 10^{-13} \\mu_{B}$,\nthat is more than one order of magnitude smaller than\nthe current experimental limits from GEMMA and Borexino.\n\n\n\nAn astrophysical bound on an effective neutrino magnetic moment (valid for both cases of\nDirac and Majorana neutrinos) is provided\n\\cite{Raffelt-Clusters:90, Viaux-clusterM5:2013, Arceo-Diaz-clust-omega:2015}\nby observations of the properties of globular cluster stars:\n$\\Big( \\sum _{i,j}\\left| \\mu_{ij}\\right| ^2\\Big) ^{1\/2}\\leq (2.2{-}2.6) \\times\n10^{-12} \\mu _B$. There is also a statement \\cite{deGouvea:2012hg}, that\nobservations of supernova fluxes  in the future largevoluem experiments like JUNO, DUNE and Hyper-Kamiokande ( see for instance\n\\cite{An:2015jdp,Giunti:2015gga,Lu:2016ipr}) may reveal the effect of  collective  spin-flavour oscillations  due to the Majorana neutrino transition moment $\\mu^{M}_\\nu \\sim 10^{-21}\\mu_B$. Other new possibilities for neutrino\nmagnetic moment visualization in extreme astrophysical environments are\nconsidered recently in \\cite{Grigoriev:2017wff,Kurashvili:2017zab}.\n\n\n\nA general and termed model-independent upper bound on the Dirac neutrino\nmagnetic moment, that can be generated by an effective theory beyond\na minimal extension of the Standard Model, has been derived in\n\\cite{Bell:2005kz}: $\\mu_{\\nu}\\leq\n10^{-14}\\mu_B$. The corresponding limit for transition moments of Majorana neutrinos is much weaker \\cite{Bell:2006wi}.\n\n\n{\\bf Neutrino dipole electric moments.} In the theoretical framework with $CP$ violation a neutrino\ncan have nonzero electric moments $\\epsilon_{ij}$. In the laboratory neutrino\nscattering experiments for searching $\\mu_{\\nu}$ (for instance, in the GEMMA experiment)\nthe electric moment $\\epsilon_{ij}$ contributions interfere with\nthose due to $\\mu_{ij}$. Thus, these kind of experiments also provide constraints\non $\\epsilon_{ij}$. The astrophysical bounds on $\\mu_{ij}$\nare also applicable for constraining $\\epsilon_{ij}$ (see \\cite{Raffelt-Clusters:90, Viaux-clusterM5:2013, Arceo-Diaz-clust-omega:2015} and \\cite{Raffelt:2000kp}).\n\n{\\bf Neutrino electric millicharge.} There are extensions of the Standard Model that allow for nonzero\nneutrino electric millicharges. This option can be provided by\nnot excluded experimentally possibilities for hypercharhge dequantization or\nanother {\\it new physics} related with an additional $U(1)$ symmetry\npeculiar for extended theoretical frameworks. Note that neutrino millicharges\nare strongly constrained on the level $q_{\\nu}\\sim 10^{-21} e_0$\n($e_0$ is the value of an electron charge) from neutrality of the hydrogen atom.\n\n A nonzero neutrino millicharge $q_{\\nu}$ would contribute to the neutrino electron scattering in the terrestrial experiments. Therefore, it is possible to get bounds on $q_{\\nu}$ in the reactor antineutrino\n experiments. The most stringent reactor antineutrino constraint\n\n $q_{\\nu}< 1.5 \\times 10^{-12} e_0 $\n\n is obtained in \\cite{Studenikin:2013my} within the free-electron approximation using the GEMMA experimental data \\cite{GEMMA:2012}. This limit is cited by the Particle Data Group since 2016 (see also \\cite{Zyla:2020zbs}).\nA certain increase in the cross section is expected in the case when instead of the free-electron approximation one accounts for\nthe so called atomic ionization effect \\cite{Chen:2014dsa}, and the obtained corresponding limit on the neutrino millicharge is\n $q_{\\nu} < 1 \\times 10^{-12} e_0$.\n\n\n The expected increasing sensitivity to the neutrino-electron scattering of the future $\\nu$GEN experiment that is aimed to reach a new  limit  for the magnetic moment would provide a possibility \\cite{Studenikin:2013my} to check the neutrino millicharge at the scale of  $ q_{\\nu} \\sim  10^{-13} e_0$.\n\nAs it has been already mentioned above, the coherent elastic neutrino-nucleus scattering \\cite{Akimov:2017ade} is a new\npowerful tool to probe the electromagnetic neutrino properties \\cite{Kosmas:2017tsq}. In the flavour basis neutrinos can have diagonal $q_{lf}$ ($l=f$, $l,f = e, \\mu, \\tau$) and transition $q_{lf}$ ($l\\neq f $) electric charges (see, for instance, \\cite{Giunti:2014ixa} and \\cite{Kouzakov:2017hbc}). Such possibilities are not excluded by theories beyond the Standard Model. Recently \\cite{Cadeddu:2019eta} from the analysis of the COHERENT data new constraints for all neutrino charges on the level of $\\sim 10^{-7} e_0$ are obtained. It follows, that the bounds for involving the electron neutrino flavour charges $q_{ee}, q_{e\\mu}$ and $q_{e\\tau}$ are not competitive with respect to constraints $\\sim 10^{-12} e_0$ obtained for the effective electron neutrino charge $q_{eff}= \\sqrt{q_{ee}^2 +q_{e\\mu}^2 +q_{e\\tau}^2}$ from the reactor antineutrino scattering experiments \\cite{Studenikin:2013my, Chen:2014dsa}. Note, that the bounds for $q_{\\mu \\mu}$ and $q_{\\mu \\tau}$ from a laboratory data are obtained in \\cite{Cadeddu:2019eta} for the first time.\n\n\nThe most recent and one of the most detailed statistical studies \\cite{Parada:2019gvy} of experimental data from the elastic neutrino-electron and coherent neutrino-nucleus scattering show that the combined inclusion of different experimental data can lead to stronger\nconstraints on $q_\\nu$ than those based on individual analysis of different experiments.\n\n\nA neutrino millicharge would have specific phenomenological consequences\nin astrophysics because of new electromagnetic processes are opened\ndue to a nonzero charge (see \\cite{Giunti:2014ixa,Raffelt:1996wa,Studenikin:2012vi}). Following this line, the most stringent astrophysical constraint on neutrino millicharges\n$q_{\\nu}< 1.3 \\times 10^{-19} e_0 $\n was obtained in \\cite{Studenikin:2012vi}. This bound\nfollows from the impact of the {\\it neutrino star turning} mechanism ($\\nu ST$) \\cite{Studenikin:2012vi} that can be considered as a {\\it new physics} phenomenon end up with a pulsar rotation frequency\nshift engendered by the motion of escaping from the\nstar neutrinos along curved trajectories due to millicharge interaction with a constant\nmagnetic field of the star. The existed other astrophysical constraints on the neutrino millicharge, however less restrictive than that of \\cite{Studenikin:2012vi}, are discussed in \\cite{Giunti:2014ixa,Parada:2019gvy}.\n\n{\\bf Neutrino cherge radius and anapole moment.} Even if a neutrino millicharge is vanishing, the electric form factor\n$f^{ij}_{Q}(q^{2})$ can still contain nontrivial information about\nneutrino electromagnetic properties. The corresponding electromagnetic characteristics is\ndetermined by the derivative of $f^{ij}_{Q}(q^{2})$ over $q^{2}$  at\n$q^{2}=0$ and is termed neutrino charge radius,\n$\\langle{r}_{ij}^{2}\\rangle\n=-\n6\n\\frac{df^{ij}_{Q}(q^{2})}{dq^{2}} \\\n_{\\mid _ {q^{2}=0}}\n$ (see \\cite{Giunti:2014ixa} for the detailed discussions).\nNote that for a massless neutrino the neutrino charge radius is the only\nelectromagnetic characteristic that can have nonzero value. In the Standard Model\nthe neutrino charge radius and the anapole moment are not defined separately,\nand there is a relation between these two values: $a = - \\frac{\\langle{r}^{2}\\rangle}{6}$.\n\nA neutrino charge radius contributes to the neutrino scattering cross section on electrons and thus\ncan be constrained by the corresponding laboratory experiments \\cite{Bernabeu:2004jr}.\nIn all but one previous studies it was claimed\n that the effect of the neutrino\ncharge radius can be included just as a shift of the vector coupling constant $g_V$\nin the weak\ncontribution to the cross section.\nHowever, as it has been recently demonstrated in \\cite{Kouzakov:2017hbc} within the direct calculations of\nthe elastic neutrino-electron scattering cross section accounting for all possible neutrino electromagnetic characteristics\nand neutrino mixing, this is not the fact. The neutrino charge radius dependence of the cross section\nis more complicated and there are, in particular, the dependence on the interference terms of the type\n$g_{V}\\langle{r}_{ij}^{2}\\rangle$ and also on the neutrino mixing.\nThe current constraints on the flavour neutrino charge radius $\\langle{r}_{e,\\mu,\\tau}^{2}\\rangle\\leq 10^{-32} - 10^{-31} \\ cm ^2$\nfrom the scattering experiments differ only by 1 to 2\norders of magnitude from the values $\\langle{r}_{e,\\mu,\\tau}^{2}\\rangle\\leq 10^{-33} \\ cm ^2$ calculated within the minimally extended Standard Model with right-handed neutrinos\n\\cite{Bernabeu:2004jr}. This indicates that the minimally extended Standard Model neutrino charge radii could be experimentally tested in the near future.\n\nNote that there is a need to re-estimate experimental constraints on\n$\\langle{r}_{e,\\mu,\\tau}^{2}\\rangle$  from the scattering experiments following\nnew derivation of the cross section \\cite{Kouzakov:2017hbc} that properly accounts for the interference of the weak and charge radius electromagnetic interactions and also for the neutrino mixing.\n\nRecently constraints on  charged radii  have been obtained\n\\cite{Caddedu:2018prd} from the analysis of the data on coherent\nelastic neutrino-nucleus scattering obtained in the COHERENT experiment\n\\cite{Akimov:2017ade,Akimov:2018vzs}. In addition to the customary diagonal\ncharge radii $\\langle{r}_{e,\\mu,\\tau}^{2}\\rangle$\nalso the neutrino transition (off-diagonal) charge radii have been constrained\nin \\cite{Caddedu:2018prd} for the first time:\n$\\left(|\\langle r_{\\nu_{e\\mu}}^2\\rangle|,|\\langle r_{\\nu_{e\\tau}}^2\\rangle|,|\\langle r_{\\nu_{\\mu\\tau}}^2\\rangle|\\right)\n< (22,38,27)\\times10^{-32}~{\\rm cm}^2$. Since 2018 these limits are included by the Particle Data Group to Review of Particle Properties (see also \\cite{Zyla:2020zbs}) and also were noted by the Editors' Suggestion as the most important results (PRD Highlights 2018) published in the journal.\n\n\n\nThe work is supported by the Russian Foundation for Basic Research under grant No. 20-52-53022-GFEN-a.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nResearch on highly accurate atomic frequency standards and their applications is making fast and steady progress. The quest for ever increased accuracy is advancing hand in hand with advances in quantum physics, with better understanding and manipulation of atomic systems, with exploration of fundamental laws of nature and with the development of important services and infrastructures for science and society. The quest for increased accuracy is also a powerful incentive to innovation in such areas as lasers, laser stabilization, low noise electronics, stable oscillators, low noise detection of optical signals, fiber devices and cold-atom based instrumentation for ground or space applications. Finally, in addition to enhancing existing applications, improved accuracy leads to new applications.\n\nThis article focuses on key achievements and trends of the last 10 years. Over this period of time, the first generation of laser-cooled standards, using the atomic fountain geometry, reached maturity and had large impact on international timekeeping. The typical accuracy of these frequency standards,  2 parts in $10^{16}$, is now permanently accessible both locally and globally. At the same time, a new generation of optical clocks showed tremendous and steady improvement, gaining more than two orders of magnitude in a decade. To date, an accuracy of $6.4\\times 10^{-18}$ \\cite{bloom2014optical} was reported. Similarly, major improvements occurred in many other aspects of optical frequency metrology, notably in optical frequency combs and optical fiber links.\n\nIn this article, we will report on developments of the LNE-SYRTE atomic clock ensemble since our 2004 report in the Special Issue of the Comptes Rendus de l'Acad\\'emie des sciences on Fundamental Metrology \\cite{Bize2004}. This work exemplifies many of the above mentioned features of research on highly accurate atomic frequency standards. We will focus on frequency standards and their impact on timescales and timekeeping, on clock comparisons, including optical-to-microwave comparisons with combs, and their applications. Several other aspects of our research are covered by other articles of the Special Issue of the Comptes Rendus de l'Acad\\'emie des Sciences (Volume 16, Issue 5), i.e. the space mission PHARAO\/ACES \\cite{cras2014onACES}, fundamental tests with clocks \\cite{cras2014onFundamentalTests}, development of technologies for space optical clocks \\cite{cras2014onSpaceOpticalClocks} and optical fibers links \\cite{cras2014onFiberLinks1,cras2014onFiberLinks2}.\n\n\n\\section{Atomic fountains} \\label{fountains}\n\n\\begin{figure*}[t]\n\t\\centering\n  \\includegraphics[width=1\\textwidth, angle=0]{Fig1_ClockEnsemble1.pdf}\n\t\\caption{Overview of the LNE-SYRTE atomic clock ensemble at the Observatoire de Paris.}\n\t\\label{fig:ClockEnsemble}\n\\end{figure*}\n\nAtomic fountains are the first generation of the laser-cooled atomic frequency standards. They use the fountain geometry where spectroscopy of the clock transition is performed onto a free-falling sample of laser-cooled atoms which is beforehand launched upwards vertically (see, for instance, \\cite{guena2012} and references therein). To date, atomic fountains using cesium provide the most accurate realization of the SI second. One important aspect of the last 10 years was to better understand systematic shifts limiting the accuracy of these devices.\n\n\\textit{Distributed cavity phase shift}-- In atomic fountains, the Ramsey interrogation is realized by the up-going and down-going passages of the atomic sample through the microwave Ramsey cavity. The spatial phase variations of the field inside the cavity, when sampled by the moving atoms, induce a frequency shift, which can be described as a residual Doppler shift. For a long time, there was a lack of both a complete and agreed model for this effect and of experiments to test it. Consequently, this effect was one of the main sources of uncertainty in atomic fountains. In \\cite{li2004,li2010}, a new approach was proposed to compute the cavity phase distribution. We performed measurements of these shifts in FO2-Cs which enabled the first quantitative comparison between theory and experiment \\cite{guena2011}. This study validated the theoretical model and lowered the distributed cavity phase uncertainty for FO2-Cs to $10^{-16}$. It also defined a method to determine this uncertainty, which was then adopted in \\cite{li2011,weyers2012} and for other SYRTE fountains.\n\n\\textit{Microwave lensing shift}-- The microwave field inside the Ramsey cavity not only excites the transition between the two internal clock states but also modifies the motion of atomic wave packets, leading to a frequency shift \\cite{Borde2002}\\cite{Wolf2004}. In \\cite{Gibble2006}, a new approach to compute the shift was proposed. It was then used in a complete model of the effect, taking into account all features of the interaction such as atomic velocity and space distributions and detection non-uniformities \\cite{li2011}\\cite{weyers2012}. This same method was applied to LNE-SYRTE fountains, for which shifts are reported in table 2 of \\cite{guena2012}.\n\n\\textit{Blackbody radiation shift}-- In 2004, conflicting measurements and calculations of this shift induced by thermal radiation bathing the atoms were reported. This led us to revisit our early accurate measurements of the Stark coefficient \\cite{simon1998}. Our new measurements at lower electric fields have been found to be in excellent agreement \\cite{rosenbusch2007}. The theory of the Stark shift developed in the 60's turned out to have a sign error for the tensor part. This led to a small change of the blackbody radiation shift correction of $7\\times 10^{-17}$ \\cite{guena2012}. Two independent high accuracy ab initio calculations further agreed with the blackbody radiation shift correction derived from our Stark measurements.\n\n\\textit{Microwave leakage and synchronous phase perturbations}-- Interaction of atoms with unintended residual microwave field and synchronous perturbation of phase of the probing field can produce shifts. We developed microwave synthesizers that can be switched without introducing phase transients and a phase transient analyzer with 1~$\\mu$rad.s$^{-1}$ resolution \\cite{santarelli2009}. Using these tools, we lowered the uncertainty related to these putative frequency shifts to less than $10^{-16}$.\n\nOur approach to deal with other systematic shifts remained as described in our last report in the Comptes Rendus \\cite{Bize2004}. Table \\ref{tab_accuracy} gives, as an example, the accuracy budget of FO2-Cs as of 2014.\n\n\\begin{table}\n\\footnotesize\n\\caption{Typical uncertainty budget of the FO2-Cs primary frequency standard (top). Uncertainty budget of the Sr1 optical lattice clock as of July 2011 (bottom). Tables give the fractional frequency correction and its Type~B uncertainty for each systematic shift, in units of $10^{-16}$. The total uncertainty is the quadratic sum of all uncertainties.}\\label{tab_accuracy}\n\\begin{center}\n\\begin{tabular}{lcc}\n\\multicolumn{3}{c}{FO2-Cs}\\\\\n\\hlin\nPhysical origin of the shift                                       & Correction   & Uncertainty  \\\\\n\\hline\n{\\small Quadratic Zeeman}               & $-1919.9$&$0.3$  \\\\\n{\\small Blackbody radiation}                  &   $168.4$&$0.6$\\\\\n{\\small Collisions and cavity pulling}        &   $201.2$&$1.5$\\\\\n{\\small Distributed cavity phase}       & $-0.9$& $1.2$   \\\\\n{\\small Microwave lensing}                     &  $-0.7$&$0.7$   \\\\\n{\\small Spectral purity \\& leakage}            & 0  &$0.5$ \\\\\n{\\small Ramsey \\& Rabi pulling}                & 0 &$0.1$  \\\\\n{\\small Relativistic effects}            & 0 & $0.05$ \\\\\n{\\small Background collisions}                 & 0  &$1.0$ \\\\\n\\hline\n{\\small Total}                               &  $-1551.9$&$2.5$  \\\\\n\\hline\n\\end{tabular}\n\\hspace{1cm}\n\\begin{tabular}{lcc}\n\\multicolumn{3}{c}{Sr1}\\\\\n\\hlin\nPhysical origin of the shift                                       & Correction   & Uncertainty  \\\\\n\\hline\n{\\small Quadratic Zeeman}               & $19.7$&$0.2$  \\\\\n{\\small Blackbody radiation}                  &   $ 53.8 $&$0.8$\\\\\n{\\small Collisions}        &   $-0.2 $&$0.5 $\\\\\n{\\small AC Stark shift lattice 1st order}       & $-0.5 $& $0.1 $   \\\\\n{\\small AC Stark shift Lattice 2nd order}      &  $0 $&$ 0.1$   \\\\\n{\\small DC Stark shift}            &  0  &$0.01 $ \\\\\n{\\small Line pulling}                & 0 &$0.5$  \\\\\n\\hline\n{\\small Total}                               &  $72.8$&$1.0$  \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{-6mm}\n\\end{table}\n\n\\normalsize\n\\vspace{1mm}\n\nAnother important achievement was the simultaneous operation with $^{87}$Rb and $^{133}$Cs of the dual fountain FO2. This was done by implementing dichroic collimators overlapping 780~nm (for Rb) and 852~nm (for Cs) radiations for all laser beams, and by adopting a time sequence enabling time resolved selective detection of the two atomic species \\cite{guena2010}. Further notable improvement relates to reliability and capability for long term unattended operation. Using 2D magneto-optic traps to load the optical molasses suppressed residual background vapor and enhanced the lifetime of the alkali sources. Also, we developed an automatic data processing system that monitors the status of all fountains, oscillators and internal links in quasi real time. This system allows rapid detection of failures. It also performs automated fountain data processing, taking account of all systematic corrections, and it continuously generates frequency measurements at the nominal uncertainty of the fountains. This capability has had major impact on timekeeping and other applications of atomic fountains (see sect.~\\ref{sec:fundphystests} and \\ref{sec:timekeeping} ).\n\n\\section{Optical lattice clocks}\n\n\\label{sec:OLC}\n\nIn optical lattice clocks (OLCs), a set of neutral cold atoms, dipole-trapped in an optical lattice, are interrogated by an ultra-stable ``clock'' laser. Because they involve probing an optical transition of a large (typically $10^4$) number of tightly confined atoms, they combine an excellent ultimate frequency stability -- only limited by the Quantum Projection Noise (QPN) -- and a high accuracy. Proposed in 2001 \\cite{Katori2001}\\cite{Katori2003a}, OLCs made tremendous progress in the last decade. OLCs have demonstrated unprecedented frequency stabilities of a few $10^{-16}\/\\sqrt{\\tau}$ and a record accuracy below $10^{-17}$ \\cite{bloom2014optical}, overcoming the best ion clocks~\\cite{letargat2013,hinkley_atomic_2013,falke2014,bloom2014optical,ushijima2014cryogenic}. With current improvement in laser stabilization, OLCs are expected to reach a QPN limited stability on the order of $10^{-17}\/\\sqrt{\\tau}$ within a few years, thus enabling even better characterization of systematic effects. OLCs with Sr, Yb, Hg and more prospectively Mg have been demonstrated. Among these atomic species, Sr is currently the most popular choice because of the accessibility of the required laser wavelengths, the possibility to cool Sr down to sub-$\\mu$K temperature using the narrow $^1S_0 \\rightarrow{}^3P_1$ inter-combination line, and the possibility it offers on the control of systematic effects, most notably concerning  the high order perturbation by the trapping light and cold collisions.\n\n\\textit{Strontium optical lattice clocks}-- LNE-SYRTE developed two OLCs using strontium atoms. The design of these clocks uses an optical cavity to enhance the optical lattice light, giving access to large trap depths. It enabled us to explore systematic effects induced by the trapping laser light. These effects are specific to OLCs, and we have demonstrated that they can be controlled to better than $10^{-17}$, even with a significant trapping depth~\\cite{Brusch2006a,westergaard2011lattice}, thus validating the concept of OLCs. In particular, we have determined the precise value for the ``magic wavelength'' for which the impact of the trapping light is canceled to first order, and resolved or upper-bounded a number of higher order effects. Because OLCs use a large number of atoms in a tightly confined space, they are subject to a significant density-dependent systematic frequency shift. Some groups have resolved a density shift on the order of $10^{-16}$ with both Sr and Yb. However, the loading technique chosen at LNE-SYRTE leads to a lower atomic density, thus dramatically reducing this effect below $10^{-17}$. The blackbody radiation shift has remained the dominant contribution to the accuracy budget, with an uncertainty around $5\\times 10^{-17}$ for both Sr and Yb, assuming a 1~K uncertainty on the temperature of the environment. Recently, precise measurements of the static polarizability of Yb and Sr, together with carefully crafted environments for the atoms has enabled a few groups to drastically reduce this uncertainty, down to the $10^{-18}$ range.\n\nComparisons between clocks are necessary to confirm their accuracy budget. The first comparisons between remote Sr OLCs were achieved by comparing to cesium clocks (see sect.~\\ref{sec:fundphystests}), but they were soon limited by the cesium accuracy. LNE-SYRTE published the first comparison between two local OLCs that confirm the accuracy budget of the clocks better than the accuracy of the cesium fountains, involving two Sr clocks with a frequency difference smaller than their combined accuracy budgets of $1.5\\times 10^{-16}$~\\cite{letargat2013}. This resolution is obtained after less than one hour of integration. Table~\\ref{tab_accuracy} gives the accuracy budget of one of the Sr OLCs at the time of this comparison.\n\n\n\\textit{Mercury optical lattice clock}-- SYRTE also started the development of a mercury OLC. Hg has the advantage of low sensitivity to blackbody radiation and to electric field (30 times lower than Sr, 15 times lower than Yb). For the $^{199}$Hg isotope, clock levels have a spin 1\/2 for which the tensor light shift sensitivity is absent. Also, because of its high vapor pressure, it does not require an oven and enables the use of a 2D magneto-optic trap as the initial source of atoms. Hg is also interesting for fundamental physics and atomic physics, because of a quite high sensitivity to a variation of $\\alpha$ (see sect.~\\ref{sec:fundphystests}) and its 7 natural isotopes. The main challenge of using Hg lies in the need for deep UV laser sources. When the potential of Hg for a highly accurate optical lattice clock arose, Hg had never been laser cooled.\n\nIn the last years, we made all the steps leading to the demonstration, for the first time, of a Hg lattice clock. Laser cooling on the 254 nm $^1S_ 0 \\rightarrow{} ^3P_1$ intercombination transition was demonstrated and studied \\cite{Petersen2008b}\\cite{Hachisu2008}\\cite{McFerran2010}. A clock laser system with thermal noise limited instability of $4\\times 10^{-16}$ was developed \\cite{Millo2009b}\\cite{Dawkins2010}. We performed the first direct laser spectroscopy of the the clock transition, firstly on atoms free-falling from a magneto-optic trap \\cite{Petersen2008a} and secondly on lattice-bound atoms, with linewidth down to 11~Hz (at 265.6~nm or 1128~THz). We performed the first experimental determination of the ``magic wavelength'' \\cite{Yi2011}, for which our best value is $362.5697 \\pm \ufffd0.0011$~nm. We performed initial measurements of the absolute frequency of the $^{199}$Hg clock transition down to an uncertainty of 5.7 parts in $10^{15}$ \\cite{McFerran2012}.\n\nSo far, the advancement of the Hg optical lattice was mainly hindered by the poor reliability of the 254~nm laser-cooling and by the modest lattice trap depth. Recent work enabled large improvements to overcome these two limitations, opening the way to in-depth systematic studies and higher accuracies.\n\n\n\n\n\\section{Optical frequency combs}\n{}\nThe development of optical frequency standards of the kind presented in the previous section is aimed at producing a laser electromagnetic field of extremely stable and accurate frequency. To realize a complete metrological chain that allows comparison with other standards operating at different wavelengths in the optical domain or in the microwave domain (such as primary frequency standards), it is necessary to use a specific device. The method of choice nowadays is the optical frequency comb based on mode-locked femto-second laser, that provides a phase coherent link spanning across the optical and microwave domains. Recently, SYRTE focused on a technology of comb based on erbium-doped fiber lasers, whose foremost asset is the capability to function for months with a very limited maintenance while performing state-of-the-art measurements. Tight phase locking (bandwidth $\\sim 1\\,$MHz) of such comb onto a 1.5~$\\mu$m ultra-stable laser sets it in the narrow-linewidth regime where beatnotes with other ultra-stable light at other wavelengths are typically $\\sim1\\,$Hz linewidth. This provides high performance simultaneous measurements of the various ultra-stable optical frequency references at SYRTE in the visible and near infrared domain.\n\nThe comb also behaves like a frequency divider: the repetition rate of the comb, $f_{\\rm rep}$, results from the coherent division of the 1.5$\\,\\mu$m laser frequency $\\nu_{\\rm L}$. By photo-detecting the train of pulses and filtering out one specific harmonic of the repetition rate, one can generate a microwave signal whose phase noise is that of the cw optical reference, divided by the large frequency ratio (typically 20000) between a 1.5$\\,\\mu$m wavelength cw laser and a $10\\,$GHz microwave signal. $10\\,$GHz signals with phase noise of -100\\,dBc\/Hz at 1\\,Hz Fourier frequency and -140\\,dBc\/Hz white noise plateau are now straightforward to produce. This low phase noise level is comparable with that of cryogenic sapphire oscillators and is thus sufficient to operate state-of-the-art microwave atomic fountains (sect.~\\ref{fountains}) with a short term stability limited only by atomic quantum projection noise. We have realized proof-of-principle experiments of such a scheme \\cite{MilloAPL2009} and are now progressing toward implementing it in an operational system. We further demonstrated several advanced techniques \\cite{ZhangAPB2012, HabouchaOL2011, ZhangOL2014} to reduce the imperfection of the frequency division and photo-detection processes. We have shown that it is now becoming possible to generate microwave signals with phase noise as low or lower than any other technology for a large range of Fourier frequencies. Applications of such extremely low noise microwave signals can be found in RADAR (civil and military) as well as very long baseline interferometry.\n\nFinally, the comb-based transfer of spectral purity between different wavelengths recently led to exciting results. This application is crucial for the future development of optical lattice clocks, whose short term stability is currently limited by the spectral purity of the ultra-stable clock laser probing the atomic transition. Several competing technologies are being explored, notably at SYRTE, to improve the performance of these lasers, all of them very challenging, some of them wavelength-specific. Being able to utilize the performance of an extremely stable laser at a given wavelength and to distribute its performance to any wavelength within reach of the frequency comb is an important milestone for the future developments of optical lattice clocks. We demonstrated such transfer from a master to a slave laser with an added instability of no more than a few $10^{-18}$ at $1\\,$s (see fig. \\ref{fig:TSP}, right), well within the requirements expected for the next several years \\cite{NicolodiNP2014}.\n\n\\begin{figure*}[ht]\n\t\\centering\n  \\includegraphics[width=0.9\\textwidth, angle=0]{Fig2_FrequencyComb.pdf}\n\t\\caption{Principle of the transfer of spectral purity (left): the optical beatnotes of the comb with a master laser on the one hand, and with a slave laser on the other hand, are rescaled and mixed before being compared to a stable synthesizer. The feedback on the offset-locked slave transfers the spectral purity of the master to the slave laser. The modified Allan deviation (right) of the noise added by the transfer itself has a level of only $2\\times10^{-18}$ at $1\\,$s, and averages down to $2 \\times 10^{-20}$ after $1000\\,$s.}\n\t\\label{fig:TSP}\n\\end{figure*}\n\n\\section{Fundamental physics tests}\\label{sec:fundphystests}\n\nOne exciting scientific application of atomic clocks with extreme uncertainties is to contribute to testing fundamental physical laws and searching for physics beyond the Standard Model of particle physics. The frequency of an atomic transition relates to parameters of fundamental interactions (strong interaction, electro-weak interaction), such as the fine-structure constant $\\alpha$, and to fundamental properties of particles like for instance the electron mass, $m_e$. Repeated highly accurate atomic clock comparisons can be used to look for a putative variation with time or with gravitational potential of atomic frequency ratios, and, via suitable atomic structure calculations, of natural constants. Clocks provide laboratory tests, independent of any cosmological model, that constrain alternative theories of gravity and quantum mechanics, thereby contributing to the quest for a unified theory of the three fundamental interactions.\n\n\n\\label{RbCsFitLin}\n\\textit{$^{87}$Rb vs $^{133}$Cs comparisons}-- Improvements of atomic fountains described in sect.~\\ref{fountains} enabled major enhancement in the number and in the quality of Rb\/Cs hyperfine frequency ratio measurements since our last report in the Comptes Rendus \\cite{Bize2004}. Measurements have been performed almost continuously since 2009. Fig.\\ref{Graphdrift}, top shows the temporal record of the variations of this ratio. Measurements extending over 14~years give stringent measurements of a putative variation with time and gravity\nof the Rb\/Cs ratio, as reported in \\cite{guena2012b}. Taking into account out most recent data, we get $d \\ln(\\nu_{\\mathrm{Rb}}\/\\nu_{\\mathrm{Cs}})\/dt=(-11.6\\pm 6.1 )\\times 10^{-17}$~yr$^{-1}$ for the time variation. For the variation scaled to the annual change of the Sun gravitational potential on Earth $U$, we get $c^2 d \\ln(\\nu_{\\mathrm{Rb}}\/\\nu_{\\mathrm{Cs}})\/dU=(7.4 \\pm 6.5)\\times10^{-7}$, which provides a differential redshift test between Rb and Cs twice more stringent than \\cite{peil2013}.\n\n\\begin{figure}\n\t\\begin{center}\n\n\t \\includegraphics[width=0.45\\textwidth]{Fig3_RbCs.pdf}\n\n\t \\vspace{3mm}\n\n \\includegraphics[width=0.48\\textwidth]{Fig3_SrCs.pdf}\n \n\t\\end{center}\n\n\t \\vspace{1mm}\n\n\\footnotesize\n\\begin{center}\n\\begin{tabular}{p{28mm} ccc}\n\\hline \\hline\n & $\\ln(\\alpha)$ &$\\ln(\\mu$)&$\\ln(m_{q}\/\\Lambda_{\\mathrm{QCD}})$ \\\\\n\\hline\n$d\/dt~ (\\times 10^{-16}\\mathrm{yr}^{-1})$& $-0.26\\pm 0.24$ &$1.1\\pm 1.4$& $59\\pm 30$\\\\\n$c^{2}d\/dU ~(\\times 10^{-6})$ & $0.27 \\pm 0.46$ & $-0.2 \\pm 2.1$ & $-2.9\\pm 5.6$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\normalsize\n\n\\caption{Top: Temporal record of fractional variations of the $\\nu_{\\mathrm{Rb}}\/\\nu_{\\mathrm{Cs}}$ hyperfine frequency ratio. The error bars are the total 1~$\\sigma$ uncertainties, dominated by the systematic uncertainties. The solid red line is the weighted fit to a line with inverse quadratic weighting. The origin of the vertical axis corresponds to the $^{87}$Rb secondary representation of the SI second recommended in 2012, with a recommended uncertainty $1.2 \\times 10^{-15}$ (grey area) \\cite{CCTF2012}. Middle: International comparisons of the $\\nu_{\\mathrm{Sr}}\/\\nu_{\\mathrm{Cs}}$ frequency ratio. A fit shows an upper bound on a drift of this ratio, as well as on a variation synchronized with the Earth's orbit around the Sun. Bottom: Results of the global analysis of accurate experimental determinations of variations of atomic frequency ratios, available as of October 2014. The table gives constraints on temporal variations and on couplings to gravitational potential for the three fundamental constants: $\\alpha$, $\\mu=m_{e}\/m_{p}$ and $m_{q}\/\\Lambda_{\\mathrm{QCD}}$.} \\label{Graphdrift}\n\\end{figure}\n\n\n\\textit{$^{87}$Sr vs $^{133}$Cs comparisons}-- Frequency ratios between optical and microwave clocks offer a different sensitivity to natural constants than hyperfine frequency ratios. International absolute frequency measurements of strontium optical lattice clocks against Cs fountain primary frequency standards over a decade (fig.\\ref{Graphdrift}, Middle) give the linear drift with time of the $\\nu_{\\mathrm{Sr}}\/\\nu_{\\mathrm{Cs}}$ ratio: $d \\ln(\\nu_{\\mathrm{Sr}}\/\\nu_{\\mathrm{Cs}})\/dt=(-2.3 \\pm 1.8)\\times 10^{-16}$~yr$^{-1}$, and of the variation with the gravitational potential: $c^2 d \\ln(\\nu_{\\mathrm{Sr}}\/\\nu_{\\mathrm{Cs}})\/dU=(-1.3 \\pm 1.5)\\times 10^{-6}$. Because the accuracy of these measurements has improved considerably over time, these bounds will be significantly improved by future measurements.\n\n\\textit{Combining with other comparisons}-- Each pair of atoms has a different sensitivity to variations of three fundamental constants $\\alpha$, $\\mu=m_e\/m_p$ and $m_{q}\/\\Lambda_{\\mathrm{QCD}}$. To set independent limits to variations of these three constants with time, one can perform weighted least-squares fit to all accurate experimental determinations of variation with time of atomic frequency ratios, available as of October 2014. This includes the above Rb\/Cs result (see sect.~\\ref{RbCsFitLin}), optical frequency measurements of H(1S-2S) \\cite{fischer2004}, Yb$^{+}$ \\cite{tamm2014}, Hg$^{+}$ \\cite{fortier2007}, Dy \\cite{leefer2013} and Sr (see above, and \\cite{letargat2013} and references therein) against Cs, and the optical-to-optical ion clock frequency ratio $\\mathrm{Al}^{+}\/\\mathrm{Hg}^{+}$ \\cite{rosenband2008}.\nThe fit yields independent constraints for the three constants given in the first row of the Table in fig.~\\ref{Graphdrift}. The constraint relative to $\\alpha$ is mainly determined by the $\\mathrm{Al}^{+}\/\\mathrm{Hg}^{+}$ comparison. In this fit, only the Rb\/Cs comparison disentangles variations of $\\mu$ and of $m_q\/\\Lambda_{\\mathrm{QCD}}$. It is therefore essential to constrain $m_q\/\\Lambda_{\\mathrm{QCD}}$. This stems from the fact that optical frequency measurements are all performed against the Cs hyperfine frequency, except $\\mathrm{Al}^{+}\/\\mathrm{Hg}^{+}$.\n\nSimilarly, we perform a global analysis for the variation with the gravitational potential exploiting all available comparisons as of October 2014 \\cite{peil2013, fortier2007, leefer2013} and the above Rb\/Cs and Sr\/Cs results.\nThe least-squares fit to these results yields independent constraints for the three couplings to gravity given in the second row of the Table in fig.~ \\ref{Graphdrift}.\n\nThe number of atomic systems contributing to improve these tests will continue to grow, e.g. with $^{88}$Sr$^+$ \\cite{Madej2012}\\cite{Barwood2014} and $^{171}$Yb \\cite{Lemke2009a}, thanks to the steady efforts of many laboratories worldwide in the field of optical frequency metrology.\n\n\n\n\\section{Advanced timekeeping}\\label{sec:timekeeping}\n\n\\textit{TAI calibration with atomic fountains}-- The International Atomic Time (TAI) which is based on approximately 400 atomic clocks, now gets its accuracy from some ten atomic fountain clocks worldwide (see e.g. \\cite{cras2014Petit}). In the last decade, the number of calibrations of TAI with atomic fountains has grown from approximately 10 per year to 4 to 6 per month in 2014, while simultaneously the accuracy improved from several $10^{-15}$ to a few $10^{-16}$, improving TAI a lot. Combining a tremendous number of monthly calibrations and a high accuracy, LNE-SYRTE atomic fountains are providing the largest contribution to the accuracy of TAI. Between 2007 and August 2014, they provided 197 calibrations out of a total of 407 calibrations worldwide, a weight of nearly 50\\%. fig.~\\ref{TAI_UTC}, Top shows these calibrations as published in \\emph{Circular T}, and the SI second (red line) which is the average over all primary calibrations computed by the BIPM on a monthly basis. This illustrates how research on laser-cooled atomic fountain started 25 years ago led to improving an important service and infrastructure for science and society.\n\n\n\n\n\\textit{UTC(OP): timescales using atomic fountain clocks}-- The UTC(OP) timescale, elaborated at SYRTE, in Observatoire de Paris, is the real time realization of UTC for France. It is a continuously operated time reference used for multiple purposes:\ndefinition of legal time disseminated in France, reference provided to French laboratories for\nsynchronization applications, pivot for French contributions to TAI, test of advanced time transfer\nmethods, link to UTC of the EGNOS system, contribution to the development of GALILEO, time reference for the ground-segment of the PHARAO\/ACES space mission \\cite{laurent2006, cacciapuoti2007,cacciapuoti2009}.\n\nProgress in the accuracy and most importantly in the reliability of SYRTE atomic fountains (see sect.~\\ref{fountains}) enabled a new implementation of UTC(OP). A new UTC(OP) algorithm based on a hydrogen maser steered by the atomic fountains was developed and\nimplemented in October 2012. The maser is predictable enough to reach a stability of $\\sim 10^{-15}$. The\natomic fountains allow the maser frequency to be calibrated with an uncertainty in the $10^{-16}$ range \\cite{guena2012, guena2014}. These features are sufficient to maintain a phase deviation of a few ns over $1-\n2$ months, which corresponds to the delay of publication of the BIPM \\emph{Circular T}.\nThis timescale is as autonomous and independent as possible, except a small long term steering to remain\nclose to UTC, and does not rely on any other timescale available in real time such as GPS time or\nother UTC(k). A timescale with these characteristics provides a powerful tool to understand current limits and eventually improve international timekeeping.\n\nPractically, UTC(OP) is realized using a microphase stepper fed by the reference maser. A frequency\ncorrection is updated every day to compensate the maser frequency and\nmaintain UTC(OP) close to UTC. This correction is the sum of two terms. The main term corresponds to the current frequency of the maser as measured\nby the fountains. The value is estimated with a linear extrapolation of the data covering the past 20~days\nto remain robust against possible interruptions of data provision or of the automatic data processing. The\nsecond term is a fine steering to maintain UTC(OP) close to UTC, compensating the\nfrequency and phase offset between UTC(OP) and UTC. It is updated monthly at the BIPM \\emph{Circular~T}\npublication. The steering correction is usually of the order of $10^{-15}$ or below.\n\nFigure \\ref{TAI_UTC}, bottom presents the comparison of three UTC(k) to UTC as published in \\emph{Circular~T}\nsince the implementation of the new UTC(OP). Over this period, UTC(OP) is one of the 3 best real time\nrealizations of UTC \\cite{rovera2013, abgrall2014}, with UTC(PTB), the pivot of time\ntransfers for international contributions to TAI, and UTC(USNO), the laboratory providing the largest number of clock data included in EAL computation. Departure between UTC(OP) and UTC remains well below 10~ns, with a rms value less than 3~ns. This is an improvement of about a factor of 5 compared to the previous realization method of the timescale. On-going instrumental upgrades shall further improve the short term stability of the timescale and the robustness of the system.\n\n\n\\begin{figure}[htb]\n\n\\includegraphics[width=0.45\\textwidth]{Fig4_GTAI4.pdf}\n\\includegraphics[width=0.45\\textwidth]{Fig4_GUTCk5.pdf}\n\\caption{Top: Calibrations of TAI by the atomic fountain PFSs. Filled symbols: contributions of SYRTE fountains. Solid red line: the SI.  Bottom: Comparisons of 3 UTC(k) to UTC: UTC(OP), UTC(PTB) and UTC(USNO). The inset shows the significant improvement achieved with the new method for generating UTC(OP) implemented at MJD 56218, compared to the previous one using a commercial Cs clock manually steered towards UTC.}\n\\label{TAI_UTC}\n\\end{figure}\n\n\\section{Toward a redefinition of the SI second}\n{}\n\nSeveral optical frequency standards are now largely surpassing Cs atomic fountains which realize the second of the international system of units (SI) and define the accuracy of TAI. This opens the inviting prospect of a redefinition the SI second. In 2001, anticipating this situation, the Consultative Committee for Time and Frequency (CCTF) of the Comit\\'e International des Poids et Mesures (CIPM) recommended that a list of Secondary Representations of the Second be established. Secondary Representations of the SI Second (SRS) are transitions which are used to realize frequency standards with excellent uncertainties, and which are measured in the SI system with accuracies close to the limit of Cs fountains. They are part of the broader list of recommended values of standard frequencies produced and maintained by the CCL-CCTF Working Group and adopted by the CIPM. Producing and maintaining these lists of recommended values is a vehicle to keep track of measurements providing the most stringent connections between the optical domain and the SI second, and to verify the level of consistency between these measurements. This is an important task to prepare for a possible redefinition of the SI second.\n\n\n\\textit{Contributions to the list of recommended values}--\nLNE-SYRTE provided several high accuracy absolute frequency measurements which contributed to the list of recommended values. The $^{87}$Rb hyperfine transition was measured repeatedly against Cs fountains, as already shown in fig.~\\ref{Graphdrift}. This transition became the first Secondary Representation of the Second proposed by the CCTF in 2004, based our early measurements, and adopted by the CIPM in 2006. After further significant progress visible in fig.~\\ref{Graphdrift}, the recommended value was revised by the 2012 CCTF and adopted in 2013. The origin of the vertical scale of the graph is the recommended value of 2012 and the gray area represents the recommended uncertainty. The weighted average of all data points (green line) gives $(2.15\\pm 1.48)\\times 10^{-16}$ is consistent with zero within the smallest overall uncertainty of the measurements ($4.4\\times 10^{-16}$). LNE-SYRTE also provided absolute frequency measurements that contributed to the establishment of the recommended values for the $^1S_0\\rightarrow{}^3P_0$ of $^{87}$Sr \\cite{Baillard2007b}, $^{88}$Sr \\cite{Baillard2007} and $^{199}$Hg \\cite{McFerran2012}. Using the transportable Cs fountain primary standard FOM, LNE-SYRTE also contributed to the establishment of the recommended value for H(1S-2S) (measured at the  Max Planck Institut f\\\"ur Quantenoptik, Garching, Germany) \\cite{parthey2011} and $^{40}$Ca$^{+}$ (at the University of Innsbruck, Austria) \\cite{chwalla2009}.\n\n$^{87}$Sr is currently the most widespread optical frequency standard. For this reason a large number of groups measured the absolute frequency of $^{87}$Sr against Cs with a remarkable degree of consistency, as can be seen in fig.~\\ref{Graphdrift}, middle. These measurements led the 2006 CCTF to recommend the $^{87}$Sr as a Secondary Representation of the SI second. The recommended value was updated by the 2012 CCTF based on measurements from 5 institutes. It has a recommended uncertainty of $1\\times 10^{-15}$. More recently, LNE-SYRTE reported a measurement of the $^{87}$Sr clock transition with an uncertainty of $3.1\\times 10^{-16}$ limited by the accuracy of atomic fountains \\cite{letargat2013}. This is the most accurate absolute measurement to date of any atomic frequency. One of the key factors for the measurement is the record stability between an optical and a microwave clock: $4.1\\times 10^{-14}\/\\sqrt{\\tau}$ against Cs (and $2.8\\times 10^{-14}\/\\sqrt{\\tau}$ against Rb). In 2014, PTB reported another absolute frequency measurement with an uncertainty of $3.9\\times 10^{-16}$ \\cite{falke2014}. These two last measurements are in excellent agreement.\n\n\n\n\n\\textit{Using a Secondary Representation of Second to calibrate TAI}-- One major application of primary frequency standards is to calibrate and steer the scale interval of the widely used International Atomic Time TAI. It is important to anticipate how a possible redefinition of the second would impact the elaboration of TAI. We used the FO2-Rb fountain to investigate how a Secondary Representation of Second could participate to TAI. Calibrations of the frequency of our reference hydrogen maser were produced with FO2-Rb, in a similar way that absolute calibrations are done with primary frequency standards. These data were then submitted to the BIPM and to the Working Group on Primary Frequency Standards. Following this submission, the BIPM and the Working Group defined how frequency standards based on Secondary Representations will be handled by the BIPM and how they will be included into the \\textit{Circular~T}. The Working Group was renamed Working Group on Primary and Secondary Frequency Standards and it was decided that calibrations produced by LNE-SYRTE with FO2-Rb could be included into \\textit{Circular~T} and, since July 2013, contribute to steering TAI. This was the first time that a transition other than the Cs hyperfine transition was used to steer TAI \\cite{guena2014}.\n\n\\textit{Absolute frequency measurement against the TAI ensemble}-- More than 40 formal calibrations of TAI with FO2-Rb have been sent, processed by the BIPM and published into \\textit{Circular~T}. These data can be used to measure the frequency Rb hyperfine transition directly against the second as realized by the TAI ensemble. This can be done with a statistical uncertainty of 1 part in $10^{-16}$, and therefore at the accuracy limit of primary frequency standards defining the scale interval of TAI \\cite{guena2014}. This illustrates how TAI provides worldwide access to the accuracy of Cs fountains. This also shows how recommended values of Secondary Representation of the Second based on optical transitions could be checked against the SI second as realized by the TAI ensemble.\n\n\n\n\\section{Prospects}\n\nIn the future, highly accurate atomic clocks and their applications will keep improving at a high pace. An important milestone in the field will be the simultaneous availability of advanced timescales, of the new generation of optical clocks and of the means to compared them remotely at unprecedented levels of uncertainty. In the coming decade, the ACES mission will allow ground-to-space comparisons to the $10^{-16}$ level and ground-to-ground comparisons to the mid $10^{-17}$ level \\cite{cras2014onACES}. Optical fiber links will allow comparisons of the most accurate optical clocks at their limit: $10^{-18}$ or better \\cite{cras2014onFiberLinks1,cras2014onFiberLinks2}. We can confidently predict major improvements in all applications of highly accurate atomic clocks. Availability of clocks and clock comparisons at the $10^{-18}$ level can further enable new applications. Clock comparisons determine Einstein's gravitational redshift between the 2 remote clock locations. For a clock at the surface of the Earth, $10^{-18}$ corresponds to an uncertainty of 1~cm in height-above-geoid, making the idea of clock-based geodesy realistic and potentially useful. Highly accurate clocks could become a new type of sensors for applications in Earth science, illustrating once again the fertilizing power of the quest for ever increased accuracy.\n\n\n\n\\section*{Acknowledgements}\n\nSYRTE is UMR CNRS 8630 between Centre National de la Recherche Scientifique (CNRS), Universit\\'e Pierre et Marie Curie (UPMC) and Observatoire de Paris. LNE, Laboratoire National de M\\'etrologie et d'Essais, is the French National Metrology Institute.\nSYRTE is a member of IFRAF, of the nanoK network of the R\\'egion \\^Ile de France and of the FIRST-TF LabeX. We acknowledge the large number of contributions of SYRTE technical services. This work is supported by LNE, CNRS, UPMC, Observatoire de Paris, IFRAF, nanoK, Ville de Paris, CNES, DGA, ERC AdOC, EMRP JRP SIB55 ITOC, EMRP JRP EXL01 QESOCAS. We are grateful to the University of Western Australia and to M.E.~Tobar for the long-lasting collaboration which gives us access to the cryogenic sapphire oscillator used in the LNE-SYRTE ultra-stable reference.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec-introduction}\n\nRobot autonomy offers great promise as a tool by which we can enhance, or restore, the natural abilities of a human partner.  For example, in the fields of assistive and rehabilitative medicine, devices such as exoskeletons and powered wheelchairs can be used to assist a human who has severely diminished motor capabilities.  However, many assistive devices can be difficult to control.  This can be due to the inherent complexity of the system, the required fidelity in the control signal, or the physical limitations of the human partner.  We can, therefore, further improve the efficacy of these devices by offloading challenging aspects of the control problem to an autonomous partner.  In doing so, the human operator is freed to focus their mental and physical capacities on important high-level tasks like path planning and interaction with the environment.  This idea forms the basis of \\textit{shared control} (see Figure~\\ref{fig-shared-control}), a paradigm that aims to produce joint human-machine systems that are more capable than either the human or machine on their own.  \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.95\\hsize]{shared_control_new.png}\n\t\\caption{Pictorial representation of a shared control paradigm.  Both the human and autonomy are capable of controlling the mechanical system, and a dynamic control allocation algorithm selects which agent is in control at any given moment.}\n\t\\label{fig-shared-control}\n\\end{figure}\n\nA primary challenge that researchers and engineers face when developing shared control paradigms for generic human-machine systems is a lack of \\textit{a priori} knowledge of the human and robot partners.  This issue is compounded by the fact that, in the real world, many users may operate the same mechanical device.  It is therefore necessary to consider solutions that generalize to a variety of potential human and machine partners.  In this work, we propose a data-driven methodology that learns all relevant information about how a given human and machine pair interact directly from observation.  We then integrate the learned model of the joint system into a single shared control paradigm.  We refer to this idea as \\textit{model-based shared control}.\n\nIn this work, we learn a model of the joint human-machine system through an approximation to the Koopman operator (\\cite{koopman1931hamiltonian}), though any machine learning approach could be used.  However, the Koopman operator is chosen specifically for this work as it has previously proven useful in human-in-the-loop systems~(\\cite{broad2017learning}) and can be computed efficiently~(\\cite{williams2015kernel}). This model is trained on observation data collected during demonstration of the human and machine interacting and therefore describes both the human's input to the system, and the robot's response to the human input and system state.  We can then integrate the portion of the learned model that specifically describes the system and control dynamics of the mechanical device into an optimal control algorithm to produce autonomous policies.  Finally, the input provided by the human and autonomous partners are integrated via a geometric signal filter to provide real-time, dynamic shared control of unknown systems.  \n\nWe validate our thesis that modeling the joint human-machine system is sufficient for the purpose of automating assistance with two human subjects studies consisting of 32 total participants.  The first study imposes a linear constraint on the modeling and control algorithms, while the second study relaxes these constraints to evaluate the more general, nonlinear case.  The linear variant of our proposed algorithm is used to validate the efficacy of our shared control paradigm and was first presented in~\\cite{broad2017learning}.  The nonlinear variant extends these results to a wider class of human-machine systems.  The results of the two studies demonstrate that the nonlinear variant has a greater impact on overall task performance than the linear methods.  We also find that our modeling technique is generalizable across users with results that suggest that individualizing the model offline, based on a user's own data, does not affect the ability to learn a useful representation of the dynamical system.  Finally, we evaluate the efficacy of our shared control paradigm in an online learning scenario, demonstrating the sample efficiency of the model-based shared control paradigm.\n\nWe provide background and related work in Section~\\ref{sec-background-and-related-work}.  We then define model-based shared control in Section~\\ref{sec-model-based-shared-control}.  In Section~\\ref{sec-experimental-validation} we describe the human subjects study we perform and detail the results in Section~\\ref{sec-results}.  We describe important takeaways in Section~\\ref{sec-discussion} and conclude in Section~\\ref{sec-conclusion}.\n\n\\section{Background and Related Work}\n\\label{sec-background-and-related-work}\n\nThis section presents background and related work in the shared control literature for human-machine systems.  We also identify alternative methods of autonomous policy generation for shared control, and provide a detailed background on the Koopman operator (\\cite{koopman1931hamiltonian}) with a particular focus on its use in learning system dynamics.\n\n\\subsection{Shared Control}\n\\label{sec-background-shared-control}\n\nIn this work, we explore the question of how automation can be used to adjust to, and account for, the specific capabilities of a human partner.  In particular, we aim to develop a methodology that allows us to \\textit{dynamically adjust} the amount of control authority given to the robot and human partners~(\\cite{hoeniger1998dynamically, hoffman2004collaboration}).  If done intelligently, and with appropriate knowledge of the individual capabilities of each team member, we can improve the overall efficiency, stability and safety of the joint system~(\\cite{lasota2017survey}).  Approaches to shared control range from pre-defined, discretely adjustable methods~(\\cite{kortenkamp2000adjustable}) to probabilistic models~(\\cite{javdani2015shared}) to policy blending~(\\cite{dragan2013policy}).  In addition to blending in the original control signal space, shared control has been researched through haptic control~(\\cite{nudehi2005shared}) and compliant control~(\\cite{kim1992force}).\n\nIn this work, we allocate control using a filter~(\\cite{tzorakoleftherakis2015controllers}) described more thoroughly in Section~\\ref{sub-sec-control-allocation}.  Our control allocation strategy is similar in practice to \\textit{virtual fixtures} and \\textit{virtual guides}, techniques that are common in the haptics literature~(\\cite{forsyth2005predictive, griffiths2005sharing}).  In particular, virtual fixtures and guides are techniques by which autonomously generated forces are \\textit{added to the control of a system} to limit movement into undesriable areas and\/or influence motion towards an optimal strategy~(\\cite{abbink2012haptic}).  These ideas have been explored most commonly in association with robotic telemanipulation~(\\cite{abbott2007haptic}), including applications like robotic surgery~(\\cite{marayong2004speed}) and robot-assisted therapy~(\\cite{noohi2016model}).  A key difference between these approaches and our own is that our control allocation method does not incorporate additional information from the autonomous partner into the control loop.  Instead, the autonomous partner simply rejects input from the operator that does not meet the proposed criteria. Our approach therefore requires no \\textit{a priori} information about (or ability to sense) the environment, and no information about the system dynamics. In contrast, virtual fixtures\/guides require information about (or the ability to detect) hard constraints in the environment, and knowledge of the system dynamics. This information is then used to compute forces---the virtual fixtures---that  counteract user-generated forces that are defined by as dangerous.  The approach in this paper does not have similar \\textit{a priori} information requirements, suggesting our approach can more easily be incorporated into novel human-machine systems. An important benefit of the methods proposed in the virtual fixtures\/guide literature is that the techniques often provide an explicit guarantee of safety for the joint human-machine system. Our approach can be extended to provide the same guarantees by incorporating information about (or the ability to sense) the environment and using control barrier functions to implement safety requirements~(\\cite{broad2018operation}).\n\nThe effects of shared control (SC) have been explored in numerous fields in which the addition of a robot partner could benefit a human operator.  For example, in assistive and rehabilitation robotics, researchers have explored the effects of shared control on teleoperation of smart wheelchairs~(\\cite{erdogan2017effect, trieu2008shared}) and robotic manipulators~(\\cite{kim2006continuous}).  Similarly, researchers have explored shared control as it applies to the teleoperation of larger mobile robots and human-machine systems, such as cars~(\\cite{dewinter11smc}) and aircraft~(\\cite{matni08acc}).  When dealing with systems of this size, safety is often a primary concern.  \n\nThe above works are conceptually similar to our own as they use automation to facilitate control of a robot by a human partner.  However, in this work, we do not augment the user's control based on an explicit model of the user.  Instead, we use observations of the user demonstrations to build a \\textit{model of the joint human-robot system}.  The effect of the human partner on the shared control system is implicitly encoded in the model learned from their interactions.\n\n\\subsection{Model-Based Reinforcement Learning}\n\\label{sec-background-model-based}\n\nModel-based shared control (MbSC) is a paradigm that generalizes shared control to generic human-machine partners~(\\cite{broad2017learning}).  That is, MbSC assumes no \\textit{a priori} knowledge of either partner and instead uses data-driven techniques to learn models of the human and\/or robot partner(s) from observation.  In addition to providing a quantitative understanding of each partner, these models can be used to generate autonomous control policies by integrating the learned system and control dynamics into an optimal control framework.  \n\nModel-based shared control is therefore highly related to model-based reinforcement learning (MbRL), a paradigm that explicitly learns a model of the system dynamics in addition to learning an effective control policy.  MbSC extends MbRL to systems that integrate control from various sources.  Early work in model-based reinforcement learning includes~(\\cite{barto1995learning}) and~(\\cite{kaelbling1996reinforcement}).  More recently, researchers have considered integrating learned system models with optimal control algorithms to produce control trajectories in a more data-efficient manner~(\\cite{mitrovic2010adaptive}).  These algorithms compute control through an online optimization process, instead of through further data collection~(\\cite{barto1995learning}).  There are of course, many viable model learning techniques that can be used to describe the system and control dynamics.  For example, Neural Networks~(\\cite{williams2017information}), Gaussian Processes~(\\cite{nguyen2009local}), and Gaussian Mixture Models~(\\cite{khansari2011learning}) have all shown great promise in this area.  Often the best choice of modeling algorithm is related specifically to the application domain.  For example, Gaussian Processes perform well in low-data regimes, but scale poorly with the size of the dataset where Neural Networks fit naturally.  In this work we explore a modeling technique that easily integrates with derivative-based optimal control algorithms.  A survey of learning for control can be found in~(\\cite{schaal2010learning}).\n\nFrom a motivational standpoint, related work also includes methods that model not only the dynamics of a robotic system, but combined human-machine systems from data. For example, researchers have explored learning control policies from user demonstrations, thereby incorporating both system dynamics and the user's desires~(\\cite{argall2009survey, celemin2019fast}).  Building on these ideas, researchers have proposed learning shared control policies directly from demonstration data using deep reinforcement learning~(\\cite{reddy2018shared}).  To improve the human partner's intuition for the interaction paradigm, researchers have also proposed learning latent spaces to allow users to control complex robots with low dimensional input devices~(\\cite{losey2019controlling}).  Relatedly, people have also proposed techniques for modeling both the dynamics of a system, and a policy for deciding when a human or autonomous partner should be in control.  One such method is to learn local approximations to the system's dynamics and only provide autonomous assistance when the system is nearby a state it has previously observed~(\\cite{peternel2016shared}).  Our approach utilizes a linear representation of the nonlinear human-robot dynamics which avoids the use of local models in exchange for a higher capacity linear model which globally represents the complex system.  This is also distinct from the virtual fixtures\/guide literature where system models are known \\textit{a priori}, and frequently nonlinear.\n\nFrom a methodological standpoint, the most closely related research is recent work that computes control trajectories by integrating learned dynamics models with model predictive control (MPC) algorithms~(\\cite{williams2017information, drews2017aggressive}).  These algorithms are defined by an iterative, receding horizon optimization process instead of using an infinite-horizon.  Similar to our own work, these researchers first collect observations from live demonstrations of the mechanical device to learn a model of the system dynamics.  They then integrate the model with an MPC algorithm to develop control policies.  Beyond methodological differences (e.g., choice of machine learning and optimal control algorithms), the key theoretical distinction between these works and our own is our focus on shared control of joint human-machine systems, instead of developing fully autonomous systems.  In particular, we learn a model of the joint system that is integrated into a shared control system to improve a human operator's control of a dynamic system.  We therefore consider the influence of the human operator both during the data-collection process and at run-time in the control of the dynamic system.\n\nIn this work, we learn a model of the system and control dynamics through an approximation to the Koopman operator~(\\cite{koopman1931hamiltonian}).  As the Koopman operator is a relatively new concept in robot learning for control, we now provide additional information on its description in the following section.\n\n\\subsection{The Koopman Operator}\n\\label{sec-background-koopman}\n\n\\begin{figure*}[!th]\n\t\\centering\n\t\\includegraphics[width=0.8\\hsize]{pipeline.png}\n\t\\caption{Pictorial depiction of the our model-based shared control paradigm.  (a) Collect observations from user interaction and learn a model of the joint human-machine system through an approximation to the Koopman operator.  This can be computed offline or online.  (b) Compute control policy of autonomous agent by solving optimal control problem using the learned model.  (c) Allocate control to integrate autonomy (gray) and user input (green\/red).}\n\t\\label{fig-pipeline}\n\\end{figure*}\n\nThe Koopman operator is an infinite-dimensional linear operator that can capture all information about the evolution of nonlinear dynamical systems.  This is possible because the operator describes a linear mapping between sequential \\textit{functions of states} instead of the state itself.  In particular, the Koopman operator acts on an infinite dimensional Hilbert space representation of the state.  To define the Koopman operator, let us consider a discrete time dynamic system $(\\mathcal{X}, t, F)$: \n\\begin{equation}\nx_{t+1} = F(x_t)\n\\label{eq-gen-dynamics}\n\\end{equation}\n\n\\noindent where $\\mathcal{X} \\subseteq \\mathbb{R}^N$ is the state space, $t \\in \\mathbb{R}$ is time and $F : \\mathcal{X} \\rightarrow \\mathcal{X}$ is the state evolution operator.  We also define $\\phi$, a nominally infinite dimensional observation function \n\\begin{equation}\ny_t = \\phi(x_t)\n\\label{fn-obs}\n\\end{equation}\n\n\\noindent where $\\phi : \\mathcal{X} \\rightarrow \\mathbb{C}$ defines the transformation from the original state space into the Hilbert space representation that the Koopman operator acts on.  The Koopman operator $\\mathcal{K}$ is defined as the composition of $\\phi$ with $F$, such that \n\n\\begin{equation}\n\\mathcal{K} \\phi = \\phi \\circ F.\n\\label{fn-koopman-eq}\n\\end{equation}\n\n\\noindent By acting on the Hilbert state representation, the \\textit{linear} Koopman operator is able to capture the complex, nonlinear dynamics described by the state evolution operator. \n\nWhile the Koopman operator is nominally infinite dimensional, recent work has demonstrated the ability to approximate a finite dimensional representation using data-driven techniques~(\\cite{rowley2009spectral, budivsic2012applied}).  In the limit of collected observation data, the approximation to the Koopman becomes exact~(\\cite{williams2015data}).  These data-driven methods have renewed an interest in using the Koopman operator in applied engineering fields.  In contemporary work, the Koopman operator has been successfully used to learn the dynamics of numerous challenging systems.  This includes demonstrations that show the Koopman operator can differentiate between cyclic and non-cyclic stochastic signals in stock market data~(\\cite{hua2016using}) and that it can detect specific signals in neural data that signify non-rapid eye movement (NREM) sleep~(\\cite{brunton2016extracting}).  More recently these systems have included physical robotics systems~(\\cite{abraham2019active, bruder2019modeling}).\n\n\\section{Model-based Shared Control}\n\\label{sec-model-based-shared-control}\n\nOur primary goal is to develop a shared control methodology that improves the skill of human-machine systems without relying on \\textit{a priori} knowledge of the relationship between the human and the machine.  To define our model-based shared control algorithm we now describe the (1) model learning process, (2) method for computing the policy of the autonomous agent (\\textit{autonomy input} in Figure~\\ref{fig-shared-control}) and (3) control allocation method (the \\textit{green box} in Figure~\\ref{fig-shared-control}).  A pictorial depiction of our model-based shared control paradigm can be found in Figure~\\ref{fig-pipeline}.  Our learning-based approach develops a model of the joint human-machine system solely from observation, and this model can be used by the policy generation method to develop autonomous control trajectories.  The control allocation method then describes how we integrate the input provided by the human partner and the autonomous agent into a single command that is sent to the dynamic system.  \n\n\\subsection{Model Learning via the Koopman Operator}\n\nWhen designing assistive shared control systems, it is important to consider both the human and autonomous partners.  To ensure that our paradigm is valid for generic human-machine systems, we learn both the \\textit{system dynamics} and information about the \\textit{user interaction} directly from data.  In this work, we develop a model of the joint human-machine through an approximation to the Koopman operator, which can be computed offline or online (discussed further in Section~\\ref{sec-study-two-results-online}).  The model learning process is depicted in Figure~\\ref{fig-pipeline}\\textcolor{red}{(a)}.  As mentioned previously, there are of course a variety of other machine learning algorithms and representions one could choose to learn the system dynamics.  In this work, we use the Koopman operator, which is particularly well suited to model-based shared control of human-machine systems for two main reasons.  First, it is possible to approximate the Koopman operator in low-data regimes (see Section~\\ref{sec-study-two-results-online}) which allows us to quickly expand the set of human-machine systems we can control under the general MbSC paradigm.  Second, there are a variety of highly efficient learning algorithms~(\\cite{williams2015data, klus2015numerical, rowley2009spectral}) that make the Koopman operator well suited to an online learning paradigm, an important feature in shared control where it is unlikely that we have \\textit{a priori} knowledge of the joint human-machine system.\n\nWe use Extended Dynamic Mode Decomposition (EDMD) to approximate the Koopman operator~(\\cite{williams2015data}).  EDMD belongs to a class of data-driven techniques known as Dynamic Mode Decomposition (DMD)~(\\cite{rowley2009spectral, schmid2010dynamic, tu2013dynamic}).  These algorithms use snapshots of observation data to approximate the Koopman modes that describe the dynamics of the observed quantities.  We now provide a mathematical treatment of the EDMD algorithm.  We start by redefining the observation function $\\phi$ from Equation~\\ref{fn-obs} as a vector valued set of basis functions chosen to compute a finite approximation to the Hilbert space representation.  We can then define the following approximation to the Koopman operator\n\n\\begin{equation}\n\\phi(x_{t+1}) = \\mathcal{K}^T\\phi(x_t) + r(x_t)\n\\end{equation}\n\n\\noindent where $r(x_t)$ is a residual term that represents the error in the model.  The Koopman operator is therefore the solution to the optimization problem that minimizes this residual error term \n\\begin{align}\n\\begin{split}\nJ & = \\frac{1}{2} \\sum_{t=1}^{T} |r(x_t)|^2\\\\\n& = \\frac{1}{2} \\sum_{t=1}^{T} |\\phi(x_{t+1}) - \\phi(x_t)K) |^2\n\\end{split}\n\\label{eq-ls}\n\\end{align}\n\n\\noindent where $T$ is the time horizon of the optimization procedure, and $|\\cdot|$ is the absolute value.  The solution to the least squares problem presented in Equation~\\eqref{eq-ls} is\n\\begin{equation*}\nK = G^\\dagger A\n\\end{equation*}\n\n\\noindent where $\\dagger$ denotes the Moore-Penrose pseudo inverse and\n\\begin{align*}\nG & = \\frac{1}{T} \\sum_{t=1}^{T} \\phi(x_t)^T\\phi(x_t) \\\\\nA & = \\frac{1}{T} \\sum_{t=1}^{T} \\phi(x_t)^T\\phi(x_{t+1})\n\\end{align*}\n\n\\subsubsection{Basis} In this work, we require that the finite basis $\\phi$ \\textit{includes both the state and control variables}~(\\cite{proctor2016generalizing}).  This ensures that the Koopman operator models both the natural dynamics of the mechanical system and the control dynamics as provided by the user demonstration.   In this work we empirically select a fixed set of basis functions to ensure that all models (across the different users in our validation study) are learned using the same basis.  Here we choose $\\phi$ such that\n\\begin{align}\n\\begin{split}\n\\phi = &[1, x_1, x_2, x_3, x_4, x_5, x_6, u_1, u_2, u_1*x_1, u_1*x_2, \\\\\n& u_1*x_3, u_1*x_4, u_1*x_5, u_1*x_6, u_2*x_1, u_2*x_2, \\\\\n& u_2*x_3, u_2*x_4, u_2*x_5, u_2*x_6, u_1*cos(x_3), \\\\\n& u_1*sin(x_3), u_2*cos(x_3), u_2*sin(x_3)].\n\\end{split}\n\\label{eq-basis}\n\\end{align}\nThese 25 basis functions were chosen to combine information about the geometry of the task (e.g., the trigonometric functions capture specific nonlinearities present in the system dynamics, see Section~\\ref{sec-experimental-environment}) with information related to how the user responds to system state. For this reason, we include terms that mix state information with control information.  To evaluate the accuracy of the learned approximation to the Koopman operator we compute the H-step prediction accuracy (see Figure~\\ref{fig-h-step-accuracy}).\n\nThere are, of course, a variety of methods that one could use to select an appropriate basis for a given dynamical system.  This step is particularly important as selecting a poor basis will quickly degrade the validity of the learned model~(\\cite{berrueta2018dynamical}).  One such method is to integrate known information about the system dynamics into the chosen basis functions, such as the relationship between the heading of the lander and the motion generated by the main thruster.  This approach works well when the system dynamics are easy to understand, however it can prove challenging when the dynamics are more complex.  For this reason, one could also choose the set of basis functions through purely data-driven techniques.  Sparsity Promoting DMD~(\\cite{jovanovic2014sparsity}) is one such algorithm.  SP-DMD takes a large initial set of randomly generated basis functions and imposes an $\\ell_1$ penalty during the learning process to algorithmically decides which basis functions are the most relevant to the observable dynamics~(\\cite{tibshirani1996regression}). An example of this purely data-driven approach being applied to human-machine systems can be found in~\\cite{broad2019highly}.\n\n\\subsection{Autonomous Policy Generation}\n\nTo generate an autonomous control policy, we can integrate the portion of the learned model that relates to the system and control dynamics into a model predictive control (MPC) algorithm.  In particular, we use Koopman operator model-based control~(\\cite{broad2017learning, abraham2017model}), which we detail now in full.  To compute the optimal control sequence, $u$, we must solve the following Model Predictive Control (MPC) problem\n\\begin{equation}\n\\begin{aligned}\n& \\underset{u}{\\text{minimize}}\n& & J = \\sum_{t=0}^{T-1} l(x_t,u_t) + l_T(x_T) \\\\\n& \\text{subject to}\n& & x_{t+1} = f(x_t, u_t), \\\\\n& & & u_t \\in U, x_t \\in X, \\forall t\n\\end{aligned}\n\\label{eqn-mpc}\n\\end{equation}\n\n\\noindent where $f(x_t,u_t)$ is the system dynamics, $l$ and $l_T$ are the running and terminal cost, and $U$ and $X$ are the set of valid control and state values, respectively.\n\nIn this work, we define\n\\begin{align*}\nl(x_t, u_t) &= \\frac{1}{2} (x_t-x_d) Q_t (x_t-x_d) + \\frac{1}{2} u_t R_t u_t \\\\\n\\textnormal{where }Q_t &= Diag[6.0, 10.0, 20.0, 2.0, 2.0, 3.0]\n\\end{align*}\n$x_t$ is the current state and $x_d$ is the desired goal state.  Additionally,\n\\begin{align*}\nl_T(x_T) &= \\frac{1}{2} (x_t-x_d) Q_T (x_t-x_d) \\\\\n\\textnormal{where }Q_T &= Diag[3.0, 3.0, 5.0, 1.0, 1.0, 1.0]\n\\end{align*}\nThese values were chosen empirically based on results observed from the system operating fully autonomously.\n\nTo integrate our learned system model, we re-write the system dynamics as such:\n\\begin{equation}\n\\phi(x_{t+1}) = f_\\mathcal{K}(x_t, u_t)\n\\label{eqn-koopman-dyn}\n\\end{equation}\n\n\\noindent where $f_\\mathcal{K} = \\mathcal{K}^T\\phi(x_t, u_t)$ is the learned system dynamics parameterized by a Koopman operator $\\mathcal{K}$.  This equation demonstrates the fact that the Koopman operator does not map directly from state to state, but rather operates on functions of state.  We can then evaluate the evolved state by recovering the portion of the basis that represents the system's state\n\\begin{equation}\nx_{t+1} = \\phi(x_{t+t})_{1:N}\n\\label{eqn-koopman-recover}\n\\end{equation}\n\\noindent where values $1:N$ are the state variables, as per our definition in Equation~\\eqref{eq-basis}, and $N$ is the dimension of the state space.  The policy generation process is depicted in Figure~\\ref{fig-pipeline}\\textcolor{red}{(b)}.\n\n\\subsubsection{Nonlinear Model Predictive Control Algorithm}\n\nWe solve Equation~\\eqref{eqn-mpc} with Sequential Action Control~(\\cite{ansari2016sequential}) (SAC).   SAC is a real-time, model-based non-linear optimal control algorithm that is designed to iteratively find a single value, a time to act, and a duration that maximally improves performance.  Other viable nonlinear optimal control algorithms include iLQR~(\\cite{li2004iterative}) and DDP~(\\cite{mayne1966second, tassa2014control}).  SAC is particularly well suited for our shared control algorithm because it searches for single, short burst actions which aligns well with our control allocation algorithm (described in detail in Section~\\ref{sub-sec-control-allocation}).   Additionally, SAC can compute closed-loop trajectories very quickly (1 kHz), an important feature for interactive human-machine systems such as the one presented in this work. \n\n\\subsubsection{Integrating the Koopman model and SAC}\nSequential Action Control is a gradient-based optimization technique and it is therefore necessary to compute derivatives of a system during the optimization process.  The linearization of the discrete time system is defined by the following equation\n\\begin{equation*}\nx_{t+1} = A  x_t +  B u_t.\\\\\n\\end{equation*}\n\n\\noindent By selecting a differentiable $\\phi$, one can compute $A$ and $B$  \n\\begin{equation}\n\\begin{aligned}\nA &= \\mathcal{K}_{1:N}^T \\frac{\\partial \\phi}{\\partial x}\\\\\nB &= \\mathcal{K}_{N:N+P}^T \\frac{\\partial \\phi}{\\partial u}\n\\label{eqn-linear-model-a_and-b}\n\\end{aligned}\n\\end{equation}\n\n\\noindent where $N$ is again the dimension of the state space, and $P$ is the dimension of the control space.  \n\n\\subsection{Control Allocation Method}\n\\label{sub-sec-control-allocation}\n\nTo close the loop on our shared control paradigm, we define a control allocation method that uses the solution from the optimal control algorithm to provide outer-loop stabilization.  We use a geometric signal filter that is capable of dynamically shifting which partner is in control at any given instant based on optimality criteria.  This technique is known as Maxwell's Demon Algorithm (MDA)~(\\cite{tzorakoleftherakis2015controllers}).  Our specific implementation of MDA is detailed in Algorithm~\\ref{mda-algorithm} where $u_h$ is the control input from the human operator, $u_a$ is the control produced by the autonomy, and $u$ is applied to the dynamic system.  \n\\begin{algorithm}[!h]\n\\caption{Maxwell's Demon Algorithm (MDA)}\n\\begin{algorithmic}\n  \\If {$\\langle u_h, u_a \\rangle \\geq 0$} \n   \\State $u = u_h$;\n  \\Else\n   \\State $u = 0$;\n  \\EndIf\n\\end{algorithmic}\n\\label{mda-algorithm}\n\\end{algorithm}\n\nWe also provide a pictorial representation of the algorithm in Figure~\\ref{fig-mda}.\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=\\hsize]{mda.png}\n\t\\caption{Maxwell's Demon Algorithm (MDA)}\n\t\\label{fig-mda}\n\\end{figure}\n\nThis control allocation method restricts the user's input to the system to be in the same half-plane as the optimal control solution, and places no other limitations on the human-machine interaction.  If the user's input is in the opposite half-plane, no input is provided to the system.  This control allocation method is lenient to the human partner, as notably, \\textit{the autonomous agent does not add any information into the system} and instead only blocks particularly bad input from the user.  Therefore, \\textit{any signal sent to the system originates from the human partner}.  We use this filter because we are motivated by assistive robotics, in which prior research has shown that there is no consensus across users on desired assistance level~(\\cite{erdogan2017effect}).  By allowing the user a high level of control freedom, the system encourages input from the human operator and restricts how much authority is granted to the autonomous partner.  This method is depicted in Figure~\\ref{fig-pipeline}\\textcolor{red}{(c)}. We use MDA in this work primarily because it has been experimentally validated in prior studies on human-machine systems for assistive robotics~(\\cite{fitzsimons2016optimal}).  Notably, this method does not guarantee optimal (or even \"good\") performance, as a human operator could theoretically always provide input orthogonal to the autonomous solution, resulting in no control ever being applied to the system.  However, this technique does allocate a large amount of control authority to the human-in-the-loop, a desirable feature in our motivating application domains.  There are also alternative methods that can be used for similar purposes, including extensions to MDA that incorporate additional information from the autonomous partner, which can be used to improve performance or safety~(\\cite{broad2018operation}).  A review paper of alternative control allocation techniques can be found in~\\cite{losey2018review}.\n\n\\section{Human Subjects Study}\n\\label{sec-experimental-validation}\n\nHere, we detail the experimental setup that we use to study three main aspects of the described system.  \n\\begin{itemize}\n\\item First, our aim is to evaluate the efficacy of model-based shared control as it relates to task success and control skill.  Concurrently, we aim to evaluate the generalizability of the learned system models with respect to a wide range of human operators.\n\\item Second, we aim to evaluate the efficacy of model-based shared control under an online learning paradigm---specifically, the sample-efficiency of the Koopman operator representation.\n\\item Finally, we aim to evaluate the impact of nonlinear modeling and policy generation techniques through a comparison to a second human-subjects study that enforces linear constraints on our model-based shared control algorithm.\n\\end{itemize}\n\n\\subsection{Experimental Environment}\n\\label{sec-experimental-environment}\n\nThe proposed shared control framework is evaluated using a simulated lunar lander (see Figure~\\ref{fig-ll-env}).  \n\n\\begin{figure}[!h]\n\t\\centering\n\t\\fbox{\\includegraphics[width=0.85\\hsize]{environment.png}}\n\t\\caption{Simulated lunar lander system.  The green circle is the goal location.  The red dots represent an engine firing.}\n\t\\label{fig-ll-env}\n\\end{figure}\n\n\\noindent We use a simulated lunar lander (rocket) as our experimental environment for a number of reasons.  This environment is challenging for a novice user, but performance can be improved (and sometimes mastered) given enough time and experience.  Similar to a real rocket, one of the main control challenges is the stability of the system.  As the rocket rotates along its yaw axis, firing the main thruster can produce nonintuitive dynamics for a novice.  Furthermore, once the rocket has begun to rotate, momentum can easily overwhelm a user who is unfamiliar with such systems.  Therefore, it is often imperative---particularly for non-expert users---to maintain a high degree of stability at all times in order to successfully complete the task.  In addition to the control challenges, we choose this environment because the simulator abstracts the system dynamics through calls to the Box2D physics engine; therefore, we do not have an exact model and thus have an \\textit{explicit need to learn one}.\n\n\\subsection{System Description}\n\\label{sub-sec-system-description}\n\nThe dynamic system is a modified version of an open-source environment implemented in the Box2D physics engine and released by OpenAI~(\\cite{brockman2016gym}).  Our modifications (1) allow for continuous-valued multi-dimensional user control via a joystick, and (2) incorporate the codebase into the open-source ROS framework.  We have made our code available online at \\url{https:\/\/github.com\/asbroad\/model_based_shared_control}.\n\nThe lunar lander is defined by a 6D state space made up of the position ($x,y$), heading ($\\theta$), and their rates of change ($\\dot{x}, \\dot{y}, \\dot{\\theta}$).  The control input to the system is a continuous two dimensional vector ($u_1, u_2$) which represents the throttle of the main and rotational thrusters.  The main engine can only apply positive force.  The left engine fires when the second input is negative, while the right engine fires when the second input is positive.  The main engine applies an impulse that acts on the center of mass of the lunar lander, while the left and right engines apply impulses that act on either side of the rocket.  We remind the reader that our goal is to learn both the system dynamics and user interaction.  For this reason, we must collect data both on the system state and also the control input.  Together, this defines an eight dimensional system: \n\\begin{equation*}\n\\mathcal{X} = [x, y, \\theta, \\dot{x}, \\dot{y}, \\dot{\\theta}, u_1, u_2]\n\\end{equation*}\n\n\\noindent where the first six terms define the lunar lander state and $u_1, u_2$ are the main and rotational thruster values, through which the user interacts with the system.\n\n\\subsection{Trial Description}\n\\label{sec-experiment-trial-description}\n\nThe task in this environment requires the user to navigate the lander from its initial location to the goal location (represented by the green circle in Figure~\\ref{fig-ll-env}) and to arrive with a heading nearly perpendicular to the ground plane and with linear and rotational velocities near zero.  A trial is considered complete either (1) when the center of an upright lunar lander is fully contained within the goal circle (i.e., the Euclidean distance between the center of the lander and the center of the goal is less than $0.9$ m) and the linear and angular velocities are near zero (i.e., the linear velocities must be less than $1.0$ m\/s and the angular velocity must be less than $0.3$ rad\/sec), or (2) when the lander moves outside the bounded environment (i.e., when the lander moves off the screen to the left or right) or crashes into the ground.\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=0.85\\hsize]{control_interface.jpeg}\n\\end{figure}\nIn each trial, the lunar lander is initialized to the same $x, y$ position ($10.0$ m, $13.3$ m), to which we added a small amount of Gaussian noise ($\\mu = 0.2$ m).  Additionally, a random force is applied at the start of each trial (uniform($-1000$ N,$1000$ N)).  The goal location ($10.0$ m, $6.0$ m) is constant throughout all trials and is displayed to the user as a green circle (see Figure~\\ref{fig-ll-env}).  \n\nThe operator uses a PS3 controller to interact with the system.  The joystick controlled by the participant's dominant hand fires the main thruster, and the opposing joystick fires the side thrusters.  As the user moves through the environment, we keep track of the full state space at each timestep (10 Hz).  We provide a video of the system, task and user interaction under shared control as part of the supplementary material.\n\n\\subsection{Analysis I : Efficacy and Generalizability of Model-based Shared Control}\n\\label{sub-sec-analysis-i}\n\n\\subsubsection{Control Conditions}\n\nTo study the efficacy and generalizability of our shared control system and the generalizability of the learned system dynamics, we compare four distinct control conditions.\n\n\\begin{itemize}\n\\item In the first condition, the user is in full control of the lander and is not assisted by the autonomy in any way; we call this approach \\textit{User Only} control.  As each user undergoes repeated trials with the same goal, this can also be considered a natural learning paradigm.  \n\\end{itemize}\n\nIn the remaining three conditions an autonomous agent provides outer-loop stabilization on the user's input as described in Section~\\ref{sec-model-based-shared-control}.  The main distinction between these three control conditions is the source of the data used to compute the model of the joint system.  \n\\begin{itemize}\n\\item In the second condition, the model is defined by a Koopman operator learned on data captured from earlier observations of the current user; we call this approach \\textit{Individual Koopman}.  \n\\item In the third condition, the model is defined by a Koopman operator learned on data captured from observations of three novice participants prior to the experiment (who were not included in our analysis); we call this approach \\textit{General Koopman}.\n\\item In the fourth condition, the model is defined by a Koopman operator learned on data captured from observations of an expert user (the first author of the paper, who has significant practice controlling the simulated system); we call this approach \\textit{Expert Koopman}.\n\\end{itemize}\n\nWe analyze the viability of model-based shared control by comparing the \\textit{User Only} condition to each of the shared control conditions.  We analyze the generalizability of the learned models by comparing the results from the \\textit{Individual Koopman}, \\textit{General Koopman} and \\textit{Expert Koopman} conditions.  All of the data we analyze to evaluate the ideas presented in this section comes from the nonlinear MbSC study.\n\n\\subsubsection{Protocol and Participants}\n\nEach experiment begins with a training period for the user to become accustomed to the dynamics of the system and the interface. This training period continues until the user is able to successfully achieve the task three times in a row or 15 minutes elapses.  During the next phase of the experiment, we collect data from 10 user-controlled trials, which we then use to develop a model.  Finally, each user performs the task under the four conditions detailed above (10 trials each).  The order in which the control paradigms are presented to the user is randomized and counter-balanced to reduce the effect of experience.\n\nThe study consisted of 16 participants (11 female, 5 male).  All subjects gave their informed consent and the experiment was approved by Northwestern University's Institutional Review Board. \n\n\\subsection{Analysis II : Online Model-based Shared Control}\n\\label{sub-sec-analysis-ii}\n\nTo study the efficacy of our model-based shared control algorithm in an online learning paradigm, we collect data from a fifth experimental condition, which we call \\textit{Online Koopman}.  \n\n\\subsubsection{Control Condition}\n\n\\begin{itemize}\n\\item The main difference between the \\textit{Online Koopman} paradigm and the three previously described shared control conditions is that the model of the joint human-machine system is learned online in real-time.  In all other control conditions, all models were trained offline from observations gathered during a data collection phase.  In the online paradigm, the model is updated continuously starting with the first collected set of observations.\n\\end{itemize}\n\nIn addition to the lack of a separate data collection phase, the online learning paradigm is distinct from the other shared control conditions because of the data that we use to learn the model.  In the shared control conditions that use a model learned offline, we use all of the observations collected from the user demonstrations to learn the model.  In the online learning paradigm we only update the model when the user input is admitted by the MDA controller.  We choose this learning paradigm because it fits well conceptually with our long term goal of using the outer-loop algorithm to provide stability and safety constraints on the shared control system.  It is important to note that at the beginning of the online learning paradigm, the MDA controller relies on randomly initialized control and system dynamics models.  For this reason, the control let through to the system will be very noisy during the first few moments of the experiment, making the system difficult to control successfully for any human-in-the-loop.  For this reason, it is important that the system dynamics and control models can be learned quickly, something we evaluate in Section~\\ref{sec-study-two-results-online}.  As soon as the learning process produces a model of any kind, the policy is computed using MPC techniques.\n\n\\subsubsection{Protocol and Participants}\n\\label{sec-sub-sub-protocol}\n\nThe online learning paradigm consists of 15 trials per user to allow us to evaluate possible learning curves.  The model is updated at the same rate as the simulator (10 Hz) and is initialized naively (i.e., all values are sampled from a uniform distribution [0,1)).  This paradigm is presented as the final experimental condition to all subjects.  The subjects are the same 16 participants as in Section~\\ref{sub-sec-analysis-i}. All of the data we analyze to evaluate the ideas presented in this section comes from the nonlinear MbSC study.\n\n\\subsection{Analysis III : Comparison of Linear and Nonlinear Model-based Shared Control}\n\\label{sub-sec-analysis-iii}\n\nTo study the impact of nonlinear modeling and policy generation techniques on our model-based shared control paradigm, we compare results from the above study to a second study (consisting of a separate group of 16 participants) that enforces linear constraints on these parts of the system.  \n\n\\subsubsection{Control Conditions}\n\nThe same four control conditions from Analysis I are evaluated.  The differences lie in (1) the choice of basis function used to approximate the Koopman operator and (2) the choice of optimal control algorithm used to generate the autonomous policy.  In this study, we use a linear basis, instead of a nonlinear basis, to approximate the Koopman operator, which consists of the first nine terms in Equation~\\eqref{eq-basis}.  We furthermore use a Linear Quadratic Regulator, instead of a nonlinear model predictive control (MPC) algorithm (Sequential Action Control (SAC)).  \n\n\\subsubsection{Protocol and Participants}\n\nThe same experimental protocol described in Section~\\ref{sub-sec-analysis-i} was used, allowing us to perform a direct comparison between the two studies.  Unlike the prior sections, the data we analyze to evaluate the ideas presented in this section comes from both the linear and nonlinear MbSC studies.  The data from the linear MbSC study was previously analyzed in~(\\cite{broad2017learning}) and was collected from a different set of 16 subjects, resulting in 32 total participants.\n\n\\subsection{Statistical Analysis} \n\\label{sec-statistical-analysis}\n\nWe analyze the results of the human-subjects studies using statistical tests to compare the performance of participants along a set of pertinent metrics under the control conditions described in Section~\\ref{sub-sec-analysis-i}.  Our analysis consists of one-way ANOVA tests conducted to evaluate the effect of the shared control paradigm on each of the dependent variables in the study.  These tests allow us to statistically analyze the effect of each condition while controlling for overinflated type I errors that are common with repeated t-tests.  Each test is computed at a significance value of 0.05.  When the omnibus F-test produces significant results, we conduct post-hoc pair-wise Student's t-tests using Holm-Bonferroni adjusted alpha values~(\\cite{wright1992adjusted}).  The post-hoc t-tests allow us to further evaluate the cause of the significance demonstrated by the ANOVA by comparing each pair of control paradigms separately.  Similar to the ANOVA test, the Holm-Bonferroni correction is used to reduce the likelihood of type I errors in the post-hoc t-tests.  \n\nIn addition to reporting the results of the statistical tests, we also use box-and-whisker diagrams to display specific metrics.  In these plots, the box represents the \\textit{interquartile range (IQR)} which refers to the data that lies between the first and third quartiles.  This area contains 50$\\%$ of the data.  The line inside the box represents the median value and the whiskers above and below the box are the minimum and maximum values inside 1.5 times the interquartile range.  The small circles are outliers.  The plots also depict the results of the reported statistical tests.  That is, if a post-hoc t-test finds statistically significant differences between two conditions, we depict these results on the box-and-whisker diagrams using asterisks to represent the significance level ($*: p < 0.05, **: p < 0.01, ***: p < 0.005$). \n\nWe note that this analysis is used for \\textit{all reported results}.  Therefore, if we present the results of a t-test, it signifies that we have previously run an ANOVA and found a statistically significant difference.  The reader can also assume that any unreported post-hoc t-tests failed to reject the null hypothesis.  \n\n\\section{Results}\n\\label{sec-results}\n\nWe now present the results of the desired analyses described in Sections~\\ref{sub-sec-analysis-i}, ~\\ref{sub-sec-analysis-ii}, and ~\\ref{sub-sec-analysis-iii}.  Our analyses support the premise that model-based shared control is a valid and effective data-driven method for improving a human operator's control of an \\textit{a priori} unknown dynamic system.  We also find the learned system models are generalizable across a population of users.  Finally, we find that these models can be learned online in a fast, data-efficient manner.\n\n\\subsection{Efficacy of Model-based Shared Control}\n\\label{sec-results-performance-metrics}\n\nTo evaluate the efficacy of our model-based shared control algorithm, we compute the average success rate under each control paradigm and examine the distribution of executed trajectories.  Our analysis compares the User Only control condition to each of the shared control conditions (Individual Koopman, General Koopman and Expert Koopman). All of the data we analyze to evaluate the ideas presented in this section comes from the nonlinear MbSC study.\n\n\\subsubsection{Task Success and User Skill}\n\nA trial is considered a success when the user is able to meet the conditions defined in Section~\\ref{sec-experiment-trial-description}.  We can interpret the success rate of a user, or shared control system, on a set of trials as a measure of skill.  The greater the skill, the higher the success rate.  By comparing the average success rate under the User Only control paradigm with the average success rate under the shared control paradigms, we can analyze the impact of the assistance provided by the autonomous partner.  \n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.8\\hsize]{results_success_nonlinear.png} \n  \\caption{Number of successful trials under each control condition.}\n  \\label{fig-results-success-nonlinear}\n\\end{figure}  \n\n\\begin{figure*}[t]\n\t\\centering\n\n\t\\includegraphics[width=0.8\\hsize]{trajectories.png}\n\t\\caption{Trajectory plots which visualize the most frequently visited parts of the state space.  The data is broken down by control condition (columns) and whether the trial was successful (rows).  The plots are generated by overlaying each trajectory with a low opacity and the intensity of the plots therefore represents more frequently visited portions of the state space.}\n\t\\label{fig-heatmaps}\n\\end{figure*}\n\nThe average number of successful trials produced in each control condition are displayed in Figure~\\ref{fig-results-success-nonlinear}. An analysis of variance shows that the choice of control paradigm has a significant effect on the success rate ($F(3, 59) = 4.58, p < 0.01$). Post-hoc t-tests find that users under the shared control conditions show statistically significant improvements in the success rate when compared to their performance under the User Only control condition ($p < 0.01$, for all cases).  No other pairings are found to be statistically distinct.  This result demonstrates that the assistance provided by the autonomous agent significantly improves the skill of the joint human-machine system, thereby validating the efficacy of model-based shared control.  This result is inline with related work that aims to provide similar assistance, such as can be found in the virtual fixtures\/guides literature~(\\cite{forsyth2005predictive, griffiths2005sharing, abbink2012haptic}).  Importantly, however, unlike these prior methods, model-based shared control does not require \\textit{a priori} knowledge of the system dynamics or the human operator.\n\nThe fact that there are no observed differences in task performance between the Individual, General and Expert cases suggests that the source of the data used to learn the model may not be important in developing helpful autonomous assistance in the shared control of dynamic systems (discussed further in Section~\\ref{sec-results-generalizability}).  An alternative interpretation of this data could be that the discrepancy of skill demonstrated by the participants in the individual, general and expert cases was not large enough to produce any potential difference in performance.  This, however, is not likely, as the expert demonstrator (the lead author) was able to easily achieve the desired goal state during every demonstration (i.e. 10 out of 10 trials).  In contrast, the average subject who provided data in the individual and general cases performed similarly to how participants performed under the User Only cases in Figure~\\ref{fig-results-success-nonlinear} (i.e. about 1 in 10 successful demonstrations).\n\n\\subsubsection{Distribution of Trajectories---Qualitative}\n\nWe further analyze the different control conditions through a comparison of the distribution of trajectories we observe in each condition.  Unlike the success metric, this analysis is not based on task performance, and is instead performed to evaluate the control skill exhibited by either the human operator alone or the joint human-machine system.  Figure~\\ref{fig-heatmaps} depicts trajectory plots which represent the most frequently occupied sections of the state space.  The plots are generated using data separated based on the control condition (columns) and whether the user was able to complete the task on a given trial (rows). \n\nThe first distinction we draw is between the User Only control condition and the three shared control conditions.  In particular, the distribution of trajectories in the User Only condition depicts both larger excursions away from the target and lower levels of similarity between individual executions.  When we focus specifically on which parts of the state space users spend the most time in (as represented by the intensity of the plots), we see two main clusters of high intensity (around the start and goal locations) in the shared control conditions, whereas we see a wider spread of high-intensity values in the User Only control condition.  This suggests more purposeful motions under the shared control conditions.\n\nThe second distinction we draw focuses on a comparison between the successful and unsuccessful trials.  Specifically, we note that trajectory plots computed from the failed trials under the shared control conditions demonstrate similar properties (e.g., the extent of the excursions away from the target, as well as two main clusters of intensity) to the trajectory plots computed from successful trials under the shared control conditions.  This suggests that users may have been closer to succeeding in these tasks than the binary success metric gives them credit for.  By comparison, the trajectory plot computed from the failed trials under the User Only control condition depicts a significantly different distribution of trajectories with less structure.  Specifically, we observe numerous clusters of intensity that represent time spent far away from the start and goal locations.  This suggests that users were particularly struggling to control the system in these cases.\n\n\\subsubsection{Distribution of Trajectories---Quantitative}\n\nThese observations are supported by an evaluation of the ergodicity~(\\cite{mathew2011metrics}) of the distributions of trajectories described above.  We find users under the shared control paradigm are able to produce trajectories that are more ergodic with respect to the goal location then users under User Only control, which means that they spend more a significantly larger proportion of their time navigating near the goal location under shared control.\n\nTo perform this comparison, we compute the ergodicity of each trajectory with respect to a probability distribution defined by a Gaussian centered at the goal location (which represents highly desirable states).  This metric can be calculated as the weighted Euclidean distance between the Fourier coefficients of the spatial distribution and the trajectory~(\\cite{miller2016ergodic}).\n\nSimilar to our qualitative analysis of the trajectory plots in Figure~\\ref{fig-heatmaps}, we first compare ergodicity between the different control conditions by analyzing \\textit{all} the trajectories observed under each condition.  An analysis of variance showed that the effect of the shared control paradigm on trajectory ergodicity is significant ($F(3, 640) = 12.97, p < 0.00001$).  Post-hoc t-tests find statistically significant differences between the performance of the users in the User Only control condition and users in the shared control conditions based on the individual, general and expert datasets ($p < 0.0005, p < 0.001, p < 0.0005$, respectively).  No other pairings demonstrate statistically distinct results. We interpret this result as additional evidence that model-based shared control improves the skill of the human partner in controlling the dynamic system.\n\nWe further analyze the ergodicity results by separating the trajectories based on whether they come from an unsuccessful or successful trial.  An analysis of variance computed over all control conditions showed that the effect of the shared control paradigm on trajectory ergodicity is significant for both unsuccessful ($F(3, 310) = 6.60, p < 0.0005$) and successful ($F(3, 325) = 7.20, p < 0.0005$) trials.  Post-hoc t-tests find statistically significant differences between the performance of the users in the User Only control condition and users in the shared control conditions ($p < 0.005$ in all unsuccessful cases, $p < 0.05$ in all successful cases).  No other pairings reject the null hypothesis.  These results suggest that the shared control paradigm is helpful in improving the user's skills even when they provide input that is ultimately unsuccessful in achieving the task.  Furthermore, our shared control paradigm is helpful, even when users are performing at their best.  Thus, for both failed and successful trials, users exhibit a greater amount of control skill than when there is no assistance. \n\n\\subsection{Generalizability of Shared Control Paradigm}\n\\label{sec-results-generalizability}\n\nWe continue the evaluation of our human subjects study with an analysis of the generalizability of the learned system models and our model-based shared control algorithm.  All of the data we analyze to evaluate the ideas presented in this section comes from the nonlinear MbSC study.  As reported in Section~\\ref{sec-results-performance-metrics}, we find no statistical evidence that the source of the data used to train the model impacts the efficacy of the shared control paradigm.  This test was again conducted using an ANOVA which can be used to evaluate differences between groups by comparing the mean and variance computed from the data collected during the experimental trials. When we compare the success rate of users in each shared control condition, we find no statistically significant difference.  However, we do find a significant difference between the user's performance under each shared control condition and the User Only condition.  The same result holds when we compare each control condition along the ergodic metric described in Section ~\\ref{sec-results-performance-metrics} and visualized by trajectory plots in Figure~\\ref{fig-heatmaps}.  Taken together, these results suggest that the efficacy of the assistance provided by the autonomous agent is \\textit{independent of the source of the data used to learn a model of the joint system}.  That is, models trained on data collected from an individual user generalize to a larger population of human partners. \n\n\n\n\\begin{figure}[h]\n  \\centering\n  \\captionsetup[subfigure]{justification=centering}\n  \\begin{subfigure}[t]{0.49\\hsize}\n  \t\\centering\n  \t\\includegraphics[width=\\hsize]{results_agreement_2_main_by_user.png} \n  \t\\caption{}\n  \t\\label{fig-results-agreement-main}\n  \\end{subfigure}\n  \\begin{subfigure}[t]{0.49\\hsize}\n    \\centering\n    \\includegraphics[width=\\hsize]{results_agreement_2_side_by_user.png} \n    \\caption{}\n    \\label{fig-results-agreement-side}\n  \\end{subfigure}\n  \\caption{Average agreement between user and optimal control algorithm as defined by the Maxwell's Demon Algorithm (Equation~\\eqref{mda-algorithm}) along the (\\subref{fig-results-agreement-main}) main and (\\subref{fig-results-agreement-side}) side thrusters.}\n  \\label{fig-results-agreement}\n\\end{figure}  \n\nTo further analyze the generalizability of the model-based shared control paradigm, we evaluate the participants' interactions with the outer-loop autonomous control.  We are interested in whether or not users agree more often with the autonomy when control signals are produced based on models learned from their personal demonstration data.  To evaluate this idea, we look at the percentage of user inputs that are let through as control to the dynamic system based on our control allocation method (MDA).  The average agreement metric is broken down by control condition and presented in Figure~\\ref{fig-results-agreement}.  \n\nAn analysis of variance shows that the effect of the source of the model data on the average agreement is not significant in either the main thruster ($F(2, 44) = 0.87, p = 0.43$) nor the side thruster ($F(2, 44) = 0.38, p = 0.69$).  These results show a uniformity in the response to system state across users and suggest that the system is able to adapt to the person, instead of requiring a personalized notion of the user and system.  \n\nWe interpret this finding as further evidence of the generalizability of our model-based shared control paradigm.  In particular, we find that it is not necessary to incorporate demonstration data from individual users when developing model-based shared control.  This result replicates findings from our analysis of data collected under a shared control paradigm that enforced a linear constraint on the model learning and policy generation techniques~(\\cite{broad2017learning}).\n\n\\subsection{Online Learning Shared Control}\n\\label{sec-study-two-results-online}\n\nWe next evaluate our model-based shared control algorithm in an online learning paradigm.  Our evaluation considers the sample complexity of our model-based learning algorithm through a comparison of the \\textit{impact each shared control paradigm has on the skill of the joint system over time}.  All of the data we analyze to evaluate the ideas presented in this section comes from the nonlinear MbSC study.  Our statistical analysis is a comparison of the percent of participants who succeed under each paradigm \\textit{by trial number}, shown in Figure~\\ref{fig-res-success-by-trial}.  We remind the reader that users participate in 15 trials of the Online Koopman condition while they participate in 10 trials of the four other experimental conditions.  For comparison we only plot the first 10 trials of the Online Koopman data, though we note that the improved success rate is sustained over the final five trials.  From this plot, we can see that users in the Online Koopman shared control condition start off performing poorly, but by around trial 7 start performing on par with the other shared control conditions.  \n\nHere, we also note the number of trials used to train the model of the system and control dynamics in each condition.  In the Individual and Expert conditions, data is collected from 10 trials to train the model.  In the General condition, data is collected from three different users who each control the system for 10 trials each, which means the model is trained from a total of 30 trials.  Finally, as discussed above, in the Online condition, the model is learned continuously over the course of 15 trials.\n\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[width=\\hsize]{success_by_trial_by_paradigm_dark.png} \n    \\caption{Average percentage success by trial for the first 10 trials by control condition.  Users under all shared control conditions using models learned offline (Individual, General, Expert) outperform the User Only control condition across all trials.  Users under the shared control condition using models learned online (Online) start off performing poorly, but quickly begin to outperform the User Only control condition and, in the end, achieve the same level of success as those under the offline shared control conditions.}\n    \\label{fig-res-success-by-trial}\n\\end{figure}\n\nTo provide quantitative evidence of this visual trend, we perform the same types of statistical analyses as in previous sections, but now include data from the \\textit{Online Koopman} as a fifth experimental condition.  For ease of discussion we refer to the \\textit{Individual}, \\textit{General} and \\textit{Expert Koopman} model-based shared control conditions as the offline learning conditions, and the \\textit{Online Koopman} model-based shared control as the online learning condition.  As users provide more data in the Online Koopman condition than in all other conditions, we perform two sets of analyses.  First, we compare the data from the first ten trials from the Online Koopman condition to all other control conditions.  We then re-perform the same tests, but use the final ten trials from the Online Koopman condition.  By comparing these results, we can evaluate the efficacy of the online learning paradigm, and also analyze the effect of the amount of data used during the learning process. \n\n\\subsubsection{Statistical Analysis of the First Ten Trials}\n\\label{sec-study-two-results-online-first-ten}\n\nAn analysis of variance finds a statistically significant difference between the various control conditions along the primary success metric ($F(4, 74) = 5.35, p < 0.001$).  Post-hoc t-tests find that all offline learning conditions significantly outperform the Online Koopman and User Only control conditions ($p < 0.05$ for all cases).  We do not find the same statistically significant difference between the User Only and Online Koopman conditions.  These results suggest that users under the online learning paradigm initially perform on par with how they perform under the user only control paradigm, but worse than under the offline control conditions.  This analysis is consistent with our expectations since, in the online condition, the model of the joint system is initialized randomly and therefore does a poor job of assisting the user.  However, it is also important that this online shared control does not degrade performance in comparison to the User Only paradigm, suggesting that there is little downside to employing the online learning paradigm during learning.  \n\n\\subsubsection{Statistical Analysis of the Final Ten Trials}\n\\label{sec-study-two-results-online-last-ten}\n\nAs a point of comparison, we now re-run the same statistical tests using the final ten trials from the Online Koopman condition.  An analysis of variance finds a statistically significant difference between the various control conditions along the primary success metric ($F(4, 74) = 3.55, p < 0.05$) (see Figure~\\ref{fig-avg-metrics-second-online-study}).  Post-hoc t-tests find that all shared control conditions (using models learned offline and online) significantly out perform the User Only control paradigm ($p < 0.01$ for all conditions).  This result is different from our analysis of the first ten trials and suggests that the learned model improves significantly with more data and now is on par with the models learned in the offline conditions.  No other pairings show statistically significant differences.  \n\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[width=0.8\\hsize]{results_success_online.png} \n    \\caption{Number of successful trials under each control condition (including an online learning paradigm).  We find statistically significant differences between the User Only condition and each shared control condition ($p < 0.01$).}\n    \\label{fig-avg-metrics-second-online-study}\n\\end{figure}\n\nThe visual trend present in Figure~\\ref{fig-res-success-by-trial} and the statistical analysis demonstrated in Figure~\\ref{fig-avg-metrics-second-online-study} suggest that the Koooman operator is able to quickly learn an actionable representation of the joint human-machine system.  These results also demonstrate the efficacy of our model-based shared control algorithm in an online learning scenario and in limited data regimes.  Here we note that follow-up studies are required to tease apart the impact of the model learning process and the user's experience controlling the dynamic system when comparing the offline paradigms to the online paradigm.  Notably, in the offline learning paradigm, users undergo 10 trials of training at the start of the experiment.  In contrast, and as stated in Section~\\ref{sec-sub-sub-protocol}, all users operate the system under the online learning paradigm as the final condition.  For this reason, we do not account for user experience in this condition and therefore highlight the data-efficiency of the model learning process instead of the overall task performance in this condition.  The main takeaway from this portion of the analysis is therefore that an actionable Koopman operator can be learned \\textit{quickly}, from significantly less data than alternative approaches commonly found in the literature, like neural networks~(\\cite{nagabandi2018neural}).\n\n\\subsection{Linear and Nonlinear Model-based Shared Control}\n\nThe final piece of analysis we perform in this work is related to the impact that nonlinear modeling and policy generation techniques have on our model-based shared control paradigm.  For this analysis we compare the User Only control condition to the three offline shared control conditions.  Unlike the prior sections, the data we analyze to evaluate the ideas presented in this section comes from both the linear and nonlinear MbSC studies.\n\n\\begin{figure}[h]\n  \\centering\n  \\captionsetup[subfigure]{justification=centering}\n  \\begin{subfigure}[t]{0.49\\hsize}\n  \t\\centering\n  \t\\includegraphics[width=\\hsize]{results_success_linear.png} \n  \t\\caption{Linear MbSC.}\n  \t\\label{fig-results-comparison-success-linear}\n  \\end{subfigure}\n  \\begin{subfigure}[t]{0.49\\hsize}\n  \\centering\n  \\includegraphics[width=\\hsize]{results_success_nonlinear.png} \n  \\caption{Nonlinear MbSC.}\n  \\label{fig-results-comparison-success-nonlinear}\n  \\end{subfigure}\n  \\caption{A comparison of the average success rate under (a) linear and (b) nonlinear model-based shared control to user only control.}\n  \\label{fig-results-comparison-success}\n\\end{figure}  \n\n\nThe average success rate of users under each control paradigm for both studies is presented in Figure~\\ref{fig-results-comparison-success}.  In the linear study~(\\cite{broad2017learning}) we observe a trend (see Figure~\\ref{fig-results-comparison-success-linear}) that suggests users perform better under the shared control paradigm, but we do not find statistically significant evidence of this observation.  In contrast, we find that model-based shared control using nonlinear modeling and policy generation techniques does statistically improve the success rate when compared to a User Only control paradigm.\n\n\nOne potential explanation for the difference we find in the results of the two studies is that the nonlinear basis produces more accurate models of the system dynamics then the linear basis.  To explore this explanation, we evaluate the predictive capabilities of a Koopman operator learned with a linear basis to one learned with a nonlinear basis.  This analysis is performed by comparing the predicted system states with ground truth data.  We evaluate the error (mean and variance) as a function of prediction horizon (a.k.a. the H-step error).  Figure~\\ref{fig-h-step-accuracy} depicts the raw error (in meters) of Koopman operators trained using linear and nonlinear bases.  \n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=0.9\\hsize]{h_step_model_prediction_accuracy_joint.png}\n\t\\caption{H-step Prediction Accuracy of Koopman operator models based on linear and nonlinear bases.  Error is computed as a combination of the Euclidean distance between the predicted $(x, y)$ values and the ground truth $(x, y)$.}\n\t\\label{fig-h-step-accuracy}\n\\end{figure}\n\nOur analysis of the predictive capabilities of the Koopman operator models demonstrates that each is highly accurate.  The nonlinear model does slightly outperform the linear model as the prediction horizon grows; however, we find that both models are able to produce single-step predictions with error on the scale of $10^{-3}$ meters.  As a reminder to the reader, the state space is bounded with $X \\in (-10, 10), Y \\in (0, 16)$.  This analysis suggests that the choice of basis function does not cause the observed difference in average success rate between the two studies.  Instead, the important design decision may be the choice of model predictive control algorithm.  In the linear study we use an infinite horizon LQR to produce autonomous control policies, whereas in the nonlinear study we use a receding-horizon Model Predictive Control (MPC) to produce autonomous control.  Our interpretation of these results is that the receding horizon nature of MPC is better suited to the visual planning approach that human operators use when solving the lunar lander task.\n\n\\section{Discussion}\n\\label{sec-discussion}\n\nIn this section, we highlight a number of main takeaways that stem from our analysis.  To begin, the results of our human-subjects studies demonstrate that our model-based shared control paradigm is able to (1) successfully learn a model of the joint human-machine system from observation data, and (2) use the learned system model to generate autonomous policies that can help assist a human partner achieve a desired goal.  We evaluate the predictive capabilities of the learned system models through a comparison to ground truth trajectory data (see Figure~\\ref{fig-h-step-accuracy}) and evaluate the impact of the assistive shared control system through a comparison of performance (success rate, see Figure~\\ref{fig-results-success-nonlinear}) with a User Only (or natural learning) control paradigm.  All analyses support the idea that MbSC can help improve the control skill of a human operator both when they are able to achieve a task on their own and when they are not.\n\nAdditional evaluations demonstrate that the learned system and control dynamics generalize across users, and suggests that, unlike in other human-machine interaction paradigms~(\\cite{sadigh2016information, macindoe2012pomcop, javdani2015shared}), personalization is not required for defining shared control paradigms of generic human-machine systems.  Specifically, we find that the demonstration data used to learn the system and control models does not need to come from an optimal, or expert, controller, and can instead come from \\textit{any} human operator.  Therefore, at a base level, the controller does not need to be personalized to each individual user as the learned model captures all necessary information.  This idea is important for application in real-world scenarios where personalization of control paradigms can be time-consuming, costly, and challenging to appropriately define, often due to the variety in preferences described by human operators~(\\cite{gopinath2016human, erdogan2017effect}).\n\nWe also demonstrate that our approach can be used in an online learning setting.  Importantly, we find that the model is able to learn very quickly, from limited amounts of data.  In the Online Koopman condition, each trial took an average of 18 seconds, and therefore provided 180 data points. From our analysis in Section~\\ref{sec-study-two-results-online-last-ten}, we find that we are able to learn an effective model of the joint system after only 5 trials (or about 900 data points). Our model learning technique is also well suited for an online learning paradigm as it is not computationally intensive and can easily run at 50Hz on a Core i7 laptop with 8 GB of RAM.  Additionally, we find that, even during the learning process, the application of the online model-based shared control algorithm does not significantly degrade the performance of the human operator.\n\nFinally, we also evaluate the impact that nonlinear modeling and policy generation techniques have on our model-based shared control algorithm~(\\cite{broad2017learning}).  In particular, we replace the nonlinear modeling and policy generation techniques with linear counterparts and compare how they impact the ability of a human operator to achieve a desired task.  This requires using a nonlinear basis when computing the approximation to the Koopman operator and using nonlinear model predictive control (SAC) to generate the autonomous policy.  We find that the nonlinear model-based shared control paradigm produces a joint human-machine system that is significantly better along the primary performance metric (task success) then users under a user only control paradigm.  The same result is not found from the data collected under a shared control paradigm that enforced linear constraints (see Figure~\\ref{fig-results-comparison-success}).\n\n\\section{Conclusion}\n\\label{sec-conclusion}\n\nIn this work, we introduce model-based shared control (MbSC).  A particularly important aspect of this work is that \\textit{we do not rely on a priori knowledge, or a high-fidelity model, of the system dynamics}.  Instead, we learn the system dynamics \\textit{and} information about the user interaction with the system directly from data.  We learn this model through an approximation to the Koopman operator, an infinite dimensional linear operator that can exactly model non-linear dynamics.  By learning the joint system dynamics through user interaction, the robot's understanding of the human is implicit to the system definition.\n\nResults from two human subjects studies (consisting of 32 total participants) demonstrate that incorporating the learned models into our shared control framework statistically improves the performance of the operator along a number of pertinent metrics.  Furthermore, an analysis of trajectory ergodicity demonstrates that our shared control framework is able encourage the human-machine system to spend a significantly greater percentage of time in desirable states.  We also find that the learned system models are able to be used in shared control systems that generalize across a population of users. Finally, we find that, using this approach, models can be efficiently learned online.  In conclusion, we believe that our approach is an effective step towards shared control of human-machine systems with unknown dynamics.  This framework is sufficiently general that it could be applied to any robotic system with a human in the loop.  Additionally, we have made our code available online at \\url{https:\/\/github.com\/asbroad\/model_based_shared_control}, and include a video depicting a user's control of the dynamic system and the impact of model-based shared control in the supplementary material.\n\n\\begin{acks}\nThis material is based upon work supported by the National Science Foundation under Grant CNS 1329891. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the aforementioned institutions.\n\\end{acks}\n\n\\bibliographystyle{SageH}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nAerial radiometric survey is a mature field.  Successful prospecting for\neconomically viable ore deposits using the radiation signal from naturally occuring\nrocks stretches back decades~\\cite{Grasty_1975}.\nSurvey systems composed of large volumes of NaI(Tl) scintillator gamma-ray detectors, as much as 20~L,\ncoupled with georeferenced position sensors (now making use of the global\nnavigation satellite system (GNSS)), record gamma energy spectra versus\nposition.  This information is later processed to produce maps of natural\npotassium, uranium and thorium concentrations.  Standards exist to guide\npractitioners in this area~\\cite{IAEA_1991,IAEA_1995} and vast regions of the\nearth have been covered~\\cite{IAEA_2010}.\nPractitioners have also developed methods to correct for terrain variation in\naerial survey~\\cite{Schwarz1992,ISHIZAKI201782}.\n\nThe emphasis in aerial radiometric survey methods until recently has been on\ndevelopment of techniques suitable for geologic sources, for which the\nsimplification of the source as an infinite and uniform sheet is reasonable in\ncomparison with the distance scales of the survey parameters (altitude, line\nspacing).  The higher an aircraft flies, the more that far-away locations\ncontribute to the detected signal, relative to locations directly underneath\nthe aircraft~\\cite{King_1912}.  This can have the\nadvantage of allowing for complete coverage in a more economical survey with\nwider line spacing.  However, detection systems at higher altitude see a signal\nwhich is effectively averaged over a larger area of the ground.  Anthropogenic\nsignals such as those resulting in case of a\nreactor accident or malicious radiological dispersal could result in hot spots\nthe concentration of which would be underestimated if averaged over a larger area.\n\nIn this paper, we present a method to deconvolve an aerial radiometric survey\nfor spatial smearing.  This kind of problem, requiring inversion of a spatial\ndistribution, is encountered frequently in geophysical surveying.  \nGeophysical spatial\ninversion problems are typically underdetermined, and one way of dealing\nwith this has been to select only those solutions which are close to some\npreconceived model~\\cite{Parker_2015}.  An approach to spatial\ndeconvolution of airborne spectrometric data which relies on an analytical\nmodel for the response function, and allows underdetermined problems, has been\npublished previously~\\cite{Minty_2016}.  A related method for spatial\ninversion to the approach presented here, but using an iterative inversion and\nneglecting uncertainties, was published recently~\\cite{Penny_2015}.  Other\ngroups are taking a similar approach to that advocated here, but applied to\nthe inversion of spectra rather than spatial\nmaps~\\cite{Mattingly_2010, Hussein_2012}.\n\nThe method which will be presented here was applied to data obtained using\ndetectors aboard a manned helicopter.  Nevertheless, it is prudent to mention\nthe proliferation of work ongoing currently in aerial radiation detection\nfrom unmanned aerial vehicle (UAV) systems.  The use of a UAV platform for \naerial survey has facilitated the advance of \naerial survey methodology.\nA good review of recent publications can be found here~\\cite{Connor_2016}.\n\nThe method which will be presented here involves simply a) determining the\ninfluence of each independent region of the earth's surface on the measured spatial\ndistribution and then b) optimizing the weight coefficients of each region of\nthe surface to obtain the best reproduction of the measured map.  The number of\npixels of the solution can be chosen such that the problem is not underdetermined.  No\npotentially biasing prior assumption about the underlying distribution is\nnecessary.  The method can handle complicated detector geometries as the\nresponse functions are determined in simulation.  The method could easily be\nextended to allow it to be applied when there is significant\nterrain variation in the source such as would be the case in an urban\nenvironment.  The method is stable under different starting\nconditions, and naturally allows for propagation of uncertainties from the\nmeasurement to the unfolded result.  \n\nWe demonstrate the application of the unfolding method by applying it first to\na synthetic data set.  This is compared with the known underlying distribution. \nWe proceed to apply\nthe unfolding method to a real aerial survey measurement acquired following\ndetonation of a radiological dispersal device~\\cite{Sinclair_RDD_2015}.\n\nThis spatial deconvolution method has been\npresented previously at conferences by this\ngroup~\\cite{RDD_CTBT_2015,NSSMIC_RDD_2015}, however, this is the first full write-up.\n\n\\section{Methods}\n\\subsection{Unfolding method}\n\\label{sec:unfolding_method}\nA measurement of surface activity concentration under the uniform infinite\nplane assumption may be denoted $g^{\\mbox{\\scriptsize MEAS}}(x,y)$ where $x$\nand $y$ represent easting and northing in geographic Cartesian coordinates.\nWe seek to determine the true underlying surface radioactivity concentration, $f(x,y)$.  $f(x,y)$ is related to  $g^{\\mbox{\\scriptsize MEAS}}(x,y)$ through\n\\begin{equation} \ng^{\\mbox{\\scriptsize MEAS}}(x,y) = S [f(x,y)],\n\\end{equation}\nwhere $S$ represents the effect of the measurement system on $f(x,y)$.\n\nWe divide space into $N^{\\mbox{\\scriptsize PAR}}$ pixels $i$, and using Monte Carlo simulation, generate uniform radioactivity distributions in each pixel, $f_i(x,y)$.\n\nThe measurement system $S$ consists of the detection system as well as the air\nand all other absorbing and scattering materials between the source and the\nproducts of scattering, and the detectors.  It is represented in the Monte\nCarlo simulation, and the emissions from the radioactive sources $f_i(x,y)$ are transported\nthrough the system $S$ to obtain the template responses\n$g_i(x,y)$, where\n\\begin{equation}\ng_i(x,y) = S[f_i(x,y)].\n\\end{equation}\n\nWe let \n\\begin{equation}\ng(x,y) = \\sum_{i=1}^{N^{\\mbox{\\tiny PAR}}} w_i g_i(x,y)\n\\end{equation}\nand fit $g(x,y)$ to $g^{\\mbox{\\scriptsize MEAS}}(x,y)$ using a $\\chi^2$ minimization~\\cite{Minuit} to extract the weighting coefficients $w_i$.\n\nTo examine this $\\chi^2$ function, let $g_j^{\\mbox{\\scriptsize MEAS}}(x,y)$ represent the $j$th measurement of the activity concentration $g^{\\mbox{\\scriptsize MEAS}}(x,y)$.  Then\n\\begin{equation}\n\\chi^2 = \\sum_{j=1}^{N^{\\mbox{\\tiny MEAS}}}\\frac{g_j^{\\mbox{\\scriptsize MEAS}}(x,y) - \\sum_{i=1}^{N^{\\mbox{\\tiny PAR}}} w_i g_i(x,y)}{e_j^2}\n\\end{equation}\nwhere there are $N^{\\mbox{\\scriptsize MEAS}}$ measurements $g_j^{\\mbox{\\scriptsize MEAS}}(x,y)$ in the problem each with uncertainty $e_j$.\n\nThe estimator of $f(x,y)$ is then the reconstructed radioactivity\nconcentration distribution $f^{\\mbox{\\scriptsize REC}}(x,y)$, where\n\\begin{equation}\nf^{\\mbox{\\scriptsize REC}}(x,y) = \\sum_{i=1}^{N^{\\mbox{\\tiny PAR}}} w_i f_i(x,y).\n\\end{equation}\n\nWe choose to require the problem to be oversampled.  That is, there is\neverywhere a greater spatial density of measurements than of fit parameters\nand $N^{\\mbox{\\scriptsize MEAS}} > N^{\\mbox{\\scriptsize PAR}}$.\nThen, provided that the uncertainties $e_j$ in the denominator of the\n$\\chi^{2}$ function encompass all of the uncertainties of the problem, the\nminimum $\\chi^2$ value will be approximately equal to the number of degrees\nof freedom of the problem.  \nAnd the MINOS algorithm~\\cite{Minuit} can be used to propagate the\n$N^{\\mbox{\\scriptsize MEAS}}$ measurement uncertainties $e_j$ through the fit procedure\nto calculate\nthe $N^{\\mbox{\\scriptsize PAR }}$ uncertainties \n$\\delta w_i$ on the weighting parameters $w_i$.\nIn practice, there are irreconcilable nonstochastic uncertainties affecting the problem which\nmust be included in the\n$e_j$ by application of a constant scaling factor\nto bring $\\chi^{2}$ per degree of freedom to one before the fit uncertainties\ncan be utilized.  These are due to the statistical uncertainties\nin the template responses $g_i(x,y)$, and the finite pixellization of the problem.\n\nUncertainties for spatial deconvolution of fallout surveys can be expected to be asymmetric owing to the\nboundary condition that the amount of deposition can not physically be less\nthan zero.  MINOS works by setting the positive and negative uncertainty for\neach parameter to the amount the parameter has to vary in each direction such\nthat $\\chi^{2}$ increases by one.  Thus MINOS naturally allows for assymetric\nuncertainties and is particularly suited to uncertainty analysis in the\nmeasurement of amount of radioactivity.\n\n\\subsection{Experimental method -- aerial survey}\n\\label{sec:exp_method}\nThe experimental methods to obtain the data to which we will apply the unfolding method have been described previously~\\cite{Sinclair_RDD_2015}.  We will repeat only the most essential points here.\nThree RDD detonations were conducted during the trial.  In the first, $\\sim$31~GBq of \\mbox{La-140} was dispersed explosively, with radioactive debris subsequently carried by wind as far as $\\sim$~2~km from the site of the detonation.\nAerial gamma-ray spectrometric surveys were conducted using two\n10.2~x~10.2~x~40.6~cm$^{3}$ NaI(Tl) crystals mounted exterior to a helicopter\nin a skid-mounted cargo expansion basket.  GNSS antennae and inertial navigation and altimetry systems\nwere also installed in the basket, to determine location.\nThe system recorded a linearized 1024-channel gamma-ray energy spectrum over the domain 0~MeV to 3~MeV, tagged with the location information, once per second.\nPost-acquisition, counts were selected from the spectra in an energy window of approximately four sigma in width around the 1.6~MeV \\mbox{La-140} photopeak. \nThese count rates were corrected for lag and dead time.  Count rates were\nalso corrected for variations in\nflight altitude to the nominal flying height making use of the Shuttle Radar\nTopography Mission~\\cite{g.2007shuttle} digital terrain model\nadjusted to the local elevation using a GNSS base station.\nBackgrounds due to the radioactivity of the earth, the\natmosphere, the aircraft and its occupants, and cosmic rays, were all\nsubtracted.  The count rates were all corrected for radioactive decay, back to\nthe time of the blast.  A coefficient to convert the measurements from counts\nper second to kBq\/m$^2$, assuming an infinite and uniform source, was obtained\nfrom experimentally validated Monte Carlo simulation.  \nFinally, measurements of radioactivity concentration in kBq\/m$^2$ for four\naerial surveys, two conducted after the first blast and one conducted after\neach of the subsequent two blasts, were presented.\n\nIn this paper, we will discuss only the data recorded during the first aerial\nsurvey after the first blast.\nThis survey was flown at a nominal 40~m flying height, with speed of\n25~m\/s and flight-line spacing of 50~m.\n\n\\subsection{Experimental method -- truckborne survey}\n\\label{sec:method_truck}\n\n\\subsubsection{Data collection -- truckborne survey}\nTruckborne surveys were driven by criss-crossing the deposited fallout in an\nextemporaneous pattern following\nthe first and third RDD blasts, restricting to the part of the fallout outside\nof a 500~m x 500~m fenced hot zone~\\cite{Green_RDD_2015,Marshall_thesis_2014}.  \nThe detection system was mounted in the bed of a pickup truck and consisted of four 10.2~x~10.2~x~40.6~cm$^{3}$\nNaI(Tl) crystals oriented vertically in a self-shielding arrangement for\nazimuthal direction measurement.\nTruckborne data following the first RDD blast will be presented here for\ncomparison with the aerial survey data.  The truckborne data has not undergone\nsufficient analysis for a full quantitative evaluation, but the shape will\nnevertheless provide an interesting comparison for interpretation of the aerial survey data.\n\n\\subsubsection{Sensitivity calculation -- truckborne survey}\n\nThe sensitivity of the truckborne system to a \\mbox{La-140} contamination on\nthe ground was determined using experimentally validated Monte Carlo\nsimulation.\nIn the simulations, the detector was placed with the centre of its sensitive\nvolume at a height of 1.2~m above ground.  \nThe NaI(Tl) crystals and their housing were represented in the simulation in \ntheir vertical arrangement. \nThe $\\sim 5000$~kg of mass of the pick-up\ntruck carrying the detector was represented by simple blocks of steel.\n\nSensitivity validation data was collected by placing a \\mbox{La-140} source of\nknown emission rate at fixed locations around the truck and detector system.\nThe dead material was adjusted in size and position in the model until an acceptable agreement with the\nsensitivity validation data was obtained.\nThe engine and other materials at the front of the truck effectively block\nradiation coming from that direction, and most of the detected counts arise\nfrom radiation originating to the side and rear of the truck where there is\ncomparatively little material.\nThe uncertainty in the estimation of the sensitivity of the truckborne\ndetector obtained in this manner is large, about 20\\% to 30\\%.\nThis level of accuracy is sufficient for illumination of the value of the spatial deconvolution applied to the aerial\ndata by comparison with data collected from a ground-based system.\n\nFig.~\\ref{fig:truck_sensitivity}~a) shows the number of energy deposits\nregistered in the detector per second as a function of the radius $R$ of a\ndisc-shaped source centered beneath the truckborne detector on the ground as\ndetermined by the simulation.\n\\begin{figure}[h]\n   \\begin{center}\n   \\begin{tabular}{c}\n   \\begin{overpic}[trim = .1cm .1cm .1cm .1cm, clip = true, height=4.7cm]{2crystal_sensivity_25.eps}\\put(18,52){\\textcolor{black}{a)}}\\end{overpic}\n   \\hspace*{-.1cm}\n   \\begin{overpic}[trim = .1cm .1cm .1cm .1cm, clip = true, height=4.7cm]{trk_scaled_25.eps}\\put(18,52){\\textcolor{black}{b)}}\\end{overpic}\n   \\end{tabular}\n   \\end{center}\n  \\caption\n \n  { \\label{fig:truck_sensitivity} \na) Sensitivity of the truckborne survey system to a disc-shaped distribution of isotropic emitters of the \\mbox{La-140} gamma spectrum, as a function of disk radius.  Black dots show EGSnrc prediction.  Solid curve shows fit of the expression $C(R,H)$ to the synthetic data.  The dashed line shows the asymptote of the fit curve and represents the sensitivity to a uniform and infinite sheet source.\nb) Comparison of the shapes of the sensitivity curves for the aerial and truckborne survey systems.  The dashed line shows the sensitivity to a disc source relative to the sensitivity to an infinite sheet as a function of disc radius for an aerial survey system at an altitude of 40~m~\\cite{Sinclair_RDD_2015}.  The solid line shows the equivalent relative sensitivity curve for the truckborne survey system.\n}\n\\end{figure} \nThe expression for the flux, $\\Phi(R,H)$, due to a surface activity concentration $S_0$ at a point an elevation $H$ above a disc-shaped source of radius $R$ can be readily calculated~\\cite{King_1912},\n\\begin{equation}\n  \\Phi(R,H) = \\frac{S_0}{2} (E_1(\\mu_{\\mbox{\\scriptsize a}}H) - E_1(\\mu_{\\mbox{\\scriptsize a}}\\sqrt{H^2+R^2})),\n\\label{eqn:flux_vs_R}\n\\end{equation}\nwhere E$_1$ is the exponential integral and $\\mu_{\\mbox{\\scriptsize a}}$ is the linear attenuation coefficient for gamma rays in \nair.\nTo determine the asymptotic sensitivity, we formed a function for the detected\ncounts as a function of the source radius, \n\\begin{equation}\n   C(R,H) = \\epsilon \\Phi(R,H),\n\\label{eqn:count_rate}\n\\end{equation}\nand fit the expression for $C(R,H)$ to the synthetic data to obtain the\ndetection efficiency $\\epsilon$.\nThe fit result is shown as the solid curve in\nFig.~\\ref{fig:truck_sensitivity}a) and the asymptotic sensitivity, shown by\nthe dashed line, is $\\sim 71$~s$^{-1}$\/(kBq\/m$^2$).\nAs mentioned, the uncertainty on this sensitivity is large due to the lack of\ndetailed representation of the shielding material of the truck in the\nsimulation.\nNevertheless, the shape of the sensitivity curve is of value, as is the shape\nof the profile\nof counts measured with the truckborne system as it traversed the deposited plume.\n\n\\subsubsection{Comparison of aerial and truckborne sensitivity curves}\nFig.~\\ref{fig:truck_sensitivity}b) shows a comparison of the shapes of the\nsensitivity curves of the ground-based and aerial systems.  Despite the tall\nnarrow shape of its detectors, which would tend to increase sensitivity to incoming radiation\nfrom the sides, for the truckborne system at ``altitude'' $H=1.2$~m, a greater\npercentage of detected gamma rays originate close to the point directly\nunderneath the detector as compared to the airborne\nsystem at $H=40$~m.\nThis leads to superior spatial precision in the results of truckborne survey.\n\n\\subsection{Aerial survey template response determination through Monte Carlo simulation}\n\\label{sec:simulation}\nThe radiation transport model EGSnrc~\\cite{EGSnrc1,EGSnrc2} was used to\ngenerate the individual uniform pixel sources $f_i(x,y)$ and to propagate the\ngenerated gamma rays and their interaction products through air and into the detection volume to create the\ntemplate responses $g_i(x,y)$.  For the solutions presented herein, the\nsimulation geometry represented the experimental setup during the first RDD\ntrial~\\cite{Sinclair_RDD_2015}.  The actual aerial survey system was described and shown\nin photographs in the previous publication and briefly reiterated in \nSect.~\\ref{sec:exp_method}.\nThe model of that system in EGSnrc is shown in\nFig.~\\ref{fig:egspp}~a).  The simulated gamma detection system included the\ntwo NaI(Tl) crystals, as well as their aluminum cladding, and felt, foam and\ncarbon fibre enclosure.  The exterior-mounted basket containing the detectors\nwas modelled in a simplified manner with 51 3~mm~x~1.5~mm bars of aluminum,\nrepresenting the basket strands, running the length of the basket in a\nsemicircle around its long axis.  Dead materials due to the photo-multiplier\ntube readout of the crystals and associated electronics, as well as the\naltimeter and GNSS receivers, were modelled as simple blocks of metal of the\nappropriate overall mass.  The ground was represented as perfectly flat with\nthe detection system at a height of 40~m.  \nThe model was validated experimentally using data from point sources of known\nemission rate placed at\nknown locations around the actual detector.  The uncertainty associated with approximation in the representation of the\nmeasurement system in the model was evaluated by variation of the arrangement\nof dead material in the model, by variation of the detector's position,\naltitude and\nattitude and by variation of the detector's energy resolution within\nreasonable limits.  This uncertainty was evaluated to be about 12\\% on the \nactivity\n  concentration and it was included the overall systematic\nuncertainty quoted in the  publication~\\cite{Sinclair_RDD_2015}.\n\n\\begin{figure}[h]\n   \\begin{center}\n   \\begin{tabular}{c}\n   \\begin{overpic}[trim=0cm -5cm 0cm 3cm, clip = true, height=5.1cm]{basket_model_4.eps}\\put(5,89){\\textcolor{white}{a)}}\\end{overpic}\n   \\hspace{1.4cm}\n   \\begin{overpic}[height=5.5cm]{K_53_resp.eps}\\put(18,70){\\textcolor{white}{b)}}\\end{overpic}\\\\\n   \\end{tabular}\n   \\end{center}\n  \\caption\n \n  { \\label{fig:egspp} \na) The aerial survey system as modelled in EGSnrc.  The two\n10.2~x~10.2~x~40.6~cm$^{3}$ NaI(Tl) crystals\nare represented in their housing, shown in purple, with steel blocks at the\nends representing the dead material of the PMTs and readout electronics.  The\ntwo crystals are mounted lengthwise with their centres 152.4~cm apart on an aluminium plate which is \nrepresented with mass and dimensions according to the engineering drawing.\nAuxilliary instrumentation is represented by an\naluminium block in the centre of the basket, of the summed instrument mass.\nThis dead material is shown in green in the figure.  The aluminum plate on\nwhich the detectors and auxiliary equipment is mounted is itself attached to a\nbasket which is mounted to\nthe skids on the outside of the helicopter.  Dead material of the basket is represented by 51 thin aluminum bars running the length of the basket, in a\nsemicircle around the basket long axis, shown in gold.  Uncertainty on the\nsensitivities due to misrepresentation of the system in the model has been\nestimated to be approximately 12\\%.\nb) Response function, $g_{53}(x,y)$ to the pixel source $f_{53}(x,y)$,\nnormalized to the number of generated events, where the $x$-axis shows Easting\nand the $y$-axis shows Northing.  The true spatial extent of pixel 53 is indicated by the black square.}\n\\end{figure}  \n\nThe simulated pixel sources, $f_i(x,y)$, consisted of uniform distributions of\nisotropic emitters of the \\mbox{La-140} spectrum of gamma rays, including all\nemission energies above 0.05\\% emission probability~\\cite{TabRad_v1}.  Gammas\nwere emitted into the full 4$\\pi$ solid angle.  Each pixel source $f_i(x,y)$\nwas square, and 50~m on a side.  Note that with the survey parameters\nmentioned previously, we have one spectrum recorded approximately every\n25~m~x~50~m in the real data.  Thus, the number of measurements in the data is\nhigher than the number of fit parameters in the simulation, the problem is over-determined, and we can expect reasonably rapid convergence of the method to a solution which is stable under different starting conditions.\n\nThe entire region to be unfolded measured 1.5~km~x~1.5~km.  To speed up\nconvergence of the fitting, we chose to parallelize the computation, breaking\nthe problem up into individual tiles, each 500~m~x~500~m.  Only eight of these\ntiles were necessary to cover the area over which the radioactivity actually\nsettled.  To knit the tiles together, we extended the fit into a border of\npixel sources 50~m wide, surrounding each tile.  Thus the fit area\ncorresponding to each tile was actually 600~m~x~600~m.  If, after background\nsubtractions, fewer than one count corresponding to a \\mbox{La-140} energy\ndeposit was measured in the detection system in the region of space\ncorresponding to one fit parameter $w_i$, then that fit parameter was assigned\na value of zero and left out of the minimization.  Thus, 144 or fewer\nfit parameters $w_i$ were determined from the inversion of each tile.\nTo merge the tiles after the individual inversions, the weighting parameters of the border pixels were simply ignored, and the central 500~m~x~500~m areas of the tiles were placed next to each other.\n\nThe simulation included one detection system for each of the 144 pixel sources, centered\nlaterally within the pixel, at 40~m height.  \nHere we have used multiple detection systems at different places at one time to\nrepresent the real world in which one detection system was in different places\nat different times.  Given the 40~m height of the detection systems above the\nsource, the 50~m spacing between them and the requirement that the deposited\nenergy lie in the highest-energy photopeak of the source, the error in this\napproximation, which would come from a single emitted gamma ray depositing energy in two\ndifferent detection systems, is negligible.\n\nThe volume of the air in which the gamma rays and electrons were tracked in each\ntile extended 1.2~km in easting (or $x$) and northing (or $y$), and 600~m up.\nA 5~m-thick concrete slab underneath the emitters and filling the lateral\ndimensions of the simulated volume, represented the earth.\n\nThe simulation included all physical interactions of the emitted gammas and of\nthe gammas, electrons and positrons resulting from those interactions.  Scattering\nand absorption in the air and ``earth'' of the simulated volume, and in the dead material\nof the basket system and housing of the NaI(Tl) crystals, was included.  All\nprocesses leading to energy deposit within the NaI(Tl) crystal from either the\ndirect gamma rays, or the products of scattering, were included.  Energy\ndeposits in the NaI(Tl) were then smeared to create simulated spectra,\naccording to the energy resolutions of the crystals determined during the experiment.\n\nFig.~\\ref{fig:egspp}~b) shows the response, $g_{53}(x,y)$, of the 144\ndetection systems of one tile, as a percentage of the number of events\ngenerated in a single one of the pixel sources, $f_{53}(x,y)$, where this\nhappens to correspond to the pixel numbered ``53''.  The extent of the source\nis indicated by the black square.  Note how the measured response extends much\nmore broadly in space.  This is the spatial smearing which will be corrected\nby the unfolding.\n\n\\section{Results}\n\n\\subsection{Results obtained by application of the method to synthetic data}\nWe begin by applying the spatial inversion method to a synthetic data set for\nwhich we know the underlying distribution $f(x,y)$.  \nAn annular region consisting of the area between two circles of radius 100~m\nand 250~m, centered at (550~m, 550~m) was uniformly populated with\n10~$\\cdot$~10$^{9}$ emitters of the \\mbox{La-140} gamma sectrum.  Considering the\nbranching ratios for the \\mbox{La-140} gamma emissions, this amounts to a source\nconcentration of 28.4~kBq\/m$^2$.  The annular region\nis indicated by the black outlines in Fig.'s~\\ref{fig:annulus_data_and_fit}~a)\nand~b).\n\\begin{figure}[h]\n   \\begin{center}\n   \\begin{tabular}{c}\n   \\begin{overpic}[trim = .1cm .1cm .1cm .1cm, clip = true, height=6.5cm, angle=270]{gdconc_K_giant_fast_99.eps}\\put(18,70){\\textcolor{white}{a)}}\\end{overpic}\n   \\hspace{.4cm}\n   \\begin{overpic}[trim = .1cm .1cm .1cm .1cm, clip = true, height=6.5cm, angle=270]{hDFIT_GIANT_ALL.eps}\\put(18,70){\\textcolor{white}{b)}}\\end{overpic}\\\\\n   \\end{tabular}\n   \\end{center}\n  \\caption\n \n  { \\label{fig:annulus_data_and_fit} \nSynthetic aerial survey and fit results, where the $x$\naxis is Easting, and the $y$ axis is Northing.  The source was of concentration \n28.4~kBq\/m$^2$ and\nannular with inner radius 100~m and outer radius 250~m, centered at\n(550~m, 550~m), as indicated by the area between the black circles.\na) Synthetic aerial survey measurement result.\nb) Result of fit of template responses $g_i(x,y)$ to the synthetic aerial survey.}\n\\end{figure}  \n\nGeneration of the synthetic dataset makes use of the\nidentical detector simulations as used in generating the template \nresponses $g_i(x,y)$ as described in Section~\\ref{sec:simulation}, however the detection systems were more narrowly spaced\nin the synthetic dataset.\nDetection systems for the synthetic dataset were located every 20~m in $x$ and\n$y$ such that there were 225 detection systems in total in each 600~m~x~600~m\ntile.\n\nThe template sources $f_i(x,y)$ and the template responses $g_i(x,y)$ utilized\nin the inversion are the same as will be used for the real data and are as\ndescribed in Section~\\ref{sec:simulation}.\nThus, the synthetic data measurement density of 20~m~x~20~m exceeds the\ndensity of the parametrization, 25~m~x~25~m, and the problem\nis overdetermined, as desired.\n\nFig.~\\ref{fig:annulus_data_and_fit}~a) shows the synthetic aerial survey\nmeasurement.  The result is broader than the underlying true source\ndistribution.  Contamination appears to extend outside of the known true\nunderlying borders.  The central area of the annulus appears to be filled with\nsignificant contamination.  The average concentration of contamination\nreconstructed within the annular region is lower than the known true concentration.  \n\nFig.~\\ref{fig:annulus_data_and_fit}~b) shows the result of the fit of the\ntemplate measured activity distributions to the synthetic aerial survey\nmeasurement.  The colour scale used in Fig.~\\ref{fig:annulus_data_and_fit}~b) is\nthe same as that of Fig.~\\ref{fig:annulus_data_and_fit}~a).\nThe first observation to note is that the tile knitting procedure\napparently works well.  The synthetic data stretches over six of the\noverlapping 600~m x\n600~m simulation tiles.  After knitting of the inverted result into adjacent \nnon-overlapping\n500~m~x~500~m tiles, there is seamless matching of the reconstructed activity\nconcentration at the tile borders.\nAlso to note is that within the limitations of the somewhat coarse pixellization of\nthe problem, the survey is well reproduced by the fit.  \nThe tendency of the\nmeasurement to extend outside of the bounds of the true source is reproduced,\nas is the tendency of the measurement to underestimate the magnitude of the\nconcentration within the source boundary.\n\nThe reconstruction of the true underlying source distribution for the\nsynthetic data is presented in Fig.~\\ref{fig:annulus_inverted}.\n\\begin{figure}[h]\n   \\begin{center}\n   \\begin{tabular}{c}\n   \\begin{overpic}[trim = .1cm .1cm .1cm .1cm, clip = true, height=6.5cm, angle=270]{hCONC_GIANT_ALL.eps}\\put(18,65){\\textcolor{white}{a)}}\\end{overpic}\\\\\n   \\begin{overpic}[trim = .1cm .1cm .1cm .1cm, clip = true, height=6.5cm, angle=270]{hpERR_GIANT_ALL.eps}\\put(18,65){\\textcolor{white}{b)}}\\end{overpic}\n   \\hspace{.4cm}\n   \\begin{overpic}[trim = .1cm .1cm .1cm .1cm, clip = true, height=6.5cm, angle=270]{hmERR_GIANT_ALL.eps}\\put(18,65){\\textcolor{white}{c)}}\\end{overpic}\\\\\n   \\end{tabular}\n   \\end{center}\n  \\caption\n \n  { \\label{fig:annulus_inverted} \nSpatially deconvolved synthetic aerial survey result, $x$ axis is Easting and\n$y$ axis is Northing.  Black circles indicate\nthe true annular source distribution.\na) The spatially deconvolved measurement.\nb) Positive statistical uncertainty on the spatially deconvolved measurement.\nc) Negative statistical uncertainty on the spatially deconvolved measurement.}\n\\end{figure}  \nAs shown in Fig.~\\ref{fig:annulus_inverted}~a), the inversion process results\nin a reconstructed source distribution which is higher in magnitude and closer\nto the true activity concentration than the\ninitial survey measurement.  The boundaries of the actual source are \nbetter reproduced after inversion, in particular the absence of radioactivity\nin the centre of the annulus is recovered.\n\nA major advantage of the spatial\ndeconvolution method presented here is that statistical uncertainties affecting\nthe measurement may be propagated through the minimization procedure by the MINOS algorithm as described\nin Sect.~\\ref{sec:unfolding_method}.\n A map\nshowing the one-sigma positive statistical uncertainty on the reconstructed surface activity\nconcentration is shown in\nFig.~\\ref{fig:annulus_inverted}~b) and the corresponding negative statistical uncertainty\nis shown in Fig.~\\ref{fig:annulus_inverted}~c) where the same colour scale is\nused for the uncertainties as for the measurement shown in \nFig.~\\ref{fig:annulus_inverted}~a).\nConsidering the uncertainties, the ability of the method to reconstruct the\ntrue underlying activity concentration is good.  The reconstructed activity\nconcentration magnitude is in agreement with the known true activity\nconcentration within uncertainties in most places.  For example, near the\ninner boundary of the annulus where the reconstructed concentration is low\ncompared to the known true concentration, further negative movement of the\nresult is not allowed by the negative uncertainty.  The positive uncertainty exceeds the\nnegative uncertainty in magnitude, and does allow for a positive movement of the\nreconstructed value.\n\n\\subsection{Real data collected following detonation of a radiological\n  dispersal device}\n\\subsubsection{Spatial deconvolution of aerial survey data}\n\nThe aerial survey measurement from the first RDD trial is shown in Fig.~\\ref{fig:data_and_fit}~a).\n\\begin{figure}[h]\n   \\begin{center}\n   \\begin{tabular}{c}\n   \\begin{overpic}[trim = .1cm .1cm .1cm .1cm, clip = true, height=6.5cm, angle=270]{gdconc_99_1.eps}\\put(18,70){\\textcolor{white}{a)}}\\end{overpic}\n   \\hspace{.4cm}\n   \\begin{overpic}[trim = .1cm .1cm .1cm .1cm, clip = true, height=6.5cm, angle=270]{hDFIT_ALL.eps}\\put(18,70){\\textcolor{white}{b)}}\\end{overpic}\\\\\n   \\end{tabular}\n   \\end{center}\n  \\caption\n \n  { \\label{fig:data_and_fit} \na) Aerial survey measurement of the distribution of fallout following detonation\n  of the radiological dispersal device.\nb) Result of fit of template histograms to the aerial survey measurement.}\n\\end{figure}\nThis result has been published previously~\\cite{Sinclair_RDD_2015} and the methods to\narrive at the result were recapitulated here in Sect.~\\ref{sec:exp_method}.\nWe observe an area of activity concentration exceeding 100~kBq\/m$^2$ near\nground zero of the detonation, with a long deposited plume of maximum width\nof 300~m to 400~m, and significant measured radioactivity extending over a\ndistance of about 2~km.\nThe total systematic uncertainty affecting this measurement was determined to\nbe around 12\\% and the statistical uncertainty peaks at 6~kBq\/m$^2$.\n\nFig.~\\ref{fig:data_and_fit}~b) shows the result of the $\\chi^2$ fit of\nthe weighting coefficients of the template response functions to the\nmeasurement of Fig.~\\ref{fig:data_and_fit}~a).  The colour scales of \nFig.'s~\\ref{fig:data_and_fit}~a) and~b) have been chosen to be equal.  \n(This colour\nscale has in fact been set to the optimal colour scale for the data from a\ntruckborne survey which will be presented in the upcoming \nSect.~\\ref{sec:truckborne}.)\nThe features of the measurement are broadly well-reproduced by the fit,\nconsidering the pixellization of the reconstruction.  In particular, the\nmagnitude, width and extent of the contamination are well represented in the\nresult of the fit.\n\nThe underlying distribution of pixel sources which gives rise to the fit\nresult of Fig.~\\ref{fig:data_and_fit}~b) is presented in\nFig.~\\ref{fig:data_inverted}~a).\n\\begin{figure}[h]\n   \\begin{center}\n   \\begin{tabular}{c}\n   \\begin{overpic}[trim = .1cm .1cm .1cm .1cm, clip = true, height=6.5cm, angle=270]{hCONC_ALL.eps}\\put(18,70){\\textcolor{white}{a)}}\\end{overpic}\\\\\n   \\begin{overpic}[trim = .1cm .1cm .1cm .1cm, clip = true, height=6.5cm, angle=270]{hpERR_ALL.eps}\\put(18,70){\\textcolor{white}{b)}}\\end{overpic}\n   \\hspace{.4cm}\n   \\begin{overpic}[trim = .1cm .1cm .1cm .1cm, clip = true, height=6.5cm, angle=270]{hmERR_ALL.eps}\\put(18,70){\\textcolor{white}{c)}}\\end{overpic}\\\\\n   \\end{tabular}\n   \\end{center}\n  \\caption\n \n  { \\label{fig:data_inverted} \na) Spatially deconvolved aerial survey measurement of fallout following\nradiological dispersal device blast.\nb) Positive statistical uncertainty on spatially deconvolved measurement.\nc) Negative statistical uncertainty on spatially deconvolved measurement.}\n\\end{figure}\n  This is the reconstructed distribution of\nthe surface activity concentration following spatial deconvolution.\nWe find that the width of the deposited plume is now much smaller than the\noriginal undeconvolved measurement, around\n50~m.  Correspondingly, the peak concentration is higher, over 400~kBq\/m$^2$.\nNote that the width of the deposited plume is small with respect to the altitude and line spacing of the\nsurvey.  This accounts for its significant overestimation when the ``infinite and\nuniform'' approximation was used to obtain a concentration measurement from the\nmeasured counts as shown in Fig.~\\ref{fig:data_and_fit}~a) and~\\cite{Sinclair_RDD_2015}.\nThe length of the deposition is\nhowever much larger than the survey parameters, so in this dimension the\n``infinite and uniform'' sheet approximation is not so bad and the original\nlength measurement is not much altered by the spatial deconvolution.\n\nThe positive and negative statistical uncertainties on the spatially\ndeconvolved deposited fallout map are shown in Fig.'s~\\ref{fig:data_inverted}~b)\nand~c) respectively.  Note that the statistical uncertainties affecting the\nmeasurement are very small, and on the colour scale of\nFig.~\\ref{fig:data_inverted}~a) would be difficult to see, so the colour scale in\nFig.'s~\\ref{fig:data_inverted}~b) and~c) is chosen to optimize the\nrepresentation of the information in Fig.~\\ref{fig:data_inverted}~c).  The\nuncertainties reveal important features of the measurement and its inversion.\nThe positive uncertainty indicates a region extending away from the measured deposited\nplume axis in which a positive quantity of activity is permitted, however at a\nvery low amount of between 5~kBq\/m$^2$ and 10~kBq\/m$^2$.  The negative\nuncertainty shows that the measurement of the presence and distribution of\nradioactivity is significantly above zero.\n\nThe MINOS error propagation includes only the stochastic uncertainties on the\nmeasurement.  There are additional uncertainties which are systematic and arise from\napproximations in the representation of the system in the simulation.  These include mis-representation of the position, particularly the altitude; the attitude (yaw, pitch and roll); the amount of shielding material in the basket containing the detectors; the energy resolution and the air density.\nThe systematic uncertainty on the (undeconvolved) radioactivity concentration distribution was \ndetermined to be about 12\\% by variation of these parameters within reasonable limits~\\cite{Sinclair_RDD_2015}.\n\nFor the spatially deconvolved measurement, some of the systematic\nuncertainties must be re-examined as they can be expected to have an effect on the shape of the reconstructed fallout distribution, as well as its overall magnitude.  These are the systematic uncertainties associated with the measurement of altitude and pitch angle.  \nIt is also interesting to examine the effect of the measurement of yaw angle\non the spatially inverted measurement as it can have no effect at all on the\noriginal undeconvolved measurement which used the infinite sheet approximation\nfor the source and therefore the detector response was invariant under changes of yaw.\n \nThe inertial navigation system\ndetermined the yaw angle \nduring the measurement to be around -30$^\\circ$.  The spatial inversion was conducted\ntwice.  For the central value of the inversion as presented in \nFig.~\\ref{fig:data_inverted}~a) the helicopter systems in the template\nhistograms were assigned a yaw of -30$^\\circ$ to match the data.  To allow for\nchanges in yaw during flight, the regression was repeated with yaw set\nmaximally different at 60$^\\circ$.  Pitch was varied from the nominal\n0$^\\circ$ to -10$^\\circ$ according to the maximum deviation of pitch recorded\nby the inertial navigation system while on line during one sortie.\nAltitude was varied from nominal by 1~m to account for approximately one sigma\nof variability in height determination.\nThese variations did not significantly alter the measurement of length and\nwidth of the deposited fallout.\nAdded in quadrature, and considering that some of the variation was already\nincluded in the original systematic uncertainty associated with the\nsensitivity, the deviations do not yield a significant additional systematic\nuncertainty.\nAlthough not a significant additional source of uncertainty for the measurements\npresented here, these sources of uncertainty are worth discussing for the\nbenefit of researchers following this approach under different operating conditions.\n\n\\subsubsection{Comparison of spatially deconvolved aerial survey data\n  with truckborne survey data}\n\\label{sec:truckborne}\n\nIn Fig.~\\ref{fig:cftruck}a) the truckborne survey result which followed the\nfirst blast is shown overlaid on\nthe undeconvolved aerial survey result from the same blast\non the same colour scale.  \n\\begin{figure}[h]\n   \\begin{center}\n   \\begin{tabular}{c}\n   \\begin{overpic}[trim = .1cm .1cm .1cm .1cm, clip = true, height=6.5cm, angle=270]{truck_on_data_1.eps}\\put(18,70){\\textcolor{white}{a)}}\\end{overpic}\n   \\hspace{.4cm}\n   \\begin{overpic}[trim = .1cm .1cm .1cm .1cm, clip = true, height=6.5cm, angle=270]{truck_on_unfold_1.eps}\\put(18,70){\\textcolor{white}{b)}}\\end{overpic}\\\\\n   \\vspace*{.5cm}\\\\   \n   \\begin{overpic}[height=6.5cm, angle=0]{RMSD_RDD.eps}\\put(95,45){\\textcolor{black}{c)}}\\end{overpic}\n   \\end{tabular}\n   \\end{center}\n  \\caption\n \n  { \\label{fig:cftruck} \na) Aerial radiation survey following detonation of radiological dispersal\ndevice, with results of radiation survey from a truck-based system overlaid.\nb) Spatially deconvolved aerial radiation survey of the RDD fallout, with\nresults of radiation survey from a truck-based system.  Colour scale is the\nsame in a) and b) and is optimized to show the range of values of the result from the truckborne survey.\nc) (top) Transects of the deposited RDD plume following the path of the truck-based survey\nsystem.  The\nsolid line shows the aerial survey result sampled at the truck location.  The dot-dashed line shows the\ntruckborne survey result and the dashed line shows the aerial survey result\nafter spatial deconvolution sampled at the truck location. (bottom)\nRoot-mean-square deviation of the seven transects versus the location of maximum\nconcentration of each transect according to the truckborne survey.  Circles\nshow the aerial survey result, squares show the truckborne survey and\ntriangles show the spatially deconvolved aerial survey result.}\n\\end{figure} \n(The colour\nscale is optimized to show the variation in the truckborne survey result.)\nThe truckborne survey reports much higher surface activity oncentrations than\nthe aerial survey and the fallout appears to be narrower in width.\n\nFig.~\\ref{fig:cftruck}b) shows the same truckborne survey result this time\noverlaid on the spatially deconvolved aerial survey measurement.  (The colour\nscale is the same as that used in Fig.~\\ref{fig:cftruck}a).)\nHere, both the width and magnitude of the concentration are in better\nagreement.\n\nThe aerial survey maps were sampled at the locations of the truck and these\nsampled activity concentration values are presented in\nFig.~\\ref{fig:cftruck}c) (top) as a function of the distance from the start of the\ntruck-driven sortie.\nAgain, it is clear that the spatially deconvolved result is generally higher\nat maximum\nmagnitude and more narrow than the undeconvolved aerial survey result.  \nThe bottom part of Fig.~\\ref{fig:cftruck}c) shows the root-mean-square\ndeviation (RMSD)\nof the curves calculated by sampling each profile at regular intervals\nfrom the maximum concentration down to the point at which the concentration\nfalls below 10\\% of the maximum.  This plot shows that the RMSD width\nof the deposition according to the undeconvolved survey is typically around\n90~m while the width of the deposition according to the\ntruckborne survey and the spatially deconvolved aerial survey tends to be\nsignificantly narrower, closer to 30~m.\nSpatially deconvolved aerial survey thus recovers the\nnarrowness of the fallout to approximately the same spectral\nprecision as the truckborne survey.\n\nTruckborne data is shown for the purpose of shape comparison only and does not\ninclude detailed error analysis.  In any event, the truckborne system has its\nown finite area of sensitivity largely caused by its ``altitude'' of just over one\nmetre, causing smearing of the measured spatial distribution.  \nA contact measurement of the deposited radioactivity can be expected to be even more\nconcentrated in places~\\cite{Erhardt_deposition_RDD_2015}.\n\n\\section{Discussion and Conclusions}\n\nRadiometric survey would be performed to map fallout following a reactor\naccident or following a malicious\nrelease.  To cover a large area quickly, the surveys are initially performed\nusing manned aircraft at some significant altitude $H$.  If there is spatial\nvariability in the fallout at distance scales much less than $H$, then\nthe map result of the survey can underestimate the quantitity of\nradioactivity on the ground in places.\n\nWe have presented here a method to deconvolve an aerial survey map for the\nspatial smearing caused by measurement at altitude, at least to the extent\npermitted by the sampling density as determined by the aircraft speed and line\nspacing.\nPerformed on synthetic data, the deconvolution method returns a distribution\nwhich is consistent with the true underlying distribution within\nmeasurement uncertainty.\nThe deconvolved distribution is more narrowly distributed, and shows regions\nof locally higher radioactivity concentration than the initial undeconvolved\nmeasurement.\nPerformed on real data acquired following detonation of a radiological\ndispersal device, the method produces a distribution which is narrower and\nshows radioactivity concentration as much as four times that of the original\nmeasurement.\n\nThe method can unfold a distribution for smearing effects up to a resolution\npermittable by the sampling frequency of the original measurement.  The\nmethod allows for propagation of stochastic measurement\nuncertainties through the unfolding to obtain the measurement uncertainties on\nthe fit parameters.\nThe method relies on application of the MINUIT and MINOS algorithms well known\nin the field of particle physics.\nWhat is perhaps not well known is that these algorithms can tolerate\noperating with hundreds of independent fit parameters, converging to a stable solution in\na reasonable amount of time from an arbitrary starting distribution.\nOur current need was to develop a method to extract the greatest possible\ninformation from a set of aerial surveys performed to improve scientific\nunderstanding of the behaviour of radiological dispersal devices.\nThe method, however, is also applicable to unfolding of any smeared distribution\nof any dimensionality.  It could find application in other fields.\n\nThe result of the unfolding is limited in spatial resolution by the requirement\nthat the density of pixellization of the answer not exceed the density of\nmeasurements as determined by the aircraft speed and line spacing.  \nNevertheless, aerial survey practitioners should be aware that there is \nimproved information about the spatial distribution of the radioactivity \ncontained in their aerial survey map that can be extracted provided good \nknowledge of the response function of the system is available.\nThe achieved spatial resolution for the particular aerial survey following RDD \ndetonation presented here approximately matched that obtained during\ntruckborne survey over the same deposition (while providing complete coverage).\nContact measurements ($H=0$~m) can be expected to reveal even greater\nlocal spatial variations than the truckborne data.\nStill, the truckborne survey ``height'' of about 1.2~m provides a salient\nbenchmark for spatial resolution as this is close to the average height of an\nadult human.\nShould humans be required to enter a possibly contaminated area guided by the\nresults of aerial survey alone, a spatially deconvolved aerial survey map could\nprovide a better predictor of the activity concentrations they will encounter\nthan the undeconvolved measurement.\n\n\\section*{Acknowledgements}\n\nThe authors gratefully acknowledge the leadership of the RDD field trials\nunder L. Erhardt, and helpful comments on the analysis from H.C.J.~Seywerd,\nP.R.B.~Saull and F.A,~Marshall.  Funding for this project was provided through\nCanada's Chemical, Biological, Radiological-Nuclear and Explosives Research\nand Technology Initiative.  This report is NRCan Contribution 20180112.\n\n\n\n\n\n\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction \\label{s.intro}}\n\nHigh-mass stars are influential in galactic evolution by dynamically affecting and ionizing the interstellar medium, and also by chemically enriching heavy elements via supernova explosions. It is of fundamental importance to understand the physical processes in the evolution of molecular clouds where high-mass stars are forming. There have been numerous works on high-mass star formation in the literature \\citep[for reviews see e.g.,][]{Zinnecker2007,Tan2014}. In spite of these works we have not yet understood how high-mass star formation takes place. One of the promising candidates where young high-mass stars are forming is the very dense and massive cores such as infrared dark clouds in the Milky Way \\citep{Peretto2013}. Another possible candidate is the compressed layer formed in cloud-cloud collisions. Observations of a few super star clusters and smaller H$\\,${\\sc ii} regions in the Milky Way have shown signs of triggered formation of high-mass stars in the collision-compressed layers \\citep[e.g.,][]{Furukawa2009,Torii2011,Torii2015,Fukui2014,Fukui2015}. Magneto-hydro-dynamical numerical simulations of two colliding molecular clouds by \\citet{Inoue2013} have shown that turbulence is excited and the magnetic field is amplified in the collision-shocked layer between the clouds. These turbulence and magnetic field increase the mass accretion rate, favoring high-mass star formation.\n\nThe difficulty in studying young high-mass stars lies in the considerably small number of young high-mass stars as compared with low-mass stars in the solar vicinity; this is in part due to the lower frequency of high-mass stars and the heavy sightline contamination in the Galactic disk. ALMA is now opening a new possibility to explore high-mass star formation in external galaxies by its unprecedented sensitivity and resolution, having a potential to revolutionize our view of high-mass star formation. The Large and Small Magellanic Clouds (LMC and SMC), at a distance of 50\\,kpc \\citep{Schaefer2008} and 61\\,kpc \\citep{Szewczyk2009}, are actively forming high-mass stars. The LMC is an ideal laboratory to see the evolution of stars and clouds thanks to the non-obscured face-on view \\citep{Subramanian2010} of all the GMCs in a single galaxy \\citep[for a review][]{Fukui2010}. A $^{12}$CO($J$=1--0) survey for GMCs with NANTEN 4m telescope \\citep{Fukui1999,Mizuno2001,Yamaguchi2001,Fukui2008} provided a sample of nearly 300 GMCs at 40\\,pc resolution and led to an evolutionary scheme from starless GMCs (Type I) to active star-forming GMCs (Type III) over a timescale of 20 Myrs \\citep{Fukui1999,Kawamura2009}. Aiming at revealing the finer-scale details of the molecular gas in the LMC, we have commenced systematic CO observations by using ALMA at sub-pc resolution.\n\nAmong the nearly 300 GMCs over the LMC obtained with NANTEN, N159 is the brightest one with H$\\,${\\sc ii} regions. Infrared studies have revealed nearly twenty young high-mass stars in N159 with Spitzer and Herschel (\\citealt{Chen2010}, \\citealt{Wong2011}, \\citealt{Carlson2012} and references therein; \\citealt{Seale2014}), where two clumps N159 East and West are active in star formation. \\citet{Mizuno2010} showed that the CO $J$=4--3\/$J$=1--0 ratio shows enhancement toward the molecular peak without a well-developed H$\\,${\\sc ii} region in N159 West (N159W). This high excitation condition suggests that N159W is possibly on the verge of high-mass star formation, and thus the initial condition of high-mass star formation may still hold. The preceding observations with Australia Telescope Array (ATCA), while low in resolution (HPBW$\\sim6\\arcsec$), presented some hint of small-scale clumps and filaments in N159W\\citep{Seale2012}. N159W is therefore the most suitable target for the purpose of witnessing the oneset of high-mass star formation. \n\nWe present the first results of the ALMA observations of N159W in this Letter mainly based on the $^{13}$CO($J$=2--1) data.\n\n\\section{Observations \\label{s.obs}}\n\nWe carried out ALMA Cycle 1 Band 3 (86--116\\,GHz) and Band 6 (211--275\\,GHz) observations toward N159W both with the main array 12m antennas and the Atacama Compact Array (ACA) 7m antennas. The observations centered at ($\\alpha_{J2000.0}$, $\\delta_{J2000.0}$) = (5$^{\\rm h}$39$^{\\rm m}$35\\fs34, -69\\arcdeg45\\arcmin33\\farcs2), were carried out between October 2013 to May 2014. The target molecular lines were $^{13}$CO($J$=1--0), C$^{18}$O($J$=1--0), CS($J$=2--1), $^{12}$CO($J$=2--1), $^{13}$CO($J$=2--1) and C$^{18}$O($J$=2--1) with a bandwidth of 58.6 MHz (15.3 kHz $\\times$ 3840 channels). We used a spectral window for the observations of the continuum emission among the four with a bandwidth of 1875.0 MHz (488.3 kHz $\\times$ 3840 channels). The radio recombination lines of H30$\\alpha$ and H40$\\alpha$ were also included in the windows. The projected baseline length of the 12m array ranges from 16\\,m to 395\\,m. The ACA covers 9\\,m to 37\\,m baselines. The calibration of the complex gains was carried out through observations of seven quasars, phase calibration on four quasars, and flux calibration on five solar system objects. For the flux calibration of the solar system objects, we used the Butler-JPL-Horizons 2012 model (https:\/\/science.nrao.edu{\\slash}facilities{\\slash}alma{\\slash}aboutALMA{\\slash}Technology{\\slash}ALMA\\_Memo\\_Series{\\slash}alma594{\\slash}abs594). The data were reduced using the Common Astronomy Software Application (CASA) package (http:\/\/casa.nrao.edu), and visibility imaged. \nWe used the natural weighting for both the Band 3 and Band 6 data, providing synthesized beam sizes of $\\sim$2\\farcs5 $\\times$ 1\\farcs8 (0.6 pc $\\times$ 0.4 pc at 50kpc) and $\\sim$1\\farcs3 $\\times$ 0\\farcs8 (0.3 $\\times$ 0.2 pc), respectively. \nThe rms noises of molecular lines of Band 3 and band 6 are $\\sim$40 mJy beam$^{-1} $and $\\sim$20 mJy beam$^{-1}$, respectively, in emission-free channels.\nThe comparison of the cloud mass derived from the ALMA observation with that of a single dish observation described in Section \\ref{s.r.filaments} suggests that the missing flux of the present ALMA observation is not significant.\n\n\\section{Results \\label{results}}\n\\subsection{A complex filamentary structure \\label{s.r.filaments}}\n\nFigure \\ref{fig1} shows the $^{13}$CO velocity integrated intensity image of the $J$=2--1 transition.  \nThe distribution of the $^{13}$CO emission is highly filamentary. \nThe filaments, having often straight or curved distribution, have a typical length of 5--10\\,pc and a width of 0.5--1.0\\,pc defined as a full-width of the emission area at the 3$\\sigma$ level of the intensity integrated over a range of 234 to 240\\,km\\,s$^{-1}$, which may be analogous to the dominance of filaments in the interstellar medium of the solar vicinity \\citep[e.g.,][]{Andre2013,Molinari2010}, possibly suggesting that filaments are ubiquitous in other galaxies as well. \nMore details of the filaments will be published separately.\nThe most active star formation is found in two regions as denoted by N159W-N and N159W-S in Figure \\ref{fig1}, both of which are associated with enhanced $^{13}$CO emission. \n\nThe cloud mass is estimated from the $^{12}$CO($J$=2--1) intensity by assuming a conversion factor from the  $^{12}$CO($J$=1--0) intensity to the column density of X(CO)=$7\\times10^{20}$cm$^{-2}$ \\citep{Fukui2008} and the typical $^{12}$CO($J$=2--1)\/$^{12}$CO($J$=1--0) ratio toward H{\\sc ii} regions of 0.85 (the ratio in the Orion-KL region of \\citealt{Nishimura2015}).\n We also assumed the absorption coefficient per unit dust mass at 1.2\\,mm and the dust-to-gas mass ratio to be 0.77\\,cm$^2$\\,g$^{-1}$ and $3.5\\times10^{-3}$, respectively to derive the gas mass from the dust emission (Herrera et al. 2013).\nIn total, the filaments have molecular mass of 2.4 $\\times$ 10$^{5}$ $M_{\\odot}$ in N159W corresponding to 35 \\% of the total mass which is estimated by the lower resolution study \\citep{Minamidani2008,Mizuno2010}. \nWe define the N159W-N and N159W-S clumps at the 5\\,$\\sigma$ level of Band 6 continuum (white contours in Figure 1), and the masses of these clumps are estimated to be $2.9\\times10^4$\\,$M_\\odot$ and $4.1\\times10^3$\\,$M_\\odot$, respectively, by assuming the dust temperature 20\\,K. Their masses derived from the CO emission are $1.5\\times10^4$\\,$M_\\odot$ and $4.2\\times10^3$\\,$M_\\odot$, respectively, as is consistent with the dust-emission estimate.\n\n\\subsection{Outflows \\label{s.r.outflows}}\n\nWe have discovered two molecular outflows having velocity span of 10--20\\,km\\,s$^{-1}$ in $^{12}$CO($J$=2--1). Figure \\ref{fig2} shows the distribution of the outflow wings. One of them corresponds to N159W-N and the other N159W-S. The N159W-S outflow has red-shifted and blue-shifted lobes which show offsets of 0.1--0.15\\,pc from the peak of the continuum emission. The outflow axis is along the east-west direction. The N159W-N outflow has only the blue-shifted lobe which shows an offset of 0.2 pc from the $^{13}$CO peak. It is possible that the complicated gas distribution around N159W-N may mask the possible red lobe. The size of the red-shifted and blue-shifted lobes is less than the beam size 0.2\\,pc $\\times$ 0.3\\,pc and the upper-limit timescale of the outflow is roughly estimated to be 10$^{4}$ yrs. This is the first discovery of extragalactic outflows associated with a single protostar.  The positions of outflows in N159W-N and N159W-S coincide with YSOs identified based on the {\\it Spitzer} data: 053937.56-694525.4 (hereafter YSO-N; Chen et al. 2010) and 053941.89-694612.0 (hereafter YSO-S; \\citealt{Chen2010}; P2 in \\citealt{Jones2005}), respectively. \n\n\\subsection{YSO characteristics \\label{s.r.yso}}\nTwo YSOs associated with outflows, YSO-N and YSO-S, have been studied extensively at near- to far-infrared, submillimeter, and radio wavelengths (\\citealt{Carlson2012} and references therein; \\citealt{Seale2014}; \\citealt{Indebetouw2004}).  Using the \\citet{Robitaille2006,Robitaille2007} YSO model grid and spectral energy distribution (SED) fitter, we model all the available data including the {\\it Spitzer} and {\\it Herschel} fluxes (1.2--500 $\\mu$m),  as well as photometry we extracted from {\\it Spitzer}\/IRS spectra  (5--37\\,$\\mu$m; \\citealt{Seale2009}), and the fit of the SEDs indicates that both YSO-N and YSO-S are Stage 0\/I YSOs. \nThe mass and luminosity are estimated to be $31\\pm8$\\,$M_\\odot$ and $(1.4\\pm0.4)\\times10^5$\\,$L_\\odot$ for YSO-N, and $37\\pm2$\\,$M_\\odot$ and $(2.0\\pm0.3)\\times10^5$\\,$L_\\odot$ for YSO-S. These results are consistent with those from Chen et al. (2010), who also used Robitaille fitter but without the Herschel constraints. \nThe dynamical ages of the two outflows are consistent with the ages output from the SED fitter  \\citep{Robitaille2006}, $\\sim$10$^{4}$ yrs.\n\nAccording to 3\\,cm radio continuum measurements YSO-N is determined to be an O5.5V star, whereas YSO-S is not detected \\citep{Indebetouw2004}, suggesting that YSO-S is in an earlier evolutionary state than YSO-N.\nThis is consistent with a non-detection of the He 2.113\\,$\\mu$m and with the weak Br$\\gamma$ in YSO-S \\citep{Testor2006}.\nThe \\citet{Testor2006} near-IR VLT data revealed that YSO-S consists of at least two sources, whereas their detailed physical properties and relation with the mid-\/far-infrared source are yet unknown.\n\n\\subsection{Filamentary collision in N159W-S \\label{s.r.col}}\n\nIn Figure \\ref{fig1}, N159W-N shows complicated $^{13}$CO distribution, whereas the source N159W-S shows relatively simple $^{13}$CO morphology. The $^{13}$CO distribution in N159W-N consists of several filaments which are elongated generally in the direction from the northeast to southwest, and N159W-S is located at the tip of a V-shaped distribution of two filaments. We shall focus on N159W-S in the following to describe the filament distribution and the high-mass young star, because the simple morphology allows us to understand the physical process unambiguously. \n\nFigure \\ref{fig3} shows the two filaments toward N159W-S (Figure \\ref{fig3}(a) the whole velocity range, Figure \\ref{fig3}(b) red-shifted filament, and Figure \\ref{fig3}(c) blue-shifted filament, respectively). The two filaments overlap toward N159W-S, where the $^{13}$CO intensity and linewidth are significantly enhanced. Figures \\ref{fig3}(d-h) show position-velocity diagrams taken along the two filaments. We see that the filaments have small velocity span of 3\\,km\\,s$^{-1}$ in the north of N159W-S which shows significantly enhanced velocity span of 8\\,km\\,s$^{-1}$ at the 15\\,\\% level of the $^{13}$CO peak in Figure\\,\\ref{fig3}(g). \nAn HST image at near infrared indicates that the red-shifted filament is extended toward the south beyond N159W-S (Carlson et al. 2015 in preparation), while no CO emission is detected there in our $^{12}$CO or $^{13}$CO observations with ALMA. We also find that the blue-shifted filament has its extension beyond N159W-S in $^{13}$CO. So, although the filaments are apparently terminated toward N159W-S, they are actually more extended, placing N159W-S in the intersection of the two filaments. \n\nIn N159W-S the longer red-shifted filament in the east is highly elongated and mildly curved, having a length of 10\\,pc, while the other blue-shifted filament in the west is straight and elongated by 5\\,pc. N159W-S clearly demonstrates that a high-mass YSO with bipolar outflow is formed toward the intersection between the two thin filaments, and the velocity dispersion is significantly enhanced in the intersection. \n\nBased on these results we set up a hypothesis that formation of N159W-S was triggered by the collision between the two filaments. We first describe a possible scenario for N159W-S and then discuss the observational constraints on the collision and high-mass star formation. The two crossing filaments overlapping toward N159W-S give direct support for the present scenario. The lower limit for the relative velocity in the collision is given by the velocity difference of the two filaments 2--3\\,km\\,s$^{-1}$. The actual collision velocity should be higher than 2--3 km s$^{-1}$ because of the projection effect. According to the magneto-hydro-dynamical numerical simulations of two colliding molecular flows by \\citet{Inoue2013}, the collision-shocked layer enhances isotropic turbulence, independent of the direction of the collision, and the velocity span in the shocked layer is similar to the relative collision velocity. The simulations by \\citet{Inoue2013} for a velocity difference of 20 km s$^{-1}$ allow us to scale the relative velocity to $\\sim$10 km s$^{-1}$ with basically the same physical process. We therefore assume the velocity span in N 159W-S, 8 km s$^{-1}$, as the actual collision velocity. This implies the relative motion of the two filaments is nearly vertical, roughly 70$^\\circ$, to the line of sight.\n\n\\section{Discussion on the high-mass star formation processes \\label{s.discussion}}\n\nSince the rest of the filaments show no sign of velocity dispersion enhancement with high-mass star formation, we assume that the non-interacting filaments retain the initial condition prior to the collision. \nThe line-mass, mass per unit length, in the filaments changes from region to region by an order of magnitude. In order to estimate the typical mass of the filaments associated with N159W-S clump for the following discussion, we pick up two segments of 1.5\\,pc and 1.8\\,pc in length and 0.7\\,pc and 0.6\\,pc in full width at a 35\\,\\% level of the $^{13}$CO peak for the red-shifted and blue-shifted filaments, respectively, as indicated in Figures 3(b) and 3(c). \n{Below the 35\\,\\% level, it is hard to estimate the line-mass of the individual filaments separately due to overlapping. Above this level, the mass sampled becomes underestimated.\nWe estimate the total mass of these two segments to be $2.9\\times10^3$\\,$M_\\odot$ from $^{12}$CO($J$=2--1) for a velocity range 234\\,--\\,240\\,km\\,s$^{-1}$. \nWe then estimate the average line-mass of these two filaments to be $8.9\\times10^2$\\,$M_\\odot$\\,pc$^{-1}$. \nThe filaments are not detected in the Band 6 continuum at the 3\\,$\\sigma$ noise level of the molecular mass density $1.6\\times10^3$\\,$M_\\odot$\\,pc$^{-1}$, which is higher than the above CO-based line-mass density of the filaments.\n \nThis suggests that the collision took place in a timescale, $\\sim$0.5\\,pc divided by 8\\,km\\,s$^{-1}$, i.e., $\\sim$6 $\\times$ 10$^{4}$ years ago. We assume that formation of the high-mass star initiated at the same time. By using the stellar mass 37 $M_{\\odot}$, the average mass accretion timescale of the star formation is given as 37\\,$M_{\\odot}$\/$6 \\times 10^{4}$\\,yrs $\\sim$6 $\\times$ 10$^{-4}$ $M_{\\odot}$ yr$^{-1}$. This rate is well in accord with the theoretical estimate around 10$^{-3}$ $M_{\\odot}$ yr$^{-1}$ and satisfies the criterion to form high-mass stars by overcoming the stellar feedback \\citep[e.g.,][]{Wolfire1986}. The small outflow timescale 10$^{4}$\\,yrs is consistent with this picture involving rapid high-mass star formation.\n\nThe present case of N159W-S has shown that the high-mass star having 37\\,$M_{\\odot}$ is formed in a turbulent condition created by the collisional shock. \nThe mass of the N159W-S clump is estimated to be $4\\times10^3$\\,$M_\\odot$ toward its CO peak. \nThere is no sign of such dense clumps over the rest of the filament according to our ALMA data, either in CS($J$=2--1) data, whose line-mass detection limit is about 150\\,$M_\\odot$\\,pc$^{-1}$, or in dust emission data, whose line-mass detection limit is about $1.6\\times10^3$\\,$M_\\odot$\\,pc$^{-1}$ by assuming a filament width of 0.6\\,pc. \nThis offers an interesting possibility that high-mass stars do not necessarily require dense cloud cores as the initial condition. Instead, high velocity colliding molecular flows are able to efficiently collect mass into a cloud core non-gravitationally. \\citet{Inoue2013} discuss that the mass flow in the collision can be efficiently converged into a shock-induced core due to the oblique shock effect, and that self-gravity is not important in the beginning of the high-mass star formation, while soon later, in the shock-collected core, self gravity will play a role to form the stellar core \\citep[see also][]{Vaidya2013}. \n\nIn the Milky Way we see increasing observational evidence for cloud-cloud collisions which trigger high-mass star formation. Four super star clusters, Westerlund2, NGC3603, RCW38 and DSB[2003]179, are found to be formed by collisions between two clouds \\citep{Furukawa2009,Ohama2010,Fukui2014,Fukui2015}. Isolated O stars with H$\\,${\\sc ii} region, M20, RCW120 etc., are also suggested to be triggered by cloud-cloud collisions \\citep{Torii2011, Torii2015}.  N159W-S is in the very early stage of star formation as indicated by the non-detection of ionized gas, as well as by the collision scenario and SED models. Therefore N159W-S is an optimal source to study filamentary collision leading to star formation. \nIt has been shown that the youngest O stars are formed coevally in a duration of $\\lesssim10^{5}$\\,yrs in NGC3603 and Westerlund1 by careful measurements of stellar ages with HST and VLT \\citep{Kudryavtseva2012}. We have here an independent estimate of the stellar age by taking the advantage of the simple cloud morphology in N159W-S, and the present time-scale estimate is consistent with that of \\citet{Kudryavtseva2012}.\n\n\\section{Conclusions \\label{s.conclusion}}\n\nIn this Letter we presented the $^{13}$CO ($J$ = 2--1) observations with ALMA on the active star-forming region N159 West in the LMC. We have found the first two extragalactic protostellar molecular outflows toward young high-mass stars, whose dynamical timescale is $\\sim$10$^{4}$ yrs. One of the two stars N159W-S is clearly located toward the intersection of two filamentary clouds. We set up a hypothesis that two filaments collided with each other $\\sim$10$^{5}$ yrs ago and triggered the formation of the high-mass star. The results demonstrate the unprecedented power of ALMA to resolve extragalactic star formation.\n\n\\acknowledgments\n\nThe authors thank the anonymous referee for his\/her helpful comments. This paper makes use of the following ALMA data: ADS\/JAO.ALMA\\#2012.1.00554.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI\/NRAO and NAOJ. This work was supported by JSPS KAKENHI grant numbers 22244014, 22540250, 22740127, 23403001, 24224005, 26247026; by JSPS and by the Mitsubishi Foundation.  MM and ON are grateful for support from NSF grant \\#1312902.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Self-injection locking with microresonators and microcombs}\nThe solution proposed in\\cite{Maleki2015} for a compact microresonator based Kerr comb used a relatively broadband (10-1000 times wider than the microresonator's resonance) single-frequency distributed feedbacak (DFB) laser. In this case, the role of the microresonator was twofold:\n1) acting as an external cavity it narrowed the linewidth of the laser via the self-injection locking effect\\cite{Vasil'ev1996,Yarovitsky1998,Donvalkar2018}, and 2) it provided a nonlinear low-threshold Kerr medium where a frequency comb appeared if the laser was appropriately detuned. Laser linewidth narrowing in this approach exploits the coupling of a free-running laser diode with the WGM microresonator\\cite{Braginsky1989} having internal and surface Rayleigh backscattering\\cite{Ilchenko1992,Ilchenko2000}. These scattering effects resonantly couple an excited traveling wave WGM with the identical but counter propagating mode which returns back to the laser, locking it to the cavity resonance. This technique was used for stabilisation of a DFB laser down to a record sub--Hz level\\cite{Liang2015}. The analytical theory of self-injection locking by a WGM cavity initially proposed in\\cite{Oraevsky} was recently revised and extended in\\cite{Kondratiev2017}. \n\nIt was previously assumed for granted that `only single-mode DFB lasers characterized with comparably high internal Q's are suitable for stable self-injection locking using multimode optical cavities'\\cite{Xie:15}. Similarly, earlier only single-frequency pre-stabilized external cavity diode lasers (ECDLs) with diffraction grating\\cite{Yarovitsky1998, Velichansky2003} were considered for self-injection locking to WGM cavity\\cite{Maleki2015, Maleki2010}. Any external cavity pre-stabilisation complicates devices and their integration. The DFB lasers, however, are not available for many wavelengths and have a limited power that in turn restricts the power of single-mode operation and generated frequency combs at milliwatt level. Meanwhile many applications require higher comb power. For example, absorption dual-comb spectroscopy using surface diffuse scattering has less than $1$\\% of light collection efficiency so that high power of incident combs must be provided\\cite{Hensley}. Evaluations in\\cite{Shchekin:18} show that dual-comb CARS spectroscopy of blood glucose provides a measurable glucose signal at about 100mW power of frequency combs with $10$~GHz mode spacing.      \n\n\n\\section*{Single-frequency lasing with a multi-frequency laser}\n\nWe revealed that the initial mode pre-selection and pre-stabilisation in laser diodes are not required to obtain stable narrow-linewidth single-frequency lasing, and a WGM microresonator can handle all of these purposes efficiently as well. Consequently, simpler and cheaper FP laser diodes with higher power may be used.  \n\nWe demonstrate for the first time an efficient (up to 50\\%) conversion of a broadband multifrequency FP laser diode, coupled to a high-Q WGM microresonator, into a narrow linewidth single-frequency light source in the 100 mW power range at optical telecom wavelength, with its subsequent transformation to a single-soliton Kerr comb oscillator. FP laser diode spectrum narrowing occurs in regime of competition between many longitudinal modes. Self-injection locking solves two critical technical problems of soliton Kerr combs: (1) thermal instability and (2) preferential excitation of multiple-soliton regimes. The soliton states are only possible at strong red detunings from the cavity resonance where the CW internal circulating power is small. That is why, the transition from a chaotic comb to a multiple solitons comb and finally to the single-soliton state via slow laser tuning leads to a fast drop of the internal power resulting in cooling of the resonator and finally, due to thermal refraction and expansion effects, to a large detuning from the required regime causing the loss of the soliton state. Detuning can be compensated by an electronic feedback which requires fast tuning (with the characteristic time of thermal relaxation of the resonator) of the laser frequency which is difficult to achieve. That is why different additional nonstationary `power kicking' methods\\cite{Kippenberg2016, Brasch2016, Yi2016} were proposed to reach the thermal equilibrium. However, the optical feedback in self-injection locking is fast enough to compensate thermal effects in real time. Additionally, slower tuning from a chaotic to soliton state, only possible with the supported thermal equilibrium, allows a transition to smaller initial numbers of circulating solitons and finally to a single-soliton state\\cite{Lobanov2016}.\n\n\n\\section*{Experimental setup} \n\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{Spectra.pdf}\n\t\\caption{\\textbf{Self-injection locking and spectral narrowing of a multi-frequency diode laser coupled to a MgF$_2$ ultra-high-Q whispering gallery microresonator.} \\textbf{(a)} The spectrum of the free-running diode laser. \\textbf{(b)} The spectrum of the diode laser stabilised by a high-Q microresonator. \\textbf{(c)} Soliton generation in self-injection locking regime. \\textbf{(d)} The FSR beat note signal of the free running multi-frequency laser, the beat note frequency corresponds to the diode chip length of 2500 $\\mu m$. \\textbf{(e)} Heterodyne signal between self-injection locked diode laser and narrow linewidth fiber laser -- blue curve, Voigt fit -- red curve. \\textbf{(f)} repetition rate signal of a single-soliton state, central frequency corresponds to a WGM cavity with 5.5 mm diameter (ESA RBW is $1$~kHz).}\n\t\\label{ris:image2}\n\\end{figure*}\n\nA schematic view and a picture of the experimental setup is presented in Fig.\\ref{ris:image1}. \nThe laser beam from a free-space multi-frequency InP diode is collimated and coupled to a MgF$_2$ WGM resonator with a glass (BK-7) prism \\cite{Gorodetsky1999}. Resonantly backscattered Rayleigh radiation returns to the diode laser and forces self-injection locking of the laser frequency to the microresonator's WGM mode. The output beam is collimated to a single-mode fiber and analyzed with an optical spectrum analyzer (OSA), on a photodiode (PD) with an oscilloscope (OSC), and an electrical spectrum analyzer (ESA). The repetition rate of the soliton pulses is monitored by a fast photodiode and ESA. The detuning of the laser frequency from an optical resonance is monitored on a PD with an oscilloscope. A narrow linewidth tunable fiber laser is used for the heterodyne linewidth measurements.\n\nFor pumping millimeter-sized MgF$_2$ resonators, ordinary packaged uncapped free-space multi-frequency laser diodes were used (Seminex, chip length $L=2500$~$\\mu$m, central wavelengths 1535~nm, 1550~nm and 1650~nm covering spectral intervals of $\\Delta \\lambda \\sim 10$~nm and a total power of $P\\sim$200~mW). Generation of the self-injection locked soliton combs with a repetition rate signal linewidth $\\sim1$~kHz was observed when the laser diode driving current was manually  adjusted to red detune the pump frequency in self-injection locked regime within a soliton-supporting high-Q cavity resonance.\n\nThe experimental results demonstrated in Fig.\\ref{ris:image2} were obtained with a MgF$_2$ resonator with a diameter of $5.5$~mm and edge curvature radius of $500 \\mu$m corresponding to the free spectral range (FSR) of $\\sim 12.5$~GHz (inverse of the pulse round-trip time in the microresonator). The group velocity dispersion (GVD) for all tested laser frequencies is anomalous allowing for the generation of DKS. The microresonator was manufactured by precise single-point diamond turning (DAC ALM lathe, see SI)\\cite{Tanabe2016}. The ultra-high intrinsic Q-factor exceeding $10^9$ was achieved by polishing with diamond slurries\\cite{Maleki2001}. Experimental results with other microresonators pumped at different wavelength are presented in SI.\n\nThe laser diode used to obtain results in Fig.\\ref{ris:image2}(a) has an optical spectrum consisting of tens of incoherent lines covering $\\sim 10$~nm with mode spacing $\\Delta f=\\frac{c}{2Ln} \\approx 17.68$~GHz around the central wavelength $1535$~nm and $200$~mW maximum output power. The intensity in the laser gain region is approximately uniformly distributed between the lines. Fig.\\ref{ris:image2}(d) shows the noisy beatnote signal from the adjacent lines of the free-running laser diode at $17.68$~GHz with $\\sim 1$~MHz full width at half maximum (FWHM) linewidth. The light back-reflected from the microresonator due to resonant Rayleigh backscattering, provided in case of crystalline materials mostly by surface inhomogeneities, leads to natural feedback for the laser diode. Back reflection is measured in our case at $10^{-3}$ intensity level (see SI for details). The backscattering intensity depends on the degree of loading (coupling efficiency) of the resonator, and thus can be regulated by changing the gap between the prism and the resonator\\cite{Yarovitsky1998}. \n\n\\section*{Self-injection locking with a multi-frequency laser diode}\n\nFig.\\ref{ris:image2}(b) demonstrates a collapse of the wide spectrum of the multi-frequency laser diode to a single-frequency line. In case of a multi-frequency diode in the self-injection locking regime, the power from multiple modes due to mode competition is transferred into a single narrow line, and its output power increases. In this way, the microresonator behaves not like a simple filter cavity but plays an active role in lasing. Fig.\\ref{ris:image2}(b) illustrates that in the case of self-injection locking to the WGM microresonator, the power in the dominant line increases by $\\sim 7$~dBm. This effect gives a significant additional advantage of using longer multi-frequency FP diode chips with higher power as compared to DFB lasers (with maximal power $\\sim 40$~mW at telecom wavelengths) \\cite{Maleki2010,Maleki2015}. The asymmetry of residual laser modes in the self-injection locking regime in Fig.\\ref{ris:image2}(b), with the dip on the high-frequency wing, is associated with the anomalous interaction of spectral modes in a semiconductor laser\\cite{Bogatov1975, ahmed2002,AhmedYamada2010}. Note that the contrast between the dominant line and residual lines of an order of 35~dB may in our case be significantly improved with an additional coupler prism as a drop port. In this case, the residual modes of the laser will be filtered out by the resonator. Recently a 446.5 nm  self-injection locked laser \\cite{Donvalkar2018} was demonstrated with sub-MHz linewidth by using a high-${Q}$ (${Q>10^9}$) WGM ${\\rm MgF_2}$ microresonator in conjunction with a multi-longitudinal-mode laser diode. The presented blue FP laser had a two peak spectrum at low driving current in the free running regime. Longitudinal mode competition and conversion efficiency of the laser power to the power of a single mode were not considered.  \n\nWe used the heterodyne technique to measure the instantaneous laser linewidth in the self-injection locking regime operating at $1550$~nm. The beatnote of the self-injection locked diode laser with the narrow linewidth tunable fiber laser (Koheras Adjustik) was analyzed on a fast PD and is shown in Fig.\\ref{ris:image2}(e). The Voigt profile\\cite{Stephan2005} fit provides a Lorentzian width contribution due to laser white frequency noise of $370$Hz, and Gaussian contribution due to flicker frequency noise of $1.7$ kHz. The theory of self-injection locking developed in\\cite{Kondratiev2017} allows  this linewidth to be estimated (see the Supplementary Information (SI) for details) with a good agreement.  In this way, an efficient and compact single-mode diode laser with a narrow linewidth at kHz range is demonstrated.\n\n\\section*{Soliton microcomb with a multifrequency laser}\n\n\\begin{figure*}[ht]\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{Pulse1.pdf}\n\t\\caption{\\textbf{Soliton generation with a laser diode.} \\textbf{(a)} Cavity mode response on oscilloscope when the frequency of the laser is swept through the WGM cavity resonance, characteristic for the self-injection locking, with step-like transition to a soliton state. \\textbf{(b)} The autocorrelation of a soliton waveform obtained from the spectrum in Fig.\\ref{ris:image2}b (blue) and the theoretical fit (red). $\\tau_R = 80 ps$ is the roundtrip time.}. \n\t\\label{ris:image3}\n\\end{figure*}\n\n\nFig.\\ref{ris:image3}(a) illustrates the typical cavity response on PD when the frequency of the laser is swept through the WGM cavity resonance, characteristic for the self-injection locking\\cite{Kondratiev2017}, with step-like transition to a soliton state. The self-injection locking range, calibrated with a fiber-loop cavity, was of an order of $1$~GHz, depending on the quality-factor of the WGM resonance. Region (a) corresponds to the free-running laser (Fig.\\ref{ris:image2}(a)), region (b) corresponds to the case of a self-injection locking regime without a frequency comb (Fig.\\ref{ris:image2}(b)), and region (c) is a soliton existence region analogous to the soliton step obtained in experiments with tunable single frequency lasers\\cite{Kippenberg2014} (Fig.\\ref{ris:image2}(c)). We use MgF$_2$ material for microresonator fabrication where thermo-optical instability is suppressed due to the same sign of thermo-refractive and thermal expansion effects and stable soliton generation is possible. In the self-injection locking regime, if the characteristic time of back-reflection is shorter than the thermal relaxation time, the laser frequency follows the cavity thermally shifted resonance and thermo-optical instabilities are suppressed. We observed partial suppressing of thermal nonlinearity while scanning the laser across the cavity resonance (Fig.\\ref{ris:image3}(a)) due to self-injection locking and we were able to tune into stable regime of soliton generation without thermal jumps.\n\nBy gradually red detuning the diode laser frequency with the current, but staying in the locked regime, we could smoothly switch the system into a soliton comb regime (predominantly single-soliton one) with a very characteristic sech$^2$(x) envelope Fig.\\ref{ris:image2}(c). The soliton  comb has a span $\\sim 30$~nm with a line spacing of $12.5$~GHz. Additional residual laser lines separated by $17.68$~GHz are visible in optical spectrum, although they are weak and could be filtered out in drop-port configuration. By stronger variation of the diode current it was also possible to switch to a different resonance of the microresonator, jumping from one resonance to another within a single mode family, thus gradually changing the central frequency of the soliton and its bandwidth (see SI Fig.S.6). \n\nThe soliton repetition rate signal in the microwave range is demonstrated in Fig.\\ref{ris:image2}(f). Fig.\\ref{ris:image3}(b) shows the result of the inverse Fourier transform of the spectrum from \\ref{ris:image2}(c) and a fit $2A^2t\/\\sinh(Bt)$ which corresponds to the soliton waveform $A{\\,\\rm sech}(Bt)$, which reveals the soliton duration of $220$~fs. The residual fringes are caused by a dispersive wave formed due to parasitic mode-crossings which one can see in Fig. \\ref{ris:image2}(c). \n\nNote, that no particular technique with amplitude and frequency manipulations and hence no additional equipment was used to control the soliton generation in the presence of thermal nonlinearities\\cite{Kippenberg2014}. This is possible due to the very fast optical feedback which compensates thermal cavity detuning upon switching. Single-soliton states lived several hours in laboratory conditions without any additional stabilisation techniques -- another convenient consequence of self-injection locking. In our experiment coupling rate to soliton resonances was around 15--25\\% due to the non-optimised geometry of the microresonator and prism coupling\\cite{bilenko2017optimisation68043753}, so we did not efficiently use available laser power to generate a wide frequency comb. Nevertheless, our result shows that this proposed method is applicable due to pump power overhead even when coupling conditions are not optimal or the Q-factor is not ultra high, e.g. for integrated microresonators.\n\nWe have checked that the Kerr soliton comb is generated inside the microresonator and is not generated or amplified in the FP laser chip by adding a beam splitter between the laser chip and the microresonator and by observing a spectrum of the light immediately after the gain chip. In this way we confirm that only single frequency lasing (corresponding to  Fig.\\ref{ris:image2}(b)) is observed at the output facet of the laser. It should be noted that in our work the FSRs of the FP diode laser and microresonator did not match.\n\nWe observe that single soliton generation is preferable although multi-soliton states are also possible. This results from a relatively slow transition from CW to soliton regime \\cite{Lobanov2016}, due to low speed of pump frequency tuning in the locked regime. Comparing to previous realisations of tuning methods to obtain soliton states (fast forward scan\\cite{Kippenberg2014}, slow backward tuning\\cite{Karpov16}, pump power kicking\\cite{Brasch2016}), in self-injection locking the frequency tuning is orders of magnitude slower. Considering realistic parameters of our system we estimate the tuning speed is $10^4$ times slower than without self-injection locking\\cite{Kondratiev2017} (see SI for details)\n\nWe have also checked and confirmed the generation of the soliton comb in the same microresonator using the traditional technique with a CW narrow linewidth tunable fiber laser (see SI for details). \n\n\\section*{Discussion and conclusion}\n\nIn conclusion, we have demonstrated a new efficient method for achieving a single-frequency narrow linewidth lasing and independently a method for generating stable Kerr soliton combs directly from multi-frequency high-gain laser diode chips using the self-injection locking effect. This result paves a way to compact, low-noise photonic microwave sources and for generating stable powerful frequency combs, which are important for spectroscopy, LIDAR application, astronomy, metrology and telecommunications.  \n\n\\section*{Methods}\n\n{In experiments we used for the self-injection locking with high-Q MgF$_2$ WGM resonators Indium Phosphide multifrequency and single latitudinal mode laser diode chips. The length of chips was 1.5--2.5 mm and the free running spectrum consisted approximately of 50 FP lines with a beatnote between adjacent diode FP lines 1--3~MHz. The output power was 100--500~mW depending of the diode length and applied current. Such chips are commercially available in wide wavelength range.}\n\n\n\n\n\\section*{Data availability statement}\nAll data used in this study are available from the corresponding authors upon reasonable request.\n\n\\section*{Authors contributions}\nExperiments were conceived by N.G.P., S.K., and M.L.G. Analysis of results was conducted by N.G.P., A.S.V, S.K., G.V.L. and M.L.G.,  N.G.P., A.S.V., S.K. and A.S.G performed measurements with diode lasers and G.V.L. with a fiber laser. G.V.L., N.G.P. and A.S.V. fabricated devices. S.V.P. and M.R. set the research direction relevant for industrial needs - comb source for wearable spectrometer (Samsung Gear). S.V.P. supervised the project from the Samsung. M.L.G. supervised the project. All authors participated in writing the manuscript.\n\n\\begin{acknowledgments}\nThis publication was supported by the Russian Science Foundation (17-12-01413). G.V.L., N.G.P. and A.S.V. were partially supported by the Samsung Research center in Moscow.  The authors gratefully acknowledge valuable discussions with Tobias Kippenberg, Kerry Vahala and Vitaly Vassiliev. The authors express gratefulness to Hong-Seok Lee and Young-Geun Roh from Samsung Advanced Institute of Technologies for help in establishing the project and its further support.\n\\end{acknowledgments}\n\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\la{intro}\nFigure \\ref{figConvergenceintro} is extracted from \\cite{Paper3} and \\cite{Andrea}. \n\\begin{figure}[h]\n\t\\centering \n\t\\includegraphics[width=\\linewidth]{FigConvergence.pdf}\n\t\\vspace{-6.2cm}\n\t\\caption{{\\textbf a)} Maximal cubic coupling showing up in the scattering of the lightest particle in a gapped theory with a single bound-state (in this channel at least)~\\cite{Paper3}. Convergence is perfect when the bound-state mass (measured in units of the lightest mass) is bigger than $\\sqrt{2}$ and quite painful otherwise. {\\textbf b)} The allowed chiral zeroes space of putative pion S-matrices associated to an $SU(2)$ chiral symmetry breaking patterns draws a beautiful peninsula like object with a sharp tip~\\cite{Andrea}.\\protect\\footnotemark \\,Convergence is great almost everywhere except close to the tip where numerics struggle. In those cases where the primal problem struggles, having a dual rigorous bound would be a blessing. This paper is about such dual bounds. }\n\t\\label{figConvergenceintro}\n\\end{figure}\n\\footnotetext{There are, at least, other two structures would benefit a dual description. One is the ``pion lake''~\\cite{Andrea}, found imposing the presence of the physical $\\rho$ resonance only. Another interesting and recent structure is the ``pion river''~\\cite{river}, found imposing additional constraints on the scattering lengths arising from $\\chi$PT and monotonicity of the relative entropy. The dual formulation would allow to rigorously define these structures excluding theories not compatible with the assumed low energy QCD behavior.}\n\nThese works explore the allowed space of physical 4D S-matrices. One parametrizes a vast family of S-matrices compatible with given physical and mathematical assumptions and maximize or minimize quantities within this ansatz to find the boundaries of what is possible. The more parameters the ansatz has, the better is the exploration. As the number of parameters become very large, one hopes that these boundaries converge towards the true boundaries of the S-matrix space. \n\nSometimes this works beautifully as illustrated in the figure; sometimes convergence is painful, to say the least, as also illustrated in the figure.\nIn those cases where convergence is a struggle, what can we do? Sometimes, it is a simple matter of improving the ansatz; sometimes it is not clear what exactly is missing. And in either case, how can we ever tell how close to converging are we  anyways?  \n\nA solution would be to develop a dual numerical procedure -- called the \\text{dual} problem -- where instead of constructing viable S-matrices we would instead rule out unphysical S-matrix space.\\footnote{Such dual bounds were attempted more than 50 years ago already in \\cite{Archeo1, Archeo2, Archeo3, Archeo4}. Would be very important to do some archeology work and revive\/translate\/re-discover\/improve those old explorations in a modern computer friendly era. A beautiful first step is currently being pursued by Martin Kruczenski and Yifei He \\cite{MartinTalkBootstrap}. The conformal bootstrap bounds are also exclusion analysis of this sort \\cite{bootstrapReview}.} Then we would approach the boundaries of the S-matrix space from two sides, dual and primal, and in this way rigorously bracket the true boundaries of the sought after S-matrix space. \nThis was recently achieved in two dimensions for simple models with a single type of particle transforming in some non-trivial global symmetry group \\cite{Monolith}.\\footnote{The primal version of these single particle studies with global symmetry was the subject of \\cite{Martin,Miguel,Lucia}; the case without global symmetry was considered in \\cite{Creutz,Paper2}.} \n\nThis paper concerns two dimensional multi-particle systems with arbitrary mass spectra from this dual perspective, clearly one  step further in the complexity ladder, closer to the full higher dimensional problem.\\footnote{Multi-particle primal problems of this kind were pioneered in \\cite{Paper4,ToAppearSUSY}.} We will also consider a different technical approach, complementary to \\cite{Monolith}, with some aspects which we hope can be more directly transposable to higher dimensions. \n\n\n\\section{Dual optimization and the S-matrix bootstrap}\n\\label{sec2}\n\n\\qquad To achieve the desired dual formulation, it is useful to revisit the S-matrix bootstrap with a slightly different perspective.\n\nIn the \\textit{primal} S-matrix bootstrap formulation\n one constructs scattering amplitudes consistent with a set of axioms, or constraints. Such amplitudes are said to be \\textit{feasible}, that is, they belong to the allowed space of theories. \nOne then optimizes physical observables, such as the interaction strength between stable particles, in the space of feasible amplitudes. The prototypical example is \\cite{Paper2,Creutz}: in a 2D theory with a single stable particle of mass $m$, what is the maximum cubic coupling $g$ consistent with a $2 \\rightarrow 2$ scattering amplitude $M$ satisfying the constraints of unitarity, extended analyticity, and crossing? \n\nIn other words, we would like to solve the optimization problem\n         \\begin{mdframed}[frametitle={Primal problem},frametitlealignment=\\centering,backgroundcolor=blue!6, leftmargin=0cm, rightmargin=0cm, topline=false,\n\tbottomline=false, leftline=false, rightline=false] \\label{primal}\n\t\\vspace{-0.6cm}\n  \\begin{align} &\\underset{\\text{in } M(s)\\text{, }g^2}{\\text{maximize}} && g^2  \\label{primal}\\\\\n        & \\text{constrained by} && \\mathcal{A}(s) \\equiv M(s) - \\Big(M_\\infty -\\frac{g^2}{s-m^2} +\\!\\!\\! \\int\\limits_{4m^2}^\\infty \\! \\frac{dz}{\\pi} \\frac{\\text{Im}M(z)}{s-z {+}i0} {+} \\left(s \\leftrightarrow 4m^2-s \\right)\\Big) =0 \\nonumber\\\\\n        & &&\\text{for } s>4m^2, \\label{analandcrossing}\\\\\n        & \\text{} && \\mathcal{U}(s) \\equiv 2\\text{ Im}M(s) - \\frac{\\lvert M(s)\\rvert^2}{2\\sqrt{s-4m^2}\\sqrt{s}} \\geq  0 \\qquad \\text{for } s>4m^2. \\label{unitarity}\n       \\end{align}\n     \\end{mdframed}\nwhere we maximize over the space of analytic functions $M$, and emphasize that one parameter in this infinite dimensional space is the residue of such functions at $s=m^2$ which is equal to~$-g^2$. \nThe first constraint (\\ref{analandcrossing}), an exact equality, imposes that feasible scattering amplitudes must respect crossing, real analyticity, and have singularities determined by physical processes: poles corresponding to one particle states, and cuts corresponding to multi-particle states.\\footnote{It turns out that there is no loss of generality in omitting subtractions from (\\ref{analandcrossing}), since a more careful analysis shows that the inclusion of those leads to the same result (\\ref{dual bootstrap}). We opt for not including subtractions in the main text for the sake of clarity -- see appendix~\\ref{analyticstuff} for a more detailed discussion.}\n We choose to impose this condition for $s > 4m^2$, but because we maximise over analytic functions, feasible amplitudes will have have this property for all $s$ in the physical sheet.\\footnote{The physical sheet is defined as the first Riemann sheet encountered after analytically continuing from physical kinematics, $s>4m^2$, using the $+ i \\epsilon$ prescription.\n } The convenience of imposing this condition for $s > 4m^2$ will become clear in time. The second constraint (\\ref{unitarity}) is the physical unitarity condition, equivalent to~$\\lvert S(s)\\rvert \\leq 1$. \n\nSince the quantity we are maximising, the objective, is a linear map in the space of analytic functions, the map that evaluates the residue at a point, and since the constraints~(\\ref{analandcrossing}),~(\\ref{unitarity}) are affine and convex respectively, the optimization problem we aim to solve is an infinite dimensional convex optimization problem. For such a simple problem, there are now two directions that can be taken. The first option is to solve the infinite dimensional problem analytically. As is well known by now, this follows from a simple application of the maximum modulus principle~\\cite{Paper2, Creutz}. The second option, available in more complicated situations, is to bring the problem to the realm of computers by maximizing our objective in some finite dimensional subspace of analytic functions. For example, one can consider analytic functions that are, up to poles, polynomial of at most degree $N_\\text{max}$ in some foliation variable $\\rho$ that trivializes the constraint (\\ref{analandcrossing}), as done in~\\cite{Paper3}. This truncated problem can be efficiently solved by a convex optimization software, for example SDPB \\cite{SDPB, scaling}. By choosing and increasing the finite dimensional subspace smartly, one obtains lower bounds to the solution of the primal problem that should converge to the correct bound with more expensive numerics.\n\nThe primal formulation suffers from two important shortcomings. First, for some problems it is hard to identify a simple ansatz, or truncation scheme, that allows for fast convergence. This is often the case in higher dimensional S-matrix bootstrap applications, or when scattering heavy particles in 2D. Second, and perhaps more importantly, one may want to add extra variables and constraints to the primal problem. In the previous example, those variables  and constraints could be, respectively, higher point amplitudes and higher point unitarity equations. It may be the case that a feasible $2 \\rightarrow 2$ amplitude in the original primal problem may no longer be feasible in the enlarged space with extra constraints. In those cases, a point in theory space previously said to be allowed becomes forbidden. It would be more satisfying if bounds on the space of theories obtained by studying some scattering subsector remained true once the full set of QFT constraints were imposed.\\footnote{Much in the same way that CFT data excluded by the numerical conformal bootstrap remains excluded once more crossing equations are included into the system.} To overcome both of this shortcomings, we introduce the dual formulation. We use the coupling maximization problem as a guiding example, before generalizing.\n\nConsider the Lagrangian\\footnote{Note $\\mathcal{A}(s)$ is actually real. }\n\\begBvR{equation}\n\\mathcal{L}(M,w,\\lambda) = g^2 + \\int_{4 m^2}^\\infty ds\\text{ } w(s) \\mathcal{A}(s) + \\lambda(s) \\mathcal{U}(s)  \\label{lagrangian}\n\\end{equation}\nwith $\\lambda(s) \\geq 0$ and define the dual functional\n\\begBvR{equation}\nd(w,\\lambda) =  \\underset{\\{M, g\\}}{\\text{sup}} \\mathcal{L}(M,w,\\lambda)\\label{dualfunctional}\n\\end{equation}\nNotice that the supremum is taken over unconstrained analytic functions $M$.\\footnote{It is useful to think of analytic functions as being defined through their independent real and imaginary parts along a line. Of course, if the dispersion (\\ref{analandcrossing}) were to hold, then those would not be independent. However, since we maximise over generic analytic functions, we are free to treat $\\text{Re }M$ and $\\text{Im } M$  for $s>4m^2$ as independent.} The dual functional $d$ is the central object in the dual formulation due to the following property: \n\n  \\begin{mdframed}[frametitle={Weak Duality},frametitlealignment=\\centering,backgroundcolor=black!10, leftmargin=0cm, rightmargin=0cm, topline=false,\n\tbottomline=false, leftline=false, rightline=false] \\label{duality}\n\t\\begBvR{equation}\n\\text{Let the solution of the primal problem be $g_*^2$. Then\t}\nd(w,\\lambda) \\geq g_*^2. \\label{weak}\n\\end{equation}\n     \\end{mdframed}\nWeak duality holds due to two observations. First, note that since\n\\begBvR{equation}\n\\underset{\\{\\lambda \\geq 0, w\\}}{\\text{inf  }} \\mathcal{L}(M,w,\\lambda) = \n\\begin{cases}\n    g^2& \\text{if } M \\text{ is feasible}\\\\\n    -\\infty              & \\text{otherwise},\n\\end{cases}\\label{since} \n\\end{equation}\nwe have that \n\\begin{equation*}\n\\normalfont g_*^2 = \\underset{\\{M, g\\}}{\\text{sup}} \\left[ \\underset{\\{\\lambda \\geq 0, w\\}}{\\text{inf  }} \\mathcal{L}(M,w,\\lambda)\\right].\n\\end{equation*}\nWeak duality then follows from the max-min inequality\n\\begBvR{equation}\nd(w,\\lambda) \\geq  \\underset{\\{\\lambda \\geq 0, w\\}}{\\text{inf  }} \\left[ \\underset{\\{M, g\\}}{\\text{sup}} \\mathcal{L}(M,w,\\lambda) \\right] \\geq  \\underset{\\{M, g\\}}{\\text{sup}} \\left[ \\underset{\\{\\lambda \\geq 0, w\\}}{\\text{inf  }} \\mathcal{L}(M,w,\\lambda)\\right] = g_*^2. \\label{maxmin}\n\\end{equation}\n\nExploring the $\\{w,\\lambda\\}$ space, the space of dual variables, we therefore obtain upper bounds on the values of $g$ allowed by the axioms and exclude regions in theory space. This, in turn, partially solves the first shortcome of the primal formulation: by providing upper limits on the coupling, it bounds how far from converging an ineffective primal truncation scheme may be. To find the best possible upper bound, we solve the\n                  \\begin{mdframed}[frametitle={Dual problem (generic)},frametitlealignment=\\centering,backgroundcolor=red!6, leftmargin=0cm, rightmargin=0cm, topline=false,\n\tbottomline=false, leftline=false, rightline=false] \n\\vspace{-0.6cm}\n  \\begin{align} &\\underset{\\text{in } w(s)\\text{, }\\lambda(s)}{\\text{minimize}} && d(w,\\lambda) \\label{dual generic}\\\\\n        & \\text{constrained by} && \\lambda(s)\\geq 0\\nonumber\n              \\end{align}\n     \\end{mdframed}\n\nThe construction of dual functionals from a primal optimization problem is standard in optimization theory, but the particularities of the problems encountered in the S-matrix bootstrap lead to important simplifications. One of these is that the analyticity of the scattering amplitude is inherited by the dual variable $w(s)$, conjugate to the analyticity constraint. In fact, let's define a ``dual scattering function\", $W(s)$\\footnote{It is worth stressing that the introduction of an analytic function $W(s)$ is not mandatory. It is possible to work with real densities $w(s)$ and follow the argument presented in this section using the same logic. This possibility is particularly useful in higher dimensions if one wants to assume no more than the proven analyticity domains~\\cite{Archeo4}.}, odd under crossing and whose absorptive part is $w(s)$: \\begBvR{equation}\nW(s) \\equiv  \\frac{1}{\\pi}\\int_{4m^2}^\\infty dz \\frac{w(z)}{s-z {+}i0} - \\left(s \\leftrightarrow 4m^2-s \\right). \\label{disp}\n\\end{equation}\n\nThen, swapping a few integrals in (\\ref{lagrangian}) and using $\\frac{1}{\\left(s-z{\\pm}i0\\right)} = \\mp i \\pi \\delta(s-z) + \\mathcal{P}\\frac{1}{(s-z)}$ leads to a very simple representation for the lagrangian as\n\\begBvR{equation}\n\\mathcal{L}(M,W,\\lambda) = g^2 \\left(1 + \\pi W(m^2)\\right)  + \\int_{4 m^2}^\\infty ds\\text{ } \\text{Im}\\left(W(s) M(s)\\right) + \\lambda(s)\\, \\mathcal{U}(s). \n\\label{eq13}\n\\end{equation}\nNote that the Lagrangian density is now manifestly local in $M$ as the Cauchy kernel from~(\\ref{analandcrossing}) has been nicely absorbed into $W$. This locality, together with the quadratic nature of the constraint equations\\footnote{Dispersions for higher point amplitudes are no longer expected to be quadratic in lower point functions due to the presence of Landau singularities.} leads to the next simplification over  generic dual optimization problems: we can perform both the maximization over $M$ in (\\ref{dualfunctional}) and the minimization over~$\\lambda$ in~\\eqref{dual generic} exactly. We now analyze those in sequence. \n\nBefore doing that, first notice, linearity of $\\mathcal{L}$ under $g^2$ implies that \n\\begBvR{equation}\nd(W, \\lambda) = + \\infty \\,\\,\\,\\, \\text{ unless } \\,\\,\\,\\,  \\pi W(m^2)=-1. \\label{normunless}\n\\end{equation}\nThis means that unless $W$ is properly normalized at $m^2$, the bounds obtained from the dual functional are vacuous. Hence, in solving the dual problem, there is no loss of generality in restricting ourselves to the space of $W$ satisfying the constrain in (\\ref{normunless}).\n \n The linear Lagrange equations with respect to variations of $M(s)$ for $s>4m^2$ results in\n \\begBvR{equation}\n M_\\text{critical}(s) = \\left[\\text{Im}(W(s))\/\\lambda(s) + i \\left(2\\lambda(s)  + \\text{Re}(W(s))\/\\lambda(s)\\right)\\right] \/(2 \\rho^2_{11}) .\n\\nonumber\n \\end{equation}\nwhere $ \\rho^2_{11} = 1\/(2 \\sqrt{s-4m^2}\\sqrt{s})$. Second order variations show that, indeed, this is a local maximum provided $\\lambda(s)>0$. It follows from the definition (\\ref{dualfunctional}) that, provided $\\pi W(m^2)=-1$,\n\n\\begBvR{equation}\nd(W, \\lambda) = \\int_{4m^2}^\\infty ds    \\left( \\frac{\\lvert W(s)\\rvert^2}{4 \\lambda(s)} + \\lambda(s) + \\text{Re}W(s))\\right)\/ \\rho^2_{11} . \\la{dWL}\n\\end{equation}\n\nNext, we minimize over $\\lambda$ leading to $\\lambda=|W(s)|\/2$. The result is $D(W) \\equiv \\underset{\\lambda \\geq 0}{\\text{inf  }} d(W,\\lambda))$ given by \n\\begBvR{equation}\nD(W) = \\int_{4m^2}^\\infty ds  \\left(\\text{Re}(W(s)) + \\lvert W(s)\\rvert \\right)\/  \\rho^2_{11}. \\label{bfunc} ,\n\\end{equation}\nin which case\\footnote{Note that unitarity is automatically saturated once we minimize in $\\lambda$.} \n\\begBvR{equation}\n M_{\\text{critical}}(s) = \\frac{i}{\\rho^2_{11}}\\left(1 + \\frac{W^*}{\\lvert W\\rvert} \\right).\\nonumber\n\\end{equation}\n\nIn sum, the dual of (\\ref{primal}) simplifies to\n\n                  \\begin{mdframed}[frametitle={Dual problem (S-matrix bootstrap)},frametitlealignment=\\centering,backgroundcolor=red!6, leftmargin=0cm, rightmargin=0cm, topline=false,\n\tbottomline=false, leftline=false, rightline=false] \n\t\\vspace{-0.4cm}\n  \\begin{align} &\\underset{\\text{in } W(s)}{\\text{minimize}} && D(W)=\\int_{4m^2}^\\infty ds  \\left(\\text{Re}(W(s)) + \\lvert W(s)\\rvert \\right)\/  \\rho^2_{11} \\label{dual bootstrap}\\\\\n        & \\text{constrained by} && \\pi W(m^2)=-1.  \\label{norm}\n      \\end{align}\n     \\end{mdframed}\n\nThe dual problem can be tackled numerically through the same strategy used for the primal problem, that is, restricting our search to a finite dimensional subspace of analytic $W$s. For example, one could use the $\\rho$ foliation variables to write the ansatz\\footnote{The Ansatz (\\ref{wansatz}) is consistent with the dispersion (\\ref{disp}). In particular, the poles in (\\ref{wansatz}) correspond to a delta function contribution in $w(s)$.}\n\\begBvR{equation}\nW_{\\text{ansatz}}(s) = \\frac{1}{s (4m^2-s)}\\sum_{n=1}^{N_\\text{max}}a_n (\\rho(s)^n - \\rho(t)^n), \\label{wansatz}\n\\end{equation}\nwhere \n\\begBvR{equation}\n\\rho(s) = \\frac{\\sqrt{2m^2 } - \\sqrt{4m^2 -s}}{\\sqrt{2m^2} + \\sqrt{4m^2 -s}},\n\\label{rhovariabledef}\n\\end{equation}\nand minimize the functional (\\ref{dual bootstrap}) in the finite dimensional space parametrized by the $a_n$'s. Note that the constraint (\\ref{norm}) is a linear constraint in this space. The functional (\\ref{bfunc}) is nonlinear, but it is convex in $W$. Performing such minimization, say, in \\texttt{Mathematica} shows that, as one increases $N_\\text{max}$, the result of the problem (\\ref{dual bootstrap}) converges to the result of the primal problem (\\ref{primal}). This is expected if our optimization problem satisfies\n\n\\begin{mdframed}[frametitle={Strong Duality},frametitlealignment=\\centering,backgroundcolor=black!10, leftmargin=0cm, rightmargin=0cm, topline=false,\n\tbottomline=false, leftline=false, rightline=false]\nThe solutions to the primal (\\ref{primal}) and dual problem (\\ref{dual bootstrap}) are identical, i.e. $g_*^2 = \\underset{\\text{in } W}{\\text{min}} \\text{   } D(W).$ In other words, the $\\ge$ symbol in (\\ref{weak}) is actually an $=$ sign.\n     \\end{mdframed}\nThis property is argued for in appendix \\ref{strong}.\n \nTo explain how the dual formulation solves the second shortcoming of the primal optimization, and in view of the applications in section \\ref{application}, let's consider a slightly different class of S-matrix Bootstrap problems. Consider a gapped theory with two real stable particles of masses $m_1$ and $m_2$ respectively, $m_1<m_2$, and suppose we were interested in maximizing the cubic coupling of particle $m_1$. Let $\\mathbb{M}_{ab} = M_{a\\to b}$. Assuming $P$ and $T$ symmetry, $\\mathbb{M}$ is a symmetric matrix. We would like to solve the problem\n                  \\begin{mdframed}[frametitle={Primal problem (matrix)},frametitlealignment=\\centering,backgroundcolor=blue!6, leftmargin=0cm, rightmargin=0cm, topline=false,\n\tbottomline=false, leftline=false, rightline=false]\n\t\\vspace{-0.6cm}\n  \\begin{align} &\\underset{\\text{in } \\mathbb{M}}{\\text{maximize}} && g^2  \\label{primal2}\\\\\n        & \\text{constrained by} && \\mathbb{A}(s) =0 &&&\\text{for } s>4m_1^2, \\label{analandcrossing2}\\\\\n        & \\text{} && \\mathbb{U}(s) \\equiv 2\\text{ Im}\\,\\mathbb{M}(s) - \\mathbb{M}^\\dagger \\text{\\outline{$\\rho$}}\\, \\mathbb{M} \\succeq 0 \\qquad \\qquad &&&\\text{for } s>4m_1^2. \\label{unitarity2}\n       \\end{align}\n     \\end{mdframed}\nwhere $\\mathbb{A}_{a b} \\equiv \\mathcal{A}_{a \\to b}$ are analogous to (\\ref{analandcrossing}) and impose the correct dispersion relations for the amplitudes $M_{a \\to b}$ (see e.g. (\\ref{AA}) in the next section). Here $\\text{\\outline{$\\rho$}}$ are the phase space factors for the intermediate states (see e.g. (\\ref{rhoMatrix}) in the next section). \nTo obtain the dual problem, we introduce the Lagrangian\n\\begBvR{equation}\n\\mathcal{L}(\\mathbb{M}, \\text{\\outline{w}} ,\\text{\\outline{$\\Lambda$}}) = g^2 + \\int_{4 m_1^2}^\\infty ds\\text{ } \\text{Tr}\\left( \\text{\\outline{w}} \\cdot  \\mathbb{A} (s) + \\text{\\outline{$\\Lambda$}} \\cdot \\mathbb{U} (s) \\right), \\label{lagrangian2}\n\\end{equation}\nwhere  $\\text{\\outline{w}}$ and $\\text{\\outline{$\\Lambda$}}$ are respectively symmetric and hermitian matrices of dual variables with~$\\text{\\outline{$\\Lambda$}}$ positive semi-definite. The new dual functional\n\\begBvR{equation}\nd(\\text{\\outline{w}}, \\text{\\outline{$\\Lambda$}}) =  \\underset{\\mathbb{M}}{\\text{sup  }} \\mathcal{L}(\\mathbb{M}, \\text{\\outline{w}} ,{\\text{\\outline{$\\Lambda$}}}) \\label{dualfunctional2}\n\\end{equation}\nsatisfies weak duality by similar arguments as those in equations (\\ref{since}-\\ref{maxmin}). The dual optimization problem is \n   \\begin{mdframed}[frametitle={Dual problem (matrix)},frametitlealignment=\\centering,backgroundcolor=red!6, leftmargin=0cm, rightmargin=0cm, topline=false,\n\tbottomline=false, leftline=false, rightline=false] \n\t\\vspace{-0.6cm}\n  \\begin{align} &\\underset{\\text{in } \\text{\\outline{w}}(s)\\text{, }\\text{\\outline{$\\Lambda$}}(s)}{\\text{minimize}} && d(\\text{\\outline{w}}, \\text{\\outline{$\\Lambda$}}) \\label{dual 2}\\\\\n        & \\text{constrained by} && \\text{\\outline{$\\Lambda$}}(s)\\succeq 0. \\nonumber\n      \\end{align}\n     \\end{mdframed}\n Note that an upper bound on the solution of the primal problem (\\ref{primal}) is obtained by  choosing minimizing $d$ in the subspace $\\text{\\outline{w}}_{ab}(s) = \\delta_a^{11} \\delta_b^{11} w(s)$, $\\text{\\outline{$\\Lambda$}}_{ab} =  \\delta_a^{11} \\delta_b^{11} \\lambda(s)$, $\\lambda\\geq0$. This is equivalent to the dual problem obtained by including only the amplitude $M_{11\\to11}$ in the bootstrap system, or primal problem.  Restricting to a scattering subsector in the dual formulation provides true bounds to the more complete optimization problem. Conversely, bounds obtained by studying some restricted space of amplitudes and constrains remain valid once extra axioms and degrees of freedom are considered. We hope it is clear that the argument provided by means of an example is generic. This solves the second shortcoming of the primal formulation.\n     \n\n\\section{An application}\n\\label{application}\n\\subsection{The setup}\nWe now turn our attention to much richer S-matrix bootstrap. We consider a theory with two particles of mass $m_1$ and $m_2>m_1$. We will \\textit{not} assume any global symmetry.  For concreteness, we will take\\footnote{Setting $m_1=1$ simply sets our units. All $m_2> \\sqrt{2}$ would then give very similar plots\/conclusions. We could also consider $m_2<\\sqrt{2}$; the plots are a little bit less eye pleasing in that case. The significance of the transition point $m_2^*=\\sqrt{2}$ is that this is the crossing invariant point for the $11\\to 11$ process; on either sign of this point residues have different signs leading to quite different optimization results.}\n\\begBvR{equation}\nm_1=1\\,, \\qquad m_2=3\/2 \\,.\\nonumber\n\\end{equation}\nThere are a priori four couplings involving these two particles: $g_{111},g_{112},g_{122},g_{222}$. They would show up as $s$-channel residues in the various scattering amplitudes:\n\\begBvR{equation}\n\\begin{array}{c|c|c}\n\\text{Amplitude} & \\text{Exchange of particle } 1 & \\text{Exchange of particle } 2  \\\\  \\hline\n11\\to 11 &  {\\color{red}g_{111}^2}  &  {\\color{blue} g_{112}^2} \\\\  \\hline\n11\\to 12 &{\\color{red}g_{111}}{\\color{blue} g_{112}}  &{\\color{blue} g_{112}}{\\color{cadmiumgreen}g_{122}} \\\\  \\hline\n12\\to 12 & {\\color{blue} g_{112}^2} & {\\color{cadmiumgreen} g_{122}^2} \\\\ \\hline\n11\\to 22 & {\\color{red} g_{111}} {\\color{cadmiumgreen} g_{122}}  & {\\color{blue} g_{112}} {\\color{magenta}g_{222}} \\\\ \\hline\n12\\to 22 & {\\color{blue} g_{112}}{\\color{cadmiumgreen} g_{122}} & {\\color{cadmiumgreen} g_{122}} {\\color{magenta} g_{222}} \\\\ \\hline\n22\\to 22 & {\\color{cadmiumgreen} g_{122}^2}  & {\\color{magenta} g_{222}^2 }\n\\end{array} \\nonumber\n\\end{equation}\nWe will not consider the full coupled system of six amplitudes. Instead we will consider a nice closed subset involving the $11\\to 11$, $11\\to 12$ and (the forward) $12\\to 12$ processes only (that is, the first three lines in the table). As such we will be insensitive to $g_{222}$. We will furthermore consider a section of the remaining three-dimensional space where $g_{122}=0$ so that the problem simplifies slightly to\\footnote{The analysis for any other fixed value of $g_{122}$ follows identically, see more at the end of this section. }\n\\begBvR{equation}\n\\begin{array}{c|c|c}\n\\text{Amplitude} & \\text{Exchange of particle } 1 & \\text{Exchange of particle } 2  \\\\  \\hline\n11\\to 11 & {\\color{red}g_{111}^2} & {\\color{blue} g_{112}^2} \\\\  \\hline\n11\\to 12 &{\\color{red}g_{111}}{\\color{blue} g_{112}} & 0 \\\\  \\hline\n12\\to 12 & {\\color{blue} g_{112}^2} & 0 \n\\end{array} \\nonumber\n\\end{equation}\nand our main goal here is to explore the allowed two dimensional $(g_{112},g_{111})$ space. A convenient way to find the boundary of this space is by shooting radially. We fix an angle $\\beta$ and define a radius $R$ as\n\\begBvR{equation}\n(g_{112},g_{111}) = R(\\cos\\beta,\\sin\\beta) \\,. \\nonumber\n\\end{equation}\nThen we find the maximum value of $R$ for each $\\beta$ choice to plot the full two-dimensional space. \n\nIn the primal language we will get larger and larger $R$'s as our ansatz is more and more complete. In the dual language we will rule out smaller and smaller $R$ as we improve our ansatz. Sandwiched between the two will be the true (two dimensional section of the) boundary of the S-matrix space. \n\nIt is equally straightforward to fix $g_{122}$ to any other value and analyze another 2d section in this way or even collect various values of $g_{122}$ to construct the full $3D$ space. We leave such detailed scans for the future when we will have more realistic setups designed to bootstrap particular relevant physical theories such as the (regular and tricritical) Ising model (perturbed by thermal and magnetic deformations) as discussed in the conclusions. \n\n\\subsection{Single Component Horn}\n\\label{Horn}\nLet us start our search for the two dimensional section of the allowed S-matrix space by focusing on the constraints arising from the single $M=M_{11\\to 11}$ component alone. \n\nThis is a warm up section and many of the results here are not new: indeed, the primal formulation of single component scattering has been the subject of \\cite{Paper2}; a minor new ingredient we will consider here is the radial search element. (The radial problem for the space of S-matrices with $O(N)$ symmetry and no bound states was introduced  in~\\cite{Monolith}.) In appendix H of \\cite{Paper4} an almost identical primal problem was solved analytically; the analytic curves in figure \\ref{figHorn} are obtained by trivially adapting the arguments therein. The dual formulation for these single component cases with several exchanges masses, however, will be novel and provide very useful intuition for the most general case.  \n\nThe primal radial problem can be compactly formulated as \n\n\\begin{mdframed}[frametitle={Primal Radial Problem for Single Component},frametitlealignment=\\centering,backgroundcolor=blue!6, leftmargin=0cm, rightmargin=0cm, topline=false,bottomline=false, leftline=false, rightline=false] \n\\vspace{-0.6cm}\n\\begin{align} &\\underset{\\text{in } {M, R^2}}{\\text{maximize}} &&  R^2\\nonumber\\\\\n& \\text{constr. by}  && \\text{Res}_{m_1^2}(M)=R^2\\sin^2\\beta, \\quad \\text{Res}_{m_2^2}(M)=R^2\\cos^2\\beta \\label{radialcondition}\\\\ \n& s\\geq4m_1^2  && \\mathcal{A}(s)=M(s){-}M_\\infty+\\left(\\frac{g_{111}^2}{s-m_1^2}{+}\\frac{g_{112}^2}{s-m_2^2}{-}\\frac{1}{\\pi}\\int_{4m_1^2}^\\infty dz\\,\\frac{\\IM M(z)}{z-s} +(s\\leftrightarrow t)\\right){=}0\\nonumber\\\\\n& s\\geq 4m_1^2&& \\mathcal{U}(s)=2\\IM M(s) -\\rho_{11}^2 |M(s)|^2 \\geq 0. \n \\label{primal bootstrap 11to11}\n\\end{align}\n\\end{mdframed}\n\nWe will now construct the dual problem. If it were not for the radial additional equality constraints~\\eqref{radialcondition} the corresponding dual problem would be given already in eq.~\\eqref{dual bootstrap}.\nIn this case we need to introduce additional Lagrange multipliers $\\nu_1$ and $\\nu_2$\nto the lagrangian~\\eqref{lagrangian}\n\\begBvR{equation}\n\\mathcal{L}=R^2+\\nu_1 (\\text{Res}_{m_1^2}(M)-R^2\\sin^2\\beta)+\\nu_2 (\\text{Res}_{m_2^2}(M)-R^2\\cos^2\\beta)+\\int_{4m_1^2}^\\infty ds\\,\\mathcal{A}(s)w(s)+\\mathcal{U}(s)\\lambda(s).\n\\label{lagrangianhorn}\n\\end{equation}\nNow we follow the logic of section \\ref{sec2} verbatin modulo a few small differences inherent to the radial nature of the primal problem which we will highlight. First of all note that the maximum of the Lagrangian with respect to $R^2$ yields a bounded result only when\n\\begBvR{equation}\n1-\\nu_1 \\sin^2\\beta-\\nu_2 \\cos^2\\beta=0.\\nonumber\n\\end{equation}\nNext, identifying $w(s)=\\IM W(s)$ with $W(s)$ given by (\\ref{disp}) as before will lead to a beautiful dual problem formulation with a totally local optimization target. Importantly \n\\begBvR{equation}\n\\int_{4m_1^2}^\\infty ds\\,\\mathcal{A}(s)w(s)= \\int_{4m_1^2}^\\infty ds\\,\\text{Im}(M(s)W(s))+ \\pi \\text{Res}_{m_1^2}(M) W(m_1^2)+\\pi  \\text{Res}_{m_2^2}(M) W(m_2^2)\\nonumber\n\\end{equation}\nso we see that the optimization with respect to the parameters $ \\text{Res}_{m_i^2}(M) $ identifies the lagrange multipliers $\\nu_i$ with the normalization of the dual functional at the stable mass values $W(m_i^2)$. All in all we therefore obtain the simple dual problem radial generalization of (\\ref{dual bootstrap}) as \n\\begin{mdframed}[frametitle={Dual Radial Problem for Single Component},frametitlealignment=\\centering,backgroundcolor=red!6, leftmargin=0cm, rightmargin=0cm, topline=false,bottomline=false, leftline=false, rightline=false] \n\\vspace{-0.3cm}\n\\begin{align} &\\underset{\\text{in } {W}}{\\text{minimize}} &&  D(W)=\\int_{4m^2_1}^\\infty ds  \\left(\\text{Re}(W(s)) + \\lvert W(s)\\rvert \\right)\/  \\rho^2_{11}\\nonumber\\\\\n& \\text{constrained by}  && 1+\\pi\\, W(m_1^2)\\sin^2\\beta+\\pi\\, W(m_2^2)\\cos^2\\beta=0.\n\\label{dual bootstrap 11to11}\n\\end{align}\n\\end{mdframed}\n\n\nNotice again the nice complementarity between the pole singularities associated to bound states in the physical amplitude and the absence of poles in the ``dual scattering function\" $W$ given by (\\ref{disp}), replaced instead by the simple normalization conditions (\\ref{dual bootstrap 11to11}). Conversely, when we maximize effective couplings in theories without bound-states the primal S-matrices have no bound-states and the dual functionals have poles \\cite{Monolith}. \n\n\\begin{figure}[t]\n\t\\centering \n\t\\includegraphics[width=\\linewidth]{Hornv1-crop.pdf}\n\t\\caption{Numerical bounds on the coupling space $\\{g_{111},g_{112}\\}$. The blue shaded regions enclose the allowed points for different $N_{\\text{max}}$ in our primal ansatz. The red shaded regions mark the points that are rigorously excluded. The thin black analytic curve is the boundary of the allowed region \\cite{Paper4}. \n\t As we increase $N_{\\max}$ from 1 to 5 in the primal problem, the blue regions enlarge, allowing for more and more points and eventually converging to touch the boundary of the permitted space (this is more evident in the ``horn'' region). In the dual strategy as we increase $N_{\\max}$ from 1 to 5 we exclude more and more points. At convergence the excluded region touches the boundary of the allowed space. We restrict the plot to the first quadrant since it is symmetric under $g \\leftrightarrow -g$.}\n\t\\label{figHorn}\n\\end{figure}\n\n\nIn figure~\\ref{figHorn} we show the numerical results for both the primal (inner blue shaded regions) and the dual problem (outer red shaded regions).\n\n\n\\subsection{Multiple Component Kinematics}\n\nNext we consider the full system with $11\\to 11$, $11\\to 12$ and \\textit{forward} $12\\to 12$ amplitudes.\\footnote{As reviewed in detail in \\cite{Paper4} when a particle of type $1$ scatters with a particle of type $2$ it can either continue straight (\\textit{forward amplitude}) or bounce back (\\textit{backward amplitude}). Here we consider the forward process only. This process is nicely crossing symmetric. (The backward process is not; instead it is related by crossing to $11\\to 22$ scattering so considering this backward process would require more scattering processes to close the system of {unitarity} equations.)} The two dimensional kinematics of the $11\\to 11$ process and of the \\textit{forward} $12\\to 12$ process are reviewed in great detail in section 2 of \\cite{Paper4} so here we will mostly focus on the new $11\\to 12$ process.\\footnote{This process was not considered in \\cite{Paper4} because it violates $\\mathbb{Z}_2$ symmetry. Here we don't have $\\mathbb{Z}_2$ symmetry so it is the first most natural process to consider after the lightest $11\\to 11$ scattering amplitude.} \nThis scattering process is a nice fully symmetric process. No matter which channel we look at it, it always describes two particles of type $1$ (in the infinite future or past) scattering into a particle of type $1$ and another of type $2$. As such \n\\begBvR{equation}\nM_{11\\to 12}(s,t,u)\\nonumber\n\\end{equation}\nis fully symmetric under any permutation of the three Mandelstam variables $s,t,u$. Of course, they are not independent. Besides \n\\begBvR{equation}\ns+t+u=3m_1^2+m_2^2 \\la{Plus}\n\\end{equation} \nwhich holds in any dimension, we have the two dimensional constraint \n\\begBvR{equation}\ns t u=m_1^2\\left(m_1^2- m_2^2\\right){}^2 \\la{Times}\n\\end{equation}\n\nEquations (\\ref{Plus}) and (\\ref{Times}) describe a curve. Its projection into real $s,t,u$ is given by the solid curved blue lines in figure \\ref{triangle}. \n\\begin{figure}[t]\n\t\\centering \n\t\\includegraphics[width=\\linewidth]{1112Triangle.pdf}\n\t\\caption{Maldelstam Triangle for $11\\to 12$ scattering. The x-axis is given by $x=(s+2 t-3 m_1^2-m_2^2)\/\\sqrt{3}$. The $11\\to 12$ scattering if fully crossing invariant and indeed so is this picture. Physical processes in 2D lie on top of the blue solid lines and outside the red lines; in higher dimensions they fill in the interior of the regions delimited by the blue solid lines as one scans over physical scattering angles. Similar triangle for $12\\to 12$ scattering can be found in \\cite{Paper4}.}\n\t\\label{triangle}\n\\end{figure}\nThere, we see four disconnected regions: three non-compact parabola like curves related by a rotation symmetry and a round triangle in the middle. The three outer curves are the three physical regions associated to the three scattering channels. The one in the top, for instance, corresponds to the $s$-channel. (Each outer curve has a left and right components which are equivalent; they are related to a simple parity transformation.) \nThe $s$-channel outer curve start at $s=(m_1+m_2)^2$ as indicated by the red solid line. That corresponds to the minimal energy necessary to produce a particle of type $1$ and a particle of mass $2$ at rest. (Recall that $2$ is heavier than $1$.) Another important energy marked by the blue dashed line in the figure occurs at $s=(2m_1)^2$ which would correspond to the minimal energy necessary to produce two particle of type $1$ at rest. This is however \\textit{not} a physical energy for this process since physical energies are those for which we can produce \\textit{both} initial \\textit{and} final state. Nonetheless, the region between $s=4m_1^2$ and $s=(m_1+m_2)^2$ is very interesting because we know precisely what are the only possible {physical} states in that energy range: they can only be two particle states involving two particles of type $1$.~\\cite{Landau} The equation which reflects this is the so called \\textit{extended} unitarity relation which in this case reads\n\\begBvR{equation}\n2\\IM M_{11\\to 12}=\\rho_{11}^2 M_{11\\to 11} M_{11\\to 12}^*, \\qquad 4 m_1^2< s < (m_1+m_2)^2 \\la{extUnit1112}\n\\end{equation}\n\nHere, since we are focusing on the top curve (which is crossing equivalent to any of the other two)  we can think of $M$ as a single function of $s$ with \n\\begin{eqnarray}\n&&t(s)=\\frac{1}{2} \\left(3\n   m_1^2+m_2^2-s-\\sqrt{\\frac{\\left(s-4 m_1^2\\right) \\left(-2 m_2^2\n   \\left(m_1^2+s\\right)+\\left(s-m_1^2\\right){}^2+m_2^4\\right)}{s}}\\right)\\label{t11to12}\\\\\n&&u(s)=\\frac{1}{2} \\left(3\n   m_1^2+m_2^2-s+\\sqrt{\\frac{\\left(s-4 m_1^2\\right) \\left(-2 m_2^2\n   \\left(m_1^2+s\\right)+\\left(s-m_1^2\\right){}^2+m_2^4\\right)}{s}}\\right)\\label{u11to12}\n\\end{eqnarray}\nAs a check, note that as $m_2 \\to m_1$ we find $u \\to 0$ and $t\\to 4m_1^2-s$ as expected for two dimensional elastic scattering of particles of equal mass. \n\nThe extended unitarity relation (\\ref{extUnit1112}) is of course part of a coupled system of equations when we consider all components at once. They can all be nicely packed into matrix form by defining \n\\begBvR{equation}\n\\mathbb{U}\\equiv 2\\IM \\mathbb{M}-\\mathbb{M}^\\dagger \\text{\\outline{$\\rho$}}\\, \\mathbb{M}  \\,,\n\\label{unitarityfullsystem}\n\\end{equation}\nwhere\n\\begBvR{equation}\n\\!\\!\\!\\! \\mathbb{M}\\equiv \\begin{pmatrix}\nM_{11\\to 11} & M_{11\\to 12} \\\\\nM_{11\\to 12} & M_{12\\to 12}\n\\end{pmatrix}, \\,\\,\\,\\,\\,\n\\text{\\outline{$\\rho$}} \\equiv \\begin{pmatrix}\n\\rho_{11}^2=\\frac{\\theta\\left(s-4m_1^2\\right)}{2\\sqrt{s-4m_1^2}\\sqrt{s}} & 0 \\\\\n0 & \\rho_{12}^2=\\frac{\\theta\\left(s-(m_1 + m_2)^2\\right)}{2\\sqrt{s-(m_1 + m_2)^2}\\sqrt{s-(m_1 - m_2)^2}} \n\\end{pmatrix} \\la{rhoMatrix}\n\\end{equation}\nThen extended unitarity is the statement that $\\mathbb{U}=\\mathbf{0}$ for $s\\in [4m_1^2,(m_1+m_2)^2]$. Above $s=(m_1+m_2)^2$ we are at physical energies and the extended unitarity relation is replaced by regular unitarity which is now nothing but the statement that $\\mathbb{U}$ is a positive semi-definite matrix $\\mathbb{U} \\succeq 0$ for $s>(m_1+m_2)^2$.\\footnote{Strictly speaking we can impose $\\mathbb{U}=\\mathbf{0}$ for a while longer in the unitarity region, more precisely until the energy where we can  produce two particles of type $2$ or three particles of type $1$. In practice, bounds we will find will saturate unitarity so this will be automatic. Because of this, in all implementations, we will actually impose $\\mathbb{U} \\succeq 0$ even in the extended unitarity region, that is for any $s>4m_1^2$. This is very convenient as it renders the problem convex. }\n\n\n\\begin{figure}[t]\n\t\\centering \n\t\\includegraphics[scale=0.5]{1112Poles.pdf}\n\t\\vspace{-1.5cm}\n\t\\caption{$t(s)$ (blue) and $u(s)$ (yellow) for $11\\to 12$ scattering and $m_2=\\tfrac{3}{2} m_1$. $u(s)$ and $t(s)$ are two branches of the same analytic function. In the extended unitarity region they are complex. As a function of $s$, all poles are located before the extended unitarity region. The grey horizontal dashed lines are equal to $m_1^2$ and $m_2^2$ and fix the position of the $t$-- and $u$-- channel poles.}\n\t\\label{poles1112}\n\\end{figure}\n\n\n\n\\begin{figure}[t]\n\t\\centering \n\t\\includegraphics[scale=0.5]{1212Poles.pdf}\n\t\\vspace{-1.5cm}\n\t\\caption{$t(s)$ (blue) and $u(s)=0$ (yellow) for $12\\to 12$ forward scattering and $m_2=\\tfrac{3}{2} m_1$. In the $s$-channel extended unitarity sit $t$-channel poles (and vice-versa). The $s$-channel poles lie before the s-channel extended unitarity region. As in the previous figure, the grey horizontal dashed lines are equal to $m_1^2$ and $m_2^2$ determine the position of $t$-channel poles. }\n\t\\label{poles1212}\n\\end{figure}\n\nFinally we have poles. These correspond to the single particle exchanges when $s$ or $t$ or $u$ are equal to either $m_1$ or $m_2$.  The poles show up in the (rounded) triangle region in the Mandelstam triangle picture \\ref{triangle} in the $11\\to 12$ process as depicted in figure \\ref{poles1112}. For $12\\to 12$, we have $u=0$ and the two $t$-channel poles lie in the extended unitarity region. Note here the important difference between unitarity and extended unitarity. In the unitarity region the amplitudes describe physical probability amplitudes, are bounded and can thus never have poles. In the extended unitarity region they can in principle. And here they do as we see in the figure.  \n\nAll in all, we can summarize the analytic structure of our amplitudes with their cuts and poles by dispersion relations as usual. \nThese can be conveniently packaged into a simple matrix statement $\\mathbb{A}=\\mathbf{0}$ with\n\\begBvR{equation}\n\\mathbb{A}\\equiv \\begin{pmatrix}\n\\mathcal{A}_{11\\to11} & \\mathcal{A}_{11\\to 12} \\\\\n\\mathcal{A}_{11\\to 12} & \\mathcal{A}_{12\\to12}\n\\end{pmatrix} \\, \\la{AA}\n\\end{equation}\nand \n\\begin{align}\n\\mathcal{A}_{11\\to11}(s)\\equiv&M_{11\\to11}(s)-M_{11\\to11}^\\infty+{\\color{red}g_{111}^2}\\left(\\frac{1}{s{-}m_1^2}+\\frac{1}{t(s){-}m_1^2}\\right)+{\\color{blue}g_{112}^2}\\left(\\frac{1}{s{-}m_2^2}+\\frac{1}{t(s){-}m_2^2}\\right)\\nonumber\\\\\n&- \\frac{1}{\\pi}\\int_{4m_1^2}^\\infty \\IM M_{11\\to11}(z)\\left(\\frac{1}{z-s}+\\frac{1}{z-t(s)}\\right)dz\\,,\\label{m11to11disp} \\\\\n\\mathcal{A}_{11\\to12}(s)\\equiv&\\,M_{11\\to 12}(s)-M_{11\\to12}^\\infty+{\\color{red}g_{111}}{\\color{blue}g_{112}}\\left(\\frac{1}{s-m_1^2}+\\frac{1}{t(s)-m_1^2}+\\frac{1}{u(s)-m_1^2}\\right)\\nonumber\\\\\n&-\\frac{1}{\\pi}\\int_{4m_1^2}^\\infty \\IM M_{11\\to12}(z)\\left(\\frac{1}{z-s}+\\frac{1}{z-t(s)}+\\frac{1}{z-u(s)}\\right)dz\\,,\\label{m11to12disp}\n\\end{align}\n\\begin{align}\n\\mathcal{A}_{12\\to12}(s)\\equiv&\\,M_{12\\to12}(s)-M_{12\\to12}^\\infty+{\\color{blue}g_{112}^2}\\left(\\frac{1}{s-m_1^2}+\\frac{1}{t(s)-m_1^2}\\right)\\nonumber\\\\\n&-\\frac{1}{\\pi}\\int_{4m_1^2}^\\infty \\IM M_{12\\to12}(z)\\left(\\frac{1}{z-s}+\\frac{1}{z-t(s)}\\right)dz\\,.\n\\label{m12to12disp}\n\\end{align}\nWe hope there will be no confusing created by the fact that $t(s)$ signifies different things depending in which equation we are since crossing is implemented differently for different components. In~(\\ref{m11to11disp}) is it $t(s)=4m_1^2-s$; in (\\ref{m11to12disp}) it is given by (\\ref{t11to12}); and in (\\ref{m12to12disp}) it is given by~$t(s)=2m_1^2+2m_2^2-s$. In what follows, it should always be clear from the context which~$t(s)$ we are talking about. \n\n\n\\subsection{Multiple Component Dual Problem} \\la{mDual}\n\nThe formulation of the dual problem for the multiple component scenario can be derived following the steps outlined in Sec.~\\ref{sec2}.\nThere are, however, two practical obstacles: one is the complicated analytic structure of the $11\\to12$ component, the other is the presence of the \\emph{extended} unitarity region. \nIn this section we shall solve both problems if we want to arrive at an elegant and efficient dual numerical setup.\n\nAs always, we start from the primal radial problem\n\\begin{mdframed}[frametitle={Primal Radial Problem for Multiple Component},frametitlealignment=\\centering,backgroundcolor=blue!6, leftmargin=0cm, rightmargin=0cm, topline=false,bottomline=false, leftline=false, rightline=false] \n\\vspace{-0.4cm}\n\\begin{align} &\\underset{\\text{in } {R^2,\\mathbb{M}}}{\\text{maximize}} &&  R^2\\nonumber\\\\\n& \\text{constr. by}  && \n0=c_1 \\equiv \\text{Res}_{m_1^2}(M_{11\\to11})- R^2\\sin^2\\beta \\,,   \\nonumber\\\\\n& && 0=c_2\\equiv \\text{Res}_{m_2^2}(M_{11\\to11})-R^2\\cos^2\\beta \\,,\\nonumber \\\\\n& &&  0=c_3 \\equiv \\text{Res}_{m_1^2}(M_{11\\to12})-R^2\\sin\\beta\\cos\\beta\\,, \\nonumber\\\\ \n& &&  0=c_4 \\equiv \\text{Res}_{m_1^2}(M_{12\\to12})-R^2\\cos^2\\beta \\,,\\nonumber\\\\\n& s > 4m_1^2  && \\mathbb{A}=0 \\qquad \\text{where $\\mathbb{A}$ is given in (\\ref{AA})} \\nonumber \\,,  \\\\\n&  s > 4m_1^2&& \\mathbb{U} \\succeq 0\\qquad \\text{where $\\mathbb{U}$ is given in (\\ref{unitarityfullsystem})}\\,.\n \\label{primal bootstrap 11to12}\n\\end{align}\n\\end{mdframed}\nIf not for the $c_i=0$ equality constraints related to the radial problem, this setup would fit~(\\ref{primal2}). \nNote also that the last constraint incorporate automatically unitarity and extended unitarity. Sometimes it is convenient to analyze it separately in the extended and regular unitarity regions corresponding to $s$ bigger\/smaller than $(m_1+m_2)^2$ respectively. \n\nWe start our path towards the dual problem with the usual Lagrangian starting point \n\\begBvR{equation}\n\\mathcal{L}{=}R^2 + \\sum_{i=1}^4 c_i \\nu _i +\n\\int_{4m_1^2}^\\infty {\\rm tr~}{(\\text{\\outline{w}} \\mathbb{A})}\\,ds\n+\\int_{4m_1^2}^\\infty{\\rm tr~}{(\\text{\\outline{$\\Lambda$}} \\mathbb{U})}\\,ds,\n\\label{fullsystlag}\n\\end{equation}\nwith $$\\text{\\outline{w}} =\\begin{pmatrix}\nw_1 & \\tfrac{1}{2}w_2\\\\\n\\tfrac{1}{2}w_2 & w_3\n\\end{pmatrix}$$  \nand $\\text{\\outline{$\\Lambda$}}$ semi-definite positive. Next we want to identify $\\text{\\outline{w}}$ as the discontinuities of full analytic functions $\\mathbb{W}$ such that the resulting lagrangian becomes manifestly local. This is still possible here but turns out to be more interesting than before because of the richer $11\\to 12$ kinematics reviewed in the previous section. The final result is \n\\begBvR{equation}\\mathbb{W} =\\begin{pmatrix}\nW_1 & \\tfrac{1}{2}W_2\\\\\n\\tfrac{1}{2}W_2 & W_3\n\\end{pmatrix} \\label{Wmat}\n\\end{equation}\nwith the dispersive representations of the three \\emph{dual scattering functions}\n\\begin{align}\n\\label{analyticWcitable}\nW_1(s)&=\\frac{1}{\\pi}\\int_{4m_1^2}^\\infty dz\\, \\IM W_1(z)\\left(\\frac{1}{z-s}-\\frac{1}{z-4m_1^2+s}\\right),\\\\\nW_2(s)&=\\frac{1}{\\pi}\\int_{4m_1^2}^\\infty dz\\,\\IM W_2(z)\\left(\\frac{1}{z-s}+\\frac{J_t(s)}{z-t(s)}+\\frac{J_u(s)}{z-u(s)}\\right), \\la{W2disp}\\\\\nW_3(s)&=\\frac{1}{\\pi}\\int_{4m_1^2}^\\infty dz\\, \\IM W_3(z)\\left(\\frac{1}{z-s}-\\frac{1}{z-(m_1+m_2)^2+s}\\right).\n\\end{align}\nNote that the first and last lines here are pretty much as before: they correspond to anti-crossing symmetric symmetric functionals $W_1$ and $W_3$. The middle line -- with its  Jacobians $J_t=dt\/ds$ and $J_u=du\/ds$ from (\\ref{u11to12},\\ref{t11to12}) -- is more interesting and more subtle. We explain its origin in full detail in appendix \\ref{W2explanation}. \n\n\nThen we have the crucial relation required to render the Lagrangian local:\n\\begin{eqnarray}\n\\int_{4m_1^2}^\\infty {\\rm tr~}{(\\text{\\outline{w}}\\,\\mathbb{A})}\\,ds &=&\\int_{4m_1^2}^\\infty \\IM {\\rm tr~}{(\\mathbb{W}\\, \\mathbb{M})}\\,ds+\\pi\\big(\\underset{m_1^2}{\\text{Res}}(M_{11\\to11}) W_1(m_1^2)+\\underset{m_2^2}{\\text{Res}}(M_{11\\to11}) W_1(m_2^2) \\nonumber\\\\\n&&+\\underset{m_1^2}{\\text{Res}}(M_{11\\to12}) W_3(m_1^2)+\\underset{m_1^2}{\\text{Res}}(M_{12\\to12}) W_2(m_1^2) \\big)\\nonumber\n\\end{eqnarray}\nOnce we plug this relation into our lagrangian (\\ref{fullsystlag}) the last line nicely combines with the first two terms there; these terms are the only terms where $R$, $\\nu_i$ and the various residues appear.\\footnote{{Recall that $R$, the residues and $M(s)$ for $s>4$ are our primal variables, while $\\nu_i$ and $W_i(s)$ are our dual variables.}} Maximization with respect to the residues will relate the various functionals $W$ evaluated at the stable particle masses to the lagrange multipliers $\\nu_i$ as before while maximization with respect to $R$ will lead to to a linear constraint involving all these functionals which plays the important role of our normalization condition. It reads: \n\\begBvR{equation}\n1+\\pi(W_1(m_1)^2 \\sin^2\\beta+W_1(m_2^2)\\cos^2\\beta+W_2(m_1^2)\\sin\\beta\\cos\\beta+W_3(m_1^2)\\cos^2\\beta)=0 \\,.\n\\label{RadialCondFull}\n\\end{equation}\nAt this point we already got rid of the lagrange multipliers, the radius and the residues; our (partially extremized) Lagrangian is now a functional of the real and imaginary parts of the amplitudes $\\mathbb{M}$ above $4m_1^4$ and of the functionals $W_i$ also for $s>4m_1^2$.  Our dual functional $d$ is therefore the maximization over the amplitudes $\\mathbb{M}$ of \n\\begBvR{equation}\nd( \\mathbb{W},\\text{\\outline{$\\Lambda$}})= \\sup_{\\mathbb{M}} \\int_{4m_1^2}^\\infty  ds \\Big( {\\rm tr~}\\!(\\IM \\mathbb{W}\\, \\mathbb{M})+{\\rm tr~}{(\\text{\\outline{$\\Lambda$}} \\mathbb{U}(\\mathbb{M}))}\\Big)\n\\label{LindaLagrangia}\n\\end{equation}\nSince we are dealing with small $2\\times 2$ matrices we found it convenient to go to components at this point and also to separate the last integral into its extended and regular unitarity contributions separately. \n\nFor example, using \n\\begBvR{equation}\n\\text{\\outline{$\\Lambda$}}=\\begin{pmatrix}\n\\lambda_1 & \\tfrac{1}{2} \\lambda_2\\\\\n\\tfrac{1}{2}\\lambda_2^* & \\lambda_3\n\\end{pmatrix}\\succeq \\mathbf{0}, \\label{Lmat} \n\\end{equation}\nand evaluating the equations of motion for $\\RE M_{12\\to12}$ and $\\IM M_{12\\to12}$ in the extended unitarity region we get\n\\begBvR{equation}\n\\RE W_3+2\\lambda_3=0,\\qquad \\IM W_3=0.\\nonumber\n\\end{equation}\nThese two equations constrain the dual scattering function associated to the $12\\to 12$ to have a discontinuity starting at $(m_1+m_2)^2$. Moreover, the semidefinite-positiveness condition on $\\text{\\outline{$\\Lambda$}}$ \nimplies\\footnote{Second order variations show that the full positive semidefiniteness of $\\text{\\outline{$\\Lambda$}}$ is required for the critical $\\mathbb{M}_c$ to be a maximum.} that \n\\begBvR{equation}\n\\lambda_3(s)\\geq 0 \\qquad \\implies \\qquad \\RE W_3(s)\\leq0, \\qquad \\text{for } 4m_1^2<s<(m_1+m_2)^2.\\nonumber\n\\end{equation}\nWe can solve all equations for all amplitude components in both the regular and extended unitarity region for the simple reason that $\\mathbb{U}$ is quadratic in $\\mathbb{M}$. In this way we get \n$d(\\mathbb{W},\\text{\\outline{$\\Lambda$}})$ which we should now minimize. Its explicit expression is in appendix \\ref{dDual}.\n\nWe can now minimize first over positive semi-definite $\\text{\\outline{$\\Lambda$}}$ to obtain our final dual functional~$D(\\mathbb{W})$. This step is quite non-trivial but leads to a very compact final result:\n\\begin{mdframed}[frametitle={Dual Radial Problem for Multiple Component},frametitlealignment=\\centering,backgroundcolor=red!6, leftmargin=0cm, rightmargin=0cm, topline=false,bottomline=false, leftline=false, rightline=false] \n\\vspace{-0.4cm}\n\\begin{align} &\\underset{\\text{in } {\\mathbb{W}}}{\\text{minimize}} \\qquad  D^{\\text{ext}}(\\mathbb{W})+D^{\\text{phys}}(\\mathbb{W}) \\label{dual bootstrap 12system}\\\\\n&D^{\\text{ext}}(\\mathbb{W})=-\\int_{4m_1^2}^{(m_1+m_2)^2}ds\\, \\frac{|W_2|^2-4 \\RE W_1 W_3+|W_2^2-4 W_1 W_3|}{4 \\rho_{11}^2 W_3}\\nonumber\\\\\n&D^{\\text{phys}}(\\mathbb{W})=\\int_{(m_1+m_2)^2}^\\infty ds\\,\\left(\\frac{\\RE W_1}{\\rho_{11}^2}+\\frac{\\RE W_3}{\\rho_{12}^2}+\\sqrt{\\frac{|W_1|^2}{\\rho_{11}^4}+\\frac{|W_3|^2}{\\rho_{12}^4}+\\frac{|W_2|^2{+}|W_2^2{-}4 W_1 W_3|}{2\\rho_{11}^2\\rho_{12}^2}} \\right) \\nonumber\\\\\n& \\text{const. by} \\quad \\RE W_3 \\leq 0,\\quad \\IM W_3=0 \\quad\\text{for}\\quad 4m_1^2\\leq s \\leq (m_1+m_2)^2 \\nonumber\\\\\n&\\text{and by}\\qquad\\, 1+\\pi(W_1(m_1)^2 \\sin^2\\beta{+}W_1(m_2^2)\\cos^2\\beta{+}W_2(m_1^2)\\sin\\beta\\cos\\beta{+}W_3(m_1^2)\\cos^2\\beta)=0 \\nonumber\n\\end{align}\n\\end{mdframed}\n\nHere, the two contributions $D^{\\text{phys}}(\\mathbb{W})$ and $D^{\\text{ext}}(\\mathbb{W})$ correspond to the contributions of regular and extended unitarity. \nThe last condition is the normalization condition (\\ref{RadialCondFull}) and the next-to-last line with the linear inequality constraint is in the end the only remnant of the positive semi-definiteness of the lagrange multiplier matrix $\\text{\\outline{$\\Lambda$}}$. All these constraints can actually be trivialized as we explain in the next section. This will lead to a unconstrained (albeit non-linear) dual minimization problem which we will then solve numerically. \n\n\n\\subsection{Numerical Results}\n\nNow we perform both a primal and a dual numerical exploration to check the correctness of problem~\\eqref{dual bootstrap 12system}. \n\nIt what follows we will propose ansatze to parametrize families of dual functionals $W_j$'s. The cleverer the ansatze, the best will the bounds be and the fastest they will converge of course. Clever or not, it is of course important to stress that any ansatze for $W_i$ leads to a totally rigorous exclusion bound. \n\nThe $11\\to 11$ dual ansatz is the same used to produce the rigorous dual bounds in figure~\\ref{figHorn}\n\\begBvR{equation}\nW_1(s)=\\frac{1}{s(4m_1^2-s)} \\sum_{n=1}^{N_{\\text{max}}}a_n\\left(\\rho(s)^n-\\rho(t)^n\\right),\n\\label{W1ansatz}\n\\end{equation}\nwhere $t=4m_1^2-s$, $a_n$ are free variables and $\\rho(s)$ is the usual $\\rho$-variable foliation introduced in~\\cite{Paper3} -- see eq.~\\eqref{rhovariabledef} with $m=m_1$.\nThis ansatz has the right branch-point discontinuities and it is manifestly anti-crossing symmetric. At infinity it decays as $W_1\\sim s^{-5\/2}$; in fact, this behavior ensures that the dual objective in~\\eqref{dual bootstrap 11to11} is integrable. \nThe poles at $s=4m_1^2$ and $s=0$ are not necessary to obtain optimal bounds, but in practice they speed up the numerical convergence.\\footnote{We have numerical evidence to believe they are the right singularities the optimal dual scattering function should have. However, it is worth noticing they do not spoil integrability at threshold. We can look at eq.~\\eqref{eq13}: the $\\int_4^\\infty \\IM (M(s) W(s)) \\,ds$ is integrable if $W(s)\\sim 1\/(s-4)$ close to threshold because the amplitude vanishes as $M(s)\\sim \\sqrt{s-4}$. }\n\nFor the $11\\to 12$ dual ansatz we use\n\\begin{align}\nW_2(s)&=\\frac{1}{\\sqrt{4m_1^2-s}\\sqrt{4m_1^2-t}\\sqrt{4m_1^2-u}}\\sum_{n=1}^{P_{\\text{max}}}b_n \\left(\\rho(s)^n+J_t(s)\\rho(t)^n+J_u(s)\\rho(u)^n\\right),\n\\label{W2ansatz}\n\\end{align}\nwhere $t$ and $u$ are respectively given in~\\eqref{t11to12} and~\\eqref{u11to12}. Recall also that $J_t=dt\/ds$ and $J_u=du\/ds$.\nAt infinity $W_2\\sim s^{-3\/2}$, therefore the dual objective function~\\eqref{dual bootstrap 12system} wouldn't be integrable at infinity. However, it is sufficient to fix two of the $b_n$'s free variables to ensure the $W_2\\sim s^{-5\/2}$ decay. \nNotice that eq.~\\eqref{W2ansatz} has branch point singularities at $s=t=u=4m_1^2$ where the extended unitarity discontinuity in the physical amplitude start.\nAt the physical threshold $s=(m_1+m_2)^2$ in principle we could add additional singularities such as a pole (similarly to~\\eqref{W1ansatz}), however it turns out that numerically\nit makes no difference.\n\n\\begin{figure}[t]\n\t\\centering \n\t\\includegraphics[width=\\linewidth]{FullPlot.pdf}\n\t\\vspace{-3cm}\n\t\\caption{Dual (red) and Primal (blue) excluded\/allowed regions once the full system of amplitudes is included.\\protect\\footnotemark  ~The multi-component improved boundary is now rigorously trapped between the primal and dual bounds. \n\tThe red dashed line is the previous single component boundary. As we now impose the full system constraints the bound improves dramatically excluding most of the horn like figure. The red star point, for instance, was allowed (feasible) before from the primal problem perspective (it was blue in figure \\ref{figHorn}) and is now excluded. Once again, we restrict the plot to the first quadrant due to $g \\leftrightarrow -g$ symmetry.}\n\t\t\\label{figDualFull}\n\\end{figure}\n\n\n\nIt is convenient to design the $12\\to 12$ dual ansatz such that it automatically satisfy the constraints $\\IM W_3=0$ and $\\RE W_3<0$ in the extended unitarity region so that our optimization is unconstrained.\nThe former is easily achieved using a $\\rho$-foliation with cut starting at $s=(m_1+m_2)^2$ such as \n\\footnotetext{The dual curves, from outer to inner corresponds $(N_\\text{max},P_\\text{max},Q_\\text{max})$ equal to $(8,8,8)$, $(10,10,10)$ and $(10,20,20)$; the primal curves from inner to outer correspond to 136, 271 and 1111 degrees of freedom in the primal ansatz for the amplitude matrix. We used splines analogous to those used in \\cite{Paper4}.}\n\\begBvR{equation}\n\\tilde\\rho(s)=\\frac{\\sqrt{(m_1+m_2)^2-2m_1^2}-\\sqrt{(m_1+m_2)^2-s}}{\\sqrt{(m_1+m_2)^2-2m_1^2}+\\sqrt{(m_1+m_2)^2-s}}.\\nonumber\n\\end{equation}\nThe latter is more subtle: we could always impose linear constraints such as $\\RE W_3(s)\\equiv W_3(s)\\leq 0$ on some grid of points in the $4m_1^2<s<(m_1+m_2)^2$ segment in our dual minimization problem, but this would make \\texttt{Mathematica}'s basic \\texttt{FindMinimum} slow and nearly unusable.\nInstead, we opt to write the ansatz\n\\begBvR{equation}\nW_3(s)=(\\tilde\\rho(t)-\\tilde\\rho(s))\\left(\\frac{1}{\\sqrt{(m_1+m_2)^2-s}}+(s{\\leftrightarrow} t) \\right)\\left(\\sum_{n=0}^{Q_{\\text{max}}}c_n( \\tilde\\rho(s)^n+\\tilde\\rho(t)^n) \\right)^2\\nonumber\n\\end{equation}\nwhere $t=2m_1^2+2m_2^2-s$.\nIt is easy to check that $W_3$ has actually definite sign in a larger region than extended unitarity: $W_3>0$ in $t((m_1+m_2)^2)=(m_1-m_2)^2<s<m_1^2+m_2^2$ \nand $W_3<0$ in $m_1^2+m_2^2<s<(m_1+m_2)^2$ which of course include the extended unitarity region. \nThis may sound too restrictive, however this is one of the advantages of the dual formulation:\nas long as the dual scattering functions satisfy the dual constraints, the bounds obtained are rigorous.\nOf course, a legitimate question is whether our ansatz is able to attain the optimal value of the dual problem.\nIt turns out that for the case we are studying this ansatz is also approximately optimal numerically. \n\n\n\n\nNow we have all the ingredients to just code the objective in~\\eqref{dual bootstrap 12system} and minimize it unconstrained.\nThe result for the $\\{g_{111}, g_{112}\\}$ space is shown in figure~\\ref{figDualFull} (red shaded regions). In the same figure, the blue shaded areas are determined running the primal problem eq.~\\eqref{primal bootstrap 11to12} -- see~\\cite{Paper4} for details about primal multiple component numerics. The red dashed line marks the single component analytic bound. The white space in between the primal and dual \nareas is the uncertainty we have in the definition of the boundary for the full coupled system. Clearly the optimal bound is almost completely trapped! \n\n\n\n\\section{Discussion}\n\nIcarus said that \\textit{all limits are self-imposed}. That is not totally true. Unitarity, crossing symmetry and analyticity clearly also impose very important bounds. \n\n\nIn this paper we initiated a general dual bootstrap program and applied it on the next-to-simplest S-matrix bootstrap scenario: Two dimensional amplitudes with more than one particle type and more than one mass.\\footnote{The simplest example was kicked off in \\cite{Monolith} for a single particle species transforming in some global symmetry group.} \n\n\n\n\n\n\nOne main goal of this paper was to set up the theory behind this physical problem and connect it with the standard language of dual and primal maximization problems as optimization problems. Indeed, a great deal of section \\ref{sec2} can bet transported from (\\textit{the continuum limit of}) business and finance department reviews of optimization problems (beautiful examples are \\cite{reviews}), or math books \\cite{mathbook,mathbook2,mathbook3}. \n\nIn the S-matrix bootstrap studied here the primal problem is linear but constrained; the dual problem is non-linear but unconstrained.\\footnote{The unconstrained nature of the dual problem is an extremely powerful and fortunate property which was not a priori guaranteed. It is the nature of the S-matrix Bootstrap problems considered up to now that allowed us to trivialize all dual constraints encountered thus far.} For the primal problem, we used the powerful \\texttt{SDPB} code to perform the optimizations. For the dual problem we used \\texttt{Mathematica}'s basic \\texttt{FindMinimum}.\\footnote{\\texttt{FindMinimum} is sometimes an art. It is not uncommon to ask for a minimization, give \\texttt{Mathematica} a viable starting point and obtain a final result bigger than the starting value. Go figure. Of course, it is a price to pay when having a one size fits all algorithm. See also next footnote.} \nEven so, the dual problem is orders of magnitude faster right now.\\footnote{The dual curves in figure \\ref{figDualFull} contain thousands of points and take about a day to generate in a regular laptop. The primal curves take a few days in a cluster. One reason why we did not use the cluster for the dual problem is that we found it useful to hotstart \\texttt{FindMinimum} by starting the minimization search at a given point using the final result of the neighbouring point.} It would be very interesting to look for more tailor made algorithms for our kind of minimizations to speed the dual even more. \n\nOf course, the main advantage of having a dual problem is not speed but the fact that the bounds whence generated are completely rigorous. What is once excluded can never be included back. This is in contradistinction with the primal formulation where more constraints will often rule out a previously feasible solution.\n In practice the best is to use both dual and primal problems at once. When they almost touch each other -- meaning the so called duality gap is closing -- we know we are reaching the very optimal bounds!\n\nHaving developed the theory and a very fast dual problem, we look forward to putting it to use in several interesting physical applications. \n\n\\begin{figure}[t]\n\t\\centering \n\t\\includegraphics[width=\\linewidth]{imageIsing2}\n\t\\vspace{-6cm}\n\t\\caption{\\textbf{Left:} Maximum couplings $g_{11j}$ for a theory with the masses of the Ising field theory deformed by magnetic field depicted by normalizing those by the Ising couplings, $\\hat g_{11j}\\equiv g_{11j}\/g_{11j}^\\texttt{Ising}$ at this $E_8$ point. \\textbf{Right:} The plot on the right is obtained by a simple rotation of the first by 45 degrees which magnify some of the nice features of the plot. (These plots were generated using the dual method developed in this paper with $N=20$; it might be possible to derive this shape analytically. We did it for one of the faces but did not pursue this further.) \nThe Ising field theory, the red dot, lies beautifully at the very tip of these horn shaped single component plots. \n\t}\n\t\\label{cusp3D}\n\\end{figure}\n\nOne goal would be to bootstrap the Ising model field theory with both thermal and magnetic deformations turned on. Let us recall why we think this is promising. The Ising field theory with pure magnetic deformation~\\cite{ZamoE8} is at the boundary of the single amplitude bound \\cite{Paper2}, see figure 12 there. What is more, it is precisely at the top of a sharp horn like 3D bound in the coupling space as depicted in figure \\ref{cusp3D}.\\footnote{In \\cite{Paper2} only the maximum $g_{111}$ coupling was plotted so it was not possible to see this cusp so sharply.} Something we clearly learned in this paper is how multiple amplitudes can truncate such horns; compare figures \\ref{figHorn} and \\ref{figDualFull}. At the magnetic Ising point this dramatic truncation can not happen. This theory exists after all, we can not rule it out. What happens is that the very special values of the masses of the stable particles of this theory allow for fine tuned cancelations in $11\\to 12$ and other amplitudes such that they completely vanish and thus do not affect the single component bound which produces the horn. In other words, the purely magnetic deformation, being precisely integrable, is very special. As soon as we move away from these special masses by turning a thermal deformation, the multiple amplitude bounds are now expected to strongly affect the single component analysis and this provides a strong improvement over the bounds in \\cite{Paper2}. This is not totally trivial to implement because close to the magnetic point, the Ising field theory has three stable particles. Exploring the space of couplings $g_{ijk}$ between these particles is hard because this space is ten dimensional. The trick here is to find a clever lower dimensional section of this multidimensional space, with good optimization targets, which could efficiently isolate the magnetic plus thermal Ising deformation. That is something we are currently exploring.\\footnote{Using the form-factor bootstrap~\\cite{formfactors} as a further complementary tool to nail down the relevant physical Ising deformation might be very powerful as well. And recent work \\cite{barak} provides valuable insight into particle production and analytic properties of the expected S-matrices away from the integrable points.}\n\nAnother interesting theory to explore would be the tri-critical Ising model. In the discussion section of \\cite{Paper4} an S-matrix bootstrap homework exercise was proposed in relation to this model. With the great speed gains from the dual technology here developed this homework seems very doable. The deformation proposed there concerns a deformation preserving $\\mathbb{Z}_2$ symmetry. The dual $\\mathbb{Z}_2$ symmetric bootstrap is discussed in appendix \\ref{Z2appendix} for the case of equal masses; the uneven masses case should be a straightforward generalization of the analysis of the main text. \n\n\\begin{figure}[t]\n\t\\centering \n\\includegraphics[width=\\linewidth]{1112analogy.pdf}\n\\vspace{-5cm}\n\t\\caption{The kinematics of the two-to-two process $11\\to 12$ is very reminiscent of two-to-three scattering of massless particles as illustrated here. Both processes are fully crossing symmetric. Particle $2$ on the left is analogous to the jet of two right-movers on the right. This two-to-three scattering process should show up in flux tube physics~\\cite{Dubovsky1,Zohar,axion} where parity is broken. Extending the flux tube S-matrix boostrap program initiated in \\cite{FluxTube} to include such processes would be extremely interesting.\n\t}\n\t\\label{analogy}\n\\end{figure}\n\nOne step up in the complexity ladder of bootstrap problems are problems whose amplitudes depend on more than a single complex variable. One example is of course higher dimensions where we have both an energy and an angle even in two-to-two scattering processes.\\footnote{It is also in higher dimensions where the tension between absence of particle production and crossing symmetry is most striking \\cite{aks,amitesasha} which is another point a dual formulation should be very helpful in clarifying.} Another example are higher point amplitudes, even in two dimensions. In fact, in a very roundabout way, we arrived at the class of problems presented in this paper precisely while starting to tackle these multi-particle problems in work in progress with J. Penedones. The point is that the $11\\to 12$ amplitudes studied in this paper are in a sense very similar to a sort of $2\\to 3$ scattering process of massless particles as illustrated in figure~\\ref{analogy}. The jet of the two right movers in the future is like particle $2$. Of course, that jet can have any sub-energy hence we have in that case a continuum of \"particles\" of type 2, hence the additional complex variable. Nonetheless, this problem seems within reach. We hope to reach it and report on it in the near future. \n\n \\section*{Acknowledgements}\nWe would like to specially thank Joao Penedones for enlightening discussions, for collaboration at the initial stages of this project, and for collaboration in several related projects. \nResearch at the Perimeter Institute is supported in part by the Government of Canada \nthrough NSERC and by the Province of Ontario through MRI. This work was additionally \nsupported by a grant from the Simons Foundation (PV: \\#488661) and FAPESP \ngrants 2016\/01343-7 and 2017\/03303-1.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nA quandle is a set $X$ together with a binary  operation\n$*:X\\times X\\to X$ satisfying certain conditions (see definition\non example \\ref{exrack} below), it  generalizes the operation\nof conjugation on a group, but also  is an algebraic structure that behaves well with respect to\nReidemeister moves, so it is very useful for defining knot\/links invariants. Knot theorists\nhave defined a cohomology theory for quandles (see \\cite{CJKS}\nand \\cite{tCES}) in such a way that 2-cocycles give rise to knot invariants by means of \nthe so-called state-sum procedure. \nBiquandles are generalizations of quandles in the sense that quandles give rise to solutions\nof the Yang-Baxter equation by setting $\\sigma(x,y):=(y,x*y)$. For biquandles there is also a cohomology theory and state-sum procedure for producing knot\/links invariants\n(see \\cite{CES}).\n\nIn this work, for a set theoretical solution of the Yang-Baxter equation $(X,\\sigma)$, we\ndefine a d.g. algebra $B=B(X,\\sigma)$, containing the semigroup algebra\n$A=k\\{X\\}\/\\langle xy=zt : \\sigma(x,y)=(z,t)\\rangle$,\n such that $k\\otimes_AB\\otimes_Ak$ and $\\mathrm{Hom}_{A-A}(B,k)$ are respectively\nthe standard homology and cohomology complexes attached to general set theoretical\nsolutions of the Yang-Baxter equation. \nWe prove that this d.g. algebra has a natural structure of d.g. {\\em bialgebra}\n(Theorem \\ref{teobialg}). Also, depending on properties of the solution $(X,\\sigma)$\n(square free, quandle type, biquandle, involutive,...) this d.g. bialgebra $B$ has \nnatural (d.g. bialgebra) quotients, giving  rise to the standard\nsub-complexes computing quandle cohomology (as sub-complex of rack homology),\n biquandle cohomology, etc.\n\nAs a first consequence of our construction, we give a very simple and  purely algebraic proof\nof the existence of a cup product in cohomology. This was known for rack cohomology\n(see \\cite{Cl}), the proof was based on topological methods, but it was unknown for biquandles\nor general solutions of the Yang-Baxter equation.\nA second consequence is the existence of a comparison map between Yang-Baxter (co)homology\nand Hochschild (co)homology of the semigroup algebra $A$. Looking carefully this comparison map\nwe prove that it factors through a complex of \"size\" $A\\otimes \\mathfrak{B}\\otimes A$, where\n$\\mathfrak{B}$ is the Nichols algebra associated to the solution $(X,-\\sigma)$.\nThis result leads to new questions, for instance when $(X,\\sigma)$ is\n involutive (that is $\\sigma^2=\\mathrm{Id}$) and the characteristic\nis zero we show that this complex is acyclic (Proposition \\ref{propinvo}), we wander if\n this is true in any other characteristic, and for non necessarily involutive solutions.\n\n\n\n{\\bf Acknowledgements:}\nThe first author wishes to thank Dominique Manchon for fruitful discussion  during a visit to\n Laboratoire de math\\'ematiques de l'Universit\\'e Blaise Pascal where  a preliminary version of \nthe bialgebra $B$ for racks came up. He also want to thanks Dennis Sullivan for very pleasant\nstay in Stony Brook where the contents of this work was discussed in detail, in particular,\nthe role of Proposition \\ref{fnormal} in the whole construction.\n\n\\subsection{Basic definitions}\nA set theoretical solution of the Yang-Baxter equation (YBeq) is a pair $(X, \\sigma)$ where $\\sigma: X\\times X\\rightarrow X\\times X$\nis a bijection satisfying \n\\[\n (\\mathrm{Id}\\times\\sigma)(\\sigma\\times \\mathrm{Id})(\\mathrm{Id}\\times\\sigma)=(\\sigma\\times \\mathrm{Id})(\\mathrm{Id}\\times\\sigma)(\\sigma\\times \\mathrm{Id}):X\\times X\\times X\\rightarrow X\\times X\\times X\n\\]\n\nIf $X=V$ is a $k$-vector space and $\\sigma$ is a linear bijective map satisfying YBeq \nthen it is called a braiding on $V$.\n\n\\begin{ex}\\label{exrack} A set $X$ with a binary operation $\\triangleleft:X\\times X\\rightarrow X\\times X$ is called a rack if \n\\begin{itemize}\n \\item $-\\triangleleft x:X\\rightarrow X$ is a bijection $\\forall x\\in X$ and \n \\item $(x\\triangleleft y)\\triangleleft z=(x\\triangleleft z)\\triangleleft (y\\triangleleft z)$ $\\forall x,y,z \\in X$. \n\\end{itemize}\n\n$x\\triangleleft y $ is usually denoted by $x^y$.\n\nIf $X$ also verifies that $x\\triangleleft x=x$ then $X$ is called a {\\em quandle}.\n\nAn important example of rack is $X=G$ a group, $x\\triangleleft y=y^{-1}xy$.\n\nIf $(X,\\triangleleft)$ is a rack, then \\[\n                             \\sigma(x,y)=(y, x\\triangleleft y)\n                            \\]\n  is a set theoretical solution of the YBeq.\n\\end{ex}  \n\n  Let $M=M_X$ be the monoid freely generated in $X$  with relations \\[\n                                                                 xy=zt\n                                                                \\]\n$\\forall x,y,z,t$ such that $\\sigma(x,y)=(z,t)$. Denote $G_X$ the group with the same generators and relations.\nFor example, when \n$\\sigma= \\text{flip}$ then $M=\\mathbb{N}_0^{(X)}$ and $G_X=\\mathbb{Z}_0^{(X)}$. If $ \\sigma=\\mathrm{Id}$ then $M$ is the free (non abelian)\nmonoid in $X$. If $\\sigma$ comes from a rack $(X,\\triangleleft)$ then $M$ is the monoid with relation   \n$xy=y(x\\triangleleft y)$ and $G_X$ is the group with relations $x\\triangleleft y=y^{-1}xy$.\n\n                                                                \n                                                                \n                                                                \n\n\n                          \n                          \n\\section{A d.g. bialgebra associated to $(X,\\sigma)$}\nLet $k$ be a commutative ring with 1. \nFix $X$ a set, and $\\sigma:X\\times X\\to X\\times X$ a solution of the YBeq.\nDenote $A_\\sigma(X)$, or simply $A$ if $X$ and $\\sigma$ are understood,\n the quotient of the free $k$ algebra on generators $X$\nmodulo the ideal generated by elements of the form $xy-zt$ whenever $\\sigma(x,y)=(z,t)$:\n\\[\nA:=k\\langle X\\rangle\/\\langle xy-zt : x,y\\in X,\\ (z,t)=\\sigma(x,y)\\rangle=k[M]\n\\]\nIt can be easily seen that\n$A$ is a $k$-bialgebra declaring $x$ to be grouplike for any $x\\in X$,\n since $A$ agrees with the semigroup-algebra on $M$ (the monoid\nfreely generated by $X$ with relations $xy\\sim zt$).\nIf one considers  $G_X$, the group freely generated by\n$X$ with relations $xy=zt$, then $k[G_X]$ is the (non commutative) localization of $A$, where one has inverted the\nelements of $X$.\nAn example of $A$-bimodule that will be used later, which is actually a $k[G_X]$-module, is\n $k$ with $A$-action determined on generators by\n\\[\nx\\lambda y=\\lambda, \\ \\forall x,y\\in X,\\ \\lambda\\in k\n\\]\nWe define $B(X,\\sigma)$ (also denoted by $B$) the algebra freely generated \nby three copies of $X$, denoted $x$, $e_x$ and $x'$,\nwith  relations as follows:\nwhenever $\\sigma(x,y)=(z,t)$ we have\n\\begin{itemize}\n\\item $ xy\\sim zt$ , $xy'\\sim z't$, $x'y'\\sim z't'$\n\\item $ xe_{y}\\sim e_zt$, $ e_xy'\\sim z'e_{t}$\n\\end{itemize}\nSince the relations are homogeneous, $B$ is a graded algebra  declaring\n \\[\n  |x|=|x'|=0,\\ \\ |e_x|=1\n \\]\n\n\\begin{teo}\\label{teobialg}\n The algebra $B$ admits the structure of a differential graded bialgebra, with $d$ the unique superderivation satisfying\n  \\[\n  d(x)=d(x')=0,\\ \\\n  d(e_x)=x-x'\n \\]\nand comultiplication determined by \n\\[\n\\Delta(x)=x\\otimes x,\\\n\\Delta(x')=x'\\otimes x',\\\n\\Delta(e_x)=x'\\otimes e_x+e_x\\otimes x\n\\]\n\\end{teo}\nBy differential graded bialgebra we mean that the differential is both a derivation with respect to multiplication, and \ncoderivation with respect to comultiplication.\n\\begin{proof}\nIn order to see that $d$ is well-defined as super derivation, one must check that the relations\n are compatible with $d$. The first relations\nare easier since\n\\[\nd(xy-zt)=\nd(x)y+xd(y)-d(z)t-zd(t)=0+0-0-0=0\n\\]\nand similar for the others\n(this implies that $d$ is $A$-linear and $A'$-linear). For the rest of the relations:\n\\[\nd(xe_{y}-e_zt)=\nxd(e_y)-d(e_z)t\n=\nx(y-y')-(z-z')t\n\\]\n\\[\n=xy-zt-(xy'-z't)=0\n\\]\n\\[\n d(e_xy'-z'e_{t})\n=(x-x')y'-z'(t-t')\n=xy'-z't -(x'y'-z't')=0\n\\]\nIt is clear now that $d^2=0$ since $d^2$ vanishes on generators.\nIn order to see that $\\Delta$ is well defined, we compute\n\n\\[\n \\Delta(xe_y-e_zt)\n =\n(x\\otimes x)( y'\\otimes e_y+e_y\\otimes y)\n-(z'\\otimes e_z+e_z\\otimes z)(t\\otimes t)\n\\]\n\\[=\nxy'\\otimes xe_y+xe_y\\otimes xy\n-z't\\otimes e_zt-e_zt\\otimes zt\n\\]\nand using the relations we get\n\\[=\nxy'\\otimes xe_y+xe_y\\otimes xy\n-xy'\\otimes xe_y-xe_y\\otimes xy=0\n\\]\nsimilarly\n\\[\n \\Delta(x'e_y-e_zt')\n =\n(x'\\otimes x')( y'\\otimes e_y+e_y\\otimes y)\n-(z'\\otimes e_z+e_z\\otimes z)(t'\\otimes t')\n\\]\n\\[\n=\nx'y'\\otimes x'e_y+x'e_y\\otimes x'y\n-z't'\\otimes e_zt'-e_zt'\\otimes zt'\n\\]\n\\[=\nx'y'\\otimes x'e_y+x'e_y\\otimes x'y\n-x'y'\\otimes x'e_y-x'e_y\\otimes x'y=0\n\\]\nThis proves that $B$ is a bialgebra, and $d$ is (by construction) a derivation.\nLet us see that it is also a coderivation:\n\\[\n(d\\otimes 1+1\\otimes d)(\\Delta(x))=\n(d\\otimes 1+1\\otimes d)(x\\otimes x)=0=\\Delta(0)=\\Delta(dx)\n\\]\nfor  $x'$ is the same. For $e_x$:\n\\[\n(d\\otimes 1+1\\otimes d)(\\Delta(e_x))=\n(d\\otimes 1+1\\otimes d)(x'\\otimes e_x+e_x\\otimes x)\n\\]\n\\[=\nx'\\otimes (x-x')+(x-x')\\otimes x\n=\nx'\\otimes x-x'\\otimes x'+x\\otimes x\n-x'\\otimes  x\n\\]\n\\[=\n-x'\\otimes x'+x\\otimes x\n=\\Delta(x-x')=\\Delta(de_x)\n\\]\n \\end{proof}\n \\begin{rem}\n  $\\Delta$ is coassociative.\n \\end{rem}\n\nFor a particular element of the form\n$b=e_{x_1}\\dots e_{x_n}$, the formula for $d(b)$ can be computed as follows:\n\\[  \nd(e_{x_1}\\dots e_{x_n})=\\sum_{i=1}^{n} (-1)^{i+1}e_{x_1}\\dots e_{x_{i-1}}d(e_{x_i}) e_{x_{i+1}}\\dots e_{x_n}\n\\]\n\\[\n=\\sum_{i=1}^{n} (-1)^{i+1}e_{x_1}\\dots e_{x_{i-1}}(x_i-x'_i)e_{x_{i+1}}\\dots e_{x_n}\n\\]\n\\[=\n\\overbrace{\\sum_{i=1}^{n} (-1)^{i+1}e_{x_1}\\dots e_{x_{i-1}}x_ie_{x_{i+1}}\\dots e_{x_n}}^{I}\n -\\overbrace{\\sum_{i=1}^{n} (-1)^{i+1}e_{x_1}\\dots e_{x_{i-1}}x'_ie_{x_{i+1}}\\dots e_{x_n}}^{II}\n\\]\nIf one wants to write it in a normal form (say, every $x$ on the right, every $x'$ on the left,\nand the $e_x$'s in the middle), then one should use the relations in $B$: this might be a very\ncomplicated formula, depending on the braiding. We give examples in some particular cases.\nLets denote $\\sigma(x,y)=(\\sigma^1\\!(x,y), \\sigma^2(x,y))$. \n\n\\begin{comment}\nUsing the relations in $B$ one has\n\\[\nI=\\sum^n_{i=1}(-1)^{i+1}e_{x_1}\\dots e_{x_{i-1}}e_{y_{i+1}^1}\\dots e_{y_n^1}y_{n,i}^2\n\\]\nwhere \n\\[\n\\begin{array}{rcl}\ny_{i+1,i}&=&(\\sigma^1\\!(x_i, x_{i+1}), \\sigma^2(x_i, x_{i+1}))\\\\\n   y_{i+2,i}&=& (\\sigma^1\\!(y_{i+1,i}^2, x_{i+2}), \\sigma^2(y_{i+1,i}^2, x_{i+2}))\\\\\n y_{i+3,i}&=&(\\sigma^1\\!(y_{i+2,i}^2,x_{i+3}),\\sigma^2(y_{i+2,i}^2,x_{i+3}))\\\\\n\\vdots&&\\vdots\\\\\n  y_{n,i}&=&(\\sigma^1\\!(y_{n-1,i}^2,x_n),\\sigma^2(y_{n-1,i}^2,x_n))\n\\end{array}\n\\]\nand similarly\n\\[\nII=\\sum^n_{i=1}(-1)^{i+1}(z_{1,i}^1)'e_{z_{1,i}^2}\\dots e_{z_{i-2,i}^2}e_{z_{i-1,i}^2}e_{x_i+1}\\dots e_{x_n}\n\\]\n\nwhere\n\n\\[\\begin{array}{rcl}\n z_{i-1,i}&=&(\\sigma^1\\!(x_{i-1},x_i),\\sigma^2(x_{i-1},x_i))\n\\\\\n z_{i-2,i}&=&(\\sigma^1\\!(x_{i-2},z_{i-1,i}^1),\\sigma^2(x_{i-2},z_{i-1,i}^1))\n\\\\\n\\vdots&&\\vdots\n\\\\\nz_{1,i}&=&(\\sigma^1\\!(x_1,z_{2,i}^1),\\sigma^2(x_1,z_{2,i}^1))\n\\end{array}\\]\n\n\\[\n \\partial f(x_1,\\dots,x_n)=f(d(e_{x_1}\\dots e_{x_n}))=\\]\n\\[\n \\sum^n_{i=1}(-1)^{i+1}\\left(f(x_1,\\dots, x_{i-1},y_{i+1,i}^1,\\dots,\n y_{n,i}^1)y_{n,i}^2-(z_{1,i}^1)'f(z_{1,i}^2,\\dots,z_{i-1,i}^2,x_{i+1},\\dots, x_n)\\right)\n \\]\n\n\\end{comment}\n \n \n \\begin{ex} In low degrees we have\n \\begin{itemize}\n  \\item $d(e_x)=x-x'$\n  \\item $d(e_xe_y)=(e_zt-e_xy)-(x'e_y-z'e_t)$, where as usual $\\sigma(x,y)=(z,t)$.\n  \\item $d(e_{x_1}e_{x_2}e_{x_3})=A_I-A_{II}$\n  where\n  \n  $A_I=e_{\\sigma^1\\!(x_1,x_2)}e_{\\sigma^1\\!(\\sigma^2(x_1,x_2),x_3)}\\sigma^2(\\sigma^2(x_1,x_2),x_3)-e_{x_1}e_{\\sigma^1\\!(x_2,x_3)}\n  \\sigma^2(x_2,x_3)+e_{x_1}e_{x_2}x_3$\n  \n  $A_{II}= x_1'e_{x_2}e_{x_3}-\\sigma^1\\!(x_1,x_2)'e_{\\sigma^2(x_1,x_2)}e_{x_3}+\n  \\sigma^1\\!(x_1,\\sigma^1\\!(x_2,x_3))'e_{\\sigma^2(x_1,\\sigma^1\\!(x_2,x_3))}e_{\\sigma^2(x_2,x_3)}$\n  \n  In particular, if $f:B\\to k$ is an $A$-$A'$ linear map, then\n  \\[\nf(d(e_{x_1}e_{x_2}e_{x_3}))=  \nf(e_{\\sigma^1\\!(x_1,x_2)}e_{\\sigma^1\\!(\\sigma^2(x_1,x_2),x_3)})\n-f(e_{x_1}e_{\\sigma^1\\!(x_2,x_3)})+f(e_{x_1}e_{x_2})\n\\]\\[-f(e_{x_2}e_{x_3})+f(e_{\\sigma^2(x_1,x_2)}e_{x_3})-\nf(e_{\\sigma^2(x_1,\\sigma^1\\!(x_2,x_3))}e_{\\sigma^2(x_2,x_3)})\n\\]\nErasing the $e$'s we notice the relation with the cohomological complex given in \\cite{CES},\nsee Theorem \\ref{teocomplejo} below.\n\n   \\end{itemize}\nIf  $X$ is a rack and $\\sigma$ the braiding defined by  $\\sigma(x,y)=(y,x\\triangleleft y)=(x,x^y)$, then:\n \\begin{itemize}\n  \\item $d(e_x)=x-x'$\n  \\item $d(e_xe_y)=\n( e_{y}x^{y}\n- e_{x}y)\n-(x'e_{y}-y'e_{x^y})$\n  \\item $d(e_xe_ye_z)=\ne_xe_yz\n-e_xe_zy^z\n+e_ye_zx^{yz}\n-x'e_ye_z\n+y'e_{x^y}e_z\n-z'e_{x^z}e_{y^z}$.\n\\item In general, expressions I and II are\n\\[\n I=\\sum_{i=1}^{n} (-1)^{i+1} e_{x_1}\\dots e_{x_{i-1}}e_{x_{i+1}}\\dots e_{x_n}x_{i}^{x_{i+1}\\dots x_{n}}\n\\]\n\\[\n II=\\sum_{i=1}^{n} (-1)^{i+1}x'_ie_{x_1^{x_i}}\\dots e_{x_{i-1}^{x_i}}e_{x_{i+1}}\\dots e_{x_n}\n\\]\n\n\nthen\n\n\\[\n \\partial f(x_1,\\dots,x_n)=f(d(e_{x_1}\\dots e_{x_n}))=\\]\n\\[\n \\sum_{i=1}^{n} (-1)^{i+1} \\left(f(x_1,\\dots, x_{i-1},x_{i+1},\\dots, x_n)x_{i}^{x_{i+1}\\dots x_{n}}-x'_if({x_1}^{x_i},\\dots\n , x_{i-1}^{x_i},x_{i+1},\\dots, x_n)\\right)\n \\]\n \nLet us consider $k\\otimes_{k[M']} B\\otimes_{k[M]} k$\nthen  $d$ represents the canonical differential of rack homology and \n$\\partial f (e_{x_1}\\dots e_{x_n})=f(d(e_{x_1}\\dots e_{x_n}))$ gives the traditional rack cohomology structure. \n\nIn particular, taking trivial coefficients:\n \\[\n \\partial f(x_1,\\dots,x_n)=f(d(e_{x_1}\\dots e_{x_n}))=\\]\n\\[\n \\sum_{i=1}^{n} (-1)^{i+1} \\left(f(x_1,\\dots, x_{i-1},x_{i+1},\\dots, x_n)-f({x_1}^{x_i},\\dots, x_{i-1}^{x_i},x_{i+1}\\dots, x_n)\\right)\n\\]\n\n \\end{itemize}\n\n\\end{ex}\n\n\n \n \\begin{teo}\\label{teocomplejo}\nTaking in $k$ the trivial $A'$-$A$-bimodule, the complexes associated to set theoretical Yang-Baxter solutions\ndefined in \\cite{CES} can be recovered as\n\\[\n (C_\\bullet(X,\\sigma), \\partial)\\simeq (k\\otimes_{A'} B_\\bullet \\otimes_{A} k, \\partial=id_k\\otimes_{A'} d\\otimes_{A}id_k)\n\\]\n\n\\[\n (C^\\bullet(X,\\sigma), \\partial^*)\\simeq (\\mathrm{Hom}_{A'-A}(B, k), \\partial^*=d^*)\n\\]\n \\end{teo}\nIn the  proof of the theorem  we will assume  first Proposition \\ref{fnormal}\nthat says that one has a left $A'$-linear and right $A$-linear\nisomorphism: \n\\[B\\cong A'\\otimes TE\\otimes A\\]\nwhere $A'=TX'\/(x'y'=z't': \\sigma(x,y)=(z,t))$ and $A=TX\/(xy=zt: \\sigma(x,y)=(z,t))$.\n We will prove \n Proposition \\ref{fnormal} later.\n\n \\begin{proof}\n\nIn this setting every expression in $x,x',e_x$, using the relations defining $B$, can be written as\n$x'_{i_1}\\cdots x'_{i_n} e_{x_1}\\cdots e_{x_k} x_{j_1}\\cdots x_{j_l}$, tensorizing leaves the expression  \n$$1\\otimes e_{x_1} \\cdots e_{x_k}\\otimes 1$$\nThis shows that $T=k\\otimes_{k[M']} B\\otimes_{k[M]} k\\simeq T\\{e_x\\}_{x\\in X}$, where $\\simeq$ means isomorphism of $k$-modules. \nThis also induces isomorphisms of complexes\n\n\\[\n (C_\\bullet(X, \\sigma), \\partial)\\simeq (k\\otimes_{A'} B_\\bullet \\otimes_{A} k, \\partial= id_k\\otimes_{A'} d\\otimes_{A}id_k)\n\\]\n\n\\[\n (C^\\bullet(X, \\sigma), \\partial^*)\\simeq (\\mathrm{Hom}_{A'-A}(B, k), d^*)\n\\]\n \\end{proof}\n \nNow we will prove Proposition \\ref{fnormal}:\nCall\n$Y=\\langle x,x',e_x\\rangle_{x\\in X}$ the free monoid in $X$ with unit 1, $k\\langle Y\\rangle$ the $k$ algebra associated to $Y$.\nLets define\n$w_1=xy'$, $w_2=xe_y$ and $w_3=e_xy'$.\nLet $S=\\{r_1,r_2,r_3\\}$ be the  reduction system  defined as follows: $r_i:k\\langle Y\\rangle\\rightarrow k\\langle Y\\rangle$ the families of \n$k$-module endomorphisms\nsuch that  $r_i$ fix all elements except \n\n$r_1(xy')=z't$,\\ \\ $r_2(xe_y)=e_zt$ and \n$r_3(e_xy')=z'e_t$. \n\nNote that $S$ has more than 3 elements, each $r_i$ is a family of reductions.\n\\\n\n\n\\begin{defi}\nA reduction $r_i$ {\\em acts trivially} on an element $a$ if $w_i$ does not apear in $a$, ie: $Aw_iB$ apears with coefficient 0.\n\\end{defi}\n\nFollowing \\cite{B}, $a\\in k\\langle Y\\rangle$ is called  {\\em irreducible} if $Aw_iB$ does not appear for $i\\in\\{1,2,3\\}$. Call\n$k_{irr}\\langle Y\\rangle$  the  $k$ submodule of irreducible elements of $k\\langle Y\\rangle$.\n A finite sequence of reductions is called {\\em final} in $a$ if $r_{i_n}\\circ \\dots \\circ r_{i_1}(a)\\in k_{irr}(Y)$.\nAn element $a\\in k\\langle Y\\rangle$ is called {\\em reduction-finite} if for every sequence of reductions $r_{i_n}$ acts trivially on\n  $r_{i_{n-1}}\\circ \\dots \\circ r_{i_1}(a)$ for sufficiently large $n$.\n   If a is reduction-finite, then any maximal sequence of reductions, such that  each $r_{i_j}$\n   acts nontrivially on $r_{i_{(j-1)}}\\dots r_{i_1}(a)$, will be finite, and hence a final \nsequence. It follows that the reduction-finite elements form \na k-submodule of $k\\langle Y\\rangle$ \n $a\\in k\\langle Y\\rangle$ is called {\\em reduction-unique} if is reduction finite and it's image under every finite\n  sequence of reductions is the same.\n  This comon value will be denoted $r_s(a)$.\n \n\n\\begin{defi}\n Given a monomial $a\\in k\\langle Y \\rangle$ we define the disorder degree of \n $a$, $\\hbox{disdeg}(a)=\\sum_{i=1}^{n_x}rp_i+\\sum_{i=1}^{n_{x'}}lp_j$, where \n $rp_i$ is the position of the $i$-th letter ``$x$'' counting from right to  left, and $lp_i$ is the position of the $i$-th letter ``$x'$'' \n counting from left to right.\n \n If $a=\\sum_{i=1}^{n} k_i a_i$ where $a_i$ are monomials in leters of $X,X', e_X$ and $k_i\\in K-\\{0\\} $,\n \\[\\hbox{disdeg}(a):=\\sum_{i=1}^{n}\\hbox{disdeg}(a_i)\\]\n\\end{defi}\n\n\n\n\\begin{ex}\\begin{itemize}\n           \\item $\\hbox{disdeg}(x_1e_{y_1}x_2z'_1x_3z'_2)=(2+4+6)+(4+6)=22$\n           \\item $\\hbox{disdeg}(xe_yz')=3+3=6$ and $\\hbox{disdeg}(x'e_yz)=1+1$\n           \\item $\\hbox{disdeg}(\\prod_{i=1}^{n}x'_i\\prod_{i=1}^{m}e_{y_i}\\prod^{k}_{i=1}z_i)=\\frac{n(n+1)}{2}+\\frac{k(k+1)}{2}$\n           \\end{itemize}\n\n           \n           \\end{ex}\n\n The reduction $r_1$ lowers  disorder degree in two and reductions $r_2$ and $r_3$ lowers disorder degree in one. \n\\begin{rem}\\begin{itemize}\n\\item $k_{irr}(Y)=\\{\\sum A'e_B C: A' \\ \\hbox{word in}\\  X', e_B \\hbox{word in}\\ e_x, C \\ \\hbox{word in}\\ X\\}$.\n\\item $k_{irr}\\simeq TX'\\otimes TE\\otimes TX$\n\\end{itemize}\n\\end{rem}\n\nTake for example $a=xe_yz'$, there are two possible sequences of final reductions: $r_3\\circ r_1\\circ r_2$ or $r_2\\circ r_1\\circ r_3$.\nThe result will be $a=A'e_B C$ and $a=D'e_E F$ respectively, where \n\n$A=\\sigma^{(1)}\\left(\\sigma^{(1)}(x,y),\\sigma^{(1)}(\\sigma^{(2)}(x,y),z)\\right)$\n\n$B=\\sigma^{(2)}\\left(\\sigma^{(1)}(x,y),\\sigma^{(1)}(\\sigma^{(2)}(x,y),z)\\right)$\n\n$C=\\sigma^{(2)}\\left(\\sigma^{(2)}(x,y),z\\right)$\n\n$D=\\sigma^{(1)}\\left(x,\\sigma^{(1)}(y,z)\\right)$\n\n$E=\\sigma^{(1)}\\left(\\sigma^{(2)}(x,\\sigma^{(1)}(y,z),\\sigma^{(2)}(y,z))\\right)$\n\n$F=\\sigma^{(2)}\\left(\\sigma^{(2)}(x,\\sigma^{(1)}(y,z),\\sigma^{(2)}(y,z))\\right)$\n\nWe have $A=D$, $B=E$ and $C=F$ as $\\sigma$ is a solution of YBeq, hence \\\\\n$r_3\\circ r_1\\circ r_2(xe_yz')=r_2\\circ r_1\\circ r_3(xe_yz')$.\n\n\nA monomial $a$ in $k\\langle Y\\rangle$ is said to have  an {\\em overlap ambiguity} of $S$ if $a=ABCDE$\nsuch that $w_i=BC$ and $w_j=CD$. We shall say the \noverlap ambiguity  is {\\em resolvable} if there exist compositions of \nreductions, $r,r'$ such that $r(Ar_i(BC)DE)=r'(ABr_j(CD)E)$.\nNotice that it is enough to take $r=r_s$ and $r'=r_s$.\n\\begin{rem}\nIn our case, there is only one type of overlap ambiguity and is the one we solved previously.\n\\end{rem}\n\\begin{proof}\n There is no rule with $x'$ on the left nor rule with $x$ on the right, so there will be no overlap ambiguity including the family $r_1$. \nThere is only one type of ambiguity involving reductions $r_2$ and $r_3$.\n \\end{proof}\n\n\nNotice that  $r_s$  is a  proyector and  $I=\\langle xy'-z't,xe_y-e_zt,e_xy'-z'e_t\\rangle$ is trivially included in the kernel. \nWe claim that it is actually equal:\n\\begin{proof}\n As $r_s$ is a proyector, an element  $a\\in \\ker$ must be $a=b-r_s(b)$ where $b\\in k\\langle Y\\rangle$. It is enough to prove it for monomials $b$.\n \\begin{itemize}\n  \\item if $a=0$ the result follows trivially.\n  \\item if not, then take a monomial  $b$ where at least one of the products $xy'$, $xe_y$ or $e_xy'$ appear. Lets suppose $b$ has a factor $xy'$\n  (the rest of the cases are analogous). \n  \n  $b=Axy'B$ where $A$ or $B$ may be empty words. $r_1(b)=Ar_1(xy')B=Az'tB$. Now we can rewrite:\n  \n  $b-r_s(b)=\\underbrace{Axy'B-Az'tB}_{\\in I}+Az'tB-r_s(b)$. \n  As $r_1$ lowers $\\hbox{disdeg}$ in two, we have $\\hbox{disdeg}(Az'tB-r_s(b))<\\hbox{disdeg}(b-r_s(b))$ then in a finite number of steps we get  $b=\\sum^{N}_{k=1} i_k$\n where $i_k\\in I$. It follows that $b\\in I$.  \n \\end{itemize}\n \n\\end{proof}\n\\begin{coro} $r_s$ induces a $k$-linear isomorphism: \n\\[k\\langle Y\\rangle\/\\langle xy'-z't,xe_y-e_zt,e_xy'-z'e_t\\rangle\\rightarrow TX'\\otimes TE\\otimes TX \\]\n\\end{coro}\n\nReturning to our bialgebra, taking quotients we obtain the following proposition:\n\n\\begin{prop}\\label{fnormal}\n$B\\simeq \\left(TX'\/(x'y'=z't')\\right)\\otimes TE\\otimes \\left(TX\/(xy=zt)\\right)$\n\\end{prop}\n\nNotice that $\\overline{x_1\\dots x_n}=\\overline{\\prod \\beta_m\\circ \\dots \\circ\\beta_1(x_1,\\dots, x_n)}$ where $\\beta_i=\\sigma^{\\pm 1}_{j_i}$,\nanalogously with $\\overline{x'_1\\dots x'_n}$.\n\n\nThis ends the proof of Theorem \\ref{teocomplejo}.\n\n\n\n\\begin{ex}\n \\item If the coeficients are trivial, $f\\in C^1(X,k)$ and we identify\n  $C^1(X,k)=k^X$, then\n    \\[\n   (\\partial f)(x,y)=f(d(e_xe_y))=-f(x)-f(y)+f(z)+f(t)\n  \\]\nwhere as usual $\\sigma(x,y)=(z,t)$ \n(If instead of considering $\\mathrm{Hom}_{A'-A}$, we consider $\\mathrm{Hom}_{A-A'}$ then \n $(\\partial f)(x,y)=f(d(e_xe_y))=f(x)+f(y)-f(z)-f(t)$ but with $\\sigma(z,t)=(x,y)$).\n\n \\item Again with trivial coefficients, and $\\Phi\\in C^2(X,k)\\cong k^{X^2}$, then\n    \\[\n   (\\partial \\Phi)(x,y,z)=\\Phi (d(e_xe_ye_z))=\\Phi\\left(\\overbrace{xe_ye_z}^{I}-\\overbrace{x'e_ye_z}^{II}-\\overbrace{e_xye_z}^{III}+\n   \\overbrace{e_xy'e_z}^{IV}+\\overbrace{e_xe_yz}^{V}-\\overbrace{e_xe_yz'}^{VI} \\right) \\]\n   \n   If considering  $\\mathrm{Hom}_{A'-A}$ then,using the relations defining $B$, the terms $I,III,IV$ and $VI$ changes leaving\n   \\[\n     \\partial \\Phi(x,y,z)=\\Phi (\\sigma^1\\!(x,y),\\sigma^1\\!(\\sigma^2(x,y),z))-\\Phi(y,z)-\\Phi(x,\\sigma^1\\!(y,z))+\\]\n     \\[\n    \\Phi(\\sigma^2(x,y),z)+\\Phi(x,y)-\\Phi(\\sigma^2(x,\\sigma^1\\!(y,z)),\\sigma^2(y,z))\n   \\]\n\n   \n   \n   \n   \n \\item If $M$ is a  $k[T]$-module (notice that $T$ need not to be invertible as in \\cite{tCES}) then\n $M$ can be viewed as an $A'-A$-bimodule via\n \\[\n  x' \\cdot m=m\n,\\ \\ m\\cdot x=Tm\n \\]\n  The actions are compatible with the relations defining $B$:\n  \n  \\[\n   (m\\cdot x)\\cdot y =T^2 m \\ ,\\ \\ (m\\cdot z)\\cdot t =T^2 m \\]\n   and \n   \\[\n  x'\\cdot (y'\\cdot m)=m \\  ,  \\ \\  z'\\cdot (t'\\cdot m)= m   \n  \\]\nUsing these coefficients we get twisted cohomology as in \\cite{tCES} but for general YB solutions.\n\n\n\n If one takes the special case of $(X,\\sigma)$ being a rack, namely $\\sigma(x,y)=(y,x\\triangleleft y)$, then the general formula\n gives\n \\[\n \\partial f(x_1,\\dots,x_n)=f(d(e_{x_1}\\dots e_{x_n}))=\\]\n\\[\n \\sum_{i=1}^{n} (-1)^{i+1} \\left(Tf(x_1,\\dots, x_{i-1},x_{i+1},\\dots, x_n)-f({x_1}^{x_i},\\dots, x_{i-1}^{x_i},x_{i+1},\\dots, x_n)\\right)\n \\]\n that agree with the differential of the twisted cohomology defined in \\cite{tCES}.\n\n\\end{ex}\n\n \n \n\\begin{rem} \nIf $c(x\\otimes y)=f(x,y)\\sigma^1\\!(x,y)\\otimes \\sigma^2(x,y)$, then $c$ is  a solution of YBeq if and only if $f$ is a 2-cocycle.\n\\end{rem}\n\n\\[\n c_1\\circ c_2\\circ c_1(x\\otimes y\\otimes z)\n\\]\n\n\\[\n =a\\overbrace{\\sigma^1\\!\\left(\\sigma^1\\!(x,y),\\sigma^1\\!(\\sigma^2(x,y),z)\\right)\\otimes \\sigma^2\\left(\\sigma^1\\!(x,y),\\sigma^1\\!(\\sigma^2(x,y),z\\right)\\otimes \\sigma^2\\left(\\sigma^2(x,y),z)\\right)}^{I}\n\\]\nwhere\n\\[\n a=f(x,y)f\\left(\\sigma^2(x,y),z\\right)f\\left(\\sigma^1\\!(x,y),\\sigma^1\\!(\\sigma^2(x,y),z)\\right)\n\\]\n\n\\[\n c_2\\circ c_1 \\circ c_2 (x\\otimes y\\otimes z)\n\\]\n\n\\[\n b\\overbrace{\\sigma^1\\!(x,\\sigma^1\\!(y,z))\\otimes \\sigma^1\\!\\left(\\sigma^2(x,\\sigma^1\\!(y,z)),\\sigma^2(y,z)\\right)\\otimes \\sigma^2\\left(\\sigma^2(x,\\sigma^1\\!(y,z),\\sigma^2(y,z))\\right)}^{II}\n\\]\nwhere\n\\[b= f(y,z)f\\left(x,\\sigma^1\\!(y,z)\\right)f\\left(\\sigma^2(x,\\sigma^1\\!(y,z)),\\sigma^2(y,z)\\right)\n\\]\n Writing YBeq with this notation leaves:\n\\begin{equation}\\label{trenza-trenza}\n\\sigma \\ is\\ a\\ braid \\Leftrightarrow    I=II\n\\end{equation}\n\n\nTake $f$ a two-cocycle, then \n\n\\[\n 0=\\partial f(x,y,z)=f(d(e_xe_ye_z))=f((x-x')e_ye_z-e_x(y-y')e_z+e_xe_y(z-z'))\n \\]\n\n is equivalent to the following equality\n \\[\n  f(xe_ye_z)+f(e_xy'e_z)+f(e_xe_yz)=f(x'e_ye_z)+f(e_xye_z)+f(e_xe_yz')\n \\]\nusing the relations defining $B$ we obtain\n\n\\[\n f\\left(e_{\\sigma^1\\!(x,y)}e_{\\sigma^1\\!(\\sigma^2(x,y),z)}\\sigma^2(\\sigma^2(x,y)z)\\right)+f\\left(\\sigma^1\\!(x,y)'e_{\\sigma^2(x,y)}e_z\\right)\n +f\\left(e_xe_yz\\right)\n      \\]\n\\[\n =f\\left(x'e_ye_z\\right)+f\\left(e_xe_{\\sigma^1\\!(y,z)}\\sigma^2(y,z)\\right)+\n f\\left(\\sigma^1\\!(x,\\sigma^1\\!(y,z))'e_{\\sigma^2(x,\\sigma^1\\!(y,z))}e_{\\sigma^2(y,z)}\\right)\n\\]\n\nIf $G$ is an abelian multiplicative group and $f:X\\times X\\rightarrow (G, \\cdotp)$ then the previous formula says \n\\[\n f\\left(e_{\\sigma^1\\!(x,y)}e_{\\sigma^1\\!(\\sigma^2(x,y),z)}\\sigma^2(\\sigma^2(x,y)z)\\right)f\\left(\\sigma^1\\!(x,y)'e_{\\sigma^2(x,y)}e_z\\right)\n f\\left(e_xe_yz\\right)\n\\]\n\\[\n =f\\left(x'e_ye_z\\right)f\\left(e_xe_{\\sigma^1\\!(y,z)}\\sigma^2(y,z)\\right)\n f\\left(\\sigma^1\\!(x,\\sigma^1\\!(y,z))'e_{\\sigma^2(x,\\sigma^1\\!(y,z))}e_{\\sigma^2(y,z)}\\right)\n\\]\nwhich is exactly the condition $a=b$.\n\n\nNotice that if the action is trivial, then the equation above simplifies giving\n\\begin{equation}\\label{2-cocycle}\n  f\\!\\left(e_{\\sigma^1\\!(x,y)}e_{\\sigma^1\\!   (\\sigma^2(x,y),z)} \\right)\\! \nf \\!\\left(e_{\\sigma^2(x,y)}e_z\\right)\\!\n f\\!\\left(e_xe_y\\right)\\!\n \\newline\n =f\\!\\left(e_ye_z\\right)\\!f\\!\\left(e_xe_{\\sigma^1\\!(y,z)}\\right)\\!\n f\\!\\left(e_{\\sigma^2(x,\\sigma^1\\!(y,z))}e_{\\sigma^2(y,z)}\\right)\n\\end{equation}\nwhich is precisely the formula on \\cite{CES} for Yang-Baxter 2-cocycles \n (with $R_1$ and $R_2$ instead of $\\sigma^1$ and $\\sigma^2$).\n\n\n\n\n\\section{1st application: multiplicative structure\non cohomology}\n\n\n\\begin{prop} $\\Delta$ induces an associative product in $\\mathrm{Hom}_{A'-A}(B,k)$ (the graded Hom).\n\\end{prop}\n \\begin{proof} It is clear that $\\Delta$ induces an associative product on\n $\\mathrm{Hom}_{k}(B,k)$ (the graded Hom), and\n $\\mathrm{Hom}_{A'-A}(B,k)\\subset\\mathrm{Hom}_k(B,k)$ is a $k$-submodule. We will show that it is in fact a subalgebra.\n\n Consider the $A'$-$A$ diagonal structure on $B\\otimes B$\n(i.e. $x_1'.(b\\otimes b').x_2=x_1'bx_2\\otimes x_1'b'x_2$)\nand denote $B\\otimes^DB$ the $k$-module $B\\otimes B$ considered as $A'-A$-bimodule in this diagonal way.\nWe claim that \n$\\Delta: B\\rightarrow B\\otimes^D B$ is a morphism of $A'-A$-modules:\n\\[\n \\Delta(x_1'yx_2)=x_1'yx_2\\otimes x_1'yx_2=x_1'(y\\otimes y)x_2\n\\]\nsame with $y'$, and with $e_x$:\n\\[\n \\Delta(x_1'e_yx_2)=(x_1'\\otimes x_1')(y'\\otimes e_y+e_y\\otimes y)(x_2\\otimes x_2)=x'_1\\Delta(e_y)x_2\n\\]\n\nDualizing $\\Delta$ one gets:\n\\[\n\\Delta^*:\\mathrm{Hom}_{A'-A}(B\\otimes^DB, k)\\rightarrow \\mathrm{Hom}_{A'-A}(B,k)\n\\]\nconsider the natural map\n$$\\iota:\\mathrm{Hom}_k(B,k)\\otimes \\mathrm{Hom}_k(B,k)\\rightarrow \\mathrm{Hom}_k(B\\otimes B, k)$$\n$$\\iota(f\\otimes g)(b_1\\otimes b_2)=f(b_1)g(b_2)$$\nand denote $\\iota|$ by\n\\[\n \\iota|=\\iota|_{\\mathrm{Hom}_{A'-A}(B,k)\\otimes \\mathrm{Hom}_{A'-A}(B,k)}\n\\]\nLet us see that \n\\[\n Im (\\iota|)\\subset \\mathrm{Hom}_{A'-A}(B\\otimes B, k)\\subset \\mathrm{Hom}_k(B\\otimes B, k)\n\\]\nIf  $f,g: B\\rightarrow k$ are two $A'-A$-module morphisms (recall $k$ has trivial actions, i.e. $x'\\lambda=\\lambda$ and $\\lambda x=x$),\nthen\n\\[\n \\iota(f\\otimes g)(x'(b_1\\otimes b_2))=f(x'b_1)g(x'b_2)=(x'f(b_1))(x'g(b_2))\\]\n \\[=f(b_1)g(b_2)=x'\\iota (f\\otimes g)(b_1\\otimes b_2)\n\\]\n\\[\n \\iota(f\\otimes g)((b_1\\otimes b_2)x)=f(b_1x)g(b_2x)=(f(b_1)x)(g(b_2)x)\n \\]\n \\[=(f(b_1)g(b_2))x=\\iota(f\\otimes g)(b_1\\otimes b_2)x\n\\]\nSo, it is possible to compose $\\iota|$ and $\\Delta$, and obtain in this way an associative multiplication in $\\mathrm{Hom}_{A'-A}(B,k)$.\n\\end{proof}\n\n\n\nNow we will describe several natural quotients of $B$, each of them\ngive rise to a subcomplex of the cohomological complex of $X$\nwith trivial coefficients that are not only subcomplexes but also subalgebras;\nin particular they are associative algebras.\n\n\\subsection{Square free case}\nA solution $(X,\\sigma)$ of YBeq satisfying $\\sigma(x,x)=(x,x)\\forall x\\in X$ is called {\\em square free}. \nFor instance, if $X$ is a rack, then this condition is equivalent to $X$ being a quandle.\n\nIn the square free situation,\nnamely when $X$ is such that $\\sigma (x,x)=(x, x)$ for all $x$, we add the condition $e_xe_x\\sim 0$.\n \n\n If $(X,\\sigma)$ is a square-free solution of the YBeq,  let us denote\n $sf$ the two sided ideal of $B$ generated by $\\{e_xe_{x}\\}_{x\\in X}$.\n\\begin{prop}\n\\label{propsf}\n$sf$ is a differential Hopf ideal. More precisely, \n\\[\nd(e_xe_{x})=0\n\\hbox{  and }\n\\Delta(e_xe_{x})=x'x'\\otimes e_xe_{x}+\ne_xe_{x}\\otimes xx.\\]\n\\end{prop}\nIn particular $B\/sf$ is a differential graded bialgebra.\nWe may identify \\\\\n$\\mathrm{Hom}_{A'A}(B\/sf,k)\\subset \n\\mathrm{Hom}_{A'A}(B,k)$ as the elements $f$ such that $f(\\dots,x,x,\\dots)=0$. If $X$ is a quandle, this\nconstruction leads to the quandle-complex.\nWe have    $\\mathrm{Hom}_{A'A}(B\/sf,k)\\subset \n\\mathrm{Hom}_{A'A}(B,k)$ is not only a subcomplex, but also a subalgebra.\n \n\n\n\\subsection{Biquandles}\n\nIn \\cite{KR}, a generalization of quandles is proposed (we recall it with different notation), \na solution $(X,\\sigma)$ is called\nnon-degenerated, or {\\em birack} if in addition, \n\\begin{enumerate}\n \\item for any $x,z\\in X$ there exists a unique $y$ such that $\\sigma^1\\!(x,y)=z$, (if this is the case,  $\\sigma^1\\!$ is called {\\em left invertible}),\n \\item for any $y,t\\in X$ there exists a unique $x$ such that $\\sigma^2(x,y)=t$, (if this is the case,  $\\sigma^2$ is called {\\em right invertible}),\n\\end{enumerate}\nA birack is called {\\em biquandle} if, given $x_0\\in X$, there exists a unique $y_0\\in X$ such that\n$\\sigma(x_0,y_0)=(x_0,y_0)$. In other words, if there exists a bijective map $s:X\\to X$ such\nthat\n\\[\n\\{(x,y):\\sigma(x,y)=(x,y)\\}=\n\\{(x,s(x)): x\\in X\\}\n\\]\n\n\\begin{rem}\n Every quandle solution is a biquandle, moreover, given a rack $(X,\\triangleleft)$, then\n $\\sigma(x,y)=(y,x\\triangleleft y)$ is a biquandle if and only if $(X,\\triangleleft)$ is a quandle.  \n\\end{rem}\n\nIf $(X,\\sigma)$ is a biquandle, for all $x\\in X$ we add in $B$ the relation $e_{x}e_{s(x)}\\sim 0$. \nLet us denote $bQ$ the two sided ideal of $B$ generated by $\\{e_xe_{sx}\\}_{x\\in X}$.\n\\begin{prop}\n\n\n\n\n\n\\label{propbQ}\n$bQ$ is a differential Hopf ideal. More precisely, \n$d(e_xe_{sx})=0$ and\n$\\Delta(e_xe_{sx})=x's(x)'\\otimes e_xe_{sx}+\ne_xe_{sx}\\otimes xs(x)$.\n\\end{prop}\nIn particular $B\/bQ$ is a differential graded bialgebra.\nWe may identify\n\\[\n\\mathrm{Hom}_{A'A}(B\/bQ,k)\\cong \n\\{f\\in \\mathrm{Hom}_{A'A}(B,k) : f(\\dots,x,s(x),\\dots)=0\\} \n\\subset \n\\mathrm{Hom}_{A'A}(B,k)\\]\nIn \\cite{CES}, the condition $f(\\dots,x_0,s(x_0),\\dots)=0$ is\n called the {\\em type 1 condition}. A consequence of the above proposition is that\n  $\\mathrm{Hom}_{A'A}(B\/bQ,k)\\subset \n\\mathrm{Hom}_{A'A}(B,k)$ is not only a subcomplex, but also a subalgebra.\nBefore proving this proposition we will review some other similar constructions.\n\n \n\\subsection{Identity case}\nThe two cases above may be generalized in the following way:\n\nConsider $S\\subseteq X\\times X$ a subset of elements verifying\n$\\sigma(x,y)=(x,y)$ for all $(x,y)\\in S$. \nDefine $idS$ the two sided ideal of $B$ given by $idS=\\langle e_xe_y\/ (x,y)\\in S\\rangle$.\n\n\\begin{prop}\n\\label{propidS}\n$idS$ is a differential Hopf ideal. More precisely, \n$d(e_xe_{y})=0$ for all $(x,y)\\in S$ and\n$\\Delta(e_xe_{y})=x'y'\\otimes e_xe_{y}+\ne_xe_{y}\\otimes xy$.\n\\end{prop}\nIn particular $B\/idS$ is a differential graded bialgebra.\\\\\nIf one identifies $\\mathrm{Hom}_{A'A}(B\/sf,k)\\subset \n\\mathrm{Hom}_{A'A}(B,k)$ as the elements $f$ such that\n\\[ f(\\dots,x,y,\\dots)=0 \\ \\forall (x,y)\\in S\\]\nWe have that   $\\mathrm{Hom}_{A'A}(B\/idS,k)\\subset \n\\mathrm{Hom}_{A'A}(B,k)$ is not only a subcomplex, but also a subalgebra.\n \n \\subsection{Flip case}\n Consider the condition $e_xe_y+e_ye_x\\sim 0$ for all pairs such that $\\sigma(x,y)=(y,x)$. For such a pair $(x,y)$ we have \n the equations $xy=yx$, $xy'=y'x$, $x'y'=y'x'$ and $xe_y=e_yx$. Note that there is no equation for $e_xe_y$. \n The two sided ideal $D=\\langle e_xe_y+e_ye_x:\\sigma(x,y)=(y,x)\\rangle$ is a differential and Hopf ideal.\n\nMoreover, the following generalization is still valid:\n\n\\subsection{Involutive case}\n\nAssume  $\\sigma(x,y)^2=(x, y)$. This case is called {\\em involutive} in \\cite{ETS}.\nDefine $Invo$ the two sided ideal of $B$ given by $Invo=\\langle e_xe_y+e_ze_t : (x,y)\\in X,\\sigma(x,y)=(z,t)\\rangle$.\n\n\\begin{prop}\n\\label{propInvo}\n$Invo$ is a differential Hopf ideal. More precisely, \n$d(e_xe_{y}+e_ze_t)=0$ for all $(x,y)\\in X$ (with $(z,t)=\\sigma(x,y)$) and\nif $\\omega=e_xe_y+e_ze_t$ then\n$\\Delta(\\omega)=x'y'\\otimes \\omega+\\omega \\otimes xy$.\n\\end{prop}\nIn particular $B\/Invo$ is a differential graded bialgebra.\nIf one identifies \\\\ \n$\\mathrm{Hom}_{A'A}(B\/Invo,k)\\subset \n\\mathrm{Hom}_{A'A}(B,k)$ then\n$\\mathrm{Hom}_{A'A}(B\/Invo,k)\\subset \\mathrm{Hom}_{A'A}(B,k)$ is not only a subcomplex, but a subalgebra.\n \n\\begin{conj}\n$B\/Invo$ is acyclic in positive degrees.\n\\end{conj}\n\n\\begin{ex} If $\\sigma=flip$ and $X=\\{x_1,\\dots,x_n\\}$ then $A=k[x_1,\\dots,x_n]=SV$, the symmetric algebra on \n$V=\\oplus _{x\\in X}kx$. In this case\n$(B\/Invo,d)\\cong (S(V)\\otimes\\Lambda V\\otimes S(V),d)$ gives the Koszul resolution of $S(V)$ as\n$S(V)$-bimodule.\n\\end{ex}\n\n\\begin{ex} If $\\sigma=Id$,  $X=\\{x_1,\\dots,x_n\\}$\nand $V=\\oplus _{x\\in X}kx$, then $A=TV$ the tensor algebra.\nIf $\\frac12\\in k$, then\n$(B\/invo,d)\\cong TV\\otimes (k\\oplus V)\\otimes TV$ gives the Koszul resolution\nof $TV$ as $TV$-bimodule. Notice that we don't really need $\\frac12\\in k$,\none could replace $invo=\\langle e_xe_y+e_xe_y:(x,y)\\in X\\times X\\rangle$ by\n $idXX=\\langle e_xe_y:(x,y)\\in X\\times X\\rangle$.\n \\end{ex}\n\n\nThe conjecture above, besides these examples, is supported by\nnext result:\n\\begin{prop}\\label{propinvo}\nIf $\\mathbb{Q}\\subseteq k$, then $B\/Invo$ is acyclic in positive degrees.\n\\end{prop}\n\\begin{proof}\nIn $B\/Invo$ it can be defined $h$ as the unique  (super)derivation such that:\n\\[\n h(e_x)=0; h(x)=e_x, h(x')=-e_x\n\\]\nLet us see that $h$ is well defined:\n\\[\n h(xy-zt)=e_xy+xe_y-e_zt-ze_t=0\n\\]\n\\[\n h(xy'-z't)=e_xy'-xe_y+e_zt-z'e_t=0\n\\]\n\\[\n h(x'y'-z't')=-e_xy'-x'e_y+e_zt'+z'e_t=0\n\\]\n\\[\n h(xe_y-e_zt)=e_xe_y+e_ze_t=0\n\\]\nNotice that in particular next equation shows\n that $h$ is not well-defined in $B$.\n\\[\n h(e_xy'-z'e_t)=e_xe_y+e_ze_t=0\n\\]\n\\[\n h(zt'-x'y)=e_zt'-ze_t+e_xy-x'e_y=0\n\\]\n\\[\nh(ze_t-e_xy)= e_ze_t+e_xe_y=0\n\\]\n\\[\n h(e_zt'-x'e_y)=e_ze_t+e_xe_y=0\n\\]\n\\[\n h(e_xe_y+e_ze_t)=0\n\\]\nSince (super) commutator of (super)derivations is again a derivation,\nwe have that $[h, d]=hd+dh$ is also a derivation. \n Computations on generators: \n\\[\nh(e_x)=2e_x,\\  h(x)=x-x', \\ h(x')=x'-x\n\\]\nor equivalently\n\\[\nh(e_x)=2e_x,\\  h(x+x')=0, \\ h(x-x')=2(x-x')\n\\]\nOne can also easily see that $B\/Invo$ is generated by\n$e_x,x_{\\pm}$, where $x_\\pm=x\\pm x'$, and that their\nrelations are homogeneous. We see that $hd+dh$ is nothing but the Euler\nderivation with respect to the grading defined by\n\\[\n\\deg e_x=2,\\\n\\deg x_+=0,\\\n\\deg x_-=2,\\]\nWe conclude automatically that the homology vanish for positive degrees of the $e_x$'s\n(and similarly for the  $x_-$'s).\n\\end{proof}\n\n Next, we generalize Propositions \n \\ref{propsf}, \\ref{propbQ}, \n \\ref{propidS} and \\ref{propInvo}.\n \n\\subsection{Braids of order $N$}\n \n Let $(x_0,y_0)\\in X\\times X$ such that \n $\\sigma^N(x_0,y_0)=(x_0,y_0)$ for some $N\\geq 1$.\n If $N=1$ we have the ''identity case'' and all  subcases, if $N=2$ we have the \n''involutive case''.\nDenote\n \\[\n (x_i,y_i):=\\sigma^i(x_0,y_0) \\ 1\\leq i \\leq N-1 \n \\]\n Notice that the following relations hold in $B$:\n\\begin{itemize}\n\\item[$\\star$] $x_{N-1}y_{N-1}\\sim x_0y_0$, \\ $x_{N-1}y'_{N-1}\\sim x'_0y_0$, \\  $x'_{N-1}y'_{N-1}=x'_0y'_0$\n\\item[$\\star$] $ x_{N-1}e_{y_{N-1}}\\sim e_{x_0}y_0$, \\ $ e_{x_{N-1}}y'_{N-1}\\sim x'_0e_{y_0}$\n\\end{itemize}\n  and for $1\\leq i \\leq N-1$:\n \\begin{itemize}\n\\item[$\\star$] $x_{i-1}y_{i-1}\\sim x_iy_i$, \\  $x_{i-1}y'_{i-1}\\sim x'_iy_i$, \\  $x'_{i-1}y'_{i-1}=x'_iy'_i$\n\\item[$\\star$] $ x_{i-1}e_{y_{i-1}}\\sim e_{x_i}y_i$, \\  $ e_{x_{i-1}}y'_{i-1}\\sim x'_ie_{y_i}$\n\\end{itemize} \nTake $\\omega=\\sum_{i=0}^{N-1}e_{x_i}e_{y_i}$, then we claim that \n\\[\nd\\omega=0\\]\nand \n\\[\\Delta\\omega\n=x_0y_0\\otimes\\omega+\\omega\\otimes x'_0y'_0\\]\nFor that, we compute\n\\[\n d(\\omega)=\\sum_{i=0}^{N-1}(x_i-x'_i)e_{y_i}-e_{x_i}(y_i-y'_i)=\n\\]\n\\[\n \\sum_{i=0}^{N-1}(x_ie_{y_i}-e_{x_i}y_i)-  \\sum_{i=0}^{N-1} (x'_ie_{y_i}-e_{x_i}y'_i)=0\n\\]\nFor the comultiplication,\nwe recall that \n\\[\\Delta(ab)=\\Delta(a)\\Delta(b)\\]\n where the product on the right hand side is defined using the Koszul sign rule:\n \\[\n (a_1\\otimes a_2)(b_1\\otimes b_2)=(-1)^{|a_2||b_1|}a_1b_1\\otimes a_2b_2\n \\]\nSo, in this case we have\n\\[\n \\Delta(\\omega)=\\sum_{i=0}^{N-1}\\Delta (e_{x_i}e_{y_i})=\n\\]\n\\[\n\\sum_{i=0}^{N-1}(x'_iy'_i\\otimes e_{x_i}e_{y_i}-x'_ie_{y_i}\\otimes e_{x_i}y_i+ e_{x_i}y'_i\\otimes x_ie_{y_i}+e_{x_i}e_{y_i}\\otimes x_iy_i)\n\\]\nthe middle terms cancel telescopically, giving\n\\[\n=\\sum_{i=0}^{N-1}(x'_iy'_i\\otimes e_{x_i}e_{y_i}+e_{x_i}e_{y_i}\\otimes x_iy_i)\n\\]\nand the relation $x_iy_i\\sim x_{i+1}y_{i+1}$ gives\n\\[\n=x'_0y'_0\\otimes( \\sum_{i=0}^{N-1}e_{x_i}e_{y_i})+\n(\\sum_{i=0}^{n-1}e_{x_i}e_{y_i})\\otimes x_0y_0\n\\]\n\\[\n=x'_0y'_0\\otimes \\omega+\n\\omega\\otimes x_0y_0\n\\]\nThen the two-sided ideal of $B$ generated by $\\omega$ is a Hopf ideal.\nIf instead of a single $\\omega$ we have several $\\omega_1,\\dots \\omega_n$, we simply remark that\nthe sum of differential Hopf ideals is also a differential Hopf ideal.\n\n\\begin{rem} If X, is finite then for every $(x_0,y_0)$ there exists $N>0$ such that\n$\\sigma^N(x_0,y_0)=(x_0,y_0)$.\n\\end{rem}\n\n\\begin{rem}\nLet us suppose $(x_0,y_0)\\in X\\times X$ is such that $\\sigma^N(x_0,y_0)=(x_0,y_0)$ and  $u\\in X$ an arbitrary element.\nConsider the element \n\\[\n((\\mathrm{Id} \\times \\sigma)( \\sigma \\times \\mathrm{Id}) (u,x_0,y_0)=(\\widetilde x_0,\\widetilde y_0,u'')\n\\]\ngraphically\n\\[\n \\xymatrix{\n u\\ar[rd]&x\\ar[ld]&y\\ar[d]\\\\\n \\widetilde x\\ar[d]&u'\\ar[rd]&y\\ar[ld]\\\\\n \\widetilde x&\\widetilde y&\\widetilde u'' \n }\n\\]\nthen $\\sigma^N(\\widetilde x_0,\\widetilde y_0)=(\\widetilde x_0,\\widetilde y_0)$.\n\\end{rem}\n\\begin{proof}\\[\n              (\\sigma^N \\times id)(\\widetilde x_0,\\widetilde y_0,u'')=(\\sigma^N\\times id)(id\\times \\sigma)(\\sigma \\times id)(u,x_0,y_0)=\n             \\]\n\\[\n (\\sigma^{N-1}\\times id)(\\sigma\\times id)(id\\times \\sigma)(\\sigma \\times id)(u,x_0,y_0)=\n\\]\nusing YBeq\n\\[\n  (\\sigma^{N-1}\\times id)(id\\times \\sigma)(\\sigma \\times id)(id\\times \\sigma)(u,x_0,y_0)=\n\\]\n\nrepeating the procedure $N-1$ times leaves\n\\[\n(id\\times \\sigma)(\\sigma \\times id)(id\\times \\sigma^N)(u,x_0,y_0)=(id\\times \\sigma)(\\sigma \\times id )(u,x_0,y_0)=(\\widetilde x_0,\\widetilde y_0,u'')\n\\]\n\n\\end{proof}\n\n\n\n\\section{$2^{nd}$ application: Comparison with Hochschild cohomology}\n\n$B$ is a differential graded algebra, and on each degree $n$\nit is isomorphic to $A\\otimes (TV)_n\\otimes A$, where $V=\\oplus_{x\\in X}ke_x$.\nIn particular  $B_n$\nis free as $A^e$-module. We \nhave {\\em for free} the existence of a comparison map\n\\[\n\\xymatrix@-2ex{\n\\cdots\\ar[r]&B_n\\ar[r]\\ar@{=}[d]&\\cdots\\ar[r]&B_2\\ar[r]^d\\ar@{=}[d]&B_1\\ar[r]^d\\ar@{=}[d]&\\ar@{=}[d]B_0\\\\\n\\cdots\\ar[r]&A'(TX)_n A\\ar@{=}[d] ^{\\cong}\\ar[r]&\\cdots\\ar[r] &\\oplus_{x,y\\in X}A'e_xe_y A\\ar@{=} ^{\\cong}[d]\\ar[r]^d&\\oplus_{x\\in X} A'e_x A\\ar[r]^d\\ar@{=}[d]&A' A\\ar@{=}[d] ^{\\cong}\\\\\n\\cdots\\ar[r]&A\\otimes V^{\\otimes n}\\otimes A\\ar[d]^{\\widetilde\\mathrm{Id}}\\ar[r]&\\cdots\\ar[r] &A\\otimes V^{\\otimes 2}\\otimes  A\\ar[d]^{\\widetilde\\mathrm{Id}}\\ar[r]^{d_2}& A\\otimes V\\otimes A\\ar[d]^{\\widetilde\\mathrm{Id}}\\ar[r]^{d_1}&A\\otimes A\\ar[d]^{\\mathrm{Id}}\\ar[r]^m& A\\ar[d]^{\\mathrm{Id}}\\ar[r]&0\\\\\n\\cdots\\ar[r]&A\\otimes A^{\\otimes n}\\otimes A\\ar[r]& \\cdots\\ar[r] &A\\otimes A^{\\otimes 2}\\otimes  A\\ar[r]^{b'}& A\\otimes A\\otimes A\\ar[r]^{b'}&A\\otimes A\\ar[r]^m& A\\ar[r]&0\\\\\n}\\]\n\n\\begin{coro}\nFor all $A$-bimodule $M$, there exists natural maps\n\n\\[\n\\widetilde\\mathrm{Id}_*: H^{YB}_\\bullet(X,M)\\to H_\\bullet(A,M)\n\\]\n\\[\n\\widetilde\\mathrm{Id}^*: H^\\bullet(A,M)\\to H_{YB}^\\bullet(X,M)\n\\]\nthat are the identity in degree zero and 1.\n\\end{coro}\n\n\n\nMoreover, one can choose an explicit map with extra properties. For that we recall some definitions: there is a set theoretical section to the canonical projection  from the\nBraid group to the symmetric group\n\\[\n\\xymatrix{\n\\mathbb{B}_n\\ar@{->>}[r]& \\mathbb{S}_n\\ar@\/_\/@{..>}[l]\n}\n\\]\n\\[\n\\xymatrix{\nT_s:=\\sigma_{i_1}\\dots\\sigma_{i_k}\n&\n s=\\tau_{i_1}\\dots \\tau_{i_k} \\ar@{|->}[l]\n}\n\\]\nwhere \n\\begin{itemize}\n \\item $\\tau\\in S_n$ are transpositions of neighboring elements $i$ and $i+1$, \nso-called simple transpositions,\n \\item $\\sigma_i$ are the corresponding generators of $\\mathbb{B}_n$,\n \\item $\\tau_{i_{1}}\\dots \\tau_{i_{k}}$ is one of the shortest words representing $s$.\n\\end{itemize}\nThis inclusion factorizes trough \n\\[\n \\mathbb{S}_n\\hookrightarrow \\mathbb{B}_n^+\\hookrightarrow  \\mathbb{B}_n\n \\]\n It is a set inclusion not preserving the monoid structure.\n \n\\begin{defi}\nThe permutation sets\n\\[\n \\mathrm{Sh}_{p_1,\\dots,p_k}:=\\left\\{ s\\in \\mathbb{S}_{p_1+\\dots+p_k}\/s(1)<\\dots<s(p_1),\\cdots, s(p+1)<\\dots < s(p+p_k) \\right\\},\n\\]\nwhere $p=p_1+\\dots+p_{k-1}$,\nare called {\\em shuffle sets}.\n\\end{defi}\n\n\\begin{rem}\nIt is well known that  a braiding $\\sigma$ gives an action of the positive braid monoid $B_n^+$ on $V^{\\otimes n}$, i.e. a monoid morphism\n\\[\n \\rho: B_n^+\\rightarrow End_\\mathbb{K}(V^{\\otimes n})\n\\]\ndefined on generators $\\sigma_i$ of $B_n^+$ by\n\\[\n \\sigma_i\\mapsto \\mathrm{Id}_V^{\\otimes(i-1)}\\otimes \\sigma\\otimes \\mathrm{Id}_V^{\\otimes(n-i+1)}\n\\]\n\nThen there exists a natural extension of a braiding in $V$ to a braiding in $T(V)$.\n\n\\[\n {\\bf \\sigma}(v\\otimes w)=(\\sigma_k \\dots \\sigma_1)\\circ\\dots\\circ(\\sigma_{n+k-2}\\dots\\sigma_{n-1})\\circ(\\sigma_{n+k-1}\\dots\\sigma_n)(vw)\\in V^{k}\\otimes V^{n}\n\\]\nfor $v\\in V^{\\otimes n}$, $w\\in V^{k}$ and $vw$ being the concatenation. \n\nGraphically\n\n\\[\n\\xymatrix{\n\\ar[rrrrrdd]&\\ar[rrrrrdd]&\\dots&\\ar[rrrrrdd]&\\otimes&\\ar[llllldd]&\\ar[llllldd]&\\dots&\\ar[llllldd]\\\\\n&&&&&&&&\\\\\n&&\\dots&&\\otimes&&&\\dots&\n}\\]\n\\end{rem}\n\n\\begin{defi}\n The quantum shuffle multiplication on the tensor space $T(V)$ of a braided vector space $(V,\\sigma)$ is the $k$-linear extension \n of the map \n\\[\n \\shuffle_\\sigma\n=\n\\shuffle_\\sigma^{p,q}:V^{\\otimes p}\\otimes V^{\\otimes q}\\rightarrow V^{\\otimes(p+q)} \n\\]\n \\[\n\\overline v\\otimes \\overline w \\mapsto \n\\overline v \\shuffle_\\sigma\\overline w :=\\sum_{s\\in Sh_{p,q}}T_s^\\sigma(\\overline{vw})\n \\]\nNotation: $T_s^{\\sigma}$ stands for the lift $T_s\\in \\mathbb{B}_n^+$     \nacting on $V^{\\otimes n}$ via the braiding $\\sigma$. \nThe algebra $Sh_\\sigma(V):=(TV,\\shuffle_\\sigma)$ \nis called the quantum shuffle algebra on $(V,\\sigma)$\n\\end{defi}\nIt is well-known that \n$\\shuffle_\\sigma$ is an associative product on $TV$(see for example\n\\cite{Le} for details) that makes it a Hopf algebra with deconcatenation\ncoproduct. \n\n\\begin{defi}\nLet $V$ be a braided vector space, then the quantum symmetrizer\nmap $\\shuffle_{\\sigma}:V^{\\otimes n}\\to V^{\\otimes n}$ defined by\n\\[\nQS_{\\sigma}(v_1\\otimes\\cdots  \\otimes v_n)=\n\\sum_{\\tau\\in \\mathbb{S}_n}T^\\sigma_\\tau(v_1\\otimes\\cdots\\otimes v_n)\n\\]\n where  $T_\\tau^\\sigma$ is the lift $T^\\sigma_\\tau\\in \\mathbb{B}_{n}^+$ \nof $\\tau$, acting on $V^{\\otimes n}$ via the braiding $\\sigma$.\n\nIn terms of shuffle products the quantum symmetrizer can be computed as\n\\[\n\\omega \\shuffle_{\\sigma} \\eta\n:=\\sum_{\\tau\\in\\mathrm{Sh}_{p,q}}T^\\sigma_\\tau(\\omega\\otimes\\eta)\n\\]\n\n\\end{defi}\n\nThe quantum symmetrizer map can also be defined as\n\\[\nQS_\\sigma(v_1\\otimes\\cdots\\otimes v_n)=\nv_1\\shuffle_\\sigma  \\cdots \\shuffle_\\sigma v_n\n\\]\nWith this notation, next result reads as follows:\n\\begin{teo}\\label{teoid}\n The  $A'$-$A$-linear  quantum symmetrizer map\n\\[\n \\xymatrix{\n A'V^{\\otimes n}A\\ar[r]^{\\widetilde \\mathrm{Id}}& A\\otimes A^{\\otimes n}\\otimes A\\\\\n a_1'e_{x_1}\\cdots e_{x_n}a_2\\ar@{|->}[r]&\n a_1\\otimes(x_1\\shuffle_{-\\sigma}  \\cdots \\shuffle_{-\\sigma} x_n)\\otimes a_2}\n\\]\nis a chain map lifting the identify. Moreover, \n$\\widetilde\\mathrm{Id}:B\\to (A\\otimes TA\\otimes A,b')$ is a differential graded algebra\nmap, where in $TA$ the product is $\\shuffle_{-\\sigma}$, and\nin $A\\otimes TA\\otimes A$ the multiplicative structure\n is not the usual tensor product algebra, but the braided one.\n In particular, this map factors through\n $A\\otimes \\mathfrak{B}\\otimes A$, where $\\mathfrak{B}$ is the Nichols algebra\n associated to the braiding $\\sigma'(x\\otimes y)= - z\\otimes t$, where\n $x,y\\in X$ and $\\sigma(x,y)=(z,t)$. \n\\end{teo}\n\n\n\\begin{rem}The Nichols algebra\n$\\mathfrak{B}$ is the quotient of $TV$ by the ideal generated by  (skew)primitives that\nare not in $V$, so the  result above explains the good behavior\nof the ideals $invo$, $idS$,\nor in general the ideal generated by\nelements\nof the form\n $\\omega=\\sum_{i=0}^{N-1}e_{x_i}e_{y_i}$ where $\\sigma(x_i,y_i)=(x_{i+1},y_{i+1})$ and $\\sigma^N(x_0,y_0)=(x_0,y_0)$.\n It would be interesting to know the properties of $A\\otimes\\mathfrak{B}\\otimes A$\n as a differential object, since it appears to be a candidate of\n Koszul-type resolution for the semigroup algebra $A$\n (or similarly the group algebra $k[G_X]$).\n \\end{rem}\n\nThe rest of the paper is devoted to the proof of \\ref{teoid}. Most of the Lemmas\nare \"folklore\" but we include them for completeness. The interested reader can\nlook at \\cite{Lebed2} and references therein.\n\n\\begin{lem}\\label{AC monoid}\n Let $\\sigma$ be a braid in the braided (sub)category that contains two associative algebras  $A$ and $C$, meaning there \n exists\n bijective functions \n \\[\n\\sigma_A:A\\otimes A\\to A\\otimes A,\\\n\\sigma_C:C\\otimes C\\to C\\otimes C,\\\n\\sigma_{C,A}:C\\otimes A\\to A\\otimes C\\]\nsuch that  \n\\[\n\\sigma_*(1,-)=(-,1)\\hbox{ and } \\sigma_*(-,1)=(1,-) \\ \\hbox{ for } *\\in \\{A,C;C,A\\}           \n\\]\n\\[\n          \\sigma_{C,A}\\circ (1\\otimes m_A)=(m_A\\otimes 1)(1\\otimes \\sigma_{C,A})(\\sigma_{C,A}\\otimes 1)\n\\]\n and \n\\[\\sigma_{C,A}\\circ ( m_C\\otimes 1)=(1\\otimes m_C)(\\sigma_{C,A}\\otimes 1)(1\\otimes \\sigma_{C,A})\\]\nDiagrammatically\n\\\n\\xymatrix{\nC\\ar[d]&A\\ar[rd]^{\\!\\!\\!m_A}&&A\\ar[ld]\\\\\n\\ar[rrd]^{\\!\\!\\!\\!\\!\\!\\! \\sigma_{C,A}}&&A\\ar[lld]&\\\\\nA&&C&\n}\n\\xymatrix{\n\\\\\n\\\\\n&=^{[*]}&\\\\}\n\\xymatrix{\nC\\ar[rrd]^{\\!\\!\\!\\!\\sigma_{C,A}}&&A\\ar[lld]&A\\ar[d]\\\\\nA\\ar[d]&&C\\ar[rd]&A\\ar[ld]\\\\\nA\\ar[rd]&&A\\ar[ld]&C\\ar[d]\\\\\n&A&&C\n}\n\\]\nand\n\\[\n\\xymatrix{\nC\\ar[rd]^{\\ \\ m_C}&&C\\ar[ld]&A\\ar[d]\\\\\n&C\\ar[rrd]^{\\!\\!\\!\\!\\!\\!\\!\\sigma_{C,A}}&&A\\ar[lld]\\\\\n&A&&C\n}\n\\xymatrix{\n\\\\\n\\\\\n&=^{[**]}&\\\\}\n\\xymatrix{\nC\\ar[d]&C\\ar[rrd]&&A\\ar[lld]\\\\\nC\\ar[rd]&A\\ar[ld]&&C\\ar[d]\\\\\nA\\ar[d]&C\\ar[rd]&&C\\ar[ld]\\\\\nA&&C&\n}\n\\]\nAssume that they\nsatisfy the braid equation with any combination of $\\sigma_A,\\sigma_C$ or $\\sigma_{A,C}$.\nThen, $A\\otimes_\\sigma C=A\\otimes C$ with product defined by\n \\[\n(m_A\\otimes m_C)\\circ(\\mathrm{Id}_A\\otimes \\sigma_{C,A}\\otimes\\mathrm{Id}_C)\\colon\n(A\\otimes C)\\otimes (A\\otimes C)\\to A\\otimes C\n\\]\nis an associative algebra. In diagram:\n\\[\n\\xymatrix{\nA\\ar[d]&&C\\ar[rd]^{\\!\\!\\!\\sigma}&A\\ar[ld]&&C\\ar[d]\\\\\nA\\ar[rd]^{\\ \\ m_A}&&A\\ar[ld]&C\\ar[rd]^{\\ \\ m_C}&&C\\ar[ld]\\\\\n&A&&&C&\n}\\]\n \n\\end{lem}\n\\begin{proof}\n\n\n\nTake $m\\circ (1\\otimes m)((a_1\\otimes c_2)\\otimes ((a_2\\otimes c_2)\\otimes (a_3\\otimes c_3))$\n use $[*]$, associativity in $A$, associativity in $C$ then $[**]$ and the result follows.\n\\end{proof}\n\n\n\n\n\n\\begin{lem}\nLet $M$ be the  monoid freely generated by $X$\nmodule the relation $xy=zt$ where $\\sigma(x,y)=(z,t)$, then,\n$\\sigma:X\\times X\\to X\\times X$ naturally extends to a braiding in  $M$ and verifies \n\n\n\n\\[\n\\xymatrix{\nM\\ar[rd]^{\\ \\ \\ m}&   &M\\ar[ld]&M\\ar[d]^\\mathrm{Id}\\\\\n   &M\\ar[rrd]^\\sigma&&M\\ar[lld]\\\\\n&M&&M\n} \n\\xymatrix{\n\\\\\n\\\\\n&=&\\\\\n} \n\\xymatrix{\nM\\ar[d]^{\\mathrm{Id}}   &M\\ar[rrd]^\\sigma&&M\\ar[lld]\\\\\n  M\\ar[rd]^{\\!\\!\\sigma}&M\\ar[ld]&&M\\ar[d]\\\\\nM\\ar[d]&M\\ar[rd]^{\\ \\ \\ m}&&M\\ar[ld]\\\\\nM&&M} \n\\]\n\n\\[\n\\xymatrix{\nM\\ar[d]&M\\ar[rd]   &&M\\ar[ld]_{m}\\\\\n  M \\ar[rrd]^\\sigma&&M\\ar[lld]\\\\\n  M&&M\n} \n\\xymatrix{\n\\\\\n\\\\\n&=&\\\\\n} \n\\xymatrix{\nM\\ar[rd]^{\\!\\!\\sigma}   &M\\ar[ld]&&M\\ar[d]\\\\\n  M\\ar[d]&M\\ar[rrd]^\\sigma&&M\\ar[ld]\\\\\nM\\ar[rd]^m&&M\\ar[ld]&M\\ar[d]^\\mathrm{Id}\\\\\n&M&&M} \n\\]\n\\end{lem}\n\n\\begin{proof}\n It is enough to prove that the extension mentioned before is well defined in the quotient. Inductively, it will be enough to\n see that\n $\\sigma(axyb,c)=\\sigma(aztb,c)$ and  $\\sigma(c,axyb)=\\sigma(c,aztb)$ where \n$ \\sigma(x,y)=(z,t)$, and this follows\n immediately from the braid equation:\n\n A diagram for the first equation is the following: \n  \\[\n\\xymatrix{\na\\ar[d]&x\\ar[rd]&y\\ar[ld]&b\\ar[rd]&c\\ar[ld]\\\\\n\\ar[d]&z\\ar[d]&t\\ar[rd]&\\ar[ld]&\\ar[d]\\\\\n\\ar[d]&\\ar[rd]&\\ar[ld]&\\ar[d]&\\ar[d]\\\\\n\\ar[rd]&\\ar[ld]&\\ar[d]&\\ar[d]&\\ar[d]\\\\\n&&\\alpha&\\beta&} \n\\xymatrix{\n\\\\\n\\\\\n&=&\\\\\n} \n\\xymatrix{\na\\ar[d]&x\\ar[d]&y\\ar[d]&b\\ar[rd]&c\\ar[ld]\\\\\n\\ar[d]&\\ar[d]&\\ar[rd]&\\ar[ld]&\\ar[d]\\\\\n\\ar[d]&\\ar[rd]&\\ar[ld]&\\ar[d]&\\ar[d]\\\\\n\\ar[rd]&\\ar[ld]&\\ar[rd]&\\ar[ld]&\\ar[d]\\\\\n&&\\alpha^*&\\beta^*&\n } \n\\]\n \n As $\\alpha\\beta=\\alpha^*\\beta^*$ the result follows.\n  \n  \n\\end{proof}\n\n\n\n\n\n\\begin{lem}\n\n$m\\circ\\sigma=m$, diagrammatically:\n\\[\n\\xymatrix{\nM\\ar[rrd]&&M\\ar[lld]\\\\\nM \\ar[rd]^{\\ \\ \\ m}  &&M\\ar[ld]\\\\\n&  M } \n\\xymatrix{\n\\\\\n\\\\\n&=&\\\\\n} \n\\xymatrix{\nM\\ar[rd]^{\\ \\ \\ m}&&M\\ar[ld]\\\\\n&  M\\ar[d]^\\mathrm{Id}\\\\\n&  M\n } \n\\]\n\n\\end{lem}\n\n\\begin{proof} Using successively that $m\\circ \\sigma_i=m$, we have: \n\\[m\\circ \\sigma(x_1\\dots x_n, y_1\\dots y_k)=m\\left((\\sigma_k\\dots \\sigma_1)\\dots(\\sigma_{n+k-1}\\dots \\sigma_n)_{(x_1\\dots x_ny_1\\dots y_k)}\\right)\\]\n\\[=m\\left((\\sigma_{k-1}\\dots \\sigma_1)\\dots(\\sigma_{n+k-1}\\dots \\sigma_n)_{(x_1\\dots x_ny_1\\dots y_k)}\\right)=\\dots\\newline\\]\n\\[\n=m(x_1\\dots x_n,y_1\\dots y_k) \n\\]\n\\end{proof}\n\n\\begin{coro}\nIf one considers  $A=k[M]$,\nthen the algebra $A$ verifies all diagrams in previous lemmas.\n \\end{coro}\n\n\\begin{lem}\nIf $T=(TA, \\shuffle_\\sigma)$ there are bijective functions \n\\[\n\\sigma_{T,A}:=\\sigma|_{T\\otimes A}: T\\otimes A\\rightarrow A\\otimes T                                                                                    \n\\]\n\\[\n\\sigma_{A,T}:=\\sigma|_{A\\otimes T}: A\\otimes T\\rightarrow T\\otimes A  \n\\]\nthat verifies the hypothesis of Lemma \\ref{AC monoid}, and the same for \n $(TA, \\shuffle_{-\\sigma})$.\n\\end{lem}\n\\begin{coro}\n $A\\otimes (TA, \\shuffle_{-\\sigma})\\otimes A$ is an algebra.\n\\end{coro}\n\n\\begin{proof}\n Use \\ref{AC monoid} twice and the result follows.\n\\end{proof}\n\n\\begin{coro}\\label{btp}\nTaking $A=k[M]$, then the standard resolution of $A$ as $A$-bimodule has a natural algebra structure\n defining the braided  tensorial product as follows: \n\\[\nA\\otimes TA\\otimes A=\nA\\otimes_\\sigma(T^cA,\\shuffle_{-\\sigma})\\otimes_\\sigma A\\]\n\\end{coro}\nRecall the differential of the standard resolution\nis defined as\n$b':A^{\\otimes n+1}\\to A^{\\otimes n}$\n\\[ b'(a_0\\otimes\\dots\\otimes a_n)= \\sum_{i=0}^{n-1}(-1)^{i}a_0\\otimes \\dots\\otimes a_ia_{i+1}\\otimes\\dots\\otimes a_n\\]\nfor all  $n\\geq 2$.\nIf $A$ is a commutative algebra then the Hochschild resolution\nis an algebra viewed as $\\oplus_{n\\geq 2}A^{\\otimes n}=A\\otimes TA\\otimes A$, with right and left \n$A$-bilinear extension  of the shuffle product  on $TA$, and $b'$ is a (super) derivation with\n respect to that product (see for instance Prop. 4.2.2 \\cite{L}). \nIn the braided-commutative case we have the analogous result:\n\\begin{lem}\n$b'$ is a derivation with respect to the product mentioned in Corollary \\ref{btp}.\n\\end{lem}\n\\begin{proof}\nRecall the commutative proof \nas in Prop. 4.2.2 \\cite{L}. \nDenote $*$ the product\n \\[\n(a_0\\otimes\\dots\\otimes a_{p+1} )*(b_0\\otimes\\dots\\otimes b_{q+1})=\na_0b_0\\otimes((a_1\\dots\\otimes a_{p} )\\shuffle (b_1\\otimes\\dots\\otimes b_{q}))\\otimes a_{p+1}b_{q+1}\n\\]\nSince $\\oplus_{n\\geq 2}A^{\\otimes n}=A\\otimes TA\\otimes A$\nis generated by $A\\otimes A$ and $1\\otimes TA\\otimes 1$, we check on generators.\nFor $a\\otimes b\\in A\\otimes A$, $b'(a\\otimes b)=0$, in particular, it satisfies Leibnitz\nrule for elements in $A\\otimes A$.\n Also, $b'$ is $A$-linear on the left, and right-linear on the right, so\n\\[\n b'\\big((a_0\\otimes a_{n+1})*(1\\otimes a_1\\otimes \\cdots \\otimes a_n\\otimes 1)\\big)=\nb'(a_0\\otimes a_1\\otimes\\cdots\\otimes a_n\\otimes a_{n+1})\n\\]\n\\[=\na_0b'(1\\otimes a_1\\otimes\\cdots\\otimes a_n\\otimes 1)a_{n+1}\n=(a_0\\otimes a_{n+1})*b'(1\\otimes a_1\\otimes\\cdots\\otimes a_n\\otimes 1)\n\\]\n\\[\n=0+(a_0\\otimes a_{n+1})*b'(1\\otimes a_1\\otimes\\cdots\\otimes a_n\\otimes 1)\n \\]\\[\n=b'(a_0\\otimes a_{n+1})*(1\\otimes a_1\\otimes\\cdots\\otimes a_n\\otimes 1)\n+(a_0\\otimes a_{n+1})*b'(1\\otimes a_1\\otimes\\cdots\\otimes a_n\\otimes 1)\n\\]\nNow consider $(1\\otimes a_1\\otimes \\dots\\otimes a_{p}\\otimes 1 )*(1\\otimes b_1\\otimes\\dots\\otimes b_{q}\\otimes 1)$,\nit is a sum\nof terms where two consecutive tensor terms can be of the form\n$(a_i,a_{i+1})$, or $(b_j,b_{j+1})$, or $(a_i,b_j)$ or $(b_j,a_i)$.\nWhen one computes $b'$, multiplication of two consecutive tensor factors will give,\nrespectively,  terms  of the form\n\\[ \\cdots\\otimes a_ia_{i+1}\\otimes \\cdots, \\  \\cdots\\otimes b_jb_{j+1}\\otimes\\cdots,\\ \\cdots\\otimes a_ib_j\\otimes\\cdots,\n\\cdots  \\otimes b_ja_i\\otimes\\cdots\n\\]\nThe first type of terms will recover $b'((1\\otimes a_1\\otimes\\cdots\\otimes a_n\\otimes 1))*\n(1\\otimes b_1\\otimes\\cdots\\otimes b_q\\otimes 1)$ and the second type of terms will recover\n$\\pm (1\\otimes a_1\\otimes\\cdots\\otimes a_n\\otimes 1)*b'((1\\otimes b_1\\otimes\\cdots\\otimes b_q\\otimes 1))$.\nOn the other hand, the difference between the third and forth type of terms is just\na single trasposition so they have different signs, while $a_ib_j=b_ja_i$ because the \nalgebra is commutative, if one take the {\\em signed} shuffle then  they cancel each other.\n\n\nIn the {\\em braided} shuffle product, the summands are indexed by the same set\nof shuffles, so we have the same type of terms, that is, when computing $b'$ of \na (signed) shuffle product, one may do the product of two elements in coming form the \nfirst factor, two elements of the second factor. \nor a mixed term. For the mixed terms, they will have the form\n\\[\n\\cdots\\otimes A_iB_j \\otimes\\cdots \\hbox{, or }\n\\cdots\\otimes \\sigma^1(A_i,B_j)\\sigma^2(A_i,B_j)\\otimes\\cdots\n    \\]\nAs in the algebra $A$ we have  $A_iB_j=\\sigma^1(A_i,B_j)\\sigma^2(A_i,B_j)$ then this \nterms will cancel leaving only the terms corresponding to \n $b'(1\\otimes a_1\\otimes\\cdots\\otimes a_p \\otimes 1)\\shuffle_{-\\sigma} (1\\otimes b_1\\otimes\\cdots\\otimes b_q\\otimes )$\nand $\\pm(1\\otimes a_1\\otimes\\cdots\\otimes a_p\\otimes 1 )\\shuffle_{-\\sigma} b'(1\\otimes b_1\\otimes\\cdots\\otimes b_q\\otimes 1)$\nrespectively.\n\n\n\n\\end{proof}\n\n\n\n\n\\begin{coro} There exists a  comparison morphism  \n$f:(B,d)\\to (A\\otimes TA\\otimes A,b')$ which is a differential graded algebra morphism, $f(d)=b'(f)$,\n simply defining it on  $e_x$ ($x\\in X$)\n and verifying $f(x'-x)=b'(f(e_x))$.\n\\end{coro}\n\\begin{proof}\nDefine $f$ on $e_x$, extend $k$-linearly to $V$, multiplicatively to $TV$, and $A'$-$A$ linearly to\n$A'\\otimes TV\\otimes A=B$. In order to see that $f$ commutes with the differential, by $A'$-$A$-linearity\nit suffices to check on $TV$, but since $f$ is multiplicative on $TV$ it is enough to check on $V$, and by $k$-linearity we check on basis, that is, we only need $f(de_x)=b'f(e_x)$.\n\\end{proof}\n\n\n\\begin{coro}\n$f|_{TX}$ is the quantum symmetrizer map, and therefore \n$\\mathrm{Ker}(f)\\cap TX\\subset B$ defines the  Nichol's ideal \nassociated to $-\\sigma$.\n\\end{coro}\n\\begin{proof}\n\\[\n f(e_{x_1}\\cdots e_{x_n})=f(e_{x_1})*\\cdots *f(e_{x_n})=(1\\otimes x_1\\otimes 1)*\\cdots *(1\\otimes x_n \\otimes 1)=1\\otimes(x_1\\shuffle \\cdots \\shuffle x_n)\\otimes 1\n\\]\n\n\\end{proof}\n\nThe previous corollary explains why $\\mathrm{Ker}(\\mathrm{Id}-\\sigma)\\subset B_2$\ngives a Hopf ideal and also ends the proof of Theorem \\ref{teoid}.\n\n\\begin{question}\n$Im(f)=A\\otimes \\mathfrak{B}\\otimes A$ is a resolution of $A$ as a $A$-bimodule? namely,\nis $(A\\otimes\\mathfrak{B}\\otimes A,d)$ acyclic?\n\\end{question}\nThis is the case for involutive solutions in characteristic zero, but\n also for $\\sigma=$flip in any characteristic, and $\\sigma=\\mathrm{Id}$ (notice this $\\mathrm{Id}$-case \ngives the Koszul resolution for the tensor algebra). If the answer to that question is yes,\nand $\\mathfrak{B}$ is  finite dimensional then $A$ have necessarily finite global dimension. \nAnother interesting question is how to relate \ngenerators for the relations defining  $\\mathfrak{B}$ and cohomology classes for $X$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe competition between thermal fluctuations, pinning and\ninteractions between vortices  leads to many novel physical\nphenomena in type-II high-temperature\nsuperconductors~\\cite{blatter}. Examples include the melting of the\nAbrikosov flux-lattice into an entangled vortex-liquid~\\cite{ns} and\nthe proposed existence of low temperature Bose-glass~\\cite{nv},\nvortex glass~\\cite{fisher_glass} and Bragg glass~\\cite{nat_sch}\nphases.\n\nMany experimental probes have been used to study these phenomena.\nThey include decoration, transport and magnetization measurements,\nneutron scattering, electron microscopy, electron holography and\nHall probe microscopes. More recently it has become possible to\nmanipulate single vortices, for example using magnetic force\nmicroscopy (MFM)~\\cite{Wadas92}. These can, in principle, measure\ndirectly many microscopic properties which have been up to now under\ndebate or assumed. The possibility of performing such experiments is\nsimilar in spirit to single molecule experiments on motor proteins,\nDNA, and RNA which have opened a window on phenomena inaccessible\nvia traditional bulk biochemistry experiments~\\cite{singlemol}.\n\nIn this spirit Olson-Reichhardt and Hastings~\\cite{ORH04} have\nproposed using MFM to wind two vortices around each other. Such an\nexperiment allows direct probing of the energetic barrier for two\nvortices to cut through each other. A high barrier for flux lines\ncrossing has important consequences for the dynamics of the\nentangled vortex phase.\n\nIn this paper we introduce and study several experiments in which a\nsingle vortex is depinned from extended defects using, for example,\nMFM. A brief account of the results can be found in\nRef.~[\\onlinecite{knp}]. First we consider a setup where MFM is used\nto pull an isolated vortex bound to common extended defects such as\na columnar pin, screw dislocation, or a twin plane in the presence\nof point disorder. Using a scaling argument, supported by numerical\nand rigorous analytical results, we derive the displacement of the\nvortex as a function of the force exerted by the tip of a magnetic\nforce microscope. We focus on the behavior near the depinning\ntransition and consider an arbitrary dimension $d$. We argue that\nthe transition can be characterized by a universal critical\nexponent, which depends {\\it only on the dimensionality of the\ndefect}. We show that unzipping experiments from a twin plane\ndirectly measures the free-energy fluctuations of a vortex in the\npresence of point disorder in $d=1+1$ dimensions. To the best of our\nknowledge, there is only one, indirect, measurement of this\nimportant quantity in Ref.~[\\onlinecite{Bolle}]. The form of the\nphase diagram in the force temperature plane is also analyzed in\ndifferent dimensions. Related results apply when a tilted magnetic\nfield is used to tear away vortex lines in the presence of point\ndisorder, which was not considered in earlier work on clean\nsystems.~\\cite{hatano}. Furthermore, we show that a possible\nexperimental application of the scaling argument is a direct measurement of the vortex line tension in an unzipping\nexperiment. As we will show in this paper, in a system of finite size, the displacement of the flux line at the transition\ndepends only on the critical force exerted on the flux line by the\nMFM tip, the flux line tension and the sample thickness. Thus\nunzipping experiments can provide valuable information on the\nmicroscopic properties of flux lines.\n\nNext we consider a setup where a single vortex is pulled out of a\nplane with many vortices. It is known that the large-scale behavior\nof vortices in a plane is characterized by a single dimensionless\nnumber, often referred to as the Luttinger liquid parameter due to\nan analogy with bosons in $d=1+1$ dimensions. We show that\nexperiments which unzip a single vortex out of the plane can be used\nto directly probe the Luttinger liquid parameter. We also discuss\nthe effects of disorder both within the defect and in the bulk with the same setup.\n\n\n\\section{Unzipping a vortex from a defect}\n\\label{Sec2}\n\n\\subsection{Review of clean case}\n\\label{sectioncleancase}\n\nWe begin by considering the unzipping of a vortex from an extended\ndefect in a clean sample. For a columnar defect the system is\ndepicted in Fig.~\\ref{fig:clean_unzip}. At the top of the sample the\nMFM applies a constant force ${\\bf f}$ which pulls the vortex away\nfrom the defect. We assume that at the bottom of the sample the\nvortex is always bound to the defect at a specific location. This\nassumption will not influence the results since below the unzipping\ntransition the flux line is unaffected by the boundary conditions at\nthe far end of the sample.\n\n\\begin{figure}[ht]\n\\center\n\\includegraphics[width=8cm]{Clean_unzip.eps}\n\\caption{A MFM tip applies a constant force ${\\bf f}$ which pulls the\nvortex away from the defect. The configuration of the vortex is\nrepresented by ${\\bf r}(\\tau)$. We assume throughout that the vortex\nis always bound to the defect at the bottom of the sample so that\n${\\bf r}(\\tau=0)=0$. }\\label{fig:clean_unzip}\n\\end{figure}\n\nIn the absence of external force, and for an external field aligned\nwith the defect, the appropriate energy for a given configuration\n${\\bf r}(\\tau)$ of the vortex is given by \\cite{blatter}:\n\\begin{equation}\nF_0\\!\\!=\\!\\!\\!\\int_0^L \\!\\!\\!\\!d \\tau \\left[ \\frac{\\gamma}{2}\n(\\partial_\\tau {\\bf r}(\\tau))^2+V({\\bf r}(\\tau)) \\right] .\n\\label{f0}\n\\end{equation}\nHere $\\gamma$ is the line tension and $L$ is the length of the\nsample along the $\\tau$ direction. The vector ${\\bf r}(\\tau)$\nrepresents the configuration of the vortex  in the $d$ dimensional\nspace and $V({\\bf r})$ is a short-ranged attractive\npotential describing the $d'$-dimensional extended defect (in\nFig.~\\ref{fig:clean_unzip} $d=3$ and $d'=1$). The effect of the the\nexternal force, exerted by the MFM, can be incorporated by adding to\nthe free energy the contribution\n\\begin{equation}\nF_1=-{\\bf f}\\cdot {\\bf r(L)}=-\\int_0^L {\\bf f}\\cdot\n\\partial_\\tau \\bf r(\\tau)\\,d\\tau\n\\label{eq:unzipfe}\n\\end{equation}\nwhere we have used ${\\bf r}(\\tau=0)={\\bf 0}$. Here ${\\bf f}$\nstands for the local force exerted by the MFM in the transverse\ndirection. The free energy of a given configuration of the vortex\nis given by\n\\begin{equation}\nF({\\bf r})=F_0({\\bf r})+F_1({\\bf r}) \\;.\n\\end{equation}\nThe problem, as stated, has been studied first in the context of\nvortices in the presence of a tilted magnetic field~\\cite{hatano}\nand the results have been applied to the related problem of DNA\nunzipping~\\cite{Lubensky}.\n\nWe note that a similar setup can be\nachieved by using a transverse magnetic field instead of the\nexternal force. See Fig.~(\\ref{fig:clean_unzip_mag}).\n\\begin{figure}[ht]\n\\center\n\\includegraphics[width=8cm]{Clean_unzip_mag.eps}\n\\caption{Same as in Fig.~\\ref{fig:clean_unzip} but with a transverse\nmagnetic field instead of the MFM force tearing the flux line away from a\ndefect.}\\label{fig:clean_unzip_mag}\n\\end{figure}\nIndeed in the free energy (\\ref{eq:unzipfe}) the external force\ncouples to the slope of the flux line $\\partial_\\tau {\\bf r}$ in the\nsame way as the external magnetic field does~\\cite{hatano}. The only\ndifference between the two setups is that there are now equal and\nopposite forces acting on the top and bottom ends of the sample.\nHowever this difference is important only in short samples, where\nthe two ends of the flux line are not independent from each other.\n\nIn this paper we focus on the thermal average of distance of the tip\nof the vortex from the extended defect $\\langle x_m (\\tau=L)\\rangle$. This quantity\nis related to the thermal average of the length of the vortex that is unzipped from the\ndefect, $\\langle \\tau_m \\rangle$, through $\\langle x_m \\rangle = f\n\\langle \\tau_m \\rangle \/ \\gamma$. Here and throughout the paper\n$\\langle \\ldots \\rangle$ denotes a thermal average while an overbar\ndenotes an average over realizations of the disorder.\n\nAs stated above the universal behavior of $\\langle \\tau_m \\rangle$\n(or equivalently $\\langle x_m \\rangle$) within this disorder-free model have been\nderived previously. Here we sketch two approaches which will be\ngeneralized to samples with quenched disorder in the rest of the paper.\n\nIn the first approach, instead of directly summing over\nconfigurations of the vortex we perform the sum in two parts by\ndividing the vortex into bound and unbound segments. The unbound\nsegment starts at the point where the vortex departs the defect\nwithout ever returning to hit it again up to the top of the sample.\nUsing Eq.~(\\ref{eq:unzipfe}) it is straightforward to integrate over\nvortex configurations to obtain for the partition function of the\nunzipped segment\n\\begin{eqnarray}\nZ_u(\\tau_m)&=&\\int {\\cal D}{\\bf r}(\\tau)\\,\\mathrm\ne^{-\\beta\\int_0^{\\tau_m}\\! d \\tau \\left[ \\frac{\\gamma}{2}\n(\\partial_\\tau {\\bf r}(\\tau))^2-{\\bf\nf}\\cdot\\partial_\\tau{\\bf r(\\tau)}\\right]} \\nonumber \\\\\n&\\propto& \\mathrm e^{\\tau_m \\beta f^2 \/2\\gamma} \\;,\n\\label{free_unzip1}\n\\end{eqnarray}\nso that the free energy associated with this conditional partition function is\n\\begin{equation}\n{\\cal F}_u(\\tau_m)=-\\beta^{-1}\\ln Z_u(\\tau_m)= -  f^2 \\tau_m\/\n2\\gamma \\;,\n\\label{free_unzip}\n\\end{equation}\nwhere $\\beta$ is the inverse temperature. Henceforth in this paper we\nset $\\beta=1$, which can be always achieved by appropriate rescaling\nof the energy units. Even though the above sum also runs over\nconfigurations which return to the defect it is easy to verify that\nthese configurations give rise to exponentially small correction in\nthe $\\tau_m$. Equation~(\\ref{free_unzip}) implies that as the force,\n${\\bf f}$, increases the free energy density of the unzipped portion\nof the vortex decreases. In contrast, the free energy density of the\nbound part is, clearly, independent of the force and given by ${\\cal\nF}_b(\\tau_m)=V_0(L-\\tau_m)$, where $V_0$ is the free energy per unit\nlength of a bound vortex and $L$ is the length of the sample along\nthe defect. The vortex will be unzipped when $f=f_c=\\sqrt{2 \\gamma\n|V_0|}$ such that the free-energy densities of the bound and\nunzipped states are equal.\n\nIn this representation the total free energy of the vortex is\ngiven by\n\\begin{equation}\n{\\cal F}(\\tau_m)={\\cal F}_u(\\tau_m)+{\\cal F}_b(\\tau_m)\\;.\n\\end{equation}\nThe unconstrained partition function of the model is given by\n\\begin{equation}\nZ=\\int_0^L d\\tau_m e^{-(f_c^2-f^2)\\tau_m\/2 \\gamma} \\;.\n\\end{equation}\nSince both results are independent of the dimensionality of the\ndefect (columnar or planar) near the transition one always finds\nin the $L \\to \\infty$ limit\n\\begin{equation}\n\\langle \\tau_m \\rangle \\sim \\frac{1}{(f_c-f)^\\nu} \\;,\n\\label{eq:clean}\n\\end{equation}\nwith $\\nu=1$. Note, that it can easily be seen that approaching the\ntransition from above the average length of the vortex which is\nbound to the defect, $\\langle (L-\\tau_m) \\rangle$, diverges in the\nsame manner~\\cite{hatano}.\n\nAn alternative approach, which will also be useful in this paper,\nuses the mapping of the problem to the physics of a fictitious\nquantum particle \\cite{NelsonBook}. The contribution of the external\nfield ${\\bf f}$ to the free energy now manifests itself as an\nimaginary vector potential acting on the particle in $d-1$\ndimensions (with the $\\tau$ axis acting as a time direction).\nExplicitly, using the standard conversion from path-integrals (see\nRef.~[\\onlinecite{hatano}] for details) one finds that the problem\ncan be described in terms of a non-Hermitian Hamiltonian:\n\\begin{equation}\n{\\cal H}={1\\over 2\\gamma} {\\bf p}^2 -{i\\over \\gamma} {\\bf f} \\cdot\n{\\bf p} +V({\\bf r}) \\;,\n\\label{H}\n\\end{equation}\nwhere ${\\bf p}=\\frac{1}{i} \\vec{\\nabla}$ is the momentum operator.\nIn this language the vortex is bound to the defect as long as there\nis a bound state in the Hamiltonian. As mentioned above $i {\\bf f}$\nis equivalent to a constant imaginary vector potential. This analogy\nmakes it apparent that solutions of the non-Hermitian problem can be\nrelated to those of the Hermitian Hamiltonian (where one sets ${\\bf\nf}=0$) by an imaginary gauge-transformation~\\cite{hatano}. In\nparticular the left $\\psi^L_n({\\bf r},{\\bf f})$ and the right\n$\\psi^R_n({\\bf r},{\\bf f})$ eigenfunctions of the non-Hermitian\nproblem can be obtained from those of the Hermitian problem,\n$\\psi_n({\\bf r},{\\bf f}={\\bf 0})$, using\n\\begin{eqnarray}\n\\psi^R_n({\\bf r},{\\bf f}) &=&  {\\cal U} \\psi_n({\\bf r},{\\bf\nf}={\\bf 0}) \\nonumber \\\\\n\\psi^L_n({\\bf r},{\\bf f}) &=&  \\psi_n({\\bf r},{\\bf f}={\\bf 0})\n{\\cal U}^{-1} \\;, \\label{eq:gauge1}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n{\\cal U}=\\mathrm e^{{\\bf f} \\cdot {\\bf r}}= \\mathrm e^{f x}\\; ;\n\\;\\;\\;\\;\\; {\\cal U}^{-1}=\\mathrm e^{-{\\bf f} \\cdot {\\bf r}}=\\mathrm\ne^{-f x} \\;.\n\\label{eq:gauge2}\n\\end{equation}\nThe universal behavior of $\\tau_m$ at the transition,\nEq.~(\\ref{eq:clean}), was obtained in Ref.~[\\onlinecite{hatano}] by noting\nthat\n\\begin{equation}\n\\langle\\tau_m\\rangle\\propto \\langle x_m\\rangle ={\\int x\\psi^R_n({\\bf\nr}) d{\\bf r}\\over \\int \\psi^R_n({\\bf r}) d{\\bf r}}\\propto{1\\over\n{f_c-f}}, \\label{tau_m}\n\\end{equation}\nwhere $\\psi^R_n({\\bf r})\\propto \\mathrm e^{-f_c r}$ at long $r$. We\nnote that the imaginary gauge transformation is justified only at\n$f<f_c$. Otherwise the integrals in Eq.~(\\ref{tau_m}) formally\ndiverge. In principle this divergence can be fixed by proper\ntreatment of the boundary conditions~\\cite{AHNS}.\n\n\nFinally we discuss the phase diagram of the model. It is natural\nthat as the temperature rises the value of the critical force\nneeded for the unzipping decreases (at very low temperatures a\npossible reentrance has been discussed in\nliterature~\\cite{Reentrant,Reentrant2}). Here we focus on the\nexistence of a critical force for any strength of the attractive\npotential. The question is completely equivalent to that of the\nexistence of a localized vortex line in the absence of the external force.\nUsing the analogy to the quantum mechanical problem it is well known\nthat in two dimensions (corresponding to a three dimensional sample)\nand below, as long as $\\int_{-\\infty}^\\infty d{\\bf r} V({\\bf r}) <0$\nthere exists a bound state. Therefore, in real samples, as long as\nthe potential satisfies this condition there is always a minimum nonzero\ncritical force required to unbind the flux line from the defect.\n\n\n\\subsection{Scaling arguments for the disordered case.}\n\\label{sectionscaling}\n\n\\subsubsection{Critical Behavior}\nWe now consider the effect of point\ndisorder. In this case the free energy of a given vortex\nconfiguration without the contribution from the force exerted by\nthe MFM is given by\n\\begin{equation}\nF_0\\!\\!=\\!\\!\\!\\int_0^L \\!\\!\\!\\!d \\tau \\left[ \\frac{\\gamma}{2}\n(\\partial_\\tau {\\bf r}(\\tau))^2+V({\\bf r}(\\tau))+\\mu({\\bf\nr}(\\tau),\\tau) \\right] . \\label{f0dis}\n\\end{equation}\nThe frozen point disorder $\\mu({\\bf r},\\tau)$ is assumed to be uncorrelated\nand Gaussian distributed with $\\overline{\\mu}=0$ and\n$\\overline{\\mu({\\bf r},\\tau)\\mu({\\bf r'},\\tau')}=\\sigma \\delta({\\bf\nr}-{\\bf r'})\\delta(\\tau-\\tau')$. As mentioned above the overbar denotes an average over realizations of the\ndisorder and the $\\langle \\dots\\rangle$ denotes averaging over\nthermal fluctuations. The contribution from the force exerted by the\nMFM retains the same form as in Eq.~(\\ref{eq:unzipfe}).\n\nThe free energy, Eq.~(\\ref{f0dis}), without the contribution from\nthe external force has been studied previously with and without the\npresence of an extended defect. Without the defect it is well known\nthat if one fixes one end of the vortex at the bottom of the sample\nthe mean-square displacement, $\\langle x^2(\\tau) \\rangle$ of the\nvortex after traveling a distance $\\tau$ into the sample behaves as\n$\\langle x^2(\\tau) \\rangle = B\\tau^{2\\zeta(d)}$, where\n$\\zeta(d=2)=2\/3$ and $\\zeta(d=3) \\approx 0.61$ is the wandering\nexponent~\\cite{Karreview}. Moreover, for a given realization of disorder there is\ntypically a single dominant trajectory of the vortex and\nrealizations with two competing minima are very rare. Finally, it is\nknown that the free energy fluctuations scale as\n\\begin{equation}\n\\delta {\\cal F} \\propto \\tau^{\\omega(d)} \\label{eq:fefluct}\n\\end{equation}\nwith $\\omega(d=2)=1\/3$, $\\omega(d=3) \\approx 0.22$. The exponents\n$\\omega$ and $\\zeta$ are not independent. They satisfy the\nscaling relation: $\\omega=2 \\zeta -1$. Note that as long as there is\nno defect the free energy fluctuations are unaltered even in the\npresence of a force acting on the vortex because this force can be\nsimply gauged away.\n\nThe universal properties of the unzipping transition with point\ndisorder can be analyzed using a simple scaling argument adapted\nfrom~[\\onlinecite{Lubensky}] for the unzipping of DNA (for an\nabbreviated account, see Ref.~[\\onlinecite{knp}]).\n\nFollowing the discussion in the previous section, which analyzed\nthe unzipping problem with no disorder, we sum over configurations\nof the unzipped part of the vortex and the zipped part of the\nvortex separately. As before we denote the free energy of the\nunzipped segment by ${\\cal F}_u$ and the free energy of the bound\nsegment by ${\\cal F}_b$. The total free energy is then given by\n\\begin{equation}\n{\\cal F}(\\tau_m)={\\cal F}_u (\\tau_m)+{\\cal F}_b(L-\\tau_m).\n\\label{fm}\n\\end{equation}\nFor large $L$ and $\\tau_m$ we can rewrite\n\\begin{equation}\n\\mathcal F_b(L-\\tau_m)=\\mathcal F_b(L)-\\mathcal F_b(\\tau_m).\n\\label{fb}\n\\end{equation}\nFor a one dimensional defect, like a columnar pin, $\\mathcal F_b(L)$\nis a constant equal to the free energy of a flux line completely\nlocalized on the pin. For higher dimensional defects like a twin\nplane, there is a subtlety. To see this, consider a zero temperature\ncase first. Then the partition function is determined by the optimal\ntrajectory of a flux line (in the presence of point disorder)\nstarting at $\\tau=0$ and ending at $\\tau=L-\\tau_m$. However, this\ntrajectory might not be a part of the optimal trajectory going all\nthe way from $\\tau=0$ to $\\tau=L$. In this case clearly the relation\n(\\ref{fb}) does not hold. Since this relation does not hold at zero\ntemperature, it should not hold at finite temperatures as well.\nHowever, in Ref.~[\\onlinecite{fisher}] it was argued that such\nsituations (where the optimal trajectories do not coincide) are very\nrare and can be neglected \\cite{footnote}. Thus ignoring the\nconstant term $\\mathcal F_b(L)$ we can rewrite Eq.~(\\ref{fm}) as\n\\begin{equation}\n{\\cal F}(\\tau_m)={\\cal F}_u(\\tau_m)-\\mathcal F_b(\\tau_m).\n\\end{equation}\nWe can identify three contributions to the free energy above. The\nfirst is due to the average free energy difference between a vortex\non the defect and in the bulk of the sample. Close to the\ntransition, similarly to the analysis in the clean case, it is\nlinear in $\\tau_m$ and behaves as $a(f_c-f)\\tau_m$, where $a\\approx\nf_c\/\\gamma$ is a positive constant. The second contribution $\\delta\n{\\cal F}_{b}$, comes from the free-energy fluctuations arising from\nthat part of the point disorder which is localized on or near the\ndefect. For a columnar defect of dimensionality $d'=1$ this\ncontribution is due to the sum of the fluctuations in the free\nenergy about the mean. The central limit theorem implies that it\nbehaves as $\\tau_m^{1\/2}$ at large $\\tau_m$. For $d'>1$ this result\nis modified and one can use known results for the free energy of a\ndirect path in a random media (see Eq.~(\\ref{eq:fefluct})), which\nleads to $\\delta {\\cal F}_{b}\\propto \\tau_m^{\\omega(d')}$, where\n$d'$ is the dimensionality of the defect \\cite{Karreview}. Finally,\nthere is a contribution to the free energy fluctuations from the\ninteraction of the unzipped part of the vortex with the bulk point\ndisorder, $\\delta {\\cal F}_{u}$. This contribution behaves similarly\nto $\\delta \\mathcal F_b$ with a different bulk exponent: $\\delta\n{\\cal F}_{u}\\propto \\tau_m^{\\omega(d)}$, where $d>d'$ is the\ndimensionality of the sample. Collecting all three terms gives:\n\\begin{equation}\n{\\cal F}(\\tau_m)=a(f_c-f)\\tau_m +\\delta\\mathcal F_{b}(\\tau_m)\n+\\delta\\mathcal F_u(\\tau_m)\\;.\n\\label{f}\n\\end{equation}\nAs discussed above, $\\omega(d)$ has been studied extensively in the\npast and it is well known that $\\omega(d')>\\omega(d)$ for any\n$d'<d$~\\cite{Karreview}. Therefore, disorder on or close to the\ndefect controls the unbinding transition for {\\it any dimension}\nwhen $\\tau_m$ is large and the problem is equivalent to unzipping\nfrom a sample with point disorder localized on the defect. In\nparticular, this result implies that unzipping from a columnar\ndefect with point disorder in the bulk is in the same universality\nclass as unzipping of a DNA molecule with a random sequence of base\npairing energies. In fact, point disorder is likely to be\nconcentrated within real twin planes and near columnar damage tracks\ncreated by heavy ion irradiation and near screw dislocations,\nstrengthening even more the conclusions of this simple argument.\n\nThe disorder averaged partition function is dominated by the minimum\nof the free energy and thus by configurations with $\\delta \\mathcal\nF_b(\\tau_m)<0$. Using ${\\cal F}(\\tau_m)\\approx a(f_c-f)\\tau_m -b\nm^{\\omega(d')}$, where $b$ is a positive constant, the partition\nfunction\n\\begin{equation}\nZ=\\int_0^L d\\tau_m e^{-{\\cal F}(\\tau_m)}\n\\end{equation}\ncan be evaluated using a saddle-point approximation. We then find the\nvalue of $\\tau_m$ at the saddle point satisfies $a(f_c-f) =\n\\omega(d') b \\tau_m^{\\omega(d')-1}$. Therefore\n\\begin{equation}\n\\overline{\\langle \\tau_m \\rangle} \\sim \\frac{1}{(f_c-f)^{\\nu}}, \\;\\;\n\\nu=[1-\\omega(d')]^{-1} \\;,\n\\label{nu}\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\nu&=&2 \\;\\;\\;\\;\\;\\;\\;\\, {\\rm for} \\;\\;\\; d'=1 \\nonumber \\\\\n\\nu&=&3\/2 \\;\\;\\;\\; {\\rm for} \\;\\;\\; d'=2\\;.\n\\label{valuenu}\n\\end{eqnarray}\nThe result for the columnar defect ($d'=1$) agrees with known results from\nDNA unzipping~\\cite{Lubensky}.\n\nTo check the scaling argument we have preformed numerical\nsimulations in $d=1+1$ dimensions and have been able to solve analytically the\nclosely related problem of vortex unzipping from a wall in $d=1+1$ dimensions. We\nhave also studied the simplified problems of unzipping from a $d'=1$\nand $d'=2$ dimensional defect with disorder localized on the defect.\nWhile the $d'=1$ problem was solved previously~\\cite{oper,Lubensky}\nbelow (in Sec.~\\ref{replica}) we describe a new method applying the replica method. Using\nsome approximations,this approach can be generalized to a $d'=2$\ndimensional defect. In the next sections we describe all these\nresults which support the simple scaling argument presented above.\n\n\\subsubsection{Finite size scaling and determination of the flux\nline tension.}\n\nAnother consequence of the result (\\ref{nu}) is the possibility to\nperform a finite size scaling analysis~\\cite{privman}. In\nparticular, near the unzipping transition the unzipping length\n$\\overline{\\langle \\tau_m\\rangle}$ should be a function of the\nreduced force $\\epsilon \\propto (f_c-f)$ and the system size $L$. If\nthe scaling ansatz (\\ref{nu}) is correct then we must have\n\\begin{equation}\n\\overline{\\langle \\tau_m\\rangle}\/L= g(\\epsilon L^{1\/\\nu}),\n\\label{scaling1}\n\\end{equation}\nwhere $g(x)$ is some scaling function. Quite generally we expect\nthat when $x\\gg 1$ finite size effects are unimportant and $g(x)\n\\sim 1\/x^\\nu$ so that we recover the scaling (\\ref{nu}) and at $x\\ll\n1$ we have $g(x)\\to g_0$, where $g_0$ is a constant. Note that the\nconstant $g_0$ does not depend on the system size. This constant is\na universal number of the order of one, which depends on the\ndimensionality of the defect. As we find below for the unzipping\nfrom a columnar pin $g_0=0.5$ and for the unzipping from a twin\nplane $g_0\\approx 0.7$. Relation (\\ref{scaling1}) can be used to\nextract the line tension $\\gamma$ through:\n\\begin{equation}\n\\gamma=f_c g_0 {L\\over \\langle x_m\\rangle}.\n\\label{gamma}\n\\end{equation}\nHere $f_c$ is the critical force and $\\langle x_m\\rangle$ is the\ndisplacement of the MFM tip at the transition. To derive\nEq.~(\\ref{gamma}) we used the fact that for the unbound\nsegment $\\langle x_m\\rangle=\\overline{\\langle\\tau_m\\rangle}\nf\/\\gamma$.\n\nWe emphasize that the unzipping transition is first order both in\nthe clean and disordered cases. Indeed, the unzipping occurs only at\nthe boundary and does not affect total free energy of the flux line\nin the thermodynamic limit. Nevertheless, similar to wetting\nphenomena near first order transitions, this transition possesses\nscaling properties characteristic of second order transitions like\ndiverging correlation lengths, finite size scaling etc.\n\n\\subsubsection{Phase diagram}\n\\label{scalingphasediagram}\n\nNext, we use scaling arguments to consider the behavior of the\ncritical force as the strength of the disorder is varied. In\nparticular we focus on the existence of a critical strength of the\ndisorder beyond which the flux line spontaneously unzips without any\nexternal force. The disorder induced unbinding transition of the\nvortex from the defect in the absence of the force has been\nconsidered previously~\\cite{Kar85,Zap91,Tang93,Bal93,Kol92,Bal94}.\nBelow we extend a scaling argument presented first by Hwa and\nNattermann in Ref.~[\\onlinecite{HwaNatter}] for a columnar pin in arbitrary\ndimensions to include also planar defects.\n\nAssume that the vortex is localized within a distance $l_\\perp$ from\na columnar pin or a twin plane. Then, it consists of uncorrelated\nsegments of length $l_\\parallel$ related to $l_\\perp$ via\n\\begin{equation}\nl_\\parallel \\propto  l_\\perp^{1\/\\zeta} \\;,\n\\end{equation}\nwhere $\\zeta$ is the wandering exponent defined above. Each of these\nsegments has a free energy excess of order $l_\\parallel^\\omega$\nhigher than the energy of the delocalized vortex. The free energy cost per\nlength of localization therefore scales as $l_\\parallel^{\\omega-1}\n\\sim l_\\perp^{(\\omega-1)\/\\zeta}$. Clearly, a strong enough pinning\npotential gives rise to a constant energy gain per unit length,\nwhich suppresses the random energy cost of localization (note that\n$\\omega<1$ in any dimension).\n\nSo far we established that a localized phase can exist. Now let us\nconsider perturbative effects of a weak attractive potential and ask\nwhether the vortex immediately becomes bound to the defect. The free\nenergy gained in the presence of the defect, $\\delta F$, can be\ninferred by perturbations in the strength of the defect pinning energy $V$.\nAssume that one end of the vortex is held on the defect. Then the\nenergy gain due to the attractive potential by the defect is\nassociated with the the return probability of the flux line back to\nthe defect. Since the root mean square displacement behaves as\n$l_\\parallel^\\zeta$  the return probability behaves as\n$l_\\parallel^{1-(d-d^\\prime)\\zeta}$. The free energy gained by\nhitting the defect $\\delta F$  therefore scales as\n$l_\\parallel^{1-(d-d^\\prime)\\zeta}$.\n\nTo determine if the pin is relevant one has to compare this energy\nto the intrinsic variations in the free energy, $\\Delta F$, which\nscale as $l_\\parallel^{\\omega(d)}$. This yields\n\\begin{equation}\ng= \\frac{\\delta F}{\\Delta F} \\propto l_\\parallel^{\\varepsilon}\n\\end{equation}\nwith\n\\begin{equation}\n\\varepsilon=1-\\zeta(d-d^\\prime)-\\omega=2-(d+2-d^\\prime)\\zeta.\n\\end{equation}\nWhen $\\varepsilon<0$ the defect potential is irrelevant, i. e. the\nsystem gains more energy by minimizing returns to the defect, while\nif $\\varepsilon>0$ it is relevant, i. e. long excursions are\nenergetically costly.\n\nAs mentioned above in $d=3$ numerical simulations indicate that\n$\\zeta \\approx 0.6$ which gives for the planar defect ($d^\\prime=2$)\n$\\varepsilon \\approx 1\/8$ and for the columnar pin ($d^\\prime=1$)\n$\\varepsilon \\approx -1\/2$. Therefore, a weak twin plane is always relevant and\nthe vortex is always bound to it. However, a weak columnar pin is\nirrelevant and then one expects an unbinding transition. In $d=2$,\nwhere there can be no twin plane, $\\zeta=2\/3$ and the columnar\ndefect is found to be marginal. As argued in Ref.~[\\onlinecite{HwaNatter}]\nit is in fact marginally {\\it relevant}.\n\nTo summarize this discussion for columnar defects in 3 dimensional samples we\nexpect there is a critical strength of the bulk disorder beyond which the\nflux line spontaneously unzips even at zero force. In contrast for a planar defect in\n3 dimensions and for columnar defects in planar 2 dimensions superconductors we expect\nthat for any strength of the disorder there is a finite non-zero\nvalue of the force needed to unzip the vortex.\n\nNext, we will check the scaling (\\ref{nu}) and the anticipated\nlocalization \/ delocalization behavior for a number of different\nsituations using both analytical methods based on the replica trick\nand numerical simulations.\n\n\n\\subsection{Unzipping from a disordered columnar pin without excursions.}\n\\label{replica}\n\nWe start our quantitative analysis from the simplest situation,\nwhere one can get exact analytical results. Namely, we consider\nunzipping from a 1D pin with disorder localized only on the pin.\nAdditionally we neglect all excursions of the vortex line from the\npin except for the unzipped region. This problem then becomes\nidentical to DNA unzipping. In Ref.~[\\onlinecite{Lubensky}] the\nauthors analyzed this problem using a Fokker-Planck approach and\nindeed derived $\\nu=2$ near the unzipping transition. Here we show\nhow the same problem can be solved using the replica trick. The\nsolution was sketched  in Ref.~[\\onlinecite{kp}]. Here we review the\nderivation for completeness and provide additional details.\n\nIgnoring excursions of the bound part of the flux line into the bulk\ngives the free energy a particularly simple form. We again write it\nas a sum over the contribution from the bound and unbound segments.\nThe bound segment contribution is given by ${\\cal\nF}_b(\\tau_m)=V_0(L-\\tau_m)+\\int_{\\tau_m}^L d \\tau_m' U(\\tau_m')$,\nwhere $V_0<0$ is the mean value of the attractive potential, $L$ is\nthe length of the columnar defect which is assumed to be very large,\nand $U(\\tau_m)$ is a random Gaussian uncorrelated potential with\nzero mean satisfying\n$\\overline{U(\\tau_{m_1})U(\\tau_{m_2})}=\\Delta\\delta(\\tau_{m_1}-\\tau_{m_2})$.\nThe contribution from the unzipped part takes the same form as in\nthe clean case (see Eq. (\\ref{free_unzip})). Collecting the two\nterms gives:\n\\begin{equation}\n\\mathcal F(\\tau_m)=\\epsilon \\tau_m+\\int_{\\tau_m}^L d \\tau_m'\nU(\\tau_m').\n\\label{fz}\n\\end{equation}\nAs before we work in the units, where $k_B T=1$. In the equation\nabove the deviation from the unzipping transition is measured by\n$\\epsilon=(f_c^2-f^2)\/2\\gamma$, where $f$ is the force applied to\nthe end of the flux line and $f_c=\\sqrt{2\\gamma |V_0|}$ is the\ncritical force. In Eq.~(\\ref{fz}) we dropped an unimportant constant\nadditive term $V_0 L$.\n\nThe statistical properties of the unzipping transition can be\nobtained by considering $n$ replicas of the partition function $Z(\\tau)=\\exp(-\\mathcal\nF(\\tau))$~\\cite{edwards anderson}:\n\\begin{equation}\n\\overline{Z^n}=\\int_0^L d\\tau_1\\ldots\\int_0^L\nd\\tau_n\\,\\overline{\\exp\\left(-\\sum_{\\alpha=1}^n \\mathcal\nF(\\tau_\\alpha)\\right)},\n\\label{Z_n}\n\\end{equation}\nwhere the overbar denotes averaging over point disorder. The\naveraging procedure can be easily done for a positive integer $n$.\nWe eventually wish to take the limit $n \\to 0$. First we order the\ncoordinates $\\tau_j$, where the $j^{th}$ replica unbinds from the\npin according to: $0\\leq \\tau_1\\leq \\tau_{2}\\leq\\dots\\leq \\tau_n$.\nThen for $\\tau\\in[0,z_1)$ there are no replicas bound to the\ncolumnar pin, for $\\tau \\in[\\tau_1,\\tau_2)$ there is one replica on\nthe pin until finally for $L \\geq \\tau\\geq \\tau_n$ all $n$ replicas\nare bound to the pin. Using this observation and explicitly\naveraging over the point disorder in Eq.~(\\ref{Z_n}) we arrive at:\n\\begin{equation}\n\\overline{Z^n}\\!=n!\\!\\int\\limits_0^L d\\tau_1.\\,.\\!\\!\\int\\limits_{\\tau_{n-1}}^L\n\\!\\!d\\tau_n\\exp\\!\\left[-\\!\\sum\\limits_{j=1}^n\\! \\epsilon \\tau_j+\n{\\Delta\\over 2} j^2(\\tau_{j+1}-\\tau_{j})\\right],\n\\label{tuam2}\n\\end{equation}\nwhere we use the convention $\\tau_{n+1}=L$. The integral above is\nstraightforward to evaluate in the $L \\to \\infty$ limit so that\n\\begin{eqnarray}\n&&\\overline{Z^n}=\\mathrm e^{n^2L\\Delta\/2}{1\\over \\epsilon_n^n}\\prod_{j=1}^n\n{1\\over 1-\\kappa_n j} \\nonumber\n\\\\\n&&=\\mathrm\ne^{n^2L\\Delta\/2}\\left({2\\over\\Delta}\\right)^n\n{\\Gamma(1+1\/\\kappa_n-n)\\over\\Gamma(1+1\/\\kappa_n)}\n\\;, \\phantom{XXX} \\label{eq:partunzip}\n\\end{eqnarray}\nwhere $\\epsilon_n=\\epsilon+\\Delta n$ and\n$\\kappa_n=\\Delta\/2\\epsilon_n$. The exponential prefactor is an\nunimportant overall contribution of the whole columnar pin while the\nrest of the expression is the ($L$ independent) contribution from\nthe unzipped region. Interestingly the restricted partition\nfunctions for the unbinding problem from a hard wall (with no\nexternal force) and for the unzipping from a 1 dimensional pin are identical\nand thus there is equivalence between the two problems (see\nRef.~[\\onlinecite{kp}] for more details.)\n\nThe disorder-averaged free energy is given by the limit\n$\\overline{\\mathcal F}=-\\lim_{n \\to 0}\n(\\overline{Z^n}-1)\/n$~[\\onlinecite{edwards anderson}]. With the help\nof Eq.~(\\ref{eq:partunzip}) one obtains\n\\begin{equation}\n\\overline{\\mathcal F}=\\ln (\\epsilon \\kappa) + \\Psi(1\/\\kappa),\n\\label{free_en}\n\\end{equation}\nwhere $\\Psi(x)$ is the digamma function and\n$\\kappa=\\Delta\/2\\epsilon$. The unzipping transition occurs at\n$\\epsilon=0$ or equivalently at $\\kappa \\to \\infty$. The expression\n(\\ref{free_en}) is identical to the one found in\nRef.~[\\onlinecite{oper}] using a Fokker-Planck equation approach,\nsupporting the validity of the analytic continuation in $n$ for this\nparticular application of the replica calculation.\n\nIt is easy to see that this free energy yields\n\\begin{equation}\n\\overline{\\langle \\tau_m\\rangle}={\\partial \\overline{\\mathcal F}\\over\n\\partial\\epsilon}={1 \\over \\kappa\\epsilon}\\Psi^{(1)}(1\/\\kappa),\n\\label{zav}\n\\end{equation}\nwhere $\\Psi^{(n)}(x)$ stands for the $n$-th derivative of the\ndigamma function. The expression above predicts a crossover from\n$\\overline{\\langle \\tau_m\\rangle}\\approx 1\/\\epsilon$ for $\\kappa\\ll\n1$ (far from the transition) to $\\overline{\\langle\n\\tau_m\\rangle}\\approx\\kappa\/\\epsilon=\\Delta\/\\epsilon^2$ for\n$\\kappa\\gg 1$ (close to the transition) similarly to the unzipping\nfrom the wall problem analyzed above. Also, it is easy to check that\n\\begin{equation}\nw=\\overline{\\langle \\tau_m^2 \\rangle - \\langle \\tau_m\n\\rangle^2}={\\partial^2 \\overline{\\mathcal F}\\over \\partial\\epsilon^2}=-{1 \\over\n(\\kappa\\epsilon)^2}\\Psi^{(2)}(1\/\\kappa). \\label{fav}\n\\end{equation}\nHere there is a crossover from $w \\approx 1\/\\epsilon^2$ for $\\kappa\n\\ll 1$ to $w \\approx 2 \\kappa\/\\epsilon^2=\\Delta\/\\epsilon^3$ for\n$\\kappa\\gg 1$. As has been noted in the context of DNA unzipping\n\\cite{Lubensky} $\\sqrt{w}\/\\overline{\\langle \\tau_m\\rangle}$ changes\nfrom being of order unity for the weakly disordered $\\kappa \\ll 1$ case to\n$\\sim \\epsilon^{1\/2}$ for $\\kappa \\gg 1$. Thus for $\\kappa \\gg 1$,\nclose to the unzipping transition, thermal fluctuations become\nnegligible and one can work in the zero temperature limit.\n\nThe simplicity of the problem also allows finding the higher moments\nof the distribution. Here we evaluate the second moment, which gives\nthe width of the distribution of $\\overline{\\langle \\tau_m\\rangle}$\ndue to different disorder realizations. Note that since the order of\naveraging over thermal fluctuations and disorder is important this\nquantity can not be extracted directly from Eq. (\\ref{fav}). To\nproceed we consider the generating function, ${\\cal W}_n(\\epsilon_j)$ defined by\n\\begin{equation}\n{\\cal W}_n(\\epsilon_j)=\\int\\limits_{0}^L\nd\\tau_1\\ldots\\int\\limits_{\\tau_{n-1}}^L d\\tau_n\\,\\mathrm\ne^{-\\sum\\limits_{j=1}^n \\epsilon_j\\tau_j+\\Delta\/2\nj^2(\\tau_{j+1}-\\tau_j)}\\!\\!. \\nonumber \\label{zm1}\n\\end{equation}\nThe second (and similarly the higher) moments can be found by\ndifferentiating ${\\cal W}_n$ with respect to $\\epsilon_j$:\n\\begin{equation}\n\\overline{\\langle \\tau_m^2\\rangle}=\\lim_{n\\to 0} \\left. {1\\over\n{\\cal W}_n(\\epsilon_j)}\\,{1\\over n}\\sum_{j=1}^n {\\partial^2 {\\cal\nW}_n(\\epsilon_j)\\over\\partial\n\\epsilon_j^2}\\right|_{\\epsilon_j=\\epsilon}. \\label{zm3}\n\\end{equation}\nUpon evaluating the integral, we find\n\\begin{equation}\n{\\cal W}_n(\\epsilon_j)=\\prod_{j=1}^n {1\\over\n\\sum_{k=1}^j\\epsilon_k\\,-\\,\\Delta j^2\/2} \\label{zm2}\n\\end{equation}\nand correspondingly\n\\begin{equation}\n\\overline {\\langle \\tau_m^2\\rangle}={1\\over \\epsilon^2}\\lim_{n\\to\n0}{1\\over n}\\sum_{j=1}^n {2\\over 1-\\kappa j}\\sum_{k=j}^n {1\\over k\n(1-\\kappa k)}.\n\\end{equation}\nThis double sum can be calculated using a trick similar to the one\ndescribed in Ref.~[\\onlinecite{Kardar}]:\n\\begin{eqnarray}\n\\overline{\\langle \\tau_m^2\\rangle}&=&{2\\kappa^2\\over \\epsilon\n^2}\\int\\!\\!\\!\\!\\!\\!\\int\\limits_{\\!\\!\\!\\!x>y>0}\\!\\!\\!\\! dx dy\n{1\\over \\mathrm e^{\\kappa x}-1}{y\\,\\mathrm e^{-y}\\over \\mathrm\ne^{\\kappa y }-1}\\left[ \\mathrm e^{\\kappa y}+\\mathrm e^{2y}\\mathrm\ne^{\\kappa\nx-x}\\right]\\nonumber\\\\\n&-&{4\\over \\kappa\n\\epsilon^2}\\Psi^{(1)}(1\/\\kappa)\\left(C+\\Psi(1\/\\kappa)\\right),\n\\label{z2}\n\\end{eqnarray}\nwhere $C\\approx 0.577$ is Euler's constant. In the limit of weak\ndisorder or high temperature $\\kappa\\ll 1$, not surprisingly, we get\n$\\overline{\\langle \\tau_m^2\\rangle }\\approx 2\/ \\epsilon^2$, which\nagrees with the Poissonian statistics of $\\tau_m$ with an average\ngiven by $\\overline{\\langle \\tau_m \\rangle}=1\/\\epsilon$. In the\nopposite limit $\\kappa\\gg 1$ one finds $\\overline{\\langle\n\\tau_m^2\\rangle }=4\\kappa^2\/ \\epsilon^2$. Note that\n$\\overline{\\langle \\tau_m\\rangle}=\\kappa\/\\epsilon$, thus the\nrelative width of the distribution ($\\delta \\tau_m\/\\overline{\\langle\n\\tau_m\\rangle}$), defined as the ratio of the variance of the\nunzipping length $\\tau_m$ to its mean is larger by a factor of\n$\\sqrt{3}$ than that in the high temperature regime. The\ndistribution thus becomes superpoissonian at large $\\kappa$. In\nfact, in the limit $\\kappa\\to\\infty$ one can derive the full\ndistribution function $P_{\\kappa\\to\\infty}(\\tau_m)$ using extreme\nvalue statistics~\\cite{Lubensky, ledoussal}:\n\\begin{equation}\n{\\cal P}_{\\kappa\\to\\infty}(\\tau_m)\\approx {\\epsilon\/ \\kappa}\\,\nG(\\tau_m\\,\\epsilon\/\\kappa)\n\\end{equation}\nwith\n\\begin{equation}\nG(x)={1\\over\\sqrt{\\pi x}}\\,\\mathrm e^{-x\/4}-{1\\over 2}{\\rm\nerfc}(\\sqrt{x}\/2),\n\\end{equation}\nwhere ${\\rm erfc}(x)$ is the complimentary error function. It is\neasy to check that this distribution indeed reproduces correct\nexpressions for the mean and the variance. We emphasize that while\nthe thermal fluctuations of the unzipping length become negligible\nnear the transition, the fluctuations due to different realizations\nof point disorder are enhanced and lead to a wider-than-Poissonian\ndistribution of $\\tau_m$.\n\nTo check these results and uncover subtleties that might\narise in experiments, we performed direct numerical simulations of\nthe partition function of the free energy (\\ref{fz}). For this\npurpose we considered a discrete version of the problem where the\npartition function is\n\\begin{equation}\nZ=\\prod_l \\mathrm e^{-\\epsilon m_l+\\sum_{l^\\prime=1}^l U(m_l)}.\n\\end{equation}\nHere $U(m_l)$ is the random potential uniformly distributed in the\ninterval $[-U_0,U_0]$ so that the disorder variance is\n$\\Delta=\\overline{U^2(m_l)}=U_0^2\/3$. For the simulations we choose\n$\\epsilon=\\ln(1.2)-0.18\\approx 0.00232$ and $U_0=0.3$, which gives\n$\\Delta=0.03$, $\\kappa\\approx 6.46$ and according to both\nEq.~(\\ref{zav}) and numerical simulations $\\overline{\\langle\n\\tau_m\\rangle}\\approx 2860$. Then we computed $\\delta\n\\tau_m\/\\overline{\\langle \\tau_m\\rangle}$ using both Eq.~(\\ref{z2})\nand performing numerical simulations. For the chosen parameters the\nequation~(\\ref{z2}) gives $\\delta \\tau_m\/\\overline{\\langle\n\\tau_m\\rangle}\\approx 1.68$, while the numerical simulations  yield\n$\\delta \\tau_m\/\\overline{\\langle \\tau_m\\rangle}\\approx 1.67$.\nClearly the results are very close to each other and the small\ndiscrepancy can be attributed to the discretization error. In\nFig.~\\ref{fig_var} we plot dependence of\n$\\delta\\tau_m\/\\overline{\\langle \\tau_m\\rangle}$ vs. system size.\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=9cm]{distrib1d6.eps}\n\\caption{Dependence of the relative width of the distribution\n$\\delta \\tau_m\/\\overline{\\langle \\tau_m\\rangle}$ on the system size.\nSymbols correspond to the actual data, the solid line is the guide\nto the eye, and the dashed line corresponds to the replica result in\nthe thermodynamic limit.}\\label{fig_var}\n\\end{figure}\nIt is obvious from the figure that in the thermodynamic limit\n$L\\to\\infty$ the replica result is in excellent agreement with\nnumerical simulations. We mention that numerical simulations of\n$\\delta \\tau_m$ show very strong finite size effects. Therefore one\nhas to go to very large $L\\gtrsim 50 \\overline{\\langle\n\\tau_m\\rangle}$ in order to approach the thermodynamic limit for the\nwidth of the distribution.\n\nDepending on the system the quantity $\\overline{\\langle\n\\tau_m^2\\rangle}$ is not always experimentally accessible. For\nexample, in the unzipping experiments it is easier to measure\nthermal average, $\\langle \\tau_m\\rangle$, in each experimental run.\nWe note that this quantity has sample to sample fluctuations only\ndue to the presence of disorder. Then the variance of the\ndistribution will be characterized by $\\overline{\\langle\n\\tau_m\\rangle^2}$. The difference between the two expectation values\nis given by $w$ found in Eq.~(\\ref{fav}). Defining $(\\delta\n\\tau_m^{T})^2=\\overline{\\langle \\tau_m\\rangle^2}-\\overline{\\langle\n\\tau_m\\rangle}^{\\,2}$ and using Eqs.~(\\ref{z2}) and (\\ref{fav}) we\nfind that $\\delta \\tau_m^T\/\\overline{\\langle \\tau_m\\rangle}\\approx\n\\sqrt{\\kappa\/2}$ in the weak disorder limit ($\\kappa\\ll 1$) and\n$\\delta \\tau_m^T\/\\overline{\\langle \\tau_m\\rangle}\\approx\n\\sqrt{3}-1\/(\\sqrt{3}\\kappa)$ in the opposite limit $\\kappa\\gg 1$. We\nplot both $\\delta \\tau_m^T$ and $\\delta \\tau_m$ versus the disorder\nparameter $\\kappa$ in Fig.~\\ref{fig_dz}.\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=9cm]{width.eps}\n\\caption{Dependence of the relative width of the\ndistribution on the disorder parameter $\\kappa$. The two curves\ncorrespond to different averaging over temperature and disorder (see\ntext for details). The horizontal line at $\\sqrt{3}$ denotes the\nasymptotic value of both $\\delta \\tau_m$ and $\\delta \\tau_m^T$ at\n$\\kappa\\to\\infty$}\\label{fig_dz}\n\\end{figure}\nThe same issue of importance of the order of thermal and disorder\naveraging appears in the calculation of the higher moments of\n$\\tau_m$, becoming irrelevant only in the limit ($\\kappa\\to\\infty$),\nwhich effectively corresponds to the zero temperature case.\n\nBefore concluding this section let us make a few remarks about the\nrebinding transition, i.e., the rezipping that occurs with decreasing force. One can consider a similar setup with a lower\nend of the flux line fixed at the bottom of the columnar pin and the\ntop end is pulled away from the pin with a force $f$. However, now\nwe will be interested in $f>f_c$. Then clearly most of the flux line\nwill be unzipped from the pin except for a portion near the bottom\nend. If $f$ is very large, the length of the bound segment $\\tilde\n\\tau_m$ near the sample boundary is small. However as $f$ decreases and approaches $f_c$ from\nabove, the length of this segment increases and finally diverges at\nthe transition. This rebinding transition can be described in a\nsimilar spirit to the unbinding. For example instead of the free\nenergy (\\ref{fz}) one has to deal with\n\\begin{equation}\n\\mathcal F(\\tilde \\tau_m)=|\\epsilon|\n\\tilde\\tau_m+\\int_{0}^{\\tilde\\tau_m} d \\tau_m' U(\\tau_m').\n\\label{fzr}\n\\end{equation}\nAs we already noted the free energies (\\ref{fz}) and (\\ref{fzr}) are\nequivalent up to an unimportant constant equal to the total disorder\npotential of the pin: $\\int_0^L d\\tau_m' U(\\tau_m')$. We conclude that the unbinding and rebinding\ntransitions for a single flux line on a disordered columnar pin are\nidentical. In other words, statistical properties of $\\tau_m$ for a\ngiven $f=f_c-\\delta f$ are identical to those of $\\tilde\\tau_m$ for\n$f=f_c+\\delta f$.\n\n\n\n\\subsection{Unzipping from a planar defect without excursions.}\n\\label{replica_2D}\n\nWe now generalize the ideas of the previous section to the more\ncomplicated problem of unzipping of a single flux line from a\ndisordered twin plane. As before we ignore excursions out of the\nplane for the bound part of the flux line. Let us consider the\nrebinding transition first. That is we assume that $f$ is slightly\ngreater than $f_c$ and we study the statistics of the\nbound part of the flux line. We again assume that the flux line is\npinned at the bottom of the plane ($\\tau=0$) and unbinds for $\\tau$ larger than some\n$\\tilde\\tau_m$.\n\nThe point disorder potential now depends on the two coordinates\n$\\tau$ and $z$ spanning the twin plane. Using Eq. (\\ref{free_unzip}) the\npartition function reads:\n\\begin{eqnarray}\n&&Z=\\int_0^L d\\tilde\\tau_m \\int Dz(\\tau^\\prime)\n\\exp\\biggl[-{f^2\\over\n2\\gamma}\\tilde\\tau_m-V\\tilde\\tau_m\\nonumber\\\\\n&&~~~-\\beta\\int_0^{\\tilde\\tau_m} d\\tau^\\prime \\left({\\gamma\\over\n2}\\left({dz\\over d\\tau^\\prime}\\right)^2+\n\\mu(\\tau^\\prime,z^\\prime)\\right)\\Biggr],\n\\end{eqnarray}\nwhere $V<0$ is the mean attractive potential of the twin plane and\nwe have dropped the unimportant $L$-dependent factors. As before, we\nassume a Gaussian random noise with zero mean and\n\\begin{equation}\n\\overline{\\mu(\\tau_1,z_1)\\mu(\\tau_2,z_2)}=\\sigma\n\\delta(\\tau_1-\\tau_2)\\delta(z_1-z_2).\n\\end{equation}\nWe also introduce $\\epsilon=-f^2\/(2\\gamma)-V$. Note that for the\nrebinding transition $\\epsilon<0$. After replicating the partition\nfunction and averaging over point disorder we find\n\\begin{eqnarray}\n&&\\overline{Z^n}=n!\\int\\limits_{0}^L\nd\\tilde\\tau_n\\int\\limits_{\\tilde\\tau_n}^{L}d\\tilde\\tau_{n-1}\\ldots\n\\int\\limits_{\\tilde\\tau_2}^L d\\tilde\\tau_1\\int\nDz_1(\\tau_1^\\prime)\\dots Dz_n(\\tau_n^\\prime)\\nonumber\\\\\n&&~~~~~~~\\exp\\left[\\sum_{\\alpha=1}^n\n\\epsilon\\tilde\\tau_\\alpha+\\!\\!\\!\\int\\limits_{\\tilde\\tau_{\\alpha+1}}^{\\tilde\\tau_{\\alpha}}\n\\!\\!\\!d\\tau_\\alpha^\\prime \\mathcal\nL_{\\alpha}[z_1(\\tau_1^\\prime),\\ldots,\nz_\\alpha(\\tau_\\alpha^\\prime)]\\right],\n\\end{eqnarray}\nwhere we define $\\tilde\\tau_{n+1}\\equiv 0$ and $\\mathcal L_\\alpha$\nis the Euclidean Lagrangian corresponding to the Hamiltonian\n($\\mathcal H_\\alpha$) of $\\alpha$ interacting particles~\\cite{Kardar}:\n\\begin{equation}\n\\mathcal H_\\alpha=-{\\sigma\\over 2}\\alpha-{1\\over\n2\\gamma}\\sum_{\\beta=1}^{\\alpha} {\\partial^2\\over\\partial\nz_\\beta^2}-\\sigma\\sum_{1\\leq\\beta<\\gamma\\leq\\alpha}\n\\delta(z_\\beta-z_\\gamma).\n\\end{equation}\nClose to the rebinding transition, we anticipate\n$\\tilde\\tau_m\\to\\infty$ and thus the mean separation between the\nrebinding times of different replicas $\\tilde\\tau_\\alpha$ and\n$\\tilde\\tau_{\\alpha-1}$ diverges. Therefore the contribution to the\npartition function coming from integration over $\\tau_\\alpha$ will\nbe dominated by the ground state of configurations with $\\alpha$\nreplicas. In this case we can significantly simplify the partition\nfunction and evaluate it analytically:\n\\begin{eqnarray}\n&&\\overline{Z^n}=n!\\int\\limits_{0}^L\nd\\tilde\\tau_n\\int\\limits_{\\tilde\\tau_n}^{L}d\\tilde\\tau_{n-1}\\ldots\n\\int\\limits_{\\tilde\\tau_2}^L d\\tilde\\tau_1\\\\\n&&\\exp\\left[\\sum_{\\alpha=1}^n  \\epsilon\\tilde\\tau_\\alpha+(\\mathcal\nE_{\\alpha}-\\mathcal E_{\\alpha-1})\n\\tilde\\tau_{\\alpha})\\right].\\nonumber\n\\end{eqnarray}\nHere $\\mathcal E_\\alpha$ is the ground state energy of $\\mathcal\nH_{\\alpha}$ with a subtracted term linear in $\\alpha$, that just\nrenormalizes $f_c$. Close to the transition $\\epsilon$ is linear\nin the difference $f-f_c$. The energy, $\\mathcal E_\\alpha$, was\ncomputed in Ref.~[\\onlinecite{Kardar}]:\n\\begin{equation}\n\\mathcal E_\\alpha=-{\\sigma^2\\gamma\\over\n12}\\alpha^3=-\\xi\\alpha^3.\n\\end{equation}\nUpon integrating over $\\tilde\\tau_\\alpha$ one obtains\n\\begin{equation}\n\\overline{Z^n}=n!\\prod_{\\alpha=1}^n {1\\over\n|\\epsilon|\\alpha-\\xi\\alpha^3}\\to\\prod_{\\alpha=1}^n {1\\over\n|\\epsilon|-\\xi\\alpha^2}\n\\label{Z_n0}\n\\end{equation}\nThe product above can be reexpressed in terms of $\\Gamma$-functions,\nwhich in turn allows for a straightforward analytic continuation to\n$n\\to 0$:\n\\begin{equation}\n\\overline{Z^n} ={1\\over\n\\xi^n}{1\\over1+n{\\sqrt{\\xi}\\over\\sqrt{|\\epsilon|}}}\n{\\Gamma\\left({\\sqrt{|\\epsilon|}\\over\\sqrt{\\xi}}-n\\right)\\over\n\\Gamma\\left({\\sqrt{|\\epsilon|}\\over\\sqrt{\\xi}}+n\\right)}.\n\\label{Z_n1}\n\\end{equation}\nUsing this expression we obtain the free energy and the mean\nlength of the localized segment:\n\\begin{equation}\n\\mathcal F=-\\lim_{n\\to 0}{\\overline{Z^n}-1\\over n}=\\ln\n\\xi+{\\sqrt{\\xi}\\over\\sqrt{|\\epsilon|}}+2\\Psi\\left({\\sqrt{|\\epsilon|}\\over\n\\sqrt{\\xi}}\\right),\n\\label{f_2d}\n\\end{equation}\n\\begin{equation}\n\\overline{\\langle \\tilde\\tau_m \\rangle}={\\partial \\mathcal\nF\\over\\partial |\\epsilon|}=-{\\sqrt{\\xi}\\over\n2|\\epsilon|^{3\/2}}+{1\\over\n\\sqrt{|\\epsilon|\\xi}}\\Psi^{(1)}\\left({\\sqrt{|\\epsilon|}\\over\n\\sqrt{\\xi}}\\right)\n\\label{tau_2d}\n\\end{equation}\nwhere as before $\\Psi^{(n)}(x)$ stands for the $n$th derivative of\nthe digamma function. This expression has the asymptotic behaviors:\n\\begin{eqnarray}\n&&\\overline{\\langle \\tilde\\tau_m \\rangle}\\to\n{1\\over\\epsilon}\\qquad\\quad~ \\xi\\ll\n|\\epsilon|\\nonumber\\\\\n&&\\overline{\\langle \\tilde\\tau_m \\rangle}\\to {\\sqrt{\\xi}\\over\n2|\\epsilon|^{3\/2}} \\quad \\xi\\gg|\\epsilon|.\n\\end{eqnarray}\nThis scaling confirms the crossover between exponents $\\nu=1$ and\n$\\nu=3\/2$ for the rebinding transition to a two-dimensional\ndisordered plane predicted by the simple scaling argument leading to\nEq.~(\\ref{valuenu}).\n\nIn a similar way one can also consider an unzipping transition with\n$f \\leq f_c$. One finds an expression for the partition function\nwhich is identical to (\\ref{Z_n0}) with the substitution\n$\\xi\\to-\\xi$. Note however, that the analytic continuation of the\nproduct (\\ref{Z_n1}) results in a complex partition function and\nhence a complex free energy. It thus appears that the analytic\ncontinuation of the product (\\ref{Z_n0}) to noninteger values of $n$\nis not unique. One can always multiply it by any periodic function\nof $n$, which is equal to unity when the argument is integer. While\nwe were able to find some real-valued analytic continuations of\n$\\overline{Z^n}$ to negative values of $\\xi$, these continuations\ndid not lead to physically sensible results.\n\nBecause of the ambiguity of the analytic continuation and some\napproximations used to derive Eqs.~(\\ref{f_2d}) and (\\ref{tau_2d})\nwe also performed numerical simulations for the vortex unzipping\nfrom a disordered twin plane.\n\nFor numerical simulations we are using the lattice version of the\nmodel, where in each step along the $\\tau$ direction the vortex can\neither move to the left or the right one lattice spacing. Note that\nbecause we neglect excursions the vortex motion occurs strictly\nwithin the plane until the vortex is unbound. Then the restricted\npartition function for the bound part of the flux line, $Z(x,\\tau)$,\nwhich sums over the weights of all path leading to $x,\\tau$,\nstarting at $x=0,\\tau=0$ satisfies the recursion\nrelation~\\cite{Kardar}\n\\begin{eqnarray}\n&& Z(x,\\tau+1)=e^{\\mu(x,\\tau+1)}\\big[J Z(x-1,\\tau) +J\nZ(x+1,\\tau)\\nonumber\\\\\n&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+(1-2J) Z(x,\\tau)\\big].\n\\label{eqz1}\n\\end{eqnarray}\nWe assume that $\\mu(x,\\tau)$ is uniformly distributed in the\ninterval $[-U_0,U_0]$ implying as before the variance $\\sigma=U_0^2\/3$. The\nvariable $J$ controls the line tension. In the continuum limit $J\\ll\n1$ and $U_0\\ll 1$ the equation (\\ref{eqz1}) reduces to the\nSchr\\\"odinger equation:\n\\begin{equation}\n{\\partial Z\\over \\partial\\tau}=-\\mathcal H Z(x,\\tau)\n\\label{Z_tauu}\n\\end{equation}\nwith the Hamiltonian given by Eq.~(\\ref{H}) with $\\gamma=2J$ and\n$f=0$ (there is no force acting on the flux line within the\nplane). We note that even if the parameters of the discrete model are not\nsmall we still expect that Eq.~(\\ref{Z_tauu}) remains valid at long\nlength and time scales. However, the relation between the\nmicroscopic parameters of the discrete model and the parameters of\nthe effective coarse-grained Hamiltonian (\\ref{H}) is more\ncomplicated.\n\nIn our simulations we evaluated numerically the free energy of the\nbound part of the vortex line for each realization of point disorder\nand used the analytical expression for the free energy of the\nunbound part, for which point disorder can be neglected. The latter\nis given by Eq.~(\\ref{free_unzip}). This free energy is controlled\nby a single parameter $f^2\/(2\\gamma)$. Use of the analytic result\n(\\ref{free_unzip}) significantly simplifies calculations of\n$\\overline{\\langle \\tau_m\\rangle}$ and allows us to perform large\nscale simulations.\n\nFirst we verify the scaling (\\ref{nu}) with $\\nu=3\/2$ at the\nunzipping transition. To do this we perform standard finite size\nscaling procedure.\n\\begin{figure}[ht]\n\\center\n\\includegraphics[width=9cm]{scaling_2d_6.eps}\n\\caption{Ratio of the unzipping length\n$\\overline{\\langle\\tau_m\\rangle}$ to the system size $L$ as a\nfunction of $f^2\/2\\gamma$ for different system sizes. Here $f$ is\nthe external force and $\\gamma$ is the line tension of the vortex\n(see Eqs.~(\\ref{free_unzip1} and (\\ref{free_unzip}))). According to\nthe scaling relation (\\ref{scaling1}) the crossing point corresponds\nto the unzipping transition. In simulations the parameters of the\nmicroscopic model (\\ref{eqz1}) are chosen to be $J=0.2$, $U_0=2$}\n\\label{fig3}\n\\end{figure}\nIn Fig.~\\ref{fig3} we show dependence of the ratio\n$\\overline{\\langle\\tau_m\\rangle}\/L$ on the parameter $f^2\/(2\\gamma)$\nfor four different sizes. As we expect from the scaling relation\n(\\ref{scaling1}) the three curves intersect at the same point\ncorresponding to the unzipping transition ($g_0\\approx 0.7$). Once\nwe determine the crossing point corresponding to the critical force\n$f_c$ we can verify the scaling relation (\\ref{scaling1}) with\n$\\nu=3\/2$.\n\\begin{figure}[ht]\n\\center\n\\includegraphics[width=9cm]{scaling_2d_1.eps}\n\\caption{Data collapse of $\\overline{\\langle\\tau_m\\rangle}\/L$ as a\nfunction of $\\epsilon L^{1\/\\nu}$ with the exponent $\\nu=3\/2$ for two\ndifferent system sizes (see Eq.~(\\ref{scaling1})). The parameters of\nthe model are the same as in Fig.~\\ref{fig3}. The inset shows\nderivative of $\\overline{\\langle\\tau_m\\rangle}$ with respect to\n$\\epsilon$ for $L=12800$. Clearly the scaling function is asymmetric\nwith respect to $\\epsilon\\to -\\epsilon$. Thus the unbinding and\nrebinding transitions are not equivalent.}\n\\label{fig:collapse2D}\n\\end{figure}\nIn Fig.~\\ref{fig:collapse2D} we plot\n$\\overline{\\langle\\tau_m\\rangle}\/L$ versus the scaling parameter\n$\\epsilon L^{1\/\\nu}$ (see Eq.~(\\ref{scaling1})) with $\\nu=3\/2$ for\ntwo different system sizes. Clearly the data collapse is nearly\nperfect, which proves the validity of the scaling (\\ref{scaling})\nwith $\\nu=3\/2$ for the unzipping of a flux line from a twin plane.\nThe inset shows the derivative of $\\overline{\\langle\\tau_m\\rangle}$\nwith respect to $\\epsilon$. Clearly this derivative is asymmetric\nwith respect to $\\epsilon\\to\\ -\\epsilon$, implying that there is no\nsymmetry between the unbinding and rebinding transitions. This is\ncontrary to the unzipping from a columnar pin with no excursions,\nwhere such a symmetry does exist.\n\nNext we turn to verifying the analytic prediction for\n$\\overline{\\langle\\tau_m\\rangle}$, Eq.~(\\ref{tau_2d}). As we argued\nabove the parameter $\\zeta$ describing the disorder strength can be\neasily extracted from microscopic parameters of the model only in\nthe continuum limit $U_0\\gg 1$, $J\\ll 1$. Unfortunately, it is not\npossible to do simulations directly in the continuum limit ($J\\ll 1$\nand $U_0\\ll 1$). Indeed as Eq.~(\\ref{tau_2d}) suggests in order to\nsee the scaling exponent $\\nu=3\/2$ one needs to go to length scales much\nlarger than $1\/\\xi$, where $\\xi=\\sigma^2 J\/12=U_0^4 J\/36$. If\n$J\\ll 1$ and especially $U_0\\ll 1$ then one has to simulate\nextremely large system sizes where $L$ is larger than $10^7$ for\n$U_0=0.1$ and $J=0.1$. Therefore we perform simulations in the regime\nwhere $J$ and especially $U_0$ are appreciable. We then\nregard $\\xi$ as a fitting parameter of the model which should be\nequal roughly to $U_0^4 J\/36$.\n\\begin{figure}[ht]\n\\center\n\\includegraphics[width=9cm]{scaling_2d_5.eps}\n\\caption{Dependence of the length of the bound part of the flux line\nto the twin plane $L-\\overline{\\langle\\tau_m\\rangle}$ on $\\epsilon$\nfor the rebinding transition. Different curves correspond to\ndifferent system sizes. The solid black line is the best single\nparameter fit using Eq.~(\\ref{tau_2d}) with $\\xi$ being the\nfitting parameter.}\n\\label{fig:replica1}\n\\end{figure}\nIn Fig.~\\ref{fig:replica1} we show results of numerical simulation\nfor the rezipping length $L-\\overline{\\langle\\tau_m\\rangle}$ on the\ndetuning parameter $\\epsilon$ for different system sizes. The solid\nblack line is the best single-parameter fit to the data using the\nanalytic expression (\\ref{tau_2d}). The fitting parameter $\\xi$\nfound from simulations is $\\xi \\approx 0.036$, while a continuum\nestimate $U_0^4 J\/36$ gives $\\xi \\approx 0.089$, which is very\nreasonable given that this estimate is valid only at $U_0\\ll 1$. We\nalso performed similar simulations for $U_0=1.5$ and got a very good\nfit with (\\ref{tau_2d}) for $\\xi=0.018$, while the continuum\nestimate gives $\\xi \\approx 0.028$. We thus see that indeed as\n$U_0$ decreases the fitting parameter $\\xi$ becomes closer to the\ncontinuum expression.\n\nWhile we were not able to derive a closed analytic expression for\n$\\overline{\\langle\\tau_m\\rangle}$ for the unbinding transition, we\nperformed numerical simulations. As the inset in\nFig.~\\ref{fig:collapse2D} suggests the transition is highly\nasymmetric. In fact this asymmetry persists in the thermodynamic\nlimit.\n\\begin{figure}[ht]\n\\center\n\\includegraphics[width=9cm]{scaling_2d_7.eps}\n\\caption{Comparison of dependences of\n$L-\\overline{\\langle\\tau_m\\rangle}$ for the rebinding transition and\n$\\overline{\\langle\\tau_m\\rangle}$ for the unbinding transition on\n$|\\epsilon|$. We used the parameters of Fig.~\\ref{fig3} with\n$L=51200$. The finite size effects are negligible on the scale of\nthe graph. Both curves interpolate between $1\/|\\epsilon|$ dependence\nat $|\\epsilon|\\gg \\xi$ and $C\/|\\epsilon|^{3\/2}$ at $|\\epsilon|\\ll\n\\xi$. However, the prefactor $C$ for the unbinding transition is\nabout three times larger than for the rebinding.}\n\\label{fig:unbind_rebind}\n\\end{figure}\nIn Fig.~\\ref{fig:unbind_rebind} we plot\n$L-\\overline{\\langle\\tau_m\\rangle}$ for the rebinding transition and\n$\\overline{\\langle\\tau_m\\rangle}$ for the unbinding versus\n$|\\epsilon|$. Both curves interpolate between $1\/|\\epsilon|$\ndependence at weak disorder $|\\epsilon|\\ll \\xi$ and\n$C\/|\\epsilon|^{3\/2}$ dependence at strong disorder $|\\epsilon|\\gg\n\\xi$. However, the prefactor $C$ in front of $1\/|\\epsilon|^{3\/2}$\nis larger for the unzipping transition.\n\n\\subsection{Unzipping from a hard wall}\n\\label{Bethe_ansatz}\n\nAs the next step we consider unzipping from  an attractive hard wall\nin $d=1+1$ dimensions with point disorder in the bulk. Our method\nis a straightforward generalization of the Bethe ansatz solution\nfound by Kardar in the absence of the external force~\\cite{Kardar}.\nThe system is illustrated in Fig. \\ref{Bethe}. Here the potential\nexperienced by the flux line, $V(x)$, has a short ranged attractive\npart and an impenetrable core at $x=0$. While the scaling argument\nis unchanged in this case, this problem has the merit of being\nexactly solvable within the replica approach. Since most details of\nthe calculation are identical to those presented in\nRef.~[\\onlinecite{Kardar}], here we only outline the solution.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.7]{unzipwall.eps}\n\\caption{\\label{Bethe}  An illustration of the setup considered in\nthe Bethe Ansatz calculation. The flux line is restricted to the\nhalf plane and a MFM tip is acting on it at the top of the sample.}\n\\end{figure}\nAfter replicating the free energy Eq.~(\\ref{f0dis}) along with the\ncontribution from the external field Eq.~(\\ref{eq:unzipfe}) and\naveraging over the disorder, the replicated sum over all path weights  connecting points $(0,0)$ and $(x,\\tau)$,\n$\\overline{Z^n(x,t)}$, can be calculated from\n\\begin{equation}\n\\partial_\\tau \\overline{Z^n(x,t)} = -{\\cal H} \\overline{Z^n(x,t)} \\;,\n\\end{equation}\nwith the initial condition $\\overline{Z^n(x,0)}=\\delta(x)$. The\nreplicated system describes $n$ {\\it attractively interacting\nbosons} with a non-Hermitian Hamiltonian ${\\cal H}$ given by\n\\begin{eqnarray}\n{\\cal H}&=& \\sum_{\\alpha=1}^n \\left[ -\\frac{1}{2\\gamma}\n \\partial^2_{x_\\alpha}-f\n\\partial_{x_\\alpha}+ V(x_\\alpha) \\right]\\nonumber\n\\\\&-&\\sigma \\sum_{\\alpha < \\beta}\n\\delta(x_\\alpha-x_\\beta) -\\frac{1}{2} \\sigma n   \\;.\n\\end{eqnarray}\n\nIn Ref.~[\\onlinecite{Kardar}] the problem was solved for $f=0$ using\nthe Bethe Ansatz. The boundary conditions were that the ground state\nwave function should vanish at large $x$ should decay as\n$\\exp(-\\lambda x)$ for the particle closest to the wall.  One then\nfinds that for the permutation ${\\bf P}$ of particles such that\n$0<x_{P1}<x_{P2}< \\ldots <x_{Pn}$ the wave function for $f=0$ is\n\\begin{equation}\n\\Psi_{f=0}\\sim\\exp \\left(-\\sum_{\\alpha=1}^n \\kappa_\\alpha x_{P\n\\alpha}\\right) \\;.\n\\end{equation}\nHere $\\kappa_\\alpha = \\lambda+2(\\alpha-1)\\kappa$, $\\kappa=\n\\sigma\\gamma\/2$. Taking the zero replica limit it was\nfound~\\cite{Kardar} that for weak disorder ($\\sigma\\gamma\/2 <\n\\lambda$) the vortex is bound to the wall while for strong\ndisorder ($\\sigma\\gamma\/2 > \\lambda$) it is unbound.\n\nThe ground state wave function for the {\\it non-zero} value of the force\ncan be obtained by noting that the non-Hermitian term acts like an\nimaginary vector potential. In particular, it can be gauged away when\nthe vortices are bound to the wall as discussed in Sec.\n\\ref{sectioncleancase} (see Eqs.~(\\ref{eq:gauge1}) and\n(\\ref{eq:gauge2})). This imaginary gauge transformation gives\n\\begin{equation}\n\\Psi_{f}=\\Psi_{f=0}\\exp\\left( \\sum_{\\alpha=1}^n fx_\\alpha \\right)\n\\;,\n\\end{equation}\nwhich implies that the solution is\n\\begin{equation}\n\\Psi_{f}=\\exp \\left(-\\sum_{\\alpha=1}^n \\tilde{\\kappa}_\\alpha x_{P\n\\alpha}\\right) \\;,\n\\end{equation}\nwith $\\tilde{\\kappa}_\\alpha = \\lambda+2(\\alpha-1)\\kappa-f$. The\neffect of the force is simply to shift all the $\\kappa_\\alpha$'s by a\nconstant. The average localization length (which satisfies near the\ntransition $\\langle x _m\\rangle \\simeq f_c \\langle \\tau_m \\rangle \/\n\\gamma$) is then given by\n\\begin{equation}\n\\langle x_m \\rangle={1\\over \\tilde Z_n n}\\int_0^\\infty\n\\prod_{j=1}^n dx_j \\left[\\sum_{j=1}^n x_j\\right]\\,\\Psi_f(x_j),\n\\label{eq:Kardarresult}\n\\end{equation}\nwhere $\\tilde Z_n=\\int_0^\\infty  \\prod_{j=1}^n dx_j \\Psi_f(x_j)$.\nNote that the normalization factor $\\tilde Z_n$ in the equation\nabove is formally equivalent to the partition function (\\ref{tuam2})\nfor the unzipping from a columnar pin without excursions if we\nidentify $\\lambda-\\kappa-f$ with $\\epsilon$ and $\\kappa$ with\n$\\Delta\/2$. This equivalence implies that $\\langle x_m\\rangle$ for\nthe unzipping from a hard wall has the same statistical properties\nas $\\overline{\\langle\\tau_m\\rangle}$ for the unbinding from a\ncolumnar pin (for more details see Ref.~[\\onlinecite{kp}]). In particular, the\nunzipping problem  has a crossover from $\\langle x_m\\rangle \\sim\n1\/(f_c-f)$ for $\\lambda-f\\gg\\kappa$ to $\\langle x_m\\rangle \\sim 1\/\n(f_c-f)^{3\/2}$ in the opposite limit.\n\nThis example confirms another prediction of\nthe simple scaling argument: the critical exponents for the\nunbinding transition are determined only by the dimensionality of\nthe defect even if the disorder is also present in the bulk of the system.\n\n\\subsection{Unzipping from a columnar pin with excursions into the bulk.}\n\\label{sec:numerics}\n\nIn this section we consider the setup similar to Sec.~\\ref{replica},\nnamely unzipping from a columnar defect in $d=1+1$ dimensions, but\nallowing excursions of the flux line to the bulk (see\nFig.~\\ref{fig:unzip_1D}).\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=9cm]{unzip.eps}\n\\caption{A setup illustrating unzipping from a\ncolumnar pin in $d=1+1$ dimensions with excursions into the bulk.}\n\\label{fig:unzip_1D}\n\\end{figure}\nUnfortunately there is no analytic solution available for this\nproblem. Therefore we present only numerical results. As in\nSec.~\\ref{replica_2D} we consider a lattice version of the model\nwhere in each step along the $\\tau$ direction the vortex can either\nmove to the left or the right one lattice spacing. The attractive\npotential was placed at $x=0$. The restricted partition function of\nthis model, $Z(x,\\tau)$, which sums over the weights of all path\nleading to $x,\\tau$, starting at $x=0,\\tau=0$ satisfies the\nrecursion relation~\\cite{Kardar}:\n\\begin{eqnarray}\n&&Z(x,\\tau+1)=\\delta_{x,0}(e^{V}-1)Z(0,\\tau) \\nonumber\\\\\n&&+e^{\\mu(x,\\tau+1)}\\left[J e^f Z(x-1,\\tau) +J\ne^{-f}Z(x+1,\\tau)\\right]. \\label{eqz}\n\\end{eqnarray}\nSimilarly to Eq.~(\\ref{eqz1}) we assume that $\\mu(x,\\tau)$ is\nuniformly distributed in the interval $[-U_0,U_0]$ implying the\nvariance $\\sigma=U_0^2\/3$. The variable $J$ controls the line\ntension, $V$ is the attractive pinning potential, and $f$ is proportional\nto the external force. In the continuum limit $J\\ll 1$, $f\\ll 1$,\nand $U_0\\ll 1$, equation (\\ref{eqz}) reduces to the Schr\\\"odinger\nequation:\n\\begin{equation}\n{\\partial Z\\over \\partial\\tau}=-\\mathcal H Z(x,\\tau)\n\\label{Z_tau}\n\\end{equation}\nwith the Hamiltonian given by Eq.~(\\ref{H}) with $\\gamma=2J$.\n\nFor the simulations we have chosen particular values of $J=0.1$ and\n$V=0.1$. As before we work in units such that $k_B T=1$. In the\nresults described below the partition function was evaluated for\neach variance of the disorder for several systems of finite width\n$w=2L_x$ averaging over the time-like direction (typically $\\tau\n\\simeq 10^6$ ``time'' steps) with the initial condition $Z(0,0)=1$\nand $Z(x,0)=0$ for $x \\neq 0$.\n\nTo analyze the numerics we performed a finite size scaling analysis. In the spirit of Eq.~(\\ref{nu}), in the vicinity of the transition we\nexpect the scaling form (compare Eq.~(\\ref{scaling1})):\n\\begin{equation}\n\\overline{\\langle\\tau_m\\rangle}=L_x \\Phi\\left[L_x(f_c-f)^\\nu\\right],\n\\label{scaling}\n\\end{equation}\nwhere $\\Phi$ is some scaling function.\nBased on the results of previous sections we anticipate a smooth\ninterpolation between scaling exponents $\\nu=1$ and $\\nu=2$ with either\nincreasing $L_x$ or increasing strength of disorder at fixed $L_x$.\nTo perform the finite size scaling we obtain for each value of $L_x$\na value for the exponent $\\nu$ from the best collapse of the\nnumerical data of two systems sizes $L_x$ and $L_x\/2$. In\nFig.~\\ref{fig1} we plot $1\/\\nu$ as a function of the system size\n$L_x$. As can be seen the data is consistent with $\\nu$ saturating\nat $\\nu=2$ for large systems. The crossover to $\\nu=2$ is much more\nrapid if the point disorder is enhanced near the columnar pin (see\nthe inset in Fig.~\\ref{fig1}), as might be expected for damage\ntracks created by heavy ion radiation.\n\\begin{figure}\n\\center\n\\includegraphics[width=8.5cm]{scaling_1dv2.eps}\n\\caption{Effective exponent $1\/\\nu$ versus $L_x$ for a fixed\nstrength of point disorder $\\sigma=0.03$. The results are\nconsistent with the general argument that this exponent should\nsaturate at $\\nu=2$ as $L_x\\to\\infty$. The inset shows the same\nexponent vs $\\sigma_c$, the variance of additional point disorder\nplaced directly on the columnar pin extracted from two system\nsizes $L_x=600$ and $L_x=1200$. It appears that $\\nu\\to 2$ as\n$\\sigma_c$ increases.} \\label{fig1}\n\\end{figure}\n\nNext, we test the behavior of the critical force as the\ndisorder strength is increased. According to our discussion in Sec.\n\\ref{scalingphasediagram}, we anticipate that in the absence of an\nexternal force the flux line is always bound to the pin in $1+1$\ndimensions. This is in contrast with the problem of unzipping from\nthe wall discussed in the previous section, where there is a\ncritical strength of the disorder, $\\sigma_c$, which leads to an\nunbinding transition for $f=0$. Note that the existence of a critical value\nof the disorder is a direct consequence (see discussion in Sec.\n\\ref{scalingphasediagram}) of the excursions of the vortex from the\ndefect which, as argued above, do not modify the critical behavior\nof the unzipping transition. The existence of a critical value of\nthe disorder is therefore strongly dependent on the dimensionality\nof the problem.\n\nIn numerical simulations for each strength of disorder we determine\nthe critical force plotting the ratio\n$\\overline{\\langle\\tau_m\\rangle}\/L_x$ for two different sizes $L_x$\nand using the scaling relation~(\\ref{scaling}). Note that this ratio\ndoes not depend on $L_x$ at $f=f_c$ (see also the discussion in\nSec.~\\ref{replica_2D}). We checked that this is indeed the case.\nUpon repeating this procedure for different disorder strengths we\nobtain the dependence $f_c(U_0)$ which is plotted in\nFig.~\\ref{fig8}.\n\\begin{figure}[ht]\n\\hspace{0.5cm}\n\\includegraphics[bb=1cm 1cm 20cm 25cm, scale=0.38, angle=90]{crit_f.eps}\n\\caption{Critical force for unzipping from a columnar defect in\n$1+1$ dimensions as a function of the disorder\nstrength.}\\label{fig8}\n\\end{figure}\nThe graph suggests that there is no unbinding transition at zero\ntilt at any strength of disorder consistent with the scaling\nargument presented in Sec.~\\ref{scalingphasediagram} and those of\nRef.~[\\onlinecite{HwaNatter}]. We point out that the strongest disorder\nshown in the graph $U_0=0.9$ required samples quite extended in the\ntime-like direction, $L_\\tau\\approx 10^8$.\n\n\n\\section{Unzipping a Luttinger liquid}\n\\label{sec:Lutunzip}\n\nWe now turn to consider the effect of interactions on the unzipping\nof single vortices. To do this we study a system where the vortices\nare preferentially bound to a thin two-dimensional slab which is\nembedded in a three-dimensional sample so that the density of\nvortices in the slab is much higher than in the bulk.\nExperimentally, this setup could be achieved using, for example, a\ntwin plane in YBCO or by inserting a thin plane with a reduced lower\ncritical field $H_{c1}$ (with, for example, molecular beam epitaxy)\ninto a bulk superconductor. The scenario we analyze is one where a\nMFM is used to pull a single vortex out of the two-dimensional slab\n(see Fig. \\ref{fig9}). The physics of the vortices confined to two\ndimensions is well understood and is analogous to a spinless Luttinger\nliquid of bosons (see, e.g. Ref.~[\\onlinecite{AHNS}]).\n\nAs we show below the dependence of the displacement of the vortex\nfrom the two-dimensional slab on the force exerted by the MFM\ndepends on the physics of the two-dimensional vortex liquid which\nresides in the slab. Specifically, the critical properties of the\nunbinding transition depend on the ``Luttinger liquid parameter''\nwhich controls the large-distance behavior of the vortex liquid. The\nexperimental setup can thus be used to probe the two-dimensional\nphysics of the vortices in the slab.\n\n\\begin{figure}[ht]\n\\includegraphics[scale=0.6]{UnzipLuttinger.eps}\n\\caption{Possible experimental setup for studying unzipping from\nLuttinger Liquid. A MFM is used to pull a single vortex out of a\nplane where the vortices are confined. The measured quantity is the\ndistance of the pulled vortex from the confining plane as a function\nof the force $f$. }\\label{fig9}\n\\end{figure}\n\\subsection{Two-dimensional vortex liquids}\n\nThe physics of vortices in two dimensions is very well understood.\nThe vortices form a one-dimensional array located at position\n$x_i(\\tau)$. The density profile of the vortices is then given by\n\\begin{equation}\n    n(x,\\tau)=\\sum_j \\delta \\left[ x-x_j(\\tau)\\right] \\;,\n\\end{equation}\nwhere $x$ and $\\tau$ denote transverse and longitudinal coordinates\nwith respect to the vortices and $i$ is an index labeling the\nvortices. By changing variable into the phonon displacement field\n$u_j$ through $x_j(\\tau)=a\\left[j+u_j(\\tau)\\right]$, where $a$ is\nthe mean distance between vortex lines the free-energy of a\nparticular configuration can be written as:\n\\begin{equation}\n    {\\cal F}_0=\\frac{a^2}{2} \\int dx d\\tau \\left[ c_{11}\n    (\\partial_x u)^2 + c_{44} (\\partial_\\tau u)^2\\right] \\;.\n\\end{equation}\nHere $c_{11}$ and $c_{44}$ are the compressional and the tilt moduli\nrespectively. After rescaling the variables $x$ and $\\tau$ according\nto\n\\begin{equation}\n    x \\to x \\left(\\frac{c_{11}}{c_{44}}\\right)^{1\/4} \\;\\; ,\n    \\; \\tau \\to \\tau \\left(\\frac{c_{44}}{c_{11}}\\right)^{1\/4} \\;,\n\\end{equation}\nthe free energy takes the isotropic form\n\\begin{equation}\n    {\\cal F}_0=\\frac{A}{2}\\int dx d\\tau\n    \\left[ (\\partial_x u)^2 + (\\partial_\\tau u)^2\\right]\n\\end{equation}\nwith $A=a^2\\sqrt{c_{11}c_{44}}$. The partition function is then\ngiven by the functional integral\n\\begin{equation}\n    Z=\\int D u(x,\\tau) e^{-S} \\;,\n\\end{equation}\nwith $S=S_0={\\cal F}_0\/T$. In the limit of large sample sizes in the\n``timelike'' direction  one can regard $Z$ as the zero temperature\npartition function of interacting bosons~\\cite{AHNS}. In this\nlanguage the imaginary time action can be written as\n\\begin{equation}\n    S_0=\\frac{\\pi}{2g}\\int dx d\\tau\n    \\left[ (\\partial_x u)^2 + (\\partial_\\tau u)^2\\right] \\;.\n    \\label{freeaction}\n\\end{equation}\nHere we set $\\hbar=1$ and identified the Luttinger-liquid parameter,\n$g$, as\n\\begin{equation}\n    g=\\frac{\\pi T}{A} \\;.\n    \\label{Lutpara}\n\\end{equation}\nThe Luttinger-liquid parameter controls the long-distance properties\nof the model. For vortices $g$ it is a\ndimensionless combination of the compressional and tilt moduli, the\ndensity of vortices and temperature.\n\nVarious properties of Luttinger liquids are well understood. For\nexample, the correlation function for the density fluctuations\n$\\delta n(x,\\tau)=n(x,\\tau)-n_0$, where $n_0=1\/a$ is the mean\ndensity, obeys\n\\begin{equation}\n    \\langle \\delta n(x,\\tau) \\delta n(0,0) \\rangle \\simeq\n    \\frac{\\cos\\left( 2 \\pi n_0 x \\right)}{(x^2+\\tau^2)^g} \\;.\n\\end{equation}\nThere is quasi long-range order in the system and the envelope of\nthe density correlation function decays as a power law with the exponent\ndepending only on $g$. As we show below, $g$ can be probed by\nunzipping a single vortex out of a plane which contains a $(1+1)$-dimensional vortex liquid.\n\nIn what follows we also consider the case where there is point\ndisorder present in the sample. The behavior will be strongly influenced by the behavior of the vortices\nin two dimensions in the presence of disorder. This problem has been\nstudied in some detail in the past (see e.g. Ref.~[\\onlinecite{pkn}] and\nreferences therein). Here we briefly review features which will be\nimportant in analyzing the unzipping problem. The most relevant (in\nthe renormalization group sense) contributions to the action from\nthe point disorder is\n\\begin{equation}\n    S_{PD}=2\\int dx d\\tau R(x,\\tau)\n    \\cos \\left[2 \\pi u(x,\\tau) +\\beta(x,\\tau) \\right] \\;,\n\\end{equation}\nwhere positive (negative) $R$ implies a repulsive (attractive)\npotential between the vortices and the quenched random disorder. We assume, for simplicity, that $\\beta(x,\\tau)$ is\ndistributed uniformly between $0$ and $2 \\pi$ and $R(x,\\tau)$ has a\nan uncorrelated Gaussian distribution with the variance $\\Delta_0$:\n\\begin{equation}\n    \\overline{R(x_1,\\tau_1)R(x_2,\\tau_2)}=\n    \\Delta_0 \\delta(x_1-x_2)\\delta(\\tau_1-\\tau_2) \\;,\n\\end{equation}\nwhere the overbar, as before, represents averaging over disorder.\n\nTo analyze the disordered problem, similar to the single vortex case,\nwe use the replica trick. Then the replicated noninteracting part of\nthe action becomes\n\\begin{equation}\n    S_0=\\frac{\\pi}{2g} \\sum_{\\alpha,\\beta} \\int\n    \\int dx d\\tau \\left[ \\frac{\\partial u_\\alpha}{\\partial \\tau}\n    \\frac{\\partial u_\\beta}{\\partial \\tau} +\\frac{\\partial u_\\alpha}{\\partial x}\n    \\frac{\\partial u_\\beta}{\\partial x}  \\right] \\left[ \\delta_{\\alpha,\\beta}\n    - \\frac{\\kappa}{g}  \\right] \\;.\n\\end{equation}\nHere $u_\\alpha(x,\\tau)$ is the replicated phonon field and $\\kappa$\nis an off-diagonal coupling which is zero in the bare model but is\ngenerated by the disorder. It plays the role of a quenched random\n``chemical potential'' which is coupled to the first derivative of\nthe phonon field $u$. The replica indices, $\\alpha$ and $\\beta$ run\nfrom $1$ to $n$ and at the end of the calculation one takes the\nlimit $n \\to 0$. After replication the contribution from the point\ndisorder becomes\n\\begin{equation}\n    S_{PD}=-\\Delta_0 \\sum_{\\alpha,\\beta} \\int \\int dx d\\tau\n    \\cos 2 \\pi \\left[ u_\\alpha (x,\\tau) - u_\\beta (x,\\tau)  \\right] \\;.\n\\end{equation}\nThe combined action can be treated within the renormalization group\nusing a perturbation series near $g=1$ where a phase transition\nbetween a vortex liquid and a vortex glass\noccurs~\\cite{fisher_v_glass}. By continuously eliminating degrees of\nfreedom depending on frequency and momentum within the shell\n$\\Lambda - \\delta \\Lambda < \\sqrt{\\omega^2+q^2} < \\Lambda$, one\nobtains the following renormalization group equations~\\cite{Cardy,\npkn}\n\\begin{eqnarray}\n    \\frac{dg}{dl}&=&0 \\\\\n    \\frac{d \\Delta}{dl}&=&2(1-g) \\Delta - 2 C \\Delta^2 \\\\\n    \\frac{d \\kappa}{dl}&=&C^2 \\Delta^2\n\\end{eqnarray}\nHere $l$ is the flow parameter $\\Lambda(l)=\\Lambda e^{-l}$.  $C$ is\na non-universal constant which depends on the cutoff $\\Lambda$. The\nequations are subject to the initial conditions $\\kappa(l=0)=0$ and\n$\\Delta(l=0)=\\Delta_0$. Note that the Luttinger liquid parameter is\nnot renormalized.  Analyzing the flow equations it has been shown\nthat in the vortex liquid phase ($g>1$) the correlation of the\ndensity fluctuation behaves in the vortex liquid phase as\n\\begin{equation}\n    \\langle \\delta n(x,\\tau) \\delta n(0,0) \\rangle\n    \\simeq \\frac{1}{(x^2+\\tau^2)^{g+\\tilde{\\kappa}\/2}} \\;,\n\\end{equation}\nwhere $\\tilde{\\kappa}$ is a nonuniversal exponent. In the glass\nphase ($g<1$) correlations decay faster than a power law, with\n\\begin{equation}\n    \\langle \\delta n(x,\\tau) \\delta n(0,0) \\rangle\n    \\simeq \\exp \\left( -(1-g)^2 \\ln^2 \\sqrt{x^2+\\tau^2}\\right) \\;.\n\\end{equation}\n\nIn what follows we consider a setup in which a two dimensional array\nof vortices, whose properties have been described above, is embedded\nin a three dimensional bulk sample. As shown below when a single vortex\nis unzipped into the bulk in a clean sample the critical properties\nof the unzipping transition yield information on the properties of\nthe two dimensional vortex liquid. In particular, they provide a\ndirect measure of the Luttinger-liquid parameter. In the same setup\nin a disordered sample we will show that the critical properties of\nthe unzipping transition will be modified. In particular, they can\nyield information on the on the three-dimension wandering exponent\nof a single vortex in a disordered sample.\n\n\n\\subsection{Unzipping a Luttinger liquid: The clean case}\nConsider first an experiment where an attractive two-dimensional\npotential holds vortices confined to it. A MFM then pulls a {\\it\nsingle} vortex out of the plane (see Fig. \\ref{fig9}). We assume\nthroughout that the density of vortices in the three dimensional bulk\nis so small that we can neglect interactions between the vortex that\nis pulled out of the sample and vortices in the three dimensional\nbulk. In this subsection only the clean case (no point disorder) will be studied.\n\nWe assume the MFM exerts a force ${\\bf f}=f \\hat{x}$. As in the\nunzipping experiments discussed above we expect that for large\nforces $f>f_c$ the vortex will be completely pulled out of the two\ndimensional slab. Similar to the case of the unzipping of a single\nvortex we write the free energy of the vortex as a sum of two\ncontributions. The first, ${\\cal F}_u(\\tau_m)$, arises from the part\nof the vortex that is outside the two dimensional slab. The second\n${\\cal F}_b(\\tau_m)$ is the change in the free-energy of the\nvortices that remain inside the two dimension slab. As before\n$\\tau_m$ is the length along the $\\tau$ direction which is unbound\nfrom the two-dimensional slab. The free-energy of the unzipped part\nis clearly identical to that calculated in Eq.~\\ref{free_unzip} or\nexplicitly\n\\begin{equation}\n    {\\cal F}_u(\\tau_m)= -  f^2 \\tau_m\/ 2\\gamma \\;.\n    \\label{eq:unzupfeagain}\n\\end{equation}\n\nThe calculation of the free-energy, ${\\cal F}_b(\\tau_m)$, is\nsomewhat more involved. Clearly there is a linear contributions due\nto the length $\\tau_m$ removed from the attractive potential of the\nslab. However, in addition there is an extra contribution from the\nenergy of the dislocation, ${\\cal F}_d(\\tau_m)$, (see Fig.\n\\ref{fig9}) created in the two dimensional vortex array. This\ncontribution to the free-energy, as we show below, is {\\it\nnon-linear} and controlled by the Luttinger liquid parameter $g$.\nThis non-linearity results, near the unzipping transition, in a\nsensitivity of the critical properties to the value of $g$.\n\nWe leave the details of the calculation of the dislocation energy to\nAppendix~\\ref{App:dislocation} and present here only the key steps\nof derivation.\n\nIn order to satisfy boundary conditions near the interface one can\nuse the method of images (see Fig.~(\\ref{fig11})).  The free energy\nof this dislocation pair can be calculated by standard methods (see\ndetails in Appendix~\\ref{App:dislocation}). In particular, at large\n$\\tau_m$ it behaves logarithmically (see e.g. Ref.~[\\onlinecite{chakin}]):\n\\begin{equation}\n    {\\cal F}_d=\\frac{T}{4g} \\ln(\\tau_m\/a_0),\n    \\label{free_en_dis}\n\\end{equation}\nwhere $a_0$ is the short range cutoff of the order of the distance\nbetween flux lines. We note that the free energy of the dislocation\nnear the interface (\\ref{free_en_dis}) is one half of the free\nenergy of a dislocation pair.\n\nWith the energy of the dislocation in hand we can now analyze the\nproperties of the unzipped length near the transition using the\nmethods used for analyzing the single vortex unzipping experiments.\nThe contributions to the free energy are from the unzipped part of\nthe vortex and the energy of the dislocation. Collecting all the\nrelevant terms, near the transition the free energy is given by\n\\begin{equation}\n  {\\cal F}(\\tau_m)={\\cal F}_u(\\tau_m)+{\\cal F}_b(\\tau_m)=\n  \\epsilon\\tau_m+\\frac{T}{4g}\\ln(\\tau_m\/a_0)\\;.\n\\end{equation} \nThe probability of finding a certain value of $\\tau_m$ is then given\nby\n\\begin{equation}\n    P(\\tau_m) \\propto e^{-F(\\tau_m)\/T}=\\frac{C}{\\tau_m^{1\/(4g)}}e^{-\\epsilon\\tau_m}.,\n\\end{equation}\nwhere $C$ is the normalization constant. At the transition\n$\\epsilon=0$ the distribution becomes a pure power law in $\\tau_m$.\nTherefore, the average value of $\\tau_m$ is very sensitive to the\nvalue of $g$. In particular, for $g>1\/4$ (i.e. for weakly\ninteracting flux lines) the behavior of $\\langle\\tau_m\\rangle$ near\nthe transition is identical to that of a single vortex in the\nabsence of interactions with other vortices\n\\begin{equation}\n    \\langle \\tau_m \\rangle \\sim {1\\over\\epsilon} \\;.\n\\end{equation}\nIn contrast, for $1\/8 < g < 1\/4$ (stronger interactions) there is a\ncontinuously varying exponent governing the transition\n\\begin{equation}\n\\langle\\tau_m\\rangle\\sim {1\\over \\epsilon^{2-1\/4g}} \\;.\n\\end{equation}\nAnd finally, for $g<1\/8$ (strongly interacting flux lines) we find\nthat $\\langle \\tau_m\\rangle$ does not diverge near the transition.\nNote that even though in this regime the mean displacement remains\nconstant at the transition the higher moments of $\\tau_m$ diverge\nand are thus sensitive to $\\epsilon$. The reason for this is at the\ntransition the distribution of $\\tau_m$ is a power law.\n\n\n\\subsection{Unzipping from a twin plane with point disorder}\n\\label{3c}\n\nWe now consider the problem of unzipping a vortex from a\nplane with many vortices in the presence of disorder. In the spirit\nof the treatments presented in this paper, one needs to calculate the\nfree-energy of the unzipped part of the vortex ${\\cal F}_u(\\tau_m)$,\nthe free-energy of the bound part of the vortex ${\\cal F}_b(\\tau_m)$\nand the {\\it fluctuations} in both quantities averaged over\nrealizations of disorder. This can be done perturbatively near $g=1$. We again relegate details of the derivation of\nthe dislocation energy to\nAppendix~\\ref{App:dislocation1}. One conclusion from our\ncalculations is that the mean free energy of the dislocation near\nthe boundary is not affected by the disorder and is given by\nEq.~(\\ref{free_en_dis}). Another important conclusion is that the\nfluctuations of the free energy also depend logarithmically on\n$\\tau_m$:\n\\begin{equation}\n    \\overline{\\delta {\\cal F}^2_d(\\tau_m)}= T^2\\frac{\\kappa(\\infty)}{8g^2} \\ln(\\tau_m\/a_0)\n\\end{equation}\nfor $g>1$ and\n\\begin{equation}\n    \\overline{\\delta {\\cal F}^2_d(\\tau_m)}=T^2\\frac{(1-g)^2}{4} \\ln^2(\\tau_m\/a_0)\n\\end{equation}\nfor $g<1$.\n\\begin{figure}[ht]\n\\includegraphics[scale=0.6]{UnzipLuttingerdis.eps}\n\\caption{Possible experimental setup for studying unzipping from\nLuttinger Liquid in the presence of disorder.}\\label{fig10}\n\\end{figure}\nWe note that in the case of many flux lines there is a weak\nlogarithmic dependence of free energy fluctuations on $\\tau_m$ as\nopposed to strong power law dependence in the case of a single flux\nline (compare Eq.~(\\ref{eq:fefluct})). This somewhat surprising\nresult is a consequence of the screening of strong power-law\nfluctuations by other flux lines. We note that if the pinning of\nflux lines by disorder is extremely strong so that tearing a single\nflux line does not affect positions of other lines in the duration\nof experiment, we are back to the single flux line physics and\n$\\overline{\\delta \\mathcal F_d^2}\\propto \\tau_m$.\n\nTo complete the analysis, we need to consider the free-energy\ncontribution from the unzipped part. Of particular of importance are\nthe free-energy fluctuations due to the disorder in the bulk of the\nsample. As discussed in Sec. \\ref{Sec2}, in a three dimensional\nsample these grow as $\\delta {\\cal F}_u \\propto m^{\\omega(3)}$ with\n$\\omega(3) \\simeq 0.22$. This contribution grows much quicker than\nthe contribution from the fluctuations in the free-energy of the\ndislocation. Therefore following the ideas of Sec. \\ref{Sec2} the\ntotal free-energy is given by\n\\begin{equation}\n{\\cal F}(m)=a(f_c-f)\\tau_m -b\\tau_m^{\\omega(3)}\\;.\n\\label{fplane}\n\\end{equation}\nwhere $a$ and $b$ are positive constants. Minimizing Eq. (\\ref{fplane}) gives for the critical properties in\nthis case\n\\begin{equation}\n    s \\sim \\frac{1}{(f_c-f)^{1.28}} \\;.\n\\end{equation}\nThus screening disorder fluctuations in the plain by other flux\nlines effectively enhances the role of disorder in the bulk. As the\nresult these unzipping experiments can serve as a probe of the\nthree-dimensional anomalous wandering exponent.\n\n\\acknowledgements\n\nYK was supported by the Israel Science Foundation and thanks the\nBoston University visitors program for hospitality. YK and DRN were\nsupported by the Israel-US Binational Science Foundation. Research\nby DRN was also supported by the National Science Foundation,\nthrough grant DMR 0231631 and through the Harvard Materials Research\nScience and Engineering center through grant DMR0213805. AP was\nsupported by AFOSR YIP.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Learning with Adversarial Contexts}\n\\label{sec:adversarial}\n\n\nIn this section, we focus on the adversarial contexts case, where at time $t\\in [T]$, the contexts $x_{t,1},\\cdots,x_{t,K}$ can be arbitrarily chosen by an adversary who observes all past contexts and rewards, with $\\|x_{t,a}\\|_2\\le 1$ for any $a\\in [K]$. \nWe first state the main results in Section~\\ref{subsec:adversarial_main}, which\ncharacterize the upper and lower bounds of the regret.\nWe then give a UCB-based algorithm in the sequential batch setting in Section \\ref{subsec:adversarial_UCB} and describe several important aspects of the algorithm, including a variant that is used for theoretical bound purposes. \nNext, in Section \\ref{subsec.UCB_upperbound}, we show that the proposed sequential batching algorithm achieves the regret upper bound in Theorem \\ref{thm.adversarial} . \nFinally, we prove the regret lower bound in Section \\ref{subsec.adversarial} and therefore establish that the previous upper bound is close to be tight. \n\n\\subsection{Main Results}\\label{subsec:adversarial_main}\n\\begin{theorem}\\label{thm.adversarial}\n\tLet $T$, $M$ and $d$ be the learning horizon, number of batches and each context's dimension, respectively. Denote by $\\mathsf{polylog}(T)$ all the poly-logarithmic factors in $T$.\n\t\\begin{enumerate}\n\t\t\\item Under Assumption~\\ref{aspn.TKd}, there exists a sequential batch learning algorithm \\textbf{Alg}= $({\\mathcal{T}}, \\pi)$, where ${\\mathcal{T}}$ is a uniform grid  defined by $t_m = \\lfloor \\frac{mT}{M}\\rfloor$  and $\\pi$ is explicitly defined in Section \\ref{subsec:adversarial_UCB},\n\t\tsuch that:\n\t\t\\begin{align*}\n\t\t\\sup_{\\theta^\\star: \\|\\theta^\\star\\|_2\\le 1} \\mathbb{E}_{\\theta^\\star}[R_T\\left(\\textbf{Alg}\\right)] \\le \\mathsf{polylog}(T)\\cdot \\left(\\sqrt{dT} + \\frac{dT}{M}\\right).\n\t\t\\end{align*}\n\t\t\\item  Conversely, for $K=2$ and any sequential batch learning algorithm, we have:\n\t\t\\begin{align*}\n\t\t\\sup_{\\theta^\\star: \\|\\theta^\\star\\|_2\\le 1} \\mathbb{E}_{\\theta^\\star}[R_T\\left(\\textbf{Alg}\\right)] \\ge c\\cdot \\left(\\sqrt{dT} + \\left(\\frac{T\\sqrt{d}}{M}\\wedge \\frac{T}{\\sqrt{M}}\\right)\\right),\n\t\t\\end{align*}\n\t\twhere $c>0$ is a universal constant independent of $(T,M,d)$. \n\t\\end{enumerate}\n\\end{theorem}\n\nOur subsequent analysis easily gives high-probability regret upper bounds. However, for simplicity and to highlight more clearly the matching between the upper and lower bounds, \nwe stick with presenting results on expected regret.\nTheorem \\ref{thm.adversarial} shows a polynomial dependence of the regret on the number of batches $M$ under adversarial contexts, and the following corollary is immediate. \n\\begin{corollary}\\label{cor.adversarial}\n\tUnder adversarial contexts, $\\Theta(\\sqrt{dT})$ batches achieve the fully online regret $\\tilde{\\Theta}(\\sqrt{dT})$. \n\\end{corollary}\n\nAccording to Corollary \\ref{cor.adversarial}, $T$ batches are not necessary to achieve the fully online performance under adversarial contexts: $\\Theta(\\sqrt{Td})$ batches suffice. Since we are \\text{not} in the high-dimensional regime (per Assumption~\\ref{aspn.TKd}, $d \\le \\sqrt{T}$), the number of batches needed without any performance suffering is at most $O(T^{0.75})$, a sizable reduction from $O(T)$. Further, in the low-dimensional regime (i.e. when $d$ is a constant), only $O(\\sqrt{T})$ batches are needed to achieve fully online performance.\nNevertheless, $O(\\sqrt{dT})$ can still be a fairly large number. \nIn particular, if only a constant number of batches are available, then the regret is linear. The lower bound indicates that not much better can be done in the adversarial contexts. \nThis is because the power of the adversary under adversarial contexts is too strong when the learner only has a few batches: the adversary may simply pick any batch and choose all contexts anterior to this batch to be orthogonal with the contexts within this batch, such that the learner can learn nothing about the rewards in any given batch. \n\n\n\n\\subsection{A Sequential Batch UCB Algorithm}\\label{subsec:adversarial_UCB}\nThe overall idea of the algorithm is that, at the end of every batch, the learner computes an estimate $\\hat{\\theta}$ of the unknown parameter $\\theta^\\star$ via ridge regression as well as a confidence set that contains $\\theta^\\star$ with high probability. Then, whenever the learner enters a new batch, at each time $t$ he simply picks the action with the largest upper confidence bound. Finally, we choose the uniform grid, i.e., $t_m = \\lfloor \\frac{mT}{M}\\rfloor$ for each $m\\in [M]$. The algorithm is formally illustrated in Algorithm \\ref{algo.ucb}.\n\n\\begin{algorithm}[h!]\n\t\\DontPrintSemicolon  \n\t\\SetAlgoLined\n\t\\BlankLine\n\t\\caption{Sequential Batch UCB (SBUCB) \\label{algo.ucb}}\n\t\\textbf{Input:} time horizon $T$; context dimension $d$; number of batches $M$; tuning parameter $\\gamma>0$.\n\t\n\t\\textbf{Grid choice:} ${\\mathcal{T}} = \\{t_1,\\cdots,t_M\\}$ with $t_m = \\lfloor \\frac{mT}{M}\\rfloor$. \n\t\n\t\\textbf{Initialization:} $A_0 = I_d\\in \\mathbb{R}^{d\\times d}$, $\\hat{\\theta}_0={\\bf 0}\\in \\mathbb{R}^d$, $t_0 = 0$.\n\t\n\t\\For{$m \\gets 1$ \\KwTo $M$}{\n\t\t\t\\For{$t\\gets t_{m-1}+1$ \\KwTo $t_m$}{\n\t\t\t\tChoose $a_t = \\arg\\max_{a\\in [K]} x_{t,a}^\\top \\hat{\\theta}_{m-1} + \\gamma\\sqrt{x_{t,a}^\\top A^{-1}_{m-1} x_{t,a}}$ (break ties arbitrarily). \\\\\n\t\t\t}\n\t\tReceive rewards in the $m$-th batch: $\\{r_{t,a_t}\\}_{t_{m-1}+1 \\le t \\le t_m}$. \n\t\t\n\t\t$A_m =  A_{m-1} + \\sum_{t=t_{m-1}+1}^{t_m}  x_{t,a_t}x_{t,a_t}^\\top$. \\\\\n\t\t$\\hat{\\theta}_m = A^{-1}_m\\sum_{t=t_{m-1}+1}^{t_m} r_{t,a_t}x_{t,a_t}$.\n\t}\n\\end{algorithm}\n\n\\begin{remark}\nNote that when $M=T$ (i.e. the fully online setting), Algorithm~\\ref{algo.ucb} degenerates to the standard LinUCB algorithm in~\\cite{chu2011contextual}.\n\\end{remark}\t\n\nTo analyze the sequential batch UCB algorithm, we need to first show that the constructed confidence bound is feasible. By applying \\cite[Lemma 1]{chu2011contextual} to our setting, we immediately obtain the following concentration result that the estimated $\\hat{\\theta}_{m-1}$ is close to the true $\\theta^\\star$: \n\n\\begin{lemma}\\label{lemma.concentration}\n\tFix any $\\delta > 0$.\n\tFor each $m\\in [M]$, if for a fixed sequence of selected contexts $\\{x_{t,a_t}\\}_{t\\in [t_m]}$ up to time $t_m$, the (random) rewards $\\{r_{t,a_t}\\}_{t\\in [t_m]}$  are independent, then for each $t \\in [t_{m-1}+1, t_m]$,\n\twith probability at least $1-\\frac{\\delta}{T}$, the following holds for all $a\\in [K]$:\n\t\\begin{align*}\n\t|x_{t,a}^\\top (\\hat{\\theta}_{m-1} - \\theta^\\star)| \\le \\left(1+\\sqrt{\\frac{1}{2}\\log\\left(\\frac{2KT}{\\delta}\\right)}\\right)\\sqrt{x_{t,a}^\\top A_{m-1}^{-1}x_{t,a}}.\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{remark}\nLemma~\\ref{lemma.concentration} rests on an important conditional independence assumption of the rewards $\\{r_{t,a_t}\\}_{t\\in [t_m]}$. However, this assumption\ndoes not hold in the vanilla version of the algorithm as given in Algorithm~\\ref{algo.ucb}.\nThis is because a future selected action $a_t$ and hence the chosen context $x_{t,a_t}$ depends on  \nthe previous rewards. Consequently, by conditioning on $x_{t,a_t}$, previous rewards, say $r_{\\tau_1}, r_{\\tau_2}$ ($\\tau_1, \\tau_2 < t$), can become dependent. Note the somewhat subtle issue here on\nthe dependence of the rewards: when conditioning on $x_{t,a_t}$, the corresponding reward $r_t$ becomes independent of all the past rewards $\\{r_\\tau\\}_{\\tau < t}$. Despite this, when a future $x_{t^\\prime, a_{t^\\prime}}$ is revealed ($t^\\prime > t$), these rewards (i.e. $r_t$ and all the rewards prior to $r_t$) become coupled again: what was known about $r_t$ now reveals information about the previous rewards $\\{r_\\tau\\}_{\\tau < t}$, because $r_t$ itself would not determine the selection of $x_{t^\\prime, a_{t^\\prime}}$:\nall those rewards have influence over $x_{t^\\prime, a_{t^\\prime}}$. Consequently, a complicated dependence structure is thus created when conditioning on $\\{x_{t,a_t}\\}_{t\\in [t_m]}$.\n\nThis lack of independence issue will be handled with a master algorithm variant of Algorithm~\\ref{algo.ucb} discussed in the next subsection. \nUsing the master algorithm to decouple dependencies is a standard technique in contextual bandits that was first developed in~\\cite{auer2002using}. Subsequently, it has been used for the same purpose in~\\cite{chu2011contextual, li2017provably}, among others. We will describe how to adapt the master algorithm in our current sequential batch learning setting next. We end this subsection by pointing out that, strictly speaking, our regret upper bound is achieved only by this master algorithm, rather than Algorithm~\\ref{algo.ucb}. However, we take the conventional view that the master algorithm is purely used as a theoretical construct (to resolve the dependence issue) rather than a practical algorithm that should actually be deployed in practice. In practice, Algorithm~\\ref{algo.ucb} should be used instead. For that reason, we discuss the master algorithm only in the proof.\n\\end{remark}\n\n\\subsection{Regret Analysis for Upper bound}\\label{subsec.UCB_upperbound}\n\nWe start with a simple fact from linear algebra that will be useful later.\n\n\\begin{lemma}\\cite[Lemma 11]{auer2002using}\\label{lemma.eigenvalue}\n\tLet $A$ be a symmetric matrix such that $I_d\\preceq A$, and $x\\in \\mathbb{R}^d$ be a vector satisfying $\\|x\\|_2\\le 1$. Then the eigenvalues $\\lambda_1,\\cdots,\\lambda_d$ of $A$ and the eigenvalues $\\nu_1,\\cdots,\\nu_d$ of $A+xx^\\top$ can be rearranged in a way such that $\\lambda_i\\le \\nu_i$ for all $i\\in [d]$, and\n\t\\begin{align*}\n\t\\mathsf{Tr}(A^{-1}xx^\\top) \\le 10\\sum_{j=1}^d \\frac{\\nu_j - \\lambda_j}{\\lambda_j}. \n\t\\end{align*}\n\\end{lemma}\n\nWe next establish a key technical lemma that will be used in establishing our regret upper bound. \n\\begin{lemma}\\label{lemma.trace_sum}\nDefine $X_m = \\sum_{t=t_{m-1}+1}^{t_m} x_{t,a_t}x_{t,a_t}^\\top$. We have:\n\\begin{align*}\n\\sum_{m=1}^M \\sqrt{ \\mathsf{Tr}(A_{m-1}^{-1} X_m)} \\le \\sqrt{10}\\log(T+1)\\cdot \\left(\\sqrt{Md} + d\\sqrt{\\frac{T}{M}} \\right). \n\\end{align*}\n\\end{lemma} \n\n\\begin{proof}\nWe start by noting that with the above notation, we have $A_m=A_{m-1}+X_m$ for any $m\\in [M]$ with $A_0=I_d$. \nApplying Lemma \\ref{lemma.eigenvalue} repeatedly, we may rearrange the eigenvalues $\\lambda_{m,1},\\cdots,\\lambda_{m,d}$ of $A_m$ in such a way that $\\lambda_{m-1,j}\\le \\lambda_{m,j}$ for all $m\\in [M], j\\in [d]$, and \n\\begin{align}\\label{eq.reduction_eigenvalue}\n\\sum_{m=1}^M \\sqrt{ \\mathsf{Tr}(A_{m-1}^{-1} X_m)} \\le \\sqrt{10}\\cdot \\sum_{m=1}^M \\sqrt{\\sum_{j=1}^d \\frac{\\lambda_{m,j} - \\lambda_{m-1,j}}{\\lambda_{m-1,j}}}. \n\\end{align}\nNote that $\\lambda_{0,j}=1$ for all $j\\in [d]$. \nNote further that $ \\lambda_{M,j}\\le 1+T, \\forall j \\in [d]$, which follows from the \nfact that $z^\\top (A_M) z = z^\\top (I_d + \\sum_{t=1}^T x_{t,a_t}x_{t,a_t}^\\top ) z = \\|z\\|_2^2 + \\sum_{t=1}^T \\|z^\\top x_{t,a_t}\\|^2_2 \\le (T+1) \\|z\\|_2^2$, since $\\|x_{t,a_t}\\|_2 \\le 1$.\nConsequently, every eigenvalue of $A_M$ must be bounded by $T+1$.\n\nUtilizing the above two pieces of information on $\\lambda_{0,j}$ and $ \\lambda_{M,j}$, we then have the following:\n\\begin{align}\n\\sum_{m=1}^M \\sqrt{\\sum_{j=1}^d \\frac{\\lambda_{m,j} - \\lambda_{m-1,j}}{\\lambda_{m,j}}} &\\le \\sqrt{M\\sum_{m=1}^M\\sum_{j=1}^d \\frac{\\lambda_{m,j} - \\lambda_{m-1,j}}{\\lambda_{m,j}} }\n =\\sqrt{M \\sum_{j=1}^d \\sum_{m=0}^{M-1} \\frac{\\lambda_{m+1,j} - \\lambda_{m,j}}{\\lambda_{m+1,j}} }  \\nonumber \\\\\n&\\le \\sqrt{M\\sum_{j=1}^d \\int_{\\lambda_{0,j}}^{\\lambda_{M,j}} \\frac{dx}{x} } \n= \\sqrt{M\\sum_{j=1}^d \\log \\lambda_{M,j}}\n\\le \\sqrt{Md\\log(T+1)}, \\label{eq.inequality_1}\n\\end{align}\nwhere the first inequality follows from $(\\sum_{i=1}^n x_i)^2 \\le n \\sum_{i=1}^n x_i^2$, for any real numbers\n$x_1, \\dots, x_n$.\n\n\nWe now look at the difference between Equation~\\eqref{eq.reduction_eigenvalue} and Equation~\\eqref{eq.inequality_1} and have:\n\\begin{align*}\n\\sqrt{\\sum_{j=1}^d \\frac{\\lambda_{m,j} - \\lambda_{m-1,j}}{\\lambda_{m-1,j}}} - \\sqrt{\\sum_{j=1}^d \\frac{\\lambda_{m,j} - \\lambda_{m-1,j}}{\\lambda_{m,j}}} &\\stepa{\\le} \\frac{\\sum_{j=1}^d \\frac{(\\lambda_{m,j}-\\lambda_{m-1,j})^2}{\\lambda_{m,j}\\lambda_{m-1,j} }}{\\sqrt{\\sum_{j=1}^d \\frac{\\lambda_{m,j} - \\lambda_{m-1,j}}{\\lambda_{m-1,j}}}}\\\\\n & =  \\frac{\\sum_{j=1}^d \\frac{(\\lambda_{m,j}-\\lambda_{m-1,j})^{1\/2}}{\\lambda_{m-1,j}^{1\/2} }  \\cdot \n\\frac{(\\lambda_{m,j}-\\lambda_{m-1,j})^{3\/2}}{\\lambda_{m,j}\\lambda_{m-1,j}^{1\/2}}}{\\sqrt{\\sum_{j=1}^d \\frac{\\lambda_{m,j} - \\lambda_{m-1,j}}{\\lambda_{m-1,j}}}}\\\\\n  &\\stepb{\\le} \n \\frac{\\sqrt{\\sum_{j=1}^d \\frac{(\\lambda_{m,j}-\\lambda_{m-1,j})}{\\lambda_{m-1,j} }}  \\cdot \n \\sqrt{\\sum_{j=1}^d\t\\frac{(\\lambda_{m,j}-\\lambda_{m-1,j})^{3}}{\\lambda_{m,j}^2\\lambda_{m-1,j}}}}{\\sqrt{\\sum_{j=1}^d \\frac{\\lambda_{m,j} - \\lambda_{m-1,j}}{\\lambda_{m-1,j}}}}\\\\\n &= \\sqrt{\\sum_{j=1}^d \\frac{(\\lambda_{m,j}-\\lambda_{m-1,j})^3}{\\lambda_{m-1,j}\\lambda_{m,j}^2} }, \n\\end{align*}\nwhere step (a) follows from the basic inequality $\\sqrt{a}-\\sqrt{b}\\le (a-b)\/\\sqrt{a}$ for $a\\ge b\\ge 0$, and step (b) is due to Cauchy--Schwartz. \n\n\nNote further that $\\lambda_{m,j}-\\lambda_{m-1,j}\\le \\mathsf{Tr}(X_m) = \\sum_{t=t_{m-1}+1}^{t_m} \\|x_{t,a_t}\\|_2^2\\le t_m - t_{m-1} = \\frac{T}{M}$, we therefore have:\n\\begin{align}\n&\\sum_{m=1}^M \\left(\\sqrt{\\sum_{j=1}^d \\frac{\\lambda_{m,j} - \\lambda_{m-1,j}}{\\lambda_{m-1,j}}} - \\sqrt{\\sum_{j=1}^d \\frac{\\lambda_{m,j} - \\lambda_{m-1,j}}{\\lambda_{m,j}}} \\right) \\le \\sum_{m=1}^M \\sqrt{\\sum_{j=1}^d \\frac{(\\lambda_{m,j}-\\lambda_{m-1,j})^3}{\\lambda_{m-1,j}\\lambda_{m,j}^2} } \\nonumber \\\\\n& \\le \\sum_{m=1}^M \\sqrt{\\sum_{j=1}^d \\frac{(\\lambda_{m,j}-\\lambda_{m-1,j})^2}{\\lambda_{m,j}^2 }}\\cdot \\sqrt{\\frac{T}{M}}\n\\le \\sum_{m=1}^M \\sum_{j=1}^d \\frac{\\lambda_{m,j}-\\lambda_{m-1,j}}{\\lambda_{m,j} }\\cdot \\sqrt{\\frac{T}{M}} \\nonumber \\\\\n&\\le \\sqrt{\\frac{T}{M}}\\sum_{j=1}^d \\int_{\\lambda_{0,j}}^{\\lambda_{M,j}} \\frac{dx}{x}  \\nonumber \\\\\n&\\le d\\sqrt{\\frac{T}{M}}\\log(T+1) \\label{eq.inequality_2},\n\\end{align}\nwhere the second inequality follows from the fact that\n$\\lambda_{m-1,j}\\ge \\lambda_{0,j}=1$ for any $m\\in [M]$.\nNow combining \\eqref{eq.reduction_eigenvalue}, \\eqref{eq.inequality_1} and \\eqref{eq.inequality_2} completes the proof. \n\\end{proof}\n\nWe are now ready to prove the regret upper bound stated in Theorem~\\ref{thm.adversarial}.\n\\begin{proof}[Proof of Statement 1 in Theorem~\\ref{thm.adversarial}]\n\\begin{enumerate}\n\\item[]\n\\item \\textbf{Regret bound under conditional independence assumption.}\n\nFor a given $\\delta > 0$, set the hyper-parameter $\\gamma$ in Algorithm~\\ref{algo.ucb} to be $1+\\sqrt{\\frac{1}{2}\\log(\\frac{2KT}{\\delta})}$ for the entire proof.\nUnder the conditional independence assumption in Lemma~\\ref{lemma.concentration},\nby a simple union bound over all $t \\in [T]$, we have with probability at least $1 - \\delta$, the following event holds:\n\\begin{align*}\n\\forall m \\in [M], \\forall t \\in [t_{m-1}+1, t_m], \\forall a\\in [K], \\quad |x_{t,a}^\\top (\\hat{\\theta}_{m-1} - \\theta^\\star)| \\le \\gamma\\sqrt{x_{t,a}^\\top A_{m-1}^{-1}x_{t,a}}.\n\\end{align*}\nOn this high probability event (with probability $1 - \\delta$), we can bound the regret as follows:\n\\begin{align}\nR_T(\\textbf{Alg}) &= \\sum_{t=1}^T \\left( \\max_{a\\in [K]}x_{t,a}^\\top \\theta^\\star  - x_{t,a_t}^\\top \\theta^\\star \\right)\n= \\sum_{m=1}^M \\sum_{t=t_{m-1}+1}^{t_m} \\left( \\max_{a\\in [K]}x_{t,a}^\\top \\theta^\\star  - x_{t,a_t}^\\top \\theta^\\star \\right) \\nonumber \\\\\n& \\le \\sum_{m=1}^M \\sum_{t=t_{m-1}+1}^{t_m} \\left( \\max_{a\\in [K]} \\Big(x_{t,a}^\\top \\hat{\\theta}_{m-1} +  \\gamma\\sqrt{x_{t,a}^\\top A_{m-1}^{-1}x_{t,a}}\\Big) - x_{t,a_t}^\\top \\theta^\\star \\right) \\nonumber \\\\\n& = \\sum_{m=1}^M \\sum_{t=t_{m-1}+1}^{t_m} \\left( x_{t,a_t}^\\top \\hat{\\theta}_{m-1} +  \\gamma\\sqrt{x_{t,a_t}^\\top A_{m-1}^{-1}x_{t,a_t}} - x_{t,a_t}^\\top \\theta^\\star \\right) \\nonumber \\\\\n& = \\sum_{m=1}^M \\sum_{t=t_{m-1}+1}^{t_m} \\left( x_{t,a_t}^\\top (\\hat{\\theta}_{m-1} - \\theta^\\star) +  \\gamma\\sqrt{x_{t,a_t}^\\top A_{m-1}^{-1}x_{t,a_t}}  \\right) \\nonumber\\\\\n& \\le  \n\\sum_{m=1}^M \\sum_{t=t_{m-1}+1}^{t_m} 2\\gamma\\sqrt{x_{t,a_t}^\\top A_{m-1}^{-1}x_{t,a_t}} = 2\\gamma\\cdot \\sum_{m=1}^M \\sum_{t=t_{m-1}+1}^{t_m} 1\\cdot \\sqrt{x_{t,a_t}^\\top A_{m-1}^{-1} x_{t,a_t}} \\nonumber \\\\ \\label{eq.regret_ucb}\n&\\le 2\\gamma\\sqrt{\\frac{T}{M}}\\cdot \\sum_{m=1}^M \\sqrt{\\sum_{t=t_{m-1} +1}^{t_m} x_{t,a_t}^\\top A_{m-1}^{-1} x_{t,a_t}} = \n 2\\gamma\\sqrt{\\frac{T}{M}}\\cdot \\sum_{m=1}^M \\sqrt{ \\mathsf{Tr}(A_{m-1}^{-1} X_m)}, \n\\end{align}\nwhere the inequality in~\\eqref{eq.regret_ucb} follows from Cauchy--Schwartz and the choice of a uniform grid (without loss of generality we assume that $T\/M$ is an integer).\n\nNext, setting $\\delta = \\frac{1}{T}$ (and hence resulting in $\\gamma = 1+\\sqrt{\\frac{1}{2}\\log\\left(2KT^2\\right)}$) and  applying Lemma~\\ref{lemma.trace_sum} to the upper bound in~\\eqref{eq.regret_ucb}, we immediately obtain that again on this high-probability event:\n\\begin{align}\nR_T(\\textbf{Alg}) &\\le  2\\sqrt{10}\\left(\\sqrt{\\frac{1}{2}\\log\\left(2KT^2\\right)}+1\\right)\\log(T+1)\\sqrt{\\frac{T}{M}}\\left(\\sqrt{Md} + d\\sqrt{\\frac{T}{M}} \\right) \\nonumber\\\\ \n&\\label{eq.x}= \\mathsf{polylog}(T)\\cdot (\\sqrt{dT} + \\frac{dT}{M}).\n\\end{align}\nConsequently, taking the expectation of $R_T(\\textbf{Alg})$ yields the same bound as in Equation~\\eqref{eq.x}, since with probability at most $\\frac{1}{T}$, the total regret over the entire horizon is at most $T$ (each time accumulates at most a regret of $1$ by the normalization assumption).\nSince the regret bound is independent of $\\theta^\\star$, it \nimmediately follows that \n$\\sup_{\\theta^\\star: \\|\\theta^\\star\\|_2\\le 1} \\mathbb{E}_{\\theta^\\star}[R_T(\\pi)]\n\\le \\mathsf{polylog}(T)\\cdot (\\sqrt{dT} + \\frac{dT}{M}).$ \n\n\n\\item \\textbf{Building a Master algorithm that satisfies conditional independence}\n\nTo complete the proof, we need to validate the conditional independence assumption in Lemma~\\ref{lemma.concentration}. Since the length of the confidence intervals does not depend on the random rewards, this task can be done by using a master algorithm SupSBUCB (Algorithm~\\ref{algo:SupSBUCB}), which runs in $O(\\log T)$ stages at each time step $t$ similar to \\cite{auer2002using}, which is subsequently adopted in the linear contextual bandits setting~\\cite{chu2011contextual} and then in the generalized linear contextual bandits setting~\\cite{li2017provably} for the same purpose of meeting the conditional independence assumption. \nNote that SupSBUCB is responsible for selecting the actions $a_t$ and it does so by calling BaseSBUCB (Algorithm~\\ref{algo:BaseSBUCB}), which merely performs regression.\nThis master-base algorithm pair has by now become a standard trick to get around the conditional dependency in the vanilla UCB algorithm for a variety of contextual bandits problems (by sacrificing at most $O(\\log T)$ regret).\n\n\\begin{algorithm}[!h]\n\t\\DontPrintSemicolon  \n\t\\SetAlgoLined\n\t\\BlankLine\n\t\\caption{SupSBUCB \\label{algo:SupSBUCB}}\n     \\textbf{Inputs}: $T, M \\in \\mathbb{Z}_{++}$, Grid ${\\mathcal{T}}=\\{t_1,t_2,\\cdots,t_M\\}$.\n     \n     $S\\leftarrow \\log(T), \\Psi_1^s \\leftarrow \\emptyset$ for all $s\\in[S]$\n     \n\t\\For{$m=1,2,\\cdots,M$}{\n\t\tInitialize $\\Psi_{m+1}^{s^{\\prime}}\\leftarrow \\Psi_{m}^{s^{\\prime}}$ for all $s^{\\prime} \\in [S]$.\n\t\t\n\t  \\For{$t= t_{m-1}+1, \\dots, t_m$}{\n\t    $s \\leftarrow 1$ and $\\hat{A}_1 \\leftarrow [K]$\n\t\t\n\t\t\\textbf{Repeat:}\n\t\t\n\n\t\t\n\t    Use BaseSBUCB with $\\Psi_{m}^s$ to compute $\\theta_m^s$ and $A_m^s$\n\t    \n\t    For all $a \\in \\hat{A}_s$, compute $w^s_{t,a} =\\gamma \\sqrt{x_{t,a}^T (A_{m}^s)^{-1}x_{t,a}}$, $\\hat{r}_{t,a}^s = \\langle \\theta_m^s, x_{t,a} \\rangle$ \n\t\t\n\t\t\\textbf{(a)} If $w^s_{t,a}\\leq 1\/\\sqrt{T}$ for all $a\\in \\hat{A}_s$,\n\t\tchoose $a_t = \\arg\\max_{a\\in \\hat{A}_s}\\left( \\hat{r}_{t,a}^s+w^s_{t,a}\\right)$. \n\t\t\n\t\t\n\t\t\\textbf{(b)} Else if $w^s_{t,a}\\leq 2^{-s}$ for all $a \\in \\hat{A}_s$,\n\t\t$\\hat{A}_{s+1}\\leftarrow \\{a\\in \\hat{A}_s \\,\\,\\vert\\,\\,\\hat{r}_{t,a}^s+w^s_{t,a}\\geq \\max_{a^{\\prime}\\in \\hat{A}_s}(\\hat{r}_{t,a^{\\prime}}^s+w^s_{t,a^{\\prime}})-2^{1-s}\\}$, \n\t\t\n\t\t\\quad $s \\leftarrow s+1$.\n\t\t\n\t\t\\textbf{(c)} Else choose any $a_t \\in \\hat{A}_{s}$ such that $w_{t,a_t}^s >2^{-s}$, Update\n\t    $\\Phi_{m+1}^{s} \\leftarrow\n\t\t\\Phi_{m+1}^{s} \\cup \\{t\\}.$\n\t\n\t\t\n\t\t\\textbf{Until}{\\quad an action $a_t$ is found.}\n\t}\n}\n\\end{algorithm}\n\n\\begin{algorithm}[!h]\n\t\\DontPrintSemicolon  \n\t\\SetAlgoLined\n\t\\BlankLine\n\t\\caption{BaseSBUCB \\label{algo:BaseSBUCB}}\n\t\n\t\t\\textbf{Input}: $\\Psi_m$.\n\t\n\t\t$A_m = I_d+\\sum_{\\tau \\in \\Psi_m}x_{t,a_{\\tau}}x^{\\prime}_{t,a_{\\tau}}$\n\t\t\n\t\t$c_m = \\sum_{\\tau \\in \\Psi_m} r_{\\tau,a_{\\tau}}x_{\\tau,a_{\\tau}}$\n\t\n\t\t$\\theta_m = A_m^{-1} c_m$\n\t\t\n\t\t\\textbf{Return} $(\\theta_m, A_m)$.\n\\end{algorithm}\n\n\n\nMore specifically, the master algorithm developed by~\\cite{auer2002finite} has the following structure: each time step is divided into at most $\\log T$ stages. At the beginning of each stage $s$, the learner computes the confidence interval using only the previous contexts designated as belonging to that stage and selects any action whose confidence interval has a large length (exceeding some threshold). If all actions has a small confidence interval, then we end this stage, observe the rewards of the given contexts and move on to the next stage with a smaller threshold on the length of the confidence interval. In other words, conditional independence is obtained by successive manual masking and revealing of certain information. One can intuitively think of each stage $s$ as a color, and each time step $t$ is colored using one of the $\\log T$ colors (if colored at all). When computing confidence intervals and performing regression, only previous contexts that have the same color are used, instead of all previous contexts. \n\nAdapting this algorithm to the sequential batch setting is not difficult: we merely keep track\nof the sets $\\Psi_{m}^s$ ($s \\in [\\log T]$) per each batch $m$ (rather than per each time step $t$ as in the fully online learning case). Note that we are still coloring each time step $t$, the difference here lies in the frequency at which we are running BaseSBUCB to compute the confidence bounds and rewards. \nDue to great similarity we omit the details here and refer to \\cite[Section 4.3]{auer2002using}. In particular, by establishing similar results to \\cite[Lemma 15, Lemma 16]{auer2002using}, it is straightforward to show that the regret of the master algorithm SupSBUCB here is enlarged at most by a multiplicative factor of $O(\\log T)$, which leads to the upper bound in Theorem \\ref{thm.adversarial}. \n\\end{enumerate}\n\\end{proof}\n\n\n\n\n\n\\subsection{Regret Analysis for Lower bound}\\label{subsec.adversarial}\nIn this section, we establish the regret lower bound and show that for any fixed grid ${\\mathcal{T}}=\\{t_1,\\cdots,t_M\\}$ and any learner's policy on this grid, there exists an adversary who can make the learner's regret at least $\\Omega(\\sqrt{Td}+(T\\sqrt{d}\/M \\wedge T\/\\sqrt{M}))$ even if $K=2$. Since the lower bound $\\Omega(\\sqrt{Td})$ has been proved in \\cite{chu2011contextual} even in the fully online case, it remains to show the lower bound $\\Omega(T\\sqrt{d}\/M \\wedge T\/\\sqrt{M})$. Note that in the fully online case, the lower bound $\\Omega(\\sqrt{Td})$ given in \\cite{chu2011contextual} is obtained under the same assumption $d^2 \\le T$ as in Assumption~\\ref{aspn.TKd}.\n\n\\begin{proof}[Proof of Statement 2 in Theorem~\\ref{thm.adversarial}]\nFirst we consider the case where $M\\ge d\/2$, and without loss of generality we may assume that $d'=d\/2$ is an integer (if $d$ is odd, then we can take $d^\\prime = \\frac{d-1}{2}$ and modify the subsequent procedure only slightly). By an averaging argument, there must be $d'$ batches $\\{i_1,i_2,\\cdots,i_{d'}\\}\\subset [M]$ such that\n\\begin{align}\\label{eq.large_batch}\n\\sum_{k=1}^{d'} \\left(t_{i_k} - t_{i_k-1} \\right) \\ge \\frac{d'T}{M}. \n\\end{align}\nNow $\\theta^*$ is chosen as follows: Flip $d'$ independent fair coins to obtain $U_1,\\cdots,U_{d'}\\in \\{1,2\\}$, and set $\\theta^\\star = (\\theta_1,\\cdots,\\theta_d)$ with\n$\\theta_{2k-1} = \\frac{1}{\\sqrt{d'}}\\mathbbm{1}(U_k = 1), \\theta_{2k} = \\frac{1}{\\sqrt{d'}}\\mathbbm{1}(U_k = 2), \\forall k\\in [d']$.\n(If $d$ is odd, then the last component $\\theta_d$ is set to $0$.)\n\n\nNote that $\\theta^\\star$ is a random variable and clearly $\\|\\theta^\\star\\|_2=1$ (surely). Next the contexts are generated in the following manner: for $t\\in (t_{m-1},t_{m}]$, if $m=i_k$ for some $k\\in [d']$, set $x_{t,1}=e_{2k-1}, x_{t,2}=e_{2k}$, where $e_j$ is the $j$-th basis vector in $\\mathbb{R}^d$; otherwise, set $x_{t,1}=x_{t,2}={\\bf 0}$. \n\nNow we analyze the regret of the learner under this environment. Clearly, for any $k\\in [d']$, the learner has no information about whether $(\\theta_{2k-1}, \\theta_{2k}) = (1\/\\sqrt{d'},0)$ or $(0,1\/\\sqrt{d'})$ before entering the $i_k$-th batch, while an incorrect action incurs an instantenous regret $1\/\\sqrt{d'}$. Consequently, averaged over all possible coin flips $(U_1,\\cdots,U_{d'})\\in \\{1,2\\}^{d'}$, the expected regret is at least:\n\\begin{align*}\n\\frac{1}{2}\\sum_{k=1}^{d'} \\frac{t_{i_k} - t_{i_{k-1}}}{\\sqrt{d'}} \\ge \\frac{1}{2\\sqrt{2}}\\cdot \\frac{T\\sqrt{d}}{M}\n\\end{align*}\ndue to \\eqref{eq.large_batch}, establishing the lower bound $\\Omega\\left(\\frac{T\\sqrt{d}}{M}\\right)$ when $M\\ge d\/2$.\n\nNext, in the case where $M < d\/2$, choose $d^\\prime = M$.\nHere, we obviously have $\\sum_{k=1}^{d'} \\left(t_{i_k} - t_{i_k-1} \\right) = T.$ \nIn this case, again flip $d'$ independent fair coins to obtain $U_1,\\cdots,U_{d'}\\in \\{1,2\\}$, and set $\\theta^\\star = (\\theta_1,\\cdots,\\theta_d)$ with\n$\\theta_{2k-1} = \\frac{1}{\\sqrt{d'}}\\mathbbm{1}(U_k = 1), \\theta_{2k} = \\frac{1}{\\sqrt{d'}}\\mathbbm{1}(U_k = 2), \\forall k\\in [d']$.\nSet all remaining components of $\\theta$ to $0$.\nThe contexts are generated as follows: for $t\\in (t_{m-1},t_{m}], 1\\le m \\le M$, set $x_{t,1}=e_{2m-1}, x_{t,2}=e_{2m}$.\nIn this case, we again average over all possible coin flips $(U_1,\\cdots,U_{d'})\\in \\{1,2\\}^{d'}$, and the expected regret is at least:\n\\begin{align*}\n\\frac{1}{2}\\sum_{m=1}^{M} \\frac{t_{m} - t_{m-1}}{\\sqrt{d'}} = \\frac{1}{2}\\cdot \\frac{T}{\\sqrt{M}}\n\\end{align*}\n\nCombining the above two cases yields a lower bound of  $ \\Omega\\left(\\frac{T\\sqrt{d}}{M}\\wedge \\frac{T}{\\sqrt{M}}\\right)$.\n\\end{proof}\n\n\\section{Definitions and Auxiliary Results}\\label{appendix.auxiliary}\n\n\\begin{definition}\nLet $(\\mathcal{X}, \\mathcal{F})$ be a measurable space and $P$, $Q$\nbe two probability measures on $(\\mathcal{X}, \\mathcal{F})$. \n\\begin{enumerate}\n\t\\item The total-variation distance between $P$ and $Q$ is defined as:\n\t$$ \\mathsf{TV}(P,Q) = \\sup_{A \\in \\mathcal{A}} |P(A) - Q(A)|.$$\n\t\\item The KL-divergence between $P$ and $Q$ is:\n\t\\begin{equation*}\n\tD_{\\text{\\rm KL}}(P\\|Q) = \\begin{cases}\n\t\\int \\log \\frac{dP}{dQ} dP \\text{\\quad if $P << Q$} \\\\\n\t+\\infty \\text{\\quad otherwise}\n\t\\end{cases}\n\t\\end{equation*}\n\\end{enumerate}\n\n\n\\end{definition}\n\\begin{lemma}\\cite[Lemma 2.6]{Tsybakov2008}\\label{lemma.TV_KL}\n\tLet $P$ and $Q$ be any two probability measures on the same measurable space. Then\n\t\\begin{align*}\n\t1- \\mathsf{TV}(P,Q) \\ge \\frac{1}{2}\\exp\\left(-D_{\\text{\\rm KL}}(P\\|Q)\\right). \n\t\\end{align*}\n\\end{lemma}\n\n\\begin{lemma}\\cite[Theorem 6.1]{wainwright2019high}\n\t\\label{lemma.wishart}\nLet $x_1,x_2,\\cdots,x_n\\sim {\\mathcal{N}}(0,I_d)$ be i.i.d. random vectors. Then for any $\\delta>0$, \n\\begin{align*}\n\\mathbb{P}\\left(\\sigma_{\\max}\\left(\\frac{1}{n}\\sum_{i=1}^n x_ix_i^\\top\\right) \\ge 1+\\sqrt{\\frac{d}{n}}+\\delta \\right) \\le \\exp\\left(-\\frac{n\\delta^2}{2}\\right),\n\\end{align*}\nwhere $\\sigma_{\\max}(A)$ denotes the largest singular value of $A$. \n\\end{lemma}\n\n\n\\section{Proof of Main Lemmas}\n\\subsection{Proof of Lemma \\ref{lemma.equator}}\nLet $y_{t,a} = \\Sigma^{-1\/2}x_{t,a}$, then each $y_{t,a}$ is marginally distributed as ${\\mathcal{N}}(0,I_d)$. Define\n\\begin{align*}\nB \\triangleq \\frac{1}{t_m-t_{m-1}}\\sum_{t=t_{m-1}+1}^{t_m} y_{t,a_t}y_{t,a_t}^\\top.\n\\end{align*}\n\nRecall that $a_t = \\arg\\max_{a\\in [K]} x_{t,a}^\\top \\hat{\\theta} = \\arg\\max_{a\\in [K]} y_{t,a}^\\top (\\Sigma^{1\/2}\\hat{\\theta})$ for any $t\\in [t_{m-1}+1,t_m]$, and $\\hat{\\theta}$ is an estimate of $\\theta^\\star$ that is independent of all contexts in the current batch $ [t_{m-1}+1,t_m]$. By rotational invariance of ${\\mathcal{N}}(0,I_d)$, we can without loss of generality assume $\\Sigma^{1\/2}\\hat{\\theta}=ce_d$ for some $c>0$. Consequently, each $y_{t,a_t}$ follows the distribution\n$\\mu_t = {\\mathcal{N}}(0,1) \\otimes \\cdots \\otimes {\\mathcal{N}}(0,1) \\otimes \\nu_t,$\nwhere $\\nu_t$ is the probability distribution of $\\max_{a\\in [K]} Z_{t,a}$, where each $Z_{t,a}$ is a standard Gaussian and the $Z_{t,a}$'s can be correlated across different $a$'s. \n\nNow for $y=(y_1,y_2,\\cdots,y_d)\\sim \\mu_t$ and any unit vector $u\\in \\mathbb{R}^d$, we show that there exist numerical constants $c_1,c_2>0$ independent of $(d,K)$ such that\n\\begin{align}\\label{eq.large_prob_fixed_u}\n\\mathbb{P}\\left(|y^\\top u| \\ge c_1\\right) \\ge c_2.\n\\end{align}\nTo establish \\eqref{eq.large_prob_fixed_u}, we distinguish into two cases. If $|u_d|<\\frac{1}{2}$, using the fact that $\\mathbb{P}(|{\\mathcal{N}}(0,1)+t|\\ge c)$ is minimized at $t=0$ for any fixed $c>0$, we conclude that\n\\begin{align*}\n\\mathbb{P}\\left(|y^\\top u| \\ge c_1 \\right) \\ge \\mathbb{P}\\left( \\left| \\sum_{i=1}^{d-1}y_iu_i \\right| \\ge c_1 \\right) = \\mathbb{P}(|{\\mathcal{N}}(0,1-u_d^2)|\\ge c_1) \\ge \\mathbb{P}\\left(\\left|{\\mathcal{N}}(0,\\frac{3}{4})\\right|\\ge c_1\\right)\n\\end{align*}\nis lower bounded by some positive constant. If $|u_d|\\ge \\frac{1}{2}$, we have\n\\begin{align*}\n\\mathbb{P}\\left(|y^\\top u| \\ge c_1 \\right)\\ge \\frac{1}{2}\\mathbb{P}\\left(|u_dy_d| \\ge c_1 \\right) \\ge \\frac{1}{2}\\mathbb{P}\\left(|y_d| \\ge 2c_1 \\right) \\ge \\frac{1}{2}\\mathbb{P}\\left(Z_{t,1}\\ge 2c_1\\right) = \\frac{1}{2}\\mathbb{P}({\\mathcal{N}}(0,1)\\ge 2c_1),\n\\end{align*}\nwhich is again lower bounded by a numerical constant. Hence the proof of \\eqref{eq.large_prob_fixed_u} is completed. \n\nBased on \\eqref{eq.large_prob_fixed_u} and the deterministic inequality\n\\begin{align*}\nu^\\top\\cdot \\left(\\frac{1}{t_m-t_{m-1}}\\sum_{t=t_{m-1}+1}^{t_m} y_{t,a_t}y_{t,a_t}^\\top\\right)\\cdot u \\ge \\frac{c_1^2}{t_m-t_{m-1}}\\sum_{t=t_{m-1}+1}^{t_m} \\mathbbm{1}\\left(|y_{t,a_t}^\\top u| \\ge c_1 \\right),\n\\end{align*}\nthe Chernoff inequality yields that for any unit vector $u\\in \\mathbb{R}^d$, we have\n\\begin{align}\\label{eq.concentration_fixed_u}\n\\mathbb{P}\\left( u^\\top B u \\ge \\frac{c_1^2c_2}{2}\\right) \\ge 1 - e^{-c_3(t_m-t_{m-1})},\n\\end{align}\nwhere $c_3>0$ is some numerical constant. \n\nNext we prove an upper bound of $\\lambda_{\\max}(B)$, i.e., the largest eigenvalue of $B$. Since $(a+b)(a+b)^\\top \\preceq 2(aa^\\top + bb^\\top)$ for any vectors $a,b\\in\\mathbb{R}^d$, for $y_t\\sim \\mu_t$ we have\n\\begin{align*}\ny_ty_t^\\top \\preceq 2(v_tv_t^\\top + w_tw_t^\\top), \n\\end{align*}\nwhere $v_t=(v_{t,1},\\cdots,v_{t,d-1},0)$ with $v_{t,i}\\sim{\\mathcal{N}}(0,1)$, and $w_t=(0,\\cdots,0,w_{t,d})$ with $w_{t,d}\\sim \\nu_t$. By concentration of Wishart matrices (cf. Lemma \\ref{lemma.wishart}), with probability at least $1-e^{-\\Omega(t_m-t_{m-1})}$, \n\\begin{align*}\n\\lambda_{\\max}\\left(\\frac{1}{t_m-t_{m-1}}\\sum_{t=t_{m-1}+1}^{t_m} v_tv_t^\\top \\right) \\le c_4\n\\end{align*}\nholds for some numerical constant $c_4>0$. For the second term, since $w_{t,d}\\sim \\nu_t$ is the maximum of $K$ arbitrary ${\\mathcal{N}}(0,1)$ random variables, the Gaussian tail and the union bound imply that $|w_{t,d}|\\le \\sqrt{c_5\\log(KT)}$ with probability at least $1-O(T^{-5})$. Hence, with probability at least $1 - O(T^{-4})$, we have\n\\begin{align*}\n\\lambda_{\\max}\\left(\\frac{1}{t_m-t_{m-1}}\\sum_{t=t_{m-1}+1}^{t_m} w_tw_t^\\top \\right) = \\frac{1}{t_m-t_{m-1}}\\sum_{t=t_{m-1}+1}^{t_m} w_{t,d}^2 \\le c_5\\log (KT). \n\\end{align*}\nCombining all the previous results, and using $\\lambda_{\\max}(A+B)\\le \\lambda_{\\max}(A)+\\lambda_{\\max}(B)$ for symmetric matrices $A,B$, we conclude that with probability at least $1-e^{-\\Omega(t_m-t_{m-1})} - O(T^{-4})$, we have\n\\begin{align}\\label{eq.lambda_max}\n\\lambda_{\\max}(B) \\le c_6\\log (KT)\n\\end{align}\nholds for some numerical constant $c_6>0$. \n\nFinally, we are ready to prove a lower bound on $\\lambda_{\\min}(B)$ via an $\\varepsilon$-net argument. Let ${\\mathcal{N}}_d(\\varepsilon)$ be an $\\varepsilon$-net of the unit ball in $\\mathbb{R}^d$ (both in $\\ell_2$ norm) with cardinality at most $(1+\\frac{2}{\\varepsilon})^d$. Standard $\\varepsilon$-net techniques (cf. \\cite[Section 2.3.1]{tao2012topics}) give\n\\begin{align*}\n\\min_{u: \\|u\\|_2=1} u^\\top Bu \\ge \\min_{u\\in {\\mathcal{N}}_d(\\varepsilon)} u^\\top Bu - 2\\varepsilon\\lambda_{\\max}(B).\n\\end{align*}\nHence, choosing $\\varepsilon = \\frac{c_1^2c_2}{8c_6\\log (KT)}$ and combining \\eqref{eq.concentration_fixed_u}, \\eqref{eq.lambda_max} and the union bound over ${\\mathcal{N}}_d(\\varepsilon)$ gives\n\\begin{align*}\n\\mathbb{P}\\left(\\lambda_{\\min}(B) \\ge \\frac{c_1^2c_2}{4}\\right) \\ge 1 - e^{O(d\\log\\log (KT)) - \\Omega(t_m-t_{m-1})} - O(T^{-4}). \n\\end{align*}\nBy noting that $t_m - t_{m-1} = \\Omega(d\\sqrt{T})$ due to the choice of the grid in \\eqref{eq.minimax_grid}, the parameter $a$ in \\eqref{eq.a}, and the assumption $M=O(\\log\\log T)$, we conclude that $\\lambda_{\\min}(B) \\ge c_7$ for some numerical constant $c_7>0$ with probability at least $1 - O(T^{-4})$. The proof is completed by noting that\n$$\n\\frac{1}{t_m - t_{m-1}}\\sum_{t = t_{m-1}+1}^{t_m} x_{t,a_t}x_{t,a_t}^\\top = \\Sigma^{1\/2} B \\Sigma^{1\/2} \\succeq \\Sigma^{1\/2} (c_7I_d) \\Sigma^{1\/2}= c_7\\Sigma\n$$\nwhenever $\\lambda_{\\min}(B) \\ge c_7$ and the assumption $\\lambda_{\\min}(\\Sigma)\\ge \\kappa\/d$. \n\\qed\n\n\\subsection{Proof of Lemma \\ref{lemma.Q}}\nLet $v_1,\\cdots,v_d$ be an orthonormal basis of $\\mathbb{R}^d$ with $v_1=u_t$. By rotational invariance of the uniform distribution on spheres, we have $(v_1^\\top \\theta, v_2^\\top \\theta, \\cdots, v_d^\\top \\theta)\\sim \\mathsf{Unif}(\\Delta\\mathbb{S}^{d-1})$ under $Q_0$. Now recall that\n\\begin{align*}\n\\frac{dQ_1}{dQ_0}(\\theta) = \\frac{r_t\\jiao{v_1,\\theta}_+}{Z_0}, \\qquad \\frac{dQ_2}{dQ_0}(\\theta) = \\frac{r_t\\jiao{v_1,\\theta}_-}{Z_0}, \n\\end{align*}\nwe conclude that if $\\theta' = \\theta - 2(v_1^\\top \\theta)v_1$, we have\n\\begin{align*}\n\\frac{dQ_1}{dQ_0}(\\theta) = \\frac{dQ_2}{dQ_0}(\\theta'). \n\\end{align*}\nAs a result, it is equivalent to have $\\theta\\sim Q_1$ or $\\theta' = \\theta - 2(v_1^\\top \\theta)v_1\\sim Q_2$. \n\nFor the identity \\eqref{eq.Z_0}, recall that the density of $\\theta=(\\theta_1,\\cdots,\\theta_d)\\sim \\mathsf{Unif}(\\Delta\\mathbb{S}^{d-1})$ is \n\\begin{align}\\label{eq.uniform_density}\nf(\\theta) = f(\\theta_2,\\cdots,\\theta_d) = \\left(\\frac{d\\pi^{d\/2}\\Delta^{d-1}}{\\Gamma(\\frac{d}{2}+1)}\\right)^{-1}\\frac{2\\Delta}{\\sqrt{\\Delta^2 - \\theta_2^2 - \\cdots - \\theta_d^2}}\\cdot \\mathbbm{1}\\left(\\sum_{i=2}^d \\theta_i^2 \\le \\Delta^2\\right),  \n\\end{align}\nwhere $\\Gamma(t)=\\int_0^\\infty x^{t-1}e^{-x}dx$ is the Gamma function. Hence, by rotational invariance, we have\n\\begin{align*}\nZ_0 &= \\frac{r_t}{2}\\mathbb{E}_{Q_0}[|\\theta_1|] = r_t\\Delta\\cdot \\int_{\\sum_{i=2}^d \\theta_i^2 \\le \\Delta^2} \\left(\\frac{d\\pi^{d\/2}\\Delta^{d-1}}{\\Gamma(\\frac{d}{2}+1)}\\right)^{-1} d\\theta_2\\cdots d\\theta_d \\\\\n&= r_t\\Delta\\left(\\frac{d\\pi^{d\/2}\\Delta^{d-1}}{\\Gamma(\\frac{d}{2}+1)}\\right)^{-1}\\cdot \\frac{\\Delta^{d-1} \\pi^{\\frac{d-1}{2}}}{\\Gamma(\\frac{d-1}{2}+1)} = r_t\\Delta\\cdot\\begin{cases}\n\\frac{2^d}{\\pi d}\\binom{d}{d\/2}^{-1}, & \\text{if }d\\text{ is even} \\\\\n\\frac{1}{2^d}\\binom{d-1}{(d-1)\/2}, & \\text{if }d\\text{ is odd}\n\\end{cases}. \n\\end{align*}\n\nUsing Stirling's approximation $\\sqrt{2\\pi n}(\\frac{n}{e})^n\\le n!\\le e\\sqrt{n}(\\frac{n}{e})^{n}$ for any $n\\ge 1$, we have\n\\begin{align}\\label{eq.combinatorics}\n\\frac{2}{e^2}\\sqrt{\\frac{\\pi}{n}}\\le \\frac{1}{2^{2n}}\\binom{2n}{n} \\le \\frac{e}{\\pi\\sqrt{2n}}\n\\end{align}\nfor all $n\\ge 1$, and the rest of \\eqref{eq.Z_0} follows from \\eqref{eq.combinatorics}. \n\nAs for the second moment in \\eqref{eq.second_moment}, we use the spherical coordinates \n\\begin{align*}\n\\left\\{\n\\begin{array}{lr}\n\\theta_2 = r\\cos\\varphi_1, \\\\\n\\theta_3 = r\\sin\\varphi_1\\cos\\varphi_2, \\\\\n\\vdots  \\\\\n\\theta_{d-1} = r\\sin\\varphi_1\\sin\\varphi_2\\cdots\\sin\\varphi_{d-3}\\cos\\varphi_{d-2},  \\\\\n\\theta_d = r\\sin\\varphi_1\\sin\\varphi_2\\cdots\\sin\\varphi_{d-3}\\sin\\varphi_{d-2}.\n\\end{array}\n\\right.\n\\end{align*}\nto obtain\n\\begin{align*}\n\\mathbb{E}_{Q_1}[(v_1^\\top \\theta)^2] &= \\frac{r_t}{2Z_0}\\cdot \\mathbb{E}_{Q_0}[|\\theta_1|^3] \\\\\n&= \\frac{r_t\\Delta}{Z_0}\\cdot \\int_{\\sum_{i=2}^d \\theta_i^2 \\le \\Delta^2} \\left(\\frac{d\\pi^{d\/2}\\Delta^{d-1}}{\\Gamma(\\frac{d}{2}+1)}\\right)^{-1} (\\Delta^2-\\theta_2^2-\\cdots-\\theta_d^2) d\\theta_2\\cdots d\\theta_d \\\\\n&= \\frac{r_t\\Delta}{Z_0}\\cdot \\int_0^\\Delta \\int_0^\\pi \\cdots \\int_0^\\pi \\int_0^{2\\pi} \\left(\\frac{d\\pi^{d\/2}\\Delta^{d-1}}{\\Gamma(\\frac{d}{2}+1)}\\right)^{-1} (\\Delta^2-r^2) \\\\\n&\\qquad \\cdot r^{d-2}\\sin^{d-3}(\\varphi_1) \\sin^{d-4}(\\varphi_2)\\cdots \\sin(\\varphi_{d-3}) drd\\varphi_1\\cdots d\\varphi_{d-2} \\\\\n&= \\frac{r_t\\Delta}{Z_0}\\left(\\frac{d\\pi^{d\/2}\\Delta^{d-1}}{\\Gamma(\\frac{d}{2}+1)}\\right)^{-1}\\cdot \\frac{2\\Delta^{d+1}}{d^2-1}\\cdot \\frac{\\Gamma(\\frac{d-2}{2})\\Gamma(\\frac{1}{2})}{\\Gamma(\\frac{d-1}{2})}\\cdot \\frac{\\Gamma(\\frac{d-3}{2})\\Gamma(\\frac{1}{2})}{\\Gamma(\\frac{d-2}{2})}\\cdot \\cdots \\cdot \\frac{\\Gamma(1)\\Gamma(\\frac{1}{2})}{\\Gamma(\\frac{3}{2})}\\cdot 2\\pi \\\\\n&= \\frac{r_t\\Delta}{Z_0}\\left(\\frac{d\\pi^{d\/2}\\Delta^{d-1}}{\\Gamma(\\frac{d}{2}+1)}\\right)^{-1}\\cdot \\frac{2\\Delta^{d+1}}{d^2-1}\\cdot\\frac{2\\pi^{\\frac{d-1}{2}}}{\\Gamma(\\frac{d-1}{2})} \\\\\n&= \\frac{2\\Delta^2}{d+1}. \n\\end{align*}\\qed\n\n\\section{Problem Formulation}\n\\label{sec:problem_formulation}\nWe introduce the problem of sequential batch learning on finite-action linear contextual bandits.\n\n\\subsection{Notation}\nWe start by fixing some notation that will be used throughout the paper. For a positive integer $n$, let $[n]\\triangleq\\{1,\\cdots,n\\}$. For real numbers $a,b$, let $a\\wedge b\\triangleq \\min\\{a,b\\}$. For a vector $v$, let $v^\\top$ and $\\|v\\|_2$ be the transpose and $\\ell_2$ norm of $v$, respectively. For square matrices $A,B$, let $\\mathsf{Tr}(A)$ be the trace of $A$, and let $A\\preceq B$ denote that the difference $B-A$ is symmetric and positive semi-definite. We adopt the standard asymptotic notations: for two non-negative sequences $\\{a_n\\}$ and $\\{b_n\\}$, let $a_n=O(b_n)$ iff $\\limsup_{n\\to\\infty} a_n\/b_n<\\infty$, $a_n=\\Omega(b_n)$ iff $b_n=O(a_n)$, and $a_n=\\Theta(b_n)$ iff $a_n=O(b_n)$ and $b_n=O(a_n)$. We also write $\\tilde{O}(\\cdot), \\tilde{\\Omega}(\\cdot)$ and $\\tilde{\\Theta}(\\cdot)$ to denote the respective meanings within multiplicative logarithmic factors in $n$. For probability measures $P$ and $Q$, let $P\\otimes Q$ be the product measure with marginals $P$ and $Q$. If measures $P$ and $Q$ are defined on the same probability space, we denote by $\\mathsf{TV}(P,Q) = \\frac{1}{2}\\int |dP-dQ| $ and $ D_{\\text{KL}}(P\\|Q) = \\int dP\\log\\frac{dP}{dQ}$ the total variation distance and Kullback--Leibler (KL) divergences between $P$ and $Q$, respectively. \n\n\n\\subsection{Decision Procedure and Reward Structures}\nLet $T$ be the time horizon of the problem. At the beginning of each time $t\\in [T]$, the decision maker observes a set of $K$ $d$-dimensional feature vectors (i.e. contexts) $\\{x_{t,a} \\mid a \\in [K]\\} \\subseteq \\mathbb{R}^d$ corresponding to the $t$-th unit.\nIf the decision maker selects action $a \\in [K]$, then a reward $r_{t,a} \\in \\mathbb{R}$ corresponding to time $t$ is incurred (although not necessarily immediately observed).\nWe assume the mean reward is linear: that is, there exists an underlying (but unknown) parameter\n$\\theta^\\star$ such that \n $$r_{t,a} =  x_{t,a}^\\top \\theta^\\star + \\xi_t,$$\n where $\\{\\xi_t\\}_{t=0}^{\\infty}$ is a sequence of zero-mean independent sub-Gaussian random variables with a uniform upper bound on the sub-Gaussian constants. Without loss of generality and for notational simplicity,\n we assume each $\\xi_t$ is $1$-sub-Gaussian:  $\\mathbf{E}[e^{\\lambda \\xi_t}] \\le e^{\\lambda^2\/2}, \\forall t, \\forall \\lambda \\in \\mathbb{R}$.\n Further, without loss of generality (via normalization), we assume $\\|\\theta^\\star\\|_2\\le 1$.\n We denote by $a_t$ and $r_{t, a_t}$ the action chosen and the reward obtained at time $t$, respectively. \n Note that both are random variables; in particular, $a_t$ is random either because the action is randomly selected based on the contexts $\\{x_{t,a} \\mid a \\in [K]\\}$ or because the contexts $\\{x_{t,a} \\mid a \\in [K]\\}$\n are random, or both.\n\nAs there are different (but equivalent) formulations of contextual bandits, we briefly discuss the meaning of the above abstract quantities and how they arise in practice. In general, at each round $t$, an individual characterized by $v_t$ (a list of characteristics associated with that individual) becomes available.\nWhen the decision maker decides to apply action $a_t$  to this individual,\n a reward $y_t(v_t, a_t)$, which depends (stochastically) on both $v_t$ and $a_t$, is obtained. In practice, for both modelling and computational reasons, one often first featurizes the individual characteristics and the actions.\nIn particular, with sufficient generality, one assumes  $\\mathbf{E}[y_t(v_t, a_t) \\mid v_t, a_t] = g_{\\theta} (\\phi(v_t, a_t))$, \nwhere $g_{\\theta}(\\cdot)$ is the parametrized mean reward function and $\\phi(v_t, a_t)$ extracts the features from the given raw individual characteristics $v_t$ and action $a_t$. In the above formulation,\nas is standard in the literature, we assume the feature map $\\phi(\\cdot)$ is known and given and $x_{t,a} = \\phi(v_t, a)$. Consequently, we directly assume access to contexts  $\\{x_{t,a} \\mid a \\in [K]\\}$.\nNote that the linear contextual bandits setting then corresponds to $g_{\\theta}(\\cdot)$ is linear.\n\n\n\\subsection{Sequential Batch Learning}\nIn the standard online learning setting, the decision maker immediately observes the reward $r_{t, a_t}$ after selecting action $a_t$ at time $t$. Consequently, in selecting $a_t$, the decision maker can base his decision on all the past contexts $\\{x_{\\tau,a} \\mid a \\in [K],  \\tau\\le t\\}$ and all the past rewards $\\{r_{\\tau,a_\\tau} \\mid \\tau\\le t-1\\}$. \n\nIn constrast, we consider a \\textit{sequential batch learning} setting, where the decision maker is only allowed to partition the $T$ units into (at most) $M$ batches, and the reward corresponding to each unit in a batch can only be observed at the end of the batch. More specifically, given a maximum batch size $M$, the decision maker needs to choose a sequential batch learning algorithm \\textbf{Alg} that has the following two components:\n\\begin{enumerate}\n\t\\item A \\emph{grid} ${\\mathcal{T}}=\\{t_1,t_2,\\cdots,t_M\\}$, with $0 = t_0 < t_1<t_2<\\cdots<t_M=T$.\n\tIntuitively, this grid partitions the $T$ units into $M$ batches: the $k$-th batch contains units $t_{k-1}+1$ to $t_k$. Note that without loss of generality, the decision maker will always choose a grid of $M$ batches (despite being allowed to choose less than $M$ batches) since choosing less than $M$ batches will only decrease the amount of information available and hence yields worse performance. More formally, for any sequential batch learning algorithm that uses a grid of size less than $M$, there exists another sequential batch learning algorithm that uses a grid of size $M$ whose performance is no worse. \n\t\n\t\\item A sequential batch policy $\\pi = (\\pi_1, \\pi_2, \\dots, \\pi_T)$ such that each $\\pi_t$ can only use reward information from all the prior batches (as well as all the contexts that can be observed up to $t$). More specifically, for any $t \\in [T]$, define the batched history $\\mathcal{H}^t = \\{x_{\\tau,a} \\mid a \\in [K]\\}_{\\tau=1}^t \\cup \\{r_{\\tau, a_\\tau}\\}_{\\tau=1}^{j(t) - 1}$, where $j(t)$ is the unique integer satisfying\n\t$t_{j(t)-1}<t\\le t_{j(t)}$. Intuitively, there are $j(t) - 1$ batches prior to unit $t$: only the rewards of those batches have been observed. With this definition, a sequential batch policy is any policy $\\pi$ such that\n    each $\\pi_t$ is adapted to $\\mathcal{H}^t$ for each $t \\in [T]$.\n\\end{enumerate}\n\n\\begin{remark}\n$M = T$ yields the standard online learning setting, where the decision maker need not select a grid. Consequently, the sequential batch learning setting has a more complex decision space--one that entails selecting both the grid and the policy. \n\\end{remark}\n\n\n\n\n\\subsection{Performance Metric: Regret}\nTo assess the performance of a given sequential batch learning algorithm \\textbf{Alg}, we compare the cumulative reward obtained by \\textbf{Alg} to the cumulative reward obtained by an \\textbf{optimal} policy, if the decision maker had the prescient knowledge of the optimal action for each unit (i.e. an oracle that knows $\\theta^\\star$). This is formalized by the following definition of regret.\n\n\\begin{definition}\\label{def:regret}\nLet \\textbf{Alg} $= ({\\mathcal{T}}, \\pi)$ be a sequential batch learning algorithm.\nThe regret of \\textbf{Alg} is:\n\\begin{align}\\label{eq.regret}\nR_T(\\textbf{Alg}) \\triangleq \\sum_{t=1}^T \\left( \\max_{a\\in [K]}x_{t,a}^\\top \\theta^\\star  - x_{t,a_t}^\\top \\theta^\\star \\right),\n\\end{align}\nwhere $a_1, a_2, \\dots, a_T$ are actions generated by \\textbf{Alg} in the online decision making process. \t\n\\end{definition}\t\n\n\\begin{remark}\nAlthough the form of regret defined here is the same as that in standard online learning, the goal here is much more ambitious, because batches induce delays in obtaining reward feedback, and hence the decision maker cannot immediately incorporate the feedback into his subsequent decision making process.\nNevertheless, we still make the performance comparison to the oracle that is used in the standard online learning setting.   \t\nFurther, note that $R_T(\\textbf{Alg})$ is a random variable since as mentioned earlier,  $a_t$ is random.\nFor simplicity, we will mostly focus on bounding the expected regret, where the expectation is taken with respect to all sources of randomness (to be made precise later in various settings). However, high-probability regret bounds can also be obtained in our setting, and we will discuss them in relevant places throughout the paper. \n\\end{remark}\t\n\n\\subsection{Adversarial Contexts v.s. Stochastic Contexts}\nThe regret defined in Definition~\\ref{def:regret} is also a function of all the contexts that arrive over a horizon of $T$.\nIn the bandits literature, depending on how these contexts are generated, there are two main categories.\nThe first category is adversarial contexts, where at each $t$, an adversary can choose the contexts $\\{x_{t,a} \\mid a \\in [K]\\}$ that will be revealed to the decision maker. The second category is stochastic contexts, where at each $t$, $\\{x_{t,a} \\mid a \\in [K]\\}$ is \\textbf{iid} drawn from some fixed underlying distribution.\n\nIn the standard online learning case (i.e., $M=T$), the optimal regret bounds under adversarial or stochastic contexts are both $\\tilde{\\Theta}(\\sqrt{dT})$ (see~\\cite{chu2011contextual} and Theorem \\ref{thm.stochastic} below). However, it turns out that in the sequential batch setting, there is a sharp difference between the adversarial contexts case and the stochastic contexts case. \nIn particular, it turns out that the power of the adversary in choosing the contexts has a profound impact on what regret bounds can be obtained. Consequently, we divide our presentation into these two distinct regimes: we study the adversarial contexts case in Section~\\ref{sec:adversarial} and the stochastic contexts case in Section~\\ref{sec:stochastic}.\nIn each of these two settings, we give upper and lower bounds for regret.\nFinally, throughout the paper, we consider the standard low-dimensional contextual bandits regime where the action set is not too large, as made precise by the following assumption.\n\\begin{assumption}\\label{aspn.TKd}\n$K=O(\\mathsf{poly}(d))$ and $T\\ge d^2$.\n\\end{assumption}\nThat is, we are assuming that the number of actions cannot be too large (at most polynomial in the dimension of the context), for there is a phase transition on the regret when $K$ can be exponential in $d$ (see~\\cite{abe2003reinforcement}). Further, the time horizon must be large enough, otherwise one cannot even estimate the true parameter $\\theta^\\star$ within a constant $\\ell_2$ risk at the end of the time horizon $T$. \nEquivalently, we are in the low-dimensional contextual bandits setting, where the number of covariates is small compared to the number of samples we receive over the entire horizon.\n\\section{Introduction}\n\\label{sec:intro}\n\nWith the rapid advances of digitization of the economy, massive amounts of user-specific data have become increasingly available. Among its varied implications, one that holds center-stage importance is the advent of the new era of data-driven personalized decision making: equipped with such user-specific data, decision makers across a wide variety of domains are now able to personalize the service decisions based on individuals' characteristics, thereby improving the outcomes. Fundamentally, such improved outcomes are achieved by intelligently exploring and exploiting the heterogenity in a given population, which manifests itself as different individuals responding differently to the same treatments\/recommendations\/actions.\nSuch heterogenity is ubiquitous across a variety of applications, including \nmedical treatment selection in clinical trials, product recommendation in marketing, \norder provisioning in inventory management,\nhospital  staffing in operation rooms, ads selection in online advertising~\\cite{bertsimas2007learning, LCLS2010, kim2011battle, he2012timing, chapelle2014, chen2015statistical, bastani2015online, SBF2017, ferreira2018online, ban2019big}. \n\n\nSituated in this broader context and rising to materialize the value from personalization, contextual bandits have emerged to be the predominant mathematical framework that is at once rich and elegant. Its three modelling cores, contexts, actions, and rewards (representing individual features, recommendations and outcomes respectively), capture the salient aspects of personalized decision making and provide fertile ground for developing algorithms that contribute to making quality decisions at the fine-grained individual level.\n\nAs such, in the past decade, dedicated efforts have been devoted to this area, yielding a flourishing line of literature. Broadly speaking, the existing contextual bandits literature falls into two main categories: offline contextual bandits and online contextual bandits. \nIn offline contextual bandits \\cite{langford2011doubly, zhang2012estimating,zhao2012estimating, zhao2014doubly, JMLR:v16:swaminathan15a, rakhlin2016bistro, athey2017efficient, kitagawa2018should, kallus2018confounding, zhou2018offline, deep-learning-logged-bandit-feedback}, the decision maker is given a full batch dataset that has been collected from historical observations.\nThe decision maker aims to learn from this dataset an effective policy (i.e. a function that maps contexts to decisions) that will be deployed in the future and that (hopefully) yields good performance.\nAs such, in offline contextual bandits,  the decision maker is not allowed to perform active learning: selecting a policy in this setting is one-shot and the decision maker is solely concerned with identifying the best policy using available data.\n\nOn the other hand, in online contextual bandits, the decision maker actively interacts with the data-collection process: as data arrive sequentially, the decision maker can adapt his decisions based on what has been observed in the past, thereby deciding what data is collected. Typically, in such settings, a decision is made on the current individual based on all the past feedback, yielding an outcome that is immediately observed and incorporated to make the next decision. A rich literature has studied this setting--\\cite{LCLS2010,rusmevichientong2010linearly,FCGS2010, rigollet2010nonparametric, chu2011contextual, goldenshluger2013linear, AG2013a, AG2013b, RV2014, russo2016information, JBNW2017, LLZ2017,abeille2017linear,dimakopoulou2017estimation,li2017provably}, for a highly incomplete list-- and has developed online contextual bandits algorithms \n(most notably UCB-based algorithms and Thompson sampling based algorithms) \nthat effectively balance exploration with exploitation trade-off, a key challenge therein.\nSee further \\cite{BCN2012, lattimore2018bandit,slivkins2019introduction} for three articulate surveys that more systematically describe the field.\n\n\nHowever, in practice, it is often the case that neither setting provides a close approximation of the underlying reality. Specifically, that the decision maker can only perform one-shot policy learning--the key setting in offline contextual bandits--is simply too restrictive and pessimistic for almost all applications. On the other hand, assuming that the decision maker can constantly observe and incorporate feedback at a  per-individual scale--the key setting online contextual bandits--is also a over-simplification and too optimistic for many applications. In fact, reality often stands somewhere in between: decision makers across many applications are typically able to perform active learning and incorporate feedback from the past to adapt their decisions in the future; however, due to the physical and cost constraints, such adaptation is often limited to a fixed number of rounds of interaction.\nFor instance, in clinical trials~\\cite{robbins1952some,thompson1933likelihood,kim2011battle}, each trial involves applying medical treatments to a group of patients, where the medical outcomes are observed and collected for the entire group at the end of the trial.\nThe data collected from previous trials are then analyzed to design the medical treatment selection schemes for the next trial.\nNote that as the medical information from previous trials are incorporated to inform\ntreatment selection in future trials, medical decision makers do have the \n have the flexibility in adaptive learning. However, such flexibility is limited since in practice, only a limited number of trials (e.g. $3$) can be conducted, far less than the number of patients in the trials, hence rendering the standard online learning models inapplicable here. \n Another example where adaptive learning is possible but with batch constraints is product recommendation in marketing~\\cite{bertsimas2007learning, schwartz2017customer}.\n In this case, the marketer sends out product offers to a batch of customers at once when running a promotions campaign. Customers' feedback will then be batch collected at the end and analyzed to design the next round of promotions in the targeted population. Other examples include crowdsourcing~\\cite{kittur2008crowdsourcing} and simulations~\\cite{chick2009economic}, where both cases exhibit the characteristic that a single-run experiment consists of a batch of individuals.\n \n \\subsection{Related Work}\n \n Motivated by these considerations, we study the problem of sequential batch learning in linear contextual bandits in this paper, where a decision maker can adaptively learn to make decisions subject to the batch constraints that, as described above, commonly occur in practice.  \n This batch constrained bandits problem have recently been studied in the simple multi-armed bandits (MAB) case. In particular, \\cite{perchet2016batched} studied sequential batch learning in the 2-armed MAB problem,\n where they have given a successive elimination algorithm during each batch and established that \n $O(\\log \\log T)$ batches are needed in order to achieve the same regret bound as in the standard online learning setting.  Very recently, \\cite{gao2019batched} generalized the results to the $K$-armed bandit setting (despite the seeming simplicity, the generalization is not easy) and obtained a tight $\\Theta(\\log \\log T)$ result therein even when the batch sizes can be chosen adaptively. \n \n However, these initial efforts on MAB settings, despite providing interesting insights, cannot capture\n individuals' characteristics: in MABs, decisions can only be made at a population level (i.e. the decision maker aims to select an action that is the best for the entire population), rather than personalized at an individual level, which severly limits its practical applicability. In this paper, we fill in this gap by providing\n a comprehensive inquiry into sequential batch learning in linear contextual bandits with a finite number of actions, a cannonical setting where decisions are provisioned based on the individual features. \n Our goal is then to delineate, in this more general setting, how the batch constraints impact the performance of adaptive decision making and characterize in depth the fundamental limits therein, thereby shedding light on how practical adaptive decision making can be most efficiently done in practice.\n \n \nOur work is also related to but distinct from learning on bandits (either contextual or MABs) with delayed feedback~\\cite{NAGS2010, DHKKLRZ2011, JGS2013, QK2015, GST2016, bis2019, zhou2019learning}: since the decision maker is not able to observe rewards in the interim of a batch (as they only come at the batch's end), feedback are delayed from the decision maker's perspective.\n However, a key difference exists between learning in bandits and our sequential batch learning mdoel: the former setting works with exogenous delays--drawn either from some stochastic process or from some arbitrary sequence--that, completely contrary to the latter setting, is neither influenced nor known by the decision maker. The literature on learning in bandits with delayed feedback then develops adapted algorithms in the presence of delays and study how regret bounds scale as a function of the underlying delays. On the other hand, feedback delays in sequential batch learning are endogenous and arise as a consequence of the batch constraints; in particular, the decision maker in this setting is at full discretion of choosing the batch sizes which in turn determines how the feedback for each individual is delayed. Consequently, the frameworks provided by the learning-in-bandits-with-delays literature--both the aglorithms and analyses--are not applicable here. Of course, it should also be pointed out that\n when viewed through the lens of learning with delayed feedback, the delays in our setting exhibit a particular structure: if a batch has size $B$, then the reward for the first item in the batch is delayed by $M-1$ time units, the reward for the second item in the batch is delayed by $M-2$ time units, and so on, and the reward for the last item in the batch has no delays.\nConsequently our results here also do not directly imply regret bounds in that literature.\n\n \n\\subsection{Our Contributions}\n\nWe study the sequential batch learning problem in linear contextual bandits when the number of actions is finite (and not too large, to be made precise later). Our main contributions are twofold.\n\nFirst, we consider the adversarial contexts setting, where the contexts can be generated arbitrarily or even adversarially. In this setting, we provide a UCB-style algorithm that is adapted to the sequential batch setting and achieves $\\mathsf{polylog}(T)\\cdot (\\sqrt{dT} + dT\/M)$ expected regret,\nwhere $d$ is the dimension of the context, $M$ is the number of allowed batches and $T$ is the learning horizon. This regret bound highlights that in the adversarial context setting, $\\Theta(\\sqrt{dT})$ batches--rather than $T$ batches--are sufficient to achieve the $\\tilde{\\Theta}(\\sqrt{dT})$ regret bound that is minimax optimal in the standard online learning setting. Further, we characterize the fundamental limits of learning in this setting by establishing a $c\\cdot (\\sqrt{dT} + \\min \\{ T\\sqrt{d}\/M, T\/\\sqrt{M}\\})$ lower bound for the expected regret, where $c$ is some universal constant. \nThis regret lower bound highlights that at least $O(\\sqrt{T})$ batches are needed for any algorithm to achieve the optimal performance for standard online learning (when no batch constraints are present). Consequently, our regret bound is tight up to at most a factor of $O(\\sqrt{d})$ (and other polylog factors). In the common low-dimensional regime (i.e. $d$ considered a constant when compared to $T$), our regret bounds provide a complete characterization and indicates the proposed algorithm is optimal.\n\nSecond, we consider the stochastic contexts setting, where the contexts are generated \\textit{iid} from a Gaussian distribution. In this case, we reveal an interesting phenomenon that stands in sharp constrast to the adversarial contexts setting: a simple pure-exploitation algorithm alone can achieve the $\\tilde{\\Theta}(\\sqrt{dT})$ regret bound (which, as mentioned above, is minimax optimal in standard online learning) using only $O(\\log\\log(T\/d^2))$ matches, far less than what is required in the adversarial contexts setting. More specifically, we establish a $\\mathsf{polylog}(T)\\cdot \\sqrt{dT}(T\/d^2)^{{1}\/{[2(2^M-1)]}}$ upper bound \nand a $c\\cdot \\sqrt{dT}({T}\/{d^2})^{{1}\/{[2(2^M-1)]}}$ lower bound for the expected regret, respectively.  Consequently, up to polylog factors, our regret bound is minimax optimal, indicating that\n$O(\\log\\log ({T}\/{d^2}))$ batches are also necessary to achieve the optimal performance of the standard online learning setting.\n\nTo further appreciate this result, and to get some intuition into why such bounds are possible, consider the special case where $M = 3$ (not uncommon in a typical clinical trial) and the context distribution follows $\\mathcal{N}(0, {\\mathbf{I}_d}\/{d})$ (dividing the identity matrix by $d$ simply ensures that the norm of each context is bounded by $1$ in mean square). \nIn this case, our bound indicates that the optimal performance is $\\tilde{O}(d^{{5}\/{14}} T^{{4}\/{7}})$, which is already quite close to $O(\\sqrt{dT})$; further,\npure exploitation--selecting the best action given our current estimate--on each batch\nis able to achieve this regret bound.\nWhy?\nTo get a rough sense, let's allocate the $T$ units into three batches in the following way: the first batch contains $O(d^{{6}\/{7}} T^{{4}\/{7}})$ units, the second batch contains $O(d^{{2}\/{7}} T^{{6}\/{7}})$ units and the third batch contains the rest.\nNow, for the moment, let's assume that the contexts selected over time are \\textit{iid}: of course they are not  \\textit{iid}, because how one context is chosen at time $t$ depends critically on how contexts were chosen in the previous times (otherwise, there is no learning that occurs); but for simplicity, let's assume they are. \nThen the regret incurred on the first batch--since we haven't observed anything and hence know nothing--is $\\tilde{O}({d^{{6}\/{7}} T^{{4}\/{7}}}\/{\\sqrt{d}}) = \\tilde{O}(d^{{5}\/{14}} T^{{4}\/{7}})$, since each unit incurs $\\tilde{O}(1\/\\sqrt{d})$ regret (this is a consequence of the normalization as well as a separate but simple analysis of instantaneous regret).\nNow after observing the results from the first batch, and running a least squares on that, we now have an estimate of the underlying parameters that achieves a certain level of accuracy. \nHow accurate is our estimate now? \nFrom standard theory on linear regression (which says that if each covariate sample is \\textit{iid} drawn from a standard multi-variate Guassian and there are $n$ samples, then with high probability $\\|\\hat{\\theta} -\\theta^*\\|_2 = O(\\sqrt{d\/n})$), we can deduce that after observing $O(d^{{6}\/{7}} T^{{4}\/{7}})$ samples in the first batch, we would be able to achieve the following estimation accuracy:\n $\\|\\hat{\\theta} -\\theta^*\\|_2 = O_p(\\sqrt{d} \\sqrt{{d}\/{(d^{{6}\/{7}} T^{{4}\/{7}})}}  ) = O_p(d^{4\/7}T^{-2\/7})$, where the extra $\\sqrt{d}$ factor is again a result of our normalization on the covariance matrix.\n\nNow, a more accurate such estimate on $\\theta^*$ would yield smaller regret for each individual unit,\nand if each individual's regret is inversely proportional to $\\|\\hat{\\theta} -\\theta^*\\|_2$--as it turns out to be--then the total regret for the second batch is the number of units in that batch times each individual regret, yielding $\\tilde{O}(1\/\\sqrt{d}) \\times O( d^{2\/7}T^{6\/7}\/(d^{-4\/7}T^{2\/7})) = \\tilde{O}(d^{{5}\/{14}} T^{{4}\/{7}})$, where the $\\tilde{O}(1\/\\sqrt{d})$ factor is the proportionality constant that, as before, comes from the normalization factor on the covariance matrix.\nFinally, after observing all the $O(d^{2\/7} T^{6\/7})$ units in the second batch, \nour estimation accuracy would further improve to \n$\\|\\hat{\\theta} -\\theta^*\\|_2 = O_p(\\sqrt{d} \\sqrt{d\/(d^{2\/7}T^{6\/7})}  ) = O_p(d^{5\/7}T^{-3\/7})$.\nConsequently, since there are $O(T)$ units in the third batch, the total regret\nin this batch would be--by applying the same line of reasoning as in the second batch--$\\tilde{O}(1\/\\sqrt{d}) \\times O(T\/(d^{-6\/7}T^{3\/7})) = \\tilde{O}(d^{5\/14}T^{4\/7})$.\nAs such, aggregating over all three batches, we obtain $\\tilde{O}(d^{5\/14}T^{4\/7})$ total regret.\nIt then remains to complete the reasoning by showing that the selected contexts form a well-conditioned matrix--at least well-conditioned enough to ensure the same estimation rate in standard linear regression applies--despite being selected in a non-\\textit{iid} way to maximize the rewards, as it turns out to be true.\n\n \nAdditionally, we also give gap-dependent regret bounds--both upper and lower bounds--in the stochastic contexts setting (note that there is no notion of gap when the contexts can be arbitrary). These bounds are typically sharper compared to their gap-independent counterparts (to which the above-mentioned regret bounds belong) when the gap is large; in particular, the dependence on $T$ would\nbe logarithmic rather than $\\sqrt{T}$. Section~\\ref{sec:gap} provides a more detailed discussion on this, including how the pure-exploitation algorithm should be (slightly) modified in this case to achieve the minimax optimal gap-dependent regret bound.\nFinally, we mention that our analyses easily yield high-probability regret bounds as well for all settings, although for simplicity, we have chosen only to present bounds on expected regret.\n\n\n\n\\section{Learning with Stochastic Contexts}\n\\label{sec:stochastic}\n\nIn this section, we focus on the stochastic contexts case,\n where at each time $t\\in [T]$, each context $x_{t,a}$ is drawn\n from  ${\\mathcal{N}}(0,\\Sigma)$,  with a possibly unknown covariance matrix $\\Sigma$. \n Note that for each $t$, $x_{t,a}$'s can be arbitrarily corrleated across different $a$'s. \n This is a simple setting that presents\n an interesting case for study: at a population level, each one of \n the $K$ actions is equally good; in particular, if the decision maker\n is not allowed to personalize the action based on the context and hence restricted to choose a single-action policy (i.e. always choose action $1$ or action $2$ no matter what the contexts are), then all the actions perform equally well. However, as we shall see, being able\n to select different actions based on the realized contexts allows\n the decision maker to do much more. We start by making an assumption\n on the covariance matrix.\n \n \\begin{assumption}\\label{assumption:cov}\n \tThe covariance matrix $\\Sigma$ satisfies\n \t$\n \t\\frac{\\kappa}{d} \\le \\lambda_{\\min}(\\Sigma) \\le \\lambda_{\\max}(\\Sigma) \\le \\frac{1}{d}\n \t$\n \tfor some numerical constant $\\kappa>0$, where $\\lambda_{\\min}(\\Sigma), \\lambda_{\\max}(\\Sigma)$ denote the smallest and the largest eigenvalues of $\\Sigma$, respectively. \n \\end{assumption}\n \nThe upper bound $ \\lambda_{\\max}(\\Sigma) \\le 1\/d$ in Assumption \\ref{assumption:cov} ensures that $\\mathbb{E}\\|x_{t,a}\\|_2^2\\le 1$, and therefore the stochastic contexts share the similar constraint with the previous adversarial contexts. The lower bound $\\lambda_{\\min}(\\Sigma) \\ge \\kappa\/d$ ensures that each stochastic context is approximately distributed as an isotropic Gaussian random vector, with a bounded condition number no less than $\\kappa^{-1}$. We assume that $\\kappa>0$ is a fixed constant (say $0.1$) and will not optimize the dependence on $\\kappa$.  \n \nThe next theorem presents tight regret bounds for the stochastic contexts case.\n\n\\begin{theorem}\\label{thm.stochastic}\n\tLet $T$, $M=O(\\log\\log T)$ and $d$ be the learning horizon, number of batches and each context's dimension, respectively. Denote by $\\mathsf{polylog}(T)$ all the poly-logarithmic factors in $T$.\n\t\\begin{enumerate}\n\t\\item \n\tUnder Assumptions \\ref{aspn.TKd} and \\ref{assumption:cov}, there exists a sequential batch learning algorithm \\textbf{Alg}= $({\\mathcal{T}}, \\pi)$ (explicitly defined in Section \\ref{subsec.pure-exp}) such that:\n\t\\begin{align*}\n\t\\sup_{\\theta^\\star: \\|\\theta^\\star\\|_2\\le 1} \\mathbb{E}_{\\theta^\\star}[R_T(\\pi)] \\le \\mathsf{polylog}(T)\\cdot \\sqrt{\\frac{dT}{\\kappa}}\\left(\\frac{T}{d^2}\\right)^{\\frac{1}{2(2^M-1)}}.\n\t\\end{align*}\n\t\\item\n\tConversely, even when $K=2$ and contexts $x_{t,a}\\sim {\\mathcal{N}}(0,I_d\/d)$ are independent over all $a\\in [K], t\\in [T]$, for any $M\\le T$ and any sequential batch learning algorithm, we have:\n\t\\begin{align*}\n\t\\sup_{\\theta^\\star: \\|\\theta^\\star\\|_2\\le 1} \\mathbb{E}_{\\theta^\\star}[R_T(\\pi)] \\ge c\\cdot \\sqrt{dT}\\left(\\frac{T}{d^2}\\right)^{\\frac{1}{2(2^M-1)}},\n\t\\end{align*}\n\twhere $c>0$ is a numerical constant independent of $(T,M,d)$. \n\\end{enumerate}\n\\end{theorem}\n\nTheorem \\ref{thm.stochastic} completely characterizes the minimax regret for the sequential batch learning problem in linear contextual bandits with stochastic contexts, and shows a doubly exponential dependence of the optimal regret on the number of batches $M$. The following corollary is immediate. \n\\begin{corollary}\\label{cor.stochastic}\n\tUnder stochastic contexts, it is necessary and sufficient to have $\\Theta(\\log\\log (T\/d^2))$ batches to achieve the fully online regret $\\tilde{\\Theta}(\\sqrt{dT})$. \n\\end{corollary}\n\nIn contrast to Corollary \\ref{cor.adversarial}, the above corollary shows that a much smaller number of batches are capable of achieving the fully online performance, which suits better for many practical scenarios. Note that for smaller number of batches, Theorem \\ref{thm.stochastic} also gives the tight regrets within logarithmic factors, e.g., the optimal regret is $\\tilde{\\Theta}(Td^{-1\/2})$ when $M=1$, is $\\tilde{\\Theta}(T^{2\/3}d^{1\/6})$ when $M=2$, is $\\tilde{\\Theta}(T^{4\/7}d^{5\/14})$ when $M=3$, and so on. \n\n\n\\subsection{A Sequential Batch Pure-Exploitation Algorithm}\\label{subsec.pure-exp}\n\nIn contrast to the adversarial contexts, under stochastic contexts the decision maker enjoys the advantage that he can choose to learn the unknown parameter $\\theta^\\star$ from any desired direction. In other words, the exploration of the learner is no longer subject to the adversary's restrictions, and strikingly, making decisions based on the best possible inference of $\\theta^\\star$ is already sufficient.\n\n\\begin{algorithm}[h!]\n\t\\DontPrintSemicolon  \n\t\\SetAlgoLined\n\t\\BlankLine\n\t\\caption{Sequential Batch Pure-exploitation\t\\label{algo.pure-exp}}\n\t\\textbf{Input:} Time horizon $T$; context dimension $d$; number of batches $M$. \\\\\n\t\\textbf{Set} $a = \\Theta\\left( \\sqrt{T}\\cdot \\left(\\frac{T}{d^2}\\right)^{\\frac{1}{2(2^M-1)}} \\right)$\\\\\n\t\\textbf{Grid choice}: ${\\mathcal{T}} = \\{t_1,\\cdots,t_M\\}$, with $t_1 = ad, \\quad t_m = \\lfloor a\\sqrt{t_{m-1}} \\rfloor, m=2,3,\\cdots,M,.$\\\\\n\t\\textbf{Initialization:} $A = {\\bf 0}\\in \\mathbb{R}^{d\\times d}$, $\\hat{\\theta}={\\bf 0}\\in \\mathbb{R}^d$\\;\n\t\\For{$m \\gets 1$ \\KwTo $M$}{\n\n\t\t\t\\For{$t\\gets t_{m-1}+1$ \\KwTo $t_m$}{\n\t\t\t\tchoose $a_t = \\arg\\max_{a\\in [K]} x_{t,a}^\\top \\hat{\\theta}$ (break ties arbitrarily). \\\\\n\t\t\t\treceive reward $r_{t,a_t}$. \n\t\t\t}\n\t\t}\n\t\t$A\\gets A + \\sum_{t=t_{m-1}+1}^{t_m} x_{t,a_t}x_{t,a_t}^\\top$. \\\\\n\t\t$\\hat{\\theta} \\gets A^{-1}\\sum_{t=t_{m-1}+1}^{t_m} r_{t,a_t}x_{t,a_t}$.\n\\end{algorithm}\n\nThe algorithm we use in this setting is quite simple (see Algorithm~\\ref{algo.pure-exp}). Specifically, under a particularly chosen grid ${\\mathcal{T}}=\\{t_1,t_2,\\cdots,t_M\\}$,  the learner, at the beginning of each batch, uses the least squares estimate $\\hat{\\theta}$ of $\\theta^\\star$ based on the data in the previous batches, and then simply selects the action $a\\in [K]$ which maximizes the estimated reward $x_{t,a}^\\top \\hat{\\theta}$ for any time $t$ in this batch. Then at the end of each batch, the learner updates his estimate $\\hat{\\theta}$ of $\\theta^\\star$ based on the new observations from the current batch. \n\nHow do we select the grid ${\\mathcal{T}}$? Intuitively, in order to minimize overall regret, we must ensure that the regret incurred on each batch is not too large, because the overall regret is dominated by the batch that has the largest regret. Guided by this observation, we can see intuitively an optimal way of selecting the grid must ensure that each batch's regret is the same (at least orderwise in terms of the dependence of $T$ and $d$): for otherwise, there is a way of reducing the regret order in one batch and increasing the regret order in the other and the sum of the two will still have smaller regret order than before (which is dominated by the batch that has larger regret order). As we shall see later, the following grid choice satisfies this equal-regret-across-batches requirement:\n\\begin{align}\\label{eq.minimax_grid}\nt_1 = ad, \\quad t_m = \\lfloor a\\sqrt{t_{m-1}} \\rfloor, \\qquad m=2,3,\\cdots,M,\n\\end{align}\nwhere the parameter $a = \\Theta\\left( \\sqrt{T}\\cdot \\left(\\frac{T}{d^2}\\right)^{\\frac{1}{2(2^M-1)}} \\right)$ is chosen so that $t_M=T$. \n\n\n\n\n\\subsection{Regret Analysis for Upper bound}\\label{subsec.stochastic_upperbound}\nWe now turn to establishing the upper bound in Theorem \\ref{thm.stochastic}. \nWe again execute a two-step program. First, we prove that Algorithm \\ref{algo.pure-exp} with the grid ${\\mathcal{T}}=\\{t_1,\\cdots,t_M\\}$ in \\eqref{eq.minimax_grid} attains the regret upper bound in Theorem \\ref{thm.stochastic}, assuming the conditional independence assumption (cf. Lemma \\ref{lemma.difference}) holds. Second, similar to the master algorithm in the previous section, we then modify Algorithm \\ref{algo.pure-exp} slightly to validate this condition. One thing to note here is that, unlike in the adversarial contexts case, here the modification is much simpler, as we shall see later.\n\nWe start by establishing that the least squares estimator $\\hat{\\theta}$ is close to the true parameter $\\theta^\\star$ at the beginning of every batch with high probability. By the theory of least squares, this would be obvious if the chosen contexts $x_{t,a_t}$ were i.i.d. Gaussian. However, since the action $a_t$ depends on all contexts $(x_{t,a})_{a\\in [K]}$ available at time $t$, the probability distribution of $x_{t,a_t}$ may be far from isotropic. Consequently, a priori, there might be one or more directions in the context space that were never chosen, hence yielding inaccurate estimation of $\\theta^\\star$ along that (or those) direction(s). However, as we shall see next, this is not a concern: we establish that the matrix formed by the selected contexts are reasonbly well-conditioned, despite being selected in a greedy fashion.\n\n\n\\begin{lemma}\\label{lemma.equator}\n\tFor each $m\\in [M]$, with probability at least $1-O(T^{-4})$ we have\n\t\\begin{align*}\n\t\\lambda_{\\min}\\left(\\sum_{t=t_{m-1}+1}^{t_m} x_{t,a_t}x_{t,a_t}^\\top \\right) \\ge c\\cdot \\frac{\\kappa(t_m-t_{m-1})}{d},\n\t\\end{align*}\n\twhere $c>0$ is a numerical constant independent of $(K,T,d,m,\\kappa)$. \n\\end{lemma}\n\nThe proof of the above lemma is a bit long and hence deferred to the appendix. Based on Lemma \\ref{lemma.equator}, we are ready to show that the least squares estimator $\\hat{\\theta}$ is close to the true parameter $\\theta^\\star$ with high probability. For $m\\in [M]$, let $\\hat{\\theta}_m$ be the estimate at the end of $m$-th batch, and $A_m = \\sum_{t=1}^{t_m} x_{t,a_t}x_{t,a_t}^\\top$ be the regression matrix. \n\\begin{lemma}\\label{lemma.difference}\n\tFor each $m\\in [M]$, if the rewards $\\{r_{t,a_t}\\}_{t\\in [t_m]}$ up to time $t_m$ are mutually independent given the selected contexts $\\{x_{t,a_t}\\}_{t\\in [t_m]}$, then with probability at least $1-O(T^{-3})$,\n\t\\begin{align*}\n\t\\|\\hat{\\theta}_m - \\theta^\\star\\|_2 \\le Cd\\cdot \\sqrt{\\frac{\\log T}{\\kappa t_m}}\n\t\\end{align*}\n\tfor a numerical constant $C>0$ independent of $(K,T,d,m,\\kappa)$. \n\\end{lemma}\n\\begin{proof}{Proof.}\n\tBy the standard algebra of linear regression, we have:\n\t\\begin{align*}\n\t\\hat{\\theta}_m - \\theta^\\star = A_m^{-1}\\sum_{t=1}^{t_m}x_{t,a_t} (r_{t,a_t} - x_{t,a_t}^\\top \\theta^\\star). \n\t\\end{align*}\n\tHence, conditioned on the contexts $\\{x_{t,a_t}\\}_{t\\in [t_m]}$, the noise terms $r_{t,a_t} - x_{t,a_t}^\\top \\theta^\\star$ are independent by the assumption, and each noise term $r_{t,a_t} - x_{t,a_t}^\\top \\theta^\\star$ is $1$-sub-Gaussian.\n\t\n\tNext, we show that the random vector $\\hat{\\theta}_m - \\theta^\\star$ is $\\sigma^2$-sub-Gaussian conditioned on the contexts with $\\sigma^2 = \\lambda_{\\min}(A_m)^{-1}$.  To see this, we start by recalling that a centered (i.e. zero-mean) random vector $V$ is $v$-sub-Gaussian if the scalar random variable $\\langle V, u\\rangle$ is $v$-sub-Guassian for any unit vector $u$.\n\tConsequently, take any unit vector $u \\in \\mathbf{R}^d$, we have:\n\t$$\\langle\t\\hat{\\theta}_m - \\theta^\\star , u \\rangle = \\langle A_m^{-1}\\sum_{t=1}^{t_m}x_{t,a_t} (r_{t,a_t} - x_{t,a_t}^\\top \\theta^\\star), u\\rangle = \\sum_{t=1}^{t_m} u^T A_m^{-1}x_{t,a_t} (r_{t,a_t} - x_{t,a_t}^\\top \\theta^\\star).$$\n\tSince each term in the summand is $(u^T A_m^{-1}x_{t,a_t})^2$-sub-Gaussian, and since all of them are independent (after being conditioned on  $\\{x_{t,a_t}\\}_{t\\in [t_m]}$), their sum is also sub-Gaussian with the sub-Gaussian constant  equal to the sum of the sub-Guassian constants:\n\t\\begin{align*}\n\t&\\sum_{t=1}^{t_m}  (u^T A_m^{-1}x_{t,a_t})^2= \\sum_{t=1}^{t_m}  u^T A_m^{-1}x_{t,a_t} x_{t,a_t}^T A_m^{-1} u\n\t= u^T A_m^{-1}\\big( \\sum_{t=1}^{t_m}  x_{t,a_t} x_{t,a_t}^T \\big) A_m^{-1} u \\\\\n\t&= u^T A_m^{-1}A_m A_m^{-1} u =  u^T A_m^{-1} u \\le \\lambda_{\\max}(A_m^{-1}) = \\lambda_{\\min}(A_m)^{-1}. \n\t\\end{align*}\n\tSince the above inequality holds for any unit vector $u$, choosing $\\sigma^2 = \\lambda_{\\min}(A_m)^{-1}$\n\testablishes the claim.\n\t\n\t\n\tProceeding further, by Lemma \\ref{lemma.equator}, we have for each $m\\in [M]$, with probability at least $1-O(T^{-4})$ \n\t$\\lambda_{\\min}\\left(\\sum_{t=t_{m-1}+1}^{t_m} x_{t,a_t}x_{t,a_t}^\\top \\right) \\ge c\\cdot \\frac{\\kappa(t_m-t_{m-1})}{d}$. Consequently, by a union bound over all $M$ (which is at most $T$),\n\twe have with probability at least $1-O(T^{-3})$, $\\lambda_{\\min}\\left(\\sum_{t=t_{m-1}+1}^{t_m} x_{t,a_t}x_{t,a_t}^\\top \\right) \\ge c\\cdot \\frac{\\kappa(t_m-t_{m-1})}{d}$ for all $m \\in [M]$.\n\tSince $\\lambda_{\\min}(X+Y)\\ge \\lambda_{\\min}(X)+\\lambda_{\\min}(Y)$ for any symmetric matrices $X,Y$, \n\tit then follows that with probability at least $1-O(T^{-3})$:\n\t\\begin{align*}\n\t\\lambda_{\\min}(A_m) = \\lambda_{\\min}\\left(\\sum_{l=1}^m \\sum_{t=t_{l-1}+1}^{t_l} x_{t,a_t}x_{t,a_t}^\\top \\right) \\ge \\sum_{l=1}^m \\lambda_{\\min}\\left(\\sum_{t=t_{l-1}+1}^{t_l} x_{t,a_t}x_{t,a_t}^\\top \\right) \\ge \\frac{c\\kappa t_m}{d}.\n\t\\end{align*}\n\t\n\tFinally, since $\\hat{\\theta}_m - \\theta^\\star$ is a $\\frac{d}{c\\kappa t_m}$-sub-Gaussian random vector, $\\|\\hat{\\theta}_m - \\theta^\\star\\|_2^2 $\n\tis a sub-exponential random variable.\n\tTherefore, conditioned on the above event for the stochastic contexts, the sub-exponential concentration gives the claimed upper bound on $\\|\\hat{\\theta}_m - \\theta^\\star\\|_2$ with a further probability at least $1 - O(T^{-3})$ over the random noises. Finally, taking a union bound to complete the proof. \n\\end{proof}\n\nLemma~\\ref{lemma.difference} shows that given the conditional independence assumption, the estimator $\\hat{\\theta}$ given by pure exploitation essentially achieves the rate-optimal estimation of $\\theta^\\star$ even if one purely explores. This now positions us well to prove the upper bound of Theorem \\ref{thm.stochastic}. Of course, bear in mind that when using Algorithm~\\ref{algo.pure-exp}, the conditional independence assumption does not hold, for the choice of future contexts depends on the rewards in the previous batches. Therefore, we will use sample splitting to build another master algorithm to gain independence at the cost of the sample size reduction by a multiplicative factor of $M$ (recall that $M = O(\\log\\log T)$). The following proof implements these two steps; note that in this setting, the master algorithm is entirely different from and much simpler than the one given in the adversarial case.\n\n\\begin{proof}[Proof of Statement 1 in Theorem~\\ref{thm.stochastic}]\n\\begin{enumerate}\n\\item[]\n\\item \\textbf{Regret bound under conditional independence assumption.}\n\n Consider the $m$-th batch with any $m\\ge 2$, and any time point $t$ inside this batch. By the definition of $a_t$, we have $x_{t,a_t}^\\top \\hat{\\theta}_{m-1}\\ge x_{t,a}^\\top \\hat{\\theta}_{m-1}$ for any $a\\in [K]$. Consequently, \n\\begin{align*}\n\\max_{a\\in [K]} (x_{t,a} - x_{t,a_t})^\\top \\theta^\\star &\\le \\max_{a\\in [K]} (x_{t,a} - x_{t,a_t})^\\top (\\theta^\\star - \\hat{\\theta}_{m-1}) \\\\\n&\\le \\max_{a,a'\\in [K]} (x_{t,a} - x_{t,a'})^\\top (\\theta^\\star - \\hat{\\theta}_{m-1}) \\\\\n&\\le 2\\max_{a\\in [K]} |x_{t,a}^\\top (\\theta^\\star - \\hat{\\theta}_{m-1})|. \n\\end{align*}\nFor fixed $a\\in [K]$, marginally we have $x_{t,a}\\sim {\\mathcal{N}}(0,\\Sigma)$ independent of $\\hat{\\theta}_{m-1}$. Therefore, conditioning on the previous contexts and rewards, we have $x_{t,a}^\\top (\\theta^\\star - \\hat{\\theta}_{m-1})\\sim {\\mathcal{N}}(0,\\sigma^2)$ with\n$$\n\\sigma^2 = (\\theta^\\star - \\hat{\\theta}_{m-1})^\\top \\Sigma (\\theta^\\star - \\hat{\\theta}_{m-1}) \\le \\frac{\\|\\theta^\\star - \\hat{\\theta}_{m-1}\\|_2^2}{d}\n$$\nby Assumption \\ref{assumption:cov}. By a union bound over $a\\in [K]$, with probability at least $1-O(T^{-3})$ over the randomness in the current batch we have\n\\begin{align*}\n\\max_{a\\in [K]} (x_{t,a} - x_{t,a_t})^\\top \\theta^\\star \\le  2\\max_{a\\in [K]} |x_{t,a}^\\top (\\theta^\\star - \\hat{\\theta}_{m-1})| = O\\left(\\|\\theta^\\star - \\hat{\\theta}_{m-1} \\|_2 \\cdot \\sqrt{\\frac{\\log(KT)}{d}}\\right). \n\\end{align*}\nApplying Lemma \\ref{lemma.difference} and another union bound, there exists some numerical constant $C'>0$ such that with probability at least $1-O(T^{-3})$, the instanteous regret at time $t$ is at most\n\\begin{align*}\n\\max_{a\\in [K]} (x_{t,a} - x_{t,a_t})^\\top \\theta^\\star \\le C'\\sqrt{\\log(KT)\\log T}\\cdot \\sqrt{\\frac{d}{\\kappa t_{m-1}}}. \n\\end{align*}\nNow taking the union bound over $t\\in [T]$, the total regret incurred after the first batch is at most\n\\begin{align}\\label{eq.later_batch}\n\\sum_{m=2}^M C'\\sqrt{\\log(KT)\\log T}\\cdot t_m\\sqrt{\\frac{d}{\\kappa t_{m-1}}} \\le C'\\sqrt{\\frac{\\log(KT)\\log T}{\\kappa}}M\\cdot a\\sqrt{d}\n\\end{align}\nwith probability at least $1-O(T^{-2})$, where the inequality is due to the choice of the grid in \\eqref{eq.minimax_grid}. \n\nAs for the first batch, the instanteous regret at any time point $t$ is at most the maximum of $K$ Gaussian random variables ${\\mathcal{N}}(0,(\\theta^\\star)^\\top \\Sigma \\theta^\\star)$. Since $\\|\\theta^\\star\\|_2\\le 1$ and $\\lambda_{\\max}(\\Sigma)\\le 1\/d$, we conclude that the instanteous regret is at most $C''\\sqrt{\\log(KT)\/d}$ for some constant $C''>0$ with probability at least $1-O(T^{-3})$. Now by a union bound over $t\\in [t_1]$, with probability at least $1-O(T^{-2})$ the total regret in the first batch is at most\n\\begin{align}\\label{eq.first_batch}\nC''\\sqrt{\\log(KT)\/d}\\cdot t_1 = C''\\sqrt{\\log(KT)}\\cdot a\\sqrt{d}. \n\\end{align}\n\nNow combining \\eqref{eq.later_batch}, \\eqref{eq.first_batch} and the choice of $a$ in Algorithm~\\ref{algo.pure-exp} gives the desired regret bound in Theorem \\ref{thm.stochastic} with high probability (note that $M=O(\\log\\log T)$), and consequently in expectation. \n\n\\item \\textbf{Building a Master algorithm that satisfies conditional independence}\n\\begin{algorithm}[h!]\n\t\\DontPrintSemicolon  \n\t\\SetAlgoLined\n\t\\BlankLine\n\t\\caption{Batched Pure-exploitation (with sample splitting)\t\\label{algo.sample_splitting}}\n\t\\textbf{Input:} Time horizon $T$; context dimension $d$; number of batches $M$; grid ${\\mathcal{T}} = \\{t_1,\\cdots,t_M\\}$ same as in Algorithm~\\ref{algo.pure-exp}.\\;\n\t\\textbf{Initialization:} Partition each batch into $M$ intervals evenly, i.e., $(t_m,t_{m+1}]=\\cup_{j=1}^M T_m^{(j)}$. \\;\n\t\\For{$m \\gets 1$ \\KwTo $M$}{\n\t\t\\If{$m=1$}{\n\t\t\tchoose $a_t = 1$ and receives reward $r_{t,a_t}$ for any $t\\in [1,t_1]$. \n\t\t}\n\t\t\\Else{\n\t\t\t\\For{$t\\gets t_{m-1}+1$ \\KwTo $t_m$}{\n\t\t\t\tchoose $a_t = \\arg\\max_{a\\in [K]} x_{t,a}^\\top \\hat{\\theta}_{m-1}$ (break ties arbitrarily). \\\\\n\t\t\t\treceive reward $r_{t,a_t}$. \n\t\t\t}\n\t\t}\n\t\t$T^{(m)} \\gets \\cup_{m'=1}^m T_{m'}^{(m)}.$\\\\\n\t\t$A_m\\gets \\sum_{t\\in T^{(m)}} x_{t,a_t}x_{t,a_t}^\\top$. \\\\\n\t\t$\\hat{\\theta}_m \\gets A_m^{-1}\\sum_{t\\in T^{(m)}} r_{t,a_t}x_{t,a_t}$.\n\t}\n\t\\textbf{Output: resulting policy $\\pi=(a_1,\\cdots,a_T)$}.\n\\end{algorithm}\n\nWe start by proposing a sample splitting based master algorithm (see Algorithm~\\ref{algo.sample_splitting}) that ensures that when restricting to the subset of observations used for constructing $\\hat{\\theta}$, the rewards are conditionally independent given the contexts. \nThe key modification in Algorithm \\ref{algo.sample_splitting} lies in the computation of the estimator $\\hat{\\theta}_{m}$ after the first $m$ batches. Specifically, instead of using all past contexts and rewards before $t_m$, we only use the past observations inside the time frame $T^{(m)}\\subsetneq [t_m]$ to construct the estimator. The key property of the time frames is the disjointness, i.e., $T^{(1)},\\cdots,T^{(M)}$ are pairwise disjoint. Then the following lemma shows that the conditional independence condition holds within each time frame $T^{(m)}$. \n\n\n\\begin{lemma}\\label{lemma.cond_indep}\n\tFor each $m\\in [M]$, the rewards $\\{r_{t,a_t}\\}_{t\\in T^{(m)}}$ are mutually independent conditioning on the selected contexts $\\{x_{t,a_t}\\}_{t\\in T^{(m)}}$. \n\\end{lemma}\n\\begin{proof}{Proof.}\n\tFor $t\\in T^{(m)}$, the action $a_t$ only depends on the contexts $\\{x_{t,a}\\}_{a\\in [K]}$ at time $t$ and the past estimators $\\hat{\\theta}_1, \\cdots, \\hat{\\theta}_{m-1}$. However, for any $m'\\in [m-1]$, the estimator $\\hat{\\theta}_{m'}$ only depends on the contexts $x_{\\tau,a_\\tau}$ and rewards $r_{\\tau,a_\\tau}$ with $\\tau\\in T^{(m')}$. Repeating the same arguments for the action $a_\\tau$ with $\\tau\\in T^{(m')}$, we conclude that $a_t$ only depends on the contexts $\\{x_{\\tau,a}\\}_{a\\in [K],\\tau\\in \\cup_{m'\\le m-1} T^{(m')}\\cup \\{t\\}}$ and rewards $\\{r_{\\tau,a_\\tau} \\}_{\\tau\\in \\cup_{m'\\le m-1} T^{(m')}}$. Consequently, by the disjointness of $T^{(m)}$ and $\\cup_{m'\\le m-1} T^{(m')}$, the desired conditional independence holds. \n\\end{proof}\n\nBy Lemma \\ref{lemma.cond_indep}, the conditional independence condition of Lemma \\ref{lemma.difference} holds for Algorithm \\ref{algo.sample_splitting}. Moreover, the sample splitting in Algorithm \\ref{algo.sample_splitting} reduces the sample size by a multiplicative factor at most $M$ at each round, and $M=O(\\log\\log T)$, therefore all proofs in Section \\ref{subsec.pure-exp} continue to hold with a multiplicative penalty at most doubly logarithmic in $T$. As a result, Algorithm \\ref{algo.sample_splitting} achieves the regret upper bound in Theorem \\ref{thm.stochastic}. \n\\end{enumerate}\n\\end{proof}\n\n\n\\subsection{Lower bound}\\label{subsec.stochastic_lower}\nIn this section we prove the minimax lower bound of the regret under stochastic contexts for $K=2$. \nThe lower bound argument for the stochastic context case is quite involved and we start by establishing the following key lemma. \n\\begin{lemma}\\label{lemma.lower_bound}\n\tFor any fixed grid $0=t_0<t_1<\\cdots<t_M=T$ and any $\\Delta \\in [0,1]$, the following minimax lower bound holds for any policy $\\pi$ under this grid: \n\t\\begin{align*}\n\t\\sup_{\\theta^\\star: \\|\\theta^\\star\\|_2 \\le 1} \\mathbb{E}[R_T(\\pi)] \\ge \\Delta \\cdot\\sum_{m=1}^M \\frac{t_m-t_{m-1}}{10\\sqrt{d}}\\exp\\left(-\\frac{16t_{m-1}\\Delta^2}{d^2}\\right). \n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}{Proof.}\n\tLet $\\theta^\\star =\\theta\\sim \\mathsf{Unif}(\\Delta\\mathbb{S}^{d-1})$ be uniformly distributed on the $d$-dimensional sphere centered at the origin with radius $\\Delta$. Clearly $\\|\\theta^\\star\\|_2\\le 1$ surely since $\\Delta\\le 1$. Hence, \n\t\\begin{align}\\label{eq.bayesian}\n\t\\sup_{\\theta^\\star: \\|\\theta^\\star\\|_2 \\le 1} \\mathbb{E}[R_T(\\pi)] \\ge \\mathbb{E}_\\theta \\mathbb{E}[R_T(\\pi)] = \\sum_{t=1}^T \\mathbb{E}_\\theta\\left(\\mathbb{E}\\left[\\max_{i\\in \\{1,2\\}}(x_{t,i}-x_{t,a_t})^\\top \\theta \\right] \\right). \n\t\\end{align}\n\tWe will lower bound each term in the RHS of \\eqref{eq.bayesian} separately. Note that there are multiple sources of randomness involved in the expectation: the randomness in the parameter $\\theta$, in the contexts $x_{t,i}$, and in all the past rewards which determine the random action $a_t$. Throughout the proof, $\\mathbb{E}_\\theta$ denotes taking expectation with respect to $\\theta$, $\\mathbb{E}_x$ denotes taking expectation with respect to all (past and current) random contexts, and $P_{\\theta,x}^t$ denotes the distribution of all random rewards observable before time $t$ conditioned on the parameter $\\theta$ and contexts $x$, with $\\mathbb{E}_{P_{\\theta,x}^t}$ being the corresponding expectation. \n\t\n\tNote that for each $t\\in [T]$, we have\n\t$\\max_{i\\in \\{1,2\\}} (x_{t,i} - x_{t,a_t})^\\top \\theta = \\jiao{x_{t,1}-x_{t,2},\\theta}_+\\cdot \\mathbbm{1}(a_t=2) +  \\jiao{x_{t,1}-x_{t,2},\\theta}_-\\cdot \\mathbbm{1}(a_t=1),$\n\twhere we define $\\jiao{u,v}_+ = \\max\\{0,u^\\top v\\}$ and $\\jiao{u,v}_- = \\max\\{0,-u^\\top v\\}$. Taking expectations on both sides gives\n\t\\begin{align}\n\t& \\mathbb{E}_\\theta \\mathbb{E}_{P_{\\theta,x}^t} \\left[ \\max_{i\\in \\{1,2\\}} (x_{t,i} - x_{t,a_t})^\\top \\theta\\right] \\nonumber \\\\\n\t&= \\mathbb{E}_\\theta \\left[ \\jiao{x_{t,1}-x_{t,2},\\theta}_+\\cdot \\mathbb{P}_{P_{\\theta,x}^t}(a_t=2) +  \\jiao{x_{t,1}-x_{t,2},\\theta}_-\\cdot \\mathbb{P}_{P_{\\theta,x}^t}(a_t=1) \\right] \\nonumber \\\\\n\t&= Z_0\\cdot \\left( \\mathbb{E}_{\\mathbb{E}_{Q_1}P_{\\theta,x}^t}(a_t= 2) + \\mathbb{E}_{\\mathbb{E}_{Q_2}P_{\\theta,x}^t}(a_t= 1) \\right), \\label{eq.change_of_measure}\n\t\\end{align} \n\twhere in the last identity \\eqref{eq.change_of_measure} we define two new probability distributions of $\\theta$ via\n\t\\begin{align*}\n\t\\frac{dQ_1}{dQ_0}(\\theta) = \\frac{\\jiao{x_{t,1}-x_{t,2}, \\theta}_+}{Z_0}, \\qquad \\frac{dQ_2}{dQ_0}(\\theta) = \\frac{\\jiao{x_{t,1}-x_{t,2}, \\theta}_-}{Z_0}, \n\t\\end{align*}\n\twhere $Q_0=\\mathsf{Unif}(\\Delta\\mathbb{S}^{d-1})$ is the original probability measure of $\\theta$, $Z_0$ is the common normalization factor, and $\\mathbb{E}_{Q_i}P_{\\theta,x}^t$ denotes the mixture distribution of $z\\sim P_{\\theta,x}^t$ where $\\theta\\sim Q_i$, for $i\\in \\{1,2\\}$. The following lemma investigates some properties of $Q_1$ and $Q_2$.  \n\t\\begin{lemma}\\label{lemma.Q}\n\t\tLet $x_{t,1} - x_{t,2} = r_tu_t$ with $r_t\\ge 0, \\|u_t\\|_2=1$. Then $\\theta\\sim Q_1$ if and only if $\\theta-2(u_t^\\top \\theta)u_t\\sim Q_2$. Moreover, we have\n\t\t\\begin{align}\\label{eq.Z_0}\n\t\tZ_0 &=  r_t\\Delta\\cdot\\begin{cases}\n\t\t\\frac{2^d}{\\pi d}\\binom{d}{d\/2}^{-1}, & \\text{if }d\\text{ is even} \\\\\n\t\t\\frac{1}{2^d}\\binom{d-1}{(d-1)\/2}, & \\text{if }d\\text{ is odd}\n\t\t\\end{cases} \\ge \\frac{r_t\\Delta}{5\\sqrt{d}}, \\\\ \\label{eq.second_moment}\n\t\t\\mathbb{E}_{Q_1}(u_t^\\top \\theta)^2 &= \\mathbb{E}_{Q_2}(u_t^\\top \\theta)^2 = \\frac{2\\Delta^2}{d+1}.\n\t\t\\end{align}\n\t\\end{lemma}\n\tThe proof of Lemma \\ref{lemma.Q} is postponed to the appendix. Continuing from \\eqref{eq.change_of_measure}, we have\n\t\\begin{align}\n\t\\mathbb{E}_{\\mathbb{E}_{Q_1}P_{\\theta,x}^t}(a_t= 2) + \\mathbb{E}_{\\mathbb{E}_{Q_2}P_{\\theta,x}^t}(a_t= 1) &\\stepa{\\ge} 1-\\mathsf{TV}(\\mathbb{E}_{Q_1}P_{\\theta,x}^t, \\mathbb{E}_{Q_2}P_{\\theta,x}^t) \\nonumber \\\\\n\t&\\stepb{\\ge} \\frac{1}{2}\\exp\\left( - D_{\\text{KL}}(\\mathbb{E}_{Q_1}P_{\\theta,x}^t \\| \\mathbb{E}_{Q_2}P_{\\theta,x}^t ) \\right) \\nonumber \\\\\n\t&\\stepc{=} \\frac{1}{2}\\exp\\left( - D_{\\text{KL}}(\\mathbb{E}_{Q_1}P_{\\theta,x}^t \\| \\mathbb{E}_{Q_1}P_{\\theta - 2(u_t^\\top \\theta)u_t,x}^t ) \\right) \\nonumber \\\\\n\t&\\stepd{\\ge}  \\frac{1}{2}\\exp\\left( - \\mathbb{E}_{Q_1}D_{\\text{KL}}(P_{\\theta,x}^t \\| P_{\\theta - 2(u_t^\\top \\theta)u_t,x}^t ) \\right) \\label{eq.divergence},\n\t\\end{align}\n\twhere step (a) follows from Le Cam's first lemma (cf. e.g., \\cite{Tsybakov2008}), step (b) is due to Lemma \\ref{lemma.TV_KL} in Appendix \\ref{appendix.auxiliary}, step (c) follows from Lemma \\ref{lemma.Q}, and step (d) is due to the joint convexity of the KL divergence. For $t\\in (t_{m-1},t_m]$, the learner can only observe rewards up to time $t_{m-1}$ at time $t$, and therefore\n\t\\begin{align}\n\tD_{\\text{KL}}(P_{\\theta,x}^t \\| P_{\\theta - 2(u_t^\\top \\theta)u_t,x}^t ) &= \\frac{1}{2}\\sum_{\\tau=1}^{t_{m-1}} \\left[x_{\\tau,a_\\tau}^\\top[\\theta - (\\theta - 2(u_t^\\top \\theta)u_t)] \\right]^2 \\nonumber\\\\\n\t&= 2\\sum_{\\tau=1}^{t_{m-1}} (u_t^\\top \\theta)^2(u_t^\\top x_{\\tau,a_\\tau})^2. \\label{eq.KL_divergence}\n\t\\end{align}\n\t\n\tNow combining \\eqref{eq.change_of_measure} to \\eqref{eq.KL_divergence}, we arrive at\n\t\\begin{align*}\n\t\\mathbb{E}_\\theta \\mathbb{E}_{P_{\\theta,x}^t} \\left[ \\max_{i\\in \\{1,2\\}} (x_{t,i} - x_{t,a_t})^\\top \\theta\\right] &\\ge \\frac{r_t\\Delta}{10\\sqrt{d}}\\exp\\left(- \\frac{4\\Delta^2}{d+1} u_t^\\top \\left(\\sum_{\\tau=1}^{t_{m-1}} x_{\\tau,a_\\tau}x_{\\tau,a_\\tau}^\\top\\right) u_t\\right) \\\\\n\t&\\ge \\frac{r_t\\Delta}{10\\sqrt{d}}\\exp\\left(- \\frac{4\\Delta^2}{d+1} u_t^\\top \\left(\\sum_{\\tau=1}^{t_{m-1}} (x_{\\tau,1}x_{\\tau,1}^\\top+x_{\\tau,2}x_{\\tau,2}^\\top)\\right) u_t\\right). \n\t\\end{align*}\n\tFinally, we take the expectation with respect to the contexts $x$. Using the independence of $\\{x_{\\tau,i}\\}_{\\tau<t}$ and $(r_t,u_t)$, the convexity of $x\\mapsto \\exp(-x)$ and $\\mathbb{E}[x_{\\tau,i}x_{\\tau,i}^\\top] = I_d\/d$, we arrive at\n\t\\begin{align}\\label{eq.target}\n\t\\mathbb{E}_x \\mathbb{E}_\\theta \\mathbb{E}_{P_{\\theta,x}^t} \\left[ \\max_{i\\in \\{1,2\\}} (x_{t,i} - x_{t,a_t})^\\top \\theta\\right] &\\ge \\frac{\\mathbb{E}[r_t]\\Delta}{10\\sqrt{d}}\\exp\\left(-\\frac{8\\Delta^2t_{m-1}}{d(d+1)} \\right) \\nonumber\\\\\n\t&\\ge \\frac{\\Delta}{10\\sqrt{d}}\\exp\\left(-\\frac{16\\Delta^2t_{m-1}}{d^2} \\right),\n\t\\end{align}\n\twhere in the last inequality we have used\n\t\\begin{align*}\n\t\\mathbb{E}[r_t] = \\mathbb{E}\\|x_{t,1}-x_{t,2}\\|_2 \\ge \\frac{\\mathbb{E}\\|x_{t,1}-x_{t,2}\\|_1}{\\sqrt{d}} = \\frac{2}{\\sqrt{\\pi}} > 1. \n\t\\end{align*}\n\tCombining \\eqref{eq.bayesian} and \\eqref{eq.target} completes the proof of Lemma \\ref{lemma.lower_bound}. \n\\end{proof}\n\nWe are now ready to put everything together and complete the proof of the lower bound.\n\n\\begin{proof}[Proof of Statement 2 in Theorem~\\ref{thm.stochastic}]\nFor any fixed grid ${\\mathcal{T}}=\\{t_1,\\cdots,t_M\\}$, define $s=\\min\\{m\\in [M]: t_m\\ge d^{2} \\}$, which always exists due to our assumption that $T\\ge d^2$. Now choosing some candidates of $\\Delta \\in \\{1, \\frac{d}{\\sqrt{t_s}}, \\frac{d}{\\sqrt{t_{s+1}}}, \\cdots, \\frac{d}{\\sqrt{T}}\\} \\subset [0,1]$ in Lemma \\ref{lemma.lower_bound} gives\n\\begin{align}\\label{eq.minimax}\n\\sup_{\\theta^\\star: \\|\\theta^\\star\\|_2 \\le 1} \\mathbb{E}[R_T(\\pi)] \\ge c\\cdot \\max\\left\\{\\frac{t_s}{\\sqrt{d}}, t_{s+1}\\sqrt{\\frac{d}{t_s}}, t_{s+2}\\sqrt{\\frac{d}{t_{s+1}}},\\cdots, T\\sqrt{\\frac{d}{t_{M-1}}} \\right\\}\n\\end{align}\nfor some numerical constant $c>0$. After some algebra, the right-hand side of \\eqref{eq.minimax} may be further lower bounded by\n\\begin{align*}\n\\sup_{\\theta^\\star: \\|\\theta^\\star\\|_2 \\le 1} \\mathbb{E}[R_T(\\pi)] \\ge c\\sqrt{dT}\\cdot \\left(\\frac{T}{d^2}\\right)^{\\frac{1}{2(2^{M-s+1}-1)}} \\ge c\\sqrt{dT}\\cdot \\left(\\frac{T}{d^2}\\right)^{\\frac{1}{2(2^{M}-1)}}. \n\\end{align*}\n\\end{proof}\n\n\\section{Problem-Dependent Regret Bounds}\\label{sec:gap}\n\nThe regret bounds given in the previous two sections are problem-independent regret bounds (also known as gap-independent regret bounds in the bandits literature): they do not depend on the underlying parameters of the probability distribution. When the contexts are stochastic, under certain ``margin\" conditions, we can also consider problem-dependent regret bounds that can result in sharper bounds than those problem-independent ones. When the number of contexts is small (e.g., $K=2$), there could be a large margin between the performance of the optimal context and any sub-optimal contexts if $\\|\\theta^\\star\\|_2$ is bounded away from zero, raising the possibility that a problem-dependent regret bound sometimes better than the worst-case regret $\\Theta(\\sqrt{dT})$ could be obtained in sequential batch learning. The next theorem characterizes this. \n\n\\begin{theorem}\\label{thm.problem-dependent}\n\tAssume $K=2$, and let $T$, $M=O(\\log T)$, $d$ be the learning horizon, number of batches and the dimension of each context respectively. Denote by $\\mathsf{polylog}(T)$ all the poly-logarithmic factors in $T$. Assume without loss of generality $\\|\\theta^*\\|_2 > 0$. \n\t\\begin{enumerate}\n\t\t\\item \n\t\tUnder Assumptions \\ref{aspn.TKd} and \\ref{assumption:cov}, there exists a sequential batch learning algorithm \\textbf{Alg}= $({\\mathcal{T}}, \\pi)$ (explicitly defined below ) that achieves the following regret:\n\t\t\\begin{align*}\n\t   \\mathbb{E}_{\\theta^\\star}[R_T(\\pi)] \\le \\mathsf{polylog}(T)\\cdot \\frac{(d\/\\kappa)^{3\/2}}{\\|\\theta^\\star\\|_2} \\left(\\frac{T}{d^2}\\right)^{\\frac{1}{M}}.\n\t\t\\end{align*}\n\t\t\\item\n\t\tConversely, when the contexts $x_{t,a}\\sim {\\mathcal{N}}(0,I_d\/d)$ are independent over all $a\\in [K], t\\in [T]$, for any $M\\le T$ and any sequential batch learning algorithm, we have:\n\t\t\\begin{align*}\n\t\t\\sup_{\\theta^\\star: \\|\\theta^\\star\\|_2\\le 1} \\|\\theta^\\star\\|_2\\cdot  \\mathbb{E}_{\\theta^\\star}[R_T(\\pi)] \\ge c\\cdot d^{3\/2} \\left(\\frac{T}{d^2}\\right)^{\\frac{1}{M}}, \n\t\t\\end{align*}\n\t\twhere $c>0$ is a numerical constant independent of $(T,M,d)$. \n\t\\end{enumerate}\n\\end{theorem}\n\\begin{corollary}\nIn this setting, it is necessary and sufficient to have $\\Theta(\\log(T\/d^2))$ batches to achieve the optimal problem-dependent regret $\\tilde{\\Theta}(d^{3\/2} \/ \\|\\theta^\\star\\|_2)$. \nHere we are not aiming to get the tightest dependence on $\\log T$ (note that $\\tilde{\\Theta}(\\cdot)$ hides polylog factors). \n\\end{corollary}\n\nNote that the dependence on $T$ is significantly better than $\\sqrt{T}$ in the problem-dependent bound, showing that a large $\\|\\theta^\\star\\|_2$ makes learning simpler. We remark that although the problem-dependent regret in Theorem \\ref{thm.problem-dependent} only holds for $K=2$, the generalization to a generic $K$ is straightforward. Moreover, the margin between the optimal context and the sub-optimal context shrinks quickly as $K$ gets larger, and therefore the margin-based problem-dependent bound is not that useful compared with the worst-case regret bound in Theorem \\ref{thm.stochastic} for large $K$. \n\n\\subsection{Proof of the Upper Bound in Theorem \\ref{thm.problem-dependent}}\nThe sequential batch learning algorithm which achieves the claimed upper bound is exactly the batched pure-exploitation algorithm with sample splitting shown in Algorithm \\ref{algo.sample_splitting}, with a different choice of the grid: we consider a geometric grid ${\\mathcal{T}}' = \\{t_1', t_2', \\cdots, t_M'\\}$ with\n\\begin{align*}\nt_1' = bd^2, \\qquad t_m' = \\lfloor bt_{m-1}' \\rfloor, \\quad m=2,3,\\cdots,M,\n\\end{align*}\nwhere $b = \\Theta((T\/d^2)^{1\/M})$ so that $t_M' = T$. Next we show that with the above choice of the grid, Algorithm \\ref{algo.sample_splitting} attains the regret upper bound in Theorem \\ref{thm.problem-dependent}. \n\nConsider the $m$-th batch with any $m\\ge 2$, and any time point $t$ inside this batch. Define $v_t = x_{t,1} - x_{t,2}$, then our algorithm chooses the wrong arm if and only if $v_t^\\top \\theta^\\star$ and $v_t^\\top \\hat{\\theta}_{m-1}$ have different signs. Hence, the instantenous regret at time $t$ is\n\\begin{align*}\nv_t^\\top \\theta^\\star \\cdot \\mathbbm{1}(v_t^\\top \\theta^\\star \\ge 0, v_t^\\top \\hat{\\theta}_{m-1} \\le 0) - v_t^\\top \\theta^\\star \\cdot \\mathbbm{1}(v_t^\\top \\theta^\\star \\le 0, v_t^\\top \\hat{\\theta}_{m-1} \\ge 0),\n\\end{align*}\nand by the symmetry of $v_t\\sim {\\mathcal{N}}(0,2\\Sigma)$, it holds that\n\\begin{align*}\n\\mathbb{E}\\left[\\max_{a\\in \\{1,2\\}} (x_{t,a} - x_{t,a_t})^\\top \\theta^\\star \\right] = 2\\mathbb{E}\\left[v_t^\\top \\theta^\\star\\cdot \\mathbbm{1}(v_t^\\top \\theta^\\star \\ge 0, v_t^\\top \\hat{\\theta}_{m-1} \\le 0)  \\right]. \n\\end{align*}\nSet $\\delta = \\sqrt{d\\log T\/(\\kappa t_{m-1}')}$, and partition the non-negative axis $\\mathbb{R}_+$ into $\\bigcup_{i=0}^\\infty [i\\delta, (i+1)\\delta)$. Using this partition gives\n\\begin{align}\n&\\mathbb{E}\\left[v_t^\\top \\theta^\\star\\cdot \\mathbbm{1}(v_t^\\top \\theta^\\star \\ge 0, v_t^\\top \\hat{\\theta}_{m-1} \\le 0)\\cdot \\mathbbm{1}(\\|v_t\\|_2 \\le \\sqrt{10\\log T})  \\right] \\nonumber\\\\\n&= \\sum_{i=0}^\\infty \\mathbb{E}\\left[v_t^\\top \\theta^\\star\\cdot \\mathbbm{1}(v_t^\\top \\theta^\\star  \\in [i\\delta, (i+1)\\delta), v_t^\\top \\hat{\\theta}_{m-1} \\le 0)\\cdot \\mathbbm{1}(\\|v_t\\|_2 \\le \\sqrt{10\\log T}) \\right] \\nonumber\\\\\n&\\le \\sum_{i=0}^\\infty (i+1)\\delta\\cdot \\mathbb{P}\\left(v_t^\\top \\theta^\\star  \\in [i\\delta, (i+1)\\delta), v_t^\\top \\hat{\\theta}_{m-1} \\le 0,\\|v_t\\|_2 \\le \\sqrt{10\\log T} \\right) \\nonumber\\\\\n&\\le \\sum_{i=0}^\\infty (i+1)\\delta\\cdot \\mathbb{P}\\left(v_t^\\top \\theta^\\star  \\in [i\\delta, (i+1)\\delta), v_t^\\top (\\theta^\\star - \\hat{\\theta}_{m-1}) \\ge i\\delta,\\|v_t\\|_2 \\le \\sqrt{10\\log T} \\right) \\nonumber\\\\\n&\\le \\sum_{i=0}^\\infty (i+1)\\delta\\cdot \\mathbb{P}\\left(v_t^\\top \\theta^\\star  \\in [i\\delta, (i+1)\\delta) \\right)\\cdot \\mathbb{P}\\left( v_t^\\top (\\theta^\\star - \\hat{\\theta}_{m-1}) \\ge i\\delta \\big| v_t^\\top \\theta^\\star  \\in [i\\delta, (i+1)\\delta), \\|v_t\\|_2 \\le \\sqrt{10\\log T}\\right). \\label{eq.partition}\n\\end{align}\n\nWe deal with each term in \\eqref{eq.partition} separately. For $\\mathbb{P}\\left(v_t^\\top \\theta^\\star  \\in [i\\delta, (i+1)\\delta) \\right)$, note that $v_t^\\top\\theta^\\star$ is a normal random variable with variance $(\\theta^\\star)^\\top \\Sigma \\theta^\\star \\ge \\lambda_{\\min}(\\Sigma)\\|\\theta^\\star\\|_2^2\\ge \\kappa\\|\\theta^\\star\\|_2^2\/d$, thus the probability density of this random variable is upper bounded by $\\sqrt{d\/2\\pi \\kappa}\/\\|\\theta^\\star\\|_2$ everywhere. Therefore, \n\\begin{align}\\label{eq.anticoncentration}\n\\mathbb{P}\\left(v_t^\\top \\theta^\\star  \\in [i\\delta, (i+1)\\delta) \\right) \\le \\delta\\cdot \\frac{\\sqrt{d}}{\\sqrt{2\\pi\\kappa}\\|\\theta^\\star\\|_2}. \n\\end{align}\nFor the second term of \\eqref{eq.partition}, the proof of Lemma \\ref{lemma.difference} shows that the random vector $\\theta^\\star - \\hat{\\theta}_{m-1}\\in \\mathbb{R}^d$ is $d\/(c\\kappa t_{m-1}')$-subGaussian for some absolute constant $c>0$, and is also independent of $v_t$. Hence, conditioning on $\\|v_t\\|_2\\le \\sqrt{10\\log T}$, the random variable $v_t^\\top(\\theta^\\star - \\hat{\\theta}_{m-1})$ is also subGaussian with parameter $\\|v_t\\|_2^2d\/(c\\kappa t_{m-1})\\le 10d\\log T\/(c\\kappa t_{m-1}')$. Consequently, subGaussian concentration gives\n\\begin{align}\\label{eq.concentration}\n\\mathbb{P}\\left( v_t^\\top (\\theta^\\star - \\hat{\\theta}_{m-1}) \\ge i\\delta \\big| v_t^\\top \\theta^\\star  \\in [i\\delta, (i+1)\\delta), \\|v_t\\|_2 \\le \\sqrt{10\\log T}\\right) \\le \\exp\\left(-\\frac{c\\kappa i^2\\delta^2t_{m-1}'}{20d\\log T}\\right). \n\\end{align}\n\nCombining \\eqref{eq.partition}, \\eqref{eq.anticoncentration}, \\eqref{eq.concentration} and the choice of $\\delta$, we conclude that\n\\begin{align*}\n\\mathbb{E}\\left[v_t^\\top \\theta^\\star\\cdot \\mathbbm{1}(v_t^\\top \\theta^\\star \\ge 0, v_t^\\top \\hat{\\theta}_{m-1} \\le 0)\\cdot \\mathbbm{1}(\\|v_t\\|_2 \\le \\sqrt{10\\log T})  \\right]  &\\le \\frac{d^{3\/2}\\log T}{\\sqrt{2\\pi \\kappa^3}t_{m-1}'\\|\\theta^\\star\\|_2} \\sum_{i=0}^\\infty (i+1)e^{-ci^2\/20} \\\\\n&\\le C\\cdot \\frac{d^{3\/2}\\log T}{\\kappa^{3\/2}t_{m-1}'\\|\\theta^\\star\\|_2}. \n\\end{align*}\nMoreover, since $v_t^\\top \\theta^\\star \\le 2$ almost surely and $\\mathbb{P}(\\|v_t\\|_2\\ge \\sqrt{10\\log T})\\le T^{-5}$, we also have\n\\begin{align*}\n\\mathbb{E}\\left[v_t^\\top \\theta^\\star\\cdot \\mathbbm{1}(v_t^\\top \\theta^\\star \\ge 0, v_t^\\top \\hat{\\theta}_{m-1} \\le 0)\\cdot \\mathbbm{1}(\\|v_t\\|_2 > \\sqrt{10\\log T})  \\right] \\le 2T^{-5}. \n\\end{align*}\nTherefore, by the choice of the grid, the expected total regret in the $m$-th batch is at most\n\\begin{align*}\n\\left(C\\cdot \\frac{d^{3\/2}\\log T}{\\kappa^{3\/2}t_{m-1}'\\|\\theta^\\star\\|_2} + 2T^{-5}\\right)\\cdot t_m' = O\\left(\\frac{d^{3\/2}\\log T}{\\kappa^{3\/2}\\|\\theta^\\star\\|_2}\\cdot \\left(\\frac{T}{d^2}\\right)^{1\/M} \\right). \n\\end{align*}\n\nThe first batch is handled in the same way as the upper bound proof of Theorem \\ref{thm.stochastic}. Specifically, the expected total regret in the first batch is \n\\begin{align*}\nO\\left( t_1'\\cdot \\sqrt{\\frac{\\log T}{d}} \\right) = O\\left(\\sqrt{d^3\\log T}\\left(\\frac{T}{d^2}\\right)^{\\frac{1}{M}}\\right) = O\\left(\\frac{\\sqrt{d^3\\log T}}{\\kappa^{3\/2}\\|\\theta^\\star\\|_2}\\left(\\frac{T}{d^2}\\right)^{\\frac{1}{M}} \\right).\n\\end{align*}\nFinally summing up all batches $m=1,2,\\cdots,M$ completes the proof. \n\n\\subsection{Proof of the lower bound in Theorem \\ref{thm.problem-dependent}} The proof is entirely analogous to the lower bound proof of Theorem \\ref{thm.stochastic}. First we observe that by Lemma \\ref{lemma.lower_bound}, for any $\\Delta \\in [0,1]$ and fixed grid ${\\mathcal{T}} = \\{t_1, t_2, \\cdots, t_M\\}$ we have\n\\begin{align*}\n\\inf_\\pi \\sup_{\\theta^\\star: \\|\\theta^\\star\\|_2\\le 1} \\|\\theta^\\star\\|_2\\cdot  \\mathbb{E}_{\\theta^\\star}[R_T(\\pi)] &\\ge \\Delta\\cdot \\inf_\\pi \\sup_{\\theta^\\star: \\Delta\\le \\|\\theta^\\star\\|_2\\le 1} \\mathbb{E}_{\\theta^\\star}[R_T(\\pi)] \\\\\n&\\ge \\Delta^2 \\cdot\\sum_{m=1}^M \\frac{t_m-t_{m-1}}{10\\sqrt{d}}\\exp\\left(-\\frac{16t_{m-1}\\Delta^2}{d^2}\\right). \n\\end{align*}\nNow define $s = \\min\\{m\\in [M]: t_m \\ge d^2 \\}$, which always exists due to the assumption $T\\ge d^2$. Choosing $\\Delta \\in \\{1,d^2\/t_s,d^2\/t_{s+1},\\cdots,d^2\/t_M\\} \\subseteq [0,1]$ in the above inequality gives\n\\begin{align*}\n\\inf_\\pi \\sup_{\\theta^\\star: \\|\\theta^\\star\\|_2\\le 1} \\|\\theta^\\star\\|_2\\cdot  \\mathbb{E}_{\\theta^\\star}[R_T(\\pi)] \\ge c\\cdot \\max\\left\\{\\frac{t_s}{\\sqrt{d}}, \\frac{d^{3\/2}t_{s+1}}{t_s},  \\frac{d^{3\/2}t_{s+2}}{t_{s+1}},\\cdots,  \\frac{d^{3\/2}T}{t_{M-1}} \\right\\}\n\\end{align*}\nfor some absolute constant $c>0$. Finally, applying $\\max\\{a_1,\\cdots,a_n\\} \\ge \\sqrt[n]{a_1a_2\\cdots a_n}$ gives\n\\begin{align*}\n\\inf_{\\pi} \\sup_{\\theta^\\star: \\|\\theta^\\star\\|_2\\le 1} \\|\\theta^\\star\\|_2\\cdot  \\mathbb{E}_{\\theta^\\star}[R_T(\\pi)] \\ge c\\cdot d^{3\/2}\\left(\\frac{T}{d^2}\\right)^{\\frac{1}{M-s+1}} \\ge c\\cdot d^{3\/2}\\left(\\frac{T}{d^2}\\right)^{\\frac{1}{M}},\n\\end{align*}\nas claimed. \n\n\n\\section{Conclusion}\n\nAs we have shown in this paper, sequential batch learning provides an interesting and nontrivial departure from the traditional online learning setting where feedback is immediately observed and incorporated into making the next decision. We studied sequential batch learning in the linear contextual bandits setting and provided an in-depth inquiry into the algorithms and theoretical performance. An important insight here is that the nature of the contexts-adversarial or stochastic--has a significant impact on the optimal achievable performance, as well as the algorithms that would achieve the minimax optimal regret bounds.\n\nSeveral questions immediately suggest themselves.\nFirst, in the stochastic context setting, our current regret upper bound\ndepends heavily on the Gaussian assumption of the contexts. It would be interesting to see how far we can move beyond the Gaussian family. \nIt would be unlikely that the same result holds for any distribution and hence, characterizing a (hopefully large) class of distributions under which the same tight bounds are achievable would be interesting.\nAnother direction would be to look at more complex reward structures that go beyond linear bandits and see to what extent can the current set of results be generalized. We leave them for future work.   \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Datasets and their Templates}\n\\label{sec:appendix:analysis}\n\n\\subsection{Division of Crowdsourcing Instructions into Minimal Tasks}\n\\label{sec:appendix:division:screenshots}\nFig.~\\ref{fig:subtsakdivision} shows an example of how a task is divided into multiple subtasks for the MC-TACO dataset. MC-TACO has five categories (Event Duration, Event Frequency etc.). Each category contributes to 2 subtasks one for question generation and one for answer generation.\n\n\n\n\n\n\\paragraph{Number of tasks in each dataset.} \nFig.~\\ref{fig:no. of subtasks} illustrates how the number of steps in the data creation process varies across the 6 datasets. QASC and MC-TACO contain a relatively higher number of steps in the data creation process in comparison to DROP, Quoref, CosmosQA, and Winogrande. \n\n\n\\begin{figure}[H]\n\\centering\n  \\includegraphics[scale=0.28,trim=2cm 4cm 0cm 3cm]{figures\/Number_of_subtasks.pdf}\n  \\caption{Variations in the number of subtasks}\n    \\label{fig:no. of subtasks}\n\\end{figure}\n\n\\subsection{Analysis of Crowdsourcing Templates}\nWe analyzed crowdsourcing templates of 6 datasets: CosmosQA~\\cite{huang2019cosmos}, \nDROP~\\cite{dua2019drop}, \nMC-TACO~\\cite{zhou2019going}, \nQASC~\\cite{khot2020qasc}, \nQuoref~\\cite{dasigi2019quoref}, and\nWinogrande~\\cite{sakaguchi2020winogrande}. Our intention behind the analysis is to identify similarities and differences across templates and subsequently decide regarding the collection of more templates.\n\\label{appendix:analysis:templates}\n\n\n\\paragraph{Size of the instructions.} We observe significant variation in size across the 6 datasets (Fig.~\\ref{fig:size inst}). In the case of QASC, the instruction size associated with each step of the data creation process is very high, whereas for Winogrande, it is exactly the opposite-- instruction size associated with each step of the data creation process is very low. Instead, the size of the common instruction (i.e., the instruction preceding the first step of the data creation process) is high in Winogrande; this is  also seen for DROP. The major mode of instruction varies across datasets. Examples and instructions associated with each step of data creation respectively take up the majority of space in Quoref and CosmosQA.  MC-TACO relies on examples to explain the crowdsourcing task, while Winogrande and QASC depend mostly on common instructions and instructions associated with each step of the data creation process respectively, to explain the task to the crowdworker.\n\n\\paragraph{The number of positive\/negative examples.} \nVariation in the occurrence of \\textsc{Positive} and \\textsc{Negative} Examples across datasets has been illustrated in Fig.~\\ref{fig:no. of examples}. Only Winogrande provides an equal number of \\textsc{Positive} and \\textsc{Negative} Examples. \nQASC instructions do not contain any \\textsc{Negative} Examples. \nOverall, DROP instructions consist of a relatively higher number of examples than other datasets.\n\n\\begin{figure}[H]\n\\centering\n  \\includegraphics[width=0.96\\columnwidth ]{figures\/example_num.png}\n    \\caption{Variation in the number of positive and negative examples}\n\\label{fig:no. of examples}\n\\end{figure}\n\n\n\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.96\\columnwidth]{figures\/instruction_size.pdf}\n    \\caption{Variation in the number of sentences in the crowdsourcing instructions across datasets}\n    \\label{fig:size inst}\n\\end{figure}\n\n\\paragraph{Presence of reasons\/suggestions in examples.} All datasets except QASC contain both \\textsc{Positive} and \\textsc{Negative} Examples. \nHowever, Quoref is the only dataset to provide \\textsc{Reasons} for all the \\textsc{Positive} and \\textsc{Negative} Examples. There are explanations associated with each of the \\textsc{Negative} Examples, but the presence of explanations associated with \\textsc{Positive} Examples varies across datasets. Finally, Quoref is the only dataset to provide \\textsc{Suggestions} along with the \\textsc{Reasons} associated with the \\textsc{Negative} Examples.\n\n\\begin{comment}\n\\paragraph{Dimensions of Input and Output:}The input dimension of a step is defined as the number of previous step outputs that are fed as input. Parallely, the output dimension of a step is the number of distinct outputs the model needs to produce in that step-- for example, if a model has to generate both a question and an answer in a step, the output dimension will be 2. CosmosQA and QASC have relatively high dimensional instances, whereas Quoref and MC-TACO have relatively low dimensional instances.\n\\end{comment}\n\n\\begin{figure}\n\\centering\n  \\includegraphics[width=0.5\\textwidth,trim=0cm 0cm 0cm 0cm]{figures\/sub-task.pdf}\n  \\caption{\n    Dividing a data creation task into multiple subtasks for the MC-TACO dataset. \n  }\n  \\label{fig:subtsakdivision}\n\\end{figure}\n\n\n\n\n\\subsection{Qualitative Analysis}\n\\paragraph{Writing Style.} There are significant variation in writing style across the datasets, even among those datasets \nthat have the common a objective (e.g., DROP, Quoref and QASC). \nDROP instructions say \\textit{\"There is an AI running in the background which will also try to answer the question. You won't be able to submit the question if the AI gives the same response.\"} The writing style in Quoref however is different: \\textit{\"We also want you to avoid questions that can be answered correctly by someone without actually understanding the paragraph. ...\"} \n\n\\paragraph{Information.} We observe that sometimes instructions of a dataset contain information that is relevant to several other datasets, which do not contain similar instruction information. For example, Quoref, DROP and CosmosQA are datasets that are all based on reading comprehension tasks. CosmosQA contains a step in the data creation process asking users to skip passages containing inappropriate or offensive content. This information is also relevant to Quoref and DROP, but is not mentioned in their respective instructions.\n\n\n\n\n\\begin{figure}[t]\n    \\centering\n   \n    \\includegraphics[scale=0.36,trim=0.1cm 0.1cm 0.1cm 0.1cm]{figures\/Task_specification.pdf}\n   \n    \\caption{Variation in Task Specification: Quoref contains a single line instruction whereas the CosomosQA contains a detailed instruction. QASC on the other hand, contains examples along with instruction.}\n    \\label{fig:task_specification}\n\\end{figure}\n\n\n\n\\paragraph{Hardness.} In a typical crowdsourcing task, certain tasks may be harder than the others, often these are the core tasks, e.g.: question generation, adversarial data creation, etc. Additional information, especially in the form of tips is always helpful in solving these hard tasks. Figure~\\ref{fig:task_specification} illustrates that the task of question generation is stated differently in Quoref, CosmosQA, and QASC. QASC mentions an easy and detailed way to create questions, whereas CosmosQA mentions several different attributes of a good quality question. Knowing about the CosmosQA and QASC question generation processes may help with data creation for Quoref and other such question generation tasks, where less additional information is provided regarding question creation. \n\n\n\n\n\n\\subsection{Data Curation Effort}\n\\label{appendix:subsect:curation}\nTable \\ref{tab:datacuration} shows the effort distribution in the data curation process of \\textsc{Natural Instructions}{}. Step-8 which involves parsing instances is the main bottleneck in the data curation process. Table \\ref{tab:structure} shows the detailed structure of tasks in \\textsc{Natural Instructions}{}. Fig.~\\ref{fig:examplesfull} shows examples of four different tasks in \\textsc{Natural Instructions}{}.\n\n\\begin{table}[h]\n    \\centering\n    \\footnotesize\n   \n    \\begin{tabular}{m{0.5cm}p{4.5cm}p{1.5cm}}\n        \\toprule\n        step & task & time per task  \\\\ \n        \\midrule\n        1 & Identify crowdsourced dataset and engage with their authors.  & 20-30 mins \\\\\n       \n        2 & Go through the template and understand the task. & 10-15 mins \\\\ \n        3 & Manually fill fields in the schema with content from the template. & 30-45 mins \\\\ \n        4 & Iterate over the instructions to ensure their clarity while eliminating the repeated content. Fix writing issue in examples, also typos etc. \n       \n        & 2-3 hrs\\\\ \n        5 & Create negative examples if not present. Add the missing explanations to the examples. & 1-2 hrs \\\\ \n       \n        6 & Extract the input\/output instances from raw crowdsourcing annotations.  & 0.5-24 hrs \\\\ \n       \n        7 & Final inspections of the data to verify the data quality \n       \n        & 0.25- 2hrs \\\\\n        \\midrule\n         & Overall & 6-34 hrs\\\\\n        \\bottomrule\n    \\end{tabular}\n   \n    \\caption{Steps taken to curate each task in \\textsc{Natural Instructions}{} and their estimated times.\n   \n    }\n    \\label{tab:datacuration}\n\\end{table}\n\n\n\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[scale=0.75,trim=0.7cm 0.5cm 0.5cm 1.5cm]{figures\/examples_detailed.pdf}\n    \\caption{\n        Examples from \\textsc{Natural Instructions}{}. \n        Each task follows the schema provided in Fig.~\\ref{fig:schema_plate}. \n    }\n    \\label{fig:examplesfull}\n\\end{figure*}\n\n\n\\begin{table*}\n    \\centering\n    \\small\n    \\begin{adjustbox}{max width=\\textwidth}\n    \\begin{tabular}{llcc}\n        \\toprule\n        task id & title& source dataset & task category\\\\\n        \\midrule\n1 & task001\\_quoref\\_question\\_generation & Quoref & Question Generation \\\\\n2 & task002\\_quoref\\_answer\\_generation & Quoref & Answer Generation \\\\\n\\midrule \n3 & task003\\_mctaco\\_question\\_generation\\_event\\_duration & MC-TACO & Question Generation \\\\\n4 & task004\\_mctaco\\_answer\\_generation\\_event\\_duration & MC-TACO & Answer Generation  \\\\\n5 & task005\\_mctaco\\_wrong\\_answer\\_generation\\_event\\_duration & MC-TACO & Incorrect Answer Generation  \\\\\n6 & task006\\_mctaco\\_question\\_generation\\_transient\\_stationary & MC-TACO & Question Generation  \\\\\n7 & task007\\_mctaco\\_answer\\_generation\\_transient\\_stationary & MC-TACO & Answer Generation  \\\\\n8 & task008\\_mctaco\\_wrong\\_answer\\_generation\\_transient\\_stationary & MC-TACO & Incorrect Answer Generation  \\\\\n9 & task009\\_mctaco\\_question\\_generation\\_event\\_ordering & MC-TACO & Question Generation \\\\\n10 & task010\\_mctaco\\_answer\\_generation\\_event\\_ordering & MC-TACO & Answer Generation  \\\\\n11 & task011\\_mctaco\\_wrong\\_answer\\_generation\\_event\\_ordering & MC-TACO & Incorrect Answer Generation  \\\\\n12 & task012\\_mctaco\\_question\\_generation\\_absolute\\_timepoint & MC-TACO & Question Generation  \\\\\n13 & task013\\_mctaco\\_answer\\_generation\\_absolute\\_timepoint & MC-TACO & Answer Generation  \\\\\n14 & task014\\_mctaco\\_wrong\\_answer\\_generation\\_absolute\\_timepoint & MC-TACO & Incorrect Answer Generation  \\\\\n15 & task015\\_mctaco\\_question\\_generation\\_frequency & MC-TACO & Question Generation  \\\\\n16 & task016\\_mctaco\\_answer\\_generation\\_frequency & MC-TACO & Answer Generation  \\\\\n17 & task017\\_mctaco\\_wrong\\_answer\\_generation\\_frequency & MC-TACO & Incorrect Answer Generation \\\\\n18 & task018\\_mctaco\\_temporal\\_reasoning\\_presence & MC-TACO & Classification  \\\\\n19 & task019\\_mctaco\\_temporal\\_reasoning\\_category & MC-TACO & Classification  \\\\\n20 & task020\\_mctaco\\_span\\_based\\_question & MC-TACO & Classification  \\\\\n21 & task021\\_mctaco\\_grammatical\\_logical & MC-TACO & Classification  \\\\\n\\midrule \n22 & task022\\_cosmosqa\\_passage\\_inappropriate\\_binary & Cosmosqa & Classification  \\\\\n23 & task023\\_cosmosqa\\_question\\_generation & Cosmosqa & Question Generation  \\\\\n24 & task024\\_cosmosqa\\_answer\\_generation & Cosmosqa & Answer Generation  \\\\\n25 & task025\\_cosmosqa\\_incorrect\\_answer\\_generation & Cosmosqa & Incorrect Answer Generation \\\\\n\\midrule \n26 & task026\\_drop\\_question\\_generation & DROP & Question Generation  \\\\\n27 & task027\\_drop\\_answer\\_type\\_generation & DROP & Classification  \\\\\n28 & task028\\_drop\\_answer\\_generation & DROP & Answer Generation \\\\\n\\midrule \n29 & task029\\_winogrande\\_full\\_object & Winogrande & Minimal Text Modification  \\\\\n30 & task030\\_winogrande\\_full\\_person & Winogrande & Minimal Text Modification  \\\\\n31 & task031\\_winogrande\\_question\\_generation\\_object & Winogrande & Question Generation  \\\\\n32 & task032\\_winogrande\\_question\\_generation\\_person & Winogrande & Question Generation  \\\\\n33 & task033\\_winogrande\\_answer\\_generation & Winogrande & Answer Generation  \\\\\n34 & task034\\_winogrande\\_question\\_modification\\_object & Winogrande & Minimal Text Modification  \\\\\n35 & task035\\_winogrande\\_question\\_modification\\_person & Winogrande & Minimal Text Modification \\\\\n\\midrule \n36 & task036\\_qasc\\_topic\\_word\\_to\\_generate\\_related\\_fact & QASC & Minimal Text Modification \\\\\n37 & task037\\_qasc\\_generate\\_related\\_fact & QASC & Minimal Text Modification  \\\\\n38 & task038\\_qasc\\_combined\\_fact & QASC & Minimal Text Modification  \\\\\n39 & task039\\_qasc\\_find\\_overlapping\\_words & QASC & Verification  \\\\\n40 & task040\\_qasc\\_question\\_generation & QASC & Question Generation  \\\\\n41 & task041\\_qasc\\_answer\\_generation & QASC & Answer Generation \\\\\n42 & task042\\_qasc\\_incorrect\\_option\\_generation & QASC & Incorrect Answer Generation \\\\\n\\midrule \n43 & task043\\_essential\\_terms\\_answering\\_incomplete\\_questions & Essential Terms & Answer Generation  \\\\\n44 & task044\\_essential\\_terms\\_identifying\\_essential\\_words & Essential Terms & Verification  \\\\\n\\midrule \n45 & task045\\_miscellaneous\\_sentence\\_paraphrasing & Miscellaneous & Minimal Text Modification  \\\\\n46 & task046\\_miscellaenous\\_question\\_typing & Miscellaenous & Classification  \\\\\n47 & task047\\_miscellaenous\\_answering\\_science\\_questions & Miscellaenous & Answer Generation \\\\\n\\midrule \n48 & task048\\_multirc\\_question\\_generation & MultiRC & Question Generation  \\\\\n49 & task049\\_multirc\\_questions\\_needed\\_to\\_answer & MultiRC & Classification  \\\\\n50 & task050\\_multirc\\_answerability & MultiRC & Classification  \\\\\n51 & task051\\_multirc\\_correct\\_answer\\_single\\_sentence & MultiRC & Answer Generation  \\\\\n52 & task052\\_multirc\\_identify\\_bad\\_question & MultiRC & Classification  \\\\\n53 & task053\\_multirc\\_correct\\_bad\\_question & MultiRC & Minimal Text Modification  \\\\\n54 & task054\\_multirc\\_write\\_correct\\_answer & MultiRC & Answer Generation  \\\\\n55 & task055\\_multirc\\_write\\_incorrect\\_answer & MultiRC & Incorrect Answer Generation  \\\\\n56 & task056\\_multirc\\_classify\\_correct\\_answer & MultiRC & Classification  \\\\\n57 & task057\\_multirc\\_classify\\_incorrect\\_answer & MultiRC & Classification  \\\\\n58 & task058\\_multirc\\_question\\_answering & MultiRC & Answer Generation  \\\\\n\\midrule \n59 & task059\\_ropes\\_story\\_generation & ROPES & Minimal Text Modification  \\\\\n60 & task060\\_ropes\\_question\\_generation & ROPES & Question Generation  \\\\\n61 & task061\\_ropes\\_answer\\_generation & ROPES & Answer Generation \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\end{adjustbox}\n    \\caption{Detailed set of tasks included in \\textsc{Natural Instructions}{}}\n    \\label{tab:structure}\n\\end{table*}\n\n\n\\clearpage\n\\onecolumn\n\n\\changed{\n\\subsection{Qualitative Comparison to PromptSource}\n\\label{subsec:promptsource}\nWe provide a comparison between our proposed dataset and PromptSource~\\cite{sanh2021multitask}. \nPromptSource tasks are mainly focused on the common NLP downstream tasks (such as question-answering, coreference, NLI, etc).  \nHowever, since we create tasks from various steps (including the intermediate steps) in a data creation process, our instructions contain a broader variety of tasks. For example, tasks for chaining facts (task 38; Table~\\ref{tab:structure}), question typing (task 27; Table~\\ref{tab:structure}) or detecting inappropriate content (task 22; Table~\\ref{tab:structure}) are unique additions in \\textsc{Natural Instructions}{}. \nAdditionally, since our instructions were originally written by various researchers targeted for crowdworkers, they are elaborate and contain the complete definition of each task. \nThis is somewhat evident from observation that GPT3 leads to higher performance on our instructions (Table~\\ref{tab:prompt:source:gpt3:eval}). \nLast but not least, since we represent the instructions in a structured format, we are able to ablate various elements of the instructions (definition, negative\/positive examples, etc.) and empirically quantify their contributions (\\S\\ref{sec:experiments}).  \n}\n\n\\begin{table}[h]\n    \\centering\n    \\small\n    \\begin{tabular}{clcc}\n        \\toprule \n        Task & Model & PromptSource & \\textsc{Natural Instructions}{} \\\\\n        \\midrule\n        \\multirow{2}{*}{ Quoref QA (002) } & GPT3-Instruct & 43 & {\\bf 47} \\\\\n            & GPT3 & 2 & {\\bf 13} \\\\\n        \\multirow{2}{*}{ DROP QA (028) } & GPT3-Instruct & 6 & {\\bf 10} \\\\\n            & GPT3 & 2 &  {\\bf 3} \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{ \n        Comparing zero-shot performance of GPT3 on our instructions vs. PromptSource. \n        The instructions curated in this work, despite being lengthier, lead to higher performance. \n       \n        }\n    \\label{tab:prompt:source:gpt3:eval}\n\\end{table}\n\n\\begin{table*}[h]\n    \\centering\n   \n   \n    \\includegraphics[scale=0.88,trim=1.4cm 13.4cm 1.2cm 1.85cm,clip=true]{figures\/comparisonWithPromptSource-3.pdf}\n    \\caption{Qualitative comparison of the task instructions for several shared tasks among \\textsc{Natural Instructions}{} and PromptSource~\\cite{sanh2021multitask}.}\n    \\label{tab:prompt:source}\n\\end{table*}\n\n\\twocolumn\n\\clearpage\n\n\\section{Building Baselines for \\textsc{Natural Instructions}{}}\nIn this section, we provide several details on the baselines included in our work. \n\n\\subsection{Encoding of the instructions}\n\\label{appendix:subsect:encoding}\n\nAccording to our schema (\\S\\ref{subsec:schema}), each instruction $I_t$ for the $t$-th task is a set that contains the following fields:\n$$\nI_t = \\setOf{ \n        \\I{t}{title},  \n        \\I{t}{def.},  \n        \\I{t}{avoid},  \n        \\I{t}{emph.},  \n        \\I{t}{prompt},\n        \\I{t}{pos. ex.},\n        \\I{t}{neg. ex.}\n    }\n$$\n\n\nTo feed the instances to LMs, we first encoder them into plain text. \nLet $enc(I, x)$ define a function that maps a given instruction $I$ and input instance $x$ to plain text. \nEvidently, there are many choices for this function. \nIn our study, we consider the following encodings: \n\n\\paragraph{\\textsc{No-instructions} encoding.}\nThis encoding is the conventional paradigm where no instructions exist: \n\n\\begin{equation} \\label{eq1}\n    \\small\n    \\begin{split}\n        enc(I_t, x)  := & \\mathtt{\\small input:} \\; x \\\\ \n        & \\mathtt{\\small output:}  \\textnormal{''}\n    \\end{split}\n\\end{equation}\n\n\n\\paragraph{\\textsc{prompt} encoding.}\nIn this encoding, we append the prompt message before the input:\n\n\\begin{equation} \\label{eq2}\n    \\small\n    \\begin{split}\n        enc(I_t, x)  := & \\mathtt{\\small Prompt:} \\; \\I{t}{prompt}  \\\\  \n        & \\mathtt{\\small input:} \\; x \\\\ \n        & \\mathtt{\\small output:}  \\textnormal{''}\n    \\end{split}\n\\end{equation}\n\n\\paragraph{\\textsc{Prompt + Definition} encoding.}\nIn this encoding, the prompt message and the task definition appear before the input: \n\\begin{equation} \\label{eq3}\n    \\small\n    \\begin{split}\n        enc(I_t, x)  := &  \\textnormal{``}\\mathtt{\\small Definition:} \\;  \\I{t}{def.}  \\\\ \n        & \\mathtt{\\small Prompt:} \\;  \\I{t}{prompt}  \\\\  \n        & \\mathtt{\\small input:} \\; x \\\\\n        & \\mathtt{\\small output:}  \\textnormal{''}\n    \\end{split}\n\\end{equation}\nIntuitively, this encoding is more informative and more complex than ``prompt'' encoding. \n\n\\paragraph{\\textsc{Full Instructions} encoding.}\nThis encoding contains all the instruction content: \n\\begin{equation} \n    \\label{eq4}\n    \\small\n    \\begin{split}\n        enc(I_t, x)  := \n        &  \\textnormal{``}\\mathtt{\\small Definition:} \\;  \\I{t}{def.}  \\\\ \n        & \\mathtt{\\small Prompt:} \\;  \\I{t}{prompt}  \\\\  \n        & \\mathtt{\\small Things \\; to \\; Avoid:} \\;  \\I{t}{avoid.}  \\\\  \n        & \\mathtt{\\small Emphasis \\& Caution:} \\;  \\I{t}{emph.}  \\\\  \n        & \\textnormal{``}\\mathtt{\\small Negative Example1-}   \\\\ \n        & \\hspace{0.7cm} \\mathtt{\\small input:} \\; \\I{t}{pos. ex.}\\mathtt{(input)}  \\\\  \n        & \\hspace{0.7cm} \\mathtt{\\small output:} \\; \\I{t}{pos. ex.}\\mathtt{(output)}  \\\\  \n        & \\hspace{0.7cm} \\mathtt{\\small reason:} \\; \\I{t}{pos. ex.}\\mathtt{(reason)}  \\\\  \n        & \\mathtt{\\small Negative Example2-}   \\\\ \n        & \\hspace{0.7cm} \\mathtt{\\small \\hdots }  \\\\ \n        & \\textnormal{``}\\mathtt{\\small Positive Example1-}   \\\\ \n        & \\hspace{0.7cm} \\mathtt{\\small input:} \\; \\I{t}{pos. ex.}\\mathtt{(input)}  \\\\  \n        & \\hspace{0.7cm} \\mathtt{\\small output:} \\; \\I{t}{pos. ex.}\\mathtt{(output)}  \\\\  \n        & \\hspace{0.7cm} \\mathtt{\\small reason:} \\; \\I{t}{pos. ex.}\\mathtt{(reason)}  \\\\  \n        & \\mathtt{\\small Positive Example2-}   \\\\ \n        & \\hspace{0.7cm} \\mathtt{\\small \\hdots }  \\\\ \n        & \\mathtt{\\small input:} \\; x \\\\ \n        & \\mathtt{\\small output:}  \\textnormal{''}\n    \\end{split}\n\\end{equation}\n\nwhere $enc_{\\textnormal{ex}} (I_t)$ is an alternating encoding positive and negative examples. We include as many examples as possible, before exceeding the input limit. \n\n\\begin{comment}\n\\newcommand{\\mathrel{+}=}{\\mathrel{+}=}\n\\begin{equation*} \n    \\begin{split}\n        & \\mathtt{for \\; (p, n) \\; in \\; zip(} \\I{t}{pos. ex.}, \\I{t}{neg. ex.} \\mathtt{):} \\\\ \n        & \\hspace{0.5cm} enc_{\\textnormal{ex}} (I_t) \\mathrel{+}= \\\\ \n        & \\hspace{0.99cm} \\textnormal{``}\\mathtt{\\small Positive Example-}  \\\\ \n        & \\hspace{0.99cm} \\mathtt{\\small input:} \\; \\mathtt{p}_{\\textnormal{\\tiny input}} \\; \\mathtt{\\small output:} \\; \\mathtt{p}_{\\textnormal{\\tiny output}}  \\\\  \n        & \\hspace{0.99cm}  \\mathtt{\\small reason:} \\; \\mathtt{p}_{\\textnormal{\\tiny reason}}  \\\\ \n        & \\hspace{0.99cm} \\mathtt{\\small Negative Example-} \\\\ \n        & \\hspace{0.99cm} \\mathtt{\\small input:} \\; \\mathtt{n}_{\\textnormal{\\tiny input}} \\;  \\mathtt{\\small output:} \\; \\mathtt{n}_{\\textnormal{\\tiny output}} \\\\  \n        & \\hspace{0.99cm} \\mathtt{\\small reason:} \\; \\mathtt{n}_{\\textnormal{\\tiny reason}} \\;\n        \\mathtt{\\small suggestion:} \\; \\mathtt{n}_{\\textnormal{\\tiny sugg.}}  \n        \\textnormal{''} \n    \\end{split}\n\\end{equation*}\n\\end{comment}\n\n\\paragraph{\\textsc{Positive Examples} encoding.}\nThis encoding contains only positive examples of the subtask (no task description, etc).  \n\n\\begin{equation} \n    \\label{eq5}\n    \\small\n    \\begin{split}\n        enc(I_t, x)  := \n       \n        & \\hspace{0.7cm} \\mathtt{\\small input:} \\; \\I{t}{pos. ex.}\\mathtt{(input)}  \\\\  \n        & \\hspace{0.7cm} \\mathtt{\\small output:} \\; \\I{t}{pos. ex.}\\mathtt{(output)}  \\\\  \n       \n       \n        & \\hspace{0.7cm} \\mathtt{\\small \\hdots }  \\\\ \n        & \\mathtt{\\small input:} \\; x \\\\ \n        & \\mathtt{\\small output:}  \\textnormal{''}\n    \\end{split}\n\\end{equation}\nSuch example-only have been used in several recent studies in the field~\\cite{zhao2021calibrate}. \n\n\\clearpage\n\\onecolumn\n\n\\twocolumn\n\n\\section{Analysis on Baseline Results}\n\\label{sec:appendix:banalysis}\n\n\n\n\\changed{\n\\subsection{Comparison to Raw Instructions}\n\\label{subsec:efratlevycomparison}\nWe seek to understand the value of breaking the tasks into sub-tasks and mapping them into our proposed schema (\\S\\ref{sec:mapping}). \nWe compute performance of raw instructions (first sub-task of four datasets), \nin the same vein as \n\\citep{efrat2020turking}'s setup. \nWe compare this to our  \\textsc{Full Instruction - neg examples} encoding. \nThe results in Table~\\ref{tab:comparison:raw:instructions} indicate that GPT3 leads to higher performance with our encoding (2nd row) compared to raw instructions (first row). \nWeak performance of LMs on raw instructions aligns with \\citep{efrat2020turking}'s finding that ``language model performs poorly''. \n\n\\newcolumntype{R}[2]{%\n    >{\\adjustbox{angle=#1,lap=\\width-(#2)}\\bgroup}%\n    l%\n    <{\\egroup}%\n}\n\\newcommand*\\rot{\\multicolumn{1}{R{30}{1em}}\n\n\n\\begin{table}[h]\n    \\small\n    \\centering\n    \\begin{tabular}{ccccc}\n         \\toprule\n          & \\rot{Quoref} & \\rot{MCTaco} & \\rot{CosmosQA} & \\rot{QASC}  \\\\\n         \\midrule\n         \\makecell{raw instructions} & 12.5 & 5.00 & 6.9 & 3.7  \\\\\n         \\makecell{our schema} &       25.8 & 42.6 & 17.7 & 51.3 \\\\ \n         \\bottomrule\n    \\end{tabular}\n    \\caption{Comparing GPT3 performance on raw crowdsourcing instructions vs. our encoding. All numbers are ROUGE-L.} \n    \\label{tab:comparison:raw:instructions}\n\\end{table}\n\nThis might be partly due to the verbose language of the raw instructions: \nthe average length of the raw instructions is $2.5k$ tokens, in comparison to $950$ tokens for our encoding. \nWhile repetition often helps human understanding, concise instructions seem to be more effective for computers. \n}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{comment}\n\\subsection{An Ablation Study of Instructional Elements}\n\\label{sec:ablation:study}\nWe conduct an ablation study with GPT3 on 3 distinct tasks (answer generation from Winogrande; question generation from QASC; verifying temporal reasoning category of a given question from MC-TACO). \nTable~\\ref{tab:ablation:subset} (top) shows the effect of eliminating various fields in the encoding while Table~\\ref{tab:ablation:subset} (bottom) indicates the gains from adding each field. \nThe overall observation is that GPT3 benefits the most from \\emph{positive examples}, mildly from \\emph{definition}, and deteriorates with \\emph{negative examples}. \nWe hypothesize it is easier for GPT3 to mimic the patterns in positive examples while utilizing \\emph{negative examples} requires deeper understanding.  \n\n\n\n\n\n\\begin{table}[h]\n    \\centering\n   \n   \n   \n   \n   \n        \\includegraphics[scale=0.73,trim=8.7cm 5.7cm 2cm 1.9cm]{figures\/ablation-subset.pdf}\n   \n    \\caption{An ablation study of the different fields included in \\textsc{Natural Instructions}{} based on GPT3. This model benefits the most from \\textsc{positive} examples and the least from \\textsc{negative} examples.  \n    }\n    \\label{tab:ablation:subset}\n\\end{table}\n\\newpage\n\\end{comment}\n\n\n\n\\begin{comment}\n\\begin{table}[ht]\n    \\centering\n    \\small\n     \\resizebox{\\columnwidth}{!}{\n    \\begin{tabular}{lcc}\n        \\toprule\n         error type & GPT3  & BART  \\\\\n      \\midrule\n       \n       \n       \n       \n            does not follow instruction and generate an invalid question & 14 & 8\\\\\n           \n            generates a nonsensical\/vague question & 4 & 47\\\\\n            copies the given fact or a subset of it & 8 & 3 \\\\\n            explains the question after generating it & 6 & 0\\\\\n            generates a yes\/no question & 12 & 4\\\\\n            generates candidate answers as output &4 & 0\\\\\n            generates questions whose answer does not exist &4 &3\\\\\n            \\makecell[l]{generates generic questions independent\\\\ of the given context} &6 &0\\\\\n        \\bottomrule\n        \\end{tabular}\n        }\n        \\caption{\n            Percentage of  errors on QASC QG task (\\S\\ref{sec:error:analysis}). \n            The numbers do not sum to 100 since the error types are not mutually exclusive. \n           \n        }\n        \\label{Tab: Error Analysis}\n\\end{table}\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{comment}\n\\begin{table*}[t]\n    \\small \n    \\centering\n    \\resizebox{\\linewidth}{!}{\n\\begin{tabular}{lcccccc|l||cccccc|l}\n\\toprule\n& \\multicolumn{7}{c}{BART} & \\multicolumn{7}{c}{GPT3} \\\\\n\\cmidrule(r){2-8} \\cmidrule(r){9-15} \n task category \u2192 & QG & AG & CF & IAG & MM & VF & avg & QG & AG & CF & IAG & MM & VF & avg  \\\\\n\\midrule\n\\textsc{No Instruction} & 26 & 6 & 0 & 21 & 33 & 7 & 13 & - & - & - & - & - & - & - \\\\\n\\midrule\n\\textsc{prompt} & 27 & 22 & 7 & 22 & 34 & \\textbf{9} & 20  & 33 & 32 & 14 & 13 & \\textbf{73} & 16 & 30 \\\\\n{\\ \\ \\ +\\textsc{definition}} & 35 & 24 & 50 & \\textbf{25} & 36 & 7 & 30$\\uparrow$ (+50) & 36 & 35 & 40 & 14 & 70 & 16 & 35$\\uparrow$ (+17)\\\\\n{ \\ \\ \\ +\\textsc{things to avoid}} & 33 & 24 & 4 & 24 & \\textbf{58} & \\textbf{9} & 25$\\uparrow$ (+25) & 28 & 33 & 11 & 16 & 68 & 14 & 28$\\downarrow$ (-7) \\\\\n{\\ \\ \\ +\\textsc{emphasis}} & 38 & 23 & 16 & \\textbf{26} & 49 & 3 & 26$\\uparrow$ (+30) & 29 & 28 & 18 & 16 & 72 & 16 & 30 \\\\\n{\\ \\ \\ +\\textsc{pos. examp.}} & 53 & 22 & 14 & \\textbf{25} & 17 & 7 & 23$\\uparrow$ (+15) & \\textbf{43} & 49 & 29 & 21 & 70 & \\textbf{36} & 41$\\uparrow$ (+37) \\\\\n{\\ \\ \\ +\\textsc{definition+pos. examp.}} & 51 & 23 & \\textbf{56} & \\textbf{25} & 37 & 6 & 33$\\uparrow$ (+65) & \\textbf{43} & 50 & \\textbf{45} & \\textbf{23} & 70 & 32 & \\textbf{44}$\\uparrow$(+47) \\\\\n{\\ \\ \\ +\\textsc{pos, neg ex+ explan.}} & 50 & 21 & 27 & 25 & 50 & 7 & 30 $\\uparrow$ (+50) & 32 & 19 & 8 & 12 & 61 & 13 & 24$\\downarrow$(-20) \\\\\n\\textsc{pos. examp.} & \\textbf{55} & 6 & 18 & \\textbf{25} & 8 & 6 & 20 & 30 & 32 & 15 & 16 & 68 & 23 & 31$\\uparrow$(+3) \\\\\n\\midrule\n\\textsc{Full Instruction} & 46 & 25 & 52 & 25 & 35 & 7 & 32$\\uparrow$ (+60) & 33 & 18 & 8 & 12 & 60 & 11 & 24$\\downarrow$(-20) \\\\\n{\\ \\ \\ -\\textsc{ examples }} & 40 & 24 & 36 & 25 & 55 & 8 & 31$\\uparrow$ (+55) & 31 & 34 & 39 & 14 & 69 & 13 & 33$\\uparrow$(+10) \\\\\n{\\ \\ \\ - \\textsc{neg. examp.}} & 52 & \\textbf{30} & 50 & \\textbf{25} & 47 & 8 & \\textbf{35}$\\uparrow$ (+75) & \\textbf{43} & \\textbf{54} & 44 & 21 & 70 & 32 & \\textbf{44}$\\uparrow$(+47) \\\\\n\\bottomrule\n\\end{tabular}\n}\n \\caption{\n    Full BART and GPT3 results with various input encodings for different task categories, under random split (\\S\\ref{subsec:split}).\n    Both models show improved results when encoded with instructions, comparing relative gains indicated in the `avg' columns (in percentage compared to \\textsc{prompt} encoding.)\n        Category names: QG: Question Generation, AG: Answer Generation, CF: Classification, IAG: Incorrect Answer Generation, MM: Minimal Text Modification, VF: Verification.\n       \n    }\n    \\label{tab:random:splitfull2}\n\\end{table*}\n\\end{comment}\n\n\n\n\\begin{comment}\n\\begin{figure*}\n    \\centering\n    \\begin{subfigure}[b]{0.4\\textwidth}\n        \\caption{GPT3}\n        \\includegraphics[scale=0.62,trim=0cm 0cm 0cm 0.1cm ]{figures\/gains-categories-gpt.pdf}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.4\\textwidth}\n        \\caption{BART}\n        \\includegraphics[scale=0.62,trim=0cm 0cm 0cm 0.1cm ]{figures\/gains-categories-bart.pdf}\n    \\end{subfigure}\n   \n    \\caption{\n        GPT3 and BART were evaluated with various encoding and various categories of tasks. \n        The benefit of instructions to the models depends on the semantics of the task. For instance, for GPT3 (left) \\emph{minimal text modification} category benefits a lot, while the benefits to \\emph{verification} tasks are minimal.  \n    }\n    \\label{fig:gains:per:categories}\n\\end{figure*}\n\\end{comment}\n\n\\begin{comment}\n\\subsection{Generalization vs. number of positive examples}\n\\label{subsection:numberofpositiveexamples}\nFig.~\\ref{fig:GPTexample} and \\ref{fig:BARTexample} illustrates the performance variation of models with respect to the number of examples. Clearly, addition of examples is not helping GPT3 and BART. Note that model performance with just the prompt+definition encoding is 35 in case of GPT3 and 30 in case of BART. This may suggest that the effort in creating many examples can be utilized to improve other aspects of Instruction. Another demerit of larger number of examples is that they increases input token size which increases the API usage cost in case of GPT3 and training time and higher memory usage in case of BART.\n\\end{comment}\n\n\\clearpage\n\n\\begin{comment}\n\\onecolumn\n\n\\subsection{User Study to Find Important Task-Specific Instruction Fields}\n\\label{subsec:appendix:user:study}\nWe ask our quality assessment annotators to also specify which instruction fields help them understand the task and answer prompts. For each of the 12 tasks in our evaluation set, we ask: \\textit{Which instruction field helps you the most to understand the task and answer questions and why? Remember, on removing this field significant major information should get lost.} We compile these results category-wise, and present them in Table \\ref{Tab: User Study}. In particular, there are two tasks Classification (CF) and  Minimal Text Modification (MM) for which humans find only a single instruction field to be important. We find that models also find the same fields to be most important, as evinced in Table \\S\\ref{tab:random:splitfull}), where the performance of models with these fields is higher than the rest. Interestingly, this is similar to the patterns observed in the model performance (Table \\S\\ref{tab:random:splitfull}).\n\n\\begin{figure*}[]\n    \\centering\n    \\includegraphics[scale=0.55,trim=0.7cm 1cm 0.1cm 0cm]{figures\/example_variation_GPT.pdf}\n    \\caption{GPT3 performance as a function of the number of examples in its encoding. The number of examples is limited by three upperbounds: 3, 10 and 70. This shows that addition of examples is not helping GPT3.\n    }\n    \\label{fig:GPTexample}\n\\end{figure*}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.55,trim=0.7cm 1cm 0.1cm 0cm]{figures\/example_variation_BART.pdf}\n    \\caption{BART performance as a function of the number of examples in its encoding. The number of examples is limited by two upperbounds: 3 and 10. This shows that addition of examples is not helping BART. Since BART's maximum token size is 1024, it can not fit a lot examples unlike GPT3, so we did not experiment further with larger number of examples.\n    }\n    \\label{fig:BARTexample}\n\\end{figure*}\n\\end{comment}\n\n\n\n\\begin{comment}\n\\begin{table*}\n    \\centering\n   \n    \\includegraphics[scale=0.65,trim=1.8cm 6.5cm 0cm 3cm]{figures\/ablation-polished.pdf}\n   \n    \\caption{Detailed results of the encoding ablation performed on three distinct subtasks.}\n    \\label{tab:ablation:all}\n\\end{table*}\n\\end{comment}\n\n\\begin{comment}\n\\begin{table*}\n    \\centering\n    \\includegraphics[scale=0.65,trim=1.7cm 8.9cm 0cm 2cm]{figures\/results_table_det.pdf}\n   \n    \\caption{\n        Empirical results \\textsc{Natural Instructions}{}. \n        The best numbers among the four encodings are indicated with \\textbf{bold}. \n        The first row is {\\color{gray} grayed out} since it is our oracle upperbound.\n    }\n    \\label{tab:results:detail}\n\\end{table*}\n\\end{comment}\n\\subsection{Evaluating Generalization Across Tasks} \n\n\n\n\\begin{comment}\n\\paragraph{\\textsc{Natural Instructions}{} tasks splits.}\nTo evaluate generalization across subtasks, we divide \\textsc{Natural Instructions}{} into two collections: \n(i) \\emph{evaluation} tasks \\task{eval}  (35 subtasks) for test and \n(ii) \\emph{non-evaluation} tasks \\task{non-eval}  (26 subtasks) for training.\\footnote{The tasks are enumerated in the appendix.} \nIn making this collection, we ensure that the tasks included in \\task{eval} accept a relatively reliable automatic evaluation.  For example,  those tasks that have restricted answer space (like classification tasks) or those that have several gold output references. The end tasks of the source datasets which are typically the answer generation tasks are often included in the evaluation set.\nAdditionally, we ensure to have at least one representative subtask from each of the semantic categories (\\S\\ref{sec:mapping}) in the \\emph{non-evaluation} collection\nHowever, tasks within categories are very different from each other. For instance, creation of DROP questions requires understanding of numerical reasoning and reading comprehension, whereas creation of Winogrande questions requires understanding of co-reference resolution and the requirement of the task to create twin question and answer pairs.\n\n\\daniel{\n    emphasize that not any two tasks are exactly the same. Every two QG tasks are different (e.g., MC-TACO vs CosmosQA). \n}\n\\paragraph{Evaluation.}\nWe formulate three evaluation settings with different  supervision types available to a model (Table~\\ref{tab:supervision:types}). \nIn `task-specific' setting, a model is supervised with the training instances of the evaluation task -- similar to the conventional setup. \nIn `few-shot' setting, a model  only observes a  few examples of the evaluation task.\\footnote{\n    We use ``few-shot'' to refer to \\emph{any setup with a small number of labeled examples},\n   \n    regardless of whether these examples are used for fine-tuning or inference-time conditioning (no gradient updates). \n}\nIn  `generalization' setting, a model does not observe any instances from the evaluation task.\n\\end{comment}\n\n\n\n\\begin{comment}\n\\paragraph {``no-instructions''} encoding.\nThis encoding is the conventional paradigm where no instructions exist, except the input instance (Eq.~\\ref{}). \n\n\\paragraph{\\emph{``prompt''} encoding.}\nIn this encoding, we append the prompt message before the input instance. \n\n\\paragraph{\\emph{``prompt + definition''} encoding.}\nIn this encoding, the prompt message and the task \\emph{definition} appear before the input instance.\nIntuitively, this encoding is more informative and more complex than \\emph{``prompt''} only encoding. \n\n\\paragraph{\\emph{``all instructions''} encoding.}\nThis encoding contains all the instruction content.\nWe include as many examples as possible, before exceeding the token limit of LMs. \n\n\\paragraph{\\emph{``positive examples''} encoding.}\nThis encoding contains only positive examples from the task instructions. \nSuch example-only encodings have been used in several recent studies in prompting LMs~\\cite{zhao2021calibrate}. \n\n\n\\end{comment}\n\n\\begin{comment}\n\\begin{table}[h]\n    \\centering\n    \\small\n    \\resizebox{\\columnwidth}{!}{\n    \\begin{tabular}{L{1.9cm}cL{3.9cm}}\n        \\toprule\n        Setup & Evaluation & Supervision \\\\\n        \\midrule\n       \n        task-specific & $T \\in$ \\task{eval} & all the instances of $T$ \\\\ \n        \\cmidrule(r){1-3} \n        few-shot & $T \\in$ \\task{eval} & \n       \n        instructions of $T$ \\\\ \n        \\cmidrule(r){1-3} \n        generalization & $T \\in$ \\task{eval} & instructions+ instances of \\task{non-eval} tasks + instructions of $T$ \\\\ \n       \n       \n        \\bottomrule\n    \\end{tabular}\n    }\n    \\caption{\n        Different modes of supervision considered in this work, when evaluating a model on the instances of a fixed task $T \\in$ \\task{eval}. \n    }\n    \\label{tab:supervision:types}\n\\end{table}\n\\end{comment}\n\n\\begin{comment}\n\\section{Evaluating Language Models to Address \\textsc{Natural Instructions}{}} \n\nWe use generative language models BART~\\cite{lewis2019bart} and GPT-3~\\cite{brown2020language} to address tasks in \\textsc{Natural Instructions}{}. Here, we describe how we encode instructions and instances into plain text and feed them into generative language models (\\S \\ref{subsect:encoding}). We then describe the model details (\\S \\ref{subsec:models}). \nWe then explain how we use language models to encode instruction (\\S \\ref{subsect:encoding}). \\daniel{to be updated}\n\\end{comment}\n\n\\begin{comment}\n\\paragraph{The benefit from instructions heavily depends on the task at hand.}\nFigure~\\ref{fig:gains:per:categories} shows the performance of our models on our task categories, broken down into several coarse input encodings. \nSimilar to our previous observations, \\emph{all instructions} encoding \\emph{typically} performs better than other encodings. \nHowever, these gains are not uniform across task categories. \n\n\n\n\\end{comment}\n\n\\begin{comment}\n\\paragraph{Task-specific BART (oracle upper-bound estimate).}\nWe train BART on input\/output instances of each task (no instructions) and evaluate on the same task. \nThis is the conventional setup where the model is fine-tuned to solve the task only, without any instructions involved.\nSuch a model, by design, won't generalize across different tasks since it is specialized to each subtask. \nHowever, the numbers elicited from this can be viewed as the upper-bounds for each task (i.e., how well can BART perform, if it were to be trained on many instances of this particular task). \n\n\\end{comment}\n\n\\begin{comment}\n\\subsection{Task-specific Calibration of GPT-3}\n\\label{subsec:calibration}\n\nWe transform inputs (Instructions) to make it more understandable for GPT-3. We have a human-in-the loop setup to perform calibration. We do calibration in two steps (i) we develop various calibration procedure by experimenting with various type of prompts and identifying the types of prompt which help model follow instructions better. This is done on the non-eval split. (ii) we employ one of the calibration procedures by looking at few samples of the end-task in the eval split.\n\\end{comment}\n\n\\begin{comment}\n\\paragraph{A Data Creation Toolbox:} \n\\textsc{Natural Instructions}{} covers various skills beyond question generation and answer generation, such as sentence paraphrasing, verification of whether a question is in a specified category, etc that are frequently used during NLP dataset creation. A successful model trained on \\textit{Natural Instructions} will be a toolbox for dataset creation. Our intuition behind the focus on dataset creation is that any NLP task can be expressed as a step in the data creation \n\\end{comment}\n\n\\section{Introduction}\n\nWe have witnessed great progress in solving many NLP datasets through fine-tuning pre-trained language models (LMs)~\\cite{peters2018deep,brown2020language}. \nMore recent studies show tremendous promise in  generalization \\emph{within} the set of observed tasks through multi-task training and unified encoding~\\cite{khashabi2020unifiedqa,aghajanyan2021muppet}.  \nHowever, cross-task generalization -- \\emph{generalization} to  \\emph{unseen} tasks -- has generally remained under-explored.\nFor example, can we supervise a model with instances of grammar checking or question answering tasks, yet expect it to solve a different task like question typing (Fig.\\ref{fig:teaster}).\nEvidently, humans are capable of such generalizations; an average human can follow natural language \\emph{instructions} to solve a variety of problems, as evident by the success of crowdsourcing platforms (also argued in~\\citet{efrat2020turking}). In this paper, we study if models can generalize to  \\emph{unseen} tasks given their \ncrowdsourcing instructions (Fig.\\ref{fig:teaster}). \n\n\n\n\\begin{figure}[t]\n    \\centering\n   \n   \n   \n   \n   \n   \n    \\includegraphics[scale=0.9, trim=0.75cm 0.8cm 0cm 1.0cm,clip=false]{figures\/teaser1-2.pdf}\n   \n    \\caption{\n        We construct the \\textsc{Natural Instructions}{} dataset from crowdsourcing instructions and instances of different NLP datasets.  We study if models can learn from  {\\emph{\\color{blue} seen}}  tasks and generalize to  {\\emph{\\color{red} unseen}} tasks given their natural crowdsourcing instructions. \n       \n       \n       \n       \n       \n       \n       \n       \n    }\n    \\label{fig:teaster}\n\\end{figure}\n\n\n\n\\begin{figure*}\n     \\centering\n     \\begin{subfigure}[b]{0.49\\textwidth}\n         \\centering\n       \n       \n       \n            \\small\n        \\resizebox{\\linewidth}{!}{\n        \\begin{tabular}{ccc}\n            \\toprule \n             Task   & \\makecell{Instance-Level\\\\Generalization}  & \\makecell{Task-Level\\\\Generalization} \\\\\n             \\midrule \n             \\makecell{Training\\\\data}  & $X^{\\color{darkgreen} \\text{train}}, Y^{\\color{darkgreen} \\text{train}}$ & \\makecell{$(I_t, X_t^{{\\color{darkgreen} \\text{train}}}, Y_t^{{\\color{darkgreen} \\text{train}}})$ \\\\ $t \\in \\text{\\task{\\color{blue} seen}}  $ \\\\ }  \\\\ \n             \\midrule \n             Evaluation &  \\makecell{ $x \\rightarrow y$ \\vspace{0.2cm} \\\\ where: \\\\  $(x, y) \\in (X^{ \\color{purple} \\text{test}}, Y^{ \\color{purple} \\text{test}})$  \\vspace{0.3cm} }  & \\makecell{$(x, I_t) \\rightarrow y$ \\vspace{0.2cm} \\\\  where: \\\\ $(x, y) \\in (X_t^{ {\\color{purple} \\text{test}}}, Y_t^{{\\color{purple} \\text{test}}})$ \\\\ $t \\in$ \\task{\\color{red} unseen} } \\\\ \n            \\bottomrule\n        \\end{tabular}\n        }\n        \\caption{\n            A comparison of \\emph{task} vs \\emph{instance}-level generalization \n            $I_t$, $X_t$ and $Y_t$ indicate natural language instructions, input, and output sets respectively for task $t$.\n           \n            In the conventional setup, training and evaluation are done on the instances of the same task. \n            However, in task-level generalization, a model is expected to generalize to  {\\color{red} unseen} tasks, where \\task{\\color{red} unseen} $\\cap$ \\task{\\color{blue} seen}$ = \\emptyset $. \n        }\n        \\label{tab:comparison}\n     \\end{subfigure}\n     \\hfill\n     \\begin{subfigure}[b]{0.49\\textwidth}\n         \\centering\n       \n       \n       \n         \\includegraphics[scale=0.64,trim=0.2cm 0cm 0cm 1cm,clip=false]{figures\/fig5scaleup-4.pdf}\n         \\caption{BART evaluation on {\\emph{unseen}} tasks ($y$-axis is perf. on \\task{unseen}) when supervised with {\\emph{seen}} tasks ($x$-axis is $|$\\task{seen}$|$). \n           \n            \\changed{\n             A model using {\\color{purple}instructions} ($I_t$) consistently improves with more observed tasks. In contrast, models with {\\color{orange} no access to the instructions} show no sign of improved generalization. \n             }\n            \\changed{Details in \\S\\ref{subsec:supervision:size:experiment}.}\n        }\n        \\label{fig:scaling:tasks}\n    \\end{subfigure}\n    \\caption{The formal definition of generalization to unseen tasks (a) and a summary of its empirical outcome (b).  }\n   \n\\end{figure*}\n\n\n\n\nWe build \\textsc{Natural Instructions}, a dataset consisting of {\\it natural} crowdsourcing instructions for various tasks and their instances. \nTraining on {\\it seen} tasks $\\text{\\task{\\color{blue} seen}}$ in our dataset, we  build a model that learns to follow natural instructions that define a task and perform tasks  (i.e., mapping input to output).\nTesting on \\emph{unseen} tasks \\text{\\task{\\color{red} unseen}}, we evaluate if the model can perform {\\it unseen} tasks  solely from their instructions and without any task-specific labeled data   (Table~\\ref{tab:comparison}; right).  \nIn contrast to the instance-level generalization (Table~\\ref{tab:comparison}; left), our model uses instruction as additional input, and evaluations are done on tasks that were not observed in the training stage. \n\n\\changed{\nWe compile \\textsc{Natural Instructions}{} from task instructions written by researchers for crowdsourcing existing NLP datasets. \nSuch crowdsourcing instructions often elaborate a variety of details about how a task should (and should not) be done.  \nTo provide a systematic study of various elements of crowdsourcing instructions, we map them\n}\nto a unified {\\it schema} to cover the most important elements of task descriptions --- such as definition, constraints, positive and negative examples. \nWe collect tasks in  \\textsc{Natural Instructions}{}  as minimal stand-alone steps provided to crowdworkers to complete a downstream NLP task. \nFor example, tasks collected from  \n\\changed{QASC~\\cite{khot2020qasc} include sub-tasks about generating topic words or combining facts, as well as answering multi-hop questions. \nTherefore our dataset not only contains typical downstream tasks in NLP, but also the intermediate subtasks that are not well-represented in the common benchmarks. \n}\nThe unified schema and the collection of minimal subtasks enable training LMs that can generalize across different tasks by learning from instructions.\nIn total, our dataset consists of 61 distinct NLP tasks and $193k$ instances.\n\n\nOur experimental results indicate that LMs learn to leverage natural language instructions as they show improved generalization to new\ntasks. \nFor example, a BART~\\cite{lewis2019bart}  achieves a 19\\% gain in terms of cross-task generalization compared to a model not using instructions\n(\\S\\ref{sec:experiments}). \nImportantly, LMs can generalize better to unseen tasks if they observe more tasks in training (Fig.\\ref{fig:scaling:tasks}). \nThis upward trajectory suggests the potential for stronger cross-task generalizable models upon scaling up the diversity of tasks represented in a meta-dataset of task instructions. \nDespite the benefits of instructions, we observe a sizable gap between models' generalization and their estimated upperbounds (\\ref{subsec:task-specific}), encouraging the community to work on this challenging problem. \n\n\n\n\\vspace{.1cm}\n\\noindent\\textbf{Contributions:} In summary, the contributions of this work are as follows: \n(a) we introduce \\textsc{Natural Instructions}{}, a  dataset of human-authored instructions  curated from existing well-known datasets mapped to a unified schema, providing training and evaluation data for learning from instructions;\n(b) we build models that can encode instructions and show: \n(b.1) the benefit of cross-task generalization by leveraging instructions; \n(b.2) the importance of different elements of instructions in the performance;  \n(b.3) noteworthy headroom for improvement on our benchmark, which hopefully will motivate further work in this direction. \n\n\\input{related}\n\n\n\\changed{\n\\section{Defining Cross-Task Generalization}\n\\label{subsec:input:output}\nHere we formally define the problem setup for generalization across tasks. \nEach task $t$ consists of input\/output instances $(X_t, Y_t)$ and is described in terms of its natural language instructions $I_t$. \n\n\n\\vspace{-.2cm}\n\\paragraph{Task-specific models.}\nStandard supervised learning algorithms use task-specific labeled instances to learn a mapping from input $x$ to output $y$:  $M(x)=y$ for $(x,y)\\in (X_t^{\\text{train}}, Y_t^{\\text{train}})$ and is evaluated on the test instances of the same (or similar) task $(X_t^{\\text{test}}, Y_t^{\\text{test}})$. We refer to this as the \\emph{instance-level} generalization (Table~\\ref{tab:comparison}; left).\n\n\\vspace{-.2cm}\n\\paragraph{Cross-task models.} \nIn this setup, the goal is to learn a model $M$ that at inference obtains the output $y$ given the input $x$ and the task instruction $I_t$: $M(I_t, x) = y, \\;  \\mbox{for} \\ (x,y)\\in (X_t, Y_t)$.\nIn contrast to the task-specific models, no task-specific training data is used to learn the mapping $M$. We collect \\textsc{Natural Instructions}\\ (\\S\\ref{sec:construction:natural:instructions}) to study this question: can a model be trained to follow instructions via training tasks \\task{seen} and be generalized to follow instructions for a task $t' \\in$ \\task{unseen}. \nWe refer to this as a \\emph{task}-level generalization (Table~\\ref{tab:comparison}; right). \n}\n\n\\section{\\textsc{Natural Instructions}{}}\n\\label{sec:construction:natural:instructions}\n\n\\textsc{Natural Instructions}{} consists of instructions that describe a task (e.g., question answering) and instances of that task (e.g., answers extracted for a given question). \nFig.\\ref{fig:examples} shows  an example instruction for the task of `generating questions that require an understanding of event duration' accompanied with positive and negative examples \nthat contextualize the task. \nHere we introduce a schema for representing instructions  (\\S\\ref{subsec:schema}) and then describe how existing datasets (their crowdsourcing templates) are mapped into  our schema  (\\S\\ref{sec:mapping}).\n\n\n\n\n\n\\begin{figure}[t]\n    \\centering\n   \n   \n    \\includegraphics[scale=0.70, trim=0.45cm 0.8cm 0cm 0.99cm]{figures\/examples_detailed_two.pdf}\n    \\caption{\n        An example from our dataset.  \n        Note that it follows the schema provided in Fig.\\ref{fig:schema_plate}. See Fig~.\\ref{fig:examplesfull} for more examples.\n    }\n    \\label{fig:examples}\n\\end{figure}\n\n\\subsection{Instruction Schema}\n\\label{subsec:schema}\n\n\nInstructions used in crowdsourcing various datasets, are written by distinct authors for different purposes, and  they are different in a variety of ways (see  Appendix~\\ref{appendix:analysis:templates} for their differences.) We introduce a unified schema (Fig.\\ref{fig:schema_plate}) to consistently represent these diverse forms of instructions. \nOur instruction schema is the result of our pilot study conducted on a subset of  datasets. Below we describe the ingredients of this schema: \n\n\\begin{figure}[t]\n\\centering\n    \\includegraphics[width=0.97\\columnwidth,trim=0.35cm 0.8cm 0.5cm 1cm]{figures\/schema-2.pdf}\n  \\caption{The schema used for representing instruction in \\textsc{Natural Instructions}{} (\\S\\ref{subsec:schema}), shown in plate notation.\n  }\n  \\label{fig:schema_plate}\n\\end{figure}\n\n\n\\begin{itemize}[noitemsep,topsep=0pt,parsep=3pt,leftmargin=0.3cm]\n   \n    \\item \\underline{\\textsc{Title}} provides a high-level description of a task and its associated skill (such as question generation, answer generation).\n    \\item  \\underline{\\textsc{Prompt}} is a single sentence command that often appears before the input instance and connects it to the instructions.\n    \\item \\underline{\\textsc{Definition}} provides the core detailed instructions for a task. \n   \n   \n    \\item \\underline{\\textsc{Things to Avoid}} contain instructions regarding undesirable annotations that must be avoided. These help to define the scope of a task and the space of acceptable responses. \n   \n    \\item \\underline{\\textsc{Emphasis and Caution}} are short, but important statements highlighted in the crowdsourcing templates which were intended to be  emphasized or warned against.\n    \\item \\underline{\\textsc{Positive Examples}} contain inputs\/outputs similar to the input given to a worker\/system and its expected output,  helping crowdworkers better understand a task~\\cite{ali1981use}. \n    \\item \\underline{\\textsc{Negative Examples}} contain inputs\/outputs to emphasize \\textsc{Things to Avoid} by providing examples that must not be produced. \n   \n   \n   \n    \\item \\underline{\\textsc{Reason}}  provides explanations behind why an example is positive or negative.\n    \\item \\underline{\\textsc{Suggestion}} contains suggestions on how a negative example could be modified to turn it into a positive example. \n\\end{itemize}\n\n    The next section describes the process of mapping the raw instructions (designed for crowdworkers) to our instruction schema. \n\n\n\n\n\n\n\n\n\n\\subsection{Constructing \\textsc{Natural Instructions}} \n\\label{sec:mapping}\n\n\n\n\n\n\\subsubsection{Collecting Data}\n\\label{sec:datacollection}\n\\paragraph{Collecting raw instructions and instances.} \n    We use existing, widely adopted NLP benchmarks that are collected via crowdsourcing platforms and hence, come with crowdsourcing templates. \n    In the first step, we identified several datasets and engaged with their authors to get their crowdsourcing templates and raw data. \nThis yields the following datasets: \nCosmosQA~\\cite{huang2019cosmos}, \nDROP~\\cite{dua2019drop}, \nEssential-Terms~\\cite{khashabi2017learning}, \nMCTACO~\\cite{zhou2019going}, \nMultiRC~\\cite{khashabi2018looking}, \nQASC~\\cite{khot2020qasc}, \nQuoref~\\cite{dasigi2019quoref}, ROPES~\\cite{lin2019reasoning} and\nWinogrande~\\cite{sakaguchi2020winogrande}.\\footnote{\n        We only focus on textual instructions and  avoid datasets that involve visual or auditory steps, mostly focusing on QA datasets that were available to the authors. \n} \n    \n\\vspace{-.2cm}\n\\paragraph{Splitting crowdsourcing instructions into minimal tasks.} \nAlmost all the crowdworking instructions include sequences of steps to guide crowdworkers  in creating task instances.\nFor example, QASC and MCTACO include 7 and 19 steps in the data creation process, respectively. \nWe divide crowdsourcing instructions into their underlying steps and generate multiple subtasks that are minimal and standalone.\\footnote{\n    We eliminate tasks that involve model-in-the-loop. \n} Table~\\ref{tab:sample:tasks} shows subtasks extracted for Quoref and QASC. For example, the main task in Quoref is to answer a question given a context paragraph, but the crowdsourcing template consists of two sub-tasks of {\\it question generation}  and {\\it answer generation} with their separate instructions.  This process  results in a more consistent definition of tasks, enabling a successful mapping of instructions into our schema,  in contrast to the work of \\citet{efrat2020turking} that uses crowdsourcing instructions as-is. \n\n\n\\begin{table}\n    \\centering\n    \\footnotesize \n    \\begin{tabular}{ll}\n        \\toprule\n        source dataset & task  \\\\\n        \\midrule\n         \\multirow{2}{*}{\\makecell{Quoref\\\\ \\cite{dasigi2019quoref} }} & question generation \\\\ \n            & answer generation \\\\ \n        \\midrule\n         \\multirow{6}{*}{\\makecell{QASC\\\\ \\cite{khot2020qasc}} }  & topic word generation \\\\\n                & fact generation \\\\ \n                & combining facts \\\\ \n                & question generation \\\\ \n                & answer generation \\\\ \n                & incorrect answer generation \\\\ \n        \\bottomrule\n    \\end{tabular}\n    \\caption{\n        Examples of the datasets and the tasks formed from them. \n        The extracted tasks are independent annotation assignments in the crowdsourcing templates of the datasets. \n        The complete list is in Table~\\ref{tab:structure} in  Appendix. \n    }\n    \\label{tab:sample:tasks}\n\\end{table}\n\n\\begin{table}\n    \\footnotesize \n    \\begin{tabular}{lcc}\n        \\toprule\n        category  & \\# of tasks  & \\# of instances  \\\\\n        \\midrule\n        {question generation} & 13 & 38$k$ \\\\ \n        {answer generation} &  16 & 53$k$ \\\\ \n        {classification}   & 12 & 36$k$ \\\\ \n        {incorrect answer generation} & 8 & 18$k$ \\\\ \n        {minimal modification}  & 10 & 39$k$ \\\\ \n        {verification}  & 2 & 9$k$ \\\\ \n        \\midrule\n        Total & 61 & 193$k$ \\\\\n        \\bottomrule\n    \\end{tabular}\n    \\caption{Task categories and their statistics. \n   \n    }\n    \\label{tab:taskcategories}\n\\end{table}\n\n\nIn total, there are 61 tasks, which are categorized into 6 semantic categories (Table~\\ref{tab:taskcategories}). \nWe assigned these broad categories to the tasks to understand their collective behavior in the experiments. \nIt is noteworthy that, despite the apparent resemblance of the tasks included in the same category,  \nany pair of tasks are distinct. \nFor example, while \\emph{question generation} is part of Quoref, CosmosQA, and QASC, each has its own separate variant of the question generation task (see Fig.\\ref{fig:task_specification} in Appendix). \n\n\n\n\n\\subsubsection{Mapping Raw Instructions to Schema } \n\\label{subsec:maptoschema}\n     We manually fill in the fields of our instruction schema  with the content from the crowdsourcing instructions.\n    For instance, parts of the raw instructions that are highlighted for emphasis are incorporated as part of our \\emph{emphasis\/caution} field. \nThe modifications suggested in this step were applied by one author and were verified by another author.\\footnote{On average, the process of data curation for each task takes around 5 hrs-34 hrs (details in Appendix; Table~\\ref{tab:datacuration}).} \n\n\\vspace{-.2cm}\n\\paragraph{Improving description quality and consistency.}\n    We edit raw instructions to ensure their quality. Particularly, we fix  writing issues (typos, ambiguities, etc.) and  redact repetitions. \n    While repetition often helps in augmenting human understanding, short and concise instructions are often more effective for computers due to their limited attention span~\\cite{beltagy2020longformer}. \n    \n\\vspace{-.2cm}\n\\paragraph{Augmenting examples and reasons.}\n   \n There is a large variance in the number of examples provided in the raw instructions. Instructions often include more positive examples, or some instructions do not include any negative examples (e.g., QASC). \nWhenever possible, we add negative examples such that each task has at least two negative examples. \nFurthermore, not  all raw instructions contain \\textsc{reasons} or \\textsc{suggestions} for each of their examples. For example, positive examples are usually not accompanied by explanations, and most datasets do not include suggestions.\nWe add them, wherever such information is missing in the instructions. \n\n\\vspace{-.2cm}\n\\paragraph{Collecting input\/output instances for subtasks.} \nMost of our tasks are the intermediate steps in the crowdsourcing process. \n    Therefore, to extract input\/output instances for each task, we need to parse the raw annotations of crowdworkers for every step. Since each dataset stores its annotations in a slightly different format, extracting and unifying such intermediate annotations can be non-trivial. \n   \n   \n   \n   \n   \n\\vspace{-.2cm}\n\\paragraph{Verification.} \n\\changed{\nAn annotator verified the quality of the resulting data in consultation with dataset authors. \nThe annotator iterated on the authors' feedback (avg of 3 iters) until they were \nsatisfied. \n}\n\n\\vspace{-.2cm}\n\\paragraph{Quality assessment.}\nWe ask independent human annotators to answer 240 random instances (20 instances from 12 random tasks, used later for our evaluation~\\S\\ref{subsec:split}). \nThe subsequent evaluation of the human-generated responses results in more than 96\\% accuracy, which indicates that humans can effortlessly understand and execute our instructions. \n\n\n\n\n\n\n\n\n\n\n\\subsubsection{\\textsc{Natural Instructions}\\ Statistics}\n\\label{subsec:dataset:statistics}\n\nIn summary, \\textsc{Natural Instructions}\\ consists of  subtasks  each with a set of instructions and input\/output instances (Fig.\\ref{fig:examples} and \\ref{fig:schema_plate}). The complete list of instructions is included in the appendix. In total, the dataset includes 61 tasks and 193$k$ instances.\nTable~\\ref{tab:taskcategories} shows data statistics for each task category.\\footnote{We limit the number of instances in each task to $6.5k$ to avoid massive instance imbalance.} On average, instructions  contain 4.9 positive examples and 2.2 negative examples. \nThe longest element of instructions is usually \\textsc{Definitions} with 65.5 tokens and the shortest is \\textsc{title} with 8.3 tokens (more statistics in Table~\\ref{tab:schemastat}).\n\n\\begin{table}[ht]\n    \\centering\n    \\small\n    \\begin{tabular}{lc}\n        \\toprule\n        statistic & value  \\\\ \n        \\midrule\n        ``title'' length  & 8.3 tokens \\\\ \n        ``prompt'' length  & 12.6 tokens \\\\ \n        ``definition'' length & 65.5 tokens \\\\ \n        ``things to avoid'' length  & 24.1 tokens\\\\ \n        ``emphasis\/caution'' length  & 45.0 tokens\\\\\n        ``reason'' length  & 24.9 tokens\\\\ \n        ``suggestion'' length & 19.6 tokens\\\\ \n       \n       \n       \n        num of positive examples & 4.9 \\\\ \n        num of negative examples & 2.2 \\\\ \n       \n        \\bottomrule\n    \\end{tabular}\n    \\caption{\n    Statistics of \\textsc{Natural Instructions}{}\n   \n}\n    \\label{tab:schemastat}\n\\end{table}\n\n\n\n\\section{Problem Setup and Models }\n\\label{subsec:setup}\n\\changed{\nHere we define different cross-task generalization settings  (\\S \\ref{subsec:split}) and the models (\\S\\ref{subsec:models}). \n}\n\n\\begin{comment}\n\\subsection{Learning Tasks From Instructions} \n\\label{subsec:input:output}\nEvery  task $t$ in \\textsc{Natural Instructions}{} consists of an instruction $I_t$ and a set of input and output instances $D_t=\\{(x,y)|x \\in X_t,y \\in Y_t\\}.$\n\\vspace{-.2cm}\n\\paragraph{Task-specific models.}  Standard supervised learning uses task-specific training instances  to train a model that learns a mapping between input and output:  $M(x)=y$ for $(x,y)\\in D_t$. \n\n\\vspace{-.2cm}\n\\paragraph{Learning from instructions.} In this setup, the goal is to learn a model $M$ that at inference obtains the output $y$ given the input $x$ and the task instruction $I_t$: $M(I_t, x) = y, \\;  \\mbox{for} \\ (x,y)\\in D_t$.\n\nIn contrast to the task-specific models, no task-specific training data is used to learn the mapping $M$. Instead, we use \\textsc{Natural Instructions}\\ to study this question: can a model be trained to follow instructions via training tasks \\task{seen} and be generalized to follow instructions for a task $t' \\in$ \\task{unseen}. \nWe study various generalization settings with different splits of the tasks. \n\n\\end{comment}\n\n\\subsection{Task Splits and Generalizations Types}\n\\label{subsec:split}\n\n\n\n\n\\paragraph{Random split.}\nThis setup follows the common practice in benchmarking NLP models with random data splits. Here, two tasks from each task category (Table~\\ref{tab:taskcategories}) in \\textsc{Natural Instructions}{} are randomly selected for evaluation, and the rest of the tasks are used for training. This leads to 12 tasks in \\task{unseen} and 49 tasks in \\task{seen}.\\footnote{Those tasks that do not accept a relatively reliable automatic evaluation are excluded from \\task{unseen}. } \n\n\n\\paragraph{Leave-one-out generalization.}\nTo better understand the nature of cross-task generalization, we study more restrictive settings of dividing training and evaluation tasks. \n\n\\noindent \\ul{leave-one-category}: evaluates how well a model  generalizes to a task category if it is trained on others  -- no task of that category is in \\task{seen}. \n\n\n\\noindent \\ul{leave-one-dataset}: evaluates how well a model can generalize to all tasks in a particular dataset  if it is trained on all other tasks -- no task of that dataset is in \\task{seen}.\nThis split prevents any leakage across tasks that belong to the same source datasets. \n\n\\noindent \\underline{leave-one-task}: evaluates how well a model can learn a single task by training on all other tasks. \\\\\n\n\n\n\n\\subsection{Models}\n\\label{subsec:models}\nWe build models using pre-trained LMs with encoder-decoder architectures BART~\\cite{lewis2019bart} for fine-tuning and GPT3~\\cite{brown2020language} for few-shot experiments. \n \\paragraph{Encoding instructions and instances.} \n  For every problem setup, we map a given instruction $I_t$ and an input instance $x$ into a textual format and decode an output $y$ and obtain $enc(I_t, x)$. \nThis encoding function is then fed to an encoder-decoder model to predict $y$: $M:enc(I_t, x) \\rightarrow y$. \n\n\n\\begin{figure}\n\\centering\n\\begin{boxedminipage}{\\columnwidth}\n\\begin{equation*} \n    \\small\n    \\begin{split}\n        & \\mathtt{\\small Prompt:} \\;  \\I{t}{prompt}  \\\\ \n        & \\mathtt{\\small Definition:} \\;  \\I{t}{Definition}  \\\\  \n        & \\mathtt{\\small Things \\; to \\; Avoid:} \\;  \\I{t}{avoid.}  \\\\  \n        & \\mathtt{\\small Emphasis \\& Caution:} \\;  \\I{t}{emph.}  \\\\  \n        & \\textnormal{}\\mathtt{\\small Negative Example1-}   \\\\ \n        & \\hspace{0.7cm} \\mathtt{\\small input:} \\; \\I{t}{pos. ex.}, \\mathtt{\\small output:} \\; \\I{t}{pos. ex.},\n         \\mathtt{\\small reason:} \\; \\I{t}{pos. ex.}  \\\\  \n      \n        & \\mathtt{\\small Positive Example1-}   \\\\ \n        & \\hspace{0.7cm} \\mathtt{\\small input:} \\; \\I{t}{pos. ex.}, \n          \\mathtt{\\small output:} \\; \\I{t}{pos. ex.} \\mathtt{\\small reason:} \\; \\I{t}{pos. ex. }  \\\\  \n       \n        & \\mathtt{\\small input:} \\; x,  \\mathtt{\\small output:}  \\textnormal{''}\n    \\end{split}\n\\end{equation*}\n\\end{boxedminipage}\n\\caption{Encoding instruction $I_t$, where $I_t^c$ refers to the text of a component $c$ in the instruction schema.}\n\\label{fig:encoding}\n\\end{figure}\nEncoding instances follows a standard NLP paradigm of mapping an input instance to text. \nEach instruction $I_t$ consists of multiple elements as described in our instruction schema (\\S\\ref{subsec:schema}). Here, we map each element of the instruction to a textual format and append it before the input instance.  Fig.\\ref{fig:encoding} shows how we encode the full instruction. \n\nTo study the impact of each instruction element for cross-task generalization, we compare these encodings: (1) \\textsc{prompt}, (2) \\textsc{pos. examples}, (3) \\textsc{prompt + definition},  (4) \\textsc{prompt + things to avoid},  (5) \\textsc{prompt + emphasis} , (6) \\textsc{prompt + pos. examples}, (7) \\textsc{prompt + + definition + pos. examples}, and (8) \\textsc{Full instruction}.\n\\changed{\n Each of these (e.g., \\textsc{prompt} and \\textsc{pos. examples}) correspond to prompting setups in the recent literature~\\cite{scao2021many,lu2021fantastically}. \n} \n\n \\begin{table*}[t]\n    \\small \n    \\centering\n       \n        \\begin{tabular}{clcccc}\n        \\toprule\n           \n       \n       \n         \\makecell{model \u2193} & \\makecell{evaluation set \\task{unseen} \u2192}    & \\makecell{random split\\\\of tasks}   &\\makecell{leave-one-\\\\category (QG)} & \\makecell{leave-one-\\\\dataset (QASC)} & \\makecell{leave-one-\\\\task (QASC QG)} \\\\\n        \\cmidrule(lr){1-1} \\cmidrule(lr){2-2} \\cmidrule(lr){3-3} \\cmidrule(lr){4-4} \\cmidrule(lr){5-5} \\cmidrule(lr){6-6} \n       \n       \n        \\multirow{2}{*}{\\makecell{BART (fine-Tuned)} } & \\textsc{No instructions} & 13 & 6 & 37 & 20 \\\\\n        & \\textsc{Full instructions} & \\textbf{32} & \\textbf{17} & \\textbf{51} & \\textbf{56} \\\\\n        \\midrule\n        GPT3 (not fine-tuned)& \\textsc{Full instructions} & 24 & 33  & 22 & 33 \\\\\n        \\bottomrule\n        \\end{tabular}\n       \n    \\caption{Cross-task generalization of BART under various splits (\\S\\ref{subsec:split}).\n   \n    Fine-tuned BART shows improved performance when provided with instructions. \n     It also archives better performance than GPT3, despite being over $1k$ times smaller. \n   \n    \\changed{All numbers are ROUGE-L. }\n    }\n    \\label{tab:bart:generalization:all:splits}\n\\end{table*}\n\n\\vspace{-.2cm}\n\\paragraph{BART.}\n\\label{sec:bart}\nWe use  BART (base)~\\cite{lewis2019bart} which allows us to fine-tune its  model parameters. \nThis is an encoder-decoder architecture with $140m$ parameters.\nFor each setup, the input is encoded using different instruction elements,  trained  on all \\task{seen} tasks, and evaluated  on \\task{unseen} (\\S\\ref{subsec:split}). \n\n\\vspace{-.2cm}\n\\paragraph{GPT3.}\nAs a comparison, we evaluate\nGPT3~\\cite{brown2020language} which is a $175B$ parameter autoregressive LM ($\\times1.2k$ larger than BART) and has shown promising results in mimicking demonstrations provided in its prompt.\nWe cannot fine-tune the parameters of this massive model and use it as-is  \nunder its default setting on the evaluation tasks in \\task{unseen} (\\S\\ref{subsec:split}) using the  encoding introduced earlier. \n\n\\begin{comment}\n \\begin{table*}[t]\n    \\small \n    \\centering\n        \\resizebox{\\linewidth}{!}{\n        \\begin{tabular}{cl|c|cc|cc|cc}\n        \\toprule\n         model \u2193  &  & random split & \\multicolumn{6}{c}{leave-one-$x$ split} \\\\\n       \n        \\cmidrule(lr){3-3} \\cmidrule(lr){4-9}\n         & & &\\multicolumn{2}{c}{$x =$ category} & \\multicolumn{2}{c}{$x =$ dataset} & \\multicolumn{2}{c}{$x =$ task} \\\\\n        \\cmidrule(lr){4-5} \\cmidrule(lr){6-7} \\cmidrule(lr){8-9} \n        & evaluation set \\task{unseen} \u2192 & \\makecell{ALL} &\\makecell{AG} & \\makecell{QG} & \\makecell{QASC} & \\makecell{Quoref} & \\makecell{Winogrande AG } & \\makecell{QASC QG } \\\\\n        \\midrule\n       \n       \n       \n        \\multirow{4}{*}{\\makecell{\\cha{BART-Fine-Tuned}} } & \\textsc{No instructions} & 13 & 11 & 6 & 37 & 10 & 11 & 20 \\\\\n        \\cmidrule(lr){2-9}\n        & \\textsc{prompt+definition} & 30 &18 & 10 & 43 & \\textbf{39} & 11 & 22 \\\\\n        & \\textsc{prompt+pos. examp.} & 23 &18 & \\textbf{20} & 47 & 33 & 16 & 55 \\\\\n       \n       \n        & \\textsc{Full instructions} & \\textbf{32} &\\textbf{19} & 17 & \\textbf{51} & 37 & \\textbf{19} & \\textbf{56} \\\\\n        \\midrule\n       \n        \\cha{GPT3-Few-Shot}& \\textsc{Full instructions} & 24 & 18 & 33  & 22 & 24 & 10 & 33 \\\\\n        \\bottomrule\n        \\end{tabular}\n        }\n    \\caption{BART generalization under various leave-one-out splits (\\S\\ref{subsec:split}).  Encoding instructions improve cross-task generalization across all settings.  \n    \\changed{All numbers are ROUGE-L. }\n    }\n    \\label{tab:bart:generalization}\n\\end{table*}\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\\input{maintable}\n\n\n\n\n\\section{Experiments}\n\\label{sec:experiments}\n\\vspace{-.1cm}\n\\paragraph{Evaluation metrics.}\nWe treat all of our tasks as text generation problems and evaluate them with \nautomated evaluation metrics for text generation. \nIn particular, we use \nROUGE-L~\\cite{lin2004rouge} to automatically evaluate the generated outputs.\\footnote{\nOur experiments show that other metrics, e.g. BLEURT~\\cite{sellam2020bleurt} are also correlated with ROUGE-L, which has also been used in generative QA tasks.\n}\n\n\n\\vspace{-.2cm}\n\\paragraph{Implementation details.}\nFor BART, our  models are trained for 3 epochs with a learning rate of 5e-5 for a given training split and input encoding. For GPT3, we use the {\\texttt{davinci-instruct}} engine and produce outputs with greedy decoding, \ngenerating up to a maximum number of tokens of 16 (the default value). We use the default stop condition which is 2 newline tokens.\\footnote{The relevant code is available at:  \n\\url{https:\/\/github.com\/allenai\/natural-instructions-v1}\n}\n\n\n\\subsection{Generalization Under Various Task Splits}\n\\label{sec:gen:various:splits}\n\n\\changed{\nTable~\\ref{tab:bart:generalization:all:splits} reports the results of the BART model train and evaluated with various task splits (\\S\\ref{subsec:split})}. \nFor comparison, we evaluate GPT3 which uses no fine-tuning, unlike BART that is fine-tuned with the \\task{seen} tasks.\nThe first column corresponds to random split of tasks, while the remaining columns report cross-task generalization results of the BART model under \n\\changed{\nleave-one-$x$\n}\nsplits (\\S\\ref{subsec:split}). \nFor \n\\changed{\n$x =$ \\ul{category},}\nthe tasks in \\emph{question-generation} category are held out during training. \nFor \n\\changed{\n$x =$ \\ul{dataset},}\nthe tasks that were extracted from the \\emph{QASC} dataset were excluded from training. \nFor \n\\changed{\n$x =$ \\ul{task},}\nwe train a model on all tasks, except \\emph{QASC question generation} task which is used for evaluation. \n\n\n\n\\vspace{-.2cm}\n\\paragraph{Instructions benefit cross-task generalization.} \nThe results indicate that BART benefits from instructions in generalizing to new tasks, regardless of task splits.  \nFor example, under random split, the model using \\textsc{Full Instructions}  results in +19\\% gains over a model that is not using instructions.  \n This is particularly interesting for \nleave-one-\\ul{category}-out split\nsince the trained model can generalize to the tasks of a particular semantic category, without being exposed to it. \nIn comparison to GPT3, the fine-tuned BART model that utilizes instructions  achieves a stronger performance despite being $\\times 1k$ smaller than GPT3. \nFor example, a BART models using \\textsc{Full Instructions} achieves 8\\% higher performance than GPT3 under random split of tasks.  \n\n\nNote that the absolute values in leave-one-category are lower due to the difficulty of this setup compared to, for example, the random split setup. \nWhile all settings involve evaluating on tasks not seen during training, the leave-one-category setting enforces more dissimilarity among training and evaluation tasks.\n\n\n\\subsection{Generalization Under  Instruction Encoding and Task Categories}\nTable~\\ref{tab:random:splitfull2} reports the results of the BART model  \nper encodings of different instruction elements (\\S\\ref{subsec:models}) and for different task categories.\nThe table shows that encoding more elements of the instructions generally achieves better results than just using  \\textsc{prompt} or \\textsc{positive examples}. \nIt additionally shows that the benefit of the instruction elements seems to depend on the target task category. \nWe observe that the \\emph{question-generation} (QG) tasks benefit the most from \\textsc{positive examples}, whereas  in \\emph{classification} (CF),\n\\textsc{positive examples} are of little help. We hypothesis this is because it is easier to mimic question-generation based on a few examples, whereas it is difficult to define classes via a few examples, where \\textsc{definition} can be more helpful. \nThe models show little improvement in \\emph{verification} (VF). \nWe hypothesize these tasks are inherently more difficult,  partially because of  their distinctness from the rest of the tasks in the dataset. \nWe hope future work on this line will study a wider variety of tasks and will improve our understanding of such failure cases. \n\n\\subsection{Generalization vs. Number of Seen Tasks}\n\\label{subsec:supervision:size:experiment}\nFig.\\ref{fig:scaling:tasks} compares the impact of the number of seen tasks for cross-task generalization. \nFor supervision, we randomly sample a few tasks as \\task{seen} and evaluate on 6 tasks (one from each category). \n(each point in the figure is averaged over 5 random subsamples.) \nThe results\nshow that with \\textsc{no-instruction} encoding there is no tangible value in observing more tasks.  \nIn contrast, the generalization of the models that encode instructions improves with observing more tasks. \nThis is an exciting observation since it suggests that scaling up our dataset to more tasks may lead to stronger instruction-following systems. \n\n\n\n\n\n\\subsection{Analyses} \\label{subsec:task-specific}\n\n\n\\paragraph{Upperbound: Task-specific Models.}\nFor each task, we obtain a task-specific model (\\S~\\ref{subsec:input:output}) by training BART separately on each task's annotated training data. We evaluate these task-specific models to obtain a loose estimate of \\emph{upperbounds} for each task. \nOn average, task-specific models score 66\\% which is considerably higher than our models' best generalization (32\\%; Table~\\ref{tab:bart:generalization:all:splits}).\nThis indicates that { there is considerable room for improving generalization-based models} that use instructions. \n\n\\begin{comment}\n\\begin{table}\n    \\centering\n   \n    \\footnotesize\n    \\resizebox{\\columnwidth}{!}{\n    \\begin{tabular}{lcc}\n        \\toprule\n         error type & GPT3  & BART  \\\\\n       \\midrule\n       \n       \n       \n       \n           \n           \n            generates a nonsensical\/vague question & 4 & 47\\\\\n           \n            explains the question after generating it & 6 & 0\\\\\n            generates a yes\/no question & 12 & 4\\\\\n           \n           \n            \\makecell[l]{generates generic questions independent\\\\ of the given context} &6 &0\\\\\n        \\bottomrule\n        \\end{tabular}\n        }\n        \\caption{\n            Percentage of  errors on QASC QG task. \n            The numbers do not sum to 100 since the error types are not mutually exclusive. \n           \n        }\n        \\label{tab_error_analysis_main_text}\n\\end{table}\n\\subsection{Error Analysis}\nTable~\\ref{tab_error_analysis_main_text} shows the breakdown of most common error types for the QASC question generation task by analyzing 30 errors (more error analyses can be found in Appendix~\\ref{sec:error:analysis}; Table~\\ref{Tab: Error Analysis}).\n\\end{comment}\n\n\\begin{table}\n    \\centering\n    \\small\n    \\resizebox{0.99\\linewidth}{!}{\n    \\begin{tabular}{llcc}\n       \n        \\toprule\n        Model \u2193 & Split \u2193 & \\makecell{w\/ neg.\\\\examples}  & \\makecell{w\/o neg.\\\\examples} \\\\\n        \\midrule\n        \\multirow{5}{*}{BART} & random & 32 & {\\bf 35} \\\\ \n        & leave-one-$x$ \\\\ \n        & \\ $\\drsh x=$ category (AG) & 19 & {\\bf 21} \\\\ \n        & \\ $\\drsh x=$ dataset (Quoref) & 37 & 37  \\\\ \n        & \\ $\\drsh x=$ task (QASC QG) & 56 &  {\\bf 57} \\\\ \n        \\midrule\n        GPT3 & - & 24 & {\\bf 44} \\\\ \n        \\bottomrule\n    \\end{tabular}\n    }\n    \\caption{\n        Effect of excluding negative examples from \\textsc{Full Instruction} encoding. Negative instructions are surprisingly difficult for the models to learn from. \n    }\n    \\label{tab:negative:examples}\n\\end{table}\n\n\n\n\\paragraph{Impact of Negative Examples.}\nCrowdsourcing instructions often include negative examples to exemplify undesirable responses. \nWe study how negative examples in instructions affect cross-task generalization. \nOur cases study (Table~\\ref{tab:negative:examples}) indicates that the models work better \\emph{without} (w\/o) negative examples,  \ncontrary to the previously-observed benefits of other instructional elements (e.g., definition, positive examples). \nThis is aligned with the previous studies ~\\cite{xuan2020hard,lin2003bootstrapped} that discuss the challenges of learning from negative examples.\nInterestingly, GPT3's drop (44 vs 24) is more significant than BART (35 vs 32), showing that BART can partly recover through the training step. \n\n\\begin{table*}[ht]\n    \\centering\n    \\resizebox{0.78\\textwidth}{!}{\n    \\footnotesize\n    \\begin{tabular}{p{4.5cm}p{3.5cm}p{6.5cm}}\n        \\toprule\n        Category & Helpful Fields & Explanation \\\\\n        \\midrule\n        Question Generation (QG) & 1. \\textsc{Definition} & - Provides a holistic picture of the task.\\\\\n        & 2. \\textsc{Emphasis \\& Caution} & - Provides key information for solving the task.\\\\\n        & 3. \\textsc{Positive Examples} & - This gives an idea of what is expected in the output.\\\\\n        & 4. \\textsc{Negative Examples} & - Good to know the common mistakes people do.\\\\ \n        \\midrule\n        Answer Generation (AG) & \\textsc{1. Prompt} & - It limits the exploration space to question spans.\\\\\n        & \\textsc{2. Definition} & - Provides a general understanding of the task. \\\\\n        & \\textsc{3. Positive Examples} & - Reason field is very helpful.\\\\\n        \\midrule\n        Classification (CF) & \\textsc{1. Definition} & - The task is unclear without this field.\\\\\n        \\midrule\n        Incorrect Answer Generation (IAG) & \\textsc{1. Definition} & - Helps understand the utility of such a task.\\\\\n        & \\textsc{2. Emphasis \\& Caution} & - Source of some useful shortcuts.\\\\\n        & \\textsc{3. Positive Examples} & - Helps in understanding the type of questions asked.\\\\\n        \\midrule\n        Minimal Text Modification (MM) & \\textsc{1. Things to Avoid} & - Provides critical information.\\\\\n        \\midrule\n        Verification (VF) & \\textsc{1. Definition} & - Makes the task easy to understand.\\\\\n        & \\textsc{2. Things to avoid} & - Contains useful tips required for this task.\\\\\n        & \\textsc{3. Positive Examples} & - Exemplifies task understanding.\\\\ \n        & \\textsc{4. Negative examples} & - Helps avoid potential mistakes.\\\\\n        \\bottomrule\n        \\end{tabular}\n        }\n        \\caption{Results of humans' perceived importance of instruction elements. Our annotators, for example, find \\textsc{Definition} and \\textsc{Thing to Avoid} to be helpful for \\textit{Classification} and \\textit{Minimal Text Modification} tasks, respectively.}\n        \\label{Tab:User:Study}\n\\end{table*}\n\n\n\\paragraph{Error Analysis.}\nWe randomly sample 30 erroneous predictions of our fine-tuned BART on 3 distinct tasks (Winogrande answer generation; QASC question generation; MC-TACO incorrect answer generation). We categorize the errors into common patterns (Table~\\ref{Tab: Error Analysis}).\n\\begin{table}[ht]\n    \\centering\n    \\small\n   \n    \\begin{tabular}{lc}\n        \\toprule\n         error type & BART  \\\\\n      \\midrule\n            {\\color{brown} \\textit{Generates a nonsensical\/vague question}} &  47\\\\\n            {\\color{brown}\\textit{Generate an invalid question}} &  8\\\\\n            {\\color{brown}\\textit{Generates a yes\/no question}} &  4\\\\\n            {\\color{brown}\\textit{Copies the given fact or a subset of it}} &  3 \\\\\n            {\\color{brown}\\textit{Generates unanswerable questions}}  &3\\\\\n        \\bottomrule\n        \\end{tabular}\n       \n        \\caption{\n            Percentage of  errors on QASC QG task. \n            The numbers do not sum to 100 since the error types are not mutually exclusive.\n           \n        }\n        \\label{Tab: Error Analysis}\n\\end{table}\n\nUnlike GPT3 which generally suffers from generating redundant content irrelevant to instructions, our BART model provides more control, however, it can fail to generate proper output. \nHere are several erroneous predictions from our model: \n\n\n\\egbox{\n\\fontsize{8pt}{10pt}\\selectfont\n    {\\color{brown}\\textit{Generates a nonsensical\/vague question}}\\\\\n    \\textbf{BART input:}[...]input: Fact: graduations are used for measuring the volume of substances. output: ?\\\\\n    \\textbf{BART output:} How many substances are there in volume?\n}\n\n\\egbox{\n\\fontsize{8pt}{10pt}\\selectfont\n    {\\color{brown}\\textit{Copies the given fact or a subset of it}}\\\\\n    \\textbf{BART input:}[...] input: Fact: If someone is not breathing, begin with ventilation. output: ? \\\\\n    \\textbf{BART output:} If someone is breathing, begin with ventilation.\n}\n\n\n\\egbox{\n\\fontsize{8pt}{10pt}\\selectfont\n    {\\color{brown}\\textit{Generates a yes\/no question }}\\\\\n    \\textbf{BART input:}[...] input: Fact: Lack of saliva hinders the break down of food into components the body can absorb. output: ?\n    \\textbf{BART output:} Does saliva hinders the break down of food into components the body can absorb?\n}\n\n\\paragraph{Perceived Impact of Instruction Elements.}\nWe survey human  annotators to find out the value of instruction elements to humans.\nExcept for the negative examples which were shown to be difficult for models, we observe similar trends between  humans' perceived value of those elements (Table~\\ref{Tab:User:Study}) and their contributions to the model performance (Table~\\ref{tab:random:splitfull2}).  \nFor example, humans viewed  \\textsc{Definition} and \\textsc{Things to Avoid} as necessary fields for  \\emph{classification}  and \\emph{minimal text modification} categories, respectively, which is compatible with our empirical observations (e.g., \\textsc{prompt + definition} has the highest score on CF category in Table~\\ref{tab:random:splitfull2}). \n\n\n\n\n\\section{Conclusion}\n\\label{sec:discussion}\nIn this paper, we studied the goal of building models that  generalize to new tasks by encoding and understanding crowdsourcing instructions. \nWe introduced \\textsc{Natural Instructions}{}, which is built based on existing crowdsourced datasets,  that enables building such models and systematically evaluate them. \nTo the best of our knowledge, this is the first work to show the benefit of instructions towards improved cross-task generalization. \nAdditionally, we observe that our proposed task has a large room for improvement, which we believe \nwill bring more attention to building stronger models that can generalize to a wider range of tasks. \n\n\n\\begin{comment}\n\\vspace{-.2cm}\n\\paragraph{Future extensions.}\n The observations made in \\S\\ref{subsec:supervision:size:experiment} indicate that there are likely benefits to repeating our study with a larger set of datasets. \n We hope the future work expands our work with a larger and broader range of tasks.  \n \n We use  automatic evaluation, in order to facilitate the replicability of the follow-up work on \\textsc{Natural Instructions}{}. Admitting limitations of automatic evaluations, we hope future work  will provide an easy-to-reproduce human evaluation for the tasks studied here, based on the recent proposals for streamlining human evaluation of text generation models~\\cite{khashabi2021genie}. \n\\end{comment}\n\n\n\n\n\n\n\\section*{Acknowledgements}\nWe thank OpenAI for providing access to the GPT3 API, \nauthors who generously shared their dataset templates with us, Matt Peters and Nicholas Lourie for helpful input, the Beaker team for their support with experiments, and the anonymous reviewers for their helpful feedback. \nThe support of DARPA SAIL-ON,  DARPA CHESS program,  NSF IIS-2044660, ONR N00014-18-1-2826,\nand Paul G. Allen Foundation is gratefully acknowledged.\n\n\n\n\n\n\\section{Related Works}\n\\label{sec:related:work}\n\\begin{comment}\n\\paragraph{Instructions in NLP applications.} \\hanna{if no space, you can cut this paragraph}\nPrior work has studied ``instructions'' in various niches, such as\nrobotic instructions~\\cite{shridhar2020alfred, stepputtis2020language}, \ndatabases~\\cite{kim2020natural},  \nprogramming~\\cite{lin2018nl2bash,shao2020chartdialogs}, \\emph{inter alia}. \nSuch {instructions} are inherently different from ours, as they are intended to be mapped to pre-defined symbolic forms (e.g., SQL commands). \nConversely, our instructions describe general NLP tasks (no underlying grammar) for measuring task-level generalization. \n\\end{comment}\n\n\\changed{\n\\vspace{-.2cm} \\paragraph{Learning from instructions.}\nThere is recent literature on the extent to which models follow language instructions~~\\cite{hase2021can,ye2021zero,Gupta2021TowardsGP,Zhong2021AdaptingLM}.\nFor example, \\citet{efrat2020turking} examine if language models can follow crowdsourcing  instructions with no further training. On the contrary, our work is pursuing a fundamentally different goal: creating a dataset of crowdsourcing instructions and task instances and formulating cross-task generalization by training models on seen tasks and measuring generalization to the remaining unseen ones.\n\\citet{weller-etal-2020-learning} construct a crowdsourced dataset with short question-like task descriptions.  \nCompared to this work, our instructions are longer, more complex and natural since they were used to collect datasets through crowdsourcing. \n\nPromptSource and FLAN~\\cite{wei2021finetuned,sanh2021multitask} are two concurrent works that pursue a similar goal as ours. \nA key difference between our work to these works is in terms of data collection strategy. \nOur work uses natural instructions created by NLP researchers before the dataset instances were created by crowd workers, and hence it contains the complete definition of each task (definition, things to avoid, negative examples, etc.). \nOn the other hand, instructions in the concurrent work are collected retroactively based on the already-available task instances. \nOur  {\\it natural} instructions enable evaluating models on how they learn tasks given different elements of task descriptions. (See \\S\\ref{subsec:promptsource} for further comparisons.) \nNevertheless, we believe that all these approaches to constructing instructions and task categories are complementary and the community will benefit from considering both towards solving the challenging problem of cross-task generalization.\n\n\\vspace{-.2cm}\\paragraph{Prompt engineering.}\nConstructing effective discrete prompts for language models to perform NLP tasks is an active area of research~\\cite{schick2020few,reynolds2021prompt,liu2021pre}. \nSuch prompts are often extremely short and may not include a complete definition of complex tasks. \nIn contrast, our instructions encode detailed instructions as they were used to collect the datasets. \nMoreover, the goals are different:\nMost prompt-engineering approaches seek prompts with higher performance on a particular task, \ntypically through assumptions about their target task which make them non-trivial to generalize to any other task. \nHowever, our introduced meta dataset enables the measurement of generalization to unseen tasks. \n\n\n\\vspace{-.2cm}\\paragraph{Beyond standard multi-task learning.}\nMulti-task learning is a long-standing goal for AI~\\cite{caruana1997multitask} and has led to successful models that can support a wider range of tasks\n~\\cite{mccann2018natural,raffel2020exploring,khashabi2020unifiedqa,mishra2020towards,aghajanyan2021muppet,ye2021crossfit}.\nMost of the conventional setups in the multi-tasking literature  evaluate on instances that belong to the tasks that are seen, i.e., their labeled instances were observed during training (1st column of Table~\\ref{tab:comparison}). \nWe augment this setup by \nintroducing natural language instructions which enable our models to bridge to tasks that were not seen during training. \n}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Lambda Function and Motivations}\n\nThe existence of the core decomposition with Peano quotient for planar compacta \\cite{LLY-2019} enables us to associate to each compact set $K\\subset\\hat{\\bbC}$ a map $\\lambda_K:\\hat{\\bbC}\\rightarrow\\bbN\\cup\\{\\infty\\}$, called the {\\em lambda function of $K$}. This function sends all points $x\\notin K$ to zero and may take a positive value for some $x\\in K$. It ``quantifies'' certain aspects of the topological structure of $K$, that is more or less related to the property of being locally connected. In particular, a continuum $K\\subset\\hat{\\bbC}$ is locally connected if and only if $\\lambda_K(x)=0$ for all $x\\in\\hat{\\mathbb{C}}$. On the other hand, if  a continuum $K\\subset\\hat{\\bbC}$ is not locally connected at $x\\in K$ then $\\lambda_K(x)\\ge1$; but the converse is not necessarily true.\n\nThe quantification in terms of lambda function allows us to carry out a new analysis of the topology of $K$, by computing or estimating  $\\lambda_K(x)$ for specific choices of $x\\in K$. In the current paper, we will investigate an interesting phenomenon that was firstly revealed in a fundamental result by Marie Torhorst, as one of the three highlights of \\cite{Torhorst}. This result is often referred to as Torhorst Theorem \\cite[p.106, (2.2)]{Whyburn42} and reads as follows.\n\\begin{theorem*}[{\\bf Torhorst Theorem}]\nThe boundary $F$ of every complementary domain $R$ of a locally connected continuum $M\\subset\\hat{\\mathbb{C}}$ is itself a locally connected continuum.\n\\end{theorem*}\n\nWe will obtain an inequality that includes the Torhorst Theorem as a simple case.\nThe inequality is about the lambda function $\\lambda_K$. The function $\\lambda_K$ is based on the core decomposition of $K$ with Peano quotient \\cite{LLY-2019}, which is motivated by some open questions in \\cite{Curry10} and extends two earlier models of polynomial Julia sets developed in \\cite{BCO11,BCO13}. Those models, briefly called BCO models, provide efficient ways (1) to describe the topology of unshielded compacta, like polynomial Julia sets, and (2) to obtain specific factor systems for polynomials restricted to the Julia set. The BCO models are special cases of a more general model, working well for all planar compacta, that associates natural factor systems to the dynamics of  rational functions \\cite{LLY-2019,LYY-2020}.\n\nRecall that a {\\bf Peano continuum} means the image of $[0,1]$ under a continuous map. By  Hahn-Mazurkiewicz-Sierpi\\'nski Theorem\n\\cite[p.256, \\S 50, II, Theorem 2]{Kuratowski68},  a continuum is locally connected if and only if it is a Peano continuum. On the other hand,\na {\\bf Peano compactum} is defined to be a compactum having locally connected components such that  for any constant $C>0$ at most finitely many of its components are of diameter greater than $C$. Therefore, the Cantor ternary set is a Peano compactum and a Peano continuum is just a Peano compactum that is connected.  Concerning how such a definition arises from the discussions of BCO models, we refer to  \\cite[Theorems 1-3]{LLY-2019}.\n\n\n\nGiven a compactum $K\\subset\\hat{\\mathbb{C}}$, there exists an upper semi-continuous decomposition  of $K$ into sub-continua, denoted as $\\Dc_K^{PC}$, such that (1) the quotient space is a Peano compactum and (2) $\\Dc_K^{PC}$ refines every other such decomposition of $K$  \\cite[Theorem 7]{LLY-2019}.\nWe call $\\Dc_K^{PC}$ the core decomposition of $K$ with Peano quotient. The hyperspace $\\Dc_K^{PC}$ under quotient topology is called the Peano model of $K$. Every $d\\in\\Dc_K^{PC}$ is called an {\\bf atom} of $K$, or an {\\bf order-one atom}, or an atom of order $1$. Every atom of an order-one atom is called an {\\bf order-two atom}, and so on. Note that a compactum such as the pseudo-arc or Cantor's Teepee may have a non-degenerate atom of  order $\\infty$.\n\nConsidering the atoms of a compactum $K\\subset\\hat{\\mathbb{C}}$ as its structural units, we summarize the results obtained in \\cite[Theorem 7]{LLY-2019} and \\cite[Theorem 1.1]{LYY-2020} in the following way.\n\\begin{theorem*}[{\\bf Theory of Atoms}\nEvery compactum $K\\subset\\hat{\\mathbb{C}}$ is made up of atoms; all its atoms are sub-continua of $K$ and they form an upper semi-continuous decomposition, with its quotient space being a Peano compactum, that refines every other such decomposition; moreover, for any finite-to-one open map $f:\\hat{\\bbC}\\rightarrow\\hat{\\bbC}$ and any atom $d$ of $K$, each component of $f^{-1}(d)$ is an atom of $f^{-1}(K)$.\n\\end{theorem*}\n\nUsing the hierarchy formed by {\\bf atoms of atoms}, we introduce the lambda function.\n\\begin{definition*}[{\\bf Lambda Function}]\nGiven a compactum $K\\subset\\hat{\\bbC}$. Let  $\\lambda_K(x)=0$ for $x\\notin K$. Let $\\lambda_K(x)=m-1$ for any $x\\in K$, if there is a smallest integer $m\\ge1$ such that $\\{x\\}$ is an order-$m$ atom of $K$. If such an integer $m$ does not exist, we put $\\lambda_K(x)=\\infty$.\n\\end{definition*}\nWhen little is known about the topology of $K$, it is  difficult to completely determine the values of $\\lambda_K$. On the other hand, the level sets $\\lambda_K^{-1}(n)(n\\ge0)$ are ``computable''  for typical choices of $K$. In such circumstances, the lambda function $\\lambda_K$ is useful in describing certain aspects of the topology of $K$. For instance, one may check the following observations: (1) a compact set $K\\subset\\hat{\\bbC}$ is a Peano compactum if and only if $\\lambda_K(x)=0$ everywhere; (2) if $K=\\left\\{t+\\left(\\sin\\frac1t\\right){\\bf i}: 0<t\\le1\\right\\}\\cup\\{s{\\bf i}: |s|\\le 1\\}$  then $\\lambda_K^{-1}(1)=\\{s{\\bf i}: |s|\\le 1\\}$ and $\\lambda_K^{-1}(0)=\\hat{\\bbC}\\setminus \\lambda_K^{-1}(1)$; and (3) if $K=\\left\\{t+s{\\bf i}: t\\in[0,1], s\\in\\mathcal{C}\\right\\}$, where $\\mathcal{C}$ denotes Cantor's ternary set, then $\\lambda_K^{-1}(1)=K$ and $\\lambda_K^{-1}(0)=\\hat{\\bbC}\\setminus K$.\n\nThe lambda function $\\lambda_K$ helps us to {\\bf measure}, locally and globally, how far a compact set $K\\subset\\hat{\\mathbb{C}}$ is from being a Peano compactum. When $K=\\partial\\Omega$ for a planar domain $\\Omega\\subset\\hat{\\bbC}$, which is the image of a conformal map $\\varphi:\\mathbb{D}\\rightarrow\\Omega$ of the unit disk $\\mathbb{D}=\\{z\\in\\mathbb{C}: |z|<1\\}$, the study of $\\lambda_K$ is particularly interesting.\nIn order to illustrate how $\\lambda_K$ is related to the boundary behavior of $\\varphi$, we recall a theorem by Fatou \\cite[p.17, Theorem 2.1]{CL66}, which says that the radial limits $\\lim\\limits_{r\\rightarrow1}\\varphi\\left(re^{{\\bf i}\\theta}\\right)$ exist for all $\\theta\\in[0,2\\pi)$, except possibly for a set of linear measure zero.\nOne may call the map $e^{{\\bf i}\\theta}\\mapsto \\lim\\limits_{r\\rightarrow1}\\varphi\\left(re^{{\\bf i}\\theta}\\right)$ the {\\bf boundary function} of $\\varphi$, denoted  as $\\varphi^b$. Sometimes we also call  $\\varphi^b$ the {\\bf boundary of $\\varphi$}. Its domain consists of all points $e^{{\\bf i}\\theta}\\in\\partial\\mathbb{D}$ such that the limit $\\lim\\limits_{r\\rightarrow1}\\varphi\\left(re^{{\\bf i}\\theta}\\right)$ exists.\n\nRecall that the prime end at $e^{{\\bf i}\\theta}$ is of the first or the second type, if  the  limit $\\lim\\limits_{r\\rightarrow1}\\varphi\\left(re^{{\\bf i}\\theta}\\right)$  exists \\cite[p.177, Theorem 9.7]{CL66}.\nBy Carath\\'eodory's {\\bf Continuity Theorem} \\cite[p.18]{Pom92}, the boundary function $\\varphi^b$ is defined well and continuous on the whole unit circle $\\partial\\bbD$ if and only if the boundary $\\partial\\Omega$ is a Peano continuum. By \\cite[p.17,Theorem 2.1]{CL66}, if $\\partial\\Omega$ is not a Peano continuum then $\\varphi^b$ is a member of $L^\\infty(\\partial\\bbD)\\setminus C(\\partial\\bbD)$. Therefore, we are curious about two types of quantities: (1) those describing how far $\\varphi^b$ is from being continuous and (2) those  measuring how far $\\partial\\Omega$ is from being locally connected.\n\nThe first type concerns the asymptotic of $\\varphi(z)$ for $z\\in\\bbD$ as $|z|\\rightarrow 1$. The second type concerns the topology of $\\partial\\Omega$.\nThe lambda function $\\lambda_{\\partial\\Omega}$ gives rise to a quantity of the second type. This function vanishes everywhere if and only if $\\varphi^b$ belongs to $C(\\partial\\bbD)$. Among others, the questions below are of some interest: {\\em What does it mean if $\\lambda_{\\partial\\Omega}(x)\\le 1$ for all $x$? Can we say something about $\\lambda_{\\partial\\Omega}$ when $\\varphi^b$ has finitely many discontinuities?}\n\nIn particular, one may consider the case that $\\varphi(\\mathbb{D})$ coincides with the complement of the Mandelbrot set $\\M$. The set $\\mathcal{M}$ consists of all the complex parameters $c$ such that the Julia set of  $f_c(z)=z^2+c$ is connected. Its local connectedness is still an open question.\nDue to the deep works on quadratic polynomials, we know that $\\M$ is locally connected at many of its points, such as the Misiurewicz points and those lying on the boundary of a hyperbolic component. From these known results arises  a natural question: {\\bf Is it true that $\\lambda_\\M(x)=0$ for all $x$ at which $\\M$ is known to be locally connected?} Another  of some interest is, {\\bf Can we find some upper bound for the lambda function $\\lambda_\\M$?} In particular, {\\bf Can we show that $\\lambda_\\M(x)\\le1$ for all $x$?}\nVery recently, Yang and Yao \\cite{YY-2020} show that $\\lambda_{\\mathcal{M}}(x)=0$ for all points $x$ lying on the boundary of a hyperbolic component. More studies on questions of a similar nature can be expected.\n\n\n\n\\section{Main Results}\n\nIn the current paper, we want to analyze $\\lambda_K, \\lambda_L$ for compacta $L\\subset K\\subset\\hat{\\mathbb{C}}$, that satisfy specific  properties. Our analysis is connected to very basic results of topology, such as the Torhorst Theorem and the gluing lemma for continuous functions.\n\n\nWe will extend and quantify the  Torhorst Theorem by an inequality, stating that the lambda function $\\lambda_K$ of any compactum $K\\subset\\hat{\\mathbb{C}}$ is an {\\bf upper bound} of $\\lambda_{\\partial U}$ for any  complementary component $U$ of $K$.\nTo this end, let us examine how the atoms of $K$ are related to those of any compactum  $L$ lying on the boundary of a component $U$ of $\\hat{\\bbC}\\setminus K$.\n\\begin{main-theorem}\\label{compare_atoms}\nGiven a compact $K\\subset\\hat{\\mathbb{C}}$ and a component $U$ of \\ $\\hat{\\bbC}\\setminus K$. If $L\\subset\\partial U$ is a compactum then every atom of $L$ lies in a single atom of $K$. Consequently, every atom of $L$ lies in a single atom of $\\partial U$. \\end{main-theorem}\n\nWith Theorem \\ref{compare_atoms}, we compare the lambda functions  $\\lambda_K,\\lambda_L$ and obtain the following.\n\n\\begin{main-theorem\n\\label{lambda_inequality}\nGiven a compactum $K\\subset\\hat{\\mathbb{C}}$ and  a component $U$ of \\ $\\hat{\\bbC}\\setminus K$. If $L\\subset\\partial U$ is a compactum then $\\lambda_L(x)\\le \\lambda_K(x)$ for all $x\\in\\hat{\\bbC}$.\n\\end{main-theorem}\n\nLet $\\mathcal{A}$ consist of the components of $\\hat{\\mathbb{C}}\\setminus K$. The  {\\bf envelope function} of $K$ is defined as\n$\\displaystyle \\tilde{\\lambda}_K(x)=\\sup\\limits_{L\\subset\\partial U, U\\in\\mathcal{A}}\\lambda_L(x)\\ \\left(x\\in\\hat{\\bbC}\\right).$\nBy Theorems \\ref{compare_atoms} and \\ref{lambda_inequality}, we have $\\tilde{\\lambda}_K(x)=\\sup\\limits_{U\\in\\mathcal{A}}\\lambda_{\\partial U}(x)$  and $\\tilde{\\lambda}_K(x)\\le\\lambda_K(x)$ for all $x\\in\\hat{\\bbC}$. From now on we call $\\tilde{\\lambda}_K(x)\\le\\lambda_K(x)\\left(x\\in\\hat{\\mathbb{C}}\\right)$  the {\\bf\\em lambda inequality}, which is also written as $\\tilde{\\lambda}_K\\le\\lambda_K$. If $K$ is a Peano compactum then $\\lambda_K$ and hence $\\lambda_L$ vanish everywhere, for any compactum $L$ lying on the boundary of any component $U$ of $\\hat{\\mathbb{C}}\\setminus K$. That is to say, each compactum $L\\subset\\partial U$ is a Peano compactum. This includes as a special sub-case the Torhorst Theorem, in which  $K$ is assumed to be a Peano continuum and  $L=\\partial U$.\n\n\\begin{rem}\nIn the above setting, the boundary $\\partial K$  may not be locally connected, even if it is a continuum.\nSee \\cite[Example 3.2]{Luo07} or Example \\ref{bd_larger} of this paper. In Example \\ref{bd_smaller}, we will construct a continuum $L$ such that the range of $\\lambda_L-\\lambda_{\\partial L}$ is $\\{-1,0,1\\}$. \n\\end{rem}\n\nNow we consider compacta $K\\subset\\hat{\\mathbb{C}}$ for which the {\\bf Lambda Equality} $\\tilde{\\lambda}_K=\\lambda_K$ holds, so that $\\tilde{\\lambda}_K(x)=\\lambda_K(x)$ for all $x\\in\\hat{\\bbC}$.\nLet us start from a simple example.\n\\begin{exam}\\label{why_E_compactum}\nLet $U$ be the complement of the unit square $\\{a+b{\\bf i}: 0\\le a,b\\le 1\\}$. Let $K$ be the compactum consisting of $\\partial U$ and an infinite sequence of squares (only the boundary) of side length $<1$. These squares are centered at $0.5+0.5{\\bf i}$ and converge to $\\partial U$ under Hausdorff distance. Then $\\lambda_K(x)-\\tilde{\\lambda}_K(x)=\\left\\{\\begin{array}{ll}1& x\\in\\partial U\\\\ 0&\\text{otherwise}.\\end{array}\\right.$\n\n\\begin{figure}[ht] \\vskip -0.5cm \\begin{center} \\begin{tikzpicture}[x=0.2cm,y=0.2cm,scale=0.45] \\draw[black,thick] (0,0) -- (0,32) -- (32,32) -- (32,0) -- (0,0);  \\foreach \\j in {3,...,6} {    \\draw[gray, thick] (2*\\j,32-2*\\j) -- (32-2*\\j,32-2*\\j) -- (32-2*\\j,2*\\j) -- (2*\\j,2*\\j) --(2*\\j,32-2*\\j); }     \\foreach \\j in {2,...,5} {\\fill[gray] (\\j,\\j) circle(0.5ex); \\fill[gray] (32-\\j,32-\\j) circle(0.5ex); \\fill[gray] (\\j,32-\\j) circle(0.5ex); \\fill[gray] (32-\\j,\\j) circle(0.5ex); }    \\draw(0,1) node[left]{$0$}; \\draw(32,1) node[right]{$1$}; \\draw(0,31) node[left]{${\\bf i}$}; \\end{tikzpicture} \\end{center} \\vskip -1.0cm \\caption{A simplified depiction of $K$ and the small squares.}\\label{seq-squares}\\vskip -0.25cm \\end{figure}\n\\end{exam}\n\nIf $K$ is given in Example \\ref{why_E_compactum} then $\\hat{\\mathbb{C}}\\setminus K$ has infinitely many components of diameter greater than $1-\\epsilon$ for some constant $\\epsilon\\in(0,1)$. Borrowing the idea of {\\bf $E$-continuum} \\cite[p.113]{Whyburn42}, we define an {\\bf $E$-compactum} to be a planar compactum such that for any constant $\\varepsilon>0$ its complement has at most finitely many components of diameter $>\\varepsilon$. Unfortunately, the condition of $E$-compactum alone is still not sufficient for the {\\bf Lambda Equality} $\\tilde{\\lambda}_K=\\lambda_K$.  See Examples \\ref{E-compactum} and \\ref{finite-comp}.\n\nThe theorem below gives three conditions under which the Lambda Equality holds.\n\n\\begin{main-theorem}\\label{equality-case}\nGiven a compactum $K\\subset\\hat{\\mathbb{C}}$, the Lambda Equality  $\\tilde{\\lambda}_K=\\lambda_K$ holds if one of the following conditions is satisfied:\n\n(i) $K$ is an $E$-compactum such that the envelope function $\\tilde{\\lambda}_K(x)$ vanishes everywhere;\n\n(ii)  $K$ is an $E$-compactum whose complementary components have disjoint closures.\n\n(iii) $K$ is a partially unshielded compactum.\n\\end{main-theorem}\n\n\\begin{rem}\nIn (i) and (ii) of Theorem \\ref{equality-case}, the assumption  that $K$ is an $E$-compactum  can not be removed. We may set $K=[0,1]\\!\\times\\![0,{\\bf i}]\\setminus\\left(\\bigcup\\limits_1^\\infty R_n\\right)$, with $R_n=\\left(\\frac{1}{3n},\\frac{2}{3n}\\right)\\times\\left(\\frac13{\\bf i},\\frac23{\\bf i}\\right)$. If $W=\\hat{\\mathbb{C}}\\setminus[0,1]\\!\\times\\![0,{\\bf i}]$ then $\\partial W\\cap\\partial R_n=\\emptyset$ for all $n\\ge1$ and  $\\partial R_n\\cap \\partial R_m=\\emptyset$ for $n\\ne m$. See Figure \\ref{non-E} for a simple depiction of $K$.\nThe continuum $K$ is not an $E$-continuum but it satisfies the other assumptions in (i) and (ii) of Theorem \\ref{equality-case}. It has exactly one non-degenerate atom, the segment $\\left[\\frac13{\\bf i},\\frac23{\\bf i}\\right]$. Thus $\\lambda_K(x)-\\tilde{\\lambda}_K(x)=\\left\\{\\begin{array}{ll}1& x\\in\\left[\\frac13{\\bf i},\\frac23{\\bf i}\\right]\\\\ 0&\\text{otherwise}.\\end{array}\\right.$\n\n\\begin{figure}[ht]\n\\vskip -0.75cm\n\\begin{center}\n\\begin{tikzpicture}[x=5cm,y=5cm,scale=0.618]\n\\fill[gray!20,thick] (0,0) -- (0,1) -- (1,1) -- (1,0) -- (0,0);\n\\draw[gray,thick] (0,0) -- (0,1) -- (1,1) -- (1,0) -- (0,0);\n\\draw[black, ultra thick] (0,1\/3) -- (0,2\/3);\n\n\\foreach \\j in {1,...,3}\n{\n \\fill[white] (1\/3^\\j,1\/3) -- (2\/3^\\j,1\/3) -- (2\/3^\\j,2\/3) -- (1\/3^\\j,2\/3) --(1\/3^\\j,1\/3);\n    \\draw[gray, thick] (1\/3^\\j,1\/3) -- (2\/3^\\j,1\/3) -- (2\/3^\\j,2\/3) -- (1\/3^\\j,2\/3) --(1\/3^\\j,1\/3);\n}\n\\foreach \\j in {1,...,6}\n{\\fill[gray] (1\/60,0.32+0.05*\\j) circle(0.3ex);\n}\n\n\\draw(0,0.02) node[left]{$0$};\n\\draw(1,0.02) node[right]{$1$};\n\\draw(0,0.98) node[left]{${\\bf i}$};\n\n\\end{tikzpicture}\n\\end{center}\n\\vskip -0.95cm\n\\caption{A depiction for $K$ and some of the rectangles.}\\label{non-E}\n\\vskip -0.25cm\n\\end{figure}\n\\end{rem}\n\nNote that the Lambda Equality may not hold for an $E$-compactum  $K\\subset\\hat{\\mathbb{C}}$, even if it  has finitely many complementary components. See Example \\ref{finite-comp}. Also notice that the Lambda Equality under condition (i) implies the theorem below. This extends Whyburn's Theorem \\cite[p.113, (4.4)]{Whyburn42}, which says that {\\em an $E$-continuum is a Peano continuum if and only if the boundary of any of its complementary components is a Peano  continuum}.\n\\begin{theorem*}[{Extended Whyburn's Theorem}] An $E$-compactum is a Peano compactum if and only if the boundary of any of its complementary components is a Peano compactum.\n\\end{theorem*}\n\n\nTheorem \\ref{lambda_inequality} addresses how $\\lambda_K$ and $\\lambda_L$ are related when $L$ lies on the boundary of a component of $\\hat{\\bbC}\\setminus K$. There are other choices of  planar compacta $K\\supset L$ so that $\\lambda_K$ and $\\lambda_L$ are intrinsically related. A typical situation happens, if the common part of $\\overline{K\\setminus L}$ and $L$ is  a finite set.\n\n\\begin{main-theorem}\\label{gluing_lemma}\nIf $K\\supset L$ are planar compacta such that $\\overline{K\\setminus L}$ intersects $L$ at finitely many points then $\\lambda_K(x)=\\max\\left\\{\\lambda_{\\overline{K\\setminus L}}(x),\\lambda_L(x)\\right\\}$ for all $x$.  \\end{main-theorem}\n\nSetting $A=\\overline{K\\setminus L}$, we can infer that that $\\lambda_K(x)$ coincides with $\\lambda_A(x)$ for $x\\in A\\setminus L$ and with $\\lambda_L(x)$ for $x\\in L\\setminus A$, equals $\\max\\left\\{\\lambda_{A}(x),\\lambda_L(x)\\right\\}$ for $x\\in A\\cap L$, and vanishes for every $x\\notin(A\\cup L)$. Therefore, we have.\n\n\\begin{theorem*}[{Gluing Lemma for Lambda Functions}]\\label{gluing_lemma_1}\nIf in addition $\\lambda_A(x)=\\lambda_L(x)$ for all $x\\in A\\cap L$ then $\\lambda_K(x)=\\lambda_{A\\cup L}$ may be obtained by gluing $\\lambda_A$ and $\\lambda_L$, in the sense that\n\\begin{equation}\\label{form-1}\n\\lambda_{A\\cup L}(x)=\\left\\{\\begin{array}{ll}\\lambda_A(x)& x\\in A\\\\ \\lambda_L(x)& x\\in L\\\\ 0& {otherwise.}\\end{array}\\right.\n\\end{equation}\n\\end{theorem*}\n\\begin{rem}\nThe formation in Equation (\\ref{form-1}) is similar to the one illustrated in the well known gluing lemma for continuous maps. See for instance \\cite[p.69, Theorem (4.6)]{Armstrong}.\nIn certain situations, Theorem \\ref{gluing_lemma} helps us to analyze questions concerning local connectedness of polynomial Julia sets. See Question \\ref{small_julia}. However,\nthe case that $A\\cap L$ is an infinite set is more involved. In Theorem \\ref{baby_M}, we will extend Theorem 4 to such a case under additional assumptions. This extension allows one to choose $K$ to be the Mandelbrot set and $L$ the closure of a hyperbolic component. For concrete choices of $A$ and $L$ so that Equation \\ref{form-1} does not hold, we refer to Examples \\ref{cantor_combs}, \\ref{brooms} and \\ref{cup-fs}.\n\\end{rem}\n\n\n\n\n\n\n\n\nThe other parts of this paper are arranged as follows. Section \\ref{proof-c} is devoted to the proofs for Theorems \\ref{compare_atoms} to \\ref{lambda_inequality}. Section \\ref{equality} gives a proof for  Theorem \\ref{equality-case}. In Section \\ref{glue} we firstly prove Theorem \\ref{gluing_lemma} and then continue to establish Theorem \\ref{baby_M}.\nSection \\ref{examples} gives examples.\n\n\n\n\n\n\n\\section{The Lambda Inequality}\\label{proof-c}\n\nIn this section we prove Theorems \\ref{compare_atoms} and \\ref{lambda_inequality}.\n\nWe will study relations on compacta $K\\subset\\hat{\\mathbb{C}}$. Such a relation is considered as a subset of the product space $K\\times K$ and  is said to be {\\bf closed} if it is closed in $K\\times K$. Given a relation $\\Rc$ on $K$, we call $\\Rc[x]=\\{y\\in K: \\ (x,y)\\in\\Rc\\}$ the fiber of $\\Rc$ at $x$. We mostly consider  closed relations $\\Rc$ that are reflexive and symmetric, so that for all $x,y\\in K$ we have (1) $x\\in \\Rc[x]$ and (2) $x\\in\\Rc[y]$ if and only if $y\\in\\Rc[x]$. For such a relation, the {\\em iterated relation} $\\Rc^2$ is defined naturally so that\n$\\displaystyle y\\in\\Rc^2[x]$ if and only if there exist $z\\in K$ with $(x,z), (z,y)\\in \\Rc$.\n\nRecall that the {\\bf Sch\\\"onflies relation} on a planar compactum $K$ is a reflexive symmetric relation. Under this relation, two points $x_1, x_2$ are related provided that either $x_1=x_2$ or there exist two disjoint Jordan curves $J_i\\ni x_i$ such that  $\\overline{U}\\cap K$ has infinitely many components $P_n$, intersecting $J_1$ and $J_2$ both, whose limit under Hausdorff distance contains $\\{x_1,x_2\\}$. Here $U$ is the component of \\ $\\hat{\\bbC}\\setminus(J_1\\cup J_2)$ with $\\partial U=J_1 \\cup J_2$.\n\nGiven a compactum $K\\subset\\hat{\\bbC}$,  denote by $R_K$ the  Sch\\\"onflies relation on $K$ and by $\\overline{R_K}$ the closure of $R_K$. We also call $\\overline{R_K}$ the {\\bf closed Sch\\\"onflies relation}.\nLet $\\Dc_K$ be the finest upper semi-continuous decompositions of $K$ into sub-continua that splits none of the fibers $R_K[x]$. Then $\\Dc_K$ coincides with $\\Dc_K^{PC}$, the core decomposition of $K$ with Peano quotient \\cite[Theorem 7]{LLY-2019}. Therefore the elements of $\\mathcal{D}_K$ are the (order-one) atoms of $K$.\n\nEvery fiber $\\overline{R_K}[x]$ is a continuum. See\n\\cite[Theorem 1.4]{LYY-2020}. However, the compactness and connectedness of the fibers of $R_K$ remain open.\nMoreover, in order that a point $y\\ne x$ lies  in $\\overline{R_K}[x]$ it is necessary and sufficient that for small enough  $r>0$ the difference $K\\setminus(B_r(x)\\cup B_r(y))$ has infinitely many components that intersect each of the circles $\\partial B_r(x)$ and $\\partial B_r(y)$. See  \\cite[Theorem 1.3]{LYY-2020}.\nThe lemma below relates the fibers of $\\overline{R_K}^2$ to those of $\\overline{R_L}$, where $L$ is a compact subset of $K$ satisfying certain properties.\n\n\\begin{lemma}\\label{key-lemma}\nGiven a compactum $K\\subset\\hat{\\mathbb{C}}$ and  a component $U$ of \\ $\\hat{\\bbC}\\setminus K$. If $L\\subset\\partial U$ is compact then $\\overline{R_L}[x]\\subset \\overline{R_K}^2[x]$  for any $x\\in L$.\n\\end{lemma}\n\\begin{proof}\nTo obtain the containment $\\overline{R_L}[x]\\subset \\overline{R_K}^2[x]$ for any given $x\\in L$,  we may fix an arbitrary point $y\\in\\overline{R_L}[x]\\setminus\\{x\\}$ and consider the annulus $A_n=\\hat{\\mathbb{C}}\\setminus\\left(B_{1\/n}(x)\\cup B_{1\/n}(y)\\right)$, for any integer $n\\ge1$ such that $\\overline{B_{1\/n}(x)}\\bigcap\\overline{B_{1\/n}(y)}=\\emptyset$. Here $B_{1\/n}(x)$ and $B_{1\/n}(y)$ are open disks with radius $1\/n$ under spherical distance, that are respectively centered at $x$ and $y$. By \\cite[Theorem 1.3]{LYY-2020}, $A_n\\cap L$ has infinitely many components intersecting both $\\partial B_{1\/n}(x)$ and $\\partial B_{1\/n}(y)$. So we can find an infinite sequence $\\{P_i\\}$ of such components that converge to some continuum $P_\\infty$ under Hausdorff metric.\n\nSince $x,y\\in L\\subset\\partial U$, we can find an open arc $\\alpha_0\\subset U$ that connects a point on $\\partial B_{1\/n}(x)$ to one on $\\partial B_{1\/n}(y)$. Going to an appropriate sub-arc, if necessary, we may assume that $\\alpha\\subset A_n$. Then, we may slightly thicken the closed arc $\\overline{\\alpha}$ and obtain a topological disc $\\alpha^*\\subset A_n$, satisfying $\\alpha^*\\cap K=\\emptyset$. From this we see that $\\overline{A_n\\setminus\\alpha^*}$ is homeomorphic to $[0,1]^2$. We will obtain the following.\n\n\n{\\bf Claim}. $P_\\infty$ contains two points $u_n\\in \\partial B_{1\/n}(x), v_n\\in \\partial B_{1\/n}(y)$ with $v_n\\in\\overline{R_K}^2[u_n]$.\n\nThe flexibility of the large enough integers $n$ ensures that $\\lim\\limits_nu_n=x$ and $\\lim\\limits_nv_n=y$. Since $\\overline{R_K}^2$ is a closed relation, we surely obtain $y\\in \\overline{R_K}^2[x]$.  This completes our proof. Thus, the remaining issue is to verify the above claim.\n\nAs $\\overline{A_n\\setminus\\alpha^*}$ is a topological disc, we consider it to be the unit square $[0,1]^2$. Moreover, we may represent by $[0,1]\\times\\{1\\}$  the arc $l_1=\\overline{A_n\\setminus\\alpha^*}\\cap\\partial B_n(x)$ and by $[0,1]\\times\\{0\\}$ the arc $l_2=\\overline{A_n\\setminus\\alpha^*}\\cap\\partial B_n(y)$. Fix any point $z$ in $P_\\infty\\cap(0,1)^2$. For any $r>0$ that is small, let $W_r$ denote the open rectangle centered at  $z$ with diameter $r$.\nSince $P_i\\rightarrow P_\\infty$ under Hausdorff distance we may assume that every $P_i$ intersects $W_r$ and lies in $[0,1]^2$, which from now on represents $\\overline{A_n\\setminus\\alpha^*}$.\nSee Figure \\ref{key}.\n\\begin{figure}[ht]\n\\vskip -0.5cm\n\\center{\n\\begin{tikzpicture}[scale=0.8,x=1.618cm, y=0.618cm]\n\\draw(-2,0)--(-2,7);\n\\draw(-2,0)--(7,0)node[below]{$l_2\\subset \\partial B_n(y)$};\n\\draw(-2,7)--(7,7)node[above]{$l_1\\subset \\partial B_n(x)$};\n\\draw(7,0)--(7,7);\n\\draw(2,0)--(2,7)node[above]{$P_\\infty$};\n\\fill(2,3.5)circle(2pt);\n\\draw[blue,thick](-1,2)--(5,2) -- (5,5) -- (-1,5) -- (-1,2);\n\\draw(2,3.5) node[left]{$W_r\\ni z$};\n\\draw(3,0)--(3,7)node[above]{$P_{2i+1}$};\n\\draw(4.5,0)--(4.5,7)node[above]{$P_{2i-1}$};\n\\draw[dashed](3.75,0)--(3.75,7);\n\\draw (3.75,0)node[below]{$P_{2i}$};\n\\draw(3.75,3) node[left]{$a_i$};\n\\fill(3.75,3)circle(2pt);\n\\draw(4,4) node[right]{$b_i$};\n\\fill(4,4)circle(2pt);\n\\draw[red](4.25,5)--(4.25,7); \\draw(4.3,6) node[left]{$\\beta_i$};\n\\draw[red](4.25,2)--(4.25,0); \\draw(4.3,1) node[left]{$\\beta_i$};\n\\end{tikzpicture}\n}\\vskip -0.75cm\n\\caption{Relative locations of $z,l_1,l_2,W_r, P_{2i-1}, P_{2i}, P_{2i+1}$ and $a_i, b_i$.}\\label{key}\n\\vskip -0.25cm\n\\end{figure}\n\nRecall that $[0,1]^2\\setminus P_\\infty$ has two components, one containing $\\{1\\}\\times[0,1]$ and the other $\\{0\\}\\times[0,1]$. One of these components contains infinitely many $P_i$. Without losing generality we may assume that every $P_i$ lies in the one containing $\\{1\\}\\times[0,1]$, denoted  $V$. Thus $P_i$ can be connected to $\\{1\\}\\times[0,1]$ by an arc in $[0,1]^2$ that does not intersect $P_\\infty$. Moreover, rename $P_i(i\\ge1)$ so that every $P_i$ can be connected to $\\{1\\}\\times[0,1]$ by an arc in $[0,1]^2$ that does not intersect $P_j$ for $j\\ge i+1$. Therefore, each $P_i$ is ``to the right of'' $P_{i+1}$.\n\n\nFor all $i\\ge1$ let $V_i$ be the unique component of $\\hat{\\bbC}\\setminus\\left(P_{2i-1}\\cup P_{2i+1}\\cup l_1\\cup l_2\\right)$ whose boundary intersects each of $l_1$,  $l_2$, $P_{2i-1}$ and $P_{2i+1}$. Then $P_{2i}\\subset \\overline{V_i}$ for $i\\ge1$. For the previously given point $z$ in $P_\\infty\\cap(0,1)^2$, we can find for each $i\\ge1$ a point  $a_i\\in P_{2i}\\cap W_r$ such that $\\lim\\limits_{i\\rightarrow\\infty}a_i=z$. Since $P_{2i}\\subset L\\subset \\partial U$, we further find a point $b_i\\in (W_r\\cap V_i\\cap U)$ for every $i\\ge1$, such that the distance between $a_i$ and $b_i$ converges to zero as $i\\rightarrow\\infty$. Check Figure \\ref{key} for relative locations of $a_i\\in P_{2i+1}$ and $b_i\\in(W_r\\cap V_i\\cap U)$.\n\nNow, we may find arcs $\\alpha_i\\subset U$ for each $i\\ge1$ that starts from a fixed point $b_0\\in U$ and ends at $b_i$. Let $c_i$ be the last point on $\\alpha_i$ that leaves $\\partial[0,1]^2$. Let $d_i$ be the first point on $\\alpha_i$  after $c_i$ at which $\\alpha_i$ intersects $\\partial W_r$. Clearly, we have $c_i\\in(l_1\\cup l_2)$. Let $\\beta_i$ be the sub-arc of $\\alpha_i$ from $c_i$  to $d_i$. Check Figure \\ref{key} for a rough depiction of two possible locations for $\\beta_i$. Then $\\beta_i$ and $\\beta_j$ for $i\\ne j$ are contained in distinct components of $\\mathcal{A}_r\\setminus L$, where $\\mathcal{A}_r=[0,1]^2\\setminus W_r$ is topologically a closed annulus.\nSince $L\\subset K$ and $K\\cap U=\\emptyset$, the arcs $\\beta_i$ and $\\beta_j$ for $i\\ne j$ are contained in distinct components of $\\mathcal{A}_r\\setminus K$.\n\nLet $x_n$ be the only point on $l_1\\cap P_\\infty$ such that the right piece of $l_1\\setminus\\{x_n\\}$ does not intersect $P_\\infty$. Let $y_n$ be the point on $l_2\\cap P_\\infty$ such that the right piece of $l_2\\setminus\\{y_n\\}$ does not intersect $P_\\infty$. The sequence $\\{c_i\\}$ then has a limit point in $\\{x_n,y_n\\}$. We may assume that $z_r=\\lim\\limits_{i\\rightarrow\\infty}d_i$ for some point $z_r\\in\\partial W_r$. Since $\\partial[0,1]^2$ and $\\partial W_r$ are disjoint Jordan curves, from the choices of $x_n, y_n$ and $z_r$ we can infer that either $(x_n,z_r)\\in R_K$ or  $(y_n,z_r)\\in R_K$. The flexibility of $r>0$ then leads to the inclusion $z\\in\\left(\\overline{R_K}[x_n]\\cup\\overline{R_K}[y_n]\\right)$.\n\nNow consider the two closed sets $E_n=P_\\infty\\cap \\left(\\overline{R_K}[x_n]\\cup l_1\\right)$ and $F_n=P_\\infty\\cap \\left(\\overline{R_K}[y_n]\\cup l_2\\right)$, which satisfy $P_\\infty=E_n\\cup F_n$. From the connectedness of $P_\\infty$ we see that $E_n\\cap F_n\\ne\\emptyset$. Clearly, each point $w\\in (E_n\\cap F_n)$ necessarily falls into one of the following cases:\n\\begin{itemize}\n\\item[(1)] $w$ lies in $l_1\\subset\\partial B_{1\/n}(x)$ and belongs to $\\overline{R_K}[y_n]$,\n\\item[(2)] $w$ lies in $l_2\\subset\\partial B_{1\/n}(y)$  and belongs to $\\overline{R_K}[x_n]$,\n\\item[(3)] $w\\notin(l_1\\cup l_2)$ and it lies in $\\overline{R_K}[x_n]\\cap\\overline{R_K}[y_n]\\cap(0,1)^2$.\n\\end{itemize}\nIn case (1) we set $u_n=w,v_n=y_n$; in case (2) we set $u_n=x_n, v_n=w$; in case (3) we set $u_n=x_n, v_n=y_n$. Then, in cases (1) and (2) we have $v_n\\in\\overline{R_K}[u_n]\\subset\\overline{R_K}^2[u_n]$; and in case (3) we will have $v_n\\in\\overline{R_K}^2[u_n]$. This verifies the claim and completes our proof.\n\\end{proof}\n\nWith Lemma \\ref{key-lemma}, we are well prepared to prove Theorems \\ref{compare_atoms} and \\ref{lambda_inequality} as follows.\n\n\\begin{proof}[{\\bf Proof for Theorem \\ref{compare_atoms}}]\nSince $U$ is also a complementary component of $\\partial U$, we only verify that every atom of $L$ is contained in a single atom of $K$.\n\nTo this end, let $\\Dc_L^\\#$ consist of all those continua that are each a component of $d^*\\cap L$ for some $d^*\\in\\Dc_K$. By \\cite[p.44, Theorem 3.21]{Nadler92} and \\cite[p.278, Lemma 13.2]{Nadler92}, we see that $\\Dc_L^\\#$ is an upper semi-continuous decomposition of $L$. As  every fiber of $\\overline{R_K}^2$ is entirely contained in a single element of $\\Dc_K$, by Lemma \\ref{key-lemma} we know that every fiber $\\overline{R_L}[z]$ is entirely contained in a single element of $\\Dc_L^\\#$. This implies that $\\Dc_L^\\#$ is refined by $\\Dc_L$. In other words, every atom of $L$ is entirely contained in a single atom of $K$.\n\\end{proof}\n\n\\begin{proof}[{\\bf Proof for Theorem \\ref{lambda_inequality}}]\nTo obtain $\\lambda_L(x)\\le\\lambda_K(x)$ for all $x$, we only need to consider the points $x\\in L$. With no loss of generality, we may assume that $\\lambda_K(x)=m-1$ for some integer $m\\ge1$. That is to say, there exist strictly decreasing continua $d_1^*\\supset d_2^*\\supset\\cdots\\supset d_{m}^*=\\{x\\}$ such that $d_1^*$ is an atom of $K$ and $d_{i+1}^*$ an atom of $d_i^*$ for $1\\le i\\le m-1$. Here we may have $m=1$. By Theorem \\ref{compare_atoms}, the atom of $L$ containing $x$, denoted as $d_1$, is a subset of $d_1^*$. Since $d_1\\subset d_1^*$ also satisfies the assumptions of  Theorem \\ref{compare_atoms}, we can infer that the atom of $d_1$ containing $x$, denoted as $d_2$, is a subset of $d_2^*$. Repeating the same argument for $m$ times, we obtain for $1\\le i\\le m$ an order-$i$ atom $d_i$ of $L$ with $d_i\\subset d_i^*$. Here we have $d_{m}=\\{x\\}$ and hence $\\lambda_L(x)\\le m=\\lambda_K(x)$.\n\\end{proof}\n\\begin{rem}\nIn the proof for Theorem \\ref{lambda_inequality}, we know that $U$ is a component of $\\hat{\\bbC}\\setminus K$ and $L\\subset\\partial U$. Therefore, in the same way we can show that $\\lambda_L(x)\\le\\lambda_{\\partial U}(x)$ for all $x$. From this we can infer that\n$\\sup\\limits_U\\lambda_{\\partial U}(x)=\\tilde{\\lambda}_K(x)$ for all $x\\in\\hat{\\mathbb{C}}$.\n\\end{rem}\n\n\n\n\n\\section{On Lambda Equalities}\\label{equality}\n\nWe prove Theorem \\ref{equality-case}, establishing three equalities in terms of the lambda function. Two of these equalities are for $E$-compacta. The other one is for partially unshielded compacta.\n\nGiven an $E$-compactum  $K\\subset\\hat{\\mathbb{C}}$ with complementary components $U_1,U_2,\\ldots$, so that the diameters  $\\delta(U_i)$ either form a finite sequence of an infinite one converging to zero. The Torhorst Inequality requires that $\\sup\\limits_i\\lambda_{\\partial U_i}(x)\\le\\lambda_K(x)$ for all $x\\in\\hat{\\mathbb{C}}$ and for all $i\\ge1$. Since $\\lambda_K(x)=\\tilde{\\lambda}_K(x)=0$ for all $x\\in K^o\\cup\\left(\\bigcup\\limits_iU_i\\right)$, we only need to consider the points on $\\partial K$, which may not equal $\\bigcup\\limits_i\\partial U_i$.\n\nLemma \\ref{bridging_lemma} follows from \\cite[Lemma 3.3]{LLY-2019} and is useful when we prove Lemma \\ref{trivial-fiber}.\n\\begin{lemma}\\label{bridging_lemma}\nIf $A\\subset\\hat{\\bbC}$ is a closed topological annulus and $K\\subset\\hat{\\bbC}$ a compactum then the following statements are equivalent: (1) $A\\cap K$ has infinitely many components intersecting each of the two components of $\\partial A$; (2) $A\\setminus K$ has infinitely many components intersecting each of the two components of $\\partial A$.\n\\end{lemma}\n\n\n\n\\begin{lemma}\\label{trivial-fiber}\nGiven an $E$-compactum  $K\\subset\\hat{\\mathbb{C}}$ with complementary components $U_1,U_2,\\ldots$. If $\\overline{R_K}[x]$ contains a point $y\\ne x$ then $y\\in\\overline{R_{\\partial U_i}}[x]$ for some $i$.\n\\end{lemma}\n\\begin{proof}\nLet $\\rho(x,y)$ be the spherical distance between $x$ and $y$. For each $n\\ge2$ let $B_n(x)$ and $B_n(y)$ be the open disks of radius $2^{-n}\\rho(x,y)$ that are centered at $x$ and $y$ respectively. Then $A_n=\\hat{\\mathbb{C}}\\setminus\\left(B_n(x)\\cup B_n(y)\\right)$ is a topological annulus. By \\cite[Theorem 1.3]{LYY-2020}, the intersection $A_n\\cap K$ has infinitely many components that intersect $\\partial B_n(x)$ and $\\partial B_n(y)$ both. By Lemma \\ref{bridging_lemma}, the difference $A_n\\setminus K$ has infinitely many components, say $\\{P^n_j: j\\ge1\\}$, that intersect $\\partial B_n(x)$ and $\\partial B_n(y)$ both. Since the diameters of those $P^n_j$ are no less than $\\rho(x,y)\/2$ and since we assume $K$ to be an $E$-compactum, there is an integer $i(n)$ such that $U_{i(n)}$ contains infinitely many of those $P^n_j$. Here all those $P^n_j$ that are contained in $U_{i(n)}$ are each a component of $A_n\\cap U_{i(n)}$.\n\nNow, choose a subsequence $\\{Q^n_k: k\\ge1\\}$ of $\\{P^n_j: j\\ge1\\}$, with $Q_k^n\\subset U_{i(n)}$, such that $\\overline{Q^n_k}$ converges under Hausdorff distance to a continuum $M_n$. Then $M_n$ is a subset of $\\partial U_{i(n)}$ and intersects  $\\partial B_n(x)$ and $\\partial B_n(y)$ both. Fixing any $a_n$ in $M_n\\cap \\partial B_n(x)$ and $b_n$ in $M_n\\cap \\partial B_n(y)$, we will have $(a_n,b_n)\\in R_{\\partial U_{i(n)}}$. Since $K$ is an $E$-compactum, there are infinitely many integers $n$ such that $i(n)$ takes the same value, say $i$. Therefore, we have two infinite sequences $\\{c_n\\}\\subset\\{a_n\\}$ and $\\{d_n\\}\\subset \\{b_n\\}$, with $c_n,d_n\\in\\partial U_i$, such that $(c_n,d_n)\\in R_{\\partial U_i}$ for all $n\\ge2$. Since $\\lim\\limits_{n\\rightarrow\\infty}c_n=x$ and $\\lim\\limits_{n\\rightarrow\\infty}d_n=y$, we readily have $(x,y)\\in\\overline{R_{\\partial U_i}}$, or equivalently $y\\in\\overline{R_{\\partial U_i}}[x] $.\n\\end{proof}\n\nNow we are well prepared to prove parts (i) and (ii) of Theorem \\ref{equality-case}, whose results are respectively included in the next two propositions.\n\n\\begin{proposition}\\label{equality-case-1}\nIf $K$ is an $E$-compactum such that $\\tilde{\\lambda}_K(x)=0$ for all $x\\in\\hat{\\mathbb{C}}$ then $\\lambda_K(x)$ vanishes everywhere.\n\\end{proposition}\n\\begin{proof}\nAs $\\tilde{\\lambda}_K(x)$ vanishes everywhere, all the relations $\\overline{R_{\\partial U_i}}$ are trivial, in the sense that the fibers $\\overline{R_{\\partial U_i}}[x]$ are each a singleton for all $i$ and all $x\\in\\partial U_i$. Combing this with the conclusion of Lemma \\ref{trivial-fiber}, we can infer that the fiber $\\overline{R_K}[x]=\\{x\\}$ for all $x\\in K$. From this, we see that every atom of $K$ is a singleton and that $\\lambda_K(x)=0$ for all $x$.\n\\end{proof}\n\n\n\\begin{proposition}\\label{equality-case-2}\nGiven an $E$-compactum $K$. If $\\partial U_i\\cap\\partial U_j=\\emptyset$ for $i\\ne j$ then $\\lambda_K=\\tilde{\\lambda}_K$.\n\\end{proposition}\n\\begin{proof}\nLet $\\mathcal{D}_i$ denote the core decomposition of $\\partial U_i$. Since we assume that $\\partial U_i\\cap\\partial U_j=\\emptyset$ for $i\\ne j$, the collection\n$\\displaystyle \\mathcal{D}_K^*:=\\left(\\bigcup\\limits_i\\mathcal{D}_i\\right)\\cup\\left\\{\\{x\\}: x\\in K\\setminus\\left(\\bigcup\\limits_i\\partial U_i\\right)\\right\\}$\nis a partition that divides $K$ into sub-continua. It suffices to show that $\\Dc_K^*$ is the core decomposition of $K$.\n\nRecall that $\\Dc_K$ is the finest monotone decomposition such that every fiber of $\\overline{R_K}$ is contained in a single element of $\\Dc_K$. By Lemma \\ref{key-lemma}, we know that $\\Dc_K$ is refined by $\\Dc_K^*$. On the other hand, since $K$ is an $E$-compactum and since $\\partial U_i\\cap\\partial U_j=\\emptyset$ for $i\\ne j$, we can use Lemma \\ref{trivial-fiber} to infer that every fiber of $\\overline{R_K}$ is contained in a single element of $\\Dc^*_K$. Therefore, we only need to verify that $\\mathcal{D}^*_K$ is upper semi-continuous, which then indicates that $\\Dc_K^*$ is a monotone decomposition hence is refined by $\\Dc_K$.\n\nIn other words, we need to verify that the equivalence $\\sim$ determined by the partition $\\Dc_K^*$ is closed as a subset of $K\\times K$. To this end, we consider an arbitrary sequence $\\{(x_n,y_n): n\\ge1\\}$ in $K\\times K$ with $\\lim\\limits_{n\\rightarrow\\infty}(x_n,y_n)=(x,y)$ such that $x_n\\sim y_n$ for all $n\\ge1$. There are two possibilities: either $x=y$ or $x\\ne y$. In the first case, we have $(x,y)=(x,x)$, which is surely an element of $\\sim$. In the second, the assumption that $K$ is an $E$-compactum implies that there is some $U_i$ such that $\\{x_n,y_n\\}\\subset\\partial U_i$ for infinitely many $n\\ge1$. Consequently, the subset $\\{x,y\\}$ is contained in a single element of $\\Dc_{i}$, which is a sub-collection of $\\Dc_K^*$. That is to say, we have $x\\sim y$. This ends our proof.\n\\end{proof}\n\n\nThe arguments in the above proof actually imply the following.\n\\begin{theo}\\label{equal-cd}\nGiven an $E$-compactum $K$. If $\\partial U_i\\cap\\partial U_j=\\emptyset$ for $i\\ne j$ then every atom of $K$ is either an atom of some $\\partial U_i$ or a singleton $\\{x\\}$ with $x\\in K\\setminus\\left(\\bigcup_i\\partial U_i\\right)$.\n\\end{theo}\n\n\nNow we go on to consider partially unshielded compacta and obtain Theorem \\ref{equality-case}(iii).\n\n\\begin{deff}\\label{part-unshielded}\nLet $L\\subset\\hat{\\mathbb{C}}$ be an unshielded compactum, which equals the boundary $\\partial U$ of one of its complementary components $U$. A compactum $K$ formed by the union of $L$ with some complementary components of $L$ other than $U$ is called a {\\bf partially unshielded compactum} determined by $L$.\n\\end{deff}\n\nIn order to find typical examples, one may set $L$ to be a polynomial Julia set, $U$ the unbounded Fatou component, and $K$ the union of $L$ and some bounded Fatou components. The next proposition discusses the relation between the atoms of any given compactum $L\\subset\\hat{\\mathbb{C}}$ and those of a compactum $K$, where $K$  is  the union of $L$ with some (not all) components of $\\hat{\\bbC}\\setminus L$.\n\n\n\\begin{proposition}\\label{useful}\nGiven a planar compactum $L\\subset \\hat{\\mathbb{C}}$ and a family $\\{U_\\alpha:\\ \\alpha\\in I\\}$  of  components of \\ $\\hat{\\mathbb{C}}\\!\\setminus\\!L$. If $\\displaystyle K=L\\cup\\left(\\bigcup_{\\alpha\\in I}U_\\alpha\\right)$ then  $\\overline{R_K}$ is a subset of  $\\{(z,z):\\ z\\in K\\!\\setminus\\!L\\}\\cup\\overline{R_L}$. Consequently, every atom of $K$ is either a singleton lying in $K\\setminus L$ or a sub-continuum  of an atom of $L$.\n\\end{proposition}\n\n\\begin{proof}\nSince $\\displaystyle K=L\\cup\\left(\\bigcup_{\\alpha\\in I}U_\\alpha\\right)$, every point $z\\in (K\\setminus L)$ lies in some $U_\\alpha$. Thus the atom of $K$ containing $z$ is exactly the singleton $\\{z\\}$. From this it readily follows that  every atom $d^*$ of $K$ that intersects $L$ is a sub-continuum of $L$. So we have $\\overline{R_K}=\\{(z,z):\\ z\\in K\\!\\setminus\\!L\\}\\cup\\left(\\overline{R_K}\\cap L^2\\right)$. Therefore, we only need to show that $\\left(\\overline{R_K}\\cap L^2\\right)\\subset \\overline{R_L}$.\n\nIndeed, if on the contrary there were some $(x,y)\\in \\overline{R_K}\\cap L^2$ not belonging to $\\overline{R_L}$ then, for any small enough number $r>0$, the difference $L\\setminus (B_r(x)\\cup B_r(y))$ would have finitely many components intersecting $\\partial B_r(x)$ and $\\partial B_r(y)$ both. Let $A_r=\\hat{\\mathbb{C}}\\setminus (B_r(x)\\cup B_r(y))$. By  Lemma \\ref{bridging_lemma},   $A_r\\setminus L$  has at most finitely many components that intersect $\\partial B_r(x)$ and $\\partial B_r(y)$ both. As we assume that  $\\displaystyle K=L\\cup\\left(\\bigcup_{\\alpha\\in I}U_\\alpha\\right)$, every component of $A_r\\setminus K$ is also a component of $A_r\\setminus L$. Thus $A_r\\setminus K$ has at most finitely many components that intersect both $\\partial B_r(x)$ and $\\partial B_r(y)$. In other words, we have $(x,y)\\notin\\overline{R_K}$. This is absurd since we assume that $(x,y)\\in \\overline{R_K}$.\n\\end{proof}\n\n\nThere are other basic facts concerning an unshielded compactum $L$  and a partially unshielded compactum $K$ determined by $L$. Firstly, every interior point of $K$ lies in some complementary component of $L$; secondly, every boundary point of $K$ lies in $L$. Thus we always have $\\partial K=L$; moreover, every atom of $K$ that intersects the interior $K^o$ is necessarily a singleton. Therefore, in order to determine the atoms of $K$ we only need to consider those of $L$.\n\n\\begin{theo}\\label{part-2}\nLet $L\\subset\\hat{\\mathbb{C}}$ be  an unshielded compactum. Let $K$ be a partially unshielded compactum determined by $L$. Then every atom of $L$ is also an atom of $K$ and we have $\\Dc_K=\\Dc_L\\cup\\{\\{x\\}: x\\in K\\setminus L\\}$.  Consequently, $\\tilde{\\lambda}_K(x)=\\lambda_K(x)$ for all $x\\in\\hat{\\mathbb{C}}$.\n\\end{theo}\n\n\\begin{proof}\nAs $L$ is unshielded, there is a component $U$ of $\\hat{\\mathbb{C}}\\setminus L$ with $L=\\partial U$. By Lemma \\ref{key-lemma},  every atom of $L$ lies in a single atom of $K$. By Lemma \\ref{useful}, every atom of $K$ intersecting $L$ is contained in a single atom of $L$. Thus every atom of $L$ is also an atom of $K$. As any singleton $\\{x\\}$ with $x\\in K^o= K\\setminus L$ is an atom of $K$, we have $\\Dc_K=\\Dc_L\\cup\\{\\{x\\}: x\\in K\\setminus L\\}$. This indicates the Lambda Equality $\\tilde{\\lambda}_K=\\lambda_K$.\n\\end{proof}\n\n\\begin{rem}\\label{why_partially_unshielded}\nTheorem \\ref{part-2} gives a result that is slightly stronger than  Theorem \\ref{equality-case}(iii). In particular, for any full compactum $K$ we have $\\mathcal{D}_{\\partial K}\\subset\\mathcal{D}_K$. Therefore, a full compactum $K$ is a Peano compactum if and only if the boundary $\\partial K$ is. In particular, if  $G\\subset\\hat{\\mathbb{C}}$ is  a simply connected bounded domain then $\\partial G$ is locally connected if and only if $K=\\hat{\\mathbb{C}}\\setminus G$ is locally connected, or equivalently when $K$ is a Peano continuum. This basic fact has been well known, see for instance the items (iii) and (iv) of \\cite[p.20, Theorem 2.1]{Pom92}. Now, it is extended to a quantitative version in Theorem \\ref{part-2}. This extension applies to an arbitrary full continuum, that may or may not be locally connected.\n\\end{rem}\n\n\n\\section{The Gluing Lemma for Lambda Functions}\\label{glue}\n\nWe will follow the philosophy of the well known gluing lemma for continuous maps.\nSee for instance \\cite[p.69, Theorem (4.6)]{Armstrong} for the simple case and \\cite[p.70, Theorem (4.8)]{Armstrong} for the general setting. Our aim is to prove Theorem \\ref{gluing_lemma}, which deals with the lambda functions $\\lambda_K,\\lambda_L$ for planar compacta  $K\\supset L$  such that $A=\\overline{K\\setminus L}$ intersects $L$ at finitely many points $x_1,\\ldots,x_n$. In Theorem \\ref{baby_M}, we further extend Theorem \\ref{gluing_lemma} to the case that $A\\cap L$ is a countably infinite set, under additional assumptions. Notice that when $A\\cap L$ is an infinite set Theorem \\ref{gluing_lemma} may not hold. See Examples \\ref{cantor_combs}, \\ref{brooms} and \\ref{cup-fs}.\n\n\\begin{proof}[{\\bf Proof for Theorem \\ref{gluing_lemma}}]\nFor $1\\le i\\le n$, denote by $d_i^1$ the order-$1$ atom of $A$ that contains $x_i$. Similarly, denote by $e_i^1$  the atom of $L$ that contains $x_i$.\nLet $K_1=A_1\\cup L_1$, where $\\displaystyle A_1=\\bigcup_id_i^1$ and $\\displaystyle L_1=\\bigcup_ie_i^1$. Then $K_1$ has finitely many components. Let $\\Ec_1$ be the collection of these components.\n\nBy \\cite[Theorem 1.3]{LYY-2020},  a point $y\\ne x$ lies in $\\overline{R_K}[x]$\nif and only if $K\\setminus(B_r(x)\\cup B_r(y))$ has infinitely many components that intersect both $\\partial B_r(x)$ and $\\partial B_r(y)$ for small enough  $r>0$. Because of this, we can directly check that $\\overline{R_K}=\\overline{R_A}\\cup\\overline{R_L}$. Here $\\overline{R_K},\\overline{R_A},\\overline{R_L}$ are respectively  the closed Sch\\\"onflies relations on $K,A$ and $L$.  Let \\[\n\\Dc_1=\\left(\\Dc_L\\setminus\\left\\{e_1^1,\\ldots,e_n^1\\right\\}\\right)\\cup\n\\left(\\Dc_A\\setminus\\left\\{d_1^1,\\ldots,d_n^1\\right\\}\\right)\\cup\n\\Ec_1.\\]\nThen $\\Dc_1$ is an upper semi-continuous decomposition of $K$ into subcontinua. Since $\\Dc_1$ does not split the fibers of $\\overline{R_K}$, it is refined by $\\Dc_K$, the core decomposition of $K$ with Peano quotient. On the other hand, the equality $\\overline{R_K}=\\overline{R_A}\\cup\\overline{R_L}$ indicates that $\\mathcal{D}_K$ does not split the fibers of $\\overline{R_A}$ and those of $\\overline{R_L}$. Thus each atom of $A$ lies in an atom of $K$; similarly, every atom of $L$ lies in an atom of $K$. Consequently, we have.\n\\begin{lemma}\\label{gluing_atoms_a}\n$\\Dc_K=\\Dc_1$. Thus $d\\cap A$ (or $d\\cap L$) either is empty or consists of finitely many atoms of $A$ (resp. $L$) for any atom $d$ of $K$.\n\\end{lemma}\n\n\nLemma \\ref{gluing_atoms_a} ensures that $\\displaystyle \\lambda_K(x)=\\max\\left\\{\\lambda_A(x),\\lambda_L(x)\\right\\}$ for all $x\\notin K_1$.  That is to say, the equation $\\lambda_K(x)=\\max\\left\\{\\lambda_{\\overline{K\\setminus L}}(x),\\lambda_L(x)\\right\\}$ in Theorem \\ref{gluing_lemma} holds for all points $x\\notin K_1$, so that we only need to consider the points $x\\in K_1$.\n\nNotice that we have set $A=\\overline{K\\setminus L}$, $\\displaystyle A_1=\\bigcup_id_i^1$, and $\\displaystyle L_1=\\bigcup_ie_i^1$. We will need to verify that $\\displaystyle \\lambda_{K_1}(x)=\\max\\left\\{\\lambda_{A_1}(x),\\lambda_{L_1}(x)\\right\\}$ for all $x\\in K_1$, since for $x\\in A_1$ and $y\\in L_1$ we have\n\\[\n\\begin{array}{ccc}\n\\lambda_A(x)=\\left\\{\\begin{array}{ll} 0& \\{x\\}\\in\\Dc_A\\\\\n1+\\lambda_{A_1}(x)& otherwise\\end{array}\\right. &\\text{and}&\n\\lambda_L(y)=\\left\\{\\begin{array}{ll} 0& \\{y\\}\\in\\Dc_L\\\\\n1+\\lambda_{L_1}(y)& otherwise.\\end{array}\\right.\n\\end{array}\\]\nTo do that, we recall that  $\\mathcal{D}_{A_1}$ consists of all the order-$2$ atoms of $A$ lying in $A_1$. Similarly, $\\mathcal{D}_{L_1}$ consists of all the order-$2$ atoms of $L$ lying in $L_1$. Thus we may repeat the above procedure again, replacing $A$ and $L$ by $A_1$ and $L_1$. This then gives rise to two compacta $A_2\\subset A_1$ and $L_2\\subset L_1$ such that\n$\\displaystyle \\lambda_{K_1}(x)=\\max\\left\\{\\lambda_{A_1}(x),\\lambda_{L_1}(x)\\right\\}$ for all $x\\notin K_2=A_2\\cup L_2$.\n\nWe may carry out the same procedure indefinitely and obtain two decreasing sequences of compacta: (1) $A_1\\supset A_2\\supset\\cdots$ and (2) $L_1\\supset L_2\\supset\\cdots$. Setting $K_p=A_p\\cup L_p$ for $p\\ge1$, we have the following equations:\n\\begin{equation}\n\\displaystyle \\lambda_{K_{p}}(x)=\\left\\{\\begin{array}{ll}0& \\{x\\}\\in\\Dc_{K_p}\\\\\n1+\\lambda_{K_{p+1}}(x)& otherwise\\end{array}\\right. \\quad (x\\in K_{p+1}).\n\\end{equation}\n\\begin{equation}\n\\displaystyle \\lambda_{A_{p}}(x)=\\left\\{\\begin{array}{ll}0& \\{x\\}\\in\\Dc_{A_p}\\\\\n1+\\lambda_{A_{p+1}}(x)& otherwsie\\end{array}\\right. \\quad(x\\in A_{p+1})\n\\end{equation}\n\\begin{equation}\n\\displaystyle \\lambda_{L_{p}}(x)=\\left\\{\\begin{array}{ll}0& \\{x\\}\\in\\Dc_{L_p}\\\\\n1+\\lambda_{L_{p+1}}(x)& otherwise\\end{array}\\right.\\quad (x\\in L_{p+1})\n\\end{equation}\n\\begin{equation}\n\\lambda_{K_p}(x)=\\max\\left\\{\\lambda_{A_p}(x),\\lambda_{L_p}(x)\\right\\}\\quad (x\\notin K_{p+1})\n\\end{equation}\nThere are two possibilities. In the first, we have $K_p=K_{p+1}$  for some $p\\ge1$, indicating that $K_m=K_p$ for all $m\\ge p$.\nIn such a case, we have $\\lambda_{K_p}(x)=\\max\\left\\{\\lambda_{A_p}(x),\\lambda_{L_p}(x)\\right\\}$ and hence $\\lambda_{K}(x)=\\max\\left\\{\\lambda_{A}(x),\\lambda_{L}(x)\\right\\}$.\nIn the second, we have $K_p\\ne K_{p+1}$ for all $p\\ge1$. This implies that $\\lambda_K(x)=\\max\\left\\{\\lambda_A(x),\\lambda_L(x)\\right\\}=\\infty$ holds for all $x\\in K_\\infty=\\bigcap_pK_p$ and that $\\lambda_{K}(x)=p+\\lambda_{K_p}(x)=p+\\max\\left\\{\\lambda_{A_p}(x),\\lambda_{L_p}(x)\\right\\}=\n\\max\\left\\{\\lambda_{A}(x),\\lambda_{L}(x)\\right\\}$ holds for $p\\ge1$ and $x\\notin K_p\\setminus K_{p+1}$. Here $\\displaystyle K_1\\setminus K_{\\infty}=\\bigcup_{p=1}^\\infty(K_p\\setminus K_{p+1})$. This completes our proof.\n\\end{proof}\n\nLemma \\ref{gluing_atoms_a} and Theorem \\ref{gluing_lemma} are useful, when we study $\\lambda_K$ for certain choices of planar compacta $K$. For instance, we may choose $K$ to be the Julia set of a renormalizable polynomial $f(z)=z^2+c$ and $L$ the small Julia set. For the sake of convenience, we further assume that the only critical point of $f$ is recurrent and that there is no irrationally neutral cycle. Then it is possible to choose a decreasing sequence of Jordan domains $\\{U_n\\}$, with $\\overline{U_{n+1}}\\subset U_n$ and $\\displaystyle L=\\bigcap_{n=1}^\\infty U_n$, such that every $K\\cap \\partial U_n$ consists of finitely many points that are periodic or pre-periodic. See for instance \\cite[section 2.2]{Jiang00}.\nFor any $n\\ge1$ we can use \\cite[Theorems 2 and 3]{Kiwi04}  to infer that every singleton $\\{x\\}$ with $x\\in (K\\cap\\partial U_n)$ is an atom of $K$ hence is also an atom of $L_n=K\\cap\\overline{U_n}$. Combining these with Lemma \\ref{gluing_atoms_a} and Theorem \\ref{gluing_lemma},  we further see that $\\mathcal{D}_{L_{n+1}}\\subset\\mathcal{D}_{L_n}\\subset\\mathcal{D}_K$ for all $n\\ge1$. However, we are not sure whether $\\mathcal{D}_L\\subset\\mathcal{D}_K$. Similarly, it is not clear whether $\\lambda_K(x)=\\lambda_L(x)$ holds for $x\\in L$. Therefore, we propose the following.\n\n\\begin{que}\\label{small_julia}\nLet $K=L_0\\supset L_1\\supset L_2\\supset\\cdots$ a decreasing sequence of planar compacta such that  $L_n\\cap\\overline{K\\setminus L_n}$ is a finite set for all $n\\ge1$.\nSetting $L=\\bigcap_{n\\ge1}L_n$. Find conditions so that (1) $\\mathcal{D}_L\\subset\\mathcal{D}_K$ or  (2) $\\lambda_K(x)=\\lambda_L(x)$ holds for all $x\\in L$.\n\\end{que}\n\nAs a response to Question \\ref{small_julia}, we turn to study the lambda functions of two planar compact $K\\supset L$\nsuch that $K\\setminus L$ is contained in the union of at most countably many continua  $P_n\\subset K$ that satisfy the following properties:\n\\begin{itemize}\n\\item[(P1)] every $P_n$ intersects $L$ at a single point $x_n$, and\n\\item[(P2)] for any constant $C>0$ at most finitely many $P_n$ are of diameter greater than $C$.\n\\item[(P3)] $P_n\\cap P_m=\\emptyset$ for $n\\ne m$.\n\\end{itemize}\nHere $P_n\\setminus\\{x_n\\}$ might be disconnected for some of the integers $n\\ge1$. Notice that there is a special situation, when $K$ is the Mandelbrot set $\\M$. Then, in order that the above properties (P1)-(P3) be satisfied, we may choose $L$ to be  the closure of a hyperbolic component or   a {\\bf Baby Mandelbrot set}.\n\nAs an extension of Theorem \\ref{gluing_lemma}, we will obtain the following.\n\n\\begin{theo}\\label{baby_M}\nGiven two planar compacta $K\\supset L$ that satisfy (P1) to (P3),  we have \\begin{equation}\\label{baby}\n\\lambda_K(x)=\\left\\{\\begin{array}{lll} \\lambda_{P_n}(x)&x\\in P_n\\setminus\\{x_n\\}\\ {for\\ some}\\ n&({case}\\ 1)\\\\ \\lambda_L(x)& x\\in L\\setminus\\{x_n: n\\in\\mathbb{N}\\}&({case}\\ 2)\\\\ \\max\\left\\{\\lambda_L(x_n),\\lambda_{P_n}(x_n)\\right\\}& x=x_n\\ {for\\ some}\\ x_n&({case}\\ 3)\\\\\n0 &{otherwise}&(case\\ 4)\\end{array}\\right. \\end{equation}\n\\end{theo}\n\nWe just need to consider the above equation for points $x\\in K$.\nTo do that, we may\ndefine an equivalence $\\sim$ on $\\mathbb{N}$ so that $m\\sim n$ if and only if $x_m,x_n$ are  contained in the same atom of $L$. Let $\\{I_j:j\\}$ be the equivalence classes of $\\sim$.\n\nDenote by $d_n$ the atom of $P_n$ that contains $x_n$, and by $e_j$ the atom of $L$ that contains all $x_n$ with $n\\in I_j$. Moreover, set $e'_j=e_j\\cup\\left(\\bigcup_{n\\in I_j}d_n\\right)$ for every $j$.\n\nThen $\\{e'_j: j\\}$ is a collection of at most countably many continua that are pairwise disjoint.\nNow we consider the following upper semi-continuous decomposition of $K$:\n\\begin{equation}\\label{baby_M_partition}\n\\Dc_1=\\left(\\Dc_L\\setminus\\left\\{e_j: j\\right\\}\\right)\\cup\n\\left(\\bigcup_n\\Dc_{P_n}\\setminus\\{d_n\\}\\right)\\cup\n\\{e'_j: j\\}.\n\\end{equation}\nAll its elements are sub-continua of $K$ that do not split the fibers of $\\overline{R_K}$. So it is refined by $\\Dc_K$. On the other hand, by \\cite[Theorem 1.3]{LYY-2020}, we also have $\\overline{R_K}=\\overline{R_L}\\cup\\left(\\bigcup_{n\\ge1}\\overline{R_{P_n}}\\right)$. Thus $\\mathcal{D}_K$ does not split the fibers of $\\overline{R_L}$ and those of $\\overline{R_{P_n}}$ for all $n\\ge1$. Therefore, every atom of $L$ lies in an atom of $K$, so does  every atom of $P_n$. Consequently, we have.\n\\begin{lemma}\\label{gluing_atoms_b}\n$\\Dc_K=\\Dc_1$. Therefore, for any atom $d$ of $K$  the intersection $d\\cap L$ (or $d\\cap P_n$ for any $n\\ge1$) is either empty or a single atom of $L$ (respectively, $P_n$).\n\\end{lemma}\n\n\n\\begin{proof}[{\\bf Proof for Theorem \\ref{baby_M}}]\nClearly,  $\\lambda_K(x)=\\lambda_L(x)$ for all $x$ in $L\\setminus\\left(\\bigcup_je'_j\\right)$. Similarly,  $\\lambda_K(x)=\\lambda_{P_n}(x)$ for all $x$ in $P_n\\setminus d_n$. Moreover, $\\lambda_K(x_n)=\\lambda_L(x_n)=\\lambda_{P_n}(x_n)=0$ for all $x_n$ such that $\\{x_n\\}$ is an atom of $L$ and also an atom of $P_n$. Therefore, we just need to consider those $e_j'$  that are non-degenerate.\n\nLet $\\mathcal{N}_1$ be the collection of all the integers $j$ such that $e_j'$ is not a singleton.\nThen $e_{n_1}$ is a subcontinuum of $e_{n_1}'$ for any $n_1\\in\\mathcal{N}_1$, such that $e_{n_1}'\\setminus e_{n_1}$ is covered by all those $d_n$ with $n\\in I_{n_1}$. Thus the properties (P1) - (P3) are satisfied, if $K$ and $L$ are respectively replaced by $e_{n_1}'$ and $e_{n_1}$. It is then routine to check the following:\n\\begin{equation}\\label{inductive_1}\n\\lambda_K(x)=\\left\\{\\begin{array}{ll}\n\\lambda_{P_n}(x)&x\\in P_n\\setminus\\left(\\bigcup_{n_1}e_{n_1}'\\right)\\\\\n\\lambda_{L}(x)& x\\in L\\setminus\\left(\\bigcup_{n_1}e_{n_1}'\\right)\\\\\n1+\\lambda_{e_{n_1}'}(x)& x\\in e_{n_1}'\\ \\text{for\\ some}\\ n_1\\in\\mathcal{N}_1\n\\end{array}\\right.\n\\end{equation}\n\nEvery atom of $e_{n_1}'$ falls into exactly one of the following possibilities: (1) an order-two atom of $P_n$ for some $n\\in I_{n_1}$ that is disjoint from $\\{x_n\\}$, (2) an order-two atom of  $L$ that is disjoint from  $\\{x_n: n\\in I_{n_1}\\}$,  (3) a singleton $\\{x_n\\}$ for some $n\\in I_{n_1}$, which is an order-two atom of $L$ and is also an order-two atom of  $P_n$, (4) a non-singleton continuum that consists of the order-two atom of $L$ containing some $x_n$, with $n\\in I_{n_1}$, and the order-two atom of $P_n$ containing $x_n$.\n\nAn atom falling in the first three possibilities  is called an atom of {\\bf pure type}.\nWe can check that $e_{n_1}'$ has at most countably many atoms that is not of pure type. Such an atom is generally denoted as $e'_{n_1n_2}$. Similarly we can define continua $e'_{n_1n_2\\ldots n_p}$ for $p>2$. On the one hand, such a continuum  is an order-$p$ atom of $K$; on the other, it is also an atom of $e'_{n_1n_2,\\ldots n_{p-1}}$ that is not of pure type.\nBy the same arguments, that have been used in obtaining Equation (\\ref{inductive_1}), we can infer the following equation:\n\\begin{equation}\\label{inductive_2}\n\\lambda_K(x)=\\left\\{\\begin{array}{ll}\n\\lambda_{P_n}(x)&x\\in P_n\\setminus\\left(\\bigcup_{n_1,n_2}e_{n_1n_2}'\\right)\\\\\n\\lambda_{L}(x)& x\\in L\\setminus\\left(\\bigcup_{n_1,n_2}e_{n_1n_2}'\\right) \\\\\n2+\\lambda_{e_{n_1n_2}'}(x)& x\\in  e_{n_1n_2}'\\ \\text{for\\ some}\\ n_1,n_2\n\\end{array}\\right.\n\\end{equation}\nThis equation may be extended to order-$p$ atoms $e'_{n_1n_2\\ldots n_p}$ with $p\\ge2$ in the following way.\n\\begin{equation}\\label{inductive_p}\n\\lambda_K(x)=\\left\\{\\begin{array}{ll}\n\\lambda_{P_n}(x)&x\\in P_n\\setminus\\left(\\bigcup_{n_1,\\ldots,n_p}e_{n_1\\cdots n_p}'\\right)\\\\\n\\lambda_{L}(x)& x\\in L\\setminus\\left(\\bigcup_{n_1,\\ldots,n_p}e_{n_1\\cdots n_p}'\\right) \\\\\np+\\lambda_{e_{n_1\\cdots n_p}'}(x)& x\\in  e_{n_1\\cdots n_p}'\\ \\text{for\\ some}\\ n_1,\\ldots,n_p\n\\end{array}\\right.\n\\end{equation}\nNotice that Theorem \\ref{baby_M} holds for every $x\\in K$ lying in an atom  of  $e'_{n_1n_2\\ldots n_p}$ that is of pure type. Such a point $x$ does not lie in $e'_{n_1n_2\\ldots n_pn_{p+1}}$ for any choice of $n_{p+1}$ and hence falls into exactly one of the following possibilities:\n\\begin{itemize}\n\\item that $x\\in P_n\\setminus\\{x_n\\}$ for some $n\\ge1$ and $\\lambda_K(x)=\\lambda_{P_n}(x)\\ge p$.\n\\item that $x\\in L\\setminus\\{x_n: n\\ge1\\}$ and $\\lambda_K(x)=\\lambda_L(x)\\ge p$.\n\\item that $x=x_n$ for some $n\\ge1$ and $\\lambda_K(x)=\\max\\left\\{\\lambda_L(x),\\lambda_{P_n}(x)\\right\\}=p$.\n\\end{itemize}\nEvery other point $x\\in K$ necessarily lies in $e'_{n_1n_2\\ldots n_p}$ for infinitely many $p$. The continua $e'_{n_1n_2\\ldots n_p}$ decrease to a continuum $M_x$. There are three possibilities, either $x\\in L\\setminus\\{x_n\\}$, or  $x\\in P_n\\setminus\\{x_n\\}$ for some $n\\ge1$, or $x=x_n$ for some $n\\ge1$.  In the first case, we have $\\lambda_K(x)=\\lambda_L(x)=\\infty$; in the second, we have  $\\lambda_K(x)=\\lambda_{P_n}(x)=\\infty$; in the third, we have $\\lambda_K(x)=\\max\\left\\{\\lambda_L(x),\\lambda_{P_n}(x)\\right\\}=\\infty$. This completes our proof.\n\\end{proof}\n\n\n\\begin{rem}\nLet $K=L_0\\supset L_1\\supset L_2\\supset\\cdots$ be given as in Question \\ref{small_julia}. Also let  $L=\\bigcap_{n\\ge1}L_n$.  Then $L_n\\cap\\overline{K\\setminus L_n}$ is a finite set for all $n\\ge1$.\nAssume in addition that (1) every singleton $\\{x_n\\}$ is an atom of $K$ and (2) $K$ and $L$  satisfy the requirements in Theorem \\ref{baby_M}. By Lemma \\ref{gluing_atoms_a}, we see that $\\mathcal{D}_{L_{n+1}}\\subset\\mathcal{D}_{L_n}\\subset\\mathcal{D}_K$ for all $n\\ge1$; thus from Theorem \\ref{gluing_lemma} we can infer that $\\lambda_K(x)=\\lambda_{L_n}(x)$  for all $x\\in L_n$. Moreover, by Lemma \\ref{gluing_atoms_b} we have $\\mathcal{D}_L\\subset\\mathcal{D}_K$. Therefore, by Theorem \\ref{baby_M} we further infer that $\\lambda_K(x)=\\lambda_L(x)$ holds for all $x\\in L$.\n\\end{rem}\n\n\n\\section{Examples}\\label{examples}\n\nWe shall construct examples.\nThe beginning two  provide choices of compacta $A,B\\subset\\hat{\\bbC}$ such that $\\lambda_{A\\cup B}(x)\\ne\\max\\left\\{\\lambda_A(x),\\lambda_B(x)\\right\\}$ for some $x$, although $\\lambda_A(x)=\\lambda_B(x)$ for all $x\\in A\\cap B$. In the first,  $A\\cap B$ is an uncountable set; in the second,  $A\\cap B$ is a countably infinite set. Therefore, the conditions of Theorem \\ref{gluing_lemma} are not satisfied.\n\n\n\\begin{exam}\\label{cantor_combs}\nLet $A\n\\{t+s{\\bf i}: t\\in\\Kc, 0\\le s\\le1\\}$, where $\\Kc$ is the Cantor ternary set. Let\n$B\n\\{t+(1+s){\\bf i}: 0\\le t\\le 1, s\\in\\Kc\\}$. Let $A_1=A\\cup B$ and $B_1=(A+1+{\\bf i})\\cup(B+1-{\\bf i})$.\nSee Figure \\ref{not_glued} for a simplified depiction of $A, B, A_1, B_1$.\n\\begin{figure}[ht]\n\\vspace{-0.05cm}\n\\begin{tabular}{ccccc}\n\\begin{tikzpicture}[x=1cm,y=1cm,scale=0.5]\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (3*\\i\/27,0) -- (3*\\i\/27,3);\n    \\draw[gray,very thick] (6\/9+3*\\i\/27,0) -- (6\/9+3*\\i\/27,3);\n     \\draw[gray,very thick] (2+3*\\i\/27,0) -- (2+3*\\i\/27,3);\n    \\draw[gray,very thick] (2+6\/9+3*\\i\/27,0) -- (2+6\/9+3*\\i\/27,3);\n}\n\\end{tikzpicture} \\hspace{0.25cm}\n&\n\\hspace{0.25cm}\n\\begin{tikzpicture}[x=1cm,y=1cm,scale=0.5]\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (0,3*\\i\/27+3) -- (3,3*\\i\/27+3);\n    \\draw[gray,very thick] (0,6\/9+3*\\i\/27+3) -- (3,6\/9+3*\\i\/27+3);\n     \\draw[gray,very thick] (0,2+3*\\i\/27+3) -- (3,2+3*\\i\/27+3);\n    \\draw[gray,very thick] (0,2+6\/9+3*\\i\/27+3) -- (3,2+6\/9+3*\\i\/27+3);\n}\n\\draw[gray,dashed] (0,0) -- (3,0)-- (3,3)-- (0,3)-- (0,0);\n\\end{tikzpicture} \\hspace{0.25cm}\n\n&\n\\hspace{0.25cm}\n\\begin{tikzpicture}[x=1cm,y=1cm,scale=0.5]\n\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (3*\\i\/27,0) -- (3*\\i\/27,3);\n    \\draw[gray,very thick] (6\/9+3*\\i\/27,0) -- (6\/9+3*\\i\/27,3);\n     \\draw[gray,very thick] (2+3*\\i\/27,0) -- (2+3*\\i\/27,3);\n    \\draw[gray,very thick] (2+6\/9+3*\\i\/27,0) -- (2+6\/9+3*\\i\/27,3);\n}\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (0,3*\\i\/27+3) -- (3,3*\\i\/27+3);\n    \\draw[gray,very thick] (0,6\/9+3*\\i\/27+3) -- (3,6\/9+3*\\i\/27+3);\n     \\draw[gray,very thick] (0,2+3*\\i\/27+3) -- (3,2+3*\\i\/27+3);\n    \\draw[gray,very thick] (0,2+6\/9+3*\\i\/27+3) -- (3,2+6\/9+3*\\i\/27+3);\n}\n\\end{tikzpicture} \\hspace{0.25cm}\n\n&\n\\hspace{0.25cm}\n\\begin{tikzpicture}[x=1cm,y=1cm,scale=0.5]\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (3+3*\\i\/27,3) -- (3+3*\\i\/27,6);\n    \\draw[gray,very thick] (3+6\/9+3*\\i\/27,3) -- (3+6\/9+3*\\i\/27,6);\n     \\draw[gray,very thick] (3+2+3*\\i\/27,3) -- (3+2+3*\\i\/27,6);\n    \\draw[gray,very thick] (3+2+6\/9+3*\\i\/27,3) -- (3+2+6\/9+3*\\i\/27,6);\n}\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (3,3*\\i\/27) -- (3+3,3*\\i\/27);\n    \\draw[gray,very thick] (3,6\/9+3*\\i\/27) -- (3+3,6\/9+3*\\i\/27);\n     \\draw[gray,very thick] (3,2+3*\\i\/27) -- (3+3,2+3*\\i\/27);\n    \\draw[gray,very thick] (3,2+6\/9+3*\\i\/27) -- (3+3,2+6\/9+3*\\i\/27);\n}\n\\end{tikzpicture} \\hspace{0.25cm}\n\n&\n\\hspace{0.25cm}\n\\begin{tikzpicture}[x=1cm,y=1cm,scale=0.5]\n\n\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (3*\\i\/27,0) -- (3*\\i\/27,3);\n    \\draw[gray,very thick] (6\/9+3*\\i\/27,0) -- (6\/9+3*\\i\/27,3);\n     \\draw[gray,very thick] (2+3*\\i\/27,0) -- (2+3*\\i\/27,3);\n    \\draw[gray,very thick] (2+6\/9+3*\\i\/27,0) -- (2+6\/9+3*\\i\/27,3);\n}\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (0,3*\\i\/27+3) -- (3,3*\\i\/27+3);\n    \\draw[gray,very thick] (0,6\/9+3*\\i\/27+3) -- (3,6\/9+3*\\i\/27+3);\n     \\draw[gray,very thick] (0,2+3*\\i\/27+3) -- (3,2+3*\\i\/27+3);\n    \\draw[gray,very thick] (0,2+6\/9+3*\\i\/27+3) -- (3,2+6\/9+3*\\i\/27+3);\n}\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (3+3*\\i\/27,3) -- (3+3*\\i\/27,6);\n    \\draw[gray,very thick] (3+6\/9+3*\\i\/27,3) -- (3+6\/9+3*\\i\/27,6);\n     \\draw[gray,very thick] (3+2+3*\\i\/27,3) -- (3+2+3*\\i\/27,6);\n    \\draw[gray,very thick] (3+2+6\/9+3*\\i\/27,3) -- (3+2+6\/9+3*\\i\/27,6);\n}\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (3,3*\\i\/27) -- (3+3,3*\\i\/27);\n    \\draw[gray,very thick] (3,6\/9+3*\\i\/27) -- (3+3,6\/9+3*\\i\/27);\n     \\draw[gray,very thick] (3,2+3*\\i\/27) -- (3+3,2+3*\\i\/27);\n    \\draw[gray,very thick] (3,2+6\/9+3*\\i\/27) -- (3+3,2+6\/9+3*\\i\/27);\n}\n\\end{tikzpicture}\n\\\\ $A$& $B$& $A_1=A\\cup B$& $B_1$& $A_1\\cup B_1$\\end{tabular}\n\\caption{The two compacta $A, B$ and their union.}\\label{not_glued}\n\\end{figure}\nThen\n$\\lambda_A(x)=1$ for all $x\\in A$ and vanishes otherwise; similarly,  $\\lambda_B(x)=1$ for all $x\\in B$ and vanishes otherwise.\nHowever, both $A\\cap B$ and $A_1\\cap B_1$ are uncountable, thus the conditions in Theorem \\ref{gluing_lemma} are not satisfied. Moreover, we have\n\\[\\lambda_{A_1}(x)=\\lambda_{A\\cup B}(x)=\\left\\{\\begin{array}{ll}2&x\\in A\\\\ 1& B\\setminus A\\\\ 0& {otherwise}\\end{array}\\right.\\quad {and}\\quad\n\\lambda_{A_1\\cup B_1}(x)=\\left\\{\\begin{array}{ll}\\infty & x\\in (A_1\\cup B_1)\\\\  0& {otherwise}.\\end{array}\\right.\\]\n\n\\end{exam}\n\n\n\\begin{exam}\\label{brooms}\nSet $A=\\bigcup\\limits_{n\\ge0}A_n$. Here $A_0=\\{s{\\bf i}: 0\\le s\\le1\\}$ and $A_1$ is the continuum that consists of the line $\\displaystyle\\left\\{1+t{\\bf i}: 0\\le t\\le1\\right\\}$ and all those lines connecting $1+{\\bf i}$ to $\\displaystyle\\frac{k}{k+1}$ for $k\\ge1$; moreover, for $n\\ge2$, $A_n=\\displaystyle \\left\\{2^{-n+1}t+s{\\bf i}: t+s{\\bf i}\\in A_1\\right\\}$.  See  Figure \\ref{broom_comb}.\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{cc}\n\\begin{tikzpicture}[x=1.618cm,y=1cm,scale=1]\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (0,3+3*\\i\/27) -- (3,3+3*\\i\/27);\n    \\draw[gray,very thick] (0,3+6\/9+3*\\i\/27) -- (3,3+6\/9+3*\\i\/27);\n     \\draw[gray,very thick] (0,3+2+3*\\i\/27) -- (3,3+2+3*\\i\/27);\n    \\draw[gray,very thick] (0,3+2+6\/9+3*\\i\/27) -- (3,3+2+6\/9+3*\\i\/27);\n}\n\n\\draw[gray,very thick] (0,0) --(0,3);\n\\draw[gray,very thick] (3,0) --(3,3);\n\\draw[gray,very thick] (3\/2,0) --(3\/2,3);\n\\draw[gray,very thick] (3\/4,0) --(3\/4,3);\n\\draw[gray,very thick] (3\/8,0) --(3\/8,3);\n\\draw[gray,very thick] (3\/16,0) --(3\/16,3);\n\\draw[gray,very thick] (3\/32,0) --(3\/32,3);\n\\draw[gray,very thick] (3\/64,0) --(3\/64,3);\n\\draw[gray,very thick] (3\/128,0) --(3\/128,3);\n\\draw[gray,very thick] (3\/256,0) --(3\/256,3);\n\\draw[gray,very thick] (3\/512,0) --(3\/512,3);\n\n\\foreach \\i in {2,...,7}\n{\n    \\draw[gray,very thick] (3,3) -- (3-3\/\\i,0);\n}\n\\node at (2.8,0.15) {$\\ldots$};\n\n\n\\foreach \\i in {2,...,7}\n{\n    \\draw[gray,very thick] (3\/2,3) -- (3\/2-1.5\/\\i,0);\n}\n\n\n \\node at (9\/16,1.75) {$\\vdots$};\n  \\node at (9\/16,1.25) {$\\vdots$};\n\\node at (-0.1,0.2){$0$}; \\node at (3.1,0.2){$1$};\n\\node at (0,0){$\\cdot$}; \\node at (3,0){$\\cdot$};\n\\node at (3,3){$\\cdot$}; \\node at (3,6){$\\cdot$};\n\\node at (3.35,3.0){$1\\!+\\!{\\bf i}$}; \\node at (3.35,6.0){$1\\!+\\!2{\\bf i}$};\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[x=1.618cm,y=1cm,scale=1]\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,dashed] (0,3+3*\\i\/27) -- (3,3+3*\\i\/27);\n    \\draw[gray,dashed] (0,3+6\/9+3*\\i\/27) -- (3,3+6\/9+3*\\i\/27);\n     \\draw[gray,dashed] (0,3+2+3*\\i\/27) -- (3,3+2+3*\\i\/27);\n    \\draw[gray,dashed] (0,3+2+6\/9+3*\\i\/27) -- (3,3+2+6\/9+3*\\i\/27);\n}\n\n\\draw[gray,very thick] (0,3) --(3,3);\n\n\\draw[gray,very thick] (0,0) --(0,3);\n\\draw[gray,very thick] (3,0) --(3,3);\n\\draw[gray,very thick] (3\/2,0) --(3\/2,3);\n\\draw[gray,very thick] (3\/4,0) --(3\/4,3);\n\\draw[gray,very thick] (3\/8,0) --(3\/8,3);\n\\draw[gray,very thick] (3\/16,0) --(3\/16,3);\n\\draw[gray,very thick] (3\/32,0) --(3\/32,3);\n\\draw[gray,very thick] (3\/64,0) --(3\/64,3);\n\\draw[gray,very thick] (3\/128,0) --(3\/128,3);\n\\draw[gray,very thick] (3\/256,0) --(3\/256,3);\n\\draw[gray,very thick] (3\/512,0) --(3\/512,3);\n\n\\node at (-0.1,0.2){$0$}; \\node at (3.1,0.2){$1$}; \\node at (3.35,3.0){$1\\!+\\!{\\bf i}$};\n\\end{tikzpicture}\\\\ $A\\cup B$ & $d$\n\\end{tabular}\n\\end{center}\n\\vskip -0.75cm\n\\caption{The compactum $A\\cup B$ and the  atom $d$ of $A\\cup B$.}\\label{broom_comb}\n\\end{figure}\nFurther setting $B$ as in Example \\ref{cantor_combs}, we have $A\\cap B=\\{ {\\bf i}\\}\\cup\\left\\{2^{-n}+{\\bf i}: n\\ge0\\right\\}$.\nIf $x\\in A\\cap B$ then $\\lambda_A(x)=\\lambda_B(x)=1$. Let\n$\\displaystyle L_1=\\left\\{t+s{\\bf i}:  0\\le s\\le 1, t=0\\ {or}\\ 2^{-n}\\ {for\\ some}\\ n\\ge0\\right\\}.$ Then $d=L_1 \\cup\\ \\{t+{\\bf i}: 0\\le t\\le1\\}$ is an atom of $A\\cup B$ and is not locally connected at any $x\\in A_0$. Moreover, we have\n\\[\\lambda_{A}(x)=\\left\\{\\begin{array}{ll}1&x\\in L_1 \\\\  0& {otherwise}\\end{array}\\right.\\quad {and}\\quad\n\\lambda_{A\\cup B}(x)=\\left\\{\\begin{array}{ll}2& x\\in A_0\\\\ 1&x\\in (B\\cup d)\\setminus A_0\\\\  0& {otherwise}.\\end{array}\\right.\\]\n\\end{exam}\n\n\n\nThe next two examples are about $E$-continua $K\\subset\\hat{\\mathbb{C}}$ such that the lambda equality given in (i) or (ii) of Theorem \\ref{equality-case} does not hold.\n\n\n\\begin{exam}\\label{E-compactum}\nLet $X$ denote the square $[1,2]\\times[0,{\\mathbf i}]\\subset\\hat{\\mathbb{C}}$. Let $Y$ be an embedding of $[0,\\infty)$ whose closure $\\overline{Y}$ equals the union of $Y$ with $\\partial X$. See the left part of Figure \\ref{negative} for a simplified representation of \\ $\\overline{Y}$, which is depicted as \\tb{blue}.\n\\begin{figure}[ht]\n\\vskip -0.25cm\n\\begin{tabular}{ll}\n\\begin{tikzpicture}[x=0.2cm,y=0.2cm,scale=0.55]\n\\draw[gray,thick] (64,0) -- (64,32) -- (0,32) -- (0,0) -- (64,0);\n\n\\foreach \\j in {0,1}\n{ \\draw[gray, thick] (32,\\j*16) -- (32,16+\\j*16) -- (16,16+\\j*16) -- (16,\\j*16) --(32,\\j*16);\n}\n\\foreach \\j in {0,...,3}\n{\n    \\draw[gray, thick] (16,\\j*8) -- (16,8+\\j*8) -- (8,8+\\j*8) -- (8,\\j*8) --(16,\\j*8);\n}\n\\foreach \\j in {0,...,7}\n{\n    \\draw[gray, thick] (8,\\j*4) -- (8,4+\\j*4) -- (4,4+\\j*4) -- (4,\\j*4) --(8,\\j*4);\n}\n\\foreach \\j in {0,...,15}\n{\n    \\draw[gray, thick] (4,\\j*2) -- (4,2+\\j*2) -- (2,2+\\j*2) -- (2,\\j*2) --(4,\\j*2);\n}\n\\foreach \\j in {0,...,31}\n{\n    \\draw[gray, thick] (2,\\j*1) -- (2,1+\\j*1) -- (1,1+\\j*1) -- (1,\\j*1) --(2,\\j*1);\n}\n\\foreach \\j in {0,...,63}\n{\n    \\draw[gray, thick] (1,\\j*1\/2) -- (1,1\/2+\\j*1\/2) -- (1\/2,1\/2+\\j*1\/2) -- (1\/2,\\j*1\/2) --(1,\\j*1\/2);\n}\n\n\\foreach \\j in {0,...,127}\n{\n    \\draw[gray, thick] (1\/2,\\j*1\/4) -- (1\/2,1\/4+\\j*1\/4) -- (1\/4,1\/4+\\j*1\/4) -- (1\/4,\\j*1\/4) --(1\/2,\\j*1\/4);\n}\n\\foreach \\j in {0,...,255}\n{\n    \\draw[gray, thick] (1\/4,\\j*1\/8) -- (1\/4,1\/8+\\j*1\/8) -- (1\/8,1\/8+\\j*1\/8) -- (1\/8,\\j*1\/8) --(1\/4,\\j*1\/8);\n}\n\n\n\\foreach \\k in {0,1}\n{\n\\draw[gray, dashed, thick] (16+1,16*\\k+1) -- (32-1,16*\\k+1) -- (32-1, 16*\\k+16-1) -- (16+1\/2,16*\\k+16-1);\n}\n\n\n\\foreach \\k in {0,1,2,3}\n{\n\\draw[gray, dashed, thick] (8+1\/2,8*\\k+1\/2) -- (16-1\/2,8*\\k+1\/2) -- (16-1\/2, 8*\\k+8-1\/2) -- (8+1\/2,8*\\k+8-1\/2);\n}\n\n\\foreach \\i in {0,1}\n{\\foreach \\j in {2,...,6}\n{\n    \\draw[gray, thick] (16+\\j,\\j+16*\\i) -- (32-\\j,\\j+16*\\i) -- (32-\\j,16-\\j+16*\\i) -- (16+\\j-1,16-\\j+16*\\i)--(16+\\j-1,\\j-1+16*\\i);\n}\n}\n\n\\foreach \\i in {0,...,3}\n{\\foreach \\j in {2,...,6}\n{\n    \\draw[gray] (8+\\j\/2,\\j\/2+8*\\i) -- (16-\\j\/2,\\j\/2+8*\\i) -- (16-\\j\/2,8-\\j\/2+8*\\i) --\n    (8+\\j\/2-1\/2,8-\\j\/2+8*\\i)--(8+\\j\/2-1\/2,\\j\/2-1\/2+8*\\i);\n}\n}\n\\foreach \\j in {0,...,7}\n{\n \\fill(5.5,2+\\j*4)circle(1pt);\n \\fill(6,2+\\j*4)circle(1pt);\n \\fill(6.5,2+\\j*4)circle(1pt);\n}\n\\node at (0.25,-1.5){$0$};\n\\node at (32,-1.5){$1$};\n\\node at (64,-1.5){$2$};\n\\node at (0.25,33.5){${\\mathbf i}$};\n\n\\draw[blue,thick] (64,0) -- (64,32) -- (32,32) -- (32,0) -- (64,0);\n\n\\foreach \\j in {2,...,6}\n{    \\draw[blue, thick] (32+2*\\j,2*\\j) -- (64-2*\\j,2*\\j) -- (64-2*\\j,32-2*\\j) -- (32+2*\\j-2,32-2*\\j) --(32+2*\\j-2,2*\\j-2);\n}\n\n\\draw[blue, dashed, thick] (32+2,2) -- (64-2,2) -- (64-2,32-2) -- (32+1,32-2);\n\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[x=0.2cm,y=0.2cm,scale=0.55]\n\\draw[gray,thick] (64,0) -- (64,32) -- (0,32) -- (0,0) -- (64,0);\n\\foreach \\j in {0,1}\n{ \\draw[gray, thick] (32,\\j*16) -- (32,16+\\j*16) -- (16,16+\\j*16) -- (16,\\j*16) --(32,\\j*16);\n}\n\\foreach \\j in {0,...,3}\n{\n    \\draw[gray, thick] (16,\\j*8) -- (16,8+\\j*8) -- (8,8+\\j*8) -- (8,\\j*8) --(16,\\j*8);\n}\n\\foreach \\j in {0,...,7}\n{\n    \\draw[gray, thick] (8,\\j*4) -- (8,4+\\j*4) -- (4,4+\\j*4) -- (4,\\j*4) --(8,\\j*4);\n}\n\\foreach \\j in {0,...,15}\n{\n    \\draw[gray, thick] (4,\\j*2) -- (4,2+\\j*2) -- (2,2+\\j*2) -- (2,\\j*2) --(4,\\j*2);\n}\n\\foreach \\j in {0,...,31}\n{\n    \\draw[gray, thick] (2,\\j*1) -- (2,1+\\j*1) -- (1,1+\\j*1) -- (1,\\j*1) --(2,\\j*1);\n}\n\\foreach \\j in {0,...,63}\n{\n    \\draw[gray, thick] (1,\\j*1\/2) -- (1,1\/2+\\j*1\/2) -- (1\/2,1\/2+\\j*1\/2) -- (1\/2,\\j*1\/2) --(1,\\j*1\/2);\n}\n\n\\foreach \\j in {0,...,127}\n{\n    \\draw[gray, thick] (1\/2,\\j*1\/4) -- (1\/2,1\/4+\\j*1\/4) -- (1\/4,1\/4+\\j*1\/4) -- (1\/4,\\j*1\/4) --(1\/2,\\j*1\/4);\n}\n\\foreach \\j in {0,...,255}\n{\n    \\draw[gray, thick] (1\/4,\\j*1\/8) -- (1\/4,1\/8+\\j*1\/8) -- (1\/8,1\/8+\\j*1\/8) -- (1\/8,\\j*1\/8) --(1\/4,\\j*1\/8);\n}\n\n\n\\node at (0.25,-1.5){$0$};\n\\node at (32,-1.5){$1$};\n\\node at (64,-1.5){$2$};\n\\node at (0.25,33.5){${\\mathbf i}$};\n\\end{tikzpicture}\n\\end{tabular}\n\\vskip -0.25cm\n\\caption{(left): the $E$-continuum $K$; (right): the only non-degenerate atom.}\\label{negative}\n\\vskip -0.25cm\n\\end{figure}\nLet $f_1(z)=\\frac{z}{2}$ and $f_2(z)=\\frac{z+{\\mathbf i}}{2}$. Let $K_0=\\overline{Y}$. For all $n\\ge1$, let $K_n=f_1\\left(K_{n-1}\\right)\\cup f_2\\left(K_{n-1}\\right)$. Then $K_0,K_1,\\ldots$ is an infinite sequence of continua converging to the segment $[0,{\\mathbf i}]$ under Hausdorff distance. Clearly,\n\\[K=\\left(\\bigcup\\limits_{n\\ge0}K_n\\right)\\cup\\{s{\\mathbf i}: 0\\le s\\le1\\}\\]\nis an $E$-continuum. See left part of Figure \\ref{negative}.  Let $L_0=\\partial X$. For all $n\\ge1$, let $L_n=f_1\\left(L_{n-1}\\right)\\cup f_2\\left(L_{n-1}\\right)$. Then $L_0,L_1,\\ldots$ is an infinite sequence of continua converging to the segment $[0,{\\mathbf i}]$ under Hausdorff distance. Similarly, we see that\n\\[L=\\left(\\bigcup\\limits_{n\\ge0}L_n\\right)\\cup\\{s{\\mathbf i}: 0\\le s\\le1\\}\\]\nis also an $E$-continuum. See right part of Figure \\ref{negative}. Moreover, the continuum $K$ has exactly one atom of order $1$ that is not a singleton. This atom equals $L$. Thus we have\n\\[\n\\lambda_K(x)=\\left\\{\\begin{array}{ll}1&x\\in L\\\\ 0& {otherwise}\\end{array}\\right.\\ \\text{and}\\\n\\tilde{\\lambda}_K(x)=\\left\\{\\begin{array}{ll}\n1& x\\in L\\setminus[0,{\\mathbf i}]\\\\ 0& {otherwise}.\\end{array}\\right.\n\\]\n\\end{exam}\n\n\n\\begin{exam}\\label{finite-comp}\nLet $\\mathcal{C}$ denote Cantor's ternary set. Let $U_1\\subset\\hat{\\mathbb{C}}$ be the domain, not containing $\\infty$, whose boundary consists of  $[0,1]\\times{\\bf i}\\mathcal{C}=\\{t+s{\\bf i}: 0\\le t\\le 1, s\\in\\mathcal{C}\\}$ and  $\\partial\\left([0,\\frac43]\\times[0,{\\bf i}]\\right)$.\n\\begin{figure}[ht]\n\\vspace{-0.05cm}\n\\begin{center}\n\\begin{tabular}{ccc}\n\\begin{tikzpicture}[x=1cm,y=1cm,scale=0.618]\n\\draw[gray,very thick] (0,0) -- (3,0)-- (3,4)-- (0,4)-- (0,0);\n\\draw[gray,very thick] (3,0) -- (-1,0)-- (-1,-3)-- (3,-3)-- (3,0);\n\\draw[gray,very thick] (3,0) -- (3,-4)-- (6,-4)-- (6,0)-- (3,0);\n\\draw[gray,very thick] (3,0) -- (7,0)-- (7,3)-- (3,3)-- (3,0);\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (3*\\i\/27,0) -- (3*\\i\/27,3);\n    \\draw[gray,very thick] (6\/9+3*\\i\/27,0) -- (6\/9+3*\\i\/27,3);\n     \\draw[gray,very thick] (2+3*\\i\/27,0) -- (2+3*\\i\/27,3);\n    \\draw[gray,very thick] (2+6\/9+3*\\i\/27,0) -- (2+6\/9+3*\\i\/27,3);\n}\n\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (0,3*\\i\/27-3) -- (3,3*\\i\/27-3);\n    \\draw[gray,very thick] (0,6\/9+3*\\i\/27-3) -- (3,6\/9+3*\\i\/27-3);\n     \\draw[gray,very thick] (0,2+3*\\i\/27-3) -- (3,2+3*\\i\/27-3);\n    \\draw[gray,very thick] (0,2+6\/9+3*\\i\/27-3) -- (3,2+6\/9+3*\\i\/27-3);\n}\n\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (3+3*\\i\/27,0) -- (3+3*\\i\/27,-3);\n    \\draw[gray,very thick] (3+6\/9+3*\\i\/27,0) -- (3+6\/9+3*\\i\/27,-3);\n     \\draw[gray,very thick] (3+2+3*\\i\/27,0) -- (3+2+3*\\i\/27,-3);\n    \\draw[gray,very thick] (3+2+6\/9+3*\\i\/27,0) -- (3+2+6\/9+3*\\i\/27,-3);\n}\n\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (3,3*\\i\/27) -- (3+3,3*\\i\/27);\n    \\draw[gray,very thick] (3,6\/9+3*\\i\/27) -- (3+3,6\/9+3*\\i\/27);\n     \\draw[gray,very thick] (3,2+3*\\i\/27) -- (3+3,2+3*\\i\/27);\n    \\draw[gray,very thick] (3,2+6\/9+3*\\i\/27) -- (3+3,2+6\/9+3*\\i\/27);\n}\n\n\\draw(1.5,2.75) node[above]{$U_2$};\n\\draw(0.25,-1.5) node[left]{$U_3$};\n\\draw(4.5,-2.75) node[below]{$U_4$};\n\\draw(5.75,1.5) node[right]{$U_1$};\n\\draw(4.5,3.5) node[right]{$U_5$};\n\\end{tikzpicture}\n&&\\hskip 1.0cm\n\\begin{tikzpicture}[x=1cm,y=1cm,scale=0.618]\n\\draw[gray,dashed] (0,0) -- (3,0)-- (3,4)-- (0,4)-- (0,0);\n\\draw[gray,dashed] (3,0) -- (-1,0)-- (-1,-3)-- (3,-3)-- (3,0);\n\\draw[gray,dashed] (3,0) -- (3,-4)-- (6,-4)-- (6,0)-- (3,0);\n\\draw[gray,dashed] (3,0) -- (7,0)-- (7,3)-- (3,3)-- (3,0);\n\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (3*\\i\/27,0) -- (3*\\i\/27,3);\n    \\draw[gray,very thick] (6\/9+3*\\i\/27,0) -- (6\/9+3*\\i\/27,3);\n     \\draw[gray,very thick] (2+3*\\i\/27,0) -- (2+3*\\i\/27,3);\n    \\draw[gray,very thick] (2+6\/9+3*\\i\/27,0) -- (2+6\/9+3*\\i\/27,3);\n}\n\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (0,3*\\i\/27-3) -- (3,3*\\i\/27-3);\n    \\draw[gray,very thick] (0,6\/9+3*\\i\/27-3) -- (3,6\/9+3*\\i\/27-3);\n     \\draw[gray,very thick] (0,2+3*\\i\/27-3) -- (3,2+3*\\i\/27-3);\n    \\draw[gray,very thick] (0,2+6\/9+3*\\i\/27-3) -- (3,2+6\/9+3*\\i\/27-3);\n}\n\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (3+3*\\i\/27,0) -- (3+3*\\i\/27,-3);\n    \\draw[gray,very thick] (3+6\/9+3*\\i\/27,0) -- (3+6\/9+3*\\i\/27,-3);\n     \\draw[gray,very thick] (3+2+3*\\i\/27,0) -- (3+2+3*\\i\/27,-3);\n    \\draw[gray,very thick] (3+2+6\/9+3*\\i\/27,0) -- (3+2+6\/9+3*\\i\/27,-3);\n}\n\n\\foreach \\i in {0,...,3}\n{\n    \\draw[gray,very thick] (3,3*\\i\/27) -- (3+3,3*\\i\/27);\n    \\draw[gray,very thick] (3,6\/9+3*\\i\/27) -- (3+3,6\/9+3*\\i\/27);\n     \\draw[gray,very thick] (3,2+3*\\i\/27) -- (3+3,2+3*\\i\/27);\n    \\draw[gray,very thick] (3,2+6\/9+3*\\i\/27) -- (3+3,2+6\/9+3*\\i\/27);\n}\n\\end{tikzpicture}\n\\end{tabular}\n\\end{center}\n\\vskip -0.5cm\n\\caption{The continuum $K$ and the only non-degenerate atom $d\\in\\Dc_K$.}\\label{finite-comp-pic}\n\\end{figure}\nFor $2\\le j\\le 4$ let $U_j=f^{j-1}(U_1)$, where $f(z)={\\bf i}z$. See the left part of Figure \\ref{finite-comp-pic}. Then $K=\\bigcup_i\\partial U_i$ is a continuum, whose complementary components are $U_1,\\ldots, U_4, U_5$. Here $U_5$ is the one containing $\\infty$. Moreover, the only non-degenerate atom of $K$ is\n$\\displaystyle d=\\bigcup_{j=0}^3 f^j([0,1]\\times{\\bf i}\\mathcal{C})$.\nSince the continuum $d$ has a single atom, which is itself, we have\n$\\lambda_K(x)=\\left\\{\\begin{array}{ll}\\infty& x\\in d\\\\ 0&{otherwise}.\\end{array}\\right.$\nOn the other hand, by the construction of $U_1,\\ldots, U_4$ and $U_5$, we also have\n$\\tilde{\\lambda}_K(x)=\\left\\{\\begin{array}{ll}1& x\\in d\\\\ 0&{otherwise}.\\end{array}\\right.$\nConsequently, we have $\\lambda_K(x)-\\tilde{\\lambda}_K(x)=\\left\\{\\begin{array}{ll} \\infty& x\\in d\\\\ 0& {otherwise}.\\end{array}\\right.$\n\n\\end{exam}\n\n\n\n\nThen we continue to find planar continua $K$, trying to describe possible relations between $\\lambda_K$ and $\\lambda_{\\partial K}$. The first one is Peano continuum $K$ but its boundary is a continuum that is not locally connected. Therefore, $\\lambda_{\\partial K}(x)\\ge\\lambda_K(x)$ for all $x\\in\\hat{\\mathbb{C}}$ and $\\lambda_{\\partial K}(x)>\\lambda_K(x)$ for uncountably many $x$.\n\n\\begin{exam}\\label{bd_larger}\nConsider a spiral made of broken lines, lying in the open square $W=\\{t+s{\\mathbf i}: 0< t,s<1\\}\\subset\\hat{\\mathbb{C}}$, which converges to $\\partial W$. See  the left of Figure \\ref{spiral}\n\\begin{figure}[ht]\n\\vspace{-0.05cm}\n\\center{\\begin{tabular}{ccc}\n\\begin{tikzpicture}[x=0.2cm,y=0.2cm,scale=0.6]\n\n\\draw[blue, thick] (-16,0) -- (16,0) -- (16,32) -- (-16,32) --(-16,0);\n\\foreach \\j in {1,2}\n\\foreach \\k in {1,2}\n{\n\\draw[blue, ultra  thick] (-4,-4*\\j+16) -- (4*\\j,-4*\\j+16) -- (4*\\j,4*\\j+16) --(-4*\\j-4,4*\\j+16)-- (-4*\\j-4,-4*\\j+12) --(0,-4*\\j+12);\n }\n\\draw[blue,ultra thick, dashed] (0,4) -- (13.0,4);\n\n\\draw[blue,ultra thick, dashed] (12.75,4.0) -- (12.75,27.5);\n\n\\end{tikzpicture}\n\\hspace{0.25cm}\n&\n\\hspace{0.25cm}\n\\begin{tikzpicture}[x=0.2cm,y=0.2cm,scale=0.6]\n\n\n\n\n\n\n\n\\fill[gray!80] (-16,0) -- (16,0) -- (16,32) -- (-16,32) --(-16,0);\n\\draw[blue, thick] (-16,0) -- (16,0) -- (16,32) -- (-16,32) --(-16,0);\n\\foreach \\j in {1,2}\n\\foreach \\k in {1,2}\n{\n\\draw[gray!16, ultra  thick] (-4,-4*\\j+16) -- (4*\\j,-4*\\j+16) -- (4*\\j,4*\\j+16) --(-4*\\j-4,4*\\j+16)-- (-4*\\j-4,-4*\\j+12) --(0,-4*\\j+12);\n\\draw[gray!16, ultra  thick] (-4,-4*\\j+16-\\k*0.45) -- (4*\\j+\\k*0.45,-4*\\j+16-\\k*0.45) -- (4*\\j+\\k*0.45,4*\\j+16+\\k*0.45) --(-4*\\j-4-\\k*0.45,4*\\j+16+\\k*0.45)-- (-4*\\j-4-\\k*0.45,-4*\\j+12-\\k*0.45) --(0,-4*\\j+12-\\k*0.45);\n }\n\n\\draw[gray!16,ultra thick, dashed] (0,4) -- (13.2,4); \\draw[gray!16,ultra thick, dashed] (0,3.55) -- (13.2,3.55); \\draw[gray!16,ultra thick, dashed] (0,3.1) -- (13.2,3.1);\n\n\n\\end{tikzpicture}\n\\hspace{0.25cm}\n&\n\\hspace{0.25cm}\n\\begin{tikzpicture}[x=0.2cm,y=0.2cm,scale=0.6]\n\\fill[gray!80] (-16,0) -- (16,0) -- (16,32) -- (-16,32) --(-16,0);\n\\draw[blue, thick] (-16,0) -- (16,0) -- (16,32) -- (-16,32) --(-16,0);\n\\foreach \\j in {1,2}\n\\foreach \\k in {1,2}\n{\n\\draw[gray!16, ultra  thick] (-4,-4*\\j+16) -- (4*\\j,-4*\\j+16) -- (4*\\j,4*\\j+16) --(-4*\\j-4,4*\\j+16)-- (-4*\\j-4,-4*\\j+12) --(0,-4*\\j+12);\n\\draw[gray!16, ultra  thick] (-4,-4*\\j+16-\\k*0.45) -- (4*\\j+\\k*0.45,-4*\\j+16-\\k*0.45) -- (4*\\j+\\k*0.45,4*\\j+16+\\k*0.45) --(-4*\\j-4-\\k*0.45,4*\\j+16+\\k*0.45)-- (-4*\\j-4-\\k*0.45,-4*\\j+12-\\k*0.45) --(0,-4*\\j+12-\\k*0.45);\n }\n\\draw[gray!16,ultra thick, dashed] (0,3.8) -- (13.2,3.8);\n\\draw[gray!16,ultra thick, dashed] (0,3.55) -- (13.2,3.55);\n\\draw[gray!16,ultra thick, dashed] (0,3.1) -- (13.2,3.1);\n\n\n\\foreach \\j in {1,2}\n\\foreach \\k in {2}\n{\n\\draw[blue] (-4,-4*\\j+16+0.2) -- (4*\\j-0.2,-4*\\j+16+0.2) -- (4*\\j-0.2,4*\\j+16-0.2) --(-4*\\j-4+0.2,4*\\j+16-0.2)-- (-4*\\j-4+0.2,-4*\\j+12+0.2) --(0,-4*\\j+12+0.2);\n\\draw[blue] (-4,-4*\\j+16-\\k*0.45-0.3) -- (4*\\j+\\k*0.45+0.3,-4*\\j+16-\\k*0.45-0.3) -- (4*\\j+\\k*0.45+0.3,4*\\j+16+\\k*0.45+0.3) --(-4*\\j-4-\\k*0.45-0.3,4*\\j+16+\\k*0.45+0.3)-- (-4*\\j-4-\\k*0.45-0.3,-4*\\j+12-\\k*0.45-0.3) --(0,-4*\\j+12-\\k*0.45-0.3);\n }\n\n\n\\foreach \\i in {0,1,2}\n{\n\\draw[blue] (-4+3.9*\\i,12.25) -- (-4+3.9*\\i,12-1.25);\n\\draw[blue] (3.8,13.8+3*\\i) -- (5.2,13.8+3*\\i);\n\\draw[blue] (-7.8,13.8+3*\\i) -- (-9.2,13.8+3*\\i);\n\\draw[blue] (-7.8,12-1.9*\\i) -- (-9.2,12-1.9*\\i);\n}\n\n\n\\foreach \\i in {0,...,3}\n{\n\\draw[blue] (-5.6+3.6*\\i,19.8) -- (-5.6+3.6*\\i,21.2);\n\\draw[blue] (-7.8+1.2*\\i,8.2) -- (-7.8+1.2*\\i,6.8);\n\\draw[blue] (-3+1.2*\\i,8.2) -- (-3+1.2*\\i,6.8);\n\\draw[blue] (1.8+1.0*\\i,8.2) -- (1.8+1.0*\\i,6.8);\n\\draw[blue] (4.8+1.0*\\i,8.2) -- (4.8+1.0*\\i,6.8);\n\\draw[blue] (7.8,7.8+1.0*\\i) -- (9.2,7.8+1.0*\\i);\n\\draw[blue] (7.8,11.8+1.0*\\i) -- (9.2,11.8+1.0*\\i);\n\\draw[blue] (7.8,15.8+0.8*\\i) -- (9.2,15.8+0.8*\\i);\n\\draw[blue] (7.8,19.2+0.8*\\i) -- (9.2,19.2+0.8*\\i);\n\\draw[blue] (7.8,22.4+0.7*\\i) -- (9.2,22.4+0.7*\\i);\n\\draw[blue] (7.8-0.8*\\i,23.8) -- (7.8-0.8*\\i,25.2);\n\\draw[blue] (4.6-0.8*\\i,23.8) -- (4.6-0.8*\\i,25.2);\n\\draw[blue] (1.4-0.8*\\i,23.8) -- (1.4-0.8*\\i,25.2);\n\\draw[blue] (-1.8-0.8*\\i,23.8) -- (-1.8-0.8*\\i,25.2);\n\\draw[blue] (-5-0.8*\\i,23.8) -- (-5-0.8*\\i,25.2);\n\\fill[blue!62](-9-\\i,24.5) circle(1.8pt);\n}\n\n\\end{tikzpicture}\n\n\\end{tabular}\n}\n\\caption{A Peano continuum $K$ whose boundary is not locally connected.}\\label{spiral}\n\\end{figure}\nWe may thicken the spiral to an embedding $h: [0,\\infty)\\times[0,1]\\rightarrow W$, of the unbounded strip $U=[0,\\infty)\\times[0,1]$. Such an embedding may be chosen appropriately, so that $h(\\partial U)$ consists of countably many segments. Then, we obtain a continuum\n$K_0=\\overline{W}\\setminus h(U)$. See the middle part of Figure \\ref{spiral}. Clearly, the continuum $K_0$ is not locally connected on $\\partial W$; and it is locally connected at all the other points. Now, divide the thickened spiral $h(U)$ into smaller and smaller quadruples, which are depicted in the right part of Figure \\ref{spiral} as small rectangles. Let $K$ be the union of $K_0$ with the newly added bars, used in the above mentioned division. Then $K$ is locally connected everywhere hence is a Peano continuum. However, its boundary $\\partial K$ is not locally connected on $\\partial W$ and is locally connected elsewhere. Therefore, we have\n\\[\\lambda_K(x)\\equiv 0\\quad{and}\\quad\n\\lambda_{\\partial K}(x)=\\left\\{\\begin{array}{ll}1 & x\\in\\partial W\\\\\n0& x\\notin\\partial W.\\end{array}\\right.\\]\n\\end{exam}\n\n\n\n\\begin{exam}\\label{bd_smaller}\nLet the continuum $K$ be defined as in Example \\ref{bd_larger}. Let $f_j(z)=\\frac{z}{2}+\\frac{j-1}{2}{\\mathbf i}$ for $j=1,2$. For any compact set $X\\subset\\hat{\\bbC}$, put $\\Phi(X)=f_1(X)\\cup f_2(X)$. We will use the continuum $K$ and the mapping $\\Phi$ to construct a continuum $L$. See Figure \\ref{spiral-double}.\n\\begin{figure}[ht]\n\\vskip -0.05cm\n\\center{\n\\begin{tikzpicture}[x=0.2cm,y=0.2cm,scale=0.8]\n\\draw[gray,thick] (32,0) -- (0,0) -- (0,32) -- (32,32);\n\\draw[gray,thick] (32,0) -- (64,0) -- (64,32) -- (32,32) -- (32,0);\n\\foreach \\j in {0,1}\n{ \\draw[gray, thick] (32,16+\\j*16) -- (16,16+\\j*16) -- (16,\\j*16) --(32,\\j*16);\n}\n\\foreach \\j in {0,...,3}\n{\n    \\draw[gray, thick] (16,\\j*8) -- (16,8+\\j*8) -- (8,8+\\j*8) -- (8,\\j*8) --(16,\\j*8);\n}\n\\foreach \\j in {0,...,7}\n{\n    \\draw[gray, thick] (8,\\j*4) -- (8,4+\\j*4) -- (4,4+\\j*4) -- (4,\\j*4) --(8,\\j*4);\n}\n\\foreach \\j in {0,...,15}\n{\n    \\draw[gray, thick] (4,\\j*2) -- (4,2+\\j*2) -- (2,2+\\j*2) -- (2,\\j*2) --(4,\\j*2);\n}\n\\foreach \\j in {0,...,31}\n{\n    \\draw[gray, thick] (2,\\j*1) -- (2,1+\\j*1) -- (1,1+\\j*1) -- (1,\\j*1) --(2,\\j*1);\n}\n\\foreach \\j in {0,...,63}\n{\n    \\draw[gray, thick] (1,\\j*1\/2) -- (1,1\/2+\\j*1\/2) -- (1\/2,1\/2+\\j*1\/2) -- (1\/2,\\j*1\/2) --(1,\\j*1\/2);\n}\n\n\\foreach \\j in {0,...,127}\n{\n    \\draw[gray, thick] (1\/2,\\j*1\/4) -- (1\/2,1\/4+\\j*1\/4) -- (1\/4,1\/4+\\j*1\/4) -- (1\/4,\\j*1\/4) --(1\/2,\\j*1\/4);\n}\n\\foreach \\j in {0,...,255}\n{\n    \\draw[gray, thick] (1\/4,\\j*1\/8) -- (1\/4,1\/8+\\j*1\/8) -- (1\/8,1\/8+\\j*1\/8) -- (1\/8,\\j*1\/8) --(1\/4,\\j*1\/8);\n}\n\n\n\n \\draw[gray, thick] (-4,-4) -- (64+4,-4) -- (64+4,32+4) --(-3,32+4)--(-3,-3);\n \\draw[gray, thick] (-3,-3) -- (64+3,-3) -- (64+3,32+2) --(-2,32+2) -- (-2,-2);\n  \\draw[gray, thick] (-2,-2) -- (64+2,-2) -- (64+2,32+1) --(-1,32+1) -- (-1,-1);\n \\draw[gray, thick, dashed] (-1,-1)--(64+1,-1)--(64+1,30);\n\n\\node at (33,2) {$1$};  \\fill(32,0) circle(2pt);\n\\node at (63,2) {$2$};  \\fill(64,0) circle(2pt);\n\\node at (61,30) {$2\\!+\\!{\\mathbf i}$};  \\fill(64,32) circle(2pt);\n\n\\node at (48,16) {$\\partial K+1$};\n\\node at (24,8) {$f_1(\\partial K\\!+\\!1)$};\n\\node at (24,24) {$f_2(\\partial K\\!+\\!1)$};\n\n\\draw[gray, very thin] (12,4) -- (-12.5,8);\n\\node at (-15,10) {$f_1\\circ f_1(K\\!+\\!1)$};\n\\draw[gray, very thin] (12,12) -- (-12.5,16);\n\\node at (-15,18) {$f_1\\circ f_2(K\\!+\\!1)$};\n\n\n\\draw[gray, very thin] (12,20) -- (-12.5,24);\n\\node at (-15,26) {$f_2\\circ f_1(K\\!+\\!1)$};\n\\end{tikzpicture}\n}\\vskip -0.5cm\n\\caption{Relative locations of $\\partial K+1$, $\\Phi^{1}(\\partial K+1)$  and $\\Phi^{2}(K+1)$.}\\label{spiral-double}\n\\end{figure}\nThe continuum $L$ consists of five parts:\n\\begin{enumerate}\n\\item the segment $[0,{\\mathbf i}]=\\{s{\\bf i}: 0\\le s\\le1\\}$;\n\\item a spiral converging to the boundary of $[0,2]\\times[0,{\\mathbf i}]$;\n\\item $\\partial K+1=\\{z+1: z\\in \\partial K\\}$;\n\\item $\\Phi^{2n}(K+1)$ for all integers $n\\ge1$; and\n\\item $\\Phi^{2n-1}(\\partial K+1)$ for all integers $n\\ge1$.\n\\end{enumerate}\nOn the one hand, we can directly check that $L$ has a unique non-degenerate atom $d$, which consists of the following four parts: (1) the segment $[0,{\\mathbf i}]$; (2) the boundary of $[1,2]\\times[0,{\\mathbf i}]$, denoted as $A$; (3) $\\Phi^{2n-1}(A)$ for all integers $n\\ge1$; and (4) the boundary of $[2^{-2n},2^{-2n+1}]\\times[0,{\\mathbf i}]$ for all integers $n\\ge1$.\nOn the other hand, the boundary $\\partial L$ has a unique non-degenerate atom $d^*$, which is the union of $A$, the segment $[0,{\\mathbf i}]$, and $\\Phi^{n}(A)$ for all integers $n\\ge1$. See Figure \\ref{atoms} for a depiction of  $d$ and $d^*$.\n\\begin{figure}[ht]\n\\vspace{-0.25cm}\n\\center{\\begin{tabular}{cc}\n\\begin{tikzpicture}[x=0.2cm,y=0.2cm,scale=0.6]\n\\draw[gray,thick] (32,0) -- (0,0) -- (0,32) -- (32,32);\n\\draw[gray,thick] (32,0) -- (64,0) -- (64,32) -- (32,32) -- (32,0);\n\n\n\\foreach \\j in {0,1}\n{ \\draw[gray, thick] (32,\\j*16) -- (16,\\j*16) -- (16,\\j*16) --(32,\\j*16);\n}\n\n\\foreach \\j in  {0} \n{\n    \\draw[gray, thick] (16,+\\j*8) -- (16,32+\\j*8) -- (8,32+\\j*8) -- (8,\\j*8) --(16,\\j*8);\n}\n\n\n\\foreach \\j in {0,...,7}\n{\n    \\draw[gray, thick] (8,\\j*4) -- (8,\\j*4) -- (4,\\j*4) -- (4,\\j*4) --(8,\\j*4);\n}\n\n\\foreach \\j in {0} \n{\n    \\draw[gray, thick] (4,\\j*2) -- (4,32+\\j*2) -- (2,32+\\j*2) -- (2,\\j*2) --(4,\\j*2);\n}\n\n\n\\foreach \\j in {0,...,31}\n{\n    \\draw[gray, thick] (2,\\j*1) -- (2,\\j*1) -- (1,\\j*1) -- (1,\\j*1) --(2,\\j*1);\n}\n\n\\foreach \\j in {0} \n{\n    \\draw[gray, thick] (1,\\j*1\/2) -- (1,32+\\j*1\/2) -- (1\/2,32+\\j*1\/2) -- (1\/2,\\j*1\/2) --(1,\\j*1\/2);\n}\n\n\\foreach \\j in {0,...,127}\n{\n    \\draw[gray, thick] (1\/2,\\j*1\/4) -- (1\/2,1\/4+\\j*1\/4) -- (1\/4,1\/4+\\j*1\/4) -- (1\/4,\\j*1\/4) --(1\/2,\\j*1\/4);\n}\n\n\\foreach \\j in {0,...,255}\n{\n    \\draw[gray, thick] (1\/4,\\j*1\/8) -- (1\/4,1\/8+\\j*1\/8) -- (1\/8,1\/8+\\j*1\/8) -- (1\/8,\\j*1\/8) --(1\/4,\\j*1\/8);\n}\n\n\\node at (34,2) {$1$};  \\fill(32,0) circle(2pt);\n\\node at (62,2) {$2$};  \\fill(64,0) circle(2pt);\n\\node at (60,30) {$2\\!+\\!{\\mathbf i}$};  \\fill(64,32) circle(2pt);\n\n\\end{tikzpicture}\n\n& \\begin{tikzpicture}[x=0.2cm,y=0.2cm,scale=0.6]\n\\draw[gray,thick] (32,0) -- (0,0) -- (0,32) -- (32,32);\n\\draw[gray,thick] (32,0) -- (64,0) -- (64,32) -- (32,32) -- (32,0);\n\\foreach \\j in {0,1}\n{ \\draw[gray, thick] (32,16+\\j*16) -- (16,16+\\j*16) -- (16,\\j*16) --(32,\\j*16);\n}\n\\foreach \\j in {0,...,3}\n{\n    \\draw[gray, thick] (16,\\j*8) -- (16,8+\\j*8) -- (8,8+\\j*8) -- (8,\\j*8) --(16,\\j*8);\n}\n\\foreach \\j in {0,...,7}\n{\n    \\draw[gray, thick] (8,\\j*4) -- (8,4+\\j*4) -- (4,4+\\j*4) -- (4,\\j*4) --(8,\\j*4);\n}\n\\foreach \\j in {0,...,15}\n{\n    \\draw[gray, thick] (4,\\j*2) -- (4,2+\\j*2) -- (2,2+\\j*2) -- (2,\\j*2) --(4,\\j*2);\n}\n\\foreach \\j in {0,...,31}\n{\n    \\draw[gray, thick] (2,\\j*1) -- (2,1+\\j*1) -- (1,1+\\j*1) -- (1,\\j*1) --(2,\\j*1);\n}\n\\foreach \\j in {0,...,63}\n{\n    \\draw[gray, thick] (1,\\j*1\/2) -- (1,1\/2+\\j*1\/2) -- (1\/2,1\/2+\\j*1\/2) -- (1\/2,\\j*1\/2) --(1,\\j*1\/2);\n}\n\n\\foreach \\j in {0,...,127}\n{\n    \\draw[gray, thick] (1\/2,\\j*1\/4) -- (1\/2,1\/4+\\j*1\/4) -- (1\/4,1\/4+\\j*1\/4) -- (1\/4,\\j*1\/4) --(1\/2,\\j*1\/4);\n}\n\\foreach \\j in {0,...,255}\n{\n    \\draw[gray, thick] (1\/4,\\j*1\/8) -- (1\/4,1\/8+\\j*1\/8) -- (1\/8,1\/8+\\j*1\/8) -- (1\/8,\\j*1\/8) --(1\/4,\\j*1\/8);\n}\n\n\\node at (34,2) {$1$};  \\fill(32,0) circle(2pt);\n\\node at (62,2) {$2$};  \\fill(64,0) circle(2pt);\n\\node at (60,30) {$2\\!+\\!{\\mathbf i}$};  \\fill(64,32) circle(2pt);\n\n\\end{tikzpicture}\n\\end{tabular}\n}\\vskip -0.0cm\n\\caption{A depiction of $d$ and $d^*$.}\\label{atoms}\n\\end{figure}\nThe atom $d^*$ (of $\\partial L$) is a Peano continuum and contains $d$. However, the atom $d$ (of $L$) is not locally connected at points $s{\\mathbf i}$ with $0<s<1$ and is locally connected elsewhere. Therefore, we can compute their lambda functions as follows:\n\\[\n\\lambda_L(x)=\\left\\{\\begin{array}{ll}2& x\\in[0,{\\mathbf i}]\\\\\n1 & x\\in d\\setminus[0,{\\mathbf i}]\\\\\n0& {otherwise}\\end{array}\\right.\\quad{and}\\quad \\lambda_{\\partial L}(x)=\\left\\{\\begin{array}{ll}1& x\\in[0,{\\mathbf i}]\\\\\n1 & x\\in d^*\\setminus[0,{\\mathbf i}]\\\\\n0& {otherwise}.\\end{array}\\right.\n\\]\nFrom these we further infer that\n\\[\n\\lambda_L(x)-\\lambda_{\\partial L}(x)=\\left\\{\\begin{array}{ll} 1& x\\in[0,{\\mathbf i}]\\\\ -1& x\\in d^*\\setminus d\\\\ 0&{otherwise}.\\end{array}\\right.\n\\]\n\\end{exam}\n\n\n\nNote that it is still unknown {\\bf whether  there is a compactum $K\\subset\\hat{\\mathbb C}$ such that $\\lambda_K(x)\\ge\\lambda_{\\partial K}(x)$ for all $x\\in\\hat{\\mathbb{C}}$ and  $\\lambda_K(x)>\\lambda_{\\partial K}(x)$ for at least one point $x\\in\\partial K$}.\n\n\n\nTo conclude this section, we now consider  unions and intersections of specific Peano compacta in the plane. We will find concrete Peano continua in the plane, say $X$ and $Y$, such that $X\\cap Y$ is a continuum that is not locally connected. Notice that $X\\cup Y$ is always a Peano continuum.\n\n\\begin{exam}\\label{cap-peano}\nLet $M$ be the union of $[0,1]\\times\\{0\\}$ with the vertical segments $\\{0\\}\\times[0,1]$ and $\\{2^{-k}\\}\\times[0,1]$ for integers $k\\ge0$. Then $M$ is a continuum and is not locally connected at points on $\\{0\\}\\times(0,1]$; moreover, we have\n\\[\\lambda_M(x)=\\left\\{\\begin{array}{ll}1& x\\in \\{t{\\bf i}: 0\\le t\\le 1\\}\\\\ 0 & otherwise\\end{array}\\right.\n\\]\nWe will construct two Peano continua $X$ and $Y$ satisfying $X\\cap Y=M$. To this end, for all integers $k\\ge1$ we put\n\\[A_k=\\bigcup_{j=1}^{2^k-1}\\left[0,2^{-k+1}\\right]\\times\\left\\{j2^{-k}\\right\\}.\\]\nThen $X=M\\cup\\left(\\bigcup_kA_k\\right)$ is a Peano continuum.\n\\begin{figure}[ht]\n\\vskip -0.05cm\n\\begin{center}\n\\begin{tabular}{ccc}\n\\begin{tikzpicture}[x=1cm,y=1cm,scale=0.8]\n\\foreach \\i in {1,...,3}\n{\n\\draw[red] (0,1.296*\\i) -- (2.592,1.296*\\i);\n}\n\n\\foreach \\i in {1,...,7}\n{\n\\draw[red] (0,0.648*\\i) -- (1.296,0.648*\\i);\n}\n\n\\foreach \\i in {1,...,15}\n{\n\\draw[red] (0,0.324*\\i) -- (0.648,0.324*\\i);\n}\n\n\\foreach \\i in {1,...,31}\n{\n\\draw[red] (0,0.162*\\i) -- (0.324,0.162*\\i);\n}\n\n\\foreach \\i in {1,...,63}\n{\n\\draw[red] (0,0.081*\\i) -- (0.162,0.081*\\i);\n}\n\n\\draw[blue,thick] (0,0) -- (0,5.184);\n\\draw[blue,thick] (0,0) -- (5.184,0);\n\\draw[red] (2.592,2.592) -- (5.184,2.592);\n\n\\foreach \\i in {1,2,4,8,16,32}\n{\n\\draw[blue,thick] (5.184\/\\i,0) -- (5.184\/\\i,5.184);\n}\n\n\n\\fill[black] (5.184,0) circle (0.35ex); \\draw[purple] (5.184,0) node[right]{$1$};\n\\fill[black] (5.184,5.184) circle (0.35ex); \\draw[purple] (5.184,5.184) node[right]{$1+{\\bf i}$};\n\\fill[black] (0,0) circle (0.35ex); \\draw[purple] (0,0) node[left]{$0$};\n\\end{tikzpicture}\\hspace{0.25cm}\n&\n&\\hspace{0.25cm}\n\\begin{tikzpicture}[x=1cm,y=1cm,scale=0.8]\n\\foreach \\i in {1,...,2}\n{\n\\draw[red] (0,1.728*\\i) -- (5.184,1.728*\\i);\n}\n\n\\foreach \\i in {1,...,8}\n{\n\\draw[red] (0,0.576*\\i) -- (2.592,0.576*\\i);\n}\n\n\\foreach \\i in {1,...,26}\n{\n\\draw[red] (0,0.192*\\i) -- (1.296,0.192*\\i);\n}\n\n\\foreach \\i in {1,...,80}\n{\n\\draw[red] (0,0.064*\\i) -- (0.648,0.064*\\i);\n}\n\n\\draw[blue,thick] (0,0) -- (0,5.184);\n\\draw[blue,thick] (0,0) -- (5.184,0);\n\n\\foreach \\i in {1,2,4,8}\n{\n\\draw[blue,thick] (5.184\/\\i,0) -- (5.184\/\\i,5.184);\n}\n\n\n\\fill[black] (5.184,0) circle (0.35ex);       \\draw[purple] (5.184,0) node[right]{$1$};\n\\fill[black] (5.184,5.184) circle (0.35ex);   \\draw[purple] (5.184,5.184) node[right]{$1+{\\bf i}$};\n\\fill[black] (0,0) circle (0.35ex);           \\draw[purple] (0,0) node[left]{$0$};\n\\end{tikzpicture}\n\\end{tabular}\n\\end{center}\n\\vskip -0.5cm\n\\caption{\\small Two Peano continua that intersect at a non-locally connected continuum.}\\label{peano-cap}\n\\end{figure}\nSee left part of Figure \\ref{peano-cap} for a rough approximate of $X$. Similarly, if for every $k\\ge1$ we set\n\\[B_k=\\bigcup_{j=1}^{3^k-1}\\left[0,2^{-k+1}\\right]\\times\\left\\{j3^{-k}\\right\\},\\]\nthen $\\displaystyle Y=M\\cup\\left(\\bigcup\\limits_kB_k\\right)$ is also a Peano continuum. See right part of Figure \\ref{peano-cap} for a rough approximate of $Y$. Moreover, we have $X\\cap Y=M$.\n\\end{exam}\n\nWe also find Peano compacta $X$ and $Y$ such that the union $X\\cup Y$ is not a Peano compactum, although the intersection $X\\cap Y$ is always a Peano compactum. We will use {\\em fractal squares} to construct two such compacta.\nHere a {\\bf fractal square of order $n\\ge2$} is the attractor of an iterated function system\n$\\displaystyle \\Fc_\\Dc:=\\left\\{f_d(x)=\\frac{x+d}{n}: d\\in\\Dc\\right\\}$\nfor some $\\Dc\\subset\\{0,1,\\ldots,n-1\\}^2$ which contains at least $2$ and at most $n^2-1$ elements.\nFor general theory on iterated function systems, we refer to \\cite{Hutchinson81}.\n\n\\begin{exam}\\label{cup-fs}\nLet $X$ and $Y$ be the fractal squares determined by $\\Fc_{\\Dc_X}$ and $\\Fc_{\\Dc_Y}$. Here $\\Dc_X=\\{(i,0): i=0,1,2\\}\\cup\\{(0,2)\\}$ and $\\Dc_Y=\\{(i,0): i=0,1,2\\}\\cup\\{(1,2),(2,2)\\}$. See Figure \\ref{fs-cup} for relative locations of the small squares $f_d([0,1]^2)$ with $d\\in\\Dc_X$  and $d\\in\\Dc_Y$.\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\begin{tikzpicture}[x=1cm,y=1cm,scale=0.618]\n\n\\fill[purple!30] (0,0) -- (5.184,0) -- (5.184,1.728) -- (0,1.728) -- (0,0);\n\\fill[purple!30] (0,3.456) -- (1.728,3.456) -- (1.728,5.184) -- (0,5.184) -- (0,3.456);\n\n\\foreach \\i in {0,...,3}\n{\n\\draw[gray,thick] (0,1.728*\\i) -- (5.184,1.728*\\i);\n\\draw[gray,thick] (1.728*\\i,0) -- (1.728*\\i,5.184);\n}\n\\end{tikzpicture}\n&\n\\hskip 0.5cm\n&\n\\begin{tikzpicture}[x=1cm,y=1cm,scale=0.618]\n\n\\fill[purple!30] (0,0) -- (5.184,0) -- (5.184,1.728) -- (0,1.728) -- (0,0);\n\\fill[purple!30] (1.728,3.456) -- (5.184,3.456) -- (5.184,5.184) -- (1.728,5.184) -- (1.728,3.456);\n\n\\foreach \\i in {0,...,3}\n{\n\\draw[gray,thick] (0,1.728*\\i) -- (5.184,1.728*\\i);\n\\draw[gray,thick] (1.728*\\i,0) -- (1.728*\\i,5.184);\n}\n\\end{tikzpicture}\n\\end{tabular}\n\\end{center}\n\\vskip -0.5cm\n\\caption{\\small The small squares $f_d([0,1]^2)$ for $d\\in\\Dc_1$ (left part) and for $d\\in\\Dc_2$ (right part).}\\label{fs-cup}\n\\end{figure}\nThen $X$ and $Y$ are Peano compacta,  each of which contains the interval $[0,1]$, such that $X\\cup Y$ contains all the segments $[0,1]\\times\\{\\frac{2}{3^k}\\}$ for $k\\ge1$. Moreover,  $X\\cap Y$ is a Peano compactum having uncountably many components. All but one of these components are single points. The only non-degenerate component is the interval $[0,1]$. On the other hand, for all $k\\ge1$ the horizontal strip $\\bbR\\times\\left(\\frac{1}{3^k},\\frac{2}{3^k}\\right)$ is disjoint from $X\\cup Y$. This implies that $X\\cup Y$ is not a Peano compactum. Consequently, we have\n\\[\n\\begin{array}{ccc}\\lambda_{X}(x)=0 (\\forall x\\in \\hat{\\mathbb{C}}); & \\lambda_{Y}(x)= 0 (\\forall x\\in \\hat{\\mathbb{C}});  & \\lambda_{X\\cup Y}(x)=\\left\\{\\begin{array}{ll}1& x\\in [0,1]\\\\ 0 & otherwise.\\end{array}\\right.\n\\end{array}\n\\]\nNotice that $Y\\cup X+1$ is also a Peano compactum, although $Y\\cap X+1$ has uncountably many components. Thus  $\\lambda_{X+1}(x)=\\lambda_{Y}(x)=\\lambda_{Y\\cup X+1}(x)=0$ for all  $x\\in \\hat{\\mathbb{C}}$. \\end{exam}\n\n\n\n\n\n\\noindent\n{\\bf Acknowledgement}. The authors are grateful to Dr. Yi Yang at Sun Yat-sen University, for valuable discussions during the period of his phd study.\n\n\\bibliographystyle{plain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nAfter tremendous success in task automation, machine learning has recently been used to accelerate expert development. In the field of medical research, new tools to efficiently exploit knowledge database or to accelerate simulations by orders of magnitude can be of great aid. In that matter, the problem of drug design is a particularly challenging but important problem: Before a compound is commercialized, it is proposed as a candidate and will have to pass an extensive battery of tests and certifications on animals and humans. In current modern pipelines, only one candidate in ten thousand will make its way to the market, explaining the development cost (between 1 and 3 billion of dollars) and time (10 to 15 years) needed for a single new drug.\n\nWith a large quantity of freely available data, many recent works aiming at improving this ratio have been proposed by the community. Searching for a good molecule requires two things: a way to score its quality, and a method to search the ensemble of reasonable candidates. The former was addressed by supervised algorithms, trained to predict many molecular properties of interest, that could be efficiently used to accelerate the prediction while increasing its accuracy \\citep{gilmer2017neural, wu2018moleculenet}. Notably, the use of graph neural networks seems to be particularly adapted to these tasks \\citep{Hu2020, wu2018moleculenet}. Although the latter has also been explored extensively by different methods, some remaining recurring issues still limit their practical use. In particular, \\citet{gao2020synthesizability} have shown that molecules generated by deep learning models are often unrealistic, and cannot be synthesized. \n\nMolecular data can be represented in many forms, but in this work we focus on molecules represented as graphs, with atoms seen as nodes and chemical bonds seen as edges. In literature, three main approaches can be found. \nFirstly, a simple representation learning method can be used to produce an embedding of molecules in a smooth space, in which standard optimization methods can be used, such as bayesian optimization \\citep{jin2018junction}. This approach relies heavily on the quality of the embedding space, limiting the optimization performance. A second one is to explicitly train a model to produce molecules with given properties \\citep{jin2018learning}. However, this requires a large quantity of data specific to the target properties, limiting the usefulness of such methods for the elaboration of drugs with \\emph{new} effects. Finally, reinforcement learning (RL) methods have proven effective as a pure optimization method \\citep{zhou2019optimization, you2018graph, shi2020graphaf}. Most works often add a component to enforce realism regarding a dataset, such as a discriminator or pretraining, as it is often easier for the model to learn the specifics of the score function, leading to the generation of unrealistic molecules. Although one can argue that proposing new ways to optimize a score can be interesting, these models are difficult to use in practice.\n\nIn this work, we propose a different approach to enforce molecule realism, combining machine learning and traditional importance sampling. More precisely, we train a graph autoregressive generative model to learn the distribution of reasonable molecules on the ZINC dataset \\citep{irwin2005zinc}. We then alter a simple Metropolis sampling algorithm to propose molecules which are both optimized and realistic. The resulting method, that we refer to as Learned Realism Sampling (LRS) does not need score-specific data, produces realistic molecules when RL algorithms do not, and improves performance on all our baselines in the task of molecule \\emph{improvement} on a variety of different scores.\n\n\n\\section{Related work}\n\\label{related}\n\n\\subsection*{Molecule generation}\nThis work focuses mainly on graph representations of molecules.\nMolecular graphs do not have a well defined node ordering and their ensemble lacks a topology, making the problem of molecular graph generation difficult in most deep learning frameworks. Three main approaches can be found in literature. First, a standard generative framework such as a Variational Autoencoder \\citep{Kingma2013VAE} can be used to generate the graph adjacency matrix \\citep{simonovsky2018graphvae}. Although simple, these methods suffer from the fact that they are not node permutation invariant. A second popular approach relies on the junction tree representation \\citep{jin2018junction, jin2018learning}. Briefly, it consists in heuristics to represent the molecule as a tree, ie to remove the cycles. A tree is considerably simpler than a molecular graph, and recurrent methods can be efficiently applied to it. Finally, autoregressive models can be used to generate a graph node-by-node and edge-by-edge \\citep{popova2019molecularrnn, shi2020graphaf, you2018graph, zhou2019optimization, li2018multi-objective}. This has the advantage of breaking the symmetries introduced by the incomplete ordering of the nodes.\n\nMore broadly used than graph representations, SMILES and more recently SELFIES\\citep{krenn2020self} are text-based representations of molecules, on which standard sequence models can be used. The main drawback of this approach is that neighboring atoms in the molecule can be far away in the representation. In this regard, we highlight the work of \\citet{alperstein2019all} that solved the problem by using multiple SMILES representations of the molecule. Here, we also consider the work of \\cite[SMILES LSTM]{segler2018generating} in section \\ref{section: unconstrained optimisation} as another baseline.\n\n\\subsection*{Optimization}\nA first approach for property optimization is to perform representation learning to embed molecules as vectors in a smooth space. Standard algorithms such as Bayesian optimization\\citep[JTVAE]{jin2018junction}, gradient descent \\citep[All SMILES]{alperstein2019all} or hill climbing \\citep[SMILES LSTM]{segler2018generating} can then be used on those representations . Although efficient in principle, these algorithms are limited by the quality of the representation as well as by the ability of the model to reconstruct the original molecule from it.\nA second approach is to train the model in a \\emph{domain transfer} framework, that is to train the model to take a molecule with low score as input and give a similar molecule with high score as output \\citep[VJTNN or VJTNN+GAN]{jin2018learning}. These models can be effective, although they do not actually optimize properties. Notice that both these approaches rely heavily on the availability of labelled data. \n\nIn order to overcome this last issue, a standard approach is to train a model to directly maximize a score with reinforcement learning. These algorithms are very efficient for the optimization itself. Methods to enforce molecule realism regarding a dataset are added, such as pretraining \\citep[MRNN]{popova2019molecularrnn}\\citep[GraphAF]{shi2020graphaf} or a discriminator \\citep[GCPN]{you2018graph}. However, despite these efforts, they usually generate unrealistic samples, as highlighted in this work.\n\nFinally, there is a proposition of an hybrid approach between machine learning and other algorithms. Most similar to our work, we highlight two recent models based on Markov Chain Monte Carlo (MCMC). The MARS model \\citep{xie2021mars} trains and generates molecules simultaneously, adding better candidates to its training set. The generation procedure shares a lot of similarity with our work. Although the model produces impressive result in optimization, it does not attempt to enforce realism and we do not use it as a baseline. The MIMOSA model \\citep{fu2020mimosa} performs MCMC and directly models the transition probability with a pretrained neural network, as opposed to this work, where we model a dataset distribution. \n\n\n\n\n\n\n\n\\section{Method}\n\\subsection{Graph generation with an auto-regressive model}\n\\label{section:armodel}\n\nWe consider a dataset representing a distribution $\\mathcal{Q}(G)$ over molecular graphs G that we wish to approximate. We first represent a graph $G$ in a constructive fashion, as a sequence $s$ of vectors which can either represent the addition of an atom or of a chemical bond, as detailed in section \\ref{section:sequence definition}. The resulting distribution $\\mathcal{Q}(s)$ is then approximated by an autoregressive distribution $\\mathcal{P}(s)$ parameterized with neural networks, as detailed in section \\ref{section: nn}.\n\n\\subsubsection{Graphs as sequences}\n\\label{section:sequence definition}\nWe consider a molecule, seen as a complete graph $G(\\mathcal{V}, \\mathcal{E})$ where $\\mathcal{V} = 1 .. N$ is the set of nodes and $\\mathcal{E}\\subset \\mathcal{V}\\times\\mathcal{V}$ is the set of \\emph{undirected} edges, $x_v \\in \\{1..D\\}$ is the label of the node $v\\in \\mathcal{V}$ and $e_{ij} \\in \\{1..E\\}$ is the label of the edge $(i, j) \\in \\mathcal{E}$. \n\nThis graph can be represented as a sequence $s$ of vectors $V_k$ that can represent either a node or an edge. This sequence is constructed sequentially by adding each node label $x_i$ followed by all the edge labels between this node and the previously added ones $e_{ij}$:\n\\begin{equation*}\n    s = \\left(V_k\\right)_{k=1..T} = \\left(x_0, x_1, e_{01}, x_2, e_{02}, e_{12}, x_3, ...\\right)\n\\end{equation*}\nwhere $T$ is the length of the sequence. \nFor practical reasons, we define all sequences to end with a particular node type vector acting as an \\emph{end sequence} token. \nAs an illustration, the representation of propane, ie a chain of three carbons with single bonds, could be $s=$ (\\emph{carbon, carbon, single bond, carbon, no edge, single bond, end token}). Any sequence $s_k = \\left(V_u\\right)_{u=1..k-1}$ defines a valid subgraph $G_{s_k}$. \n\nDifferent sequences can represent the same molecule. Typically, the choice of the node indices $\\mathcal{V}$ is arbitrary but changing the node indices will change the sequence s. We note $seqs(G)$ the ensemble of sequences allowed by our algorithm that represent graph $G$.\n\nFrom a distribution over graphs $\\mathcal{Q}(G)$ we can then trivially obtain a distribution on sequences $\\mathcal{Q}(s)=\\mathbb{E}_{G\\sim\\mathcal{Q}(G)}\\mathcal{Q}(s|G)$ where $\\mathcal{Q}(s|G)$ is uniform over $seqs(G)$. Notice that to any sequence corresponds only one graph, so that ${\\mathcal{Q}(s) = \\frac{\\mathcal{Q}(G)}{|seqs(G)|}}$.\n\n\\subsubsection{Auto-regressive model}\n\\label{section: nn}\n\nA learned distribution over sequences $\\mathcal{P}(s)$ is then modelled in an auto-regressive fashion:\n\n\\begin{equation}\n    \\mathcal{P}(s=\\left(V_k\\right)_{k=1..T}) = \\prod_k \\mathcal{P}_\\theta(V_k|G_{s_k})\n\\end{equation}\nwhere $\\mathcal{P}_\\theta$ is a graph neural network defined below, with parameter $\\theta$.\n\nAs there is only one graph per sequence, the corresponding distribution on graphs is simply $\\mathcal{P}(G) = \\sum_{s\\in seqs(G)} \\mathcal{P}(s)$.\n\nWe explicitly model $\\mathcal{P}_\\theta(V_k|G_{s_k})$ as a categorical distribution whose probability vector is the output of a graph neural network. \nFirst, node embeddings $h_i$ are produced by $L$ layers of convolution. As in \\citet{shi2020graphaf}, we use an edge sensitive version of the standard GCN convolution:\n\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:conv}\n    h_i^{(l)} &= \\sum_{e=1}^E ReLU\\left(W_e h_i^{(l-1)} + V_e\\sum_{j \\in \\mathcal{N}_e(i)} \\frac{h_j^{(l)}}{\\sqrt{d_i^e d_j^e}} + U_e\\right)\n\\end{aligned}\n\\end{equation}\nwhere $U_e$, $V_e$ and $W_e$ are learnable weight matrices, $\\mathcal{N}_e(i) = \\left\\{j \/ (i,j)\\in E \\text{ and } e_{ij} = e\\right\\}$ is the set of neighbours of $i$ via an edge labelled $e$ and $d^e_i=|\\mathcal{N}_e(i)|$ is the degree of node $i$ in the graph containing only edges labelled $e$. \nWe set $h_i^{(0)}$ as the one hot embedding of $x_i$.\n\nA global embedding is then defined by global addition pooling, ie $H = \\sum_i h_i^{(l)}$.\n\nFinally, two independent MLPs with \\emph{softmax} activation are used to produce probability vectors for node and edge distributions:\n\n\\begin{equation}\n\\label{eq:mlp}\n\\begin{aligned}\n\\mathcal{P}(V_k=x_i|G_k) &= Cat(MLP^{node}(H)) \\\\\n\\mathcal{P}(V_k=e_{ij}|G_k) &= Cat(MLP^{edge}(H, h_i^{(l)}, h_j^{(l)}))\n\\end{aligned}\n\\end{equation}\nwhere $Cat(.)$ is the categorical distribution with probabilities defined by the outputs of the network.\n\nThe network is then trained by minimizing the cross-entropy loss of each decision over the dataset.\n\n\\subsubsection{Further improvements and modifications}\n\n\n\\nlparagraph{Conditioning the network on molecule size}\n\\label{section:size}\nIn the next sections, we use among others a version of the model conditioned on molecule size \\footnote{Additional care must be taken in training such network, as detailed in supplementary materials}. \nIn order to do so, we redefine the convolution step in eq. \\ref{eq:conv} as:\n\n\\begin{equation}\n    h_i^{(l)} = \\widehat{h}_i^{(l)} + C * \\frac{s}{S}\n\\end{equation}\nwhere $s$ is the final molecule size, $S$ is the maximal allowed molecule size, $C^{(l)}$ is a learned vector, and $\\widehat{h}_i^{(l)}$ is the original definition in eq. \\ref{eq:conv}.\n\n\n\\nlparagraph{Limiting the length of sequences}\n\\label{section:seq}\nThe length of the sequences is the major limitation of auto-regressive methods in term of time efficiency.\nMoreover, most $V_k$ correspond to $e_{ij}=\\text{\\emph{no edge}}$, which seems like a waste of resources. \nA common approach is to always present the nodes ordered as a breadth first search(BFS) traversal of the graph, without repetition and rooted randomly. \nIn that setting, there exists an upper limit $B \\ge |i-j|$ depending on the maximal width of the BFS tree in the dataset. \nIn the ZINC dataset, $B=12$. We can then omit all $e_{ij}$ if $|i-j|> B$ in $s$.\nNotice that the same advantage occurs if the nodes are instead presented in the reversed order of a BFS exploration, meaning that the last explored node is presented first in the sequence. \nAs opposed to the former case, in the latter, the last generated node is uniformly sampled among all the nodes. \n\nThese two choices correspond to two definitions of $seqs(G)$. The difference between them will be exploited in section \\ref{section:mcmc}.\n\n\\nlparagraph{Labelling the nodes for better cycle detection}\nA common issue in graph generation is the occurrence of unrealistic cycles, never seen in the dataset. \nWe believe this is due to the fact that atoms are anonymous, so that two atoms in a similar environment will have similar embeddings. Hence, the atom embedding cannot capture information about the distance to another specific atom. \nIn order to partially solve the problem, we add a random label to each node to enable the model to distinguish between close pair of nodes, that have interacted with each other through convolutions, and far-away pairs. \nWe implement that by redefining the initial node embedding: $h_i^{(0)} = (F(x_i),  \\epsilon)$ where $F(x_i)$ is the one hot embedding of $x_i$ and $\\epsilon$ is a random normal vector of fixed dimension. \nWe found empirically that, without completely solving the problem, this improved the distribution of cycles in generated molecules.\n\n\\nlparagraph{Enforcing validity}\n\\label{section:validity}\nA great advantage of auto-regressive models is that we can easily ensure that the maximal valency of each atom is respected. \nIn order to do so, we simply set the probability of edges that would break the valency constraint to zero during generation. \nIf the nodes are presented in standard BFS order, we can also ensure that the molecule remains connected during the generation process, by stopping the generation if a new node has no real bond to the rest of the molecule.\n\n\\subsection{Direct optimization in molecule manifold}\n\\label{section:mcmc}\nIn order to optimize molecule properties we use the Metropolis Hastings algorithm, with a slight adaptation in order to bias the simulated distribution, centered around the criterion to optimize, toward the learned distribution, which defines implicitly the criterion of realism. We refer to that method as Learned Realism Sampling (LRS).\n\n\\subsubsection{Algorithm}\nFirst, we define a conditional transition probability $\\mathcal{T}(G_1|G_0)$ from graph $G_0$ to graph $G_1$. \nIntuitively, $\\mathcal{T}$ is a distribution over the neighborhood of $G_0$ in the ensemble of all graphs, which is not defined. \nMoreover, in order to ensure the realism of the proposal $G_1$, we need to sample from the learned distribution $\\mathcal{P}$. \nIn order to satisfy both conditions, we simply resample the last steps of the generative algorithm. \nMore precisely, we start by sampling uniformly a generation sequence $s \\in seqs(G_0)$ of length $M$, and remove the last elements corresponding to $\\delta n\\in\\mathbb{N}$ node additions, yielding a shorter sequence that we call $s*$. We then rerun the generative model, starting from the intermediate graph $G_{s*}$. \n\nThen, given a score function $sc$ and temperature $T$, we define an acceptance probability $\\mathcal{R}$ as:\n\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{R}(G_1|G_0) &= \\min\\left(\\exp\\left(\\frac{sc(G_1) - sc(G_0)}{T}\\right), 1\\right)\n\\end{aligned}\n\\label{eq:acceptance}\n\\end{equation}\n\nFrom those two distributions and an initial graph $G_0$, we iteratively produce new graphs similarly to the Metropolis Hastings algorithm, as summed up in algorithm \\ref{algo}:\n\n\\begin{algorithm}[h]\n\\begin{enumerate}\n\\item Sample a proposal graph $G^{proposal}_{n+1} \\sim \\mathcal{T}(G_{n+1}|G_n)$\n\\item Accept the proposal graph with probability $\\mathcal{R}(G^{proposal}_{n+1}, G_n)$\n\\item If the proposal is accepted, set $G_{n+1} = G^{proposal}_{n+1}$. Else, set $G_{n+1}=G_n$.\n\\end{enumerate}\n\\caption{Learned Realism Sampling}\n\\label{algo}\n\\end{algorithm}\n\nNotice that this is \\emph{not} Metropolis Hastings algorithm as $\\mathcal{T}$ is not symmetric, which should require a correction in $r$. Hence the simulated distribution is \\emph{not} proportional to $F$. \n\n\\begin{lemma}\\ \\\\\nGiven a joint distribution $\\mathcal{P}(s, G)$ over $\\mathcal{S}\\times\\mathcal{G}$, if the conditional distribution $\\mathcal{P}(s|G)$ is the uniform distribution over $seqs(G)$, then algorithm \\ref{algo} is equivalent to Metropolis Hastings sampling for a distribution $\\pi(G) \\propto F(G)\\mathcal{P}(G)$.\n\\label{lemma}\n\\end{lemma}\nThe proof is given in supplementary materials. \nNotice that the condition is fulfilled by the dataset distribution $\\mathcal{Q}$, as detailed in \\ref{section:sequence definition}. The result is only exact in the limit where the learned distribution $\\mathcal{P}$ perfectly matches $\\mathcal{Q}$.\nThis lemma should however describe the general tendency of the algorithm even if the approximation is imperfect, as there is no consistent bias during training. \n\nAs a result, algorithm \\ref{algo} produces graphs which both have high scores and are likely with regards to the learned distribution $\\mathcal{P}$. \nThe temperature parameter can be used to tune the importance of both those aspects or to perform annealing.\n\nA great advantage of the method is that the learned distribution  is completely independent from the score calculation. \nTypically, $\\mathcal{P}$ can be learned on large public datasets, like ZINC, while the score can be learned on scarce data, using very simple models, or even on no data at all if an analytic solution exists. \nNotice that many models  optimize the score by learning, in particular reinforcement learning ones. There is then a risk that some specificity of the score is learned at the expense of neglecting the dataset statistics. This is not the case of our model as the learning phase is only performed prior to optimization.\n\n\\subsubsection{Further improvements and variations}\n\n\\nlparagraph{Controlling the sampled graph size}\nIf the initial graph $G_0$ of size $N$ has few nodes compared to the median size of the training dataset, then the graph $G_1\\sim\\mathcal{T}(G_1|G_0)$ is likely to be much larger than $G_0$, breaking the intuition that $G_1$ should be similar to $G_0$. \nTo correct that effect, a model conditioned on size can be used. We remove $\\delta n_r$ nodes and condition the output size to be $N - \\delta n_r + \\delta n_a$. We sample both quantities $\\delta n_r$ and $\\delta n_a$ from a Poisson distribution with parameter $\\lambda$.\n\n\\nlparagraph{Sampling the modified region uniformly}\nAs detailed in section \\ref{section:seq}, the nodes can be presented in different orders. \nFollowing \\citet{shi2020graphaf}, our basic model presents the nodes in a BFS order to improve efficiency. The removed atoms are the ones visited last. Thus, the probability to be removed is not the same among all nodes. For example, in a long carbon chain, the atoms in the middle will almost never be removed.\n\nIn order to circumvent that effect and without loss of efficiency, the nodes can be presented in reversed BFS order, so that the root is presented last. \nBy doing so, the first removed node is sampled uniformly among the graph. \nNotice that if $\\delta n_r > 1$, this modification is not enough to ensure that nodes have a uniform probability to be removed, as more connected nodes will be selected more frequently. \nStill, we find that this correction is enough for substantial improvement while remaining very simple.\n\n\\section{Experiments}\nWe compare our model to multiple state-of-the-art models described in more details in section \\ref{related}.\n\n\\subsection{Distribution Learning}\n\n\\begin{table}[t]\n\\caption{Distribution learning performance of all the baselines. Validity(Val.), Novelty(Nov.), Unicity(Uni.), Kl, FCD scores are presented. ($\\uparrow$) and ($\\downarrow$) indicates respectively that higher or lower is better. VJTNN and MIMOSA are missing from this table, as they cannot be trained only for distribution learning. All models have been retrained using the official released code, except for MRNN which is not available. }\n\\label{results:distribution learning}\n\\vskip 0.15in\n\\centering\n\\begin{tabularx}{\\linewidth}{@{}lXXXXX@{}}\n\\toprule\n~& Val.($\\uparrow$) & Nov.($\\uparrow$) & Uni.($\\uparrow$) & KL($\\downarrow$) & FCD($\\downarrow$)\\\\\n \\midrule\nJTVAE & 1.00 & 1.0 & 1.000 & 0.078 & 1.46\\\\\nGCPN &  0.20 & 1.0 & 1.000 & 0.604 & 12.49\\\\\nMRNN & 0.65 & 1.0 & 0.999 & - & -\\\\\nGraphAF & 0.68 &1.0 & 0.991 & 0.2294 &10.06\\\\\nSMILES LSTM & 0.87 & 0.999 & 1.000 & 0.178 & 0.515\\\\\n \\midrule\nAR & 0.97 & 1.0 & 1.000 & 0.060 & 1.48\\\\\nAR-SC & 0.97& 1.0 & 1.000 & 0.058 & 1.50\\\\\nAR-RBFS & 0.88 & 1.0 & 1.000 & 0.102 & 2.83\\\\\n \\bottomrule\n\\end{tabularx}\n\\end{table}\n\n\\newcommand{\\textwidth}{\\textwidth}\n\\newcommand{0.35}{0.35}\n\\newcommand{3}{3}\n\n\\begin{figure*}[htb]\n\\centering{\n\\begin{subfigure}[b]{0.3\\textwidth}\n\\def\\molsize}\\input{ours_reversed_0.pdf_tex}{3\\textwidth}\n\\scalebox{0.35}{\\input{basic_2x2_compressed.pdf_tex}}\n\\caption{AR}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.3\\textwidth}\n\\def\\molsize}\\input{ours_reversed_0.pdf_tex}{3\\textwidth}\n\\scalebox{0.35}{\\input{sc_2x2_compressed.pdf_tex}}\n\\caption{AR-SC}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}[b]{0.3\\textwidth}\n\\def\\molsize}\\input{ours_reversed_0.pdf_tex}{3\\textwidth}\n\\scalebox{0.35}{\\input{rbfs_2x2_compressed.pdf_tex}}\n\\caption{AR-RBFS}\n\\end{subfigure}\n}\n\\caption{Samples from the different versions of our model. No cherrypicking.}\n\\label{figure:generation}\n\\end{figure*}\n\n\n\n\n\nOur method begins by training the autoregressive models described above to learn the dataset distribution. \nThis is obviously a critical step, as the subsequent algorithm will produce samples biased toward that distribution. \n\nWe train three versions of our autoregressive model:\n\\begin{itemize}\n\\item The basic model described in \\ref{section:armodel}, simply referred to as AR.\n\\item A model conditioned on size as described in section \\ref{section:size}, referred to as AR-SC\n\\item A model conditioned on size, and with nodes presented in reversed BFS order, as in section \\ref{section:seq}, referred to as AR-RBFS\n\\end{itemize}\nThe model is trained on the ZINC250K dataset from \\cite{irwin2005zinc}.\nImplementation details can be found in supplementary materials, such as input format and hyperparameters.\n\nThe model is compared to state-of-the-art baselines using multiple metrics, all detailed in appendix \\ref{metrics definition}.\nWe use the standard validity, novelty and uniqueness metrics, measured for a set of 10,000 molecules. \nFollowing  \\citet{brown2019guacamol}, we also provide two other metrics: the Kullback-Leibler divergence between the distributions of the generated molecules and the dataset, evaluated on many molecular descriptors. The Fr\u00e9chet ChemNet Distance (FCD) \\citep{preuer2018frechet} is a molecular version of the Fr\u00e9chet Inception Distance. \nFinally, we provide images of generated molecules in figure \\ref{figure:generation}.\n\n\nThe results are displayed in table \\ref{results:distribution learning}, showing that our model achieves state-of-the-art performance, and generally better than reinforcement learning based methods. \nThe RBFS model achieves slightly lower validity, KL and FCD, which is accountable to its more complex task as it must also ensure molecule connectivity.\n\n\n\\subsection{Score optimization}\nWe then use the three models from the previous section to perform our proposed LRS methods. Our method using the AR-X model is referred to as LRS-X. In the first section, we demonstrate that optimization can produce irrealistic molecules. In the second section, we show that although our model has the additional limitation to produce realistic molecules, it can still perform as well or better than state-of-the-art methods in the task of molecule improvement. \n\n\\subsubsection{Unsafe optimization can lead to generation of unrealistic molecules}\n\\label{section: unconstrained optimisation}\n\\newcommand{8cm}{8cm}\n\\newcommand{0.25}{0.25}\n\\newcommand{0.2}{0.2}\n\\begin{figure*}[htb]\n\\begin{subtable}[b]{\\textwidth}\n\\begin{tabularx}{\\textwidth}{@{}cXccXc@{}}\n (4.52) & & GCPN (7.98) & JTVAE (5.30) & & LRS (5.42)\\\\\n\\scalebox{0.25}{\\def\\molsize}\\input{ours_reversed_0.pdf_tex}{8cm}\\input{DB_0.pdf_tex}} & & \\scalebox{0.25}{\\def\\molsize}\\input{ours_reversed_0.pdf_tex}{8cm}\\input{GCPN_0.pdf_tex}} & \\scalebox{0.25}{\\def\\molsize}\\input{ours_reversed_0.pdf_tex}{8cm}\\input{JTVAE_0.pdf_tex}} & &  \\scalebox{0.25}{\\def\\molsize}\\input{ours_reversed_0.pdf_tex}{8cm}\\input{ours_basic_0.pdf_tex}}\\\\\n(4.23) & & GraphAF (12.23) & All SMILES (12.34) & & LRS-SC (5.90)\\\\\n\n\\scalebox{0.25}{\\def\\molsize}\\input{ours_reversed_0.pdf_tex}{8cm}\\input{DB_1.pdf_tex}} & & \\scalebox{0.25}{\\def\\molsize}\\input{ours_reversed_0.pdf_tex}{8cm}\\input{GraphAF_0.pdf_tex}} & \\scalebox{0.25}{\\def\\molsize}\\input{ours_reversed_0.pdf_tex}{8cm}\\input{all_smiles_0.pdf_tex}} & & \\scalebox{0.25}{\\def\\molsize}\\input{ours_reversed_0.pdf_tex}{8cm}\\input{ours_size_1.pdf_tex}}\\\\\n(4.22) & & MRNN (8.63) & SMILES LSTM (11.87) & & LRS-RBFS (6.98)\\\\\n\n\\scalebox{0.25}{\\def\\molsize}\\input{ours_reversed_0.pdf_tex}{8cm}\\input{DB_2.pdf_tex}} & & \\scalebox{0.25}{\\def\\molsize}\\input{ours_reversed_0.pdf_tex}{8cm}\\input{MRNN_0.pdf_tex}} & \\scalebox{0.25}{\\def\\molsize}\\input{ours_reversed_0.pdf_tex}{8cm}\\input{smiles_lstm_0.pdf_tex}} & & \\scalebox{0.25}{\\def\\molsize}\\input{ours_reversed_0.pdf_tex}{8cm}\\input{ours_reversed_0.pdf_tex}}\\\\\n(a) Dataset molecules & & \\multicolumn{2}{c}{(b) Baselines} & & (c) Ours \\\\\n\n\\end{tabularx}\n\\end{subtable}\n\\caption{Candidates with largest NPLogP from the dataset (a), or sampled from the baselines (b) and our model (c).}\n\\label{figure:high NPLogP}\n\\end{figure*}\n\nA standard task is to generate molecules with the largest score possible. A common proof-of-principle score is the normalized penalized water octanol partition logP coefficient (NPlogP). This score happens to be easy to maximize, if no realism constraint is required. For example, a very long carbon chain, although unlikely in the training dataset, has a very high score. We show that many models will disregard the dataset statistics in order to optimize the score.\n\nIn general, it is hard to provide a metric, or even to evaluate, the realism of a molecule regarding a dataset. In this particular case, the molecules obtained from a model that disregarded its training set are particularly odd, so that we can attempt to quantify their quality. As in literature, we provide the top 3 molecules generated by each model. To estimate their realism, we provide their mean log-likelihood in feature space: First, we use a set of property descriptors as for the KL metric in previous section. Second, we use the ChemNet features. Details can be found in appendix. Results are grouped in table \\ref{results:property optimization}. Finally, we provide images of the top candidate for each model in figure \\ref{figure:high NPLogP}.\n\nFirst, we notice that \\emph{all} models produce NPlogP scores that are considerably above what can be found in the dataset, which is already an alarming sign. We also notice that most models produce extremely unlikely molecules. Indeed, as can be seen on figure \\ref{figure:high NPLogP}, most models only learned to repeat the same pattern many times. On the other hand, both JTVAE and our models produce more reasonable candidates (with some doubt concerning the descriptor log likelihood of LRS-SC and LRS-RBFS). We conclude that our model is indeed efficiently able to maximize a criterion while better enforcing realism of the generated candidates.\n\n\n\n\n\n\\begin{table}[hb]\n\\caption{Top three performances of NPLogP optimization for all models and log-likelihood against descriptors and ChemNet features.}\n\\label{results:property optimization}\n\\vskip 0.15in\n\\centering\n\\begin{tabularx}{\\linewidth}{@{}lXXXXX@{}}\n \\toprule\n & \\multicolumn{3}{c}{NPLogP ($\\uparrow$)} & \\multicolumn{2}{c}{Log-Likelihood ($\\uparrow$)}\\\\ \n \\cmidrule(r){2-4} \\cmidrule(l){5-6}\n & 1st & 2nd & 3rd & Descriptors & Chemnet Feat.\\\\\n \\midrule\nDataset & 4.52 & 4.23 & 4.22 & -37 & 291\\\\\n \\hline\n JTVAE & 5.30 & 4.93 & 4.49 & -38 & 283\\\\\n GCPN & 7.98 & 7.85 & 7.80 & -101 & -226 \\\\\n MRNN & 8.63 & 6.90 & 4.73 & -44 & -125\\\\\n GraphAF & 12.23 & 11.29 & 11.05 & -286 & 130\\\\\n All-SMILES & 12.34 & 12.16 & 12.05 & -210 & -333\\\\\n \\small{SMILES} LSTM & 11.87 & 11.84 & 11.70 & -164 & -171\\\\\n \\midrule\n LRS & 5.42  & 5.22 & 5.11 & -36  & 427\\\\\n LRS-SC & 5.90& 5.90 & 5.83 & -47 & 284\\\\\n LRS-RBFS & 6.98  & 6.92 & 6.69 & -51  & 293\\\\\n \\bottomrule\n\\end{tabularx}\n\\end{table}\n\n\n\n\\subsubsection{Molecule improvement}\n\n\\begin{table}[t]\n\\caption{NPlogP score improvement for a given similarity threshold $\\delta$. Results for JTVAE are taken from \\citet{jin2018learning}.}\n\\label{results:molecule improvement NPlogP}\n\\vskip 0.15in\n\\centering\n\\begin{tabularx}{\\linewidth}{@{}lXXX@{}}\n \\toprule\n & \\multicolumn{3}{c}{NPlogP}\\\\\n \\cmidrule(l){2-4}\n & $\\delta = 0.2$ & $\\delta = 0.4$ & $\\delta = 0.6$\\\\\n \\midrule\n JTVAE & -  & $1.03 \\pm 1.39$ & $0.28 \\pm 0.79$ \\\\\n VJTNN & -  & $3.55  \\pm 1.67$ & $2.33  \\pm 1.24$ \\\\\n GCPN & $4.12  \\pm 1.19$& $2.49 \\pm 1.30$ & $0.79  \\pm 0.63$ \\\\\n GraphAF & $5.00 \\pm 1.37$& $3.72 \\pm 1.19$ & $1.94 \\pm 1.00$ \\\\\n MIMOSA & $3.48 \\pm 1.41$& $3.44 \\pm 1.41$ & $\\mathbf{3.24 \\pm 1.48}$ \\\\\n \\midrule\n LRS & $4.68 \\pm 1.48$ & $3.63 \\pm 1.23$ & $2.29 \\pm 1.12$ \\\\\n LRS-SC & $5.49 \\pm 1.37$ & $4.14 \\pm 1.08$& $2.81 \\pm 0.92$ \\\\\n LRS-RBFS & $\\mathbf{6.23  \\pm 1.28}$  & $\\mathbf{4.44  \\pm 1.15}$ & $2.80 \\pm 1.09$ \\\\\n \\bottomrule\n\\end{tabularx}\n\\end{table}\n\nWe then evaluate our model in the context of molecule improvement. Given an initial molecule $m_0$ and a threshold $\\delta$, the objective is to produce a molecule $m^*$ with maximal score so that the similarity $sim(m^*, m_0)$ between $m^*$ and $m_0$ is larger than $\\delta$. As in our baselines, the similarity is defined as the Tanimoto similarity between the Morgan fingerprints with radius of 2 of $m^*$ and $m_0$. We simply set the score of a molecule with $sim(m^*, m_0) < \\delta$ to be $-\\infty$. \n\nAs a first task, we adopt the same setup as most previous works. More precisely, we optimize the NPlogP scores of 800 molecules with initially low score in ZINC test set, and report the average score improvement for different values of $\\delta$. \nThe results are presented in table \\ref{results:molecule improvement NPlogP}.\nOur model outperforms all baselines by a large margin, except MIMOSA in the case $\\delta=0.6$.\n\n\n\n\\begin{table}[t]\n\\caption{QED and DRD2 score improvement and success for a similarity threshold $\\delta = 0.4$.}\n\\label{results:molecule improvement QED and DRD2}\n\\begin{tabularx}{\\linewidth}{@{}lXXXX@{}}\n \\toprule\n & \\multicolumn{2}{c}{QED}&\\multicolumn{2}{c}{DRD2}\\\\\n \\cmidrule(l){2-3}\n \\cmidrule(l){4-5}\n & Improvement & Success& Improvement & Success\\\\\n \\cmidrule(l){1-3}\n \\cmidrule(l){4-5}\nVJTNN + GAN & $0.11(0.08)$ &  $79.3\\%$& $0.61(0.39)$&$\\mathbf{97.3\\%}$\\\\\nMIMOSA & $0.10(0.05)$&$16.3\\%$& $0.28(0.30)$&$25.8\\%$\\\\\n \\cmidrule(l){1-3}\n \\cmidrule(l){4-5}\nLRS-SC& $0.17(0.04)$& $80.0\\% $&$0.73(0.31)$&$79.1\\%$\\\\\nLRS-RBFS& $\\mathbf{0.18(0.03)}$ & $\\mathbf{94.5\\%}$ &$\\mathbf{0.76(0.26)}$&$83\\%$\\\\\n \\bottomrule\n\\end{tabularx}\n\\end{table}\n\nThen, following \\citet{jin2018learning}, we optimize the QED score and the DRD2 score, which is the affinity with dopamine receptor of type 2, as predicted by a model from \\citet{olivecrona2017molecular}, and report the average score improvement and success rate. In the former case, we optimize a set of 800 molecules with QED scores between 0.7 and 0.8. An optimization is considered a success if a candidate with a score greater than 0.9 is found. In the latter case, we optimize a set of 1000 molecules with DRD2 scores lower than 0.05. An optimization is considered successful if a candidate with  a score above 0.5 is found. For those experiments, we use a threshold $\\delta = 0.4$. The results are presented in table \\ref{results:molecule improvement QED and DRD2}.  In general, our method produces larger improvements.\n\n\\section{Conclusion}\n\nIn this work, we proposed a method for molecular property optimization that only produces realistic candidates with respect to a given dataset. In order to do so, we start by learning the dataset distribution using a generative graph autoregressive model. Based on this learned model, we then perform sampling akin to Metropolis Hastings, implicitly forcing the exploration to focus on regions with higher probability under the training distribution.\n\nWe demonstrate that the generated molecules are visually varied and realistic, compared to all reinforcement learning based baselines. Furthermore, our model outperforms all baselines in the task of molecule \\emph{improvement}, showing that enforcing realism is not only desirable but also beneficial. \n\nNotably, the optimization step in our model is not based on learning, so that no data specific to the optimized score is required, which is essential for practical applications where data is scarce.\n\nThese results prove that the combination of distribution learning and traditional importance sampling is promising to improve current molecule design methods. Other than Markov Chain Monte Carlo methods, like ours, we believe that similar ideas could be promising when applied to genetic algorithms, which are prominent in modern pipelines. We leave this perspective to future work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\nThroughout the years,\nthe use of predictive computational models has become standard practice\nin many professional fields such as logistics, economics or R\\&D\n\\cite{generic0, generic1, generic2, generic3, generic4}, with computational\nchemistry being no exception.\nPredictive models in this field often approximate\nthe potential energy surface (PES) and derived properties for a given chemical system,\nand can be broadly categorized based on their level of theory:\nQuantum mechanical (QM) approaches\nsuch as wave function or density functional theory\nexplicitly model the electronic structure,\ngenerally resulting in a high accuracy and broad applicability.\nHowever, explicit electronic treatments come with a high computational price tag,\nmaking calculations unfeasible for larger systems.\nThis limitation can be overcome by introducing more empiricism to the model\non the electronic (\\textit{e.g. } tight binding), atomic (\\textit{e.g. } molecular mechanics) or \nmolecular (\\textit{e.g. } coarse-graining)\nlevel.\nSuch empirical models attempt to strike the balance between speed and prediction accuracy by approximating the PES description through the introduction of parameters.\nProminent examples include approximate functionals in\ndensity functional theory (DFT) \\cite{becke3, lyp, b3lyp, wB97XD},\ndensity functional tight binding (DFTB) \\cite{dftb1, dftb2, dftb3, xtb1, xtb2},\nmachine learning (ML) potentials \\cite{behler, schnet, ani1x, pinn},\nand force fields (FF). \\cite{amber, charmm, reax1, reax2}\n\nWhile some of the best empirical models can closely approximate higher-level QM theories\nat only a fraction of the computational time \\cite{ani1ccx, gfn2xtb},\ntheir quality strongly varies with different sets of parameters\n\\cite{anna,ogolem,forcebalance,potfit,garfield}.\nAdditionally, many empirical models can only deliver accurate results for a comparably limited chemical space,\neither due to their functional form or parameters being specific to certain (combinations of) elements.\nThese limitations can lead to the existence of multiple parameter sets\nbased on the desired application, \\textit{e.g. } ReaxFF has distinct families of combustion or condensed phase parameterizations. \\cite{reax_npj}\nSuch lack of general parameters gives rise to the research field of parameter fitting\n\\cite{fitting1, fitting2, fitting3, fitting4, anna},\nwhere the task is to find an optimal parameter set, given training data\nconstructed from chemical systems and their properties of interest.\nAlthough the fitting process can be an appealing solution to the above shortcomings,\nits practical implementation remains hardly accessible to the broader audience\nand instead is almost exclusively carried out by specialized research groups.\nIn our experience, the majority of researchers,\nalthough being interested in individual parameter fitting,\nare discouraged by the high barrier that comes with it.\nThe main reason for this being a lack of generalization and transparency:\nTraining data often comes in a variety of formats, optimizers expect a different input all together and the format in which parameters are stored is specific to each method. \\cite{reax_manual, forcebalance_manual, ogolem_manual, posmat}\nThe combination of these oftentimes results in works that can be hardly comprehended and reproduced by third parties.\nIn an effort to address the above issues, we introduce the ParAMS scripting package for Python. \nThe following section briefly summarizes the architecture of ParAMS and we refer to the documentation for further technical details. \\cite{doc}\nThe Results section demonstrates how the package can be used to i) generate density functional-based tight binding (DFTB) two-body repulsive potentials for an ionic material, and ii) reparameterize a ReaxFF reactive force field \\cite{reax1, reax2} for organic disulfides.\nStep-by-step Jupyter notebooks for the two case studies are provided as supporting information, as well as on GitHub \\cite{si}.\nAdditional application examples are available in the package's documentation. \\cite{doc}\nThe final section concludes with a summary and an outlook on future work.\n\n\n\\section{Implementation}\nParAMS follows a modular package structure\nwith well-defined application programming interfaces (APIs),\nwhich allows components to be treated independently.\nThis is essential for future development, as individual sub-modules can\nbe easily worked on and extended.\nWe describe the main components and their functionality below.\nFor a mathematical description of the functionality as well as \nadditional explanation of the syntax, please refer to section S1 of the supporting information.\n\n\\textbf{Job Collection and Data Set}\nclasses are responsible for the input \/ output (IO) of relevant data.\nIn the context of ParAMS, these will be collections of chemical systems, properties and settings alongside optional metadata.\nA Job Collection clearly defines job entries by combining systems and settings.\nThe Data Set defines which properties of a job are relevant to the optimization and stores\nthe reference values of all entries in a vector $\\bm{y}$.\nReference values can be added from any source: Experimental \/ external results, or a high-level calculation.\nTo ensure reproducibility and ease-of-use, ParAMS makes use of the\nYAML data-serialization format \\cite{yaml} as the default for all IO operations.\n\n\\textbf{Extractors} tell ParAMS how to \nextract a property of interest $P$ from a calculated job.\nTechnically, they are small standalone Python modules that read the native\n(\\textit{e.g. } stream, text or binary file) output of the Amsterdam Modeling Suite\\cite{ams, scm} into Python variables.\nExamples for implemented extractors are:\nInteratomic distances, valence angles, dihedral angles, atomic charges, reaction energies,\nlinear transit energy profiles, lattice vectors, bulk moduli, atomic forces, stress tensors, Hessian matrices and vibrational frequencies.\nNew extractors are easily written by the user,\neffectively allowing any  property that can be calculated with AMS\nto be fitted within the scope of ParAMS.\nExtractors also support more elaborate cases that require additional processing before a comparison. This is for example needed when computing the minimal root-mean-square deviation of atomic positions.\n\n\\textbf{Loss Functions} implement various metrics that describe the distance between two vectors.\nIn the context of parameter fitting, a vector consisting of all reference values \n$\\bm{y}$ has to be compared to the predictions vector $\\bm{\\hat{y}}$,\nas generated given a specific set of parameters,\nin order to measure the quality of the fit.\nImplemented metrics are least absolute error (LAE), mean absolute error (MAE),\nroot-mean-square error (RMSE) and residual sum of squares (RSS).\nAdditionally, user-defined metrics are supported.\n\n\\textbf{Optimizers} provide a unified interface to a variety of optimization algorithms.\nCurrently, the following are supported:\nCovariance Matrix Adaptation Evolution Strategy (CMA-ES) \\cite{cma1,cma2,cma_py},\nAdaptive Rate Monte Carlo (ARMC) \\cite{armc} and\noptimizers available through the Nevergrad \\cite{nevergrad} and SciPy \\cite{scipy} packages. \nTo guarantee parameterization support for a wider range of models,\nthe current version of ParAMS is designed to work with gradient-free optimization algorithms only.\n\n\\textbf{Parameter interfaces} translate the parameter vector $\\bm{x}$ into the native\nformat of the empirical model (\\textit{e.g. } a file on disk).\nAny existing parameter interface can be parameterized.\nAt the time of writing, ParAMS supports interfaces to\nReaxFF \\cite{reax_ADF}, SCC-DFTB repulsive potentials, GFN1-xTB \\cite{xtb1}, and  Lennard-Jones potentials.\n\n\\textbf{Callbacks} allow the interaction with a parameterization at runtime.\nSuch interactions can be progress loggers, timeouts, early stopping criteria or plotting functions \\cite{matplotlib}.\n\nThe \\textbf{main script} is a command line interface to ParAMS for users who do not wish\nto spend much time writing their own parameterization scripts.\nIt allows the setup and execution of the most common tasks through a configuration file.\n\nFigure \\ref{fig:loop} shows the general parameterization loop and highlights the main input-output relationships.\n\\begin{figure}\n \\centering\n \\includegraphics[width=.4\\textwidth]{fig\/flowchart.png}\n \\caption{\n A schematic representation of the ParAMS parameterization loop.\n Highlighting the interplay between the three main components:\n Optimizer, Job Collection and Data Set.\n Every parameter set suggested by the optimizer produces different job results,\n which are evaluated by the Data Set based on a unique \\textit{jobID}.\n The results are then compared with the reference values,\n weighted and combined into the loss function value.\n A more detailed description is available in the package documentation.\n }\n \\label{fig:loop}\n\\end{figure}\nThe implementation allows users already experienced in other data science packages to prepare inputs and process results in a familiar way.\nTechniques like training- and validation set splitting,\ncross-validation, early stopping, batching or outlier detection are supported out of the box.\nParameter constraints can be further used to limit the search space of a problem.\nIn addition to regular box constraints,\nusers can express inequality constraints involving multiple parameters (\\textit{e.g. } $x_1 \\leq x_2$), as will be demonstrated in the ReaxFF example application.\nParAMS implements two levels of parallelism.\nMultiple parameter vectors and multiple jobs per vector can be evaluated at the same time,\nresulting in workloads that can be distributed effectively based on the training set size and optimization algorithm.\nMoreover, since the signatures of all classes in a submodule are the same,\ndifferent models and optimizers can be effortlessly compared and deployed (\\textit{e.g. } comparison of different levels of theory in Tab. \\ref{tab:repulsive}).\nDaily regression tests are performed to guarantee an error-free functionality of the package.\n\n\n\n\\section{Results}\n\\subsection{DFTB two-body repulsive potential parameterization}\\label{sec:dftb}\nHere, we illustrate how the ParAMS package can be used to train a two-body SCC-DFTB repulsive potential \\cite{dftb1}. \nFor simplicity, we use ZnO, for which several previous parameterizations already exist in the literature \\cite{znopt}. We will approximately follow the approach from Ref. \\citenum{znopt}, in which the authors reused the electronic parameters from the znorg-0-1 DFTB parameter set \\cite{znorg} and reparametrized the two-body Zn-O repulsive potential to reference data calculated for the wurtzite and rocksalt polymorphs of ZnO. With the znorg-0-1 parameters, the rocksalt polymorph is predicted to be more stable than the wurtzite polymorph, but in experiments and DFT calculations, the opposite is true, which motivates the reparameterization.\n\nThe entire code needed for the parameterization is provided as supporting information \\cite{si}. It fits the Zn-O pairwise repulsive potential $V^\\text{rep}$ as a tapered double exponential function of the form\n\\begin{equation}\n\\begin{split}\n    &V^\\mathrm{rep}(r) = \\\\ &[A_0\\exp(-A_1 r) + A_2\\exp(-A_3 r)]f^\\mathrm{cut}(r) \n\\end{split}\n\\end{equation}\nwhere $A_0$, $A_1$, $A_2$, $A_3$ are the parameters and $f^\\text{cut}(r)$ is a tapering function of the form\n\\begin{equation}\n    f^\\text{cut}(r) = \\frac{1}{2}\\left(\\cos(\\frac{\\pi r}{r_\\text{cut}})+1\\right)\n\\end{equation}\nwith the cutoff distance $r_\\text{cut} = 5.67$ bohr.\n\nWith ParAMS, it is possible to either directly define the reference values for the training set (for example, from literature values) or to automatically calculate them if no reference values have been given. Here, we illustrate the second approach, and perform the reference calculations using the periodic DFT code BAND\\cite{band} in the Amsterdam Modeling Suite\\cite{ams} (AMS). \n\nThe training set comprises the $a$ and $c$ lattice parameters of wurtzite ZnO, the bulk modulus $B_0$ of wurtzite ZnO, as well as the relative energies of the wurtzite and rocksalt polymorphs of ZnO,\n$\\Delta E = E_\\text{wurtzite}-E_\\text{rocksalt}$ (per ZnO formula unit). We do not need to specify the reference values themselves, as they will automatically be calculated by ParAMS.\n\nThe job collection contains two jobs: lattice optimizations of the wurtzite and rocksalt polymorphs. From these two jobs, the $a$, $c$, $B_0$, and $\\Delta E$ quantities can be extracted. For wurtzite, $B_0$ can be extracted by requesting that the elastic tensor be calculated at the end of the lattice optimization.\n\nThe reference DFT calculations were run with the PBE exchange-correlation functional, a triple-$\\zeta$ (TZP) basis set, and ``Good'' numerical quality (dense k-space and integration grids). For the parametrized DFTB engine, a ``Good'' (dense) k-space grid was also used, since the results of lattice optimizations can be quite sensitive to the k-space grid.\n\n\nThe optimization was done with the Nelder-Mead algorithm \\cite{neldermead} from scipy, with a sum-of-squared-errors loss function. The smallest loss function value was obtained for $A_0 = 0.45$, $A_1 = 1.01$, $A_2 = 0.25$, $A_3 = 0.40$. The resulting repulsive potential is shown in Figure \\ref{fig:repulsive}. Typical Zn-O distances in wurtzite (``w\") and rocksalt (``rs\") are indicated with gray lines at 3.8 bohr and 4.1 bohr, respectively. With znorg-0-1 (red line), the repulsive potential decays very rapidly between the typical wurtzite and rocksalt distances. This decrease is not as pronounced with either the potential in this work (black line) or znopt (blue line). This affects the relative stability of the wurtzite and rocksalt ZnO polymorphs. \n\nTable \\ref{tab:repulsive} compares the resulting ZnO properties from the parameterization in this work to the DFT reference data, as well as previous DFTB ZnO parameterizations (note: znorg-0-1 was not parametrized to the DFT data in Table~\\ref{tab:repulsive}, and znopt was parametrized to also reproduce some adsorption energies). The repulsive potential in this work closely reproduces the wurtzite lattice parameters $a$ and $c$ and the relative energy $\\Delta E$, and provides a good estimate of the bulk modulus, compared to the DFT reference to which it was trained. \n\nWith ParAMS it is additionally possible to evaluate the loss function, and individual training set entries, using any of the engines in the Amsterdam Modeling Suite. For comparison, Table~\\ref{tab:repulsive} also gives the corresponding quantities for the UFF (Universal Force Field) engine, which performs significantly worse than the DFTB parameterizations for these quantities.\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{fig\/fig-repulsive.pdf}\n    \\caption{\n    ZnO parameterization data.\n    (a-b) pictures of the unit cells of ZnO wurtzite and rocksalt polymorphs. Zn is shown in blue and O in red. (c) DFTB Zn-O pairwise repulsive potentials for the znorg-0-1 \\cite{znorg} (red), znopt \\cite{znopt} (blue), and refitted repulsive potential from this work (black). The gray lines mark typical Zn-O distances in the wurtzite (w) and rocksalt (rs) polymorphs, respectively.} \n    \\label{fig:repulsive}\n\\end{figure}\n\n\\begin{table*}\n    \\centering\n    \\begin{tabular}{l|llll}\n        \n         & $a$ (\\AA) & $c$ (\\AA) & $B_0$ (GPa) & $\\Delta E$ (eV) \\\\\n         \\hline\n         DFT (this work) & 3.30 & 5.32 & 126 & $-$0.24 \\\\\n         DFTB (this work) & 3.28 & 5.34 & 146 & $-$0.24 \\\\\n         DFT \\cite{znopt} & 3.29 & 5.31 & 129 & $-$0.30 \\\\\n        \n         DFTB, znopt \\cite{znopt} & 3.21 & 5.25 & 161 & $-$0.32 \\\\\n         DFTB, znorg-0-1 \\cite{znorg} & 3.29 & 5.38 & 161 & +0.14 \\\\\n         UFF & 2.90 & 4.73 & 199 & $-$16.2 \\\\\n    \\end{tabular}\n    \\caption{Calculated ZnO wurtzite lattice constants $a$ and $c$, bulk modulus $B_0$, and energy relative to the rocksalt polymorph $\\Delta E = E_\\text{wurtzite}-E_\\text{rocsksalt}$ (per ZnO formula unit)}\n    \\label{tab:repulsive}\n\\end{table*}\n\n\n\n\\subsection{ReaxFF parameterization} \\label{sec:application}\nReaxFF is another contemporary example of an empirical model. \\cite{reax1,reax2}.\nThis formalism has been applied to a wide range of chemical problems,\nand consequently has seen a lot of new parameter development\n\\cite{reax_RDX, reax_Silica, muellerhartke}\n(for a general overview of ReaxFF and its development, see Senftle \\textit{et al.}\\cite{reax_npj}).\nIn this section, we demonstrate the parameterization of ReaxFF with a training set\npreviously published by M\u00fcller and Hartke (MH) \\cite{muellerhartke}. The optimized\nparameter vector $\\bm{x}^*$, as found by MH, is called Mue2016.\nAn overview of the training set can be found in Table S1 of the supporting information.\nIt features a total of 231 geometries needed for the computation of 4875 chemical properties. Additionally, MH included a validation set to check for overfitting.\nExamples of three structures included in the data are depicted in Fig. S1 of the supporting information, showing cyclopentathione, diphenyl disulfide and \ndimethyl disulfide.\nPrior to the parameter optimization, we evaluate\nthe training and validation sets with the Mue2016 ReaxFF parameters and\nreport sum-of-squared-errors (SSE) losses of 14441 and 14451 respectively.\nNote that in the original publication,  MH report a training set\nloss of 12400\\cite{muellerhartke}, while in a more recent work, Shchygol \\textit{et al. } calculate a loss of 16300\\cite{anna}.\nSuch differences are expected, because numerical instabilities inherent to ReaxFF and software improvements (mostly related to geometry optimization) may result in different optimized geometries. \\cite{anna, tapered_bond_orders}\n\nIn our setup, we use the covariance matrix adaptation evolution strategy (CMA-ES)\n\\cite{cma1, cma2, cma_py} as the optimization algorithm with Mue2016 as the initial point.\nCMA-ES is gradient-free, and relies on a population to sample\nnew parameter vectors from an adapted, n-dimensional Gaussian.\nIt does not require additional hyperparameters other than a population size and an initial\nwidth of the Gaussian distribution, $\\sigma$.\nHere, we use a population size of 36 and an initial $\\sigma$\nof 0.3.\nFurthermore, we limit the optimization to 24 hours and set up an \nearly stopping mechanism based on the validation set.\nThe optimization is set up to stop early only if there has been no\nimprovement in the validation set error for the last 6000 evaluations.\n\nRather than optimizing the same 87 parameters as MH,\nwe perform a one-dimensional scan on all\nparameters and select the 35 most sensitive with respect to the training set:\nAlthough Mue2016 is a set of 701 parameters in total,\nonly a subset of these significantly affects the overall cost function value.\nThis is for example the case when a model includes parameters for each chemical element (\\textit{e.g. } C, H, O),\nbut the total training set of systems $R$ can be constructed from fewer elements (\\textit{e.g. } C, H).\nIn such cases, the dimensionality of the problem can be reduced by scanning for a\nrelevant parameter subset which yields the biggest change\nin the cost function value.\nThe simplest setting, which we used in this case study, only modifies one parameter at a time to determine its influence on the objective function.\nIt is also possible to scan all parameter combinations, to discover coupling between parameters, albeit at a highly increased computational cost.\nOut of the 35 parameters selected this way, 16 have also been optimized by MH.\nWe list all optimized parameters in the provided Python notebooks\\cite{si}.\n\nParameter bounds are set to be relative to the initial values such that \n$\\bm{x}_\\pm = \\bm{x}_0 \\pm 0.2|\\bm{x}_0|$.\nIn addition to box constraints, ParAMS enables a definition of inequality constraints.\nAs the ReaxFF formalism works with bond orders, we limit the parameters responsible for the covalent radii of\n$\\sigma$, $\\pi$ and $\\pi\\pi$ bonds to\n$r_0^\\sigma \\geq r_0^\\pi \\geq r_0^{\\pi\\pi}$\nfor every atom and atom pair defined in the force field.\nThis approach effectively limits the search space and is available in combination with all optimizers.\n\nA summary of all settings is provided in Table S2 in the supporting information.\nTo compensate for the randomness of CMA-ES, we repeat the optimization\nset-up nine times.\nFor the best solution, we report improved training and validation set losses of 11877 and 5377 respectively.\nWe make this work's optimized parameter set available through the supporting information under the title MueParAMS.\nCorrelation plots between reference and predicted values for the\nnew parameters are presented in Figure \\ref{fig:corrplot}, showing very good agreement to the reference data.\nMoreover,\nFigure \\ref{fig:pes} compares the S-S dissociation curve of\ndiphenyl disulfide, as computed with Mue2016 and the new MueParAMS parameters, showing an improved agreement to the reference data for this case.\n\n\\begin{figure}\n  \\includegraphics[width=.5\\textwidth]{fig\/corrplots.png}\n  \\centering\n  \\caption{\n    Training data correlation plots. Showing\n    (a,b) energy differences, (c,d) atomic forces, (e,f) atomic distances, (g,h) interatomic angles and (i,k) dihedral angles\n    as calculated with the Mue2016 (left) and MueParAMS (right) parameters. Reaction energies and internal coordinates are compared after geometry optimization.\n    X and Y axes depict reference and predicted values respectively.\n    }\n  \\label{fig:corrplot}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=.45\\textwidth]{fig\/ssopscan.png}\n  \\caption{\n    S-S dissociation curve of diphenyl disulfide. Computed with the\n    Mue2016 and MueParAMS ReaxFF parameterizations. For details about the reference data, see Ref. \\citenum{muellerhartke}.\n    }\n  \\label{fig:pes}\n\\end{figure}\n\n\n\\section{Summary and Outlook}\nWith ParAMS, we have presented a modern Python package, supporting versatile parameterization workflows with minimal effort.\nIts integration with the Amsterdam Modeling Suite adds a high amount of\nflexibility through the number of properties that can be fitted\nalongside the support for multiple codes when it comes to the model, optimization algorithm and reference data selection.\nFeatures such as highly customizable optimizations,\nsupport for multiple validation sets, or the\nintuitive processing of data\naim to make ParAMS accessible to both, advanced and less experienced users.\nAt the same time, developers can easily extend existing functionality.\nWe showed how an SCC-DFTB repulsive potential could easily be parameterized for the inorganic crystal ZnO. The reference data was calculated automatically using a DFT engine within AMS.\nThis example application also demonstrates how ParAMS can be used\nto compare the accuracy of different chemical simulation packages given the same\ntraining set.\nWe also demonstrated how the package can be used to easily process, set up and start a fitting procedure for ReaxFF.\nUsing previously published data by M\u00fcller and Hartke,\nwe were able to find parameters that produce a considerably lower error for the validation set\nwhile maintaining a similar accuracy in the training data.\nIn future, we hope to extend the number of empirical models that can be fitted with ParAMS\nand further improve the ease of use through the introduction of additional shortcut functions for training set building.\nWe also expect additions in other functionality such as optimization algorithms or extractors\nbased on user feedback and wishes as the project matures.\nThe package is included in all AMS releases since 2020.\n\n\\textbf{Supporting Information Available}\nPDF file with mathematical description of the optimization problem, summary of the reference data published by M\u00fcller and Hartke used in the ReaxFF example, settings used for the ReaxFF parameterization and visualization of some structures in the M\u00fcller and Hartke training set.\n\n\n\\textbf{Data and Software Availability}\nAll data needed to reproduce the examples is available\nat \\url{www.doi.org\/10.5281\/zenodo.4629706}.\nThe package's documentation is available at\n\\url{www.scm.com\/doc.trunk\/params}.\nParAMS is distributed with the Amsterdam Modeling Suite, for which a free trial can be requested at \\url{www.scm.com}.\n\n\n\\begin{acknowledgement}\nWe thank Micha\u0142 Handzlik for the initial design of the \nParAMS package and Dr.\\ Tom\u00e1\u0161 Trnka for the implementation of the AMSWorker interface,\nresulting in a considerable computational speed-up.\nThis project has received funding from the European Union's Horizon\n2020 research and innovation programme under grant agreement No 814143 (L.K. \\& T.V.), No 798129 (M.H).\nT.V. also acknowledges funding of the research board of Ghent University.\nR.R. and M.H. have received funding from the Netherlands Enterprise Agency (RVO) and Stimulus under the MIT R\\&D Collaboration programme, project number PROJ-02612.\nThe computational resources (Stevin Supercomputer Infrastructure) and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by Ghent University, FWO and the Flemish Government \u2013 department EWI.\n\\end{acknowledgement}\n\n\\textbf{Author Contributions}\nL.K. and R.R. developed the ParAMS Python package.\nL.K. and M.H. performed the case studies.\nL.K., M.H. and T.V. wrote the paper.\nT.V. oversaw the project.\nAll authors read and approved the final manuscript.\n\n\\textbf{Competing Interests}\nAuthors L.K., R.R. and M.H. were employed by the company Software for Chemistry and Materials (SCM). SCM develops and commercializes the Amsterdam Modeling Suite, of which ParAMS is a new module.\n\n\\footnotesize{\n\n\\section{The Parameterization Problem}\n\nThis section provides a mathematical framework for the parameterization problem. \nWe assume that the training data can be defined as a set of physico-chemical properties for a number of isolated or periodic systems.\nExamples for relevant properties are energy differences, nuclear gradients or system geometries.\nIn the context of ParAMS, we define an arbitrary property $P$ \nthat can be expressed as the output of a computational job.\nWhen working with multiple jobs as part of a training set, a job function\ncan be defined as\n\n\\begin{equation}\n    \\label{eq:jobcollection} \n    J(R_j, S_j, M) =\n        \\left( R'_j, P_j^n\\right)\n        \\quad \\forall\n            \\; j\\in \\{1\\ldots N_\\text{job}\\},\n            \\; n\\in \\{1\\ldots N_\\text{prop}(j)\\},\n\\end{equation}\n\ncalculating the output geometry $R'_j$ and all properties $P_j^n$ of a job $j$.\nThe input for every job in $J$ consists of the input geometry $R_j$, \nthe job settings $S_j$ (\\textit{e.g. } geometry optimization and frequencies)\nand the computational model $M$.\nNote that a parametric model is additionally a function of the parameter vector $\\bm{x}$,\nin which case the outputs of the above equation can be denoted with the hat operator\n(\\textit{i.e. } $ \\hat{R}'_j, \\hat{P}_j^n$),\nas to distinguish between reference properties and properties predicted by the parametric model.\nTraining set entries can be constructed, for example, from a linear combination of multiple properties\n\\begin{align}\n    \\label{eq:dsentry}\n    y_i &=\n        \\sum_{k=1}^{N_\\text{lc}(i)} c_{i,k} P_{j(i,k)}^{n(i,k)}\n        \\quad \\forall\\; i\\in \\{1\\ldots N_\\text{data}\\},\n\\end{align}\nwhere $c_{i,k}$ is the coefficient for term $k$ of training set entry $i$ and $N_\\text{lc}(i)$ is the total number of terms\nper entry.\nNon-linear combinations of properties to construct $y_i$ are also possible.\nSuch a formulation offers a high degree of flexibility for the construction of a training set.\nOne example is the combination of multiple system energies into one reaction energy.\nIt should be noted that a training set entry, as defined in Eq.\\ \\ref{eq:dsentry},\ndoes not have to originate from the results of computational jobs.\nThe reference value can instead be provided directly, making it easy to work with experimental or external data.\n\nWhile training set entries $\\bm{y}$ have to be defined only once,\ntheir predicted counterpart $\\bm{\\hat{y}}$ has to be re-calculated every time the model parameters\nchange.\nFor this purpose, we introduce a Data Set function\n\\begin{equation}\n    \\label{eq:dataset}\n    DS(\\bm{x} | \\bm{y}) = \\bm{y} - \\hat{\\bm{y}},\n\\end{equation}\nwhich extracts all properties needed for the calculation of $\\bm{\\hat{y}}$\nbased on a parameter set $\\bm{x}$ and returns the respective vector of residuals.\nA metric in the form of a loss function $L(\\bm{y} - \\hat{\\bm{y}}, \\bm{w})$\nis then applied to the residuals for a qualitative measure of how close\nreference and predicted values  are.\nThe additional weights vector vector $\\bm{w}$ can be used to balance\npossibly different orders of magnitude in the data set or\nmake certain entries more relevant for the fitting process than others.\n\nFinally, the optimization algorithm can be defined as a function that minimizes $L$ with respect to the parameters\n\\begin{equation}\n    \\label{eq:optimizer}\n    O(\\bm{x}_0, L) = \n    \\underset{\\bm{x}} {\\mathop{\\mathrm{arg\\,min}}}\\,\n        L = \\bm{x}^*,\n\\end{equation}\nfinding an optimal solution $\\bm{x}^*$ from an initial point $\\bm{x}_0$.\n\n\n\n\n\n\\section{Additional Display Items}\n\\begin{table}\n\\centering\n\\caption{\nComposition of the reference data published by M\\\"uller and Hartke\\cite{muellerhartke},\nsplit by the computational tasks Single Point (SP) and Geometry Optimization (GO).\nFor each of the two sets, the upper part describes the chemical systems,\nwhile the lower breaks down the individual entries in the training and validation sets.\nNote that some entries might be a function of multiple chemical systems,\nmeaning that the sum of SP+GO is not necessarily equal to the total\nnumber of entries for that row (\\textit{cf. } Sec.\\ 3.1 in the main text).\n}\n\\begin{tabular}{lrrr}\n\n    \\textbf{Training Set}&\n    \\textbf{SP}&\n    \\textbf{GO}&\n    \\textbf{Total}\n    \\\\\n    \\hline\n    \n    Number of systems&\n    222&\n    9&\n    231\n    \\\\\n    \n    Mean system size (atoms)&\n    6.6&\n    11.4&\n    6.8\n    \\\\\n    \n    Std. dev. (atoms)&\n    2.9&\n    7.7&\n    3.3\n    \\\\\n    \n    \n    \\\\\n    Total number of entries&\n    4620&\n    317&\n    4875\n    \\\\\n    \n    Energies&\n    219&\n    62&\n    219\n    \\\\\n    \n    Forces&\n    4401&\n    0&\n    4401\n    \\\\\n    \n    Atomic distances&\n    0&\n    94&\n    94\n    \\\\\n    \n    Angles&\n    0&\n    85&\n    85\n    \\\\\n    \n    Dihedrals&\n    0&\n    76&\n    76\n    \\\\\n   \n\n    \\\\\n    \\textbf{Validation Set}&\n    &\n    &\n    \\\\\n    \\hline\n    \n    Number of systems&\n    200&\n    24&\n    224\n    \\\\\n    \n    Mean system size (atoms)&\n    24.0&\n    12.7&\n    22.8\n    \\\\\n    \n    Std. dev. (atoms)&\n    0.0&\n    5.9&\n    4.0\n    \\\\\n\n    \n    \\\\    \n    Total number of entries&\n    199&\n    771&\n    970\n    \\\\\n    \n    Energies&\n    199&\n    0&\n    199\n    \\\\\n    \n    Forces&\n    &\n    &\n    0\n    \\\\\n    \n    Atomic distances&\n    0&\n    281&\n    281\n    \\\\\n    \n    Angles&\n    0&\n    257&\n    257\n    \\\\\n    \n    Dihedrals&\n    0&\n    233&\n    233\n\\label{tab:trainingset}\n\\end{tabular}\n\\end{table}\n\n\n\n\\begin{table}\n\\centering\n\\caption{Summary of relevant ParAMS settings used for the re-parameterization of Mue2016.}\n\\begin{tabular}{lr}\n    \\textbf{Setting}&\n    \\textbf{Value}\n    \\\\\n    \\hline\n\n    Number of optimizations&\n    9\n    \\\\\n    \n    Number of parameters to optimize&\n    35\n    \\\\\n    \n    Lower \/ upper parameter bounds&\n    $\\bm{x}_0 \\pm 0.2|\\bm{x}_0|$\n    \\\\\n    \n    Optimization timeout&\n    24 hours\n    \\\\\n    \n    CMA-ES population size&\n    36\n    \\\\\n    \n    CMA-ES sigma&\n    0.3\n    \\\\\n    \n    Loss function&\n    sum of squared errors\n    \\\\\n    \n    Early stopping patience&\n    6000 evaluations\n    \\\\\n    \n    Constraints&\n    $r_0^\\sigma \\geq r_0^\\pi$\n    and\n    $r_0^\\pi \\geq r_0^{\\pi\\pi}$\n\\label{tab:params_setup}\n\\end{tabular}\n\\end{table}\n\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=.45\\textwidth]{fig\/compounds.pdf}\n  \\caption{\n    From top to bottom: Example structures of\n    cyclopentathione, diphenyl disulfide and dimethyl disulfide, \n    containing S (yellow), C (black), and H (white),\n    included in the data provided by MH.\n    The fitted properties include bond distances, angles, relative\n    energies and atomic forces.\n    }\n  \\label{fig:compounds}\n\\end{figure}\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nMilky Way (MW) satellite galaxies have long anchored our understanding of low-mass mass galaxy formation.  Their number counts, spatial distributions, and structural properties are used to constrain dark matter cosmology on small scales \\citep[e.g.,][]{bullock2017, simon2019}.  Their star formation histories (SFHs) and chemical content provide insight into cosmic reionization and the baryonic processes that uniquely shape low-mass galaxy evolution \\citep[e.g.,][]{kirby2011, brown2014, weisz2014a}. More recently, their orbital histories, as measured by the Hubble Space Telescope (HST) and Gaia, reveal the complex effects of central galaxies on the evolution of low-mass satellites \\citep[e.g.,][]{sohn2012, fritz2018, simon2018}.\n\nAt the same time, there is growing evidence that the MW satellites may not be representative of low-mass satellites in general. Compared to the MW, satellite systems throughout the local Universe show varying luminosity functions, stellar populations, quenching properties, and spatial configurations, often in excess of cosmic variance \\citep[e.g.,][]{mcconnachie2006, brasseur2011, Tollerud:2011wd, geha2017, muller2018, pawlowski2018, smercina2018, pawlowski2019}. Even in our nearest neighbor, M31, there are hints that the internal \\citep[e.g., kinematics, stellar content;][]{dacosta1996, collins2010, collins2014, martin2017} and global \\citep[e.g., ``plane of satellites'';][]{ibata2013, pawlowski2018} properties of M31 and MW satellites are different. Thus, it is unclear whether the fundamental insights established in MW satellites are applicable to all low-mass systems or stem from the specific accretion history of the MW.\n\nIn this Letter, we present the first uniform SFH measurements of many faint M31 satellites.  We use sub-horizontal branch (HB) HST imaging, which has been presented in two previous papers in this series \\citep{martin2017, weisz2019a}, to measure their SFHs, and we compare our measurements to literature SFHs of MW satellites.   Relative to the MW satellites, the SFHs we measure for M31 satellites have coarser age resolution, because the M31 CMDs do not reach the oldest main sequence turn off (MSTO).  We will acquire oldest MSTO photometry for all known M31 satellites during HST Cycle 27 as part of HST-GO-15902 (PI D.~Weisz), which will strengthen the results we preview in this Letter.\n\n\n\\begin{figure*}[ht!]\n\\begin{center}\n\\includegraphics[scale=0.65]{m31_example_fit.png}\n\\caption{An example BASTI-based SFH measurement using And~XXI.  Panel (a) shows the observed Hess diagram. Panel (b) shows the best fit model Hess diagram for the BaSTI stellar evolution models. Panel (c) shows the residual significance diagram, i.e., (data-model)\/model, in units of standard deviation.  The lack of systematic structures in Panel (c), i.e., large contiguous regions of black or white points, indicates a good fit.  Panel (d) shows the cumulative SFH, i.e., the fraction of stars formed before a given epoch.  The solid black line is the best fit SFH for the BaSTI models and the grey shaded envelope are the total (random plus systematic) uncertainties for the 68\\% confidence interval.  The thin colored lines are the best fit SFHs for different stellar evolution models.  Overall, the shape of the SFH is similar between the models, but the ages of SFH features can shift by $\\sim 2$~Gyr owing to differences in the underlying stellar physics.  The grey shaded region captures the 1-$\\sigma$ scatter in the different SFHs reasonably well.}\n\\label{fig:4panel}\n\\end{center}\n\\end{figure*}\n\n\\section{The Data} \\label{sec:data}\n\nThe observations and data reduction used in this program are described in detail in \\citet{martin2017}. Here, we provide a brief summary.  \n\nThrough HST-GO-13699 (PI N.~Martin), we observed 16 faint M31 satellites with the Advanced Camera for Surveys (ACS) \n\nthat had no previous HST imaging. Each galaxy was observed for a single orbit with equal integration times in the F606W and F814W filters.  We added archival F606W and F814W ACS data of comparable depth for And~XVIII (HST-SNAP-13442; PI B.~Tully).  We also added archival observations of And~XI, And~XII, and And~XIII (HST-GO-11084; PI D.~Zucker) which were taken in F606W and F814W with the Wide Field Planetary Camera 2 (WFPC2).\n  \nIn total, our sample has 20 systems with $-12 \\lesssim M_V \\lesssim -6$, including 7 ultra-faint dwarf galaxies (UFDs, $M_{V} > -7.7$; \\citealt{simon2019}).\n\nFor each galaxy, we performed point spread function photometry with \\texttt{DOLPHOT}, a widely-used package for reducing observations of resolved stellar populations with HST-specific modules \\citep{dolphin2000b}. We adopted the \\texttt{DOLPHOT} parameters recommended in \\citet{williams2014}. The raw photometric catalogs were culled to include only good stars as described in \\citet{martin2017}.  We ran $\\sim 10^5$ artificial star tests (ASTs) for each galaxy to quantify completeness and photometric uncertainties.  The 50\\% completeness limit for a typical galaxy in our sample is $F606W \\sim 27.1$ and $F814W \\sim 26.2$.\n\nFigure \\ref{fig:4panel} illustrates the quality of our data by showing a Hess diagram, i.e., a binned CMD, for And~XXI ($M_{V} = -9.2$). The CMD shows a clear red giant branch (RGB), red clump (RC), and a predominantly red HB, as described in \\citet{martin2017}.  The faint limit of the CMD is set to the 50\\% completeness limit of $F814W =26.3$, which is $\\sim$1.5 magnitudes fainter than the HB. \n\n\\section{Methodology} \\label{sec:methods}\n\nWe measure the SFHs of the 20 galaxies in our sample using \\texttt{MATCH} \\citep{dolphin2002}, a software package that forward models the CMD of a galaxy in order to measure its SFH as described in \\citet{weisz2014a}.  Here, we provide a brief summary of the pertinent details.  \n\n\nFor this analysis, we adopt a Kroupa IMF \\citep{kroupa2001}, a binary fraction of 0.35, HB-based distances for the ACS data \\citep{weisz2019a} and self-consistent TRGB-based distances for the WFPC2 data \\citep{weisz2014a}, and foreground extinction values from \\citet{schlafly2011}.  \n\nWe fit each entire CMD with five different stellar evolution libraries: Dartmouth \\citep{dotter2008}, Padova \\citep{girardi2010}, PARSEC \\citep{bressan2012}, MIST \\citep{choi2016}, and BaSTI \\citep{hidalgo2018}. We find that for all CMDs analyzed in this paper the BaSTI 2018 models provide the best overall fits in terms of visual inspection of the residuals and through comparison of likelihood ratios between models.  Thus, we adopt the BaSTI models for this paper.  \n\nWe adopt a metallicity grid that ranges from $-2.3 \\le$ [M\/H] $\\le -0.5$ with a resolution of 0.1 dex and an age grid that ranges from $9.00 \\le \\log(t\/{\\rm yr}) \\le 10.15$ with a resolution of $\\log(t\/{\\rm yr}) = 0.05$ dex. We find that including ages younger than $\\log(t\/{\\rm yr}) =9.0$ did not change the SFHs (as these galaxies have no young populations) but increased the computational time. Thus, we simply exclude ages with $\\log(t\/{\\rm yr}) < 9.0$ from our CMD modeling.\n\nFinally, given that our CMDs do not reach the oldest MSTO, we follow \\citet{weisz2011a} and\n\\citet{weisz2014a} and adopt a prior on the age-metallicity relationship that requires the metallicity to increase monotonically with time, with a modest dispersion allowed at each age.  This choice helps to mitigate some of the age-metallicity degeneracy on the RGB and RC.   We compute random (which account for the finite number of stars on the a CMD) and systematic uncertainties (which are a proxy for uncertainties in the physics of the underlying stellar models) as described in \\citet{weisz2014a}.  Finally, as detailed in \\S 3.6 and Figure 6 of \\citet{weisz2014a}, SFHs measured from CMDs that include the HB but do not include the oldest MSTO have larger systematic uncertainties, but are consistent with SFHs measured from oldest MSTO-depth CMDs.\n\n\nFigure \\ref{fig:4panel} illustrates the CMD modeling process.  Panel (a) shows the observed Hess diagram of And~XXI and panel (b) shows the best fit model for the BaSTI stellar library.  A visual comparison of the model and data indicates good overall agreement.  Panel (c) is a residual significance diagram, i.e., (model-data)\/model, which quantifies the level of (dis)agreement.  The majority of populated pixels are consistent within $\\sim$ 1-$\\sigma$, while only a handful of pixels are highly discrepant (i.e., $>$ 3-$\\sigma$). There are no signs of poorly fit regions of the CMD (e.g., large swaths of only black or white pixels), which indicates that the model is a good fit to the data.\n\n\\begin{figure*}[ht!]\n\\begin{center}\n\\includegraphics[scale=0.7]{m31_shallow_basti_sfhs_v2.png}\n\\caption{The cumulative SFHs of 20 faint M31 satellites ordered by increasing luminosity.  The black solid line is the best fit BaSTI SFH.  The grey and purple shaded envelopes reflect the 68\\% confidence intervals for the random and total uncertainties, respectively. Fainter M31 satellites generally form the bulk of their stellar mass at earlier time compared to the brighter systems. All galaxies appear to have quenching times between 3 and 9 Gyr ago.    \\label{fig:sfhs}}\n\\end{center}\n\\end{figure*}\n\nPanel (d) shows the cumulative SFH of And~XXI, i.e., the fraction of total stellar mass formed prior to a given epoch.  The solid black line is the best fit BaSTI solution and the grey shaded band reflects the 68\\% confidence interval of the total (i.e., random plus systematic) uncertainties. The random uncertainties are negligibly small compared to the systematic uncertainties.  \n\nThe thin colored lines in panel (d) are the best fit SFHs from the other four stellar libraries.  These SFHs are similar in shape to the BaSTI solution, though particlar features can be shifted by up to $\\sim 2$ Gyr, which is due to differences in the underlying stellar physics \\citep[e.g.,][]{Gallart:2005qy}.  As intended, the scatter in the SFHs from different models is well-captured by the grey-shaded error envelope. \n\nThe small amount of apparent star formation $\\sim$ 1-3 Gyr ago is likely due to a handful of blue stragglers which occupy similar portions of the CMD as younger main sequence stars \\citep[e.g.,][]{monelli2012}.  \n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[scale=0.7]{m31_mw_t50_t90_all_lines.png}\n\\caption{The lookback time (Gyr ago) at which 50\\% of the stellar mass formed ($\\tau_{50}$) versus the time at which 90\\% of the stellar mass formed ($\\tau_{90}$), i.e., the quenching time.  The top panel includes the 20 M31 satellites from this paper and 6 from \\citet{skillman2017}.  Points are color-coded by luminosity and their relative sizes reflect their half-light radii.  The grey point indicates a size of 500~pc.  The black lines illustrate a constant and exponentially declining SFHs.  The bottom panels shows results from literature SFHs of MW satellites.  The area enclosed by the blue dotted line contains half the M31 sample, but no MW satellites. The smaller uncertainties for the \\citet{skillman2017} M31 dSphs are indicative of what can be expected from the forthcoming cycle 27 observations. \\label{fig:t50_t90}}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\\section{Results and Discussion} \\label{sec:discussion}\n\nFigure \\ref{fig:sfhs} shows the cumulative SFHs of 20 faint M31 satellites plotted in order of increasing luminosity from upper left to lower right.  The solid black lines are the best fit SFHs, while the grey and purple shaded envelopes reflect the 68\\% confidence intervals for the random and total (random plus systematic) uncertainties, respectively.  These values are tabulated in Table \\ref{tab:sfhs}.\n\nThis figure reveals both a diversity of SFHs among the M31 faint satellite population and some broad trends.  For example,  galaxies with $M_V \\gtrsim -8.5$ tend to have formed $\\gtrsim 50$\\% of their stars prior to $\\sim 10-12$~Gyr ago, compared to 6-9~Gyr ago for more luminous systems.  Interestingly, galaxies such as And~XXIX, Per~I, and Lac~I appear to have formed $\\lesssim$ 10\\% of their stellar mass prior to $\\sim 10$~Gyr ago, which is unusually low when put into context with our knowledge of LG dwarf galaxy SFHs \\citep[e.g.,][]{weisz2014a, gallart2015}. \n\nAnother interesting feature is the quenching times.  That is, very few systems have either very early ($>$10-12 Gyr ago) or very late ($\\lesssim 3$~Gyr ago) quenching times.  Instead, the vast majority of the systems stopped forming stars $\\sim 3-6$ Gyr ago, almost independent of luminosity.  \n\nThe top panel of Figure \\ref{fig:t50_t90} consolidates the SFHs of the 20 faint M31 satellites from this paper and the 6 systems from \\citet{skillman2017} into a more digestible form. Here, we plot the quenching time\\footnote{Following the literature \\citep[e.g.,][]{weisz2015a, skillman2017}, we adopt the time at which 90\\% of the total stellar mass formed as a proxy for the quenching time to avoid ambiguity due to blue stragglers.} ($\\tau_{90}$) versus the time at which 50\\% of the total stellar mass formed ($\\tau_{50}$).  Points are color-coded by luminosity, point sizes are proportional to half-light radius, and the error bars reflect the total (random plus systematic) uncertainties.  To guide the eye, we overplot black lines that illustrate cases of a constant (solid) and exponentially declining SFHs ($\\tau_{SFH}=10$~Gyr, dashed lined; $\\tau_{SFH}=2$~Gyr, dot-dashed lined).  \n\nThis plot shows several interesting trends.  First, although there are several predominantly ancient galaxies (i.e., $\\tau_{50}$ $>12$~Gyr), there are very few systems with $\\tau_{90}$ $>12$~Gyr. Instead, the predominantly ancient systems have a range of $\\tau_{90}$\\ values that extend from 3 to 10 Gyr ago.  This is particularly interesting in the context of reionization, in which the prevailing view is that the lowest-mass galaxies have star formation shutdown by reionization in the very early Universe \\citep[e.g.,][]{ bullock2000, benson2002, ricotti2005}.  However, we caution against over-interpretation of this finding, as (i) we lack a complete census of UFDs around M31 and (ii) current SFHs are uncertain at the oldest ages.\n\nSecond, there are no galaxies that quenched within the last 3 Gyr, i.e., $\\tau_{90}$\\ $<3$~Gyr ago.  Instead, most galaxies have quenching times concentrated at intermediate ages, with a notable grouping at $\\tau_{90}$$\\sim$ 3-6~Gyr ago.  \n\nThird, there appears to be some degree of synchronicity in the quenching of some systems.  Most notably, the over-density of M31 satellites at $\\tau_{50}$ $\\sim 6-9$~Gyr ago also all seem to have $\\tau_{90}$\\ $\\sim 3-6$~Gyr ago, with no clear trends in galaxy size or luminosity, in agreement with the results of the representative sample from \\citet{skillman2017}.   \n\n\n\n\nFourth, compared to the over-plotted fiducial SFH models (i.e., constant SFH, exponentially declining) there are two distinct groups. Eight galaxies fall tightly on the $\\tau_{SFH}=2$~Gyr track while 15 are to the right of the constant SFH track (i.e., they have rising SFHs). Only three galaxies exist between these groups.  \n\n\n\nThe age information provided by our SFHs may help model the complex formation and accretion history of the M31 and its satellites.  The M31 halo hosts rich stellar substructures (e.g., streams, over-densities) that suggest an active history of mergers in M31 (see \\citealt{mcconnachie2018} and references therein) and its stellar halo and outer disk have large populations of intermediate age stars, as revealed by oldest MSTO-depth CMD analysis \\citep[e.g.,][]{brown2006a, bernard2015b}.  Several models have posited a major merger between M31 and (what would have been) the third largest member of the LG $\\sim$2-4 Gyr ago \\citep[e.g.,][]{hammer2018, dsouza2018}.  These models can qualitatively explain some observed features of M31, such as a global burst of star formation 2-4 Gyr ago \\citep[e.g.,][]{bernard2015b, williams2017} and the metal-rich inner halo.  For example, \\citet{dsouza2018} hypothesize such an interaction between M31 and M32p (the putative progenitor of M32) could explain the metal-rich component of M31's halo and the unusually compact nature of M32.  This model implies that M32p had a $M_{\\star} \\sim 2.5\\times 10^{10}$ $M_{\\odot}$ prior to its interaction with M31, making it the third largest member of the LG just a few Gyr ago.  One implication of this scenario may be that the large number of satellites with $\\tau_{90}$ $\\sim$ 3-6~Gyr ago may have been environmentally quenched during the merger. A second speculative angle is that the dichotomy of SFHs in the top panel of Figure \\ref{fig:t50_t90}, i.e., rising SFHs vs exponentially declining, may be due to the presence of two difference satellite populations, i.e., one set from M31, the other from M32p.  Though speculative, we use these examples to illustrate the potential of our data for deciphering the formation history of M31's halo and emphasize that more rigorous analysis is clearly warranted.\n\n\n\nWe also consider the relationship between our SFHs and sub-structures in the M31 system, e.g., the plane of satellites from \\citet{ibata2013}.  We find no clear evidence for a correlation with membership in structures  identified in \\citet{ibata2013} and \\citet{santossantos2019}.  However, given the large uncertainties and unclear theoretical expectations between sub-structures and SFHs, the lack of a clear correlation is challenging to interpret.\n\nFigure \\ref{fig:t50_t90} also summarizes differences in the formation histories of M31 and MW satellites.  In the bottom panel, we plot $\\tau_{90}$\\ \\emph{vs.} $\\tau_{50}$\\ for the MW satellites using literature SFHs \\citep[e.g.,][]{brown2014, weisz2015a} and global properties \\citet[e.g., luminosity, size;][]{mcconnachie2012}.  \n\nIt is striking that the M31 and MW satellite populations do not share many similar trends.  The M31 satellites fill out intermediate values of $\\tau_{50}$\\ and $\\tau_{90}$, i.e., $6 \\lesssim$ $\\tau_{50}$\\ $\\lesssim 12$~Gyr ago and $3 \\lesssim$ $\\tau_{90}$\\ $\\lesssim 6$~Gyr ago (the dotted blue box in Figure \\ref{fig:t50_t90}, whereas there are essentially no MW satellites in that range.  In terms of quenching, the MW satellites Fornax, Carina, and Leo~I (galaxies located in the upper right region of the lower panel) all ceased star formation within the most recent $\\sim 1-3$~Gyr, whereas none of the M31 satellites did. The faintest MW satellites ($M_V \\gtrsim -7$) all quenched $\\gtrsim 12$~Gyr ago, presumably due to reionization \\cite[e.g.,][]{brown2014}, but some of the comparably faint M31 satellites appear to have more extended SFHs. \n\nThis may indicate that the evolution of the satellites are coupled to the accretion history of the host galaxy.  By extension, it may be that the MW satellites do not cover the full range of intrinsic formation histories of low-mass galaxies.\n\n\\begin{deluxetable}{lcccc}\n\\tabletypesize{\\footnotesize}\n\\tablecaption{Summary statistics for SFHs of faint M31 satellites. Values of $\\tau_{50}$\\ and $\\tau_{90}$\\ are from the best fit SFHs measured in this paper.  Error bars are the 68\\% confidence intervals for the total uncertainties (i.e., random plus systematic).  Values for the galaxies below the horizontal lines are taken directly from \\citet{skillman2017}. }\n\\label{tab:sfhs}\n\\tablehead{\n\\colhead{Name} & \\colhead{$M_V$} & \n\\colhead{$r_h$} & \\colhead{$\\tau_{50}$} & \n\\colhead{$\\tau_{90}$} \\\\ \n\\colhead{} & \\colhead{(mag)} & \n\\colhead{(pc)} & \\colhead{(Gyr ago)} & \n\\colhead{(Gyr ago)} \\\\\n\\colhead{(1)} & \\colhead{(2)} & \n\\colhead{(3)} & \\colhead{(4)} & \n\\colhead{(5)} \n} \n\\startdata\nCas~III & $-12.6$ & 1640 & 7.9$_{-1.6}^{+1.6}$ & 4.1$_{-1.5}^{+2.5}$\\\\\nLac~I & $-11.5$ & 967 & 8.1$_{-1.7}^{+0.8}$ & 4.9$_{-1.7}^{+1.7}$\\\\\nCas~II & $-11.2$ & 275 & 9.8$_{-1.1}^{+3.4}$ & 7.2$_{-3.4}^{+2.8}$\\\\\nPer~I & $-10.2$ & 384 & 7.9$_{-1.8}^{+1.4}$ & 4.0$_{-1.6}^{+2.6}$\\\\\nAnd~XXIII & $-10.0$ & $1277$ & 6.8$_{-0.9}^{+1.9}$ & 5.1$_{-2.8}^{+1.5}$\\\\\nAnd~XXV & $-9.3$ & 679 & 8.7$_{-1.1}^{+2.8}$ & 5.8$_{-1.3}^{+2.6}$\\\\\nAnd~XXI & $-9.2$ & 1033 & 8.3$_{-1.9}^{+1.2}$ & 5.8$_{-2.5}^{+0.9}$\\\\\nAnd~XVIII & $-9.2$ & 262 & 8.5$_{-1.6}^{+2.0}$ & 4.6$_{-2.1}^{+1.7}$\\\\\nAnd~IX & $-9.0$ & 444 & 7.2$_{-0.3}^{+2.5}$ & 5.1$_{-2.0}^{+1.8}$\\\\\nAnd~XIV & $-8.6$ & 379 & 8.7$_{-0.4}^{+4.5}$ & 4.8$_{-0.7}^{+5.2}$\\\\\nAnd~XXIX & $-8.5$ & 397 & 7.6$_{-0.7}^{+3.1}$ & 5.2$_{-1.2}^{+2.2}$\\\\\nAnd~XVII & $-8.2$ & $339$ & 13.2$_{-0.3}^{+0.0}$ & 10.5$_{-5.0}^{+2.1}$\\\\\nAnd~XXIV & $-7.9$ & $579$ & 12.9$_{-3.3}^{+0.3}$ & 5.4$_{-3.1}^{+4.4}$\\\\\nAnd~X & $-7.5$ & 239 & 9.5$_{-0.2}^{+3.6}$ & 6.5$_{-2.1}^{+4.8}$\\\\\nAnd~XII & $-7.0$ & 420 & 12.9$_{-6.3}^{+0.3}$ & 3.4$_{-0.2}^{+2.6}$\\\\\nAnd~XXII & $-6.7$ & 253 & 11.5$_{-0.8}^{+2.0}$ & 6.8$_{-2.5}^{+5.8}$\\\\\nAnd~XX & $-6.7$ & 110 & 10.2$_{-0.9}^{+2.6}$ & 6.9$_{-2.1}^{+4.6}$\\\\\nAnd~XIII & $-6.5$ & $130$ & 9.1$_{-0.4}^{+4.1}$ & 6.5$_{-0.1}^{+3.5}$\\\\\nAnd~XI & $-6.3$ & 120 & 13.2$_{-0.3}^{+0.0}$ & 7.4$_{-1.4}^{+2.4}$\\\\\nAnd~XXVI & $-6.1$ & 228 & 12.9$_{-0.3}^{+0.3}$ & 9.1$_{-6.0}^{+2.9}$\\\\\n\\hline \n\\hline\nAnd~II & $-12.6$ & 965 & 11.5$_{-2.1}^{+0.8}$ & 6.3$_{-0.6}^{+0.5}$\\\\\nAnd~I & $-12.0$ & 815 & 12.6$_{-3.9}^{+0.3}$ & 7.4$_{-0.7}^{+0.9}$ \\\\\nAnd~III & $-10.1$ & 405 & 11.7$_{-1.3}^{+1.1}$ & 8.7$_{-0.6}^{+1.5}$\\\\\nAnd~XXVIII & $-8.8$ & 265 & 12.6$_{-0.8}^{+0.3}$ & 7.6$_{-0.3}^{+1.7}$\\\\\nAnd~XV & $-8.4$ & 230 & 12.9$_{-0.9}^{+0.3}$ & 9.3$_{-0.8}^{+3.3}$\\\\\nAnd~XVI & $-7.5$ & 130 & 9.8$_{-1.1}^{+1.4}$ & 5.9$_{-0.6}^{+0.4}$\n\\enddata\n\\end{deluxetable}\n\nThere are several caveats with the present analysis. First, while SFHs from MW satellites are all measured from CMDs that reach the oldest MSTO, our new M31 data are much shallower.  Consequently, we are left with large uncertainties that may hide various trends in the data. Moreover, our SFHs are based primarily on the HB morphology, which is not as well an understood phase of stellar evolution as the MSTO \\citep[e.g.,][]{Gallart:2005qy}. However, we note that comparisons of SFHs measured from different depths (i.e., HB vs. oldest MSTO) generally show good agreement \\citep[e.g.,][]{weisz2014a} as previously described. We urge appropriate caution against over-interpreting this generation of M31 SFHs, particularly at ancient epochs.  \nSecond, there are various selection effects that we have not explicitly considered.  One is the size of the HST field of view relative to the size of a galaxy.  In some cases, this can lead to $\\sim 1$~Gyr biases in the measured SFH relative to the true global SFH \\citep[e.g.,][]{graus2019b}.  Another is the lack of many known UFDs in the M31 ecosystem.  Detecting faint systems is quite challenging at the distance of M31, given the paucity of bright stars and the high level of contamination.  \n\nDespite these challenges, we are optimistic about prospects of placing the M31 satellites onto equal observational footing with their MW counterparts.  The 244 orbit cycle 27 HST Treasury program (HST-GO-15902; PI D.~Weisz) will obtain MSTO depth imaging across the entire M31 satellite system, which will significantly reduce the uncertainties on the SFHs and establish a first epoch for proper motion measurements. \n\nIn our view, among the most important next steps for M31 satellites is to identify UFDs around M31. If UFDs around M31 are found to have substantially extended SFHs, then our picture of how reionization affects low-mass galaxy formation, halo occupation, etc.\\ may fundamentally change.  Finding and characterizing UFDs around M31 requires dedicated imaging and spectroscopic efforts, as well as the power of HST and\/or JWST for measuring their star formation and orbital histories.  \n\n\\section*{Acknowledgements}\n\nSupport for HST program GO-13699 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. These observations are associated with program HST-SNAP-13442 and HST-GO-13699. DRW acknowledges support from an Alfred P. Sloan Fellowship and an Alexander von Humboldt Fellowship. SMA is supported by the National Science Foundation Graduate Research Fellowship under Grant DGE 1752814. This research has made use of the NASA\/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. \n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{{Introduction \\& Related Work}}\n  The spatial degrees of freedom offered by multiple antenna arrays are a valuable interference mitigation resource.  Advanced signal processing techniques are currently  employed to boost the performance of the multi-antenna transmitters without compromising the complexity of single antenna receivers.\nThese   beamforming (or equivalently precoding) techniques efficiently manage the co-channel interferences to achieve the targeted service requirements  (Quality of Service--$\\mathrm{QoS}$ targets). As a result,   the available spectrum can be aggressively reused towards increasing the system throughput.\n\nThe optimal downlink transmission strategy in the sense of minimizing the total transmit power whilst guaranteing specific $\\mathrm{QoS}$ targets at each user,    was given in \\cite{Bengtsson2001,bengtsson1999}.\n{Therein, the  tool of Semi-Definite Relaxation ($\\mathrm{SDR}$) reduced the non-convex quadratically constrained quadratic  problem ($\\mathrm{QCQP}$)  into a relaxed semi-definite programming  instance by changing the optimization variables and disregarding the unit-rank constraints over the new variable. The solution of the relaxed problem was proven to be optimal.  {The multiuser downlink beamforming problem in terms of maximizing the minimum $\\mathrm{SINR}$,  was optimally solved in \\cite{Schubert2004}. The goal of the later  formulation is to increase the fairness of the system by boosting the  $\\mathrm{SINR}$ of the  user that is further away from a targeted performance. Hence,  the problem is commonly referred to as \\textit{max--min fair}.  In \\cite{Schubert2004}, this problem was solved using the principles of uplink\/downlink duality. Therein,  \\textit{Schubert and Boche} developed a strongly convergent iterative alternating optimization algorithm for the equivalent uplink problem.   In the same work, the power minimization problem of \\cite{Bengtsson2001} was also solved  by acknowledging its inherent connection with the max-min fair problem. Consequently, a significantly less complex  framework to solve the optimal beamforming problem was established. }Extending these works, the practical per-antenna  power constraints $\\mathrm{  (PAC)}$ were considered  in \\cite{Yu2007}. {Generalized power constraints, including sum power, per-antenna power and per-antenna array power constraints were considered in \\cite{Dartmann2013}, where the proposed max-min fair solution was  derived on an  extended duality framework. This framework accounted for both instantaneous and long term channel state information ($\\mathrm{CSI}$).} $\\mathrm{PAC}$s are motivated from the practical implementation of systems that rely on precoding. The lack of flexibility in sharing energy resources amongst the antennas of the transmitter is usually the case, since a  common practice in multi-antenna systems is the use of individual amplifiers per  antenna. Despite the fact that flexible amplifiers could be incorporated in multi-antenna transmitters, specific communication systems cannot afford this design. Typical per antenna power limited systems can be found in  multibeam satellite communications \\cite{Christopoulos2013AIAA}, where flexible on board payloads are difficult to implement and in cooperative  multicell systems (also known  as distributed antenna systems, $\\mathrm{DAS}$), where the  physical co-location of the transmitting elements is not a requisite and hence power sharing might be infeasible.\n\n\nA fundamental  consideration of the aforementioned works is that  independent  data is addressed to multiple users. However, the new generation of multi-antenna communication standards has to adapt  the physical layer design to the needs of the higher network  layers.  Examples of  such cases include  highly demanding applications (e.g. video broadcasting) that stretch the throughput limits  of multiuser broadband systems.\nIn this direction, physical layer ($\\mathrm{PHY}$) multicasting\nhas the potential to efficiently address the nature of future traffic demand and  has become part of the new generation of communication standards. $\\mathrm{PHY}$ multicasting is also relevant for the application of beamforming  without changing the framing structure of standards.{Such a scenario can be found in satellite communications where the communication standards are  optimized to cope with  long propagation delays and guarantee scheduling efficiency by framing multiple users per transmission \\cite{Christopoulos2013AIAA,Christopoulos2014_ASMS}.}\n\nIn \\cite{Sidiropoulos2006},   the NP-hard multicast problem was accurately approximated by $\\mathrm{SDR}$ and Gaussian randomization.\nThe natural  extension of the multicast concept lies in assuming multiple interfering  groups of users.\nA unified framework for physical layer multicasting to multiple co-channel groups, where independent sets of common data are transmitted to  groups of users by the multiple antennas, was given in \\cite{Karipidis2005CAMSAP,Karipidis2008}. Therein, the  $\\mathrm{QoS}$ and the fairness problems were formulated, proven NP-hard and solved for  the sum power constrained multicast multigroup case.  In parallel to \\cite{Karipidis2005CAMSAP}, the independent work   of \\cite{Gao2005}  involved complex dirty paper coding methods. Also, a convex approximation method was proposed in \\cite{Pesavento2012b} that exhibits superior performance as the number of users per group grows. Finally, in \\cite{Silva2009}\nthe multicast multigroup problem under $\\mathrm{SPC}$, was solved based on approximations and uplink-downlink duality \\cite{Schubert2004}.\n{In the context of coordinated multicast multicell systems{\\footnote{{ Coordinated multicell networks consist of connected base stations ($\\mathrm{BS}$), with each $\\mathrm{BS}$ serving a single multicast group, a case tackled in \\cite{Xiang2013}. Extending this,  the methods presented herein can be applied in cooperative multicell  systems where all $\\mathrm{BS}$s will jointly transmit to several multicast groups\\cite{Chatzinotas_JWCOM}.}}},  max--min fair beamforming with per base-station ($\\mathrm{BS}$) constraints has been considered in \\cite{Xiang2013} where  each $\\mathrm{BS}$ transmits to a single multicast group.   Hence, a power constraint over each precoder was imposed while  no optimization weights were considered.This formulation still considers power sharing amongst the multiple antennas at each transmitter.}\n\n\n\n\n\n\nTowards deriving the optimal multigroup multicast precoders when a maximum limit is imposed on the  transmitted power of each antenna, a new optimization problem with  one constraint per transmit antenna needs to be formulated.\n{Amid the extensive literature on multigroup multicast beamforming, the $\\mathrm{PAC}$s have only been considered in \\cite{Christopoulos2014_ICC}, where an   equally fair multicast multigroup solution is presented.} Extending these considerations, the present work accounts optimization weights.  Therefore,  a consolidated solution for the weighted max--min fair multigroup multicast beamforming under $\\mathrm{PAC}$s is hereafter presented. The contributions of the present work  are summarized as follows\n{\\begin{itemize}\n\\item The $\\mathrm{PAC}$ weighted fair multigroup multicast beamforming problem is formulated and solved.\n\\item Practical system design insights are given by examining the implications of the $\\mathrm{PAC}$s on  multigroup multicast distributed antenna systems ($\\mathrm{DAS}$), modulation constrained systems and  uniform linear array  ($\\mathrm {ULA}$) transmitters.\n\\item A robust to erroneous $\\mathrm{CSI}$ multigroup multicast design under  $\\mathrm{PAC}$s is proposed.\n\\item The {performance  of the solution} is evaluated through extensive numerical results under various system setups.\\end{itemize}\n}\n\nThe rest of the paper is structured as follows.   The multigroup multicast system model is presented in Sec. \\ref{sec: System Model} while the weighted  fair problem is formulated and solved in Sec. \\ref{sec: problem}. In Sec. \\ref{sec: performance}, the performance of the design is evaluated for various system setups along with a robust extension of the derived algorithm and a weighted multigroup multicast application paradigm. Finally,  Sec. \\ref{sec: conclusions} concludes the paper.\n\n\n\n\n\n\n\n\n\n\n\n\n\n{\\textit{Notation}: In the remainder of this paper, bold face lower case and upper case characters denote column vectors  and matrices, respectively. The operators \\(\\left(\\cdot\\right)^\\text{T}\\), \\(\\left(\\cdot\\right)^\\dag\\), $|\\cdot|$, ${\\mathrm{Tr}\\left(\\cdot\\right)}$ and \\(||\\cdot||_2, \\) correspond to   the transpose, the conjugate transpose,  the absolute value, the {trace} and the Frobenius norm operations,  while $[\\cdot]_{ij}  $  denotes the $i, j$-th element of a matrix.  The principal eigenvalue of a matrix $\\mathbf X$ are denoted as $\\lambda_{max}(\\mathbf X)$. Calligraphic indexed characters denote sets}.\n\n\\section{System Model }\\label{sec: System Model}\n Herein, the focus is on a multi-user ($\\mathrm{MU}$) multiple input single output ($\\mathrm{MISO}$) multicast system. Assuming a single transmitter, let   $N_t$ denote the number of transmitting elements  and  $N_{u}$ the  total number of users served.  The input-output analytical expression  will read as $y_{i}= \\mathbf h^{\\dag}_{i}\\mathbf x+n_{i},$\nwhere \\(\\mathbf h^{\\dag}_{i}\\) is a \\(1 \\times N_{t}\\) vector composed of the channel coefficients (i.e. channel gains and phases) between the \\(i\\)-th user and the  \\(N_{t}\\) antennas of the transmitter, \\(\\mathbf x\\) is the \\(N_{t} \\times 1\\)  vector of the transmitted symbols and  \\(n_{i}\\) is the independent complex circular symmetric (c.c.s.) independent identically distributed (i.i.d) zero mean  Additive White Gaussian Noise ($\\mathrm{AWGN}$)  measured at the \\(i\\)-th user's receive antenna.\nFocusing  in a multigroup multicasting scenario,  let there be a total of $1\\leq G \\leq N_{u}$ multicast groups with  $\\mathcal{I} = \\{\\mathcal{G}_1, \\mathcal{G}_2, \\dots  \\mathcal{G}_G\\}$ the collection of   index sets and $\\mathcal{G}_k$ the set of users that belong to the $k$-th multicast group, $k \\in \\{1\\dots G \\}$. Each user belongs to only one group, thus $\\mathcal{G}_i\\cap\\mathcal{G}_j=$\\O ,$  \\forall i,j \\in \\{1\\cdots G\\}$. Let $\\mathbf w_k \\in \\mathbb{C}^{N_t \\times 1}$ denote the precoding weight vector applied to the transmit antennas to beamform towards the $k$-th group.\nThe assumption of independent data transmitted to different groups renders the symbol streams $\\{s_k\\}_{k=1}^G$ mutually uncorrelated and the total power radiated from the antenna array is\n\\begin{align}\nP_{tot} = \\sum_{k=1}^ G \\mathbf w_k{^\\dag} \\mathbf w_k\n\\end{align}\nThe power radiated by each\nantenna element is  a  linear combination of all precoders\n\\cite{Yu2007}:\\begin{align}\\label{eq: PAC}\nP_n = \\left[\\sum_{k=1}^G \\mathbf w_k \\mathbf w_k^\\dag \\right]_{nn}\n\\end{align}\nwhere $n \\in \\{1\\dots  N_t\\}$ is the antenna index.\nThe fundamental difference between the $\\mathrm{SPC}$  of \\cite{Karipidis2008} and the proposed $\\mathrm{PAC}$ is clear in  \\eqref{eq: PAC}, where instead of one,  $N_t$ constraints are realized, each one involving all the precoding vectors.  A more general constraint formulation to model power flexibility amongst groups of antennas can be found in \\cite{Zheng2011a}.\n\\section{Multicast Multigroup Beamforming with Per Antenna Power Constraints}\\label{sec: problem}\n\\subsection{Weighted Max-Min Fair Formulation}\nThe $\\mathrm{PAC}$ weighted max-min fair  problem   is defined as\n\\begin{empheq}[box=\\fbox]{align}\n\\mathcal{F:}\\    \\max_{\\  t, \\ \\{\\mathbf w_k \\}_{k=1}^{G}}  &t& \\notag\\\\\n\\mbox{subject to } & \\frac{1}{\\gamma_i}\\frac{|\\mathbf w_k^\\dag \\mathbf h_i|^2}{\\sum_{l\\neq k }^G |\\mathbf w_l^\\dag\\mathbf h_i|^2+\\sigma_i^2 }\\geq t, &\\label{const: F SINR}\\\\\n&\\forall i \\in\\mathcal{G}_k, k,l\\in\\{1\\dots G\\},\\notag\\\\\n \\text{and to }\\ \\ \\ \\ & \\left[\\sum_{k=1}^G  \\mathbf w_k\\mathbf w_k^\\dag  \\right]_{nn}  \\leq P_n, \\label{eq: max-min fair power const1 }\\\\\n &\\forall n\\in \\{1\\dots N_{t}\\},\\notag\n \\end{empheq}\n where $\\mathbf w_k\\in \\mathbb{C}^{N_t}$ and $t \\in \\mathbb{R}^{+}$. Different service levels between the users can be acknowledged in this weighted  formulation.   Problem $ \\mathcal{F}$ receives as inputs the  $\\mathrm{PAC}$s vector $\\mathbf p = [P_1, P_2\\dots P_{N_t}]$ and the target $\\mathrm{SINR}$s vector $\\mathbf g = [\\gamma_1,\\gamma_2, \\dots \\gamma_{N_u}]$. {Its goal is to maximize the slack variable $t$ while keeping all $\\mathrm{SINR}$s above this value. Thus, it constitutes a max-min problem that guarantees fairness amongst users.}   Following the common  in the literature notation for ease of reference, the  optimal objective value of $ \\mathcal{F}$ is denoted as $t^*=\\mathcal{F}(\\mathbf g,  \\mathbf p)$ and the associated optimal point as $\\{\\mathbf w_k^\\mathcal{F}\\ \\}_{k=1}^{G}$. {Of particular interest is the case where the co-group users  share the same target i.e. $\\gamma_i = \\gamma_{k},\\ \\forall i \\in\\mathcal{G}_k, k\\in\\{1\\dots G\\}$.}\n\n{\\textit{Remark 1}: {The difference of the present formulation with respect to the weighted max-min fair problem with $\\mathrm{SPC}$ presented in   \\cite{Sidiropoulos2006,Karipidis2008} lies in  the $N_t$ power constraints over each individual radiating element. Additionally, this formulation differs from the coordinated multicell multicasting Max-Min formulation of \\cite{Xiang2013} since {the constraint  is imposed on  the $n$-th   diagonal element of the summation of the correlation matrices of all precoders, while weights on each users } $\\mathrm{SINR}$ are also inserted. On the contrary, in \\cite{Xiang2013}, the imposed per base station constraints are translated to one power constraint per each precoder.\nIn the present work, weights to differentiate the $\\mathrm{QoS}$ targets between users are also proposed.}\n\n\\subsection{Per-antenna power  minimization}\n{ The relation between the fairness  and the power minimization problems for the multicast multigroup case was firstly established in \\cite{Karipidis2008}. As a result, by bisecting  the solution of the  $\\mathrm{QoS}$ optimization, a solution to the weighted fairness problem can be derived. Nevertheless, fundamental differences between the existing formulations and problem $\\mathcal{F}$ complicate the solution. In more detail, the per-antenna constraints are not necessarily met with equality (a discussion on this is also given in Sec. \\ref{sec: power consumption}). Therefore, the fairness problem is no longer equivalent to the sum power minimization under  $\\mathrm{QoS}$ constraints problem. Since the absence of a related, solvable  problem prohibits the immediate application of bisection,  a novel equivalent per-antenna power  minimization problem is proposed as}\n\\begin{empheq}[box=\\fbox]{align}\n\\mathcal{Q:} \\min_{\\ r, \\ \\{\\mathbf w_k \\}_{k=1}^{G}}  &r& \\notag\\\\\n\\mbox{subject to } & \\frac{|\\mathbf w_k^\\dag \\mathbf h_i|^2}{\\sum_{l\\neq k }^G | \\mathbf w_l^\\dag\\mathbf h_i|^2 + \\sigma^2_i}\\geq \\gamma_i, \\label{const: Q SINR}\\\\\n&\\forall i \\in\\mathcal{G}_k, k,l\\in\\{1\\dots G\\},\\notag\\\\\n\\text{and to} \\ \\ \\ \\ \\ & \\frac{1}{P_n} \\left[\\sum_{k=1}^G  \\mathbf w_k\\mathbf w_k^\\dag \\right]_{nn} \\leq  r,\\label{eq: PAC Q}\\\\\n& \\forall n\\in \\{1\\dots N_{t}\\}, \\notag\n \\end{empheq}\n with $r\\in\\mathbb{R^+}$. Problem $\\mathcal{Q }$ receives as input  $\\mathrm{SINR}$ constraints for  all users, defined   before as $\\mathbf g $,  as well as the per antenna power constraint vector $\\mathbf p$ of \\eqref{eq: max-min fair power const1 }. {The introduction of the slack-variable  $r$,  a common practice in convex optimization \\cite{convex_book}, constraints the power consumption of each and every antenna.   Subsequently, at the optimum $r^*$,   the maximum  power consumption out of all antennas is minimized} and this solution is denoted as $r^*=\\mathcal{Q}(\\mathbf g, \\mathbf p)$. {The generic difference of the present min-max formulation and the formulation proposed in \\cite{Xiang2013} lies in the per antenna constraint \\eqref{eq: PAC Q}. Instead of constraining the  power of each antenna, the authors of \\cite{Xiang2013} impose a constraint over each precoder that serves a common multicast group. In the case tackled herein, the number of constraints is increased from one to $N_t$, while  each constraint is a function of all multigroup precoders as the summation in \\eqref{eq: PAC Q} reveals.}     The following claim reveals the relation between the described problems.\n\n \\textit{Claim 1}: Problems $\\mathcal{F}\\ $ and  $\\mathcal{Q}\\ $ are related as follows\n\\begin{align}\n1 = \\mathcal{Q}\\left(\\mathcal{F}\\left( \\mathbf g, \\mathbf p\\right)\\cdot\\mathbf g, \\mathbf p \\right)\\label{eq: equivalence 1}\\\\\nt = \\mathcal{F}\\left(\\mathbf g, \\mathcal{Q}\\left( t\\cdot \\mathbf g,\\mathbf p\\right)\\cdot\\mathbf p \\right)\\label{eq: equivalence 2}\n \\end{align}\n \\textit{Proof: }{Similar to the line of reasoning in \\cite{Xiang2013} the above claims will be proven by contradiction.} Starting with   \\eqref{eq: equivalence 1},  let $t^*=\\mathcal{F}(\\mathbf g,\\ \\mathbf p)$ denote the optimal value of $\\mathcal{F}$ with associated variable $\\{\\mathbf w_k^F \\}_{k=1  }^{G}$.\nAlso, let $\\hat r=\\mathcal{Q}\\left( t^*\\cdot\\mathbf g,\\ \\mathbf p\\right)$ be the optimal value of $\\mathcal{Q}$ at the point  $ \\{\\mathbf w_k^Q \\}_{k=1  }^{G}$. Then, assuming that $\\hat r > 1$, the vectors $ \\{\\mathbf w_k^F \\}_{k=1  }^{G}$ satisfy the feasibility criteria of  $\\mathcal{Q}$ and produce a lower optimal value  thus contradicting the optimality of  $ \\{\\mathbf w_k^Q \\}_{k=1  }^{G}$ and opposing the hypothesis. Alternatively, assuming that $\\hat r < 1$ then the solutions $ \\{\\mathbf w_k^Q \\}_{k=1  }^{G}$  can be scaled by the non-negative $ \\hat r $. The vectors $ \\{ \\hat r \\cdot\\mathbf{ w}_k^Q \\}_{k=1  }^{G}$ are feasible solutions to $\\mathcal{F }$  which provide the same optimal objective value with however some remaining power budget. Therefore, the power could be scaled up until at least one of the $\\mathrm{PAC}$s is satisfied with equality and a higher objective value would be derived thus again contradicting the hypothesis. Consequently,  $\\hat r = 1$.\nThe same line of reasoning is followed to prove \\eqref{eq: equivalence 2}. Let $r^*=\\mathcal{Q}(t\\cdot\\mathbf g,\\ \\mathbf p)$ denote the optimal value of $\\mathcal{Q}$ with associated solution $ \\{\\mathbf w_k^Q \\}_{k=1  }^{G}$. Assuming that the optimal value of $\\mathcal{F}$ under constraints scaled by the solution of $\\mathcal{Q}$ is different, i.e. $\\hat t= \\mathcal{F}\\left(\\mathbf g , \\ \\mathcal{Q}\\left(t\\cdot \\mathbf g ,\\ \\mathbf p\\right)\\cdot\\mathbf p\\right)$ with $ \\{\\mathbf w_k^F \\}_{k=1}^{G}$, the following contradictions arise. In the case where $\\hat t < t $, then the precoders   $ \\{\\mathbf w_k^Q \\}_{k=1 }^{G}$  are feasible solutions to  $\\mathcal{F}$ which lead to a higher minimum $\\mathrm{SINR}$, thus contradicting the optimality of $\\hat t$. Alternatively, if $\\hat t > t $ then the solution set $ \\{\\mathbf w_k^F \\}_{k=1   }^{G}$ can be scaled by a positive constant $ c = t \/\\hat t<1$. The new solution  $ \\{c \\mathbf w_k^F \\}_{k=1   }^{G}$ respects the feasibility conditions of $\\mathcal{Q}$ and provides a lower optimal value, i.e. $c\\cdot r^*$, thus again contradicting the hypothesis. As a result, $\\hat t = t\\ \\square$.\n\n \\subsection{Semidefinite Relaxation}\\label{sec: SDR}\n Problem  $\\mathcal{Q}$ belongs in the general class of non-convex $\\mathrm{QCQP}$s for which the $\\mathrm{SDR}$ technique is proven to be a powerful and computationally efficient approximation technique \\cite{Luo2010}. The relaxation is based on the observation that\n$|\\mathbf w_k^\\dag \\mathbf h_i|^2 = \\mathbf w_k^\\dag \\mathbf h_i \\mathbf h_i^\\dag \\mathbf w_k = {\\mathrm{Tr}}(\\mathbf w_k^\\dag \\mathbf h_i \\mathbf h_i^\\dag \\mathbf w_k) = {\\mathrm{Tr}}(\\mathbf w_k \\mathbf w_k^\\dag \\mathbf h_i \\mathbf h_i^\\dag )$. With the change of variables $\\mathbf X_i = \\mathbf w_i \\mathbf w_i^\\dag $,\n $\\mathcal{Q}$ can be relaxed to\n $\\mathcal{Q}_r$\n\n\\begin{empheq}[box=\\fbox]{align}\n\\mathcal{Q}_r:\\min_{r,\\ \\{\\mathbf X_k \\}_{k=1}^{G}}  &r& \\notag\\\\\n\\mbox{subject to } & \\frac{\\mathrm{Tr}\\left(\\mathbf Q_i \\mathbf X_k\\right)}{\\sum_{l\\neq k }^G \\mathrm{Tr}\\left(\\mathbf Q_l \\mathbf X_k\\right) +\\sigma_i^2}\\geq \\gamma_i, \\label{const: Q_r SINR}\\\\\n&\\forall i \\in\\mathcal{G}_k, k,l\\in\\{1\\dots G\\},\\notag\\\\\n \\text{and to} \\ \\ \\ \\ \\ &\\frac{1}{P_n} \\left[\\sum_{k=1}^G \\mathbf X_{k}\\right]_{nn} \\leq  r   \\label{const: Q_r Power}\\\\\n \\text{and to}  \\ \\ \\ \\ \\ &\\ \\mathbf X_k\\succeq0,\\label{const: Q_r SDF}\n\\ \\forall n\\in \\{1\\dots N_{t}\\},\\notag\n \\end{empheq}\nwhere $\\mathbf Q_i = \\mathbf h_i\\mathbf h_i^\\dag$, $r \\in \\mathbb{R}^{+}$ , while the constraint $\\text{rank}(\\mathbf X_i)  = 1$ is dropped. Now the relaxed  $\\mathcal{Q}_r$ is convex, thus solvable to an arbitrary accuracy. {This relaxation can be  interpreted as a Lagrangian bi-dual of the original problem  \\cite{convex_book}. }The weighted max-min fair optimization is  also relaxed as\n\\begin{empheq}[box=\\fbox]{align}\n\\mathcal{F}_r:\\max_{t, \\ \\{\\mathbf w_k \\}_{k=1}^{G}}&t& \\notag\\\\\n\\mbox{subject to } & \\frac{1}{\\gamma_i}\\frac{\\mathrm{Tr}\\left(\\mathbf Q_i \\mathbf X_k\\right)}{\\sum_{l\\neq k }^G \\mathrm{Tr}\\left(\\mathbf Q_l \\mathbf X_k\\right)+ \\sigma_i^2}\\geq t, \\label{const: F_r SINR}\\\\\n&\\forall i \\in\\mathcal{G}_k, k,l\\in\\{1\\dots G\\}, \\notag\\\\\n\\mbox{and to }\\ \\ \\ \\ &\\left[\\sum_{k=1}^G  \\mathbf X_k \\right]_{nn} \\leq  P_n,\\\\\n  &\\forall n\\in \\{1\\dots N_{t}\\},\\notag \\\\\n \\mbox{and to }\\ \\ \\ \\ &  \\mathbf X_k\\succeq0,\n\\end{empheq}\n{which, however, remains non-convex due to  \\eqref{const: F_r  SINR}, as in detail explained in \\cite{Karipidis2008}}. However, this obstacle can  be overcome by  the following observation.\n\n\\textit{ \\textit{Claim 2}: Problems $\\mathcal{F}_r$ and  $\\mathcal{Q}_r$ are related as follows}\n\\begin{align}\n1 = \\mathcal{Q}_r\\left(\\mathcal{F}_{r}\\left( \\mathbf g, \\mathbf p\\right)\\cdot\\mathbf g, \\mathbf p \\right)\\label{eq: equivalence 3}\\\\\nt = \\mathcal{F}_r\\left(\\mathbf g, \\mathcal{Q}_{r}\\left( t\\cdot \\mathbf g,\\mathbf p\\right)\\cdot\\mathbf p \\right)\\label{eq: equivalence 4}\n \\end{align}\n\\textit{Proof:} Follows the steps of the proof of \\textit{Claim 1} and is therefore omitted. $\\square$\n\\subsection{Gaussian Randomization} \\label{sec: Gaussian randomization}\nDue to the NP-hardness of the multicast problem,   the relaxed problems do not necessarily yield unit rank matrices. Consequently, one can  apply a rank-1 approximation over $\\mathbf X^{*}$.\nMany types of rank-1 approximations are possible depending on the nature of the original problem.\nThe solution with the highest provable accuracy for the multicast case is given by the {Gaussian randomization method \\cite{Luo2010}.} In more detail, let  $\\mathbf X^{*}$ be a symmetric positive semidefinite solution of the relaxed problem. Then, a candidate solution to the original problem can be generated as a Gaussian random variable with zero mean and covariance equal to $\\mathbf X^{*}$, i.e.  $\\hat{\\mathbf{w}} \\backsim\\mathbb{C}\\mathbb{N}(0, \\mathbf X^{*} )$.\nAfter generating a predetermined number of candidate solutions, the one that yields the highest objective value of the original problem can be chosen. The accuracy of this approximate solution is measured by the distance of the approximate objective value and the optimal value of the relaxed problem and it increases with the predetermined number of  randomizations {\\cite{Luo2010,Karipidis2008}}.\nNonetheless, an intermediate problem dependent step between generating a Gaussian instance with the statistics obtained from the relaxed solution and creating a feasible candidate instance of the original problem still remains, since the feasibility of the original problem is not yet guaranteed.\n\n\\subsection{Feasibility Power Control }\\label{sec: power control}\n\nAfter generating a random instance of a Gaussian variable with statistics defined by the relaxed problem, an additional step comes in play to guarantee the feasibility of the original problem. { In \\cite{Sidiropoulos2006},   the feasibility of the candidate     solutions, as given by the Gaussian randomization, was guaranteed by a simple  power rescaling.} Nevertheless, since in the multigroup case  an interference scenario is dealt with,   a simple rescaling does not guarantee feasibility. Therefore, an additional  optimization step is proposed in \\cite{Karipidis2008} to re-distribute the power amongst the candidate precoders. To  account for the inherently different $\\mathrm{PAC}$s,  a novel power control problem with per antenna power constraints is  proposed. Given a set of  Gaussian instances,  $\\{\\mathbf{\\hat w}_k\\}_{k=1 }^G$, the \\textit{Multigroup Multicast Per Antenna power Control} ($\\mathrm{MMPAC}$) problem reads as\n\\begin{empheq}[box=\\fbox]{align}\n\\mathcal{S^{\\mathcal{F}}:} \\max_{t, \\ \\{  p_{k} \\}_{k=1}^{G}}  &t& \\notag\\\\\n\\mbox{subject to } & \\frac{1}{\\gamma_i}\\frac{|\\mathbf{\\hat  w}_k^\\dag \\mathbf{  h}_i|^2 p_{k}}{\\sum_{l\\neq k }^G |\\mathbf{\\hat  w}_l \\mathbf{ h}_i |^2  p_{l} +\\sigma_i^2}\\geq t, \\label{const: S F SINR}\\\\\n&\\forall i \\in\\mathcal{G}_k,k,l\\in\\{1\\dots G\\}\\notag \\\\\n\\mbox{and  to}  \\ \\ \\ \\ \\ & \\left[\\sum_{k=1}^G \\mathbf{\\hat  w}_k\\mathbf{\\hat  w}_k^\\dag  p_k\\right]_{nn} \\leq   P_{n,} \\label{eq: MMPAC PACs}\\\\\n&\\forall n\\in \\{1\\dots N_{t}\\},\\notag\n \\end{empheq}\nwith $\\{  p_{k} \\}_{k=1}^{G}\\in\\mathbb{R}^{+}$. Problem $\\mathcal{S^{\\mathcal{F}}}$ receives as input the $\\mathrm{PAC}$s as well as the  $\\mathrm{SINR }$ targets and returns the maximum scaled worst $\\mathrm{SINR}$ $t^{*} = \\mathcal{S}(\\mathbf g, \\mathbf p)$ and is also non-convex like $ \\mathcal{F}$.\n{{The difference of this problem compared to \\cite{Karipidis2008} lies in \\eqref{eq: MMPAC PACs}.\n}\n\\textit{\\\\ Remark 2:} A very important observation is clear in the formulation of the power control problem. The optimization variable $\\mathbf p $ is of size $G$, i.e. equal to the number of groups, while the power constraints are equal to the number of antennas, $N_t$. In each constraint, all the optimization variables contribute. This fact prohibits the total exploitation of the available power at the transmitter. Once at least one of the $N_t$ constraints is satisfied with equality and remaining power budget, then the rest can not be scaled up since this would lead to at least one constraint exceeding the maximum permitted value.}\n\n\\subsection{Bisection}\\label{sec: bisec}\n\n{The establishment of claims 1 and 2, allows for the application of the bisection method, as developed in \\cite{Sidiropoulos2006,Karipidis2008}.} The solution of $\nr^* = \\mathcal{Q}_r \\left ( \\frac{L+U}{2}\\mathbf g, \\mathbf p\\right)\n$ is obtained by bisecting the interval $[L, U]$  as defined by the minimum and maximum $\\mathrm{SINR}$ values. Since $t=(L+U)\/2$ represents the $\\mathrm{SINR}\n, it will always be positive or zero. Thus, $L = 0.$ Also, if the system was interference  free while all the users had the channel of the best user, then the maximum worst $\\mathrm{SINR}$ would be attained, thus $U = \\max_i\\{P_{tot}\\mathbf Q_i\/\\sigma_i \\}. $ If $r^*<1$,   then the lower bound of the interval is updated with this value. Otherwise the value is assigned to the upper bound of the interval. Bisection is iteratively performed until an the interval size is reduced to a pre-specified value $\\epsilon$.\nThis value needs to be dependent on the magnitude of $L \\text{ and } U$ so that the accuracy of the solution is maintained regardless of the region of operation.\n After a finite number of iterations the optimal value of $\\mathcal{F}_r$ is given as the resulting value for which $L \\text{ and } U$ become almost identical. This procedure provides an accurate solution to the  non-convex $\\mathcal{F}_r$.\nFollowing this, for each and every solution $\\{\\mathbf{\\hat w}_k\\}_{k=1 }^G$, the power of the precoders needs to be controlled.\n{Consequently, problem $\\mathcal{S}^F$ can be solved using the well established framework of  bisection \\cite{convex_book} over its convex equivalent problem, which reads as}\n\\begin{empheq}[box=\\fbox]{align}\n\\mathcal{S^{\\mathcal{Q}}:} \\min_{r, \\ \\{ p_k \\}_{k=1}^{G}}  &r& \\notag\\\\\n\\mbox{subject to } & \\frac{|\\mathbf{\\hat  w}_k^\\dag \\mathbf h_i|^2 p_{k}}{\\sum_{l\\neq k }^G | \\mathbf{\\hat  w}_l^\\dag\\mathbf h_i|^2  p_{l} +\\sigma_i^2}\\geq \\mathbf \\gamma_i, \\label{const: F SINR}\\\\\n&\\forall i \\in\\mathcal{G}_k, k,l\\in\\{1\\dots G\\},\\notag \\\\\n\\mbox{and  to}  \\ \\ \\ \\ \\ & \\frac{1}{  P_{n}} \\left[\\sum_{k=1}^G  \\mathbf{\\hat  w}_k\\mathbf{\\hat  w}_k^\\dag  p_k\\right]_{nn} \\leq  r, \\\\\n&\\forall n\\in \\{1\\dots N_{t}\\},\\notag\n \\end{empheq}\n{ Problem $\\mathcal{S^\\mathcal{Q}}$ is an instance of a linear programming (LP) problem.}\n\n{\\textit{Remark\n3}:\nFor  completeness,\nthe\npossible\nreformulation of the non-convex problem\n$\\mathcal{S^{\\mathcal{F}}} $ into the following geometric problem ($\\mathrm{GP}$)   is considered, thus surpassing the need for bisection:}\n\\begin{empheq}[box=\\fbox]{align}\n\\mathcal{S^{\\mathcal{F}}_{\\mathcal{GP}}}:& \\min_{t, \\ \\{  p_{k} \\}_{k=1}^{G}}  t^{-1}& \\notag\\\\\n\\mbox{s. t. }& {\\sum_{l\\neq k }^G |\\mathbf{\\hat  w}_l \\mathbf{ h}_i |^2  p_{l} +\\sigma_i^2}\\leq \\frac{t^{-1}}{\\gamma_i}|\\mathbf{\\hat  w}_k^\\dag \\mathbf{  h}_i|^2 p_{k}, \\label{const: S F SINR}\\\\\n&\\forall i \\in\\mathcal{G}_k,k,l\\in\\{1\\dots G\\}\\notag \\\\\n\\mbox{and  to} &  \\left[\\sum_{k=1}^G \\mathbf{\\hat  w}_k\\mathbf{\\hat  w}_k^\\dag  p_k\\right]_{nn} \\leq   P_{n,} \\label{eq: MMPAC PACs GP}\n\\forall n\\in \\{1\\dots N_{t}\\},\\notag\n \\end{empheq}\n\\subsection{Complexity}\\label{sec: complexity}\n An important discussion involves the complexity of the employed techniques  to approximate a solution of  the highly complex, NP-hard  multigroup multicast problem under $\\mathrm{PAC}$s.   Focusing on the proposed algorithm (cf. Alg. 1), the main complexity burden originates from the solution of a $\\mathrm{SDP}$. The present work relies on  the CVX tool \\cite{convex_book} which calls  numerical  solvers such as SeDuMi to solve  semi-definite programs. The complexity of the $\\mathrm{SDR}$  technique has been  exhaustively discussed in  \\cite{Luo2010} and the references therein.    To calculate the total worst case complexity of the  solution proposed in the present work, the following are considered.\n\nInitially, a  bisection search is performed over  $\\mathcal{Q}_r$ to  obtain the relaxed solution.  This bisection runs for $N_{iter} = \\lceil\\log_2\\left(U_{1}-L_{1}\\right)\/\\epsilon_{1}\\rceil$ where $\\epsilon_1 $ is the desired accuracy of the search.  Typically   $\\epsilon_1 $ needs to be at least three orders of magnitude below the magnitudes of $U_{1}, L_{1}$ for sufficient accuracy. In each iteration of the bisection search, problem $\\mathcal{Q}_r$ is solved.  This $\\mathrm{SDP}$ has $G$ matrix variables of $N_t\\times N_t$ dimensions and $N_{u}+N_t$ linear constraints.   The interior point methods employed to solve this $\\mathrm{SDP}$  require at most { $\\mathcal{O}\\left(\\sqrt{GN_t}\\log(1\/\\epsilon) \\right)$ } iterations, where $\\epsilon $ is the desired numerical accuracy of the solver. Moreover, in each iteration not more than $\\mathcal{O}({G^3 N_t^6 +GN_{t}^{3} +N_{u}GN_t^2})$ arithmetic operations will be performed.  The increase in complexity stems from increasing the number of constraints, i.e. $N_t+N_{u}$  constraints are  considered instead of only $N_{u}$ as in \\cite{Karipidis2008}. However, this increase is not significant, since  the order of the polynomial with respect to the number of transmit antennas is not increased.   The solver used also exploits the specific structure of matrices hence the actual running time is reduced. Next,  a fixed number of Gaussian random instances with covariance given by the previous solution are generated. {The complexity burden of this step is given by the following considerations. For each randomization, a second bisection search is performed this time over the $\\mathrm{LP}$ $\\mathcal{S}^Q$. An $\\epsilon-$optimal solution of this problem can be generated with a worst case complexity of  $\\mathcal O(G^{3.5}\\log(1\/\\epsilon))$ \\cite{ye2011interior} . The second bisection runs for $N_{iter} = \\lceil\\log_2\\left(U_{2}-L_{2}\\right)\/\\epsilon_{2}\\rceil$ iterations, which are significantly reduced   since the upper bound $U_{2}$ is now the optimal value of the relaxed problem.\n Moreover, the Gaussian randomization  is executed for a fixed number of iterations. The accuracy of the solution increases with the number of randomizations \\cite{Karipidis2008,Sidiropoulos2006,Luo2010}. Finally, the complexity burden can be further reduced by the reformulation of the non-convex\n $ \\mathcal{S}^{\\mathcal{F}} $\n into the $\\mathrm{GP}$, $\\mathcal{S^{\\mathcal{F}}_{\\mathcal{GP}}}$ which is efficiently solved by successive approximations of primal-dual interior point numerical methods \\cite{convex_book}. Thus the need for the second bisection can be surpassed.  }\n\\begin{algorithm\n \\SetAlgoLined\n \\KwIn{$ N_{rand}, \\mathbf p, \\mathbf g, \\mathbf Q_i, \\sigma_i^2 \\ \\forall i \\in\\{1\\dots G\\} $ }\n \\KwOut{   $ \\{\\mathbf w_{k}^{opt}\\}_{k=1}^G$, $t^*_{opt}$ of $\\mathcal{F}$ $ \\{\\mathbf w_{k}^{out}\\}_{k=1}^G$   $t^*_{out}$  \n \\Begin{\n \\textbf{\\textit{\\textit{\\uline{Step 1:}}}} Solve $t_{opt} = \\mathcal{F}_r\\left(\\mathbf g, \\mathbf p \\right)$ by bisecting  $\\mathcal{Q}_r\\left(\\frac{L+U}{2}\\mathbf g, \\mathbf p \\right)$, (see Sec. \\ref{sec: bisec}).\nLet  the associated point be $\\{\\mathbf w_{k}^{opt}\\}_{k=1}^G$.\\\\\n \\eIf{$\\mathrm{rank}(\\mathbf X_{k}^{opt} )= 1, \\forall \\ k \\in\\{1\\dots G\\} $}\n{$ \\{\\mathbf w_{k}^{out}\\}_{k=1}^G$ is given by $\\lambda_{max}(\\mathbf X^{opt})$.  }\n{\n\\textbf{\\textit{\\uline{Step 2:}}}\n: Generate $N_{rand}$  precoding vectors\n$\\{ \\mathbf{\\hat{w}}_{k}\\}_{k=1}^G$, (see Sec. \\ref{sec: Gaussian randomization} ).\n$t^{*}_{(0)}= 0$\\;\n\\For{$i=1\\dots N_{rand}$}{\n\\textbf{\\textit{\\uline{Step 3:}}}\nSolve $\\mathcal{S}^{\\mathcal{F}}\\left(\\mathbf g, \\mathbf p \\right)$ by bisecting $\\mathcal{S}^{\\mathcal{Q}}\\left(\\frac{L+U}{2}\\mathbf g, \\mathbf p \\right)$ to get a $\\{\\mathbf w_{k}^{can}\\}_{k=1}^G$ with $t_{(i)}^*$.\\\\\n \\If {$t_{(i)}^* > t_{(i-1)}$ }{ $t^{*}_{out}=t^*_{(i)}, \\{\\mathbf w_{k}^{out}\\}_{k=1}^G = \\{\\mathbf w_{k}^{can}\\}_{k=1}^G$}\n}\n}\n\n \\caption{ Fair multigroup multicasting under $\\mathrm{PAC}$s.}\n\\end{algorithm}\n\n\n\\section{ Performance Evaluation \\& Applications} \\label{sec: performance}\n\\subsection{Multigroup multicasting over Rayleigh Channels}\\label{sec: performance rayleigh}\nThe performance of linear multicast multigroup beamforming under per antenna power constraints is examined for a system with $N_t = 5$ transmit antennas, $G = 2 $ groups and  $N_u = 4  $ users. Rayleigh fading is considered, thus the channels are generated as Gaussian complex variable instances with unit variance and zero mean. For every channel instance, the  approximate solutions of the max-min fair $\\mathrm{SPC}$   and the proposed $\\mathrm{PAC}$ problems are evaluated using $N_{rand} = 100$ Gaussian randomizations\\cite{Karipidis2008}.\nThe results are averaged over one hundred channel realizations, while the noise variance is normalized to one for all receivers and all $\\mathrm{SINR}$ targets are  assumed equal to one.\n\n {The achievable  minimum rate\n is plotted for the $\\mathrm{SPC}$ and the $\\mathrm{PAC  }$ optimization  in Fig. \\ref{fig:  power}\n with respect to the total transmit power in dBWs. Noise   is assumed normalised to one.} For fair comparison, the total power constraint $P_{tot}$~[Watts] is equally distributed amongst the transmit antennas when $\\mathrm{PAC}$s  are considered, hence each antenna can radiate at most $P_{tot}\/N_t $~[Watts].  The  accuracy of the approximate solutions for both problems, {given by comparing the actual solution to the relaxed upper bound \\cite{Sidiropoulos2006,Karipidis2008},}     is clear across a wide range of $\\mathrm{SNR}$. Nevertheless, the accuracy due to the $\\mathrm{PAC  }$s is slightly reduced. This accuracy degradation  is intuitively justified. A Gaussian randomization instance is less likely to approach the optimal point when the number of constraints is increased while the same number of Gaussian randomizations are performed ($N_\\mathrm{rand} = 100$).\n{Towards quantifying the gains of the proposed solution, the performance of the $\\mathrm{SPC  }$ solution re-scaled to respect the $\\mathrm{PAC  }$s is also included in Fig. 1. Re-scaling is achieved by multiplying each line of the precoding matrix with the square root of the inverse level of power over satisfaction of the corresponding antenna. In Fig. 1 it is clear that more than 1 dB of gain can be obtained by the proposed method over the suboptimal re-scaling approach.}\n\nA significant issue for the $\\mathrm{SDR}$ techniques in multicast applications is the tightness of the approximate solution  versus an increasing number of receivers per multicast. In the extreme case of one user per group, it was proven in\\cite{Bengtsson2001} that the relaxation provides an optimal solution. Thus the solution is no longer approximate but exact. However, the increasing number of users per group degrades the solution, as depicted in Fig. \\ref{fig:  users} for both problems. It is especially noticed that the $\\mathrm{PAC}$ system suffers more than  the $\\mathrm{SPC}$ of \\cite{Karipidis2008} as the number of users per multicast group increases.  An attempt to solve this inaccuracy, but only under sum power constraints, is presented in \\cite{Pesavento2012b}.\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width=0.8\\columnwidth]{rate_Gauss_100_revised2}\\\\\n \\caption{Minimum user rate with $\\mathrm{SPC}$ and $\\mathrm{PAC}$.\n Results for $N_{u} = 4$ users, $N_t = 5$ antennas, $L= 2$ groups and   $\\rho = 2$ users per group.\n}\\label{fig:  power}\n \\end{figure}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{PAC_user_Gauss_100_prop}\\\\\n\\caption{Minimum $\\mathrm{SINR}$ with $\\mathrm{SPC}$ and $\\mathrm{PAC}$  versus an increasing  ratio of users per group $ \\rho = N_u \/ G  $,  for  ${P}_{tot} = 10$~dBW.\n}\\label{fig:  users}\n\\end{figure}\n\\subsection{Power Consumption in DAS }\\label{sec: power consumption}\nThe main difference between the $\\mathrm{SPC}$ and the $\\mathrm{PAC}$  optimization problems is the utilization of the available on board power in each system architecture.\nIn \\cite{Karipidis2008},  the sum power constraint is always satisfied with equality, since any remaining power budget can be equally distributed to the precoding vectors and the solution is further maximized.\nOn the contrary, the $\\mathrm{PAC }$ system includes $N_t$  constraints which are coupled via the precoders.   According to the relation between $\\mathcal{F}$  and $\\mathcal{Q}$, i.e. \\eqref{eq: equivalence 1}, the ratio of transmitted power over the power constraint (i.e. $r$) is one. Since this ratio applies for at least one of the $N_t$ power constraints, if one  is met with equality and the remaining $N_t - 1$ are not, then no more power can be allocated to the precoders.\nLet us assume a  channel matrix with one compromised transmit antenna, i.e. $\\mathbf{H}=$\n\\begin{align}\n \\begin{bmatrix}\\notag\n2.94\\angle41^\\circ & 11\\angle{-25}^\\circ & 4.4\\angle{50}^\\circ &6.6\\angle{-4}^\\circ \\\\\n 13.2 \\angle{-150}^\\circ & 4.8\\angle{14}^\\circ & 15.2\\angle{-7}^\\circ &4.8\\angle{-37}^\\circ \\\\\n 12 \\angle{-155}^\\circ&1.5\\angle{163}^\\circ & 13.5\\angle{-105}^\\circ &3.9\\angle{-46}^\\circ \\\\\n 0.02 \\angle{-53}^\\circ & 0.03\\angle{-66}^\\circ & 0.03\\angle{120}^\\circ &0.03\\angle{-129}^\\circ \\\\\n 5.66 \\angle{137}^\\circ &   9.2\\angle{49}^\\circ & 13\\angle{-175}^\\circ & 2.45\\angle{126}^\\circ  \\\\\n\\end{bmatrix} ^\\text T,\n \\end{align}\nwhere $4$ users, divided into $2$ groups,  are served by $5$ antennas. One of the antennas (the $4$-th antenna) has severely degraded gains towards all users. This practical case can  appear in a $\\mathrm{DAS }$ where the physical separation of the transmit antennas not only imposes per antenna constraints but can also justify highly unbalanced channel conditions around the environment the antennas.\nThe power utilization of the solution of the optimization for each of the two problems is defined as the total transmitted power over the total available power $P_{tot}$, that is $P_u =\\left(\\sum_{k=1}^ G \\mathbf w_k{^\\dag} \\mathbf w_k\\right)\/{P_{tot}}$,\n and is  plotted versus an increasing power budget in Fig. \\ref{fig: power consumption}. It is clear that in the low power regime the available power is not fully utilized. As the available power increases, however, the power consumption of the $\\mathrm{PAC}$ increases. This result is in accordance with the optimality of equal power allocation in the high power regime and renders  the $\\mathrm{PAC}$ formulation relevant for power limited systems.\nFurther insights for this $\\mathrm{PAC}$ system are given in Fig. \\ref{fig: power bar}, where   the power utilization of each antenna is shown,  for different total power budgets.\nInterpreting these  results, it can be concluded that the $\\mathrm{PAC }$ problem is highly relevant for power-over-noise limited systems. Otherwise, in the high  power regime, the solution of the $\\mathrm{SPC }$ problem with less constraints could be also used as an accurate approximation.\n \\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.8\\columnwidth]{power_cons_Gauss_100_prop2}\\\\\n \\caption{Total power consumption of a   $\\mathrm{PAC}$  system versus available power.}\\label{fig:  power consumption}\n \\end{figure}\n \\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.8\\columnwidth]{power_cons_bar}\\\\\n \\caption{Per-antenna  consumption  in a  $\\mathrm{PAC}$ system  versus   transmit power.\n}\\label{fig:  power bar}\n\\end{figure}\n\\subsection{Weighted Fairness Paradigm}\nTo the end of establishing the importance of the weighted optimization,  a simple paradigm is elaborated herein. Under the practical assumption of a modulation constrained system, the weighted fair design can be exploited for  rate allocation towards increasing the total system throughput.\n More specifically, the considered system employs adaptive modulation and allocates binary phase shift keying ($\\mathrm{BPSK}$) modulation  if the minimum $\\mathrm{SINR_i}$  in the $k$-th group is less than the ratio for which the maximum modulation constrained spectral efficiency is achieved. This ratio is simply given by $ \\log_2 M$, where $M $ is the modulation order. Hence for $\\mathrm{BPSK}$, $\\gamma_2 = 0$~dB, and so forth. If for some group $k$,  $\\min_i{\\mathrm{SINR_i}} \\geq \\gamma_{ \\mathrm{2}},\\ \\forall i \\in\\mathcal{G}_k, $ then quaternary phase shift keying ($\\mathrm{QPSK}$) is used  for all users in the group. Forward error correction is not assumed.  Let there be a two antenna transmitter that serves four users   grouped into two groups. The considered channel matrix reads as\n \\begin{align}\\mathbf{H }=\n \\begin{bmatrix}\\notag\n 0.2 \\angle106^\\circ & 90\\angle{-69}^\\circ & 0.5\\angle{-99}^\\circ &0.5\\angle{61}^\\circ \\\\\n 0.8 \\angle111^\\circ &   120 \\angle{-112}^\\circ & 1\\angle{127}^\\circ & 1.5\\angle{49}^\\circ  \\\\\n\\end{bmatrix} ^\\text T.\n \\end{align}\nThe attributes of the specific channel matrix depict one possible instance of the system where one user with a good channel state (i.e. user two) is in the same group with a jeopardized user, namely user one. On the other hand, the second group contains relatively balanced users in terms of channel conditions. For an un-weighted optimization (i.e. $\\mathbf g = [1\\ 1\\ 1\\ 1]$) the  spectral efficiency of each user is shown in  Fig. \\ref{fig: spf bar}. Baring in mind  that each user is constrained by the minimum group rate, the actual rate at which  all users will receive data is 0.52 [bps\/Hz]. Both groups  achieve the same spectral efficiency since the minimum $\\mathrm{SINR}$s and hence the minimum rates are balanced between the groups.   Subsequently, a modulation constrained  multicast transmitter  will  employ $\\mathrm{BPSK}$ for all users.\n By heuristically choosing the constraint vector  to be  $\\mathbf g = [1\\ 1\\ 5.3\\ 5.3]$ each user rate is modified. As depicted in Fig. \\ref{fig: spf bar} both users in the second group are achieving adequate   $\\mathrm{SINR}$ to support a higher order modulation. This gain is achieved at the expense of the rates of the users of the first group.\nFollowing this paradigm, the weight optimization can lead to an improved modulation assignment and thus higher throughput in practical systems. Hence, the weighted formulation offers the substantial degrees of freedom to maximize the total throughput of a modulation constrained multicast system by properly allocating the rates amongst the groups.\n \\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.8\\columnwidth]{SINR_bar_weights}\\\\\n  \\caption{Modulation constrained paradigm.}\n\\label{fig: spf bar}\n\\end{figure}\n\\subsection{{Uniform Linear Arrays}}\nTo the end of investigating the sensitivity of the proposed algorithm with respect to the angular separation of co-group users, a uniform linear array ($\\mathrm{ULA}$) transmitter is considered. Assuming far-field, line-of-sight conditions, the user  channels  can be  modeled using Vandermonde matrices. For this important special case, the $\\mathrm{SPC}$ multicast multigroup problem was reformulated into a convex optimization problem and solved in \\cite{Karipidis2006,Karipidis2007}. These results where motivated by the observation that in  $\\mathrm{ULA}$ scenarios, the relaxation consistently yields rank one solutions. Thus, for such cases, the $\\mathrm{SDR}$ is essentially optimal \\cite{Sidiropoulos2006}. The fact that the $\\mathrm{SDR}$ of the sum power minimization problem   is tight for Vandermonde channels was established in \\cite{Karipidis2007}.\nLet us consider a $\\mathrm{ULA}$ serving $4$ users allocated to $2$ distinct groups.  In Fig. \\ref{fig: ULA pos}, its radiation pattern for co-group angular separation $\\theta_a = 35^\\circ$  is plotted. The symmetricity due to the inherent ambiguity of the $\\mathrm{ULA}$ is apparent. Clearly, the multigroup multicast beamforming optimizes the lobes  to reduce interferences between the two groups. The $\\mathrm{SPC}$ solution, re-scaled to respect the $\\mathrm{PAC}$s are also included in Fig. \\ref{fig: ULA pos}. The superiority of the proposed solution is apparent.\n\nIn Fig. \\ref{fig: ULA sr}, the performance in terms of minimum user rate over the area with respect to an increasing angular separation is investigated. When co-group users are collocated, i.e. $\\theta_a = 0^\\circ$, the highest performance is attained.  As the separation increases, the performance is reduced reaching the minimum when users from different groups are placed in the same position, i.e. $\\theta_a = 45^\\circ$. In Fig. \\ref{fig: ULA sr}, the tightness of the relaxation for the  $\\mathrm{SPC}$ problem \\cite{Karipidis2007} is clear. However, the same does not apply for the proposed $\\mathrm{PAC}$. As co-group  channels tend to become  orthogonal, the approximation becomes less tight. Nevertheless, $N_\\mathrm{rand} = 200$ randomizations are sufficient to maintain the solution above the re-scaled $\\mathrm{SPC}$, as shown in Fig.  \\ref{fig: ULA sr}. Consequently, the proposed solution outperforms a re-scaled  to respect the per-antenna constraints, $\\mathrm{SPC}$ solution, over the span of the angular separations.\n\n\\textit{Remark 4:} The semidefinite relaxation of the per-antenna power minimization problem in $\\mathrm{ULA}$ transmitters is not always tight.\n\nFor every optimum high rank set of matrices $\\{\\mathbf { X}_k ^{opt}\\}_{k=1}^G$, there exists a set of rank one positive semidefinite matrices  $\\{\\mathbf {\\bar X}_k^{opt}\\}_{k=1}^G$, i.e. $\\mathrm{rank}(\\mathbf {\\bar X}_k ^{opt})=1, \\forall k \\in \\{1\\dots G\\}  $,  which is equivalent with respect to the power received at each user, i.e $\\mathrm{Tr}(\\mathbf { X}_k ^{opt}\\mathbf { Q}_i)=\\mathrm{Tr}(\\mathbf {\\bar X}_k ^{opt}\\mathbf { Q}_i) , \\forall i \\in\\mathcal{G}_k, k,l\\in\\{1\\dots G\\}$. This result is based on the Riesz-F\\'ejer theorem on real valued complex trigonometric polynomials \\cite{Karipidis2007}. Therefore, the Vandermonde channels impose a specific structure to the $\\mathrm{SPC}$ solution that allows for a convex reformulation.\nThe difference in the case tackle herein lies in the $N_t$  $\\mathrm {PAC}$s, i.e. $\\left[\\sum_{k=1}^G  \\mathbf X_k \\right]_{nn} \\leq  P_n, \\forall n\\in \\{1\\dots N_{t}\\}$, in which the channel structure is not involved.    Thus, a rank-1 matrix is equivalent in terms of per user received power \\cite{Karipidis2007} but not necessarily in terms of per-antenna consumed power, as shown herein.\n \\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.8\\columnwidth]{ULA_tot}\\\\\n  \\caption{ULA beampattern for $\\mathrm{PAC}$ and re-scaled $\\mathrm{SPC}$ solutions.}\n\\label{fig: ULA pos}\n\\end{figure}\n \\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.8\\columnwidth]{ULA_final3}\\\\\n  \\caption{ $\\mathrm{ULA}$ performance for increasing co-group user angular separation.\n\\label{fig: ULA sr}\n\\end{figure}\n\\subsection{{Robust Design under $\\mathrm{PAC}$s}}\\label{sec: robustness}\n When beamforming under uncertainty is considered, three different designs can be realized\\cite{Gershman2010}. Namely,  the probabilistic design, where acceptable performance is guaranteed for some percentage of time, the expectation based design that requires knowledge of the second order channel statistics but cannot guarantee any outage performance and the worst-case design. The latter approach guarantees a minimum $\\mathrm{QoS}$ requirement  for any error realization.\n\nFocusing on a worst-case design, let us assume an elliptically bounded error vector. In this context, the actual channel is given as $\\mathbf h_{i} = \\bar{ \\mathbf h}_{i} +\\mathbf e_{i}$  where $\\bar{\\mathbf h_{i}}$ is the channel available at the transmitter and $\\mathbf e_{i}$ is an error vector  bounded by $\\mathbf e_{i}^\\dag \\mathbf C_i \\mathbf e_{i}\\leq 1$. The hermitian positive definite matrix $\\mathbf C_{i}$ defines the shape and size of the ellipsoidal bound. For  $\\mathbf C_{i} = 1\/\\sigma_\\epsilon^2\\mathbf I_{N_t}$, then  $||\\mathbf e_i||^2_2\\leq \\mathbf \\sigma_\\epsilon^2$ and  the error remains in a spherical region of radius $ \\mathbf \\sigma_\\epsilon$ \\cite{Shenouda2007}. This spherical error model is mostly relevant when the feedback quantization error of a uniform  quantizer at the receiver is considered \\cite{Jindal2004}.\nThe proposed design is formulated as  \\begin{empheq}[box=\\fbox]{align}\n\\mathcal{F_{RB}:}&\\max_{\\  t, \\ \\{\\mathbf w_k \\}_{k=1}^{G}}  t\\notag\\\\\n\\mbox{s. t. }  &\\frac{1}{\\gamma_i}\\frac{|\\mathbf w_k^\\dag \\left(\\bar{ \\mathbf h}_{i} +\\mathbf e_{i}\\right)|^2}{\\sum_{l\\neq k }^G |\\mathbf w_l^\\dag\\left(\\bar{ \\mathbf h}_{i} +\\mathbf e_{i}\\right)|^2+\\sigma_i^2 }\\geq t, \\label{const: FRB SINR}\\\\\n&\\forall i \\in\\mathcal{G}_k, k,l\\in\\{1\\dots G\\},\\notag\\\\\n \\text{and to }&  \\left[\\sum_{k=1}^G  \\mathbf w_k\\mathbf w_k^\\dag  \\right]_{nn}  \\leq P_n, \\forall n\\in \\{1\\dots N_{t}\\}\\label{const: FRB PAC},\n \\end{empheq}\nand involves the channel imperfections  only in the $\\mathrm{SINR}$ constraints. The novelty of $\\mathcal{F_{RB}}$ over existing robust multicast formulations lies in \\eqref{const: FRB PAC}.\nThe $\\mathrm{SINR}$ constraints of $\\mathcal{F_{RB}}$, i.e. \\eqref{const: FRB SINR}, are over all possible error realizations and cannot be handled.  However, by applying the S-lemma \\cite{convex_book},  the error vector in   \\eqref{const: FRB SINR} can be eliminated.  This procedure is analytically described in  \\cite{Chen2012}. Thus, $\\mathcal{F_{RB}}$ can be converted to a $\\mathrm{SDP}$ and solved efficiently using the methods described in Sec. \\ref{sec: problem}.\nThe performance gain  of the proposed  robust design for a $\\mathrm{ULA}$ with $N_t = 2$ transmit antennas, serving $N_{u}=6$ users is given in Fig. \\ref{fig: robust SR config}, versus an increasing error radius $ \\mathbf \\sigma_\\epsilon$, for different user per group   configurations,  $\\rho$. These results exhibit the significant gains of the proposed technique as the error and   the group sizes increase.\n \\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.8\\columnwidth]{Robust_prop2}\\\\\n  \\caption{Robust performance for various user per group configurations.\n}\n\\label{fig: robust SR config}\n\\end{figure}\n\n { To establish the importance of the novel formulation, the performance in terms of  minimum user rate   over 1000 error realizations is given in Fig. \\ref{fig: robust SR}, versus a wide range of the error radius $ \\mathbf \\sigma_\\epsilon$ for the proposed $\\mathcal{F_{RB}}$ as well as the existing $\\mathrm{SPC}$ solutions re-scaled to respect the per-antenna constrains. For this figure, a $\\mathrm{ULA}$ with     $N_{t} = 3 $ transmit antennas is considered,  serving $N_{u}=6$ users partitioned into $L=2$ multicast groups.  The co-group angular separation is $\\theta_a = 10^\\circ$ and the number of Gaussian randomizations chosen is $N_\\mathrm{rand} = 200$ and $N_\\mathrm{rand} = 1000$ for the high and low precision curves respectively.  According to Fig. \\ref{fig: robust SR},  the proposed robust $\\mathrm{PAC}$ formulation (i.e. $\\mathcal{F_{RB}}$) outperforms existing solutions, in a per-antenna power constrained setting, for a wide range of channel error radius.\n However, as the error radius increases, a slight performance degradation is noted, especially for the low precision results.\nTo further investigate on this result, the following remark is given.}\n\n{\\textit {Remark 5}: The semidefinite relaxation of robust multigroup multicasting under $\\mathrm{PAC}$s yields non rank-1 solutions with higher probability as the channel errors increase.  }\n\n{The accuracy of the minimum rate results of Fig. 9, is presented in   Fig. 10.  The accuracy is measured by the distance of the randomized solution from the upper bound given by the relaxation, following  the standards of Sec. IV-A and \\cite{Sidiropoulos2006,Karipidis2008}. In Fig. 10, the results are also normalized by the value of the upper bound.   According to these results,  the  probability for the $\\mathrm{SDR}$  to yield rank-1 solutions is reduced as the error radius increases, for all problems. The accuracy reduction  of the $\\mathrm{SDR}$ technique as the channel errors increase  was also reported via simulations in \\cite{Zheng2009_TSP}, but  for unicast scenarios. What is more, $\\mathcal{F_{RB}}$ yields non rank-1 solutions as the   errors increase, with higher probability than the  $\\mathrm{SPC}$ problem. However, 1000 randomizations are sufficient to reduce the inaccuracy of all solutions to less than 7\\%, as illustrated in Fig. 10. }\n{\n  It is therefore concluded that although  the relaxation of the robust formulations does not consistently yield rank-1 solutions, especially for higher values of error radius, the Gaussian randomization can provide solutions with adequate accuracy. Finally, the proposed solutions surpass the performance of existing approaches, in practical per-antenna power constrained settings.    }\n\n \\begin{figure}[h]\n \\centering\n \\includegraphics[width=1\\columnwidth]{Robust_round2_prop3}\\\\\n\\caption{{Minimum user rate versus increasing $\\mathrm{CSI}$ error.} }\n\\label{fig: robust SR}\n\\end{figure}\n \\begin{figure}[h]\n \\centering\n \\includegraphics[width=1\\columnwidth]{Robust_round2_accu_prop3}\\\\\n  \\caption{{Accuracy of the semidefinite relaxation versus an increasing $\\mathrm{CSI}$ error}.\n}\n\\label{fig: robust accuracy}\n\\end{figure}\n\n\\section{Conclusions} \\label{sec: conclusions}\n\\textit{}In the present work, optimum linear precoding vectors are derived under per antenna power constraints, when independent sets of common information are transmitted by an antenna array to distinct  co-channel sets of users. The novel weighted max--min fair multigroup multicast problem under $\\mathrm{PAC}$s  is formulated. An  approximate solution for this NP-hard  problem is presented based on the well established methods of semidefinite relaxation.\nThe performance of the weighted max--min fair multigroup multicast optimization is examined under various system parameters and important insights on the system design are gained. Moreover, an application paradigm of the new system design is described while robust to imperfect $\\mathrm{CSI}$ extensions are given. \nConsequently, an important practical constraint towards the implementation of physical layer multigroup multicasting is alleviated.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nMotivated by the equations describing the steady motion of\ngeneralized Newtonian fluids we study the following fully\ninhomogeneous system \n\\begin{equation} \\label{eq:main_problem} \\begin{split}\n- \\div \\S(\\vec{Dv}) + \\vec{v} \\cdot \\nabla \\vec{v} + \\nabla \\pi &= \\vec{f}, \\\\\n\\div \\vec{v} &= g_1, \\\\\n\\vec{v}_{\\vert \\del \\Omega} &= \\vec{g_2}.\n\\end{split} \\end{equation} In this setting, $\\S$ is an extra stress\ntensor with $p$-$\\delta$-structure, $\\vec{v}$ is the velocity field\nwith its symmetric gradient $\\vec{Dv}$, $\\pi$ is the pressure,\n$\\vec{f}$ is the external force and $g_1$ and $\\vec{g_2}$ are data\non a sufficiently regular bounded domain\n$\\Omega \\subset \\Rd$ of dimension $d \\in \\{2, 3\\}$.\n\nSince \\eqref{eq:main_problem}$_1$ leads to a pseudomonotone and\ncoercive operator in the homogeneous case $g_1 = 0$,\n$\\vec{g}_2=\\vec{0}$ and $p >\\frac {3d}{d+2}$ (cf.~\\cite{lions-quel})\nand in the shear-thickening case $p>2$ (cf.~\\cite{r-mol-inhomo}), the\nexistence of weak solutions $(\\vec{v}, \\pi)$ to\n\\eqref{eq:main_problem} follows directly from the theory of\npseudomonotone operators in these cases. This approach can be adapted to the situation of homogeneous data and very low values of $p$: if $g_1 = 0$,\n$\\vec{g}_2=\\vec{0}$ and $p \\in (\\frac{2d}{d+2}, \\frac{3d}{d+2}]$, one can construct approximate solutions by the theory of pseudomonotone operators and prove their convergence with the Lipschitz truncation method (cf. \\cite{dms}, \\cite{fms2}). In the case $p=2$, we have to\ndeal with the fully inhomogeneous steady Navier-Stokes equations which\nare studied intensively (cf.~\\cite{Galdi}) and where the existence of\nsolutions is known under appropriate smallness conditions.  In the\nshear-thinning, inhomogeneous case, i.e.~if $p \\in (1, 2)$ and the\ndata $g_1$, $\\vec{g_2}$ do not vanish, the coercivity of the elliptic\nterm is weaker than the growth of the convective term, i.e.~we are in\nthe supercritical\ncase.\nThis situation is treated in \\cite{mikelic}, \\cite{Sin} for\n$g_1=0$. In \\cite{Sin} even the case of electrorheological fluids is\ncovered. The result there is based on a nice smallness argument (\\cite[Lemma\n3.2]{Sin}), which is applied to estimate the convective term. \nSince we did not understand the application of this lemma in detail,\nwe give a different proof of local coercivity here. Our main result\nshows the existence of weak solutions of the fully inhomogeneous\nproblem \\eqref{eq:main_problem} in the shear-thinning case under appropriate smallness conditions\ninvolving higher regularity of the data.\n\nThe paper is organised as follows: by representing the inhomogeneous\ndata by a fixed function $\\vec{g}$ (Subsection \\ref{sub:div_eq}),\n\\eqref{eq:main_problem} turns into a homogeneous problem. We\ninvestigate the newly formed elliptic and convective terms in\nSubsections \\ref{sub:extra_stress} and \\ref{sub:conv_term}. Then we\nconclude properties and local coercivity of the whole system and prove\nexistence of solutions (Subsection \\ref{sub:existence}).  In the case\n$p \\in (\\frac{2d}{d+2}, \\tfrac{3d}{d+2})$ we use the Lipschitz\ntruncation method in order to establish convergence of approximate\nsolutions. This step is presented as an abstract statement, Theorem~\\ref{thm:ident_limits}, which should fit to more general situations.\nIn contrast to \\cite{Sin}, we had to require additional regularity of\nthe data in our proof of local coercivity. We discuss this issue in\nSubsection \\ref{sub:less_regularity} and prove that the additional\nregularity assumption is necessary in the framework of pseudomonotone\noperator theory.\n\nThe results presented here are based on the thesis \\cite{MA} of the\nfirst author.\n\n\\section{Preliminaries}\n\n\\subsection{Notation} \\label{sub:notation} We work on a bounded\nLipschitz domain $\\Omega \\subset \\Rd$, $d \\in \\{2, 3\\}$, with\npossesses an exterior normal $\\vec{\\nu}$.  Points and scalar-valued\nquantities are written in normal letters whereas vector- and matrix-valued functions, variables and operators are denoted in bold\nletters. The space of symmetric quadratic matrices is denoted as\n$\\R_\\textit{sym}^{d\\times d}$.\n\nWe use standard Lebesgue measure and integration theory. For a ball $B$, we\ndenote the ball with the same center and the double radius\nby~$2B$. The characteristic function of a set $S \\subset \\Rd$ is\ncalled $\\chi_S$.\n\nWe use standard notation for Lebesgue and Sobolev spaces. Due to\n\\cite{Gagliardo}, there exists a well-defined, surjective trace\noperator $W^{1,p}(\\Omega) \\to W^{1-\\frac{1}{p}, p}(\\del \\Omega)$ that\nassigns boundary values to a Sobolev function.  We denote by\n$L_0^p(\\Omega)$ the subspace of $L^p(\\Omega)$ of functions with mean\nvalue zero and by $V_p$ the subspace of $ W_0^{1,p}(\\Omega)$ of vector\nfields with zero boundary values and zero divergence.  For a\nvector-valued function $\\vec{v} \\in W^{1,p}(\\Omega)$, the definition\nof the (weak) gradient field follows the convention\n$(\\nabla \\vec{v})_{ij} = \\del_j v_i$ and the symmetric gradient is\ndefined as\n$\\vec{Dv} := \\frac{1}{2} (\\nabla\\vec{v}+\\nabla\\vec{v}^\\top)$.  On\n$W_0^{1,p}(\\Omega)$ and on $V_p$, we may work with the symmetric\ngradient norm $\\norm{\\vec{D} \\cdot}_p$, thanks to Poincar\\'e's and\nKorn's inequalities.\n\nThe dual of some Banach space $X$ is denoted as $X\\d\\!$ and\n$\\fp{\\cdot}{\\cdot}$ denotes their canonical dual pairing.  For an\nexponent $p \\in [ 1, \\infty ]$, we define its conjugate exponent\n$p' \\in [ 1, \\infty ]$ via $\\frac{1}{p} + \\frac{1}{p'} = 1$ and use\nthe duality $L^{p'}(\\Omega) = L^p(\\Omega)\\d$ for $p <\n\\infty$. Finally, we define the critical Sobolev exponent\n$p\\d := \\frac{pd}{d-p} \\in (p, \\infty)$ for~$p < d$.\n\n\\subsection{The divergence equation} \\label{sub:div_eq} In order to\nfulfil the boundary and divergence conditions in\n\\eqref{eq:main_problem}, we follow the usual ansatz\n$\\vec{v} = \\vec{u} + \\vec{g}$, where $\\vec{u} \\in V_p$ and\n$\\vec{g} \\in W^{1,p}(\\Omega)$ fulfils the boundary and divergence\ndata, i.e.~the vector field $\\vec{g} \\in W^{1,p}(\\Omega)$ solves\n\\begin{equation} \\label{eq:g_problem}\n\\begin{split}\n\\div \\vec{g} &= g_1, \\\\\n\\vec{g}_{\\vert \\del \\Omega} &= \\vec{g_2}.\n\\end{split}\n\\end{equation}\nFor the corresponding homogeneous system, we have the fundamental\nresult due to Bogovski\\u{\\i} (cf.~\\cite{bo1}, \\cite{bo2}, \\cite{Galdi}):\n\\begin{thm}[Bogovski\\u{\\i} operator]\\label{thm:bogovski}\n\tLet $\\Omega \\subset \\Rd$ be a bounded Lipschitz domain with\n\t$d \\geq 2$ and $p \\in (1,\\infty)$. Then there exists a linear and\n\tbounded operator $\\Bog \\colon L_0^p(\\Omega) \\to W_0^{1,p}(\\Omega)$\n\tand a constant $c_{Bog} = c(\\Omega, p)$ such that\n\t\\begin{align*}\n\t\t\\div \\Bog f &= f, \\\\\n\t\t\\norm{\\Bog f}_{1,p} &\\leq c_{Bog} \\norm{f}_p\n\t\\end{align*}\n\tfor all $f \\in L_0^p(\\Omega)$. \n\\end{thm}\n\nFor the inhomogeneous system, we combine Bogovski\\u{\\i}'s Theorem and\nthe fact that $ W^{1-\\frac{1}{p}, p}(\\del \\Omega)$ is precisely the\nspace of boundary values of $W^{1,p}(\\Omega)$-functions:\n\\begin{lem}[The inhomogeneous divergence equation] \\label{lem:div_eq}\n\tLet $\\Omega \\subset \\Rd$ be a bounded Lipschitz domain with\n\t$d \\geq 2$ and $p \\in (1, \\infty)$. Suppose $g_1 \\in L^p(\\Omega)$\n\tand $\\vec{g_2} \\in W^{1-\\frac{1}{p}, p}(\\del \\Omega)$ satisfy\n\t$\\int_{\\Omega} g_1\\, dx = \\int_{\\del \\Omega} \\vec{g_2} \\cdot\n\t\\vec{\\nu}\\, do$.\n\tThen there exists a solution $\\vec{g} \\in W^{1,p}(\\Omega)$ of\n\tproblem \\eqref{eq:g_problem} that satisfies\n\t\\begin{equation*}\n\t\t\\norm{\\vec{g}}_{1,p} \n\t\t\\leq c_{\\textit{lift}} \\left( 1 + c_{Bog} \\right) \\norm{\\vec{g_2}}_{1-\\frac{1}{p}, p} \n\t\t+ c_{Bog} \\norm{g_1}_p\n\t\\end{equation*}\n\twith constants $c_\\text{lift}$ and $c_\\text{Bog}$ from the trace\n\tlifting and the Bogovski\\u{\\i} operator.\n\\end{lem}\n\\begin{proof}\n\tDue to \\cite{Gagliardo}, there exists a trace lifting\n\t$\\vec{\\hat{g}} \\in W^{1,p}(\\Omega)$ of the boundary values\n\t$\\vec{g_2}$. By integration by parts, we see that the function\n\t$g_1 - \\div \\vec{\\hat{g}}$ has mean value zero.  Thus, we may apply the\n\tBogovski\\u{\\i} operator and directly obtain that\n\t$\\vec{g} := \\vec{\\hat{g}} + \\Bog(g_1 - \\div \\vec{\\hat{g}}) \\in\n\tW^{1,p}(\\Omega)$ solves \\eqref{eq:g_problem}. The estimate of\n\t$\\vec{g}$ follows from the boundedness of the trace lifting and the\n\tBogovski\\u{\\i} operator.\n\\end{proof}\n\n\\subsection{Local coercivity}\nWe will work with the following notion of local coercivity: \n\\begin{defi}[local coercivity]\\label{defi:loc_coercive}\n\tLet $X$ be a Banach space. An operator $A \\colon X \\to X\\d$ is\n\tcalled \\emph{locally coercive with radius} $R$ if there exists a\n\tpositive real number $R$ such that\n\t\\begin{equation*}\n\t\t\\fp{Ax}{x} \\geq 0\n\t\\end{equation*}\n\tholds for all $x \\in X$ with $\\norm{x}_X = R$.\n\\end{defi}\n\nLocal coercivity is precisely the condition that allows to apply\nBrouwer's fixed point theorem in order to obtain approximate solutions\nin the proof of Br\\'ezis' theorem about pseudomonotone operators\n\\cite[Thm.~27.A]{Zeidler2B}. Therefore, we get a generalized version\nof Br\\'ezis' theorem that can be proved along the lines of the\nstandard version. It can also be regarded as a special case of the\nexistence theorem of Hess and Kato \\cite[Thm.~27.B]{Zeidler2B}.\n\n\\begin{thm}[Existence theorem for pseudomonotone operators]\\label{thm:brezis}\n\tLet $X$ be a reflexive and separable Banach space and\n\t$A \\colon X \\to X^\\ast$ be a pseudomonotone, demicontinuous and\n\tbounded operator that is locally coercive with radius $R$. Then\n\tthere exists a solution $u \\in X$ of the problem\n\t\\begin{equation*}\n\t\tAu = 0\n\t\\end{equation*}\n\tthat  satisfies $\\norm{u}_X \\leq R$.\n\\end{thm}\n\n\\subsection{The extra stress tensor and its induced\n\toperator} \\label{sub:extra_stress}\n\nThe stress tensor describes the mechanical properties \nof the fluid in dependence on the strain rate $\\vec{Dv}$. In Newtonian fluid\ndynamics, the viscosity is a constant $\\kappa \\in \\R$ which induces\nthe linear operator $- \\div \\S(\\vec{Dv}) = - \\kappa \\Delta \\vec{v}$\ndescribing the viscous part of the stress tensor. The general\nsituation of  non-Newtonian fluids can be modeled in various ways\n(cf.~\\cite{Saramito}, \\cite{BoyerFabrie}).  Here, we consider the\nclass of fluids with extra stress tensor having\n$p$-$\\delta$-structure. This class includes and generalizes power law fluids,\nwhere the constitutive relation is given by \n\\begin{equation*}\n\t\\S(\\vec{Dv}) = \\mu_0 \\vec{Dv} + \\mu (\\delta + \\abs{\\vec{Dv}})^{p-2} \\vec{Dv}\n\\end{equation*}\nwith material constants $p \\in (1, \\infty)$,\n$\\mu_0, \\mu, \\delta \\geq 0$ (cf.~\\cite{Cetraro}).\n\n\\begin{defi}[extra stress tensor] \\label{defi:stress_tensor} An operator\n\t$\\S \\colon \\mathbb{R}_\\textrm{sym}^{d \\times d} \\to\n\t\\mathbb{R}_{\\textrm{sym}}^{d \\times d}$ is called an \\emph{extra\n\t\tstress tensor} with $p$-$\\delta$-structure if it is continuous,\n\tsatisfies $\\S(\\vec{0}) = \\vec{0}$ and if there exist constants\n\t$p \\in (1, \\infty)$, $\\delta \\geq 0$ and $C_1(\\S)$, $C_2(\\S) > 0$\n\tsuch that \n\t\\begin{equation} \\begin{split} \\label{eq:pdelta_growth1}\n\t(\\S(\\vec{A}) - \\S(\\vec{B}))\\cdot (\\vec{A}-\\vec{B}) &\\geq C_1(\\S)\n\t\\, (\\delta + \\abs{\\vec{B}} + \\abs{\\vec{A}-\\vec{B}})^{p-2}\n\t\\abs{\\vec{A}-\\vec{B}}^2,\n\t\\\\ \n\t\\abs{\\S(\\vec{A}) - \\S(\\vec{B})} &\\leq C_2(\\S) \\, (\\delta +\n\t\\abs{\\vec{B}} + \\abs{\\vec{A}-\\vec{B}})^{p-2}\n\t\\abs{\\vec{A}-\\vec{B}}\n\t\\end{split}\\end{equation}\n\tholds for all $\\vec{A}, \\vec{B} \\in \\mathbb{R}_\\text{sym}^{d \\times\n\t\td}$. The constants $C_1(\\S), C_2(\\S)$ and $p$ are called the\n\tcharacteristics of $\\S$. \n\\end{defi}\n\n\n\\begin{lem} [{\\cite{Cetraro}}] \\label{lem:pdelta_growth}\n\tLet $\\S$ be an extra stress tensor with $p$-$\\delta$-structure. Then, it holds\n\t\\begin{equation*}\n\t\t\\fp{\\S(\\vec{Dv}) - \\S(\\vec{Dw})}{\\vec{Dv}-\\vec{Dw}}\n\t\t\\geq C_3(\\S) \\int_{\\Omega} \\int_0^{\\abs{\\vec{Dv}-\\vec{Dw}}} (\\abs{\\vec{Dw}} + \\delta + s)^{p-2}s \\, ds\\, dx\n\t\\end{equation*}\n\tfor $\\vec{v}, \\vec{w} \\in W^{1,p}(\\Omega)$ with a constant $C_3(\\S)$\n\tthat only depends on the characteristics of $\\S$.\n\\end{lem}\n\n\\begin{comment}\n\\begin{defi}[extra stress tensor]\\emph{\\cite{Cetraro}}\nA map $\\S \\colon \\mathbb{R}^{d \\times d} \\to \\mathbb{R}_{\\textrm{sym}}^{d \\times d}$ is called an \\emph{extra stress tensor} if it is continuous on $\\mathbb{R}^{d \\times d}$, differentiable on $\\mathbb{R}^{d \\times d} \\setminus \\{ \\vec{0} \\}$, satisfies $\\S(\\vec{0}) = \\vec{0}$ and if it depends only on the symmetric part of the argument, i.e., $\\S(\\vec{A}) = \\S(\\tfrac{1}{2}(\\vec{A}+\\vec{A}^\\top))$.\n\nAn extra stress tensor $\\S$ has $p$-$\\delta$-structure if there exist constants $p \\in (1, \\infty)$, $\\delta \\geq 0$ and $C_1, C_2 > 0$ such that\n\\begin{align}\n\\sum_{i,j,k,l = 1}^{d} \\del_{kl} \\S_{ij}(\\vec{A}) \\vec{B}_{ij} \\vec{B}_{kl} &\\geq C_1 (\\delta + \\abs{\\vec{A}})^{p-2} \\abs{\\vec{B}}^2, \\label{eq:pdelta_cond1} \\\\\n\\abs{\\del_{kl} \\S_{ij} (\\vec{A})} &\\leq C_2 (\\delta + \\abs{\\vec{A}})^{p-2} \\label{eq:pdelta_cond2}\n\\end{align}\nholds for all $\\vec{A}, \\vec{B} \\in \\mathbb{R}_\\text{sym}^{d \\times d}$ with $\\vec{A} \\neq \\vec{0}$ and all $i,j,k,l \\in \\{1, ..., d\\}$.\n\nThe constants $C_1, C_2$ and $p$ are called the characteristics of $\\S$.\n\\end{defi}\n\nThe following Lemma shows how the growth behavior of the extra stress\ntensor $\\S$ is determined by the $p$-$\\delta$-structure:\n\\begin{lem} \\emph{\\cite{Cetraro}}\nLet $\\S$ be an extra stress tensor with\n$p$-$\\delta$-structure. Then, we have the following relations, where\nthe constants only depend on the characteristics of $\\S$:\n\\begin{enumerate}\n\\item For all\n$\\vec{A}, \\vec{B} \\in \\mathbb{R}_{\\textrm{sym}}^{d \\times d}$, we\nhave\n\\begin{equation}\n\\begin{split}\n\\abs{\\S(\\vec{A}) - \\S(\\vec{B})} &\\sim (\\delta + \\abs{\\vec{B}}\n+ \\abs{\\vec{A}-\\vec{B}})^{p-2} \\abs{\\vec{A}-\\vec{B}},  \\\\\n(\\S(\\vec{A}) - \\S(\\vec{B}))\\cdot (\\vec{A}-\\vec{B}) &\\sim\n(\\delta + \\abs{\\vec{B}} + \\abs{\\vec{A}-\\vec{B}})^{p-2}\n\\abs{\\vec{A}-\\vec{B}}^2.\n\\end{split}\n\\end{equation}\n\\item For $\\vec{v}, \\vec{w} \\in W^{1,p}(\\Omega)$, we have\n\\begin{equation*}\n\\fp{\\S(\\vec{Dv}) - \\S(\\vec{Dw})}{\\vec{Dv}-\\vec{Dw}}\n\\sim \\int_{\\Omega} \\int_0^{\\abs{\\vec{Dv}-\\vec{Dw}}} (\\abs{\\vec{Dw}} + \\delta + s)^{p-2}s \\, ds\\, dx.\n\\end{equation*}\n\\end{enumerate}\n\\end{lem}\n\\end{comment}\n\nSince we represented the inhomogeneous data in \\eqref{eq:main_problem}\nby a fixed function $\\vec{g}$ and since we want to solve\n\\eqref{eq:main_problem} by the ansatz $\\vec{v}=\\vec{u}+\\vec{g}$ with\n$\\vec{u} \\in V_p$, we shall work with a shifted version of the viscous\nstress tensor. Therefore, we define the \\emph{induced} operator\n$\\vec{S}\\colon W_0^{1,p}(\\Omega) \\to W_0^{1,p}(\\Omega)\\d$  via\n\\begin{equation} \\label{eq:defi_S}\n\\fp{\\vec{S}(\\vec{v})}{\\vec{\\phi}}\n:= \\fp{\\S(\\vec{Dv}+\\vec{Dg})}{\\vec{D\\phi}}\n\\end{equation}\nfor $\\vec{v}, \\vec{\\phi} \\in W_0^{1,p}(\\Omega)$.\n\n\\begin{lem}[Properties of $\\vec{S}$] \\label{lem:properties_S} Let $\\S$\n\tbe an extra stress tensor with $p$-$\\delta$-structure,\n\t$p \\in (1, 2]$ and $\\vec{g} \\in W^{1,p}(\\Omega)$. Then the induced\n\toperator~$\\vec{S}$ defined in \\eqref{eq:defi_S} is well-defined,\n\tbounded and continuous.\n\\end{lem}\n\\begin{proof}\n\tUsing \\eqref{eq:pdelta_growth1}$_2$ with $\\vec{A} = \\vec{Dw}$ and\n\t$\\vec{B} = \\vec{0}$, we obtain\n\t\\begin{equation*}\n\t\t\\abs{\\S(\\vec{Dw})}^{p'} \n\t\t\\leq C_2(\\S)^{p'} \\left[ (\\abs{\\vec{Dw}}+\\delta)^{p-2} \\abs{\\vec{Dw}} \\right]^{p'}\n\t\t\\leq C_2(\\S)^{p'} (\\abs{\\vec{Dw}}+\\delta)^p\n\t\\end{equation*}\n\tand consequently \n\t\\begin{equation} \\label{eq:S_bdd}\n\t\\norm{\\S(\\vec{Dw})}_{p'} \\leq\n\tC_2(\\S) \\bignorm{\\abs{\\vec{Dw}}+\\delta}_p^{p-1}\n\t\\end{equation}\n\tfor any $\\vec{w} \\in W^{1,p}(\\Omega)$.  From this, we deduce that\n\t$\\vec{S}\\colon W_0^{1,p}(\\Omega) \\to W_0^{1,p}(\\Omega)\\d$ is\n\twell-defined and bounded.\n\t\n\tIn order to prove continuity, let\n\t$\\vec{v^n} \\to \\vec{v} \\in W_0^{1,p}(\\Omega)$ be a convergent\n\tsequence. Then, by the H\\\"older inequality and by\n\t\\eqref{eq:pdelta_growth1}$_2$ we get \n\t\\begin{align*}\n\t\t\\Vert\\vec{S}(\\vec{v^n}) - \\vec{S}(\\vec{v})\n\t\t&\\Vert_{W_0^{1,p}(\\Omega)\\d}\n\t\t\\leq \\norm{\\S(\\vec{Dv^n}+\\vec{Dg}) - \\S(\\vec{Dv}+\\vec{Dg})}_{p'} \\\\\n\t\t&\\leq C_2(\\S) \\norm{\\left(\\delta + \\abs{\\vec{Dv}+\\vec{Dg}} +\n\t\t\t\\abs{\\vec{Dv^n}-\\vec{Dv}}\\right)^{p-2}  \n\t\t\t\\abs{\\vec{Dv^n}-\\vec{Dv}}}_{p'} \\\\\n\t\t&\\leq C_2(\\S) \\norm{\\vec{Dv^n}-\\vec{Dv}}_p^\\frac{p}{p'}\n\t\t\\xrightarrow{n \\to \\infty} 0.\n\t\t\\\\[-12mm]\n\t\\end{align*}\n\\end{proof}\n\nOur next goal is to describe coercivity properties of the operator\n$\\vec{S}$. For the proof of a good lower bound of $\\vec{S}$, we\nprove an auxiliary algebraic result.\n\n\\begin{lem} \\label{lem:estim_young_fct}\n\tLet $a, t \\geq 0$ and $p \\in (1,2 ] $. Then it holds\n\t\\begin{equation*}\n\t\t\\int_{0}^{t} (a+s)^{p-2} s \\, ds \\geq \\tfrac{1}{p} t^p - ta^{p-1}.\n\t\\end{equation*}\n\\end{lem}\n\\begin{proof}\n\tThe statement becomes trivial if $a = 0$, so we may assume $a >\n\t0$. For all $s \\geq 0$, it holds\n\t$\\frac{a}{(a+s)^{2-p}} \\leq \\frac{a}{a^{2-p}} = a^{p-1}$.  We\n\testimate\n\t\\begin{equation*}\n\t\t\\frac{s}{(a+s)^{2-p}} = (a+s)^{p-1} - \\frac{a}{(a+s)^{2-p}} \\geq\n\t\t(a+s)^{p-1} - a^{p-1} \\geq s^{p-1} - a^{p-1} \n\t\\end{equation*}\n\tand by integration we obtain the result.\n\\end{proof}\n\n\\begin{comment}\nFor $p \\in (1,2)$, the function $x \\mapsto x^{p-1}$ is not\nconvex. However, we can still get a Jensen-type inequality:\n\\marginpar{still needed?}%\n\\begin{lem} \\label{lem:rev_jensen}\nFor $x, y \\geq 0$ and $p \\in (1,2 ] $, it holds\n\\begin{equation*}\n(x+y)^{p-1} \\leq x^{p-1} + y^{p-1}.\n\\end{equation*}\n\\end{lem}\n\\begin{proof}\nThe assertion becomes obvious in the case $y = 0$. Since the claimed\ninequality is homogeneous, we may fix $y = 1$ without loss of\ngenerality.  The auxiliary function\n$f \\colon \\R _{\\geq 0} \\to \\R _{\\geq 0}\\colon x \\mapsto x^{p-1} + 1\n- (x+1)^{p-1}$ is continuous on $\\R _{\\geq 0} $, satisfies\n$f(0) = 0$ and it is differentiable on $\\R _{ x > 0 }$ with a\nnon-negative derivative. Thus $f(x)$ is non-negative for all\n$x \\geq 0$, which yields the result.\n\\end{proof}\n\\end{comment}\n\nWith this tool, we are able to prove a lower bound for $\\vec{S}$: \n\\begin{lem}[Lower bound for $\\vec{S}$] \\label{lem:S_estimate} For a\n\tgiven extra stress tensor $\\S$ with $p$-$\\delta$-structure,\n\t$p \\in (1, 2 ] $, and a function $\\vec{g} \\in W^{1,p}(\\Omega)$, the\n\tinduced operator\n\t$\\vec{S}\\colon W_0^{1,p}(\\Omega) \\to W_0^{1,p}(\\Omega)\\d$, defined\n\tin \\eqref{eq:defi_S}, satisfies  the lower bound \n\n\n\n\n\n\n\t\\begin{equation*}\n\t\t\\fp{\\vec{S}(\\vec{v})}{\\vec{v}} \n\t\t\\geq \\tfrac{C_3(\\S)}{p} \\norm{\\vec{Dv}}_p^p - \\big ( C_2(\\S) +\n\t\tC_3(\\S) \\big) \\bignorm{\\abs{\\vec{Dg}}+\\delta}_p^{p-1}\n\t\t\\norm{\\vec{Dv}}_p \n\t\\end{equation*}\n\tfor all $\\vec{v} \\in W_0^{1,p}(\\Omega)$.\n\n\\end{lem}\n\n\\begin{proof}\n\tWe apply Lemma \\ref{lem:pdelta_growth} with\n\t$\\vec{v} = \\vec{v}+\\vec{g}$ and $\\vec{w}=\\vec{g}$ and Lemma\n\t\\ref{lem:estim_young_fct} to estimate\n\t\\begin{align*}\n\t\t\\fp{\\S(\\vec{Dv}+\\vec{Dg}) - \\S(\\vec{Dg})}{\\vec{Dv}}\n\t\t&\\geq C_3(\\S) \\int_{\\Omega} \\int_{0}^{\\abs{\\vec{Dv}}}\n\t\t(\\abs{\\vec{Dg}} + \\delta + s)^{p-2} s \\, ds \\, dx\n\t\t\\\\\n\t\t&\\geq C_3(\\S) \\int_{\\Omega} \\tfrac{1}{p} \\abs{\\vec{Dv}}^p -\n\t\t\\abs{\\vec{Dv}} (\\abs{\\vec{Dg}} + \\delta)^{p-1} \\, dx.\n\t\\end{align*}\n\tThis, the H\\\"older inequality and \\eqref{eq:S_bdd} with\n\t$\\vec{w} = \\vec{g}$\n\t\\begin{align*}\n\t\t\\fp{\\vec{S}(\\vec{v})}{\\vec{v}}\n\t\t&= \\fp{\\S(\\vec{Dv}+\\vec{Dg}) - \\S(\\vec{Dg})}{\\vec{Dv}} +\n\t\t\\fp{\\S(\\vec{Dg})}{\\vec{Dv}} \\nonumber\n\t\t\\\\\n\t\t&\\geq \\tfrac{C_3(\\S)}{p} \\norm{\\vec{Dv}}_p^p \n\t\t- \\left[C_3(\\S)\\, \\big\\Vert\n\t\t\\left(\\abs{\\vec{Dg}}+\\delta\\right)^{p-1} \\big\\Vert_{p'} +\n\t\t\\norm{\\S(\\vec{Dg})}_{p'}\\right] \\norm{\\vec{Dv}}_p\n\t\t\\\\ \n\t\t&\\geq \\tfrac{C_3(\\S)}{p} \\norm{\\vec{Dv}}_p^p \n\t\t- \\big (C_2(\\S) + C_3(\\S)\\big )\n\t\t\\bignorm{\\abs{\\vec{Dg}}+\\delta}_p^{p-1}\n\t\n\t\t\\norm{\\vec{Dv}}_p , \n\t\\end{align*}\n\twhich is the assertion.\n\\end{proof}\n\nIn the treatment of the inhomogeneous problem \\eqref{eq:main_problem},\nwe will have to deal with the shifted extra stress tensor\n$\\vec{A} \\mapsto \\S(\\vec{A} + \\vec{G})$ for some constant symmetric\nmatrix $\\vec{G}$. In order to get a precise description of the growth\nbehavior of this mapping, we introduce the notion of locally uniform\nmonotonicity:\n\n\\begin{defi}[Locally uniform monotonicity] \\label{defi:uni_monotone}\n\tLet $X$ be a reflexive Banach space and $A\\colon X \\to X\\d$ an\n\toperator.  The operator $A$ is called \\emph{locally uniformly\n\t\tmonotone on $X$}\n\tif for every $y\\in X$ there exists a strictly monotonically\n\tincreasing function $\\rho_y\\colon [0, \\infty) \\to [0, \\infty)$ with\n\t$\\rho_y(0) = 0$ such that for all $x\\in X$ holds\n\t\\begin{equation} \\label{eq:uni_monotone}\n\t\\fp{Ax-Ay}{x-y} \\geq \\rho_y(\\norm{x-y}_X)\\,.\n\t\\end{equation}\n\n\\end{defi}\n\nBy the lower bound \\eqref{eq:pdelta_growth1}$_1$, we obtain that\n(possibly shifted) extra stress tensors are locally uniformly\nmonotone.\n\n\\begin{lem} \\label{lem:uni_monotonicity}\n\tLet $\\S\\colon \\R_{\\text{sym}}^{d\\times d}\\to \\R_{\\text{sym}}^{d\\times\n\t\td}$ be an extra stress tensor with $p$-$\\delta$-structure and\n\t$\\vec{G} \\in \\R_{\\text{sym}}^{d\\times d}$ be a symmetric\n\tmatrix. Then the shifted extra stress tensor\n\t$\\vec{A} \\mapsto \\S(\\vec{A}+\\vec{G})$ is a locally uniformly\n\tmonotone operator on $\\R_{\\text{sym}}^{d\\times d}$.\n\\end{lem}\n\\begin{proof}\n\tBy \\eqref{eq:pdelta_growth1}$_1$, we obtain for any $\\vec{A},\n\t\\vec{B} \\in \\R_{\\text{sym}}^{d\\times d}$ \n\t\\begin{align*}\n\t\t&(\\S(\\vec{A}\\!+\\!\\vec{G}) \\! -\\! \\S(\\vec{B}\\!+\\!\\vec{G})) \\cdot\n\t\t(\\vec{A}\\!-\\!\\vec{B})\n\t\t\\geq C_1(\\S)  \\big(\\delta \\!+\\! \\abs{\\vec{B}\\!+\\!\\vec{G}}\\! +\\!\n\t\t\\abs{\\vec{A}\\!-\\!\\vec{B}}\\big)^{p-2} \\abs{\\vec{A}\\!-\\!\\vec{B}}^2.\n\t\\end{align*}\n\tFor any $\\vec{B}$, the function\n\t$\\rho_{\\vec{B}}(t) := C_1(\\S) (\\delta + \\abs{\\vec{B}+\\vec{G}} +\n\tt)^{p-2} t^2$ is non-negative, satisfies $\\rho_{\\vec{B}}(0) = 0$,\n\tand it is strictly monotonically increasing since for its derivative\n\tit holds\n\t\\begin{equation*}\n\t\t\\rho'_{\\vec{B}}(t) \n\t\t= C_1(\\S) (\\delta + \\abs{\\vec{B}+\\vec{G}} + t)^{p-3} t (2\\delta +\n\t\t2\\abs{\\vec{B}+\\vec{G}} + pt) > 0 \n\t\\end{equation*}\n\tfor all $t > 0$. Therefore, it fulfils the requirements from\n\tDefinition \\ref{defi:uni_monotone}.\n\\end{proof}\n\n\\subsection{Properties of the convective term} \\label{sub:conv_term}\n\nSince we fixed a function $\\vec{g}$ that expresses the inhomogeneous\ndata in \\eqref{eq:main_problem}, we shall work with a \"shifted\"\nversion of the convective term\n$\\fp{(\\vec{u}+\\vec{g}) \\, \\cdot \\nabla (\\vec{u}+\\vec{g})}{\\vec{\\phi}}$\nthat is integrable and thus well-defined even for\n$p > \\tfrac{2d}{d+2}$ and sufficiently regular $\\vec{\\phi}$ and\n$\\vec{g}$.  Therefore, we set\n\\begin{equation}\\label{eq:s}\ns=s(p) := \\max \\Big\\{ p, \\Big(\\frac{p\\d}{2}\\Big)' \\Big\\}\n= \\left\\{ \\begin{array}{cl}\np &\\text{if }p > \\frac{3d}{d+2},\n\\\\\n\\left(\\frac{p\\d}{2}\\right)' &\\text{if } p \\leq \\frac{3d}{d+2}\n\\end{array} \\right.\n\\end{equation}\nfor $p \\in \\big(\\tfrac{2d}{d+2}, 2\\big)$ and define the convective\nterm $\\vec{T}\\colon V_p \\to W_0^{1,s}(\\Omega)\\d$ via\n\\begin{equation} \\label{eq:defi_T}\n\\fp{\\vec{T}(\\vec{u})}{\\vec{\\phi}}\n:= - \\fp{(\\vec{u}+\\vec{g}) \\otimes (\\vec{u}+\\vec{g})}{\\vec{D\\phi}} -\n\\fp{(\\div \\vec{g})(\\vec{u}+\\vec{g})}{\\vec{\\phi}}\n\\end{equation}\nfor $\\vec{u} \\in V_p$ and $\\vec{\\phi} \\in W_0^{1,s}(\\Omega)$.\n\n\\begin{lem}[Properties of the convective term] \\label{lem:t2_str_contin}\n\tFor $p \\in \\big(\\frac{2d}{d+2}, 2\\big)$ let $s $ be defined in\n\t\\eqref{eq:s}\n\tand let $q \\in \\R$ satisfy $q \\geq s$ and $q > (\\frac{p\\d}{2})'$. Then, for any given function\n\t$\\vec{g} \\in W^{1,s}(\\Omega)$, the operator $\\vec{T}$ defined in \n\t\\eqref{eq:defi_T} is formally equivalent to\n\t$\\fp{(\\vec{u}+\\vec{g}) \\, \\cdot \\nabla\n\t\t(\\vec{u}+\\vec{g})}{\\vec{\\phi}}$. It is well-defined and bounded\n\tfrom $V_p$ to $W_0^{1,s}(\\Omega)\\d$ and also from $V_p$ to\n\t$W_0^{1,q}(\\Omega)\\d$. The operator  $\\vec{T}$ is continuous from $V_p$ to\n\t$W_0^{1,s}(\\Omega)\\d$ and strongly continuous from\n\t$V_p$ to $W_0^{1,q}(\\Omega)\\d$.  It fulfils the estimate\n\t\\begin{align}\\label{eq:econv}\n\t\t\\begin{split}\n\t\t\t\\abs{\\fp{\\vec{T}(\\vec{u})}{\\vec{u}}} &\\leq c_\\text{Sob}\\,\n\t\t\tc_\\text{Korn}^2 \\left( \\norm{ \\vec{Dg}}_s + \\tfrac{1}{2}\n\t\t\t\\norm{\\div \\vec{g}}_s \\right) \\norm{\\vec{Du}}_p^2\n\t\t\t\\\\\n\t\t\t&\\quad + c_\\text{Sob}\\, \\big( \\norm{\\vec{g}}_{1,s}^2 +\n\t\t\tc_\\text{Korn} \\norm{\\div \\vec{g}}_s \\norm{\\vec{g}}_{1,s} \\big)\n\t\t\t\\norm{\\vec{Du}}_p\n\t\t\\end{split}\n\t\\end{align}\n\tfor all $\\vec{u} \\in V_q$, where\n\t$c_\\text{Sob}$ are Sobolev embedding constants and\n\t$c_\\text{Korn}$ is the constant in the  Korn inequality for $\\Omega$.\n\\end{lem}\n\n\\begin{proof}\n\tThe formal equivalence follows from a straightforward computation\n\twith integration by parts.  We abbreviate $g_1 := \\div \\vec{g} \\in\n\tL^s(\\Omega)$ and use the continuous Sobolev embeddings\n\t$W^{1,s}(\\Omega) \\injto W^{1,p}(\\Omega) \\injto\n\tL^{p\\d}(\\Omega)$. The definition of $s$ implies $\\frac{1}{p\\d} +\n\t\\frac{1}{p\\d}+ \\frac{1}{s} \\leq\n\t1$, so both well-definedness of $\\vec{T}(\\vec{u}) \\in\n\tW_0^{1,s}(\\Omega)\\d$ for $\\vec{u} \\in\n\tV_p$ and boundedness follow by the H\\\"older inequality.\n\t\n\tIn view of the continuous embedding\n\t$W_0^{1,s}(\\Omega)\\d \\injto W_0^{1,q}(\\Omega)\\d$, we immediately\n\tobtain well-definedness and boundedness if $\\vec{T}$ is considered\n\tas an operator from $V_p$ to $W_0^{1,q}(\\Omega)\\d$.\n\t\n\tSince $\\frac{1}{p\\d} + \\frac{1}{p\\d}+ \\frac{1}{q} < 1$, there is\n\tsome $\\tau < p\\d$ such that\n\t$\\frac{1}{\\tau} + \\frac{1}{p\\d}+ \\frac{1}{q} = 1$. Let\n\t$\\vec{u^n} \\weakto \\vec{u} \\in V_p$~be a weakly convergent\n\tsequence. The Sobolev embedding\n\t$W^{1,p}(\\Omega) \\injto\\injto L^{\\tau}(\\Omega)$ is compact, so\n\t$\\vec{u^n} \\to \\vec{u} \\in L^{\\tau}(\\Omega)$ converges strongly.\n\tThus, we estimate\n\t\\begin{align*}\n\t\t&\\sup_{\\norm{\\vec{D\\phi}}_q \\leq 1} \n\t\t\\abs{\\fp{(\\vec{u^n} +\\vec{g}) \\otimes (\\vec{u^n}+\\vec{g}) \n\t\t\t\t- (\\vec{u}+\\vec{g}) \\otimes (\\vec{u}+\\vec{g})}{\\vec{D\\phi}}} \n\t\t\\\\\n\t\t&\\quad = \\sup_{\\norm{\\vec{D\\phi}}_q \\leq 1} \\big \\vert \\langle\\vec{u^n} \\otimes (\\vec{u^n}-\\vec{u}) \n\t\t+ (\\vec{u^n}-\\vec{u}) \\otimes \\vec{u} \\\\\n\t\t&\\qquad\\qquad\\qquad \t+ \\vec{g} \\otimes (\\vec{u^n}-\\vec{u}) \n\t\t+ (\\vec{u^n}-\\vec{u}) \\otimes \\vec{g}, \\vec{D\\phi}\\rangle \\big \\vert \n\t\t\\\\\n\t\t&\\quad \\leq \\norm{\\vec{u^n}}_{p\\d} \\norm{\\vec{u^n}-\\vec{u}}_\\tau \n\t\t+ \\norm{\\vec{u^n}-\\vec{u}}_\\tau \\norm{\\vec{u}}_{p\\d}\n\t\t+ 2 \\norm{\\vec{g}}_{p\\d} \\norm{\\vec{u^n}-\\vec{u}}_\\tau\n\t\t\\xrightarrow{\\,n \\to \\infty\\,} 0.\n\t\\end{align*}\t\n\tSimilarly, we obtain\n\t\\begin{equation*}\n\t\t\\sup_{\\norm{\\vec{D\\phi}}_q \\leq 1} \\abs{\\fp{g_1(\\vec{u^n}-\\vec{u})}{\\vec{\\phi}}}\n\t\t\\leq C \\norm{g_1}_s \\norm{\\vec{u^n}-\\vec{u}}_\\tau\n\t\t\\xrightarrow{n \\to \\infty} 0.\n\t\\end{equation*}\n\tThus, we proved $\\vec{T}(\\vec{u^n}) \\to \\vec{T}(\\vec{u})$ in\n\t$W_0^{1,q}(\\Omega)\\d$, i. e.\n\t$\\vec{T}\\colon V_p \\to W_0^{1,q}(\\Omega)\\d$ is strongly continuous.\n\t\n\tAnalogously, we prove continuity for\n\t$\\vec{T} \\colon V_p \\to W_0^{1,s}(\\Omega)\\d$ using\n\t$\\vec{u^n} \\to \\vec{u} \\in V_p$ and the continuous embedding\n\t$W^{1,p}(\\Omega) \\injto L^{p\\d}(\\Omega)$.\n\t\n\tFor the bound \\eqref{eq:econv} of $\\vec{T}$, we use \n\t\\begin{equation} \\label{eq:t2_eq1}\n\t\\fp{\\vec{u}\\otimes\\vec{u}}{\\vec{Du}} = 0\n\t\\end{equation}\n\twhich follows by integration by parts, since $\\vec{u}$ has zero\n\tdivergence and zero boundary values.  In the same way, we see\n\t\\begin{equation} \\label{eq:t2_eq2}\n\t\\fp{\\vec{u} \\otimes\n\t\t\\vec{g}}{\\nabla \\vec{u}} = - \\fp{\\vec{u} \\otimes\n\t\t\\vec{u}}{\\vec{Dg}}\n\t\\end{equation}\n\tand\n\t\\begin{equation} \\label{eq:t2_eq3}\n\t\\fp{\\vec{u} \\otimes\n\t\t\\vec{g}}{\\nabla \\vec{u}^T} = -\\tfrac{1}{2}\\, \\fp{g_1\n\t\t\\vec{u}}{\\vec{u}}.\n\t\\end{equation}\n\tUsing \\eqref{eq:t2_eq1}, \\eqref{eq:t2_eq2} and \\eqref{eq:t2_eq3}\n\tin the definition of $\\vec{T}$, we obtain the following expression\n\tfor the convective term:\n\t\\begin{align}\n\t\t\\langle \\vec{T} (\\vec{u}), \\vec{u}\\rangle\n\t\t&= \\fp{\\vec{u} \\otimes \\vec{u}}{\\vec{Du}} + \\fp{\\vec{u} \\otimes\n\t\t\t\\vec{g}}{\\nabla \\vec{u}} \n\t\t+ \\fp{\\vec{u} \\otimes \\vec{g}}{\\nabla \\vec{u}^T} + \\fp{\\vec{g}\n\t\t\t\\otimes \\vec{g}}{\\vec{Du}} \\nonumber\n\t\t\\\\\n\t\t&\\quad + \\fp{g_1 \\vec{u}}{\\vec{u}} + \\fp{g_1 \\vec{g}}{\\vec{u}}\n\t\t\\nonumber\n\t\t\\\\\n\t\t&= - \\fp{\\vec{u} \\otimes \\vec{u}}{\\vec{Dg}} + \\tfrac{1}{2}\\,\n\t\t\\fp{g_1 \\vec{u}}{\\vec{u}}   + \\fp{\\vec{g} \\otimes\n\t\t\t\\vec{g}}{\\vec{Du}} + \\fp{g_1\\vec{g}}{\\vec{u}}. \\label{eq:t2_eq4} \n\t\\end{align}\n\tIn order to estimate this expression, we use the Sobolev embedding\n\t$W^{1,s}(\\Omega) \\injto L^{2p'}(\\Omega)$. In fact, in the case\n\t$s \\geq d$ this follows directly. If $\\frac{3d}{d+2} < p < 2$ and\n\t$s<d$, we have $s\\d = p\\d > 2p'$, due to a straightforward\n\tcomputation. If $\\frac{2d}{d+1} < p \\leq \\frac{3d}{d+2}$ and\n\t$s < d$, we get\n\t$s\\d \\geq \\big(\\big(\\frac{p\\d}{2}\\big)'\\big)\\d = \\frac{pd}{pd-2d+p}\n\t\\geq 2p'$. Finally, if $\\frac{2d}{d+2} < p \\leq \\frac{2d}{d+1}$,\n\tthen it holds $s \\geq \\big(\\frac{p\\d}{2}\\big)' \\geq d$.  Applying\n\tthe H\\\"older and the Korn inequality and the embeddings\n\t$W^{1,s}(\\Omega) \\injto L^{2p'}(\\Omega)$ and\n\t$W^{1,p}(\\Omega) \\injto L^{p\\d}(\\Omega)$ to \\eqref{eq:t2_eq4}, the \n\tclaimed estimate follows.\n\\end{proof}\n\n\\subsection{Lipschitz truncation} \\label{sub:lip_trunc}\nIn case of a small growth parameter $p$, this means\n$p \\in \\big( \\frac{2d}{d+2}, \\frac{3d}{d+2} \\big)$, a function\n$\\vec{v} \\in W^{1,p}(\\Omega)$ does not have enough integrability to be\nchosen as a test function in operators like\n$\\fp{\\vec{v}\\otimes \\vec{v}}{\\vec{D\\phi}}$.  Hence, we use\nsufficiently smooth approximations of the test functions in the limit\nprocess of the existence proof, which are given by the Lipschitz\ntruncation method. The existence of Lipschitz truncations is\nguaranteed by the following result proved in \\cite{fms2}, \\cite{dms}, \\cite{Cetraro}:\n\\begin{thm}[Lipschitz truncation] \\label{thm:lip_trunc}\n\tLet $\\Omega$ be a bounded domain with Lipschitz continuous\n\tboundary, let $p \\in (1, \\infty)$ and let \n\t$(\\vec{v^n})_{n \\in \\N} \\subset W_0^{1, p}(\\Omega)$ be a sequence\n\tsuch that   $\\vec{v^n} \\weakto \\vec{0}$ weakly.\n\t\n\tThen, for all $j, n \\in \\N$, there exists a function\n\t$\\vec{v_j^n} \\in W_0^{1, \\infty}(\\Omega)$ and a number\n\t$\\lambda_j^n \\in \\big[ 2^{2^j}, 2^{2^{j+1}} \\big]$ such that\n\t\\begin{align} \\label{eq:lip_trunc1} \\begin{split}\n\t\t\t\\lim\\limits_{n \\to \\infty} \\big( \\sup\\limits_{j \\in \\N}\n\t\t\t\\norm{\\vec{v_j^n}}_{\\infty} \\big) &= 0,\n\t\t\t\\\\ \n\t\t\t\\norm{\\nabla \\vec{v_j^n}}_{\\infty} &\\leq c \\, \\lambda_j^n,\n\t\t\t\\\\\n\t\t\t\\limsup_{n \\to \\infty} \\left( \\lambda_j^n \\right)^p \\abs{\\{\n\t\t\t\t\\vec{v_j^n} \\neq \\vec{v^n} \\}} &\\leq c \\, 2^{-j},\n\t\t\t\\\\ \n\t\t\t\\limsup_{n \\to \\infty} \\norm{\\nabla \\vec{v_j^n} \\, \\chi_{\\{\n\t\t\t\t\t\\vec{v_j^n} \\neq \\vec{v^n} \\}}}_p^p &\\leq c \\, 2^{-j}\n\t\\end{split} \\end{align}\n\tholds with a uniform constant $c = c(d, p, \\Omega)$. \n\t\n\tMoreover, for fixed $j \\in \\N$ and $r \\in [1, \\infty)$, we have\n\t\\begin{align} \\label{eq:lip_trunc2}\n\t\t\\begin{split}\n\t\t\t\\nabla \\vec{v_j^n} \\weakto \\vec{0} &\\quad \\text{ in }\n\t\t\tL^r(\\Omega),\n\t\t\t\\\\\n\t\t\t\\nabla \\vec{v_j^n} \\weakstarto \\vec{0} &\\quad \\text{ in }\n\t\t\tL^{\\infty}(\\Omega) \n\t\t\\end{split}\n\t\\end{align}\n\tas $n \\to \\infty$.\n\\end{thm}\n\nThe following lemma shows how Lipschitz truncation can be used to get\na connection between weak and almost everywhere convergence. Statement\nand proof are close to \\cite[Lemma 2.6]{dms}, only the\nassumptions on the operator $\\S$ have been reduced for the reasons\ndiscussed in the previous Subsection~\\ref{sub:extra_stress}.\n\n\\begin{lem}[Almost everywhere convergence for the Lipschitz truncation\n\tmethod] \\label{lem:ae_equality2} Let $\\Omega$ be a bounded domain,\n\t$p \\in (1, \\infty)$,\n\t$(\\vec{u^n})_{n \\in \\N} \\subset W^{1,p}(\\Omega)$ be a weakly\n\tconvergent sequence with limit $\\vec{u} \\in W^{1,p}(\\Omega)$.  Let\n\t$\\A\\colon \\R_{\\text{sym}}^{d\\times d}\\to \\R_{\\text{sym}}^{d \\times\n\t\td}$ be a locally uniformly monotone operator on $\\R_{\\text{sym}}^{d\\times d}$ such that the induced\n\toperator $\\vec{w} \\mapsto \\A(\\vec{Dw})$ is well-defined and\n\tbounded in $W^{1,p}(\\Omega) \\to L^{p'}(\\Omega)$.\n\n\n\n\n\n\n\n\t\n\tNow let $B$ be a ball with $2B \\subset \\subset \\Omega$ and\n\t$\\xi \\in C_0^\\infty (\\Omega)$ be a cutoff function such that \n\t$\\chi_B \\leq \\xi \\leq \\chi_{2B}$. We set\n\t$\\vec{v^n} := \\xi (\\vec{u^n}- \\vec{u})$ and let $\\vec{v_j^n}$ be the\n\tLipschitz truncation of $\\vec{v^n}$ with respect to the domain $2B$\n\tas described in Theorem \\ref{thm:lip_trunc}. If we have\n\t\\begin{equation} \\label{eq:ae_equality_assum}\n\t\\lim_{n \\to \\infty} \\abs{\\fp{\\A(\\vec{Du^n}) -\n\t\t\t\\A(\\vec{Du})}{\\vec{Dv_j^n}}} \\leq \\delta_j \n\t\\end{equation}\n\tfor all $j \\in \\N$ and some sequence $(\\delta_j)_{j \\in \\N}$ with\n\t$\\lim_{j \\to \\infty} \\delta_j = 0$, then a subsequence of\n\t$\\vec{Du^n}$ converges to $\\vec{Du}$ almost everywhere in $B$.\n\\end{lem}\n\n\\begin{rem}\\label{rem:S}\n\tIn Lemma \\ref{lem:uni_monotonicity} and lemma \\ref{lem:properties_S}\n\twe have seen that for an extra stress tensor\n\t$\\S\\colon \\R_{\\text{sym}}^{d\\times d}\\to \\R_{\\text{sym}}^{d\\times d}$\n\twith $p$-$\\delta$-structure and a given vector field\n\t$\\vec {g} \\in W^{1,p}(\\Omega)$ the operator\n\t$\\A(\\vec{B}):=\\S(\\vec{B} + \\vec{Dg})$ fulfils the requirements in\n\tLemma~\\ref{lem:ae_equality2}.\n\\end{rem}\n\n\\begin{proof}[Proof of Lemma \\ref{lem:ae_equality2}]\n\tLet $\\theta \\in (0, 1)$. Making use of the properties of $\\A$, we\n\tobtain strong convergence\n\t$\\left[(\\A(\\vec{Du^n})-\\A(\\vec{Du}))\\cdot\n\t(\\vec{Du^n}-\\vec{Du})\\right] ^\\theta \\to 0$ in $L^1(B)$ as\n\t$n \\to \\infty$ along the lines of \\cite[Lemma 2.6]{dms}. We switch to a\n\tsubsequence that converges almost everywhere.\n\tBy the definition of locally uniform monotonicity, there exists a\n\tstrictly monotonically increasing function\n\t$\\rho_x\\colon [0, \\infty) \\to [0, \\infty)$ with\n\t\\begin{align*}\n\t\t&(\\A(\\vec{Du^n}(x))-\\A(\\vec{Du}(x)))\\cdot\n\t\t(\\vec{Du^n}(x)-\\vec{Du}(x))\n\t\t\\geq \\rho_x   (\\abs{\\vec{Du^n}(x)-\\vec{Du}(x)}) \n\t\\end{align*}\n\tfor all $n\\in\\N$ and almost every $x\\in B$ ($\\rho_x$~depends on\n\t$\\vec{Du}(x)$). Utilizing the almost everywhere convergence of the\n\tleft-hand side and the non-negativity of the right-hand side, we\n\tobtain a subsequence that fulfils\n\t$\\rho_x (\\abs{\\vec{Du^n}(x)-\\vec{Du}(x)}) \\to 0$ almost\n\teverywhere. Thus, it holds $\\vec{Du^n}(x) \\to \\vec{Du}(x)$ as\n\t$n\\to\\infty$ for this subsequence and almost every~$x\\in B$.\t\n\\end{proof}\n\nBy applying a covering argument and taking the diagonal sequence we\nobtain a global version of Lemma \\ref{lem:ae_equality2}\n(cf.~\\cite[Cor. 3.32]{Cetraro}):\n\n\\begin{cor}\\label{cor:ae_equality3}\n\tAssume that the assumptions of Lemma \\ref{lem:ae_equality2} are fulfilled\n\tfor all balls $B$ with $2B \\subset \\subset \\Omega$ (with sequences\n\t$\\delta_j$ that may depend on the ball $B$). Then $\\vec{Du^n}$\n\tconverges to $\\vec{Du}$ almost everywhere on $\\Omega$ for a suitable\n\tsubsequence.\n\\end{cor}\n\nUsing the almost everywhere convergence established in Corollary\n\\ref{cor:ae_equality3}, we may prove a general statement about the\nlimit process with the Lipschitz truncation method in existence\nproofs:\n\n\\begin{thm}[Identification of limits using the Lipschitz truncation\n\tmethod] \\label{thm:ident_limits} Let $\\Omega \\subset \\Rd$ be a\n\tbounded domain, $p \\in (1, \\infty)$ and $\\vec{g}\\in W^{1,p}(\\Omega)$\n\tbe given. Let\n\t$\\A\\colon \\R_{\\text{sym}}^{d\\times d}\\to \\R_{\\text{sym}}^{d \\times\n\t\td}$ be a continuous, locally uniformly monotone operator on\n\t$\\R_{\\text{sym}}^{d\\times d}$ such that the induced operator\n\t$\\vec{w} \\mapsto \\A(\\vec{Dw})$ is well-defined and bounded\n\tin $W^{1,p}(\\Omega) \\to L^{p'}(\\Omega)$. Let there be an operator\n\t$\\vec{B} \\colon V_p \\to V_s\\d$ for some $s \\in [p, \\infty)$ and a\n\tspace $X$ such that $X \\injto V_p$ embeds continuously and such that\n\t$\\vec{B}$ is well-defined as an operator $X \\to X\\d$.  Assume we\n\thave a sequence of operators $\\vec{A_n}\\colon X \\to X\\d$ and\n\tsolutions $\\vec{u^n} \\in X$ to\n\t\\begin{equation} \\label{eq:lem_ident_appr}\n\t\\fp{\\A(\\vec{Du^n})}{\\vec{D\\phi}} +\n\t\\fp{\\vec{B}(\\vec{u^n})}{\\vec{\\phi}} +\n\t\\fp{\\vec{A_n}(\\vec{u^n})}{\\vec{\\phi}} = 0\n\t\\end{equation}\n\twith test functions $\\vec{\\phi} \\in X\\!$.\n\t\n\tIn addition, assume that for some $r \\in (1, \\infty)$ the embedding\n\t$V_r \\injto X$ is   continuous and dense, that $\\vec{B}$ is strongly\n\tcontinuous as an operator $V_p \\to W_0^{1,r}(\\Omega)\\d$ and that we have\n\tconvergences\n\t\\begin{align} \\label{eq:ident_assum}\n\t\t\\begin{split}\n\t\t\t\\vec{u^n} \\weakto \\vec{u} &\\quad \\text{weakly in } V_p,\\\\\n\t\t\t\\vec{A_n}(\\vec{u^n}) \\to \\vec{0}  &\\quad \\text{strongly in } W_0^{1,r}(\\Omega)\\d\n\t\t\\end{split}\n\t\\end{align}\n\tas $n \\to \\infty$.\n\t\n\tThen $\\vec{u}$ is a solution of the limit equation\n\t\\begin{equation} \\label{eq:lem_ident_limit}\n\t\\fp{\\A(\\vec{Du})}{\\vec{D\\phi}} + \\fp{\\vec{B}(\\vec{u})}{\\vec{\\phi}} = 0\n\t\\end{equation}\n\tfor all $\\vec{\\phi} \\in V_s$.\n\\end{thm}\n\n\\begin{rem}\n\tThe operator $\\A$ represents a (possibly shifted) extra stress\n\ttensor (cf.~Remark \\ref{rem:S}) and\n\t$\\vec{B}$ may be chosen as the convective term. Typical choices for\n\tthe space $X$ are $X = V_q$ or $X = V_p \\cap L^q(\\Omega)$ with\n\tcoercive operators\n\t$\\fp{\\vec{A_n}(\\vec{v})}{\\vec{\\phi}} = \\fp{\\abs{ \\vec{Dv}}^{q-2}\n\t\t\\vec{Dv}}{\\vec{D\\phi}}$ and\n\t$\\fp{\\vec{A_n}(\\vec{v})}{\\vec{\\phi}} = \\fp{\\abs{\\vec{v}}^{p-2}\n\t\t\\vec{v}}{\\vec{\\phi}}$ respectively.\n\t\n\tThe inclusions $X\\injto V_p$, $V_s \\injto V_p$ and $V_r \\injto X$\n\tguarantee the well-definedness of $\\vec{A_n}$ and of the operator\n\twhich is induced by $\\A$.\n\\end{rem}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:ident_limits}]\n\tThe proof of Theorem \\ref{thm:ident_limits} follows and generalizes\n\tthe procedure in \\cite{dms}, \\cite{Cetraro}. First, we check the assumptions of\n\tLemma \\ref{lem:ae_equality2}\/ Corollary \\ref{cor:ae_equality3} in\n\torder to obtain almost everywhere convergence\n\t$\\vec{Du^n} \\to \\vec{Du}$, then we use this to prove\n\t\\eqref{eq:lem_ident_limit}.\n\t\n\tAs in Lemma \\ref{lem:ae_equality2} we let $B$ be a ball with\n\t$2B \\subset \\subset \\Omega$ and $\\xi \\in C_0^\\infty (\\Omega)$ be a\n\tcutoff function such that $\\chi_B \\leq \\xi \\leq \\chi_{2B}$. We set\n\t$\\vec{v^n} := \\xi (\\vec{u^n}- \\vec{u})$ and let $\\vec{v_j^n}$ be the\n\tLipschitz truncation of $\\vec{v^n}$ with respect to the domain $2B$\n\tfrom Theorem \\ref{thm:lip_trunc}. Since the functions $\\vec{v_j^n}$\n\tare in general not divergence-free, we have to introduce correction\n\tterms in order to use them as test functions in\n\t\\eqref{eq:lem_ident_appr}. We use the Bogovski\\u{\\i} operator\n\t$\\Bog \\colon L_0^r(\\Omega) \\to W_0^{1,r}(\\Omega)$ and set\n\t\\begin{equation} \\label{eq:defi_eta}\n\t\\vec{\\psi_j^n} := \\Bog(\\div \\vec{v_j^n}) \\in W_0^{1,r}(\\Omega)\n\t\\quad\\text{and}\\quad \\vec{\\eta_j^n} := \\vec{v_j^n} -\n\t\\vec{\\psi_j^n} \\in V_r. \n\t\\end{equation}\n\tBy \\eqref{eq:lip_trunc2}$_1$, we get\n\t$\\nabla \\vec{v_j^n} \\weakto \\vec{0}$ in $L^r(\\Omega)$ for each\n\t$j \\in \\N$ as $n\\to\\infty$. Since both the divergence and the\n\tBogovski\\u{\\i} operator are linear and continuous, we get the\n\tconvergence\n\t\\begin{align} \\label{eq:conv_psi1}\n\t\t\\vec{\\psi_j^n} \\weakto \\vec{0} &\\quad \\text{ weakly in } W_0^{1,r}(\\Omega)\n\t\n\t\\end{align}\n\tas $n \\to \\infty$ for every $j \\in \\N$.  By a well-known fact, we\n\tknow $\\nabla \\vec{v_j^n} = \\nabla \\vec{v^n}$ on the set\n\t$\\{\\vec{v_j^n} = \\vec{v^n}\\}$ (cf.~\\cite{Maly}). Thus, we obtain\n\t$\\div \\vec{v^n} = \\nabla \\xi \\cdot (\\vec{u^n}-\\vec{u})$ by the\n\tproduct rule and get\n\t$\\div\\vec{v_j^n} = \\chi_{\\{\\vec{v^n}\\neq\\vec{v_j^n} \\}} \\div\n\t\\vec{v_j^n} + \\chi_{\\{\\vec{v^n}=\\vec{v_j^n} \\}} \\nabla \\xi \\cdot\n\t(\\vec{u^n}-\\vec{u})$. Together with the continuity of the\n\tBogovski\\u{\\i} operator and the $W^{1, \\infty}(\\Omega)$-boundedness\n\tof the cutoff function~$\\xi$, this implies\n\t\\begin{equation} \\label{eq:estim_psi1}\n\t\\norm{\\vec{\\psi_j^n}}_{1,p}\n\t\\leq c \\norm{\\div \\vec{v_j^n}}_p \\leq c\n\t\\big\\Vert\\chi_{\\{\\vec{v^n}\\neq\\vec{v_j^n} \\}} \\nabla \\vec{v_j^n}\n\t\\big\\Vert_p + c\\,(\\xi) \\norm{\\vec{u^n}-\\vec{u}}_p\\!.\n\t\\end{equation}\n\tFurthermore, due to the assumption $\\vec{u^n} \\weakto \\vec{u}$ in\n\t$W_0^{1,p}(\\Omega)$ and the compact embedding\n\t$W_0^{1,p}(\\Omega) \\injto \\injto L^p(\\Omega)$, we have strong\n\tconvergence $\\vec{u^n} \\to \\vec{u}$ in $L^p(\\Omega)$. Applying\n\t\\eqref{eq:lip_trunc1}$_4$ and this strong convergence in\n\t\\eqref{eq:estim_psi1}, we obtain\n\t\\begin{equation} \\label{eq:estim_psi2}\n\t\\limsup_{n \\to \\infty} \\norm{\\vec{\\psi_j^n}}_{1,p} \\leq c\\, 2^{\\frac{-j}{p}}\n\t\\end{equation}\n\tfor all $j \\in \\N$.\n\t\n\tFrom \\eqref{eq:lip_trunc2}$_1$, \\eqref{eq:conv_psi1} and the compact\n\tembedding $W_0^{1,r}(\\Omega) \\injto \\injto L^r(\\Omega)$, we conclude\n\t\\begin{align} \\label{eq:conv_eta1}\n\t\t\\vec{\\eta_j^n} \\weakto \\vec{0}\n\t\t&\\quad \\text{ weakly in } W_0^{1,r}(\\Omega)\n\t\n\t\\end{align}\n\tfor all $j \\in \\N$ as $n \\to \\infty$.\n\t\n\tSince $\\vec{B}\\colon V_p \\to W_0^{1,r}(\\Omega)\\d$ is strongly\n\tcontinuous and $\\vec{u^n} \\weakto \\vec{u}$ in $V_p$, we obtain the\n\tconvergence $\\vec{B}(\\vec{u^n}) \\to \\vec{B}(\\vec{u})$ in\n\t$W_0^{1,r}(\\Omega)\\d$. This and \\eqref{eq:conv_eta1} imply\n\t\\begin{equation} \\label{eq:lip2}\n\t\\lim_{n \\to \\infty} \\fp{\\vec{B}(\\vec{u^n})}{\\vec{\\eta_j^n}} = 0.\n\t\\end{equation}\n\tSimilarly, we obtain\n\t\\begin{equation} \\label{eq:lip3}\n\t\\lim_{n \\to \\infty} \\fp{\\vec{A_n}(\\vec{u^n})}{\\vec{\\eta_j^n}} = 0\n\t\\end{equation}\n\tfrom \\eqref{eq:ident_assum}$_2$ and \\eqref{eq:conv_eta1}.\n\tFurthermore, \\eqref{eq:lip_trunc2}$_1$ implies\n\t$\\vec{v_j^n} \\weakto \\vec{0}$ in $W_0^{1,p}(\\Omega)$ and\n\t\\begin{equation} \\label{eq:lip4}\n\t\\lim_{n \\to \\infty} \\fp{\\A(\\vec{Du})}{\\vec{Dv_j^n}} \n\t= 0\n\t\\end{equation}\n\tfor all $j \\in \\N$.\n\t\n\tBy \\eqref{eq:defi_eta} and equation\n\t\\eqref{eq:lem_ident_appr} we have\n\t\\begin{align*}\n\t\t&\\langle \\A (\\vec{Du^n}) - \\A(\\vec{Du}), \\vec{Dv_j^n} \\rangle\n\t\t\\\\\n\t\t&= - \\fp{\\vec{B}(\\vec{u^n})}{\\vec{\\eta_j^n}} - \\fp{\\vec{A_n}(\\vec{u^n})}{\\vec{\\eta_j^n}} \n\t\t+ \\fp{\\A(\\vec{Du^n})}{\\vec{D\\psi_j^n}} - \\fp{\\A(\\vec{Du})}{\\vec{Dv_j^n}}.\n\t\\end{align*}\n\tWe use the convergences \\eqref{eq:lip2}, \\eqref{eq:lip3},\n\t\\eqref{eq:estim_psi2} and \\eqref{eq:lip4} in this identity and obtain\n\t\\begin{equation*}\n\t\t\\limsup_{n \\to \\infty} \\abs{\\fp{\\A(\\vec{Du^n}) -\n\t\t\t\t\\A(\\vec{Du})}{\\vec{Dv_j^n}}} \\leq c\\, 2^{\\frac{-j}{p}}\\!. \n\t\\end{equation*}\n\tSince $2^{\\frac{-j}{p}} \\to 0$ as $j \\to \\infty$, we may apply\n\tCorollary \\ref{cor:ae_equality3} and conclude\n\t$\\vec{Du^n} \\to \\vec{Du}$ almost everywhere in $\\Omega$ up to some\n\tsubsequence. By the continuity of $\\A$, it follows\n\t$\\A(\\vec{Du^n}) \\to \\A(\\vec{Du})$ almost everywhere in $\\Omega$.\n\t\n\tBy assumption, the mapping $\\vec{v} \\mapsto \\A(\\vec{Dv})$ defines a\n\tbounded operator \\linebreak$W^{1,p}(\\Omega) \\to L^{p'}(\\Omega)$,\n\tand thus the sequence \n\t$(\\A(\\vec{Du^n}))_{n \\in \\N}$ is bounded. We may extract a weakly convergent\n\tsubsequence $\\A(\\vec{Du^n}) \\weakto \\vec{\\chi}$\n\tin~$L^{p'}(\\Omega)$. The combination of almost everywhere\n\tconvergence $\\A(\\vec{Du^n}) \\to \\A(\\vec{Du})$ and weak convergence\n\t$\\A(\\vec{Du^n}) \\weakto \\vec{\\chi}$ (for some subsequences) implies\n\t$\\A(\\vec{Du}) = \\vec{\\chi}$ by a well-known convergence principle\n\t(cf.~\\cite{Gajewski}); in particular, it follows\n\t\\begin{equation} \\label{eq:conv_s}\n\t\\A(\\vec{Du^n}) \\weakto \\A(\\vec{Du}) \\quad \\text{weakly in } L^{p'}(\\Omega).\n\t\\end{equation}\n\tWe pass to the limit for $n \\to \\infty$ in\n\t\\eqref{eq:lem_ident_appr} and use \\eqref{eq:conv_s}, the strong\n\tcontinuity of $\\vec{B}$ and \\eqref{eq:ident_assum}$_2$ to obtain\n\t\\begin{align*}\n\t\t0 &= \\lim_{n \\to \\infty} \\fp{\\A(\\vec{Du^n})}{\\vec{D\\phi}} +\n\t\t\\fp{\\vec{B}(\\vec{u^n})}{\\vec{\\phi}} +\n\t\t\\fp{\\vec{A_n}(\\vec{u^n})}{\\vec{\\phi}}\n\t\t\\\\ \n\t\t&= \\fp{\\A(\\vec{Du})}{\\vec{D\\phi}} + \\fp{\\vec{B}(\\vec{u})}{\\vec{\\phi}}\n\t\\end{align*}\n\tfor $\\vec{\\phi} \\in V_p \\cap V_r \\cap X = V_r$ and therefore, by\n\tdensity, for all $\\vec{\\phi} \\in V_s$.\n\\end{proof}\n\n\\section{Existence of weak solutions} \\label{sec:existence}\n\n\\subsection{Smallness condition and main result}\nAs mentioned in the introduction, our ansatz for proving existence requires smallness of the boundary and the divergence data which is necessary for proving local coercivity. In order to formulate a precise smallness condition, we define the following dependent constants:\n\nFor a domain $\\Omega$, an extra stress tensor $\\S$ with\n$p$-$\\delta$-structure,  $s = \\max \\big\\{ p, \\big (\\frac{p\\d}{2}\\big )' \\big\\}$, a functional $\\vec{f} \\in W_0^{1,p}(\\Omega)\\d$\nand a function $\\vec{g} \\in W^{1,s}(\\Omega)$ we define \n\\begin{equation} \\begin{split} \\label{eq:g123} G_1 &:= \\tfrac{1}{p} \\,\nC_3(\\S),\n\\\\\nG_2 &:= c_\\text{Sob} c_\\text{Korn}^2 \\left[ \\norm{\\vec{Dg}}_{s} +\n\\tfrac{1}{2} \\norm{\\div \\vec{g}}_s \\right],\n\\\\\nG_3 &:= \\left( C_2(\\S) + C_3(\\S) \\right)\n\\bignorm{\\abs{\\vec{Dg}}+\\delta}_p^{p-1} + c_\\text{Sob}\n\\norm{\\vec{g}}_{1,s}^2\n\\\\\n&\\qquad + c_\\text{Sob} c_\\text{Korn} \\norm{\\div \\vec{g}}_s\n\\norm{\\vec{g}}_{1,s} + c_\\text{Korn}\n\\norm{\\vec{f}}_{W_0^{1,p}(\\Omega)\\d}\n\\end{split} \\end{equation} with constants $c_\\text{Korn}$,\n$c_\\text{Sob}$, $C_2(\\S)$ and $C_3(\\S)$ that do only depend on\n$\\Omega$ and the characteristics of~$\\S$.\n\nWith these constants, we impose a smallness condition on the data\n$g_1$ and $\\vec{g_2}$:\n\n\\begin{assum} \\label{assum:smallness}\n\tWe assume that $g_1 \\in L^s(\\Omega)$ and\n\t$\\vec{g_2} \\in W^{s-\\frac{1}{s}, s}(\\del \\Omega)$ satisfy the\n\tcompatibility condition\n\t$\\int_{\\Omega} g_1\\,dx = \\int_{\\del \\Omega} \\vec{g_2} \\cdot\n\t\\vec{\\nu}\\, do$\n\tand that their norms are so small that a solution\n\t$\\vec{g} \\in W^{1,s}(\\Omega)$ of the corresponding inhomogeneous\n\tdivergence equation (see Lemma \\ref{lem:div_eq}) satisfies\n\t\\begin{equation} \\label{eq:g_smallness}\n\t(2-p)^{2-p} (p-1)^{p-1} G_1 \\geq G_2^{p-1}G_3^{2-p}\n\t\\end{equation}\n\tfor the constants $G_1, G_2, G_3$ from \\eqref{eq:g123}.\n\\end{assum}\n\nUnder that condition, we are able to prove the following existence result:\n\n\\begin{thm}[Existence] \\label{thm:main_thm} Let $\\Omega \\subset \\Rd$\n\tbe a bounded Lipschitz domain with $d \\in \\{2, 3\\}$. Let~$\\S$ be an\n\textra stress tensor with $p$-$\\delta$-structure,\n\t$p \\in \\big( \\frac{2d}{d+2}, 2 \\big)$,\n\t$s := \\max \\big\\{ p, \\big (\\frac{p\\d}{2}\\big )' \\big\\}$ and\n\t$\\vec{f} \\in W_0^{1,p}(\\Omega)\\d$. For any $g_1 \\in L^s(\\Omega)$ and\n\t$\\vec{g_2} \\in W^{s-\\frac{1}{s}, s}(\\del \\Omega)$ that satisfy\n\tAssumption \\ref{assum:smallness}, there exists a weak solution\n\t$(\\vec{v}, \\pi) \\in W^{1,p}(\\Omega) \\times L^{s'}(\\Omega)$\n\tof~\\eqref{eq:main_problem}.\n\\end{thm}\n\n\\subsection{Existence proof} \\label{sub:existence}\nTo get a formulation of \\eqref{eq:main_problem}$_1$, we use the\ndefinitions \\eqref{eq:defi_S} and \\eqref{eq:defi_T} of the operators\n$\\vec{S}$ and $\\vec{T}$ and define the \"full\" operator\n$\\vec{P} \\colon V_p \\to W_0^{1,s}(\\Omega)\\d$ via\n\\begin{equation} \\begin{split} \\label{eq:defi_P}\n\\fp{\\vec{P}(\\vec{v})}{\\vec{\\phi}} := \\,\n&\\fp{\\vec{S}(\\vec{v}) + \\vec{T}(\\vec{v}) - \\vec{f}}{\\vec{\\phi}} \\\\\n= \\,&\\fp{\\S(\\vec{Dv}+\\vec{Dg})}{\\vec{D\\phi}} - \\fp{(\\vec{v}+\\vec{g}) \\otimes (\\vec{v}+\\vec{g})}{\\vec{D \\phi}} \\\\\n&- \\fp{(\\div \\vec{g}) (\\vec{v}+\\vec{g})}{\\vec{\\phi}}\n- \\fp{\\vec{f}}{\\vec{\\phi}}\n\\end{split} \\end{equation}\nfor $\\vec{v} \\in V_p$ and $\\vec{\\phi} \\in W_0^{1,s}(\\Omega)$.\n\nWe collect our results on $\\vec{S}$ and $\\vec{T}$ to deduce properties\nof $\\vec{P}$: \n\\begin{cor} \\label{cor:t_estim}\n\tFor $p$, $q$ and $s$ as in Lemma\n\t\\ref{lem:t2_str_contin}, the operator $\\vec{P}$ defined in\n\t\\eqref{eq:defi_P} is well-defined, bounded and continuous on\n\t$V_p \\to W_0^{1,s}(\\Omega)\\d$. It is well-defined, bounded,\n\tcontinuous and pseudomonotone on $V_q \\to V_q\\d$ and it fulfils the\n\testimate\n\t\\begin{equation*}\n\t\t\\fp{\\vec{P}(\\vec{u})}{\\vec{u}} \\geq G_1 \\norm{\\vec{Du}}_p^p - G_2\n\t\t\\norm{\\vec{Du}}_p^2 - G_3 \\norm{\\vec{Du}}_p \n\t\\end{equation*}\n\tfor $\\vec{u} \\in V_q$. If furthermore\n\t\\begin{equation} \\label{eq:smallness2}\n\t(2-p)^{2-p} (p-1)^{p-1} G_1 \\geq G_2^{p-1}G_3^{2-p},\n\t\\end{equation}\n\tthen\n\t\\begin{equation*}\n\t\t\\fp{\\vec{P}(\\vec{u})}{\\vec{u}} \\geq 0\n\t\\end{equation*}\n\tholds for all $\\vec{u} \\in V_q$ with $\\norm{\\vec{Du}}_p = R :=\n\t\\left[\\frac{G_3}{(2-p)\\,G_1}\\right]^\\frac{1}{p-1}$. \n\\end{cor}\n\\begin{proof}\n\tWell-definedness, boundedness and continuity follow similarly to the\n\tproperties of $\\vec{S}$ and $\\vec{T}$ in Lemmas\n\t\\ref{lem:properties_S} and \\ref{lem:t2_str_contin}.\n\t\n\tThe continuity of $\\S$ and its monotonicity, which follows from\n\t\\eqref{eq:pdelta_growth1}$_1$, yield that $\\vec{S}\\colon V_q \\to V_q\\d$ is\n\tpseudomonotone. Lemma \\ref{lem:t2_str_contin} shows that\n\t$\\vec{T}\\colon V_q \\to V_q\\d$ is strongly continuous and thus\n\tpseudomonotone. Therefore, the sum\n\t$\\vec{P} = \\vec{S} + \\vec{T} - \\vec{f}$ is also pseudomonotone.\n\t\n\tBy Lemmas \\ref{lem:S_estimate} and \\ref{lem:t2_str_contin} and by\n\tthe definition of the constants $G_1$, $G_2$, $G_3$\n\tin~\\eqref{eq:g123}, we have\n\t\\begin{align} \\label{eq:t_estim1}\n\t\t\\fp{\\vec{P}(\\vec{u})}{\\vec{u}}\n\t\t&\\geq \\fp{\\vec{S}(\\vec{u})}{\\vec{u}} -\n\t\t\\abs{\\fp{\\vec{T}(\\vec{u})}{\\vec{u}}}\n\t\t- \\abs{\\fp{\\vec{f}}{\\vec{u}}} \\nonumber\n\t\t\\\\\n\t\t&\\geq G_1 \\norm{\\vec{Du}}_p^p - G_2 \\norm{\\vec{Du}}_p^2 - G_3\n\t\t\\norm{\\vec{Du}}_p\n\t\\end{align}\n\tfor any $\\vec{u} \\in V_q$. \n\t\n\tNow assume that \\eqref{eq:smallness2} holds. It follows\n\t$\\left[\\frac{(p-1)\\,G_1}{G_2}\\right]^{p-1} \\geq \\left[\n\t\\frac{G_3}{(2-p)G_1}\\right]^{2-p}$, so we may define\n\t$R := \\left[\\frac{G_3}{(2-p)G_1}\\right]^\\frac{1}{p-1}$ and obtain\n\t$\\frac{(p-1)\\,G_1}{G_2} \\geq R^{2-p}$. Together, we get\n\t\\begin{equation} \\label{eq:t_estim2}\n\tG_1 R^p = (p-1)G_1 R^p + (2-p)G_1 R^p \\geq G_2 R^2 + G_3 R.\n\t\\end{equation}\n\tWe use $\\norm{\\vec{Du}}_p = R$, insert \\eqref{eq:t_estim2} into\n\t\\eqref{eq:t_estim1} and obtain the result. \n\\end{proof}\n\\begin{rem}\n\n\t(i)  Note that the dependence of the constants $G_i$, $i=1,2,3$, on\n\t$\\norm{\\vec{g}}_{1,s}$ stems only from the estimate of the\n\tconvective term.\\\\[-3mm]\n\t\n\t(ii)  In order to prove $G_1 R^p - G_2 R^2 - G_3 R \\geq 0$, we have split\n\tthe positive summand $G_1 R^p$ into two parts using the weights\n\t$p-1$ and $2-p$.  By considering the weights as a free parameter,\n\tone can show that this choice is optimal.\n\\end{rem}\n\nNow we are ready to complete the proof of Theorem\n\\ref{thm:main_thm}. Since the proof requires an approximation process\nonly if $p \\in {\\big ( \\frac{2d}{d+2}, \\frac{3d}{d+2} \\big ]} $, we\nhandle the two cases separately.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main_thm} in the case $p \\in\n\t\\big (\\frac{3d}{d+2}, 2\\big )$]\n\tIn this case we have  $s \n\n\tp$. By Lemma \\ref{lem:div_eq}, we  find a function\n\t$\\vec{g} \\in W^{1,p}(\\Omega)$ that solves the corresponding\n\tinhomogeneous divergence equation \\eqref{eq:g_problem}.\n\t\n\tWe consider the corresponding operator $\\vec{P}$ defined in\n\t\\eqref{eq:defi_P} and prove existence of a function\n\t$\\vec{u} \\in V_p$ which satisfies $\\vec{P}(\\vec{u}) = \\vec{0}$.  The\n\tspace $V_p$ is reflexive and separable as a closed subspace of\n\t$W^{1,p}(\\Omega)$. In Corollary \\ref{cor:t_estim} (with\n\t$q = s = p$), we proved that $\\vec{P}$ is well-defined, bounded,\n\tcontinuous and pseudomonotone on $V_p \\to V_p\\d$ and we concluded\n\tfrom assumption \\eqref{eq:g_smallness} that there exists a positive\n\tnumber~$R$ such that $\\vec{P}$ is locally coercive with radius $R$.\n\tSo we may apply the main theorem on pseudomonotone operators,\n\tTheorem \\ref{thm:brezis}, to the operator $\\vec{P}$ on the space\n\t$V_p$ and obtain a weak solution $\\vec{u} \\in V_p$ of\n\t$\\vec{P}(\\vec{u}) = \\vec{0}$.\n\t\n\tBy a standard characterization of weak gradient fields (cf.~\\cite{Sohr}),\n\tthis is equivalent to the existence of a pressure $\\pi$ such that\n\t$(\\vec{u}+\\vec{g}, \\pi) \\in W^{1,p}(\\Omega) \\times L^{p'}(\\Omega)$\n\tsolves the original system \\eqref{eq:main_problem}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main_thm} in the case\n\t$p \\in {\\big ( \\frac{2d}{d+2}, \\frac{3d}{d+2} \\big ]} $]\n\tIn this case we have  $s = \\big (\\frac{p\\d}{2}\\big\n\t)' $. By assumption \\ref{assum:smallness}, there is a function\n\t$\\vec{g} \\in W^{1,s}(\\Omega)$ which solves the inhomogeneous\n\tdivergence equation \\eqref{eq:g_problem} and satisfies\n\t\\eqref{eq:g_smallness}. We prove the existence of a function\n\t$\\vec{u} \\in V_p$ that solves $\\vec{P}(\\vec{u}) = \\vec{0}$ for the\n\tcorresponding operator $\\vec{P}$ from \\eqref{eq:defi_P}.  For\n\tregularization, we choose some $q > s$ with $q > 2$ and consider the\n\tsymmetric $q$-Laplacian $\\vec{A} \\colon V_q \\to V_q\\d$ defined via\n\t\\begin{equation*}\n\t\t\\fp{\\vec{A}(\\vec{u})}{\\vec{\\phi}} := \\fp{\\abs{ \\vec{Du}}^{q-2}\n\t\t\t\\vec{Du}}{\\vec{D\\phi}}. \n\t\\end{equation*}\n\tThe operator $\\vec{A}$ is well-defined, bounded, continuous and\n\tmonotone.\n\t\n\tWe work in the reflexive and separable spaces $V_q^n$, which are\n\tdefined as the space $V_q$ equipped with the equivalent norms\n\t$\\lVert {\\vec{u}} \\rVert _{q, n} := \\max \\left\\{ n^\\frac{-2}{2q-1}\n\t\\norm{\\vec{Du}}_q, \\norm{\\vec{Du}}_p \\right\\}$.  For sufficiently\n\tlarge $n$, we want to establish the existence of solutions $\\vec{u^n} \\in V_q^n$ to\n\tthe equation\n\t\\begin{equation} \\label{eq:tq}\n\t\\fp{\\vec{P}(\\vec{u^n})}{\\vec{\\phi}} + \\tfrac{1}{n}\n\t\\fp{\\vec{A}(\\vec{u^n})}{\\vec{\\phi}} = 0 \n\t\\end{equation}\n\tfor $\\vec{\\phi} \\in V_q$, which shall approximate a solution of the\n\toriginal equation.  The operator $\\vec{P}$ is pseudomonotone,\n\tcontinuous and bounded by Corollary~\\ref{cor:t_estim} and the same\n\tholds for $\\vec{A}$ and their sum $\\vec{P}+\\frac{1}{n}\\vec{A}$.  We\n\tprove local coercivity of $\\vec{P} + \\frac{1}{n} \\vec{A}$ with\n\tradius $R := \\left[\\frac{G_3}{(2-p)G_1}\\right]^\\frac{1}{p-1}$. So,\n\tlet $\\lVert {\\vec{u}} \\rVert _{q, n} = R$. If\n\t$n^\\frac{-2}{2q-1} \\norm{\\vec{Du}}_q \\leq \\norm{\\vec{Du}}_p$, we\n\thave $\\norm{\\vec{Du}}_p = \\lVert {\\vec{u}} \\rVert _{q, n} = R$ and we get\n\t$\\fp{\\vec{P}(\\vec{u})}{\\vec{u}} \\geq 0$ by assumption\n\t\\eqref{eq:g_smallness} and Corollary \\ref{cor:t_estim}.  Otherwise,\n\tsuppose $n^\\frac{-2}{2q-1} \\norm{\\vec{Du}}_q > \\norm{\\vec{Du}}_p$,\n\tso $R = n^\\frac{-2}{2q-1} \\norm{\\vec{Du}}_q$ and\n\t$R > \\norm{\\vec{Du}}_p$. This, Corollary \\ref{cor:t_estim} and the\n\tSobolev embedding $V_q \\injto V_p$ imply\n\t\\begin{align*}\n\t\t\\fp{\\vec{P}(\\vec{u})}{\\vec{u}} + \\tfrac{1}{n}\\fp{\\vec{A}(\\vec{u})}{\\vec{u}} \n\t\t&\\geq G_1 \\norm{\\vec{Du}}_p^p - G_2 \\norm{\\vec{Du}}_p^2 - G_3\n\t\t\\norm{\\vec{Du}}_p + \\tfrac{1}{n}\\norm{\\vec{Du}}_q^q\n\t\t\\\\ \n\t\t&> -G_2 R^{\\,2} - G_3 R + n^\\frac{1}{2q-1} R^{\\,q}\\!.\n\t\\end{align*}\n\tAs $n^\\frac{1}{2q-1}$ grows to infinity, the latter expression\n\tbecomes positive for any $R > 0$ and sufficiently large $n$.\n\t\n\tThus, the existence Theorem \\ref{thm:brezis} gives us solutions\n\t$\\vec{u^n} \\in V_q^n$ of \\eqref{eq:tq} with\n\t\\begin{equation} \\label{eq:bound_un} \\max \\left\\{ n^\\frac{-2}{2q-1}\n\t\\norm{ \\vec{Du}}_q\\!, \\norm{\\vec{Du}}_p \\right\\} =\n\t\\lVert {\\vec{u^n}} \\rVert _{q, n} \\leq R\n\t\\end{equation}\n\tand the bound $R$ holds uniformly with respect to $n$.\n\t\n\tWe switch to a weakly convergent (and renamed) subsequence\n\t$\\vec{u^n}\\weakto \\vec{u}$ in $V_p$. The bound \\eqref{eq:bound_un}\n\timplies  $\\norm{n^{-1}\\vec{A}(\\vec{u^n})}_{q'} = n^{-1}\n\t\\norm{\\vec{Du^n}}_{q}^{q-1} \\leq n^{\\frac{-1}{2q-1}} R^{\\,q-1} \\to\n\t0$ as $n \\to \\infty $.\n\t\n\tWe apply Theorem \\ref{thm:ident_limits} with the shifted\n\textra stress tensor $\\A(\\cdot) := \\S(\\cdot+\\vec{Dg})$,\n\t$\\vec{B} := \\vec{T} - \\vec{f}\\colon V_p \\to V_s\\d$,   $r := q$, $X := V_q$ and\n\t$\\vec{A_n} := n^{-1}\\vec{A}\\colon V_q \\to V_q\\d\\!$ and obtain that\n\t$\\vec{u}$ solves $\\vec{P}(\\vec{u}) = \\vec{0}$ weakly.\n\t\n\tSimilarly to the proof in the first case, we obtain a pressure $\\pi$\n\tsuch that the pair\n\t$(\\vec{u}+\\vec{g}, \\pi) \\in W^{1,p}(\\Omega) \\times L^{s'}(\\Omega)$\n\tis a weak solution of \\eqref{eq:main_problem}.\n\\end{proof}\n\n\\subsection{Less regular data} \\label{sub:less_regularity}\n\nIn Theorem \\ref{thm:main_thm}, we demanded additional regularity of\nthe data: we required $\\vec{g} \\in W^{1,s}(\\Omega)$ with\n$s = \\big (\\frac{p\\d}{2}\\big )' > p$ in the case\n$p \\in \\big (\\frac{2d}{d+2}, \\frac{3d}{d+2}\\big )$, while the solution\n$\\vec{v}$ is sought only in $W^{1,p}(\\Omega)$. Thus, we want to\ndiscuss whether this assumption is really necessary or if it may be\nremoved, perhaps for the price of more regular test functions. This\nquestion only arises if\n$p \\in \\big (\\frac{2d}{d+2}, \\frac{3d}{d+2}\\big )$, since one has\n$s = p$ and $\\vec{g} \\in W^{1,s}(\\Omega) = W^{1,p}(\\Omega)$ in the\nother case.\n\nIn the proof of Theorem \\ref{thm:main_thm}, we used the additional\nregularity of our data to get more convenient estimates of the\nconvective term in Lemma \\ref{lem:t2_str_contin}. Since these\nestimates are mainly based on the H\\\"older inequality, one has to use\na stronger norm of $\\vec{u}$ if only $\\vec{g}\\in W^{1,p}(\\Omega)$ is\npresumed. By \\eqref{eq:t2_eq4} and the H\\\"older inequality, one\nobtains the following result similar to Lemma \\ref{lem:t2_str_contin}:\n\n\\begin{lem}[Properties of $\\vec{T}$] \\label{lem:t2_str_contin2}\n\tLet $p \\in \\big (\\frac{2d}{d+2}, \\frac{3d}{d+2}\\big )$ and $q$ be so\n\tlarge that it holds both $q > s = \\big (\\frac{p\\d}{2}\\big )'$ and\n\t$\\frac{1}{q'} \\geq \\frac{1}{p}+\\frac{1}{p\\d}$. For any given\n\tfunction $\\vec{g} \\in W^{1,p}(\\Omega)$, the operator $\\vec{T}$\n\thas the upper bound\n\t\\begin{align*}\n\t\t\\abs{\\fp{\\vec{T}(\\vec{u})}{\\vec{u}}}\n\t\t&\\leq c_\\text{Sob} c_\\text{Korn}^2 \\left( \\norm{\\vec{Dg}}_p +\n\t\t\\tfrac{1}{2} \\norm{\\div \\vec{g}}_p \\right) \\norm{\\vec{Du}}_p\n\t\t\\norm{\\vec{Du}}_q\n\t\t\\\\ \n\t\t&\\quad + c_\\text{Sob} \\left( \\norm{\\vec{g}}_{1,p}^2 +\n\t\tc_\\text{Korn} \\norm{\\div \\vec{g}}_p \\norm{\\vec{g}}_{1,p} \\right)\n\t\t\\norm{\\vec{Du}}_q \n\t\\end{align*}\n\tfor all $\\vec{u} \\in V_q$, where $C_\\text{Sob}$ are Sobolev\n\tembedding constants.\n\\end{lem}\n\nThis and Lemma \\ref{lem:S_estimate} yield an alternative lower bound\nof $\\vec{P}$: \n\n\\begin{cor}[Alternative estimate of $\\vec{P}$] \\label{cor:t_estim2}\n\tLet $p \\in \\big (\\frac{2d}{d+2}, \\frac{3d}{d+2}\\big )$ and $q$ be so\n\tlarge that it holds both $q > s = \\big (\\frac{p\\d}{2}\\big )'$ and\n\t$\\frac{1}{q'} \\geq \\frac{1}{p}+\\frac{1}{p\\d}$. For any given\n\tfunction $\\vec{g} \\in W^{1,p}(\\Omega) \\setminus \\{\\vec{0}\\}$, there\n\tare constants $F_1, F_2, G_1 \\geq 0$ with $F_1, G_1 > 0$ such that\n\tit holds\n\t\\begin{equation*}\n\t\t\\fp{\\vec{P}(\\vec{u})}{\\vec{u}} \\geq G_1 \\norm{\\vec{Du}}_p^p -\n\t\tF_1 \\norm{\\vec{Du}}_q - F_2 \\norm{\\vec{Du}}_p\n\t\t\\norm{\\vec{Du}}_q \n\t\\end{equation*}\n\tfor all $\\vec{u} \\in V_q$.\n\\end{cor}\n\nTo the authors' knowledge, a substantial improvement of the\nestimates in Lemma~\\ref{lem:t2_str_contin2} and Corollary\n\\ref{cor:t_estim2} is not available. So we ask whether the proof of\nTheorem \\ref{thm:main_thm} can be modified such that it works out with\n\\emph{these} estimates.\n\nThe main theorem on pseudomonotone operators, Theorem\n\\ref{thm:brezis}, which was used to obtain approximate solutions in\nsome smoother space $X = V_p \\cap Y$ already gave a priori estimates\nfor these approximate solutions. These a priori bounds were needed to\nestablish a weak accumulation point.  The next Lemma shows that it is\nimpossible to obtain approximate solutions which are coming with an a\npriori bound in $V_p$, by Brouwer's fixed point theorem\/the main\ntheorem on pseudomonotone operators.\n\n\\begin{lem}[Limits for the applicability of pseudomonotone operator\n\ttheory for solving~\\eqref{eq:main_problem}] \\label{lem:no_brouwer2}\n\tLet $F_1, G_1, R, 1<p<2<q$ be arbitrary, positive constants,\n\t$Y \\subset L^1(\\Omega)$ be a Banach space with norm $\\norm{\\cdot}_Y$\n\tand assume there is no continuous embedding $V_p \\injto Y\\!$. Define\n\tthe operators $P_n\\colon V_p \\cap Y \\to \\R$ via\n\t\\begin{equation*}\n\t\tP_n(\\vec{u}) := G_1 \\norm{\\vec{Du}}_p^p + \\tfrac{1}{n}\n\t\t\\norm{\\vec{u}}_Y^q - F_1 \\norm{\\vec{u}}_Y\\!. \n\t\\end{equation*}\n\tConsider Banach spaces $Y_n := (V_p \\cap Y, \\norm{\\cdot}_{Y_n})$\n\twhich satisfy $Y_n\\injto Y$ and\n\t${\\norm{\\vec{D}\\cdot}_p \\leq \\norm{\\cdot}_{Y_n}}$ for all $n \\in\n\t\\N$. Then, for all sufficiently large $n$ there exists a function\n\t$\\vec{u^n} \\in V_p \\cap Y$ with $\\norm{\\vec{u^n}}_{Y_n}=R$ and\n\t\\begin{equation} \\label{eq:eq1}\n\tP_n(\\vec{u^n}) < 0.\n\t\\end{equation}\n\\end{lem}\n\n\\begin{rem}[Discussion of the assumptions in Lemma\n\t\\ref{lem:no_brouwer2}] \\label{rem:lem24_assumptions}\n\tThe fact that the space $V_p \\cap Y$, where we seek for approximate\n\tsolutions, needs to be strictly smoother than $V_p$ is caught up in\n\tthe assumption that $V_p$ does not embed continuously into $Y\\!$.\n\t\n\tThe term $\\frac{1}{n} \\norm{\\vec{u}}_Y^q$ in the definition of~$P_n$\n\tmay be equivalently replaced by any operator that is coercive on $Y$\n\tdamped by some factor which is decreasing in $n$.\n\t\n\tThe existence of embeddings $Y_n \\injto Y$ is a natural assumption\n\tfor the intersection of Banach spaces. The requirement\n\t$\\norm{\\vec{D} \\cdot}_p \\leq \\norm{\\cdot}_{Y_n}$ for all $n$ implies\n\tthe existence of embeddings $Y_n \\injto V_p$ and further means that\n\ta-priori estimates of the form $\\norm{\\vec{u^n}}_{Y_n}< c$ imply\n\tuniform boundedness in the weaker norm\n\t$\\norm{\\vec{D} \\vec{u^n}}_p < c$ which is a necessary element in the\n\tproof of Theorem \\ref{thm:main_thm}.\n\t\n\tTypically, one chooses $Y=V_q$ or $Y=L^q(\\Omega)$ for some large\n\tnumber $q \\in\\R$ and works with the weighted sum norm\n\t$\\norm{\\cdot}_{Y_n} = \\norm{\\vec{D} \\cdot}_p + \\tfrac{1}{n}\n\t\\norm{\\cdot}_Y$. Obviously, the assumptions from Lemma\n\t\\ref{lem:no_brouwer2} are fulfilled for such choices.\n\\end{rem}\n\n\\begin{proof}[Proof of Lemma \\ref{lem:no_brouwer2}]\n\tSince $Y \\subset L^1(\\Omega)$, the intersection $V_p \\cap Y$ is\n\twell-defined.  For each $n\\in\\N$, we define\n\t$f(n) := \\inf_{u\\in Y_n\\setminus\\{0\\} }\n\t\\frac{\\norm{\\vec{u}}_{Y_n}}{\\norm{\\vec{u}}_Y}$.  The embedding\n\t$Y_n\\injto Y$ and the assumption on $\\norm{\\cdot}_{Y_n}$ imply that\n\t$f(n)$ is strictly positive and that\n\t\\begin{equation} \\label{eq:eq2}\n\t\\max \\left\\{ \\norm{\\vec{D} \\vec{u}}_p, f(n) \\norm{\\vec{u}}_Y \\right\\}\n\t\\leq R\n\t\\end{equation}\n\tholds for all $\\vec{u} \\in Y_n$ with $\\norm{\\vec{u}}_{Y_n}=R$. \n\t\n\tWe define the constant\n\t$c_s := \\sup_{\\vec{u}\\in Y_n\\setminus\\{0\\} } \\frac{\\norm{\\vec{D}\n\t\t\t\\vec{u}}_p}{\\norm{\\vec{u}}_Y} \\in (0, \\infty]$ and use the\n\tconvention $\\frac{t}{\\infty}=0$ for any $t\\in\\R$.\n\t\n\tOur first step is the indirect proof of an upper bound on\n\t$f$. Therefore, we choose $c_1\\in\\R$ so large that\n\t$\\frac{R^{\\,q-1}}{c_1} - \\frac{F_1}{2} < 0$ is satisfied.\n\t\n\t\\emph{Step 1: We prove that \\eqref{eq:eq1} is true for those\n\t\t$n \\in \\N$ with $f(n)^{q-1} \\geq c_1 n^{-1}$.}\n\t\n\tBy \\eqref{eq:eq2}, we obtain\n\t\\begin{equation*}\n\t\t\\frac{1}{n} \\norm{\\vec{u}}_Y^q - \\frac{F_1}{2} \\norm{\\vec{u}}_Y\n\t\t\\leq \\norm{\\vec{u}}_Y \\left[ \\frac{R^{\\,q-1}}{n f(n)^{q-1}} - \\frac{F_1}{2} \\right]\n\t\t\\leq \\norm{\\vec{u}}_Y \\left[ \\frac{R^{\\,q-1}}{c_1} - \\frac{F_1}{2} \\right]\n\t\t< 0\n\t\\end{equation*}\n\tfor any $\\vec{u}\\in Y_n$ with $\\norm{\\vec{u}}_{Y_n}=R$. Since there\n\tis no embedding $V_p \\injto Y\\!$, we may find some\n\t$\\vec{u^n} \\in Y_n$ with $\\norm{\\vec{u^n}}_{Y_n}=R$ and\n\t$G_1 \\norm{\\vec{D} \\vec{u^n}}_p^p < \\frac{F_1}{2}\n\t\\norm{\\vec{u^n}}_Y$. Together, we obtain\n\t$G_1 \\norm{\\vec{D} \\vec{u^n}}_p^p + \\frac{1}{n} \\norm{\\vec{u^n}}_Y^q\n\t- F_1 \\norm{\\vec{u^n}}_Y < 0$, which is \\eqref{eq:eq1}.\n\t\n\tIn the following, we only consider those $n \\in \\N$ with\n\t$f(n)^{q-1} < c_1 n^{-1}$.\n\t\n\t\\emph{Step 2: Computation of suitable norms for functions\n\t\t$\\vec{u^n}$ such that \\eqref{eq:eq1} becomes true.}\n\t\n\tWe choose $c_2 > 1$ so large that\n\t$\\frac{c_2\\, R^{\\,q-1}}{F_1} \\geq c_1$ and define the auxiliary\n\tfunctions $t_n\\colon \\R\\to\\R$ via\n\t\\begin{equation*}\n\t\tt_n(x) := \\frac{c_2}{n} x^{q-1} - F_1.\n\t\\end{equation*}\n\tWe claim that for all sufficiently large $n$, the equation\n\t$t_n(x)=0$ has a solution  $y_n \\in \\big ( \\frac{R}{c_s}, \\frac{R}{f(n)} \\big )$.\n\tSince the upper bound on $f$ implies $f(n) \\to 0$ as $n \\to\\infty$,\n\tit holds $\\frac{R}{c_s} < \\frac{R}{f(n)}$ for sufficiently large $n$\n\tand the interval $\\big ( \\frac{R}{c_s}, \\frac{R}{f(n)} \\big )$ is\n\tnot empty.  We have\n\t$t_n\\! \\big (\\frac{R}{c_s}\\big ) = \\frac{c_2 R^{\\,q-1}}{n\\,\n\t\tc_s^{q-1}} - F_1 < 0$ for sufficiently large $n$ and the\n\tdefinitions of $c_1$ and $c_2$ imply\n\t$\\frac{c_2 R^{\\,q-1}}{F_1} \\geq c_1 > n f(n)^{q-1}$, thus\n\t$t_n\\! \\big (\\frac{R}{f(n)}\\big ) = \\frac{c_2\\, R^{\\,q-1}}{n\\,\n\t\tf(n)^{\\,q-1}} - F_1 > 0$. The existence of zeroes $y_n$ then\n\tfollows from the mean value theorem.\n\t\n\tRight from the definition of $y_n$, we obtain\n\t$\\frac{1}{n}\\, y_n^{\\,q} - \\frac{1}{c_2} F_1 y_n = 0$ and\n\t\\begin{equation*}\n\t\t\\left[1-\\frac{1}{c_2}\\right] F_1 y_n =\n\t\t\\left[1-\\frac{1}{c_2}\\right] F_1 \\left[ \\frac{nF_1}{c_2}\n\t\t\\right]^{\\frac{1}{q-1}} > G_1 R^p \n\t\\end{equation*}\n\tfor sufficiently large $n$.  Together, it follows\n\t\\begin{equation*}\n\t\tG_1 R^p + \\frac{1}{n} y_n^q - F_1 y_n   = G_1 R^p -\n\t\t\\left[1-\\frac{1}{c_2}\\right] F_1 y_n + \\frac{1}{n} y_n^q -\n\t\t\\frac{1}{c_2} F_1 y_n    < 0.\n\t\\end{equation*}\n\tThus, any function $\\vec{u^n} \\in V_p \\cap Y$ with\n\t$\\norm{\\vec{u^n}}_{Y_n}=R$ and $\\norm{\\vec{u^n}}_{Y}=y_n$ satisfies\n\t\\begin{equation*}\n\t\tP_n(\\vec{u^n}) \\leq G_1 R^p + \\frac{1}{n} y_n^q - F_1 y_n < 0.\n\t\\end{equation*}\n\t\n\t\\emph{Step 3: Construction of functions $\\vec{u^n}\\in V_p \\cap Y$\n\t\twith prescribed norms.}\n\t\n\tIt remains to prove that for any\n\t$y_n \\in \\big ( \\frac{R}{c_s}, \\frac{R}{f(n)} \\big )$, there is a\n\tfunction $\\vec{u^n} \\in V_p \\cap Y$ with $\\norm{\\vec{u^n}}_{Y_n}=R$\n\tand $\\norm{\\vec{u^n}}_{Y}=y_n$.\n\t\n\tAssume that $\\norm{\\vec{u}}_{Y}>y_n$ for all $\\vec{u} \\in Y_n$ with\n\t$\\norm{\\vec{u}}_{Y_n}=R$. Since\n\t$y_n \\in \\big( \\frac{R}{c_s}, \\frac{R}{f(n)} \\big)$, it holds\n\t$c_s > \\frac{R}{y_n}$. By the definition of $c_s$, this implies the\n\texistence of a function $\\vec{u} \\in Y_n$ such that\n\t$\\norm{\\vec{D} \\vec{u}}_p > \\frac{R}{y_n} \\norm{\\vec{u}}_Y$. Without\n\tloss of generality, we may scale $\\vec{u}$ such that\n\t$\\norm{\\vec{u}}_{Y_n}=R$. We compile these estimates of $\\vec{u}$\n\tand apply \\eqref{eq:eq2} to obtain\n\t\\begin{equation*}\n\t\tR < \\frac{R}{y_n} \\norm{\\vec{u}}_Y \n\t\t< \\norm{\\vec{D} \\vec{u}}_p\n\t\t\\leq R,\n\t\\end{equation*}\n\twhich is impossible.\n\t\n\tNow assume that $\\norm{\\vec{u}}_{Y}<y_n$ for all\n\t$\\vec{u} \\in Y_n$ with $\\norm{\\vec{u}}_{Y_n}=R$. As above, we obtain\n\t$\\frac{R}{y_n} > f(n)$. This and the definition of $f(n)$ imply the\n\texistence of a function $\\vec{u} \\in Y_n$ such that it holds\n\t$\\norm{\\vec{u}}_{Y_n} < \\frac{R}{y_n} \\norm{\\vec{u}}_Y$. We may\n\tscale $\\vec{u}$ such that $\\norm{\\vec{u}}_{Y_n} = R$ and it follows\n\t\\begin{equation*}\n\t\tR = \\norm{\\vec{u}}_{Y_n} < \\frac{R}{y_n} \\norm{\\vec{u}}_Y < R, \n\t\\end{equation*}\n\twhich is again a contradiction. \n\t\n\tHence, there are $\\vec{v^n}, \\vec{w^n} \\in Y_n$ with\n\t$\\norm{\\vec{v^n}}_{Y_n} = \\norm{\\vec{w^n}}_{Y_n} = R$ and\n\t$\\norm{\\vec{v^n}}_{Y} \\leq y_n \\leq \\norm{\\vec{w^n}}_{Y}$ for every\n\t$n\\in\\N$. By assumption, the mapping\n\t$\\vec{u}\\mapsto\\norm{\\vec{u}}_Y$ is continuous on $Y_n$. We apply\n\tthe mean value theorem on the (path-connected) sphere\n\t$\\left\\{ \\norm{\\vec{\\cdot}}_{Y_n} = R \\right\\}$ and obtain functions\n\t$\\vec{u^n} \\in V_p \\cap Y$ with $\\norm{\\vec{u^n}}_{Y_n}=R$ and\n\t$\\norm{\\vec{u^n}}_{Y} = y_n$. In Step 2 we proved that such an\n\telement $\\vec{u^n}$ solves \\eqref{eq:eq1} for all sufficiently large\n\t$n$.\n\\end{proof}\n\nLemma \\ref{lem:no_brouwer2} shows that, without assuming additional\nregularity, it is impossible to find a radius $R$ such that local\ncoercivity is fulfilled and Brouwer's fixed point theorem becomes\napplicable. Consequently, the authors view the existence proof in\n\\cite[Theorem 1.3]{Sin} with suspicion; in particular, the requirements for\nBrouwer's fixed point theorem do not seem to be satisfied in our eyes.\nWe conclude from Lemma \\ref{lem:no_brouwer2} that it is impossible to\nmodify the proof of existence theorem \\ref{thm:main_thm} such that it\navoids the critical regularity assumption within the framework of\npseudomonotone operator theory.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{The domain of complex metrics}\n\nA Riemannian metric on a manifold $M$ is a positive-definite symmetric bilinear form $g: T_x \\times T_x \\to \\mathbb R$ on the tangent space $T_x$ at each point $x \\in M$. The metrics we shall consider will be defined by symmetric $\\mathbb R$-bilinear maps $g: T_x \\times T_x \\to \\mathbb C$ at each point, with an appropriate generalization of the positivity condition.\n\nTo see what condition we should require, let us consider the simplest example of a field theory: a free real scalar field of mass $m$. Then the space of `fields' $\\Phi_M$ is the vector space C$^{\\infty}(M;\\mathbb R)$ of smooth functions, and in the exponent of the path-integral we have the quadratic form\n\\begin{eqnarray*}\n{\\rm i} S_g(\\phi)  \\ \\  &=& \\  \\ \\frac{1}{2}\\int_M ( d\\phi\\wedge*d\\phi + m^2 \\phi\\wedge *\\phi) \\\\ &=& \\  \\ \\frac{1}{2}\\int_M \\left\\{\\sum g^{ij}\\frac{\\partial \\phi}{\\partial x^i}\\frac{\\partial \\phi}{\\partial x^j} + m^2\\phi^2 \\right\\} ( \\det g)^{1\/2} |dx^1 \\ldots dx^d|.\n\\end{eqnarray*}\nHere $(g^{ij})$ denotes the inverse of the matrix $g = (g_{ij})$, and $*$ is the Hodge star-operator  defined by the metric, which takes differential forms of degree $p$ to  forms of degree $d-p$ twisted by the orientation bundle.  (We shall not assume the space-time $M$ is orientable.) In particular the star-operator takes the constant function 1 to the volume element \n$$*1 \\ = \\ {\\rm vol}_g \\ = \\ ({\\rm det} g)^{1\/2}|dx^1 \\ldots dx^d| \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (3)$$\nNotice that for a Lorentzian metric $g$ the volume element $*1$ is pure imaginary. This agrees with the fact that the `action' $S_g$ should be real for a Lorentzian manifold. We want the real part of the quadratic form i$S_g$ to be positive-definite for all the complex metrics we allow. This imposes two conditions. First,  we need the real part of the twisted $d$-form vol$_g$ defined by the formula (3) to be a positive volume-form on $M$. We therefore require that $\\det g$, which is invariantly defined up to multiplication by a positive real number, is \\emph{not} real and negative, and  we choose $(\\det g)^{1\/2}$ to have positive real part. \n\nThe second condition we need is that the real part of the matrix (det$g)^{1\/2}g^{-1}$ --- or equivalently of the inverse matrix $({\\rm det}g)^{-1\/2}g$ --- is positive-definite.  The two conditions together would give us a domain whose  Shilov boundary (like that of the Siegel generalized half-plane) contains indefinite real quadratic forms of all signatures, and not only the Lorentzian ones. But we shall impose further conditions. A clue to what more is needed comes from the theory of the electromagnetic field on $M$, with its field-strength given by a real 2-form $F$ on $M$, and with the action-functional\n$${\\rm i} S_g(F) \\ = \\ \\frac{1}{2}\\int_M F \\wedge * F.$$\nThe Hodge $*$-operator makes sense for a complex metric: for a $p$-form $\\alpha$ we define a twisted $(d-p)$-form $*\\alpha$  by taking the inner-product of $\\alpha$ with vol$_g \\ = \\ *1$, using the complex inner-product $g$.  We regard vol$_g$ as an element of the complex line $|\\wedge^d(T^*_x)|_{\\mathbb C}$, where $|\\wedge^d(T^*_x)|$ is the tensor product of $\\wedge^d(T^*_x)$ with the real orientation line of $T_x$, and $|\\wedge^d(T^*_x)|_{\\mathbb C}$ is its complexification, \\emph{but} with the convention that the orientation-reversing automorphisms of $T_x$ act \\emph{antilinearly}. We say that an element of the real part of the line is positive if it is a positive volume-element.\n\nFor the electromagnetic field we need the real part of the quadratic form\n$$\\wedge^2(T_x^*)  \\ \\longrightarrow \\ |\\wedge^d(T_x^*)|_{\\mathbb C}$$ \ngiven by $F \\mapsto F\\wedge *F$ to be positive-definite. \n\nThis makes it natural, if we are going to consider space-time manifolds $M$ of all dimensions, to propose \n\n\\bigskip\n\n\\noindent{\\bf Definition 2.1} \\ \\  {\\it On a $d$-dimensional real vector space $V$ a quadratic form $g:V \\to \\mathbb C$ is called an {\\rm allowable} complex metric if, for all degrees $p \\geq 0$, the real part of the quadratic form \n$$\\wedge^p(V^*) \\ \\longrightarrow \\ |\\wedge^d(V^*)|_{\\mathbb C}$$\ngiven by  $\\alpha \\mapsto \\alpha \\wedge * \\alpha$ is positive-definite.} \n\n\\bigskip\n\nFortunately, this definition has an equivalent formulation which is much more explicit and illuminating.\n\n\\bigskip\n\n\n\\noindent{\\bf Theorem 2.2} \\ \\ {\\it Definition 2.1 is equivalent to: there is a basis of the real vector space $V$ in which the quadratic form $g$ can be written\n$$\\lambda_1 y_1^2 + \\lambda_2 y_2^2 + \\ldots + \\lambda_d y_d^2,$$\nwhere the $y_i$ are coordinates with respect to the basis, and the $\\lambda_i$ are non-zero complex numbers, not on the negative real axis, such that}\n$$|\\arg (\\lambda_1) | + |\\arg (\\lambda_2) | + \\ldots + |\\arg (\\lambda_d)| < \\pi. \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (4)$$ \n\n\\bigskip\n\nThe complex-valued quadratic forms  $g:V \\to \\mathbb C$ on a real vector space $V$ which satisfy the conditions of (2.1) or (2.2) form an open subset  $Q_{\\mathbb C}(V)$ of the complex vector space $S^2(V^*_{\\mathbb C})$.   It  follows from Theorem 2.2 that the real inner products with signature $(d-1,1)$ --- but not those with other signatures --- lie on the boundary of the domain $Q_{\\mathbb C}(V)$. For if the metric is real then each $|\\arg(\\lambda_i)|$ is either 0 or $\\pi$, and the inequality (4)  shows that at most \\emph{one} of the $|\\arg(\\lambda_i)|$ can become $\\pi$ on the boundary. \n\nAnother consequence of (4) is that \n$$\\max \\arg \\lambda_i \\ - \\ \\min \\arg \\lambda_i \\ < \\ \\pi,$$\nwhich shows  that when $v$ runs through $V$ the complex numbers $g(v)$ form a closed convex cone in $\\mathbb C$ disjoint from the open negative real axis. In particular, $g(v)$ can never be real and negative.\n  \n  Using the criterion mentioned at the end of Section 1 we see that the real Lorentzian metrics --- and no other nondegenerate metrics ---  belong to the \\emph{Shilov} boundary of $Q_{\\mathbb C}(V)$, when it is regarded as a bounded domain in an affine variety (cf. the proof of 2.7 below).  Indeed if $g = \\sum \\lambda_j y_j^2$ is a complex metric for which the inequality (4) becomes an equality, and at least two of the eigenvalues $\\lambda_j$ and $\\lambda_k$ are not on the negative real axis, then (after rescaling the basis vectors $e_j$ and $e_k$ so that $|\\lambda_j | = |\\lambda_k| = 1$) we get a holomorphic curve through $g$, in the closure of $Q_{\\mathbb C}(V)$, by changing $\\lambda_j$ to $(\\lambda_j)^{1+z}$ and $\\lambda_k$ to $(\\lambda_k)^{1-\\varepsilon z}$, where $\\varepsilon$ is $+1$ or $-1$ according as the arguments of $\\lambda_j$ and $\\lambda_k$ have the same or opposite signs. \n  \n  In fact the Shilov boundary of $Q_{\\mathbb C}(V)$ contains \\emph{two} disjoint copies of the space of Lorentzian metrics on $V$, for an eigenvalue $\\lambda$ can approach the negative real axis either from above or from below. The two copies are interchanged by the complex-conjugation map on $Q_{\\mathbb C}(V)$. Because of our choice to make the orientation-reversing elements of GL$(V)$  act antilinearly on the orientation-line of $V$, we can say that the nondegenerate points of the Shilov boundary of $Q_{\\mathbb C}(V)$ are the \\emph{time-oriented} Lorentzian metrics.\n\n\\bigskip\n\n\nWe define the space Met$_{\\mathbb C}(M)$ of allowable complex metrics on a smooth manifold $M$ as the space of smooth sections of the bundle on $M$ whose fibre at $x$ is $Q_{\\mathbb C}(T_x)$. \n\n\\bigskip\n\nBefore giving the surprisingly simple proof of Theorem 2.2\nlet us say what motivated  the two different-looking  conditions. The desire to make the real parts of natural quadratic action functionals positive-definite hardly needs further comment, but choosing to focus on the  `higher abelian gauge field' actions $\\alpha \\wedge  * \\alpha$ --- the `Ramond-Ramond' fields of superstring theory ---  may well seem arbitrary. Why not allow other kinds of tensor fields? Our conditions do not imply that they will be positive-definite. Witten has kindly suggested to us a justification for our focus, based on properties of the classical energy-mometum tensor explained in [WW]. Including the higher gauge theories does, in any case, impose an \\emph{upper} bound on the class of complex metrics we can allow, for the  partition functions of these theories on a  $d$-dimensional torus $M$ with a flat Riemannian metric $g$ are explicitly known (cf. [Ke], [Sz](4.4)), and we can see to which complex metrics they can be analytically continued. The gauge-equivalence classes of fields form an infinite-dimensional Lie group which is a product of a torus, a lattice, and an infinite-dimensional real vector space, and the partition function is the product of three corresponding factors.  More precisely, an abelian gauge $(p-1)$-field $A$ has a \\emph{field-strength} $F_A$, a closed $p$-form on $M$ with integral periods, which determines $A$ up to the finite-dimensional torus $H^{p-1}(M;\\mathbb T)$ of flat gauge fields with $F_A = 0$. The space of fields is therefore a product\n$$H^{p-1}(M;\\mathbb T) \\ \\times \\ \\Phi_p \\ \\times \\ \\Gamma_p,$$\nwhere $\\Phi_p$ is the vector space of exact $p$-forms on $M$, and $\\Gamma_p \\cong {\\rm Harm}_{\\mathbb Z}^p(M) \\cong H^p(M;\\mathbb Z)$ is the finite-dimensional lattice of harmonic (and hence constant) $p$-forms with integral periods. The partition function is a Gaussian integral on this product: the torus of flat fields contributes its volume (for an appropriate metric determined by the geometry of $M$), the lattice $\\Gamma_p$  of harmonic $p$-forms contributes its theta-function $$\\sum_{\\alpha \\in \\Gamma_p}  \\ \\exp \\left(-\\frac{1}{2} \\int_M\\ \\alpha \\wedge *\\alpha \\right ) \\ ,$$\nwhile the vector space $\\Phi_p$ contributes an `analytic torsion' which is a power of the determinant of the Laplace operator acting on smooth functions on $M$ (with the zero-eigenvalue omitted) --- an analogue of the Dedekind eta-function, but with the lattice of characters of the torus $M$ replacing the lattice $\\mathbb Z + \\tau \\mathbb Z \\subset \\mathbb C$. Of these three  factors, the first clearly extends holomorphically to the space of all flat complex metrics on $M$, and the analytic torsion can be continued to a non-vanishing holomorphic function in the open set of complex metrics $g$ for which (det  $g)^{-1\/2}g$ belongs to the Siegel domain $\\mathbb U(V))$; but the theta-function cannot be continued beyond those metrics for which the real part of the form $\\int \\alpha \\wedge *\\alpha$ is positive. \n\n\n\\bigskip\n\nApproaching from the opposite direction, the inequality (4) is motivated by \nthe traditional analytical continuation of vacuum expection values to an open subset of the $k$-fold product of complexified Minkowski space $\\mathbb M_{\\mathbb C}$. The Wightman axioms imply that the expectation values extend holomorphically to a domain $\\mathcal U_k$ called the `permuted extended tube'\\footnote{A set of points $x_1, \\ldots , x_k$ belongs to $\\mathcal U_k$ if, after ordering them suitably, there is an element $\\gamma$ of the complexified Lorentz group such that the imaginary part of $\\gamma(x_i - x_{i+1})$ belongs to the forward light-cone for each $i$. }, which is functorially associated to $\\mathbb M_{\\mathbb C}$ with its $\\mathbb C$-bilinear metric.  It is a basic result in the Wightman theory (cf. [SW], or [Ka](2.1)) that $\\mathcal U_k$ contains the configuration space Conf$_k(\\mathbb E)$ of all $k$-tuples of \\emph{distinct} points of the standard Euclidean subspace $\\mathbb E \\subset \\mathbb M_{\\mathbb C}$. For a $d$-dimensional real vector space $V$ with a complex metric the complexification $V_{\\mathbb C}$ is isomorphic to $\\mathbb M_{\\mathbb C}$, uniquely up to a complex Lorentz transformation, and so the domain $\\mathcal U_k(V)$ is well-defined in $(V_{\\mathbb C})^k$. In  the next section we shall give a definition of a quantum field theory on space-times $M$ with complex metrics: it implies that the expectation values are smooth functions on the configuration spaces Conf$_k(M)$ of distinct $k$-tuples in $M$.  That makes it natural to\nask which (constant) complex metrics on $V$  have the property that the configuration space Conf$_k(V)$  is contained in the holomorphic envelope of $\\mathcal U_k(V)$, i.e.  the largest Stein manifold to which all holomorphic functions on $\\mathcal U_k(V)$ automatically extend. The original motivation of condition (4) was\n\n\\bigskip\n\n\\noindent{\\bf Proposition 2.3} \\ {\\it If a complex metric on a $d$-dimensional real vector space $V$ satisfies condition (4)  then ${\\rm Conf}_k(V)$ is contained in the holomorphic envelope of $\\mathcal U_k(V)$.}\n\n\\bigskip\n\n We shall postpone the  proof of this result to an appendix at the end of this section. \n\n\\bigskip\n     \n\\noindent{\\it Proof of Theorem 2.2} \\ \\  The first point is to show that a quadratic form which satisfies the conditions of Definition 2.1 can be written in the diagonal form $\\sum \\lambda_j y_j^2$ with respect to  real coordinates $y_j$ on $V$. To diagonalize a complex form $g = A+ {\\rm i} B$ with respect to a real basis is to diagonalize its real and imaginary parts simultaneously, which is possible if either $A$ or $B$ --- or, more generally, a real linear combination of them such as the real part of $({\\rm det}g)^{-1\/2}g$ --- is positive-definite. But 2.1, applied when  $p=1$, implies that the real part of $({\\rm det}g)^{-1\/2}g$ is positive.\n\nSuppose now that $g$ is diagonalized  with respect to a basis $\\{e_i\\}$ of $V$. Then the form $\\alpha \\mapsto \\alpha \\wedge * \\alpha$ on $\\wedge^p(V^*)$ is diagonal with respect to the basis $\\{e_S^* = e^*_{i_1}\\wedge \\ldots \\wedge e^*_{i_p}\\}$, where $\\{e_i^*\\}$ is the dual basis to $\\{e_i\\}$, and $S$ runs through $p$-tuples $S = (i_1, \\ldots , i_p)$. The value of the form  $\\alpha \\wedge * \\alpha$ on the basis element $e^*_S$ is\n$$(\\lambda_1 \\ldots \\lambda_d)^{1\/2}\\prod_{i \\in S}\\lambda_i^{-1},$$\nwhich has positive real part if its argument\n$$\\frac{1}{2}\\left\\{\\sum_{i \\in S} {\\rm arg}(\\lambda_i) - \\sum_{i \\not \\in S} {\\rm arg}(\\lambda_i)\\right\\}$$\nlies in the open interval $(-\\pi\/2,\\pi\/2)$. But to say that this is true for every subset $S$ of $\\{1, \\ldots , d\\}$ is precisely condition $(4)$.  $\\spadesuit$\n\n\\bigskip\n\n\nThe proof of Theorem 2.2 shows that to give an element $g$ of $Q_{\\mathbb C}(V)$ is the same as to give a finite sequence $\\theta_1 \\geq \\theta_2 \\geq \\ldots \\geq \\theta_m$ in the interval $(-\\pi, \\pi)$ together with a  decomposition\n $$V = V_1 \\oplus \\ldots \\oplus V_m$$\n such that\n $$\\sum_k \\dim V_k\\cdot |\\theta_k|<\\pi. $$ Thus on $V_k$ the bilinear form $g$ is ${\\rm e}^{{\\rm i} \\theta_k}$ times a real positive-definite form.  The only ambiguity in this description is that if, say, $\\theta_k = \\theta_{k+1}$ we can  replace $V_k$ by $V_k \\oplus V_{k+1}$ and omit $\\theta_{k+1}$ and $V_{k+1}$. This means that the subspace $P = \\bigoplus {\\rm e}^{-{\\rm i} \\theta_k\/2}V_k$ of the complexification $V_{\\mathbb C}$ of $V$ is \\emph{canonically} associated to the form $g$. On the real subspace $P$ the complex bilinear form $g$ is real and positive-definite. Our argument  gives us  canonical isomorphisms \n $$V \\ = \\ \\exp({\\rm i}\\pi\\Theta\/2)(P) \\ \\subset \\ P_{\\mathbb C} \\ = \\ V_{\\mathbb C},$$\n where $\\Theta:P \\to P$ is the self-adjoint operator which is multiplication by $\\theta_k$ on $P_k = {\\rm e}^{-{\\rm i}\\theta_k\/2} V_k$. Condition $(4)$ becomes the assertion that $\\Theta$ has trace-norm\\footnote{The trace-norm is the sum of the absolute values of the eigenvalues.} $||\\Theta ||_1 < 1$. \nThis shows that the space $Q_{\\mathbb C}(V)$ is parametrized by the pairs $(g_0, \\Theta)$, where $g_0$ is a positive-definite inner-product on $V$ and $\\Theta$ belongs to the  convex open set $\\Pi(V,g_0)$ of operators in $V$ which are self-adjoint with respect to $g_0$ and satisfy $||\\Theta||_1 < 1$, i.e. the interior of the convex hull of the rank 1 orthogonal projections in $V$.  In fact we have proved\n \n \\bigskip\n \n \\noindent{\\bf Proposition 2.4} \\ \\ $Q_{\\mathbb C}(V)$ {\\it is a fibre-bundle over the space of positive-definite inner products on $V$ whose fibre at a point $g_0$ is $\\Pi(V,g_0)$. Equivalently, choosing a reference inner-product on $V$, we have\n $$Q_{\\mathbb C}(V) \\ \\cong \\ {\\rm GL}(V) \\times_{{\\rm O}(V)} \\Pi(V).$$\n In particular, $Q_{\\mathbb C}(V)$ is contractible.}\n \n \\bigskip\n\nIt is an important fact that an allowable complex metric on $V$ remains allowable when restricted to any subspace $W$ of $V$. This follows from an analogous property of the trace-norm, but we shall give a direct proof, as its point of view on the angles $\\theta_i$ as critical values  helps give a feeling for allowable complex metrics.\n\n\\bigskip \n\n \\noindent{\\bf Proposition 2.5} \\  \\ {\\it If $g \\in Q_{\\mathbb C}(V)$ and $W$ is any vector subspace of $V$ then $g|W$ belongs to $Q_{\\mathbb C}(W)$. }\n \n \\bigskip\n\n\\noindent{\\it Proof} \\ \\ \\  For any $g \\in Q_{\\mathbb C}(V)$ the function $v \\mapsto {\\rm arg}(g(v))$ is a smooth map from the real projective space $\\mathbb P(V)$ to the open interval $(-\\pi, \\pi) \\subset \\mathbb R$.  By rescaling the basis elements $\\{e_k\\}$  we can write $g$ as $\\sum{\\rm e}^{{\\rm i}\\theta_k}y_k^2$. The numbers $\\theta_k$ are precisely the critical values of arg$(g)$.  We shall order the basis elements so that\n$$\\pi \\ > \\ \\theta_1 \\ \\geq \\ \\theta_2 \\ \\geq \\ \\ldots \\ \\geq \\ \\theta_d \\ > \\ -\\pi.$$\n\n\nFor each vector subspace $A$ of $V$  let us write $\\theta^A$ and $\\theta_A$ for the supremum and infimum of arg$(g)$ on $\\mathbb P(A)$.  Then we have\n$$\\theta_k  \\ \\ = \\ \\ {\\rm sup}\\{\\theta_A: {\\rm dim}(A) = k\\} \\ \\ = \\ \\ {\\rm inf}\\{\\theta^A: {\\rm dim}(A) = d-k+1\\}.$$\nIt is enough to prove Proposition 2.5 when $W$ is a subspace of $V$ of codimension 1. In that case the preceding characterization of the critical values shows that if \n$\\theta'_1 \\geq \\ldots \\geq \\theta'_{d-1}$\nare the critical values of arg$(g|W)$  we have $\\theta_k \\geq \\theta'_k \\geq \\theta_{k+1}$. The critical values for $g|W$ therefore \\emph{interleave} those for $g$:\n$$\\theta_1 \\geq \\theta'_1 \\geq \\theta_2 \\geq \\theta'_2 \\geq \\ldots \\geq \\theta_{d-1} \\geq \\theta'_{d-1} \\geq \\theta_d.$$\nThis implies that $\\sum |\\theta'_k| \\leq \\sum |\\theta_k| < \\pi$, as we want. $ \\ \\spadesuit$\n\n\n  \n\\bigskip\n\nIn Section 5 we shall need the following variant of the preceding formulation. \nSuppose that $Z$ is a $d$-dimensional complex vector space with a nondegenerate quadratic form $g$. (Any such pair $(Z,g)$ is isomorphic to $\\mathbb C^d$ with the standard form $\\sum z^2_k$.) Let $\\mathcal R(Z)$ denote the space of all $d$-dimensional real subspaces $A$ of $Z$ such that $g|A$ belongs to $Q_{\\mathbb C}(A)$. This is an open subset of the Grassmannian of all real subspaces of $Z$. If $Z_{\\mathbb R}$ is any $d$-dimensional real vector subspace of $Z$ for which $g|z_{\\mathbb R}$ is real and positive-definite then  the projection $A \\subset Z \\to Z_{\\mathbb R}$ is an isomorphism, for any non-zero element of its kernel would have the form i$v$ with $v \\in Z_{\\mathbb R}$, and so $g({\\rm i}v)$ would be real and negative, which cannot happen if $g|A$ is allowable.\n\n\\bigskip\n\n\\noindent{\\bf Proposition 2.6}  \\ \\ {\\it The space $\\mathcal R(Z)$ is contractible, and is  isomorphic to}\n$$ {\\rm O_{\\mathbb C}}(Z) \\times_{{\\rm O}(Z_{\\mathbb R})} \\Pi(Z_{\\mathbb R}).$$\n\n\\bigskip\n\n\\noindent{\\it Proof} \\ \\ This is essentially a reformulation of what has been said, but it may be helpful to relate the spaces $Q_{\\mathbb C}(V)$ and $\\mathcal R(Z)$ by considering, for a complex quadratic vector space  $(Z,g)$ as above, the intermediate space $\\mathcal R(V;Z)$ of $\\mathbb R$-linear embeddings $f:V \\to Z$ of the real vector space $V$  such that $f^*(g)$ is allowable. This space has two connected components, corresponding to the orientation of the projection $V \\to Z_{\\mathbb R}$.\n\nThe groups GL$(V)$ and O$_{\\mathbb C}(Z)$ act by right- and left-composition on $\\mathcal R(V;Z)$, and each action is free. Thus $\\mathcal R(V;Z)$ is at the same time a principal GL$(V)$-bundle with base $\\mathcal R(Z)$ and a principal O$_{\\mathbb C}(Z)$-bundle with base $Q_{\\mathbb C}(V)$. But the Lie groups GL$(V)$ and O$_{\\mathbb C}(Z)$ are homotopy equivalent to their maximal compact subgroups, i.e. in both cases to the compact orthogonal group O$_d$. More precisely, the contractibility of $Q_{\\mathbb C}(V)$ implies that $\\mathcal R(V;Z)$ is homotopy-equivalent to the fibre O$_{\\mathbb C}(Z)f$ for any  $f\\in \\mathcal R(V;Z)$. If we choose $f$ so that $f^*(g)$ is a positive-definite real form on $V$ this gives us a homotopy-equivalence $ {\\rm O}(V) \\to {\\rm O}_{\\mathbb C}(Z)f \\to \\mathcal R(V;Z)$. But O$(V)$ is also contained in and equivalent to the fibre $f{\\rm GL}(V)$ of the other fibration $\\mathcal R(V;Z) \\to \\mathcal R(Z)$, which implies the contractibility of its base $\\mathcal R(Z)$. \\ $\\spadesuit$\n\n\\bigskip\n\nThe last property of $Q_{\\mathbb C}(V)$ which we shall record briefly, for the sake of experts, is\n\n\\bigskip\n \n \n\\noindent{\\bf Proposition 2.7} \\  \\ {\\it The domain $Q_{\\mathbb C}(V)$ is holomorphically convex, i.e.\\ a `domain of holomorphy'.}\n\n\\bigskip\n\n\\noindent{\\it Proof} \\ \\ The Siegel domain $\\mathbb U(V)$ of  complex-valued inner products with positive-definite real part on a real vector space $V$ is known to be a domain of holomorphy in $S^2(V^*_{\\mathbb C})$. So therefore is the product\n$$\\prod_{0 \\leq p \\leq d\/2} \\mathbb U(\\wedge^p(V))$$\ninside its ambient complex vector space. The space $Q_{\\mathbb C}(V)$ is the intersection of this product domain with the affine variety which is the natural embedding of $S^2(V^*_{\\mathbb C})$ in this ambient vector space, and so it too is a domain of holomorphy. \\ $\\spadesuit$\n\n\\bigskip\n\n\n\\bigskip\n \n\\noindent {\\bf The two-dimensional case}\n\n\\bigskip\n\nThe case $d = 2$ is especially simple because then the matrix $(\\det g)^{- 1\/2}g$ depends only on the conformal structure, and decouples from  the volume element.\n\nA non-degenerate complex inner product $g$ on a 2-dimensional real vector space $V$ is determined up to a scalar multiple by its two distinct null-directions in the complexified space $V_{\\mathbb C}$. We can think of these as two points of the Riemann sphere $\\mathbb P(V_{\\mathbb C})$. Then  $(\\det g)^{-1\/2}g$ has positive real part precisely when the two points lie one in each of the open hemispheres of the sphere $\\mathbb P(V_{\\mathbb C})$ separated by the real equatorial circle $\\mathbb P(V)$. When the two points move to distinct points of the equator we get a Lorentzian inner product, with its two light-directions in $\\mathbb P(V)$.\n\nA point of the sphere $\\mathbb P(V_{\\mathbb C})$ not on the equator can be regarded as a complex structure on the real vector space $V$, and the two hemispheres correspond to the two possibilities for the orientation which a complex structure defines. On a smooth surface $\\Sigma$ any almost-complex structure is integrable, so a point of Met$_{\\mathbb C}(\\Sigma)$ is a pair of complex structures on $\\Sigma$ of opposite orientations, together with a complex volume element. The Riemannian metrics are those for which the two complex structures are complex-conjugate to each other, and the volume element is real.\n\nWhen $d=2$ the domain $Q_{\\mathbb C}(V)$ is thus a 3-dimensional polydisc, one disc for each of the complex structures, and the third for the volume-element.\n\n\\bigskip\n\n\\newpage\n\n\\noindent{\\bf The one-dimensional case: electric circuits}\n\n\\bigskip\n\nOur concept of an allowable complex metric does not at first look interesting in the one-dimensional case, but if we allow \\emph{singular} 1-manifolds --- identified with finite graphs $M$ --- we find that complex metrics arise naturally in electrical circuit theory. A Riemannian metric on $M$ is determined (up to isometry) by the assignment of a positive real number to each edge of the graph, and can be interpreted as its \\emph{resistance} when the edge is regarded as a wire in an electrical circuit. A state of the system (perhaps with current entering or leaving at each node) is determined by a continuous potential function $\\phi:M \\to \\mathbb R$ which is smooth on each closed edge, and whose gradient is the current flowing in the circuit. Because $\\phi$ is determined only up to adding a constant we shall normalize it by $\\int_M \\phi = 0.$  The energy of a state is \n$$\\frac{1}{2}\\int_M ||\\nabla \\phi ||^2 {\\rm d}s,$$\nand so the system can be regarded as a free massless field theory on the graph: in particular the vacuum expectation value $\\langle \\phi(x)\\phi(y)\\rangle$, when $x$ and $y$ are two nodes of the graph, is the ratio of the potential-difference $\\phi(x) - \\phi(y)$ to the current flowing in at $x$ and out at $y$ when no current is allowed to enter or leave at other nodes.\n\nWe encounter complex metrics when we consider a circuit in which  an alternating current with frequency $\\omega$ is flowing, and in which each branch has not only a resistance $R$ but also a positive inductance $L$ and a positive capacitance $C$. In that situation the volume element $\\sqrt g = R$ is replaced by the \\emph{impedance} $$\\sqrt g = R + {\\rm i}\\omega L + 1\/{\\rm i} \\omega C,$$ \na complex number which defines an allowable metric because Re$\\sqrt g > 0$.\n\nQuite apart from electric circuitry, however, singular one-dimensional manifolds with allowable complex metrics can arise in quantum field theory as the Gromov-Hausdorff limits of non-singular space-times of higher dimension. For example, if we embed a smooth graph $M$ in $\\mathbb R^3$, then for almost all sufficiently small $\\varepsilon >0$ the boundary of the $\\varepsilon$-neighbourhood of $M$ is a smooth surface $M_{\\varepsilon}$ whose limit is $M$ as $\\varepsilon \\to 0$: this is one way of viewing the passage from closed  string theory to quantum field theory.\n\n\\bigskip\n\n\\newpage\n\n\\noindent{\\bf Appendix to Section 2: proof of 2.3}\n\n\\bigskip\n\nIf $V$ is a real vector space with an allowable complex metric then  the preceding discussion shows that it can be identified with the subspace\n$$V \\ = \\ \\exp({\\rm i} \\Theta \/2)(\\mathbb E)$$\nof $\\mathbb M_{\\mathbb C}$. Here $\\mathbb E = \\mathbb R^d$ is the standard Euclidean subspace of $\\mathbb M_{\\mathbb C}$, and  $\\Theta$ is a real diagonal matrix whose entries $\\theta_1, \\ldots , \\theta_d$ belong to the `generalized octahedron' $\\Pi_0 \\subset \\mathbb R^d$ consisting of those $\\Theta$ whose diagonal entries $\\theta_1, \\ldots , \\theta_d$ satisfy  the inequality (4).  We want to prove  that $\\exp({\\rm i} \\Theta \/2)$ maps each $k$-tuple ${\\bf x} = \\{x_1,\\ldots , x_k\\}$ of distinct points of $\\mathbb E$ to a point of the holomorphic envelope $\\hat \\mathcal U_k$ of the Wightman permuted extended tube $\\mathcal U_k$. In fact we shall prove the stronger statement that $\\exp(\\rm i\\Theta\/2)({\\bf x}) \\in \\hat \\mathcal U_k$ when $\\Theta$ is a \\emph{complex} diagonal matrix with Re$(\\Theta) \\in \\Pi_0$.\n\n\n\nThe crucial fact is that $\\Pi_0$ is the convex hull of its intersection $\\Pi_{00}$ with the coordinate axes in $\\mathbb R^d$,  (i.e.\\ $\\Pi_{00}$ consists of the diagonal matrices with only one entry  $\\theta_r$  non-zero, and  $-\\pi < \\theta_r < \\pi$). Our strategy is to show that  $\\exp(\\rm i \\Theta \/2)({\\bf x}) \\in \\mathcal U_k$ when Re$(\\Theta) \\in \\Pi_{00}$, and to deduce that the same is true when Re$(\\Theta)$ belongs to the convex hull $\\Pi_0$.  The essential tool  is Bochner's `tube theorem' ([H\\\"or] Thm 2.5.10), which asserts that if $P$ is a connected open subset of $\\mathbb R^d$ then a holomorphic function defined in the tube domain $P\\times \\rm i\\mathbb R^d$ extends holomorphically to the tube domain $P' \\times \\rm i\\mathbb R^d$, where $P'$ is the convex hull of $P$.\n\nHaving fixed a $k$-tuple ${\\bf x}$ in $\\mathbb M_{\\mathbb C}$, let us first show that if Re$(\\Theta) \\in \\Pi_{00}$ then $\\exp({\\rm i} \\Theta \/2)({\\bf x})$ is contained in $\\mathcal U_k$. Suppose that the non-zero diagonal element of $\\Theta$ is in the $r^{\\rm th}$ place. Because $\\mathcal U_k$ in invariant under the orthogonal group O$(\\mathbb E)$ we can assume that the $r^{\\rm th}$ basis vector $e_r$ of $\\mathbb E$ is the Wick-rotated time-axis of $\\mathbb M$, so that $e_r$ belongs to $\\rm i C$, where $C$ is the forward light-cone in $\\mathbb M$. With respect to the real structure $\\mathbb M_{\\mathbb C} = \\mathbb M \\oplus \\rm i \\mathbb M$ the imaginary part of the $k$-tuple $${\\bf y} \\ =  \\ \\exp(\\rm i\\Theta\/2)({\\bf x})$$  lies on the line $\\mathbb R e_r$, and so, after ordering the points appropriately, ${\\bf y}$ will belong to the forward tube in $\\mathbb M_{\\mathbb C}$ providing the points of ${\\bf x}$ have distinct $r^{\\rm th}$ coordinates. But if the $r^{\\rm th}$ coordinates of Im$({\\bf y})$ are not distinct, we can make them so by choosing a unit vector $e \\in \\mathbb E$ perpendicular to $e_r$ such that the coordinates $\\langle {\\bf x}, e \\rangle$ are distinct, and rotating the $k$-tuple ${\\bf y}$ by a small amount in the $\\{e,e_r\\}$-plane, again using the O$(\\mathbb E)$-invariance of $\\mathcal U_k$.\n\nWe now know that $\\mathcal U_k$ contains an open neighbourhood of $\\Pi_{00} \\times \\rm i\\mathbb R^d$ in $\\mathbb C^d$. To apply Bochner's  theorem we need to know that the envelope $\\hat \\mathcal U_k$ contains a \\emph{tube}  $P \\times \\rm i\\mathbb R^d$,  where $P$ is an open neighbourhood of $\\Pi_{00}$ in $\\mathbb R^d$.  In fact it is enough, by induction, to treat the case  $d = 2$, for that case, together with Bochner's theorem, implies that a function holomorphic in a neighbourhood of $(\\Pi_0(\\mathbb R^r) \\cup \\Pi_{00}(\\mathbb R^{d-r}) \\times \\rm i\\mathbb R^d$ is holomorphic in a neighbourhood of $(\\Pi_0(\\mathbb R^{r+1}) \\cup \\Pi_{00}(\\mathbb R^{d-r-1})) \\times \\rm i\\mathbb R^d$.\n\nTo reduce the  $d=2$ case to the standard Bochner theorem it is enough to prove the following\n\n\\bigskip\n\n\\noindent{\\bf Lemma 2.8} \\ {\\it Let $L$ be the L-shaped subset $(\\{0\\} \\times [0,1)) \\cup ([0,1) \\times \\{0\\})$ of the quadrant $(\\mathbb R_+)^2$. Then any holomorphic function $F$ defined in a neighbourhood of $L \\times \\rm i\\mathbb R^2 \\subset \\mathbb C^2$ can be extended  holomorphically to  $P \\times \\rm i\\mathbb R^2$, where $P$ is the intersection of $(\\mathbb R_+)^2$ with a neighbourhood of $L$ in $\\mathbb R^2$.}\n\n\\bigskip\n\n\\noindent{\\it Proof} \\ For any $t \\in (0,1\/2)$ we define $D_t$ as the intersection of the two  unit discs $\\{z \\in \\mathbb C: |z - (1-t)| \\leq 1\\}$ and $\\{z \\in \\mathbb C: |z + (1-t)| \\leq 1 \\}$. Then we define $f:D_t \\to \\mathbb C^2$ by\n$$f(z) \\ = \\ (-\\log ((1-t)-z), -\\log ((1-t) +z).$$\nThe map $f$ is a holomorphic embedding in a neighbourhood of $D_t$ in $\\mathbb C$, and Re $f(\\partial D_t)$ is contained in the coordinate axes of $\\mathbb R^2$. If we choose \n $T = (1 - {\\rm e}^{-1})\/2$ then Re $f(\\partial D_T)$ is precisely the closure of $L$.\n  \nFor any $\\eta \\in \\mathbb R^2$, define $f_{\\eta}:D_T \\to \\mathbb C^2$ by $ f_{\\eta}(z) = f(z)+ \\rm i\\eta$. Then the holomorphic map $F$ is defined in a neighbourhood of the  curve $f_{\\eta}(\\partial D_T)$, and if we can show that $F \\circ f_{\\eta}$ extends holomorphically over $D_T$ then we shall have continued $F$ analytically to the tube domain $f(\\mathring D_T) + \\rm i\\mathbb R^2$, and the proof will be complete.\n\nWhen  a function $F$ is holomorphic in an open domain containing the boundary of a holomorphically-embedded disc  --- in this case $f_{\\eta}(D_T)$ --- then to show that $F$ can be extended over the whole disc  the standard method is to show that the disc can be moved holomorphically, keeping its boundary within the domain of $F$, until the whole disc is contained in the domain of $F$; the Cauchy integral formula then defines the desired extension. In our case we can deform  $f_{\\eta}(D_T)$ through the family $f_{\\eta}(D_t)$ as $t$ decreases from $T$ towards 0. As $t \\downarrow 0$ the domain $D_t$ shrinks to the origin in $\\mathbb C$, and $f_{\\eta}(D_t) \\to \\rm i\\eta$, which is contained in the domain of $F$. \\ \\ $\\spadesuit$ \n\n\n\n\\newpage\n\n\\section{Quantum field theories as functors}\n\nThe traditional Wightman approach to quantum field theory is not  well-adapted to important examples such as gauge theories, especially when the space-time is not flat. Another formulation --- potentially more general --- views a $d$-dimensional field theory as something more like a group representation, except that the group is replaced by a {\\it category} $\\mathcal C_d^{\\mathbb C}$ of space-time manifolds. The guiding principle of this approach is to preserve as much as possible of the path-integral intuition. We shall present it very briefly here, with minimal motivation.\n\nRoughly, the objects of the category $\\mathcal C_d^{\\mathbb C}$ are compact smooth $(d-1)$-dimensional manifolds $\\Sigma$ equipped with complex metrics $g \\in {\\rm Met}_{\\mathbb C}(\\Sigma)$. A morphism from $\\Sigma_0$ to $\\Sigma_1$ is a cobordism $M$ from $\\Sigma_0$ to $\\Sigma_1$, also with a complex metric. We shall write $M:\\Sigma_0 \\leadsto \\Sigma_1$ to indicate a cobordism. Composition of morphisms is by concatenation of the cobordisms.  The reason for the word `roughly' is that, because there is no canonical way to give a smooth structure to the concatenation of two smooth cobordisms, we must modify the definition slightly so that an object of $\\mathcal C_d^{\\mathbb C}$ is not a $(d-1)$-manifold but rather is a {\\it germ} of a $d$-manifold along a given $(d-1)$-manifold $\\Sigma$ --- i.e. $\\Sigma$ is given as a closed submanifold of a $d$-manifold $U$, but any two open neighbourhoods of $\\Sigma$ in $U$ define the same object of $\\mathcal C_d^{\\mathbb C}$. We require $\\Sigma$ to be \\emph{two-sided} in $U$, and equipped with a \\emph{co-orientation} which tells us which side is incoming and which is outgoing. (Nevertheless, we shall usually suppress  the thickening $U$, the co-orientation, and the complex metric $g$ from the notation.) Furthermore, two morphisms $M$ and $M'$ from $\\Sigma_0$ to $\\Sigma_1$ are identified if there is an isometry $M \\to M'$ which is the identity on the germs $\\Sigma_0$ and $\\Sigma_1$. (We shall return below to the question of the existence of identity morphisms in the cobordism category.)\n\n\n\\bigskip\n\nIn terms of the category $\\mathcal C_d^{\\mathbb C}$ we make the \n\n\\bigskip\n\n\\noindent{\\bf Definition} \\ {\\it A $d$-dimensional field theory is a \\emph{holomorphic} functor from $\\mathcal C_d^{\\mathbb C}$ to the category of Fr\\'echet topological vector spaces and \\emph{nuclear} (i.e. trace-class) linear maps which takes disjoint unions to tensor products.}\n\n\\bigskip\n\nUnfortunately, almost every word in this definition requires further explication.\n\n We shall write $E_{\\Sigma}$ for the vector space associated to an object $\\Sigma$, and $Z_M: E_{\\Sigma_0} \\to E_{\\Sigma_1}$ for the linear map associated to a cobordism $M: \\Sigma_0 \\leadsto \\Sigma_1$. To say that the functor is `holomorphic' means that, for a given smooth manifold-germ $\\Sigma \\subset U$, the topological vector spaces $E_{\\Sigma}$ form a locally trivial holomorphic vector bundle on the complex manifold Met$_{\\mathbb C}(U)$ of complex metrics on $U$, and that the maps $Z_M:E_{\\Sigma_0} \\to E_{\\Sigma_1}$ define a morphism of holomorphic vector bundles on the manifold Met$_{\\mathbb C}(M)$ (to which the bundles $\\{E_{\\Sigma_0}\\}$ and $\\{E_{\\Sigma_1}\\}$ are pulled back).\n\nIn practice, theories are usually defined on cobordism categories where the manifolds are required to have additional structure such as an orientation or a spin-structure. These can easily be included, but are not relevant to  our account. For the same reason we do not mention that, for a theory including fermions, the vector spaces $E_{\\Sigma}$ will have a mod 2 grading, and the usual sign-conventions must be applied when we speak of their tensor products.\n\n\\bigskip\n\nBecause our objects $\\Sigma \\subset U$ are really germs of $d$-manifolds, we automatically have a family of cobordisms $\\Sigma' \\leadsto \\Sigma$ embedded in $U$, each diffeomorphic to the trivial cobordism $\\Sigma \\times [0,1]$ with the outgoing boundary $\\Sigma \\times \\{1\\}$ corresponding to $\\Sigma \\subset U$. These cobordisms can be ordered by inclusion, giving us a direct system of objects $\\Sigma'$ with cobordisms to $\\Sigma$. Similarly, looking downstream rather than upstream, we have a family of cobordisms $\\Sigma \\leadsto \\Sigma''$ contained in $U$, giving us an inverse system of objects $\\Sigma''$ to which $\\Sigma$ maps.  For any field theory, therefore, there are natural maps\n$$\\lim_{\\rightarrow} E_{\\Sigma'} \\ \\  \\to \\ \\ E_{\\Sigma} \\ \\ \\to \\ \\ \\lim_{\\leftarrow} E_{\\Sigma''},$$\nwhich we shall write\n$$\\check E_{\\Sigma} \\ \\to \\ E_{\\Sigma} \\ \\to \\ \\hat E_{\\Sigma},$$\nintroducing the notations $\\check E_{\\Sigma} = \\varinjlim E_{\\Sigma'}$ and $\\hat E_{\\Sigma} = \\varprojlim E_{\\Sigma''}$ for the upstream and downstream limits. \n\nWe shall assume the functor has the \\emph{continuity} property that each of these maps is injective with dense image.  The space $\\hat E_{\\Sigma}$, being the inverse-limit of a countable sequence of nuclear maps of Fr\\'echet spaces, is a \\emph{nuclear} Fr\\'echet space\\footnote {A very useful concise account of nuclear spaces can be found in [C].}. The other space $\\check E_{\\Sigma}$ is also nuclear, but  usually not metrizable: it is the dual of the nuclear Fr\\'echet space $\\hat E_{\\Sigma^*}$, where $\\Sigma^*$ denotes the germ $\\Sigma$ with its co-orientation reversed. As this is such a basic point, we have included a proof as an Appendix at the end of this section. \n\nWhen we have a cobordism $M:\\Sigma_0 \\leadsto \\Sigma_1$ we automatically get maps $\\check E_{\\Sigma_0} \\to \\check E_{\\Sigma_1}$ and $\\hat E_{\\Sigma_0} \\to \\hat E_{\\Sigma_1}$.\nThe space $E_{\\Sigma}$ with which we began plays only a provisional role in the theory, serving to construct the fundamental nuclear spaces between which it is sandwiched. \n\n \n\n\\bigskip\n\nThe essential requirement we place on the functor is that it takes disjoint unions to tensor products, i.e., we are given an isomorphism of functors\n$$\\check E_{\\Sigma} \\otimes \\check E_{\\Sigma'} \\ \\to \\ \\check E_{\\Sigma \\sqcup \\Sigma'} ,$$\nwhich is associative and commutative in terms of the usual isomorphisms for the disjoint union and tensor product. There is a unique natural concept of tensor product here, because all the vector spaces are nuclear, and $\\check E_{\\Sigma} \\otimes \\check E_{\\Sigma'} \\cong \\check E_{\\Sigma \\sqcup \\Sigma'}$ is equivalent to $\\hat E_{\\Sigma} \\otimes \\hat E_{\\Sigma'} \\cong \\hat E_{\\Sigma \\sqcup \\Sigma'}$.  The functoriality means that we are assuming \n$$Z_M \\otimes Z_{M'} \\ = \\ Z_{M \\sqcup M'}$$\nfor two cobordisms $M$ and $M'$. \n\n\\bigskip\n\nThe tensoring assumption implies that $E_{\\emptyset} = \\mathbb C$, where $\\emptyset$ denotes the empty $(d-1)$-manifold.  Thus for a \\emph{closed} $d$-manifold $M$ we have a \\emph{partition function} $Z_M \\in {\\rm End}(E_{\\emptyset}) = \\mathbb C$. The whole structure of the theory is a way of expressing the sense in which the number $Z_M$ depends \\emph{locally} on $M$.\n\n\n\\bigskip \n\nIn this discussion we have still committed an abuse of language: the ``category\" $\\mathcal C_d^{\\mathbb C}$ is not really a category because it does not have identity maps. We could deal with this by agreeing that an isomorphism $\\Sigma_0 \\to \\Sigma_1$ is a cobordism of zero length, but then these degenerate cobordisms are represented by operators which are not nuclear. The true replacement for the missing identity operators is our assumption that the maps $\\check E_{\\Sigma} \\to \\hat E_{\\Sigma}$ are injective with dense image. To avoid the abuse of language we can say that  a field theory is a functor $\\Sigma \\mapsto E_{\\Sigma}$ from $(d-1)$-manifolds and isomorphisms to vector spaces, together with a transformation $Z_M: E_{\\Sigma_0} \\to E_{\\Sigma_1}$ for each cobordism. Whichever line we take, we must  assume that an isomorphism $f:\\Sigma_0 \\to \\Sigma_1$ of germs of $d$-manifolds  induces an isomorphism $f_*: E_{\\Sigma_0} \\to E_{\\Sigma_1}$ which depends smoothly on $f$, in the sense that for any family $P \\times \\Sigma_0 \\to \\Sigma_1$ parametrized by a finite-dimensional manifold $P$ the induced map $P \\times E_{\\Sigma_0} \\to E_{\\Sigma_1}$ is smooth.\n \n\n \n \\bigskip\n\n\n\\bigskip\n\nLet us explain briefly \nhow to get from this functorial picture to the traditional language of local observables and vacuum expectation values. For a point $x$ of a $d$-manifold $M$ we define the vector space $\\mathcal O_x$ of observables at $x$ as follows. We consider the family of all closed discs $D$ smoothly embedded in $M$ which contain $x$ in the interior $\\mathring D$. If $D' \\subset \\mathring D$ then $D \\setminus \\mathring D'$ is a cobordism  $\\partial D' \\leadsto \\partial D$ and gives us a trace-class map $E_{\\partial D'} \\to E_{\\partial D}$. We therefore have an inverse system $\\{E_{\\partial D}\\}$ indexed by the discs $D$, and we define $\\mathcal O_x$ as its inverse-limit.\n\nNow suppose that $M$ is closed, and that $x_1, \\ \\ldots \\  x_k$ are distinct points of $M$. Let $D_1, \\ \\ldots \\ D_k$ be disjoint discs in $M$ with $x_i \\in \\mathring D_i$. Then $M' = M \\setminus \\bigcup \\mathring D_i$ is a cobordism from $\\bigsqcup \\partial D_i$ to the empty $(d-1)$-manifold $\\emptyset$, and defines $Z_{M'}:E_{\\sqcup \\partial D_i} \\to E_{\\emptyset} =\\mathbb C$. Using the tensoring property  we can write this\n$$Z_{M'} : \\bigotimes E_{\\partial D_i} \\  \\longrightarrow \\ \\mathbb C,$$\nand then we can pass to the inverse-limits to get the expectation-value map\n$$\\bigotimes \\mathcal O_{x_i} \\ \\longrightarrow \\ \\mathbb C.$$\n\n\n\nWe might prefer the language of ``field operators\" to that of vacuum expectation values. If the space-time $M$ is a cobordism $\\Sigma_0 \\leadsto \\Sigma_1$, then for any $x$ in the interior of $M$ --- say $x \\in \\mathring D \\subset M$ --- the cobordisms $M \\setminus \\mathring D$ from $\\partial D \\sqcup \\Sigma_0$ to $\\Sigma_1$ define maps \n$$\\mathcal O_x \\  \\to \\ {\\rm Hom}_{nucl}(E_{\\Sigma_0}; E_{\\Sigma_1}),$$\nwhile if $x$ lies on a hypersurface $\\Sigma$ an observable at $x$ defines a map $\\check E_{\\Sigma} \\to \\hat E_{\\Sigma}$, i.e. it acts on $E_{\\Sigma}$ as an unbounded operator. But on a Lorentzian space-time $M$ we sometimes want to make the observables at all points $x \\in M$ act on a \\emph{single} vector space, and to ask whether they commute when space-like separated. We shall postpone that discussion to Section 5.\n\n\\bigskip\n\nOne observable which we should mention is the \\emph{energy-momentum tensor}. If we think of a field theory as analogous to a group representation then the energy-momentum tensor is the analogue of the induced representation of Lie algebras: for every cobordism $M:\\Sigma_0 \\leadsto \\Sigma_1$ it is the derivative of the operator $Z_M$ with respect to the metric of $M$. This makes it a distributional symmetric tensor-density $T^{ij}$ on $\\mathring M$ with values in Hom$_{nucl}(E_{\\Sigma_0};E_{\\Sigma_1})$. If we cover $M$ with small balls $D_i$, then by using a partition of unity we can write an infinitesimal change in the metric as the sum of contributions supported in the interiors of the $D_i$, and so the change in $Z_M$ is the sum of contributions coming from the spaces $E_{\\partial D_i}$, and hence from a field operators placed at the centres of the balls $D_i$.  But to develop this picture properly needs much more discussion, which we shall not embark on here;  it probably requires the assumption that the theory is asymptotically conformal at short distances.  The case of a 2-dimensional conformal theory is treated fully in Section 9 of [Se2].\n\n\\bigskip\n \n\n\n\n\\noindent {\\bf Lorentzian manifolds}\n\n\\bigskip\n\n\nThere is a category  $\\mathcal C^{\\rm Lor}_d$ which at first sight looks more relevant to quantum field theory than $\\mathcal C_d^{\\mathbb C}$.  Its objects are compact Riemannian manifolds of dimension $(d-1)$ and its morphisms are $d$-dimensional cobordisms equipped with real Lorentzian metrics. Fredenhagen and his coworkers (cf. [BF]) have developed the theory of quantum fields in curved space-time using a version of this category.  The category $\\mathcal C^{\\rm Lor}_d$ lies ``on the boundary\" of the category $\\mathcal C_d^{\\mathbb C}$. In section 5 we shall discuss the sense in which a representation of $\\mathcal C_d^{\\mathbb C}$ has a ``boundary value\" on $\\mathcal C_d^{\\rm Lor}$, at least if it is \\emph{unitary}. \n\n\\bigskip\n\n\n\n\\noindent {\\bf Unitarity}\n\n\n\n\\nopagebreak\n\n\n\\bigskip\n\nSo far we have not asked for an inner product on the topological vector space $E_{\\Sigma}$ associated to a $(d-1)$-manifold $\\Sigma$. Our main concern in this work is with {\\it unitary} theories, even though not  all interesting quantum field theories are unitary.\n\nTo define unitarity in our context, recall that,  if $\\Sigma^*$ denotes the manifold germ $\\Sigma$ with its co-orientation reversed, then $\\check E_{\\Sigma^*}$ is the dual topological vector space to $\\hat E_{\\Sigma}$. \nFurthermore, a cobordism $M: \\Sigma_0 \\leadsto \\Sigma_1$  can also be regarded as a cobordism from $\\Sigma^*_1$ to $\\Sigma^*_0$, and the two maps $E_{\\Sigma_0} \\to E_{\\Sigma_1}$ and $E_{\\Sigma_1^*} \\to E_{\\Sigma_0^*}$ are automatically algebraic transposes of each other. Thus $\\Sigma \\mapsto \\Sigma^*$ is a {\\it contravariant} functor.\n\n\nIn a unitary theory we shall not expect the vector space $E_{\\Sigma}$ to have an inner product for every $(d-1)$-manifold $\\Sigma$. A complex metric $g \\in {\\rm Met}_{\\mathbb C}(\\Sigma)$ has a complex conjugate $\\bar g$.\nIf we write $\\bar \\Sigma$ for $\\Sigma$ with the metric $\\bar g$ but with its co-orientation unchanged\\footnote{If our theory is defined on a category of \\emph{oriented} space-time manifolds, we must give $\\bar \\Sigma$  the opposite orientation to $\\Sigma$, although the same co-orientation in $U$.} then $\\Sigma \\mapsto \\bar \\Sigma$ is a \\emph{covariant} functor. It is natural to require that there is an antilinear involution\n$$E_{ \\bar\\Sigma} \\cong \\bar E_{\\Sigma}. \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (5)$$\n\n\n\n\\bigskip\n\nFor a theory satisfying condition (5) the conjugate dual of the vector space $\\check E_{\\Sigma}$ is $\\hat E_{\\bar\\Sigma^*}$.  We expect $\\check E_{\\Sigma}$ to have an inner product only when $\\Sigma \\cong \\bar\\Sigma^*$, i.e. when the $d$-manifold germ $\\Sigma \\subset U$ admits a reflection with fixed-point set $\\Sigma$ which reverses the co-orientation and changes the metric to its complex conjugate.\nSuch a hypersurface-germ $\\Sigma$ will be called {\\it time-symmetric}. Its metric is real and Riemannian when restricted to the $(d-1)$-dimensional hypersurface $\\Sigma$ itself.\n\n\\bigskip\n\nWe can now define a {\\it unitary} theory as one which satisfies two conditions:\n\n\\bigskip\n\n(i) \\ \\ the reality condition (5), and\n\n\\bigskip\n\n(ii) \\ \\ {\\it reflection-positivity}, in the sense that when we have a time-symmetric hypersurface $\\Sigma \\cong \\bar \\Sigma^*$ the hermitian duality between $\\check E_{\\Sigma}$ and $\\check E_{\\bar \\Sigma}$ is positive-definite. \n\n\\bigskip\n\nFor a unitary theory, when we have a time-symmetric germ $\\Sigma$ we can complete the pre-Hilbert space $\\check E_{\\Sigma}$ to obtain a Hilbert space $E^{Hilb}_{\\Sigma}$ with\n$$\\check E_{\\Sigma}  \\ \\to \\ E_{\\Sigma}^{Hilb} \\ \\to \\ \\hat E_{\\Sigma}.$$ \n\n\n\n\n\n\n\n\n\n\n\\noindent{\\bf The theory on flat tori}\n\n\\bigskip\n\nThe partition function of a theory on oriented flat Riemannian tori already gives us a lot of information about the theory.  The moduli space of such tori is the double-coset space$${\\rm O}_d \\backslash {\\rm GL}_d(\\mathbb R)\/{\\rm SL}_d(\\mathbb Z) \\ \\cong \\ Q(\\mathbb R^d)\/{\\rm SL}_d(\\mathbb Z),$$\nwhere $Q(\\mathbb R^d) = {\\rm O}_d\\backslash {\\rm GL}_d(\\mathbb R)$ is the space of positive-definite real $d \\times d$ matrices. This space is an orbifold, so the partition function is best described as a smooth function $Z: Q(\\mathbb R^d) \\to \\mathbb C$ which is invariant under SL$_d(\\mathbb Z)$. Our axioms imply that $Z$ extends to a \\emph{holomorphic} function\n$$Q_{\\mathbb C}(\\mathbb R^d) \\ \\to \\ \\mathbb C,$$\nbut they also imply very strong constraints beyond that. Notably, each choice of a surjection $\\mathbb Z^d = \\pi_1(M) \\to \\mathbb Z$ gives us a way of writing the torus $M$ as a cobordism $\\tilde M : \\Sigma \\leadsto \\Sigma$ from a $(d-1)$-dimensional torus $\\Sigma$ to itself, and then we have $Z(M) = {\\rm trace} (Z_{\\tilde M})$, where \\mbox{$Z_{\\tilde M}:E_{\\Sigma} \\to E_{\\Sigma}$} is a nuclear operator in the vector space $E_{\\Sigma}$, which is graded by the characters $\\chi$ of the translation-group $T_{\\Sigma}$ of $\\Sigma$. More explicitly, $M$ is constructed from the product manifold $\\tilde M \\times [0,t]$ by attaching the ends to each other after translating by a vector $\\xi \\in T_{\\Sigma}$, and  we  have\n$$Z(A,t,\\xi) \\ = \\ \\sum_{i,\\chi} n_{i,\\chi }\\chi(\\xi) \\  {\\rm e}^{-\\lambda_i t},$$\nwhere $\\{\\lambda_i = \\lambda_i(A) \\}$ is the sequence (tending to $+\\infty$) of eigenvalues of the Hamiltonian operator on  $E_{\\Sigma}$, and the $n_{i,\\chi}$ are positive integers which, for each $i$, vanish for all but finitely many characters $\\chi$. \n\n\\bigskip\n\n\n\\noindent{\\bf Appendix  to Section 3: \\ The duality $(\\check E_{\\Sigma})^* \\ \\cong \\ \\hat E_{\\Sigma^*}$}\n\n\\bigskip\n\nTo keep things as general as possible, we suppose that $\\Sigma \\mapsto E_{\\Sigma}$ is a functor from the $d$-dimensional cobordism category to a category of metrizable topological vector spaces and nuclear maps. We suppose also that the category of vector spaces is equipped with a  tensor product functor\\footnote{For example, we could work with the category of Hilbert spaces with the natural Hilbert space tensor product.} which is coherently associative and commutative, and that we are given natural isomorphisms $E_{\\Sigma_1} \\otimes E_{\\Sigma_2} \\ \\to \\ E_{\\Sigma_1 \\sqcup \\Sigma_2}$.\n\n\\bigskip\n\nComposable cobordisms\n$\\Sigma_1 \\leadsto \\Sigma_2 \\leadsto \\Sigma_3$ \n  give us  maps\n$$E_{\\Sigma_1} \\to E_{\\Sigma_2} \\to E_{\\Sigma_3}. \\ \\qquad  \\ \\qquad \\ (6)$$\nBy reinterpreting  $\\Sigma_1 \\leadsto \\Sigma_2$ as a cobordism $\\Sigma_1 \\sqcup \\Sigma_2^* \\leadsto \\emptyset$ we get a map $E_{\\Sigma_1} \\otimes E_{\\Sigma_2^*} \\to \\mathbb C$, and hence $E_{\\Sigma_1} \\to (E_{\\Sigma_2^*})^*$. Similarly, we can reinterpret $\\Sigma_2 \\leadsto \\Sigma_3$ as $\\emptyset \\leadsto \\Sigma_2^* \\sqcup \\Sigma_3$, which gives $(E_{\\Sigma_2^*})^* \\to E_{\\Sigma_3}$. It is easy to see that the composite $E_{\\Sigma_1} \\to (E_{\\Sigma_2^*})^* \\to E_{\\Sigma_3}$ coincides with $E_{\\Sigma_1} \\to E_{\\Sigma_2} \\to E_{\\Sigma_3}$. \n\nYet again, performing the reinterpretations in the reverse order, we get maps\n$$(E_{\\Sigma^*_1})^* \\to E_{\\Sigma_2} \\to (E_{\\Sigma^*_3})^*$$\nwhose composite  is the transpose of the map induced by the composite cobordism $\\Sigma_3^* \\leadsto \\Sigma_1^*$.\n\n\\bigskip\n\nNow suppose that we have an infinite sequence of cobordisms \n$$\\ldots \\leadsto \\Sigma_{i+1} \\leadsto \\Sigma_{i} \\leadsto \\Sigma_{i-1} \\leadsto \\ldots \\ , \\qquad \\qquad (7)$$ indexed by $i \\geq 0$, which form the downstream tail of a manifold-germ $\\Sigma$, i.e. the sequence which we used above to define the space $\\hat E_{\\Sigma} = \\lim_{\\leftarrow} E_{\\Sigma_i}$. \nLet us perform the two manipulations that we performed on (6) alternately on the sequence (7), thereby obtaining a sequence whose even terms are $E_{\\Sigma_{2i}}$ and whose odd terms are $(E_{\\Sigma^*_{2i+1}})^*$.  The inverse-limit of the whole sequence is the same as that of any cofinal subsequence. Considering the cofinal subsequence of even terms shows that the inverse-limit is $\\hat E_{\\Sigma}$. But the inverse-limit of the cofinal sequence of odd terms is\n$$\\lim_{\\leftarrow} \\ (E_{\\Sigma_{21+1}^*})^* \\ = \\ (\\lim_{\\rightarrow} E_{\\Sigma^*_{21+1}})^*.$$ This  shows that $\\hat E_{\\Sigma} \\cong (\\check E_{\\Sigma^*})^*$. But, because $\\hat E_{\\Sigma}$ is automatically a nuclear Fr\\'echet space, we can dualize again and conclude that $(\\hat E_{\\Sigma})^* \\cong \\check E_{\\Sigma^*}$ also.\n\n\\bigskip \n\n\n\\section{Some analogies from representation theory}\n\n\\nopagebreak\n\nThe relation between representations of the category $\\mathcal C_d^{\\mathbb C}$ and of the Lorentzian category $\\mathcal C_d^{\\rm Lor}$ which lies ``on its boundary\"  follows a pattern familiar in the representation theory of many Lie groups which have a special class of unitary representations characterized as  the boundary values of holomorphic representations of a complex semigroup by contraction operators. The essential features can all be seen in the simplest example.\n\n\\bigskip\n\nThe group $G = {\\rm PSL}_2(\\mathbb R)$ is the group of M\\\"obius transformations of the Riemann sphere $\\Sigma = \\mathbb C \\cup \\infty$ which map the open upper half-plane $\\mathbb U$ to itself. It lies on the boundary of the complex sub-semigroup of $G_{\\mathbb C} = {\\rm PSL}_2(\\mathbb C)$ consisting of M\\\"obius transformations which map the closure of $\\mathbb U$ into its own interior. It is natural, however, to consider a slightly larger semigroup $G^<_{\\mathbb C}$ by including the degenerate M\\\"obius transformations which collapse $\\mathbb U$ to a single point in $\\mathbb U$ --- these correspond to complex $2 \\times 2$ matrices of rank one. The resulting semigroup is then a contractible open subset of the 3-dimensional complex projective space formed from the $2 \\times 2$ matrices.  The topological boundary of $G^<_{\\mathbb C}$ consists of the M\\\"obius transformations which take $\\mathbb U$ to a disc or point in the upper  half-plane which touches the real axis, and the Shilov boundary consists of the group $G$ of real M\\\"obius transformations ---  an open solid torus --- compactified by its 2-torus boundary, which is the hyperboloid  det$(A) = 0$ in $\\mathbb P^3_{\\mathbb R}$ consisting of the degenerate real M\\\"obius transformations. (Thus the complete Shilov boundary is the part of $\\mathbb P^3_{\\mathbb R}$ where det$(A) \\geq 0$.)\n\n\n\n\\bigskip\n\nThe irreducible unitary representations of the group $G = {\\rm PSL}_2(\\mathbb R)$ are essentially\\footnote{We shall ignore the  ``supplementary\" series, which is of measure zero in the space of representations.} of two kinds, the {\\it principal series} and the {\\it discrete series}. The best-known principal series representation is the action of $G$ on the Hilbert space of $1\/2$-densities on the circle $\\mathbb P^1_{\\mathbb R}$ which is the boundary of $\\mathbb U$ --- the general member of the series is the action on densities of complex degree $s$ with Re$(s) = 1\/2$. The best-known discrete series representation is the action of $G$ on the square-summable holomorphic 1-forms on $\\mathbb U$,  with the natural norm \n$$\\parallel \\alpha \\parallel ^2 = {\\rm i}\\int_{\\mathbb U} \\alpha \\wedge \\bar \\alpha$$\n --- more generally, for each positive integer $p$ we have the action on holomorphic $p$-forms $\\alpha = f(z)(dz)^{\\otimes p}$, when one must divide $\\alpha \\wedge \\bar \\alpha$ by the $(p-1)^{\\rm st}$ power of the $G$-invariant area form on the Poincar\\'e plane $\\mathbb U$ to define the norm. \n \n The discrete series representations obviously extend to bounded holomorphic representations of the semigroup $G^<_{\\mathbb C}$ by contraction operators. They are singled out by this `positive energy' property: the principal series representations cannot extend to $G^<_{\\mathbb C}$, because when $|a| < 1$ the element $w \\mapsto aw$ (here $w = (z-{\\rm i})\/(z+{\\rm i})$ is the coordinate in the unit-disc model $|w|<1$ of $\\mathbb U$) of the semigroup $G^<_{\\mathbb C}$ would be represented by an operator whose eigenvalues are $a^n$ for \\emph{all} $n \\in \\mathbb Z$.  But let us notice that, though  the discrete series representations are unitary on the boundary group $G = {\\rm PSL}_2(\\mathbb R)$, the degenerate elements of $G^<_{\\mathbb C}$, which collapse $\\mathbb U$ to a point $p \\in \\mathbb U$, are represented by bounded operators of rank 1. So these unitary representations of PSL$_2(\\mathbb R)$ do not extend unitarily to the whole Shilov boundary: the degenerate elements correspond to unbounded rank 1 operators $\\xi \\mapsto \\langle \\zeta , \\xi \\rangle \\eta$, where $\\eta$ and $\\zeta$ are ``non-normalizable elements\" of the Hilbert space --- i.e. they belong to an appropriate completion of it.\n  \n\\bigskip \n\nThe group $G$ is a subgroup of the group Diff$^+(S^1)$ of orientation-preserving diffeomorphisms of the circle. This infinite-dimensional Lie group does not possess a complexification, though its Lie algebra, the space of smooth vector fields on the circle, can of course be complexified. The beginning of the present work was the observation, made in the 1980s quite independently by the two authors and also by Yu. Neretin ([N], [Se1]), that there is an infinite-dimensional complex semigroup $\\mathcal A$ which has exactly the same relation to Diff$^+(S^1)$ as $G_{\\mathbb C}^<$ has to $G = {\\rm PSL}_2(\\mathbb R)$.  Its elements are complex annuli with parametrized boundary circles: one can think of them as `` exponentiations\" of outward-pointing complex vector fields defined on a circle in the the complex plane. The annuli form a complex semigroup when concatenated as cobordisms, and the lowest-weight or ``positive-energy\" representations of Diff$^+(S^1)$ --- and of loop groups --- which arise in 2-dimensional conformal field theory are precisely those which are boundary values of holomorphic representations of the semigroup $\\mathcal A$ by trace-class operators.\n\n\\bigskip\n\nThe discussion of PSL$_2(\\mathbb R)$ generalizes to the symplectic group $G = {\\rm Sp}(V) \\cong {\\rm Sp}_{2n}(\\mathbb R)$ of a real symplectic vector space $V$ of dimension $2n$. The role of the upper half-plane $\\mathbb U$ is played by the Siegel `generalized  upper half-plane' --- the domain $\\mathbb U(V)$ of positive  Lagrangian subspaces of the complexification $V_{\\mathbb C}$  described in Section 1. The group $G$ lies on the boundary of a semigroup $G^<_{\\mathbb C}$  which is the Siegel domain $\\mathbb U(\\tilde V \\oplus V)$, where $\\tilde V$ denotes $V$ with sign of its symplectic form reversed. A generic element of this domain is the graph of a complex symplectic transformation of $V_{\\mathbb C}$ which maps the closure of $\\mathbb U(V)$ into its own interior, but, just as was the case with PSL$_2(\\mathbb C)$, there are degenerate elements which map $\\mathbb U(V)$ non-injectively into itself.\nThe complex semigroup $G_{\\mathbb C}^<$ has been carefully studied by Roger Howe [H], who called it the {\\it oscillator semigroup}.  \n\nThe Shilov boundary of $G^<_{\\mathbb C}$ is the Grassmannian of real Lagrangian subspaces of $\\tilde V \\oplus V$: generically, these are the graphs of elements of the real  group $G = {\\rm Sp}(V)$, but this group is compactified by the addition of Lagrangian subspaces which intersect the axes of $\\tilde V \\oplus V$ nontrivially, and thus correspond to Lagrangian correspondences from $V$ to $V$ which are not actual maps $V \\to V$. Once again, whereas Sp$^<(V_{\\mathbb C})$ is a genuine semigroup, the composition-law of the real group Sp$(V)$ does not extend to the compactification.\n\n\\bigskip\n\n\n\\bigskip\n\n\n\nThe group $G = {\\rm Sp}(V)$ has a discrete series of unitary representations generalizing those of PSL$_2(\\mathbb R)$. The most important   is the \\emph{metaplectic representation} --- actually a representation of a double covering $\\tilde G$ of Sp$(V)$ --- which is the action on the {\\it quantization} $\\mathcal H_V$ of the symplectic space $V$.  The Hilbert space $\\mathcal H_V$ is characterized by the property that it contains a copy of the ray $(\\bigwedge^n (W))^{\\otimes (1\/2)}$ for each point $W$ of the domain $\\mathbb U(V)$ ---  the square-root of the natural hermitian holomorphic line bundle $\\{\\bigwedge^n(W)\\}$ on $\\mathbb U(V)$ is canonical up to multiplication by $\\pm 1$, and is holomorphically embedded in $\\mathcal H_V$. It is acted on by $\\tilde G$ rather than $G$.\n\nThe action of $\\tilde G$ on $\\mathcal H_V$ is the boundary-value of a holomorphic projective representation of the oscillator semigroup $G_{\\mathbb C}^<$. For $G_{\\mathbb C}^<$ is just the domain $\\mathbb U(\\tilde V \\oplus V)$, each point of which defines a ray in\n$$\\mathcal H_{\\tilde V \\oplus V} \\ \\cong \\  \\mathcal H_V^* \\otimes \\mathcal H_V \\ \\cong \\  {\\rm End}_{HS}(\\mathcal H_V),$$\nwhere End$_{HS}$ denotes the Hilbert-Schmidt endomorphisms.  (A more careful discussion shows that $G^<_{\\mathbb C}$ is represented by operators of trace class.)\n\n\\bigskip\n\nWhen $n = 1$ the group Sp$(V)$ is SL$_2(\\mathbb R)$, a double covering of the group PSL$_2(\\mathbb R)$ of M\\\"obius transformations we considered before. To relate the  cases of PSL$_2(\\mathbb R)$ and Sp$(V)$, recall that PSL$_2(\\mathbb C)$ is an open subspace of the complex projective space $\\mathbb P^3_{\\mathbb C}$ formed from the vector space of $2 \\times 2$ matrices: in fact it is the complement of the quadric $Q^2_{\\mathbb C} \\cong \\mathbb P^1_{\\mathbb C} \\times \\mathbb P^1_{\\mathbb C}$  defined by the vanishing of the determinant, i.e. by the matrices of rank 1.   The double covering group SL$_2(\\mathbb C)$ sits inside the Grassmannian of complex Lagrangian subspaces of $\\mathbb C^4$, which is a quadric 3-fold $Q^3_{\\mathbb C}$ in $\\mathbb P^4_{\\mathbb C}$: it is a non-singular hyperplane section (corresponding to the Lagrangian condition) of the Klein quadric formed by all the lines in $\\mathbb P^3(\\mathbb C)$. The quadric $Q^3_{\\mathbb C}$ is the branched double-covering of the projective space $\\mathbb P^3_{\\mathbb C}$ of $2 \\times 2$ matrices, branched along the quadric $Q^2_{\\mathbb C}$ of rank 1 matrices. The contractible semigroup SL$^<_2(\\mathbb C)$ is the open subset of the Lagrangian Grassmannian of $\\mathbb C^4$ consisting of the positive Lagrangian subspaces, and it is a double covering of PSL$^<_2(\\mathbb C)$.\n\n\\bigskip\n \n\n\\section{Unitarity and global hyperbolicity}\n\nIn the previous section we saw how a holomorphic representation of a complex semigroup by contraction operators on a Hilbert space can give rise --- on passing to the boundary --- to a unitary representation of a group which is a dense open subset of the Shilov boundary of the semigroup. The remaining points of the Shilov boundary are not represented by unitary operators; the representation extends to them only in  some ``weak\" sense. We now come to the analogue of this phenomenon in quantum field theory, where the Lorentzian cobordism category $\\mathcal C_d^{\\rm Lor}$ lies on the boundary of $\\mathcal C_d^{\\mathbb C}$, and the role of the open dense subgroup of the Shilov boundary is played by the subcategory of \\emph{globally hyperbolic} cobordisms which we shall define below. We should  mention, however, that although the category of globally hyperbolic cobordisms is very natural, the category $\\mathcal C_d^{{\\rm Lor}}$ may be smaller than the optimal category we could put on the boundary of $\\mathcal C_d^{\\mathbb C}$. For example, the Lorentzian cobordisms could possibly be allowed to contain `black holes' surrounded by horizons, rather analogous  to the `cobordisms-with-boundaries' used to describe two-dimensional theories with both open and closed strings. We shall not pursue such speculations here.\n\n\\bigskip\n\nWhen we have a theory defined on $\\mathcal C_d^{\\mathbb C}$ let us first consider how to extend the assignment $\\Sigma \\mapsto E_{\\Sigma}$ to  a \\emph{Lorentzian} germ\n $\\Sigma \\subset U$,  with $\\Sigma$ co-oriented in $U$.  We can identify $U$ with $\\Sigma \\times (-\\varepsilon, \\varepsilon)$ by exponentiating the geodesic curves emanating perpendicularly from $\\Sigma$.  The metric then takes the form $h_t - {\\rm d}t^2$, where $t \\mapsto h_t$ is a smooth map from $(-\\varepsilon, \\varepsilon)$ to the manifold of Riemannian metrics on $\\Sigma$. If the germ is time-symmetric then we can define $E_{\\Sigma}$ by replacing the Lorentzian metric by the `Wick rotated' Riemannian metric  $h_{{\\rm i}t} + {\\rm d}t^2$, which makes sense because if $h_t = h_{-t}$ then $h_t$ is a function of $t^2$, so that $h_{{\\rm i}t}$ is defined and real. But this does not help for a general hypersurface, and in any case seems rather arbitrary: we shall return to this point in Remark 5.3 below.\n\n\\bigskip\n\nIt is  less easy to assign an operator $Z_M:E_{\\Sigma_0} \\to E_{\\Sigma_1}$ to a Lorentzian cobordism $M:\\Sigma_0 \\leadsto \\Sigma_1$. Even if $M$ is a cylinder topologically, it can be complicated in its ``causal\" structure. Consider, for example, a 2-dimensional cylindrical space-time. We saw in Section 2 that, up to a conformal multiplier, a complex metric on a surface is a pair of complex structures with opposite orientations. At the Shilov boundary the complex structures degenerate to the foliations by the left- and right-moving light-lines of a Lorentzian surface. If each light-line which sets out from the incoming boundary circle of the cylinder eventually reaches the outgoing boundary circle then each family of light-lines gives us a diffeomorphism from the incoming to the outgoing boundary. In fact (cf. [Se2] p.8 and p.16) the isomorphism classes of Lorentzian cylinders of this kind are determined up to conformal equivalence by the pair of diffeomorphisms together with a positive integer which counts the number of times that the left- and right-moving lines emanating from a given point of the incoming circle cross before hitting the outgoing circle. This agrees with the well-known fact that the Hilbert space associated to a circle in 2-dimensional conformal field theory comes with a projective unitary representation of the group Diff$^+(S^1) \\times {\\rm Diff}^+(S^1)$.\n\nBut the light-lines from the incoming circle can behave in a more complicated way. For example, one set of light-lines may spiral closer and closer to a closed limit-cycle of the foliation, a light-line which is a circle parallel to the incoming boundary circle of the annulus. That set of lines will then never reach the outgoing circle. One might think of this phenomenon as  akin to a black hole in the space-time, though, unlike a black hole, the Lorentzian metric here has no singularity. The ``blocked'' foliation is conformally the same as the ``degenerate annulus'' obtained by collapsing the closed light-line to a point, i.e.   a pair of discs with their centre-points identified. This is usually regarded as an ``annulus of infinite length\", and it acts on an irreducible positive-energy representation of Diff$^+(S^1)$ by a projection operator of rank one, like the action of a degenerate complex M\\\"obius transformation  in a discrete-series representation of PSL$_2(\\mathbb R)$.\n\n\n\n\\bigskip\n\n\n\nIn works on general relativity a Lorentzian cobordism $M:\\Sigma_0 \\leadsto \\Sigma_1$ between Riemannian manifolds is called {\\it globally hyperbolic} if every maximally-extended time-like geodesic in $M$ travels from $\\Sigma_0$ to $\\Sigma_1$. Such an $M$ must be diffeomorphic to $\\Sigma_0 \\times [0,1]$.  It is only for globally hyperbolic manifolds that, for example, the  Cauchy problem for the wave-equation on $M$ is soluble.\n\nOf course here we are only considering {\\it compact} cobordisms, which are not the usual focus in relativity theory. In the compact situation we can take the definition of global hyperbolicity to be the existence of a smooth time-function $t: M \\to [0,1]$ whose gradient is everywhere in the positive light-cone,  and which is therefore a fibration with Riemannian fibres. From $t$ we obtain a diffeomorphism $M \\to \\Sigma_0 \\times [0,1]$  by following the orthogonal trajectories to the time-slices.\n\nThe existence of a time-function on a compact Lorentzian cobordism is clearly an open condition, and so the globally hyperbolic cobordisms form an open subcategory $\\mathcal C_d^{\\rm gh}$ of $\\mathcal C_d^{\\rm Lor}$ which should play the role of the real Lie group to which the holomorphic contraction representations of Section 4 can be extended (though the result (5.2) we prove below is unfortunately weaker).\n\n\\bigskip\n\nFor a globally hyperbolic cobordism equipped with a time-function, the metric, in terms of the  diffeomorphism $M \\to \\Sigma_0 \\times [0,1]$, takes  the form $h_t  + c^2{\\rm d}t^2$ for some function $c: \\Sigma_0 \\times [0,1]\\to {\\rm i}\\mathbb R$. A small deformation $\\delta c$ of $c$ into the right half-plane  changes the Lorentzian metric into an allowable complex metric, and we could hope to define $Z_M$ in the Lorentzian case as the limit of the operators associated to such deformations. That, however, encounters the problem that the deformed metric depends not only on the choice of the deformation $\\delta c$, but, more importantly, on the choice of the time-function, which should be irrelevant to the operator $U_M$. Happily, there is a better point of view, which also shows why the boundary-value of a semigroup of contraction operators is a \\emph{unitary} representation. There is, after all, no obvious reason why the concatenation of a Lorentzian cobordism with its reverse should be represented by the identity operator --- quite unlike what happens with Riemannian cobordisms.  (A possible analogy is the process of making a based loop-space  into a topological group by collapsing paths which retrace their steps.) \n\n\\bigskip\n\nThe passage from $\\mathcal C_d^{\\mathbb C}$ to $\\mathcal C_d^{\\rm Lor}$ is already interesting when $d=1$, i.e. for quantum mechanics rather than quantum field theory --- the case when the Euclidean path-integral can be treated by traditional measure-theory. It is worthwhile to spell out the argument in this case, before passing to higher dimensions. \n\nWe began this work with the relation of positive energy to 1-parameter contraction semigroups. Our first task now is to understand why a holomorphic representation of the category $\\mathcal C_1^{\\mathbb C}$ is just such a 1-parameter semigroup, where the parameter runs through the open half-plane $\\mathbb C_+ = \\{z \\in \\mathbb C: {\\rm Re}(z)>0\\}$. Whereas a Riemannian structure on a closed interval is completely determined by its length, the allowable complex metrics on the interval have an infinite-dimensional moduli-space.\n\nAny complex metric on  $I = [0,1]$ can be pulled back from the holomorphic quadratic differential ${\\rm d}z^2$ on $\\mathbb C$ by means of a smooth embedding $f: I \\to \\mathbb C$ such that $f(0) = 0$ and Re $f'(t) > 0$ for all $t \\in I$. In fact the space Emb$(I;\\mathbb C)$ of such embeddings is isomorphic to Met$_{\\mathbb C}(I)$ {\\it as a complex manifold}. If $f'(t) = 1$ when $t$ is sufficiently close to the ends of the  interval $I$ then the pulled-back metric defines a morphism $I_f: P \\to P$ in the category $\\mathcal C_1^{\\mathbb C}$, where $P$ denotes the object defined by the germ of the standard metric on the line $\\mathbb R$ at the origin. \n\nThe crucial observation is that the operator $Z_f: E_P \\to E_P$ defined by $I_f$ depends only on the point $f(1) \\in \\mathbb C_+$. It is as if $Z_f$ were the `contour integral' of a holomorphic differential on $\\mathbb C$ along the path $f$. The argument is as follows. First, $Z_f$ does not change if $f$ is replaced by $\\tilde f = f \\circ \\phi$ where $\\phi$ is any diffeomorphism $I \\to I$ which is the identity near the ends of the interval. This means that $Z_f$ does not change if $f$ moves along a curve in Emb$(I;\\mathbb C)$ whose tangent vector at each point is the action of an element of the Lie algebra Vect$(\\mathring I)$ of compactly supported vector fields on the interior of $I$. But then --- because $Z_f$ depends holomorphically on $f$ ---  it does not change if each tangent vector is the action of an element of the \\emph{complexified} Lie algebra Vect$_{\\mathbb C}(\\mathring I)$. Finally, if $f,\\tilde f \\in {\\rm Emb}(I;\\mathbb C)$ define two morphisms $P \\to P$ and have $f(1) = \\tilde f(1)$, the tangent vectors to the obvious linear path from $f$ to $\\tilde f$  are given by the action of elements of Vect$_{\\mathbb C}(\\mathring I)$. \n\nWe can therefore write $Z_f = u(z)$, where $z = f(1)$. Obviously we have $u(z_1)u(z_2) = u(z_1 + z_2)$ for any $z_1,z_2 \\in \\mathbb C_+$. Furthermore, because the object $P$ of $\\mathcal C_1^{\\rm gh}$ is time-symmetric, the vector space $\\check E_P$ is a pre-Hilbert space, and the unitarity condition tells us that $u(\\bar z)$ is the hermitian transpose of $u(z)$.\n\nThe desired unitary semigroup $\\{u({\\rm i} T)\\}_{T \\in \\mathbb R}$, which will act on the triple $\\check E_P \\to E_P^{Hilb} \\to \\hat E_P$, can now be defined as follows. As explained in Section 3, any vector $\\xi \\in \\check E_P$ can be written $\\xi = u(\\varepsilon )\\eta$ for some $\\varepsilon > 0$ and some $\\eta \\in E_P$. We define $u({\\rm i} T)\\xi = u(\\varepsilon + {\\rm i}T)\\eta$, which is plainly independent of $\\varepsilon$. Finally, $u({\\rm i} T)$ is unitary because\n\\begin{eqnarray*}\nu(-{\\rm i}T)u({\\rm i}T) \\xi \\ &=&  \\ u(-{\\rm i} T)u(\\varepsilon + {\\rm i} T)\\eta\\\\\n&=& \\ u(-{\\rm i} T)u(\\varepsilon\/2)u(\\varepsilon\/2 + {\\rm i} T)\\eta\\\\\n&=& \\ u(\\varepsilon\/2 - {\\rm i} T)u(\\varepsilon\/2 + {\\rm i} T)\\eta\\\\\n&=& \\ u(\\varepsilon)\\eta \\ = \\ \\xi.\n\\end{eqnarray*}\n\n\\bigskip\n\nTo pass from $d=1$ to higher-dimensional cobordisms we observe that the essential step in our argument was the first case of the following \n\n\\bigskip\n\n\\noindent{\\bf Principle 5.1} \\ \\ \\ {\\it If a  $d$-dimensional cobordism $M$  is a real submanifold of a complex $d$-manifold $M_{\\mathbb C}$, and $M$ has an allowable complex metric induced from a  holomorphic symmetric form $g$ on the tangent bundle $TM_{\\mathbb C}$,   then the linear map $Z_M$ does not change when $M$ is moved around smoothly inside $M_{\\mathbb C}$ (leaving its ends fixed), providing the restriction of $g$ to $M$ remains an allowable complex metric.}\n\n\\bigskip\n\nAs in the $d=1$ case, this principle holds because any infinitesimal movement of $M$ inside $M_{\\mathbb C}$ is given by a complex vector field on $M$, while $Z_M$ depends holomorphically on $M$ and, being invariant under the action of \\mbox{Diff$(M$ rel $\\partial M)$}, does not change when $M$ moves in a direction given by the action of a complexified tangent vector to this group. \n\n\\bigskip\n\nUnfortunately, to  use  the principle we need the cobordism $M$ to be embedded in a complexification $M_{\\mathbb C}$, and the only natural way to ensure this  is to pass from the smooth Lorentzian category $\\mathcal C_d^{\\rm Lor}$ to the corresponding \\emph{real-analytic} cobordism category $\\mathcal C_d^{\\rm Lor,\\omega}$, where both the manifolds and their metrics are assumed real-analytic. Inside this category there is the subcategory $\\mathcal C_d^{\\rm gh,\\omega}$ of globally hyperbolic cobordisms: we shall also assume that the time-function $\\tau:M \\to {\\rm i}[0,1]$ is real-analytic, though that could be avoided, because any smooth function can be approximated real-analytically. \n\n\\bigskip\n\nThere are two ways of thinking about restricting to real-analytic cobordisms. One might think that the smooth cobordism category is the natural object, and try to eliminate the analyticity hypothesis. But one could also think that that the natural allowable space-times really do come surrounded by a thin holomorphic thickening, within which the choice of a smooth totally-real representative is essentially arbitrary. In any case, we can prove the following theorem.\n\n\\bigskip\n\n\\noindent{\\bf Theorem 5.2} \\ \\  {\\it A unitary quantum field theory as defined in Section 3 on the category $\\mathcal C_d^{\\mathbb C}$ induces a functor from $\\mathcal C_d^{\\rm gh, \\omega}$ to topological vector spaces. The functor takes time-symmetric objects to Hilbert spaces, and takes cobordisms between them to unitary operators.} \n\n\\bigskip\n\nTo be quite precise: the theorem asserts that if $\\Sigma$ is a time-symmetric $(d-1)$-manifold germ then there is a Hilbert space $E_{\\Sigma}^{Hilb}$ with\n$$\\check E_{\\Sigma} \\ \\ \\subset \\ \\ E_{\\Sigma}^{Hilb} \\ \\ \\subset \\hat E_{\\Sigma},$$\nand a real-analytic globally hyperbolic cobordism $\\Sigma_0 \\leadsto \\Sigma_1$ between time-symmetric hypersurfaces induces a unitary isomorphism $E_{\\Sigma_0}^{Hilb} \\to E_{\\Sigma_1}^{Hilb}$ which also maps $\\check E_{\\Sigma_0}$ to $\\check E_{\\Sigma_1}$ and $\\hat E_{\\Sigma_0} $ to $\\hat E_{\\Sigma_1}$.\n\n\\bigskip\n\n\n\n\n\n\n\\bigskip\n\n \n  \n\\noindent {\\it Proof of 5.2} \\ \\   Given  a real-analytic globally hyperbolic  cobordism  \\mbox{$M:\\Sigma_0 \\leadsto \\Sigma_1$} we choose a time function $t:M \\to [0,1]$ whose level surfaces foliate $M$ by Riemannian manifolds, and, following the orthogonal trajectories to the foliation, we  identify $M$ with $\\Sigma_0 \\times [0,1]$ as before. \n  \n  Using the real-analyticity assumptions, we can find a complexification $M_{\\mathbb C}$ of $M$ to which both $t$ and $g$ can be extended holomorphically, and we can assume that $\\tau = {\\rm i}t:M_{\\mathbb C} \\to U \\subset \\mathbb C$ is a holomorphic fibre bundle over a neighbourhood $U$ of the interval ${\\rm i} [0,1]$. Furthermore, the isomorphism $\\Sigma_0 \\times [0,1] \\to M$ extends to a holomorphic trivialization of the bundle $M_{\\mathbb C} \\to U$.  For any smooth curve $f:[0,1] \\to U$ such that $f(0) = 0$ and ${\\rm Re} \\    f'(s) > 0$ for $s \\in [0,1]$ this gives us a totally real submanifold $M_f$ of $M_{\\mathbb C}$ sitting over the curve. We can use the morphism associated to the cobordism $M_f$ in exactly the way we used $Z_f$ in  discussing the 1-dimensional case, to obtain a unitary operator $Z_M$ associated to the Lorentzian cobordism.\n  \n  It is important  that $Z_M$ does not depend on the choice of the time-function $t$ defining the foliation. For two choices of $t$ are linearly homotopic, and changing from one to the other amounts to deforming the totally-real embedding $\\Sigma_0 \\times [0,1] \\to M_{\\mathbb C}$ by a real-analytic diffeomorphism of $\\Sigma_0 \\times [0,1]$.\n  \n  \\bigskip\n \n\\noindent{\\bf Remark 5.3} \\ \\ \\   We can apply the principle 5.1  to understand better how a theory defined on $\\mathcal C_d^{\\mathbb C}$ assigns a vector space $E_{\\Sigma}$ to a Lorentzian germ $\\Sigma \\subset U$. \n \n If the Lorentzian metric on $U$ is real-analytic then the complex theory gives us a holomorphic bundle $\\{\\hat E_f\\}$ on the space \n\n$\\mathcal J$   of germs of embeddings $f:(-\\varepsilon,\\varepsilon) \\to \\mathbb C$ such that $f(0)=0$ and Re $f'(t) >0$ for all $t$. In particular, for $\\lambda \\in \\mathbb C_+$ we have the radial paths $f_{\\lambda} \\in \\mathcal J$ for which $f_{\\lambda}(t) = \\lambda t$. But recall that $\\hat E_f$ is the inverse-limit of a sequence of spaces associated to the germs of $f$ at the points $f(t_k)$, for any sequence $\\{t_k \\downarrow 0\\}$. \n\nNow consider two neighbouring rays $f_{\\lambda},f_{\\lambda'}$ with $|\\lambda| = |\\lambda'|$, and choose a sequence $\\{t'_k \\downarrow 0 \\}$ which interleaves $\\{t_k \\}$, i.e. $t_k > t'_k >t_{k+1}$. We can choose a path $f \\in \\mathcal J$ which lies in the sector bounded by the rays $f_{\\lambda}$  and $f_{\\lambda '}$  and coincides with them alternately in the neighbourhoods of the points $\\lambda t_k$ and $\\lambda't'_k$. This $f$ gives us a family of cobordisms from the germ at $\\lambda' t'_k$ to the germ at $\\lambda t_k$, and from the germ at $\\lambda t_{k+1}$ to the germ at $\\lambda 't'_k$. Putting these together, we obtain  inverse canonical isomorphisms between $\\hat E_{f_{\\lambda}}$ and $\\hat E_{f_{\\lambda '}}$. The coherence of these isomorphisms when we consider three nearby rays also follows from the principle 5.1.\n\nBy this means we see that we could have chosen \\emph{any} smooth path $f$ to define $\\hat E_{\\Sigma}$. However the family $\\hat E_f$ has the property that $\\hat E_{\\bar f}$ is the complex-conjugate space to $\\hat E_f$, so that reversing the complex time-direction conjugates the identification of $\\hat E_{\\Sigma}$ with the Euclidean choice $\\hat E_{f_1}$. If the Lorentzian germ $\\Sigma \\subset U$ is time-symmetric --- but not otherwise --- the arguments we have already used will give us a hermitian inner product on $\\check E_{\\Sigma}$.\n \n  \n \n\\bigskip\n\n\\bigskip\n\n\\noindent{\\bf Field operators}\n\n\\nopagebreak\n\n\\bigskip\n\nFinally, we come to the Wick rotation of field operators, though our account will be sketchy. The first step is to understand how the vector space $\\mathcal O_x$ of observables at a point $x$ of a space-time $M$ behaves as the metric of $M$ passes from complex to Lorentzian. We shall continue to assume that $M$ and its Lorentzian metric are real-analytic.\n\nIn Section 3 we associated a space  $\\mathcal O_x$ to a germ at $x$ of a complex metric on a manifold containing $x$: it is the fibre of a bundle on the space Met$_{\\mathbb C}(\\hat x)$ of such germs. If we embed a Lorentzian $M$ in a complexification $M_{\\mathbb C}$ there will be a holomorphic exponential map from a neighbourhood of $0$ in the complexified tangent space $T_x^{\\mathbb C} = T_xM \\otimes \\mathbb C$ to $M_{\\mathbb C}$. Inside $T_x^{\\mathbb C}$ we can consider the $d$-dimensional real vector subspaces $V$ on which the metric induced  from the complex bilinear form of $T_x^{\\mathbb C}$ is allowable. We saw in  (2.6) that these $V$ form a contractible open subset $\\mathcal U$ of the real Grassmannian Gr$_d(T_x^{\\mathbb C})$.  Exponentiating $V$ will give us a germ of a $d$-manifold with a complex metric, and hence a map $\\mathcal U \\to {\\rm Met}_{\\mathbb C}(\\hat x)$. Pulling back the bundle of observables by this map gives us a bundle on $\\mathcal U$, which, using  the principle (5.1) as we did in (5.3), we see to be trivial. Identifying its fibres gives us our definition of $\\mathcal O_x$ for Lorentzian $M$.\n\n\\bigskip\n\nWe need no new ideas to see that for any Lorentzian cobordism $M:\\Sigma_0 \\leadsto \\Sigma_1$ and any $x \\in \\mathring M$ an element $\\psi \\in \\mathcal O_x$ acts as an operator $E_{\\Sigma_0} \\to E_{\\Sigma_1}$. Furthermore, if $x$ lies on a time-slice $\\Sigma$ we get an operator $\\psi \\in {\\rm Hom}( \\check E_{\\Sigma}; \\hat E_{\\Sigma})$, i.e. an unbounded operator in $E_{\\Sigma}$, simply by considering the cobordisms corresponding to a sequence of successively thinner collars of $\\Sigma$. Indeed the same argument shows that if $x_1, \\ldots , x_k$ are distinct points on $\\Sigma$,  we have a map\n$$\\mathcal O_{x_1} \\otimes \\ldots \\otimes \\mathcal O_{x_k} \\ \\ \\to \\ \\  {\\rm Hom}( \\check E_{\\Sigma}; \\hat E_{\\Sigma})$$\nwhich does not depend on choosing an ordering of the points. \n\n\\bigskip\n\nIn the introduction we  mentioned the Wightman axiom  that field operators at space-like separated points must commute. We can now see how this follows from our framework, at least in a globally hyperbolic space-time. For the spaces $\\check E_{\\Sigma_t} \\subset E_{\\Sigma_t}^{Hilb} \\subset E_{\\Sigma_t}$ for all  times $t_0 \\leq t \\leq t_1$ can be identified with those at time $t_0$ by the unitary propagation $Z_{t,t'}$ from time $t$ to a later time $t'$ to get a single rigged Hilbert space $\\check E \\subset E^{Hilb} \\subset \\hat E$, and we can define an unbounded operator\n$$\\tilde \\psi \\ \\ = \\ \\ Z_{t_0,t}^{-1}\\circ \\psi \\circ Z_{t_0,t}:\\check E \\to \\hat E$$\nfor any $\\psi \\in \\mathcal O_x$ with $x \\in \\Sigma_t$. Furthermore, if we change the choice of time-function on the cobordism, so that $x$ lies on a different time-slice, then $\\tilde \\psi$ will not change.\n\nThe fact that two observables $\\psi,\\psi'$ situated at space-like separated points $x,x'$ give rise to operators $\\tilde \\psi, \\tilde \\psi'$ which are composable, and commute, is now clear. For if $x$ and $x'$ are space-like separated we can choose a single time-slice $\\Sigma_t$ which contains them both, and we see that the composed operator, in either order, is  $Z_{t_0,t}^{-1}\\circ ( \\psi \\otimes \\psi') \\circ Z_{t_0,t}$.\n\n\\bigskip\n\n\n\n\\noindent{\\bf The domain of holomorphicity of the vacuum expectation values}\n\n\\bigskip\n\n\nWe end with a conjecture about a question arising in the traditional treatment of field theories defined in the standard Minkowski space $\\mathbb M = \\mathbb R^{d-1,1}$. There, the Wightman axioms imply that the vacuum expectation values, initially  defined as distributions or other generalized functions on the $k$-fold products $\\mathbb M \\times \\ldots \\times \\mathbb M$, are  boundary values of holomorphic functions defined in an open domain $\\mathcal U_k$ in the complexified space $\\mathbb M_{\\mathbb C}\\times \\ldots \\times \\mathbb M_{\\mathbb C}$. The definition of $\\mathcal U_k$,  known as the `permuted extended tube',  was given in Section 2. Recall that $\\mathcal U_2$ consits of all pairs of points $x,y$ such that $||x-y||^2$ is not real and $\\leq 0$.\n\nIf $k > 2$, however,  $\\mathcal U_k$ is known not to be holomorphically convex, so it cannot be the largest complex manifold to which the expectation values can be analytically continued. It is an old problem to describe this largest manifold $\\mathcal V_k$, or even the  holomorphic envelope of $ \\mathcal U_k$.\n\nThe ideas of this paper suggest  a candidate for $\\mathcal V_k$. It sits over the open subset $\\check \\mathcal V_k$ of all $k$-tuples ${\\bf x} = \\{x_1, \\ldots ,x_k\\}$ of distinct points in $\\mathbb M_{\\mathbb C}$ which lie on some totally-real submanifold $M$ with two properties:\n\n(i) \\ the metric on $M$ induced from $\\mathbb M_{\\mathbb C}$ is allowable, and\n\n(ii) \\ $M$ projects surjectively onto the usual real Euclidean subspace $\\mathbb E = \\mathbb R^d$ of $\\mathbb M_{\\mathbb C} = \\mathbb E \\oplus \\rm i\\mathbb E$.\n\nNotice that, by the remark before Prop.\\ 2.6, the projection $M \\to \\mathbb E$ is a local diffeomorphism if the metric of $M$ is allowable, so (ii) implies that $M$ is the graph of a smooth map $\\mathbb E \\to \\rm i\\mathbb E$.\n\nLet $\\mathcal F_k$ denote the space of all pairs $(M,{\\bf x})$ satisfying the above conditions. It is an infinite-dimensional complex manifold projecting to the open subset $\\check \\mathcal V_k$ of Conf$_k(\\mathbb M_{\\mathbb C})$, and it is easy to see that the map $\\pi: \\mathcal F_k \\to \\check \\mathcal V_k$ is open.  We define $\\mathcal V_k$  as the largest Hausdorff quotient manifold of $\\mathcal F_k$ through which $\\pi$ factorizes and which maps to $\\check V_k$ by a local diffeomorphism. Thus two points $(M,{\\bf x}),\\, ( M',\\bf x)$ of the fibre $\\mathcal F_{k,\\bf x}$  of $\\pi$ at $\\bf x$ have the same image in $\\mathcal V_k$ if they are in the same connected component of the fibre, but --- as that equivalence relation need not give a Hausdorff quotient --- also if there are paths $\\gamma,\\gamma'$ from $(M,\\bf x)$ and $(M',\\bf x)$ to a third point\n$(M'',\\bf x'')$ of $\\mathcal F_k$ which cover the same path from $\\bf x$ to $\\bf x''$ in $\\check V_k$.\n\n\\bigskip\n\n\n\n\n\n\n \n\nTo motivate the  definition of $\\mathcal V_k$ we must enlarge our framework to allow Lorentzian space-times whose time-slices are not compact. The simplest way to do this is to introduce the cobordism category in which a morphism is the part of a  $d$-dimensional allowable submanifold $M$ of $\\mathbb M_{\\mathbb C}$ cut off between two time-symmetric hypersurfaces.\n\nA field theory defined and holomorphic on this category, if it has a Lorentz-invariant vacuum state in a natural sense, will have vacuum expectation values which are holomorphic functions $\\mathcal E_k$ on the spaces $\\mathcal F_k$ of  pairs $(M,{\\bf x})$.  Strictly speaking, $\\mathcal E_k$ is a holomorphic section of a bundle on $\\mathcal F_k$, but we can use the local diffeomorphism $M \\to \\mathbb E$ to trivialize the bundle, giving us a holomorphic function\n$$\\mathcal E_k: \\mathcal F_k \\  \\to  \\ {\\rm Hom}(\\mathcal O^{\\otimes k};\\mathbb C),$$\nwhere $\\mathcal O$ is the space of observables at a point of $\\mathbb E$.\n\nOur much-used Principle 5.1 tells us that the value of the function $\\mathcal E_k$ does not change if, while holding the marked points {\\bf x} fixed in $\\mathbb M_{\\mathbb C}$, we move $M$ smoothly in the allowable class. So in fact we have a holomorphic function on  $\\mathcal F_k$ which is constant on the connected components  of the fibres of $\\mathcal F_{k,\\bf x}$ of the map $\\pi$, i.e. to the isotopy classes of allowable manifolds conatining $\\bf x$.\n\n\\bigskip  \n\nUnfortunately we have no proof that $\\mathcal V_k$ is a domain of holomorphy, but  at least we can assert\n\n\\bigskip\n\n\\noindent {\\bf Proposition 5.4} \\ {\\it $\\mathcal V_k$ contains the Wightman domain $\\mathcal U_k$.}\n\n\\bigskip\n\nFurthermore, we saw in Proposition 2.3 that the holomorphic envelope of $\\mathcal U_k$ contains $\\mathcal V_k^{flat}$, the part of $\\mathcal V_k$ represented by flat affine submanifolds $M \\subset \\mathbb M_{\\mathbb C}$.\n\n\\bigskip\n\n\\noindent{\\it Proof of 5.4} \\ First, $\\mathcal V_k$ is invariant under the complex orthogonal group of $\\mathbb M_{\\mathbb C}$, and under reorderings of the points ${\\bf x} = \\{x_1, \\ldots ,x_k\\}$. So it is enough to consider ${\\bf x}$ such that the imaginary part of $x_{i+1} - x_i$ belongs to the forward light-cone $C \\subset \\mathbb M$ for each $i$.\n\nSmoothing the obvious polygonal path joing the points, we can thus assume that the $x_i$ lie on a curve $x: \\mathbb R \\to \\mathbb M_{\\mathbb C}$ whose derivative  Im$(x'(t))$ belongs to $C$ for all $t$. But then we can choose, smoothly in  $t$, a set of $d-1$ orthonormal vectors $e_j(t)$ in $\\mathbb M_{\\mathbb C}$ which are all orthogonal to $x'(t)$. Let $M_t$ be the \\emph{real} vector subspace of $\\mathbb M_{\\mathbb C}$ spanned by the vectors $e_j(t)$. The points ${\\bf x}$ lie on the $d$-dimensional real ruled manifold $M$ swept out by the  affine $d-1$-planes  $x(t)+ M_t$, and the metric of $M$ is clearly allowable. \\ \\ $\\spadesuit$ \n\n\n\n\n\\bigskip\n\n\n\\noindent{\\bf References}\n\n\\bigskip\n\n\\begin{enumerate}\n\\item[[BF]] Brunetti, R.,   and K. 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Amer.\\ Math.\\ Soc. 1999.\n\\item[[Ke]] Kelnhofer, Gerald, \\, {\\it Functional integration and gauge ambiguities in generalized abelian gauge theories.} \\, J.\\ Geom.\\ Physics {\\bf 59} (2009) \\, 1017--1035. (arXiv:0711.4085 [hep-th])\n\\item[[N]] Neretin, Yu. A., \\, {\\it Holomorphic continuations of representations of the group of diffeomorphisms of the circle.} \\, Mat. Sb. {\\bf 180} \\, (1989), 635--57. \\ (English translation Math.\\ USSR-Sb. {\\bf 67} (1990).\n\\item[[PS]] Pressley, A., and G. Segal, \\, {\\it Loop Groups}. Oxford U.P. 1986.\n\\item[[Se1]] Segal, Graeme, \\, {\\it The definition of conformal field theory}. \\, In: {\\it Differential geometrical methods in theoretical physics. (Como 1987)}\\, NATO Adv.\\ Sci.\\ Inst. Ser C, Math Phys. Sci. {\\bf 250}, 165 -- 171, Kluwer 1988.\n\\item[[Se2]] Segal, Graeme, \\, {\\it The definition of conformal field theory.} \\, In: {\\it Topology, Geometry, and Conformal Field Theory}, \\, ed. U. Tillmann, \\, London Math.\\ Soc. Lecture Notes {\\bf 308} (2004), 421 -- 577.\n\\item[[Sz]] Szabo, R., \\, {\\it Quantization of higher abelian gauge theory in generalized differential cohomology.} \\, In: {\\it Proc. 7$^{\\rm th}$ Internat.\\ Conf.\\ on Math.\\ Methods in Physics} \\, (ICMP2012)  (arXiv:1209.2530 [hep-th]).\n\\item[[SW]] Streater, R. F.,  and A. S. Wightman, \\, {\\it PCT, Spin and Statistics, and all that.} \\, Princeton U.P. 2000. \\ ($1^{\\rm st}$ edn Benjamin 1964)\n\\item[[WW]] Weinberg, S., and E. Witten, \\, {\\it  Limits on massless particles.} \\, Phys. Lett. B{\\bf 96} (1980), 59 -- 62.\n\n\n\n\n\\end{enumerate}\n\n\\\n\n\\textit{E-mail addresses}: \\texttt{maxim@ihes.fr},  \\texttt{ graeme.segal@all-souls.ox.ac.uk}\n\n   \n\n\n\\end{document}\n\n \n  \n  \n\n\n\n\n\n  \n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection*{Results}\n\nThe combined UED\/UEDS experiments are based on a pump-probe scheme, and were performed with a radio-frequency compressed ultrafast electron scattering instrument described in detail elsewhere~\\cite{Chatelain2012, Otto2017}. Briefly, \\SI{35}{\\femto\\second} pump pulses at \\SI{800}{\\nano\\meter} (\\SI{1.55}{\\electronvolt}) drive vertical electronic transitions at $t=t_0$, photodoping electrons and holes into the band valleys near $\\Gamma$, $\\frac{2}{3}Y$ and $\\frac{3}{4}Z$~\\cite{Li2015,Zhao2016,Melendez2018,Wei2019} (Fig. \\ref{FIG:structure}c). The effects of this photoexcitation on both the lattice structure and phonon system are measured with UED and UEDS simultaneously at a time $t = t_0 + \\tau$. Time-series of the changes in electron scattering intensity at all scattering vectors, $\\vec{q}$, were assembled by scanning the pump-probe time delay, $\\tau$. The experiments were repeated over a range of pump-fluences at \\SI{300}{\\kelvin}.\n\nThe total measured electron scattering intensity $I(\\vec{q}, \\tau)$ can be decomposed into $I(\\vec{q}, \\tau) = I_0(\\vec{q}, \\tau) + I_1(\\vec{q}, \\tau) + ...$, where $I_0$ corresponds to the Bragg peaks of conventional diffraction and $I_1$ is known as \\emph{single-phonon diffuse scattering}. The intensity of Bragg peaks at scattering vector $\\vec{G}$, $I_0(\\vec{q}=\\vec{G},\\tau)$, reports on the lattice constants, unit cell structure, coherent modulation~\\cite{Chatelain2014, Sie2019} and\/or transient Debye-Waller (DW) suppression~\\cite{Siwick2003, Ernstorfer2009, Waldecker2016} of peak intensity. Diffraction signals can be $10^5$--$10^8$ times more intense than phonon diffuse scattering signals~\\cite{Stern2018,RenedeCotret2019}. The expression for single-phonon diffuse scattering (PDS) is given by:\n\\begin{equation}\n\tI_1(\\vec{q}, \\tau) \\propto \\sum_\\lambda \\left| a_{\\lambda \\vec{k}}\\right|^2 \\left| F_{1\\lambda}\\left(\\vec{q}, \\left\\{ e_{\\lambda\\vec{k}}\\right\\} \\right) \\right|^2 \\label{EQ:diffuse}\n\\end{equation}\nwhere the label $\\lambda$ indicates the specific phonon branch, $\\vec{q}$ is the electron scattering vector, $\\vec{k}$ is the reduced phonon wavevector (i.e. $\\vec{k}$ = $\\vec{q}$ - $\\vec{G}$, where $\\vec{G}$ is the closest Bragg peak), $a_{\\lambda\\vec{k}}$ is the vibrational amplitude of mode $\\lambda$, and $F_{1\\lambda}$ are known as the one-phonon structure factors. $I_1$ provides momentum-resolved information on the nonequilibrium distribution of phonons across the entire Brillouin zone, since $I_1(\\vec{q}, \\tau)$ depends only on phonon modes with wavevector $\\vec{k}$ = $\\vec{q}$ - $\\vec{G}$ (Fig. \\ref{FIG:data}a). The one-phonon structure factors $F_{1\\lambda}$ are geometrical weights that depend sensitively on the phonon mode atomic polarization vectors $\\left\\{ e_{\\lambda\\vec{k}}\\right\\}$~\\cite{RenedeCotret2019}. Most importantly, $F_{1\\lambda}\\left(\\vec{q}, \\left\\{ e_{\\lambda\\vec{k}}\\right\\} \\right)$ are relatively large when the phonon mode $\\lambda$ is polarized parallel to the scattering vector $\\vec{q}$. Terms of higher-order than $I_1$ have lower cross-sections and do not contribute significantly to the interpretation of the low-order Brillouin zone (BZ) signals reported on here~\\cite{Wang2013,Zacharias2021a,Zacharias2021b}. \n\n\\begin{figure*}\n    \\centering\n\t\\includegraphics{diffuse.pdf}\n\t\\caption{Ultrafast electron diffraction and diffuse scattering signals from photodoped SnSe. \\textbf{a} Equilibrium scattering pattern of SnSe oriented along the $[100]$ direction with key vectors, the square BZ ($\\vec{b}^\\star$--$\\vec{c}^\\star$ plane) and high symmetry points indicated. \\textbf{b} Regions of interest for scattering as described in the text, shown around reflection $(00\\bar{2})$ as an example; (1) Bragg intensity, (2) small-wavevector phonons ($\\SI{0.114}{\\per\\angstrom} <|\\vec{k}| < \\SI{0.228}{\\per\\angstrom}$) and (3) larger wavevector phonons ($|\\vec{k}| > \\SI{0.228}{\\per\\angstrom}$)  \\textbf{c} Line cut across the horizontal line shown in panel b). The Bragg peak lineshape is fit with a Voigt profile (solid black line) with a full-width at half-max of \\SI{0.158}{\\per\\angstrom}. \\textbf{d} Transient (photoinduced) ultrafast electron scattering intensity changes in several regions of the BZ shown in a) and b). The decrease of intensity directly on the Bragg peaks shows transient DW decays that are strongly anisotropic. The fast ($\\sim \\SI{300}{\\femto\\second}$) decay component is maximized in Bragg peaks along $\\vec{c}^\\star$ (black), and only the slow components ($\\sim \\SI{4}{\\pico\\second}$) is observed in peaks perpendicular to $\\vec{c}^\\star$ (red). Transient diffuse intensity also shows pronounced anisotropy and $\\vec{k}$-dependence. A fast rise in diffuse intensity (purple) is only observed for $|\\vec{k}| < \\SI{0.228}{\\per\\angstrom}$ (region 2, panel b) in BZs that show the fast DW dynamics. A slow increase in diffuse intensity (orange) is observed at all zone boundary high-symmetry points (see Fig. S4) and elsewhere in the BZ for all reflections $\\vec{G}$ (e.g. region (3), panel b). Error bars represent the standard error in the mean of intensity before time-zero, but are generally smaller than the markers.}\n\t\\label{FIG:data}\n\\end{figure*}\n\nFig. \\ref{FIG:data}a shows an equilibrium diffraction pattern of SnSe at room temperature along the $[100]$ zone axis with the rectangular in-plane ($\\vec{b}^\\star$--$\\vec{c}^\\star$) BZ of the $Pnma$ phase indicated. This pattern contains both Bragg scattering (at zone-center positions) and PDS contributions at all scattering vectors.  Following photoexcitation, Bragg peaks show transient decreases in intensity whose dynamics are well-described by a biexponential decay with time-constants (\\SI{400 \\pm 130}{\\femto\\second} and \\SI{4 \\pm 1}{\\pico\\second}), as shown in Fig. \\ref{FIG:data}d.  The fast component of these dynamics are profoundly anisotropic, with a maximum contribution in the $\\vec{c}^\\star$ direction and below detection along $\\vec{b}^\\star$ (Fig. \\ref{FIG:data}d).  Given the DW factor ($\\exp(-\\frac{1}{2}\\langle \\vec{q} \\cdot \\vec{u} \\rangle^2$), this indicates that there are at least two distinct processes that contribute to increasing the atomic displacement $\\vec{u}$ following photoexcitation.  The ultrafast dynamics of the PDS intensity following photoexcitation provides a clear perspective on these distinct processes. At all high-symmetry BZ boundary positions ($Z$, $Y$ and $T$) we find that PDS intensity increases with a single exponential time constant of \\SI{4 \\pm 1}{\\pico\\second} (Fig. S4) that is the complement of the slow time-constant observed in the Bragg peak dynamics. In fact, within experimental uncertainties an identical time-constant is determined for increases in PDS observed at all BZ positions $|\\vec{k}| > \\SI{0.228}{\\per\\angstrom}$ (i.e. far from Bragg peaks), as is shown in Fig. \\ref{FIG:data}d (yellow) and Fig. \\ref{FIG:polaron}e. \n\nPrevious work has identified a number of soft and strongly-coupled optical phonons in the zone-center region of the $Pnma$ phase~\\cite{Chattopadhyay1986,Li2015,Gong2020,Lanigan2020}.  As $\\vec{k}$ approaches zone-center, the PDS contribution overlaps with the Bragg peak lineshape.  Thus, Bragg peak intensity must be subtracted to accurately determine the differential PDS from small-wavevector phonons following photoexcitation. As part of this analysis we investigated whether photoexcitation resulted in measurable time-dependence of Bragg peak positions and widths.  Fig. S3 demonstrates that the Bragg peak center positions and widths are shot noise limited and neither parameter shows a measurable time-dependence;  i.e. any photoinduced change to in-plane lattice constants (which shift Bragg peak positions) or in-plane long-range strain (which broaden and skew the width of peaks) over the range of delay times investigated here are at a level that is below our signal-to-noise ratio. The Bragg peak lineshapes are effectively constant, but have integrated intensities that vary according to the observed transient Debye-Waller dynamics. Panels b and c of Fig. \\ref{FIG:data} show two regions of interest around every Bragg peak. We define an area at the BZ center associated with strongest Bragg scattering (region (1), $|\\vec{k}| \\leq \\SI{0.114}{\\per\\angstrom}$) and a surrounding region associated with wavevectors in the range $\\SI{0.114}{\\per\\angstrom} < |\\vec{k}| \\leq \\SI{0.228}{\\per\\angstrom}$ (region (2)). By integrating scattered intensity in these areas separately, we assemble time-series that probe both the Bragg (region 1) and the small wavevector ($|\\vec{k}| \\sim \\Gamma$) PDS (region 2) after subtraction of Bragg intensity.  The results for reflections parallel to $\\vec{c}^\\star$ is shown in Fig. \\ref{FIG:data}d (purple), demonstrating that the fast component of the DW dynamics is exclusively associated with a rapid increase in the amplitude of phonons near zone-center. This signal shows the same $\\vec{b}^\\star$\/$\\vec{c}^\\star$ anisotropy as the Bragg peaks (Fig. S6) and is \\emph{insensitive} to the precise definition of the indicated regions, as demonstrated below.  Determination of the relatively small differential PDS signals directly under the Bragg peak (region 1) is obviously subject to large uncertainties and is not reported here. These observations corroborate computational work~\\cite{Caruso2019,Ma2018} which found that the electron-phonon coupling is profoundly anisotropic in SnSe, with carriers coupling very strongly to polar zone-center modes and only much more weakly elsewhere. \n\nThe PDS in intermediate regions of the BZ is a mixture of the fast and slow dynamics shown in Fig. \\ref{FIG:data}d, with the magnitude of these components varying as a function of $|\\vec{k}|$ as shown in Fig. \\ref{FIG:polaron}. In a conventional weakly polar semiconductor like GaAs, these dynamics can be described in terms of a model for the re-equilibration of the photodoped carriers with the phonon-system via intra- and inter-valley inelastic electron-phonon scattering processes based on the electron and phonon bandstructures of the material~\\cite{Sjakste2018}.  The dynamics observed here for SnSe cannot be described in these terms as is explained further in the discussion below.  Polaron formation, however, provides a robust description of these observations.      \n\n\\begin{figure*}\n    \\centering\n    \\includegraphics{polaron.pdf}\n    \\caption{Polaron formation in SnSe visualized with UEDS. \\textbf{a} Wavevector-dependent scattering intensity for a Gaussian displacement field model of the polaron lattice distortion as a function of size (polaron FWHM), as described in the text. \\textbf{b} Measured change in diffuse intensity at \\SI{1}{\\pico\\second} (black triangles) and \\SI{5}{\\pico\\second} (orange circles) fit to the Gaussian displacement model above (solid curves). Best-fit FWHM polaron dimensions are $f=\\SI{18.7 \\pm 0.3}{\\angstrom}$ (\\SI{1}{\\pico\\second}) and $f=\\SI{4.2 \\pm 0.1}{\\angstrom}$ (\\SI{5}{\\pico\\second}).  The larger polaron is consistent with uniaxial displacements along $\\vec{c}^\\star$ and the smaller polaron with uniform displacements in the $\\vec{b}$--$\\vec{c}$ plane. The boundary between regions (2) and (3) from Fig. \\ref{FIG:data}b are indicated. The shaded region represents the standard error in the mean of intensity across the integration region. \\textbf{c} Differential scattering intensity across the BZ due to the large polaron lattice distortion. \\textbf{d} Differential scattering intensity across the BZ due to the small polaron lattice distortion. \\textbf{e} The measured differential diffuse intensity across the BZ at $\\tau=\\SI{5}{\\pico\\second}$ is in excellent agreement with the predicted scattering vector dependence of the model.}\n    \\label{FIG:polaron}\n\\end{figure*}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{polaron-realspace.pdf}\n    \\caption{Real-space visualization of the in-plane ($\\vec{b}$ -- $\\vec{c}$) atomic displacement due to the large and small polarons in Fig. \\ref{FIG:polaron}. The unperturbed in-plane dimensions of the unit cell are marked by solid lines. \\textbf{a} Large (\\SI{18.7}{\\angstrom} FWHM) one-dimensional polaron aligned to the $c$-axis \\textbf{b} Small (\\SI{4.2}{\\angstrom} FWHM) three-dimensional polaron. In both subpanels, the blue-shaded background represents the FWHM of the atomic displacement field $\\vec{u}(\\vec{r})$. The magnitude of the atomic displacements has been exaggerated for visual clarity.}\n    \\label{FIG:polaron-realspace}\n\\end{figure}\n\nThe local lattice distortions associated with a polaron in real-space can be modelled as an atomic displacement field $\\vec{u}(\\vec{r})$, where atoms are displaced as $\\vec{r}_j \\to \\vec{r}_j + \\vec{u}(\\vec{r}_j)$ \\cite{Guzelturk2021}. For small displacement fields, the contribution of such a localized lattice distortion to the total scattering amplitude, $f^p$, is given by:\n\\begin{equation}\n    f^p(\\vec{q}) \\approx - i \\sum_j f_{e,j}(\\vec{q}) e^{-i \\vec{q} \\cdot \\vec{r}_j} \\left( \\vec{G} \\cdot \\vec{u}_j \\right)\n\\end{equation}\nwhere $\\left\\{ \\vec{r}_j \\right\\}$ are the atomic positions, $\\left\\{ f_{e,j} \\right\\}$ are the atomic form factors for electron scattering, and $\\vec{G}$ is the Bragg reflection nearest to $\\vec{q}$ as in Fig. \\ref{FIG:data} (see SI). We consider two specific displacement fields due to the anisotropic observations described above; an effectively one-dimensional displacement field directed along the $c$-axis ($\\vec{u}_c(\\vec{r}) \\propto e^{-|\\vec{r}|^2\/r_p^2} ~ \\hat{\\vec{r}} \\cdot \\hat{\\vec{c}}$), and a three-dimensional displacement field ($\\vec{u}(\\vec{r}) \\propto e^{-|\\vec{r}|^2\/r_p^2} \\hat{\\vec{r}}$) (see SI). In both cases, $2 \\sqrt{2 \\ln(2)} ~ r_p$ is the full-width at half-maximum (FWHM) of the local lattice distortion associated with the polaron. The effect of both displacement fields is identical in terms of the impact on the diffuse scattering measured within a BZ, shown as a function of polaron FWHM in Fig. \\ref{FIG:polaron} a).  The one-dimensional displacement field, however, has the same $\\vec{b}^\\star$\/$\\vec{c}^\\star$ anisotropy in scattering intensity as our measurements.\n\nThis model was fit to the measured photoinduced differential PDS signals at \\SI{1}{\\pico\\second} and at \\SI{5}{\\pico\\second}, to capture changes due to the fast and slow dynamics respectively. The fast PDS dynamics are in excellent agreement with the formation of a \\SI{18.7 \\pm 0.3}{\\angstrom} polaron displacement field. The slow PDS dynamics, by contrast, are in excellent agreement with the displacement field associated with a \\SI{4.2 \\pm 0.1}{\\angstrom} FWHM polaron. A qualitative real-space representation of the two polaron modes is presented in Fig. \\ref{FIG:polaron-realspace}. We tentatively assign the larger polaron to the electron and the smaller polaron to the hole, in analogy to the work on Sio \\emph{et al.}~\\cite{Sio2019} on other polar materials. However, these observations are also consistent with electron and hole polarons being a similar size in SnSe (i.e. \\SI{4.2 \\pm 0.1}{\\angstrom} FWHM) and formation occurring in two steps.  We discuss these results within the context of both interpretations below.  \n\n\\subsection*{Discussion}\n\n The carriers generated through photoexcitation in these experiments localize via their interactions with the phonon system.  These interactions create a local potential that minimizes the free energy of the system according to the standard picture of polaron formation ~\\cite{Franchini2021} shown schematically in Fig. 1 d) - f). A polaron quasiparticle is thus formed as a localized carrier dressed by phonons, potentially in several phonon branches and over a range of wavevectors~\\cite{Sio2019}; a local lattice distortion in real-space is equivalent to a distribution of lattice normal modes in reciprocal space. These details are revealed by the UED and UEDS data. As recent ab initio work by Giustino and colleagues has demonstrated in other polar lattices~\\cite{Sio2019}, polarons require the recruitment of phonon modes across the entire BZ when the dimensions of the polaron approaches those of the lattice constants.  In SnSe, we propose that this manifests in the bimodal formation dynamics reported here due to the profoundly anisotropic (momentum-dependent) nature of electron-phonon coupling in the material.  Strong Fr\\\"ohlich coupling to near zone-center polar optical phonons ($A_g$, $B_u$, and $B_g$) results in the rapid ($\\sim\\SI{300}{\\femto\\second}$) formation of relatively large (electron) polarons, since polarons of this size only require the recruitment of strongly coupled small-wavevector phonon modes.  The formation time of the smaller (hole) polarons is an order of magnitude longer ($\\sim \\SI{4}{\\pico\\second}$) due to the relatively weak coupling to the large-wavevector phonons that must be recruited to form polarons of this size~\\cite{Sio2019}. \n\nAlternatively, the observed bimodal polaron formation dynamics are also consistent with a two-step process that has features in common with Onsager's inverse snowball effect, often discussed in the context of the theory of solvation~\\cite{neria1992simulations}.  Here the rapid but weak localization of the charge carriers is provided by the 1D ferroelectric-like lattice distortions along the $c$-axis.  The slower but strong localization is provided by the subsequent 3D distortions.  Polaron formation dynamically proceeds from the outside (long range) inwards (short range), not via layer accumulation from the inside out like a typical snowball.    \n\nThese measurements alone cannot identify precisely how vibrational excitation is distributed over the phonon branches near zone-center by $\\sim \\SI{1}{\\pico\\second}$; however, the increase in mean-square atomic displacements can be determined from the Bragg peak DW decays and can be used to estimate the fraction of excitation energy that has left the carrier system in the form of the polaronic lattice distortion by $\\sim \\SI{1}{\\pico\\second}$. This quantity is shown in Fig. \\ref{FIG:msd}, and is linear with pump fluence over the range investigated.  The observed increase, $\\Delta \\langle u_c^2\\rangle$, is consistent with a nearly complete transfer ($>85\\%$) of the excess photodoped carrier energy to zone center phonons (see SI and Fig. S7). Thus, the coupled electron-phonon system at $\\sim$\\SI{1}{\\pico\\second} is well described as a state in which the photodoped carriers in each valley have reached an equilibrium with only this limited set of strongly coupled small-wavevector phonons.  This equilibrium is well described by the formation of polaron quasiparticles (Fig. \\ref{FIG:polaron} b) rather than the simple heating of the phonon system as has been observed in other systems, like graphite where rapid electron cooling through a pre-equilibrium with specific strongly coupled modes has been observed ~\\cite{Stern2018,RenedeCotret2019}.  A similarly rapid coupling of electronic excitation energy to polaronic lattice distortions is thought to be present in methylammonium lead iodide perovskites~\\cite{Niesner2016}.\n\nPolaron formation is the simplest \\emph{self-consistent} explanation of these data. The isotropic slow-rise in diffuse scattering observed across the entire BZ specifically precludes an understanding of these results in terms of the conventional semiconductor picture of carrier relaxation through intervalley scattering mediated by large-wavevector phonons~\\cite{Sjakste2018,Waldecker2017,Stern2018,RenedeCotret2019,Otto2021}.  Based on the electronic dispersion of SnSe calculated by Wei \\emph{et al.}~\\cite{Wei2019}, we modelled the decay of hot electrons and holes, mediated by phonons, via energy-allowed and momentum-conserving pathways. This relaxation mechanism imprints the structure of the electronic dispersion onto the PDS, including a pronounced anisotropy between the $\\vec{b}^\\star$ and $\\vec{c}^\\star$ directions as shown in Fig. S8. This is ruled out by our measurements, which are azimuthally symmetric in reciprocal space (Fig. \\ref{FIG:polaron}e). Neither does the anharmonic decay of phonons provide an explanation for these data, given that phonon lifetimes are estimated to be almost an order of magnitude longer (\\SIrange{15}{30}{\\pico\\second})~\\cite{Chandrasekhar1977,Li2015,Lanigan2020} than the timescales observed herein. Moreover, the anharmonic decay of phonons measured in the time-domain displays an imprint of the phonon dispersion due to energy- and momentum-conservation rules~\\cite{Stern2018,RenedeCotret2019}, which is not seen in our data (Fig. \\ref{FIG:polaron}e).\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{displacement.pdf}\n    \\caption{Increase in mean-square-displacement of all atoms along the $c$ axis, $\\Delta \\langle u_c^2 \\rangle$, due to the change in vibrational amplitude of the strongly-coupled zone-center modes exclusively. The associated photocarrier concentration $N_{\\gamma}$ for a sample of dimensions $\\SI{50 x 50 x 0.045}{\\micro\\meter}$ is shown above the plot. Boxes are used to represent error bars along both axes.}\n    \\label{FIG:msd}\n\\end{figure}\n\nThere are a number of important connections between these observations and an understanding of the thermoelectric properties of SnSe.  First, the rate of large polaron formation is very rapid and appears to be at or near the limit imposed by the period of the highest frequency phonons in SnSe ($\\sim \\SI{200}{\\femto\\second}$) due to the strong Fr\\\"ohlich coupling in the polar lattice, even at the high carrier density operating conditions of optimally doped SnSe for thermoelectric device applications~\\cite{Sootsman2009,Zhao2016,Fan2018}.  At this doping level, these results indicate that SnSe is well described as a dense polaron system, since the density of polarons overlapping at a distance equal to their FWHM is  \\SI{3.2e21}{\\per\\cubic\\centi\\meter} for the smaller (hole) polarons, overlapping with this optimal doping range.  \nThe polaronic nature of the charge carriers in SnSe likely plays an important role in preserving the electron-crystal, phonon-glass conditions that are important for high-performance thermoelectric materials~\\cite{Rowe1995}. The dressed charges are better screened from scattering mechanisms that could otherwise deteriorate mobility at high carrier density and temperatures approaching a structural phase transition, where $ZT$ in SnSe is highest.  Several open questions remain regarding the nature of polarons in SnSe. Our measurements directly probe the in-plane lattice distortions, but do not provide information on the inter-layer (or $a$-axis) dependence. There is also the important question regarding the potential ferroelectric nature of polarons in SnSe, as has been discussed in the context of lead halide perovskites~\\cite{Frost2017,Joshi2019}.  Monolayer SnSe has been investigated as a possible platform for ultrathin ferroelectrics~\\cite{Fei2016}, and polar nanodomain formation and interlayer coupling may contribute to the dynamics we have observed. The in-plane polarization within a single layer can be modulated simply through changes in the angle of the Sn-Se bonds relative to the $a$-axis ~\\cite{Fei2016} in a manner that is consistent with the local polaronic displacements shown schematically in Fig. \\ref{FIG:polaron-realspace}.\n\nElectron-phonon coupling is not normally considered to be an important contributor to lattice thermal conductivity, however, previous work has shown that these interactions can play a role in suppressing thermal conductivity and enhancing thermoelectric performance in Si~\\cite{Liao2015} and SiGe~\\cite{Fan2018} at high carrier doping.  While a giant lattice anharmonicity and 3-phonon scattering processes seems to be sufficient to explain the ultralow thermal conductivity of undoped SnSe~\\cite{Zhao2014, Li2015}, a quantitative understanding of electron-phonon interactions and their impact on both electronic and lattice thermal conductivity (in addition to electrical conductivity) in this strongly coupled, high-carrier density regime is important for the further development of thermoelectrics and higher $ZT$.  The rate of electron-phonon scattering with the strongly coupled polar modes is more than an order of magnitude higher than 3-phonon scattering processes at the carrier densities investigated here~\\cite{Lanigan2020}.\n\nWe are currently in an excellent position to make significant progress understanding this complex, strongly-coupled regime across many material classes. The recent parallel development of ab initio approaches for both electron-phonon coupling~\\cite{Ponce2016} and polaron formation~\\cite{Sio2019} and time- and momentum-resolved measurement techniques like ultrafast electron\/xray diffuse scattering that are capable of interrogating electron-phonon interactions in exquisite detail and directly visualizing polaron formation.  Future work that combines these approaches together with more well established ultrafast spectroscopic methods are likely to yield significant insights.\n\n\\subsection*{Synthesis and sample preparation} \n\nA SnSe ingot (\\SI{20}{\\gram}) was synthesized by mixing appropriate ratios of high purity starting materials (Sn chunk, 99.999\\%, American Elements, USA and Se shot, 99.999\\%, 5N Plus, Canada) in \\SI{13}{\\milli\\meter} diameter quartz tube. The tube was flame-sealed at a residual pressure of $\\SI{1e-4}{\\mmHg}$, then slowly heated to \\SI{1223}{\\kelvin} over \\SI{10}{\\hour}, soaked at this temperature for \\SI{6}{\\hour} and subsequently furnace cooled to room temperature. The obtained ingot was crushed into powder and flame-sealed in a quartz tube, which was placed into another, bigger, flame-sealed quartz tube. A crystal with dimensions of $\\sim$\\SI{13}{\\milli\\meter} (diameter) $\\times$ \\SI{20}{\\milli\\meter} (length) was obtained.\n\nSeven samples were used for the ultrafast electron scattering measurements, to ensure reproducibility. Six samples were ultramicrotomed with a \\ang{35} diamond blade, while one sample was mechanically exfoliated. Three of the ultramicrotomed samples were cut from a first SnSe mother flake at a thickness of \\SI{90}{\\nano\\meter}. The remaining three ultramicrotomed samples were cut to a thickness of \\SI{70}{\\nano\\meter} from a different mother flake, synthesized separately from the first. The exfoliated sample was prepared from the second mother crystal, with a final thickness of \\SI{45}{\\nano\\meter} as estimated based on the relative ratio of intensities, compared to thicker ultramicrotomed samples.\n\n\\subsection*{Ultrafast electron scattering experiments} \nElectron scattering experiments employed \\SI{35}{\\femto\\second} pulses of \\SI{800}{\\nano\\meter} light at a repetition rate of \\SI{1}{\\kilo\\hertz}. Part of these light pulses is upconverted to \\SI{266}{\\nano\\meter} pulses via third-harmonic generation, which is then used to generate bunches of $10^6$ -- $10^7$ electrons from a bulk copper photocathode. These electrons are then accelerated to \\SI{90}{\\kilo\\electronvolt} in a DC electric field. Electron bunches are compressed with a radio-frequency cavity to counterbalance space-charge repulsion, resulting in a time-resolution of $\\SI{130}{\\femto\\second}$. Electrons are transmitted through the samples before being collected by an electron camera. The other part of the \\SI{800}{\\nano\\meter} pulses is used to photoexcite the sample almost co-linearly with the electron propagation axis ($\\sim \\ang{5}$). Experiments were repeated for up to \\SI{72}{\\hour} to maximize signal-to-noise, which was made possible by improvements in RF-laser synchronization~\\cite{Otto2017}. Detailed descriptions of this instrument are presented elsewhere~\\cite{Chatelain2012, Otto2017}. The samples were thinner than than the optical depth of $>\\SI{100}{\\nano\\meter}$ at \\SI{800}{\\nano\\meter}~\\cite{Barrios2014, Makinistian2009}, ensuring that the entire sample was photoexcited. The samples were oriented in the $\\langle 100 \\rangle$ direction, giving UEDS measurements a full view of lattice dynamics in the plane spanned by $\\vec{b}$ and $\\vec{c}$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIn recent years, the studies on black holes in higher dimensions have attracted much attention. \nSome of these studies show that they have much more complicated and richer structure than 4-dimensional ones.\nEspecially, the study of Kaluza-Klein (KK) black holes is important \nin association with our apparent 4-dimensional spacetime. \nThe Gregory-Laflamme instability~\\cite{Gregory:1993vy} \n(see also \\cite{Harmark:2007md} and references therein) and black hole\/black string phase transition \n(see {\\it e.g.} the reviews \\cite{Kol:2004ww}) \nmotivate many studies on thermodynamic aspects of KK black holes. \nNow, the thermodynamic properties and the first law for asymptotically flat KK black holes \nare widely investigated \\cite{Kol:2003if}-\\cite{Kastor:2007wr}. \n\n\nIn 5-dimensional Einstein-Maxwell theory, there is an analytic solution representing an \nelectrically charged black hole with squashed horizons \n\\cite{Ishihara:2005dp} as a generalization of the solution given in \\cite{Dobiasch:1981vh} and \\cite{Gibbons:1985ac}. \nIntriguingly, the spacetime far from the black hole is \nlocally a product of the 4-dimensional Minkowski spacetime and $S^1$. \nIn this sense, the black hole resides in KK spacetime and \nis worth to be named a KK black hole. \n\n\nThe black hole has an interesting property that various definitions of mass take different values, \nwhich means that the black hole gives an opportunity to investigate \nthe differences in various definitions of mass. \nIn this paper, we show some expressions for the first law of black hole thermodynamics \nsatisfied by those masses and discuss the differences from the viewpoint of thermodynamics. \n\n\n\\section{Kaluza-Klein black hole}\n\n\nLet us review the KK black holes  with squashed horizons~\\cite{Ishihara:2005dp}, \nwhich is a solution of the 5-dimensional Einstein-Maxwell theory. \nThe action is given by \n\n \\begin{eqnarray}\n I = \\frac{1}{16\\pi G} \\int_M d^5 x \\sqrt{-g} \n         \\left[ R - F^{\\mu\\nu} F_{\\mu\\nu} \\right]+\\frac{1}{8\\pi G} \\int_{\\partial M} \n K\\sqrt{-h}d^4x, \n \\end{eqnarray}\n\n where $G$ is the 5-dimensional Newton constant, $g_{\\mu\\nu}$ is the metric, $R$ is scalar curvature, \n $F_{\\mu\\nu} = \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu $ is \n the field strength of the 5-dimensional $U(1)$ gauge field $A_\\mu$. \nThe second term is so-called Gibbons-Hawking term, in which $h$ is determinant of the induced metric and \n $K$ is trace of the extrinsic curvature of the boundary $\\partial M$, respectively. \nThe metric of the black hole is given by \n\\begin{eqnarray}\n ds^2 &=& - V(\\rho) d\\tau^2 + \\frac{B (\\rho)}{V(\\rho)} d\\rho^2 \n + \\rho^2 B (\\rho) d\\Omega^2  \n + \\frac{r_{\\infty}^2}{4 B (\\rho)} \\left(d\\psi + \\cos \\theta d\\phi \\right)^2 ,\n\\label{eq:metric-Trho}\n\\end{eqnarray}\nwhere $d\\Omega^2 = d\\theta^2 + \\sin^2 \\theta d\\phi^2$ is the metric of the unit two-dimensional sphere and \n\\begin{eqnarray}\nV(\\rho)=  \\left( 1- \\frac{\\rho_{+}}{\\rho}\\right)\\left( 1-\\frac{\\rho_{-}}{\\rho} \\right) , \\ \\ \nB(\\rho) =   1+ \\frac{\\rho_0}{\\rho}, \\ \\ \nr_{\\infty} = 2\\sqrt{(\\rho_+ +\\rho_0 )(\\rho_- + \\rho_0 )}. \n\\end{eqnarray}\nHere, the coordinate ranges are \n$0 \\leq \\theta <2\\pi ,\\   0 \\leq \\phi < 2\\pi ,\\  0 \\leq \\psi < 4\\pi $. \nThe gauge potential is given by\n\\begin{eqnarray}\nA = \\mp   \\frac{\\sqrt{3}}{2}  \\frac{\\sqrt{{\\rho_+}{\\rho_-}}}{ \\rho_0}\n  \\left(1+\\frac{\\rho_0}{\\rho} \\right) d\\tau.\n \\end{eqnarray}\n\n\nIt is easy to see the apparent singularity at $\\rho_{+}$ corresponds to the outer horizon of the black hole.\nThe inner horizon $\\rho_{-}$ is analogous to that of the Reissner-Nordstr\\\"om (RN) black holes.\nThe spatial infinity corresponds to a limit $\\rho \\to \\infty$. \nIt should be noted that the shape of the horizon is a squashed sphere \nas was discussed in~\\cite{Ishihara:2005dp}.\nFrom the metric (\\ref{eq:metric-Trho}), \nit is seen that the $S^1$ circle parametrized by a coordinate $\\psi$ has finite size even at the spatial infinity. \nThe non-trivial twisting of the $S^1$ circle fibrated over the $S^2$ base space \nleads a 4-dimensional U(1) gauge field by KK reduction.\nActually, in no horizon limit $\\rho_+, \\rho_- \\to 0$, the black hole spacetime becomes the KK monopole \nspacetime \\cite{Gross-Perry}\\cite{Sorkin}. \nIn the limit $r_\\infty \\to \\infty$, the KK monopole becomes 5-dimensional Minkowski spacetime and \nthe black hole reduces to the 5-dimensional RN black hole.\nWe term this limit the spherically symmetric limit. \n\n\nGiven the metric (\\ref{eq:metric-Trho}), we can calculate various physical quantities.\nThe surface gravity is calculated as\n\\begin{eqnarray}\n   \\kappa_{+} = \\frac{\\rho_{+} -\\rho_{-}}{2\\rho_{+} \\sqrt{{\\rho_{+}}({\\rho_{+} + \\rho_0})}},\n   \\label{eq:surfacegravity}\n\\end{eqnarray}\nwhich gives the Hawking temperature of the black hole $T= \\kappa_+\/2\\pi$ \\cite{Hawking:1974sw}.\nWe assume that the entropy of the black hole is given by the Bekestein-Hawking formula \n\\begin{eqnarray}\nS = \\frac{A_+}{4G} = \\frac{4\\pi^2}{  G}  \\rho_+  (\\rho_+ + \\rho_0) \\sqrt{\\rho_+(\\rho_- + \\rho_0 ) },\n \\label{eq:entropy}\n\\end{eqnarray}\nwhich is consistent with the Wald's entropy formula \\cite{Wald:1993nt} \\cite{Iyer:1994ys}. \nThe electric charge and electrostatic potential of the black hole are also calculated as \\cite{Ishihara:2005dp}\n\\begin{eqnarray}\n  Q = \n    \\pm \\frac{\\sqrt{3} \\pi}{G} r_\\infty \\sqrt{\\rho_{+} \\rho_{-} }, \\quad \n\\Phi \n= \\pm \\frac{\\sqrt{3}}{2} \\sqrt{\\frac{\\rho_-}{\\rho_+} }. \n\\end{eqnarray}\n\n\n\\section{Mass and free energy}\n\nThere are several definitions of mass for the black hole spacetime.\nCai {\\it et al.} \\cite{Cai:2006td} discussed mass of the black hole defined by the counter-term method for\nasymptotically locally flat spacetime \\cite{Mann:1999pc}-\\cite{Kraus:1999di}. \nUsing the counter-term mass $M_{ct}$, they investigated the first law of black hole thermodynamics and \nsuggested the existence of a new work term in the first law. \nThe direct calculation reveals \n\\begin{eqnarray}\n M_{ct} = M_{AD} = \\frac{\\pi}{2G}r_{\\infty} \\left( 2\\rho_+ + 2\\rho_- +\\rho_0 \\right), \n\\label{eq:ct-mass-BH}\n\\end{eqnarray}\nwhere $M_{AD}$ is the Abbott-Deser (AD) definition of mass~\\cite{Abbott:1981ff} \nevaluated on a product $S^1$ bundle over 4-dimensional Minkowski spacetime, which is a completely \nflat spacetime\\footnote{ Cai {\\it et al.} showed that $M_{ct}$ equals the AD mass evaluated on a \"twisted\" \n$S^1$ bundle over 4-dimensional Minkowski spacetime.  \nHowever, this background spacetime is not a solution of the vacuum Einstein equation \nand we can not obtain finite free energy and Hamiltonian of the black hole on it. \nThus, we consider the flat background.\n}.\nHereafter, we term this spacetime flat background, shortly.\nTherefore, $M_{AD}$ satisfies the same equations as $M_{ct}$. \nThe Komar mass is also meaningful as a mass of black holes which possess a timelike Killing vector.\nUsing the timelike Killing vector $\\partial \/\\partial \\tau $ normalized at the spatial infinity, \nwe can calculate the Komar mass for the black hole (\\ref{eq:metric-Trho}) as \n\\begin{eqnarray}\n   M_K  =  \\frac{3\\pi }{4 G} r_{\\infty} \\left( \\rho_{+} + \\rho_{-} \\right) ,\n   \\label{eq:Komar}\n\\end{eqnarray}\nwhere the integral is taken over the 3-dimensional sphere at the spatial infinity.\nSmarr-type formula was shown generally in~\\cite{Gauntlett:1998fz} for the Komar mass \n\\begin{eqnarray}\n M_K -Q\\Phi = \\frac{3}{2} TS,\n\\end{eqnarray}\nwhich is sometimes called integrated expression for the first law. \nThis implies that we would have a differential expression for the first law using the Komar mass.\nClearly, since $M_K$ does not equal $M_{ct}$, then \nthe expressions for the first law satisfied by these masses should be different. \n\n\n\nIn order to deduce the work term suggested by Cai {\\it et al.}, \nwe note the fact that, far from the black hole, the geometry locally looks like the black string. \nIt is known that, in the case of the black p-branes or black string without twisting, \nthe so-called gravitational tension and the size of the extra-dimension \ncontribute to the first law~\\cite{Kol:2003if}\\cite{Townsend:2001rg}\\cite{Harmark:2003eg}\\cite{Traschen:2001pb}. \nOne may think that, also in the case of the squashed KK black hole, \nthese quantities contribute to the first law as a work term. \nThe gravitational tension which can be applied to non-asymptotically flat spacetime was \ndefined by using the Hamiltonian formalism to a foliation of the spacetime \nalong asymptotically translationally-invariant spatial direction~\\cite{Harmark:2004ch}. \nThe definition requires some reference spacetime in order to give finite gravitational tension. \nAs a reference background, we choose the flat background.\nThen, using the definition given in \\cite{Harmark:2004ch}, we can calculate the gravitational tension \nassociated with the direction $\\partial_{\\psi}$ as\n\\begin{eqnarray}\n\\mathcal{T} = \\frac{1}{4 G}\\left(\\rho_+ +\\rho_- +2\\rho_0\\right). \n\\end{eqnarray}\nThe conjugate variable to $\\mathcal{T}$ is \nthe size of the extra-dimension at the spatial infinity, $L:=2\\pi r_{\\infty}$. \nWith these quantities, $M_{AD}$ is related to the entropy and the electric charge \nvia the following expression for the first law: \n\\begin{eqnarray}\ndM_{AD} &=& TdS + \\Phi dQ +\\mathcal{T} dL.\n\\label{eq:first-law-AD}\n\\end{eqnarray}\nThus, the last term $\\mathcal{T}dL$ is  \nthought of as a work term suggested by Cai {\\it et al.}~\\cite{Cai:2006td}. \nThe expression for the first law (\\ref{eq:first-law-AD}) shows that \n$M_{AD}$ is a thermodynamic potential as a function of $(S,Q,L)$. \nIt means that $M_{AD}$ is relevant to the thermodynamic system with natural variables $(S,Q,L)$. \nThe same holds true for $M_{ct}$, because of $M_{ct}=M_{AD}$.\n\n\nThe free energy of the black hole is obtained from the classical Euclidean action $I_E$ as \n\\begin{eqnarray}\nF =T {I_E} = \\frac{\\pi}{2G}r_{\\infty} (\\rho_+ + \\rho_0 ),\n\\end{eqnarray}\nwhich was calculated by the traditional background subtraction method \nwith the flat background~\\cite{Gibbons:1976ue}.\nThe free energy $F$ equals one evaluated by the counter-term method.\nThus, in this case, \nthe counter-term method is equivalent to background subtraction method with the flat background.\nIn the calculation of the free energy by the background subtraction method, we fixed the temperature, \nthe electro-static potential and the size of the extra-dimension at the boundary of the spacetime. \nIt follows that the free energy satisfies the following differential relation: \n\\begin{eqnarray}\ndF = -SdT -Qd\\Phi +\\mathcal{T} dL.\n\\label{eq:first-law-F}\n\\end{eqnarray}\nThus, the free energy $F$ has natural variables $(T, \\Phi,L)$. \nThe equations (\\ref{eq:first-law-AD}) and (\\ref{eq:first-law-F}) suggest the following relation with the AD mass:\n\\begin{eqnarray} \nF=M_{AD} - Q\\Phi -TS,\n\\label{eq:Legendre-F-AD}\n\\end{eqnarray}\nwhich can be easily shown by direct calculation. \nThis relation (\\ref{eq:Legendre-F-AD}) is nothing but Legendre transformation between \n$F$ and $M_{AD}$. \n\n\n\nFor asymptotically flat electrically charged black holes, it is known that Hamiltonian \ndoes not equal ADM mass and these two quantities differ by $Q\\Phi$~\\cite{Hawking:1995ap}.\nThe black hole we consider here is not asymptotically flat but asymptotically locally flat.\nIn spite of the difference in asymptotic structure, \nif we regard the AD mass as a counterpart of ADM mass, \nthe same is true for the case of the squashed black hole; \nThe Hamiltonian of the black hole evaluated on the flat background can be related to the AD mass as\n~\\cite{Hawking:1998jf}\n\\begin{eqnarray}\nH = \\frac{\\pi}{2G} r_{\\infty} \\left( 2\\rho_+ - \\rho_- +\\rho_0 \\right)\n=M_{AD} -Q\\Phi.\n\\label{eq:Hamiltonian}\n\\end{eqnarray}\nOften, Hamiltonian for solution without electric charge is called Hawking-Horowitz (HH) mass \\cite{Hawking:1995fd}.\nIn the case of $Q=0$, the HH mass of the black hole is equal to the AD mass \nas is in the case of asymptotically flat black hole. \nFrom the relation (\\ref{eq:Hamiltonian}), the first law can take the form with the Hamiltonian as \n\\begin{eqnarray}\ndH &=& TdS - Qd\\Phi+\\mathcal{T} dL.\n\\label{eq:first-law-flat-H}       \n\\end{eqnarray}\nEquivalently, \nthe relation (\\ref{eq:Hamiltonian}) is a Legendre transformation between the Hamiltonian and the AD mass. \nThe equation (\\ref{eq:first-law-flat-H}) shows \nthat the Hamiltonian is the thermodynamic potential with natural variables $(S, \\Phi, L)$. \nThe Hamiltonian is also the Legendre transform of the free energy with respect to $TS$; \n$F = H-TS$.\nTherefore, \nthe AD mass and the Hamiltonian are related to the free energy $F$ via Legendre transformations and \ncan be interpreted as different thermodynamic potentials. \n\n\nHowever, it can be shown that the Komar mass does not have $\\mathcal{T}$ or $L$ as a natural variable. \nNow, let us obtain the differential expression for the first law by use of the Komar mass. \nIn order to do so, we require that the Komar mass should be related to the free energy via Legendre transformations.\nIn stead of $(L, \\mathcal{T})$, we introduce a couple of new variables $(\\epsilon, \\Sigma)$ satisfying  \n\\begin{eqnarray}\n F=M_K - TS -Q\\Phi + \\epsilon \\Sigma,\n\\quad \ndM_K = TdS + \\Phi dQ -\\epsilon d \\Sigma.\n\\label{eq:first-law-Komar}\n\\end{eqnarray}\nThen, $(\\epsilon, \\Sigma)$ is determined up to a constant $C$ as\n\\begin{eqnarray}\n\\epsilon = C (2\\pi r_{\\infty})^2=CL^2, \\quad \n\\Sigma =\\frac{1}{16 \\pi GC r_{\\infty}}  \\left(\\rho_+ +\\rho_-+2\\rho_{0}\\right)=  \\frac{\\mathcal{T}}{2CL}. \n\\end{eqnarray}\nThus, the Komar mass has this quantity $\\Sigma$ as a natural variable as well as $S$ and $Q$.\nThe pair of variables $(\\epsilon, \\Sigma)$ contributes not only to the differential relation \nfor the Komar mass \nbut also to the following:\n\\begin{eqnarray}\ndF = -SdT -Qd\\Phi +\\Sigma d\\epsilon,\\ \ndH = TdS -Qd\\Phi +\\Sigma d\\epsilon, \\ dM_{AD} = TdS +\\Phi dQ +\\Sigma d\\epsilon.\n\\label{eq:first-law-H-AD-epsilon}\n\\end{eqnarray}\nTherefore, the expression for the first law including the free energy, \n the Hamiltonian or the AD mass is not unique. \nThese expressions are consistent with the interpretation that $H$ or $M_{AD}$ is the thermodynamic potential \nwith natural variables $(S,\\Phi, L)$ or $(S, Q,L)$, because \nsystem with fixed $L$ \nis equivalent to that with fixed $\\epsilon$ due to the relation $\\epsilon \\propto L^2$.\n\n\n\nIn order to clarify the relation with the case of the 5-dimensional RN black hole,\nlet us consider the spherically symmetric (SS) limit $r_{\\infty}\\to \\infty$.\nHowever, the free energy, the Hamiltonian and the AD mass evaluated on the flat background diverge in the limit.\nAs an alternative background, we consider the KK monopole spacetime. \nThe free energy and the Hamiltonian on the KK monopole background are calculated as \n\\begin{eqnarray}\n\\tilde{F} = \\frac{\\pi}{2G} r_{\\infty} \n \\left(\\rho_+ + \\rho_0 -  \\frac{r_{\\infty}}{2} \\right), \\quad \n\\tilde{H} = \\frac{\\pi}{2G} r_{\\infty} \\left(2\\rho_+ -\\rho_- +\\rho_0 -\\frac{r_{\\infty}}{2} \\right).\n\\end{eqnarray}\nIt is easily checked that the difference between $F$ and $\\tilde{F}$ is the free energy of the KK monopole \non the flat background; $F-\\tilde{F}=\\frac{\\pi}{4G}r_\\infty^2$, \nwhich equals the free energy calculated by the counter-term method \\cite{Mann:2005cx}.\nThe gravitational tension calculated on the KK monopole background is \n\\begin{eqnarray}\n\\tilde{\\mathcal{T}} &=& \\frac{1}{4 G} \\left( \\rho_+ +\\rho_- + 2\\rho_0 -r_{\\infty}\\right).\n\\end{eqnarray}\nWith this tension, \nthe free energy and the Hamiltonian satisfy \n\\begin{eqnarray}\nd\\tilde{F} = -SdT -Qd\\Phi + \\tilde{\\mathcal{T}} dL, \\quad \nd\\tilde{H} = TdS -Qd\\Phi + \\tilde{\\mathcal{T}} dL.\n\\end{eqnarray}\nAs before, $\\tilde{\\mathcal{T}}$ or $L$ can not be a natural variable for the Komar mass.\nAs in the previous case, we can obtain a couple of thermodynamic variables $(\\epsilon, \\tilde{\\Sigma})$ satisfying\n\\begin{eqnarray}\n\\tilde{F}=M_K-TS-Q\\Phi+\\epsilon \\tilde{\\Sigma}, \\quad  dM_K = TdS + \\Phi dQ -\\epsilon d \\tilde{\\Sigma}.\n\\label{eq:first-Komar-KKm}\n\\end{eqnarray}\nThe result is \n\\begin{eqnarray}\n\\epsilon ={C} (2\\pir_\\infty)^2 = {C}L^2, \\quad \n\\tilde{\\Sigma} = \n\\frac{1}{16\\pi G {C} r_{\\infty}}  \\left(\\rho_+ +\\rho_-+2\\rho_{0}-r_\\infty \\right)=\\frac{\\tilde{\\mathcal{T}}}{2{C}L},\n\\end{eqnarray}\nwhere $\\tilde{\\Sigma}$ is different from $\\Sigma$ by a constant $(16 \\pi GC)^{-1}$. \nTherefore, the two quantities $\\Sigma$ and $\\tilde{\\Sigma}$ are essentially the same, \nand the differential expressions (\\ref{eq:first-law-Komar}) \nand (\\ref{eq:first-Komar-KKm}) are equivalent.\nThis is consistent with the fact that the Komar mass does not depend on the choice of background spacetime.\nIn the SS limit, the products $\\epsilon\\tilde{\\Sigma}$ and $L\\tilde{\\mathcal{T}}$ become zero, \nand $M_K$, $\\tilde{F}$ and $\\tilde{H}$ \nbecome those of 5-dimensional RN black hole evaluated on the 5-dimensional Minkowski background.\nThus, this formulation for the squashed black hole \nincludes usual thermodynamic formulation for 5-dimensional RN black hole as a limit.\nIn this sense, it \nis a generalized formulation of thermodynamics for 5-dimensional electrically charged static black holes. \n\n\nBecause $\\tilde{\\Sigma}$ and $\\tilde{\\mathcal{T}}$ vanish when the outer horizon is a perfectly-round three-sphere,\none may think that\nthe thermodynamic variable $\\tilde{\\Sigma}$ or $\\tilde{\\mathcal{T}}$ can be interpreted as \na quantities representing the deformation in the shape of the horizon. \nIn fact, the variables $\\tilde{\\Sigma}$ and $\\tilde{\\mathcal{T}}$ can be rewritten as \n\\begin{eqnarray} \n\\tilde{\\Sigma}  = \\frac{1}{32\\pi G C}\\frac{1}{R_+ \\ell_+} \\left({\\ell_+} - R_+ \\right)^2, \\quad \n\\tilde{\\mathcal{T}} = \\frac{1}{8G}\\frac{ r_{\\infty}}{R_+ \\ell_+} \\left( \\ell_+ - R_+ \\right)^2,\n\\label{eq:Sigma-squashing}\n\\end{eqnarray}\nwhere we have defined new parameters $R_+$ and $\\ell_+$ as follows:\n \\begin{eqnarray}\nR_+ := \\sqrt{\\rho_+(\\rho_+ + \\rho_0 )}, \\quad {\\ell_+} := \\sqrt{\\rho_+ (\\rho_- +\\rho_0)}, \n \\end{eqnarray}\nwhich denote \nthe circumference radius of the $S^2$ base space at the outer horizon and \nthat of the twisted $S^1$ fibre there, respectively. \nIn this way, $\\tilde{\\Sigma}$ and $\\tilde{\\mathcal{T}}$ measure the squashing of the outer horizon.\n\n\n\\section{Summary}\n\n\nAs shown in this paper, \nthe Abbott-Deser mass which equals the counter-term mass, the Komar mass and the Hamiltonian \ncontribute to different expressions for the first law \nand are related to each other by the Legendre transformations. \nEach mass can be interpreted as a thermodynamic potential with its own natural variables.\nThe consistent set of natural variables for each mass has been revealed, \nand we have obtained a more general thermodynamic formulation for electrically charged black holes\nin 5-dimensional Einstein-Maxwell system.\n\n\nNow, we discuss the relation between $(L, \\mathcal{T})$ and \nthe pair of new quantities ($\\epsilon, \\Sigma$).\nLet us begin by considering the case of the free energy. \nIn the evaluation of the free energy, the size of the extra-dimension at spatial infinity $L$ was fixed, \nso that $L$ is a natural variable of the free energy. \nIn general, $L$ can be replaced by any monotonic function of $L$, say $f(L)$, as a thermodynamics variable. \nThis may be trivial because thermodynamic environment characterized by fixing the size $L$ is equivalent to\nthat by fixing $f(L)$. \nOne may write the differential relation for the free energy as \n\\begin{eqnarray}\ndF = -SdT -Qd\\Phi + W d f(L),\n\\label{eq:first-law-fL}\n\\end{eqnarray}\nwhere $W$ is conjugate to $f(L)$. \nSince the last work term can be rewritten as $Wdf(L) =W f'(L) dL$,  \nthe relation (\\ref{eq:first-law-fL}) is equivalent to (\\ref{eq:first-law-F}) \nif the conjugate variable $W$ satisfies \n\\begin{eqnarray}\nW= \\frac{\\mathcal{T}}{f'(L)}.\n\\end{eqnarray}\nIn this way, the work term with the form $\\mathcal{T} dL$ can be replaced by $W d f(L)$.\nThe first law (\\ref{eq:first-law-fL}) is equivalent to (\\ref{eq:first-law-H-AD-epsilon}),\nif $f(L) = CL^2 = \\epsilon$, \nand the relation between $\\Sigma$ and $\\mathcal{T}$ is given as\n\\begin{eqnarray}\n\\Sigma = \\frac{\\mathcal{T}L}{2\\epsilon}\\  \\propto\\  \\frac{\\mathcal{T}}{2L}.\n\\label{eq:Sigma-T-relation}\n\\end{eqnarray}\nThe Hamiltonian and the AD mass are Legendre transforms of the free energy with respect to \n$TS$ or $TS+Q\\Phi$ respectively, so that the expressions in (\\ref{eq:first-law-H-AD-epsilon}) are \nequivalent to (\\ref{eq:first-law-AD}) and (\\ref{eq:first-law-flat-H}). \nTherefore, the work term in the first law for the Hamiltonian and the AD mass is not unique. \n\n\nHowever, the quantities $\\mathcal{T}$ and $\\Sigma$, which are conjugate to $L$ and $\\epsilon$, \nare different in the sense of thermodynamics: \nthermodynamic environment characterized by fixing $\\Sigma$ is one by fixing $\\mathcal{T}\/2L$, as shown in \n(\\ref{eq:Sigma-T-relation}). \nThus, the pair ($\\epsilon, \\Sigma$) is thermodynamically different from ($L, \\mathcal{T}$). \nIt is natural that thermodynamic potentials suitable for different environments are different. \nThe Komar mass is a thermodynamic potential for environment characterized by ($S, Q, \\Sigma$), \nwhile the AD mass is a thermodynamic potential for that by ($S, Q, \\epsilon$) or ($S, Q, L$). \nIn this way, we can interpret the difference of masses from thermodynamical viewpoint. \n\n\nIt is interesting to investigate thermodynamic properties and stability of the black hole in each environment \nand to compare the black hole with black string. \nIt will be reported in a future publication. \n\n\n\n\n\\begin{acknowledgments} \nY.K. was partially supported by the Yukawa memorial foundation and is also supported by \nthe 21st Century COE \"Constitution of wide-angle mathematical basis focused on knots\". \nThis work is supported by the Grant-in-Aid for Scientific Research Fund of the Ministry of Education, Science and \nCulture of Japan No. 13135208 and No. 14540275. \n\\end{acknowledgments} \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Introduction} \n\nMulticomponent quantum gases constitute an ideal playground for investigating a plethora of many-body (MB) \nprocesses~\\cite{Bloch2008,Cazalilla2011} \nincluding in particular the generation of quasiparticles~\\cite{landau1933bewegung,pekar1946autolocalization} such as polarons. \nQuasiparticle formation can be studied owing to the unprecedented experimental tunability of the impurity-medium \ninteraction strength, via Feshbach \nresonances~\\cite{Ospelkaus2006,Zaccanti2006,chin2010feshbach},  while systems containing few particles \ncan be realized especially in one spatial dimension~\\cite{serwane2011deterministic,wenz2013few}. \nDepending on the quantum statistics of the host, these quasiparticles are known \nas Bose~\\cite{grusdt2015new,rath2013field} and Fermi~\\cite{schmidt2018universal,massignan2014polarons} polarons respectively. \nTheir existence and a variety of their properties have already been experimentally probed in both \nBose~\\cite{jorgensen2016observation,hu2016bose,yan2020bose} and \nFermi~\\cite{schirotzek2009observation,kohstall2012metastability,koschorreck2012attractive,cetina2016ultrafast,scazza2017repulsive} \ngases, e.g. via employing injection spectroscopy~\\cite{cetina2016ultrafast,jorgensen2016observation,hu2016bose}. \nThe progress regarding the understanding of the quasiparticles features has also been corroborated by an extensive \ntheoretical activity revealing different aspects of their underlying dressing mechanism such as their effective \nmass~\\cite{grusdt2017bose,ardila2015impurity}, lifetime~\\cite{kohstall2012metastability}, induced \ninteractions~\\cite{dehkharghani2018coalescence,mistakidis2020many,mistakidis2019repulsive}, and bound \nstates termed bipolarons~\\cite{camacho2018bipolarons,schmidt2018universal} or trimerons~\\cite{nishida2015polaronic,alhyder2020impurity,naidon2018two}. \n\nAccordingly, the interaction of the impurities with their surrounding leads to deformations of the latter in the vicinity of the former being manifested \nas impurity-medium bound states~\\cite{ardila2019strong} for strong attractions as well as sound wave emission~\\cite{marchukov2020shape} \nand phase-separation~\\cite{mistakidis2020many,mistakidis2019quench} for repulsive interactions. \nThese phenomena are a direct imprint of the inevitable entangled nature of these systems~\\cite{mistakidis2019quench} \nwhose non-equilibrium dynamics is far less appreciated~\\cite{skou2020non}. \nThe impurity dynamics holds the premise of unveiling even more complex \nprocesses that will shape our understanding on these settings and may be exploited in future technological applications. \nTo date remarkable demonstrations of the impurities' non-equilibrium dynamics include the spontaneous generation \nof nonlinear excitations~\\cite{grusdt2017bose,mistakidis2019correlated}, collision induced pattern \nformation~\\cite{mistakidis2019dissipative,burovski2014momentum,gamayun2018impact,knap2014quantum,tajima2019collisional}, their \nmediated correlations~\\cite{kwasniok2020correlated,mistakidis2020many,mistakidis2020induced} and relaxation \nprocesses~\\cite{mistakidis2020pump,lausch2018prethermalization,boyanovsky2019dynamics}, as well as their \ntransport properties in optical lattices~\\cite{johnson2011impurity,theel2020entanglement,keiler2020doping,keiler2019interaction}. \nIt is also important to emphasize that the above-mentioned investigations have predominantly considered a single impurity \nwhilst the effect of larger impurity concentrations leading to enhanced correlation-induced phenomena is until now largely unexplored. \nFor these latter settings, the interplay of the quantum statistics between the impurities and the host is of \nimportance especially for the induced impurity-impurity correlations. \n\nIn this context, a very promising candidate is a fermionic environment containing two bosonic impurities which can \ninteract via direct $s$-wave scattering. \nIndeed, most of the experimental and theoretical endeavors of Fermi polarons have been mainly focused on the \nlimiting case of a strongly spin imbalanced Fermi gas ~\\cite{Schmidt2012,Vlietinck2013,Mora2009,Trefzger2012,Massignan2013,Pilati2010,Schmidt2011,Sanner2012}, while the situation of bosonic impurities in a Fermi sea is arguably much less studied~\\cite{De2014,Fratini2012,cetina2016ultrafast,huber2019medium}. \nIn this setting it is very interesting to reveal the presence and nature of induced impurity-impurity interactions which are known to be suppressed for fermionic impurities~\\cite{mistakidis2019correlated,dehkharghani2018coalescence,mistakidis2019repulsive}. \nThe study of the competition between induced and direct $s$-wave interactions, with the latter being naturally absent for fermionic impurities, is \nan intriguing prospect. \nAn additional perspective is the possible emergence of impurity-impurity and impurity-medium bound states for strong attractions. \nCertainly, the identification of the above properties in the dynamical response of the system e.g. subjected to an \nimpurity-medium interaction quench~\\cite{volosniev2015real,mistakidis2019effective} as well as the characterization of the respective pattern formation especially of the host is desirable. \nIn order to address these questions we consider, as a paradigmatic setup, a one-dimensional harmonically trapped Bose-Fermi (BF) mixture consisting of two bosonic impurities immersed in a few-body fermionic environment. \nTo track the stationary properties and importantly the quantum dynamics of this impurity setting we resort to the multi-layer multi-configuration time-dependent Hartree method for atomic mixtures (ML-MCTDHX)~\\cite{cao2017unified,cao2013multi} being a variational approach that allows us to capture all the relevant correlations of the BF mixture. \n\nFor the ground state we find that the impurities and the fermionic environment \nphase-separate for strong impurity-medium repulsions~\\cite{mistakidis2019correlated,Viverit2000,lous2018probing}, while \nthey exhibit a localization \ntendency close to the trap center for large attractions. \nInterestingly, attractive induced impurity-impurity interactions~\\cite{huber2019medium} mediated by the fermionic host \nare revealed for the case of non-interacting bosons for increasing impurity-medium repulsion and attraction. \nHowever, for repulsively interacting impurities we unveil that the \ninduced interactions dominate the direct $s$-wave ones for increasing impurity-medium attractions. \n\nA quench from zero to finite repulsive impurity-medium interactions triggers a breathing motion~\\cite{boudjemaa2020breathing,kiehn2019spontaneous}, \nin each component, with an interaction dependent frequency and amplitude for the impurities. \nMoreover, a dynamical impurity-bath phase-separation takes place for quenches to strong repulsions. \nImportantly, induced impurity-impurity correlations mediated by the host are identified during the evolution of two \nnon-interacting impurities and become more pronounced for quenches to stronger repulsions. \nHowever, in the case of repulsively interacting impurities a competition of induced and direct interactions is evident \nwith the latter dominating the former and enforcing the impurities to reside in a two-body superposition. \n\nQuenching to attractive impurity-medium interactions gives rise to \na beating pattern~\\cite{mistakidis2020many} on the single-particle level which originates from the participation of two breathing \nfrequencies in the dynamics of the impurities due to the dominant presence of their attractive induced interactions. \nThe impurities show a spatial localization tendency around the trap center leading to a density accumulation of the Fermi sea at \ntheir instantaneous location. \nThe strength of the attractive induced interactions is larger compared to the reverse quench scenario and it is possible to overcome \nthe direct impurities coupling for large post-quench attractions~\\cite{mistakidis2020induced,mistakidis2020many}. \nIn all cases, we show that the degree of impurity-medium entanglement is appreciable, and exhibits a hierarchy. \nFor instance, it is larger for fixed impurity interaction and increasing \nquench amplitude. \n\n\nThis work is structured as follows. \nSection~\\ref{theory} introduces the setup under consideration [Sec.~\\ref{setup}], the employed many-body variational approach [Sec.~\\ref{method}] \nand the main observables [Sec.~\\ref{observables}] utilized for the characterization of the ground state and the dynamics of the BF mixture. \nIn section~\\ref{Ground state} we address the ground state properties of the BF mixture \nwith a particular focus on the impurity-impurity induced interactions [Sec.~\\ref{corel_ground}]. \nThe non-equilibrium dynamics upon considering a quench of the impurity-medium coupling to either repulsive [Sec.~\\ref{repulsive_quench}] \nor attractive [Sec.~\\ref{attractive quench}] interaction regimes is discussed in Sec.~\\ref{quench_dynamics}. \nThe emergent entanglement dynamics is presented in Sec.~\\ref{entanglemet_dynamics}. \nWe summarize our results and give an outlook in Sec.~\\ref{conclusion}. \nAppendix~\\ref{convergence} elaborates further on the details of our variational method and \ndelineates the convergence of the presented results exemplarily. \n\n\n\n\\section{Theoretical Background}\\label{theory} \n\n\\subsection{Setup and Hamiltonian}\\label{setup}\n\nWe consider a particle-imbalanced ultracold BF mixture containing $N_B=2$ bosonic impurities and $N_F=6$ spin-polarized fermions constituting the environment. \nThe mixture is assumed to be mass-balanced i.e. $M_B=M_F\\equiv M$ and both species are confined in the same one-dimensional (1D) harmonic trap namely $\\omega_B=\\omega_F\\equiv \\omega=0.1$. \nThis 1D geometry can be experimentally realized by imposing a strong transverse confinement ($\\omega_{\\perp}$) compared to the longitudinal ($\\omega_{\\parallel}$) \none obeying $\\omega=\\omega_{\\parallel}\/\\omega_{\\perp} \\ll 1$~\\cite{katsimiga2020observation,serwane2011deterministic,wenz2013few}. \nThe individual species of such an approximately mass-balanced BF mixture correspond, for instance, to bosonic and fermionic isotopes of the same element e.g. $^{7}$Li-$^{6}$Li~\\cite{Kempen2004, Delehaye2015} or $^{171}$Yb-$^{172}$Yb~\\cite{Honda2002}. \nThe underlying MB Hamiltonian of the above-described system reads  \n\\begin{equation}\\label{1}\n\\begin{split}\n&H = \\sum_{\\sigma = F, B}^{}\\sum_{i = 1}^{N_\\sigma} \\bigg [ -\\frac{\\hbar^2}{2M}\\bigg (\\frac{\\partial}{\\partial x^{\\sigma}_i} \\bigg)^2  \n+   \\frac{1}{2}M\\omega^2(x^\\sigma_i)^2 \\bigg ] \\\\ &  + g_{BB}\\sum_{ i < j }^{} \\delta(x^{B}_i - x^{B}_j) + g_{BF}\\sum_{ i = 1 }^{N_F} \n\\sum_{j = 1}^{N_B}\\delta(x^{F}_i - x^{B}_j).\n\\end{split}\n\\end{equation}\nOperating in the ultracold regime, $s$-wave scattering constitutes the dominant two-body interaction process and hence interparticle interactions can be modeled by a short-range contact potential~\\cite{olshanii1998atomic}. \nNote that for the spin-polarized fermions $s$-wave scattering is forbidden due to the Pauli exclusion principle~\\cite{pethick2008bose,pitaevskii2003international} and therefore \ntheir intraspecies interactions vanish. \nAccordingly, the boson-boson and boson-fermion (alias impurity-medium) 1D effective coupling constants~\\cite{olshanii1998atomic} are $g_{BB} = 4 \\hbar^2a_{BB}\/(M a^2_{\\perp}) [1 - C a_{BB}\/a^2_{\\perp,B}]^{-1}$ and $g_{BF} = 4 \\hbar^2a_{BF}\/(M a^2_{\\perp}) [1 - Ca_{BF}\/a^2_{\\perp,B}]^{-1}$ respectively. \nHere, $a_{BB}$ ($a_{BF}$) is the three-dimensional boson-boson (boson-fermion) $s$-wave scattering length and $C \\approx 1.4603$. \nThe parameter $a_\\perp = \\sqrt{\\hbar\/M\\omega_\\perp}$ denotes the transversal confinement length scale, with $\\omega_{\\perp}$ being the transversal trapping frequency. \nImportantly, the boson-boson and boson-fermion interaction strengths $g_{BB}$ and $g_{BF}$ can be experimentally tuned \neither by means of ${a^s_{BB}}$, ${a^s_{BF}}$ using Feshbach resonances~\\cite{kohler2006production,chin2010feshbach} or \nvia adjusting ${{\\omega_\\bot}}$ by employing confinement-induced resonances~\\cite{olshanii1998atomic}. \n\nBelow, we rescale the MB Hamiltonian of Eq.~(\\ref{1}) in terms of $\\hbar \\omega_\\perp$. \nAs a consequence, the length, time and interaction strengths are expressed in units of $\\sqrt{\\hbar\/M\\omega_\\perp}\\equiv a_{\\perp}$, $\\omega^{-1}_\\perp$ and $\\sqrt{\\hbar^3 \\omega_\\perp\/M}$, respectively. \nIt is also worth mentioning that a BF mixture with $N_B \\ll N_F$, as the one considered herein, features supressed \nthree-body recombination particle losses, since their rate is known \nto be proportional to $N^2_B N_F$~\\cite{Helfrich2010}. \n\nIn the following, we characterize the ground-state properties of the highly particle-imbalanced BF mixture particularly focusing on the emergent correlation patterns and unveiling, for instance, phase-separation processes as well as identify impurity-impurity induced interactions for varying boson-boson and impurity-medium interaction strengths, see Sec.~\\ref{Ground state}. \nRecall that in the absence of an external confinement the two species are miscible by means that they spatially overlap when $g^2_{BF} < g_{BB}$, otherwise they phase-separate~\\cite{mistakidis2018correlation,Viverit2000,Roth2002,lous2018probing,mistakidis2019correlated}. \nIn the presence of an external trap and also away from the thermodynamic limit the above-mentioned relation is modified, i.e. $g_{BF}$ should \nbecome substantially larger than $g_{BB}$ in order to achieve the phase-separation. \nSubsequently, we trigger the non-equilibrium dynamics of the BF mixture by applying a quench of the impurity-medium interaction strength ($g_{BF}$) from zero to either repulsive [Sec.~\\ref{repulsive_quench}] or attractive [Sec.~\\ref{attractive quench}] couplings. \nImportantly, within these latter post-quench interaction regimes impurity-impurity correlations are finite whilst they vanish for the initial state. \nThus, the system is driven towards regions of finite impurity-impurity interactions aiming at exploring their dynamical fate, the consequent pattern formation and the associated build-up of correlations.   \n\n\n\n\\subsection{Variational wavefunction ansatz and quantum dynamical approach} \\label{method} \n\nTo investigate the ground-state and most importantly the quench dynamics of the particle-imbalanced BF mixture we solve the underlying MB Schr{\\\"o}dinger \nequation using the variational ML-MCTDHX approach~\\cite{cao2017unified,cao2013multi}.   \nIt is based on expanding the MB wavefunction in terms of a time-dependent and variationally optimized basis. \nThis asset enables us to capture both the inter- and the intraspecies correlations of the binary system in a computationally efficient manner compared to methods relying on a time-independent basis set.  \n\nThe MB wavefunction, $\\Psi_{\\rm MB}$, is initially expressed in the form of a truncated Schmidt decomposition of rank $D$~\\cite{Horodecki2009}. \nNamely \n\\begin{equation} \\label{4}\n\\Psi_{\\rm MB}(\\vec{x}^B, \\vec{x}^F;t) = \\sum_{k = 1}^{D} \\sqrt{\\lambda_k(t)}\\Psi^B_k(\\vec{x}^B;t)\\Psi^F_k(\\vec{x}^F;t). \n\\end{equation}\nThe values of the Schmidt coefficients, $\\lambda_k(t)$, characterize the degree of entanglement of the binary system. \nIn decreasing order they are also known as natural species populations of the $k$-th species function. Evidently, the \nsystem is entangled~\\cite{roncaglia2014,Horodecki2009,mistakidis2018correlation} in the case that more than a single \ncoefficients $\\lambda_k(t)$ exhibit an non-zero population. \nThen, the many-body state [Eq.~(\\ref{4})] is a superposition of the respective species states instead of being a direct product of only two states (non-entangled case). \n\nAs a next step, each of the above-mentioned species functions is expanded in terms of the determinants and permanents of $d_\\sigma$ distinct time-dependent fermionic and bosonic single particle functions (SPFs) respectively. \nTherefore, each $\\Psi^{\\sigma}_k(\\vec{x}^{\\sigma};t)$ reads  \n\\begin{equation}\\label{5}\n\\begin{split}\n&\\Psi_k^{\\sigma}(\\vec x^{\\sigma};t) = \\sum_{\\substack{l_1,\\dots,l_{d_{\\sigma}} \\\\ \\sum l_i=N}} C_{k,(l_1,\n\\dots,l_{d_{\\sigma}})}(t)\\sum_{i=1}^{N_{\\sigma}!} \\big ( {\\rm sign} (\\mathcal{P}_i) \\big ) ^{\\zeta} \\\\ & \\times \\mathcal{P}_i\n \\left[ \\prod_{j=1}^{l_1} \\varphi_1^{\\sigma}(x_j;t) \\cdots \\prod_{j=1}^{l_{d_{\\sigma}}} \\varphi_{d_{\\sigma}}^{\\sigma}(x_{K(d_{\\sigma})+j};t) \\right]. \n  \\end{split}\n\\end{equation} \nIn this expression, $C_{k,(l_1,....., l_{d_{\\sigma}})}(t)$ denote the time-dependent expansion coefficients of a particular \ndeterminant for fermions or permanent for bosons and $n_i(t)$ is the occupation number of the SPF, $\\varphi_i(x;t)$. \nThe index $\\zeta = 0, 1 $ for bosons and fermions respectively and $\\mathcal{P}$ is the permutation operator exchanging the particle configuration $x_{\\nu}^{\\sigma}$, $\\nu=1,\\dots,N_{\\sigma}$ within the SPFs. \nAlso, $\\rm sign(\\mathcal{P}_i)$ is the sign of the corresponding permutation and $K(r)\\equiv l_1+l_2+\\dots+l_{r-1}$,\nwhere $l_{r}$ is the occupation of the $r$-th SPF and $r\\in\\{1,2,\\dots,d_{\\sigma}\\}$. \nWe remark that the bosonic subsystem is termed intraspecies correlated if more than one SPF is occupied otherwise it is fully coherent~\\cite{lode2020colloquium}, see also the discussion below. \nOn the other hand, the fermionic species exhibit beyond non-trivial Hatree-Fock correlations when more than $N_F$ eigenvalues possess a macroscopic population~\\cite{mistakidis2019correlated,erdmann2019phase}. \n\nThe time-evolution of the $(N_B+N_F)$-body wavefunction obeying the MB Hamiltonian of Eq.~(\\ref{1}) is determined by calculating the corresponding ML-MCTDHX equations of motion~\\cite{cao2013multi}. \nThe latter are found by performing e.g. the Dirac-Frenkel variational principle~\\cite{Dirac193,Frenkel1934} for the MB ansatz provided by Eqs.~\\eqref{4} and \\eqref{5}. \nAs a result we obtain a set of $D^2$ linear differential equations of motion for the coefficients $\\lambda_k(t)$ being coupled to $D[$ ${N_B+d_B-1}\\choose{d_B-1}$+${d_F}\\choose{N_F}$] non-linear integro-differential equations for the species functions and $d_F+d_B$ integro-differential equations for the SPFs. \nFinally, let us mention in passing that the variational ML-MCTDHX ansatz can be easily reduced to different levels of approximation. \nAs a case example, the corresponding mean-field wavefunction ansatz of the BF mixture corresponds to the case of $D = d_B = 1$ and $d_F = N_F$ while the respective mean-field equations of motion are retrieved \nby following a variational principle, see e.g. for details \\cite{lode2020colloquium,kohler2019dynamical}. \n\n\n\\subsection{Observables and analysis}\\label{observables}\n\nIn the following, we briefly introduce the basic observables that will be employed in the remainder of our work in order to characterize \nboth the stationary properties and the non-equilibrium dynamics of the BF mixture. \nA particular emphasis is paid on the impurities subsystem. \nTo visualize the spatial distribution of the $\\sigma=B,F$ species, i.e. the impurities and the medium respectively, on the single-particle \nlevel we invoke the corresponding one-body reduced density matrix \n\\begin{equation}\\label{1BD}\n\\rho^{(1)}_{\\sigma}(x, x';t)=\\langle \\Psi_{\\rm MB}(t)| \\hat{\\Psi}_\\sigma^{\\dagger}(x) \\hat{\\Psi}_{\\sigma}(x')  | \\Psi_{\\rm MB}(t) \\rangle.\n\\end{equation}\nHere, $\\hat{\\Psi}_{B}(x)$ [$\\hat{\\Psi}_{F}(x)$] is the so-called bosonic [fermionic] field operator acting on position $x$ and satisfying the \nstandard commutation [anti-commutation] relations~\\cite{pethick2008bose,pitaevskii2003international}. \nThe diagonal of $\\rho^{(1)}_{\\sigma}(x, x';t)$ is the well-known one-body density of the $\\sigma$-species \ni.e. $\\rho^{(1)}_{ \\sigma}(x;t) = \\rho^{(1)}_ {\\sigma}(x, x' = x;t)$~\\cite{lode2020colloquium}. \nThe latter is accessible in ultracold atom experiments using the single-shot absorption imaging technique~\\cite{sakmann2016single,Bloch2008} \nand especially for few atoms can be retrieved by averaging over a sample of single-shots~\\cite{mistakidis2018correlation,Klein2005,mistakidis2019dissipative}. \nWe remark that the eigenfunctions of the $\\sigma$-species one-body reduced density matrix are known as the $\\sigma$-species natural orbitals, namely $\\phi^{\\sigma}_i$. \nIn this sense, when more than $N_F$ (one) fermionic (bosonic) natural orbitals are significantly populated the corresponding subsystem is called \nfragmented or intraspecies correlated~\\cite{mistakidis2018correlation,lode2020colloquium}. \nAccordingly, the underlying degree of fragmentation can be quantified via measuring $F_F(t) = 1 -\\sum_{i = 1}^{N_F}n^F_i(t)$ and $F_B(t) = 1 - n^B_1(t)$ for the fermionic and the bosonic subsystems respectively. \nHere we consider that the population of the total number of orbitals are normalized to unity i.e. \n$\\sum_{i=1}^{d_F}n^{F}_i(t) = 1$ and $\\sum_{i=1}^{d_B}n^{B}_i(t)=1$. \nRecall that in the MF limit of the BF mixture~\\cite{pitaevskii2003international,mistakidis2019correlated,karpiuk2004soliton} where \n$\\Psi_{\\rm MB}(\\vec{x}^F, \\vec{x}^B;t) \\rightarrow \\Psi_{MF}(\\vec{x}^{F}, \\vec{x}^B ; t) $ the natural populations \nof the fermionic and the bosonic species satisfy the constraints $\\sum_{i=1}^{N_F}n^{F}_i(t) = 1$, $n^F_{i > N_F}(t) = 0$, \nand $n^{B}_1(t) = 1$, $n^{B}_{i >1}(t) = 0$. \n\nThe emergence of impurity-medium entanglement can be identified by calculating the Schmidt coefficients, $\\lambda_k(t)$, participating in the MB wavefunction ansatz as described by Eq. (\\ref{4}). \nIndeed, in the case that more than one coefficients are populated, i.e. $\\lambda_{k>1}(t)\\neq 0$, then the MB wavefunction is not a single product state and the system is entangled~\\cite{Horodecki2009,mistakidis2018correlation}. \nThe Schmidt coefficients are essentially the eigenvalues of the species reduced density matrix namely  \n$\\rho^{N_{\\sigma}} (\\vec{x}^{\\sigma}, \\vec{x}'^{\\sigma};t)=\\int d^{N_{\\sigma'}} x^{\\sigma'} \\Psi^*_{\\rm MB}(\\vec{x}^{\\sigma}, \n\\vec{x}^{\\sigma'};t) \\Psi_{\\rm MB}(\\vec{x}'^{\\sigma},\\vec{x}^{\\sigma'};t)$, with $\\vec{x}^{\\sigma}=(x^{\\sigma}_1, \\cdots,x^{\\sigma}_{N_{\\sigma-1}})$, and $\\sigma\\neq \\sigma'$. \nConsequently, in order to determine the degree of the impurity-medium entanglement we use the Von-Neumann entropy~\\cite{Catani2009, Horodecki2009} given by \n\\begin{equation}\\label{VN}\nS_{VN}(t) = - \\sum_{k = 1}^{D} \\lambda_k(t) \\ln[\\lambda_k(t)].\n\\end{equation}\nIt becomes apparent that $S_{VN}(t) \\geq 0$ only when $\\lambda_{k>1}(t)\\neq 0$ meaning that entanglement is present. \nFor instance, in the mean-field limit where $\\lambda_1 (t) = 1$, $\\lambda_{k>1}(t)= 0$ and entanglement is absent it holds that $S_{VN}(t)=0$. \n\nTo infer the role of impurity-impurity and fermion-fermion two-body correlation processes in the ground state as well as \nin the dynamics of the BF mixture in a spatially resolved manner we resort to the diagonal of the two-body reduced \ndensity matrix~\\cite{mistakidis2018correlation,lode2020colloquium,sakmann2008reduced}\n\\begin{equation}\\label{2BD}\n\\begin{split}\n \\rho^{(2)}_{\\sigma \\sigma}(x, x';t) = & \\langle \\Psi_{\\rm MB}(t)| \\hat{\\Psi}^{\\dagger}_{\\sigma}(x')\n \\hat{\\Psi}^{\\dagger}_{\\sigma}(x) \\hat{\\Psi}_{\\sigma}(x') \\\\ &\n \\times \\hat{\\Psi}_\\sigma(x) | \\Psi_{\\rm MB}(t) \\rangle.\n \\end{split}\n\\end{equation} \nThis measure refers to the probability of detecting simultaneously one impurity $\\sigma=B$ (fermionic, $\\sigma=F$) particle located at $x$ and another one at $x'$. \nIn that light, it reveals the occurrence of impurity-impurity (fermion-fermion) two-body correlations and thus provides insights on how \nthe two bosons (fermions) behave with respect to one another~\\cite{mistakidis2019correlated,erdmann2019phase,mistakidis2020many,mistakidis2020induced}.  \n\nTo estimate the strength of the effective interactions between the two bosonic impurities \nwe utilize their relative distance~\\cite{mistakidis2019correlated,mistakidis2020many,mistakidis2019repulsive} defined as\n\\begin{equation} \\label{7}\n\\mathcal{D}_{\\rm rel}(t) = \\frac{\\int dx_1 dx_2 \\abs{x_1 -x_2} \\rho^{(2)}_{BB} (x_1, x_2; t)}{\\langle \\Psi_{\\rm MB}(t) | \\hat{N}_B(\\hat{N}_B -1) | \n\\Psi_{\\rm MB}(t) \\rangle}.\n\\end{equation}\nHere, $\\hat{N}_B$ is the bosonic number operator and $\\rho^{(2)}_{BB} (x_1, x_2; t)$ denotes the \ntwo-body density matrix [Eq.~(\\ref{2BD})] of the bosonic impurities subsystem. \nThe relative distance can be experimentally accessed using {\\it in-situ} spin-resolved single-shot measurements~\\cite{bergschneider2018spin}, \nwhere in particular the actual shape of $\\mathcal{D}_{\\rm rel}(t)$ can be retrieved by averaging over a sample of the individually obtained images. \n\n\n\n\\section{Ground state properties of two bosonic impurities in a fermionic environment}\\label{Ground state} \n\nWe consider $N_B = 2$ bosonic impurities in a fermionic finite-sized medium composed of $N_F=6$ spin-polarized fermions. \nRecall that an one-dimensional Fermi sea with $N_F>5$ atoms approaches the behavior of a many-body fermionic environment, see for instance \nRef.~\\cite{wenz2013few} for a corresponding experimental verification. \nIn our setting we have checked that our results, to be presented below, regarding both the ground state and the dynamics remain qualitatively the same also for $N_F=8$ (not shown here for brevity). \nThe system is mass-balanced and both species are trapped in the same harmonic oscillator of frequency $\\omega = 0.1$, unless it is stated otherwise. \nBelow, we examine the ground state characteristics of the composite system with a particular focus on the impurities properties for attractive and repulsive impurity-medium interactions. \nIn order to discriminate between direct and effective impurity-impurity interaction effects we analyze both the cases of non-interacting and interacting impurities. \nThe impact of the impurities mass on their induced interactions mediated by the fermionic environment is also discussed. \nAnother objective of our analysis is to unveil the spatial distributions of each species, discuss possibly emerging phases of the BF mixture as well as their associated correlation properties for varying impurity-medium interactions. \nTo obtain the ground state of the BF mixture governed by Eq.~\\eqref{1} we employ either the imaginary-time propagation or the improved relaxation \nmethod within ML-MCTDHX~\\cite{cao2017unified}.  \n\n\n\\subsection{Single-particle density distribution}\\label{ground_state_density} \n\nLet us first inspect the spatial configuration of the ground state of the bosonic impurities and the fermionic sea for varying impurity-medium interaction strength $g_{BF}$. \nFor this reason, we employ the corresponding single-particle densities $\\rho^{(1)}_{F}(x)$ and $\\rho^{(1)}_{B}(x)$ with respect to $g_{BF}$ [Fig.~\\ref{spd_g}] \nfor both the cases of two non-interacting $g_{BB}=0$ [Figs.~\\ref{spd_g}($a_1$), ($b_1$)] and two repulsively interacting \nwith $g_{BB}=1$ [Figs. \\ref{spd_g} ($a_2$), ($b_2$)] bosonic impurities. \nOverall, we observe that the behavior of both the impurities and the medium depends strongly on the value of $g_{BF}$. \nAlso $\\rho^{(1)}_{F}(x)$ exhibits six shallow local density maxima [Figs.~\\ref{spd_g}($c$)-($h$)] almost irrespectively of $g_{BF}$, which \nindicates the presence of six fermions~\\cite{kwasniok2020correlated}, see \nalso the remark in \\footnote{Note that for increasing attraction, i.e. $g_{BF}<-2.5$, a major portion of $\\rho^{(1)}_{F}(x)$ resides around \n$x=0$ and its two central local maxima come very close to each other and eventually merge for very strong attractions.}. \nInterestingly, the shape of $\\rho^{(1)}_{F}(x)$ for fixed $g_{BF}$ remains almost unchanged between the $g_{BB}=0$ and the $g_{BB}=1$ cases, \nsee Figs.~\\ref{spd_g}($a_1$) and ($a_2$) as well as Figs.~\\ref{spd_g}($c$)-($h$). \nOn the other hand, $\\rho^{(1)}_{B}(x)$ at a certain value of $g_{BF}$ is affected by the direct impurity-impurity interactions since \nfor $g_{BB}=1$ it becomes slightly broader than for $g_{BB}=0$ especially when $-1.5<g_{BF}<1.5$, see Figs.~\\ref{spd_g}($d$)-($g$). \nAs we shall argue below, this difference is caused by the presence of attractive induced impurity-impurity interactions mediated by the Fermi sea and become more \npronounced when $g_{BB}=0$. \nA similar effect has also been discussed in the context of two bosonic impurities in a BEC bath~\\cite{mistakidis2020many,mistakidis2020pump}. \nThe impact of $g_{BB}$ on the behavior of the impurities and the interplay between direct and  attractive impurity-impurity induced interactions \nwill be discussed in detail in Sec.~\\ref{corel_ground}.  \n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{Fig1.eps}\n\\caption{Ground state single-particle density of ($a_1$), ($a_2$) the six fermions and ($b_1$),($b_2$) the two bosons for varying impurity-medium interaction strength $g_{BF}$ when [($a_1$), ($b_1$)] $g_{BB}=0$ and [($a_2$), ($b_2$)] $g_{BB}=1$. \n($c$)-($h$) Density profiles $\\rho^{(1)}_{B}(x)$ and $\\rho^{(1)}_{F}(x)$ at specific $g_{BF}$ values and for both non-interacting and repulsively interacting impurities (see legend). \nThe BF mixture consists of $N_B = 2$ bosonic impurities and $N_F = 6$ fermions both trapped in a harmonic oscillator with $\\omega = 0.1$.}  \n\\label{spd_g}\n\\end{figure}\n\nIn particular, for $-1.5<g_{BF}<1.5$ the Fermi sea exhibits a broad spatial distribution as identified by $\\rho^{(1)}_{F}(x)$ enclosing the \nbosonic impurities whose $\\rho^{(1)}_{B}(x)$ is located around the trap center [Figs.~\\ref{spd_g}($d$)-($f$)]. \nTherefore, for these impurity-medium interactions the bosons and the Fermi sea show a miscible behavior~\\cite{mistakidis2019correlated} independently of $g_{BB}$, \nsee Figs. \\ref{spd_g} ($a_1$), ($a_2$). Turning to attractive $g_{BF}<-1.5$ we observe a drastically altered behavior of both the impurities \nand the fermionic ensemble. \nMore specifically, the impurities are strongly localized in the vicinity of the trap center since $\\rho^{(1)}_{B}(x)$ shows a peaked structure \nat $-1<x<1$ having a sech-like shape [Fig.~\\ref{spd_g}($b_1$), ($b_2$)]. \nThis spatial localization tendency of the impurities signals the formation of an attractively bound pair~\\cite{mistakidis2020many,mistakidis2020pump} \nas a consequence of the presence of strong attractive induced impurity-impurity interactions \\cite{mistakidis2020many,mistakidis2020induced}, \nsee for more details Sec.~\\ref{corel_ground}. \nSimultaneously $\\rho^{(1)}_{F}(x)$ majorly accumulates at the same location by developing a density hump, see for instance \nFigs.~\\ref{spd_g}($a_1$), ($a_2$) and ($c$), while its background is significantly reduced when compared to smaller $g_{BF}$ values. \nThe latter behavior of $\\rho^{(1)}_{F}(x)$ indicates that the fermions of the medium lie very close to the impurities suggesting \nthat for even larger attractions a bound state between an impurity and an atom of the Fermi sea may be formed~\\cite{mistakidis2019dissipative}, a phenomenon which is not further examined herein. \nOf course, the fermions of the medium can not reside exactly at the same location due to the Pauli exclusion \nprinciple~\\cite{pethick2008bose,pitaevskii2003international}, see also the discussion in Sec.~\\ref{corel_ground}. \n\nOn the other hand, for strong repulsive impurity-bath interactions such that $g_{BF}>1.5$ the spatial configuration of the system and especially of the Fermi sea is significantly changed compared to smaller values of $g_{BF}$. \nIndeed, a local density dip builds upon $\\rho^{(1)}_{F}(x)$ around $x\\approx 0$ [Fig.~\\ref{spd_g}($g$)] which becomes more pronounced for increasing $g_{BF}$ and for $g_{BF}>2$ $\\rho^{(1)}_{F}(x)$ is segregated into two fragments residing in the left and right side with respect to $x=0$ [Figs. \\ref{spd_g} ($a_1$), ($h$)]. \nNote that each of the fragments has three local density maxima indicating that predominantly three fermions populate each of them and also reflects the fact that the first six lowest-lying single-particle eigenstates of the harmonic trap majorly contribute to the fermionic MB wavefunction. \nThe impurities density $\\rho^{(1)}_{B}(x)$ lies in between the two fragments of $\\rho^{(1)}_{F}(x)$ and therefore an impurity-medium phase-separation process takes place~\\cite{mistakidis2018correlation}, see e.g. Figs. \\ref{spd_g} ($a_1$), ($a_2$) and ($h$). \nThis procedure is identified by the small spatial overlap among the components~\\cite{mistakidis2018correlation} which becomes suppressed for increasing $g_{BF}$. \nWe remark that the phase-separation region is shifted to larger $g_{BF}$ values when $g_{BB}$ is finite, compare in particular Figs. \\ref{spd_g} ($a_1$) and ($a_2$). \nIndeed, phase-separation occurs when the interspecies interaction energy overcomes the intraspecies one and thus a larger $g_{BF}$ is required for \nincreasing $g_{BB}$~\\cite{mistakidis2018correlation} in order to accomplish this process. \nIt is also worth mentioning at this point that a system of two fermionic impurities immersed in a bosonic bath exhibits a similar phase-separation behavior at repulsive $g_{BF}$ but in this case the impurities reside at the edges of \nthe bosonic medium~\\cite{mistakidis2019correlated}. \n\nThe above-described phase-separation process as well as the localization tendency of the components taking place at large repulsive and attractive impurity-medium interactions respectively can be intuitively understood in terms of an effective potential approach~\\cite{mistakidis2020many,mistakidis2019quench,kiehn2019spontaneous}. \nFor this picture one can consider an effective potential for the impurities [Fermi sea] constructed by superimposing the single-particle density of the Fermi sea [impurities] to the external harmonic trap, namely $V_{\\rm eff}^{B}(x) = \\frac{1}{2}m \\omega^2 x^2 + g_{BF}\\rho^{(1)}_{F}(x)$ [$V_{\\rm eff}^{F}(x) = \\frac{1}{2}m \\omega^2 x^2 + g_{BF}\\rho^{(1)}_{B}(x)$]. \nReferring to the impurities subsystem at strong repulsive $g_{BF}$ their effective potential $V_{\\rm eff}^{B}(x)$ corresponds to a deformed harmonic trap due to $\\rho^{(1)}_{F}(x)$, see e.g. Fig. \\ref{spd_g} ($h$). \nIn this sense the impurities reside around the trap center possessing a Gaussian-like spatial distribution [Fig.~\\ref{spd_g}($h$)]. \nOn the other hand, for $g_{BF}>0$ the corresponding $V_{\\rm eff}^{F}(x)$ has a double-well like structure where the role of the potential barrier at $x=0$ is played by $\\rho^{(1)}_{B}(x)$. \nIn turn, this $V_{\\rm eff}^{F}(x)$ enforces the splitting of $\\rho^{(1)}_{F}(x)$ into two fragments, see e.g. Fig.~\\ref{spd_g}($h$). \nNote also here that for $g_{BB}=1$ the maximum of $\\rho^{(1)}_{B}(x)$ is smaller compared to the $g_{BB}=0$ case [Fig.~\\ref{spd_g}($g$)]. \nThis gives rise to a shallower double-well effective potential for fixed $g_{BF}$ and thus the barrier height that allows for phase-separation is achieved for larger values of $g_{BF}$ when $g_{BB}$ is finite. \nA similar argumentation can also be applied for attractive $g_{BF}$ where, for instance, the aforementioned localization tendency of $\\rho^{(1)}_{B}(x)$ is essentially determined by the hump structure building upon $\\rho^{(1)}_{F}(x)$ [Fig. \\ref{spd_g} ($c$)] and vice versa due to back-action. \nFor more details on the range of applicability of this effective potential picture we refer the interested \nreader to Refs.~\\cite{mistakidis2020many,mistakidis2019dissipative,mistakidis2019quench,kiehn2019spontaneous}. \n\n\n\n\\subsection{Impurity-impurity induced interactions}\\label{corel_ground}\n\nThe impurities being immersed in the Fermi sea are dressed by its excitations forming quasiparticles, herein Fermi \npolarons~\\cite{schmidt2018universal,massignan2014polarons,mistakidis2019repulsive}. \nAn intriguing property of the generated quasiparticles is the emergence of attractive induced interactions among them mediated by their host~\\cite{dehkharghani2018coalescence,mistakidis2020many,mistakidis2019repulsive} and that they can possibly form a bound pair for strong impurity-medium attractions~\\cite{schmidt2018universal,massignan2014polarons}. \nTo identify such quasiparticle related mechanisms in the ground state of the BF mixture we subsequently inspect the relative \ndistance $\\mathcal{D}_{\\rm rel}$ [Eq.~(\\ref{7})] and the spatially resolved two-body reduced density matrix $\\rho^{(2)}_{BB}(x_{1}, x_2)$ of \nthe bosonic impurities~\\cite{mistakidis2020induced} for different impurity-medium interaction strengths, see \nFigs.~\\ref{Relative_dis}, \\ref{2bd_g0} and \\ref{2bd_g1}. \nRecall that $\\rho^{(2)}_{BB}(x_1,x_2)$ measures the probability of finding a boson at position $x_1$ while the second one is located at $x_2$. \nImportantly, the combination of the behavior of $\\mathcal{D}_{\\rm rel}$ and $\\rho^{(2)}_{BB}(x_{1}, x_2)$ enables us to infer the presence \nand strength of the attractive induced interactions as well as the spatial configuration of the impurities~\\cite{mistakidis2020many}.   \nBelow, we discuss the cases of both non-interacting ($g_{BB}=0$) and repulsively interacting ($g_{BB}=1.0$) impurities as well as the \neffect of a mass-imbalance between the impurities and their bath. \n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{Fig2.eps}\n\\caption{Relative distance $\\mathcal{D_{\\rm rel}}$ between the two bosonic impurities in the ground state of the BF mixture with \nrespect to the impurity-medium interaction strength $g_{BF}$. The relative distance is presented for the cases of two \nnon-interacting ($g_{BB} = 0$), and  interacting ($g_{BB} = 1$) impurities in a mass-balanced as well as a mass-imbalanced system (see legend). \nThe medium consists of $N_F = 6$ fermions, while the BF mixture is confined in a harmonic trap with frequency $\\omega = 0.1$.}  \n\\label{Relative_dis}\n\\end{figure}\n\nThe corresponding relative distance $\\mathcal{D}_{\\rm rel}$ between the impurities and their two-body density $\\rho^{(2)}_{BB}(x_{1}, x_2)$ \nfor a mass-balanced BF system containing non-interacting impurities are presented in Fig.~\\ref{Relative_dis} and \nFigs.~\\ref{2bd_g0}($a_1$)-($a_5$) respectively as a function of the impurity-medium interaction strength $g_{BF}$ ranging \nfrom attractive to repulsive values. \nIt becomes apparent that $\\mathcal{D}_{\\rm rel}$ gradually decreases, when compared to its value for $g_{BF}=0$, as $\\abs{g_{BF}}$ is \nincreased towards the attractive or the repulsive interaction regime. This overall decreasing behavior of $\\mathcal{D}_{\\rm rel}$ for \nlarger $\\abs{g_{BF}}$ indicates that for finite attractive or repulsive impurity-medium interactions the impurities move close to each other as compared to the $g_{BF}=0$ scenario. \nThe latter tendency suggests the emergence of attractive impurity-impurity induced interactions~\\cite{huber2019medium,mistakidis2020many}. \nInterestingly, $\\mathcal{D}_{\\rm rel}$ tends to approach a constant value which is different for strong repulsions ($g_{BF}>3$) and \nattractions ($g_{BF}<-3$). \nIndeed, the saturation value of $\\mathcal{D}_{\\rm rel}$ for strong repulsions is somewhat larger when compared to the corresponding \nvalue for strong attractions. \nThis means that for attractive $g_{BF}$ the impurities are substantially closer with respect to one another than in the repulsive case. \nConcluding, $\\mathcal{D}_{\\rm rel}$ signals the presence of induced impurity-impurity interactions, which are manifested to be attractive \nin general, irrespectively of the sign of the impurity-medium coupling~\\cite{huber2019medium,mistakidis2020many}. \n\n\nTo confirm the existence of attractive impurity-impurity induced interactions when $g_{BB}=0$ we next rely on the impurities two \nparticle density $\\rho^{(2)}_{BB}(x_1, x_2)$, see Figs.~\\ref{2bd_g0} ($a_1$)-($a_5$). \nThis quantity allows us to explicitly identify the spatial distribution of impurities. \nAs it can be seen, irrespectively of the value of $g_{BF}$ the two non-interacting bosons prefer to reside together close to the \ntrap center since $\\rho^{(2)}_{BB}(x_1, x_2)$ shows a maximum value in the vicinity of $x_1=0$, $x_2=0$, see Figs.~\\ref{2bd_g0}($a_1$)-($a_5$). \nIn particular, for $g_{BF} = 0$ [Fig.~\\ref{2bd_g0}($a_3$)] $\\rho^{(2)}_{BB}(x_1, x_2)$ has a circularly symmetric shape in the ($x_1,x_2$)-plane while showing a peak around $x_1=x_2=0$. \nThis can be understood by the fact that in the absence of any correlation with the majority species there is no induced interaction among the bosons. \nHence, the probability of finding the two bosons together at $x_1=x_2$ or one at $x_1$ and the other at $x_2=-x_1$ is the same and becomes maximal at the trap minimum i.e. at $x_1=x_2=0$. \nHowever, for a finite $g_{BF}$ the shape of $\\rho^{(2)}_{BB}(x_1, x_2)$ is significantly altered when compared to the $g_{BF}=0$ case \nsince predominantly the diagonal $\\rho^{(2)}_{BB}(x_1,x_2=x_1)$ is populated. \nIn fact, $\\rho^{(2)}_{BB}(x_1, x_2)$ becomes more elongated along the diagonal ($x_1 = x_2$) with increasing $\\abs{g_{BF}}$, while it \nshrinks across its anti-diagonal ($x_2 = -x_1$), see Figs.~\\ref{2bd_g0}($a_4$)-($a_5$) and Figs.~\\ref{2bd_g0}($a_1$)-($a_2$). \nThis means that the probability of detecting the two bosons at two different positions is substantially smaller than that of being \nclose together. \nTherefore, an effective attractive interaction between the impurities is established and occurs for both attractive and repulsive \nimpurity-medium interactions. \nImportantly, within the attractive $g_{BF}$ regime the shrinking of the anti-diagonal of $\\rho^{(2)}_{BB}(x_1, x_2)$ is much more \npronounced than on the repulsive $g_{BF}$ side, compare in particular Figs.~\\ref{2bd_g0}($a_1$)-($a_2$) with Figs.~\\ref{2bd_g0}($a_4$)-($a_5$). \nThe latter observation supports our previous statement in terms of $\\mathcal{D}_{\\rm rel}$ of a much stronger effective attraction \nbetween the impurities for negative than positive $g_{BF}$, a result that also holds for Bose polarons as it has been \ndemonstrated in Ref.~\\cite{mistakidis2020many}. \nMoreover, the pronounced elongation along the diagonal accompanied by the strong suppression of the anti-diagonal of $\\rho^{(2)}_{BB}(x_1, x_2)$ e.g. for $g_{BF}=-3$ is indicative of a bound state being formed between the impurities known as a bipolaron state~\\cite{mistakidis2020many,camacho2018bipolarons,schmidt2018universal}.     \n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{Fig3.eps}\n\\caption{Reduced two-body ($a_1$)-($a_5$) boson-boson $\\rho^{(2)}_{BB}(x_1, x_2)$, and ($b_1$)-($b_5$) fermion-fermion $\\rho^{(2)}_{FF}(x_1, x_2)$ \ndensity in the ground state of the BF mixture for selective impurity-medium interaction strengths (see legend).  \nThe system contains $N_B =2$ non-interacting ($g_{BB}=0$) bosonic impurities and $N_F = 6$ fermions. \nIt is further confined in a harmonic trap of $\\omega = 0.1$. }  \n\\label{2bd_g0}\n\\end{figure}\n\nNext, we turn our attention to repulsively interacting bosonic impurities with $g_{BB}=1$ aiming to investigate the competition between attractive induced interactions and direct $s$-wave ones. \nTo this end, we measure the impurities relative distance [Fig.~\\ref{Relative_dis}] and their two-particle density [Fig.~\\ref{2bd_g1}($a_1$)-($a_5$)] for distinct values of $g_{BF}$. \nAs expected, in the absence of direct impurity-impurity interactions i.e. $g_{BB}=0$ the impurities distance $\\mathcal{D}_{\\rm rel}$ is in general smaller than the corresponding for two repulsively interacting ones with $g_{BB} = 1$. \nThis difference becomes maximal for zero impurity-medium interactions, namely $g_{BF}=0$. \nIndeed, $\\mathcal{D}_{\\rm rel}$ decreases for a larger positive or negative $g_{BF}$ tending to approach a constant value which is smaller for attractive $g_{BF}$ interactions. \nConsequently, also the difference $\\mathcal{D}_{\\rm rel}(g_{BB} = 1)-\\mathcal{D}_{\\rm rel}(g_{BB}=0)$ gradually decreases and becomes constant for increasing $\\abs{g_{BF}}$. \nA direct comparison between $\\mathcal{D}_{\\rm rel}(g_{BB}=1)$ and $\\mathcal{D}_{\\rm rel}(g_{BB}=0)$ reveals that the \ndistance saturates at relatively larger (smaller) positive (negative) $g_{BF}$ values when $g_{BB}=1$. \nFor instance, the decreasing rate of $\\mathcal{D}_{\\rm rel}$ in the attractive $g_{BF}$ regime is much larger in the $g_{BB}=1$ scenario before showcasing a saturation tendency around $g_{BF}=-4$ towards $\\mathcal{D}_{\\rm rel} \\approx 0.5$. \nThe above-described overall behavior of $\\mathcal{D}_{\\rm rel}$ for varying $g_{BF}$ suggests the occurence of attractive induced interactions for a finite $g_{BF}$ despite the existence of direct $s$-wave ones. \nNevertheless, as we shall explicate in the following the direct interactions dominate the induced ones at least for repulsive impurity-medium \ncouplings, a result which reveals that the strength of induced interactions is larger in the negative $g_{BF}$ regime.   \n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{Fig4.eps}\n\\caption{Reduced two-body ($a_1$)-($a_5$) impurity-impurity $\\rho^{(2)}_{BB}(x_1, x_2)$, and ($b_1$)-($b_5$) fermion-fermion $\\rho^{(2)}_{FF}(x_1,x_2)$ \ndensity in the ground state of the BF mixture for different values of the boson-medium coupling constant $g_{BF}$ (see legend). \nThe system consists of $N_B =2$ interacting with $g_{BB}=1$ bosonic impurities and $N_F = 6$ fermions while it is trapped in a harmonic oscillator with frequency $\\omega = 0.1$.}  \n\\label{2bd_g1}\n\\end{figure}\n\nIndeed by inspecting the impurities two-body density $\\rho^{(2)}_{BB}(x_1,x_2)$, illustrated in Figs.~\\ref{2bd_g1}($a_1$)-($a_5$), the following conclusions can be immediately drawn. \nThe circularly symmetric pattern of $\\rho^{(2)}_{BB}(x_1,x_2)$ occurring for $g_{BB}=0$ when $g_{BF}=0$ [Fig.~\\ref{2bd_g0}($a_3$)] is completely modified for repulsively interacting impurities [Fig.~\\ref{2bd_g1}($a_3$)]. \nThis modification favors a pattern whose diagonal is depleted, giving rise to a correlation hole~\\cite{erdmann2019phase,kwasniok2020correlated}, whilst the anti-diagonal develops two symmetric lobes with respect to $x_1=x_2$ and it is predominantly populated. \nThis is an explicit imprint of the direct $s$-wave interaction among the impurities and means that the probability of finding two bosons exactly at the same position is vanishingly \nsmall in contrast to the situation where each boson resides on a separate side in terms of the trap center. \nSwitching on $\\abs{g_{BF}}$ introduces deformations in the shape of $\\rho^{(2)}_{BB}(x_1, x_2)$ and in particular in the position of its anti-diagonal lobes which suggests that the induced interactions set in. \nReferring to repulsive impurity-medium interactions [Figs.~\\ref{2bd_g1}($a_4$-($a_5$)] it is apparent that for increasing $g_{BF}$ the anti-diagonal lobes of $\\rho^{(2)}_{BB}(x_1, x_2)$ approach the diagonal. \nTherefore, the two bosons get closer due to the presence of their attractive induced interactions mediated by the fermionic environment. \nNotice, however, that the two lobe structure of $\\rho^{(2)}_{BB}(x_1, x_2)$ is maintained also at $g_{BF}=3$ [Fig.~\\ref{2bd_g1}($a_5$)], indicating that the $s$-wave interactions dominate the induced ones. Turning to the attractive $g_{BF}$ regime we observe that for weak $g_{BF}$ values the anti-diagonal of $\\rho^{(2)}_{BB}(x_1, x_2)$ shrinks and thus the bosons come closer when compared to the $g_{BF}=0$ case due to the existence of attractive induced interactions. \nImportantly, this behavior of $\\rho^{(2)}_{BB}(x_1, x_2)$ is drastically changed for large attractive impurity-medium interactions. \nMore precisely, the two-lobed anti-diagonal structure related to the dominant repulsive contact interaction is changed into a circularly symmetric pattern, see e.g. $\\rho^{(2)}_{BB}(x_1,x_2)$ at $g_{BF}=-3$ depicted in Fig.~\\ref{2bd_g1}($a_1$). \nRecall that the appearance of such a circularly symmetric structure in $\\rho^{(2)}_{BB}(x_1, x_2)$ occurs in the case of zero effective interactions between the two bosons when $g_{BB}=0$ and $g_{BF}=0$ [Fig.~\\ref{2bd_g0}($a_3$)]. \nThis observation suggests that the attractive induced interactions nullify the direct repulsive contact ones for large attractive impurity-medium couplings, a phenomenon that is absent in the repulsive $g_{BF}$ regime. \nSummarizing, attractive impurity-medium couplings lead to stronger induced interactions than repulsive ones. \n\nSubsequently, we study the impact of the impurities mass on the strength of the induced interactions by  invoking as an appropriate measure the impurities relative distance [Fig. \\ref{Relative_dis}]. \nFor this investigation we consider a harmonically trapped mass-imbalanced BF mixture consisting of a $^{40}$K fermionic environment and two $^{87}$Rb bosonic impurities~\\cite{Fratini2012}. \nEvidently, the overall phenomenology of $\\mathcal{D}_{\\rm rel}$ for varying $g_{BF}$ is similar to the mass-balanced scenario for both the $g_{BB}=0$ and the $g_{BB}=1$ cases. \nMoreover, $\\mathcal{D}_{\\rm rel}$ is always reduced when compared to the mass-balanced system, thus suggesting that heavier impurities prefer to stay closer to each other than lighter ones~\\cite{kwasniok2020correlated,mistakidis2020many}. Accordingly, we can deduce that an \nincreasing impurity mass allows for stronger attractive induced interactions. \n\n\n\\subsection{Two-body correlations of the fermionic medium}\\label{corel_ground_bath} \n\nHaving explicated the existence of attractive induced impurity-impurity correlations we then analyze \nthe two-particle distributions of the fermionic environment for different impurity-medium interactions. \nOur main objective here is to expose the back-action of the impurities onto their host~\\cite{mukherjee2020pulse,mistakidis2019dissipative}. \nRegarding the system with two non-interacting impurities ($g_{BB}=0$) $\\rho^{(2)}_{FF}(x_1,x_2)$ is presented in Figs. \\ref{2bd_g0} ($b_1$)-($b_5$) for specific values of $g_{BF}$. \nA depleted diagonal is observed irrespectively of $g_{BF}$ due to the Pauli exclusion principle, namely two fermions cannot occupy the same spatial region. \nAt $g_{BF}=0$ two fermions can be found at any two distinct positions within the interval $[-10, 10]$, see Fig.~\\ref{2bd_g0}($b_3$), possessing a slightly larger probability to reside close to the trap center either on the same side or at different ones with respect to $x=0$. \nInterestingly, even the presence of a very small number of bosonic impurities is able to significantly alter the properties of the Fermi sea if $g_{BF}\\neq 0$. \nFor repulsive $g_{BF}$, the fermions exhibit a tendency to stay away from the trap center, \ne.g. $\\rho^{(2)}_{FF}(x_1=5,x_2=-5)\\approx 0.14$ at $g_{BF} = 1$ in Fig. \\ref{2bd_g0} ($b_4$). \nThis behavior is manifested by the appearance of relatively low density stripes along the lines $x_1=0$ and $x_2=0$ e.g. for $g_{BF}=1$ [Fig.~\\ref{2bd_g0}($a_4$)] which are transformed into completely depleted density regions e.g. for $g_{BF} = 3$ [Fig. \\ref{2bd_g0}($a_5$)]. \nTurning to the attractive $g_{BF}$ regime [Figs.~\\ref{2bd_g0}($b_1$)-($b_2$)], the distribution of $\\rho^{(2)}_{FF}(x_1,x_2)$ is changed significantly. \nIndeed, the probability of finding two fermions at different positions in the vicinity of the trap center is the dominant contribution to $\\rho^{(2)}_{FF}(x_1,x_2)$ especially for larger attractions. \nFor instance, at $g_{BF} = -1$, the two particle density shown in Fig.~\\ref{2bd_g0}($b_2$) is higher close to the trap \ncenter [e.g. $\\rho^{(2)}_{FF}(x_1=1,x_2=-1)\\approx 0.16$] compared to the edges [e.g. $\\rho^{(2)}_{FF}(x_1=5,x_2=-14)\\approx 0.09$]. \nAlso, for $g_{BF} = -3$, the spatial region apart from the one close to the trap center is almost completely depleted [see Fig.~\\ref{2bd_g0}($b_1$)], as identified by the relevant cross-like pattern building upon $\\rho^{(2)}_ {FF}(x_1,x_2)$. \n\nComparing now $\\rho^{(2)}_{FF}(x_1, x_2)$ between the cases of $g_{BB}=1$ [Figs.~\\ref{2bd_g1}($b_1$)-($b_5$)] and $g_{BB}=0$ [Figs.~\\ref{2bd_g0}($b_1$)-($b_5$)] we can easily deduce that their shapes at specific $g_{BF}$ values are to a great extent similar. \nA slight difference occurs for moderate repulsive interactions e.g. $g_{BF}=1$ where the two-body density stripes imprinted along the lines $x_1=0$ and $x_2=0$ for $g_{BB}=0$ [Fig.~\\ref{2bd_g0}($b_4$)] \nare not noticeable for $g_{BB}=1$ [Fig.~\\ref{2bd_g1}($b_4$)]. \nAlso, for attractive impurity-medium interactions $g_{BF}<0$ the regions away from the trap center are relatively \nstronger populated for $g_{BB}=1$, compare Figs. \\ref{2bd_g1} ($b_1$)-($b_2$) with Figs. \\ref{2bd_g0} ($b_1$)-($b_2$). \nAs a case example, for $g_{BF}=-3$ it holds that $\\rho^{(2)}_{FF}(x_1=4,x_2=4) \\approx 0.1$ when $g_{BB} = 0$ \nwhile $\\rho^{(2)}_{FF}(x_1=4,x_2=-4)\\approx 0.12$ for $g_{BB} = 1$, see Fig.~\\ref{2bd_g0}($b_1$) and \nFig. ~\\ref{2bd_g1}($b_1$) respectively.  \n\n \n\\section{Quench Dynamics}\\label{quench_dynamics}\n\nUp to now we have discussed the ground state properties of the harmonically trapped particle imbalanced BF mixture \nwith $N_F=6$ and $N_B=2$ for different impurity-medium interaction strengths ranging from attractive to repulsive values. \nImportantly, we have identified the presence of attractive induced interactions for the non-interacting impurities and \nanalyzed the competition between the direct $s$-wave repulsive interactions with the induced ones. \nAlso, in all cases we have quantified the back-action of the impurities to their fermionic environment. \n\nBelow, we explore the corresponding non-equilibrium dynamics of the impurities and the Fermi sea. \nThe mixture is prepared in its ground-state configuration, as already discussed in Sec.~\\ref{Ground state}, with zero impurity-medium coupling strength. \nThe dynamics is triggered by applying a quench of this coupling towards either the repulsive [Sec.~\\ref{repulsive_quench}] or the \nattractive [Sec.~\\ref{attractive quench}] regime of interactions~\\cite{volosniev2015real,mistakidis2019effective}. \nOur main objective is to inspect the dynamical emergence of induced impurity-impurity correlations and the pattern formation of the fermionic environment as a result of the impurities motion. \nIn the subsequent analysis we first study the dynamics of two non-interacting impurities and then contrast our findings to the case of two repulsively interacting ones. \n\n\n\\subsection{Quench to repulsive interactions}\\label{repulsive_quench} \n\nWe focus first on the correlated dynamics of the BF mixture induced by a quench from a vanishing to repulsive impurity-medium interactions. \nThe emergent dynamics is firstly analyzed by employing the corresponding single-particle density evolution of the participating \ncomponents [Sec.~\\ref{density_evol_repul}] and then by inspecting their two-body density matrix [Sec.~\\ref{two_body_evol_repul}] in \nthe course of the evolution. \nThese observables enable us to gain an overview of the dynamical evolution and importantly shed light on \nthe existence of impurity-impurity and bath correlations respectively. \n\n\n\\subsubsection{Single-particle density evolution}\\label{density_evol_repul}\n\nTo gain an overview of the spatially resolved quench-induced dynamics of the BF mixture we show the corresponding single-particle density evolution of the impurities $\\rho^{(1)}_{B}(x;t)$ and the Fermi sea $\\rho^{(1)}_{F}(x;t)$ in Fig.~\\ref{spd_dr} for different impurity-medium and impurity-impurity interaction strengths. \nNaturally, we commence our discussion on the system containing non-interacting impurities which provides the most clear signatures of induced correlations. \nReferring to weak post-quench interactions namely $g_{BF}=0.8$ the fermionic environment performs an overall \nbreathing motion~\\cite{huang2019breathing,boudjemaa2020breathing} manifested as a small amplitude periodic expansion and contraction of its cloud, see Fig.~\\ref{spd_dr}($a_1$). \nThe frequency of this global breathing mode is $\\omega_F^{br}\\approx 0.193\\approx2\\omega$ which is indeed in accordance to the corresponding theoretical predictions~\\cite{bauch2009quantum,abraham2012quantum}. \nMoreover $\\rho^{(1)}_{F}(x;t)$ exhibits at each time-instant of the evolution six in total shallow local maxima, namely three on the left and other three on the right side with respect to $x=0$, while a shallow density dip occurs around the trap center $x=0$. \nThese local maxima are indicative of the six fermions present in the system and also the fact that majorly the first six single-particle eigenstates of the trap participate in the dynamics. \nOn the other hand, the shallow dip of $\\rho^{(1)}_{F}(x=0;t)$ is caused by the presence of the impurities at the same location. \nThe impurities density $\\rho^{(1)}_{B}(x;t)$ undergoes a very weak amplitude breathing dynamics [see Fig.~\\ref{spd_dr}($a_3$)] characterized by a predominant \nfrequency $\\omega_{B}^{br}\\approx 0.24$. \nNotice here that $\\omega_B^{br}$ is slightly larger than $\\omega_F^{br}$ since the impurities experience an effective potential, created by the external trap and the density of the Fermi sea, which possesses a larger than the trap frequency. \nMoreover, $\\rho^{(1)}_{B}(x;t)$ completely overlaps with $\\rho^{(1)}_{F}(x;t)$ throughout the time-evolution, thus indicating the miscible character of the dynamics. \nRecall that the two components are also miscible in the ground state of the system for $g_{BF}=0.8$ as discussed in Sec.~\\ref{ground_state_density} and demonstrated in Figs. \\ref{spd_g} ($a_1$),($b_1$). \n\nTurning to repulsively interacting impurities with $g_{BB}=1$ for the same quench amplitude i.e. from $g_{BF} = 0$ to $g_{BF} = 0.8$ we observe that a qualitatively similar to the \nabove-described dynamics takes place when $g_{BB}=1$ for both the fermionic environment [Fig.~\\ref{spd_dr}($a_5$)] and the \nbosons [Fig.~\\ref{spd_dr}($a_7$)]. \nA notable difference is that $\\rho^{(1)}_{B}(x;t)$ is broader in the $g_{BB}=1$ case due to the presence of the direct $s$-wave interaction, compare in particular Fig.~\\ref{spd_dr}($a_3$) with Fig.~\\ref{spd_dr}($a_7$). \nNote that this broadening of $\\rho^{(1)}_{B}(x;t)$ for finite $g_{BB}$ occurs also in the ground state of the system, see Figs.~\\ref{spd_g}($b_1$), ($b_2$). \nAs expected, also the breathing amplitude of the impurities is larger when $g_{BB}=1$ while the frequency of this motion remains almost \nthe same ($\\omega_{B}^{br}\\approx 0.22$) with the $g_{BB}=0$ case. \nThis small deviation in the value of $\\omega_{B}^{br}$ is attributed to interaction effects~\\cite{schmitz2013quantum,kiehn2019spontaneous}. Consequently, due to the broader $\\rho^{(1)}_{B}(x;t)$ when $g_{BB}=1$ the central dip in $\\rho^{(1)}_{F}(x=0;t)$ occurring for $g_{BB}=0$ [Fig.~\\ref{spd_dr}($a_1$)] becomes very shallow and almost disappears for $g_{BB}=1$ [Fig.~\\ref{spd_dr}($a_3$)]. \nOtherwise, the inclusion of direct $s$-wave impurity-impurity interactions does not alter the characteristics of the system's dynamics at least on the single-particle level. \n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{Fig5.eps}\n\\caption{One-body density evolution of [($a_1$), ($a_2$), ($a_5$), ($a_6$)] the Fermi sea $\\rho^{(1)}_{F}(x,t)$ and [($a_3$), ($a_4$), ($a_7$), ($a_8$)] the bosonic impurities $\\rho^{(1)}_{F}(x,t)$ upon considering an impurity-medium interaction quench from $g_{BF}=0$ to a finite repulsive value (see legend). \nThe impurities are ($a_1$)-($a_4$) either free i.e. $g_{BB} = 0$ or ($a_5$)-($a_8$)  repulsively interacting with $g_{BB} = 1$. \nThe system is confined in a harmonic trap with $\\omega=1$ and comprises of $N_B = 2$ bosons immersed in Fermi sea of $N_F = 6$ fermions. \nIt is initialized in its ground state with $g_{BF}=0$ and either $g_{BB}=0$ or $g_{BB}=1$.}  \n\\label{spd_dr}\n\\end{figure} \n\nIncreasing the post-quench interaction strength e.g. to $g_{BF}=2.5$ gives rise to a much more intricate dynamics for both the non-interacting impurities [Fig.~\\ref{spd_dr}($a_4$)] and the fermionic medium [Fig.~\\ref{spd_dr}($a_2$)] when compared to the $g_{BF}=0.8$ quench amplitude. \nWe remark that such a difference is already expected from the ground state properties of the system since for $g_{BF}=0.8$ the components are spatially overlapping (miscible) and become immiscible for $g_{BF}=2.5$, see also Figs. \\ref{spd_g} ($a_1$), ($b_1$). \nIn particular, the cloud of the Fermi sea exhibits a breathing oscillation with almost the same frequency $\\omega_F^{br}\\approx0.195$ \nas for the quench to $g_{BF}=0.8$. \nHowever, the amplitude of the contraction and expansion dynamics of $\\rho^{(1)}_{F}(x;t)$ [Fig.~\\ref{spd_dr}($a_2$)] is larger when compared to the smaller $g_{BF}=0.8$ [Fig.~\\ref{spd_dr}($a_1$)] leading to a comparatively more excited medium in the former case. \nAccordingly, $\\rho^{(1)}_{F}(x;t)$ appears to be in general wider for $g_{BF}=2.5$, a result that can again be traced back to the ground state density of the Fermi sea [Fig.~\\ref{spd_g}($a_1$)]. \nMoreover, the local density humps building upon $\\rho^{(1)}_{F}(x;t)$ [Fig.~\\ref{spd_dr}($a_2$)] are found to be shallower (deeper) during the expansion (contraction) of the fermionic cloud for $g_{BF}=2.5$ than for $g_{BF}=0.8$. \nImportantly, the density dip of $\\rho^{(1)}_{F}(x;t)$  around the trap center is substantially deeper when $g_{BF}=2.5$. \nThis is a direct consequence of the emergent phase-separation being anticipated already from the ground state of the system for such strongly repulsive impurity-medium interactions, see also Figs.~\\ref{spd_g} ($a_1$), ($b_1$). \n\nOf course, most of the above-described features of $\\rho^{(1)}_{F}(x;t)$ are intimately connected with the corresponding behavior of the single-particle density of the bosons $\\rho^{(1)}_{B}(x;t)$ [Fig.~\\ref{spd_dr}($a_4$)] since the two components are inevitably interdependent due to their mutual finite coupling $g_{BF}$. \nSpecifically, the impurities density $\\rho^{(1)}_{B}(x;t)$ shows a relatively larger localization tendency [Fig.~\\ref{spd_dr}($a_4$)] than for $g_{BF}=0.8$ [Fig.~\\ref{spd_dr}($a_3$)] which is expected due to the aforementioned phase-separated behavior among the components. \nMoreover, $\\rho^{(1)}_{B}(x;t)$ exhibits a weaker amplitude and \nlarger frequency $\\omega_{B}^{br}\\approx 0.36$ breathing motion for $g_{BF}=2.5$ than for $g_{BF}=0.8$. \nThe alteration of the impurities breathing frequency for $g_{BF}=2.5$ can in turn be explained within an effective potential picture. \nIndeed, as already argued in the ground state of the system the impurities can be viewed as trapped in the potential formed by \nthe harmonic trap with a superimposed density of their Fermi sea. \nSince $\\rho^{(1)}_{F}(x;t)$ is wider for increasing $g_{BF}$ also the impurities effective trapping frequency being r\nelated to the breathing one is larger. \n\nThe dynamics of interacting impurities with $g_{BB}=1$ following a quench to $g_{BF}=2.5$ as captured by  $\\rho^{(1)}_{F}(x;t)$ \n[Fig.~\\ref{spd_dr}($a_6$)] and $\\rho^{(1)}_{B}(x;t)$ [Fig.~\\ref{spd_dr}($a_8$)] is more involved than the $g_{BB}=0$ case, especially for long evolution times $t>40$. \nEvidently, the impurities exhibit a significantly broader density distribution for $g_{BB}=1$ [Fig.~\\ref{spd_dr}($a_8$)] than \nfor $g_{BB}=0$ [Fig.~\\ref{spd_dr}($a_4$)], while performing a breathing motion of a larger amplitude and smaller \nfrequency $\\omega_{B}^{br}\\approx 0.33$ in the former case. \nFurthermore, $\\rho^{(1)}_{B}(x;t)$ initially ($t=0$) having a Gaussian profile deforms already within the initial stages of the dynamics ($t>5$) \nby developing three shallow humps being more pronounced during expansion and coming very close at the contraction points [Fig.~\\ref{spd_dr}($a_8$)]. \nThis behavior of $\\rho^{(1)}_{B}(x;t)$ essentially indicates that for $t<5$ the interacting impurities dominantly occupy the lowest-lying \nsingle-particle eigenstate of their external potential while as time evolves the contribution of higher-lying eigenstates becomes significant \nand excitations are formed. \nThis statement is also supported by the population of the individual bosonic orbitals $\\phi_i^{B}(x,t)$ with $i=1,2,\\dots,12$ \n(see also the discussion following Eq.~(\\ref{1BD})) from which the first eight have a non-negligible population during the evolution (results not shown). \nThe above-mentioned differences, regarding mainly the breathing mode and the structure of $\\rho^{(1)}_{B}(x;t)$ for $g_{BB}=1$ compared to \nthe $g_{BB}=0$ case are attributed to the presence of the direct $s$-wave repulsive interaction between the impurities. \nAs expected, the features of $\\rho^{(1)}_{B}(x;t)$ are also imprinted to a certain extent in the density of the fermionic environment \n$\\rho^{(1)}_{F}(x;t)$ [Fig.~\\ref{spd_dr}($a_6$)] due to the finite $g_{BF}$. \nNotably, $\\rho^{(1)}_{F}(x;t)$ shows an arguably suppressed central dip compared to the $g_{BB}=0$ case due to \nthe broader distribution of the impurities for $g_{BB}=1$. \nThis in turn gives rise to an almost vanishing degree of phase-separation in the latter $g_{BB}=1$ case. \nOther properties, such as the amplitude and the frequency of the breathing mode remain almost the same as in the $g_{BB}=0$ scenario. \n\nConcluding this section, it is important to emphasize that the quench dynamics of non-interacting and interacting bosonic impurities \ndiffers noticeably already on the single-particle level, especially for large post-quench interaction strengths. \nThis impact of the direct $s$-wave interaction of the impurities is also imprinted in the Fermi sea leading to changes in its \npattern formation. \nAs we shall explicate below, the origin of the above-mentioned differences is the presence of impurity-impurity induced interactions. \n\n\n\\subsubsection{Dynamics of impurity-impurity correlations}\\label{two_body_evol_repul} \n\nTo track the spatially resolved dynamics of the two bosonic impurities with respect to one another we next invoke their \ntwo-particle density, $\\rho^{(2)}_{BB}(x_1,x_2);t)$ [Eq.~(\\ref{2BD})], which essentially provides the probability of measuring \nsimultaneously one particle at position $x_1$ and the other at $x_2$. \nAs a complementary measure of the impurities position we also calculate their relative distance $\\mathcal{D}_{\\rm rel}(t)$ [Eq.~(\\ref{7})] during the time-evolution. \nThis observable will allow us to identify whether the impurities interact among each other via induced correlations mediated by their host or \nthey move independently~\\cite{mistakidis2019correlated,mistakidis2020many}. \nSnapshots of $\\rho^{(2)}_{BB}(x_1, x_2;t)$ at specific time-instants of the evolution upon considering a quench from $g_{BF}=0$ to $g_{BF}=2.5$ for the cases of $g_{BB}=0$ and $g_{BB}=1$ are presented in Figs.~\\ref{2B_den_r}($a_1$)-($a_4$) and Figs.~\\ref{2B_den_r}($c_1$)-($c_4$) respectively. \nMoreover, the corresponding $\\mathcal{D}_{\\rm rel}(t)$ when $g_{BB}=0$ [Fig.~\\ref{2B_den_r}($d$)] and $g_{BB}=1$ [Fig.~\\ref{2B_den_r}($e$)] is demonstrated for different post-quench interactions providing an overview of the impurity-impurity correlations. \n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{Fig6.eps}\n\\caption{Snapshots of the two-body reduced density of ($a_1$)-($a_4$) two non-interacting bosons $\\rho^{(2)}_{BB}(x_1,x_2)$, ($b_1$)-($b_4$) two fermions of the medium $\\rho^{(2)}_{FF}(x_1,x_2)$ and ($c_1$)-($c_4$) two repulsively interacting bosons $\\rho^{(2)}_{BB}(x_1,x_2)$ at specific time-instants of the evolution (see legends). \nThe system consists of $N_F=6$ fermions and $N_B=2$ bosons while it is confined in a harmonic trap with $\\omega=0.1$. \nIt is initialized in its ground state with $g_{BF}=0$ and either $g_{BB}=0$ or $g_{BB}=1$. \nTo trigger the dynamics an interaction quench is performed from $g_{BF} = 0$ to $g_{BF} = 2.5$. \nTime-evolution of the relative distance $\\mathcal{D}_{\\rm rel}(t)$ between the two bosonic impurities with ($d$) $g_{BB} = 0$ and ($e$) $g_{BB} =1$ at distinct post-quench $g_{BF}$ values (see legend).}  \n\\label{2B_den_r}\n\\end{figure}\n\nFor non-interacting bosonic impurities, $\\rho^{(2)}_{BB}(x_1,x_2;t=0)$ has a circular shape in the ($x_1,x_2$)-plane [Fig.~\\ref{2B_den_r}($a_1$)] with a peak around $x_1,x_2\\in [-2,2]$. \nTherefore, the bosons are likely to reside in this spatial region close to the trap center. \nHowever, in the course of the dynamics this shape of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ is drastically altered exhibiting an elongated diagonal and a suppressed anti-diagonal, see Figs.~\\ref{2B_den_r}($a_2$)-($a_4$). \nNote that the anti-diagonal of the two-particle density of the impurities dictates their relative distance $\\mathcal{D}_{\\rm rel}(t)$ [Eq.~(\\ref{7})]. \nThe latter is illustrated in Fig.~\\ref{2B_den_r}($d$) for a variety of post-quench $g_{BF}$ values. \nAs it can be seen, in all cases $\\mathcal{D}_{\\rm rel}(t)$ undergoes a decaying amplitude oscillatory motion characterized by \ntwo dominantly participating frequencies which essentially correspond to the center-of-mass and relative coordinate \nbreathing modes~\\cite{schmitz2013quantum}, e.g. $\\omega_1\\approx 0.19$, $\\omega_2\\approx 0.24$ for $g_{BF}=0.8$ and \n$\\omega_1\\approx 0.19$, $\\omega_2\\approx 0.36$ when $g_{BF}=2.5$. \nIndeed, the oscillatory behavior of $\\mathcal{D}_{\\rm rel}(t)$ reflects the breathing motion of the impurities cloud already identified in \nthe dynamics of $\\rho^{(1)}_{B}(x;t)$ [Fig.~\\ref{spd_dr}($a_3$)]. \nThis is also imprinted in the modulated shape of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ [Figs.~\\ref{2B_den_r}($a_2$-($a_4$)], e.g. \nthe anti-diagonal of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ is more expanded at $t=55$ compared to $t=130$. \nAlso, the oscillation amplitude of $\\mathcal{D}_{\\rm rel}(t)$ and as a consequence of the breathing mode is enhanced \nfor a larger post-quench $g_{BF}$, see also Figs.~\\ref{spd_dr}($a_3$), ($a_7$). \nImportantly the decaying amplitude in time of $\\mathcal{D}_{\\rm rel}(t)$, and thus the elongated shape \nof $\\rho^{(2)}_{BB}(x_1,x_2;t)$ across its diagonal, implies that the impurities tend to approach \neach other during the dynamics and since they are non-interacting ($g_{BB}=0$) they experience an effective \nattraction mediated by the fermionic environment. \n\nProceeding we examine the role played by the direct $s$-wave repulsive contact interaction between the impurities and its competition with the induced interactions on the quench dynamics. \nDue to the finite impurity-impurity repulsion, herein $g_{BB}=1$, the bosons initially ($t=0$) reside one in the left ($x<0$) and the other in the right ($x>0$) side of the trap, see the pronounced anti-diagonal distribution of $\\rho^{(2)}_{BB}(x_1,x_2=-x_1;t=0)$ in Fig.~\\ref{2B_den_r}($c_1$). \nIn contrast, after the quench ($t>0$) three distinct segments develop in $\\rho^{(2)}_{BB}(x_1,x_2;t)$ [Figs.~\\ref{2B_den_r}($c_2$)-($c_4$)]. \nNamely the impurities are either very close to each other around the trap center [see the diagonal of $\\rho^{(2)}_{BB}(x_1,x_2;t)$] or they remain spatially separated with one of them located in the left and the other in the right side of the trap with respect to $x=0$ [see the anti-diagonal of $\\rho^{(2)}_{BB}(x_1,x_2;t)$]. \nThis two-body superposition is a consequence of the competition between the direct repulsive and induced \nattractive interactions~\\cite{mistakidis2020many,mistakidis2020induced}. \nInspecting now $\\mathcal{D}_{\\rm rel}(t)$ for different post-quench values of $g_{BF}$ [Fig.~\\ref{2B_den_r}($d$)] we can \nreadily see that it performs oscillations possessing two predominant frequencies, for instance \n$\\omega_1\\approx 0.22$, $\\omega_2\\approx 0.19$ when $g_{BF}=0.8$ and $\\omega_1\\approx 0.33$, $\\omega_2\\approx 0.19$ if $g_{BF}=2.5$. \nThese frequencies are again related to the center-of-mass and relative coordinate breathing modes respectively. \nInterestingly, the oscillation amplitude of $\\mathcal{D}_{\\rm rel}(t)$ e.g. for $g_{BF}=0.8$ and $g_{BF}=2.5$ is almost constant while for $g_{BF}=4$ it shows a decaying tendency. \nThis means that in the latter case the induced attraction tends to surpass the impurities direct repulsion. \nFinally, we remark that the oscillation amplitude (decay rate) of $\\mathcal{D}_{\\rm rel}(t)$ for fixed $g_{BF}$ is in general larger (smaller) when $g_{BB}=1$ [Fig.~\\ref{2B_den_r}($e$)] compared to $g_{BB}=0$ [Fig.~\\ref{2B_den_r}($d$)], thus evincing the presence of the direct impurities repulsion. \n\n\n\\subsubsection{Correlations of the fermionic environment}\\label{two_body_evol_repul_bath} \n\nTo complement our study we then investigate the correlation patterns of the Fermi sea as encoded in its two-body \ndensity $\\rho^{(2)}_{FF}(x_1,x_2;t)$ shown in Figs.~\\ref{2B_den_r}($b_1$)-($b_4$) at specific time-instants after a \nquench from $g_{BF}=0$ to $g_{BF}=2.5$ for $g_{BB}=0$. \nA correlation hole occurs along the diagonal of $\\rho^{(2)}_{FF}(x_1,x_2;t)$ throughout the evolution due to the Pauli exclusion principle. \nAlso, an expansion [Fig.~\\ref{2B_den_r}($b_2$)] and contraction [Fig.~\\ref{2B_den_r}($b_4$)] of the anti-diagonal \nof $\\rho^{(2)}_{FF}(x_1,x_2;t)$ \ntakes place which manifest the collective breathing motion of the fermionic cloud~\\cite{kwasniok2020correlated}, see \nalso Fig.~\\ref{spd_dr}($a_2$). \nMoreover, a depletion along the $x_1=0$ and $x_2=0$ spatial regions is observed indicating that it is more likely for \none fermion to be located in the vicinity of a density hump at $x<0$ and the other one being symmetrically placed \nwith respect to the trap center, see also Fig.~\\ref{spd_dr}($a_2$). \n\n\n\n\\subsection{Quench to attractive interactions}\\label{attractive quench} \n\nIn the following, we shall study the dynamical response of the impurities and the fermionic bath after a quench from $g_{BF}=0$ to \nthe attractive ($g_{BF}<0$) impurity-medium interaction regime. \nTo quantify the arising distinctive dynamical features we analyze the single-particle density [Sec.~\\ref{density_evol_attract}] \nand the corresponding two-body density [Sec.~\\ref{two_body_evol_attract}] evolution of the participating components. \nAs in the previous section, we first discuss the time-evolution of two non-interacting ($g_{BB} = 0$) impurities and subsequently \ncompare to the case of two repulsively interacting ($g_{BB} = 1.0$) ones. \n\n\n\\subsubsection{Density evolution}\\label{density_evol_attract}\n\nThe spatio-temporal evolution of $\\rho^{(1)}_{F}(x;t)$ and $\\rho^{(1)}_{B}(x;t)$ after a quench from $g_{BF}=0$ to $g_{BF}=-0.8$ \nfor $g_{BB}=0$ is presented in Fig.~\\ref{s_den_a} ($a_1$) and Fig.~\\ref{s_den_a} ($a_3$) respectively. \nAs a consequence of the interaction quench both the impurities and the fermionic clouds undergo a collective weak amplitude \nbreathing motion identified by their contraction and expansion dynamics~\\cite{huang2019breathing,mistakidis2019correlated}. \nThe breathing frequency of the fermionic bath is $\\omega_F^{br}\\approx 0.208\\approx2\\omega$ while for the impurities it corresponds to \n$\\omega_B^{br} \\approx 0.251$ since they experience a modified external potential composed of the harmonic oscillator and the density of their host, see for details Refs.~\\cite{mistakidis2020many,kiehn2019spontaneous}. \nImportantly, the attractive impurity-medium coupling results in the formation of a shallow density hump in $\\rho^{(1)}_{F}(x;t)$ [Fig.~\\ref{s_den_a}($a_1$)] at the instantaneous location of the impurities [Fig.~\\ref{s_den_a}($a_3$)] i.e. around the trap center. \nTurning to repulsively interacting impurities where $g_{BB}=1$ we observe that a similar to the above-described dynamics takes place, see Figs.~\\ref{s_den_a} ($a_5$), ($a_7$). \nHowever, the expansion amplitude of $\\rho^{(1)}_{B}(x;t)$ is larger and the breathing frequency $\\omega_B^{br}\\approx 0.226$ is slightly smaller compared to the $g_{BB}=0$ scenario due to the inclusion of direct $s$-wave repulsive interactions. \nAlso, since $\\rho^{(1)}_{B}(x;t)$ is relatively wider than for $g_{BB}=0$ the density hump developed in $\\rho^{(1)}_{F}(x;t)$ in \nthe latter case almost disappears for $g_{BB}=1$.   \n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{Fig7.eps}\n\\caption{Time-evolution of the one-body density of [($a_1$), ($a_2$), ($a_5$), ($a_6$)] the fermionic medium $\\rho^{(1)}_{F}(x,t)$ and [($a_3$), ($a_4$), ($a_7$), ($a_8$)] the bosonic impurities $\\rho^{(1)}_{B}(x,t)$ after an interaction quench of the boson-fermion coupling constant from $g_{BF}=0$ to different attractive values (see legend). \nThe impurities are considered to be ($a_1$)-($a_4$) free i.e. $g_{BB} = 0$ and ($a_5$)-($a_8$) repulsively interacting with $g_{BB} = 1$. \nThe harmonically trapped mixture with $\\omega=1$ consists of $N_F = 6$ fermions and $N_B = 2$ bosons while it is prepared in its ground state with $g_{BF}=0$ and either $g_{BB}=0$ or $g_{BB}=1$.} \n\\label{s_den_a}\n\\end{figure} \n\nFollowing a quench to stronger impurity-medium interactions, e.g. $g_{BF}=-2.5$, leads to a more intricate response of both components than for $g_{BF}=-0.8$, see Figs.~\\ref{s_den_a}($a_2$), ($a_4$), ($a_6$), ($a_8$). \nReferring to the system containing non-interacting impurities [Figs.~\\ref{s_den_a}($a_2$), ($a_4$)] we observe that $\\rho^{(1)}_{B}(x;t)$ has a pronounced spatial localization tendency around the trap center while performing an ``irregular'' weak amplitude breathing dynamics. \nThe latter is characterized by two dominant frequencies, namely $\\omega_{B_1}^{br}=0.234$ and $\\omega_{B_2}^{br}=0.263$ corresponding to the \ncenter-of-mass and relative coordinate breathing mode respectively. \nSince these frequencies are close the dynamics of $\\rho^{(1)}_{B}(x;t)$ is reminiscent of a beating pattern. \nWe remark that a similar time-evolution takes place also for bosonic impurities immersed in a bosonic background~\\cite{mistakidis2020many}.\nAccordingly, as a result of this sharply peaked distribution of $\\rho^{(1)}_{B}(x;t)$ in the vicinity of $x=0$ there is an accumulation \nof the fermionic density at the same location due to the finite $g_{BF}$. \nIndeed, a prominent density hump builds upon $\\rho^{(1)}_{F}(x;t)$ [Fig.~\\ref{s_den_a}($a_2$)] which otherwise exhibits collective \nbreathing oscillations of a frequency $\\omega_F^{br}\\approx 0.211$. \nThe dynamical response is somewhat changed when considering interacting impurities ($g_{BB}=1$) as depicted in Figs.~\\ref{s_den_a}($a_6$), ($a_8$). \nThe impurities possess a comparatively wider density distribution than for $g_{BB}=0$ as a consequence of their finite repulsion, $g_{BB}=1$. \nAlso, the amplitude of their breathing motion is slightly larger compared to the one of $g_{BB}=0$ and the participating frequencies \n$\\omega_{B_1}^{br}=0.355$ and $\\omega_{B_2}^{br}=0.376$ are very close thereby producing a beating pattern (hardly visible \nin Fig.~\\ref{s_den_a}($a_8$)) manifested by the periodic \nappearance of sharp peaks in $\\rho^{(1)}_{B}(x;t)$ as a result of its contraction. \nThe differences in the dynamics of non-interacting and interacting impurities for fixed post-quench $g_{BF}$ will be further analyzed in the next section. \nThis beating pattern is also imprinted in $\\rho^{(1)}_{F}(x;t)$ which as a back-action develops humps which follow the location of the impurities. \nOtherwise, $\\rho^{(1)}_{F}(x;t)$ performs a breathing mode of almost the same amplitude and equal \nfrequency $\\omega_{F}^{br}\\approx0.2$ compared to the $g_{BB}=0$ case. \n\n\n\n\\subsubsection{Evolution of impurity-impurity correlations}\\label{two_body_evol_attract} \n\nTo identify and consequently characterize the nature of the impurity-impurity correlations in the course of the dynamics we monitor the two-body density $\\rho^{(2)}_{BB}(x_1,x_2;t)$ and relative distance $\\mathcal{D}_{\\rm rel}(t)$ of the impurities depicted in Fig.~\\ref{2B_den_a} for different quench amplitudes. \nFigures~\\ref{2B_den_a}($a_1$)-($a_4$) show snapshots of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ upon considering a quench of two non-interacting impurities from $g_{BF}=0$ to $g_{BF}=-2.5$. \nInitially, $t=0$, the impurities lie in the vicinity of the trap center since $\\rho^{(2)}_{BB}(x_1,x_2;t)$ is non-zero within the spatial region $x_1,x_2\\in [-2,2]$ [Fig.~\\ref{2B_den_a}($a_1$)]. However, as time evolves, the two bosons start to occupy a smaller spatial region and therefore approach each other, a tendency that becomes evident by the gradual shrinking of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ across its diagonal accompanied by the depression of its anti-diagonal, see Figs.~\\ref{2B_den_a}($a_2$)-($a_4$). \n\nAs it has already been discussed in Sec.~\\ref{two_body_evol_repul} the shape of the anti-diagonal of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ is well captured by $\\mathcal{D}_{\\rm rel}(t)$ which is presented in Fig.~\\ref{2B_den_a}($d$) for distinct post-quench values of $g_{BF}$. \nEvidently $\\mathcal{D}_{\\rm rel}(t)$ oscillates irrespectively of $g_{BF}$, a behavior that corresponds to the breathing motion of the impurities and can also be inferred from the weak expansion [Fig.~\\ref{2B_den_a}($a_3$)] and contraction [Fig.~\\ref{2B_den_a}($a_2$)] of $\\rho^{(2)}_{BB}(x_1,x_2=-x_1;t)$. \nIts evolution contains a multitude of frequencies whose number and value depend on $g_{BF}$ and refer to the underlying \nbreathing motion, e.g. for $g_{BF}=-2.5$ the dominantly involved frequencies are $\\omega_1=0.234$ and $\\omega_2=0.263$ respectively. \nAlso, the oscillation amplitude of $\\mathcal{D}_{\\rm rel}(t)$ is smaller for a larger $\\abs{g_{BF}}$ which is in accordance to the localization tendency of the impurities for quenches to stronger impurity-medium attractions. \nImportantly, $\\mathcal{D}_{\\rm rel}(t)$ exhibits a decaying tendency in time which is more pronounced for increasing $\\abs{g_{BF}}$ and shows a saturation behavior for quite strong attractions, e.g. $g_{BF}=-4$ here, and long evolution times $t>80$. \nThis latter decaying behavior is again a manifestation of the presence of attractive induced impurity-impurity interactions. \nIt is also worth commenting at this point that the suppression of the anti-diagonal of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ or equivalently \nthe decay of $\\mathcal{D}_{\\rm rel}(t)$ is significantly more pronounced for quenches towards the attractive interaction regime than \nin the repulsive one, e.g. compare Fig.~\\ref{2B_den_r}($d$) and Fig.~\\ref{2B_den_a}($d$). \nTherefore, we can infer the generation of stronger attractive induced interactions for quenches in the attractive than in the repulsive \nimpurity-medium interaction regime~\\cite{mistakidis2020induced}, a result that also holds for the ground state of the system as \nexplicated in Sec.~\\ref{corel_ground}. \n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{Fig8.eps} \n\\caption{Instantaneous profiles of the two-body reduced density of ($a_1$)-($a_4$) two non-interacting bosons $\\rho^{(2)}_{BB}(x_1,x_2)$, ($b_1$)-($b_4$) two fermions of the bath $\\rho^{(2)}_{FF}(x_1,x_2)$ and ($c_1$)-($c_4$) two repulsively interacting bosons $\\rho^{(2)}_{BB}(x_1,x_2)$. \nThe BF mixture contains $N_F = 6$ fermions and $N_B =  2$ bosons. \nIt is confined in a harmonic trap with $\\omega = 0.1$ and it is prepared in its ground state with $g_{BF} = 0$ and either $g_{BB}=0$ or $g_{BB}=1$. \nThe dynamics is triggered upon considering an impurity-medium interaction quench from $g_{BF} = 0$ to $g_{BF} = -2.5$. \nTemporal-evolution of the boson-boson relative distance $\\mathcal{D}_{\\rm rel}(t)$ for ($d$) $g_{BB} = 0$ and ($e$) $g_{BB} =1$ at specific post-quench $g_{BF}$ couplings (see legend).}  \n\\label{2B_den_a}\n\\end{figure} \n\nOn the other hand, the two-body dynamics of two repulsively interacting impurities, here $g_{BB}=1$, subjected to a quench \nfrom $g_{BF}=0$ to $g_{BF}=-2.5$ showcases quite different characteristics from the $g_{BB}=0$ case, compare in particular \nFigs.~\\ref{2B_den_a}($c_1$)-($c_4$) with Figs.~\\ref{2B_den_r}($c_1$)-($c_4$). \nIndeed, even for the ground state of the system ($t=0$) the impurities, since $g_{BB}$ is finite, are spatially separated with the one residing \nat $x<0$ and the other at $x>0$ as can be inferred from the correlation hole of $\\rho^{(2)}_{BB}(x_1,x_2=x_1;t=0)$ in Fig.~\\ref{2B_den_a}($c_1$). \nAfter the quench, they oscillate between two distinct configurations. \nNamely they either stay separated, see the elongated anti-diagonal of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ in Figs.~\\ref{2B_den_a}($c_2$), ($c_3$), or they move close to each other, see their bunching tendency in the region $x_1,x_2\\in [-2,2]$ in Fig.~\\ref{2B_den_a}($c_4$). \nThis behavior is caused by the competition of their inherent repulsive contact interaction and the induced attraction mediated by the fermionic environment~\\cite{huber2019medium}. \n\nTo understand better the aforementioned competing mechanism we present the time-evolution of the impurities relative distance $\\mathcal{D}_{\\rm rel}(t)$ for a variety of post-quench interactions $g_{BF}$ in Fig.~\\ref{2B_den_a}($e$). \nAs it can be directly seen, the response of $\\mathcal{D}_{\\rm rel}(t)$ depends crucially on $g_{BF}$. Indeed for quenches to weak attractions, e.g. $g_{BF}=-0.8$, $\\mathcal{D}_{\\rm rel}(t)$ oscillates with an almost constant amplitude and a \ndominant frequency $\\omega_1=0.226$. \nHowever, by increasing the quench amplitude e.g. to $g_{BF}=-2.5$ $\\mathcal{D}_{\\rm rel}(t)$ performs ``irregular'' oscillations characterized by multiple frequencies and importantly a decaying amplitude. \nThis decay is more pronounced for larger attractions e.g. $g_{BF}=-4$ where $\\mathcal{D}_{\\rm rel}(t)$ drops at the early stages of the dynamics and saturates to a fixed value for $t>50$. \nAs a consequence, we can deduce that the attractive induced interactions become stronger for quenches towards larger impurity-medium attractions and gradually dominate with respect to the impurities direct repulsion. \nNote here that such a mechanism is also present for quenches to repulsive impurity-medium interactions, see Fig.~\\ref{2B_den_r}($e$), but it is apparently less effective compared to the attractive $g_{BF}$ quench scenario. \n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=\\textwidth]{Fig9.eps}\n\\caption{Temporal-evolution of the Von-Neumann entropy $S_{VN}(t)$ for different post-quench ($a$)-($b$) attractive and ($c$)-($d$) repulsive impurity-medium interaction strengths (see legends). \nThe bosonic impurities are considered to be either [($a$), ($c$)] non-interacting $g_{BB}=0$ or [($b$), ($d$)] interacting with $g_{BB}=1$. \n($e$) Time-averaged Von-Neumann entropy $\\bar{S}_{VN}$ for distinct post-quench $\\abs{g_{BF}}$ attractive and repulsive values (see legend) when $g_{BB}=0$. \nThe solid and dashed lines provide a guide to the eye. \nThe BF mixture consists of $N_B=2$ bosons and $N_F=6$ fermions with equal masses and are confined in the same harmonic trap with $\\omega = 0.1$.}  \n\\label{fig:vn_ent}\n\\end{figure*}\n\n\n\\subsubsection{Correlations of the Fermi bath}\\label{two_body_evol_bath_attract}\n\nTurning to the Fermi sea we observe the appearance of completely different correlation patterns building upon $\\rho^{(2)}_{FF}(x_1,x_2;t)$ when compared to the repulsive quench scenario [Figs.~\\ref{2B_den_r}($c_1$)-($c_4$)] as demonstrated in Figs.~\\ref{2B_den_a}($c_1$)-($c_4$) for a quench to $g_{BF}=-2.5$ in the system with two non-interacting ($g_{BB}=0$) impurities. \nAs expected, a correlation hole exists for the entire time-evolution due to the fermionic character of the bath~\\cite{erdmann2019phase,kwasniok2020correlated}. \nInitially, $t=0$, the fermions are symmetrically placed with respect to $x=0$ and predominantly reside one at $x>0$ and the other \nat $x<0$, see the bright spots close to the diagonal in Fig.~\\ref{2B_den_a}($c_1$). \nFollowing the quench a cross-like correlation pattern appears in $\\rho^{(2)}_{BB}(x_1,x_2;t)$ which becomes more elongated across the spatial regions lying in the vicinity of $x_1=0$ and $x_2=0$ [Figs.~\\ref{2B_den_a}($c_3$), ($c_4$)]. \nThis cross-like correlation pattern is the two-body analogue of the accumulation of the Fermi density $\\rho^{(1)}_{F}(x;t)$ [Fig.~\\ref{spd_dr}($a_2$)] around the trap center and more precisely in the vicinity of the position of the impurities due to the attractive $g_{BF}$. \nThus it is a direct imprint of the impurities motion into their host, see also the elongated diagonal of $\\rho^{(2)}_{BB}(x_1,x_2;t)$ in Figs.~\\ref{2B_den_a}($a_2$)-($a_4$), evincing that for strong attractive $g_{BF}$ the fermions move close to the trap center, a mechanism that competes with their inherent Fermi pressure. \n\n\n\\subsection{Entanglement dynamics}\\label{entanglemet_dynamics} \n\nTo further unveil the degree of impurity-medium correlations during the quench dynamics of the BF mixture we employ the time-evolution of the Von-Neumann entropy $S_{VN}(t)$ [Eq.~\\eqref{VN}]. \nThis quantity provides a measure of the overall build up of the impurity-medium entanglement~\\cite{mukherjee2020pulse,theel2020entanglement,kwasniok2020correlated} and also reveals the complexity of the time-evolved post-quench state of the system. \nThe dynamics of $S_{VN}(t)$ after a quench to attractive (repulsive) interactions for the system of two non-interacting and interacting impurities is shown in Figs.~\\ref{fig:vn_ent}($a$) and ($b$) [Figs.~\\ref{fig:vn_ent}($c$) and ($d$)] respectively. \n\nFocusing on the attractive post-quench interaction regime [Figs.~\\ref{fig:vn_ent}($a$), ($b$)] we observe that independently of the inclusion \nof direct $s$-wave impurity-impurity interactions $S_{VN}(t=0)=0$ and hence the components are initially non-entangled. \nHowever, directly after the quench an appreciable impurity-medium entanglement generation takes place in all cases \nsince $S_{VN}(t)\\neq 0$~\\cite{mistakidis2019correlated,mistakidis2019quench,mukherjee2020pulse}. \nMore precisely, an almost ballistic linear growth of $S_{VN}(t)$ is manifested at the very early stages of the dynamics ($t<5$) accompanied by a fluctuating behavior of $S_{VN}(t)$ around a fixed $g_{BF}$-dependent value at later evolution times. \nNotice that for both $g_{BB}=0$ and $g_{BB}=1$ the response of $S_{VN}(t)$ shows a hierarchy in terms of $g_{BF}$, namely it acquires \nlarger values for stronger attractions. \nAlso, the temporal fluctuations of $S_{VN}(t)$ deep in the evolution are suppressed for quenches to weak attractions, e.g. compare $S_{VN}(t)$ for $g_{BF}=-0.8$ and $g_{BF}=-2.5$ in Figs.~\\ref{fig:vn_ent}($a$), ($b$). \nThe latter means that for larger post-quench attractions the system is in a more complicated many-body superposition involving a \nlarger amount of states [see also Eq.~(\\ref{4})] than for smaller negative $g_{BF}$ values. \nThis situation holds equal for fixed $g_{BF}$ but increasing $g_{BB}$. \nIndeed, it becomes apparent by inspecting $S_{VN}(t)$ for fixed $g_{BF}$ between the $g_{BB}=0$ and $g_{BB}=1$ cases that in the latter case the temporal fluctuations of $S_{VN}(t)$ are enhanced, especially for a larger $\\abs{g_{BF}}$. \nWe remark that the saturating tendency of $S_{VN}(t)$ for long times can be attributed to the finite size of the system~\\cite{Calabrese_2005}, i.e. if the system would have been infinite then $S_{VN}(t)$ should increase linearly in time throughout the time-evolution. \n\nA similar to the above-described phenomenology regarding the entanglement dynamics takes place also during the unitary evolution of the system for quenches towards the repulsive impurity-medium interaction regime for both non-interacting [Fig.~\\ref{fig:vn_ent}($c$)] and interacting [Fig.~\\ref{fig:vn_ent}($d$)] impurities. \nIndeed, the sudden increase of $g_{BF}$ leads to entanglement formation since $S_{VN}(t)\\neq 0$ while $S_{VN}(t=0)=0$. \nAs for $g_{BF}<0$, here also $S_{VN}(t)$ increases linearly for $t<5$ and subsequently oscillates around a mean value, see Figs.~\\ref{fig:vn_ent}($c$), ($d$). \nInterestingly, the degree of entanglement is larger for quenches to the repulsive than the attractive interaction regimes, e.g. compare Fig.~\\ref{fig:vn_ent}($a$) with Fig.~\\ref{fig:vn_ent}($c$). \nThis fact evinces that a larger amount of dynamical impurity-medium entanglement is established in the repulsive interaction regime. \nTo support our argument we exemplarily showcase the time-averaged Von-Neumann entropy, defined as $\\bar{S}_{VN} = (1\/T)\\int_{0}^{T}dtS_{VN}(t)$ with $T$ being the considered evolution time, in Fig.~\\ref{fig:vn_ent}($e$) for varying post-quench repulsive ($g_{BF}>0$) and attractive ($g_{BF}<0$) impurity-medium interactions in the system containing the non-interacting impurities. \nAs it can be readily seen, irrespectively of the quench direction $\\bar{S}_{VN}$ increases monotonously with increasing magnitude of $g_{BF}$. \nHowever, it is also apparent that $\\bar{S}_{VN}$ is in general slightly larger for quenches to repulsive than to attractive interactions at a specific post-quench $\\abs{g_{BF}}$. \n\n\n\\section{Conclusions}\\label{conclusion} \n\nWe have unraveled the role of induced correlations and pattern formation in the ground state and the non-equilibrium quantum dynamics of two bosonic impurities embedded in a fermionic environment. \nThe one-dimensional Bose-Fermi mixture is harmonically trapped and the time-evolution is initiated upon considering a quench of the \nimpurity-medium coupling from a vanishing towards the repulsive or the attractive interaction regime. \nInspecting both one- and two-body observables enables us to expose correlation-induced phenomena mediated by the host, analyze the \ncompetition of induced interactions and direct $s$-wave ones, the emergent phase-separation processes and the underlying entanglement dynamics. \n\nReferring to the ground state of two non-interacting bosonic impurities it is shown that on the single-particle level they phase-separate \nwith the Fermi sea for strong repulsions and accumulate at the trap center together with their environment for large attractions, otherwise they are miscible. \nIn the system of two repulsively interacting impurities the boundaries of the aforementioned regions are shifted to larger interactions. \nImportantly, we identify the presence of induced impurity-impurity interactions mediated by the fermionic environment, in the system with non-interacting bosons, for either increasing impurity-medium repulsion or attraction. \nFor repulsively interacting impurities we elaborate on the competition of induced and direct interactions with the latter (former) dominating for \nrepulsive (attractive) impurity-medium couplings, evincing that the strength of induced interactions is larger for attractive impurity-bath interactions. \nInspecting the two-body correlation function of the Fermi sea we showcase that two fermions are likely to remain far apart (approach each other) for larger impurity-medium repulsions (attractions). \n\nWe trigger the dynamics by suddenly changing the impurity-medium interaction strength from zero to finite repulsive or attractive values. \nA quench to repulsive interactions induces in both components a collective breathing motion. \nThe impurities breathing frequency and amplitude depend on the post-quench coupling and their interacting nature. \nMoreover, a dynamical phase-separation occurs for quenches to large repulsions with the impurities residing at the origin and the \nfermionic environment splitting into two symmetric density branches with respect to the trap center. \nHere, two fermions are likely to lie one on the left and the other on the right density branch. \nInterestingly, induced impurity-impurity correlations mediated by the host are manifested in the course of the evolution of two \nnon-interacting impurities and become more pronounced for quenches to stronger repulsions. \nOn the other hand, monitoring the dynamics of repulsively interacting impurities we showcase the competition of induced and \ndirect interactions with the latter prevailing and enforcing the impurities to be in a two-body superposition. \nThe impact of induced interactions is also captured by the decaying amplitude in time of the impurities relative distance, which is \nclearly more prominent for non-interacting bosonic impurities. \n\nFor quenches to attractive impurity-medium interactions both components perform an overall breathing motion, whose amplitude and frequency regarding the impurities are impacted by the considered impurity-impurity and post-quench impurity-medium couplings. \nRemarkably, a beating pattern appears on the single-particle level stemming from the involvement of two nearly resonant breathing frequencies in the dynamics of the impurities due to the dominant nature of their attractive induced interactions. \nFurthermore, the impurities exhibit a spatial localization tendency around the trap center causing a density accumulation of the Fermi sea at their instantaneous location. \nThis mechanism becomes more pronounced for quenches to larger attractions and it is imprinted as a cross-like correlation pattern in the Fermi sea and dictates the dominant presence of attractive induced interactions whose strength is enhanced for quenches to larger attractions. \nIndeed they can even gradually surpass the direct impurity-impurity repulsive coupling, a result that is also evident by the prominent decaying amplitude of the impurities relative distance during the time-evolution. \n\nMoreover, by measuring the Von-Neumann entropy we explicate that in all cases the impurity-medium entanglement rises in a linear manner at the initial stages of the dynamics and afterwards it exhibits a fluctuating \nbehavior around a constant value. \nAlso, the entanglement exhibits a hierarchy by means that it is larger for fixed impurity (post-quench impurity-medium) interaction \nand increasing quench amplitude (impurity coupling). \n\nThere are several research directions that can be pursued in future endeavors. \nAn intriguing perspective is to study the robustness of the discussed phenomena in the presence of finite \ntemperature effects~\\cite{tajima2019thermal} and in particular their impact on the impurities induced interactions. \nCertainly, the generalization of our findings to higher dimensional settings as well as to larger impurity concentrations is desirable. \nAnother interesting direction would be to consider an additional long-range interparticle interaction potential~\\cite{kain2014polarons} \nand unravel the corresponding quench induced dynamics. \nThe emergent quasiparticle properties~\\cite{schmidt2018universal,mistakidis2019repulsive} such as the lifetime, residue, \neffective mass and induced interactions are of particular interest. \n\n\n\n\\begin{acknowledgments} \nK.M. acknowledges a research fellowship (Funding ID no 57381333) from the Deutscher Akademischer Austauschdienst (DAAD). \nS. I. M. gratefully acknowledges financial support in the framework of the Lenz-Ising Award of the University of  Hamburg. \nP.S. is grateful for financial support by the Deutsche Forschungsgemeinschaft (DFG) in the framework \nof the SFB 925 ``Light induced dynamics and control of correlated quantum systems''. \n\\end{acknowledgments} \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsubsection{Formulation}\n\nUnlike pulsed lasers, pulsation in optical microresonators requires neither active nor passive mode locking elements (e.g. modulators or saturable absorbers)\\cite{haus2000mode, kutz2006mode}. Rather, pulsed states arise naturally from a simple damped driven nonlinear Schroedinger equation known as the Lugiato-Lefever equation (LLE) \\cite{lugiato1987spatial, haelterman1992dissipative, matsko2011mode, coen2013modeling, chembo2013spatiotemporal}. Two categories of stable pulsed state solutions have been identified for the LLE: stable modulational instability (MI, also known as hyper-parametric oscillations or Turing rolls) and stable cavity solitons \\cite{matsko2012hard, erkintalo2014coherence}. Turing rolls arise from the intra-cavity equilibrium field through modulational instability of vacuum fluctuations and usually have multiple-FSR (free spectral range) spacing between their adjacent teeth, while the word soliton is used to refer to coherent combs with single-FSR spacing.  Experimental and theoretical studies have suggested that soliton states are not accessible from the continuous wave (CW) intra-cavity field without seeding \\cite{taheri2015soliton, lobanov2015generation}, changing the pump frequency or power \\cite{matsko2012hard, lamont2013route, herr2014temporal}, or a suitable input pulse \\cite{leo2010temporal}. Owing to the low phase noise and exceedingly stable frequency spacing of the comb teeth in Turing rolls and solitons, chip-scale pure low-phase-noise radio frequency (RF) sources \\cite{liang2015high} and coherent communication with speeds in excess of 100 Gbit\/s per comb line have been demonstrated \\cite{pfeifle2014coherent, pfeifle2015optimally}.\\newline\n\\indent In addition to the generation of a frequency comb with equidistant teeth, temporal pulse generation requires mutual phase locking of the complex amplitudes. Phase locking in optical microresonators has been studied in terms of the cascaded emergence of phase-locked triplets \\cite{coillet2014robustness}. Injection locking of overlapping bunched combs has been explained using the Adler equation \\cite{del2014self}. Few-mode models have explained the phase offset between the pumped mode and the rest of the comb teeth \\cite{loh2014phase, taheri2015anatomy}. More recently, Wen \\emph{et al.} \\cite{wen2014self} have emphasized the link between oscillator synchronization---most famously described by the Kuramoto model \\cite{strogatz2000kuramoto}---and the onset of pulsing behavior. However, while stable ultrashort pulses have been demonstrated in a variety of microresonator platforms \\cite{herr2014temporal, saha2013modelocking, brasch2016photonic, vahala2015soliton}, the underlying phase locking mechanism is still unknown. As a result, features of microcomb phase spectra revealed in recent measurements \\cite{del2015phase} are yet not understood.\\newline\n\\indent In this paper, we introduce a reduced phase model which governs the nonlinear mode interactions responsible for spontaneous creation of pulsed states in the LLE which result from a balance between Kerr nonlinearity, dispersion (or, in the spatial case, diffraction), parametric gain, and cavity loss \\cite{grelu2012dissipative}.  The model interactions are \\emph{ternary} (that is, they involve three-variable combinations) rather than binary, as in typical phase models.  Our model admits attracting solutions which, interpreted in the context of nonlinear optical cavities, correspond to stable cavity solitons and Turing patterns, and provides an explanation of recent observations of phase steps in optical frequency combs. Moreover, our model clarifies the role of MI and chaos in the generation and stability of Turing rolls and solitons. \\newline\n\\indent The LLE is a nonlinear partial differential equation in time and the azimuthal angle around the whispering-gallery mode resonator \\cite{chembo2013spatiotemporal}, or, equivalently, in a slow and a fast time variable \\cite{haelterman1992dissipative,coen2013modeling}. Equivalently, a set of coupled nonlinear ordinary differential equations (ODEs) can be used to study resonator-based optical frequency comb generation \\cite{chembo2010modal}. The generalized spatiotemporal LLE in normalized form\n\\begin{equation}\\label{eq:LLE}\n\\pdifdisp{\\psi}{\\tau}=-(1+\\imi\\alpha)\\psi-\\imi \\frac{d_2}{2} \\frac{\\partial^2\\psi}{\\partial\\theta^2}+\\imi|\\psi|^2\\psi+F,\n\\end{equation}\nand its corresponding ODEs\n\\begin{equation}\\label{eq:CNODE}\n\\difdisp{\\tilde{a}_\\eta}{\\tau}=-(1+\\imi\\alpha)\\tilde{a}_\\eta+\\imi\\frac{d_2}{2}\\eta^2\\tilde{a}_\\eta+\\imi\\sum_{l,\\, m,\\, n}\\tilde{a}_l \\tilde{a}_m^*\\tilde{a}_n \\, \\delta_{\\eta_{lmn}\\eta}+\\tilde{F}_\\eta,\n\\end{equation}  \nare Fourier transform pairs, the conjugate variables of the transform being the azimuthal angle around the resonator $\\theta$ and comb mode number $\\eta$, see the Supplemental Material (SM). The number of ODEs equals the number of modes comprising the frequency comb; each equation follows the temporal evolution of the complex amplitude (magnitude and phase) of a single mode. In the ODEs picture, each optical comb tooth can be thought of as a nonlinear oscillator coupled to other oscillators (comb teeth). In Eq.~(\\ref{eq:LLE}), $\\psi(\\theta, \\tau)$ is the normalized field envelope, $\\tau = t\\Delta\\omega_0\/2$ is the normalized time with $\\Delta\\omega_0$ the resonance linewidth for the cavity mode closest to the pump (the pumped resonance) and $t$ the laboratory time, $\\alpha = -2(\\omega_{\\mathrm{P}}-\\omega_0)\/\\Delta\\omega_0$ is the normalized detuning between the pump laser frequency $\\omega_{\\mathrm{P}}$ and the cold-cavity pumped resonance frequency $\\omega_0$, $d_2 = -2D_2\/\\Delta\\omega_0$ is the normalized second-order dispersion parameter, $D_2$ being the cavity second-order dispersion coefficient, and $F$ is the normalized pump amplitude. The field envelop $\\psi$ and the pump amplitude $F$ are normalized to the sideband generation threshold such that the comb generation threshold in Eq.~(\\ref{eq:LLE}) is equal to unity \\cite{chembo2010modal}. In Eq.~(\\ref{eq:CNODE}), $\\tilde{a}_\\eta=a_\\eta(\\tau) \\exp[\\imi\\phi_\\eta(\\tau)]$ is the complex-valued comb tooth amplitude for mode $\\eta$ with magnitude $a_\\eta(\\tau)$ and phase $\\phi_\\eta(\\tau)$, $\\tilde{F}_\\eta(\\tau)$ is the Fourier transform of $F$ and equals $\\delta_{0\\eta}F_{\\mathrm{P}}\\exp(\\imi\\phi_{\\mathrm{P}})$ for CW pumping, $\\delta_{pq}$ (for integers $p$ and $q$) is the Kronecker delta, and $\\eta_{lmn}=l-m+n$. All mode numbers $\\eta$ are define relative to the pumped mode. For a soliton, $\\mathord{\\eta\\in\\{0, \\pm 1, \\dots, \\pm N\\}}$ while for Turing rolls $\\mathord{\\eta\\in\\{0, \\pm\\mu, \\pm2\\mu, \\dots, \\pm N\\mu\\}}$, where the integer $\\mu \\geq 1$ is the mode number at which MI gain peaks.\\newline\n\\indent When driven by a CW pump, experiments and numerical simulations suggest that for stable solutions, the magnitude of the pumped mode is much larger than that of the other modes and that in the absence of third- and higher-order dispersion, the magnitude spectrum of these solutions are symmetric with respect to the pumped mode $\\eta=0$ \\cite{saha2013modelocking,herr2014temporal} (see, e.g., the inset curves $a_\\eta^2$ vs. mode number in Fig.~\\ref{fig:prelim:phasealigned}). Therefore, we exploit the symmetry of the magnitude spectrum, adopt a perturbative approach (with $a_\\eta$ for $\\eta\\ne0$ as the small perturbation parameters), and following Ref.~\\cite{wen2014self} simplify Eq.~(\\ref{eq:CNODE}) by keeping only terms with at least one contribution from the pumped mode $a_0$ in the triple summations \\cite{taheri2015anatomy}. The magnitude and phase equations for the pumped mode include no linear contributions from $a_{\\eta\\ne0}$ (corrections are proportional to $a_\\eta^2$, $\\eta\\ne0$), and their solutions settle on a fast time scale to the equilibrium intra-cavity field $\\psi_\\mathrm{e}=a_0\\exp(\\imi\\phi_0)$; subsequently, $a_0$ and $\\phi_0$ can be treated as constants (\\cite{taheri2015anatomy}, also see SM).\\newline\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{phasealigned}\n\\caption[Phase alignment in (a) solitons and (b) Turing rolls.]{\\label{fig:prelim:phasealigned}Phase alignment in (a) solitons and (b) Turing rolls seen in the steady-state solutions of Eqs.~(\\ref{eq:LLE}) and (\\ref{eq:CNODE}). The inset curves in red (top corners) show the spatiotemporal waveforms and those in black (bottom corners) are the frequency spectra. For both solitons and rolls the phases lie on straights lines of arbitrary slope. Parameter values are (a) $\\alpha=2,\\, d_2=-0.0124,\\, F=1.41$, and (b) $\\alpha=0,\\, d_2=-0.0124,\\, F=1.63$. The phase profile has been unwrapped in (b).}\n\\end{figure}\n\\indent Equations of motion for the magnitudes $a_\\eta(\\tau)$ and phases $\\phi_\\eta(\\tau)$ are readily found from Eq.~(\\ref{eq:CNODE}). The equation for the temporal evolution of the centered phase averages $\\zeta_\\eta=\\bar{\\phi}_\\eta-\\phi_0$, where the phase average $\\bar{\\phi}_\\eta=(\\phi_\\eta+\\phi_{-\\eta})\/2$ is centered to the pumped mode phase $\\phi_0$, can be found using the equations for $\\phi_{\\pm\\eta}$ and $\\phi_0$. To lowest non-zero order in $a_{\\eta\\ne 0}$, this equation can be integrated directly to give\n\\begin{equation}\\label{eq:antisym}\n\\tan{\\zeta_\\eta}=\\sqrt{\\abs*{\\frac{C+2}{C}}}\\tanh[\\sqrt{|C(C+2)|}a_0^2(\\tau-\\tau_0)].\n\\end{equation}\nHere $C=d_2\\eta^2\/2a_0^2-F_\\mathrm{P}\\sin(\\phi_\\mathrm{P}-\\phi_0)\/a_0^3$, $\\phi_\\mathrm{P}$ and $F_\\mathrm{P}$ are the phase and normalized magnitude of the pump, and $\\tau_0$ accounts for constants of integration (or initial conditions). Equation~(\\ref{eq:antisym}) holds when $|2a_0^2-\\alpha+d_2\\eta^2\/2|<a_0^2$, a condition that is automatically satisfied when MI gain exists (see SM).  Because the hyperbolic function approaches unity as $\\mathord{\\tau\\to\\infty}$, $\\bar{\\phi}_\\eta$ reaches the same constant irrespective of the initial conditions.  Since $\\bar{\\phi}_\\eta$ is fixed, each pair of phases $\\phi_{\\pm \\eta}$ must take values symmetrically located relative to the same value. We will refer to this as ``\\emph{anti-symmetrization}'' of the phases, following the terminology used in \\cite{wen2014self}.  Once established, phase anti-symmetrization means each centered phase average $\\zeta_\\eta$ can be treated as a constant to first order in $a_{\\eta\\ne0}$. \\newline\n\\indent The equations of motion for the phase differences $\\Delta_\\eta=(\\phi_\\eta-\\phi_{-\\eta})\/2$, \n\\begin{equation}\\label{eq:pl}\n\\dot{\\Delta}_\\eta=\\frac{a_0}{a_\\eta}\\sum\\nolimits_l K(l,\\eta) \\sin(\\Delta_l+\\Delta_{\\eta-l}-\\Delta_\\eta),\n\\end{equation}\nare found by combining the equations for each $\\pm\\eta$ mode pair (see SM). In Eq.~(\\ref{eq:pl}), the over-dot indicates time derivative, and $K(l,\\eta)=a_l a_{\\eta-l} \\{2 \\sin(\\zeta_\\eta-\\zeta_{\\eta-l}+\\zeta_l)+\\sin(\\zeta_\\eta-\\zeta_{\\eta-l}-\\zeta_l)\\}$ is the coupling coefficient for the pump--non-degenerate interaction of comb teeth $\\tilde{a}_0$, $\\tilde{a}_l$, $\\tilde{a}_{\\eta-l}$, and $\\tilde{a}_\\eta$. A family of fixed point solutions of Eq.~(\\ref{eq:pl}) is $\\Delta_\\eta=s_0\\eta+k\\pi$, where $s_0$ is a constant and $k$ an integer.  These solutions imply that the phases have aligned:  the slope of the line passing through the phases of any pair of comb teeth $\\eta$ and $-\\eta$ will be the same and equal to $s_0$, that is $(\\phi_\\eta-\\phi_{-\\eta})\/2\\eta=s_0$. Figure~\\ref{fig:prelim:phasealigned} shows two examples, in solitons and Turing rolls, where Eq.~(\\ref{eq:LLE}) has been integrated numerically using the split-step Fourier transform method for a typical microresonator.\n\nNext, we consider the stability of these states.  For simplicity, we take $k=0$.  (Stability analysis for $k\\ne0$ follows in a similar way.)  The linear stability of a frequency comb with $2N+1$ phase-locked teeth, with mode numbers $\\mathord{\\eta\\in\\{0,\\pm 1,\\pm 2,\u2026,\\pm N\\}}$, is found using Eq.~(\\ref{eq:pl}). We temporarily ignore the dependence of the comb teeth magnitudes on the mode number and take $a_\\eta=a$. (The effect of the mode number dependence of the comb teeth magnitudes will be included shortly.) After phase locking, the centered phase averages $\\zeta_\\eta$ reach a steady-state value which is independent of mode number $\\eta$ since the phases $\\phi_\\eta$ lie on a straight line. Therefore, the coupling coefficients in Eq.~(\\ref{eq:pl}) are all equal, i.e., $K(l,\\eta)=K > 0$. The Jacobian matrix $\\bm{\\mathrm{J}}$ and its eigenvalues can be expressed in closed form for any $N$ (see SM). Except for one zero eigenvalue forced by the rotational symmetry of the LLE, all of the eigenvalues are negative and real, indicating asymptotic stability of the synchronized state. Figure~\\ref{fig:prelim:eigs}(a) shows the non-zero eigenvalues of the equilibrium for increasing comb span ($2N+1$). It is seen that the eigenvalue closest to zero grows more negative with increasing comb span. Hence, for the constant comb amplitude case, a wider comb demonstrates superior stability.\n\nTo investigate the effect of a non-constant comb amplitude profile, we set $\\mathord{a_\\eta\\propto\\exp(-k_0|\\eta|)}$. This profile assumes a linear decay (in logarithmic scale) of the comb teeth magnitude with slope $-20k_0$ dB per increasing mode number by unity, (see, e.g., the insets $a_\\eta^2$ vs. mode number in Fig.~\\ref{fig:prelim:phasealigned}).  Though not analytically tractable, we find numerically that the eigenvalues of $\\bm{\\mathrm{J}}$ all have negative real part (except for the single zero eigenvalue forced by symmetry). Figure~\\ref{fig:prelim:eigs}(b) shows the eigenvalue spectrum vs. increasing combs span for $\\mathord{a_\\eta\\propto\\exp(-k_0|\\eta|)}$. Note that as the comb span increases, the smallest magnitude eigenvalue becomes bounded and almost independent of $N$ (black curve in Fig.~\\ref{fig:prelim:eigs}(b)). Therefore, the stability of the comb \\emph{does not improve}---nor does it degrade---with increasing comb span when the mode number dependence of the comb teeth magnitude is taken into account. Pfeifle \\emph{et al}. \\cite{pfeifle2015optimally} showed that in the presence of pump magnitude and frequency noise, solitons are less robust than Turing rolls in the same microresonator with comparable pump power. Our results suggest that the superior stability of Turing rolls does not originate from their smaller number of comb teeth compared to solitons. Rather, it is linked to the presence of MI gain, which is responsible for the generation of Turing rolls from vacuum fluctuations. We note that Eq.~(\\ref{eq:pl}) does not explicitly include the effect of MI gain; this influence is reflected through the coupling coefficients $K(l,\\eta)$.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{eigs}\n\\caption{\\label{fig:prelim:eigs}Non-zero eigenvalues of the equilibrium (the Jacobian matrix $\\bm{\\mathrm{J}}$) versus comb span for Eq.~(\\ref{eq:pl}) for (a) uniform and (b) mode-number--dependent comb teeth magnitude profile of $\\mathord{a_\\eta\\propto\\exp(-k_0|\\eta|)}$, ($k_0=0.1$). The negative eigenvalue of smallest magnitude (black curve) increases in size with increasing comb span for constant magnitudes, but reaches a constant for the realistic comb magnitude profile.}\n\\end{figure}\n\\begin{figure*}[htbp]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{numsol}\n\\caption[Numerical solutions of Eq.~(\\ref{eq:pl}) for uniform as well as mode-number--dependent comb teeth magnitude spectra, and the emergence of phase steps.]{\\label{fig:prelim:numsol}Numerical solutions of Eq.~(\\ref{eq:pl}) for uniform and mode-number--dependent comb teeth magnitude spectra, and the emergence of $\\pi$ phase steps. (a) Sample steady-state solution of Eq.~(\\ref{eq:pl}) for a comb with 101 teeth, with uniform magnitudes and random initial conditions for the phase differences (shown in (e)). (b) The temporal evolution of the phase differences shown in (a). Most of the phase differences settle to integer multiples of $\\pi$, but some deviations may arise (dotted red circles). (c,d) Same as (a,b) but for comb magnitude profile of $\\mathord{a_\\eta\\propto\\exp(-k_0|\\eta|)}$, ($k_0=0.1$). All of the phase differences $\\Delta_\\eta$ settle to integer multiples of $\\pi$. In (a,c), only the $\\pi$ phase steps are physically significant. The $2\\pi$ steps have not been removed (e.g., through unwrapping the phases) to better illustrate the correspondence of (a,c) with (b,d). (e) The phase differences at the onset of integration (initial conditions) for (a-d). (f) Schematic illustrating $\\pi$ steps in the phase profile $\\phi_\\eta$ of a comb resulting from steps in the phase differences $\\Delta_\\eta$. Comb teeth symmetrically positioned around the pumped mode ($\\eta$ and $-\\eta$) will show $\\pi$ phase steps, with one phase increasing as its counterpart decreases. Such phase steps do not change the phase averages $\\bar{\\phi}_\\eta$.}\n\\end{figure*}\n\nWe consider next numerical solutions of Eq.~(\\ref{eq:pl}). Our numerous runs of numerical integration for different comb spans ($N$ from 5 to 1000) and random initial phase differences taken from a uniform distribution over the range $(-\\pi, \\pi]$ and for uniform comb magnitude spectrum, typically lead to $\\Delta_\\eta=k\\pi$, ($k$ an integer). While a steady-state is always obtained, other steady-state solutions are also possible. A mode-number-dependent comb magnitude spectrum, in contrast, always leads to phase differences equal to integer multiples of $\\pi$, even for those cases in which steady-state phase differences for uniform comb magnitude spectra are not equal to integer multiples of $\\pi$. The reason is that a non-uniform comb magnitude profile places more strict constraints on the steady-state solution of Eq.~(\\ref{eq:pl}). In Figs.~\\ref{fig:prelim:numsol}(a-d), we show sample solutions found by numerically integrating Eq.~(\\ref{eq:pl}) for $N=50$ phase differences (a comb with $2N+1=101$ teeth) for constant comb teeth magnitudes (Fig.~\\ref{fig:prelim:numsol}(a,b)) and the non-constant magnitude profile of $\\mathord{a_\\eta\\propto\\exp(-k_0|\\eta|)}$ (Fig.~\\ref{fig:prelim:numsol}(c,d)). We show the steady-state solutions $\\Delta_\\eta$ at the end of the simulation time vs. mode number as well as the the evolution of the phase differences with time. The initial values of $\\Delta_\\eta$, $\\mathord{\\eta\\in\\{1,2,\u2026,50\\}}$, in both cases is shown in Fig.~\\ref{fig:prelim:numsol}(e). While most of the steady-state phase differences in Fig.~\\ref{fig:prelim:numsol}(a) are integer multiples of $\\pi$, some of them deviate from these values (the dotted red circles). For the non-uniform magnitude spectrum, however, steady-state phase differences are all integer multiples of $\\pi$, as seen in Fig.~\\ref{fig:prelim:numsol}(c). \\newline\n\\indent The $\\pi$ phase steps in the phase differences $\\Delta_\\eta$ imply similar steps in the phases $\\phi_\\eta$. To show this, we assume a set of solutions $\\Delta_\\eta=s_0\\eta$ is known and try to find another set based on it. It may seem that any constant $x$ can be added to the phases of comb teeth symmetrically positioned with respect to the pumped mode (i.e., $\\phi_{\\pm\\eta}\\to\\phi_{\\pm\\eta}+x$) without affecting the solution. Unfortunately, this alters the phase averages $\\bar{\\phi}_\\eta$ and so invalidates the stability analysis presented earlier. However, we can generate new stable phase-locked solutions by considering anti-symmetric changes of the phases, i.e., $\\phi_\\eta\\to\\phi_\\eta\\pm x$ and $\\phi_{-\\eta}\\to\\phi_{-\\eta}\\mp x$, which means $2\\Delta_\\eta=\\phi_\\eta-\\phi_{-\\eta}\\pm 2x=2s_0\\eta +2k\\pi$, and hence $x=\\pm \\pi$ (recall that $k$ is an integer). This demonstrates that the appearance of $\\pi$ steps in the phase spectrum of stable phase-locked frequency combs is permissible, as shown schematically in Fig.~\\ref{fig:prelim:numsol}(f). Indeed, such phase steps have been observed experimentally \\cite{del2015phase} (see, e.g., Fig.~\\ref{fig:prelim:steps}(a)) but have, to the best of our knowledge, remained unexplained until now. \\newline\n\\indent Besides $\\pi$ phase steps, $\\pi\/2$ phase steps \\cite{del2015phase} have also been observed in experiments. These phase steps also can be explained within the framework of our model, as follows. A comb with $\\pi\/2$ phase steps is in fact two interleaved \\emph{non-interacting} combs, each of which has $\\pi$ steps in its phase spectrum \\cite{del2015phase}. These two combs do not interact as a result of the $\\pi\/2$ offset between their phase spectra because this phase offset causes the coupling coefficients between their comb teeth, $K(l, \\eta)$, to vanish. To clarify this point, we refer to Fig.~\\ref{fig:prelim:steps}(c,d) where comb teeth labeled with $\\mathord{\\eta_\\mathrm{A}\\in\\{0,2,4,6,\u2026,14\\}}$ (red) share a constant phase, while those with mode numbers $\\mathord{\\eta_\\mathrm{B}\\in\\{1,3,5,\u2026,13\\}}$ (blue) share another phase, different from that of the former group by $\\pi\/2$, such that $\\zeta_1=\\bar{\\phi}_1-\\phi_0=\\pi\/2$ and $\\zeta_{\\eta_\\mathrm{A}}-\\zeta_{\\eta_\\mathrm{B}}=\\pi\/2$ (recall that the value of $\\zeta_{\\eta}$ is independent of the mode number for a stable comb, cf. Fig.~\\ref{fig:prelim:numsol}(f)). As a result, the coupling coefficient $K(\\eta, \\eta+1)$ is zero because $\\zeta_{\\eta+1}\\pm\\zeta_{\\eta}=\\pm\\pi\/2$, and so there is no coupling between modes $\\eta$ and $\\eta+1$ of the comb for any $\\eta$ (see Eq.~(\\ref{eq:pl})). It is worth noting that the frequency combs with phase steps in \\cite{del2015phase} were obtained through tuning the laser pump into resonance, which alters the MI gain profile and sweeps its peak.\\newline\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{steps}\n\\caption[Experimental data showing steps in the measured phase spectra of optical frequency combs.]{\\label{fig:prelim:steps}Experimental data showing steps in the measured phase spectra of optical frequency combs. (a,c) are the phase spectra while (b,d) depict the power profiles of the combs. (a,b) $\\pi$-steps in the phase spectrum of a stable comb; The green dot corresponds to the pumped mode phase. (c,d) $\\pi$\/2-steps in the phase spectrum of a stable comb; This comb is recognized as two interleaved combs (red and blue) with phases offset by $\\pi\/2$, each exhibiting $\\pi$-phase steps as well (cf. Fig.~\\ref{fig:prelim:numsol}(f)). The $\\pi\/2$-phase offset leads to the decoupling of the two combs, indicated by vanishing coupling coefficients, i.e., $K(l, \\eta)=0$ in Eq.~(\\ref{eq:pl}). The green dots correspond to the phases of the stronger comb teeth. (These plots are reproduced using data originally presented in~\\cite{del2015phase}.)}\n\\end{figure}\nIt has been argued that passing through the chaotic state is necessary for microcomb soliton formation \\cite{lamont2013route}. The foregoing analysis suggests a way of understanding this: passage through the chaotic state serves to provide the system with a large pool of initial conditions, which increases the odds of getting peaks that will then grow into solitons. Numerical simulations of Eq.~(\\ref{eq:pl}) suggest that with increasing comb span and non-uniform comb magnitude spectrum, chances of getting groups of phase-locked comb teeth, or weak pulses, will increase. These weak pulses will then grow into the modes of the nonlinear system (i.e., the solitons). \\newline\n\\indent Finally, although we have compared our theoretical results with microresonator-based frequency comb experiments, they should also apply to mode-locked laser systems.  In 2002, Gordon and Fisher developed a many-body statistical mechanical theory to describe the onset of laser pulsations as a first order phase transition, treating the modes as the elementary degrees of freedom \\cite{gordon2002phase}.  Their ordered collective state is analogous to our synchronized dynamical attractor.  Now, Eq.~(\\ref{eq:pl}) roots in the cubic nonlinear term in the LLE, and the same nonlinearity appears in the master equation for passive mode locking based on a saturable absorber, which approximates the absorber with a cubic nonlinearity \\cite{haus2000mode}. We therefore expect the same dynamical mechanism to be responsible for the creation of sharp pulses in passively mode-locked lasers, despite the different physical source of optical gain (population inversion and stimulated emission rather than parametric amplification).  What matters is the fundamental link between spatiotemporal pulse formation and mode synchronization. \\newline\n\\indent The generic reduced nonlinear oscillator model introduced in this work clearly demonstrates the fundamental link between mode synchronization and spatiotemporal pulse formation in Kerr-nonlinear media. This model admits attracting fixed point solutions corresponding to stable cavity solitons and Turing patterns, permits analyzing their stability in a unified scheme, and explains phase jumps observed in recent phase measurements of stable optical frequency combs. It also provides insight into the role of chaos and parametric gain in the generation of solitons and Turing rolls. This insight can be utilized towards devising novel techniques for controlled formation of robust pulses in optical microresonators.\n\\begin{acknowledgments}\nK.W. and H.T. thank Brian Kennedy for useful discussions.  K.W. also thanks Henry Wen and Steve Strogatz for generously discussing the details of their results reported in \\cite{wen2014self}.  H.T. was supported by the Air Force Office of Scientific Research Grant No.~2106DKP.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIn the last years a focused ion beam (FIB) of Ga$^+$-ions for etching\n\\cite{mat91} has attracted the attention of the community as an\nalternative and flexible method to produce micro- and nanostructures\nof materials, especially where the use of conventional methods\nappears to be limited. This FIB technique has been successfully used\nfor nanostructuring different materials like magnetic and\nsuperconducting \\cite{gie05,tru,tse05} or more recently to study the\nconduction behavior in metallic constrictions \\cite{fer08}. One of\nthe advantages of this technique is its versatility; the use of any\nresist appears, a priory, unnecessary. Nowadays, FIB devices are also\nused for deposition of metallic or insulating materials with the help\nof the same Ga$^+$-ions. These ions induce a decomposition of a\nchemical metal precursor over the surface in question, a technique\ncalled ion-beam induced or assisted deposition (IBID,IBAD)\n\\cite{mel87,mat96,lan02}. The main advantage of IBID\/EBID is the\ndeposition of the desired patterns of a material without the need of\na mask or a pre-structured pattern using optical or e-beam\nlithography (EBL). Also the possibility to modify only parts of the\npatterns in electronic devices within nanometer dimensions is other\nof the FIB advantages.\n\nThe modification of different properties of different materials\n has been studied in the past, as e.g. in\n magnetic La$_{0.7}$Sr$_{0.3}$MnO$_3$ thin films \\cite{pal08} or\n Co\/Pt multilayers  \\cite{hyn01}. However, a fundamental\n problem of FIB was less discussed in\nliterature, namely the modification of the sample near surface region\nand to a certain extent also its interior and their influence in the\ntransport properties by the use of Ga$^+$ions of energies up to\n30~keV and fluences below $10^{12}$~cm$^{-2}$ in usual devices. We\nnote that before cutting or depositing material, the use of FIB\nrequires the precise alignment of the Ga-beam and this is done taking\na picture of the region in question irradiating it with the same\nGa$^+$-ions. Depending on the surface properties of the material in\nquestion, in general Ga$^+$ fluences $\\gtrsim 10^{11}~$cm$^{-2}$ are\nused.  As we demonstrate below, these Ga$^+$-fluences necessary for\nthe first alignment may already affect seriously the intrinsic\nproperties of the material of interest  and can lead to wrong\ninterpretations of the effects that a reduction of the sample\ngeometry may produce.\n\nThe influence of Ga irradiation during the FIB preparation processes\nwas not yet quantitatively studied, neither in situ nor after\nirradiation for fluences below $10^{12}$~cm$^{-2}$.  Specially when\nmaterials are selected to investigate their transport properties\nwhile their size is being reduced, care should be taken since the\nelectrical transport can be sensitive to lattice defects as well as\nto the produced Ga contamination. The aim of this paper  is to report\non the changes in the electrical resistivity measured in-situ and its\ntemperature $T$ and magnetic field $B$ dependence after irradiation\nof three different thin film materials to show the influence of the\nGa irradiation used in FIB devices and at fluences as low as the ones\nused for beam alignment. The experiments were realized in two stages.\nIn the first stage we have done in-situ measurements of the\nelectrical resistance during irradiation and in the second stage its\nresistance was measured as  a function of $T$ and $B$ to compare with\nthe corresponding virgin states. Using graphite as a test material\nbecause its electrical resistance is extremely sensitive to lattice\ndefects \\cite{arn09}, we provide in this work a possible solution\nthat can be used to strongly diminish the effects due to the\nGa$^+$-irradiation on different materials.\n\n\\section{Experimental Details} \\label{details}\n\n\n\\begin{table}[b]\n  \\caption{Samples used, their dimensions (total\n  length $\\times$ width  $\\times$ thickness) and the Ga$^+$-fluence\n  irradiated on the samples. The fluence number in brackets refers to the total\n  fluence after the second irradiation. The corresponding data of the Co-sample \\#2 is\n  given in the Table. The Co\\#1 sample  had a thickness of\n  57~nm. Other dimensions can be taken from Fig.~\\protect\\ref{samples_2}(c).}\n  \\begin{tabular}[htbp]{@{}llll@{}}\n    \\hline\n    Sample & graphite  & La$_{0.7}$Ca$_{0.3}$MnO$_3$ & Co\\#2 \\\\\n    \\hline\n    Dimensions ($\\mu$m)& $11\\times 2 \\times 0.015$ & $52 \\times 7.3 \\times 0.035$ & $18 \\times 0.9 \\times 0.022$ \\\\\n    Fluence ($10^{11}$cm$^{-2}$) & 5 (10)  & $\\geqslant 2.2$ & $\\geqslant 2.2$ \\\\\n\\hline\n  \\end{tabular}\n  \\label{table}\n\\end{table}\n\n\nWe have used the FIB capabilities of  a FEI NanoLab XT 200 Dual Beam\nmicroscope (DBM). The acceleration voltage was fixed in all our\nexperiments to 30~kV. The ion current and the area to be irradiated\nwere changed in order to obtain different Ga$^+$-fluences, see\nTable~\\ref{table}.  The selected samples were a crystalline graphite\nflake prepared by a rubbing and ultrasonic process and\npre-characterized with electron backscattering diffraction (EBSD) and\nRaman scattering \\cite{bar08,arn09},  a La$_{0.7}$Ca$_{0.3}$MnO$_3$\n(LCMO) thin film prepared by plasma laser deposition (PLD) and\nmicro-structured by EBL and wet etching process \\cite{barpat}, and\nthermal evaporated Co thin films (\\#1 and \\#2) previously structured\nby EBL, see Fig.~\\ref{samples_2}(a-c) and Table~\\ref{table}.\n\nLow-noise four-wires resistance measurements  (two for the input\ncurrent and two for the voltage measurement, important to eliminates\ncontributions of the lead resistance) have been performed with the AC\ntechnique (Linear Research LR-700 Bridge with 8 channels LR-720\nmultiplexer) with ppm resolution and in some cases also with the DC\ntechnique (Keithley 2182 with 2001 Nanovoltmeter and Keithley 6221\ncurrent source).\n\nThe Au\/Pt lead contacts used for all samples were prepared by EBL\nprocess using a e-beam resist PMMA 950K of $\\sim 200~$nm thickness.\nThe lithography process was done with the Raith ELPHY Plus system\nincluded in our microscope. The Au\/Pt deposition of the contact\nelectrodes was done by evaporation in a high-vacuum chamber with a\nnominal thickness of 25 and 9~nm, respectively. The in-situ\nresistance measurements performed before and during the irradiation\nof the sample  were done using a self-made sample holder fixed inside\nthe microscope chamber. The temperature and magnetic field dependence\nmeasurements were performed using a commercial cryostat with a\ntemperature stability of 0.1~mK at 100~K. The magnetic field\ngenerated by a superconducting solenoid was always applied normal to\nthe sample and input current.\n\nTo avoid or diminish irradiation effects we protected the graphite\nsample with a negative electron beam resist (AR-N 7500) of thickness\nof 300~nm.  In order to test the effectiveness of the resist film to\navoid contamination during irradiation, part of the graphite flake\nwas covered with the above mentioned resist in an additional process\nafter the Au\/Pt leads were deposited on the sample, see\nFig.~\\ref{samples_2}(a). This resin protects  the graphite sample in\nthe region of the three upper electrodes, allowing us to compare the\nchange in voltage in the unprotected, protected and intermediate\nregions.\n\nThe advantages of this resist is that it allows us the patterning\nby EBL in the desired shape, it is sufficiently robust in the used\n temperature range and it is a very bad electrical\nconductor. The penetration depth of the Ga$^+$-ions in this resist as\nwell as in the samples and the distribution of the density of\nvacancies as a function of sample depth were estimated taking into\naccount their density and using Monte Carlo simulations given by the\nstopping and range of ions in matter (SRIM) \\cite{ziegler,Ziegler2}\ntaking into account the energy of the Ga$^+$-ions, see\nFig.~\\ref{SRIM}.\n\n\n\n\\begin{figure}[htb]%\n\\includegraphics[width=\\textwidth]{samples_22.eps}\n\\caption{Scanning electron microscope images of: (a) the graphite\nflake (dashed line denotes its borders) with the six Au\/Pt contacts.\n(b) The LCMO film with the two inner voltages electrodes and one of\nthe input current electrodes. (c) The Co microwire \\#1 with\nelectrodes at different positions. The irradiation has been made in\nthe whole region and the electrical resistance was measured between\nthe third and second electrodes from right.} \\label{samples_2}\n\\end{figure}\n\n\n\n\n\n\n\\section{Results and discussion}\\label{results}\n\n\\subsection{In-situ transport measurements}\n\n\\begin{figure}[htb]%\n\\includegraphics[width=0.9\\textwidth]{Fig2.eps}\n\\caption{Resistance as a function of time before, during and after\nGa$^+$ irradiation inside the FIB chamber for the samples (a)\ncrystalline graphite flake (first irradiation with a fluence of $5\n\\times 10^{11}$~cm$^{-2}$, (b) LCMO film and (c) Co film \\#1. For\nthese two samples the used fluences are written in the figures.\nAll these measurements were done in-situ and at room temperature.\n} \\label{rgra}\n\\end{figure}\n\n\\begin{figure}[htb]%\n\\includegraphics[width=0.6\\textwidth]{SRIM.eps}\n\\caption{(a) Qualitative distribution of implanted Ga ions as a\nfunction of penetration depth into the sample for the three studied\nmaterials. (b) Similarly for the vacancy density but taking into\naccount fixed fluences. The curves were obtained using the stopping\nand range of ions in matter SRIM or IIS\n\\protect\\cite{ziegler,Ziegler2} taking into account the energy of the\nGa$^+$-ions and the material densities.} \\label{SRIM}\n\\end{figure}\n\nA detailed  study of the electrical resistance of the above mentioned\nmaterials was realized  {\\em in-situ} during the Ga$^+$-ion\nirradiation in the microscope chamber. In the case of the graphite\nsample we measured simultaneously the resistance of the covered and\nuncovered parts before, during and after irradiation.\nFigure~\\ref{rgra} shows the changes observed in this sample during\nand after the first irradiation of  fluence  $5 \\times 10\n^{11}~$cm$^{-2}$. This fluence produces nominally $10^3$~ppm vacancy\nconcentration inside the sample whereas the Ga concentration\nimplanted is less than 1~ppm, see Fig.~\\ref{SRIM}. The disorder\nproduced by the irradiation increases the resistance of the uncovered\npart of this sample by a factor $> 4$.\n\nThe resistance of the covered part remains unchanged within $10^{-4}$\nrelative change indicating that the 300~nm thick resist  was enough\nto stop the Ga$^+$-ions as expected since according the SRIM\ncalculations the maximal penetration of the Ga$^+$-ions in the resist\nshould be $\\simeq 75~$nm. Immediately after irradiation the\nresistance of the uncovered part starts to decay exponentially with\ntwo characteristic relaxation times, as has been also observed after\nproton irradiation at room temperature \\cite{arn09}. This time\nrelaxation is observed for all samples just after the irradiation\nfinishes. This decay is related to local thermal relaxation process\nand  to the diffusion of carbon interstitials and vacancies\n\\cite{niw95,lee05_vac}.\n\nQualitatively speaking, similar resistance changes during the\nirradiation process are also observed in the two other samples, see\nFig.~\\ref{rgra}(b,c). In the case of the LCMO sample the maximal\npenetration of the ions is $\\sim 45~$nm, see Fig.~\\ref{SRIM}(a).\nBecause the thickness of this sample is 35~nm, part of the\nGa$^+$-ions are expected to be implanted and the rest to go through\nthe sample generating a considerable amount of defects as can be seen\nin the calculated curves, see Fig.~\\ref{SRIM}(b). In the case of the\nCo wire \\#1 the maximal ion penetration is $\\sim 30~$nm~$ < 57~$nm\nthickness, see Fig.~\\ref{SRIM}, therefore the produced defects plus\nGa implantation are responsible for the relatively small increase in\nthe electrical resistance, see Fig.~\\ref{rgra}(c). To study the\ninfluence on the transport properties mainly due to the produced\ndefects by the Ga$^+$ irradiation, a second Co~\\#2 sample with less\nthickness has been studied and its results are discussed below.\n\nThe effect of the Ga irradiation to sample volume expansion\n(thickness swelling) as well as milling (thickness reduction) depends\non the ion fluence, ion energy and target material. As an example we\nrefer to the work done in Ref.~\\cite{pal08} where such studies were\ndone on La$_{0.7}$Sr$_{0.3}$MnO$_3$ thin films. According to this\nwork and taking into account our used fluences, any thickness\nincrease as well as any milling are completely negligible and do not\naffect the changes measured in the resistance.\n\n\\subsection{Temperature and magnetic field dependence before and after irradiation}\n\n1.{\\em Graphite}: The special lattice structure of graphite and\nthe weak coupling between graphene layers make graphite a quasi-2D\nsemimetal, which carrier density depends strongly on the lattice\ndefects like vacancies and\/or impurities. In a recent work it was\ndemonstrated that the electrical resistance of thin graphite\ncrystals of micrometer size changes after inducing less than 1~ppm\nvacancy concentration by ion irradiation \\cite{arn09}. This makes\ngraphite a extraordinary sensor for testing the efficiency of the\nresin cover as well as to show the dramatic changes produced by a\nrelatively weak Ga$^+$-ion irradiation fluence.\n\n\\begin{figure}[htb]%\n\\includegraphics[width=0.50\\textwidth]{Fig3.eps}\n\\caption{Normalized resistance as a function of temperature for the\ncrystalline graphite flake in the: (1) as-prepared state, (2) after\nthe first ($5 \\times 10^{11}~$cm$^{-2}$)  and (3) second ($1 \\times\n10^{12}~$cm$^{-2}$) Ga$^+$ irradiation.} \\label{rtgra}\n\\end{figure}\n\nFigure~\\ref{rtgra} shows the temperature dependence of the graphite\nsample in its three states, as prepared and after the two\nirradiations. The temperature dependence in the as-prepared state has\na semiconducting behavior above 50~K and metallic below. The\nsemiconducting behavior is mainly due to the increase in carrier\nconcentration with temperature because most of the carriers are\nthermally activated and the Fermi energy increases linearly with\ntemperature \\cite{arn09}. The metallic behavior below 50~K is not\nintrinsic of graphite but comes from internal interfaces between\ncrystalline regions parallel to the graphene layers but of slightly\ndifferent orientation\\cite{bar08}. The mentioned interfaces originate\nduring the pyrolysis process \\cite{interfaces}.\n\nThe first irradiation increased the resistance in all the temperature\nrange without changing strongly the relative change with temperature,\nsee Fig.~\\ref{rtgra}. The metallic part was shifted to below 25~K and\nits $T$-dependence gets weaker, suggesting that the irradiation also\naffected the properties of the internal interface(s). After the\nsecond irradiation the metallic region vanishes completely and the\nresistance decreases rather linearly with $T$, see Fig.~\\ref{rtgra}.\nWe note that the absolute value of the resistance is fairly\nproportional to the used fluence. Between the virgin state and first\nirradiation we have an increase in the resistance of a factor of 3.5.\nDoubling the fluence we expect an increase of a factor $\\sim 7$ of\nthe resistance of the as-received state, a factor in agreement with\nthe experimental observation.\n\nAs shown in \\cite{arn09} the increase in resistance is due to the\ndecrease in the mean free path that overwhelms the increase in the\ncarrier density that these irradiations produce inside the graphite\nstructure. This behavior is related to difference weights the carrier\ndensity $n$ and the mean free path $l$ have in the 2D resistivity,\ni.e. $R \\propto 1\/(n^{1\/2}l)$ in  contrast with the 3D resistivity\nequation $R \\propto 1\/(nl)$. Note also that the second irradiation\nproduces already a vacancy density that implies a vacancy distance of\nless than 2~nm in the graphene plane. At distances smaller than $\\sim\n3~$nm we do not expect a large increase in the carrier density with\nfurther irradiation \\cite{arn09}.\n\n\\begin{figure}[htb]%\n\\includegraphics[width=0.95\\textwidth]{Fig4.eps}\n\\caption{Magnetoresistance as a function of field applied normal to\nthe graphene planes for the sample in its virgin state (a), after the\nfirst (b) and second (c) irradiation. } \\label{mrgra}\n\\end{figure}\n\n\n\n\n\n\nFigure~\\ref{mrgra} shows the magnetoresistance (MR) vs. applied field\nat different constant temperatures for the graphite sample in the\nvirgin (a), first (b) and second (c) irradiated state. The MR in the\nvirgin state agrees with previous reports \\cite{bar08}. Note that the\nMR shows a quasi linear field behavior at low temperatures. Also\nanomalous is the systematic increase of the MR below 100~K. This is\nrelated to the decrease in the resistance with decreasing\ntemperature, see Fig.~\\ref{rtgra}. The MR reaches a value of 16 at\n8~T and 2~K. After the first irradiation the MR decreased a factor 16\nand remains practically temperature independent. After the second\nirradiation the MR decreased further a factor of 6 and shows a\nsimilar temperature independence. The data reveal that the\nShubnikov-de Haas (SdH) oscillations increase their amplitude and\nstart to be measurable at lower fields after the first irradiation\n\\cite{arn09}. This behavior is related to the increase in the carrier\ndensity due to the creation of defects, whereas the decrease in the\nMR is due to the decrease in the mean free path.\n\n\\begin{figure}[htb]%\n\\includegraphics[width=0.7\\textwidth]{Fig4b.eps}\n\\caption{First derivative of the resistance on field vs. inverse\nfield for the first (continuous line) and second (dotted line)\nirradiation. The data have been multiplied by a factor in order to\nshow both derivatives in the same $y-$axis scale.} \\label{sdh}\n\\end{figure}\n\nNote that after the second irradiation the SdH oscillations do not\nchange qualitatively (their absolute amplitude changes due to the\nlarge change of the MR), see Fig.~\\ref{sdh}. After the second\nirradiation we observe the first low-field oscillation at a similar\nfield and a similar oscillation period in reciprocal field as after\nthe first irradiation, see Fig.~\\ref{sdh}. These results indicate\nthat no further change in the carrier density has been produced for a\nvacancy distance less than 2~nm. Taking into account that 3~nm is of\nthe order of the range of modification of the electronic structure\nproduced by, e.g. a single vacancy \\cite{rufi00}, then a saturation\nof the carrier density is reached decreasing the vacancy distance\nbelow $\\sim 3~$nm but keeping the graphene structure. As expected,\nthe covered part of the sample does not show any change after the\nfirst or second irradiation, as can be seen in Fig.~\\ref{MRcovered}.\n\n\n\n\\begin{figure}[htb]%\n\\includegraphics[width=0.6\\textwidth]{MRC.eps}\n\\caption{Magnetoresistance of the upper covered part of the graphite\nflake, see Fig.~\\ref{samples_2}(a), at $T = 4~$K and in the virgin\nstate (continuous line) and after the second Ga$^+$ irradiation\n($\\circ$).} \\label{MRcovered}\n\\end{figure}\n\n\n\n\\begin{figure}[htb]%\n\\includegraphics[width=0.9\\textwidth]{Fig5.eps}\n\\caption{(a) Temperature dependence of the normalized resistance\nof the LCMO thin film before and after irradiation. The inset\nshows the same data but in a double logarithmic scale. (b) The MR\nvs. applied field for the as-prepared state. } \\label{tlaca}\n\\end{figure}\n\n\n\\begin{figure}[htb]%\n\\includegraphics[width=0.9\\textwidth]{Fig6.eps}\n\\caption{(a) The MR of the LCMO film after irradiation. Compare\nthese results with those in Fig.~\\protect\\ref{tlaca}(b) and note\nthe induced changes in the MR by the irradiation. (b)\nMagnetoresistance as a function of field at 75~K before and after\nirradiation in a reduced field range to show the clear increase in\ncoercivity. This increase is observed at all temperatures in the\nferromagnetic state.} \\label{hlaca}\n\\end{figure}\n2.{\\em LaCaMnO film}: The manganite sample undergoes a paramagnetic\ninsulator to ferromagnetic metal  transition leading to a sharp peak\nin the resistance near its Curie temperature as shown in\nFig.~\\ref{tlaca}(a). For our sample this peak is observed at $T_c =\n106~$K. Interestingly, after irradiation the temperature dependence\ndoes change only in the ferromagnetic state of the sample, which\nshows now a $T_c = 95~$K, see Fig.~\\ref{tlaca}(a). The measured MR of\nthis sample in the as-prepared state agrees with published literature\nand shows hysteretic behavior in the ferromagnetic state whereas no\nhysteresis in the paramagnetic state above $T_c$, see\nFig.~\\ref{tlaca}(b).\n\nThe influence of ion irradiation on the magnetic and transport\nproperties of manganites were studied in the past but mainly at much\nhigher ion-energies, see e.g.\nRefs.~\\cite{str97,wol01,cha05,ram09,pal08}. In general at fluences\nabove $10^{12}~$cm$^{-2}$ the ion irradiation decreases the\nmetal-insulator transition temperature and the magnetic hysteresis\ngets broader reflecting the increase of the pinning of the domain\nwalls by the induced defects. A similar behavior is observed in our\nsample, see Fig.~\\ref{hlaca}(a).  Figure~\\ref{hlaca}(b) shows in\ndetail the MR between -2~T and 2~T at 75~K. The irradiation induces\nan increase in the coercivity $H_c$, from 0.1~T to 0.42~T after\nirradiation, defined at the maxima of the MR, as well as in the\noverall hysteresis width.\n\nThese results indicate that care should be taken when changes in\nthe transport properties of ferromagnetic oxides are observed\nafter microstructuring the samples with FIB. The observed results\nafter patterning might not be due to the change of the sample\ndimension but to the induced structural changes by the Ga$^+$\nirradiation.\n\n\n\n\\begin{figure}[htb]%\n\\includegraphics[width=0.6\\textwidth]{Fig7.eps}\n\\caption{Scanning electron microscope image of the Co sample ~\\#2\nwith its four electrodes for resistance measurement. The thickness\nof the wire was 22~nm.} \\label{Co}\n\\end{figure}\n\n\n3.{\\em Co film}: Figure~\\ref{Co} shows a SEM picture of the\nCo-sample \\#2 with the four electrodes for the resistance\nmeasurement. The resistance as a function of temperature and\nmagnetic field before and after a Ga$^+$ irradiation is shown in\nFig.~\\ref{Co-T-B}(a) and (b). Due to the smaller thickness of this\nsample the irradiation at 30~kV produces mainly defects instead of\ndoping since the Ga$^+$ ions stop beyond the sample thickness,\ni.e. inside the substrate. The influence of the induced defects in\nthe Co sample can be clearly recognized by the increase in the\nabsolute value of the resistance. For example at 275~K the\nresistance of the Co-sample~\\#2 increases from $36~\\Omega$ to\n$175~\\Omega$ flattening the temperature dependence, see\nFig.~\\ref{Co-T-B}(a). The MR shown in Fig.~\\ref{Co-T-B}(b)\nindicates a clear change in the hysteresis indicating a change in\npinning of the domain walls. These results are qualitatively\nsimilar to those obtained for the manganite shown above, see\nFig.~\\ref{hlaca}.\n\n\n\n\n\\begin{figure}[htb]%\n\\includegraphics[width=0.9\\textwidth]{Fig8.eps}\n\\caption{(a) The normalized resistance of the Co~\\#2 wire as a\nfunction of temperature for the as-prepared and after irradiation\nstates. (b) MR of this sample for the two states at 4~K. The\nGa$^+$-irradiation fluence was $2.2 \\times 10^{11}~$cm$^{-2}$.\nNote the change in the MR at the coercivity field.} \\label{Co-T-B}\n\\end{figure}\n\n\n\\section{Conclusion}\\label{con}\n\nIn conclusion our studies indicate clearly that care should be taken\nwith the change of the intrinsic properties of the materials when FIB\ndevices are used for patterning, cutting or just for deposit other\nmaterials for contacts, for example. Our work demonstrates that\nalready the usual Ga$^+$ fluences needed for a precise alignment of\nthe Ga$^+$ beam before really using it, induce relevant changes in\nthe transport properties of the three different materials studied\nhere. Using a thin crystalline graphite flake we were able to\ndemonstrate also that covering the sample with a sufficiently thick\nresist film one can avoid the irradiation damage completely. This\nindicates that in principle one can use this technique to protect\ncertain parts and produce defined changes in other parts of the\nsample of interest. This method might be used to  induce  changes in\nthe hysteretic properties of ferromagnetic micro and nano-structures\nor in the electronic density in graphite or multigraphene in specific\nparts of the sample, for example.\n\n\n\n\n\n\\ack This work has been possible with the support of the DFG grant\nDFG ES86\/16-1. The authors S.D. and G.B. are supported by the\nGraduate school BuildMona and the Collaborative Research Center (SFB\n762) ``Functionality of Oxide Interfaces'', respectively.\n\n\n\n\n\\bibliographystyle{unsrt}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLoop quantum gravity (for lecture books, see \\cite{rovelli2007quantum,thiemann2007modern,Gambini:2011zz}) proposes a non-perturbative and background independent framework for quantum gravity. Based on a 3+1 splitting of space-time distinguishing time from the 3d space, it realizes a canonical quantization of the general relativity reformulated as a gauge field theory. The canonically conjugate fields are the triad, defining the local 3d frame, and the Ashtekar-Barbero $\\mathrm{SU}(2)$ connection \\cite{PhysRevLett.57.2244,Barbero:1994ap,Immirzi:1996di}. Quantum states of geometry, called spin network states, define the excitations of those fields over the kinematical Ashtekar-Lewandowski vacuum of vanishing triad and connection (corresponding to the ``nothing''-state of a degenerate vanishing metric). Their dynamics is implemented through the Hamiltonian constraints, ensuring the invariance of the theory under space-time diffeomorphisms.\nSeveral explicit proposals for the quantum dynamics exist, either in the pure canonical formalism from the original Thiemann definition of the Hamiltonian constraint operator \\cite{Thiemann:1996ay,Thiemann:1996aw,Thiemann:1996av} to more recent constructions \\cite{Thiemann:2003zv,Alesci:2011aj,Alesci:2015wla,Assanioussi:2015gka} (for a review see \\cite{Bonzom:2011jv}), or in a covariant path integral approach with transition amplitudes defined from spinfoam models as in the EPRL model  \\cite{Engle:2007wy,Geloun:2010vj,rovelli2014quantum} and variations \\cite{Dupuis:2011fz,Speziale:2012nu,Wieland:2013cr} (for a review of the spinfoam framework see \\cite{Livine:2010zx,Perez:2012wv,Bianchi:2012nk}).\n\nThe main challenges in this context are the correct definition of the quantum dynamics and the coarse-graining of the theory. These two issues are intimately intertwined in that the proper quantum gravity dynamics should address and solve the perturbative non-renormalisability of general relativity, and more generally because a consistent theory of quantum gravity should give us the effective dynamics of the geometry at all scale of length and energy and provide us with a flow from the probably discrete and quantum dynamics at the Planck scale to the classical dynamics of a classical space-time manifold prescribed by general relativity.\nThe main difficulty in realizing this program in the loop quantum gravity  framework is to proceed to a coarse-graining of the gravitational degrees of freedom in a background independent theory with no a priori length or energy scale and no a priori regular geometrical background on which to define a coarse-graining procedure.\nThis translates into the problem of defining new vacuum states representing non-degenerate metrics and geometries (such as the flat Minkowski space-time) and working out how to expand the quantum gravity theory, initially defined above the ``nothing''-state,   around them with an explicit dictionary between the fundamental geometry excited states and the new effective gravity excitations.\n\nSome progress in this direction has been achieved by Koslowski and Sahlmann in \\cite{Koslowski:2011vn}, where they define new vacuum states for loop quantum gravity peaked on some classical field configuration for the triad and connection and describe the spin network excitations above them. Another approach by Dittrich and Geiller was to define another vacuum state, given as the flat connection vacuum of a topological BF theory, and to work out a Hilbert space better suited to account for curvature defects  \\cite{Dittrich:2014wpa,Bahr:2015bra}. Another line of research in the spinfoam framework is the promising  renormalisation program for tensor models and group field theories  \\cite{Oriti:2013aqa,Rivasseau:2011hm,Rivasseau:2013uca,Carrozza:2013mna,Carrozza:2014rya}, but this deviates from the canonical point of view that we will pursue in the present paper. Indeed, we would like to propose another path to define the effective loop quantum gravity dynamics on non-trivial background quantum geometries following the previous work on the coarse-graining of spin network states \\cite{Livine:2006xk,Livine:2013gna}.\n\n\\medskip\n\nIn loop quantum gravity, quantum states of geometry are wave-functions of the Ashtekar-Barbero connection. More precisely, they are defined as cylindrical functionals of that connection, in that they realize a finite sampling of the connection. Indeed each wave-function is constructed with respect to a graph $\\Gamma$ (embedded in the canonical hypersurface) and depends on the holonomies of the connection along the graph edges (defined as $\\mathrm{SU}(2)$ group elements). The set of wave-functions is obtained by taking the union over all embedded graphs with the requirement of cylindrical consistency\\footnotemark, that is a wave-function initially defined over a given graph $\\Gamma$ is considered as equivalently defined over any refinement of that graph (i.e. any graph containing $\\Gamma$ as a subgraph). This union quotiented by the cylindrical consistency is rigorously defined as a projective limit \\cite{Ashtekar:1994mh,Ashtekar:1994wa}. It leads to the Hilbert space of quantum states of geometry of loop quantum gravity and has been shown to be the $L^{2}$ space of functionals of the connection with respect to the Ashtekar-Lewandowski measure \\cite{Ashtekar:1993wf}.\n\\footnotetext{The definition of cylindrical consistency can be generalized and extended to a mapping or identification between two wave-functions living on a given graph and a refinement of it. Formulated as such, it becomes equivalent to the choice of a coarse-graining procedure for the quantum states of geometry. This logic has been used to construct new Hilbert spaces for loop quantum states describing the excited states of geometry above non-trivial vacua \\cite{Koslowski:2011vn}.\n}\n\n\n\n\\begin{figure}\n\n\n\\begin{subfigure}[t]{.3\\linewidth}\n\\begin{tikzpicture}[scale=1]\n\\coordinate(A) at (0,0);\n\\coordinate(B) at (2,0);\n\\coordinate(C) at (2,-2);\n\\coordinate(D) at (0,-2);\n\n\\draw (A) to[bend left] node[midway,sloped]{$>$} node[midway,above]{$\\ell_1$} (B);\n\\draw (B) to[bend left] node[midway,sloped]{$>$} node[midway,right]{$\\ell_2$} (C);\n\\draw (C) to[bend left] node[midway,sloped]{$>$} node[midway,above]{$\\ell_3$} (D);\n\\draw (D) to[bend right] node[midway,sloped]{$>$} node[midway,left]{$\\ell_4$} (A);\n\n\\draw (A) to[bend left] node[midway,sloped]{$>$} ++(-1,0.5) node[above]{$j_1$};\n\\draw (B) to[bend right] node[midway,sloped]{$>$} ++(1,0.5) node[above]{$j_2$};\n\\draw (C) to[bend right] node[midway,sloped]{$>$} ++(1,-0.5) node[right]{$j_3$};\n\\draw (D) to[bend left] node[midway,sloped]{$>$} ++(-1,-0.5) node[left]{$j_4$};\n\n\\draw (A) node {$\\bullet$} node[above]{$i_1$};\n\\draw (B) node {$\\bullet$} node[above]{$i_2$};\n\\draw (C) node {$\\bullet$} node[right]{$i_3$};\n\\draw (D) node {$\\bullet$} node[left]{$i_4$};\n\\end{tikzpicture}\n\\caption{Spin networks are graphs coloured with spins at the edges and intertwiners at the vertices.}\n\\label{fig:spinnetwork_a}\n\\end{subfigure}\n\\hspace{2mm}\n\\begin{subfigure}[t]{.3\\linewidth}\n\\begin{tikzpicture}[scale=1]\n\\coordinate (O) at (0,0,0);\n\n\\coordinate (A) at (0,1.061,0);\n\\coordinate (B) at (0,-0.354,1);\n\\coordinate (C) at (-0.866,-0.354,-0.5);\n\\coordinate (D) at (0.866,-0.354,-0.5);\n\n\\draw[blue] (A) -- (B);\n\\draw[blue] (A) -- (C);\n\\draw[blue] (A) -- (D);\n\\draw[blue] (B) -- (C);\n\\draw[dashed,blue] (C) -- (D);\n\\draw[blue] (D) -- (B);\n\n\\draw[dotted] (O) -- ++(0,-0.531,0);\n\\draw (0,-0.531,0) -- ++(0,-0.531,0);\n\n\\draw[dotted] (O) -- ++(0,0.177,-0.5);\n\\draw[dashed] (0,0.177,-0.5) -- ++(0,0.177,-0.5);\n\n\\draw[dotted] (O) -- ++(0.433,0.177,0.25);\n\\draw (0.433,0.177,0.25) -- ++(0.433,0.177,0.25);\n\n\\draw[dotted] (O) -- ++(-0.433,0.177,0.25);\n\\draw (-0.433,0.177,0.25) -- ++(-0.433,0.177,0.25);\n\n\\draw (O) node{$\\bullet$};\n\\draw[blue] (A) node{$\\bullet$};\n\\draw[blue] (B) node{$\\bullet$};\n\\draw[blue] (C) node{$\\bullet$};\n\\draw[blue] (D) node{$\\bullet$};\n\\end{tikzpicture}\n\n\\caption{Each node of a spin network can naturally be interpreted as a polyhedron (in blue on the figure), each face corresponding to an edge of the graph.}\n\\label{fig:spinnetwork_b}\n\\end{subfigure}\n\\hspace{2mm}\n\\begin{subfigure}[t]{.3\\linewidth}\n\\begin{tikzpicture}[scale=1]\n\\coordinate (A1) at (0,0,-2);\n\\coordinate (B1) at (0,1,0);\n\\coordinate (C1) at (-0.866,-0.5,0);\n\\coordinate (D1) at (0.866,-0.5,0);\n\n\\fill[blue!50] (B1) -- (C1) -- (D1) -- cycle;\n\\fill[blue!50] (B1) -- (D1) -- (A1) -- cycle;\n\n\\draw[blue] (A1) -- (B1);\n\\draw[blue,dashed] (A1) -- (C1);\n\\draw[blue] (A1) -- (D1);\n\\draw[blue] (B1) -- (C1);\n\\draw[blue] (C1) -- (D1);\n\\draw[blue] (D1) -- (B1);\n\n\n\\draw[blue] (A1) node{$\\bullet$};\n\\draw[blue] (B1) node{$\\bullet$};\n\\draw[blue] (C1) node{$\\bullet$};\n\\draw[blue] (D1) node{$\\bullet$};\n\n\\coordinate (A2) at (0,0,0.5);\n\\coordinate (B2) at (0,-1,0);\n\\coordinate (C2) at (-0.866,0.5,0);\n\\coordinate (D2) at (0.866,0.5,0);\n\n\\fill[red!50] (A2) -- (B2) -- (D2) -- cycle;\n\\fill[red!50] (A2) -- (C2) -- (D2) -- cycle;\n\\fill[red!50] (A2) -- (B2) -- (C2) -- cycle;\n\n\\draw[red] (A2) -- (B2);\n\\draw[red] (A2) -- (C2);\n\\draw[red] (A2) -- (D2);\n\\draw[red] (B2) -- (C2);\n\\draw[red] (C2) -- (D2);\n\\draw[red] (D2) -- (B2);\n\n\\draw[red] (A2) node{$\\bullet$};\n\\draw[red] (B2) node{$\\bullet$};\n\\draw[red] (C2) node{$\\bullet$};\n\\draw[red] (D2) node{$\\bullet$};\n\\end{tikzpicture}\n\n\\caption{Spin network geometry is not Regge geometry since the shapes of the faces of the glued polyhedra do not have to match.}\n\\label{fig:spinnetwork_c}\n\\end{subfigure}\n\n\n\\caption{The geometrical interpretation of spin networks}\n\\label{fig:spinnetwork}\n\\end{figure}\n\n\n\nSpin networks provide basis states of this Hilbert space. They are introduced as diagonalizing the area and volume operators, both shown to have discrete spectra \\cite{Rovelli:1994ge,Rovelli:1995ac}. A spin network state, as drawn on fig.\\ref{fig:spinnetwork_a}, lives on a given graph $\\Gamma$ (and therefore lives on any refinement of that graph by cylindrical consistency) and is defined by spins (as half-integers determining an irreducible representation of the Lie group $\\mathrm{SU}(2)$) on each edge and intertwiners (as singlet states) on each vertex or node of the graph. Spins define quanta of area while intertwiners give the quanta of volume. This hints towards a geometrical interpretation of spin networks as discrete geometries. This can be realized explicitly and  spin networks have been shown to be the quantization of twisted geometries \\cite{Freidel:2010aq,Dupuis:2012yw}, which generalize 3d Regge geometries to account for some torsion.\nNodes of the graph represent polyhedra, which are glued along faces dual to the graph edges, as illustrated on fig.\\ref{fig:spinnetwork_b}.\nThe precise face matching of Regge geometries is relaxed to a simpler area face matching, and the resulting potential shape mismatch is interpreted as torsion along the graph edge, as depicted on fig.\\ref{fig:spinnetwork_c}. \n\nThe Hamiltonian operators of loop quantum gravity then act on both algebraic and combinatorial data, meaning that they modify both  spins and intertwiners living on a given graph and the graph itself. Thus the loop quantum gravity (LQG) dynamics seems to be a careful balance between fixed-graph dynamics and graph-changing dynamics, which reflects the classical dynamics of general relativity as both dynamics on a given space-time manifold and of the space-time geometry itself. This mixture between these two types of dynamics render analytical and numerical calculations extremely difficult.\nThe usual strategy for discrete systems on fixed graphs, as in condensed matter theory, is to coarse-grain the theory, that is, to integrate the microscopic degrees of freedom inside bounded regions, thus assimilated to points, and to write effective theories for the relevant macroscopic degrees of freedom. This process of coarse-graining ultimately leads to the continuum limit of the theory.\nOne can also study the statistical physics of a varying graph, for instance using matrix models for 2d quantum gravity. Putting these two ingredients together, for example to study matter coupled to 2d quantum gravity through condensed matter models on random lattices, is much more involved. Results in this direction, like the KPZ conjecture  \\cite{Knizhnik:1988ak}, mostly relie on conformal field theory techniques in the continuum limit.\n\n\\medskip\n\nIn the loop quantum gravity context,  through the logic of coarse-graining, we would like to map the varying graphs dynamics onto a fixed graph dynamics.\nThe rational behind this is the following: starting from a base graph, each node will correspond to a varying coarse-grained region. This means that we will add some extra internal degrees of freedom to the graph vertices in the effective theory, which should reflect that each vertex represent in fact a possibly varying subgraph itself.  This method should mimic a development around this base graph considered as a skeleton graph for the gravity excitations. \nDigging deeper in the structure of the spin networks, the wave-functions carry non-trivial curvature around the loops of the graph (when the composition of the holonomies along the edges of the loops does not yield the identity). \nWhen coarse-graining, curvature should build up and the effective coarse-grained vertices should naturally describe locally curved geometries. \nAs illustrated in fig.\\ref{fig:newstructure}, we will need additional data at each vertex to encode the coarse-grained curvature now located at the vertices. \nWe will realize this program using the coarse-graining through gauge fixing procedure already developed in  \\cite{Freidel:2002xb,Livine:2013gna}. The idea is that the gauge fixing procedure collapses a subgraph into a single vertex plus some additional (self-)loops connecting that vertex to itself. These self-loops carry the curvature initially carried by the loops of the subgraph. In this picture, coarse-graining a spin network state thus leads to a state living on a coarser graph with many little loops living at each vertex.\n\nReversing the logic of this procedure, we propose to fix a background graph and define the Fock space of ``loopy spin networks'' above that graph. The base states will be spin network states living on the background graph itself, while excitations will be spin networks living on the graph plus an arbitrary number of little loops attached to its vertices. These little loops allow to represent excitations of the curvature and take into account that each vertex of the background graph is in fact a coarse structure which could be unfolded into a non-trivial, possibly complex, subgraph. Taking into account these little loops as localized excitations of the quantum geometry allow to project the graph changing LQG dynamics onto a fixed background graph but with dynamical curvature excitations living at each vertex.\nIn some sense, the underlying full graph is still dynamical and changes, but we always coarse-grain it to the same skeleton graph plus some little loops. The dynamics then affect the little loops without touching the skeleton graph and of course change the algebraic  data -spin and intertwiners- carried by the edges and vertices of the graph.\nThis explicitly realizes  the idea proposed in the conclusion of \\cite{Livine:2013gna}.\n\nSo we are here proposing an expansion of the theory around an arbitrary non-trivial background graph, in some sense truncating the dynamics by a coarse-graining procedure to keep the quantum states living on that chosen background graph with some localized excitations of the geometry. This is rather different from the expansion around a continuous background metric, proposed up to now in loop quantum gravity as in \\cite{Koslowski:2011vn}, and it will be necessary to later compare these two approaches  in a continuum limit of spin networks.\n\n\n\n\n\n\\begin{figure}\n\\centering\n\n\\begin{tikzpicture}[scale=0.85]\n\\coordinate(A) at (0,0);\n\\coordinate(B) at (2,0);\n\\coordinate(C) at (2,-2);\n\\coordinate(D) at (0,-2);\n\n\\draw (A) -- (B) node[midway,above] {$\\ell_1$};\n\\draw (B) -- (C) node[midway,right] {$\\ell_2$};\n\\draw (C) -- (D) node[midway,below] {$\\ell_3$};\n\\draw (D) -- (A) node[midway,left] {$\\ell_4$};\n\\draw (B) -- (D) node[midway,sloped,above] {$\\ell_5$};\n\n\\draw (A) -- ++(-1,0.5) node[above] {$j_1$};\n\\draw (B) -- ++(1,0.5) node[above] {$j_2$};\n\\draw (C) -- ++(1,-0.5) node[below] {$j_3$};\n\\draw (D) -- ++(-1,-0.5) node[below] {$j_4$};\n\n\\draw (A) node {$\\bullet$} node[above]{$i_1$};\n\\draw (B) node {$\\bullet$} node[above]{$i_2$};\n\\draw (C) node {$\\bullet$} node[below]{$i_3$};\n\\draw (D) node {$\\bullet$} node[below]{$i_4$};\n\n\\draw[gray,dashed] (1,-1) circle(2);\n\n\\draw[->,>=stealth,very thick] (3.5,-1) -- (5.5,-1);\n\n\\coordinate(O) at (8,-1);\n\\draw (O) -- ++(-2,1) node[above] {$j_1$};\n\\draw (O) -- ++(2,1) node[above] {$j_2$};\n\\draw (O) -- ++(2,-1) node[below] {$j_3$};\n\\draw (O) -- ++(-2,-1) node[below] {$j_4$};\n\n\\draw[fill=lightgray] (O) circle(0.5);\n\\draw[scale=2] (O) node{\\textbf{?}};\n\n\\end{tikzpicture}\n\n\\caption{When coarse-graining, the curvature carried by the loops in the collapsed bounded region leads to curvature. Coarse-grained vertices thus need a new structure to be described.}\n\\label{fig:newstructure}\n\\end{figure}\n\n\\medskip\n\nThis paper is organized as follows.\nThe first section will review the Hilbert space of spin networks for loop quantum gravity, underlining the cylindrical consistency requirement, and the ``coarse-graining through gauge-fixing'' of these quantum states of geometry. From this perspective, we will introduce a hierarchy of possible extensions of spin networks encoding extra information at the graph nodes: folded, loopy and tagged spin networks, from the finer to coarser objects.\nFolded spin networks allow for an arbitrary number of little (self-)loops at every node of the graph and moreover contain the data of a circuit at each node, that is a tree linking  the little loop ends to the edges attached to the node. Loopy spin networks forget about the local circuit data, while tagged spin networks simply trace out all the little loop data and only retains the resulting closure defect at each node.\n\nThe second section is dedicated to the definition and investigation of loopy spin networks. We discuss the holonomy operator, which is the elementary brick of the loop quantum gravity formalism, and check the compatibility of our definition with the cylindrical consistency conditions. We apply this construction to the topological BF theory and solve for the physical state of flat connections in our new Hilbert of loopy spin networks.\nWe define the BF Hamiltonian constraints as holonomy constraints peaking the group elements along the little loops on the identity. We show that these are (unexpectedly) not enough to enforce the uniqueness of the physical state and lead to an infinite-dimensional space of (almost)-flat states defined from higher derivatives of the $\\delta$-distribution. This is due to the highly non-trivial structure of the intertwiner space recoupling the loops. We supplement these constraints with new Laplacian constraints, which decouple the loop and trivialize the intertwiner, finally leading to a unique flat state defined by the $\\delta$-distribution.\n\nIn a third section, we explore the possibility of endowing the little loops at the graph's nodes  with bosonic statistics and define the symmetrized Fock space of loopy spin networks. We discuss the interplay between loop creation, annihilation and spin shift in the definition of the holonomy operator. This leads us to define the Hamiltonian constraints for BF theory in terms of creation and annihilation operators.\n\nThe fourth section develops tagged spin networks and shows how they provide a basis for reduced density matrix when  coarse-graining spin networks.\nWe conclude this paper with a discussion on the potential applications of this new framework, for instance using fixed skeleton graph as background lattices or to the coarse-graining of loop quantum gravity dynamics.\n\n\\section{Spin networks and their coarse-graining}\n\\label{sec:overview}\n\n\\subsection{Spin networks as projective limits}\n\\label{ProjectiveLimits}\n\nLoop quantum gravity is based on the first order reformulation of general relativity in terms of the Ashtekar-Barbero variables\\cite{PhysRevLett.57.2244}. The fundamental variables of the theory on the canonical hypersurface  are the densitized triad and the Ashtekar-Barbero connection\\footnotemark, which are endowed with the following symplectic structure:\n\\begin{equation}\n\\{E^a_i(x),A^j_b(y)\\} = \\frac{\\beta \\kappa}{2} \\delta^a_b \\delta^j_i \\delta(x-y)\\,,\n\\end{equation}\nwith all other brackets being zero.\nThe indices $a,b,c,..$  denote space coordinates while the indices $i,j,k,...$ refer to tangent space coordinates.\nThe canonical fields are the densitized triad $E^a_i(x)$ and  the Ashtekar-Barbero $\\mathrm{SU}(2)$ connection $A^j_b(x)$. The Poisson backet coupling is given in terms of the gravitational constant $\\kappa = 16\\pi G$ and  the Immirzi parameter $\\beta$ \\cite{Barbero:1994ap,Immirzi:1996di,Holst:1995pc}. \n\\footnotetext{The Ashtekar-Barbero connection is only a space connection defined on the canonical hypersurface and is not generically the pull-back of a space-time connection \\cite{Samuel:2000ue,Alexandrov:2001wt,Geiller:2011cv,Geiller:2011bh,Charles:2015rda}, except in the case of the self-dual and anti-self dual connections given by the purely imaginary choice of Immirzi paramater $\\beta=\\pm i$.} \nThe classical theory is defined by imposing on this phase space a set of seven first class constraints: the three Gauss constraints generating the local $\\mathrm{SU}(2)$ gauge invariance and the four constraints generating space-time diffeomorphisms. \n\n\n\\begin{figure}[t!]\n\\centering\n\\begin{tikzpicture}\n\\coordinate(A) at (0,0);\n\\coordinate(B) at (2,0);\n\\coordinate(C) at (2,-2);\n\\coordinate(D) at (0,-2);\n\n\\draw (A) to[bend left] node[midway,sloped]{$>$} node[midway,above]{$g_e$} node[midway,below]{e} (B);\n\\draw[lightgray] (B) to[bend left] node[midway,sloped]{$>$} (C);\n\\draw[lightgray] (C) to[bend left] node[midway,sloped]{$>$} (D);\n\\draw[lightgray] (D) to[bend right] node[midway,sloped]{$>$} (A);\n\n\\draw[lightgray] (A) to[bend left] node[midway,sloped]{$>$} ++(-1,0.5);\n\\draw[lightgray] (B) to[bend right] node[midway,sloped]{$>$} ++(1,0.5);\n\\draw[lightgray] (C) to[bend right] node[midway,sloped]{$>$} ++(1,-0.5);\n\\draw[lightgray] (D) to[bend left] node[midway,sloped]{$>$} ++(-1,-0.5);\n\n\\draw (A) node {$\\bullet$} node[above]{$s(e)$};\n\\draw (B) node {$\\bullet$} node[above]{$t(e)$};\n\\draw[lightgray] (C) node {$\\bullet$};\n\\draw[lightgray] (D) node {$\\bullet$};\n\\end{tikzpicture}\n\\caption{We consider cylindrical wavefunctions: the function only depends on a support graph with oriented edges. Each edge $e$ carries a colouring $g_e$ which is a group element and corresponds to the (open) holonomy along the corresponding path.}\n\\label{fig:graph}\n\\end{figure}\nAt the quantum level, we consider cylindrical wave-functions of the Ashtekar-Barbero connection $A$. We choose an arbitrary oriented graph $\\Gamma$ embedded in the canonical hypersurface and consider functions of the holonomies of the connection along the links or edges $e$ of the graph, as illustrated in fig.\\ref{fig:graph}:\n\\begin{equation}\n\\psi_{\\Gamma}[A]\n\\,\\equiv\\,\n\\psi\\left(\n\\{U_{e}[A]\\}_{e\\in\\Gamma}\n\\right)\\,,\n\\qquad\nU_{e}[A]\\in\\mathrm{SU}(2)\\,.\n\\end{equation} \nSuch functionals realize a finite sampling of the connection along the considered graph. We require these functionals to solve the Gauss law, that is to be  invariant under local $\\mathrm{SU}(2)$ gauge transformations. These acts at the end points of the holonomies, that is at the nodes or vertices $v$ of the graph $\\Gamma$:\n\\begin{equation}\n\\label{gaugeinv}\n\\forall h_{v}\\in\\mathrm{SU}(2)^{\\times V}\\,,\n\\quad\n\\psi\\left(\n\\{U_{e}[A]\\}_{e\\in\\Gamma}\n\\right)\n\\,=\\,\n\\psi\\left(\n\\{h_{s(e)}^{-1}\\,U_{e}[A]\\,h_{t(e)}\\}_{e\\in\\Gamma}\n\\right)\\,,\n\\end{equation}\nwhere $V$ is the number of vertices of the graph $\\Gamma$, while $s(e)$ and $t(e)$ respectively denote the source and target vertices of the oriented edge $e$. The Hilbert space of states of the fixed graph $\\Gamma$ is defined by endowing this set of wave-functions with the natural scalar product induced by the Haar measure on $\\mathrm{SU}(2)$:\n\\begin{equation}\n{\\mathcal H}_{\\Gamma}\\,\\equiv\\,L^{2}\\left(\\mathrm{SU}(2)^{E}\/\\mathrm{SU}(2)^{V}\\right)\\,,\\nonumber\n\\end{equation}\n\\begin{equation}\n\\forall \\Psi\\,,\\widetilde{\\Psi}\\,\\in\\,{\\mathcal H}_{\\Gamma}\\,,\\quad\n\\langle \\Psi | \\widetilde{\\Psi} \\rangle\n\\,=\\,\n\\int_{\\mathrm{SU}(2)^{E}} \\prod_{e=1}^{E} dg_e\\,\\,\n\\overline{\\Psi(g_1,...,g_{E})} \\widetilde{\\Psi}(g_1,...,g_{E})\\,,\n\\end{equation}\nwhere $E$ counts the number of edges in the graph. A basis of this space is provided by the spin networks with support on the graph $\\Gamma$. Technically, these are obtained through the Peter-Weyl decomposition of $L^{2}$ functions on the Lie group $\\mathrm{SU}(2)$ in terms of the orthogonal Wigner matrices in the irreducible representations of $\\mathrm{SU}(2)$. As a result, a spin network state is labeled by a spin on each edge, which is a half-integer $j_{e}\\in{\\mathbb N}\/2$ determining the corresponding irreducible $\\mathrm{SU}(2)$-representation ${\\mathcal V}^{j_{e}}$ of dimension $(2j_{e}+1)$,  and an intertwiner $i_{v}$ at each vertex $v$, which is an invariant tensor in the tensor product of the representations living on the incoming and outgoing edges attached to the vertex $v$:\n\\begin{equation}\ni_{v}:\\bigotimes_{e|s(e)=v}{\\mathcal V}^{j_{e}}\\longrightarrow\\bigotimes_{e|t(e)=v}{\\mathcal V}^{j_{e}}\\,.\n\\end{equation}\nThe spin network function is defined by contracting the chosen intertwiners with the Wigner matrices of the holonomies living along the graph edges:\n\\begin{eqnarray}\n\\psi_{\\Gamma}^{\\{j_{e,i_{v}}\\}}\n\\Big{(}\\{g_{e}\\}_{e\\in\\Gamma}\\Big{)}\n&=&\n{\\mathrm{Tr}}\\,\\left[\\bigotimes_{e} D^{j_{e}}(g_{e}) \\otimes \\bigotimes_{v} i_{v}\\right]\n\\nonumber\\\\\n&=&\n\\prod_{e}\\langle j_{e}m_{e}^{s}|\\,g_{e}\\,|j_{e}m_{e}^{t}\\rangle\\,\n\\prod_{v}\\langle \\otimes_{e|t(e)=v}j_{e}m_{e}^{t}|\\,i_{v\\,}|\\otimes_{e|s(e)=v}j_{e}m_{e}^{s}\\rangle\\,,\n\\end{eqnarray}\nwith an implicit sum over all the $m_{e}^{s,t}$ indices,  where we have introduced the usual spin basis $|j,m\\rangle$ of the Hilbert space ${\\mathcal V}^{j}$ with the index $m$ running from $-j$ to $+j$ by integer steps.\nThat spin network functional is automatically gauge-invariant due to the $\\mathrm{SU}(2)$-invariance of the intertwiners at each vertex. Intertwiners $i_{v}$ at the vertex give the volume excitations, thus representing chunks of volume dual to each vertex, while the spin $j_{e}$ living on the edge $e$ linking two vertices give the area quanta of the (quantum) surface boundary between the corresponding two chunks of space.\nThis endows spin networks with a natural interpretation as discrete quantum geometries.\n\n\n\nFrom here, the loop quantum gravity programme proceeds in two steps. First, one sums over all possible graphs $\\Gamma$ imposing cylindrical consistency. This yields the kinematical Hilbert space of spin network states. Second, one imposes the Hamiltonian constraints generating the space-time diffeomorphisms at the quantum level to define the physical Hilbert space of loop quantum gravity.\n\n\\medskip\n\nIndeed, in order to consider the full space of connections and not just its finite sampling on a fixed graph $\\Gamma$, we will consider all possible graphs and sums of cylindrical functions over different graphs. To this purpose, one needs to compare wave-functions with support on different graphs, take their sum and scalar product. This is achieved through requiring cylindrical consistency. A function $\\psi$ on a graph $\\Gamma$ is considered as equivalent to another function $\\widetilde{\\psi}$ defined on a larger graph $\\widetilde{\\Gamma}$, containing $\\Gamma$ as a subgraph, if the finer function does not depend on the group elements living on the extra edges and coincides with the original coarser function $\\psi$ on the subgraph:\n\\begin{equation}\n\\Gamma\\subset\\widetilde{\\Gamma},\\qquad\n\\psi_{\\Gamma}\\,\\sim\\,\\widetilde{\\psi}_{\\widetilde{\\Gamma}}\n\\quad\\Leftrightarrow\\quad\n\\forall g_{e}\\in\\mathrm{SU}(2)\\,\\quad \\widetilde{\\psi}(\\{g_{e}\\}_{e\\in\\widetilde{\\Gamma}})=\\psi(\\{g_{e}\\}_{e\\in\\Gamma})\\,.\n\\end{equation}\nThis means that any wave-function defined on a graph $\\Gamma$ is automatically extended to live on any refinement of $\\Gamma$. Now, when summing two wave-functions living on a priori different graphs or taking their scalar product, one will refine the two graphs, say $\\Gamma$ and $\\Gamma'$, to a larger and finer graph $\\widetilde{\\Gamma}$ containing both of them as subgraphs: the two wave-functions will then be compared on that larger graph $\\widetilde{\\Gamma}$ and their sum will be be defined as living on it.\nA precise and rigorous treatment of these projective limit techniques can be found in \\cite{Ashtekar:1993wf,Ashtekar:1994mh,Ashtekar:1994wa}. The set of wave-functions is defined by the union of the sets of wave-functions with support on every graph quotiented by the cylindrical consistency equivalence relation, and the resulting kinematical Hilbert space for loop quantum gravity is obtained by the sum of all graphs also quotiented by the cylindrical consistency:\n\\begin{equation}\n\\mathcal{F}= \\left.\\left(\\bigcup_\\Gamma \\mathcal{F}_\\Gamma\\right)\\right\/\\sim\\,,\n\\qquad\n\\mathcal{H}_\\textrm{kin} = \\left.\\left(\\bigoplus_\\Gamma \\mathcal{H}_\\Gamma\\right)\\right\/\\sim\\,.\n\\end{equation}\nThis Hilbert space, defined as a projective limit, was shown to be the space of $L^{2}$-functionals of the connection with respect to the Ashtekar-Lewandowski measure \\cite{Ashtekar:1994wa}.\nGoing to the spin network basis, the cylindrical consistency corresponds to identifying graphs with edges $e$ carrying a vanishing spin $j_{e}=0$ to the same graphs without those edges. Thus, if we went to pick a specific representative for every equivalence class, we could simply choose all spin networks carrying no vanishing spins:\n\\begin{equation}\n\\mathcal{H}_\\textrm{kin} = \\bigoplus_\\Gamma \\widetilde{\\mathcal{H}}_\\Gamma\\,,\n\\qquad\n\\widetilde{\\mathcal{H}}_\\Gamma = \\bigoplus_{\\{j_e \\neq 0,i_v\\}} \\mathbb{C}|j_e,i_v\\rangle\n\\end{equation}\n\n\n\n\\begin{figure}[t!]\n\\centering\n\n\\begin{tikzpicture}[scale=0.5]\n\\coordinate(A) at (0,0);\n\\coordinate(B) at (2,0);\n\\coordinate(C) at (4,0);\n\\coordinate(D) at (0,-2);\n\\coordinate(E) at (2,-2);\n\\coordinate(F) at (4,-2);\n\n\\draw[decorate, decoration=snake] (A) to[bend left] (B);\n\\draw[decorate, decoration=snake] (B) to[bend right] (C);\n\\draw[decorate, decoration=snake] (D) to[bend left] (E);\n\\draw[decorate, decoration=snake] (E) to[bend right] (F);\n\n\\draw[decorate, decoration=snake] (A) -- (D);\n\\draw[decorate, decoration=snake] (B) -- (E);\n\\draw[decorate, decoration=snake] (C) -- (F);\n\n\\draw (A) node {$\\bullet$};\n\\draw (B) node {$\\bullet$};\n\\draw (C) node {$\\bullet$};\n\\draw (D) node {$\\bullet$};\n\\draw (E) node {$\\bullet$};\n\\draw (F) node {$\\bullet$};\n\n\\coordinate(O1) at (-4,6);\n\\coordinate(A1) at ($(O1)+(0,0)$);\n\\coordinate(B1) at ($(O1)+(2,0)$);\n\\coordinate(C1) at ($(O1)+(4,0)$);\n\\coordinate(D1) at ($(O1)+(0,-2)$);\n\\coordinate(E1) at ($(O1)+(2,-2)$);\n\\coordinate(F1) at ($(O1)+(4,-2)$);\n\n\\draw[<-,>=stealth,very thick] (1,1) -- (-1,3);\n\n\\draw[decorate, decoration=snake] (A1) to[bend left] (B1);\n\\draw[gray,dashed,decorate, decoration=snake] (B1) to[bend right] (C1);\n\\draw[decorate, decoration=snake] (D1) to[bend left] (E1);\n\\draw[gray,dashed,decorate, decoration=snake] (E1) to[bend right] (F1);\n\n\\draw[decorate, decoration=snake] (A1) -- (D1);\n\\draw[decorate, decoration=snake] (B1) -- (E1);\n\\draw[gray,dashed,decorate, decoration=snake] (C1) -- (F1);\n\n\\draw (A1) node {$\\bullet$};\n\\draw (B1) node {$\\bullet$};\n\\draw[gray] (C1) node {$\\bullet$};\n\\draw (D1) node {$\\bullet$};\n\\draw (E1) node {$\\bullet$};\n\\draw[gray] (F1) node {$\\bullet$};\n\n\\coordinate(O2) at (4,6);\n\\coordinate(A2) at ($(O2)+(0,0)$);\n\\coordinate(B2) at ($(O2)+(2,0)$);\n\\coordinate(C2) at ($(O2)+(4,0)$);\n\\coordinate(D2) at ($(O2)+(0,-2)$);\n\\coordinate(E2) at ($(O2)+(2,-2)$);\n\\coordinate(F2) at ($(O2)+(4,-2)$);\n\n\\draw[<-,>=stealth,very thick] (3,1) -- (5,3);\n\n\\draw[gray,dashed,decorate, decoration=snake] (A2) to[bend left] (B2);\n\\draw[decorate, decoration=snake] (B2) to[bend right] (C2);\n\\draw[gray,dashed,decorate, decoration=snake] (D2) to[bend left] (E2);\n\\draw[decorate, decoration=snake] (E2) to[bend right] (F2);\n\n\\draw[gray,dashed,decorate, decoration=snake] (A2) -- (D2);\n\\draw[decorate, decoration=snake] (B2) -- (E2);\n\\draw[decorate, decoration=snake] (C2) -- (F2);\n\n\\draw[gray] (A2) node {$\\bullet$};\n\\draw (B2) node {$\\bullet$};\n\\draw (C2) node {$\\bullet$};\n\\draw[gray] (D2) node {$\\bullet$};\n\\draw (E2) node {$\\bullet$};\n\\draw (F2) node {$\\bullet$};\n\\end{tikzpicture}\n\n\\caption{To define the scalar product in the continuum, we use cylindrical consistency: wavefunctions with trivial dependancy on some edges are identified with functions on coarser graphs (with the gray dashed edges removed from the graph). As a consequence, two coarse graphs can always be considered as being embedded in another finer graph on which the scalar product is well-defined.}\n\\label{fig:cylindrical}\n\\end{figure}\n\n\\medskip\n\nThis projective limit technique was introduced for graphs embedded in the canonical hypersurface, but was also shown to work for equivalence classes of graphs under (spatial) diffeomorphisms, and can be directly extended to abstract graphs defined purely combinatorially without reference to an embedding in a specific manifold. In the following, we will not make direct use of the embedding of spin network states, so our definition and procedures can be applied to any of those cases. Nevertheless, since we do not discuss the coarse-graining from an embedding point of view, it is simpler to consider all our definitions as for abstract graphs.\n\n\n\\subsection{Coarse-graining by gauge-fixing}\n\nLet us now discuss the main context of this paper: the coarse-graining of spin networks for  loop quantum gravity. The idea of coarse-graining is to integrate out the microscopic degrees of freedom, by an iterative procedure, up to some given energy or length scale to get the effective dynamics of the macroscopic degrees of freedom.\nIn condensed matter models, one typically works on a regular lattice with degrees of freedom living on its edges and\/or nodes and one can decimate consistently the variables, integrating out one node out of two for example, and thus derive an effective Hamiltonian on the coarser lattice. The length scale is set by the lattice spacing. In quantum field theory, the renormalisation group scheme integrates out quantum fluctuations of the field of high momentum and energy to derive an effective dynamics on the low momentum degrees of freedom.\nIn general relativity, the main difficulty is that the space-time geometry itself has become dynamical thus leading to some serious obstacles: in a background independent context, we face the problems of defining consistently a length or energy scale and of properly localizing perturbations and degrees of freedom both in position and momentum. These issues persist in the quantum theory.\n\nIn loop quantum gravity, one could think that the natural graph structure of the theory makes it simpler to tackle the coarse-graining of the theory. However, even putting aside the huge complication of fluctuating graphs and graph superpositions, working out   the coarse-graining of loop quantum gravity on a fixed graph still faces the problem of localizing and determining the energy scale of the geometry fluctuations. Indeed, a natural coarse-graining procedure on a fixed graph is to subdivide it into a partition of bounded (usually connected) regions and to collapse those subgraphs to single points. The internal geometrical information carried by the spin network state on those subgraphs would be coarse-grained to some effective data living at the new node of the coarser graph, as illustrated on fig.\\ref{fig:coarsegraining}.  Integrating over these local degrees of freedom would lead to new effective dynamics on the coarser graph. Such a  procedure would then be iterated  to obtain a tower of effective theories \\textit{\\`a la} Wilson for loop quantum gravity towards a large scale limit.\n\\begin{figure}[h!]\n\n\\centering\n\n\\begin{tikzpicture}[scale=0.3]\n\\coordinate(A) at (0,0);\n\\coordinate(B) at (2,0);\n\\coordinate(C) at (2,-2);\n\\coordinate(D) at (0,-2);\n\n\\coordinate(E) at (6,0);\n\\coordinate(F) at (6,-2);\n\\coordinate(G) at (8,0);\n\\coordinate(H) at (0,-6);\n\\coordinate(I) at (2,-6);\n\\coordinate(J) at (2,-8);\n\\coordinate(K) at (6,-6);\n\n\\draw (A) -- (B);\n\\draw (B) -- (C);\n\\draw (C) -- (D);\n\\draw (D) -- (A);\n\\draw (B) -- (D);\n\n\\draw (A) -- ++(-1,1);\n\\draw (B) -- (E);\n\\draw (C) -- (F);\n\\draw (C) -- (I);\n\\draw (D) -- (H);\n\n\\draw (G) -- ++(1,1);\n\\draw (E) -- (F) -- (G) -- (E);\n\n\\draw (H) -- ++(-1,-1);\n\\draw (J) -- ++(-1,-1);\n\\draw (H) -- (I) -- (J) -- (H);\n\n\\draw (I) -- (K);\n\\draw (J) -- (K);\n\\draw (K) -- (F);\n\n\\draw (K) -- ++(1,-1);\n\n\\draw (A) node {$\\bullet$};\n\\draw (B) node {$\\bullet$};\n\\draw (C) node {$\\bullet$};\n\\draw (D) node {$\\bullet$};\n\\draw (E) node {$\\bullet$};\n\\draw (F) node {$\\bullet$};\n\\draw (G) node {$\\bullet$};\n\\draw (H) node {$\\bullet$};\n\\draw (I) node {$\\bullet$};\n\\draw (J) node {$\\bullet$};\n\\draw (K) node {$\\bullet$};\n\n\\draw[gray,dashed] (1,-1) circle(2);\n\\draw[gray,dashed] (1.3,-6.7) circle(1.7);\n\\draw[gray,dashed] (6.7,-0.7) circle(1.7);\n\\draw[gray,dashed] (K) circle(1);\n\n\\draw[->,>=stealth,very thick] (9,-4) -- (13,-4);\n\n\\coordinate(O1) at (15,-1);\n\\coordinate(O2) at (15.3,-6.7);\n\\coordinate(O3) at (20.7,-0.7);\n\\coordinate(O4) at (20,-6);\n\\draw (O1) -- ++(-1,1);\n\\draw (O2) -- ++(-1.7,-0.5);\n\\draw (O2) -- ++(-0.5,-1.7);\n\\draw (O3) -- ++(1,1);\n\\draw (O4) -- ++(1,-1);\n\n\\draw (O1) to[bend left] (O2);\n\\draw (O1) to[bend right] (O2);\n\\draw (O1) to[bend left] (O3);\n\\draw (O1) to[bend right] (O3);\n\\draw (O2) to[bend left] (O4);\n\\draw (O2) to[bend right] (O4);\n\\draw (O4) -- (O3);\n\n\n\\draw[fill=lightgray] (O1) circle(0.3);\n\\draw[fill=lightgray] (O2) circle(0.2);\n\\draw[fill=lightgray] (O3) circle(0.2);\n\\draw[fill=lightgray] (O4) circle(0.1);\n\\end{tikzpicture}\n\n\\caption{\\label{fig:coarsegraining}\nWe coarse-grain a graph by partionning it into disjoint connected subgraphs. We will reduce each of these bounded region of space by a single vertex of the coarser graph. Since each of these regions of space had some internal geometrical structure and were likely carrying curvature, the natural question is whether spin network vertices carry each data to account for these internal structure and curvature. We will see that standard spin network vertices can be interpreted as flat and that we need to introduce some new notion of ``curved vertices'' carrying extra algebraic information and define new extensions of spin network states more suitable to the process of coarse-graining loop quantum gravity.}\n\n\n\\end{figure}\n\n\nIt remains to decide which partition to choose in practice, which one is the most ``coarse-grainable''. We need to identify the regions of the spin network state whose geometry has the smallest (quantum) fluctuations. Since the algebraic data -spins and intertwiners- living on the graph determine the discrete geometry defined by the spin network state, the combinatorial data of the subgraph is not enough to decide if it is to be coarse-grained. One actually needs to find a suitable scale function -length or energy or another geometrical observable such as curvature- and to use an optimization algorithm running through all possible bounded regions and partitions of the graph in order to find the correct partition to coarse-grain at each step. In simpler words, the obstacle is that the geometry corresponding to the considered graph is not entirely determined by the combinatorial definition of the graph but crucially depends on the algebraic data living on it and carried by the spin network state. And on top of this difficulty remains to find a consistent way to deal with graph fluctuations and superpositions.\n\n\\medskip\n\nWe propose a truncation of the theory re-introducing a background lattice through coarse-graining. From the point of view of a given observer, one chooses a lattice, which defines the network of points whose geometry the observer will probe. The lattice is not considered as the fundamental graph underlying the physical spin network state. Instead, since the observer is assumed to have a finite resolution, its nodes represent bounded regions of space whose internal geometry can fluctuate. Then, if we consider a spin network states based on a graph with a very fine structure, we will coarse-grain it onto our chosen lattice. Such a scheme allows to take into account \ngraph fluctuations and superpositions while actually working on a fixed lattice. Indeed, considering a superposition of graphs, it will live by cylindrical consistency on a finer graph containing both graphs. Then we will coarse-grain the quantum geometry state on the finer graph until it lives on our reference lattice.\n\nA key step of this procedure is the coarse-graining of subgraphs to nodes. We use the ``coarse-graining through gauge-fixing'' procedure introduced in \\cite{Livine:2006xk, Livine:2013gna} and also exploited in \\cite{Dittrich:2014wpa,Bahr:2015bra} to reformulate the algebra of geometrical observables in loop quantum gravity. This is based on the  gauge-fixing  for spin networks defined earlier in \\cite{Freidel:2002xb}, which allows to collapse an arbitrary subgraph to a {\\it flower} , that is a single vertex with self-loops -or petals- attached to it. These loops account  for the building-up of the curvature and thus of the gravitational energy density within these microscopic bounded regions which we will coarse-grain to single points on the measurement lattice chosen by the observer.\n\n\\medskip\n\nLet us give a closer look to this gauge-fixing procedure and the resulting coarse-graining of spin networks.\nAt the classical level, a spin network state is given by the graph dressed with discrete holonomy-flux data: each oriented edge carries a $\\mathrm{SU}(2)$ group element $g_{e}\\,\\in\\mathrm{SU}(2)$ while each edge's extremity around  a vertex is colored with a vector $X^{v}_{e}\\in{\\mathbb R}^{3}$. So one edge carries two vectors, one living at its source vertex and the other living at its target vertex, respectively $X^{s,t}_{e}\\equiv X^{s,t(e)}_{e}$. The group element gives the parallel transport of the vectors along the edges, that is $X^{t}_{e}=-\\,g_{e}\\triangleright X^{s}_{e}$ with the action of $g_{e}$ as a $\\mathrm{SO}(3)$-rotation on the flat 3d space.\nThis obviously forces the two vectors to have equal norm, $|X^{t}_{e}|=|X^{s}_{e}|$, which is called the (area-)matching constraint.\nOne requires another set of constraints: we impose the closure constraint at each vertex $v$, so that the sum of the fluxes around the vertex vanishes, $\\sum_{e\\ni v} X^{v}_{e}=0$. This holonomy-flux data can be interpreted as some discrete geometry in the framework of twisted geometries\n\\cite{Freidel:2010aq,Freidel:2013bfa}.\nThis is achieved through Minkowski's theorem stating that the closure constraint determines a unique convex polyhedron in flat 3d space dual to each vertex $v$, such that the fluxes $X^{v}_{e}$ are the normal vectors to the polyhedron faces.\n\nCurvature appears as non-trivial holonomies around loops ${\\mathcal L}$ of the graph, when $\\overrightarrow{\\prod_{e\\in{\\mathcal L}}}g_{e}\\,\\ne{\\mathbb{I}}$. As pointed out in \\cite{Livine:2013gna}, coarse-graining a subgraph carrying non-trivial curvature leads to an effective vertex breaking the closure constraint. This underlines the fact that a generalization of spin network states is required in order to properly carry out a coarse-graining procedure: we need an extended structure allowing for {\\it curved} vertices.\n\nWe illustrate this in fig.\\ref{fig:nonclosure}, in 2d instead of 3d. Let us consider a bounded region in space and the normals to its boundary.\nDue to gauge-invariance, if the region contains a single vertex, the sum of the normals will sum up to zero. But if there are loops inside the region, the parallel transport around these loops might introduce non-trivial rotations. And indeed, as soon as the parallel transport around the loops is non-trivial, the sum of the normals is no longer zero, leading to a closure defect \\cite{Livine:2013gna}. This is natural and translates the fact that curvature is carried by the loops of the spin network. And this must be taken  into account when coarse-graining.\n\n\\begin{figure}[h!]\n\n\\centering\n\n\\begin{tikzpicture}\n\n\\def 1.2 {1.2}\n\\def 0.2 {0.2}\n\n\\coordinate (A1) at (1*1.2,0.2,0);\n\\coordinate (A2) at (0.5*1.2,0.2,0.866*1.2);\n\\coordinate (A3) at (-0.5*1.2,0.2,0.866*1.2);\n\\coordinate (A4) at (-1*1.2,0.2,0);\n\\coordinate (A5) at (-0.5*1.2,0.2,-0.866*1.2);\n\\coordinate (A6) at (0.5*1.2,0.2,-0.866*1.2);\n\\coordinate (B) at (0,0.8+0.2,0);\n\n\\draw (A6) -- (A1) -- (A2) -- (A3) -- (A4);\n\\draw[dashed] (A4) -- (A5) -- (A6);\n\n\\draw[blue,thick] (A1) -- (B);\n\\draw (A2) -- (B);\n\\draw (A3) -- (B);\n\\draw (A4) -- (B);\n\\draw[dashed] (A5) -- (B);\n\\draw (A6) -- (B);\n\n\\draw (A1) node {$\\bullet$};\n\\draw (A2) node {$\\bullet$};\n\\draw (A3) node {$\\bullet$};\n\\draw (A4) node {$\\bullet$};\n\\draw (A5) node {$\\bullet$};\n\\draw (A6) node {$\\bullet$};\n\\draw (B) node {$\\bullet$};\n\n\\draw[->,>=stealth,thick] (2,0.5) -- (4,0.5);\n\n\\coordinate (P) at (6.5,0.5);\n\n\\def 50 {50}\n\\def 1.4 {1.4}\n\n\\coordinate (C1) at ($(P) + (0*50:1.4)$);\n\\coordinate (C2) at ($(P) + (1*50:1.4)$);\n\\coordinate (C3) at ($(P) + (2*50:1.4)$);\n\\coordinate (C4) at ($(P) + (3*50:1.4)$);\n\\coordinate (C5) at ($(P) + (4*50:1.4)$);\n\\coordinate (C6) at ($(P) + (5*50:1.4)$);\n\\coordinate (C7) at ($(P) + (6*50:1.4)$);\n\n\\fill[lightgray] (C7) -- (P) -- (C1) -- cycle;\n\\draw[red,thick,dashed] (C1) -- (C7);\n\n\\draw[blue,thick] (P) -- (C1);\n\\draw (P) -- (C2);\n\\draw (P) -- (C3);\n\\draw (P) -- (C4);\n\\draw (P) -- (C5);\n\\draw (P) -- (C6);\n\\draw[blue,thick] (P) -- (C7);\n\n\\draw (C1) -- (C2) -- (C3) -- (C4) -- (C5) -- (C6) -- (C7);\n\n\\draw[red,thick] ($(P) + (6*50:0.3*1.4)$) arc (6*50:360:0.3*1.4);\n\n\\def 0.01 {0.01}\n\\def 0.9 {0.9}\n\n\\draw[->,>=stealth] ($(P) + (0.5*50:1.4*0.9)$) -- ++(0.5*50:50*0.01);\n\\draw[->,>=stealth] ($(P) + (1.5*50:1.4*0.9)$) -- ++(1.5*50:50*0.01);\n\\draw[->,>=stealth] ($(P) + (2.5*50:1.4*0.9)$) -- ++(2.5*50:50*0.01);\n\\draw[->,>=stealth] ($(P) + (3.5*50:1.4*0.9)$) -- ++(3.5*50:50*0.01);\n\\draw[->,>=stealth] ($(P) + (4.5*50:1.4*0.9)$) -- ++(4.5*50:50*0.01);\n\\draw[->,>=stealth] ($(P) + (5.5*50:1.4*0.9)$) -- ++(5.5*50:50*0.01);\n\n\\draw[->,>=stealth,red,thick,dashed] ($(P) + ({180+3*50}:1.2)$) -- ++({180+3*50}:{360*0.01-6*50*0.01});\n\n\\draw (P) node {$\\bullet$};\n\n\\draw (C1) node {$\\bullet$};\n\\draw (C2) node {$\\bullet$};\n\\draw (C3) node {$\\bullet$};\n\\draw (C4) node {$\\bullet$};\n\\draw (C5) node {$\\bullet$};\n\\draw (C6) node {$\\bullet$};\n\\draw (C7) node {$\\bullet$};\n\n\\end{tikzpicture}\n\n\n\\caption{On this figure, we represented the dual graph of a 2d trivalent graph. The curvature at the vertex manifests itself as a defect in the closure condition. This can be seen by flattening the triangulation, which amounts to gauge-fix the variables. The curvature manifests itself as a gap (in gray on the figure) at some edge (in blue on the figure) in the flattened manifold. The closure defect can be seen as the missing normal coming from the closure of the flattened polygon (in red on the figure).}\n\\label{fig:nonclosure}\n\n\\end{figure}\n\n\nA rigorous way to make this explicit is to gauge-fix the spin network state, following the procedure devised in \\cite{Freidel:2002xb}.\nLet us consider a bounded region of a larger spin network, defined as a finite connected subgraph $\\gamma$ of the larger graph $\\Gamma$,  as in fig.\\ref{fig:GaugeFix}.\nThe procedure goes as follow:\n\\begin{enumerate}\n\n\\item Choose arbitrarily a root vertex $v_{0}$ of the subgraph and select a maximal tree $T$ of the region: \n\nThe subgraph being connected, the maximal tree  goes through every vertex of the region and defines a unique path of edges from the root vertex $v_{0}$ to any vertex of the subgraph.\n\n\\item Gauge-fix iteratively all the group elements along the edges of the tree $g_{e\\in T}={\\mathbb{I}}$:\n\nUsing the gauge-invariance of the wave-functions as given by \\eqref{gaugeinv} with gauge transformations acting at every vertex by $\\mathrm{SU}(2)$ group elements $h_{v}$ as $g_{e}\\rightarrow h_{s(e)}^{-1}g_{e}h_{t(e)}$, we can start from the root of the tree $v_{0}$ and progress through the tree until we reach the boundary of our subgraph. We define the appropriate gauge transformations $h_{v}$ at every vertex in order to fix all the group elements along the edges of the tree to the identity ${\\mathbb{I}}$. The absence of loops in the tree, by definition, guarantees the consistency of this gauge-fixing. We can somewhat interpret this maximal tree as a synchronization network: we set all the parallel transports along the tree edges to the identity, thus synchronizing the reference frames at all the vertices and identifying them to a single reference frame living at the root of the subgraph. This realizes the coarse-graining of the subgraph $\\gamma$ to its chosen root vertex $v_{0}$.\nThe action of $\\mathrm{SU}(2)$ gauge transformations inside the region is not entirely gauge-fixed and we are still left with the $\\mathrm{SU}(2)$ gauge transformations at the root vertex.\n\n\n\\item Having collapsed the subgraph $\\gamma$ to its root vertex $v_{0}$, the edges of the subgraph $\\gamma$ which are not in the tree, $e\\in\\gamma\\setminus T$ label all the (independent) loops of the subgraph and lead to self-loops attached to the $v_{0}$:\n\nAs illustrated on fig.\\ref{fig:GaugeFix}, these self-loops or little loops carry the holonomies around the loops of the original subgraph $\\gamma$, that is the curvature living in the bounded region. The flux-vectors living on the boundary edges, linking the region to the outside bulk, generically do not satisfy the closure constraint anymore since the effective vertex does satisfy a closure constraint which takes into account the flux-vectors of those boundary edges but also of the internal loops. The closure defect, induced by the little loops, thus reflects the non-trivial internal structure  of the coarse-grained subgraph and curvature developed in the corresponding region of the spin network state. The interested reader can find details and proof in the previous work \\cite{Livine:2013gna}.\n\n\n\n\\end{enumerate}\n\n\n\\begin{figure}[h!]\n\n\\begin{tikzpicture}[scale=0.8]\n\\coordinate(A) at (0,0);\n\\coordinate(B) at (2,0);\n\\coordinate(C) at (2,-2);\n\\coordinate(D) at (0,-2);\n\n\\draw[blue,thick] (A) -- (B) node[midway,above] {$\\ell_1$};\n\\draw[blue,thick] (B) -- (C) node[midway,right] {$\\ell_2$};\n\\draw[red,very thick] (C) -- (D) node[midway,below] {$\\ell_3$};\n\\draw[red,very thick] (D) -- (A) node[midway,left] {$\\ell_4$};\n\\draw[red,very thick] (B) -- (D) node[midway,sloped,above] {$\\ell_5$};\n\n\\draw (A) -- ++(-1,0.5) node[above] {$j_1$};\n\\draw (B) -- ++(1,0.5) node[above] {$j_2$};\n\\draw (C) -- ++(1,-0.5) node[below] {$j_3$};\n\\draw (D) -- ++(-1,-0.5) node[below] {$j_4$};\n\n\\draw (A) node {$\\bullet$} node[above]{$i_1$};\n\\draw (B) node {$\\bullet$} node[above]{$i_2$};\n\\draw (C) node {$\\bullet$} node[below]{$i_3$};\n\\draw[red] (D) node[scale=2] {$\\bullet$} node[below]{$i_4$};\n\n\\draw[gray,dashed] (1,-1) circle(2);\n\n\\draw[->,>=stealth,very thick] (3.5,-1) -- (5.5,-1);\n\n\\coordinate(O) at (8,-1);\n\n\\draw (O) -- ++(-2,1) node[above] {$j_1$};\n\\draw (O) -- ++(2,1) node[above] {$j_2$};\n\\draw (O) -- ++(2,-1) node[below] {$j_3$};\n\\draw (O) -- ++(-2,-1) node[below] {$j_4$};\n\\draw[blue,thick,scale=3] (O) to[loop] (O);\n\\draw[blue,thick] (O) ++(0,1) node {$k_1$};\n\\draw[blue,thick,scale=3,rotate=180] (O) to[loop] (O);\n\\draw[blue,thick] (O) ++(0,-1) node {$k_2$};\n\n\\draw[red] (O) node[scale=2] {$\\bullet$} ++(0.35,0) node{$i$};\n\\end{tikzpicture}\n\n\\caption{Coarse-graining via gauge-fixing: we can gauge-fix the subgraph using a maximal subtree (in red). The remaining edges (in blue) correspond to loops on the coarse-grained vertex. There is a residual gauge-freedom at the coarse-grained vertex that corresponds to the action of the gauge group at the root of the tree (red vertex on the figure).}\n\\label{fig:GaugeFix}\n\\end{figure}\n\n\nThis gauge-fixing procedure allows to clearly identify and distinguish between the degrees of freedom of the internal geometry of the considered bounded region of space to coarse-grain. The tree encodes the internal combinatorial structure of the region and describes the network of points and links within: they provide the bulk structure on which we can create curvature.  The little loops and the $\\mathrm{SU}(2)$ group elements coloring them are the  excitations of the parallel transport and curvature. Together, tree and little loops attached to a vertex describe all its internal structure and are the extra data needed to define \\textit{curved vertices} for the effective coarse-grained theory. These curvature excitations create a closure defect for the flux-vectors living on the boundary edges linking the coarse-grained vertex -the root vertex- to the rest of the spin network (obtained by the actually satisfied closure constraint between boundary edges and little loops)\n\nWhen coarse-graining in practice, we do not want to retain all the information about the internal geometry, but only want to retain the degrees of freedom most relevant to the dynamics and interaction with the exterior geometry. In the next section, we will therefore introduce a hierarchy of extensions of spin network states with {\\it curved vertices}, from the finest notion of spin networks decorated with both trees and little loops to the coarser notion of spin networks with a simple tag at each vertex recording the induced closure defect.\n\n\n\n\\subsection{A hierarchy of coarse-grained spin network structures}\n\nIn loop quantum gravity, we start with spin network states, which are graphs decorated with spins on the edges and intertwiners at the vertices:\n\\begin{equation}\n{\\mathcal{H}}_\\Gamma = \\bigoplus_{\\{j_e,i_v\\}} \\mathbb{C}|j_e,i_v\\rangle\\,.\n\\end{equation}\nCurvature is carried loops of the graph.\nWe have argued that coarse-graining these networks should naturally lead to extended spin networks that can carry localized curvature excitations at the vertices. Following the coarse-graining through gauge-fixing procedure\\footnotemark, we propose a hierarchy of three possible extensions of the spin network states, which depend on how much extra information and structure are added to each vertex:\n\\footnotetext{\nAnother approach is to define spin networks made of intertwiners directly interpretable as dual to polyhedron in a curved space. These has been developed in the framework of spin networks for loop quantum gravity with a non-vanishing cosmological constant and is based on a quantum deformation of the $\\mathrm{SU}(2)$ gauge group \\cite{Dupuis:2013haa,Bonzom:2014wva,Dupuis:2014fya,Charles:2015lva,Haggard:2014xoa,Haggard:2015ima,Haggard:2015yda}. However, it is not yet clear how, if possible, to depart from a homogeneous curvature and glue pieces carrying a different curvature, thus obtaining actual spin networks with variable curvature.\nOne possible link with our present framework would be to show that these curved and quantum-deformed intertwiners can be obtained in a continuum limit as a vertex with an infinite number of little loops creating a constant homogeneous curvature excitation (for instance, triangulating a hyperbolic tetrahedron with finer and finer tetrahedra which can be considered as flat in an infinite refinement limit).\n}\n\\begin{figure}\n\\begin{subfigure}[t]{.33\\linewidth}\n\\centering\n\\begin{tikzpicture}[scale=0.7]\n\\coordinate(O1) at (0,0);\n\\coordinate(O2) at (2.5,3.5);\n\\coordinate(A) at ($(O2)+(0,0)$);\n\\coordinate(B) at ($(O2)+(1,0)$);\n\\coordinate(C) at ($(O2)+(1,-1)$);\n\\coordinate(D) at ($(O2)+(0,-1)$);\n\n\\draw (O1) -- ++(-2,1) node[above] {$j_1$};\n\\draw (O1) -- ++(2,1) node[above] {$j_2$};\n\\draw (O1) -- ++(2,-1) node[below] {$j_3$};\n\\draw (O1) -- ++(-2,-1) node[below] {$j_4$};\n\\draw[blue,thick] (O1) to[loop,scale=3] (O1) ++(0,1) node {$k_1$};\n\\draw[blue,thick] (O1) to[loop,scale=3,rotate=180] (O1) ++(0,-1) node {$k_2$};\n\n\\draw[red] (O1) node[scale=2] {$\\bullet$} ++(0.35,0) node{$i$};\n\n\\draw[gray,dashed] (O1) circle (0.5) ++(45:0.5) -- (45:3) ++(45:1.2) coordinate (O3) circle (1.2);\n\n\n\\clip (O3) circle (1.2);\n\n\\draw[blue,thick] (A) -- (B);\n\\draw[blue,thick] (B) -- (C);\n\\draw[red,very thick] (C) -- (D);\n\\draw[red,very thick] (D) -- (A);\n\\draw[red,very thick] (B) -- (D);\n\n\\draw (A) -- ++(-2,1);\n\\draw (B) -- ++(2,1);\n\\draw (C) -- ++(2,-1);\n\\draw (D) -- ++(-2,-1);\n\n\\draw (A) node {$\\bullet$};\n\\draw (B) node {$\\bullet$};\n\\draw (C) node {$\\bullet$};\n\\draw[red] (D) node[scale=2] {$\\bullet$};\n\\end{tikzpicture}\n\n\\caption{All the information can be preserved by carrying the $SU(2)$ labels and an unfolding tree describing the inner details of the coarse-grained vertex.}\\label{fig:loopy_a}\n\\end{subfigure}%\n\\hspace{2mm}\n\\begin{subfigure}[t]{.28\\linewidth}\n\\centering\n\\begin{tikzpicture}[scale=0.7]\n\\coordinate(O1) at (0,0);\n\n\\draw (O1) -- ++(-2,1) node[above] {$j_1$};\n\\draw (O1) -- ++(2,1) node[above] {$j_2$};\n\\draw (O1) -- ++(2,-1) node[below] {$j_3$};\n\\draw (O1) -- ++(-2,-1) node[below] {$j_4$};\n\\draw[blue,thick] (O1) to[loop,scale=3] (O1) ++(0,1) node {$k_1$};\n\\draw[blue,thick] (O1) to[loop,scale=3,rotate=180] (O1) ++(0,-1) node {$k_2$};\n\n\\draw[red] (O1) node[scale=2] {$\\bullet$} ++(0.35,0) node{$i$};\n\\end{tikzpicture}\n\\caption{The particular subgraph can be forgotten and only the $SU(2)$ information is preserved.}\\label{fig:loopy_b}\n\\end{subfigure}\n\\hspace{2mm}\n\\begin{subfigure}[t]{.28\\linewidth}\n\\centering\n\\begin{tikzpicture}[scale=0.7]\n\\coordinate(O1) at (0,0);\n\n\\draw (O1) -- ++(-2,1) node[above] {$j_1$};\n\\draw (O1) -- ++(2,1) node[above] {$j_2$};\n\\draw (O1) -- ++(2,-1) node[below] {$j_3$};\n\\draw (O1) -- ++(-2,-1) node[below] {$j_4$};\n\\draw[blue,thick] (O1) to ++(0,0.5) node[above]{$j,m$};\n\n\\draw[red] (O1) node[scale=2] {$\\bullet$} ++(0,-0.3) node{$i'$};\n\n\\end{tikzpicture}\n\\caption{Everything except the closure defect is forgotten. Only a ``tag'' remains.}\\label{fig:loopy_c}\n\\end{subfigure}\n\\caption{The hierarchy of possible coarse-graining frameworks}\\label{fig:loopy}\n\\end{figure}\n\\begin{enumerate}\n\n\\item {\\bf Folded spin networks~:}\n\nIn the first scenario, we follow the gauge-fixing procedure but we do a minimal coarse-graining, retaining as much information as possible on the original state.\nEach vertex is allowed with an arbitrary number of little loops attached to it and is endowed with a tree connecting the ends of the external edges and of the internal loops, as represented in fig.\\ref{fig:loopy_a}. This tree can be seen as a circuit telling us how to unfold the vertex, reversing the gauge-fixing procedure and recovering the original (finer) graph.\nThis Hilbert space ${\\mathcal H}_{\\Gamma}^\\mathrm{folded}$ can be written formally as:\n\\begin{equation}\n{\\mathcal H}_{\\Gamma}^\\mathrm{folded}\n= \\bigoplus_{\\{j_e,j^{(v)}_{\\ell},i_v,\\mathcal{T}_v\\}} \\mathbb{C}\\,|j_e,j^{(v)}_{\\ell},i_v,\\mathcal{T}_v\\rangle\\,.\n\\end{equation}\n $\\mathcal{T}_v$ is the unfolding tree for each vertex, $j^{(v)}_{\\ell}$ are the spins carried by the additional loops labeled by the index $\\ell$ and the intertwiners $i_{v}$ now lives in the tensor product of the spins $j_{e}$ of the edges linking to the other neighboring vertices and (twice) the spins $j^{(v)}_{\\ell}$ living on the internal loops (because each loop has its two ends at the vertex).\n \n With such an internal space at each vertex, we actually lose no information at all on the internal degrees of freedom. Starting with a spin network state living on a finer graph $\\widetilde{\\Gamma}$, we simply gauge-fix it to a spin network on our coarser graph $\\Gamma$. And we can follow the reverse path. Using the tree at each vertex, we can fully reconstruct the original finer graph $\\widetilde{\\Gamma}$ thus simply perform generic gauge transformations to recover the fully gauge-invariant spin network state.\n\nThus the chosen graph $\\Gamma$ can be considered as a skeleton graph, to which we can add extra information to represent spin network states living on any (finer) graph. In a sense, we have not done any coarse-graining yet. The truncation of the theory will happen when defining the dynamics on the folded spin network Hilbert space, distinguishing actual edges and spins of our  skeleton lattice -the background- from spins and edges on the unfolding trees and little loops, when the fundamental dynamics would have considered them on equal footing.\n\n\\item {\\bf Loopy spin networks~:}\n\nIn a second scenario, we coarse-grain the internal structure of the effective vertices by discarding the unfolding trees. We keep the curvature excitations living on the little loops, but we discard the combinatorial information of the internal subgraph: we forget that the  vertex effectively represents an actual extended region of space and we localize all the internal curvature degrees of freedom on that coarse-grained vertex. This leads to loopy spin networks, with an arbitrary number of loops at each vertex but no unfolding tree data:\n\\begin{equation}\n\\mathcal{H}^{\\mathrm{loopy}}_{\\Gamma} = \\bigoplus_{\\{j_e,j^{(v)}_{\\ell},i_v\\}} \\mathbb{C}|j_e,j^{(v)}_{\\ell},i_v\\rangle\\,,\n\\end{equation}\nwhere the $j^{(v)}_{\\ell}$ are the spins living on the little loops attached to the vertex $v$ and the intertwiners $i_{v}$ live again in the tensor product of the spins carried by the graph edges attached to the vertex $v$ and the spins carried by its little loops.\n\nNow our chosen graph $\\Gamma$ for loopy spin network states is to be considered as a background graph. The little loops are explicit local excitations of the gravitational fields located at each vertex of the graph. A given loopy spin network comes from the coarse-graining of several possible finer spin network states living on finer graph, but we lack the unfolding tree information to recover the original more fundamental state.\n\nThe truncation of full theory is clear. Spin network states on the ``loopy graphs'' living on top on $\\Gamma$, that is the base graph $\\Gamma$ plus an arbitrary number of self-loops at every vertices, are already in the Hilbert space of the loop quantum gravity, although we do not usually focus on such graphs. Restricting ourselves to this subset of states is a clear truncation of the full Hilbert space. The difference with the standard interpretation is that we think here of the base graph $\\Gamma$ as embedded in the space manifold, while the little loops are abstract objects decorating the base graph vertices.\n\nSince we have local degrees of freedom, carried by the little loops, we need to discuss their statistics, which leads to a few variations of this theme:\n\n\\begin{enumerate}\n\\item {\\it Distinguishable loops~:}\nFirst, it is natural to consider that the loops are distinguishable as they come from a substructure. The loops do come from different edges of a finer graph and create curvature excitations  at different places within the coarse-grained bounded region. As a result, we should distinguish them and allow to number and order them.\n\n\\item {\\it Undistinguishable bosonic loops~:}\nA second possibility is to push further along the logic  of coarse-graining  and to consider that the loops undistinguishable since we do not have access anymore to the specific substructure. This should lead to bosonic statistics, as expected for gravitational field exicitations. Formally, this can be written as the identification:\n\\begin{equation}\n|j_e,i_v,j^{(v)}_{\\ell}\\rangle = |j_e,i_v,j^{(v)}_{\\sigma_{v}(\\ell)}\\rangle\n\\end{equation}\nfor any permutation $\\sigma_{v}\\in\\,S_{\\#\\ell}$ in the symmetric group of order $\\#\\ell$ when the vertex $v$ has $\\#\\ell$ loops. This point of view is compatible with considering the action of space diffeomorphisms on the little loops around the vertex as gauge transformations.\n\n\\item  {\\it Anyonic statistics~:} \nWe can easily imagine other statistics, for instance by allowing for a phase in the equality above (i.e a non-trivial representation of the permutation group). In fact, instead of thinking of the vertex as a mere point, we can represent the boundary of the bounded region as a sphere and consider the little loops as living on a sphere around it. Then the diffeomorphism invariance on the sphere will lead to an action of the braiding group leading to interesting anyonics statistics, similarly to the punctures of a Chern-Simons theory as  already explored  in the case of black holes in loop quantum gravity \\cite{Pithis:2014uva}.\n\n\\end{enumerate}\n\n\\item {\\bf Tagged spin networks~:}\n\nIn this third and last scenario, we fully coarse-grain the internal geometry of the bounded region now reduced to a graph vertex. We discard the unfolding tree, used in the gauge-fixing and unfixing procedure, and we integrate out the little loops attached to the vertex. All we retain is the closure defect induced by the non-trivial holonomies and spins carried by those little loops. The fact that coarse-graining spin networks, or their classical counterpart of twisted geometries, leads to closure defect, accounting for the presence of a non-trivial curvature within the coarse-grained region was already pointed out in \\cite{Livine:2013gna}.\nHere, the simplest method to see how this comes about is to use the intermediate spin decomposition of the intertwiner at the vertices, as illustrated on fig.\\ref{fig:intermediatespin}, introducing a fiducial link separating the external edges from the internal loops:\n\\begin{equation}\n\\mathrm{Inv}_{\\mathrm{SU}(2)}\\,\\Big{[}\n\\bigotimes_{e}{\\mathcal V}^{j_{e}}\n\\otimes\n\\bigotimes_{\\ell}\\big{(}{\\mathcal V}^{j_{\\ell}}\\otimes{\\mathcal V}^{j_{\\ell}}\\big{)}\n\\Big{]}\n\\,=\\,\n\\bigoplus_{J}\n\\,\n\\mathrm{Inv}_{\\mathrm{SU}(2)}\\,\\Big{[}\n{\\mathcal V}^{J}\n\\otimes\n\\bigotimes_{e}{\\mathcal V}^{j_{e}}\n\\Big{]}\n\\otimes\n\\mathrm{Inv}_{\\mathrm{SU}(2)}\\,\\Big{[}\n{\\mathcal V}^{J}\n\\otimes\n\\bigotimes_{\\ell}\\big{(}{\\mathcal V}^{j_{\\ell}}\\otimes{\\mathcal V}^{j_{\\ell}}\\big{)}\n\\Big{]}\\,.\n\\end{equation}\n\\begin{figure}[h!]\n\n\\centering\n\n\\begin{tikzpicture}[scale=0.8]\n\\coordinate(O1) at (0,0);\n\n\\draw (O1) -- ++(-2,1);\n\\draw (O1) -- ++(-2,0.5);\n\\draw (O1) -- ++(-2,-0.5);\n\\draw (O1) -- ++(-2,-1);\n\\draw (O1) to[in=-25,out=25,loop,scale=3] (O1);\n\\draw (O1) to[in=30,out=80,loop,scale=3] (O1);\n\\draw (O1) to[in=-80,out=-30,loop,scale=3] (O1);\n\n\\draw (O1) node {$\\bullet$} ++(-0.15,0.5) node{$i_v$};\n\n\\draw[->,>=stealth,very thick] (3,0) -- (5,0);\n\n\\coordinate(O2) at (8,0);\n\\coordinate(O3) at (10,0);\n\n\\draw (O2) -- ++(-2,1);\n\\draw (O2) -- ++(-2,0.5);\n\\draw (O2) -- ++(-2,-0.5);\n\\draw (O2) -- ++(-2,-1);\n\n\\draw[in=-25,out=25,scale=3] (O3) to[loop] (O3);\n\\draw[in=30,out=80,scale=3] (O3) to[loop] (O3);\n\\draw[in=-80,out=-30,scale=3] (O3) to[loop] (O3);\n\n\\draw[red] (O2) -- (O3) node[midway,below]{$J$};\n\\draw (O2) node {$\\bullet$} ++(0.12,0.4) node{$i_v^{J}$};\n\\draw (O3) node {$\\bullet$} ++(-0.2,0.4) node{$\\tilde{i}_v^{J}$};\n\n\\end{tikzpicture}\n\n\\caption{We represent a loopy vertex $v$, here with three little loops attached to it. The intertwiner $i_{v}$ can be decomposed onto the intermediate spin basis, where we introduce a fiducial edge between the external legs and the internal loops. This orthogonal basis is labeled by the intermediate spin $J$, and two intertwiners $i^{J}_{v}$ and $\\tilde{i}^{J}_{v}$ intertwining between that intermediate spin  and respectively the external legs or the internal loops.} \n\\label{fig:intermediatespin}\n\n\\end{figure}\nThis spin $J_{v}$ living at the vertex $v$ encodes the closure defect and is the only extra information with which we decorate the graph.\nWe call it the {\\it tag} and amounts to adding an open leg to every vertex of the graph. This open edge is colored with the spin $J_{v}$ and a vector in that $\\mathrm{SU}(2)$ representation. Using the standard spin basis labeled by magnetic moment number $M$, the Hilbert space of {\\it tagged spin networks} on the base graph $\\Gamma$ is then formally defined as:\n\\begin{equation}\n\\mathcal{H}_\\Gamma^{\\mathrm{tag}} = \\bigoplus_{\\{j_e,J_v,M_v,i_v\\}} \\mathbb{C}|j_e,J_v,M_v,i_v\\rangle\\,,\n\\end{equation}\nwhere the intertwiner $i_{v}$ at the vertex $v$ now lives in the tensor product of the spins $j_{e\\ni v}$ on the external edges $e$ attached to the vertex and of the vertex tag $J_{v}$.\n\nThe state $|J_v,M_v\\rangle$ is the quantized version of the closure defect vector. Indeed, at the classical level, as shown in \\cite{Livine:2013gna}, the sum of the flux-vectors living on the external edges $e\\ni v$ does not vanish anymore and should be balanced by the sum of the flux-vectors living on the internal loops. This defect vector means that there is  no convex polyhedron dual to the vertex, as usual in twisted geometries. One way to go is to try to open the polyhedron somehow, which wouldn't have a clear geometrical interpretation. Instead we propose to interpret it as the dual convex polyhedron should not be embedded in flat space but in a (homogeneous) curved space, the curvature radius depending on the actual value of the closure defect. Progress in this direction has been achieved in the study of hyperbolic and spherical tetrahedra \\cite{Bonzom:2014wva,Charles:2015lva,Haggard:2015ima} but we do not yet have an explicit  embedding and formula relating the curvature to the norm of the defect. It would ultimately be enlightening to relate this tag $J_{v}$ to the spectrum of some quasi-local energy operator in loop quantum gravity (e.g. \\cite{Yang:2008th}), which would allow to view it as a measure of the gravitational energy density within the bounded coarse-grained region.\n\n\\end{enumerate}\n\n\\medskip\n\nThese three extended spin network structures are the heart of our present proposal for studying effective truncations for the coarse-graining of loop quantum gravity. The goal would be to reformulate the dynamics of loop quantum gravity on these new structures and study their renormalisation flow under coarse-graining. An important point is that these folded, loopy and tagged spin networks sidestep the problem of fluctuating graph dynamics and allow to project the whole dynamics on a fixed background graph, or skeleton, interpreted as the lattice postulated by the observer\\footnotemark. We then have local excitations of the geometry, representing the internal fluctuations of the gravitational field in the coarse-grained regions, living at the graph vertices and represented by the new information attached to them, respectively unfolding trees, little loops or tags.\n\\footnotetext{\nThe background lattice can then be adapted to the studied models. We could choose a regular lattice or a much simpler graph, such as a flower with a single vertex and an arbitrary number of little loops. Such simple graphs could reveal useful in the study of highly symmetric problems as is the case in cosmology or in the study of Einstein-Rosen waves \\cite{Korotkin:1997ps,Ashtekar:1996cm}.\n}\nThe use of a background lattice, which might be regular, would simplify greatly the setting of a systematic coarse-graining of loop quantum gravity.\n\nThe folded spin networks are mathematically a simple gauge-fixing of spin networks onto the skeleton graph. In the following sections, we will focus on providing a clean mathematical definition of loopy and tagged spin networks and exploring the definition of a Fock space of loopy spin networks with bosonic statistics for the little loops living at every graph vertex.\n\n\n\n\n\\section{Loopy spin networks}\n\n\nHere we  would like to define properly loopy spin networks and investigate their propreties.\nChoosing a fixed graph $\\Gamma$ with $E$ edges, and given numbers of little loops $N_{v}$ at each vertex $v$, we consider the following space of wave-functions on $\\mathrm{SU}(2)^{\\times\\,(E+\\sum_{v}N_{v})}$ invariant under $\\mathrm{SU}(2)$  gauge transformations acting at every vertices:\n\\begin{equation}\n\\psi\\Big{(}\\{g_{e}\\,,\\,h^{v}_{\\ell}\\}_{e,v\\in\\Gamma}\\Big{)}\n\\,=\\,\n\\psi\\Big{(}\\{a_{s(e)}g_{e}a_{t(e)}^{-1}\\,,\\,a_{v}h^{v}_{\\ell}a_{v}^{-1}\\}\\Big{)}\n\\,,\\qquad\n\\forall a_{v}\\in\\mathrm{SU}(2)^{\\times V}\\,.\n\\end{equation}\nThe $\\mathrm{SU}(2)$ gauge transformations act as usual on the edges $e$ of the graph, while they act by conjugation as expected on the little loops.\nA basis is provided by the spin decomposition on functions in $L^{2}(\\mathrm{SU}(2))$ as with standard spin networks. The loopy spin network basis states are labeled with a spin $j_{e}$ on each edge $e$, a spin $k^{v}_{\\ell}$ on each little loop $\\ell$ attached to a vertex $v$, and an intertwiners $i_{v}$ at each vertex leaving in the tensor product of the attached edges and of the loop spins:\n\\begin{equation}\ni_{v}\\in\\,\\mathrm{Inv}_{\\mathrm{SU}(2)}\\,\n\\Big{[}\n\\bigotimes_{e\\ni v}{\\mathcal V}^{j_{e}}\n\\,\\otimes\\,\n\\bigotimes_{\\ell \\ni v}({\\mathcal V}^{k^{v}_{\\ell}}\\otimes\\bar{{\\mathcal V}}^{k^{v}_{\\ell}})\n\\Big{]}\\,\n\\end{equation}\nso that the Hilbert space of loopy spin networks on the graph $\\Gamma$ with given number $N_{v}$ of little loops at every vertex is, as announced in the previous section presenting the hierarchy of extended spin network structures:\n\\begin{equation}\n{\\mathcal H}_{\\Gamma,\\{N_{v}\\}}^{\\mathrm{loopy}}\n\\,=\\,\nL^{2}\\big{(}\n\\mathrm{SU}(2)^{\\times\\,(E+\\sum_{v}N_{v})}\n\\,\/\\,\n\\mathrm{SU}(2)^{\\times V}\n\\big{)}\n\\,=\\,\n\\bigoplus_{\\{j_{e},k^{v}_{\\ell},i_{v}\\}}\\,\n{\\mathbb C}\\,|j_{e},k^{v}_{\\ell},i_{v}\\rangle\\,.\n\\end{equation}\nWhat needs to be properly defined and analyzed is the Hilbert space of states with arbitrary number of little loops, allowing $N_{v}$ to run all over ${\\mathbb N}$ and summing over all these possibilities. To this purpose, the full graph structure $\\Gamma$ does not intervene and we can ignore it and focus on the space of little loops around a single vertex. Thus, for the sake of simplifying the discussion, we will focus on a single vertex with no external, but with an arbitrary umber of little loops attached to it. This is the {\\it flower} graph.\n\nIn this section, we will assume the little loops to be distinguishable. We define the spin network states with a given number of loops -the flower graph with fixed number of petals- and we then discuss the whole Hilbert space of states with arbitrary number of excitations by a projective limit. We define and analyze the holonomy operators acting on that space and we finally  implement the BF theory dynamics on that space as a first application of our framework and a consistency check.\nWe will tackle the case of indistinguishable little loops in the next section, imposing bosonic statistics and defining the holonomy operator on symmetrized  states.\n\n\\subsection{Loopy intertwiners}\n\\label{sec:LoopyIntertwiners}\n\nLet us start with the flower graph with a fixed number $N$ of petals, that is a single vertex with $N$ little loops attached to it as drawn on fig.\\ref{fig:singlevertex}. We are going to define the wave-functions on that graph, the corresponding decomposition on the spin and intertwiner basis and the action of the holonomy operators.\n\\begin{figure}[h!]\n\\centering\n\\begin{tikzpicture}\n\\coordinate(O1) at (0,0);\n\n\\draw (O1) to[in=-30,out=+30,loop,scale=3,rotate=90] (O1) ++(0,1.2) node {$k_2$};\n\\draw (O1) to[in=-30,out=+30,loop,scale=3,rotate=30] (O1) ++(1,0.7) node {$k_3$};\n\\draw (O1) to[in=-30,out=+30,loop,scale=3,rotate=-30] (O1) ++(1,-0.7) node {$k_4$};\n\\draw (O1) to[in=-30,out=+30,loop,scale=3,rotate=150] (O1) ++(0,-1.2) node {$k_5$};\n\\draw (O1) to[in=-30,out=+30,loop,scale=3,rotate=-90] (O1) ++(-1,0.7) node {$k_{1}$};\n\n\\draw (O1) node[scale=1] {$\\bullet$} ++(-0.32,-0.18) node{${\\mathcal I}$};\n\n\\end{tikzpicture}\n\\caption{We consider the class of special graph, flowers, with a single vertex and an arbitrary number $N$ of little loops attached to it. Here we have drawn a flower with $N=5$ petals. The spin network states on such graphs are labeled by a spin on each loop, $k_{\\ell=1..N}$, and an intertwiner ${\\mathcal I}$ living in the tensor product $\\bigotimes_{\\ell=1}^{N} ({\\mathcal V}^{k_{\\ell}}\\otimes\\bar{{\\mathcal V}}^{k_{\\ell}})$.}\n\\label{fig:singlevertex}\n\\end{figure}\n\nWave-functions are gauge-invariant functions of $N$ group elements, that is functions on $\\mathrm{SU}(2)^{\\times N}$ invariant under the global action by conjugation:\n\\begin{equation}\n\\Psi(h_1,...,h_N) = \\Psi(gh_1g^{-1},...,gh_Ng^{-1})\\,.\n\\end{equation}\nThe scalar product is defined by integration with respect to the Haar measure on $\\mathrm{SU}(2)$ and the resulting Hilbert space is: \n\\begin{equation}\n{\\mathcal H}_{N}=\nL^2\\,\\Big{(}\\mathrm{SU}(2)^{\\times N}\/\\mathrm{Ad}\\,\\mathrm{SU}(2)\\Big{)}\\,.\n\\end{equation}\nA basis of this space is provided as usual by the spin network states, labeled by a spin on each loop, $k_{\\ell=1..N}\\,\\in\\frac\\N2$, and an intertwiner ${\\mathcal I}$ living in the tensor product $\\bigotimes_{\\ell=1}^{N} ({\\mathcal V}^{k_{\\ell}}\\otimes\\bar{{\\mathcal V}}^{k_{\\ell}})$ and invariant under the action of $\\mathrm{SU}(2)$:\n\\begin{equation}\n\\Psi^{\\{k_{\\ell},{\\mathcal I}\\}}\\big{(}\\{h_{\\ell}\\}_{\\ell=1..N}\\big{)}\n\\,=\\,\n\\langle h_{\\ell}\\,|\\,k_{\\ell},{\\mathcal I}\\rangle\n\\,=\\,\n{\\mathrm{Tr}}\\,\\Big{[}\n{\\mathcal I}\\otimes\\bigotimes_{\\ell=1}^{N}D^{k_{\\ell}}(h_{\\ell})\n\\Big{]}\\,,\n\\end{equation}\nwhere the trace is taken over the tensor product $\\bigotimes_{\\ell=1}^{N} ({\\mathcal V}^{k_{\\ell}}\\otimes\\bar{{\\mathcal V}}^{k_{\\ell}})$. To underline that each spin repreentation is doubled and that ${\\mathcal I}$ is an intertwiner between the loops around the vertex, we can dub it a {\\it loopy intertwiner}\n\nThe holonomy operator is the basic gauge-invariant operator of loop quantum gravity. It can shift and increase the spins along the edges on which it acts and so is used in practice as a creation operator. We define the holonomy operators $\\hat{\\chi}_\\ell$ along the loops around the vertex as acting by multiplication on the wave-functions in the group representation:\n\\begin{equation}\n(\\hat{\\chi}_\\ell \\triangleright\\Psi)\\,(h_1,...,h_N)\n\\,=\\,\n\\chi_\\frac{1}{2}(h_\\ell) \\Psi(h_1,...,h_N)\n\\end{equation}\nwhere $\\chi_\\frac{1}{2}$ is the trace operator in the fundamental two-dimensional representation of $\\mathrm{SU}(2)$. We can of course also consider holonomy operators that wrap around several loops around the flower:\n\\begin{equation} \n(\\hat{\\chi}_{i,j,k,l,...}  \\triangleright\\Psi)\\,(h_1,...,h_N)\n\\,=\\,\n\\chi_\\frac{1}{2}(h_i h_j h_k h_l ...) \\Psi(h_1,...,h_N)\\,,\n\\end{equation}\nwhere the $i,j,k,l,..$ indices label loops. These operators are obviously still gauge-invariant, and we can further take the inverse or arbitrary powers of each group element.\nThere are two remarks we should do about these multi-loop operators. First, they can be decomposed as a composition of single loop operators combining both holonomy operators and grasping operators (action of the ${\\mathfrak{su}}(2)$ generators as a quantization of the flux-vectors) by iterating the following 2-loop identity:\n\\begin{equation}\n\\chi_\\frac{1}{2}(h_i h_j)\n\\,=\\,\n\\f12\\,\\left[\n\\chi_\\frac{1}{2}(h_i)\\chi_\\frac{1}{2}(h_j)\n+\\sum_{a=1}^{3}\\chi_\\frac{1}{2}(h_i\\sigma_{a})\\chi_\\frac{1}{2}(h_j\\sigma_{a})\n\\right]\\,,\n\\end{equation}\nwhere the $\\sigma_{a}$'s are the three Pauli matrices, normalized such that their square is equal to the identity matrix.\nSecond, if the loopy spin network state comes from the gauge fixing of a more complicated graph down to a single vertex, we had chosen a particular maximal tree on that graph to define the gauge-fixing procedure. The loops around the coarse-grained vertex correspond to the edges that didn't belong to the folding tree. Changing the tree actually maps the single loop holonomies onto multi-loop holonomies \\cite{Livine:2013gna}. So, from the coarse-graining perspective, there is no special reason to prefer single loops over multi-loop operators.\n\n\n\n\\subsection{Superposition of number of loops}\n\\label{sec:loopyprojective}\n\nWe would like to allow for an arbitrary number of loops $N$, with possibly an infinite number of loops, and superpositions of number of loops. We will apply the usual projective limit techniques used in loop quantum gravity, as briefly reviewed in  section \\ref{ProjectiveLimits}.\nWe assume here that the little loops are all distinguishable, so we avoid all symmetrization issue. The case of indistinguishable loops will be dealt with in the next section \\ref{sec:bosonisation}. We discuss the countable infinity of loops around the vertex, so we can number them using the integers ${\\mathbb N}$. The point, as with standard spin networks, is that a state with a spin-0 on an edge does not actually depend on the group element carried by that edge and is thus equivalent to a state on the flower without that edge. Reversing this logic, a state built on a finite number of loops is equivalent to a state with an arbitrary larger number of loops carrying a spin-0 on all the extra edges , which will allow to define it in the projective limit as a state on the flower with an infinite number of loops.\n\n\\begin{figure}[h!]\n\\centering\n\\begin{tikzpicture}\n\n\\coordinate(O1) at (0,0);\n\n\\draw (O1) to[in=-30,out=+30,loop,scale=3,rotate=90] (O1) ++(0,1.2) node {$k_2$};\n\\draw (O1) to[in=-30,out=+30,loop,scale=3,rotate=-90] (O1) ++(0,-1.2) node {$k_5$};\n\\draw (O1) to[in=-30,out=+30,loop,scale=3,rotate=30] (O1) ++(1,0.7) node {$k_3$};\n\\draw[dashed] (O1) to[in=-30,out=+30,loop,scale=3,rotate=-30] (O1) ++(1,-0.7) node {$k_4$};\n\\draw[dashed] (O1) to[in=-30,out=+30,loop,scale=3,rotate=150] (O1) ++(-1,0.7) node {$k_{1}$};\n\\draw[black!50,dashed] (O1) to[in=-30,out=+30,loop,scale=3,rotate=-150] (O1) ++(-1,-0.7) node {$k_6$};\n\n\\draw (O1) node[scale=1] {$\\bullet$} ;\n\n\\end{tikzpicture}\n\n\\caption{We consider a loopy spin network state with a varying number of loops as a superposition of states with support over different loops.}\n\\label{fig:variousloops}\n\n\\end{figure}\n\nLet us consider  the set $\\mathcal{P}_{<\\infty}(\\mathbb{N})$  of all finite subsets of $\\mathbb{N}$. A flower with a finite number of loops corresponds to a finite subset $E\\in \\,\\mathcal{P}_{<\\infty}(\\mathbb{N})$ of indices labeling its loops. Since we keep the loops distinguishable, we do not identify all the subsets with same cardinality and keep on distinguishing them. We define the Hilbert space of gauge-invariant wave-functions on the flower corresponding to $E$:\n\\begin{equation}\n{\\mathcal H}_{E}\n\\,=\\,\nL^2\\,\\Big{(}\\mathrm{SU}(2)^{E}\/\\mathrm{Ad}\\,\\mathrm{SU}(2)\\Big{)}\\,,\n\\qquad\n\\Psi(\\{h_{\\ell}\\}_{\\ell\\in E})=\\Psi(\\{gh_{\\ell}g^{-1}\\}_{\\ell\\in E})\n\\quad\n\\forall g\\in\\mathrm{SU}(2)\\,.\n\\end{equation}\nWe would like to consider arbitrary superpositions of states with support on arbitrary subsets $E$ of loops, but we do not wish to brutally consider the direct sum over all $E$'s. We still require cylindrical consistency. Indeed, a function on $\\mathrm{SU}(2)^{E}$ which actually  does not depend at all on the loop $\\ell_{0}\\in E$ can legitimately be considered as a function on $\\mathrm{SU}(2)^{E\\setminus\\ell_{0}}$. We introduce the equivalence relation making this explicit. For two subsets $E\\subset F$, and two functions $\\Psi$ and $\\widetilde{\\Psi}$ respectively on  $\\mathrm{SU}(2)^{E}$ and  $\\mathrm{SU}(2)^{F}$, the two wave-functions are defined as equivalent if:\n\\begin{equation}\nE\\subset F\\,,\n\\quad\n\\Psi:\\mathrm{SU}(2)^{E}\\rightarrow{\\mathbb C}\\,,\n\\quad\n\\widetilde{\\Psi}:\\mathrm{SU}(2)^{F}\\rightarrow{\\mathbb C}\\,,\n\\qquad\n\\Psi\\sim\\widetilde{\\Psi}\n\\quad\\Leftrightarrow\\quad\n\\widetilde{\\Psi}(\\{h_{\\ell}\\}_{\\ell\\in F})\n\\,=\\,\n{\\Psi}(\\{h_{\\ell}\\}_{\\ell\\in E})\\,,\n\\end{equation}\nthat is the function $\\widetilde{\\Psi}$ on the larger set $F$ does not depend on the group elements $h_{\\ell}$ for $\\ell\\in F\\setminus E$ and coincides with the function $\\Psi$ on the smaller set $E$. More generally, when the two subsets $E$ and $F$ do not contain one or the other, we transite trough their intersection $E\\cap F$.\n\nThe space of wave-functions in the projective limit  is defined as the union over all subsets $E$ of functions on  $\\mathrm{SU}(2)^{E}$, quotiented by this equivalence. We similarly define the projective limit of the integration measure over $\\mathrm{SU}(2)$. We use this measure to define the Hilbert space ${\\mathcal H}^{\\mathrm{loopy}}$ of states on the flower with an arbitrary number of loops. All the rigorous mathematical definitions and proofs are given in the appendix  \\ref{app:ProjectiveLimit}.\n\nThe practical way to see this Hilbert space is to use the spin network basis and understand that a loop carrying a spin-0 means that the wave-function actually does not depend on the group element living on that loop. For every state, we can thus reduce its underlying graph to the minimal possible one removing all the loops with trivial dependency. Following this logic, for every subset $E$, we define the space of proper states living on $E$, that is without any spin-0 on its loops. This amounts to removing all possible 0-modes:\n\\begin{equation}\n\\mathcal{H}_{E}^0 \n\\,=\\,\n\\Bigg{\\{}\n\\Psi\\in{\\mathcal H}_{E}\n\\,:\\,\n\\forall \\ell_{0}\\in E\\,,\\,\\,\\int_{\\mathrm{SU}(2)}\\mathrm{d}h_{\\ell_{0}}\\,\\Psi =0\n\\Bigg{\\}}\\,.\n\\end{equation}\nWe can decompose the Hilbert space of states on the subset $E\\subset{\\mathbb N}$ of loops onto proper states:\n\\begin{prop}\n\\label{proper}\nThe Hilbert space ${\\mathcal H}_{E}$ on loopy intertwiners on the set of loops $E$ decomposes as a direct sum of the Hilbert spaces of proper states with support on every subset of $E$:\n\\begin{equation}\n\\mathcal{H}_{E} \\simeq \\bigoplus_{F \\subset E} \\mathcal{H}_{F}^0\n\\,.\n\\end{equation}\nThis isomorphism is realized through the projections $f_{F}=P_{E,F}f\\in{\\mathcal H}^{0}_{F}$, acting on wave-functions $f\\in{\\mathcal H}_{E}$, defined for an arbitrary subset $F\\subset E$:\n\\begin{equation}\nf_{F}\\big{(}\n\\{h_{\\ell}\\}_{\\ell\\in F}\n\\big{)}\n\\,=\\,\n\\sum_{\\widetilde{F}\\subset  F}\n(-1)^{\\#\\widetilde{F}}\n\\int \\prod_{\\ell\\in E\\setminus F}\\mathrm{d}g_{\\ell}\n\\prod_{\\ell\\in\\widetilde{F}}\\mathrm{d}k_{\\ell}\\,\nf\\big{(}\n\\{h_{\\ell}\\}_{\\ell\\in F\\setminus \\widetilde{F}},\n\\{k_{\\ell}\\}_{\\ell\\in\\widetilde{F}},\n\\{g_{\\ell}\\}_{\\ell\\in E\\setminus F}\n\\big{)}\\,.\n\\end{equation}\nThese projections realize a combinatorial transform of the state $f\\in{\\mathcal H}_{E}$:\n\\begin{equation}\nf=\\sum_{F\\subset E} f_{F}\\,,\n\\qquad\nf_{F}\\in{\\mathcal H}_{F}^{0}\\,,\n\\qquad\n\\forall\\ell\\in F\\,,\\quad\\int \\mathrm{d}h_{\\ell}\\,f_{F}=0\\,.\n\\end{equation}\n\\end{prop}\nThis decomposition is straightforward to prove. It will also be crucial in the case of undistinguishable loops and  symmetrized states, as we will see in the next section \\ref{sec:bosonisation}.\nThen, as we show in the appendix  \\ref{app:ProjectiveLimit},  the Hilbert space of loopy spin networks on the flower, with an arbitrary number of distinguishable loops, defined as the projective limit of the Hilbert spaces ${\\mathcal H}_{E}$ is realized as  the direct sum of those spaces of proper states:\n\\begin{equation}\n\\mathcal{H}^{\\mathrm{loopy}} \\simeq \\bigoplus_{F \\in \\mathcal{P}_{<\\infty}(\\mathbb{N})} \\mathcal{H}_{F}^0\\,,\n\\qquad\n\\mathcal{H}_{F}^0 = \\bigoplus_{j_{\\ell \\in F} \\neq 0, {\\mathcal I}} \\mathbb{C} |j_{\\ell \\in F}, {\\mathcal I}\\rangle\\,.\n\\end{equation}\n\n\n\\subsection{Holonomy operators as creation and annihilation operators}\n\nWe can revisit the definition of the holonomy operators\\footnotemark on our Hilbert space ${\\mathcal H}$ of states with arbitrary number of loops. Let us consider the loop $\\ell_{0}\\in{\\mathbb N}$ and define the corresponding holonomy operator $\\hat{\\chi}_{\\ell_{0}}$. Looking at its action on a state $\\Psi$ with finite number of loops living in the Hilbert space ${\\mathcal H}_{E}$, we have two possibilities: either the loop $\\ell_{0}$ belongs to the subset $E$ or it doesn't. If the acting loop $\\ell_{0}$ is already a loop of our state $\\Psi$, then the holonomy operator acts on as before by multiplication:\n\\footnotetext{\nIn order to identify a complete set of operators acting on the Hilbert space ${\\mathcal H}^{\\mathrm{loopy}}$, we should further consider multi-loops holonomy operators or grasping operators or deformation operators such as $\\mathrm{U}(N)$ operators \\cite{Borja:2010rc}, but in the first exploration we propose, in this paper, we decide to focus on the single-loop holonomy operator.}\n\\begin{equation}\n\\ell_{0}\\in E\\,,\n\\quad\n\\Psi\\in{\\mathcal H}_{E}\\,,\n\\quad\n\\hat{\\chi}_{\\ell_{0}}\\,\\Psi\\in{\\mathcal H}_{E}\\,,\n\\qquad\n(\\hat{\\chi}_{\\ell_{0}}\\Psi)\\,(\\{h_{\\ell}\\}_{\\ell\\in E})\n\\,=\\,\n\\chi_{\\frac{1}{2}}(h_{\\ell_{0}})\\,\\Psi\\,(\\{h_{\\ell}\\}_{\\ell\\in E})\\,.\n\\end{equation}\nIf the acting loop doesn't belong to the initial subset $E$, we use the cylindrical consistency equivalence relation and we embed both the new loop and the initial loops in a larger graph, say $E\\cup\\{\\ell_{0}\\}$,\n\\begin{equation}\n\\ell_{0}\\notin E\\,,\n\\quad\n\\Psi\\in{\\mathcal H}_{E}\\,,\n\\quad\n\\hat{\\chi}_{\\ell_{0}}\\,\\Psi\\in{\\mathcal H}_{E\\cup\\{\\ell_{0}\\}}\\,,\n\\qquad\n(\\hat{\\chi}_{\\ell_{0}}\\Psi)\\,(\\{h_{\\ell}\\}_{\\ell\\in E})\n\\,=\\,\n\\chi_{\\frac{1}{2}}(h_{\\ell_{0}})\\,\\Psi\\,(\\{h_{\\ell}\\}_{\\ell\\in E})\\,,\n\\end{equation}\nwith the holonomy operator $\\hat{\\chi}_{\\ell_{0}}$ acting as a creation operator, creating a new loop and curvature excitation.\nSince the $\\mathrm{SU}(2)$ character $\\chi_{\\frac{1}{2}}$ is real and bounded by two, $|\\chi_{\\frac{1}{2}}|\\le 2$, we can check that the holonomy operators $\\hat{\\chi}_{\\ell}$ are Hermitian, bounded and thus essentially self-adjoint.\n\nThe holonomy operator $\\hat{\\chi}_{\\ell}$  is Hermitian and has a component acting as a creation operator. It must have an annihilation counterpart. The best way to see this explicitly is to write its action on proper states, consistently removing the zero-modes. Indeed, if a loop carries a spin $\\f12$, then it gets partly annihilated by the holonomy operator:\n\\begin{equation}\n\\ell_{0}\\in E\\,,\n\\quad\n\\Psi\\in{\\mathcal H}_{E}^{0}\\,,\n\\quad\n\\hat{\\chi}_{\\ell_{0}}\\,\\Psi\\in{\\mathcal H}_{E}^{0}\\oplus{\\mathcal H}_{E\\setminus\\{\\ell_{0}\\}}^{0}\\,, \\nonumber\n\\end{equation}\n\\begin{equation}\n(\\hat{\\chi}_{\\ell_{0}}\\Psi)\\,(\\{h_{\\ell}\\}_{\\ell\\in E})\n\\,=\\,\n\\underset{\\in\\,{\\mathcal H}_{E}^{0}}{\\underbrace{\\Bigg{[}\\chi_{\\frac{1}{2}}(h_{\\ell_{0}})\\Psi\\,(\\{h_{\\ell}\\}_{\\ell\\in E})\n-\\int\\mathrm{d}h_{\\ell_{0}}\\chi_{\\frac{1}{2}}(h_{\\ell_{0}})\\Psi\\,(\\{h_{\\ell}\\}_{\\ell\\in E})\\Bigg{]}}}\n+\\underset{\\in\\,{\\mathcal H}_{E\\setminus\\{\\ell_{0}\\}}^{0}}\n{\\underbrace{\\Bigg{[}\\int\\mathrm{d}h_{\\ell_{0}}\\chi_{\\frac{1}{2}}(h_{\\ell_{0}})\\Psi\\,(\\{h_{\\ell}\\}_{\\ell\\in E})\\Bigg{]}}}\n\\,.\n\\end{equation}\nThis way, it is clear that the holonomy operator $\\hat{\\chi}_{\\ell_{0}}$ creates transition adding and removing one loop. This proper state decomposition of the holonomy operator will become essential when defining it on the Fock space of symmetrized loopy spin networks in the next section \\ref{sec:bosonisation}.\n\n\\subsection{Imposing BF dynamics on loopy spin networks}\n\nNow that we have describe the whole kinematics of loopy spin networks, with distinguishable loops, we would like to tackle the issue of the dynamics and imposing the Hamiltonian constraints on the Hilbert space of loopy states ${\\mathcal H}^{\\mathrm{loopy}}$. The final goal of our proposal is to write the Hamiltonian constraints of loop quantum gravity on  ${\\mathcal H}^{\\mathrm{loopy}}$, such that it allows explicitly for local degrees of freedom,  study its renormalization group flow under the coarse-graining and extract its large scale or continuum limit.\nHere we will instead describe the much simpler BF dynamics. BF theory can be considered as a consistency check for all attempts and methods to define of dynamics in (loop) quantum gravity\\footnotemark. Its physical states are well-known and the Hamiltonian constraints project onto flat connection states. It is furthermore a topological theory with no local degrees of freedom -they are pure gauge.\nFinally it has a trivial renormalization flow. Indeed the flatness constraint behaves very nicely under coarse-graining, as illustrated on fig.\\ref{fig:Flat}~: considering a spin network graph, imposing the flatness of the connection on all small loops  garanties that  larger loops will be flat too.\nAll these features must reflect in any proposal for the quantum dynamics of BF theory. \n\\footnotetext{\nOnce the dynamics of BF theory is properly implemented and well under control in a certain framework, one usually use it as a starting point for imposing the true gravity dynamics, with local degrees of freedom, relying on the reformulation of general relativity as a BF theory with constraints. This is for instance the logic behind the construction of spinfoam models for a quantum gravity path integral \\cite{Livine:2010zx,Perez:2012wv,Bianchi:2012nk}.\n}\n\n\\begin{figure}[h!]\n\n\\centering\n\n\\begin{tikzpicture}\n\\coordinate(A) at (0,0);\n\\coordinate(A1) at (0.2,-0.2);\n\\coordinate(A2) at (0.4,-0.2);\n\\coordinate(B) at (2,0);\n\\coordinate(B1) at (1.8,-0.2);\n\\coordinate(B2) at (2.2,-0.2);\n\\coordinate(B3) at (2.4,-0.2);\n\\coordinate(C) at (4,0);\n\\coordinate(C1) at (3.8,-0.2);\n\\coordinate(D) at (0,-2);\n\\coordinate(D1) at (0.2,-1.8);\n\\coordinate(D2) at (0.2,-0.6);\n\\coordinate(D3) at (0.2,-0.4);\n\\coordinate(E) at (2,-2);\n\\coordinate(E1) at (1.8,-1.8);\n\\coordinate(E2) at (2.2,-1.8);\n\\coordinate(E3) at (2.2,-0.6);\n\\coordinate(E4) at (2.2,-0.4);\n\\coordinate(F) at (4,-2);\n\\coordinate(F1) at (3.8,-1.8);\n\\coordinate(O1) at (1,-1);\n\\coordinate(O2) at (3,-1);\n\n\n\\draw[dashed] (A) -- (B) -- (E) -- (D) -- (A);\n\\draw[dashed] (B) -- (C) -- (F) -- (E);\n\n\\draw (A) node {$\\bullet$};\n\\draw (B) node {$\\bullet$};\n\\draw (C) node {$\\bullet$};\n\\draw (D) node {$\\bullet$};\n\\draw (E) node {$\\bullet$};\n\\draw (F) node {$\\bullet$};\n\n\n\\draw[blue,rounded corners,thick] (A2) -- (B1) -- (E1) -- (D1) -- (D2);\n\\draw[blue,thick,->,>=stealth] (D2) -- (D3);\n\\draw[blue,rounded corners,thick] (B3) -- (C1) -- (F1) -- (E2) -- (E3);\n\\draw[blue,thick,->,>=stealth] (E3) -- (E4);\n\n\\draw[blue] (O1) node[scale=2] {\\textbf{${\\mathbb{I}}$}};\n\\draw[blue] (O2) node[scale=2] {\\textbf{${\\mathbb{I}}$}};\n\n\n\\draw[->,>=stealth,very thick] (5,-1) -- (7,-1);\n\n\\coordinate(P) at (8,0);\n\\coordinate(A0) at ($(P)+(0,0)$);\n\\coordinate(A01) at ($(P)+(0.2,-0.2)$);\n\\coordinate(A02) at ($(P)+(0.4,-0.2)$);\n\\coordinate(B0) at ($(P)+(2,0)$);\n\\coordinate(B01) at ($(P)+(1.8,-0.2)$);\n\\coordinate(B02) at ($(P)+(2.2,-0.2)$);\n\\coordinate(B03) at ($(P)+(2.4,-0.2)$);\n\\coordinate(C0) at ($(P)+(4,0)$);\n\\coordinate(C01) at ($(P)+(3.8,-0.2)$);\n\\coordinate(D0) at ($(P)+(0,-2)$);\n\\coordinate(D01) at ($(P)+(0.2,-1.8)$);\n\\coordinate(D02) at ($(P)+(0.2,-0.6)$);\n\\coordinate(D03) at ($(P)+(0.2,-0.4)$);\n\\coordinate(E0) at ($(P)+(2,-2)$);\n\\coordinate(E01) at ($(P)+(1.8,-1.8)$);\n\\coordinate(E02) at ($(P)+(2.2,-1.8)$);\n\\coordinate(E03) at ($(P)+(2.2,-0.6)$);\n\\coordinate(E04) at ($(P)+(2.2,-0.4)$);\n\\coordinate(F0) at ($(P)+(4,-2)$);\n\\coordinate(F01) at ($(P)+(3.8,-1.8)$);\n\\coordinate(O0) at ($(P)+(2,-1)$);\n\n\\draw[dashed] (A0) -- (B0);\n\\draw[dashed,gray] (B0) -- (E0);\n\\draw[dashed] (E0) -- (D0) -- (A0);\n\\draw[dashed] (B0) -- (C0) -- (F0) -- (E0);\n\n\\draw (A0) node {$\\bullet$};\n\\draw (B0) node {$\\bullet$};\n\\draw (C0) node {$\\bullet$};\n\\draw (D0) node {$\\bullet$};\n\\draw (E0) node {$\\bullet$};\n\\draw (F0) node {$\\bullet$};\n\n\\draw[blue,rounded corners,thick] (A02) -- (C01) -- (F01) -- (D01) -- (D02);\n\\draw[blue,thick,->,>=stealth] (D02) -- (D03);\n\n\\draw[blue] (O0) node[scale=2] {\\textbf{${\\mathbb{I}}$}};\n\n\\end{tikzpicture}\n\n\\caption{In BF theory, holonomies behave very nicely under coarse-graining. If each small loops is flat, large loops are flat too. In other words, the physical state of BF theory is a flat space, which is flat at all scales.}\n\\label{fig:Flat}\n\n\\end{figure}\n\nConsidering the full space of loopy spin networks on some arbitrary graph $\\Gamma$, we would like the BF Hamiltonian constraints to project onto the flat connection state(s), that is impose flatness around all the loops of the graph $\\Gamma$ and also kill all the local excitations represented by the little loops at every vertex. Flatness around the loops of the background graph is the standard result for BF constraints. So here we will focus on the fate of the little loops, that we introduced. To this purpose, it suffices to focus on a single vertex, that is to work on the flower graph.\n\nConsidering the flower graph with arbitrary number of loops, as we have defined above, we introduce  the following set of constraints:\n\\begin{equation}\n\\forall \\ell \\in \\mathbb{N},\n\\quad\n\\big{(}\\hat{\\chi}_\\ell-2\\big{)} |\\Psi\\rangle \\,=\\, 0\\,.\n\\end{equation}\nWe impose one constraint for every (possible) loop by imposing that the corresponding holonomy operator saturates its bound and projects on its highest eigenvalue. These constraints all commute with each other.\nLet us underline the dual role of Hamiltonian constraints. As first class constraints, we need to solve them and identify their solution space, but they also generate gauge transformations and we need to gauge out their action. Here, the holonomy constraint operators both impose the flatness of the connection, but they also imply that the little loops are pure gauge, so that their action can change the number of loops to arbitrary values.\nWe will see below that these one-loop holonomy constraints are almost enough to fully constrain the theory to the single flat state on the flower graph.\n\n\\bigskip\n\nLet us solve these constraints and consider a loop $\\ell_{0}$ and its action of its holonomy operator $\\widehat{\\chi}_{\\ell_{0}}$ on a wave-function $\\Psi\\in{\\mathcal H}_{E}$ with support on the finite subset $E\\subset{\\mathbb N}$ of loops. A first case is when $\\ell_{0}\\in E$ belongs to the subset, in which case we have a simple functional equation on $\\mathrm{SU}(2)^{E}$:\n$$\n(\\hat{\\chi}_{\\ell_{0}}\\Psi)\\,(\\{h_{\\ell}\\}_{\\ell\\in E})\n\\,=\\,\n\\chi_{\\frac{1}{2}}(h_{\\ell_{0}})\\Psi\\,(\\{h_{\\ell}\\}_{\\ell\\in E})\n\\,=\\,\n2\\,\\Psi\\,(\\{h_{\\ell}\\}_{\\ell\\in E})\\,.\n$$\nThe second case is when the considered loop $\\ell_{0}\\notin E$  doesn't belong to the subset. The holonomy operator $\\hat{\\chi}_{\\ell_{0}}$ then creates a loop, making a transition from ${\\mathcal H}_{E}^{0}$ to the orthogonal space ${\\mathcal H}_{E\\cup \\{\\ell_{0}\\}}^{0}$. This illustrates that the flow generated by those Hamiltonian constraints can arbitrarily shift the number of loops and therefore the little loops become pure gauge at the dynamical level in BF theory. This also means that there is no solution to all holonomy constraints with support on a finite subset $E$ and a physical state must have support on all possible loops.\n\n\\medskip\n\nTo be rigorous, we need to go to the dual space  $({\\mathcal H}^{\\mathrm{loopy}})^*$ and solve the holonomy constraints on the space of distribution defined in the projective limit.  We are looking for a  family of distributions $\\varphi_{E}$ on $\\mathrm{SU}(2)^{E}$, that is continuous linear forms over smooth functions on  $\\mathrm{SU}(2)^{E}$ (see appendix \\ref{app:distribution} for a discussion of the definition of distributions over $\\mathrm{SU}(2)$). The cylindrical consistency means that their evaluations on two cylindrically equivalent smooth functions must be equal:\n$$\n\\forall E\\subset \\widetilde{E}\\,,\\,\\,\nf_{E}\\sim f_{\\widetilde{E}}\\quad\n\\Rightarrow\n\\varphi_{E}(f_{E})=\\int_{\\mathrm{SU}(2)^{E}}\\varphi_{E}f_{E}\n\\,=\\,\n\\int_{\\mathrm{SU}(2)^{\\widetilde{E}}}\\varphi_{\\widetilde{E}}f_{\\widetilde{E}}=\\varphi_{\\widetilde{E}}(f_{\\widetilde{E}})\\,.\n$$\nThen the holonomy constraints read:\n$$\n\\forall \\ell\\in{\\mathbb N}\\,,\n\\forall E \\ni \\ell\\,,\n\\forall f_{E}\\in{\\cal C}^{\\infty}_{\\mathrm{SU}(2)^{E}}\\,,\n\\,\\,\n\\int_{\\mathrm{SU}(2)^{E}}\\ \\varphi_{E} (\\chi_{\\ell}-2)f_{E}\n\\,=\\,0\\,,\n$$\nwhere we have considered by default that the loop $\\ell$ belongs to the wave-function support $E$. Indeed, if $\\ell$ didn't belong to $E$, then we could enlarge the subset $E$ to $E\\cup \\{\\ell\\}$ by cylindrical consistency and consider both the test function $f$ and the distribution $\\varphi$ as living on that larger subset.\nOur goal is to show that the unique solution to these equations is the flat state, i.e. that there exists $\\lambda\\in{\\mathbb C}$ such that $\\varphi_{E}=\\lambda\\,\\delta^{\\otimes E}$:\n\\begin{equation}\n\\forall f_{E}\\in{\\cal C}^{\\infty}_{\\mathrm{SU}(2)^{E}}\\,,\n\\varphi_{E}(f_{E})=\\lambda \\delta_{E}(f_{E})\n=\n\\lambda\\int_{\\mathrm{SU}(2)^{E}}\\prod_{\\ell\\in E}\\delta(h_{\\ell}) f_{E}(\\{h_{\\ell}\\}_{\\ell\\in E})\n=\n\\lambda f_{E}({\\mathbb{I}},..,{\\mathbb{I}})\\,.\n\\end{equation}\nCylindrical consistency simply requires that the factor $\\lambda$ does not depend on the subset $E$.\nSo we are led to solve the holonomy constrain on every finite subset $E$. Thus, let us consider the functional equation on $\\mathrm{SU}(2)^{N}$:\n\\begin{equation}\n\\forall 1\\le\\ell\\le N\\,,\\,\\,\n\\left(\\hat{\\chi}_\\ell - 2\\right)\\varphi = 0\\,,\n\\end{equation}\nwhere we drop the subset label $E$.\n\n\\subsubsection{Holonomy constraint on $\\mathrm{SU}(2)$}\n\nLet us start with the one-loop case and solve for distributions $\\varphi$ on $\\mathrm{SU}(2)$ the equation:\n\\begin{equation}\n\\forall h\\in\\mathrm{SU}(2)\\,,\\,\\,\n\\chi_{\\f12}(h)\\varphi(h)=2\\varphi(h)\\,.\n\\end{equation}\nSince the character $\\chi_{\\f12}$ is smooth and reaches its maximum value $2$ at a single point, the identity ${\\mathbb{I}}$, it seems natural that the $\\varphi$ must be a distribution peaked at the identity. We therefore expect that the only solution be the $\\delta$-distribution on $\\mathrm{SU}(2)$, $\\varphi=\\delta$. However, since the identity is actually an extremum of $\\chi_{\\f12}$ and that the first derivatives of the character thus vanishes at this point, this equation admit more solutions: the first derivatives of the $\\delta$-distribution. This clearly came as a surprise for us.\n\nLet us first assume that $\\varphi$ is gauge-invariant, i.e. invariant under conjugation. Its Fourier decomposition on $\\mathrm{SU}(2)$ involves only the characters in all spins:\n$$\n\\varphi=\\sum_{j\\in\\frac\\N2}\\varphi_{j}\\chi_{j}\\,.\n$$\nAs well known, the holonomy constraint leads to a recursion relation on the coefficients $\\varphi_{j}$:\n\\begin{equation}\n\\chi_{\\f12}\\chi_{j}=\\chi_{j-\\f12}+\\chi_{j+\\f12}\n\\quad\\Rightarrow\\qquad\n2 \\varphi_{0}=\\varphi_{\\f12}\\,,\\quad\n2\\varphi_{j\\ge\\f12}=\\varphi_{j-\\f12}+\\varphi_{j+\\f12}\\,.\n\\end{equation}\nOnce the initial condition $\\varphi_{0}$ is fixed, these lead to a unique solution:\n\\begin{equation}\n\\varphi_{j}=(2j+1)\\varphi_{0}\\,,\n\\qquad\n\\varphi=\\varphi_{0}\\sum_{j}(2j+1)\\chi_{j}=\\varphi_{0}\\delta\\,.\n\\end{equation}\nWhen solving such functional equations in the Fourier basis, one should nevertheless be very careful to work with well-defined distributions. These are characterized by Fourier coefficients $\\varphi^{j}$ growing at most polynomially with the spin $j$. This ensures that evaluations $\\int f\\varphi$ of the distribution $\\varphi$ on smooth test functions $f$ are convergent series. The $\\delta$-distribution is clearly a good solution. But, as an example, solving for eigenvectors of the holonomy operator associated to (real) eigenvalues (strictly) larger than 2 would lead to exponentially growing Fourier coefficients, which are too divergent to define a proper distribution. The interested reader will find more details in the appendix \\ref{app:distribution}.\n\n\\medskip\n\nOn the space of  functions invariant under conjugation, everything works as expected. Let us now consider the general case dropping the requirement of gauge-invariance. The $\\delta$-distribution is obviously still a solution:\n\\begin{equation}\n\\forall f\\in{\\mathcal C}^{\\infty}_{\\mathrm{SU}(2)}\\,,\n\\,\\,\n\\int f\\, (\\chi_{\\f12}-2)\\,\\delta\n=\n(\\chi_{\\f12}({\\mathbb{I}})-2)f({\\mathbb{I}})\n=0\\,.\n\\end{equation}\nBut, now the first derivatives of the $\\delta$-distributions are also  solutions:\n\\begin{equation}\n\\forall f\\in{\\mathcal C}^{\\infty}_{\\mathrm{SU}(2)}\\,,\n\\,\\,\n\\int f\\, (\\chi_{\\f12}-2)\\,\\partial_{x}\\delta\n=\n\\left.-(\\partial_{x}\\chi_{\\f12})\\,f-(\\chi_{\\f12}-2)\\,f\\right|_{{\\mathbb{I}}}\n=0\\,,\n\\label{ppdelta}\n\\end{equation}\nwhere $x\\in{\\mathbb R}^{3}$ indicates the direction of the derivative and the derivatives of the character vanish at the identity since it is a extremum.\nWe remind the reader that the right-derivative $\\partial_{x}^{R}$ on $\\mathrm{SU}(2)$ is a anti-Hermitian operator ($i\\partial$ is Hermitian) defined by the infinitesimal action of the ${\\mathfrak{su}}(2)$ generator $\\vec{x}\\cdot\\vec{J}$ (where the $\\vec{J}$ in the fundamental spin-$\\f12$ representation are simply half the Pauli matrices):\n\\begin{equation}\n\\partial_{x}^{R}f(h)=\n\\lim_{\\epsilon\\rightarrow 0}\\frac{f(h e^{i\\epsilon \\vec{x}\\cdot \\vec{J}})-f(h)}{\\epsilon}\n=f(h\\,  x)\\,,\\quad\n\\textrm{with}\\quad x= \\vec{x}\\cdot \\vec{J}\\,.\n\\end{equation}\nWe usually differentiate along the three directions in ${\\mathbb R}^{3}\\sim{\\mathfrak{su}}(2)$ leading to the insertion of the generators $J_{a=1,2,3}$:\n\\begin{equation}\n\\partial_{a}^{R}f(h)=if(h J_{a})\\,,\n\\quad\n\\partial_{a}^{L}f(h)=if(J_{a}h)\\,.\n\\end{equation}\nActing on the $\\delta$-distribution gives the following Fourier decomposition for its derivatives $\\partial_{a}^{L}\\delta=\\partial_{a}^{R}\\delta=\\partial_{a}\\delta$:\n\\begin{equation}\n\\partial_{a}\\delta(h)\n=i\\sum_{j} (2j+1)D^{j}_{nm}(J_{a})\\,D^{j}_{mn}(h)\\,,\n\\end{equation}\nwhere we use the Wigner matrices for the group element $h$ and the ${\\mathfrak{su}}(2)$ generators.\n\nWe can actually generate a whole tower of higher derivative solutions to the holonomy constraints. We simply need to identify the differential operators whose action on the spin-$\\f12$ character vanishes at the identity. Thus, at second order, we get five new independent solutions given by the following operators:\n\\begin{equation}\n\\partial_{1}\\partial_{2}\n\\,,\\,\\,\n\\partial_{1}\\partial_{3}\n\\,,\\,\\,\n\\partial_{2}\\partial_{3}\n\\,,\\,\\,\n(\\partial_{1}\\partial_{1}-\\partial_{2}\\partial_{2})\n\\,,\\,\\,\n(\\partial_{1}\\partial_{1}-\\partial_{3}\\partial_{3})\n\\,,\n\\end{equation}\nthat is the $\\partial_{a}\\partial_{b}$ and $(\\partial_{a}\\partial_{a}-\\partial_{b}\\partial_{b})$ for $a\\ne b$. Following this logic, we will get 7 new independent solutions at third order, and so on with $(2n+1)$ independent differential operators at order $n$, for a total of $(n+1)^{2}$ independent solutions to the holonomy constraints given by differential operators of order at most $n$ acting on the $\\delta$-distribution.\n\nSuch as in the conjugation-invariant case, it is enlightening to switch to the Fourier decomposition and translate the holonomy constraint into a recursion relation on the Fourier coefficients. The difference is that we had one Fourier coefficient $\\varphi^{j}$ for each spin $j$ in the gauge-invariant case while in the general case $\\varphi^{j}$ is a $(2j+1)\\times(2j+1)$ matrix. Implementing the recursion, we start from spin 0 and work the way up to higher spins. The problem is that the recursion relations determine only $(2j)^{2}$ matrix elements of  $\\varphi^{j}$ in terms of the lower spins coefficients, leaving $(2j+1)^{2}-(2j)^{2}=(4j+1)$ matrix elements free to be specified as initial conditions. This leads to an infinite number of solutions to the recursion relations, which reproduces the tower of higher order derivative solutions.\nThe interested reader will find all of the details on the recursion relations in appendix \\ref{app:recursion}. \n\n\\subsubsection{Introducing the Laplacian constraint on $\\mathrm{SU}(2)$}\n\\label{derivativesolution1}\n\nIf we work with a single loop, a single petal on the flower, then the wave-function is obviously gauge-invariant and we do not have to deal with these extra solutions to the holonomy constraint\\footnotemark.\n\\footnotetext{\nIndeed, since we already proved that the $\\delta$-distribution is the only gauge-invariant solution to the holonomy constraint,  all the derivative solutions can not be gauge-invariant. We can also prove this directly.\nTo get a solution invariant under conjugation, we need to contract the derivative indices together. Now a fundamental theorem on rotational invariants states that all $\\mathrm{SO}(3)$-invariant polynomial of $n$ 3d-vectors $\\vec{v}_{i=1..n}$ are generated by scalar products $\\vec{v}_{i}\\cdot\\vec{v}_{j}$ and triple products $\\vec{v}_{i}\\cdot(\\vec{v}_{j}\\wedge\\vec{v}_{k})$. We simply have to check that the Laplacian and triple-grasping of the $\\delta$-distribution are not solutions of the holonomy constraints:\n$$\n\\int f (\\chi_{\\f12}-2)\\partial_{a}\\partial_{a}\\delta\n=\n\\Delta \\chi_{\\f12}({\\mathbb{I}})\\,f({\\mathbb{I}})\n=\n\\frac{-3}2\\,f({\\mathbb{I}})\\ne 0\n\\,,\n\\quad\n\\int f (\\chi_{\\f12}-2)\\epsilon^{abc}\\partial_{a}\\partial_{b}\\partial_{c}\\delta\n=\n\\frac{-i^{3}}{2^{3}}\\epsilon^{abc}\\chi_{\\f12}(\\sigma_{c}\\sigma_{b}\\sigma_{a})\\,f({\\mathbb{I}})\n=\n\\frac{3}2\\,f({\\mathbb{I}})\\ne 0\\,.\n$$\n}\nHowever, as soon as we add external legs attached to the vertex (linking the flower to other vertices in the graph) or add more loops, then we have to find a way to suppress those derivative solutions, in $\\partial_{a}\\delta$ and so on, which would lead to extra degrees of freedom as some kind of polarized flat states.\n\nSince we want to ensure the full flatness of the holonomy, the most natural proposal is to constrain all the components of the group element living on the loop and not only its trace:\n$$\n\\forall m,n=\\pm\\f12\\,,\\,\\,\nD^{\\f12}_{mn}(h) \\,\\varphi(h)=\\delta_{mn}\\,\\varphi(h)\\,.\n$$\nOne can indeed check, both from the differential calculus point of view or the recursion relations in Fourier space, that these equations admit the $\\delta$-distribution as unique solutions. We can also go beyond multiplicative operators and insert some differential operators. Then supplementing the trace holonomy constraint with the other constraints $\\chi_{\\f12}\\partial_{a}\\,\\varphi =2\\partial_{a}\\varphi$ for $a=1,2,3$ also ensures a unique flat solution. However, these constraints are not gauge-invariant: the constraint operators map wave-functions invariant under conjugation to non-invariant functions.\n\nIn order to keep gauge-invariant constraints, we go to the second derivatives and consider the Laplacian operator. Actually, we introduce the right-Laplacian $\\Delta\\equiv \\sum_{a}\\partial_{a}^{R}\\partial_{a}^{R}$ and a mixed Laplacian operator $\\widetilde{\\Delta}\\equiv \\sum_{a}\\partial_{a}^{L}\\partial_{a}^{R}$, and we propose a new constraint\\footnotemark:\n\\begin{equation}\n\\Delta \\varphi =\\widetilde{\\Delta} \\varphi\\,,\n\\end{equation}\n\\footnotetext{\nAs an example, we can see how $\\Delta$ and $\\widetilde{\\Delta}$ differ through their action on the coupled character $\\chi(h_{1}h_{2})$:\n$$\n\\Delta_{1}\\chi_{\\f12}(h_{1}h_{2})=-\\f14\\chi_{\\f12}(h_{1}\\sigma_{a}\\sigma_{a}h_{2})=-\\f34\\chi_{\\f12}(h_{1}h_{2})\n\\,,\n\\quad\n\\widetilde{\\Delta}_{1}\\chi_{\\f12}(h_{1}h_{2})=-\\f14\\chi_{\\f12}(\\sigma_{a}h_{1}\\sigma_{a}h_{2})\n=-\\f14\\,\\big{(}2\\chi_{\\f12}(h_{1})\\chi_{\\f12}(h_{2})-\\chi_{\\f12}(h_{1}h_{2})\\big{)}\\,,\n$$\nwhich are of course equal at $h_{1}={\\mathbb{I}}$.\n}\nAt the classical level, the differential operator $\\partial_{a}$ represents the flux vector $X_{a}$: the right derivative represents the flux $\\vec{X}^{s}$ at the source of the loop while the left derivative is the flux $\\vec{X}^{t}$ at the target of the loop. The target flux is equal to the source flux parallely transported around the loop by the holonomy $h$. The Laplacian constraint is the equality of the scalar product $\\vec{X}^{t}\\cdot\\vec{X}^{s}$ with the squared norm $\\vec{X}^{s}\\cdot\\vec{X}^{s}$ and therefore means that the two flux are equal, $\\vec{X}^{s}=\\vec{X}^{t}$. This implies the flatness of the group element $h$ (up to the $\\mathrm{U}(1)$ stabilizer of the flux vector). \n\nAt the quantum level, the Laplacian constraint turns out to play a different role. It implies the invariance of the wave-function by conjugation:\n\\begin{equation}\n\\Delta \\varphi =\\widetilde{\\Delta} \\varphi\n\\quad\\Rightarrow\\quad\n\\forall h,g\\in\\mathrm{SU}(2)\\,,\\,\\,\n\\varphi(h)=\\varphi(ghg^{-1})\\,.\n\\end{equation}\nWe rigorously prove this statement in the appendix \\ref{app:Laplacian} solving explicitly the recursion relations implied by the Laplacian constraint on the Fourier coefficients of $\\varphi$. Another way to understand the relation of the Laplacian constraint to the invariance under conjugation is to think in terms of spin recoupling. Let us call $\\vec{J}^{L,R}$ respectively the ${\\mathfrak{su}}(2)$ generators living at the two ends of the loop and defining the left and right derivations. The two Casimirs, given by the two scalar products $\\vec{J}^{L}\\cdot\\vec{J}^{L}$ and $\\vec{J}^{R}\\cdot\\vec{J}^{R}$, are equal and their (eigen)value is $j(j+1)$ is the loop carries the spin $j$. Then the Laplacian constraint means that their recoupling is trivial:\n\\begin{equation}\n0=\\vec{J}^{R}\\cdot\\vec{J}^{R}-\\vec{J}^{R}\\cdot\\vec{J}^{L}=\\f12(\\vec{J}^{R}-\\vec{J}^{L})^{2}\\,,\n\\end{equation}\nso that the two ends of the loop recouple to the trivial representation, i.e. the spin-0. As illustrated on fig.\\ref{fig:Laplacian}, this also allows to show that the Laplacian constraint operator $(\\widetilde{\\Delta}-\\Delta)$ is positive and its spectrum is $k(k+1)\/2$ where $k$ is an integer running from 0 to $(2j)$ if the loop carries the spin $j$.\n\\begin{figure}[h!]\n\n\\centering\n\n\\begin{tikzpicture}[scale=0.8]\n\\coordinate(O2) at (-2,0);\n\\coordinate(O3) at (0,0);\n\n\\draw (O3) to[in=-45,out=+45,loop,scale=5] (O3)++(1.6,0) node {$j$};\n\n\\draw[red] (O2) -- (O3) node[midway,below]{$k$};\n\\draw (O3) node {$\\bullet$} ++(0.2,0.6) node{$\\vec{J}^{L}$} ++(0,-1.2) node{$\\vec{J}^{R}$};\n\n\\end{tikzpicture}\n\n\\caption{The left and right derivations respectively  act as graspings at the source and target of the loop, inserting ${\\mathfrak{su}}(2)$ generators in the wave-functions. The Laplacian operator $(\\widetilde{\\Delta}-\\Delta)$ then measures the difference between the two scalar products $\\vec{J}^{R}\\cdot\\vec{J}^{R}$ and $\\vec{J}^{R}\\cdot\\vec{J}^{L}$, or equivalently the Casimir $(\\vec{J}^{R}-\\vec{J}^{L})^{2}\/2$ of the recoupling of the spins at the two ends of the loop. Assuming that the loop carries the spin $j$ then recoupling $j$ with itself gives a spin $k$ running from 0 to $(2j)$.\n} \n\\label{fig:Laplacian}\n\n\\end{figure}\nOne can also see that the derivatives of the $\\delta$-distribution are eigenstates of $(\\widetilde{\\Delta}-\\Delta)$ with non-vanishing eigenvalues. For example, we compute:\n\\begin{equation}\n\\int f(\\widetilde{\\Delta}-\\Delta)\\partial_{a}^{R}\\delta\n\\,=\\,\ni^{3}\\sum_{b}\n\\big{[}\nf(J_{a}J_{b}J_{b}) -f(J_{b}J_{a}J_{b})\n\\big{]}\n\\,=\\,\n-if(J_{a})\n\\,=\\,\n+\\int f \\partial_{a}^{R}\\delta\n\\,,\n\\end{equation}\nand so on with higher order differential operators. In particular, the derivative distribution $\\partial_{a}\\delta$ corresponds to the eigenvalue $k(k+1)\/2$ for $k=1$. Higher order derivatives will explore higher eigenvalues.\n\nTo conclude, the original holonomy constraint, supplemented with the new Laplacian constraint, acting on functions on $\\mathrm{SU}(2)$ admit the $\\delta$-distribution as unique solution: the Laplacian constraint imposes invariance under conjugation while the holonomy constraint then imposes the flatness of the group element along the loop.\n\n\\begin{prop}\nThere is a unique solution (up to a numerical factor) as a distribution over $\\mathrm{SU}(2)$ to the holonomy and Laplacian constraints:\n\\begin{equation}\n\\left|\n\\begin{array}{l}\n(\\widehat{\\chi}-2)\\,\\varphi=0\\\\\n(\\Delta-\\widetilde{\\Delta})\\,\\varphi=0\n\\end{array}\n\\right.\n\\quad\\Longrightarrow\\quad\n\\exists \\lambda\\in{\\mathbb C}\\,,\\,\\, \\varphi(h)=\\lambda\\,\\delta(h)\n\\,.\n\\end{equation}\n\\end{prop}\n\nBelow, we look at the generic case of an arbitrary number of loops. We will show that we can supplement the holonomy constraints around each loop  either with Laplacian constraints for each loop or with  multi-loop holonomy constraints (that still act by multiplication) wrapping around several loops at once.\n\n\\subsubsection{Holonomy constraints on $\\mathrm{SU}(2)^{N}$ for $N\\ge 2$}\n\\label{derivativesolution2}\n\nWe now turn to the holonomy constraints on $\\mathrm{SU}(2)^{N}$:\n$$\n\\forall 1\\le\\ell\\le N, \\,\\,(\\widehat{\\chi}_{\\ell}-2)\\varphi=0\\,,\n$$\nwith the requirement of invariance under simultaneous conjugation of all the arguments $h_{\\ell}$. Since we do not require the invariance under the individual action of conjugation on each little loop, the gauge invariance is not enough to kill the spurious solution identified above.\nAs proposed above, we can reach the uniqueness of the physical state by further imposing the Laplacian constraint on each loop:\n\\begin{equation}\n\\forall \\ell\\in{\\mathbb N}\\,,\\,\\,\n(\\widetilde{\\Delta}_{\\ell}-\\Delta_{\\ell})\\,\\varphi=0\\,.\n\\end{equation}\nThis now  implies the invariance of the wave-function under the individual action of conjugation on each loop. In terms of spin recoupling, each little loop is linked to the vertex by a spin-0, as illustrated on fig.\\ref{fig:0spincoupling},  this effectively trivializes the intertwiner space living at the vertex and the loops can be thought of as decoupled from one another.\nThe holonomy constraints then impose that the only solution state is the $\\delta$-distribution. \n\\begin{figure}[h!]\n\\centering\n\\begin{tikzpicture}[scale=0.55]\n\n\\coordinate(O) at (0,0);\n\\coordinate(O1) at (2,0);\n\\coordinate(O2) at (1,1.7);\n\\coordinate(O3) at (-1,1.7);\n\\coordinate(O4) at (-2,0);\n\\coordinate(O5) at (-1,-1.7);\n\\coordinate(O6) at (1,-1.7);\n\n\n\n\\draw (O) node[scale=1] {$\\bullet$} ;\n\\draw (O1) node[scale=1] {$\\bullet$} ;\n\\draw (O2) node[scale=1] {$\\bullet$} ;\n\\draw (O3) node[scale=1] {$\\bullet$} ;\n\\draw (O4) node[scale=1] {$\\bullet$} ;\n\\draw (O5) node[scale=1] {$\\bullet$} ;\n\\draw (O6) node[scale=1] {$\\bullet$} ;\n\n\\draw[red] (O) -- (O1)  ++(-0.9,-0.3) node{$k_{1}$};\n\\draw[red] (O) -- (O2) node[midway,right]{$k_{2}$};\n\\draw[red] (O) -- (O3) node[midway,right]{$k_{3}$};\n\\draw[red] (O) -- (O4) node[midway,above]{$k_{4}$};\n\\draw[red] (O) -- (O5) ++(0,0.8) node{$k_{5}$};\n\\draw[red] (O) -- (O6) ++(-0.75,0.8) node{$k_{6}$};\n\n\n\\draw[in=-45,out=45,scale=5] (O1)  to[loop] (O1) ++(0.35,0) node {$j_{1}$};\n\\draw[in=-45,out=45,scale=5,rotate=60] (O2)  to[loop] (O2) ++(0.35,0) node {$j_{2}$};\n\\draw[in=-45,out=45,scale=5,rotate=120] (O3)  to[loop] (O3) ++(0.35,0) node {$j_{3}$};\n\\draw[in=-45,out=45,scale=5,rotate=180] (O4)  to[loop] (O4) ++(0.35,0) node {$j_{4}$};\n\\draw[in=-45,out=45,scale=5,rotate=-120] (O5)  to[loop] (O5) ++(0.35,0) node {$j_{5}$};\n\\draw[in=-45,out=45,scale=5,rotate=-60] (O6)  to[loop] (O6) ++(0.35,0) node {$j_{6}$};\n\n\\end{tikzpicture}\n\n\\caption{ The Laplacian constraint on a loop $\\ell$ constraint the spin $j_{\\ell}$ carried by the loop to recouple with itself into the trivial representation with vanishing spin $k_{\\ell}=0$. Imposing this constraint on every loop, the vertex then recouples a collection of spin-0, the intertwiner is thus trivial and the loops are totally decoupled.}\n\\label{fig:0spincoupling}\n\n\\end{figure}\n\n\n\\medskip\n\nInstead of imposing the Laplacian constraints, another way to proceed is to introduce multi-loop holonomy constraints. To prove this, let us start by describing the gauge-invariant derivative solutions to the holonomy constraints. The general structure is as follows. One acts with arbitrary derivatives on the $\\delta$-distribution $\\prod_{\\ell=1}^{N}\\delta(h_{\\ell})$. Then to ensure invariance under simultaneous conjugation, one must contract all the indices with a $\\mathrm{SO}(3)$-invariant tensor ${\\mathcal I}$:\n\\begin{equation}\n\\varphi^{{\\mathcal I}}(\\{h_{\\ell}\\})=\n\\sum_{\\{a_{i}^{\\ell}\\}_{i=1..n_{\\ell}}}\n{\\mathcal I}^{a^{1}_{1}..a^{N}_{n_{N}}}\\,\n\\prod_{\\ell=1}^{N}\\partial_{a^{\\ell}_{1}}..\\partial_{a^{\\ell}_{n_{\\ell}}}\\delta(h_{\\ell})\\,,\n\\end{equation}\nwhere $n_{\\ell}$ is the order of the differential operator acting on the loop $\\ell$, for an overall order $n=\\sum_{\\ell}n_{\\ell}$, and ${\\mathcal I}$ is a rotational invariant tensor defining the contraction of the differential indices $a$'s, i.e. it is an intertwiner between $n$ spin-1 representations.\n\nTo be explicit, for $n=2$ differential insertions, there is a single invariant tensor: ${\\mathcal I}^{ab}=\\delta^{ab}$. Either we act with the two derivatives on the same group elements, but then we already know that $\\Delta\\delta$ is not a solution to the holonomy constraint, or we act on two different loops getting the non-trivial distribution $\\sum_{a}\\partial_{a}\\delta(h_{1})\\partial_{a}\\delta(h_{2})$ (here we put aside all the other loops, where no differential operator act):\n\\begin{equation}\n\\langle \\,\\sum_{a}\\partial_{a}\\delta_{1}\\partial_{a}\\delta_{2}\\,|f\\rangle=\n-f(J_{a},J_{a})\\,,\n\\end{equation}\nwhich yields the evaluation $f(J_{a},J_{a})$ of the spin network state obtained by acting with the double grasping $J_{a}\\otimes J_{a}$ on the test wave-function $f$. \nWe easily check that this provides a solution to the individual one-loop holonomy constraints:\n\\begin{equation}\n\\forall f\\in {\\mathcal C}^{\\infty}_{\\mathrm{SU}(2)^{2}}\\,,\\,\\,\n\\int f (\\chi_{\\f12}(h_{1})-2)\\,\\sum_{a}^{3}\\partial_{a}\\delta(h_{1})\\partial_{a}\\delta(h_{2})\n=\n0\\,,\n\\end{equation}\nThe double grasping, as shown on fig.\\ref{fig:doublegrasping}, couples the two loops. The goal is to suppress such coupling between the two loops in order to get as unique solution the factorized flat state $\\delta^{\\otimes N}$ where all the loops are entirely decoupled.\n\\begin{figure}[h!]\n\\centering\n\\begin{tikzpicture}[scale=1.5]\n\n\\coordinate(O1) at (0,0);\n\\draw (O1) node[scale=2] {$\\bullet$} ;\n\n\\draw (O1)   to[in=-45,out=+45,loop,scale=3,rotate=90] node[very near end](A){}   (O1)++(0,1) node {$h_1$};\n\\draw (O1) to[in=-45,out=+45,loop,scale=3,rotate=-30] node[very near end](B){} (O1) ++(0.8,-0.7) node {$h_2$};\n\\draw (O1) to[in=-45,out=+45,loop,scale=3,rotate=-150] (O1) ++(-0.8,-0.7) node {$h_3$};\n\n\\draw[dotted,thick] (A) to[bend left] node[very near start,above,right](C){}    (B) ++(-0.1,-0.25) node{$\\partial_{a}$};\n\\draw (A) node[scale=1,red!50] {$\\bullet$} ++(0.22,-0.05)node{$\\partial_{a}$};\n\\draw (B) node[scale=1,red!50] {$\\bullet$} ;\n\n\\end{tikzpicture}\n\n\\caption{We act with derivatives $\\partial_{a}$ on the group elements $h_{1}$ and $h_{2}$ and contract  the indices, which translates graphically as a double grasping linking the two loops.}\n\\label{fig:doublegrasping}\n\n\\end{figure}\nTo make the system more rigid, the natural constraint to introduce is a two-loop holonomy constraint, which would kill any correlation between the two loops:\n\\begin{equation}\n(\\widehat{\\chi}_{12}-2)\\varphi (h_{1},h_{2})\\equiv\n(\\chi_{\\f12}(h_{1}h_{2})-2)\\varphi (h_{1},h_{2})\n=0\\,.\n\\end{equation}\nWe check that this two-loop constraint eliminates the coupled solution proposed above:\n\\begin{equation}\n\\int f (\\chi_{\\f12}(h_{1}h_{2})-2)\\,\\sum_{a}^{3}\\partial_{a}\\delta(h_{1})\\partial_{a}\\delta(h_{2})\n=\n\\left.f\\Delta\\chi_{\\f12}\\right|_{{\\mathbb{I}}}\n=\n\\f32\\,f({\\mathbb{I}},{\\mathbb{I}})\\,\\ne0\\,.\n\\end{equation}\n\nFor $n=3$ differential insertions, we still have a unique intertwiner, given by the completely antisymmetric tensor $\\epsilon^{abc}$. This corresponds a triple grasping. The three derivatives can all act on the same loop, in which case we do not get a solution of the one-loop holonomy constraint, or they can act on two different loops, in which case it is not a solution of the two-loop holonomy constraints we have just introduced, or they can act on three different loops in which case we need to introduce a three-loop holonomy constraint to discard it:\n\\begin{equation}\n\\int f (\\chi_{\\f12}(h_{1}h_{2}h_{3})-2)\\epsilon^{abc}\\partial_{a}\\delta(h_{1})\\partial_{b}\\delta(h_{2})\\partial_{c}\\delta(h_{3})\n=\n-\\frac i2^{3}f({\\mathbb{I}})\\epsilon^{abc}\\chi_{\\f12}(\\sigma_{a}\\sigma_{b}\\sigma_{c})\n=\n-\\frac{3i}2\\,f({\\mathbb{I}})\\,\\,\\ne0\\,.\n\\end{equation}\n\nFor an arbitrary number $n$ of differential insertions acting on the $N$ loops, the grasping will potentially couple the $N$ loops. In order to kill all those coupled solutions, we introduce all multi-loop holonomy constraints:\n\\begin{equation}\n\\forall E\\subset \\{1,..,N\\}\\,,\n\\,\\,\n\\Big{[}\n\\chi_{\\f12}\\big{(}\n\\prod_{\\ell\\in E}h_{\\ell}\n\\big{)}\n-2\n\\Big{]}\\,\\varphi=0\n\\,.\n\\end{equation}\nThe ordering of the group elements is important of course for the precise definition of the multi-loop operator but is irrelevant to ensure that the action of the corresponding constraint operator on the coupled derivative distributions does not vanish.\nIn fact, looking deeper into the structure of $\\mathrm{SO}(3)$-invariant tensors, \na fundamental theorem on rotational invariants states that all $\\mathrm{SO}(3)$-invariant polynomial of $n$ 3d-vectors $\\vec{v}_{i=1..n}$ are generated by scalar products $\\vec{v}_{i}\\cdot\\vec{v}_{j}$ and triple products $\\vec{v}_{i}\\cdot(\\vec{v}_{j}\\wedge\\vec{v}_{k})$. This means that we only need the two-loop and three-loop holonomy constraints to ensure that the flat state, defined as the $\\delta$-distribution, is the only solution to the Hamiltonian constraints.\n\n\n\n\n\\subsubsection{The full Hamiltonian constraints for BF theory on loopy spin networks}\n\\label{derivativesolution3}\n\nTo summarize the implementation of BF theory on loopy spin networks, we have introduced individual holonomy constraints on each little loop around each vertex of the background graph. This is the usual procedure, for instance when constructing spinfoam amplitudes for BF theory from a canonical point of view. Surprisingly, these constraints are not strong enough to fully constraint the theory to the single flat state and kill al the little loop excitations. This can be backtracked to the simple fact that the identity ${\\mathbb{I}}$ is an extremum  of the $\\mathrm{SU}(2)$-character $\\chi_{\\f12}$ and thus the derivative of the character vanishes at that point. As a result, the $\\delta$-function on $\\mathrm{SU}(2)$ is not the unique solution to the holonomy constraints, but its first derivative are also solutions. While all the solution distributions are peaked on the identity and vanish elsewhere, we are allowed  grasping operators coupling the loops together. To forbid such such coupling and force to have a unique physical state, we have showed that we can supplement the original one-loop holonomy constraints with either one-loop Laplacian constraints or with multi-loop holonomy constraints, which leads us to two proposals for the Hamiltonian constraints for BF theory on loopy intertwiners:\n\\begin{itemize}\n\\item We  impose on each loop two gauge-invariant constraints, the holonomy constraint that acts by multiplication and the Laplacian constraint which acts by differentiation:\n\\begin{equation}\n\\forall \\ell\\,,\\quad\n\\widehat{\\chi}_{\\ell}\\,\\varphi=2\\varphi\\,,\\quad\n\\Delta_{\\ell}\\,\\varphi=\\widetilde{\\Delta}_{\\ell}\\,\\varphi\\,.\n\\end{equation}\n\n\\item We impose all multi-loop holonomy constraints, requiring not only that the group elements $h_{\\ell}$ on each loop $\\ell$ is the identity ${\\mathbb{I}}$ but also that all their products remain flat. This means one constraint for each finite subset $E$ of the set of all loops:\n\\begin{equation}\n\\forall E\\subset {\\mathbb N},\\quad\n\\widehat{\\chi}_{E}\\,\\varphi=2\\varphi\\,,\n\\qquad\n\\widehat{\\chi}_{E}\\,\\varphi (\\{h_{\\ell}\\}_{\\ell\\in{\\mathbb N}})=\n\\chi_{\\f12}\\Big{(}\\prod_{\\ell \\in E}h_{\\ell}\\Big{)}\\,\\varphi (\\{h_{\\ell}\\}_{\\ell\\in{\\mathbb N}})\\,.\n\\end{equation}\nThe ordering of the group elements does not matter in order to impose the flatness. These multi-loop constraints kill any correlation or entanglement between the loops. It is actually  sufficient to impose only the two-loop and three-loop holonomy constraints.\n\n\\end{itemize}\n\nIf we only impose the one-loop holonomy constraints, then the totally flat state defined by the $\\delta$-distribution is not the only physical state. We get an infinite-dimensional  space of physical states, obtained by the action of first order grasping operators on the $\\delta$-distribution, allowing for non-trivial coupling and correlations between the little loops. It would be interesting to understand the geometrical meaning of those states and if they play a special role\\footnotemark in the spinfoam models for BF theory (the Ponzano-Regge and Turaev-Viro models for 3d BF theory and the Crane-Yetter model for 4d BF theory). Maybe those local excitations could provide a first extension of the topological BF theory to a field theory with local degrees of freedom.\n\\footnotetext{\nAs an example, we have in mind the recursion relation satisfied by the 6j symbol, which is understood to be the expression of the action of the holonomy operator on the flat state on the tetrahedron graph \\cite{Bonzom:2009zd,Bonzom:2011hm,Bonzom:2014bua}. Our results suggest that the double and triple graspings on the 6j symbol might be other solutions to this recursion relation.That would be specially interesting since the triply grasped 6j symbols is understood to be the first order correction of the q-deformed 6j-symbol \\cite{Freidel:1998ua}.\n}\n\n\\medskip\n\nOn the other hand, imposing the full set of Hamiltonian constraints proposed above leads to a unique physical state for BF theory: the flat state $\\varphi_{\\mathrm{BF}}=\\delta$.\nThis physical state  is clearly not normalizable. But since it is unique, it is not a big problem to define the scalar product on this final one-dimension Hilbert space\\footnotemark.\n\\footnotetext{\nIf we had more solutions to the constraints, for example if we only impose the holonomy constraints without the Laplacian constraints, we would propose to render the distributions normalizable by modulating them by a damping operator $\\exp(-\\tau\\Delta)$ (heat kernel) or similar.  Expanding all the functions and distributions in Fourier modes, all the scalar products would become finite and we could then compute expectation values of operators on the resulting physical Hilbert space, for instance:\n$$\n\\int \\delta e^{-\\tau\\Delta}\\delta=\\sum_{j}(2j+1)^{2}e^{-\\tau j(j+1)} <+\\infty\n\\,,\\qquad\n\\int \\delta e^{-\\tau\\Delta}\\partial_{a}\\delta=i\\sum_{j}(2j+1)^{2}e^{-\\tau j(j+1)}\\chi_{j}(J_{a})=0\\,.\n$$\nBut we haven't investigated this line of research further.\n}\nThe physical scalar product on the initial Hilbert space of loopy spin networks is defined by projecting on this physical state, which amounts at the end of the day to simply evaluate the wave-functions at the identity i.e. on flat connections:\n\\begin{equation}\n\\forall f,\\tilde{f}\\in{\\mathcal H}^{\\mathrm{loopy}}\\,,\\quad\n\\langle f|\\tilde{f}\\rangle_{\\mathrm{phys}}\n\\,=\\,\n\\langle f|\\varphi_{BF}\\rangle\\,\\langle\\varphi_{BF}|\\tilde{f}\\rangle\n\\,=\\,\n\\overline{\\langle\\varphi_{BF}|f\\rangle}\\,\\langle\\varphi_{BF}|\\tilde{f}\\rangle\n\\,=\\,\n\\overline{f({\\mathbb{I}})}\\,\\tilde{f}({\\mathbb{I}})\\,.\n\\end{equation}\nAs expected, we are left with a single physical state on the flower, the little loops have been projected out and all local degrees of freedom have disappeared.\n\n\\medskip\n\nNow that we have checked that loopy spin networks allow for a correct implementation of BF theory's topological dynamics, we would like to later introduce Hamiltonian constraints allowing for local degrees of freedom. We wouldn't want to kill the little loops as happens for BF theory. The goal would be to have dynamics coupling the little loops to the spins living on the links of the background graph, in such a way that it reproduces the propagation of the local geometry excitations of general relativity in a continuum limit. The strategy would be to slightly modify the BF dynamics -``constrain the BF theory''- most likely following the approaches for the dynamics of discrete\/twisted geometries \\cite{Bonzom:2011hm,Bonzom:2011nv,Bonzom:2013tna} or of EPRL spinfoam models \\cite{Engle:2007wy,Geloun:2010vj,Bianchi:2012nk}.\n\n\n\n\n\n\\section{The Fock space of loopy spin networks}\n\\label{sec:bosonisation}\n\nUp to now, we have introduced the loopy spin network states, on a fixed background graph with an arbitrary number of little loop excitations attached to each vertex. Here we would like to tackle the issue of endowing these local loops with bosonic statistics and define the Fock space of loop spin networks over a background skeleton graph with indistinguishable little loops living at its vertices.\n\nFrom the perspective of coarse-graining, the little loops represent curvature excitations within the bounded region coarse-grained to a single vertex. Keeping these little loops distinguishable amounts to remembering that they are excitations of different parts of the internal geometry of that region, while fully coarse-graining the region should erase any memory of internal localization and the little loops should be considered as indistinguishable: incoming energy at the vertex would then equally excite any of those loops, irrespective to their a priori different localization on the internal subgraph that we coarse-grained.\n\nIn this section, we will thus symmetrize our spin network states over the little loops attached at each vertex. The difficulty reside in the compatibility of the symmetrization with the cylindrical consistency. Indeed, a little loop carrying a spin-0 is considering as a non-existing loop, and vice-versa. We cannot symmetrize these non-existing loops with the other loops carrying non-trivial spins: since we would like to allow for an infinite number of loops, the symmetrization operation would be ill-defined. In the framework of usual quantum field theory, this would be similar to considering particles carrying a vanishing momentum as non-existent. We would have to update the definition of the symmetrization to take this new fact into account. Here, we will show how to systematically subtract the 0-modes components of the loopy spin network states, symmetrize over non-trivial little loops and define an appropriate holonomy operator acting on symmetrized states. A resulting subtlety is that we will be led to distinguish three components of the holonomy operator, that respectively conserves the number of loops, acts as a creation operator adding one little loop or as an annihilation operator removing a loop.\n\n\n\\subsection{Bosonics statistics for loops}\n\nWe would like to define symmetrized loopy intertwiner states in ${\\cal H}^{\\mathrm{loopy}}$. A direct way would be to work directly on states with an arbitrary number of loops\\footnotemark, but imposing invariance under the symmetry group for a infinite number of loops introduces a lof of technicalities. So we follow a more constructive approach and work with finite number of loops, symmetrize and then allow for  varying number of loops.\n\\footnotetext{\nWe would use an extension of the finite symmetry groups $S_{n}$ to the group of permutations of integers which only act non-trivially on a finite subset:\n$$\nS_\\infty = \\{f : \\mathbb{N} \\rightarrow \\mathbb{N}, f \\textrm{ bijective and }\\exists n \\in \\mathbb{N}, \\forall m>n, f(m)=m\\}\\,.\n$$\nWe would use the canonical action of $S_\\infty$  on the Hilbert spaces of loopy spin networks ${\\mathcal H}_{E}$ with finite number of loops:\n$$\n\\sigma : {\\mathcal H}_{E} \\rightarrow {\\mathcal H}_{\\sigma(E)}\\,,\n\\qquad\n(\\sigma \\triangleright f)(\\{h_{e_i}\\}) = f(\\{h_{\\sigma^{-1}(e_i)}\\})\\,.\n$$\nThis action is compatible with the cylindrical consistency conditions and naturally extends to the projective limit.\nHowever, requiring invariance of states $|\\Psi\\rangle \\in{\\cal H}^{\\mathrm{loopy}}$ under permutations, $\\forall \\sigma \\in S_\\infty,~\\sigma \\triangleright |\\Psi\\rangle = |\\Psi\\rangle$, only provides non-normalizable states. This forces us to work on the dual space to define symmetrized states and creates unnecessary technicalities for our present purpose.\n}\n\nWe start from the definition of the loopy states in terms of proper states, ${\\cal H}^{\\mathrm{loopy}}=\\bigoplus_{E\\in\\mathcal{P}_{<\\infty}(\\mathbb{N})}{\\mathcal H}_{E}^{0}$. This decomposition has removed all spin-0 and avoids all of the redundancies due to the cylindrical consistency. We can now symmetrize the states. For each number of loops $N$, we consider gauge-invariant wave-functions, symmetric under the exchange of the $N$ loops and such that no loop carries a vanishing spin. The full symmetrized Hilbert space ${\\cal H}^{\\mathrm{sym}}$ will then be the direct sum over $N$ of all the finite symmetrized states.\n\nLet us realize this programme explicitly. We start with the Hilbert space ${\\cal H}^{\\mathrm{sym}}_{N}$ of wave-functions, $f\\in L^{2}(\\mathrm{SU}(2)^{\\times N})$, gauge-invariant and  symmetrized  on $N$ loops:\n\\begin{eqnarray}\n\\forall k \\in \\mathrm{SU}(2),&&\nf(h_1,...,h_N) = f(kh_1k^{-1},...,kh_N k^{-1})\\,,\\nonumber\\\\\n\\forall \\sigma \\in S_N, &&\nf(h_1,...,h_N) = f(h_{\\sigma(1)},...,h_{\\sigma(N)})\\,.\n\\end{eqnarray}\nWe define the subspace of proper states, removing the 0 mode:\n\\begin{equation}\n{\\mathcal H}_{N}^{0}=\n\\Bigg{\\{}\nf\\in{\\cal H}^{\\mathrm{sym}}_{N}\n\\,:\\,\n\\int \\mathrm{d}h_{1}\\,f(h_1,...,h_N) =0\n\\Bigg{\\}}\\,.\n\\end{equation}\nWe only need to impose one integration condition, since the function is invariant under permutation of its arguments.\nWe have a simplified version of the decomposition onto proper states given in lemma \\ref{proper}:\n\\begin{lemma}\n\\label{propersym}\nThe Hilbert space of symmetrized states on $N$ loops decomposes as a direct sum of the Hilbert spaces of proper symmetrized states on at most $N$ loops:\n\\begin{equation}\n{\\cal H}^{\\mathrm{sym}}_{N}=\\bigoplus_{n=0}^{N}{\\mathcal H}_{n}^{0}\\,.\n\\end{equation}\nThis isomorphism is realized through a combinatorial transform of the wave-functions:\n\\begin{equation}\n\\label{symresum}\n\\forall f\\in{\\cal H}^{\\mathrm{sym}}_{N}\\,,\\quad\nf=\\sum_{n=0}^{N}\\,\\,\n\\sum_{1\\le i_{1}<..<i_{n}\\le N}f_{n}\n\\big{(}\nh_{i_{1}},..,h_{i_{n}}\n\\big{)}\n\\,,\\quad\nf_{n}\\in{\\mathcal H}_{n}^{0}\n\\end{equation}\n\\begin{equation}\nf_{n}(h_{1},..,h_{n})\n\\,=\\,\n\\sum_{m=0}^{n}\n(-1)^{n-m}\n\\int\\prod_{i=m+1}^{n}\\mathrm{d}k_{i}\\,\\prod_{i=n+1}^{N}\\mathrm{d}g_{i}\\,\n\\sum_{1\\le i_{1}<..<i_{m}\\le n}\nf\\big{(}\nh_{i_{1}},..,h_{i_{m}},k_{m+1},..,k_{n-m},g_{n+1},..,g_{N}\n\\big{)}\\,.\n\\end{equation}\nThe scalar product is given by the integration with respect to the Haar measure. The integral condition (absence of 0-mode) for the proper states implies that two proper states with different support are immediately orthogonal:\n\\begin{equation}\n\\label{scalarN}\n\\forall f,\\widetilde{f}\\,\\in{\\cal H}^{\\mathrm{sym}}_{N}\\,,\\quad\n\\langle f|\\widetilde{f}\\rangle_{N}\n\\,=\\,\n\\int \\prod_{i=1}^{N}\\mathrm{d}h_{i}\\,\\overline{f(h_{1},..,h_{N})}\\,\\widetilde{f}(h_{1},..,h_{N})\n\\,=\\,\n\\sum_{n=0}^{N} \\binomial{N}{n}\\,\\langle f_{n}|\\widetilde{f}_{n}\\rangle_{n}\\,.\n\\end{equation}\n\\end{lemma}\nIn the resummation formula \\eqref{symresum} above, the sum over labels  $1\\le i_{1}<..<i_{n}\\le N$ corresponds to the sum over all subsets with $n$ elements -or $n$-uplets- among the first $N$ integers $\\{1,..,N\\}$. And the injection of the proper state Hilbert space ${\\mathcal H}_{n}^{0}$ in the larger symmetrized space ${\\cal H}^{\\mathrm{sym}}_{N}$ requires this sum over all possible choices of $n$-uplets. This leads to the binomial coefficient in the scalar product formula \\eqref{scalarN}. This is a clear remnant of having distinguishable loops. Once the little loops are assumed to be bosonic and fully indistinguishable, there is no reason to distinguish a state $f_{n}(h_{a_{1}},..,h_{a_{n}})$ from $f_{n}(h_{b_{1}},..,h_{b_{n}})$ with different choice of $n$-uplets. \n\nTherefore, to define bosonic states in the projective limit $N\\rightarrow \\infty$, we will keep the decomposition as a direct sum of vector spaces ${\\cal H}^{\\mathrm{sym}}_{N}=\\bigoplus_{n=0}^{N}{\\mathcal H}_{n}^{0}$ defining the tower of symmetrized states, but we will modify the scalar product to remove its dependence on $N$ and make it compatible with the projective limit:\n\\begin{equation}\n\\label{bosonicN}\n\\langle f|\\widetilde{f}\\rangle_{N}^{\\mathrm{bosonic}}\n\\,=\\,\n\\sum_{n=0}^{N} \\langle f_{n}|\\widetilde{f}_{n}\\rangle_{n}\\,.\n\\end{equation}\nThis is achieved by simply including the symmetrizing factor in the definition of the injection $I_{N,N+1}:\\,{\\cal H}^{\\mathrm{sym}}_{N}\\hookrightarrow{\\cal H}^{\\mathrm{sym}}_{N+1}$ of wave-functions of $N$ loops seen as wave-functions of $(N+1)$ loops:\n\\begin{equation}\nf\\,\\in{\\cal H}^{\\mathrm{sym}}_{N}\n\\,\\mapsto\\,\nI_{N,N+1}f\\,\\in{\\cal H}^{\\mathrm{sym}}_{N+1}\\,,\n\\qquad\n\\big{(}I_{N,N+1}f\\big{)}(h_{1},..,h_{N+1})\n\\,=\\,\n\\f1{N+1}\\,\n\\sum_{i=1}^{N+1}f(h_{1},..,\\widehat{h_{i}},..,h_{N+1})\\,,\n\\end{equation}\nwhere the element $\\widehat{h_{i}}$ means that we omit it from the list of arguments. This generalizes to injections ${\\cal H}^{\\mathrm{sym}}_{N}\\hookrightarrow{\\cal H}^{\\mathrm{sym}}_{N+p}$ using the binomial coefficients:\n\\begin{equation}\nf\\,\\in{\\cal H}^{\\mathrm{sym}}_{N}\n\\,\\mapsto\\,\nI_{N,N+p}f\\,\\in{\\cal H}^{\\mathrm{sym}}_{N+p}\\,,\n\\qquad\n\\big{(}I_{N,N+p}f\\big{)}(h_{1},..,h_{N+p})\n\\,=\\,\n\\binomial{N+p}{N}^{-1}\\,\n\\sum_{1\\le i_{1}<..<i_{N}\\le N+p}f(h_{i_{1}},..,h_{i_{N}})\\,.\n\\end{equation}\nThese factors compensate the binomial factors from the scalar product formula \\eqref{scalarN}.\nAs we will see a little bit further, this scalar product $\\langle f|\\widetilde{f}\\rangle_{N}^{\\mathrm{bosonic}}$  on symmetric states is the one which makes the holonomy operator(s) Hermitian.\nThen we can define the Fock space of loopy spin networks with bosonic little loop excitations.\n\\begin{definition}\nThe full Fock space of symmetrized loop states is defined as the projective limit of the Hilbert spaces ${\\cal H}^{\\mathrm{sym}}_{N}$, endowed with the bosonic scalar product \\eqref{bosonicN}, which amounts to the direct sum of the spaces of proper states:\n\\begin{equation}\n{\\cal H}^{\\mathrm{sym}}\\,\\equiv\\,\\bigoplus_{N\\in{\\mathbb N}}{\\mathcal H}_{N}^{0}\\,,\\qquad\n\\forall f,\\widetilde{f}\\,\\in{\\cal H}^{\\mathrm{sym}}\\,,\\quad\n\\langle f|\\widetilde{f}\\rangle\n\\,=\\,\n\\sum_{N=0}^{\\infty} \\langle f_{N}|\\widetilde{f}_{N}\\rangle\\,.\n\\end{equation}\n\\end{definition}\nThis describes bosonic excitations of the holonomy at each vertex of the base graph for the loopy spin network states.\nThis Fock space of little loops at a vertex have states for an arbitrary number of indistinguishable loops, that can be created and annihilated, each of them carrying a spin $j_{\\ell}\\in{\\mathbb N}\/2$ encoding the corresponding excitation of the geometry (area quanta). The spin carried by a loop is similar to the momentum carried by a particle. One must nevertheless keep in mind two differences with the usual Fock space construction used in standard quantum field theory:\n\\begin{itemize}\n\n\\item {\\it 0-modes are pure gauge:}\nFirst, we have implemented explicitly the cylindrical consistency requirement in the definition of the Fock space of loopy spin networks. A little loop carrying a spin-0 is identified to a vanishing excitation, i.e. a non-existing loop, so we have systematically removed them using proper states. This is similar to removing particle states with 0-momentum.\n\n\\item {\\it Non-trivial intertwiner structure:}\nSecond, for a given number of loops carrying some given spins, the loopy spin network state still contains more information: the state requires the data of an intertwiner linking all these little loops together (and to the external legs of the vertex). Each time we create a loop, the intertwiner space at the vertex is further enlarged. This extra structure implies that factorized states do not constitute a basis of the Fock space of loopy intertwiners.\n\n\\end{itemize}\nAfter describing factorized states below, we will define the holonomy operators acting on the Fock space of symmetrized states and show how they shift the number of loops and become the basic creation and annihilation operators.\n\n\\subsection{Factorized states and $\\delta$-distribution}\n\nIt is interesting to check how factorized state, with no correlations between the loops, get decomposed onto proper states. Let us consider a integrable function $\\varphi$ on $\\mathrm{SU}(2)$. We assume it to be invariant under conjugation, so that it can be decomposed over the $\\mathrm{SU}(2)$-characters for all spins:\n\\begin{equation}\n\\forall h,g\\,\\in\\mathrm{SU}(2)\\,,\\quad\n\\varphi(h)=\\varphi(ghg^{-1})\\,,\n\\qquad\n\\varphi(h)=\\sum_{j\\in\\frac\\N2}\\varphi_{j}\\chi_{j}(h)\\,.\n\\end{equation}\nWe consider the $N$-loop symmetric  state $\\varphi^{\\otimes N}$ and check its proper state decompositon by the combinatorial formula given above in the lemma \\ref{propersym}:\n\\begin{eqnarray}\n\\varphi^{\\otimes N}_{0}&=&\\left(\\int\\varphi\\right)^{N}=\\varphi_{0}^{N}\\,, \\\\\n\\varphi^{\\otimes N}_{1}(h)&=&\n\\varphi_{0}^{N-1}\n\\big{[}\n\\varphi(h)-\\varphi_{0}\n\\big{]}\n\\,,\\nonumber\\\\\n\\varphi^{\\otimes N}_{2}(h_{1},h_{2})&=&\n\\varphi_{0}^{N-2}\n\\big{[}\n\\varphi(h_{1})\\varphi(h_{2})\n-\\varphi_{0}\\varphi(h_{1})\n-\\varphi_{0}\\varphi(h_{2})\n+\\varphi_{0}^{2}\n\\big{]}\n\\,=\\,\n\\varphi_{0}^{N-2}\n\\big{[}\n\\varphi(h_{1})-\\varphi_{0}\n\\big{]}\n\\big{[}\n\\varphi(h_{2})-\\varphi_{0}\n\\big{]}\n\\,,\\nonumber\\\\\n\\varphi^{\\otimes N}_{n}(h_{1},..,h_{n})&=&\n\\varphi_{0}^{N-n}\n\\,\\sum_{m=0}^{n}\n(-1)^{n-m}\\sum_{1\\le i_{1}<..<i_{m}\\le n}\n\\varphi_{0}^{n-m}\\,\\varphi(h_{i_{1}})..\\varphi(h_{i_{m}})\n=\n\\varphi_{0}^{N-n}\\prod_{i=1}^{n}\n\\big{[}\n\\varphi(h_{i})-\\varphi_{0}\n\\big{]}\n\\,.\\nonumber\n\\end{eqnarray}\nWe can check the scalar product formula \\eqref{scalarN}:\n$$\n\\Bigg{(}\\int |\\varphi|^2\\Bigg{)}^{N}\n\\,=\\,\n\\Bigg{(} |\\varphi_{0}|^2+\\int\\big{|}\\varphi- \\varphi_{0}\\big{|}^2\\Bigg{)}^{N}\n\\,=\\,\n\\sum_{n}^{N}\\binomial{N}{n}|\\varphi|_{0}^{2(N-n)}\\,\\Bigg{(}\\int\\big{|}\\varphi- \\varphi_{0}\\big{|}^{2}\\Bigg{)}^n\n\\,=\\,\n\\sum_{n}^{N}\\binomial{N}{n} \\int \\big{|}\\varphi_{n}^{\\otimes N}\\big{|}^2\n$$\n\nFirst, we notice that the proper state projections are still factorized. We are merely consistently removing the spin-0 component from all the loops, without creating any correlation during  the process. Second, if we normalize the one-loop wave-function $\\varphi_{0}=\\int \\varphi=1$, then the projections of the factorized state do not depend anymore on the number of loops $N$ and we can take the projective limit. We can define a factorized state $\\varphi^{\\otimes \\infty}$ with support on an infinite number of loops by taking the limit $N\\rightarrow\\infty$. We define its components, dropping the useless $\\infty$ label:\n\\begin{equation}\n\\varphi=1+\\widetilde{\\varphi}\\,,\\quad\n\\int\\widetilde{\\varphi}=0\\,,\\qquad\n\\varphi_{0}=1\\,,\n\\quad\n\\varphi_{n}=\\widetilde{\\varphi}^{\\otimes n}\\,.\n\\end{equation}\nWe can for instance apply this to the $\\delta$-distribution and define the flat holonomy state in our Fock space of symmetrized states:\n\\begin{equation}\n\\label{delta-def} \n\\widetilde{\\delta}=\\delta-1\n=\\sum_{j\\ne 0}(2j+1)\\chi_{j}\n\\,,\\qquad\n\\delta_{0}=1\\,,\\quad\n\\delta_{n}=\\widetilde{\\delta}^{\\otimes n}\\,.\n\\end{equation}\n\n\\subsection{Holonomy operator on symmetrized states}\n\nNow that we have defined the Fock space of loopy intertwiners, we would like to have the basic  operators creating and annihilating loop excitations. This is naturally achieved by the (one-loop) holonomy operator. We start, as in the case of distinguishable loops, with multiplying wave-functions by the spin-$\\f12$ character $\\chi(h_{\\ell})$ applied to the group element $h_{\\ell}$ living on a little loop $\\ell$. Then we will to distinguish three cases: the loop $\\ell$ does not belong the existing loops and the operator creates a new loop, or the loop $\\ell$ is already excited, in which case it can act on a spin-$\\f12$ excitation and actually annihilate the loop, or the operator will generically act on all other spin excitations by simple multiplication. This leads us to defining three components of the holonomy operator $\\hat{\\chi}$ acting on symmetrized states:\n\\begin{definition}\n\\label{AAB-def}\nWe define three operators $A,\\widetilde{A},B$ acting on the Fock space of symmetrized loopy intertwiners ${\\cal H}^{\\mathrm{sym}}$. They act on an arbitrary state $(f_{N})_{N\\in{\\mathbb N}}$ as:\n\\begin{equation}\n(Af)_{N}(h_{1},..,h_{N})\n\\,=\\,\n\\int \\mathrm{d}k\\,\\chi_{\\f12}(k)\\,f_{N+1}(h_{1},..,h_{N},k)\\,,\n\\end{equation}\n\\begin{equation}\n(Bf)_{0}=2f_{0}\\,,\n\\quad\n\\forall N>0\\,,\\,\\,\n(Bf)_{N}(h_{1},..,h_{N})\n\\,=\\,\n\\f1N\\sum_{i=1}^{N}\\Bigg{[}\n\\chi_{\\f12}(h_{i})f_{N}(h_{1},..,h_{N})\n-\\int \\mathrm{d}k_{i}\\,\\chi_{\\f12}(k_{i})\\,f_{N}(h_{1},..,k_{i},..,h_{N})\n\\Bigg{]}\n\\end{equation}\n\\begin{equation}\n(\\widetilde{A} f)_{0}=0\\,,\n\\quad\n\\forall N>0\\,,\\,\\,\n(\\widetilde{A} f)_{N}(h_{1},..,h_{N})\n\\,=\\,\n\\f1{N}\\sum_{i=1}^{N}\\chi_{\\f12}(h_{i})f_{N-1}(h_{1},..,\\widehat{h_{i}},..,h_{N})\n\\end{equation}\nThe operator $B$ is the usual action of the holonomy operator by multiplication by the character up to the subtraction of the resulting spin-0 component.\nThe operator $A$ is the annihilation operator, removing one loop, while the operator $\\widetilde{A}$ creates a new loop. We have the following relations on ${\\cal H}^{\\mathrm{sym}}$:\n\\begin{equation}\n\\widetilde{A}=A^{\\dagger}\\,,\\qquad\nB=B^{\\dagger}\n\\,.\n\\end{equation}\nFinally the one-loop holonomy operator for spin $\\f12$  is defined as the sum of these three components and is self-adjoint:\n\\begin{equation}\n\\widehat{\\chi}_{\\f12}\\,\\equiv\\,\\f12\\big{(}A+\\widetilde{A}+B\\big{)}\\,.\n\\end{equation}\n\\end{definition}\nThe convention $(Bf)_{0}=2f_{0}$ follows the logic that the 0-component $f_{0}$, with no loop, represents by default a flat holonomy and thus should be multiplied by $\\chi_{\\f12}({\\mathbb{I}})=2$.\nHere is the proof for the Hermicity relations:\n\\begin{proof}\nWe compare the action of $A$ and $\\widetilde{A}$:\n$$\n\\langle \\phi|A\\psi\\rangle\n\\,=\\,\n\\sum_{N\\in{\\mathbb N}}\n\\int [\\mathrm{d}h_{i}]_{i=1}^{N}\\mathrm{d}k\\,\n\\chi_{\\f12}(k)\\,\\overline{\\phi_{N}(h_{1},..,h_{N})}\\,\\psi_{N+1}(h_{1},..,h_{N},k)\\,,\n$$\n$$\n\\langle \\widetilde{A} \\phi|\\psi\\rangle\n\\,=\\,\n\\sum_{N>0}\n\\int [\\mathrm{d}h_{i}]_{i=1}^{N}\n\\f1N\\sum_{i=1}^{N}\\chi_{\\f12}(h_{i})\\,\n\\overline{\\phi_{N-1}(h_{1},..,\\widehat{h_{i}},..,h_{N})}\\,\n\\psi_{N}(h_{1},..,h_{N})\\,.\n$$\nWe shift the sum over $N$ in $\\langle \\widetilde{A} \\phi\\,|\\,\\psi\\rangle$ and we use the invariance of $\\psi_{N}$ under permutation of its arguments to conlude that these two expressions coincides, $\\langle \\phi|A\\psi\\rangle=\\langle \\widetilde{A} \\phi|\\psi\\rangle$. As for the operator $B$, we compute:\n\\begin{eqnarray}\n\\langle \\phi|B\\psi\\rangle\n&=&\n2\\overline{\\phi_{0}}\\,\\psi_{0}\n\\,+\\,\n\\sum_{N>0}\n\\int [\\mathrm{d}h_{i}]_{i=1}^{N}\n\\f1N\\sum_{i}^{N}\n\\chi_{\\f12}(h_{i})\\overline{\\phi_{N}(h_{1},..,h_{N})}\\,\\psi_{N}(h_{1},..,h_{N})\\nonumber\\\\\n&&\n-\\,\\int [\\mathrm{d}h_{i}]_{i=1}^{N}\n\\f1N\\sum_{i}^{N}\n\\int \\mathrm{d}k_{i}\\,\\chi_{\\f12}(k_{i})\\overline{\\phi_{N}(h_{1},..,h_{i},..,h_{N})}\\,\\psi_{N}(h_{1},..,k_{i},..,h_{N})\\,.\\nonumber\n\\end{eqnarray}\nThe last term vanishes due to the absence of 0-mode, $\\int \\mathrm{d}h_{i}\\,\\phi_{N}=0$. This ensures that $\\langle \\phi|B\\psi\\rangle=\\langle B\\phi|\\psi\\rangle$ and thus $B$ is a Hermitian operator.\n\\end{proof}\n\nTo ensure that the operators $A$ and $B$ are well-defined and that the holonomy operator $\\widehat{\\chi}_{\\f12}$ is  self-adjoint, it is enough to check that it is bounded. And we show below that it is indeed bounded by 2 as in the usual framework.\n\\begin{lemma}\nThe two parts of the holonomy operators are both bounded by 2, that is for all states $\\phi\\in{\\cal H}^{\\mathrm{sym}}$, we have the two inequalities:\n\\begin{equation}\n|\\langle \\phi|\\,(A+\\widetilde{A})\\,|\\phi\\rangle|\\,\\le\\,2\\langle \\phi|\\phi\\rangle\n\\,,\\qquad\n|\\langle \\phi|B|\\phi\\rangle|\\,\\le\\,2\\langle \\phi|\\phi\\rangle\\,.\n\\end{equation}\nThis ensures that they are both self-adjoint. The holonomy operator $\\widehat{\\chi}_{\\f12}$ is then also bounded by 2 and self-adjoint.\n\\end{lemma}\n\n\\begin{proof}\nLet us start with the operator $B$. The analysis is simpler since it doesn't shift the number of loops:\n$$\n\\langle \\phi|B|\\phi\\rangle=2|\\phi_{0}|^{2}+\\sum_{N>0}B_{N}\\,,\n\\quad\nB_{N}=\n\\f1N\\sum_{i}^{N}\n\\int [\\mathrm{d}h_{i}]_{i=1}^{N}\\chi_{\\f12}(h_{i})\\overline{\\phi_{N}}(h_{1},..,h_{N})\\,\\phi_{N}(h_{1},..,h_{N})\\,.\n$$\nThe extra term in the action of $B$ on the state $\\phi$ vanishes as earlier due to the integral condition on proper states, $\\int \\mathrm{d}h_{i}\\,\\phi_{N}=0$ for all $i$'s. Since the character $\\chi_{\\f12}$ is bounded by 2, it is direct to conclude:\n$$\n|B_{N}|\\le \\f2N\\sum_{i}^{N}\\int |\\phi_{N}|^{2}= 2\\int |\\phi_{N}|^{2}\\,,\n\\quad\n\\langle \\phi|B|\\phi\\rangle\n\\le 2|\\phi_{0}|^{2}+2\\sum_{N>0}\\int |\\phi_{N}|^{2}=2\\langle \\phi|\\phi\\rangle\\,.\n$$\nWe can proceed similarly with the operator $A+\\widetilde{A}$:\n$$\n\\langle \\phi|\\,(A+\\widetilde{A})\\,|\\phi\\rangle=\n\\langle \\phi|A|\\phi\\rangle+\\langle \\phi|A^{\\dagger}|\\phi\\rangle=\\langle \\phi|A|\\phi\\rangle+\\overline{\\langle \\phi|A|\\phi\\rangle}\\,,\\quad\n|\\langle \\phi|\\,(A+\\widetilde{A})\\,|\\phi\\rangle|\\le 2 |\\langle \\phi|A|\\phi\\rangle|\\,,\n$$\n\\begin{equation}\n\\langle \\phi|A|\\phi\\rangle=\\sum_{N}A_{N}\\,,\\quad\nA_{N}=\\int \\prod_{i=1}^{N}\\mathrm{d}h_{i}\\,\\mathrm{d}k\\,\n\\chi_{\\f12}(k)\\,\\overline{\\phi_{N}}(h_{1},..,h_{N})\\,\\phi_{N+1}(h_{1},..,h_{N},k)\n\\end{equation}\nAs long as the components $\\phi_{N}$'s are square-integrable, we can use the Cauchy-Schwarz inequality to bound these integrals:\n$$\n\\left|\nA_{N}\n\\right|\n\\le\n\\sqrt{\\int  \\prod_{i}^{N}\\mathrm{d}h_{i}\\,\\mathrm{d}k\\,\\chi_{\\f12}(k)^{2}\\,\\big{|}\\phi_{N}(h_{1},..,h_{N})\\big{|}^{2}}\\,\n\\sqrt{\\int  \\prod_{i}^{N}\\mathrm{d}h_{i}\\,\\mathrm{d}k\\,\\big{|}\\phi_{N+1}(h_{1},..,h_{N},k)\\big{|}^{2}}\\,.\n$$\nWe use that the $\\mathrm{SU}(2)$ character is normalized, $\\int \\chi_{\\f12}^{2}=1$,  and then apply the inequality bounding a product $ab\\le (a^{2}+b^{2})\/2$:\n\\begin{equation}\n\\left|\nA_{N}\n\\right|\n\\le\n\\f12{\\int  \\prod_{i}^{N}\\mathrm{d}h_{i}\\,\\big{|}\\phi_{N}(h_{1},..,h_{N})\\big{|}^{2}}\n+\\f12{\\int  \\prod_{i}^{N}\\mathrm{d}h_{i}\\,\\mathrm{d}k\\,\\big{|}\\phi_{N+1}(h_{1},..,h_{N},k)\\big{|}^{2}}\n=\\f12\\,\\Big{[}\\langle \\phi_{N}|\\phi_{N}\\rangle+\\langle \\phi_{N+1}|\\phi_{N+1}\\rangle\\Big{]}\\,.\n\\end{equation}\nSumming over $N\\in{\\mathbb N}$, this allows us to conclude that $|\\langle \\phi|A|\\phi\\rangle|\\le \\langle\\phi|\\phi\\rangle-\\f12|\\phi_{0}|^{2}\\le \\langle\\phi|\\phi\\rangle$ and thus reproduces the expected bound $|\\langle \\phi|\\,(A+\\widetilde{A})\\,|\\phi\\rangle|^{2}\\le 2\\langle\\phi|\\phi\\rangle$.\n\\end{proof}\n\nAlthough we consider the holonomy operator $\\widehat{\\chi}_{\\f12}$ to be the averaged sum of the two self-adjoint components $(A+A^{\\dagger})$ and $B$, each of these is a legitimate operator in itself. We could push this logic further and state that we have defined two different holonomy operators, \non the one hand, a holonomy operator $(A+A^{\\dagger})$ that acts as a ladder operator, creating and annihilating loops, and on the other hand, a holonomy operator $B$ which acts as spin shifts on existing loops (i.e. modifies the area quanta carried by each loop).\n\nThe important consistency check, which will be essential for the analysis of the BF theory dynamics, is that the flat state is an eigenvector of the one-loop holonomy operator:\n\\begin{prop}\n\\label{flat-prop1}\nThe flat state $\\delta$, defined in \\eqref{delta-def} by its proper state projections, $\\delta_{0}=1$ and $\\delta_{N}=(\\delta-1)^{\\otimes N}$ for $N\\ge 1$, is an eigenvector of the spin-$\\f12$ one-loop holonomy operator  $\\widehat{\\chi}_{\\f12}$ with the highest eigenvalue on ${\\cal H}^{\\mathrm{sym}}$:\n\\begin{equation}\n\\widehat{\\chi}_{\\f12}\\,\n|\\delta\\rangle\n\\,=\\, 2\\,|\\delta\\rangle\n\\,.\n\\end{equation}\nThis distributional flat state is also an eigenvector of the loop annihilation operator $A$ and of the loop creation operator $(B+A^{\\dagger})$:\n\\begin{equation}\nA|\\delta\\rangle\n\\,=\\,\n(B+A^{\\dagger})|\\delta\\rangle\n\\,=\\,\n2\\,|\\delta\\rangle\n\\,.\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nWe compute the action of the three parts of the holonomy operators acting on the flat state defined explicitly as\n$$\n\\delta_{0}=1\\,,\\quad\n\\delta_{N}(h_{1},..,h_{N})\n=\\prod_{i}^{N}\\big{[}\\delta(h_{i})-1\\big{]}\\,.\n$$\nFor the no-loop component, we get:\n$$\n(A\\delta)_{0}=\\int \\mathrm{d}k\\,\\chi_{\\f12}(k)\\,\\big{[}\\delta(k)-1\\big{]}=2\n\\,,\\quad\n(A^{\\dagger}\\delta)_{0}=0\n\\,,\\quad\n(B\\delta)_{0}=2\\,,\n$$\nwhile we compute for all other components:\n\\begin{eqnarray}\n(A\\delta)_{N}(h_{1},..,h_{N})&=& 2\\prod_{i}^{N}\\big{[}\\delta(h_{i})-1\\big{]}=2\\delta_{N}(h_{1},..,h_{N})\n\\\\\n(A^{\\dagger}\\delta)_{N}(h_{1},..,h_{N})&=& \n\\f1N\\sum_{i}^{N}\\chi_{\\f12}(h_{i})\\,\\prod_{\\ell\\ne i}^{N}\\big{[}\\delta(h_{\\ell})-1\\big{]}\n\\\\\n(B\\delta)_{N}(h_{1},..,h_{N})&=&\n2\\prod_{i}^{N}\\big{[}\\delta(h_{i})-1\\big{]}\n-\\f1N\\sum_{i}^{N}\\chi_{\\f12}(h_{i})\\,\\prod_{\\ell\\ne i}^{N}\\big{[}\\delta(h_{\\ell})-1\\big{]}\n\\end{eqnarray}\nAdding these three contributions, we get as expected for all number of loops $(\\widehat{\\chi}_{\\f12}\\,\\delta)_{N}=2\\delta_{N}$.\n\n\\end{proof}\n\n\\smallskip\n\nSince we have three operators built in the holonomy operator, it is natural to investigate their commutation algebra. It actually involves higher spin operators. We generalize the definition of the operators $A$, $A^{\\dagger}$ and $B$ to arbitrary spins: one simply replaces in their definition \\ref{AAB-def} the character in the fundamental representation $\\chi_{\\f12}$ by the higher spin character $\\chi_{j}$ for any $j\\in{\\mathbb N}^{*}\/2$, thus producing new operators $A_{j}$ annihilating a loop excitation of spin $j$,  $A^{\\dagger}_{j}$ creating a new loop carrying a spin $j$ and $B_{j}$ acting with a spin $j$ excitation on an existing loop.\n\nThen acting on a an arbitrary state $f$, we get:\n$$\n(ABf)_{N}=\\frac{N}{N+1}(BAf)_{N}+\\f1{N+1}(A_{1}f)_{N}\n\\,,\\quad\n(BA^{\\dagger}f)_{N}=\\frac{N-1}{N}(A^{\\dagger}Bf)_{N}+\\f1{N}(A_{1}^{\\dagger}f)_{N}\\,,\n$$\n$$\n(AA^{\\dagger}f)_{N}\n=\\frac{N}{N+1}(A^{\\dagger}Af)_{N}+\\f1{N+1}f_{N}\\,.\n$$\nRemembering that the number of loops $N$ is not constant on the Fock space of loopy intertwiners and should be treated as an operator $\\hat{N}$, these translate into commutation relations, being careful about the operator ordering:\n\\begin{equation}\n\\hat{N} B = B\\hat{N}\n\\,,\\quad\n(\\hat{N}+1)A=A\\hat{N}\n\\,,\\quad\n\\hat{N} A^{\\dagger}= A^{\\dagger}(\\hat{N}+1)\\,,\n\\end{equation}\n\\begin{equation}\n\\label{commAB1}\nAB\\hat{N}=\\hat{N} BA+A_{1}\n\\,,\\quad\n\\hat{N} B A^{\\dagger}=A^{\\dagger}B\\hat{N}+A_{1}^{\\dagger}\n\\,,\\quad\nA\\hat{N} A^{\\dagger}={\\mathbb{I}}+A^{\\dagger}A\\hat{N}={\\mathbb{I}}+\\hat{N} A^{\\dagger}A\n\\,.\n\\end{equation}\nThese generalize to the whole tower of higher spin operators, for all spins $a,b\\in\\frac{{\\mathbb N}^{*}}2$:\n\\begin{equation}\n\\label{commAB2}\nA_{a}B_{b}\\hat{N}=\\hat{N} B_{b}A_{a}+A_{a\\otimes b}\n\\,,\\quad\n\\hat{N} B_{b} A_{a}^{\\dagger}=A_{a}^{\\dagger}B_{b}\\hat{N}+A_{a\\otimes b}^{\\dagger}\n\\,,\\quad\nA_{a}\\hat{N} A_{b}^{\\dagger}=\\delta_{ab}{\\mathbb{I}}+A_{b}^{\\dagger}A_{a}\\hat{N}=\\delta_{ab}{\\mathbb{I}}+\\hat{N} A_{b}^{\\dagger}A_{a}\n\\,,\n\\end{equation}\nwhere we use the (natural) convention of the tensor product of spins for the annihilation operator:\n\\begin{equation}\nA_{a\\otimes b}\\,\\equiv\\,\n\\sum_{c=|a-b|}^{a+b}A_{c}\\,.\n\\end{equation}\nWe give the last commutation relation:\n\\begin{equation}\n\\hat{N}\\,[B_{a},B_{b}]\n\\,=\\,\nA^{\\dagger}_{b}A_{a}-A^{\\dagger}_{a}A_{b}\\,.\n\\end{equation}\n\n\nWe can combine these higher spin creation and annihilation operators to define a spin-$j$ holonomy operator $\\widehat{\\chi}_{j}$ as the average sum of those operators as for the fundamental representation:\n\\begin{equation}\n\\widehat{\\chi}_{j}=\\f12\\,\\big{(}\nA_{j}+A^{\\dagger}_{j}+B_{j}\n\\big{)}\\,.\n\\end{equation}\nThis rather natural definition unfortunately doesn't ensure that the operators $\\widehat{\\chi}_{j}$'s for different spins $j$'s commute with each other. Using the algebra computed above, the commutator of two holonomy opertaors $\\widehat{\\chi}_{a}$ and $\\widehat{\\chi}_{b}$ actually looks like a mess.\nNevertheless we can simplify the expressions by introducing suitable number of loops factors. Inserting the operator $\\hat{N}$ in the character, we find:\n\\begin{equation}\n\\big{[}\n\\hat{N}\\widehat{\\chi}_{a},\\hat{N}\\widehat{\\chi}_{b}\n\\big{]}\n=\\frac{\\hat{N}}2\\,(A^{\\dagger}_{b}A_{a}-A^{\\dagger}_{a}A_{b})\n=\\frac{\\hat{N}^{2}}2\\,[B_{a},B_{b}]\\,.\n\\end{equation}\nThis combination $\\hat{N}\\widehat{\\chi}_{a}$ isn't Hermitian, but this can be easily remedied to by considering $\\sqrt{\\hat{N}}\\widehat{\\chi}_{a}\\sqrt{\\hat{N}}$ instead. This commutator doesn't vanish, but we can easily find other combinations of the creation and annihilation operators that do:\n\\begin{equation}\n\\big{[}\n\\hat{N}(B_{a}+A_{a}^{\\dagger}-A_{a}),\\hat{N}(B_{b}+A_{b}^{\\dagger}-A_{b})\n\\big{]}\n\\,=\\,\n0\n\\,.\n\\end{equation}\nThis suggests using the operators $A_{a}$ and $(B_{a}+A^{\\dagger}_{a})$ as more fundamental as the holonomy operators. Although they are not Hermitian, the flat state is an eigenvector of both operators and we will exploit this fact in defining flatness constraints for BF theory in the  following section \\ref{BFsym}.\n\n\\medskip\n\nThe other way to proceed to defining higher spin holonomy operators is to reproduce the classical algebra of the $\\mathrm{SU}(2)$ characters. For instance, a spin-1 is obtained from the tensor product of two spin-$\\f12$ representations:\n$$\n\\chi_{1}(h)=\\chi_{\\f12}(h)^{2}-1\\,.\n$$\nWe propose to promote these relations to the quantum level:\n\\begin{equation}\n\\widehat{\\chi}_{1}^{\\,\\mathrm{full}}\n\\,\\equiv\\,\n\\widehat{\\chi}_{\\f12}^{\\,2}-1\n=\n\\f14\\Big{[}\nA^{2}+AB+BA+AA^{\\dagger}+A^{\\dagger}A+B^{2}+A^{\\dagger}B+BA^{\\dagger}+A^{\\dagger}{}^{2}\n\\Big{]}-1\n\\,.\n\\end{equation}\nThis new spin-1 holonomy operator is already a multi-loop operator: it has a component $A^{2}$ annihilating two loops and its adjoint component $A^{\\dagger}{}^{2}$ creating two loops, and so on.\nWe then define all the other spin-$j$ holonomy operators by recursion as polynomials of the fundamental $\\widehat{\\chi}_{\\f12}$ operator:\n\\begin{equation}\n\\chi_{\\f12}^{4}=\\chi_{2}+3\\chi_{1}+2\n\\,\\,\\Rightarrow\\,\\,\n\\widehat{\\chi}_{2}^{\\,\\mathrm{full}}\n\\,\\equiv\\,\n\\widehat{\\chi}_{\\f12}^{\\,4}-3\\widehat{\\chi}_{\\f12}^{\\,2}+1\\,,\n\\quad\\dots\n\\end{equation}\nand so on with $\\widehat{\\chi}_{j}^{\\,\\mathrm{full}}$ constructed from $\\widehat{\\chi}_{\\f12}^{2j}$ and smaller powers.\nThis construction clearly ensures that all the holonomy operators commute with each other. This method closely intertwines the definition of higher spin operators with  multi-loop holonomies. These multi-loop operators create spins $\\f12$ (and then higher spins too) excitations on several loops at once. \n\n\\medskip\n\nTo conclude the exploration of the basic loop quantum gravity operators, we should also deal with the symmetrized flux operators (and scalar products) with the ${\\mathfrak{su}}(2)$ generators acting as derivations on the wave-functions, and check their commutation relations with our new holonomy operators.\nThe flux and grasping operators are especially important since they allow to explore the intertwiner structure at the vertices. Indeed, acting with one-loop holonomy operators will only create decoupled loops at the vertex, while a generic intertwiner will couple them. So, even though we postpone the detailed analysis of the action of flux operators on loopy spin network to future investigation, we discuss below multi-loop holonomy operators that allow for coupled loops and thus explore the space of (loopy) intertwiners at the vertex.\n\nFor instance, considering two loops with group elements $h_{1}$ and $h_{2}$, we would like to excite the overall holonomy instead of the two independent holonomies, that is act with $\\chi(h_{1}h_{2})$ instead of $\\chi(h_{1})\\chi(h_{2})$. Proceeding similarly to the one-loop holonomy operator, we define a two-loop holonomy operator $\\widehat{\\chi}_{\\f12}^{(2)}$, which creates and annihilates pairs of coupled loops:\n\n\\begin{definition}\n\\label{C-def}\nWe define the following five operators $C_{-2,-1,0,+1,+2}$ on the Fock space of symmetrized loopy intertwiners ${\\cal H}^{\\mathrm{sym}}$. They act on an arbitrary state $(f_{N})_{N\\in{\\mathbb N}}$ as:\n\n\\begin{equation}\n(C_{-2}f)_{N}(h_{1},..,h_{N})\n\\,=\\,\n\\int \\mathrm{d}k \\mathrm{d}k\\,\\chi_{\\f12}(k\\tilde{k})\\,f_{N+2}(h_{1},h_{N},k,\\tilde{k}) \n\\end{equation}\n\\begin{equation}\n(C_{-1}f)_{N}(h_{1},..,h_{N})\n\\,=\\,\n\\f1N\\sum_{i}^{N}\n\\Bigg{[}\n\\int \\mathrm{d}k\\,\\chi_{\\f12}(kh_{i})\\,f_{N+1}(h_{1},..,h_{N},k)\n-\\int \\mathrm{d}k\\, \\mathrm{d}k_{i}\\,\\chi_{\\f12}(kk_{i})\\,f_{N+1}(h_{1},..,k_{i},..,h_{N},k)\n\\Bigg{]}\n\\end{equation}\n\\begin{eqnarray}\n(C_{0}f)_{N}(h_{1},..,h_{N})\n\\,=\n&&\n\\f2{N(N-1)}\\sum_{i<j}^{N}\\Bigg{[}\n\\chi_{\\f12}(h_{i}h_{j})f_{N}(h_{1},..,h_{N})\n+\\int \\mathrm{d}k_{i} \\mathrm{d}k_{j}\\,\\chi_{\\f12}(k_{i}k_{j})\\,f_{N}(h_{1},..,k_{i},..,k_{j},..,h_{N}) \\nonumber\\\\\n&&-\\int \\mathrm{d}k_{i}\\,\\chi_{\\f12}(k_{i}h_{j})\\,f_{N}(h_{1},..,k_{i},..,h_{N})\n-\\int \\mathrm{d}k_{i}\\,\\chi_{\\f12}(h_{i}k_{j})\\,f_{N}(h_{1},..,k_{j},..,h_{N})\n\\Bigg{]}\n\\end{eqnarray}\n\\begin{equation}\n(C_{+1}f)_{N}(h_{1},..,h_{N})\n\\,=\\,\n\\f1{N(N-1)}\\sum_{i\\ne j}^{N}\n\\Bigg{[}\n\\chi_{\\f12}(h_{i}h_{j})\nf_{N-1}(h_{1},..,\\widehat{h_{i}},..,h_{N})\n-\\int \\mathrm{d}k_{j}\\,\\chi_{\\f12}(h_{i}k_{j})\nf_{N-1}(h_{1},..,\\widehat{h_{i}},..,k_{j},..,h_{N})\n\\Bigg{]}\n\\end{equation}\n\\begin{equation}\n(C_{+2}f)_{N}(h_{1},..,h_{N})\n\\,=\\,\n\\f2{N(N-1)}\\sum_{i<j}^{N}\n\\chi_{\\f12}(h_{i}h_{j})f_{N-2}(h_{1},..,\\widehat{h_{i}},..,\\widehat{h_{j}},..,h_{N})\n\\end{equation}\nWe complete this definition with the ``initial conditions'' for $N=0$ and $N=1$:\n\\begin{equation}\n(C_{-1}f)_{0}=\\int \\chi_{\\f12}f_{1}\n\\,,\\quad\n(C_{0}f)_{0}=2f_{0}\n\\,,\\quad\n(C_{+1}f)_{0}=(C_{+2}f)_{0}=0\\,,\n\\end{equation}\n\\begin{equation}\n(C_{0}f)_{1}(h)=\\chi_{\\f12}(h)f_{1}(h)-\\int \\chi_{\\f12}f_{1}\n\\,,\\quad\n(C_{+1}f)_{1}(h)=\\chi_{\\f12}(h)f_{0}\n\\,,\\quad\n(C_{+2}f)_{0}=0\\,.\n\\end{equation}\nThey satisfy the Hermiticity relations:\n\\begin{equation}\nC_{-2}=C_{2}^{\\dagger}\\,,\\,\\,\nC_{-1}=C_{1}^{\\dagger}\\,,\\,\\,\nC_{0}=C_{0}^{\\dagger}\n\\,,\n\\end{equation}\nand the bounds for the $L^{2}$-norm:\n\\begin{equation}\n||C_{-2}+C_{+2}||_{2}\\le 2\\,,\\quad\n||C_{-1}+C_{+1}||_{2}\\le 2\\,,\\quad\n||C_{0}||_{2}\\le 2\\,.\n\\end{equation}\nFinally the two-loop holonomy operator $\\widehat{\\chi}_{\\f12}^{(2)}$ for spin $\\f12$  is defined as the sum of these five components:\n\\begin{equation}\n\\widehat{\\chi}_{\\f12}^{(2)}\\,\\equiv\\,\n\\f14\n\\big{(}C_{-2}+2C_{-1}+C_{0}+2C_{+1}+C_{+2}\\big{)}\\,.\n\\end{equation}\nThis operator is essentially self-adjoint and bounded by 2.\n\\end{definition}\n\nSo the spectrum of the two-loop holonomy operator is once again bounded by 2. An important consistency check is that this bound is saturated by the flat state. The proof is a straightforward computation, with special care to the initial conditions for $N=0$ and $N=1$.\n\\begin{prop}\nThe flat state $\\delta$, defined by $\\delta_{0}=1$ and $\\delta_{N\\ge 1}=(\\delta-1)^{\\otimes N}$, is an eigenvector of the spin-$\\f12$ two-loop holonomy operator  $\\widehat{\\chi}_{\\f12}^{(2)}$ with the highest eigenvalue on ${\\cal H}^{\\mathrm{sym}}$:\n\\begin{equation}\n\\widehat{\\chi}_{\\f12}^{(2)}\\,\n|\\delta\\rangle\n\\,=\\, 2\\,|\\delta\\rangle\n\\,.\n\\end{equation}\n\\end{prop}\n\n\nWe could then similarly define an operator $\\chi(h_{i}h_{j}^{-1})$ with a loop reversal or multi-loop operators $\\chi(h_{i_{1}}..h_{i_{n}})$ taking care of properly symmetrizing the group elements. We can also generalize our construction replacing the spin-$\\f12$ character by an arbitrary spin $j$ and define the spin-$j$ two-loop holonomy operator $\\widehat{\\chi}^{(2)}_{j}$, and so on for more loops.\nWe will not go into these details, although we do not foresee any obstacle (beside the inflation of indices and sums). \n\n\nWe now turn to the first application of the holonomy operators discuss below the Hamiltonian constraints for BF theory on the Fock space of loopy spin networks.\n\n\n\n\\subsection{Revisiting the BF constraints as creation and annihilation of loops}\n\\label{BFsym}\n\n\n\nLet us see how to implement the flatness constraint on our Fock space of loopy spin networks with bosonic statistics for the little loops. As earlier, we do not discuss the flatness constraints around loops of the base graph $\\Gamma$, which are implemented as usual by using the standard holonomy operators around those loops. Here, we will focus on the fate of the little loop excitations at every vertex of the background graph $\\Gamma$. For this purpose, we can focus on a single vertex and we can restrict ourselves to the flower graph, i.e. to the Fock space of loop intertwiners around a unique vertex. As we have constructed the holonomy operator in the previous section, we propose to use it as the Hamiltonian constraints for BF theory and simply impose:\n\\begin{equation}\nH^{\\mathrm{BF}}=\\widehat{\\chi}_{\\f12}-2\\,.\n\\end{equation}\nThis is a self-adjoint operator and imposing this constraint amounts to projecting onto the highest eigenvalue of the holonomy operator. Since  $\\widehat{\\chi}_{\\f12}$ creates and annihilates loops by construction, $H^{\\mathrm{BF}}$ shifts the number of loops and its flow should imply that the number of loops becomes pure gauge. Let us look at the space of physical states solving this flatness constraint.\nBy  proposition \\ref{flat-prop1}, we already know that the flat state, defined as the factorized $\\delta$-distribution state, saturates the holonomy bound, $H^{\\mathrm{BF}}\\,|\\delta\\rangle=0$ .\nThe natural question is whether the flat state is the only solution to this constraint.\n\nWe will run into the same problem as in the case of distinguishable loops of higher derivative solutions to the holonomy constraint. In order to deal with this potential infinite-dimensional space of solutions, we will introduce as before a Laplacian constraint and multi-loop holonomy operators. However we will ultimately show that we require only a finite number of constraint operators (three to be exact) to impose full flatness and the uniqueness of the physical state despite the infinite number of loop excitation modes that need to be constrained. \n\n\\medskip\n\nMore precisely, the holonomy constraint amounts to solving  functional recursion relations, relating $f_{N+1}$, $f_{N}$ and $f_{N-1}$ at each step. The problem is that this relation doesn't entirely fix $f_{N+1}$ in terms of $f_{N}$ and $f_{N-1}$, even assuming that these functions are invariant under permutations of their arguments and invariant under conjugation. Indeed it only fixes the integral $\\int\\mathrm{d}k \\chi_{\\f12}(k)\\,f_{N+1}(h_{1},..,h_{N},k)$. This condition seems to fix only the spin-$\\f12$ component of the function, so we face two obstacles: the non-trivial internal intertwiner structure and arbitrary higher spin excitations on each loop. We explain below how to get rid of all those modes by introducing constraints on the creation and annihilation of loops together with a Laplacian constraint.\n\n\nBefore treating the general case, we explore two simplified cases. First, factorized states avoid the problem of possible non-trivial intertwiner structure. It turns out that the spin-$\\f12$ one-loop holonomy constraint is enough to constrain all the higher spin excitations and lead to the flat state as the unique physical state.  Second we consider the larger class of states with decoupled loops, defined mathematically as the wave-functions which are invariant under conjugation of its individual arguments (and not simply under the simultaneous conjugation of all its arguments as required by gauge invariance). In this case, the spin-$\\f12$ constraint is not enough anymore and we need to explicitly introduce explicit constraints for all the higher spin excitations. We summarize these two cases in the following two propositions.\n\n\n\\begin{prop}\nLet us consider a factorized state $\\varphi\\in{\\cal H}^{\\mathrm{sym}}$, that $\\varphi_{0}=1$ and $\\varphi_{N}=F^{\\otimes N}$ for an integrable $F$ invariant under conjugation, $F(h)=F(ghg^{-1})$. Then the constraint $\\widehat{\\chi}_{\\f12}\\,\\varphi=\\,2\\varphi$ has a unique solution, which is the flat state, $\\varphi=\\delta$ and $F(h)=\\delta(h)-1$.\n\\end{prop}\n\\begin{proof}\nLet us look at the eigenvector equation on factorized states $\\varphi$ defined as $\\varphi_{0}=1$ and $\\varphi_{N}=F^{\\otimes N}$:\n$$\n\\big{(}A+A^{\\dagger}+B\\big{)}\\,F^{\\otimes N}\n\\,=\\,\n4\\,F^{\\otimes N}\\,.\n$$\nFor $N=0$, this gives an integral condition on the one-loop wave-function $F$:\n$$\n\\int \\mathrm{d}k\\,\\chi_{\\f12}(k)F(k)=2.\n$$\nThen, for $N=1$, we get a functional equality:\n$$\n\\chi_{\\f12}F+\\chi_{\\f12}-2=2F\\,.\n$$\nLet us decompose $F$ on the spin basis. Since it is invariant under conjugation, it decomposes onto the characters $F=\\sum_{j\\ne 0} F_{j}\\chi_{j}$. The $N=1$ equation translates into a recursion relation on the coefficients $F_{j}$ while the $N=0$ equation sets its initial condition:\n\\begin{equation}\nF_{\\f12}=2\\,,\\quad\nF_{1}+1=2F_{\\f12}\\,,\\quad\n\\forall {j\\ge 1}\\,,\\,\\,\nF_{j+\\f12}+F_{j-\\f12}=2F_{j}\\,.\n\\end{equation}\nThis has a unique solution $F_{j}=(2j+1)$, which translates to $F=\\delta-1$. The constraint equation for $N\\ge 2$ automatically follows.\n\\end{proof}\n\nThe case of factorized state works because the holonomy operator couples the creation of loops and the exploration of the higher spin components of the one-loop wave-function.\nNext, we move to the larger class of functions which are invariant under conjugation of its individual arguments. Then the functions $\\phi_{N}$ decompose on the character basis. Imposing the one-loop holonomy constraints for all spins leads to functional recursion equations such that the flat state is solution to the holonomy constraint. \n\\begin{lemma}\nConsidering a state invariant under conjugation of each of its arguments,\n$$\n\\phi_{N}(h_{1},..,h_{N})=\\phi_{N}(g_{1}h_{1}g_{1}^{-1},..,g_{N}h_{N}g_{N}^{-1})\\,,\\quad\\forall g_{i}\\in\\mathrm{SU}(2)^{N}\\,,\n$$\nit decomposes on the character basis:\n$$\n\\phi_{N}(h_{1},..,h_{N})=\\sum_{j_{1},..,j_{N}}\\phi_{N}^{j_{1},..,j_{N}}\\prod_{i}^{N}\\chi_{j_{i}}(h_{i})\\,.\n$$\nAssuming that the $\\phi_{N}$'s are all symmetric under permutations of their arguments and that they have no 0-modes, $\\int dh_{1}\\phi_{N}=0$ for all $N\\ge1$, then\nthe only such solution to the set of holonomy constraints $\\widehat{\\chi}_{j}\\,\\phi\\,=\\,(2j+1)\\phi$ for all spins $j\\in\\frac{{\\mathbb N}^{*}}2$ is  the flat state $\\phi_{N}=(\\delta-1)^{\\otimes N}$ (up to a global factor).\n\\end{lemma}\n\\begin{proof}\nThe proof is straightforward by recursion. For $N=0$, the constraint gives $\\phi_{1}$ in terms of the no-loop mode $\\phi_{0}$:\n\\begin{equation}\n\\forall j\\ge \\f12\\,,\\,\\,\n\\int dk\\,\\chi_{j}(k)\\phi_{1}(k)=(2j+1)\\phi_{0}\\,,\n\\end{equation}\nwhich gives $\\phi_{1}^{j}=(2j+1)\\phi_{0}$ for all non-vanishing spins $j$ while $\\phi_{1}^{0}=0$ by hypothesis. This way, if we fix the initial normalization to $\\phi_{0}=1$,  we recover $\\phi_{1}=(\\delta-1)$.\nThen the constraint equations for $N\\ge1$ reads:\n\\begin{eqnarray}\n\\forall j\\ge\\f12\\,,\n\\,\\,\n2(2j+1)\\phi_{N}(h_{1},..,h_{N})\n&=&\n\\int \\mathrm{d} k\\,\\chi_{j}(k)\\phi_{N+1}(h_{1},..,h_{N},k)+\\f1N\\sum_{i=1}^{N}\\phi_{N-1}(h_{1},..,\\widehat{h_{i}},..,h_{N}) \\nonumber\\\\\n&&\n+\\,\n\\f1N\\sum_{i=1}^{N}\\Bigg{[}\n\\chi_{j}(h_{i})\\phi_{N}(h_{1},..,h_{N})-\\int \\mathrm{d} k_{i}\\,\\chi_{j}(k_{i})\\phi_{N}(h_{1},..,{k_{i}},..,h_{N})\n\\Bigg{]}\n\\,.\n\\end{eqnarray}\nWe can solve this equation by recursion, determining the Fourier coefficients of $\\phi_{N+1}$ in terms of $\\phi_{N}$ and $\\phi_{N-1}$. The coefficients $\\phi_{N}^{j_{1}..j_{N}}$ vanish by assumption if one of the spins $j_{i}$ is zero. When none of the spins vanishes, we show that \n\\begin{equation}\n\\phi_{N}^{j_{1}..j_{N}}=\\prod_{i=1}^{N}(2j_{i}+1)\\,.\n\\end{equation}\n\\end{proof}\nComparing to the case of distinguishable loops, in this case where the loops are individually gauge-invariant and thus decoupled, we have traded the infinity of holonomy constraints, one for each distinguishable loop, for the infinite tower of one holonomy constraint per spin mode for indistinguishable loops. Exploiting further the Fock space structure for the bosonic little loops, we can nevertheless reduce this infinity of holonomy constraints to a pair of constraints.\nIndeed, checking the details of the proof of proposition \\ref{flat-prop1} on the action of holonomy operators on the $\\delta$-state, we propose to use  non-Hermitian constraints and characterize the flat state as an eigenvector of the loop annihilation operator $A$ and the loop creation operator $(B+A^{\\dagger})$:\n\\begin{lemma}\n\\label{lemmaAB}\nConsidering a state $\\phi$ invariant under conjugation of each of its arguments, and with no 0-modes, we introduce the pair of non-Hermitian constraint operators defined by the spin-$\\f12$ annihilation and creation operators acting on ${\\cal H}^{\\mathrm{sym}}$:\n\\begin{equation}\nA\\,|\\phi\\rangle=2\\,|\\phi\\rangle\n\\,,\\quad\n(B+A^{\\dagger})\\,|\\phi\\rangle=2\\,|\\phi\\rangle\n\\,.\n\\end{equation}\nThen the only solution to all these constraints is the flat state  $|\\phi\\rangle=|\\delta\\rangle$.\n\\end{lemma}\n\\begin{proof}\nLet us write explicitly the eigenvalue equations for the state $\\phi$:\n\\begin{equation}\n\\forall N\\ge 0,\\,\\,\n\\int \\mathrm{d}k\\,\\chi_{\\f12}(k)\\phi_{N+1}(h_{1},..,h_{N},k)\n\\,=\\,\n2\\phi_{N}(h_{1},..,h_{N})\n\\end{equation}\n\\begin{equation}\n\\forall N\\ge 1,\\,\\,\n\\sum_{\\ell=1}^{N}\n\\Bigg{[}\n\\chi_{\\f12}(h_{\\ell})\\,\\Big{[}\\phi_{N-1}(h_{1},..,\\widehat{h_{\\ell}},..,h_{N})\n+\\phi_{N}(h_{1},..,h_{N})\\Big{]}\n\\,-\\,\\int \\mathrm{d}k_{\\ell}\\,\\chi_{\\f12}(k_{\\ell})\\phi_{N}(h_{1},..,{k_{\\ell}},..,h_{N})\n\\Bigg{]}\n\\,=\\,\n2N\\,\\phi_{N}(h_{1},..,h_{N})\n\\nonumber\n\\end{equation}\nwith the initial conditions equation at $N=0$ for the creation operator  $(B+A^{\\dagger})$ trivially satisfied.\nWe could translate these equations into recursion relations on the Fourier coefficients, but there is actually a simpler and more direct route.\nThe first equation (for $A$) can be injected in the second equation turning it into a functional recursion:\n\\begin{equation}\n\\sum_{\\ell=1}^{N}\n(2-\\chi_{\\f12}(h_{\\ell}))\\,\n\\Big{[}\n\\phi_{N-1}(h_{1},..,\\widehat{h_{\\ell}},..,h_{N})\n+\\phi_{N}(h_{1},..,h_{N})\n\\Big{]}\n\\,=\\,\n0\\,.\n\\end{equation}\nFor $N=1$, this relates the one-loop wave-function $\\phi_{1}$ to the no-loop normalization $\\phi_{0}$:\n$$\n\\forall h\\in\\mathrm{SU}(2),\\quad\n(2-\\chi_{\\f12}(h))\\,(\\phi_{0}+\\phi_{1}(h))\\,=\\,0\\,.\n$$\nSince $\\phi_{1}$ is invariant under conjugation, this holonomy constraint has a unique distributional solution up to an arbitrary factor, $(\\phi_{0}+\\phi_{1})\\propto\\delta$. The integral condition, $\\int \\chi_{\\f12}\\phi_{1}=2\\phi_{0}$, fixes this factor and we recover $\\phi_{1}=(\\delta -1)$ as expected as we fix the normalization $\\phi_{0}=1$.\n\nWe then proceed by recursion, fixing the number of loops $N\\ge 2$ and assuming that $\\phi_{n}=(\\delta-1)^{\\otimes n}$ for all $n\\le (N-1)$. Let us now prove this statement holds for $n=N$. Using the identity $(2-\\chi_{\\f12}(h))\\delta(h)=0$ , we start by checking that:\n$$\n\\sum_{\\ell}^{N}\n(2-\\chi_{\\f12}(h_{\\ell}))\\,\n\\Big{[}\n\\prod_{i}(\\delta(h_{i})-1)-\\prod_{i\\ne\\ell}(\\delta(h_{i})-1)\n\\Big{]}\n\\,=\\,0\\,.\n$$\nThis implies that:\n$$\n\\Bigg{(}\n\\sum_{\\ell}^{N} (2-\\chi(h_{\\ell})\n\\Bigg{)}\n\\,\n\\Bigg{(}\n\\phi_{N}(h_{1},..,h_{N})-\\prod_{i}(\\delta(h_{i})-1)\n\\Bigg{)}\n\\,=\\,0\\,.\n$$\nSince every holonomy operator $(2-\\widehat{\\chi}_{\\ell})$ is Hermitian positive, this means that the holonomy constraint holds for each loop individually:\n\\begin{equation}\n\\forall \\ell \\le N\\,,\\,\\,\n\\sum_{\\ell}^{N} (2-\\chi(h_{\\ell})\n\\,\n\\Bigg{(}\n\\phi_{N}(h_{1},..,h_{N})-\\prod_{i}(\\delta(h_{i})-1)\n\\Bigg{)}\n\\,=\\,0\\,.\n\\end{equation}\nSince we have assumed that the wave-function is invariant under conjugation individually for each of its arguments, the only distribution solution to this equation is the product of $\\delta$-function up to a global factor:\n$$\n\\phi_{N}(h_{1},..,h_{N})=\\prod_{i}(\\delta(h_{i})-1)+\\alpha \\prod_{i}\\delta(h_{i})\\,,\n$$\nfor some factor $\\alpha$ to be determined. Checking this identity against the integral condition $\\int \\chi_{\\f12}\\phi_{N}=\\phi_{N-1}$ yields $\\alpha=0$ thus proving the proposition.\n\\end{proof}\nWe see that the requirement of the invariance under conjugation for each loop individually (stronger than gauge-invariance requiring the invariance under global conjugation) is crucial in the last step of the proof. Else we would have to deal with derivative solutions, in $\\partial\\delta$ and so on, as in the case of distinguishable loops.\n\n\\medskip\n\nOur proposal amounts to adding another constraint along side the Hermitian holonomy constraint $\\widehat{\\chi}_{\\f12}=2$. Instead of taking the average of the two operators $A$ and $(B+A^{\\dagger})$ and defining the holonomy operator, we subtract them and get the other constraint $B+(A^{\\dagger}-A)=0$. This new constraint operator has a Hermitian part $B$ and a anti-Hermitian part $(A^{\\dagger}-A)$, such that the overall structure can be interpreted as a holomorphic constraint, similar to the annihilation operator $a=\\hat{x}-i\\hat{p}$ for the harmonic oscillator. \nFrom this perspective, eigenvectors of this ``holomorphic'' operator $B+(A^{\\dagger}-A)$ can be considered as coherent states, which is pretty natural since we are looking into coherent superpositions of any number of loops summing over $N$, and the $\\delta$-state, as a null eigenvector of that operator, can be considered as a ground state.\n\n\nThe trick why these two  constraint operators $A$ and $(B+A^{\\dagger})$ are enough to kill all the degrees of freedom and lead to a single physical state is that they do not commute and their commutators actually generate higher spin constraints:\n\\begin{lemma}\nImposing the two constraints with $A$ and $(B+A^{\\dagger})$ on ${\\cal H}^{\\mathrm{sym}}$ implies a tower of constraints with all the higher spin annihilation operators:\n\\begin{equation}\nA\\,|\\phi\\rangle=(B+A^{\\dagger})\\,|\\phi\\rangle=2\\,|\\phi\\rangle\n\\qquad\\Longrightarrow\\qquad\n\\forall j\\ge \\f12\\,,\\,\\,\nA_{j}\\,|\\phi\\rangle=(2j+1)\\,|\\phi\\rangle\\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nLet us look at the commutator $(\\hat{N}+1)\\,[A_{j},(B+A^{\\dagger})]$. This commutator will vanish on solution states $|\\phi\\rangle$. Using the commutation relations computed earlier \\eqref{commAB1} and \\eqref{commAB2}, we get for $j=\\f12$:\n$$\n(\\hat{N}+1)\\,[A,(B+A^{\\dagger})]=A_{1}+{\\mathbb{I}}-(B+A^{\\dagger})A\\,,\n\\quad\nA_{1}\\,|\\phi\\rangle=\\Big{[}(B+A^{\\dagger})A-{\\mathbb{I}}\\Big{]}\\,|\\phi\\rangle=(4-1)\\,|\\phi\\rangle=3\\,|\\phi\\rangle\\,,\n$$\nthen  for higher spins $j\\ge 1$ the commutation relation $(\\hat{N}+1)\\,[A_{j},(B+A^{\\dagger})]=A_{j+\\f12}+A_{j-\\f12}{\\mathbb{I}}-(B+A^{\\dagger})A_{j}$ implies:\n\\begin{equation}\nA_{j+\\f12}\\,|\\phi\\rangle=\\Big{[}(B+A^{\\dagger})A_{j}-A_{j-\\f12}\\Big{]}\\,|\\phi\\rangle\n=\\Big{[}2(2j+1)-2j\\Big{]}\\,|\\phi\\rangle=(2j+2)\\,|\\phi\\rangle\\,.\n\\end{equation}\n\\end{proof}\n\nOur two (non-Hermitian) constraints do not commute and generate an infinite number of constraints killing all the higher spin excitations, leaving us at the end of the day with the single totally flat state.\nIn some sense, this pair of annihilation and creation constraint operators can be considered as the generators of the algebra of holonomy operators on the Fock space of loopy spin networks.\n\n\\medskip\n\nLet us move on to the general case. Working with states invariant under conjugation for each loop individually amounts to considering states created loop by loop, only by the action of one-loop holonomy operators. This leads to decoupled loops and unfortunately  does not explore the whole space of intertwiners: we still need to reach all the states globally invariant under conjugation but not invariant under conjugation of the individual arguments, such as $\\chi(h_{1}h_{2}..)$. In the spin decomposition of the wave-functions, this corresponds to the fact that the modes are not simply $\\phi_{N}^{j_{1},..,j_{N}}$ but should be labelled as $\\phi_{N}^{j_{1},..,j_{N},{\\mathcal I}}$: they do not depend only on the spins $j_{i=1..N}$ but further depend on the data of a (loopy) intertwiner ${\\mathcal I}$ between (two copies of) all the spins. This leads to the existence of the derivative solutions to the holonomy constraints, defined by applying differential operators (graspings) to the $\\delta$-distribution.\n\nThe intertwiner structure is hard to constraint completely. One way to go is to not only use higher spin operators but introduce multi-loop holonomy constraints, as in the case of disitinguishable loops. Indeed, since we would like to freeze all the spin excitations on the possible infinity of loops, it is natural to introduce one constraint operator per mode\\footnotemark. \n\\footnotetext{\nThis leads us to conjecture a set of complete holonomy constraints for BF theory. Considering all the multi-loop holonomy operators for arbitrary spins acting on the Fock space of loopy intertwiners ${\\cal H}^{\\mathrm{sym}}$:\n$$\n\\forall j\\in\\frac{{\\mathbb N}^{*}}2\\,,\\quad\n\\forall n\\in{\\mathbb N}^{*}\\,,\\quad\n\\widehat{\\chi}^{(n)}_{j}\\,|\\phi\\rangle=\\,(2j+1)\\,|\\phi\\rangle\\,,\n$$\nthen the only solution to all these constraints is the flat state $\\phi=\\delta$.\nWe have checked this conjecture up to the three-loop component of the state, $N=3$, but we haven't gone further. This would require explicitly and carefully defining the multi-loop holonomy operators. We should also take special care of working with legitimate states, controlling the convergence\/divergence of the series in $j$ and $N$ to ensure that the states are distributions.\n}\n\nWe propose to take a different route in order to keep a finite number of (primary) constraints. We introduce a Laplacian constraint to project onto the space of wave-functions invariant under conjugation and use the creation and annihilation operators for loops to impose flatness:\n\\begin{prop}\nWe consider the pair of non-Hermitian constraint operators defined by the spin-$\\f12$ annihilation and creation operators acting on ${\\cal H}^{\\mathrm{sym}}$:\n\\begin{equation}\nA\\,|\\phi\\rangle=2\\,|\\phi\\rangle\n\\,,\\quad\n(B+A^{\\dagger})\\,|\\phi\\rangle=2\\,|\\phi\\rangle\n\\,.\n\\end{equation}\nWe supplement these constraints with the Laplacian constraint:\n\\begin{equation}\n(\\widetilde{\\Delta}-\\Delta)\\,|\\phi\\rangle=0\n\\qquad\\textrm{with}\\quad\n(\\Delta\\varphi)_{N}=\\f1N\\sum_{\\ell}^{N}\\Delta_{\\ell}\\,\\varphi_{N}\\,,\n\\quad\n(\\widetilde{\\Delta}\\varphi)_{N}=\\f1N\\sum_{\\ell}^{N}\\widetilde{\\Delta}_{\\ell}\\,\\varphi_{N}\\,.\n\\end{equation}\nImposing these three eigenvalue equations leads to a unique solution (up to a global factor), the flat state $|\\phi\\rangle=|\\delta\\rangle$ defined by $\\phi_{N}=(\\delta-1)^{\\otimes N}$.\n\\end{prop}\n\\begin{proof}\nWe start with the Laplacian constraints:\n$$\n\\sum_{\\ell}(\\widetilde{\\Delta}_{\\ell}-\\Delta_{\\ell})\\phi_{N}=0\\,.\n$$\nSince every Laplacian constraint operator $(\\widetilde{\\Delta}_{\\ell}-\\Delta_{\\ell})$ on each loop $\\ell$ is Hermitian and positive, this imposes that each of them vanish on the wave-function, i.e. for all $\\ell$ we have $(\\widetilde{\\Delta}_{\\ell}-\\Delta_{\\ell})\\phi_{N}=0$. This implies that $\\phi_{N}$ is invariant under conjugation of each of its argument. Then we apply lemma \\ref{lemmaAB} to prove the uniqueness of the solution state.\n\n\\end{proof}\n\nOn the one hand, the Laplacian constraint fixes how every loop is attached to the vertex, through a trivial spin-0. Each loop is invariant under conjugation on its own, states are collections of bosonic loops, each carrying a spin and with a trivial intertwiner between them. On the other hand, the constraints $A$ and $(B+A^{\\dagger})$ realize explicitly the idea that BF dynamics impose that the creation and annihilation of loops are pure gauge. \n\n\\medskip\n\nTo conclude this section, we would like to underline the similarities and differences between the case of distinguishable loops and the Fock space of indistinguishable little loop excitations. When working with little loops endowed with bosonic statistics, one must take a special care to consistently remove the spin-0 modes on every loop to implement the cylindrical consistency of the wave-functions. This leads to a (spin-$\\f12$) holonomy operator also creating and annihilating loops . We explicitly separate its components respectively creating and annihilating loops and use them as legitimate constraint operators for BF theory. This is different from distinguishable loops where holonomy operators are defined as attached to a loop: a holonomy operator acts on a given loop, exciting and shifting the spin carried by the loop.\n\nNevertheless, the issue of the intertwiner space living at vertex and coupling the loops is the same in both frameworks. We have identified an infinity of solutions to the holonomy constraints, constructed as differential operators acting on the $\\delta$-distribution (as graspings on the spin network wave-function). These are still peaked on the identity group element, but they potentially define an infinity of gauge-invariant local degrees of freedom living at the vertex. To get rid of these ``spurious'' solutions, we have introducing a Laplacian constraint that forces each loop excitation to be invariant under conjugation, thus linking it trivially to the vertex. This allows to kill all local intertwiner excitation. Then we take as  Hamiltonian constraints for BF theory this combination on holonomy and Laplacian constraints, which lead as wanted to a unique physical state, the flat $\\delta$-state.\n\n\n\n\\section{Tagged spin networks}\n\n\n\nAfter folded spin networks, which retains the internal combinatorial structure inside coarse-grained regions, and loopy spin networks, which keep local curvature excitations as little loops attached to the vertices of the base graph and which we have explored in great details in the previous sections, we move on to the third and last step of coarse-grained structures: {\\it tagged spin networks}.\n\nWhen integrating out the connection group elements inside a bounded region, as discussed in \\cite{Livine:2013gna,Dittrich:2014wpa,Bahr:2015bra} and reviewed in the first section \\ref{sec:overview}, and coarse-graining the region to a single vertex, we naturally break the local gauge invariance at the resulting coarse-grained vertex.\nThis  also happens as soon as we introduce fermionic matter fields, which act as sources for loop quantum gravity's Gauss law and thus create non-trivial closure defects.\nAt the classical level, this is reflected by a non-vanishing closure defect: the sum of the flux vectors around the effective vertex does not vanish anymore and is actually balanced by the internal fluxes living on the loops living inside the bounded region and carrying non-trivial holonomies. Overall, the gauge invariance is restored if we take into account the internal degrees of freedom of the region however, once we have coarse-grained it, the breaking of the gauge invariance reflects the geometry excitations which have developed in the region's bulk and which we have traced out.\nAt the quantum level, the closure defect becomes a tag, attached to each vertex, as drawn on fig.\\ref{tagged}. This internal degree of freedom is defined as an extra spin coupling to the actual spin living on the links and edges attached to the effective vertex and connecting the coarse-grained region to its exterior. This tag allows to relax the gauge invariance in a controlled way.\n\nMathematically, we are thus led to consider the whole space of non-gauge-invariant cylindrical functionals of the connection on a given fixed background graph $\\Gamma$. The tagged spin networks will provide a basis of that space, with the tags record how much the local gauge-invariance is broken: when the tags vanish, we recover the usual gauge-invariant spin network basis states. This allows to account for graph changing dynamics in an effective manner. Even though the graph changes and might get more complexed as the geometry evolves, we keep on coarse-graining the state projecting it onto the fixed base graph (chosen by the observer), then the internal degrees of freedom and non-trivial curvature developed inside the coarse-grained regions gets translated into excitations of the effective tag degree of freedom attached to the base graph vertices.\n\nThe space of tagged spin networks is naturally quite simple and we believe it offers a useful framework for the study of coarse-graining of the loop quantum gravity dynamics.\n\n\\begin{figure}[h!]\n\n\\centering\n\n\\begin{tikzpicture}[scale=0.7]\n\\coordinate(O1) at (0,0);\n\n\\draw (O1) -- ++(-2,1) node[above] {$j_1$};\n\\draw (O1) -- ++(2,1) node[above] {$j_2$};\n\\draw (O1) -- ++(2,-1) node[below] {$j_3$};\n\\draw (O1) -- ++(-2,-1) node[below] {$j_4$};\n\\draw[thick,red] (O1) to ++(0,0.5) node[above]{$j,m$};\n\n\\draw (O1) node {$\\bullet$} ++(0,-0.3) node{$i$};\n\n\\end{tikzpicture}\n\n\\caption{We consider tagged vertices: a vertex with the additional tag corresponding to a closure defect. The representations $j_{1},..,j_{4}$ living on the graph edges linked to the vertex do not form an intertwiner on their own, but they recouple to the spin $j$ defining the tag.\n\\label{tagged}}\n\n\\end{figure}\n\n\\subsection{The tagged spin network basis}\n\nWe consider the space ${\\mathcal H}_{\\Gamma}^{tag}$ of non-gauge-invariant wave-functions on the (oriented and connected) graph $\\Gamma$. This is simply the space of $L^{2}$ functions on $\\mathrm{SU}(2)^{\\times E}$, where $E$ is the number of edges or links of $\\Gamma$, with no further assumption. Considering such a function, we can project onto the usual space of gauge-invariant states by group averaging:\n\\begin{equation}\n\\psi(\\{g_{e}\\}_{e\\in\\Gamma})\n\\,\\in{\\mathcal H}_{\\Gamma}^{tag}\n\\quad\\mapsto\\quad\n\\psi_{0}(\\{g_{e}\\}_{e\\in\\Gamma})\n\\,=\\,\n\\int_{\\mathrm{SU}(2)^{V}}\\mathrm{d} h_{v}\n\\psi(\\{h_{s(e)}^{-1}g_{e}h_{t(e)}\\}_{e\\in\\Gamma})\n\\,\\,\\in{\\mathcal H}_{\\Gamma}\\,.\n\\end{equation}\nWe can generalize this projection to non-trivial recouplings at every vertex and get an exact decomposition of the full non-invariant state:\n\\begin{equation}\n\\psi\n\\,=\\,\n\\sum_{\\{J_{v}\\}\\in\\frac\\N2}\n\\prod_{v}(2J_{v}+1)\\,P_{\\{J_{v}\\}}\\psi\\,,\n\\quad\nP_{\\{J_{v}\\}}\\psi\\,(\\{g_{e}\\}_{e\\in\\Gamma})\n\\,=\\,\n\\int \\mathrm{d} h_{v}\\,\n\\prod_{v}\\chi_{J_{v}}(h_{v})\\,\n\\psi(\\{h_{s(e)}^{-1}g_{e}h_{t(e)}\\}_{e\\in\\Gamma})\\,.\n\\end{equation}\nThe spin $J_{v}$ is the tag living at the vertex $v$ and provides a measure of how much gauge-invariance is relaxed at that vertex. It is the variable conjugate to the group averaging variable $h_{v}$. \nFollowing this logic, we can make all states in   ${\\mathcal H}_{\\Gamma}^{tag}$ gauge-invariant by adding  $h_{v}$ as an actual argument of the wave-function. This provides a isomorphism between  ${\\mathcal H}_{\\Gamma}^{tag}=L^{2}(\\mathrm{SU}(2)^{\\times E})$ and ${\\mathcal H}_{\\Gamma}^{ext}=L^{2}(\\mathrm{SU}(2)^{\\times E}\\times\\mathrm{SU}(2)^{\\times V}\/\\mathrm{SU}(2)^{\\times V})$ where $ext$ stands for ``extended'':\n\\begin{equation}\n\\Psi(\\{g_{e},h_{v}\\}_{e,v\\in\\Gamma})\n\\,=\\,\n\\Psi(\\{k_{s(e)}^{-1}g_{e}k_{t(e)},k_{v}^{-1}h_{v}\\}_{e,v\\in\\Gamma})\n\\quad\\mapsto\\quad\n\\psi(\\{g_{e}\\}_{e\\in\\Gamma})\n\\,=\\,\n\\Psi(\\{g_{e},h_{v}={\\mathbb{I}}\\}_{e,v\\in\\Gamma})\\,.\n\\end{equation}\nWe define tagged spin networks as basis states for ${\\mathcal H}_{\\Gamma}^{ext}$ thus providing through this gauge-fixing map a basis for generic non-gauge-invariant states. These generalizations of spin networks are labeled by spins $j_{e}$ on every edge $e$, the tag spin $J_{v}$ an magnetic momentum $M_{v}$ at every vertex, as well as an intertwiner ${\\mathcal I}_{v}$ recoupling at each vertex between the tag  and the spins on the edges attached to that vertex:\nthe incoming and outgoing edges attached to the vertex $v$:\n$$\n{\\mathcal I}_{v}:{\\mathcal V}^{J_{v}}\\otimes\\bigotimes_{e|s(e)=v}{\\mathcal V}^{j_{e}}\\longrightarrow\\bigotimes_{e|t(e)=v}{\\mathcal V}^{j_{e}}\\,,\n$$\n\\begin{equation}\n\\label{tagbasis}\n\\Psi_{\\{j_{e},J_{v},M_{v},{\\mathcal I}_{v}\\}}\n(\\{g_{e},h_{v}\\})\n\\,\\equiv\\,\n\\prod_{v}D^{J_{v}}_{m_{v}M_{v}}(h_{v})\\,\n\\prod_{e}\\langle j_{e}m_{e}^{s}|\\,g_{e}\\,|j_{e}m_{e}^{t}\\rangle\\,\n\\prod_{v}\\langle \\otimes_{e|t(e)=v}j_{e}m_{e}^{t}|\\,{\\mathcal I}_{v}\\,|J_{v}m_{v}\\otimes_{e|s(e)=v}j_{e}m_{e}^{s}\\rangle\\,,\n\\end{equation}\nwith an implicit over the magnetic momenta $m_{e}^{v}$ and $m_{v}$. In simple words, we work with spin network on graphs with an extra open edge at every vertex. The spins carried by those open edges are the tags.\n\n\\medskip\n\nThe whole question is the physical interpretation of these tags, which we added to the usual spin network states. It is mathematically clear how the closure defects arise from coarse-graining and that the tags reflect non-trivial holonomies around the loops of the subgraph within the coarse-graining regions. The next challenge would be to show that they can be related to some physical notions of (quasi-)local energy density or mass (see e.g. \\cite{Yang:2008th} for a definition of the quasi-local energy operator in loop quantum gravity).\n\n\\subsection{Tags from coarse-graining and tracing out little loops}\n\nLet us show how starting from a loopy spin network and tracing out the little loops attached to the vertices leads naturally to a reduced density matrix defined in terms of tagged spin networks. So we consider a gauge-invariant loopy state defined on the base graph $\\Gamma$ with a certain number of loops $n_{v}$ attached to each vertex $v$:\n$$\n\\phi(\\{g_{e},h^{v}_{\\ell}\\})\n\\,=\\,\n\\phi(\\{k_{s(e)}^{-1}g_{e}k_{t(e)},k_{v}^{-1}h^{v}_{\\ell}k_{v}\\})\\,\\,\n\\forall k_{v}\\in\\mathrm{SU}(2)^{\\times V}\\,,\n$$\nwhere the group elements $h^{v}_{\\ell}$ live on the little loops $\\ell$ attached to the vertex $v$, and we integrate out the loops:\n\\begin{equation}\n\\rho(\\{g_{e},\\tilde{g}_{e}\\}_{e\\in\\Gamma})\n\\,=\\,\n\\int \\prod_{v}\\prod_{\\ell=1}^{n_{v}}\\mathrm{d} h_{\\ell}^{v}\\,\\,\n\\overline{\\phi(\\{g_{e},h^{v}_{\\ell}\\})}\\,\n\\phi(\\{\\tilde{g}_{e},h^{v}_{\\ell}\\})\\,.\n\\end{equation}\nLet us compute the reduced density matrix using the natural loopy spin network basis. We focus on the little loops attached to single vertex, say $v_{0}$, and drop the index $v$ from the little loop group elements for the sake of simplicity.\nWe consider the loopy states defined by basis intertwiners defined by two intertwiners, one recoupling the spins living on the edges linked to the vertex $v_{0}$ and one recoupling the little loops attached to that vertex, glued through an intermediate spin $J_{v_{0}}$, as drawn on fig.\\ref{fig:intermediatespin2}:\n\\begin{eqnarray}\n\\Phi_{\\{j_{e},J_{v_{0}},i_{v},j_{\\ell},\\tilde{i}_{v_{0}}\\}}\n(\\{g_{e},h_{\\ell}\\})\n&=&\n\\prod_{e}\\langle j_{e}m_{e}^{s}|\\,g_{e}\\,|j_{e}m_{e}^{t}\\rangle\\,\n\\prod_{\\ell}\\langle j_{\\ell}m_{\\ell}^{s}|\\,h_{\\ell}\\,|j_{\\ell}m_{\\ell}^{t}\\rangle\\,\n\\prod_{v\\ne v_{0}}\\langle \\otimes_{e|t(e)=v}j_{e}m_{e}^{t}|\\,i_{v\\,}|\\otimes_{e|s(e)=v}j_{e}m_{e}^{s}\\rangle\\,\n\\nonumber\\\\\n&&\n\\langle \\otimes_{e|t(e)=v_{0}}j_{e}m_{e}^{t}|\\,i_{v_{0}}\\,|J_{v_{0}}M_{v_{0}}\\otimes_{e|s(e)=v}j_{e}m_{e}^{s}\\rangle\\,\n\\langle J_{v_{0}}M_{v_{0}} \\otimes_{\\ell}j_{\\ell}m_{\\ell}^{t}|\\,\\tilde{i}_{v_{0}}\\,|\\otimes_{\\ell}j_{\\ell}m_{\\ell}^{s}\\rangle\n\\,,\n\\end{eqnarray}\nwith an implicit sum over all magnetic moment labels.\nWe have assumed, as announced, that only the vertex $v_{0}$ has little loops attached to it, so all other vertices are thought as having a vanishing intermediate spin $J_{v\\ne v_{0}}=0$.\n\\begin{figure}\n\\begin{tikzpicture}[scale=0.8]\n\n\\coordinate(O2) at (8,0);\n\\coordinate(O3) at (10,0);\n\n\\draw (O2) -- ++(-2,1) node[left]{$j_{e}$};\n\\draw (O2) -- ++(-2,0.4);\n\\draw (O2) -- ++(-2,-0.4);\n\\draw (O2) -- ++(-2,-1);\n\n\n\\draw[in=-25,out=25,scale=3] (O3) to[loop] (O3);\n\\draw[in=30,out=80,scale=3] (O3)  to[loop] node[pos=0.6,right,above]{$j_{\\ell}$} (O3);\n\\draw[in=-80,out=-30,scale=3] (O3) to[loop] (O3);\n\n\\draw[red] (O2) -- (O3) node[midway,below]{$J_{v}$};\n\\draw (O2) node {$\\bullet$} ++(0.12,0.4) node{$i_v$};\n\\draw (O3) node {$\\bullet$} ++(-0.2,0.4) node{$\\tilde{i}_v$};\n\n\\end{tikzpicture}\n\n\\caption{We introduce the intermediate spin basis for the vertices of loopy spin networks: intertwiners will decompose into two intertwiners, the first one $i_{v}$ recoupling the spins living on the base graph edges and the other $\\tilde{i}_{v}$ recoupling the spins living on the little loops attached to the vertex, which are linked together by the intermediate spin $J_{v}$. When we coarse-grain by tracing over the little loops, the only remaining information is this intermediate spin $J_{v}$, which becomes the tag measuring the closure defect at the vertex. It is the remnant of the curvature fluctuations and internal geometry within the vertex.} \n\\label{fig:intermediatespin2}\n\n\\end{figure}\n\n\nA loopy state will decompose onto that basis, $|\\phi\\rangle=\\phi_{\\{j_{e},J_{v_{0}},i_{v},j_{\\ell},\\tilde{i}_{v_{0}}\\}}\n\\,|\\Phi_{\\{j_{e},J_{v_{0}},i_{v},j_{\\ell},\\tilde{i}_{v_{0}}\\}}\\rangle$, and we easily compute the resulting reduced density matrix using the orthonormality of the Wigner matrices with respect to the Haar measure on $\\mathrm{SU}(2)$ and find that it naturally decompose onto the tagged spin network basis introduced $\\Psi_{\\{j_{e},J_{v},M_{v},{\\mathcal I}_{v}\\}}$ above in \\eqref{tagbasis}:\n\\begin{equation}\n\\rho\n\\,=\\,\n{\\mathrm{Tr}}_{\\{h_{\\ell}\\}}|\\phi\\rangle\\langle\\phi|\n\\,=\\,\n\\Big{(}\n\\sum_{\\{m^{s,t}_{\\ell}\\}}\n\\big{|}\n\\langle J_{v_{0}}M_{v_{0}} \\otimes_{\\ell}j_{\\ell}m_{\\ell}^{t}|\\,\\tilde{i}_{v_{0}}\\,|\\otimes_{\\ell}j_{\\ell}m_{\\ell}^{s}\\rangle\n\\big{|}^{2}\n\\Big{)}\n\\,\n|\\Psi_{\\{j_{e},J_{v_{0}},M_{v_{0}},i_{v}\\}}\\rangle\\langle\\Psi_{\\{j_{e},J_{v_{0}},M_{v_{0}},i_{v}\\}}|\\,.\n\\end{equation}\nThus the intermediate spins of the loopy spin networks, which recouple between the base graph edges and the little loop excitations, become the tags of the tagged spin network basis after tracing out the holonomies living on the little loops. This concludes the coarse-graining of the geometry of a bounded region to a single vertex plus one extra degree of freedom -the tag- registering the  excitations of geometry and curvature within that region's bulk.\n\n\n\n\n\\subsection{Revisiting BF theory: the non-trivial space of flat spin network states}\n\n\n\nThe (totally) flat state on a closed and connected graph $\\Gamma$ is defined by a $\\delta$-distribution on every loop of the graph. This is generically redundant, so we choose a set of independent loops\\footnotemark, that is $L\\equiv (E-V+1)$ loops generating all the loops of $\\Gamma$, and define the flat state as the product of the $\\delta$-distribution on those independent loops.\n\\footnotetext{\nThe simplest way to proceed is, as in the gauge fixing procedure, to choose a maximal tree on $\\Gamma$, then to associate one  loop to each edge which does not belong to the tree. This ensures that each of those loops contains one edge that no other loop contain and are thus independent. \n}\nNow we would like to impose holonomy operator constraints along all the (independent) loops of the graph in order to this flat state as single physical states. We will encounter the same obstacle of the derivative solutions to the holonomy constraints as  with loopy spin networks.\n\n\\begin{figure}[h!]\n\\centering\n\\begin{tikzpicture}[scale=1]\n\n\\coordinate(A) at (-4,0);\n\\coordinate(B) at (-2,0);\n\n\\draw(A) -- node[midway,above]{2} (B) ;\n\\draw (A) to[bend left=80] node[midway,above]{1}(B);\n\\draw (A) to[bend right=80] node[midway,above]{3}(B);\n\n\\coordinate(a) at (0,0);\n\\coordinate(b) at (1.5,-0.3);\n\\coordinate(c) at (2,1);\n\\coordinate(d) at (2.3,-1.4);\n\n\\draw[->,>=stealth](a) --  node[midway,above]{2} (b) ;\n\\draw[->,>=stealth](a) -- node[midway,above]{3} (c) ;\n\\draw[->,>=stealth](a) -- node[midway,below]{1} (d) ;\n\\draw[->,>=stealth](b) -- node[midway,left]{4} (c) ;\n\\draw[->,>=stealth](c) -- node[midway,right]{5} (d) ;\n\\draw[->,>=stealth](d) -- node[midway,left]{6} (b) ;\n\n\n\n\\end{tikzpicture}\n\n\\caption{The $\\Theta$-graph on the left and the tetrahedron graph on the right.}\n\\label{fig:thetatetra}\n\n\\end{figure}\n\nLet us, for the sake of simplicity, look at the $\\Theta$-graph, made of two vertices connected by three edges, as in fig.\\ref{fig:thetatetra}. The flat state is:\n\\begin{equation}\n\\delta_{\\Theta}(g_{1},g_{2},g_{3})\n\\,=\\,\n\\delta(g_{1}g_{2}^{-1})\\delta(g_{1}g_{3}^{-1})\n\\,=\\,\n\\delta(g_{1}g_{2}^{-1})\\delta(g_{2}g_{3}^{-1})\n\\,=\\,\n\\delta(g_{1}g_{3}^{-1})\\delta(g_{2}g_{3}^{-1})\n\\,,\n\\end{equation}\nwhere we give the three possible  sets of independent loops, corresponding to the three choices of tree on the $\\Theta$-graph.\nThis is clearly a solution to the three holonomy constraints:\n$$\n\\Big{(}\\chi_{\\f12}(g_{1}g_{2}^{-1})-2\\Big{)}\\,\\delta_{\\Theta}\n\\,=\\,\n\\Big{(}\\chi_{\\f12}(g_{2}g_{3}^{-1})-2\\Big{)}\\,\\delta_{\\Theta}\n\\,=\\,\n\\Big{(}\\chi_{\\f12}(g_{3}g_{1}^{-1})-2\\Big{)}\\,\\delta_{\\Theta}\n\\,=\\,\n0\\,.\n$$\nHowever, any first derivative give also a solution to those holonomy constraints. As an example, if we differentiate along the first group element $g_{1}$, we ge the distribution \n$\\partial_{a}^{(1)}\\delta_{12}\\delta_{23}\n=\n\\partial_{a}^{(1)}\\delta(g_{1}g_{2}^{-1})\\delta(g_{2}g_{3}^{-1})\n=\ni\\delta(g_{1}J_{a}g_{2}^{-1})\\delta(g_{2}g_{3}^{-1})$\n also satisfying:\n\\begin{equation}\n\\Big{(}\\chi_{\\f12}(g_{1}g_{2}^{-1})-2\\Big{)}\\,\\partial_{a}^{(1)}\\delta_{12}\\delta_{23}\n\\,=\\,\n\\Big{(}\\chi_{\\f12}(g_{2}g_{3}^{-1})-2\\Big{)}\\,\\partial_{a}^{(1)}\\delta_{12}\\delta_{23}\n\\,=\\,\n\\Big{(}\\chi_{\\f12}(g_{3}g_{1}^{-1})-2\\Big{)}\\,\\partial_{a}^{(1)}\\delta_{12}\\delta_{23}\n\\,=\\,\n0\n\\,.\n\\end{equation}\nAnd we can go on constructing an infinity of higher order derivative solutions, as explained in the framework of loopy spin networks in sections \\ref{derivativesolution1} and \\ref{derivativesolution2}.\nIn some sense, the totally flat state, defined by the $\\delta$-distribution, is the primary state. Then we act on it with differentiation operators and generate a whole space of solutions. All these states are technically still flat, since they are peaked exclusively on the identity group element ${\\mathbb{I}}$. They seem to be pure excitations of the triad conjugate to the connection (similar to quantum excitations of the electric field at vanishing magnetic field).  This non-trivial space of flat states over the flat state sounds similar in spirit to the construction by Dittrich and collaborators \\cite{Dittrich:2014wpa,Bahr:2015bra} where they build a Hilbert space representation for loop quantum gravity with the flat state as ``ground state''. It would be interesting to check if an explicit relation with our present construction could be identified.\n\n\\medskip\n\nAs the first derivative are not gauge-invariant distributions, it is reasonable to wonder if restricting to gauge-invariant spin network states by imposing vanishing tags allows to kill all those higher order flat solutions and leave the original $\\delta$-distribution as sole physical state. We show below that this is indeed the case, but with the very important subtlety that we need to impose the holonomy constraints along all the loops of the graph and not only a set of independent loops.\n\n\nFirst, we impose the tags to vanish by a Laplacian constraint at every vertex:\n\\begin{equation}\n\\Delta_{v}\n\\,\\equiv\\,\n\\big{(}\n\\sum_{e|v=t(e)}\\partial_{a}^{(e)R}\n-\\sum_{e|v=s(e)}\\partial_{a}^{(e)L}\n\\big{)}^{2}\\,.\n\\end{equation}\nRequiring $\\Delta_{v}\\,\\varphi=0$ implies that the recoupling of the spins $j_{e}$ on the edges attached to the vertex $v$ is trivial, i.e. that the tag $J_{v}$ is the trivial representation, i.e that the wave-function $\\varphi$ satisfies the  gauge-invariance at the vertex $v$. Imposing this constraint at all the vertices allows to project from the tagged spin networks down to the usual gauge-invariant spin network states.\n\n\\medskip\n\nSecond, let us revisit the flatness constraints to impose on standard spin networks in order  to implement BF theory and its projector on the flat state. Imposing the holonomy constraints on a set of independent loops turns out not to be enough to get a unique solution state and we have gauge-invariant remnants of the derivative solutions.  To remedy this, it is necessary and sufficient to impose the holonomy constraints on all possible loops of the graph. This is the counterpart of the multi-loop holonomy constraints for imposing the flatness of loopy intertwiners as explained in sections \\ref{derivativesolution2} and \\ref{derivativesolution3}.\n\nFor instance, on the $\\Theta$-graph, a set of independent loops is provided by the two loops, $g_{1}g_{2}^{-1}$ and $g_{2}g_{3}^{-1}$. Classically, imposing that these two group elements vanish implies that the flatness of the third ``composite'' loop, $g_{1}g_{3}^{-1}$. At the quantum level however, we can construct an infinity of derivative solutions to the holonomy constraints $\\widehat{\\chi}_{12}$ and $\\widehat{\\chi}_{23}$ by acting with grasping operators, as an example:\n\\begin{eqnarray}\n\\varphi=\\sum_{a}\\partial_{a}^{(1)}\\delta(g_{1}g_{2}^{-1})\\partial_{a}^{(3)}\\delta(g_{3}g_{2}^{-1})\n\\quad\\Rightarrow\n&&\n\\Big{(}\\chi_{\\f12}(g_{1}g_{2}^{-1})-2\\Big{)}\\,\\varphi\n\\,=\\,\n\\Big{(}\\chi_{\\f12}(g_{2}g_{3}^{-1})-2\\Big{)}\\,\\varphi\n=0\n\\nonumber\\\\\n\\quad\\textrm{but}&&\n\\int f (\\chi_{\\f12}(g_{1}g_{3}^{-1})-2)\\,\\varphi\n\\,=\\,\n\\left.-\\f14f\\Delta\\chi_{\\f12}\\right|_{{\\mathbb{I}}}\n=-\\f32\\,f({\\mathbb{I}})\\ne 0\\,.\n\\end{eqnarray}\nThese distributions will actually not be solution to the third holonomy constraint $\\widehat{\\chi}_{13}$ and, so imposing directly the three holonomy constraints together determines the $\\delta$-state $\\delta_{\\Theta}$ as unique solution.\n\nAnother interesting case is on the tetrahedron graph, see fig.\\ref{fig:thetatetra}, where imposing the holonomy constraints around three triangles $(g_{1}g_{6}g_{2}^{-1})$, $(g_{2}g_{4}g_{3}^{-1})$ and $(g_{3}g_{5}g_{1}^{-1})$ does not imply the holonomy constraint around the fourth triangle $(g_{4}g_{5}g_{6})$. This is realized by a triple grasping around the vertex $(123)$ acting on the $\\delta$-distribution, which gives the following gauge-invariant state:\n\\begin{equation}\n\\varphi\n=\n\\epsilon^{abc}\\partial_{a}^{(1)L}\\delta(g_{1}g_{6}g_{2}^{-1})\n\\partial_{b}^{(2)L}\\delta(g_{2}g_{4}g_{3}^{-1})\n\\partial_{c}^{(3)L}\\delta(g_{3}g_{5}g_{1}^{-1})\\,.\n\\end{equation}\nApplying this distribution against a gauge-invariant test function $f(g_{1},..,g_{6})$, we compute the action of the holonomy operators around the four 3-cycles of the tetrahedron graph:\n\\begin{equation}\n\\int f (\\chi_{\\f12}(g_{1}g_{6}g_{2}^{-1})-2)\\,\\varphi\n\\,=\\,\n\\int f (\\chi_{\\f12}(g_{2}g_{4}g_{3}^{-1})-2)\\,\\varphi\n\\,=\\,\n\\int f (\\chi_{\\f12}(g_{3}g_{5}g_{1}^{-1})-2)\\,\\varphi\n\\,=\\,\n 0\\,,\\nonumber\n\\end{equation}\n\\begin{eqnarray}\n\\int f (\\chi_{\\f12}(g_{4}g_{5}g_{6})-2)\\,\\varphi\n&=&\n\\int f({\\mathbb{I}},{\\mathbb{I}},{\\mathbb{I}},g_{4},g_{5},g_{6})\\,\n(\\chi_{\\f12}(g_{4}g_{5}g_{6})-2)\\,\n\\partial_{a}\\delta(g_{6})\n\\partial_{b}\\delta(g_{4})\n\\partial_{c}\\delta(g_{5}) \\nonumber\\\\\n&=&\n\\frac i8\\epsilon^{abc}\\chi_{\\f12}(\\sigma_{a}\\sigma_{b}\\sigma_{c})\\,f({\\mathbb{I}})\\,\\ne 0\\,.\n\\end{eqnarray}\nSo this triple-grasped flat state is a solution to three holonomy constraints, but the flatness around these three loops does not imply the flatness around the composite loop at the quantum level.\n\nAt the end of the day, to fully impose the flatness of the physical state, we require ``redundant'' holonomy constraints: imposing the holonomy constraints on an independent set of loops, as expected at the classical level, is not sufficient anymore at the quantum level to kill all the potential geometry excitations. This is especially relevant for the coarse-graining of the dynamics of loop quantum gravity. A common scenario is that we impose the flatness of the smallest loops, at the ``fundamental'' Planck scale, and that these will induce the flatness of the larger loops at larger scale. However, we see that it is not enough: there exist flat states (distributions peaked on the identity), solutions to the fundamental holonomy constraints, but which couple and entangle the fundamental loops through some grasping operators in such a way that they are not solutions anymore to the holonomy constraints around larger loops. In order to impose the complete flatness  of the state at all scales, we need to impose the flatness of all the loops at all scales, allowing for holonomy constraints around loops of arbitrary size similarly to the construction of Ising-like states for loop quantum gravity defined in \\cite{Feller:2015yta}.\n\n\n\n\\section*{Conclusion \\& Outlook}\n\nFollowing the logic of coarse-graining the quantum geometry of loop quantum gravity, we have extended spin network states by enriching \nthe structure of its vertices: we have attached to the vertices new local degrees of freedom so that they effectively represent coarse-grained regions of space with non-trivial gravitational field and geometry fluctuations. To this purpose, we have introduced a hierarchy of three generalization of spin network states, based on the coarse-graining through gauge-fixing approach developed in \\cite{Livine:2013gna}: folded, loopy and tagged spin networks.\n\nFolded spin networks on a graph $\\Gamma$ are spin network states with an arbitrary number of additional little loops attached to each vertex  of the graph and the extra information of a circuit, or folding tree, for each vertex describing how these little loops are connected to each other and to the edges of the graph. This is a mathematical reformulation of gauge-fixed spin networks where the original states live on finer graphs which have been coarse-grained down to $\\Gamma$.\nLoopy spin networks forget about the folding trees and describe spin networks on the base graph $\\Gamma$ plus the little loops attached to its vertices. These little loops describe local excitations of curvature and geometry, which can then propagate on top of the background geometry defined by the base graph. We have payed special attention to describing the cases of distinguishable and indistinguishable little loops, leading to a definition of a Fock space for loopy spin network with bosonic little loop excitations.\nTagged spin networks are the last step of coarse-graining and define a basis for non-gauge-invariant spin network states. Each vertex carries an extra spin, or tag, represented as living on an open leg attached the vertex, which defines the closure defect, that is how much the local gauge-invariance is broken. From the perspective of coarse-graining, this closure defect provides an overall measure of the non-trivial holonomies which have developed within the coarse-grained region or along the little loops attached to the vertex. Ultimately we would like to interpret this tag as some quasi-local mass or energy density for the (quantum) gravitational field fluctuations inside the coarse-grained region.\n\n\\medskip\n\nThese structures define a new framework for  loop quantum gravity, where we can implement and study its graph-changing dynamics while working on a fixed background graph. Indeed, starting from a given base graph $\\Gamma$, we represent any spin network states on a finer graph as a loopy spin network on the base graph plus little loops taking into account the more complex structure of the original graph. In a  way, we constantly coarse-grain spin networks to our chosen background graph and the little loops represent all the finer geometry excitations. This proposes a truncation of loop quantum gravity where the little loops are effective local degrees of freedom, which can propagate and interact on and with the base graph $\\Gamma$. For instance, we could choose as background graph, a regular 3d cubic lattice (e.g. as for defining Bianchi I cosmology as a truncation of loop quantum gravity\\cite{Alesci:2013xd}) or any other base graph suited for the case at study, and consider all the finer geometry fluctuations from an effective point of view as little loop inhomogeneities propagating on that base graph, as illustrated on fig.\\ref{fig:lattice}.\nThe little loops are the extra information carried by the loopy spin networks compared to a lattice formulation of loop quantum gravity. They encode an infinity of degrees of freedom attached to each vertex of the graph and describing excitations and fluctuations of the gravitational field.\n\\begin{figure}[h!]\n\\centering\n\\begin{tikzpicture}[scale=0.6]\n\n\\coordinate(a1) at (0,1);\n\\draw (a1) node {$\\bullet$};\n\\draw[in=-30,out=30,scale=1.8,rotate=45] (a1)  to[loop] (a1);\n\n\\coordinate(a2) at (1,2);\n\\draw (a2) node {$\\bullet$};\n\\draw[in=-30,out=30,scale=1.8,rotate=45] (a2)  to[loop] (a2);\n\\draw[in=-30,out=30,scale=1.8,rotate=135] (a2)  to[loop] (a2);\n\n\\coordinate(a3) at (3,3);\n\\draw (a3) node {$\\bullet$};\n\\draw[in=-30,out=30,scale=1.8,rotate=-135] (a3)  to[loop] (a3);\n\n\n\\coordinate(a4) at (3,0);\n\\draw (a4) node {$\\bullet$};\n\\draw[in=-30,out=30,scale=1.8,rotate=-135] (a4)  to[loop] (a4);\n\\draw[in=-30,out=30,scale=1.8,rotate=135] (a4)  to[loop] (a4);\n\n\\coordinate(a5) at (2,1);\n\\draw (a5) node {$\\bullet$};\n\\draw[in=-30,out=30,scale=1.8,rotate=135] (a5)  to[loop] (a5);\n\\draw[in=-30,out=30,scale=1.8,rotate=-135] (a5)  to[loop] (a5);\n\\draw[in=-30,out=30,scale=1.8,rotate=45] (a5)  to[loop] (a5);\n\n\\coordinate(a6) at (0,0);\n\\draw (a6) node {$\\bullet$};\n\\draw[in=-30,out=30,scale=1.8,rotate=135] (a6)  to[loop] (a6);\n\n\n\\draw (0,-.5) --(0,3.5);\n\\draw (1,-.5) --(1,3.5);\n\\draw (2,-.5) --(2,3.5);\n\\draw (3,-.5) --(3,3.5);\n\n\\draw (-.5,0) --(3.5,0);\n\\draw (-.5,1) --(3.5,1);\n\\draw (-.5,2) --(3.5,2);\n\\draw (-.5,3) --(3.5,3);\n\n\\end{tikzpicture}\n\n\\caption{Loopy spin networks on a 2d square lattice: the little loop excitations attached to its vertices represent finer graph structures which have been coarse-grained and allow to take into account graph changing dynamics while working on a fixed base graph. }\n\\label{fig:lattice}\n\n\\end{figure}\n\nRe-introducing in such a way a background structure  offers a perfect setting for studying the coarse-graining of the dynamics of loop quantum gravity.\nIn a sense, we have split the gravitational field degrees of freedom into a dynamical background geometry on a fixed graph and localized fluctuations of geometry, which can be though of as higher energy or finer scale excitations. As we coarse-grain, structures of the base graph will become little loops.\n\n\\medskip\n\nWe have also studied in great detail how to implement the dynamics of BF theory and defined suitable Hamiltonian constraints that select ultimately the flat state as unique physical state. In particular, it should kill any local degree of freedom and project out all the potential little loop excitations (or non-trivial tags). We faced two subtleties. First, the holonomy operators of loop quantum gravity now act also as creation and annihilation operators for the little loops. Second we identified an infinity of solutions to the holonomy constraints, defined as higher derivative of the $\\delta$-distribution or equivalently by acting with grasping operators on the totally flat state. These are still peaked exclusively on the flat connection, but introduce some non-trivial correlation and entanglement between the little loop excitations. This comes from the fact that the dimension of the intertwiner space at a vertex grows with the number of little loops and these ``spurious'' solutions can be interpreted as non-trivial intertwiners between flat little loops. We have introduced Laplacian constraints to supplement the holonomy constraints and decouple the loops, allowing us to get rid of this tower of higher order flat states and get finally as wanted a unique physical state. Nevertheless, this Hilbert space of ``grasped flat states'' could prove an interesting sector of loop quantum gravity, with an infinity of  degrees of freedom, especially as a toy model to investigate the continuum limit of the theory.\n\n\\begin{figure}[h!]\n\\centering\n\n\\begin{tikzpicture}[scale=1]\n\n\n\n\\node[draw,circle,dotted,fill=gray!30,scale=2] (A) at (0,0) {};\n\\node[draw,circle,dotted,fill=gray!30,scale=2] (B) at (2,0) {};\n\\draw[->] (A) to[bend left=20] (B);\n\\draw[->] (A) to[bend right=20] (B);\n\n\\draw[->,>=stealth,thick] (3,0) -- (4,0);\n\n\\node[draw,circle,dotted,fill=gray!30,scale=2] (C) at (5,0) {};\n\\node[draw,circle,dotted,fill=gray!30,scale=2] (D) at (7,0) {};\n\n\\coordinate(O) at (6,0);\n\\draw[red] (O) node {$\\bullet$};\n\n\\draw(C) -- (O) node[midway,below]{$j_{e}^{s}$};\n\\draw[->] (O) -- (D) node[midway,below]{$j_{e}^{t}$};\n\\draw[red,thick] (O) -- ++ (0,0.3) node[above] {$k_{e}$};\n\n\\draw[->,>=stealth,thick] (-1,0) -- (-2,0);\n\n\\node[draw,circle,dotted,fill=gray!30,scale=2] (E) at (-5,0) {};\n\\node[draw,circle,dotted,fill=gray!30,scale=2] (F) at (-3,0) {};\n\n\\coordinate(P) at (-4,0);\n\\draw[red] (P) node {$\\bullet$};\n\n\\draw(E) -- (P) node[midway,below]{$j_{e}^{s}$};\n\\draw[->] (P) -- (F) node[midway,below]{$j_{e}^{t}$};\n\\draw[in=-40,out=40,scale=1.8,rotate=90,red,thick] (P)  to[loop] node[midway,above] {$k_{\\ell}$}(P);\n\n\\end{tikzpicture}\n\n\\caption{Two regions, to be coarse-grained, can be related by several links. These links should in turn be coarse-grained into a single edge. However since the holonomies along the original links might have been different, the effective edge should be able to carry some data reflecting this non-trivial curvature. Identifying a spin network edge as a bivalent vertex, it is natural to introduce tagged edges, where the tag $k_{e}$ creates a trivalent intertwiner with the spins at the source and target of the edge, $j_{e}^{s}$ and $j_{e}^{t}$, which can now be different. Another natural possibility is to attach a loop or tadpole to the edge to account for the non-trivial curvature between the two regions, which would also create a defect along the edge leading to different spins at the source and target of the edge. The difference between these two generalizations of spin network edges is that the loop implies that the midway vertex now has a non-trivial volume  while the tag does not.\n}\n\\label{fig:tagged-edge}\n\n\\end{figure}\n\nAfter having set in the present work the kinematics of loopy and tagged spin networks and shown how to implement the topological dynamics of BF theory, the next step will be to define some non-topological loop quantum gravity dynamics, coupling the degrees of freedom on the base graph to the little loops, and study its coarse-graining flow.\n\nThere is however another generalization of spin networks worth investigating before moving on to the dynamics of the theory. We have focussed up to now on coarse-graining bounded regions of spaces into effective vertices and therefore introduced the notions of loopy and tagged spin network states with extra structure and data attached to the vertices. We have dressed the graph's nodes, so shouldn't we also consider dressing the its links? Comparing to Feynman diagrams in quantum field theory, we have renormalized the interaction vertices but we should also describe how to renormalize the propagator. Indeed, after partitioning the graph spanning the 3d space into bounded regions and coarse-graining each region to a single vertex, there is generically several edges linking these effective vertices. We need to bundle them together into a single new effective edge. Since those edges to coarse-grain link the same two regions and thus form loops, there is naturally non-trivial holonomies which can formed between the two regions and therefore the new effective edge should be able to carry some notion of  curvature. One way to proceed is to consider all the edges of a spin network state as bivalent intertwiners, recoupling between the two spins living at its source and at its target. Such intertwiners are trivial and the source and target spins are identified. But as curvature builds up, it is natural to allow this intertwiner to acquire a tag or little loops, accounting for the curvature excitations carried by the edge. For instance, as shown on fig.\\ref{fig:tagged-edge}, a tag would turn the bivalent intertwiner into a trivalent intertwiner allowing the source and target spins to differ. It would be interesting to develop the notion of spin networks with tagged links. And it could be relevant to compare such spin networks with both tagged vertices and edges to projected $\\mathrm{SL}(2,{\\mathbb C})$ spin networks\\cite{Livine:2002ak,Dupuis:2010jn}, which allow for both features of non-vanishing closure defect at vertices and non-matching spins along edges, and which are the basic states and building blocks for the EPRL-FK spinfoam models for loop gravity path integrals\\cite{Engle:2007wy,Geloun:2010vj,rovelli2014quantum}.\n\nFinally, tags and little loops closely resemble particle insertions on spin network states and it would be enlightening to understand if they can truly be interpreted as matter field degrees of freedom, especially from the perspective of working out the continuum limit of loop quantum gravity as a quantum field theory.\n \n\n\\section*{Acknowledgments}\n\nC.C. would like to thank Michel Fruchart and Dimitri Cobb for their keen insights and useful discussions with them.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Section I}\n\n\tQuantum theory as a theory of measurement was promulgated by Bohr\\cite{Bohr}, and this point of view has its adherents today\\cite{Peres-Fuchs}.  But, the success of the theory suggests that it could be more, a theory of reality.  Einstein wrote Schr\\\"odinger\\cite{Einsteinletter} that, if there were no more complete description of nature than that envisaged by Bohr, ``...then physics could only claim the interest of shopkeepers and engineers; the whole thing would be a wretched bungle.\"  Bell made much the same point, writing\\cite{Bell} `` To restrict quantum mechanics to be exclusively about piddling laboratory experiments is to betray the great enterprise.\"\n\t\n\tThe lack in standard quantum theory has been called the ``measurement problem,\" but I prefer to call it the ``reality problem.\"  I believe it is most precisely characterized as a failure to give a satisfactory response to the following operational requirement for a well-defined quantum theory of reality. You are given, at time $t=0$, a  state vector for an arbitrary physical system, together with its Hamiltonian, but no other information.  For $t>0$, specify (i.e., give a procedure for how to identify) the realizable states and their probabilities of realization.  \n\t\t\n\tNot only Bohr's version of quantum theory, but more recent sophisticated attempts, such as many worlds\\cite{lev}, environmental decoherence\\cite{zehzurek}, consistent histories\\cite{griffithshartlegellmannomnes}, have not succeeded so far at providing such a specification\\cite{mekent}.  They are successfully applied to many important and interesting examples, but these examples require additional information ({\\it this} is the nature of the system,  {\\it that} is the apparatus,  {\\it there} is the environment...) which lie outside the theory.   \n\t\n\tThe reality problem {\\it has} been solved by enlarging quantum theory so that it describes state vector collapse as a physical process.  The Continuous Spontaneous Localization (CSL) model\\cite{me, GPR, myreview}  provides a modified Schrodinger equation.   It describes how, in addition to their usual interactions, the particles of the world interact with a posited classical field $w({\\bf x}, t)$ resulting in a nonunitary state vector evolution (which changes the norm of the state vector). What are the realizable states?  Each such state vector, corresponding to a different \n$w({\\bf x}, t)$ (drawn from the class of white noise ``functions), is realizable.  What are the associated probabilities?  CSL provides a second equation, the prescription that the probability of occurrence of a state vector (the probability of occurrence of its associated $w({\\bf x}, t)$) is proportional to its squared norm.  In this straightforward manner, the operational requirement cited above is satisfied.  Each of these CSL states accurately describes (except for occasional brief moments) the macroscopic world we see around us.  The predictions agree with all presently performed experiments (but there are testable differences from standard quantum theory).  The CSL model is reviewed in section II.  \n\n\tI  have previously presented a model\\cite{Ways, mekent, myreview}  called the ``Index\" model,  but  here  called  the ``Completely Quantized Collapse\" (CQC) model.  It is reviewed in section III.  It utilizes a formalism invented to model continuous measurements\\cite{contmeas}.   CQC is completely equivalent to CSL in its specification of the realizable states and their probabilities, in the following manner.\n\t\t\n\tIn CQC, the classical field is replaced by a quantized field $W({\\bf x}, t)$ which, like the classical field $w({\\bf x}, t)$ of CSL, commutes with itself, $[W({\\bf x}, t), W({\\bf x}', t')]=0$.  There is a conjugate field, $\\Pi({\\bf x}, t)$,  which also commutes with itself, but which satisfies $[W({\\bf x}, t), \\Pi ({\\bf x}', t')]=\ni\\delta(t-t')\\delta({\\bf x}-{\\bf x}')$.  The basis of eigenstates of $W({\\bf x}, t)$ for every ${\\bf x}$, $t$, is denoted $|w\\rangle$.   It satisfies $W({\\bf x}, t)|w\\rangle=w({\\bf x}, t)|w\\rangle$, i.e., there is a different eigenstate for every possible $w({\\bf x}, t)$ function. There is a single state vector, called here the ``ensemble vector,\"  which evolves unitarily (the Hamiltonian is given).   When the ensemble vector is expanded as a Schmidt decomposition in the ``preferred basis\" $|w\\rangle$, the particle state in the direct product with $|w\\rangle$  is the CSL state associated with $w({\\bf x}, t)$: schematically, \n\n\\[ |\\hbox{ensemble vector}, t\\rangle=\\sum_{\\hbox{all}\\thinspace\\thinspace w}|w\\rangle|\\psi_{\\hbox{csl}}, t\\rangle_{w}.  \n\\]\n\n\\noindent  The usual quantum rule for probabilities (the squared norm of a state in the superposition) gives the CSL probability of realization of that state.  \n\n\tThe measurement (reality) problem in standard quantum theory has often been phrased in terms of difficulties associated with the ill-defined collapse postulate, nowhere more succinctly than Bell's Boolean phrase``And is not Or,\" i.e., Schr\\\"odinger's equation gives a sum of terms but we observe one term or another.  CSL provides a resolution of the problem by individually, dynamically, providing each term we can observe.  However, CQC gives a sum of terms: how can that be considered as a resolution of the problem?   \n\t\n\tA colleague once observed to me that construction of CQC goes against everything I have worked for, the resolution of the reality problem by dynamical collapse.  However, what I have worked for is to make a well-defined quantum theory of reality\\cite{Pearle'67}, i.e., one which satisfies the aforementioned operational requirement.  CSL satisfies this requirement, but so does CQC.  The terms in the superposition which make up the ensemble vector are precisely defined (by the orthogonal preferred basis $|w\\rangle$), do not interfere (they obey a superselection rule), \nare macroscopically sensible (because the associated CSL states are), and occur with the correct amplitudes (which give the correct probabilities). In his famous ``cat paradox\" paper\\cite{Schr}, Schr\\\"odinger referred to the state vector as the ``prediction catalog\" \nor the  ``expectation catalog\" because, in Bohr's quantum theory, it lists what can be expected to occur if a measurement is performed.  The CQC state vector may, likewise, be called a ``reality catalog.\"\n\nHowever, the justification here is greater than Schr\\\"odinger's.  His usage requires judgment {\\it outside} the theory as to {\\it when} the state vector has evolved sufficiently to be considered a catalog, and as to {\\it which} are the appropriate states in the superposition (i.e., which basis) to consider as making up the catalog.   In CQC, these criteria lie {\\it inside} the theory, since the ensemble-vector may {\\it always} be considered a catalog, and the appropriate states are {\\it labeled} by the preferred basis $|w\\rangle$.  Roughly speaking, CSL provides a vertical catalog (Or), and CQC provides a horizontal catalog (And), but it is the same catalog.  \nIf one has a precise rule for identifying a superposition of states at time $t$ which never interfere thereafter,  it is a matter of indifference whether one removes them from the superposition and follows their separate evolutions that way, or leaves them in the superposition and follows their joint evolution that way.\n\nAs  previously pointed out\\cite{mekent}, one may regard CQC as a model for a satisfactory resolution of other attempted interpretations of quantum theory\\cite{GS}.  The superposed states making up the ensemble vector could be thought of as many worlds (although it would seem that there would then be no reason to insist upon such an interpretation).  If the $W$-field is regarded as the environment,  then CQC could be regarded  as  describing  environmental decoherence.  The evolving superposed states making up the ensemble vector could be taken to represent a single family of consistent histories\\cite{kent}.   The Schmidt decomposition of the ensemble vector is precisely what the modal interpretation requires. What is currently missing from these approaches, and what CQC provides, is a subsystem with a well-defined ``universal basis\" (here the $|w\\rangle$ basis vectors) whose Schmidt decomposition partners make macroscopic sense.  One might hope that such a basis may be found arising in a natural way from a  physical entity, whence perhaps CSL\/CQC and all these interpretations may merge (see the last two paragraphs of this introduction for further remarks on this). \n\n\tNow I turn to the topics which will occupy the rest of this paper. Because collapse narrows wave functions,  the energy of particles in CSL increases with time.  This growth of energy of particles provides for experimental tests of the model\\cite{exptlpapers} but, from a theoretical point of view, it is desirable to have a conserved energy.  Recently, I showed how to define energy associated with the $w$-field in CSL so that the first moment of the total energy (particles plus $w$-field) is conserved\\cite{energy}.  However, energy conservation requires all moments of the energy to be conserved.  The formalism of CQC facilitates that demonstration (which can be converted to CSL language with less transparency).  Because the ensemble vector evolution in CQC is a Hamiltonian evolution, the Hamiltonian is a conserved energy associated with the ensemble. \n\n\tThe Hamiltonian does not commute with $W({\\bf x}, t)$  since it is composed of the $W$ {\\it and} $\\Pi$-field operators. Thus one cannot associate a conserved energy with an individual realizable state labeled by eigenvalues of $W$ . Nonetheless, it is of interest to see how energy fares for the ensemble of  possible states, for example, how the ensemble energy of the $W$-field can go negative to compensate for the energy increase of the particles whose wavefunctions are narrowed by collapse.  The discussion of energy conservation takes place in section IV.\n\n\tAnother example of the utility of the CQC model is considered in section V.   Because the  $W$-field energy has the whole real line for its spectrum, it is possible to construct its self-adjoint conjugate, a  time-operator $T$, built solely out of the $W$ and $\\Pi$-field operators. One may consider that time should be intimately related to the realization of physical states, since without \tsuch a realization there are no events and without events there is no time.  This, and the fact that, like the Matterhorn, it is ``there\" (i.e., very few viable physical theories possess an unbounded Hamiltonian) motivate looking at $T$.\t\n\t\t\n\t$T$'s ensemble probability distribution is examined. $T$'s variance diminishes with time.   $T$'s mean, roughly speaking, is a ``center of time,\"  i.e., the average time over all space, and over time from 0 to $t$, weighted by the square of the particle number density.  Thus, if the particle Hamiltonian vanishes so the only particle evolution is collapse, which does not change the ensemble spatial particle density, the mean of $T$ is $t\/2$.  If, however, the particles have a high clumping rate (for example, due to gravitational interaction), the spatial average of the square of the particle number density is largest most recently, and the mean of $T$ is closer to $t$. \n\n\tOne of the motivations of this work is to provide a bridge from dynamical collapse models to other interpretations which seek a well-defined quantum theory based upon the standard quantum formalism alone (although they may be satisfied with just accounting for observation, not reality), rather than the altered  formalism of collapse models. Now, what I consider to be a pressing problem associated with CSL is that it is phenomenological. (Some have said ``ad hoc,\" but that is not appropriate terminology since ad hoc means ``for this case only,\" whereas CSL applies to all cases, all physical situations: in fact, the label ``ad hoc\" more appropriately belongs to these other interpretations  to the extent that  they require additional information appropriate to each physical situation). It could be that CSL dynamics may arise from a larger structure which generates quantum theory itself, as has been suggested by Adler\\cite{Adler}.  In this paper, we consider another possibility.\n\t\n\tCSL depends upon interaction of a posited field's interaction with particles, but no identification of that field with a physical entity has been made.  A purpose in discussing CQC  is that, since it is formulated completely quantum mechanically, that may make it easier to identify the universal $W$-field with a quantized physical entity which arises naturally in another context.  In that case, CQC would appear as based upon the standard quantum formalism alone.  \n\t\n\tHowever, it is not so easy to find a candidate for a universal fluctuating field with the desired properties. There is experimental evidence\\cite{exptlpapers} that the coupling of  $W$ is to the mass-density of particles, which strongly suggests a gravitational connection, a proposal which has a long history\\cite{gravity, pearlesquires}.  This sugggests looking for a mechanism whereby gravitational excitations could naturally act like the $W$-field.  \n\n\tBy way of illustration, Section VI contains two models for $W$ with gravitational overtones, neither to be taken too seriously.   In one, $W$ is constructed out of many scalar quantum fields (of Planck mass), each interacting only for a brief interval  (the Planck time)  with the particle's mass-density.  The other utilizes aspects of a previous gravitational classical model\\cite{pearlesquires} of $W$ and crudely pays homage to spin-network models\\cite{smolinseth}.  It  assumes that space consists of Planck-volume cells, each containing a bound spin 1\/2 entity with two possible energy states equal to $\\pm$Planck mass.  A spin occasionally breaks free to briefly interact with a thermal bath, as well as gravitationally with the particle's mass-density, before it is bound once more.  In this case,  $W({\\bf x},t)$ is identified with the free mean spin, in a small volume surrounding ${\\bf x}$,  during the small time interval surrounding $t$,.  These examples suggest that what is required for $W$ is a quantity which \nhas the freedom to vary independently over all space-time and for which the state vector describes its prior space-time values . This is unlike a conventional field, which can vary independently (in magnitude and time derivative) only on a single spacelike hypersurface, and whose present values are all that is described by the state vector.  \n\n\t\t\t\n\\section{CSL}\n\\subsection{The Simplest Model}\n\tThe simplest CSL model, which will supply most of the illustrative calculations in this paper, describes collapse toward an eigenstate of the operator $A$. The state vector evolution, governed by a function $w(t)$ of white noise class, is given by \n\\begin{equation}\\label{1} \n|\\psi, t\\rangle_{S}={\\cal T} e^{-\\int_{0}^{t}dt'\\{iH_{A}+(4\\lambda)^{-1}[w(t')-2\\lambda A]^{2}\\}}|\\phi\\rangle.\n\\end{equation}\n\\noindent In Eq.(1), ${\\cal T}$ is the time-ordering operator, $H_{A}$ is the (usual) Hamiltonian of the system, $|\\phi\\rangle=|\\psi,0\\rangle$ is the initial state vector, the subscript $S$ denotes the Schr\\\"odinger-picture state vector, and the parameter $\\lambda$ governs the collapse rate.  Eq. (1) is the solution of the modified Schr\\\"odinger equation:\n\\begin{equation}\nd|\\psi, t\\rangle_{S}\/dt=\\{-iH_{A}-(4\\lambda)^{-1}[w(t)-2\\lambda A]^{2}\\}|\\psi, t\\rangle_{S}.\\nonumber\n\\end{equation}\n\\noindent However, only the solution form (1) shall be utilized in this paper.  \n\n\tCSL is completed by specifying the probability that a particular $w$, lying in the interval $(w(t), w(t)+dw),$ is realized in nature:\t\n\\begin{equation}\\label{2}\n\tP_{t}(w)Dw=_{S}\\negthinspace\\langle\\psi,t|\\psi,t\\rangle_{S}Dw.\n\\end{equation}\n\\noindent  In Eq.(2), and for all integrations, one may discretize time, i.e., with $t_{j}\\equiv jdt$, each $w(t_{j})$ is considered to be an independent variable ($-\\infty<w(t_{j})<\\infty$), and \n$Dw\\equiv\\prod_{j=1}^{t\/dt}\\{dw(t_{j})\/[dt\/2\\pi\\lambda]^{1\/2}\\}$, with the eventual limit $dt\\rightarrow 0$.  \n\n\tIn  the ``collapse interaction picture,\" which is what is mostly employed in this paper,  the state vector is \n\\begin{subequations}\n\\begin{eqnarray}\\label{3} \n|\\psi, t\\rangle&\\equiv&\\exp{iH_{A}t}|\\psi, t\\rangle_{S}\\\\\n&=&{\\cal T} e^{-(4\\lambda)^{-1}\\int_{0}^{t}dt'[w(t')-2\\lambda A(t')]^{2}}|\\phi\\rangle, \n\\end{eqnarray}\n\\end{subequations}\t\n\\noindent where (3b) follows from Eq.(1), and $A(t')\\equiv\\exp(iH_{A}t)A\\exp(-iH_{A}t)$. \nFrom Eq.(3a), the probability rule, Eq.(2), is the same, except that the subscript $S$ is dropped.   \n\n\tThe density matrix describing the ensemble in the interaction picture follows from Eqs.(2), (3b):\n\\begin{subequations}\\label{4} \n\\begin{eqnarray}\n\\rho(t)&&\\equiv\\int P_{t}(w)Dw\\frac{|\\psi, t\\rangle\\langle\\psi, t|}{\\langle\\psi, t|\\psi, t\\rangle}\n=\\int Dw|\\psi, t\\rangle\\langle\\psi, t| \\\\\n&&={\\cal T} e^{-(\\lambda\/2)\\int_{0}^{t}dt'[A_{L}(t')-A_{R}(t')]^{2}}|\\phi\\rangle\\langle\\phi |\n\\end{eqnarray}\n\\end{subequations}\t\n\\noindent where, in an expansion of the exponent in Eq.(4b), $A_{L}(t')$ $(A_{R}(t')$) appears to the left (right) of $|\\phi\\rangle\\langle\\phi |$, and is  time-ordered (reverse-time-ordered) by ${\\cal T}$. \n\\subsection{Proof of Collapse When $H_{A}=0$} \n\t\n\tIf $H_{A}=0$ (or if the collapse rate is so fast that, over the time interval $(0,t)$, we can well approximate $A(t)\\approx A$), then the state vector dynamically collapses, as $t\\rightarrow \\infty$, to one of the eigenstates of A. Let $|\\phi\\rangle=\\sum_{n}\\alpha_{n}|a_{n}\\rangle$, where \n$A|a_{n}\\rangle=a_{n}|a_{n}\\rangle$.  An intimation of the collapse behavior in this case may be seen from the density matrix (4b):\n\\begin{equation}\n\\rho(t)=\\sum_{n,m} \\alpha_{n}\\alpha^{*}_{m}e^{-(\\lambda t\/2)(a_{n}-a_{m})^{2}}|a_{n}\\rangle\\langle a_{m}|,\\nonumber\n\\end{equation}\n\\noindent since the off-diagonal elements vanish as $t\\rightarrow \\infty$, and the diagonal element magnitudes are the ones obtained from $|\\phi\\rangle$ using the ``collapse postulate\" of standard quantum theory.  However, this ensemble density matrix behavior does not prove collapse in the individual case, since many different ensembles can have the same density matrix. \n \n\tHere is a proof of collapse not previously given. Eqs.(3),(2) become respectively:\n\\begin{subequations}\\label{5} \n\\begin{eqnarray}\t\n|\\psi, t\\rangle&=&\\sum_{n} \\alpha_{n}e^{-(4\\lambda)^{-1}\\int_{0}^{t}dt'[w(t')-2\\lambda a_{n}]^{2}}|a_{n}\\rangle\\\\\nP_{t}(w)&=&\\sum_{n} |\\alpha_{n}|^{2}e^{-(2\\lambda)^{-1}\\int_{0}^{t}dt'[w(t')-2\\lambda a_{n}]^{2}}\n\\end{eqnarray}\n\\end{subequations}\n\n\t It shall now be shown that the only $w(t)$'s of non-zero measure are those for which the asymptotic ($t\\rightarrow\\infty$) time-average of $w(t)$ is $2\\lambda a_{n}$.  To see this, consider the joint probability of $w$ and $a\\equiv (2\\lambda t)^{-1}\\int_{0}^{t}dt'w(t')$:\n\\begin{equation}\\label{6} \nP_{t}(w,a)Dwda=2\\lambda tda\\delta\\big[\\int_{0}^{t}dt'w(t')-2\\lambda ta\\big]P_{t}(w)Dw,\n\\end{equation}\n\\noindent where $P_{t}(w)Dw$ is the probability of $w$ and $2\\lambda tda\\delta$ is the conditional probability, given $w$, that $w$'s time-average is $2\\lambda a$.  To find the probability distribution  $P_{t}(a)$ of  $a$, one must integrate (6) over $Dw$.  To do so, make the replacement  \n$\\delta(z)=(2\\pi)^{-1}\\int d\\omega \\exp i\\omega z$, and then the resulting gaussian integrals over $Dw$ \nmay be performed.  Next, the (gaussian) integral over $\\omega$ can be performed, resulting in:\n\\begin{equation}\\label{7} \nP_{t}(a)da=\\sum_{n} |\\alpha_{n}|^{2}\\frac{e^{-2\\lambda t (a- a_{n})^{2}}}{\\sqrt{\\pi\/2\\lambda t}}da\\quad_\n{\\overrightarrow{t\\rightarrow\\infty} }\\quad\\sum_{n} |\\alpha_{n}|^{2}\\delta (a- a_{n})da.\n\\end{equation}\n\\noindent  Thus,  the asymptotic ensemble probability is completely accounted for by $a$ taking on \nonly the values $a_{n}$, and the probability of an $a_{n}$ is $|\\alpha_{n}|^{2}$.  \n\n\tIt remains to show that (5a) asymptotically describes collapse.  Let $w(t)$ satisfy the constraint  $C_{a}$ defined as $\\lim_{T\\rightarrow\\infty}(2\\lambda T)^{-1}\\int_{0}^{T}dt'w(t')= a$.  Taking the asymptotic limit of (6) and then integrating over an infinitesimal range about $a$ followed by integrating over $w$, it follows from (6), (7) that \t\n\\[  |\\alpha_{n}|^{2}\\delta_{a,a_{n}}= \n \\int Dw\\int_{a-\\epsilon}^{a+\\epsilon}da P_{\\infty}(w,a)=\\int_{C_{a}} Dw P_{\\infty}(w).\n\\]\n\t\n\tApplying this result to (5b) yields\n\\[  \\int_{C_{a}} Dw e^{-(2\\lambda)^{-1}\\int_{0}^{\\infty}dt'[w(t')-2\\lambda b]^{2}}=\\delta_{a,b}.\n\\]\n\\noindent Since each term in the integral here is non-negative, if $a\\neq b$, for {\\it each} $w(t')$ obeying \n the constraint $C_{a}$ (i.e., for {\\it each} $w(t')$ which asymptotically equals $2\\lambda a$ plus a \n fluctuating term which averages out to 0), then $\\ exp-(2\\lambda)^{-1}\\int_{0}^{\\infty}dt'[w(t')-2\\lambda \n b]^{2}=0$.  \n \n \tThis result can be applied to Eq.(5a), where the gaussians are of the same form (except $4\\lambda$ replaces  $2\\lambda$).  Thus, for each such $w(t)$ which obeys the constraint $C_{a}$, with $a\\neq a_{n}$, the state vector asymptotically vanishes:  that doesn't matter since, by Eq.(5b), the probability of its occurrence vanishes also.  When $w(t)$ obeys the constraint $C_{a_{m}}$, all terms  in (5a) but the $m$th asymptotically vanish, $|\\psi,\\infty \\rangle\\rightarrow\\sim |a_{m}\\rangle$: collapse occurs. \n \n\\subsection{Full-Blown CSL}\n\n\tThe CSL proposal for a serious collapse model, applicable to all nonrelativisitic systems, involves multiple $A$'s, each with its own $w$.  That is, collapse is toward the joint eigenstates of an operator $A({\\bf x})$ for all ${\\bf x}$, and $w(t)$ is replaced by the random field $w({\\bf x},t)$.  The choice of $A$ is\\cite{me,pearlesquires}:\n\\begin{equation}\\label{8} \nA({\\bf x})\\equiv \\sum_{i}\\frac{m_{i}\/m_{0}}{(\\pi a^{2})^{3\/4}}\\int d{\\bf z}e^{-(2a^{2})^{-1}({\\bf x}-{\\bf z})^{2}}\\xi_{i}^{\\dagger}({\\bf z})\\xi_{i} ({\\bf z}).\n\\end{equation}\n\\noindent In Eq.(8), $\\xi_{i} ({\\bf z})$ is the annihilation operator at location ${\\bf z}$ for a particle of mass $m_{i}$ ($m_{0}$ is the proton mass), and $a$ (and $\\lambda$) are parameters from the seminal GRW model\\cite{GRW}, with chosen values $a\\approx 10^{-5}$cm and  $\\lambda\\approx 10^{-16}$sec$^{-1}$.  Thus,  since $\\xi_{i}^{\\dagger}({\\bf z})\\xi_{i} ({\\bf z})$ is the particle number density of the $i$th particle type at ${\\bf z}$, $A({\\bf x})$ is essentially  proportional to the mass density inside a sphere of radius $a$ centered at {\\bf x}.  The CSL dynamical equation which replaces Eq.(3b) is:\n\\begin{equation}\\label{9} \n|\\psi, t\\rangle={\\cal T} e^{-(4\\lambda)^{-1}\\int_{0}^{t}dt'd{\\bf x}[w({\\bf x},t')-2\\lambda \nA({\\bf x},t')]^{2}}|\\phi\\rangle\n\\end{equation}\n\\noindent (where $A({\\bf x},t')\\equiv\\exp(iH_{A}t)A({\\bf x})\\exp(-iH_{A}t)$), i.e., time is replaced by space-time.  The probability rule is unchanged, $P_{t}(w)Dw=\\langle \\psi,t|\\psi, t\\rangle Dw$, but space-time is discretized for integration, so \n$Dw\\equiv\\prod_{j, k}\\{dw({\\bf x}_{k},t_{j})\/[dtd{\\bf x}\/2\\pi\\lambda]^{1\/2}\\}$,\n\n\\section{CQC}\n\n\\subsection{Simplest Model}\n\n\tStart by introducing creation and annihilation operators $b^{\\dagger}(\\omega)$, $b(\\omega)$ for $-\\infty<\\omega<\\infty$, which obey the usual commutation rules, \n\\begin{equation}\n[b(\\omega),b(\\omega ')]=0, \\quad [b^{\\dagger}(\\omega),b^{\\dagger}(\\omega)']=0, \\quad\n[b(\\omega),b^{\\dagger}(\\omega ')]=\\delta(\\omega-\\omega ')\\nonumber.\n\\end{equation}\n\\noindent Then, the operator $W(t)$ (the quantized equivalent of the CSL $w(t)$) and its conjugate \n$\\Pi (t)$ may be defined as\n\\begin{subequations}\\label{10} \n\\begin{eqnarray}\nW(t)&\\equiv&(\\lambda\/2\\pi)^{1\/2}\\int_{-\\infty}^{\\infty}d\\omega [e^{-i\\omega t}b(\\omega)+\ne^{i\\omega t}b^{\\dagger}(\\omega)],\\\\\n\\Pi(t)&\\equiv&i\/(8\\pi\\lambda)^{1\/2}\\int_{-\\infty}^{\\infty}d\\omega [-e^{-i\\omega t}b(\\omega)+\ne^{i\\omega t}b^{\\dagger}(\\omega)]\n\\end{eqnarray}\n\\end{subequations}\n\\noindent which are readily shown to satisfy\n\\begin{equation}\\label{11}\n[W(t),W(t')]=0, \\quad [\\Pi(t),\\Pi(t')]=0, \\quad  [W(t),\\Pi(t')]= i\\delta(t-t')\n\\end{equation}\n\\noindent The vanishing of the first two commutators in Eqs.(11) result from cancellation of \nthe positive frequency contribution in Eq.(10) by the negative frequency contribution.  (Incidentally, a free tachyonic \nfield also has equal positive and negative energy spectra, so it too can be quantized to have vanishing self-commutator\\cite{tachyon}).  \n\t\n\tThe joint eigenstate of $W(t)$ ($\\Pi (t)$) is denoted $|w\\rangle$ ($|\\pi\\rangle$), where the label represents a particular function $w(t)$ ($\\pi(t)$) of white noise class.    It satisfies $W(t)|w\\rangle=w(t)|w\\rangle$ ($\\Pi(t)|\\pi\\rangle=\\pi(t)|\\pi\\rangle$). The range of $t$ is  \n$(-\\infty,\\infty)$, and the states are normalized so that $\\int_{-\\infty}^{\\infty}D'w|w\\rangle\\langle w|=1$ ($\\int_{-\\infty}^{\\infty}D'\\pi|\\pi\\rangle\\langle \\pi|=1$).  $D'$ differs from  the $D$  employed for CSL in that it involves the product of $dw(t)$'s for the infinite range of $t$.  To discretize,  write\n $|w\\rangle=\\prod_{j=-\\infty}^{\\infty}|w(t_{j}\\rangle$, with \n $ \\langle w(t_{j})| w'(t_{j})\\rangle=\\delta[w(t_{j})-w'(t_{j})]\\sqrt{2\\pi\\lambda\/dt}$.\n\n\tIt follows from Eqs.(11) that, for an arbitrary vector  $|\\Phi\\rangle$, \n\\begin{equation}\\label{12}\n\\langle w|\\Pi(t)|\\Phi\\rangle=\\frac{1}{i}\\frac{\\delta}{\\delta w(t)}\\langle w|\\Phi\\rangle.\n\\end{equation}\n\\noindent Here, $\\delta\/\\delta w(t)$ is the functional derivative.  In time-discretized calculations, \n$\\delta\/\\delta w(t)\\rightarrow (dt)^{-1}\\partial \/\\partial w(t_{j})$.\n\n\tThe vacuum state $|0\\rangle$ satisfies $b(\\omega)|0\\rangle=0$ as usual.  Since, from Eqs.(10), \n$ W(t)+2i\\lambda\\Pi(t)$ is an integral over $b(\\omega)$,  Eqs.(10), (12) imply  \n\\begin{equation}\n\\langle w|W(t)+2i\\lambda\\Pi(t)|0\\rangle=[w(t)+2\\lambda\\delta\/\\delta w(t)]\\langle w|0\\rangle=0\\nonumber\n\\end{equation}\n\\noindent and a similar equation for $\\langle \\pi|0\\rangle$, whose solutions are \n\\begin{equation}\\label{13}\n\\langle w|0\\rangle=e^{-(4\\lambda)^{-1}\\int_{-\\infty}^{\\infty}dt'w^{2}(t')},\\quad \n\\langle \\pi|0\\rangle=e^{-\\lambda\\int_{-\\infty}^{\\infty}dt'\\pi^{2}(t')}.  \n\\end{equation}\n\\noindent The normalization in Eqs.(13) ensure that $\\int D'w|\\langle w|0\\rangle|^{2}=\n\\int D'\\pi |\\langle \\pi|0\\rangle|^{2}=\\langle 0|0\\rangle=1$.\n\t\n\tThe $W$-field Hamiltonian, the energy operator which generates time-translations of $W(t)$, $\\Pi (t)$, is \n\\begin{equation}\\label{14}\nH_{w}\\equiv\\int_{-\\infty}^{\\infty}d\\omega \\omega b^{\\dagger}(\\omega)b(\\omega)=\n\\int_{-\\infty}^{\\infty}dt'\\dot{W}(t')\\Pi(t'), \n\\end{equation}\n\\noindent which follows from Eqs.(10).  (By adding an infinitesimal imaginary part to $\\omega$ \nin Eqs.(10), one can arrange that $W(t)$, $\\Pi(t)$ vanish at $t\\rightarrow \\pm\\infty$, so one can freely integrate Eq.(14) by parts, to put the time derivative on $\\Pi$.  One can also change the \norder of $\\dot{W}(t')$ and $\\Pi(t')$ in Eq.(14), since $\\int dt'[\\dot{W}(t'),\\Pi(t')]=0$.)\n\n\tThe ``ensemble vector\" of CQC in the collapse interaction picture is defined as:\n\\begin{subequations}\n\\begin{eqnarray}\\label{15}\n|\\psi,t\\rangle&\\equiv&{\\cal T}e^{-2i\\lambda\\int_{0}^{t}dt''A(t'')\\Pi(t'')}|0\\rangle|\\phi\\rangle\\\\\n&=&\\int D'w|w\\rangle\\langle w|\n{\\cal T}e^{-2i\\lambda\\int_{0}^{t}dt''A(t'')\\Pi(t'')}|0\\rangle|\\phi\\rangle\\\\ \n&=&\\int D'w|w\\rangle\n{\\cal T}e^{-2\\lambda\\int_{0}^{t}dt''A(t'')(\\delta\/\\delta w(t''))}\ne^{-(4\\lambda)^{-1}\\int_{-\\infty}^{\\infty}dt'w^{2}(t')}|\\phi\\rangle\\\\\n&=&\\int D'w|w\\rangle \n {\\cal T}e^{-(4\\lambda)^{-1}\\int_{0}^{t}dt'[w(t')-2\\lambda A(t')]^{2}}|\\phi\\rangle\ne^{-(4\\lambda)^{-1}\\{ \\int_{-\\infty}^{0}dt'w^{2}(t')\n+\\int_{t}^{\\infty}dt'w^{2}(t')\\}}\n\\end{eqnarray}\n\\end{subequations}\n\n\tA way to look at this is as a continuous measurement sequence.  The \nchange of the wave function at time $t$ over the interval $dt$, according to (15a),  resides solely in\n\\[\ne^{-2i\\lambda A(t)i^{-1}\\partial\/\\partial w(t)}e^{-(4\\lambda)^{-1}dtw^{2}(t)}=e^{-(4\\lambda)^{-1}dt [w(t)-2\\lambda A(t)]^{2}}, \n\\]\n\\noindent which has the form of the von Neumann measurement of $A(t)$ by the pointer $w(t)$. However, here the pointer uncertainty $\\sim (dt)^{-1} $ is quite large.  It is only the accumulation of a large number of measurements \twhich can distinguish the eigenvalues of $A(t)$.  One can visualize the collection of pointers for all t (and for all $({\\bf x}, t)$ in the case of full-blown CSL), each waiting, patiently, \nfor its brief opportunity to perform its measurement. \n\n\\subsection{Equivalence of CSL and CQC}\n\n\tThe time-ordered term in Eq.(15d) is precisely the CSL expression (3b) for \nthe state vector which evolves under the classical noise $w(t')$ for $0\\leq t'\\leq t$.  \nThe remaining exponentials are unaltered parts of the vacuum state, one for $-\\infty< t'<0$ which \nnever changes, and  one for $t<t'<\\infty$ which evolves.  \n\n\tIn standard quantum theory, a superselection rule\\cite{super} obtains at time $t$ if thereafter the state vector may be written as a \nsuperposition of orthogonal states, each evolving independently of \nall the others (i.e., the matrix elements of the Hamiltonian between \nthese states vanishes).  In such a case, the state vector may be regarded \nas a \"catalog,\" as discussed in the Introduction, merely a convenient device for listing the different states and their probabilities, since any state's evolution after time t can be separately described.\n\n\tIn CQC, the ensemble vector admits a superselection rule at any time.  Consider the superselected states at time $t$. The labels which distinguish them are the values of $w(t')$ for $0\\leq t'\\leq t$.  The differently labeled states evolve independently thereafter.  Each state corresponds to a different CSL state.  The values of $w(t')$ for $-\\infty<t'< 0$ and  $t< t'\\leq \\infty$ are irrelevant as far as the particle physics over the interval $(0,t)$ is concerned. The former never becomes relevant. For the latter, as time increases, say from $t$ to $t_{1}$, the values of $w(t')$ for $t<t'< t_{1}$ become relevant: a superselected state at time $t$ splits into many superselected states over the interval $(t,t_{1})$ (in a way that a many-worlds advocate might admire).  Thus, as far as the particle physics is concerned, the ensemble vector may be regarded \nas merely a convenient device for listing the different CSL states, and this is the sense in which CSL and CQC are equivalent.  \n\n\tThe evolution (15) is unitary and guarantees conservation of probability \nif the usual quantum rule is employed, that the probability of realization of the state $\\sim |w\\rangle\\langle w|\\psi,t\\rangle$ is $D'w|\\langle w|\\psi,t\\rangle|^{2}$.\n As far as the particle physics is concerned at time $t$, one may integrate the probability of a superselected set over $w(t')$ for \n$t'$ outside the range $0\\leq t'\\leq t$: the result is the CSL probability $P_{t}(w)Dw$.\n\n\\subsection{Full-Blown CQC}\n\n\tFull-blown CQC corresponding to full-blown CSL described in Eqs.(8),(9), simply replaces \ntime integrals by space-time integrals, e.g., \n\\begin{equation}\n W(x)\\equiv(\\lambda)^{1\/2}\/(2\\pi)^{2}\\int d^{4}k [e^{-ik\\cdot x}b(k)+\ne^{ik\\cdot x}b^{\\dagger}(k)]\\nonumber\n\\end{equation}\n\\begin{equation}\n|\\psi,t\\rangle\\equiv{\\cal T}e^{-2i\\lambda\\int_{0}^{t}d^{4}x'A(x')\\Pi(x')}|0\\rangle|\\phi\\rangle\\nonumber\n\\end{equation}\n\\noindent replace Eqs.(10a), (15a), etc.\n\n\\subsection{Nonwhite Noise CQC}\nA CSL collapse evolution which generalizes Eq.(3b)\\cite{nonwhite} is\n\\[\n|\\psi, t\\rangle={\\cal T} e^{-(4\\lambda)^{-1}\\int_{0}^{t}dt'dt''[w(t')-2\\lambda A(t')]g(t'-t'')[w(t\")-2\\lambda A(t\")]} |\\phi\\rangle, \n\\]\n\\noindent where $g(t)$ is real, symmetric in $t$ and has a positive spectrum, i.e., $g(t)=\\int d\\omega \n\\exp (i\\omega t)h^{2}(\\omega)$ with $h(\\omega)=h(-\\omega)=h^{*}(\\omega)$.  \n\n\tA  CQC counterpart requires defining $W(t)$ and $\\Pi (t)$ as in Eqs.(10), except that \nthe square bracket in the integral of Eq.(10a)  (Eq.(10b)) is multiplied by $h^{-1}(\\omega)$ ($h(\\omega)$).  This replaces the equation before (13) and Eq.(13a) respectively by \n\\[\n\\big[\\int_{-\\infty}^{\\infty}dt'g(t-t')w(t')+2\\lambda\\delta\/\\delta w(t)\\big]\\langle w|0\\rangle=0,\\quad\n\\langle w|0\\rangle=e^{-(4\\lambda)^{-1}\\int_{-\\infty}^{\\infty}dt'dt''w(t')g(t'-t'')w(t'')}\n\\]\n\n\tWith the evolution equation the same as Eq.(15a), Eq.(15d) is replaced by \n\\[\n|\\psi,t\\rangle=\n {\\cal T}e^{-(4\\lambda)^{-1}\\int_{-\\infty}^{\\infty}dt'dt''[w(t')-2\\lambda A(t')\\Theta_{0,t}(t')]g(t'-t'')[w(t'')-2\\lambda A(t'')\\Theta_{0,t}(t'')]}|\\phi\\rangle\n\\]\n\\noindent where $\\Theta_{0,t}(t')=1$ for $0\\leq t'\\leq t$ and $ = 0$ otherwise.  For general $h(\\omega)$, this equation is {\\it not} the same as its CSL counterpart written above, except in the S-matrix limit where the interval $(0,t)$ is replaced by \n$(-\\infty, \\infty)$.  This generalization shall not be considered any further in this paper.     \n\n\\section{Energy}\n\n\\subsection{Pictures}\n\tThe CSL Schr\\\"odinger picture state vector (1) has a corresponding CQC ensemble vector:\n\\begin{subequations}\\label{16}\n\\begin{eqnarray}\n|\\psi, t\\rangle_{S}&\\equiv&e^{-iH_{A}t}|\\psi, t\\rangle\\\\\n&=&{\\cal T} e^{-i\\int_{0}^{t}dt'[H_{A}+2\\lambda A\\Pi(t')]}|0\\rangle|\\phi\\rangle, \n\\end{eqnarray}\n\\end{subequations}\n\\noindent  where (16b)  follows from putting (15a) into (16a).  Although $A$'s  time dependence in (15a)   is removed in (16b), this is still not the Schr\\\"odinger  picture for CQC's ensemble vector  (nonetheless, we'll keep the $S$ subscript on it), since the $\\Pi$ operator still has time dependence.  It can be removed in what we'll call the sub-Schr\\\"odinger picture:\n\\begin{subequations}\\label{17}\n\\begin{eqnarray}\n|\\psi, t\\rangle_{SS}&\\equiv&e^{-iH_{w}t}|\\psi, t\\rangle_{S}\\\\\n&=& e^{-it[H_{w}+H_{A}+2\\lambda A\\Pi(0)]}|0\\rangle|\\phi\\rangle,\n\\end{eqnarray}\n\\end{subequations}\n\n\tThus, since this is a unitary evolution, there is a conserved energy, the Hamiltonian in Eq.(17b). In the collapse-interaction picture it is \t\n\\[ H(t)=e^{i(H_{A}+H_{W})t}[H_{A}+H_{W}+2\\lambda A \\Pi (0)]e^{-i(H_{A}+H_{W})t}=\nH_{A}+H_{W}+2\\lambda A(t) \\Pi (t).\n\\]\n\t\n\tIt should be emphasized that, since $H_{w}$ and the interaction term depend upon $\\Pi$  which has nonvanishing matrix elements between different eigenstates of $W$, the conserved energy belongs to the ensemble of realizable states, not to an individual state. A single collapse to the realized state generally will not preserve the energy distribution. (Of course, this is also the case in standard quantum theory with the collapse postulate.)   Nonetheless, it is interesting to consider the probability distributions of the ensemble total energy and its parts.  A conserved quantity gives insight to the behavior of the ensemble of possibilities.  \n\t\n\tFor example, one may see how the the ensemble average increase of particle energy, due to the localization mechanism of CSL, is compensated by the ensemble average decrease of the random field energy.  And, laboratory experiments often do involve an ensemble of (approximately) identical initial states, for which therefore energy is (approximately) conserved for the ensemble of outcome states.  Also interesting  are circumstances where one can speak of conservation of energy associated with the ensemble of states that collapses to a distinctly identifiable outcome. For example, when considering a situation when the initial state vector is a superposition of particles in widely separated locations,  the ongoing orthogonality of the differently located particle states ensures energy conservation for the ensemble of states which describe collapse to particles in one of the locations.   \n\t\t\n\\subsection{Simplest Model, $H_{A}=0$}\n\n\tWe display here generating functions and probability distributions for the total energy and its components, for the model of section IIa, in the simplest case, $H_{A}=0$.  \n\t\n\tUsing the expression for $|\\psi, t\\rangle$ given in Eq.(15a),  \nthe generating function for the moments of the total energy is\n\\begin{subequations}\\label{18}\n\\begin{eqnarray}\n\\langle\\psi,t| e^{-iH(t)\\beta}|\\psi, t\\rangle&=&\\sum_{n}|\\alpha_{n}|^{2}\n\\langle 0| e^{2i\\lambda a_{n}\\int_{0}^{t}dt' \\Pi(t')]}e^{-i[H_{w}+2\\lambda a_{n}\\Pi(t)]\\beta}\ne^{-2i\\lambda a_{n}\\int_{0}^{t}dt' \\Pi(t')]}|0\\rangle\\\\\n&=&\\sum_{n}|\\alpha_{n}|^{2}\n\\langle 0| e^{-i[H_{w}+2i\\lambda a_{n}\\int_{0}^{t}dt'i\\dot{\\Pi}(t')+2\\lambda a_{n}\\Pi(t)]\\beta}|0\\rangle\\\\\n&=&\\sum_{n}|\\alpha_{n}|^{2}\n\\langle 0| e^{-i[H_{w}+2\\lambda a_{n}\\Pi(0)]\\beta}|0\\rangle\\\\\n&=&\\sum_{n}|\\alpha_{n}|^{2}\n\\langle 0| e^{-iH_{w}\\beta}e^{-2i\\lambda a_{n}\\int_{0}^{\\beta}dt'\\Pi(t')}|0\\rangle\\\\\n&=&\\sum_{n}|\\alpha_{n}|^{2}\\int D' \\pi e^{-2\\lambda\\int_{-\\infty}^{\\infty}\\pi^{2}(t')}\ne^{-2i\\lambda a_{n}\\int_{0}^{\\beta}dt'\\pi(t')}\\\\\n&=&\\sum_{n}|\\alpha_{n}|^{2} e^{-(\\lambda\/2)a_{n}^{2}|\\beta |}.\n\\end{eqnarray}\n\\end{subequations}\t \n\\noindent (18b) is the result of performing the unitary transformation on $e^{-iH(t)\\beta}$ in (18a), the integration in the exponent of (18b) results in (18c) which confirms that the generating function is time-independent, (18d) expresses the unitary operator in (18c) in a different form, (18e) follows from $H_{w}|0\\rangle=0$ and insertion of the identity $1=\\int D'\\pi |\\pi\\rangle\\langle \\pi |$, (18f) follows by discretization and performance of the gaussian integrals.\n\n\tThe moments of the energy follow from Eq.(18f) using the formal Taylor series expansion, \n$|\\beta|=0+\\beta\\epsilon (0)+(\\beta^{2}\/2!)2\\delta(0)+(\\beta^{3}\/3!)2\\delta '(0)+...$  \n(where $\\epsilon (x)\\equiv x\/|x|$ for $x\\neq 0$ and $\\epsilon (0)\\equiv 0$).  \nThus the mean energy is 0 and the higher moments of the energy are infinite.  The associated probability distribution more transparently entails this result:\n\\begin{subequations}\\label{19}\n\\begin{eqnarray}\n{\\cal P}(E)&\\equiv&\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}e^{iE\\beta}\\langle\\psi,t| e^{-iH(t)\\beta}|\\psi, t\\rangle\\\\\n&=&\\frac{1}{\\pi}\\sum_{n}|\\alpha_{n}|^{2}\\frac{\\lambda a_{n}^{2}\/2}{E^{2}+(\\lambda a_{n}^{2}\/2)^{2}}.\n\\end{eqnarray}\n\\end{subequations}\t \n\n\tAlthough the total energy probability distribution is time-independent, the \nenergy $H_{w}$ is not.  The generating function here is \n\\begin{subequations}\\label{20}\n\\begin{eqnarray}\n\\langle\\psi,t| e^{-iH_{w}\\beta}|\\psi, t\\rangle&=&\\sum_{n}|\\alpha_{n}|^{2}\n\\langle 0| e^{2i\\lambda a_{n}\\int_{0}^{t}dt' \\Pi(t')}e^{-2i\\lambda a_{n}\\int_{0}^{t}dt' \\Pi(t'-\\beta)}|0\\rangle\\\\\n&=&\\sum_{n}|\\alpha_{n}|^{2}\\int D' \\pi e^{-2\\lambda\\int_{-\\infty}^{\\infty}\\pi^{2}(t')}\ne^{2i\\lambda a_{n}[\\int_{0}^{t} dt-\\int_{-\\beta}^{t-\\beta}]dt'\\pi(t')}\\\\\n&=&\\sum_{n}|\\alpha_{n}|^{2}\\big\\{[\\Theta(\\beta-t) + \\Theta (-t-\\beta)]e^{-\\lambda a_{n}^{2}t} \\nonumber\\\\\n&&\\qquad\\qquad\\qquad\\qquad+\\Theta (t-\\beta)\\Theta (t+\\beta)e^{-\\lambda a_{n}^{2}|\\beta |}\\big\\} .\n\\end{eqnarray}\n\\end{subequations}\t \t \n\\noindent where (20a) utilizes (15a) and the time-translation capability of $H_{w}$, (20b) involves going to the $|\\pi\\rangle$ representation, and (20c) follows from performing the gaussian integrals.  From the Taylor series expansion of (20c), one sees that the mean value of $H_{w}$ is zero and the higher moments are infinite.  \n\n\tThe probability distribution following from Eq.(20c) confirms this result:  \n \\begin{subequations}\\label{21}\n\\begin{eqnarray}\n{\\cal P}_{w}(E)&\\equiv&\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}e^{iE\\beta}\\langle\\psi,t| e^{-iH_{w}\\beta}|\\psi, t\\rangle\\\\\n&=&\\frac{1}{\\pi}\\sum_{n}|\\alpha_{n}|^{2}\\bigg \\{e^{-\\lambda a_{n}^{2}t}\\bigg [\\pi\\delta(E) -\n\\frac{\\sin Et}{E}+\\frac{E\\sin Et-\\lambda a_{n}^{2}\\cos Et}{E^{2}+(\\lambda a_{n}^{2})^{2}}\\bigg ] \\nonumber \\\\\n&&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n+\\frac{\\lambda a_{n}^{2}}{E^{2}+(\\lambda a_{n}^{2})^{2}}\\bigg \\}. \n\\end{eqnarray}\n\\end{subequations}\t\n\\noindent Eq. (21) shows how the $W$-field energy, starting at $t=0$ with the vacuum energy $E= 0$, evolves toward a stationary energy distribution associated with the completely collapsed state.  (Note that the asymptotic value of (21b) is {\\it not} the same as (19b), although it is similar in form.)\n\n\tThe interaction energy's probability distribution is \n \\begin{subequations}\\label{22}\n\\begin{eqnarray}\n{\\cal P}_{I}(E)&\\equiv&\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}e^{iE\\beta}\\langle\\psi,t| e^{-iH_{I}\\beta}|\\psi, t\\rangle\\\\\n&=&\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}e^{iE\\beta}\n\\sum_{n}|\\alpha_{n}|^{2}\\int D' \\pi e^{-2\\lambda\\int_{-\\infty}^{\\infty}dt'\\pi^{2}(t')}\ne^{-2i\\lambda a_{n}\\pi(t)\\beta}\\\\\n&=&\\sum_{n}\\frac{|\\alpha_{n}|^{2}}{\\sqrt{2\\pi \\lambda a_{n}^{2}\/dt}}e^{-E^{2}\/(2\\lambda a_{n}^{2}\/dt)}\n\\end{eqnarray}\n\\end{subequations}\n\\noindent Since the interaction energy is proportional to the white noise operator $\\Pi(t)$, it has a white noise energy spectrum, with all energies equally likely.   \n \t\t \n\\subsection{Simplest Model, $H_{A}\\neq 0$}\n\\subsubsection{Moment Generating Functions}\n\tAs in the previous section, also in this more general case, the random field variables can be integrated out of the expressions for the moment generating functions.  For the total energy, corresponding to Eq.(18), one obtains:\n\\begin{subequations}\\label{23}\n\\begin{eqnarray}\n\\langle\\psi,t| e^{-iH(t)\\beta}|\\psi, t\\rangle&=&\n\\langle \\phi|\\langle 0| e^{-i[H_{A}+H_{w}+2\\lambda\\Pi(0)A]\\beta}|0\\rangle|\\phi\\rangle\\\\\n&=&\\langle \\phi| \\langle 0|e^{-i[H_{A}+H_{w}]\\beta}{\\cal T}e^{-2i\\lambda \\int_{0}^{\\beta}dt'\\Pi(t')A(t')}|0\\rangle|\\phi\\rangle\\\\\n&=&\\langle \\phi| e^{-iH_{A}\\beta} \\int D'\\pi e^{-2\\lambda\\int_{-\\infty}^{\\infty}dt'\\pi^{2}(t')}{\\cal T}e^{-2i\\lambda \\int_{0}^{\\beta}dt'\\pi(t')A(t')}|\\phi\\rangle\\\\\n&=&\\langle \\phi| e^{-iH_{A}\\beta} {\\cal T}e^{-(\\lambda\/2) \\int_{0}^{\\beta}dt'A^{2}(t')}|\\phi\\rangle\\\\\n&=&\\langle \\phi| e^{-[iH_{A} +(\\lambda\/2)A^{2}(0)]\\beta}|\\phi\\rangle. \n\\end{eqnarray}\n\\end{subequations}\n\\noindent ((23a) is expressed in the manifestly time-independent sub-Schr\\\"odinger picture, and the remaining steps are ones taken before.)\n\tFor $H_{A}$, the moment generating function is \n\\begin{subequations}\\label{24}\n\\begin{eqnarray}\n\\langle\\psi,t| e^{-iH_{A}\\beta}|\\psi, t\\rangle&=&\t \t \n\\langle \\phi|\\langle 0|{\\cal T}e^{2i\\lambda \\int_{0}^{t}dt'\\Pi(t')A_{L}(t')} e^{-iH_{A}\\beta}\ne^{-2i\\lambda \\int_{0}^{t}dt'\\Pi(t')A_{R}(t')}|0\\rangle|\\phi\\rangle\\\\\n&=&\\langle\\phi |{\\cal T}\\big\\{e^{ -(\\lambda\/2)\\int_{0}^{t}dt'[A_{L}(t')-A_{R}(t')]^{2}}e^{-iH_{A}\\beta}\\big\\}|\\phi\\rangle,\n\\end{eqnarray}\n\\end{subequations}\n\\noindent where $A_{R}$ ($A_{L}$) acts to the right (left) of $\\exp-iH_{A}\\beta$, and is time-ordered (time--reverse-ordered) by ${\\cal T}$.  \n\n\tFor $H_{w}$, the calculation is  \n\\begin{subequations}\\label{25}\n\\begin{eqnarray}\n\\langle\\psi,t| e^{-iH_{w}\\beta}|\\psi, t\\rangle&&=\t \t \n\\langle \\phi|\\langle 0|{\\cal T}e^{2i\\lambda \\int_{0}^{t}dt'\\Pi(t')A_{L}(t')} e^{-iH_{w}\\beta}\ne^{-2i\\lambda \\int_{0}^{t}dt'\\Pi(t')A_{R}(t')}|0\\rangle|\\phi\\rangle\\\\\n&&=\\langle \\phi|\\langle 0|{\\cal T}e^{2i\\lambda \\int_{0}^{t}dt'\\Pi(t'+\\beta\/2)A_{L}(t')}e^{-2i\\lambda \\int_{0}^{t}dt'\\Pi(t'-\\beta\/2)A_{R}(t')}|0\\rangle|\\phi\\rangle\\\\\n&&=\\langle \\phi|\\int D' \\pi e^{-2\\lambda\\int_{-\\infty}^{\\infty}dt'\\pi^{2}(t')}     \\notag      \\\\ \n&&\\cdot {\\cal T}e^{2i\\lambda \\int_{\\beta\/2}^{t+\\beta\/2}dt'\\pi(t')A_{L}(t'-\\beta\/2)}e^{-2i\\lambda\\int_{-\\beta\/2}^{t-\\beta\/2}dt'\\pi(t')A_{R}(t'+\\beta\/2)}|\\phi\\rangle\\\\\n&&=\\langle \\phi|{\\cal T}e^{ -(\\lambda\/2)\\int_{0}^{t}dt'[A_{L}^{2}(t')+A_{R}^{2}(t')]}\n\\big[\\Theta (\\beta -t)+\\Theta (-\\beta -t) \\nonumber\\\\\n&&\\qquad+\\Theta (t-\\beta)\\Theta (t+\\beta)e^{\\lambda\n\\int_{|\\beta | \/2}^{t-|\\beta | \/2}dt' A_{L}(t'-\\beta\/2)A_{R}(t'+\\beta\/2)}\\big]|\\phi\\rangle\n\\end{eqnarray}\n\\end{subequations}\t \n\n \\subsubsection{First Moments}\n \n \tHere is shown conservation of energy in the mean, reproducing more simply a previously obtained result (all that that had been proved about conservation of energy for CSL\\cite{energy}).   \n \tEquating the terms proportional to the first power of $\\beta$ in Eq,(24b) produces\t\n\\begin{equation}\\label{26}\n \\langle\\psi,t| H_{A}|\\psi, t\\rangle=\t \t \n\\langle\\phi |{\\cal T}e^{ -(\\lambda\/2)\\int_{0}^{t}dt'[A_{L}(t')-A_{R}(t')]^{2}}H_{A}|\\phi\\rangle.\n\\end{equation}\t\t\n\tTo find the first moment of $H_{w}$,  the exponent in (25d) must be written for small $\\beta$:\n\\begin{eqnarray}\\label{27}\n&&\\int_{|\\beta | \/2}^{t-|\\beta | \/2}dt' A_{L}(t'-\\beta\/2)A_{R}(t'+\\beta\/2)=\n\\int_{0}^{t}dt' A_{L}(t')A_{R}(t')\\nonumber\\\\\n&&\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\n-(|\\beta| \/2)[A_{L}(t)A_{R}(t)+A_{L}(0)A_{R}(0)]\n+(\\beta \/2)\\int_{0}^{t}dt' [A_{L}(t')\\dot{A}_{R}(t')- \\dot{A}_{L}(t')A_{R}(t')]+...\n\\end{eqnarray}\n \\noindent The zeroth and first order terms in the Taylor series expansion of (27) are the same as (27),  except that the second term vanishes, since $d|\\beta|\/dt|_{\\beta=0}=\\epsilon(0)=0$. The step-functions in (25d) likewise do not contribute to the Taylor series expansion's first two terms, so we get from (25d):\n\\begin{subequations}\\label{28}\n\\begin{eqnarray}\n \\langle\\psi,t| H_{w}|\\psi, t\\rangle&=&i(\\lambda\/2)  \t \t \n\\langle\\phi |{\\cal T}e^{ -(\\lambda\/2)\\int_{0}^{t}dt'[A_{L}(t')-A_{R}(t')]^{2}}\\nonumber\\\\ \n&&\\qquad\\qquad\\cdot \\int_{0}^{t}dt' [ A_{L}(t')\\dot{A}_{R}(t')- \\dot{A}_{L}(t')A_{R}(t')] |\\phi\\rangle\\\\\n&=&(\\lambda\/2)\\langle\\phi |{\\cal T}\\int_{0}^{t}dt' e^{ -(\\lambda\/2)\\int_{0}^{t'}dt''[A_{L}(t'')-A_{R}(t'')]^{2}}[A(t'),i\\dot{A}(t')]\n |\\phi\\rangle\\\\\n&=&(\\lambda\/2)\\langle\\phi |{\\cal T}\\int_{0}^{t}dt' e^{ -(\\lambda\/2)\\int_{0}^{t'}dt''[A_{L}(t'')-A_{R}(t'')]^{2}}[A(t'),[{A}(t'),H_{A}]] |\\phi\\rangle\n\\end{eqnarray}\n\\end{subequations}\t   \n \\noindent In going from (28a) to (28b) we have utilized  \n\\[ \n{\\cal T}e^{ -(\\lambda\/2)\\int_{0}^{t}dt'[A_{L}(t')-A_{R}(t')]^{2}}B_{L}(t_{1})C_{R}(t_{1})=\n{\\cal T}e^{ -(\\lambda\/2)\\int_{0}^{t_{1}}dt'[A_{L}(t')-A_{R}(t')]^{2}}B(t_{1})C(t_{1})\n\\]\n\\noindent for $t_{1}<t$ and arbitrary operators $B_{L}(t_{1})$, $C_{R}(t_{1})$ (the subscripts $R$, $L$ may be dispensed with when operators having the same time argument are adjacent and in the correct order).  \n\n\tSince the first moment of the \ninteraction energy vanishes, from Eqs.(26), (28c) follows conservation of energy in the mean:\n\\begin{eqnarray}\\label{29}\n(d\/dt) \\langle\\psi,t| H_{w}|\\psi, t\\rangle&=&-(d\/dt)\\langle\\psi,t| H_{A}|\\psi, t\\rangle\\nonumber\\\\\n&=&(\\lambda\/2)\\langle\\phi |{\\cal T} e^{ -(\\lambda\/2)\n\\int_{0}^{t}dt'[A_{L}(t')-A_{R}(t')]^{2}}[A(t),[{A}(t),H_{A}]] |\\phi\\rangle\n\\end{eqnarray}\n\n\\subsubsection{Large t}\n\n\tThe large t-behavior of the probability density distribution of $H_{w}$ is due to the \nthird term in the bracket of Eq.(25d).  Because this term's exponent is largest for small $\\beta$, and also because $\\exp iE\\beta$ oscillates rapidly for large $\\beta$, the small $\\beta$ approximation of (25d) should be a good approximation to the large t limit. Putting (27) into (25d) results in\n\\begin{subequations}\\label{30}\n\\begin{eqnarray}\n&&{\\cal P}_{w}(E)=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}e^{iE\\beta}\\langle\\psi,t| e^{-iH_{w}\\beta}|\\psi, t\\rangle\\\\\n&&\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\n\\approx\\frac{1}{2\\pi}\\int_{-t}^{t}d\\beta e^{iE\\beta}\\langle\\phi|{\\cal T} \ne^{ -(\\lambda\/2)\\int_{0}^{t}dt'[A_{L}(t')-A_{R}(t')]^{2}}e^{-(\\lambda \/2)|\\beta|[A^{2}(t)+A^{2}(0)]}\ne^{-i(\\lambda \/2)\\beta\\int_{0}^{t}dt'C_{LR}(t')}|\\phi\\rangle\\\\\n&&\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\n=\\langle\\phi|{\\cal T} \ne^{ -(\\lambda\/2)\\int_{0}^{t}dt'[A_{L}(t')-A_{R}(t')]^{2}} \n\\frac{1}{\\pi}\n\\frac{\\lambda [A^{2}(t)+A^{2}(0)]\/2}{[E-(\\lambda \/2)\\int_{0}^{t}dt'C_{LR}(t')]^{2}+[\\lambda (A^{2}(t)+A^{2}(0))\/2]^{2}}|\\phi\\rangle\n \\nonumber\\\\\n\\end{eqnarray}\n\\end{subequations}\t \t \n\\noindent where $C_{LR}(t')\\equiv A_{L}(t')[A_{R}(t'),H_{A}]-[A_{L}(t'),H_{A}]A_{R}(t')$.  Note that,  when $H_{A}=0$,\nthe large t-limit of (30c)  is the same as (21b). Also, note  that, when $H_{A}\\neq0$,\n the mean value of $E$ given by (30c) is the same as (28c).\n \n \tFor example, in the case of a free particle of mass $m$ which collapses toward a position eigenstate,  where $H=p^{2}\/2m$ and $A=x$, the term in (30c) $[E-(\\lambda \/2)\\int_{0}^{t}dt'C_{LR}(t')]^{2}=\n[E+\\lambda\\hbar t\/2m]^{2}$.  Because the ensemble energy is conserved, the mean $W$-field energy decreases with time to compensate for the  increase of the particle's mean energy $=\\lambda\\hbar t\/2m$ due to the narrowing of wavepackets by the collapse mechanism.  \n \n\t\\section{Time Operator}\n\t\n\tUnlike the usual Hamiltonian which is semi-bounded\\cite{timeref},  $H_{w}$ has an unbounded spectrum, providing an opportunity to define a self-adjoint  time operator:\n\t\\begin{subequations}\\label{31}\n\\begin{eqnarray}\nT&\\equiv&\\frac{ \\int_{-\\infty}^{\\infty}d\\omega b^{\\dagger}(\\omega)\\frac{1}{i}\\frac{\\partial}{\\partial \\omega}b(\\omega)}{\n \\int_{-\\infty}^{\\infty}d\\omega b^{\\dagger}(\\omega)b(\\omega)}\\equiv \\frac{B}{N}\\\\\n &=&\\frac{ \\int_{-\\infty}^{\\infty}dt' t'[(4\\lambda)^{-1}W^{2}(t')+\\lambda \\Pi^{2}(t')]} \n { \\int_{-\\infty}^{\\infty}dt' [(4\\lambda)^{-1}W^{2}(t')+\\lambda \\Pi^{2}(t')]},\n \\end{eqnarray}\n\\end{subequations}\t\n\\noindent where $[B,N]=0$.  It is simple to show that $[H_{w},T]=i$, where $H_{w}$ is given by Eq.(14).  The action of $T$ on $|0\\rangle$, is undefined, but one can\ntake $T=B\/(N+\\epsilon)$ so that $T |0\\rangle=0\/\\epsilon =0$. Of course, $T$ plus any function of $H_{w}$ and $N$ is also \nconjugate to $H_{w}$, but (31)  is the simplest  choice.\n\n\t\\subsection{Eigenfunctions} \n\t\n\t$T$'s eigenvalues are highly degenerate.  A complete set of (unnormalized) eigenfunctions in the $|w\\rangle$ basis is:\n\\begin{eqnarray}\n\\langle w| \\prod_{t}[\\int_{-\\infty}^{\\infty}d\\omega e^{i\\omega t}b^{\\dagger}(\\omega)]^{n(t)}|0\\rangle\n&\\sim& \\prod_{t}[ w(t)-2\\lambda\\delta\/\\delta\\omega ]^{n(t)}e^{-(4\\lambda)^{-1}\\int_{-\\infty}^{\\infty}dt' w^{2}(t')}\n\\nonumber\\\\\n&\\sim& \\prod_{t}H_{n(t)}[w(t)(dt\/2\\lambda)^{1\/2}]e^{-(4\\lambda)^{-1}\\int_{-\\infty}^{\\infty}dt' w^{2}(t')}\\nonumber\n\\end{eqnarray}\n\\noindent where $n(t)=0,1,2...$ and the $H_{n}$ are Hermite polynomials.  The associated \neigenvalues $\\sum_{t=-\\infty}^{\\infty}t n(t)\/\\sum_{t=-\\infty}^{\\infty}n(t)$ have a ``center of time\" form.\n\n\\subsection{Moment Generating Function}\n\n\tTo find the moment generating function for $T$, first observe, using (15a) and (10b), that \n\n\\begin{eqnarray}\n &&e^{-i\\beta T}|\\psi, t\\rangle=e^{-i\\beta T}{\\cal T}e^{(\\lambda\/2\\pi)^{1\/2}\\int_{0}^{t}dt'\n A_{R}(t')\\int _{-\\infty}^{\\infty}d\\omega e^{i\\omega t'} b^{\\dagger}(\\omega)}|0\\rangle \n e^{-(\\lambda\/2)\\int_{0}^{t}dt' A^{2}_{R}(t')}|\\phi\\rangle\\nonumber \\\\\n &&\\quad= {\\cal T}\\bigg[1+\\sum_{n=1}^{\\infty}e^{-i\\beta B\/n}\\frac{(\\lambda\/2\\pi)^{n\/2}}{n!}\n \\big[\\int_{0}^{t}dt' A_{R}(t')\\int _{-\\infty}^{\\infty}d\\omega e^{i\\omega t'} b^{\\dagger}(\\omega)\\big]^{n}\\bigg]\n |0\\rangle \n e^{-(\\lambda\/2)\\int_{0}^{t}dt' A^{2}_{R}(t')}|\\phi\\rangle. \\nonumber \n\\end{eqnarray}\t\n\\noindent Since $\\exp -(i\\beta B\/n) b^{\\dagger}(\\omega)\\exp (i\\beta B\/n)= b^{\\dagger}\n(\\omega+\\beta\/n)$, one gets:\n\\begin{subequations}\\label{32}\n\\begin{eqnarray}\n\\langle\\psi,t | e^{-i\\beta T}|\\psi, t\\rangle&=&\\langle\\phi | {\\cal T} \ne^{-(\\lambda\/2)\\int_{0}^{t}dt' [A^{2}_{R}(t')+A^{2}_{L}(t')]}\\langle 0| \n\\bigg[1+\\sum_{n=1}^{\\infty}\\frac{(\\lambda\/2\\pi)^{n\/2}}{n!}\n \\big[\\int_{0}^{t}dt' A_{L}(t')\\int _{-\\infty}^{\\infty}d\\omega e^{-i\\omega t'} b(\\omega)\\big]^{n}\\bigg]\\nonumber\\\\\n&&\\quad\\cdot\\bigg[1+\\sum_{n=1}^{\\infty}\\frac{(\\lambda\/2\\pi)^{n\/2}}{n!}\n \\big[\\int_{0}^{t}dt' A_{R}(t')\\int _{-\\infty}^{\\infty}d\\omega e^{i(\\omega-n^{-1}\\beta )t'} b^{\\dagger}(\\omega)\\big]^{n}\\bigg]|0\\rangle|\\phi\\rangle\\\\\n&=&\\langle\\phi | {\\cal T} \ne^{-(\\lambda\/2)\\int_{0}^{t}dt' [A^{2}_{R}(t')+A^{2}_{L}(t')]}\n\\bigg[1+\\sum_{n=1}^{\\infty}\\frac{(\\lambda)^{n}}{n!}\n \\big[\\int_{0}^{t}dt' A_{L}(t')A_{R}(t')e^{-i\\beta t'\/n}\\big]^{n}\\bigg]|\\phi\\rangle\n\\end{eqnarray}\n\\end{subequations}\t\n\\noindent\t As a check, when $\\beta=0$, (32b) becomes:\n\\[\n\\langle\\phi | {\\cal T} \ne^{-(\\lambda\/2)\\int_{0}^{t}dt' [A^{2}_{R}(t')+A^{2}_{L}(t')]}\ne^{\\lambda\\int_{0}^{t}dt' [A_{R}(t')A_{L}(t')]}|\\phi\\rangle=\n\\langle\\phi | {\\cal T} \ne^{-(\\lambda\/2)\\int_{0}^{t}dt' [A_{R}(t')-A_{L}(t')]^{2}}\n|\\phi\\rangle=1.\n\\]\n\n\\subsection{First Moments}\n\tNow, extract the first two moments of $T$ from (32b).  From the coefficient of $\\beta$ in (32b) we obtain:\n\\begin{eqnarray}\\label{33}\n\\langle\\psi,t | T|\\psi, t\\rangle&=&\\langle\\phi | {\\cal T} \\frac{\\int_{0}^{t}dt' t'A_{L}(t')A_{R}(t')}\n{\\int_{0}^{t}dt' A_{L}(t')A_{R}(t')}\\nonumber\\\\\n&&\\qquad\\cdot\\big[e^{-(\\lambda\/2)\\int_{0}^{t}dt' [A_{L}(t')-A_{R}(t')]^{2}}\n-e^{-(\\lambda\/2)\\int_{0}^{t}dt' [A^{2}_{L}(t')+A^{2}_{R}(t')]}\\big]|\\phi\\rangle\n\\end{eqnarray}\n\t\n\tTo get a feel for the meaning of (33), consider the case $H_{A}=0$:\n\\begin{eqnarray}\\label{34}\n\\langle\\psi,t | T|\\psi, t\\rangle=\\sum_{k}|\\alpha_{k}|^{2}\\frac{\\int_{0}^{t}dt't'}{\\int_{0}^{t}dt}\n[1-e^{-\\lambda t a_{k}^{2}}]=\n\\frac{t}{2}[1-\\sum_{k}|\\alpha_{k}|^{2}e^{-\\lambda t a_{k}^{2}}].\n\\end{eqnarray}\n\\noindent Thus, asymptotically, $ \\langle\\psi,t | T|\\psi, t\\rangle$ approaches \nthe ``center of time,\" a weighted average (here uniformly weighted) of the \ntime over the interval $(0,t)$.  For $H_{A}\\neq 0$ much the same behavior prevails, according to Eq.(33).  For example, set $A_{R}(t')=A_{L}(t')=At'^{s}$.  Then, from (33), asymptotically, \n$ \\langle\\psi,t | T|\\psi, t\\rangle\\rightarrow t(2s+1)\/(2s+2)$: for $s$ large, the mean value of $T$ \nis almost $t$.\n\n\tFor full-blown CQC, the expression that replaces (33) is  the same expression  \nwith $A(t')\\rightarrow A({\\bf x},t')$ and $dt'\\rightarrow d{\\bf x}dt'$.   $A$ (given by \nEq. (8)) is $\\approx$ the mass density in a sphere of radius $a$.  Then the appropriate \n``center of time \" expression appearing in the equivalent of (33) is \n\\[ \n\\int_{0}^{t}dt' d{\\bf x}t'A_{L}({\\bf x},t')A_{R}({\\bf x},t')\n\/\\int_{0}^{t}dt' d{\\bf x}A_{L}({\\bf x},t')A_{R}({\\bf x},t'), \n\\]\n\\noindent i.e., $t'$, roughly speaking,  is weighted by \nthe squared mass density integrated over all space and over time from 0 to $t$.  Should there \nbe a high rate of agglomeration of matter, e.g., due to gravitational clumping, \nthen $ \\langle\\psi,t | T|\\psi, t\\rangle$ will approach $t$.  It should be emphasized that gravitational \ncollapse has this effect, not wave function collapse, as Eq. (34) attests.  \n\n\tThe second moment of $T$ is likewise found from (32b):\t\n\n\\begin{eqnarray}\\label{35}\n \\langle\\psi,t | T^{2}|\\psi, t\\rangle&=&\\langle\\phi | {\\cal T}\\bigg\\{\\bigg[ \\frac{\\int_{0}^{t}dt' t'A_{L}(t')A_{R}(t')}\n{\\int_{0}^{t}dt' A_{L}(t')A_{R}(t')}\\bigg]^{2}\\nonumber\\\\\n&&\\quad\\cdot\\big[ e^{-(\\lambda\/2)\\int_{0}^{t}dt' [A_{L}(t')-A_{R}(t')]^{2}}-\ne^{-(\\lambda\/2)\\int_{0}^{t}dt' [A^{2}_{L}(t')+A^{2}_{R}(t')]}[1+f(Z_{LR})]\\big]\\nonumber\\\\\n&&\\quad+ \\frac{\\int_{0}^{t}dt' t'^{2}A_{L}(t')A_{R}(t')}\n{\\int_{0}^{t}dt' A_{L}(t')A_{R}(t')}e^{-(\\lambda\/2)\\int_{0}^{t}dt' [A^{2}_{R}(t')+A^{2}_{L}(t')]}f(Z_{LR})\\bigg\\}|\\phi\\rangle,\n\\end{eqnarray}\n\\noindent where $Z_{LR}\\equiv \\lambda\\int_{0}^{t}dt' A_{L}(t')A_{R}(t')$ and\n $f(z)\\equiv\\int_{0}^{z}dz'[\\exp z' -1]\/z'$.\t\n\\noindent For the example $A_{R}(t')=A_{L}(t')=At'^{s}$ ($H_{A}=0$ corresponds to $s=0$), Eq.(35) asymptotically becomes (using $f(z)\\rightarrow z^{-1}\\exp z$):\n\\begin{equation}\\label{36}\n\\langle\\psi,t | T^{2}|\\psi, t\\rangle\\rightarrow\\bigg[\\frac{t(2s+1)}{2s+2}\\bigg]^{2}\n\\bigg[1+\\frac{1}{\\lambda t^{2s+1}}\\frac{(2s+2)^{2}}{2s+3}\\sum_{k}\\frac{|\\alpha_{k}|^{2}}{a_{k}^{2}}\\bigg].\n\\end{equation}\n\\noindent  so the fractional deviation decreases over time, $\\overline{\\Delta T^{2}}\/\\overline{T}^{2}\\sim t^{-(2s+1)}$.  \n \t\n\\subsection{Large t}\n\n\tMore generally, an approximation to the large $t$ probability distribution of $T$ may be had \nby employing the small $\\beta$ approximation in  (32b),\n\\begin{eqnarray}\n[\\int_{0}^{t}dt' A_{L}(t')A_{R}(t')e^{-i\\beta t'\/n}]^{n}&\\approx &[\\int_{0}^{t}dt' A_{L}(t')A_{R}(t')(1-i\\beta t'\/n)]^{n}\\nonumber\\\\\n&\\approx &[\\int_{0}^{t}dt' A_{L}(t')A_{R}(t')]^{n}e^{-i\\beta\\int_{0}^{t}dt't' A_{L}(t')A_{R}(t')\/\\int_{0}^{t}dt' A_{L}(t')A_{R}(t')},\\nonumber\n\\end{eqnarray}\n\\noindent with the result\n\\begin{eqnarray}\n{\\cal P}(\\tau)&\\equiv &(2\\pi)^{-1}\\int_{-\\infty}^{\\infty}d\\beta e^{i\\beta \\tau}\\langle\\psi,t | e^{-i\\beta \\tau}|\\psi, t\\rangle\\nonumber\\\\\n&\\approx&\\langle\\phi | {\\cal T}e^{-(\\lambda\/2)\\int_{0}^{t}dt' [A_{L}(t')-A_{R}(t')]^{2}}\n\\delta\\bigg[ \\tau- \\frac{ \\int_{0}^{t}dt't' A_{L}(t')A_{R}(t')}{\\int_{0}^{t}dt' A_{L}(t')A_{R}(t')}\\bigg]|\\phi\\rangle.  \n\\end{eqnarray}\n\n\tTo summarize, there is a natural clock associated with the random  field, \nwhose time reading depends upon global dynamics and whose deviance decreases with time.  In future work, rather than collapse depending upon time, one may look to express the attractive idea that the notion of time may depend upon collapse, since no collapse implies no events implies no time.  \n\t\n\\section{Two examples}\n\tCollapse models are phenomenological, a guess (based upon a perceived flaw in standard quantum theory) that nature utilizes this description \nof dynamical behavior.  To be generally accepted, such models need \nexperimental verification\\cite{exp} and\ngrounding within a larger physical framework.  One of the main aims of this paper has been to emphasize that CSL can be formulated  as CQC, within an essentially quantum framework, in order to encourage thoughts about how collapse might arise from a physical structure which has a natural quantum theory expression.  In that spirit,  \nthis section contains two examples of CQC dynamics arising from other quantum conceptual structures.\n\n \\subsection{Quantum Fields}\n \n\t Consider a collection of scalar quantum fields, associated with particles of large mass $M$  \nwhich, for definiteness, take to be  the Planck mass, so large that \n$\\sqrt{{\\bf k}^{2}+M^{2}}\\approx M$ for any so-far experimentally achievable ${\\bf k}$.  Each field  (more precisely, its time derivative, which is its conjugate field) interacts with the mass density $A({\\bf x},t)$  (given in Eq.(8) et. seq.) over a brief interval which, for definiteness, we shall imagine is the Planck time $\\tau=M^{-1}$.  Call the field $\\Phi_{t}({\\bf x})$ (and call its conjugate $\\Pi_{t}({\\bf x})$) which is  responsible for the interaction over the interval centered at time $t$;\n\\begin{subequations}\\label{38}\n\\begin{eqnarray}\n\\Phi_{t}({\\bf x})&\\equiv& C\\frac{1}{(2\\pi)^{3\/2}\\sqrt{2M}}\\int {\\bf dk}[e^{i({\\bf k}\\cdot{\\bf x}-Mt)} \nb_{t}({\\bf k})+e^{-i({\\bf k}\\cdot{\\bf x}-Mt)} b_{t}^{\\dagger}({\\bf k})]\\\\\n\\Pi_{t}({\\bf x})&\\equiv& C^{-1}\\frac{i}{(2\\pi)^{3\/2}}\\sqrt{\\frac{{M}}{{2}}}\\int {\\bf dk}[-e^{i({\\bf k}\\cdot{\\bf x}-Mt)} \nb_{t}({\\bf k})+e^{-i({\\bf k}\\cdot{\\bf x}-Mt)} b_{t}^{\\dagger}({\\bf k})]\n\\end{eqnarray}\n\\end{subequations}\t\n\\noindent where the constant $C$ is yet to be chosen.  \n\n\tDefine the eigenstates by $\\Phi_{t}({\\bf x})|\\varphi_{t}\\rangle \n=\\varphi_{t}({\\bf x})|\\varphi_{t}\\rangle $ and the associated vacuum state by $b_{t}({\\bf k})|0\\rangle_{t}=0$. Then, Eqs.(38) imply \n\\begin{equation}\\label{39}\n\\big[\\frac{M}{C^{2}}\\varphi_{t}({\\bf x})+\\frac{\\delta}{\\delta \\varphi_{t}({\\bf x})}\\big]\\langle \\varphi_{t}({\\bf x})|0\\rangle_{t}=0, \\quad\\langle\\varphi_{t}({\\bf x})|0\\rangle_{t}=e^{-\\frac{M}{2C^{2}}\\int d{\\bf x}\\varphi_{t}({\\bf x})^{2}}.\n\\end{equation}\n\\noindent The joint vacuum state for all the fields is $|0\\rangle\\equiv\\prod_{t}|0\\rangle_{t}$ and a joint eigenstate of all the fields is $|\\varphi\\rangle\\equiv\\prod_{t}|\\varphi_{t}({\\bf x})\\rangle_{t}$. To make their scalar product close to  the scalar product  in Eq.(13) (suitably generalized to space-time),  choose $C=M\\sqrt{2\\lambda}$ so, from (39), \n\\begin{equation}\\label{40}\n\\langle\\varphi |0\\rangle=e^{-(4\\lambda)^{-1}\\sum_{t}\\tau\\int d{\\bf x}\\varphi_{t}({\\bf x})^{2}}\n\\approx e^{-(4\\lambda)^{-1}\\int_{-\\infty}^{\\infty}dt\\int d{\\bf x}\\varphi_{t}({\\bf x})^{2}}\n\\end{equation}\n\n\tThe evolution equation is chosen as:  \n\\begin{subequations}\\label{41}\n\\begin{eqnarray}\n\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\n\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\\negthinspace\n\\langle\\varphi | \\psi,t\\rangle&\\equiv&{\\cal T}\ne^{-2i\\lambda\\sum_{t'=0}^{t}\\int d{\\bf x}'\\Pi_{t'}({\\bf x})A({\\bf x}', t')}|0\\rangle|\\phi\\rangle \\approx\ne^{-2i\\lambda\\int_{t'=0}^{t}dt'\\int d{\\bf x}'M\\Pi_{t'}({\\bf x}')A({\\bf x}', t')}|0\\rangle|\\phi\\rangle\\\\\n&=&{\\cal T}e^{-(4\\lambda)^{-1}\\sum_{t'=0}^{t}\\tau \\int d{\\bf x}'[\\varphi_{t'}({\\bf x}')-2\\lambda A({\\bf x}',t')]^{2}}|\\phi\\rangle\ne^{-(4\\lambda)^{-1}[ \\sum_{t'=-\\infty}^{0}+ \\sum_{t'=t}^{\\infty}] \\tau \\int d({\\bf x}')\\varphi_{t'}^{2}({\\bf x}')}\n\\end{eqnarray}\n\\end{subequations}\t\n\n\tTo complete the demonstration that this model is equivalent to CQC,  identify the fields\n\\begin{equation}\\label{42}\n\\tilde{W}({\\bf x},t)\\equiv \\Phi_{t}({\\bf x}),\\quad \\tilde{\\Pi}({\\bf x},t)\\equiv M \\Pi_{t}({\\bf x})\n\\end{equation}\n\\noindent from which follows \n\\[\n[\\tilde{W}({\\bf x},t), \\tilde{\\Pi}({\\bf x}',t')]=\\delta({\\bf x}-{\\bf x}')\\delta_{t,t'}\/\\tau\\approx \\delta({\\bf x}-{\\bf x}')\\delta(t-t').\n\\]\n\\noindent It is immediately seen, using Eq.(42) for  $\\tilde{\\Pi}({\\bf x},t)$, that the statevector evolution equation (41a) is identical to Eq.(15a) (generalized to space-time).\n\n\tTo see that the expressions for $\\tilde{W}$ and $\\tilde{\\Pi}$ in terms of particle creation and annihilation operators closely approximate the similar expressions for $W$ and $\\Pi$,  note that \n\\begin{equation}\\label{43}\n[e^{-iMt}b_{t}({\\bf k})\/\\sqrt{\\tau}, e^{iMt}b_{t}^{\\dagger}({\\bf k})\/\\sqrt{\\tau}]=\\delta({\\bf k}-{\\bf k}')\n\\delta_{t,t'}\/\\tau\\approx\\delta({\\bf k}-{\\bf k}')\\delta(t-t').\n\\end{equation}\n\\noindent Therefore,  define \n\\begin{equation}\\label{44}\n\\tilde{b}({\\bf k}, \\omega)\\equiv(2\\pi)^{-1\/2}\\int dt' e^{i\\omega t'}e^{-iMt'}b_{t'}({\\bf k}')\/\\sqrt\\tau, \n\\end{equation}\n\\noindent so $[\\tilde{b}({\\bf k}, \\omega),\\tilde{b}^{\\dagger}({\\bf k}', \\omega ']=\\delta({\\bf k}-{\\bf k}')\\delta(\\omega-\\omega ')$  follows from Eqs.(43),(44).  Inserting the inverse of Eq.(44), \nfor $b_{t}({\\bf k})$ in terms of $\\tilde{b}({\\bf k}, \\omega)$, into Eqs.(38),(42) gives for  $\\tilde{W}$ and $\\tilde{\\Pi}$ expressions whose form is identical to (10)'s expressions (generalized to space-time) for $W$ and $\\Pi$.  \n\n\\subsection{Many Spins} \n\t \n\tThe model described here borrows from a gravitationally based \nsemi-classical collapse model\\cite{pearlesquires} and spin-network models of \nspace-time\\cite{smolinseth}, without being faithful to either.  Suppose that space consists \nof ``Planck cells\" (each of volume $\\ell^{3}$, where $\\ell$ is the Planck length), each containing a \nspin which is normally rigidly enmeshed with the other spins, but which occasionally breaks loose.  Then,  \n for a duration equal to the Planck time $\\tau$, it interacts both with a thermal bath,  \n(characterized by the energy $\\beta^{-1}$) and gravitationally \nwith nearby particles (more on this shortly).  Following this brief dynamics it is \nfrozen, no longer interacting with either bath or particles, once more enmeshed with other spins.\n\t\n\tCharacterize the spin in cell i located at ${\\bf x}_{i}$, which interacts over the interval $(t-\\tau,t)$ and then freezes at time $t$, by $\\sigma_{i,t}$ (a Pauli z-spin matrix) and mass $\\mu\\sigma_{i,t},$ where $\\mu$ is the Planck mass.  Suppose that a spin-mass at ${\\bf x}_{i}$ interacts gravitationally only with the particles in a sphere of radius $a$ around it,  and that the interaction is as if the particles are uniformly smeared over that sphere, with resulting mass density represented by the operator $\\rho ({\\bf x}_{i})$.  For simplicity,  take the particle Hamiltonian to vanish, so that $\\rho$ is independent of $t$.  Then, since \nany initial statevector for the particles can be written as the sum of joint eigenstates of $\\rho ({\\bf x}_{i})$ \nfor all ${\\bf x}_{i}$,  calculations for the statevector need consider only the special case of one such term, for which $\\rho ({\\bf x}_{i})$ is a c-number, with the more general case being reconstructed afterward as the sum.  Then, for one such term,  the Hamiltonian for the gravitational interaction of all the spins which have \ninteracted with the particles over the interval $(0,t)$ (up to a numerical factor which, for simplicity, is taken equal to 1)  is \n\\begin{equation}\\label{45}\nH= -G\\mu a^{2}\\sum_{i,t'=\\tau}^{t}\\sigma_{i,t'}\\rho ({\\bf x}_{i}). \n\\end{equation}\n\n\tBecause all the spins act independently,  focus upon \na group of spins excited over the time interval $(t', t'+\\Delta t)$ which lie within a volume $\\Delta V$.  Take $\\Delta t$ and $\\Delta V$ to be large ($\\Delta t\/\\tau>>1$, $\\Delta V\/\\ell^{3}>>1$), large enough to contain $N$ interacting spins where $N>>1$, but still small on the scale of particle physics (e.g., \n$\\rho ({\\bf x}_{i})$ is essentially constant over $\\Delta V$).  Denote by $p$ the probability that \na spin in a cell is freed up to interact over a time interval $\\tau$ so, on average, \n\\begin{equation}\\label{46}\nN=p\\frac{\\Delta V}{\\ell^{3}}\\frac{\\Delta t}{\\tau}.\n\\end{equation}\n\\noindent For simplicity, we shall take (46) to be the exact expression for $N$, not just for its average.   \n\n\tNow consider the (interaction picture) statevector $|\\psi,t\\rangle$.  It may be written as the direct product of  states associated with all volumes $\\Delta V$ and all time intervals \n$\\Delta t$  within $(0, t)$. The state which is the contribution of one such volume and interval has the form \n\\begin{equation}\\label{47}\n|\\chi\\rangle=\\sum_{s_{1}=-1}^{1}...\\sum_{s_{N}=-1}^{1}|s_{1}...s_{N}\\rangle|\\hbox {bath},s_{1}...s_{N}\\rangle C_{s_{1}...s_{N}}(\\rho).\n\\end{equation}\n\\noindent The sum in Eq.(47) contains all the spin states entangled with the associated bath states which interacted with the spins and brought them to thermal equilibrium:  assume for simplicity that the bath states are mutually orthogonal.  \n \n\tThen, $|\\chi\\rangle$'s contribution to the density matrix describing the spins alone  is\n\\begin{equation}\n\\hbox{Tr}_{\\hbox{bath}}|\\chi\\rangle\\langle\\chi |=\\sum_{s_{1}=-1}^{1}...\\sum_{s_{N}=-1}^{1}|s_{1}...s_{N}\\rangle \\langle s_{1}...s_{N}| |C_{s_{1}...s_{N}}(\\rho)|^{2}.\\nonumber \n\\end{equation}\n\\noindent On the other hand, because these states are all in thermal equilibrium,  using Eq.(45)'s Hamiltonian for these spins, the thermal density matrix is:\n\\begin{equation}\n\\sum_{s_{1}=-1}^{1}...\\sum_{s_{N}=-1}^{1}|s_{1}...s_{N}\\rangle \\langle s_{1}...s_{N}| \n\\frac{e^{\\beta G\\rho \\mu a^{2}\\sum_{i=1}^{N}s_{i}}}{\\sum_{s_{1}=-1}^{1}...\\sum_{s_{N}=-1}^{1}e^{\\beta G\\rho \\mu a^{2}\\sum_{i=1}^{N}s_{i}}}.\\nonumber \n\\end{equation}\t\nThus, by comparison of these two equations, \n\\begin{equation}\\label{48}\n|C_{s_{1}...s_{N}}(\\rho)|^{2}=\n\\frac{e^{\\beta G\\rho \\mu a^{2}\\sum_{i=1}^{N}s_{i}}}{\\sum_{s_{1}=-1}^{1}...\\sum_{s_{N}=-1}^{1}e^{\\beta G\\rho \\mu a^{2}\\sum_{i=1}^{N}s_{i}}}.\n\\end{equation}\n\n\tEventually, $W$ associated with this space-time region will be taken $\\sim S\\equiv \\sum_{i=1}^{N}\\sigma_{i}$.  The statevector which is the sum of all states in (47) which have the same eigenvalue of $S$ \nand which is normalized to 1 is:  \n\\begin{equation}\n|s\\rangle\\equiv\\sum_{\\{s_{i}\\}, \\sum s_{i}=s}|s_{1}...s_{N}\\rangle|\\hbox {bath},s_{1}...s_{N}\\rangle \\frac{C_{s_{1}...s_{N}}(\\rho)}{\\sqrt{\\sum_{\\{s_{i}\\},\\sum s_{i}=s}|C_{s_{1}...s_{N}}|^{2}}}\\nonumber, \n\\end{equation}\nso, using (48), \n\\begin{equation}\\label{49}\n\\langle s|\\chi\\rangle=\\bigg[\\sum_{\\{s_{i}\\}, \\sum s_{i}=s}|C_{s_{1}...s_{N}}(\\rho)|^{2}\\bigg]^{1\/2}=\n\\bigg[\\frac{\\sum_{\\{s_{i}\\},\\sum s_{i}=s}e^{\\beta G\\rho \\mu a^{2}\\sum_{i=1}^{N}s_{i}}}{\\sum_{\\{s_{i}\\}}e^{\\beta G\\rho \\mu a^{2}\\sum_{i=1}^{N}s_{i}}}\\bigg]^{1\/2}.\n\\end{equation}\t\n\n\tEvaluation of (49) is well known but, for completeness, it is sketched here.  The \npartition function in the denominator of (49) is (writing $C= G\\rho \\mu a^{2}$ and taking $N$ even)\n\\begin{equation}\\label{50}\n\\hbox{Tr}e^{\\beta CS} =\\sum_{k=0}^{N}e^{\\beta C(N-2k)}\\frac{N!}k!{(N-k)!}=(2\\cosh\\beta C)^{N}, \n\\end{equation}\t\n\\noindent so the expression in (49) under the square root is \t\n\\begin{subequations}\\label{51}\n\\begin{eqnarray}\n\\sum_{\\{s_{i}\\}, \\sum s_{i}=s}|C_{s_{1}...s_{N}}(\\rho)|^{2}&=&\\frac{e^{\\beta Cs}}{(2\\cosh\\beta C)^{N}}\\frac{N!}{[(N-s)\/2]![(N+s)\/2]!}\\\\\n&\\approx&\\frac{1}{\\sqrt{2\\pi N \\cosh^{2}\\beta C}}e^{-[s-N \\tanh\\beta C]^{2}\/2N\\cosh^{2}\\beta C}\\\\\n&\\approx&\\frac{1}{\\sqrt{2\\pi N}}e^{-[s-N\\beta C]^{2}\/2N}.\n\\end{eqnarray}\n\\end{subequations}\t\t\n\\noindent In (51b) the well-known gaussian approximation to the binomial distribution has been employed\\cite{Feller} \nwhich is valid for large $N$ and for s within a few standard deviations of its mean.  \nThe approximation in (51c) depends upon validity of the inequality\n\\begin{equation}\\label{52}\n\\beta G\\rho \\mu a^{2}<<1\n\\end{equation}\n\\noindent which shall need to be checked.  \n\n\tNow it is possible to put together the contributions of all volumes $\\Delta V$ and all time intervals \n$\\Delta t$ within $(0, t)$ to construct the statevector $|\\psi,t\\rangle$.  \nDefine $|\\tilde{s}\\rangle\\equiv\\prod_{\\Delta V, \\Delta t}|s\\rangle_{i,j}$,  the normalized basis vector which is the joint eigenvector of $S({\\bf x}_{i},t_{j})$ for all ${\\bf x}_{i}$ and for $0\\leq t_{j}\\leq t$.  The scalar \nproduct  $\\langle \\tilde{s} |\\psi,t\\rangle$ is the direct product of the expressions (49) so, from (51c) (without the normalization factors, which are tucked into the element of integration):\n\\begin{subequations}\\label{53}\n\\begin{eqnarray}\n\\langle \\tilde{s} |\\psi,t\\rangle&=&e^{-(4N)^{-1}\\sum_{{\\bf x}_{i},t_{j}=0}^{t}[s_{i,j}-N\\beta G\\rho \\mu a^{2}]^{2}}\\\\\n&=&e^{-4^{-1}N(\\beta G \\mu a^{2})^{2}\\sum_{{\\bf x}_{i},t_{j}=0}^{t}[s_{i,j}\/N\\beta G \\mu a^{2}-\\rho_{i}]^{2}}\\\\\n&\\approx&e^{-4^{-1}(p\/\\ell^{3}\\tau)(\\beta G \\mu a^{2})^{2}\\int d{\\bf x}\\int_{0}^{t}dt'[s'(\\bf {x},t')-\\rho(\\bf {x})]^{2}},\n\\end{eqnarray}\n\\end{subequations}\n\\noindent where (53c) follows from (46), and we have defined:\n\\begin{equation}\\label{54}\ns'({\\bf x},t)\\equiv s_{i,j}\/N\\beta G \\mu a^{2}.  \n\\end{equation}\n\tNow,  compare (53c) with CQC's comparable expression.  Note from (8) \nthat $A({\\bf x})=m_{0}^{-1}a^{3\/2}\\rho(\\bf {x})$ (up to a numerical factor which shall be ignored) so the CQC expression comparable to Eq.(9) is\n\\begin{equation}\\label{55}\n\\langle w |\\psi,t\\rangle=e^{-(4\\lambda)^{-1}\\int d{\\bf x}\\int_{0}^{t}dt'[w({\\bf x},t')-2\\lambda m_{0}^{-1}a^{3\/2} \\rho(\\bf {x})]^{2}}.  \n\\end{equation}\n\\noindent  The model's Eq.(53c) and the CQC Eq.(55) are identical if   \n\\begin{subequations}\\label{56}\n\\begin{eqnarray}\nW({\\bf x},t)&=&\\frac{2\\lambda a^{3\/2}}{m_{0}}S'({\\bf x},t)=\\frac{2\\lambda}\n {(\\beta m_{0}c^{2})\\ell a^{1\/2}}\\frac{S({\\bf x},t)}{N},\\\\\n4(\\lambda \\tau)(\\ell\/a)&=& p(\\beta m_{0}c^{2})^{2}, \n \\end{eqnarray}\n\\end{subequations}\t\n\\noindent where $G\\mu=\\ell c^{2}$ has ben used, and $S'{\\bf x},t)$ is the  operator whose eigenvalues are $s'({\\bf x},t)$ given in Eq.(54).\n\n\tThus, this model produces CSL dynamics.  \n\t\n\tIt remains to check whether the numerical values in Eqs.(52), (56) are reasonable.\nConsider two possible temperatures for the bath, the cosmic radiation temperature $\\beta^{-1}\\approx 2\\cdot 10^{-4}$ eV. and the Planck temperature $\\beta^{-1}\\approx 10^{28}$ eV.  For normal mass \ndensities, $\\rho\\approx 1 \\hbox{gm\/cc}\\approx 5\\cdot 10^{33}\\hbox{eV\/cc}$, one gets\nrespectively $\\beta G\\rho \\mu a^{2}\\approx 10^{-6}$  and $10^{-38}$, so the inequality of \nEq.(52) is satisfied for both.  \n\n\tFor these two temperatures, (56b) gives $p\\approx 10^{-112} $ and $10^{-49}$ respectively.  \nThe second number is too small for the conceptual picture we have outlined since, after \n$10^{49}\\tau\\approx 10^{5}$sec, the spin in every cell is likely to have interacted and frozen, so the \nprocess would cease.  However, one may change the conceptual picture.  It is possible for a spin in a cell to interact repeatedly, without changing the dynamical equation because the bath states \n preserve the past orthogonality of the realizable states.   Although they are represented by bath states, they are labeled by the past values of W({\\bf x},t).  \nEach of these orthogonal realizable states at time $t$ evolves independently of the other states over the next $\\Delta t$, via the CSL evolution.  \n\n\tLastly, look at Eq.(56a).  Note that  $W\\sim S\/N$ is intensive, proportional to the mean \nvalue of the spin in a cell, so this result is independent of the size of $\\Delta {\\bf x}$, $\\Delta t$ \nas it should be.  For the numerical coefficient in (56a), it is more informative to discuss $S'({\\bf x},t)$'s relation to  $S({\\bf x},t)$ rather than \n$W({\\bf x},t)$'s relation to either,  because $S'({\\bf x},t)$'s  mean value is simply  $\\rho$ (see (53c)).  One obtains from Eq.(56a), for a cosmic bath and for a Planck bath respectively,  \n$S'\\approx (S\/N)10^{39}$eV\/cc and $S'\\approx (S\/N)10^{71}$eV\/cc.  Both factors are large compared \nto normal mass density $\\approx 10^{34}$eV\/cc, so that the mean spin value $S\/N$ \n for such a density  is respectively $10^{-5}$ and $10^{-37}$.  \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOver the past few years much attention has been devoted to \nthe study of traffic flow. \nSince the seminal work of Lighthill and Whitham \nin the middle of the 50's \\cite{LIGHT_1} many attempts have been\nmade to construct more and more sophisticated models which incorporate\nvarious phenomena occurring in real \ntraffic (for an overview see \\cite{WOLF}).\nRecently, a new class of models, based on the idea of cellular\nautomata, has been proven to describe traffic dynamics\nin a very efficient way \\cite{NASCH_1}.\nEspecially the transition from free flow to jammed\ntraffic with increasing car density could be investigated\nvery accurately.\nNevertheless, besides various indications \\cite{CSANYI_1}, \nno unique description for a dynamical transition has \nbeen found (see for instance \\cite{SASVARI_1,LUEB_6} and\nreferences therein).\nIn this article we consider a method of analysis\nwhich allows us to identify the different phases of the system\nand to describe the phase transition, i.e., considering the \nfluctuations which drive the transition,\nand determining the phase diagram.\n\n\nWe consider a one-dimensional cellular automaton of\nlinear size $L$ and $N$ particles.\nEach particle is associated the integer values \n$v_i\\in\\{0,1,2,...,v_{\\rm max}\\}$ and $d_i\\in\\{0,1,2,3,...\\}$,\nrepresenting the velocity and the distance to the next\nforward particle \\cite{NASCH_1}.\nFor each particle, the following \nupdate steps representing the acceleration, the slowing down, the noise,\nand the motion of the particles are done in parallel:\n(1) if $v_i < d_i$ then $v_i \\to \\mbox{Min}\\{v_i+1, v_{\\rm max}\\}$,\n(2) if $v_i > d_i$ then $v_i \\to d_i$,\n(3) with probability $P\\\/$ $v_i \\to \\mbox{Max}\\{v_i-1, 0\\}$, \nand\n(4) $r_i \\to r_i+v_i$, \nwhere $r_i$ denotes the position of the $i$-th particle.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Simulation and Results}\n\n\n\n\\begin{figure}[b]\n \\begin{center}\n \\epsfxsize=8.6cm\n \\epsfysize=7.0cm\n \\epsffile{space_time_dia.eps} \n \\end{center}\n \\vspace{-0.8cm}\n \\caption{Space-time plot for $v_{\\rm max}=5$, $P=0.5$, and $\\rho_g>\\rho_c$.\n          Note the separation of the system in high and low density\n\t  regions.}\n \\label{space_time_dia} \n\\end{figure}\n\n\n\n\n\n\nFigure~\\ref{space_time_dia} shows a space-time plot of the system.\nEach dot corresponds to a particle at a given time step.\nThe global density $\\rho_g=N\/L$ exceeds the critical density and \njams occur.\nTraffic jams are characterized by a high local density of the\nparticles and by a backward movement of shock waves \\cite{LIGHT_1}.\nOne can see from Fig.~\\ref{space_time_dia} that in the jammed regime \nthe system is inhomogeneous, \ni.e., traffic jams with a high local density and free flow regions with\na low local density coexist.\nIn order to investigate this transition one has to take this\ninhomogeneity into account.\n\nTraditionally one determines the so-called fundamental diagram, \ni.e.,~the diagram of the flow vs the density.\nThe global flow is given by, \n$\\Phi \\; = \\; \\rho_g \\, \\langle v \\rangle$,\nwhere $\\langle v \\rangle$ denotes the averaged velocity of the \nparticles.\nThis non-local measurements are not sensitive to the inhomogeneous\ncharacter of the system, i.e.,~the information\nabout the two different coexisting phases is lost.\nIn the following we consider\na method of analysis which is based on the measurement of \nthe local density distribution $p(\\rho)$~\\cite{LUEB_6}.\nThe local density $\\rho$ is measured on a section of the \nsystem of size $\\delta$ according to\n\\begin{equation}\n\\rho \\; = \\; \\frac{1}{\\rho_g \\delta} \\, \\sum_{i=1}^{N} \\,\n\\theta(\\delta -r_i).  \n\\label{eq:density_dist}\n\\end{equation}\n\n\n\n\n\n\\begin{figure}[p]\n \\begin{center}\n \\epsfxsize=8.0cm\n \\epsfysize=8.0cm\n \\epsffile{density_dist.eps} \n \\end{center}\n \\vspace{-0.9cm}\n \\caption{The local density distribution $p(\\rho)$ for various\n\t  values of the global density, $v_{\\rm max}=5$, $P=0.5$\n\t  and $\\delta=256$. \n\t  The dashed line corresponds to the characteristic density \n\t  of the free flow phase.}\n \\label{density_dist} \n \\vspace{0.5cm}\n \\begin{center}\n \\epsfxsize=7.4cm\n \\epsfysize=7.4cm\n \\epsffile{b_8192_0128_2.ps} \n \\end{center}\n \\vspace{-0.0cm}\n \\caption{The local density distribution $p(\\rho_g,\\rho)$ as a \n\t  function of the global density (horizontal axis) and \n\t  local density (vertical axis), respectively.\n\t  The colors correspond to the values of the probability \n\t  $p(\\rho_g,\\rho)$, increasing from black to red.}\n \\label{density_dist_2} \n\\end{figure}\n\n\n\n\n\nThe local density distribution $p(\\rho)$ is plotted \nfor various values of the global \ndensity $\\rho_g$ in Fig.~\\ref{density_dist}.\nIn the case of small values of $\\rho_g$, see Fig.~\\ref{density_dist}a, \nthe particles can\nbe considered as independent (see below) and the \nlocal density distribution is simply Gaussian with\nthe mean values $\\rho_g$ and a width which scales \nwith $\\sqrt{\\delta}$. \nIncreasing the global density, jams occur\nand the distribution displays two different \npeaks (Fig.~\\ref{density_dist}c).\nThe first peak corresponds to the density of free particles and\nin the phase coexistence regime the position of this peak\ndoes not depend on the global density (see the dashed lines\nin Fig.~\\ref{density_dist}).\nThe second peak is located at larger densities and characterizes\nthe jammed phase.\nWith increasing density the second peak occurs in the vicinity\nof the critical density $\\rho_c$ (Fig.~\\ref{density_dist}b)\nand grows further (Fig.~\\ref{density_dist}c) until it dominates\nthe distribution in the sense that the first peak \ndisappears (Fig.~\\ref{density_dist}d).\nThe two peak structure of the local density distribution clearly \nreflects the coexistence of the free flow and \njammed phase above the critical value $\\rho_c$.\nIn Fig.~\\ref{density_dist_2} we present the \nprobability distribution as function of the\nglobal and local density.\nAbove a certain value of the global density $\\rho_g$  \nthe two peak structure occurs.\nThe behavior of the first peak yields a criterion to \ndetermine the transition point~\\cite{LUEB_6} and one gets\n$\\rho_c=0.0695\\pm 0.0007$ for $P=0.5$ and $v_{\\rm max}$,\nrespectively.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[t]\n \\begin{center}\n \\epsfxsize=8.6cm\n \\epsfysize=7.5cm\n \\epsffile{struc_5.eps} \n \\vspace{-0.8cm}\n \\caption{The structure factor $S(k)$ for $P=0.5$, $v_{\\mbox{max}}=5$\n\t  and for various values of the\n\t  density $\\rho$. The dashed line marks the characteristic\n\t  wavelength $\\lambda_0$ of the free flow phase.}\n \\label{struc_5} \n \\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\n\nIn order to describe the spatial decomposition of the\ncoexisting phases we measured the steady state structure factor \\cite{SCHMITT_1}\n\\begin{equation}\nS(k) \\; = \\; \\frac{1}{L} \\left \\langle \\left | \\, \\sum_{r=1}^{L} \\,\n\\eta(r) \\, e^{i k r} \\right|^2 \\right \\rangle,\n\\label{eq:structure_factor}\n\\end{equation}\nwhere $\\eta(r)=1$ if the lattice site $r$ is occupied and\n$\\eta(r)=0$ otherwise.\nIn Fig.~\\ref{struc_5} we plot the structure factor $S(k)$\nfor the same values of the global density as in Fig.~\\ref{density_dist}, \ni.e., below, in the vicinity, above and far away of the transition\npoint.\nIt is remarkable that $S(k)$ exhibits a maximum for all considered\nvalues of the global density at $k_0 \\approx 0.72$ (dashed lines in\nFig.~\\ref{struc_5}). \nThis value correspondence to the characteristic wave length \n$\\lambda_0=\\frac{2 \\pi}{k_0}$ of the density fluctuations \nin the free flow phase.\nThe steady state structure factor is related to the \nFourier transform of the real space density-density\ncorrelation function.\nThe wave length $\\lambda_0$ corresponds to a maximum of the\ncorrelation function, i.e., $\\lambda_0$ describes the \nmost likely distance of two particles in the free flow phase.\nFor low densities the structure factor is almost independent\nof the density and displays a minimum\nfor small $k$ values indicating the lack of long-range correlations.\nCrossing the transition point the smallest mode \n$S(k=\\frac{2 \\pi}{L})$ increases quickly. \nThis suggests that the jammed phase is characterized by \nlong-range correlations which decay in the limit $\\rho_g \\gg \\rho_c$ \nalgebraically as one can see from the log-log plot \nin Fig.~\\ref{struc_5}d.\n\n\n\\begin{figure}[t]\n \\begin{center}\n \\epsfxsize=8.6cm\n \\epsfysize=7.0cm\n \\epsffile{phase_dia.eps} \n \\end{center}\n \\vspace{-1.0cm}\n \\caption{The phase diagram of the Nagel-Schreckenberg model.\n\t  Note that in the non-deterministic region $0<P<1$ \n\t  the density of the maximum flow exceeds \n\t  the density of the transition point.}\n \\label{phase_dia}\n\\end{figure}\n\n\n\n\n\nUp to now we only considered the case $P=0.5$.\nThe phase diagram in Fig.~\\ref{phase_dia} shows the $P$ dependence \nof the transition density $\\rho_c$.\n{\\it f} denotes the free flow phase and\n{\\it f+j} corresponds to the coexistence region\n where the system separates in the\nfree flow and jammed phase.\nThe dashed line displays the $P$ dependence of the maximum\nflow obtained from an analysis of the fundamental\ndiagram \\cite{EISEN_1}.\nThe critical densities $\\rho_c$, where the phase transition \ntakes place, are lower than \nthe density values of the maximum flow.\nMeasurements of the relaxation time, which is expected to\ndiverge at a transition point \\cite{CSANYI_1},\nconfirm this result \\cite{EISEN_2} (see Fig.~\\ref{phase_dia}).\nBut one has to mention that the determination of the critical\ndensity via relaxation times leads in the coexistence regime\n{\\it f+j} to unphysical results, in the sense\nthat the relaxation time becomes negative \\cite{EISEN_1,EISEN_2}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\n\n\nIn conclusion we have studied numerically the Nagel-Schreckenberg \ntraffic flow model using a local density analysis.\nCrossing the critical line of the system\na phase transition takes place from a \nhomogeneous regime (free flow phase)\nto an inhomogeneous regime which is characterized by a \ncoexistence of two phases (free flow traffic and jammed traffic).\nThe decomposition in the phase coexistence regime\nis driven by density fluctuations, provided they exceed\na critical wavelength $\\lambda_c$.\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe divergences in quantum field theory (QFT) are currently generally dealt with by using the techniques of regularization and renormalization.  Two of the principal methods of regularization are Pauli-Villars regularization and dimensional regularization. Both of these methods involve modifying a divergent integral to form an integral which exists in a manner depending on a parameter where the parameter is a momentum cutoff $\\Lambda$ for Pauli-Villars regularization or a perturbation $\\epsilon>0$ of the space-time dimension $D=4-\\epsilon$ for dimensional regularization. The parameter is then varied towards the value that it would have if the divergent integral existed ($\\Lambda\\rightarrow\\infty$ for Pauli-Villars, $\\epsilon\\rightarrow0$ for dimensional) and the badly behaved contributions (e.g. terms of the order of $\\log(\\Lambda)$ or $\\epsilon^{-1}$) are subtracted out or ignored to obtain finite answers which can be compared with experiment. The process of removing the badly behaved contributions is called renormalization (e.g. minimal subtraction).\n\nMany of the initial developers of QFT such as Dirac and Feynman were not happy with the fact that many of the integrals in QFT, in particular, those involving fermion loops, do not exist in the mathematical sense but more recently, especially through the important work of Wilson and others on the renormalization group, the parameterized variation with scale has been seen to have physical significance and not just the result of a mathematical artifice. In particular the concept of running coupling constant and the comparison of its behavior in QED and QCD, which is asymptotically free, is seen to be of great physical significance. The significance has propagated into other areas of physics such as solid state physics and critical phenomena in condensed matter physics. The exact renormalization group equations (ERGE) of both Wilson and Polchinski involve cutoffs (a smooth ultraviolet regulator in the work of Polchinski).\n\nWe believe, nevertheless, that it would be desirable to have well defined initial equations or principles as a starting point for physical theory which are such that concepts such as the running coupling constant would follow from these basic principles. We have shown in a previous paper (Mashford, 2017b) how one can, through a brief formal argument, consider the problematical objects in QFT as being Lorentz invariant Borel complex measures (more generally $K$ invariant ${\\bf C}^{4\\times4}$ valued measures) on Minkowski space. We will repeat this derivation in the present paper for the case of the contraction of the vacuum polarization tensor. Having given a definition of the objects as well defined mathematical objects one can proceed and analyze these objects, computing the consequences of assuming them, without infinities or ill-definedness propagating through the calculations.\n\nIt can be shown (see Appendix 3) that any Lorentz invariant Borel complex measure on Minkowski space has a certain spectral representation.\nAn important part of this paper is the presentation of a spectral calculus whereby the spectrum of a causal Lorentz invariant Borel measure on Minkowski space can be calculated, where by causal is meant that the support of the measure is contained in the closed future null cone of the origin.\n\nIf, using the spectral calculus, one can obtain a spectrum for a causal Lorentz invariant Borel measure which is a continuous function (or, more generally, a sufficiently well behaved measurable function) then, as we will show, one can compute an equivalent density for the measure with respect to Lebesque measure on ${\\bf R}^4$ which can be used in QFT calculations. \n\nWe will show, generally, how to convolve or form products of causal Lorentz invariant Borel measures using their spectral representations. This is to be compared to the work of Scharf and others, dating back to the paper of Epstein and Glaser, (1973)  on forming products of causal distributions.\n\nThe concept of spectral representation in QFT dates back to the work of K\\\"{a}llen (1952) and Lehmann (1954) who, independently, proposed the representation\n\\begin{equation}\n<0|[\\phi(x),\\phi^{\\dagger}(y)]|0>=i\\int_0^{\\infty}d{m^{\\prime}}^2\\sigma({m^{\\prime}}^2)\\Delta_{m^{\\prime}}(x-y),\n\\end{equation}\nfor the commutator of interacting fields where $\\Delta_{m^{\\prime}}$ is the Feynman propagator corresponding to mass $m^{\\prime}$. Itzikson and Zuber (1980) state, with respect to $\\sigma$, ``In general this is a positive measure with $\\delta$-function singularities.\" While K\\\"{a}llen, Lehmann and others propose and use this decomposition they do not present a way to compute the spectral measure $\\sigma$. As mentioned above one of the main results of the present paper is a presentation of the spectral calculus which enables one to compute the spectral function of a causal Lorentz invariant Borel measure on Minkowski space. This spectral calculus is quite easy to use in practice but it is somewhat tedious to prove rigorously its validity (see Appendix 6).\n\nIn Section 3 of the paper we use the spectral calculus and other methods to compute the spectrum of the measure $\\Omega_m*\\Omega_m$ which is a convolution of the standard Lorentz invariant measure on the mass $m$ mass shell (i.e the Feynman propagator corresponding to mass $m$ on the space of positive energy functions) with itself, where $m>0$. In Section 4 we use general arguments to compute the spectrum of $\\Omega_{im}*\\Omega_{im}$ where $\\Omega_{im}$ is standard Lorentz invariant measure on the imaginary mass hyperboloid corresponding to mass $im$, $m>0$. These computations form practice for the main application of the paper which is an investigation in Section 7 of vacuum polarization, i.e. the self energy of the photon.\n\nIn Section 7 we compute the spectral function and hence the density associated with the complex measure obtained by contracting the vacuum polarization tensor. This is used to define our spectral vacuum polarization function. Our function is seen to agree with a high degree of accuracy (up to finite renormalization) with the vacuum polarization function obtained using regularization\/renormalization. \n\nWe follow Weinberg and others' method for the computation of the Uehling contribution to the Lamb shift in the H atom. Ours differs because we have a different vacuum polarization function in the imaginary mass regime. We compute using the Born approximation a value for the Uehling effect of $\\approx-28.7$ MHz for the hydrogen atom.\n\nWe compute and display the running coupling constant for 1 loop QED. This computation is shown to be convergent when the spectral vacuum polarization function is used while the standard method using the vacuum polarization function obtained using regularization\/renormalization is shown to be divergent for all non-zero energies. \n\n\\section{A spectral calculus of Lorentz invariant measures}\n\nConsider the following general form of a complex measure $\\mu$ on Minkowski space.\n\\begin{equation} \\label{eq:invariant1}\n\\mu(\\Gamma)=c\\delta(\\Gamma)+\\int_{m=0}^{\\infty}\\Omega^{+}_m(\\Gamma)\\,\\sigma_1(dm)+\\int_{m=0}^{\\infty}\\Omega_m^{-}(\\Gamma)\\,\\sigma_2(dm)+\\int_{m=0}^{\\infty}\\Omega_{im}(\\Gamma)\\,\\sigma_3(dm),\n\\end{equation}\nwhere $c\\in{\\bf C}$ (the complex numbers), $\\delta$ is the Dirac delta function (measure), $\\sigma_1,\\sigma_2,\\sigma_3:{\\mathcal B}([0,\\infty))\\rightarrow{\\bf C}$ are Borel complex measures (where ${\\mathcal B}([0,\\infty))$ denotes the Borel algebra of $[0,\\infty)$), $\\Omega_m^{+}$ is the standard Lorentz invariant measure concentrated on the mass shell $H_m^{+}$ (see (Mashford, 2017b)), $\\Omega_m^{-}$ is the standard Lorentz invariant measure concentrated on the mass shell $H_m^{-}$ and $\\Omega_{im}$ is the standard Lorentz invariant measure on the imaginary mass hyperboloid $H_{im}$. Then $\\mu$ is a Lorentz invariant measure. Conversely we have the following.\n\\newtheorem{theorem}{Theorem}\n\\begin{theorem}{The Spectral Theorem.}\nLet $\\mu:{\\mathcal B}({\\bf R}^4)\\rightarrow{\\bf C}$ be a Lorentz invariant Borel complex measure. Then $\\mu$ has the form of Eq.~\\ref{eq:invariant1} for some $c\\in{\\bf C}$ and  Borel spectral measures $\\sigma_1,\\sigma_2$ and $\\sigma_3$.\n\\end{theorem}\nThe proof of this theorem is given in Appendix 3.\n\nIf $\\sigma_2=\\sigma_3=0$ then $\\mu$ will be said to be {\\em causal} or a type I measure. If $\\sigma_1=\\sigma_3=0$ then $\\mu$ will be said to be a type II measure and if $c=0$ and $\\sigma_1=\\sigma_2=0$ then $\\mu$ will be said to be a type III measure. Thus any Lorentz invariant measure is a sum of a type I measure, a type II measure and a type III measure. In particular, any measure of the form\n\\begin{equation} \\label{eq:invariant2}\n\\mu(\\Gamma)=\\int_{m=0}^{\\infty}\\sigma(m)\\Omega^{+}_m(\\Gamma)\\,dm,\n\\end{equation} \nwhere $\\sigma$ is locally integrable function and the integration is carried out with respect to the Lebesgue measure, is a causal Lorentz invariant Borel complex measure. If $\\sigma$ is polynomially bounded then $\\mu$ is a tempered measure.\n\nThe spectral calculus that we will now explain is a very simple way to compute the spectrum $\\sigma$ of a Lorentz invariant measure $\\mu$ if we know that $\\mu$ can be written in the form of Eq.~\\ref{eq:invariant2} and $\\sigma$ is continuous.\n\nFor $m>0$ and $\\epsilon>0$ let $S(m,\\epsilon)$ be the hyperbolic (hyper-)disc defined by\n\\begin{equation}\nS(m,\\epsilon)=\\{p\\in{\\bf R}^4:p^2=m^2,|{\\vct p}|<\\epsilon, p^0>0\\},\n\\end{equation}\nwhere, as usual in QFT, $p^2=\\eta_{\\mu\\nu}p^{\\mu}p^{\\nu}=(p^0)^2-(p^1)^2-(p^2)^2-(p^3)^2$ and ${\\vct p}=\\pi(p)=(p^1,p^2,p^3)$. For $a,b\\in{\\bf R}$ with $0<a<b$ let $\\Gamma(a,b,\\epsilon)$ be the hyperbolic cylinder defined by\n\\begin{equation}\n\\Gamma(a,b,\\epsilon)=\\bigcup_{m\\in(a,b)}S(m,\\epsilon).\n\\end{equation}\nNow suppose that we have a measure in the form of Eq.~\\ref{eq:invariant2} where $\\sigma$ is continuous. Then we can write (using the notation of (Mashford, 2017b))\n\\begin{eqnarray}\n\\mu(\\Gamma(a,b,\\epsilon)) & = & \\int_{m=0}^{\\infty}\\sigma(m)\\Omega_m(\\Gamma(a,b,\\epsilon))\\,dm \\nonumber \\\\\n    & = & \\int_{m=0}^{\\infty}\\sigma(m)\\int_{\\pi(\\Gamma(a,b,\\epsilon)\\cap H_m^{+})}\\frac{d{\\vct p}}{\\omega_m({\\vct p})}\\,dm \\nonumber \\\\\n    & = & \\int_a^b\\sigma(m)\\int_{B_{\\epsilon}({\\vct 0})}\\frac{d{\\vct p}}{\\omega_m({\\vct p})}\\,dm \\nonumber \\\\\n    & \\approx & \\frac{4}{3}\\pi\\epsilon^3\\int_a^b\\frac{\\sigma(m)}{m}\\,dm.  \n\\end{eqnarray}\nwhere $B_{\\epsilon}({\\vct 0})=\\{{\\vct p}\\in{\\bf R}^3:{|\\vct p}|<\\epsilon\\}$.\n\nThe approximation $\\approx$ in the last line comes about because the hyper-cylinder $\\Gamma(a,b,\\epsilon)$ is not exactly equal to the cylinder $(a,b)\\times B_{\\epsilon}({\\vct 0})$ (they become equal in the limit $\\epsilon\\rightarrow0$).\n\nThus if we define \n\\begin{equation} \\label{eq:limit1}\ng_a(b)=g(a,b)=\\lim_{\\epsilon\\rightarrow0}\\epsilon^{-3}\\mu(\\Gamma(a,b,\\epsilon)),\n\\end{equation}\nthen we can retreive $\\sigma$ using the formula\n\\begin{equation} \n\\sigma(b)=\\frac{3}{4\\pi}bg_a^{\\prime}(b).\n\\end{equation}\nThus we have proved the following fundamental theorem of the spectral calculus of causal Lorentz invariant measures.\n\\begin{theorem} \\label{theorem:fundamental}\nSuppose that $\\mu$ is a causal Lorentz invariant measure with continuous spectrum $\\sigma$. Then $\\sigma$ can be calculated from the formula\n\\begin{equation}\n\\sigma(b)=\\frac{3}{4\\pi}bg_a^{\\prime}(b),\n\\end{equation}\nwhere, for $a,b\\in{\\bf R}, 0<a<b, g_a:(a,\\infty)\\rightarrow{\\bf R}$ is given by Eq.~\\ref{eq:limit1}.\n\\end{theorem} \nTo make the proof of this theorem rigorous we prove the following.\n\\newtheorem{lemma}{Lemma}\n\\begin{lemma}\nLet $a,b\\in{\\bf R}, 0<a<b$. Then\n\\begin{equation}\n\\lim_{\\epsilon\\rightarrow0}\\epsilon^{-3}\\int_{B_{\\epsilon}(0)}\\frac{d{\\vct p}}{\\omega_m({\\vct p})}=\\frac{4\\pi}{3}\\frac{1}{m},\n\\end{equation}\nuniformly for $m\\in[a,b]$.\n\\end{lemma}\n{\\bf Proof} \nDefine\n\\begin{equation}\nI=I(m,\\epsilon)=\\int_{B_{\\epsilon}(0)}\\frac{d{\\vct p}}{\\omega_m({\\vct p})}.\n\\end{equation}\nThen\n\\begin{equation}\nI=\\int_{r=0}^{\\epsilon}\\frac{4\\pi r^2\\,dr}{(r^2+m^2)^{\\frac{1}{2}}}.\n\\end{equation}\nNow\n\\[ I_1<I<I_2, \\]\nwhere\n\\[ I_1=\\int_{r=0}^{\\epsilon}\\frac{4\\pi r^2\\,dr}{(\\epsilon^2+m^2)^{\\frac{1}{2}}}=\\frac{4\\pi}{(\\epsilon^2+m^2)^{\\frac{1}{2}}}\\frac{1}{3}\\epsilon^3, \\]\n\\[ I_2=\\int_{r=0}^{\\epsilon}\\frac{4\\pi r^2\\,dr}{m}=\\frac{4\\pi}{m}\\frac{1}{3}\\epsilon^3. \\]\nTherefore\n\\[ \\frac{4\\pi}{3(\\epsilon^2+m^2)^{\\frac{1}{2}}}<\\epsilon^{-3}I<\\frac{4\\pi}{3m}. \\]\nThus\n\\[ \\frac{4\\pi}{3m}-\\frac{4\\pi}{3(\\epsilon^2+m^2)^{\\frac{1}{2}}}>\\frac{4\\pi}{3m}-\\epsilon^{-3}I>0. \\]\nHence\n\\begin{equation}\n\\left|\\epsilon^{-3}I-\\frac{4\\pi}{3m}\\right|<\\frac{4\\pi}{3m}-\\frac{4\\pi}{3(\\epsilon^2+m^2)^{\\frac{1}{2}}}.\n\\end{equation}\nWe have\n\\begin{eqnarray}\n\\frac{4\\pi}{3m}-\\frac{4\\pi}{3(\\epsilon^2+m^2)^{\\frac{1}{2}}} & = & \\frac{4\\pi}{3}\\frac{(\\epsilon^2+m^2)^{\\frac{1}{2}}-m}{m(\\epsilon^2+m^2)^{\\frac{1}{2}}} \\nonumber \\\\\n    & = & \\frac{4\\pi}{3}\\frac{\\epsilon^2}{m(\\epsilon^2+m^2)^{\\frac{1}{2}}((\\epsilon^2+m^2)^{\\frac{1}{2}}+m)} \\nonumber \\\\\n    & < &  \\frac{4\\pi}{3}\\frac{\\epsilon^2}{2m^3} \\nonumber \\\\\n    & \\leq & \\frac{4\\pi}{3}\\frac{\\epsilon^2}{2a^3}, \\mbox{ for all }m\\in[a,b]. \\nonumber\n\\end{eqnarray}\nTherefore\n\\begin{equation}\n\\left|\\epsilon^{-3}I-\\frac{4\\pi}{3m}\\right|<\\frac{4\\pi}{3}\\frac{\\epsilon^2}{2a^3},\n\\end{equation}\nfor all $m\\in[a,b]$\n\n $\\Box$\n\n This lemma justifies the step of taking the limit under the integral sign (indicated by the symbol  $\\approx$) in the proof of Theorem~\\ref{theorem:fundamental}.\n\nMore generally, suppose that $\\mu:{\\mathcal B}({\\bf R}^4)\\rightarrow{\\bf C}$ is a causal Lorentz invariant Borel measure on Minkowski space with spectrum $\\sigma$. Then, by the Lebesgue decomposition theorem there exist unique measures $\\sigma_c,\\sigma_s:{\\mathcal B}([0,\\infty))\\rightarrow{\\bf C}$ such that $\\sigma=\\sigma_c+\\sigma_s$ where $\\sigma_c$, the continuous part of the spectrum of $\\mu$, is absolutely continuous with respect to Lebesque measure and $\\sigma_s$, the singular part of the spectrum of $\\mu$, is singular with respect to $\\sigma_c$.  \n\nIt is straightforward to prove the following.\n\\begin{theorem} \\label{Theorem:Th2}\nSuppose that $a^{\\prime},b^{\\prime}\\in{\\bf R}$ are such that $0<a^{\\prime}<b^{\\prime}$, $\\sigma_c|_{(a^{\\prime},b^{\\prime})}$ is continuous. Then for all $a,b\\in{\\bf R}$ with $a^{\\prime}<a<b<b^{\\prime}$, $g_a(b)$ defined by Eq.~\\ref{eq:limit1} exists and is continuously differentiable. Furthermore $\\sigma_c|_{(a^{\\prime},b^{\\prime})}$ can be computed using the formula\n\\begin{equation} \\label{eq:retrieve_sigma_c}\n\\sigma_c(b)=\\frac{3}{4\\pi}bg_a^{\\prime}(b),\n\\end{equation}\nand\n\\begin{equation}\n\\sigma_s(E)=0, \\forall \\mbox{ Borel }E\\subset(a^{\\prime},b^{\\prime}).\n\\end{equation}\nConversely suppose that $a^{\\prime},b^{\\prime}\\in{\\bf R}$ are such that $0<a^{\\prime}<b^{\\prime}$ and for all $a,b\\in{\\bf R}$ with $a^{\\prime}<a<b<b^{\\prime}$, $g_a(b)$ defined by Eq.~\\ref{eq:limit1} exists and is continuously differentiable. Then $\\sigma_c|_{(a^{\\prime},b^{\\prime})}$ is continuous and can be retrieved using the formula of Eq.~\\ref{eq:retrieve_sigma_c}.\n\\end{theorem}\n\n\\section{Investigation of the measure defined by the convolution $\\Omega_m*\\Omega_m$}\n\n\\subsection{Determination of some properties of $\\Omega_m*\\Omega_m$}\n\nConsider the measure defined by\n\\begin{equation}\n\\mu(\\Gamma)=(\\Omega_m*\\Omega_m)(\\Gamma)=\\int\\chi_{\\Gamma}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq),\n\\end{equation}\nwhere, for any set $\\Gamma$, $\\chi_{\\Gamma}$ denotes the characteristic function of $\\Gamma$ defined by\n\\begin{equation}\n\\chi_{\\Gamma}(p)=\\left\\{\\begin{array}{l}\n1\\mbox{ if }p\\in\\Gamma \\\\\n0\\mbox{ otherwise.}\\end{array}\\right.\n\\end{equation}\n$\\mu$ exists as a Borel measure because as $|p|,|q|\\rightarrow{\\infty}$ with $p,q\\in H_m^{+}$, $(p+q)^0\\rightarrow{\\infty}$ and so $p+q\\mbox{ is eventually }{\\slas \\in}\\Gamma$ for any compact set $\\Gamma\\subset{\\bf R}^4$. Now\n\\begin{eqnarray}\n\\mu(\\Lambda(\\Gamma)) & = & \\int\\chi_{\\Lambda(\\Gamma)}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq) \\nonumber \\\\\n    & = & \\int\\chi_{\\Gamma}(\\Lambda^{-1}p+\\Lambda^{-1}q)\\,\\Omega_m(dp)\\,\\Omega_m(dq) \\nonumber \\\\\n    & = & \\int\\chi_{\\Gamma}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq) \\nonumber \\\\\n    & = & \\mu(\\Gamma),\n\\end{eqnarray}\nfor all $\\Lambda\\in O(1,3)^{+\\uparrow}, \\Lambda\\in{\\mathcal B}({\\bf R}^4)$. Thus $\\mu$ is a Lorentz invariant measure.\n\nWe will now show that $\\mu$ is concentrated in the set \n\\begin{equation}\nC_m=\\{p\\in{\\bf R}^4:p^2\\ge4m^2,p^0>0\\},\n\\end{equation}\nand therefore, that $\\mu$ is causal.\nLet $U\\subset{\\bf R}^4$ be open. Then\n\\begin{equation}\n\\mu(U)=\\int_{{\\bf R}^3}\\int_{{\\bf R}^3}\\chi_U(\\omega_m({\\vct p})+\\omega_m({\\vct q}),{\\vct p}+{\\vct q})\\,\\frac{d{\\vct p}}{\\omega_m({\\vct p})}\\,\\frac{d{\\vct q}}{\\omega_m({\\vct q})}.\n\\end{equation} \nTherefore, using continuity, it follows that\n\\begin{eqnarray}\n\\mu(U)>0 & \\Leftrightarrow & (\\exists p\\in U,{\\vct q}_1,{\\vct q}_2\\in{\\bf R}^3)\\mbox{ } p=(\\omega_m({\\vct q}_1)+\\omega_m({\\vct q}_2),{\\vct q}_1+{\\vct q}_2). \\nonumber\n\\end{eqnarray}\nSuppose that $p\\in\\mbox{supp}(\\mu)$ (the support of the measure $\\mu$) i.e $p$ is such that $\\mu(U)>0$ for all open neighborhoods $U$ of $p$. Let $U$ be an open neighborhood of $p$. Then, since $\\mu(U)>0$, there exists $q\\in U, {\\vct q}_1,{\\vct q}_2\\in{\\bf R}^3$ such that $q=(\\omega_m({\\vct q}_1)+\\omega_m({\\vct q}_2),{\\vct q}_1+{\\vct q}_2)$.  Clearly $q^0\\ge2m$. Since this is true for all neighborhoods $U$ of $p$  it follows that $p^0\\ge2m$. By Lorentz invariance we may assume without loss of generality that ${\\vct p}=0$. Therefore $p^2\\ge 4m^2$. Thus supp$(\\mu)\\subset C_m$. \n\nFor the converse, let $p=(\\omega_m({\\vct p}),{\\vct p}), q= (\\omega_m({\\vct p}),-{\\vct p})\\in H_m^{+}$ for ${\\vct p}\\in{\\bf R}^3$. As ${\\vct p}$ ranges over ${\\bf R}^3$, $p+q=(2\\omega_m({\\vct p}),{\\vct 0})$ ranges over $\\{(m^{\\prime},{\\vct 0}):m^{\\prime}\\geq2m\\}$. It follows using Lorentz invariance that supp$(\\mu)\\supset C_m$.\n\nTherefore the support supp$(\\mu)$ of $\\mu$ is $C_m$. Therefore by the spectral theorem $\\mu$ has a spectral representation of the form\n\\begin{equation} \\label{eq:convolution1}\n\\mu(\\Gamma)=\\int_{m^{\\prime}=2m}^{\\infty}\\Omega_{m^{\\prime}}(\\Gamma)\\,\\sigma(dm^{\\prime}),\n\\end{equation}\nfor some Borel measure $\\sigma:{\\mathcal B}([2m,\\infty))\\rightarrow{\\bf C}$. \n\n\\subsection{Computation of the spectrum of $\\Omega_m*\\Omega_m$ using the spectral calculus}\n\nLet $a,b\\in{\\bf R}$ with $0<a<b$. Let \n\\begin{equation}\ng_a(b,\\epsilon)=\\mu(\\Gamma(a,b,\\epsilon)).\n\\end{equation}\nWe would like to calculate\n\\begin{equation}\ng_a(b)=\\lim_{\\epsilon\\rightarrow0}\\epsilon^{-3}g_a(b,\\epsilon),\n\\end{equation}\nand then retreive the spectral function as\n\\begin{equation}\n\\sigma(b)=\\frac{3}{4\\pi}b g^{\\prime}(b).\n\\end{equation}\nTo this effect we calculate\n\\begin{eqnarray}\ng(a,b,\\epsilon) & = & \\mu(\\Gamma(a,b,\\epsilon)) \\nonumber \\\\\n    & = & \\int\\chi_{\\Gamma(a,b,\\epsilon)}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq) \\nonumber \\\\\n    & \\approx & \\int\\chi_{(a,b)\\times B_{\\epsilon}({\\vct 0})}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq) \\nonumber \\\\\n    & = & \\int\\chi_{(a,b)}(\\omega_m({\\vct p})+\\omega_m({\\vct q}))\\chi_{B_{\\epsilon}({\\vct 0})}({\\vct p}+{\\vct q})\\,\\frac{d{\\vct p}}{\\omega_m({\\vct p})}\\frac{d{\\vct q}}{\\omega_m({\\vct q})} \\nonumber \\\\\n    & = & \\int\\chi_{(a,b)}(\\omega_m({\\vct p})+\\omega_m({\\vct q}))\\chi_{B_{\\epsilon}({\\vct 0})-{\\vct q}}({\\vct p})\\,\\frac{d{\\vct p}}{\\omega_m({\\vct p})}\\frac{d{\\vct q}}{\\omega_m({\\vct q})} \\nonumber \\\\\n    & \\approx & \\int\\chi_{(a,b)}(2\\omega_m({\\vct q}))\\frac{\\frac{4}{3}\\pi\\epsilon^3}{\\omega_m({\\vct q})^2}\\,d{\\vct q}. \\nonumber\n\\end{eqnarray}\nWe will call this argument Argument 1. See Appendix 6 for a rigorous justification of Argument 1. Now \n\\begin{eqnarray}\na<2\\omega_m({\\vct q})<b & \\Leftrightarrow & \\left(\\frac{a}{2}\\right)^2-m^2<{\\vct q}^2<\\left(\\frac{b}{2}\\right)^2-m^2 \\nonumber \\\\\n    & \\Leftrightarrow & mZ(a)<|{\\vct q}|<mZ(b), \\nonumber\n\\end{eqnarray}\nwhere\n\\begin{equation} \\label{eq:Z_def}\nZ(m^{\\prime})=(\\frac{m^{\\prime2}}{4m^2}-1)^{\\frac{1}{2}},\\mbox{ for }m^{\\prime}\\ge2m.\n\\end{equation}\nThus\n\\begin{equation}\ng(a,b,\\epsilon)\\approx\\frac{16\\pi^2}{3}\\epsilon^3\\int_{r=mZ(a)}^{mZ(b)}\\frac{r^2}{m^2+r^2}\\,dr.\n\\end{equation}\nHence\n\\begin{equation}\ng_a(b)=\\frac{16\\pi^2}{3}\\int_{r=mZ(a)}^{mZ(b)}\\frac{r^2}{m^2+r^2}\\,dr.\n\\end{equation}\nTherefore $g_a$ is continuously differentiable and so Theorem~\\ref{Theorem:Th2} applies.\nUsing the fundamental theorem of calculus\n\\begin{equation}\ng_a^{\\prime}(b)=\\frac{16\\pi^2}{3}\\frac{m^2Z^2(b)}{m^2+m^2Z^2(b)}mZ^{\\prime}(b)=\\frac{16\\pi^2}{3}\\frac{mZ(b)}{b}.\n\\end{equation}\nTherefore we compute the spectrum of $\\mu$ as\n\\begin{equation}\n\\sigma(b)=4\\pi mZ(b) \\mbox{ for } b\\ge2m.\n\\end{equation}\n\n\\section{Investigation of the measure defined by the convolution $\\Omega_{im}*\\Omega_{im}$ \\label{spacelike_convolution}}\n\nDefine the measure $\\Omega^{+}_{im}$ by\n\\begin{equation}\n\\Omega_{im}^{+}(\\Gamma)=\\int_{\\pi(\\Gamma\\cap H_{im}^{+})}\\frac{dp}{\\omega_{im}({\\vct p})} \\mbox{ for }\\Gamma\\in{\\mathcal B}({\\bf R}^4),\n\\end{equation}\nwhere\n\\begin{equation}\nH_{im}^{+}=\\{p\\in{\\bf R}^4:p^2=-m^2, p^0\\geq0\\}.\n\\end{equation}\n$\\Omega_{im}^{+}$ is a measure concentrated on the positive time imaginary mass hyperboloid $H_{im}^{+}$ corresponding to mass $im$. There is also a measure $\\Omega_{im}^{-}$ on $H_{im}^{-}$ and we may define $\\Omega_{im}=\\Omega_{im}^{+}+\\Omega_{im}^{-}$, for $m>0$. $\\Omega_{im}$ is a Lorentz invariant measure on $H_{im}=\\{p\\in{\\bf R}^4:p^2=-m^2\\}$.\n\nDefine, for $m\\in{\\bf C}$\n\\begin{equation}\nJ_m^{+}=\\{p\\in{\\bf C}^4:p^2=m^2,\\mbox{Re}(p^0)\\geq0, \\mbox{Im}(p^0)\\geq0\\},\n\\end{equation}\nwhere $p^2=\\eta_{\\mu\\nu}p^{\\mu}p^{\\nu}$. Then, for $m>0$,\n\\begin{equation}\nJ_m^{+}\\cap{\\bf R}^4=\\{p\\in{\\bf R}^4:p^2=m^2, p^0\\geq0\\}=H_m^{+},\n\\end{equation}\n\\begin{eqnarray} \\label{eq:cx_hyp_property}\nJ_m^{+}\\cap(i{\\bf R}^4) & = & \\{p\\in i{\\bf R}^4:p^2=m^2, \\mbox{Re}(p^0)\\geq0, \\mbox{Im}(p^0)\\geq0\\} \\nonumber \\\\\n    & = & \\{iq:q\\in{\\bf R}^4, q^2=-m^2, q^0\\geq0\\} \\nonumber \\\\\n    & = & iH_{im}^{+}. \n\\end{eqnarray}\n\nOne may consider the measure $\\Omega_m^{+}$ to be defined on $i{\\bf R}^4$ as well as ${\\bf R}^4$ and for all $m\\in{\\bf C}$ according to\n\\begin{equation}\n\\Omega_{m}^{+}({\\Gamma})=\\int_{\\pi(\\Gamma\\cap J_{m}^{+})}\\frac{d{\\vct p}}{\\omega_{m}({\\vct p})},\n\\end{equation}\nwhere, for $m\\in{\\bf C}$,\n\\begin{equation}\n\\omega_m:{\\bf C}^3\\rightarrow{\\bf C}, \\omega_m({\\vct p})=({\\vct p}^2+m^2)^{\\frac{1}{2}}, \\mbox{ where }{\\vct p}^2=\\delta_{jk}p^jp^k.\n\\end{equation}\nThen from Equation~\\ref{eq:cx_hyp_property}\n\\begin{equation}\n\\Omega_m^{+}(i\\Gamma)=\\int_{i\\pi(\\Gamma\\cap H_{im}^{+})}\\frac{d{\\vct p}}{\\omega_m({\\vct p})}.\n\\end{equation}\nNow make the substitution ${\\vct p}=i{\\vct q}$. Then $d{\\vct p}=-id{\\vct q}$.  Also\n\\[ \\omega_{m}({\\vct q})=(m^2+{\\vct q}^2)^{\\frac{1}{2}}=(m^2-{\\vct p}^2)^{\\frac{1}{2}}=(-((im)^2+{\\vct p}^2))^{\\frac{1}{2}}=i\\omega_{im}({\\vct p}). \\]\nThis is true for all $m\\in{\\bf C}$. Therefore\n\\begin{equation}\ni\\omega_m({\\vct p})=\\omega_{(-im)}({\\vct q})=\\omega_{im}({\\vct q}),\n\\end{equation}\nand hence\n\\begin{equation}\n\\omega_m({\\vct p})=-i\\omega_{im}({\\vct q}).\n\\end{equation}\nThus\n\\begin{equation}\n\\Omega_m^{+}(i\\Gamma)=\\int_{\\pi(\\Gamma\\cap H_{im}^{+})}\\frac{-id{\\vct q}}{-i\\omega_{im}({\\vct q})}=\\Omega_{im}^{+}(\\Gamma). \n\\end{equation}\nDefine\n\\begin{equation}\n{\\mathcal B}_0({\\bf R}^4)=\\{\\Gamma\\in{\\mathcal B}({\\bf R}^4):\\Gamma\\mbox{ is relatively compact}\\}.\n\\end{equation}\nNow suppose that\n\\begin{equation}\n\\psi=\\sum_k c_k\\chi_{E_k},\n\\end{equation}\nwhere $c_i\\in{\\bf C}$ and $E_k\\in{\\mathcal B}_0({\\bf R}^4)$, is a simple function. Then\n\\begin{eqnarray}\n\\int_{{\\bf R}^4}\\psi(p)\\,\\Omega_{im}(dp) & = & \\sum_k c_k\\Omega_{im}(E_k) \\nonumber \\\\\n    & = & \\sum_k c_k\\Omega_m(iE_k) \\nonumber \\\\\n    & = & \\sum_k c_k\\int_{i{\\bf R}^4}\\chi_{iE_k}(p)\\,\\Omega_m(dp) \\nonumber \\\\\n    & = & \\sum_k c_k\\int_{i{\\bf R}^4}\\chi_{E_k}(\\frac{p}{i})\\,\\Omega_m(dp) \\nonumber \\\\\n    & = & \\int_{i{\\bf R}^4}\\psi(\\frac{p}{i})\\,\\Omega_m(dp). \\\\\n\\end{eqnarray}\nSince this is true for every such simple function $\\psi$ it follows that\n\\begin{equation}\n\\int_{{\\bf R}^4}\\psi(p)\\,\\Omega_{im}(dp)=\\int_{i{\\bf R}^4}\\psi(\\frac{p}{i})\\,\\Omega_m(dp),\n\\end{equation}\nfor every locally integrable function $\\psi$. Therefore\n\\begin{eqnarray} \\label{eq:convolve}\n(\\Omega_{im}*\\Omega_{im})(\\Gamma) & = & \\int_{({\\bf R}^4)^2}\\chi_{\\Gamma}(p+q)\\,\\Omega_{im}(dp)\\,\\Omega_{im}(dq) \\nonumber \\\\\n    & = & \\int_{(i{\\bf R}^4)^2}\\chi_{\\Gamma}\\left(\\frac{p+q}{i}\\right)\\,\\Omega_m(dp)\\,\\Omega_m(dq) \\nonumber \\\\\n    & = & \\int_{(i{\\bf R}^4)^2}\\chi_{i\\Gamma}(p+q)\\Omega_m(dp)\\Omega_m(dq) \\nonumber \\\\\n    & = & (\\Omega_m*\\Omega_m)(i\\Gamma).\n\\end{eqnarray}\n\nNow in general, suppose that a measure $\\mu$ has a causal spectral representation of the form\n\\begin{equation}\n\\mu(\\Gamma)=\\int_{m^{\\prime}=0}^{\\infty}\\Omega_{m^{\\prime}}^{+}(\\Gamma)\\,\\sigma(m^{\\prime}),\n\\end{equation}\nfor some Borel spectral measure $\\sigma:[0,\\infty)\\rightarrow{\\bf C}$. \nThen $\\mu$ extends to a measure  defined on $i{\\bf R}^4$ by\n\\begin{equation}\n\\mu(i\\Gamma)=\\int_{m=0}^{\\infty}\\Omega_m^{+}(i\\Gamma)\\,\\sigma(dm)=\\int_{m=0}^{\\infty}\\Omega_{im}^{+}(\\Gamma)\\,\\sigma(dm),\n\\end{equation}\nfor $\\Gamma\\in{\\mathcal B}({\\bf R}^4)$.\nTherefore since, as we have determined above, $\\Omega_m^{+}*\\Omega_m^{+}$ is a causal spectral measure with spectrum\n\\begin{equation} \\label{eq:imaginary_spectrum}\n\\sigma(m^{\\prime})=4\\pi mZ(m^{\\prime}) \\mbox{ for } m^{\\prime}\\ge2m,\n\\end{equation}\nit follows that\n\\begin{equation}\n(\\Omega_m^{+}*\\Omega_m^{+})(i\\Gamma)=\\int_{m=0}^{\\infty}\\Omega_{im^{\\prime}}^{+}(\\Gamma)\\,\\sigma(dm^{\\prime}).\n\\end{equation}\nTherefore using Eq.~\\ref{eq:convolve} $\\Omega_{im}^{+}*\\Omega_{im}^{+}$ is a measure with spectral representation\n\\begin{equation}\n(\\Omega_{im}^{+}*\\Omega_{im}^{+})(\\Gamma)=\\int_{m^{\\prime}=2m}^{\\infty}\\Omega_{im^{\\prime}}^{+}(\\Gamma)\\,\\sigma(m^{\\prime})\\,dm^{\\prime},\n\\end{equation}\nwhere $\\sigma$ is the spectral function given by Eq.~\\ref{eq:imaginary_spectrum}.\n Note that $\\Omega_{im}^{+}*\\Omega_{im}^{+}$ is not causal, it is a type III measure, and\n\\begin{equation}\n\\mbox{supp}(\\Omega_{im}^{+}*\\Omega_{im}^{+})=\\{p\\in{\\bf R}^4:p^2\\leq-4m^2, p^0\\geq0\\}.\n\\end{equation}\n\n\\section{Determination of the density defining a causal Lorentz invariant measure from its spectrum}\n\nSuppose that $\\mu$ is of the form of Eq.~\\ref{eq:invariant2} where $\\sigma$ is a well behaved (e.g. locally integrable) function. We would like to see if $\\mu$ can be defined by a density with respect to the Lebesgue measure, i.e. if there exists a function $g:{\\bf R}^4\\rightarrow{\\bf C}$ such that\n\\begin{equation}\n\\mu(\\Gamma)=\\int_{\\Gamma}g(p)\\,dp.\n\\end{equation}\nWell we have that\n\\begin{equation}\n\\mu(\\Gamma)=\\int_{m=0}^{\\infty}\\sigma(m)\\Omega_m^{+}(\\Gamma)\\,dm=\\int_{m=0}^{\\infty}\\sigma(m)\\int_{\\pi(\\Gamma\\cap  H_m^{+})}\\frac{d{\\vct p}}{\\omega_m({\\vct p})}\\,dm.\n\\end{equation}\nNow \n\\begin{eqnarray}\n{\\vct p}\\in\\pi(\\Gamma\\cap H_m^{+}) & \\Leftrightarrow & (\\exists p\\in{\\bf R}^4){\\vct p}=\\pi(p),p\\in H_m^{+}, p\\in\\Gamma \\nonumber \\\\\n    & \\Leftrightarrow & (\\omega_m({\\vct p}),{\\vct p})\\in\\Gamma \\nonumber \\\\\n    & \\Leftrightarrow & \\chi_{\\Gamma}(\\omega_m({\\vct p}),{\\vct p})=1. \\nonumber\n\\end{eqnarray}\nTherefore\n\\begin{equation}\n\\mu(\\Gamma)=\\int_{m=0}^{\\infty}\\sigma(m)\\int_{{\\bf R}^3}\\chi_{\\Gamma}(\\omega_m({\\vct p}),{\\vct p})\\frac{1}{\\omega_m({\\vct p})}\\,d{\\vct p}\\,dm.\n\\end{equation}\nNow consider the transformation defined by the function $h:(0,\\infty)\\times{\\bf R}^3\\rightarrow{\\bf R}^4$ given by\n\\begin{equation}\nh(m,{\\vct p})=(\\omega_m({\\vct p}),{\\vct p}).\n\\end{equation}\nLet \n\\begin{equation}\nq=h(m,{\\vct p})=(\\omega_m({\\vct p}),{\\vct p})=((m^2+{\\vct p}^2)^{\\frac{1}{2}},{\\vct p}).\n\\end{equation}\nThen\n\\begin{equation}\n\\frac{\\partial q^0}{\\partial m}=m\\omega_m({\\vct p})^{-1}, \\frac{\\partial q^0}{\\partial p^j}= p^j\\omega_m({\\vct p})^{-1},  \\frac{\\partial q^i}{\\partial m}=0, \\frac{\\partial q^i}{\\partial p^j}=\\delta_{ij}, \n\\end{equation}\nfor $i,j=1,2,3$. Thus the Jacobian of the transformation is\n\\begin{equation}\nJ(m,{\\vct p})=m\\omega_m({\\vct p})^{-1}.\n\\end{equation}\nNow $q=(\\omega_m({\\vct p}),{\\vct p})$. Therefore $q^2=\\omega_m({\\vct p})^2-{\\vct p}^2=m^2$. So $m=(q^2)^{\\frac{1}{2}},q^2>0$.\nThus\n\\begin{eqnarray}\n\\mu(\\Gamma) & = & \\int_{q\\in{\\bf R}^4,q^2>0,q^0>0}\\chi_{\\Gamma}(q)\\frac{\\sigma(m)}{\\omega_m({\\vct p})}\\frac{dq}{J(m,{\\vct p})} \\nonumber \\\\\n    & = & \\int_{q^2>0,q^0>0}\\chi_{\\Gamma}(q)\\frac{\\sigma(m)}{m}\\,dq.\n\\end{eqnarray}\nHence\n\\begin{eqnarray}\n\\mu(\\Gamma) & = & \\int_{q^2>0,q^0>0}\\chi_{\\Gamma}(q)\\frac{\\sigma((q^2)^{\\frac{1}{2}})}{(q^2)^{\\frac{1}{2}}}\\,dq \\nonumber \\\\\n    & = & \\int_{\\Gamma}g(q)\\,dq, \\nonumber\n\\end{eqnarray}\nwhere $g:{\\bf R}^4\\rightarrow{\\bf C}$ is defined by\n\\begin{equation}\ng(q)=\\left\\{\\begin{array}{l}\n(q^2)^{-\\frac{1}{2}}\\sigma((q^2)^{\\frac{1}{2}})\\mbox{ if } q^2>0,q^0>0 \\\\\n0\\mbox{ otherwise.}\n\\end{array}\\right.\n\\end{equation}\nWe have therefore shown how, given a spectral representation of a causal measure in which the spectrum is a complex function, one can obtain and equivalent representation of the measure in terms of a density with respect to Lebesgue measure.\n\n\\section{Convolutions and products of causal Lorentz invariant Borel measures}\n\n\\subsection{Convolution of measures}\n\nLet $\\mu$ and $\\nu$ be causal Lorentz invariant Borel measures. Then there exist Borel spectral measures $\\sigma,\\rho:{\\mathcal B}([0,\\infty))\\rightarrow{\\bf C}$ such that\n\\begin{eqnarray}\n\\mu & = & \\int_{m=0}^{\\infty}\\Omega_m\\,\\sigma(dm), \\nonumber \\\\\n\\nu & = & \\int_{m=0}^{\\infty}\\Omega_m\\,\\rho(dm).\n\\end{eqnarray}\nThe convolution of $\\mu$ and $\\nu$, if it exists, is given by\n\\begin{equation}\n(\\mu*\\nu)(\\Gamma)=\\int\\chi_{\\Gamma}(p+q)\\,\\mu(dp)\\,\\nu(dq).\n\\end{equation}\nNow let $\\psi=\\sum_i c_i\\chi_{E_i}$ with $c_i\\in{\\bf C}, E_i\\in{\\mathcal B}_0({\\bf R}^4)$ be a simple function. Then\n\\begin{eqnarray}\n\\int\\psi(p)\\,\\mu(dp) & = & \\int\\sum_i c_i\\chi_{E_i}\\,\\mu(dp) \\nonumber \\\\\n    & = & \\sum_i c_i\\mu(E_i) \\nonumber \\\\\n    & = & \\sum_i c_i\\int_{m=0}^{\\infty}\\Omega_m(E_i)\\,\\sigma(dm) \\nonumber \\\\\n    & = & \\sum_i c_i\\int_{m=0}^{\\infty}\\int_{{\\bf R}^4}\\chi_{E_i}(p)\\,\\Omega_m(dp)\\,\\sigma(dm) \\nonumber \\\\\n    & = & \\int_{m=0}^{\\infty}\\int_{{\\bf R}^4}\\psi(p)\\,\\Omega_m(dp)\\,\\sigma(dm). \\nonumber\n\\end{eqnarray}\nTherefore for any sufficiently well behaved (e.g Schwartz) measurable function $\\psi:{\\bf R}^4\\rightarrow{\\bf C}$\n\\begin{equation}\n\\int\\psi(p)\\mu(dp)=\\int\\psi(p)\\,\\Omega_m(dp)\\,\\sigma(dm).\n\\end{equation}\n(Note that the integral exists because $\\sigma$ is a Borel measure.) Hence\n\\begin{eqnarray} \\label{eq:convolve1}\n(\\mu*\\nu)(\\Gamma) & = & \\int\\chi_{\\Gamma}(p+q)\\,\\mu(dp)\\,\\nu(dq) \\nonumber \\\\\n    & = & \\int\\chi_{\\Gamma}(p+q)\\,\\Omega_m(dp)\\,\\sigma(dm)\\,\\Omega_{m^{\\prime}}(dq)\\,\\rho(dm^{\\prime}) \\nonumber \\\\\n    & = & \\int\\chi_{\\Gamma}(p+q)\\,\\Omega_m(dp)\\,\\Omega_{m^{\\prime}}(dq)\\sigma(dm)\\,\\rho(dm^{\\prime}),  \n\\end{eqnarray}\nby Fubini's theorem, as long as the integral\n\\begin{equation}\n\\int\\chi_{\\Gamma}(p+q)\\,\\Omega_m(dp)\\,\\Omega_{m^{\\prime}}(dq)|\\sigma|(dm)\\,|\\rho|(dm^{\\prime}),\n\\end{equation}\nexists where $|\\sigma|,|\\rho|$ are the total variations of the measures $\\sigma,\\rho$. \n\nSuppose that $\\Gamma\\subset{\\bf R}^4$ is compact. Then there exists $a,R\\in(0,\\infty)$ such that $\\Gamma\\subset(-a,a)\\times B_R({\\vct 0})$, where $B_R({\\vct 0})=\\{{\\vct p}\\in{\\bf R}^3:|{\\vct p}|<R\\}$. Now\n\\begin{equation}\n\\int\\chi_{\\Gamma}(p+q)\\,\\Omega_m(dp)=\\int_{\\Gamma-q}\\Omega_m(dp)=\\Omega_m(\\Gamma-q)<\\infty,\n\\end{equation}\nfor all $q\\in{\\bf R}^4$ because $\\Omega_m$ is Borel and $\\Gamma$ is compact. \n\nNow suppose that $m,m^{\\prime}>a$. Then\n\\begin{equation}\np\\in H_m^{+},q\\in H_{m^{\\prime}}^{+}\\Rightarrow(p+q)^0=p^0+q^0\\geq m+m^{\\prime}>2a\\Rightarrow (p+q){\\slas\\in}\\Gamma.\n\\end{equation}\nThus\n\\begin{equation}\n\\int\\chi_{\\Gamma}(p+q)\\,\\Omega_m(dp)\\,\\Omega_{m^{\\prime}}(dq)=0.\n\\end{equation}\n\nTherefore since $\\sigma$ and $\\rho$ are Borel, $(\\mu*\\nu)(\\Gamma)$ exists, is finite and is given by Eq.~\\ref{eq:convolve1}. \n\nNow let $\\Lambda\\in O(1,3)^{+\\uparrow}$, $\\psi:{\\bf R}^4\\rightarrow{\\bf C}$ be a measurable function of compact support. Then\n\\begin{eqnarray}\n<\\mu*\\nu,\\Lambda\\psi> & = & \\int\\psi(\\Lambda^{-1}(p+q))\\,\\Omega_m(dp)\\,\\Omega_{m^{\\prime}}(dq)\\,\\sigma(dm)\\,\\rho(dm^{\\prime}) \\nonumber \\\\\n    & = & \\int\\psi(p+q)\\,\\Omega_m(dp)\\,\\Omega_{m^{\\prime}}(dq)\\,\\sigma(dm)\\,\\rho(dm^{\\prime}). \\nonumber \\\\\n    & = & <\\mu*\\nu,\\psi> \\nonumber\n\\end{eqnarray}\nTherefore $\\mu*\\nu$ is Lorentz invariant. It can be shown, by an argument similar to that used for the case $\\Omega_m*\\Omega_m$ that $\\mu*\\nu$ is causal.\n\nWe have therefore shown that the convolution of two causal Lorentz invariant Borel measures exists an is a causal Lorentz invariant Borel measure.\n\n\\subsection{Product of measures}\n\nWe now turn to the problem of computing the product of two causal Lorentz invariant Borel measures. The problem of computing the product of measures or distributions is difficult in general and has atracted a large amount of research (Colombeau, 1984; Oberguggenberger, 1992). In such work one generally seeks a definition of product of measures or distributions which agrees with the ordinary product when the measures or distributions are functions (i.e. densities with respect to Lebesgue measure). The most common approach is to use the fact that, for Schwartz functions $f,g\\in{\\mathcal S}({\\bf R}^4)$ multiplication in the spatial domain corresponds to convolution in the frequency domain, i.e. $(fg)^{\\wedge}=f^{\\wedge}*g^{\\wedge}$ (where $\\wedge$ denotes the Fourier transform operator). Thus one defines the product of measures or distributions $\\mu,\\nu$ as\n\\begin{equation}\n\\mu\\nu=(\\mu^{\\wedge}*\\nu^{\\wedge})^{\\vee}.\n\\end{equation}\nHowever this definition is only successful when the convolution that it involves exists which may not be the case in general. If $\\mu,\\nu$ are tempered measures then $\\mu^{\\wedge}$ and $\\nu^{\\wedge}$ exist as tempered distributions, however they are generally not causal, even if $\\mu,\\nu$ are causal. \n\nWe will therefore not use the ``frequency space\" approach to define the product of measures but will use a different approach. Our approach is just as valid as the frequency space approach because our product will coincide with the usual function product when the measures are defined by densities. Furthermore, our approach is useful for the requirements of QFT because measures and distributions in QFT are frequently Lorentz invariant and causal.\n\nLet int$(C)=\\{p\\in{\\bf R}^4:p^2>0,p^0>0\\}$. Suppose that $f:\\mbox{int}(C)\\rightarrow{\\bf C}$ is a Lorentz invariant locally integrable function. Then it defines a causal Lorentz invariant Borel measure $\\mu_f$ which, by the spectral theorem, must have a representation of the form\n\\begin{equation}\n\\mu_f(\\Gamma)=\\int_{\\Gamma}f(p)\\,dp=\\int_{m=0}^{\\infty}\\Omega_m(\\Gamma)\\,\\sigma(dm),\n\\end{equation}\nfor some spectral measure $\\sigma:{\\mathcal B}([0,\\infty))\\rightarrow{\\bf C}$. Since $\\mu_f$ is absolutely continuous with respect to Lebesgue measure it follows that $\\sigma$ must be non singular, i.e. a function.\nBy the result of the previous section a density defining $\\mu_f$ is ${\\tld f}:\\mbox{int}(C)\\rightarrow{\\bf C}$ defined by\n\\begin{equation}\n{\\tld f}(p)=(p^2)^{-\\frac{1}{2}}\\sigma((p^2)^{\\frac{1}{2}}), p\\in\\mbox{ int}(C).\n\\end{equation}\nWe must have that ${\\tld f}=f$ (almost everywhere). Therefore (almost everywhere on int$(C)$)\n\\begin{equation}\nf(p)=(p^2)^{-\\frac{1}{2}}\\sigma((p^2)^{\\frac{1}{2}}).\n\\end{equation}\n$f(p)$ depends only on $p^2$. Therefore $\\sigma(m)=mf(p)$ for all $p\\in\\mbox{int}(C)$ such that $p^2=m^2$. In particular\n\\begin{equation}\n\\sigma(m)=mf((m,{\\vct 0})^{T}), \\forall m>0.\n\\end{equation}\nNow we are seeking a definition of product which has useful properties. Two such properties would be that it is distributive with respect to generalized sums such as integrals and also that it agrees with the ordinary product when the measures are defined by functions. Suppose that we had such a product. Let $f,g:\\mbox{int}(C)\\rightarrow{\\bf C}$ be Lorentz invariant locallly integrable functions. Let $\\mu,\\nu:{\\mathcal B}(\\mbox{int}(C))\\rightarrow{\\bf C}$ be the associated measures with spectra $\\sigma,\\rho$. Then\n\\begin{eqnarray}\n\\mu\\nu & = & \\int_{m=0}^{\\infty}\\Omega_m\\,\\sigma(dm)\\int_{m^{\\prime}=0}^{\\infty}\\Omega_{m^{\\prime}}\\,\\rho(dm^{\\prime}) \\nonumber \\\\\n    & = & \\int_{m=0}^{\\infty}\\Omega_m\\,mf((m,{\\vct 0})^{T})\\,dm\\int_{m^{\\prime}=0}^{\\infty}\\Omega_{m^{\\prime}}\\,m^{\\prime}g((m^{\\prime},{\\vct 0})^{T})\\,dm^{\\prime} \\nonumber \\\\\n    & = & \\int_{m=0}^{\\infty}\\int_{m^{\\prime}=0}^{\\infty}\\Omega_m\\Omega_{m^{\\prime}}mf((m,{\\vct 0})^{T})m^{\\prime}g((m^{\\prime},{\\vct 0})^{T})\\,dm\\,dm^{\\prime}. \\nonumber \n\\end{eqnarray}\nNow we want this to be equal to\n\\begin{equation}\n\\int_{m=0}^{\\infty}\\Omega_mm(fg)((m,{\\vct 0})^{T})\\,dm\n\\end{equation}\nThis will be the case (formally) if we have\n\\begin{equation}\n\\Omega_m\\Omega_{m^{\\prime}}=\\frac{1}{m}\\delta(m-m^{\\prime})\\Omega_m, \\forall m,m^{\\prime}>0.\n\\end{equation}\nPhysicists will be familiar with such a formula (e.g. the equal time commutation relations). Rather than attempting to define its meaning in a rigorous way we will simply carry out the following formal computation for general Lorentz invariant Borel measures $\\mu,\\nu$ with spectra $\\sigma,\\rho$\n\\begin{eqnarray}\n\\mu\\nu & = & \\int_{m=0}^{\\infty}\\Omega_m\\,\\sigma(dm)\\int_{m^{\\prime}=0}^{\\infty}\\Omega_{m^{\\prime}}\\,\\rho(dm^{\\prime}) \\nonumber \\\\\n    & = & \\int_{m=0}^{\\infty}\\int_{m^{\\prime}=0}^{\\infty}\\Omega_m\\Omega_{m^{\\prime}}\\sigma(m)\\rho(m^{\\prime})\\,dm\\,dm^{\\prime} \\nonumber \\\\\n    & = & \\int_{m=0}^{\\infty}\\int_{m^{\\prime}=0}^{\\infty}\\frac{1}{m}\\Omega_m\\delta(m-m^{\\prime})\\sigma(m)\\rho(m^{\\prime})\\,dm^{\\prime}\\,dm \\nonumber \\\\\n    & = & \\int_{m=0}^{\\infty}\\frac{1}{m}\\Omega_m\\sigma(m)\\rho(m)\\,dm. \\nonumber\n\\end{eqnarray}\nTherefore we can simply define the product $\\mu\\nu$ in general by\n\\begin{equation}\n\\mu\\nu=\\int_{m=0}^{\\infty}\\frac{1}{m}\\Omega_m\\,(\\sigma\\rho)(dm).\n\\end{equation}\nWe have therefore reduced the problem of computing the product of measures on int$(C)$ to the problem of computing the product of their 1D spectral measures. The problem of multiplying 1D measures is somewhat less problematical than the problem of multiplying 4D measures. A large class of 1D measures is made up of measures which are of the form of a function plus a finite number of ``atoms\" (singularities of the form $c\\delta_a$ where $c\\in{\\bf C},a\\in[0,\\infty)$, where $\\delta_a$ is the Dirac delta function (measure) concentrated at $a$). There are other pathological types of 1D measure but these may not be of interest for physical applications. \n\nIn the general non-pathological case, if $\\mu,\\nu$ are causal Lorentz invariant Borel measures with spectra $\\sigma(m)=\\xi(m)+\\sum_{i=1}^{k}c_i\\delta(m-a_i), \\rho(m)=\\zeta(m)+\\sum_{j=1}^l d_j\\delta(m-b_j)$ where $\\xi,\\zeta:[0,\\infty)\\rightarrow{\\bf C}$ are locally integrable functions, $c_i,d_j\\in{\\bf C},a_i,b_j\\in[0,\\infty)$ are such that $a_i\\neq b_j,\\forall i,j$ then we may define the product of $\\mu$ and $\\nu$ to be the causal Lorentz invariant measure $\\mu\\nu$ given by\n\\begin{equation}\n\\mu\\nu=\\int_{m=0}^{\\infty}\\Omega_m\\tau(dm),\n\\end{equation}\nwhere \n\\begin{equation}\n\\tau(m)=\\frac{1}{m}(\\xi(m)\\zeta(m)+\\sum_{i=1}^k\\sum_{j=1}^l c_id_j\\delta(m-a_i)\\delta(m-b_j)).\n\\end{equation}\n\nThis definition will suffice for many of the requirements of QFT, and has the properties that we desire.\n\n\\section{Vacuum polarization}\n\n\\subsection{Definition of the contraction of the vacuum polarization tensor as a Lorentz invariant tempered complex measure $\\Pi$}\n\nThe vacuum polarization tensor is written as\n\n\\begin{equation}\n\\Pi^{\\mu\\nu}(k)=-e^2\\int\\frac{dp}{(2\\pi)^4}\\mbox{Tr}(\\gamma^{\\mu}\\frac{1}{{\\slas p}-m+i\\epsilon}\\gamma^{\\nu}\\frac{1}{{\\slas p}-{\\slas k}-m+i\\epsilon}),\n\\end{equation}\n(Itzikson and Zuber, 1980, p. 319). This can be rewritten as\n\\begin{equation}\n\\Pi^{\\mu\\nu}(k)=-\\frac{e^2}{(2\\pi)^4}\\int\\frac{\\mbox{Tr}(\\gamma^{\\mu}({\\slas p}+m)\\gamma^{\\nu}({\\slas p}-{\\slas k}+m))}{(p^2-m^2+i\\epsilon)((p-k)^2-m^2+i\\epsilon)}\\,dp.\n\\end{equation}\nTherefore, contracting with the Minkowski space metric tensor, the ``function\" that we are interested in computing is\n\\begin{equation} \\label{eq:Pi1}\n\\Pi(k)=-\\frac{e^2}{(2\\pi)^4}\\int\\frac{\\mbox{Tr}(\\eta_{\\mu\\nu}\\gamma^{\\mu}({\\slas p}+m)\\gamma^{\\nu}({\\slas p}-{\\slas k}+m))}{(p^2-m^2+i\\epsilon)((p-k)^2-m^2+i\\epsilon)}\\,dp.\n\\end{equation}\nAs is well known, the integral defining this ``function\" is divergent for all $k\\in{\\bf R}^4$ and all the machinery of regularization and renormalization has been developed to get around this problem.\n\nWe propose that the object defined by Eq.~\\ref{eq:Pi1} exists when viewed as a measure on Minkowski space. To show this, suppose that $\\Pi$ were a density for a measure which we also denote as $\\Pi$. Then we may make the following formal computation.\n\\begin{eqnarray}\n\\Pi(\\Gamma) & = & \\int_{\\Gamma}\\Pi(k)\\,dk \\nonumber \\\\\n    & = & \\int\\chi_{\\Gamma}(k)\\Pi(k)\\,dk \\nonumber \\\\\n    & = & -\\frac{e^2}{(2\\pi)^4}\\int\\chi_{\\Gamma}(k)\\frac{\\mbox{Tr}(\\eta_{\\mu\\nu}\\gamma^{\\mu}({\\slas p}+m)\\gamma^{\\nu}({\\slas p}-{\\slas k}+m))}{(p^2-m^2+i\\epsilon)((p-k)^2-m^2+i\\epsilon)}\\,dp\\,dk \\nonumber \\\\\n   & = & -\\frac{e^2}{(2\\pi)^4}\\int\\chi_{\\Gamma}(k)\\frac{\\mbox{Tr}(\\eta_{\\mu\\nu}\\gamma^{\\mu}({\\slas p}+m)\\gamma^{\\nu}({\\slas p}-{\\slas k}+m))}{(p^2-m^2+i\\epsilon)((p-k)^2-m^2+i\\epsilon)}\\,dk\\,dp \\nonumber \\\\ \n   & = & -\\frac{e^2}{(2\\pi)^4}\\int\\chi_{\\Gamma}(k+p)\\frac{\\mbox{Tr}(\\eta_{\\mu\\nu}\\gamma^{\\mu}({\\slas p}+m)\\gamma^{\\nu}(-{\\slas k}+m))}{(p^2-m^2+i\\epsilon)(k^2-m^2+i\\epsilon)}\\,dk\\,dp \\nonumber \\\\ \n   & = & \\frac{e^2}{(2\\pi)^4}\\int\\chi_{\\Gamma}(k+p)\\frac{\\mbox{Tr}(\\eta_{\\mu\\nu}\\gamma^{\\mu}({\\slas p}+m)\\gamma^{\\nu}({\\slas k}-m))}{(p^2-m^2+i\\epsilon)(k^2-m^2+i\\epsilon)}\\,dk\\,dp. \\nonumber\n\\end{eqnarray} \nNow the propagators in QFT can be viewed in a rigorous fashion as measures on Minkowski space and we make the identification\n\\begin{equation}\n\\frac{1}{p^2-m^2+i\\epsilon}\\rightarrow -\\pi i\\Omega_m^{\\pm}(p), m\\ge0,\n\\end{equation}\n(see (Mashford, 2017b) for explanation). Therefore the outcome of our formal computations is that\n\\begin{equation} \\label{eq:Pi_gen}\n\\Pi(\\Gamma)=-\\frac{e^2}{16\\pi^2}\\int\\chi_{\\Gamma}(k+p)\\mbox{Tr}(\\eta_{\\mu\\nu}\\gamma^{\\mu}({\\slas p}+m)\\gamma^{\\nu}({\\slas k}-m))\\,\\Omega_m^{\\pm}(dk)\\,\\Omega_m^{\\pm}(dp).\n\\end{equation}\nWe will consider the case\n\\begin{equation} \\label{eq:Pi2}\n\\Pi(\\Gamma)=-\\frac{e^2}{16\\pi^2}\\int\\chi_{\\Gamma}(k+p)\\mbox{Tr}(\\eta_{\\mu\\nu}\\gamma^{\\mu}({\\slas p}+m)\\gamma^{\\nu}({\\slas k}-m))\\,\\Omega_m(dk)\\,\\Omega_m(dp), m>0.\n\\end{equation}\n(We use the symbol $\\Omega_m$ to denote $\\Omega_m^{+}$ if $m>0$ or $\\Omega_m^{-}$ if $m<0$.)\n\nThe important thing is that the object defined by Eq.~\\ref{eq:Pi2} exists as a Borel complex measure (i.e. when its argument $\\Gamma$ is a relatively compact Borel set in ${\\bf R}^4$). This is because\n\\begin{equation}\n\\int\\chi_{\\Gamma}(k+p)|\\mbox{Tr}(\\eta_{\\mu\\nu}\\gamma^{\\mu}({\\slas p}+m)\\gamma^{\\nu}({\\slas k}-m))|\\,\\Omega_m(dk)\\,\\Omega_m(dp)<\\infty, \n\\end{equation}\nfor all $\\Gamma\\in{\\mathcal B}_0({\\bf R}^4)$.\n\nIt also exists as a tempered distribution since\n\\begin{equation} \\label{eq:Pi3}\n\\int\\psi(k+p)\\mbox{Tr}(\\eta_{\\mu\\nu}\\gamma^{\\mu}({\\slas p}+m)\\gamma^{\\nu}({\\slas k}-m))\\,\\Omega_m(dk)\\,\\Omega_m(dp),\n\\end{equation}\nis convergent for any Schwartz function $\\psi\\in{\\mathcal S}({\\bf R}^4,{\\bf C})$. The basic reason for both these facts is that as $|p|,|q|\\rightarrow{\\infty}$ with $p,q\\in H_m^{+}$, $(p+q)^0\\rightarrow{\\infty}$.\n\nThus $\\Pi$ exists as a tempered measure. Hence we have in a few lines of formal argument arrived at an object which has a well defined existence and can investigate the properties of this object $\\Pi$ without any further concern about ill-definedness or the fear of propagating ill-definedness through our calculations.\n\nBy (Mashford, 2017b, Theorem 5) the ${\\bf C}^{4\\times4}$ valued measure defined by\n\\begin{equation} \\label{eq:Pi4}\n\\Phi(\\Gamma)=-\\frac{e^2}{16\\pi^2}\\int\\chi_{\\Gamma}(k+p)(\\eta_{\\mu\\nu}\\gamma^{\\mu}({\\slas p}+m)\\gamma^{\\nu}({\\slas k}-m))\\,\\Omega_m(dk)\\,\\Omega_m(dp),\n\\end{equation}\nis $K$ invariant for all $m>0$ (see (Mashford, 2017b) for a definition of the group $K$ and of the notion of $K$ invariance). Also we have the following. \n\\begin{theorem}\nIf $\\Psi:{\\mathcal B}_0({\\bf R}^4)\\rightarrow{\\bf C}^{4\\times4}$ is a $K$ invariant measure then the object $\\mbox{Tr}(\\Psi):{\\mathcal B}_0({\\bf R}^4)\\rightarrow{\\bf C}$ defined by\n\\begin{equation}\n(\\mbox{Tr}(\\Psi))(\\Gamma)=\\mbox{Tr}(\\Psi(\\Gamma)),\n\\end{equation}\nis a Lorentz invariant complex measure.\n\\end{theorem}\n{\\bf Proof} It is straightforward to show that Tr$(\\Psi)$ is countably subadditive. By definition of $K$ invariance (Mashford, 2017b) we have that\n\\begin{equation}\n\\Psi(\\Lambda(\\kappa)(\\Gamma))=\\kappa\\Psi(\\Gamma)\\kappa^{-1}, \\forall\\kappa\\in K,\\Gamma\\in{\\mathcal B}_0({\\bf R}^4),\n\\end{equation}\nwhere $\\Lambda(\\kappa)\\in O(1,3)^{+\\uparrow}$ is the Lorentz transformation corresponding to $\\kappa\\in K$. Therefore\n\\begin{equation}\n\\mbox{Tr}(\\Psi)(\\Lambda(\\kappa)(\\Gamma))=\\mbox{Tr}(\\kappa\\Psi(\\Gamma)\\kappa^{-1})=\\mbox{Tr}(\\Psi(\\Gamma)), \\forall\\kappa\\in K,\\Gamma\\in{\\mathcal B}_0({\\bf R}^4),\n\\end{equation}\nand hence\n\\begin{equation}\n\\mbox{Tr}(\\Psi)(\\Lambda(\\Gamma))=\\mbox{Tr}(\\Psi(\\Gamma))=(\\mbox{Tr}(\\Psi))(\\Gamma), \\forall\\Lambda\\in O(1,3)^{+\\uparrow},\\Gamma\\in{\\mathcal B}_0({\\bf R}^4). \n\\end{equation}\n$\\Box$\n\nIt follows that the measure $\\Pi$ that we have defined is a Lorentz invariant measure.\n\nUsing an argument similar to that for $\\Omega_m*\\Omega_m$ it can be shown that the support supp$(\\Pi)$ of $\\Pi$ is a subset of $C_m$. \n\n\\subsection{Application of the spectral calculus to determine the spectrum of $\\Pi$}\n\nWe have shown that $\\Pi$ is a Lorentz invariant tempered complex measure with support contained in $C_m$. Therefore by the spectral theorem  $\\Pi$ must have a spectral representation of the form\n\\begin{equation} \\label{eq:vp_spectral1}\n\\Pi(\\Gamma)=\\int_{m^{\\prime}=2m}^{\\infty}\\sigma(dm^{\\prime})\\Omega_{m^{\\prime}}(\\Gamma).\n\\end{equation}\nWe would like to compute the spectral measure $\\sigma$. First we have\n\\begin{eqnarray}\n&  & \\mbox{Tr}(\\eta_{\\mu\\nu}\\gamma^{\\mu}({\\slas p}+m)\\gamma^{\\nu}({\\slas k}-m)) \\nonumber \\\\\n&  & = \\eta_{\\mu\\nu}p_{\\alpha}k_{\\beta}\\mbox{Tr}(\\gamma^{\\mu}\\gamma^{\\alpha}\\gamma^{\\nu}\\gamma^{\\beta})+0+0-\\mbox{Tr}(\\gamma^{\\mu}\\gamma_{\\mu}m^2) \\nonumber \\\\\n& & = \\eta_{\\mu\\nu}p_{\\alpha}k_{\\beta}\\mbox{Tr}(\\gamma^{\\mu}\\gamma^{\\alpha}\\gamma^{\\nu}\\gamma^{\\beta})-16m^2 \\nonumber \\\\\n& & = 4p_{\\alpha}k_{\\beta}\\eta_{\\mu\\nu}(\\eta^{\\mu\\alpha}\\eta^{\\nu\\beta}-\\eta^{\\mu\\nu}\\eta^{\\alpha\\beta}+\\eta^{\\mu\\beta}\\eta^{\\alpha\\nu})-16m^2 \\nonumber \\\\\n& & = 4\\eta_{\\mu\\nu}(p^{\\mu}k^{\\nu}-\\eta^{\\mu\\nu}p.k+k^{\\mu}p^{\\nu})-16m^2 \\nonumber \\\\\n& & = 4(p.k-4p.k+p.k-4m^2) \\nonumber \\\\\n& & = -8(p.k+2m^2), \\nonumber \n\\end{eqnarray}\nwhere we have used in the second line the fact that the trace of a product of an odd number of gamma matrices vanishes.\n\nWe now compute in a fashion similar to that used when determining the spectrum of $\\Omega_m*\\Omega_m$ (which can be justified in a fashion similar to the justification of Argument 1) as follows.\n\\begin{eqnarray}\ng(a,b,\\epsilon) & = & \\Pi(\\Gamma(a,b,\\epsilon)) \\nonumber \\\\\n    & = & -\\frac{e^2}{16\\pi^2}\\int\\chi_{\\Gamma(a,b,\\epsilon)}(k+p)\\mbox{Tr}(\\eta_{\\mu\\nu}\\gamma^{\\mu}({\\slas p}+m)\\gamma^{\\nu}({\\slas k}-m))\\,\\Omega_m(dk)\\,\\Omega_m(dp) \\nonumber \\\\\n    & = &  \\frac{e^2}{2\\pi^2}\\int\\chi_{\\Gamma(a,b,\\epsilon)}(k+p)(p.k+2m^2)\\,\\Omega_m(dk)\\,\\Omega_m(dp) \\nonumber \\\\\n    & \\approx & \\frac{e^2}{2\\pi^2}\\int\\chi_{(a,b)}(\\omega_m({\\vct k})+\\omega_m({\\vct p}))\\chi_{B_{\\epsilon}({\\vct 0})}({\\vct k}+{\\vct p})(\\omega_m({\\vct p})\\omega_m({\\vct k})-{\\vct p}.{\\vct k}+2m^2)\\, \\nonumber \\\\\n    & & \\frac{d{\\vct k}}{\\omega_m({\\vct k})}\\,\\frac{d{\\vct p}}{\\omega_m({\\vct p})} \\nonumber \\\\\n & = & \\frac{e^2}{2\\pi^2}\\int\\chi_{(a,b)}(\\omega_m({\\vct k})+\\omega_m({\\vct p}))\\chi_{B_{\\epsilon}({\\vct 0})-{\\vct p}}({\\vct k})(\\omega_m({\\vct p})\\omega_m({\\vct k})-{\\vct p}.{\\vct k}+2m^2)\\, \\nonumber \\\\\n    & & \\frac{d{\\vct k}}{\\omega_m({\\vct k})}\\,\\frac{d{\\vct p}}{\\omega_m({\\vct p})} \\nonumber \\\\\n    & \\approx & \\frac{e^2}{2\\pi^2}\\int\\chi_{(a,b)}(2\\omega_m({\\vct p}))(3m^2+2{\\vct p}^2)\\,\\frac{d{\\vct p}}{\\omega_m({\\vct p})^2}(\\frac{4}{3}\\pi\\epsilon^3) \\nonumber \n\\end{eqnarray}\nTherefore \n\\begin{eqnarray}\ng_a(b) & = & \\lim_{\\epsilon\\rightarrow0}\\epsilon^{-3}g(a,b,\\epsilon) \\nonumber \\\\\n    & = & \\frac{e^2}{2\\pi^2}\\int\\chi_{(a,b)}(2\\omega_m({\\vct p}))(3m^2+2{\\vct p}^2)\\,\\frac{d{\\vct p}}{\\omega_m({\\vct p})^2}(\\frac{4}{3}\\pi) \\nonumber \\\\\n    & = & \\frac{2e^2}{\\pi}\\int_{r=mZ(a)}^{mZ(b)}(3m^2+2r^2)\\frac{r^2}{m^2+r^2}\\,dr(\\frac{4}{3}\\pi). \\nonumber\n\\end{eqnarray}\nThus we compute the spectum of $\\Pi$ as follows.\n\\begin{eqnarray}\n\\sigma(b) & = & \\frac{3}{4\\pi}bg_a^{\\prime}(b) \\nonumber \\\\\n    & = & \\frac{2e^2}{\\pi}b(3m^2+2m^2Z^2(b))\\frac{m^2Z^2(b)}{m^2+m^2Z^2(b)}\\frac{b}{4mZ(b)} \\nonumber \\\\\n    & = & \\frac{2}{\\pi}e^2m^3Z(b)(3+2Z^2(b)), \\nonumber\n\\end{eqnarray}\nwhere $Z:[2m,\\infty)\\rightarrow[0,\\infty)$ is given by Eq.~\\ref{eq:Z_def}.\n\nThe spectrum has this value $\\sigma(b)$ for $b\\ge2m$ and the value 0 for $b\\leq2m$. One can now see that $\\Pi$ is a Borel measure in the ordinary sense of the word, i.e. $[0,\\infty]$ valued countably subadditive function on ${\\mathcal B}({\\bf R}^4)$ which is finite on compact sets. (It is clearly defined on the larger sigma algebra of Lebesgue measurable sets.) $\\Pi$ is finite on compact sets and when evaluated on test functions of rapid decrease, it is not divergent.\n\n\\subsection{The vacuum polarization function $\\pi$}\n\nWe know that $q\\mapsto\\Pi(q)$ does not exist pointwise as a function, the integral defining it is divergent. However, pretend for the moment that $\\Pi$ did exist as a function. Then we can define a measure which we also denote by $\\Pi$ by\n\\begin{equation}\n\\Pi(\\Gamma)=\\int_{\\Gamma}\\Pi(q)\\,dq.\n\\end{equation}\nThus the function $\\Pi$ is the density defining the measure $\\Pi$. Now we know that in fact $\\Pi$ exists as a tempered measure with density\n\\begin{equation}\n\\Pi(q)=\\left\\{\\begin{array}{l}\n(q^2)^{-\\frac{1}{2}}\\sigma((q^2)^{\\frac{1}{2}})\\mbox{ if } q^2>0,q^0>0 \\\\\n0\\mbox{ otherwise,}\n\\end{array}\\right.\n\\end{equation}\nwhere \n\\begin{equation} \\label{eq:pi_spectrum}\n\\sigma(b)=\\frac{2}{\\pi}e^2m^3Z(b)(3+2Z^2(b)), b\\ge0,\n\\end{equation} \nis the spectrum of the measure $\\Pi$. Thus we may think of $\\Pi$ the function as being defined to be equal to this density.\n\nWe define the vacuum polarization function $\\pi:\\{q\\in{\\bf R}^4:q^2>0,q^0>0\\}\\rightarrow{\\bf R}$ by\n\\begin{equation} \\label{eq:pi_def}\n\\pi(q)=\\frac{\\Pi(q)}{q^2},\n\\end{equation}\n(Weinberg (2005, p. 478) states that $\\Pi^{\\rho\\sigma}$ has the form $\\Pi^{\\rho\\sigma}(q)=(q^2\\eta^{\\rho\\sigma}-q^{\\rho}q^{\\sigma})\\pi(q^2)$ from which, contracting with $\\eta_{\\rho\\sigma}$, it would follow that $\\pi(q)=(3q^2)^{-1}\\Pi(q)$. However Eq. 11.2.23 of (Weinberg, 2005, p. 480) is consistent with $\\pi$ having the form of Eq.~\\ref{eq:pi_def}). \n\nThen our spectral vacuum polarization is\n\\begin{equation}\n\\pi(q)=\\frac{\\Pi(q)}{q^2}=\\left\\{\\begin{array}{l}\n(q^2)^{-\\frac{3}{2}}\\sigma((q^2)^{\\frac{1}{2}})\\mbox{ if } q^2>4m^2,q^0>0 \\\\\n0\\mbox{ otherwise,}\n\\end{array}\\right.\n\\end{equation}\nfor $q\\in{\\bf R}^4$.\n$\\pi$ is a function on ${\\bf R}^4$ supported on $C_m$ but its value for argument $q$ only depends on $q^2$. Therefore, with no fear of confusion, one may define the vacuum polarization function $\\pi:[2m,\\infty)\\rightarrow[0,\\infty)$ by\n\\begin{equation} \\label{eq:polarization_function}\n\\pi(s)=s^{-3}\\sigma(s)=\\frac{2}{\\pi}s^{-3}e^2m^3Z(s)(3+2Z^2(s)),\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:Z_def_1}\nZ(s)=(\\frac{s^2}{4m^2}-1)^{\\frac{1}{2}}.\n\\end{equation}\n\n\\subsection{Definition of $\\pi(q)$ for $q^2<0$}\n\n\\subsubsection{On the non-positivity of $q^2$ for momentum transfer $q$}\n\nConsider a scattering process involving a particular particle of mass $m$ with $|\\mbox{in}>$ momentum $p$ and $|\\mbox{out}>$ momentum $p^{\\prime}$ with $q=p^{\\prime}-p$ the momentum transferred. Let $p=(p^0,{\\vct p})=(\\omega_m({\\vct p}),{\\vct p}), p^{\\prime}=(p^{\\prime0},{\\vct {p^{\\prime}}})=(\\omega_m({\\vct {p^{\\prime}}}),{\\vct {p^{\\prime}}})$. (So the incoming and outgoing particles are ``on shell\".) Suppose that\n\\begin{equation}\n|{\\vct p}|=\\alpha m, |{\\vct {p^{\\prime}}}|=\\beta m, \\alpha,\\beta\\in[0,\\infty).\n\\end{equation}\nThen\n\\begin{eqnarray}\nq^2 & = & (p^{\\prime}-p)^2 \\nonumber \\\\\n    & = & p^2+p^{\\prime2}-2p.p^{\\prime} \\nonumber \\\\\n    & = & 2(m^2-\\omega_m({\\vct p})\\omega_m({\\vct {p^{\\prime}}})+{\\vct p}.{\\vct {p^{\\prime}}}) \\nonumber \\\\\n    & = & 2(m^2-(m^2+\\alpha^2 m^2)^{\\frac{1}{2}}(m^2+\\beta^2m^2)^{\\frac{1}{2}}+{\\vct p}.{\\vct {p^{\\prime}}}) \\nonumber \\\\\n    & = & 2m^2(1+\\alpha\\beta\\cos(\\theta)-(1+\\alpha^2)^{\\frac{1}{2}}(1+\\beta^2)^{\\frac{1}{2}}), \\nonumber\n\\end{eqnarray}\nwhere $\\theta$ is the angle between ${\\vec p}$ and ${\\vec {p^{\\prime}}}$. Hence\n\\begin{equation}\nq^2<2m^2(1+\\alpha\\beta\\cos(\\theta)-\\alpha\\beta)=2m^2(1-\\alpha\\beta(1-\\cos(\\theta)).\n\\end{equation}\n If $\\theta\\neq0$ and $\\beta>0$ then $q^2<0$ for $\\alpha$ sufficiently large and $\\rightarrow-\\infty$ as $\\alpha\\rightarrow\\infty$. A similar statement applies for $\\beta$. Furthermore when considering the non-relativistic approximation, $q^2$ is invariably spacelike.\n\n\\subsubsection{Definition of $\\pi$ in the spacelike (imaginary mass) domain}\n\nWe have found that the vacuum polarization function $q\\mapsto\\pi(q)$ that we have defined is zero when its argument $q$ is such that $q^2<0$. However we will find shortly that we need to consider $\\pi(q)$ for values of $q$ which are such that $q^2<0$ (since the $q$ values are momentum transfer values). In this case we may consider the vacuum polarization function to be defined in the spacelike, or imaginary mass, domain by making the substitution $\\Omega_m*\\Omega_m\\rightarrow\\Omega_{im}*\\Omega_{im}$. Then as in Section 4 we may consider the type III measure with spectrum $\\sigma$ given by Eq.~\\ref{eq:pi_spectrum} and the vacuum polarization function given by\n\\begin{equation}\n\\pi(q)=\\left\\{\\begin{array}{l}\n(-q^2)^{-\\frac{3}{2}}\\sigma((-q^2)^{\\frac{1}{2}})\\mbox{ if } q^2<0,q^0>0 \\\\\n0\\mbox{ otherwise,}\n\\end{array}\\right.\n\\end{equation}\nand the associated function $s\\mapsto\\pi(s)$ given by Eq.~\\ref{eq:polarization_function} where $s>0$ now represents a spacelike label.\n\n\\subsection{Comparison of the spectral vacuum polarization function with the renormalized vacuum polarization function}\n\nRegularization and renormalization are techniques invented by physicists to control the infinities in divergent integrals in quantum field theory to obtain finite answers which can be compared with experiment. The answers obtained using these methods are in close agreement with experiment so there is clearly great merit in the approach. However many mathematicians are confused by these methods since they do not seem to make mathematical sense (e.g. introducing infinite ``counterterms\" into Lagrangians to cancel infinities produced when carrying out integrations implied by these Lagrangians or perturbing the dimension D of space-time to $D=4-\\epsilon, \\epsilon>0$ because everything blows up when $D=4$ and then later ignoring or subtracting out terms proportional to $\\epsilon^{-1}$ before taking the limit as $\\epsilon$ tends to $0$ to obtain the answers which are compared with experiment (dimensional regularization\/renormalization)). \n\nThe vacuum polarization function is generally computed in QFT using the dimensional regularization approach with the result\n\\begin{equation}\n\\pi_r(k^2)=-\\frac{2\\alpha}{\\pi}\\int_0^1dz\\,z(1-z)\\log(1-\\frac{k^2z(1-z)}{m^2}),\n\\end{equation}\n(Mandl and Shaw, 1991, p. 229) where $m>0$ is the mass of the electron and (in natural units) $\\alpha=(4\\pi)^{-1}e^2$ is the fine structure constant in which $e>0$ is the magnitude of the electron charge. $\\pi_r$ is defined for all $k\\in{\\bf R}^4$ for which $k^2\\leq4m^2$. The integral can be performed leading to the analytic expression\n\\begin{equation} \\label{eq:pi_Itzikson}\n\\pi_r(k^2)=-\\frac{\\alpha}{3\\pi}\\{\\frac{1}{3}+2(1+\\frac{2m^2}{k^2})[(\\frac{4m^2}{k^2}-1)^{\\frac{1}{2}}\\mbox{arcot}(\\frac{4m^2}{k^2}-1)^{\\frac{1}{2}}-1]\\}.\n\\end{equation} \n(See Appendix 1 for a proof of this.)\n\nThus \n\\begin{equation}\n\\pi_r(k)=-\\frac{\\alpha}{\\pi}\\frac{1}{3}(\\frac{1}{3}+(Y^2+3)(Y\\mbox{arcot}(Y)-1)),\n\\end{equation}\nwhere\n\\begin{equation}\nY=Y(k)=\\left(\\frac{4m^2}{k^2}-1\\right)^{\\frac{1}{2}}.\n\\end{equation}\nThis expression for $\\pi_r$ is defined on $\\{k\\in{\\bf R}^4:0<k^2\\leq4m^2\\}$.\n\n$\\pi_r(k)$ depends only on the value $k^2$ of its argument $k$. Therefore we define (without fear of confusion) $\\pi_r:(0,2m]\\rightarrow[0,\\infty)$ by\n\\begin{equation}\n\\pi_r(q)=-\\frac{\\alpha}{\\pi}\\frac{1}{3}(\\frac{1}{3}+(Y^2+3)(Y\\mbox{arcot}(Y)-1)),\n\\end{equation}\nwhere\n\\begin{equation}\nY=Y(q)=\\left(\\frac{4m^2}{q^2}-1\\right)^{\\frac{1}{2}}.\n\\end{equation}\nLet $\\pi_s$ denote our $\\pi$ calculated using spectral calculus. $\\pi_s(q)$ is defined for $q\\geq2m$ while $\\pi_r(q)$ is defined for $0<q\\leq2m$. To compare them we note that\n\\begin{equation}\nZ(q)=\\left(\\frac{q^2}{4m^2}-1\\right)^{\\frac{1}{2}}=\\frac{q}{2m}\\left(1-\\frac{4m^2}{q^2}\\right)^{\\frac{1}{2}}=\\frac{qi}{2m}\\left(\\frac{4m^2}{q^2}-1\\right)^{\\frac{1}{2}}=\\frac{qi}{2m}Y(q),\n\\end{equation}\nwhere $Z:[2m,\\infty)\\rightarrow[0,\\infty)$ is the function defined by Eq.~\\ref{eq:Z_def_1}.\n\nBoth (Weinberg, 2005, p. 475)  and (Itzikson and Zuber, 1980, p. 322) in their highly complex manipulations to compute $\\pi_r$ do a rotation in the complex plane. Thus we will compare our $\\pi_s$ with their $\\pi_r$ using the assignment\n\\begin{equation}\nY\\mapsto iY.\n\\end{equation}\n\nAs mentioned above, Weinberg (2005) seems to be assuming at some points that $\\pi(q)=(3q^2)^{-1}\\Pi(q)$. Itzikson and Zuber (1980)  p. 322 write (paraphrasing) ``$\\Pi_{\\rho\\sigma}=-i(\\eta_{\\rho\\sigma}q^2-q_{\\rho}q_{\\sigma})\\pi(q^2)$\" from which it would follow that $\\pi(q)=(-3iq^2)^{-1}\\Pi(q)$. We find that, for the purposes of comparison, $\\pi_r(q)$ should be rescaled by a factor of $9$ and then shifted by an amount of $\\frac{2}{\\pi^2}$. These parameters may be traced to some of the complex manipulations of Weinberg and others with infinite quantities (Weinberg, 2005, p. 479) and may be described as finite renormalization constants. It is to be emphasized that these renormalization constants are finite.   \n\nHence to compare our spectral vacuum polarization function with the vacuum polarization function computed by dimensional regularization \/ renormalization we implement the following C++ code where we have omitted the common factor of $-e^2$ from the values for $\\pi_r(q)$ and $\\pi_s(q)$.\n\\begin{verbatim}\n\/\/---------------------------------------------------------------------------\n\n#include \"iostream.h\"\n#include \"math.h\"\n\nconst int N_display = 10000;\nconst double Lambda = 4.0;\nconst double pi = 4.0*atan(1.0);\n\nint main(int argc, char* argv[])\n{\n    double delta = Lambda\/N_display;\n    int i;\n    for(i=1;i<=N_display;i++)\n    {\n        double rho = 2.0+i*delta; \/\/ q\/m\n        double Z = sqrt(rho*rho\/4.0-1.0);\n        double v = 1.0\/(rho*rho*rho);\n        double pi_spectral;\n        pi_spectral = v*Z*(3.0+2.0*Z*Z);\n        pi_spectral *= (2.0\/pi);\n        double xi = 1.0\/9.0; \/\/ factor to rescale pi_spectral\n        pi_spectral *= xi;\n        double Y = 2.0*Z\/rho;\n        double pi_renormalized;\n        pi_renormalized = (1.0\/3.0)+(Y*Y+3.0)*(Y*atan(1.0\/Y)-1.0);\n        pi_renormalized += (8.0\/3.0); \/\/ term to shift pi_renormalized so that\n                                      \/\/ pi_renormalized = 0 when rho = 2\n        pi_renormalized \/= 3.0;\n        pi_renormalized \/= (4.0*pi*pi);\n        cout << rho << '\\t' << pi_renormalized << '\\t'\n             << pi_spectral  << endl;\n\n    }\n    return(0);\n}\n\n\/\/---------------------------------------------------------------------------\n\n\\end{verbatim}\n\nThe graph of the output produced by this program is given in Figure 1. Thus we have shown that\n\\begin{equation}\n\\pi_s(\\rho)\\approx\\frac{2}{\\pi^2}+9\\pi_r(\\rho), \\forall\\rho\\in[2,\\infty).\n\\end{equation}\nIt can be seen that (up to finite renormalization) the difference between the spectral vacuum polarization and that obtained using dimensional regularization \/ renormalization is very small even though they are defined by completely different analytic expressions and derived by totally different approaches.\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=15cm]{vp_r_vp_spectral.jpg}\n\\caption{vp using renormalization and spectral vp versus rho = q\/m} \\label{fig:Fig1}\n\\end{figure}\n\n\n\\section{The Uehling contribution to the Lamb shift for the H atom}\n\nFollowing Weinberg (2005) we carry out a {\\em gedankenexperiment} in which an electron is scattered off a proton to compute, using the Born approximation, the Uehling contributions to the Lamb shift i.e. the result of including a Feynman diagram with a single fermion loop in addition to the diagram associated with M\\\"{o}ller scattering, to compute the effective potential of the H atom for determination of the Uehling contribution to the Lamb shift. \n\nThe Feynman diagram associated with M\\\"{o}ller scattering contributes a scattering matrix of (Itzikson and Zuber, 1980, p. 277)\n\\begin{equation}\nS_{afi}=-ie_1e_2(2\\pi)^4\\delta(p_1^{\\prime}+p_2^{\\prime}-p_1-p_2){\\mathcal M}_a,\n\\end{equation}\nwhere $e_1=-e$ and $e_2=e$ are the charges of the electron and proton respectively,\n\\begin{equation}\n{\\mathcal M}_{a,\\alpha_1^{\\prime}\\alpha_2^{\\prime}\\alpha_1\\alpha_2}(p_1^{\\prime},p_2^{\\prime},p_1,p_2) =-\\frac{{\\mathcal M}_{0,\\alpha_1^{\\prime}\\alpha_2^{\\prime}\\alpha_1\\alpha_2}(p_1^{\\prime},p_2^{\\prime},p_1,p_2) }{(p_2-p_2^{\\prime})^2+i\\epsilon},\n\\end{equation}\nin which\n\\begin{equation} \\label{eq:M_0_def}\n{\\mathcal M}_{0,\\alpha_1^{\\prime}\\alpha_2^{\\prime}\\alpha_1\\alpha_2}(p_1^{\\prime},p_2^{\\prime},p_1,p_2)  =  \\overline{u}( p_1^{\\prime},\\alpha_1^{\\prime})\\gamma^{\\rho}u(p_1,\\alpha_1)\\eta_{\\rho\\sigma}\\overline{u}(p_2^{\\prime},\\alpha_2^{\\prime})\\gamma^{\\sigma} u(p_2,\\alpha_2), \n\\end{equation}\nand $u(p,\\alpha)$ are Dirac spinors corresponding to $p\\in H_{m}^{+}$ for $\\alpha\\in\\{0,1\\}$,\n(see Appendix 4). (There is another contributing diagram obtained by making the substitution $p_1^{\\prime}\\leftrightarrow p_2^{\\prime}$).\n\nThe Uehling contribution to the vacuum polarization contributes a scattering matrix given by\n\\begin{equation}\nS_{bfi}=-ie_1e_2(2\\pi)^4\\delta(p_1^{\\prime}+p_2^{\\prime}-p_1-p_2){\\mathcal M}_b.\n\\end{equation}\nTo compute the contribution ${\\mathcal M}_b$ of the vacuum polarization Feynman amplitude we need to form the product of 3 measures. In fact we have\n\\begin{eqnarray} \\label{eq:Uehling_amplitude}\n{\\mathcal M}_{b\\alpha_1^{\\prime}\\alpha_2^{\\prime}\\alpha_1\\alpha_2}(p_1^{\\prime},p_2^{\\prime},p_1,p_2) & = & -\\overline{u}(p_1^{\\prime},\\alpha_1^{\\prime})\\gamma^\\rho u(p_1,\\alpha_1)D_{F\\rho\\sigma}(p_2-p_2^{\\prime})\\Pi^{\\mu\\nu}(p_2-p_2^{\\prime}) \\nonumber \\\\\n    &  &  D_{F\\mu\\nu}(p_2-p_2^{\\prime})\\overline{u}(p_2^{\\prime},\\alpha_2^{\\prime})\\gamma^{\\sigma}u(p_2,\\alpha_2), \\nonumber \\\\\n\\end{eqnarray}\nwhere \n\\begin{equation}\nD_{F\\alpha\\beta}(q)=\\frac{-\\eta_{\\alpha\\beta}}{q^2+i\\epsilon},\n\\end{equation}\nis the photon propagator. (The minus sign in Eq.~\\ref{eq:Uehling_amplitude} is associated with the fermion loop.)\nTherefore since \n\\begin{equation}\n\\eta_{\\mu\\nu}\\Pi^{\\mu\\nu}(q)=q^2\\pi(q), \\forall q\\in{\\bf R}^4,\n\\end{equation}\nwe have\n\\begin{equation}\n{\\mathcal M}_{b\\alpha_1^{\\prime}\\alpha_2^{\\prime}\\alpha_1\\alpha_2}(p_1^{\\prime},p_2^{\\prime},p_1,p_2)=-\\frac{{{\\mathcal M}_{0\\alpha_1^{\\prime}\\alpha_2^{\\prime}\\alpha_1\\alpha_2}}(p_1^{\\prime},p_2^{\\prime},p_1,p_2)\\pi(p_2-p_2^{\\prime})}{(p_2-p_2^{\\prime})^2+i\\epsilon}.\n\\end{equation}\nFor this calculation we are multiplying measures by multiplying the density functions corresponding to them. $\\pi$ has a well defined density function determined from the calculations of the previous section. We are taking the density function for the Feynman photon propagator $D_{F\\alpha\\beta}$ to be the function $q\\mapsto-(q^2)^{-1}\\eta_{\\alpha\\beta}$. (We can not multiply the measures by multiplying their spectra because we have a multiplicity of atoms at $m=0$ corresponding to trying to compute the product $\\Omega_0\\Omega_0$.)\n\nThe total scattering matrix is given by\n\\begin{equation}\nS_{a+bfi}=S_{afi}+S_{bfi}=-ie_1e_2(2\\pi)^4\\delta(p_1^{\\prime}+p_2^{\\prime}-p_1-p_2){\\mathcal M}_{a+b},\n\\end{equation}\nwhere\n\\begin{eqnarray}\n{\\mathcal M}_{a+b\\alpha_1^{\\prime}\\alpha_2^{\\prime}\\alpha_1\\alpha_2}(p_1^{\\prime},p_2^{\\prime},p_1,p_2) & = & -\\frac{{\\mathcal M}_{0,\\alpha_1^{\\prime}\\alpha_2^{\\prime}\\alpha_1\\alpha_2}(p_1^{\\prime},p_2^{\\prime},p_1,p_2) }{(p_2-p_2^{\\prime})^2+i\\epsilon}(1+\\pi(p_2-p_2^{\\prime})) \\nonumber \\\\\n     & = & -\\frac{{\\mathcal M}_{0,\\alpha_1^{\\prime}\\alpha_2^{\\prime}\\alpha_1\\alpha_2}(p_1^{\\prime},p_2^{\\prime},p_1,p_2) }{q^2+i\\epsilon}(1+\\pi(q)), \\label{eq:Feynman_amplitude}\n\\end{eqnarray}\nwhere $q$ is the momentum transfer.\n\nThe nature of momentum space is influenced by the convention used for the definition of the Fourier transform operator ${\\mathcal F}$. There are two main possibilities (with space-time dimension = 4):\n\\begin{enumerate}\n\\item $({\\mathcal F}f)(p)={\\chp f}(p)=(2\\pi)^{-2}\\int f(x)e^{-ip.x}\\,dx$\n\\item $({\\mathcal F}f)(p)={\\chp f}(p)=\\int f(x)e^{-ip.x}\\,dx$\n\\end{enumerate}\nThen ${\\mathcal F}$ (Convention 2) $=(2\\pi)^2{\\mathcal F}$ (Convention 1). Convention 1 is such that ${\\mathcal F}^{-1}={\\mathcal F}^{{\\large \\mbox{$*$}}}$ and so ${\\mathcal F}$ is unitary. Thus Convention 1 is, in a sense, canonical. Thus we will assume that the vacuum polarization tensor is defined with respect to Convention 1. This is consistent with the formula:\n\\begin{eqnarray}\n\\Pi^{\\mu\\nu}(k) & = & -e^2\\int \\mbox{Tr}(\\gamma^{\\mu}({\\slas p}+m)\\gamma^{\\nu}({\\slas p}-{\\slas k}+m)) \\nonumber \\\\\n    & & (2\\pi)^{-2}\\frac{1}{p^2-m^2+i\\epsilon}(2\\pi)^{-2}\\frac{1}{(p-k)^2-m^2+i\\epsilon}\\,dp. \\nonumber\n\\end{eqnarray}\nHowever calculations in QFT are usually carried out using Convention 2. Therefore we convert $\\Pi$ to Convention 2 by making the transformation $\\Pi\\rightarrow(2\\pi)^2\\Pi$.\n\nNow using the Born approximation and Eq.~\\ref{eq:Feynman_amplitude}, the change in potential as a result of the Uehling correction is given by\n\\begin{equation}\n\\Delta V(q)=\\pi(q)V_0(q)= \\pi(q)A_0(q).(1,{\\vct 0}),\n\\end{equation}\nwhere $A_0$ is the 4-potential associated with the Coulomb potential and $V_0(q)=A_0(q).(1,{\\vct 0})$ is the scalar potential. By Maxwell's equations \n\\begin{equation}\n({\\large \\mbox{$\\Box$}} A_0)(x)=j(x), \n\\end{equation}\nwhere $j(x)=(j({\\vct x}),0)$ with $j({\\vct x})=Ze\\delta({\\vct x})$ is the 4-current associate with a stationary point charge of magnitude $Ze$. Thus, in momentum space, we have\n\\begin{equation}\nA_0(q)=-\\frac{j(q)}{q^2},\n\\end{equation}\nand so\n\\begin{equation}\n(\\Delta V)(q)=-\\pi(q)\\frac{j(q)}{q^2}.(1,{\\vct 0}).\n\\end{equation}\nThus, as just discussed, we set\n\\begin{equation}\n(\\Delta V)(q)=-(2\\pi)^2\\pi(q)\\frac{j(q)}{q^2}.(1,{\\vct 0}).\n\\end{equation}\nSince we are using a non-relativistic approximation (the Born approximation) we take $\\pi$ to be defined by its spacelike form\n\\begin{equation}\n\\pi(q)=\\left\\{\\begin{array}{l}\n\\pi((-q^2)^{\\frac{1}{2}}) \\mbox{ if } q^2<0,q^0>0, \\\\\n0 \\mbox{ otherwise}.\n\\end{array}\\right.\n\\end{equation}\nIn configuration space,\n\\begin{eqnarray}\n(\\Delta V)(x) & = & (\\Delta V)(t,{\\vct x}) \\nonumber \\\\\n    & = & -(2\\pi)^{-2}\\int\\frac{\\pi(q)}{q^2}e^{iq.(x-x^{\\prime})}\\,dq\\,j({\\vct {x^{\\prime}}})\\,d{\\vct {x^{\\prime}}}\\,dt^{\\prime} \\nonumber \\\\\n    & = & -(2\\pi)^{-2}Ze\\int\\frac{\\pi(q)}{q^2}e^{-i{\\vct q}.{\\vct x}}e^{iq^0(t-t^{\\prime})}\\,dt^{\\prime}\\,dq^0\\,d{\\vct q} \\nonumber \\\\\n    & = & -(2\\pi)^{-2}Ze\\int_{q^2<0,q^0>0}\\frac{\\pi((-q^2)^{\\frac{1}{2}})}{q^2}e^{-i{\\vct q}.{\\vct x}}e^{iq^0(t-t^{\\prime})}\\,dt^{\\prime}\\,dq^0\\,d{\\vct q} \\nonumber \\\\\n\\end{eqnarray}\nTherefore\n\\begin{equation}\n(\\Delta V)(t,{\\vct x})=-(2\\pi)^{-2}Ze\\int I(t,{\\vct q})e^{-i{\\vct q}.{\\vct x}}\\,d{\\vct q},\n\\end{equation}\nwhere\n\\begin{eqnarray}\nI(t,{\\vct q}) & = & \\int_{(q^0)^2<{\\vct q}^2, q^0>0}\\frac{\\pi((-q^2)^{\\frac{1}{2}})}{q^2}e^{iq^0(t-t^{\\prime})}\\,dt^{\\prime}\\,dq^0 \\nonumber \\\\\n    & = & \\int_{(q^0)^2<{\\vct q}^2,q^0>0}\\frac{\\pi((-q^2)^{\\frac{1}{2}})}{q^2}e^{-iq^0t^{\\prime}}\\,dt^{\\prime}\\,dq^0.\n\\end{eqnarray}\nNow\n\\begin{eqnarray}\nI^{*}(t,{\\vct q}) & = & \\int_{(q^0)^2<{\\vct q}^2, q^0>0}\\int\\frac{\\pi((-q^2)^{\\frac{1}{2}})}{q^2}e^{iq^0t^{\\prime}}\\,dt^{\\prime}\\,dq^0 \\nonumber \\\\\n    & = & \\int_{(q^0)^2<{\\vct q}^2,q^0<0}\\frac{\\pi((-q^2)^{\\frac{1}{2}})}{q^2}e^{-iq^0t^{\\prime}}\\,dt^{\\prime}\\,dq^0. \\nonumber \\\\\n\\end{eqnarray}\nTherefore\n\\begin{equation}\n\\mbox{Re}(I(t,{\\vct q}))=\\frac{1}{2}\\int_{(q^0)^2<{\\vct q}^2}\\frac{\\pi((-q^2)^{\\frac{1}{2}})}{q^2}e^{-iq^0t^{\\prime}}\\,dt^{\\prime}\\,dq^0.\n\\end{equation} \nBut\n\\begin{equation}\nq^0\\mapsto\\frac{\\pi((-q^2)^{\\frac{1}{2}})}{q^2},\n\\end{equation}\nis even. Hence $I(t,{\\vct q})$ is real. Therefore\n\\begin{eqnarray}\n(\\Delta V)(t,{\\vct x})  & = & -(2\\pi)^{-2}Ze\\frac{1}{2}\\int_{q^2<0}\\frac{\\pi((-q^2)^{\\frac{1}{2}})}{q^2}e^{-i{\\vct q}.{\\vct x}}e^{-iq^0t^{\\prime}}\\,dt^{\\prime}\\,dq^0\\,d{\\vct q} \\nonumber \\\\\n    & = & -(2\\pi)^{-1}Ze\\frac{1}{2}\\int_{q^2<0}\\frac{\\pi((-q^2)^{\\frac{1}{2}})}{q^2}e^{-i{\\vct q}.{\\vct x}}\\delta(q^0)\\,dq^0\\,d{\\vct q} \\nonumber \\\\\n    & = & (2\\pi)^{-1}Ze\\frac{1}{2}\\int\\frac{\\pi(|{\\vct q}|)}{{\\vct q}^2}e^{-i{\\vct q}.{\\vct x}}\\,d{\\vct q} \\nonumber \\\\\n    & = & (2\\pi)^{-1}Ze\\frac{1}{2}\\int_{s=0}^{\\infty}\\int_{\\theta=0}^{\\pi}\\frac{\\pi(s)}{s^2}e^{-irs\\cos(\\theta)}s^2(2\\pi)\\sin(\\theta)\\,d\\theta\\,ds \\nonumber \\\\\n    & = & Ze\\frac{1}{2}\\int\\pi(s)\\frac{1}{irs}\\left.{e^{irsu}}\\right|_{u=-1}^1\\,ds \\nonumber \\\\\n    & = & \\frac{Ze}{r}\\int\\frac{\\pi(s)}{s}\\sin(rs)\\,ds, \\nonumber\n\\end{eqnarray}\nwhere $r=|{\\vct x}|$ (${\\vct x}\\mapsto(\\Delta V)(t,{\\vct x})$ is invariant under orthogonal transformations for all $t\\in{\\bf R}$). Thus\n\\begin{equation}\n(\\Delta V)(x)=(\\Delta V)(t,{\\vct x})=(\\Delta V)({\\vct x})=(\\Delta V)(r),\n\\end{equation}\nwhere\n\\begin{equation}\n(\\Delta V)(r)=\\frac{Ze}{r}\\int\\frac{\\pi(s)}{s}\\sin(rs)\\,ds.\n\\end{equation}\nFor vacuum polarization $\\pi=\\pi_s$ in the spacelike domain determined by the spectral calculus is given by\n\\begin{equation}\n\\pi(s)=\\frac{\\sigma(s)}{s^3},\n\\end{equation}\nwhere $\\sigma$ is the spectrum of $\\pi$ in the timelike domain. Therefore\n\\begin{equation}\n(\\Delta V)(r)=\\frac{Ze}{r}\\int\\frac{\\sigma(s)}{s^4}\\sin(rs)\\,ds.\n\\end{equation}\n \nApplying first order perturbation theory we compute the Uehling contribution to the Lamb shift to be\n\\begin{equation}\n\\Delta E=<\\psi|-e(\\Delta V)|\\psi>=-e\\int\\psi^2({\\vct x})(\\Delta V)({\\vct x})\\,d{\\vct x},\n\\end{equation}\nwhere $\\psi$ is the 2s wave function for the hydrogenic atom. Therefore our prediction for the Uehling contribution to the Lamb shift is\n\\begin{eqnarray}\n\\Delta E & = & -4\\pi Ze^2\\int\\frac{\\sigma(s)}{s^4}r\\psi^2(r)\\sin(rs)\\,ds\\,dr \\nonumber \\\\\n    & = & -4\\pi Ze^2\\int\\frac{\\sigma(s)}{s^4}r\\psi^2(r)\\sin(rs)\\,dr\\,ds. \\label{eq:Delta_E}\n\\end{eqnarray}\nThe notation and argument that we have used to compute $\\Delta E$ given the vacuum polarization function $\\pi$ is somewhat formal but is entirely consistent with standard usage in the physics literature. Any issues that may arise in presenting it in a  rigorous fashion will be discussed in a separate publication.\n\nWe calculate $\\Delta E$ for the H atom using numerical integration based on Eq.~\\ref{eq:Delta_E} with the result that $\\Delta E\\approx-28.7$ MHz. The C++ code for the program to carry out this computation can be found in Appendix 2 and a graph of the convergence of the integral can be found in Figure 2.\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=15cm]{convergence.jpg}\n\\caption{Convergence of integral for $\\Delta E$ = Uehling contribution to Lamb shift for H atom} \\label{fig:Fig3}\n\\end{figure}\n\n\\section{The running coupling constant}\n\nThe total equivalent potential for the electron-proton system (H atom) in the Born approximation is \n\\begin{equation}\nV(r)=-\\frac{e^2}{4\\pi r}-\\frac{e^2}{r}\\int\\frac{\\pi(s)}{s}\\sin(rs)\\,ds.\n\\end{equation}\n\nAt range $r$ the potential is equivalent to that produced by an effective charge or running coupling constant $e_r$ given by\n\\begin{eqnarray}\n-\\frac{e_r^2}{4\\pi r} & = & -\\frac{e^2}{4\\pi r}(1+4\\pi\\int\\frac{\\pi(s)}{s}\\sin(rs)\\,ds) \\nonumber \\\\\n    & = & -\\frac{e^2}{4\\pi r}(1+4\\pi\\int\\pi(\\frac{s}{r})\\frac{\\sin(s)}{s}\\,ds). \\nonumber \n\\end{eqnarray}\nTherefore the running fine structure ``constant\" at energy $\\mu$ is given by\n\\begin{equation} \\label{eq:running_coupling}\n\\alpha(\\mu)=\\alpha(0)(1+4\\pi\\int\\pi(\\mu s)\\mbox{sinc}(s)\\,ds).\n\\end{equation}\n\n$\\alpha(0)\\approx1\/137$ and $\\alpha$ increases with increasing energy having been measured to have a value of $\\alpha(\\mu)\\approx1\/127$ for $\\mu=90$ GeV. Given this explicit expression for the running coupling it is not neccessary to use the techniques of the renormalization group equation involving a beta function to investigate its behavior. \n\nOur expression for the running coupling only involves the vacuum polarization contribution. Other contributions such as the electron self energy and higher order Feynman diagrams need to be considered in order to determine the complete running coupling behavior.\n\n\\subsection{Determination of the behavior of the running coupling constant in one loop QED when using the renormalized vacuum polarization function $\\pi=\\pi_r$ \\label{section:running_cc_renormalization}}\n\nIn this case we have\n\\begin{equation}\n\\pi(\\mu s)=\\pi_r(\\mu s),\n\\end{equation}\nwhere\n\\begin{equation} \n\\pi_r(s)=-\\frac{\\alpha}{3\\pi}(\\frac{1}{3}+(3-W^2)(W\\mbox{arcoth}(W)-1)),\n\\end{equation}\nand $W$ is given by\n\\begin{equation} \nW=W(s)=(1+\\frac{4m^2}{s^2})^{\\frac{1}{2}}, s\\in(0,\\infty),\n\\end{equation}\n(see Appendix 1). Note that we are using $\\pi_r$ as defined in the imaginary mass domain because we are considering $s$ corresponding to spacelike $q$.\n\n\\begin{theorem}\nThe integral Eq.~\\ref{eq:running_coupling} defing the running coupling constant is divergent at all non zero energies when $\\pi=\\pi_r$.\n\\end{theorem}\n{\\bf Proof}\nLet $\\mu>0$ Now\n\\begin{equation}\nW(\\mu s)=(1+\\frac{4m^2}{\\mu^2 s^2})^{\\frac{1}{2}}, \\forall s>0.\n\\end{equation}\nTherefore\n\\begin{equation}\ns=\\frac{2m}{\\mu(W^2(\\mu s)-1)^{\\frac{1}{2}}}.\n\\end{equation}\nNow as $s\\rightarrow\\infty$, $W(\\mu s)\\rightarrow1^{+}$. We will now show that \n\\begin{equation} \\label{eq:frac_limit1}\n\\frac{\\pi(\\mu s)}{s}\\rightarrow\\infty,\n\\end{equation}\nas $s\\rightarrow\\infty$. Terms in $\\pi$ that have a finite limit as $s\\rightarrow\\infty$ vanish in the limit of Eq.~\\ref{eq:frac_limit1}. Therefore we are interested in the limiting behavior of\n\\[ s\\mapsto\\mbox{atanh}(\\frac{1}{W(\\mu s)})(W^2(\\mu s)-1)^{\\frac{1}{2}}, \\] \nas $s\\rightarrow\\infty$. This is the same as the limiting behavior of\n\\[ W\\mapsto\\mbox{atanh}(\\frac{1}{W})(W-1)^{\\frac{1}{2}}(W+1)^{\\frac{1}{2}}, \\]\nas $W\\rightarrow1^{+}$, which is the same as the limiting behavior of\n\\[ x\\mapsto \\mbox{atanh}(\\frac{1}{x+1})x^{\\frac{1}{2}}=\\frac{x^{\\frac{1}{2}}}{f(x)}, \\]\nas $x\\rightarrow0^{+}$, where \n\\begin{equation}\nf(x)=\\frac{1}{\\mbox{atanh}(\\frac{1}{x+1})}.\n\\end{equation}\nNow, $x^{\\frac{1}{2}}\\rightarrow0^{+},f(x)\\rightarrow0^{+}$ as $x\\rightarrow0^{+}$. Therefore, by L'H\\^{o}pital's rule\n\\begin{equation}\n\\lim_{x\\rightarrow0^{+}}\\frac{x^{\\frac{1}{2}}}{f(x)}=\\lim_{x\\rightarrow0^{+}}\\frac{\\frac{1}{2}x^{-\\frac{1}{2}}}{f^{\\prime}(x)},\n\\end{equation}\nif the limit exists. Now\n\\begin{eqnarray}\n\\lim_{x\\rightarrow0^{+}}f^{\\prime}(x) & = & \\lim_{x\\rightarrow0^{+}}[-(\\mbox{atanh}(\\frac{1}{x+1}))^{-2}\\frac{1}{1-(\\frac{1}{x+1})^2}(-(x+1)^{-2})] \\nonumber \\\\\n    & = & \\lim_{x\\rightarrow0^{+}}[(\\mbox{atanh}(\\frac{1}{x+1}))^{-2}\\frac{1}{x(x+2)}] \\nonumber \\\\\n    & = & \\lim_{x\\rightarrow0^{+}}\\frac{g(x)}{x(x+2)}, \\nonumber \\nonumber \\\\\n\\end{eqnarray}\nwhere\n\\begin{equation}\ng(x)=(\\mbox{atanh}(\\frac{1}{x+1}))^{-2}.\n\\end{equation}\nNow\n\\[ g^{\\prime}(x)=-2(\\mbox{atanh}(\\frac{1}{x+1}))^{-3}(-(x+1)^{-2}). \\]\nTherefore\n\\begin{equation}\n\\lim_{x\\rightarrow0^{+}}f^{\\prime}(x)=\\lim_{x\\rightarrow0^{+}}\\frac{2\\mbox{atanh}(\\frac{1}{x+1}))^{-3}}{2x+2}=0.\n\\end{equation}\n Thus\n\\begin{equation}\n\\lim_{x\\rightarrow0^{+}}\\frac{x^{\\frac{1}{2}}}{f(x)}=\\infty.\n\\end{equation}\nTherefore the integrand of the integral Eq.~\\ref{eq:running_coupling} defining the running coupling constant is oscillatory with ever increasing amplitude and hence the integral divergent for all non zero energies.\n$\\Box$\n\nThis is to be compared with the work of Landau and others relating to the Landau pole or ``ghost\" pole in the solution of the renormalization group equations in QED ``the possible existence of which leads to a serious contradiction with a number of general principles of the theory\" (Bogoliubov and Shirkov, 1980, p. 517). \n\n\\subsection{Determination of the behavior of the running coupling constant in one loop QED when using the spectral vacuum polarization function $\\pi=\\pi_s$}\n\nIn this case we have\n\\begin{equation}\n\\pi(\\mu s)=\\pi_s(\\mu s),\n\\end{equation}\nwhere\n\\begin{equation}\n\\pi_s(s)=\\frac{2}{\\pi}s^{-3}e^2m^3Z(s)(3+2Z^2(s)),\n\\end{equation}\nand\n\\begin{equation} \nZ(s)=(\\frac{s^2}{4m^2}-1)^{\\frac{1}{2}},\n\\end{equation}\nfor $s\\in(0,\\infty)$.\n\nAs $s\\rightarrow\\infty$, $s^{-1}Z(s)\\rightarrow(2m)^{-1}$ and so $\\pi_s(s)\\rightarrow\\frac{e^2}{2\\pi}$. Thus\n\\begin{equation}\n\\pi(\\mu s)\\rightarrow\\frac{e^2}{2\\pi}, \\mbox{ as }s\\rightarrow{\\infty},\n\\end{equation}\nwhich is a finite limit. \n\\begin{theorem}\nThe integral given by Eq.~\\ref{eq:running_coupling} is convergent for all energies $\\mu\\geq0$ when $\\pi=\\pi_s$.\n\\end{theorem}\n{\\bf Proof}\nConsider the case when $\\mu=1$. All other values of $\\mu$ can be dealt with similarly. We want to show that the integral\n\\[ \\int\\mbox{sinc}(s)\\pi_s(s)\\,ds \\]\nis convergent. It is sufficient to show that the integral\n\\begin{equation} \\label{eq:rcc_theorem2_integral}\n\\int_{s=2m}^{\\infty}\\mbox{sinc}(s)\\frac{Z^3(s)}{s^3}\\, ds,\n\\end{equation}\nis convergent. Let\n\\begin{equation}\nL=\\lim_{s\\rightarrow\\infty}\\frac{Z^3(s)}{s^3}=\\frac{1}{(2m)^3}.\n\\end{equation}\nThen the integral \\ref{eq:rcc_theorem2_integral} will converge if $\\frac{Z^3(s)}{s^3}\\rightarrow L$ fast enough. Let\n\\begin{equation}\n\\epsilon_n=\\mbox{sup}\\{\\left|\\frac{Z^3(s)}{s^3}-L\\right|:s\\geq2\\pi (n-1)\\}, \\mbox{ for }n=1,2,\\ldots.\n\\end{equation}\nIf we define\n\\begin{equation}\nI_n=\\sum_{i=1}^n(I_i^{+}-I_i^{-}),\n\\end{equation}\nwhere\n\\begin{equation}\nI_i^{+}=\\int_{2\\pi(i-1)}^{2\\pi(i-1)+\\pi}\\frac{\\sin(s)}{s}\\,ds,\n\\end{equation}\nand\n\\begin{equation}\nI_i^{-}=-\\int_{2\\pi(i-1)+\\pi}^{2\\pi i}\\frac{\\sin(s)}{s}\\,ds,\n\\end{equation}\nthen, as is well known,\n\\begin{equation}\nI_n\\rightarrow\\frac{\\pi}{2}, \\mbox{ as }n\\rightarrow\\infty.\n\\end{equation}\nNow define\n\\begin{equation}\nJ_i^{+}=\\int_{2\\pi(i-1)}^{2\\pi(i-1)+\\pi}\\frac{\\sin(s)}{s}\\frac{Z^3(s)}{s^3}\\,ds,\n\\end{equation}\nand\n\\begin{equation}\nJ_i^{-}=-\\int_{2\\pi(i-1)+\\pi}^{2\\pi i}\\frac{\\sin(s)}{s}\\frac{Z^3(s)}{s^3}\\,ds,\n\\end{equation}\nand let\n\\begin{equation}\nS_n=\\sum_{i=1}^n(J_i^{+}-J_i^{-}), n=1,2,\\ldots.\n\\end{equation}\nWe want to show that $S_n$ converges to a finite limit as $n\\rightarrow\\infty$. We have\n\\begin{eqnarray}\nS_n & \\in & (\\sum_{i=1}^n((L-\\epsilon_i)I_i^{+}-(L+\\epsilon_i)I_i^{-}),\\sum_{i=1}^n((L+\\epsilon_i)I_i^{+}-(L-\\epsilon_i)I_i^{-})) \\nonumber \\\\\n    & = & (\\sum_{i=1}^n L(I_i^{+}-I_i^{-})-\\epsilon_i(I_i^{+}+I_i^{-}),\\sum_{i=1}^n L(I_i^{+}-I_i^{-})+\\epsilon_i(I_i^{+}+I_i^{-})) \\nonumber\n\\end{eqnarray}\nClearly if $\\epsilon_i\\rightarrow0$ fast enough then $S_n$ is convergent. Well we have\n\\begin{eqnarray}\n\\left|\\frac{Z^3(s)}{s^3}-L\\right| & = & \\left|(\\frac{1}{4m^2}-\\frac{1}{s^2})^{\\frac{3}{2}}-\\frac{1}{(2m)^3}\\right| \\nonumber \\\\\n    & = & \\left|\\frac{(\\frac{1}{4m^2}-\\frac{1}{s^2})^{3}-\\frac{1}{(2m)^6}}{(\\frac{1}{4m^2}-\\frac{1}{s^2})^{\\frac{3}{2}}+\\frac{1}{(2m)^3}}\\right| \\nonumber \\\\\n    & \\leq & (2m)^3\\left|(\\frac{1}{4m^2}-\\frac{1}{s^2})^{3}-\\frac{1}{(2m)^6}\\right| \\nonumber \\\\\n    & = & \\frac{(2m)^3}{s^2}\\left|\\frac{3}{4m^2s^2}-\\frac{3}{(2m)^4}-\\frac{1}{s^4}\\right| \\nonumber \n\\end{eqnarray}\nNow if $s$ is sufficiently large then\n\\begin{eqnarray}\n|\\frac{3}{4m^2s^2}-\\frac{3}{(2m)^4}-\\frac{1}{s^4}| & = & \\frac{3}{(2m)^4}-\\frac{3}{4m^2s}+\\frac{1}{s^4} \\nonumber \\\\\n    & < & \\frac{3}{(2m)^4}+\\frac{1}{s^4} \\nonumber \\\\\n    & \\leq & \\frac{3}{(2m)^4}+\\frac{1}{(2m)^4} \\nonumber \\\\\n    & = & \\frac{1}{4m^4}. \\nonumber\n\\end{eqnarray} \n Therefore\n\\begin{equation}\n\\mbox{sup}\\{\\left|\\frac{Z^3(s)}{s^3}-L\\right|:s\\geq a\\}\\leq\\frac{2}{ma^2}, \\mbox{ for $a$ sufficiently large}.\n\\end{equation}\nHence\n\\begin{equation}\n\\epsilon_i\\leq\\frac{2}{m(2\\pi(i-1))^2}, \\mbox{ for $i$ sufficiently large}.\n\\end{equation}\nThus\n\\begin{equation}\n\\epsilon_i(I_i^{+}+I_i^{-})\\leq\\frac{2}{m(2\\pi(i-1))^2}\\int_{2\\pi(i-1)}^{2\\pi i}\\frac{1}{s}\\,ds\\leq\\frac{2}{m(2\\pi(i-1))^2}\\frac{1}{2\\pi(i-1)}2\\pi,\n\\end{equation}\nfor $i$ sufficiently large. Therefore the sequence $S_n$ is convergent as $n\\rightarrow\\infty$.\n$\\Box$\n\nA graph of the running fine structure constant versus $\\mu^{-1}=\\frac{r}{2\\pi}$ for $r\\in(0,\\frac{1}{10}a_0)$ where $a_0=$ the first Bohr radius of the H atom in natural units is shown in Figure \\ref{fig:Fig3}.\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=15cm]{vp_rcc.jpg}\n\\caption{QED running fine structure constant on the basis of vacuum polarization} \\label{fig:Fig3}\n\\end{figure}\n\n\\section{Conclusion}\n\nWe have presented a spectral calculus for the computation of the spectrum of causal Lorentz invariant Borel complex measures on Minkowski space and shown how this enables one to compute the density for such a measure with respect to Lebesgue measure. This has been applied to the case of the contraction of the vacuum polarization tensor resulting in a spectral vacuum polarization function which has very close agreement with the vacuum polarization function computed using dimensional regularization \/ renormalization in the domain of real mass.\n\nUsing the Born approximation together with the spectral vacuum polarization function the Uehling effect contribution to the Lamb shift for the H atom is computed to be $\\approx-28.7$ MHz. \nWith the spectral vacuum polarization function we obtain a well defined convergent running coupling function whereas the running coupling function generated using dimensional regularization \/ renormalization is shown to be divergent at all non-zero energies. \n\nIn subsequent work we will apply the spectral calculus to the electron self energy and generally to all renormalization issues arising in the QFT of the electroweak force. In addition QCD will be formulated in the context of locally conformally flat space-times (M\\\"{o}bius structures) (Mashford, 2017a and b) and the running coupling constant for QCD will be computed with a view to proving, or deriving, the asymptotic freedom of QCD.\n\n\\section*{Acknowledgements}\nThe author thanks Randolf Pohl and Christopher Chantler for very helpful discussions. \n\n\\section*{References}\n\n\\vskip .1in\\par\\sloppy\\hangindent=1pc\\hangafter=1 \\noindent Bogoliubov, N. N., Logunov, A. A. and Todorov, I. T., {\\em Introduction to Axiomatic Quantum Field Theory}, Benjamin, 1975.\n\n\\vskip .1in\\par\\sloppy\\hangindent=1pc\\hangafter=1 \\noindent Bogoliubov, N. N., and Shirkov, D. V., {\\em Introduction to the theory of quantized fields}, Third edition, Wiley, New York, 1980.\n\n\\vskip .1in\\par\\sloppy\\hangindent=1pc\\hangafter=1 \\noindent Choquet-Bruhat, Y., DeWitte-Morette, C., and Dillard-Bleick, M., {\\em Analysis, manifolds and physics}, North-Holland, Amsterdam, 1982.\n\n\\vskip .1in\\par\\sloppy\\hangindent=1pc\\hangafter=1 \\noindent Colombeau, J. F., {\\em New generalized functions and multiplication of distributions}, North Holland, Amsterdam, 1984.\n\n\\vskip .1in\\par\\sloppy\\hangindent=1pc\\hangafter=1 \\noindent Epstein, H. and Glaser, V., ``Role of locality in perturbation theory\", {\\em Annales de l'Institut Henri Poincar\\'{e} Section A Physique Th\\'{e}orique} 19(3), 211-295, 1973.\n\n\\vskip .1in\\par\\sloppy\\hangindent=1pc\\hangafter=1 \\noindent Halmos, P. R., {\\em Measure theory}, Springer-Verlag, New York, 1988.\n\n\\vskip .1in\\par\\sloppy\\hangindent=1pc\\hangafter=1 \\noindent Itzykson, C. and Zuber, J.-B., {\\em Quantum Field Theory}, McGraw-Hill, New York, 1980.\n\n\\vskip .1in\\par\\sloppy\\hangindent=1pc\\hangafter=1 \\noindent Mandl, F. and Shaw, G., {\\em Quantum Field Theory}, Wiley, Chichester, 1991.\n\n\\vskip .1in\\par\\sloppy\\hangindent=1pc\\hangafter=1 \\noindent Mashford, J. S., ``A non-manifold theory of space-time\", {\\em Journal of Mathematical Physics} 22(9), 1981, 1990-1993.\n\n\\vskip .1in\\par\\sloppy\\hangindent=1pc\\hangafter=1 \\noindent Mashford, J. S., {\\em Invariant  measures and M\\\"{o}bius structures: A framework for field theory}, PhD thesis, University of Melbourne, 2005.\n\n\\vskip .1in\\par\\sloppy\\hangindent=1pc\\hangafter=1 \\noindent Mashford, J. S., ``An approach to classical quantum field theory based on the geometry of locally conformally flat space-time\", \n{\\em Advances in Mathematical Physics} (2017), Article ID 8070462, 15 pages, https:\/\/doi.org\/10.1155\/2017\/8070462, 2017a. \n\n\\vskip .1in\\par\\sloppy\\hangindent=1pc\\hangafter=1 \\noindent Mashford J. S., ``Second quantized quantum field theory based on invariance properties of locally conformally flat space-times\",  arXiv:1709.09226, 2017b.\n\n\\vskip .1in\\par\\sloppy\\hangindent=1pc\\hangafter=1 \\noindent Oberguggenberger, M., {\\em Multiplication of distributions and applications to partial differential equations}, Longman, 1992.\n\n\\vskip .1in\\par\\sloppy\\hangindent=1pc\\hangafter=1 \\noindent Pohl, R., ``Laser Spectroscopy of Muonic Hydrogen and the Puzzling Proton\", Journal of the Physical Society of Japan 85, 091003 (2016).\n\n\\vskip .1in\\par\\sloppy\\hangindent=1pc\\hangafter=1 \\noindent Scharf, G., {\\em Finite Quantum Electrodynamics: The Causal Approach}, Springer-Verlag, Berlin, 1995.\n\n\\vskip .1in\\par\\sloppy\\hangindent=1pc\\hangafter=1 \\noindent Weinberg, S., {\\em The quantum theory of fields}, Vol. 1, Cambridge University Press, 2005.\n\n\\vskip .1in\\par\\sloppy\\hangindent=1pc\\hangafter=1 \\noindent Wilson, K., ``The renormalization group and critical phenomena\", {\\em Reviews of Modern Physics}, 55(3), 1983.\n\n\\section*{Appendix 1: Derivation of closed form solution forregularized\/renormalized vacuum polarization}\n\nThe standard formula for the vacuum polarization function $\\pi_r$ as obtained using regularization and renormalization is\n\\begin{equation} \\label{eq:appendix1_vp1}\n\\pi_r(k^2)=-\\frac{2\\alpha}{\\pi}\\int_0^1d\\beta\\,\\beta(1-\\beta)\\log\\left(1-\\frac{k^2\\beta(1-\\beta)}{m^2}\\right),\n\\end{equation}\n(Mandl and Shaw, 1991, p. 229) where $m>0$ is the mass of the electron and (in natural units) $\\alpha=(4\\pi)^{-1}e^2$ is the fine structure constant in which $e>0$ is the magnitude of the charge of the electron. $\\pi_r$ is defined for all $k\\in{\\bf R}^4$ for which $k^2<4m^2$. This integral can be performeed leading to the closed form solution (see Itzikson and Zuber (1980, p. 323)) \n\\begin{equation} \\label{eq:pi_Itzikson}\n\\pi_r(k^2)=-\\frac{\\alpha}{3\\pi}\\left\\{\\frac{1}{3}+2\\left(1+\\frac{2m^2}{k^2}\\right)\\left[\\left(\\frac{4m^2}{k^2}-1\\right)^{\\frac{1}{2}}\\mbox{arcot}\\left(\\frac{4m^2}{k^2}-1\\right)^{\\frac{1}{2}}-1\\right]\\right\\}.\n\\end{equation}\nThe function defined by Eq.~\\ref{eq:pi_Itzikson} is only defined for $0<k^2<4m^2$ unless one allows the functions $z\\mapsto z^{\\frac{1}{2}}$ and $z\\mapsto\\mbox{arcot}(z)$ to be defined on complex domains. It is useful for the work of the present paper to write down the derivation of this result and to consider the answer when $k^2<0$.\n\nLet $m>0$ and $I:\\{k\\in{\\bf R}^4:k^2<4m\\}\\rightarrow(-\\infty,0)$ be defined by\n\\begin{equation}\nI(k)=2\\int_{\\beta=0}^1d\\beta\\,\\beta(1-\\beta)\\log(1-\\frac{k^2\\beta(1-\\beta)}{m^2}),\n\\end{equation}\nThen\n\\begin{eqnarray}\nI(k) & = & 2\\int_{\\beta=0}^1d(\\frac{1}{2}\\beta^2-\\frac{1}{3}\\beta^3)\\log(1-\\frac{k^2\\beta(1-\\beta)}{m^2}) \\nonumber \\\\\n    & = & -2\\int_{\\beta=0}^1\\frac{\\frac{1}{2}\\beta^2-\\frac{1}{3}\\beta^3}{1-\\beta(1-\\beta)m^{-2}k^2}\\frac{k^2}{m^2}(2\\beta-1)\\,d\\beta \\nonumber \\\\\n    & = & 2\\int_{\\beta=0}^1\\frac{(\\frac{1}{3}\\beta^3-\\frac{1}{2}\\beta^2)(2\\beta-1)}{m^2(k^2)^{-1}-\\beta(1-\\beta)}\\,d\\beta \\nonumber\n\\end{eqnarray}\nNow\n\\begin{equation}\n\\frac{m^2}{k^2}-\\beta(1-\\beta)=\\frac{m^2}{k^2}-\\beta+\\beta^2=(\\beta-\\frac{1}{2})^2-\\frac{1}{4}+\\frac{m^2}{k^2}.\n\\end{equation}\nTherefore, changing variables,\n\\begin{eqnarray}\nI(k) & = & 2\\int_{\\beta=-\\frac{1}{2}}^{\\frac{1}{2}}\\frac{(\\frac{1}{3}(\\beta+\\frac{1}{2})^3-\\frac{1}{2}(\\beta+\\frac{1}{2})^2)2\\beta}{\\beta^2-\\frac{1}{4}+m^2(k^2)^{-1}}\\,d\\beta \\label{eq:appendix1_vp2} \\\\\n    & = & 2\\int_{\\beta=-\\frac{1}{2}}^{\\frac{1}{2}}\\frac{(\\frac{1}{3}(\\beta+\\frac{1}{2})^3-\\frac{1}{2}(\\beta+\\frac{1}{2})^2)2\\beta}{\\beta^2+X^2}\\,d\\beta\n\\end{eqnarray}\nwhere\n\\begin{equation}\nX=X(k)=\\frac{1}{2}(\\frac{4m^2}{k^2}-1)^{\\frac{1}{2}}\\in(0,\\infty), \\mbox{ for } 0<k^2<4m^2.\n\\end{equation}\nLet $\\beta=X\\tan(u)$. Then $\\beta^2+X^2=X^2\\sec^2(u)$ and $d\\beta=X\\sec^2(u)\\,du$. Also\n\\begin{equation}\n\\frac{1}{3}(\\beta+\\frac{1}{2})^3-\\frac{1}{2}(\\beta+\\frac{1}{2})^2=\\frac{1}{3}(\\beta+\\frac{1}{2})^2(\\beta-1).\n\\end{equation}\nThus \n\\begin{eqnarray}\nI(k) & = & \\frac{4}{3}\\frac{1}{X}\\int_{\\beta=-\\frac{1}{2}}^{\\frac{1}{2}}(\\beta+\\frac{1}{2})^2(\\beta-1)\\beta\\,du \\nonumber \\\\\n    & = & \\frac{4}{3}\\frac{1}{X}\\int_{\\beta=-\\frac{1}{2}}^{\\frac{1}{2}}\\beta^4-\\frac{3}{4}\\beta^2\\,du,\n\\end{eqnarray}\n(the integral of the odd powers of $\\beta$ vanishes). Therefore\n\\begin{eqnarray}\nI(k) & = & \\frac{4}{3}\\frac{1}{X}\\int_{\\beta=-\\frac{1}{2}}^{\\frac{1}{2}}X^4\\tan^4(u)-\\frac{3}{4}X^2\\tan^2(u)\\,du \\nonumber \\\\\n    & = & \\frac{4}{3}(X^3(\\frac{1}{3}\\tan^3(u)-\\tan(u)+u)-\\frac{3}{4}X(\\tan(u)-u))\\left|_{\\beta=-\\frac{1}{2}}^\\frac{1}{2}\\right. \\nonumber \\\\\n    & = & \\frac{1}{3}(\\frac{1}{3}+(4X^2+3)(2X\\mbox{arcot}(2X)-1)), \\label{eq:appendix1_vp_result1}\n\\end{eqnarray}\nwhich leads directly to the required result.\n\nClearly, when $k^2=0$, $I(k)$ defined by Eq.~\\ref{eq:appendix1_vp1} has the value $I(k)=0$. Now consider the case when $k^2<0$. Then we proceed with the same steps up to Eq.~\\ref{eq:appendix1_vp2} but now we write\n\\begin{equation}\nI(k)=2\\int_{\\beta=-\\frac{1}{2}}^{\\frac{1}{2}}\\frac{(\\frac{1}{3}(\\beta+\\frac{1}{2})^3-\\frac{1}{2}(\\beta+\\frac{1}{2})^2)2\\beta}{{\\beta^2-W^2}}\\,d\\beta,\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:appendix1_W_def}\nW=W(k)=(\\frac{1}{4}+\\frac{m^2}{-k^2})^{\\frac{1}{2}}=\\frac{1}{2}(1-\\frac{4m^2}{k^2})^{\\frac{1}{2}}.\n\\end{equation}\nThen make the substitution $\\beta=W\\mbox{tanh}(u)$ so that\n\\[ \\beta^2-W^2=-W^2\\mbox{sech}^2(u), d\\beta=W\\mbox{sech}^2(u)\\,du. \\]\nTherefore\n\\begin{eqnarray}\nI(k) & = & -\\frac{4}{3}\\frac{1}{W}\\int_{\\beta=-\\frac{1}{2}}^{\\frac{1}{2}}\\beta^4-\\frac{3}{4}\\beta^2\\,du, \\nonumber \\\\\n       & = & -\\frac{4}{3}\\frac{1}{W}\\int_{\\beta=-\\frac{1}{2}}^{\\frac{1}{2}}W^4\\mbox{tanh}^4(u)-\\frac{3}{4}W^2\\mbox{tanh}^2(u)\\,du \\nonumber\n\\end{eqnarray}\nNow\n\\[ \\int\\mbox{tanh}^2(u)\\,du=u-\\mbox{tanh}(u)+c, \\]\n\\[ \\int\\mbox{tanh}^4(u)\\,du=u-\\mbox{tanh}(u)-\\frac{1}{3}\\mbox{tanh}^3(u)+c. \\]\nThus\n\\begin{eqnarray}\nI(k) & = & -\\frac{4}{3}\\frac{1}{W}\\int_{\\beta=-\\frac{1}{2}}^{\\frac{1}{2}}W^4\\mbox{tanh}^4(u)-\\frac{3}{4}W^2\\mbox{tanh}^2(u)\\,du \\nonumber \\\\\n    & = & -\\frac{4}{3}(W^3(-\\frac{1}{3}\\mbox{tanh}^3(u)-\\mbox{tanh}(u)+u)-\\frac{3}{4}W(-\\mbox{tanh}(u)+u))\\left|_{\\beta=-\\frac{1}{2}}^\\frac{1}{2}\\right. \\nonumber \\\\\n    & = & \\frac{1}{3}(\\frac{1}{3}+(3-4W^2)(2W\\mbox{arcoth}(2W)-1)). \\label{eq:appendix_vp_result2}\n\\end{eqnarray}\nThis result may be obtained more directly by noting that\n\\begin{equation}\nX=\\frac{1}{2}\\left(\\frac{4m^2}{k^2}-1\\right)^{\\frac{1}{2}}=i\\frac{1}{2}\\left(1-\\frac{4m^2}{k^2}\\right)^{\\frac{1}{2}}=iW,\n\\end{equation}\nwhen $k^2<0$ and\n\\begin{equation}\n\\mbox{arcot}(2X)=\\mbox{atan}(\\frac{1}{2X})=-\\mbox{atan}(\\frac{i}{2W})=-i\\mbox{atanh}(\\frac{1}{2W})=-i\\mbox{arcoth}(2W).\n\\end{equation}\nand then using Eq.~\\ref{eq:appendix1_vp_result1}. Thus the renormalized vacuum polarization when $k^2<0$ is given by\n\\begin{equation} \n\\pi_r(k^2)=-\\frac{\\alpha}{3\\pi}(\\frac{1}{3}+(3-4W^2)(2W\\mbox{arcoth}(2W)-1)),\n\\end{equation}\nwhere $W:\\{k\\in{\\bf R}^4:k^2<0\\}\\rightarrow(0,\\infty)$ is given by Eq.~\\ref{eq:appendix1_W_def}.\n\nThus, in other words, \n\\begin{equation} \\label{eq:pi_Itzikson1}\n\\pi_r(k^2)=-\\frac{\\alpha}{3\\pi}(\\frac{1}{3}+(3-W^2)(W\\mbox{arcoth}(W)-1)),\n\\end{equation}\nwhere $W$ is given by\n\\begin{equation} \\label{eq:appendix1_W_def1}\nW=W(k)=(1-\\frac{4m^2}{k^2})^{\\frac{1}{2}}.\n\\end{equation}\n\n\\section*{Appendix 2: C++ code to compute the value of Uehling effect }\n\n\\begin{verbatim}\n\n\/\/---------------------------------------------------------------------------\n\n#pragma hdrstop\n\n#include \"iostream.h\"\n#include \"fstream.h\"\n#include \"math.h\"\n\nconst double pi = 4.0*atan(1.0);\nconst double m_electron = 9.10938356e-31; \/\/ electron mass in Kg mks\nconst double c = 2.99792458e8;  \/\/ speed of light m\/s mks\nconst double e = 1.6021766208e-19; \/\/ electron charge in Coulombs mks\nconst double h = 6.626070040e-34; \/\/ Planck constant mks\nconst double h_bar = h\/(2.0*pi);\nconst double epsilon_0 = 8.854187817e-12; \/\/ permittivity of free space mks\nconst double e1 = e\/sqrt(epsilon_0); \/\/ electron charge in rationalized units\n\/\/const double e1 = e\/sqrt(4*pi*epsilon_0); \/\/ electron charge in Gaussian units\nconst double alpha = e1*e1\/(4*pi*h_bar*c); \/\/ fine structure constant\n\nconst double e_Tiny = 1.0e-2;\nconst double e_Big = 1.0e2;\nconst double Tiny = 1.0e-10;\n\ndouble a_0,a_0_natural;\ndouble m = m_electron;\ndouble m_natural = m*c*c\/e;\ndouble psi(double);\ndouble sigma(double);\n\n\/\/---------------------------------------------------------------------------\n\n#pragma argsused\nint main(int argc, char* argv[])\n{\n    a_0 = 4.0*pi*h_bar*h_bar\/(m*e1*e1);\n        \/\/ Bohr radius of the Hydrogen atom in meters\n    a_0_natural = 1.0\/(m_natural*alpha);\n        \/\/ a_0 in natural units eV^{-1}\n    cout << \"electron mass = \" << m_natural << \" eV\" << endl;\n    cout << \"electron mass in Kg = \" << m << \" Kg\" << endl;\n    cout << \"Inverse fine structure constant 1\/alpha = \" << 1\/alpha << endl;\n    cout << \"Bohr radius of hydrogen atom = \"\n         << a_0 << \" m\" << endl;\n    cout << \"Bohr radius in natural units = \" << a_0_natural\n         << \" eV^{-1}\" << endl;\n    const int N_int = 10000;\n    double Lambda_int = 50.0*m_natural;\n    double delta_int = Lambda_int\/N_int;\n    double integral = 0.0;\n    int i,j;\n    for(i=1;i<N_int;i++)\n    {\n        double s = 2.0*m_natural+i*delta_int;\n        double integral_1 = 0.0;\n        double Lambda_int_1 = 2.0*pi*100.0\/s;\n        if(Lambda_int_1<100.0*a_0_natural) Lambda_int_1 = 100*a_0_natural;\n        double delta_int_1 = 2.0*pi\/(100.0*s);\n        if(delta_int_1>0.001*a_0_natural) delta_int_1 = 0.001*a_0_natural;\n        int N_int_1 = Lambda_int_1\/delta_int_1;\n        for(j=1;j<N_int_1;j++)\n        {\n            double r = j*delta_int_1;\n            integral_1 += r*psi(r)*psi(r)*sin(r*s);\n        }\n        integral += integral_1*delta_int_1*sigma(s)\/(s*s*s*s);\n    }\n    integral *= -delta_int*4.0*pi*alpha*4.0*pi;\n    cout << \"answer = \" << integral << \" eV = \"\n         << integral*e\/(1.0e06*h) << \" MHz\"\n         << endl;\n    return(0);\n}\n\ndouble psi(double r)\n{\n\/\/ Hydrogen atom wave function for 2s orbital\n    double answer;\n    double v = r\/(2.0*a_0_natural);\n    answer = (2.0-r\/a_0_natural)*exp(-v);\n    answer \/= (4.0*sqrt(2.0*pi)*a_0_natural*sqrt(a_0_natural));\n    return(answer);\n}\n\ndouble sigma(double s)\n{\n    if(s<2.0*m_natural) return(0.0);\n    double Z = sqrt(s*s\/(4.0*m_natural*m_natural)-1.0);\n    double zeta = Z*(3.0+2.0*Z*Z);\n    zeta *= 4.0*pi*alpha;\n    zeta *= 2.0\/pi;\n    zeta *= m_natural*m_natural*m_natural;\n    return(zeta);\n}\n\n\/\/---------------------------------------------------------------------------\n\n\\end{verbatim}\n\n\\section*{Appendix 3: Proof of the Spectral Theorem}\n\nThe action of the proper orthochronous Lorentz group $O(1,3)^{+\\uparrow}$ on Minkowski space has 5 classes of orbits each corresponding to a particular isotropy subgroup (little group). Firstly there is the distinguished orbit $\\{0\\}$ consisting of the origin. Secondly there are the positive mass hyperboloids $\\{p\\in{\\bf R}^4:p^2=m^2, p^0>0\\}$ with little group isomorphic to $SO(3)$. Then there are the negative mass hyperboloids, the positive open null cone, the negative open null cone and the imaginary mass hyperboloids. The spectral theorem is proved by considering separately each class of orbit. We will prove it for the space $X=\\{p\\in{\\bf R}^4:p^2>0,p^0>0\\}$ consisting of the union of all positive mass hyperboloids. The other cases can be proved similarly. We will prove the spectral theorem first for Lorentz invariant Borel measures $\\mu:{\\mathcal B}(X)\\rightarrow[0,\\infty]$ and then generalize the theorem later to Lorentz invariant Borel complex measures.\n\nLet Rotations $\\subset O(1,3)^{+\\uparrow}$ be defined by\n\\begin{equation}\n\\mbox{Rotations}=\\left\\{\\left(\\begin{array}{ll}\n1 & 0 \\\\\n0 & A\n\\end{array}\\right):A\\in SO(3)\\right\\},\n\\end{equation}\nand Boosts $\\subset O(1,3)^{+\\uparrow}$ be the set of pure boosts. Then it can be shown that for every $\\Lambda\\in O(1,3)^{+\\uparrow}$ there exist unique $B\\in$ Boosts and $R\\in$ Rotations such that\n\\[ \\Lambda = BR. \\]\nThus there exist maps $\\pi_1:O(1,3)^{+\\uparrow}\\rightarrow$ Boosts and $\\pi_2:O(1,3)^{+\\uparrow}\\rightarrow$ Rotations such that for all $\\Lambda\\in O(1,3)^{+\\uparrow}$\n\\begin{eqnarray}\n&  & \\Lambda = \\pi_1(\\Lambda)\\pi_2(\\Lambda), \\nonumber \\\\\n&  & \\Lambda = BR \\mbox{ with } B\\in\\mbox{Boosts and }R\\in\\mbox{ Rotations }\\Rightarrow B=\\pi_1(\\Lambda), R=\\pi_2(\\Lambda). \\nonumber\n\\end{eqnarray}\nFor $m>0$, define $h_m:\\mbox{Boosts}\\rightarrow H_m$ by\n\\begin{equation}\nh_m(B)=B(m,{\\vct 0})^{T}.\n\\end{equation}\nWe will show that $h_m$ is a bijection. Let $p\\in H_m$. Choose $\\Lambda\\in O(1,3)^{+\\uparrow}$ such that $p=\\Lambda(m,{\\vct 0})^{T}$. Then $p=\\pi_1(\\Lambda)\\pi_2(\\Lambda)(m,{\\vct 0})^{T}=\\pi_1(\\Lambda)(m,{\\vct 0})^{T}\\in h(\\mbox{Boosts})$. Therefore $h_m$ is surjective. Now suppose that $h(B_1)=h(B_2)$. Then $B_1(m,{\\vct 0})^{T}=B_2(m,{\\vct 0})^{T}$. Thus $B_2^{-1}B_1(m,{\\vct 0})^{T}=(m,{\\vct 0})^{T}$. Hence $B_2^{-1}B_1=R$ for some $R\\in$ Rotations. Therefore $B_1=\\pi_1(B_1)=\\pi_1(B_2R)=\\pi_1(B_2)=B_2$. Therefore $h_m$ is a bijection.\n\nNow there is an action $\\rho_m:O(1,3)^{+\\uparrow}\\times H_m\\rightarrow H_m$ of $O(1,3)^{+\\uparrow}$ on $H_m$ defined by\n\\begin{equation}\n\\rho_m(\\Lambda,p)=\\Lambda p.\n\\end{equation}\n$\\rho_m$ induces an action ${\\tld \\rho}_m:O(1,3)^{+\\uparrow}\\times$Boosts$\\rightarrow$Boosts according to\n\\begin{eqnarray}\n{\\tld \\rho}_m(\\Lambda,B) & = & h_m^{-1}(\\rho_m(\\Lambda,h_m(B))) \\nonumber \\\\\n    & = & h_m^{-1}(\\Lambda B(m,{\\vct 0})^{T}) \\nonumber \\\\\n    & = & h_m^{-1}(\\pi_1(\\Lambda B)\\pi_2(\\Lambda B)(m,{\\vct 0})^{T}) \\nonumber \\\\\n    & = & h_m^{-1}(\\pi_1(\\Lambda B)(m,{\\vct 0})^{T}) \\nonumber \\\\\n    & = & \\pi_1(\\Lambda B).\n\\end{eqnarray}\nNote that the induced action is independent of $m$ for all $m>0$.\n\n Let\n\\begin{equation}\nX=\\bigcup_{m>0}H_m=\\{p\\in{\\bf R}^4:p^2>0,p^0>0\\}.\n\\end{equation}\nDefine the action $\\rho:O(1,3)^{+\\uparrow}\\times X\\rightarrow X$ by $\\rho(\\Lambda,p)=\\Lambda p$. Then $\\rho$ induces an action ${\\tld\\rho}:O(1,3)^{+\\uparrow}\\times(0,\\infty)\\times\\mbox{Boosts}\\rightarrow(0,\\infty)\\times\\mbox{Boosts}$ according to\n\\begin{equation} \\label{eq:action_Boosts}\n{\\tld\\rho}(\\Lambda,m,B)={\\tld\\rho}_m(B)=\\pi_1(\\Lambda B).\n\\end{equation}\nDefime, for each $m>0$, $f_m:O(1,3)^{+\\uparrow}\\rightarrow H_m\\times\\mbox{Rotations}$ by\n\\begin{equation}\nf_m(\\Lambda)=(h_m(\\pi_1(\\Lambda)),\\pi_2(\\Lambda)).\n\\end{equation}\nThen each $f_m$ is a bijection.\nThe map $g:(0,\\infty)\\times\\mbox{Boosts}\\rightarrow X$ defined by\n\\begin{equation}\ng(m,B)=h_m(B)=B(m,{\\vct 0})^{T},\n\\end{equation}\nis a bijection. Define $f:(0,\\infty)\\times O(1,3)^{+\\uparrow}\\rightarrow X\\times\\mbox{Rotations}$ by\n\\begin{equation}\nf(m,\\Lambda)=f_m(\\Lambda).\n\\end{equation}\n$f$ is a bijection so we can push forward or pull back measures using $f$ at will. \n\nSuppose that $\\mu:{\\mathcal B}(X)\\rightarrow[0,\\infty]$ is a Borel measure on $X$ (by Borel measure we mean a measure defined on ${\\mathcal B}(X)$ which is finite on compact sets) and that $\\mu$ is invariant under the action . Let $\\mu_R$ be the measure on Rotations induced by Haar measure on $SO(3)$. Let $\\nu$ be the product measure $\\nu=\\mu\\times\\mu_R$ whose existence and uniqueness is guaranteed by the Hahn-Kolmogorov theorem and the fact that both $X$ and Rotations are $\\sigma$-finite. Let $\\nu\\#f^{-1}$ denote the pull back of $\\nu$ by $f$ (i.e. the push forward of $\\nu$ by $f^{-1}$). Then\n\\begin{equation}\n(\\nu\\#f^{-1})(\\Gamma)=\\nu(f(\\Gamma)), \\forall\\Gamma\\in{\\mathcal B}((0,\\infty)\\times O(1,3)^{+\\uparrow}).\n\\end{equation}\nConsider the action $\\tau:O(1,3)^{+\\uparrow}\\times(0,\\infty)\\times O(1,3)^{+\\uparrow}\\rightarrow(0,\\infty)\\times O(1,3)^{+\\uparrow}$ defined by\n\\begin{equation}\n\\tau(\\Lambda,m^{\\prime},\\Lambda^{\\prime})=(m^{\\prime},\\Lambda\\Lambda^{\\prime}).\n\\end{equation}\n$\\tau$ induces an action ${\\tld \\tau}:O(1,3)^{+\\uparrow}\\times X\\times\\mbox{Rotations}\\rightarrow X\\times\\mbox{Rotations}$ so that if $p^{\\prime}\\in X$ with $p^{\\prime}=B^{\\prime}(m^{\\prime},{\\vct 0})^{T}$ with $m^{\\prime}\\in(0,\\infty), B^{\\prime}\\in\\mbox{Boosts}$ and $R^{\\prime}\\in\\mbox{Rotations}$ then\n\\begin{eqnarray}\n{\\tld \\tau}(\\Lambda,(p^{\\prime},R^{\\prime})) & = & {\\tld \\tau}(\\Lambda,B^{\\prime}(m^{\\prime},{\\vct 0})^{T},R^{\\prime}) \\nonumber \\\\\n    & = & f_{m^{\\prime}}(\\Lambda f_{m^{\\prime}}^{-1}(h_{m^{\\prime}}(B^{\\prime}(m^{\\prime},{\\vct 0})^{T},R^{\\prime})) \\nonumber \\\\\n    & = & f_{m^{\\prime}}(\\Lambda f_{m^{\\prime}}^{-1}(h_{m^{\\prime}}(\\pi_1(\\Lambda^{\\prime})),\\pi_2(\\Lambda^{\\prime})) \\nonumber \\\\\n    & = & f_{m_{\\prime}}(\\Lambda\\Lambda^{\\prime}) \\nonumber \\\\\n    & = & (h_{m^{\\prime}}(\\pi_1(\\Lambda\\Lambda^{\\prime})),\\pi_2(\\Lambda\\Lambda^{\\prime})), \\nonumber\n\\end{eqnarray}\nwhere $\\Lambda^{\\prime}=B^{\\prime}R^{\\prime}$. We will now show that the measure $\\nu$ is an invariant measure on $X\\times\\mbox{Rotations}$ with respect to the action ${\\tld \\tau}$. To this effect let $E_1^{\\prime}\\subset\\mbox{Boosts}, E_2^{\\prime}\\subset\\{(m^{\\prime},{\\vct 0}):m^{\\prime}\\in(0,\\infty)\\}$ and $E_3^{\\prime}\\subset\\mbox{Rotations}$ be Borel sets. Then\n\\begin{eqnarray}\n\\nu({\\tld\\tau}(\\Lambda,E_1^{\\prime}E_2^{\\prime},E_3^{\\prime})) & = & \\nu(\\pi_1(\\Lambda E_1^{\\prime})E_2^{\\prime}\\times\\pi_2(\\Lambda E_3^{\\prime})) \\nonumber \\\\\n    & = & \\mu(\\pi_1(\\Lambda E_1^{\\prime})E_2^{\\prime})\\mu_R(\\pi_2(\\Lambda E_3^{\\prime})) \\nonumber \\\\\n    & = & \\mu(\\pi_1(\\Lambda E_1^{\\prime}))\\mu_R(\\pi_2(\\Lambda)\\pi_2(E_3^{\\prime})) \\nonumber \\\\\n    & = & \\mu(E_1^{\\prime}E_2^{\\prime})\\mu_R(E_3^{\\prime}) \\nonumber \\\\\n    & = & \\nu(E_1^{\\prime}E_2^{\\prime},E_3^{\\prime}), \\nonumber\n\\end{eqnarray}\n (here we have used the notation of juxtaposition of sets to denote the set of all products i.e. $ S_1S_2=\\{xy:x\\in S_1, y\\in S_2\\}$, also $xS=\\{xy:y\\in S\\}$). Therefore the measure $\\nu\\#f^{-1}$ is an invariant measure on $O(1,3)^{+\\uparrow}$ with respect to the action $\\tau$. Therefore for each Borel set $E\\subset(0,\\infty)$ the measure $(\\nu\\#f^{-1})_E:{\\mathcal B}(O(1,3)^{+\\uparrow})\\rightarrow[0,\\infty]$ defined by\n\\begin{equation}\n(\\nu\\#f^{-1})_E(\\Gamma)=(\\nu\\#f^{-1})(E,\\Gamma),\n\\end{equation}\nis a translation invariant measure on the group $O(1,3)^{+\\uparrow}$. Therefore since, $O(1,3)^{+\\uparrow}$ is a locally compact second countable topological group there exists, by the uniqueness part of Haar's theorem, a unique $c=c(E)\\ge0$ such that \n\\begin{equation}\n(\\nu\\#f^{-1})_E=c(E)\\mu_{O(1,3)^{+\\uparrow}},\n\\end{equation} \nwhere $\\mu_{{O(1,3}^{+\\uparrow}}$ is the Haar measure on $O(1,3)^{+\\uparrow}.$ Denote by $\\sigma$ the map $\\sigma:{\\mathcal B}((0,\\infty))\\rightarrow[0,\\infty]$ defined by $\\sigma(E)=c(E)$.\n\nWe will now show that $\\sigma$ is a measure on $(0,\\infty)$. We have that for any $\\Gamma\\in{\\mathcal B}(O(1,3)^{+\\uparrow}),E\\in{\\mathcal B}((0,\\infty))$\n\\begin{equation}\n\\sigma(E)\\mu_{O(1,3)^{+\\uparrow}}(\\Gamma)=(\\nu\\#f^{-1})_E(\\Gamma)=\\nu(f(E,\\Gamma))=\\nu(\\pi_1(\\Gamma)E\\times\\pi_2(\\Gamma)).\n\\end{equation}\nChoose $\\Gamma\\in{\\mathcal B}(O(1,3)^{+\\uparrow})$ such that $\\mu_{O(1,3)^{+\\uparrow}}(\\Gamma)\\in(0,\\infty)$.\n\nThen \n\\begin{equation}\n\\sigma(\\emptyset)\\mu_{O(1,3)^{+\\uparrow}}(\\Gamma)=\\nu(\\pi_1(\\Gamma)\\emptyset\\times\\pi_2(\\Gamma))=\\nu(\\emptyset)=0.\n\\end{equation}\nTherefore \n\\begin{equation}\n\\sigma(\\emptyset)=0.\n\\end{equation}\nAlso let $\\{E_n\\}_{n=1}^{\\infty}\\subset{\\mathcal B}((0,\\infty))$.  Then\n\\begin{eqnarray}\n\\sigma(\\bigcup_{n=1}^{\\infty}(E_n) & = & \\mu_{O(1,3)^{+\\uparrow}}(\\Gamma)^{-1}\\nu(\\pi_1(\\Gamma)\\bigcup_{n=1}^{\\infty}E_n\\times\\pi_2(\\Gamma)) \\nonumber \\\\\n     & = & \\mu_{O(1,3)^{+\\uparrow}}(\\Gamma)^{-1}\\nu(\\bigcup_{n=1}^{\\infty}(\\pi_1(\\Gamma)E_n\\times\\pi_2(\\Gamma))) \\nonumber \\\\\n     & = & \\sum_{n=1}^{\\infty}\\mu_{O(1,3)^{+\\uparrow}}(\\Gamma)^{-1}\\nu((\\pi_1(\\Gamma)E_n\\times\\pi_2(\\Gamma))) \\nonumber \\\\\n     & = & \\sum_{n=1}^{\\infty}\\sigma(E_n).\n\\end{eqnarray}\nThus $\\sigma$ is a measure. \n\nThe above argument holds for all invariaant measures $\\mu:{\\mathcal B}(X)\\rightarrow[0,\\infty]$. Therefore, in particular, it is true for $\\Omega_m$ for $m\\in(0,\\infty)$. Hence there exists a measure $\\sigma_{\\Omega_m}:{\\mathcal B}((0,\\infty))\\rightarrow[0,\\infty]$ such that\n\\begin{equation}\n((\\Omega_m\\times\\mu_{SO(3)})\\#f^{-1})(E,\\Gamma)=\\sigma_{\\Omega_m}(E)\\mu_{O(1,3)^{+\\uparrow}}(\\Gamma),\n\\end{equation}\nfor $E\\in{\\mathcal B}((0,\\infty)), \\Gamma\\in{\\mathcal B}(O(1,3)^{+\\uparrow}$. But\n\\begin{eqnarray}\n((\\Omega_m\\times\\mu_{SO(3)})\\#f^{-1})(E,\\Gamma) & = & (\\Omega_m\\times\\mu_{SO(3)})(f(E,\\Gamma)) \\nonumber \\\\\n    & = &  (\\Omega_m\\times\\mu_{SO(3)})(\\pi_1(\\Gamma)(E\\times\\{{\\vct 0}\\}),\\pi_2(\\Gamma)) \\nonumber \\\\\n    & = & \\Omega_m(\\pi_1(\\Gamma)(E\\times\\{{\\vct 0}\\}))\\mu_{SO(3)}(\\pi_2(\\Gamma)) \\nonumber \\\\\n    & = & \\Omega_m(\\pi_1(\\Gamma)(m,{\\vct 0})^{T})\\mu_{SO(3)}(\\pi_2(\\Gamma))\\delta_E(m) \\nonumber \\\\\n\\end{eqnarray}\nwhere $\\delta_m$ is the Dirac measure concentrated on $m$. Thus\n\\begin{equation}\n\\sigma_{\\Omega_m}(E)=\\mu_{O(1,3)^{+\\uparrow}}(\\Gamma)^{-1}\\Omega_m(\\pi_1(\\Gamma)(m,{\\vct 0})^{T})\\mu_{SO(3)}(\\pi_2(\\Gamma))\\delta_m(E),\n\\end{equation}\nfor any $E\\in{\\mathcal B}((0,\\infty)),\\Gamma\\in{\\mathcal B}(O(1,3)^{+\\uparrow})$ such that $\\mu_{O(1,3)^{+\\uparrow}}(\\Gamma)\\in(0,\\infty).$ Choose any $\\Gamma\\in{\\mathcal B}(O(1,3)^{+\\uparrow})$ such that $\\mu_{O(1,3)^{+\\uparrow}}(\\Gamma)\\in(0,\\infty)$ and define $\\sigma_{\\Omega}:(0,\\infty)\\rightarrow(0,\\infty)$ by\n\\begin{equation}\n\\sigma_{\\Omega}(m)=\\mu_{O(1,3)^{+\\uparrow}}(\\Gamma)^{-1}\\Omega_m(\\pi_1(\\Gamma)(m,{\\vct 0})^{T})\\mu_{SO(3)}(\\pi_2(\\Gamma)).\n\\end{equation}\nThen\n\\begin{equation}\n\\sigma_{\\Omega_m}=\\sigma_{\\Omega}(m)\\delta_m, \\forall m\\in(0,\\infty).\n\\end{equation}\n\nReturning now to the general invariant measure $\\mu:{\\mathcal B}(X)\\rightarrow[0,\\infty]$ we will now show that $\\mu$ can be written as a product $\\mu=\\sigma\\times\\mu_B$ for some measure $\\mu_B:{\\mathcal B}(\\mbox{Boosts})\\rightarrow[0,\\infty]$ relative to the identification $g:(0,\\infty)\\times\\mbox{Boosts}\\rightarrow X$. We have\n\\begin{equation}\n\\nu(\\Gamma\\times F)=\\mu(\\Gamma)\\mu_{SO(3)}(F), \\forall\\Gamma\\in{\\mathcal B}(X),F\\in{\\mathcal B}(SO(3)).\n\\end{equation}\nTherefore\n\\begin{equation}\n\\mu(\\Gamma)=\\mu_{SO(3)}(F)^{-1}\\nu(\\Gamma\\times F),\\forall\\Gamma\\in{\\mathcal B}(X),F\\in{\\mathcal B}(SO(3)),\\mbox{ such that }\\mu_{SO(3)}(F)>0.\n\\end{equation}\nChoose $F\\in{\\mathcal B}(SO(3)),\\mbox{ such that }\\mu_{SO(3)}(F)>0$. Then for all $\\Gamma\\in{\\mathcal B}(X)$\n\\begin{eqnarray}\n\\mu(\\Gamma) & = & \\mu_{SO(3)}(F)^{-1}\\nu(\\Gamma\\times F) \\nonumber \\\\\n    & = & \\nu_{SO(3)}(F)^{-1}\\nu(f(f^{-1}(\\Gamma\\times F))) \\nonumber \\\\\n    & = & \\mu_{SO(3)}(F)^{-1}\\nu(f(f^{-1}(B(E\\times\\{{\\vct 0}\\})\\times F))) \\nonumber \\\\\n    & = & \\mu_{SO(3)}(F)^{-1}\\nu(f((E\\times\\{{\\vct 0}\\})\\times BF)) \\nonumber \\\\\n    & = & \\mu_{SO(3)}(F)^{-1}\\sigma(E)\\mu_{O(1,3)^{+\\uparrow}}(BF) \\nonumber \\\\\n    & = & \\sigma(E)\\mu_B(B), \\nonumber\n\\end{eqnarray} \nwhere $\\Gamma=g(E\\times B)=B(E\\times\\{{\\vct 0}\\})$ and\n\\begin{equation}\n\\mu_B(B)=\\mu_{SO(3)}(F)^{-1}\\mu_{O(1,3)^{+\\uparrow}}(BF).\n\\end{equation}\nIt is straightforward to show that $\\mu_B$ is a well defined Borel measure.\n\nTherefore for any measurable function $\\psi:X\\rightarrow[0,\\infty]$\n\\begin{eqnarray}\n<\\mu,\\psi> & = & \\int\\psi(p)\\,\\mu(dp)\\nonumber \\\\\n    & = & \\int\\psi(g(m,B))\\,\\mu_B(dB)\\,\\sigma(dm) \\nonumber \\\\\n    & = & \\int<M_m,\\psi>\\,\\sigma(dm), \\label{eq:spectral_rep}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n<M_m,\\psi>=\\int\\psi(g(m,B))\\mu_B(dB).\n\\end{equation}\nIt is straightforward to show that for all $m\\in(0,\\infty)$ $M_m$ defines a Borel measure on $X$ with supp$(M_m)=H_m$. Therefore by the above argument, there exists $c=c(m)\\in(0,\\infty)$ such that $M_m=c_m\\Omega_m$. This fact, together with the spectral representation Eq.~\\ref{eq:spectral_rep} establishes (rescaling $\\sigma$) that there exists a Borel measure $\\sigma:{\\mathcal B}((0,\\infty))\\rightarrow[0,\\infty]$ such that\n\\begin{equation} \\label{eq:spectral_decomposition1}\n\\mu(\\Gamma)=\\int_{m=0}^{\\infty}\\Omega_m(\\Gamma)\\,\\sigma(dm),\n\\end{equation}\nas desired. \n\nNow suppose that $\\mu:{\\mathcal B}(X)\\rightarrow{\\bf R}$ is a Borel signed measure which is Lorentz invariant. Then by the Jordan decomposition theorem $\\mu$ has a decomposition $\\mu=\\mu^{+}-\\mu^{-}$ where $\\mu^{+},\\mu^{-}:{\\mathcal B}({\\bf R}^4)\\rightarrow[0,\\infty]$ are measures. $\\mu^{+}$ and $\\mu^{-}$ must be Borel (finite on compact sets). In fact if $P,N\\in{\\mathcal B}(X)$ is a Hahn decomposition of $X$ with respect to $\\mu$ then\n\\begin{equation}\n\\mu^{+}(\\Gamma)=\\mu(\\Gamma\\cap P), \\mu^{-}(\\Gamma)=\\mu(\\Gamma\\cap N), \\forall\\Gamma\\in{\\mathcal B}(X).\n\\end{equation}\nNow let $\\Lambda\\in O(1,3)^{+\\uparrow}$ Then since $(\\Lambda P)\\cup(\\Lambda N)=\\Lambda(P\\cup N)=X$, $(\\Lambda P)\\cap (\\Lambda N)=\\Lambda(P\\cap N)=\\emptyset$, $\\mu((\\Lambda P)\\cap\\Gamma)=\\mu((\\Lambda P)\\cap(\\Lambda\\Lambda^{-1}\\Gamma))=\\mu(P\\cap(\\Lambda^{-1}\\Gamma))\\ge0$ and, similarly, $\\mu((\\Lambda N)\\cap\\Gamma)\\le0$ $\\Lambda P$ and $\\Lambda N$ form a Hahn decomposition of $\\mu$. Therefore \n\\begin{equation}\n\\mu^{+}(\\Lambda\\Gamma)=\\mu(\\Lambda P\\cap(\\Lambda\\Gamma))=\\mu(\\Lambda(P\\cap\\Gamma))=\\mu(P\\cap\\Gamma)=\\mu^{+}(\\Gamma).\n\\end{equation}\nHence $\\mu^{+}$ is a Lorentz invariant Borel measure. Therefore it has a spectral decomposition of the form of Eq.~\\ref{eq:spectral_decomposition1}. Sinilarly $\\mu^{-}$ is a Lorentz invariant Borel measure and so it has a spectral decomposition of the form of Eq.~\\ref{eq:spectral_decomposition1}. Thus $\\mu$ has a spectral decomposition of the form of Eq.~\\ref{eq:spectral_decomposition1} where $\\sigma:{\\mathcal B}((0,\\infty)\\rightarrow{\\bf R}$ is a Borel signed measure. \n\nFinally suppose that $\\mu:{\\mathcal B}(X)\\rightarrow{\\bf C}$ is a Lorentz invariant Borel complex measure. Define Re$(\\mu):{\\mathcal B}(X)\\rightarrow{\\bf R}$ and Im$(\\mu):{\\mathcal B}(X)\\rightarrow{\\bf R}$ by\n\\begin{equation}\n(\\mbox{Re}(\\mu))(\\Gamma)=\\mbox{Re}(\\mu(\\Gamma)), (\\mbox{Im}(\\mu))(\\Gamma)=\\mbox{Im}(\\mu(\\Gamma)), \\forall\\Gamma\\in{\\mathcal B}(X).\n\\end{equation}\nThen for all $\\Lambda\\in O(1,3)^{+\\uparrow}$\n\\begin{equation}\n(\\mbox{Re}(\\mu))(\\Lambda\\Gamma)=\\mbox{Re}(\\mu(\\Lambda\\Gamma))=\\mbox{Re}(\\mu(\\Gamma))=(\\mbox{Re}(\\mu))(\\Gamma).\n\\end{equation}\nThus Re$(\\mu)$ is a Lorentz invariant Borel signed measure and so has a representation of the form of Eq.~\\ref{eq:spectral_decomposition1} for some Borel signed measure $\\sigma$. Similarly Im$(\\mu)$ has such a representation. Therefore $\\mu$ has a representation of this form for some Borel complex spectral measure $\\sigma:{\\mathcal B}((0,\\infty))\\rightarrow{\\bf C}$. This completes the proof of the spectral theorem.\n\n\\section*{Appendix 4: Dirac spinors}\n\n\\subsection*{Construction of the Dirac spinors}\n\nDirac spinors are usually obtained by seeking solutions to the Dirac equation of the form\n\\begin{eqnarray}\n\\psi^{+}(x) & = & e^{-ik.x}u(k), \\mbox{ positive energy} \\nonumber \\\\\n\\psi^{-}(x) & = & e^{ik.x}v(k), \\,\\,\\,\\mbox{ negative energy},\n\\end{eqnarray}\n(Itzikson and Zuber, 1980, p. 55). \n\nThus, in general, we are seeking solutions to the Dirac equation of the form\n\\begin{equation} \\label{eq:Appendix4_1}\n\\psi(x)=e^{-ip.x}u,\n\\end{equation}\nfor some $p\\in{\\bf R}^4,u\\in{\\bf C}^4$. If $u=0$ the the Dirac equation is trivially satisfied, so assume that $u\\neq0$. Now if $\\psi$ is of this form then\n\\begin{equation}\n(i{\\slas\\partial}-m)\\psi=0\\Leftrightarrow({\\slas p}-m)u=0.\n\\end{equation}\nIf this is the case then\n\\begin{equation}\n0=({\\slas p}+m)({\\slas p}-m)u=(p^2-m^2)u.\n\\end{equation}\nTherefore we must have that $p^2=m^2$, i.e. that $p\\in H_{\\pm m}$.\nThus we are seeking $p\\in H_{\\pm m}, u\\in{\\bf C}^4\\backslash\\{0\\}$ such that $({\\slas p}-m)u=0$, i.e. $u\\in\\mbox{Ker}({\\slas p}-m)$. \n\nLet $p\\in H_{\\pm m}$. Choose $\\Lambda\\in O(1,3)^{+\\uparrow},\\kappa\\in K$ such that \n\\begin{equation}\n\\Lambda p=(\\pm m,{\\vct 0})^{T}, \\Lambda=\\Lambda(\\kappa),\n\\end{equation}\n(see (Mashford, 2017a)). Then\n\\begin{eqnarray}\n\\mbox{Ker}({\\slas p}-m) & = & \\kappa^{-1}\\mbox{Ker}(\\kappa({\\slas p}-m)\\kappa^{-1}) \\nonumber \\\\\n    & = & \\kappa^{-1}\\mbox{Ker}(\\Sigma(\\Lambda p)-m) \\nonumber \\\\\n    & = & \\kappa^{-1}\\mbox{Ker}(\\Sigma((\\pm m,{\\vct 0})^{T})-m), \\nonumber\n\\end{eqnarray}\nwhere $\\Sigma$ denotes the map $p\\mapsto{\\slas p}$. \n\nWe will use the Dirac representation for the gamma matrices in which \n\\begin{equation}\n\\gamma^0=\\left(\\begin{array}{cc}\n1 & 0 \\\\\n0 & -1\n\\end{array}\\right).\n\\end{equation}\nWith respect to the metric $g=\\gamma^0$ the vectors $\\{e_{\\alpha}\\}_{\\alpha=0}^3$ form an orthonormal basis where\n\\begin{equation}\n(e_{\\alpha})_{\\beta}=\\delta_{\\alpha\\beta},\n\\end{equation}\ni.e.\n\\begin{equation}\n\\overline{e}_{\\alpha}e_{\\beta}=e_{\\alpha}^{\\dagger}\\gamma^0e_{\\beta}=\\gamma^0_{\\alpha\\beta},\\forall\\alpha,\\beta\\in\\{0,1,2,3\\}.\n\\end{equation}\nNow\n\\begin{equation}\n\\Sigma((\\pm m,{\\vct 0})^{T})-m=\\left(\\begin{array}{cc}\n\\pm m-m & 0 \\\\\n0 & \\mp m-m\n\\end{array}\\right),\n\\end{equation}\nTherefore, if $u=(u_1,u_2)^{T}$ then\n\\begin{equation}\nu\\in\\mbox{Ker}(\\Sigma((\\pm m,{\\vct 0})^{T})-m) \\Leftrightarrow \\left(\\begin{array}{cc}\n\\pm m-m & 0 \\\\\n0 & \\mp m-m\n\\end{array}\\right)\\left(\\begin{array}{l}\nu_1 \\\\\nu_2\n\\end{array}\\right)=0 \\nonumber\n\\end{equation}\nIn the positive energy case, i.e. when $p\\in H_m$ this is equivalent to \n\\begin{equation}\nu_1=\\mbox{ arbitrary}, u_2=0.\n\\end{equation}\nHence Dim$(\\mbox{Ker}({\\slas p}-m))=2$. In other words fermions have 2 polarization states. A basis for $\\mbox{Ker}({\\slas p}-m)$ is \n\\begin{equation}\nu_0=\\kappa^{-1}e_0, u_1=\\kappa^{-1}e_1,\n\\end{equation}\nand we may describe $u_0,u_1$ as being Dirac spinors associated with $p\\in H_m$ ($u_0,u_1$ are not unique because the choice of $\\kappa$ is not unique).\n\nSimilarly, in the negative energy case, i.e. when $p\\in H_{-m}$ a basis for  $\\mbox{Ker}({\\slas p}-m)$ is \n\\begin{equation}\nv_0=\\kappa^{-1}e_2, v_1=\\kappa^{-1}e_3.\n\\end{equation}\n\nNow let $v\\in{\\bf C}^4$. Then clearly $({\\slas p}+m)v\\in\\mbox{Ker}({\\slas p}-m)$. Therefore the space $<({\\slas p}+m)e_{\\alpha},\\alpha=0,1,2,3>$ is a subspace of Ker$({\\slas p}-m)$. We will show that in fact it is equal to Ker$({\\slas p}-m)$. We have\n\\begin{eqnarray}\n({\\slas p}+m) & = & \\kappa^{-1}\\kappa({\\slas p}+m)\\kappa^{-1}\\kappa \\nonumber \\\\\n    & = & \\kappa^{-1}(\\Sigma((\\pm m,{\\vct 0})^{T}+m)\\kappa  \\nonumber \\\\\n    & = & \\kappa^{-1}\\left(\\begin{array}{ll}\n\\pm m+m & 0 \\\\\n0 & \\mp m+m\n\\end{array}\\right)\\kappa \\nonumber \\\\\n & = & \\kappa^{-1}\\left(\\begin{array}{cccc}\n\\pm m+m & 0 & 0 & 0 \\\\\n0 & \\pm m+m & 0 & 0 \\\\\n0 & 0 & \\mp m+m & 0 \\\\\n0 & 0 & 0 & \\mp m+m\n\\end{array}\\right)\\kappa. \\nonumber\n\\end{eqnarray}\nThus, in the positive energy case,\n\\begin{equation}\n({\\slas p}+m)=2m\\kappa^{-1}(e_0,e_1,0,0)\\kappa,\n\\end{equation}\nand in the negative energy case\n\\begin{equation}\n({\\slas p}+m)=2m\\kappa^{-1}(0,0,e_2,e_3)\\kappa.\n\\end{equation}\nTherefore\n\\begin{equation}\n\\frac{{\\slas p}+m}{2m}=(u_0,u_1,0,0)\\kappa,\n\\end{equation}\n(positive energy) and\n\\begin{equation}\n\\frac{{\\slas p}+m}{2m}=(0,0,v_0,v_1)\\kappa,\n\\end{equation}\n(negative energy).\n\nLet $w_{\\alpha}=\\kappa^{-1}e_{\\alpha}, \\alpha=0,1,2,3$. $\\{w_{\\alpha}\\}_{\\alpha=0}^3$ forms an orthonormal basis for ${\\bf C}^4$ with respect to the metric $g=\\gamma^0$. Then\n\\begin{equation}\n\\frac{{\\slas p}+m}{2m}w_{\\alpha}=u_{\\alpha},\\mbox{ for }\\alpha=0,1,\n\\end{equation}\n(positive energy) and\n\\begin{equation}\n\\frac{{\\slas p}+m}{2m}w_{\\alpha+2}=v_{\\alpha},\\mbox{ for }\\alpha=0,1,\n\\end{equation}\n(negative energy).\n\nSince $\\{(2m)^{-1}({\\slas p}+m)w_{\\alpha},\\alpha=0,1\\}$ is a basis for Ker$({\\slas p}-m)$ it follows that $\\{(2m)^{-1}({\\slas p}+m)e_{\\alpha},\\alpha=0,1,2,3\\}$ spans Ker$({\\slas p}-m)$.\n\nIt is straightforward to show that the Dirac spinors that we have constructed satisfy the usual normalization properties (Itzikson and Zuber, 1980, p. 696).\n\n\\subsection*{Dirac bilinears in the non-relativistic approximation}\n\nIn the non-relativistic approximation we have\n\\begin{equation}\n\\frac{p^0}{m}\\approx1,\\frac{p^j}{m}\\approx0,\\mbox{ for } j=1,2,3.\n\\end{equation}\nTherefore\n\\begin{equation}\n\\kappa\\approx I.\n\\end{equation}\nTherefore we can take, in the positive energy case,\n\\begin{equation}\n(u_0,u_1,0,0)=\\frac{{\\slas p}+m}{m}\\kappa^{-1}=\\Sigma((1,{\\vct 0})^{T})+1=\\left(\\begin{array}{cccc}\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \n\\end{array}\\right).\n\\end{equation}\nThus \n\\begin{equation}\nu_0(p)=e_0,u_1(p)=e_1,\\forall p\\in H_m.\n\\end{equation}\nTherefore\n\\begin{equation}\n\\overline{u}_{\\alpha}(p^{\\prime})\\gamma^0u_{\\beta}(p)=u^{\\dagger}_{\\alpha}(p^{\\prime})\\gamma^0\\gamma^0u_{\\beta}(p)=e_{\\alpha}^{\\dagger}e_{\\beta}=\\delta_{\\alpha\\beta},\\forall\\alpha,\\beta\\in\\{0,1\\},p,p^{\\prime}\\in H_m.\n\\end{equation}\nAlso\n\\begin{eqnarray}\n\\overline{u}_{\\alpha}(p^{\\prime})a_{j}\\gamma^{j}u_{\\beta}(p) & = & u^{\\dagger}_{\\alpha}(p^{\\prime})\\gamma^{0}a_{j}\\gamma^{j}u_{\\beta}(p) \\nonumber \\\\\n    & = & e_{\\alpha}^{\\dagger}\\left(\\begin{array}{cccc}\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & -1 & 0 \\\\\n0 & 0 & 0 & -1\n\\end{array}\\right)\\left(\\begin{array}{cccc}\n0 & 0 & a_3 & a_1-ia_2 \\\\\n0 & 0 & a_1+ia_2 & -a_3 \\\\\n-a_3 & -a_1+ia_2 & 0 & 0 \\\\\n-a_1-ia_2 & a_3 & 0 & 0\n\\end{array}\\right)e_{\\beta} \\nonumber \\\\\n    & = 0, \\nonumber\n\\end{eqnarray}\nfor all $\\alpha,\\beta\\in\\{0,1\\},a\\in{\\bf R}^4,p,p^{\\prime}\\in H_m$. Therefore\n\\begin{equation}\n\\overline{u}_{\\alpha}(p^{\\prime})\\gamma^{j}u_{\\beta}(p)=0,\\forall\\alpha,\\beta\\in\\{0,1\\},j\\in\\{1,2,3\\},p,p^{\\prime}\\in H_m.\n\\end{equation}\n\n\\section*{Appendix 6: Rigorous justification of Argument 1}\n\nWe want to show that if $g(a,b,\\epsilon)$ is defined by $g(a,b,\\epsilon)=\\mu(\\Gamma(a,b,\\epsilon))$ then the following formal argument\n\\begin{eqnarray}\ng(a,b,\\epsilon) & = & \\mu(\\Gamma(a,b,\\epsilon)) \\label{eq:Argument1} \\\\\n    & = & \\int\\chi_{\\Gamma(a,b,\\epsilon)}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq) \\nonumber \\\\\n    & \\approx & \\int\\chi_{(a,b)\\times B_{\\epsilon}(0)}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq) \\nonumber \\\\\n    & = & \\int\\chi_{(a,b)}(\\omega_m({\\vct p})+\\omega_m({\\vct q}))\\chi_{B_{\\epsilon}(0)}({\\vct p}+{\\vct q})\\,\\frac{d{\\vct p}}{\\omega_m({\\vct p})}\\frac{d{\\vct q}}{\\omega_m({\\vct q})} \\nonumber \\\\\n    & = & \\int\\chi_{(a,b)}(\\omega_m({\\vct p})+\\omega_m({\\vct q}))\\chi_{B_{\\epsilon}(0)-{\\vct q}}({\\vct p})\\,\\frac{d{\\vct p}}{\\omega_m({\\vct p})}\\frac{d{\\vct q}}{\\omega_m({\\vct q})} \\nonumber \\\\\n    & \\approx & \\int\\chi_{(a,b)}(2\\omega_m({\\vct q}))\\frac{\\frac{4}{3}\\pi\\epsilon^3}{\\omega_m({\\vct q})^2}\\,d{\\vct q}, \\nonumber\n\\end{eqnarray}\nis justified in the sense that\n\\begin{equation}\n\\lim_{\\epsilon\\rightarrow0}\\epsilon^{-3}g(a,b,\\epsilon)=\\frac{4}{3}\\pi\\int\\chi_{(a,b)}(2\\omega_m({\\vct q}))\\frac{1}{\\omega_m({\\vct q})^2}\\,d{\\vct q}.\n\\end{equation}\nThere are 2 $\\approx$ signs that we have to consider. The first is between lines 2 and 3 and arises because we are approximating the hyperbolic cylinder between $a$ and $b$ with an ordinary cylinder of radius $\\epsilon$. We will show that the error is of order greater than $\\epsilon^3$. Let $\\Gamma$ be the aforementioned hyperbolic cylinder. Then\n\\begin{equation}\n\\Gamma=\\bigcup_{m\\in(a,b)}S(m,\\epsilon).\n\\end{equation}\nNow \n\\begin{equation}\n\\Gamma=\\Gamma^{\\prime}\\sim\\Gamma^{\\prime-}\\cup\\Gamma^{\\prime+},\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\Gamma^{\\prime} & = & \\bigcup_{m\\in(a,b)}\\{m\\}\\times B_{\\epsilon}({\\vct 0}) \\nonumber \\\\\n\\Gamma^{\\prime-} & = & \\bigcup_{m\\in(a,a^{+})}(\\{m\\}\\times B_{\\epsilon}({\\vct 0})\\sim S(m,\\epsilon))\\subset\\bigcup_{m\\in(a,a^{+})}(\\{m\\}\\times B_{\\epsilon}({\\vct 0})) \\nonumber \\\\\n\\Gamma^{\\prime+} & = & \\bigcup_{m\\in(b^{-},b)}(\\{m\\}\\times B_{\\epsilon}({\\vct 0})\\sim S(m,\\epsilon))\\subset\\bigcup_{m\\in(b^{-},b)}(\\{m\\}\\times B_{\\epsilon}({\\vct 0})), \\nonumber\n\\end{eqnarray}\nin which\n\\begin{equation}\na^{+}=(a^2+\\epsilon^2)^{\\frac{1}{2}}, b^{-}=(b^2-\\epsilon^2)^{\\frac{1}{2}}, \\epsilon<b.\n\\end{equation}\nIt is straightforward to show that \n\\begin{eqnarray}\n& & |\\int\\chi_{\\Gamma_1\\cup\\Gamma_2}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq)-\\int\\chi_{\\Gamma_1}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq)| \\leq \\nonumber \\\\\n& & \\int\\chi_{\\Gamma_2}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq) \\nonumber \\\\\n& & \\mbox{and} \\nonumber \\\\\n& & |\\int\\chi_{\\Gamma_1\\sim\\Gamma_2}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq)-\\int\\chi_{\\Gamma_1}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq)| \\leq \\nonumber \\\\\n& & \\int\\chi_{\\Gamma_2}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq), \\nonumber\n\\end{eqnarray}\nfor all $\\Gamma_1,\\Gamma_2\\in{\\mathcal B}({\\bf R}^4)$. Therefore\n\\begin{eqnarray}\n& & |\\int\\chi_{\\Gamma}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq)-\\int\\chi_{\\Gamma^{\\prime}}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq)| \\leq \\nonumber \\\\\n& & \\int\\chi_{\\Gamma^{\\prime-}}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq)+\\int\\chi_{\\Gamma^{\\prime+}}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq) \\nonumber\n\\end{eqnarray} \nWe will show that\n\\begin{equation}\n\\lim_{\\epsilon\\rightarrow0}(\\epsilon^{-3}\\int\\chi_{\\Gamma^{+-}}(p+q)\\,\\Omega_m(d{\\vct p})\\Omega_m(d{\\vct q}))=0,\n\\end{equation}\nIt suffices to consider the $+$ case.\nWe have\n\\begin{eqnarray} \\label{eq:first_approx1} \n\\int\\chi_{\\Gamma^{+}}(p+q)\\,\\Omega_m(d{\\vct p})\\Omega_m(d{\\vct q}) & = & \\int\\chi_{(a,a^{+})\\times B_{\\epsilon}(0)}(p+q)\\,\\Omega_m(dp)\\,\\Omega_m(dq) \\nonumber \\\\\n    & = & \\int\\chi_{(a,a^{+})}(\\omega_m({\\vct p})+\\omega_m({\\vct q}))\\chi_{B_{\\epsilon}(0)-{\\vct q}}({\\vct p})\\,\\frac{d{\\vct p}}{\\omega_m({\\vct p})} \\nonumber \\\\\n    & & \\frac{d{\\vct q}}{\\omega_m({\\vct q})}. \n\\end{eqnarray}\nWe will come back to this equation later but will now return to the general argument~\\ref{eq:Argument1} and consider the second and final $\\approx$. This $\\approx$ arises because we are approximating ${\\vct p}$ by $-{\\vct q}$ since ${\\vct p}$ ranges over a ball of radius $\\epsilon$ centred on $-{\\vct q}$.\n\nSuppose that ${\\vct p}$ and ${\\vct q}$ are such that $\\chi_{B_{\\epsilon}(0)-{\\vct q}}({\\vct p})=1$. Then $|{\\vct p}+{\\vct q}|<\\epsilon$. Thus $||{\\vct p}|-|{\\vct q}||<\\epsilon$. Hence\n\\begin{eqnarray}\n|\\omega_m({\\vct p})-\\omega_m({\\vct q})| & = & |({\\vct p}^2+m^2)^{\\frac{1}{2}}-({\\vct q}^2+m^2)^{\\frac{1}{2}})| \\nonumber \\\\\n    & =  & \\left|\\frac{{\\vct p}^2-{\\vct q}^2}{({\\vct p}^2+m^2)^{\\frac{1}{2}}+({\\vct q}^2+m^2)^{\\frac{1}{2}})}\\right| \\nonumber \\\\\n    & \\leq & \\frac{|{\\vct p}^2-{\\vct q}^2|}{2m} \\nonumber \\\\\n    & = & \\frac{||{\\vct p}|-|{\\vct q}||\\times||{\\vct p}|+|{\\vct q}||}{2m} \\nonumber \\\\\n    & < & \\frac{\\epsilon}{2m}||{\\vct p}|+|{\\vct q}||. \\nonumber\n\\end{eqnarray}\nWe have $|{\\vct p}|\\in(|{\\vct q}|-\\epsilon,|{\\vct q}|+\\epsilon)$. Therefore $|{\\vct p}|+|{\\vct q}|<2|{\\vct q}|+\\epsilon)$. Thus \n\\[ |\\omega_m({\\vct p})-\\omega_m({\\vct q})| < \\frac{\\epsilon}{2m}(2|{\\vct q}|+\\epsilon). \\]\nTherefore\n\\begin{eqnarray}\n|\\omega_m({\\vct p})+\\omega_m({\\vct q})| & = & |\\omega_m({\\vct p})-\\omega_m({\\vct q})+\\omega_m({\\vct q})+\\omega_m({\\vct q})| \\nonumber \\\\\n    & \\leq & |\\omega_m({\\vct p})-\\omega_m({\\vct q})|+2\\omega_m({\\vct q}) \\nonumber \\\\\n    & < & 2\\omega_m({\\vct q})+ \\frac{\\epsilon}{2m}(2|{\\vct q}|+\\epsilon). \\nonumber\n\\end{eqnarray}\nNow let\n\\begin{eqnarray}\nI(\\epsilon) & = & \\int\\chi_{(a,b)}(\\omega_m({\\vct p})+\\omega_m({\\vct q}))\\chi_{B_{\\epsilon}(0)-{\\vct q}}({\\vct p})\\,\\frac{d{\\vct p}}{\\omega_m({\\vct p})}\\frac{d{\\vct q}}{\\omega_m({\\vct q})} \\nonumber \\\\\nJ(\\epsilon) & = & \\int\\chi_{(a,b)}(2\\omega_m({\\vct q}))\\chi_{B_{\\epsilon}(0)-{\\vct q}}({\\vct p})\\,\\frac{d{\\vct p}}{\\omega_m({\\vct p})}\\frac{d{\\vct q}}{\\omega_m({\\vct q})} \\nonumber \\\\  \nK(\\epsilon) & = &  \\int\\chi_{(a,b)}(2\\omega_m({\\vct q}))\\frac{\\frac{4}{3}\\pi\\epsilon^3}{\\omega_m({\\vct q})^2}\\,d{\\vct q} \\nonumber\n\\end{eqnarray}\nWe will show that \n\\begin{equation} \\label{eq:Appendix6_limits1}\n\\lim_{\\epsilon\\rightarrow0}\\epsilon^{-3}(I(\\epsilon)-J(\\epsilon))=0,\n\\mbox{ and }\n\\lim_{\\epsilon\\rightarrow0}\\epsilon^{-3}(J(\\epsilon)-K(\\epsilon))=0.\n\\end{equation}\nConcerning the first limit we note that $\\chi_{(a,b)}(\\omega_m({\\vct p})+\\omega_m({\\vct q}))$ differs from $\\chi(2\\omega_m({\\vct q}))$ if and only if $\\omega_m({\\vct p})+\\omega_m({\\vct q})\\in(a,b)$ but $2\\omega_m({\\vct q})\\leq a$ or else $2\\omega_m({\\vct q})\\in(a,b)$ but $\\omega_m({\\vct p})+\\omega_m({\\vct q})\\geq b$. Thus\n\\begin{equation}\n|I(\\epsilon)-J(\\epsilon)|=I_1(\\epsilon)+I_2(\\epsilon),\n\\end{equation}\nwhere\n\\begin{eqnarray}\nI_1(\\epsilon) & = & \\int\\chi_{(a,b)}(\\omega_m({\\vct p})+\\omega_m({\\vct q}))\\chi_{(-\\infty,a]}(2\\omega_m({\\vct q}))\\chi_{B_{\\epsilon}(0)-{\\vct q}}({\\vct p})\\, \\nonumber \\\\\n    & & \\frac{d{\\vct p}}{\\omega_m({\\vct p})}\\frac{d{\\vct q}}{\\omega_m({\\vct q})} \\nonumber \\\\\nI_2(\\epsilon) & = & \\int\\chi_{[b,\\infty)}(\\omega_m({\\vct p})+\\omega_m({\\vct q}))\\chi_{(a,b)}(2\\omega_m({\\vct q}))\\chi_{B_{\\epsilon}(0)-{\\vct q}}({\\vct p})\\, \\nonumber \\\\\n    & & \\frac{d{\\vct p}}{\\omega_m({\\vct p})}\\frac{d{\\vct q}}{\\omega_m({\\vct q})}\n\\end{eqnarray}\nWe will show that\n\\begin{equation}\n\\lim_{\\epsilon\\rightarrow\\infty}\\epsilon^{-3}I_1(\\epsilon)=0.\n\\end{equation}\n$I_2$ can be dealt with similarly. \n\nConsider $f:[0,\\infty)\\rightarrow(0,\\infty)$ defined by\n\\[ f(p)=\\omega_m(p)^{-1}=(p^2+m^2)^{-\\frac{1}{2}}. \\]\nThen\n\\[ f^{\\prime}(p)=-p(p^2+m^2)^{-\\frac{3}{2}}. \\]\nTherefore, by Taylor's theorem\n\\[ f(p)-f(q)=-r(r^2+m^2)^{-\\frac{3}{2}}(p-q), \\]\nfor some $r$ between $q$ and $p$. Thus\n\\[ |f(p)-f(q)|\\leq (q+\\epsilon)((q+\\epsilon)^2+m^2)^{-\\frac{3}{2}}\\epsilon, \\]\nif $|p-q|<\\epsilon$ and so\n\\[ f(p)\\leq f(q)+(q+\\epsilon)((q+\\epsilon)^2+m^2)^{-\\frac{3}{2}}\\epsilon=(q^2+m^2)^{-\\frac{1}{2}}+(q+\\epsilon)((q+\\epsilon)^2+m^2)^{-\\frac{3}{2}}\\epsilon. \\]\nTherefore\n\\[ \\frac{1}{\\omega_m({\\vct p})}\\leq ({\\vct q}^2+m^2)^{-\\frac{1}{2}}+(|{\\vct q}|+\\epsilon)((|{\\vct q}|+\\epsilon)^2+m^2)^{-\\frac{3}{2}}\\epsilon. \\]\nAlso\n\\[ \\omega_m({\\vct p})+\\omega_m({\\vct q})\\in(a,b)\\mbox{ and }2\\omega_m({\\vct q})\\leq a\\Rightarrow a-\\frac{\\epsilon}{2m}(2|{\\vct q}|+\\epsilon)<2\\omega_m({\\vct q})\\leq a. \\]\nTherefore\n\\begin{eqnarray}\nI_1(\\epsilon) & \\leq & \\int\\chi_{(a-(2|{\\vct q}|+\\epsilon)\\epsilon\/(2m),a)}(2\\omega_m({\\vct q}))\\chi_{B_{\\epsilon}(0)-{\\vct q}}({\\vct p})\\, \\nonumber \\\\\n    & & (({\\vct q}^2+m^2)^{-\\frac{1}{2}}+(|{\\vct q}|+\\epsilon)((|{\\vct q}|+\\epsilon)^2+m^2)^{-\\frac{3}{2}}\\epsilon)d{\\vct p}\\frac{d{\\vct q}}{\\omega_m({\\vct q})} \\nonumber \\\\\n& = & \\frac{4}{3}\\pi\\epsilon^3\\int\\chi_{(a-(2|{\\vct q}|+\\epsilon)\\epsilon\/(2m),a)}(2\\omega_m({\\vct q}))\\, \\nonumber \\\\\n    & & (({\\vct q}^2+m^2)^{-\\frac{1}{2}}+(|{\\vct q}|+\\epsilon)((|{\\vct q}|+\\epsilon)^2+m^2)^{-\\frac{3}{2}}\\epsilon)\\frac{d{\\vct q}}{\\omega_m({\\vct q})} \\nonumber \\\\\n\\end{eqnarray}\nHence\n\\begin{eqnarray}\n\\epsilon^{-3}I_1(\\epsilon) & \\leq & \\frac{4}{3}\\pi\\int\\chi_{(a-(2|{\\vct q}|+\\epsilon)\\epsilon\/(2m),a)}(2\\omega_m({\\vct q}))\\, \\nonumber \\\\\n    & & (({\\vct q}^2+m^2)^{-\\frac{1}{2}}+(|{\\vct q}|+1)((|{\\vct q}|+1)^2+m^2)^{-\\frac{3}{2}})\\frac{d{\\vct q}}{\\omega_m({\\vct q})},\n\\end{eqnarray}\nfor all $\\epsilon<1$. The integrand vanishes outside the compct set\n\\[ C = \\{{\\vct q}\\in{\\bf R}^3:2\\omega_m({\\vct q})\\leq a\\}, \\]\nis dominated by the integrable function\n\\[ g({\\vct q})=(({\\vct q}^2+m^2)^{-\\frac{1}{2}}+(|{\\vct q}|+1)((|{\\vct q}|+1)^2+m^2)^{-\\frac{3}{2}})\\frac{1}{\\omega_m({\\vct q})}, \\]\nand converges pointwise to $0$ everywhere on $C$ as $\\epsilon\\rightarrow0$ except on the set $\\{{\\vct q}\\in{\\bf R}^3:2\\omega_m({\\vct q})=a\\}$ which is a set of measure $0$. Therefore by the dominated convergence theeorem\n\\begin{equation}\n\\lim_{\\epsilon\\rightarrow\\infty}\\epsilon^{-3}I_1(\\epsilon)=0,\n\\end{equation}\nas required.\n\nPart of the argument that we have given above to establish the correctness of the first limit in Eq.~\\ref{eq:Appendix6_limits1} can be used to establish the second limit in that equation.\n\nWe have therefore dealt with the second $\\approx$ in Eq.~\\ref{eq:Argument1}. To finish dealing with the first $\\approx$ we have, from Eq.~\\ref{eq:first_approx1} and subsequent calculations that\n\\begin{eqnarray}\n\\lim_{\\epsilon\\rightarrow0}\\epsilon^{-3}\\int\\chi_{\\Gamma^{+}}(p+q)\\,\\Omega_m(d{\\vct p})\\Omega_m(d{\\vct q}) & = & \\lim_{\\epsilon\\rightarrow0}\\epsilon^{-3}\\int\\chi_{(a,a^{+})}(\\omega_m({\\vct p})+\\omega_m({\\vct q}))\\chi_{B_{\\epsilon}(0)-{\\vct q}}({\\vct p})\\, \\nonumber \\\\\n    &  & \\frac{d{\\vct p}}{\\omega_m({\\vct p})}\\frac{d{\\vct q}}{\\omega_m({\\vct q})} \\nonumber \\\\\n    & = & \\frac{4}{3}\\pi\\int\\chi_{(a,a^{+})}(2\\omega_m({\\vct q}))\\frac{d{\\vct q}}{\\omega_m({\\vct q})^2} \\nonumber \\\\\n    & = & \\frac{4}{3}\\pi(4\\pi)\\int_{r^2\\in(a^2,a^{+2})}r^2(2(r^2+m^2)^{\\frac{1}{2}})\\frac{dr}{r^2+m^2} \\nonumber \\\\\n    &\\leq & \\frac{4}{3}\\pi(4\\pi)\\int_{r^2\\in(a^2,a^{+2})}a^{+2}(2(a^{+2}+m^2)^{\\frac{1}{2}})\\frac{dr}{m^2}, \\nonumber    \n\\end{eqnarray}\n$\\rightarrow0$ as $\\epsilon\\rightarrow0$ since $a^{+}\\rightarrow a$ as $\\epsilon\\rightarrow0$. This completes the proof of the validity of Argument 1.\n\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{$\\text{Introduction}$}\n\n General Relativity (GR) as the standard model of gravity theory is tested well, especially in the solar system.   But it also confronts some unanswered questions, such as the problems on  the dark matter \\cite{DM1,DM2},  the late accelerating universe \\cite{SN-acc1,SN-acc2}, the inflation \\cite{inflation1,inflation2}, and the quantization, etc. Maybe, the gravity does not work in the framework of the GR theory at the large scales and the high-energy region. In order to explore the \"ultimate\" theory that adequately describe gravitational interaction, modified or extended gravity theories have been widely investigated \\cite{mg1,mg2,mg3,mg4,mg5,mg6,mg7,mg8,mg9,mg10}, such as the $f(R)$ theory \\cite{fr-review1,fr-review2,fr-3,fr-4}, the $f(T)$ theory \\cite{ft1,ft2}, the $f(G)$ theory \\cite{fg1,fg2}, the  Brans-Dicke (BD)  theory  \\cite{original-BD}, and so on.\n\n\n In the BD theory, the Newton gravity constant $G(t)=1\/\\phi(t)$ is considered as a function of time. Thus, a scalar field $\\phi$ can be introduced naturally into the action. Some observational constraints on $G(t)$  can be found in Refs. \\cite{VG-MNRAS-2004-dwarf,VG-PRD-2004-white,VG-APJ,VG-PRD-2002-SN,VG-PRL-1996-neutron,vg-prl-constraint,vg-constraint-lu-prd,vg-constraint-lu-epjc}. The action of the BD theory has a form \\cite{bd-action}\n \\begin{equation}\n S_{BD}=\\int \\sqrt{-g}d^{4}x[\\frac{1}{16\\pi}(\\phi R-\\frac{\\omega}{\\phi}\\partial_{\\mu}\\phi\\partial^{\\mu}\\phi)+L_{m}],\\label{action-BD}\n \\end{equation}\n where $\\omega$ is the coupling constant. For observational and theoretical  motivations, several extended versions of the BD theory have been developed, such as adding a potential term to the original BD theory \\cite{BD-potential},   assuming the coupling constant $\\omega$ to be variable with respect to time \\cite{BD-omegat1,BD-omegat2}, generalizing $\\phi$ to be a function $f(\\phi)$ in the coupling term $\\phi R$ \\cite{BD-ST,BD-ST1}, choosing a higher-dimension geometry in the BD theory \\cite{Ma-5DBD}, etc.  The applications of these extended BD theories have been investigated widely, such as at the aspects of cosmology \\cite{GBD-cosmic1,GBD-cosmic2,GBD-cosmic3}, weak-field approximation \\cite{GBD-weak}, observational constraints \\cite{GBD-constraint1,GBD-constraint2}, and so on   \\cite{BD-widely1,BD-widely2,BD-widely3}.\n\n\n Recently, a different strategy was proposed  to modify the BD theory  (called GBD)  in Refs. \\cite{GBD-L,GBD}  by generalizing the Ricci scalar $R$ to be an arbitrary function $f(R)$ in the BD action.  Comparing with other modified theories,  one can find  that the GBD theory in the metric formalism have some interesting properties or solve some problems existing in other theories  \\cite{GBD-L,GBD}. Several results could be exhibited  briefly as follows. (1)  The state parameter of geometrical dark energy in the GBD model can cross over the phantom boundary $w=-1$ as achieved in the quintom model,  without bearing the problems existing in the quintom model \\cite{GBD} (in the double scalar-fields quintom model, it is required to include both the canonical quintessence field and the non-canonical phantom field in order to make the state parameter to cross over $w=-1$ \\cite{quintom,quintom1}, while several fundamental problems are associated with phantom field, such as the problem of negative kinetic term and the fine-tuning problem, etc). (2) One knows that the metric-formalism $f(R)$ theories are equivalent to the BD theory with a potential (abbreviated as BDV) for taking a specific value of the coupling  parameter $\\omega=0$ \\cite{fr-review2}, where the specific choice: $\\omega=0$ is quite exceptional, and it is hard to understand the corresponding absence of the kinetic-energy term for the field in the action. However,  the BD field in the GBD theory  owns the non-disappeared  kinetic term in the action \\cite{GBD-L,GBD}. (3) Using the method of the weak-field approximation Ref. \\cite{GBD-L} showed that the GBD theory could  solve the problem of $\\gamma$ value emerging in $f(R)$ modified gravity (i.e. the inconsistent problem between the observational $\\gamma$ value and the theoretical $\\gamma$ value \\cite{fr-review2,ppn-fr1,ppn-fr2}), without introducing the so-called chameleon mechanism. Here $\\gamma$  is the parametrized post-Newtonian (PPN) parameter. Furthermore, the GBD theory tends to investigate the physics from the viewpoint of geometry, while the BDV or the two scalar-fields quintom model tends to solve physical problems from the viewpoint of matter. It is possible that several special characteristics of scalar field could be revealed through studies of the gravitational geometry in the GBD theory.\n\n\n It is well known that  some assumptions  have to be taken at prior for developing modified gravity theories of GR. For example: (1) which one or ones of the dynamical variables  (the metric, the connection,  or  the tetrad, etc.) should be chosen to describe the gravitational interaction? (2) As a covariant action theory, what is the constructed Lagrangian form for the gravitational system, and  which corresponding space-time geometry  (the Riemann, the Weitzenb$\\ddot{o}$ck, the Riemann-Cartan, or the higher-dimension geometry, etc.) should be used? Then based on the constructed Lagrangian quantity,  one can derive the field equations by using the variational principle. And the surviving theories should pass through the tests of experiments from the solar system, the astrophysics, the cosmology, etc.\n\n\n As shown in some references, the  gravity theories defining in the Riemann geometry are often depicted in two formalisms: the metric formalism \\cite{fr-review2,metric-variable} and the Palatini formalism \\cite{Pala-variable,Pala-variable1}. In the metric formalism  (or the standard formalism) the Levi-Civta connection is related to metric, while in the Palatini formalism the metric and the connection are considered as the independent dynamical variables.  For a general non-linear $f(R)$ function, the different field equations can be obtained for the metric-$f(R)$ theories and the Palatini-$f(\\tilde{R})$ theories \\cite{fr-review2}, respectively. Comparing to the metric formalism, some  advantages of the Palatini approach could be found. For example: (1) the field equations  in the metric formalism are the fourth-order PDE (partial differential equation), while the field equations  in the Palatini formalism are the second-order PDE  which is easier to solve  and interpret \\cite{Pala-merit1}; (2) Palatini-formalism theory raises the effective cutoff of the theory without introducing additional degrees of freedom below the Planck scale \\cite{Pala-merit2};  (3)  Palatini-formalism theory could lead to different  inflationary predictions, which opens a door to test the nature of gravity by using the future cosmological observations \\cite{Pala-merit3}; (4) Palatini-formalism theory could lead to different interactions among the Standard Model particles and the Higgs field  in the large field regime, with a potential impact on the entropy production process following the end of inflation \\cite{Pala-merit4}, etc. \\cite{Pala-merit5,Pala-merit6}.\n\n\n\n In this paper, we investigate the GBD theory in the framework of  Palatini formalism.  The field equations and the linearized equations are derived in the Palatini-GBD theory.   Comparing to the metric formalism of GBD theory  \\cite{GBD-L}, some new results can be found in the Palatini-formalism of GBD theory. (1) The geometrical scalar field  in the Palatini-formalism of GBD theory is massless and source-free. These properties of geometric scalar field are different from the results given in the metric-formalism of GBD theory  \\cite{GBD-L}; (2) The parameterized post Newton parameter in the Palatini-GBD theory has a form:  $\\gamma=\\frac{2\\omega-F_{0}}{2\\omega+F_{0}}$, which could pass through the experimental test: $ |\\omega| > 40000$. Comparing to the Ref.  \\cite{GBD-L}, we can see a different expression  of $\\gamma$ in the metric-GBD theory, where $\\gamma$  depends on the parameters: $\\omega$, $F_{0}$ and $m_{s}$. (3) Given that investigating the polarization of  gravitational waves (GWs) can serve to discriminate the different gravitational theories, we also study the polarization modes of GWs in the Palatini-GBD theory. The  Newman-Penrose (NP) method \\cite{gw-polar4,gw-polar5} and the geodesic deviation (GD) method  \\cite{gw-polar6} are used to explore the polarization modes of GWs in the GBD theory.  It is observed  that  the polarization modes of GWs in the Palatini-GBD theory (the three  polarization types) are different from that in the metric-GBD theory (the four polarization types). The results in the GBD theory also show that   the extra scalar field $F$  and the BD scalar field generate the new polarizations  (the scalar polarization modes) for GWs  which are not present in the standard GR or the Palatini-$f(R)$ theory.\n\n\n The structure of our paper are as follows. In section II, we derive the basic equations in the  Palatini-GBD theory. In section III,  the linearized field equations are gained by using the method of the weak-field approximation.  Section IV  investigate the polarization modes of gravitational waves  in the Palatini-GBD theory.  Section V discuss the parameterized post Newton parameter (PPN). Section VI is the conclusions.\n\n\n\n\n\\section{$\\text{Field equations  in the Palatini-formalism of GBD theory}$}\n\n\nThis section is devoted to derive the basic equations in the Palatini-GBD theory. The action of the GBD theory in the Palatini formalism read as,\n\\begin{equation}\nS=S_g(g_{\\mu \\nu },\\widetilde{\\Gamma}^{\\lambda}_{\\mu\\nu},\\phi)+S_{\\phi}(g_{\\mu \\nu },\\phi)+S_m(g_{\\mu \\nu },\\psi )=\\frac{1}{2}\\int d^4x{\\cal L}_{T},\\label{action}\n\\end{equation}\nwith the total Lagrange quantity\n\\begin{equation}\n{\\cal L}_T=\\sqrt{-g}[\\phi f(\\tilde{R})- \\frac{\\omega}{\\phi}\\partial _\\mu \\phi \\partial ^\\mu \\phi+\\frac{16\\pi }{c^4}{\\cal L}_m].\\label{lagrange}\n\\end{equation}\nHere the metric $g_{\\mu \\nu }$ and the connection $\\widetilde{\\Gamma}^{\\lambda}_{\\mu\\nu}$ are considered as the independent dynamical variables, $g$ denotes the determinant of $g_{\\mu\\nu}$, and ${\\cal L}_m$ denotes the matter Lagrangian  associated with the matter field $\\psi$ and $g_{\\mu\\nu}$. $f(\\tilde{R})$ is an arbitrary  function of Ricci scalar: $\\tilde{R}=g_{\\mu\\nu}\\tilde{R}^{\\mu\\nu}$, and the Ricci tensor $\\tilde{R}_{\\mu\\nu}$ is defined by the independent Palatini connection $\\widetilde{\\Gamma}^{\\lambda}_{\\mu\\nu}$\n\\begin{equation}\n\\tilde{R}_{\\mu\\nu}=\\tilde{R}^{\\alpha}_{\\mu\\alpha\\nu}=\\partial_{\\lambda}\\widetilde{\\Gamma}^{\\lambda}_{\\mu\\nu}-\\partial_{\\mu}\\widetilde{\\Gamma}^{\\lambda}_{\\lambda\\nu}\n+\\widetilde{\\Gamma}^{\\lambda}_{\\mu\\nu}\\widetilde{\\Gamma}^{\\rho}_{\\rho\\lambda}-\\widetilde{\\Gamma}^{\\lambda}_{\\nu\\rho}\\widetilde{\\Gamma}^{\\rho}_{\\mu\\lambda}\n\\label{Ricci-Tensor-Pala}.\n\\end{equation}\n Using the variational principle,  we can derive the evolutional equations of the dynamical fields in the Palatini-formalism of GBD theory.  Varying the action (\\ref{action}) with respect to $g_{\\mu\\nu}$ and $\\phi$, we gain two field  equations   as follows\n\\begin{eqnarray}\n\\phi \\left[ F(\\tilde{R})\\tilde{R}_{\\mu \\nu }-\\frac{1}{2}f(\\tilde{R})g_{\\mu \\nu }\\right]+ \\frac{1}{2}\\frac{\\omega}{\\phi}g_{\\mu\\nu}\\partial_\\sigma\\phi\\partial^\\sigma\\phi\n-\\frac{\\omega}{\\phi}\\partial_\\mu\\phi\\partial_\\nu\\phi = 8\\pi T_{\\mu \\nu },\\label{gravitational-eq-Pala}\n\\end{eqnarray}\n\\begin{equation}\nf(\\tilde{R})+2\\omega\\frac{\\Box \\phi}{\\phi} -\\frac{\\omega}{\\phi^{2}}\\partial _\\mu \\phi \\partial ^\\mu \\phi=0,\\label{BD-scalar-eq}\n\\end{equation}\nwhere $F(\\tilde{R})\\equiv\\partial f(\\tilde{R})\/\\partial \\tilde{R}$, $\\Box \\equiv \\nabla ^\\mu \\nabla _\\mu $ and  $T_{\\mu \\nu }=\\frac{-2}{\\sqrt{-g}}\\frac{\\delta S_m}{\\delta g^{\\mu \\nu }}$ is the energy-momentum tensor of matter. The trace of Eq.(\\ref{gravitational-eq-Pala}) is\n\\begin{eqnarray}\nF(\\tilde{R})\\tilde{R}-2f(\\tilde{R})+\\frac{\\omega}{\\phi^{2}}\\partial_\\mu\\phi\\partial^\\mu\\phi = \\frac{8\\pi T}{\\phi}.\\label{trace-grav-Pala}\n\\end{eqnarray}\nVarying the action with respect to $\\widetilde{\\Gamma}^{\\lambda}_{\\mu\\nu}$ gives\n\\begin{eqnarray}\n\\widetilde{\\nabla}_{\\lambda}(\\sqrt{-g}\\phi F(\\tilde{R})g^{\\mu\\nu})=0,\\label{connection-eq}\n\\end{eqnarray}\nwhere $\\widetilde{\\nabla}$ is the covariant derivative with respect to the Palatini connection.  Eq.(\\ref{connection-eq})  implies that the connection can be represented as the Christoffel symbol associated with the metric $h_{\\mu\\nu}$ by defining $h_{\\mu\\nu}=\\phi F(\\tilde{R})g_{\\mu\\nu}$. Then we can  have a following relation\n\\begin{equation}\n\\widetilde{\\Gamma}^{\\lambda}_{\\mu\\nu}=\\Gamma^{\\lambda}_{\\mu\\nu}+\\frac{1}{2\\phi F}[-g_{\\mu\\nu}\\partial^{\\lambda}(\\phi F)+\\delta^{\\lambda}_{\\nu}\\partial_{\\mu}(\\phi F)+\\delta^{\\lambda}_{\\mu}\\partial_{\\nu}(\\phi F)],\\label{connection-relation}\n\\end{equation}\nwhere $\\Gamma^{\\lambda}_{\\mu\\nu}$ is the Livi-Civita connection associated with the metric $g_{\\mu\\nu}$. Thus, by using Eq.(\\ref{Ricci-Tensor-Pala}) the Ricci tensor and the Ricci scalar in the Palatini formalism are rewritten as\n\\begin{equation}\n\\tilde{R}_{\\mu\\nu}=R_{(g)\\mu\\nu}+\\frac{3}{2(\\phi F)^{2}}\\nabla_{\\mu}(\\phi F)\\nabla_{\\nu}(\\phi F)-\\frac{1}{\\phi F}\\nabla_{\\mu}\\nabla_{\\nu}(\\phi F)-\\frac{1}{2\\phi F}g_{\\mu\\nu}\\Box(\\phi F),\\label{Ricci-Tensor-metric}\n\\end{equation}\n\\begin{equation}\n\\tilde{R}=R_{(g)}+\\frac{3}{2(\\phi F)^{2}}\\nabla^{\\sigma}(\\phi F)\\nabla_{\\sigma}(\\phi F)-\\frac{3}{\\phi F}\\Box(\\phi F),\\label{Ricci-scalar-metric}\n\\end{equation}\nwhere $R_{(g)\\mu\\nu}$ and $R_{(g)}$ denotes the Ricci tensor and the  Ricci  scalar defining in the metric formalism, and all covariant derivatives are taken with respect to the metric $g_{\\mu\\nu}$. Combining above equations, the modified Einstein equation is derived as\n\\begin{equation}\nG_{\\mu\\nu}=R_{(g)\\mu\\nu}-\\frac{1}{2}R_{(g)}g_{\\mu\\nu}=\\frac{8\\pi T_{\\mu\\nu}}{\\phi F}+8\\pi T_{\\mu\\nu}^{eff}\\label{Einstain-Tensor-Pala}\n\\end{equation}\nwith\n\\begin{equation}\n\\begin{split}\n8\\pi T_{\\mu\\nu}^{eff}=-\\frac{\\omega}{2\\phi^2F}g_{\\mu\\nu}\\partial_{\\sigma}\\phi\\partial^{\\sigma}\\phi+\\frac{f}{2F}g_{\\mu\\nu}+\\frac{\\omega}{\\phi^{2} F}\\partial_{\\mu}\\phi\\partial_{\\nu}\\phi-\\frac{3}{2(\\phi F)^{2}}\\nabla_{\\mu}(\\phi F)\\nabla_{\\nu}(\\phi F)   \\\\\n+\\frac{1}{\\phi F}\\nabla_{\\mu}\\nabla_{\\nu}(\\phi F)-\\frac{1}{2}g_{\\mu\\nu}\\tilde{R}+\\frac{3}{4(\\phi F)^{2}}g_{\\mu\\nu}\\nabla^{\\sigma}(\\phi F)\\nabla_{\\sigma}(\\phi F)-\\frac{1}{\\phi F}g_{\\mu\\nu}\\Box(\\phi F).\\label{re-gravitation-eq}\n\\end{split}\n\\end{equation}\nWhen $f(\\tilde{R})$ linear in $\\tilde{R}$, Eq.(\\ref{Einstain-Tensor-Pala}) is identical to the field equation in the metric formalism of the GBD theory \\cite{GBD}. The trace of the gravitational field equation and the BD field equation have the forms:\n\\begin{equation}\n\\Box (\\phi F)=\\frac{8\\pi T}{3}-\\frac{\\omega}{3\\phi}\\partial_{\\mu}\\phi\\partial^{\\mu}\\phi+\\frac{2\\phi f(\\tilde{R})}{3}-\\frac{2\\phi F}{3}\\tilde{R}+\\frac{1}{2\\phi F}\\nabla_{\\mu}(\\phi F)\\nabla^{\\mu}(\\phi F)+\\frac{\\phi F}{3}R_{(g)},\\label{trace-gravity-eq-re}\n\\end{equation}\n\\begin{equation}\n\\Box\\phi-\\frac{\\partial_{\\mu}\\phi\\partial^{\\mu}\\phi}{4\\phi}=\\frac{1}{4\\omega}[8\\pi T-\\phi F R_{(g)}-\\frac{3}{2\\phi F}\\nabla_{\\mu}(\\phi F)\\nabla^{\\mu}(\\phi F)+3\\Box (\\phi F)].\\label{re-BD-equation}\n\\end{equation}\n We can see that Eqs. (\\ref{trace-gravity-eq-re}) and (\\ref{re-BD-equation}) describe the dynamics of  the two scalar fields: $\\phi$ and $F$ in the Palatini-GBD theory, which are different from the results in the  Palatini-$f(R)$ theory, where the scalar field carries no dynamics of its own  \\cite{fr-review2}.\n\nIn this section, we derived the field equations of the  GBD theory by using the non-standard Palatini approach, where the connection was treated as an independent dynamical variable. The gravitational field equations in  this theory were gained by performing variations of action with respect to the metric and the connection, respectively. Variation with respect to the metric gave  new  field equation containing $F(\\tilde{R})$, and variation with respect to the connection gave  the Riemann connection associated with metric $h_{\\mu\\nu}$ via appropriate conformal transformation. Based on the above field equations, we derive the linearized equations in the Palatini-GBD theory in the following.\n\n\n\n\n\\section{$\\text{ Linearized field equations in the Palatini-GBD theory}$}\n\n Modified  gravitational  theory should have the correct weak-field limit at the Newtonian and the post-Newtonian levels.  In this section, we derive the linearized field equations in the Palatini-GBD theory via the weak-field  approximation method. And then in the following two sections, we solve the linearized field equations for two cases: the vacuum case and the static point-mass case, respectively.\n\n\n As a begining, we discuss the weak-field approximations of GBD theory in the Palatini formalism  via\n\\begin{eqnarray}\ng_{\\mu\\nu}=\\eta_{\\mu\\nu}+b_{\\mu\\nu},~~~~~~\\phi=\\phi_{0}+\\varphi,~~~~~~F=F_{0}+\\delta F,\\label{weak-conditions}\n\\end{eqnarray}\nwhere $\\eta_{\\mu\\nu}$ denotes the Minkowski metric,  $\\phi$  and $F$  are two scalar fields, and the following three relations are required: $|b_{\\mu\\nu}|\\ll 1$,  $|\\varphi|\\ll \\phi_{0}$ and $|\\delta F|\\ll F_{0}$. Using Eqs.(\\ref{Einstain-Tensor-Pala}-\\ref{weak-conditions}),   linearized field equations in the Palatini-GBD theory are derived as\n \\begin{eqnarray}\n\\bar{R}_{(g)\\mu\\nu}-\\frac{\\bar{R}_{(g)}}{2}\\eta_{\\mu\\nu}=\n\\partial_{\\mu}\\partial_{\\nu}\\frac{\\delta F}{F_{0}}+\\partial_{\\mu}\\partial_{\\nu}\\frac{\\varphi}{\\phi_{0}}\n-\\eta_{\\mu\\nu}\\bar{\\Box}_{p}\\frac{\\delta F}{F_{0}}-\\eta_{\\mu\\nu}\\bar{\\Box}_{p}\\frac{\\varphi}{\\phi_{0}}+\\frac{8\\pi T_{\\mu\\nu}}{\\phi_{0}F_{0}},\\label{eq-weak-gravity}\n\\end{eqnarray}\n \\begin{eqnarray}\n\\bar{\\Box}_{p}\\varphi=\\frac{3}{4\\omega-3F_{0}}[8\\pi T-\\phi F_{0}\\bar{R}_{(g)}+3\\phi_{0}\\bar{\\Box}_{p}\\delta F],\\label{eq-weak-phi}\n\\end{eqnarray}\n \\begin{eqnarray}\n\\bar{\\Box}_{p}\\frac{\\delta F}{F_{0}}=\\frac{8\\pi T}{3\\phi_{0}F_{0}}-\\frac{\\bar{\\Box}_{p}\\varphi}{\\varphi_{0}}+\\frac{\\bar{R}_{(g)}}{3},\\label{eq-weak-Phi}\n\\end{eqnarray}\nwhere $\\bar{\\Box}_{p}=\\partial^{\\mu}\\partial_{\\mu}$. $\\bar{R}_{(g)\\mu\\nu}$ and $\\bar{R}_{(g)}$  denote  the linearized quantities, and they can be rewritten as\n\\begin{eqnarray}\n\\bar{R}_{(g)\\mu\\nu}=\\frac{1}{2}(-2\\partial_{\\mu}\\partial\\nu b_{f}+2\\partial\\mu\\partial\\nu\\frac{\\varphi}{\\phi_{0}}-\\bar{\\Box}_{p}\\theta_{\\mu\\nu}+\\frac{\\eta_{\\mu\\nu}}{2}\\bar{\\Box}_{p}\\theta-\\eta_{\\mu\\nu}\\bar{\\Box}_{p} b_{f}+\\eta_{\\mu\\nu}\\bar{\\Box}_{p}\\frac{\\varphi}{\\phi_{0}}),\\label{linear-Rmunu}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\bar{R}_{(g)}=-3\\bar{\\Box}_{p} b_{f}+3\\bar{\\Box}_{p}\\frac{\\varphi}{\\phi_{0}}+\\frac{\\bar{\\Box}_{p}\\theta}{2}\\label{linear-R}\n\\end{eqnarray}\nvia introducing a new tensor $\\theta_{\\mu\\nu}=b_{\\mu\\nu}-\\frac{1}{2}\\eta_{\\mu\\nu}b-\\eta_{\\mu\\nu}\\frac{\\varphi}{\\phi_{0}}+\\eta_{\\mu\\nu}b_{f}$, where $b_{f}\\equiv \\frac{\\delta F}{F_{0}}$, $b=\\eta^{\\mu\\nu}b_{\\mu\\nu}$ and   $\\theta=\\eta^{\\mu\\nu}\\theta_{\\mu\\nu}$.\nAfter choosing a so-called Lorenz gauge or the Harmonic gauge: $\\partial^{\\nu}\\theta_{\\mu\\nu}=0$ and using Eqs. (\\ref{eq-weak-gravity}-\\ref{linear-R}),  we get the linearized gravitational field equation and the linearized scalar-field equations in the Palatini-GBD theory as follows\n\\begin{eqnarray}\n\\bar{\\Box}_{p}\\theta_{\\mu\\nu}=-\\frac{16\\pi T_{\\mu\\nu}}{\\phi_{0}F_{0}},\\label{eq-box-theta}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\bar{\\Box}_{p}\\varphi=\\frac{4\\pi T}{\\omega},\\label{eq-box-varphi}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\bar{\\Box}_{p} b_{f}=0,\\label{eq-box-Phi}\n\\end{eqnarray}\nwith $T=\\eta^{\\mu\\nu}T_{\\mu\\nu}$. Comparing the linearized field equation (\\ref{eq-box-Phi}) in the Palatini-GBD theory with that in the metric-GBD theory: $\\bar{\\Box}_{p} b_{f}-m_{s}^{2}b_{f}=\\frac{16\\pi\\omega T}{3\\phi_{0}F_{0}(2\\omega+3F_{0})}$, we can read some different properties for the scalar field $b_{f}$. In the  Palatini-formalism of GBD theory,  the scalar field $b_{f}$  is massless, while  in the metric-formalism GBD the  scalar field is massive.  We also can see that the scalar field $b_{f}$ is source-free in the Palatini-formalism GBD, which is different from the result in the metric-formalism GBD.\n\n\n\\section{$\\text{Gravitational waves polarization  in the Palatini-GBD theory}$}\n\n  Gravitational-waves (GWs) physics is an important aspect for probing the viable gravitational theory.   Studying on the polarization modes of GWs is also useful for exploring the valuable information on the early universe \\cite{GW-polar}. How many additional polarization modes are detected   in  GWs experiments could instruct us to study which theories of gravity.  Given that more accurate observational data on  GWs  will be received in the future \\cite{GW-observations-future}, it is  worthwhile to investigate GWs physics in alternative theories of gravity,  especially in the Palatini-formalism of modified gravity. The weak-field approximation method provides a natural way to study the GWs. And in some references, the authors have applied this method to discuss the polarization of GWs in different theories \\cite{GW-other1,GW-other2,GW-other3,GW-other4,GW-other5,GW-other6,GW-other7,GW-other8}.\n\n   Considering GWs which propagate along the $z$-direction, we have $k^{\\alpha}=\\varpi(1,0,0,1)$ with the angular frequency  $\\varpi$. And let us consider an observer detecting the gravitational radiation described by a unit timelike vector: $u^{\\alpha}=(1,0,0,0)$. In the vacuum, we solve the wave Eqs. (\\ref{eq-box-theta}-\\ref{eq-box-Phi}) in the Palatini-formalism GBD to get\n\\begin{equation}\n\\theta_{\\mu\\nu}=A_{\\mu\\nu}(\\vec{p})\\exp(i k_{\\alpha}x^{\\alpha}),\\label{solution-theta}\n\\end{equation}\n\\begin{equation}\n\\varphi=c(\\vec{p})\\exp(i p_{\\alpha}x^{\\alpha}),\\label{solution-varphi}\n\\end{equation}\n\\begin{equation}\nb_{f}=d(\\vec{p})\\exp(i q_{\\alpha}x^{\\alpha}).\\label{solution-hf}\n\\end{equation}\nWhere $k_{\\alpha}$  denotes the four-wavevector, and it is a null vector with $\\eta_{\\mu\\nu}k^{\\mu}k^{\\nu}=0$. Eq.(\\ref{solution-theta}) denotes the plane-wave solution of gravitational radiation, while Eqs. (\\ref{solution-varphi}) and (\\ref{solution-hf}) denote the  plane-wave solutions for the massless BD-field perturbation $\\varphi$ and the massless geometry-field perturbation $b_{f}$, respectively.\n\n\n\n\n\n\n In theory, several methods have been developed to analyze the polarization  of GWs \\cite{gw-polar4,gw-polar5,gw-polar6,gw-polar1,gw-polar2,gw-polar3,GW-polar1a,GW-polar2a,GW-polar3a,GW-polar4a}, such as  the Newman-Penrose (NP) method \\cite{gw-polar4,gw-polar5}, the  geodesic deviation (GD) method \\cite{gw-polar1,gw-polar6}, etc. In the following,  we investigate  the polarization modes of GWs in the Palatini-GBD theory by using these two methods. In a local proper reference frame, the equation of geodesic deviation can be described as\n \\begin{equation}\n \\ddot{x}^{i}=-R^{i}_{~0k0}x^{k},\\label{eq-GD}\n \\end{equation}\n here $i$ and $k$ can be taken as $\\{1,2,3\\}$, respectively. $R^{i}_{~0k0}$  denotes the so-called \"electric\" components of the Riemann tensor  with its expression as follows \\cite{GW-polar3a}\n  \\begin{equation}\n R^{(1)}_{i0j0}=(h_{i0,0j}+h_{0j,i0}-h_{ij,00}-h_{00,ij}),\\label{linear-Riemann}\n \\end{equation}\n where $h_{\\mu\\nu}$ denotes the linear perturbation. Using Eqs. (\\ref{eq-GD}) and (\\ref{linear-Riemann}), we gain\n   \\begin{eqnarray}\n \\ddot{x}(t)=-(xh_{11,00}+yh_{12,00}),~~~~~~~~\\nonumber\\\\\n \\ddot{y}(t)=-(xh_{12,00}+yh_{11,00}),~~~~~~~~\\nonumber\\\\\n \\ddot{z}(t)=(2h_{03,03}-h_{33,00}-h_{00,33})z.\\label{xyzdott}\n \\end{eqnarray}\n Using solution (\\ref{solution-theta}) and Eqs. (\\ref{xyzdott}), we obtain\n   \\begin{eqnarray}\n \\ddot{x}(t)=k_{0}^{2}[\\hat{\\epsilon}^{(+)}(k_{0})x+\\hat{\\epsilon}^{(\\times)}(k_{0})y]\\exp(ik_{\\alpha}x^{\\alpha})+c.c.,~~\\nonumber\\\\\n \\ddot{y}(t)=k_{0}^{2}[-\\hat{\\epsilon}^{(+)}(k_{0})y+\\hat{\\epsilon}^{(\\times)}(k_{0})x]\\exp(ik_{\\alpha}x^{\\alpha})+c.c.,\\nonumber\\\\\n \\ddot{z}(t)=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\label{xyzdott-tensor}\n \\end{eqnarray}\nwhich describe the two standard plus and cross polarization modes of GR with the frequency $k_{0}$. For case of massless scalar field $\\varphi$ we have\n  \\begin{eqnarray}\n \\ddot{x}(t)=-p_{0}^{2}xc(\\vec{p})\\exp(ip_{\\alpha}x^{\\alpha})+c.c.,~~~~\n \\ddot{y}(t)=-p_{0}^{2}yc(\\vec{p})\\exp(ip_{\\alpha}x^{\\alpha})+c.c.,~~~~\n \\ddot{z}(t)=0,\\label{xyzdott-bf}\n \\end{eqnarray}\nand for case of massless scalar field  $b_{f}$ we obtain\n   \\begin{eqnarray}\n \\ddot{x}(t)=-q_{0}^{2}xd(\\vec{p})\\exp(iq_{\\alpha}x^{\\alpha})+c.c.,~~~~\n \\ddot{y}(t)=-q_{0}^{2}yd(\\vec{p})\\exp(iq_{\\alpha}x^{\\alpha})+c.c.,~~~~\n \\ddot{z}(t)=0.\\label{xyzdott-varphi}\n \\end{eqnarray}\nObviously, Eqs.(\\ref{xyzdott-bf}) and (\\ref{xyzdott-varphi}) indicate a breathing type of GWs polarization, which have two oscillation modes with the frequency $q_{0}$ and frequency $p_{0}$, respectively.  The same results can  also be obtained by using the Newman-Penrose (NP) method \\cite{GW-polar3a,NP-tetrad}. Following the method shown in Refs. \\cite{GW-polar3a,GW-polar4a,GNP-polar-six1,GNP-polar-six2,GNP-polar-six3,GNP-polar-six4,GNP-polar-six5,GNP-polar-six6,NP-polar-six}, the amplitudes of six polarizations in the Palatini-GBD theory can be calculated as follows\n \\begin{eqnarray}\n p^{(l)}_{1}=-\\frac{1}{6}R_{0303}=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\nonumber\\\\\n p^{(x)}_{2}=-\\frac{1}{2}R_{0301}=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\nonumber\\\\\n p^{(y)}_{3}=\\frac{1}{2}R_{0302}=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\nonumber\\\\\n p^{(+)}_{4}=-R_{0101}+R_{0202}=-2k_{0}^{2}\\hat{\\epsilon}^{(+)}(k_{0})\\exp(ik_{\\alpha}x^{\\alpha})+c.c.,~~~~~~~~~~~~~~~~~~~~~\\nonumber\\\\\n p^{(\\times)}_{5}=2R_{0102}=2k_{0}^{2}\\hat{\\epsilon}^{(\\times)}(k_{0})\\exp(ik_{\\alpha}x^{\\alpha})+c.c.,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\nonumber\\\\\n p^{(b)}_{6}=-R_{0101}-R_{0202}=2p_{0}^{2}c(\\vec{p})\\exp(ip_{\\alpha}x^{\\alpha})+2q_{0}^{2}d(\\vec{p})\\exp(iq_{\\alpha}x^{\\alpha})+c.c..\n \\label{p-np}\n \\end{eqnarray}\nHere the six polarizations modes of GWs are: the longitudinal scalar mode $p_{1}^{(l)}$, the vector-$x$ model $p_{2}^{(x)}$, the vecotr-$y$ mode $p_{3}^{(y)}$, the plus tensorial  mode $p_{4}^{(+)}$, the cross  tensorial  mode $p_{5}^{(\\times)}$, and the breathing scalar mode $p_{6}^{(b)}$, respectively. Form expressions (\\ref{p-np}), we can read the  plus tensor polarization mode, the cross tensor polarization mode, and a breathing scalar mode with two oscillation in the Palatini-GBD theory.    It indicates that the extra scalar field $F$  and the BD scalar field generate the new polarizations of GWs which are not present in the standard GR or the Palatini-$f(\\tilde{R})$ theory. In the GR and the Palatini-$f(\\tilde{R})$ theory, both of them  predict two  tensorial polarization modes: $+$ and $\\times$, not have any scalar modes \\cite{GW-polar}.\n\n Comparing our results with other theoretical results in Refs. \\cite{GW-polar,GW-polar1a,GW-polar2a,GW-polar3a,GW-polar4a}, it is observed that the polarization modes of GWs in the Palatini-GBD theory are  different from the results given in some other gravitational theories. For example, in the massive BD theory  \\cite{GW-polar}, it has  two standard tensorial modes of GR and  two scalar modes (the longitudinal and the breathing modes); In the massless BD theory it owns two standard  tensorial modes and one breathing scalar mode \\cite{GW-polar}; In the $f(R)$ theories, there are two tensorial modes and a massive scalar mode that is a mix of the longitudinal and the transverse breathing polarization \\cite{GW-polar1a,GW-polar2a};  In the $f(T,B)$ theory of teleparallel gravity (it is equivalent to $f(R)$ gravity by linearized the field equations in the weak field limit approximation), there are three polarizations \\cite{GW-polar3a}: the two standard of general relativity and an additional massive scalar mode, where the boundary term $B$ excites the extra scalar polarization; In the higher order local and non-local theories of gravity, they have three state of polarization and $n + 3$ oscillation modes \\cite{GW-polar4a} (concretely, they are the two transverse tensor ($+$) and ($\\times$) standard polarization modes of frequency $\\omega_{1}$, and the $n+1$ further scalar modes of frequency $\\omega_{2}, ..., \\omega_{n+2}$, each of which has the same mixed polarization, partly longitudinal and partly transverse).\n\n\n\n We also compare the results of GWs  polarization in the Palatini-GBD theory with that in the metric-GBD theory. The plane-wave solutions in the  metric-GBD theory can be expressed as \\cite{GBD-L}: $\\theta_{\\mu\\nu}=A_{\\mu\\nu}(\\vec{p})\\exp(i k_{\\alpha}x^{\\alpha})$, $\\varphi=a(\\vec{p})\\exp(i p_{\\alpha}x^{\\alpha})$, and $b_{f}=b(\\vec{p})\\exp(i q_{\\alpha}x^{\\alpha})$. Here,  $\\varphi$ denotes the massless BD-field perturbation and $b_{f}$ denotes the massive geometry-field perturbation, respectively. For the massive plane wave propagating along $z-$direction, we have $q_{\\alpha}=(q_{0},0,0,q_{3})$ with $ m^{2}=q_{0}^{2}-q_{3}^{2}\\neq  0$. Originally, the NP formalism was applied to work out for massless waves. Recently, it was  also generalized  to explore the massive waves propagating along non-null geodesics \\cite{GW-polar3a,NP-tetrad}. Using this method,  the non-zero amplitudes of polarizations for the metric-GBD theory in Ref. \\cite{GBD-L} are calculated as\n \\begin{eqnarray}\n p^{(l)}_{1}=\\frac{1}{6}(-q_{3}^{2}+q_{0}^{2})b(\\vec{p})exp(iq^{\\alpha}x_{\\alpha})+c.c.,~~~~~~~~~~~~~~~~~~\\nonumber\\\\\n p^{(+)}_{4}=-\\sqrt{2}k_{0}^{2}\\hat{\\epsilon}^{(+)}(k_{0})\\exp(ik_{\\alpha}x^{\\alpha})+c.c.,~~~~~~~~~~~~~~~~~~\\nonumber\\\\\n p^{(\\times)}_{5}=\\sqrt{2}k_{0}^{2}\\hat{\\epsilon}^{(\\times)}(k_{0})\\exp(ik_{\\alpha}x^{\\alpha})+c.c.,~~~~~~~~~~~~~~~~~~~~\\nonumber\\\\\n p^{(b)}_{6}=2p_{0}^{2}a(\\vec{p})\\exp(ip_{\\alpha}x^{\\alpha})+2q_{0}^{2}b(\\vec{p})\\exp(iq_{\\alpha}x^{\\alpha})+c.c..\n \\label{p-np-metric}\n \\end{eqnarray}\n Obviously, Eqs.(\\ref{p-np-metric})  show that there are four polarizations modes for  GWs in the metric-GBD theory: the two standard tensorial modes ($+$ and $\\times$), a scalar  breathing  mode with frequency $p_{0}$, and  a massive scalar mode that is a mix of the longitudinal and the transverse breathing polarization with frequency $q_{0}$.\n\n\n\n\\section{$\\text{ PPN parameter in the Palatini-GBD theory}$}\n\n\nIn this section, we  derive the theoretical expressions of the parametrized post-Newtonian (PPN) parameter $\\gamma$  in the Palatini-GBD theory by using the weak-field approximation method.  Considering a static point-mass source, we have the form of the energy-momentum tensor: $T_{\\mu\\nu}=m_{p}\\delta(\\vec{r})diag(1,0,0,0)$. Obviously, the point particle is located at $\\vec{r}=0$. Solving Eqs. (\\ref{eq-box-theta}) and (\\ref{eq-box-varphi}), we get the perturbation solutions: $\\theta_{00}=\\frac{4m_{p}}{\\phi_{0}F_{0}}\\frac{1}{r}$ and $\\varphi(r)=\\frac{m_{p}}{\\omega}\\frac{1}{r}$. Combining relations: $b_{\\mu\\nu}=\\theta_{\\mu\\nu}-\\eta_{\\mu\\nu}\\frac{\\theta}{2}+\\eta_{\\mu\\nu}b_{f}-\\eta_{\\mu\\nu}\\frac{\\varphi}{\\phi_{0}}$ and $\\theta=\\eta^{\\mu\\nu}\\theta_{\\mu\\nu}=-\\frac{4m_{p}}{\\phi_{0}F_{0}}\\frac{1}{r}$, we gain  the non-vanishing components of the metric perturbation\n\\begin{equation}\nb_{00}=\\frac{2m_{p}}{\\phi_{0}F_{0}r}+\\frac{m_{p}}{\\phi_{0}\\omega r}-b_{f},\\label{h00}\n\\end{equation}\n\\begin{equation}\nb_{ij}=\\frac{2m_{p}}{\\phi_{0}F_{0}r}-\\frac{m_{p}}{\\phi_{0}\\omega r}+b_{f}.\\label{hij}\n\\end{equation}\nHere $i,j =1,2,3$ is the space index. The term $\\frac{m_{p}}{\\phi_{0}\\omega r}$ in above two equations reflect the effect  of scalar field $\\phi$. Considering that $b_{f}$ is  negligible for a point-mass case, then the concrete form of the PPN  parameter $\\gamma$  in the Palatini-GBD theory are derived as follows\n\\begin{equation}\n\\gamma=\\frac{b_{ii}}{b_{00}}=\\frac{2\\omega-F_{0}}{2\\omega+F_{0}}.\\label{gamma}\n\\end{equation}\n From Eq. (\\ref{gamma}), we can see the dependence of the PPN parameter $\\gamma$ with respect to model parameters: $\\omega$ and $F_{0}$. It is well known that a gravity theory alternative to GR should be tested by the well-founded experimental results. Some observations can be directly applied to constrain the value of the PPN parameter  $\\gamma$. In Ref. \\cite{bound-omega-gamma},  the experimental bound on $\\gamma$ is: $|\\gamma-1|<2.3*10^{-5}$. For the Palatini-GBD theory, then we have that $\\gamma\\sim 1$ requires $\\omega \\gg F_{0}$, which can be consistent with the observational constraint: $ |\\omega| > 40000$ \\cite{bound-omega-gamma}.\n\n\n\n\\section{$\\text{Conclusions}$}\n\nSeveral observational and theoretical  problems motivate us to investigate the  modified or alternative theories of GR.  Lots of modified gravity theories have been proposed and widely studied.  In the BD modified theory,  the scalar field can be introduced by considering a time-variable Newton gravity constant. Many extended versions of the BD theory have been explored and developed. As one of the generalized BD theories,  some interesting properties have been found in the metric-formalism of GBD theory \\cite{GBD,GBD-L}. For examples: (1)  In the GBD model, the state parameter of geometrical dark energy can cross over the phantom boundary $w=-1$ as achieved in the quintom model,  without bearing the problems existing in the two-fields quintom model, such as the problem of negative kinetic term and the fine-tuning problem, etc. (2) It is well known that the  metric-$f(R)$ theories are equivalent to the BD theory with a potential (abbreviated as BDV) for taking a specific value of the BD parameter $\\omega=0$, where the specific choice: $\\omega=0$ for the BD parameter is quite exceptional, and it is hard to understand the corresponding absence of the kinetic term for the field. However, for the GBD theory, it is similar to the double scalar-fields model, and it owns the non-disappeared  kinetic term in the action.  In addition, the GBD theory tends to investigate the physics from the viewpoint of geometry, while the BDV or the two scalar-fields quintom model tends to solve physical problems from the viewpoint of matter. It is possible that several special characteristics of scalar field  could be revealed through studies of geometrical gravity in the GBD.\n\n\nIn this paper, we  studied the generalized Brans-Dicke theory in the Palatini formalism. Firstly, we derived the  field equations for  the gravitational field, the independent connection  and the BD scalar field, respectively. Secondly, using the weak-field method we obtained the  linearized gravitational field equation and the linearized scalar-fields equations. We showed various properties of the geometrical scalar field in the Palatini-formalism of GBD theory: it is massless and source-free, which are different from the results given in the metric-formalism of GBD theory.  According to the weak-field equations, we investigated the parameterized post Newton parameter in the Palatini-GBD theory by using the point-source method. It was shown that  $\\gamma=\\frac{2\\omega-F_{0}}{2\\omega+F_{0}}\\sim 1$ requires $\\omega \\gg F_{0}$, which can be consistent with the observational constraint: $ |\\omega| > 40000$.  Comparing to the Ref.  \\cite{GBD-L}, we can see that the difference for expressions of $\\gamma$ between the Palatini-GBD theory and the metric-GBD theory. In the metric-GBD theory  \\cite{GBD-L},   $\\gamma$  depends on the parameters: $\\omega$, $F_{0}$ and $m_{s}$. Thirdly, we discussed the gravitational waves physics in the Palatini-GBD theory.  The properties of  GWs in the modified gravity theory have recently attracted lots of attention \\cite{gw-mg1,gw-mg2,gw-mg3,gw-mg4,gw-mg5}, since investigating the polarization modes of GWs can serve to discriminate the different gravitational theories.\nThe results in the Palatini-formalism of GBD showed that   the extra scalar field $F$  and the BD scalar field generated the new polarizations for GWs which were not present in the standard GR or the Palatini-$f(R)$ theory, where both  theories (GR and Palatini-$f(\\tilde{R})$) predicted two  tensorial polarization modes: $+$ and $\\times$, not have any scalar modes. Concretely, we can read that there are three modes of polarization and four oscillation for GWs in the  Palatini-GBD theory, i.e. the plus tensor polarization mode, the cross tensor polarization mode, and a breathing scalar mode with two oscillation. The  results of GWs polarization in the Palatini-GBD theory are different from that in the metric-GBD theory. In the  metric-GBD theory, there are four polarizations modes: the two standard tensorial modes ($+$ and $\\times$), a scalar  breathing  mode, and  a massive scalar mode that is a mix of the longitudinal and the breathing polarization.\n\n\n\n\n\\textbf{\\ Acknowledgments }\n The research work is supported by   the National Natural Science Foundation of China (11645003,11705079), and supported by  LiaoNing Revitalization Talents Program.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n An intuitive understanding of a prototype of non-Fermi liquid (NFL), a multichannel Kondo model, was proposed by Nozi\\`eres and Blandin about 25 years ago.\\cite{NB} The understanding of the multichannel Kondo model was obtained by various methods, such as Bethe-ansatz\\cite{Wiegman, Andrei}, boundary conformal field theory (BCFT)\\cite{AffLud1,AffLud2,AffLud3,AffLud4,2IK}, numerical renormalization group (NRG)\\cite{Cragg,Pang}, and so on. In particular, we can compare the NRG finite size spectra and those of the BCFT even if we have NFL spectra and the agreements are excellent.\\cite{Kim2,AffLud4,2IK} \n\n In the realistic model of diluted f-electron systems, a candidate for a two-channel Kondo (2CK) model is the quadrupolar Kondo model\\cite{Cox} for U and Ce based alloys, in which non-Kramers doublet in $f^2$ configuration plays an important role. Several models that have a NFL property were proposed, in which the $\\Gamma_8$ conduction electrons interact with a localized $f^1$ or $f^2$ crystalline-electric-field state under cubic symmetry\\cite{Koga1,KusuKura,Koga2,Kim1}. These models may be relevant for dilute alloys of Ce$^{3+}$ or U$^{4+}$ ions, such as La$_{1-x}$Ce$_x$Cu$_2$Si$_2$\\cite{Andraka}, UCu$_{5-x}$Pd$_x$\\cite{Andrade} among others. In general, NFL behaviors are observed in real materials as transient phenomena, which are controlled by temperature, pressure, alloying and so on.\n\n In this Letter, we reexamine an impurity Kondo model in which a localized f-electron with $\\Gamma_8$ symmetry interacts with $\\Gamma_8$ conduction electrons under cubic symmetry. This model was proposed as that of Ce$_x$La$_{1-x}$B$_6$\\cite{Koga1}. The earlier NRG calculations confirmed the existence of a NFL fixed point that is unstable  against the particle-hole (PH) symmetry breaking. Indeed, Ce$_x$La$_{1-x}$B$_6$ is not located near this NFL fixed point, so that the property of this NFL may not be relevant to the case of Ce$_x$La$_{1-x}$B$_6$. However, the NFL would be realized in the system with small quadrupolar interactions that break PH symmetry. Theoretically, the origin of the NFL was thought to be mysterious from the conventional BCFT point of view. The NRG energy spectrum was similar to that of 2CK model but the origin was unknown.\n\nAlthough it is difficult to control PH asymmetry in experiments, for both the pursuit of new materials and theoretical interest, it is worthwhile to study the detailed properties of this NFL by BCFT. It is noted that a BCFT approach is a powerful tool for answering {\\it why} NFL behaviors emerge in various impurity problems. The same situation occurs in the case of the two-impurity Kondo model (the NFL is unstable against PH asymmetry of conduction electrons), which has been extensively investigated so far\\cite{Jones}. Roughly speaking, the quantum critical phenomena near the antiferromagnetic critical point at zero temperature\\cite{Stewart} can be seen as a kind of a lattice generalization of the two-impurity Kondo model (however, it is not so simple). The PH asymmetry in a generalized 2CK model\nwas also discussed\\cite{Pan2}. In this case, in the presence of a double tensor interaction that breaks the PH symmetry, the system flows to a Fermi liquid. \n\n\n We assume the following points for the local f-electron state: (i) cubic symmetry, (ii) $f^1$ $\\Gamma_8$ ground state configuration, (iii) the lowest excited state is f$^0$ and\/or $f^2$ $\\Gamma_1$ singlet configuration. \n Under these assumptions, we can map the $\\Gamma_8$ index to a pseudospin-$3\/2$ representation as $|\\Gamma_{8,{\\pm \\frac{3}{2}}}\\rangle=\\mp(| \\pm \\frac{3}{2}\\rangle+\\sqrt{5}|\\mp \\frac{5}{2}\\rangle)\/\\sqrt{6}$ and $|\\Gamma_{8,{\\pm \\frac{1}{2}}}\\rangle=\\pm| \\pm \\frac{1}{2}\\rangle$, where $|j_z\\rangle$ is a state with the total angular momentum $J=5\/2$ and its z-component $j_z$.\n\n\n\nWe start with the following pseudospin-$3\/2$ Kondo model (see the derivation in ref. 13) in 1-dimension considering the s-wave scattering at the impurity site\\cite{AffLud2}:\n\\begin{eqnarray}\nH&=&H_0+\\sum_{m=\\rm dip,quad,oct}H_{m},\\label{H}\\\\\nH_0&=&\\frac{iv_F}{2\\pi}\\sum_{\\mu=\\pm \\frac{3}{2},\\pm\\frac{1}{2}}\\int dx\\psi_{\\mu}^{\\dagger}(x)\\partial \\psi_{\\mu}(x),\\\\\nH_{m}&=& J_{m}\\sum_{\\mu,\\nu=\\pm \\frac{3}{2},\\pm\\frac{1}{2}}\\psi_{\\mu}^{\\dagger}(0)({\\bf x}_c^m)_{\\mu\\nu}\\psi_{\\nu}(0)\\cdot {\\bf X}_I^m,\n\\end{eqnarray}\nwhere we introduce left-moving fermion annihilation operators $\\psi_{\\mu}(x)$, Fermi velocity $v_F$ and spin $3\/2$ dipolar (${\\bf x}_c^{\\rm dip}={\\bf s}_c$), quadrupolar (${\\bf x}_c^{\\rm quad}={\\bf q}_c$) and octupolar (${\\bf x}_c^{\\rm oct}={\\bf t}_c$) matrices of the conduction electron (similar definitions for ${\\bf X}_I^m$ of the impurity).\\cite{Relation} $J_{m}$ is the coupling constant of each multipoles. It is noted that under the assumptions (i)-(iii) the interactions are isotropic in the pseudospin space.\n\n\nThe ``conformal embedding'' often found in literatures is SU(4)$\\to$ SU(2)$_{10}\\otimes$ SU(2)$_4$.\\cite{Kim2}\n The SU(2)$_{10}$ corresponds to a spin current $ {{ \\mbox{\\boldmath $ \\mathcal J$}}}(x)$\\cite{J}, i.e., {\\it dipole}. The other SU(2) current, SU(2)$_4$, is an axial charge (AC) current $\\mbox{\\boldmath $\\mathcal Q$}(x)$\\cite{Q}. In this embedding, $H_0$ can be written in the following Sugawara form:\n\\begin{eqnarray}\n\\frac{l}{\\pi v_F}H_0=\\sum_{n=-\\infty}^{\\infty}\\Big(\\frac{1}{4}:\\mbox{\\boldmath $\\mathcal Q$}_{-n}\\cdot\\mbox{\\boldmath $\\mathcal Q$}_{n}: + \\frac{1}{12}:{\\mbox{\\boldmath $\\mathcal J$}}_{-n}\\cdot \n{\\mbox{\\boldmath $\\mathcal J$}}_{n}:\\Big), \\label{Hboson}\n\\end{eqnarray}\nwhere \n${\\mbox{\\boldmath $\\mathcal J$}}_{n}\\ {\\rm and}\\ {\\mbox{\\boldmath $\\mathcal Q$}}_{n}$ are the Fourier components of ${\\mbox{\\boldmath $\\mathcal J$}}(x)\\ {\\rm and}\\ {\\mbox{\\boldmath $\\mathcal Q$}}(x)$, respectively. We set the system size to $2l$. $:A:$ indicates the normal ordering of the operator $A$. The energy eigenvalues of the right hand side of eq. (\\ref{Hboson}) for primary states that form conformal towers\\cite{Itz} are given by\n\\begin{eqnarray}\nE(q,j)=\\frac{q(q+1)}{4}+\\frac{j(j+1)}{12},\\label{E0}\n\\end{eqnarray}\n where $(q=0,1\/2, 1)$ and $(j=0,1\/2, \\cdots, 5)$. The energies of the descendant states\\cite{Itz} are given by $E(q,j)+n$, where $n$ is a positive integer and indicates the PH excitations from the ground state.\\cite{AffLud3} The descendant states generally have different quantum numbers from those of the corresponding primary state in the same conformal tower.\n The form of eq. (\\ref{Hboson}) is suitable to the case of $J_{\\rm quad}=J_{\\rm oct}=0$. In this case, the impurity spin can be absorbed into ${\\mbox{\\boldmath $\\mathcal J$}}_{n}$\\cite{AffLud1} and the predicted NFL spectra are in complete agreement with the NRG results\\cite{Kim2,Koga1,Koga2}.\n\nIn the presence of quadru- and octupolar interactions, the above SU(2)$_{10}$ absorption is not applicable.\nAs noted by Wu {\\it et al.} the spin $3\/2$ fermionic system has an exact SO(5) symmetry under some conditions\\cite{Wu1}. In the following, we transform the Hamiltonian (\\ref{H}) into the SO(5) language, i.e., the embedding is SU(4)$\\to $SO(5)$_2\\otimes$ SU(2)$_4$.\n\nFirst, quadrupolar matrices of $s_c=3\/2$ have the following forms:\n\\begin{eqnarray}\nq_c^{3z^2-r^2}&\\equiv&\\frac{1}{\\sqrt{3}}(3s_c^zs_c^z-\\frac{3}{2}\\frac{5}{2}\\hat{\\bf 1})\\equiv \\sqrt{3}\\Gamma^4 \\label{qz},\\\\\nq_c^{x^2-y^2}&\\equiv&\\frac{1}{\\sqrt{2}}(s_c^xs_c^x-s_c^ys_c^y)\\equiv-\\sqrt{3}\\Gamma^5,\\\\\nq_c^{xy}&\\equiv&s_c^xs_c^y+s_c^ys_c^x\\equiv -\\sqrt{3}\\Gamma^1,\\\\\nq_c^{yz}&\\equiv&s_c^ys_c^z+s_c^zs_c^y\\equiv\\sqrt{3}\\Gamma^3,\\\\\nq_c^{zx}&\\equiv&s_c^zs_c^x+s_c^xs_c^z\\equiv\\sqrt{3}\\Gamma^2,\\label{qzx}\n\\end{eqnarray}\nwhere $\\hat{\\bf 1}$ is an identity matrix and the spin $3\/2$ operators $s_c^i\\ (i=x,y,z)$ are constructed on the basis of $^{\\dagger}(\\psi_{\\frac{3}{2}},\\psi_{\\frac{1}{2}},\\psi_{\\frac{-1}{2}},\\psi_{\\frac{-3}{2}})$. We have introduced five Dirac matrices $\\Gamma^a\\ (1\\le a \\le 5)$.\nFrom eqs. (\\ref{qz})-(\\ref{qzx}), we can define ten SO(5) generators $\\Gamma^{ab}$ as $\\Gamma^{ab}\\equiv \\frac{1}{2i}[\\Gamma^a, \\Gamma^b]$. These generators satisfy the SO(5) commutation relations:\n\\begin{eqnarray}\n[\\Gamma^{ab}, \\Gamma^{cd}]=-2i(\\delta_{bc}\\Gamma^{ad}-\\delta_{ac}\\Gamma^{bd}-\\delta_{bd}\\Gamma^{ac}+\\delta_{ad}\\Gamma^{bc}).\n\\end{eqnarray}\n\nThe important point is that $H_{\\rm dip}$ and $H_{\\rm oct}$ are written by the ten $\\Gamma^{ab}$ when the condition $J_{\\rm dip}=J_{\\rm oct}$ is satisfied. Thus, under this condition, each multipolar Hamiltonian can be written in the SO(5)'s vector $n^a_I$ ($n_c^a$) and generators $L^{ab}_I$ ($L_c^{ab}$) of the impurity (conduction electron), as follows:\n\\begin{eqnarray}\nH_{\\rm quad}&=& 12J_{\\rm quad}\\sum_{a=1}^5n_c^a(0)n^a_I,\\label{Hq}\\\\\nH_{\\rm dip}+H_{\\rm oct}&=& 5J_{\\rm dip}\\sum_{a=1}^5\\sum_{b>a}^5L_c^{ab}(0)L^{ab}_I,\\label{Hdo}\n\\end{eqnarray}\n where \n\\begin{eqnarray}\nn_c^a(x)&\\equiv& \\frac{1}{2}\\sum_{\\mu\\,\\nu=\\pm \\frac{3}{2},\\pm\\frac{1}{2}}\\psi_{\\mu}^{\\dagger}(x)(\\Gamma^a)_{\\mu\\nu}\\psi_{\\nu}(x),\\label{nc}\\\\\nL_c^{ab}(x)&\\equiv&-\\frac{1}{2}\\sum_{\\mu,\\nu=\\pm \\frac{3}{2},\\pm\\frac{1}{2}}\\psi^{\\dagger}_{\\mu}(x)(\\Gamma^{ab})_{\\mu\\nu}\\psi_{\\nu}(x).\\label{Lc}\n\\end{eqnarray}\n\\begin{table}[t]\n\t\\begin{tabular}{cccc}\n         \\hline\n         $q$ & $j$ & {\\bf u} & $E(q,{\\bf u}) \\ (E(q,j))$\\\\\n         \\hline\n         \\hline\n         $0$ & $0$ & $\\bf 1$ & $0$\\\\\n         \\hline\n         $1\/2$ & $3\/2$ & $\\bf 4$ & $1\/2$\\\\\n         \\hline\n         $1$ & $2$ & $\\bf 5$ & $1$\\\\\n         \\hline\n         $0$ & $3$ & $\\bf 10$ & $1$\\\\\n         $0$ & $1$ &      & $1$\\\\\n         \\hline\n         $1$ & $0$ & $\\bf 1$  & $1$\\\\\n         \\hline\n         $3\/2$ & $3\/2$ & $\\bf 4$  & $3\/2$\\\\\n         $1\/2$ & $3\/2$ & $\\bf 4$  & $3\/2$\\\\\n         \\hline\n         $1\/2$ & $7\/2$ & $\\bf 16$ & $3\/2$\\\\\n         $1\/2$ & $5\/2$ &      & $3\/2$\\\\\n         $1\/2$ & $1\/2$ &      & $3\/2$\\\\\n         \\hline\n        \\end{tabular}\n\\caption{Low-energy spectrum of free Hamiltonian for nondegenerate ground state. The first and the third columns are the AC and SO(5) indices, respectively. The second column is the eigenvalue of the spin current in eq. (\\ref{E0}). $E(q,{\\bf u})$ and $E(q,j)$ are measured from the ground state using eqs. (\\ref{E1}) and (\\ref{E0}), respectively.}\n\\label{tbl-1}\n\\end{table}\n\n\nThe free Hamiltonian $H_0$ in (\\ref{H}) is expressed using the SO(5) generators\n\\begin{eqnarray}\n\\frac{l}{\\pi v_F}H_0=\\sum_{n=-\\infty}^{\\infty}\\Big(\\frac{1}{4}:\\mbox{\\boldmath $\\mathcal Q$}_{-n}\\cdot\\mbox{\\boldmath $\\mathcal Q$}_{n}: + \\frac{1}{8}:{\\mbox{\\boldmath $\\mathcal L$}}_{-n}\\cdot \n{\\mbox{\\boldmath $\\mathcal L$}}_{n}:\\Big), \\label{Hboson2}\n\\end{eqnarray}\nwhere $({\\mbox{\\boldmath $\\mathcal L$}}_{n})^{ab}$ is the n-th Fourier component of SO(5) current $L_c^{ab}(x)$: ${\\mbox{\\boldmath $\\mathcal L$}_n}=({\\mathcal L^{12}_{n}}, {\\mathcal L^{13}_{n}},\\cdots ,{\\mathcal L^{35}_{n}}, {\\mathcal L^{45}_{n}})$ and satisfies a level 2 SO(5) Kac-Moody algebra. The energy eigenvalues $E(q,{\\bf u})$ for the primary states are expressed by the Casimir operators in each sector\n\\begin{eqnarray}\nE(q,{\\bf u})=\\frac{q(q+1)}{4}+\\frac{\\bar{c}_{\\bf u}}{8},\\label{E1}\n\\end{eqnarray}\nwhere $q=0,1\/2, 1$ for the AC sector (this is as same as in eq. (\\ref{E0})), and $\\bf u=1\\ {\\rm(identity)},\\ 4\\ {\\rm (spinor)}\\ ,5\\ {\\rm (vector)}$ for the SO(5) sector. ${\\bf u}$ indicates the dimension of the representation in the SO(5), see Fig. \\ref{fig-1}. There are three primary fields in both the AC and the SO(5) sectors.\nThe values of $\\bar{c}_{\\bf u}$ are $\\bar{c}_{\\bf 1}=0,\\ \\bar{c}_{\\bf 4}=5\/2\\ {\\rm and }\\  \\bar{c}_{\\bf 5}=4$. The noninteracting energy spectra for the nondegenerate ground state are shown in Table \\ref{tbl-1} together with the indices of $j$ in eq. (\\ref{E0}). We can see that both eqs. (\\ref{E0}) and (\\ref{E1}) can reproduce the spectrum of free $s_c=3\/2$ fermions.\n\n\n\\begin{figure}[t]\n  \\begin{center}\n    \\includegraphics[width=.25\\textwidth]{.\/fig1.eps}\n  \\end{center}\n\\caption{$H_1-H_2$ diagrams associated with the SO(5) multiplies that enter the low-energy spectrum, where $2H_1\\equiv \\Gamma^{15}$ and $2H_2\\equiv \\Gamma^{23}$ forming the Cartan subalgebra of SO(5) group. In (a), the corresponding components of the spinor are indicated.}\n\\label{fig-1}\n\\end{figure}\n\n\nNext, we consider eqs. (\\ref{Hq}), (\\ref{Hdo}) and (\\ref{Hboson2}) together. It is noted that the total Hamiltonian (\\ref{H}) is written as\n\n\\begin{eqnarray}\n\\frac{l}{\\pi v_F}H&=&\\sum_{n=-\\infty}^{\\infty}\\Big(\\frac{1}{4}:\\mbox{\\boldmath $\\mathcal Q$}_{-n}\\cdot\\mbox{\\boldmath $\\mathcal Q$}_{n}: + \\frac{1}{8}:{\\mbox{\\boldmath $\\mathcal L$}}_{-n}^{'}\\cdot \n{\\mbox{\\boldmath $\\mathcal L$}}_{n}^{'}:\\Big)\\nonumber\\\\\n&&+\\frac{12lJ_{\\rm quad}}{\\pi v_F}\\sum_{a=1}^5n_c^a(0)n^a_I+\\rm const.,\\label{HH}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n{\\mathcal L'}_{n}^{ab}\\equiv{\\mathcal L}_{n}^{ab}+\\frac{20lJ_{{\\rm dip}}}{\\pi v_F}L_I^{ab}.\n\\end{eqnarray}\nThe values at possible fixed points, the $J_{\\rm dip}^*$, $J^*_{\\rm quad}$ and $J^*_{\\rm oct}$ are determined (assumed) as\n\\begin{eqnarray}\n   J_{\\rm dip}^*=J_{\\rm oct}^*=\\frac{\\pi v_F}{20l}{\\rm,\\ \\ \\ \\ and}\\ \\ \\ J^*_{\\rm quad}=0.\\label{Jc}\n\\end{eqnarray}\n The case in which $J_{\\rm quad}\\not=0$ is discussed later. At these values of the couplings, the impurity SO(5) ``superspin'' can be absorbed into the conduction electron ``superspin'' current, generating the new SO(5) ``superspin'' $ {\\mbox{\\boldmath $\\mathcal L$}}_{n}^{'}$. This impurity absorption does not affect the SO(5) Kac-Moody algebra, except the gluing conditions in Table \\ref{tbl-1}, which is much the same as in the case of the multichannel Kondo model\\cite{AffLud1}. \n\n\n At this stage, there is a need to introduce a suitable fusion rule to generate the nontrivial spectrum. Because the interaction is only in the SO(5) sector, any fusion in the AC sector is unphysical. An important point is that the impurity is described in the spin $3\/2$ representation (i.e., dimension 4). We find that the desired fusion is a direct product of $\\bf 4$ representation in the SO(5) sector to each state in the spectra of the free Hamiltonian. To execute this, the following formulae are useful (easily deducible from Fig. \\ref{fig-1}): $\n\\bf 1\\otimes 4= \\bf 4,\\ \n\\bf 4\\otimes 4= \\bf 1\\oplus 5\\oplus 10\\ {\\rm and}\\ \n\\bf 4\\otimes 5= \\bf 4\\oplus 16$.\n\nThus, even after absorbing the impurity ``superspin'', we can calculate the energy spectra at this fixed point by using eq. (\\ref{E1}\n. \nThe obtained spectra are shown in Table \\ref{tbl-2}(a). As shown by previous NRG studies\\cite{Koga1, KusuKura}, the NFL spectra are the same as those of the 2CK NFL if the AC sector in the present model and the spin sector in the 2CK model are interchanged. Indeed, the existence of a SO(5) symmetry in the 2CK model has been pointed out before in literatures.\\cite{AffLud4}\n\n\\begin{table}[t]\n\\begin{center}\n\t\\begin{tabular}{cccccccc}\n         \\hline\n         (a)&$q$  & {\\bf u} & $E(q,{\\bf u})$  & (b) &  $q$ & {\\bf u} & $\\Delta$\\\\\n         \\hline\n         \\hline\n         &$1\/2$ & $\\bf 1$ & $0$       &     &  $0$ & $\\bf 1$ & $0$     \\\\\n         &$0$ &   $\\bf 4$ & $1\/8$     &     &  $0$ & $\\bf 5$ & $1\/2$   \\\\\n         &$1\/2$ & $\\bf 5$ & $1\/2$     &     &  $1\/2$ & $\\bf 4$ & $1\/2$ \\\\\n         &$1$ & $\\bf 4$   & $5\/8$     &     &  $1\/2$ & $\\bf 4$ & $1\/2$ \\\\\n         &$3\/2$ & $\\bf 1$ & $1$       &     &  $1$   & $\\bf 1$ & $1\/2$ \\\\\n         &$1\/2$ & $\\bf 10$ & $1$      &     &  $1$ & $\\bf 5$  & $1$    \\\\\n         &$1\/2$ & $\\bf 1$ & $1$       &     &  $0$ & $\\bf 10$  & $1$   \\\\\n         \\hline\n        \\end{tabular}\n\\caption{(a) Low energy spectrum of NFL fixed point with $E(q,{\\bf u})\\le 1$. In the third column, the ground state energy $3\/16$ is subtracted. (b) Operator contents at the NFL fixed point. We show the operators with $\\Delta \\le 1$ where $\\Delta$ means the scaling dimension of the corresponding operator. \n}\n\\label{tbl-2}\n\\end{center}\n\\end{table}\n\n\nThe operator contents at the NFL fixed point are obtained by applying a double fusion\\cite{AffLud1} in the SO(5) sector (see Table \\ref{tbl-2}(b)). Again, we obtain the same scaling dimensions $\\Delta$ as those at the 2CK fixed point with the above-mentioned interchange. The operator with $(q, {\\bf u})=(0,{\\bf 5})$ is the SO(5) vector $\\mbox{\\boldmath $\\mathcal \\phi$}_{\\rm SO(5)}$, which corresponds to the quadrupolar operator in the original spin $3\/2$ representation. This operator has $\\Delta=1\/2$, so that the corresponding local quadrupolar susceptibility diverges logarithmically at low temperatures. This is consistent with the result of NRG studies \\cite{Koga1, KusuKura}. The local charge and pair field susceptibility would diverge at low temperatures, because the operator $(q, {\\bf u})=(1,{\\bf 1})$, $\\mbox{\\boldmath $\\mathcal \\phi$}_{\\rm AC}$, which is the AC vector, has $\\Delta=1\/2$. The dipolar susceptibility in addition to the octupolar one, is classified in $\\bf 10$ representation with $\\Delta=1$. This means that the dipolar and the octupolar susceptibilities $\\chi$ are not singular, which is also consistent with the NRG calculation\\cite{Koga1,KusuKura}.\n\n\nNext we discuss the stability of the NFL fixed point against various perturbations.\n\na) In the presence of a uniaxial distortion, the term $h_Qn_I^4$, which breaks the SO(5) symmetry, appears in the effective Hamiltonian. This term allows $\\mbox{\\boldmath $\\mathcal \\phi$}_{\\rm SO(5)}$ with $\\Delta=1\/2$ to appear. Thus, the conjugate field $h_Q$ becomes a relevant perturbation and the NFL fixed point becomes unstable.\n\nb) In the presence of a potential scattering at the impurity site $V\\sum_{\\mu}\\psi_{\\mu}^{\\dagger}(0)\\psi_{\\mu}(0)=VQ_z(0)+\\ \\rm const.$, the SU(2) symmetry in the AC sector is broken. In this case, $\\phi_{\\rm AC}^3$, the component $q_z=0$ of $\\mbox{\\boldmath $\\mathcal \\phi$}_{\\rm AC}$ with $\\Delta=1\/2$, can appear in the effective Hamiltonian. Again, the NFL fixed point becomes unstable. The PH symmetry can also be broken by the quadrupolar interaction of eq. (\\ref{Hq}). Indeed, the system flows into the Fermi liquid fixed point of SU(4) Coqblin-Schrieffer model without PH symmetry\\cite{Koga1}. \n\nc) The exchange anisotropy in the SO(5) sector is irrelevant, because the marginal operator $(q,{\\bf u})=(0,{\\bf 10})$ cannot appear under the time-reversal symmetry as discussed in Ref. 8 (that is, our assumption $J_{\\rm dip}=J_{{\\rm oct}}$, does not affect the present result).\\cite{COMMENT}\n\n\\begin{figure}[t!]\n  \\begin{center}\n    \\includegraphics[width=.4\\textwidth]{.\/fig2.eps}\n  \\end{center}\n\\caption{$C_{\\rm imp}\/T$ vs $T$ and ${\\rm Im}\\chi_c(\\omega)$ vs $\\omega$. $V$ is the strength of potential scattering at the impurity site. The parameters used are $J_{\\rm dip}=J_{\\rm oct}=0.1D$ and $J_{\\rm quad}=0.0D$, where $D$ is the half of the bandwidth of conduction electrons.}\n\\label{fig-2}\n\\end{figure}\n\nIn early NRG studies\\cite{Koga1,KusuKura}, the NFL fixed point was considered to be similar to the 2CK fixed point. But the question arises, `` Is the present NFL really equivalent to the 2CK ?'' To verify this, we consider the effective Hamiltonian near the NFL fixed point. If the present NFL and the NFL of the 2CK were equivalent, the leading irrelevant operator would be $\\mbox{\\boldmath $\\mathcal Q$}_{-1}\\cdot\\mbox{\\boldmath $\\mathcal \\phi$}_{\\rm AC}$ with $\\Delta=1\/2+1=3\/2$. This operator, however, is not physically adequate because the impurity absorption occurs in the SO(5) sector. The low-energy effective Hamiltonian should be made by using operators in the SO(5) sector and these should transform as a singlet\\cite{AffLud1}. \n\nBecause ${\\mathcal L}^{ab}_{-1}\\phi^c_{\\rm SO(5)}\\ (1\\le a,b,c\\le 5)$ type operators with $\\Delta=3\/2$ cannot become a singlet, the leading irrelevant operator should be the energy-momentum operator $\\sum_{ab}:L_c^{ab}(0)L_c^{ab}(0):$ with $\\Delta=2$. This leads us to an important conclusion: {\\it impurity specific heat does not diverge at low temperatures} unlike in the 2CK case, {\\it but shows a linear temperature behavior}. As we mentioned above, $\\chi$ is not singular, so the Wilson ratio $R_W=(\\delta\\chi\/\\chi)\/(\\delta C\/C)$ is calculated using the conformal charge in each of sectors $c_{\\rm SO(5)}$ and $c_{\\rm AC}$ as $R_W=(c_{\\rm SO(5)}+c_{\\rm AC})\/c_{\\rm SO(5)}=(5\/2+3\/2)\/(5\/2)=8\/5$($C$: specific heat).\\cite{AffLud3} This result is different from that of the 2CK case: $R_W=8\/3$.\n\nFinally, to check the new results above, we show the results of the NRG calculations of the impurity specific heat $C_{\\rm imp}$ and the z-component of the dynamical AC susceptibility for localized conduction electrons $\\chi_c(\\omega)$ at zero temperature in Fig. {\\ref{fig-2}}. Note that $\\chi_c(\\omega)$ is the dynamical charge susceptibility of the conduction electrons at the impurity site. We used logarithmic discretization parameter $\\Lambda=3$ and kept 400 states at each NRG step. It was confirmed that $C_{\\rm imp}\/T$ is constant at low temperatures and that Im$\\chi(0)\\not=0$ indicates the divergence of the conduction electron's charge susceptibility. We can see that the divergence of the charge susceptibility is suppressed when PH symmetry is broken ($V\\not=0$ case). We also obtained the residual impurity entropy $S_0=k_B\\log \\sqrt{2}$, which is the same value as that at the 2CK NFL fixed point. An increase in the data $V\/D=10^{-4}$ of the $C_{\\rm imp}\/T$ is the natural consequence of the entropy release from $k_B\\log \\sqrt{2}$ to $0$.\n\nIn summary, we have investigated a multipolar Kondo model with $S_I=3\/2$ and $s_c=3\/2$ using BCFT and NRG. The 2CK-like NFL fixed point observed in the earlier NRG calculation is explicitly derived using the ``superspin'' absorption in a hidden symmetry SO(5). We find that the leading irrelevant operator at the NFL fixed point is ``Fermi liquid-like'' in contrast to its 2CK-like NFL spectra. All predictions are consistent with earlier works and the present NRG calculation. In particular, the low temperature impurity specific heat is proportional to temperature, the Wilson ratio $R_W=8\/5$ and the local charge susceptibility of conduction electrons diverges at zero temperature. These results remarkably distinguish the present NFL from the 2CK model.\n\n\\vspace{.5cm}\nThe author would like to thank A. Yotsuyanagi, T. Takimoto, H. Kohno and K. Miyake for valuable comments. This work was supported by the 21st Century COE Program (G18) of the Japan Society for the Promotion of Science.\nThe author is supported by the Research Fellowships of JSPS for Young Scientists.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\nIn the face of the proliferation of real-time IoT applications, fog computing has come as a promising complement to cloud computing by extending cloud to the edge of the network to meet the stringent latency requirements and intensive computation demands of such applications \\cite{xiao2017qoe}.\n\nA typical fog computing system consists of a set of geographically distributed fog nodes\nwhich are deployed at the network periphery with elastic resource provisioning such as storage, computation, and network bandwidth\\cite{yi2015fog}. \nDepending on their distance to IoT devices, fog nodes are often organized in a hierarchical fashion, with each layer as a \\textit{fog tier}.\nIn such a way, resource-limited IoT devices, when heavily loaded, can delegate workloads via wireless links to nearby fog nodes, \\textit{a.k.a.}, \\textit{workload offloading}, \nto reduce the power consumption and accelerate workload processing; \nmeanwhile, each fog node can offload workloads to nodes in its upper fog tier. \nHowever, along with all the benefits come the extended latency and extra power consumption.   \nGiven such a power-latency tradeoff, two interesting questions arise.\nOne is \\textit{where} and \\textit{how much workloads} to offload between successive fog tiers. \nThe other is how to \\textit{allocate resources} for workload processing and offloading.\nThe timely decision making regarding these two questions is critical but challenging, due to temporal variations of system dynamics in wireless environment, uncertainty in the resulting offloading latency, and the unknown traffic statistics.\n\nWe summarize the main challenges of dynamic offloading and resource allocation in fog computing as follows: \n\\begin{enumerate}\n\t\\item[$\\diamond$] \\textbf{Characterization of system dynamics and the power-latency tradeoff}: In practice, a fog system often consists of multiple tiers, with complex interplays between fog tiers and the cloud, not to mention the constantly varying dynamics and intertwined power-latency tradeoffs therein.\n\t\tA model that accurately characterizes the system and tradeoffs is the key to the fundamental understanding of the design space.\n\t\\item[$\\diamond$] \\textbf{Efficient online decision making:}\n\t\tThe decision making must be computationally efficient, so as to minimize the overheads. The difficulties often come from the uncertainties of traffic statistics, online nature of workload arrivals, and intrinsic complexity of the problem.\n\t\\item[$\\diamond$] \\textbf{Understanding the benefits of predictive offloading:} One natural extension to online decision making is to employ predictive offloading to further reduce latencies and improve quality of service. For example, Netflix preloads videos onto users' devices based on user behavior prediction\\cite{NetflixPred}. Despite the wide applications of such approaches, the fundamental limits of predictive offloading in fog computing still remain unknown.\n\\end{enumerate}\n\n\n\\begin{table*}[!h]\n\\centering\n\\caption{Comparisons of related works}\n\\label{table: related works}\n\\begin{threeparttable}\t\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n & D2D-enabled IoT & IoT-Fog\\tnote{1} & Fog-Fog\\tnote{2} & Fog-Cloud\\tnote{3} & Dynamic & Prior Arrival Distribution & Prediction \\\\\n\\hline\n\\cite{xiao2017qoe} &  & \\checkmark & \\checkmark & \\checkmark &  & -- &  \\\\\n\\hline\n\\cite{wang2019cooperative} &  & \\checkmark &  & \\checkmark &  & -- &  \\\\\n\\hline\n\\cite{liu2018socially} &  & \\checkmark &  & \\checkmark & \\checkmark & Poisson & \\\\\n\\hline \n\\cite{misra2019detour} &  & \\checkmark &  &  &  & -- &  \\\\\n\\hline\n\\cite{lei2019joint} &  & \\checkmark &  &  & \\checkmark & Poisson &  \\\\\n\\hline\n\\cite{mao2016power} &  & \\checkmark &  &  & \\checkmark & Not Required &  \\\\\n\\hline\n\\cite{chen2019energy} &  & \\checkmark &  &  & \\checkmark & Not Required &  \\\\\n\\hline\n\\cite{gao2019dynamic} & \\checkmark & \\checkmark &  &  & \\checkmark & Not Required &  \\\\\n\\hline\n\\cite{zhang2019near} &  & \\checkmark &  &  & \\checkmark & Not Required &  \\\\\n\\hline\nOurs &  &  & \\checkmark & \\checkmark & \\checkmark &  Not Required & \\checkmark \\\\\n\\hline\n\\end{tabular}\n\\begin{tablenotes}\n\t\\footnotesize\n\t \\item[1,2,3] ``IoT-Fog'' means offloading from IoT devices to fog, ``Fog-Fog'' means offloading between fog tiers, while ``Fog-Cloud'' means offloading from fog to cloud. \n\\end{tablenotes}\n\\end{threeparttable}\n\t\\vspace{-0.5cm} \n\\end{table*}\n\nIn this paper, we focus on the workload offloading problem for multi-tiered fog systems. We address the above challenges by developing a fine-grained queueing model that accurately depicts such systems and proposing an efficient online scheme that proceeds the offloading on a time-slot basis. \nTo the best of our knowledge, we are the first to conduct systematic study on predictive offloading in fog systems. \nOur key results and main contributions are summarized as follows:\n\\begin{enumerate}\n\t\\item[$\\diamond$] \\textbf{Problem Formulation:} We formulate the problem of dynamic offloading and resource allocation as a stochastic optimization problem, aiming at minimizing the long-term time-average expectation of total power consumptions of fog tiers with queue stability guarantee.\n\t\\item[$\\diamond$] \\textbf{Algorithm Design:} \n\t\tThrough a non-trivial transformation, \n\t\twe decouple the problem into a series of subproblems over time slots. By exploiting their unique structures, we propose PORA, an efficient scheme that exploits predictive scheduling to make decisions in an online manner.\n\t\\item[$\\diamond$] \\textbf{Theoretical Analysis and Experimental Verification:} We conduct theoretical analysis and trace-driven simulations to evaluate the effectiveness of PORA. \n\t\tThe results show that PORA achieves a tunable power-latency tradeoff while effectively reducing the average latency with only mild-value of predictive information, even in the presence of prediction errors.\n\t\\item[$\\diamond$] \\textbf{New Degree of Freedom in the Design of Fog Computing Systems:} We systematically investigate the fundamental benefits of predictive offloading in fog computing systems, with both theoretical analysis and numerical evaluations.\n\\end{enumerate}\n\n\nWe organize the rest of the paper as follows. \nSection \\ref{sec: related work} discusses the related work. \nNext, in Section \\ref{sec: motivating example}, we provide an example that motivates our design for dynamic offloading and resource consumption in fog computing systems. \nSection \\ref{sec: model} presents the system model and problem formulation, followed by the algorithm design of PORA and performance analysis in Section \\ref{sec: algorithm}.\nSection \\ref{sec: simulation} analyzes the results from trace-driven simulations, while Section \\ref{sec: conclusion} concludes the paper.  \n\n\n\\section{Related Work}\\label{sec: related work}\n\nIn recent years, a series of works have been proposed to optimize the performance fog computing systems from various aspects \\cite{xiao2017qoe, taneja2017resource, chen2019dynamic, wang2019cooperative, liu2018socially, misra2019detour, lei2019joint, mao2016power, chen2019energy, gao2019dynamic, zhang2019near}.\nAmong such works, the most related are those focusing on the design of effective offloading schemes. \nFor example, \nby adopting alternating direction method of multipliers (ADMM) methods,\nXiao \\textit{et al.}\\cite{xiao2017qoe} and Wang \\textit{et al.}\\cite{wang2019cooperative} proposed two offloading schemes for cloud-aided fog computing systems to minimize average task duration and average service response time under different energy constraints, respectively. \nLater, Liu \\textit{et al.} \\cite{liu2018socially} took the social relationships among IoT users into consideration and developed a socially aware offloading scheme by advocating game theoretic approaches. \nMisra \\textit{et al.} \\cite{misra2019detour} studied the problem in software-defined fog computing systems and proposed a greedy heuristic scheme to conduct multi-hop task offloading with offloading path selection. \nLei \\textit{et al.} \\cite{lei2019joint} considered the joint minimization of delay and power consumption over all IoT devices; they formulated the problem under the settings of continuous-time Markov decision process and solved it via approximate dynamic programming techniques.\nThe above works, despite their effectiveness, generally assume the availability of the statistical information on task arrivals in the systems which is usually unattainable in practice with highly time-varying system dynamics \\cite{zhang2017resource}.\n\n\n\nIn the face of such uncertainties, a number of works have applied stochastic optimization methods such as Lyapunov optimization techniques to online and dynamic offloading scheme design \n\\cite{mao2016power, chen2019energy, gao2019dynamic, zhang2019near}. \nFor instance, Mao \\textit{et al.}\\cite{mao2016power} investigated the tradeoff between the power consumption and execution delay, then developed a dynamic offloading scheme for energy-harvesting-enabled IoT devices.\nChen \\textit{et al.} \\cite{chen2019energy} designed an adaptive and efficient offloading scheme to minimize the transmission energy consumption with queueing latency guarantee.\nGao \\textit{et al.} \\cite{gao2019dynamic} investigated efficient offloading and social-awareness-aided network resource allocation for device-to-device-enabled (D2D-enabled) IoT users. \nZhang \\textit{et al.} \\cite{zhang2019near} designed an online rewards-optimal scheme for the computation offloading of energy harvesting-enabled IoT devices based on Lyapunov optimization and Vickrey-Clarke-Groves auction.\nDifferent from such works that focus on fog computing systems with flat or two-tiered architectures, \nour solution is applicable to general multi-tiered fog computing systems with time-varying wireless channel states and unknown traffic statistics. \nMoreover, to the best of our knowledge, \nour solution is also the first to proactively leverage the predicted traffic information to optimize the system performance with theoretical guarantee. \nWe are also the first to investigate the fundamental benefits of predictive offloading in fog computing systems.\nWe compare our work with the above mentioned works \nin TABLE \\ref{table: related works}.\n\n\n \n\\section{Motivating Example}\\label{sec: motivating example}\n\nIn this section, we provide a motivating example to show the potential power-latency tradeoff in multi-tiered fog computing systems. \nThe objective is to achieve low power consumptions and short average workload latency (in the unit of packets). \n\nFigure \\ref{figure: motivating example} shows an instance of time-slotted fog computing system with two fog tiers, \\textit{i.e.}, edge fog tier and central fog tier. \nWithin each fog tier resides one fog node, \\textit{i.e.}, an edge fog node (EFN) in edge fog tier and a central fog node (CFN) in central fog tier. \nThe EFN connects to the CFN via a wireless link, while the CFN connects to the cloud data center over wired links. \nEach fog node maintains one queue to store packets. \nFigure \\ref{figure: motivating example}(a) shows that during time slot $t_{0}$, both the EFN and the CFN store $8$ packets in their queues. \n\nWe assume that each fog node sticks to one policy all the time to handle packets, \\textit{i.e.}, either \\textit{processing packets locally} or \\textit{offloading them to its next tier}.\nThe local processing capacities of EFN and CFN are $1$ and $8$ packets per time slot, respectively.\nThe transmission capacities from EFN to CFN and from CFN to cloud are $4$ and $5$ packets per time slot, respectively. \nThe power consumption is assumed linearly proportional to the number of processed\/transmitted packets. \nIn particular, processing one packet locally consumes $1$ mW power, while transmitting one packet over wireless link consumes $0.5$ mW. \nWe ignore the processing latency in the cloud due to its powerful processing capacity.  \n\nTABLE \\ref{table: motivating example} lists the total power consumptions and average packet latencies under all four possible settings. \nFigures \\ref{figure: motivating example}(b)-\\ref{figure: motivating example}(d) show the case when EFN sticks to offloading and CFN sticks to local processing. \nIn time slot $(t_{0}+1)$, EFN offloads four packets to CFN at its full transmission capacity, while CFN processes all the eight packets locally. \nIn time slot $(t_{0}+2)$, EFN offloads the rest four packets to CFN; meanwhile, CFN locally processes the four packets that arrive in previous time slot. \nIn time slot $(t_{0}+3)$, CFN finishes processing the rest four packets. \nIn this case, the system consumes $16$ mW power in local processing and $4$ mW power in transmission, with an average packet latency of $1.75$ time slots. \n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[scale=0.44]{figures1\/motivatingExample}\n\t\\caption{Motivating example of dynamic offloading and resource consumption in multi-tiered fog computing systems.}\n\t\\label{figure: motivating example}\n\\end{figure}\n\n\\begin{table}[!h]\n\\centering\n\\caption{Performance under different offloading policies}\n\\label{table: motivating example}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nPolicy of & Policy of & Total Power & Average Packet \\\\ \nEFN & CFN & Consumptions (mW) & Latency (time slot) \\\\ \\hline\nLocal & Local & 16 & 2.75 \\\\ \\hline\nLocal & Offload & 8 & 2.9375 \\\\ \\hline\nOffload & Local & 20 & 1.75 \\\\ \\hline\nOffload & Offload & 4 & 2.125 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nFrom TABLE \\ref{table: motivating example}, we conclude that: \nFirst, when EFN sticks to offloading and CFN sticks to local processing, the system achieves the lowest average packet latency of $1.75$ slots but the maximum power consumption of $20$mW. \nSecond, with the same offloading policy on EFN, there is a tradeoff between the total power consumptions and the average packet latency when CFN sticks to different policies. \nThe reason is that offloading to the cloud can not only reduce power consumptions but  also prolong latency as well. \nThird, when CFN sticks to local processing, there is a power-latency tradeoff with different policies at EFN, in that offloading to CFN can induce lower processing latency but at the cost of even higher power consumption for wireless transmissions.\n\n\\section{Model and Problem Formulation}\n\\label{sec: model}\n\nWe consider a multi-tiered fog computing system, as shown in Figure \\ref{figure: hierarchical fog system}.\nThe system evolves over time slots indexed by $t \\in \\{0, 1, 2,...\\}$.\nEach time slot has a length of $\\tau_{0}$.\nInside the edge fog tier (EFT) are a set of edge fog nodes (EFNs) that offer low-latency access to IoT devices.\nOn the other hand, the central fog tier (CFT) comprises of central fog nodes (CFNs) with greater processing capacities than EFNs.\nWe assume that the workload on each EFN can be offloaded to and processed by any of its accessible CFNs, and that each CFN can offload its workload to the cloud.\nIn our model, we do not consider the power consumptions and latencies within the cloud. We mainly focus on the power consumptions and latencies within fog tiers, as shown in TABLE \\ref{table: model}. First, the power consumptions we consider include two parts: processing power and transmit power. The processing power consumption is induced by the workload processing on both EFT and CFT. The transmit power is induced by the transmissions from EFT to CFT. We do not consider the transmit power consumption from CFT to cloud because we assume that the CFT communicates with the cloud through wireline connections. Second, the latencies we consider include three parts: queueing latency, processing latency and transmit latency. We focus on the queueing latency on both EFT and CFT. We assume that the workload processing in each time slot can be completed by the end of the same time slot, and then we can ignore the processing latency. Since the EFT communicates with the  CFT through high-speed  wireless connections and the  CFT communicates with the cloud through high-speed wireline connections, we assume that transmission latencies from both EFT to CFT  and CFT to Cloud are negligible.\n\n\\begin{table}[!h]\n\\centering\n\\caption{Performance Metrics in Our Model}\n\\label{table: model}\n\\begin{tabular}{|p{1.2cm}<{\\centering}|p{1.1cm}<{\\centering}|p{0.9cm}<{\\centering}|p{1.0cm}<{\\centering}|p{1.1cm}<{\\centering}|p{0.9cm}<{\\centering}|}\n\\hline\n\\multirow{2}{*}{} & \\multicolumn{2}{c|}{Power Consumption} & \\multicolumn{3}{c|}{Latency} \\\\ \\cline{2-6} \n& Processing & Transmit & Queueing & Processing & Transmit \\\\ \\hline\nEFT & \\checkmark &  & \\checkmark & &  \\\\ \\hline\nEFT2CFT &  & \\checkmark &  &  &  \\\\ \\hline\nCFT & \\checkmark &  & \\checkmark & &  \\\\ \\hline\nCFT2Cloud &  &  &  &  &  \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\nIn the following, we first introduce the basic settings in Section \\ref{subsec: basic setting}, then elaborate the queueing models in Section \\ref{subsec: local queue}. \nNext, we define the optimization objective in Section \\ref{subsec: power} while proposing the problem formulation in Section \\ref{subsec: formulation}.\nWe summarize the key notations in TABLE \\ref{table: key notations}.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[scale=0.35]{figures1\/edgeCentralFog}\n\t\\caption{An example of fog computing systems with two fog tiers.}\n\t\\label{figure: hierarchical fog system}\n\\end{figure}\n\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Key notations}\n\\label{table: key notations}\n\\centering\n\\begin{tabular}{p{0.89cm} l}\n    \\hline\\hline\n    Notation & Description  \\\\ \\hline\n    $\\tau_{0}$ & Length of each time slot  \\\\ \\hline\n    $\\mathcal{N}$ & $\\mathcal{N}$ is the set of EFNs with $|\\mathcal{N}|\\triangleq N$ \\\\ \\hline\n    $\\mathcal{M}$ & $\\mathcal{M}$ is the set of CFNs with $|\\mathcal{M}|\\triangleq M$  \\\\ \\hline\n    $\\mathcal{N}_{j}$ & Set of accessible EFNs from CFN $j$  \\\\ \\hline    \n    $\\mathcal{M}_{i}$ & Set of accessible CFNs from EFN $i$  \\\\ \\hline\n\t$A_{i}(t)$ & Amount of workload arriving to EFN $i$ in time slot $t$  \\\\ \\hline\n\t$\\lambda_{i}$ & Average workload arriving rate on EFN $i$, $\\lambda_{i}\\triangleq \\mathbb{E}\\{A_{i}(t)\\}$  \\\\ \\hline\n\t$W_{i}$ & Prediction window size of EFN $i$  \\\\ \\hline\n\t$A_{i,-1}(t)$ & Arrival queue backlog of EFN $i$ in time slot $t$  \\\\ \\hline\n\t\\multirow{2}*{$A_{i,w}(t)$} & Prediction queue backlog of EFN $i$ in time slot $t$, such that  \\\\ \n\t& $0\\leq w\\leq W_{i}-1$  \\\\ \\hline\n\t$Q_{i}^{(e,a)}(t)$ & Integrate queue backlog of EFN $i$ in time slot $t$  \\\\ \\hline\n\t$Q_{i}^{(e,l)}(t)$ & Local processing queue backlog of EFN $i$ in time slot $t$  \\\\ \\hline\n\t$Q_{i}^{(e,o)}(t)$ & Offloading queue backlog of EFN $i$ in time slot $t$  \\\\ \\hline\n\t$b_{i}^{(e,l)}(t)$ & Amount of workload to be sent to $Q_{i}^{(e,l)}(t)$ in time slot $t$ \\\\ \\hline\n\t$b_{i}^{(e,o)}(t)$ & Amount of workload to be sent to $Q_{i}^{(e,o)}(t)$ in time slot $t$ \\\\ \\hline\n\t$f_{i}^{(e)}(t)$ & CPU frequency of EFN $i$ in time slot $t$ \\\\ \\hline\n\t$H_{i,j}(t)$ & Wireless channel gain between EFN $i$ and CFN $j$ \\\\ \\hline\n\t$p_{i,j}(t)$ & Transmit power from EFN $i$ to CFN $j$ in time slot $t$ \\\\ \\hline\n\t$R_{i,j}(t)$ & Transmit rate from EFN $i$ to CFN $j$ in time slot $t$ \\\\ \\hline\n\t$Q_{j}^{(c,a)}(t)$ & Arrival queue backlog of CFN $j$ in time slot $t$  \\\\ \\hline\n\t$Q_{j}^{(c,l)}(t)$ & Local processing queue backlog of CFN $j$ in time slot $t$  \\\\ \\hline\n\t$Q_{j}^{(c,o)}(t)$ & Offloading queue backlog of CFN $j$ in time slot $t$  \\\\ \\hline\n\t$b_{j}^{(c,l)}(t)$ & Amount of workload to be sent to $Q_{j}^{(c,l)}(t)$ in time slot $t$ \\\\ \\hline\n\t$b_{j}^{(c,o)}(t)$ & Amount of workload to be sent to $Q_{j}^{(c,o)}(t)$ in time slot $t$ \\\\ \\hline\n\t$f_{j}^{(c)}(t)$ & CPU frequency of CFN $j$ in time slot $t$ \\\\ \\hline\n\t$P(t)$ & Total power consumptions in time slot $t$ \\\\ \\hline\n\t\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Basic Settings}\\label{subsec: basic setting}\n\nThe fog computing system consists of $N$ EFNs in EFT and $M$ CFNs in CFT. \nLet $\\mathcal{N}$ and $\\mathcal{M}$ be the sets of EFNs and CFNs.\nEach EFN $i$ has access to a subset of CFNs in their proximities. We denote the subset by $\\mathcal{M}_{i}\\subset\\mathcal{M}$. \nFor each CFN $j$, $\\mathcal{N}_{j}\\subset\\mathcal{N}$ denotes the set of its accessible EFNs. \nAccordingly, for any $i\\in\\mathcal{N}_{j}$ we have $j\\in\\mathcal{M}_{i}$. \n\n\\subsection{Queueing Model for Edge Fog Node}\\label{subsec: queue for EFN}\n\nDuring time slot $t$, there is an amount $A_{i}(t)$ ($\\le A_{\\text{max}}$ for some constant $A_{\\text{max}}$) of workload generated from IoT devices arrive to be processed on EFN $i$ such that $\\mathbb{E}\\{ A_{i}(t)\\} =\\lambda_{i}$.\nWe assume that such arrivals are independent over time slots and different EFNs. \nEach EFN $i$ is equipped with a learning module\\footnote{We do not specify any particular learning method in this paper, since our work aims to explore the \\textit{fundamental} benefits of predictive offloading. \nIn practice, one can leverage machine learning techniques such as time-series prediction methods \\cite{ahmed2010empirical} for workload arrival prediction.}\nthat can predict the future workload within a \\textit{prediction window} of size $W_{i}$, \\textit{i.e.} workload will arrive in the next $W_{i}$ time slots. The predicted arrivals are pre-generated and recorded, then arrive to EFN $i$ for pre-serving. Once the predicted arrivals actually arrive after pre-serving, they will be considered finished.\n\nOn each EFN, as Figure \\ref{figure: queue} shows, there are four types of queues:\nprediction queues with the backlogs as $A_{i,0}(t)$, ..., $A_{i,W_{i}-1}(t)$, \narrival queue $A_{i,-1}(t)$, \nlocal processing queue $Q_{i}^{(e,l)}(t)$, and offloading queue $Q_{i}^{(e,o)}(t)$.\nIn time slot $t$, prediction queue $A_{i,w}(t)$ ($0\\leq w\\leq W_{i}-1$) stores untreated workload that will arrive in time slot $(t+w)$. \nWorkload that actually arrives at EFN $i$ is stored in the arrival queue $A_{i,-1}(t)$, \nawaiting being forwarded to the local processing queue $Q_{i}^{(e,l)}(t)$ or the offloading queue $Q_{i}^{(e,o)}(t)$. \nWorkload in $Q_{i}^{(e,l)}(t)$ will be processed locally by EFN $i$, \nwhile workload in $Q_{i}^{(e,o)}(t)$ will be offloaded to CFNs in set $\\mathcal{M}_{i}$.\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[scale=0.3]{figures1\/model}\n\t\\caption{Queueing model of the system.}\n\t\\label{figure: queue}\n\\end{figure}\n\n\\subsubsection{Prediction Queues and Arrival Queues in EFNs}\n\nWithin each time slot $t$, in addition to the current arrivals in the arrival queue, EFN $i$ can also forward future arrivals in the prediction queues.\nWe define $\\mu_{i,w}(t)$ as the amount of output workload from $A_{i,w}(t)$, for $w\\in\\{-1,0,...,W_{i}-1\\}$.\nSuch workload should be distributed to the local processing queue and offloading queue.\nWe denote the amounts of workloads to be distributed to the local processing queue and offloading queue as $b_{i}^{(e,l)}(t)$ and $b_{i}^{(e,o)}(t)$, respectively, such that\n\\begin{equation}\n\t0\\leq b^{(e,\\beta)}_{i}\\left(t\\right)\\leq b^{(e,\\beta)}_{i,\\text{max}},\\ \\forall \\beta\\in\\{l,o\\} \\label{constraint: EFN offloading decision 1}\n\\end{equation}\nwhere each $b^{(e,\\beta)}_{i,\\text{max}}$ is a positive constant.\nAs a result, we have\n\\begin{equation}\\label{constraint: pre-serve rates}\n\t\\sum_{w=-1}^{W_{i}-1}\\mu_{i,w}\\left(t\\right)= b^{(e,l)}_{i}\\left(t\\right)+b^{(e,o)}_{i}\\left(t\\right).\n\\end{equation}\n\nNext, we consider the queueing dynamics for different types of queues in EFN, respectively.\n\nRegarding $A_{i,w}(t)$, it is updated whenever pre-service is finished and the lookahead window moves one slot ahead at the end of each time slot. \nTherefore, we have\n\\begin{enumerate}[(i)]\n\t\\item If $w=W_{i}-1$, then\n\t\\begin{equation}\\label{update: fog node prediction queue update 1}\n\t\tA_{i,W_{i}-1}\\left(t+1\\right)=A_{i}\\left(t+W_{i}\\right).\n\t\\end{equation}\n\t\\item If $0\\leq w\\leq W_{i}-2$, then\n\t\\begin{equation}\\label{update: fog node prediction queue update 2}\n\t\tA_{i,w}(t+1)=[A_{i,w+1}(t)-\\mu_{i,w+1}(t)]^{+},\n\t\\end{equation}\n\\end{enumerate}\nwhere $[x]^{+}\\triangleq \\max\\{x,0\\}$ for $x \\in \\mathbb{R}$. \nIn time slot $(t+1)$, the amount of workload that will arrive after $(W_{i}-1)$ time slots is $A_{i}(t+W_{i})$ and it remains unknown until time slot $(t+1)$. \n\nRegarding the arrival queue $A_{i,-1}(t)$, it records the actual backlog of EFN $i$ with the update equation as follows:\n\\begin{equation}\\label{update: fog node true queue update}\n\tA_{i,-1}(t+1)\\!=\\![A_{i,-1}(t)-\\mu_{i,-1}(t)]^{+}\\!+\\![A_{i,0}(t)-\\mu_{i,0}(t)]^{+}.\n\\end{equation}\nNote that $\\mu_{i,-1}(t)$ denotes the amount of distributed workload that have already being in $A_{i,-1}(t)$. \n\nNext, we introduce an integrate queue with a backlog size as the sum of all prediction queues and the arrival queue on EFN $i$, denoted by $Q_{i}^{\\left(e,a\\right)}\\left(t\\right)\\triangleq \\sum_{w=-1}^{W_{i}-1}A_{i,w}\\left(t\\right)$ . \nUnder \\textit{fully-efficient} \\cite{huang2016predictive} service policy,  $Q_{i}^{\\left(e,a\\right)}\\left(t\\right)$ is updated as\n\\begin{multline}\\label{update: prediction sum queue update}\n\tQ_{i}^{\\left(e,a\\right)}\\left(t+1\\right)=[Q_{i}^{\\left(e,a\\right)}\\left(t\\right)-(b^{(e,l)}_{i}\\left(t\\right)+b^{(e,o)}_{i}\\left(t\\right))]^{+}\\\\\n\t+A_{i}\\left(t+W_{i}\\right).\n\\end{multline}\nThe input of integrate queue $Q_{i}^{(e,a)}(t)$ consists of the predicted workload that will arrive at EFN $i$ in time slot $(t+W_{i})$, while its output consists of workloads being forwarded to the local processing queue and the offloading queue. \nNote that $b^{(e,l)}_{i}(t)+b^{(e,o)}_{i}(t)$ is the output capacity of integrate queue $Q_{i}^{(e,a)}(t)$ in time slot $t$. If the capacity is larger than the queue backlog size, the true output amount will be smaller than $b^{(e,l)}_{i}(t)+b^{(e,o)}_{i}(t)$.\n \n\n\\subsubsection{Offloading Queues in EFNs}\n\nIn time slot $t$, workload in queue $Q^{(e,o)}_{i}(t)$ will be offloaded to CFNs in set $\\mathcal{M}_{i}$. \nThe transmission capacities are determined by the transmit power decisions $(p_{i,j}(t))_{j\\in\\mathcal{M}_{i}}$, where $p_{i,j}(t)$ is the transmit power from EFN $i$ to CFN $j$. \nThe transmit power is nonnegative and the total transmit power of each EFN is upper bounded, \\textit{i.e.}, \n\\begin{align}\n\t&p_{i,j}\\left(t\\right)\\geq0,\\ \\forall i\\in\\mathcal{N},j\\in\\mathcal{M}_{i}\\text{ and }t, \\label{constraint: power allocation 1} \\\\\n\t&\\sum_{j\\in\\mathcal{M}_{i} }p_{i,j}\\left(t\\right)\\leq p_{i,\\text{max}},\\ \\forall i\\in\\mathcal{N}\\text{ and }t.\\label{constraint: power allocation 2}\n\\end{align}\nAccording to Shannon's capacity formula \\cite{gallager2008principles}, the transmission capacity from EFN $i$ to CFN $j$ is\n\\begin{equation}\\label{equation: offload rate to central fog}\n\tR_{i,j}(t)\\!\\triangleq\\!\\hat{R}_{i,j}(p_{i,j}(t))\\!=\\!\\tau_{0}B\\log_{2}\\left(1\\!+\\!\\frac{p_{i,j}(t)H_{i,j}(t)}{N_{0}B}\\right),\n\\end{equation}\nwhere $\\tau_{0}$ is the length of each time slot, $B$ is the channel bandwidth, $H_{i,j}(t)$ is the wireless channel gain between EFN $i$ and CFN $j$, and $N_{0}$ is the system power spectral density of the additive white Gaussian noise.\nNote that $H_{i,j}(t)$ is an uncontrollable environment state with positive upper bound $H_{\\text{max}}$.\nWe do not consider the interferences among fog nodes and tiers. \nBy adjusting the transmit power $p_{i,j}(t)$, we can offload different amounts of workload from EFN $i$ to CFN $j$ in time slot $t$. \nAccordingly, the update equation of offloading queue $Q^{(e,o)}_{i}(t)$ is\n\\begin{equation}\\label{update: edge offloading queue update}\n\tQ^{(e,o)}_{i}\\left(t+1\\right)\\leq [Q^{(e,o)}_{i}\\left(t\\right)\\!-\\!\\sum_{j\\in\\mathcal{M}_{i}}R_{i,j}\\left(t\\right)]^{+}+b^{(e,o)}_{i}\\left(t\\right),\n\\end{equation}\nwhere $\\sum_{j \\in \\mathcal{M}_{i}} R_{i,j}(t)$ is the total transmission capacity to EFN $i$ in time slot $t$.\nThe inequality here means that the actual arrival of $Q_{i}^{(e,o)}(t)$ may be less than $b_{i}^{(e,o)}(t)$, because $b^{(e,o)}_{i}(t)$ is the transmission capacity from integrate queue $Q_{i}^{(e,a)}(t)$ to offloading queue $Q_{i}^{(e,o)}(t)$ instead of the amount of truly transmitted workload. \nRecall that we assume the transmission latency from EFT to CFT is negligible compared to the length of each time slot, the workload transmission in each time slot can be accomplished by the end of that time slot.\n\n\\subsection{Queueing Model for Central Fog Node}\\label{subsec: queue for CFN}\n\nFigure \\ref{figure: queue} also shows the queueing model on CFN.\nEach CFN $j\\in\\mathcal{M}$ maintains three queues: \nan arrival queue $Q_{j}^{(c,a)}(t)$, a local processing queue $Q_{j}^{(c,l)}(t)$, and an offloading queue $Q_{j}^{(c,o)}(t)$. Similar to EFNs, workload offloaded from the EFT will be firstly stored in the arrival queue,\nthen distributed to $Q_{j}^{(c,l)}(t)$ for local processing and to $Q_{j}^{(c,o)}(t)$ for further offloading.\n\n\n\\subsubsection{Arrival Queues in CFNs}\n\nThe arrivals on CFN $j$ consist of \nworkloads offloaded from EFNs in the set $\\mathcal{N}_{j}$. \nWe denote the amounts of workloads distributed to the local processing queue and offloading queue in time slot $t$ as $b_{j}^{(c,l)}(t)$ and $b_{j}^{(c,o)}(t)$, respectively, such that\n\\begin{equation}\n\t0\\leq b^{(c,\\beta)}_{j}\\left(t\\right)\\leq b^{(c,\\beta)}_{j,\\text{max}},\\ \\forall \\beta\\in\\{l,o\\}, \\label{constraint: CFN offloading decision 1} \n\\end{equation}\nwhere each $b^{(c,\\beta)}_{j,\\text{max}}$ is a positive constant. Accordingly, $Q_{j}^{(c,a)}(t)$ is updated as follows:\n\\begin{multline}\\label{update: central arrival queue update}\n\tQ_{j}^{(c,a)}(t+1)\\\\\n\t\\leq [Q_{j}^{(c,a)}(t)-(b_{j}^{(c,l)}(t)+b_{j}^{(c,o)}(t))]^{+}\\!+\\!\\sum_{i\\in\\mathcal{N}_{j}}R_{i,j}(t).\n\\end{multline}\n\n\\subsubsection{Offloading Queues in CFNs}\n\nFor each CFN $j\\in\\mathcal{M}$, its offloading queue $Q^{(c,o)}_{j}(t)$ stores the workload to be offloaded to the cloud. \nWe define $D_{j}(t)$ as the transmission capacity of the wired link from CFN $j$ to the cloud during time slot $t$, \nwhich depends on the network state and is upper bounded by some constant $D_{\\text{max}}$ for all $j$ and $t$. Then we have the following update function for $Q^{(c,o)}_{j}(t)$:\n\\begin{equation}\\label{update: central offloading queue update}\n\tQ^{(c,o)}_{j}\\left(t+1\\right)\\leq [Q^{(c,o)}_{j}\\left(t\\right)-D_{j}\\left(t\\right)]^{+}+b^{(c,o)}_{j}\\left(t\\right).\n\\end{equation}\nNote that the amount of actually offloaded workload to the cloud is $\\min\\{Q_{j}^{(c,o)}(t),D_{j}(t)\\}$.\n\n\\subsection{Local Processing Queues on EFNs and CFNs}\\label{subsec: local queue}\n\nWe assume that all fog nodes are able to adjust their CPU frequencies in each time slot, by applying \\textit{dynamic voltage and frequency scaling} (DVFS) techniques\\cite{mao2017survey}. \nNext, we define $L_{k}^{(\\alpha)}$ as the number of CPU cycles that fog node $k\\in\\mathcal{N}\\cup\\mathcal{M}$ requires to process one bit of workload, where $\\alpha$ is an indicator of fog node $k$'s type ($\\alpha=e$ if $k$ is an EFN, and $\\alpha=c$ if $k$ is a CFN). \n$L_{k}^{(\\alpha)}$ is assumed constant and can be measured offline \\cite{miettinen2010energy}. \nTherefore, the local processing capacity of fog node $k$ is $f_{k}^{(\\alpha)}(t)\/L_{k}^{(\\alpha)}$.\nThe local processing queue on fog node $k$ evolves as follows:\n\\begin{equation}\\label{update: local processing queue update}\n\tQ_{k}^{(\\alpha,l)}(t+1)\\!\\leq \\![Q_{k}^{\\left(\\alpha,l\\right)}\\left(t\\right)\\!-\\tau_{0}f_{k}^{\\left(\\alpha\\right)}\\left(t\\right)\\!\/\\!L_{k}^{\\left(\\alpha\\right)}]^{+}\\!+b_{k}^{\\left(\\alpha,l\\right)}\\left(t\\right).\n\\end{equation}\nAll CPU frequencies are nonnegative and finite:\n\\begin{equation}\\label{constraint: CPU frequency}\n\t0\\leq f_{k}^{(\\alpha)}\\left(t\\right)\\leq f^{(\\alpha)}_{k,\\text{max}},\\ \\forall k\\in\\mathcal{N}\\cup\\mathcal{M}\\text{ and }t,\n\\end{equation}\nwhere each $f^{(\\alpha)}_{k,\\text{max}}$ is a positive constant. \n\n\\subsection{Power Consumptions}\\label{subsec: power}\n\nThe total power consumptions $P(t)$ of fog tiers in time slot $t$\nconsist of the processing power consumption and wireless transmit power consumption. \nGiven a local CPU with frequency $f$, its power consumption per time slot is  $\\tau_{0}\\varsigma f^{3}$, where $\\varsigma$ is a parameter depending on the deployed hardware and is measurable in practice \\cite{kim2018dual}. \nThus $P(t)$ is defined as follows:\n\\begin{multline}\\label{equation: total fog power consumptions}\n\tP\\left(t\\right)\\triangleq\\hat{P}\\left(\\boldsymbol{f}\\left(t\\right),\\boldsymbol{p}\\left(t\\right)\\right)=\\sum_{i\\in\\mathcal{N}}\\tau_{0}\\varsigma(f_{i}^{\\left(e\\right)}\\left(t\\right))^{3}\\\\\n\t+\\sum_{j\\in\\mathcal{M}}\\tau_{0}\\varsigma(f_{j}^{\\left(c\\right)}\\left(t\\right))^{3}+\\sum_{i\\in\\mathcal{N}}\\sum_{j\\in\\mathcal{M}_{i}} \\tau_{0}p_{i,j}\\left(t\\right),\n\\end{multline}\nwhere $\\boldsymbol{f}(t)\\triangleq((f_{i}^{(e)}(t))_{i\\in\\mathcal{N}},(f_{j}^{(c)}(t))_{j\\in\\mathcal{M}})$ is the vector of all CPU frequencies, and $\\boldsymbol{p}(t)\\triangleq(\\boldsymbol{p}_{i}(t))_{i\\in\\mathcal{N}}$ in which $\\boldsymbol{p}_{i}(t)=(p_{i,j}(t))_{j\\in\\mathcal{M}_{i}}$ is the transmit power allocation of EFN $i$. \n\n\n\\subsection{Problem Formulation}\\label{subsec: formulation}\n\nWe define the long-term time-average expectation of total power consumptions $\\bar{P}\n$ and total queue backlog $\\bar{Q}$ as follows:\n\\begin{align}\n\t&~~~~~~~~~~~~~~\\bar{P}\\triangleq\\limsup_{T\\rightarrow\\infty}\\frac{1}{T}\\sum_{t=0}^{T-1}\\mathbb{E}\\left\\{ P\\left(t\\right)\\right\\}, \\label{definition: time average exp power}\\\\\n\t&\\bar{Q}\\triangleq\\limsup_{T\\rightarrow\\infty}\\frac{1}{T}\\sum_{t=0}^{T-1}\\sum_{\\beta\\in\\{ a,l,o\\} }(\\sum_{i\\in\\mathcal{N}}\\mathbb{E}\\{ Q_{i}^{\\left(e,\\beta\\right)}(t)\\}\\nonumber \\\\\n\t&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\\sum_{j\\in\\mathcal{M}}\\mathbb{E}\\{ Q_{j}^{(c,\\beta)}(t)\\}) \\label{definition: time average exp backlog}.\n\\end{align}\nIn this paper, we aim to minimize the long-term time-average expectation of total power consumptions $\\bar{P}$, while ensuring the stability of all queues in the system, \\textit{i.e.}, $\\bar{Q}<\\infty$. \nThe problem formulation is given by\n\\begin{equation}\\label{problem: general}\n\\begin{array}{cl}\n\\underset{\\{\\boldsymbol{b}(t),\\boldsymbol{f}(t),\\boldsymbol{p}(t)\\}_{t}}{\\text{Minimize}}\n\t&\\displaystyle \\bar{P}\\\\\n\t\\text{Subject to}&\\displaystyle (\\ref{constraint: EFN offloading decision 1})\n\t(\\ref{constraint: power allocation 1})\n\t(\\ref{constraint: power allocation 2})(\\ref{constraint: CFN offloading decision 1})(\\ref{constraint: CPU frequency}),\\\\\n\t&\\displaystyle \\bar{Q}<\\infty.\n\\end{array}\n\\end{equation}\n\n\n\\section{Algorithm Design}\\label{sec: algorithm}\n\n\\subsection{Predictive Algorithm}\n\nTo solve problem (\\ref{problem: general}), we adopt Lyapunov optimization techniques\\cite{huang2016predictive}\\cite{neely2010stochastic} to decouple the problem into a series of subproblems over time slots. We show the detail of this process in Appendix \\ref{appendix: pora design}. \nBy solving each of these subproblems during each time slot, \nwe propose PORA, an efficient and predictive scheme conducts workload offloading in an online and distributed manner. \nWe show the pseudocode of PORA in Algorithm \\ref{algorithm: pora}. \nNote that symbol $\\alpha \\in \\{ e, c \\}$ indicates the type of fog node. \nSpecifically, for each fog node $k$, $\\alpha=e$ if $k$ is an EFN and CFN otherwise.\n\\begin{algorithm}[t]\n\\caption{Predictive Offloading and Resource Allocation (PORA) in One Time Slot}\n\\label{algorithm: pora}\n\\begin{algorithmic}[1]\n  \\STATE Initialize $\\boldsymbol{b}(t)\\leftarrow \\boldsymbol{0}$, $\\boldsymbol{f}(t)\\leftarrow\\boldsymbol{0}$, $\\boldsymbol{p}(t)\\leftarrow\\boldsymbol{0}$.\n  \\FOR {each fog node $k\\in\\mathcal{N}\\cup\\mathcal{M}$}\n    \\STATE \\%\\%\\textit{Make Offloading Decisions}\n    \\IF {$Q_{k}^{(\\alpha,a)}(t)> Q_{k}^{(\\alpha,l)}(t)$}\n      \\STATE Set $b_{k}^{(\\alpha,l)}(t)\\leftarrow b^{(\\alpha,l)}_{k,\\text{max}}$.\n    \\ENDIF\n    \\IF {$Q_{k}^{(\\alpha,a)}(t)> Q_{k}^{(\\alpha,o)}(t)$}\n      \\STATE Set $b_{k}^{(\\alpha,o)}(t)\\leftarrow b^{(\\alpha,o)}_{k,\\text{max}}$.   \n    \\ENDIF\n    \\STATE \\%\\%\\textit{Local CPU Resource Allocation}\n    \\STATE Set $ f^{(\\alpha)}_{k}\\left(t\\right)\\leftarrow\\min\\{ \\sqrt{Q_{k}^{(\\alpha,l)}(t)\/3V\\varsigma L_{k}^{(\\alpha)}},f^{(\\alpha)}_{k,\\text{max}}\\}$.\n  \\ENDFOR\n  \\STATE \\%\\%\\textit{Transmit Power Allocation}\n  \\FOR {each EFN $i\\in\\mathcal{N}$}\n    \\STATE Set $\\lambda_{\\text{min}}\\leftarrow 0$.\n    \\STATE Set $\\lambda_{\\text{max}}\\leftarrow \\max_{j\\in\\mathcal{M}_{i}}\\frac{(Q_{i}^{(e,o)}-Q_{j}^{(c,a)})H_{i,j}(t)}{N_{0}}-V$.\n\\WHILE{$\\lambda_{\\text{max}}-\\lambda_{\\text{min}}> \\varepsilon$}\n  \\STATE \\%\\%\\textit{Water Filling with Bisection Method}\n  \\STATE Set $\\lambda^{*}\\leftarrow (\\lambda_{\\text{min}}+\\lambda_{\\text{max}})\/2$ \n  \\STATE Set $p_{i,j}(t)\\!\\leftarrow\\!B\\left[\\frac{Q_{i}^{(e,o)}(t)-Q_{j}^{(c,a)}(t)}{V+\\lambda^{*}}-\\frac{N_{0}}{H_{i,j}(t)}\\right]^{+}$.\n  \\IF {$\\sum_{j\\in\\mathcal{M}_{i}}p_{i,j}(t)> p_{i,\\text{max}}$}\n      \\STATE Set $\\lambda_{\\text{max}}\\leftarrow\\lambda^{*}$.\n  \\ELSE\n      \\STATE Set $\\lambda_{\\text{min}}\\leftarrow\\lambda^{*}$.\n  \\ENDIF\n\\ENDWHILE\n\\ENDFOR\n  \\STATE Enforce scheduling decisions $\\boldsymbol{b}(t)$, $\\boldsymbol{f}(t)$, and $\\boldsymbol{p}(t)$.\n\\end{algorithmic}\n\\end{algorithm}\nNext, we introduce PORA in detail.\n\n\n\\subsubsection{Offloading Decision}\n\nIn each time slot $t$, under PORA, each fog node $k\\in\\mathcal{N}\\cup\\mathcal{M}$ decides the amounts of workload scheduled to the offloading queue and the local processing queue, denoted by $b_{k}^{(\\alpha,l)}(t)$ and $b_{k}^{(\\alpha,o)}(t)$, respectively. \nSuch decisions are obtained by solving the following problem:\n\\begin{equation}\\label{problem: offloading decision}\n\\begin{array}{cl}\n\\underset{0\\leq b_{k}^{(\\alpha,\\beta)}\\leq b^{(\\alpha,\\beta)}_{k,\\text{max}}}{\\text{Minimize}}\n\t\\displaystyle \\left(Q^{(\\alpha,\\beta)}_{k}\\left(t\\right)-Q_{k}^{(\\alpha,a)}\\left(t\\right)\\right)b_{k}^{(\\alpha,\\beta)},\n\\end{array}\n\\end{equation}\nwhere $\\beta\\in\\{l,o\\}$. \nAccordingly, the optimal solution to (\\ref{problem: offloading decision}) is\n\\begin{equation}\\label{equation: optimal offload decision}\n\tb_{k}^{(\\alpha,\\beta)}\\left(t\\right)=\\begin{cases}\nb^{(\\alpha,\\beta)}_{k,\\text{max}}, & \\text{if }Q_{k}^{(\\alpha,\\beta)}\\left(t\\right)<Q_{k}^{(\\alpha,a)}(t),\\\\\n0, & \\text{otherwise}.\n\\end{cases}\n\\end{equation}\nFrom (\\ref{equation: optimal offload decision}), \nwe see that, to determine the optimal solutions $b_{i}^{(e,l)}(t)$ and $b_{i}^{(e,o)}(t)$, \neach EFN $i$ would compare its integrate queue backlog size $Q_{i}^{(e,a)}(t)$ with its local processing queue backlog size $Q_{i}^{(e,l)}(t)$ and offloading queue backlog size $Q_{i}^{(e,o)}(t)$, respectively.\nParticularly, \nif there is too much workload in its integrate queue compared to its local queue ($Q_{i}^{(e,l)}\\left(t\\right)<Q_{i}^{(e,a)}(t)$),\nthen it will offload as much workload (up to $b^{(e,l)}_{i,\\text{max}}$) as possible to its local queue.  \nLikewise, if its integrate queue is loaded with more workload than its offloading queue ($Q_{i}^{(e,o)}\\left(t\\right)<Q_{i}^{(e,a)}(t)$), it will offload up to an amount of $b^{(e,o)}_{i,\\text{max}}$ workload to its offloading queue. \n\nNotably, if the backlog size of the EFN $i$'s integrate queue is larger than both its local queue and offloading queue, then the EFN will transmit the workload one by one unit (\\textit{e.g.} packets); \neach unit of workload is either sent to the EFN $i$'s local queue or its offloading queue, such that the amounts of workload distributed to such two queues are no greater than $b_{i,\\text{max}}^{(e,l)}$ and $b_{i,\\text{max}}^{(e,o)}$, respectively.\nIn practice, the workload distributing strategy is left as a degree of freedom to be specified in the implementation of PORA. \nIn our simulation, we adopt the following distributing strategy.\nWhen an EFN $i$'s integrate queue backlog size is greater than both its local queue and its offloading backlog size, then it will transmit workload to its local queue until the amount of transmitted workload reaches $b_{i,\\text{max}}^{(e,l)}$. \nThen the rest workload in the integrate queue is transmitted to the offloading queue until the amount of distributed workload reaches $b_{i,\\text{max}}^{(e,o)}$. \nSuch a process terminates whenever the integrate queue becomes empty.\n\nThe decision making process is similar for CFNs.\nSpecifically, each CFN $j$ determines $b_{j}^{(c,l)}(t)$ and $b_{j}^{(c,o)}(t)$ by comparing its arrival queue backlog size $Q_{j}^{(c,a)}(t)$ with its local processing queue backlog size $Q_{j}^{(c,l)}(t)$ and offloading queue backlog size $Q_{j}^{(c,o)}(t)$, respectively. \n\n \n\\textbf{Remark:}\nFor each EFN, we can view the difference between the backlog sizes of its integrate queue and its local processing\/offloading queue as its willingness of workload transmission. \nIf such willingness is positive, then the EFN will transmit as much workload as possible from its integrate queue; \notherwise, the EFN will leave the workload not distributed in the current time slot. \nIn such a way, PORA always endeavors to balance the integrate queue backlog and the local\/offloading queue backlog.\nLikewise, under PORA, each CFN determines its offloading decisions upon the difference between the backlog sizes of its arrival queue and its local processing\/offloading queue to ensure the queue stability.\n\n\\subsubsection{Local CPU Frequency Allocation}\nUnder PORA, in each time slot $t$, each fog node $k\\in\\mathcal{N}\\cup\\mathcal{M}$ sets its local CPU frequency $f_{k}^{(\\alpha)}(t)$ by solving the following subproblem:\n\\begin{equation}\\label{problem: CPU frequency}\n\\begin{array}{cl}\n\\underset{0\\leq f_{k}^{(\\alpha)}\\leq f^{(\\alpha)}_{k,\\text{max}}}{\\text{Minimize}}\n\t\\displaystyle V\\varsigma(f_{k}^{(\\alpha)})^{3}-Q_{k}^{\\left(\\alpha,l\\right)}\\left(t\\right)f_{k}^{(\\alpha)}\/L_{k}^{(\\alpha)}.\n\\end{array}\n\\end{equation}\nBy setting the second derivative of the objective function in (\\ref{problem: CPU frequency}) to zero, \nwe can obtain the optimal CPU frequency $f_{k}^{(\\alpha)}(t)$ to be set by fog node $k$ as\n\\begin{equation}\\label{equation: optimal CPU frequency}\nf_{k}^{(\\alpha)}(t)=\\min\\left\\{\\sqrt{Q_{k}^{\\left(\\alpha,l\\right)}\\left(t\\right)\/3V\\varsigma L_{k}^{(\\alpha)}},f_{k,\\text{max}}^{(\\alpha)}\\right\\}.\n\\end{equation}\nWe prove the optimality of (\\ref{equation: optimal CPU frequency}) in Appendix \\ref{proof: CPU frequency}.\n\n\\textbf{Remark:}\nWhen $f^{(\\alpha)}_{k}(t) < f_{k,\\text{max}}^{(\\alpha)}$, the allocated CPU frequency $f_{k}^{(\\alpha)}(t)$ is proportional to the square root of the backlog size of local processing queue $Q_{k}^{(\\alpha,l)}(t)$ and the inverse of the value of parameter $V$. \nThis shows that, on the one hand, PORA would allocate as much CPU frequency as possible to process the workload in the queues. \nOn the other hand, the value of parameter $V$ determines the tradeoff between power consumption and the backlog sizes of queues: a small value of $V$ will encourage the fog node to allocate more CPU frequency to process the workload and hence a small queue backlog size; in contrast, a large value of $V$ will make the fog node more conservative to allocate resources, leading to less power consumptions but a large queue backlog size as well. \nIn practice, the choice of the value of $V$ is dependent on the system design objective. \n\n\\subsubsection{Power Allocations for EFNs}\nIn each time slot $t$, under PORA, each EFN $i\\in\\mathcal{N}$ determines its allocated transmit power $\\boldsymbol{p}_{i}(t)$ by solving the following optimization problem.\n\\begin{equation}\\label{problem: power allocation}\n\\begin{split}\n\t\\underset{\\boldsymbol{p}_{i}}{\\text{Minimize}}\n\t&\\displaystyle\\sum_{j\\in\\mathcal{M}_{i} }\\Big[Vp_{i,j}-m_{i,j}(t)\\log_{2}(1+l_{i,j}(t)p_{i,j})\\Big]\\\\\n\t\\text{Subject to}&\\displaystyle\\sum_{j\\in\\mathcal{M}_{i}}p_{i,j}\\leq p_{i,\\text{max}},\\\\\n\t &\\displaystyle\\ p_{i,j}\\geq 0,\\ \\forall j\\in\\mathcal{M}_{i},\n\\end{split}\n\\end{equation}\nwhere $m_{i,j}(t)\\triangleq (Q_{i}^{(e,o)}(t)-Q_{j}^{(c,a)}(t))B$ and $l_{i,j}(t)\\triangleq\\frac{H_{i,j}(t)}{N_{0}B}$.\nBy applying \\textit{water-filling} algorithm\\cite{boyd2004convex}, we obtain the optimal solution to problem (\\ref{problem: power allocation}) as\n\\begin{equation}\\label{equation: Pij*}\n\tp_{i,j}(t)=[m_{i,j}(t)\/(V+\\lambda^{*})-1\/l_{i,j}(t)]^{+},\\ \\forall j\\in\\mathcal{M}_{i},\n\\end{equation}\nwhere $\\lambda^{*}$ is the optimal Lagrangian variable that satisfies\n\\begin{equation}\\label{equation: lambda}\n\t\\sum_{j\\in\\mathcal{M}_{i}}[m_{i,j}\\left(t\\right)\/(V+\\lambda^{*})-1\/l_{i,j}\\left(t\\right)]^{+}=p_{i,\\text{max}}.\n\\end{equation} \nThe optimality of such solutions is proven in Appendix \\ref{proof: transmit power}. \nWe adopt bisection method (line 15-25 in Algorithm \\ref{algorithm: pora}) to obtain the value of $\\lambda^{*}$ with its lower and upper bounds as $\\lambda_{\\text{min}}$ and $\\lambda_{\\text{max}}$, respectively. \nNote that the value of $\\lambda^{*}$ converges asymptotically to the optimum $\\lambda^{\\text{opt}}$ as the tolerance parameter $\\varepsilon$ approaches zero, such that $|\\lambda^{*}-\\lambda^{\\text{opt}}|\\leq\\varepsilon\/2$.\n\n\\textbf{Remark:}\nPORA tends to allocate more transmit power to the CFN with smaller arrival queue backlog size $Q_{j}^{(c,a)}(t)$ for load balancing. When $Q_{j}^{(c,a)}(t)\\geq Q_{i}^{(e,o)}(t)$, we have $m_{i,j}(t)\\leq 0$ and $p_{i,j}(t)=0$, \\textit{i.e.}, EFN $i$ allocates no transmit power to CFN $j$ unless the backlog size of the arrival queue on CFN $j$ is greater than that of the offloading queue on EFN $i$.\nBy increasing the value of $V$, transmit power consumption will be reduced but the backlog size will increase as well.\n\n\n\n\\subsection{Computational Complexity of PORA}\n\nDuring each time slot, part of the computational complexity concentrates on the calculation for CPU frequency settings and offloading decision makings. \nSince the calculation (line 3-11) requires only constant time for each fog node, the total complexity of these steps is $O(N+M)$. \nNext, each EFN $i$ applies the bisection method (line 15-26) to calculate the optimal dual variable, with a complexity of $O(\\log_{2}((\\lambda_{\\text{max}}-\\lambda_{\\text{min}}\/\\varepsilon)+|\\mathcal{M}_{i}|)$. \nAfter that, EFN $i$ determines the transmit power to each CFN in the set $\\mathcal{M}_{i}$. In the worst case, each EFN is potentially connected to all CFNs, thus the total complexity of PORA algorithm is $O(M \\times N )$.\n\n\n\n\\subsection{Performance Analysis}\n\nWe conduct theoretical analysis on the relationship between the average power consumption $\\bar{P}$ and queue backlog $\\bar{Q}$ under PORA scheme in the non-predictive case ($W_{i}=0,\\ \\forall i\\in\\mathcal{N}$),\nand then analyze the benefits of predictive offloading in terms of latency reduction.\n\n\\subsubsection{Time-average Power Consumption and Queue Backlog}\n\nLet $P^{*}$ be the achievable minimum of $\\bar{P}$ \nover all feasible non-predictive polices. \nWe have the following theorem.\n\\begin{theorem}\\label{theorem: performance}\n\\textit{Assume the system arrival lies in the interior of the capacity region and $\\boldsymbol{Q}(0)<\\infty$. Under PORA, without prediction, there exist constants $\\theta>0$ and $\\epsilon>0$ such that\n\\begin{equation*}\n\t\\bar{P}\\leq \\theta\/V+P^{*},\\ \\bar{Q}\\leq (\\theta+VP_{\\text{max}})\/\\epsilon,\n\\end{equation*}\nwhere $\\bar{P}$ and $\\bar{Q}$ are defined in (\\ref{definition: time average exp power}) and (\\ref{definition: time average exp backlog}), respectively.}\n\\end{theorem}\nThe proof is quite standard and hence omitted here.\n\n\\textbf{Remark:}\nBy \\textit{Little's} theorem\\cite{leon2017probability}, the average queue backlog size is proportional to the average queueing latency. \nTherefore, Theorem \\ref{theorem: performance} implies that by adjusting parameter $V$, PORA can achieve an $[O(1\/V),O(V)]$ power-latency tradeoff in the non-predictive case. \nFurthermore, the average power consumption $\\bar{P}$ approaches the optimum $P^{*}$ asymptotically as the value of $V$ increases to infinity. \n\n\\subsubsection{Latency Reduction}\nWe analyze the latency reduction induced by PORA under perfect prediction compared to the non-predictive case. \nIn particular, we denote the prediction window vector $(W_{i})_{i\\in\\mathcal{N}}$ by $\\boldsymbol{W}$ and the corresponding delay reduction by $\\eta(\\boldsymbol{W})$. \nFor each unit of workload on EFN $i$, let $\\pi_{i,w}$ denote the steady-state probability that it experiences a latency of $w$ time slots in $A_{i,-1}(t)$. \nWithout prediction, \nthe average latency on its \\textit{arrival queues} is $\n\td=\\sum_{i\\in\\mathcal{N}}\\lambda_{i}\\sum_{w\\geq1}w\\pi_{i,w}\/\\sum_{i\\in\\mathcal{N}}\\lambda_{i}$. Then we have the following theorem.\n\\begin{theorem}\\label{theorem: delay}\n\\textit{Suppose the system steady-state behavior depends only on the statistical behaviors of the arrivals and service processes. Then the latency reduction $\\eta(\\boldsymbol{W})$ is\n\\begin{multline}\\label{theorem 2: result 1}\n\t\\eta\\left(\\boldsymbol{W}\\right)\\\\\n\t=\\frac{\\sum_{i\\in\\mathcal{N}}\\lambda_{i}\\!\\left(\\!\\sum_{1\\leq w\\leq W_{i}}\\!w\\pi_{i,w}\\!+\\!W_{i}\\!\\sum_{w\\geq1}\\!\\pi_{i, w+W_{i}}\\right)}{\\sum_{i\\in\\mathcal{N}}\\lambda_{i}}.\n\\end{multline}\nFurthermore, if $d<\\infty$, as $\\boldsymbol{W}\\rightarrow\\infty$, \\textit{i.e.}, \nwith inifinite predictive information, we have\n\\begin{equation}\\label{theorem 2: result 2}\n\t\\lim_{\\boldsymbol{W}\\rightarrow\\infty}\\eta\\left(\\boldsymbol{W}\\right)=d.\n\\end{equation}\n}\n\\end{theorem}\n\nWe relegate the proof of Theorem \\ref{theorem: delay} to Appendix \\ref{proof: delay}.\n\n\\textbf{Remark:}\nTheorem \\ref{theorem: delay} implies that predictive offloading conduces to a shorter workload latency; \nin other words, with predicted information, PORA can break the barrier of $[O(1\/V),O(V)]$ power-latency tradeoff.\nFurthermore, the latency reduction induced by PORA is proportional to the inverse of the prediction window size, and approaches zero as prediction window sizes go to infinity. \nIn our simulations, we see that PORA can effectively shorten the average arrival queue latency with only mild-value of future information.\n\n\n\\subsection{Impact of Network Topology}\n\nFog computing systems generally proceed in wireless environments, thus the network topology of such systems is usually dynamic and may change over time slots.\nHowever, at the beginning of each time slot, the network topology is observed and deemed fixed by the end of the time slot. \nTherefore, in the following, we put the focus of our discussion on the impact of network topology within each time slot.\n\nRecall that in our settings, each EFN has access to only a subset of CFNs in its vicinity. For each EFN $i$, the subset of its accessible EFNs is denoted by $\\mathcal{M}_{i}$ with a size of $|\\mathcal{M}_{i}|$. From the perspective of graph theory, we can view the interconnection among fog nodes of different tiers as a directed graph, in which each vertex corresponds to a fog node and each edge indicates a directed connection between nodes. Hence, the value of $|\\mathcal{M}_{i}|$ can be regarded as the out-degree of EFN $i$, which is an important parameter of network topology that measures the number of directed connections originating from EFN $i$. \nDue to time-varying wireless dynamics, the out-degree of each fog node may vary over time slots; \nconsequentially, the resulting topology would significantly affect the system performance. In the following, we discuss such impacts under two channel conditions, respectively.\n\nOn the one hand, within each time slot, poor channel conditions (\\textit{e.g.} in terms of low SINR) would often lead to unreliable or even unavailable connections among fog nodes and hence a network topology with a relatively smaller out-degree of nodes. In this case, each fog node may have a very limited freedom to choose the best target node to offload its workloads, further leading to backlog imbalance among fog nodes or even overloading in its upper tier with a large cumulative queue backlog size. Besides, poor channel conditions may also require more power consumptions to ensure reliable communication between successive fog nodes.\n\nOn the other hand, within each time slot, good channel conditions allow each fog node to have a broader access to the fog nodes in its upper tier, resulting a network topology with a relatively larger out-degree of nodes. In this case, each fog node is able to conduct better decision-making with more freedom in choosing the fog nodes in its upper fog tier, thereby achieving a better tradeoff between power consumptions and backlog sizes.\n\n\\begin{table}[!t]\n\\centering\n\\begin{threeparttable}\n\\caption{Simulation Settings}\n\\label{table: simulation parameters}\n\\begin{tabular}{|c|c|}\n\\hline\nParameter & Value \\\\\n\\hline\n$B$ & $2$ MHz \\\\\n\\hline\n$H_{i,j}(t),\\forall i\\in\\mathcal{N},j\\in\\mathcal{M}$ & $24\\log_{10}d_{i,j}+20\\log_{10}5.8$+60 \\tnote{a} \\\\\n\\hline\n$N_{0}$ & $-174$ dBm\/Hz \\\\\n\\hline\n$P_{i,\\text{max}},\\forall i\\in\\mathcal{N}$ & $500$ mW \\\\\n\\hline\n$L^{(e)}_{i}\\forall i\\in\\mathcal{N}$, $L^{(c)}_{j}\\forall j\\in\\mathcal{M}$ & $297.62$ cycles\/bit \\\\\n\\hline\n$f^{(e)}_{i,\\text{max}},\\forall i\\in\\mathcal{N}$ & $4$ G cycles\/s \\\\\n\\hline\n$f^{(c)}_{j,\\text{max}},\\forall j\\in\\mathcal{M}$ & $8$ G cycles\/s \\\\\n\\hline\n$\\varsigma$ & $10^{-27}$ W$\\cdot$s$^{3}$\/cycle$^{3}$ \\\\\n\\hline\n$b^{(e,l)}_{i,\\text{max}},b^{(e,o)}_{i,\\text{max}},\\forall i\\in\\mathcal{N}$ & $6$ Mb\/s \\\\\n\\hline\n$b^{(c,l)}_{j,\\text{max}},b^{(c,o)}_{j,\\text{max}},\\forall j\\in\\mathcal{M}$ & $12$ Mb\/s \\\\\n\\hline\n$D_{j}(t),\\forall j\\in\\mathcal{M},t$ & $6$ Mb\/s \\\\\n\\hline\n\\end{tabular}\n\\begin{tablenotes}\\footnotesize\n\\item [a] $d_{i,j}$ is the distance between EFN $i$ and CFN $j$.\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{table}\n\n\n\\subsection{Use Cases}\nIn practice, PORA can be applied as a theoretical framework to design the offloading schemes for fog computing systems under various use cases, such as public safety systems, intelligent transportation, and smart healthcare systems.\nFor example, in a public safety system, each street is usually deployed with multiple smart cameras (IoT devices). \nAt runtime, such smart cameras would upload real-time vision data to one of their accessible EFNs. Each EFN aggregates such data to extract or even analyze the instant road conditions within multiple streets. \nSuch EFNs can upload some of the workload to their upper-layered CFNs (each taking charge of one community consisting of several streets) with greater computing capacities. \nEach CFN can further offload the workload to the cloud via optical fiber links. \nFor latency-sensitive applications, the real-time vision data will be processed locally on EFNs or offloaded to CFNs. \nFor latency-insensitive applications with intensive computation demand, the data will be offloaded to the cloud through the fog nodes. \nPORA conduces to the design of dynamic and online offloading and resource allocation schemes to support such fog systems with various applications.\n\n\n\\section{Numerical Results}\\label{sec: simulation}\n\n\nWe conduct extensive simulations to evaluate PORA and its variants.\nThe parameter settings in our simulation are based on the commonly adopted wireless environment settings that have been used in \\cite{liu2017latency, du2017computation}.\nThe simulation is conducted on a MacBook Pro with 2.3 GHz Intel Core i5 processor and 8GB 2133 MHz LPDDR3 memory, and the simulation program is implemented using Python 3.7.\nThis section firstly presents the basic settings of our simulations, and then provides the key results under perfect and imperfect prediction, respectively.\n\n\n\n\\subsection{Basic Settings}\n\nWe simulate a hierarchal fog computing system with $80$ EFNs and $20$ CFNs. All EFNs have a uniform prediction window size $W$, which varies from $0$ to $30$. Note that  $W=0$ refers to the case without prediction. \nFor each EFN $i$, its accessible CFN set $\\mathcal{M}_{i}$ is chosen uniformly randomly from the power set of the CFN set with size $|\\mathcal{M}_{i}|=5$. \nWe set the time slot length $\\tau_{0}=1$ second. \nDuring each time slot, workload arrives to the system in the unit of packets, each with a fixed size of $4096$ bits. \nThe packet arrivals are drawn from previous measurements \\cite{benson2010network}, \nwhere the average flow arrival rate is $538$ flows\/s, \nand the distribution of flow size has a mean of $13$ Kb. \nGiven these settings, the average arrival rate is about $7$ Mbps. \nAll results are averaged over $50000$ time slots. \nWe list all other parameter settings in TABLE \\ref{table: simulation parameters}.\n\n\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.8\\linewidth]{figures1\/decisions_V}\n\t\\caption{Offloading decisions when $W=10$.}\n\t\\label{figure: offloading decisions vs. V}\n\\end{figure}\n\n\n\\begin{figure}[!t]\n  \\centering\n  \\subfigure[Queue backlogs.]{\n    \\label{subfig: backlog-W}\n    \\includegraphics[width=0.8\\linewidth]{figures1\/backlog_W}}\n  \\subfigure[Power consumptions.]{\n    \\label{subfig: power-W}\n    \\includegraphics[width=0.8\\linewidth]{figures1\/power_W}}\n  \\caption{Performance of PORA vs. $W$ when $V=10^{11}$.}\n  \\label{figure: performance vs. window size}\n\\end{figure}\n\n\\subsection{Evaluation with Perfect Prediction}\n\nUnder the perfect prediction settings, we evaluate how the values of parameter $V$ and prediction window size $W$ influence the performance of PORA, respectively.\n\n\n\\textbf{System Performance under Different Values of $V$:}\nFigure \\ref{figure: offloading decisions vs. V} shows the impact of parameter $V$ on the offloading decisions of PORA: \nWhen the value of $V$ is around $10^{10}$, the time-average amount of locally processed workload on EFNs reaches the bottom of the curve, \nwhile other offloading decisions induce the peak workload. \nThe reason is that the offloading decisions are not only determined by the value of $V$, \nbut also influenced by the queue backlog sizes.\n\nFigure \\ref{figure: performance vs. V} presents the impact of the value of $V$ on different types of queues and power consumptions in the system, respectively. \nAs the value of $V$ increases, we see a rising trend in the sizes of all types of queue backlogs, and a roughly falling trend in all types of power consumptions. \n\n\n\n\n\\begin{figure}[!t]\n  \\centering\n  \\subfigure[Queue backlogs.]{\n    \\label{subfig: backlog vs. V}\n    \\includegraphics[width=0.8\\linewidth]{figures1\/backlog_V}}\n  \\subfigure[Power consumptions.]{\n    \\label{subfig: power vs. V}\n    \\includegraphics[width=0.8\\linewidth]{figures1\/power_V}}\n  \\caption{Performance of PORA when $W=10$.}\n  \\label{figure: performance vs. V}\n\\end{figure}\n\n\n\\begin{figure*}[!t]\n  \\centering\n  \\subfigure[Total queue backlogs.]{\n    \\label{subfig: backlog-V-variant}\n    \\includegraphics[width=0.4\\linewidth]{figures1\/backlog_V_variant}}\n  \\subfigure[Total power consumptions.]{\n    \\label{subfig: power-V-variant}\n    \\includegraphics[width=0.4\\linewidth]{figures1\/power_V_variant}}\n  \\caption{Performance of variants of PORA.}\n  \\label{figure: performance v.s. variants}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n  \\centering\n  \\subfigure[Total queue backlogs.]{\n    \\label{subfig: backlog v.s. time}\n    \\includegraphics[width=0.4\\linewidth]{figures1\/backlog_time}}\n  \\subfigure[Total power consumptions.]{\n    \\label{subfig: power v.s. times}\n    \\includegraphics[width=0.4\\linewidth]{figures1\/power_time}}\n  \\caption{Comparison between PORA and baselines.}\n  \\label{figure: comparison}\n\\end{figure*}\n\n\n\\textbf{System Performance with Different Values of Prediction Window Size $W$:}\nFigures \\ref{subfig: backlog-W} and \\ref{subfig: power-W} show the system performance with the prediction window size $W$ varying from $0$ to $30$. \nWith perfect prediction, PORA effectively shortens the average queueing latencies on EFN arrival queues -- eventually close to zero with no extra power consumption and only a mild-value of prediction window size ($W=20$ in this case).\n\n\n\\textbf{PORA vs. PORA-$d$ (Low-Sampling Variant):}\nIn practice, since PORA requires to sample system dynamics across various fog nodes, it may incur considerable sampling overheads.\nBy adopting the idea of randomized load balancing techniques \\cite{mitzenmacher2001power}, we propose PORA-$d$, a variant of PORA that reduces the sampling overheads by \nprobing $d$ ($d\\in\\{1,2,3,4\\}$) \\footnote{When $d=1$, the scheme degenerates to uniform random sampling.} CFNs and conducting resource allocation on which are uniformly chosen for each EFN from its accessible CFN set. \n\nFigure \\ref{figure: performance v.s. variants} compares the performance of PORA with PORA-$d$. \nWe observe that PORA achieves the smallest queue backlog size. The result is reasonable since each EFN has access to 5 CFNs under PORA, more than the $d\\leq 4$ CFNs under PORA-$d$. As a result, each EFN has more chance to access to the CFNs with better wireless channel condition and processing capacity under PORA when compared with PORA-$d$. The observation that the queue backlog size increases as $d$ decreases further verifies our analysis. In fact, we can view $d$ as the degree of each EFN in the network topology. As $d$ decreases, the system performance degrades.\nHowever, when the value of $V$ is sufficiently large, PORA-$d$ achieves the similar power consumptions as PORA and the ratio of increment in the backlog size is small. For example, when $V=2\\times 10^{11}$, PORA-$4$ achieves $4.3$\\% larger backlog size than PORA, and PORA-$3$ achieves $10.9$\\% larger backlog size than PORA.\nIn summary, PORA-$d$ (when $d=2,3,4$) can reduce the sampling overheads by trading off only a little performance degradation under large $V$.\n\n\n\\textbf{Comparison of PORA and Baselines:}\nWe introduce four baselines to evaluate the performance of PORA: (1) NOL (No Offloading): All nodes in the EFT process packets locally. (2) O2CFT (Offload to CFT): All packets are offloaded to the CFT and processed therein. (3) O2CLOUD (Offload to Cloud): All packets are offloaded to the cloud. (4) RANDOM: Each fog node randomly chooses to offload each packet or process it locally with equal chance. \nNote that all above baselines are also assumed capable of pre-serving future workloads in the prediction window.\nFigure \\ref{figure: comparison} compares the instant total queue backlog sizes and power consumptions over time slots under the five schemes (PORA, NOL, O2CFT, O2CLOUD, RANDOM), \nwhere $W=10$ and $V\\in\\{10^{9},10^{11}\\}$.\n\n\nWe observe that scheme O2CLOUD achieves the minimum power consumptions, but incurs constantly increasing queue backlog sizes over time. \nThe reasons are shown as follows. \nOn one hand, in our settings, the mean power consumption for transmitting workload from EFT to CFT is smaller than the mean power consumption of processing the same amount of workload on fog nodes; under scheme O2CLOUD, only wireless transmit power is consumed and hence the minimum is achieved.  \nOn the other hand, all the workload must travel through all fog tiers before being offloaded to the cloud, which results in network congestion within fog tiers and thus workload accumulation with increasing queue backlogs.\n\nAs Figure \\ref{figure: comparison} illustrates, PORA achieves the maximum power consumptions but the smallest backlog size when $V=10^{9}$. \nUpon convergence of PORA, the power consumptions under all these schemes reach the same level, but the differences between their queue backlog sizes become more obvious:\nPORA ($V=10^{9}$) reduces $96$\\% of the queue backlog when compared with NOL and RANDOM.\nThe results demonstrate that with the appropriate choice of the value of $V$,\nPORA can achieve less latency than the four baselines under the same power consumptions.\n\n\\subsection{Evaluation with Imperfect Prediction}\n\nIn practice, prediction errors are inevitable. \nHence, we investigate the performance of PORA in the presence of prediction errors \\cite{chen2017timely}. \nParticularly, we consider two kinds of prediction errors: false alarm and missed detection.\nA packet is falsely alarmed if it is predicted to arrive but it does not arrive actually.\nA packet is missed to be detected if it will arrive but is not predicted.\nWe assume that all EFNs have the uniform false-alarm rate $p_{1}$ and missed-detection rate $p_{2}$.\nIn our simulation, we consider different pairs of values of $(p_{1},p_{2})$: $(0.0, 0.0)$, $(0.05, 0.05)$, $(0.5, 0.05)$, $(0.05, 0.25)$, and $(0.5, 0.25)$. \nNote that $(p_{1},p_{2})=(0.0,0.0)$ corresponds to the case when the prediction is perfect. \n\n\\begin{figure}[!t]\n  \\centering\n  \\subfigure[Total queue backlogs.]{\n    \\label{subfig: backlog under imperfect prediction}\n    \\includegraphics[width=0.8\\linewidth]{figures1\/backlog_V_error}}\n  \\subfigure[Total power consumptions.]{\n    \\label{subfig: power under imperfect prediction}\n    \\includegraphics[width=0.8\\linewidth]{figures1\/power_V_error}}\n  \\caption{Performance of PORA under imperfect prediction.}\n  \\label{figure: performance under imperfect prediction}\n\\end{figure}\n\nFigure \\ref{figure: performance under imperfect prediction} presents the results under prediction window size $W=10$.\nWe observe when $V\\leq 7.5\\times 10^{10}$, both the total queue backlog sizes and power consumptions under imperfect prediction are larger than that under perfect prediction. The reason for this performance degradation is twofold:\nFirst, arrivals that are missed to be detected cannot be pre-served, thus leading to larger queue backlog sizes.\nSecond, PORA allocates redundant resources to handle the falsely predicted arrivals, thus causing more power consumptions.\nAs the value of $V$ increases, this performance degradation becomes negligible. Taking the total queue backlog under $(p_{1},p_{2})=(0.25,0.5)$ as an example, when compared with the case under perfect prediction, it increases by $4.72\\%$ at $V=10^{11}$, and increases by $2.24\\%$ at $V=2\\times 10^{11}$. \nMoreover, there is no extra power consumption under imperfect prediction when $V\\geq 7.5\\times 10^{10}$ since PORA tends to reserve resources to reduce power consumptions under large $V$.\n\nIn summary, there will be performance degradation in both total queue backlog sizes and power consumptions in the presence of prediction errors. However, as the value of $V$ increases, this degradation decreases and becomes negligible.\nThough a large value of $V$ can improve the robustness of PORA and achieve small power consumptions, it brings long workload latencies. \nIn practice, the choice of the value of $V$ depends on how the system designer trades off all these criterions.\n\n\\section{Conclusion}\\label{sec: conclusion}\n\nIn this paper, we studied the problem of dynamic offloading and resource allocation with prediction in a fog computing system with multiple tiers. \nBy formulating it as a stochastic network optimization problem, we proposed PORA, an efficient online scheme that exploits predictive offloading to minimize power consumption with queue stability guarantee. \nOur theoretical analysis and trace-driven simulations showed that PORA achieves a tunable power-latency tradeoff, while effectively shortening latency with only mild-value of future information, even in the presence of prediction errors. \nAs for future work, our model can be further extended to more general settings such that the instant wireless channel states may be unknown by the moment of decision making or the underlying system dynamics is non-stationary.\n\n\n\n\n\n\n\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\nThe recently proposed model of quantum gravity  by  Ho\\v{r}ava \\cite{horava1,horava2,horava3} has recently attracted much attention, and many aspects of it\n have been extensively analyzed, ranging from formal developments, cosmology, dark energy\n and dark matter, spherically symmetric solutions, gravitational waves, and its viability with observational constraints; for a full list of references see, e.g., \\cite{cinesi,Bom,Wang,Carloni}.\nSuch a  theory admits  Lifshitz's  scale invariance: $\\mathbf{x} \\rightarrow b \\mathbf{x}, \\quad t \\rightarrow b^{q} t$, and, after this, it is referred to as Ho\\v{r}ava-Lifshitz (HL) gravity.  Actually, it has anistropic scaling in the short distances domain (UV), since it is $q=3$, while isotropy is recovered at large distances (IR).\n\nOne of the key features of the theory is its good UV behavior, since it is power-counting renormalizable; for a discussion of the renormalizability beyond power counting arguments, see  \\cite{Orla}. However, in its original formulation, it experiences some problems: for instance, it leads to a non-zero cosmological constant with the wrong sign, in order to be in agreement with the observations \\cite{Soda,horatius,visser1}. To circumvent these issues, it was suggested to abandon the principle of ``detailed balance'' \\cite{Calca1,Calca2}, initially introduced by Ho\\v{r}ava  in his model to restrict the number of possible parameters. As a consequence, phenomenologically viable extensions of the theory were proposed \\cite{visser1,visser2}. It was also shown that HL gravity can reproduce General Relativity (GR) at large distances \\cite{pope,KS}; for other solutions non-asymptotically flat see \\cite{Cai1,Cai2}. However, there is still an ongoing discussion on the consistency of HL gravity, since it seems that modes arise which develop exponential instabilities at short distances, or become strongly coupled \\cite{padilla,blas}. Moreover, according to  \\cite{Miao}, the constraint algebra does not form a closed structure. Perturbative instabilities affecting HL gravity have been pointed out in  \\cite{Saridakis}.\n\nActually, it is important to stress that, up to now, in HL gravity the gravitational field is purely geometrical: in other words, the way matter has to be embedded still needs to be studied. Nevertheless, there are interesting vacuum solutions that can be studied, such as the static spherically symmetric solution found by Kehagias and Sfetsos (hereafter KS) \\cite{KS}. Such a  solution is the analog of Schwarzschild solution of GR and, moreover, it asymptotically reproduces the usual behavior of Schwarzschild spacetime. It is interesting to point out that it is obtained without requiring the projectability condition, assumed in the original HL theory, while spherically symmetric solutions with the projectability condition are however available \\cite{tang,Wang}. {Nonetheless, because of its simplicity, it is possible to consider  KS solution as toy model useful to better understand some phenomenological implications of HL gravity.  Actually, in \\cite{Tib} it was shown that KS solution  is in agreement with the classical tests of GR, while in a previous paper \\cite{HLIR} we studied the corrections to the general relativistic Einstein's pericentre precession determined by this solution and compared the theoretical predictions to\nthe latest determinations of the corrections  to the standard Newtonian\/Einsteinian planetary perihelion precessions recently estimated with the EPM2008  ephemerides.  We found that the KS dimensionless parameter is constrained from the bottom at $\\omega_0\\geq 10^{-12}-10^{-24}$ level depending on the planet considered.\n\nIn our analysis, we assumed that particles followed geodesics of KS metric: however, it is important to point out that this is true if matter is minimally and universally coupled to the metric, which is not necessarily true in HL gravity, where, as we said above, the role of matter has not been yet clarified. In this paper, starting from the same assumption, we focus on the effects induced by the examined solution on the orbital period $P_{\\rm b}$ of a test particle, on an extra solar system environment. We will explicitly work out the consequent correction $P_{\\omega_0}$ to the usual third Kepler law in Section \\ref{3kep}. In Section \\ref{osiris} we compare it with the observations of the transiting extrasolar planet HD209458b \\virg{Osiris}. {We point out that the resulting constraints are to be considered as preliminary and just order-of-magnitude figures because, actually, the entire data set of HD209458b should be re-processed again by explicitly modeling the effect of the KS gravity; however, this is outside the scopes of the present paper.} Section \\ref{conclu} is devoted to the conclusions.\n\\section{KS corrections to the third Kepler law}\\lb{3kep}\nAs shown in \\cite{HLIR},\nfrom \\cite{Brum}\n\\begin{eqnarray} \\ddot x^i &=& -\\frac{1}{2}c^2 h_{00,i} - \\frac{1}{2}c^2 h_{ik}h_{00,k}+h_{00,k}\\dot x^k\\dot x^i \\nonumber \\\\ &+& \\left(h_{ik,m}-\\frac{1}{2}h_{km,i}\\right)\\dot x^k\\dot x^m,\\ i=1,2,3,\\end{eqnarray}\nit is possible to obtain the following radial acceleration acting upon a test particle at distance $d$ from a central body of mass $M$\n\\eqi \\vec{A}_{\\omega_0}\\approx \\frac{4 (GM)^4}{\\omega_0 c^6 d^5}\\hat{d},\\lb{accel}\\eqf valid up to terms of order $\\mathcal{O}(v^2\/c^2)$.\nIts effect on the pericentre of a test particle have been worked out in  \\cite{HLIR}; here we want to look at a different orbital feature affected by  \\rfr{accel} which can be compared to certain observational determinations.\n\nThe mean anomaly is defined as\n\\eqi \\mathcal{M}\\doteq n(t-t_p);\\eqf in it $n=\\sqrt{GM\/a^3}$ is the Keplerian mean motion, $a$ is the semimajor axis and $t_p$ is the time of pericentre passage.\nThe anomalistic period $P_{\\rm b}$ is the time elapsed between two consecutive pericentre passages; for an unperturbed Keplerian orbit it is $P_{\\rm b}=2\\pi\/n$. Its modification due to a small perturbation of the Newtonian monopole can be evaluated  with  standard perturbative approaches. The Gauss equation for the variation of the mean anomaly is, in the case of a radial perturbation $A_d$ to the Newtonian monopole \\cite{Ber},\n\\eqi \\frac{d{\\mathcal{M}}}{dt}=n-\\rp{2}{na}A_d\\left(\\rp{d}{a}\\right)+\\rp{(1-e^2)}{nae}A_d\\cos f,\\lb{Gaus}\\eqf where $e$ is the eccentricity and $f$ is the true anomaly counted from the pericentre position.\nThe right-hand-side of \\rfr{Gaus} has to be valuated onto the unperturbed Keplerian orbit given by (see \\cite{Roy})\n\\eqi d=\\rp{a(1-e^2)}{1+e\\cos f}.\\eqf\nBy using (see \\cite{Roy})\n\\eqi df = \\left(\\rp{a}{d}\\right)^2(1-e^2)^{1\/2}d\\mathcal{M}\\eqf\nand\n\\eqi \\int_0^{2\\pi} (1+e\\cos f)^2\\left[2-\\rp{(1+e\\cos f)}{e}\\cos f\\right]df=\\pi\\left(1+\\rp{5}{4}e^2\\right),\\eqf it is possible to work out the correction to the Keplerian period due to \\rfr{accel}; it is\n\\eqi P_{\\omega_0}=\\rp{4\\pi (GM)^4\\left(1+\\rp{5}{4}e^2\\right)}{\\omega_0 c^6 n^3 a^6(1-e^2)^{5\/2}}=\\rp{4\\pi (GM)^{5\/2}\\left(1+\\rp{5}{4}e^2\\right)}{\\omega_0 c^6  a^{3\/2}(1-e^2)^{5\/2}}.\\lb{PHL}\\eqf\nNote that \\rfr{PHL} retains its validity in the limit $e\\rightarrow 0$ becoming equal to\n\\eqi P_{\\omega_0}\\rightarrow\\rp{4\\pi (GM)^{5\/2}}{\\omega_0 c^6  d^{3\/2}}\\lb{circo},\\eqf where $d$ represents now the fixed radius of the circular orbit. It turns out that \\rfr{circo} is equal to the expression that can be easily obtained by equating the centripetal acceleration $\\Omega^2 d$, where $\\Omega$ is the particle's angular speed, to the total gravitational acceleration $GM\/d^2 - 4(GM)^4\/\\omega_0c^6 d^5$ with the obvious assumption that the Newtonian monopole is the dominant term in the sum.\n\\section{Confrontation with the observations}\\lb{osiris}\nIn the scientific literature there is a large number of papers (see, e.g., \\cite{Tal,Wri,Ove,Jae,Rey,Ser1,Ser2,Ior,Bro,Peri,Mof,Adl,Page}) in which the authors use the third Kepler law to determine, or, at least,  constrain un-modeled dynamical effects of mundane, i.e. due to the standard Newtonian\/Einsteinian laws of gravitation, or non-standard, i.e. induced by putative modified models of gravity.\nAs explained below, in many cases such a strategy has been, perhaps, followed in a self-contradictory way, so that the resulting  constraints on, e.g., new physics, may be regarded as somewhat \\virg{tautologic}.\n\nLet us briefly recall that the orbital period $P_{\\rm b}$ of two point-like bodies of mass $m_1$ and $m_2$ is, according to the third Kepler law,\n\\eqi P^{\\rm Kep}=2\\pi\\sqrt{\\rp{a^3}{G{M}}},\\eqf\nwhere $a$ is the relative semi-major axis and ${M}=m_1+m_2$ is the total mass of the system.\nLet us consider an unmodeled dynamical effect which induces a non-Keplerian (NK) correction to the third Kepler law, i.e.\n\\eqi P_{\\rm b}= P^{\\rm Kep} + P^{\\rm NK},\\eqf\nwhere\n\\eqi P^{\\rm NK}=P^{\\rm NK}(M,a,e; p_j),\\eqf\nis the analytic expression of the correction to the third Kepler law  in which $p_j$, $j=1,2,...$N are the parameters of the NK effect to be determined or constrained. Concerning standard physics, $P_{\\rm NK}$ may be due to the centrifugal oblateness of the primary, tidal distortions, General Relativity; however, the most interesting case is that in which $P_{\\rm NK}$ is due to some putative modified models of gravity. {As a first, relatively simple step to gain insights into the NK effect one can act as follows.}\nBy comparing the measured orbital period to the computed Keplerian one it is possible, in principle, to obtain preliminary information on the dynamical effect investigated from $\\Delta P\\doteq P_{\\rm b}-P^{\\rm Kep}$. {Actually, one should re-process the entire data set of the system considered by explicitly modeling the non-standard gravity forces, and simultaneously solving for one or more dedicated parameter(s) in a new global solution along with the other ones routinely estimated. Such a procedure would be, in general, very time-consuming and should be repeated for each models considered. Anyway, it is outside the scopes of the present paper, but it could be pursued in further investigations.}\n\nConcerning our simple approach, in order to meaningfully solve for $p_j$ in\n\\eqi\\Delta P=P^{\\rm NK}\\eqf\nit is necessary that\n\\begin{itemize}\n  \\item In the system considered a measurable quantity which can be identified with the orbital period  and directly measured independently of the third Kepler law itself, for example from spectroscopic or photometric measurements, must exist. This is no so obvious as it might seem at first sight; indeed, in a N-body system like, e.g., our solar system a directly measurable thing like an \\virg{orbital period} simply does not exist  because the orbits of the planets are not closed due to the non-negligible mutual perturbations. Instead, many authors use values of the \\virg{orbital periods} of the planets which are retrieved just from the third Kepler law itself.\n      Examples of systems in which there is a measured orbital period are many transiting exoplanets, binaries and, e.g,  the double pulsar.\n      Moreover, if the system considered follows an eccentric path one should be careful in identifying the measured orbital period with the predicted sidereal or  anomalistic periods. A work whose authors are aware of such issues is \\cite{Capoz}.\n \n \n  \\item  The quantities entering $P^{\\rm Kep}$, i.e. the relative semimajor axis $a$ and the total mass $M$, must be known independently of the third Kepler law. Instead, in many cases values of the masses obtained by applying just the third Kepler law itself are used.\n      Thus, for many exoplanetary systems the mass $m_1\\doteq M_\\bigstar$ of the hosting star should be taken from stellar evolution models and the associated scatter should be used to evaluate the uncertainty $\\delta M_\\bigstar$ in it, while for the mass $m_2=m_p$ of the planet a reasonable range of values should be used instead of straightforwardly taking the published value because it comes  from the mass function which is just another form of the third Kepler law.\n       {Some extrasolar planetary systems represent good scenarios because it is possible to know many of the parameters entering $P^{\\rm Kep}$ independently of the third Kepler law itself, thanks to the redundancy offered by the various techniques used.}\n       \n\nSuch issues have been accounted for in several astronomical and astrophysical scenarios in, e.g.,  \\cite{IorWD,IorIJMPA,IorCyg,IorReg,IorNA}.\n\\end{itemize}\n\n\\section{The transiting exoplanet HD209458b}\nLet us consider HD 209458b \\virg{Osiris}, which is the first exoplanet\\footnote{See on the WEB http:\/\/www.exoplanet.eu\/} discovered with the transit method \\cite{Cha,Hen}. Its orbital period $P_{\\rm b}$ is known with a so high level of accuracy that it was proposed  to use it for the first time to test General Relativity in a planetary system different from ours \\cite{osi}; for other proposals to test General Relativity with different orbital parameters of other exoplanets, see  \\cite{ada,ung1,ung2,RAGO}.\n\nIn the present case, the system's parameters entering the Keplerian period\ni.e. the relative semimajor axis $a$, the mass $M_{\\bigstar}$ of the host star and the mass $m_p$ of the planet, can be determined independently of the third Kepler law itself, so that it is meaningful to compare the photometrically measured orbital period $P_{\\rm b}=3.524746$ d \\cite{exo} to the computed Keplerian one $P^{\\rm Kep}$: their difference can be used to put genuine constraints on KS solution which predicts the corrections of \\rfr{PHL} to the third Kepler law.\nIndeed, the mass $M_{\\bigstar}=1.119\\pm 0.033$M$_{\\odot}$ and the radius $R_{\\bigstar}=1.155^{+0.014}_{-0.016}$R$_{\\odot}$ of the star \\cite{exo}, along with other stellar properties, are fairly straightforwardly estimated by matching direct spectral observations with stellar evolution models since for HD 209458 we have also the Hipparcos parallax $\\pi_{\\rm Hip}=21.24\\pm 1.00$ mas \\cite{Perry}.\nThe semimajor axis-to-stellar radius ratio $a\/R_{\\bigstar}=8.76\\pm 0.04$ is estimated from the photometric light curve, so that $a=0.04707^{+0.00046}_{-0.00047}$AU \\cite{exo}. The mass $m_p$ of the planet can be retrieved from the parameters of the photometric light curve and of the spectroscopic one entering the formula for the planet's surface gravity $g_p$  (eq.(6) in \\cite{exo}). As a result, after having computed the uncertainty in the Keplerian period by summing in quadrature the errors due to $\\delta a,\\delta M_{\\bigstar},\\delta m_p$, it turns out\n \\eqi \\Delta P\\doteq P_{\\rm b}-P^{\\rm Kep} = 204\\pm 5354\\ {\\rm s};\\lb{exop}\\eqf the uncertainties $\\delta M_{\\bigstar}$, $\\delta a$,  $\\delta m_p$  contribute  4484.88 s, 2924.77 s, 2.66 s, respectively to $\\delta(\\Delta P)=5354$ s.\n\n The discrepancy $\\Delta P$ between $P_{\\rm b}$ and $P^{\\rm Kep}$ of \\rfr{exop} is statistically compatible with zero; thus, \\rfr{exop} allows to constrain the parameter $\\omega_0$ entering $P_{\\omega_0}$.\n Since\n \\eqi P^{\\rm NK}\\doteq P_{\\omega_0}=\\rp{\\mathcal{K}}{\\omega_0},\\eqf\n with\n \\eqi \\mathcal{K}\\doteq\\rp{4\\pi(GM)^{5\/2}}{c^6 d^{3\/2}}=8\\times 10^{-15}\\ {\\rm s},\\lb{cazza}\\eqf\n by equating the non-Keplerian correction $P_{\\omega_0}$ to the measured $\\Delta P$ one has\n \\eqi \\omega_0=\\rp{\\mathcal{K}}{\\Delta P}.\\lb{vaffa}\\eqf\n Since $\\Delta P$ is statistically compatible with zero, the largest value of $\\omega_0$ is infinity; from \\rfr{vaffa} a lower bound on $|\\omega_0|$ can be obtained amounting to\\eqi |\\omega_0|\\geq 1.4\\times 10^{-18}.\\lb{lowerbo}\\eqf\n A confrontation with the solar system constraints\\footnote{To avoid confusions with the perihelion $\\omega$, the KS parameter is dubbed $\\psi_0$ in  \\cite{HLIR}.}  Our previous paper \\cite{HLIR} shows that such a lower bound is at the level of those from Jupiter and Saturn, while it contradicts the possibility that values of $\\omega_0$ as small as those allowed by Uranus, Neptune and Pluto ($|\\omega_0|\\geq 10^{-24}-10^{-22}$) may exist.\n However, tighter constraints are established by the inner planets for which $|\\omega_0|\\geq 10^{-15}-10^{-12}$.\n\n\n\n\n\\section{Conclusions}\\lb{conclu}\nWe have investigated how the third Kepler law is modified by  the KS solution,\nwhose Newtonian and lowest order post-Newtonian limits coincides with those of GR, by using the standard Gauss perturbative approach. The resulting expression for $P_{\\omega_0}$, obtained from the Gauss equation of the variation of the mean anomaly $\\mathcal{M}$, in the limit $e\\rightarrow 0$ reduces to the simple formula which can be derived by equating the centripetal acceleration to the Newton$+$KS gravitational acceleration for a circular orbit.\n\nThen, after having discussed certain subtleties connected, in general, with a meaningful use of the third Kepler law to put on the test alternative theories of gravity, we compared our explicit expression for $P_{\\omega_0}$ to the discrepancy $\\Delta P$ between the phenomenologically determined orbital periods $P_{\\rm b}$ and the computed Keplerian ones $P^{\\rm Kep}$ for the transiting extrasolar planet HD209458b \\virg{Osiris}. Since $\\Delta P$ is statistically compatible with zero, it has been possible to {preliminary} obtain the lower bound $|\\omega_0|\\geq 1.4 \\times 10^{-18}$ on the  dimensionless KS parameter. {However, the entire data set of HD209458b should be re-processed by including KS gravity as well, and a dedicated, solve-for parameter should be estimated as well. The previously reported } constraint rules out certain smaller values allowed by the lower bounds obtained from the perihelia of Uranus, Neptune and Pluto ($|\\omega_0|\\geq 10^{-24}-10^{-22}$). On the other hand, our exoplanet bound still leaves room for values of $\\omega_0$ too small according to the constraints from the perihelia of Mercury, Venus and the Earth ($|\\omega_0|\\geq 10^{-14}-10^{-12}$).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\paragraph{Planar maps.}\n\nMaps, i.e. gluings of polygons forming an orientable surface, have been the object of extensive research in the last decades, both from the combinatorial and probabilistic viewpoints. The most popular category of maps are planar maps, i.e. maps homeomorphic to the sphere. Their combinatorial study goes back to Tutte in the 60s, e.g.~\\cite{Tutte63}, who gave explicit formulas for the enumeration of various classes of planar maps using a generating functions approach. More recently, bijective approaches have been developped such as the Cori--Vauquelin--Schaeffer bijection for quadrangulations~\\cite{Sch98} and its generalization, the Bouttier--di Francesco--Guitter bijection~\\cite{BDG04}.\n \nOn the other hand, much attention has been given in the last 20 years to asymptotic properties of large random planar maps picked uniformly in certain classes. These asymptotic properties are usually understood by proving the convergence of random maps in some sense when the size goes to infinity. Two different notions of limits are commonly used: scaling and local limits. Scaling limits, which we will not study in this work, consist in renormalizing the distances in order to build continuous objects. In particular, many discrete models are known to have the Brownian map as a scaling limit~\\cite{LG11, Mie11, Mar16}. The theory of scaling limits of planar maps shares deep links with other random geometry models such as Liouville Quantum Gravity~\\cite{MS16b}. On the other hand, local limits, on which the present work focuses, study the neighbourhood of a typical point in a map in order to obtain an infinite but discrete object in the limit. In the context of planar maps, this was first considered by Angel and Schramm who proved the convergence of large uniform triangulations towards the Uniform Infinite Planar Triangulation (UIPT)~\\cite{AS03}. The study of the UIPT using Markovian explorations called \\emph{peeling} explorations was then initiated by Angel~\\cite{Ang03}. More general models have followed since, such as general planar maps with Boltzmann weights on the face degrees~\\cite{St18}. The bipartite case, which will be of particular interest for us, is investigated in~\\cite{Bud15, BC16}, see also~\\cite{C-StFlour} for a complete survey.\n\n\\paragraph{Maps of higher genus.}\nIt seems natural to try to extend the combinatorial and probabilistic study of planar maps to maps of higher genus. On the combinatorial side, the enumeration of maps with any genus is a very rich topic, with links to irreducible representations of the symetric group and integrable hierarchies~\\cite{MJD00,Okounkov00}. In particular, double recursions on both the size and the genus are known for the counting of maps, see~\\cite{GJ08} for triangulations and \\cite{Lo19} for general classes of bipartite maps. However, explicit enumeration formulas are lacking. Asymptotics can be obtained when the genus is fixed and the size goes to infinity~\\cite{BC86}, but are still missing when the genus goes to infinity as well.\n\nSimilarly, on the probabilistic side, higher genus versions of random surface models have been constructed such as Brownian surfaces~\\cite{Bet16} or Liouville quantum gravity on complex tori \\cite{DRV16}. However, their behaviour when the genus goes to infinity is still poorly understood. Finally, a regime that is much easier to handle is the regime where the genus is not constrained, and the faces are simply glued uniformly at random \\cite{Gam06, CP16, BCP19}. In this case, the genus is concentrated very close to its maximal possible value.\n\nMore recently, some progress was made in the study of high genus maps, namely when the genus grows linearly in the size of the map. In this case, the Euler formula shows that maps satisfy a discrete notion of \"negative average curvature\", which suggests that the neighbourhood of a typical vertex should look hyperbolic. The first category of maps that was investigated in this setting were uniform unicellular maps (i.e. maps with one face). See \\cite{ACCR13} for the proof of local convergence to a supercritical Galton--Watson tree, and \\cite{Ray13a} for the study of more global properties such as logarithmic diameter.\n\nShortly after, Curien introduced a one-parameter family of random hyperbolic triangulations of the plane \\cite{CurPSHIT}, following the work of Angel and Ray in the half-planar case \\cite{AR13}. More precisely, random maps of this family are called Planar Stochastic Hyperbolic Triangulations (PSHT) $\\left( \\mathbb{T}_{\\lambda} \\right)_{0<\\lambda \\leq (12\\sqrt{3})^{-1}}$ and they are the only random triangulations satisfying the following spatial Markov property: for any finite triangulation $t$ with $|t|$ vertices in total and a hole of perimeter $p$, we have\n\\begin{equation}\n\\P \\left( t \\subset \\mathbb{T}_{\\lambda} \\right) = C_{p}(\\lambda) \\lambda^{|t|}.\n\\end{equation}\nIn particular, such a triangulation exists if and only if $\\lambda \\in \\left( 0, \\frac{1}{12\\sqrt{3}} \\right]$. Except for the critical case $\\lambda=\\frac{1}{12\\sqrt{3}}$ (which corresponds to the UIPT), these objects exhibit hyperbolic properties \\cite{CurPSHIT, B18}. Benjamini and Curien conjectured in~\\cite{CurPSHIT} that they are the local limits of uniform high genus triangulations. \n\nIn a recent paper \\cite{BL19}, the authors of the present work proved this conjecture. Asymptotics for the enumeration of high genus triangulations up to subexponential factors were derived as a byproduct.\n\n\\paragraph{Infinite Boltzmann Planar Maps.}\nThe goal of the present work is to generalize the results of~\\cite{BL19} to a much wider family of models, where faces do not have to be triangles. For combinatorial reasons\\footnote{More precisely, the enumeration results in the planar case are simpler for bipartite maps, and the recursion of~\\cite{Lo19} holds only for bipartite maps.}, we will restrict ourselves to bipartite maps, which means that the face degrees have to be even. The limiting objects appearing in the limits are the \\emph{Infinite Boltzmann Bipartite Planar Maps} (IBPM) introduced in~\\cite[Appendix C]{B18these} as an analog of the PSHT for bipartite maps. We also refer to~\\cite[Chapter 8]{C-StFlour} for the study of basic properties of these objects.\n\nThe IBPM are characterized by a spatial Markov property similar to the one satisfied by the PSHT. If $m$ is a finite map with one hole, we write $m \\subset M$ if $M$ can be obtained by filling the hole of $m$, possibly with a map with a nonsimple boundary\\footnote{More precisely, we use the sense introduced by Budd in~\\cite{Bud15}, i.e. the sense corresponding to the \"lazy\" peeling process, as opposed to the one introduced by Angel in~\\cite{Ang03}. See Section~\\ref{subsec_univ_defns} for precise definitions.}. Let $\\mathbf{q}=(q_j)_{j \\geq 1}$ be a sequence of nonnegative numbers. An infinite random planar map $M$ is called a $\\mathbf{q}$-IBPM if there are numbers $\\left( C_p \\right)_{p \\geq 1}$ such that, for any finite map $m$ with one hole of perimeter $2p$, we have\n\\[ \\P \\left( m \\subset M \\right) = C_p \\times \\prod_{f \\in m} q_{\\mathrm{deg}(f)\/2}, \\]\nwhere the product is over all internal faces of $m$. In particular, they generalize the infinite \\emph{critical} bipartite Boltzmann maps defined and studied by Budd in \\cite{Bud15}. It was proved in~\\cite[Appendix C]{B18these} that there is at most one $\\mathbf{q}$-IBPM, that we denote by $\\mathbb{M}_{\\mathbf{q}}$. Moreover \\cite[Appendix C]{B18these} provides both necessary conditions and sufficient ones on $\\mathbf{q}$ for the existence of $\\mathbb{M}_{\\mathbf{q}}$, but no explicit characterization (these results are recalled in Section~\\ref{subsec_IBPM} below). The present work improves on these results by giving an explicit parametrization of the weight families $\\mathbf{q}$ for which $M_{\\mathbf{q}}$ exists. More precisely, as stated in Theorem~\\ref{thm_prametrization_rootface} below, such families $\\mathbf{q}$ can be parametrized by the law of the degree of the root face of $\\mathbb M_{\\mathbf{q}}$, which may be any law on $\\{2,4,6,\\dots\\}$, and an additional hyperbolicity parameter $\\omega \\in [1, +\\infty)$. The critical maps of~\\cite{Bud15} correspond to the case $\\omega=1$, and are already known to be local limits of \\emph{planar} maps. On the other hand, the case $\\omega>1$ has a hyperbolic flavour.\n\n\\paragraph{Local limits of high genus bipartite maps.}\nThe main result of this work is that uniform maps with high genus and prescribed face degrees converge locally to the $\\mathbf{q}$-IBPM when the size goes to infinity. This can be seen as a \\textit{universality} result in the domain of high genus maps, in the sense that regardless of the precise model of maps, the same phenomenon is observed.\n\nMore precisely, we will use the notation $\\mathbf{f}=(f_j)_{j \\geq 1}$ for face degree sequences (i.e. $f_j \\geq 0$ for all $j$, and $f_j=0$ eventually). For such a sequence $\\mathbf{f}$, we set $|\\mathbf{f}|=\\sum_{j \\geq 1} j f_j$, which describes the number of edges of a map with $f_j$ faces of degree $2j$ for all $j \\geq 1$. For $g \\geq 0$, we also write\n\\begin{equation}\\label{defn_v_f_g}\nv(\\mathbf{f}, g) = 2-2g+\\sum_{j \\geq 1} (j-1)f_j.\n\\end{equation}\nBy the Euler formula, a bipartite map with genus $g$ and face degrees described by $\\mathbf{f}$ exists if and only if $v(\\mathbf{f}, g) \\geq 2$, and in this case $v(\\mathbf{f}, g)$ is the number of vertices of such a map. For such $\\mathbf{f}$ and $g$, we denote by $M_{\\mathbf{f},g}$ a uniform bipartite map with genus $g$ and $f_j$ faces of degree $2j$ for all $j \\geq 1$.\n\n\\begin{thm}\\label{univ_main_thm}\nLet $(\\mathbf{f}^{n})_{n \\geq 1}$ be a sequence of face degree sequences, and let $(g_n)$ be a sequence such that $v(\\mathbf{f}^{n}, g_n) \\geq 2$ for all $n \\geq 1$. We assume that $|\\mathbf{f}^n| \\to +\\infty$ when $n \\to +\\infty$ and that $\\frac{f^n_j}{|\\mathbf{f}^n|} \\to \\alpha_j$ for all $j \\geq 1$, where $\\sum_{j \\geq 1} j \\alpha_j=1$.\nWe also assume $\\frac{g_n}{|\\mathbf{f}^n|} \\to \\theta$, where $0 \\leq \\theta < \\frac{1}{2} \\sum_{j \\geq 1} (j-1) \\alpha_j$.\nFinally, assume that $\\sum_{j \\geq 1} j^2 \\alpha_j  <+\\infty$.\n\nThen we have the convergence in distribution\n\\[ M_{\\mathbf{f}^n, g_n} \\xrightarrow[n \\to +\\infty]{(d)} \\mathbb M_{\\mathbf{q}}\\]\nfor the local topology, where the weight sequence $\\mathbf{q}$ depends only on $\\theta$ and $\\left( \\alpha_j \\right)_{j \\geq 1}$, in an injective way.\n\\end{thm}\n\nLet us now make a few comments on the various assumptions of the main theorem.\n\\begin{itemize}\n\\item[$\\bullet$]\nWe first note that the assumption that $\\sum_{j \\geq 1} j \\alpha_j=1$ means that the proportion of the edges that are incident to a face with degree larger than $A$ goes to $0$ as $A \\to +\\infty$, uniformly in $n$. This is equivalent to saying that the root face stays almost surely finite in the limit, so this assumption is necessary to obtain a local limit with finite faces. If this assumption is waived, we expect to obtain different limit objects with infinitely many infinite faces.\n\\item[$\\bullet$]\nThe assumption $\\theta < \\frac{1}{2} \\sum_{j \\geq 1} (j-1) \\alpha_j$ means that the number of vertices is roughly proportional to $|\\mathbf{f}^n|$, so that the average degree of the vertices stays bounded. Therefore, it is also necessary in order to have a proper local limit with finite vertex degrees. Note that this assumption also implies $\\alpha_1<1$, i.e. it is not possible that almost all faces are $2$-gons.\n\\item[$\\bullet$]\nThe assumption $\\sum_{j \\geq 1} j^2 \\alpha_j  <+\\infty$ means that the \\emph{expected} degree of the root face stays finite in the limit. We do not expect this assumption to be necessary. However, one of the steps of our proof (the \"two-holes argument\" of Section~\\ref{sec_arg_deux_trous}) crucially requires a bound on the tail of the degrees of the faces.\n\\end{itemize}\nFinally, the application associating the weight sequence $\\mathbf{q}$ given by Theorem~\\ref{univ_main_thm} to $\\left( \\theta, \\left( \\alpha_j \\right)_{j \\geq 1}\\right)$ is surjective in the sense that every IBPM $\\mathbb M_{\\mathbf{q}}$ for which the degree of the root face has finite expectation can be obtained as a local limit through Theorem~\\ref{univ_main_thm}.\n\n\\paragraph{The heavy tail case.}\nAlthough we could not remove the assumption $\\sum_{j \\geq 1} j^2 \\alpha_j$ in Theorem~\\ref{univ_main_thm}, most of the steps of the proof do not require this assumption. In particular, we can still obtain the following partial result in the general case.\n\n\\begin{thm}\\label{thm_main_more_general}\nLet $(\\mathbf{f}^{n})_{n \\geq 1}$ be a sequence of face degree sequences, and let $(g_n)$ be a sequence such that $v(\\mathbf{f}^{n}, g_n) \\geq 2$ for all $n \\geq 1$. We assume that $|\\mathbf{f}^n| \\to +\\infty$ when $n \\to +\\infty$ and that $\\frac{f^n_j}{|\\mathbf{f}^n|} \\to \\alpha_j$ for all $j \\geq 1$, where $\\sum_{j \\geq 1} j \\alpha_j=1$.\nWe also assume $\\frac{g_n}{|\\mathbf{f}^n|} \\to \\theta$, where $0 \\leq \\theta < \\frac{1}{2} \\sum_{j \\geq 1} (j-1) \\alpha_j$.\n\nThen the sequence of random maps $\\left( M_{\\mathbf{f}^n, g_n} \\right)_{n \\geq 1}$ is tight for the local topology. Moreover, all its subsequential limits are of the form $\\mathbb M_{\\mathbf{Q}}$, where $\\mathbf{Q}$ is a random Boltzmann weight sequence.\n\\end{thm}\n\n\\paragraph{A parametrization of Infinite Bipartite Boltzmann Planar Maps.}\nAs explained briefly above, we also provide a new parametrization of the Boltzmann weight families $\\mathbf{q}$ associated to an IBPM: instead of directly using the Boltzmann weights $q_j$, we parametrize them according to the law of the degree of the root face.\n\n\\begin{thm}\\label{thm_prametrization_rootface}\nLet $\\boldsymbol{\\alpha}$ be a probability measure on $\\mathbb N^*$. Then the set of IBPM for which the half-degree of the root face has law $\\boldsymbol{\\alpha}$ forms a one parameter family $\\left( \\mathbb M_{\\mathbf{q}^{(\\omega)}} \\right)_{\\omega \\geq 1}$. Moreover, $\\mathbf{q}^{(\\omega)}$ is critical if and only if $\\omega=1$, and the vertex degrees in $\\mathbb M_{\\mathbf{q}^{(\\omega)}}$ go to infinity when $\\omega \\to +\\infty$.\n\\end{thm}\n\nIn particular, the existence of hyperbolic Boltzmann maps with arbitrarily heavy-tailed face degrees answers a question from~\\cite{C-StFlour}, that was not settled by the results of \\cite[Appendix C]{B18these}. Moreover, we can think of $\\mathbf{q}^{(\\omega)}$ as interpolating between a critical non-hyperbolic map, and a degenerate map with infinite vertex degrees.\n\n\\paragraph{Asymptotic enumeration.}\nLike for triangulations, the most natural way to try to prove Theorem~\\ref{univ_main_thm} would be to obtain precise asymptotics on the counting of maps with prescribed genus and face degrees, in order to mimic classical arguments going back to~\\cite{AS03}. However, such asymptotics are not available and seem difficult to obtain. On the other hand, just like in~\\cite{BL19} for triangulations, once Theorem~\\ref{univ_main_thm} is proved, applying the arguments of~\\cite{AS03} \"backwards\" allows to obtain a result about convergence of the ratio when we add one face of fixed degree. We denote by $\\beta_g(\\mathbf{f})$ the number of bipartite maps of genus $g$ with face degrees prescribed by $\\mathbf{f}$.\n\n\\begin{corr}\\label{prop_cv_ratio}\nLet $\\left( \\mathbf{f}^{n} \\right)_{n \\geq 0}$ and $\\left( g_n \\right)_{n \\geq 0}$ be such that $\\frac{g_n}{n}\\to\\theta$ and $\\frac{f^n_j}{|\\mathbf{f}^n|}\\to \\alpha_j$ for all $j \\geq 1$. We assume that $\\sum j \\alpha_j=1$, that $0 \\leq \\theta < \\frac{1}{2} \\sum (j-1) \\alpha_j$ and that $\\sum j^2 \\alpha_j < +\\infty$. We recall that by Theorem~\\ref{univ_main_thm}, there is a weight sequence $\\mathbf{q}$ such that $M_{\\mathbf{f}^{n},g_n}$ converges locally to $\\mathbb{M}_{\\mathbf{q}}$. Then for all $j \\geq 1$, we have\n\\begin{equation}\\label{eq_cv_ratio}\n\\frac{\\beta_{g_n}(\\mathbf{f}^{n}-\\mathbf{1}_j)}{\\beta_{g_n}(\\mathbf{f}^{n})} \\xrightarrow[n \\to +\\infty]{} C_2(\\mathbf{q})q_j.\n\\end{equation}\n\\end{corr}\n\nWe also believe that the following is true:\n\n\\begin{conj}\\label{thm_univ_asympto}\nLet $(\\mathbf{f}^{n})_{n \\geq 1}$ be a sequence of face degree sequences, and let $(g_n)$ be a sequence such that $v(\\mathbf{f}^{n}, g_n) \\geq 2$ for all $n \\geq 1$. We assume that $|\\mathbf{f}^n| \\to +\\infty$ when $n \\to +\\infty$ and that $\\frac{f^n_j}{|\\mathbf{f}^n|} \\to \\alpha_j$ for all $j \\geq 1$, where $\\sum_{j \\geq 1} j \\alpha_j=1$.\nWe also assume $\\frac{g_n}{|\\mathbf{f}^n|} \\to \\theta$, where $0 \\leq \\theta \\leq \\frac{1}{2} \\sum_{j \\geq 1} (j-1) \\alpha_j$.\nFinally, assume that $\\sum_{j \\geq 1} j^2 \\alpha_j  <+\\infty$.\nThen\n\\[\\beta_{g_n}(\\mathbf{f}^{n})= |\\mathbf{f}^n|^{2g_n} \\exp \\left( \\varphi \\left( \\theta,\\left( \\alpha_j \\right)_{j \\geq 1} \\right) |\\mathbf{f}^n| + o \\left( |\\mathbf{f}^n| \\right) \\right),\\]\nwhere $\\varphi$ is some function.\n\\end{conj}\n\nMore precisely, in~\\cite{BL19}, the proof consists of first using the analog of Corollary~\\ref{prop_cv_ratio} to estimate the ratio between triangulations with any genus and triangulations with a genus close to maximal (say with $\\varepsilon |\\mathbf{f}|$ vertices). To count such triangulations, we contracted a spanning tree to reduce the problem to triangulations with only one vertex, for which explicit formulas are known. This \"contraction\" is the step that is difficult to extend to our setting here: while for triangulations we simply obtained a triangulation with less faces, here the impact on the face degrees may become much more complex. This is why we leave the question as open.\n\n\\paragraph{Sketch of the proof of Theorem~\\ref{univ_main_thm}: common points and differences with the triangular case.}\nThe proof is a combination of combinatorial and probabilistic ideas. It follows the same global strategy as in \\cite{BL19}, which shows the robustness of the approach of~\\cite{BL19}. However, new difficulties arise at each of the steps, which makes the overall proof much longer. \n\nMore precisely, the first step consists of showing the tightness of $M_{\\mathbf{f}^n, g_n}$. This follows from a \\emph{bounded ratio lemma} (Lemma~\\ref{lem_BRL}), stating that under certain assumptions the ratio $\\frac{\\beta_g \\left(\\mathbf{f}+\\mathbf{1}_j \\right)}{\\beta_g \\left(\\mathbf{f} \\right)}$ is bounded. As in~\\cite{BL19}, this lemma is established using surgery operations to remove a face, but this surgery can affect a larger region than in the triangular case, which makes it more elaborated. The second step is to prove that any subsequential limit is planar and one-ended. This relies on the recurrence proved by the second author in~\\cite{Lo19}, and only requires minor adaptations compared to~\\cite{BL19}. We then notice that any subsequential limit enjoys a weak spatial Markov property, which implies that it must be of the form $\\mathbb M_{\\mathbf{Q}}$, for some random weight sequence $\\mathbf{Q}$. This part is also similar to~\\cite{BL19}, although additional technicalities arise. All these arguments do not use any assumption on the tail of the degrees of the faces, and prove Theorem~\\ref{thm_main_more_general}.\n\nThe end of the proof consists in showing that $\\mathbf{Q}$ is actually deterministic. As in~\\cite{BL19}, this step relies on a surgery argument called the \\emph{two holes argument}, for which we need to explore two pieces of maps with the exact same boundary length. This is where the assumption $\\sum j^2 \\alpha_j<+\\infty$ is crucial: without it, when we explore a piece of map \"face by face\", the perimeter makes large positive jumps and misses too many values. Another major difference with \\cite{BL19} is in the last step, where we match the average degree in finite models (computed with the Euler formula) and in infinite ones. In particular, we need to argue that a weight sequence $\\mathbf{q}$ is determined by the law of the root face of $\\mathbb{M}_{\\mathbf{q}}$ and the average vertex degree. While for triangulations we were able to obtain an explicit formula for the average vertex degree, this is not the case here. Therefore, we need to develop new arguments making use of the local limit results obtained earlier in the paper. This is also the reason why the link between $\\theta$, $\\left( \\alpha_j\\right)$ and $\\mathbf{q}$ in Theorem~\\ref{univ_main_thm} is not explicit.\n\n\\paragraph{Weakly Markovian bipartite maps.}\nJust like in~\\cite{BL19}, the argument showing that a subsequential limit is a mixture of IBPM is a result of independent interest, so we give its statement here. Let $M$ be a random infinite, one-ended, bipartite planar map. We call $M$ \\emph{weakly Markovian} if for any finite map $m$ with one hole, the probability $m \\subset M$ only depends on the perimeter of $m$ and on the family of degrees of its internal faces. We denote by $\\mathcal{Q}_h$ the set of weight sequences $\\mathbf{q}$ for which $\\mathbb{M}_{\\mathbf{q}}$ exists, and by $\\mathcal{Q}_f \\subset \\mathcal{Q}_h$ the set of those $\\mathbf{q}$ for which the expected degree of the root face in $\\mathbb{M}_{\\mathbf{q}}$ is finite (this will be useful to handle the last assumption in Theorem~\\ref{univ_main_thm}).\n\n\\begin{thm}\\label{thm_weak_Markov_general}\nLet $M$ be a weakly Markovian infinite, one-ended, bipartite random planar map. Then there is a random weight sequence $\\mathbf{Q} \\in \\mathcal{Q}_h$ such that $M$ has the same distribution as $\\mathbb M_{\\mathbf{Q}}$. Moreover, if the degree of the root face of $M$ has finite expectation, then $\\mathbf{Q} \\in \\mathcal{Q}_f$ almost surely.\n\\end{thm}\n\n\\paragraph{Structure of the paper.}\nIn Section~\\ref{sec_univ_prelim}, we review basic definitions on maps, and previous combinatorial results that will be used in all the paper. We also introduce the IBPM and describe various parametrizations of the set of IBPMs, and in particular prove Theorem~\\ref{thm_prametrization_rootface}. In Section~\\ref{sec_univ_tight}, under the assumptions of Theorem~\\ref{thm_main_more_general} (i.e. without the assumption on the tail of face degrees), we prove that the maps $M_{\\mathbf{f}^n, g_n}$ are tight for the local topology, and that any subsequential limit is a.s. planar and one-ended. Section~\\ref{sec_univ_markov} is devoted to the proof of Theorem~\\ref{thm_weak_Markov_general}, which implies that any subsequential limit of $M_{\\mathbf{f}^n, g_n}$ is an IBPM with random parameters. This is sufficient to prove Theorem~\\ref{thm_main_more_general}. In Section~\\ref{sec_univ_end}, we conclude the proof of Theorem~\\ref{univ_main_thm} by showing that the parameters are deterministic and depend only on $\\theta$ and $\\left( \\alpha_j \\right)_{j \\geq 1}$. In Section~\\ref{sec_univ_asymp}, we deduce the combinatorial estimate of Corollary~\\ref{prop_cv_ratio} from Theorem~\\ref{univ_main_thm}. Finally, the Appendices contain the proofs of some technical results.\n\n\\tableofcontents\n\n\\newpage\n\n\\section*{Index of notations} \\label{sec_univ_index}\n\n\\addcontentsline{toc}{section}{\\protect\\numberline{}Index of notations}\n\nIn general, we will we will use lower case letters such as $m$ to denote deterministic objects or quantities, upper case letters such as $M$ for random objects and $\\mathtt{mathcal}$ letters such as $\\mathcal{M}$ for sets of objects. We will use $\\mathtt{mathbf}$ characters such as $\\mathbf{q}$ for sequences, and normal characters such as $q_j$ for their terms.\n\n\\begin{itemize}\n\\item\n$\\mathbf{f}=(f_j)_{j \\geq 1}$: denotes a face degree sequence ($\\mathbf{F}$ will denote a random face degree sequence).\n\\item\n$g$: will denote the genus.\n\\item\n$\\mathcal{B}_{g}(\\mathbf{f})$: set of finite bipartite maps with genus $g$ and $f_j$ faces of degree $2j$ for all $j \\geq 1$.\n\\item\n$\\beta_{g}(\\mathbf{f})$: cardinality of $\\mathcal{B}_{g}(\\mathbf{f})$.\n\\item\n$M_{\\mathbf{f},g}$: uniform random map in $\\mathcal{B}_{g}(\\mathbf{f})$.\n\\item\n$|\\mathbf{f}|=\\sum_{j \\geq 1} j f_j$ (i.e. the number of edges of a map in $\\mathcal{B}_{g}(\\mathbf{f})$).\n\\item\n$v(\\mathbf{f}, g) = 2-2g+\\sum_{j \\geq 1} (j-1)f_j$ (i.e. the number of vertices of a map in $\\mathcal{B}_{g}(\\mathbf{f})$).\n\\item\n$\\overline{\\mathcal{B}}$: space of finite or infinite bipartite maps with finite vertex degrees, equipped with the local distance $d_{\\mathrm{loc}}$.\n\\item\n$\\overline{\\mathcal{B}}^*$: space of finite or infinite bipartite maps with finite or infinite vertex degrees, equipped with the dual local distance $d_{\\mathrm{loc}}^*$.\n\\item\n$\\theta$: limit value of $\\frac{g}{|\\mathbf{f}|}$ when $|\\mathbf{f}| \\to +\\infty$.\n\\item\n$\\mathbf{q}=(q_j)_{j \\geq 1}$: denotes a weight sequence ($\\mathbf{Q}$ denotes a random weight sequence).\n\\item\n$\\mathbb M_{\\mathbf{q}}$: the infinite bipartite Boltzmann planar map with weight sequence $\\mathbf{q}$.\n\\item\n$W_p(\\mathbf{q})$: partition function of finite Boltzmann bipartite maps of the $2p$-gon with weights $\\mathbf{q}$.\n\\item\n$\\mathcal{Q}=[0,1]^{\\mathbb N^*}$.\n\\item\n$\\mathcal{Q}^*=\\left\\{ \\mathbf{q}=(q_j)_{j \\geq 1} \\in \\mathcal{Q} | \\exists j \\geq 2, q_j>0 \\right\\}$.\n\\item\n$\\mathcal{Q}_a$: set of admissible families of Boltzmann weights $\\mathbf{q}$, i.e. such that $W_p(\\mathbf{q})<+\\infty$.\n\\item\n$\\mathcal{Q}_h$: set of Boltzmann weights for which $\\mathbb M_{\\mathbf{q}}$ exists. We have $\\mathcal{Q}_h \\subset \\mathcal{Q}_a \\cap \\mathcal{Q}^*$.\n\\item\n$\\mathcal{Q}_f$: set of Boltzmann weights $\\mathbf{q} \\in \\mathcal{Q}_h$ for which the expectation of the degree of the root face of $\\mathbb M_{\\mathbf{q}}$ is finite.\n\\item\n$c_{\\mathbf{q}}$: for $\\mathbf{q} \\in \\mathcal{Q}_a$, denotes the solution of the equation \\[\\sum_{j \\geq 1} q_j \\frac{1}{4^{j-1}} \\binom{2j-1}{j-1} c_{\\mathbf{q}}^{j-1} = 1-\\frac{4}{c_{\\mathbf{q}}}.\\]\n\\item\n$\\nu_{\\mathbf{q}}(i)= \\left\\{ \n\\begin{array}{ll}\nq_{i+1} \\, c_{\\mathbf{q}}^{i} & \\mbox{ if $i \\geq 0$,} \\\\\n2 W_{-1-i}(\\mathbf{q}) \\, c_{\\mathbf{q}}^i & \\mbox{ if $i \\leq -1$.}\n\\end{array} \\right.$. Step distribution of the random walk associated to the perimeter process of a finite $\\mathbf{q}$-Boltzmann planar map.\n\\item\n$\\omega_{\\mathbf{q}} \\geq 1$: for $\\mathbf{q} \\in \\mathcal{Q}_h$, denotes the solution (other than $1$, unless $\\mathbf{q}$ is critical) of \\[\\sum_{i \\in \\mathbb Z} \\omega^i \\nu_{\\mathbf{q}}(i)=1.\\]\n\\item\n$\\widetilde{\\nu}_{\\mathbf{q}}(i)=\\omega_{\\mathbf{q}}^i \\nu_{\\mathbf{q}}(i)$. Step distribution of the random walk on $\\mathbb Z$ associated to the peeling process of $\\mathbb M_{\\mathbf{q}}$.\n\\item\n$C_p(\\mathbf{q})$: for $\\mathbf{q} \\in \\mathcal{Q}_h$ and $p \\geq 1$, constants such that \\[\\P \\left( m \\subset \\mathbb M_{\\mathbf{q}} \\right)=C_p(\\mathbf{q}) \\times \\prod_{f \\in m} q_{\\mathrm{deg}(f)\/2}\\] for every finite map $m$ with one hole of perimeter $2p$.\n\\item\n$h_p(\\omega)=\\sum_{i=0}^{p-1} (4 \\omega)^{-i} \\binom{2i}{i}$. We have $C_p(\\mathbf{q})= (c_{\\mathbf{q}} \\omega_{\\mathbf{q}})^{p-1} h_p(\\omega_{\\mathbf{q}})$. Also, the perimeter process associated to a peeling exploration of $\\mathbb M_{\\mathbf{q}}$ is a Doob transform of the $\\widetilde{\\nu}_{\\mathbf{q}}$-random walk by the harmonic function $\\left( h_p(\\omega_{\\mathbf{q}}) \\right)_{p \\geq 1}$.\n\\item\n$a_j(\\mathbf{q})=\\frac{1}{j} \\P \\left( \\mbox{the degree of the root face of $\\mathbb M_{\\mathbf{q}}$ is $2j$}\\right)$ for all $j \\geq 1$.\n\\item\n$\\alpha_j$: denotes a possible value of $a_j(\\mathbf{q})$, or the limit of the ratio $\\frac{f_j}{|\\mathbf{f}|}$ when $|\\mathbf{f}| \\to +\\infty$. We will always have $\\sum_{j \\geq 1} j \\alpha_j=1$ and $\\alpha_1<1$.\n\\item\n$\\mathbf{q}^{(\\omega)}$: once $\\left( \\alpha_j \\right)_{j \\geq 1}$ has been fixed, denotes the weight sequence for which $\\omega_{\\mathbf{q}}=\\omega$, and the law of the degree of the root face is described by $\\left( \\alpha_j \\right)_{j \\geq 1}$.\n\\item\n$\\mathcal{A}$: denotes a peeling algorithm.\n\\item\n$\\mathcal{E}_t^{\\mathcal{A}}(m)$: explored map after $t$ filled-in peeling steps on the map $m$ using algorithm $\\mathcal{A}$. This is a finite map with holes.\n\\item\n$P_t, V_t$: denote respectively the perimeter (i.e. the half-length of the boundary of the hole) and the volume (i.e. the total number of edges) of the explored map after $t$ steps during a peeling exploration.\n\\item\n$d(\\mathbf{q})=\\mathbb E \\left[ \\left( \\mbox{degree of the root vertex in $\\mathbb M_{\\mathbf{q}}$} \\right)^{-1} \\right]$.\n\\item\n$r_j(\\mathbf{q})=\\left( c_{\\mathbf{q}} \\omega_{\\mathbf{q}} \\right)^{j-1} q_j=\\lim_{t \\to +\\infty} \\frac{1}{t} \\sum_{i=0}^{t-1} \\mathbbm{1}_{P_{i+1}-P_i=j-1}$ for $\\mathbf{q} \\in \\mathcal{Q}_h$ and $j \\geq 1$, where $P$ is the perimeter process associated to a peeling exploration of $\\mathbb M_{\\mathbf{q}}$ (see Proposition~\\ref{prop_q_as_limit}).\n\\item\n$r_{\\infty}(\\mathbf{q}) = \\frac{ \\left(\\sqrt{\\omega_{\\mathbf{q}}}-\\sqrt{\\omega_{\\mathbf{q}}-1} \\right)^2}{2 \\sqrt{\\omega_{\\mathbf{q}}(\\omega_{\\mathbf{q}}-1)}} = \\lim_{n \\to +\\infty} \\frac{V_n-2P_n}{n}$ for $\\mathbf{q} \\in \\mathcal{Q}_h$ and $j \\geq 1$, where $P$ and $V$ are the perimeter and volume processes associated to a peeling exploration of $\\mathbb M_{\\mathbf{q}}$ (see Proposition~\\ref{prop_q_as_limit}).\n\\end{itemize}\n\n\\newpage\n\n\\section{Preliminaries}\\label{sec_univ_prelim}\n\nOur purpose in this section is to recall basic definitions related to maps, local topology and peeling explorations as well as combinatorial results from previous works, and to introduce precisely the infinite objects that will appear in this paper.\n\n\\subsection{Definitions: maps and local topology}\n\\label{subsec_univ_defns}\n\\label{subsec_local_topology_univ}\n\n\\paragraph{Maps.} A (finite or infinite) \\emph{map} $M$ is a way to glue a finite or countable collection of finite oriented polygons, called the \\emph{faces}, along their edges in a connected way. By forgetting the faces of $M$ and looking only at its vertices and edges, we obtain a graph $G$ (if $M$ is infinite, then $G$ may have vertices with infinite degree). The maps that we consider will always be \\emph{rooted}, i.e. equipped with a distinguished oriented edge called the \\emph{root edge}. The face on the right of the root edge is the \\emph{root face}, and the vertex at the start of the root edge is the \\emph{root vertex}.\n\nThe \\emph{dual map} of a map $m$ is the map $m^*$ whose vertices are the faces of $m$, and whose edges are the dual edges to those of $m$. We root $m^*$ at the oriented edge crossing the root edge of $m$ from left to right (see Figure~\\ref{fig_map_dual}).\nIf the number of faces is finite, then $M$ is always homeomorphic to an orientable topological surface, so we can define the genus of $M$ as the genus of this surface. In particular, we call a map \\emph{planar} if it has genus $0$.\n\nA \\emph{bipartite map} is a rooted map where it is possible to color every vertex in black or white without any monochromatic edge. By convention, we may assume that the root is always oriented from white to black, and each edge of the map has a natural orientation from white to black. In a bipartite map, all faces have an even degree.  In what follows, we will only deal with bipartite maps (except when dealing with dual maps). Therefore, we will not always specify that the map we consider is bipartite.\n\nFor every $\\mathbf{f}=(f_j)_{j \\geq 1}$ and $g\\geq 0$, we will denote by $\\mathcal{B}_g(\\mathbf{f})$ the set of bipartite maps of genus $g$ with exactly $f_j$ faces of degree $2j$ for all $j\\geq 1$.\nA map of $\\mathcal{B}_g(\\mathbf{f})$ has $|\\mathbf{f}| =\\sum_{j\\geq 1} j f_j$ edges, $\\sum_{j\\geq 1} f_j$ faces and $v(\\mathbf{f}, g)=2-2g+\\sum_{j\\geq 1} (j-1)f_j$ vertices by the Euler formula. In particular, such a map exists if and only if $v(\\mathbf{f},g) \\geq 2$. We will denote by $\\beta_g(\\mathbf{f})$ the cardinality of $\\mathcal{B}_g(\\mathbf{f})$.\n\n\\begin{figure}\n\\center\n\\includegraphics[scale=0.5]{carte_duale}\n\\caption{A map (in black) and its dual (in blue). The arrows mark the roots.}\\label{fig_map_dual}\n\\end{figure}\n\n\\paragraph{Maps with boundaries.} We will need to consider two different notions of bipartite maps with boundaries, that we call \\emph{maps with holes} and \\emph{maps of multi-polygons}. Roughly speaking, the first ones will be used to describe a small neighbourhood of the root in a larger map, and the second ones to describe the complement of this neighbourhood. Note that, since we will only consider bipartite maps in this work, we assume in both definitions that the maps are bipartite.\n\n\\begin{defn}\nA \\emph{map with holes} is a finite, bipartite map with a set of marked faces (called \\emph{holes}) such that:\n\\begin{itemize}\n\\item\nthe boundary of each hole is a simple cycle,\n\\item\nthe boundaries of the different holes are vertex-disjoint,\n\\item\nthe adjacency graph of the \\emph{internal faces} (i.e. the faces that are not holes) is connected,\n\\item\nthe root edge may be any oriented edge of the map.\n\\end{itemize}\nBy convention, the map consisting of two vertices joined by a single edge is a map with one hole and no internal face.\nIf $m$ is a map with holes, we denote by $\\partial m$ its boundary, i.e. the union of the boundaries of its holes.\n\\end{defn}\n\n\\begin{defn}\nLet $\\ell \\geq 1$ and $p_1, p_2, \\dots, p_{\\ell} \\geq 1$. A \\emph{map of the $(2p_1, \\dots, 2p_{\\ell})$-gon} is a finite or infinite bipartite map with $\\ell$ marked oriented edges $(e_i)_{1 \\leq i \\leq \\ell}$, such that:\n\\begin{itemize}\n\\item\n$e_1$ is the root edge,\n\\item\nthe faces on the right of the $e_i$ are distinct,\n\\item\nfor all $1 \\leq i \\leq \\ell$, the face on the right of $e_i$ has degree $2p_i$.\n\\end{itemize}\nThe faces on the right of the marked edges are called \\emph{external faces}, and the other ones are called the \\emph{internal faces}.\nWe denote by $\\mathcal{B}^{(p_1,p_2,\\dots,p_{\\ell})}_g(\\mathbf{f})$ the set of bipartite maps of the $(2p_1,2p_2,\\dots,2p_{\\ell})$-gon of genus $g$ with interior faces given by $\\textbf{f}$. We also denote by $\\beta^{(p_1,p_2,\\dots,p_{\\ell})}_g(\\mathbf{f})$ its cardinal, with the convention that $\\beta_g^{(0)}(\\mathbf{f})$ is $1$ if $g=0$ and $\\mathbf{f}=\\mathbf{0}$, and $0$ otherwise.\n\\end{defn}\n\nNote that, in this second definition, we do not ask that the boundaries are simple or disjoint. The convention for the $0$-gon can be interpreted as saying that the only map of the $0$-gon is the map with $1$ vertex, no edge and no internal face.\n\n\n\\paragraph{Map inclusion.}\nGiven a map $m$, let $m^*$ be its dual map. Let $\\mathfrak{e}$ be a finite, connected subset of edges of $m^*$ such that the root vertex of $m^*$ is incident to $\\mathfrak{e}$. To $\\mathfrak{e}$, we associate the map $m_{\\mathfrak{e}}$ that is obtained by gluing the faces of $m$ corresponding to the vertices of $m^*$ incident to $\\mathfrak{e}$ along the dual of the edges of $\\mathfrak{e}$ (see Figure~\\ref{fig_univ_lazy_inclusion}). Note that $m_{\\mathfrak{e}}$, once rooted at the root edge of $m$, is a map with holes. We will refer to $m_{\\mathfrak{e}}$ as the \\emph{submap of $m$ spanned by $\\mathfrak{e}$}.\n\nIf $m'$ is a map with holes and $m$ is a (finite or infinite) map, we write\n\\[m' \\subset m\\]\nif $m'$ can be obtained from $m$ by the procedure described above. By convention, we also write $m' \\subset m$ if $m'$ is the trivial map with two vertices and one edge, or if $m'$ consists of a simple cycle with the same perimeter as the root face of $m$ (which corresponds to the case $\\mathfrak{e} = \\emptyset$).\n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[scale=1]{lazy_inclusion}\n\\caption{Inclusion of bipartite maps, on an example. On the right, the map $m$ and, in red, the set of dual edges $\\mathfrak{e}$. On the left, the map $m_{\\mathfrak{e}}$.}\\label{fig_univ_lazy_inclusion}\n\\end{figure}\n\nEquivalently, we have $m' \\subset m$ if $m$ can be obtained from $m'$ by gluing one or several maps of multipolygons in the holes of $m'$. We highlight that this definition of map inclusion is taken from \\cite{C-StFlour} and is tailored for the \\emph{lazy peeling process} of \\cite{Bud15}. More precisely, maps of multipolygons may not have simple nor disjoint boundaries, so if $m' \\subset m$, it is possible that two boundary edges of $m'$ actually coincide in $m$.\n\n\\paragraph{Local convergence and dual local convergence.}\nThe goal of this paragraph is to recall the definition of local convergence in a setting that is not restricted to planar maps. We denote by $\\overline{\\mathcal{B}}$ the set of finite or infinite bipartite maps in which all the vertices have finite degrees. A map $m$ is naturally equipped with a \\emph{graph distance} $d_m$ on the set of its vertices. If $m \\in \\overline{\\mathcal{B}}$, for every $r \\geq 1$, we denote by $B_r(m)$ the submap of $m$ spanned by the duals of those edges of $m$ which have an endpoint at distance $d_m$ at most $r-1$ from the root vertex. The map $B_r(m)$ is then a map with holes. We also write $B_0(m)$ for the trivial bipartite map consisting of only one edge.\n\nFor any two maps $m,m' \\in \\overline{\\mathcal{B}}$, we write\n\\[d_{\\mathrm{loc}}(m,m')=\\left( 1+\\max \\{r \\geq 0 | B_r(m)=B_r(m')\\} \\right)^{-1}.\\]\nThis is the \\emph{local distance} on $\\overline{\\mathcal{B}}$. As in the planar case, the space $\\overline{\\mathcal{B}}$ is a Polish space and is the completion of the space of finite bipartite maps for $d_{\\mathrm{loc}}$. However, this space is not compact, since $B_1(m)$ may take infinitely many values.\n\nIn our tightness argument, it will be more convenient to first work with a weaker notion of convergence which we call the \\emph{dual local convergence}. We denote by $\\overline{\\mathcal{B}}^*$ the set of finite or infinite bipartite maps (regardless of whether vertex degrees are finite or not).\nLet $m \\in \\overline{\\mathcal{B}}^*$, and let $d_{m^*}$ be the graph distance on its dual. For $r \\geq 1$, we denote by $B_r^{*}(m)$ the submap of $m$ spanned by those edges of $m^*$ which are incident to a face of $m$ lying at distance $d_{m^*}$ at most $r-1$ from the root face of $m$. By convention, let also $B_0^*(m)$ be the map consisting of a simple cycle with the same length as the boundary of the root face. Like $B_r(m)$, the \"ball\" $B^*_r(m)$ is a finite map with holes. For any $m,m' \\in \\overline{\\mathcal{B}}^*$, we write\n\\[d_{\\mathrm{loc}}^*(m,m')=\\left( 1+\\max \\{r \\geq 0 | B_r^*(m)=B_r^*(m')\\} \\right)^{-1}.\\]\nWe call $d^*_{\\mathrm{loc}}$ the \\emph{dual local distance}. Then $\\overline{\\mathcal{B}}^*$ is a Polish space for $d_{\\mathrm{loc}}^*$ and is the completion of the set of finite bipartite maps.\n\n\n\n\n\n\n\n\n\nThe reason why we introduced $d_{\\mathrm{loc}}^*$ is that it will be very easy to obtain tightness for this distance. This will allow us to work directly on infinite objects and deduce tightness for $d_{\\mathrm{loc}}$ later. Tightness for $d_{\\mathrm{loc}}^*$ will be deduced from the next result.\n\n\\begin{lem}\\label{lem_tight_degree_in_ball}\nLet $A(\\cdot)$ be a function from $(0,1)$ to $\\mathbb N$ and let $r \\geq 1$. There is a function $A_r(\\cdot)$ from $(0,1)$ to $\\mathbb N$ with the following property. Let $G$ be a stationary (for the simple random walk) random graph such that, for all $\\varepsilon>0$, we have\n\\[ \\P \\left( \\mathrm{deg}_G(\\rho)>A(\\varepsilon) \\right) \\leq \\varepsilon,\\]\nwhere $\\rho$ is the root vertex. Then for all $\\varepsilon>0$, we have\n\\[ \\P \\left( \\max_{x \\in B_r(G)} \\mathrm{deg}_G(x)>A_r(\\varepsilon) \\right) \\leq \\varepsilon,\\]\nwhere $B_r(G)$ is the ball of radius $r$ centered at the root vertex in $G$.\n\\end{lem}\n\n\\begin{proof}\nThis result goes back to~\\cite{AS03}. More precisely, although not stated explicitely as such, it is proved by induction on $r$ in the proof of tightness of uniform triangulations for the local topology~\\cite[Lemma 4.4]{AS03}. See also~\\cite[Theorem 3.1]{BLS13} for a general statement with minimal assumptions.\n\\end{proof}\n\nFrom here, we easily obtain tightness for $d_{\\mathrm{loc}}^*$ in our setting.\n\n\\begin{lem}\\label{lem_easy_dual_convergence}\nLet $\\mathbf{f}^{n}$ be face degree sequences such that $\\frac{1}{\\left| \\mathbf{f}^{n} \\right|} \\sum_{j> A} j f_j^{n}\\rightarrow 0$ as $A \\to +\\infty$ uniformly in $n$, and let $(g_n)$ be any sequence such that $\\mathcal{B}_{g_n}(\\mathbf{f}^{n}) \\ne \\emptyset$ for every $n$. Recall that $M_{\\mathbf{f}^n, g_n}$ is a uniform map in $\\mathcal{B}_{g_n}(\\mathbf{f}^{n})$. Then $(M_{\\mathbf{f}^n, g_n})$ is tight for $d_{\\mathrm{loc}}^*$.\n\\end{lem}\n\n\\begin{proof}\nLet $M_{\\mathbf{f}^n, g_n}^*$ be the dual map of $(M_{\\mathbf{f}^n, g_n})$. Since $M_{\\mathbf{f}^n, g_n}$ is invariant under rerooting at a uniform edge, the probability that the root vertex of $M_{\\mathbf{f}^n, g_n}^*$ has degree $2j$ is equal to $\\frac{j f^{n}_j}{\\left| \\mathbf{f}^{n} \\right|}$. Therefore, it follows from the assumption of the lemma that the root degree of $M_{\\mathbf{f}^n, g_n}^*$ is tight. Moreover $M_{\\mathbf{f}^n, g_n}^*$ is invariant by rerooting along the simple random walk. Therefore, by Lemma~\\ref{lem_tight_degree_in_ball}, for every $r \\geq 1$, the maximal degree in the ball of radius $r$ centered at the root in $M_{\\mathbf{f}^n, g_n}^*$ is tight. This implies tightness for $d_{\\mathrm{loc}}^*$.\n\\end{proof}\n\nFinally, as in \\cite{BL19}, tightness for $d_{\\mathrm{loc}}$ will be deduced from tightnesss for $d^*_{\\mathrm{loc}}$ using the following result (the proof is the same as for triangulations, and is therefore omitted).\n\n\\begin{lem}\\label{lem_dual_convergence_univ}\nLet $(m_n)$ be a sequence of maps of $\\overline{\\mathcal{B}}$. Assume that\n\\[ m_n \\xrightarrow[n \\to +\\infty]{d_{\\mathrm{loc}}^*} m,\\]\nwith $m \\in \\overline{\\mathcal{B}}$ (i.e. with finite vertex degrees). Then $m_n \\to m$ for $d_{\\mathrm{loc}}$ as $n \\to +\\infty$.\n\\end{lem}\n\n\\subsection{The lazy peeling process of bipartite maps}\n\\label{subsec_lazy_peeling}\n\nWe now recall the definition of the \\emph{lazy peeling process} of maps introduced in \\cite{Bud15} (see also \\cite{C-StFlour} for an extensive study). We will make heavy use of this notion in our proofs.\n\nA \\emph{peeling algorithm} is a function $\\mathcal{A}$ that takes as input a finite bipartite map $m$ with holes, and that outputs an edge $\\mathcal{A}(m)$ on $\\partial m$ (i.e. on the boundary of one of the holes). Given an infinite, planar, one-ended bipartite map $m$ and a peeling algorithm $\\mathcal{A}$, we can define an increasing sequence $\\left( \\mathcal{E}_t^{\\mathcal{A}}(m) \\right)_{t \\geq 0}$ of maps with one hole, such that $\\mathcal{E}_t^{\\mathcal{A}}(m) \\subset m$ for every $t$, in the following way. First, the map $\\mathcal{E}_0^{\\mathcal{A}}(m)$ is the trivial map consisting of the root edge only. For every $t \\geq 0$, we call the edge $\\mathcal{A} \\left( \\mathcal{E}_t^{\\mathcal{A}}(m) \\right)$ the \\emph{peeled edge at time $t$}. Let $F_t$ be the face of $m$ on the other side of this peeled edge (i.e. the side incident to a hole in $\\mathcal{E}_t^{\\mathcal{A}}(m)$). There are two possible cases, as summed up on Figure~\\ref{fig_univ_lazy_peeling}:\n\\begin{itemize}\n\\item either $F_t$ doesn't belong to $\\mathcal{E}_t^{\\mathcal{A}}(m)$, and then $\\mathcal{E}_{t+1}^{\\mathcal{A}}(m)$ is the map obtained from $\\mathcal{E}_t^{\\mathcal{A}}(m)$ by gluing a simple face of size $\\mathrm{deg}(F_t)$ along $\\mathcal{A} \\left( \\mathcal{E}_t^{\\mathcal{A}}(m) \\right)$;\n\\item or $F_t$ belongs to $\\mathcal{E}_t^{\\mathcal{A}}(m)$. In that case, by planarity, there exists an edge $e_t \\in \\mathcal{E}_t^{\\mathcal{A}}(m)$ on the same hole as $\\mathcal{A} \\left( \\mathcal{E}_t^{\\mathcal{A}}(m) \\right)$ such that $e_t$ and $\\mathcal{A} \\left( \\mathcal{E}_t^{\\mathcal{A}}(m) \\right)$ are actually identified in $m$. The map $\\mathcal{E}_{t+1}^{\\mathcal{A}}(m)$ is then obtained from $\\mathcal{E}_t^{\\mathcal{A}}(m)$ by gluing $\\mathcal{A} \\left( \\mathcal{E}_t^{\\mathcal{A}}(m) \\right)$ and $e_t$ together and, if this creates a finite hole, by filling it in the same way as in $m$.\n\\end{itemize}\nSuch an exploration is called \\emph{filled-in}, because all the finite holes are filled at each step.\n\nLet us now discuss two different ways to define peeling explorations on finite or nonplanar maps. We first note that, if we do not fill the region in the second case, then the definition of a peeling exploration still makes sense for any finite or infinite map, with the only difference that the explored part may now have several holes. This is what we will call a \\emph{non-filled-in} peeling exploration, and this will only be used briefly in Section \\ref{subsec_planarity}.\n\nFinally, for a finite map $m$, we define a filled-in exploration using the following convention. Assume that the peeled edge at time $t$ is glued to another boundary edge of $\\mathcal{E}^{\\mathcal{A}}_t(m)$ and forms two holes:\n\\begin{itemize}\n\\item\nif these two holes are connected in $m \\backslash \\mathcal{E}^{\\mathcal{A}}_t(m)$ (which may occur if $m$ is not planar), we stop the exploration at time $t$;\n\\item\nif not, we obtain $\\mathcal{E}^{\\mathcal{A}}_{t+1}(m)$ by filling completely the hole which contains the smallest number of edges in $m$.\n\\end{itemize}\nNote that with these conventions, the map $\\mathcal{E}^{\\mathcal{A}}_t(m)$ always have exactly one hole. This definition will be used to compare peeling explorations of finite and infinite maps in Section \\ref{sec_univ_end}. At this point, the local planarity results from Section \\ref{sec_univ_tight} will allow us to assume that with high probability, the explorations are not stopped too early.\n\n\\begin{figure}\n\\center\n\\includegraphics[scale=1]{lazy_peeling}\n\\caption{The lazy peeling on an example. The peeled edge is in red. Either a new face is discovered (center case), or the chosen edge is glued to another boundary edge (right case, the glued edge is in blue and the filled part in pink).}\\label{fig_univ_lazy_peeling}\n\\end{figure}\n\n\\subsection{Combinatorial enumeration}\\label{subsec_prelim_combi}\n\n\\paragraph{Partition functions for Boltzmann planar maps.}\nBefore describing infinite Boltzmann models in detail, we recall well-known enumeration results in the finite, planar case. We write $\\mathcal{Q}=[0,1]^{\\mathbb N^*}$. Fix a sequence $\\mathbf{q} \\in \\mathcal{Q}$. The partition function of bipartite, planar maps of the $2p$-gon with Boltzmann weights $\\mathbf{q}$ is defined as\n\\[W_p(\\mathbf{q})=\\sum_{m} \\prod_{f \\in m} q_{\\mathrm{deg}(f)\/2},\\]\nwhere the sum spans over all planar bipartite maps $m$ of the $2p$-gon, and the product is over internal faces of $m$. We also denote by $W_p^{\\bullet}(\\mathbf{q})$ the \\emph{pointed} partition function, i.e. the sum obtained by multiplying the weight of a map $m$ by its total number of vertices. Note that $W_1(\\mathbf{q})$ can also be interpreted as the partition function of maps of the sphere.\n\nWe recall from \\cite{MM07} the classical necessary and sufficient condition for the finiteness of these partition functions. Given a weight sequence $\\mathbf{q} \\in \\mathcal{Q}$, let \n\\begin{equation*}\nf_{\\mathbf{q}}(x)=\\sum_{j \\geq 1} q_{j}\\binom{2 j-1}{j-1} x^{j-1}.\n\\end{equation*}\nIf the equation \n\\begin{equation}\\label{eq_univ_admissible}\nf_{\\mathbf{q}}(x)=1-\\frac{1}{x}\n\\end{equation}\nhas a positive solution $Z_{\\mathbf{q}}$ we call $\\mathbf{q}$ \\emph{admissible}, and write $c_{\\mathbf{q}}=4Z_{\\mathbf{q}}$. Then by results from \\cite{MM07}, for all $p \\geq 1$, the partition functions $W_p(\\mathbf{q})$ and $W_p^{\\bullet}(\\mathbf{q})$ are finite if and only if $\\mathbf{q}$ is admissible. Moreover, in this case, for $p \\geq 0$, we have\n\\begin{equation}\\label{eqn_exact_pointed_partition_function}\nW_p^{\\bullet}(\\mathbf{q}) = c_{\\mathbf{q}}^p \\times \\frac{1}{4^p} \\binom{2p}{p}.\n\\end{equation}\nIt is also possible to derive simple integral formulas for $W_p(\\mathbf{q})$ in terms of $c_{\\mathbf{q}}$ but this will not be needed here, see \\cite{C-StFlour} for more details. We denote by $\\mathcal{Q}_a$ the set of admissible weight sequences.\n\nFinally, let $\\mathcal{Q}^*$ be the set of those $\\mathbf{q}=(q_j)_{j \\geq 1} \\in \\mathcal{Q}$ for which there exists $j \\geq 2$ such that $q_j>0$ (which ensures $W_p(\\mathbf{q})>0$ for all $p \\geq 1$). For $\\mathbf{q} \\in \\mathcal{Q}^* \\cap \\mathcal{Q}_a$, we define the \\emph{Boltzmann distribution} with weights $\\mathbf{q}$ on finite planar bipartite maps of the $2p$-gon as\n\\[\\P(m)=\\frac{1}{W_p(\\mathbf{q})} \\prod_{f \\in m} q_{\\mathrm{deg}(f)\/2}\\]\nfor all bipartite planar map $m$ of the $2p$-gon.\n\n\\paragraph{A general recursion for bipartite maps.}\nAs in \\cite{BL19}, we are lacking precise asymptotics on the enumeration of maps when both the genus and the size go to infinity. The following recurrence formula, proved in \\cite{Lo19}, will play the same role as the Goulden--Jackson formula for triangulations \\cite{GJ08}. We set the convention $\\beta_g(\\mathbf{0})=0$ for all $g$. Then, for every $g\\geq 0$ and every face degree sequence $\\mathbf{f}$, we have\n\\begin{equation}\\label{rec_biparti_genre_univ}\n\\binom{|\\mathbf{f}|+1}{2}\\beta_g(\\mathbf{f})=\\hspace{-0.5cm}\\sum_{\\substack{\\mathbf{h}^{(1)}+\\mathbf{h}^{(2)}=\\mathbf{f}\\\\g^{(1)}+g^{(2)}+g^*=g}}\\hspace{-0.5cm}(1+|\\mathbf{h}^{(1)}|)\\binom{v \\left( \\mathbf{h}^{(2)},g^{(2)} \\right) }{2g^*+2}\\beta_{g^{(1)}}(\\mathbf{h}^{(1)}) \\beta_{g^{(2)}}(\\mathbf{h}^{(2)})+\\sum_{g^*\\geq 0}\\binom{v \\left( \\mathbf{f},g \\right) +2g^*}{2g^*+2}\\beta_{g-g*}(\\mathbf{f}),\n\\end{equation}\nwhere we recall that $|\\mathbf{f}|=\\sum_{j \\geq 1} j f_j$ and $v(\\mathbf{f},g)=2-2g+\\sum_j (j-1) f_j$ (i.e. it is the number of vertices of a map with face degrees $\\mathbf{f}$ and genus $g$).\n\n\\subsection{Infinite Boltzmann bipartite planar maps}\n\\label{subsec_IBPM}\n\n\\paragraph{Definition of the models.}\nOur goal here is to recall the definition of infinite Boltzmann bipartite planar maps introduced in \\cite[Appendix C]{B18these} (and earlier in \\cite{Bud15} in the critical case). We also refer to \\cite[Chapter 8]{C-StFlour} for some basic properties of these objects that we will state below.\n\nLet $\\mathbf{q}=(q_j)_{j \\geq 1}$ be a sequence of nonnegative real numbers that we will call the \\emph{Boltzmann weights}. A random infinite bipartite planar map $M$ is called $\\mathbf{q}$-Boltzmann if there are constants $\\left( C_p(\\mathbf{q}) \\right)_{p \\geq 1}$ such that, for every finite bipartite map $m$ with one hole of perimeter $2p$, we have\n\\[ \\P \\left( m \\subset M \\right)=C_p(\\mathbf{q}) \\prod_{f \\in m} q_{\\mathrm{deg}(f)\/2},\\]\nwhere the product is over all internal faces of $m$.\n\nWe will see that given $\\mathbf{q}$, such a map does not always exist, but when it does, it is unique, i.e. the constants $C_p(\\mathbf{q})$ are determined by $\\mathbf{q}$, which justifies the notation $C_p(\\mathbf{q})$. More precisely, as noted in \\cite[Appendix C]{B18these}, if a $\\mathbf{q}$-Boltzmann map exists, then the partition function of maps of a $2$-gon with Boltzmann weights $\\mathbf{q}$ must be finite, which is equivalent to the admissibility criterion \\eqref{eq_univ_admissible}. Moreover, with the notation of Section~\\ref{subsec_univ_defns}, we call $\\mathbf{q}$ \\emph{critical} if $f'_{\\mathbf{q}}(Z_{\\mathbf{q}})=\\frac{1}{Z_{\\mathbf{q}}^2}$ and \\emph{subcritical} if this is not the case.\n\nFinally, we define a measure $\\nu_{\\mathbf{q}}$ on $\\mathbb Z$ as follows:\n\\begin{equation}\\label{eqn_defn_nu}\n\\nu_{\\mathbf{q}}(i)= \\left\\{ \n\\begin{array}{ll}\nq_{i+1} \\, c_{\\mathbf{q}}^{i} & \\mbox{ if $i \\geq 0$,} \\\\\n2 W_{-1-i}(\\mathbf{q}) \\, c_{\\mathbf{q}}^i & \\mbox{ if $i \\leq -1$.}\n\\end{array} \\right.\n\\end{equation}\nAs noted in \\cite{Bud15}, this is the step distribution of the random walk on $\\mathbb Z$ describing the evolution of the perimeter of a finite $\\mathbf{q}$-Boltzmann map with a large perimeter (see also \\cite[Chapter 5.1]{C-StFlour}). Then previous results about the existence of $\\mathbf{q}$-IBPM can be summed up as follows.\n\n\\begin{thm}\\label{thm_rappel_these}\n\\begin{enumerate}\n\\item\nIf a $\\mathbf{q}$-IBPM exists, it is unique (in distribution), so we can denote it by $\\mathbb M_{\\mathbf{q}}$.\n\\item\nIf $\\mathbf{q} \\notin \\mathcal{Q}^* \\cap \\mathcal{Q}_a$, then $\\mathbb M_{\\mathbf{q}}$ does not exist.\n\\item\nIf $\\mathbf{q} \\in \\mathcal{Q}^* \\cap \\mathcal{Q}_a$ is critical, then $\\mathbb M_{\\mathbf{q}}$ exists and $C_p(\\mathbf{q})=c_{\\mathbf{q}}^{p-1} \\times \\frac{2p}{4^p} \\binom{2p}{p}$.\n\\item\nIf $\\mathbf{q} \\in \\mathcal{Q}^* \\cap \\mathcal{Q}_a$ is subcritical, then $\\mathbb M_{\\mathbf{q}}$ exists if and only if the equation\n\\begin{equation}\\label{boltzmann_equation_omega}\n\\sum_{i \\in \\mathbb Z} \\nu_{\\mathbf{q}}(i) \\omega^i =1\n\\end{equation}\nhas a solution $\\omega_{\\mathbf{q}}>1$.\n\\item\nIn this case, the solution $\\omega_{\\mathbf{q}}$ is unique and, for every $p \\geq 1$, we have\n\\begin{equation}\\label{boltzmann_formula_cp}\nC_p(\\mathbf{q})= \\left( c_{\\mathbf{q}} \\omega_{\\mathbf{q}} \\right)^{p-1} \\sum_{i=0}^{p-1} (4\\omega_{\\mathbf{q}})^{-i} \\binom{2i}{i}.\n\\end{equation}\n\\end{enumerate}\n\\end{thm}\nThe third point is from \\cite{Bud15}, and the others are from \\cite[Appendix C]{B18these}\\footnote{We have fixed a small mistake from \\cite[Appendix C]{B18these}, where $c_{\\mathbf{q}}$ was omitted in the formula for $C_p(\\mathbf{q})$.}. When it exists, we will call the map $\\mathbb M_{\\mathbf{q}}$ the \\emph{$\\mathbf{q}$-IBPM} (for \\emph{Infinite Boltzmann Planar Map)}.\nWe denote by $\\mathcal{Q}_h \\subset \\mathcal{Q}^* \\cap \\mathcal{Q}_a$ the set of weight sequences $\\mathbf{q}$ for which $\\mathbb M_{\\mathbf{q}}$ exists. We also note that the formula for $C_p(\\mathbf{q})$ in the critical case is a particular case of the subcritical one where $\\omega=1$. Since this function will appear many times later, for $\\omega \\geq 1$ and $p \\geq 1$, we write:\n\\begin{equation}\\label{eqn_defn_homega}\nh_p(\\omega)=\\sum_{i=0}^{p-1} (4\\omega)^{-i} \\binom{2i}{i}.\n\\end{equation}\nIn particular, if $\\omega=1$, then $h_p(\\omega)=\\frac{2p}{4^p} \\binom{2p}{p} \\sim \\frac{2}{\\sqrt{\\pi}} \\sqrt{p}$ as $p \\to +\\infty$. If $\\omega>1$, then $h_p(\\omega) \\to \\sqrt{\\frac{\\omega}{\\omega-1}}$ as $p \\to +\\infty$.\n\n\\paragraph{The random walk $\\widetilde{\\nu}_{\\mathbf{q}}$.}\nTo study the $\\mathbf{q}$-IBPM, we define the measure $\\widetilde{\\nu}_{\\mathbf{q}}$ on $\\mathbb Z$ by $\\widetilde{\\nu}_{\\mathbf{q}}(i)=\\omega_{\\mathbf{q}}^i \\nu_{\\mathbf{q}}(i)$, where $\\omega_{\\mathbf{q}}$ is given by \\eqref{boltzmann_equation_omega} if $\\mathbf{q}$ is subcritical, and $\\omega=1$ if $\\mathbf{q}$ is critical. The random walk with step distribution $\\widetilde{\\nu}_{\\mathbf{q}}$ plays an important role when studying $\\mathbb M_{\\mathbf{q}}$. We first note that, if $\\mathbf{q}$ is not critical, then this walk has a positive drift. Indeed, denoting by $F_{\\mathbf{q}}$ the generating function of $\\nu_{\\mathbf{q}}$, we have\n\\[ \\sum_{i \\in \\mathbb Z} i \\, \\widetilde{\\nu}_{\\mathbf{q}}(i) = F'_{\\mathbf{q}}(\\omega_{\\mathbf{q}})>0,\\]\nsince $F_{\\mathbf{q}}$ is convex and takes the value $1$ both at $1$ and at $\\omega_{\\mathbf{q}}>1$. Note also that it is possible that the drift is $+\\infty$.\n\n\\paragraph{Lazy peeling explorations of the $\\mathbf{q}$-IBPM.}\nWe now perform a few computations related to lazy peeling explorations of the $\\mathbf{q}$-IBPM. For this, we fix a peeling algorithm $\\mathcal{A}$, and consider a filled-in exploration of $\\mathbb M_{\\mathbf{q}}$ according to $\\mathcal{A}$. We recall that $\\mathcal{E}_t^{\\mathcal{A}}(\\mathbb M_{\\mathbf{q}})$ is the explored region after $t$ steps, and we denote by $\\left( \\mathcal{F}_t \\right)_{t \\geq 0}$ the filtration generated by this exploration. We denote by $P_t$ (resp. $V_t$) the half-perimeter (resp. total number of edges) of $\\mathcal{E}_t^{\\mathcal{A}}(\\mathbb M_{\\mathbf{q}})$. We will call $P$ and $V$ the \\emph{perimeter and volume processes} associated to a peeling exploration of $\\mathbb M_{\\mathbf{q}}$.\n\nIt follows from the definition of $\\mathbb M_{\\mathbf{q}}$ that $(P_t, V_t)_{t \\geq 0}$ is a Markov chain on $\\mathbb N^2$ and that its law does not depend on the algorithm $\\mathcal{A}$. More precisely $P$ is a Doob transform of the random walk with step distribution $\\widetilde{\\nu}_{\\mathbf{q}}$, i.e. it has the following transitions:\n\\begin{equation}\\label{peeling_transitions}\n\\P \\left( P_{t+1}=P_t+i | \\mathcal{F}_t \\right)= \\widetilde{\\nu}_{\\mathbf{q}}(i) \\frac{h_{P_t+i}(\\omega_{\\mathbf{q}})}{h_{P_t}(\\omega_{\\mathbf{q}})},\n\\end{equation}\nwhere $h_p(\\omega)$ is given by \\eqref{eqn_defn_homega}. As noted in \\cite{C-StFlour}, this implies that $\\left( h_p(\\omega_{\\mathbf{q}}) \\right)_{p \\geq 1}$ is harmonic on $\\{1,2, \\dots\\}$ for the random walk with step distribution $\\widetilde{\\nu}_{\\mathbf{q}}$, and that for $\\mathbf{q}$ subcritical $P$ has the distribution of this random walk, conditioned to stay positive (if $\\mathbf{q}$ is critical, the conditioning is degenerate, but this can still be made sense of).\n\n\\paragraph{IBPM with finite expected degree of the root face.}\nWe denote by $\\mathcal{Q}_f$ the set of $\\mathbf{q} \\in \\mathcal{Q}_h$ such that the degree of the root face of $\\mathbb M_{\\mathbf{q}}$ has finite expectation. Since our Theorem~\\ref{univ_main_thm} only holds if the expected degree of the root face is finite in the limit, we will need to gather a few consequences of this assumption on $\\mathbf{q}$ and the peeling process of $\\mathbb M_{\\mathbf{q}}$. Note that, for all $\\mathbf{q} \\in \\mathcal{Q}_h$, the degree of the root face is determined by the first peeling step on $\\mathbb M_{\\mathbf{q}}$. More precisely, by \\eqref{peeling_transitions}, we have for all $j \\geq 1$:\n\\begin{equation}\\label{eqn_walk_to_rootface}\n\\P \\left( \\mbox{the root face of $\\mathbb M_{\\mathbf{q}}$ has degree $2j$} \\right)=\\frac{h_j(\\omega_{\\mathbf{q}})}{h_1(\\omega_{\\mathbf{q}})} \\, \\widetilde{\\nu}_{\\mathbf{q}}(j-1) = \\frac{h_j(\\omega_{\\mathbf{q}})}{h_1(\\omega_{\\mathbf{q}})} \\left( c_{\\mathbf{q}} \\omega_{\\mathbf{q}} \\right)^{j-1} q_j.\n\\end{equation}\nIf $\\mathbf{q}$ is critical, the right hand-side is equivalent to $\\frac{2}{\\sqrt{\\pi}}\\sqrt{j} c_{\\mathbf{q}}^{j-1} q_j$ as $j \\to +\\infty$, so $\\mathbf{q} \\in \\mathcal{Q}_f$ if and only if\n\\begin{equation}\\label{eqn_finite_32_moment}\n\\sum_{j \\geq 1} j^{3\/2} c_{\\mathbf{q}}^j q_j <+\\infty.\n\\end{equation}\nOn the other hand, we recall (see e.g. \\cite[Chapter 5.2]{C-StFlour}) that a critical weight sequence $\\mathbf{q}$ is called \\emph{of type $\\frac{5}{2}$}, or \\emph{critical generic}, if\n\\[ \\sum_{j \\geq 1} (j-1)(j-2) \\binom{2j-1}{j-1} q_j \\left( \\frac{c_{\\mathbf{q}}}{4} \\right)^{j-3} <+\\infty, \\]\nwhich is clearly equivalent to \\eqref{eqn_finite_32_moment}.\nIn the subcritical case, by \\eqref{eqn_walk_to_rootface}, $\\mathbf{q} \\in \\mathcal{Q}_f$ is equivalent to\n\\[\\sum_{j \\geq 1} j \\widetilde{\\nu}_{\\mathbf{q}}(j) <+\\infty,\\] i.e. the drift of $\\widetilde{\\nu}$ is finite. To sum up:\n\\begin{itemize}\n\\item\nIn the critical case, $\\mathbf{q} \\in \\mathcal{Q}_f$ if and only if $\\mathbf{q}$ is critical generic, which means that the perimeter process $(P_n)$ converges to a $3\/2$-stable L\u00e9vy process with no positive jump, conditioned to be positive (see \\cite[Theorem 10.1]{C-StFlour}). This basically means that $\\mathbf{q}$-Boltzmann finite maps for $\\mathbf{q} \\in \\mathcal{Q}_f$ lie in the domain of attraction of the Brownian map \\cite{MM07}.\n\\item\nIn the subcritical case, $\\mathbf{q} \\in \\mathcal{Q}_f$ if and only if the measure $\\widetilde{\\nu}$ has finite expectation. Since the perimeter process $P$ has the law of a $\\widetilde{\\nu}$-random walk conditioned on an event of positive probability, this means that $P$ has linear growth (instead of super-linear if the expectation of $\\widetilde{\\nu}$ was infinite).\n\\end{itemize}\n\n\\subsection{Four ways to describe Boltzmann weights}\n\n\\paragraph{Four parametrizations of $\\mathcal{Q}_h$.}\nIn this work, we will make use of four different \"coordinate systems\" to navigate through the spaces $\\mathcal{Q}_h$ and $\\mathcal{Q}_f$, each with its own advantages. The goal of this section is to define these parametrizations and to establish some relations between them. In particular, we will prove Theorem \\ref{thm_prametrization_rootface}.\n\nOur first coordinate system, already used in the last pages, consists in using directly the Boltzmann weights $q_j$ for $j \\geq 1$. It is the simplest way to define the model $\\mathbb M_{\\mathbf{q}}$ and gives the simplest description of its law.\n\nThe second parametrization we will use is the one given by Proposition~\\ref{prop_q_as_limit} below: we describe $\\mathbf{q}$ by parameters $r_j(\\mathbf{q}) \\in [0,1)$ for $j \\geq 1$ and $r_{\\infty}(\\mathbf{q}) \\in (0,+\\infty]$. Here $r_j(\\mathbf{q})$ describes the proportion of peeling steps where we discover a face of degree $2j$ during a peeling exploration of $\\mathbb M_{\\mathbf{q}}$, and $r_{\\infty}(\\mathbf{q})$ comes from a comparison between the volume and perimeter growths. The advantage of these parameters is that they allow to \"read\" $\\mathbf{q}$ as an almost sure observable on a peeling exploration of the map $\\mathbb M_{\\mathbf{q}}$. This will be useful in Section~\\ref{sec_arg_deux_trous}.\n\nThe third parametrization consists in using on the one hand the law of the root face, and on the other hand the average degree of the vertices. More precisely, for $j \\geq 1$, we write\n\\[a_j(\\mathbf{q})=\\frac{1}{j} \\P \\left( \\mbox{the root face of $\\mathbb M_{\\mathbf{q}}$ has degree $2j$} \\right).\\]\nWe note that $\\sum_{j \\geq 1} j a_j(\\mathbf{q})=1$ and that $a_1(\\mathbf{q})<1$ since a map consisting only of $2$-gons would have vertices with infinite degrees.\nWe also write $d(\\mathbf{q})=\\mathbb E \\left[ \\frac{1}{\\mathrm{deg}_{\\mathbb M_{\\mathbf{q}}}(\\rho)} \\right]$, where $\\rho$ is the root vertex. The advantage of this parametrization is that the analogues of these quantities are easy to compute if we replace $\\mathbb M_{\\mathbf{q}}$ by a finite uniform map with prescribed genus and face degrees. These parameters are our only way to link the finite and infinite models, and will therefore be useful in the end of the proof of Theorem~\\ref{univ_main_thm}. However, it is not obvious at all that $\\left( a_j(\\mathbf{q}) \\right)_{j \\geq 1}$ and $d(\\mathbf{q})$ are sufficient to characterize $\\mathbf{q}$. We will actually prove this in the end of the paper, only for $\\mathbf{q} \\in \\mathcal{Q}_f$, as a consequence of local convergence arguments (Proposition~\\ref{prop_monotonicity_deg}).  \n\nFinally, the fourth coordinate system is the one from Theorem \\ref{thm_prametrization_rootface}: it is a variant of the third one where we replace $d(\\mathbf{q})$ by $\\omega_{\\mathbf{q}}$, which makes it easier to handle. This one is useful as an intermediate step towards the third one. Moreover, contrary to the third one, we can prove rather quickly (Theorem \\ref{thm_prametrization_rootface}) that it provides a nice parametrization of the whole space $\\mathcal{Q}_h$.\n\n\\paragraph{Recovering $\\mathbf{q}$ from explorations of $\\mathbb M_{\\mathbf{q}}$.}\nWe now describe more precisely our second parametrization of $\\mathcal{Q}_h$. The next result basically states that we can recover the weight sequence $\\mathbf{q}$ by just observing the perimeter and volume processes defined above (we recall that the volume is measured by the total number of edges).\n\n\\begin{prop}\\label{prop_q_as_limit}\nLet $\\mathbf{q} \\in \\mathcal{Q}_h$, and let $P$ and $V$ be the perimeter and volume processes associated to a peeling exploration of $\\mathbb M_{\\mathbf{q}}$. We have the following almost sure convergences:\n\\begin{equation}\\label{limit_number_jgons}\n\\frac{1}{t} \\sum_{i=0}^{t-1} \\mathbbm{1}_{P_{i+1}-P_i=j-1} \\xrightarrow[t \\to +\\infty]{a.s.} \\left( c_{\\mathbf{q}} \\omega_{\\mathbf{q}} \\right)^{j-1} q_j =: r_j(\\mathbf{q}) \\in [0,1)\n\\end{equation}\nfor every $j \\geq 1$, and\n\\begin{equation}\\label{limit_volume_growth}\n\\frac{V_t-2P_t}{t} \\xrightarrow[t \\to +\\infty]{a.s.} \\frac{ \\left(\\sqrt{\\omega_{\\mathbf{q}}}-\\sqrt{\\omega_{\\mathbf{q}}-1} \\right)^2}{2 \\sqrt{\\omega_{\\mathbf{q}}(\\omega_{\\mathbf{q}}-1)}} =: r_{\\infty}(\\mathbf{q}) \\in (0,+\\infty].\n\\end{equation}\nMoreover, the weight sequence $\\mathbf{q}$ is a measurable function of the numbers $r_j(\\mathbf{q})$ for $j \\in \\mathbb N^* \\cup \\{\\infty\\}$.\n\\end{prop}\n\n\\begin{proof}\nIn the subcritical case, the second convergence is Proposition 10.12 of \\cite{C-StFlour}. In the critical case, we have $\\omega_{\\mathbf{q}}=1$ so the right-hand side of \\eqref{limit_volume_growth} is infinite, and the result follows from Lemma 10.9 of \\cite{C-StFlour}.\n\nLet us now prove the first convergence. For this, we first note that we have $P_t \\to +\\infty$ almost surely as $t \\to +\\infty$. Indeed, this again follows from \\cite[Proposition 10.12]{C-StFlour} in the subcritical case and from \\cite[Lemma 10.9]{C-StFlour} in the critical case. On the other hand, given the asymptotics for $h_p(\\omega)$ right after \\eqref{eqn_defn_homega}, for any fixed $j \\geq 1$, we have $\\frac{h_{p+j-1}(\\omega_{\\mathbf{q}})}{h_p(\\omega_{\\mathbf{q}})} \\to 1$ as $p \\to +\\infty$. It follows that\n\\[ \\P \\left( P_{t+1}-P_t=j-1 | \\mathcal{F}_t \\right) \\xrightarrow[t \\to +\\infty]{a.s.} \\widetilde{\\nu}_{\\mathbf{q}}(j-1) = \\left( c_{\\mathbf{q}} \\omega_{\\mathbf{q}} \\right)^{j-1} q_j, \\]\nand the first convergence follows by the law of large numbers.\n\nFinally, the function $\\omega \\to \\frac{ \\left(\\sqrt{\\omega}-\\sqrt{\\omega-1} \\right)^2}{2 \\sqrt{\\omega(\\omega-1)}}$ is a decreasing homeomorphism from $[1,+\\infty)$ to $(0,+\\infty]$, so $\\omega_{\\mathbf{q}}$ is a measurable function of $r_{\\infty}(\\mathbf{q})$. Moreover, by the\ndefinition of $c_{\\mathbf{q}}$~\\eqref{eq_univ_admissible}, we have\n\\[ 1-\\frac{4}{c_{\\mathbf{q}}} = \\sum_{j \\geq 1} \\binom{2j-1}{j-1} q_j \\left( \\frac{c_{\\mathbf{q}}}{4} \\right)^{j-1} = \\sum_{j \\geq 1} \\frac{1}{4^j} \\binom{2j-1}{j-1} \\frac{r_j(\\mathbf{q})}{\\omega_{\\mathbf{q}}^{j-1}},\\]\nwhich implies that $c_{\\mathbf{q}}$ is a measurable function of $\\omega_{\\mathbf{q}}$ and the numbers $r_j(\\mathbf{q})$ for $j \\in \\mathbb N^*$. Finally, given $c_{\\mathbf{q}}$ and the $r_j(\\mathbf{q})$, we easily recover the $q_j$ from~\\eqref{limit_number_jgons}.\n\\end{proof}\n\n\\paragraph{Weight sequences corresponding to a given distribution of the root face.}\nWe now prove Theorem~\\ref{thm_prametrization_rootface} by showing that our fourth parametrization is indeed bijective. We first state the precise version of Theorem~\\ref{thm_prametrization_rootface}.  We recall that for $\\mathbf{q} \\in \\mathcal{Q}_h$, the numbers $a_j(\\mathbf{q})$ satisfy $\\sum_{j \\geq 1} j a_j(\\mathbf{q})=1$ and $\\alpha_1<1$, and we have $\\omega_{\\mathbf{q}} \\geq 1$.\n\n\\begin{prop}\\label{prop_third_parametrization}\nLet $(\\alpha_j)_{j \\geq 1}$ be such that $\\sum_{j \\geq 1} j \\alpha_j=1$ and $\\alpha_1 < 1$, and let $\\omega \\geq 1$. Then there is a unique $\\mathbf{q} \\in \\mathcal{Q}_h$ such that\n\\[ \\omega_{\\mathbf{q}}=\\omega \\mbox{ and } \\forall j \\geq 1, \\, a_j(\\mathbf{q})=\\alpha_j.\\]\nMoreover, this weight sequence $\\mathbf{q}$ is given by\n\\begin{equation}\\label{eqn_qjomega_univ}\nq_j=\\frac{j \\alpha_j}{\\omega^{j-1}h_{\\omega}(j)} \\left( \\frac{1-\\sum_{i \\geq 1} \\frac{1}{4^{i-1}} \\binom{2i-1}{i-1} \\frac{i \\alpha_i}{\\omega^{i-1} h_{\\omega}(i)} }{4} \\right)^{j-1}.\n\\end{equation}\n\\end{prop}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop_third_parametrization}]\nWe start with uniqueness. We note that\n\\begin{equation}\\label{eqn_alpha_from_q}\na_j(\\mathbf{q})=\\frac{1}{j} C_j(\\mathbf{q}) q_j = \\frac{1}{j} \\left( c_{\\mathbf{q}} \\omega_{\\mathbf{q}}\\right)^{j-1} h_{j}(\\omega_{\\mathbf{q}}) q_j,\n\\end{equation}\nso $q_j$ can be obtained as a function of $a_j(\\mathbf{q})=\\alpha_j$, $\\omega_{\\mathbf{q}}$ and $c_{\\mathbf{q}}$. Moreover, by the definition~\\eqref{eq_univ_admissible} of $c_{\\mathbf{q}}$, we have\n\\begin{equation}\\label{eqn_g_function_of_aj}\n1-\\frac{4}{c_{\\mathbf{q}}}=\\sum_{i \\geq 1} \\frac{1}{4^{i-1}} \\binom{2i-1}{i-1} q_i c_{\\mathbf{q}}^{i-1} = \\sum_{i \\geq 1} \\frac{1}{4^{i-1}} \\binom{2i-1}{i-1} \\frac{i a_i(\\mathbf{q})}{\\omega_{\\mathbf{q}}^{i-1} h_{i}(\\omega_{\\mathbf{q}})},\n\\end{equation}\nso $c_{\\mathbf{q}}$, and therefore $q_j$ for all $j \\geq 1$, can be deduced from $\\omega_{\\mathbf{q}}$ and $\\left( a_j(\\mathbf{q}) \\right)_{j \\geq 1}$. More precisely, we obtain the formula~\\eqref{eqn_qjomega_univ}, which in particular proves the uniqueness.\n\nTo prove the existence, it is enough to check that, for all $\\omega \\geq 1$ and $(\\alpha_j)_{j \\geq 1}$ with $\\sum j \\alpha_j=1$ and $\\alpha_1<1$, the sequence $\\mathbf{q}$ given by \\eqref{eqn_qjomega_univ} is indeed in $\\mathcal{Q}_h$, with $\\omega_{\\mathbf{q}}=\\omega$ and $a_j(\\mathbf{q})=\\alpha_j$ for all $j$. \nFollowing \\eqref{eqn_g_function_of_aj}, we first write\n\\begin{equation}\\label{eqn_c_of_alpha_omega}\nc=\\frac{4}{1-\\sum_{i \\geq 1} \\frac{1}{4^{i-1}} \\binom{2i-1}{i-1} \\frac{i \\alpha_i}{\\omega^{i-1} h_{i}(\\omega) }},\n\\end{equation}\nand check that $\\mathbf{q}$ is admissible with $c_{\\mathbf{q}}=c$. First $\\omega^{i-1} h_{i}(\\omega)$ is a polynomial in $\\omega$ with nonnegative coefficients so $\\omega^{i-1} h_{i}(\\omega) \\geq h_i(1)=\\frac{2i}{4^i} \\binom{2i}{i}$. From here, we get\n\\[ \\sum_{i \\geq 1} \\frac{1}{4^{i-1}} \\binom{2i-1}{i-1} \\frac{i \\alpha_i}{\\omega^{i-1} h_{i}(\\omega) } \\leq \\sum_{i \\geq 1} \\alpha_i < \\sum_{i \\geq 1} i \\alpha_i = 1 \\]\nbecause $\\alpha_1 <1$. Therefore, the numbers $q_j$ are nonnegative and $c>0$, and we can rewrite \\eqref{eqn_qjomega_univ} as\n\\[q_j=\\frac{j \\alpha_j}{(\\omega c)^{j-1} h_{j}(\\omega)}. \\]\nFrom here, we get\n\\[ \\sum_{i \\geq 1} \\frac{1}{4^{i-1}} \\binom{2i-1}{i-1} q_i c^{i-1} = 1-\\frac{4}{c}\\]\nimmediately by the definition of $c$, which proves $\\mathbf{q} \\in \\mathcal{Q}_a$ and $c_{\\mathbf{q}}=c$.\nAlso, we know that $\\alpha_1<1$ so there is $j \\geq 2$ with $\\alpha_j>0$, which implies $q_j>0$, so $\\mathbf{q} \\in \\mathcal{Q}^*$.\n\nWe now prove $\\mathbf{q} \\in \\mathcal{Q}_h$ with $\\omega_{\\mathbf{q}}=\\omega$, which is equivalent to proving\n\\[ \\sum_{i \\in \\mathbb Z} \\nu_{\\mathbf{q}}(i) \\omega^i = 1,\\]\nwhere we recall that $\\nu_{\\mathbf{q}}$ is defined by \\eqref{eqn_defn_nu}. For this,\ninspired by similar arguments in the critical case (see e.g. \\cite[Lemma 5.2]{C-StFlour}), the basic idea will be to show that $\\left( \\omega^i h_{i}(\\omega) \\right)_{i \\geq 1}$ is harmonic for $\\nu_{\\mathbf{q}}$. More precisely, the equality $\\sum_{i \\geq 1} i \\alpha_i=1$ can be interpreted as a harmonicity relation at $1$: setting $h_i(\\omega)=0$ for $i \\leq -1$, we have\n\\begin{equation}\\label{eqn_harmo_at_one}\n\\sum_{i \\in \\mathbb{Z}} h_{i+1}(\\omega) \\omega^i \\nu_{\\mathbf{q}}(i) = \\sum_{j \\geq 1} \\omega^{j-1} h_{j}(\\omega) c^{j-1} q_j = \\sum_{j \\geq 1} j \\alpha_j = 1 = h_{1}(\\omega),\n\\end{equation}\nwhere in the beginning we do the change of variables $j=i+1$. On the other hand, we know that $h_{p}(\\omega)=\\sum_{i=0}^{p-1} \\omega^{-i} u(i)$, where $u(i)=\\frac{1}{4^i} \\binom{2i}{i}$ for $i \\geq 0$ (and we set the convention $u(i)=0$ for $i \\leq -1$). But the same function $u$ plays an important role in the description of the law of the peeling process of finite Boltzmann maps. In particular, we know that $u$ is $\\nu_{\\mathbf{q}}$-harmonic on positive integers for any admissible weight sequence $\\mathbf{q}$ (this can be found in the proof of Lemma 5.2 in \\cite{C-StFlour}). That is, for all $j \\geq 1$, we have\n\\[ u(j)=\\sum_{i \\in \\mathbb{Z}} \\nu_{\\mathbf{q}}(i) u(i+j).\\]\nMultiplying by $\\omega^{-j}$ and summing over $1 \\leq j \\leq p-1$, we get, for all $p \\geq 1$:\n\\[ h_{p}(\\omega)-h_{1}(\\omega) = \\sum_{i \\in \\mathbb Z} \\omega^i \\nu_{\\mathbf{q}}(i) \\left( h_{p+i}(\\omega) - h_{i+1}(\\omega) \\right). \\]\nSumming this with \\eqref{eqn_harmo_at_one} and dividing by $h_{p}(\\omega)$, we obtain\n\\[ \\sum_{i \\in \\mathbb Z} \\omega^i \\nu_{\\mathbf{q}}(i) \\frac{h_{p+i}(\\omega)}{h_{p}(\\omega)} =1. \\]\nfor all $p \\geq 1$. When $p \\to +\\infty$, we have that\n$h_{p}(\\omega)$ has a positive limit if $\\omega>1$ and is equivalent to $\\frac{2}{\\sqrt{\\pi}}\\sqrt{p}$ if $\\omega=1$, so $\\frac{h_{p+i}(\\omega)}{h_{p}(\\omega)} \\to 1$ in every case. Therefore, by dominated convergence, we get\n\\[ \\sum_{i \\in \\mathbb Z} \\nu_{\\mathbf{q}}(i) \\omega^i=1,\\]\nwhere the domination $\\sum_{i \\in \\mathbb Z} \\nu_{\\mathbf{q}}(i) \\omega^i <+\\infty$ is immediate for negative values of $i$ since $\\omega \\geq 1$, and comes from the convergence of the sum \\eqref{eqn_harmo_at_one} for positive values of $i$. This proves $\\mathbf{q} \\in \\mathcal{Q}_h$ with $\\omega_{\\mathbf{q}}=\\omega$, and from here $a_j(\\mathbf{q})=\\alpha_j$ is immediate using \\eqref{eqn_alpha_from_q}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm_prametrization_rootface}]\nIt is clear from Proposition \\ref{prop_third_parametrization} that weight sequences with a given root face distribution are parametrized by $\\omega \\in [1,+\\infty)$. We denote by $\\mathbf{q}^{(\\omega)}$ the unique weight sequence for which the law of the root face is given by $(\\alpha_j)_{j \\geq 1}$ and for which $\\omega_{\\mathbf{q}^{(\\omega)}}=\\omega$. Then $q^{(1)}$ is critical by definition. Moreover, using \\eqref{eqn_qjomega_univ} and \\eqref{eqn_c_of_alpha_omega}, we get for $i \\geq 0$:\n\\begin{equation}\\label{eqn_omega_infinite}\n\\widetilde{\\nu}_{\\mathbf{q}^{(\\omega)}}(i)=q^{(\\omega)}_{i+1} \\omega^i c_{\\mathbf{q}^{(\\omega)}}^i \\xrightarrow[\\omega \\to +\\infty]{} (i+1) \\alpha_{i+1}.\n\\end{equation}\nThe sum over $i \\geq 0$ of the right-hand side is equal to $1$, so $\\widetilde{\\nu}_{\\mathbf{q}^{(\\omega)}} \\left( (-\\infty,-1] \\right) \\to 0$ as $\\omega \\to +\\infty$. By \\eqref{peeling_transitions}, this means that the probability of peeling cases decreasing the perimeter goes to $0$. Since these are the cases creating cycles in the dual, the dual of $\\mathbb M_{\\mathbf{q}^{(\\omega)}}$ becomes close to a tree when $\\omega \\to +\\infty$, and the vertex degrees in $\\mathbb M_{\\mathbf{q}^{(\\omega)}}$ go to infinity.\n\\end{proof}\n\n\\paragraph{Two technical results on the dependance in $\\omega$.}\nWe conclude this section with two technical results that we will need in the end of the proof (Section~\\ref{subsec_last_step}). Both deal with the way that some quantities depend on the parameter $\\omega$. We fix $(\\alpha_j)_{j \\geq 1}$ such that $\\sum_{j \\geq 1} j \\alpha_j = 1$ and $\\alpha_1<1$. By Proposition~\\ref{prop_third_parametrization}, we can denote by $\\mathbf{q}^{(\\omega)}$ the unique weight sequence for which the law of the root face is given by $(\\alpha_j)_{j \\geq 1}$ and $\\omega_{\\mathbf{q}^{(\\omega)}}=\\omega$.\n\nThe first technical lemma states that we can recover $\\mathbf{q}$ from the law of the root face $(\\alpha_j)_{j \\geq 1}$ and a single weight $q_j$, provided $j \\geq 2$.\n\\begin{lem}\\label{lem_qj_is_monotone}\nFor every $j \\geq 1$, the function $\\omega \\to q_j^{(\\omega)}$ is nonincreasing. Moreover, if $j \\geq 2$ and $\\alpha_j>0$, this function is decreasing.\n\\end{lem}\nSince the proof is not particularly enlightening, we postpone it to Appendix \\ref{appendix_qj_monotone}.\n\nOur second technical lemma is a reinforcement of a part of Proposition~\\ref{prop_q_as_limit} above. It states that the second convergence result~\\eqref{limit_volume_growth} is uniform in $\\omega$ as long as $\\omega$ is bounded away from $1$ and $+\\infty$.\n\n\\begin{lem}\\label{lem_unif_volume}\nLet $\\left( P_t^{(\\omega)} \\right)_{t \\geq 0}$ and $\\left( V_t^{(\\omega)}  \\right)_{t \\geq 0}$ denote respectively the perimeter and volume processes associated to a peeling exploration of $\\mathbb M_{\\mathbf{q}^{(\\omega)}}$.\nThe convergence in probability\n\\[ \\frac{V_t^{(\\omega)}-2P_t^{(\\omega)}}{t} \\xrightarrow[t \\to +\\infty]{P)} \\frac{ \\left(\\sqrt{\\omega}-\\sqrt{\\omega-1} \\right)^2}{2 \\sqrt{\\omega(\\omega-1)}} \\]\nis uniform in $\\omega$ over any compact subset $K$ of $(1,+\\infty)$ in the sense that for all $\\varepsilon>0$, there is $t_0>0$ such that, for all $t \\geq t_0$ and $\\omega \\in K$:\n\\[ \\P \\left( \\left| \\frac{V_t^{(\\omega)}-2P_t^{(\\omega)}}{t} - \\frac{ \\left(\\sqrt{\\omega}-\\sqrt{\\omega-1} \\right)^2}{2 \\sqrt{\\omega(\\omega-1)}} \\right| >\\varepsilon \\right) < \\varepsilon. \\]\n\\end{lem}\n\nThe proof of Lemma~\\ref{lem_unif_volume} is an adaptation of the proof of~\\eqref{limit_volume_growth} in~\\cite{C-StFlour}, but using a uniform weak law of large numbers. It is delayed to Appendix~\\ref{subsec_unif_volume}.\n\n\\section{Tightness, planarity and one-endedness}\\label{sec_univ_tight}\n\nIn all this section, we will work in the general setting of Theorem \\ref{thm_main_more_general}, i.e. we do not assume $\\sum_{j \\geq 1} j^2 \\alpha_j<+\\infty$.\n\n\\begin{prop}\\label{prop_tightness_dloc_univ}\nLet $(\\mathbf{f}^n, g_n)_{n \\geq 1}$ be as in Theorem \\ref{thm_main_more_general}. Then the sequence $\\left( M_{\\mathbf{f}^n, g_n} \\right)_{n \\geq 1}$ is tight for $d_{\\mathrm{loc}}$, and every subsequential limit is a.s. planar and one-ended.\n\\end{prop}\n\nOur strategy to prove Proposition~\\ref{prop_tightness_dloc_univ} will be similar to \\cite{BL19}, and in particular relies on a Bounded ratio Lemma (Lemma \\ref{lem_BRL}). Sections \\ref{subsec_BRL}, \\ref{subsec_good_sets} and \\ref{subsec_proof_BRL} are devoted to the proof of the Bounded ratio Lemma, which is significantly more complicated than in \\cite{BL19}. In Section \\ref{subsec_planarity}, we prove that any subsequential limit of $\\left( M_{\\mathbf{f}^n, g_n} \\right)_{n \\geq 1}$ for $d^*_{\\mathrm{loc}}$ (which exist by Lemma~\\ref{lem_easy_dual_convergence}) is planar and one-ended. Finally, in Section \\ref{subsec_finite_degrees}, we finish the proof of Proposition \\ref{prop_tightness_dloc_univ} using Lemma \\ref{lem_dual_convergence_univ} and the Bounded ratio Lemma.\n\n\\subsection{The Bounded ratio Lemma}\n\\label{subsec_BRL}\n\nThe Bounded ratio Lemma below means that, as long as the faces are not too large and the number of vertices remains proportional to the number of edges (i.e. basically under the assumptions of Theorem~\\ref{thm_main_more_general}), removing a face of degree $2j_0$ changes the number of maps by at most a constant factor, provided the faces of degree $2j_0$ represent a positive proportion of the faces. We recall from \\eqref{defn_v_f_g} that $|\\mathbf{f}|$ and $v(\\mathbf{f},g)$ are respectively the number of edges and of vertices of a map with genus $g$ and face degrees given by $\\mathbf{f}$. For $j \\geq 1$, we denote by $\\mathbf{1}_j$ the face degree sequence consisting of a single face of degree $2j$, i.e. $\\left( \\mathbf{1}_j \\right)_i$ is $1$ if $i=j$ and $0$ otherwise.\n\n\\begin{lem}[Bounded ratio Lemma]\\label{lem_BRL}\nWe fix $\\kappa, \\delta>0$ and a function $A:(0,1]\\rightarrow \\mathbb N$. Let $\\mathbf{f}$ be a face degree sequence, and let $g \\geq 0$. We assume that\n\\begin{equation}\\label{eq_petites_faces}\nv(\\mathbf{f}, g) > \\kappa |\\mathbf{f}| \\quad \\mbox{ and } \\quad \\forall \\varepsilon>0, \\, \\sum_{i>A(\\varepsilon)} i f_i< \\varepsilon |\\mathbf{f}|.\n\\end{equation}\nLet also $j_0 \\geq 1$ be such that $j_0 f_{j_0}>\\delta |\\mathbf{f}|$. Then the ratio\n\\[\\frac{\\beta_g(\\mathbf{f})}{\\beta_g(\\mathbf{f}-\\mathbf{1}_{j_0})}\\]\nis bounded by a constant depending only on $\\delta, \\kappa$ and the function $A$.\n\\end{lem}\n\nWe will not try to obtain an explicit constant. As in \\cite{BL19}, we will use the Bounded ratio Lemma to estimate the probability of certain events during peeling explorations, so we will need versions with a boundary. Here are the precise versions that we will need later in the paper.\n\n\\begin{corr}\\label{lem_BRL_boundaries}\nLet $\\kappa, \\delta>0$ and $A(\\cdot)$ be as in Lemma \\ref{lem_BRL}. Then there is a constant $C$ such that the following holds.\n\\begin{enumerate}\n\\item\nLet $p,p',j\\geq 1$. Then there is $N$ such that, for all $\\mathbf{f}$ and $g$ satisfying \\eqref{eq_petites_faces} and $j f_j > \\delta |\\mathbf{f}|$ and $|\\mathbf{f}|>N$, we have\n\\[\\frac{\\beta_g^{(p,p')}(\\mathbf{f})}{\\beta_g^{(p,p')}(\\mathbf{f}-\\mathbf{1}_j)}< C\\]\nand in particular\n\\begin{equation}\\label{eqn_BRL_one_boundary}\n\\frac{\\beta_g^{(p)}(\\mathbf{f})}{\\beta_g^{(p)}(\\mathbf{f}-\\mathbf{1}_j)}< 2C.\n\\end{equation}\n\\item\nLet $p_1, p_2 \\geq 1$ and $i_1, i_2 \\geq 0$. Then there is $N$ such that, for all $\\mathbf{f}$ and $g$ satisfying \\eqref{eq_petites_faces} and $|\\mathbf{f}|>N$, we have\n\\[ \\frac{\\beta_g^{(p_1+i_1,p_2+i_2)}(\\mathbf{f})}{\\beta_g^{(p_1,p_2)}(\\mathbf{f})} < C^{i_1+i_2}. \\]\n\\end{enumerate}\n\\end{corr}\n\nSince this will be very important later, we highlight that the constant $C$ does not depend on $p, p', j$ but that $N$ does. The inequality~\\eqref{eqn_BRL_one_boundary} will be used in the tightness argument just like in \\cite{BL19}, whereas the statements with two boundaries will be needed in the two hole argument (Section \\ref{subsec_same_perimeter}).\n\n\\begin{proof}\nWe first claim that we have the identity\n\\begin{equation}\\label{eqn_add_boundary}\n\\beta_g^{(p,p')}(\\mathbf{f})=\\frac{2p(f_{p}+1)p'(f_{p'}+\\mathbbm{1}_{p \\ne p'})}{2(|\\mathbf{f}|+p+p')}\\beta_g(\\mathbf{f}+\\mathbf{1}_{p}+\\mathbf{1}_{p'}).\n\\end{equation}\nIndeed, the factor $p(f_p+1)$ corresponds to the number of ways to add a second root to a map of $\\mathcal{B}_g(\\mathbf{f}+\\mathbf{1}_p)$ such that this second root has a face of degree $2p$ on its right (with respect to the canonical white to black orientation of edges). The factor $p'(f_{p'}+\\mathbbm{1}_{p \\ne p'})$ corresponds to the number of ways of adding a third root next to a face of degree $2p'$ so that the two root faces are distinct. The $(|\\mathbf{f}|+p+p')$ in the denominator corresponds to forgetting the original root. Moreover, if $(\\mathbf{f},g)$ satisfy the assumptions of Lemma \\ref{lem_BRL} for $\\delta, \\kappa, A(\\cdot)$ and $|\\mathbf{f}|$ is large enough, then $(\\mathbf{f}+\\mathbf{1}_p+\\mathbf{1}_{p'},g)$ also satisfies the assumptions of Lemma \\ref{lem_BRL} for $\\frac{\\delta}{2}, \\frac{\\kappa}{2}, A \\left( \\frac{\\cdot}{2}\\right)$. Therefore, the first point of the corollary follows from Lemma \\ref{lem_BRL} and \\eqref{eqn_add_boundary}. To deduce \\eqref{eqn_BRL_one_boundary}, just take $p'=1$ and use the identity $\\beta_g^{(p,1)}(\\mathbf{f})=|\\mathbf{f}|\\beta_g^{(p)}(\\mathbf{f})$ (adding a $2$-gon is equivalent to marking an edge) and the fact that $|\\mathbf{f}|$ is large enough.\n\nFor the second point, we first note that it is sufficient to prove it for $\\{i_1, i_2\\}=\\{0,1\\}$. Since $C$ does not depend on $(p_1, p_2)$, the general case easily follows by induction on $i_1+i_2$. Without loss of generality, we assume $i_1=1, i_2=0$.\n\nWe now note that there is $\\delta, j_1>0$ depending only on $A(\\cdot)$ such that, if \\eqref{eq_petites_faces} is satisfied, then there is $2 \\leq j \\leq j_1$ such that $jf_j > \\delta |\\mathbf{f}|$ (we can assume $j \\geq 2$ because if there are too many $2$-gons, then the number of vertices cannot be macroscopic). We fix such a $j$.\nThen, by the injection that consists in gluing a $2j$-gon on the first boundary as on Figure~\\ref{fig_reducing_boundary}, we have \n\\[\\beta_g^{(p_1+1,p_2)}(\\mathbf{f}) \\leq \\beta_g^{(p_1,p_2)}(\\mathbf{f}+\\mathbf{1}_j) \\leq C \\beta_g^{(p_1,p_2)}(\\mathbf{f}),\\]\nwhere the last inequality uses the first item of the Corollary. This proves the second point.\n\\begin{figure}\n\\center\n\\includegraphics[scale=0.8]{reducing_boundary}\n\\caption{Reducing the size of a boundary by $2$ by adding a $2j$-gon (here, the boundary is in blue, and $j=3$).}\\label{fig_reducing_boundary}\n\\end{figure}\n\\end{proof}\n\n\\paragraph{Outline of the proof of Lemma \\ref{lem_BRL}.}\nThe general idea is the same as in \\cite{BL19}, namely building an injection that removes a small piece of a map (here, we would like to remove a face of degree $2j_0$). Just like in \\cite{BL19}, this implies to merge vertices, so we will try to bound the degrees of the vertices involved, so that the number of ways to do the surgery backwards is not too high. However, since we work in a more general setting, several new constraints appear. First, the degrees of the faces are not bounded, so we must make sure that our surgery operations do not involve faces of huge degrees. This is the purpose of finding \"very nice edges\" in Section \\ref{subsec_good_sets} below. Also, we will not always be able to remove a face of degree exactly $2j_0$. We will therefore remove either a face with degree higher than $2j_0$, or several faces which combined are \"larger\" than a face of degree $2j_0$. We will then use the two (easy) Lemmas~\\ref{lem_grande_face} and~\\ref{lem_petites_faces} to conclude.\n\n\\begin{lem}\\label{lem_grande_face}\nIf $p\\geq j_0 \\geq 1$, then\n\\[ |\\mathbf{f}| \\beta_g \\left( \\mathbf{f}-\\mathbf{1}_{j_0} \\right) \\geq j_0 f_{j_0} \\, \\beta_g \\left( \\mathbf{f}-\\mathbf{1}_p \\right).\\]\nIn particular, if $j_0 f_{j_0} \\geq \\delta |\\mathbf{f}|$, then\n\\begin{equation}\\label{eq_transfer_grande_face}\n\\beta_g(\\mathbf{f}-\\mathbf{1}_{j_0}) \\geq \\delta \\beta_g(\\mathbf{f}-\\mathbf{1}_p)\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nThe second point is immediate from the first. For the first point, the right-hand side counts maps in $\\mathcal{B}_g \\left( \\mathbf{f}-\\mathbf{1}_p \\right)$ with a marked edge such that the face on its right has degree $2j_0$. The left-hand side counts maps in $\\mathcal{B}_g \\left( \\mathbf{f}-\\mathbf{1}_{j_0} \\right)$ with a marked edge, so it is enough to build an injection from the first set to the second. Take a map $m$ in $\\mathcal{B}_g \\left( \\mathbf{f}-\\mathbf{1}_p \\right)$ and mark an edge $e$ of $m$ with a face of degree $2j_0$ on its right. We glue a path of $p-j_0$ edges to the starting point of $e$, just on the right of $e$ as on Figure~\\ref{fig_plus_grande_face}. One obtains a map of $\\mathcal{B}_g \\left( \\mathbf{f}-\\mathbf{1}_{j_0} \\right)$ with a marked edge, and going backwards is straightforward.\n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[scale=0.7]{plus_grande_face}\n\\caption{The injection of Lemma~\\ref{lem_grande_face} (here with $j_0=3$ and $p=5$). The marked edge is in red.}\\label{fig_plus_grande_face}\n\\end{figure}\n\\end{proof}\n\n\\begin{lem}\\label{lem_petites_faces}\nIf $1<d_1, d_2, \\ldots,d_k<j_0$ and ${\\sum_{i=1}^k (d_i-1) \\geq j_0-1}$, then \n\\[2|\\mathbf{f}| \\beta_g \\left( \\mathbf{f}-\\mathbf{1}_{j_0} \\right) \\geq 2j_0 f_{j_0} \\beta_g \\left( \\mathbf{f}-\\sum_{i=1}^k \\mathbf{1}_{d_i} \\right).\\]\nIn particular, if $j_0 f_{j_0} \\geq \\delta |\\mathbf{f}|$, then\n\\begin{equation}\\label{eq_transfer_petites_faces}\n\\beta_g  \\left( \\mathbf{f}-\\mathbf{1}_{j_0} \\right) \\geq \\delta \\beta_g \\left( \\mathbf{f}-\\sum_{i=1}^k\\mathbf{1}_{d_i} \\right).\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nThe proof is very similar to Lemma~\\ref{lem_grande_face}. This time, consider a map $m$ in $\\mathcal{B}_g \\left( \\mathbf{f}-\\sum_{i=1}^k\\mathbf{1}_{d_i} \\right)$ with a marked oriented edge $e$ that has a face of degree $2j_0$ on its right. Let $d=1+\\sum_{i=1}^k {(d_i-1)}$, and note that $d \\geq j_0$. If $d>j_0$, transform the face of degree $2j_0$ into a face of degree $2d$ by adding a path of $d-j_0$ edges like in the previous proof. Then tessellate this face of degree $2d$ as on Figure~\\ref{fig_plus_petites_faces}. We obtain a map of $\\mathcal{B}_g \\left( \\mathbf{f}-\\mathbf{1}_{j_0} \\right)$ with a marked edge, and this operation is also injective.\n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[scale=0.8]{plus_petites_faces}\n\\caption{The injection of Lemma~\\ref{lem_petites_faces} (here with $j_0=5$ and $(d_1,d_2,d_3)=(2,3,3)$).}\\label{fig_plus_petites_faces}\n\\end{figure}\n\\end{proof}\n\n\\subsection{Good sets of edges}\n\\label{subsec_good_sets}\n\nThe injection we will build to prove the Bounded ratio Lemma takes as input a pair $(m, E)$, where $m$ is a map and $E$ is a set of edges of $m$ satisfying the properties we will need to perform some surgery around $E$. We will call such a set a \\emph{good set}. Our goal in this subsection is to define a good set and to prove that any map contains a linear number of good sets (Proposition~\\ref{prop_good_sets}). We recall that we consider that the edges are oriented from white to black, and therefore it makes sense to define the left or right side of an edge.\n\nThroughout this section, we work under the assumptions of Lemma~\\ref{lem_BRL}. Let $m \\in \\mathcal{B}_g(\\mathbf{f})$. Let $A_1:=2A \\left( \\min \\left( \\frac{\\kappa}{32},\\delta \\right) \\right)$.\n\n\\begin{rem}\\label{rem_causal_graph}\nWe will have several different constants (depending on $A(\\cdot)$, $\\delta$ and $\\kappa$) defined in terms of each other in this subsection. To help convince the reader there is no circular dependency between them, we provide a \"causal graph\" of all the involved constants.\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[scale=0.8]{causal_graph}\n\\end{figure}\n\\end{rem}\n\nWe say that an edge $e$ of $m$ is \\emph{nice} if it is not incident to a face of degree larger than $2A_1$.\n\\begin{fact} At least $\\left( 1-\\frac{\\kappa}{16} \\right) |\\mathbf{f}|$ of the edges in $m$ are nice.\n\\end{fact}\n\\begin{proof}\nDraw an edge $e$ of $m$ uniformly at random. The face $f$ sitting to the right of $e$ is drawn at random with a probability proportional to its degree. By the second assumption of \\eqref{eq_petites_faces}, the probability that $f$ has degree larger than $2A_1$ is less than $\\frac{\\kappa}{32}$. The same is true for the face sitting to the left of $e$.\n\\end{proof}\n\nWe will need to bound the degrees not only of the faces incident to an edge, but also of the faces close to this edge for the dual distance. More precisely, we will define the \\emph{dual distance between two edges $e_1, e_2$ of $m$} as the dual distance between the face on the right of $e_1$ and the face on the right of $e_2$. We fix a value $r$ (depending on $A_1$ and $\\kappa$) that we will specify later. Let $A_r$ be the function given by Lemma~\\ref{lem_tight_degree_in_ball} for $A(\\cdot)$ and $r$, and let $A_2=A_r \\left( \\frac{\\kappa}{16} \\right)$. We will call an edge $e$ of $m$ \\emph{very nice} if it is nice and no edge at dual distance $r$ or less from $e$  is incident to a face of degree larger than $2A_2$. By applying Lemma~\\ref{lem_tight_degree_in_ball} to the stationary random graph obtained by rooting the dual map $m^*$ at a uniform edge, the proportion of edges of $m$ at dual distance $r$ or less from a face larger than $2A_2$ is at most $\\frac{\\kappa}{16}$. Hence, we get the following observation.\n\n\\begin{fact}At least $\\left( 1-\\frac{\\kappa}{8} \\right) |\\mathbf{f}|$ of the edges of $m$ are very nice.\n\\end{fact}\n\nLet $D=\\frac{4}{\\kappa}$. By the first assumption of \\eqref{eq_petites_faces}, we know that $D$ is larger than twice the average vertex degree in $m$. Since at most half of the vertices have degree at least twice the average degree, and since there are more than $\\kappa |\\mathbf{f}|$ vertices, we have the following.\n\n\\begin{fact}\nThere are at least $\\frac{\\kappa}{4}|\\mathbf{f}|$ vertices of the same colour with degree less than $D$ in $m$.\n\\end{fact}\n\nWithout loss of generality, assume that this colour is white (we recall that the vertices are coloured black and white so that each edge joins two vertices of different colours and the root is oriented from white to black).  We say that a white vertex is \\emph{fine} if it has degree at most $D$, and that an edge is \\emph{fine} if it is incident to a fine white vertex and the face on its right is not of degree $2$. By the previous fact, and since every vertex is incident to at least a face of degree $>2$, there are at least $\\frac{\\kappa}{4} |\\mathbf{f}|$ fine edges in $m$, incident to $\\frac{\\kappa}{4}|\\mathbf{f}|$ distinct white vertices.\nAn edge is said to be \\textit{good} if it is both very nice and fine. Summing up the last results, we have the following.\n\n\\begin{lem}\\label{lem_good_edges}\nThere are at least $\\frac{\\kappa }{8} |\\mathbf{f}|$ good edges in $m$, incident to $\\frac{\\kappa}{8} |\\mathbf{f}|$ distinct white vertices.\n\\end{lem}\n\nWe now fix the value of $r$ at $r=\\frac{16A_1}{\\kappa}+1$ (which is possible since $A_1$ does not depend on $r$, see Remark~\\ref{rem_causal_graph} above). We call a set $S$ of $A_1$ edges of $m$ a \\textit{good set} if all the edges of $S$ are good, they are incident to distinct white vertices and they are all at dual distance less than $2r$ from each other. Our next goal is to find a large number of good sets in the map $m$. Note that these good sets do not need to be disjoint.\n\n\\begin{prop}\\label{prop_good_sets}\nThere are at least $\\frac{\\kappa}{16} |\\mathbf{f}|$ good sets of edges in $m$.\n\\end{prop}\n\n\\begin{proof}\nThe proof follows the argument from~\\cite{BL19}.\nLet $G$ be a set of $\\frac{\\kappa }{8} |\\mathbf{f}|$ good edges incident to distinct white vertices given by Lemma~\\ref{lem_good_edges}. In this proof, the balls $B_r^*(e)$ that we will consider will be for the dual distance. We can assume that for every $e\\in G$, the ball $B_r^*(e)$ does not contain all the edges of $m$, since otherwise the proposition is obviously true.\n\nIn that case, for all $e\\in G$, since $m^*$ is connected we must have $|B^*_r(e)|>r$.\nWe are going to find a collection of distinct good sets $(S_i)$. For this, we build by induction a decreasing sequence of sets of good edges $(G_i)$, such that for each $i$, the set $G_{i+1}$ is obtained from $G_i$ by removing one element. We set $G_0=G$. Let $0 \\leq i<\\frac{\\kappa }{16} |\\mathbf{f}|$, and assume that we have built $G_0, G_1, \\dots, G_i$. Then $|G_i|=|G|-i$, so\n\\begin{align*}\n\\sum_{e\\in G_i} |B_r^*(e)|> \\left( |G|-i \\right)r >\\frac{\\kappa }{16} |\\mathbf{f}| r> A_1 |\\mathbf{f}|\n\\end{align*}\nby our choice of $r$. Therefore, there must be an \"$A_1$-overlap\", i.e. there exist $A_1$ edges whose balls of radius $r$ have a nonempty intersection. Thus they are all at distance at most $2r$ of each other, and we just found a good set $S_{i+1}$. Choose $e_{i+1} \\in S_{i+1}$ arbitrarily, and let $G_{i+1}=G_{i}\\setminus \\{e_{i+1}\\}$. This way we can build $G_i$ and $S_i$ for $1 \\leq i < \\frac{\\kappa}{16} |\\mathbf{f}|$, which proves the lemma.\n\\end{proof}\n\n\\subsection{Proof of the Bounded ratio Lemma: the injection}\n\\label{subsec_proof_BRL}\n\nWe now prove the Bounded ratio Lemma (Lemma~\\ref{lem_BRL}). We start with the easy case $j_0=1$: a marked digon can be contracted into a marked edge (see Figure~\\ref{fig_digon}), and if $f_1>\\delta |\\mathbf{f}|$, we have\n\\[|\\mathbf{f}|\\beta_g(\\mathbf{f}-\\mathbf{1}_1)>\\delta |\\mathbf{f}|  \\beta_g(\\mathbf{f})\\]\nwhich yields the result. \n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=0.5]{digon}\n\\caption{Contraction of a digon.}\\label{fig_digon}\n\\end{figure}\n\nWe can now assume that $j_0>1$ and $j_0 f_{j_0} > \\delta |\\mathbf{f}|$. The injection we will build takes as input a map of $\\mathcal{B}_g(\\mathbf{f})$ with a marked good set of edges, and outputs a map of $\\mathcal{B}_g(\\mathbf{\\tilde{f}})$ with a marked edge and some finite information (i.e. with values in a finite set whose size depends only on $\\delta, \\kappa$ and the function $A$), with $\\mathbf{\\tilde{f}}$ of the form:\n\\begin{enumerate}\n\\item[(i)]\neither\n\\begin{equation}\\label{brl_ftilde_first_condition}\n\\mathbf{\\tilde{f}}=\\mathbf{f}-\\mathbf{1}_p \\mbox{ where } j_0 \\leq p < A_1,\n\\end{equation}\n\\item[(ii)]\nor\n\\begin{equation}\\label{brl_ftilde_second_condition}\n\\mathbf{\\tilde{f}}=\\mathbf{f}-\\sum_{i=1}^k\\mathbf{1}_{d_i} \\mbox{ where } k \\leq A_1 \\mbox{ and } 1<d_i<j_0 \\mbox{ for all } i \\mbox{ but } \\sum_{i=1}^k d_i\\geq j_0+k-1.\n\\end{equation}\n\\end{enumerate}\nSince the number of possibilities for $\\mathbf{\\tilde{f}}$ is bounded in terms of $A_1$, by \\eqref{eq_transfer_grande_face} and~\\eqref{eq_transfer_petites_faces}, such an injection will prove Lemma~\\ref{lem_BRL}.\n\nThe surgery operation is quite complicated and, contrary to \\cite{BL19}, some intermediate steps affect the topology of the map (although the genus remains unchanged in the end). It is broken down into four steps for better understanding.\n\nBefore describing this operation in details, we first need to give a few definitions.\nConsider a map $m$ with a good set $S$ of edges, and a distinguished edge $e^*\\in S$ called the \\textit{anchor} (the way to choose $e^*$ will be specified later). For $e\\in S\\setminus\\{e^*\\}$, let $p_{e}$ be the leftmost shortest path\\footnote{the leftmost shortest path is constructed by, at each step, taking the leftmost face that gets strictly closer to the target.} in the dual $m^*$ from the face on the right of $e^*$ to the face on the right of $e$. We denote by $P(S)$ the union of all the paths $p_e$ for $e\\in S\\setminus\\{e^*\\}$ (see the left of Figure~\\ref{fig_path_carving}). With this definition $P(S)$ forms a tree. Since the edges of $S$ are at dual distance at most $2r$ from each other, we also know that the number of dual edges in $P(S)$ satisfies $|P(S)|<2A_1r$.\n\nWe will now describe four injective operations (four \"steps\"). To make things less cumbersome, we will use the term \\emph{finite} in lieu of \"bounded by a constant that depends only on $\\delta,\\kappa$ and the function $A$\" (without trying to make the bounds explicit). These steps will involve marked faces of possibly high but finite degree that we will call \\emph{megafaces}. Since one of the steps might temporarily disconnect the map, we also precise that by a \\emph{possibly non-connected map}, we mean a finite gluing of polygons which is not necessarily connected, and where only one of the connected components bears a root edge.\n\nAfter having defined the four steps, we will explain how to apply them to a map of $\\mathcal{B}_g(\\mathbf{f})$ with a marked good set to obtain the desired injection. In particular, some of the steps depend on additional parameters (such as the anchor edge in the good set $S$ for step $1$), and the way to choose the parameters will be specified in the end.\n\n\\paragraph{Step 1: Carving.} \n\\textit{This step inputs a map $m\\in \\mathcal{B}_g(\\mathbf{f})$, with a distinguished good set $S$, anchored at some edge $e^*$, and outputs a map $m_1$ of genus $g$ with a marked face of finite degree, $A_1$ distinguished fine vertices incident to this face, and some finite information.}\n\nStart from a map $m\\in \\mathcal{B}_g(\\mathbf{f})$, with a distinguished good set $S$, anchored at some edge $e^*$. Draw $P(S)$ on $m^*$ and cut each edge of $m$ that is crossed by $P(S)$. For every such edge, mark the two corners corresponding to the ends of this edge (see Figure~\\ref{fig_path_carving}).\nThe tree structure of $P(S)$ ensures that the map $m_1$ that we obtain is still connected and has genus $g$. Moreover $m_1$ has a distinguished face called the \\emph{megaface}, resulting from merging several faces of $m$. The $A_1$ white vertices incident to the edges of $S$ will be the $A_1$ distinguished vertices output by Step 1. These vertices are fine (i.e. have degree less than $\\frac{4}{\\kappa}$) by definition of a good set.\n\nMoreover, we claim that there are at most $2A_1 r$ marked corners of each colour in the megaface, and that the megaface is of degree less than $4A_1A_2r$.\nIndeed, each time an edge is deleted, two faces merge, and the megaface is the consequence of the mergings of the faces created by cutting edges. The number of marked corners is twice the number of edges cut, which is less than $|P(S)|$. Let $2j_1,2j_2,\\ldots,2j_{\\ell}$ be the degrees of the faces of $m$ that were merged in the carving process. Then the megaface of $m_1$ has degree $2F=2(j_1+j_2+\\ldots+j_{\\ell}-\\ell+1)$. By definition of a good set, $\\ell\\leq |P(S)|+1\\leq 2A_1r$, and also for all $i$, we have $j_i<A_2$, which implies $2F<4A_1A_2r$. Therefore, given the map $m_1$ and its megaface, the number of ways to choose the marked corners is bounded by $\\binom{4A_1 A_2 r}{2 A_1 r}$, so the marked corners are finite information, which will allow us to forget about them in the next steps.\n\nFinally, note that step 1 is injective up to finite information: given the map $m_1$, its megaface and the marked corners, there is a finite number of ways to pair the marked corners in a planar way inside the megaface.\n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[scale=0.7]{face_carving}\n\\caption{The carving operation, from $m$ (on the left) to $m_1$ (on the right). The good set $S$ of edges is in purple with the anchor in red), and the paths $P(S)$ are in blue. Marked corners in $m_1$ are represented as dangling half-edges.}\\label{fig_path_carving}\n\\end{figure}\n\n\\paragraph{Step 2: Vertex deletion.}\n\\textit{This step inputs a map $m_1$ of genus $g$, with a marked face of finite degree and $A_1$ distinguished fine vertices incident to this face, as well as a number $d<A_1$. It outputs (together with some finite information) a possibly disconnected map $m_2$ with $K$ connected components, of total genus $g' \\leq g$ with $g-g'$ finite, with $v(\\mathbf{f},g)-d$ vertices in total and with $M$ marked faces of finite total degree, such that there is at least one marked face in each connected component, and we have\n\\begin{equation}\\label{eq_degenerescence}\nM \\leq (g-g')+K.\n\\end{equation}\n}\n\nStart with a map $m_1$ of genus $g$, with a marked megaface to which $A_1$ marked fine vertices are incident, as well as a number\\footnote{the number $d$ is a parameter that will be fixed when defining the injection.} $d<A_1$. Pick $d$ of the marked good vertices arbitrarily, and delete them as well as all their incident edges (see Figure~\\ref{fig_vertex_deletion}). We obtain a map $m_2$ that might be disconnected (see right before Step 1 for a definition). \n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[scale=0.8]{vertex_deletion}\n\\caption{Deleting a vertex.}\\label{fig_vertex_deletion}\n\\end{figure}\n\nWe now precise which are the marked faces. Everytime an edge is deleted in the process, three cases may happen:\n\\begin{itemize}\n\\item two faces are merged,\n\\item or a face is split in two and the genus decreases by one,\n\\item or a face is split in two and the number of connected components increases by one.\n\\end{itemize}\n\nThe marked faces (or megafaces) of $m_2$ are then the result of the consecutive splittings and mergings. In particular, the last two cases explain \\eqref{eq_degenerescence}. Moreover, in the third case, if a connected component is split in two, the marked face gets split between both components, so in the end there is at least one marked face on each connected component.\n\nWe now bound the total degree of the megafaces. The edges incident to the megafaces of $m_2$ come from the megaface of $m_1$, plus the \"small\" faces we merged with it in the process. By definition of a good set, all these small faces were of degree $A_2$ or less, and the number of mergings is bounded by the total number of edges deleted in the operation, which itself is at most $DA_1$ ($d<A_1$ vertices of degree bounded by $D$ were deleted). Hence the total degree of the megafaces is bounded by $DA_1 A_2 + 2F$, where $2F$ is the degree of the megaface of $m_1$. In particular, this total degree is finite. On the other hand, the genus variation $g-g'$ is bounded by the number of removed edges, so it is finite as well.\n\nFinally, we argue that this step is injective up to finite information. To go back to $m_1$, one only needs to recreate the white vertices and reattach them to the right black corners. The number of edges to add back is finite (bounded by $D A_1$), and the black corners to which we may attach them are on the megafaces, so their number is also finite. Therefore, up to adding finite information in the output, Step 2 is injective.\n\n\\paragraph{Step 3: Reconstruction.} \n\\textit{This step inputs a possibly disconnected map $m_2$ with $v(\\mathbf{f},g)-d$ vertices in total for some $d$, of genus $g' \\leq g$ with $g-g'$ finite, with $K$ connected components and $M$ marked faces of finite total degree, satisfying~\\eqref{eq_degenerescence} and such that there is at least one marked face in each connected component. It outputs a connected map $m_3$ with $v(\\mathbf{f},g)-d+1$ vertices, of genus $g$, with one marked face of finite degree and a marked edge lying in that face, plus some finite information.}\n\nWe start with a map $m_2$ as above, and denote by $2F$ the total degree of its megafaces. We create one new white vertex $v$, select arbitrarily a black corner in each megaface $f$ of $m_2$, and draw an edge between $v$ and these black corners (see the left part of Figure~\\ref{fig_reconstruction}). We keep track of one of these newly created edges and call it the \\emph{special edge}. We obtain a map $m'_2$ that is now connected and has a unique megaface of degree at most $2F+2M$ (because we added $M$ edges). This megaface is incident to the vertex $v$ and to the special edge. Let $g''$ be the genus of $m'_2$. We note that when we add one by one the edges joining $v$ to the megafaces of $m_2$, the genus increases by one everytime we connect a megaface which is not the first one in its connected component of $m_2$. Hence $g''-g'=M-K$, so~\\eqref{eq_degenerescence} implies $g''\\leq g$. If $g''=g$, we set $m_3=m'_2$. If $g''<g$, we attach a pair of edges to the right of the special edge as on the right of Figure~\\ref{fig_reconstruction}. Now the resulting map has still only one megaface and is of genus $g''+1$. Repeat if necessary to obtain a map $m_3$ of genus $g$ with one megaface of degree bounded by $2F+2K+4(g-g')$ (because we added at most $2(g-g')$ edges to recover the genus). Using~\\eqref{eq_degenerescence} again, the degree of the megaface is bounded by $2F+4M \\leq 6F$ so it is finite. Moreover, the only vertex we have added is $v$, so $m_3$ has $v(\\mathbf{f},g)-d+1$ vertices.\n\n\\begin{figure}[!ht]\n\\center\n\\includegraphics[scale=0.4]{reconnexion}\n\\hspace{1cm}\n\\includegraphics[scale=0.4]{remonter_genre}\n\\caption{The reconstruction. Left: reconnecting the map. The megafaces are in pink, and the rest of the map is in turquoise. Marked corners are represented as dangling half-edges. Right: recovering the genus. The special edge is in red, and the added pair of edges is in blue.}\\label{fig_reconstruction}\n\\end{figure}\n\nFinally, to recover $m_2$ from $m_3$, one only has to remember the value of $g-g''$. This is finite information since $g-g'' \\leq g-g'$ is finite. We can then simply remove the $2(g-g'')$ edges on the right of the special edge to recover $m'_2$, and then delete the vertex $v$ (i.e. the white vertex incident to the marked edge of $m_3$), together with all its incident edges, to recover $m_2$. Therefore, Step 3 is injective up to finite information.\n\n\\paragraph{Step 4: Filling of the megaface.} \n\\textit{This step inputs a map $m_3$ in $\\mathcal{B}_g(\\mathbf{f'})$ for some $\\mathbf{f'}$ with a marked face of finite degree $2F$ and a marked edge incident to that face. We also assume that $\\mathbf{f'}$ satisfies the following assumptions:\n\\begin{enumerate}\n\\item\nwe have $\\mathbf{f'}=\\mathbf{f}+\\mathbf{1}_F-\\mathbf{1}_{j_1}-\\mathbf{1}_{j_2}-\\ldots-\\mathbf{1}_{j_{\\ell}}$, where $\\sum_{i=1}^{\\ell} j_i$ is finite;\n\\item\n$v(\\mathbf{f'},g) = v(\\mathbf{f},d)-d+1$ for some finite $d \\geq 1$;\n\\item\nthere is an index $k< \\ell$ such that \n\\begin{equation}\\label{eq_condition_BRL}\n(j_1-1)+(j_2-1)+\\ldots+(j_k-1)=d-1.\n\\end{equation}\n\\end{enumerate}\nIt outputs a map $m_4\\in \\mathcal{B}_g (\\mathbf{\\tilde{f}} )$ with a marked edge, where $\\mathbf{\\tilde{f}} \\leq \\mathbf{f}$ and $\\left| \\mathbf{f}-\\mathbf{\\tilde{f}} \\right|$ is finite.}\n\nWe start with a map $m_3 \\in \\mathcal{B}_g(\\mathbf{f'})$ as above, and let $d$ and $j_1, \\dots, j_{\\ell}$ be as in the assumptions on $\\mathbf{f'}$. By the Euler formula on $\\mathbf{f}$ and $\\mathbf{f'}$, we must have \n\\begin{equation}\\label{eqn_megaface_decomposition}\nF-1=(j_1-1)+(j_2-1)+\\ldots+(j_{\\ell}-1)-(d-1).\n\\end{equation}\nCombining this with the assumption~\\eqref{eq_condition_BRL}, we obtain\n\\[F-1=(i_{k+1}-1)+(i_{k+2}-1)+\\ldots+(i_{\\ell}-1).\\]\nHence, we can tessellate the megaface of degree $2F$ into faces of degrees $2j_{k+1}, \\dots, 2j_{\\ell}$ as in the proof of \\eqref{lem_grande_face}. We obtain a map $m_4$ (keeping the same marked edge) with face degrees given by $\\mathbf{\\tilde{f}}=\\mathbf{f}-\\sum_{i=1}^k \\mathbf{1}_{j_i}$, in an injective way.\n\n\\vspace{1cm}\nNow we can describe our injection in terms of the different operations mentioned above. Roughly speaking, we will successively apply our four steps. We will specify the choice of some of the parameters in the first three steps in such a way that the face degree distribution $\\mathbf{f'}$ after Step $3$ satisfies all the assumptions required by Step $4$, and the face degree distribution $\\mathbf{\\tilde{f}}$ obtained after Step $4$ is of one of the two forms given by~\\eqref{brl_ftilde_first_condition} and~\\eqref{brl_ftilde_second_condition}.\n\nMore precisely, we start with a map $m \\in \\mathcal{B}_g(\\mathbf{f})$ with a marked good set $S$. For all $e\\in S$, let $d_e$ be the half-degree of the face sitting on the right of $e$. We treat two cases separately (which will in the end correspond to~\\eqref{brl_ftilde_first_condition} and to~\\eqref{brl_ftilde_second_condition}).\n\n\\paragraph{Case 1:} Assume there exists $e^*$ in $S$ such that $d_{e^*}\\geq j_0$. Take $e^*$ as the anchor, and apply Step 1. Note that the face of degree $2d_{e^*}$ sitting to the right of $e^*$ is one of the faces that have been destroyed. We then apply Step 2 with $d=d_{e^*}$, i.e. we remove $d$ good vertices, including the one that was incident to $e^*$. We then apply Step $3$ to obtain a map $m_3$, and denote by $\\mathbf{f'}$ the face degree sequence of $m_3$. Note that $m_3$ has $d-1$ vertices less than $m$, which ensures that the $d$ as defined in the assumptions of Step $4$ is the same as the one used in Step 2 and 3. Moreover, it follows from Steps 1, 2 and 3 that $\\mathbf{f'}$ is of the form $\\mathbf{f}+\\mathbf{1}_F-\\mathbf{1}_{j_1}-\\mathbf{1}_{j_2}-\\ldots-\\mathbf{1}_{j_{\\ell}}$, where $2F$ is the degree of the megaface, and $2j_1, \\dots, 2j_{\\ell}$ are the degrees of the faces destroyed in Step 1. In particular $\\sum_{i=1}^{\\ell} j_i$ is finite and, up to reordering the $j_i$'s, we may assume $j_1=d=d_{e^*}$. This implies that~\\eqref{eq_condition_BRL} is satisfied for $k=1$, so we can apply Step 4 to obtain a map $m_4$. Finally, the face degree sequence of $m_4$ is $\\mathbf{\\tilde{f}}=\\mathbf{f}-\\mathbf{1}_d$, where $d=d_{e^*} \\geq j_0$, and $d<A_1$ by definition of a nice edge. Hence $\\mathbf{\\tilde{f}}$ is of the form given by~\\eqref{brl_ftilde_first_condition}.\n\n\\paragraph{Case 2:} We now assume $d_e<j_0$ for all $e \\in S$. Take any arbitrary ordering $e_1,e_2, \\ldots, e_{A_1}$ of the edges of $S$, and for all $i$ let $d_i=d_{e_i}$ be the half-degree of the face on the right of $e_i$. We first claim that there is an index $k$ such that\n\\begin{equation}\\label{eqn_brl_case_2}\nj_0-1<\\sum_{i=1}^k (d_i-1)<A_1-1\n\\end{equation}\nIndeed, by definition of a fine edge, we have $d_i \\geq 2$ for all $i$ so $\\sum_{j=1}^{A_1}(d_j-1)\\geq A_1>j_0-1$. Therefore, we can consider the first index $k$ such that $\\sum_{i=1}^k (d_i-1)>j_0-1$. Then we have $d_k < j_0$ by assumption, so $\\sum_{i=1}^k (d_i-1) < 2j_0-1$. Finally, by definition of $A_1$, we have $A_1 \\geq 2A(\\delta) \\geq 2j_0$ (the second inequality follows from the assumption $j_0 f_{j_0} > \\delta|\\mathbf{f}|$), so~\\eqref{eqn_brl_case_2} is indeed satisfied for this $k$.\n\nWe now choose the anchor of $S$ arbitrarily, and apply Step 1. We then apply Step 2 with $d=1+\\sum_{i=1}^k (d_i-1)$, i.e. we delete $d$ good vertices, including the ones incident to $e_1, \\ldots, e_k$. Note that at this point, all the faces that were incident to the edges $e_1, \\dots, e_k$ have been destroyed. We then apply Step 3 and obtain a map $m_3$ with face degree sequence $\\mathbf{f'}$. The first two assumptions on $\\mathbf{f'}$ in Step 4 are satisfied for the same reason as in Case 1, with $\\mathbf{f'}$ of the form $\\mathbf{f}+\\mathbf{1}_F-\\mathbf{1}_{j_1}-\\ldots-\\mathbf{1}_{j_{\\ell}}$. Moreover, since the faces incident to $e_1, \\dots, e_k$ have been destroyed previously, up to reordering the $j_i$'s, we may assume $j_i=d_i$ for $1 \\leq i \\leq k$. Therefore, by our choice of $d$, the third assumption~\\eqref{eq_condition_BRL} of Step 4 is also satisfied. After applying Step 4, we obtain a map $m_4$ with face degree sequence\n\\[\\mathbf{\\tilde{f}}=\\mathbf{f}-\\sum_{i=1}^{k} d_i.\\]\nFinally, we have $1<d_i<j_0$ for all $i$ by the assumption of Case 2, and $\\sum_{i=1}^k d_i \\geq j_0+k-1$ by~\\eqref{eqn_brl_case_2}, so $\\mathbf{\\tilde{f}}$ is of the form given by~\\eqref{brl_ftilde_second_condition}.\n\n\\begin{proof}[Conclusion of the proof of Lemma~\\ref{lem_BRL}]\nConsider the injection that we have just built. By Proposition~\\ref{prop_good_sets}, the number of inputs $m$ is at least $\\frac{\\kappa}{16} |\\mathbf{{f}}|\\beta_g(\\mathbf{f})$, whereas the number of outputs $m_4$ is at most $\\sum_{\\mathbf{\\tilde{f}}} |\\mathbf{\\tilde{f}}| \\beta_g(\\mathbf{\\tilde{f}})$, where the sum is over face degree sequences $\\mathbf{\\tilde{f}}$ of the form either~\\eqref{brl_ftilde_first_condition} of~\\eqref{brl_ftilde_second_condition}.\n\nWhen we apply successively Steps 1, 2, 3 and 4, we note that each step is injective up to finite information. This means that there is a constant $c$ depending only on $\\delta$, $\\kappa$ and $A(\\cdot)$ such that\n\\[ \\frac{\\kappa}{16}|\\mathbf{{f}}| \\beta_g(\\mathbf{f}) \\leq c \\sum_{\\mathbf{\\tilde{f}}} |\\mathbf{\\tilde{f}}| \\beta_g(\\mathbf{\\tilde{f}}) \\leq c |\\mathbf{f}| \\sum_{\\mathbf{\\tilde{f}}} \\beta_g(\\mathbf{\\tilde{f}}), \\]\nwhere the second inequality uses $\\mathbf{\\tilde{f}} \\leq \\mathbf{f}$.\nMoreover, the number of possible sequences $\\mathbf{\\tilde{f}}$ is bounded in terms of $\\delta$, $\\kappa$ and $A(\\cdot)$. Hence, there is a sequence $\\mathbf{\\tilde{f}}$ of the form~\\eqref{brl_ftilde_first_condition} or~\\eqref{brl_ftilde_second_condition} such that\n\\[ \\beta_g(\\mathbf{f}) \\leq c' \\beta_g(\\mathbf{\\tilde{f}}), \\]\nwhere $c'$ depends only on $\\delta$, $\\kappa$ and $A(\\cdot)$. We can finally conclude using Lemma~\\ref{lem_grande_face} if $\\mathbf{\\tilde{f}}$ is of the form~\\eqref{brl_ftilde_first_condition}, or Lemma~\\ref{lem_petites_faces} if $\\mathbf{\\tilde{f}}$ is of the form~\\eqref{brl_ftilde_second_condition}.\n\\end{proof}\n\n\\subsection{Planarity and One-Endedness}\n\\label{subsec_planarity}\n\nWe fix $(\\mathbf{f}^{n})_{n \\geq 1}$ and $(g_n)_{n \\geq 1}$ satisfying the assumptions of Theorem~\\ref{thm_main_more_general}. We recall that by Lemma~\\ref{lem_easy_dual_convergence} $(M_{\\mathbf{f}^{n}, g_n})$ is tight for $d_{\\mathrm{loc}}^*$. In all this subsection, we will denote by $M$ a subsequential limit in distribution. Since $|\\mathbf{f}^{n}| \\to +\\infty$, it must be an infinite map. To prove Proposition~\\ref{prop_tightness_dloc_univ}, we will first prove that $M$ is a.s. planar and one-ended (Corolaries~\\ref{cor_planar} and \\ref{cor_OE}). Like in \\cite{BL19}, the proofs will rely heavily on a recursion counting maps. More precisely, the formula~\\eqref{rec_biparti_genre_univ} proved in \\cite{Lo19} will play the role of the Goulden--Jackson formula in \\cite{BL19}. This step of the proof only differs from \\cite{BL19} by minor adaptations, so we will not give the full details. The proofs of Lemmas \\ref{lem_calcul_planar} and \\ref{lem_calcul_OE} using~\\eqref{rec_biparti_genre_univ} can be found in the Appendix~\\ref{sec_univ_estim_lemmas}.\n\nAs in \\cite{BL19}, to establish planarity, we want to bound, for any non-planar finite map $m$, the probability $\\P \\left( m \\subset M_{\\mathbf{f}^{n},g_n} \\right)$ for $n$ large. For this, we will need the following combinatorial estimate.\n\n\\begin{lem}\\label{lem_calcul_planar}\nWe fix $(\\mathbf{f}^{n})_{n \\geq 1}$ and $(g_n)_{n \\geq 1}$ which satisfy the assumptions of Theorem~\\ref{thm_main_more_general}. We also fix $\\mathbf{h}^{(0)}$ a face degree sequence such that $\\mathbf{h}^{(0)} \\leq\\mathbf{f}^{n}$ for $n$ large enough.  Let $k\\geq 2$, numbers $\\ell_1, \\ell_2,\\ldots,\\ell_k$ and perimeters $p_i^j$ for $1\\leq j\\leq k$ and $1\\leq i \\leq \\ell_j$. Then\n\\[\\sum_{\\substack{\\mathbf{h}^{(1)}+\\mathbf{h}^{(2)}+\\ldots+\\mathbf{h}^{(k)}=\\mathbf{f}^{n}-\\mathbf{h}^{(0)} \\\\ g^{(1)}+g^{(2)}+\\ldots+g^{(k)}=g_n-1-\\sum_j(\\ell_j-1)}}\\prod_{j=1}^k \\beta_{g^{(j)}}^{(p^j_1,p^j_2,\\ldots,p^j_{\\ell_j})}(\\mathbf{h}^{(j)})=o\\left(\\beta_{g_n}(\\mathbf{f}^{n})\\right)\\]\nas $n\\rightarrow \\infty$.\n\\end{lem}\n\n\n\\begin{corr}\\label{cor_planar}\nLet $(\\mathbf{f}^{n})_{n \\geq 1}$ and $(g_n)_{n \\geq 1}$ satisfy the assumptions of Theorem~\\ref{thm_main_more_general}. Then every subsequential limit $M$ of $\\left( M_{\\mathbf{f}^n}, g_n \\right)$ for $d^*_{\\mathrm{loc}}$ is a.s. planar.\n\\end{corr}\n\n\\begin{proof}\nThe proof is basically the same as~\\cite[Corollary 7]{BL19}, with the exception of the assumption $\\mathbf{h}^{(0)} \\leq\\mathbf{f}^{n}$ in Lemma~\\ref{lem_calcul_planar} (this assumption was automatically satisfied in~\\cite{BL19}). More precisely, it is sufficient to prove $\\P \\left( m \\subset M \\right)=0$ for every finite map $m$ with holes and of genus $1$. We fix such a map $m$.\n\nWe also note that almost surely, for every face $f$ of $M$, we have $\\alpha_{\\mathrm{deg}(f)\/2}>0$.  Indeed, this is true for the root face since the root face of $M$ has degree $j$ with probability $j \\alpha_j$ for all $j \\geq 1$, and this can be extended to all faces using stationarity with respect to the simple random walk on the dual of $M$. Therefore, if $m$ has a face of degree $2j$ with $\\alpha_j=0$, then $\\P \\left( m \\subset M \\right)=0$.\n\nIf not, let $\\mathbf{h}^{(0)}$ be the internal face degree sequence of $m$. Then $f^n_j \\to +\\infty$ as $n \\to +\\infty$ for every $j$ such that $h^{(0)}_j>0$, so $\\mathbf{h}^{(0)} \\leq\\mathbf{f}^{n}$ for $n$ large enough. In particular, we are in position to use Lemma~\\ref{lem_calcul_planar}. The proof is now exactly the same as in~\\cite{BL19}: we use the fact that $\\P \\left( m \\subset M \\right)$ can be expressed using the number of ways to fill the holes of $m$ with maps of multipolygons, which is given by the left-hand side of Lemma~\\ref{lem_calcul_planar}.\n\\end{proof}\n\nWe now move on to one-endedness. The proof is quite similar, and relies on the following estimate.\n\n\\begin{lem}\\label{lem_calcul_OE}\nWe fix $(\\mathbf{f}^{n})_{n \\geq 1}$ and $(g_n)_{n \\geq 1}$ which satisfy the assumptions of Theorem~\\ref{thm_main_more_general}. We also fix $\\mathbf{h}^{(0)}$ a face degree sequence such that $\\mathbf{h}^{(0)} \\leq\\mathbf{f}^{n}$ for $n$ large enough.\n\\begin{itemize}\n\\item\nLet $k\\geq 1$, numbers $\\ell_1, \\ell_2,\\ldots,\\ell_k$ \\textbf{not all equal to $1$} and perimeters $p_i^j$ for $1\\leq j\\leq k$ and $1\\leq i\\leq \\ell_j$. Then\n\\[\\sum_{\\substack{\\mathbf{h}^{(1)}+\\mathbf{h}^{(2)}+\\ldots+\\mathbf{h}^{(k)}=\\mathbf{f}^n-\\mathbf{h}^{(0)} \\\\ g^{(1)}+g^{(2)}+\\ldots+g^{(k)}=g_n-\\sum_j(\\ell_j-1)}}\\prod_{j=1}^k \\beta_{g^{(j)}}^{(p^j_1,p^j_2,\\ldots,p^j_{\\ell_j})}(\\mathbf{h}^{(j)})=o\\left(\\beta_{g_n}(\\mathbf{f}^n)\\right)\\]\nas $n\\rightarrow \\infty$.\n\\item \nLet $k\\geq 1$ and perimeters $p_1,\\ldots,p_k$. There is a constant $C$ (that may depend on everything above) such that for every $a$ and $n$ large enough we have\n\\[\\sum_{\\substack{\\mathbf{h}^{(1)}+\\mathbf{h}^{(2)}+\\ldots+\\mathbf{h}^{(k)}=\\mathbf{f}^n-\\mathbf{h}^{(0)} \\\\ g^{(1)}+g^{(2)}+\\ldots+g^{(k)}=g_n\\\\  |\\mathbf{h}^{(1)}|, |\\mathbf{h}^{(2)}|>a}} \\, \\prod_{j=1}^k \\beta_{g_j}^{(p_j)}(\\mathbf{h}^{(j)})\\leq \\frac{C}{a} \\beta_{g_n}(\\mathbf{f}^n).\\]\n\\end{itemize}\n\\end{lem}\n\n\\begin{corr}\\label{cor_OE}\nLet $(\\mathbf{f}^{n})_{n \\geq 1}$ and $(g_n)_{n \\geq 1}$ satisfy the assumptions of Theorem~\\ref{thm_main_more_general}. Every subsequential limit $M$ of $\\left( M_{\\mathbf{f}^n}, g_n \\right)$ for $d^*_{\\mathrm{loc}}$ is a.s. one-ended in the sense that, for every finite map $m$ with holes such that $m \\subset M$, only one hole of $m$ is filled with infinitely many faces\\footnote{This is a \"weak\" definition of one-endedness. For example, it does not prevent $m$ to be the dual of a tree. However, once we will have proved that $M$ has finite vertex degrees, this will be equivalent to the usual definition.}.\n\\end{corr}\n\n\\begin{proof}\nThe proof given Lemma~\\ref{lem_calcul_OE} is exactly the same as~\\cite[Corollary 9]{BL19}, except for the additional assumption in Lemma~\\ref{lem_calcul_OE} that $\\mathbf{h}^{(0)} \\leq\\mathbf{f}^{n}$ for $n$ large enough. We take care of it in the same way as in the proof of Corollary~\\ref{cor_planar}.\n\\end{proof}\n\n\\subsection{Finiteness of the root degree}\n\\label{subsec_finite_degrees}\n\nWe now finish the proof of tightness for $d_{\\mathrm{loc}}$ (Proposition~\\ref{prop_tightness_dloc_univ}). Let $M$ be a subsequential limit of $\\left( M_{\\mathbf{f}^n, g_n} \\right)$ for $d_{\\mathrm{loc}}^*$. By Lemma~\\ref{lem_dual_convergence_univ}, to get tightness for $d_{\\mathrm{loc}}$, we need to show that almost surely, all the vertices of $M$ have finite degree. Our argument is now very similar to \\cite{BL19} and inspired by \\cite{AS03}: we will first study the degree of the root vertex by using the Bounded ratio Lemma, and then extend finiteness by using invariance under the simple random walk.\n\n\\begin{lem}\\label{lem_root_degree_is_finite_univ}\nThe root vertex of $M$ has a.s. finite degree.\n\\end{lem}\n\n\\begin{proof}\nFollowing the approach of \\cite{AS03}, we perform a filled-in lazy peeling exploration of $M$. Note that we already know by Corollary~\\ref{cor_planar} that the explored part will always be planar, so no peeling step will merge two different existing holes. Moreover, by Corollary~\\ref{cor_OE}, if a peeling step separates the boundary into two holes, then one of them is finite and will be filled with a finite map. Therefore, at each step, the explored part will have only one hole.\n\nThe peeling algorithm $\\mathcal{A}$ that we use is the following: if the root vertex $\\rho$ belongs to $\\partial m$, then $\\mathcal{A}(m)$ is the edge on $\\partial m$ on the left of $\\rho$. If $\\rho \\notin \\partial m$, then the exploration is stopped. Let $\\tau$ be the time at which the exploration is stopped. Since only finitely many edges incident to $\\rho$ are added at each step, it is enough to prove $\\tau<+\\infty$ a.s.. We recall that $\\mathcal{E}_t^{\\mathcal{A}}(M)$ is the explored part at time $t$.\n\nWe will prove that at each step, conditionally on $\\mathcal{E}_t^{\\mathcal{A}}(M)$, the probability to swallow the root and finish the exploration in a bounded amount of time is bounded from below by a positive constant. We fix $j^* \\geq 2$ with $\\alpha_{j^*}>0$. Note that such a $j^*$ exists because of the assumption $\\theta<\\frac{1}{2} \\sum_{j \\geq 1} (j-1) \\alpha_j$. For every map $m$ with one hole such that $\\rho \\in \\partial m$, we denote by $m^+$ the map constructed from $m$ as follows (see Figure~\\ref{fig_swallowing_root_univ}):\n\\begin{itemize}\n\\item\nwe first glue a \"face\" of degree $2j^*$ to $m$ along the edge of $\\partial m$ on the left of $\\rho$;\n\\item\nwe then glue together the two edges of the boundary incident to $\\rho$ together;\n\\item\nduring the next $j^*-2$ steps, at each step, we pick two consecutive edges of the boundary according to some fixed convention and glue them together.\n\\end{itemize}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.7]{swallow_root}\n\\caption{The construction of $m^+$ from $m$. In gray, the map $m$. In red, the root vertex. In blue, the new face. Here, $|\\partial m|=j^*=3$.}\n\\label{fig_swallowing_root_univ}\n\\end{figure}\n\nNote that $m^+$ is a planar map with the same perimeter as $m$ but one more face (of degree $2j^*$). By the choice of our peeling algorithm, if we have $\\tau \\geq t$ and $\\mathcal{E}_t^{\\mathcal{A}}(M)^+ \\subset M$, then we have $\\tau \\leq t+2$. Hence it is enough to prove that the quantity\n\\[ \\P \\left( m^+ \\subset M | m \\subset M \\right)\\]\nis bounded from below over finite, planar maps $m$ with one hole such that $\\rho \\in \\partial m$.\n\nWe fix such an $m$, with half-perimeter $p$ and internal face degrees given by $\\mathbf{h}$. Along some subsequence, we have $M_{\\mathbf{f}^{n},g_{n}} \\to M$ in distribution (for $d_{\\mathrm{loc}}^*$). Along the same subsequence, it holds that\n\\[ \\P \\left( m^+ \\subset M | m \\subset M \\right)\\hspace{-0.1cm}=\\hspace{-0.1cm}\\lim_{n \\to +\\infty}\\hspace{-0.1cm} \\frac{\\P \\left( m^+ \\in M_{\\mathbf{f}^{n},g_{n}} \\right)}{\\P \\left( m\\in M_{\\mathbf{f}^{n},g_{n}} \\right)} \\hspace{-0.1cm} = \\hspace{-0.1cm}\\lim_{n \\to +\\infty}\\hspace{-0.1cm}  \\frac{\\beta^{(p)}_{g_n} \\left( \\mathbf{f}^n-\\mathbf{h}-\\mathbf{1}_{j^*} \\right)}{\\beta^{(p)}_{g_n} \\left( \\mathbf{f}^n-\\mathbf{h} \\right)}. \\]\nBy our choice of $j^*$, we have $f^n_{j^*} \\geq \\frac{\\alpha_{j^*}}{2} |\\mathbf{f}^n|$ for $n$ large enough, so we can apply the Bounded ratio Lemma, which concludes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop_tightness_dloc_univ}]\nLet $M$ be a subsequential limit of $\\left( M_{\\mathbf{f}^{n},g_{n}} \\right)$. We recall that for all $n$, the map $\\left( M_{\\mathbf{f}^{n},g_{n}} \\right)$ is stationary for the simple random walk on its vertices. Therefore, by Lemma~\\ref{lem_root_degree_is_finite_univ} and the same argument as in \\cite{AS03} (see also the proof of Lemma~\\ref{lem_easy_dual_convergence} above), almost surely all the vertices of $M$ have finite degree. By Lemma~\\ref{lem_dual_convergence_univ}, this guarantees that $\\left( M_{\\mathbf{f}^{n},g_{n}} \\right)$ is tight for $d_{\\mathrm{loc}}$.\n\nThe a.s. planarity of $M$ is proved in~Corollary \\ref{cor_planar}. Finally, it easy to check that for maps with finite vertex degrees, the weak version of one-endedness proved in Corollary~\\ref{cor_OE} implies the usual one. Indeed, if $V$ is a finite set of vertices of $M$, one can consider a finite, connected submap of $M$ containing all the faces and edges incident to vertices of $V$. Then Corollary~\\ref{cor_planar} ensures that this submap does not separate $M$ into two infinite maps.\n\\end{proof}\n\n\\section{Weakly Markovian bipartite maps}\\label{sec_univ_markov}\n\nOur goal in this Section is to prove Theorem~\\ref{thm_weak_Markov_general}.\n\n\\paragraph{Weakly Markovian bipartite maps.}\nFor a finite, bipartite map $m$ with one hole, we denote by $|\\partial m|$ the half-perimeter of the hole of $m$. For all $j \\geq 1$, we also denote by $v_j(m)$ the number of internal faces of $m$ with degree $2j$.\n\n\\begin{defn}\\label{defn_weak_Markov}\nLet $M$ be a random infinite, one-ended, bipartite planar map. We say that $M$ is \\emph{weakly Markovian} if for every finite map $m$ with one hole, the probability $\\P \\left( m \\subset M \\right)$ only depends on $|\\partial m|$ and $\\left( v_j(m) \\right)_{j \\geq 1}$.\n\\end{defn}\n\nLet $\\mathcal{V}$ be the set of sequences $\\mathbf{v}=(v_j)_{j \\geq 1}$ such that $v_j=0$ for $j$ large enough. If $M$ is weakly Markovian and $\\mathbf{v} \\in \\mathcal{V}$, we will denote by $a^p_{\\mathbf{v}}$ the probability $\\P \\left( m \\subset M \\right)$ for a map $m$ with $|\\partial m|=p$ and $v_j(m)=v_j$ for all $j$. Note that this only makes sense if there is such a map $m$, which is equivalent to\n\\begin{equation}\\label{eqn_good_pvv}\np \\leq 1+\\sum_{j \\geq 1} (j-1)v_j.\n\\end{equation}\nTherefore, if $p \\geq 1$, we will denote by $\\mathcal{V}_p \\subset \\mathcal{V}$ the set of those $\\mathbf{v}$ satisfying \\eqref{eqn_good_pvv}. Note that $\\mathcal{V}_1=\\mathcal{V}$.\nIn particular, by definition, for $\\mathbf{q} \\in \\mathcal{Q}_h$, the $\\mathbf{q}$-IBPM is weakly Markovian, and the corresponding constants $a^p_{\\mathbf{v}}$ are:\n\\[a^p_{\\mathbf{v}}(\\mathbf{q}):= C_p(\\mathbf{q}) \\mathbf{q}^{\\mathbf{v}},\\]\nwhere $\\mathbf{q}^{\\mathbf{v}} := \\prod_{j \\geq 1} q_j^{v_j}$. Therefore, if $M$ is of the form $\\mathbb M_{\\mathbf{Q}}$ for some random weight sequence $\\mathbf{Q}$, we have $a^p_{\\mathbf{v}} = \\mathbb E[ C_p(\\mathbf{Q}) \\mathbf{Q}^{\\mathbf{v}} ]$.\n\n\\paragraph{Sketch of the proof of Theorem~\\ref{thm_weak_Markov_general}.}\nWe first note that the second point of Theorem~\\ref{thm_weak_Markov_general} is immediate once the first point is proved. Indeed, let us write $\\mathrm{Rootface}(m)$ for the degree of the root face of a map $m$. If $\\mathrm{Rootface}(M)$ has finite expectation, then\n\\[ \\mathbb E \\left[ \\mathbb E \\left[ \\mathrm{Rootface}(\\mathbb M_{\\mathbf{Q}}) | \\mathbf{Q} \\right] \\right] = \\mathbb E \\left[ \\mathrm{Rootface}(\\mathbb M_{\\mathbf{Q}}) \\right] <+\\infty, \\]\nso $\\mathbb E \\left[ \\mathrm{Rootface}(\\mathbb M_{\\mathbf{Q}}) | \\mathbf{Q} \\right]<+\\infty$ a.s., so $\\mathbf{Q} \\in \\mathcal{Q}_f$ a.s..\n\nThe first point of Theorem~\\ref{thm_weak_Markov_general} is the natural analogue of Theorem~2 of \\cite{BL19}, where triangulations are replaced by more general maps. The proof will rely on similar ideas: we fix a weakly Markovian map $M$ with associated constants $a^p_{\\mathbf{v}}$, and we would like to find a random $\\mathbf{Q}$ such that $a^p_{\\mathbf{v}} = \\mathbb E[ C_p(\\mathbf{Q}) \\mathbf{Q}^{\\mathbf{v}} ]$ for all $p$ and $\\mathbf{v}$. We will use peeling equations to establish inequalities between the $a^p_{\\mathbf{v}}$, and the existence of $\\mathbf{Q}$ will follow from the Hausdorff moment problem. However, compared to~\\cite{BL19}, two new difficulties arise:\n\\begin{itemize}\n\\item[$\\bullet$]\nthe random weights $\\mathbf{q}$ form a family of real numbers instead of just one real number;\n\\item[$\\bullet$]\nin the triangular case, with the notations of Definition~\\ref{defn_weak_Markov}, it was immediate that all the numbers $a^p_{\\mathbf{v}}$ are determined by the numbers $a^1_{\\mathbf{v}}$. This is not true anymore.\n\\end{itemize}\nThe first issue can be handled by using the multi-dimensional version of the Hausdorff moment problem. The second one, on the other hand, will make the proof a bit longer than in \\cite{BL19}. More precisely, the Hausdorff moment problem will now provide us, for every $p \\geq 1$, a $\\sigma$-finite measure $\\mu_p$ on the set of weight sequences, which describes $a^p_{\\mathbf{v}}$ for all $\\mathbf{v}$. We will then use the peeling equations to prove that all the $\\mu_p$ are actually determined by $\\mu_1$.\n\nBecause of the condition \\eqref{eqn_good_pvv}, we will need to find a measure with suitable $\\mathbf{v}$-th moments for all $\\mathbf{v} \\in \\mathcal{V}_p$, which is slightly different than the usual Hausdorff moment problem where $\\mathbf{v} \\in \\mathcal{V}$. Therefore, we first need to state a suitable version of the moment problem, which will follow from the usual one. This is done in the next subsection.\n\n\\subsection{The incomplete Hausdorff moment problem}\n\nTo state our version of the moment problem (Proposition~\\ref{prop_moment_Hausdorff} below), we will need to consider the space of sequences $\\left( u_{\\mathbf{v}} \\right)_{\\mathbf{v} \\in \\mathcal{V}_p}$. For $j \\geq 1$, we denote by $\\Delta_j$ the discrete derivation operator on the $j$-th coordinate on this space. That is, if $u=(u_{\\mathbf{v}})$, we write\n\\[ \\left( \\Delta_j u \\right)_{\\mathbf{v}} = u_{\\mathbf{v}}-u_{\\mathbf{v}+\\mathbf{1}_j}.\\]\nIt is easy to check that the operators $\\Delta_j$ commute with each other. For all $\\mathbf{k}=(k_j)_{j \\geq 1}$ such that $k_j=0$ for $j$ large enough (say for $j \\geq j_0$), we define the operator $\\Delta^{\\mathbf{k}}$ by\n\\[\n\\Delta^{\\mathbf{k}} u=  \\Delta_1^{k_1} \\Delta_2^{k_2} \\dots \\Delta_{j_0}^{k_{j_0}} u.\n\\]\nIn other words, we have\n\\[\\Delta^{\\mathbf{k}} u= \\sum_{\\mathbf{i}} \\left( \\prod_{j \\geq 1} (-1)^{i_j} \\binom{k_j}{i_j} \\right) u_{\\mathbf{v}+\\mathbf{i}}, \\]\nwhere the sum is over families $\\mathbf{i}=(i_j)_{j \\geq 1}$, and the terms with a nonzero contribution are those for which $0 \\leq i_j \\leq k_j$ for every $j \\geq 1$. The \"usual\" Hausdorff moment problem is then the following.\n\n\\begin{thm}\\label{thm_hausdorff_usual}\nLet $(u_{\\mathbf{v}})_{\\mathbf{v} \\in \\mathcal{V}}$ be such that, for any $\\mathbf{v} \\in \\mathcal{V}$ and any $\\mathbf{k} \\geq \\mathbf{0}$, we have\n\\[ \\Delta^{\\mathbf{k}} u_{\\mathbf{v}} \\geq 0.\\]\nThen there is a unique measure $\\mu$ on $\\mathcal{Q}=[0,1]^{\\mathbb N^*}$ (equipped with the product $\\sigma$-algebra) such that, for all $\\mathbf{v} \\in \\mathcal{V}$, we have\n\\[ u_{\\mathbf{v}}=\\int \\mathbf{q}^{\\mathbf{v}} \\mu(\\mathrm{d} \\q). \\]\nIn particular $\\mu$ is finite, with total mass $u_{\\mathbf{0}}$.\n\\end{thm}\nMore precisely, this is the infinite-dimensional Hausdorff moment problem, which can be deduced immediately from the finite-dimensional one by the Kolmogorov extension theorem.\n\nFor $p \\geq 1$, we recall that $\\mathcal{V}_p \\subset \\mathcal{V}$ is the set of $\\mathbf{v} \\in \\mathcal{V}$ that satisfy $\\sum_{j \\geq 1} (j-1) v_j \\geq p-1$. We also denote by $\\mathcal{V}_p^*$ the set of $\\mathbf{v} \\in \\mathcal{V}_p$ for which there is $j \\geq 2$ such that $v_j>0$ and $\\mathbf{v}-\\mathbf{1}_j \\in \\mathcal{V}_p$. In other words $\\mathcal{V}_p^*$ can be thought of as the \"interior\" of $\\mathcal{V}_p$. Finally, we recall that\n\\[ \\mathcal{Q}^*=\\{ \\mathbf{q} \\in [0,1]^{\\mathbb N^*} | \\exists j \\geq 2, q_j>0\\}.\\]\n\n\\begin{prop}\\label{prop_moment_Hausdorff}\nFix $p \\geq 1$, and let $\\left( u_{\\mathbf{v}} \\right)_{\\mathbf{v} \\in \\mathcal{V}_p}$ be a family of real numbers. We assume that for all $\\mathbf{v} \\in \\mathcal{V}_p$ and all $\\mathbf{k} \\geq \\mathbf{0}$, we have\n\\[ \\Delta^{\\mathbf{k}} u_{\\mathbf{v}} \\geq 0.\\]\nThen there is a $\\sigma$-finite measure $\\mu$ on $\\mathcal{Q}^*$ such that, for all $\\mathbf{v} \\in \\mathcal{V}_p^*$, we have\n\\[ u_{\\mathbf{v}}=\\int \\mathbf{q}^{\\mathbf{v}} \\mu(\\mathrm{d} \\q). \\]\nMoreover, if $p=1$, then $\\mu$ is finite and $\\mu(\\mathcal{Q}^*) \\leq u_\\mathbf{0}$.\n\\end{prop}\n\nNote that this version is \"weaker\" than Theorem~\\ref{thm_hausdorff_usual} in the sense that it is not always possible to have $u_{\\mathbf{v}}=\\int \\mathbf{q}^{\\mathbf{v}} \\mu(\\mathrm{d} \\q)$ for $\\mathbf{v} \\in \\mathcal{V}_p \\backslash \\mathcal{V}^*_p$. A simple example of this phenomenon in dimension one is that the sequence $\\left( \\mathbbm{1}_{i=1} \\right)_{i \\geq 1}$ has all its discrete derivatives nonnegative. However, there is no measure on $[0,1]$ with first moment $1$ and all higher moments $0$. On the other hand, we can assume an additional property of our measure $\\mu$, namely it is supported by $\\mathcal{Q}^*$ instead of $\\mathcal{Q}$ in Theorem~\\ref{thm_hausdorff_usual}.\n\n\\begin{proof}\nWe start with the case $p=1$. Then $\\mathcal{V}_1=\\mathcal{V}$, so by Theorem~\\ref{thm_hausdorff_usual}, there is a measure $\\widetilde{\\mu}$ on $\\mathcal{Q}$ such that, for all $\\mathbf{v} \\in \\mathcal{V}_1$, we have\n\\[u_{\\mathbf{v}} = \\int_{\\mathcal{Q}} \\mathbf{q}^{\\mathbf{v}} \\widetilde{\\mu}(\\mathrm{d} \\q). \\]\nLet $\\mu$ be the restriction of $\\widetilde{\\mu}$ to $\\mathcal{Q}^*$. If $\\mathbf{v} \\in \\mathcal{V}_1^*$ and $\\mathbf{q} \\in \\mathcal{Q} \\backslash \\mathcal{Q}^*$, then there is $j \\geq 2$ such that $\\mathbf{v}_j >0$ but $q_j=0$, so $\\mathbf{q}^{\\mathbf{v}}=0$. It follows that, for all $\\mathbf{v} \\in \\mathcal{V}_1^*$, we have\n\\[ \\int_{\\mathcal{Q}^*} \\mathbf{q}^{\\mathbf{v}} \\mu(\\mathrm{d} \\q) = \\int_{\\mathcal{Q}} \\mathbf{q}^{\\mathbf{v}} \\widetilde{\\mu}(\\mathrm{d} \\q) = u_{\\mathbf{v}}.\\]\nMoreover, the total mass of $\\mu$ is not larger than the total mass of $\\widetilde{\\mu}$, so it is at most $u_{\\mathbf{0}}$.\n\nWe now assume $p \\geq 2$. Let $\\mathbf{v} \\in \\mathcal{V}_p$. Then $\\mathbf{v}+\\mathbf{w} \\in \\mathcal{V}_p$ for all $\\mathbf{w} \\in \\mathcal{V}$, so the sequence $\\left( u_{\\mathbf{v}+\\mathbf{w}} \\right)_{\\mathbf{w} \\in \\mathcal{V}}$ satisfies the assumptions of Theorem~\\ref{thm_hausdorff_usual}. Therefore, there is a finite measure $\\mu_{\\mathbf{v}}$ on $\\mathcal{Q}$ such that\n\\[ u_{\\mathbf{v}+\\mathbf{w}} = \\int \\mathbf{q}^{\\mathbf{w}} \\mu_{\\mathbf{v}}(\\mathrm{d} \\q)\\]\nfor all $\\mathbf{w} \\in \\mathcal{V}$. Now let $\\mathbf{v}, \\mathbf{v}' \\in \\mathcal{V}_p$. For all $\\mathbf{w}$, we have\n\\[ \\int \\mathbf{q}^{\\mathbf{v}'} \\mathbf{q}^{\\mathbf{w}} \\mu_{\\mathbf{v}}(\\mathrm{d} \\q)= u_{\\mathbf{v}+\\mathbf{v}'+\\mathbf{w}}=\\int \\mathbf{q}^{\\mathbf{v}} \\mathbf{q}^{\\mathbf{w}} \\mu_{\\mathbf{v}'}(\\mathrm{d} \\q).\\]\nIn other words, the measures $\\mathbf{q}^{\\mathbf{v}'} \\mu_{\\mathbf{v}}(\\mathrm{d} \\q)$ and $\\mathbf{q}^{\\mathbf{v}} \\mu_{\\mathbf{v}'}(\\mathrm{d} \\q)$ have the same moments, so by uniqueness in Theorem~\\ref{thm_hausdorff_usual}\n\\begin{equation}\\label{eqn_consistence_muv}\n\\mathbf{q}^{\\mathbf{v}'} \\mu_{\\mathbf{v}}(\\mathrm{d} \\q)=\\mathbf{q}^{\\mathbf{v}} \\mu_{\\mathbf{v}'}(\\mathrm{d} \\q).\n\\end{equation}\nIn particular, for all $\\mathbf{v} \\in \\mathcal{V}_p$, we can consider the $\\sigma$-finite measure \\[\\widetilde{\\mu}_{\\mathbf{v}}(\\mathrm{d} \\q)= \\frac{\\mu_{\\mathbf{v}}(\\mathrm{d} \\q)}{\\mathbf{q}^{\\mathbf{v}}}\\] defined on $\\{ \\mathbf{q}^{\\mathbf{v}}>0\\}$. Then \\eqref{eqn_consistence_muv} implies that, for any $\\mathbf{v}, \\mathbf{v}' \\in \\mathcal{V}_p$, the measures $\\widetilde{\\mu}_{\\mathbf{v}}$ and $\\widetilde{\\mu}_{\\mathbf{v}'}$ coincide on $\\{ \\mathbf{q}^{\\mathbf{v}}>0 \\} \\cap \\{ \\mathbf{q}^{\\mathbf{v}'}>0\\}$. Therefore, there is a measure $\\mu$ on $\\bigcup_{\\mathbf{v} \\in \\mathcal{V}_p} \\{\\mathbf{q}^{\\mathbf{v}}>0\\} = \\mathcal{Q}^*$ such that, for all $\\mathbf{v} \\in \\mathcal{V}_p$, we have\n\\begin{equation}\\label{eqn_mu_and_muv_weak}\n\\mu_{\\mathbf{v}}(\\mathrm{d} \\q) = \\mathbf{q}^{\\mathbf{v}} \\mu(\\mathrm{d} \\q) \\quad \\mbox{ on } \\quad \\{\\mathbf{q}^{\\mathbf{v}}>0\\}.\n\\end{equation}\nSince $\\mu$ is finite on $\\{q_j>\\varepsilon\\}$ for all $\\varepsilon>0$ and $j \\geq 2$, the measure $\\mu$ is $\\sigma$-finite. We would now like to extend the equality~\\eqref{eqn_mu_and_muv_weak} to all $\\mathcal{Q}^*$ under the condition $\\mathbf{v} \\in \\mathcal{V}_p^*$.\n\nFor this, let $\\mathbf{v} \\in \\mathcal{V}_p^*$, and let $j \\geq 2$ be such that $v_j>0$ and $\\mathbf{v}-\\mathbf{1}_j \\in \\mathcal{V}_p$. We have $p \\mathbf{1}_j \\in \\mathcal{V}_p$, so we can apply \\eqref{eqn_consistence_muv} to $\\mathbf{v}$ and $p \\mathbf{1}_j$. We obtain, on $\\{ q_j>0 \\}$:\n\\[ \\mu_{\\mathbf{v}}(\\mathrm{d} \\q) = \\mathbf{q}^{\\mathbf{v}} \\frac{\\mu_{p \\mathbf{1}_j}(\\mathrm{d} \\q)}{q_j^p} = \\mathbf{q}^{\\mathbf{v}} \\mu(\\mathrm{d} \\q), \\]\nusing also \\eqref{eqn_mu_and_muv_weak} for $p \\mathbf{1}_j$. In other words, \\eqref{eqn_mu_and_muv_weak} holds on $\\{ q_j>0 \\}$.\n\nOn the other hand, for $\\mathbf{v}$ and $\\mathbf{v}-\\mathbf{1}_p$, we can obtain a stronger version of \\eqref{eqn_consistence_muv}. More precisely, for all $\\mathbf{w}$, we have\n\\[  \\int q_j \\mathbf{q}^{\\mathbf{w}} \\mu_{\\mathbf{v}-\\mathbf{1}_j}(\\mathrm{d} \\q) = u_{\\mathbf{v}+\\mathbf{w}} = \\int \\mathbf{q}^{\\mathbf{w}} \\mu_{\\mathbf{v}}(\\mathrm{d} \\q),\\]\nso the measures $q_j \\mu_{\\mathbf{v}-\\mathbf{1}_j}(\\mathrm{d} \\q)$ and $\\mu_{\\mathbf{v}}(\\mathrm{d} \\q)$ have the same moments, so they coincide. But the first one is $0$ on $\\{q_j=0\\}$, so it is also the case for the second. Therefore, \\eqref{eqn_mu_and_muv_weak} holds on $\\{ q_j=0 \\}$, with both sides equal to $0$.\n\nTherefore, we have proved that \\eqref{eqn_mu_and_muv_weak} holds on $\\mathcal{Q}$. By integrating over $\\mathcal{Q}^*$ and using that the total mass of $\\mu_{\\mathbf{v}}$ is $u_{\\mathbf{v}}$ and is supported by $\\mathcal{Q}^*$, we get the result.\n\\end{proof}\n\n\\subsection{Proof of Theorem~\\ref{thm_weak_Markov_general}}\n\nAs in \\cite{BL19}, we start by writing down the peeling equations, which are linear equations between the numbers $a^p_{\\mathbf{v}}$ together. For every $p \\geq 1$ and $\\mathbf{v} \\in \\mathcal{V}_p$, we have\n\\begin{equation}\\label{eqn_peeling_equation_bipartite}\na^p_{\\mathbf{v}}= \\sum_{j \\geq 1} a^{p+j-1}_{\\mathbf{v}+\\mathbf{1}_j}+2\\sum_{i=1}^{p-1} \\sum_{\\mathbf{w} \\in \\mathcal{V}} \\beta_0^{(i-1)}(\\mathbf{w}) a^{p-i}_{\\mathbf{v}+\\mathbf{w}},\n\\end{equation}\nwhere we recall that $\\beta_0^{(i-1)}(\\mathbf{w})$ is the number of planar, bipartite maps of the $2(i-1)$-gon with exactly $w_j$ internal faces of degree $2j$ for all $j \\geq 1$. These equations, together with the facts that $a^1_\\mathbf{0}=1$ and $a^p_{\\mathbf{v}} \\geq 0$, characterize the families $(a^p_{\\mathbf{v}})$ of numbers that may arise from a weakly Markovian map. In order to be able to use the Hausdorff moment problem, we now need to check that the discrete derivatives of $(a^p_{\\mathbf{v}})$ are nonnegative.\n\n\\begin{lem}\\label{lem_abs_monotone}\nLet $M$ be a weakly Markovian bipartite map, and let $\\left( a^p_{\\mathbf{v}} \\right)$ be the associated constants. For every $\\mathbf{k} \\geq \\mathbf{0}$, $p \\geq 1$ and $\\mathbf{v} \\in \\mathcal{V}_p$, we have\n\\[ \\left( \\Delta^{\\mathbf{k}} a^p \\right)_{\\mathbf{v}} \\geq 0.\\]\n\\end{lem}\n\n\\begin{proof}\nThe proof is similar to the proof of Lemma 16 in \\cite{BL19}, with the following modification: in \\cite{BL19}, it was useful that in the same peeling equation, we had $a^p_v$ appearing on the left and $a^p_{v+1}$ on the right. However, in \\eqref{eqn_peeling_equation_bipartite} $a^p_{\\mathbf{v}+\\mathbf{1}_j}$ does not appear in the right-hand side (this is because we are using the lazy peeling process of \\cite{Bud15} instead of the simple peeling of \\cite{Ang03}). Therefore, instead of using directly the peeling equation, we will need to use the \\emph{double peeling equation}, which corresponds to performing two peeling steps, instead of one in~\\eqref{eqn_peeling_equation_bipartite}.\n\nMore precisely, the peeling equation \\eqref{eqn_peeling_equation_bipartite} gives an expansion of $a^p_{\\mathbf{v}}$. The \\emph{double peeling equation} is obtained from \\eqref{eqn_peeling_equation_bipartite} by replacing all the terms in the right-hand side by their expansion given by \\eqref{eqn_peeling_equation_bipartite}. Note that this indeed makes sense because if $\\mathbf{v} \\in \\mathcal{V}_p$, then $\\mathbf{v}+\\mathbf{1}_j \\in \\mathcal{V}_{p+j-1}$ for all $j \\geq 1$, and $\\mathbf{v}+\\mathbf{w} \\in \\mathcal{V}_{p-i}$ for all $i \\geq 1$ and $\\mathbf{w} \\in \\mathcal{V}$.\n\nThe equation we obtain is of the form\n\\begin{equation}\\label{eqn_double_peeling_equation}\na^p_{\\mathbf{v}}=\\sum_{i \\in \\mathbb Z, \\, \\mathbf{w} \\in \\mathcal{V}} c^{p,i}_{\\mathbf{v},\\mathbf{w}} a^{p+i}_{\\mathbf{v}+\\mathbf{w}},\n\\end{equation}\nwhere the coefficients $c^{p,i}_{\\mathbf{v},\\mathbf{w}}$ are nonnegative integers. An explicit formula for these could be computed in terms of the $\\beta_0^{(i)}(\\mathbf{w})$, but this will not be needed. Here are the facts that will be useful:\n\\begin{enumerate}\n\\item\nthe coefficients $c^{p,i}_{\\mathbf{v},\\mathbf{w}}$ actually do not depend on $\\mathbf{v}$, so we can write them $c^{p,i}_{\\mathbf{w}}$,\n\\item\\label{item_coeff_equal_one}\nwe have $c^{p,2j-2}_{2\\cdot\\mathbf{1}_j}=1$ for every $j \\geq 1$,\n\\item\\label{item_coeff_positive}\nwe have $c^{p,0}_{\\mathbf{1}_j} \\geq 1$ for every $j \\geq 2$.\n\\end{enumerate}\nThe first item follows from the fact that at each time, the available next peeling steps do not depend on the internal face degrees of the explored region.\nThe second item expresses the fact that, for a given peeling algorithm, there is a unique way to obtain a map with half-perimeter $p+2j-2$ with internal faces $\\mathbf{v}+2\\cdot\\mathbf{1}_j$ in two peeling steps. This way is to discover a unique face of degree $2j$ at both steps. The third item means that it is possible (not necessarily in a unique way) to obtain in two peeling steps a map with the same perimeter but one more face of degree $2j$. This is achieved by discovering a new face of degree $2j$ at the first step, and gluing all but two sides of this face two by two at the second step (see Figure~\\ref{figure_pconstant_vincrease}). \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.7]{one_more_face}\n\\caption{In two peeling steps, the perimeter stays constant and one face with degree $2j$ is added (here $j=3$).}\\label{figure_pconstant_vincrease}\n\\end{center}\n\\end{figure}\n\nWe now prove the lemma by induction on $|\\mathbf{k}|=\\sum_{j \\geq 1} k_j$. First, the case $|\\mathbf{k}|=0$ just means that $a^p_{\\mathbf{v}} \\geq 0$ for all $p \\geq 1$ and $\\mathbf{v} \\in \\mathcal{V}_p$, which is immediate. Let us now assume that the lemma is true for $\\mathbf{k}$ and prove it for $\\mathbf{k}+\\mathbf{1}_j$, where $j \\geq 1$. We will first treat the case where $j \\geq 2$.\nUsing the double peeling equation \\eqref{eqn_double_peeling_equation} for $(p, \\mathbf{v}+\\mathbf{i})$ for different values of $\\mathbf{i}$, we have\n\\[ \\left( \\Delta^{\\mathbf{k}} a^p \\right)_{\\mathbf{v}} = \\sum_{i \\in \\mathbb Z, \\, \\mathbf{w} \\in \\mathcal{V}} c^{p,i}_{\\mathbf{w}} \\left(  \\Delta^{\\mathbf{k}} a^{p+i}\\right)_{\\mathbf{v}+\\mathbf{w}}. \\]\nTherefore, using the induction hypothesis and Item~\\ref{item_coeff_equal_one} above, we can write \n\\begin{align*}\n0 & \\leq \\left( \\Delta^{\\mathbf{k}} a^{p+2j-2} \\right)_{\\mathbf{v}+2\\cdot\\mathbf{1}_j}\\\\\n&= c^{p,2j-2}_{2\\cdot\\mathbf{1}_j} \\left( \\Delta^{\\mathbf{k}} a^{p+2j-2} \\right)_{\\mathbf{v}+2\\cdot\\mathbf{1}_j}\\\\\n&=  \\left( \\Delta^{\\mathbf{k}} a^{p} \\right)_{\\mathbf{v}} - \\sum_{(i,\\mathbf{w}) \\ne (2j-2, 2\\cdot\\mathbf{1}_j)} c^{p,i}_{\\mathbf{w}} \\left( \\Delta^{\\mathbf{k}} a^{p+i} \\right)_{\\mathbf{v}+\\mathbf{w}}.\n\\end{align*}\nUsing the induction hypothesis again, we can remove all the terms in the last sum except the one where $(i,\\mathbf{w})=(0,\\mathbf{1}_j)$. Moreover, by Item~\\ref{item_coeff_positive} above, we can replace the coefficient $c^{p,0}_{\\mathbf{1}_j}$ by $1$. We obtain\n\\[0 \\leq \\left( \\Delta^{\\mathbf{k}} a^{p} \\right)_{\\mathbf{v}} - \\left( \\Delta^{\\mathbf{k}} a^{p} \\right)_{\\mathbf{v}+\\mathbf{1}_j}= \\left( \\Delta^{\\mathbf{k}+\\mathbf{1}_j} a^p \\right)_{\\mathbf{v}}, \\]\nwhich proves the induction step for $j \\geq 2$. If $j=1$, Item~\\ref{item_coeff_positive} is not true anymore (it is not possible to add only one face of degree $2$ in $2$ steps without changing the perimeter). Therefore, instead of \\eqref{eqn_double_peeling_equation}, we use the simple peeling equation \\eqref{eqn_peeling_equation_bipartite} like in \\cite{BL19}. More precisely, in the induction step, we fix $j' \\geq 2$ and write, using \\eqref{eqn_peeling_equation_bipartite}:\n\\begin{align*}\n0 &\\leq \\left( \\Delta^{\\mathbf{k}} a^{p+j'-1} \\right)_{\\mathbf{v}+\\mathbf{1}_{j'}}\\\\\n&= \\left( \\Delta^{\\mathbf{k}} a^{p} \\right)_{\\mathbf{v}} - \\sum_{j'' \\ne j'} \\left( \\Delta^{\\mathbf{k}} a^{p+j''-1} \\right)_{\\mathbf{v}+\\mathbf{1}_{j''}} -2 \\sum_{i=0}^{p-1} \\sum_{\\mathbf{w}} \\beta_0^{(i-1)}(\\mathbf{w}) \\left( \\Delta^{\\mathbf{k}} a^{p-i} \\right)_{\\mathbf{v}+\\mathbf{w}}.\n\\end{align*}\nEach term in the two sums is nonnegative by the induction hypothesis, so we can remove the second sum and keep only the term $j''=1$ in the first one to obtain\n\\[0  \\leq \\left( \\Delta^{\\mathbf{k}} a^{p} \\right)_{\\mathbf{v}} - \\left( \\Delta^{\\mathbf{k}} a^{p} \\right)_{\\mathbf{v}+\\mathbf{1}_1} = \\left( \\Delta^{\\mathbf{k}+\\mathbf{1}_1} a^{p} \\right)_{\\mathbf{v}}.\\]\nThis concludes the proof of the lemma.\n\\end{proof}\n\nBy Lemma~\\ref{lem_abs_monotone} and Proposition~\\ref{prop_moment_Hausdorff}, for all $p \\geq 1$, there is a $\\sigma$-finite measure $\\mu_p$ on $\\mathcal{Q}^*$ such that, for all $\\mathbf{v} \\in \\mathcal{V}^*_p$,\n\\begin{equation}\\label{eqn_apv_as_moment}\na^p_{\\mathbf{v}}=\\int_{\\mathcal{Q}^*} \\mathbf{q}^{\\mathbf{v}} \\mu_p(\\mathrm{d} \\q)\n\\end{equation}\nand furthermore $\\mu_1(\\mathcal{Q}^*) \\leq a^1_{\\mathbf{0}}=1$. We now replace $a_{\\mathbf{v}}^p$ by this expression in the peeling equation \\eqref{eqn_peeling_equation_bipartite}. We get\n\\begin{align*}\n\\int \\mathbf{q}^{\\mathbf{v}} \\, \\mu_p(\\mathrm{d} \\mathbf{q}) \\hspace{-0.1cm}&= \\hspace{-0.1cm}\\sum_{j \\geq 1} \\int \\mathbf{q}^{\\mathbf{v}+\\mathbf{1}_j} \\, \\mu_{p+j-1}(\\mathrm{d} \\q) + 2\\sum_{i=1}^{p-1} \\sum_{\\mathbf{w} \\in \\mathcal{V}} \\beta_0^{(i-1)}(\\mathbf{w}) \\int \\mathbf{q}^{\\mathbf{v}+\\mathbf{w}} \\, \\mu_{p-i}(\\mathrm{d} \\q)\\\\\n&=\\hspace{-0.1cm} \\int \\mathbf{q}^{\\mathbf{v}} \\left( \\sum_{j \\geq 1} q_j \\, \\mu_{p+j-1}(\\mathrm{d} \\q) + 2 \\sum_{i=1}^{p-1} W_{i-1}(\\mathbf{q}) \\, \\mu_{p-i}(\\mathrm{d} \\q) \\right),\n\\end{align*}\nwhere we recall that $W_{i-1}(\\mathbf{q})$ is the partition function of Boltzmann bipartite maps of the $2(i-1)$-gon with Boltzmann weights $\\mathbf{q}$.\nIn particular, the right-hand side for $i=2$ must be finite, which means that $\\mu_p$ is supported by the set $\\mathcal{Q}_a$ of admissible weight sequences. Moreover, the last display means that the two measures\n\\[ \\mu_p(\\mathrm{d} \\q) \\mbox{ and } \\nu_p(\\mathrm{d} \\q) = \\sum_{j \\geq 1} q_j \\, \\mu_{p+j-1}(\\mathrm{d} \\q) + 2 \\sum_{i=1}^{p-1} W_{i-1}(\\mathbf{q}) \\, \\mu_{p-i}(\\mathrm{d} \\q) \\]\nhave the same $\\mathbf{v}$-th moment for all $\\mathbf{v} \\in \\mathcal{V}^*_p$. In particular, if we fix $j \\geq 2$, this is true as soon as $v_j \\geq p$, so the measures $q_j^p \\mu_p(\\mathrm{d} \\q)$ and $q_j^p \\nu_p(\\mathrm{d} \\q)$ have the same moments so they are equal, so $\\mu_p$ and $\\nu_p$ coincide on $\\{q_j>0\\}$. Since this is true for all $j \\geq 2$ and $\\mu_p, \\nu_p$ are defined on $\\mathcal{Q}^*=\\bigcup_{j \\geq 2} \\{q_j>0\\}$, the measures $\\mu_p$ and $\\nu_p$ are the same, that is,\n\\begin{equation}\\label{peeling_equation_mu}\n\\mu_p(\\mathrm{d} \\q) = \\sum_{j \\geq 1} q_j \\, \\mu_{p+j-1}(\\mathrm{d} \\q) + 2 \\sum_{i=1}^{p-1} W_{i-1}(\\mathbf{q}) \\, \\mu_{p-i}(\\mathrm{d} \\q).\n\\end{equation}\nWe now note that this equation is very similar to the one satisfied by the constants $C_p(\\mathbf{q})$ used to define the $\\mathbf{q}$-IBPM. More precisely, we fix a finite measure $\\mu$ such that all the $\\mu_p$ are absolutely continuous with respect to $\\mu$ (take e.g. $\\mu (\\mathrm{d} \\q)=\\sum_{p \\geq 1} \\frac{g_p(\\mathbf{q}) \\mu_p(\\mathrm{d} \\q)}{2^p}$, where $g_p(\\mathbf{q})>0$ is such that the total mass of $g_p(\\mathbf{q}) \\mu_p(\\mathrm{d} \\q)$ is at most $1$). We denote by $f_p(\\mathbf{q})$ the density of $\\mu_p$ with respect to $\\mu$. Then \\eqref{peeling_equation_mu} becomes\n\\[ f_p(\\mathbf{q}) = \\sum_{j \\geq 1} q_j f_{p+j-1}(\\mathbf{q}) + 2 \\sum_{i=1}^{p-1} W_{i-1}(\\mathbf{q}) f_{p-i}(\\mathbf{q}) \\]\nfor $\\mu$-almost every $\\mathbf{q} \\in \\mathcal{Q}^*$. In other words, $\\left( f_p(\\mathbf{q}) \\right)_{p \\geq 1}$ satisfies the exact same equation as $\\left( C_p(\\mathbf{q}) \\right)_{p \\geq 1}$ in \\cite[Appendix C]{B18these}. These equations have a nonzero solution if and only if $\\mathbf{q} \\in \\mathcal{Q}_h$, so the measures $\\mu_p$ are actually supported by $\\mathcal{Q}_h$. Moreover, by uniqueness of the solution (up to a multiplicative constant), we have\n\\[ f_p(\\mathbf{q})=\\frac{C_p(\\mathbf{q})}{C_1(\\mathbf{q})} f_1(\\mathbf{q}) = C_p(\\mathbf{q}) f_1(\\mathbf{q})\\]\nfor $\\mu$-almost every $\\mathbf{q}$, so $\\mu_p(\\mathrm{d} \\q)=C_p(\\mathbf{q}) \\mu_1(\\mathrm{d} \\q)$. Now let $\\alpha \\leq 1$ be the total mass of the measure $\\mu_1$, and let $\\mathbf{Q}$ be a random variable with distribution $\\alpha^{-1} \\mu$. We then have, for all $p \\geq 1$ and $\\mathbf{v} \\in \\mathcal{V}^*_p$, if $m$ is a map with half-perimeter $p$ and face degrees $\\mathbf{v}$:\n\\begin{equation}\\label{eqn_apv_expectation}\n\\P \\left( m \\subset M \\right) = a^p_{\\mathbf{v}} = \\int \\mathbf{q}^{\\mathbf{v}} \\mu_p(\\mathrm{d} \\q) = \\alpha \\mathbb E \\left[ C_p(\\mathbf{Q}) \\mathbf{Q}^{\\mathbf{v}} \\right] = \\alpha \\P \\left( m \\subset \\mathbb M_{\\mathbf{Q}} \\right).\n\\end{equation}\nNote that $\\mathbf{Q}$ is not well-defined if $\\alpha=0$, but in this case $\\mu_p=0$ for all $p$ so \\eqref{eqn_apv_expectation} remains true for any choice of $\\mathbf{Q}$. To conclude that $M$ has the law of $\\mathbb M_{\\mathbf{Q}}$, all we have left to prove is that $\\alpha=1$ and that \\eqref{eqn_apv_expectation} can be extended to any $\\mathbf{v} \\in \\mathcal{V}_p$. For this, we will show that, when we explore $M$ via a peeling exploration, the perimeter and volumes of the explored region at time $t$ satisfy $\\mathbf{v} \\in \\mathcal{V}_p^*$ for $t$ large enough.\n\nMore precisely, if $\\mathcal{A}$ is a peeling algorithm, we recall that $\\mathcal{E}^{\\mathcal{A}}_t(M)$ is the explored part of $M$ after $t$ steps of a filled-in peeling exploration according to $\\mathcal{A}$. We denote by $P_t$ the half-perimeter of the hole of $\\mathcal{E}^{\\mathcal{A}}_t(M)$ and by $\\mathbf{V}_t$ the sequence of degrees of its internal faces (that is, $V_{t,j}$ is the number of internal faces of $\\mathcal{E}^{\\mathcal{A}}_t(M)$ with degree $2j$). Since $M$ is weakly Markovian, the process $\\left( P_t, \\mathbf{V}_t \\right)_{t \\geq 0}$ is a Markov chain whose law does not depend on the peeling algorithm $\\mathcal{A}$.\n\n\\begin{lem}\\label{lem_volumes_star}\nWe have\n\\[ \\P \\left( \\mathbf{V}_{t} \\in \\mathcal{V}^*_{P_t} \\right) \\xrightarrow[t \\to +\\infty]{} 1.\\]\n\\end{lem}\n\n\\begin{proof}\nSince the probability in the lemma does not depend on $\\mathcal{A}$, it is sufficient to prove the result for a particular peeling algorithm. Therefore, we can assume that $\\mathcal{A}$ has the following property: if the root face of $m$ and its hole have a common vertex $m$, then the peeled edge $\\mathcal{A}(m)$ is incident to such a vertex. We will prove that for this algorithm, we have a.s. $\\mathbf{V}_{t} \\in \\mathcal{V}^*_{P_t}$ for $t$ large enough.\n\nMore precisely, since the vertex degrees of $M$ are a.s. finite and by definition of $\\mathcal{A}$, all the vertices incident to the root face will eventually disappear from the boundary of the explored part. Therefore, for $t$ large enough, no vertex incident to the root face is on $\\partial \\mathcal{E}^{\\mathcal{A}}_t(M)$. We now fix $t$ with this property. If we denote by $\\mathrm{Inn}(m)$ the number of internal vertices of a map $m$ with a hole and by $2J$ the degree of the root face of $M$, this implies $\\mathrm{Inn} \\left( \\mathcal{E}^{\\mathcal{A}}_t(M) \\right) \\geq 2J$ for $t$ large enough.\n\nOn the other hand, the total number of edges of $\\mathcal{E}^{\\mathcal{A}}_t(M)$ is $p+\\sum_{j \\geq 1} j V_{t,j}$, so by the Euler formula\n\\begin{align*}\n\\mathrm{Inn} \\left( \\mathcal{E}^{\\mathcal{A}}_t(M) \\right) &= 2 + \\left( P_t+ \\sum_{j \\geq 1} j V_{t,j} \\right) - \\left( 1+\\sum_{j \\geq 1} V_{t,j} \\right) -2P_t \\\\&= 1-P_t+\\sum_{j \\geq 1} (j-1) V_{t,j}. \n\\end{align*}\nTaking $t$ large enough to have $\\mathrm{Inn} \\left( \\mathcal{E}^{\\mathcal{A}}_t(M) \\right) \\geq 2J$, we obtain\n\\[ \\left( \\sum_{j \\geq 1} (j-1) V_{t,j} \\right) -(J-1) \\geq \\left( 2J +P_t -1 \\right) -(J-1) = P_t+J > P_t-1,\\]\nso $V_{t,J}>0$ and $\\mathbf{V}-\\mathbf{1}_{J} \\in \\mathcal{V}_{P_t}$. This proves $\\mathbf{V}_{t} \\in \\mathcal{V}^*_{P_t}$ for $t$ large enough.\n\\end{proof}\n\nWe now conclude the proof of Theorem~\\ref{thm_weak_Markov_general} from \\eqref{eqn_apv_expectation}. We consider a finite map $m_0$ with a hole and a peeling algorithm $\\mathcal{A}$ that is consistent with $m_0$ in the sense that $m_0$ is a possible value of $\\mathcal{E}_{t_0}^{\\mathcal{A}}$ for some $t_0 \\geq 0$. We note that $\\mathcal{E}^{\\mathcal{A}}_{t_0}(M)=m_0$ if and only if $m_0 \\subset M$. Indeed, the direct implication is immediate. The indirect one comes from the fact that, if $m_0 \\subset M$, then all the peeling steps until time $t_0$ must be consistent with $m_0$, so $m_0 \\subset M$ determines the first $t_0$ peeling steps. We now take $t \\geq t_0$. We sum \\eqref{eqn_apv_expectation} over all possible values $m$ of $\\mathcal{E}^{\\mathcal{A}}_t(M)$ such that $m_0 \\subset m$ and the half-perimeter $p$ and internal face degrees $\\mathbf{v}$ of $m$ satisfy $\\mathbf{v} \\in \\mathcal{V}_p^*$. We get\n\\[ \\P \\left( m_0 \\subset M \\mbox{ and } \\mathbf{V}_t \\in \\mathcal{V}^*_{P_t} \\right) = \\alpha \\P \\left( m_0 \\subset \\mathbb M_{\\mathbf{Q}} \\mbox{ and } \\mathbf{V}^{\\mathbf{Q}}_t \\in \\mathcal{V}^*_{P_t^{\\mathbf{Q}}} \\right), \\]\nwhere $P_t^{\\mathbf{Q}}$ and $\\mathbf{V}_t^{\\mathbf{Q}}$ are the analogues of $P_t$ and $\\mathbf{V}_t$ for $\\mathbb M_{\\mathbf{Q}}$ instead of $M$. Since $\\mathbb M_{\\mathbf{Q}}$ is weakly Markovian, we can apply Lemma~\\ref{lem_volumes_star} to both $M$ and $\\mathbb M_{\\mathbf{Q}}$. Therefore, letting $t \\to +\\infty$ in the last display, we get \\[\\P \\left( m_0 \\subset M  \\right) = \\alpha \\P \\left( m_0 \\subset \\mathbb M_{\\mathbf{Q}} \\right)\\] for all $m_0$. In particular, if $m_0$ is the trivial map consisting only of the root edge, we get $\\alpha=1$, so $M$ and $\\mathbb M_{\\mathbf{Q}}$ have the same law. This proves Theorem~\\ref{thm_weak_Markov_general}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm_main_more_general}]\nBy Proposition~\\ref{prop_tightness_dloc_univ}, any subsequential limit $M$ of $\\left( M_{\\mathbf{f}^n, g_n} \\right)$ is planar and one-ended. Moreover, let $m$ be a map with one hole of half-perimeter $p$ and $v_{j}$ faces of degree $2j$ for all $j \\geq 1$. Then\n\\[\\P \\left( m \\subset M \\right) = \\lim_{n \\to +\\infty} \\P \\left( m \\subset M_{\\mathbf{f}^n, g_n} \\right) = \\lim_{n \\to +\\infty} \\frac{\\beta_{g_n}^{(p)}(\\mathbf{f}^n-\\mathbf{v})}{\\beta_{g_n}(\\mathbf{f}^n)},\\]\nwhere the limits are along some subsequence. In particular, the dependence in $m$ is only in $p$ and $\\mathbf{v}$, so $M$ is weakly Markovian and the result follows by Theorem~\\ref{thm_weak_Markov_general}.\n\\end{proof}\n\n\\section{The parameters are deterministic}\\label{sec_univ_end}\n\n\\subsection{Outline}\\label{sec_arg_deux_trous}\n\nOur goal is now to prove Theorem~\\ref{univ_main_thm}. We fix face degree sequences $\\mathbf{f}^n$ and genuses $g_n$ for $n \\geq 0$ satisfying the assumptions of Theorem~\\ref{univ_main_thm} (in particular, we now assume $\\sum_{j} j^2 \\alpha_j <+\\infty$ until the end of the paper). By Theorem~\\ref{thm_main_more_general}, up to extracting a subsequence, we can assume $M_{\\mathbf{f}^n, g_n}$ converges to $\\mathbb M_{\\mathbf{Q}}$, where $\\mathbf{Q}$ is a random variable with values in $\\mathcal{Q}_h$. Moreover, the law of the degree of the root face in $M_{\\mathbf{f}^n, g_n}$ converges in distribution to $\\left( j \\alpha_j \\right)_{j \\geq 1}$, which has finite expectation. By the last point of Theorem~\\ref{thm_weak_Markov_general}, we have $\\mathbf{Q} \\in \\mathcal{Q}_f$ a.s.. To prove Theorem~\\ref{univ_main_thm}, it is enough to prove that $\\mathbf{Q}$ is deterministic, and only depends on $\\left( \\alpha_j \\right)_{j \\geq 1}$ and $\\theta$.\n\n\\paragraph{Sketch of the end of the proof.}\nSince we will follow similar ideas, let us first recall the strategy of \\cite{BL19}. If $e_n$ is the root edge of $M_{\\mathbf{f}^n, g_n}$, the parameters $\\mathbf{Q}$ can be observed on a large neighbourhood of $e_n$ in $M_{\\mathbf{f}^n, g_n}$ for $n$ very large.\nThe first step of the proof (Proposition~\\ref{prop_two_holes_argument}) roughly consists of showing that once $M_{\\mathbf{f}^n, g_n}$ is picked, the weights $\\mathbf{Q}$ do not depend on the choice of $e_n$. This is proved by the \\emph{two holes argument}: if $e^1_n$ and $e_n^2$ are two roots chosen uniformly on $M_{\\mathbf{f}^n, g_n}$, we swap two large neighbourhoods of $e^1_n$ and $e^2_n$ in $M_{\\mathbf{f}^n, g_n}$. We then remark that if the weights observed around the two roots are too different, then the map obtained after swapping does not look like a map of the form $\\mathbb M_{\\mathbf{Q}}$.\nThe second step consists of noticing that the average value over all choices of the root of some functions of $\\mathbf{Q}$ is fixed by $\\left( \\alpha_j \\right)_{j \\geq 1}$ and $\\theta$ (Corollary~\\ref{corr_main_minus_monotonicity}).\nFinally, in the third step we prove that these functions are sufficient to characterize $\\mathbf{Q}$ completely (Proposition~\\ref{prop_monotonicity_deg}).\n\nHowever, two important difficulties appear here compared to~\\cite{BL19}:\n\\begin{itemize}\n\\item\nin the first step, we need to find two large pieces around $e_n^1$ and $e_2^n$ with the exact same perimeter, in order to be able to swap them. This was easy for triangulations since the perimeter process associated to a peeling exploration takes all values. This is not true anymore in our general setting. This part will make crucial use of the assumption $\\sum_j j^2 \\alpha_j < +\\infty$ (this is actually the only place in the paper where we will use it). A consequence of this difficulty is that instead of performing the swapping operation \\emph{with high probability}, we will perform it \\emph{with positive probability}.\n\\item\nIn the third step, one of the parameters that we control is the average vertex degree. For triangulations, it followed from an explicit formula that the average degree characterizes $\\mathbf{Q}$. We do not have such a formula here, so our argument will be more involved, and rely on the partial results obtained so far in the present paper.\n\\end{itemize}\n\n\\paragraph{Intermediate results.}\nLet $\\left( M_n, e_n^1, e_n^2 \\right)$ be a uniform, bi-rooted map with face degrees $\\mathbf{f}^n$ and genus $g_n$ (i.e. $e_n^1$ and $e_n^2$ are picked uniformly and independently among the edges of $M_n$). We highlight that we will write $M_n$ instead of $M_{\\mathbf{f}^n, g_n}$ in this section to make notations lighter.\nUp to extracting a subsequence, we can assume the joint convergence\n\\[ \\left( (M_n, e_n^1), (M_n, e_n^2) \\right) \\xrightarrow[n \\to +\\infty]{(d)} \\left( \\mathbb M^1_{\\mathbf{Q}^1}, \\mathbb M^2_{\\mathbf{Q}^2} \\right) \\]\nfor the local topology, where $\\mathbf{Q}^1$ and $\\mathbf{Q}^2$ have the same distribution as $\\mathbf{Q}$. Moreover, by the Skorokhod representation theorem, we can assume this joint convergence is almost sure. \\emph{We will stay in this setting in Section~\\ref{subsec_same_perimeter} and~\\ref{subsec_two_holes}.} The first step of the proof will consist of proving the following.\n\n\\begin{prop}\\label{prop_two_holes_argument}\nWe have $\\mathbf{Q}^1=\\mathbf{Q}^2$ almost surely.\n\\end{prop}\n\nFrom here, the second step will be to deduce the next result. For $\\mathbf{q} \\in \\mathcal{Q}_h$, we recall that $j a_j(\\mathbf{q})$ is the probability that the root face of $\\mathbb M_{\\mathbf{q}}$ has degree $2j$, and that $d(\\mathbf{q})=\\mathbb E \\left[ \\frac{1}{\\mathrm{deg}_{\\mathbb M_{\\mathbf{q}}} (\\rho)} \\right]$. \n\n\\begin{corr}\\label{corr_main_minus_monotonicity}\nUnder the assumptions of Theorem~\\ref{univ_main_thm}, let $\\mathbb M_{\\mathbf{Q}}$ be a subsequential limit. Then almost surely, we have\n\\[ d(\\mathbf{Q})=\\frac{1}{2} \\left( 1-2\\theta-\\sum_i \\alpha_i \\right) \\mbox{ and, for all $j \\geq 1$, } a_j(\\mathbf{Q})=\\alpha_j.\\] \n\\end{corr}\n\n\\paragraph{Structure of the section.}\nIn Section~\\ref{subsec_same_perimeter}, we will address the issue of finding two large neighbourhoods of the two roots with exactly the same perimeters. In Section~\\ref{subsec_two_holes}, we use this to prove Proposition~\\ref{prop_two_holes_argument} and Corollary~\\ref{corr_main_minus_monotonicity}. Finally, Section~\\ref{subsec_last_step} is devoted to the end of the proof of Theorem~\\ref{univ_main_thm}, and consists mostly of showing that $d(\\mathbf{q})$ and $\\left( a_j(\\mathbf{q}) \\right)_{j \\geq 1}$ are sufficient to characterize $\\mathbf{q}$.\n\n\\subsection{Finding two pieces with the same perimeter}\n\\label{subsec_same_perimeter}\n\nAs explained above, given the uniform bi-rooted map $\\left( M_n, e^1_n, e^2_n \\right)$, we want to find two neighbourhoods of $e^1_n$ and $e^2_n$ with the same large perimeter $2p$. For this, we will perform a peeling exploration around the two roots and stop it when the perimeter of the explored region is exactly $2p$. However, since the perimeter process has a positive drift, it can make large positive jumps, we cannot guarantee that both perimeters will hit the value $p$ with high probability. We will therefore show a weaker result: roughly speaking, the probability that the perimeter processes around $e_n^1$ and $e_n^2$ both hit $p$ is bounded from below, even if we condition on $\\mathbf{Q}^1$ and $\\mathbf{Q}^2$. \n\nMore precisely, we fix a deterministic peeling algorithm $\\mathcal{A}$, and let $p,v_0 \\geq 1$. We recall from the end of Section~\\ref{subsec_lazy_peeling} that we can make sense of a filled-in peeling exploration on the finite map $M_n$ around $e^1_n$ or $e_n^2$. We perform the following exploration:\n\\begin{itemize}\n\\item\nwe explore the map $M_n$ around $e^1_n$ according to the algorithm $\\mathcal{A}$ until the number of edges in the explored region is larger than $v_0$, or the perimeter of the explored region is exactly $2p$, and denote by $\\tau^1_n$ the time at which we stop;\n\\item\nwe do the same thing around $e^2_n$ and denote by $\\tau^2_n$ the stopping time.\n\\end{itemize}\nWe write $\\mathcal{S}_{n,p,v_0}$ for the event where both $\\tau^1_n$ and $\\tau^2_n$ occur because the perimeter hits $2p$, and where the two regions explored around $e_n^1$ and $e_n^2$ are face-disjoint (the dependence of $\\mathcal{S}$ in $\\mathcal{A}$ will stay implicit). We note right now that $(M_n, e_n^1)$ has a planar, one-ended local limit. Hence, with probability $1-o(1)$ as $n \\to +\\infty$, the exploration is not stopped before $\\tau_n^1$ or $\\tau_n^2$ for the reason stated in the end of Section~\\ref{subsec_lazy_peeling}.\n\nThe goal of this subsection is to prove the next result. We recall that the functions $r_j(\\mathbf{q})$ for $j \\in \\mathbb N^* \\cup \\{ \\infty \\}$ and $\\mathbf{q} \\in \\mathcal{Q}_h$ are defined in Proposition~\\ref{prop_q_as_limit}.\n\n\\begin{prop}\\label{prop_finding_regions_to_cut}\nLet $(M_n, e^1_n, e_2^n)$ and $\\mathbf{Q}^1, \\mathbf{Q}^2$ be as in Section~\\ref{sec_arg_deux_trous}.\nWe fix $j \\in \\mathbb N^* \\cup \\{ \\infty \\}$, and $\\varepsilon>0$. Then there is $\\delta>0$ with the following property. For every $p \\geq 1$ large enough, there is $v_0$ such that, for $n$ large enough:\n\\[ \\mbox{if } \\P \\left( | r_j(\\mathbf{Q}^1)-r_j(\\mathbf{Q}^2) |>\\varepsilon \\right) \\geq \\varepsilon,\\]\n\\[ \\mbox{then }\\P \\left( | r_j(\\mathbf{Q}^1)-r_j(\\mathbf{Q}^2)|>\\frac{\\varepsilon}{2} \\mbox{ and } (M_n, e_n^1, e_n^2) \\in \\mathcal{S}_{n,p,v_0} \\right) \\geq \\delta. \\]\n\\end{prop}\n\nWe recall that we have used the Skorokhod theorem to couple the finite and infinite maps together, so the last event makes sense.\n\nHere is why Proposition~\\ref{prop_finding_regions_to_cut} seems reasonable: we know that conditionally on $(\\mathbf{Q}^1, \\mathbf{Q}^2)$, the perimeters of the explored region along a peeling exploration of $\\mathbb M^1_{\\mathbf{Q}^1}$ and $\\mathbb M^2_{\\mathbf{Q}^2}$ are random walks conditioned to stay positive. Moreover, since $\\mathbf{Q}^1, \\mathbf{Q}^2 \\in \\mathcal{Q}_f$, these random walks do not have a too heavy tail, so each of them have a reasonable chance of hitting exactly $p$. However, there is no reason a priori why $\\mathbb M^1_{\\mathbf{Q}^1}$ and $\\mathbb M^2_{\\mathbf{Q}^2}$ should be independent conditionally on $\\mathbf{Q}^1$ and $\\mathbf{Q}^2$, so it might be unlikely that \\emph{both} processes hit $p$. Therefore, the sketch of the proof will be the following:\n\\begin{itemize}\n\\item\nwe fix a large constant $C>0$ ($C$ will be much smaller than $p$),\n\\item\nwe prove that both walks have a large probability to hit the interval $[p,p+C]$ before the explored volume exceeds $v_0(p)$ (Lemma~\\ref{lem_RW_hits_interval}),\n\\item\nonce both perimeter processes around $e_n^1$ and $e_n^2$ in $M_n$ have hit $[p,p+C]$, we use the Bounded ratio Lemma (Lemma~\\ref{lem_BRL}, item 2) to show that, with probability bounded from below by roughly $e^{-C}$, both perimeters fall to exactly $p$ in at most $C$ steps. This will prove the proposition with $\\delta \\approx \\varepsilon e^{-C}$ and $v_0=v_0(p)$.\n\\end{itemize}\nThe point of replacing $p$ by $[p,p+C]$ is to deal with events of large probability, so that we don't need any independence to make sure that two events simultaneously happen.\n\nFor this, consider the peeling exploration of $\\mathbb M^1_{\\mathbf{Q}^1}$ according to $\\mathcal{A}$. We denote by $\\sigma^{1,\\infty}_{[p,p+C]}$ the first time at which the half-perimeter is in $[p,p+C]$ (this stopping time might be infinite). We define $\\sigma^{1,n}_{[p,p+C]}$ (resp. $\\sigma^{2, \\infty}_{[p,p+C]}$, $\\sigma^{2,n}_{[p,p+C]}$) as the analogue quantity for the exploration in $M_n$ around $e_n^1$ (resp. in $\\mathbb M^2_{\\mathbf{Q}^2}$, in $\\left( M_n, e_n^2 \\right)$).\n\n\\begin{lem}\\label{lem_RW_hits_interval}\nWe have\n\\[ \\lim_{C \\to +\\infty} \\liminf_{p \\to +\\infty} \\P \\left( \\sigma_{[p,p+C]}^{1,\\infty} <+\\infty \\right) =1.\\]\n\\end{lem}\n\n\\begin{proof}\nWe know that $\\mathbf{Q}^1 \\in \\mathcal{Q}_f$ a.s.. Hence, it is enough to prove that, for any $\\mathbf{q} \\in \\mathcal{Q}_f$, we have\n\\begin{equation}\\label{eqn_RW_hits_interval_infinite}\n\\lim_{C \\to +\\infty} \\liminf_{p \\to +\\infty} \\P \\left( \\sigma^{1,\\infty}_{[p,p+C]}<+\\infty \\big| \\mathbf{\\mathbf{Q}}=\\mathbf{q} \\right) =1.\n\\end{equation}\nThe lemma then follows by taking the expectation and using Fatou's lemma. Conditionally on $\\mathbf{Q}^1=\\mathbf{q}$, the law of $\\mathbb M^1_{\\mathbf{Q}^1}$ is the law of $\\mathbb M_\\mathbf{q}$. In particular, the process $P$ describing the half-perimeter of the explored region has the law of $X$ conditioned to stay positive, where $X$ is a random walk with step distribution $\\widetilde{\\nu}_{\\mathbf{q}}$.\n\nTo prove \\eqref{eqn_RW_hits_interval_infinite}, we distinguish two cases: the case where $\\mathbf{q}$ is critical, and the case where it is not. We start with the second one. Then by the results of Section~\\ref{subsec_IBPM}, the walk $X$ satisfies $\\mathbb E \\left[ |X_1| \\right]<+\\infty$ and $\\mathbb E \\left[ X_1 \\right]>0$, so the conditioning to stay positive is non degenerate. Therefore, it is enough to prove\n\\begin{equation}\\label{eqn_RW_hits_interval_conditioned}\n\\lim_{C \\to +\\infty} \\liminf_{p \\to +\\infty} \\P \\left( \\mbox{$X$ hits $[p,p+C]$} \\right) =1.\n\\end{equation}\nThis follows from standard renewal arguments: if we denote by $(H_i)_{i \\geq 0}$ the ascending ladder heights of $P$, then $(H_i)$ is a renewal set with density $\\mathbb E[H_1]= \\frac{1}{\\mathbb E[X_1]}>0$. Let $I_p$ be such that $H_{I_p} < p \\leq H_{I_{p+1}}$. Then the law of $H_{I_{p+1}}-H_{I_p}$ converges as $p \\to +\\infty$ to the law of $H_1$ biased by its size, so\n\\begin{align*}\n\\P \\left( \\mbox{$P$ does not hit $[p,p+C]$} \\right) &\\leq \\P \\left( H_{I_{p+1}} \\notin [p,p+C] \\right)\\\\& \\leq \\P \\left( H_{I_{p+1}}-H_{I_p} > C \\right) \\xrightarrow[p \\to +\\infty]{} \\frac{\\mathbb E \\left[ H_1 \\mathbbm{1}_{H_1>C} \\right]}{\\mathbb E[H_1]},\n\\end{align*} \nand this last quantity goes to $0$ as $C \\to +\\infty$.\n\nWe now tackle the case where $\\mathbf{q}$ is critical, which by the results in the end of~\\ref{subsec_IBPM} implies $\\sum_{i \\geq 1} i^{3\/2} \\widetilde{\\nu}_{\\mathbf{q}}(i)<+\\infty$. This case is more complicated since renewal arguments are not available anymore, and the conditioning is now degenerate, so absolute continuity arguments between $P$ and $X$ become more elaborate. On the other hand, the growth is now slower and the nonconditionned walk $X$ with step distribution $\\widetilde{\\nu}_{\\mathbf{q}}$ is now recurrent, so it seems more difficult to jump over a large interval. And indeed, we will prove\n\\[ \\lim_{p \\to +\\infty} \\P \\left( \\mbox{$P$ hits $p$} \\right) =1,\\]\nwhich is a much stronger version of \\eqref{eqn_RW_hits_interval_conditioned}.\n\nFor this, our strategy will be the following: let $\\chi_p$ be the first time at which $P$ is at least $p$.\n\\begin{itemize}\n\\item\nThe scaling limit of $P$ is a process with no positive jump, so ${P_{\\chi_p}=p+o(p)}$ in probability as $p \\to +\\infty$.\n\\item\nBetween time $\\chi_p$ and $\\tau_p+o(p^{2\/3})$, the process $P$ looks a lot like a nonconditioned random walk $X$ started from $P_{\\tau_p}$.\n\\item\nIf $X$ is started from $p+o(p)$, the time it takes to first hit $P$ is $o(p^{2\/3})$. This is a stronger version of the recurrence of $X$, and will follow from a local limit theorem for random walks.\n\\end{itemize}\nLet us now be more precise. By Theorem 3 of \\cite{Bud15} (see also \\cite[Chapter 10]{C-StFlour}), we have the convergence\n\\[ \\left( \\frac{P_{nt}}{n^{2\/3}} \\right)_{t \\geq 0} \\xrightarrow[n \\to +\\infty]{(d)} \\left( b_{\\mathbf{q}} S^+_t \\right)_{t \\geq 0} \\]\nfor the Skorokhod topology, where $S^+$ is a $3\/2$-stable L\u00e9vy process with no positive jump conditioned to stay positive, and $b_{\\mathbf{q}}>0$ (the precise value will not matter here). Since this limiting process has no positive jump, we have $P_{\\tau_p}-p=o(p)$ in probability. Hence, there is a deterministic function $f(p)$ with $\\frac{f(p)}{p} \\to 0$ when $p \\to +\\infty$ such that, for any $\\varepsilon>0$,\n\\[ \\P \\left( P_{\\tau_p}-p \\geq \\varepsilon f(p) \\right) \\xrightarrow[p \\to +\\infty]{} 0. \\]\nWe now fix $\\varepsilon>0$, and condition on $P_{\\tau_p}=p'$ for some $p \\leq p' \\leq p+\\varepsilon f(p)$. We claim that then $\\left( P_{\\tau_p+i}-p' \\right)_{0 \\leq i \\leq f(p)^{3\/2}}$ can be coupled with $(X_i)_{0 \\leq i \\leq f(p)^{3\/2}}$ in such a way that both processes are the same with probability $1-o(1)$. For this, recall from~\\eqref{peeling_transitions} that $P$ can be described as a Doob $h$-transform of $X$, where $h$ is given by \\eqref{eqn_defn_homega}. Hence, the Radon--Nikodym derivative of the first process with respect to the second is\n\\begin{equation}\\label{critical_Radon_Nikodym}\n\\frac{h_{p'+X_{f(p)^{3\/2}}}(1)}{h_{p'}(1)}.\n\\end{equation}\nSince $\\frac{X_{f(p)^{3\/2}}}{f(p)}$ converges in distribution, we have $\\frac{X_{f(p)^{3\/2}}}{p} \\to 0$ in probability. By using the fact that $\\frac{p'}{p} \\to 0$ uniformly in $p'$ and that $h_1(x)\\sim c\\sqrt{x}$ for some $c>0$ (see Section~\\ref{subsec_IBPM}), we conclude that \\eqref{critical_Radon_Nikodym} goes to $1$ as $p \\to +\\infty$, uniformly in $p' \\in \\left[ p,p+\\varepsilon f(p) \\right]$. This proves our coupling claim. Note that under this coupling, the time where $P$ hits exactly $p$ is $\\tau_p$ plus the time where $X$ hits $p-p'$.\n\nWe will now show that, if $p$ is large enough, for any $k \\in [-\\varepsilon f(p),0]$, we have\n\\begin{equation}\\label{eqn_critical_quantitative_recurrence}\n\\P \\left( \\mbox{$X$ hits $k$ before time $f(p)^{3\/2}$} \\right) \\geq 1-\\delta(\\varepsilon),\n\\end{equation}\nwhere $\\delta(\\varepsilon) \\to 0$ as $\\varepsilon \\to 0$. Together with our coupling result, this will imply that the probability for $P$ to hit $p$ before time $\\tau_p+f(p)^{3\/2}$ is at least $1-\\delta (\\varepsilon)-o(1)$ as $p \\to +\\infty$. Since this is true for any $\\varepsilon>0$, this will conclude the proof of Lemma~\\ref{lem_RW_hits_interval} in the critical case.\n\nThe proof of \\eqref{eqn_critical_quantitative_recurrence} relies on the Local Limit Theorem (this is e.g. Theorem 4.2.1 of \\cite{IL71}). This theorem (in the case $\\alpha=3\/2$) states that\n\\[ \\sup_{k \\in \\mathbb Z} \\left| n^{2\/3} \\P (X_n=k)-g \\left( \\frac{k}{n^{2\/3}} \\right) \\right| \\xrightarrow[n \\to +\\infty]{} 0, \\]\nwhere $g$ is a continuous function (the density of a $3\/2$-stable variable).\nOn the other hand, let us denote $t=f(p)^{3\/2}$. By the strong Markov property, for all $k \\in \\mathbb Z$, we have\n\\begin{align*}\n\\mathbb E_0 \\left[ \\sum_{i=0}^{t} \\mathbbm{1}_{X_i=k} \\right] &\\leq \\P_0 \\left( \\mbox{$X$ hits $k$ before time $t$} \\right) \\mathbb E_k \\left[  \\sum_{i=0}^{t} \\mathbbm{1}_{X_i=k} \\right]\\\\\n&= \\P_0 \\left( \\mbox{$X$ hits $k$ before time $t$} \\right) \\mathbb E_0 \\left[  \\sum_{i=0}^{t} \\mathbbm{1}_{X_i=0} \\right].\n\\end{align*}\nTherefore, using the local limit theorem, we can write, for $-\\varepsilon f(p) \\leq k \\leq 0$ and $p$ large (the $o$ terms are all uniform in $k$):\n\\begin{align*}\n\\P_0 \\left( \\mbox{$X$ hits $k$ before $t$} \\right) &\\geq \\frac{\\sum_{i=0}^{t} \\P_0 \\left( X_i=k \\right)}{\\sum_{i=0}^{t} \\P_0 \\left( X_i=0 \\right)}\\\\\n&= \\frac{\\sum_{i=1}^{t} \\left( \\frac{1}{i^{2\/3}} g \\left( \\frac{k}{i^{2\/3}} \\right) + o \\left( \\frac{1}{i^{2\/3}} \\right) \\right)}{1+\\sum_{i=1}^{t} \\left( \\frac{1}{i^{2\/3}} g (0) + o \\left( \\frac{1}{i^{2\/3}} \\right) \\right)}\\\\\n&\\geq \\frac{-\\varepsilon t^{1\/3} + \\sum_{i=\\varepsilon t}^{t} \\left( \\frac{1}{i^{2\/3}} \\min_{[-\\varepsilon^{1\/3},0]} g +o \\left( \\frac{1}{i^{2\/3}} \\right)\\right)}{3t^{1\/3} g(0) + \\varepsilon t^{1\/3}}\\\\\n&\\geq \\frac{-2\\varepsilon t^{1\/3}+ \\left( 3t^{1\/3}-3\\varepsilon^{1\/3} t^{1\/3}\\right) \\min_{[-\\varepsilon^{1\/3},0]} g}{\\left( 3g(0)+\\varepsilon \\right)t^{1\/3}}\\\\\n&= \\frac{-2\\varepsilon +3(1-\\varepsilon^{1\/3}) \\min_{[-\\varepsilon^{1\/3},0]} g}{3g(0)+\\varepsilon},\n\\end{align*}\nwhere the third line uses that, for any index $i \\geq \\varepsilon t$, we have\n\\[0 \\geq \\frac{k}{i^{2\/3}} \\geq -\\frac{\\varepsilon f(p)}{(\\varepsilon t)^{2\/3}}=-\\varepsilon^{1\/3}. \\]\nWe obtain a lower bound that goes to $1$ as $\\varepsilon \\to 0$, so this proves \\eqref{eqn_critical_quantitative_recurrence}, and Lemma~\\ref{lem_RW_hits_interval}.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop_finding_regions_to_cut}]\nThe subtlety in the proof is that we would like to say something about the finite maps $M_n$ conditionally on the values of $\\mathbf{Q}^1$ and $\\mathbf{Q}^2$, but $\\mathbf{Q}^1$ and $\\mathbf{Q}^2$ are defined in terms of the infinite limits. However, we can condition on the maps explored at the time when the perimeters of the explored parts hit $[p,p+C]$ for the first time. Then Proposition~\\ref{prop_q_as_limit} guarantees that from these explored parts, we can get good approximations of $\\mathbf{Q}^1$ and $\\mathbf{Q}^2$ if $p$ is large enough.\n\nIn this proof, we will use a shortened notation for our peeling explorations. For $i \\in \\{1,2\\}$ and $t \\geq 0$, we will write $\\mathcal{E}^{n,i}_t=\\mathcal{E}_t^{\\mathcal{A}} \\left( M_n, e_n^i \\right)$ and $\\mathcal{E}^{\\infty,i}_t=\\mathcal{E}_t^{\\mathcal{A}} \\left( \\mathbb M^i_{\\mathbf{Q}^i} \\right)$.\n\nWe fix $\\varepsilon>0$. By Lemma~\\ref{lem_RW_hits_interval}, let $C$ be a constant (depending only on $\\varepsilon$) such that\n\\[ \\liminf_{p \\to +\\infty} \\P \\left( \\sigma_{[p,p+C]}^{1,\\infty} <+\\infty \\right) > 1-\\frac{\\varepsilon}{20}.\\]\nFor $p$ large enough (where \"large enough\" may depend on $\\varepsilon$), there is $v_0=v_0(p)$ such that\n\\begin{equation}\\label{eqn_sigma_reasonable}\n\\P \\left( \\sigma^{\\infty,1}_{[p,p+C]} \\leq v_0 \\mbox{ and } \\left| \\mathcal{E}^{\\infty,1}_{\\sigma^{\\infty,1}_{[p,p+C]}} \\right| \\leq v_0 \\right)>1-\\frac{\\varepsilon}{20},\n\\end{equation}\nwhere $|m|$ is the number of edges of a map $m$.\nOn the other hand, let us fix $j \\in \\mathbb N^* \\cup \\{\\infty\\}$. Proposition~\\ref{prop_q_as_limit} provides a function $\\widetilde{r}_j$ on the set of finite maps with a hole such that $\\widetilde{r}_j(\\mathcal{E}^{\\infty, 1}_t) \\to r_j(\\mathbf{Q}^1)$ almost surely as $t \\to +\\infty$. Let $\\eta<1$ be a small constant, which will be fixed later and will only depend on $\\varepsilon$. For $p$ large enough, we have\n\\begin{equation}\\label{eqn_fq_approximation}\n\\P \\left( \\sigma^{\\infty,1}_{[p,p+C]} \\leq v_0(p) \\mbox{ but } \\left| \\widetilde{r}_j \\left( \\mathcal{E}^{\\infty, 1}_{\\sigma^{\\infty,1}_{[p,p+C]}} \\right) - r_j(\\mathbf{Q}^1) \\right| \\geq \\frac{\\varepsilon}{8} \\right) < \\eta \\frac{\\varepsilon}{20}.\n\\end{equation}\nFrom now on, we take $p$ large enough so that both \\eqref{eqn_sigma_reasonable} and \\eqref{eqn_fq_approximation} hold. By almost sure local convergence and \\eqref{eqn_sigma_reasonable}, for $n$ large enough (where \"large enough\" may depend on $\\varepsilon$ and $p$), we have\n\\[ \\P \\left( \\sigma^{n,1}_{[p,p+C]}, \\sigma^{n,2}_{[p,p+C]} \\leq v_0 \\mbox{ and } \\left| \\mathcal{E}^{n,1}_{\\sigma^{n,1}_{[p,p+C]}} \\right|, \\left| \\mathcal{E}^{n,2}_{\\sigma^{n,2}_{[p,p+C]}} \\right| \\leq v_0 \\right) > 1-\\frac{\\varepsilon}{10}.\\]\nBy the assumption that $|r_j(\\mathbf{Q}^1)-r_j(\\mathbf{Q}^2)|>\\varepsilon$ with probability at least $\\varepsilon$ and by \\eqref{eqn_fq_approximation}, we deduce that\n\\[ \\P \\left(\\begin{array}{c} \\sigma^{n,1}_{[p,p+C]}, \\sigma^{n,2}_{[p,p+C]} \\leq v_0 \\mbox{ and } \\left| \\mathcal{E}^{n,1}_{\\sigma^{n,1}_{[p,p+C]}} \\right|, \\left| \\mathcal{E}^{n,2}_{\\sigma^{n,2}_{[p,p+C]}} \\right| \\leq v_0\\\\ \\mbox{and } \\left| \\widetilde{r}_j \\left( \\mathcal{E}^{\\infty, 1}_{\\sigma^{\\infty,1}_{[p,p+C]}} \\right) - \\widetilde{r}_j \\left( \\mathcal{E}^{\\infty, 2}_{\\sigma^{\\infty,2}_{[p,p+C]}} \\right) \\right| \\geq \\frac{3}{4}\\varepsilon\\end{array} \\right) > \\frac{4}{5} \\varepsilon.\\]\n\n\n\nNote that if this last event occurs but the two regions $\\mathcal{E}^{n,1}_{\\sigma^{n,1}_{[p,p+C]}}$ and $\\mathcal{E}^{n,2}_{\\sigma^{n,2}_{[p,p+C]}}$ have a common face, then the dual graph distance between the two roots is bounded by $2v_0$. However, by Proposition~\\ref{prop_tightness_dloc_univ}, the volume of the ball of radius $2v_0$ around $e^1_n$ is tight as $n \\to +\\infty$, so the probability that this happens goes to $0$ as $n \\to +\\infty$. Hence, for $n$ large enough:\n\\begin{equation}\\label{eqn_two_distinct_regions}\n\\P \\left( \\begin{array}{c}\\mathcal{E}^{n,1}_{\\sigma^{n,1}_{[p,p+C]}}, \\mathcal{E}^{n,2}_{\\sigma^{n,2}_{[p,p+C]}} \\mbox{ are well-defined, face-disjoint, have at}\\\\\\mbox{most $v_0$ edges and } \\left| \\widetilde{r}_j \\left( \\mathcal{E}^{\\infty, 1}_{\\sigma^{\\infty,1}_{[p,p+C]}} \\right) - \\widetilde{r}_j \\left( \\mathcal{E}^{\\infty, 2}_{\\sigma^{\\infty,2}_{[p,p+C]}} \\right) \\right| \\geq \\frac{3}{4}\\varepsilon \\end{array} \\right) > \\frac{4}{5} \\varepsilon.\n\\end{equation}\n\nNow assume that this last event occurs and condition on the $\\sigma$-algebra $\\mathcal{F}_{\\sigma}$ generated by the pair $\\left( \\mathcal{E}^{n,1}_{\\sigma^{n,1}_{[p,p+C]}}, \\mathcal{E}^{n,2}_{\\sigma^{n,2}_{[p,p+C]}} \\right)$ of explored regions. Then, let $I_1, I_2 \\in [0,C]$ be such that the perimeters of the two explored regions are $2p+2I_1$ and $2p+2I_2$. Then the complementary map is a uniform map of the $\\left( 2p+2I_1, 2p+2I_2 \\right)$-gon with genus $g_n$ and face degrees given by $\\mathbf{\\widetilde{F}}^n$ as follows. If $F_j^n$ is the number of internal faces of degree $2j$ in $\\mathcal{E}^{n,1}_{\\sigma^{n,1}_{[p,p+C]}} \\cup \\mathcal{E}^{n,2}_{\\sigma^{n,2}_{[p,p+C]}}$, then $\\widetilde{F}_j^n=f_j^n-F_j^n$. We now perform $I_1$ peeling steps according to $\\mathcal{A}$ around $\\mathcal{E}^{n,1}_{\\sigma^{n,1}_{[p,p+C]}}$, followed by $I_2$ peeling steps according to $\\mathcal{A}$ around $\\mathcal{E}^{n,2}_{\\sigma^{n,2}_{[p,p+C]}}$. We call a peeling step \\emph{nice} if it consists of gluing together two boundary edges, which decreases the perimeter by $2$. The number of possible values of the map $M_n \\backslash \\left( \\mathcal{E}^{n,1}_{\\sigma^{n,1}_{[p,p+C]}} \\cup \\mathcal{E}^{n,2}_{\\sigma^{n,2}_{[p,p+C]}} \\right)$ is\n\\[\\beta_{g}^{(p+I_1,p+I_2)}(\\mathbf{\\widetilde{F}}^n).\\]\nOn the other hand, if the $I_1+I_2$ additional peeling steps are all good and the regions around $e_n^1$ and $e_n^2$ are still disjoint after these steps, the number of possible complementary maps is\n\\[\\beta_{g}^{(p,p)} (\\mathbf{\\widetilde{F}}^n). \\]\nIt follows that\n\\[ \\P \\left( \\mbox{the $I_1+I_2$ peeling steps are all nice} | \\mathcal{F}_{\\sigma} \\right) = \\frac{\\beta_{g}^{(p,p)} (\\mathbf{\\widetilde{F}}^n)}{\\beta_{g}^{(p+I_1,p+I_2)}(\\mathbf{\\widetilde{F}}^n)}. \\]\nSince $|\\mathbf{F}^n|$ is bounded by $v_0(p)$, for $n$ large enough (where \"large enough\" may depend on $p$), the Bounded ratio Lemma applies to $\\mathbf{\\widetilde{F}}^n$. Therefore, by the Bounded ratio Lemma (more precisely, by Corollary~\\ref{lem_BRL_boundaries}, item 2), the last ratio is always larger than a constant $\\eta$ depending on $\\varepsilon$. More precisely $\\eta$ may depend on $I_1$ and $I_2$, but $0 \\leq I_1, I_2 \\leq C(\\varepsilon)$, so $(I_1, I_2)$ can take finitely many values given $\\varepsilon$, so $\\eta$ only depends on $\\varepsilon$ (and not on $p$). This is the value of $\\eta$ that we choose for \\eqref{eqn_fq_approximation}. For $i \\in \\{1,2\\}$, we write $\\tau_p^{n,i}=\\sigma_{[p,p+C]}^{n,i}+I_i$. If the last $I_1+I_2$ peeling steps are nice, then after they are performed, both explored regions have perimeter $2p$. Therefore, it follows from the last computation and from \\eqref{eqn_two_distinct_regions} that, for $n$ large enough, we have\n\\[ \\P \\left( \\begin{array}{c}\\mathcal{E}^{n,1}_{\\tau_p^{n,1}} \\mbox{ and } \\mathcal{E}^{n,2}_{\\tau_p^{n,2}} \\mbox{ are both face-disjoint, have perimeter $2p$}\\\\\\mbox{and volume $\\leq v_0$, and } \\left| \\widetilde{r}_j \\left( \\mathcal{E}^{\\infty, 1}_{\\sigma^{\\infty,1}_{[p,p+C]}} \\right) - \\widetilde{r}_j \\left( \\mathcal{E}^{\\infty, 2}_{\\sigma^{\\infty,2}_{[p,p+C]}} \\right) \\right| \\geq \\frac{3}{4}\\varepsilon\\end{array} \\right) \\geq \\frac{4}{5} \\varepsilon \\eta.\\]\nFinally, we can use \\eqref{eqn_fq_approximation} to replace back the approximations $\\widetilde{r}_j \\left( \\mathcal{E}^{\\infty, i}_{\\sigma^{\\infty,i}_{[p,p+C]}} \\right)$ by $r_j(\\mathbf{Q}^i)$. We obtain\n\\[ \\P \\left( \\begin{array}{c}\n\\mathcal{E}^{1,n}_{\\tau_p^{n,i}} \\mbox{ and } \\mathcal{E}^{2,n}_{\\tau_p^{n,2}} \\mbox{ are both face-disjoint, have perimeter $2p$ }\\\\ \\mbox{and $\\leq v_0$ edges, and } \\left| r_j(\\mathbf{Q}^1)-r_j(\\mathbf{Q}^2) \\right| \\geq \\frac{1}{2}\\varepsilon\n\\end{array} \\right) \\geq \\frac{3}{5} \\varepsilon \\eta.\\]\nOn this event, we have $(M_n, e_n^1, e_n^2) \\in \\mathscr{S}_{n,p,v_0}$. Therefore, this concludes the proof of the proposition, with $\\delta=\\frac{3}{5} \\eta \\varepsilon$.\n\\end{proof}\n\n\\subsection{The two holes argument: proof of Proposition~\\ref{prop_two_holes_argument} and Corollary~\\ref{corr_main_minus_monotonicity}}\n\\label{subsec_two_holes}\n\nNow that we have Proposition~\\ref{prop_finding_regions_to_cut}, the proof of Proposition~\\ref{prop_two_holes_argument} is basically the same as two holes argument in~\\cite{BL19} (i.e. the proof of Proposition 18). Therefore, we will not write the argument in full details, but only sketch it. We first stress two differences:\n\\begin{itemize}\n\\item\nThe first one is that the involution obtained by (possibly) swapping the two explored parts is now non-identity on a relatively small set of maps (but still on a positive proportion). The only consequence is that in the end, instead of contradicting the almost sure convergence of Proposition~\\ref{prop_q_as_limit} on an event of probability $\\varepsilon$, we will contradict it on an event of probability $\\delta<\\varepsilon$, where $\\delta$ is given by Proposition~\\ref{prop_finding_regions_to_cut}.\n\\item\nThe other difference is that in \\cite{BL19}, the only observable we were using to approximate the Boltzmann weights was the ratio between perimeter and volume, which corresponds to our function $r_{\\infty}$. Here we also need to deal with the functions $r_j$ for $j \\in \\mathbb N^*$. For this, we simply need the observation that, if $q$ is much larger than $p$, the proportion of peeling steps before $\\tau_q$ where we discover a new face of perimeter $2j$ depends almost only on the part of the exploration between $\\tau_p$ and $\\tau_q$.\n\\end{itemize}\n\n\n\\begin{proof}[Sketch of proof of Proposition~\\ref{prop_two_holes_argument}]\nFix $j \\in \\mathbb N^* \\cup \\{ \\infty\\}$. Let $\\varepsilon>0$, and assume\n\\begin{equation}\\label{eqn_q1_ne_q2}\n\\P \\left( |r_j(\\mathbf{Q}^1)-r_j(\\mathbf{Q}^2)|>\\varepsilon \\right)>\\varepsilon.\n\\end{equation}\nLet $\\delta>0$ be given by Proposition~\\ref{prop_finding_regions_to_cut}. Consider $p$ large (depending on $\\varepsilon$) and let $v_0$ be given by Proposition~\\ref{prop_finding_regions_to_cut}. We assume that the peeling algorithm $\\mathcal{A}$ that we work with has the property that the edge peeled at time $t$ for $t \\geq \\tau_p$ only depends on $\\mathcal{E}_t^{\\mathcal{A}}(m) \\backslash \\mathcal{E}_{\\tau_p}^{\\mathcal{A}}(m)$ (see~\\cite{BL19} for a more careful description of this property). This is not a problem since Proposition~\\ref{prop_finding_regions_to_cut} is independent of the choice of $\\mathcal{A}$.\n\nWe define an involution $\\Phi_n$ on the set of bi-rooted maps with genus $g_n$ and face degrees $\\mathbf{f}^n$ as follows: if $m \\in \\mathscr{S}_{n,p,v_0}$, then $\\Phi_n(m)$ is obtained from $m$ by swapping the two regions $\\mathcal{E}^{1,n}_{\\tau_p^{1,n}}$ and $\\mathcal{E}^{2,n}_{\\tau_p^{2,n}}$. If $m \\notin \\mathscr{S}_{n,p,v_0}$, then $\\Phi_n(m)=m$. Note that $\\Phi_n(M_n,e^1_n,e^2_n)$ is still uniform on bi-rooted maps with prescribed genus and face degrees. This map rooted at $e_1^n$ converges to a map $\\widehat{M}$, with either\n\\[\\widehat{M}=M^1_{\\mathbf{Q}^1}\\]\nor\n\\[\\mathcal{E}^{\\mathcal{A}}_{\\tau^1_p}(\\widehat{M}) = \\mathcal{E}^1_{\\tau^1_p} \\quad \\mbox{ and } \\quad \\widehat{M} \\backslash \\mathcal{E}^{\\mathcal{A}}_{\\tau^1_p}(\\widehat{M}) = M^2_{\\mathbf{Q}^2} \\backslash \\mathcal{E}^2_{\\tau^2_p}. \\]\n\nMoreover, by Proposition~\\ref{prop_finding_regions_to_cut}, if $p$ has been chosen large enough, then with probability at least $\\delta$, we are in the second case and furthermore $|r_j(\\mathbf{Q}^1)-r_j(\\mathbf{Q}^2)|>\\frac{\\varepsilon}{2}$. Now assume that this last event occurs and that $q \\gg p \\gg 1$. Then we have\n\\begin{equation}\\label{eqn_rtilde_and_r_badcase}\n\\widetilde{r}_j \\left( \\mathcal{E}^{\\mathcal{A}}_{\\tau^1_p}(\\widehat{M}) \\right) \\approx r_j(\\mathbf{Q}^1) \\quad \\mbox{ and } \\quad \\widetilde{r}_j \\left( \\mathcal{E}^{\\mathcal{A}}_{\\widehat{\\tau}_q}(\\widehat{M}) \\right) \\approx r_j(\\mathbf{Q}^2),\n\\end{equation}\nwhere $\\widehat{\\tau}_q$ is the first step where the perimeter of the explored part of $\\widehat{M}$ is at least $q$. The approximations of \\eqref{eqn_rtilde_and_r_badcase} can be made arbitrarily precise if $p$ and $q$ were chosen large enough, so for $p$ large enough and $q$ large enough (depending on $p$), we have\n\\begin{equation}\\label{eqn_rj_different}\n\\P \\left( \\left| \\widetilde{r}_j \\left( \\mathcal{E}^{\\mathcal{A}}_{\\tau^1_p}(\\widehat{M})  \\right) - \\widetilde{r}_j \\left( \\mathcal{E}^{\\mathcal{A}}_{\\widehat{\\tau}_q}(\\widehat{M}) \\right) \\right| > \\frac{\\varepsilon}{4} \\right) \\geq \\delta.\n\\end{equation}\nOn the other hand $\\widehat{M}$ is a local limit of finite uniform maps, so by Theorem~\\ref{thm_main_more_general} it has to be a mixture of Boltzmann infinite planar maps. But then~\\eqref{eqn_rj_different} contradicts the almost sure convergence of Proposition~\\ref{prop_q_as_limit}, so~\\eqref{eqn_q1_ne_q2} cannot be true. Therefore, we must have $r_j(\\mathbf{Q}^1)=r_j(\\mathbf{Q}^2)$ a.s.. Since this is true for all $j \\in \\mathbb N^* \\cup \\{\\infty\\}$, by the last point of Proposition~\\ref{prop_q_as_limit}, we have $\\mathbf{Q}^1=\\mathbf{Q}^2$, which concludes the proof.\n\\end{proof}\n\nThe passage from Proposition~\\ref{prop_two_holes_argument} to Corollary~\\ref{corr_main_minus_monotonicity} does also not require any new idea compared to~\\cite{BL19}, so we do not write it down completely.\n\n\\begin{proof}[Sketch of the proof of Corollary~\\ref{corr_main_minus_monotonicity}]\nThe proof is basically the same as the end of the proof of the main theorem in \\cite{BL19}. The only difference is that we could not prove directly that $d(\\mathbf{q})$ and $\\left( a_j(\\mathbf{q}) \\right)_{j \\geq 1}$ are sufficient to characterize the weight sequence $\\mathbf{q}$, so the result that we obtain is only Corollary~\\ref{corr_main_minus_monotonicity} and not Theorem~\\ref{univ_main_thm}.\n\nMore precisely, by the Euler formula, any map with genus $g_n$ and face degrees $\\mathbf{f}^n$ has exactly $|\\mathbf{f}^n|$ edges and $|\\mathbf{f}^n|-\\sum_{j \\geq 1} f_j^n +2-2g_n$ vertices, so, by invariance of $M_n$ under uniform rerooting, we have\n\\[ \\mathbb E \\left[ \\frac{1}{\\mathrm{deg}_{M_n}(\\rho)} \\right] = \\frac{|\\mathbf{f}^n|-\\sum_{j \\geq 1} f_j^n +2-2g_n}{2|\\mathbf{f}^n|} \\xrightarrow[n \\to +\\infty]{} \\frac{1}{2} \\left( 1-2\\theta - \\sum_j \\alpha_j \\right). \\]\nBy the exact same argument as in \\cite{BL19}, we deduce from Proposition~\\ref{prop_two_holes_argument} that if $M_n \\to \\mathbb M_{\\mathbf{Q}}$, then $d(\\mathbf{Q})=\\frac{1}{2} \\left( 1-2\\theta - \\sum_j \\alpha_j \\right)$ a.s.. Similarly, by invariance under rerooting, for all $j \\geq 1$, we have\n\\[ \\P \\left( \\mbox{the root face of $M_n$ has degree $2j$} \\right) = \\frac{2j f_j^n}{2n} \\xrightarrow[n \\to +\\infty]{} j \\alpha_j. \\]\nBy the same argument as for the mean vertex degree, we obtain $a_j(\\mathbf{Q})=\\alpha_j$ a.s..\n\\end{proof}\n\n\\subsection{Monotonicity of the mean inverse degree}\n\\label{subsec_last_step}\n\nTo conclude the proof of the main theorem, given Corollary~\\ref{corr_main_minus_monotonicity}, it is enough to show that if $\\sum_{j \\geq 1} j^2 a_j(\\mathbf{q}) <+\\infty$, the weight sequence $\\mathbf{q}$ is completely determined by $\\left( a_j(\\mathbf{q}) \\right)_{j \\geq 1}$ and $d(\\mathbf{q})$. For all this subsection, we fix a sequence $(\\alpha_j)_{j \\geq 1}$ such that $\\sum_j j \\alpha_j =1$ and $\\sum_j j^2 \\alpha_j <+\\infty$ and $\\alpha_1<1$. We recall from Proposition~\\ref{prop_third_parametrization} that the weight sequences $\\mathbf{q}$ such that $a_j(\\mathbf{q})=\\alpha_j$ for all $j \\geq 1$ form a one-parameter family $\\left( \\mathbf{q}^{(\\omega)} \\right)_{\\omega \\geq 1}$ given by\n\\[ q_j^{(\\omega)}=\\frac{j \\alpha_j}{\\omega^{j-1}h_j(\\omega)} c_{\\mathbf{q}^{(\\omega)}}^{-(j-1)}, \\quad \\mbox{where} \\quad c_{\\mathbf{q}^{(\\omega)}} = \\frac{4}{1-\\sum_{i \\geq 1} \\frac{1}{4^{i-1}} \\binom{2i-1}{i-1} \\frac{i \\alpha_i}{\\omega^{i-1} h_i(\\omega)}}. \\]\nTo prove Theorem~\\ref{univ_main_thm}, it is sufficient to prove the following.\n\n\\begin{prop}\\label{prop_monotonicity_deg}\nUnder the assumption $\\sum_{j \\geq 1} j^2 \\alpha_j<+\\infty$, the function $\\omega \\to d(\\mathbf{q}^{(\\omega)})$ is strictly decreasing.\n\\end{prop}\n\nSince we were not able to establish this result by a direct argument, we will prove it using Corollary~\\ref{corr_main_minus_monotonicity}.\n\n\\subsubsection{Basic properties of the mean inverse degree function}\n\nBefore moving on to the core of the argument, we start with some basic properties of the function $\\omega \\to d(\\mathbf{q}^{(\\omega)})$.\n\n\\begin{lem}\\label{lem_degreefunction_basic}\n\\begin{itemize}\n\\item[$\\bullet$]\nThe function $\\omega \\to d(\\mathbf{q}^{(\\omega)})$ is continuous on $[1,+\\infty)$ and analytic on $(1,+\\infty)$.\n\\item[$\\bullet$]\nWe have $d(\\mathbf{q}^{(\\omega)})>0$ for all $\\omega$ and $\\lim_{\\omega \\to +\\infty} d(\\mathbf{q}^{(\\omega)}) = 0$.\n\\item[$\\bullet$]\nWe have $d(\\mathbf{q}^{(\\omega)}) \\leq 1-\\sum_{j \\geq 1} \\alpha_j$ for all $\\omega \\geq 1$, with equality if and only if $\\omega=1$.\n\\end{itemize}\n\\end{lem}\n\n\\begin{proof}\nThe proof of the analyticity and continuity on $(1,+\\infty)$ is a bit long and delayed to Appendix~\\ref{subsec_analyticity}. The third item will follow from results of Angel, Hutchcroft, Nachmias and Ray~\\cite{AHNR16}. The other properties are quite easy.\n\nWe start with the continuity statement in the first item. The analyticity proved in Appendix~\\ref{subsec_analyticity} implies continuity on $(1,+\\infty)$ so it is sufficient to prove the continuity at $\\omega=1$. By the monotone convergence theorem, the function $\\omega \\to c_{\\mathbf{q}^{(\\omega)}}$ is continuous at $\\omega=1$, so $q_j^{(\\omega)}$ is continuous at $\\omega=1$ for all $j$. Therefore, for every finite map $m$ with one hole, we have\n\\[ \\P \\left( m \\subset \\mathbb M_{\\mathbf{q}^{(\\omega)}} \\right) \\xrightarrow[\\omega \\to 1]{} \\P \\left( m \\subset \\mathbb M_{\\mathbf{q}^{(1)}} \\right), \\]\nso $\\mathbb M_{\\mathbf{q}^{(\\omega)}} \\to \\mathbb M_{\\mathbf{q}^{(1)}}$ in distribution for the local topology. Since the inverse degree of the root vertex is bounded and continuous for the local topology, the function $\\omega \\to d(\\mathbf{q}^{(\\omega)})$ is continuous at $1$.\n\nWe now prove the second item: $d(\\mathbf{q}^{(\\omega)})>0$ is immediate by finiteness of vertex degrees and $d(\\mathbf{q}^{(\\omega)}) \\to 0$ is equivalent to proving $\\mathrm{deg}_{\\mathbb M_{\\mathbf{q}^{(\\omega)}}}(\\rho) \\to +\\infty$ in probability when $\\omega \\to +\\infty$. For this, we notice (see~\\eqref{eqn_omega_infinite} above) that when $\\omega \\to +\\infty$, we have $h_{i}(\\omega) \\to 1$ for all $i \\geq 1$ and\n\\[\\widetilde{\\nu}_{\\mathbf{q}^{(\\omega)}}(i) \\xrightarrow[\\omega \\to +\\infty]{} \\begin{cases}\n0 \\mbox{ if $i \\leq -1$,}\\\\\n(i+1) \\alpha_{i+1} \\mbox{ if $i \\geq 0$}.\n\\end{cases}\n\\]\nIn other words, the probability of any peeling step swallowing at least one boundary vertex goes to $0$ when $\\omega \\to +\\infty$. Therefore, if we perform a peeling exploration where we peel the edge on the right of $\\rho$ whenever it is possible, the probability to complete the exploration of the root in less than $k$ steps goes to $0$ for all $k$. It follows that the root degree goes to $+\\infty$ in probability.\n\nFinally, we move on to the third item. Since $\\mathbb M_{\\mathbf{q}}$ is stationary, if we denote by $\\mathbb M_{\\mathbf{q}}^{\\mathrm{uni}}$ a map with the law of $\\mathbb M_{\\mathbf{q}}$ biased by the inverse of the root vertex degree, then $\\mathbb M_{\\mathbf{q}}$ is unimodular. A simple computation shows that $d(\\mathbf{q}) \\leq 1-\\sum_{j \\geq 1} \\alpha_j$ is equivalent to $\\mathbb E \\left[ \\kappa_{\\mathbb M_{\\mathbf{q}}^{\\mathrm{uni}}}(\\rho) \\right] \\geq 0$ where, if $v$ is a vertex of a map $m$:\n\\[ \\kappa_m(v)=2\\pi-\\sum_{f} \\frac{\\mathrm{deg}(f)-2}{\\mathrm{deg}(f)}\\pi,\\]\nand the sum is over all faces that are incident to $v$ in $m$, counted with multiplicity.\nMoreover, we have equality if and only if $\\mathbb E[\\kappa_{\\mathbb M_{\\mathbf{q}}^{\\mathrm{uni}}}(\\rho)] = 0$. The fact that $\\mathbb E[\\kappa_{\\mathbb M_{\\mathbf{q}}^{\\mathrm{uni}}}(\\rho)] \\geq 0$ is then a consequence of \\cite[Theorem 1]{AHNR16}. Moreover, \\cite{AHNR16} shows the equivalence between 17 different definitions of hyperbolicity. In particular, we have $\\mathbb E[\\kappa_{\\mathbb M_{\\mathbf{q}}^{\\mathrm{uni}}}(\\rho)] > 0$ (Definition 1 in \\cite{AHNR16}) if and only if $p_c<p_u$ with positive probability for bond percolation on $\\mathbb M_{\\mathbf{q}}^{\\mathrm{uni}}$. This is equivalent to $\\P \\left( p_c<p_u \\right)>0$ for bond percolation on $\\mathbb M_{\\mathbf{q}}$, which is equivalent to $\\mathbf{q}$ being critical (i.e. $\\omega=1$) by \\cite[Theorem 12.9]{C-StFlour}\\footnote{More precisely \\cite[Theorem 12.9]{C-StFlour} is about half-plane supercritical maps. Here is a way to extend it to full-plane maps: there is a percolation regime on the half-plane version of $\\mathbb M_{\\mathbf{q}}$ such that with positive probability, there are infinitely many infinite clusters. For topological reasons, at most two of them intersect the boundary infinitely many times. Hence, by changing the colour of finitely many edges, with positive probability there are two infinite clusters that do not touch the boundary. Since there is a coupling in which the half-plane version of $\\mathbb M_{\\mathbf{q}}$ is included in the full-plane version, we have with positive probability two disjoint infinite clusters in $\\mathbb M_{\\mathbf{q}}$ in a certain bond percolation regime, so $p_c<p_u$ with positive probability.}.\n\\end{proof}\n\n\\subsubsection{Proof of Proposition~\\ref{prop_monotonicity_deg}}\\label{subsubsec_final_argument}\n\n\\paragraph{Sketch of the argument.}\nRoughly speaking, the idea behind the proof of Proposition~\\ref{prop_monotonicity_deg} is the following observation. We fix $j \\geq 2$ and recall that $M_{\\mathbf{f}, g}$ stands for a uniform map in $\\mathcal{B}_g(\\mathbf{f})$. If $\\mathbf{f}$ is a face degree sequence with $f_j \\geq 1$, we write $\\mathbf{f}^-=\\mathbf{f}-\\mathbf{1}_j$. We can describe the law of $M_{\\mathbf{f}^-, g}$ in terms of the law of $M_{\\mathbf{f}, g}$. Indeed, let $m^0_j$ be the map of Figure~\\ref{fig_m_0_j} with a hole of perimeter $2$ and only one internal face which has degree $2j$. Then $M_{\\mathbf{f}^-, g}$ has the law of $M_{\\mathbf{f}, g} \\backslash m^0_j$ conditioned on $m^0_j \\subset M_{\\mathbf{f}, g}$, where the two boundary edges have been glued together.\n\nTherefore, if we know for some suitable face degree sequences $\\mathbf{f}^k$ that $M_{\\mathbf{f}^k, g_k}$ converges to an infinite map of the form $\\mathbb M_{\\mathbf{Q}}$, we can deduce that $M_{\\mathbf{f}^{k,-}, g_k}$ converges to a map $\\mathbb M_{\\mathbf{Q}^-}$ and express $\\mathbf{Q}^-$ in terms of $\\mathbf{Q}$. A simple computation (Equation~\\ref{eqn_radon_nikodym_omega} below) shows that $\\mathbf{Q}^-$ has the law of $\\mathbf{Q}$ biased by $Q_j$. By Lemma~\\ref{lem_qj_is_monotone}, this means that if $\\mathbf{Q}$ is not deterministic, then $\\mathbf{Q}^-$ is \"strictly less hyperbolic\" than $\\mathbf{Q}$ (in the sense that $\\omega_{\\mathbf{Q}^-}<\\omega_{\\mathbf{Q}}$). On the other hand, the maps $M_{\\mathbf{f}^k, g_k}$ and $M_{\\mathbf{f}^{k,-}, g_k}$ have the same genus but the second one is smaller, so it would be natural to expect it to be \"more hyperbolic\" than the first. We will derive a contradiction from this paradox by considering, for a fixed genus $g$, the smallest value $k_0(g)$ of $k$ such that the expected $\\omega$ corresponding to $M_{\\mathbf{f}^k, g_k}$ is smaller than a certain threshold.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=1]{small_piece}\n\\end{center}\n\\caption{The map $m_j^0$.}\\label{fig_m_0_j}\n\\end{figure}\n\nTo make this sketch precise, we need to define precisely several objects. We will first build the face degree sequences $\\mathbf{f}^k$, then explain how to approximate $\\omega_{\\mathbf{Q}}$ on finite maps, and finally give the precise definition of $k_0(g)$ (which will actually depend on an additional parameter $t$).\n\n\\paragraph{The face degree sequences $\\mathbf{F}^k$.}\nSince we want our face degree sequences to respect the proportions $(\\alpha_j)$, it makes sense to build the face degree sequences $\\mathbf{f}^k$ randomly, by adding at each step a face of degree $2j$ with probability proportional to $\\alpha_j$. This roughly means that we take $j$ random in the sketch above.\nMore precisely, we build our family $(\\mathbf{F}^k)_{k \\geq 0}$ of face degree sequences as follows: let $(J_k)_{k \\geq 1}$ be i.i.d. random variables on $\\mathbb N^*$ with\n\\[ \\P \\left( J_k=j \\right)=\\frac{\\alpha_j}{\\sum_{i \\geq 1} \\alpha_i}. \\] \nWe write $F^k_j=\\sum_{i=1}^k \\mathbbm{1}_{J_i=j}$, so that $\\mathbf{F}^k$ is a random face degree sequence with $k$ faces in total.\n\n\\paragraph{Approximating $\\omega$ on finite maps.}\nIn order to define $k_0(g)$, we would like to build variables $\\Omega_{k,g}$ such that if $M_{\\mathbf{F}^k,g}$ is close to $\\mathbb M_{\\mathbf{q}^{(\\Omega)}}$ for the local topology, then $\\Omega_{k,g}$ is close to $\\Omega$. For this, we will rely on Lemma~\\ref{lem_unif_volume}. We fix an arbitrary peeling algorithm $\\mathcal{A}$ throughout this section. We recall that Lemma~\\ref{lem_unif_volume} gives the convergence\n\\[ \\frac{V_t^{(\\omega)} - 2 P_t^{(\\omega)}}{t} \\xrightarrow[t \\to +\\infty]{(P)} r(\\omega),\\]\nwhere $V_t^{(\\omega)}=\\left| \\mathcal{E}^{\\mathcal{A}}_t \\left( \\mathbb M_{\\mathbf{q}^{(\\omega)}} \\right) \\right|$ and $P_t^{(\\omega)}=\\left| \\partial \\mathcal{E}^{\\mathcal{A}}_t \\left( \\mathbb M_{\\mathbf{q}^{(\\omega)}} \\right) \\right|$ are respectively the half-perimeter and the total number of edges of $\\mathcal{E}^{\\mathcal{A}}_t \\left( \\mathbb M_{\\mathbf{q}^{(\\omega)}} \\right)$, and $r(\\omega)=\\frac{ \\left(\\sqrt{\\omega}-\\sqrt{\\omega-1} \\right)^2}{2 \\sqrt{\\omega(\\omega-1)}}$ is a homeomorphism from $(1,+\\infty]$ to $[0,+\\infty)$. Therefore, for $k,g \\geq 0$ and $t \\geq 1$, we define\n\\[ \\Omega_{k,g}^t = r^{-1} \\left( \\frac{ \\left| \\mathcal{E}_t^{\\mathcal{A}} \\left( M_{\\mathbf{F}^k,g} \\right) \\right| - 2 \\left| \\partial \\mathcal{E}_t^{\\mathcal{A}} \\left( M_{\\mathbf{F}^k,g} \\right) \\right|}{t} \\right),\\]\nwith the convention $\\Omega_{k,g}^t=+\\infty$ if $M_{\\mathbf{F}^k,g}$ does not exist or if the peeling exploration of $M_{\\mathbf{F}^k,g}$ using $\\mathcal{A}$ is stopped before time $t$. Note that with this definition we always have $\\Omega^t_{k,g} \\in [1,+\\infty]$.\n\n\\paragraph{Defining $k_0^t(g)$.}\nWe now fix $1<\\omega_0<\\omega_1<+\\infty$. As explained above, for $g \\geq 0$ we want to consider the smallest $k$ for which the average value of $\\Omega^t_{k,g}$ is smaller than $\\omega_0$. Because we will need the uniform convergence of Lemma~\\ref{lem_unif_volume} which holds only for $\\omega$ bounded away from $1$ and $+\\infty$, we first exclude artificially the values of $k$ that are too small or too large. More precisely, let $\\varepsilon>0$ be small enough to satisfy\n\\begin{equation}\\label{eqn_choice_eps}\n\\varepsilon < \\min_{\\omega \\geq \\min(\\omega_0, \\omega_1)} \\left( d(\\mathbf{q}^{(1)}) - d(\\mathbf{q}^{(\\omega)}) \\right) \\quad \\mbox{and} \\quad \\varepsilon < \\min_{1 \\leq \\omega \\leq \\max(\\omega_0, \\omega_1)} d(\\mathbf{q}^{(\\omega)}).\n\\end{equation}\nNote that the existence of such an $\\varepsilon$ is guaranteed by the second and third items of Lemma~\\ref{lem_degreefunction_basic}. For $g \\geq 0$, we write\n\\[ k_{\\min}(g) = \\frac{2 \\sum_{j \\geq 1} \\alpha_j}{1-\\varepsilon-\\sum_{j \\geq 1} \\alpha_j} g \\quad \\mbox{and} \\quad k_{\\max}(g)=\\frac{1}{\\varepsilon} \\left( \\sum_{j \\geq 1} \\alpha_j \\right) g.\\]\nSince $\\varepsilon$ will be fixed until the end, we omit the dependence in $\\varepsilon$ in the notation.\nThese values were chosen so that a map with face degrees $\\mathbf{F}^{k_{\\min}(g)}$ and genus $g$ has average degree of order $\\frac{1}{\\varepsilon}$ (in particular such a map exists), whereas in a map with face degrees $\\mathbf{F}^{k_{\\max}(g)}$ and genus $g$ the genus is about $\\varepsilon$ times the size.\n\nWe now set, for $t \\geq 1$ and $g \\geq 0$:\n\\begin{align}\\label{eqn_defn_k0tg}\nk_0^t(g) &= \\min \\left\\{ k \\in \\left[ k_{\\min}(g), k_{\\max}(g) \\right] \\big| \\mathbb E \\left[ \\left( \\Omega^t_{k,g} \\right)^{-1} \\right] \\geq \\omega_0^{-1} \\right\\},\\\\\nk_1^t(g) &= \\min \\left\\{ k \\in \\left[ k_{\\min}(g), k_{\\max}(g) \\right] \\big| \\mathbb E \\left[ \\left( \\Omega^t_{k,g} \\right)^{-1} \\right] \\geq \\omega_1^{-1} \\right\\}.\\nonumber\n\\end{align}\nThe only reason why we use $\\omega_0^{-1}$ instead of $\\omega_0$ in the definition is to have a bounded quantity in the expectation, and pass easily from convergences in distribution to convergences of the expectation later. We first prove that this definition indeed makes sense and that the map $M_{\\mathbf{F}^{k_0^t(g)}, g}$ is well-defined with high probability.\n\n\\begin{lem}\\label{lem_k0_tg_reasonable}\nThere is $t_0$ depending only on $\\omega_0, \\omega_1$ such that for all $t \\geq t_0$:\n\\begin{enumerate}\n\\item\nfor $g$ large enough the number $k_0^t(g)$ is well defined;\n\\item\nfor $g$ large enough we have $k_0^t(g)>k_{\\min}(g)$;\n\\item\nwith probability $1-o(1)$ as $g \\to +\\infty$, we have $v \\left( \\mathbf{F}^{k_0^t(g)}, g \\right) \\geq \\frac{\\varepsilon}{2} \\left| \\mathbf{F}^{k_0^t(g)}\\right|$, and the same is true with $k_0^t(g)-1$ instead of $k_0^t(g)$.\n\\end{enumerate}\nMoreover, the same is true if we replace $k_0^t(g)$ with $k_1^t(g)$.\n\\end{lem}\n\n\\begin{proof}\nWe first recall that\n\\[v \\left( \\mathbf{F}^k, g \\right) = 2-2g+\\sum_{j \\geq 1} (j-1) F^k_j.\\]\nMoreover, by the law of large numbers and the definition of $\\mathbf{F}^k$, we have\n\\[\\frac{1}{k} \\sum_{j \\geq 1} (j-1) F^k_j \\xrightarrow[k \\to +\\infty]{a.s.} \\frac{1}{\\sum_{i \\geq 1} \\alpha_i} \\sum_{j \\geq 1} (j-1) \\alpha_j \\quad \\mbox{and} \\quad \\frac{1}{k} \\left| \\mathbf{F}^k \\right| \\xrightarrow[k \\to +\\infty]{a.s.} \\frac{1}{\\sum_{i \\geq 1} \\alpha_i}.  \\]\nFrom here, by the choice of $k_{\\min}(g)$, it follows easily that almost surely, for $g$ large enough, we have $v \\left( \\mathbf{F}^k, g \\right) \\geq \\frac{\\varepsilon}{2} \\left| \\mathbf{F}^k \\right|$ for all $k \\geq k_{\\min}(g)$. In particular, the third item of the Lemma will follow from the first and the second.\n\nWe now prove the first item. We need to prove that the minimum in the definition of $k_0^t(g)$ is over a nonempty set, so it is enough to prove that, if $t$ is larger than some $t_0$, we have\n\\begin{equation}\\label{eqn_omega_kmax}\n\\mathbb E \\left[ \\left( \\Omega^t_{k_{\\max}(g), g} \\right)^{-1} \\right]> \\omega_0^{-1}\n\\end{equation}\nfor $g$ large enough. We know that $v \\left( \\mathbf{F}^k, g \\right) \\geq \\frac{\\varepsilon}{2} \\left| \\mathbf{F}^k \\right|$ for $g$ large enough, so by Theorem~\\ref{thm_main_more_general} and Corollary~\\ref{corr_main_minus_monotonicity}, up to extracting a subsequence in $g$, we have the local convergence\n\\begin{equation}\\label{eqn_subseqlimit_kmax}\nM_{\\mathbf{F}^{k_{\\max}(g)}, g} \\xrightarrow[g \\to +\\infty]{(d)} \\mathbb M_{q^{(\\Omega_{\\max})}},\n\\end{equation}\nwhere $\\Omega_{\\max}$ is a random variable on $(1,+\\infty)$. On the other hand, by the law of large numbers and the definition of $k_{\\max}(g)$, we have\n\\[ \\frac{1}{g} v \\left( \\mathbf{F}^{k_{\\max}(g)}, g \\right) \\xrightarrow[g \\to +\\infty]{a.s.} -2+ \\frac{1}{\\varepsilon} \\sum_{j \\geq 1} (j-1)\\alpha_j \\quad \\mbox{and} \\quad \\frac{1}{g}\\left| \\mathbf{F}^{k_{\\max}(g)}\\right| \\xrightarrow[g \\to +\\infty]{a.s.} \\frac{1}{\\varepsilon}.\\]\nTherefore, the average degree in $M_{\\mathbf{F}^{k_{\\max}(g)}, g}$ tends to $-\\varepsilon+\\frac{1}{2} \\left( 1-\\sum_{j} \\alpha_j \\right) = d \\left( \\mathbf{q}^{(1)}\\right)-\\varepsilon$. By Corollary~\\ref{corr_main_minus_monotonicity}, we must have\n\\begin{equation}\\label{eqn_degree_omegamax}\nd \\left( \\mathbf{q}^{(\\Omega_{\\max})}\\right) = d \\left( \\mathbf{q}^{(1)}\\right)-\\epsilon\n\\end{equation}\nalmost surely. By the first inequality in our choice~\\eqref{eqn_choice_eps} of $\\varepsilon$, this implies $\\Omega_{\\max}<\\omega_0$ a.s., so $\\mathbb E \\left[ \\Omega_{\\max}^{-1} \\right] > \\omega_0^{-1}$.\n\nOn the other hand, let $t \\geq 1$. Since the explored map at time $t$ of a peeling exploration is a local function, the convergence~\\eqref{eqn_subseqlimit_kmax} implies\n\\begin{equation}\\label{eqn_convergence_omegamax_t}\n\\Omega^t_{k_{\\max}(g), g} \\xrightarrow[g \\to +\\infty]{(d)} r^{-1} \\left( \\frac{V_t^{(\\Omega_{\\max})} - 2 P_t^{(\\Omega_{\\max})}}{t} \\right).\n\\end{equation}\nMoreover, by the third item and the continuity in Lemma~\\ref{lem_degreefunction_basic}, we also get that $\\Omega_{\\max}$ is supported on a compact subset of $(1,+\\infty)$ depending only on $\\varepsilon$ (and not on the subsequence in $g$ that we are working with). Therefore, by the uniform convergence result of Lemma~\\ref{lem_unif_volume}, the right-hand side of~\\eqref{eqn_convergence_omegamax_t} converges in probability to $\\Omega_{\\max}$ as $t \\to +\\infty$, at a speed independent from the subsequence. Hence, remembering $\\mathbb E \\left[ \\Omega_{\\max}^{-1} \\right] > \\omega_0^{-1}$, there is $t_0$ depending only on $\\varepsilon$ such that for $t \\geq t_0$, we have\n\\[ \\mathbb E \\left[ r^{-1} \\left( \\frac{V_t^{(\\Omega_{\\max})} - 2 P_t^{(\\Omega_{\\max})}}{t} \\right)^{-1} \\right] > \\omega_0^{-1}. \\]\nCombined with~\\eqref{eqn_convergence_omegamax_t}, this implies $\\mathbb E \\left[ \\left( \\Omega^t_{k_{\\max}(g), g} \\right)^{-1} \\right] > \\omega_0^{-1}$ for $g$ large enough. We have proved that for $t \\geq t_0$, every subsequence in $g$ contains a subsubsequence along which~\\eqref{eqn_omega_kmax} holds for $g$ large enough, so~\\eqref{eqn_omega_kmax} holds for $g$ large enough. This proves the first item of the lemma.\n\nTo prove the second item, we need to show that there is $t'_0$ such that if $t \\geq t'_0$, then\n\\[ \\mathbb E \\left[ \\left( \\Omega^t_{k_{\\min}(g), g} \\right)^{-1} \\right] < \\omega_0^{-1} \\]\nfor $g$ large enough.\nThe proof is very similar to the proof of~\\eqref{eqn_omega_kmax}, so we do not write it in full details. The only difference is the computation of the average degree: this time, if $\\Omega_{\\min}$ plays the role of $\\Omega_{\\max}$ for the first item, using the definition of $k_{\\min}(g)$ we get $d \\left( q^{(\\Omega_{\\min})} \\right)=\\frac{\\varepsilon}{2}$ a.s.. Hence, by the second inequality of~\\eqref{eqn_choice_eps}, this implies $\\Omega_{\\min} > \\omega_0$, and the end of the proof is the same.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop_monotonicity_deg}]\nLet $t$ be larger than the $t_0$ from Lemma~\\ref{lem_k0_tg_reasonable}. By the third item of Lemma~\\ref{lem_k0_tg_reasonable} and Theorem~\\ref{thm_main_more_general}, we know that $\\left( M_{\\mathbf{F}^{k_0^t(g)},g} \\right)_{g \\geq 0}$ is tight and any subsequential limit is of the form $\\mathbb M_{\\mathbf{Q}}$. Moreover by the law of large numbers, for all $j \\geq 1$ we have $\\frac{F^k_j}{\\left| \\mathbf{F}^k_j \\right|} \\to \\alpha_j$ a.s. when $k \\to +\\infty$. Therefore, by  Corollary~\\ref{corr_main_minus_monotonicity}, any subsequential limit $\\mathbb M_{\\mathbf{Q}}$ must satisfy $a_j(\\mathbf{q})=\\alpha_j$ for all $j \\geq 1$, so $\\mathbf{Q}$ is of the form $\\mathbf{q}^{(\\Omega)}$ for some random $\\Omega$. Moreover, the same holds if $k_0^t(g)$ is replaced by $k_0^t(g)-1$, or $k_1^t(g)$, or $k_1^t(g)-1$. Therefore, for all $t \\geq t_0$, we can fix a subsequence $S^t$ (depending on $t$) such that when $g \\to +\\infty$ along $S^t$ the following convergences hold jointly:\n\\begin{align*}\nM_{\\mathbf{F}^{k_0^t(g)},g} & \\xrightarrow[g \\to +\\infty]{(d)} \\mathbb M_{\\mathbf{q}^{(\\Omega_0^t)}}, & M_{\\mathbf{F}^{k_1^t(g)},g} & \\xrightarrow[g \\to +\\infty]{(d)} \\mathbb M_{\\mathbf{q}^{(\\Omega_1^t)}},\\\\\nM_{\\mathbf{F}^{k_0^t(g)-1},g} & \\xrightarrow[g \\to +\\infty]{(d)} \\mathbb M_{\\mathbf{q}^{(\\Omega_0^{t,-})}}, & M_{\\mathbf{F}^{k_1^t(g)-1},g} & \\xrightarrow[g \\to +\\infty]{(d)} \\mathbb M_{\\mathbf{q}^{(\\Omega_1^{t,-})}},\\\\\n\\Omega^t_{k_0^t(g),g} & \\xrightarrow[g \\to +\\infty]{(d)} \\widetilde{\\Omega}^t_0, & \\Omega^t_{k_1^t(g),g} & \\xrightarrow[g \\to +\\infty]{(d)} \\widetilde{\\Omega}^t_1,\\\\\n\\Omega^t_{k_0^t(g)-1} & \\xrightarrow[g \\to +\\infty]{(d)} \\widetilde{\\Omega}^{t,-}_0, & \\Omega^t_{k_1^t(g)-1} & \\xrightarrow[g \\to +\\infty]{(d)} \\widetilde{\\Omega}^{t,-}_1,\\\\\n\\frac{g}{\\left| \\mathbf{F}^{k_0^t(g)} \\right|} & \\xrightarrow[g \\to +\\infty]{(d)} \\theta_0^t, & \\frac{g}{\\left| \\mathbf{F}^{k_1^t(g)} \\right|} & \\xrightarrow[g \\to +\\infty]{(d)} \\theta_1^t,\n\\end{align*}\nwhere the first four convergences are for the local topology. We highlight that for our purpose it is sufficient to consider \\emph{one} subsequence $S^t$, and that we will not try to understand \\emph{all} subsequential limits. In what follows, we will always consider $g$ in the subsequence $S^t$ and omit to precise it. Note that we have $\\Omega_0^t, \\Omega_0^{t,-} \\in [1,+\\infty)$ and $\\widetilde{\\Omega}_0^t, \\widetilde{\\Omega}_0^{t,-} \\in [1,+\\infty]$, since we have no bound a priori on $\\Omega^t_{k_0^t(g),g}$.\nWe also have $\\theta_0^t \\in \\left[ \\varepsilon, \\frac{1}{2} \\left( 1-\\varepsilon-\\sum_j \\alpha_j \\right) \\right]$ by the inequality $k_{\\min}(g) \\leq k_0^t(g) \\leq k_{\\max}(g)$ and the law of large numbers to estimate $\\mathbf{F}^{k_{\\min}(g)}$ and $\\mathbf{F}^{k_{\\max}(g)}$. By Corollary~\\ref{corr_main_minus_monotonicity}, we also know that almost surely\n\\begin{equation}\\label{eqn_omega0_in_compact}\nd \\left( \\mathbf{q}^{(\\Omega_0^t)} \\right) = d \\left( \\mathbf{q}^{(\\Omega_0^{t,-})} \\right) = \\frac{1}{2} \\left( 1-2\\theta_0^t-\\sum_{j \\geq 1} \\alpha_j \\right).\n\\end{equation}\nIn particular $d \\left( \\mathbf{q}^{(\\Omega_0^t)} \\right)$ and $d \\left( \\mathbf{q}^{(\\Omega_0^{t,-})} \\right)$ are bounded away from $0$ and $d(\\mathbf{q}^{(1)})$. By Lemma~\\ref{lem_degreefunction_basic}, this implies that $\\Omega_0^t$ and $\\Omega_0^{t,-}$ take their values in a compact subset of $(1,+\\infty)$ that depends only on $\\varepsilon$. Therefore, there is a subsequence $S$ such that, when $t \\to +\\infty$ along $S$ the following convergences hold jointly: \n\\begin{align*}\n\\Omega_0^t & \\xrightarrow[t \\to +\\infty]{(d)}  \\Omega_0, &  \\Omega_1^t & \\xrightarrow[t \\to +\\infty]{(d)}  \\Omega_1,\\\\\n\\Omega_0^{t,-} & \\xrightarrow[t \\to +\\infty]{(d)}  \\Omega_0^{-} , &  \\Omega_1^{t,-} & \\xrightarrow[t \\to +\\infty]{(d)}  \\Omega_1^{-},\\\\\n\\widetilde{\\Omega}_0^t & \\xrightarrow[t \\to +\\infty]{(d)}  \\widetilde{\\Omega}_0, &  \\widetilde{\\Omega}_1^t & \\xrightarrow[t \\to +\\infty]{(d)}  \\widetilde{\\Omega}_1,\\\\\n\\widetilde{\\Omega}_0^{t,-} & \\xrightarrow[t \\to +\\infty]{(d)}  \\widetilde{\\Omega}_0^{-} , &  \\widetilde{\\Omega}_1^{t,-} & \\xrightarrow[t \\to +\\infty]{(d)}  \\widetilde{\\Omega}_1^{-},\\\\\n\\theta_0^t & \\xrightarrow[t \\to +\\infty]{} \\theta_0, & \\theta_1^t & \\xrightarrow[t \\to +\\infty]{} \\theta_1,\n\\end{align*}\nwhere $\\Omega_0, \\Omega_0^- \\in (1,+\\infty)$ and $\\widetilde{\\Omega}_0, \\widetilde{\\Omega}_0^- \\in [1,+\\infty]$ and $\\theta_0 \\in \\left[ \\varepsilon, \\frac{1}{2} \\left( 1-\\varepsilon-\\sum_j \\alpha_j \\right) \\right]$. From now on, we will always consider $t \\geq t_0$ with $t$ in this subsequence $S$, and we will omit to precise it. \n\nThe rough sketch of the argument is to show that\n\\begin{equation}\\label{eqn_cyclic_ineq_omega}\n\\mathbb E \\left[ \\left( \\widetilde{\\Omega}^{-}_0 \\right)^{-1} \\right] \\leq \\omega_0^{-1} \\leq \\mathbb E \\left[ \\left( \\widetilde{\\Omega}_0 \\right)^{-1} \\right] = \\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right] \\leq \\mathbb E \\left[ \\left( \\Omega_0^{-} \\right)^{-1} \\right] = \\mathbb E \\left[ \\left( \\widetilde{\\Omega}^{-}_0 \\right)^{-1} \\right],\n\\end{equation}\nand deduce from the equality in the third inequality that $\\Omega_0$ is deterministic and therefore equal to $\\omega_0$.\n\nMore precisely, by definition of $k_0^t(g)$ and since $k_0^t(g)>k_{\\min}(g)$, we have\n\\[ \\mathbb E \\left[ \\left( \\Omega^t_{k_0^t(g)-1} \\right)^{-1} \\right] < \\omega_0^{-1} \\leq  \\mathbb E \\left[ \\left( \\Omega^t_{k_0^t(g)} \\right)^{-1} \\right].\\]\nBy letting $g \\to +\\infty$ (along $S^t$) and then $t \\to +\\infty$ (along $S$), this implies\n\\begin{equation}\\label{eqn_ineq_omegatilde}\n\\mathbb E \\left[ \\left( \\widetilde{\\Omega}^{-}_0 \\right)^{-1} \\right] \\leq \\omega_0^{-1} \\leq  \\mathbb E \\left[ \\left( \\widetilde{\\Omega}_0 \\right)^{-1} \\right],\n\\end{equation}\nwhich are the first and second inequalities in~\\eqref{eqn_cyclic_ineq_omega}.\n\nOn the other hand, the third inequality will be obtained by the argument sketched in the beginning of this Subsection~\\ref{subsubsec_final_argument}. Let us first look at the relation between $M_{\\mathbf{F}^{k}, g}$ and $M_{\\mathbf{F}^{k-1}, g}$.\nWe recall that $\\mathbf{F}^{k}=\\mathbf{F}^{k-1}+\\mathbf{1}_{J_k}$, where $\\P (J_k=j)=\\frac{\\alpha_j}{\\sum_{i \\geq 1} \\alpha_i}$ for all $j \\geq 1$. If we condition on $\\mathbf{F}^{k-1}$ and $\\mathbf{F}^{k}$, then the law of $M_{\\mathbf{F}^{k-1}, g}$ is the law of $M_{\\mathbf{F}^{k}, g} \\backslash m_{J_k}^0$, conditioned on $m_{J_k}^0 \\subset M_{\\mathbf{F}^{k}, g}$, where $m^0_{J_k}$ is the map with perimeter $2$ of Figure~\\ref{fig_m_0_j}. Therefore, for any map $m$ with one hole, we have\n\\[ \\P \\left( m \\subset M_{\\mathbf{F}^{k-1}, g} | J_k=j \\right) = \\P \\left( m+m_j^0 \\subset M_{\\mathbf{F}^{k}, g} | J_k=j, \\, m_j^0 \\subset  M_{\\mathbf{F}^{k}, g} \\right), \\]\nwhere $m+m_{j}^0$ is the map obtained from $m$ by replacing the root edge of $m$ by a copy of $m_j^0$. By summing over $j$, we obtain\n\\[\n\\P \\left( m \\subset M_{\\mathbf{F}^{k-1}, g} \\right) = \\frac{1}{\\sum_{i \\geq 1} \\alpha_i} \\sum_{j \\geq 1} \\alpha_j \\frac{\\P \\left( m+m_j^0 \\subset M_{\\mathbf{F}^{k}, g} \\right)}{\\P \\left( m_j^0 \\subset M_{\\mathbf{F}^{k}, g} \\right)}.\n\\]\nWe now take $k=k_0^t(g)$ and let $g \\to +\\infty$ (along $S^t$) to replace $M_{\\mathbf{F}^{k-1}, g}$ and $M_{\\mathbf{F}^{k}, g}$ by respectively $\\mathbb M_{\\mathbf{q}^{(\\Omega_0^{t,-})}}$ and $\\mathbb M_{\\mathbf{q}^{(\\Omega_0^{t})}}$. We note that $m_j^0+m$ has the same perimeter as $m$ but one more internal face of degree $2j$. We obtain, for every finite map $m$ with one hole,\n\\[ \\mathbb E \\left[ C_{|\\partial m|} \\left( \\mathbf{q}^{(\\Omega_0^{t,-})} \\right) \\prod_{f \\in m} q^{(\\Omega_0^{t,-})}_{\\mathrm{deg}(f)\/2} \\right] = \\frac{1}{\\sum_{i \\geq 1} \\alpha_i} \\sum_{j \\geq 1} \\alpha_j \\frac{\\mathbb E \\left[ C_{\\partial m} \\left( \\mathbf{q}^{(\\Omega_0^{t})} \\right) \\times \\prod_{f \\in m} q^{(\\Omega_0^{t})}_{\\mathrm{deg}(f)\/2} \\times q^{(\\Omega_0^{t})}_j \\right]}{ \\mathbb E \\left[ C_1 \\left( \\mathbf{q}^{(\\Omega_0^{t})} \\right) q_j^{(\\Omega_0^{t})} \\right]}. \\]\nThis can be interpreted as a Radon--Nikodym derivative, i.e. the map $\\mathbb M_{\\mathbf{q}^{(\\Omega_0^{t,-})}}$ has the law of $\\mathbb M_{\\mathbf{q}^{(\\Omega_0^{t})}}$ biased by\n\\begin{equation}\\label{eqn_radon_nikodym_omega}\n\\frac{1}{\\sum_{i \\geq 1} \\alpha_i} \\sum_{j \\geq 1} \\alpha_j \\frac{q_j^{(\\Omega_0^{t})}}{\\mathbb E \\left[ q_j^{(\\Omega_0^{t})} \\right]},\n\\end{equation}\nusing the fact that $C_1(\\mathbf{q})=1$. Since $\\Omega$ is a measurable function of the map $\\mathbb M_{\\mathbf{q}^{(\\Omega)}}$ by Proposition~\\ref{prop_q_as_limit}, it follows that $\\Omega_0^{t,-}$ has the law of $\\Omega_0^t$ biased by \\eqref{eqn_radon_nikodym_omega}. In particular, we have\n\\[\n\\mathbb E \\left[ \\left( \\Omega_0^{t,-} \\right)^{-1} \\right] = \\frac{1}{\\sum_{i \\geq 1} \\alpha_i} \\sum_{j \\geq 1} \\alpha_j \\frac{\\mathbb E \\left[ q_j^{(\\Omega_0^t)} \\, \\left( \\Omega_0^t \\right)^{-1} \\right]}{\\mathbb E \\left[ q_j^{(\\Omega_0^t)}\\right]}.\n\\]\nWe can now let $t \\to +\\infty$ (along $S$) to get\n\\begin{equation}\\label{eqn_expectation_omega_inverse}\n\\mathbb E \\left[ \\left( \\Omega_0^{-} \\right)^{-1} \\right] = \\frac{1}{\\sum_{i \\geq 1} \\alpha_i} \\sum_{j \\geq 1} \\alpha_j \\frac{\\mathbb E \\left[ q_j^{(\\Omega_0)} \\, \\left( \\Omega_0 \\right)^{-1} \\right]}{\\mathbb E \\left[ q_j^{(\\Omega_0)}\\right]}.\n\\end{equation}\nBut by Lemma~\\ref{lem_qj_is_monotone}, we have $\\mathbb E \\left[ q_j^{(\\Omega_0)} \\, \\left( \\Omega_0 \\right)^{-1} \\right] \\geq \\mathbb E \\left[ q_j^{(\\Omega_0)} \\right] \\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right]$ for all $j \\geq 1$, so the last display implies $\\mathbb E \\left[ \\left( \\Omega_0^{-} \\right)^{-1} \\right] \\geq \\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right]$, which is the third inequality of the sketch~\\eqref{eqn_cyclic_ineq_omega}.\n\nWe now move on to the two equalities of~\\eqref{eqn_cyclic_ineq_omega}. For this, we need to argue that $\\widetilde{\\Omega}_0^t$ is a good approximation of $\\Omega_0^t$ for $t$ large. By definition of $\\widetilde{\\Omega}_0^t$ and by local convergence, we have (the limits in $g$ are along $S^t$)\n\\begin{align}\\label{eqn_omegatilde_t_as_limit}\n\\mathbb E \\left[ \\left( \\widetilde{\\Omega}_0^t \\right)^{-1} \\right] &= \\lim_{g \\to +\\infty} \\mathbb E \\left[ \\left( \\Omega_{k_0^t(g),g}^t \\right)^{-1} \\right] \\nonumber \\\\\n&= \\lim_{g \\to +\\infty} \\mathbb E \\left[ \\left( r^{-1} \\left( \\frac{ \\left| \\mathcal{E}_t^{\\mathcal{A}} \\left( M_{\\mathbf{F}^{k_0^t(g)},g} \\right) \\right| - 2 \\left| \\partial \\mathcal{E}_t^{\\mathcal{A}} \\left( M_{\\mathbf{F}^{k_0^t(g)},g} \\right) \\right|}{t} \\right) \\right)^{-1} \\right] \\nonumber\n\\\\\n&= \\mathbb E \\left[ \\left( r^{-1} \\left( \\frac{V_t^{(\\Omega_0^t)}-2 P_t^{(\\Omega_0^t)}}{t} \\right) \\right)^{-1} \\right].\n\\end{align}\nWhen $t \\to +\\infty$ (along $S$), the left-hand side of~\\eqref{eqn_omegatilde_t_as_limit} goes to $\\mathbb E \\left[ \\left( \\widetilde{\\Omega}_0 \\right)^{-1} \\right]$. On the other hand, we recall that~\\eqref{eqn_omega0_in_compact} implies that $\\Omega_0^t$ lies in a compact subset of $(1,+\\infty)$ depending only on $\\varepsilon$. Since $\\Omega_0^t \\to \\Omega_0$ along $S$, by Lemma~\\ref{lem_unif_volume} we have the convergence (along $S$)\n\\[ \\frac{V_t^{(\\Omega_0^t)} -2P_t^{(\\Omega_0^t)}}{t} \\xrightarrow[t \\to +\\infty]{(P)} r(\\Omega_0). \\]\nTherefore, when $t \\to +\\infty$ (along $S$), the right-hand side of~\\eqref{eqn_omegatilde_t_as_limit} goes to $\\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right]$. This proves the first equality of~\\eqref{eqn_cyclic_ineq_omega}. The second one is proved in the exact same way, using $\\Omega^{t,-}_0$, $\\widetilde{\\Omega}^{t,-}_0$ instead of $\\Omega^{t}_0$, $\\widetilde{\\Omega}^{t}_0$.\n\nWe have therefore proved all of~\\eqref{eqn_cyclic_ineq_omega}, so all the inequalities must be equalities. In particular, \\eqref{eqn_expectation_omega_inverse} becomes\n\\[ \\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right] = \\frac{1}{\\sum_{i \\geq 1} \\alpha_i} \\sum_{j \\geq 1} \\alpha_j \\frac{\\mathbb E \\left[ q_j^{(\\Omega_0)} \\, \\left( \\Omega_0 \\right)^{-1} \\right]}{\\mathbb E \\left[ q_j^{(\\Omega_0)}\\right]}. \\]\nHowever, we also know by Lemma~\\ref{lem_qj_is_monotone} that $\\mathbb E \\left[ q_j^{(\\Omega_0)} \\, \\left( \\Omega_0 \\right)^{-1} \\right] \\geq \\mathbb E \\left[ q_j^{(\\Omega_0)} \\right] \\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right]$ for all $j$, so for all $j \\geq 1$ we must have the equality\n\\[ \\alpha_j \\mathbb E \\left[ q_j^{(\\Omega_0)} \\left( \\Omega_0 \\right)^{-1} \\right] = \\alpha_j \\mathbb E \\left[ q_j^{(\\Omega_0)} \\right] \\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right].\\]\nIn particular, we fix $j \\geq 2$ such that $\\alpha_j>0$ (such a $j$ exists because $\\alpha_1<1$). Then $\\mathbb E \\left[ q_j^{(\\Omega_0)} \\left( \\Omega_0 \\right)^{-1} \\right] = \\mathbb E \\left[ q_j^{(\\Omega_0)} \\right] \\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right]$. Since $\\omega \\to \\omega^{-1}$ and $\\omega \\to q_j^{(\\omega)}$ are two decreasing functions (by Lemma~\\ref{lem_qj_is_monotone}), this is only possible if $\\Omega_0$ is deterministic. But then~\\eqref{eqn_cyclic_ineq_omega} yields $\\mathbb E \\left[ \\left( \\Omega_0 \\right)^{-1} \\right] = \\omega_0^{-1}$, so $\\Omega_0=\\omega_0$ a.s..\n\nWe can now finish the proof. By the exact same argument as for $\\Omega_0$, we also have $\\Omega_1=\\omega_1$ a.s.. We recall that $\\omega_0<\\omega_1$. By letting $t \\to +\\infty$ (along $S$) in~\\eqref{eqn_omega0_in_compact} and using the continuity result of Lemma~\\ref{lem_degreefunction_basic}, we get\n\\begin{equation}\\label{eqn_fin_omega0_theta0}\nd \\left( \\mathbf{q}^{(\\omega_0)} \\right) = \\frac{1}{2} \\left( 1-2\\theta_0-\\sum_{j \\geq 1} \\alpha_j \\right)\n\\end{equation}\nand similarly\n\\begin{equation}\\label{eqn_fin_omega1_theta1}\nd \\left( \\mathbf{q}^{(\\omega_1)} \\right) = \\frac{1}{2} \\left( 1-2\\theta_1-\\sum_{j \\geq 1} \\alpha_j \\right).\n\\end{equation}\nOn the other hand, by the definition~\\eqref{eqn_defn_k0tg} of $k_0^t(g)$ and $k_1^t(g)$, since $\\omega_0<\\omega_1$, we have\n\\[k_0^t(g) \\geq k_1^t(g)\\]\nfor all $t$ and $g$. Therefore, we have $\\left| \\mathbf{F}^{k_0^t(g)} \\right| \\geq \\left| \\mathbf{F}^{k_1^t(g)} \\right|$. By letting $g \\to +\\infty$ (along $S^t$) we deduce $\\theta_0^t \\leq \\theta_1^t$. Letting $t \\to +\\infty$ (along $S$) we get $\\theta_0 \\leq \\theta_1$. Combining this with~\\eqref{eqn_fin_omega0_theta0} and~\\eqref{eqn_fin_omega1_theta1}, this proves $d \\left( \\mathbf{q}^{(\\omega_0)} \\right) \\geq d \\left( \\mathbf{q}^{(\\omega_1)} \\right)$, so the function $\\omega \\to d \\left( \\mathbf{q}^{(\\omega)} \\right)$ is nonincreasing on $(1,+\\infty)$. Since it is nonconstant (for example by the second item of Lemma~\\ref{lem_degreefunction_basic}) and analytic (first item of Lemma~\\ref{lem_degreefunction_basic}), it is decreasing on $(1,+\\infty)$. Finally, we extend the result to $[1,+\\infty)$ by continuity at $1$ (first item of Lemma~\\ref{lem_degreefunction_basic}).\n\\end{proof}\n\n\\section{Asymptotic enumeration: convergence of the ratio}\\label{sec_univ_asymp}\n\n\\begin{proof}[Proof of Corollary \\ref{prop_cv_ratio}]\nFix $j \\geq 1$, and let $m_j^0$ be the map of Figure~\\ref{fig_m_0_j} with a hole of perimeter $2$. On the one hand, we have\n\\[\\P \\left( m_j^0 \\subset \\mathbb{M}_{\\mathbf{q}} \\right)=C_2(\\mathbf{q})q_j.\\]\nOn the other hand, \n\\[\\P \\left( m_j^0 \\subset M_{\\mathbf{f}^{n},g_n} \\right)=\\frac{\\beta_{g_n}(\\mathbf{f}^{n}-\\mathbf{1}_j)}{\\beta_{g_n}(\\mathbf{f}^{n})}.\\]\nThe last equality is proved by contracting $m_j^0$ in $M_{\\mathbf{f}^{n},g_n}$ into the root edge of a map with face degrees given by $\\mathbf{f}^{n}-\\mathbf{1}_j$. The corollary follows by letting $n \\to +\\infty$.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\setcounter{equation}{0}\n\\citeAPY{Berryman:1990:VCE} found that classical variational principles could be used to obtain information about the\nconductivity inside a body from electrical measurements on the exterior. In this paper our main focus is on using classical\nvariational principles and known bounds on the response of periodic composites to bound\nthe volume fraction of one phase in a two-phase body $\\Omega$ from measurements on the exterior of the body. Of course if one\nknows the mass densities of the two phases, the easiest way to do this is just to weigh the body. However this \nmay not always be practical, or the densities of the two phases may be very close. \n\nTwo types of boundary conditions are most natural: what we call special Dirichlet conditions where affine Dirichlet\nconditions are imposed on the boundary of $\\Omega$ (which would render the field inside $\\Omega$ uniform if the body were\nhomogeneous) or what we call special Neumann conditions where Neumann conditions are imposed that would render the field inside $\\Omega$\nuniform if the body were homogeneous. Bounds on the electrical and elastic response of the body to these special boundary conditions were obtained \nby Nemat-Nasser and Hori (\\citeyearNP{Nemat-Nasser:1993:MOP}, \\citeyearNP{Nemat-Nasser:1995:UBO}), and were extended to piezoelectricity\nby \\citeAPY{Hori:1998:UBE}. They called these bounds universal because they did not depend on any assumption about the microgeometry in the body.\nThey obtained both elementary universal bounds based on the classical variational principles, and reviewed below in section 3, and \nuniversal bounds based on the Hashin-Shtrikman variational principles (Hashin and Shtrikman \\citeyearNP{Hashin:1962:VAT}, \\citeyearNP{Hashin:1963:VAT}).\nThe latter bounds were obtained under the assumption that $\\Omega$ is either an ellipsoid or a parallelopiped, but we will see here that they\ncan easily be improved and generalized to bodies $\\Omega$ of arbitrary shape. The key is to consider an assemblage of copies of $\\Omega$ packed to\nfill all space, and then to use the bounds of \\citeAPY{Huet:1990:AVC} which relate the effective tensor of this composite to the\nresponses of $\\Omega$ under the special boundary conditions. Then existing bounds on the effective tensor [as surveyed in the books\nof \\citeAPY{Nemat-Nasser:1993:MOP}, \\citeAPY{Cherkaev:2000:VMS}, \\citeAPY{Allaire:2002:SOH}, \\citeAPY{Milton:2002:TC}, and \\citeAPY{Tartar:2009:GTH}]\ncan be directly applied to bound the responses of $\\Omega$ under special boundary conditions (see sections 5,6,7, and 8).\nSince these bounds involve the volume fractions of the phases\n(and the moduli of the phases), they can be used in an inverse fashion to bound the volume fraction. As shown by \\citeAPY{Kang:2011:SBV}\nthe volume fraction bounds thus obtained \nfor electrical conductivity generalize those obtained by Capdeboscq and Vogelius (\\citeyearNP{Capdeboscq:2003:OAE}, \\citeyearNP{Capdeboscq:2004:RSR})\nfor the important case when the volume fraction is asymptotically small.\n\n Given the close connection between bounds on effective tensors\nand bounds on the responses of $\\Omega$ under special boundary condition, a natural question to ask is whether methods that have been used\nto derive bounds on effective tensors could be directly used to derive bounds on the response of $\\Omega$ under more general boundary conditions.\nOne such method is the Hashin-Strikman (\\citeyearNP{Hashin:1962:VAT}, \\citeyearNP{Hashin:1963:VAT}) variational method and this lead\nNemat-Nasser and Hori to their bounds for ellipsoidal or parallelopipedic $\\Omega$. Another particularly successful method is the translation method \n(\\citeAY{Tartar:1979:ECH}; Lurie and Cherkaev \\citeyearNP{Lurie:1982:AEC}, \\citeyearNP{Lurie:1984:EEC}; \\citeAY{Murat:1985:CVH}; \\citeAY{Tartar:1985:EFC}; \\citeAY{Milton:1990:CSP})\nand indeed as shown in a companion paper (\\citeAY{Kang:2011:SBV}) this method yields upper and lower bounds on the volume fraction in a\ntwo-phase body with general boundary conditions for two-dimensional conductivity without making any assumption on the shape of $\\Omega$.\nFor special boundary conditions the bounds thus derived reduce to the ones derived here. \n\nWe also provide (in section 4) some new conductivity bounds which just involve the results of just one (flux, voltage) pair\nmeasured at the boundary of $\\Omega$, and which improve upon the elementary bounds of \\citeAPY{Nemat-Nasser:1993:MOP}.\nAgain these new bounds can be used in an inverse fashion to bound the volume fraction. Other volume fraction\nbounds using one measurement were derived by  \\citeAPY{Kang:1997:ICP}\n\\citeAPY{Ikehata:1998:SEI}, \\citeAPY{Alessandrini:1998:ICP}, \\citeAPY{Alessandrini:2000:OSE}, and \\citeAPY{Alessandrini:2002:DCE}.\nThese other bounds involve constants which are not easy to determine, making it difficult to make a general comparison with\nour new bounds. \n\nThe various bounds on the volume fraction we have derived are too numerous to summarize in this introduction. \nHowever we want to draw attention to the bounds \\eq{3.12} and \\eq{3.21}  which are the natural extension of the\nfamous \\citeAPY{Hashin:1962:VAT} conductivity bounds to this problem. Also of particular note is the bound \\eq{5.18ag},\nwhich is one natural generalization of the bulk modulus bounds of \\citeAPY{Hashin:1963:VAT} and \\citeAPY{Hill:1963:EPR},\nand implies that a bound on the volume fraction can be obtained by simply immersing the body in a water filled \ncylinder with a piston at one end and measuring the change in water pressure when the piston is displaced by a known\nsmall amount.\n\n\n\\section{The conductivity response tensors with special Dirichlet and special Neumann boundary conditions}\n\\setcounter{equation}{0}\n\n\nIn electrical impedance tomography in a body $\\Omega$ containing two isotropic components with (positive, scalar) \nconductivities $\\sigma_1$ and $\\sigma_2$ the potential $V$ satisfies \n\\begin{equation} \\nabla \\cdot\\sigma\\nabla V=0, \\quad {\\rm where}\\quad \\sigma({\\bf x})=\\chi({\\bf x})\\sigma_1+(1-\\chi({\\bf x}))\\sigma_2,\n\\eeq{1.1}\nand $\\chi({\\bf x})$ is the indicator function of component $1$, taking the value $1$ in component and $0$ in component\n$2$. Equivalently, in terms of the current field ${\\bf j}({\\bf x})$ and electric field ${\\bf e}({\\bf x})$ we have\n\\begin{equation} \\nabla \\cdot{\\bf j}=0,\\quad{\\bf j}=\\sigma{\\bf e},\\quad {\\bf e}=-\\nabla V. \\eeq{1.2} \nLet us assume the components have been labeled so that $\\sigma_1\\geq\\sigma_2$. We are given a set of Cauchy data, i.e. \nmeasurements of pairs $(V_0,q)$, where $V_0({\\bf x})$ and $q({\\bf x})$ are the boundary values of the voltage $V({\\bf x})$ and\nand flux $q({\\bf x})=-{\\bf n}\\cdot{\\bf j}({\\bf x})$ at the boundary $\\partial\\Omega$ of $\\Omega$, in which ${\\bf n}({\\bf x})$ is the outwards \nnormal to the boundary. From this boundary information we can immediately determine, using integration by parts, \nvolume averages such as\n\\begin{eqnarray} \\langle {\\bf e}\\cdot{\\bf j}\\rangle &= & \\frac{1}{|\\Omega|}\\int_{\\partial\\Omega}-V_0({\\bf j}\\cdot{\\bf n})= \\frac{1}{|\\Omega|}\\int_{\\partial\\Omega}V_0q, \\nonumber \\\\\n      \\langle {\\bf e}\\rangle & = & \\frac{1}{|\\Omega|}\\int_{\\partial\\Omega}-V_0{\\bf n}, \\nonumber \\\\\n      \\langle {\\bf j}\\rangle & = & \\frac{1}{|\\Omega|}\\int_{\\partial\\Omega}-{\\bf x} q,\n\\eeqa{1.3}\nwhere the angular brackets denote the volume average, i.e.\n\\begin{equation} \\langle g \\rangle=\\frac{1}{|\\Omega|}\\int_{\\Omega} g, \\eeq{1.4}\nfor any quantity $g({\\bf x})$. From such averages our objective is to bound the\nvolume fraction $f_1=\\langle \\chi\\rangle$ of component 1 (and hence also the volume fraction $f_2=1-f_1$ of component\n2).\n\nTo obtain good estimates of the volume fraction $f_1$ it makes physical sense to use\nmeasurements where the fields ${\\bf e}({\\bf x})$ and ${\\bf j}({\\bf x})$ probe well into the interior of $\\Omega$. In this\nconnection two sets of measurements are most natural. We could apply special Dirichlet boundary conditions\n\\begin{equation} V_0=-{\\bf e}_0\\cdot{\\bf x}, \\eeq{1.5}\nand measure ${\\bf j}_0=\\langle{\\bf j}\\rangle$. Here, according to \\eq{1.3}, ${\\bf e}_0$ equals $\\langle{\\bf e}\\rangle$. Since\n${\\bf j}_0$ is linearly related to ${\\bf e}_0$ we can write\n\\begin{equation} {\\bf j}_0=\\bfm\\sigma^D{\\bf e}_0, \\eeq{1.6}  \nwhich defines the conductivity tensor $\\bfm\\sigma^D$ ($D$ for Dirichlet). To determine $\\bfm\\sigma^D$ \nin dimension $d=2,3$ it of course suffices to measure ${\\bf j}_0$ for $d$ linearly independent values of ${\\bf e}_0$. \nAlternatively we could apply the special  Neumann boundary conditions \n\\begin{equation} q={\\bf j}_0\\cdot{\\bf n}, \\eeq{1.7}\nand measure ${\\bf e}_0=\\langle{\\bf e}\\rangle$. Again according to \\eq{1.3}, ${\\bf j}_0=\\langle{\\bf j}\\rangle$. The linear relation\nbetween ${\\bf e}_0$ and ${\\bf j}_0$,\n\\begin{equation} {\\bf e}_0=(\\bfm\\sigma^N)^{-1}{\\bf j}_0 \\eeq{1.8}\ndefines the resistivity tensor $(\\bfm\\sigma^N)^{-1}$ and hence the conductivity tensor $\\bfm\\sigma^N$ ($N$ for Neumann):\nwe will see later that $(\\bfm\\sigma^N)^{-1}$ is invertible. To determine $\\bfm\\sigma^N$ it suffices to measure ${\\bf e}_0$ \nfor $d$ linearly independent values of ${\\bf j}_0$.  With either of these two sorts of boundary conditions (but \nnot in general) \\citeAPY{Hill:1963:EPR} has shown that\n\\begin{equation} \\langle {\\bf e}\\cdot{\\bf j}\\rangle=\\langle{\\bf e}\\rangle\\cdot\\langle{\\bf j}\\rangle, \\eeq{1.9}\nas follows by substituting \\eq{1.5} or \\eq{1.7} in the first of equations \\eq{1.3}.\nUsing this relationship, and its obvious generalizations, it is easy to check that both $\\bfm\\sigma^D$ and \n$\\bfm\\sigma^N$ are self-adjoint. Thus if ${\\bf e}'({\\bf x})$ and ${\\bf j}'({\\bf x})$ denote the electric and current fields\nassociated with the boundary conditions \\eq{1.5}, with ${\\bf e}_0$ replaced by ${\\bf e}_0'$, while keeping\nthe same conductivity $\\sigma({\\bf x})$ then\n\\begin{equation} {\\bf e}_0'\\cdot\\bfm\\sigma^D{\\bf e}_0=\\langle{\\bf e}'\\cdot{\\bf j}\\rangle=\\langle{\\bf e}'\\sigma{\\bf e}\\rangle\n=\\langle{\\bf e}\\cdot{\\bf j}'\\rangle={\\bf e}_0\\cdot\\bfm\\sigma^D{\\bf e}_0',\n\\eeq{1.10}\nwhich implies $\\bfm\\sigma^D$ is self-adjoint. By similar argument $\\bfm\\sigma^N$ is self-adjoint.\n\n\n\n\\section{Known elementary bounds}\n\\setcounter{equation}{0}\n\nThis section reviews the elementary bounds on  $\\bfm\\sigma^D$ and $\\bfm\\sigma^N$ obtained by\n\\citeAPY{Nemat-Nasser:1993:MOP} and by Willis\nin a 1989 private communication to Nemat-Nasser and Hori. Their implications for bounding the volume fraction\nwill be studied. We will make use of two classical variational principles: the Dirichlet variational principle that\n\\begin{equation} \\min_{\\matrix{{\\underline{{\\bf e}}}({\\bf x})=-\\nabla{\\underline{V}}({\\bf x}) \n\\cr {\\underline{V}}({\\bf x})=V_0({\\bf x})~{\\rm on}~\\partial\\Omega}}\\int_{\\Omega}\\underline{{\\bf e}}\\cdot\\sigma\\underline{{\\bf e}}\n=\\int_{\\partial\\Omega}-V_0q,\n\\eeq{2.1}\nwhich is attained when $\\underline{V}({\\bf x})=V({\\bf x})$, and the Neumann variational principle that\n\\begin{equation} \\min_{\\matrix{{\\underline{{\\bf j}}}({\\bf x})\\cr \\nabla \\cdot{\\underline{{\\bf j}}}({\\bf x})=0 \\cr \n     {\\bf n}\\cdot{\\underline{{\\bf j}}}({\\bf x})=-q({\\bf x})~{\\rm on}~\\partial\\Omega}}\\int_{\\Omega}\\underline{{\\bf j}}\\cdot\\sigma^{-1}\\underline{{\\bf j}}\n=\\int_{\\partial\\Omega}V_0q,\n\\eeq{2.2}\nwhich is attained when $\\underline{j}({\\bf x})={\\bf j}({\\bf x})$. With the special Dirichlet boundary conditions\n\\eq{1.5} the Dirichlet variational principle implies \n\\begin{equation} \\min_{\\matrix{{\\underline{{\\bf e}}}({\\bf x})=-\\nabla{\\underline{V}}({\\bf x}) \n\\cr {\\underline{V}}({\\bf x})=-{\\bf e}_0\\cdot{\\bf x}~{\\rm on}~\\partial\\Omega}}\\langle\\underline{{\\bf e}}\\cdot\\sigma\\underline{{\\bf e}}\\rangle\n={\\bf e}_0\\cdot\\bfm\\sigma^D{\\bf e}_0.\n\\eeq{2.3}\nTaking a trial potential $\\underline{V}=-{\\bf e}_0\\cdot{\\bf x}$ produces the elementary upper bound on ${\\bf e}_0\\cdot\\bfm\\sigma^D{\\bf e}_0$\n\\begin{equation} {\\bf e}_0\\cdot\\bfm\\sigma^D{\\bf e}_0\\leq \\langle\\sigma\\rangle{\\bf e}_0\\cdot{\\bf e}_0,\n\\eeq{2.4}\ngiven by \\citeAPY{Nemat-Nasser:1993:MOP}.\nTo obtain a lower bound observe, following a standard argument, that the left hand side of \\eq{2.3} is surely\ndecreased if we take the minimum over a larger class of trial fields. Since the constraints on \n${\\underline{{\\bf e}}}({\\bf x})$ imply $\\langle{\\underline{{\\bf e}}}\\rangle={\\bf e}_0$ let us replace them by this weaker constraint to \nobtain the inequality\n\\begin{equation} {\\bf e}_0\\cdot\\bfm\\sigma^D{\\bf e}_0\\geq \\min_{\\matrix{{\\underline{{\\bf e}}}({\\bf x})\\cr\\langle{\\underline{{\\bf e}}}\\rangle={\\bf e}_0}}\n\\langle\\underline{{\\bf e}}\\cdot\\sigma\\underline{{\\bf e}}\\rangle,\n\\eeq{2.5}\nwhere the minimum is now over fields ${\\underline{{\\bf e}}}$ which are not necessarily curl-free. Using Lagrange\nmultipliers one finds that the minimum is attained when ${\\underline{{\\bf e}}}=\\sigma^{-1}\\langle\\sigma^{-1}\\rangle^{-1}{\\bf e}_0$\nand so we obtain the lower bound \n\\begin{equation} {\\bf e}_0\\cdot\\bfm\\sigma^D{\\bf e}_0\\geq \\langle\\sigma^{-1}\\rangle^{-1}{\\bf e}_0\\cdot{\\bf e}_0\n\\eeq{2.6}\nof \\citeAPY{Nemat-Nasser:1993:MOP}.\nTaken together, \\eq{2.4} and \\eq{2.6} imply the lower and upper bounds\n\\begin{equation} \\left(\\frac{{\\bf e}_0\\cdot\\bfm\\sigma^D{\\bf e}_0}{{\\bf e}_0\\cdot{\\bf e}_0}-\\sigma_2\\right)\/(\\sigma_1-\\sigma_2)\n\\leq f_1 \\leq \\left(\\sigma_2^{-1}-\\frac{{\\bf e}_0\\cdot{\\bf e}_0}{{\\bf e}_0\\cdot\\bfm\\sigma^D{\\bf e}_0}\\right)\/(\\sigma_2^{-1}-\\sigma_1^{-1})\n\\eeq{2.7}\non the volume fraction $f_1$. These bounds give useful information even if we only know ${\\bf j}_0=\\bfm\\sigma^D{\\bf e}_0$\nfor only one value of ${\\bf e}_0$, i.e. if we only take one measurement. These bounds \\eq{2.7} are sharp in \nthe sense that the lower bound is approached artitrarily closely if $\\Omega$ is filled with a periodic laminate\nof components 1 and 2, oriented with the normal to the layers orthogonal to ${\\bf e}_0$ and we let the period\nlength go to zero, while the upper bound is approached artitrarily closely for the same geometry, but \noriented with the normal to the layers parallel to ${\\bf e}_0$. If the full tensor $\\bfm\\sigma^D$ is known,\nfrom $d=2,3$ measurements of pairs $({\\bf e}_0,{\\bf j}_0)$  \nthen we can take the intersection of the bounds \\eq{2.7} as ${\\bf e}_0$ is varied, and so obtain \n\\begin{equation} (\\lambda^D_+-\\sigma_2)\/(\\sigma_1-\\sigma_2)\n\\leq f_1 \\leq (1\/\\sigma_2-1\/\\lambda^D_-)\/(\\sigma_2^{-1}-\\sigma_1^{-1}),\n\\eeq{2.8}\nwhere $\\lambda^D_+$ and $\\lambda^D_-$ are the maximum and minimum eigenvalues of $\\bfm\\sigma^D$. However\nwe will see in the next section that an additional and typically sharper upper bound on $f_1$ can be obtained.\n\nWith the special Neumann boundary conditions \\eq{1.7} the variational principle \\eq{2.2} implies \n\\begin{equation} \\min_{\\matrix{{\\underline{{\\bf j}}}({\\bf x})\\cr \\nabla \\cdot{\\underline{{\\bf j}}}({\\bf x})=0 \\cr \n     {\\bf n}\\cdot{\\underline{{\\bf j}}}({\\bf x})={\\bf n}\\cdot{\\bf j}_0~{\\rm on}~\\partial\\Omega}}\n\\langle\\underline{{\\bf j}}\\cdot\\sigma^{-1}\\underline{{\\bf j}}\\rangle={\\bf j}_0\\cdot(\\bfm\\sigma^N)^{-1}{\\bf j}_0.\n\\eeq{2.9}\nBy taking a constant trial field ${\\underline{{\\bf j}}}({\\bf x})={\\bf j}_0$ or alternatively by taking the minimum over the larger\nclass of trial fields satisfying only $\\langle{\\underline{{\\bf j}}}\\rangle={\\bf j}_0$ we \nobtain the bounds\n\\begin{equation} \\langle\\sigma\\rangle^{-1}{\\bf j}_0\\cdot{\\bf j}_0\\leq{\\bf j}_0\\cdot(\\bfm\\sigma^N)^{-1}{\\bf j}_0\\leq \\langle\\sigma^{-1}\\rangle{\\bf j}_0\\cdot{\\bf j}_0\n\\eeq{2.10}\nof \\citeAPY{Nemat-Nasser:1993:MOP} which imply\n\\begin{equation} \\left(\\frac{{\\bf j}_0\\cdot{\\bf j}_0}{{\\bf j}_0\\cdot(\\bfm\\sigma^N)^{-1}{\\bf j}_0}-\\sigma_2\\right)\/(\\sigma_1-\\sigma_2)\n\\leq f_1 \\leq \\left(\\sigma_2^{-1}-\\frac{{\\bf j}_0\\cdot(\\bfm\\sigma^N)^{-1}{\\bf j}_0}{{\\bf j}_0\\cdot{\\bf j}_0}\\right)\/(\\sigma_2^{-1}-\\sigma_1^{-1}).\n\\eeq{2.11}\nThese bounds are applicable even if we only know ${\\bf e}_0=(\\bfm\\sigma^N)^{-1}{\\bf j}_0$\nfor only one value of ${\\bf j}_0$. For comparison, with these special Neumann boundary conditions \\eq{1.7},\nthe bounds in Theorem 3.1 of \\citeAPY{Kang:1997:ICP} coupled with the improvement in proposition 0\nof \\citeAPY{Ikehata:1998:SEI},\nwith $\\sigma_1=k>1$, $\\sigma_2=1$ and ${\\bf j}_0\\cdot{\\bf j}_0=1$, imply\n\\begin{equation} \n\\frac{1}{k-1}(1-{\\bf j}_0\\cdot(\\bfm\\sigma^N)^{-1}{\\bf j}_0)\n\\leq f_1 \\leq \\frac{k}{k-1}(1-{\\bf j}_0\\cdot(\\bfm\\sigma^N)^{-1}{\\bf j}_0).\n\\eeq{2.12}\nIn this case it is easy to check that the upper bounds in \\eq{2.11} and \\eq{2.12} coincide while the lower bound\nin \\eq{2.11} is tighter. The bounds \\eq{2.11} are each approached arbitrarily closely if $\\Omega$ is filled with a periodic laminate\nof components 1 and 2, oriented with ${\\bf j}_0$ either parallel or orthogonal to the layers\nand we let the period length go to zero.\n\nIn summary, \\eq{2.4},\\eq{2.6} and \\eq{2.10} imply the matrix inequalities\n\\begin{equation}  \\langle\\sigma^{-1}\\rangle^{-1}{\\bf I}\\leq\\bfm\\sigma^D\\leq \\langle\\sigma\\rangle{\\bf I},\\quad\\quad\n \\langle\\sigma^{-1}\\rangle^{-1}{\\bf I}\\leq\\bfm\\sigma^N\\leq \\langle\\sigma\\rangle{\\bf I}\n\\eeq{2.13}\nof \\citeAPY{Nemat-Nasser:1993:MOP}.\n\nFor artitrary boundary conditions, i.e. for any ${\\bf e}$ and ${\\bf j}$ satisfying \\eq{1.2} within $\\Omega$,\nwe have the bounds \n\\begin{equation} \\langle{\\bf e}\\cdot{\\bf j}\\rangle \\geq {\\bf e}_0\\cdot\\bfm\\sigma^N{\\bf e}_0,\\quad\n\\langle{\\bf e}\\cdot{\\bf j}\\rangle \\geq {\\bf j}_0(\\bfm\\sigma^D)^{-1}{\\bf j}_0,\n\\eeq{2.14}\nwhere ${\\bf e}_0=\\langle{\\bf e}\\rangle$ and ${\\bf j}_0=\\langle{\\bf j}\\rangle$,\ndue to Willis in a 1989 private communication to Nemat-Nasser and Hori, and presented by \\citeAPY{Nemat-Nasser:1993:MOP}. In conjunction with \\eq{2.13} they imply\nthe volume fraction bounds,\n\\begin{equation} \\left(\\frac{{\\bf j}_0\\cdot{\\bf j}_0}{\\langle{\\bf e}\\cdot{\\bf j}\\rangle}-\\sigma_2\\right)\/(\\sigma_1-\\sigma_2)\n\\leq f_1 \\leq \\left(\\sigma_2^{-1}-\\frac{{\\bf e}_0\\cdot{\\bf e}_0}{\\langle{\\bf e}\\cdot{\\bf j}\\rangle}\\right)\/(\\sigma_2^{-1}-\\sigma_1^{-1}).\n\\eeq{2.15}\n\n\n\\section{New bounds with one measurement}\n\\setcounter{equation}{0}\n\n\nIf we have measurements of $\\langle{\\bf e}\\cdot{\\bf j}\\rangle$ and both vectors ${\\bf e}_0$ and ${\\bf j}_0$ for arbitrary boundary \nconditions then the bounds \\eq{2.14} and \\eq{2.15} can be improved. The classical variational principle\n\\eq{2.1} implies \n\\begin{equation}\n\\min_{\\matrix{{\\underline{{\\bf e}}}({\\bf x})=-\\nabla{\\underline{V}}({\\bf x}) \n\\cr {\\underline{V}}({\\bf x})=V_0({\\bf x})~{\\rm on}~\\partial\\Omega \\cr\n\\langle\\sigma\\underline{{\\bf e}}\\rangle={\\bf j}_0}}\\langle\\underline{{\\bf e}}\\cdot\\sigma\\underline{{\\bf e}}\\rangle,\n=\\langle{\\bf e}\\cdot{\\bf j}\\rangle\n\\eeq{2.16}\nwhere we have chosen to add the constraint that $\\langle\\sigma\\underline{{\\bf e}}\\rangle={\\bf j}_0$ since we know that\nwithout this constraint the minimizer $\\underline{{\\bf e}}={\\bf e}$ satisfies $\\langle\\sigma{\\bf e}\\rangle={\\bf j}_0$. We surely\nobtain something lower if take the minimum over the larger class of fields satisfying only\n$\\langle\\underline{{\\bf e}}\\rangle={\\bf e}_0$ and $\\langle\\sigma\\underline{{\\bf e}}\\rangle={\\bf j}_0$. Thus we obtain the inequality\n\\begin{equation} \n\\min_{\\matrix{{\\underline{{\\bf e}}}({\\bf x})\\cr \\langle\\underline{{\\bf e}}\\rangle={\\bf e}_0 \\cr \\langle\\sigma\\underline{{\\bf e}}\\rangle={\\bf j}_0}}\n\\langle\\underline{{\\bf e}}\\cdot\\sigma\\underline{{\\bf e}}\\rangle\\leq\\langle{\\bf e}\\cdot{\\bf j}\\rangle.\n\\eeq{2.17}\nBy introducing two vector valued Lagrange multipliers associated with the two vector valued constraints we find that\nthe minimum is attained when\n\\begin{equation} \\underline{{\\bf e}}({\\bf x})\n={\\bf e}_0+(\\langle\\sigma^{-1}\\rangle-\\sigma^{-1}({\\bf x}))(\\langle\\sigma\\rangle\\langle\\sigma^{-1}\\rangle-1)^{-1}({\\bf j}_0-\\langle\\sigma\\rangle{\\bf e}_0).\n\\eeq{2.18}\nSubstituting this back in \\eq{2.17} gives the bound\n\\begin{equation} (\\langle\\sigma\\rangle\\langle\\sigma^{-1}\\rangle-1)(\\langle{\\bf e}\\cdot{\\bf j}\\rangle-{\\bf e}_0\\cdot{\\bf j}_0)\n\\geq ({\\bf j}_0-\\langle\\sigma\\rangle{\\bf e}_0)\\cdot(\\langle\\sigma^{-1}\\rangle{\\bf j}_0-{\\bf e}_0).\n\\eeq{2.19}\nIf, with general boundary conditions, we are interested in bounding the volume fraction $f_1$ given measured values of \n$\\langle{\\bf e}\\cdot{\\bf j}\\rangle$, ${\\bf e}_0$ and ${\\bf j}_0$ then the difference between the left hand side and right hand side\nof \\eq{2.19} is a quadratic in $f_1$ whose two roots give upper and lower bounds on $f_1$. (Unless the roots happen\nto be complex, in which case there is no configuration of the two phases within $\\Omega$ which produce the measured\n$\\langle{\\bf e}\\cdot{\\bf j}\\rangle$, ${\\bf e}_0$ and ${\\bf j}_0$, indicating the presence of other phases or indicating\nan error in measurements.)\n\n\nIn the particular cases of either special Dirichlet or special Neumann boundary conditions, \n\\eq{1.5} or \\eq{1.7}, the left hand\nside of \\eq{2.19} vanishes (see \\eq{1.9}) and we obtain the reduced bounds\n\\begin{equation} 0\\geq ({\\bf j}_0-\\langle\\sigma\\rangle{\\bf e}_0)\\cdot(\\langle\\sigma^{-1}\\rangle{\\bf j}_0-{\\bf e}_0),\n\\eeq{2.20}\nwhich are in fact implied by the matrix inequalities \\eq{2.13}. This bound \\eq{2.20} is optimal.\nFor any given fixed ${\\bf e}_0$, and fixed volume fraction $f_1$, the vector ${\\bf j}_0$ \nhas an endpoint which is constrained by \\eq{2.20} to lie within a\nsphere (disk in two dimensions) centered at $(\\langle\\sigma\\rangle+\\langle\\sigma^{-1}\\rangle^{-1}){\\bf e}_0\/2$.\nWhen $\\Omega$ is filled with a periodic laminate of the two phases\nwith interfaces orthogonal to some unit vector ${\\bf m}$, and we let the period length go to zero,\nthen the endpoint of the vector ${\\bf j}_0$ covers the entire surface of this sphere (disk) as ${\\bf m}$\nranges over all unit vectors. These bounds are the analogs, for arbitrary bodies $\\Omega$, of \nbounds on possible $({\\bf e}_0,{\\bf j}_0)$ pairs\nfor composites derived by \\citeAPY{Raitum:1983:QES} and \\citeAPY{Tartar:1995:RHM}. If we are given ${\\bf e}_0$ and ${\\bf j}_0$\nand want to bound $f_1$ then we should find the range of $f_1$ where the sphere (or disk)\ncontains the endpoint of the vector ${\\bf j}_0$. The endpoints of this range are the roots\nof the right hand side of \\eq{2.20} which is a quadratic function of $f_1$.  \n\nKnowledge of ${\\bf e}_0$ and ${\\bf j}_0$ is equivalent to knowledge of $\\langle{\\bf e}\\cdot{\\bf v}\\rangle$\nand $\\langle{\\bf v}\\cdot{\\bf j}\\rangle$ for all constant fields ${\\bf v}$. A more general alternative is to use\nthe information about  \n\\begin{eqnarray} a_i=\\langle {\\bf e}\\cdot{\\bf j}_i\\rangle &= & \\frac{1}{|\\Omega|}\\int_{\\partial\\Omega}-V_0({\\bf j}_i\\cdot{\\bf n}),\n\\nonumber \\\\\n b_k=\\langle \\nabla V_k\\cdot{\\bf j}\\rangle &= & \\frac{1}{|\\Omega|}\\int_{\\partial\\Omega}-V_k({\\bf j}\\cdot{\\bf n}),\n\\eeqa{2.21}\nfor a given set of ``comparison flux fields'' ${\\bf j}_i({\\bf x})$ satisfying $\\nabla \\cdot{\\bf j}_i=0$,\n$i=1,2,\\ldots n$ and ``comparison potentials'' $V_k({\\bf x}),~k=1,2,\\ldots m$. Suppose, for\nexample, that we have just one comparison flux field ${\\bf j}_1$. We have the variational\nprinciple\n\\begin{equation}\n\\min_{\\matrix{{\\underline{{\\bf e}}}({\\bf x})=-\\nabla{\\underline{V}}({\\bf x}) \n\\cr {\\underline{V}}({\\bf x})=V_0({\\bf x})~{\\rm on}~\\partial\\Omega \\cr\n\\langle \\underline{{\\bf e}}\\cdot{\\bf j}_1\\rangle=a_1}}\\langle\\underline{{\\bf e}}\\cdot\\sigma\\underline{{\\bf e}}\\rangle\n=\\langle{\\bf e}\\cdot{\\bf j}\\rangle,\n\\eeq{2.22}\nwhere we have chosen to add the constraint that $\\langle\\underline{{\\bf e}}\\cdot{\\bf j}_1\\rangle=a_1$ since we know that\nwithout this constraint the minimizer $\\underline{{\\bf e}}={\\bf e}$ \nsatisfies $\\langle{\\bf e}\\cdot{\\bf j}_1\\rangle=a_1$. This implies the inequality\n\\begin{equation} \\langle{\\bf e}\\cdot{\\bf j}\\rangle\\geq \\min_{\\matrix{{\\underline{{\\bf e}}}({\\bf x})\\cr \n\\langle \\underline{{\\bf e}}\\cdot{\\bf j}_1\\rangle=a_1}}\\langle\\underline{{\\bf e}}\\cdot\\sigma\\underline{{\\bf e}}\\rangle.   \n\\eeq{2.23}\nBy introducing a Lagrange multiplier associated with the constraint $\\langle{\\bf e}\\cdot{\\bf j}_1\\rangle=a_1$\nwe see the minimum occurs when\n\\begin{equation} \\underline{{\\bf e}}=a_1\\sigma^{-1}{\\bf j}_1\/\\langle{\\bf j}_1\\cdot\\sigma^{-1}{\\bf j}_1\\rangle,\n\\eeq{2.24}\ngiving the inequality\n\\begin{equation} \\langle{\\bf e}\\cdot{\\bf j}\\rangle\\geq a_1^2\/\\langle{\\bf j}_1\\cdot\\sigma^{-1}{\\bf j}_1\\rangle. \\eeq{2.25a}\nThis inequality gives information about $\\sigma({\\bf x})$ through $\\langle{\\bf j}_1\\cdot\\sigma^{-1}{\\bf j}_1\\rangle$.\nIf we only want bounds which involve the volume fraction we should choose ${\\bf j}_1({\\bf x})$\nwith\n\\begin{equation} |{\\bf j}_1({\\bf x})|=1\\quad {\\rm for~all~}{\\bf x}\\in\\Omega.\n\\eeq{2.25} \nThere are many divergence free fields ${\\bf j}_1({\\bf x})$ which satisfy this constraint. For example\nin two dimensions we can take\n\\begin{equation} {\\bf j}_1=({\\partial \\phi\/\\partial x_2,-\\partial \\phi\/\\partial x_1}),~~\n{\\rm with}~|\\nabla\\phi({\\bf x})|=1\\quad {\\rm for~all~}{\\bf x}\\in\\Omega.\n\\eeq{2.26}\nThus $\\phi({\\bf x})$ satisfies an Eikonal equation, and we could take $\\phi({\\bf x})$ to be the shortest\ndistance between ${\\bf x}$ and a curve outside $\\Omega$. Once \\eq{2.25} is satisfied \\eq{2.25a}\nimplies the volume fraction bound\n\\begin{equation} f_1\\leq \n \\left(\\sigma_2^{-1}-\\frac{a_1^2}{\\langle{\\bf e}\\cdot{\\bf j}\\rangle}\\right)\/(\\sigma_2^{-1}-\\sigma_1^{-1}).\n\\eeq{2.27}\nIn the special case when ${\\bf j}_1={\\bf e}_0\/|{\\bf e}_0|$ this reduces to the upper bound on \n$f_1$ given by \\eq{2.15}. \n\nAn important question is whether this new bound is sharp, and if so for what $\\sigma({\\bf x})$?\nThe new bound will be sharp when ${\\bf e}=\\underline{{\\bf e}}$ where $\\underline{{\\bf e}}$ is\ngiven by \\eq{2.24}. In that case\n\\begin{equation} {\\bf j}({\\bf x})=a_1{\\bf j}_1\/\\langle{\\bf j}_1\\cdot\\sigma^{-1}{\\bf j}_1\\rangle\n\\eeq{2.28}\nhas zero divergence because it is proportional to ${\\bf j}_1$. \nLet us impose the Neumann boundary condition\n\\begin{equation} {\\bf j}({\\bf x})\\cdot{\\bf n}={\\bf j}_1\\cdot{\\bf n}\\quad {\\rm for~all}~{\\bf x}\\in\\partial\\Omega,\n\\eeq{2.29}\nand look for a $\\sigma({\\bf x})$ so ${\\bf j}({\\bf x})={\\bf j}_1({\\bf x})$ and ${\\bf e}({\\bf x})=\\sigma^{-1}{\\bf j}_1({\\bf x})$\nis curl-free. Now as schematically represented by figure \\fig{0}, choose\n$\\sigma({\\bf x})$ to correspond to a finely layered composite with layers\northogonal to the streamlines of ${\\bf j}_1({\\bf x})$, and with phase 1 occupying\na local volume fraction $p({\\bf x})$. This composite will support a\ncurrent field ${\\bf j}({\\bf x})={\\bf j}_1({\\bf x})$ and an electric field ${\\bf e}({\\bf x})=\\sigma^{-1}{\\bf j}_1({\\bf x})$\nprovided \n\\begin{equation} \\nabla \\times{\\bf e}_0=0,\\quad{\\bf e}_0\\equiv[\\sigma_2^{-1}-p({\\bf x})(\\sigma_2^{-1}-\\sigma_1^{-1})]{\\bf j}_1({\\bf x}).\n\\eeq{2.30}\nHere ${\\bf e}_0({\\bf x})$ is the weak limit (local volume average) of ${\\bf e}({\\bf x})$ as the\nlayer spacing goes to zero. In two dimensions, given ${\\bf j}_1({\\bf x})$ we could look for solutions\nfor $p({\\bf x})$ such that \\eq{2.30} is satisfied and $0\\leq p({\\bf x})\\leq 1$ in $\\Omega$. We expect such\nsolutions to exist for a wide class of fields ${\\bf j}_1({\\bf x})$. This example shows that non-constant\n``comparison flux fields'' can lead to sharp bounds on the volume fraction. In three\ndimensions we only expect to find a solution of the vector equation \\eq{2.30}\nfor the scalar field $p({\\bf x})$ if ${\\bf j}_1({\\bf x})$ satisfies some additional conditions. \n  \n\n\\begin{figure}\n\\vspace{2in}\n\\hspace{1.0in}\n{\\resizebox{2.0in}{1.0in}\n{\\includegraphics[0in,0in][6in,3in]{wavelam.eps}}}\n\\vspace{0.1in}\n\\caption{A schematic of the type of layered microstructure achieving the volume fraction bound \\eq{2.27}, where the black regions denote one phase, and the\nwhite regions the other phase. The layer widths should be much finer than the size of $\\Omega$. }\n\\labfig{0}\n\\end{figure}\n\n\n\n\\section{Relationship to bounding effective tensors of composites}\n\\setcounter{equation}{0}\n\nConsider a periodic composite obtained by taking the unit cell boundaries outside $\\Omega\\equiv\\Omega_1$\nand almost filling the rest of the unit cell by non-intersecting\nrescaled and translated copies $\\Omega_i$, $i=2,\\ldots, n$ of\n$\\Omega$, as illustrated in figure \\fig{1}. The remainder of the unit cell is filled by phase 2 with\nconductivity $\\sigma_2$. The unit cell structure is periodically repeated to fill all space.\nLet $\\sigma_C({\\bf x})$ ($C$ for composite) denote this effective conductivity, i.e. in \nthe unit cell\n\\begin{eqnarray} \\sigma_C({\\bf x}) &=& \\sigma({\\bf x}\/a_i+{\\bf b}_i)~{\\rm in}~\\Omega_i,~~i=1,2,\\ldots, N \\nonumber \\\\\n                 &=& \\sigma_2~{\\rm elsewhere~outside}~\\cup_{i}\\Omega_i,\n\\eeqa{3.1}\nwhere the scaling constants $a_i$ and translation vectors ${\\bf b}_i$ (with $a_1=1$ and ${\\bf b}_1=0$)\nare determined by the size and\nposition of each copy $\\Omega_i$, so that ${\\bf x}\/a_i+{\\bf b}_i$ is on the boundary of $\\Omega$ if and only if\n${\\bf x}$ is on the boundary of $\\Omega_i$. Let $p_n$ denote the volume fraction in the unit cell\noccupied by the material with conductivity $\\sigma_2$. Let $\\bfm\\sigma^*_n$ denote the (matrix valued) \neffective conductivity of this composite, which in general depends upon the relative positions\nof the copies $\\Omega_i$ within the unit cell. \n\n\n\\begin{figure}\n\\vspace{2in}\n\\hspace{1.0in}\n{\\resizebox{2.0in}{1.0in}\n{\\includegraphics[0in,0in][6in,3in]{omegacopies.eps}}}\n\\vspace{0.1in}\n\\caption{A period cell containing rescaled copies of $\\Omega$. }\n\\labfig{1}\n\\end{figure}\n\nWe have the classical variational inequality \n\\begin{equation} {\\bf e}_0\\cdot\\bfm\\sigma^*_n{\\bf e}_0\\leq \\langle\\underline{{\\bf e}}_C\\cdot\\sigma_C\\underline{{\\bf e}}_C\\rangle,\n\\eeq{3.2}\nwhich holds for any trial electric field $\\underline{{\\bf e}}_C$ satisfying\n\\begin{equation} \\nabla \\times\\underline{{\\bf e}}_C=0,\\quad\\underline{{\\bf e}}_C~{\\rm periodic},\\quad\n\\langle\\underline{{\\bf e}}_C\\rangle={\\bf e}_0,\n\\eeq{3.3}\nwhere now the volume averages are over the entire unit cell, rather than just $\\Omega$. \nIn particular, letting ${\\bf e}({\\bf x})$ denote the electric field within $\\Omega$ when the special\nDirichlet boundary conditions \\eq{1.5} are applied, we may take in the unit cell\n\\begin{eqnarray} \\underline{{\\bf e}}_C({\\bf x}) &=& {\\bf e}({\\bf x}\/a_i+{\\bf b}_i)~{\\rm in}~\\Omega_i,~~i=1,2,\\ldots, N \\nonumber \\\\\n                 &=& {\\bf e}_0~{\\rm elsewhere~outside}~\\cup_{i}\\Omega_i,\n\\eeqa{3.4}\nand periodically extend it. Then we get\n\\begin{eqnarray}  \\langle\\underline{{\\bf e}}_C\\cdot\\sigma_C\\underline{{\\bf e}}_C\\rangle & = & p_n\\sigma_2{\\bf e}_0\\cdot{\\bf e}_0 \\nonumber \\\\\n&~& +(1-p_n)\\sum_{i=1}^N\\langle{\\bf e}({\\bf x}\/a_i+{\\bf b}_i)\\cdot\\sigma({\\bf x}\/a_i+{\\bf b}_i){\\bf e}({\\bf x}\/a_i+{\\bf b}_i)\\rangle_{\\Omega_i} \\nonumber \\\\\n&=& p_n\\sigma_2{\\bf e}_0\\cdot{\\bf e}_0+(1-p_n)\\langle{\\bf e}({\\bf x})\\cdot\\sigma({\\bf x}){\\bf e}({\\bf x})\\rangle_{\\Omega} \\nonumber \\\\\n&=& p_n\\sigma_2{\\bf e}_0\\cdot{\\bf e}_0+(1-p_n){\\bf e}_0\\cdot\\bfm\\sigma^D{\\bf e}_0,\n\\eeqa{3.5}\nwhere $\\langle\\cdot\\rangle_{\\Omega_i}$ denotes an average over the region $\\Omega_i$.\nCombined with the variational inequality \\eq{3.2} this implies the bound\n\\begin{equation} {\\bf e}_0\\cdot\\bfm\\sigma^*_n{\\bf e}_0\\leq p_n\\sigma_2{\\bf e}_0\\cdot{\\bf e}_0+(1-p_n){\\bf e}_0\\cdot\\bfm\\sigma^D{\\bf e}_0\n\\leq {\\bf e}_0\\cdot\\bfm\\sigma^D{\\bf e}_0,\n\\eeq{3.6}\nwhere we have used the inequality $\\bfm\\sigma^D\\geq\\sigma_2{\\bf I}$ implied by \\eq{2.13}. Thus we get\n\\begin{equation} \\bfm\\sigma^*_n\\leq \\bfm\\sigma^D.\n\\eeq{3.7}\nThis composite has a volume fraction $f_1'=(1-p_n)f_1$ of phase 1.\nThus any bound ``from below'' on the effective conductivity $\\bfm\\sigma^*_n$, applicable to composites\nhaving a volume fraction $f_1'$ of phase 1, immediately translates into\nbound  ``from below'' on $\\bfm\\sigma^D$. Now consider what happens as we increase $N$, inserting\nmore and more regions $\\Omega_i$, while leaving undisturbed the regions $\\Omega_i$ already in place,\nso that $p_n\\to 0$ as $n\\to\\infty$. We are assured that this is possible. Rescaled copies\nof any shaped region can be packed to fill all space: see, for example, Theorem A.1\nin \\citeAPY{Benveniste:2003:NER}. Define\n\\begin{equation} \\bfm\\sigma^*=\\lim_{n\\to\\infty}\\bfm\\sigma^*_n.\n\\eeq{3.7a}\nWe are assured this limit exists since if we change the geometry in some\nsmall volume then the effective conductivity (assuming $\\sigma_1$ and $\\sigma_2$ are strictly positive and finite)\nis perturbed only by a small amount (\\citeAY{Zhikov:1994:HDO}).\nWe will call $\\bfm\\sigma^*$ the effective conductivity tensor of an assemblage of rescaled copies\nof $\\Omega$ packed to fill all space. Then \\eq{3.7} implies\n\\begin{equation} \\bfm\\sigma^*\\leq \\bfm\\sigma^D, \\eeq{3.7b}\nwhich is essentially the bound of \\citeAPY{Huet:1990:AVC} applied to this assemblage.\nAssume the bound is continuous with respect\nto $f_1'$ at the point $f_1'=f_1$, as expected. Then taking the limit $n\\to\\infty$ the \n``lower bound'' on the effective tensor of composites having\nvolume fraction $f_1$ must also be a lower bound on $\\bfm\\sigma^D$.\n\nIn particular, the harmonic mean bound \n$\\bfm\\sigma^*\\geq\\langle\\sigma^{-1}\\rangle^{-1}{\\bf I}$ translates into the elementary bound \n$\\bfm\\sigma^D\\geq\\langle\\sigma^{-1}\\rangle^{-1}{\\bf I}$ of Nemat-Nasser and Hori, obtained before. \nAdditionally, in our two-phase composite, the effective conductivity $\\bfm\\sigma^*$ satisfies\nthe Lurie-Cherkaev-Murat-Tartar bound (Lurie and Cherkaev \\citeyearNP{Lurie:1982:AEC}, \\citeyearNP{Lurie:1984:EEC}; \\citeAY{Murat:1985:CVH}; \\citeAY{Tartar:1985:EFC})\n\\begin{equation} f_1{\\rm Tr}[(\\bfm\\sigma^*-\\sigma_2{\\bf I})^{-1}]\\leq d\/(\\sigma_1-\\sigma_2)+f_2\/\\sigma_2,\n\\eeq{3.8}\n[which are a generalization of the bounds of \\citeAPY{Hashin:1962:VAT}] where $d=2,3$ is the dimensionality\nof the composite. Since $\\bfm\\sigma^D\\geq\\bfm\\sigma^*\\geq \\sigma_2{\\bf I}$ it follows that \n\\begin{equation} (\\bfm\\sigma^*-\\sigma_2{\\bf I})^{-1}\\geq (\\bfm\\sigma^D-\\sigma_2{\\bf I})^{-1}, \\eeq{3.9}\nand so \\eq{3.8} implies the new bound\n\\begin{equation}  f_1{\\rm Tr}[(\\bfm\\sigma^D-\\sigma_2{\\bf I})^{-1}]\\leq d\/(\\sigma_1-\\sigma_2)+f_2\/\\sigma_2.\n\\eeq{3.10}\nBy multiplying this inequality by $\\sigma_2^2$ and adding $df_1\\sigma_2$ to both sides we see that it  \ncan be rewritten in the equivalent form \n\\begin{equation} f_1{\\rm Tr}[(\\sigma_2^{-1}{\\bf I}-(\\bfm\\sigma^D)^{-1})^{-1}]\\leq d\/(\\sigma_2^{-1}-\\sigma_1^{-1})-(d-1)f_2\\sigma_2.\n\\eeq{3.10a}\nAs $d^2\/{\\rm Tr}({\\bf A})\\leq {\\rm Tr}({\\bf A}^{-1})$ for any positive definite matrix ${\\bf A}$ we also obtain\nthe weaker bound\n\\begin{equation} \\frac{1}{d}{\\rm Tr}[(\\bfm\\sigma^D)^{-1}]\\leq \\sigma_2^{-1}-\\frac{f_1 d}{d\/(\\sigma_2^{-1}-\\sigma_1^{-1})-(d-1)f_2\\sigma_2},\n\\eeq{3.11}\nwhich is a particular case of the universal bounds first derived by \nNemat-Nasser and Hori (\\citeyearNP{Nemat-Nasser:1993:MOP}, \\citeyearNP{Nemat-Nasser:1995:UBO}), see equation (5.4.9) in their 1995 paper,\nobtained under the assumption that $\\Omega$ is ellipsoidal or parallelpipedic.\n(which we see is not needed).\n\nIf one is interested in bounds on the volume fraction $f_1$ then \\eq{3.10} implies the upper bound\n\\begin{equation} f_1\\leq \\frac{1\/\\sigma_2+d\/(\\sigma_1-\\sigma_2)}{1\/\\sigma_2+{\\rm Tr}[(\\bfm\\sigma^D-\\sigma_2{\\bf I})^{-1}]}.\n\\eeq{3.12}\n\nTo obtain lower bounds on $f_1$, we consider the same periodic composite\nand apply the dual variational inequality\n\\begin{equation} {\\bf j}_0\\cdot(\\bfm\\sigma^*_n)^{-1}{\\bf j}_0\\leq \\langle\\underline{{\\bf j}}_C\\cdot\\sigma_C^{-1}\\underline{{\\bf j}}_C\\rangle\n\\eeq{3.13}\nvalid for any trial current field $\\underline{{\\bf j}}_C$ satisfying\n\\begin{equation} \\nabla \\cdot\\underline{{\\bf j}}_C=0,\\quad\\underline{{\\bf j}}_C~{\\rm periodic},\\quad\n\\langle\\underline{{\\bf j}}_C\\rangle={\\bf j}_0.\n\\eeq{3.14}\nLetting ${\\bf j}({\\bf x})$ denote the current field within $\\Omega$ when the special\nNeumann boundary conditions \\eq{1.7} are applied, we may take in the unit cell\n\\begin{eqnarray} \\underline{{\\bf j}}_C({\\bf x}) &=& {\\bf j}({\\bf x}\/a_i+{\\bf b}_i)~{\\rm in}~\\Omega_i,~~i=1,2,\\ldots, N \\nonumber \\\\\n                 &=& {\\bf j}_0~{\\rm elsewhere~outside}~\\cup_{i}\\Omega_i,\n\\eeqa{3.15}\nand periodically extend it. \nSubstituting this trial field in \\eq{3.13} gives the bound \n\\begin{equation} (\\bfm\\sigma^*_n)^{-1}\\leq p_n\\sigma_2^{-1}{\\bf I}+(\\bfm\\sigma^N)^{-1}, \\eeq{3.15a}\nwhich in the limit $n\\to\\infty$ implies\n\\begin{equation}  \\bfm\\sigma^*\\geq \\bfm\\sigma^N, \\eeq{3.16}\nwhich is essentially the bound of \\citeAPY{Huet:1990:AVC} applied to the\nassemblage of rescaled copies of $\\Omega$ packed to fill all space.\n\nThus any bound ``from above'' on the effective conductivity $\\bfm\\sigma^*_n$ of composites having a volume \nfraction $f_1'$ immediately \ntranslates into a bound  ``from above'' on $(p_n\\sigma_2^{-1}{\\bf I}+(\\bfm\\sigma^N)^{-1})^{-1}$. Taking\nthe limit $n\\to\\infty$\nand assuming continuity of the bound at $f_1'=f_1$ the ``upper bound'' \non the effective tensor of composites having volume fraction $f_1$ must also be an upper bound\non $\\bfm\\sigma^N$.\nIn particular, the other Murat-Tartar-Lurie-Cherkaev bound \n\\begin{equation} f_2{\\rm Tr}[(\\sigma_1{\\bf I}-\\bfm\\sigma^*)^{-1}]\\leq d\/(\\sigma_1-\\sigma_2)-f_1\/\\sigma_1,\n\\eeq{3.17}\nimplies\n\\begin{equation} f_2{\\rm Tr}[(\\sigma_1{\\bf I}-\\bfm\\sigma^N)^{-1}]\\leq d\/(\\sigma_1-\\sigma_2)-f_1\/\\sigma_1. \n\\eeq{3.18}\nAgain using the inequality $d^2\/{\\rm Tr}({\\bf A})\\leq {\\rm Tr}({\\bf A}^{-1})$ for ${\\bf A}>0$, we obtain\nthe weaker bound\n\\begin{equation}  \\frac{1}{d}{\\rm Tr}(\\bfm\\sigma^N)\\leq \\sigma_1-\\frac{f_2 d}{d\/(\\sigma_1-\\sigma_2)-f_1\/\\sigma_1}\n\\eeq{3.20}\nwhich is a particular case of the universal bounds derived by Nemat-Nasser and Hori (\\citeyearNP{Nemat-Nasser:1993:MOP}, \\citeyearNP{Nemat-Nasser:1995:UBO}), see \nequation (5.3.11) in their 1995 paper,\nobtained under the assumption that $\\Omega$ is ellipsoidal or parallelpipedic\n(which we see is not needed).\n\nFrom \\eq{3.18} we directly obtain the volume fraction bound\n\\begin{equation} f_2\\leq \\frac{d\/(\\sigma_1-\\sigma_2)-1\/\\sigma_1}{\\{{\\rm Tr}[(\\sigma_1{\\bf I}-\\bfm\\sigma^N)^{-1}]\\}-1\/\\sigma_1},\n\\eeq{3.21}\ngiving a lower bound on the volume fraction $f_1=1-f_2$. \n\nIn the asymptotic limit as the volume fraction goes to zero\nthe volume fraction bounds \\eq{3.12} and \\eq{3.21} reduce to those of Capdeboscq and Vogelius (\\citeyearNP{Capdeboscq:2003:OAE}, \\citeyearNP{Capdeboscq:2004:RSR}),\nas shown in the two dimensional case by \\citeAPY{Kang:2011:SBV}. The paper of Kang, Kim and Milton also tests the bounds numerically, and their (two-dimensional) results \nshow the bound \\eq{3.12} is typically close to the actual volume fraction for a variety of inclusions of phase 1 in a matrix of phase 2. Similarly we can expect\nthat the bound \\eq{3.21} will be typically close to the actual volume fraction for a variety of inclusions of phase 2 in a matrix of phase 1. \n\n\n\n\\section{Coupled bounds in two-dimensions}\n\\setcounter{equation}{0}\nThe tensors $\\bfm\\sigma^D$ and $\\bfm\\sigma^N$ obviously depend on $\\sigma_1$ and $\\sigma_2$, i.e. \n$\\bfm\\sigma^D=\\bfm\\sigma^D(\\sigma_1,\\sigma_2)$ and $\\bfm\\sigma^N=\\bfm\\sigma^N(\\sigma_1,\\sigma_2)$. Let us assume we have\nmeasurements of these tensors for an additional pair of conductivities $(k_1, k_2)$,\n(which could be obtained, say from thermal, magnetic permeability, or diffusivity\nmeasurements) and let ${\\bf k}^D$ and ${\\bf k}^N$ denote these tensors,\n\\begin{equation} {\\bf k}^D=\\bfm\\sigma^D(k_1,k_2),\\quad {\\bf k}^N=\\bfm\\sigma^N(k_1,k_2).\n\\eeq{4.1}\nWe still let $\\bfm\\sigma^D$ and $\\bfm\\sigma^N$ denote the tensors associated with the first\npair of conductivities $(\\sigma_1, \\sigma_2)$, with $\\sigma_1>\\sigma_2$. From \\eq{3.7b} and\n\\eq{3.16} we have the inequalities\n\\begin{eqnarray}  \\sigma_2{\\bf I}\\leq\\bfm\\sigma^N\\leq\\bfm\\sigma^*\\leq \\bfm\\sigma^D\\leq \\sigma_1{\\bf I}, \\nonumber \\\\\nk^-\\leq {\\bf k}^N\\leq{\\bf k}^*\\leq {\\bf k}^D\\leq k^+{\\bf I},\n\\eeqa{4.2}\nwhere $k^-=\\min\\{k_1,k_2\\}$ and $k^+=\\max\\{k_1,k_2\\}$ and\n${\\bf k}^*$ is the effective conductivity the composite considered\nin the previous section when  $\\sigma_1$ and $\\sigma_2$ are replaced by $k_1$ and $k_2$. (It can\neasily be checked that these inequalities still hold if $k_2>k_1$.) \n\nFor two dimensional conductivity from duality \n(\\citeAY{Keller:1964:TCC}; \\citeAY{Dykhne:1970:CTD})\nwe know the functions \n$\\bfm\\sigma^D=\\bfm\\sigma^D(\\sigma_1,\\sigma_2)$ and $\\bfm\\sigma^N=\\bfm\\sigma^N(\\sigma_1,\\sigma_2)$ satisfy\n\\begin{eqnarray} \\bfm\\sigma^D(\\sigma_2,\\sigma_1)& = &\\sigma_1\\sigma_2{\\bf R}_\\perp^T[\\bfm\\sigma^N(\\sigma_1,\\sigma_2)]^{-1}{\\bf R}_\\perp, \\nonumber \\\\\n \\bfm\\sigma^N(\\sigma_2,\\sigma_1)& = &\\sigma_1\\sigma_2{\\bf R}_\\perp^T[\\bfm\\sigma^D(\\sigma_1,\\sigma_2)]^{-1}{\\bf R}_\\perp,\n\\eeqa{4.3}\nwhere \n\\begin{equation} {\\bf R}_\\perp=\\pmatrix{0 & 1 \\cr -1 & 0}\n\\eeq{4.4}\nis the matrix for a $90^\\circ$ rotation.\nSo if we know these tensors for the conductivity pair $(k_1, k_2)$, we also know them for the \nconductivity pair $(k_2, k_1)$. Hence, by making such an interchange if necessary, we may assume without\nloss of generality that $k_1>k_2$, i.e. that $k^+=k_1$ and $k^-=k_2$. Finally, by \ninterchanging $k$ with $\\sigma$ if necessary,\nwe may assume without loss of generality that \n\\begin{equation} \\sigma_1\/\\sigma_2\\geq k_1\/k_2>1. \\eeq{4.5}\n\n\n\nOptimal bounds on all possible matrix pairs $(\\bfm\\sigma^*,{\\bf k}^*)$  for composites having a prescribed\nvolume fraction $f_1$ of phase 1 have been derived by \\citeAPY{Cherkaev:1992:ECB}, \nand extended to an arbitrary number of effective conductivity function values by  \\citeAPY{Clark:1995:OBC}.\nHowever it seems\ndifficult to extract bounds on $f_1$ from these optimal bounds. Instead we consider a\npolycrystal checkerboard with conductivities \n\\begin{equation} \\bfm\\sigma({\\bf x})={\\bf R}^T({\\bf x})\\bfm\\sigma^*{\\bf R}({\\bf x}),\\quad {\\bf k}({\\bf x})={\\bf R}^T({\\bf x}){\\bf k}^*{\\bf R}({\\bf x}),\n\\quad {\\rm with}~{\\bf R}^T({\\bf x}){\\bf R}({\\bf x})={\\bf I},\n\\eeq{4.6}\nin which the rotation field ${\\bf R}({\\bf x})$ is ${\\bf I}$ in the ``white squares'' and\n${\\bf R}_\\perp$ in the ``black squares''. By a result of \\citeAPY{Dykhne:1970:CTD} this material\nhas effective conductivities $(\\sigma_*{\\bf I},k_*{\\bf I})$ where\n\\begin{equation} \\sigma_*=\\sqrt{\\det\\bfm\\sigma^*},\\quad k_*=\\sqrt{\\det{\\bf k}^*}.\n\\eeq{4.7}\nNow we replace the ``white squares'' by the limiting composite considered\nin the previous section (with structure much smaller than the size of the\nsquares) and we replace the ``black squares'' by the limiting composite considered\nin the previous section, rotated by $90^\\circ$. The resulting material is an isotropic\ncomposite of phases 1 and 2 and so the pair $(\\sigma_*,k_*)$ satisfies the bounds\nof \\citeAPY{Milton:1981:BTO},\n\\begin{equation} u(k_*)\\leq \\sigma_* \\leq v(k_*), \\eeq{4.10}\nwhich are attained when the composite is an assemblage of doubly coated disks, where\n\\begin{eqnarray} \n v(k_*) & = &\\sigma_1-\\frac{2f_2\\sigma_1(\\sigma_1^2-\\sigma_2^2)}{(f_2\\sigma_1+f_1\\sigma_2+\\sigma_1)(\\sigma_1+\\sigma_2)\n+(\\sigma_1-\\sigma_2)^2\\alpha_1(k_*)}, \\nonumber \\\\\nu(k_*) & \\equiv &\\sigma_2+\n\\frac{2f_1\\sigma_2(\\sigma_1^2-\\sigma_2^2)}{(f_2\\sigma_1+f_1\\sigma_2+\\sigma_2)(\\sigma_1+\\sigma_2)\n+(\\sigma_1-\\sigma_2)^2\\alpha_2(k_*)},\n\\eeqa{4.11}\nand\n\\begin{eqnarray}\n\\alpha_1(k_*)& = &\\frac{(k_1+k_2)[2f_2k_1(k_1-k_2)\/(k_1-k_*)-(f_2k_1+f_1k_2+k_1)]}{(k_1-k_2)^2}, \\nonumber \\\\\n\\alpha_2(k_*)& = &\\frac{(k_1+k_2)[2f_1k_2(k_1-k_2)\/(k_*-k_2)-(f_2k_1+f_1k_2+k_2)]}{(k_1-k_2)^2}.\n\\eeqa{4.12}\nNow for any two symmetric matrices ${\\bf A}$ and ${\\bf B}$ with ${\\bf A}\\geq{\\bf B}>0$ we \nhave ${\\bf B}^{-1\/2}{\\bf A}{\\bf B}^{-1\/2}\\geq{\\bf I}$, and so $\\det({\\bf B}^{-1\/2}{\\bf A}{\\bf B}^{-1\/2})\\geq 1$\nimplying $\\det({\\bf A})>\\det({\\bf B})$. Thus \\eq{4.2} and \\eq{4.7} imply\n\\begin{equation} \\sigma_2\\leq\\sigma_N\\leq\\sigma_*\\leq \\sigma_D\\leq\\sigma_1, \n\\quad k_2\\leq k_N\\leq k_*\\leq k_D\\leq k_1,\n\\eeq{4.13}\nwhere we define\n\\begin{equation} \\sigma_N=\\sqrt{\\det\\bfm\\sigma_N},\\quad \\sigma_D=\\sqrt{\\det\\bfm\\sigma_D},\\quad\n k_N=\\sqrt{\\det{\\bf k}_N},\\quad k_D=\\sqrt{\\det{\\bf k}_D}.\n\\eeq{4.13a}\nThe Hashin-Shtrikman bounds (\\citeAY{Hashin:1962:VAT}; \\citeAY{Hashin:1970:TCM}),\n\\begin{equation} k_1-\\frac{2f_2k_1(k_1-k_2)}{f_2k_1+f_1k_2+k_1}\\geq k_*\n\\geq  k_2+\\frac{2f_1k_2(k_1-k_2)}{f_2k_1+f_1k_2+k_2},\n\\eeq{4.14}\nimply that both $\\alpha_1(k_*)$ and $\\alpha_2(k_*)$ are non-negative. Hence the denominators\nin \\eq{4.11} are positive and so \\eq{4.10} implies\n\\begin{eqnarray} \n(\\sigma_1-\\sigma_*)[(f_2\\sigma_1+f_1\\sigma_2+\\sigma_1)(\\sigma_1+\\sigma_2)\n+(\\sigma_1-\\sigma_2)^2\\alpha_1(k_*)]\\geq 2f_2\\sigma_1(\\sigma_1^2-\\sigma_2^2),\n\\nonumber \\\\\n(\\sigma_*-\\sigma_2)[(f_2\\sigma_1+f_1\\sigma_2+\\sigma_2)(\\sigma_1+\\sigma_2)\n+(\\sigma_1-\\sigma_2)^2\\alpha_2(k_*)]\\geq 2f_1\\sigma_2(\\sigma_1^2-\\sigma_2^2).\n\\eeqa{4.15}\nSince $\\alpha_1(k_D)\\geq\\alpha_1(k_*)$ and $\\alpha_2(k_N)\\geq\\alpha_2(k_*)$, we get using\n\\eq{4.13},\n\\begin{eqnarray}\n(\\sigma_1-\\sigma_N)[(f_2\\sigma_1+f_1\\sigma_2+\\sigma_1)(\\sigma_1+\\sigma_2)\n+(\\sigma_1-\\sigma_2)^2\\alpha_1(k_D)]\\geq 2f_2\\sigma_1(\\sigma_1^2-\\sigma_2^2),\n\\nonumber \\\\\n(\\sigma_D-\\sigma_2)[(f_2\\sigma_1+f_1\\sigma_2+\\sigma_2)(\\sigma_1+\\sigma_2)\n+(\\sigma_1-\\sigma_2)^2\\alpha_2(k_N)]\\geq 2f_1\\sigma_2(\\sigma_1^2-\\sigma_2^2).\n\\nonumber \\\\ ~\n\\eeqa{4.16}\nAs $\\alpha_1(k_D)$ and $\\alpha_2(k_N)$ depend linearly on $f_1$ and $f_2=1-f_1$,\nthe equations \\eq{4.16} readily yield bounds on the volume fraction. Eunjoo Kim\nhas used an integral equation solver [as described by \\citeAPY{Kang:2011:SBV}]\nto compare the bounds \\eq{4.16} with the bounds \\eq{3.12} and \\eq{3.21}. Her results\nare presented in figures \\ref{3}, \\ref{4}, and \\ref{5}. More numerical results testing\nthe bounds \\eq{3.12} and \\eq{3.21} are in the paper by \\citeAPY{Kang:2011:SBV}.\n\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\epsfig{figure=LU_DYDNa6.3.eps,width=10cm}\n\\end{center}\n\\caption{The first figure shows the circular body $\\Omega$ containing an ellipse of phase 1 surrounded by phase 2.\nThe second figure shows the results for the bounds\n\\eq{3.12} and \\eq{3.21} while the third figure shows the results \nfor the bounds \\eq{4.16}.\nThe bounds are for increasing $\\sigma_1$, with $\\sigma_2=1$\nand (for the third figure) the pairs $(\\sigma_1,k_1)$ are taken as\n$(1.1,1.05)$, $(1.2,1.1)$, $(1.5,1.2)$, $(2,1.5)$, $(3,2)$, $(5,3)$, $(10,5)$ and $(20,10)$,\nwith $\\sigma_2=k_2=1$.\nHere $U(\\sigma_1)$ and $L(\\sigma_1)$ are the upper and \nlower bounds on the volume fraction, and the true volume fraction is $f_1=0.08$.\nFigure supplied courtesy of Eunjoo Kim.}\\label{3}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\epsfig{figure=LU_DYDNa5.3.eps,width=10cm}\n\\end{center}\n\\caption{The same as for figure \\ref{3} but with the elliptical \ninclusion moved closer to the boundary of $\\Omega$. Figure supplied courtesy of Eunjoo Kim.}\\label{4}\n\\end{figure}\n\n\n\n \n\\begin{figure}[htbp]\n\\begin{center}\n\\epsfig{figure=LU_DNDNa4.3.eps,width=10cm}\n\\end{center}\n\\caption{The same as for figure \\ref{3} but with a non-elliptical inclusion of phase 1 in a square region $\\Omega$. The true volume\nfraction is $f_1=0.0673$. Figure supplied courtesy of Eunjoo Kim. }\\label{5}\n\\end{figure}\n\n\\section{Coupled bounds in three-dimensions}\n\\setcounter{equation}{0}\n\nWe can also derive coupled bounds in three dimensions. Let us assume the phases\nhave been labeled so that\n\\begin{equation} \\sigma_1 k_1\\geq \\sigma_2 k_2, \\quad {\\rm i.e.}~\\sigma_1\/\\sigma_2\\geq k_2\/k_1,\n\\eeq{4.17}\nand by interchanging $\\sigma$ with $k$ if necessary let us assume\n\\begin{equation} \\sigma_1\/\\sigma_2\\geq k_1\/k_2.\n\\eeq{4.18}\nThese two inequalities imply $\\sigma_1\/\\sigma_2>1$ as before. We want to use the\ninequalities \\eq{4.2} to derive bounds on the volume fraction. As in the \ntwo-dimensional case the idea is to first construct an isotropic\npolycrystal, where the polycrystal has the conductivities \\eq{4.6}\nin which the rotation field ${\\bf R}({\\bf x})$ is constant within grains\nwhich we take to be spheres. These spheres fill all space, and the crystal\norientation varies randomly from sphere to sphere so that the composite\nhas isotropic conductivities $(\\sigma_*{\\bf I},k_*{\\bf I})$. We use the effective medium formula\n(\\citeAY{Stroud:1975:GEM}; \\citeAY{Helsing:1991:ECA})\nwhich gives \n\\begin{equation} \\sigma_*=g(\\bfm\\sigma_*), \\quad k_*=g({\\bf k}_*),\n\\eeq{4.19}\nwhere for any positive definite symmetric $3\\times 3$ matrix ${\\bf A}$, $g=g({\\bf A})$ is taken to be the\nunique positive root of\n\\begin{equation} \\frac{\\lambda_1-g}{\\lambda_1+2g}+\\frac{\\lambda_2-g}{\\lambda_2+2g}+\\frac{\\lambda_3-g}{\\lambda_3+2g}=0,\n\\eeq{4.20}\nin which $\\lambda_1$, $\\lambda_2$, and $\\lambda_3$ are the eigenvalues of ${\\bf A}$. This effective\nmedium formula is realizable (\\citeAY{Milton:1985:TCP}; \\citeAY{Avellaneda:1987:IHD})\nin the sense that it corresponds to a limiting composite\nof spherical grains with hierarchical structure (where any pair of grains of \ncomparable size are well separated from each other, relative to their diameter). \nNote that the left hand side of side of \\eq{4.20} increases if any of the eigenvalues\n$\\lambda_i$ increase, and decreases if $g$ increases. So $g({\\bf A})$ must increase if\nany or all of the eigenvalues of ${\\bf A}$ increase. It follows that $g({\\bf B})\\geq g({\\bf A})$\nif ${\\bf B}\\geq{\\bf A}>0$. Hence the inequalities \\eq{4.2} imply\n\\begin{equation} \\sigma_2\\leq\\sigma_N\\leq\\sigma_*\\leq \\sigma_D\\leq\\sigma_1, \n\\quad k^-\\leq k_N\\leq k_*\\leq k_D\\leq k^+,\n\\eeq{4.20a}\nwhere now\n\\begin{equation}  \\sigma_N=g(\\bfm\\sigma_N),\\quad \\sigma_D=g(\\bfm\\sigma_D),\\quad k_N=g({\\bf k}_N),\\quad k_D=g({\\bf k}_D).\n\\eeq{4.20b}\n\nWe next replace the material in each sphere by the appropriately oriented \nlimiting composite considered in the previous section (with structure much \nsmaller than the sphere diameter) to obtain a two-phase isotropic composite with\n$(\\sigma_*,k_*)$ as its conductivities. Thus $\\sigma_*$ must satisfy the upper bound of Bergman (\\citeyearNP{Bergman:1976:VBS},\\citeyearNP{Bergman:1978:DCC})\n\\begin{equation}\n\\sigma_*\\leq  f_1\\sigma_1+f_2\\sigma_2\n-\\frac{f_1f_2(\\sigma_1-\\sigma_2)^2}{3\\sigma_2+(\\sigma_1-\\sigma_2)\\gamma(k_*)},\n\\eeq{4.23}\nwhere\n\\begin{equation} \\gamma(k_*)=\\frac{f_1f_2(k_1-k_2)}{f_1k_1+ f_2k_2 -k_*}-\\frac{3k_2}{k_1-k_2},\n\\eeq{4.24}\nand the lower bound\n\\begin{equation} \\sigma_*\\geq \\sigma_2+\n\\frac{3f_1\\sigma_2(\\sigma_1-\\sigma_2)(\\sigma_2+2\\sigma_1)}{(f_2\\sigma_1+f_1\\sigma_2+2\\sigma_2)(\\sigma_2+2\\sigma_1)\n+(\\sigma_1-\\sigma_2)^2\\beta(k_*)},\n\\eeq{4.21}\nwhere\n\\begin{equation}\n\\beta(k_*) = \\frac{(k_2+2k_1)[3f_1k_2(k_1-k_2)\/(k_*-k_2)-(f_2k_1+f_1k_2+2k_2)]}{(k_1-k_2)^2}.\n\\eeq{4.22}\nThis lower bound was first conjectured by \\citeAPY{Milton:1981:BTO}. A proof was proposed\nby \\citeAPY{Avellaneda:1988:ECP} which was corrected by \\citeAPY{Nesi:1991:MII} and \\citeAPY{Zhikov:1991:EHM}.\n\nThe lower bound \\eq{4.21} is sharp, being attained for two-phase assemblages of doubly\ncoated spheres (\\citeAY{Milton:1981:BTO}). The upper bound \\eq{4.23} is attained at $5$ values of $\\gamma(k_*)$\nnamely when $\\gamma(k_*)=f_2, 3f_2\/2, 3f_2, 3-3f_1\/2,$ and $3-f_1$ (\\citeAY{Milton:1981:BCP}).\n\nThe Hashin-Shtrikman bound (\\citeAY{Hashin:1962:VAT}),\n\\begin{equation} (k_*-k_2)\/(k_1-k_2)\\geq 3f_1k_2\/(f_2k_1+f_1k_2+2k_2),\n\\eeq{4.25}\nimplies that $\\beta(k_*)$ is non-negative.  Hence the denominator\nin \\eq{4.21} is positive and so the inequality implies\n\\begin{equation} (\\sigma_*-\\sigma_2)[(f_2\\sigma_1+f_1\\sigma_2+2\\sigma_2)(\\sigma_2+2\\sigma_1)\n+(\\sigma_1-\\sigma_2)^2\\beta(k_*)]\\geq 3f_1\\sigma_2(\\sigma_1-\\sigma_2)(\\sigma_2+2\\sigma_1).\n\\eeq{4.26}\nThe Hashin-Shtrikman bounds can also be rewritten in the form\n\\begin{equation}\nf_2k_1+f_1k_2+2k^- \\leq \\frac{f_1f_2(k_1-k_2)^2}{f_1k_1+ f_2k_2 -k_*}\\leq f_2k_1+f_1k_2+2k^+.\n\\eeq{4.27}\nThese inequalities imply\n$\\gamma(k_*)$ lies between $f_2$ and $3-f_1$. Hence the denominator in \\eq{4.23} is positive and\nthe inequality can be rewritten as \n\\begin{equation}\n(f_1\\sigma_1+f_2\\sigma_2-\\sigma_*)(3\\sigma_2+(\\sigma_1-\\sigma_2)\\gamma(k_*))\\geq f_1f_2(\\sigma_1-\\sigma_2)^2.\n\\eeq{4.28}\nWhen $k_1\\geq k_2$ \\eq{4.20a} implies  $\\beta(k_N)\\geq\\beta(k_*)$ and $\\gamma(k_D)\\geq\\gamma(k_*)$, and \nhence \n\\begin{eqnarray}\n(\\sigma_D-\\sigma_2)[(f_2\\sigma_1+f_1\\sigma_2+2\\sigma_2)(\\sigma_2+2\\sigma_1)\n+(\\sigma_1-\\sigma_2)^2\\beta(k_N)]& \\geq & 3f_1\\sigma_2(\\sigma_1-\\sigma_2)(\\sigma_2+2\\sigma_1),\\nonumber \\\\\n(f_1\\sigma_1+f_2\\sigma_2-\\sigma_N)(3\\sigma_2+(\\sigma_1-\\sigma_2)\\gamma(k_D))& \\geq & f_1f_2(\\sigma_1-\\sigma_2)^2. \\nonumber \\\\\n\\eeqa{4.29}\nOn the other hand when $k_1\\leq k_2$ then \\eq{4.20a} implies $\\beta(k_D)\\geq\\beta(k_*)$ and\n$\\gamma(k_N)\\geq\\gamma(k_*)$, and hence \n\\begin{eqnarray}\n(\\sigma_D-\\sigma_2)[(f_2\\sigma_1+f_1\\sigma_2+2\\sigma_2)(\\sigma_2+2\\sigma_1)\n+(\\sigma_1-\\sigma_2)^2\\beta(k_D)]& \\geq & 3f_1\\sigma_2(\\sigma_1-\\sigma_2)(\\sigma_2+2\\sigma_1),\\nonumber \\\\\n(f_1\\sigma_1+f_2\\sigma_2-\\sigma_N)(3\\sigma_2+(\\sigma_1-\\sigma_2)\\gamma(k_N))&\\geq& f_1f_2(\\sigma_1-\\sigma_2)^2. \\nonumber \\\\\n\\eeqa{4.30}\nSince $\\beta(k_N)$ and $\\beta(k_D)$ depend linearly on the volume fractions $f_1$ and $f_2=1-f_1$, \nthe first inequalities\nin \\eq{4.29} and \\eq{4.30} also depend linearly on the volume fraction and easily yield\nbounds on the volume fraction. On the other hand, finding bounds on the volume fraction\nfrom the second inequalities in \\eq{4.29} and \\eq{4.30}, involves solving a cubic equation in $f_1$. So instead\nof analytically computing the roots of this cubic it is probably better to numerically\nsearch for the range of values of $f_1$ where the second inequalities in \\eq{4.29} and \\eq{4.30}\nare satisfied. \n\n\n\n\n\n\n\n\\section{Bounds for elasticity}\n\\setcounter{equation}{0}\n\nLet us consider solutions to the linear elasticity equations \n\\begin{equation} \\bfm\\tau({\\bf x})={\\bfm{\\cal C}}({\\bf x})\\bfm\\epsilon({\\bf x}),\\quad\\nabla \\cdot\\bfm\\theta=0,\\quad\\bfm\\epsilon=(\\nabla{\\bf u}+(\\nabla{\\bf u})^T)\/2,\n\\eeq{5.1}\nwithin $\\Omega$, where ${\\bf u}({\\bf x})$, $\\bfm\\epsilon({\\bf x})$ and $\\bfm\\tau({\\bf x})$, are the displacement\nfield, strain field, and stress field, and ${\\bfm{\\cal C}}({\\bf x})$ is the fourth\norder elasticity tensor field\n\\begin{equation} {\\bfm{\\cal C}}({\\bf x})=\\chi({\\bf x}){\\bfm{\\cal C}}^1+(1-\\chi({\\bf x})){\\bfm{\\cal C}}^2,\n\\eeq{5.2}\nin which ${\\bfm{\\cal C}}^1$ and ${\\bfm{\\cal C}}^2$ are the elasticity tensors of the phases, assumed\nto be isotropic with elements,\n\\begin{equation} {\\cal C}_{ijk\\ell}^h\n=\\mu_h(\\delta_{ik}\\delta_{j\\ell}+\\delta_{i\\ell}\\delta_{jk})+(\\kappa_h-2\\mu_h\/d)\\delta_{ij}\\delta_{k\\ell},\\quad h=1,2,\n\\eeq{5.3}\nin which $d=2$ or 3 is the dimensionality, and \n$\\mu_1,\\mu_2$ and $\\kappa_1,\\kappa_2$ are the shear and bulk moduli of the\ntwo phases. From boundary information on the displacement ${\\bf u}_0({\\bf x})={\\bf u}({\\bf x})$\nand traction ${\\bf f}({\\bf x})=\\bfm\\tau({\\bf x})\\cdot{\\bf n}$\nwe can immediately determine, using integration by parts, \nvolume averages such as\n\\begin{eqnarray} \\langle \\bfm\\epsilon:\\bfm\\tau\\rangle &= & \\frac{1}{|\\Omega|}\\int_{\\partial\\Omega}{\\bf u}\\cdot{\\bf f}, \\nonumber \\\\\n      \\langle \\bfm\\epsilon\\rangle & = & \\frac{1}{|\\Omega|}\\int_{\\partial\\Omega}({\\bf n}{\\bf u}^{T}+{\\bf u}{\\bf n}^{T})\/2, \\nonumber \\\\\n      \\langle \\bfm\\tau\\rangle & = & \\frac{1}{|\\Omega|}\\int_{\\partial\\Omega}{\\bf x}{\\bf f}^T,\n\\eeqa{5.3a}\nin which $\":\"$ denotes a contraction of two indices.\n\nThere are two natural sets of boundary conditions. \nFor any symmetric matrix $\\bfm\\epsilon_0$ we could\nprescribe the special Dirichlet boundary conditions \n\\begin{equation} {\\bf u}({\\bf x})=\\bfm\\epsilon_0{\\bf x},\\quad {\\rm for}~{\\bf x}\\in\\partial\\Omega, \n\\eeq{5.4}\nand measure $\\bfm\\tau_0=\\langle\\bfm\\tau\\rangle$. Here, according to \\eq{5.3a}, $\\bfm\\epsilon_0$ equals $\\langle\\bfm\\epsilon\\rangle$. Since\n$\\bfm\\tau_0$ is linearly related to $\\bfm\\epsilon_0$ we can write\n\\begin{equation} \\bfm\\tau_0={\\bfm{\\cal C}}^D\\bfm\\epsilon_0,\n\\eeq{5.5}  \nwhich defines the elasticity tensor $\\bfm\\sigma^D$ ($D$ for Dirichlet). Alternatively\nfor any symmetric matrix $\\bfm\\tau_0$ we could\nprescribe the special Neumann boundary conditions \n\\begin{equation} \\bfm\\tau({\\bf x})\\cdot{\\bf n}=\\bfm\\tau_0\\cdot{\\bf n},\\quad {\\rm for}~{\\bf x}\\in\\partial\\Omega, \n\\eeq{5.6}\nand measure $\\bfm\\epsilon_0=\\langle\\bfm\\epsilon\\rangle$. Here, according to \\eq{5.3a}, $\\bfm\\tau_0$ equals $\\langle\\bfm\\tau\\rangle$. Since\n$\\bfm\\epsilon_0$ is linearly related to $\\bfm\\tau_0$ we can write\n\\begin{equation} \\bfm\\epsilon_0=({\\bfm{\\cal C}}^N)^{-1}\\bfm\\tau_0, \n\\eeq{5.7}  \nwhich defines the elasticity tensor $\\bfm\\sigma^N$ ($D$ for Dirichlet). It is easy to check\nthat ${\\bfm{\\cal C}}^D$ and ${\\bfm{\\cal C}}^N$ satisfy all the usual symmetries of elasticity tensors.\n\nDirectly analogous to \\eq{2.13} we have the bounds \n\\begin{equation}  \\langle{\\bfm{\\cal C}}^{-1}\\rangle^{-1}\\leq{\\bfm{\\cal C}}^D\\leq \\langle{\\bfm{\\cal C}}\\rangle,\\quad\\quad\n \\langle{\\bfm{\\cal C}}^{-1}\\rangle^{-1}\\leq{\\bfm{\\cal C}}^N\\leq \\langle{\\bfm{\\cal C}}\\rangle\n\\eeq{5.8}\nof \\cite{Nemat-Nasser:1993:MOP}, and directly analogous to \\eq{2.14} \nfor any boundary condition (not just the\nspecial boundary conditions \\eq{5.4} and \\eq{5.6}) we have the bounds\n\\begin{equation} \\langle\\bfm\\epsilon\\cdot\\bfm\\tau\\rangle \\geq \\bfm\\epsilon_0\\cdot\\bfm\\sigma^N\\bfm\\epsilon_0,\\quad\n\\langle\\bfm\\epsilon\\cdot\\bfm\\tau\\rangle \\geq \\bfm\\tau_0(\\bfm\\sigma^D)^{-1}\\bfm\\tau_0,\n\\eeq{5.9}\nwhere $\\bfm\\epsilon_0=\\langle\\bfm\\epsilon\\rangle$ and $\\bfm\\tau_0=\\langle\\bfm\\tau\\rangle$,\ndue to Willis in 1989 private communication to Nemat-Nasser and Hori and presented by \\citeAPY{Nemat-Nasser:1993:MOP}.\n\nAlso directly analogous to \\eq{3.7b} and \\eq{3.16} we have the bounds\n\\begin{equation} {\\bfm{\\cal C}}^N\\leq{\\bfm{\\cal C}}^*\\leq {\\bfm{\\cal C}}^D,\n\\eeq{5.10}\nwhere ${\\bfm{\\cal C}}^*$ is the effective elasticity tensor of any assemblage of\nrescaled copies of $\\Omega$ packed to fill all space. These are \nessentially the bounds of \\citeAPY{Huet:1990:AVC} applied to this assemblage. Thus ``lower\nbounds'' on ${\\bfm{\\cal C}}^*$ directly give ``lower bounds'' on ${\\bfm{\\cal C}}^D$ and\n``upper bounds'' on ${\\bfm{\\cal C}}^*$ directly give ``upper bounds'' on ${\\bfm{\\cal C}}^N$.\nIn particular, in two dimensions lower and upper bounds on $\\bfm\\epsilon_0\\cdot{\\bfm{\\cal C}}^*\\bfm\\epsilon_0$ \nhave been obtained by \\citeAPY{Gibiansky:1984:DCPa} (for the equivalent plate equation) and also by \\citeAPY{Allaire:1993:EOB}.\nAssuming that the phases have been labeled so that $\\mu_1\\geq\\mu_2$ ($\\kappa_1-\\kappa_2$ could be either positive\nor negative)\nand letting $\\epsilon_1$ and $\\epsilon_2$ denote the two eigenvalues $\\bfm\\epsilon_0$,\nthe bounds imply\n\\begin{eqnarray}\n&~& \\bfm\\epsilon_0\\cdot{\\bfm{\\cal C}}^D\\bfm\\epsilon_0  \\geq\n(\\epsilon_1+\\epsilon_2)^2\/(f_1\/\\kappa_1+f_2\/\\kappa_2)\n+(\\epsilon_1-\\epsilon_2)^2\/(f_1\/\\mu_1+f_2\/\\mu_2), \\nonumber \\\\\n&~& \\quad \\quad {\\rm if~~} |\\kappa_1-\\kappa_2|(f_1\\mu_2+f_2\\mu_1)|\\epsilon_1+\\epsilon_2|\\leq\n |\\mu_1-\\mu_2|(f_1\\kappa_2+f_2\\kappa_1)|\\epsilon_1-\\epsilon_2|; \\nonumber \\\\\n&~& \\nonumber \\\\ &~&\n\\bfm\\epsilon_0\\cdot{\\bfm{\\cal C}}^D\\bfm\\epsilon_0 \\geq (\\epsilon_1+\\epsilon_2)^2(f_1\\kappa_1+f_2\\kappa_2)\n+(\\epsilon_1-\\epsilon_2)^2(f_1\\mu_1+f_2\\mu_2) \\nonumber \\\\\n&~&~~~~~~~~~~~~~~~~~~~-f_1f_2\\frac{[|\\kappa_1-\\kappa_2||\\epsilon_1+\\epsilon_2|+|\\mu_1-\\mu_2||\\epsilon_1-\\epsilon_2|]^2}\n{f_1(\\mu_2+\\kappa_2)+f_2(\\mu_1+\\kappa_1)}, \\nonumber \\\\\n&~&\\quad \\quad {\\rm if~~} (\\mu_2+f_1\\kappa_2+f_2\\kappa_1)|\\epsilon_1-\\epsilon_2| \\geq\nf_2|\\kappa_1-\\kappa_2||\\epsilon_1+\\epsilon_2| \\nonumber \\\\\n&~& \\quad \\quad{\\rm and~~}  |\\kappa_1-\\kappa_2|(f_1\\mu_2+f_2\\mu_1)|\\epsilon_1+\\epsilon_2|\\geq\n |\\mu_1-\\mu_2|(f_1\\kappa_2+f_2\\kappa_1)|\\epsilon_1-\\epsilon_2|; \\nonumber \\\\\n&~& \\nonumber \\\\ &~&\n \\bfm\\epsilon_0\\cdot{\\bfm{\\cal C}}^D\\bfm\\epsilon_0 \\geq\n\\mu_2(\\epsilon_1-\\epsilon_2)^2+\n\\frac{\\kappa_1\\kappa_2+\\mu_2(f_1\\kappa_1+f_2\\kappa_2)}{\\mu_2+f_1\\kappa_2+f_2\\kappa_1}(\\epsilon_1+\\epsilon_2)^2, \\nonumber \\\\\n&~ & \\quad \\quad{\\rm if~~} (\\mu_2+f_1\\kappa_2+f_2\\kappa_1)|\\epsilon_1-\\epsilon_2| \\leq\nf_2|\\kappa_1-\\kappa_2||\\epsilon_1+\\epsilon_2|;\n\\eeqa{5.12}\nand\n\\begin{eqnarray}\n&~& \\bfm\\epsilon_0\\cdot{\\bfm{\\cal C}}^N\\bfm\\epsilon_0  \\leq (\\epsilon_1+\\epsilon_2)^2(f_1\\kappa_1+f_2\\kappa_2)\n+(\\epsilon_1-\\epsilon_2)^2(f_1\\mu_1+f_2\\mu_2) \\nonumber \\\\\n&~&~~~~~~~~~~~~~~~~~~~-f_1f_2\\frac{[|\\kappa_1-\\kappa_2||\\epsilon_1+\\epsilon_2|-|\\mu_1-\\mu_2||\\epsilon_1-\\epsilon_2|]^2}\n{f_1(\\mu_2+\\kappa_2)+f_2(\\mu_1+\\kappa_1)}, \\nonumber \\\\\n&~&\\quad \\quad {\\rm if~~~} f_1|\\kappa_1-\\kappa_2||\\epsilon_1+\\epsilon_2|\\leq (\\mu_1+f_1\\kappa_2+f_2\\kappa_1)|\\epsilon_1-\\epsilon_2| \\nonumber \\\\\n&~&\\quad \\quad {\\rm and~~} f_+|\\mu_1-\\mu_2||\\epsilon_1-\\epsilon_2|\\leq (\\kappa_++f_1\\mu_2+f_2\\mu_1)|\\epsilon_1+\\epsilon_2|; \\nonumber \\\\\n&~& \\nonumber \\\\ &~&\n\\bfm\\epsilon_0\\cdot{\\bfm{\\cal C}}^N\\bfm\\epsilon_0  \\leq \\mu_1(\\epsilon_1-\\epsilon_2)^2+\\frac{\\kappa_1\\kappa_2+\\mu_1(f_1\\kappa_1+f_2\\kappa_2)}{\\mu_1+f_1\\kappa_2+f_2\\kappa_1}(\\epsilon_1+\\epsilon_2)^2, \\nonumber \\\\\n&~&\\quad \\quad {\\rm if~~~} f_1|\\kappa_1-\\kappa_2||\\epsilon_1+\\epsilon_2|\\geq (\\mu_1+f_1\\kappa_2+f_2\\kappa_1)|\\epsilon_1-\\epsilon_2|; \\nonumber \\\\\n&~& \\nonumber \\\\ &~&\n\\bfm\\epsilon_0\\cdot{\\bfm{\\cal C}}^N\\bfm\\epsilon_0  \\leq \\kappa_+(\\epsilon_1+\\epsilon_2)^2+\\frac{\\mu_1\\mu_2+\\kappa_+(f_1\\mu_1+f_2\\mu_2)}{\\kappa_++f_1\\mu_2+f_2\\mu_1}(\\epsilon_1-\\epsilon_2)^2, \\nonumber \\\\\n&~&\\quad \\quad {\\rm if~~~} f_+|\\mu_1-\\mu_2||\\epsilon_1-\\epsilon_2|\\geq (\\kappa_++f_1\\mu_2+f_2\\mu_1)|\\epsilon_1+\\epsilon_2|,\n\\eeqa{5.12a}\nwhere $\\kappa_+$ is the maximum of $\\kappa_1$ and $\\kappa_2$ and $f_+$ is the volume fraction of the material corresponding to $\\kappa_+$.\n\n\nThe corresponding three-dimensional bounds follow directly from\n\\eq{5.10} and the bounds of \\citeAPY{Allaire:1993:OBE}, but are not so explicit. Assuming\nthat the Lame moduli\n\\begin{equation} \\lambda_1=\\kappa_1-2\\mu_1\/3~~{\\rm and}~~\\lambda_2=\\kappa_2-2\\mu_2\/3 \n\\eeq{5.13}\nof both phases are positive, and that the bulk and shear moduli of the two phases are well-ordered with\n\\begin{equation} \\kappa_1>\\kappa_2>0 {\\rm ~~and~~}\\mu_1>\\mu_2>0, \n\\eeq{5.13a}\nthese bounds are\n\\begin{eqnarray}\n\\bfm\\epsilon_0:{\\bfm{\\cal C}}^D\\bfm\\epsilon_0 & \\geq & \\bfm\\epsilon_0:{\\bfm{\\cal C}}_2\\bfm\\epsilon_0\n+f_1\\max_{\\bfm\\eta}[2\\bfm\\epsilon_0:\\bfm\\eta-\\bfm\\eta:({\\bfm{\\cal C}}_1-{\\bfm{\\cal C}}_2)^{-1}\\bfm\\eta-f_2g(\\bfm\\eta)], \\nonumber \\\\\n\\bfm\\epsilon_0:{\\bfm{\\cal C}}^N\\bfm\\epsilon_0 & \\geq & \\bfm\\epsilon_0:{\\bfm{\\cal C}}_1\\bfm\\epsilon_0\n+f_2\\min_{\\bfm\\eta}[2\\bfm\\epsilon_0:\\bfm\\eta+\\bfm\\eta:({\\bfm{\\cal C}}_1-{\\bfm{\\cal C}}_2)^{-1}\\bfm\\eta-f_1h(\\bfm\\eta)],\n\\eeqa{5.14}\nwhere $g(\\bfm\\eta)$ and $h(\\bfm\\eta)$ are function of the eigenvalues $\\eta_1, \\eta_2$, and $\\eta_3$ \nof the symmetric matrix $\\bfm\\eta$.\nAssuming that\nthese are labeled with\n\\begin{equation} \\eta_1\\leq\\eta_2\\leq\\eta_3, \n\\eeq{5.15}\nwe have\n\\begin{eqnarray}\ng(\\bfm\\eta)&=&\n\\frac{(\\eta_1-\\eta_3)^2}{4\\mu_2}+\\frac{(\\eta_1+\\eta_3)^2}{4(\\lambda_2+\\mu_2)}  ~~{\\rm if}~~\n\\eta_3\\geq\\frac{\\lambda_2+2\\mu_2}{2(\\lambda_2+\\mu_2)}(\\eta_1+\\eta_3)\\geq\\eta_1, \\nonumber \\\\\ng(\\bfm\\eta)&=&\n\\frac{\\eta_1^2}{\\lambda_2+2\\mu_2}  ~~{\\rm if}~~\n\\eta_1>\\frac{\\lambda_2+2\\mu_2}{2(\\lambda_2+\\mu_2)}(\\eta_1+\\eta_3), \\nonumber \\\\\ng(\\bfm\\eta)&=&\n\\frac{\\eta_3^2}{\\lambda_2+2\\mu_2}  ~~{\\rm if}~~\n\\eta_3<\\frac{\\lambda_2+2\\mu_2}{2(\\lambda_2+\\mu_2)}(\\eta_1+\\eta_3),\n\\eeqa{5.16}\nand\n\\begin{equation} h(\\bfm\\eta)=\\frac{1}{\\lambda_1+2\\mu_1}\\min\\{\\eta_1^2,\\eta_2^2,\\eta_3^2\\}.\n\\eeq{5.17}\n\nThe bounds \\eq{5.12}, \\eq{5.12a} and \\eq{5.14} can be used in an inverse way to bound\nthe volume fraction $f_1=1-f_2$, for a single experiment when for special Dirichlet conditions\n$\\bfm\\epsilon_0$ is prescribed and $\\bfm\\tau_0$ ($={\\bfm{\\cal C}}^D\\bfm\\epsilon_0$) is measured, or when for special Neumann conditions\n$\\bfm\\tau_0$ is prescribed and $\\bfm\\epsilon_0$ ($={\\bfm{\\cal C}}^N\\bfm\\epsilon_0$) is measured. \\citeAPY{Allaire:1993:OBE} also derive\nbounds on the complementary energy and these imply\n\\begin{equation} \\bfm\\tau_0:({\\bfm{\\cal C}}^N)^{-1}\\bfm\\tau_0  \\geq  \\bfm\\tau_0:{\\bfm{\\cal C}}_1^{-1}\\bfm\\tau_0\n+f_2\\max_{\\bfm\\zeta}[2\\bfm\\tau_0:\\bfm\\zeta-\\bfm\\zeta:({\\bfm{\\cal C}}_2^{-1}-{\\bfm{\\cal C}}_1^{-1})^{-1}\\bfm\\zeta-f_1\\bfm\\zeta:{\\bfm{\\cal C}}_1\\bfm\\zeta+f_1h({\\bfm{\\cal C}}_1\\bfm\\zeta)], \\nonumber \\\\\n\\eeq{5.17aa}\nand\n\\begin{equation}\n\\bfm\\tau_0:({\\bfm{\\cal C}}^D)^{-1}\\bfm\\tau_0  \\leq  \\bfm\\tau_0:{\\bfm{\\cal C}}_2^{-1}\\bfm\\tau_0\n+f_1\\min_{\\bfm\\zeta}[2\\bfm\\tau_0:\\bfm\\zeta+\\bfm\\zeta:({\\bfm{\\cal C}}_2^{-1}-{\\bfm{\\cal C}}_1^{-1})^{-1}\\bfm\\zeta-f_2\\bfm\\zeta:{\\bfm{\\cal C}}_2\\bfm\\zeta+f_2g({\\bfm{\\cal C}}_2\\bfm\\zeta)].\n\\eeq{5.17ab}\n\nThe bound in \\eq{5.17aa} is particularly useful when $\\bfm\\tau_0=-p{\\bf I}$, corresponding to immersing the body $\\Omega$ in a fluid\nwith pressure $p$. Then from measurements of the resulting volume change of the body one can determine $\\bfm\\tau_0:({\\bfm{\\cal C}}^N)^{-1}\\bfm\\tau_0=-p\\mathop{\\rm Tr}\\nolimits\\bfm\\epsilon_0$.\nLet us assume $\\lambda_1>0$ and set $\\bfm\\zeta=\\alpha{\\bf I}+{\\bf A}$, with ${\\bf A}$ being a trace free matrix with eigenvalues $a_1$, $a_2$ and $a_3$. Then \nwe have $\\bfm\\eta={\\bfm{\\cal C}}_1\\bfm\\zeta=2\\mu_1(k{\\bf I}+{\\bf A})$ where $k=\\alpha[1+3\\lambda_1\/(2\\mu_1)]$. Substitution gives\n\\begin{eqnarray} &~&[\\bfm\\zeta:{\\bfm{\\cal C}}_1\\bfm\\zeta-h({\\bfm{\\cal C}}_1\\bfm\\zeta)]-\\alpha^2[{\\bf I}:{\\bfm{\\cal C}}_1{\\bf I}-h({\\bfm{\\cal C}}_1{\\bf I})] \\nonumber \\\\\n&~&~=2\\mu_1\\left[a_1^2+a_2^2+a_3^2-\\frac{2\\mu_1}{\\lambda_1+2\\mu_1}\\min\\{(k+a_1)^2-k^2,(k+a_2)^2-k^2,(k+a_3)^2-k^2\\}\\right] \\nonumber \\\\\n&~&~\\geq 2\\mu_1[a_1^2+a_2^2+a_3^2-\\min\\{2a_1k+a_1^2,2a_2k+a_2^2,2a_3k+a_3^2\\}],\n\\eeqa{5.17ac}\nwhich is surely positive since $\\min\\{2a_1k+a_1^2,2a_2k+a_2^2,2a_3k+a_3^2\\}\\leq a_j^2$ where $j$ is such that $ka_j$ is non-positive. (Note that\n$a_1$, $a_2$ and $a_3$ cannot all have the same sign since they sum to zero). Consequently when $\\bfm\\tau_0=-p{\\bf I}$\nthe maximum over $\\bfm\\zeta$ in \\eq{5.17aa} is achieved when ${\\bf A}=0$ and taking the maximum over $\\alpha$ gives\n\\begin{equation} -p\\mathop{\\rm Tr}\\nolimits\\bfm\\epsilon_0 \\geq p^2\\left[\\frac{1}{\\kappa_1}+\\frac{f_2}{\\frac{\\kappa_1\\kappa_2}{\\kappa_1-\\kappa_2}+\\frac{4f_1\\mu_1\\kappa_1}{3\\kappa_1+4\\mu_1}}\\right],\n\\eeq{5.17ad}\nor equivalently\n\\begin{equation} -p\/(\\mathop{\\rm Tr}\\nolimits\\bfm\\epsilon_0)\\leq \\kappa_{HSH}^+\\equiv\\kappa_1-\\frac{f_2}{1\/(\\kappa_1-\\kappa_2)-f_1\/(\\kappa_1+4\\mu_1\/3)}, \\eeq{5.17ae}\nwhere $\\kappa_{HSH}^+$ is the upper bulk modulus bound of \\citeAPY{Hashin:1963:VAT} and\n\\citeAPY{Hill:1963:EPR}. The inequality \\eq{5.17ad} can be \nrewritten as\n\\begin{equation} \\frac{f_2}{-(\\mathop{\\rm Tr}\\nolimits\\bfm\\epsilon_0\/p)-1\/\\kappa_1}\\leq \\frac{\\kappa_1\\kappa_2}{\\kappa_1-\\kappa_2}+\\frac{4f_1\\mu_1\\kappa_1}{3\\kappa_1+4\\mu_1},\n\\eeq{5.17af}\nwhere we have used the fact that $-(\\mathop{\\rm Tr}\\nolimits\\bfm\\epsilon_0\/p)-1\/\\kappa_1$ is positive (since $\\bfm\\tau_0:({\\bfm{\\cal C}}^N)^{-1}\\bfm\\tau_0\\geq \\bfm\\tau_0:({\\bfm{\\cal C}}_1)^{-1}\\bfm\\tau_0$\nby \\eq{5.8}). This then yields the volume fraction bound\n\\begin{equation} f_2\\leq \\frac{\\frac{\\kappa_1\\kappa_2}{\\kappa_1-\\kappa_2}+\\frac{4\\mu_1\\kappa_1}{3\\kappa_1+4\\mu_1}}{\\frac{1}{-(\\mathop{\\rm Tr}\\nolimits\\bfm\\epsilon_0\/p)-1\/\\kappa_1}+\\frac{4\\mu_1\\kappa_1}{3\\kappa_1+4\\mu_1}},\n\\eeq{5.18ag}\nwhich we expect to be closest to the actual volume fraction when phase 2 (the softer phase) \nis the inclusion phase. Thus the\nbound may be particularly effective for estimating the volume of cavities in a body. Note that\nif some granules of phase 1 lie within these cavities, then such granules will not\ncontribute to this volume fraction estimate, but will contribute to the overall weight.\nIf the weight of the body has been measured (and the density of phase 1 is known)\nthis provides a way of estimating the volume of granules of phase 1 which lie within the\ncavities. \n\nWhen multiple experiments have been\ndone, and the full tensor ${\\bfm{\\cal C}}^D$ or ${\\bfm{\\cal C}}^N$ has been determined, then the ``trace bounds'' of\nZhikov(\\citeyearNP{Zhikov:1988:ETA}, \\citeyearNP{Zhikov:1991:EHM}) and \\citeAPY{Milton:1988:VBE} can be used. \n(These generalize the well known Hashin-Shtrikman (\\citeyearNP{Hashin:1963:VAT})\nbounds to anisotropic elastic composites.) Define the two traces\n\\begin{equation} \\mathop{\\rm Tr}\\nolimits_h{\\bfm{\\cal A}}=A_{iijj}\/d,\\quad\\quad Tr_s{\\bfm{\\cal A}}=A_{ijij}-(A_{iijj}\/d),\n\\eeq{5.17a}\nfor any fourth order tensor ${\\bfm{\\cal A}}$ with elements $A_{ijk\\ell}$ in spatial dimension $d$.\nThen, assuming the moduli of the two-phases are well ordered satisfying \\eq{5.13a},\ntheir lower and upper ``bulk modulus type bounds''\nimply, through \\eq{5.10}, the universal bounds\n\\begin{eqnarray} f_1\\mathop{\\rm Tr}\\nolimits_h[({\\bfm{\\cal C}}^D-{\\bfm{\\cal C}}_2)^{-1}]\n& \\leq & \\frac{1}{d(\\kappa_1-\\kappa_2)}+\\frac{f_2}{d\\kappa_2+2(d-1)\\mu_2}, \\nonumber \\\\\nf_2\\mathop{\\rm Tr}\\nolimits_h[({\\bfm{\\cal C}}_1-{\\bfm{\\cal C}}^N)^{-1}]\n& \\leq & \\frac{1}{d(\\kappa_1- \\kappa_2)}-\\frac{f_1}{d\\kappa_1+2(d-1)\\mu_1},\n\\eeqa{5-18}\nwhile their lower and upper ``shear modulus type bounds,'' imply the universal bounds\n\\begin{eqnarray}  f_1\\mathop{\\rm Tr}\\nolimits_s[({\\bfm{\\cal C}}^D-{\\bfm{\\cal C}}_2)^{-1}] & \\leq &\n\\frac{(d-1)(d+2)}{4(\\mu_1-\\mu_2)}+\\frac{d(d-1)(\\kappa_2+2\\mu_2)f_2}{2\\mu_2(d\\kappa_2+2(d-1)\\mu_2)},\\nonumber \\\\\nf_2\\mathop{\\rm Tr}\\nolimits_s[({\\bfm{\\cal C}}_1-{\\bfm{\\cal C}}^N)^{-1}] & \\leq &\n\\frac{(d-1)(d+2)}{4(\\mu_1-\\mu_2)}-\\frac{d(d-1)(\\kappa_1+2\\mu_1)f_1}{2\\mu_1(d\\kappa_1+2(d-1)\\mu_1)}.\n\\eeqa{5.19}\nSince these inequalities depend linearly on $f_1=1-f_2$ they can easily be inverted to obtain\nbounds on $f_1$ given ${\\bfm{\\cal C}}^D$ or ${\\bfm{\\cal C}}^N$.\n\nAs noted by \\citeAPY{Milton:1988:VBE} the lower and upper ``bulk modulus type bounds'' are tighter than those obtained by \\citeAPY{Kantor:1984:IRB} and\n\\citeAPY{Francfort:1986:HOB}, which imply\n\\begin{eqnarray} \\mathop{\\rm Tr}\\nolimits_h{\\bfm{\\cal C}}^N&\\leq& d\\kappa_1-\\frac{f_2}{\\frac{1}{d(\\kappa_1-\\kappa_2)}-\\frac{f_1}{d\\kappa_1+2(d-1)\\mu_1}}, \\nonumber \\\\\n1\/\\mathop{\\rm Tr}\\nolimits_h[({\\bfm{\\cal C}}^D)^{-1}]&\\geq& d\\kappa_2+\\frac{f_1}{\\frac{1}{d(\\kappa_1-\\kappa_2)}+\\frac{f_2}{d\\kappa_2+2(d-1)\\mu_2}}.\n\\eeqa{5.20}\nFor bodies $\\Omega$ of ellipsoidal or parallelopipedic shape the universal bounds \\eq{5.20} were \nobtained by Nemat-Nasser and Hori (\\citeyearNP{Nemat-Nasser:1993:MOP}, \\citeyearNP{Nemat-Nasser:1995:UBO}):\nsee the equations (4.3.9) and (4.4.8), with $I=1$, in their 1995 paper. Their other bounds, with $I=2$,\nwhich incorporate the ``shear responses'' of the tensors ${\\bfm{\\cal C}}^N$ and ${\\bfm{\\cal C}}^D$ are improved upon by the bounds \\eq{5.19}\nas can be seen using the inequality \n\\begin{equation} \\mathop{\\rm Tr}\\nolimits_s{\\bfm{\\cal A}}^{-1}\\geq (d-1)^2(d+2)^2\/(4\\mathop{\\rm Tr}\\nolimits_s{\\bfm{\\cal A}}), \\eeq{5.21}\nwhich holds for any positive definite fourth-order tensor ${\\bfm{\\cal A}}$. \n\n\\section*{Acknowledgements}\nEunjoo Kim is deeply thanked for generously providing figures \\ref{3}, \\ref{4}, and \\ref{5}, and for doing the numerical \nsimulations which generated them. Additionally the author is grateful to Hyeonbae Kang and \nMichael Vogelius for stimulating\nhis interest in this problem, and for their comments on an initial draft of the manuscript. \nThe author is most thankful for support from the Mathematical Sciences Research Institute and the Simons foundation, \nthrough an Eisenbud fellowship, and from\nNational Science Foundation through grant DMS-0707978. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Appendix: Volterra equation for the cavity amplitude}\n\\normalsize\nWe start from the Hamiltonian (1) of the main article and derive the Heisenberg operator equations (limit of zero temperature), for the cavity and spin operators, $\\dot a=i [{\\cal H},a]-\\kappa a$, $\\dot \\sigma_k^-=i [{\\cal H},\\sigma_k^-]-\\gamma \\sigma_k^-$, respectively. Here $\\kappa$ and $\\gamma$ stand for the total cavity and spin losses, respectively. We then write a set of equations for the expectation values in the frame rotating with the probe frequency $\\omega_p$, using the commonly used Holstein-Primakoff-approximation, $\\langle \\sigma_k^z \\rangle \\approx -1$, which is valid if the number of the excited spins is small compared to the ensemble size (which is the case for all experimental results reported in the main article). Denoting $A(t)\\equiv \\langle a(t)\\rangle$ and $B_k(t)\\equiv\\langle\\sigma_k^-(t)\\rangle$, we end up with the following set of first-order ODEs with respect to the cavity and spin amplitudes\n\\begin{subequations}\n\\begin{eqnarray}\n\\label{Eq_a_Volt}\n\\dot{A}(t) & = & -\\left[\\kappa-i(\\omega_c-\\omega_p)\\right]A(t) + \\sum_k\ng_k  B_k(t)-\\eta(t), \\\\\n\\label{Eq_bk_Volt}\n\\dot{B}_k(t) & = & -\\left[\\gamma+i(\\omega_k-\\omega_p)\\right] B_k(t) - g_k A(t).\n\\end{eqnarray}\n\\end{subequations}\nNote, that the size of our spin ensemble is very large (typically $N\\sim 10^{12}$) and individual spins are distributed around a certain mean frequency $\\omega_s$. We can thus go to the continuum limit by introducing the continuous spectral density as $\\rho(\\omega)=\\sum_k g_k^2 \\delta(\\omega-\\omega_k)\/\\Omega^2$ (see, e.g. \\onlinecite{Diniz2011}), where $\\Omega$ is the collective coupling strength of the spin ensemble to the cavity and $\\int d\\omega\\rho(\\omega)=1$. In what follows we will replace any discrete function $F(\\omega_k)$ by its continuous counterpart, $F(\\omega)$: $F(\\omega_k) \\rightarrow \\Omega^2 \\int d\\omega \\rho(\\omega) F(\\omega)$. By integrating Eq.~(\\ref{Eq_bk_Volt}) in time, each individual spin amplitude, $B_k(t)$, can formally be expressed in terms of the cavity amplitude, $A(t)$. By plugging the resulting equation into Eq.~(\\ref{Eq_a_Volt}) and assuming that initially all spins are in the ground state, $B_k(t=0)=0$, we arrive at the following integro-differential Volterra equation for the cavity amplitude ($\\omega_c=\\omega_s$)\n\\begin{eqnarray}\n\\dot A(t)=-\\kappa A(t)-\\Omega^2 \\int d\\omega \\rho(\\omega) \\int\\limits_{0}^t d\\tau\ne^{-i(\\omega-\\omega_c-i\\gamma)(t-\\tau)}A(\\tau)-\\eta(t), \n\\label{Eq_rigor}\n\\end{eqnarray}\nNote that in the $\\omega_p$-rotating frame the rapid oscillations presented in the original Hamiltonian (1) are absent, so that the time variation of $\\eta(t)$ in Eq.~(\\ref{Eq_rigor}) is much slower as compared to $1\/\\omega_p$. \n\nFor a proper description of the resulting dynamics, it is essential to capture the form of the spectral density $\\rho(\\omega)$ realized in the experiment as accurately as possible. Following \\onlinecite{Sandner2012}, we take the $q$-Gaussian function for that purpose \n\\begin{eqnarray}\n\\label{rho_w_Eq}\n\\rho(\\omega)=C\\cdot\\left[1-(1-q)\\dfrac{(\\omega-\\omega_s)^2}{\\Delta^2}\\right]^{\n\\dfrac{1}{1-q}},\n\\end{eqnarray}\ncharacterized by the dimensionless shape parameter $1<q<3$ which yields the form of a Lorentzian and Gaussian distribution, for $q=2$ and for $q\\rightarrow 1$, respectively.  Here $C$ is a normalization constant which is easily obtained numerically; the full-width at half-maximum (FWHM) of $\\rho(\\omega)$ is given by $\\gamma_q=2\\Delta\\sqrt{\\dfrac{2^q-2}{2q-2}}$.\n\nAfter integrating Eq.~(\\ref{Eq_rigor}) in time, performing some algebraic manipulations and assuming that the cavity is initially empty, $A(t=0)=0$, we derive the following equation for the cavity amplitude\n\\begin{eqnarray}\n\\label{Volt_eq}\nA(t)=\\int\\limits_0^t d\\tau {\\cal K}(t-\\tau) A(\\tau)+{\\cal F}(t),\n\\end{eqnarray}\nwhich contains the kernel function ${\\cal K}(t-\\tau)$,\n\\begin{eqnarray}\n\\label{Volt_eq_K}\n{\\cal K}(t-\\tau)=\\Omega^2 \\bigintsss\\!\\!\\! d\\omega\\,\n\\dfrac{\\rho(\\omega) \\left[e^{-i (\\omega-\\omega_c-i (\\gamma-\\kappa))(t-\\tau)}-1\\right]\n}{i (\\omega-\\omega_c-i (\\gamma-\\kappa))}\\cdot e^{-\\kappa(t-\\tau)},\n\\end{eqnarray}\nand the function ${\\cal F}(t)$,\n\\begin{eqnarray}\n\\label{Volt_eq_F}\n{\\cal F}(t)=\\int\\limits_0^t d\\tau\\, \\eta(\\tau)\\cdot e^{-\\kappa(t-\\tau)}.\n\\end{eqnarray}\nDespite its seemingly simple form, Eq.~(\\ref{Volt_eq}) is not trivial to solve in practice, even numerically. The reasons are twofold: First, the result of the integration for $A(t)$ at time $t$ depends on the amplitude $A(\\tau)$ calculated at all earlier times, $\\tau<t$ (memory effect). Second, the kernel function ${\\cal K}(t-\\tau)$ contains the integration with respect to frequency, which is costly in terms of computational time. (Note that such an integration has to be performed for each $t$ and $\\tau<t$.) The smallest possible time scale in our problem is given by $T=2\\pi\/\\omega_p\\sim 0.4\\,$ns. To achieve a very good accuracy of the calculations for the results presented in Figs.~2,4 from the main article, we solve the equation on a mesh with uniform spacing, choosing a time step $dt \\sim 0.05\\,$ns (see e.g. \\onlinecite{NumRec} for more details about the method). The direct discretization of ${\\cal K}(t-\\tau)$ on the time interval of the order of $\\upmu$s (typical time of measurements) leads to a high-dimensional matrix (of a size typically exceeding $10^4 \\times 10^4$), which, together with the integration with respect to frequency, makes the problem computationally intractable by way of a direct numerical solution. To overcome this problem and to speed up the calculations drastically, we divide the whole time integration into many successive subintervals, $T_{ n}\\leq t \\leq T_{n+1}$, with $n=1,2,...$. Such a time division might, in principle, be implemented arbitrarily but we choose it to be adapted to our experimental realization. Specifically, the driving amplitude is unchanged within each subinterval, so that in our case it is given by\n\\begin{equation} \n\\eta_n=\n\\begin{cases}\n\\eta & n=1,3,5,...\\\\ -\\eta & n=2,4,6,...\n\\end{cases}.\n\\end{equation}\n\nIn this way the result of integration at the $n$-th time interval, $A^{(n)}(T_{n+1})$, enters as an initial condition for the integration during the $(n+1)$-th time interval, $A^{(n+1)}(T_{n+1})$. Finally, we end up with the following recurrence relation (time runs within $T_{ n}\\leq t \\leq T_{n+1}$ for $n=1,2,3,...$)\n\\begin{eqnarray}\n\\label{Volt_eq_itter}\nA^{(n)}(t)=\\int\\limits_{T_n}^t d\\tau {\\cal K}(t-\\tau) A^{(n)}(\\tau)+{\\cal F}^{(n)}(t),\n\\end{eqnarray}\nwhere the kernel function ${\\cal K}(t-\\tau)$ is defined by Eq.~(\\ref{Volt_eq_K}) and\n\\begin{eqnarray}\n\\nonumber\n{\\cal F}^{(n)}(t)&=&A^{(n-1)}(T_n)e^{-\\kappa(t-T_n)}+\\Omega^2 e^{-\\kappa(t-T_n)}\n\\bigintsss\\!\\!\\! d\\omega\\,\n\\dfrac{\\rho(\\omega) \\left[e^{-i (\\omega-\\omega_c-i  (\\gamma-\\kappa))(t-T_n)}-1\\right]\n}{i (\\omega-\\omega_c-i (\\gamma-\\kappa))}\\cdot {\\cal I}_n(\\omega)+\n\\\\\n&&\\dfrac{{\\cal\\eta}_n}{\\kappa}\\cdot\\left[1-e^{-\\kappa(t-T_n)} \\right]\n\\end{eqnarray}\nRemarkably, the memory about previous events enters both through the amplitude $A^{(n-1)}(T_n)$ and through the function  \n\\begin{eqnarray}\n{\\cal I}_n(\\omega)=e^{-i(\\omega-\\omega_p-i\\gamma)(T_n-T_{n-1})}{\\cal I}_{n-1}(\\omega)+\n\\int\\limits_{T_{n-1}}^{T_n} d\\tau e^{-i(\\omega-\\omega_p-i\\gamma)(T_n-\\tau)}A^{(n-1)}(\\tau).\n\\end{eqnarray}\nIn accordance with the initial conditions introduced above ($t=T_1=0$), $A(T_1)=0$ and $ {\\cal I}_1(\\omega)=0$.\n\nThe above technique allows us to solve Eq.~(\\ref{Volt_eq}) accurately while being very efficient in terms of computational time. We have tested the accuracy of our numerical results by varying the discretization both in time and frequency in a wide range obtaining excellent agreement with the experimental results shown in Figs.~2,4 of the main paper and thereby confirming the accuracy of our method.\n\n\n\\textbf{~\\\\Acknowledgements }\n\\label{par:ack}\nWe would like to thank C. Koller, F. Mintert, P. Rabl, H. Ritsch, K. Sandner, and M. Trupke for helpful discussions. \nThe experimental effort has been supported by the TOP grant of TU Vienna. S.P. acknowledges support by the Austrian Science Fund (FWF) through Project No.~W1243 (Solids4Fun), D.K. and S.R. through Projects No. F25-P14 (SFB IR-ON), No. F49-P10 (SFB NextLite).\n\n\\textbf{~\\\\Author contributions }\n\\label{par:contrib}\nS.P., R.A., A.V., T.N., J.S, and J.M. designed and set up  the experiment. S.P., R.A., and A.V. carried out the measurements with supervision by J.M.. D.O.K. and S.R. devised the theoretical framework and performed the calculations. S.P., D.O.K., S.R. and J.M. wrote the first manuscript draft to which all authors suggested improvements.\n\n\\label{biblio}\n\\bibliographystyle{plain}\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction}\n\\label{sec:intro}\n\nIn the era of big data, private data are often scattered among local clients (e.g., companies, mobile devices), giving arise to the problem of isolated data islands\\cite{yang2019FLbook}. To fully capitalize on the value of big data while protecting the data privacy and security, federated learning (FL) has attracted significant interests from both academic and industry sides\\cite{mcmahan17,yang2019federated,kairouz2019advances,yang2019FLbook,Li20FL}. Similar to the traditional distributed learning system, a typical architecture of FL consists of a parameter server (PS) and a number of distributed clients, with the key difference that local training data in FL are usually prohibited from sharing among the clients. The general goal of FL is to jointly train a global model that has high generalization ability than that only trained on local data. While FL systems allow clients to keep their private data local, a significant vulnerability arises when a subset of clients - which are modeled as Byzantine clients \\cite{Lampor82byzantine,Blanchard17Byz,Bhagoji19adverlens} - aim to prevent successful training of global model or make some targeted samples misclassified during inference phase\\cite{kairouz2019advances,lyu20survey, lyu20privacy_robust,boli21datasecurity}. It has been shown that mitigating Byzantine model poisoning attacks is crucial for robust FL and other distributed learning\\cite{shenTS16auror,Blanchard17Byz,Fang20Local}. On the other hand, distributed implementation of gradient-based learning algorithms\\cite{lecun1998gd} are increasingly popular for training large-scale models on distributed datasets, e.g., deep neural networks for human face identification and news sentimental analysis \\cite{krizhevsky2012imagenet,hinton2012speech,dean2012large}. Therefore, many efforts have been devoted to developing robust \\textbf{\\underline{g}}radient \\textbf{\\underline{a}}ggregation \\textbf{\\underline{r}}ules (GAR)\\cite{kairouz2019advances} to achieve Byzantine-robust FL algorithms.\n\nIn this paper, we focus on gradient-based FL systems and propose a novel defense that is capable of detecting state-of-the-art Byzantine model poisoning attacks. We note that Byzantine clients can send arbitrary model update vectors to the PS, which may significantly poison the training process if not identified and removed by PS. This can be seen through a simple example shown in Fig.~\\ref{fig:example} with $1$ PS and $n$ benign clients as well as $m$ Byzantine clients. During federated learning, the selected clients (including both benign and Byzantine ones) pull the updated model from the PS, then compute and send local gradients to the PS in parallel, while the PS aggregates the collected gradients to update the global model\\cite{mcmahan17,yang2019federated}. For full participation setting, PS can also directly broadcast the aggregated global gradient back to the clients to update their local parameters. When the benign clients are the majority, theoretically we can always distinguish the minority as outliers, i.e., malicious gradients, according to the principle of majority-vote\\cite{bernstein18sign,Yin18optimalrate}. For partial participation setting, we need to assume that the selected benign clients are always the majority in every communication round, otherwise it's impossible to reliably distinguish the honest gradients from the malicious ones without other information. Thus, we mainly focus on the full participation setting, where the PS collects gradient from all clients in each round. And we do not assume any trusty gradient or any other validation data available in PS. In the rest of this paper, we will use Byzantine and malicious clients interchangeably.\n\n\\begin{figure}[htbp]\t\n\t{\t\n\t\t\\begin{center}\n\t\t\n\t\t\t\\includegraphics[width=1.0\\columnwidth,clip=true]{.\/figs\/example_intro.pdf}\n\t\t\\end{center}\n\t}\n\n\t\\caption{Federated learning system: one parameter server with $n$ clients, in which a attacker controls $m$ Byzantine clients to attack the learning system.}\n\n\t\\label{fig:example}\n\\end{figure}\n\nRecently, much research attention has focused on mitigating Byzantine attack either by leveraging statistic-based outlier detection techniques \\cite{Chen17BGD,Yin18optimalrate} or by utilizing auxiliary labeled data collected by PS to verify the correctness of received gradients \\cite{Xie19Zeno,CaoL19arbitraryByz}. While in some FL systems, it is possible to collect a small amount of data from volunteer clients or external datasets to approximate the true data distribution \\cite{cao20FLTrust}, we note that auxiliary data sufficiently capturing the global data distribution may not always be practicable to PS. Therefore, we focus on the statistic-based approaches that leverage various types of statistics, e.g. magnitude and similarity, to identify and remove malicious gradients sent by Byzantine clients. Meanwhile, recent works have shown that existing statistic-based aggregation rules are vulnerable to well-crafted model poisoning attacks\\cite{BaruchBG19LIE,shejwalkar2021manipulating}, which are indistinguishable in Euclidean distance such that they can circumvent most defenses.\n\n\\noindent \\textbf{Our Method.} We propose a robust gradient aggregation framework, namely SignGuard, to enable Byzantine-robust federated learning. SignGuard leverages a new technique of sign-gradient filtering to identify malicious gradients and can be integrated with existing gradient aggregation rules, such as trimmed-mean\\cite{Yin18optimalrate}. In particular, we define \\emph{sign-gradient} as the element-wise sign of a gradient vector. The key idea of SignGuard is that the sign distribution of {sign-gradient} can provide valuable information in detecting advanced model poisoning attacks, which would otherwise evade state-of-the-art statistic-based detection methods such as Krum and Bulyan\\cite{Blanchard17Byz,Mhamdi18Bulyan}. SignGuard is inspired by our theoretical analysis of \\emph{Little is Enough} (LIE) attack \\cite{BaruchBG19LIE}, and the general good performance of signSGD\\cite{bernstein18sign} in distributed learning tasks. In\\cite{bernstein18sign} the authors show that even if PS only collects the sign of gradient, the model training can still converge with small accuracy degradation and keep the training process fault-tolerant. This fact tells us that the sign of gradient plays an vital role in model updating. Our novel analysis on the LIE attack reveals that gradient manipulation can cause {significant variation of sign distribution}, which turns out to be a breakthrough against such well-crafted attacks. We also empirically find that even the simplest \\textit{sign statistics}\\footnote{By default, we use the ``sign statistics'' to denote the proportions of positive, negative and zero signs.} can expose most of the attacks. These observations provide a new perspective towards Byzantine attack mitigation and directly inspire the design of our SignGuard framework. The core of our approach is extracting robust features of received gradients and using unsupervised clustering method to remove the anomalous ones. We find this simple strategy is able to detect suspicious gradients effectively. \n\n\\noindent \\textbf{Our Work.} To the best of our knowledge, this is the first work to utilize sign statistics of gradients for Byzantine-robust federated learning. SignGuard employs well-designed filtering techniques to identify and eliminate the suspicious gradients to favor existing aggregators. Our theoretical analysis proves that SignGuard can guarantee training convergence on both IID and non-IID training data, while introducing no extra overhead for local computation or auxiliary data collection. Further, instead of detecting suspicious clients and removing them completely from the training process, our proposed SignGuard checks the received gradients directly at each round and is able to utilize updates from Byzantine clients when they behavior honestly, which is crucial for fast convergence in the non-IID settings. In particular, for a system with $n$ clients including $m$ Byzantine clients satisfying $n\\ge 2m+1$, we quantify the gradient bias induced by ignoring $m$ suspicious gradients and show that the parameters enjoy a similar update rule as in safe training, thus the convergence analysis could be performed similarly. Finally, SignGuard is evaluated on various real-world image and text classification tasks through extensive experiments by changing the attack method and the percentage of malicious clients. Our evaluation results demonstrate the effectiveness of our SignGuard in protecting the FL system from Byzantine poisoning attacks and meanwhile achieving high model accuracy.\n\n\\noindent \\textbf{Our Contributions.} To summarize, we make the following key contributions: \n\\begin{itemize}[leftmargin=3ex]\n\\item A novel framework called SignGuard is proposed for Byzantine-robust federated learning, which leverages the sign statistics of gradient to defend against model poisoning attacks. SignGuard does not require any auxiliary data and can mitigate state-of-the-art attacks.\n\n\\item We provide theoretical analysis of the harmfulness and stealthiness of the state-of-the-art \\textit{Little is Enough} attack, and also propose a new hybrid attack strategy.\n\n\\item The convergence of SignGuard is proven with appropriate choice of learning rate. In particular, we show that Byzantine clients inevitably affect the convergence error in non-IID setting even if all malicious gradients are removed.\n\n\\item SignGuard is verified through extensive experiments on MNIST, Fashion-MNIST, CIFAR-10 datasets for image classification tasks and AG-News dataset for text classification task under various Byzantine attacks. Compared with existing approaches, our SignGuard exhibits superiority in both IID and non-IID settings.\n\\end{itemize}\n\n\n\n\\subsection{More Details of Experiments}\\label{appendix:exp}\n\n\\subsubsection{Architecture of CNN}\\label{appendix:cnn} \nWe construct a plain CNN model on MNIST dataset and Fashion-MNIST dataset. The architecture of CNN model is presented in the following table:\n\\begin{table}[htbp]\\footnotesize\n\t\\centering\n\t\\caption{Model Architecture for (Fashion-)MNIST}  \n\t\\label{tab:architecture} \n\t\\renewcommand\\arraystretch{1.2}\n\t\\begin{tabular}{ c  c  c  c  c }\n\t\t\\hline\n\t\tLayer Type & \\# of Channels & Filter Size & Stride & Activation\\\\\n\t\t\\hline\n\t\tConv & 8 & 5  & 1 & ReLU\\\\\n\t\t\\hline\n\t\tMaxPool & 8 & 2  & 2 & --\\\\\n\t\t\\hline\n\t\tConv & 16 & 3  & 1 & ReLU\\\\\n\t\t\\hline\n\t\tMaxPool & 16 & 2  & 2 & --\\\\\n\t\t\\hline\n\t\tConv & 32 & 3  & 1 & ReLU\\\\\n\t\t\\hline\n\t\tMaxPool & 32 & 2  & 2 & --\\\\\n\t\t\\hline\n\t\tFC & 120 & --  & -- & ReLU\\\\\n\t\t\\hline\n\t\tFC & 84 & --  & -- & ReLU\\\\\n\t\t\\hline\n\t\tFC & 10 & --  & -- & Softmax\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\subsubsection{More Details of Training Settings}\\label{appendix:train}  \nTable~\\ref{tab:datasets} summarizes the default federated learning system settings and hyper-parameter details.\n\\vspace{-3ex}\n\\begin{table}[htbp]\\footnotesize\n\t\\centering\n\t\\caption{Training settings on benchmark datasets}  \n\t\\label{tab:datasets} \n\t\\renewcommand\\arraystretch{1.2}\n\t\\begin{tabular}{| c | c | c | c | c |}\n\t\t\\hline\n\t\t\\textbf{} & \\textbf{MNIST} & \\textbf{F-MNIST} & \\textbf{CIFAR-10} & \\textbf{AG-News}\\\\\n\t\t\\hline\n\t\t\\# Total clients & 50 & 50  & 50 & 50\\\\\n\t\t\\hline\n\t\t\\# Byzantine clients & 10 & 10  & 10 & 10\\\\\n\t\t\\hline\n\t\tGlobal Model & CNN & CNN & ResNet-18 & TextRNN\\\\\n\t\t\\hline\n\t\t\\# Parameters & 57,620 & 57,620 & 11,173,962  & 3,837,212\\\\\n\t\t\\hline\n\t\tBatch-size & 60 & 60  & 50 & 20\\\\\n\t\t\\hline\n\t\tInitial learning rate & 0.1 & 0.1 & 0.1 & 0.1\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\\vspace{-1ex}\n\\subsubsection{More Details of Our Methods}\nFor SignGuard-Sim algorithm, we clip the similarity feature values to the range of $[0, +med]$, where $med$ is the median of similarity values, and then rescale them into the range of $[0, 0.2]$. We do not directly employ the similarity values, because the absolute true values are too small compared with sign statistics and thus problematic for clustering algorithm. Besides, in the first 3 epochs, we use the median of similarities across other gradients as the similarity feature. Afterwards, we directly compute the cosine-similarity with the aggregation result from previous iteration to alleviate computation cost. For SignGuard-Dist algorithm, we compute a distance score for each gradient as that in Krum and use the median of scores as a threshold. Then for each gradient, the distance feature is set to 0 if the score is less than the threshold, otherwise will be set to 0.2 instead.\n\n\n\\subsection{Proof of Proposition 1} \n\\label{appendix:prop1} \nNotice that the standard deviation is estimated on distributed gradients, that is:\n\n\\begin{equation*}\n\\|std(g^{\\{i\\in [n]\\}})\\|^2 = \\frac{1}{n}\\sum_{i=1}^{n}\\|g^{(i)}-\\frac{1}{n}\\sum_{j=1}^{n}g^{(j)}\\|^2\n\\end{equation*}\nso we have:\n\\begin{equation*}\n\\begin{aligned}\n&\\mathbb{E}[\\|g_m-\\tilde{g}\\|^2] =\\mathbb{E}[\\|z\\cdot std(g^{\\{i\\in [n]\\}})\\|^2]\\\\\n&=\\mathbb{E}[\\frac{z^2}{n}\\sum_{i=1}^{n}\\|g^{(i)}-\\frac{1}{n}\\sum_{j=1}^{n}g^{(j)}\\|^2] \\\\\n&=\\mathbb{E}[\\frac{z^2}{n}\\sum_{i=1}^{n}\\|g^{(i)}-\\nabla F(\\mathbf{x})+\\nabla F(\\mathbf{x})-\\frac{1}{n}\\sum_{j=1}^{n}g^{(j)}\\|^2] \\\\\n&\\le\\mathbb{E}[\\frac{z^2}{n}\\sum_{i=1}^{n}\\|g^{(i)}-\\nabla F(\\mathbf{x})\\|^2+\\|\\nabla F(\\mathbf{x})-\\frac{1}{n}\\sum_{j=1}^{n}g^{(j)}\\|^2] \\\\\n&\\le\\left(1+\\frac{1}{n}\\right)z^2 \\sigma^2\n\\end{aligned}\n\\end{equation*}\nand:\n\\begin{equation*}\n\\begin{aligned}\n&\\mathbb{E}[\\left\\|g_m-\\nabla F(\\mathbf{x})\\right\\|^2]= \\mathbb{E}[\\|g_m-\\tilde{g} + \\tilde{g}-\\nabla F(\\mathbf{x})\\|^2]\\\\\n&\\le \\mathbb{E}[\\|g_m-\\tilde{g}\\|^2 + \\|\\tilde{g}-\\nabla F(\\mathbf{x})\\|^2]\\\\\n&\\le\\left(1+\\frac{1}{n}\\right)z^2 \\sigma^2 + \\frac{1}{n}\\sigma^2\\\\\n&\\le\\left(z^2+\\frac{1+z^2}{n}\\right)\\sigma^2\n\\end{aligned}\n\\end{equation*}\nTaking the total expectations of averaged gradient on local sampling, we have\n\\begin{equation*}\n\\begin{aligned}\n&\\mathbb{E}_t[F(\\mathbf{x}_{t+1})] - F(\\mathbf{x}_t) \\\\\n&\\leq -\\eta \\left\\langle \\nabla F(\\mathbf{x}_t),\\mathbb{E}_t\\left[{g}_{m,t}\\right]\\right\\rangle\n+\\frac{L\\eta^2}{2}\\mathbb{E}_t\\left[\\left\\|{g}_{m,t}-\\nabla F(\\mathbf{x}_t)\\right\\|^2\\right]\\\\\n&+ L\\eta^2\\mathbb{E}_t\\left\\langle \\nabla F(\\mathbf{x}_t),{g}_{m,t}-\\nabla F(\\mathbf{x}_t)\\right\\rangle+\\frac{L\\eta^2}{2}\\mathbb{E}_t\\left[\\left\\|\\nabla F(\\mathbf{x}_t)\\right\\|^2\\right]\\\\\n&(\\text{substitute}\\quad \\tilde{g_{t}}-z\\cdot std(g_t^{\\{i\\in [n]\\}})\\quad \\text{for} \\quad g_{m,t})\\\\\n&\\le -\\eta(1-\\frac{L\\eta}{2})\\left\\|\\nabla F(\\mathbf{x}_t)\\right\\|^2 +\\frac{L\\eta^2}{2}\\left(z^2+\\frac{1+z^2}{n}\\right)\\sigma^2 \\\\\n&\\qquad + \\eta(1-{L\\eta})\\left\\langle \\nabla F(\\mathbf{x}_t),\\mathbb{E}_t\\left[z\\cdot std(g_t^{\\{i\\in [n]\\}})\\right]\\right\\rangle\\\\\n\\end{aligned}\n\\end{equation*}\nApplying the basic inequality $2\\mathbf{a}\\cdot \\mathbf{b}\\leq \\mathbf{a}^2+\\mathbf{b}^2$, we have\n\\begin{equation*}\n\\begin{aligned}\n&\\mathbb{E}_t[F(\\mathbf{x}_{t+1})] - F(\\mathbf{x}_t) \\\\\n&\\leq -\\eta(1-\\frac{L\\eta}{2})\\left\\|\\nabla F(\\mathbf{x}_t)\\right\\|^2 +\\frac{L\\eta^2}{2}\\left(z^2+\\frac{1+z^2}{n}\\right)\\sigma^2 \\\\ &\\qquad+\\frac{\\eta(1-{L\\eta})}{2}\\left\\|\\nabla F(\\mathbf{x}_t)\\right\\|^2+\\frac{\\eta(1-{L\\eta})}{2}\\left(1+\\frac{1}{n}\\right)z^2 \\sigma^2\\\\\n&\\leq-\\frac{\\eta}{2}\\left\\|\\nabla F(\\mathbf{x}_t)\\right\\|^2 +\\frac{\\eta}{2}\\left(1+\\frac{1}{n}\\right)z^2\\sigma^2+\\frac{L\\eta^2}{2n}\\sigma^2\n\\end{aligned}\n\\end{equation*}\nTaking total expectation and rearranging the terms, we get\n\\begin{equation*}\n\\begin{aligned}\n\\frac{\\eta}{2}\\mathbb{E}[\\left\\|\\nabla F(\\mathbf{x}_t)\\right\\|^2]\\leq & \\mathbb{E}[F(\\mathbf{x}_{t})-F(\\mathbf{x}_{t+1})]\\\\\n&+\\frac{\\eta}{2}\\left(1+\\frac{1}{n}\\right)z^2\\sigma^2+\\frac{L\\eta^2}{2n}\\sigma^2\\\\\n\\end{aligned}\n\\end{equation*}\nTaking summation and dividing by $\\frac{\\eta}{2}T$, then we get\n\\begin{equation*}\n\\begin{aligned}\n\\dfrac{1}{T}\\sum_{t=0}^{T-1}\\mathbb{E}[\\left\\|\\nabla F(\\mathbf{x}_t)\\right\\|^2] \\leq &\\frac{2(F(\\mathbf{x}_0)-F^*)}{\\eta T}\\\\\n&+\\frac{L\\eta \\sigma^2}{n} +{\\left(1+\\frac{1}{n}\\right)z^2\\sigma^2}\n\\end{aligned}\n\\end{equation*}\nwhich completes the proof.\n\n\n\\subsection{Proof of Proposition 2}\\label{appendix:prop2} \nIn Appendix~\\ref{appendix:prop1}, we already show that:\n\n\\begin{equation*}\n\\begin{aligned}\n   \\mathbb{E}[\\|g_m-\\tilde{g}\\|^2] \\le\\left(1+\\frac{1}{n}\\right)z^2 \\sigma^2\n\\end{aligned}\n\\end{equation*}\nand it's easy to see that:\n\\begin{equation*}\n\\begin{aligned}\n&\\mathbb{E}[\\|g^{(i)}-\\tilde{g}\\|^2]=\\mathbb{E}[\\|g^{(i)}-\\frac{1}{n}\\sum_{j=1}^{n}g^{(j)}\\|^2] \\\\\n&=\\mathbb{E}[\\|g^{(i)}-\\nabla F(\\mathbf{x})+\\nabla F(\\mathbf{x})-\\frac{1}{n}\\sum_{j=1}^{n}g^{(j)}\\|^2] \\\\\n&\\le\\mathbb{E}[\\|g^{(i)}-\\nabla F(\\mathbf{x})\\|^2+\\|\\nabla F(\\mathbf{x})-\\frac{1}{n}\\sum_{j=1}^{n}g^{(j)}\\|^2] \\\\\n&\\le\\left(1+\\frac{1}{n}\\right) \\sigma^2\n\\end{aligned}\n\\end{equation*}\nHence, given small enough $z$, it's possible for the malicious gradient to have smaller distance from true averaged gradient than that of some honest gradients.\n\nNext, we can express the cosine-similarity between malicious gradient and true averaged gradient as well as that of honest gradient as follows:\n\n\n\\begin{equation*}\n\\cos(g_m,\\tilde{g})=\\frac{\\left\\|g_m\\right\\|^2+\\left\\|\\tilde{g}\\right\\|^2-\\left\\|g_m-\\tilde{g}\\right\\|^2}{2\\left\\|g_m\\right\\|\\left\\|\\tilde{g}\\right\\|}\n\\end{equation*}\n\n\\begin{equation*}\n\\cos(g^{(i)},\\tilde{g})=\\frac{\\left\\|g^{(i)}\\right\\|^2+\\left\\|\\tilde{g}\\right\\|^2-\\left\\|g^{(i)}-\\tilde{g}\\right\\|^2}{2\\left\\|g^{(i)}\\right\\|\\left\\|\\tilde{g}\\right\\|}\n\\end{equation*}\n\nWe can prove that it's possible for the norm of malicious gradient and the norm of certain honest gradient to have following relations:\n\\begin{equation*}\n\t\\begin{aligned}\n\t&\\left\\|g_m\\right\\| = \\xi_m \\left\\|\\tilde{g}\\right\\| ,~~\\|g^{(i)}\\| = \\xi_i \\left\\|\\tilde{g}\\right\\|, ~~1\\le \\xi_i < \\xi_m\n\t\\end{aligned}\n\\end{equation*}\nBy Jensen inequality, we have:\n\\begin{equation*}\n\\left\\|\\tilde{g}\\right\\| = \\left\\|\\frac{1}{n}\\sum_{i=1}^{n}g^{(i)}\\right\\| \\le \\frac{1}{n}\\sum_{i=1}^{n} \\left\\|g^{(i)}\\right\\| \\le \\max\\{\\|g^{(i)}\\|\\}\n\\end{equation*}\nwhich means the norm of true averaged gradient is smaller than the averaged norm of honest gradient, so some honest gradients could have bigger norm than $\\tilde{g}$, i.e. $\\xi_i\\ge 1$.\n\nAnd a appropriate value of $z$ can make $\\xi_m > \\xi_i$. It's easy to see that:\n\\begin{equation*}\n\\begin{aligned}\n&\\left\\|g_m\\right\\|^2 > \\xi_i^2 \\left\\|\\tilde{g}\\right\\|^2 \\\\\n&\\Longleftrightarrow ~~ \\sum_{j=1}^{d}(\\mu_j-z\\sigma_j)^2 > \\xi_i^2 \\sum_{j=1}^{d}(\\mu_j)^2 \\\\\n&\\Longleftrightarrow ~~ \\sum_{j=1}^{d}(\\mu_j^2-2z\\mu_j\\sigma_j+z^2\\sigma_j^2) > \\xi_i^2 \\sum_{j=1}^{d}(\\mu_j)^2 \\\\\n&\\Longleftrightarrow ~~ \\sum_{j=1}^{d}(z^2\\sigma_j^2) > \\xi_i^2 \\sum_{j=1}^{d}(2z\\mu_j\\sigma_j)\\\\\n&\\Longleftrightarrow ~~ z > \\xi_i^2 \\frac{\\sum_{j=1}^{d}(2\\mu_j\\sigma_j)}{\\sum_{j=1}^{d}(\\sigma_j^2)}\n\\end{aligned}\n\\end{equation*}\nTherefore, given appropriate $z$, there exists $i$, such that $1\\le \\xi_i < \\xi_m$. By using these gradient norm relations, we can get:\n\n\\begin{equation*}\n\t\\begin{aligned}\n\t&\\cos(g_m,\\tilde{g}) - \\cos(g^{(i)},\\tilde{g})\\\\\n\t&= \\frac{(\\xi_m^2+1)\\left\\|\\tilde{g}\\right\\|^2-\\left\\|g_m-\\tilde{g}\\right\\|^2}{2\\xi_m\\left\\|\\tilde{g}\\right\\|^2} - \\frac{(\\xi_i^2+1)\\left\\|\\tilde{g}\\right\\|^2-\\left\\|g^{(i)}-\\tilde{g}\\right\\|^2}{2\\xi_i\\left\\|\\tilde{g}\\right\\|^2}\\\\\n\t&> \\left(\\frac{(\\xi_m^2+1)}{2\\xi_m}-\\frac{(\\xi_i^2+1)}{2\\xi_i}\\right)+\\frac{\\left\\|g_m-\\tilde{g}\\right\\|^2}{2\\left\\|\\tilde{g}\\right\\|^2}\\left(\\frac{1}{\\xi_i}-\\frac{1}{\\xi_m}\\right)\\\\\n\t&=\\frac{(\\xi_m-\\xi_i)(\\xi_m \\xi_i -1)}{2\\xi_m \\xi_i}+\\frac{(\\xi_m-\\xi_i)\\left\\|g_m-\\tilde{g}\\right\\|^2}{2\\xi_m \\xi_i\\left\\|\\tilde{g}\\right\\|^2}\\\\\n\t&>0\n\t\\end{aligned}\n\\end{equation*}\nHence, it's possible for the malicious gradient to have bigger cosine-similarity with true averaged gradient than that of some honest gradients.\n\n\n\n\n\\subsection{Proof of Lemma 2}\n\\label{appendix:lemm2} \nGiven a arbitrary subset of clients $\\mathcal{G}$ with $|\\mathcal{G}|=(1-\\beta)n$ and $\\beta<0.5$.\nLet $\\mathbf{A}=\\sum\\limits_{i\\notin \\mathcal{G}}\\left(g_{t}^{(i)}-\\nabla F(\\mathbf{x}_{t})\\right)$, $\\mathbf{B}=\\sum\\limits_{j\\in \\mathcal{G}}\\left(g_{t}^{(j)}-\\nabla F(\\mathbf{x}_{t})\\right)$, then $\\mathbf{A}$ and $\\mathbf{B}$ are independent. We have $\\mathbb{E}[\\mathbf{A}+\\mathbf{B}]=\\mathbf{0}$.\nRecall that $\\sigma^2$ is the bounded local variance for local gradient and $\\kappa^2$ is bounded deviation between local and global gradient. Applying the Jensen inequality, we have\n\\begin{equation*}\n\\begin{aligned}\n\\left\\|\\mathbb{E}\\left[\\mathbf{A}\\right]\\right\\|^2 &\\leq \\beta n\\sum\\limits_{i\\notin \\mathcal{G}}\\left\\|\\nabla F_i(\\mathbf{x}_{t})-\\nabla F(\\mathbf{x}_{t})\\right\\|^2 \\leq \\beta^2n^2\\kappa^2\\\\\n\\left\\|\\mathbb{E}\\left[\\mathbf{B}\\right]\\right\\|^2 &\\leq (1-\\beta)n\\sum\\limits_{i\\in \\mathcal{G}}\\left\\|\\nabla F_i(\\mathbf{x}_{t})-\\nabla F(\\mathbf{x}_{t})\\right\\|^2 \\leq (1-\\beta)^2n^2\\kappa^2\\\\\n\\end{aligned}\n\\end{equation*}\nNotice that $\\mathbb{E}[\\mathbf{A}]=-\\mathbb{E}[\\mathbf{B}]$, thus\n\\begin{equation*}\n\\left\\|\\mathbb{E}\\left[\\mathbf{A}\\right]\\right\\|^2=\\left\\|\\mathbb{E}\\left[\\mathbf{B}\\right]\\right\\|^2\\leq \\min\\{\\beta^2 n^2\\kappa^2, (1-\\beta)^2n^2\\kappa^2\\} = \\beta^2 n^2\\kappa^2\n\\end{equation*}\nUsing the basic relation between expectation and variance, we have\n\\begin{equation*}\n\\begin{aligned}\n&\\mathbb{E}\\left\\|\\mathbf{A}\\right\\|^2=\\left\\|\\mathbb{E}[\\mathbf{A}]\\right\\|^2+\\text{var}[\\mathbf{A}]\\leq\\left\\|\\mathbb{E}[\\mathbf{A}]\\right\\|^2+\\beta n\\sigma^2\\\\\n&\\mathbb{E}\\left\\|\\mathbf{B}\\right\\|^2=\\left\\|\\mathbb{E}[\\mathbf{B}]\\right\\|^2+\\text{var}[\\mathbf{B}]\\leq\\left\\|\\mathbb{E}[\\mathbf{B}]\\right\\|^2+(1-\\beta) n\\sigma^2\n\\end{aligned}\n\\end{equation*}\nwhich leads to\n\\begin{equation*}\n\\begin{aligned}\n\\mathbb{E}\\left\\|\\mathbf{B}\\right\\|^2 \\le \\beta^2 n^2\\kappa^2 +(1-\\beta)n\\sigma^2\n\\end{aligned}\n\\end{equation*}\nThen, we directly have\n\\begin{equation*}\n\\begin{aligned}\n&\\mathbb{E}\\left[\\left\\|\\frac{1}{|\\mathcal{G}|}\\sum\\limits_{i\\in \\mathcal{G}}\\left(g_{t}^{(i)}\\right)-\\nabla F(\\mathbf{x}_{t})\\right\\|^2\\right]=\\frac{1}{(1-\\beta)^2n^2}\\mathbb{E}\\left\\|\\mathbf{B}\\right\\|^2\\\\\n&\\leq { \\frac{\\beta^2\\kappa^2}{(1-\\beta)^2}}+\\frac{\\sigma^2}{(1-\\beta)n}\n\\end{aligned}\n\\end{equation*}\nIt completes the proof of Lemma 2.\n\n\n\\subsection{Proof of Theorem 1} \n\\label{append:theorem1}\n\nTaking the total expectations of averaged gradient on local sampling and randomness in aggregation rule, we have\n\\begin{equation*}\n\\begin{aligned}\n&\\mathbb{E}_t[F(\\mathbf{x}_{t+1})] - F(\\mathbf{x}_t) \\\\\n&\\leq -\\eta \\left\\langle \\nabla F(\\mathbf{x}_t),\\mathbb{E}_t\\left[\\hat{g}_t\\right]\\right\\rangle+\\frac{L\\eta^2}{2}\\mathbb{E}_t\\left[\\left\\|\\hat{g}_t\\right\\|^2\\right]\\\\\n&= -\\eta \\left\\langle \\nabla F(\\mathbf{x}_t),\\mathbb{E}_t\\left[\\hat{g}_t-\\tilde{g}_t+\\tilde{g}_t-\\nabla F(\\mathbf{x}_t)+\\nabla F(\\mathbf{x}_t)\\right]\\right\\rangle\\\\\n&\\qquad+\\frac{L\\eta^2}{2}\\mathbb{E}_t\\left[\\left\\|\\hat{g}_t-\\nabla F(\\mathbf{x}_t)+\\nabla F(\\mathbf{x}_t)\\right\\|^2\\right]\\\\\n&\\leq -\\eta \\left\\langle \\nabla F(\\mathbf{x}_t),\\mathbb{E}_t\\left[\\hat{g}_t-\\tilde{g}_t\\right]\\right\\rangle -\\eta \\left\\langle \\nabla F(\\mathbf{x}_t),\\mathbb{E}_t\\left[\\tilde{g}_t-\\nabla F(\\mathbf{x}_t)\\right]\\right\\rangle\\\\\n&\\qquad-\\eta\\left\\|\\nabla F(\\mathbf{x}_t)\\right\\|^2 +L\\eta^2\\left\\|\\nabla F(\\mathbf{x}_t)\\right\\|^2+L\\eta^2\\mathbb{E}_t\\left[\\left\\|\\hat{g}_t-\\nabla F(\\mathbf{x}_t)\\right\\|^2\\right]\\\\\n\\end{aligned}\n\\end{equation*}\nFrom Assumption 1 \\& 2, we have\n\\begin{equation*}\n\\begin{aligned}\n\\left[\\mathbb{E}\\left\\|\\hat{g}_t-\\bar{g}_t\\right\\|\\right]^2\n\\leq c{\\delta}\\sup_{i,j\\in \\mathcal{G}}\\mathbb{E}[\\|{g_t^{(i)}}-{g_t^{(j)}}\\|^2]\\leq 2c{\\delta}(\\sigma^2+\\kappa^2)\n\\end{aligned}\n\\end{equation*}\nthen by Young's Inequality with $\\rho=2$, we can get\n\\begin{equation*}\n\\begin{aligned}\n&-\\eta \\left\\langle \\nabla F(\\mathbf{x}_t),\\mathbb{E}_t\\left[\\hat{g}_t-\\tilde{g}_t\\right]\\right\\rangle\\\\\n&\\leq \\eta \\left\\|\\nabla F(\\mathbf{x}_t)\\right\\| \\cdot \\mathbb{E}_t\\left\\|\\hat{g}_t-\\tilde{g}_t\\right\\|\\\\\n& \\leq\\frac{\\sqrt{\\delta}\\eta}{2\\rho}\\left\\|\\nabla F(\\mathbf{x}_t)\\right\\|^2 + \\frac{\\rho}{2}\\cdot 2\\sqrt{\\delta}\\eta c(\\sigma^2+\\kappa^2)\\\\\n&\\leq\\frac{\\sqrt{\\delta}\\eta}{4}\\left\\|\\nabla F(\\mathbf{x}_t)\\right\\|^2 + 2\\sqrt{\\delta}\\eta c(\\sigma^2+\\kappa^2)\n\\end{aligned}\n\\end{equation*}\nCombining with Lemma 2, we get\n\\begin{equation*}\n\\begin{aligned}\n&-\\eta \\left\\langle \\nabla F(\\mathbf{x}_t),\\mathbb{E}_t\\left[\\tilde{g}_t-\\nabla F(\\mathbf{x}_t)\\right]\\right\\rangle\\\\\n&\\leq \\eta \\left\\|\\nabla F(\\mathbf{x}_t)\\right\\| \\cdot \\mathbb{E}_t\\left\\|\\tilde{g}_t-\\nabla F(\\mathbf{x}_t)\\right\\|\\\\\n&\\leq\\frac{\\beta\\eta}{2}\\left\\|\\nabla F(\\mathbf{x}_t)\\right\\|^2 + \\frac{\\beta\\eta\\kappa^2}{2(1-\\beta)^2}\n\\end{aligned}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{aligned}\n&\\mathbb{E}_t\\left[\\left\\|\\hat{g}_t-\\nabla F(\\mathbf{x}_t)\\right\\|^2\\right] \\\\\n&= \\mathbb{E}_t\\left[\\left\\|\\hat{g}_t-\\bar{g}_t + \\bar{g}_t - \\nabla F(\\mathbf{x}_t)\\right\\|^2\\right]\\\\\n&\\leq 2\\mathbb{E}_t\\left[\\left\\|\\hat{g}_t-\\bar{g}_t\\right\\|^2\\right]+2\\mathbb{E}_t\\left[\\left\\|\\bar{g}_t - \\nabla F(\\mathbf{x}_t)\\right\\|^2\\right]\\\\\n&=2\\left[\\mathbb{E}\\left\\|\\hat{g}_t-\\bar{g}_t\\right\\|\\right]^2+2\\text{var}\\left\\|\\hat{g}_t\\right\\|+2\\mathbb{E}_t\\left[\\left\\|\\bar{g}_t - \\nabla F(\\mathbf{x}_t)\\right\\|^2\\right]\\\\\n&\\leq \\quad \\begin{matrix} \\underbrace{ 4c\\delta(\\sigma^2+\\kappa^2)+2b^2+\\frac{2\\beta^2\\kappa^2}{(1-\\beta)^2}+\\frac{2\\sigma^2}{(1-\\beta)n} } \\\\ =\\Delta_1 \\end{matrix}\n\\end{aligned}\n\\end{equation*}\nIn the above derivations, the basic inequality $2\\mathbf{a}\\cdot \\mathbf{b}\\leq \\mathbf{a}^2+\\mathbf{b}^2$ is applied. Taking total expectation and rearranging the terms, we get\n\\begin{equation*}\n\\begin{aligned}\n&\\eta\\left(\\frac{4-\\sqrt{\\delta}-2\\beta}{4}-L\\eta \\right)\\mathbb{E}[\\left\\|\\nabla F(\\mathbf{x}_t)\\right\\|^2]\\leq \\mathbb{E}[F(\\mathbf{x}_{t})-F(\\mathbf{x}_{t+1})]\\\\\n&\\qquad\\qquad+2\\sqrt{\\delta}\\eta c(\\sigma^2+\\kappa^2)+\\frac{\\beta\\eta\\kappa^2}{2(1-\\beta)^2}+L\\eta^2\\Delta_1\\\\\n\\end{aligned}\n\\end{equation*}\nAssume that $\\eta \\le (2-\\sqrt{\\delta}-2\\beta)\/(4L)$, thus $\\left(\\frac{4-\\sqrt{\\delta}-2\\beta}{4}-L\\eta \\right)\\geq\\dfrac{1}{2}$. Taking summation and dividing by $\\eta\\left(\\frac{4-\\sqrt{\\delta}-2\\beta}{4}-L\\eta \\right)T$, then we finally get\n\\begin{equation*}\n\\begin{aligned}\n&\\dfrac{1}{T}\\sum_{t=0}^{T-1}\\mathbb{E}[\\left\\|\\nabla F(\\mathbf{x}_t)\\right\\|^2]\\leq \\frac{2(F(\\mathbf{x}_0)-F^*)}{\\eta T}+2L\\eta\\Delta_1\\\\\n&\\qquad\\qquad\\qquad\\qquad+\\quad\n\\begin{matrix} \\underbrace{ 4\\sqrt{\\delta} c(\\sigma^2+\\kappa^2)+\\frac{\\beta\\kappa^2}{(1-\\beta)^2} } \\\\ =\\Delta_2 \\end{matrix}\n\\end{aligned}\n\\end{equation*}\nwhich completes the proof.\n\\section{Background and Related Work}\n\\label{secRelated} \n\n\\subsection{Safety \\& Security in Federated Learning}\n\nThe model safety and data security are essential principles of federated learning due to the concern of privacy risks and adversarial threats \\cite{yang2019federated, kairouz2019advances, lyu20survey, ma20safeFL}, especially in the\nage of emerging privacy regulations such as General Data\nProtection Regulation (GDPR) \\cite{sharma2019data}. In the context of FL, instead of raw data, the gradient information are shared to jointly train a model. More advanced technologies such as secure multiparty computation or differential privacy are also employed to enhance the privacy guarantees\\cite{abadi2016deepdp,Bonawitz17secureML,Gao19FTL}. Meanwhile, the learning systems are vulnerable to various kinds of failures, including non-malicious faults and malicious attacks. Data poisoning attacks and model update poisoning attacks (aka. untargeted attacks) aim to degrade or even fully break the global model during training phase, while backdoor attacks (aka. targeted attacks) make the model misclassify certain samples during inference phase \\cite{kairouz2019advances}.\nIn particular, the Byzantine threats can be viewed as  worst-case attacks, in which corrupted clients can produce arbitrary outputs and are allowed to collude. In many studies, the Byzantine attacker is assumed to be omniscient and have capability to access white-box model parameters and all honest gradients to conduct strong attacks \\cite{Fang20Local}. As pointed in many works, appropriately crafted attacks can give significant impact on the model performance while circumventing most of current defenses \\cite{BaruchBG19LIE}. However, security mechanisms to protect privacy inevitably make it a more challenging task to successfully detect those failures and attacks, such as secure aggregation where the server can not directly see any individual client updates but an aggregate result \\cite{Bonawitz17secureML,kairouz2019advances}. Thus, the trade-off between privacy assurance and system robustness needs more investigations.\n\n\\subsection{Existing Defense Strategies}\n\n\\noindent \\textbf{Statistic-based.} This is also known as majority-vote based strategy, requiring the percentage of Byzantine clients less than 50\\%. This kind of methods use the $\\ell_p$-norm distance or cosine-similarity to measure the confidence for received gradients. The {Krum} as well as extended {Multi-Krum} are pioneering work towards Byzantine-robust learning\\cite{Blanchard17Byz}. In \\cite{Yin18optimalrate}, the convergence rate and error rate of {trimmed-mean (TrMean)} and {coordinate-wise median (Median)} is rigorously studied. Moreover, \\cite{Mhamdi18Bulyan} has shown that the Krum and median defenses are vulnerable to $\\ell_p$-attack and developed a meta-method called {Bulyan} on top of other robust aggregation methods. Specially, some works only aggregate the sign of gradient to mitigate the Byzantine effect\\cite{bernstein18sign, li2019rsa}. Recently, a method called {Divider and Conquer} is proposed to tackle strong attacks \\cite{shejwalkar2021manipulating}.\n\n\\noindent \\textbf{Validation-based.} The most straightforward approach to evaluate whether a particular gradient is honest or not, is utilizing the auxiliary data in PS to validate the performance of updated model. {Zeno} \\cite{Xie19Zeno} use a stochastic descendant score to evaluate the correctness of each gradient and choose those with highest scores. Fang \\cite{Fang20Local} use error rate based and loss function based rejection mechanism to reject gradients that have bad impact on model updating. In \\cite{cao20FLTrust}, the authors utilize the ReLU-clipped cosine-similarity between each received gradient and standard gradient as weight to get robust aggregation. The main concern of such approaches is the accessibility of auxiliary data.\n\n\\noindent \\textbf{History-aided.} If the one-to-one correspondence between gradient and client entity is knowable for PS, then it's possible to utilize historical data to trace the clients' behaviors. Some studies show that malicious behavior could be revealed from the gradient trace by designing advanced filter techniques \\cite{Alistarh18Byz, zhu20safeguard}. In\\cite{Mu19AFA}, the authors propose a Hidden Markov Model to learn the quality of model updates and discard the bad or malicious updates. Besides, the momentum SGD can also be considered as history-aided method and can help to alleviate the impact of Byzantine attacks\\cite{Karimireddy20history,Mahdi21momentum}.\n\n\\noindent \\textbf{Redundancy-based.} In the context of traditional distributed training, it's possible to assign each node with redundant data and use this redundancy to eliminate the effect of Byzantine failures. In \\cite{Chen18Draco}, the authors present a scalable framework called {DRACO} for robust distributed training using ideas from coding theory. In \\cite{DataSD21dataencoding}, a method based on data encoding and error correction techniques over real numbers is proposed to combat adversarial attacks. In \\cite{Rajput19Detox}, a framework called {DETOX} is proposed by combing computational redundancy and hierarchical robust aggregation to filter out Byzantine gradients.\n\n\\noindent \\textbf{Learning-based.} In \\cite{li20byzautoencoder}, the authors use VAE as spectral anomaly detection model to learn the representation of honest gradients and use reconstruction error in each round as detection threshold.\nIn \\cite{Pan20Justinian}, a method called Justinian's GAAvernor is proposed to learn a robust gradient aggregation policy against Byzantine attacks via reinforcement learning. In \\cite{regatti20bygars}, the authors use auxiliary data in PS to learn the coefficient in weighted average aggregation for each received gradient.\n\n\\noindent \\textbf{Ensemble-learning.} Another line of work leverage the ensemble learning approach to provably guarantee the predicted label for a testing example is not affected by Byzantine clients, in which multiple global models are trained and each of them is learned by using a randomly selected subset of clients \\cite{cao2021provably,qiao21provebackdoor}. However, such ensemble-learning methods significantly enlarge computational overhead and storage cost.\n\n\n\n\\section{Rethink of Recent Attacks}\n\\label{secAttackAnalysis}\n\nIn this section, we first give the threat model and then present our theoretical analysis along with empirical evidence of the\\emph{ Little is Enough (LIE)} attack \\cite{BaruchBG19LIE} to demonstrate the limitation of existing median- and distance-based defenses.\n\n\\noindent \\textbf{Threat Model.} Similar to the threat models in previous works \\cite{Blanchard17Byz,BaruchBG19LIE,Fang20Local,shejwalkar2021manipulating}, we assume that there exists an attacker that controls some malicious clients to perform model poisoning attacks. The malicious clients could be fake clients that injected by the attacker or genuine ones but corrupted by the attacker. Specially, we assume the attacker has full knowledge on all benign gradients, and model parameters, and the corrupted clients can collude to conduct strong attacks. However, the attacker cannot corrupt the server and the proportion of malicious clients $\\beta$ is less than half. For a system with $n$ clients, without loss of generality, we assume that the first $m$ clients are corrupted and $\\beta=\\frac{m}{n}<0.5$.\n\n\\vspace{1ex}\n\\noindent \\textbf{{LIE} Attack.} Byzantine clients first estimate coordinate-wise mean ($\\mu_j$) and standard deviation ($\\sigma_j$), and then send malicious gradient vector with elements crafted as follows:\n\\begin{equation}\\label{eq:lie}\n(g_m)_j = \\mu_j - z\\cdot\\sigma_j, ~ j \\in [d]\n\\end{equation}\nwhere the positive attack factor $z$ depends on the total number of clients and Byzantine fraction. The design mechanism behind this attack is circumventing the coordinate-wise median and trimmed-mean methods. As advised by original paper, the $z$ can be determined by using cumulative standard normal function $\\phi(z)$:\n\n\\begin{equation}\nz_{max} =  max_z \\left(\\phi(z)<\\frac{n-\\left\\lfloor\\frac{n}{2}+1\\right\\rfloor}{n-m}\\right)\n\\end{equation}\n\nIn the following, we will show why this attack is harmful and hard to detect. From an optimization point of view, we can check the upper bound of non-convex distributed optimization problem before and after \\textit{LIE} attack, where we assume the distributed data are IID for simplicity. Lemma~\\ref{lemma:SGD-1} gives out general upper bound when no attack and no defense are performed \\cite{bottou2018optim,yu2019on}, from which we can see that the objective function will converge to a critical point given large iterations $T$ and small learning rate $\\eta$. Applying similar analysis method, we can get a new upper bound when \\textit{LIE} attack and coordinate-wise median defense are conducted as presented in Proposition~\\ref{proposition:SGD-LIE}, where we assume the training can converge.\n\\begin{lemma}\n\tFor a distributed non-convex optimization problem $F(\\mathbf{x})$ with $n$ benign workers, suppose the data are IID and the gradient variance is bounded by $\\sigma^2$. Empoly the SGD with a fixed learning rate $\\eta \\le 1\/L$ and assume $F^*=\\min_{\\mathbf{x}}F(\\mathbf{x})$, then we have the following convergence result\\footnote{In this paper, $\\|\\cdot\\|$ denotes the $\\ell_2$ norm.}:\n\t\\begin{equation}\n\t\t\\dfrac{1}{T}\\sum_{t=0}^{T-1}\\mathbb{E}[\\left\\|\\nabla F(\\mathbf{x}_t)\\right\\|^2] \\leq \\frac{2(F(\\mathbf{x}_0)-F^*)}{\\eta T}+\\frac{L\\eta \\sigma^2}{n}\n\t\\end{equation}\n\t\\label{lemma:SGD-1}\n\\end{lemma}\n\\vspace{-2ex}\n\\begin{proposition}\n\tFor a distributed non-convex optimization problem $F(\\mathbf{x})$ with $(n-m)$ benign workers and $m$ malicious workers conducting \\textit{LIE} attack with appropriate $z$, suppose the data are IID and the gradient variance is bounded by $\\sigma^2$. Employ the Median-SGD with a fixed learning rate $\\eta \\le 1\/L$, and assume $F^*=\\min_{\\mathbf{x}}F(\\mathbf{x})$, then we have the upper bound $B$ of averaged gradient norm square:\n\t\\begin{align}\n\t\tB \\leq  \\frac{2(F(\\mathbf{x}_0)-F^*)}{\\eta T} + \\frac{L\\eta \\sigma^2}{n} +{\\left(1+\\frac{1}{n}\\right)z^2\\sigma^2}\n\t\\end{align}\n\t\\label{proposition:SGD-LIE}\n\\end{proposition}\n\\vspace{-2ex}\n\\begin{proof}\n\tDetailed proof is in Appendix \\ref{appendix:prop1}.\n\\end{proof}\n\nCompared with the result in Lemma~\\ref{lemma:SGD-1}, there exists an extra constant term in Proposition~\\ref{proposition:SGD-LIE}, which does not diminish even with decreasing learning rate, enlarging the convergence error and even making the model training totally collapsed. When no defense is employed, just replace $z$ with ${\\beta z}$, because the $m\\cdot z\\sigma$ could be averaged across all $n$ workers, resulting in a smaller upper bound than median-based defense. This also explains the phenomenon that naive Mean aggregation even has better results than median-based and distance-based defenses in some cases as shown in \\cite{BaruchBG19LIE} and experimental results in this paper. Then, we turn to the coordinate point of view to further analyze why this type of crafted gradient is harmful for model training. \nRecall that signSGD can achieve good model accuracy by only utilizing the sign of gradient, which illuminates a fact that the sign of gradient plays an crucial role in model updating. Therefore, it's worthy to check the sign of gradient for this type attack. The crafting rule of \\textit{LIE} attack is already shown in Eq. (\\ref{eq:lie}), from which we can see that $(g_m)_j $ could have opposite sign with $\\mu_j$ when $\\mu_j>0$. For coordinate-wise median and $\\mu_j>0$, we assume this aggregation rule results in $\\tilde{g} = g_m$, then we have:\n\\begin{equation}\nif ~~ z > \\frac{\\mu_j}{\\sigma_j}, ~~then ~~sign(\\tilde{g}_j) \\ne sign(\\mu_j)\n\\end{equation} For mean aggregation rule and $\\mu_j>0$, if $\\mu_j$ and $\\sigma_j$ are estimated on benign clients, then the $j$-th element becomes:\n\\begin{equation}\n\\tilde{g}_j = \\frac{1}{n}[m\\cdot (g_m)_j + (n-m)\\mu_j] = \\mu_j - z\\cdot \\beta \\cdot\\sigma_j\n\\end{equation}\nand in this case a bigger $z$ is needed to reverse the sign:\n\\begin{equation}\nif ~~ z > \\frac{n\\mu_j}{m\\sigma_j}, ~~then ~~sign(\\tilde{g}_j) \\ne sign(\\mu_j)\n\\end{equation}\n\nEmpirical results in \\cite{BaruchBG19LIE} show that mostly coordinate-wise standard deviation turns out to be bigger than the corresponding gradient element, thus a small $z$ could turn a large number of positive elements into negative, leading to incorrect model updating. To verify this theoretical result, we adopt default training setting in Section~\\ref{secExperimentSetup} to train a CNN on MNIST dataset and ResNet-18 on CIFAR-10 dataset under no attack, and calculate averaged sign statistics across all workers as well as the sign statistics of a virtual gradient that crafted as Eq. (\\ref{eq:lie}). We plot the sign statistics over iterations as Fig.~\\ref{fig:sign_byz}, which convincingly supports our theoretical analysis. \n\n\\begin{figure}[ht]\n\t\\centering\n\t\\subfigure[Honest Gradient of CNN]{\n\t\t\\includegraphics[width=0.46\\columnwidth]{.\/figs\/mnist_honest.pdf}\n\t}\n\n\t\\subfigure[Malicious Gradient of CNN]{\n\t\t\\includegraphics[width=0.46\\columnwidth]{.\/figs\/mnist_lie.pdf}\n\t}\n\n\t\\subfigure[Honest Gradient of ResNet18]{\n\t\t\\includegraphics[width=0.46\\columnwidth]{.\/figs\/cifar_honest.pdf}\n\t}\n\n\t\\subfigure[Malicious Gradient of ResNet18]{\n\t\t\\includegraphics[width=0.46\\columnwidth]{.\/figs\/cifar_lie.pdf}\n\t}\n\t\\caption{Sign statistics of honest and malicious gradient. }\n\t\\label{fig:sign_byz}\n\\end{figure}\n\nNext, we present the following Proposition~\\ref{proposition:Safe-LIE} to explain why \\textit{LIE} attack is hard to detect, in which we compare the distance to averaged true gradient $\\tilde{g}=\\frac{1}{n}\\sum_{i=1}^{n}g^{(i)}$ and similarity with $\\tilde{g}$ for malicious gradient and honest gradient, respectively.\n\n\\begin{proposition}\n\tFor a distributed non-convex optimization problem $F(\\mathbf{x})$ with $(n-m)$ benign workers and $m$ malicious workers conducting \\textit{LIE} attack, suppose the data are IID and the gradient variance is bounded by $\\sigma^2$. Given small enough $z$, then the distance between malicious gradient and true averaged gradient could be smaller than that of certain honest gradient:\n\t\\begin{equation}\n\t\\exists ~i, ~s.t. ~~ \\mathbb{E}[\\left\\|g_m-\\tilde{g}\\right\\|^2] < \\mathbb{E}[\\|g^{(i)}-\\tilde{g}\\|^2]\n\t\\end{equation}\n\tand the cosine-similarity between malicious gradient and true averaged gradient could be bigger than that of certain honest gradient:\n\t\\begin{equation}\n\t\\exists ~i, ~s.t. ~~cos(g_m,\\tilde{g}) > cos(g^{(i)},\\tilde{g})\n\t\\end{equation}\n\t\\label{proposition:Safe-LIE}\n\\end{proposition}\n\\begin{proof}\n\tDetailed proof is in Appendix \\ref{appendix:prop2}.\n\\end{proof}\n\nFrom the above results we can see that it's possible for the malicious gradient to be more ``safe'' when evaluated by Krum and Bulyan methods. Hence, it's almost impossible to detect the malicious gradient from the distance and cosine-similarity perspectives. Instead, checking the sign statistics is a novel and promising perspective to detect abnormal gradients. Similar analysis is also valid for the recent proposed Min-Max\/Min-Sum attacks as well as the adaptive attack that uses different perturbation vectors \\cite{shejwalkar2021manipulating}, along with which a new method called \\textit{Divide and Conquer (DnC)} is also proposed to tackle those attacks. However, this method makes the assumption that malicious gradients are in the direction of largest singular vector of gradient matrix, and would fail when multiple attacks exist simultaneously or in non-IID settings. \n\n\\vspace{1ex}\n\\noindent \\textbf{New Hybrid Attack.} In this work, we extend the OFOM attack in\\cite{chang19cronus} and propose a type of hybrid attack called \\textbf{ByzMean} attack, which makes the mean of gradients be arbitrary targeted malicious gradient. More specifically, the malicious clients are divided into two sets, one set with $m_1$ clients chooses a arbitrary gradient value $g_{m_1}=*$, and the other set with $m_2=m-m_1$ clients chooses the gradient value $g_{m_2}$ such that the average of all gradients is exactly the $g_{m_1}$, just as follows:\n\\begin{equation}\ng_{m_1} = *, ~ g_{m_2}=\\frac{(n-m_1)g_{m_1}-\\sum_{i=m+1}^{n}g^{(i)}}{m_2}\n\\label{eq:byzMean}\n\\end{equation}\nAll existing attacks can be integrated into this ByzMean attack, making this hybrid attack even stronger than all single attacks. For example, we can set $g_{m_1}$ as random gradient or even the gradient crafted by \\textit{LIE }attack. In that case, all existing defense methods including DnC will be broken.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Our SignGuard Framework}\n\\label{secFramework} \n\nIn this section, we present formal problem formulation and introduce our SignGuard framework for Byzantine-robust federated learning. And some theoretical analysis on training convergence is also provided.\n\n\\subsection{System Overview and Problem Setup}\nOur federated learning system consists of a parameter server and a number of benign clients along with a small portion of Byzantine clients. We assume there exists an attacker or say adversary that aims at poisoning global model and controls the Byzantine clients to perform malicious attacks. We first give out the following definitions of benign and Byzantine clients, along with the attacker's capability and defense goal.\n\n\\begin{definition}\n\t\\textbf{(Benign Client)} A benign client always sends honest gradient to the server, which is an unbiased estimation of local true gradient at each iteration.\n\\end{definition}\n\n\\begin{definition}\n\t\\textbf{(Byzantine Client)} A Byzantine client may act maliciously and can send arbitrary message to the server.\n\\end{definition}\n\n\\noindent \\textbf{Attacker's Capability:}\nAs mentioned in the threat model in Section~\\ref{secAttackAnalysis}, the attacker has full knowledge on all benign gradients and the corrupted clients can collude to conduct various kinds of attacks. However, the attacker cannot compromise the server and the proportion of Byzantine clients is less than 50\\%.\n\\vspace{-1ex}\n\n\\noindent \\textbf{Defender's Capability:} As in previous studies \\cite{Fang20Local,cao20FLTrust}, We consider the\ndefense is performed on the server side. The parameter server does not have access to the raw training data on the clients, and the server does not know the exact number of malicious clients. However, the server has full access to the global model as well as the local model updates (i.e., local gradients) from all clients in each iteration. Specially, we further assume the received gradients are anonymous, which means the behavior of each client is untraceable. In consideration of privacy and security, we think this assumption is reasonable in the context of federated learning.\n\n\\noindent \\textbf{Defense Goal:}\nAs mentioned in \\cite{cao20FLTrust}, an ideal defense method should give consideration to the following three aspects: Fidelity, Robustness and Efficiency. We hope the defense method achieves Byzantine-robustness against various malicious attacks without sacrificing the model accuracy. Moreover, the defense should be computationally cheap such that does not affect the overall training efficiency.\n\n\\noindent \\textbf{Problem Formulation:}\nWe focus on federated learning on IID settings and then extend our algorithm into non-IID settings. We assume that training data are distributed over a number of clients in a network, and all clients jointly train a shared model based on disjoint local data. Mathematically, the underlying distributed optimization problem can be formalized as follows:\n\\begin{equation}\\label{eq:objective}\n\\min_{\\mathbf{x}\\in R^d}{F(\\mathbf{x})}=\\frac{1}{n}\\sum_{i=1}^{n}\\mathbb{E}_{\\xi_i\\sim D_i}\\left[F(\\mathbf{x};\\xi_i)\\right]\n\\end{equation}\nwhere $n$ is the total number of clients, $D_i$ denotes the local dataset of \\textit{i}-th client and could have different distribution from other clients, and $F(\\mathbf{x};\\xi_i)$ denotes the local loss function given shared model parameters $\\mathbf{x}$ and training data $\\xi_i$ sampled from $D_i$. We make all clients initialize to the same point $\\mathbf{x_0}$, then FedAvg \\cite{mcmahan17} can be employed to solve the problem. At each iteration, the \\textit{i}-th benign client draws $\\xi_i$ from $D_i$, and computes local stochastic gradient with respect to global shared parameter $\\mathbf{x}$, while Byzantine clients can send arbitrary gradient message:\n\\begin{equation}\n\\begin{aligned}\ng_{t}^{(i)} = \\begin{cases} \n\\nabla F(\\mathbf{x}_{t};\\xi_i) ,&\\text{if \\textit{i}-th client is benign}\n\\\\\n   arbitrary , &\\text{if \\textit{i}-th client is Byzantine}\n\\end{cases}\n\\end{aligned}\n\\end{equation} \nThe parameter server collects all the local gradients and employs robust gradient aggregation rule to get a global model update:\n\\begin{equation}\n\\mathbf{x_{t+1}} = \\mathbf{x_{t}} -\\eta_{t}\\cdot \\textsl{GAR}(\\{g_{t}^{(i)}\\}_{i=1}^{n})\n\\end{equation}\nIn a synchronous and full participation setting, the result will be broadcast to all clients to update their local models and start a new iteration. In a partial participation setting, the model update is finished in PS and the updated model will be sent to the selected clients for next round. This process will repeat until the stop condition is satisfied.\n\n\nTo characterize the impact of Byzantine attack, we define the following two metrics:\n\\begin{definition}\n\t\\textbf{(Attack Success Rate)} The averaged proportion of malicious gradients that were selected by the detection-based GAR throughout the training iterations.\n\\end{definition}\n\\begin{definition}\n\t\\textbf{(Attack Impact)} The model accuracy drop compared with benchmark result that under no attack and no defense.\n\\end{definition}\n\nBased on above metrics, we can measure the effect of Byzantine attack by calculating the accuracy drop due to model poisoning and measure the validity of detection-based defense by calculating the attack success rate.\n\n\\begin{figure*}[htbp]\n\t\\centering\n\t{\n\t\n\t\t\\includegraphics[width=2.0\\columnwidth,clip=true]{.\/figs\/illustration_framework.pdf}\n\t}\t\n\t\\vspace{-2ex}\n\t\\caption{Illustration of the workflow of proposed SignGuard. The collected gradients are anonymous and sent into multiple filters, after which the intersection of multiple outputs are selected as trusted gradients.}\n\n\t\\label{fig:illustration}\n\\end{figure*}\n\n\\subsection{Our Proposed Solution}\n\nThe proposed SignGuard framework is described in Algorithm~\\ref{alg1}-\\ref{alg2} and the workflow is illustrated in Fig.~\\ref{fig:illustration}. On a high level, we pay attention to the magnitude and direction of the received gradients. At each iteration, the collected gradients are sent into multiple filters, including norm-based thresholding filer and sign-based clustering filter, etc. \\textbf{Firstly}, for the norm-based filter, the median of gradient norms is utilized as reference norm as the median always lies in benign set. Considering that small magnitude of gradients do less harm to the training while significantly large one is definitely malicious, we will perform a loose lower threshold and a strict upper threshold. \\textbf{Secondly}, for the sign-based clustering filter, we extract some statistics of gradients as features and using Mean-Shift \\cite{meanshif} algorithm as unsupervised clustering model with adaptive number of cluster classes, while the cluster with largest size is selected as the trusted set. In this work, the proportions of positive, zero and negative signs are computed as basic features, which are sufficient for a number of attacks, including LIE attack. \n\\begin{algorithm}[t] \n\t\\setstretch{1}\n\t\\caption{~SignGuard-based Robust Federated Learning} \n\t\\begin{algorithmic}[1] \n\t\t\\State \\textbf{Input:} learning rate $\\eta$, total iteration $T$, total client number $n$\n\t\t\\State \\textbf{Initial:} $\\mathbf{x}_0\\in R^d$\n\t\t\\For{$t=0, 1, ..., T-1$} \n\t\t\\State \\textbf{On each client \\textit{i} :}\n\t\t\\State Sample a mini-batch of data to compute gradient  $\\displaystyle g_{t}^{(i)}$\n\t\n\t\t\\State Send $\\displaystyle g_{t}^{(i)}$ to the parameter server\n\t\t\\State Wait for global gradient $\\tilde{g}_{t}$ from server\n\t\t\\State Update local model:  $\\displaystyle \\mathbf{x}_{t+1}=\\mathbf{x}_{t}-\\eta\\tilde{g}_{t}$ \n\t\t\\vspace{1ex}\n\t\t\\State \\textbf{On server:}\n\t\t\\State Collect gradients from all clients\n\t\n\t\t\\State Obtain global gradient: $\\displaystyle\\tilde{g}_{t}=SignGuard(\\{g_{t}^{(i)}\\}_{i=1}^{n})$\n\t\t\\State Send $\\tilde{g}_{t}$ to all clients\n\t\t\\EndFor \n\t\\end{algorithmic} \n\t\\label{alg1}\n\\end{algorithm}\n\n\\begin{algorithm}[t] \n\t\\setstretch{1}\n\t\\caption{~SignGuard Function} \n\t\\begin{algorithmic}[1] \n\t\t\\State \\textbf{Input:} Set of received gradients $S_t=\\{g_{t}^{(i)}\\}_{i=1}^{n}$, lower and upper bound $L,R$ for gradient norm\n\t\n\t\t\\State \\textbf{Initial:} $S_1 = S_2 = \\emptyset $\n\t\t\\State \\quad Get $l_2$-norm and element-wise sign of each gradient\n\t\t\n\t\t\\State \\textbf{Step 1:} Norm-threshold Filtering\n\t\t\\State \\quad Get the median of norm $M = med(\\{\\|g_{t}^{(i)}\\|\\}_{i=1}^{n})$\n\t\t\\vspace{1ex}\n\t\t\\State \\quad Add the gradient that satisfies $L \\leq  \\dfrac{\\|g_{t}^{(i)}\\|}{M} \\leq R $ into $S_1$\n\t\t\n\t\t\\State \\textbf{Step 2:} Sign-based Clustering\n\t\t\\State \\quad Randomly select a subset of gradient coordinates\n\t\t\\State \\quad Compute sign statistics on selected coordinates for each gradient as features\n\t\t\\State \\quad Train a Mean-Shift clustering model\n\t\t\\State \\quad Choose the cluster with most elements as $S_2$\n\t\t\\State \\textbf{Step 3:} Aggregation\n\t\t\\State \\quad Get trusted set: $S'_t=S_1 \\cap S_2$ \n\t\t\\State \\quad Get $\\displaystyle\\tilde{g}_{t}=\\frac{1}{|S'_t|}\\sum_{i\\in S'_t}g_t^{(i)}$ \n\t\t\n\t\t\\State \\textbf{Output:} Global gradient: $\\displaystyle\\tilde{g}_{t}$\n\t\\end{algorithmic} \n\t\\label{alg2}\n\\end{algorithm}\n\n\nHowever, those features only consider the overall statistics and lose sight of local properties. Take a toy example, when the amounts of positive and negative elements are approximate (just as ResNet-18), the naive sign statistics may be insufficient to detect sign-flipped gradients \\cite{Rajput19Detox} or those well-crafted attacks that have similar sign statistics. To mitigate this problem, we introduce randomized coordinate selection and add a similarity metric as additional feature in our algorithm, such as cosine-similarity or Euclidean distance between each received gradient and a ``correct'' gradient. However, without the help of auxiliary data in PS, the ``correct'' gradient is not directly available. A practical way is to compute pairwise similarities between all the other gradients and take the median as the similarity with ``correct'' gradient. Or more efficiently, just utilize the aggregated gradient from previous iteration as the ``correct'' gradient. Intuitively, it is promising to distinguish those irrelevant gradients and helps to improve the robustness of anomaly detection. What's challenging is, as shown in Section~\\ref{secAttackAnalysis}, the Euclidean distance or cosine-similarity metrics are not reliable for the state-of-the-art attacks, and even affect the judgment of SignGuard as we found in experiments. In this work, the plain ``SignGuard\" only uses sign statistics in default, and the enhanced variants that add cosine-similarity feature or Euclidean distance feature are called ``SignGuard-Sim\" and ``SignGuard-Dist\", respectively. We will provide some comparative results of them. We emphasize that the SignGuard is a sort of flexible approach and more advanced features could be further extracted to enhance the effectiveness of anomaly detection. And how to design a more reliable similarity metric is left as an open problem for future work. \n\n\nAfter filtering, the server eventually selects the intersection of multiple filter outputs as trusted gradient set, and obtains a global gradient by robust aggregation, e.g. trimmed-mean. In this work, we use the mean aggregation with magnitude normalization. It is worth noting that a small fraction of honest gradients could also be filter out due to gradient diversity, especially in the non-IID settings, depending on the variance of honest gradients and the closeness to malicious gradients.\n\n\n\\subsection{Convergence Analysis}\n\nIn this part, we provide some theoretical analysis of the security guarantee by SignGuard and the convergence of non-convex optimization problem, jointly considering the IID and non-IID data. We first claim that high separability can be achieved when the distributions of test statistics for malicious and honest gradients have negligible overlap.\n\n\\begin{claim}\n\tSuppose all honest gradients are computed with global model parameters and same batch size, and assume the test statistics of honest and malicious gradients follow two finite covariance distributions $P$ and $Q$. For $0<\\beta<1\/2$, let $U=(1-\\beta)P+\\beta Q$ be a mixture of sample points from $P$ and $Q$, denote $f(\\mathbf{x})$ and $g(\\mathbf{x})$ the PDFs of $P$ and $Q$. Then, there exists a  algorithm that separates data points with low probability of error if the total variation distance satisfies: $TV(f,g)=1-o(1)$.\n\\end{claim}\n\\begin{remark}\n\tNote that the Byzantine clients have an inevitable trade-off between the attack impact and the risk of exposure by manipulating the gradient deviation. Therefore, under our detection-based SignGuard framework, the malicious gradient either have limited attack impact or become obvious to get detected, depending on the discrepancy between $P$ and $Q$.\n\\end{remark}\n\nTo conduct convergence analysis, we also make the following basic assumption, which is commonly used in the literature \\cite{yu2019on,bottou2018optim,Karimireddy19error_fix} for convergence analysis of distributed optimization.\n\\begin{assumption}\n\tAssume that problem (\\ref{eq:objective}) satisfies:\n\t\n\t\\textbf{1. Smoothness}: The objective function $F(\\cdot)$ is smooth with Lipschitz constant $L>0$, which means $\\forall \\mathbf{x}, \\forall \\mathbf{y},~\\left\\| \\nabla F(\\mathbf{x})-\\nabla F(\\mathbf{y})\\right\\| \\leq L\\left\\| \\mathbf{x}-\\mathbf{y}\\right\\|$.\n\tIt implies that:\n\t\\begin{equation}\n\tF(\\mathbf{x})-F(\\mathbf{y}) \\leq \\nabla F(\\mathbf{x})^{T}(\\mathbf{y}-\\mathbf{x})+\\frac{L}{2}\\left\\| \\mathbf{x}-\\mathbf{y}\\right\\|^2\n\t\\end{equation}\n\t\n\t\\textbf{2. Unbiased local gradient}: For each worker with local data, the stochastic gradient is locally unbiased:\n\t\\begin{equation}\n\t\\mathbb{E}_{\\xi_i\\sim D_i}\\left[\\nabla F(\\mathbf{x};\\xi_i)\\right] = \\nabla F_i(\\mathbf{x})\n\t\\end{equation}\n\t\n\t\\textbf{3. Bounded variances}: The stochastic gradient of each worker has a bounded variance uniformly, satisfying:\n\t\\begin{equation}\n\t\\mathbb{E}_{\\xi_i\\sim D_i}[\\left\\|\\nabla F(\\mathbf{x};\\xi_i)-\\nabla F_i(\\mathbf{x})\\right\\|^2] \\leq \\sigma^2\n\t\\end{equation}\n\tand the deviation between local and global gradient satisfies:\n\t\\begin{equation}\n\t\\left\\|\\nabla F_i(\\mathbf{x})-\\nabla F(\\mathbf{x})\\right\\|^2 \\leq \\kappa^2\n\t\\end{equation}\n\t\n\t\\label{as:1}\n\\end{assumption}\n\n\nFor SignGuard framework, the trusted gradients attained by filters may still contain a part of malicious gradients. In this case, any gradient aggregation rule necessarily results in an error to the averaged honest gradient \\cite{LaiRV16agnostic,Karimireddy20history}. Here we make another assumption on the capability of aggregation rule:\n\\begin{assumption}\n\tFor problem (\\ref{eq:objective}) with $(1 - \\beta)n$ benign clients (denoted by $\\mathcal{G}$) and $\\beta n$ Byzantine clients, suppose that at most $\\delta n$ Byzantine clients can circumvent SignGuard at each iteration. We assume that the robust aggregation rule in SignGuard outputs $\\hat{g}_t$ such that for some constant $c$ and constant $b$,\n\t\\begin{equation}\n\t\\begin{aligned}\n\t&\\textbf{1. Bounded Bias:}~~\\left[\\mathbb{E}\\left\\|\\hat{g}_t-\\bar{g}_t\\right\\|\\right]^2\n\t\\leq c{\\delta}\\sup_{i,j\\in \\mathcal{G}}\\mathbb{E}[\\|{g_t^{(i)}}-{g_t^{(j)}}\\|^2]\\\\\n\t&\\textbf{2. Bounded Variance:}~~ \\text{var}\\left\\|\\hat{g}_t\\right\\|\n\t\\leq b^2\n\t\\end{aligned}\t\n\t\\end{equation}\n\twhere $\\bar{g}_t=\\frac{1}{|\\mathcal{G}|}\\sum_{i\\in \\mathcal{G}}g_t^{(i)}$ and $0\\le \\delta < \\beta<0.5~$.\n\t\\label{as:2}\n\\end{assumption}\n\n\\begin{remark}\nWhen $\\delta=0$, it's possible to exactly recover the averaged honest gradient. For most aggregation rules such as Krum, the output is deterministic and thus has $b^2=0$. For clustering-based rules, the output is randomized and could have negligible variance if the clustering algorithm is robust.\n\\end{remark}\n\nWhen $\\beta n$ Byzantine clients exist and act maliciously, the desired gradient aggregation result is the average of $(1 - \\beta)n$ honest gradients, which still has a deviation to the global gradient of no attack setting. We give the following lemma to characterize the deviation:\n\n\\begin{lemma}\n\tSuppose the training data are non-IID under Assumption 1, then the deviation between averaged gradient of $(1-\\beta)n$ clients $\\bar{g}$ and the true global gradient $\\nabla F(\\mathbf{x})$ can be characterized as follows:\n\t\\begin{equation}\n\t\\mathbb{E}\\left[\\left\\|\\bar{g}-\\nabla F(\\mathbf{x})\\right\\|^2\\right]\n\t\\leq \\frac{\\beta^2\\kappa^2}{(1-\\beta)^2}+\\frac{\\sigma^2}{(1-\\beta)n}\n\t\\end{equation}\n\\end{lemma}\n\\begin{proof}\n\tDetailed proof is in Appendix \\ref{appendix:lemm2}.\n\\end{proof}\n\n\nGiven above assumptions and lemma, extending the analysis techniques in \\cite{bottou2018optim,yu2019on, Karimireddy19error_fix, Karimireddy20history}, now we can characterize the convergence of SignGuard by the following theorem. \n\\begin{theorem}\n\tFor problem (\\ref{eq:objective}) under Assumption 1, suppose the SignGuard satisfying Assumption 2 is employed with a fixed learning rate $\\eta \\le (2-\\sqrt{\\delta}-2\\beta)\/(4L)$ and $F^*=\\min_{\\mathbf{x}}F(\\mathbf{x})$, then we have the following convergence result:\n\t\\begin{equation}\n\t\\dfrac{1}{T}\\sum_{t=0}^{T-1}\\mathbb{E}[\\left\\|\\nabla F(\\mathbf{x}_t)\\right\\|^2] \\leq \\frac{2(F(\\mathbf{x}_0)-F^*)}{\\eta T}+2L\\eta\\Delta_1 + \\Delta_2\n\t\\end{equation}\n\twhere the constant terms are $\\Delta_1=4c\\delta(\\sigma^2+\\kappa^2)+2b^2+\\frac{2\\beta^2\\kappa^2}{(1-\\beta)^2}+\\frac{2\\sigma^2}{(1-\\beta)n}$ and $\\Delta_2=4c\\sqrt{\\delta}(\\sigma^2+\\kappa^2)+\\frac{\\beta\\kappa^2}{(1-\\beta)^2}$.\n\t\\label{theorem:signGuard}\n\\end{theorem}\n\\begin{proof}\n\tDetailed proof is in Appendix \\ref{append:theorem1}.\n\\end{proof}\n\n\\begin{remark}\n\tThe terms $\\Delta_1$ and $\\Delta_2$ arise from the existence of Byzantine clients and are influenced by the capability of aggregation rule. When no Byzantine client exists ($\\beta=0$ and thus $\\delta=0$), we have $\\Delta_2=0$ and the convergence is guaranteed with sufficiently small learning rate. If Byzantine clients exist ($\\beta>0$), even the defender is capable to remove all malicious gradients ($\\delta=0$), we still have $\\Delta_2>0$ due to non-IID data and may result in some model accuracy gaps to benchmark results. \n\\end{remark}\n\n\n\\section{Experimental Setup}\\label{secExperimentSetup}\nThe proposed SignGuard framework is evaluated on various datasets for image and text classification tasks. We mainly implement the learning tasks in IID fashion, and investigate the performance of different defenses in non-IID settings as well. The models that trained under no attack and no defense are used as benchmarks. All evaluated attack and defense algorithms are implemented in PyTorch.\n\n\\subsection{Datasets and Models}\n\n\\noindent \\textbf{MNIST.} MNIST is a 10-class digit image classification dataset, which consists of 60,000 training samples and 10,000 test samples, and each sample is a grayscale image of size 28 \u00d7 28. For MNIST, we construct a convolutional neural network (CNN) as the global model (see Appendix~\\ref{appendix:cnn}).\n\n\\noindent \\textbf{Fashion-MNIST.} Fashion-MNIST\\cite{xiao17fmnist} is a clothing image classification dataset, which has exactly the same image size and structure of training and testing splits as MNIST, and we use the same CNN as global model.\n\n\\noindent \\textbf{CIFAR-10.} CIFAR-10 \\cite{cifar10\/100} is a well-known color image classification dataset with 60,000 32 \u00d7 32 RGB images in 10 classes, including 50,000 training samples and 10,000 test samples. We use ResNet-18 \\cite{he2016residual} as the global models\\footnote{We use open-source implementation of ResNet-18, which is available at https:\/\/github.com\/kuangliu\/pytorch-cifar}.\n\n\\noindent \\textbf{AG-News.} AG-News is a 4-class topic classification dataset. Each class contains 30,000 training samples and 1,900 testing samples. The total number of training samples is 120,000 and 7,600 for test. We use a TextRNN that consists of two-layer bi-directional LSTM network \\cite{LiuQH16textRNN} as the global model.\n\n\\subsection{Evaluated Attacks}\nWe consider various popular model poisoning attacks in literature as well as recently proposed state-of-the-art attacks as introduced in Section~\\ref{secRelated}, and we assume the attacker knows all the benign gradients and the GAR in server. \n\n\\textbf{Random Attack.} The Byzantine clients send gradients with randomized values that generated by a multi-dimensional Gaussian distribution $\\mathcal{N}(\\mu,\\sigma^2 \\textbf{I})$. In our experiments, we take $\\mu = (0,...,0) \\in \\mathbb{R}^d\\ $and $\\sigma=0.5$ to conduct random attacks.\n\n\\textbf{Noise Attack.} The Byzantine clients send noise perturbed gradients that generated by adding Gaussian noise into honest gradients: $ g_{m} = g_{b} + \\mathcal{N}(\\mu,\\sigma^2 \\textbf{I}) $. We take the same Gaussian distribution parameters as random attack.\n\n\\textbf{Sign-Flipping.} The Byzantine clients send reversed gradients without scaling: $ g_{m} = -g_{b}$. This is a special case of reversed gradient attack \\cite{Rajput19Detox} or empire attack \\cite{Xie19Empires}.\n\n\\textbf{Label-Flipping.} The Byzantine clients flip the local sample labels during training process to generate faulty gradient. This is also a type of data poisoning attack. In particular, the label of each training sample in Byzantine clients is flipped from $l$ to $C-1-l$, where $C$ is the total categories of labels and $l\\in \\{0,1,\\cdots,C-1\\}$.\n\n\\textbf{Little is Enough.} As in \\cite{BaruchBG19LIE}, the Byzantine clients send malicious gradient vector with elements crafted as Eq.~(\\ref{eq:lie}). We set $z=0.3$ for default training settings in our experiments.\n\n\\textbf{ByzMean Attack.} As introduced in Section~\\ref{secAttackAnalysis}, we set $m_1=\\lfloor 0.8m \\rfloor$ and $m_2=m-m_1$, and set $g_{m_1}$ as LIE attack in all experiments.\n\n\\textbf{Min-Max\/Min-Sum.} As in \\cite{shejwalkar2021manipulating}, the malicious gradient is a perturbed version of the benign aggregate as Eq.~(\\ref{eq:gstd}), where $\\nabla^p$ is a perturbation vector and $\\gamma$ is a scaling coefficient, and those two attacks are formulated in Eq.~(\\ref{eq:minmax})-(\\ref{eq:minsum}). The first Min-Max attack ensures that the malicious gradients lie close to the clique of the benign gradients, while the Min-Sum attack ensures that the sum of squared distances of the malicious gradient from all the benign gradients is upper bounded by the sum of squared distances of any benign gradient from the other benign gradients. To maximize the attack impact, all malicious gradients keep the same. By default, we choose $\\nabla^p$ as $-std(g^{\\{i\\in [n]\\}})$, i.e., the inverse standard deviation.\n\\begin{equation}\ng_m = f_{avg}(g^{\\{i\\in [n]\\}})+\\gamma \\nabla^p\n\\label{eq:gstd}\n\\end{equation}\n\\begin{equation}\n\t\\mathop{\\arg\\max}\\limits_{\\gamma} ~ \\mathop{\\max}\\limits_{i\\in [n]}\\|g_m-g^{(i)}\\|\\leq \\mathop{\\max}\\limits_{i,j\\in [n]}\\|g^{(i)}-g^{(j)}\\|\n\t\\label{eq:minmax}\n\\end{equation}\n\\begin{equation}\n\t\\mathop{\\arg\\max}\\limits_{\\gamma} ~ \\mathop{\\sum}\\limits_{i\\in [n]}\\|g_m-g^{(i)}\\|^2\\leq \\mathop{\\max}\\limits_{i\\in [n]}\\mathop{\\sum}\\limits_{j\\in [n]}\\|g^{(i)}-g^{(j)}\\|^2\n\t\\label{eq:minsum}\n\\end{equation}\n\n\nSpecially, we investigate the fixed and randomized attacking behaviors respectively. In fixed settings, all corrupted clients play the role of Byzantine nodes and always perform the predefined attack method during the whole training process. In randomized settings, all corrupted clients will change their collusion attack strategy at each training epoch.\n\n\\subsection{Training Settings}\nBy default, we assume there are $n = 50$ clients in total for each task, 20\\% of which are Byzantine nodes with fixed attack method, and the training data are IID among clients. To verify the resilience and robustness, we will also evaluate the impact of different fractions of malicious clients for different attacks and defenses. Furthermore, our approach will also be evaluated in non-IID settings. In all experiments, we set the lower and upper bounds of gradient norm as $L = 0.1$ and $R = 3.0$, and randomly select 10\\% of coordinates to compute sign statistics in our SignGuard-based algorithms. Each training procedure is run for 60 epochs for MNIST\/Fashion-MNIST\/AG-News and 160 epochs for CIFAR-10, and local iteration is always set to 1. We employ momentum in PS side and the momentum parameter is set to 0.9, and weight decay is set to 0.0005. More details on some key hyper-parameters are described in Appendix~\\ref{appendix:train}\n\n\n\\subsection{Performance Metrics}\nWe train the models for a fixed number of epochs and use the test accuracy to evaluate the model performance. Considering the instability of model training and the fluctuation of model accuracy under strong attacks, we test the training model at the end of each training epoch and take the best test accuracy during the whole training process to assess the efficacy of defenses. We repeat each experiment for three times and report the average results. When a certain defense is performed, the accuracy gap to the baseline can be utilized to evaluate the efficacy of defense under various attacks, and smaller gap indicates more effective defense method. \n\n\\section{Evaluation Results}\\label{secExperimentResult}\n\n\n\\begin{table*}[ht] \\footnotesize\n\t\\centering\n\t\\caption{Comparison of defenses under various model poisoning attacks}  \n\t\\label{tab:main_result_iid} \n\t\\renewcommand\\arraystretch{1.1}\n\t\\newcommand{\\tabincell}[2]{\\begin{tabular}{@{}#1@{}}#2\\end{tabular}} \n\t\\begin{tabular}{| c | c | p{1.08cm}<{\\centering} | p{1.08cm}<{\\centering} | p{1.08cm}<{\\centering} | c | p{1.08cm}<{\\centering} | p{1.08cm}<{\\centering} | p{1.08cm}<{\\centering} | p{1.08cm}<{\\centering} | p{1.08cm}<{\\centering} |}\n\t\t\\hline\n\t\t\\multirow{2}{*}{\\tabincell{c} {Dataset\\\\(Model)}} & \n\t\t\\multirow{2}{*}{GAR} & \n\t\t\\multirow{2}{*}{No Attack} & \n\t\t\\multicolumn{3}{c|}{Simple Attacks}&\n\t\t\\multicolumn{5}{c|}{State-of-the-art Attacks}\\\\\n\t\t\\cline{4-5} \n\t\t\\cline{5-6} \n\t\t\\cline{6-7}\n\t\t\\cline{7-8}\n\t\t\\cline{8-9} \n\t\t\\cline{9-10}  \n\t\t\\cline{10-11}    \n\t\t& & & {Random} & {Noise} & {Label-flip} & {ByzMean} & {Sign-flip} & {LIE} & {Min-Max} & {Min-Sum}\\\\\n\t\t\\hline\n\t\t\n\t\t\\multirow{10}{*}{\\tabincell{c} {MNIST\\\\(CNN)}}&\n\t\tMean & 99.23 & 84.84 & 90.48 & 99.05 & 31.98 & 98.42 & 84.49 & 68.89 & 34.46 \\\\\n\t\t& TrMean & 98.23 & 98.63 & 98.53 & 95.31 & 58.87 & 98.44 & 94.50 & 34.48 & 43.89\\\\\n\t\t& Median & 97.46 & 94.18 & 97.45 & 93.84 & 40.04 & 97.73 & 74.37 & 26.11 & 38.13 \\\\\n\t\t& GeoMed & 93.21 & 82.77 & 78.68 & 86.20 & 45.02 & 74.78 & 34.37 & 15.62 & 20.53 \\\\\n\t\t& Multi-Krum & 99.20 & 98.98 & 99.11 & 99.06 & 83.26 & 98.82 & 90.04 & 52.77 & 27.27 \\\\\n\t\t& Bulyan & 99.10 & 99.17 & {99.12} & 99.15 & 98.58 & 98.81 & 98.86 & 52.45 & 51.95 \\\\\n\t\t& DnC & 99.09 & 99.07 & 99.08 & {99.17} & 82.25 & 98.73 & {99.12} & 98.97 & 81.04 \\\\\n\t\t& SignGuard & 99.11 & {99.09} & {98.97} & \\textbf{99.18} & \\textbf{99.02} & \\textbf{99.13} & 99.15 & \\textbf{99.18} & {99.15} \\\\\n\t\t& SignGuard-Sim & 99.16 & \\textbf{99.18} & 99.16 & 99.07 & {98.91} & {99.06} & \\textbf{99.22} & {99.08} & {99.13} \\\\\n\t\t& SignGuard-Dist & 98.95 & 99.05 & \\textbf{99.18} & 99.11 & 98.93 & 98.86 & 98.96 & 99.01 & \\textbf{99.19} \\\\\n\t\t\\hline \\hline\n\t\t\n\t\t\\multirow{10}{*}{\\tabincell{c} {Fashion-MNIST\\\\(CNN)}}&\n\t\tMean & 89.51 & 69.88 & 31.83 & 89.37 & 16.31 & 86.68 & 79.78 & 47.73 & 45.12 \\\\\n\t\t& TrMean & 87.02 & 87.81 & 87.45 & 79.58 & 62.66 & 87.45 & 54.28 & 45.71 & 42.96 \\\\\n\t\t& Median & 80.77 & 82.96 & 82.59 & 77.41 & 47.46 & 82.52 & 45.14 & 47.43 & 50.83 \\\\\n\t\t& GeoMed & 76.51 & 79.96 & 78.93 & 78.16 & 40.51 & 70.65 & 10.00 & 73.75 & 66.63 \\\\\n\t\t& Multi-Krum & 87.89 & 89.12 & 88.94 & 89.27 & 69.95 & 87.59 & 72.22 & 40.08 & 47.36 \\\\\n\t\t& Bulyan & 88.80 & 89.31 & 89.32 & 89.21 & 88.72 & 87.52 & 88.64 & 59.65 & 43.63 \\\\\n\t\t& DnC & 89.21 & 88.89 & 88.14 & 88.85 & 70.15 & 87.58 & 71.82 & 88.43 & 88.94 \\\\\n\t\t& SignGuard & 89.48 & \\textbf{89.34} & \\textbf{89.32} & 89.12 & 89.35 & 88.69 & 89.34 & \\textbf{89.48} & \\textbf{88.51} \\\\\n\t\t& SignGuard-Sim & 89.43 & {89.24} & 89.21 & \\textbf{89.33} & {89.28} & {89.08} & \\textbf{89.36} & {89.04} & {88.18}\\\\\n\t\t& SignGuard-Dist & 89.37 & 88.87 & 89.30 & 89.31 & \\textbf{89.39} & \\textbf{89.21} & \\textbf{89.36} & 89.34 & 88.38 \\\\\n\t\t\\hline \\hline\n\t\t\n\t\t\\multirow{10}{*}{\\tabincell{c} {CIFAR-10\\\\(ResNet-18)}}&\n\t\tMean & 93.16 & 44.53 & 46.34 & {91.98} & 17.18 & 79.63 & 55.86 & 23.84 & 18.17 \\\\\n\t\t& TrMean & 93.15 & 89.61 & 89.47 & 85.15 & {30.13} & 85.54 & 43.76 & 24.81 & 23.36 \\\\\n\t\t& Median & 74.18 & 68.27 & 71.42 & 71.19 & 23.47 & 70.75 & 27.35 & 20.46 & 22.74 \\\\\n\t\t& GeoMed & 65.62 & 70.41 & 69.35 & 70.76 & 24.86 & 67.82 & 23.55 & 50.36 & 45.23 \\\\\n\t\t& Multi-Krum & 93.14 & \\textbf{92.88} & \\textbf{92.91} & 92.26 & 50.41 & 92.36 & 42.58 & 21.17 & 38.24 \\\\\n\t\t& Bulyan & 92.78 & 91.87 & 92.47 & 92.24 & 81.33 & 90.12 & 74.52 & 29.87 & 37.79 \\\\\n\t\t& DnC & 92.73 & 88.01 & 88.25 & 92.05 & 36.56 & 84.76 & 47.37 & 52.94 & 35.36 \\\\\n\t\t& SignGuard & 93.03 & \\textbf{92.78} & 92.52  & 92.28 & \\textbf{92.46} & 88.61 & \\textbf{92.93} & 92.56 & {92.47} \\\\\n\t\t& SignGuard-Sim & 93.19 & 92.51 & 91.38  & 92.26 & {92.26} & \\textbf{92.48} & 92.62 & {92.63} & 92.75 \\\\\n\t\t& SignGuard-Dist & 92.76 & 92.64 & 92.26 & \\textbf{92.51} & 92.42 & 91.69 & 92.36 & \\textbf{92.82} & \\textbf{92.93} \\\\\n\t\t\\hline \\hline\n\t\t\\multirow{10}{*}{\\tabincell{c} {AG-News\\\\(TextRNN)}}&\n\t\tMean & 89.36 & 28.18 & 28.41 & 86.72 & 25.05 & 84.18 & 79.34 & 27.32 & 25.24 \\\\\n\t\t& TrMean & 87.57 & 88.33 & 88.72 & 85.50 & 37.51 & 84.84 & 66.95 & 30.05 & 30.28 \\\\\n\t\t& Median & 84.57 & 84.52 & 84.59 & 82.08 & 28.99 & 81.10 & 32.39 & 30.28 & 29.71 \\\\\n\t\t& GeoMed & 82.38 & 77.63 & 77.18 & 78.42 & 27.36 & 81.64 & 31.57 & 74.82 & 71.48 \\\\\n\t\t& Multi-Krum & 88.86 & 89.18 & 89.22 & 86.89 & 68.53 & \\textbf{87.42} & 72.98 & 53.51 & 32.46 \\\\\n\t\t& Bulyan & 88.22 & 88.86 & 88.93 & 85.54 & 85.80 & 86.55 & 85.49 & 47.76 & 51.25 \\\\\n\t\t& DnC & 89.13 & 86.42 & 86.28 & 86.72 & 31.47 & 86.30 & 76.58 & 88.45 & 89.05 \\\\\n\t\t& SignGuard  & 89.29 & \\textbf{89.22} & 89.23 & 86.78 & \\textbf{89.24} & 86.53 & 89.26 & 89.23 & 89.27 \\\\\n\t\t& SignGuard-Sim & 89.24 & 89.13 & \\textbf{89.29} & 87.05 & 89.36 & 86.76 & \\textbf{89.33} & \\textbf{89.27} & \\textbf{89.37}\\\\\n\t\t& SignGuard-Dist & 89.23 & 89.16 & 89.23 & \\textbf{87.25} & 89.31 & 87.30 & 89.17 & 89.22 & 89.35 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table*}\n\n\n\nIn this section, we conduct extensive experiments with various attack-defense pairs on both IID and non-IID data settings. We compare our methods with several existing defense methods, including TrMean, Median, GeoMed, Multi-Krum, Bulyan and DnC. The numerical results demonstrate the efficacy and superiority of our proposed SignGuard framework.\n\n\\subsection{Main Results in IID Settings}\nThe main results of best achieved test accuracy during training process under different attack and defense methods in IID setting are collected in Table~\\ref{tab:main_result_iid}. The results of naive \\textit{Mean} aggregation under \\textit{No Attack} are used as benchmarks. Note that we favor other defenses by assuming the defense algorithms know the fraction of Byzantine clients, which is somewhat unrealistic but intrinsically required by existing defenses. However, we do not use the Byzantine fraction information in our SignGuard-type methods, including plain SignGuard, SignGuard-Sim and SignGuard-Dist.\n\n\\vspace{1ex}\n\\noindent \\textbf{Sign Statistics are Powerful.} Test results on four datasets consistently show that our SignGuard-type methods can leverage the power of sign statistics and similarity features to filter out most malicious gradients and achieve comparable test accuracy as general distributed SGD under no attack. Consistent with original papers \\cite{BaruchBG19LIE,shejwalkar2021manipulating}, the state-of-the-art attacks, such as LIE and Min-Max\/Min-Sum, can circumvent the median-based and distance-based defenses, preventing successful model training. Take the results of Multi-Krum on ResNet-18 as example, it can be seen that when no attack is performed, Multi-Krum has negligible accuracy drop (less than 0.1\\%). However, the best test accuracy drops to 42.58\\% under LIE attack and even less than 40\\% under Min-Max\/Min-Sum attacks. Similar phenomena can also be found in model training under TrMean, Median and Bulyan methods. Besides, even under no attack, the Median and GeoMed methods are only effective in simple tasks, such as CNN for digit classification on MNIST and TextRNN for text classification on AG-News. When applied to complicated model training, such as ResNet-18 on CIFAR-10, those two methods have high convergence error and result in significant model degradation. While Muti-Krum and Bulyan suffer from well-crafted attacks, they perform well on naive attacks and even better than our plain SignGuard in mitigating random noise and sign-flip attack. Though the DnC method has extraordinary effectiveness under many attacks, we found it is unstable during training and can be easily broken by our proposed ByzMean attack. In contrast, our proposed SignGuard-type methods is able to distinguish most of those well-crafted malicious gradients and achieve satisfactory model accuracy under various types of attacks. Considering that the local data of Byzantine clients also contribute to global model when no attack is performed, it's not surprising to see that even the best defense against Byzantine attack will still result in small gap to the benchmark results.\n\n\n\\begin{figure*}[htbp]\n\t\\centering\n\t\\subfigure[CNN trained on Fashion-MNIST]{\n\t\t\\includegraphics[width=2.1\\columnwidth]{.\/figs\/fmnist_byznum.pdf}\n\t}\n\n\t\\subfigure[ResNet-18 trained on CIFAR-10]{\n\t\t\\includegraphics[width=2.1\\columnwidth]{.\/figs\/cifar_byznum.pdf}\n\t}\n\n\t\\caption{Accuracy drop comparison under various attacks and different percentage of Byzantine clients. SignGuard has the smallest gap to the baseline. }\n\t\\label{fig:acc_byznum}\n\\end{figure*}\n\n\n\\vspace{1ex}\n\\noindent \\textbf{Sign Statistics are Insufficient.} Table~\\ref{tab:rate_iid} reports the average selected rate of both benign and Byzantine clients during the training process of ResNet-18. We notice that the SignGuard-type methods inevitably exclude part of honest gradients, and select some malicious gradients under the sign-flip attack, even with the help of similarity feature. The reason is that the proportions of positive and negative elements in normal gradient are approximate for ResNet-18, even after randomized downsampling of gradient elements. Consequently, the ratios of positive and negative signs remain approximate in the sign-flipped gradient. Therefore, the simple sign statistics are insufficient to make a distinction between those honest gradients and sign-flipped ones. We also notice that although SignGuard-Sim is resilient to all kinds of attacks and achieves high accuracy results, it only selects less than 80\\% honest gradients during training. One possible reason is that the cosine-similarity feature also has some diversity across honest gradients.\n\n\\begin{table}[htbp] \\footnotesize\n\t\\centering\n\t\\caption{Selected Rate of Honest and Malicious Gradients}  \n\t\\label{tab:rate_iid} \n\t\\renewcommand\\arraystretch{1.3}\n\t\\newcommand{\\tabincell}[2]{\\begin{tabular}{@{}#1@{}}#2\\end{tabular}} \n\t\\begin{tabular}{| c | c | c | c | c | c | c |}\n\t\t\\hline\n\t\t\\multirow{2}{*}{\\tabincell{c} {\\textbf{Attack}}} & \n\t\t\\multicolumn{2}{c|}{\\textbf{SignGuard}} &\n\t\t\\multicolumn{2}{c|}{\\textbf{SignGuard-Sim}} &\n\t\t\\multicolumn{2}{c|}{\\textbf{SignGuard-Dist}} \\\\\n\t\t\\cline{2-3} \n\t\t\\cline{3-4} \n\t\t\\cline{4-5}\n\t\t\\cline{5-6}\n\t\t\\cline{6-7}\n\t\t& {H} & {M} & {H} & {M} & {H} & {M}\\\\\n\t\t\\hline\n\t\tByzMean &  0.9625 & 0 & 0.7791 & 0 & 0.9272 & 0.0003\\\\\n\t\t\\hline\n\t\tSign-flip & 0.6870 & 0.3908 & 0.7639 & 0.0981 & 0.7570 & 0.2440\\\\\n\t\t\\hline\n\t\tLIE &  0.9532 & 0 & 0.7727 & 0 & 0.9151 & 0\\\\\n\t\t\\hline\n\t\tMin-Max &  0.9650 & 0 & 0.7866 & 0.0003 & 0.9105 & 0.0009\\\\\n\t\t\\hline\n\t\tMin-Sum &  0.9640 & 0 & 0.7752 & 0 & 0.9111 & 0\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\vspace{1ex}\n\\noindent \\textbf{Percentage of Byzantine Clients.} We also evaluate the performance of signGuard-Sim with different percentages of Byzantine clients. In this part, we conduct experiments of CNN trained on the Fashion-MNIST dataset and ResNet-18 trained on CIFAR-10 dataset. We keep the total number of clients be 50 and vary the fraction of Byzantine clients from 10\\% to 40\\% to study the impact of Byzantine percentage for different defenses. We use the default training settings, and experiments are conducted under various state-of-the-art attacks. Particularly, we compare the results of SignGuard-Sim with Median, TrMean, Multi-Krum and DnC as shown in Fig.~\\ref{fig:acc_byznum}. It can be seen that our approach can effectively filter out malicious gradients and result in slight accuracy drop regardless of the high percentage of Byzantine clients, while other defense algorithms suffer much more attack impact with increasing percentage of Byzantine clients. In particular, we also find that Multi-Krum can mitigate sign-flip attack well in ResNet-18 training, possibly because the exact percentage of Byzantine clients is provided.\n\n\\vspace{1ex}\n\\noindent \\textbf{Time-varying Attack Strategy.} Further, we test different defense algorithms under time-varying Byzantine attack strategy. We still use the default system setting, and change attack method randomly at each epoch (including no attack scenario). The test accuracy curves of CNN on Fashion-MNIST and ResNet-18 on CIFAR-10 are presented in Fig.~\\ref{fig:acc_random_attack}, where the baseline is training under no attack and no defense, and we only test the State-of-the-art defenses. It can be found that our SignGuard could ensure successful model training and closely follow the baseline, while other defenses resulted in significant accuracy fluctuation and model deterioration. For CNN, the training process even collapsed eventually for other defenses\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\\subfigure[CNN on Fashion-MNIST]{\n\t\t\\includegraphics[width=0.46\\columnwidth]{.\/figs\/fmnist_attack.pdf}\n\t}\n\n\t\\subfigure[ResNet-18 on CIFAR-10]{\n\t\t\\includegraphics[width=0.46\\columnwidth]{.\/figs\/cifar_attack.pdf}\n\t}\n\n\t\\caption{Defense effect comparison under time-varying attacks. SignGuard can ensure safe training and achieve decent model accuracy. }\n\t\\label{fig:acc_random_attack}\n\\end{figure}\n\n\n\\begin{figure*}[htbp]\n\t\\centering\n\t\\subfigure[CNN on Fashion-MNIST]{\n\t\t\\includegraphics[width=2.0\\columnwidth]{.\/figs\/fmnist_noniid.pdf}\n\t}\n\n\t\\subfigure[CifarNet on CIFAR-10]{\n\t\t\\includegraphics[width=2.0\\columnwidth]{.\/figs\/cifar_noniid.pdf}\n\t}\n\t\\caption{Model accuracy comparison under various attacks and different degrees of non-IID. SignGuard has the best performance compared with other start-of-the-art defenses. }\n\t\\label{fig:acc_noniid}\n\\end{figure*}\n\n\n\\subsection{Main Results in Non-IID Settings}\n\nThe Byzantine-mitigation in non-IID FL settings has been a well-known challenging task due to the diversity of gradients. We evaluate our SignGuard-Sim method in synthetic non-IID partition of Fashion-MNIST and CIFAR-10 datasets. As previous works, we simulate the non-IID data distribution between clients by allocating $s$-fraction of dataset in a IID fashion and the remaining (1-$s$)-fraction in a sort-and-partition fashion. Specifically, we first randomly select $s$-proportion of the whole training data and evenly distribute them to all clients. Then, we sort the remaining data by labels and divide they into multiple shards, while data in the same shard has the same label, after which each client is randomly allocated with 2 different shards. The parameter $s$ can be used to measure the skewness of data distribution and smaller $s$ will generate more skewed data distribution among clients. We consider three levels for the skewness with $s$ = 0.3, 0.5, 0.8, respectively. \n\n\\vspace{1ex}\n\\noindent \\textbf{Efficacy on Non-IID Data.} We compare the SignGuard-Sim with various start-of-the-art defenses. As shown in Fig.~\\ref{fig:acc_noniid}, our method still works well under strong attacks in non-IID settings, achieving satisfactory accuracy results in various scenarios. In contrast, TrMean and Multi-Krum could not defend LIE attack and ByzMean attack, making them not reliable any more. Bulyan has good performance on CNN trained on Fashion-MNIST, but is ineffective under LIE attack on ResNet-18 trained on CIFAR-10. DnC can defend against sign-flip attack well, but performs poorly on the other scenarios. Those results in non-IID settings further demonstrate the general validness of sign statistics.\n\n\n\n\\subsection{Computational Overhead Comparison}\nThe following Table~\\ref{tab:aggtime} reports the averaged aggregation time of different defenses on training ResNet-18, where we omit the TrMean and Median since they induce negligible computation cost. It can be seen that SignGuard resulted in shortest time compared with GeoMed, Multi-Krum and Bulyan, which means our method can achieve efficiency and robustness simultaneously. For the other two variants, we found the pairwise similarity\/distance calculation is time-consuming and using the previous aggregate as correct gradient to compute similarity\/distance could alleviate this issue.\n\n\\begin{table}[htbp]\n\\centering\n\\caption{Averaged Aggregation Time}  \n\\label{tab:aggtime} \n\\renewcommand\\arraystretch{1.2}\n\\newcommand{\\tabincell}[2]{\\begin{tabular}{@{}#1@{}}#2\\end{tabular}} \n\\begin{tabular}{| c | c | c | c | c |}\n\t\\hline\n\t\\textbf{Method} & GeoMed & Multi-Krum & Bulyan & SignGuard\\\\\n\t\\hline\n    \\textbf{Time(s)} & 0.39314 & 0.29847 & 0.29629 & 0.04706\\\\\n\t\\hline \\hline\n\t\\multirow{2}{*}{\\tabincell{c} {\\textbf{Method}}}&\n\t\\multicolumn{2}{c|}{SignGuard-Sim}&\n\t\\multicolumn{2}{c|}{SignGuard-Dist}\\\\\n\t\\cline{2-3} \n\t\\cline{3-4} \n\t\\cline{4-5}\n\t& pairwise & previous & pairwise & previous \\\\\n\t\\hline\n\t\\textbf{Time(s)} & 0.73887 & 0.07686 & 0.39117 & 0.07834\\\\\n\t\\hline\n\\end{tabular}\n\\end{table}\n\n\\section{Discussions}\\label{secDiscussion}\n\nFrom the previous sections we can see that the SignGuard approach performs well in many scenarios and can mitigate a number of attacks effectively. Extensive experiments demonstrate that sign statistics are powerful in malicious gradient detection but also insufficient in some cases. In this section, we present some discussions of our approach.\n\n\n\\subsection{Strength and Limitation of SignGuard} \n\nOur SignGuard approach mainly leverages the sign statistics to distinguish malicious gradients from honest gradients, which can overcome the drawbacks of distance and cosine-similarity based detection methods and reveals a new evaluation criteria for gradient correctness. The feasibility of our algorithm depends on the fact that sign statistics of honest gradients gather in a compact range, which is short of theoretical explanation at the moment. The experimental results show that the sign statistics are capable of detecting recent state-of-the-art attacks, which are distance-indistinguishable but do not take the variation of sign statistics into account. This is the key strength of SignGuard, however, also reveals the main limitation that it essentially depends on the distinguishability of sign statistics. As shown in evaluation results, if the ratio of positive and negative sign are approximate, then it's hard for . If the attacker keeps the values of gradient elements unmodified but shuffles the order of gradient elements, then the gradient norm and sign statistics will remain unchanged, that's why random coordinate selection and similarity feature are required in the design of SignGuard algorithm.\n\nAnother advantage is that SignGuard has good extensibility, not only sign statistics but also sophisticated distance and similarity features can be extracted as features, and more advanced clustering algorithms can be applied to separate benign and malicious gradients. And the final aggregation rule can also be replaced by TrMean or Multi-Krum to mitigate those malicious gradients that evade detection filters. Another limitation is also obvious that we only design 2-class K-Means algorithm in this work, which limits its wider applicability. The K-Means algorithm has its own drawbacks for it makes the assumption that clusters are convex and isotropic, and using fixed cluster number cannot tackle the case with dynamic varieties of attack. Therefore, more advanced and adaptive clustering algorithm should be developed to improve the flexibility and robustness.\n\n\\subsection{Possible Improvement for SignGuard} \n\nThe SignGuard framework in this work is a preliminary attempt that levarages sign-gradient to solve the Byzantine attack problem. We believe that there exist more robust characteristics of gradient that can reflect the malicious manipulation, rather than naive sign statistics. And the design of SignGuard framework needs to be improved as well. First, as shown in previous sections, similarity feature is not always helpful and can even hides sign statistics distinction for features in K-Means are isotropic, so it's hard to decide whether to use the similarity feature or not. One possible solution is to apply more advanced clustering models such as spectral clustering\\cite{} and Gaussian mixture models (GMM) with expectation-maximization (EM) algorithm\\cite{}. Second, when a variety of malicious gradients exist simultaneously, it's impossible to specify the number of clusters beforehand. In that case, the desired clustering model should be able to find suitable number of clusters automatically, such as hierarchical clustering \\cite{} and DBSCAN\\cite{}. Third, we find the sign statistics of randomly selected subset of gradient elements are almost consistent with that of original gradient, making the sign-flipped gradient still hard to detect when the ratios of positive and negative sign are approximate. Thus, possibly the algorithm can discard part of coordinates where all gradients have positive (or negative) sign, enlarging the difference between those two ratios to improve the cluster separability. What's more, it is promising to improve SignGuard by cooperating with existing defense algorithms. Because no single defense method can mitigate all possible attacks, and different defense algorithms have their own advantages and weaknesses. For example, Multi-Krum is more robust to sign-flip attack than SignGuard in both IID and non-IID settings as shown in previous sections, hence we are encouraged to adopt the design ideas behind Multi-Krum to calculate useful distance feature and take it into consideration during malicious gradient filtering.\n\n\\subsection{Trade-off Among Safety, Security and Fairness} \n\nTo ensure the safety of model training, we hope for more detailed information about gradient or even raw training data, to evaluate the correctness of each received gradient. If a small set of raw training data can be collected from voluntary clients, we can directly use them to validate the gradients. Otherwise, we should extract many statistical features, including gradient norm, distance, similarity and sign statistics, to perform anomaly detection. However, not only raw training data, but also raw gradient has the risk of privacy leakage. Hence, encryption algorithm and secure multiparty computing are also considered for data security protection, which may not support many detection algorithms and makes the Byzantine gradient detection especially challenging. Our SignGuard mainly makes use of the sign-gradient to achieve effective anomaly detection, having the potential for keeping users data security to a great extent. However, our experimental results also show that sign statistics and sign-similarity are insufficient in some cases, requiring original gradient values to improve robustness. This is a important trade-off between safety and security when developing federated learning systems. Moreover, honest gradients could be filter out as well, which may lead to some algorithm unfairness, especially in non-IID scenarios.\n\n\n\n\n\n\n\n\n\\section{Conclusion and Future Work}\\label{secConclusion}\n\nIn this work, we proposed a novel Byzantine attack detection framework, namely SignGuard, to mitigate malicious gradients in federated learning systems. It can overcome the drawbacks of median- and distance-based approaches which are vulnerable to well-crafted attacks and unlike validation-based approaches that require extra data collection in PS. And it also does not depend on historical data or other external information, only utilizing magnitude and robust sign statistics from current local gradients, making it a practical way to defend most kinds of model poisoning attacks. Extensive experimental results on image and text classification tasks verify our theoretical and empirical findings, demonstrating the extraordinary effectiveness of our proposed SignGuard-type algorithms. We hope this work can provide a new perspective on the Byzantine attack problems in machine learning security. Future directions include developing strategies to defend dynamic and hybrid model poisoning attacks as well as backdoor attacks in more complex federated learning scenarios. And how to design more effective and robust filters in the SignGuard framework for real-world learning systems is also left as an open problem.\n\n\\section{Acknowledgment}\\label{secAcknowledgment}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{}\nIn many of statistical physics applications the entropy notion enters as\na  basic concept suitable for characterize the behaviour of macroscopic\nsystems \\cite{entropy_economic,tsallis2004,entropy_ecology,Franzosi_EPL15,\nFranzosi_PRE16,Felice_PhysA_2018}.\nIn the present manuscript, we address the problem of the correct definition\nof the microcanonical entropy for classical systems. In fact, the latter\nconcern has recently become a matter of a debate where it has been discussed\nwhich one between the Boltzmann and the Gibbs definition provides the correct\nentropy. \n\nA mechanically and adiabatically isolated system, at the equilibrium and\ncomposed of a macroscopic number of interacting particles is statistically\ndescribed with the microcanonical ensemble.\nIn this statistic description the relevant thermodynamical quantities are derived\nfrom the entropy $S$ through suitable thermodynamic relations.\nNow, there are -at least- two accepted definitions for the microcanonical entropy,\nthe ones commonly referred to as Boltzmann entropy and Gibbs entropy.\nThe former is proportional to the logarithm\nof the density of microstates at a given ``energy shell'', whereas the latter \nis proportional to the logarithm of the number of microstates up to a given\nenergy.\nThe debate as to which of these definitions of entropy is the correct one dates back\nto many years ago\n\\cite{Hertz10,Einstein11,Schluter48,Jaynes,Munster_1987,Pearson85,Berdichevsky91,Adib04,\nLavis2005245,Campisi05}. \n\nVery recently \\cite{Dunkel2013,Hilbert_PRE_2014,Hanggi_15}, it has been argued that\nthe Gibbs entropy yields a consistent thermodynamics, and they have been discussed some\nconsistency issues that, the microcanonical statistical mechanics founded on the\nBoltzmann entropy, would unveil \n\\cite{Dunkel2013,Hilbert_PRE_2014,Sokolov_2014,DunkelHilbertRep1,DunkelHilbertRep2,\nCampisi_2015,Campisi_2016}. These and other related arguments \n\\cite{Romero-Rochin,Treumann_2014,Treumann_2014a} have\nbeen contended \\cite{Vilar_2014, Frenkel_2015, Schneider_2014,Wang_2015, Cerino_2015, Swendsen_Wang_Physicaa_2016,Puglisi_PhysRep_2017,\nBaldovin_JStatMech_2017},\nin what has become a lively debate.\nAlthough this may seem a marginal issue, it has crucial consequences about the foundations of statistical mechanics.\nFor instance the negative temperatures notion wouldn't make sense, since\nthey are a well founded concept in the Boltzmann description, whereas, they\nare forbidden in the case of the Gibbs entropy since the number of microstates\nwith energy below a given value $E$ is a non-decreasing function of $E$.\nEven if we do not share the point of view of authors of Refs.\n\\cite{Dunkel2013,Hilbert_PRE_2014,Hanggi_15, Campisi_2015,Campisi_2016},\nas we have clarified in Refs. \\cite{Buonsante_AoP_2016, Buonsante_2015}\nwhere we have shown that the Boltzmann entropy provides a consistent description\nof the microcanonical ensemble, in our opinion these authors must be given\ncredit for having raised this key question.\n\nA further issue raised by the authors of Refs.\n\\cite{Dunkel2013,Hilbert_PRE_2014,Hanggi_15,\nCampisi_2015,Campisi_2016} pertains to the fact that the caloric equation\nof state, for instance in the simple case of an isolated ideal gas system,\nderived with the Boltzmann entropy is not strictly extensive.\nAbout this point, in Ref. \\cite{Buonsante_AoP_2016,Swendsen_2017}, we have shown that the correction to the extensive behaviour, is of the order of $1\/(nd)$, and therefore it vanishes in the limit of infinite degrees of freedom.\nAlthough in the case of a macroscopic system (as the ones more often considered in\nstatistical mechanics) this is not an issue and it represents just an aesthetical\nmathematical problem, it pose a relevant  matter when microcanonical thermodynamics\nis applied to systems that for their nature do not admit the thermodynamic limit.\nExamples of the latter class include proteins, DNA helix, nanosystems.\n\n\nIn the present manuscript we propose a modified version of the Boltzmann\nentropy that overcomes all of these issues. In fact, this entropy reproduces\nthe same results as the Boltzmann entropy for systems with a macroscopic\nnumber of particles and predicts the correct extensivity for the caloric\nequation in the case of small systems.\nLet $H(x)$ be a classical Hamiltonian describing an autonomous many-body system\nof $n$ interacting particles in $d$ spatial dimensions, whose\ncoordinates and canonical momenta $(q_1\\ldots, p_1 ,\\ldots)$ are represented as \n$N$-component vectors $x\\in \\mathbb{R}^{N}$, with $N=2nd$.\nMoreover, we assume that no other conserved\nquantities do exist in addition to the total energy $H$ \\cite{Franzosi_JSP11,Franzosi_PRE12}.\nLet \n$\nM_E = \\left\\{x\\in \\mathbb{R}^{N} | H(x) \\leq E \\right\\}\n$\nbe the set of phase-space states with total energy less than or equal to $E$.\nThe Gibbs entropy for this system is\n\\begin{equation}\nS_G (E) = \\kappa_B \\ln \\Omega(E) \\, ,\n\\label{gibbs}\n\\end{equation}\nwhere $\\kappa_B$ is the Boltzmann constant and\n\\begin{equation}\n\\Omega(E) = \\dfrac{1}{h^{nd}} \\int d^N x \\Theta(E-H(x)) \\, ,\n\\label{OmegaE}\n\\end{equation}\nis the number of states with energy below $E$.  $h$ is the Planck constant and\n$\\Theta$ is the Heaviside function.\n\nThe Boltzmann entropy concerns the energy level sets \n$\n\\Sigma_E = \\left\\{x\\in \\mathbb{R}^{N} | H(x) = E \\right\\} \\, ,\n$\nand is given in terms of $\\omega(E) = \\partial \\Omega\/\\partial E$, according to\n\\begin{equation}\nS_B (E) = \\kappa_B \\ln \\left(\\omega(E)\\Delta \\right) \\, ,\n\\label{boltzmann}\n\\end{equation}\nwhere the constant $\\Delta$ with the dimension of energy makes the argument of the logarithm dimensionless,\nand\n\\begin{equation}\n\\omega(E) = \\dfrac{1}{h^{nd}} \\int d^N x \\delta(E-H(x)) \\, ,\n\\label{omegaE}\n\\end{equation}\nis expressed in terms of the Dirac $\\delta$ function. Remarkably, in the case\nof smooth level sets $\\Sigma_E$, $\\omega(E)$\ncan be cast in the following form \n\\cite{RughPRL97,Franzosi_JSP11,Franzosi_PRE12}\n\\begin{equation}\n\\omega(E) =  \\dfrac{1}{h^{nd}} \n\\int_{\\Sigma_E} \\dfrac{m^{N-1}(\\Sigma_E)}{\\Vert\\nabla H(x) \\Vert}  \\, ,\n\\label{omegaEdiff}\n\\end{equation}\nwhere $m^{N-1}(\\Sigma_E)$ is the metric induced from $\\mathbb{R}^N$ on the\nhypersurface $\\Sigma_E$ and $\\Vert\\nabla H(x) \\Vert$ is the norm of the gradient\nof $H$ at $x$. \n\nThe entropy that we propose here is\n\\begin{equation}\nS (E) = \\kappa_B \\ln \\left( \\sigma(E) \\Delta^{1\/2} \\right) \\, ,\n\\label{enew}\n\\end{equation}\nwhere\n\\begin{equation}\n\\sigma(E) =\\dfrac{1}{h^{nd}}  \\int_{\\Sigma_E} m^{N-1}(\\Sigma_E)  \\, .\n\\label{sigmaEdiff}\n\\end{equation}\nIn the case of a system of identical particles, to avoid the Gibbs\nparadox it is in order to introduce a factor 1\/n! in the definitions of $\\Omega$,\n$\\omega$ and $\\sigma$, Eqs. \\eqref{OmegaE}, \\eqref{omegaE}, \\eqref{omegaEdiff}\nand \\eqref{sigmaEdiff}, as we will do in the following.\n\n\nThe entropy is the fundamental thermodynamic potential of the microcanonical ensemble from which\nsecondary thermodynamic quantities are obtained by derivatives with respect to the control\nparameter: the total energy $E$, the occupied volume $V$ and, possibly, further Hamiltonian\nparameters $A_\\mu$ (in the following we omit to indicate explicitly the dependence by $A_\\mu$ in \norder to simplify the notation).\nThe inverse temperatures $\\beta=(\\kappa_B T)^{-1}$ is derived from the\nthe entropy according to $\\beta = (\\partial S\/\\partial E)\/\\kappa_B$, thus in the\nthree cases under consideration we have\n\\begin{eqnarray}\n\\beta_G &=& \n\\dfrac{\\Omega^\\prime}{\\Omega} \\, , \\\\\n\\beta_B &=& \n\\dfrac{\\omega^\\prime}{\\omega} \\, , \\\\\n\\beta &=& \n\\dfrac{\\sigma^\\prime}{\\sigma} \\, ,\n\\end{eqnarray}\nwhere the symbol $^\\prime$ denotes the partial derivative of the corresponding\nterm with respect to energy $E$.\n\nA basic requisite for $S$ is to allow the measure of temperature \nand the other secondary\nthermodynamic quantities via microcanonical averages.\nIn terms of the microscopic dynamics, from the Liouville theorem it follows that the\ninvariant measure $d \\mu$ for the dynamics on each energy level-set $\\Sigma_E$ is\n$d \\mu = {m^{N-1}(\\Sigma_E)}\/{\\Vert \\nabla H \\Vert}$.\nIn the case of the Boltzmann entropy, the temperature definition meets the\nmentioned requisite since\n\\begin{equation}\n\\beta_B =\\left\\langle \n\\nabla \\left( \\frac{\\nabla H}{\\Vert \\nabla H \\Vert^2} \\right) \\right\\rangle \\, ,\n\\label{betaB}\n\\end{equation}\nwhere $\\langle \\rangle$ indicates the microcanonical average\n\\begin{equation}\n\\langle \\phi \\rangle = \\dfrac{1}{\\omega} \\int_{\\Sigma_E} \\phi d\\mu \\, .\n\\end{equation} \nEq. \\eqref{betaB} is derived in Ref. \\cite{RughPRL97}\nfor the case of many-particle systems for which the energy is the only conserved quantity, and in Refs. \n\\cite{Franzosi_JSP11,Franzosi_PRE12} for the general case of two or more\nconserved quantities. On the contrary, the Gibbs definition of \ntemperature does not meet such important requisite as diffusely\ndiscussed in Ref. \\cite{Buonsante_AoP_2016}.\nBy using by the Federer-Laurence derivation formula \\cite{Federer_1969,Laurence_1989,Franzosi_JSP11,Franzosi_PRE12}, \nin the case of the proposed entropy we get\n\\begin{equation}\n\\beta = \\dfrac{\\sigma^\\prime}{\\sigma} =  \\dfrac{\\sigma^\\prime\/\\omega}{\\sigma\/\\omega} =\n\\dfrac{\\langle \\nabla \\left( \\frac{\\nabla H}{\\Vert \\nabla H \\Vert} \\right) \\rangle}\n{\\langle \\Vert \\nabla H \\Vert \\rangle} \\, .\n\\label{beta}\n\\end{equation}\nThis shows that also $S$, besides $S_B$, satisfies the requirement to provide secondary thermodynamic\nquantities measurable as microcanonical averages. In passing, we note that\nunder the hypothesis of ergodicity, the\naverages of each dynamical observable of the system can be equivalently measured along the dynamics.\n\nAs a simple test let us consider a classical ideal gas in $d$-spatial\ndimensions composed of $n$ \nidentical particles of mass $m$ for which it is easy matter to verify that\n\\begin{eqnarray}\n\\Omega (E,V) &=& \\dfrac{V^n (2\\pi m)^{nd\/2}}{\\Gamma(\\frac{nd}{2} +1) n! h^{nd} } E^{nd\/2} \\, ,\n\\\\\n\\omega (E,V) &=& \\dfrac{V^n (2\\pi m)^{nd\/2}}{\\Gamma(\\frac{nd}{2}) n! h^{nd} } E^{nd\/2-1} \\, ,\n\\\\\n\\sigma (E,V) &=& \\dfrac{2 V^n (2\\pi m)^{nd\/2}}{\\Gamma(\\frac{nd}{2}) n! h^{nd} } E^{(nd-1)\/2} \\, ,\n\\end{eqnarray}\nwhere the factor $1\/n!$ is introduced in order to avoid the Gibbs paradox.\nFrom these formulas one finds the following expression of the caloric \nequation\n\\begin{eqnarray}\n\\beta^{-1}_G &=& \\dfrac{E}{{nd}\/{2}}  \\, ,\\\\\n \\beta^{-1}_B &=& \\dfrac{E}{\\left({nd}\/{2}-1\\right)} \\, , \\\\\n\\beta^{-1} &=& \\dfrac{E}{{(nd-1)}\/{2}} \\, .\n\\end{eqnarray}\nIn the count for the degrees of freedom for a system of free particles,\njust the kinetic term contributes. Thus, in $d$ spatial dimensions a\nsystem of $n$ particles  have $nd$ degrees of freedom and, by setting\nthe energy  $E$ to a given value we are left with $nd-1$ degrees of\nfreedom.\nTherefore, among these only the latter expression is exactly extensive\nand, hence, rigorously satisfies the equipartition theorem for any $n$.\nWith an analogous calculation, it is easy matter to show that for a system\nof $n$ independent identical harmonic oscillators, of mass $m$ and\nfrequency $\\nu$ in $d$ spatial dimensions the caloric equations\nderived from the three entropies are\n\\begin{eqnarray}\n\\beta^{-1}_G &=& \\dfrac{E}{nd}  \\, ,\\\\ \n\\beta^{-1}_B &=& \\dfrac{E}{\\left({nd}-1\\right)} \\, , \\\\\n\\beta^{-1} &=& \\dfrac{E}{{(2nd-1)}\/{2}}  \\, .\n\\end{eqnarray}\nIn this case, either the coordinates and the motional degrees of freedom\ncontribute to the count of the degrees of freedom of the system.\nThus, when the energy has a fixed value $E$, the number of degrees\nof freedom are $2nd-1$ and only $S$ brings to the correct equipartition\nformula.\n\n\\emph{In addition to lead up the correct relation between total energy\nand true number of degrees of freedom, the entropy we propose rigorously\nsatisfies the postulate of equal a-priory probability which is a very\nfoundations of the equilibrium microcanonic statistical mechanics.\nAs a matter of fact, for a generic isolated physical-system at the\nequilibrium, a given thermodynamic state is completely determined when\nwe know the values of the macroscopic parameters as energy, volume, and\npossibly further external parameters, that characterize such system.\nIn this way, from a thermodynamic point of view we do not distinguish\nbetween the states of the system represented by different points on\nthe same energy level and consistent with the further constraints.\nThis is just what Eq. \\eqref{omegaEdiff} does, it ``counts the number\u00b4\u00b4\nof microstates satisfying the macroscopic constraint $H=E$,\nconsistently to the above mentioned postulate.\nOn the contrary, the standard\nBoltzmann entropy adopts a place-dependent weight\n$1\/\\Vert \\nabla H \\Vert$.}\n\nIn order to better clarify the connection between the Boltzmann\nentropy and that one we propose, let us perform the following\nrough calculation.\nFor a system with $N$ degrees of freedom, if $\\Delta E \\ll E$,\napproximatively we have\n\\begin{equation}\n\\Omega(E+\\Delta E) - \\Omega(E) \\approx \\omega(E) \\Delta E + O(\\Delta E^{2})\\, ,\n\\end{equation}\non the other hand, for the Cavalieri's principle, we have\n\\begin{equation}\n\\Omega(E+\\Delta E ) - \\Omega(E) \\approx \n\\left(\\sigma(E)\\Delta^{1\/2} \n\\right) \\dfrac{\\Delta E}{\\Delta}  + O(\\Delta E^{2})\n\\, .\n\\end{equation}\nHence it results\n$\n\\sigma (E)\\Delta^{1\/2} = \\omega(E)\\Delta + O(N^{2})\n$\nand, consequently\n\\begin{equation}\n\\lim_{N\\to  \\infty} \\dfrac{1}{N} \\left(\\ln (\\sigma \\Delta^{1\/2})-\\ln(\\omega \\Delta)\n\\right)\n= 0 \\, .\n\\label{bigN}\n\\end{equation}\nThis makes evident that in the limit of large number of degrees of freedom,\nthe proposed entropy predicts the same results as the Boltzmann entropy,\nwhereas, in the case of systems with small $N$ the two\nentropies differ from each other.\n\nIn order to verify our assumption, we have tested the proposed\nentropy on two systems: the two dimensional $\\Phi^4$ model and\na one dimensional model of rotors.\n\nThe $\\phi^4$ model \\cite{Franzosi_PRE99,Franzosi_PRL00,BPV_PRB04,Franzosi_PRA10} is defined by the Hamiltonian\n\\begin{equation}\nH = \\sum_{\\bf j} \\dfrac{1}{2} \\pi^2_{\\bf j}  + V(\\phi)\n\\label{Hphi4}\n\\end{equation}\nwhere\n\\begin{equation}\nV(\\phi) =\\sum_{\\bf j} \\left[ \n \\dfrac{\\lambda}{4!} \\phi^4_{\\bf j} - \\dfrac{\\mu^2}{2}\\phi^2_{\\bf j} +\n\\dfrac{J}{4} \\sum_{{\\bf k}\\in I({\\bf j})} (\\phi_{\\bf j} - \\phi_{\\bf k})^2\n \\right] \\, ,\n\\label{Vphi4}\n\\end{equation}\n$\\pi_{\\bf j}$ is the conjugate momentum of the variable $\\phi_{\\bf j}$\nthat defines the field at ${\\bf j}^{th}$ site. Indeed,\n${\\bf j} = (j_1,j_2)$ denotes a site of a two dimensional latte\nand\n$I({\\bf j})$ are the nearest neighbour lattice sites of the\n${\\bf j}^{th}$ site. The coordinates of the sites are integer numbers\n$j_k =1,\\ldots,N_k$, $k=1,2$, so that the total number of sites\nin the lattice is $N=N_1\\,N_2$. Furthermore periodic boundary conditions\nare assumed.\nThe local potential displays a double-well shape whose minima are located \nat $\\pm \\sqrt{{3! \\mu^2}\/{\\lambda}}$ and to which it corresponds \nthe ground-state energy per particle $e_0 = - 3! \\mu^4\/(2 \\lambda)$.\nAt low-energies the system is dominated by an ordered phase where the time \naverages of the local field are not vanishing. By increasing the system\nenergy the system undergoes a second order phase-transition and\nthe local $\\mathbb{Z}_2$ symmetry is restored. In fact, at high\nenergies the time averages of the local field go to zero.\n\nThe second model \\cite{Cerino_2015} is composed by $N$ rotators with\ncanonical coordinates\n$\\phi_1,\\ldots,\\phi_N,\\pi_1,\\ldots,\\pi_N$ and Hamiltonian\n\\begin{equation}\nH=\\sum^N_{j=1} [1-\\cos(\\pi_j)] + \\epsilon \n\\sum^N_{j=1}[1-\\cos(\\phi_j-\\phi_{j-1})] \\, ,\n\\label{Hrot}\n\\end{equation}\nwhere is assumed $\\phi_0 = 0$. The form of kinetic and potential terms\nin \\eqref{Hrot} makes the energy bounded either from above and from below\nand such Hamiltonian implies the existence of negative Boltzmann temperatures\n\\cite{Cerino_2015}.\n\nWe have numerically integrated the equation of motion associated to the \nHamiltonian of both the models, by using a third order symplectic algorithm\nand starting from initial conditions corresponding to different values\nof the system total energy $E$. We have measured along the dynamics the time averages of the relevant quantities that appear in \\eqref{betaB} \nand \\eqref{beta} and, then we have derived the curves $\\beta_B(E)$ and\n$\\beta(E)$ for the two models.\n\\begin{figure}[h]\n \\includegraphics[height=5.cm]{phi4_beta-e.eps}\n\\caption{The figure compares $\\beta_B(E\/N)$ (dotted line) and\n$\\beta(E\/N)$ (continuous line) numerically computed for a lattice\nof $128\\times 128$ sites for the\n$\\Phi^4$-model. The agreement is astonishing,\nin fact the two curves are indistinguishable. In the inset we report\na zoom in order to show the two curves.\n\\label{fig1}}\n\\end{figure} \n\\begin{figure}[h]\n \\includegraphics[height=5.cm]{rot_1d_beta-e.eps}\n\\caption{The figure compares $\\beta_B(E\/N)$ (dotted line) and\n$\\beta(E\/N)$ (continuous line) numerically computed for an array of $512$\nrotors. Also here the agreement is astonishing,\nthe two curves are indistinguishable thus we report the inset with\na zoom that shows the two curves.\n\\label{fig2}}\n\\end{figure} \nFigs. 1 and 2 clearly show the remarkable agreement between the curves\n$\\beta_B(E\/N)$ and $\\beta(E\/N)$, for both the models studied.\n\nIn conclusion we have proposed a novel definition of the microcanonical\nentropy for classical systems.\nWe have shown that this definition definitely resolve the debate\non the correct definition of the microcanonical entropy. In fact, we have\nshown that this entropy definition fixes the issue inherent the full\nextensivity of the caloric equation.\nFurthermore, we have given evidence by investigating\ntwo different models, that this entropy reproduces results which are in\nagreement with the ones predicted with standard Boltzmann entropy in the\ncase of macroscopic systems. \nSince the differences between the predictions of Boltzmann entropy and\nof the one here proposed, are more evident in systems with small number of\ndegrees of freedom,  we conclude that the Boltzmann entropy (with the our one)\nprovides a correct description for macroscopic systems whereas\nextremely small systems should be described with the entropy that we\nhave proposed in order to avoid, for instance, issues with the\nextensivity of the caloric equation.\n\n \n\n\n\n\n\n\n\n\n\n\n\n\\begin{acknowledgments}\nWe are grateful to A. Smerzi and P. Buonsante for useful discussions.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Sample Section Title}\n\\label{sec:sample1}\n\n\nLorem ipsum dolor sit amet, consectetur adipiscing \\citep{Fabioetal2013} elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud \\citet{Blondeletal2008} exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit \\citep{Blondeletal2008,FabricioLiang2013} anim id est laborum.\n\nLorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum see appendix~\\ref{sec:sample:appendix}.\n\n\n\\section{Sample Section Title}\n\\label{sec:sample1}\n\n\nLorem ipsum dolor sit amet, consectetur adipiscing \\citep{Fabioetal2013} elit, sed do eiusmod tempor incididunt ut labore et dolore magna \\citet{Blondeletal2008} aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit \\citep{Blondeletal2008,FabricioLiang2013} anim id est laborum.\n\nLorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum see appendix~\\ref{sec:sample:appendix}.\n\n\n\\section{Introduction}\n\\label{sec1}\nCompound refractive lens assemblies (CRLs) are frequently used as objective lenses in X-ray microscopes \\cite{lengeler99}, or as upstream condensers \\cite{schroer05,vaughan11}. Both applications are extremely sensitive to lateral misalignment. The full focusing procedure of a CRL requires five degrees of freedom. Independent translations along the $x,$ and $y$ axes, in addition to two rotations $r_x$ and $r_y$ about those axes, produce a lateral alignment. The fifth degree of freedom is a translation along the $z$ axis, which locates the classical position of the focus (see Figure~\\ref{hugh_image}). \n\nOur principal motivation for this work is the task of laterally aligning compound refractive lens assemblies (CRLs) at  X-ray free electron laser facilities (XFELs). XFELs are a new class of X-ray sources that produce the shortest duration and brightest X-ray pulses currently attainable, paving the way for experiments that were previously not possible \\cite{Yabashi2017}. A new approach to the alignment of CRLs is necessary, in large part due to the novel amplification process to generate X-ray pulses. At XFEL facilities,  Self-Amplification of Spontaneous Emission (SASE) causes the beam position, spatial mode, propagation direction (pointing), and intensity to fluctuate stochastically \\cite{Emma2010,Schneidmiller2016}. The proper lateral alignment of focusing optics is crucial to produce the highest resolution and smallest focal spots, as required for many modern X-ray experiments.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=5cm]{CRL_diagram.eps}\n    \\caption{This example diagram depicts a CRL-based imaging configuration along an optical axis $z$. A CRL is aligned to the optical axis by four independent motors. Two control translations along the perpendicular $x$ and $y$ axes, and two control rotations about the $x$ and $y$ axes.}\n    \\label{hugh_image}\n\\end{figure}\n\nThe orientation of the CRL is controlled by four motorized stages: two that translate the optic along the $x$ and $y$ axes, and two that rotate about them. To align this optic, a detector is placed behind the exit surface of the CRL to measure the transmitted X-ray beam. Typically, these sensors are either charge-coupled device (CCD) cameras or different implementations of a photodiode (e.g. ion chamber). The beam-line scientist's task is to laterally align the CRL by maximizing X-ray transmitted light that reaches the sensor as a function of the four positions. \n\nIn more formal language, let $f\\colon \\mathbb{R}^4 \\to \\mathbb{R}$ denote an idealized model of a noiseless, steady (in space and amplitude) X-ray transmission through a CRL as a function of the orientation. In \\cite{simons17}, Simons et al. demonstrated that for a convex region $\\Omega \\subset \\mathbb{R}^4$, that transmission $f$ can be modelled with a Gaussian distribution. In this idealized case, the alignment procedure reduces to the trivially convex optimization problem\n\\begin{equation*}\n\\min_{{\\bf x} \\in \\Omega} -f({\\bf x}).\n\\end{equation*}\n\nGiven the simplicity of the problem, it is common at scientific beam-lines with more-stable amplitudes to establish lateral alignment through a simple manual process. A recent study automated this task by using a modified stochastic simplex method on the lateral alignment of CRL assemblies at synchrotron facilities \\cite{Breckling2021}. In lieu of automating, the usual approach is to perform a rough initial alignment, then select two of the four dimensions of $\\Omega$ and perform a raster scan of the transmission, logging a detector's response at each particular orientation. The ``best\" position from that 2D scan is selected, and the micro controllers are driven to that position. The alternate dimensions are then selected, and the procedure repeats until alignment is satisfactory. \n\nFor many scientific beam-lines, this dead-reckoning approach is sufficient. Unfortunately, the SASE process for generating X-rays at XFEL facilities introduces an unpredictable time-dependent intensity drift, as well as stochastic perturbations of the beam's propagation axes. This, of course, is in addition to the usual sources of measurement noise. These complications prevent the reliable success of a direct implementation of the simplex-based approach seen in \\cite{Breckling2021}. Given that it is only possible to record X-ray transmission for a single orientation at a single moment in time, an orderly raster-like scan of the transmission at an XFEL facility is not likely to see a distribution that strongly agrees with the Gaussian model developed in \\cite{simons17}. As a result, it remains the common practice to rely heavily on the intuition of the beamline scientist to interpret such scans, substantially extending the time required to produce an acceptable initial lateral alignment, and realignment. Given that time is an extremely limited resource at XFELs, an alternative technique to quickly and reliably expedite this procedure is sought. \n\nIn this paper we propose a technique to estimate the gradient of the transmission function that accounts for both time-dependent amplitude fluctuations, and instrumentation noise. If successful, such a gradient could be utilized in a classic steepest descent algorithm. Given that stochastic descent-based approaches have been successful in automating similar optical alignment and focusing tasks, including the control of directed energy sources \\cite{belen2007laboratory}, aligning line-of-sight communication arrays \\cite{Raj2010}, and the alignment of two-mirror telescopes \\cite{Li20}, we suspect that these corrections will allow for expedient and accurate alignments in our application. \n\nLet $t \\in \\mathbb{R}^+$ represent time,  $T\\colon \\mathbb{R}^+ \\to \\mathbb{R}$ be an arbitrarily smooth function which denotes the intensity of the beam over time, $\\varsigma$ be the aggregate of all additive stochastic noise, and $\\Theta$ denote stochastic perturbations to the beam's orientation. We then formulate our estimate of the transmission function as $G({\\bf x},t)=-T(t)f({\\bf x}+\\Theta) + \\varsigma$. Our alignment procedure then looks like the optimization problem\n\\begin{equation}\\label{exp_prob}\n\\min_{{\\bf x} \\in \\Omega} E\\left[G({\\bf x},t)\\right], \\text{ for all } t \\geq 0,\n\\end{equation}\nwhere $E$ is the expected value. While this problem does admit an optimal solution, common stochastic steepest-descent methods are not amenable to finding it without directly addressing the amplitude fluctuations \\cite{Spall}. \n\nWe propose an approach to this problem that substitutes the usual finite difference method with one which corrects for the non-steady amplitude. The method systematically intertwines the usual spatial samples for the gradient with additional samples from a fixed central location. We demonstrate through error asymptotics and numerical benchmarks that these additional samples can, when collected at sufficient rate, can sufficiently account for amplitude changes in intensity over time. Thus, an amplitude-corrected gradient, when paired with a standard stochastic descent algorithm, becomes well-suited for minimization problems like \\eqref{exp_prob}.\n\nThe remainder of the paper is organized as follows: We formally introduce the amplitude-correcting scheme in Section~\\ref{DiffScheme}, along with notation, and asymptotic error estimates. We provide two numerical benchmarks in Section~\\ref{NumericalExperiments}. There, we first develop asymptotic error estimates for stochastic gradient descent (SGD) schemes using the amplitude-correction, along with a demonstration of the resulting convergence rates. We then demonstrate the efficacy of our amplitude-correcting gradient on a modified version of the Rosenbrock valley benchmark. Section~\\ref{XFEL} outlines how our method shows promise in automating the lateral alignment of CRLs at X-ray experimental facilities. There, we provide a proof-of-concept implementation of our full optimization scheme against a synthetic cost function modelled to behave appreciably similar to one used at a genuine XFEL facility. Finally, we provide remarks in summary in Section~\\ref{conclusions}.\n\n\\section{Constructing the Amplitude-Correcting Differencing Scheme} \\label{DiffScheme}\nGiven a function $f\\colon \\mathbb{R}^n \\to \\mathbb{R}$, we are primarily concerned with computing estimates of $\\nabla f$. To this end, we assume a high degree of smoothness, i.e., $f$ is sufficiently G\\^{a}teux and Fr\\'{e}chet differentiable to satisfy the necessary conditions of our estimates to follow. The gradient of a function at a particular point ${\\bf x}_c \\in \\mathbb{R}^n$ is typically estimated by sampling that function $\\mathcal{O}(N)$ $(N \\in \\mathbb{N}, N > n)$ times in a local region around ${\\bf x}_c$. We consider an $n-$dimensional ball of radius $\\delta>0$ centered at ${\\bf x}_c$, denoted $\\mathcal{B}_\\delta({\\bf x}_c)$, and define $\\Omega$ to be an open, connected, bounded set containing $\\mathcal{B}_\\delta({\\bf x}_c)$ within the interior. \n\nFor our application, $n=4$, given the degrees of freedom for lateral alignment. Additionally, the function $f$ can only be evaluated at one particular position ${\\bf x} \\in \\mathbb{R}^n$ at a time. Sampling another position ${\\bf x}' \\in \\mathbb{R}^n$ requires a discrete amount of time $h>0$ to elapse. Given a particular starting time $t_0 > 0$, we denote the interval of time required to compute a gradient using our technique defined below to be \n\\begin{equation*}\n    \\mathcal{T}_{h,N} =: [t_0,t_0 + (4N + 1)h].\n\\end{equation*}\nThough in the interest of brevity, we may refer to $\\mathcal{T}_{h,N}$ as simply $\\mathcal{T}$. Finally, we assume that our amplitude function $T\\colon \\mathbb{R}^+ \\to \\mathbb{R}$ is at least four-times differentiable, i.e., $T \\in \\mathcal{C}^4(\\mathcal{\\mathcal{T})}.$\n\nLet $E$ and $V$ denote the expectation and variance of a time series over $\\mathcal{T}$. Our source of additive noise is assumed to be normal and i.i.d. such that $E(\\varsigma)$ = 0 and $V(\\varsigma) = \\sigma^2$. Our smooth and additive noise-corrupted functions are written:\n\\begin{eqnarray*}\n\tF({\\bf x},t) &=& T(t)f({\\bf x}), \\\\\n\tG({\\bf x},t) &=& F({\\bf x},t) + \\varsigma(t).\n\\end{eqnarray*}\n\nTo organize our scheme, we arrange our sample indices serially in terms of the position in $\\mathcal{B}_\\delta({\\bf x}_c)$ and time $t_k \\in \\mathcal{T}$. Let ${\\bf e}$ be an arbitrary unit vector in $\\mathbb{R}^n$. For the noise-free case, we write:\n\\begin{center}\n\t\\begin{tabular}{rclcl}\n\t\t$F({\\bf x}_c,t_k)$                       & = & $F_k^c$                 & = & $T_k f^c$, \\\\\n\t\t$F({\\bf x}_c \\pm \\delta {\\bf e}, t_k \\pm h)$ & = & $F_{k\\pm1}^{{\\bf e}^{\\pm}} $& = & $T_{k\\pm1} f^{{\\bf e}^{\\pm}}$.\n\t\\end{tabular}\n\\end{center}\n\\noindent Similarly, our noise corrupted case is written:\n\\begin{center}\n\t\\begin{tabular}{rclcl}\n\t\t$G({\\bf x}_c,t_k)$                       & = & $G_k^c$                 & = & $F_k^c + \\varsigma_k$, \\\\\n\t\t$G({\\bf x}_c \\pm \\delta {\\bf e}, t_k \\pm h)$ & = & $G_{k\\pm1}^{{\\bf e}^{\\pm}} $& = & $F_{k\\pm1}^{{\\bf e}^{\\pm}} + \\varsigma_{k\\pm1}$,\n\t\\end{tabular}\n\\end{center}\nwhere ${\\bf e}$ is a unit vector in the selected direction.\n\nWe use the over-bar shorthand to denote time-averaged terms, e.g.,\n\\begin{equation*}\n\\bar{F}_k^c = \\frac{F_{k-1}^c + F_{k+1}^c}{2}.\n\\end{equation*}\nWe make use of the usual norm notation, i.e., $|| \\cdot ||_2$ denotes an $L^2$ norm; though the subscript is dropped in the context of Euclidean vectors. When discussing discretized approximations to the usual gradient operator $\\nabla$, we use $\\nabla_\\delta$ to denote the uncorrected differencing scheme provided in Definition \\ref{basic_grad}, and $\\nabla_{\\delta,h}$ for the amplitude-correcting gradient estimate developed further below. Directional derivative operators and their approximations are then written as $({\\bf e} \\cdot \\nabla)$, $({\\bf e} \\cdot \\nabla_\\delta)$, and $({\\bf e} \\cdot \\nabla_{\\delta,h})$ respectively. \n\n\\begin{definition}[A Linear Regression-Based Gradient Estimate]\\label{basic_grad} Let $\\delta > 0$, and $\\Omega \\subset \\mathbb{R}^n$ contain the open ball $\\mathcal{B}_{\\delta}({\\bf x}_c)$. Further, let the points $\\lbrace {\\bf x}_i \\rbrace_{i=1}^{N}$ be a collection of $N$ unique points on the surface of the ball $\\mathcal{B}_{\\delta}({\\bf x}_c)$ such that $N>n$. For a given function $f :\\Omega \\rightarrow \\mathbb{R}$, sample each point on the ball, collecting each sample in the vector ${\\bf F} = \\lbrace f_i \\rbrace_{i=1}^{N}$.  Use the matrix ${\\bf X} = \\lbrace 1, {\\bf x}_i \\rbrace_{i=1}^{N}$ and corresponding samples ${\\bf F}$ to assemble the linear regression problem \n\\begin{equation*}\n     {\\boldsymbol \\eta}  = ({\\bf X}^T {\\bf X})^{-1} {\\bf X}^T {\\bf F}.\n\\end{equation*}\nThe solution ${\\boldsymbol \\eta} = \\lbrace \\eta_i \\rbrace_{i=1}^{n+1}$ determines the gradient estimate\n\\begin{equation*}\n    \\nabla f({\\bf x}_c) \\approx \\nabla_\\delta f({\\bf x}_c) =: \\lbrace \\eta_k \\rbrace_{k=2}^{n+1}.\n\\end{equation*}\n\\end{definition}\n\n\\subsection{The Differencing Scheme}\n\\label{sec:diffscheme}\nThe definition below is assembled similarly to that seen in Definition \\eqref{basic_grad}, but coordinates all sampling according to a uniformly-discretized time series. A example diagram is provided in Figure \\ref{ball}. If the sampling distance $\\delta > 0$ remains uniform, it is assumed that the time required to visit each point within the sequence is uniform. While this isn't a necessary limitation in practice, this assumption simplifies the analysis provded in Appendix \\ref{AccuracyEst}.\n\n\\begin{figure} \n\t\\begin{center}\n\t\t{\\bf An Example 6-Point Stencil in 2D} \\\\\n\t\t\\begin{tikzpicture}[fill = white]\n\t\t\\draw[blue!20,thick,dashed] (0,0) circle (2cm);\n\t\t\\path (0,0)       node(a) [circle, draw, fill] {${\\bf x}_c$}\n\t\t(2.0,0.0)   node(b) [circle, draw, fill] {${\\bf x}_1$}\n\t\t(1, 1.7320) node(c) [circle, draw, fill] {${\\bf x}_3$}\n\t\t(-1,1.7320) node(d) [circle, draw, fill] {${\\bf x}_5$}\n\t\t(-2.0,0)    node(e) [circle, draw, fill] {${\\bf x}_2$}\n\t\t(-1,-1.7320)node(f) [circle, draw, fill] {${\\bf x}_4$}\n\t\t( 1,-1.7320)node(g) [circle, draw, fill] {${\\bf x}_6$};\n\t\t\n\t\t\\draw[blue!100,thick,-{Straight Barb[left]}]     (node cs:name=a, angle=8    )  --  (node cs:name=b, angle=180-8);\n\t\t\\draw[blue!100,  thick,-{Straight Barb[left]}]     (node cs:name=b, angle=180+8)  --  (node cs:name=a, angle=-8   ); \n\t\t\n\t\t\\draw[purple!100,thick,-{Straight Barb[left]}]     (node cs:name=a, angle=60+8    )  --  (node cs:name=c, angle=180+60-8);\n\t\t\\draw[purple!100,  thick,-{Straight Barb[left]}]     (node cs:name=c, angle=180+60+8)  --  (node cs:name=a, angle=60-8   ); \t\t\t\n\t\t\n\t\t\\draw[green!100,thick,-{Straight Barb[left]}]     (node cs:name=a, angle=120+8    )  --  (node cs:name=d, angle=180+120-8);\n\t\t\\draw[green!100,  thick,-{Straight Barb[left]}]     (node cs:name=d, angle=180+120+8)  --  (node cs:name=a, angle=120-8   ); \t\n\t\t\n\t\t\\draw[blue!100,thick,-{Straight Barb[left]}]     (node cs:name=a, angle=180+8    )  --  (node cs:name=e, angle=180+180-8);\n\t\t\\draw[blue!100,  thick,-{Straight Barb[left]}]     (node cs:name=e, angle=180+180+8)  --  (node cs:name=a, angle=180-8   ); \t\t\t\n\t\t\n\t\t\\draw[purple!100,thick,-{Straight Barb[left]}]     (node cs:name=a, angle=240+8    )  --  (node cs:name=f, angle=180+240-8);\n\t\t\\draw[purple!100,  thick,-{Straight Barb[left]}]     (node cs:name=f, angle=180+240+8)  --  (node cs:name=a, angle=240-8   ); \t\t\t\n\t\t\n\t\t\\draw[green!100,thick,-{Straight Barb[left]}]     (node cs:name=a, angle=300+8    )  --  (node cs:name=g, angle=180+300-8);\n\t\t\\draw[green!100,  thick,-{Straight Barb[left]}]     (node cs:name=g, angle=180+300+8)  --  (node cs:name=a, angle=300-8   ); \t\t\t\n\t\t\\end{tikzpicture}\n\t\\end{center}\n\t\\caption{\\label{ball} This depicts an example six-point ($N=3$) sampling stencil  for a two-dimensional search space. The procedure requires a total of 13 samples. Begin by sampling at ${\\bf x}_c.$ Next, sample at ${\\bf x}_1$, then return and sample ${\\bf x}_c$. Repeat this process sequentially for the remaining ${\\bf x}_i$. }\n\\end{figure}\n\n\n\\begin{definition}[Amplitude-Correcting Gradient Estimate]\\label{def_grad} Let $\\delta, h > 0,$ and $\\Omega \\subset \\mathbb{R}^n$ contain the open ball $\\mathcal{B}_{\\delta}({\\bf x}_c)$. Let the points $\\lbrace {\\bf x}_i \\rbrace_{i=1}^{N}$ be a collection of $N$ unique points on the surface of the ball $\\mathcal{B}_{\\delta}({\\bf x}_c)$, along with the $N$ corresponding antipodal points $\\lbrace {\\bf x}_i' \\rbrace_{i=1}^N$, such that $N>n$. Let $\\mathcal{T}_{h,N}$ be the uniform discretization of the time interval $\\mathcal{T}.$ For the function $f:\\Omega \\rightarrow \\mathbb{R}$, the amplitude function $T:\\mathcal{T} \\rightarrow \\mathbb{R}$, and additive noise $\\varsigma: \\mathcal{T} \\rightarrow \\mathbb{R}$, we write the given function $G : \\Omega \\times \\mathcal{T} \\rightarrow \\mathbb{R}$ such that $G({\\bf x},t) = T(t)f({\\bf x}) + \\varsigma(t)$. Let $\\mu_T = E\\left[T\\left(\\mathcal{T}_{h,N} \\right) \\right]$, and ${\\bf e}_k^+$ be the unit vector in the direction ${\\bf x}_k - {\\bf x}_c.$ Index our uniformly discretized time-steps as $k=1,\\ldots,4N+1$. Sample ${\\bf x}_c$ at odd values of $k,$  i.e.  $k = 2k'-1$, collecting each sample as $G_{k}^c.$ Time-average these values gives $\\bar{G}_{2k'}^c$.  For even values of $k,$ i.e. $k= 2k'$, alternate sampling ${\\bf x}_{k'}$ and its antipodal counterpart ${\\bf x}_{k'}'$, collecting each sample as $G_{k}^{{\\bf e}_{k}^+}.$  We organize the matrix ${\\bf X}$ such that\n\\begin{equation*}\n    {\\bf X} = \\begin{bmatrix}\n   1 & {\\bf x}_1  \\\\\n   1 & {\\bf x}_1' \\\\\n   \\vdots & \\vdots \\\\\n   1 & {\\bf x}_N \\\\\n   1 & {\\bf x}_N'\n\\end{bmatrix}, \n\\end{equation*}\nthe sample matrix ${\\bf G}$ such that\n\\begin{equation*}\n    {\\bf G} = \\frac{1}{\\mu_T}\\lbrace G_{2i}^{{\\bf e}_{2i}^+} - \\bar{G}_{2i}^c , \\rbrace_{i=1}^{2N}\n\\end{equation*}and the linear regression problem \n\\begin{equation*}\n    {\\boldsymbol \\eta} = ({\\bf X}^T {\\bf X})^{-1} {\\bf X}^T {\\bf G}.\n\\end{equation*}\nThe solution ${\\boldsymbol \\eta} = \\lbrace \\eta_i \\rbrace_{i=1}^{n+1}$ determines our gradient estimate\n\\begin{equation*}\n    \\nabla f({\\bf x}_c) \\approx \\nabla_{\\delta,h}G({\\bf x}_c, \\mathcal{T}) =: \\lbrace \\eta_i \\rbrace_{i=2}^{n+1}.\n\\end{equation*}\n\\end{definition}\n\nIf we use the following short-hand for the standard central-differencing stencil (in spatial coordinates), directional derivatives can be written \n\\begin{equation*}\n\\left({\\bf e}_k \\cdot \\nabla_{\\delta}\\right)f({\\bf x}_c) = \\frac{1}{2\\delta} \\left(f^{{\\bf e}_k^+} - f^{{\\bf e}_k^-}\\right).\n\\end{equation*}\nOur convention of selecting antipodal points in sequence allows us to utilize these directional derivative stencils directly. Since each observation of $G({\\bf x},t)$ results in an independent noise term $\\varsigma$, combining like-terms results in\n\\begin{align*}\n\t\\left({\\bf e}_k \\cdot \\nabla_{\\delta,h}\\right)G({\\bf x}_c,\\mathcal{T})  =  \\frac{1}{2\\mu_T\\delta} & \\left[\\left(G_{2k}^{{\\bf e}_k^+} - \\bar{G}_{2k}^{c} \\right) - \\left(G_{2k+2}^{{\\bf e}_k^-} - \\bar{G}_{2k+2}^{c}\\right)\\right] \\\\\n\t =  \\frac{1}{2\\mu_T\\delta} & \\Big[\\left(F_{2k}^{{\\bf e}_k^+} - \\bar{F}_{2k}^{c} \\right) - \\left(F_{2k+2}^{{\\bf e}_k^-} - \\bar{F}_{2k+2}^{c}\\right) \\\\\n\t & + \\frac{\\varsigma_{1,k}}{2} + \\varsigma_{2,k} + \\varsigma_{3,k} + \\frac{\\varsigma_{4,k}}{2}\\Big].\n\\end{align*}\n\n\\begin{thm}[Error Estimate on Noise-Free Functions]\\label{NoiseFreeThm} Let $F({\\bf x},t) = T(t)f({\\bf x})$ where $T$ and $f$ are at least $\\mathcal{C}^4(\\mathcal{T})$ and $\\mathcal{C}^3(\\Omega)$ respectively. We sample $N$ antipodal pairs such that the resulting sampling is unbiased, and quasi-uniform. For ${\\bf x}_c \\in \\Omega$, and $\\delta > 0$ such that $\\mathcal{B}_\\delta({\\bf x}_c)$ is in the interior of $\\Omega$, we let ${\\bf e}_k$ be the unit vector associated with the $k^{th}$ antipodal pair of points. Further, we let $\\mu_T$ be the known expectation of $T(t)$ over $\\mathcal{T}$. Selecting $h$ such that $h^3 < \\delta$ guarantees that there exists a constant $C^*(\\delta, h, N, T,T',T^{(4)},f, \\nabla f)  > 0$ such that, \n\t\\begin{equation*}\n\t\\left|\\left|  \\nabla f({\\bf x}_c) - \\nabla_{\\delta,h} F({\\bf x}_c,\\mathcal{T})\\right|\\right| \\leq C^* \\left(h + \\delta^2 \\right).\n\t\\end{equation*}\n\\end{thm}\n\n\\noindent A similar result is provided for the case when additive i.i.d. noise is present. \n\n\\begin{thm}[Error Estimate on Noisy Functions]\\label{NoisyThm} Let $G({\\bf x},t) = F({\\bf x},t) + \\varsigma(t)$. Under the same assumptions as Theorem \\ref{NoiseFreeThm}, the total contribution of error from stochastic sources can be written\n\t\\begin{eqnarray*}\n\t\t{\\varepsilon} &:=& \\nabla_{\\delta,h} G({\\bf x}_c, \\mathcal{T}) - \\nabla_{\\delta,h} F({\\bf x}_c, \\mathcal{T}) \\\\\n\t\t& =& \\left\\lbrace \\sum_{k=1}^N \\frac{\\hat{{\\bf e}}_i^T \\cdot {\\bf e}_k}{2 \\mu_T N \\delta} \\left[\\frac{\\varsigma_{1,k}}{2} + \\varsigma_{2,k} + \\varsigma_{3,k} + \\frac{\\varsigma_{4,k}}{2}\\right]\\right\\rbrace_{i=1}^n.\n\t\\end{eqnarray*} \n\tThen it follows that\n\t\\begin{equation*}\n\t    E\\left[||\\varepsilon||\\right] \\leq 4 \\frac{\\sigma}{\\mu_T \\delta} \\sqrt{\\frac{n}{N}},\n\t\\end{equation*}\n\tand for $p \\in (0,1)$\n\t\\begin{equation*}\n\t    \\mathcal{P} \\left[ ||\\varepsilon|| \\leq 4 \\frac{\\sigma}{\\mu_T \\delta} + 2 \\frac{\\sigma}{\\mu_T \\delta}\\sqrt{\\frac{\\log(1\/p)}{N}}\\right] \\geq 1-p.\n\t\\end{equation*}\n\\end{thm}\n\n\n\n\\section{Numerical Demonstrations} \\label{NumericalExperiments}\nGiven that our motivation is to employ the amplitude-correcting gradient in steepest descent methods, our demonstrations will focus on that application. We begin by presenting two accelerated versions of the classic SGD algorithm, differing only by which gradient estimation technique utilized. A full discussion on proper choices for $\\alpha$ and $\\beta$ can be found in \\cite{Nesterov}.\n\n\\begin{alg}[Accelerated SGD] \\label{Alg2} Choose a suitable initial condition ${\\bf x}_0 \\in \\mathbb{R}^n$, step-size $\\alpha_i > 0$, $\\alpha_i \\rightarrow 0$ as $i \\rightarrow \\infty$, and $\\beta \\in [0,1)$. Additionally, choose a radius $\\delta > 0$ for the gradient estimator. Indexing our steps with $i=0,1,\\ldots$ we proceed such that\n\t\\begin{eqnarray*}\n\t\t{\\bf y}_{i+1} &=& \\beta {\\bf y}_i - \\nabla_{\\delta}f({\\bf x}_{i+1}), \\\\\n\t\t{\\bf x}_{i+1} &=& {\\bf x}_i - \\alpha_i {\\bf y}_{i+1}.\n\t\\end{eqnarray*}\n\\end{alg}\n\n\\begin{alg}[Dynamic Amplitude-Corrected Accelerated SGD] \\label{Alg1} Choose a suitable initial condition ${\\bf x}_0 \\in \\mathbb{R}^n$, step-size $\\alpha_i > 0$, $\\alpha_i \\rightarrow 0$ as $i \\rightarrow \\infty$, and $\\beta \\in [0,1)$. Additionally, prescribe a spatial radius and time-step $\\delta, h > 0$ for the gradient estimator. Indexing our steps with $i=0,1,\\ldots$ we proceed such that\n\t\\begin{eqnarray*}\n\t\t{\\bf y}_{i+1} &=& \\beta {\\bf y}_i - \\nabla_{\\delta,h}G({\\bf x}_{i+1},t), \\\\\n\t\t{\\bf x}_{i+1} &=& {\\bf x}_i - \\alpha_i {\\bf y}_{i+1}.\n\t\\end{eqnarray*}\n\\end{alg}\n\nIn our first demonstration, we seek a direct comparison of the classic SGD algorithm with the amplitude-correcting version. In order for such a comparison to be salient, we consider two functions: Rosenbrock's valley with and without a time-varying amplitude. We then demonstrate, for well-selected parameters, that Algorithm \\ref{Alg2}'s performance on the steady-amplitude function qualitatively matches Algorithm \\ref{Alg1}'s performance on the non-steady version. When both simulations are successful against minimization problems that are otherwise formulated identically, we can conclude that the amplitude-corrections encoded into the online gradient estimate effectively overcome the variations.\n\nIn the second numerical experiment, we show that the error asymptotics provided in Theorems \\ref{NoiseFreeThm} and \\ref{NoisyThm} can be seen in SGD executions. We cite two theorems that respectively provide sufficient conditions for the convergence of Algorithm \\ref{Alg2} with probability 1, and asymptotic error estimates. We then construct a noisy, time-varying function that otherwise adheres to those conditions, then prove that well-selected parameters guarantee Algorithm \\ref{Alg1} also converges. This is numerically verified by isolating each source of error to see if the analytic rates match those encountered numerically. \n\n\\subsection{A Quake in Rosenbrock's Valley}\\\nRosenbrock's Valley \\cite{Rosenbrock} is a polynomial on $\\mathbb{R}^2$ defined as\n\\begin{equation}\\label{rosenbrock}\nf(x,y) = (1-x^2) + 100(y-x^2)^2.\n\\end{equation}\nThis polynomial has a global minimum value of $f(1,1) = 0$, and is locally convex around that point. However, the downward slope along the minimal ridge is quite low in the parabolic valley. It is this feature that made Rosenbrock's Valley a popular benchmark, since many steepest descent algorithms tend to reach the ridge quite quickly, but struggle to reach the optimal answer due to the oscillations spurred from the large values of $|\\nabla f(x,y)|$ for $(x,y)$ not precisely on the ridge path. In the interest of clarity, we will refer to these as \\emph{spatial oscillations.}\n\nThe classic benchmark nonlinear programming problem is typically presented as \n\\begin{equation}\\label{nlprog1}\n{\\bf x}^* = \\text{argmin}_{{\\bf x} \\in \\mathbb{R}^2} f(x,y).\n\\end{equation}\nWe complicate matters by including the amplitude function $T(t)$ such that\n\\begin{equation}\\label{nlprog2}\n{\\bf x}^* = \\text{argmin}_{{\\bf x} \\in \\mathbb{R}^2} E\\left[T(t)f(x,y)\\right].\n\\end{equation}\nwhere \n\\begin{equation}\\label{dynamic_amp_T1}\nT(t) = 1 + \\frac{3}{4} \\cos{(2 \\pi t)}.\n\\end{equation}\nAgain, for the sake of clarity, we shall refer to oscillations caused by a dynamic amplitudes like \\eqref{dynamic_amp_T1} as \\emph{temporal oscillations}.\n\nIn our first experiment, we attempt to solve our temporally oscillating problem  \\eqref{nlprog2} with the standard gradient descent method (Algorithm \\ref{Alg2}.)  We initialize at ${\\bf x}_0 = (-1.2,1)$, fix $\\alpha_i = \\delta =  1\/500$, $\\beta = 0$, enforce a step-size maximum $||{\\bf x}_{i+1} - {\\bf x}_{i}|| \\leq 1 \/ 4$, and a maximum iteration count of $i_{\\text{max}} = 1200$. The gradient is computed by a uniform sampling of $N=15$ antipodal pairs. In Figure \\ref{rosenbrockfigs_noise}, we see that the gradient estimates are erroneous far beyond what can be tolerated by the standard algorithm. The figure only depicts steps up to $i_{200}$, since the full path eventually diverges. Increasing the momentum value $\\beta$ has no appreciable impact on this outcome.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=10cm]{diverge.eps} \\\\\n\t\t\\hspace{0.5cm} 0 \\ \\includegraphics[width=6cm]{colorbar.eps} \\ 1800\n\t\\end{center}\n\t\\caption{\\label{rosenbrockfigs_noise}This figure demonstrates the failure of Algorithm \\ref{Alg2} to solve the temporally-fluctuating problem \\eqref{nlprog2}. Our plot only considers the first 200 steps, due to an eventual divergence. Each step is depicted by a red dot, connected in sequence by a white line. The white cross in each plot depicts the optimal solution at $(1,1)$. The spatial coordinates and color axis are all non-dimensionalized.}\n\\end{figure}\n\nIn the second experiment, we seek to demonstrate that our amplitude correcting gradient estimate is effective in overcoming the temporal oscillations imposed by \\eqref{dynamic_amp_T1}. We accomplish this by comparing the performance of Algorithm \\ref{Alg1}, which utilizes the dynamic amplitude correction, on the temporally-oscillating problem \\eqref{nlprog2} to the performance of classic gradient descent method in Algorithm \\ref{Alg2} on the non temporally-oscillating problem in \\eqref{nlprog1}. For each execution we initialize at ${\\bf x}_0 = (-1.2,1)$, selecting $\\alpha_i = \\delta =  1\/500$, $\\beta = 0$, enforce a step-size maximum $||{\\bf x}_{i+1} - {\\bf x}_{i}|| \\leq 1 \/ 4$, and a maximum iteration count of $i_{\\text{max}} = 1200$. The gradient is computed by a uniform sampling of $N=15$ antipodal pairs. In the temporally oscillating problem, we prescribe a time-step of $h = 1\/16$. We provide comparisons with, and without momentum in Figure~\\ref{rosenbrockfig}.\n\nIn the first row of Figure \\ref{rosenbrockfig} we see that without momentum ($\\beta = 0$) neither implementation manages to overcome the spatial oscillations. By iteration count $i_{\\text{max}} = 1200$, both executions seem to terminate in roughly the same position. In the second row, we see that when momentum is included, both methods overcome the spatial oscillations and reach the global minimum position. When considering the apparent qualitative similarity between these outcomes, in conjunction with the failure demonstrated in Figure \\ref{rosenbrockfigs_noise}, we posit that the amplitude corrections are effective in mitigating the temporal oscillations imposed on \\eqref{nlprog2}.\n\n\\begin{figure} \n\t\\begin{center}\n\t\t\\includegraphics[width=10cm]{comparison_plot.eps}\\\\\n\t\t0 \\ \\includegraphics[width = 6cm]{colorbar.eps} \\ 600\n\t\\end{center}\n\t\\caption{\\label{rosenbrockfig} The top row compares the results from Algorithms \\ref{Alg1} and \\ref{Alg2} to problems \\eqref{nlprog2} and \\eqref{nlprog1} respectively, with no momentum term ($\\beta = 0$). The bottom row makes the same comparison, but selects a momentum term $\\beta = 0.75$. Each step is depicted by a red dot, connected in sequence by a white line. The white cross in each plot depicts the optimal solution at $(1,1)$. The spatial coordinates and color axis are again all non-dimensionalized.}\n\\end{figure}\n\n\\subsection{A Convergence Study}\nThe following theorem provides conditions sufficient for the convergence of the standard differencing gradient in SGD (Algorithm \\ref{Alg2}), as well as an error estimate. Proof can be found in \\cite{Nguyen2019}.\n\n\\begin{thm}[Convergence of SGD with Probability One]\\label{sgd_converge} Under the following assumptions,\n\\begin{enumerate}\n    \\item[{\\bf 1.)}] The objective function $f: \\mathbb{R}^n \\rightarrow \\mathbb{R}$ is $\\mu-$strongly convex, i.e., there exists $\\mu > 0$ such that\n    \\begin{equation*}\n        f({\\bf x}) - f({\\bf x}') \\geq \\nabla f({\\bf x}')^T \\cdot ({\\bf x} - {\\bf x}') + \\frac{\\mu}{2}||{\\bf x} - {\\bf x}'||^2.\n    \\end{equation*} \n    \\item[{\\bf 2.)}] For particular realizations of ${\\bf \\varsigma}$, the noise-corrupted objective function $\\hat{f}({\\bf x}) = f({\\bf x}) + \\varsigma$ is L-smooth, i.e., there exists an $L > 0$ such that for any ${\\bf x}',{\\bf x} \\in \\mathbb{R}^n$,\n    \\begin{equation*}\n        ||\\nabla_\\delta \\hat{f}({\\bf x}) - \\nabla_\\delta \\hat{f}({\\bf x}')|| \\leq L||{\\bf x}-{\\bf x}'||.\n    \\end{equation*}\n    \\item[{\\bf 3.)}] The noise-corrupted cost function $\\hat{f}$ is convex for every realization of $\\varsigma$, i.e., for any ${\\bf x}, {\\bf x}' \\in \\mathbb{R}^n$\n    \\begin{equation*}\n        \\hat{f}({\\bf x}) - \\hat{f}({\\bf x}') \\geq \\nabla_\\delta \\hat{f}({\\bf x}')^T \\cdot ({\\bf x}-{\\bf x}').\n    \\end{equation*}\n\\end{enumerate}\nThen considering Algorithm \\ref{Alg2} with step sizes \n\\begin{equation*}\n    0 < \\alpha_i < \\frac{1}{2L}, \\ \\sum_{i=0}^\\infty \\alpha_i = \\infty \\ \\text{and} \\ \\sum_{i=0}^{\\infty} \\alpha_i^2 < \\infty,\n\\end{equation*}\nthe following holds with probability 1 (almost surely)\n\\begin{equation*}\n    ||{\\bf x} - {\\bf x}^* ||^2 \\rightarrow 0,\n\\end{equation*}\nwhere ${\\bf x}^* = \\text{\\emph{argmin}}_{{\\bf x} \\in \\mathbb{R}^n} f({\\bf x})$.\n\\end{thm}\n\nThe following result presents convergence of the stochastic gradient descent method in terms of the error seen in the gradient estimates of the cost function.  \n\\begin{cor}\\label{sgd_conv_rate} Under the same assumptions of Theorem \\ref{sgd_converge}, let $\\mathcal{E} = \\frac{4L}{\\mu}$. Initialize Algorithm \\ref{Alg2} with step size $\\alpha_i = \\frac{2}{\\mu(t+\\mathcal{E})} \\leq \\alpha_0 = \\frac{1}{2L}.$ Then,\n\\begin{equation*}\n    E\\left[||{\\bf x} - {\\bf x}^* ||^2 \\right] \\leq \\frac{16M}{\\mu^2} \\frac{1}{(t-\\tau+\\mathcal{E})},\n\\end{equation*}\nfor \n\\begin{equation*}\n    t \\geq \\tau = \\frac{4L}{\\mu} \\text{\\emph{max}}\\left\\lbrace \\frac{L\\mu}{M}||{\\bf x}_0 - {\\bf x}^*||^2,1 \\right\\rbrace - \\frac{4L}{\\mu},\n\\end{equation*}\nwhere $M = 2E\\left[||\\nabla_\\delta \\hat{f}({\\bf x}^*) ||^2\\right]$ and ${\\bf x}^* = \\text{\\emph{argmin}}_{{\\bf x} \\in \\mathbb{R}^n} \\hat{f}({\\bf x})$.\n\\end{cor}\n\nWe now look to numerically verify the convergence of Algorithm \\ref{Alg1} on the problem:\n\\begin{equation}\\label{last_nlp}\n    \\min_{{\\bf x} \\in \\mathbb{R}^3} E\\left[G({\\bf x},t)\\right],\n\\end{equation}\nwhere the cost function\n\\begin{equation}\\label{cost_f}\n    G({\\bf x},t) = -\\left(1 + \\frac{3}{4}\\cos{\\left(2\\sqrt{2} \\pi t\\right)}\\right)\\left({\\bf x}^T \\mathbf{\\Sigma} {\\bf x} \\right) + \\varsigma(t),\n\\end{equation}\nwith $\\mathbf{\\Sigma}$ given by\n\\begin{equation*}\n    \\mathbf{\\Sigma} = \\begin{bmatrix}\n   2 & -0.5 & 0 \\\\\n   -0.5 & 2 & -0.5 \\\\\n   0 & -0.5 & 2\n\\end{bmatrix}.\n\\end{equation*}\n\nWe proceed by first demonstrating that an \\emph{a priori} accuracy of the SGD algorithm can be written in terms of the asymptotic error of our gradient estimate developed in Theorem \\ref{NoisyThm}. This allows us to formalize a parameterization of the error developed in Algorithm \\ref{Alg1} as a function of $\\delta, h, \\sigma,$ and $N,$ and to test the error rates. As before, the additive noise term $\\varsigma(t)$ is i.i.d. and $\\mathcal{N}(0,\\sigma)$. The asymptotic error of the dynamic amplitude-correcting gradient estimates of $G$ are given, in expectation, in Corollary \\ref{grad_cor}. Proof of the following comes directly from Theorems \\ref{NoiseFreeThm}, \\ref{NoisyThm}, and Young's inequality.\n\n\n\\begin{cor} \\label{grad_cor} The gradient of the cost function $G({\\bf x},t)$ in \\eqref{cost_f} can be estimated such that given $\\sigma, \\delta, h> 0,$ where $h^3 < \\delta$, there exists positive constants $c_1, c_2,$ and $c_3$ such that\n\\begin{equation*}\n    E\\left[||\\nabla_{\\delta,h}G({\\bf 0},t)||^2\\right] \\leq c_1(1 + N^2)h^2  + c_2 \\delta^4 + c_3 \\frac{1}{N} \\frac{\\sigma^2}{\\delta^2}.\n\\end{equation*}\n\\end{cor}\n\nWhen we ignore the time-dependent amplitude of the cost function $G$ from \\eqref{cost_f}, we note that it was constructed to satisfy the assumptions from Theorem \\ref{sgd_converge}. In particular, Assumption 1 is satisfied with $\\mu = 2.$ In the calculations to follow, the initial position is ${\\bf x}_0 := (1,1,1)$, hence the total distance we intend Algorithm \\ref{Alg1} to travel is $|| {\\bf x}_0 - {\\bf 0}|| = \\sqrt{3}$. We also note that since $L$ is sensitive to $\\varsigma(t)$, it is not precisely known. For appropriately converging step-sizes $\\lbrace \\alpha_i \\rbrace_{i = 1}^{\\infty}$, we will see\n\\begin{equation*}\n    || {\\bf x}_i - {\\bf 0}||^2\\rightarrow 0,\n\\end{equation*}\nwith probability 1 and when \n\\begin{equation*}\n    i > \\tau =  \\max \\left\\lbrace \\frac{2\\sqrt{3}L^2}{E\\left[||\\nabla_{\\delta,h}G({\\bf 0},t)||^2\\right]}, L \\right\\rbrace,\n\\end{equation*}\nwe see\n\\begin{eqnarray} \\label{alg_Error}\n    E \\left[|| {\\bf x}_i - {\\bf 0}||^2 \\right] & \\leq & E\\left[||\\nabla_{\\delta,h}G({\\bf 0},t)||^2\\right] \\frac{1}{i - \\tau} \\nonumber \n    \\\\ & \\leq & c_1(1 + N^2)h^2  + c_2 \\delta^4 + c_3 \\frac{1}{N} \\frac{\\sigma^2}{\\delta^2}.\n\\end{eqnarray}\n\\noindent Thus, for steps $i > \\tau$, the error seen in \\eqref{alg_Error} is proportional to that seen for the gradient estimate in Corollary \\ref{grad_cor}. \n\nThese error estimates are verified in a series of Monte Carlo studies. For each parametrization, we repeat and store the results from 30 executions of Algorithm \\ref{Alg1}, storing the results in\n\\begin{equation*}\n    \\text{err}(\\delta,h,\\sigma,N) = \\left\\lbrace||{\\bf x}_{i_{\\text{max}}, k}||^2 \\right\\rbrace_{k=1}^{30}.\n\\end{equation*}\nIn the first experiment, we fix $\\delta$, $\\sigma$, and $N$ such that their contributions to the error in \\eqref{alg_Error} are several orders of magnitude below our choices for $h$. We further assume that $L \\approx ||\\mathbf{\\Sigma}||_2 = 2 + \\sqrt{2}\/2,$ which gives for $h$ sufficiently small, that our critical algorithm step $\\tau$ is $\\mathcal{O}(h^{-2}).$ Selecting a fixed step-size $\\alpha_i = \\delta$, with a fixed stopping point, trivially satisfies the convergence requirements of Theorem \\ref{sgd_converge}. In addition, given our estimate of $\\tau$, and the minimum travel distance required, selecting $N=5$, and $\\delta =$1\/100, we find a choice of $i_{\\text{max}} = 500$ to be appropriate. The results of this test confirm the \\emph{a priori} rate estimate of $\\mathcal{O}(h^2)$, and are presented in terms of the average result over the 30 simulations in Table \\ref{h_conv_sim}. \n\\begin{table} \n\\caption{\\label{h_conv_sim} Monte Carlo simulations were used to estimate the accuracy of 500 steps from Algorithm \\ref{Alg1}, in terms of $h$. Corollary \\ref{grad_cor} suggests we should see error converge at a rate of 2. We fix $N$=5, $\\delta$ = 1\/100, and $\\sigma$ = 1E-5. The other sources of error begin to dominate for choices of $h \\leq 1\/512$.}\n\\begin{center}\n\\begin{tabular}{|ccc|} \n     \\hline\n      $h$ & $\\text{AVG}\\left(\\text{err}(h)\\right)$ &  Rate \\\\\n     \\hline\n     1\/16& 1.7E-4 & - \\\\\n     1\/32& 3.2E-5 & 2.41 \\\\\n     1\/64& 2.2E-6 & 3.87 \\\\\n     1\/128& 5.2E-7 & 2.07 \\\\\n     1\/256& 1.1E-7 & 2.28 \\\\\n     1\/512& 3.7E-8 & 1.53 \\\\\n     \\hline \n\\end{tabular}\n\\end{center}\n\\end{table}\n\nWe proceed similarly in the second experiment. We fix $h = 1\/1024$, $N=10$, and $\\delta=1\/100$, varying $\\sigma$. Noting that smallest choice for $\\sigma = 1\/2560$, we again estimate the critical time-step as $\\tau =$ 500. The optimal rates are observed and presented in Table \\ref{sig_conv_sim}.\n\n\\begin{table} \n\\caption{\\label{sig_conv_sim} Monte Carlo simulations were used to estimate the accuracy of 500 steps from Algorithm \\ref{Alg1}, in terms of $\\sigma$. We fix $N$=10, $\\delta$ = 1\/100, and $h$ = 1\/1024.  Corollary \\ref{grad_cor} suggests we should see error converge at a rate of 2. The other sources of error begin to dominate for choices of $\\sigma \\leq 1\/2560$.}\n\\begin{center}\n\\begin{tabular}{|ccc|} \n     \\hline\n      $\\sigma$ & $\\text{AVG}\\left(\\text{err}(\\sigma)\\right)$ &  Rate \\\\\n     \\hline\n     1\/80    & 4.0E-4 & - \\\\\n     1\/160   & 1.2E-4 & 1.73 \\\\\n     1\/320   & 4.0E-5 & 2.04 \\\\\n     1\/640   & 9.7E-6 & 2.03 \\\\\n     1\/1280  & 2.1E-6 & 2.19 \\\\\n     1\/2560  & 5.8E-7 & 1.87 \\\\\n     \\hline \n\\end{tabular}\n\\end{center}\n\\end{table}\n\nFor the third, we fix $N$=256, $\\sigma$ = 1\/2048, and $h$ = 1\/2048, varying $\\delta.$ We maintain our choice of $i_{\\text{max}}$ = 500, presenting the results in Table \\ref{delta_conv_sim}. We see rates comparable to the $\\mathcal{O}(\\delta^4)$ rate. \n\n\\begin{table} \n\\caption{\\label{delta_conv_sim} Monte Carlo simulations were used to estimate the accuracy of 5000 steps from Algorithm \\ref{Alg1}, in terms of $\\delta$. We fix $N$=256, $\\sigma$ = 1\/2048, and $h$ = 1\/2048. Corollary \\ref{grad_cor} suggests we should see error converge at a rate of 4. The other sources of error begin to dominate for choices of $\\delta \\leq 1\/100.$}\n\\begin{center}\n\\begin{tabular}{|ccc|} \n     \\hline\n      $\\delta$ & $\\text{AVG}\\left(\\text{err}(\\delta)\\right)$ &  Rate \\\\\n     \\hline\n     0.300    & 5.4E-2 & - \\\\\n     0.210    & 1.98E-2 & 2.90 \\\\\n     0.149    & 5.480E-3 & 3.70 \\\\\n     0.105    & 1.38E-3 & 3.98 \\\\\n     0.074    & 3.32E-4 & 4.11 \\\\\n     \\hline \n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIn our final experiment, we test the linear convergence rate of the sampling count parameter $N.$ Fixing $h$ = 1\/1024, $\\delta$ = 1\/100, and $\\sigma$ = 0.64, varying $N$. We select $i_{\\text{max}} = 500$. In Table \\ref{N_conv_sim} we see convergence at a rate slightly better than the expected $\\mathcal{O}(N^{-1})$ rate.  \n\n\\begin{table} \n\\caption{\\label{N_conv_sim} Monte Carlo simulations were used to estimate the accuracy of 500 steps from Algorithm \\ref{Alg1}, in terms of the inverse sample count $N^{-1}$. We fix $\\delta$ = 1\/100, $h$ = 1\/1024, and $\\sigma = $0.64.  Corollary \\ref{grad_cor} suggests we should see error converge at a rate of 1.}\n\\begin{center}\n\\begin{tabular}{|ccc|} \n     \\hline\n      $N$ & $\\text{AVG}\\left(\\text{err}(N)\\right)$ &  Rate \\\\\n     \\hline\n     8    & 6.2E-1 & - \\\\\n     16   & 3.6E-1 & 0.77 \\\\\n     32   & 1.7E-1 & 1.05 \\\\\n     64   & 1.0E-2 & 1.86 \\\\\n     128  & 3.3E-3 & 1.67 \\\\\n     256  & 1.1E-3 & 1.58 \\\\\n     \\hline \n\\end{tabular}\n\\end{center}\n\n\\end{table}\n\n\\section{Compound Refractive Lens Alignment on  Simulated XFEL Experimental Beamline}\\label{XFEL}\nWhat follows is a proof-of-concept implementation of Algorithm \\ref{Alg1} by simulating the alignment of a CRL assembly on a scientific beam-line with a highly dynamic intensity. We begin by developing a model X-ray transmission function from data collected at the Advanced Photon Source (APS) at Argonne National Laboratory \\cite{Breckling2021}. This steady-amplitude model is then augmented with a time-dependent intensity function, recorded during an experiment performed at the Pohang Accelerator Laboratory's XFEL facility (PAL-XFEL).\n\nGiven that access to XFEL beam-lines is competitive and limited, our goal is to demonstrate the feasibility of our amplitude-correcting SGD approach to overcome the beam intensity fluctuations inherent to XFEL facilities. We break this effort into two parts: the construction of our model cost function, and the results of our implementation of Algorithm \\ref{Alg1} using that cost function in settings similar to those seen at PAL-XFEL. \n\n\\subsection{Developing a Model Cost Function}\nLet $\\Omega_{max} \\subset \\mathbb{R}^4$ denote the travel limits for the four stepper motors that determine the orientation of the CRL. For a given orientation ${\\bf x} = (x,y,r_x,r_y) \\in \\Omega$, let the resulting image deposited on the detector panel be denoted as $I({\\bf x}),$ or simply $I$ when convenient; see Figure \\ref{hugh_image}. Further, we describe position of a given pixel by its indices $I_{i,j}.$ Example detector images are shown in Figures \\ref{sensor_imgs}(a) and (b).\n\nLet $\\xi(I)$, $\\mu(I)$ and $\\sigma(I)$ denote the median, mean, and standard deviation, respectively, of the pixel values of the image $I$. We then constrain\n$I$ to a selected region of interest (ROI)\ndefined as\n$$\\hat{I}_M := \\left\\lbrace I_{i,j} \\in I \\ \\Big| \\ |I_{i,j}-\\xi(I)| > M \\times \\sigma(I) \\right\\rbrace,$$\nwhere $M > 0$ is a user-selected threshold parameter. In practice, we found $M=2$ to be a good choice. Figure \\ref{sensor_imgs} (c) and (d) highlight the corresponding ROIs, $\\hat{I}_M.$ \n\n\\begin{figure} \n    \\begin{center}\n    \\includegraphics[width=2cm]{not_well_aligned.eps} \\ \n    \\includegraphics[width=2cm]{well_aligned.eps} \\ \n    \\includegraphics[height=5.75cm]{vert_colorbar.eps} \\hspace{1cm} \n    \\includegraphics[width=2cm]{not_well_aligned_support.eps} \\ \n    \\includegraphics[width=2cm]{well_aligned_support.eps} \\ \n    \\includegraphics[height=5.75cm]{vert_binarybar.eps}\n    \\caption{Figure (a) is a cropped region collected from the imaging sensor when the CRL was poorly aligned. Figure (b) is the same cropped region, but shows the result from a well-aligned CRL. The images are shown on the same color axis, after feature normalizing against the maximum pixel value recorded. Figures (c) and (d) are binary images depicting the pixels identified in the ROI for Figures (a) and (b) respectively. \\label{sensor_imgs}}\n    \\end{center}\n\\end{figure}\n\nIn the synchrotron experiments performed at the Advanced Photon Source in \\cite{Breckling2021}, a set of coordinates found by manual alignment were defined as the ground-truth to provide the ``well-aligned'' position of the CRL. We denote that position as ${\\bf x}^* = (x^*, y^*, r_x^*, r_y^*)$. This ground truth served two purposes. First, we were then able to define a feature scaling such that our metric of X-ray transmission, in terms of CRL orientation, \n\\begin{equation}\\label{crude_cost}\n    f({\\bf x}; M) := \\mu\\left(\\hat{I}_{M}({\\bf x})\\right),\n\\end{equation}\nhad a maximal value of 1. Second, the ground-truth position allowed us to establish a four-dimensional rectangular region $\\hat{\\Omega} \\subset \\Omega_{max}$ around the best point that contained the support of $f$ above the noise floor. With this ground-truth and 4D window, we then collected several raster scans of $f(\\hat{\\Omega}; M=2)$. We make use of a full four-dimensional scan, and two high-resolution, independent, 2-dimensional raster scans of $\\hat{\\Omega}$.\n\n\\begin{figure}\n    \\begin{center}\n    \\includegraphics[width=4cm]{XrY_raster.eps} \\ \n    \\includegraphics[width=4cm]{YrX_raster.eps} \\\\\n    0 \\ \\includegraphics[width = 6cm]{colorbar.eps} \\ 1\n    \\caption{Two 2D raster scans of $f$ in $\\hat{\\Omega}$ are depicted above. Both figures are mutually min-max normalized, and plotted on the same color axis.\\label{intensity_plots}}\n    \\end{center}\n\\end{figure}\n\nAssuming a steady beam amplitude, it follows from the model developed by Simons \\textit{et al.} that an idealized transmission function $f:\\mathbb{R}^4 \\rightarrow \\mathbb{R}^+$ is given by a 4-variate Gaussian distribution \\cite{simons17}. We generalize that model as \n\\begin{equation} \\label{regression}\n    f_{\\text{Simons}}({\\bf x}; a,b,\\mathbf{A}, \\hat{{\\bf x}}) = a\\exp{\\left(-({\\bf x}-\\hat{{\\bf x}})^T  \\mathbf{A}  ({\\bf x}-\\hat{{\\bf x}}) \\right)} + b,\n\\end{equation}\nwhere $a,b \\in \\mathbb{R}$, the matrix $\\mathbf{A} \\in \\mathbb{R}^{4\\times4}$ is symmetric, and $\\hat{{\\bf x}} \\in \\mathbb{R}^4$ is the position associated with optimal lateral alignment. Fitting the four-dimensional raster scan data $f(\\hat{\\Omega}; M=2)$ to Simons' model \\eqref{regression} gives the idealized X-ray transmission $f_{\\text{Simons}}^*({\\bf x})$. We present two, two-dimensional slice views of $f_{\\text{Simons}}^*({\\bf x})$ in Figure \\ref{f_simons_plots}. \n\n\\begin{figure}\n    \\begin{center}\n    \\includegraphics[width=4cm]{XrY_Gauss.eps} \\ \n    \\includegraphics[width=4cm]{YrX_Gauss.eps} \\\\\n    0 \\ \\includegraphics[width = 6cm]{colorbar.eps} \\ 1\n    \\caption{Depicted here are 2D slices selected from $f_{\\text{Simons}}^*(\\hat{\\Omega})$. The aspect ratios were selected to agree with Figures (a) and (b) from Figure \\ref{intensity_plots}. \\label{f_simons_plots}}\n    \\end{center}\n\\end{figure}\n\nTo model the noise functions that are characteristic to the XFEL light sources, we include additive measurement noise as $\\varsigma_\\Omega(t)$. Let $\\text{diam}(\\hat{\\Omega})$ denote the maximal \\text{diam}eter of the set $\\hat{\\Omega}$. We collected a sampling $\\mathcal{S} = \\lbrace {\\bf x}_i \\rbrace_{i=1}^{500}$ such that for every orientation ${\\bf x}_i$, $||{\\bf x}_i - {\\bf x}^*|| > \\text{diam}(\\hat{\\Omega})$.  We found that $\\sigma(f(\\mathcal{S};M=2)) \\approx 4.5 \\times 10^{-3}.$ We then model the time-series of additive noise $\\varsigma_\\Omega(t)$ as i.i.d. and $\\mathcal{N}(0, 4.5 \\times 10^{-3})$. \n\nWe additionally consider fluctuations that occur because of pointing jitter (from the SASE generation scheme) \\cite{kang2017hard}. We assume the position and direction of the beam may randomly fluctuate as a function of the beam's divergence profile, which was estimated at the APS to be 6.5$\\times 10^{-3}$ Radians. We account for jitter in our model as random perturbations of the orientation vector ${\\bf x}$ in the $r_x$ and $r_y$ directions. Further, we expect that the beam will jitter randomly within 10\\% of the beam-divergence. In doing so, we define \n\\begin{equation*}\n    \\Theta(t) = (0, 0, \\theta_x(t), \\theta_y(t))\n\\end{equation*}\nwhere $\\theta_x$ and $\\theta_y$ are respectively i.i.d and $\\mathcal{N}(0,6.5\\times 10^{-4})$.\n\nWe lastly introduce the fluctuating intensity of the beam over time. To this end, we utilize the measured shot-to-shot intensity values recorded at the PAL-XFEL facility, which was recorded using a quadrant beam position monitor (QBPM) at 30 Hz \\cite{DresselhausMarais2020}. We feature-scale the raw pulse-to-pulse time-series data by normalizing the full signal against the mean recorded value. This scaled signal is written as $T_{\\text{PAL}}(t,\\kappa)$ where $\\kappa$ determines the number of pulses averaged during a data collection event. In Figure \\ref{intensity_over_time} we show $T_{\\text{PAL}}(t,1)$ in dark gray, $T_{\\text{PAL}}(t,8)$ in light gray, and $T_{\\text{PAL}}(t,264)$ in red. The mollified signals at $\\kappa=8$ and $\\kappa=264$ respectively represent the average beam intensity over a sampling interval, and the amount of time required to collect all samples necessary to compute the amplitude-correcting gradient. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=12cm]{T_plot.eps}\n    \\caption{This figure depicts a two-minute interval of the signals $T_{\\text{PAL}}(t,\\kappa=1)$ in dark gray, $T_{\\text{PAL}}(t,\\kappa=8)$ in light gray, and $T_{\\text{PAL}}(t,\\kappa=264)$ in red. All signals are feature normalized by their mean value. The data was recorded at the PAL-XFEL facility. {\\color{green} \\cite{DresselhausMarais2020}} \\label{intensity_over_time} }\n\\end{figure}\n\nOur full cost function model is hence written and evaluated as\n\\begin{equation}\\label{xfel_cost}\n    G_{\\text{XFEL}}({\\bf x},t; \\kappa) = -T_{\\text{PAL}}(t,\\kappa) f_{\\text{Simons}}^*({\\bf x} + \\Theta(t)) + \\varsigma_\\Omega(t).\n\\end{equation}\n\n\\subsection{Solving the CRL Alignment Problem}\nWe now endeavor to study the performance of Algorithm \\ref{Alg1} on our model of the CRL alignment problem\n\\begin{equation*}\n    \\min_{{\\bf x} \\in \\hat{\\Omega}} E\\left[G_{\\text{XFEL}}({\\bf x},t;\\kappa = 8)\\right], \\ \\forall t > 0.\n\\end{equation*}\nOur goal is to identify a range of nominal parameter choices for Algorithm \\ref{Alg1} that can be implemented as a starting point at an XFEL facility. \n\nWe begin by noting that when sampling \\eqref{xfel_cost} to estimate $\\nabla f_{\\text{Simons}}^*({\\bf x})$ as per Definition \\ref{def_grad}, we consider $N=8$ quasi-uniformly distributed antipodal pairs in our differencing stencil. We select our effective integration time interval for the camera to be $h_{\\text{cam}} = 8\/30$ seconds, and establish the full time interval required to complete the scheme as $\\mathcal{T}_{h,N} := [t_0, (4N+1)h + t_0].$ Further, we make use of the estimate\n\\begin{equation*}\n    \\mu_T = T_{\\text{PAL}}(t_0 + 264\/30,264),\n\\end{equation*}\nwhere $t_0$ is the moment we began estimating the gradient.\n\nFor each execution of Algorithm \\ref{Alg1} that follows, the stopping condition is established to be a maximal iteration count $i_{\\text{max}}.$ No other stopping conditions are considered. Additionally, we conceptualize our initial gradient sphere radius $\\alpha_0$ as some multiple $C r,$ where $r = ||{\\bf x}_0 - \\hat{{\\bf x}}||,$ though we don't expect users to know what $r$ is \\textit{a priori.} At each step $i$, the gradient sampling radius $\\alpha_0$ is scaled by a cooling factor such that\n\\begin{equation*}\n    \\alpha_i = \\frac{\\alpha_0}{(1 + i)^\\gamma},\n\\end{equation*}\nwhere $\\gamma > 0$ and fixed. Further, we enforce a maximum step size $||{\\bf x}_i - {\\bf x}_{i-1}|| \\leq \\delta_i = \\alpha_i.$ Given that the true distance $r$ is unknown upon initialization, the executions that follow are intended to identify a performance relationship between $\\alpha_0$ with respect to $r$, $\\gamma$, and the stopping condition.\n\nWe demonstrate a single execution of Algorithm \\ref{Alg1} with an initial position ${\\bf x}_0$ selected randomly a distance of $r = 0.4$ from $\\hat{{\\bf x}}.$ We note that this Euclidean distance is significantly further away from $\\hat{{\\bf x}}$ than the positions selected during the manually-tuned rough alignments completed during data collections at the more stable synchrotron source at APS \\cite{Breckling2021}. We fix $\\gamma = 0.3$, select $\\alpha_0$ according to the distance scalar $C = 3.0$, assign a momentum term $\\beta = 0.15,$ and set the stopping condition to $i_{\\text{max}} = 100$ iterations. In addition to the time required to collect the image data from the camera sensor $h_{\\text{cam}}$, we need to include an estimate of the time required to move the four stepper motors, and process the data. We assume $h_{\\text{move}}=5\/30s$. Given the full time interval time interval $h_{\\text{total}} = h_{\\text{cam}} + h_{\\text{move}} = 13\/30s$, the total execution time assumed necessary to reach 100 iterations is \n\\begin{equation*}\n    (4N+1) \\times i_{\\text{max}} \\times h_{\\text{total}} = 1430s,\n\\end{equation*}\n or 23.8 minutes. A figure depicting the particular route taken is presented in the 2D projections shown in Figure \\ref{single_exec}.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=9cm]{SingleExecution.eps} \\\\\n\\ \\ -1 \\includegraphics[width=6cm]{colorbar.eps} 0 \\\\\n$-f_{\\text{Simons}}^*({\\bf x})$\n\\caption{\\label{single_exec\nA single execution of Algorithm \\ref{Alg1} with an initial position ${\\bf x}_0$ selected randomly at a distance $r = ||{\\bf x}_0 - \\hat{{\\bf x}}|| = 0.4.$ The initial step-size $\\alpha_0$ is fixed to $3r = 1.20,$ and scaled with each step by the cooling parameter $\\gamma = 0.3.$ The slices depicted in (a) and (b) use the optimal off-axis values in $\\hat{{\\bf x}}$. The blue dots depict the initial position projected onto the respective 2D planes, the white crosses depict the optimum alignment coordinates $\\hat{{\\bf x}}$, while the red dots depict the 100 positions ${\\bf x}_i$. Each position is connected sequentially by a white line.}\n\\end{figure}\n\nNext, we present the result of three Monte Carlo experiments. We maintain the parameter choices established in the demonstration execution above, varying only the stopping condition $i_{\\text{max}} = 50, 100,$ and $200$. Each Monte Carlo executes Algorithm \\ref{Alg1} to completion 100 times, varying the initial position randomly on  $\\partial \\mathcal{B}_r({\\bf x}_0)$ where $r = 0.4.$ The results depicted in Figure \\ref{final_MC_1} demonstrate the expected convergence behavior for those well-selected parameters. \n\n\\begin{figure}\n    \\centering\n    $i_{\\text{max}} = 50; (11.9 \\ \\text{Minutes})$\\\\\n    \\includegraphics[width=7cm]{itermaxmax_50.eps}\\\\\n    $i_{\\text{max}} = 100; (23.8 \\ \\text{Minutes})$ \\\\\n    \\includegraphics[width=7cm]{itermaxmax_100.eps}\\\\\n    $i_{\\text{max}} = 200; (47.7 \\ \\text{Minutes})$ \\\\\n    \\includegraphics[width=7cm]{itermaxmax_200.eps}\\\\\n    \\caption{\\label{final_MC_1}Depicted above are the results of three Monte Carlo simulations, wherein Algorithm \\ref{Alg1} is executed 100 times to solve the synthetic CRL alignment problem, varying the stopping condition $i_{\\text{max}}$. The point $\\hat{{\\bf x}}$ is depicted in each figure as a red cross. Figures (a) and (b) show the spatial distribution of results when $i_{\\text{max}} = 50$. Similarly, Figures (c) and (d) denote the results when $i_{\\text{max}} = 100$, and Figures (e) and (f) show $i_{\\text{max}} = 200$. The result of a particular execution is shown as a dark blue dot. The blue ellipses highlight the 99.3\\% uncertainty region.}\n\\end{figure}\n\nWhat remains to be assessed is performance as a function of the user's choice of step size, and cooling parameter. We consider two values for the cooling parameter $\\gamma$,  five initial step-size scales $C$, and six stopping conditions $i_{\\text{max}}.$ For each particular set of parameters, Algorithm \\ref{Alg1} is executed 100 times, where the starting position ${\\bf x}_0$ is again sampled randomly from $\\partial \\mathcal{B}_r(\\hat{{\\bf x}})$. These regions demonstrate a collection of parameters that tend to reliably converge under an hour ($i_{\\text{max}} < 200$ iterations.)  \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=5cm]{gamma_0_3.eps} \\ \n    \\includegraphics[width=5cm]{gamma_0_4.eps}\\\\\n    \\caption{Depicted above are the results from 60 Monte Carlo studies varying $i_{\\text{max}}$, the cooling parameter $\\gamma$, and gradient sphere-radius interval $\\alpha_0$. We vary $\\alpha_0$ as a multiple of the initial position's distance from ground truth $C||{\\bf x}_0 - \\hat{{\\bf x}}|| = Cr.$ Figures (a) and (b) fix $\\gamma$ as 0.3 and 0.4, respectively. Each Monte Carlo simulation executes Algorithm \\ref{Alg1} a total of 100 times; the average value of which is depicted normalized by $r$, and depicted as the vertical height. The shaded regions above and below the interpolated lines represent the standard deviation trend, in terms of the 4D Euclidean distance. \\label{final_all}}\n\\end{figure}\n\nWhile additional parameters remain to be thoroughly studied, namely the momentum term $\\beta,$ we found that choices of $\\beta > 0.15$ tended to perform poorly over longer periods of computation time. In particular, when $i_{\\text{max}} > 50$ we saw no apparent improvement to performance. Given that the settings identified above demonstrate convergence that tends to improve with additional computation, an attractive behavior for an unsupervised optimization method, we advise being conservative with $\\beta.$ We additionally note that our choice to equate the maximal step-size with the gradient \\text{diam}eter was born out of observation. Choices of $\\alpha_i$ substantially larger than $\\delta_i$ frequently resulted in failure over longer time intervals. Lastly, we observed that selecting $\\gamma$ too large tended to collapse $\\alpha_i$ too quickly, which was also detrimental to long-time performance. Conversely, selecting $\\gamma$ too small tends to result in slow convergence. \n\n\\section{Conclusions}\\label{conclusions}\nThe motivation for this work was encountered while attempting to automate the task of laterally aligning optics at an X-ray Free-Election Laser (XFEL) facility. These facilities, in aggregate, are capable of generating extremely bright pencil beams of X-ray light, but from moment to moment that brightness fluctuates in time. If not for the stochastic noise sources and the apparent intensity fluctuations, the task of orienting beam-line optics reduces to a rather simple minimization problem \\cite{simons17}. While the apparent level of stochastic measurement noise is certainly tractable for many stochastic descent methods, the intensity fluctuations are so severe that they required a separate, independent treatment. \n\nIn this paper, we introduced a differencing scheme to estimate the gradient of a cost functional potentially corrupted by both stochastic noise and independent amplitude fluctuations. We assume that only one position in the search space can be measured at any particular moment in time. Thus, any finite differencing scheme is going to require procedurally moving from point to point, recording each intensity along with the corresponding position and time. In this scheme, we account for the fluctuating amplitude by introducing additional samples at a single, fixed location central to the differencing stencil. By alternating these samples sequentially in time, separating the resulting data post-hoc provides a proportional estimate of the functional's amplitude. This additional signal is then interpolated along the time axis, and subtracted from the corresponding signal generated sequentially by the finite differencing stencil. When well-sampled in space and time, this method is effective at detrending those measurements. \n\nWe included a detailed error analysis of this amplitude-correcting gradient estimate, as well as numerical benchmarking of its performance in nonlinear programming problems solved with SGD. Additionally, given that access to XFEL facilities are highly limited, we included a proof-of-concept implementation of an amplitude-correcting SGD method that we believe shows promise. In doing so, we identified regions of parameter choices that will likely be effective in a similarly-configured apparatus. \n\n\\section*{Acknowledgments} \nThis manuscript has been authored in part by Mission Support and Test Services, LLC, under Contract No. DE-NA0003624 with the U.S. Department of Energy, National Nuclear Security Administration (DOE-NNSA), NA-10 Office of Defense Programs, and supported by the Site-Directed Research and Development Program. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published content of this manuscript, or allow others to do so, for United States Government purposes. The U.S. Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http:\/\/energy.gov\/downloads\/doe-public-access-plan). The views expressed in the article do not necessarily represent the views of the U.S. Department of Energy or the United States Government. DOE\/NV\/03624--1406.\n\nPortions of this work were performed at High Pressure Collaborative Access Team (HPCAT; Sector 16), Advanced Photon Source (APS), Argonne National Laboratory. HPCAT operations are supported by the DOE-NNSA's Office of Experimental Sciences.  The Advanced Photon Source is a DOE Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357\n\nSunam Kim, Sangsoo Kim, and Daewoong Nam would like to acknowledge support from the National Research Foundation of Korea (NRF), specifically NRF-2019R1A6B2A02098631 and NRF-2021R1F1A1051444.\n\nPart of this work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. We also acknowledge the support of the Lawrence Fellowship in this work.\n\n \\bibliographystyle{elsarticle-num} \n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Introduction}\n\nThe inherent randomness of the wireless medium can be utilized for extracting a shared secret, since wireless channels exhibit the feature of \\emph{reciprocity}. This approach is referred to as channel-reciprocity based key generation (CRKG). The underlying assumption is that an eavesdropper (Eve) cannot obtain the same channel state, and thus cannot compute the key. The general feasibility of the approach has been reported by several early works in the literature~\\cite{src:li2006securing,src:wilson2007channel},\nwhich have been extended by subsequent studies related to practical key agreement~\\cite{src:liu2012exploiting,src:pierrot2013}. In particular, there have been some works that deal with the removal of temporal correlation, by methods like principal component analysis (PCA)~\\cite{src:chen2011secret}, beamforming~\\cite{src:madiseh2012applying} or linear prediction~\\cite{src:mcguire2014bounds}.   \n\nThroughout the paper, we use \\textit{cross-correlation}, \\textit{mutual information}, and \\emph{secret-key rates} as performance metric. The theoretical foundation of secret-key rates has been established by Maurer~\\cite{src:maurer1993} and Ahlswede et al.~\\cite{src:ahlswede1993}. They coined the information-theoretic \\emph{source-type model}, where Alice, Bob and Eve have access to a jointly random source, and derived bounds on the secret-key \\emph{capacity}. Their result is used in a large body of research, especially for Gaussian channels, e.g., reference~\\cite{src:wallace2009key} for a multi-observation model or \\cite{src:wilson2007channel} for the application to UWB channels. \n\nHowever, some of the popular beliefs regarding the capabilities of the eavesdropper have to be challenged. Many previous works, e.g.,~\\cite{DBLP:conf\/mobicom\/MathurTMYR08sh,DBLP:conf\/mobicom\/JanaPCKPK09sh}, have relied on the assumption that the channel of Alice-to-Bob gets uncorrelated to that of Eve, as long as Eve is positioned more than half a wavelength away from Alice and Bob, commonly referred to as Jake's model~\\cite[Chapter 3.2.1]{src:goldsmith2005wireless}. In the literature, this is usually referred to as \\emph{spatial decorrelation}~\\cite{src:zhang2016key}.  A study~\\cite{src:pierrot2013} has questioned this assumption by practical evaluation. Recently, a comprehensive study~\\cite{src:he2016toward} has shown that for many popular correlation models of scattering environments, the eavesdropper might obtain largely correlated observations, especially if Eve is located within the line-of-sight beam of Alice and Bob. \n\nIn this work, we intend  and shed more light on the threats for CRGK from passive eavesdropping. \nAs a consequence, we extend the work of~\\cite{src:he2016toward} by providing more elaborated practical measurements. We quantify the leakage of Alice and Bob in relation to Eve with respect to the distance, especially for low ranges that introduce near-field effects. The measurement setup is designed with the objective to generate \\emph{reproducible} results, such that we can justify\n\\emph{stationary} random processes. This is a fundamental necessity in order to obtain meaningful results, which has sometimes been overlooked in previous work. The cross-correlation and achievable secret-key rate serve as the performance metrics that indicate the common randomness available to Alice and Bob, and likewise, the information loss to Eve. We evaluate the metrics for the original data and the processed versions after down-sampling or decorrelation. The results demonstrate that the close physical presence of Eve in the communication setting significantly changes the channel statistics. This phenomenon is so far not covered by conventional channel models for CRKG.\n\nSection~\\ref{sec:systemmodel} introduces the system model and elaborates on both the processing of the measured data and the performance metrics on security. The measurement setup is described in section~\\ref{sec:measurements}. The evaluation and results of the measurement campaign are presented in section~\\ref{sec:evaluation}. Finally, section~\\ref{sec:conclusion} concludes the paper. \n\n\\section{System model} \n\\label{sec:systemmodel}\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[\nblock\/.style={draw, drop shadow, fill=white, rectangle, minimum height=0.75cm, minimum width=3.5em},\npublicch\/.style={draw, drop shadow, fill=white, rectangle, rounded corners,minimum height=1.5em, minimum width=2.5em},\n]\n\\node[block] (alice) at (0,0) {Alice};\t\n\\node[block] (eve) at ($(alice)+(2.5,-1.25)$) {Eve};\n\\node[block] (bob) at ($(alice)+(5,0)$) {Bob};\n\n\\draw[thick,->] ($(alice)+(1,0.2)$) -- node[above] {$h_{ab,k}$} ($(bob)+(-1,0.2)$);\n\\draw[thick,->] ($(bob)+(-1,-0.1)$) -- node[below] {$h_{ba,k}$} ($(alice)+(1,-0.1)$);\n\\draw[thick,->,dashed] (alice) -- node[below,pos=0.5] {$h_{ae,k}$} (eve);\n\\draw[thick,->,dashed] (bob) -- node[below,pos=0.3] {$h_{be,k}$} (eve);\n\\end{tikzpicture}\n\\caption{Overview of the system model.} \\label{fig:systemmodel}\n\\end{figure}\nAs depicted in Figure~\\ref{fig:systemmodel}, we consider Alice, Bob and Eve measuring the channel $h_{ab,k}\\in\\mathbb{R}$, $h_{ba,k}\\in\\mathbb{R}$, $h_{ae,k}\\in\\mathbb{R}$ and $h_{be,k}\\in\\mathbb{R}$, which represent the state of Alice-to-Bob, Bob-to-Alice, Alice-to-Eve and Bob-to-Eve channels, respectively, and $k$ denotes a discrete time instant. We model these variables as joint stationary and ergodic random processes. In general, Eve gets two channel states $\\left(h_{ae,k},h_{be,k}\\right)$, however, in this study we focus on $h_{ae,k}$ only. In the following, we use the labels $x_k:=h_{ba,k}$ for Alice, $y_k:=h_{ab,k}$ for Bob, and $z_k:=h_{ae,k}$ for Eve. Furthermore, we define the vector process\n$\\mybold{v}_k:=\\left(x_{k}, y_{k}, z_{k}\\right)^T$.\n\n\\subsection{Processing}\nFor different $k$, the random vectors $\\mybold{v}_k$ are likely to exhibit correlation in time, since the wireless channel is varying only slowly in indoor environments. In order to remove the temporal dependencies, we perform two alternative options of processing, namely either downsampling or decorrelation. We show both options for $x_{k}$ only, since we have the same processing for $y_{k}$ and $z_{k}$. \n\n\\subsubsection{Downsampling}\nIf we keep only every $N_m$th variable of the process $x_{k}$, we effectively\ndownsample by factor $N_m$ and obtain\n\\begin{align}\n\\label{eq:donwsampled}\nx_{k}^{\\text{ds}}=x_{kN_m}.\n\\end{align} \nThe generated $x^{\\text{ds}}_k$ can be assumed independent under the condition that the process does not exhibit any dependence after an interval of $N_m$ variables. Subsequently, we assume that the $\\mybold{v}^{\\text{ds}}_k=\\left(x_{k}^{\\text{ds}},y_{k}^{\\text{ds}},z_{k}^{\\text{ds}}\\right)^T$ are identically and independently distributed (i.i.d.) for different $k$. \n\n\\subsubsection{Decorrelation}\n We need to provide an estimator for the autocorrelation function \n\\begin{align}\n\\label{eq:autocorrest}\n\\hat{r}_{xx}[l] = \\frac{1}{N-l} \\sum_{i=0}^{N-l-1}x_{i}x_{i+l}.\n\\end{align}\nThis estimator is unbiased if the process is correlation-ergodic. The linear forward predictor for $x_{k}$ of order $N_m$ is given by\n\\begin{align}\n\\label{eq:predictor}\n\\hat{x}_{k} = \\sum_{i=1}^{N_m} a_i x_{k-i},\n\\end{align}\nwhere $a_i\\in\\mathbb{R}$ are parameter coefficients, which can be computed by Levinson-Durbin recursion based on Yule-Walker equations~\\cite{src:vaidyanathan2007the}. We define \n\\begin{align}\n\\label{eq:decorr}\nx_{k}^{\\text{de}} = x_{k}-\\hat{x}_{k}\n\\end{align}\nas \\emph{innovation sequence}, which is orthogonal to past $h_{ab,k-i}$ for $i>0$. However, orthogonal (or uncorrelated if zero-mean) variables do not necessarily imply independence, especially not \\emph{joint} independence of $\\mybold{v}^{\\text{de}}_k=\\left(x_{k}^{\\text{de}},y_{k}^{\\text{de}},z_{k}^{\\text{de}}\\right)^T$ for different $k$. Decorrelation is practically more relevant than downsampling (even if no i.i.d. can be achieved), since the information loss is significantly lower.\n\n\\subsection{Performance metrics}\nThroughout the paper, we use (1) the Pearson correlation and (2) secret-key rates as performance metrics for security.\n\n\\subsubsection{Pearson correlation}\nThe Pearson correlation provides a measure of linear dependence between two data series. The values span between $-1$ and $1$, where $1$ refers to absolute correlation, $0$ to no correlation, and $-1$ to perfect inverse correlation. It is a wide-used metric for secrecy of practical secret-key generation~\\cite{src:he2016toward}. Given a finite collection of $N$ pairs $\\left(x_{k},y_{k}\\right)$ from the process, we use the estimator\n\\begin{align}\n\\label{eq:pearson}\n\\rho_{xy}=\\frac{\\sum\\limits_{i=0}^{N-1}\\left(x_{i}-\\bar{x}\\right)\\left(y_{i}-\\bar{y}\\right)}{\\sqrt{\\sum\\limits_{i=0}^{N-1}\\left(x_{i}-\\bar{x}\\right)^2}\\sqrt{\\sum\\limits_{i=0}^{N-1}\\left(y_{i}-\\bar{y}\\right)^2}},\n\\end{align}\nwhere $\\bar{x}=\\frac{1}{N}\\sum_{j=0}^{N-1}x_{j}$ and $\\bar{y}=\\frac{1}{N}\\sum_{j=0}^{N-1}y_{j}$ are the sample means.\n\n\n\n\\subsubsection{Secret-key rate} \n\\label{sec:sk}\nWe introduce the information-theoretic secret-key rate and use the downsampled process~\\eqref{eq:donwsampled}.\nRecall that the $\\mybold{v}^{\\text{ds}}_k$ are i.i.d. We characterize $\\mybold{v}^{\\text{ds}}_k$ by the joint probability density function $f_{\\mybold{v}^{\\text{ds}}_k}$. We apply a lower bound on secret-key capacity based on the source-type model, under the following conditions:\n\\begin{enumerate}\n\\item The joint probability density function $f_{\\mybold{v}^{\\text{ds}}_k}$ is known a priori at all terminals.\n\\item Alice and Bob exchange messages over an authenticated, public channel with unlimited communication capacity.\n\\item Eve remains passive at all times.\n\\end{enumerate}\nSubsequently, the asymptotic bound is given by~\\cite{src:ahlswede1993}\n\\begin{align}\n\\label{eq:sklower}\nC_{\\text{sk}} &\\geq \\mui\\left(x_{k}^{\\text{ds}};y_{k}^{\\text{ds}}\\right) \\notag\\\\\n& \\qquad-\\min\\left[ \\mui\\left(x_{k}^{\\text{ds}};z_{k}^{\\text{ds}}\\right), \\mui\\left(y_{k}^{\\text{ds}};z_{k}^{\\text{ds}}\\right) \\right]=:R_{\\text{sk}}\n\\end{align}\nfor each $k$, since the process is stationary. Since the actual probability distributions are unknown in practice, we evaluate the lower bound~\\eqref{eq:sklower} by estimations, based on a finite number of measured samples. We utilize a $k$-nearest neighbor estimator (NNE) for the mutual information, which is based on the idea and implementation of~\\cite{src:kraskov2004}. Mutual information is a function of joint and marginal probability densities. For  a measure of the joint density, the estimator computes the distance between a tuple of samples and its $k$th-next neighbor. A similar approach is provided for the marginal densities. To best of our knowledge, the reliability of the NNE has not been studied systematically. However, results in~\\cite{src:kraskov2004} indicate that at least for multivariate Gaussian variables, the estimation error is very low if $N>10^4$ samples are used for the estimation.\n\nNote that the bound~\\eqref{eq:sklower} could have been defined with the original $\\mybold{v}_k$ or the decorrelated processes~\\eqref{eq:decorr}, such that less information is discarded than in case of downsampling. However, in order to obtain an accurate estimation of~\\eqref{eq:sklower}, we require i.i.d. samples for the two following reasons: \n\\begin{enumerate}\n\\item The bound~\\eqref{eq:sklower} has been derived under the assumption of an unlimited number of i.i.d. observations from a random source. Therefore, a value of $R_{\\text{sk}}$ measured in bits per observation, is meaningful only if the time series is i.i.d. as well.\n\\item The NNE of~\\cite{src:kraskov2004} requires i.i.d. samples, since it relies on Khinchin's theorem~\\cite[p. 277]{src:papoulis2002probability}. If the time series of samples exhibits some dependence in time, the estimator might induce an undesired bias.\n\\end{enumerate}\n \nTherefore, if we apply the process $\\mybold{v}_k$ or its decorrelated modification~\\eqref{eq:decorr}, we have an approximation of the lower bound $R_{\\text{sk}}$~\\eqref{eq:sklower} only. While approximating the common information of Alice and Bob is a rather \"safe\" option, we need to be cautious regarding Eve. In order to minimize the risk of underestimating Eve, we verify our results obtained from $\\mybold{v}_k$ or the decorrelated version~\\eqref{eq:decorr} by comparing them with the downsampling approach, since it provides a more accurate description of the information leakage to Eve. Unfortunately, by removing samples from the estimation, the NNE gets more biased.\n\n\n\n\n\\begin{figure}[h]\n\t\t\\centering\n\\includegraphics[width=0.5\\textwidth]{figures\/raumplan\/Raumplan_Setup2.pdf}\n  \t\t\\caption{The testbed includes several experimental setups for performance evaluation as well as for security analysis. \n  \t\tAlice (X), Bob (Y) and Eve (Z) are mounted on a automated antenna positioning system.}\n        \\label{fig:setup}\n\\end{figure}\n\n\\section{Measurements}\n\\label{sec:measurements}\nThe testbed is applied at the premises of our research group, which is an office area in a university building. \nAlice is positioned at a predestined access point position. Bob and Eve are mounted on an automated antenna positioning setup, which is located at several predestined \"end-device\" positions (cf.\\ Figure~\\ref{fig:setup}). For this, we choose positions which are representative for security-related IoT devices, such as doorknobs (keyless entry systems), window frames (perimeter fence intrusion sensor), and wall (motion detectors) positions. Due to a lack of space, in this version of the paper we restrict  ourselves to a description of one representative realization of all experiments. We will also provide a full version of the paper with results of $23$ further positionings in the building.\n\n\\begin{table}\n\\caption{Parameters of the measurement setup}\n\\begin{center}\n\\begin{tabular}{|l | c | l| }\n\t\\hline\n\tParameter &  Variable & Value \\\\\n\t\\hline\\hline\n\tSampling interval & $T_s$    & $100$ msec  \\\\\\hline\n\tProbing duration & $T_p$    & $<5$ msec  \\\\\\hline\n\tStep size & $\\Delta_d$    & $5$ mm  \\\\\\hline\n\tAccuracy of step size & $\\hat{\\Delta}_d$    & $\\pm 0.05$ mm  \\\\\\hline\n\tGeometrical distance Bob-Eve & $\\Delta_{BE}$    & $[0,30]$ cm  \\\\\\hline\t\n\tGeometrical distance Alice-Bob & $\\Delta_{AB}$    & $5$ m  \\\\\\hline\t\n\tSamples per step & $N$    & $3\\cdot 10^5$   \\\\\\hline\n\\end{tabular}\n\\end{center}\n\n\\label{tab:parameters}\n\\end{table}\n\nWe perform mobile, long-time narrow-band channel measurements on $2.4$~GHz (wavelength $12.5$~cm). The data exchange protocol is implemented on three Raspberry Pi $2$ platforms (credit-card sized computer). \nAll devices are equipped with a CC$2531$ USB enabled IEEE $802.15.4$ communication interface\\footnote{http:\/\/www.ti.com\/tool\/cc2531emk}. The CC$2531$ is a true SoC solution for IEEE $802.15.4$ applications, that is compatible to network layer standards for resource-constrained devices: ZigBee, WirelessHART, and 6LoWPAN. The platform is equipped with proprietary PCB antennas, i.e., \\textit{Meandered Inverted-F antenna} (MIFA), with the size of $5\\times 12$~mm. These antennas provide good performance with a small form factor. The platform and antenna design are widely used in commercial products and suited for systems where ultra-low-power consumption is required. \n\nIn order to establish common channel probing, Alice periodically sends data frames to Bob and waits for acknowledgments. Eve also receives these request-response pairs. When receiving a probe, all three devices extract Received Signal Strength Indicators (RSSI) values and, thus, can measure a channel-dependent sequence over time. For evaluation of the channel measurements, we store and process the realizations of  $\\mybold{v}_k:=\\left(x_k, y_k, z_k\\right)^T$, locally on a monitoring laptop\n\n\nTable~\\ref{tab:parameters} lists the relevant parameters of our measurement setup. We obtain a complete realization of $\\mybold{v}_k$ on every sampling interval $T_s=100$~msec. The protocol ensures that Alice, Bob, and Eve can probe the channel within a probing duration $T_p<5$~msec. We want to analyze the joint statistical properties of the samples with respect to the position of Eve in the scene. As a consequence, we apply an automated antenna positioning system, which is constructed from a low-reflective material, cf. Figure~\\ref{fig:setup}. \nIt moves the antenna of Eve on a linear guide towards the fixed antenna of Bob in step size $\\Delta_d=5$~mm with accuracy $\\hat{\\Delta}_d=\\pm 0.05$~mm. The variable distance $\\Delta_{BE}$ ranges from $0$ to $30$~cm in order to provide $60$ different locations. Alice's antenna is placed orthogonal to the linear guiding at a fixed distance $\\Delta_{AB}=5$~m. For each position of Eve's antenna, we record at least $N$ samples.  \n\nAlice and Bob extract the common randomness $x_k$ and $y_k$ from a time-varying channel. Since we aim for meaningful and reproducible results, we have to create an environment which provides the joint stationarity to the random process. \nTherefore, with a distance of $10$~cm to Alice's antenna, we deploy  a curtain of $30 \\times 30$~cm aluminum strips that continuously rotates at $ \\approx 0.1$~rotations per second, cf. Figure~\\ref{fig:setup}. However, the rotation itself inserts a deterministic component into the channel. The evolution of the self-dependence of channel gains  --- we show exemplary $x^{\\text{ds}}_k$ --- is illustrated in Figure~\\ref{fig:MI_cyclic}. It shows that the mutual information decays rapidly and vanishes after four samples, corresponding to approximately $400$~ms. However, due to the continuously rotating curtain of aluminum strips, we discover strong stochastical dependencies after $96$ samples, corresponding to approximately $9.6$~s. Therefore, we adapt a random source (Unix file \\texttt{\/dev\/urandom}) to the motor controller and program the instrument to rotate with random speed between $0.240$ rad\/s and $1$~rad\/s in random direction and with random interval lengths $0^\\circ, 1^\\circ, \\ldots 60^\\circ$ (uniformly distributed).\n Figure~\\ref{fig:MI_cyclic} shows that no strong stochastical dependencies are given anymore.\n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[%\nwidth=7cm,\nheight=4cm,\nscale only axis,\nxmin=0,\nxmax=120,\nxlabel={$l$},\nymin=0,\nymax=0.6,\nylabel={$I(x_0;x_l)$ [bits\/observation]},\nlegend style={draw=black,fill=white,legend cell align=left,at={(0.7,0.9)}}\n]\n\\addplot [color=blue,solid,very thick,dashed]\n  table[]{figures\/muiVsDelayAliceOld-1.tsv};\n\\addlegendentry{Continuous rotation}; \n\\addplot [color=red,solid,very thick]\n  table[]{figures\/muiVsDelayAliceNew-1.tsv};\n\\addlegendentry{Random rotation}; \n\\end{axis}\n\\end{tikzpicture}%\n\t\\caption{Self-dependence of channel gains with respect to time delay. Setup is equipped with aluminum strips of either continuous or random rotation.}\n\t\\label{fig:MI_cyclic}\n\\end{figure}\n\n\n\\section{Evaluation and Results}\n\\label{sec:evaluation}\nWe now use the experimental measurements to evaluate and compare the results of the Pearson correlation~\\eqref{eq:pearson}, mutual information, as well as the achievable bound of the secret-key capacity~\\eqref{eq:sklower}, as a function of attacker's distance $\\Delta_{BE}$ to Bob. We interpret the original measurements as realizations of $\\mybold{v}_k$. In addition, we have the decorrelated and downsampled outcomes, denoted by the processes $\\mybold{v}^{\\text{de}}_k$ and $\\mybold{v}^{\\text{ds}}_k$, respectively. The decorrelated samples are obtained by a linear prediction of order $N_m = 30$. To generate the i.i.d. random vectors $\\mybold{v}^{\\text{ds}}_k$ we downsample $\\mybold{v}_k$ by the factor $N_m = 30$. In subsubsection~\\ref{sec:sk}, we have already outlined the necessity of i.i.d. random vectors to obtain accurate estimations. This is not given for $\\mybold{v}_k$ and $\\mybold{v}^{\\text{de}}_k$, however, they provide valid approximations, as the results indicate later on.\nWe present three~Figures~\\ref{fig:original}, \\ref{fig:DS}, \\ref{fig:decorr} with three Subfigures a)-c) each, which are arranged in a 3x3 matrix on the next page. The \\emph{rows} denote the Figures as follows.\n\\begin{enumerate}\n\\item Fig.~\\ref{fig:original} illustrates the results for the \\emph{original} process $\\mybold{v}_k$.\n\\item Fig.~\\ref{fig:DS} shows the results for the \\emph{downsampled} process $\\mybold{v}^{\\text{ds}}_k$ of~\\eqref{eq:donwsampled}.\n\\item Fig.~\\ref{fig:decorr} depicts the results for the \\emph{decorrelated} process $\\mybold{v}^{\\text{de}}_k$ of~\\eqref{eq:decorr}.\n\\end{enumerate}\nThe \\emph{columns} constitute Subfigures as follows. For convenience, we introduce generic labels $X\\in\\left\\lbrace x_k,x^{\\text{de}}_k,x^{\\text{ds}}_k \\right\\rbrace$ for Alice, $Y\\in\\left\\lbrace y_k,y^{\\text{de}}_k,y^{\\text{ds}}_k \\right\\rbrace$ for Bob and $Z\\in\\left\\lbrace z_k,z^{\\text{de}}_k,z^{\\text{ds}}_k \\right\\rbrace$ for Eve.\n\\begin{enumerate}\n\\item Subfigures a) show the Pearson correlation~\\eqref{eq:pearson} vs. geometrical distance $\\Delta_{BE}$ between the three pairs (Alice$\\leftrightarrow$Bob $\\rho_{XZ}$, Alice$\\leftrightarrow$Eve $\\rho_{XY}$, Bob$\\leftrightarrow$Eve $\\rho_{YZ}$).\n\\item Subfigures b) zoom into the correlation $\\rho_{XY}$ of Alice$\\leftrightarrow$Bob.\n\\item Subfigures c) depict the three mutual information results ($I(X;Y)$, $I(X;Z)$, $I(Y;Z)$) and the secret-key rate $R_{\\text{sk}}$ of ~\\eqref{eq:sklower} vs. geometrical distance $\\Delta_{BE}$.\n\\end{enumerate}\n\nMost of the practical key generation schemes use downsampling or decorrelation on the original observations $\\mybold{v}_k$. We introduce the Figs. \\ref{fig:original}, \\ref{fig:DS} and \\ref{fig:decorr} in order to analyze whether downsampling and decorrelation obscure certain features of the channel that are important for the security evaluation of the system. We start with a comparison of the cross-correlation behavior between Alice and Bob, as well as to a potential attacker. By comparing Figure~\\ref{fig:original} (a-b) and Figure~\\ref{fig:DS} (a-b) we see that no significant differences in $\\rho_{XY}$ and $\\rho_{XZ}$ occur after downsampling. (Further, $\\rho_{XZ}$ and $\\rho_{YZ}$ are almost identical due to channel reciprocity between Alice and Bob.) The high similarity is due to the fact that even the process $\\mybold{v}_k$ does not exhibit much dependency in time, as already hinted in Figure~\\ref{fig:MI_cyclic}. As a consequence, the results obtained for $\\mybold{v}_k$ expose a valid approximation of the cross-correlation. As it can be seen from Figure~\\ref{fig:DS}, in case of downsampling the results are more noisy, since much fewer samples are available for the estimations.\n\n\n\\begin{figure*}[htp!]\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given.}\n\t\\label{fig:original}\n\\end{figure*}\n\nAfter decorrelation, the results (see Figure~\\ref{fig:decorr}) show that (unlike in case of downsampling) the correlation decreases on average by $\\approx 0.05$, which can have a significant negative impact on the performance of a potential quantization scheme, cf.~\\cite[Figure 3]{WiComSec-Phy-QuantAna}. \nFurthermore, the difference between the minimum and maximum value significantly decreases. Whereas in the original (and downsampled) signal the difference is $0.995-0.98=0.015$, the difference is $0.97-0.89=0.08$ for the decorrelated signal. \nThis probably stems from errors of the autocorrelation estimate~\\eqref{eq:autocorrest}, which is necessary for the linear forward prediction. Another reason might be the Pearson correlation where single outliers (e.g., strong peaks) significantly influence the result. Analyzing the impact of decorrelation techniques on the reciprocity and security in detail is left for future work.\n\n\n\\begin{figure*}[htp!]\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given.}\n\t\\label{fig:DS}\n\\end{figure*}\n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/results\/R_sk_new\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given.}\n\t\\label{fig:decorr}\n\\end{figure*}\n\n\n\n\n\n\nBy analyzing the attacker's opportunity, we observe a wavelength dependent behavior of the correlation between $z_k$ and $x_k$ (or $y_k$), as illustrated in Subfigures a). The following findings hold for all three processes: $\\mybold{v}_k$, $\\mybold{v}^{\\text{ds}}_k$, $\\mybold{v}^{\\text{de}}_k$. The correlation vs. distance function $\\rho_{XZ}$ (and $\\rho_{YZ}$) looks similar to the channel diversity function known from Jake's model~\\cite{src:goldsmith2005wireless}, which is a zero-order Bessel function\\footnote{A zero-order Bessel function is expected for the cross-correlation behavior of two receivers if uniformly distributed scatterers are given. According to Jake's model the first zero correlation is given after $\\approx 0.4\\lambda$, where $\\lambda$ is the wavelength of the carrier~\\cite{src:goldsmith2005wireless,DBLP:books\/daglib\/0025266}.} (cf. Figure~\\ref{fig:bessel}). However, the highest correlation is not at distance $\\Delta_{BE} =0$, where the correlation is only $0.2$. The highest cross-correlation is given at a distance of $\\Delta_{BE} \\approx 12.5$~cm, which is the wavelength of the $2.4$~GHz carrier. The first correlation of zero is given at a distance of $4$~cm. \n\n\\begin{figure}[htp]\n\t\t\\centering\n\\includegraphics[width=0.375\\textwidth]{figures\/bessel\/bessel.pdf}\n  \t\t\\caption{Bessel function versus distance.}\n        \\label{fig:bessel}\n\\end{figure}\n\n\n\n\n\nNote that the cross-correlation behavior of $x_k$ to $y_k$ is not independent of Eve's antenna position. Figure~\\ref{fig:original}(b) illustrates the correlation behavior in detail. The correlation has an \"oscillating\" behavior with a wavelength of approximately $11$~cm, whereby at a distance of $5$~cm the curve decreases rapidly to the lowest level of $\\approx 0.98$. The reason for that might be the non-perfect uniformly distributed scatterers in the environment, which are the basis of Jake's model. \nThe oscillating behavior in Alice's and Bob's original observation is also given in the downsampled and decorrelated versions, cf. Figure~\\ref{fig:DS}(b) and Figure~\\ref{fig:decorr}(b).\nThis behavior is contradictory to theoretical approaches based on Jake's Doppler spectrum~\\cite{DBLP:books\/daglib\/0025266}. The reason might be because the narrow band fading models do not include coupling and near field effects between both antennas for the spatial evaluation of autocorrelation, cross-correlation, and power spectral density (cf. \\cite[Chapter 3.2]{src:goldsmith2005wireless}). \n\n\nThe boundary $B$ between the near field zone and the far field zone can usually be determined by the following relationship: $B\\geq \\frac{2D^2}{\\lambda}$, where $D$ is the largest antenna size~\\cite{DBLP:journals\/comnet\/DlugoszT10}. We estimated the size of our antenna to be $6$~cm. Therefore, the boundary is $\\approx 5.7$~cm. \nAnalyzing near field boundaries in detail is left for future work. \n\n\n\n\nCompared to the cross-correlation behavior between the i.i.d. samples   $x^{\\text{ds}}_k$ and $y^{\\text{ds}}_k$ (after downsampling), both mutual information $I(X;Y)$ and $R_{sk}$ have very similar oscillating behavior, shown in Subfigures c). The (minimum, maximum) values of the correlation are ($0.980$, $0.995$) and the ones of the mutual information are ($2.1$, $2.75$). \nBy analyzing Eve's observation, we see only a slight similarity between the mutual information $I(X;Z)$ (and $I(Y;Z)$) to the correlation behavior of her observation $\\rho_{XZ}$ (and $\\rho_{YZ}$). The similarity can be found by comparing the maximum absolute values. For instance, the highest correlation occurs at $10$~cm with a value of $0.5$, and corresponds to the highest mutual information of $0.5$~bits per sample. \nHowever, the Bessel-like behavior is not evident. Notably is the fact that the attackers observation $z_k$ does not significantly impact $R_{sk}$. Our results show that $R_{sk}$ is mainly dependent on $x_k$ and $y_k$. \nHowever, Eve's antenna affects Alice's and Bob's observation and, therefore, affects $R_{sk}$. Table~\\ref{tab:results} summarizes our results.\n\n\n\\begin{table}\n\\caption{Averaged results of our experiment.}\n\\begin{center}\n\\begin{tabular}{|l | c | c | c | }\n    \\hline\n    & $\\mybold{v}_k$ &  $\\mybold{v}^{\\text{ds}}_k$ &\n$\\mybold{v}^{\\text{de}}_k$ \\\\\n    \\hline\\hline\n    $\\rho_{x_k,y_k}$ & $\\approx 0.99$  & $\\approx 0.99$  & $0.94$  \\\\\\hline\n    $\\rho_{y_k,z_k}$ & $\\approx 0.09$  & $\\approx 0.09$  & $ \\approx\n0.07$  \\\\\\hline\n    $I(X;Y)$         & $\\approx 2.92$  & $\\approx 2.89$  & $\\approx \n2.31$  \\\\\\hline\n    $I(Y;Z)$         & $\\approx 0.26$  & $\\approx 0.27$  & $0.10$  \\\\\\hline\n    $R_{\\text{sk}}$ & $\\approx 2.67$  & $\\approx 2.63$  & $\\approx 2.22$  \\\\\\hline\n\n\\end{tabular}\n\\end{center}\n\n\\label{tab:results}\n\\end{table}\n\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this work, we have provided an important pillar to bridge the gap between theory and practice-oriented approaches for CRKG. Our experimental study helps to provide a better understanding of channel statistics in wireless environments for security applications. \nWe present reproducible results based on a relevant environment which justifies the joint stationarity of a random process.  \nWe show results of cross-correlation, mutual information and secret-key rates, which are dependent on attacker's (or third device's) position. \nAs a result, we discovered that the \\textit{observer effect} occurs, which most probably originates from near field distortions. \nWe believe the effect needs to be considered in the future. Common channel models like Jake's model for channel diversity need to be extended in order to be valid for key generation setups. Furthermore, it might be pertinent, for instance, to detect the proximity of Eve. Basing on our results two bidirectionally communicating nodes might recognize a third device, its relative position, and its motion in the proximity. Further studies might use complex-valued channel profiles to analyze  third party positioning based and motion based influences.\n\n\n\n\n\\section{Full Measurement}\n\\begin{figure*}\n\t\t\\centering\n\\includegraphics[width=0.75\\textwidth]{figures\/raumplan\/Raumplan_Setup.pdf}\n  \t\t\\caption{The testbed includes several experimental setups for performance evaluation as well as for security analysis. \tAlice (X), Bob (Y) and Eve (Z) are mounted on a automated antenna positioning system.}\n        \\label{fig:setup_ur}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 0.}\n\t\\label{fig:app_original_0}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 0.}\n\t\\label{fig:app_ds_0}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-05_Nachtmessung_Paar_Office\/results\/03-02-2016_11-44-39\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 0.}\n\t\\label{fig:app_decorr_0}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 1.}\n\t\\label{fig:app_original_1}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 1.}\n\t\\label{fig:app_ds_1}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Nachtmessung_Paar_Office\/results\/03-02-2016_11-54-10\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 1.}\n\t\\label{fig:app_decorr_1}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 2.}\n\t\\label{fig:app_original_2}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 2.}\n\t\\label{fig:app_ds_2}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-06_Tagmessung_Paar_Office\/results\/03-02-2016_12-06-20\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 2.}\n\t\\label{fig:app_decorr_2}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 3.}\n\t\\label{fig:app_original_3}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 3.}\n\t\\label{fig:app_ds_3}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-07_TagNachtMessung_Paar_Office\/results\/03-02-2016_12-13-48\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 3.}\n\t\\label{fig:app_decorr_3}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 4.}\n\t\\label{fig:app_original_4}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 4.}\n\t\\label{fig:app_ds_4}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Tagmessung_Paar_Office\/results\/03-02-2016_14-47-03\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 4.}\n\t\\label{fig:app_decorr_4}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 5.}\n\t\\label{fig:app_original_5}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 5.}\n\t\\label{fig:app_ds_5}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-08_Wochenendmessung_Paar_Office_und_Serverraum\/results\/03-02-2016_16-00-49\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 5.}\n\t\\label{fig:app_decorr_5}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 6.}\n\t\\label{fig:app_original_6}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 6.}\n\t\\label{fig:app_ds_6}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Nachtmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-15-42\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 6.}\n\t\\label{fig:app_decorr_6}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 7.}\n\t\\label{fig:app_original_7}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 7.}\n\t\\label{fig:app_ds_7}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-11_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-27-43\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 7.}\n\t\\label{fig:app_decorr_7}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 8.}\n\t\\label{fig:app_original_8}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 8.}\n\t\\label{fig:app_ds_8}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Nachtmessung_Irmgard_und_Serverraum\/results\/03-02-2016_17-35-29\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 8.}\n\t\\label{fig:app_decorr_8}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 9.}\n\t\\label{fig:app_original_9}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 9.}\n\t\\label{fig:app_ds_9}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-12_Tagmessung_Archiv_und_Serverraum\/results\/03-02-2016_17-44-50\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 9.}\n\t\\label{fig:app_decorr_9}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 10.}\n\t\\label{fig:app_original_10}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 10.}\n\t\\label{fig:app_ds_10}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-13_Nachtmessung_Serverraum_und_Tim\/results\/03-02-2016_17-52-17\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 10.}\n\t\\label{fig:app_decorr_10}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 11.}\n\t\\label{fig:app_original_11}\n\\end{figure*}\n\n\\clearpage\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 11.}\n\t\\label{fig:app_ds_11}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Nachtmessung_Serverraum_und_Falk\/results\/03-02-2016_18-02-23\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 11.}\n\t\\label{fig:app_decorr_11}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 12.}\n\t\\label{fig:app_original_12}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 12.}\n\t\\label{fig:app_ds_12}\n\\end{figure*}\n\n\\clearpage\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-14_Tagmessung_Serverraum\/results\/03-02-2016_18-11-21\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 12.}\n\t\\label{fig:app_decorr_12}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 13.}\n\t\\label{fig:app_original_13}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 13.}\n\t\\label{fig:app_ds_13}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_18-19-03\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 13.}\n\t\\label{fig:app_decorr_13}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 14.}\n\t\\label{fig:app_original_14}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 14.}\n\t\\label{fig:app_ds_14}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-15_Wochenendmessung_Serverraum_und_Christian\/results\/03-02-2016_18-25-55\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 14.}\n\t\\label{fig:app_decorr_14}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 15.}\n\t\\label{fig:app_original_15}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 15.}\n\t\\label{fig:app_ds_15}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Nachtmessung_Serverraum_und_Kueche\/results\/03-02-2016_19-06-14\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 15.}\n\t\\label{fig:app_decorr_15}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 16.}\n\t\\label{fig:app_original_16}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 16.}\n\t\\label{fig:app_ds_16}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-18_Tagmessung_Serverraum_und_Archiv\/results\/03-02-2016_19-18-10\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 16.}\n\t\\label{fig:app_decorr_16}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 17.}\n\t\\label{fig:app_original_17}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 17.}\n\t\\label{fig:app_ds_17}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Nachtmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-24-32\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 17.}\n\t\\label{fig:app_decorr_17}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 18.}\n\t\\label{fig:app_original_18}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 18.}\n\t\\label{fig:app_ds_18}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-19_Tagmessung_Serverraum_und_Tim\/results\/03-02-2016_19-32-16\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 18.}\n\t\\label{fig:app_decorr_18}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 19.}\n\t\\label{fig:app_original_19}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 19.}\n\t\\label{fig:app_ds_19}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Nachtmessung_Serverraum_und_Tobias\/results\/03-02-2016_19-38-45\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 19.}\n\t\\label{fig:app_decorr_19}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 20.}\n\t\\label{fig:app_original_20}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 20.}\n\t\\label{fig:app_ds_20}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-20_Tagmessung_Serverraum_und_Lounge\/results\/03-02-2016_19-50-33\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 20.}\n\t\\label{fig:app_decorr_20}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 21.}\n\t\\label{fig:app_original_21}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 21.}\n\t\\label{fig:app_ds_21}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Nachtmessung_Serverraum_und_Erik\/results\/03-02-2016_19-55-50\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 21.}\n\t\\label{fig:app_decorr_21}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 22.}\n\t\\label{fig:app_original_22}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 22.}\n\t\\label{fig:app_ds_22}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-21_Tagmessung_Serverraum_und_Erik\/results\/03-02-2016_20-08-10\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 22.}\n\t\\label{fig:app_decorr_22}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/before_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/before_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/before_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 23.}\n\t\\label{fig:app_original_23}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/after_ds_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=0.5cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/aftere_ds_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=2.2cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/after_ds_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{ds}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 23.}\n\t\\label{fig:app_ds_23}\n\\end{figure*}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\subfloat[]{\\includegraphics[trim=1.4cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/after_decorr_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/aftere_decorr_AB_Corr.pdf}}\n\t\\subfloat[]{\\includegraphics[trim=1.8cm 0.1cm 3.5cm 1.6cm, clip=true, height=0.224\\textwidth]{figures\/apendix_results\/2016-01-22_Tagmessung_Serverraum_und_Christian\/results\/03-02-2016_20-15-07\/after_decorr_SC.pdf}}\n\t\\caption{Evaluation results of $\\mybold{v}^{\\text{de}}_k$. In (a) and (b) the cross-correlations is given; in (c) the mutual information as well as $R_{\\text{sk}}$ is given. Position 23.}\n\t\\label{fig:app_decorr_23}\n\\end{figure*}\n\n\n\\end{appendices}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nHigher-order derivative terms play important roles in the several contexts, e.g., inflation models, modified gravity, renormalization of gravity, and so on. From a phenomenological and theoretical viewpoint, their embeddings into supersymmetry (SUSY) or supergravity (SUGRA) are also interesting. In particular, there exist many non-renormalizable terms in SUGRA and it is quite natural to consider the extension including higher-order derivative terms and the effects of them on cosmology and particle phenomenology. The higher-order derivative terms of a chiral superfield in 4D SUSY or SUGRA and their cosmological applications have been investigated so far, e.g., in Refs.~\\cite{Khoury:2010gb,Khoury:2011da,Baumann:2011nm,Farakos:2012je,Koehn:2012ar,Farakos:2012qu,Koehn:2012te,Farakos:2013zya,Gwyn:2014wna,Aoki:2014pna,Aoki:2015eba,Ciupke:2015msa,Bielleman:2016grv}. \n\n\nThe Dirac-Born-Infeld (DBI) action \\cite{Born:1934gh,Dirac:1962iy} includes such higher-order derivative terms. It was first proposed as a nonlinear generalization of Maxwell theory. The DBI action is also motivated by string theory, which is a promising candidate for a unified theory including gravity. In the context of string theory, an effective action of D-brane is described by a DBI-type action, which consists of Maxwell terms $F_{\\mu \\nu }$ as well as the ones of scalar fields $\\partial _{\\mu }\\phi^i\\partial _{\\nu }\\phi^jg_{ij}$ and a two-form $B_{\\mu \\nu }$ in general, \n\\begin{align}\nS_{\\rm{DBI}}=\\int d^Dx \\sqrt{-g}\\left( 1-\\sqrt{{\\rm{det}}(g_{\\mu \\nu }+ \\partial _{\\mu }\\phi ^i\\partial _{\\nu }\\phi ^jg_{ij}+B_{\\mu \\nu }+F_{\\mu \\nu })} \\right) . \\label{string DBI}\n\\end{align}\n\nSUSY Dp-brane actions in $D$ dimension are also important for the effective theory of superstring. With a component formalism, such actions have also been discussed in many literature. For example, in Refs.~\\cite{Aganagic:1996pe,Aganagic:1996nn}, the authors construct SUSY Dp-brane actions with local kappa symmetry based on a component formalism in 10 dimensional spacetime. In a similar way, the p-brane action in various dimensions has also been discussed in Ref.~\\cite{Bergshoeff:2013pia}. In Refs.~\\cite{Howe:1996mx,Cederwall:1996pv,Cederwall:1996ri,Bergshoeff:1996tu}, the SUSY Dp-brane in SUGRA background is constructed by considering the background super-vielbein on the brane and couplings between them. \n\nAn approach based on superfields is useful for constructing a manifestly SUSY invariant action and generalizing it. Within the formalism, such 4D $\\mathcal{N}=1$ SUSY extensions of the DBI action have been known partially. The DBI action of a vector superfield, which corresponds to the case with $\\phi^i=B_{\\mu \\nu }=0$ in Eq. $\\eqref{string DBI}$, is constructed in Refs.~\\cite{Cecotti:1986gb,Bagger:1996wp,Rocek:1997hi,Kuzenko:2002vk,Kuzenko:2005wh}. In particular, in Refs.~\\cite{Bagger:1996wp,Rocek:1997hi}, it is shown that such an action appears from the partial breaking of 4D $\\mathcal{N}=2$ SUSY. Its SUGRA embedding has also been discussed in Refs.~\\cite{Cecotti:1986gb,Kuzenko:2002vk,Kuzenko:2005wh,Abe:2015nxa}. Its application to inflation models has been investigated in Ref.~\\cite{Abe:2015fha}. Furthermore, in global SUSY,  multiple $U(1)$~\\cite{Ferrara:2014oka,Ferrara:2014nwa} and massive~\\cite{Ferrara:2015ixa}  extensions of the DBI action have been discussed. In particular, for the case with multiple U(1) vector multiplets, linear actions ~\\cite{Andrianopoli:2014mia}, general conditions for partial SUSY breaking ~\\cite{Andrianopoli:2015wqa,Andrianopoli:2015rpa}, and c-maps ~\\cite{Andrianopoli:2016eub} have also been discussed.\n \nFor the DBI action of scalar fields, which corresponds to the case with $F_{\\mu \\nu }=B_{\\mu \\nu }=0$ in Eq. $\\eqref{string DBI}$, its SUSY extension has been done via partially broken $\\mathcal{N}=2$ SUSY theory, where the Goldstino multiplet is an $\\mathcal{N}=1$ real linear superfield \\cite{Rocek:1997hi,Bagger:1997pi,GonzalezRey:1998kh}. However, there has never been the SUGRA extension of the DBI action of a real linear superfield. In this paper, we discuss the embedding of the DBI action of a real linear superfield into SUGRA. The action of a chiral superfield can be found in Ref.~\\cite{Koehn:2012ar}. In general, it is known that the action with a chiral superfield can be rewritten in terms of the one with a real linear superfield, and vice versa (via linear-chiral duality \\cite{Siegel:1979ai}). Therefore, our action, which will be discussed in this paper, would be equivalent to that derived in Ref.~\\cite{Koehn:2012ar} through the duality transformation. We will discuss this point and the differences between their result and ours.\n\nIn Refs. \\cite{Rocek:1997hi,Bagger:1997pi,GonzalezRey:1998kh}, the DBI action of a real linear multiplet is realized with a chiral multiplet, which is constrained by a specific $\\mathcal{N}=1$ SUSY constraint. We will investigate the corresponding constraint which is a key for the construction of DBI action, in SUGRA. To achieve this, we use a formulation based on conformal SUGRA \\cite{Kaku:1978nz,Kugo:1982cu,Kugo:1983mv}\\footnote{We will use the superconformal tensor calculus~\\cite{Kaku:1978nz,Kugo:1982cu,Kugo:1983mv}. See also another formulation, conformal superspace~\\cite{Butter:2009cp,Kugo:2016zzf}.}, where  one can treat off-shell SUGRA with different sets of auxiliary fields in a unified manner. Because of the restrictions on the SUGRA embedding of the ${\\cal N}=1$ constraint, we will find that the DBI action of a real linear superfield can be realized only in the so-called new minimal formulation of SUGRA. Furthermore, we will extend the DBI action to the matter coupled version of it.\n\nThe remaining parts of this paper are organized as follows. First, we will briefly review the SUSY DBI action of a real linear superfield in Sec.~\\ref{review}. There, we will find that the constraint imposed between a chiral and real linear superfield is important for the construction. Then, we will extend the constraint to that in conformal SUGRA in Sec.~\\ref{extension}. After a short review of conformal SUGRA, we will also review the concept of the {\\it{u-associated}} derivative which is crucial for the superconformal extension. Using this {\\it{u-associated}} derivative, we will complete the embedding and find that the constraint can be consistently realized in the new minimal SUGRA. With the constraint, we will construct the corresponding action in the new minimal SUGRA, and write down the bosonic component action in Sec.~\\ref{Component action}. The linear -chiral duality and the matter coupled extension will be also discussed there. Finally, we will discuss the correspondence and differences between results in related works and ours in Sec.~\\ref{discussion}, and summarize this paper in Sec.~\\ref{summary}. In Appendix.~\\ref{explicit}, the explicit components of the multiplet including the {\\it{u-associated}} derivative are shown.\n\nIn this paper, we use the unit $M_P=1$ where $M_P=2.4\\times 10^{18}$ GeV is the reduced Planck mass, and follow the conventions of \\cite{Wess:1992cp} in Sec.~\\ref{review} and of \\cite{Freedman:2012zz} in other parts. $a,b\\cdots$ denote Minkowski indices and $\\mu,\\nu\\cdots$ denote curved indices.\n\n\\section{Review of DBI action in global SUSY}\\label{review}\nIn this section, we briefly review the DBI action of a real linear superfield in global SUSY \\cite{Bagger:1997pi}.  We use a chiral superfield $X$ and a real linear superfield $L$ which satisfy the conditions,\n\\begin{align}\n\\bar{D}_{\\dot{\\alpha }} X=0, \\ \\ \\ D^2L=\\bar{D}^2L=0,\n\\end{align}\nwhere $D_{\\alpha }$ and $\\bar{D}_{\\dot{\\alpha }}$ are a SUSY spinor derivative and its complex conjugate.\nTo construct the DBI action for $L$, we consider the following constraint between $X$ and $L$,\n\\begin{align}\nX-\\frac{1}{4}X\\bar{D}^2\\bar{X}-\\bar{D}_{\\dot{\\alpha }}L\\bar{D}^{\\dot{\\alpha }}L=0 , \\label{global constraint}\n\\end{align}\nwhere $\\bar{X}$ is a complex conjugate of $X$ \\footnote{In Ref.~\\cite{Bagger:1997pi}, the constraint $\\eqref{global constraint}$ has been obtained from the tensor multiplet in $\\mathcal{N}=2$ SUSY through partial breaking of it. Here, we do not discuss its origin and we just use the constraint as a guideline to obtain the DBI action. In Sec.~\\ref{discussion}, we will briefly comment on the relation between the partial breaking of ${\\cal N}=2$ SUSY and our construction.}. The equation. $\\eqref{global constraint}$ can be solved with respect to $X$ and we obtain\n\\begin{align}\nX=\\bar{D}_{\\dot{\\alpha }}L\\bar{D}^{\\dot{\\alpha }}L +\\frac{1}{2}\\bar{D}^2\\Biggl[ \\frac{D^{\\alpha }LD_{\\alpha }L\\bar{D}_{\\dot{\\alpha }}L\\bar{D}^{\\dot{\\alpha }}L }{1-\\frac{1}{2}A+\\sqrt{1-A+\\frac{1}{4}B^2}}\\Biggr],\\label{solution for global constraint}\n\\end{align}\nwhere\n\\begin{align}\nA\\equiv \\frac{1}{2}\\{ D^2( \\bar{D}_{\\dot{\\alpha }}L\\bar{D}^{\\dot{\\alpha }}L)+{\\rm{h.c.}}  \\}, \\ \\ \\ B\\equiv \\frac{1}{2}\\{ D^2( \\bar{D}_{\\dot{\\alpha }}L\\bar{D}^{\\dot{\\alpha }}L)-{\\rm{h.c.}}  \\}.\n\\end{align}\nUsing this solution $\\eqref{solution for global constraint}$, we can construct the SUSY DBI action as \n\\begin{align}\n\\mathcal{L}=\\int d^2 \\theta X(L) +{\\rm{h.c.}}. \\label{global DBI}\n\\end{align}\nOne can check that the bosonic part of the Lagrangian $\\eqref{global DBI}$ produces,\n\\begin{align}\n\\mathcal{L}_B=1-\\sqrt{1-B\\cdot B+\\partial C\\cdot \\partial C-(B\\cdot \\partial C)^2} ,\\label{component form of global DBI}\n\\end{align}\nwhere $C$ and $B_a$ are a real scalar and a constrained vector satisfying $\\partial ^a B_a=0$, in the real linear superfield, and we use the notation $B\\cdot \\partial C \\equiv B^a \\partial _aC $. It is known that, through the linear-chiral duality, Eq. $\\eqref{component form of global DBI}$ produces the DBI action of a complex scalar, which can be interpreted as the 4D effective D3-brane action. \nWe call Eq. $\\eqref{component form of global DBI}$ the DBI action of a real linear superfield in this paper.\n\nIt is worth noting that Eq. $\\eqref{solution for global constraint}$ satisfies the nilpotency condition, i.e., $X^2=0$, due to the Grassmann property of the SUSY spinor derivative, $\\bar{D}_{\\dot{\\alpha }}$. This reflects the underlying Volkov-Akulov SUSY \\cite{Volkov:1972jx,Rocek:1978nb}. Instead of writing the action like Eq. $\\eqref{global DBI}$, we can also rewrite the same system imposing the constraint $\\eqref{global constraint}$ by a chiral superfield Lagrange multiplier $\\Lambda $,\n\\begin{align}\n\\mathcal{L}=\\int d^2 \\theta \\biggl[ X +\\Lambda \\left( X-\\frac{1}{4}X\\bar{D}^2\\bar{X}-\\bar{D}_{\\dot{\\alpha }}L\\bar{D}^{\\dot{\\alpha }}L\\right) +\\tilde{\\Lambda }X^2\\biggr]  +{\\rm{h.c.}}. \\label{global DBI with constraint}\n\\end{align}\nHere we have introduced another Lagrange multiplier $\\tilde{\\Lambda }$, which ensures the nilpotency of $X$. Indeed, we need not require this condition in the Lagrangian since $X$ satisfies $X^2=0$ after integrating out $\\Lambda$ first and solving $X$ with respect to $L$, but the condition is still consistent and makes the calculation simple as far as we focus on the bosonic part of the action, as we will see in the following section.\n\\section{Extension to 4D $\\mathcal{N}=1$ conformal SUGRA}\\label{extension}\nIn this section, we generalize the SUSY DBI action $\\eqref{global DBI with constraint}$ discussed in Sec.~\\ref{review} to that in SUGRA.  \n\n\\subsection{Review of conformal SUGRA} \\label{conformal SUGRA}\nTo construct the action in SUGRA, we use conformal SUGRA formulation. Then, let us briefly review the basics of the conformal SUGRA before proceeding to the specific construction of the DBI action. \n\nIn this formulation, there are extra gauge symmetries such as dilatation, $U(1)_A$ symmetry, S-SUSY and conformal boost in addition to translation, Lorentz transformation and SUSY. The commutation and anti-commutation relations are governed by the superconformal algebra and its representation $\\Phi$ called a superconformal multiplet has the following components, \n\\begin{align}\n\\Phi =\\{ \\mathcal{C}, \\mathcal{Z},\\mathcal{H},\\mathcal{K},\\mathcal{B}_a,\\Lambda ,\\mathcal{D}\\} ,\\label{general multiplet}\n\\end{align}\nwhere $\\mathcal{Z}$ and $\\Lambda $ are spinors; $\\mathcal{B}_a$ is a vector; the others are complex scalars.  We also denote the superconformal multiplet $\\Phi$ by its first component $\\mathcal{C}$,\n\\begin{align}\n\\Phi =\\langle \\mathcal{C} \\rangle ,\n\\end{align}\nwhere $\\langle  ...\\rangle $ represents the superconformal multiplet which has $\\mathcal{C}$ as the first component. $\\mathcal{C}$ must be invariant under the transformations of S-SUSY and conformal boost in order for $\\Phi=\\langle \\mathcal{C}\\rangle$ to be a superconformal multiplet \\cite{Kugo:1983mv}.\n \nA superconformal multiplet is characterized by the charge $(w,n)$ under dilatation and $U(1)_A$ symmetry called the Weyl weight and the chiral weight, respectively. For example, a chiral multiplet $X$ has $(w,w)$, in order to satisfy\n\\begin{align}\n\\bar{\\mathcal{D}}_{\\dot{\\alpha}}X=0, \\label{dbar}\n\\end{align}\nwhere $\\bar{\\mathcal{D}}_{\\dot{\\alpha}}$ is a spinor derivative \\cite{Kugo:1983mv}. For a real linear multiplet $L$ defined by, \n\\begin{align}\n\\Sigma L =\\bar{\\Sigma } L=0,\n\\end{align}\nwhere $\\Sigma $ ($\\bar{\\Sigma }$) is a (anti-) chiral projection operator, the values of each weight are determined as $(w,n)=(2,0)$. We will discuss these operators, $\\mathcal{D}_\\alpha$ and $\\Sigma $, more precisely in the following subsections.\n\nThe chiral multiplet consists of the following components, $\\{z,P_L\\chi,F\\}$, where $z$ and $F$ are complex scalars and $P_L\\chi$ is a chiral spinor; $P_L=(1+\\gamma_5)\/2$ is a left-handed projection operator. It is embedded into a general superconformal multiplet $\\eqref{general multiplet}$ as\n\\begin{align}\n\\{ z,-\\sqrt{2}iP_L\\chi , -F,iF,iD_az,0,0  \\} , \\label{embedding chiral}\n\\end{align}\nwhere $D_a$ is a superconformal covariant derivative. On the other hand, a real linear multiplet has components, $\\{C,Z,B_a\\}$, where $C$ is a real scalar, $Z$ is a Majorana spinor and $B_a$ is a constrained vector which satisfies $D^aB_a=0$. A real linear multiplet is embedded into a general superconformal multiplet $\\eqref{general multiplet}$ as\n\\begin{align}\n\\{ C,Z,0,0,B_a,-\\slash{D}Z,-\\Box C\\}, \\label{embedding linear}\n\\end{align}\nwhere $\\slash{D} \\equiv \\gamma ^a D_a$.\n\nFor later convenience, we also introduce a multiplication rule for superconformal multiplets. For a function of multiplets $f(\\mathcal{C}^I)$, where $I$ classifies different multiplets, we have\n\\begin{align}\n\\nonumber \\langle f({\\cal C}^I)\\rangle=\\biggl[ &f,f_I\\mathcal{Z}^I, f_I\\mathcal{H}^I-\\frac{1}{4}f_{IJ}\\bar{\\mathcal{Z}}^J\\mathcal{Z}^I, f_I\\mathcal{K}^I+\\frac{i}{4}f_{IJ}\\bar{\\mathcal{Z}}^J\\gamma _5\\mathcal{Z}^I, f_I\\mathcal{B}_a^I-\\frac{i}{4}f_{IJ}\\bar{\\mathcal{Z}}^J\\gamma_a \\gamma _5\\mathcal{Z}^I,\\\\\n\\nonumber &f_I\\Lambda^I-\\frac{i}{2}\\gamma _5\\left( \\mathcal{K}^I-\\slash{\\mathcal{B}}^I-i\\gamma _5\\slash{D}\\mathcal{C}^I+i\\gamma _5\\mathcal{H}^I\\right) f_{IJ}\\mathcal{Z}^J-\\frac{1}{4}\\left( \\bar{\\mathcal{Z}}^J\\mathcal{Z}^I\\right) \\mathcal{Z}^Kf_{IJK},\\\\\n\\nonumber &f_I\\mathcal{D}^ I+\\frac{1}{2}f_{IJ}\\left( \\mathcal{K}^I\\mathcal{K}^J+\\mathcal{H}^I\\mathcal{H}^J-\\mathcal{B}^{aI}\\mathcal{B}_a^J-D_a\\mathcal{C}^ID^a\\mathcal{C}^J-2\\bar{\\mathcal{Z}}^J\\Lambda^I-\\bar{\\mathcal{Z}}^J\\slash{D}\\mathcal{Z}^I\\right)\\\\\n&-\\frac{1}{4}f_{IJK}\\bar{\\mathcal{Z}}^J(\\mathcal{H}^K-i\\gamma _5\\mathcal{K}^K-i\\slash{\\mathcal{B}}^K\\gamma _5)\\mathcal{Z}^I+\\frac{1}{16}f_{IJKL}(\\bar{\\mathcal{Z}}^J\\mathcal{Z}^I)(\\bar{\\mathcal{Z}}^K\\mathcal{Z}^L) \\biggr] , \\label{formulaF}\n\\end{align}\nwhere $f_{IJ\\cdots }$ is $\\partial f\/\\partial \\mathcal{C}^I\\partial\\mathcal{C}^J\\cdots$ and $\\bar{\\mathcal{Z}}\\equiv \\mathcal{Z}^T\\hat{C}$ ($\\hat{C}$ is a charge conjugation matrix).\n\nWe also need action formulas to construct a superconformal action. For a chiral multiplet $X=\\{z,P_L\\chi,F\\}$ with its weight $(3,3)$, there exists the so-called F-term formula~\\cite{Kugo:1982cu}, \n\\begin{align}\n[X]_F=\\int d ^{4}x\\sqrt{-g}{\\rm{Re}} \\biggl[ F+\\frac{1}{\\sqrt{2}}  \\bar {\\psi }_{\\mu}\\gamma ^{\\mu}P _{L}\\chi +\\frac{1}{2}z\\bar {\\psi }_{\\mu}\\gamma ^{\\mu \\nu}P _{R}\\psi _{\\nu} \\biggr] , \\label{Fformula}\n\\end{align}\nwhere $\\psi _{\\mu}$ is a gravitino.\nFor a real multiplet $\\phi =\\{C,Z,H,K,B_a,\\Lambda,D\\}$ with its weight $(2,0)$, we can apply the following D-term formula~\\cite{Kugo:1982cu},\n\\begin{align}\n\\nonumber  [\\phi ]_D= \\int d^{4}x\\sqrt{-g}\\biggl[ &D-\\frac{1}{2}i\\bar{\\psi}\\cdot \\gamma \\gamma _{5}\\lambda  -\\frac{1}{3}CR+\\frac{1}{3}(C\\bar{\\psi }_{\\mu}\\gamma ^{\\mu \\rho \\sigma }-i\\bar{Z }\\gamma ^{\\rho \\sigma }\\gamma _{5})D_{\\rho }\\psi _{\\sigma }\\\\\n    &+\\frac{1}{4} \\varepsilon ^{abcd}\\bar {\\psi }_{a}\\gamma _{b}\\psi _{c}\\left(B _{d}-\\frac{1}{2}\\bar{\\psi }_{d}Z \\right)\\biggr] . \\label{Dformula}\n\\end{align}\nHere, all the components of $\\phi$ are real (Majorana).\n\nUsing these superconformal multiplets, the multiplication rule $\\eqref{formulaF}$, and the action formulas $\\eqref{Fformula}$ and $\\eqref{Dformula}$, we can construct superconformal invariant actions. Finally, we fix some parts of the extra gauge symmetries by imposing the condition to one of the superconformal multiplets $\\Phi_0$ called a compensator multiplet, and obtain the $\\rm{Poincar\\acute{e}}$ SUGRA action.\n\n\n\n\\subsection{{\\it{u-associated}} derivative} \\label{u-associated derivative}\nNow, we have prepared the tool for constructing the DBI action in SUGRA. Within the conformal SUGRA formulation, we will discuss a constraint corresponding to that in global SUSY, \n\\begin{align}\nX-\\frac{1}{4}X\\bar{D}^2\\bar{X}-\\bar{D}_{\\dot{\\alpha }}L\\bar{D}^{\\dot{\\alpha }}L=0 , \\label{global constraint 2}\n\\end{align}\nin the following. However, it seems to be a nontrivial task to extend the term including SUSY spinor derivatives,\n\\begin{align}\n\\bar{D}_{\\dot{\\alpha }}L\\bar{D}^{\\dot{\\alpha }}L \\label{global derivative}\n\\end{align}\n to that in conformal SUGRA. \n \nTo treat the term $\\eqref{global derivative}$ in conformal SUGRA, we need the spinor derivative defined as a superconformal operation. In Ref.~\\cite{Kugo:1983mv}, it is pointed out that the spinor derivative in conformal SUGRA, $\\mathcal{D}_{\\alpha }$ ($\\bar{\\mathcal{D}}_{\\dot{\\alpha }}$), cannot be defined on a superconformal multiplet $\\Phi$ unless $\\Phi$ satisfies a specific weight condition, $w=-n$ ($w =n$). This is because $\\mathcal{D}_{\\alpha }\\Phi $ ($\\bar{\\mathcal{D}}_{\\dot{\\alpha }}\\Phi$) is not generically a superconformal multiplet, i.e., the first component of it is S-SUSY and conformal boost inert only when $w=-n$ ($w=n$) is satisfied. \nThen, it is obvious that we cannot define $\\bar{\\cal D}_{\\dot{\\alpha}}L$ as a superconformal multiplet since $L$ has the weight with $(2,0)$.\n\nHowever, the authors in Ref.~\\cite{Kugo:1983mv} also proposed an improved spinor derivative operation, which can be defined on any supermultiplet. They introduced another multiplet, ${\\bf{u}}$, called a {\\it{u-associated}} multiplet, \n\\begin{align}\n{\\bf{u}} =\\{ \\mathcal{C}_u, \\mathcal{Z}_u,\\mathcal{H}_u,\\mathcal{K}_u,\\mathcal{B}_{au},\\Lambda _u,\\mathcal{D}_u \\} ,\n\\end{align}\nin order to force the first component of ${\\cal D}_\\alpha \\Phi$ to be invariant under S-SUSY and conformal boost. To be specific, they defined the {\\it{u-associated}} spinor derivative as \n\\begin{align}\n\\mathcal{D}^{({\\bf{u}})}_{\\alpha }\\Phi =\\langle (P_L\\mathcal{Z})_{\\alpha }+i(n+w)\\lambda _{\\alpha }\\mathcal{C}\\rangle ,\\ \\ \\ \\lambda _{\\alpha } \\equiv \\frac{i(P_L\\mathcal{Z}_u)_{\\alpha }}{(w_u+n_u)\\mathcal{C}_u}, \\label{def u}\n\\end{align}\nwhere $w_u$ and $n_u$ are the Weyl and chiral weight of a {\\it{u-associated}} multiplets, respectively. Unless $w_u+n_u=0$, we can choose any multiplet as the {\\it{u-associated}} multiplet. \nThen, we can define the spinor derivative for an arbitrary superconformal multiplet by this {\\it{u-associated}} spinor derivative.\n\nFor our purpose, we need the {\\it{u-associated}} spinor derivative acting on a real linear multiplet, $\\mathcal{D}^{({\\bf{u}})}_{\\alpha }L $. More generally, we can consider \n\\begin{align}\n\\mathcal{D}^{({\\bf{u}}_1)}_{\\alpha }({\\bf{u}}_2L), \\label{u-derivative}\n\\end{align}\nwhere ${\\bf{u}}_1$ is a {\\it{u-associated}} multiplet and ${\\bf{u}}_2$ is an additional multiplet. These multiplets must satisfy ${\\bf{u}}_1\\neq {\\bf{u}}_2$, since $\\mathcal{D}^{({\\bf{u}})}_{\\alpha }{\\bf{u}}$ is identically zero obviously from the definition $\\eqref{def u}$.\\footnote{As we will discuss, we choose ${\\bf{u}}_1$ and ${\\bf{u}}_2$ as compensators, which become some parts of the gravity multiplet after superconformal gauge fixings. In the global SUSY expression $\\eqref{global derivative}$, all the fields in the gravitational multiplet decouple from it. Therefore, it is natural to consider a possibility that a compensator appears as in Eq. $\\eqref{u-derivative}$.} Using this {\\it{u-associated}} spinor derivative, Eq. $\\eqref{global derivative}$ can be generalized to the one in conformal SUGRA as\n\\begin{align}\n\\frac{1}{{\\bf{u}}_3}\\bar{\\mathcal{D}}^{({\\bf{u}}_1)}(\\bar{\\bf{u}}_2L)\\bar{\\mathcal{D}}^{({\\bf{u}}_1)}(\\bar{\\bf{u}}_2L) , \\label{u-derivative part}\n\\end{align}\nwhere we have introduced a new multiplet ${\\bf{u}}_3$\\footnote{We will refer all of ${\\bf {u}}_i$ as {\\it u-associated} multiplets.} for generality and omitted the spinor index, $\\dot{\\alpha }$, and we have also defined the conjugate of a {\\it{u-associated}} derivative as $\\bar{\\mathcal{D}}_{\\dot{\\alpha }}^{\\bf{u}} \\Phi = (\\mathcal{D}_{\\alpha }^{\\bf{u}}(\\Phi )^*)^*$ following Ref. \\cite{Kugo:1983mv}.\n\nLet us comment on the weight of the multiplet $\\eqref{u-derivative part}$. The operator $\\bar{\\mathcal{D}}^{({\\bf{u}})}_{\\dot{\\alpha} }$ has the weight $(1\/2,3\/2)$, then the total weight of Eq. $\\eqref{u-derivative part}$ is $(2w_2-w_3+5,2n_2-n_3+3)$, where $w_i$ and $n_i$ with $i=1,2,3$ are the Weyl and chiral weights of ${\\bf{u}}_i$, respectively. \n\nFurthermore, Eq. $\\eqref{global constraint 2}$ is a ``chiral\" constraint since the first and second term in Eq. $\\eqref{global constraint 2}$ are chiral multiplets. Then, we require a condition that the multiplet $\\eqref{u-derivative part}$ is a chiral multiplet, that is, \n\\begin{align}\n\\bar{\\mathcal{D}}\\biggl[ \\frac{1}{{\\bf{u_3}}}\\bar{\\mathcal{D}}^{({\\bf{u_1}})}(\\bar{\\bf{u_2}}L)\\bar{\\mathcal{D}}^{({\\bf{u_1}})}(\\bar{\\bf{u_2}}L)  \\biggr]  =0 . \\label{condition for chiral}\n\\end{align}\nTo apply $\\bar{\\mathcal{D}}$ for Eq. $\\eqref{u-derivative part}$, the Weyl and chiral weight of Eq. $\\eqref{u-derivative part}$ must satisfy $w=n$ as mentioned before,\n\\begin{align}\n2w_2-w_3+5=2n_2-n_3+3. \\label{weight condition between w3n3 and w2n2}\n\\end{align}\nThe condition $\\eqref{condition for chiral}$ implies that \n\\begin{align}\nP_R \\mathcal{Z}'=0, \\label{condition for chiral2}\n\\end{align}\nwhere $P_R= (1-\\gamma _5)\/2$ is a right-handed projection operator and $\\mathcal{Z}'$ is the second component of the multiplet $\\eqref{u-derivative part}$. The equation $\\eqref{condition for chiral2}$ can be written explicitly as\n\\begin{align}\n\\nonumber &\\bar{\\tilde{\\mathcal{Z}}}_2^cP_R\\tilde{\\mathcal{Z}}_2^c\\biggl[P_R\\tilde{Z}+k   P_R\\tilde{\\mathcal{Z}}_1^c- P_R\\tilde{\\mathcal{Z}}_3\\biggr] \n+\\bar{\\tilde{Z}}P_R\\tilde{Z}\\biggl[P_R\\tilde{\\mathcal{Z}}_2^c+k P_R\\tilde{\\mathcal{Z}}_1^c- P_R\\tilde{\\mathcal{Z}}_3\\biggr] \\\\\n\\nonumber &-k  \\bar{\\tilde{\\mathcal{Z}}}_1^cP_R\\tilde{\\mathcal{Z}}_1^c\\biggl[\\left( 1-2k \\right) \\left( P_R\\tilde{Z}+P_R\\tilde{\\mathcal{Z}}_2^c\\right) +P_R\\tilde{\\mathcal{Z}}_3\\biggr] \\\\\n\\nonumber &-2k \\biggl[\\bar{\\tilde{\\mathcal{Z}}}_2^cP_R\\tilde{\\mathcal{Z}}_1^c \\left( 2P_R\\tilde{Z}-P_R\\tilde{\\mathcal{Z}}_3\\right)+\\bar{\\tilde{Z}}P_R\\tilde{\\mathcal{Z}}_1^c\\left( 2P_R\\tilde{\\mathcal{Z}}_2^c-P_R\\tilde{\\mathcal{Z}}_3\\right) \\biggr] \\\\\n&-2i\\biggl[i\\tilde{\\mathcal{H}}_2^*+\\tilde{\\mathcal{K}}_2^*-k \\left( i\\tilde{\\mathcal{H}}_1^*+\\tilde{\\mathcal{K}}_1^*\\right) \\biggr] \\biggl[P_R\\tilde{\\mathcal{Z}}_2^c+P_R\\tilde{Z}-k P_R\\tilde{\\mathcal{Z}}_1^c\\biggr] \n-2\\bar{\\tilde{\\mathcal{Z}}}_2^cP_R\\tilde{Z}P_R\\tilde{\\mathcal{Z}}_3=0,\\label{explicit condotion for chiral} \n\\end{align}\nwhere \n\\begin{align}\n&{\\bf{u}}_i =\\{ \\mathcal{C}_i, \\mathcal{Z}_i,\\mathcal{H}_i,\\mathcal{K}_i,\\mathcal{B}_{ai},\\Lambda _i,\\mathcal{D}_i \\} ,\\ \\ \\ (i=1,2,3),\\\\\n&\\tilde{Z}\\equiv \\frac{1}{C}Z, \\ \\ \\ \\tilde{\\mathcal{Z}}_i\\equiv \\frac{1}{C_i}\\mathcal{Z}_i, \\ \\ \\ \\tilde{\\mathcal{H}}_i(\\tilde{\\mathcal{K}}_i)\\equiv \\frac{1}{C_i}\\mathcal{H}_i(\\mathcal{K}_i), \\label{tilde}\\\\\n&k \\equiv \\frac{w_2+n_2+2}{w_1+n_1},\n\\end{align}\nand $``c\"$ denotes the charge conjugation for spinors. \n\nAs a summary, we find that the superconformal realization of Eq. $\\eqref{global derivative}$ is the multiplet $\\eqref{u-derivative part}$ satisfying the conditions $\\eqref{weight condition between w3n3 and w2n2}$ and $\\eqref{explicit condotion for chiral}$. \n\n\\subsection{Old minimal versus New minimal}\nWe have found, in the previous subsection ~\\ref{u-associated derivative}, the conditions for extending Eq. $\\eqref{global derivative}$ to that in conformal SUGRA. Here, we will choose a conformal compensator $\\Phi_0$ as {\\it{u-associated}} multiplets, ${\\bf{u}}_i$. Then, we have two choices of compensators; one of them is a chiral compensator $S_0$ realizing the old minimal SUGRA and the other is a real linear compensator $L_0$ realizing the new minimal SUGRA.\\footnote{We do not discuss the case of the   non-minimal formulation which is realized with a complex linear compensator.}  \n\nNow, we will examine what forms of ${{\\bf{u}}_i}$ with both compensators are allowed. Let us start from the old minimal SUGRA realized with a chiral compensator,\n\\begin{align}\nS_0=\\{ z_0,-\\sqrt{2}iP_L\\chi_0 , -F_0,iF_0,iD_az_0,0,0  \\},\n\\end{align}\nwith its weight $(1,1)$. Here we assume that the multiplets ${\\bf{u}}_i$ take the following form \n\\begin{align}\n{\\bf{u}}_i=S_0^{p_i}\\bar{S}_0^{q_i},\\ \\ \\ (i=1,2,3), \\label{s0s0}\n\\end{align}\nwhere $p_i$ and $q_i$ are the power of $S_0$ and $\\bar{S}_0$, and satisfy $p_1\\neq 0$ since $w_1+n_1=(p_1+q_1)+(p_1-q_1)=2p_1$ must be nonzero by a definition of the {\\it{u-associated}} multiplet. Here we have to stress that Eq. $\\eqref{s0s0}$ is the most general form except for the case including derivative operators on a compensator,\\footnote{For example, $S_0\\Sigma \\bar{S}_0$ could be considered.} which might produce higher-derivative terms of gravity. Using Eq. $\\eqref{embedding chiral}$ and the multiplication rule $\\eqref{formulaF}$, the components of the multiplet in Eq. $\\eqref{s0s0}$ are written as \n\\begin{align}\n\\nonumber &\\{ \\mathcal{C}_i, \\mathcal{Z}_i,\\mathcal{H}_i,\\mathcal{K}_i,\\mathcal{B}_{ai},\\Lambda _i,\\mathcal{D}_i \\} \\\\\n\\nonumber  &= \\{ z_0^{p_i}z_0^{*q_i},\\sqrt{2}iz_0^{p_i-1}z_0^{*q_i-1}(q_iz_0P_R\\chi_0 -p_iz_0^*P_L\\chi_0 ),\\\\\n\\nonumber &z_0^{p_i-2}z_0^{*q_i-2}\\left( -q_iz_0^2z_0^*F_0^*-p_iz_0z_0^{*2}F_0+\\frac{1}{2}q_i(q_i-1)z_0^2\\bar{\\chi}_0P_R\\chi _0+\\frac{1}{2}p_i(p_i-1)z_0^{*2}\\bar{\\chi}_0P_L\\chi _0\\right) ,\\\\\n\\nonumber &z_0^{p_i-2}z_0^{*q_i-2}\\left( -iq_iz_0^2z_0^*F_0^*+ip_iz_0z_0^{*2}F_0+\\frac{i}{2}q_i(q_i-1)z_0^2\\bar{\\chi}_0P_R\\chi _0-\\frac{i}{2}p_i(p_i-1)z_0^{*2}\\bar{\\chi}_0P_L\\chi _0\\right) , \\\\\n&...,...,...\\}, \\label{explicit s0s0}\n\\end{align}\nwhere we have omitted the components, $\\mathcal{B}_{ai},\\Lambda _i$ and $\\mathcal{D}_i $, which are not necessary to evaluate  Eq. $\\eqref{explicit condotion for chiral}$.  \nOne finds that Eq. $\\eqref{explicit condotion for chiral}$ cannot be satisfied by Eq. $\\eqref{s0s0}$ by the following reason: Terms including ${\\cal H}_i$ and ${\\cal K}_i$ must vanish by themselves since any other terms cannot cancel them. After substituting Eq. $\\eqref{explicit s0s0}$ into such a part, we obtain\n\\begin{align}\n\\nonumber &i\\mathcal{\\tilde{H}}_2^*+\\tilde{\\mathcal{K}}_2^*-k \\left( i\\tilde{\\mathcal{H}}_1^*+\\tilde{\\mathcal{K}}_1^*\\right) =2iF_0^*z_0^{*-1}+i\\bar{\\chi}_0P_R\\chi _0z_0^{*-2}(p_2^2-p_2p_1-p_1+1).\n\\end{align}\nApparently, the first term cannot be eliminated no matter how we choose the parameters $p_i$ and $q_i$, and the other terms in Eq. $\\eqref{explicit condotion for chiral}$ cannot eliminate it because they do not contain $F_0^*$. \nTherefore, we find that Eq. $\\eqref{s0s0}$ cannot be a solution of Eq. $\\eqref{explicit condotion for chiral}$. This means that Eq. $\\eqref{u-derivative part}$ cannot be realized as a chiral constraint in the old minimal SUGRA.\n\nNext, we examine the case in the new minimal SUGRA with a real linear compensator\n\\begin{align}\nL_0=\\{ C_0,Z_0,0,0,B_{0a},-\\slash{D}Z_0,-\\Box C_0\\}\n\\end{align}\nwith its weight $(2,0)$. In the same way as the old minimal case, we assume the general form of ${\\bf{u}}_i$ as\n\\begin{align}\n{\\bf{u}}_i=L_0^{r_i},\\ \\ \\ (i=1,2,3), \\label{l0}\n\\end{align}\nwhose components are \n\\begin{align}\n\\nonumber &\\{ \\mathcal{C}_i, \\mathcal{Z}_i,\\mathcal{H}_i,\\mathcal{K}_i,\\mathcal{B}_{ai},\\Lambda _i,\\mathcal{D}_i \\} \\\\\n &= \\{ C_0^{r_i}, r_iC_0^{r_i-1}Z_0,-\\frac{1}{4}r_i(r_i-1)C_0^{r_i-2}\\bar{Z}_0Z_0, \\frac{i}{4}r_i(r_i-1)C_0^{r_i-2}\\bar{Z}_0\\gamma _5Z_0,...,...,...\\} . \\label{explicit l0}\n\\end{align}\nHere we have used Eq. $\\eqref{embedding linear}$ and Eq. $\\eqref{formulaF}$. Then, after substituting Eq. $\\eqref{explicit l0}$ into Eq. $\\eqref{explicit condotion for chiral}$ with the Fierz rearrangement, Eq. $\\eqref{explicit condotion for chiral}$ is summarized as\n\\begin{align}\n(2r_2-r_3+1)\\left\\{ CP_RZ\\bar{Z}_0P_RZ_0+C_0P_RZ_0\\bar{Z}P_RZ\\right\\}=0. \\label{L0 condition}\n\\end{align}\nTo satisfy Eq. $\\eqref{L0 condition}$, the coefficient must be zero,\n\\begin{align}\n2r_2-r_3+1=0.\\label{L0 condition2}\n\\end{align}\nThen, we find that the chiral condition $\\eqref{explicit condotion for chiral}$ is satisfied as long as the {\\it{u-associated}} multiplets follow the condition $\\eqref{L0 condition2}$.\n\nNoting that $w_i=2r_i$ and $n_i=0$ in the ansatz $\\eqref{l0}$, the weight condition $\\eqref{weight condition between w3n3 and w2n2}$ which the chiral multiplet should obey is now reduced to\n\\begin{align}\n2r_2-r_3+1=0. \\label{L0 condition22}\n\\end{align}\nThis is nothing but Eq. $\\eqref{L0 condition2}$ which is satisfied automatically.\n\nTherefore, we conclude that one can make a multiplet in Eq. $\\eqref{u-derivative part}$ a chiral one with the real linear compensator if Eq. $\\eqref{L0 condition2}$ is satisfied. Here and hereafter, we focus on the case of the new minimal SUGRA with $r_1=r_3=1$ and $r_2=0$ for simplicity. In this case, the multiplet in Eq. $\\eqref{u-derivative part}$ becomes\n\\begin{align}\n\\frac{1}{L_0}\\bar{\\mathcal{D}}^{(L_0)}L\\bar{\\mathcal{D}}^{(L_0)}L . \\label{Lo derivative part}\n\\end{align}\nWe present the components of this chiral multiplet $\\eqref{Lo derivative part}$ explicitly in Appendix A.\n\n\\subsection{Embedding the constraint into conformal SUGRA} \\label{embedding}\nLet us consider the remaining terms, $X$ and $X\\bar{D}^2\\bar{X}$ in Eq. $\\eqref{global constraint 2}$. For $X$, we just regard it as a superconformal chiral multiplet with the weight $(w,w)$. In order to extend the second one, $X\\bar{D}^2\\bar{X}$, to a superconformal multiplet, we replace it with $X\\Sigma \\bar{X}$, where $\\Sigma $ is a chiral projection operator in conformal SUGRA. However, $\\Sigma $ cannot always be applied for any multiplet $\\Phi$ in the same way as the spinor derivative $\\mathcal{D}$. It can be applied only when $\\Phi$ satisfies the following weight condition, \n\\begin{align}\nw_{\\Phi }=n_{\\Phi }+2. \\label{condition for projection}\n\\end{align}\nTherefore, we compensate the weight of $\\bar{X}$, which has the weight $(w,-w)$, by the real linear compensator multiplet $L_0^s$, where $s$ is the power of $L_0$, \n\\begin{align}\nX\\Sigma \\left( \\frac{1}{L_0^s}\\bar{X} \\right) . \\label{compensated second term}\n\\end{align}\nHere, the term, $\\frac{1}{L_0^s}\\bar{X}$, has the weight $(-2s+w,-w)$. According to Eq. $\\eqref{condition for projection}$, $s$ must satisfy the condition,\n\\begin{align}\ns=w-1. \\label{weight condition of v1}\n\\end{align}\nTaking into account this condition and the fact that $\\Sigma $ raises the weight by (1,3), Eq. $\\eqref{compensated second term}$ has the weight $(3,3)$, which is correct  for a chiral multiplet. Since the total weight of Eq. $\\eqref{compensated second term}$ must be the same as the first term $X$, the value of $w$ is determined as \n\\begin{align}\nw=3. \\label{weight condition of v2}\n\\end{align}\nThen, we find $s=2$ from Eq.  $\\eqref{weight condition of v1}$, and Eq. $\\eqref{compensated second term}$ becomes  \n\\begin{align}\nX\\Sigma \\left( \\frac{1}{L_0^2}\\bar{X} \\right) . \\label{compensated second term2}\n\\end{align}\nFinally, the weight of the multiplet in Eq. $\\eqref{u-derivative part}$ with that in Eq. $\\eqref{l0}$ is $(3,3)$ as long as Eq. $\\eqref{L0 condition22}$ is satisfied, then Eq. $\\eqref{Lo derivative part}$ is automatically satisfied. \n\nTherefore, we find the complete embedding of a global SUSY expression $\\eqref{global constraint 2}$,\n\\begin{align}\nX+\\frac{1}{2}X\\Sigma \\left( \\frac{1}{L_0^2}\\bar{X} \\right) +\\frac{1}{4L_0}\\bar{\\mathcal{D}}^{(L_0)}L\\bar{\\mathcal{D}}^{(L_0)}L =0,\\label{local constraint}\n\\end{align}\nwhere $X$ is a chiral multiplet with $(3,3)$, $L$ is a real linear multiplet with $(2,0)$, and $L_0$ is a real linear compensator with $(2,0)$.\n\n\n  \n\n\n\\section{Component action}\\label{Component action}\nIn this section, we derive the DBI action based on the constraint $\\eqref{local constraint}$ in the new minimal SUGRA.\n\n\\subsection{Minimal action} \\label{Minimal action}\nWe first consider the minimal extension of the action $\\eqref{component form of global DBI}$. The action corresponding to Eq. $\\eqref{global DBI with constraint}$ is expected to be\n\\begin{align}\nS=&[2X]_F+\\Biggl[2\\Lambda \\left\\{ X+\\frac{1}{2}X\\Sigma \\left(  \\frac{\\bar{X}}{L_0^2}\\right) +\\frac{1}{4L_0}\\bar{\\mathcal{D}}^{(L_0)}L\\bar{\\mathcal{D}}^{(L_0)}L \\right\\} \\Biggr] _F+[\\tilde{\\Lambda }X^2]_F +\\Biggl[\\frac{3}{2}L_0V_R\\Biggr] _D, \\label{SL0}\n\\end{align}\nwhere $V_R\\equiv \\log \\frac{L_0}{S\\bar{S}}$, $S$ is a chiral multiplet with $(1,1)$, and we have assigned the weights of the Lagrange multiplier chiral multiplet $\\Lambda$ to $(0,0)$ and also $\\tilde{\\Lambda }$ to $(-3,-3)$ in such a way that the total weight is equal to $(3,3)$.\nThe last term in Eq. $\\eqref{SL0}$ is responsible for the kinetic term of the gravitational multiplet. Note that this term is invariant under the transformation $S\\to Se^{i\\Theta}$ where $\\Theta$ is a chiral multiplet with the weight $(0,0)$ since $[L_0(\\Theta+\\bar{\\Theta})]_D\\equiv0$ by the nature of a real linear multiplet. Due to this additional gauge invariance, we have gauge degrees of freedom other than superconformal ones. After imposing the gauge fixing condition for this additional gauge symmetry as $S=\\{1,0,0\\}$, the bosonic part of $\\eqref{SL0}$ is given by\n\\begin{align}\n\\nonumber S_B=& \\int d^4 x \\sqrt{-g}\\Biggl[ \\Biggl( F_X (1+\\Lambda )-\\frac{|F_X|^2\\Lambda }{C_0^2}-\\frac{\\Lambda}{4C_0}(B_a-i\\hat{D}_aC)^2\\\\\n\\nonumber &+\\frac{C\\Lambda}{2C_0^2}(B_a-i\\hat{D}_aC)(B_0^a-i\\hat{D}^aC_0)\n-\\frac{C^2\\Lambda}{4C_0^3}(B_{0a}-i\\hat{D}_aC_0)^2 +{\\rm{h.c.}}\\Biggr) \\\\\n&-\\frac{3}{2}\\hat{\\Box} C_0\\log C_0-\\frac{3}{2}\\hat{\\Box} C_0-\\frac{3}{4C_0}(B_0\\cdot B_0+\\hat{D}C_0\\cdot \\hat{D}C_0) \n+3A\\cdot B_0 \\Biggr] , \\label{SBfull}\n\\end{align}\nwhere $\\Lambda$ and $F_X$ are a scalar component of the chiral multiplet $\\Lambda$ and an auxiliary field of $X$, and $\\hat{D}_{\\mu}$ is a superconformal covariant derivative only including bosonic fields, for example,\n\\begin{align}\n\\hat{D}_{\\mu}C=\\partial_{\\mu}C-2b_{\\mu}C,\n\\end{align}\nwhere $b_{\\mu}$ is the gauge field of dilatation. The third term in Eq. $\\eqref{SL0}$, $\\tilde{\\Lambda }X^2$, imposes the nilpotency condition for $X$. Thanks to this, we can drop the scalar component of the chiral multiplet $X$ since the first scalar component can be represented as a fermion bilinear after solving $X^2=0$. That is why, we have inserted this term into the action from the beginning. \nIntegrating out the gauge field of $U(1)_A$ symmetry $A_\\mu$, we obtain    \n\\begin{align}\nB_{0a}=0.\n\\end{align}\nTo eliminate the dilatation symmetry and conformal boost symmetry, we impose the following $D$-gauge and $K$-gauge conditions, \n\\begin{align}\nC_0=1,\\ \\ \\ \\ b_{\\mu}=0.\n\\end{align}\nThese conditions simplify the action $\\eqref{SBfull}$, which becomes\n\\begin{align}\n\\nonumber S_B=\\int d^4 x \\sqrt{-g}\\Biggl[ &\\frac{1}{2}R+\\Bigl( F_X (1+\\Lambda ) -|F_X|^2\\Lambda \\\\\n&-\\frac{\\Lambda }{4}(B\\cdot B-2iB\\cdot \\partial C-\\partial C\\cdot \\partial C)+{\\rm{h.c.}}\\Bigr) \\Biggr] . \\label{FX}\n\\end{align}\nThen, eliminating the auxiliary field $F_X$ leads to\n\\begin{align}\nS_B=\\int d^4 x \\sqrt{-g}\\Biggl[ &\\frac{1}{2}R+\\frac{1}{2\\lambda }\\Bigl(  (\\lambda +1)^2+\\chi ^2\\Bigr) -\\frac{1}{2}(B\\cdot B-\\partial C\\cdot \\partial C)\\lambda -B\\cdot \\partial C \\chi \\Biggr] , \\label{FXX}\n\\end{align}\nwhere $\\lambda={\\rm Re}\\Lambda$ and $\\chi={\\rm Im}\\Lambda$.\nFinally, we obtain the following conditions from the E.O.Ms for $\\lambda $ and $\\chi $,\n\\begin{align}\n&\\frac{\\chi }{\\lambda }=B\\cdot \\partial C ,\\\\\n&\\frac{1}{\\lambda ^2}=1-(B\\cdot \\partial C)^2-B\\cdot B+\\partial C\\cdot \\partial C .\n\\end{align}\nSubstituting them into the action $\\eqref{FXX}$, we obtain the on-shell DBI action of a real linear multiplet,\n\\begin{align}\n S_B=\\int d^4 x \\sqrt{-g}\\Biggl[ \\frac{1}{2}R+1-\\sqrt{1-B\\cdot B+\\partial C\\cdot \\partial C-(B\\cdot \\partial C)^2}\\Biggr] .  \\label{DBI}\n\\end{align}\nThis is almost the same form as Eq. $\\eqref{component form of global DBI}$ except for that our action $\\eqref{DBI}$ is formulated in curved background.\n\nBefore closing this subsection, let us discuss the linear-chiral duality. It is known that the action of a real linear multiplet can be rewritten in terms of that of a chiral multiplet. However, in the case with the action including derivative terms such as Eq. $\\eqref{SL0}$, it is nontrivial to take this duality transformation in a manifestly SUSY way.\\footnote{In global SUSY, the dual action has been obtained at the level of superfield in Ref.~\\cite{GonzalezRey:1998kh}.} Then, we focus only on the bosonic part $\\eqref{DBI}$ and discuss this duality at the component level of bosonic part.\n\nWe start from the following Lagrangian which is the relevant part in the action $\\eqref{DBI}$,\n\\begin{align}\n\\mathcal{L}=1-\\sqrt{1-B\\cdot B+\\partial C\\cdot \\partial C-(B\\cdot \\partial C)^2}. \\label{Ori}\n\\end{align}\nTo rewrite this Lagrangian $\\eqref{Ori}$ in terms of the complex scalar of a chiral multiplet, we first relax the constraint on the vector field $B_a$. We impose it by the E.O.M for a scalar field $\\ell$, that is, we use \n\\begin{align}\n\\mathcal{L}=1-\\sqrt{1-B\\cdot B+\\partial C\\cdot \\partial C-(B\\cdot \\partial C)^2}+B\\cdot \\partial \\ell ,\\label{LL}\n\\end{align}\nwhere $B_a$ is now an unconstrained vector. The Lagrangian $\\eqref{LL}$ is equivalent to the original one $\\eqref{Ori}$ since the variation with respect to $\\ell$ leads to the constraint, $\\partial _a B^a=0$.\nInstead of $\\ell$, varying with respect to $B_a$ gives\n\\begin{align}\n\\partial ^a\\ell + (\\partial ^a C B\\cdot \\partial C+B^a) \\{1-B\\cdot B+\\partial C\\cdot \\partial C-(B\\cdot \\partial C)^2\\} ^{-1\/2} =0. \\label{du}\n\\end{align}\nOur task is now to solve this equation $\\eqref{du}$ with respect to $B_a$.\nBy taking scalar products of Eq. $\\eqref{du}$ with $B_a, \\partial _a C$ and $\\partial _a \\ell $, we obtain three independent equations and can solve them with respect to $B^2$, $B\\cdot\\partial C$, and $B\\cdot \\partial \\ell$. The solutions are\n\\begin{align}\n&B^2=\\frac{(\\partial \\ell) ^2(1+(\\partial C)^2)^2-(\\partial C\\cdot \\partial \\ell )^2(2+(\\partial C)^2)}{Y^2},\\\\\n&B\\cdot \\partial C=-\\frac{\\partial C\\cdot \\partial \\ell }{Y},\\\\\n&B\\cdot \\partial \\ell =\\frac{-(\\partial \\ell) ^2(1+(\\partial C)^2)+(\\partial C\\cdot \\partial \\ell )^2}{Y},\n\\end{align}\nwhere\n\\begin{align}\nY \\equiv \\{ (1+(\\partial C)^2)(1+(\\partial \\ell) ^2)-(\\partial C\\cdot \\partial \\ell )^2\\}^{1\/2}.\n\\end{align}\nSubstituting these solutions into the Lagrangian $\\eqref{LL}$, we obtain the dual action,\n\\begin{align}\n\\nonumber \\mathcal{L}&=1-\\sqrt{1+(\\partial C)^2+(\\partial \\ell) ^2+(\\partial C)^2(\\partial \\ell) ^2-(\\partial C\\cdot \\partial \\ell )^2}\\\\\n&=1-\\sqrt{1+\\partial \\phi \\cdot \\partial \\bar{ \\phi }-\\frac{1}{4}(\\partial \\phi)^2 (\\partial \\bar{ \\phi })^2+\\frac{1}{4}(\\partial \\phi \\cdot \\partial \\bar{ \\phi })^2} ,\\label{LC}\n\\end{align}\nwhere we have defined a complex scalar $\\phi =\\ell + iC$. The Lagrangian $\\eqref{LC}$ can be written as the DBI form\n\\begin{align}\n\\mathcal{L}=1-\\sqrt{{\\rm{det}} \\left( g_{ab}+\\frac{1}{2}\\partial_a \\phi \\partial_b \\bar{\\phi} \\right)}.\\label{LC2}\n\\end{align}\nThis Lagrangian $\\eqref{LC2}$ agrees with the one constructed in Ref.~\\cite{Koehn:2012ar} using a chiral multiplet directly. \n  \n\n\\subsection{Matter coupled extension} \\label{matter}\nFinally, we discuss the matter coupled DBI action given by\n\\begin{align}\nS=&[2f(\\Phi^{I})X]_F+\\left[2\\Lambda \\left\\{ X+\\frac{1}{2}X\\Sigma\\left(\\frac{\\bar{X}}{M(L_0,\\Phi^I,\\bar{\\Phi}^{\\bar{J}})}\\right)+\\frac{1}{4L_0}\\bar{\\cal D}^{(L_0)}L\\bar{\\cal D}^{(L_0)}L\\right\\}\\right]_F\\nonumber\\\\\n&+[{\\cal F}(L_0,\\Phi^I,\\bar{\\Phi}^{\\bar{J}})]_D+[\\tilde{\\Lambda}X^2]_F\\label{mDBI},\n\\end{align}\nwhere $\\Phi^I$ ($\\bar{\\Phi}^{\\bar{J}}$) is a (anti-) chiral matter multiplet; $f(\\Phi)$ is a holomorphic function of $\\Phi^I$ with $(0,0)$; $M(L_0,\\Phi^I,\\bar{\\Phi}^{\\bar{J}})$ and ${\\cal F}(L_0,\\Phi^I,\\bar{\\Phi}^{\\bar{J}})$ are real functions of $\\Phi^I,\\bar{\\Phi}^{\\bar{J}}$ and  $L_0$ with $(4,0)$ and $(2,0)$, respectively. Note that we have omitted superpotential term $[W(\\Phi^I)]_F$, where $W(\\Phi^I)$ is a holomorphic function of $\\Phi^I$ with the weight $(w,n)=(3,3)$, since the term is irrelevant to the following discussion. Taking into account the nilpotency condition on $X$, the bosonic component of the action~(\\ref{mDBI}) is given by\n\\begin{align}\nS_B=&\\int d^4x \\sqrt{-g}\\Biggl[ \\Biggl( F_X(f+\\Lambda) -\\frac{\\Lambda |F_X|^2}{M}-\\frac{\\Lambda}{4C_0}(B_a-i\\hat{D}_aC)^2 \\nonumber\\\\\n&+\\frac{C\\Lambda}{2C_0^2}(B_a-i\\hat{D}_aC)(B_0^a-i\\hat{D}^aC_0)-\\frac{C^2\\Lambda}{4C_0^3}(B_0^a-i\\hat{D}^aC_0)^2+{\\rm h.c.}\\Biggr)+{\\cal L}_m\\Biggr],\n\\end{align}\nwhere\n\\begin{align}\n{\\cal L}_m=&-\\frac{1}{3}({\\cal F}-{\\cal F}_{C_0}C_0)R(b)+\\frac{1}{2}{\\cal F}_{C_0C_0}(\\hat{D}C_0\\cdot \\hat{D}C_0-B_0\\cdot B_0)\\nonumber\\\\\n&+2{\\cal F}_{I\\bar{J}}(F^I\\bar{F}^{\\bar{J}}-\\hat{D}\\Phi^I \\cdot \\hat{D}\\bar{\\Phi}^{\\bar{J}})+\\left(-i{\\cal F}_{C_0I}B_0\\cdot \\hat{D}\\Phi^I+{\\rm h.c.}\\right)\\label{defLm}.\n\\end{align}\nIn the above expression, $\\Phi^I$ ($\\bar{\\Phi}^{\\bar{J}}$) and $F^I$ ($\\bar{F}^{\\bar{J}}$) represent the scalar and auxiliary components of the (anti-) chiral matter multiplet, and subscripts denote the derivative with respect to the corresponding scalar. $R(b)$ becomes a Ricci scalar when $b_{\\mu}=0$ is imposed as the $K$-gauge condition.\n\nBefore setting superconformal gauge conditions, we integrate out the auxiliary field $F_X$ and the Lagrange multiplier $\\Lambda$. We can easily solve the E.O.M for $F_X$ and obtain\n\\begin{align}\nS_B=&\\int d^4x \\sqrt{-g}\\Biggl[\\frac{M}{2\\lambda}\\left\\{(\\lambda+p)^2+(\\chi+q)^2\\right\\}-\\frac{\\lambda}{2C_0}(B\\cdot B-\\hat{D}C\\cdot \\hat{D}C)\\nonumber\\\\\n&-\\frac{\\chi}{C_0}B \\cdot \\hat{D}C+\\frac{C\\lambda}{C_0^2}(B_0\\cdot B-\\hat{D}C_0\\cdot \\hat{D}C)+\\frac{C\\chi}{C_0^2}(B_0\\cdot \\hat{D}C+B\\cdot \\hat{D}C_0)\\nonumber\\\\\n&-\\frac{C^2\\lambda}{2C_0^3}(B_0 \\cdot B_0-\\hat{D}C_0\\cdot \\hat{D}C_0)-\\frac{C^2\\chi}{C_0^3}B_0\\cdot \\hat{D}C_0+{\\cal L}_m\\Biggr],\\label{mDBI2}\n\\end{align}\nwhere $\\lambda={\\rm Re}\\Lambda$, $\\chi={\\rm Im}\\Lambda$, $p={\\rm Re}f$, and $q={\\rm Im}f$. Note that, at this stage, the matter Lagrangian ${\\cal L}_m$ is not affected by the DBI sector. Next, we eliminate $\\lambda$ and $\\chi$ by using their E.O.Ms, which are given by\n\\begin{align}\n&-\\frac{M}{2\\lambda^2}\\left\\{(\\lambda+p)^2+(\\chi+q)^2\\right\\}+\\frac{M}{\\lambda}(\\lambda+p)+\\mathcal{A}=0,\\\\\n&\\frac{M}{\\lambda}(\\chi+q)+\\mathcal{B}=0,\n\\end{align}\nwhere\n\\begin{align}\n&\\mathcal{A}\\equiv -\\frac{1}{2C_0}(B\\cdot B-\\hat{D}C\\cdot \\hat{D}C)+\\frac{C}{C_0^2}(B_0\\cdot B-\\hat{D}C_0\\cdot \\hat{D}C)-\\frac{C^2}{2C_0^3}(B_0\\cdot B_0-\\hat{D}C_0\\cdot \\hat{D}C_0),\\\\\n&\\mathcal{B}\\equiv -\\frac{1}{C_0}B\\cdot \\hat{D}C+\\frac{C}{C_0^2}(B_0\\cdot \\hat{D}C+B\\cdot \\hat{D}C_0)-\\frac{C^2}{C_0^3}B_0\\cdot \\hat{D}C_0.\n\\end{align}\nSolutions for them are\n\\begin{align}\n&\\lambda|_{\\rm sol}^{-1}=\\frac{1}{p}\\sqrt{1+\\frac{2\\mathcal{A}}{M}-\\frac{\\mathcal{B}^2}{M^2}},\\\\\n&\\chi|_{\\rm sol}=-q-\\frac{\\lambda|_{\\rm sol}}{M}\\mathcal{B}.\n\\end{align} \nSubstituting the above solutions into the action~(\\ref{mDBI2}), we obtain a relatively simple form\n\\begin{align}\nS_B=\\int d^4x\\sqrt{-g}\\left[Mp\\left(1-\\sqrt{1+\\frac{2\\mathcal{A}}{M}-\\frac{\\mathcal{B}^2}{M^2}}\\right)-q\\mathcal{B}+{\\cal L}_m\\right].\\label{mDBI3}\n\\end{align}\n\nThe remaining issue is the elimination of auxiliary fields $B_0^a$ and $A_a$. However, it is difficult to do it because of the presence of nonlinear terms of $B_0^a$ contained in the first term in Eq.~(\\ref{mDBI3}). In addition, ${\\cal L}_m$ has $A_aA^a$ as well as mixing terms between $B_0^a$ and $A_a$ in general cases. Therefore, integration of those auxiliary fields is technically difficult and we cannot obtain the complete on-shell action.\\footnote{The general matter coupled system in the new minimal SUGRA not including higher-order derivative terms can be found in Ref.~\\cite{Ferrara:1983dh}.}\n\nAlthough a general case is difficult to complete the remaining task, we can continue our discussion for the following special case. Let us consider the following choice of ${\\cal F}(L_0,\\Phi^I,\\bar{\\Phi}^{\\bar{J}})$,\n\\begin{align}\n{\\cal F}=L_0\\log \\left(\\frac{L_0G(\\Phi^i,\\bar{\\Phi}^{\\bar{j}})}{S\\bar{S}}\\right), \\label{special form}\n\\end{align}\nwhere $\\Phi^i$ is a matter chiral multiplet with its weight $(0,0)$, $G(\\Phi^i,\\bar{\\Phi}^{\\bar{j}})$ is a real function of $\\Phi^i$ and $\\bar{\\Phi}^{\\bar{j}}$, and $S$ is a chiral multiplet with $(1,1)$. This action is also invariant under the transformation $S\\to Se^{i\\Theta}$ in the same way as the last term in Eq. $\\eqref{SL0}$, which characterizes the new minimal SUGRA.\n\nWe use the $D$-gauge condition to make the Ricci scalar term canonical. From Eq.~(\\ref{defLm}), we can find an appropriate $D$-gauge choice~\\cite{Ferrara:1983dh}\n\\begin{align}\n{\\cal F}-{\\cal F}_{C_0}C_0=-\\frac{3}{2}.\n\\end{align}\nAs the choice of the additional gauge, we set ${\\cal F}_{C_0}=0$~\\cite{Ferrara:1983dh}. Then, we can solve these gauge conditions with respect to $C_0$ and $S$ and obtain\n\\begin{align}\nS\\bar{S}=&\\frac{3}{2}eG,\\\\\nC_0=&\\frac{3}{2}.\n\\end{align} \nUsing the $K$-gauge, we also set a condition $b_\\mu=0$.\n\nUnder these conditions, ${\\cal L}_m$ becomes\n\\begin{align}\n{\\cal L}_m=&\\frac{1}{2}R+2{\\cal F}_{i\\bar{j}}(F^i\\bar{F}^{\\bar{j}}-\\partial_a\\Phi^i\\partial^a\\bar{\\Phi}^{\\bar{j}})-\\frac{1}{2}B_0^aB_{0a}\\nonumber\\\\\n&+(-i{\\cal F}_{C_0i}B_0^a\\partial_a \\Phi^i+{\\rm h.c.})+(iB_0^a\\partial_a\\log S+{\\rm h.c.})+2B_0^aA_a,\n\\end{align}\nwhere $A_a$ is the $U(1)_A$ gauge field mentioned above. We find that the E.O.M for $A_a$ gives a constraint $B_0^a=0$ and the difficulty due to the nonlinear term of $B_0^a$ is circumvented in this case. This result is irrelevant to other parts of the action (\\ref{mDBI3}) since they do not contain terms of $A_a$. $F^i$ can be eliminated by their E.O.Ms, and we finally obtain the following on-shell action,\n\\begin{align}\nS_B=\\int d^4x\\sqrt{-g}\\left[Mp\\left(1-\\sqrt{1+\\frac{2\\mathcal{A}}{M}-\\frac{\\mathcal{B}^2}{M^2}}\\right)-q\\mathcal{B}+\\frac{1}{2}R-2{\\cal F}_{i\\bar{j}}\\partial_a\\Phi^i\\partial^a\\bar{\\Phi}^{\\bar{j}}\\right], \\label{special matter}\n\\end{align}\nwith\n\\begin{align}\n\\mathcal{A}=\\frac{1}{3}(\\partial C\\cdot \\partial C-B\\cdot B),\\ \\ \\ \\mathcal{B}=-\\frac{2}{3}B\\cdot \\partial C.\n\\end{align}\nHere, the real function $M$ should be understood as $M|_{C_0=3\/2}$.\nNote that, in this case, we cannot add superpotential terms of $\\Phi^i$ by the following reason: To obtain the constraint $B_0^a=0$, we assumed that only $S$ has the weight $(w,n)=(1,1)$ and a special form of ${\\cal F}$ giving ${\\cal F}_{S\\bar{S}}=0$, otherwise such a constraint does not appear. For the superconformal invariance, the superpotential $W$ should have $(3,3)$. From the weight condition, a possible form is $W=S^3g(\\Phi^i)$ but this term is forbidden by the symmetry under $S\\to Se^{i\\Theta}$ which the D-term part $[{\\cal F}]_D$ has. Therefore, we cannot add any superpotential terms of matter multiplets.\n\\section{Relation between our results and other works}\\label{discussion}\nHere, we comment on the differences between ours and the results in Ref.~\\cite{Koehn:2012ar}, in which the DBI action of a chiral multiplet is constructed in the old minimal SUGRA. As we mentioned before, the DBI action of a real linear multiplet can be rewritten in terms of a chiral multiplet through the linear-chiral duality and the whole action of a chiral multiplet is obtained in global SUSY in terms of superfield \\cite{GonzalezRey:1998kh}. The authors of Ref.~\\cite{Koehn:2012ar} embedded the dual chiral multiplet action into the old minimal SUGRA. On the other hand, our starting point is the action of a real linear multiplet, more precisely, the constraint $\\eqref{global constraint}$ imposed upon it. This constraint has its origin in the tensor multiplet of $\\mathcal{N}=2$ SUSY \\cite{Rocek:1997hi,Bagger:1997pi,GonzalezRey:1998kh}. Indeed, in global SUSY case, the real linear multiplet corresponds to a Goldstino multiplet for the broken SUSY. From such a viewpoint, our construction is important since it makes the connection with the partial breaking of $\\mathcal{N}=2$ SUSY much clearer . \n\nAlthough the ways of construction are different, our action would realize their result. Indeed, at the bosonic component level, we have found the correspondence between the result in Ref.~\\cite{Koehn:2012ar} and ours. However, we also found that the action cannot be realized in the old minimal SUGRA when we do not consider the case including higher-derivative terms of a chiral compensator, which may contradict the result of Ref.~\\cite{Koehn:2012ar}. Unlike the DBI action of a real linear multiplet, that of a vector multiplet can be constructed in both of the old and new minimal SUGRA \\cite{Abe:2015nxa}. The difference originates from the necessity of {\\it u-associated} derivatives in the DBI action of a real linear multiplet. For a vector superfield case, we can construct the DBI action only with the chiral projection operator $\\Sigma$, which does not require {\\it u-associated} multiplet to make the operand superfield a primary superfield~\\cite{Kugo:1983mv,Butter:2009cp,Kugo:2016zzf}. It is interesting to explore these reasons and we expect that the direct derivation of the constraint $\\eqref{global constraint}$ and also DBI action from $\\mathcal{N}=2$ SUGRA are necessary to understand this issue, which would be our future work \\footnote{For the DBI action of a vector multiplet, such attempts have been recently discussed \\cite{Kuzenko:2015rfx}. There, the partial breaking of ${\\cal N}=2$ SUSY in some ${\\cal N}=1$ SUSY background has been discussed.}.\n\n\n\n\n\\section{Summary}\\label{summary}\nIn this paper, we have discussed superconformal generalization of a DBI action of a real linear superfield known in global SUSY. \n\nTo achieve this, we have focused on the constraint $\\eqref{global constraint}$ between a chiral multiplet and a real linear multiplet, which comes from the partial breaking of 4D $\\mathcal{N}=2$ SUSY \\cite{Bagger:1997pi}. However, it is a nontrivial task to embed this constraint into conformal SUGRA due to the existence of the SUSY spinor derivative, which in general, cannot be applied for arbitrary multiplets in conformal SUGRA. Instead of using an original spinor derivative, we have adopted the {\\it{u-associated}} spinor derivative, proposed in Ref.~\\cite{Kugo:1983mv}. We obtained the condition $\\eqref{weight condition between w3n3 and w2n2}$ and $\\eqref{explicit condotion for chiral}$ by requiring that the corresponding constraint $\\eqref{u-derivative part}$ in conformal SUGRA becomes a chiral constraint. Surprisingly, we have found that these conditions can be realized only in the new minimal formulation of SUGRA when we choose the general power function of compensator as the {\\it{u-associated}} multiplet. Then, we have derived the condition $\\eqref{L0 condition2}$ which {\\it{u-associated}} multiplets must satisfy.\n\nAfter embedding the constraint into the new minimal SUGRA, we have shown the component action which is formulated in curved spacetime. We have also discussed the linear-chiral duality at the level of bosonic components and rewritten the action from a complex scalar field of a chiral multiplet. Finally, we have constructed the action where matter multiplets are directly coupled to the DBI sector. Due to the appearance of nonlinear terms for vector field $B_{0a}$, we have restricted the discussion to the special form of matter function $\\eqref{special form}$ and derived the bosonic action $\\eqref{special matter}$.  \n\nIn this paper, we have shown that the DBI action of a real linear multiplet cannot be realized in the old minimal SUGRA as a naive embedding of the constraint $\\eqref{global constraint}$, which may contradict the result of Ref.~\\cite{Koehn:2012ar}. The duality relation between the old and new minimal SUGRA \\cite{Ferrara:1983dh} is generically not obvious when there exist higher-derivative terms. For example, the non-minimal coupling of gravity is realized only in new minimal SUGRA \\cite{Farakos:2012je} as in the case of the DBI action we discussed here. Such an issue may be revealed with the help of deep understanding of SUGRA system with higher-order derivative terms.  \n\nTo investigate our model further, we need the direct derivation of the constraint from $\\mathcal{N}=2$ SUGRA. And also, the remaining part in Eq. $\\eqref{string DBI}$, i.e., a term including $B_{\\mu \\nu }$, and possible combinations of the Maxwell, scalar and 2-form parts have not been constructed. We leave them for future work.\n\n\n\n\n\\section*{Acknowledgment}\nThe authors would like to thank Taichiro Kugo for helpful discussions and comments. YY would like thank also to Hiroyuki Abe and Yutaka Sakamura for useful discussion and collaboration in the related work. The work of YY is supported by JSPS\nResearch Fellowships for Young Scientists No. 26-4236 in Japan.\n\\begin{appendix}\n\\section{The components of {\\it{u-associated}} spinor derivative multiplet}\\label{explicit}\nHere we show the explicit component form of \n\\begin{align}\n\\frac{1}{L_0}\\bar{\\mathcal{D}}^{(L_0)}L\\bar{\\mathcal{D}}^{(L_0)}L. \\label{Lo derivative part2}\n\\end{align}\nAs we have seen in Sec.~\\ref{extension}, Eq. $\\eqref{Lo derivative part2}$ is a chiral multiplet with weight $(3,3)$. The components of this multiplet, $\\{ z',P_L\\chi' , F'\\}$, are  \n\\begin{align}\nz'&=\\frac{C^2}{C_0} \\left(\\bar{\\tilde{Z}}-\\bar{\\tilde{Z}}_0  \\right) P_R\\left(   \\tilde{Z}-\\tilde{Z}_0  \\right),\\\\\n\\nonumber P_L\\chi' &=\\frac{\\sqrt{2}C^2}{C_0}P_L\\biggl[ \\left(  \\tilde{\\slash {B}}-i\\slash{D}\\tilde{C}-\\tilde{\\slash {B}}_0+i\\slash{D}\\tilde{C}_0\\right) \\left( \\tilde{Z}-\\tilde{Z}_0 \\right) -\\frac{3i}{2}\\tilde{Z}_0\\bar{\\tilde{Z}}_0P_R\\tilde{Z}_0\\\\\n&-\\frac{i}{2}\\tilde{Z}_0\\bar{\\tilde{Z}}P_R\\tilde{Z}+\\frac{i}{4}\\gamma ^a\\tilde{Z}_0\\bar{\\tilde{Z}}\\gamma _a\\gamma _5 \\tilde{Z}+i\\tilde{Z}\\bar{\\tilde{Z}}_0P_R\\tilde{Z}_0-\\frac{i}{2}\\gamma ^a\\tilde{Z}\\bar{\\tilde{Z}}_0\\gamma _a\\gamma _5 \\tilde{Z}_0\\biggr] ,\\\\\n\\nonumber F'&=\\frac{C^2}{C_0}\\biggl[ -\\left( \\tilde{B}_a-i D_a\\tilde{C}\\right) ^2 +2\\left( \\tilde{B}_a-i D_a\\tilde{C} \\right) \\left( \\tilde{B}^a-i D^a\\tilde{C} \\right) -\\left( \\tilde{B}_{0a}-i D_{0a}\\tilde{C}\\right) ^2\\\\\n\\nonumber &+i\\bar{\\tilde{Z}}_0\\gamma _5\\left(  \\tilde{\\slash {B}}-i\\slash{D}\\tilde{C} \\right)\\left( \\tilde{Z}-\\tilde{Z}_0  \\right)+\\frac{i}{2}\\bar{\\tilde{Z}} \\gamma _5\\left(  \\tilde{\\slash {B}}_0-i\\slash{D}\\tilde{C} _0\\right)\\tilde{Z}\\\\\n\\nonumber &-2i\\bar{\\tilde{Z}} \\gamma _5\\left(  \\tilde{\\slash {B}}_0-i\\slash{D}\\tilde{C} _0\\right)\\tilde{Z}_0+\\frac{3i}{2}\\bar{\\tilde{Z}}_0 \\gamma _5\\left(  \\tilde{\\slash {B}}_0-i\\slash{D}\\tilde{C} _0\\right)\\tilde{Z}_0\\\\\n\\nonumber &+2\\left( \\bar{\\tilde{Z}}-\\bar{\\tilde{Z}}_0 \\right)P_R \\slash{D}\\left( \\tilde{Z}-\\tilde{Z}_0 \\right) +\\frac{1}{2}\\bar{\\tilde{Z}}_0P_R\\tilde{Z}_0\\bar{\\tilde{Z}}\\tilde{Z}+\\frac{1}{2}\\bar{\\tilde{Z}}P_R\\tilde{Z}\\bar{\\tilde{Z}}_0\\tilde{Z}_0\\\\\n&+2\\bar{\\tilde{Z}}P_R\\tilde{Z}_0\\bar{\\tilde{Z}}\\tilde{Z}_0-3\\bar{\\tilde{Z}}P_R\\tilde{Z}_0\\bar{\\tilde{Z}}_0\\tilde{Z}_0-3\\bar{\\tilde{Z}}\\tilde{Z}_0\\bar{\\tilde{Z}}_0P_R\\tilde{Z}_0+\\frac{1}{2}\\bar{\\tilde{Z}}_0P_R\\tilde{Z}_0\\bar{\\tilde{Z}}_0\\tilde{Z}_0\\biggr] ,\n\\end{align}\nwhere the fields with $\\tilde{}$ are divided by the first components of the multiplet they belong to, in the same way as Eq. $\\eqref{tilde}$, and the superconformal derivative $D_a$ is understood to act only on the numerator but not on the denominator, e.g., $D^a\\tilde{C} \\equiv D^aC\/C =D^a \\log C$.\n\n\n\\end{appendix}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe famous Hasse--Minkowski theorem gives precise conditions for a quadratic form to have a non-trivial zero. For the solubility of a system of two quadratic forms there are still many open questions. In this paper we restrict to the study of smooth intersections of two quadrics in five variables. Such a system describes a geometric object $X \\subseteq \\mathbb P^4$ called a \\textit{del Pezzo surfaces of degree $4$}.\n\nAs in the Hasse--Minkowski theorem, a natural first step is to study the inclusion $X(\\mathbb{Q}) \\subseteq X(\\mathbb{A}_\\mathbb{Q})$ of the rational points into the \\textit{adelic points}. Since we are considering homogeneous equations we can identify  $X(\\mathbb{A}_\\mathbb{Q})$ with $\\prod_{p \\leq \\infty} X(\\mathbb{Q}_p)$, that is, an adelic solution consists of local solutions in each completion of $\\mathbb{Q}$. We say that a class of varieties satisfies the \\textit{Hasse principle} if for each member of the family the existence of local solutions implies the existence of a global solution. In general the Hasse principle can fail. One such example was the quartic del Pezzo surface\n$$\n\\begin{cases}\nx^2-5y^2=uv;\\\\\nx^2-5z^2=(u+v)(u+2v),\n\\end{cases}\n$$\nby Birch and Swinnerton-Dyer \\cite{BSD}.\n\nThis failure can be explained by the Brauer--Manin obstruction, as introduced by Manin \\cite{ManinICM}; any element $\\mathcal{Q}$ in the Brauer group of $X$ determines an intermediate set\n\\[\nX(\\mathbb{Q}) \\subseteq X(\\mathbb{A}_\\mathbb{Q})^{\\mathcal{Q}} \\subseteq X(\\mathbb{A}_\\mathbb{Q})\n\\]\nwhich can be used in some cases to show that $X(\\mathbb{Q})=\\emptyset$ even if there are adelic points on $X$. It has been conjectured by Colliot-Th\\'el\\`ene and Sansuc \\cite{CTSansuc} that for varieties such as del Pezzo surfaces, the Brauer--Manin obstruction is the \\textit{only obstruction to the Hasse principle}, that is, $X(\\mathbb{Q})$ is empty if and only if $X(\\mathbb{A}_{\\mathbb{Q}})^{\\Br} := \\bigcap_{\\mathcal{Q} \\in \\Br X} X(\\mathbb{A}_{\\mathbb{Q}})^{\\mathcal{Q}}$ is.\n\nPioneering work of Swinnerton-Dyer shows that for quartic del Pezzo surfaces the Brauer group $\\Br X\/\\Br_0 X$ is isomorphic to $\\left(\\mathbb{Z}\/2\\mathbb{Z} \\right)^i$ for $i \\in\\{0,1,2\\}$ \\cite{SDBrauerGroupCubicSurfaces}. Furthermore, the Brauer group can be explicitly computed from the equations \\cite{BBFL}, \\cite{VAV}.\n\nThe conjecture by Colliot-Th\\'el\\`ene and Sansuc was proven by Wittenberg \\cite{WittenbergBook} for quartic del Pezzo surface with a trivial Brauer group subject to the Schinzel hypothesis and the conjectured finiteness of Tate--Shafarevich groups of elliptic curves. These techniques were then applied by V\u00e1rilly-Alvarado and Viray for certain surfaces with a Brauer group of order $2$.\n\nWe will study the arithmetic of quartic del Pezzo surfaces with a Brauer group of order $4$, using fibration $X \\dashrightarrow \\mathbb P^1$ into curves. First we show that the existence of a commonly studied fibration implies the Hasse principle.\n\n\\begin{thm}[Thm.~\\ref{thm:conicfibrations}]\\label{thm:thm1}\nLet $X$ be a quartic del Pezzo surface with $\\# \\Br X\/\\Br_0 X =4$ over a number field $K$. If $X$ admits a conic fibration $X \\dashrightarrow \\mathbb P^1$ then $X(K)\\neq \\emptyset$.\n\\end{thm}\n\nThe work of V\u00e1rilly-Alvarado and Viray in the case of a smaller Brauer group, however uses maps $X \\dashrightarrow \\mathbb P^1$ obtained by embedding $X$ anticanonically and projecting away from a plane. They ask if a Brauer group of order $4$ can be ``vertical'' with respect to such maps. The second result of this paper is that this is not the case.\n\n\\begin{thm}[Thm.~\\ref{thm:nonewpoints}]\nLet $X \\subseteq \\mathbb P^4$ be an anticanonically embedded quartic del Pezzo surface over a number field $K$ with $\\# \\Br X\/\\Br_0 X =4$ and $X(K)=\\emptyset$. Then $\\Br X$ is not vertical with respect to a map $f \\colon X \\dashrightarrow \\mathbb P^1$ obtained by projecting away from a plane.\n\\end{thm}\n\nThis shows that the techniques of \\cite{CTSSD}, \\cite{WittenbergBook} and \\cite{VAV} cannot be directly applied to prove that the Brauer--Manin obstruction is the only one to the Hasse principle for quartic del Pezzo surfaces with a Brauer group of order $4$.\n\nSurfaces with two independent classes in the Brauer group have rarely shown up in the literature. Jahnel and Schindler \\cite{JS} prove that quartic del Pezzo surfaces with a Brauer group of order $2$ (and even those for which the conjecture is true) are Zariski dense in the moduli space of all quartic del Pezzo surfaces. In contrast, Mitankin and Salgado study a subfamily where infinitely many members have a Brauer group of order $4$, but all of these surfaces admit a conic fibrations and hence have a rational point in the light of Theorem~\\ref{thm:thm1}.\n\nWe will restrict to a different subfamily in which each member has a Brauer group of order $4$, which we will show to be non-empty.\n\n\\begin{thm}[Thm.~\\ref{thm:failureWA}, Thm.~\\ref{thm:curlyABorC}]\\label{thm:intro2}\nLet $X$ be a quartic del Pezzo surface with $\\#\\Br X\/\\Br_0 X =4$ given by a system\n\\[\n\\begin{cases}\ny^2-pz^2 = (A_1u+B_1v)(C_1u+D_1v);\\\\\nz^2-pz^2 = (A_2u+B_2v)(C_2u+D_2v),\n\\end{cases}\n\\]\nwhere $p$ is an odd prime. Then $X$ fails weak approximation and there is at most one element $\\mathcal{Q} \\in \\Br X\/\\Br_0 X$ for which $X(\\mathbb{A}_{\\mathbb{Q}})^{\\mathcal{Q}}=\\emptyset$.\n\\end{thm}\n\nNote the contrast with the result that if $X(\\mathbb{A}_\\mathbb{Q})^{\\Br} = \\emptyset$ then there exists an $\\mathcal{Q} \\in \\Br X$ such that $X(\\mathbb{A}_{\\mathbb{Q}})^{\\mathcal{Q}}=\\emptyset$ for any quartic del Pezzo surface \\cite[Rem.~2 following Lem.~3.4]{CTPoonen}.\n\nFor every quartic del Pezzo surface $X$ with $\\Br X\/\\Br_0 X$ has order $4$, we can write down two explicit generators $\\mathcal{A}_X$ and $\\mathcal{B}_X$, where $\\mathcal{A}_X$ is uniquely defined and $\\mathcal{B}_X$ is any other non-trivial element. We produce examples in which either generator obstructs the Hasse principle.\n\n\\begin{thm}\nConsider the surfaces\n$$\nY\\colon\\begin{cases}\ny^2-13x^2=uv;\\\\\nz^2-13x^2=(2u-13v)(u-6v),\n\\end{cases}\n$$\nand\n$$\nS \\colon \\begin{cases}\ny^2-13x^2=uv;\\\\\nz^2-13x^2=(u+v)(153u+179v).\n\\end{cases}\n$$\n\\begin{enumerate}\n\\item[(a)] The surfaces $Y$ and $S$ are everywhere locally soluble.\n\\item[(b)] The Brauer groups $\\Br Y\/\\Br_0 Y$ and $\\Br S\/\\Br_0 S$ are both isomorphic to $(\\mathbb Z\/2\\mathbb Z)^2$.\n\\item[(c)] We have $Y(\\mathbb{A}_\\mathbb{Q})^{\\mathcal{A}_Y} = \\emptyset$ for $\\mathcal{A}_Y = \\left(13,\\frac{u-6v}v\\right) \\in \\Br Y$, and $S(\\mathbb{A}_\\mathbb{Q})^{\\mathcal{B}_S} = \\emptyset$ for $\\mathcal{B}_S = \\left(13,\\frac{y+z}u\\right) \\in \\Br S$.\n\\end{enumerate}\n\\end{thm}\n\nUsing Theorem~\\ref{thm:intro2} we see that $\\mathcal{A}_Y$ and $\\mathcal{B}_S$ are the unique elements in respectively $\\Br Y$ and $\\Br S$ with these properties.\n\nTo prove for a general quartic del Pezzo surface with a Brauer group of order $4$ that $X(\\mathbb{Q})\\ne\\emptyset$ when neither $\\mathcal A_X$, $\\mathcal B_X$ nor $\\mathcal{A}_X+\\mathcal{B}_X$ obstructs the Hasse principle is still open.\n\n\\subsection*{Acknowledgements}\n\nThis paper is a product of an internship of the second author at IST Austria under the supervision of the first author.\n\n\\section{Preliminaries}\n\nLet us fix our terminology.\n\n\\begin{defi}\nLet $k$ be a field. A $k$-scheme $X$ is called \\textit{nice} if it is proper, geometrically integral and smooth over $k$. A \\textit{surface} will be a nice $2$-dimensional $k$-scheme.\n\\end{defi}\n\nAll varieties under consideration in this paper will be nice $\\mathbb Q$-surfaces. So by the properness we can identify $X(\\mathbb{A}_\\mathbb{Q})$ with $\\prod_{p\\leq \\infty} X(\\mathbb{Q}_p)$ where we write $\\mathbb{Q}_\\infty := \\mathbb R$. We have the diagonal embedding $X(\\mathbb{Q}) \\hookrightarrow X(\\mathbb{A}_\\mathbb{Q})$.\n\nWe will say that a class $\\mathcal S$ of varieties satisfies the \\textit{Hasse principle} when $X(\\mathbb{Q})=\\emptyset$ if and only if $X(\\mathbb{A}_\\mathbb{Q}) = \\emptyset$ for all $X \\in \\mathcal S$. If $X(\\mathbb{Q})$ is dense in $X(\\mathbb{A}_\\mathbb{Q})$ we say that X satisfies \\textit{weak approximation}.\n\n\\subsection*{The Brauer--Manin obstruction}\n\nThe failure of the Hasse principle or weak approximation can be explained using the Brauer group. So let us consider $\\Br X := \\H^2(X_{\\text{\\'et}},\\mathbb{G}_m)$ and its natural subgroup $\\Br_0 X := \\Im(\\Br \\mathbb{Q} \\to \\Br X)$. Since $X$ is nice we have an inclusion $\\Br X \\hookrightarrow \\Br \\kappa(X)$, and we will only be interested in elements $\\mathcal Q \\in \\Br X$ of order $2$. Hence such elements are always represented by a quaternion algebra $(f,g)$ over the function field $\\kappa(X)$ of $X$.\n\nRecall the \\textit{invariant map} $\\inv_v \\mathcal Q \\colon X(\\mathbb{Q}_v) \\to \\mathbb{Q}\/\\mathbb{Z}$ of an element $\\mathcal Q=(f,g) \\in \\Br X[2]$ at a place $v$ \\cite[Thm.~1.5.36]{Poonen}, which are constant for $\\mathcal Q \\in \\Br_0 X$. At a point $P \\in X(\\mathbb{Q})$ for which $f,g\\in \\mathcal O^\\times_{X,P}$ the invariant $\\inv_v \\mathcal Q(P)$ coincides with the Hilbert symbol $(f(P),g(P)) \\in \\{\\pm 1\\}$ after identifying the groups $\\{\\pm 1\\}$ and $\\frac12\\mathbb{Z}\/\\mathbb{Z}$. By abuse of terminology we will say that $\\inv_v\\mathcal Q$ is \\textit{surjective} if $\\# \\Im(\\inv_v \\mathcal Q) = 2$.\n\nFor $\\mathcal Q \\in \\Br X$ we can consider\n\\[\nX(\\mathbb{A}_\\mathbb{Q})^{\\mathcal Q}  := \\{(P_v)_v \\in X(\\mathbb{A}_\\mathbb{Q})\\mid \\sum_{v \\leq \\infty} \\inv_v \\mathcal Q(P_v) =0\\}.\n\\]\nBy global reciprocity we have $X(\\mathbb{Q}) \\subseteq X(\\mathbb{A}_\\mathbb{Q})^{\\mathcal{Q}} \\subseteq X(\\mathbb{A}_\\mathbb{Q})$ or even $X(\\mathbb{Q}) \\subseteq X(\\mathbb{A}_\\mathbb{Q})^{\\Br} \\subseteq X(\\mathbb{A}_\\mathbb{Q})$ for $X(\\mathbb{A}_\\mathbb{Q})^{\\Br} := \\bigcap X(\\mathbb{A}_\\mathbb{Q})^{\\mathcal{Q}}$.\n\nIf $X(\\mathbb{A}_\\mathbb{Q}) \\ne \\emptyset$, but $X(\\mathbb{A}_\\mathbb{Q})^{\\Br} = \\emptyset$, then the Hasse principle fails for $X$ and we say there is a \\textit{Brauer--Manin obstruction to the Hasse principle}. If $X(\\mathbb{A}_\\mathbb{Q})^{\\Br} \\subsetneq X(\\mathbb{A}_\\mathbb{Q})$, then $X$ cannot satisfy weak approximation, and there is a \\textit{Brauer--Manin obstruction to weak approximation}.\n\nNote that if an invariant map of $\\mathcal Q$ is surjective, then there is a Brauer--Manin obstruction to weak approximation, but not necessarily to the Hasse principle.\n\n\\subsection*{Quartic del Pezzo surfaces}\n\nWe will be interested in a particular type of surface, namely del Pezzo surface of degree $4$. For a general treatise on general del Pezzo surfaces and their arithmetic the reader is referred to \\cite{arithmeticdps}. We will give the following characterisation of quartic del Pezzo surfaces, which we will take as the definition.\n\n\\begin{defi}\nA \\textit{del Pezzo surface of degree $4$} is a surface $X \\subseteq \\mathbb P^4$, hence in particular smooth, given by two quadratic forms $Q=0=\\tilde Q$.\n\\end{defi}\n\nIt is known that the Brauer group of a quartic del Pezzo surface modulo constants is either trivial, of order $2$ or isomorphic to the Klein four-group. Algorithms to determine this isomorphism class and explicit generators are given in both \\cite{BBFL} and \\cite{VAV}. Let us introduce some notation from the latter algorithm to determine if we are dealing with a Brauer group of order $4$.\n\nRepresent the quadratic forms $Q$ and $\\tilde Q$ by symmetric matrices $M$ and $\\tilde M$. For a point $T=(\\kappa \\colon \\lambda) \\in \\mathbb P^1(\\bar{\\mathbb{Q}})$ we define the symmetric matrix $M_T = \\kappa M + \\lambda \\tilde M$, with associated quadratic form $Q_T$ over $\\kappa(T)$. Let $f$ be the discriminant of $M_T$, then $\\mathscr{S}=V(f)\\subseteq\\mathbb{P}^1$ is the locus of the degenerate fibres. For the quadratic forms $Q_T$ of rank $4$ we will need the determinant $\\varepsilon_T$ of its restriction to a linear subspace of codimension $1$ on which it is nondegenerate. The invariant $\\varepsilon_T$ is well defined up to squares.\n\n\\begin{pro}[{\\cite[\\textsection 4.1]{VAV}}]\\label{pro:Brauergrouporder4}\nLet $X \\subseteq \\mathbb P^4$ be a quartic del Pezzo surface. Then $\\#\\Br X\/\\Br_0 X =4$ if and only if there exists an $\\varepsilon\\not\\in\\mathbb{Q}^{\\times, 2}$, and three degree 1 points $T_i \\in \\mathscr{S}$ such that each $Q_{T_i}$ has rank $4$ and satisfies $\\varepsilon\\cdot\\varepsilon_{T_i}\\in\\mathbb{Q}^{\\times, 2}$.\n\\end{pro}\n\n\\section{A Brauer group of order $4$}\n\nFrom now on $X\/\\mathbb Q$ will be a quartic del Pezzo surface anticanonically embedded in $\\mathbb P^4$. Such a surface is the intersection of two quadrics. We will be particularly interested in the $X$ for which the Brauer group modulo constants has order $4$. \n\n\\begin{pro}\\label{pro:gen_sys}\nA quartic del Pezzo surface $X$ over $\\mathbb Q$ satisfies $\\Br X\/\\Br \\mathbb Q \\cong (\\mathbb Z\/2\\mathbb Z)^2$ precisely if it can be given by a system of equations of the form\n\\begin{equation}\\label{gen_sys}\n\\begin{cases}\nd_0y^2-\\varepsilon x^2=a_0u^2+2b_0uv+c_0v^2;\\\\\nd_1z^2-\\varepsilon x^2=a_1u^2+2b_1uv+c_1v^2,\n\\end{cases}\n\\end{equation}\nsuch that\n\\begin{enumerate}\n\\item $\\varepsilon\\not\\in\\mathbb{Q}^{\\times, 2}$,\n\\item $d_0=b_0^2-a_0c_0$ and $d_1=b_1^2-a_1c_1$,\n\\item $\\varepsilon d_0d_1d_2\\in \\mathbb{Q}^{\\times, 2}$ where $d_2:=(b_1-b_0)^2-(a_1-a_0)(c_1-c_0)$, and\n\\item the quadratic forms $a_iu^2+2b_iuv+c_iv^2$ do not have a common projective root.\n\\end{enumerate}\n\\end{pro}\n\n\\begin{proof}\nProposition~4.2 in \\cite{VAV} and Proposition~\\ref{pro:Brauergrouporder4} yield that after a linear change of variables the matrices of $Q_{T_i}$ have the following form\n$$\nM_{T_0}=\n\\begin{pmatrix}\na_0 & b_0 & 0 & 0 & 0\\\\\nb_0 & c_0 & 0 & 0 & 0\\\\\n0 & 0 & m_0 & 0 & 0\\\\\n0 & 0 & 0 & n_0 & 0\\\\\n0 & 0 & 0 & 0 & 0\n\\end{pmatrix},\\ \nM_{T_1}=\n\\begin{pmatrix}\na_1 & b_1 & 0 & 0 & 0\\\\\nb_1 & c_1 & 0 & 0 & 0\\\\\n0 & 0 & m_1 & 0 & 0\\\\\n0 & 0 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0 & k_1\n\\end{pmatrix}\n\\ \n\\text{ and }\n\\ \nM_{T_2}=\n\\begin{pmatrix}\na_2 & b_2 & 0 & 0 & 0\\\\\nb_2 & c_2 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0 & 0\\\\\n0 & 0 & 0 & n_2 & 0\\\\\n0 & 0 & 0 & 0 & k_2\n\\end{pmatrix}.\n$$\nWe get that\n$$\n\\varepsilon_{T_0}=-d_0m_0 n_0,\\quad\\varepsilon_{T_1}=-d_1m_1 k_1 \\quad \\text{ and } \\quad \\varepsilon_{T_2}=-d_2n_2k_2,\n$$\nwhere $d_i:=b_i^2-a_ic_i$. Since the $M_{T_i}$ are linearly dependent we can assume $M_{T_2}=M_{T_1}-M_{T_0}$. Without loss of generality we can set $m_0=m_1$ equal to $\\varepsilon$. We can still change $n_0$ and $k_1$ independently up to squares, and the condition $\\varepsilon\\cdot\\varepsilon_{T_i}\\in\\mathbb{Q}^{\\times,2}$ allows us to take $n_0=-d_0$ and $k_1=-d_1$. The condition $\\varepsilon d_0d_1d_2\\in\\mathbb{Q}^{\\times,2}$ is then automatically satisfied. The last condition is equivalent to the projective scheme defined by \\eqref{gen_sys} is smooth.\n\nFrom Proposition~\\ref{pro:Brauergrouporder4} it is easy to check that any such system defines a surface with a Brauer group of order $4$.\n\\end{proof}\n\nThe algorithm in \\cite{VAV} even allows us to represent the elements of the Brauer group by explicit quaternion algebras. Given the explicit element of the Brauer group we can now discuss whether the Brauer--Manin obstruction to Hasse principle is the only one. For simplicity we restrict to surfaces over the rational numbers, but the results and proofs in this and the last section equally apply to quartic del Pezzo surfaces with a Brauer group of order $4$ over any number fields. \n\nFor quartic del Pezzo surfaces with a large Brauer group which admit conic fibrations there cannot be a Brauer--Manin obstruction to the Hasse principle.\n\n\\begin{thm}\\label{thm:conicfibrations}\nLet $X\/\\mathbb Q$ be a quartic del Pezzo surface with $\\# \\Br X\/\\Br_0 X =4$. If $X$ admits a conic fibration $X \\dashrightarrow \\mathbb P^1$ then $X(\\mathbb Q)\\neq \\emptyset$.\n\\end{thm}\n\n\\begin{proof}\nSince $X$ is a nice variety over a number field we conclude from the Hochschild--Serre spectral sequence \\cite[Cor.~6.7.8]{Poonen} that $\\Br X\/\\Br_0 X \\cong \\H^1(G_{\\mathbb Q}, \\Pic \\bar X)$. We will need to understand the Galois action on the geometric Picard group.\n\nFor a general quartic del Pezzo surface the action of the absolute Galois group $G_{\\mathbb Q}$ on $\\Pic \\bar X$ factors through a subgroup $H \\subseteq W_5$. Here $W_5$ is the finite group Weyl group which acts on $\\Pic \\bar X$, see \\cite[\\textsection 1.5]{arithmeticdps}. Since $\\Pic \\bar X$ is torsion-free we conclude from the inflation-restriction sequence that $\\H^1(G_{\\mathbb Q}, \\Pic \\bar X) \\cong \\H^1(H, \\Pic \\bar X)$. For each of the 196 different subgroups of $W_5$ (up to conjugation) we can compute the first cohomology group directly and we find that only four subgroups yield a Brauer group of order $4$.\n\nAgain using the Hochschild--Serre spectral sequence we see that $\\Pic X \\hookrightarrow (\\Pic \\bar X)^{G_{\\mathbb Q}}$ is injective. For the four relevant Galois actions on $\\Pic \\bar X$ we see that $(\\Pic \\bar X)^{G_{\\mathbb Q}}$ has rank $1$ in three cases, and rank $2$ in one case.\n\nIn the case of rank $1$ we conclude that $\\Pic X$ is free of rank $1$, since it contains the canonical class. If $X$ would admit a conic fibration then the class of the fibre should be a (rational) multiple of the canonical class, and have self-intersection $0$. Clearly a contradiction.\n\nIn the case of rank $2$ we find that the action of Galois permutes two exceptional curves, whose intersection pairing equals $1$. Hence their intersection point is defined over the base field.\n\\end{proof}\n\n\\begin{rema} More precisely, in the latter case there are multiple pairs of such intersecting lines which sum to the same class in $\\Pic X$. One can conclude from this that among the quartic del Pezzo surfaces with a Brauer group of order $4$ the following statements are equivalent.\n\\begin{enumerate}\n\\item $\\rk \\Pic X =2$.\n\\item $X$ contains two conjugate intersecting lines.\n\\item $X$ admits a conic fibration.\n\\end{enumerate}\n\\end{rema}\n\nFor general maps $f\\colon X\\dashrightarrow \\mathbb{P}^1$ the following notion will be very useful. We say that $\\Br X$ is \\textit{vertical} with respect to $f$ if $\\Br X\\subseteq f^*(\\Br \\kappa(\\mathbb{P}^1))$.\n\nFor quartic del Pezzo surfaces with vertical Brauer groups there is a method to prove that $X(\\mathbb{A}_\\mathbb{Q})^{\\Br}\\neq\\emptyset$ \nimplies $X(\\mathbb{Q})\\neq\\emptyset$ (see \\cite{WittenbergBook} and \\cite{CTSSD}). V\u00e1rilly-Alvarado and Viray proved in \\cite{VAV} that the Brauer group is vertical for certain quartic del Pezzo surfaces, and they concluded that the Brauer--Manin obstruction to Hasse principle is the only one for these surface of ``BSD-type''.\n\nNext they considered quartic del Pezzo surface with a Brauer group of order $4$. First they noted that if $X(\\mathbb{Q})\\neq \\emptyset$, then $\\Br X$ is vertical with respect to a specific map. Next they asked if $X(\\mathbb{Q})= \\emptyset$ whether the Brauer group can be vertical, in particular for rational maps $f \\colon X \\dashrightarrow \\mathbb P^1$ obtained from projecting away from a plane. They suggested the following approach.\n\nLet $T_i$ be the three degree 1 points of $\\mathscr{S}$, see Proposition~\\ref{pro:Brauergrouporder4}. For $P_i\\in V(Q_{T_i})(\\mathbb{Q})$, denote $M(P_0,P_1,P_2)$ the matrix whose $(i,j)$th entry is $\\frac{\\partial Q_{T_i}}{\\partial x_j}(P_i)$. Then take a hyperplane $H\\subseteq \\mathbb{P}^4$ which does not pass through the vertices of $V(Q_{T_i})$, and define\n$$\nY^{(X)}=\\{(P_0,P_1,P_2)\\in H^3\\mid \\rnk M(P_0,P_1,P_2)\\le 2 \\text{ and } Q_{T_i}(P_i)=0\\}.\n$$\nThere is an inclusion $X(\\mathbb{Q})\\hookrightarrow Y^{(X)}(\\mathbb{Q})$, which is described in \\cite{VAV}.\n\nIf we can find a rational point on $Y^{(X)}$, then we obtain the verticality of $\\Br X\/\\Br\\mathbb{Q}$ with respect to a certain map $f$. We already know that if $X(\\mathbb{Q})\\neq \\emptyset$, then $Y^{(X)}(\\mathbb{Q})\\neq\\emptyset$. The question is, whether there is a point in $Y^{(X)}(\\mathbb{Q})$ which does not come from a point in $X(\\mathbb{Q})$ (see \\cite{VAV}, Question~6.3). Here we show that the answer is no.\n\n\\begin{defi}\nWe define a point $P=(u \\colon v \\colon x \\colon y \\colon z) \\in \\mathbb P^4(\\mathbb{Q})$ to be equivalent to the points $(u \\colon v \\colon \\pm x \\colon \\pm y \\colon \\pm z)$. This induces an equivalence relation $\\sim$ on $Y^{(X)}(\\mathbb{Q})$.\n\\end{defi}\n\nIn other words, two points $\\mathcal P = (P_0,P_1,P_2)$ and $\\mathcal P' = (P'_0,P'_1,P'_2)$ are equivalent precisely if the $P_i$ and $P'_i$ are the same projective point up to a possible change of sign in the last three coordinates.\n\n\nConsider a point $\\mathcal P \\in Y^{(X)}(\\mathbb{Q})$. Although $\\mathcal P$ need not lie the image of the map $X(\\mathbb{Q})\\hookrightarrow Y^{(X)}(\\mathbb{Q})$, we now show that it is equivalent to a point that does.\n\n\\begin{thm}\\label{thm:nonewpoints}\nThe composite map\n$$\nX(\\mathbb{Q})\\hookrightarrow Y^{(X)}(\\mathbb{Q})\\longrightarrow Y^{(X)}(\\mathbb{Q})\/\\sim\n$$\nis surjective.\n\\end{thm}\n\n\\begin{proof}\nWe will represent $X$ by a system of equations as in \\eqref{gen_sys}. In addition, we can assume without loss of generality $a_0 \\ne 0$ and hence after completing the square that $b_0=0$. This implies that $d_0 = -a_0c_0$ and we conclude that also $c_0 \\ne 0$.\n\nConsider the point $(P_0,P_1,P_2)\\in Y^{(X)}(\\mathbb{Q})$ where $P_i=(u_i:v_i:x_i:y_i:z_i)$. Our goal is to prove that $(P_1,P_2,P_3)\\sim (P,P,P)$ for some $P\\in X(\\mathbb{Q})$. By the definition of $Y^{(X)}$ using $T_0=(1:0)$, $T_1=(0:1)$ and $T_2=(-1:1)$ as in the proof of Proposition~\\ref{pro:gen_sys}, we have\n$$\n\\mathrm{rank\\,}\n\\begin{pmatrix}\n2a_0u_0                & 2c_0v_0                & 2\\varepsilon x_0  & -2d_0y_0  & 0    \\\\\n2a_1u_1+2b_1v_1        & 2b_1u_1+2c_1v_1        & 2\\varepsilon x_1  & 0      & -2d_1z_1\\\\\n2(a_1-a_0)u_2+2b_1v_2  & 2b_1u_2+2(c_1-c_0)v_2  & 0            & 2d_0y_2   & -2d_1z_2\n\\end{pmatrix}\n=2.\n$$\nWrite $\\ell_i \\in\\mathbb{Q}^5$ for the rows of the above matrix. The condition on the rank implies the existence of a non-trivial relation $\\kappa\\ell_0+\\lambda \\ell_1+\\mu \\ell_2=0$. After multiplying the coordinates of $P_0$, $P_1$ and $P_2$ by respectively $\\kappa$, $-\\lambda$ and $\\mu$, we obtain $\\ell_0-\\ell_1+\\ell_2=0$. This implies \n\\begin{equation}\\label{xyz}\nx_0=x_1,\\ y_0=y_2\\ \\text{ and }\\ z_1=z_2,\n\\end{equation} \nand the first two columns give us the relations\n\\begin{equation}\\label{subst}\nu_0=\\frac{1}{a_0}\\Bigl(a_1u_1+b_1v_1-(a_1-a_0)u_2-b_1v_2\\Bigr)\n\\quad \\text{ and } \\quad\nv_0=\\frac{1}{c_0}\\Bigl(b_1u_1+c_1v_1-b_1u_2-(c_1-c_0)v_2\\Bigr).\n\\end{equation}\nCondition $Q_{T_i}(P_i)=0$ gives us\n\\begin{multline}\\label{quadr}\n(a_0u_0^2+c_0v_0^2)-(a_1u_1^2+2b_1u_1v_1+c_1v_1^2)+((a_1-a_0)u_2^2+2b_1u_2v_2+(c_1-c_0)v_2^2)=\\\\\n(d_0y_0^2-\\varepsilon x_0^2)-(d_1z_1^2-\\varepsilon x_1^2)+(d_1z_2^2-d_0y_2^2)=0.\n\\end{multline}\nSubstituting \\eqref{subst} into \\eqref{quadr}, we obtain the following:\n\\begin{multline}\\label{quad_form}\n(a_0^2c_0+b_0^2a_0-a_1a_0c_0)(u_1-u_2)^2+2(a_1b_1c_0+b_1c_1a_0-b_1a_0c_0)(u_1-u_2)(v_1-v_2)+\\\\\n(b_1^2c_0+c_1^2a_0-c_1a_0c_0)(v_1-v_2)^2=0.\n\\end{multline}\nThis is a quadratic form in $u_1-u_2$ and $v_1-v_2$ whose determinant equals \n$$\n-a_0c_0(b_1^2-a_1c_1)(b_1^2-(a_1-a_0)(c_1-c_0))=d_0d_1d_2.\n$$ \nFrom $\\varepsilon\\notin\\mathbb{Q}^{\\times,2}$ and $\\varepsilon d_0d_1d_2\\in\\mathbb{Q}^{\\times,2}$ we conclude $d_0d_1d_2\\notin\\mathbb{Q}^{\\times,2}$, so \n\\eqref{quad_form} implies $u_1-u_2=0=v_1-v_2$. Then \\eqref{subst} implies that $u_0=u_1=u_2$ and $v_0=v_1=v_2$.\n\nFinally, conditions \\eqref{xyz} and $Q_{T_i}(P_i)=0$ give us $x_0=x_1=\\pm x_2$, $y_0=y_2=\\pm y_1$ and $z_1=z_2=\\pm z_0$, \nwhich means that $P_i \\in X(\\mathbb{Q})$ and $P_0 \\sim P_1 \\sim P_2$ which proves the proposition.\n\\end{proof}\n\n\nWe deduce that the answer to Question~6.3 in \\cite{VAV} is no. We conclude that one cannot directly apply to machinery of \\cite{CTSSD} and \\cite{WittenbergBook} to prove there are rational points on quartic del Pezzo surfaces with a Brauer group of order $4$.\n\n\\section{A subfamily}\n\nWe will study the arithmetic of quartic del Pezzo surfaces with a Brauer group of order $4$. In \\cite{MS} such surfaces appeared, but those all had a rational point for obvious geometrical reasons; on these surfaces there is a pair of intersecting lines which, as a pair, are fixed by the Galois action. We will restrict to a subfamily for which such behaviour does occur.\n\n\\begin{defi}\nLet $\\mathcal X$ be the set of isomorphism classes of quartic del Pezzo surfaces for which the two quadratic forms $Q_i=a_iu^2+b_iuv+c_iv^2$ split over $\\mathbb Q$, and $\\varepsilon$ is an odd prime number $p$.\n\\end{defi}\n\n\\begin{pro}\\label{pro:sys_subfam}\nAny $X \\in \\mathcal X$ can be written as\n\\begin{equation}\\label{sys}\n\\begin{cases}\ny^2-px^2=Muv;\\\\\nz^2-px^2=(Au+Bv)(Cu+Dv),\n\\end{cases}\n\\end{equation}\nfor $A,B,C,D,M,N\\in\\mathbb{Z}$ which satisfy\n\\begin{enumerate}\n\\item[\\textbf{(C1)}] $(AD+BC-M)^2-4ABCD=pN^2$, and\n\\item[\\textbf{(C2)}] $ NM(AD-BC)\\neq 0$.\n\\end{enumerate}\n\\end{pro}\n\nCondition (C1) is a restatement of $\\varepsilon d_0d_1d_2\\in\\mathbb{Q}^{\\times,2}$ and (C2) is equivalent to $X$ being smooth. \n\n\n\\begin{proof}\nIt is an immediate consequence of Proposition \\ref{pro:gen_sys}.\n\\end{proof}\n\nTo study the members of this family which fail the Hasse principle we will first consider the question of local solubility. For places $v$ such that $p\\in\\mathbb{Q}_v^{\\times,2}$ we have $(0\\colon 0\\colon 1\\colon \\sqrt{p}\\colon \\sqrt{p})\\in X(\\mathbb{Q}_v)$, so $X(\\mathbb{Q}_v)\\neq\\emptyset$. The following proposition covers all but two of the other places.\n\n\n\\begin{pro}\\label{pro:loc_sol_subfam}\nConsider $X=X_{p,A,B,C,D,M,N}$. Let $v\\not \\in \\{2,p\\}$ be a place such that $p\\notin\\mathbb{Q}_v^{\\times,2}$ and $v\\nmid N$. Then $X(\\mathbb{Q}_v)\n\\neq\\emptyset$.\n\\end{pro}\n\n\\begin{proof}\nObviously $v\\neq \\infty$, so it corresponds to a prime number $q$. The Chevalley--Warning theorem yields that the system \\eqref{sys} has a non-trivial solution $(u,v,x,y,z)\\in\\mathbb{F}_q^5$. Let $(u_0,v_0,x_0,y_0,z_0)$ be an arbitrary lift to $\\mathbb{Z}_q^5$. We distinguish four cases.\n\n\\noindent\\textbf{Case 1.} $q \\nmid y_0,z_0$. In this case take \n$$\ny_1=\\sqrt{Mu_0v_0+px_0^2}\\in\\mathbb{Z}_q,\\quad z_1=\\sqrt{(Au_0+Bv_0)(Cu_0+Dv_0)+px_0^2}\\in\\mathbb{Z}_q,\n$$ \nthen $(u_0\\colon v_0\\colon x_0\\colon y_1\\colon z_1)\\in X(\\mathbb{Q}_q)$.\n\n\n\n\\noindent\\textbf{Case 2.} $q \\mid y_0$ and $q \\nmid x_0,z_0$. Take \n$$\nx_1=\\sqrt{-\\tfrac{M}{p}u_0v_0}\\in\\mathbb{Z}_q,\\quad z_1=\\sqrt{(Au_0+Bv_0)(Cu_0+Dv_0)+px_1^2}\\in\\mathbb{Z}_q,\n$$ \nthen $(u_0\\colon v_0\\colon x_1\\colon 0\\colon z_1)\\in X(\\mathbb{Q}_q)$.\n\n\\noindent\\textbf{Case 3.} $q \\mid x_0, y_0$ and $q \\nmid z_0$. We deduce that $q \\mid u_0v_0$. If $q \\mid u_0$, take $(0\\colon v_0\\colon 0\\colon 0\\colon \\sqrt{BDv_0^2})\\in X(\\mathbb{Q}_q)$. If on the other hand $q \\mid v_0$, take $(u_0\\colon 0\\colon 0\\colon 0\\colon \\sqrt{ACu_0^2})\\in X(\\mathbb{Q}_q)$.\n\n\\noindent\\textbf{Case 4.} $q \\mid y_0, z_0$. Subtracting the two equations in \\eqref{sys} yields\n$$\nACu^2+(AD+BC-M)uv+BDv^2\\equiv 0\\mod q,\n$$\nwhich can be rewritten to\n$$\n(2ACu+(AD+BC-M)v)^2 \\equiv (AD+BC-M)^2v^2-4ABCDv^2 \\equiv pN^2v^2 \\mod q,\n$$\nand equivalently\n$$\n(2BDv+(AD+BC-M)u)^2 \\equiv pN^2u^2 \\mod q.\n$$\nFrom $\\left(\\frac{p}{q}\\right)=-1$ and $q\\nmid N$ we obtain $u\\equiv v\\equiv 0\\mod q$, and \\eqref{sys} gives us $x\\equiv 0\\mod q$. Contradicting the non-triviality of $(u,v,x,y,z)\\in\\mathbb{F}_q^5$.\n\\end{proof}\n\n\n\n\n\nMost of our examples will satisfy $N=1$ so $X$ will be locally soluble everywhere away from $2$ and $p$.\n\nThe algorithm in \\cite[\\textsection 4.1]{VAV} allows to write down the explicit elements of $\\Br X\/\\Br \\mathbb{Q}$. Choosing $P_{T_0}=(0\\colon 1\\colon 0\\colon 0\\colon 0)\\in Q_{T_0}(\\mathbb{Q})$, $P_{T_1}=(-B\\colon A\\colon 0\\colon 0\\colon 0)\\in Q_{T_1}(\\mathbb{Q})$ and $P_{T_2}=(0\\colon 0\\colon 0\\colon 1\\colon 1)\\in Q_{T_2}(\\mathbb{Q})$, we obtain\n\\begin{equation}\\label{Br_subfam}\n\\Br X\/\\Br \\mathbb{Q}=\\left\\{\\id, \\left(p,\\frac{u}{Au+Bv}\\right), \\left(p,\\frac{z-y}{u}\\right), \\left(p,\\frac{Au+Bv}{z-y}\n\\right)\\right\\}.\n\\end{equation}\n\n\n\\begin{defi}\\label{defi:ABC}\nFor any $X\\in\\mathcal{X}$ given by \\eqref{sys}, we will write $\\mathcal{A}=\\left(p,\\frac{u}{Au+Bv}\\right)$, $\\mathcal{B}=\\left(p,\\frac{z-y}{u}\\right)$ and $\\mathcal{C}=\\left(p,\\frac{Au+Bv}{z-y}\\right)$ and assume the dependency on $X$ to be understood.\n\\end{defi}\n\n\\begin{rema}\nThe above notation actually depends on the representation \\eqref{sys} we chose for the equivalence class $X\\in\\mathcal{X}$. One can see that the change of variables $\\tilde{u}=\\frac{Au+Bv}{AD-BC}$, $\\tilde{v}=\\frac{Cu+Dv}{AD-BC}$, $\\tilde{y}=z$, $\\tilde{z}=y$ provides another representation of $X$ in the form \\eqref{sys}, but with possibly different coefficients, and $\\mathcal{B}$ for the new representation is exactly $\\mathcal{C}$ for the old one. \nHowever, one can show that $\\mathcal{A}$ does not depend on the representation of $X$.\n\n\n\n\\end{rema}\n\n\n\n\\begin{rema}\nOne can see that the classes $\\mathcal{A}$ and $\\mathcal{B}$ are also represented by\n\\begin{equation}\\label{Br_gr_rewr}\n\\mathcal{A}=\\left(p,\\frac{Mv}{Au+Bv}\\right) \\quad \\text{ and } \\quad\n\\mathcal{B}=\\left(p,\\frac{AC(z+y)}{u}\\right).\n\\end{equation}\n\\end{rema}\n\n\n\n\n\n\\section{Simultaneous obstructions to the Hasse principle}\n\nWe use the notation from the previous section. In particular, $X$ is a quartic del Pezzo surface given by explicit equations as in Proposition~\\ref{pro:sys_subfam}. We then see from \\eqref{Br_subfam} and Definition~\\ref{defi:ABC} that\n\\[\n\\Br X\/\\Br \\mathbb Q = \\{1,\\mathcal A, \\mathcal B, \\mathcal C\\}.\n\\]\nWe will concern ourselves with the Brauer--Manin obstruction to weak approximation and the Hasse principle. The first results is on weak approximation.\n\n\\begin{thm}\\label{thm:failureWA}\nAny $X \\in \\mathcal X$ fails weak approximation.\n\\end{thm}\n\nThe second result is about the Hasse principle and seems related to the following fact by Colliot-Th\\'el\\`ene and Poonen \\cite[Rem.~2 following Lem.~3.4]{CTPoonen} on quartic del Pezzo surfaces: suppose that $X(\\mathbb A_\\mathbb{Q})^{\\Br} = \\emptyset$ then there exists an element $\\mathcal Q \\in \\Br X$ such that\n $X(\\mathbb A_\\mathbb{Q})^{\\mathcal Q} = \\emptyset$. However, neither result implies the other.\n\n\\begin{thm}\\label{thm:curlyABorC}\nOnly one of $\\mathcal A$, $\\mathcal B$ and $\\mathcal C$ can give an obstruction to the Hasse principle on $X \\in \\mathcal X$.\n\\end{thm}\n\nTheorem~\\ref{thm:failureWA} and Theorem~\\ref{thm:curlyABorC} are direct consequences of the following proposition.\n\n\\begin{pro}\\label{prop:surjectiveinvariantmap}\nConsider an $X\\in\\mathcal{X}$. Assume $X(\\mathbb{Q}_p)\\neq\\emptyset$, then the invariant map at $p$ of $\\mathcal{A}$, $\\mathcal{B}$ or $\\mathcal{C}$ is surjective.\n\\end{pro}\n\nWe will need the following lemmas.\n\n\\begin{lem}\\label{lem:quadres}\nLet $p\\equiv 1\\mod 4$ be a prime. Consider the set $S(a,b)=\\{a+by\\mid y\\in\\mathbb{F}_p^{\\times,2}\\}$ for $a,b\\in\\mathbb{F}_p^\\times$. Denote $S(a,b)_\\zeta=\\{x\\in S(a,b)\\mid \\left( \\frac xp\\right)=\\zeta\\}$. \n\\begin{enumerate}\n\\item[(a)] If $a,b\\in\\mathbb{F}_p^{\\times,2}$, then $|S(a,b)_0|=1$, $|S(a,b)_1|=\\dfrac{p-5}{4}$ and $|S(a,b)_{-1}|=\\dfrac{p-1}{4}$.\n\\item[(b)] If $a\\in\\mathbb{F}_p^{\\times,2}$, $b\\notin\\mathbb{F}_p^{2}$, then $|S(a,b)_0|=0$, $|S(a,b)_1|=\\dfrac{p-1}{4}$ and $|S(a,b)_{-1}|=\\dfrac{p-1}{4}$.\n\\end{enumerate}\nIn particular, for $a\\in\\mathbb{F}_p^{\\times,2}$ we conclude that $S(a,b)$ contains a non-square.\n\\end{lem}\n\n\\begin{proof}\nBoth parts follow from the identity $\\sum_{x\\in\\mathbb{F}_p} (\\frac{a+bx^2}{p})=-(\\frac{b}{p})$ where $a,b\\in\\mathbb{F}_p^\\times$.\n\\end{proof}\n\n\n\\begin{lem}\\label{lem:quadres_table}\nLet $p\\equiv 1\\mod 4$ be a prime and fix $a,b,c,d\\in\\mathbb{F}_p^{\\times,2}$. Then there exists an $y_0\\in\\mathbb{F}_p^\\times$ such that either\n\\begin{enumerate}\n\\item[(i)] $a+by_0^2,\\ c+dy_0^2\\in\\mathbb{F}_p^{\\times}\\setminus\\mathbb{F}_p^{\\times,2}$, or\n\\item[(ii)] $a+by_0^2\\in \\mathbb{F}_p^{\\times}\\setminus\\mathbb{F}_p^{\\times,2}$ and $c+dy_0^2=0$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nApply the statement (a) of Lemma~\\ref{lem:quadres} for $S(a,b)$ and $S(c,d)$.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:surjectiveinvariantmap}]\nWe will first make a few reductions.\n\nFirst suppose that $p \\mid M, A, C$. Then the system \\eqref{sys} is equivalent to the one with $M$, $A$ and $B$ divided by $p$. Also if $p^2 \\mid M, A, C$ then we obtain an equivalent system by dividing these three coefficients by $p^2$. This shows that we can simply to the cases where\n\\begin{equation}\\label{gcd_cond_1}\n p \\nmid \\gcd(A,C,M) \\text{ and } p \\nmid \\gcd(B,D,M),\n\\end{equation}\nand\n\\begin{equation}\\label{gcd_cond_2}\np^2\\nmid\\gcd(A,B,M) \\text{ and } p^2\\nmid \\gcd(C,D,M).\n\\end{equation}\n\nConsider the case $p \\equiv 3 \\mod 4$. For a point $P=(u\\colon v \\colon x \\colon y \\colon z) \\in X(\\mathbb Q_p)$ we compute the invariant map at $P$ and $P'=(u\\colon v \\colon x \\colon -y \\colon -z)$. We see that\n\\[\n\\inv_p \\mathcal{B}(P) \\ne \\inv_p \\mathcal{B}(P')\n\\]\nand this proves the proposition in this case.\n\nNow we can assume \n\\begin{equation}\\label{p_1_mod4}\np\\equiv 1\\mod 4.\n\\end{equation}\nCondition (C1) implies $(p,ABCD)_p=1$. Consider the case $(p,AC)_p=(p,BD)_p=-1$. For a point $P=(u\\colon v\\colon x\\colon y\\colon z)\\in X(\\mathbb{Q}_p)$, we define $P'=(u\\colon v\\colon x\\colon -y\\colon z)\\in X(\\mathbb{Q}_p)$. One can see that $\\inv_p\\mathcal{B}(P)\\neq\\inv_p\\mathcal{B}(P')$ using \\eqref{Br_gr_rewr}, and this proves the proposition in this case.\n\nSo now we can assume\n\\begin{equation}\\label{hilbert_cond}\n(p,AC)_p=(p,BD)_p=1.\n\\end{equation}\n\nFor equations satisfying (C1) and (C2), and the additional assumptions \\eqref{gcd_cond_1} up to \\eqref{hilbert_cond} we consider different cases depending on the valuation of $M$ and $m = \\max\\{v_p(A),v_p(B),v_p(C),v_p(D)\\}$. Without loss of generality we can assume that $p^m \\mathrel{\\|} A$ and $p^{m+1} \\nmid B,C,D$.\n\nIn the first two cases we show that the invariant map of $\\mathcal{B}$ at $p$ is surjective.\n\\begin{enumerate}\n\\item[\\textbf{Case 1.}] $\\mathrm{v}_p(M)=0$ and $m\\geq 1$\n\\item[\\textbf{Case 2.}] $\\mathrm{v}_p(M)=1$ and $m=1$\n\\end{enumerate}\n In the next two cases we show that the invariant map of $\\mathcal{A}$ at $p$ is surjective.\n\\begin{enumerate}\n\\item[\\textbf{Case 3.}] $\\mathrm{v}_p(M)=0$ and $m=0$\n\\item[\\textbf{Case 4.}] $\\mathrm{v}_p(M)$ is odd and $m=0$\n\\end{enumerate}\nIn the last four cases we can make a change of variables to reduce to one of the previous cases.\n\\begin{enumerate}\n\\item[\\textbf{Case 5.}] $\\mathrm{v}_p(M)=1$ and $m\\geq 2$\n\\item[\\textbf{Case 6.}] $\\mathrm{v}_p(M) \\geq 3$ is odd and $m \\geq 1$\n\\item[\\textbf{Case 7.}] $\\mathrm{v}_p(M) \\geq 2$ is even and $m=0$\n\\item[\\textbf{Case 8.}] $\\mathrm{v}_p(M) \\geq 2$ is even and $m\\geq 1$\n\\end{enumerate}\n\n\\noindent\n\\textbf{Surjectivity for $\\mathcal{B}$}.\nTo prove $\\inv_p\\mathcal B$ is surjective we will find points $P_i=(u_i\\colon v_i\\colon x_i\\colon y_i\\colon z_i)\\in X(\\mathbb{Q}_p)$ such that \n$$\n\\left(\\frac{u_1(z_1-y_1)}{p}\\right)\\left(\\frac{u_2(z_2-y_2)}{p}\\right)=-1,\n$$ \nsince this implies $\\inv_p\\mathcal{B}(P_1)+\\inv_p\\mathcal{B}(P_2)=\\frac{1}{2}$.\n\n\\noindent\\textbf{Case 1.} Since $p \\mid A$, condition (C1) implies that \n$\nBC\\equiv M\\not\\equiv 0\\mod p.\n$\nHence $\\mathrm{v}_p(B)=\\mathrm{v}_p(C)=0$.\n\n\\textbf{Case 1a.} If $\\mathrm{v}_p(D)=0$ then \\eqref{p_1_mod4} and \\eqref{hilbert_cond} imply $-BD\\in\\mathbb{Z}_p^{\\times, 2}$.\nChoose $r\\in\\mathbb{Z}_p^{\\times}$ such that $r\\sqrt{-BD}\\notin\\mathbb{Z}_p^{\\times, 2}$. Consider $y_2=\\frac{1}{2}\\left(\\frac{BD}{r}-r\n\\right)$ and note that\n\\begin{equation*}\n-\\frac{DM}{C}\\equiv -BD\\mod p\n\\text{\\quad and\\quad}\n\\left(\\frac{A}{M}y_2^2+B\\right)\\left(\\frac{C}{M}y_2^2+D\\right)\\equiv y_2^2 + BD\\equiv \\frac{1}{4}\\left(r+\\frac{BD}{r}\\right)^2 \\not\\equiv 0\\mod p.\n\\end{equation*}\nThen take \n$$\nP_1=\\left(-\\frac DC \\colon 1 \\colon 0\\colon \\sqrt{-\\dfrac{DM}C}\\colon 0\\right) \\; \\text{ and } \\; P_2=\\left(\\frac{y_2^2}{M}\\colon 1\\colon 0\\colon y_2\\colon \\sqrt{\\left(\\frac{A}{M}y_2^2+B\\right)\\left(\\frac{C}{M}y_2^2+D\\right)}\\right)\\in X(\\mathbb{Q}_p).\n$$\n\n\\textbf{Case 1b.} If $\\mathrm{v}_p(D)\\geq 1$ then take $y_1\\in\\mathbb{Z}_p^{\\times, 2}$ and $y_2\\notin \\mathbb{Z}_p^{\\times, 2}$.\nThen\n\\begin{equation}\n\\left(\\frac{A}{M}y_i^2+B\\right)\\left(\\frac{C}{M}y_i^2+D\\right)\\equiv \\frac{BC}{M}y_i^2\\equiv y_i^2\\mod p,\n\\end{equation}\nso we can choose $z_i\\in\\mathbb{Z}_p$ such that $z_i\\equiv -y_i\\mod p$ and $z_i^2=\\left(\\frac{A}{M}y_i^2+B\\right)\\left(\\frac{C}{M}y_i^2+D\\right)$.\nNow take \n$$\nP_i=\\left(1\\colon \\frac{y_i^2}{M}\\colon 0\\colon y_i\\colon z_i\\right)\\in X(\\mathbb{Q}_p).\n$$\n\n\n\n\\noindent\\textbf{Case 2.} In this case we have $p \\mathrel{\\|} M$ and $p \\mathrel{\\|} A$ and $p^2 \\nmid B,C,D$. From \\eqref{gcd_cond_1} we get $\\mathrm{v}_p(C)=0$, so condition (C1) implies $p \\mid B$. From $p^2 \\nmid B$, \\eqref{gcd_cond_1} and (C1) we conclude in order that $\\mathrm{v}_p(B)=1$, $\\mathrm{v}_p(D)=0$ and $\\mathrm{v}_p(N)\\ge 1$.\n\nIntroduce $A=pA'$, $B=pB'$, $M=pM'$ and $N=pN'$. Then (C1) becomes $(A'D+B'C-M')^2-4A'B'CD=pN'^2$ and we conclude $\\mathrm{v}_p(A'D+B'C-M)=0$. \nOur system is equivalent to\n$$\n\\begin{cases}\n-x^2+\\dfrac{y^2}{p}=M'uv;\\\\\n\\dfrac{4A'C}{p}(z^2-y^2)=\\bigl(2A'Cu+(B'C+A'D-M')v\\bigr)^2-pN'^2v^2.\n\\end{cases}\n$$\n\\indent\\textbf{Case 2a.} Suppose that $M'\\cdot\\frac{B'C+A'D-M'}{2A'C}\\in\\mathbb{Z}_p^{\\times 2}$. \nChoose $r_1\\in\\mathbb{Z}_p^{\\times 2}$ and $r_2\\not\\in\\mathbb{Z}_p^{\\times 2}$ and define \n$$\ny_i=\\frac{1}{2}\\left(\\frac{A'C}{r_i}-r_i\\right)N',\\quad z_i=\\frac{1}{2}\\left(\\frac{A'C}{r_i}+r_i\\right)N'\\quad \\text{ and } \\quad x_i=\\sqrt{M'(B'C+A'D-M')\\cdot 2A'C+py_i^2}.\n$$ \nNow we can just take\n$\nP_i=(-(B'C+A'D-M')\\colon 2A'C\\colon x_i\\colon py_i\\colon pz_i)\\in X(\\mathbb{Q}_p).\n$\n\n\\textbf{Case 2b.} Now suppose that $M'\\cdot\\frac{B'C+A'D-M'}{2A'C}\\notin\\mathbb{Z}_p^{\\times 2}$. Let $(u:v:x:y:z)\\in X(\\mathbb{Q}_p)$ be represented by a primitive tuple in $\\mathbb{Z}_p$. Obviously, $\\mathrm{v}_p(y),\\mathrm{v}_p(z)\\ge 1$ so write $y=py_1$ and $z=pz_1$. Then \n$$\n(2A'Cu+(B'C+A'D-M')v)^2-pN'^2v^2=4A'Cp(z_1^2-y_1^2),\n$$ \nhence $u\\equiv -\\frac{B'C+A'D-M'}{2A'C} v\\mod p$. This turns the first equation into $M'\\cdot\\frac{B'C+A'D-M'}{2A'C} \nv^2\\equiv x^2\\mod p$. It is possible only if $u\\equiv v\\equiv x\\equiv 0\\mod p$. This contradicts the assumption $X(\\mathbb{Q}_p) \\ne \\emptyset$.\n\\bigskip\n\n\\noindent\\textbf{Surjectivity for $\\mathcal{A}$.}\nIn Cases 3 and 4 we apply the same technique for proving that $\\inv_p \\mathcal A$ is surjective.\n\n\\noindent\\textbf{Case 3.} By assumption $p \\nmid MABCD$. Since $(p,AC)_p=(p,BD)_p=1$, we have $AC,BD\\in \\mathbb{Z}_p^{\\times, 2}$.\n\n\n\\textbf{Case 3a.} If $ABM\\notin \\mathbb{Z}_p^{\\times, 2}$ then $\\mathcal A$ has different invariants at the points\n$$\nP_1=(1\\colon 0\\colon 0\\colon 0\\colon \\sqrt{AC})\\in X(\\mathbb{Q}_p)\\quad \\text{ and } \\quad P_2=(0\\colon 1\\colon 0\\colon 0\\colon \\sqrt{BD})\\in X(\\mathbb{Q}_p).\n$$\nas one can see using \\eqref{Br_gr_rewr}.\n\n\n\\textbf{Case 3b.} Now let $ABM\\in\\mathbb{Z}_p^{\\times, 2}$. It follows from Lemma \\eqref{lem:quadres_table} that there exists a $y_0\\in \\mathbb{Z}_p$ such that $1+\\frac{B}{MA}y_0^2$, $\\frac{C}{A}+\\frac{D}{MA}y_0^2\\not\\in\\mathbb{Z}_p^{\\times, 2}$, or $\\frac{C}{A}+\\frac{D}{MA}y_0^2 \\equiv 0 \\mod p$ and $1+\\frac{B}{MA}y_0^2\\not\\in\\mathbb{Z}_p^{\\times, 2}$. In the first case we consider\n$$\nP_1=(1\\colon 0\\colon 0\\colon 0\\colon \\sqrt{AC})\\quad \\text{ and } \\quad\nP_2=\\left(1\\colon \\frac{y_0^2}{M}\\colon 0\\colon y_0\\colon A\\sqrt{\\left(1+\\frac{B}{MA}y_0^2\\right)\\left(\\frac{C}{A}+\\frac{D}{MA}y_0^2\\right)}\\right)\n\\in X(\\mathbb{Q}_p).\n$$\nIn the second case we choose a $y_1\\in\\mathbb{Z}_p$ such that $y_1\\equiv y_0\\mod p$ and $\\frac{C}{A}+\\frac{D}{MA}y_1^2=0$. We then work with\n$$\nP_1=(1\\colon 0\\colon 0\\colon 0\\colon \\sqrt{AC})\\quad \\text{ and } \\quad  P_2=\\left(1\\colon \\frac{y_1^2}{M}\\colon 0\\colon y_1\\colon 0\\right)\\in X(\\mathbb{Q}_p).\n$$\n\n\\noindent\\textbf{Case 4.} Let $M=p^{2k+1}M'$ with $k\\geq 0$ and $M' \\in \\mathbb Z_p^\\times$. From (C1) we obtain that $AD\\equiv BC\\mod p$. Lemma~\\ref{lem:quadres}(b) for $S\\left(1,\\frac{B}{M'A}\\right)$ implies the existence of an $x_0\\in\\mathbb{Z}_p^\\times$ such that $1-\\frac{B}{M'A}x_0^2\\in\\mathbb{Z}_p^\\times\\setminus\\mathbb{Z}_p^{\\times 2}$. Then \n$$\n1-\\frac{D}{M'C}\nx_0^2\\equiv 1-\\frac{B}{M'A}x_0^2\\mod p,\n$$ \nso we can take \n$$\nP_1=(1:0:0:0:\\sqrt{AC})\\, \\text{ and }\\, P_2=\\left(1:-\\frac{x_0^2}{M'}:p^kx_0:0:\\sqrt{AC\\left(1-\\frac{Bx_0^2}{M'A}\\right)\\left(1-\\frac{Dx_0^2}{M'C}\\right)+p^{2k+1}x_0^2}\\right).\n$$\n\n\\noindent\\textbf{Case 5.} We have $p\\mathrel{\\|} M$ and $p^2 \\mid A$. Conditions (C1) and \\eqref{gcd_cond_1} imply $p\\nmid C$, $p \\mid B$, $p \\nmid D$ and $p \\mid N$. Then \\eqref{sys} is isomorphic to the system with coefficients $(p^{-2}A,p^{-1}B,C,pD,p^{-1}M,p^{-1}N)$ under the morphism $(u\\colon v \\colon x \\colon y \\colon z) \\mapsto (pu\\colon v \\colon x\\colon y \\colon z)$. For the new system we have $\\mathrm{v}_p(M)=0$ and $m\\geq 1$, which was considered in Case 1.\n\n\\noindent\\textbf{Case 6.} From $p^{2k+1} \\mathrel{\\|} M$ and $p\\mid A$ we get, as in the previous case, $p \\mid B,N$ and $p \\nmid C,D$. Considering (C1) modulo $p^{2k+1}$ we get $(AD-BC)^2 \\equiv pN^2 \\mod p^{2k+1}$. From this we conclude that $p^{k+1} \\mid AD-BC$ and $p^k \\mid N$. In particular, $p^2 \\mid A$ if and only if $p^2 \\mid B$. So we conclude $p\\mathrel{\\|} A,B$ from \\eqref{gcd_cond_2}. This implies $2\\mathrm{v}_p(AD-BC+M) = \\mathrm{v}_p(pN^2-4ABCD) = 2$ and hence $\\mathrm{v}_p(AD+BC)=1$. From this we deduce that\n\\[\n2\\mathrm{v}_p(AD-BC)=\\mathrm{v}_p(pN^2 +2(AD+BC)M-M^2) = 2k+2.\n\\]\nHence $p^{k+1} \\mathrel{\\|} AD-BC$.\n\nOur system of equations is equivalent to \\eqref{sys} with the coefficients\n\\[\n(p^{-2k}MD,-p^{-2k-1}MB,-C,p^{-1}A,p^{-2k-1}(AD-BC)^2)\n\\]\nunder the morphism $(u\\colon v \\colon x \\colon y \\colon z) \\mapsto (\\frac{Au+Bv}{AD-BC} \\colon p\\frac{Cu+Dv}{AD-BC} \\colon p^{-k}x \\colon p^{-k}y \\colon p^{-k}z)$. This new system satisfied $\\mathrm{v}_p(M)=1$ and $m=1$, which was discussed in Case 2.\n\n\\noindent\\textbf{Case 7.} Now we know that $p^{2k} \\mathrel{\\|} M$ and $p\\nmid ABCD$. Condition (C1) implies $AD\\equiv BC\\mod p$, so $\\mathrm{v}_p(AD+BC)=0$. From $\\mathrm{v}_p(M)=2k$ and (C1) we obtain that $\\mathrm{v}_p(AD-BC)=k$. Now consider the coefficients\n\\[\n(p^{-2k}MD,-p^{-2k}MB,-C,A,p^{-2k}(AD-BC)^2)\n\\]\nwith variables $(\\frac{Au+Bv}{AD-BC}\\colon \\frac{Cu+Dv}{AD-BC}\\colon p^{-k}x\\colon p^{-k}y\\colon p^{-k}z)$ for the system \\eqref{sys}. This is precisely Case 3.\n\n\\noindent\\textbf{Case 8.} Now we assume $p^{2k} \\mathrel{\\|} M$ and $m=1$. Without loss of generality $p \\mathrel{\\|} A$ and as in the previous cases we can prove $p \\mathrel{\\|} B$, $p\\nmid CD$, $\\mathrm{v}_p(AD+BC)=1$ and $\\mathrm{v}_p(AD-BC)\\ge k+1$. The system \\eqref{sys} with coefficients $(p^{-2k}MD,-p^{-2k-1}MB,-C,p^{-1}A,p^{-2k-1}(AD-BC)^2)$ with variables $(\\frac{Au+Bv}{AD-BC}\\colon p\\frac{Cu+Dv}{AD-BC}\\colon p^{-k}x\\colon p^{-k}y\\colon p^{-k}z)$ is isomorphic to our system. This new system satisfies $\\mathrm{v}_p(M)$ is odd and $m=0$, which is precisely Case~4.\n\\end{proof}\n\n\\section{Explicit examples}\n\nAll known examples of Brauer--Manin obstructions for quartic del Pezzo surfaces occur on surfaces with a Brauer group modulo constants of order $2$. The first such obstruction was described by Birch and Swinnerton-Dyer in \\cite{BSD}. Work by Jahnel and Schindler \\cite{JS} showed that the locus of quartic del Pezzo surfaces with a Brauer group of order $2$ for which the Brauer--Manin obstruction is the only one, is dense in the moduli space of all del Pezzo surfaces of degree $4$.\n\nMitankin and Salgado \\cite{MS} restricted to the subfamily in which a positive proportion has a Brauer group of order $4$. Such surfaces can be written as\n$$\n\\begin{cases}\nx_0x_1-x_2x_3=0;\\\\\na_0x_0^2+a_1x_1^2+a_2x_2^2+a_3x_3^2+a_4x_4^2=0,\n\\end{cases}\n$$\nwhere $a_0a_1,a_2a_3,-a_0a_2\\in\\mathbb{Q}^{\\times,2}$ and $-a_0a_4(a_0a_1-a_2a_3)\\notin\\mathbb{Q}^{\\times,2}$. The Hasse principle, however, trivially holds for such surfaces, because of the rational point $(1:0:\\sqrt{-a_0\/a_2}:0:0)$.\n\nIn this section we will exhibit families of quartic del Pezzo surfaces with a Brauer group of order $4$ with a Brauer--Manin obstruction to the Hasse principle.\n\n\\subsection{Obstructions coming from $\\mathcal A$}\n\nLet us define our first subfamily. Consider a prime number $p\\equiv 1\\mod 4$ and integers $a,b$ such that $ab=p-1$. The surface $Y_{p,a,b}$ is given by the system\n$$\n\\begin{cases}\ny^2-px^2=uv;\\\\\nz^2-px^2=(au-pv)(u-bv).\n\\end{cases}\n$$\nObviously, $Y_{p,a,b}=X_{p,a,-p,1,-b,1,1}$, and one can check that conditions (C1) and (C2) are satisfied with $N=1$.\n\nFor such surfaces we have $\\mathcal{A}=\\left(p,\\frac{u}{au-pv}\\right)$, $\\mathcal{B}=\\left(p,\\frac{z-y}{u}\\right)$ and $\\mathcal{C}=\\left(p,\\frac{au-pv}{z-y}\\right)$.\n\n\\begin{pro}\\label{pro:ex1_inv_B}\nConsider the surface $Y=Y_{p,a,b}$.\n\\begin{enumerate}\n\\item[(a)] The variety $Y$ is everywhere locally soluble.\n\\item[(b)] The invariant map of $\\inv_p\\mathcal{B} \\colon Y(\\mathbb A_\\mathbb{Q}) \\to \\frac12\\mathbb{Z}\/\\mathbb{Z}$ is surjective.\n\\end{enumerate}\n\\end{pro}\n\n\\begin{proof}\nLocal solubility at all places $v\\not\\in \\{2,p\\}$ has already been proven in Proposition \\ref{pro:loc_sol_subfam}. Local solubility at $p$ and the surjectivity of the invariant map of $\\mathcal{B}$ at $p$ follow from Case 1 in the proof of Proposition~\\ref{prop:surjectiveinvariantmap}. For local solubility at $2$ one can simply consider all possibilities of $a$, $b$ and $p$ modulo $8$.\n\\end{proof}\n\nSo we might expect an obstruction coming from $\\mathcal{A}$.\n\n\\begin{pro}\\label{pro:ex1_inv_A}\nFor all $(s_v)_v\\in Y_{p,a,b}(\\mathbb{A}_\\mathbb{Q})$ we have: $\\inv_v \\mathcal{A}(s_v)=0$ for $v\\neq p$, and:\n$$\n\\inv_p \\mathcal{A}(s_p)=\\begin{cases}\n0 & \\text{ if }\\left(\\dfrac{a}{p}\\right)=1;\\\\\n\\dfrac{1}{2} & \\text{ if }\\left(\\dfrac{a}{p}\\right)=-1.\n\\end{cases}\n$$\n\\end{pro}\n\n\\begin{proof}\nOne can prove this by direct computation. However, we will apply results from \\cite{Beffeval} and \\cite{CTS13} to conclude that $\\inv_v$ is constant for $v \\ne p$ and compute it at a single point in $X(\\mathbb{Q}_v)$ to conclude $\\inv_v$ is identically zero.\n\nFinally, consider the case $v=p$. Let $s_p=(u\\colon v\\colon x\\colon y\\colon z)$ with $(u,v,x,y,z)$ a primitive tuple in $\\mathbb{Z}^5_p$. Suppose $p \\mid u$, then immediately $p \\mid y,z$. Denote $u=pu'$,  then \n$$\nu'v\\equiv -x^2\\equiv (au'-v)(pu'-bv)\\equiv u'v+bv^2\\mod p\n$$\nwhich means that $p \\mid v$ and then $p \\mid x$, which contradicts the primitivity of $(u,v,x,y,z)$. Thus, $\n\\mathrm{v}_p(u)=0$, so $\\left(p,\\frac{u}{au-pv}\\right)_p=\\left(\\frac{u(au-pv)}{p}\\right)=\\left(\\frac{a}{p}\\right)$. This proves the proposition \nin the case $v=p$.\n\\end{proof}\n\n\nWe can now compute the Brauer--Manin obstruction coming from $\\mathcal{A}$.\n\n\\begin{thm}\\label{thm_ex1}\nConsider the surface $Y_{p,a,b}$. Note that from $ab=p-1$ and $p\\equiv 1\\mod 4$ we conclude $(\\frac{a}{p})=(\\frac{b}{p})$. \nAlso, $(\\frac{a}{p})=(\\frac{b}{p})=-1$ holds if and only if $p\\equiv 5\\mod 8$, and $a$ and $b$ are both even.\n\\begin{enumerate}\n\\item[(a)] If $(\\frac{a}{p})=(\\frac{b}{p})=-1$ then $Y_{p,a,b}(\\mathbb{Q})=\\emptyset$ and this failure of the Hasse principle is \nexplained by the Brauer--Manin obstruction coming from $\\mathcal{A}$.\n\\item[(b)] Otherwise, we have\n\\[\nY_{p,a,b}(\\mathbb A_\\mathbb{Q})^{\\Br X} \\ne \\emptyset.\n\\]\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\nThe first observation comes from the following facts: for $p\\equiv 1\\mod 4$ we have $(\\frac{-1}{p})=1$ and $(\\frac{q}{p})=1$ for odd prime divisors $q$ of $p-1$. Also, $(\\frac{2}{p})=1$ if $p\\equiv 1\\mod 8$ and $(\\frac{2}{p})=-1$ if $p\\equiv 5\\mod 8$.\n\nIf $(\\frac{a}{p})=(\\frac{b}{p})=-1$ then Proposition \\ref{pro:ex1_inv_A} implies that $\\sum_v\\inv_v\\mathcal{A}(s_v)=\\frac{1}{2}$ for all $(s_v)_v\\in Y_{p,a,b}(\\mathbb{A}_\\mathbb{Q})$, so $Y_{p,a,b}(\\mathbb{A}_\\mathbb{Q})^\\mathcal{A}=\\emptyset$.\n\nIf $(\\frac{a}{p})=(\\frac{b}{p})=1$ then Proposition \\ref{pro:ex1_inv_A} and Proposition \\ref{pro:ex1_inv_B} imply that \n$\\sum_v\\inv_v\\mathcal{A}(s_v)=0$ for all ${(s_v)_v\\in Y_{p,a,b}(\\mathbb{A}_\\mathbb{Q})}$, and invariant maps of $\\mathcal{B}$ and $\\mathcal{C}$ at $p$ are \nsurjective. Thus, $Y_{p,a,b}(\\mathbb{A}_\\mathbb{Q})^{\\Br}\\neq\\emptyset$.\n\\end{proof}\n\n\n\n\n\\begin{exa}\nConsider a surface $Y_{13,2,6}$ given by\n$$\n\\begin{cases}\ny^2-13x^2=uv;\\\\\nz^2-13x^2=(2u-13v)(u-6v)\n\\end{cases}\n$$\nTheorem \\ref{thm_ex1} shows that $Y_{13,2,6}(\\mathbb{Q})=\\emptyset$.\n\\end{exa}\n\n\\begin{exa}\nConsider a surface $Y_{13,1,12}$ given by\n$$\n\\begin{cases}\ny^2-13x^2=uv;\\\\\nz^2-13x^2=(u-13v)(u-12v)\n\\end{cases}\n$$\nTheorem \\ref{thm_ex1} shows that there is no Brauer--Manin obstruction for $Y_{13,1,12}$ to the Hasse principle, and it has a trivial rational point $(1:0:0:0:1)\\in Y_{13,1,12}(\\mathbb{Q})$.\n\\end{exa}\n\n\n\\begin{exa}\nConsider a surface $Y_{13,12,1}$ given by\n$$\n\\begin{cases}\ny^2-13x^2=uv;\\\\\nz^2-13x^2=(12u-13v)(u-v)\n\\end{cases}\n$$\nTheorem \\ref{thm_ex1} shows that there is no Brauer--Manin obstruction for $Y_{13,12,1}$, and it has a rational point $(1:-3:2:7:16)\\in Y_{13,12,1}(\\mathbb{Q})$ which is actually non-trivial to find.\n\\end{exa}\n\n\\begin{rema}\nNote that we can change $a$ and $b$ up to squares without changing the isomorphism class of the surface. Unless $(a,b)$ equals either $(1,p-1)$ or $(-1,-(p-1))$, up to squares, there does not seem to be an obvious rational point. However, for each explicit surface we studied we were always able to find such a point.\n\nThe machinery from \\cite{CTSSD} and \\cite{WittenbergBook} seems ideally suited to proving such a statement by making $X$ into an elliptic fibration using the projection $f \\colon X \\dashrightarrow \\mathbb P^2$ away from a plane studied in the beginning of this paper. Unfortunately the ``condition (D)'' in those works is not satisfied, which relates to the fact that the Brauer group of $X$ is not vertical.\n\\end{rema}\n\n\\begin{rema}\nAlso, note that the technique described above would not just establish the existence of a rational point, but even prove that these are Zariski dense. We already know these two statements to be equivalent by work of Salberger and Skorobogatov \\cite{SalbergerSkorobogatov91}.\n\\end{rema}\n\n\\subsection{Obstructions coming from $\\mathcal B$}\n\nWe will now consider a second subfamily for which $\\mathcal B$ might give an obstruction.\n\n\\begin{defi}\\label{def_ex_B}\nLet $p$ be an odd prime, and $a$ and $b$ integers such that\n\\begin{enumerate}\n\\item $(a+b-1)^2-4ab=p$,\n\\item $4a\\equiv 4b\\equiv 1\\mod p$,\n\\item $a\\equiv b\\equiv 1\\mod 2$, and\n\\item $a\\equiv 1\\mod 8$.\n\\end{enumerate}\nLet $S_{p,a,b}$ be the surface given by\n\\begin{equation}\\label{sys_ex_B}\n\\begin{cases}\ny^2-px^2=uv;\\\\\nz^2-px^2=(u+v)(au+bv).\n\\end{cases}\n\\end{equation}\n\\end{defi}\n\nSuch triples $(p,a,b)$ do exist, as one can see from considering $p \\equiv 5 \\mod 8$, an integer $t \\equiv 3\\frac{p-1}{4}\\mod 8$ and\n\\[\na_t=t^2p^2-tp-\\frac{p-1}{4} \\quad \\text{ and } \\quad  b_t=t^2p^2+tp-\\frac{p-1}{4}.\n\\]\n\nOne can also check that $S_{p,a,b}=X_{p,1,1,a,b,1,1}$ satisfies conditions (C1) and (C2).\n\n\\begin{pro}\n\\\n\\begin{enumerate}\n\\item[(a)] All $S_{p,a,b}$ are everywhere locally soluble.\n\\item[(b)] For all $S_{p,a,b}$ the invariant map of $\\mathcal{A}$ at $p$ is surjective.\n\\end{enumerate}\n\\end{pro}\n\n\\begin{proof}\nProposition \\ref{pro:loc_sol_subfam} addresses the local solubility at all places $v\\not\\in\\{2,p\\}$. Case 3 in the proof of Proposition~\\ref{prop:surjectiveinvariantmap} establishes the local solubility at $p$ and the surjectivity of the invariant map of $\\mathcal{A}$ at $p$. The local solubility at $2$ follows from the point $(1:0:0:0:\\sqrt{a})\\in X(\\mathbb{Q}_2)$, since $a\\equiv 1\\mod 8$.\n\\end{proof}\n\nWe will now consider the obstruction coming from $\\mathcal{B}$.\n\n\\begin{thm}\\label{thm_ex_B}\nWe have $S_{p,a,b}(\\mathbb A_\\mathbb{Q})^\\mathcal{B}=\\emptyset$ and in particular $S_{p,a,b}(\\mathbb{Q})=\\emptyset$.\n\\end{thm}\n\n\\begin{proof}\nWe can prove that $\\inv_v$ is identically zero for $v \\ne p$ as we did in the proof of Proposition~\\ref{pro:ex1_inv_A}.\n\nFinally, consider the case $v=p$. Let $s_p=(u:v:x:y:z)$ such that $(u,v,x,y,z)$ is primitive in $\\mathbb{Z}_p$, hence in particular $\\mathrm{v}_p(u)\\mathrm{v}_p(v)=0$. Assume $\\mathrm{v}_p(u)=0$; the other case is similar. The system \\eqref{sys_ex_B} together with the condition $4a\\equiv 4b\\equiv 1\\mod p$ gives us\n$$\nv\\equiv \\frac{y^2}{u}\\mod p\\quad \\text{ and } \\quad z^2\\equiv \\frac{1}{4}(u+v)^2\\mod p.\n$$\nLet us write $z \\equiv \\frac{\\varepsilon}2(u+v) \\mod p$ with $\\varepsilon = \\pm 1$. Then we get\n\\begin{align*}\nz\\pm y \\equiv \\frac{\\varepsilon}2(u+v) \\pm y  & \\equiv \\frac{\\varepsilon}2 \\frac{y^2}u \\pm y + \\frac{\\varepsilon}2u\\\\\n & \\equiv \\frac{\\varepsilon}2u\\left(\\frac{y^2}{u^2} \\pm 2 \\varepsilon \\frac yu + 1 \\right)\\\\\n& \\equiv \\frac{\\varepsilon}2u\\left(\\frac yu \\pm \\varepsilon\\right)^2 \\mod p.\n\\end{align*}\n\nSo at least one of $\\eta = \\frac u{z\\pm y}$ is non-zero modulo $p$ at $s_p$, and we have $(p,\\eta)=-1$ since $(\\frac{2}{p})=-1$. We conclude $\\inv_p\\mathcal{B}(s_p)=\\frac{1}{2}$.\n\\end{proof}\n\n\n\n\n\\begin{exa}\nConsider $S_{13,153,179}$ given by\n$$\n\\begin{cases}\ny^2-13x^2=uv;\\\\\nz^2-13x^2=(u+v)(153u+179v).\n\\end{cases}\n$$\nTheorem \\ref{thm_ex_B} shows that $S_{13,153,179}(\\mathbb{Q})=\\emptyset$ and hence this surface does not satisfy the Hasse principle.\n\\end{exa}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the last decade, hybrid organic-inorganic materials with the perovskite structure, most notably CH$_3$NH$_3$Pb(I,Cl)$_3$, have become the most promising light-harvesting active layer for the implementation of high efficiency and low cost solar cells \\cite{Graetzel_2014,Hodes_2013,Xiao_2016,Lee_2012,Heo_2013,You_2014,Zhou_2014,Nanni_2014}. As a matter of fact, the power conversion efficiency of perovskite solar cells (PSCs) has shown a very fast growth, increasing from 3.8\\% \\cite{Kojima_2009} in 2009 to 22.1\\%  \\cite{NREL_eff_chart} in 2016. The ease of fabrication and the high radiative efficiency have made perovskites also extremely attractive for the development of bright LEDs and lasers \\cite{Veldhuis_2016,Sutherland_2016,Palma_2016,Palma_2016_2}.\n\nNevertheless, a major drawback originates from the poor material stability\\cite{Xu_2016,Xiao_2016}: decomposition after exposure to moisture, cracks and defects generated by thermal stresses, crystal phase transition, UV-light exposure represent the principal causes of aging\\cite{Xu_2016,Xiao_2016}.\nMoreover, a high density of trap states, both on surfaces and in grain boundaries \\cite{Xiao_2016,Draguta_2016,Wu_2015,Kim_2014}, are present in polycrystalline perovskites, used for solar cells and light emitters. \nIndeed, even though theoretical predictions show that deep trap states are not generally formed inside perovskites grains, the opposite is observed at the grain boundaries and at the surfaces\\cite{Xiao_2016,Park_2016_book}.\nTherefore, sophisticated passivation strategies are essential for increasing the efficiency of PSCs and light emitters \\cite{Son_2016,Giordano_2016,deQuilettes_2016,Park_2016_book}.\n\nFor CH$_3$NH$_3$PbI$_3$ (MAPI) perovskites, a correlation between the solar cell efficiency and the grain morphology was recently demonstrated \\cite{Shao_2016}. Results of local short-circuit photocurrent, open-circuit voltage and dark drift current within individual grains correlate these quantities to different crystal facets, as a consequence of a facet-dependent density of trap states \\cite{Leblebici_2016} and it was also proven that the structural order of the electron transport layer (ETL) impacts the overall cell performance \\cite{Shao_2016}. Moreover, the nature of grain boundaries was shown to affect the carrier recombination kinetics because of non-radiative pathways that would also play a role in the process of charge separation and collection \\cite{deQuilettes_2016}.\n\n\n\nIt turns out that the possibility of controlling the morphology of the perovskite thin film and the understanding if different realizations of the ETL can modify the morphology of the grains are of the utmost relevance to further improvements of perovskite based solar cells.\n\nRecently, graphene and related two-dimensional materials have been introduced in the device structure in order to improve the charge injection and\/or collection at the electrodes: an enhancement of the power conversion efficiency\\cite{Acik_2016} and a long-term stability\\cite{Agresti_2016} was obtained.\nAs a matter of fact, interfaces between perovskite and transport layers have been recently demonstrated to dramatically affect the charge recombination processes and material instability within the working device.\\cite{Capasso_2016}\nIn fact, when free charges are fast injected from perovskite to the electron transport layer, the perovskite degradation is slowed down and the non-radiative recombination is reduced\\cite{Ahn_2016}. \nIn particular, the insertion of graphene flakes into the mesoporous-TiO$_2$ layer (mTiO$_2$) and of lithium-neutralized graphene oxide (GO-Li) as interlayer at perovskite\/mTiO$_2$ interface showed enhanced conversion efficiency and stability on both small and large area devices by demonstrating the crucial role of graphene interface engineering in perovskite-based devices\\cite{Agresti_2017}. Thus, the influence of mesoscopic-graphene modified substrates onto the perovskite film need to be investigated more in details to finely control the photovoltaic performance of complete devices.\n\nGiven the typical size of the grains (a few hundreds of nanometers), high resolution techniques as TEM (Transmission Electron Microscopy), AFM (Atomic-Force Microscopy), SNOM (Scanning Near-field Optical Microscopy), etc.~are employed to investigate the grain morphology but a difficult task is to correlate physical properties at the nanoscale with the device performance measuring, for instance, the $I$-$V$ curve of the cell\\cite{Leblebici_2016}.\nThus, it would be of  extreme relevance to extract information on the active film morphology with much easier techniques on a length scale of tens\/hundreds of microns and even larger so to assess the homogeneity of the film deposition and the reliability of the synthesis protocol and post-deposition treatments. Moreover, the high spatial resolution analysis can explore a limited thickness of the film, and therefore it appears difficult to get information in the case of a real device.\nPhotoluminescence (PL) spectroscopy is an effective tool to investigate the film quality: in fact, from the comparison of samples with different ETLs, it is possible to extract quantitative information on the carrier capture and transport and, by the spectral shape, identify the crystalline phase of the active layer and evaluate the density of traps\/defects.\nMoreover, given the limited thickness (few hundreds of nm) of the perovskite film in solar cells and given the steep behavior of the absorption coefficient in MAPI\\cite{Loper_2015}, by varying the excitation wavelength in the range 300-700\\,nm, it is possible to probe thicknesses of the film from a hundred of nm to the whole film layer.\n\nIn this paper we aim to establish the effects of different graphene-based\nETLs in sensitized MAPI solar cells. In particular we will study the ETL\neffects on the carrier collection efficiency and on the MAPI morphology along\nthe thickness.\nBy picosecond time-resolved measurements we correlate the carrier recombination dynamics to the crystalline quality of the active material in presence\/absence of the ETL. We will find an increase of the electron collection efficiency up to a factor 3 with respect to standard mTiO$_2$.\nTaking advantage of the absorption coefficient dispersion, we are able to assess the film morphology along the thickness. In fact, by tuning the excitation wavelength, we investigate a thickness range from 150 to 400\\,nm and we can highlight the morphology changes induced by different ETLs.\nOur results will indicate that, when a graphene doped mesoporous TiO$_2$ (G+mTiO$_2$) with the addition of a GO-Li interlayer is used as ETL, the morphology of the MAPI film embedded in the mesoporous layer is frozen in the tetragonal phase, regardless of the temperature. In addition, the defect concentration is about one order of magnitude lower than that found with the other ETLs.\n\n\n\n\n\n\n\n\n\\section{Results and discussion}\n\nFour types of samples were prepared using different combinations of ETLs: mTiO$_2$, G+mTiO$_2$, mTiO$_2$ plus GO-Li interlayer and G+mTiO$_2$ plus GO-Li interlayer. \nThe different sample structures are schematically shown in Fig.\\,\\ref{Fig1} and listed in Tab.\\,\\ref{Tab1}. For clear comparison a sample of MAPI on FTO without ETL (reference sample) was also investigated. \nDetails on the sample preparation are reported in the Experimental Section.\nIt is worth noting that PL analysis was carried out on simple photo-electrodes, lacking of the hole collecting layer and the bottom contact: this allows us to focus only on the electron collection and transport after the electron-hole pairs creation due to the photon absorption. Moreover, PL experiments can be realized illuminating the samples either from the perovskite film side (side A) or the FTO side (side B). Varying the excitation wavelength and the excitation side, we can differently penetrate into the MAPI film and selectively probe spatial regions few tens of nm near the ETL or far from it.\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{Fig1.pdf}\n\\caption{Structures of the investigated samples. The ETLs are indicated in red.}\n\\label{Fig1}\n\\end{figure}\n\n\n\\begin{table}[b]\n\\caption{List of the investigated samples and description of their corresponding ETLs.}\n\\label{Tab1}\n\\addtolength{\\tabcolsep}{3.8mm}\n\\begin{tabular*}{\\columnwidth}{l@{\\hspace{22mm}}l}\n\\hline\nSample    & ETL  \\\\\n\\hline\nReference & No ETL       \\\\\nETL 1         & mTiO$_2$   \\\\\nETL 2         & G+mTiO$_2$  \\\\\nETL 3         & mTiO$_2$ plus GO-Li\\\\\nETL 4         & G+mTiO$_2$ plus GO-Li \\\\\n\\hline\n\\end{tabular*}\n\\end{table}\n\n\nIn Fig.\\,\\ref{Fig2} PL decays at room temperature, after excitation with photons of 2.06\\,eV, are compared for the different samples, while in the inset a PL spectrum at room temperature is shown: typical spectra of the tetragonal phase are observed\\cite{Milot_2015,Dar_2016}. \nBy exciting from side A (Fig.\\,\\ref{Fig2}a), the PL decays are identical for all samples and no effect related to the presence of the ETL is detected. \nIn contrast there is a significant difference in the PL decay by exciting from side B depending on the sample, as shown in Fig.\\,\\ref{Fig2}b: faster decays are observed in presence of ETL, especially when graphene and\/or GO-Li are used.\nIt is worth noting that the PL decay of reference sample excited from side B is slower with respect to the decays of side A. This can be attributed to non-radiative states at the surface of the uncovered perovskite film (side A).\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{Fig2.pdf}\n\\caption{PL decay (at the PL peak energy) and PL spectra (in the inset) at room temperature, after ps excitation at 2.06\\,eV with an average intensity of 10\\,W\/cm$^2$. a) From side A. b) From side B.}\n\\label{Fig2}\n\\end{figure}\n\nTo extract information about the efficiency in the carrier injection from perovskite to ETL, we first fitted the decays of Fig.\\,\\ref{Fig2}b with a double exponential\\cite{Son_2016,Bi_2016}, taking into account the laser pulse repetition period \\cite{Warren_2013}. The fitting function $I(t)$ can be written as\n\\begin{equation}\nI(t) = f(t)+g(t)\n\\end{equation}\nwhere $f(t)$ is the original double exponential and $g(t)$ the correction term due to the intrinsic periodic nature of TCSCP measurements\\cite{Warren_2013}:\n\\begin{equation}\nf(t)=\\Theta(t-t_0)[Ce^{-(t-t_0)\/\\tau_1}+ \n(1-C)e^{-(t-t_0)\/\\tau_2}]\n\\end{equation}\n\\begin{equation}\ng(t)=C\\frac{e^{-(t-t_0)\/\\tau_1}}{e^{T\/\\tau_1}-1}+ \n(1-C)\\frac{e^{-(t-t_0)\/\\tau_2}}{e^{T\/\\tau_2}-1} \n\\end{equation}\nIn the previous equations $\\theta$(t) is the Heaviside function, $\\tau_1$ and $\\tau_2$ are the decay time constants, $T$ is the laser pulse repetition period (13.15\\,ns), $C$ is the contribution of $\\tau_1$ exponential to the fit and $t_0$ is a constant. \nThe results obtained by the fitting procedure are reported in Tab.\\,\\ref{Tab2}.\nInserting graphene-based ETLs in the samples reduces $\\tau_1$ from 25\\,ns to 15\\,ns while the reduction of $\\tau_2$ depends on the ETL.\nIn the literature\\cite{Son_2016,Bi_2016} the longer decay constant ($\\tau_1$) is ascribed to the radiative recombination in MAPI while the shorter decay constant can be attributed to the carrier removal from MAPI layer towards the ETL.\n\nTo get rid of the local inhomogeneities in the samples, which can give rise to variation of the TI-PL intensity and considering that the PL time evolution does not depend on the detection spot, we estimate the TI-PL intensity from the PL decay. It turns out that we can express the integrated PL intensity ($I_\\mathrm{PL}$) as\n\\begin{equation}\nI_\\mathrm{PL}=\\eta P,\n\\end{equation}\nwhere $\\eta$ is the radiative efficiency and $P$ is the pump intensity, equals to 10\\,W\/cm$^2$ for all the TR-PL measurements.\nAssuming a unitary radiative efficiency for the reference sample, we can evaluate the change in $\\eta$ for the different ETLs (see Tab.\\,\\ref{Tab2}). Lower $\\eta$ is obtained in case of ETLs 2, 3 and 4: thus the insertion of graphene and\/or GO-Li interlayer improves the electron capture from perovskite to ETL by a factor between two and three with respect to ETL 1.\nWe want to remark that this result does not necessarily imply an increase of the solar cell short circuit current density ($J_\\mathrm{sc}$) of the same amount. In fact hole collection by the hole transport layer\\cite{Pydzinska_2016} and non-radiative recombination in the ETL have to be taken into account. However, as shown in the Supporting Information, a significant increase of $J_\\mathrm{sc}$ is measured for complete cell when ETL 4 is used.\n\n\\begin{table}\n\\caption{Results of the fits of data of Fig.\\,\\ref{Fig2}b: $\\tau_1$ and $\\tau_2$ are the decay time constants, $C$ is the contribution of $\\tau_1$ to the fit and $\\eta$ is the radiative efficiency.}\n\\label{Tab2}\n\\addtolength{\\tabcolsep}{3.8mm}\n\\begin{tabular*}{\\columnwidth}{lcccc}\n\\hline\nSample    & $\\tau_1$ (ns) & $\\tau_2$ (ns) & $C$  &$\\eta$ \\\\\n\\hline\nReference & 25       & --    & 1.00 &1 \\\\\nETL 1         & 25       & 2.05   & 0.43 & 0.48 \\\\\nETL 2         & 15       & 1.99   & 0.36 & 0.27\\\\\nETL 3         & 15       & 1.24   & 0.20 &0.16\\\\\nETL 4        & 15       & 1.30   & 0.34 &0.24\\\\\n\\hline\n\\end{tabular*}\n\\end{table}\n\nMore insight in the role of the ETL can be obtained by PL spectra at low temperature ($T=11$\\,K). In Fig.\\,\\ref{Fig3} PL spectra, obtained by exciting the samples from side A (Fig.\\,\\ref{Fig3}a) and side B (Fig.\\,\\ref{Fig3}b), are reported.\nThe PL spectra from side A show two peaks. As expected for $T < 150$\\,K in MAPI perovskite \\cite{Milot_2015,Kong_2015}, the peak at about 1.65\\,eV is attributed to the orthorhombic phase of MAPI. But the major contribution to the spectra comes from the other peak, centered at 1.55\\,eV. This emission is likely due to the sum of two contributions: the radiative recombination arising from the residual tetragonal phase at 1.56\\,eV and the radiative recombination from localized states below 1.52\\,eV. In the literature, these localized states are identified with radiative traps\\cite{Wu_2015,Fang_2015} or, recently, to methylammonium-disordered domains in orthorhombic phase of MAPI\\cite{Dar_2016}.\nWe want to remark that, in both interpretations of the low energy side emission as radiative traps or disordered domains, a carrier localization is present. More relevant is the fact that the low energy states are radiative and do not produce a loss of photogenerated carriers.\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{Fig3.pdf}\n\\caption{TI-PL spectra at 11\\,K for the various samples after excitation at 2.06\\,eV  with an average intensity of 10\\,W\/cm$^2$. a) From side A. b) From side B.}\n\\label{Fig3}\n\\end{figure}\n\nConsidering that the absorption length of the MAPI film at about 2\\,eV is roughly 200\\,nm \\cite{Loper_2015} and that the thickness of the perovskite layer in our samples is about 350\\,nm (see Fig.\\,S5 in the Supporting Information), we can conclude that the emission, exciting from side A, comes mostly from the MAPI film and that no effect related to the presence of the ETL is detected. \nOn the contrary, the  excitation from side B can reveal the ETL effect on the MAPI. As a matter of fact, relevant differences are observed between the emissions from side B (Fig.\\,\\ref{Fig3}b). First of all we observe a smaller  contribution of the orthorhombic phase for all the samples with respect to the excitation from side A. Moreover, in the case of GO-Li plus G+mTiO$_2$ (ETL 4), we detect a strong reduction of the radiative traps at 1.52\\,eV and the dominance of the emission from the tetragonal phase at 1.56\\,eV.\n\nSuch results suggest that an incomplete phase transition occurs for the perovskite wrapped into the mesoporous ETL layer. In particular, in the case of ETL 4, where the PL lineshape shows a negligible low energy tail and a smaller linewidth with respect to the other samples, we can argue that the crystallization of the MAPI film is very good, as also confirmed by the scanning electron microscopy (SEM) image reported in Fig.\\,S4 of the Supporting Information. However, the interaction of MAPI and ETL inhibits the phase transition, at least for a thickness of about 200\\,nm (see SEM cross section in Fig.\\,S5 of the Supporting Information). In order to confirm and test our hypothesis we performed PL measurements as a function of temperature, excitation wavelength and excitation power.\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{Fig4.pdf}\n\\caption{Temperature-dependent measurement on ETL 4 by exciting with photon energy of 2.06\\,eV with an average intensity of 10\\,W\/cm$^2$. The labels D, T and O stand for defects (trap states), tetragonal and orthorhombic, respectively. a) PL spectra from 10 to 300\\,K after excitation on side A. b) PL spectra from 10 to 300\\,K after excitation on side B. c) Position of the PL peaks of the spectra in Fig.\\,\\ref{Fig4}a and Fig.\\,\\ref{Fig4}b as a function of temperature. The excitation side is indicated in brackets. d) FWHM of the PL peaks of Fig.\\,\\ref{Fig4}a and Fig.\\,\\ref{Fig4}b as a function of temperature. The excitation side is indicated in brackets. Red solid line shows the fitting of FWHM through Eq.\\,\\eqref{Eq_Gamma}.}\n\\label{Fig4}\n\\end{figure}\n\nIn Fig.\\,\\ref{Fig4}a we report the emission spectra of ETL 4 exciting from side A, varying the sample temperature from 10 to 300\\,K. As already shown before, at low temperature we observe two bands, one at 1.65\\,eV corresponding to the orthorhombic phase and one at about 1.55 eV corresponding to the sum of the emission from optically active trap states and the emission from a residual tetragonal phase.\nAs expected\\cite{Kong_2015,Fang_2015,Dar_2016,Wright_2016}, increasing the temperature, the orthorhombic phase emission shifts at higher energy (see Fig.\\,\\ref{Fig4}c), showing a monotonic increase of its full width at half maximum (FWHM) (see Fig.\\,\\ref{Fig4}d), and it disappears above 150\\,K, where the phase transition of MAPI from orthorhombic to tetragonal phase occurs.\nBy increasing the temperature, the low energy band shows instead the typical S shape both in the peak emission energy (see Fig.\\,\\ref{Fig4}c) and its FWHM (see Fig.\\,\\ref{Fig4}d). This is an indication of the phase transition from orthorhombic to tetragonal phase, with the concurrent lower contribution of the traps.\nAbove 150\\,K the PL spectrum has only one peak, arising from the tetragonal phase emission, which continues to monotonically blue shift increasing the temperature (see Fig.\\,\\ref{Fig4}c), as expected\\cite{Kong_2015,Fang_2015,Dar_2016,Wright_2016}. \nApart from different relative weights of the emission bands, for all samples we find PL spectra very similar to the one of the sample with ETL 4 when the excitation is performed from side A.\n\nA very similar trend with temperature, as shown in Fig.\\,\\ref{Fig4}a for ETL 4, is found for ETL 1, 2 and 3, irrespective of the excitation side, and this trend is commonly reported in the literature\\cite{Dar_2016}. On the contrary, relevant differences are observed in case of ETL 4 exciting from side B (Fig.\\,\\ref{Fig4}b). By increasing the temperature, the PL spectrum shows a single band, corresponding to the tetragonal phase, with a monotonic increase of the emission energy (see Fig.\\,\\ref{Fig4}c) and the FWHM (see Fig.\\,\\ref{Fig4}d). Such behavior indicates  that the  MAPI film embedded in the mesoporous ETL side remains in the tetragonal phase even down to 10\\,K.\n\nThe FWHM of the PL bands as a function of temperature can be fitted taking into account the temperature-independent inhomogeneous broadening and the interaction between carriers and acoustic and longitudinal optical (LO) phonons\\cite{Dar_2016,Wright_2016}, using the following equation:\n\\begin{equation}\n\\Gamma(T)=\\Gamma_0+\\gamma_{\\mathrm{ac}}T + \\frac{\\gamma_{\\mathrm{LO}}}{e^{E_{\\mathrm{LO}}\/k_\\mathrm{B}T}-1},\n\\label{Eq_Gamma}\n\\end{equation}\nwhere $\\Gamma_0$ is the inhomogeneous broadening, $\\gamma_{\\mathrm{ac}}$ and $\\gamma_\\mathrm{LO}$ are the acoustic and LO phonon-carrier coupling strengths, respectively, and $E_\\mathrm{LO}$ is the LO phonon energy.\nWe fitted the FWHM data extracted from the PL from side A: in particular we considered the orthorhombic phase from 10\\,K to 150\\,K and the tetragonal phase from 150\\,K up to room temperature. If the two set of data can be fitted with a single function, we can conclude that the acoustic and optical phonons causing the PL line broadening have very similar energies for the two phases. The solid line in Fig.\\,\\ref{Fig4}d shows the best fitting curve and a good agreement is reached between the data and the model with the fitting parameters $\\Gamma_0 = (23\\pm1)$\\,meV, $\\gamma_\\mathrm{ac} = (30\\pm5)$\\,\\micro eV\/K, $\\gamma_\\mathrm{LO} = (75\\pm5)$\\,meV, $E_\\mathrm{LO} = (19\\pm1)$\\,meV. These values  well agree with  data in the literature \\cite{Dar_2016,Wright_2016}.\n\nA further evidence proving that the emission band at low temperature in ETL 4 must be attributed to a tetragonal crystalline phase of MAPI with good quality, is given in Fig.\\,\\ref{Fig5}a. We report two spectra, already shown before, from side A (red curve) and side B (black curve), acquired at the same average excitation intensity $I_0=10$\\,W\/cm$^2$.\nDecreasing the excitation intensity by one order of magnitude, the emission from side B changes completely, showing a spectrum (blue curve) similar to the side A (red curve), with the dominant contribution of the traps and a small signal from the orthorhombic phase. This is explained by the fact that, lowering the power density, only the trap energy levels close to the band gap tail are filled.\nIn addition, the PL spectrum of the sample ETL 4, obtained by exciting from side B with an intensity of $I_0\/10$ (blue curve in Fig.\\,\\ref{Fig5}a), is very similar to the spectra of the ETLs 1, 2 and 3 (Fig.\\,\\ref{Fig3}b) by exciting from side B with an higher intensity $I_0$. This means that the trap density in the mesoporous region of MAPI in ETL 4 is lower, of about one order of magnitude, with respect to the other samples. This result agrees with the better crystal quality found by SEM analysis for ETL 4 (see Fig.\\,S4 in the Supporting Information).\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{Fig5.pdf}\n\\caption{a) Normalized PL spectra of ETL 4 at $T=11$\\,K, after excitation at 2.06\\,eV, for different excitation densities and excitation sides (in brackets). $I_0$ corresponds to an average intensity of 10\\,W\/cm$^2$. b) Normalized PL spectra of ETL 4 at $T=11$\\,K for different excitation photon energies and excitation sides (in brackets).}\n\\label{Fig5}\n\\end{figure}\n\nAs stated previously, to confirm that the crystalline nature of the film changes when in contact with the ETL, in particular in presence of GO-Li plus G+mTiO$_2$, we probed the ETL 4 along the thickness exploiting the different absorption coefficient of MAPI varying the excitation photon energies.\nWe spanned the range from 1.73 to 3.1\\,eV, where FTO and cTiO$_2$ have a low and nearly constant absorption.\nThe result of this experiment, performed at 11\\,K, is reported in Fig.\\,\\ref{Fig5}b. Let us focus on side B. At high photon energy excitation, 3.1\\,eV (blue curve) and 2.06\\,eV (orange curve), the MAPI is excited for a few tens of nanometers close to the ETL and, as already observed above, the spectra show only the tetragonal phase (which should not be observed at this temperature since the only stable phase below 150\\,K is the orthorhombic). Decreasing the excitation photon energy down to 1.73\\,eV, the absorption coefficient of MAPI decreases and therefore the sample is excited more uniformly in depth. The resulting spectrum (red curve) shows an increase of the contribution of the orthorhombic phase.\nFor comparison, a spectrum from side A with an excitation of 2.06\\,eV is reported (black curve). The comparison of this spectrum with that at 1.73\\,eV from side B (red curve) shows, apart from a difference in the intensity of the orthorhombic phase, exactly the same contribution from the radiative traps and the tetragonal phase. The observed behavior proves that the crystalline nature of MAPI at low temperature is influenced by the interaction with the ETL, which inhibits the MAPI phase change into the orthorhombic form.\n\nOur results demonstrate a substantial improvement of the active layer morphology of ETL 4 with an efficient carrier capture from the ETL.\nThis is confirmed by a remarkable increase in power conversion efficiency (PCE) for complete devices, obtained by $I$-$V$ characterization (see Fig.\\,S6, S7 and Tab.\\,S1 in the Supporting Information), which is mainly ascribed to an improved short circuit current density ($J_\\mathrm{SC}$).\n\n\n\\section{Conclusions}\n\nWe investigated the effects of different graphene-based ETLs in sensitized MAPI photoelectrodes. In particular we have compared four different samples with the following ETLs: mTiO$_2$, G+mTiO$_2$, mTiO$_2$ plus GO-Li interlayer and G+mTiO$_2$ plus GO-Li interlayer. \nWe have studied the ETL effects on the carrier collection efficiency and on the MAPI morphology and quality along the thickness.\nIn presence of ETL, we found faster PL decays by exciting on FTO side with respect to the MAPI side which is explained by efficient electron removal from MAPI layer due to the ETL. In particular an increase of the electron collection efficiency up to a factor 3 with respect to standard mTiO$_2$ is reported.\n\nMoreover, the MAPI layer embedded in G+mTiO$_2$ plus GO-Li ETL shows a crystalline quality much better than the other samples, with a trap density about one order of magnitude lower.\nExploiting the dispersion of the MAPI absorption coefficient, we could probe the sample along the thickness, finding that the morphology of the MAPI film embedded in the G+mTiO$_2$ plus GO-Li ETL is frozen in the tetragonal phase, regardless of the temperature.\nMoreover, the observed morphology improvement of the MAPI encapsulated in the mTiO$_2$ plus GO-Li layer supports the increased efficiency measured for the complete devices.\n\nFinally, our results show that graphene based ETLs significantly improve both the carrier collection and the crystalline quality of the active material, opening new routes to the development of efficient and stable MAPI solar cells.\n\n\n\\section{Experimental section}\n\n\\textbf{Sample preparation.} \nSolar cells photoelectrodes were prepared on Fluorine-doped Tin Oxide (FTO) conductive glass (Pilkington TEC 8, 8\\,\\ohm\/$\\square$, 25\\,mm${}\\times{}$25\\,mm). \nThe substrates were cleaned in an ultrasonic bath, using three sequential steps: detergent with de-ionized water, acetone and 2-Propanol (10\\,min for each step).\nThe substrates were covered by a compact layer of TiO$_2$ (cTiO$_2$).\nA solution of acetylacetone (2\\,mL), titanium diisopropoxide (3\\,mL) and ethanol (45\\,mL) was deposited onto the FTO substrates by Spray Pyrolysis Deposition at 450\\,\\C. The final thickness of the cTiO$_2$ layer was measured about 50\\,nm by a Dektak Veeco 150 profilometer.\n\nThe mTiO$_2$ layer was obtained starting by an ethanol solution of 18NR-T titania paste (Dyesol) dissolved in pure ethanol (1:5 by weight), stirred overnight. \nThe graphene-doped mTiO$_2$ was obtained by adding sonicated graphene ink (1\\% in vol.),\nprepared by dispersing 5\\,g of graphite flakes (+100 mesh, $\\ge$75\\%, Sigma Aldrich) in 500\\,mL of N-methyl-2-pyrrolidone, NMP (Sigma Aldrich). The initial dispersion was ultrasonicated (VWR) for 6 hours and subsequently ultracentrifuged using a SW32Ti rotor in a Beckman-Coulter Optima XPN ultracentrifuge at 10000\\,rpm ($\\sim$12200 g) for 30 mins at 15\\,$^\\circ$C. After ultracentrifugation, the upper 80\\% supernatant was extracted by pipetting. The concentration of the graphitic flakes is calculated from the OAS (see Fig.\\,S1 in the Supporting Information), giving a concentration of 0.25\\,g L$^{-1}$. The morphology of the flakes, i.e., lateral size and thicknesses are characterized by transmission electron microscopy (TEM) and atomic force microscopy (AFM), respectively (see Fig.\\,S2 in the Supporting Information) giving a lateral size distribution of 150\\,nm and thickness of 1.7\\,nm. Raman spectroscopy data are found in the Supporting Information (see Fig.\\,S3).\nBoth standard mTiO$_2$ and G+mTiO$_2$ dispersions were sonicated 10\\,min prior to be spin-coated in air at 1700\\,rpm for 20\\,s onto the cTiO$_2$ surface, followed by a calcination step at 450\\,\\C\\ for 30\\,min.\n\nThe GO-Li interlayer was realized by spin coating (2000\\,rpm for 10\\,s) 200\\,\\micro L of GO-Li dispersion in ethanol\/H$_2$O (3:1) prepared as reported in Ref.\\,\\onlinecite{Agresti_2016b}. After the deposition, the substrates were annealed at 110\\,\\C\\ for 10\\,min.\n\nThe photo-electrodes were completed by depositing the perovskite active layer in dry conditions (relative humidity less than 30\\%) by a double step method: a lead iodide solution (PbI$_2$ in N,N-dimethylformamide, 1\\,M, heated at 70\\,\\C) was spin coated at 6000\\,rpm for 10\\,s on heated substrates (50\\,\\C) which were then dipped into a CH$_3$NH$_3$I (Dyesol) in anhydrous 2-propanol solution (10\\,mg\/mL) for 15\\,min. Finally, the samples were heated at 80\\,\\C\\ for 20\\,min in air. The MAPI absorbing layer has a typical thickness of 350\\,nm with a perovskite penetration into mesoporous layer of about 200\\,nm.\nThe samples were not encapsulated. \nComplete solar cells were realized with the same structure of the investigated samples to compare the PL results with the current-voltage ($I$-$V$) characteristics.\nIn the case of complete devices doped spiro-OMeTAD (73.5\\,mg\/mL) in chlorobenzene solution doped with tert-butylpyridine (TBP 26.77\\,\\micro L\/mL), lithium bis(trifluoromethanesulfonyl)imide (LiTFSI 16.6\\,\\micro L\/mL), and cobalt(III) complex (FK209 from Lumtec, 7.2\\,\\micro L\/mL) is spin coated (2000\\,rpm for 20\\,s) onto the tested photo-electrodes. The final devices are completed by Au counter-electrode thermal evaporation (100\\,nm). $I$-$V$ characteristics are recorded under AM1.5G solar simulator Solar Constant from KHS at 1000\\,W\/m$^2$ (1\\,sun). Results on the solar cells are shown in the Supporting Information.\n\n\\textbf{Optical Absorption Spectroscopy (OAS).}\nThe OAS of the as-produced inks was carried out in the 300-1000\\,nm range with a Cary Varian 5000i UV-vis-NIR spectrometer. The absorption spectra was acquired using a 1\\,mL quartz glass cuvette. The ink is diluted to 1:7 in NMP. The NMP solvent baseline was subtracted. The concentration of graphitic flakes is determined from the extinction coefficient at 660\\,nm, using $A = \\alpha l c$ where $l$ [m] is the light path length, $c$ [gL$^{-1}$] is the concentration of dispersed graphitic material, and $\\alpha$ [Lg$^{-1}$m$^{-1}$] is the absorption coefficient, with $\\alpha \\sim 1390$\\,Lg$^{-1}$m$^{-1}$ at 660\\,nm.\\cite{Lotya_2009}\n\n\\textbf{Transmission electron microscopy (TEM).}\nThe exfoliated flakes morphology is characterized by using a TEM JOEL JEM 1011, using an acceleration voltage of 100\\,kV. The sample preparation was performed diluting the ink  in NMP (1:10). 20\\,\\micro L of the diluted sample were drop cast on copper grids (200 mesh), and dried in vacuum overnight.  Statistical analyses are fitted with log-normal distributions.\n\n\\textbf{Atomic force microscopy (AFM).}\nThe dispersions are diluted 1:30 in NMP. 100\\,\\micro L of the dilutions are drop-casted onto Si\/SiO$_2$ wafers. AFM images are acquired with Bruker Innova AFM in tapping mode using silicon probes (frequency = 300\\,kHz, spring constant = 40\\,Nm$^{-1}$). Statistical analysis are fitted with log-normal distributions.\n\n\\textbf{Raman spectroscopy.}\nThe graphene inks are drop-cast onto Si\/SiO$_2$ wafers (LDB Technologies Ltd.) and dried under vacuum. Raman measurements are collected with a Renishaw inVia confocal Raman microscope using an excitation line of 514\\,nm with a 100X objective lens, and an incident power of $\\sim 1$\\,mW on the sample. 20 spectra are collected for each sample. Peaks are fitted with Lorentzian functions.\n\n\\textbf{Scanning electron microscopy (SEM).}\nElectrodes are imaged by aim of a field-emission scanning electron microscope FE-SEM (JOEL JSM-7500 FA). The acceleration voltage is set at 5\\,kV. Images are collected using the in-lens sensors (secondary electron in-lens image, SEI) and the secondary electron sensor (lower secondary electron image, LEI). No coating is applied.\n\n\n\\textbf{Photoluminescence spectroscopy (PL).}\nPL experiments were performed, in a quasi back-scattering geometry, keeping the samples in a closed cycle cryostat and the temperature was changed from 10 to 300\\,K. Time integrated photoluminescence (TI-PL) measurements were performed exciting the samples by different mode-locked ps laser sources: a tunable (700--850\\,nm and 350--425\\,nm with the second harmonic generator) Ti-Sapphire laser operating at 81.3\\,MHz repetition rate with 1.2\\,ps pulses and a 4\\,ps Rhodamine 6G dye laser synchronously pumped by the second harmonic of a mode-locked Nd-YAG laser, operating at 76\\,MHz. The PL signal was spectrally dispersed by a 50\\,cm monochromator providing a spectral resolution of 1\\,meV and detected by a microchannel plate photomultiplier.\nTime resolved (TR-PL) measurements were carried out exciting the samples by the ps dye laser operating at 600\\,nm and using time-correlated single photon counting technique (TCSPC) with a temporal resolution of about 60\\,ps.\n\n\n\n\\begin{acknowledgments}\nFB acknowledges funding from the Italian Ministry for Education, University and Research within the Futuro in Ricerca (FIRB) program (project DeLIGHTeD, Protocollo RBFR12RS1W). We warmly acknowledge Franco Bogani for fruitful discussions. This work was partially supported by ENTE CARIFI grant n.\\,2015\/11162 and from the European Union's Horizon 2020 research and innovation programme under grant agreement n. 696656 - GrapheneCore1.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe doping phase diagram near the underdoped region is one of the\nimportant and long-debated issues with high-$T_{c}$ cuprates\n\\cite{LeeRMP06}.\nSince the parent compound is antiferromagnetic (AFM) Mott insulator,\nthe AFM correlation plays a significant role in the emergence of\nsuperconductivity by doping charge carries.\nThe intrinsic proximity of the superconducting (SC) phase with the\nAFM phase is also shared by the phase diagrams of other SC\nmaterials, such as iron pnictides \\cite{StewartRMP11} and heavy\nfermion superconductors \\cite{NicklasPRB07}.\nThe multi-layered cuprate superconductors exhibit the coexistence of\nAFM and SC states at underdoping discovered by nuclear magnetic\nresonance measurements \\cite{MukudaJPSJ08,MukudaJPSJ09}.\nHowever, the AFM phase and the SC phase never coexist in the phase\ndiagram of single-layered cuprates such as\nLa$_{2-x}$Sr$_{x}$CuO$_{4}$ \\cite{KeimerPRB92} and\nBi$_{2}$Sr$_{2}$CuO$_{6+\\delta}$ \\cite{KatoJSSC97}.\nIn particular, Bi$_{2}$(Sr$_{2-x}$La$_{x}$)CuO$_{6+\\delta}$ systems\nshows that the three-dimensional AFM region, separated by the SC\nphase, even survives until a high underdoping level\n\\cite{KawasakiPRL10}.\n\nThe existence of the coexisting state has been found by analytical\nand numerical approaches in Hubbard$-$type models\n\\cite{ReissPRB07,JarrellEPL01,AichhornPRB07,DSPRL05,KobayashiPhysicaC10,CaponePRB06,KancharlaPRB08}\nand $t-J-$type models\n\\cite{InabaPhysicaC96,HimedaPRB99,YamasePRB04,ShihPRB04,ShihLTP05,PathakPRL09,WatanabePhysicaC10}.\nThey seem to contribute the underlying mechanism to the coexisting\nstate observed in multi-layered cuprates.\nHowever, a proper mechanism to explain why these two phases do not\nlike to coexist in single-layered cuprates remains needed.\nInterestingly, some previous studies proposed the spin-bag mechanism\nfor superconductivity since two spin bags would attract each other\nto form a Cooper pair and lower the total energy\n\\cite{SchriefferPRB89,WengPRB90,EderPRB94}.\nAs for doping more holes, therefore, it is necessary to re-examine\nhow the local distortion of the AFM background around holes\ninfluences AFM order and SC order.\n\nOn the other hand, one of the most exciting experimental results is\nthe observation of quantum oscillations in the hole-doped cuprates\nwhich pointed to electron pockets \\cite{LeBoeufNat07,LeBoeufPRB11}.\nIn particular, they proposed that these electron pockets probably\noriginate from the Fermi surface reconstruction caused by the onset\nof a density-wave phase, e.g. the AFM phase.\nUnfortunately the electron-like Fermi pockets have never been found\nin most of hole-doped cuprates using angle-resolved photoemission\nspectroscopy (ARPES) \\cite{YangNat08,MengNat09,LuARCMP12}.\nThus, to comprehend the loss of the electron pocket observed by\nARPES experiments, we inquire to what extent into the electronic\ncorrelations ignored in mean-field calculations.\n\nIn this work, we study Gutzwiller's trial wave functions with the\ncoexistence of AFM order and SC order by means of variational Monte\nCarlo (VMC) method.\nTo improve the trial state, we further consider the off-site\ncorrelations between two electrons by applying suitable Jastrow\ncorrelators.\nSurprisingly, the long-range AFM order is strongly enhanced due to\nthe local ferromagnetic (FM) Jastrow correlation, or precisely local\nAFM distortion, giving rise to the disappearance of the coexisting\nstate near underdoping in the phase diagram.\nBesides, the spin-spin correlation in the non-coexisting state\ntransfers the quasiparticle spectral weight from the antinodal\nelectron pockets to the lower AFM band, and also the nodal hole\npockets can remain until superconductivity occurs.\nTherefore, it is expected that the signal of the electron pockets\naround antinodes cannot be found in many hole-doped compounds by\nusing ARPES.\n\n\\section{Theory}\nLet us begin by the Hamiltonian on a square lattice of size\n$16\\times16$,\n\\begin{eqnarray}\nH=-\\sum_{i,j,\\sigma}t_{ij}\\tilde{c}_{i\\sigma}^{\\dag}\\tilde{c}_{j\\sigma}+J\\sum_{\\langle\ni,j\\rangle}\\left(\\mathbf{S}_{i}\\cdot\\mathbf{S}_{j}-\\frac{1}{4}n_{i}n_{j}\\right),\n\\label{e:equ1}\n\\end{eqnarray}\nwhere the hopping $t_{ij}=t$, $t'$, and $t''$ for sites i and j\nbeing the nearest, second-nearest, and third-nearest neighbors,\nrespectively.\nOther notations are standard. We restrict the electron creation\noperators $\\tilde{c}_{i\\sigma}^{\\dag}$ to the subspace without\ndoubly-occupied sites.\nIn the following, the bare parameters $(t',t'',J)\/t$ in the\nHamiltonian are set to be in the hole-doped regime:\n$(-0.3,0.15,0.3)$.\nIn order to understand how AFM order and SC order compete in\nvariational phase diagram, we choose the mean-field ground state\nincluding both AFM order and SC order (AFSC) as a starting point,\n\\begin{eqnarray}\n|\\Psi_{AFSC}\\rangle=\\prod'_{\\bf{k},s=\\{a,b\\}}\\gamma_{\\bf{k}\\uparrow}^{s}\\gamma_{-\\bf{k}\\downarrow}^{s}|0\\rangle,\\label{e:equ2}\n\\end{eqnarray}\nwhere the prime means the product only includes momenta inside the\nmagnetic zone boundary (MZB).\nNote that $s$ represents the quasiparticle coming from the upper AFM\nband ($s=b$) or the lower AFM band ($s=a$).\nThe Bogoliubov's quasiparticle operators $\\gamma_{\\bf{k}\\sigma}^{s}$\nare defined as\n\\begin{eqnarray}\n\\gamma_{\\bf{k}\\sigma}^{s}=u_{\\bf{k}}^{s}\\hat{s}_{\\bf{k}\\sigma}-\\sigma\nv_{\\bf{k}}^{s}\\hat{s}_{-\\bf{k}\\bar{\\sigma}}^{\\dag}.\\label{e:equ5}\n\\end{eqnarray}\nThe coefficients $u_{\\bf{k}}^{s}$ and $v_{\\bf{k}}^{s}$ are the BCS\ncoherence factor of AFM quasiparticles corresponding to the $s$\nband,\n\\begin{eqnarray}\n(u_{\\bf{k}}^{s})^{2}&=&\\frac{1}{2}\\left(1+\\frac{\\xi_{\\bf{k}}^{s}}{\\sqrt{(\\xi_{\\bf{k}}^{s})^{2}+\\Delta_{\\bf{k}}^{2}}}\\right),\\nonumber\\\\\n(v_{\\bf{k}}^{s})^{2}&=&1-(u_{\\bf{k}}^{s})^{2},\\label{e:equ6}\n\\end{eqnarray}\nwhere the AFM band dispersion\n$\\xi_{\\bf{k}}^{b\/a}=\\epsilon_{\\bf{k}}^{+}\\pm\\sqrt{(\\epsilon_{\\bf{k}}^{-})^{2}+m^{2}}$\nand\n$\\epsilon_{\\bf{k}}^{\\pm}\\equiv\\left(\\varepsilon_{\\bf{k}}\\pm\\varepsilon_{\\bf{k+Q}}\\right)\/2$.\nHere $\\varepsilon_{\\bf{k}}$ is the normal-state dispersion.\n$\\Delta_{\\bf{k}}$($=2\\Delta\\left(\\cos\\bf{k_{x}}-\\cos\\bf{k_{y}}\\right)$)\nis $d$-wave pairing amplitude and $m$ AFM order parameter.\nThe annihilation operators for AFM bands, $\\hat{s}_{\\bf{k}\\sigma}$,\nare given by\n\\begin{eqnarray}\n\\left(\n  \\begin{array}{c}\n    a_{\\bf{k}\\sigma} \\\\\n    b_{\\bf{k}\\sigma} \\\\\n  \\end{array}\n  \\right)=\\left(\n          \\begin{array}{cc}\n            \\alpha_{\\bf{k}} & \\sigma\\beta_{\\bf{k}} \\\\\n            -\\sigma\\beta_{\\bf{k}} & \\alpha_{\\bf{k}} \\\\\n          \\end{array}\n        \\right)\\left(\n                 \\begin{array}{c}\n                   c_{\\bf{k}\\sigma} \\\\\n                   c_{\\bf{k+Q}\\sigma} \\\\\n                 \\end{array}\n               \\right),\\label{e:equ3}\n\\end{eqnarray}\nwith $\\bf{Q}=(\\pi,\\pi)$ and the coefficients\n\\begin{eqnarray}\n\\alpha_{\\bf{k}}^{2}&=&\\frac{1}{2}\\left(1-\\frac{\\epsilon_{\\bf{k}}^{-}}{\\sqrt{(\\epsilon_{\\bf{k}}^{-})^{2}+m^{2}}}\\right),\\nonumber\\\\\n\\beta_{\\bf{k}}^{2}&=&1-\\alpha_{\\bf{k}}^{2}.\\label{e:equ4}\n\\end{eqnarray}\n\nIn order to introduce more correlations in the mean-field wave\nfunction, we first formulate the trial wave function fixing the\nnumber of electrons $\\hat{P}_{N_{e}}$ with on-site Gutzwiller\nprojector\n$\\hat{P}_{G}(=\\prod_{i}\\left(1-\\hat{n}_{i\\uparrow}\\hat{n}_{i\\downarrow}\\right))$\nand charge-charge Jastrow correlator ($\\hat{P}_{J}^{CC}$)\n\\cite{CPC08,CPC12},\n\\begin{eqnarray}\n|\\Psi_{CC}\\rangle=\n\\hat{P}_{N_{e}}\\hat{P}_{G}\\hat{P}_{J}^{CC}|\\Psi_{AFSC}\\rangle.\\label{e:equ7}\n\\end{eqnarray}\nMore importantly, we also consider the correlation between spins by\nusing spin-spin Jastrow correlator ($\\hat{P}_{J}^{SS}$),\n\\begin{eqnarray}\n|\\Psi_{CCSS}\\rangle=\\hat{P}_{J}^{SS}|\\Psi_{CC}\\rangle.\\label{e:equ82}\n\\end{eqnarray}\nThe Jastrow correlator is constructed by classical Boltzmann\noperator, $\\hat{P}_{J}^{i}=e^{\\hat{H}_{i}}$, encoding the intersite\ncorrelations.\nFor the sake of simplicity, $\\hat{H}_{i}$ depicting charge ($i=CC$)\nand spin ($i=SS$) parts are chosen to be diagonal in real-space\nconfiguration.\nThe charge-charge Jastrow correlator describes the short- and\nlong-range correlations between holes in the lattice system.\nThus,\n\\begin{eqnarray}\n\\hat{H}_{CC}=\\sum_{i<j}\\eta_{ij}\\hat{n}_{i}^{h}\\hat{n}_{j}^{h},\\label{e:equ81}\n\\end{eqnarray}\nwith\n$\\eta_{ij}\\equiv\\ln(r_{ij}^{\\alpha}v_{\\gamma}^{\\delta_{j,i+\\gamma}})$.\nHere $r_{ij}$ is the chord length of $|\\vec{r}_{i}-\\vec{r}_{j}|$ and\n$\\hat{n}_{i}^{h}=1-\\sum_{\\sigma}\\hat{n}_{i\\sigma}$.\nWe consider three parameters $v_{\\gamma}$, the nearest ($\\gamma=1$),\nsecond-nearest ($\\gamma=2$) and third-nearest ($\\gamma=3$)\nneighbors, standing for short-range hole-hole repulsion if\n$v_{\\gamma}<1$.\nThe factor $r_{ij}^{\\alpha}$ denotes attractive long-range\n($r_{ij}>1$) and repulsive short-range ($r_{ij}<1$) correlations\nbetween holes if $\\alpha>0$.\n\nA similar formalism to the spin-spin correlation has been considered\nat half-filling \\cite{HuesPRL88}.\nWe further imitate the formalism described above to write down the\nspin-spin Jastrow correlator,\n\\begin{eqnarray}\n\\hat{H}_{SS}=\\sum_{i<j}\\kappa_{ij}\\hat{S}_{i}^{z}\\hat{S}_{j}^{z},\\label{e:equ8}\n\\end{eqnarray}\nwhere\n$\\kappa_{ij}\\equiv\\ln(r_{ij}^{\\beta}w_{\\gamma}^{\\delta_{j,i+\\gamma}})$\nand $\\hat{S}_{i}^{z}$ the spin operator along $z$ direction at site\n$i$.\nThe only difference from the charge counterpart is that the\nsign of $\\hat{S}_{i}^{z}\\hat{S}_{j}^{z}$ determines the type of\nmagnetic correlations.\nIn other words, it will be the FM (AFM) correlation if\n$\\hat{S}_{i}^{z}\\hat{S}_{j}^{z}>0$ ($<0$).\nIn addition to the parameter $\\beta$ controlling the long-range spin\ncorrelations, we consider the other three parameters\n$w_{\\gamma=1,2,3}$ for the neighboring spin-spin correlations.\nFor example of the FM case, the short-range correlation would be\nsuppressed when $w_{\\gamma}<1$.\nOn the other hand, the factor $r_{ij}^{\\beta}$ control the\nlong-range ($r_{ij}>1$) and short-range ($r_{ij}<1$) correlations.\nIn the long-range case of $\\beta<0$, for instance, $r_{ij}^{\\beta}$\nwould decrease the FM correlation but conversely increase the AFM\ncorrelation.\n\nIn addition to the ground state, we also propose a trial wave\nfunction for the low-lying excitation of the Gutzwiller-projected\ncoexisting state simply generated by Gutzwiller projecting the\nmean-field excited state\n\\begin{eqnarray}\n|\\Psi_{AFSC}^{\\bf{k}\\sigma\ns}\\rangle=\\left(\\gamma_{\\bf{k}\\sigma}^{s}\\right)^{\\dag}|\\Psi_{AFSC}\\rangle.\\label{e:equ9}\n\\end{eqnarray}\nHere we have applied the particle-hole transformation\n\\cite{YokoyamaJPSJ88,CPCPRB12} into Eq.(\\ref{e:equ9}) to avoid the\ndivergence from the nodes of the mean-field wave function.\nThe Gutzwiller-projected excited state with both AFM order and SC\norder fixing to $N_{e}-1$ electrons is written as\n\\begin{eqnarray}\n|\\Psi_{\\bf{k}\\sigma}^{s}\\rangle=\n\\hat{P}_{N_{e}-1}\\hat{P}_{G}\\hat{P}_{J}^{CC}\\hat{P}_{J}^{SS}|\\Psi_{AFSC}^{\\bf{k}\\sigma\ns}\\rangle.\\label{e:equ10}\n\\end{eqnarray}\nHence we can compute the excitation energies\n$E_{k}(\\equiv\\langle\\Psi_{\\bf{k}\\sigma}^{s}|H|\\Psi_{\\bf{k}\\sigma}^{s}\\rangle-\\langle\\Psi_{0}|H|\\Psi_{0}\\rangle)$\nfor either upper ($s=b$) or lower ($s=a$) AFM quasiparticles.\nFurthermore, the quasiparticle spectral weight measured from ARPES\ncan be obtained by calculating\n\\begin{eqnarray}\nZ_{\\bf{k}}^{-}\\equiv\\frac{\\left|\\langle\\Psi_{\\bf{k}\\sigma}^{s}|c_{-\\bf{k}\\bar{\\sigma}}|\\Psi_{0}\\rangle\\right|^{2}}{\\langle\\Psi_{\\bf{k}\\sigma}^{s}|\\Psi_{\\bf{k}\\sigma}^{s}\\rangle\\langle\\Psi_{0}|\\Psi_{0}\\rangle}.\\label{e:equ11}\n\\end{eqnarray}\nSome details in the VMC calculation should be noticed.\nThe boundary condition we use is periodic along both directions.\nIn order to achieve a reasonable acceptance ratio, the simulation\nconsists of a combination of one-particle moves and two-particle\nmoves.\nThe variational parameters of the Gutzwiller-projected coexisting\nstate are optimized by using the stochastic reconfiguration method\n\\cite{SorellaPRB01}.\nAll physical quantities are evaluated using the optimized\nparameters.\nWe also take a sufficient number of samples ($=2\\times10^{5}$) to\nreduce the statistical errors, and keep the sampling interval\n($\\sim40$) long enough to ensure statistical independence between\nsamples.\n\n\\section{Results}\n\n\\begin{figure}[t]\n\\begin{center}\\rotatebox{0}{\\includegraphics[height=3.5in,width=3in]{fig1.png}}\\end{center}\n\\caption{(a) Variational phase diagram plotted by staggered\nmagnetization $M_{s}$ (squares) and superconducting order parameter\n$\\Delta_{SC}$ (circles). Filled and empty symbols represent\n$|\\Psi_{CC}\\rangle$ and $|\\Psi_{CCSS}\\rangle$, respectively. (b) The\ndifference of the energy components between $|\\Psi_{CCSS}\\rangle$\nand $|\\Psi_{CC}\\rangle$ as a function of hole doping $\\delta$ in\n$16\\times16$ lattice.}\\label{fig1}\n\\end{figure}\n\nWe first consider the trial state with only the charge-charge\nJastrow correlator to better demonstrate the variational phase\ndiagram.\nThen we further include the spin-spin Jastrow correlator to\nsee how the phase diagram changes.\nOrder parameters shown in the phase diagram are determined by the\nstaggered magnetization\n\\begin{eqnarray}\nM_{s}=\\frac{1}{N}\\sum_{i}\\langle\\hat{S}_{i}^{z}\\rangle\ne^{i\\bf{Q}\\cdot\\bf{R}_{i}}\\label{e:mag}\n\\end{eqnarray}\nand the long-range pair-pair correlation function\n\\begin{eqnarray}\nC_{PP}(R)=\\frac{1}{N}\\sum_{i,\\alpha,\\alpha'}\\lambda_{\\alpha,\\alpha'}\\langle\\Delta_{i,\\alpha}^{\\dag}\\Delta_{i+R,\\alpha'}\\rangle.\\label{e:ppcf}\n\\end{eqnarray}\nThe creation operator\n$\\Delta_{i,\\alpha}^{\\dag}$($\\equiv\\tilde{c}_{i\\uparrow}^{\\dag}\\tilde{c}_{i+\\alpha\\downarrow}^{\\dag}-\\tilde{c}_{i\\downarrow}^{\\dag}\\tilde{c}_{i+\\alpha\\uparrow}^{\\dag}$)\ncreates a singlet on the bond $(i,i+\\alpha)$, $\\alpha=x,y$.\nThe factor $\\lambda_{\\alpha,\\alpha'}$ describes $d$-wave symmetry:\n$\\lambda_{\\alpha,\\alpha'}=1$($-1$) as\n$\\alpha=\\alpha'$($\\alpha\\neq\\alpha'$).\n\nIn Fig.\\ref{fig1}(a), without the spin-spin Jastrow correlators as\nindicated by $|\\Psi_{CC}\\rangle$, there exists a region showing the\ncoexistence of AFM order and SC order within doping\n$\\delta\\lesssim0.125$ in the phase diagram\n\\cite{ShihLTP05,PathakPRL09,WatanabePhysicaC10}, where $M_{s}$ and\n$\\Delta_{SC}$($\\equiv\\sqrt{C_{PP}(R>2)}$) are finite.\nLet us turn to the case with both charge-charge and spin-spin\nJastrow correlators denoted by $|\\Psi_{CCSS}\\rangle$.\nObviously the coexisting region disappears and a clear boundary\nseparating the AFM phase and the SC phase shows up at doping\n$\\delta=0.156$.\nNote that near the boundary the spin-spin Jastrow correlator can\ngreatly improve the ground-state energies from $0.3\\%$ to $0.7\\%$.\nFrom the numerical optimization, we find the spin-spin Jastrow\ncorrelator can provide a conduit to vary the mean-field AFM order in\n$|\\Psi_{AFSC}\\rangle$.\nSurprisingly, the optimized spin-spin Jastrow parameters slightly\ndisplay short-range FM correlations in the AFM background (e.g. at\n$\\delta=0.156$ the spin-spin Jastrow weights $w_{\\gamma}$ for\n$\\gamma=1$, $2$ and $3$ would be increased to $1.12$, $1.02$ and\n$1.01$, respectively).\nThe local FM correlation introduced by the Jastrow factors is\nharmful to the mean-field AFM order.\nTo make them balance, it is inevitable to largely enhance the AFM\nbackground in $|\\Psi_{AFSC}\\rangle$.\nThe surprising competition between AFM order and SC order near the\nphase boundary is mainly due to the hugely enhanced AFM order\nfurther leading to the diminished SC order.\n\nTo further demonstrate the energy competition, we analyze the\ndifference of the energy components in the Hamiltonian between\n$|\\Psi_{CCSS}\\rangle$ and $|\\Psi_{CC}\\rangle$ shown in\nFig.\\ref{fig1}(b).\nOur data clearly show that within $0.04<\\delta<0.2$ the spin Jastrow\ncorrelator helps the trial mean-field state gain much more energy\nfrom the second-nearest-neighbor hopping term.\nOn the other hand, the competing energy primarily comes from the\nspin-spin superexchange interaction.\nFrom real-space point of view, holes prefer to move along diagonal\ndirection in strong AFM background so that the hopping energy from\nthe second nearest neighbors ($t'$) is likely to compete with the\nsuperexchange energy ($J$).\n\n\\begin{figure}[t]\n\\begin{center}\\rotatebox{0}{\\includegraphics[height=2.4in,width=3.4in]{fig2.png}}\\end{center}\n\\caption{The difference of the momentum distribution function\nbetween $|\\Psi_{CCSS}\\rangle$ and $|\\Psi_{CC}\\rangle$ for doping (a)\n$\\delta=0.125$, (b) $\\delta=0.156$ and (c) $\\delta=0.188$ plotted in\nthe first Brillouin zone. (d) The next-nearest-neighbor energy\ncomponent, $-4t'\\cos(k_{x})\\cos(k_{y})$. The black diamond is the\nhalf-filled Fermi surface. White (Purple) regions present the\npositive (negative) values. Red lines mean zero.}\\label{fig2}\n\\end{figure}\n\nIn momentum space, it is apparent that the $t'$ energy gain would\ninfluence how the band dispersion evolves from Fermi pocket to Fermi\nsurface as increasing doping.\nIn Fig.\\ref{fig2}(a)-(c), the difference of the momentum\ndistribution function between $|\\Psi_{CCSS}\\rangle$ and\n$|\\Psi_{CC}\\rangle$ shows how electrons distribute in the band\nstructure.\nAt $\\delta=0.125$ (Fig.\\ref{fig2}(a)), obviously electrons in the\nsystem would prefer to stay around \"hot spots\" rather than living\nnear nodes and antinodes, which hole pockets and electron pockets\nseem to be observed as well.\nThe hot spot is defined as the momenta along the MZB that can be\nconnected by ($\\pi,\\pi$) momentum scattering.\nOnce doping is increased to $0.156$ which is the phase boundary\n(Fig.\\ref{fig2}(b)), hole pockets become larger and electron pockets\nslightly shrink.\nNow that electrons like to circle just outside the electron pockets,\nthey attempt to form a large Fermi surface.\nIndeed, as further increasing doping to $0.188$ where the long-range\nAFM order almost disappears (Fig.\\ref{fig2}(c)), a clear Fermi\nsurface in which electrons cluster together can be seen.\nSo far, we also understand the reason why the system gain much\nenergy from $t'$ term since the hot spots are located right at the\npurple region shown in Fig.\\ref{fig2}(d).\n\n\\begin{figure}[t]\n\\begin{center}\\rotatebox{0}{\\includegraphics[height=3.7in,width=2.4in]{fig3.png}}\\end{center}\n\\caption{(a) Spin-spin, (b) hole-hole and (c) pair-pair correlation\nfunctions for the optimized state $|\\Psi_{CCSS}\\rangle$\n($|\\Psi_{CC}\\rangle$), denoted by red circle (black square) symbols.\nThere are 40 doped holes in $16\\times16$ lattice\n($\\delta=0.156$).}\\label{fig3}\n\\end{figure}\n\nIn Fig.\\ref{fig3}, we compute the spin-spin, hole-hole and pair-pair\ncorrelation functions (already shown in Eq.(\\ref{e:ppcf})) defined\nas,\n\\begin{eqnarray}\nC_{CC}(\\bf{R})&=&\\frac{1}{N}\\sum_{i}\\langle\\hat{n}_{i}^{h}\\hat{n}_{i+\\bf{R}}^{h}\\rangle,\\\\\nC_{SS}(\\bf{R})&=&\\frac{1}{N}\\sum_{i}\\langle\n\\hat{S}_{i}^{z}\\hat{S}_{i+\\bf{R}}^{z}\\rangle\ne^{i\\bf{Q}\\cdot\\bf{R}}.\\label{e:equ12}\n\\end{eqnarray}\nThe doping density we choose to present is $0.156$.\nFigure \\ref{fig3}(a) illustrates that the spin-spin Jastrow\ncorrelators indirectly induce the stronger AFM background showing a\nconstant tail in the staggered spin-spin correlation function which\nimplies a clear AFM order.\nNote that The enhancement of the AFM order mainly arises from the\nmean-field wave function $|\\Psi_{AFSC}\\rangle$.\nFurthermore, we find in Fig.\\ref{fig3}(b) that the hole-hole\ncorrelation function makes no difference even if including the\nspin-spin Jastrow correlators, except that the short-range part\nbecomes less staggered.\nFor spin and charge, there is no correlation for their long-range\nbehavior.\nFinally, we can also see in Fig.\\ref{fig3}(c) that as considering\n$\\hat{P}_{J}^{SS}$ the pair-pair correlation almost vanishes at\nlarge distances so that the SC properties is not available.\n\nNext, it would be interesting to examine the low-lying\nsingle-particle excitation spectra near the phase boundary.\nIn Fig.\\ref{fig4}, by applying the ansatz (Eq.(\\ref{e:equ10})) to\nthe single-particle excitation, we calculate two quasiparticle band\ndispersions ($s=a,b$) and their corresponding spectral weight for\nremoving one particle defined by Eq.(\\ref{e:equ11}).\nIn order to compare with the excitations with\/without spin-spin\nJastrow correlators $\\hat{P}_{J}^{SS}$, we plot their excitation\nenergy $E_{\\bf{k}}$ along the high symmetric momenta in\nFig.\\ref{fig4}(a).\nIn the case where the trial state only includes the charge-charge\nJastrow factors, its optimized mean-field parameters $\\Delta\\gg m$.\nDue to large $d$-wave BCS pairing contribution, the dispersions thus\nshow convex around the antinodes and almost zero gap between the two\nbands at nodes.\nEspecially, the upper AFM band is beneath the lower AFM band near\nthe antinodal regions, and hence there is a clear signal of electron\npockets arising from the upper AFM band shown in Fig.\\ref{fig4}(c).\n\n\\begin{figure}[t]\n\\begin{center}\\rotatebox{0}{\\includegraphics[height=2in,width=3.4in]{fig4.png}}\\end{center}\n\\caption{(a) The quasi-particle excitation dispersion $E_{\\bf{k}}$\nfor different optimized states (denoted in the legend of (c)) along\nhigh symmetric momenta at $\\delta=0.156$. Empty (Filled) symbols\nrepresent the upper (lower) AFM band and squares (circles) the trial\nstate $|\\Psi_{CC}\\rangle$ ($|\\Psi_{CCSS}\\rangle$). Due to much\nsmaller $\\Delta$ than $m$ for the trial state $|\\Psi_{CCSS}\\rangle$,\nwe simply plot the lower AFM band (red circles) below the Fermi\nlevel (pink line) except the nodal regions for clear demonstration.\nThe quasiparticle spectral weight $Z_{\\bf{k}}^{-}$ are obtained from\n(b) the lower AFM band and (c) the upper AFM band.}\\label{fig4}\n\\end{figure}\n\nWhen further considering spin-spin Jastrow correlators, the\noptimized mean-field parameters $m\\gg\\Delta$.\nSuch a huge AFM parameter $m$ gives rise to a typical AFM band\ndispersion and opens a AFM gap between these two bands at nodes, as\nindicated by red circles in Fig.\\ref{fig4}(a).\nInterestingly, Fig.\\ref{fig4}(c) shows that near antinodes the\nquasiparticle spectral weight of the upper AFM band disappear and\ntransfer to almost entire lower AFM band (see Fig.\\ref{fig4}(b)).\nIn particular, a clear hole pocket of the lower AFM band centering\naround $\\bf{Q}\/2$ is also observed in Fig.\\ref{fig4}(b).\nThe Gutzwiller and Jastrow correlators arising from electronic\ncorrelation firmly influence the low-lying quasiparticle excitation\nspectra of the mean-field state $|\\Psi_{AFSC}\\rangle$.\nTherefore, the loss of the electron pockets due to electron\ncorrelations provides a route to figure out why electron pockets\nhave never been found in most of hole-doped cuprates measured by\nARPES.\n\n\\section{Conclusions}\nSumming up, by using VMC approach we have studied the coexisting\nstate with both AFM order and SC order simultaneously underneath the\nGutzwiller's projection and Jastrow correlators.\nWe have thereby re-examined the variational ground-state phase\ndiagram and found that the AFM phase competes with the SC phase as\nfurther considering off-site spin correlations.\nThe reasoning for the competition is that the mean-field AFM order\nis considerably enhanced due to short-range FM correlation\nintroduced by the spin-spin Jastrow factors, further leading to the\nvanished SC order.\nAs well, we have first investigated the Gutzwiller-projected\nquasiparticle excitations of the coexisting state.\nBased on the Gutzwiller ansatz, passing through the boundary between\nAFM and SC phases, we have observed the loss of electron pockets\nnear antinodes coming from the upper AFM band and the occurrence of\nhole pockets near nodes arising from the lower AFM band as long as\nthe spin-spin Jastrow correlators are included.\nTherefore, such a strongly correlated electron system needs to be\ncarefully inspected in the explanation for the low-lying\nquasiparticle excitations observed by ARPES experiments.\n\n\\section{Acknowledgments}\n\\label{Acknowledgment} Greatly thanks S.-M. Huang, W. Ku and T.-K.\nLee for helpful discussions. This work is supported by the\nPostdoctoral Research Abroad Program sponsored by National Science\nCouncil in Taiwan with Grant No. NSC 101-2917-I-564-010 and by CAEP\nand MST. All calculations are performed in the National Center for\nHigh-performance Computing in Taiwan.\n\\\\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\n\n\n\nThis paper is part of a program aimed to understand equivariant and $K$-theoretic invariants for Artin groups.\nThese groups are ubiquitious in geometric group theory, as they comprehend many different subfamilies of groups (braid groups, right-angled, spherical type, extra-large, free groups...), provide examples and counterexamples for many interesting phenomena, and at the same time they are not well understood from a global point of view: it is not even known if every Artin group is torsion-free. See \\cite{Par09} for a good survey. In this context, Azzali  \\emph{et al.} have recently computed explicitly both sides of Baum-Connes for pure braid groups \\cite{ABGRW}; in joint work of the second author with J. Gonz\\'alez-Meneses \\cite{FlGo18} it is computed the minimal dimension of a model of ${\\underline{\\underline{E}}} G$ for braid groups; J. and virtually-cyclic dimensions of mapping class groups (which include in particular certain Artin groups) have been recently investigated by Aramayona \\emph{et al.}  (\\cite{AJT18}, \\cite{ArMa}) and by Petrosyan and Nucinkis \\cite{NuPe18}.\n\n \n\nIn the present paper we consider the case of Artin groups of dihedral type, denoted by $A_n$, which are the groups defined by\n$$A_n=\\gp{a,b}{\\mathrm{prod}(a,b;n)=\\mathrm{prod}(b,a;n)}.$$\nHere  $\\mathrm{prod}(x,y;n)$ denotes the word of length $n$ that alternates $x$ and $y$ and starts with $x$.\n These groups   are one-relator and torsion-free, they  have small geometric dimension and its group-theoretic structure is quite well understood. These features makes them strongly appropriate for computations, and in fact have been recently studied from different angles, as growth series \\cite{MaMa06}, systems of equations \\cite{CHR20} or geodesics \\cite{Wal09}.\n\nWe compute here Bredon homology groups of the classifying space of these groups with respect to the family of virtually cyclic groups. Bredon homology was first described by Glen Bredon in the sixties \\cite{Bre67}, and it is a $G$-equivariant homology theory that takes into account the action over the target space of the subgroups of $G$ that belong to a predefined family. Since their appearance, Bredon homology groups have played a prominent role in Homotopy Theory and Group Theory, particularly in relation with finiteness conditions, dimension theory for groups and classifying spaces, and in the framework of the Isomorphism Conjectures. In fact, our choice of coefficients in the $K$-theory of a ring $G$ has been done with an eye in possible applications of the computations to the left-hand side of Farrell-Jones Conjecture (see below), although we believe that the methods of the present paper may be useful in a more general context, switching the $K$-theoretic coefficients to a general coefficient module.\n\nOur main technical result is the following:\n\n\n\n\\noindent\\textbf{Theorem \\ref{Thm:Bredon}.} Let $A_n$ be an Artin group of dihedral type and $R$ a ring, $R$ a ring, $K_q(R[-])$ the covariant module over the orbit category $O_{\\mathcal F}(G)$ that sends every (left) coset  $G\/H$ to the $K$-theory group $K_q(R[H])$. Then we have the following:\n\\begin{enumerate}\n  \\setlength{\\itemindent}{-2em}\n\\item $H_i^{vc}({\\underline{\\underline{E}}} A_n,K_q (R[-]))=\\{0\\}$ for $i\\geq 4$.\n\\item $H_3^{vc}({\\underline{\\underline{E}}} A_n,K_q (R[-]))=\\begin{cases}\\bigoplus_{[H]\\neq [Z(A_n)]} K_q(R) & \\text{$n$ odd}\\\\\n\\ker g_2^2 & \\text{$n$ even.}\n\\end{cases}$\n\n\\item $H_2^{vc}({\\underline{\\underline{E}}} A_n,K_q (R[-]))= \\ker g_2^1$.\n\n\\item $H_1^{vc}({\\underline{\\underline{E}}} A_n,K_q (R[-]))= \\emph{coker } g_2^1 = \\begin{cases}(\\bigoplus_{[H]\\neq [Z(A_n)]}N_q^{[H]})\\oplus T_1(K_q(R))\\oplus T_2(K_q(R)) & \\text{$n$ odd}\\\\\n\\bigoplus_{[H]} N_q^{[H]}\n  & \\text{$n$ even.}\n\\end{cases}$%\n\\item $H_0^{vc}({\\underline{\\underline{E}}} A_n,K_q (R[-]))= \\emph{ coker } g_2^0= \\begin{cases}(\\bigoplus_{[H]\\neq [Z(A_n)]} N_q^{[H]})\\oplus K_q(R)\\oplus\\overline{C}(K_q(R)) & \\text{$n$ odd}\\\\\n(\\bigoplus_{[H]} N_q^{[H]})\\oplus K_q(R)\n  & \\text{$n$ even.}\n  \\end{cases}$%\n\\end{enumerate}\n\n\nHere $g^j_i$ stands for a homomorphism in the Degrijse-Petrosyan exact sequence  {\\cite[Section 7]{DP}}, $[H]$ for the commensurability class of a {non-trivial} cyclic subgroup $H$, and $N_q^{[H]}$, $T_i(K_q(R))$ and $\\overline{C}(K_q(R))$ for groups that depend on $H$ and the Bass-Heller-Swan decomposition of $K_q(R[\\Z])$. See Section \\ref{Sect:BredonArtin} for details.\n\n\n\n\nThe main tool used in the proof of Theorem \\ref{Thm:Bredon} is an exact sequence in Bredon homology \\cite{DP}, which is in turn the Mayer-Vietoris sequence associated to the push-out that defines the L\\\"{u}ck-Weiermann model for ${\\underline{\\underline{E}}} G$ (\\cite{LW}, see also Section \\ref{Sect:Prelim} below). The knowledge about the group-theoretic structure of the groups includes a complete understanding of the commensurators, which is crucial in the computations. Using the theorem, we are able to describe with precision the Bredon homology of $A_n$ with coefficients in the $K$-theory of several rings, both regular and non-regular (see Section \\ref{Sect:concrete}). \n\nNext we describe the implications of our work in relation with the Farrell-Jones Conjecture. Recall that given a group $G$, a ring $R$ and $n\\in\\mathbb{Z}$, Farrell-Jones stated in \\cite{FaJo93} the existence of an assembly map:\n\n$$H_n^G({\\underline{\\underline{E}}} G,\\mathbf{K} (R))\\rightarrow K_n(RG),$$\nwhere ${\\underline{\\underline{E}}} G$ is the classifying space of $G$ with respect to the family of virtually cyclic groups,\n$H_*^G(-,\\mathbf{K} (R))$ is the $G$-homology theory defined in Section 1 of \\cite{FaJo93} and $K_n(RG)$ stands for the $n$-th group of algebraic $K$-theory of the group ring $RG$.\nThe Farrell-Jones conjecture predicts that the assembly map is an isomorphism.\nThe conjecture has been verified for a big family of groups (see \\cite{LR05} for an excellent survey), and no counterexample has been found so far.\nThe philosophy in this context is, for a group for which the assembly map is known to be an isomorphism, to perform computations in the topological side in order to extract information about the algebraic $K$-theory of the group ring. It is remarkable that the latter are difficult to compute, and at the same time encode fundamental invariants of manifolds, including obstructions to the existence of cobordisms and information about groups of pseudoisotopies. Explicit calculations in this context can be found in \\cite{BuSa16}, \\cite{DQR11}, \\cite{KLL21} or \\cite{SaVe18}, for example.\n\nThe left-hand side of the conjecture can be approached by means of a $G$-equivariant version of the Atiyah-Hirzebruch spectral sequence, which  converges to the Farrell-Jones $K$-homology, and whose $E_2$-page is  the Bredon homology of the classifying space ${\\underline{\\underline{E}}} G$ with coefficients in the $K$-theory of the group rings of the virtually cyclic subgroups of $G$; example of such calculations in the Farrell-Jones framework can be found in \\cite{BJV14} \\cite{LuRo14}. In this setting, Theorem \\ref{Thm:Bredon} and the examples of Section \\ref{Sect:concrete} can be interpreted as an explicit computation of such $E_2$-page, in the case of Artin groups of dihedral type. We remark that in the case of $R$ regular, the left-hand side of the conjecture for these Artin groups can be deduced from \\cite[Lemma 16.12]{Luc21}, using previously a splitting result of L\\\"{u}ck-Steimle \\cite{LS16} (see the end of Section \\ref{Sect:concrete} for details), so our results provide new information in this context for a non-regular $R$; in this sense, we expect that Examples 5.3-5.5 will be useful. It is worth to point out that the Atiyah-Hirzebruch spectral sequence collapses at most at the $E_5$-page in this context, so an (at least partial) computation of the differentials may not be completely out of sight.\n\nWe finish by pointing out that Artin groups of dihedral type are free-by-cyclic, and the Farrell-Jones conjecture has been recently verified for this class of groups \\cite{BFW21}. Hence, all the computations in the left-hand side can be read in terms of algebraic $K$-theory of $RG$.\n\n\n\n\\textbf{Summary of contents}. In Section \\ref{Sect:Prelim} we recall the main definitions about classifying spaces for families and Bredon homology, with special emphasis in the L\\\"{u}ck-Weiermann model and its associated Mayer-Vietoris sequence. In Section \\ref{Sect:Artin} the main properties of Artin groups that will be used on the rest of the paper are studied. Then, in Section \\ref{Sect:BredonArtin}, we carefully analyze the homomorphisms in the exact sequence and prove Theorem \\ref{Thm:Bredon}; and in final Section \\ref{Sect:concrete} we apply our results to describe different concrete examples.\n\n\n\\section{Preliminaries}\n\\label{Sect:Prelim}\n\nIn this section we state some notions of $G$-equivariant homotopy that will frequently appear in the rest of the paper.\nThe exposition will be sketchy and very focused to our goals; the reader interested in a thorough treatment of the subject is referred to \\cite{TDieck} for the theory of $G$-$CW$-complexes and actions on them, to \\cite{Luc05} for the theory of classifying spaces and to the first part of \\cite{MiVa03} for Bredon homology.\n\n\\subsection{Classifying spaces for families}\n\\label{Sect:classify}\nIn this section we will briefly recall the notion of classifying space for a family of subgroups, which is the central object in the topological side of the Isomorphism Conjectures. Then we will review L\\\"{u}ck-Weiermann model and the definition of commensurator, which will be crucial in our computations.\n\n\\begin{defn}\n\nLet $G$ be a discrete group, and $\\mathcal{F}$ be a family of subgroups of $G$  closed under passing to subgroups and conjugation. A $G$-CW-complex $X$ is a \\emph{classifying space for the family} $\\mathcal{F}$ if for every $H\\in \\mathcal{F}$ the fixed-point set $X^H$ is contractible, and empty otherwise.\n\n\\end{defn}\n\nThe classifying space for the family $\\mathcal{F}$ is usually denoted by $E_{\\mathcal{F}}G$.\nMoreover,  two models for $E_{\\mathcal{F}}G$ are $G$-homotopy equivalent. A point is always a model for $E_{\\mathcal{F}}G$ if $G\\in \\mathcal{F}$, and the closeness under subgroups implies that $E_{\\mathcal{F}}G$ is always a contractible space.\n\nIf there is a family ${\\mathcal{F}}$ of subgroups of $G$ with the previous closeness properties and a subgroup $H\\leqslant G$, we denote by ${\\mathcal{F}}\\cap H$ the family whose elements are the intersections $F\\cap H$, with $F\\in {\\mathcal{F}}$.\nThe family ${\\mathcal{F}}\\cap H$ of subgroups of $H$ is again closed under $H$-conjugation and taking subgroups.\nIn these conditions the action of $H$ over $E_{{\\mathcal{F}}}G$ by restriction turns $E_{{\\mathcal{F}}}G$ into a model for $E_{{\\mathcal{F}}\\cap H}H$.\n\n\nThe most important families of subgroups in this context are the trivial family $\\mathcal F_{\\{1\\}}$, the family $\\mathcal{F}_{Fin}$ of finite groups and the family $\\mathcal{F}_{vc}$ of virtually cyclic groups of $G$; the classifying spaces for these families are respectively denoted by $EG$, $\\underline{E} G$ and ${\\underline{\\underline{E}}} G$. Observe that $\\mathcal F_{\\{1\\}}\\subseteq \\mathcal{F}_{Fin}\\subseteq \\mathcal{F}_{vc}$, and that $\\mathcal F_{\\{1\\}}=\\mathcal{F}_{Fin}$ if and only if $G$ is torsion-free.\nIt is  also a standard argument to show that torsion-free virtually cyclic groups are cyclic (see for example Lemma 3.2 in \\cite{Mac96}).\nThen, for torsion-free groups, $\\mathcal F_{vc}$ is the set of cyclic subgroups.\n\nFrom now on we will describe the model of ${\\underline{\\underline{E}}} G$ developed by L\\\"{u}ck-Weiermann in \\cite{LW}, for the special families we are interested (the construction is indeed more general). Given a group $G$, and two subgroups $H$ and $K$,  we consider the equivalence relation generated by $H\\sim K$ if $H\\cap K$ has finite index in both $H$ and $K$.\nObserve that if $H$ is in ${\\mathcal F}_{vc}\\setminus \\mathcal{F}_{Fin}$ then $H\\sim K$ if and only if $K$ is virtually cyclic and $H\\cap K$ is infinite.\nAlso remark  that if $H\\in \\mathcal{F}_{Fin}$, then $H\\sim K$ if and only if $K\\in \\mathcal{F}_{Fin}$.\nThe equivalence class of $H$ will be denoted by $[H]$.\nThis equivalence relation is preserved by conjugation, it can be defined $g^{-1}[H]g$ as $[g^{-1}Hg]$ for any $H\\leqslant G$ and for any $g\\in G$.\n\nNow we can define the notion of \\emph{commensurator}, central in this model and  in our paper:\n\n\\begin{defn}\n\\label{defn:comm}\nGiven an equivalence class $[H]$ of the relation $\\sim$, the \\emph{commensurator} of $[H]$ in $G$ is defined as the subgroup\n$$ \\textrm{Comm}_G[H]=\\{g\\in G\\:|\\ g^{-1}[H]g=[H]\\}.$$\n\\end{defn}\n\n\nIn \\cite{LW}, it is also defined a family of subgroups of $\\textrm{Comm}_G[H]$ as:\n$${\\mathcal{F}} [H]:=\\{K<{ \\textrm{Comm}_G[H]} \\: |\\ K\\sim H \\text{ or } |K|<\\infty \\}.$$\nIf $H$ is virtually cyclic, it is easy to check that this family is closed under taking subgroups and conjugation in {$\\textrm{Comm}_G[H]$}. {We remark that the commensurator of $[H]$ is sometimes denoted by $N_G[H]$ in the literature}.\n\nNow we have all the ingredients needed for building L\\\"{u}ck-Weiermann model:\n\n\\begin{thm}\n\\label{maintheorem}{\\rm (\\cite{LW}, Theorem 2.3)} We denote by $I$ a complete set of representatives of the $G$-orbits (under conjugation) of equivalence classes $[H]$ of infinite virtually cyclic subgroups of $G$, and we choose, for every $[H]\\in I$, models for the classifying spaces $\\underline{E}  \\Comm_G[H]$ and $E_{{\\mathcal{F}} [H]}\\Comm_G[H]$.\nWe also choose a model for $\\underline{E}  G$. Consider the $G$-pushout:\n$$\n\\xymatrix{  \\coprod_{[H]\\in I}G\\times_{\\Comm_G[H]}\\underline{E}  \\Comm_G[H] \\ar[r]^{\\hspace{2cm} i} \\ar[d]^{\\coprod_{[H]\\in I}id_G\\times_{\\Comm_G[H]}f_{[H]}}  & \\underline{E} G \\ar[d] \\\\\n\\coprod_{[H]\\in I}G\\times_{\\Comm_G[H]}E_{{\\mathcal{F}} [H]}\\Comm_G[H]  \\ar[r] & X }\n$$\nwhere  $f_{[H]}$ is a cellular $\\Comm_G[H]$-map for every $[H]\\in I$ and $i$ is the inclusion.\n In these conditions, $X$ is a model for ${\\underline{\\underline{E}}} G$.\n\\end{thm}\n\n\nIn practice, this theorem implies that the existence of good models for the proper classifying space of the commensurators and $G$, and also of the classifying spaces with respect to the families ${\\mathcal{F}} [H]$ will lead to the knowledge of good models for ${\\underline{\\underline{E}}} G$. Moreover, the push-out implies the existence of a long exact sequence in Bredon homology, and dimensional consequences that we will analyze in next section.\n\n\n\n\\subsection{Bredon homology}\n\\label{Sect:Bredon}\n\n{In this subsection we will briefly review the main definitions concerning Bredon homology. We follow the topological concise approach from \\cite{San08}, which we use in our computations}.\n\n{Consider a discrete group $G$, $\\mathcal{F}$ a family of groups which is closed under conjugation and taking subgroups.  Let $O_{\\mathcal{F}}(G)$ be the \\emph{orbit category}  whose objects are the homogeneous spaces $G\/K$, $K\\subset G$ with $K\\in\\mathcal{F}$, and whose morphisms are the $G$-equivariant maps.\nThen a \\emph{left Bredon module} $N$ over $O_{\\mathcal{F}}(G)$ is a covariant functor $$N:O_{\\mathcal{F}}(G)\\rightarrow \\textbf{Ab},$$ where $\\textbf{Ab}$ is the category of abelian groups}.\n\n{Let $N$ be  a left Bredon module and $X$ a $G$-CW-complex, and assume that all the stabilizers of the $G$-action belong to the family $\\mathcal{F}$. Then the \\emph{Bredon chain complex} $(C_n^{\\mathcal{F}}(X,N),\\Phi_n)$ can be defined in the following way.\nFor every $d\\geq 0$, consider a set $\\{e_i^d\\}_{i\\in I}$ of representatives of orbits of $d$-cells in $X$, and denote by $\\stab(e_i^d)$ the stabilizer of $e_i^d$.\nThen we define the \\emph{n-th group of Bredon chains} as $C_n^{\\mathcal{F}}(X,N)=\\bigoplus_{i\\in I} N(G\/\\stab(e_i^d))$}.\n\n{Consider now a $(d-1)$-face of $e_i^d$, which can be given as $ge$ for a certain $(d-1)$-cell $e$.\nThen we have an inclusion of stabilizers $g^{-1}\\stab(e_i^d)g\\subseteq \\stab(e)$.\nAs $g^{-1}\\stab(e_i^d)g$ and $stab(e_i^d)$ are isomorphic, the previous inclusion induces an equivariant $G$-map $f\\colon G\/\\stab(e_i^d)\\rightarrow G\/\\stab(e)$.\nIn turn, as $N$ is a functor, we have an induced homomorphism $N(f)\\colon N(G\/\\stab(e_i^d))\\rightarrow N(G\/\\stab(e))$.\nTaking into account that the boundary of $e_i^d$ can be written as $\\partial e^d_i=\\sum_{j=1}^n g_j e_j^{d-1}$ for certain $g_j\\in G$ and using linear extension to all representatives of equivariant $d$-cells, we obtain a differential $\\Phi_d\\colon C_d^{\\mathcal{F}}(X,N)\\rightarrow C_{d-1}^{\\mathcal{F}}(X,N)$ for every $d>0$.\nSo we have the following definition:}\n\n\\begin{defn}\n{The homology groups of the chain complex $(C_i^{\\mathcal{F}}(X,N),\\Phi_i)$ will be denoted by $H_i^{\\mathcal{F}}(X,N)$ and called \\emph{Bredon homology groups} of $X$ with coefficients in $N$ with respect to the family $\\mathcal{F}$}.\n\nWe define $H_i^{\\mathcal{F}}(G,N)$, the {\\emph Bredon homology groups} of $G$ with coefficients in $N$  with respect to the family $\\mathcal{F}$ as $H_i^{{\\mathcal{F}}}(E_{\\mathcal{F}}G,M)$.\n\\end{defn}\n\n{These groups are preserved under $G$-equivariant homotopy equivalence}.\n\n\n\\textbf{Notation}. When $\\mathcal{F}$ is the family of finite groups, we use indistinctly the notations $\\underline{E}G$ or $E_{\\mathcal{F}}G$, and similarly when $\\mathcal{F}$ is the family of virtually cyclic groups and ${\\underline{\\underline{E}}} G$ and $E_{\\mathcal{F}}G$ notations for the corresponding classifying space. If $\\mathcal{F}$ is the family that only contains the trivial group, the superindex in the Bredon homology will be supressed, as it is ordinary homology in this case.\nIt is worth noticing that there is an algebraic definition of $H_*^{\\mathcal{F}}(G,M)$, however there is an isomorphism $H_*^{\\mathcal{F}}(G,M)\\simeq H_*^{\\mathcal{F}}(E_{\\mathcal{F}}G,M)$ between the algebraic and the topological definitions of Bredon homology \\cite[page 15]{MiVa03}.\nTo not overload the paper with unnecessary notation, we commonly denote this homology groups by $H_*^{\\mathcal{F}}(G,M)$, although as said above, we mainly deal with the topological definition.\n\nThe cyclic group of order $n$ will be denoted by $C_n$. When we want to consider its ring structure we might use  $\\Z \/{\\bf n}$. When $n$ is prime, we might use $\\mathbb{F}_n$ to emphasize its field structure.\n\n\n\n\\section{Artin groups of dihedral type}\n\\label{Sect:Artin}\nIn this section we present the main features of the Artin groups of dihedral type that we will need in the remaining of the paper. We start with the definition of the groups:\n\n\\begin{defn}\n\\label{Defn:Artindih}\nLet $n\\geq 1$. By $\\mathrm{prod}(x,y;n)$ we denote the word of length $n$ that alternates $x$ and $y$ and starts with $x$.\nFor example, $\\mathrm{prod}(x,y;3)=xyx$ and $\\mathrm{prod}(x,y;4)=xyxy$. With this notation, a {\\it dihedral Artin group of type $n$} is the group $$A_n=\\gp{a,b}{\\mathrm{prod}(a,b;n)=\\mathrm{prod}(b,a;n)}.$$\n\\end{defn}\nThe name ``dihedral\" comes from the associated Coxeter group, $$\\gp{a,b}{a^2=b^2=1, \\mathrm{prod}(a,b;n)=\\mathrm{prod}(b,a;n)}$$ which is the dihedral group of order $2n$. Dihedral Artin groups are torsion-free, even more $A_n\\cong F_{n-1}\\rtimes \\Z$, where $F_k$ is a free group of rank $k$. To see this, one can check that the kernel of $A_n\\to \\Z$, $a,b\\mapsto 1$ is free on rank $n-1$. In particular, Dihedral Artin groups satisfy the Farrell-Jones conjecture \\cite{BFW21}.\n\nWe are interested on understanding the commensurators of the virtually cyclic subgroups of $A_n$, for that we will use that $A_n$ is also a central extension of a virtually free group.\n\nAs indicated above an important ingredient of our calculations will be the description of some commensurators inside these Artin groups of subgroups from the family of virtually cyclic groups (which in this case turn to be just cyclic groups). We start by observing that for any  virtually cyclic subgroup $H$ of a group $G$, and any $h\\in H$ of` infinite order, we have that\n$$\\Comm_G[H]=\\Comm_G(\\gen{h})=\\{g\\in G\\mid gh^mg^{-1}=h^n \\text{ for some }n,m\\in \\Z-\\{0\\}\\}.$$\n\nWe will be interested in computing commensurators up to isomorphism, and we will use the fact that commensurators of conjugated subgroups are conjugated.\nLet us denote by $\\pi$ the natural\nprojection map $\\pi\\colon A_{n}\\to \\ol{A_{n}}\\mathrel{\\mathop\\mathchar\"303A}\\mkern-1.2mu= A_{n}\/Z(A_{n})$. Then, for every $g\\in A_{n}$,\n$$\\pi(\\Comm_{A_{n}}[\\gen{g}])\\leqslant \\Comm_{\\ol{A_{n}}}[\\gen{\\pi(g)}],$$\nand\n$$Z(A_{n})\\leqslant \\Comm_{A_{n}}[\\gen{g}].$$\n\nWe will prove the following.\n\\begin{lem}\\label{lem:commensurators}\nLet $A_n$ be a dihedral Artin group and $g\\in A_n$ of infinite order. Then\n\\begin{enumerate}\n\\item If $\\langle g\\rangle \\cap  Z(A_n)\\neq \\{1\\}$ then $\\Comm_{A_n}[\\langle g\\rangle]= A_n$ and $Z(A_n)\\in [H]$,\n\\item If $\\langle g\\rangle \\cap Z(A_n)= \\{1\\}$ then $\\Comm_{A_n}[\\langle g\\rangle]\\cong \\mathbb{Z}^2$ and there is $\\langle g'\\rangle \\in [\\langle g\\rangle]$  that is a direct factor of $\\Comm_{A_n}[\\langle g\\rangle]$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n\nIt is a well-known fact that the centers of the dihedral Artin groups have different shape depending on the parity of $n$. Hence, we should divide the study of the commensurators of their cyclic subgroups in two subcases, even and odd.\n\n\n{\\bf Case $n$ even: }\nIn the group $$A_{2n}=\\gp{a,b}{\\mathrm{prod}(a,b;2n)=\\mathrm{prod}(b,a;2n)},$$\nit is known that $Z(A_{2n})=\\gen{(ab)^{n}}$.\nMoreover, $A_{2n}\\cong \\gp{x,y}{x^{-1}y^nx=y^n}$ via the isomorphism defined by $x\\to b$, $y\\mapsto ba$.\nTherefore $A_{2n}\/Z(A_{2n})=\\gp{x,y}{x^{-1}y^nx=y^n,y^n =1}=\\gp{\\ol{x},\\ol{y}}{\\ol{y}^n=1}\\cong C_{\\infty}*C_{n}$.\n\n\nSince $\\ol{A_{2n}}$ is hyperbolic (even more, virtually free), the commensurator of an infinite order element  is virtually cyclic. (See for example \\cite[Theorem 2]{Arz} bearing in mind that infinite cyclic subgroups of hyperbolic groups are quasi-convex).\n\nSuppose first that $\\pi(g)$ has infinite order. Then $\\Comm_{\\overline{A_{2n}}}[\\langle \\pi(g)\\rangle]$ is infinite and virtually cyclic and hence $\\pi (\\Comm_{A_{2n}}[\\gen{g}])$ is infinite and virtually cyclic.\nSince all infinite virtually  cyclic subgroups of $\\ol{A_{2n}}$ are infinite cyclic  we have that  $\\pi (\\Comm_{A_{2n}}[\\gen{g}])$ is infinite cyclic. Therefore,\n$\\Comm_{A_{2n}}[\\gen{g}]$ is a central extension of $\\Z$ by $\\Z$ and hence $\\Comm_{A_n}[\\gen{g}]\\cong \\Z^2$.\nWe can take $g'$ as a pre-image of a generator of $\\pi (\\Comm_{A_{2n}}[\\gen{g}])$.\n\nSuppose now that $\\pi(g)$ has finite order.\nThus $\\langle g \\rangle \\cap Z(A_{2n})$ is non-trivial and infinite,  and thus $[\\langle g \\rangle]= [Z(A_{2n})]$ and $\\Comm_{A_{2n}}[\\gen{g}]=\\Comm_{A_{2n}}Z(A_n)=A_{2n}.$\n\n\n\n{\\bf Case $n$ odd:}\nIn the group $$A_{2n+1}=\\gp{a,b}{\\mathrm{prod}(a,b;2n+1)=\\mathrm{prod}(b,a;2n+1)}$$ we have that $Z(A_{2n+1})=\\gen{(ab)^{2n+1}}$.\nMoreover, $A_{2n+1}\\cong \\gp{x,y}{xy^n=y^{n+1}x^{-1}}$ via the isomorphism defined by $x\\to b$, $y\\mapsto ab$.\nTherefore\n\\begin{align*}\nA_{2n+1}\/Z(A_{2n+1})&=\\gp{x,y}{xy^nx=y^{n+1},y^{2n+1}=1}=\\gp{x,y}{(xy^n)^2=1=y^{2n+1}}\\\\\n&=\\gp{\\ol{z},\\ol{y}}{\\ol{z}^2=1=\\ol{y}^{2n+1}}\\cong C_{2}*C_{2n+1},\n\\end{align*}\nwhere $\\ol{z}$ denotes the class of $xy^nZ(A_{2n+1})$.\n\n\n\nSince $\\ol{A_{2n+1}}$ is hyperbolic and infinite virtually cyclic subgroups of $\\ol{A_{2n+1}}$ are infinite cyclic we get, arguing as above, that if $\\pi(g)$ has infinite order then $\\Comm_{A_{2n+1}}[\\gen{g}]\\cong \\Z^2$.\n\nSuppose now that $\\pi(g)$  has finite order. Arguing as above, $\\Comm_{A_{2n+1}}[\\gen{g}] = \\Comm_{A_{2n+1}}[Z(A_{2n+1})] = A_{2n+1}$.\n\\end{proof}\n\n\n\nWe  recall  the ordinary homology of these groups, which will be important in the remaining sections of the paper.\n\n\\begin{prop}\n\\label{OrdHom}\nFor $n\\geq 2$.\n\nIf $n$ is even, we have  $H_0(A_n)=\\mathbb{Z}$, $H_1(A_{n})=\\mathbb{Z}\\oplus \\mathbb{Z}$, $H_2(A_{n})=\\mathbb{Z}$ and $H_i(A_{n})=0$ for $i>2$.\n\n If $n$ is odd, we have $H_0(A_n)=\\mathbb{Z}$, $H_1(A_{n})=\\mathbb{Z}$ and $H_i(A_{n})=0$ for $i>1$.\n\n\\end{prop}\n\n\\begin{proof}\nAs dihedral Artin groups are one-relator groups, the Cayley complex (associated to the one-relator presentation) gives a 2-dimensional model for $K(A_n,1)$ (see \\cite{Lyndon} or \\cite{CMW04} for more general Artin groups).\n As $A_n$ is not free, this implies that $\\textrm{gd }A_n=\\textrm{cd }A_n=2$ for every $n$, and then in particular $H_i(A_n)=0$ for $i\\geq 3$. Moreover, as $K(A_n,1)$ is connected, $H_0(A_n)=\\mathbb{Z}$ for every $n$.\nThe formulae for $H_1$ follow from performing abelianization to the groups. Finally, the results for the Schur multiplier $H_2$ are a consequence of Theorem 3.1 in \\cite{CE09}.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Bredon homology of Artin groups of dihedral type}\n\\label{Sect:BredonArtin}\n\nIn this section we start with the computation of homological invariants of the Artin groups of dihedral type, which is the main goal of this paper. The computation of the Farrell-Jones homology in the case of a general ring is usually very complicated. A powerful tool to undertake this computation is the $G$-equivariant version of the Atiyah-Hirzebruch spectral sequence, which is a spectral sequence of the 1st and 4th quadrant. In our case of interest, the $E_2$-page of this sequence is given by Bredon homology of the classifying space ${\\underline{\\underline{E}}} G$, and the $E_{\\infty}$-page encodes the Farrell-Jones homology. Let us make the last statement more precise.\n\nLet $G$ be a discrete group, $\\mathcal{F}$ its family of virtually cyclic subgroups, $R$ a ring. For every $q\\in\\mathbb{Z}$, denote by $K_q(R[-])$ the covariant module over the orbit category $O_{\\mathcal F}(G)$ that sends every (left) coset  $G\/H$ to the $K$-theory group $K_q(R[H])$. In these conditions, the $E_2$-page of the Atiyah-Hirzebruch  spectral sequence that we will use is defined as $E_2^{p,q}=H_p^{\\mathcal{F}}({\\underline{\\underline{E}}} G,K_q(R[-]))$, for every $p\\geq 0$ and $q\\in\\mathbb{Z}$, and converges to $H^G_{p+q}({\\underline{\\underline{E}}} G,\\mathbf{K}(R))$ the $(p+q)$-th group of Farrell-Jones homology of ${\\underline{\\underline{E}}} G$ with coefficients in $R$. An excellent source for more information about this sequence is \\cite{LR05}.\n\nAlthough the path {to} the computation is very clear, in general it is difficult to obtain explicit formulae for the Bredon\nhomology of these classifying spaces. Different reasons for this are the complexity of the models for ${\\underline{\\underline{E}}} G$ and the fact that the exact values of $K_q(R[H])$ are only known for very special instances of $R$ and $H$. In fact, taking $R=\\mathbb{Z}$ and $H$ the trivial group, the groups $K_q(\\mathbb{Z})$ are not completely listed, as their value in some cases depend on the solution of the Vandiver conjecture, which remains unsolved. See \\cite{Wei05}.\n\n\nIn this section we show that for Artin groups of dihedral type it is possible to describe to some extent the aforementioned Bredon homology groups, with coefficients in $K_q(R[-])$ for a general ring $R$ (see Theorem \\ref{Thm:Bredon}); of course, the result strongly depends on the concrete shape of the $K$-theory groups of the group rings involved. The key to our computations is the description of the commensurators of the virtually cyclic subgroups of the groups $A_n$ (see previous section), and mainly the Mayer-Vietoris sequence associated to the push-out of L\\\"{u}ck-Weiermann model (Theorem \\ref{maintheorem}), that was explicitly stated by Degrijse-Petrosyan in Section 7 of \\cite{DP}, for the cohomological case. We offer here the homological version, which is the one needed in our context. To avoid confusions, we denote Bredon homology with respect to the family of virtually cyclic groups as $H_*^{vc}$ from now on.\n\n\\begin{prop}{\\rm (\\cite[Proposition 7.1]{DP}, see also \\cite[Theorem 2.3]{LW})}\n\\label{mainMV} Let  $M$ be a left Bredon module over $O_{{\\mathcal{F}}_{vc}}(A_n)$.\nThere is an exact sequence:\n\n$$\\ldots \\stackrel{}{\\rightarrow} H^{vc}_{i+1}(A_n,M) \\stackrel{g_1^{i+1}}{\\rightarrow} \\bigoplus_{[H]\\in I}H_i^{Fin\\cap \\Comm_{A_n}[H]}(\\Comm_{A_n}[H],M) \\stackrel{g_2^i}{\\rightarrow} $$\n\n$$\\stackrel{g_2^i}{\\rightarrow} (\\bigoplus_{[H]\\in I}H_i^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M))\\oplus H_i^{Fin}(A_n,M) \\stackrel{g_3^i}{\\rightarrow} H^{vc}_i(A_n,M) \\rightarrow \\ldots$$\n\n\\end{prop}\n\nBefore we undertake our computations of the homology groups, we will compute their geometric dimension ${\\underline{\\underline{\\textrm{gd}}}\\ }$ with respect to the family of virtually cyclic groups.\n\n\\begin{prop}\n\\label{gd}\n\nThe geometric dimension of $A_n$ with respect of the family of virtually cyclic groups is 3.\n\n\\end{prop}\n\n\\begin{proof}\nAs seen in Lemma \\ref{lem:commensurators}, $A_n=\\langle a,b\\,|\\,\\textrm{prod}(a,b;n)=\\textrm{prod}(b,a;n) \\rangle$, contains subgroups isomorphic to $\\mathbb{Z}\\oplus\\mathbb{Z}$ (for example $\\Comm_{A_n}(\\langle a\\rangle))$. According to Example 5.21 in \\cite{LW}, this implies that ${\\underline{\\underline{\\textrm{gd}}}\\ } A_n\\geq 3.$\n\n\nOn the other hand, as the Artin groups $A_n$ are one-relator, Corollary 3 in \\cite{Deg16} implies that ${\\underline{\\underline{\\textrm{gd}}}\\ } A_n\\leq 3$, and then ${\\underline{\\underline{\\textrm{gd}}}\\ } A_n=3$.\n\\end{proof}\n\nAs stated, the strategy to compute $H_i^{vc}(A_n,M)$ goes through describing the different elements of the exact sequence of Proposition \\ref{mainMV}.\nWe first understand the terms $H_i^{\\mathcal{F}[H]}$ on Subsection \\ref{HiF[H]}.\nThen, in Subsection \\ref{Ktheory}, we give a more concrete description of the different homology groups when we take coefficients in the $K$-theory.\nFinally, in Subsection \\ref{morphism} {we study the homomorphisms} $g_1$ and $g_2$ of the exact sequence of Proposition \\ref{mainMV}  and prove our main theorems.\n\n\\subsection{Computing {the homology of the commensurators}}\\label{HiF[H]}\nLet us start the main calculations of this section.\nIt is clear from the previous Mayer-Vietoris sequence that the computations of the ordinary homology of the commensurators\n(which includes the homology of $A_n$, as the center of $A_n$ is virtually cyclic)\n and the homology of the commensurators with coefficients in $\\mathcal{F}[H]$ will give us valuable information about the Bredon homology of $A_n$ with respect to the family of virtually cyclic subgroups,\nso we will perform these calculations in the sequel.\nWe start with the ordinary homology, which is straightforward and depends on the shape of the commensurators:\n\n\\begin{itemize}\n\n\\item {If} $\\textrm{Comm}_{A_n}[H]\\simeq\\mathbb{Z}\\oplus \\mathbb{Z}$, we have $H_0(\\textrm{Comm}_{A_n}[H])=H_2(\\textrm{Comm}_{A_n}[H])=\\mathbb{Z}$, $H_1(\\textrm{Comm}_{A_n}[H])=\\mathbb{Z}\\oplus \\mathbb{Z}$, $H_i(\\textrm{Comm}_{A_n}[H])=0$ for $i>2$.\n\n\\item {If} $\\textrm{Comm}_{A_n}[H]\\simeq A_n$, see Proposition \\ref{OrdHom}.\n\n\n\\end{itemize}\n\n\nWe concentrate now in the case of $\\mathcal{F}[H]$, which we recall is the family of subgroups of $\\Comm_{A_n}[H]$ that are either finite or commensurable with $H$. Concretely, we intend to compute $H_i^{\\mathcal{F}[H]}(\\textrm{Comm}_{A_n}[H],M)$ for every non-trivial cyclic subgroup $H$ of $A_n$.\nWe have two cases, either $H\\cap Z(A_n)$ is trivial or not. We will use the following Convention throughout the paper without explictly refering to it.\n\n\\begin{conv}\nFor $[H]$ a class of infinite cyclic subgroups, we can always take a representative $H$ that is normal in $\\Comm_{A_n}[H]$.\n\n\nIf $H\\cap Z(A_n)$ is trivial, then Lemma \\ref{lem:commensurators} states  that $H\\unlhd \\Comm_{A_n}[H]$. Moreover, by  Lemma \\ref{lem:commensurators} we assume that  $H$ is a direct factor of  $\\Comm_{A_n}[H]$.\n\n\nIf $H\\cap Z(A_n)$ is non-trivial, then $[H]=[Z(A_n)]$ and we will assume in this case that we choose $Z(A_n)$ as the representative of $[H]$, and thus  $H\\unlhd \\Comm_{A_n}[H]$ again.\n\nWith this convention,  the projection $\\pi\\colon\\textrm{Comm}_{A_n}[H]\\rightarrow \\textrm{Comm}_{A_n}[H]\/H$ is well-defined and $\\Comm_{A_n}[H]\/H$ is isomorphic to $C_\\infty$, $C_\\infty*C_n$ or $C_2*C_{2n+1}$.\n\\end{conv}\n\n\nLet now $M$ be a module over the orbit category of $\\textrm{Comm}_{A_n}[H]$ with respect to $\\mathcal{F}[H]$, and $\\pi^{-1}M$ the induced module over the orbit category of $\\Comm_{A_n}[H]\/H$ with respect to the family of finite groups.\nThat is, for $K\\leqslant \\textrm{Comm}_{A_n}[H]\/H$ finite {it is defined}\n$$\\pi^{-1}M((\\Comm_{An}[H]\/H)\/K)\\mathrel{\\mathop\\mathchar\"303A}\\mkern-1.2mu= M(\\Comm_{A_n}[H]\/\\pi^{-1}(K)).$$ {Observe that this assignation gives rise to a natural transformation of functors $\\pi^{-1}M\\rightarrow M$. In the next result, which is essentially \\cite[Lemma 4.2]{DP}, but stated for homology instead of cohomology, we will see that this natural transformation induces an isomorphism in Bredon homology}. It will be a powerful tool in our computations.\n\\begin{prop}\n\\label{induced}\nLet $H$ be an infinite cyclic subgroup of $A_n$ normal in its commensurator.\nFor every $n\\geq 0$, and every module $M$ over the orbit category of $\\Comm_{A_n}[H]$ there is an isomorphism\n{$$H_i^{Fin}(\\Comm_{A_n}[H]\/H,\\pi^{-1} M)\\simeq H_i^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M).$$}\nMoreover, every model for $\\uE(\\Comm_{A_n}[H]\/H)$ is a model for $E_{\\mathcal{F}[H]}\\Comm_{A_n}H$, with the action induced by the quotient map $\\Comm_{A_n}[H]\\to \\Comm_{A_n}[H]\/H$.\n\\end{prop}\n\\begin{proof}\nThe argument here is taken from the proof of \\cite[Lemma 4.2]{DP}. We write it here to make our paper more self-contained.\n\nThe projection $\\pi\\colon \\Comm_{A_n}[H]\\to \\Comm_{A_n}[H]\/H$ maps the family $\\mathcal{F}[H]$ onto the family $Fin$ of finite subgroups of the quotient $\\textrm{Comm}_{A_n}[H]\/H$.\nMoreover, the pre-image $\\pi^{-1}(K)$ for any finite group $K$ of $\\Comm_{A_n}[H]\/H$ lies in $\\mathcal{F}[H]$.\nTherefore, $\\uE \\Comm_{A_n}[H]\/H$ is a model for $E_{\\mathcal{F}[H]}\\textrm{Comm}_{A_n}[H]$, with the action induced by the projection $\\pi$.\n\nConsider now the spectral sequence associated to the short exact sequence\n$1\\to H\\to \\Comm_{A_n}[H]\\stackrel{\\pi}{\\to} \\Comm_{A_n}[H]\/H\\to 1$ for homology \\cite{Mar02}.\nFor every module $M$ over the orbit category of $\\Comm_{A_n}[H]$ we have\n$$E^{p,q}_2 (M)= H_p^{{Fin}}(\\Comm_{A_n}[H]\/H, H_q^{\\mathcal{F}[H]\\cap \\pi^{-1}(-)}(\\pi^{-1}(-), M)),  $$\nwhich converges to $E^{p,q}_{\\infty} (M)=H_{p+q}^{\\mathcal{F}[H]}(\\Comm_{A_n}[H], M)$.\n\nObserve that $E^{p,q}_2$ is trivial for $q\\geq 1$, as for every finite subgroup $K<\\Comm_{A_n}[H]\/H$, $\\pi^{-1}(K)$ belongs to the family  $\\mathcal{F}[H]\\cap \\pi^{-1}(K)$, and then $H_q^{\\mathcal{F}[H]\\cap \\pi^{-1}(-)}(\\pi^{-1}(-),M)$ is zero. Thus, as $E^{i,0}_2 (M)=H_i^{Fin}(\\Comm_{A_n}[H]\/H,\n\\pi^{-1}M)$\nin the 0-th row of the sequence, we have  {$$H_i^{Fin}(\\Comm_{A_n}[H]\/H,\\pi^{-1} M)\\simeq H_i^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M).$$} for every $i\\geq 0.$\nWe are done.\n\\end{proof}\n\n The family of finite subgroups is easier to deal with than the family $\\mathcal{F}[H]$,  taking account of the previous proposition, the next step is to construct the corresponding classifying spaces for proper actions of the commensurators in $A_n$ modulo the corresponding subgroups. We have the following:\n\n\\begin{itemize}\n\n\n\\item If $\\textrm{Comm}_{A_n}[H]\\simeq\\mathbb{Z}\\oplus \\mathbb{Z}$, then there is a representative of the class $[H]$ that can be identified with one of these copies of $\\Z$; hence, we may assume that the inclusion $H\\hookrightarrow \\textrm{Comm}_{A_n}[H]=\\mathbb{Z}\\oplus \\mathbb{Z}$ is the inclusion of the first factor, and so $\\textrm{Comm}_{A_n}[H]\/H$ is isomorphic to $\\mathbb{Z}$. Then a model for $\\underline{{E}}(\\textrm{Comm}_{A_n}[H]\/H)$ is the straight line, and the action is by shifting.\n\n\\item If $\\textrm{Comm}_{A_n}[H]\\simeq A_n$, then $Z(A_n)$ is a representative of the class $[H]$.\nWe have seen that $\\textrm{Comm}_{A_n}[H]\/H=A_n\/Z(A_n)$ is an amalgamated product of two cyclic groups, depending its concrete shape on the parity of $n$.\nIn this case a tree model for ${\\uE}(\\textrm{Comm}_{A_n}[H]\/H)$ can be explicitly constructed.\n\nIf $n$ is even, we have that $A_{n}\/Z(A_{n})\\cong C_{\\infty}*C_{n}$.\nDenote $C_\\infty*C_{n}$ by $\\overline{A_n}$.\nLet $s$ be a generator of $C_\\infty$.\nThen Bass-Serre theorem guarantees that the graph with vertex set $\\overline{A_n}\/C_n$, edge set $\\overline{A_n}$ and incidence maps $\\iota(g) = gC_\\infty$ and $\\tau(g)=gsC_\\infty$ is a $\\overline{A_n}$-equivariant oriented tree.\n\n\nIf $n$ is odd, we have that $A_{n}\/Z(A_{n})\\cong C_{2}*C_{n}$. Denote $C_2*C_{n}$ by $\\overline{A_n}$.\nThen Bass-Serre theorem guarantees that the graph with vertex set $\\overline{A_n}\/C_2 \\sqcup \\overline{A_n}\/C_n$, edge set $\\overline{A_n}$ and incidence maps $\\iota(g) = gC_2$ and $\\tau(g)=gC_n$ is a $\\overline{A_n}$-equivariant oriented tree.\n\nNote that in both cases, the isotropy groups are the finite subgroups of $\\overline{A_n}$ and they fix exactly a vertex; and therefore these are  models for $\\uE \\overline{A_n}$.\n\\end{itemize}\n\n\n\n\nAs the classifying spaces of the commensurators that we have described are all 1-dimensional, we will use the following result of Mislin:\n\n\n\\begin{lem}[\\cite{MiVa03}, Lemma 3.14]\n\\label{1dim}\nSuppose that for a family $\\mathcal{F}$ of subgroups of a group $G$ there is a tree model $T$ for $E_{\\mathcal{F}}G$. Let $S_e$ be the stabilizer of the edge $e\\in T$ and $S_v$ the stabilizer of a vertex. Let $N$ be a coefficient module. Then $H_i^{\\mathcal{F}}(G,N)=0$ for $i>1$ and there is an exact sequence:\n\n$$ 0\\rightarrow H_1^{\\mathcal{F}}(G,N)\\rightarrow \\bigoplus_{[e]} N(G\/S_e) \\rightarrow \\bigoplus_{[v]} N(G\/S_v)\\rightarrow H_0^{\\mathcal{F}}(G,N)\\rightarrow 0,$$ where $[e]$ and $[v]$ run over the $G$-orbits of edges and vertices of $T$, respectively.\n\\end{lem}\n\n\\begin{rem}\n\\label{diff}\nIt is interesting to remark that the middle map $\\bigoplus_{[e]} N(G\/S_e) \\rightarrow \\bigoplus_{[v]} N(G\/S_v)$ is induced by the (formal) border operation defined by $\\partial[e]=[v_1]-[v_2]$, being $v_1$ and $v_2$ vertices of a representative $e$ of $[e]$. The tree is assumed to be oriented.\n\\end{rem}\n\nIn particular we obtain:\n\n\\begin{cor}\n\\label{zerohomology}\nFor every $n\\geq 2$, $i\\geq 2$, $H\\leqslant  A_n$ infinite cyclic and every module $M$ over the orbit category with respect to the family $\\mathcal{F}[H]$, we have $H_i^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M)=0$.\n\\end{cor}\n\\begin{proof}\nRecall that by Proposition \\ref{induced} $H_i^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M)\\simeq H_i^{Fin}(\\Comm_{A_n}[H]\/H,\\pi^{-1}M)$. As $\\uE\\Comm_{A_n}[H]\/H$ is 1-dimensional, the result follows by Lemma \\ref{1dim}.\n\\end{proof}\n\nSo it remains to compute $H_0^{\\mathcal{F}[H]}(\\textrm{Comm}_{A_n}[H],M)$ and $H_1^{\\mathcal{F}[H]}(\\textrm{Comm}_{A_n}[H],M)$. We assume here a general coefficient module $M$, but the reader may keep in mind that our case of interest is $M(A_n\/-\n)=K_q (R[-])$, $q\\in\\Z$.\n\n\n\\begin{prop}\\label{homology0and1}\nLet $A_n =\\langle a, b \\,\\textrm{ }|\\textrm{ } \\mathrm{prod}(a,b;n) = \\mathrm{prod}(b,a;n)\\rangle$\n be a dihedral Artin group, let $H\\leqslant A_n$ be an infinite cyclic group that is normal in its commensurator, let $\\pi\\colon \\Comm_{A_n} [H]\\to \\Comm_{A_n}[H]\/H$ be the natural projection.\nLet $M$ be a module over the orbit category of $\\Comm_{A_n}[H]$.\nThen\n\\begin{enumerate}\n\\item[(i)] If $H\\cap Z(A_n)$ is trivial, then\n$$H_0^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M)=H_1^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M)=M(\\Comm_{A_n}[H]\/H).$$\n\\item[(ii)] If $H=Z(A_{n})$, $n=2k+1$  and if $f\\colon M(A_{n}\/Z(A_n))\\to M(A_n\/\\langle b(ab)^k\\rangle ) \\oplus M(A_n\/\\langle ab \\rangle)$ {is} induced by the  natural projections,  we have\n$$H_0^{\\mathcal{F}[H]}(A_{n},M)= \\mathrm{coker} f\\quad \\text { and } \\quad H_1^{\\mathcal{F}[H]}(A_{n},M)= \\ker  f.$$\n\n\\item[(iii)] If $H=Z(A_n)$ and $n=2k$, we have\n $$H_0 ^{\\mathcal{F}[H]}(A_{n},M)=M(A_{n}\/\\langle ab \\rangle)\\quad \\text{  and  }\\quad H_1^{\\mathcal{F}[H]}(A_{n},M)= M(A_{n}\/Z(A_n)).$$\n\\end{enumerate}\nNote that in cases (ii) and (iii), $A_n=\\Comm_{A_n}[H]$.\n\\end{prop}\n\n\\begin{proof}\nIn the case $H\\cap Z(A_n)$ is trivial, the classifying space $\\uE\\Comm_{A_n}[H]\/H$ is the real line,  there is one orbit of edges and one orbit of vertices, and the action is free. Then by Remark \\ref{diff}, the central map in that exact sequence is trivial and by Lemma \\ref{1dim},\n$$H_1^{Fin}(\\Comm_{A_n}[H]\/H,\\pi^{-1}M)= \\pi^{-1}M((\\Comm_{A_n}[H]\/H)\/\\{1\\})$$ and\n$$H_0^{Fin}(\\Comm_{A_n}[H]\/H,\\pi^{-1}M)=\\pi^{-1}M((\\Comm_{A_n}[H]\/H)\/\\{1\\}).$$\nUsing  Proposition \\ref{induced}, case (i) follows.\n\nLet us discuss the case $H = Z(A_n)$. Now $\\Comm_{A_n}[H]\/H=A_n\/Z(A_n)$ is an amalgamated product of two finite cyclic groups or of a finite cyclic and an infinite cyclic group, depending on the parity of $n$. In both cases, there is a tree model for the classifying space for proper actions of the commutator modulo $H$. Denote $\\Comm_{A_n}[H]\/H$ by $\\overline{A_n}$ for brevity.\n\nWe consider first the case $n=2k+1$.\nHere $\\overline{A_{n}}=S*L$, with $S=\\langle b(ab)^k Z(A_{n})\\rangle \\cong C_2$ and $L=\\langle ab Z(A_{n}) \\rangle \\cong C_{n}$.\nThe Bass-Serre tree has two orbits of vertices, with stabilizers conjugated to $S$ or $L$, and one free orbit of edges.\nThen, by Lemma \\ref{1dim}\n\n\\begin{equation}\n\\label{MVodd}\nH_1^{Fin}(\\overline{A_n},\\pi^{-1} M)\\hookrightarrow \\pi^{-1}M(\\overline{A_{n}}\/\\{1\\})\\stackrel{f}{\\rightarrow}\n \\end{equation}\n$$\\stackrel{f}{\\rightarrow} \\pi^{-1}M(\\overline{A_{n}}\/S)\\oplus \\pi^{-1}M(\\overline{A_{n}}\/L)\\twoheadrightarrow H_0^{Fin}(\\overline{A_{n}},\\pi^{-1}M).\n$$\n Hence, we obtain that $$H_0^{Fin}(\\overline{A_{n}},\\pi^{-1}M)=(\\pi^{-1}M(\\overline{A_{n}}\/S)\\oplus \\pi^{-1}M(\\overline{A_{n}}\/L))\/\\textrm{Im }f$$ and $$H_1^{Fin}(\\overline{A_{n}},\\pi^{-1}M)=\\textrm{Ker }f.$$\n  Observe that the two components of $f$ are induced by the images of the projections $\\overline{A_{n}}\/\\{1\\}\\rightarrow \\overline{A_{n}}\/S$ and $\\overline{A_{n}}\/\\{1\\}\\rightarrow \\overline{A_{n}}\/L$ by the functor $\\pi^{-1}M$. The case (ii) follows by applying Proposition \\ref{induced}.\n\n\n\n\nConsider now the  case  $n=2k$.\nIn this situation $\\overline{A_{n}}=A_{n}\/Z(A_{n})$ is a free product $G=S\\ast L$, with $S=\\langle b Z(A_n) \\rangle \\simeq C_{\\infty}$ and $L=\\langle ab Z(A_n) \\rangle\\simeq C_n$.\nWe denote by $s$  the element  $b Z(A_n)$.\nThe Bass-Serre tree for this group has\none orbit of vertices with stabilizers conjugate to $L$ and one orbit of edges with trivial stabilizers.\n\nThe exact sequence of Proposition \\ref{1dim} has the following form:\n\\begin{equation}\n\\label{MVeven}\nH_1^{Fin}(\\overline{A_n},\\pi^{-1}M)\\hookrightarrow M(\\overline{A_{n}}\/\\{1\\}) \\stackrel{f}{\\rightarrow}  \\pi^{-1}M(\\overline{A_{n}}\/L) \\twoheadrightarrow H_0^{Fin}(\\overline{A_{n}},\\pi^{-1}M).\n\\end{equation}\nThe function $f$ is induced by  $g\\{1\\}\\in A_{2n}\/\\{1\\} \\mapsto gsL- gL$ by the functor $M$.\nAnd hence $f$ is trivial.\n\nHence, we deduce that $$H_1^{Fin}(\\overline{A_{n}},\\pi^{-1}M)= \\pi^{-1}M(\\overline{A_{n}}\/\\{1\\})$$ and $$H_0^{Fin}(\\overline{A_{n}},\\pi^{-1}M)=\\pi^{-1}M(\\overline{A_{n}}\/L).$$\nThe result now follows by applying Proposition \\ref{induced}.\n\\end{proof}\n\nObserve that taking into account that $\\overline{A_{2n}}$ is one-relator, this computation agrees with the result of Corollary 3.23 in \\cite{MiVa03}, in the case of $M=R_\\mathbb{C}$, the complex representation ring.\n\n\n\\subsection{ Coefficients in  $K$-theory groups}\n\\label{Ktheory}\nFrom the point of view of Farrell-Jones conjecture, the case of interest is the coefficients in the $K$-theory $K_q(R)$ of the group ring. We now describe all the homology groups of the exact sequence of Proposition \\ref{mainMV} with  coefficients in this module, except the groups $H^{vc}(A_n,K_q(R[-]))$ which will be studied in the next section.\n\nWe begin recalling the following definiton.\n\\begin{defn}\\label{defn:regular}\nA {\\it regular} ring is a\n commutative noetherian ring such that in the localization at every prime ideal, the Krull dimension of the maximal ideal is equal to the cardinal of a minimal set of generators.\n\\end{defn}\nExamples of regular rings include fields (of dimension zero) and Dedekind domains. If $R$ is regular then so is the polynomial ring $R[x]$, with dimension one greater than that of $R$.\n\n\n\n Before stating our results, we also need to recall the \\emph{Bass-Heller-Swan decomposition} in $K$-theory, which permits to {decompose} the $K$-theory of $R[\\Z]$. For thorough approaches to algebraic $K$-theory the reader is referred to \\cite{Bas68}, \\cite{Wei13} or \\cite{Luc21}.\n\n\\begin{thm}[\\cite{Sri91}, Theorem 9.8]\n\\label{BHS}\nGiven a ring $R$ and $q\\in\\Z$, there exists an isomorphism $$K_q(R[\\Z])\\simeq K_q(R)\\oplus K_{q-1}(R)\\oplus NK_q(R)\\oplus NK_q(R),$$ which is natural in the ring $R$.\n\\end{thm}\n\n{The additional terms $NK_q(R)$ are called the Nil-terms, and $NK_n(R)$ is defined as the kernel of the homomorphism induced in $K_q$ by the homomorphism $R[t]\\rightarrow R$ which sends $t$ to $1$. These terms vanish for a regular ring $R$, see \\cite[Section 9]{Sri91}.}\n\n{In our computations it will be important to understand the endomorphism $ind_n$ of $K_q(R[\\Z])$ induced by multiplication by $n$ in $\\Z$. The references for the sequel are \\cite[Section 2]{HaLu12}, \\cite{Sti82} and \\cite{Wei81}.}\n\nAccording to Bass-Heller-Swan-decomposition, $ind_n$ can be seen as a homomorphism\n$$ind_n\\colon K_q(R)\\oplus K_{q-1}(R)\\oplus NK_q(R)\\oplus NK_q(R)\\rightarrow K_q(R)\\oplus K_{q-1}(R)\\oplus NK_q(R)\\oplus NK_q(R).$$\n\\noindent{Here, by naturality of the decomposition, the image of $ind_n|_{K_q(R)}$ lies inside $K_q(R)$, the image of $ind_n|_{K_{q-1}(R)}$ lies inside $K_{q-1}(R)$ and analogously for the Nil-terms. Now, the restriction of  $ind_n$ to $K_q(R)$ is the identity and the restriction of  $ind_n$ to $K_{q-1}(R)$ is multiplication by $n$.  By Farrell's trick \\cite{Far77}, $ind_n$ admits a transfer $res_n$ such that $res_n\\circ ind_n$ is multiplication by $n$.\nThis implies the following:}\n\n{\\begin{prop} \\label{kernel}\nIn the previous notation, the kernel of $ind_n$ is isomorphic to a direct sum $T_1(K_q(R))\\oplus T_2(K_q(R))$, where $T_1(K_q(R))$ is the $n$-torsion subgroup of $K_{q-1}(R)$ and $T_2(K_q(R))$ is a subgroup of the  $n$-torsion subgroup of $NK_q(R)\\oplus NK_q(R).$\n\\end{prop}}\n\n{It should be remarked that the restriction of $ind_n|_{NK_q(R)}$ (sometimes call the \\emph{Frobenius map} in the literature) is related to the action of big Witt vectors in $NK_q(R)$, and is hard to describe in general. Anyhow, the previous proposition will be important when computing the Bredon homology of Artin groups of dihedral type.}\n\n\n\nThe following  propositions\n{particularize} our previous results when the coefficient module is $K$-theory.\n\n For the sake of clarity, we maintain the notation from Proposition \\ref{mainMV}, although we are aware that sometimes can be found redundant.\n\n\\begin{prop}\n\\label{Anordinary}\n\nLet us consider an Artin group $A_n$ of dihedral type, $q\\in\\mathbb{Z}$, $R$ a ring. Then we have:\n\n\\begin{itemize}\n\n\\item $H_0^{Fin}(A_n,K_q(R[-]))=K_q(R)$.\n\n\\item $H_1^{Fin}(A_n,K_q(R[-]))=K_q(R)\\oplus K_q(R)$ if $n$ is even\n\n\\item $H_1^{Fin}(A_n,K_q(R[-]))=K_q(R)$ if $n$ is odd.\n\n\\item $H_2^{Fin}(A_n,K_q(R[-]))=K_q(R)$ if $n$ is even\n\\item $H_2^{Fin}(A_n,K_q(R[-]))=0$ if $n$ is odd.\n\n\\item $H_i^{Fin}(A_n,K_q(R[-]))=0$ if $i\\geq 3$.\n\n\\end{itemize}\n\n\\end{prop}\n\n\\begin{proof}\n\nAs $A_n$ is torsion-free, Bredon homology with respect to the family of finite groups is in fact ordinary homology. The proposition is then obtained by applying Universal Coefficient Theorem \\cite[Theorem 3A.3]{Hat02} to the ordinary homology groups of $A_n$ (Proposition \\ref{OrdHom}), taking into account that the latter are torsion-free and that the $K$-theory groups are abelian.\n\\end{proof}\n\nWe consider now {the} Bredon homology of the commensurators.\n\n\\begin{prop}\n\\label{FinComm}\n\nLet us consider an Artin group $A_n$ of dihedral type, $q\\in\\mathbb{Z}$, $R$ a ring. Let $\\Comm_{A_n}[H]$ be the commensurator of a virtually cyclic group in $A_n$. Then:\n\n\\begin{itemize}\n\n\n\\item If $\\Comm_{A_n}[H]=\\mathbb{Z}\\oplus\\mathbb{Z}$, then\n\n$$H_i^{Fin\\cap \\Comm_{A_n}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=\\begin{cases} K_q(R) & \\text{ if $i=0,2$}\\\\\nK_q(R)\\oplus K_q(R) & \\text{ if i =1}\\\\\n0 & \\text{otherwise}. \\end{cases}$$\n\n\n\\item If $\\Comm_{A_n}[H]=A_n$, then $$H_i^{Fin\\cap \\Comm_{A_n}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=H_i^{Fin}(A_n,K_q(R))$$ and this case was described in the previous proposition.\n\n\n\\end{itemize}\n\n\\end{prop}\n\n\\begin{proof}\n\nGiven a commensurator $\\textrm{Comm}_{A_n}[H]$, its Bredon homology with respect to the family $Fin\\cap \\textrm{Comm}_{A_n}[H]$ is its ordinary homology. Then, {as in the previous proposition}, the result follows by applying the Universal Coefficient Theorem to the ordinary homology groups of the commensurators.\n\\end{proof}\n\nWe now undertake the remaining case. In item (3) we follow the notation of Proposition \\ref{kernel}.\nMoreover, we use the following notation.\n\\begin{nt}\\label{not: CKR}\nFor $n$ odd, we denote by $C(K_q(R))$ the cokernel of the homomorphism $$\\tilde{f}\\colon K_q(R[\\Z])\\to K_q(R[\\Z])\\oplus K_q(R[\\Z])$$ induced by\n$f\\colon R[\\Z]\\rightarrow R[\\Z]\\oplus  R[\\Z]$ induced in the first component by multiplication by 2 and in the second by multiplication by $n=2k+1$.\n\\end{nt}\nBy Bass-Heller-Swan decomposition, $C(K_q(R))$ this is a quotient of $K_q(R)\\oplus K_q(R)\\bigoplus(\\oplus_{i=1}^2 K_{q-1}(R))\\bigoplus (\\oplus_{i=1}^4 NK_q(R))$. Moreover, $\\tilde{f}$ restricts to the component corresponding to  $K_q(R)$ of the Bass-Heller-Swan decomposition of $K_q(R[\\Z])$ to the diagonal map to $K_q(R)\\oplus K_q(R)$, a component  of $K_q(R[\\Z])\\oplus K_q(R[\\Z])$.\nThus, $C(K_q(R))$ can be viewed as a quotient of $K_q(R)\\bigoplus(\\oplus_{i=1}^2 K_{q-1}(R))\\bigoplus (\\oplus_{i=1}^4NK_q(R))$ that can be identified in many cases, see Section \\ref{Sect:concrete}.\n\n\n\n\\begin{prop}\n\\label{FHcomm}\nLet $A_n$  be an Artin group of dihedral type, $q\\in\\mathbb{Z}$, $R$ a ring.\nConsider for every non-trivial virtually cyclic $H\\leqslant A_n$ and the family $\\mathcal{F}[H]$. Then:\n\n\n\n\\begin{enumerate}\n\n\n\\item If $i\\geq 2$, then $$H_i^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=0.$$\n\n\\item If $H\\cap Z(A_n)$ is trivial, then $$H_0^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=H_1^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=K_q(R[\\mathbb{Z}]).$$\n\n\\item If $n$ is odd and $\\Comm_{A_n}[H]=A_n$, then $$H_0^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=C(K_q(R))$$ and $$H_1^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=T_1(K_q(R))\\oplus T_2(K_q(R)).$$\n\n\\item If $n$ is even and $\\Comm_{A_n}[H]=A_n$, then $$H_0^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=H_1^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],K_q(R[-]))=K_q(R[\\mathbb{Z}]).$$\n\n\\end{enumerate}\n\n\\end{prop}\n\n\n\\begin{proof}\n\nWe will check every item separately.\n\n\\begin{enumerate}\n\n\n\n\\item It follows straightforward from Corollary \\ref{zerohomology}.\n\n\\item The claim follows from item (i) in Proposition \\ref{homology0and1}, taking into account that $H$ is infinite cyclic an then $K_q(R[H])=K_q(R[\\mathbb{Z}])$.\n\n\\item Let $n=2k+1$. Consider the homomorphism $f\\colon M(A_{n}\/Z(A_n))\\to M(A_n\/\\langle b(ab)^k\\rangle ) \\oplus M(A_n\/\\langle ab \\rangle)$ defined in item (2) of Proposition \\ref{homology0and1}.\nTaking $M(A_n\/-)=K_q(R[-])$, we obtain a homomorphism $f_K:K_q(R[Z(A_n)])\\to K_q(R[\\langle b(ab)^k\\rangle])\\oplus K_q(R[\\langle ab \\rangle])$.\nObserve that the two components of $f_K$ are respectively induced by the inclusions $Z(A_n)\\hookrightarrow \\langle b(ab)^k\\rangle$ and $Z(A_n)\\hookrightarrow \\langle ab \\rangle$, which are both inclusions $\\Z\\hookrightarrow \\Z$ given respectively by multiplication by 2 and multiplication by $n$.\nThen, $f_K$ can be seen as the homomorphism $K_q(R[\\Z])\\rightarrow K_q(R[\\Z])\\oplus K_q(R[\\Z])$ induced by each multiplication in the corresponding component.\nAccording to Proposition \\ref{kernel}, the kernel of the first component of this homomorphism is equal to $T_1(K_q(R))\\oplus T_2(K_q(R))$, while its cokernel is $C(K_q(R))$ by definition.\nThe result now follows from item (2) in Proposition \\ref{homology0and1}.\n\n\n\n\\item  In this case $A_n=\\Comm_{A_n}[H]$. The claim follows from item (3) in Proposition \\ref{homology0and1}, taking into account that $H$ is infinite cyclic and  then $K_q(R[H])=K_q(R[\\mathbb{Z}])$.\n\\end{enumerate}\n\\end{proof}\n\n\\subsection{Understanding the homomorphisms of Proposition \\ref{mainMV}}\n\\label{morphism}\nIn our way to describe the Bredon homology of $A_n$ with respect to the family of virtually cyclic groups, we should describe to some extent the homomorphisms that appear in the Mayer-Vietoris sequence of Proposition \\ref{mainMV}.\n\n In the following we will use without explicit mention the previous three propositions, which identify the terms of the exact sequence. We also maintain the name of the homomorphisms in the sequence.\nNote that the superscript of the homomorphism $g^k_i$ specifies the degree of the homology in the source of $g^k_i$.\nMoreover,  when we need to refer to the $j$-th component of $g_i^k$, we will write $g_{ij}^k$.\nFor example, $g_{22}^1$ is the second component of the homomorphism $g_2^1$ defined over the first homology group.\n\nTo prove our results, we need to analyze in detail the following homomorphism\n\\begin{equation}\\label{g2}\n\\bigoplus_{[H]}H_i^{Fin\\cap \\Comm_{A_n}[H]}(\\Comm_{A_n}[H],M) \\stackrel{g_2^i}{\\rightarrow} \\end{equation}\n$$ \\stackrel{g_2^i}{\\rightarrow}\\left(\\bigoplus_{[H]}H_i^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M)\\right)\\oplus H_i^{Fin}(A_n,M).\n$$\n\n\nThe homomorphism $g_{21}^i$ is induced by  the  vertical left arrow $\\sqcup_{[H]\\in I}\\mathrm{id}_{A_n}\\times_{\\Comm_{A_n}[H]}f_{[H]}$ of L\\\"{u}ck-Weiermann push-out  \\eqref{eq:Luck-Weiermann} below (see also Theorem \\ref{maintheorem}).\nIn turn, the homomorphism $g_{22}^i$ is induced by the inclusion in the upper horizontal arrow of the push-out  \\eqref{eq:Luck-Weiermann}\n\\begin{equation}\n\\label{eq:Luck-Weiermann}\n\\xymatrix{  \\coprod_{[H]\\in I}A_n\\times_{\\Comm_{A_n}[H]}\\uE \\Comm_{A_n}[H] \\ar[r]^{\\hspace{2cm} i} \\ar[d]^{\\coprod_{[H]\\in I}id_{A_n}\\times_{\\Comm_{A_n}[H]}f_{[H]}}  & \\uE A_n \\ar[d] \\\\\n\\coprod_{[H]\\in I}A_n\\times_{\\Comm_{A_n}[H]}E_{{\\mathcal{F}} [H]}\\Comm_{A_n}[H]  \\ar[r] & X. }\n\\end{equation}\n\n\n\n\nWe will decompose the homomorphism $g_2^i$ of Equation \\eqref{g2} into homomorphisms $g_{2[H]}^i$ which are the restriction of $g_2^i$ to the factor of the domain corresponding to $[H]$. We will write\n\n\\begin{equation}\n\\label{eq:g}\ng_{2[H]}^i\\colon H_i^{Fin\\cap \\Comm_{A_n}[H]}(\\Comm_{A_n}[H],M)\\to H_i^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M)\\oplus H_i^{Fin}(A_n,M).\n\\end{equation}\nNote that we are making a slight abuse of notation, as the real codomain of $g_{2[H]}^i$ is the same as {that of} $g_2^i$, but we have chosen to write only the subgroup where the image of $g_{2[H]}^i$ lies.\nMoreover, when needed, we will further decompose $g^i_{2[H]}$ into $g^i_{21[H]}\\oplus g^i_{22[H]}$ indicating the different factors of the image of $g^i_{2[H]}$. Observe that this notation is coherent with the previous one.\n\n\nAs all the commensurators are torsion-free, $H_i^{Fin\\cap \\Comm_{A_n}[H]}(\\Comm_{A_n}[H],M)$ is ordinary homology with coefficients in $M(\\Comm_{A_n}[H]\/\\{1\\})$.\n\n\n\n\nWe now analyze the homomorphism $g_2^i$ on the case $M=K_q(R[-])$. Before that, we introduce the following notation.\n\\begin{nt}\nWe will denote\n$N_q^{[H]} =K_{q-1}(R)\\oplus NK_q(R)\\oplus NK_q(R)$, where the superindex means that this group is associated to a concrete commensurability class $[H]$.\n\\end{nt}\nWe now will give more detailed descriptions of the homomorphism $g_2^i$ and the cokernels introduced above.\n\n\\subsubsection{{\\bf The homomorphism $g_2^2$ when $n$ odd:}}\n\\label{NqH}\nIn this case we have that the codomain of $g_2^2$ is  $(\\bigoplus H_2^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M))\\oplus H_2^{Fin}(A_n,M)=\\{0\\}$, and therefore for all commensurability classes of infinite  cyclic subgroups $[H]$ the homomorphisms $g_{2[H]}^2$ are trivial.\n\n\\subsubsection{{\\bf The homomorphism $g_2^2$ when $n$ even:}}\nIn this case we have that the codomain of $g_2^2$ is  $(\\bigoplus H_2^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M))\\oplus H_2^{Fin}(A_n,M)=(\\bigoplus_{[H]}\\{0\\})\\oplus K_q(R)$.\nMoreover, by Proposition \\ref{Anordinary} and Proposition \\ref{FinComm},\nthe domain of all $g_{2[H]}^2$ are the same, {and} we have\n$$g_{2[H]}^2 \\colon K_q(R)\\to \\{0\\}\\oplus K_q(R).$$\nNote that $g_{22[H]}^2$ is induced by the inclusion given by the upper arrow in the push-out.\nWhen $H=Z(A_n)$, we have that $\\Comm_{A_n}[H]=A_n$ and therefore this inclusion is the identity.\nSince $g_{22[Z(A_n)]}^2$ is surjective, we have that $g_2^2$ is surjective.\n\n\n\n\n\\subsubsection{{\\bf The homomorphism $g_2^1$ when $n$ is odd:}}\n\\label{g21}\n{Before we describe this case, we need to make some considerations.\nRecall that we assume that $H$ is normal in $\\textrm{Comm}_{A_n}[H]$.\nObserve that in the models described in Section \\ref{HiF[H]} for $\\underline{E}(\\textrm{Comm}_{A_n} [H]\/H)$, the stabilizers of the edges are trivial for every $H$.\nThis means that if we consider the space $\\underline{E}(\\textrm{Comm}_{A_n} [H]\/H)$ as a model for $E_{{\\mathcal{F}} [H]}\\Comm_{A_n}[H]$ (see Proposition \\ref{induced}) the stabilizers of the edges are always isomorphic to $H$.\nOn the other hand, in any model of $\\underline{E}\\textrm{Comm}_{A_n} [H]$ the stabilizer of the edges should be trivial, as the action of $\\textrm{Comm}_{A_n} [H]$ is free.\nThen, the homomorphism induced by $f_{[H]}:\\underline{E}\\textrm{Comm}_{A_n} [H]\\rightarrow E_{{\\mathcal{F}} [H]}\\Comm_{A_n}[H]$ in the first chain group of the Bredon complex with coefficients in a module $M$ takes every copy of $M(\\textrm{Comm}_{A_n} [H]\/{1})$ to a copy of $M(\\textrm{Comm}_{A_n}[H]\/H)$ with the homomorphism induced by the inclusion of the trivial group in $H$. In particular, if $M=K_q(R[-])$ for some $q$, the corresponding homomorphism $K_q(R)\\rightarrow K_q(R[\\mathbb{Z}])$ is given by the inclusion of $K_q(R)$ in the corresponding piece of the Bass-Heller-Swan decomposition of $K_q(R[\\mathbb{Z}])$. This fact will be very useful in the sequel}.\n\nIn the following we use the notation of Proposition \\ref{kernel} when needed.\nWe have that $$g_{2[Z(A_n)]}^1\\colon K_q(R)\\to T_1(K_q(R))\\oplus T_2(K_q(R))\\oplus K_q(R)$$ and for $H$ nontrivial, $[H]\\neq [Z(A_n)]$\n$$g_{2[H]}^1\\colon  K_q(R)\\oplus K_q(R)\\to   K_q(R[\\Z]) \\oplus K_q(R).$$\n\n\n\n\nNote that there are infinitely many commensurability classes of infinity cyclic subgroups different from the $[Z(A_n)]$ and hence infinitely many terms of this  kind.\nWe now examine the cases $[H]\\neq [Z(A_n)]$ and $[H]=[Z(A_n)]$ separately.\n\nIf $[H]=[Z(A_n)]$ then we have $\\Comm_{A_n}[H]=A_n$.\n\n In order to describe $g^{1}_{21[Z(A_n)]}$, consider the homomorphism $$C_1^{Fin}(\\underline{E}\\textrm{Comm}_{A_n}[H],K_q)\\rightarrow C_1^{{\\mathcal{F}} [H]}(E_{{\\mathcal{F}} [H]}\\Comm_{A_n}[H],K_q)$$ at the level of Bredon chains that induces $g^{1}_{21[Z(A_n)]}$ in homology. According to the previous considerations, this homomorphism is given by the inclusion $i:K_q(R)\\hookrightarrow K_q(R[H])=K_q(R[\\Z])$ given by Bass-Heller-Swan decomposition. But by item (3) of Proposition \\ref{FHcomm}, the image of this homomorphism is trivial in $H_1^{{\\mathcal{F}} [H]}(E_{{\\mathcal{F}} [H]}\\Comm_{A_n}[H],K_q)$, and hence $g^{1}_{21[Z(A_n)]}$ is also trivial. In turn, $g^{1}_{22[Z(A_n)]}$ is the identity.\n\n\n\nIf $[H]\\neq[Z(A_n)]$ then $\\Comm_{A_n}[H]=\\Z^2$, and $\\Comm_{A_n}[H]\/H$ is isomorphic to $\\Z$.\nIn this case $g_{21 [H]}^1$ is given by {a} homomorphism $K_q(R)\\oplus K_q(R)\\rightarrow K_q(R[\\Z])$, where we assume that the first component of the domain corresponds to $H$ and the second to $Z(A_n)$. As the homomorphism $g_{21 [H]}^1$ is induced in homology by the quotient homomorphism $\\textrm{Comm}_{A_n}[H]\\rightarrow \\textrm{Comm}_{A_n}[H]\/H$, the previous results imply that the first component of $g_{21 [H]}^1$ is trivial, while the second, which corresponds to the center, identifies the copy of $K_q(R)$ in the Bass-Heller-Swan decomposition of $K_q(R[\\Z])$.\n\n\n\n On the other hand,  $g_{22[H]}^1\\colon (\\Z\\oplus\\Z)\\otimes K_q(R)\\rightarrow \\Z\\otimes K_q(R)$  is defined by the abelianization of the inclusion $H_1(\\Comm_{A_n}[H])\\rightarrow H_1(A_n)$ in the first component of the tensor product and by the identity in the second.\n\nNow since the image of $g^1_{2[Z(A_n)]}$ is precisely given by the copy of $K_q(R)$ that corresponds to $H_1(A_n,K_q)$, the previous computations imply that the cokernel of $g^1_2$ is then equal to $(\\bigoplus_{[H]\\neq [Z(A_n)]}N_q^{[H]})\\oplus T_1(K_q(R))\\oplus T_2(K_q(R))$.\n\n\\subsubsection{ {\\bf The homomorphism $g_2^1$ when $n$ is even:}}\n\n\nThis homomomorphism is defined in equation \\eqref{eq:g} and, according to Propositions {\\ref{Anordinary}, \\ref{FinComm} and \\ref{FHcomm}}, its first component is given by\n$$g_{2[H]}^1 \\colon K_q(R)\\oplus K_q(R)\\to K_q(R[\\Z]) \\oplus ( K_q(R)\\oplus K_q(R))$$\nfor every commensurability class $[H]$. Let us describe this component with more detail.\n\n\n\n\nWe consider first the case $[H] \\neq [Z(A_n)]$. Here, the same argument as in the odd case proves that the component of $g_{21[H]}^1$ given by the center is the inclusion $K_q(R)\\hookrightarrow K_q(R[\\Z])$ via Bass-Heller-Swan decomposition, and the other component is trivial.\n\nAlso when $[H]\\neq [Z(A_n)]$ the homomorphism $g_{22[H]}^1$ is identified (via Proposition \\ref{FinComm} and the Universal Coefficient Theorem) with a homomorphism $(\\Z\\oplus \\Z) \\otimes K_q(R)\\rightarrow (\\Z\\oplus\\Z)\\otimes K_q(R)$, which comes, as above, from tensoring with $K_q(R)$ the homomomorphism $H_1(\\Comm_{A_n}[H])\\rightarrow H_1(A_n)$ given by abelianization of the inclusion of the commensurator in $A_n$.\n\nNow we consider $[H]=[Z(A_n)]$. As in the previous case, the homomorphism $g_{21[H]}^1$ can be described as:\n$$H_1^{Fin}(A_n,K_q(-))\\to H_1^{{\\mathcal{F}} [H]}(A_n,\\pi^{-1} K_q(-)) \\cong H_1^{Fin}(A_n\/Z(A_n),\\pi^{-1} K_q(-)).$$\nNow recall from Proposition \\ref{homology0and1} that\n$$H_1^{Fin}(A_n\/Z(A_n),\\pi^{-1} K_q(-))=\\pi^{-1}K_q(R[\\{1\\}])=K_q(R[Z(A_n)])=K_q(R[\\Z]),$$\nwhere $Z(A_n)$ is interpreted here as the stabilizer of the unique $A_n$-class of edges in the model of $E_{{\\mathcal{F}} [H]}A_n$ described in Section \\ref{HiF[H]}.\nOn the other hand, the two copies of $K_q(R)$ in $H_1(A_n,K_q(R))$ come from taking values of the module $K_q(R[-])$ on the trivial group, interpreted as the stabilizer of two different $A_n$-classes of edges in a model of $EA_n$. Again taking into account our previous considerations about stabilizers, the two components of $g_{21[Z(A_n)]}^1 \\colon K_q(R)\\oplus K_q(R)\\to K_q(R[\\Z])$ induce inclusion of the $K_0(R)$ via Bass-Heller-Swan decomposition. On the other hand, it is clear that $g_{22[Z(A_n)]}^1$ is the identity.\n\nAs $H_1(A_n)$ is a free abelian group of rank 2 generated by the images of $a$ and $b$ under abelianization, we may assume that the two copies of $K_q(R)$ in the image of $g_{2[H]}^1$ correspond respectively to these two copies of $\\Z$, after tensoring with $K_q(R)$. Now observe that if $H=\\langle a\\rangle$, the image of the restriction of  $g_{22[H]}^1$ to the homology of its commensurator is exactly the first of the two copies of $K_q(R)$, while the image of the restriction of $g_{22[H]}^1$ to the homology of the commensurator of $H=\\langle b\\rangle$ is the other copy. As the restrictions of $g_{21[H]}^1$ to the homology of these commensurators are trivial, we obtain that $\\{0\\}\\oplus K_q(R)\\oplus K_q(R)$ lies in the image of $g_{2[H]}^1$. In fact, the description of $g_{21[H]}^1$ for $H=Z(A_n)$ implies that the copy of $K_q(R)$ inside $K_q(R[\\Z])$ is also in the image, and now it is easy to conclude that the image of $g_{2[H]}^1$ is in fact $(\\bigoplus_{[H]}) K_q(R))\\oplus K_q(R)\\oplus K_q(R)$, corresponding the big direct sum to the copies of $K_q(R)$ included in each copy of $K_q(R[\\Z])$, and the remaining two copies corresponding to the ordinary homology of $A_n$. In particular, we have that $\\textrm{coker } g_2^1=\\bigoplus_{[H]} N_q^{[H]}.$\n\n\n\n\n\n\\subsubsection{{\\bf The homomorphism $g_2^0$ for every $n$:}}\n\nSimilar considerations to those of the beginning of Section \\ref{g21} hold here.\nIf $x$ is a vertex of $\\underline{E}\\textrm{Comm}_{A_n}[H]$ such that the stabilizer of $f_{[H]}(x)$ is infinite cyclic (here $f_{[H]}$ is the function of \\eqref{eq:Luck-Weiermann}), then the induced map $K_q(R)\\rightarrow K_q(R[\\Z])$ induces the injection in the correspondent component of Bass-Heller-Swan decomposition. This is always the case except when $n$ is odd, $H=Z(A_n)$ and $f_{[H]}(x)$ has the shape $gC_{\\infty}$ if we consider $\\underline{E}(\\textrm{Comm}_{A_n} [H]\/H)$ as a $\\textrm{Comm}_{A_n} [H]\/H$-complex.\nThis will be enough to describe $g_2^0$ to the extent we need.\n\nWe begin describing $g_{2[H]}^0$ according to the different prossibilities of $[H]$ and $n$.\n\nWhen $H\\neq Z(A_n)$, the results of Propositions \\ref{Anordinary}, \\ref{FinComm} and \\ref{FHcomm} imply that $g_{2[H]}^0$ is defined in the following way:\n$$g_{2[H]}^0\\colon K_q(R)\\to K_q(R[\\Z])\\oplus K_q(R).$$\nBy the previous considerations, the homomorphism $g_{21[H]}^0$ identifies $K_q(R)$ as the corresponding direct summand of $K_q(R[\\mathbb{Z}])$ in the Bass-Heller-Swan decomposition, while $g^{0}_{22[H]}$ is induced by the inclusion $\\textrm{Comm}_{A_n} [H]\\hookrightarrow A_n$, counts the number of connected components of the classifying space, and then is the identity.\n\nWhen $n$ is even and $H=Z(A_n)$, we have that $H_0^{{\\mathcal{F}} [H]}(\\textrm{Comm}_{A_n} [H],K_q(-))=K_q(R[\\Z])$ by Proposition \\ref{FHcomm}, and this copy of $\\Z$ corresponds to a stabilizer which is isomorphic to $C_n$, a cyclic group of order $n$. Then, by the previous considerations, the homomorphism $g_{2[H]}^0$ behaves as in the case $n$ even and $H\\neq Z(A_n)$.\n\nIn the case $n$ odd and $H=Z(A_n)$, again by Proposition \\ref{FHcomm} we have that $ H_0^{{\\mathcal{F}} [H]}(\\textrm{Comm}_{A_n} [H],K_q(-))=C(K_q(R))$, where $C(K_q(R))$ was defined in Notation \\ref{not: CKR}.\nThen $g_{21[H]}^0$ is defined as a homomorphism\n$$g_{2[H]}^0\\colon K_q(R)\\to C(K_q(R))\\oplus K_q(R).$$\nRemark from the definition of $C(K_q(R))$ that this group is the direct sum of $K_q(R)$ with quotients of $\\oplus_{i=1}^2 K_{q-1}(R)$ and $\\oplus_{i=1}^4 NK_q(R)$,\nand that this copy of $K_q(R)$ comes from the identification of the respective copies of $K_q(R)$ that appear in the Bass-Heller-Swan decomposition of $K_q(R[\\langle b(ab)^k\\rangle])$ and $K_q(R[\\langle ab\\rangle])$.\nThese copies correspond respectively to the stabilizers $C_2$ and $C_{2k+1}$ in $\\underline{E}(\\textrm{Comm}_{A_n} [H]\/H)$ (as a model for the $\\textrm{Comm}_{A_n} [H]\/H$-action). Then, the previous considerations about stabilizers imply that $g_{21[H]}^0$ maps this $K_0(R)$ isomorphically to the mentioned copy of itself inside $C(K_q(R))$.\nIn turn, $g_{22[H]}^0$ is again the identity, arguing as in the even case.\n\n\nWith the previous information, we can describe $\\mathrm{coker}g_2^0$.\n\nNow if $n$ is even, observe that the source of $g_2^0$ is $\\oplus_{[H]} K_q(R)$ and the codomain is $(\\oplus_{[H]}K_q(R[\\Z]))\\oplus K_q(R)$.\nMoreover, the image of every $g_{2[H]}^0$ consists in a copy of $K_q(R)$ inside $K_q(R[\\Z])$ (again the copy that appears in the Bass-Heller-Swan decomposition), and the copy of $K_q(R)$ that corresponds to $H_0^{Fin}(A_n,M)$, which is fixed for any choice of $H$. Hence, the cokernel of the homomorphism $g_2^0$ is isomorphic to the quotient of a direct sum of copies of $K_q(R[\\Z])$ (indexed by $[H]$) by the identification of all the copies $K_q(R)$ which are the images of the homomorphisms $g^0_{22[H]}$. Observe that  the copies of $K_q(R[\\Z])$ are ``glued\" by the copy of $K_q(R)$ that corresponds to $H_0^{Fin}(A_n,M)$, and hence we have $$\\mathrm{ coker } g_2^0=(\\bigoplus_{[H]} N_q^{[H]})\\oplus K_q(R).$$\n\nWhen $n$ is odd, we only need to take into account that in the case of $H=Z(A_n)$ the role of $K_q(R[\\Z])$ in the codomain is played by $C(K_q(R))$.\nThen, using the same argument as in the previous case and denoting by $\\overline{C}(K_q(R))$ the quotient of $C(K_q(R))$ under the copy of $K_q(R)$ in the Bass-Heller-Swan decomposition, we obtain that: $$\\mathrm{ coker } g_2^0=(\\bigoplus_{[H]\\neq [Z(A_n)]} N_q^{[H]})\\oplus K_q(R)\\oplus\\overline{C}(K_q(R)).$$\n\nObserve that $g_2^0$ is a monomorphism for every $n$, as the inclusions of $K_q(R)$ in $K_q(R[\\Z])$ and $C(K_q(R))$ are so.\n\n\\vspace{0.5cm}\n\n\nNow we can describe the Bredon homology of $A_n$ with respect to the family of virtually cyclic groups.\nObserve that the (co)kernels of the statement have been previously described, and that the groups $N_q^{[H]}$ were defined in Section \\ref{NqH}.\n\n\n\n\\begin{thm}\n\\label{Thm:Bredon}\nLet $A_n$ be an Artin group of dihedral type. In the previous notation, we have the following:\n\\begin{enumerate}\n  \\setlength{\\itemindent}{-2em}\n\\item $H_i^{vc}(A_n,K_q (R[-]))=\\{0\\}$ for $i\\geq 4$.\n\\item $H_3^{vc}(A_n,K_q (R[-]))=\\begin{cases}\\bigoplus_{[H]\\neq [Z(A_n)]} K_q(R) & \\text{$n$ odd}\\\\\n\\ker g_2^2 & \\text{$n$ even.}\n\\end{cases}$\n\n\\item $H_2^{vc}(A_n,K_q (R[-]))= \\ker g_2^1$.\n\n\\item $H_1^{vc}(A_n,K_q (R[-]))= \\emph{coker } g_2^1= \\begin{cases}(\\bigoplus_{[H]\\neq [Z(A_n)]}N_q^{[H]})\\oplus T_1(K_q(R))\\oplus T_2(K_q(R)) & \\text{$n$ odd}\\\\\n\\bigoplus_{[H]} N_q^{[H]}\n  & \\text{$n$ even.}\n\\end{cases}$%\n\\item $H_0^{vc}(A_n,K_q (R[-]))= \\emph{ coker } g_2^0= \\begin{cases}(\\bigoplus_{[H]\\neq [Z(A_n)]} N_q^{[H]})\\oplus K_q(R)\\oplus\\overline{C}(K_qR) & \\text{$n$ odd}\\\\\n(\\bigoplus_{[H]} N_q^{[H]})\\oplus K_q(R)\n  & \\text{$n$ even.}\n  \\end{cases}$%\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\n\nThe proof is based on the sequence of Proposition \\ref{mainMV} and the previous homological computations. We will check the five items in a separate way.\n\n\\begin{enumerate}\n\\item By Proposition \\ref{Anordinary}, Proposition \\ref{FinComm}  and Proposition \\ref{FHcomm} the sequence of Proposition \\ref{mainMV} is identically trivial to the left of $g_3^3$.\n\n\\item As the exact sequence of Proposition \\ref{mainMV} is identically trivial to the left of $g_3^3$, the map $g_3^3$ is trivial and by exactness, $H_3^{vc}(A_n, K_q(R[-]))$ is isomorphic to the kernel of $g_2^2$. This completes the even case.\n\n\nFor the case $n$ odd, we claim that $g_1^3$ is an isomorphism.\nIndeed, from Proposition \\ref{Anordinary} and Proposition \\ref{FHcomm}, the term $(\\bigoplus H_2^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M))\\oplus H_2^{Fin}(A_n,M)=\\{0\\}$ and in particular  $\\textrm{Im}(g_2^2)=\\{0\\}$.\nSince $g_1^3$ is injective, the claim follows.\nBy Proposition \\ref{FinComm}, the specific description of the odd case follows.\n\n\\item There are two different arguments depending if $n$ is odd or even.\n\nFor $n$ odd, as the term  $(\\bigoplus H_2^{\\mathcal{F}[H]}(\\Comm_{A_n}[H],M))\\oplus H_2^{Fin}(A_n,M)$  is trivial, the statement is a direct consequence of the exactness of the sequence of Proposition \\ref{mainMV}.\n\nFor $n$ even, we have seen in the discussion above about $g_2^2$ that this map is surjective, and hence $g_3^2$ is the trivial map.\nThis implies that $H_2^{vc}(A_n, K_q(R[-]))$ is the image of $g_1^2$, or equivalently, the kernel of $g_2^1$.\n\n\\item The previous description of the homomorphisms in the Mayer-Vietoris sequence proves that $g_2^0$ is a monomorphism, and hence $g_3^1$ is surjective and $H_1^{vc}(A_n,K_q( R[-]))$ the cokernel of $g_2^1$, which has been described above in terms of the summands $N_q^{[H]}$.\n\n\\item Since the Mayer-Vietoris sequence {ends} at $H_0^{vc}(A_n ,K_q( R[-]))$ this term is equal to the image of $g_3^0$ which is equal to the cokernel of $g_2^0$ that was  described above.\n\\end{enumerate}\n\\end{proof}\n\n\nAs said in the introduction, this theorem opens the door to concrete computations of Bredon homology of $A_n$ with respect to the family of virtually cyclic groups, provided there is available information about the $K$-theory of the coefficient ring. In particular, when $R$ is a regular ring the absence of Nil-terms and negative K-theory groups make the calculations easier and more precise.\nFor instance, we have\n\\begin{cor}\\label{cor:K_0}\nLet $A_n$, $n>2$ be an Artin group of dihedral type. Let $R$ be a regular ring. Then $K_0(RA_n)=K_0(R)$.\n\\end{cor}\n\\begin{proof}\nAs $R$ is regular, $K_{i}(R)$ vanishes for $i<0$ and also the Nil-Terms of the Bass-Heller-Swan decomposition.\nIn particular, by Theorem \\ref{Thm:Bredon}, $H_0^{vc}(A_n, K_0(R[-]))= K_0(R)$.\nMoreover, as $K_{i}(R)$ vanishes for $i<0$ we see  by Theorem \\ref{Thm:Bredon}, that $H_j^{vc}(A_n,K_i(R[-]))=\\{0\\}$ for $i<0$.\nThen the $E_2$-page of the Atiyah-Hirzebruch spectral sequence is concentrated in the non-negative part of the 0th, 1st, 2nd and 3rd columns, and $E_{\\infty}^{0,0}=E_2^{0,0}= K_0(R)$.\n\\end{proof}\n\nIn particular, this implies that every finitely dominated $CW$-complex whose fundamental group is $A_n$ has the homotopy type of a finite $CW$-complex.\n\nIn the following section we compute these Bredon homology groups for different choices of the ring, including some non-regular ones.\n\n\\section{Computations of $H_i^{vc}(A_n,K_q (R[-]))$ for several coefficient rings}\n\\label{Sect:concrete}\n\nIn this section we use Theorem \\ref{Thm:Bredon} to describe $H_i^{vc}(A_n,K_q (R[-]))$ for some instances of the ring $R$, both regular ($\\mathbb{Z}, \\mathbb{F}_q$) and non-regular ($\\mathbb{Z}[\\mathbb{Z}\/{\\bf 2}], \\mathbb{Z}[\\mathbb{Z}\/{\\bf 2}\\times \\mathbb{Z}\/{\\bf 2}]$ and $\\mathbb{Z}[\\mathbb{Z}\/{\\bf 4}]$).\nWe recall that all these groups give information about the $E^2$-term of the corresponding Atiyah spectral sequence. We point out that in the regular cases many groups can be computed, because the $K$-theory of $\\Z$ is nearly known and the $K$-theory of $\\mathbb{F}_q$ is known (see the corresponding examples below).\nIn the non-regular framework, by contrast, it is very difficult to find concrete descriptions of $K_q(R)$ for $q>1$, or of the corresponding Nil-terms.\n\nRecall that the groups $H_i^{vc}(A_n,K_0 (R[-]))$ are trivial for $i\\geq 4$ and any ring $R$.\n\n\\begin{ex}\n\\label{KZ}\n\nFirst we compute the Bredon homology of $A_n$ with respect to the family of virtually cyclic subgroups, taking as coefficients $K_q(\\Z [-])$, for $q=0,1,2$.\n\nWe need in our computations the lower algebraic $K$-theory groups of the integers. The groups $K_q(\\Z)$ are known for all $q\\leq 7$ and all $q\\geq 8$ such that $q \\not\\equiv 0 \\mod 4$,  see  \\cite[page 2]{Wei05}.\nFor example,  the first values of $K_q(\\Z)$ are given in the following table:\n\\begin{center}\\begin{tabular}{c|c|c|c|c|c|c|c|c|c}\n\\hline\n$q$ & $<0$ &0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\\n\\hline\n$K_q(\\Z)$ & 0 & $\\Z$ & $\\Z\/{\\bf 2}$ & $\\Z\/{\\bf 2}$ & $\\Z\/{\\bf 48}$ & 0 & $\\Z$ & 0 & $\\Z\/{\\bf 240}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\nObserve that, as $\\Z$ is regular, the Nil-terms in the Bass-Heller-Swan decomposition are trivial, and then for every $i\\in \\mathbb{N}$, $K_i(\\Z[\\Z])=K_i(\\Z)\\oplus K_{i-1}(\\Z)$ and $N_q^{[H]}=K_{q-1}(\\Z)$. Moreover, also by regularity, $K_i(\\Z)=0$ if $i\\leq -1$.\n\nTaking into account of all these considerations, the previous theorem implies the following.\n\nFor $q=0$, we have:\n\n\\begin{enumerate}\n\n\\item $H_3^{vc}(A_n,K_0 (\\Z[-]))\\simeq H_2^{vc}(A_n,K_0 (\\Z[-])) \\simeq \\bigoplus_{\\aleph_0} \\Z$.\n\\item {$H_1^{vc}(A_n,K_0 (\\Z[-]))=0$.}\n\\item $H_0^{vc}(A_n,K_0 (\\Z[-]))=\\Z$.\n\n\\end{enumerate}\n\nNow for $q=1$,\n\n\\begin{enumerate}\n\n\n\n\\item $H_3^{vc}(A_n,K_1 (\\Z[-]))\\simeq H_2^{vc}(A_n,K_1 (\\Z[-])) \\simeq \\bigoplus_{\\aleph_0} \\Z \/{\\bf 2}$.\n\\item {$H_1^{vc}(A_n,K_1 (\\Z[-]))=\\bigoplus_{\\aleph_0} \\Z$}.\n\\item {$H_0^{vc}(A_n,K_1 (\\Z[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus \\Z \/{\\bf 2}$}.\n\n\\end{enumerate}\n\nAnd finally, for $q=2$, all these groups are isomorphic to $\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2}$.\n\n\nLet us briefly explain these computations.\nWe consider first $q=0$.\nFor $H_3^{vc}(A_n,K_0 (\\Z[-]))$, the odd case is immediate.\nIn the even case, $H_3^{vc}(A_n,K_1 (\\Z[-]))$ is the kernel of a homomorphism  $\\bigoplus_{\\aleph_0} \\Z\\rightarrow \\Z$, and then isomorphic to $\\bigoplus_{\\aleph_0} \\Z$.\nThis easy argument will be frequently used in the examples of this section without express mention.\n\nWe now deal with $H_2^{vc}(A_n,K_0 (\\Z[-]))$.\nWe only deal with the odd case, the even one is very similar.\nFirst observe that the kernel of $g_2^1$ is isomorphic to $\\bigoplus_{\\aleph_0} \\Z$.\nMore precisely, taking account the description of Section \\ref{g21} and the values of the $K_i(\\Z)$, $g_2^1$ (for $n$ odd) is defined in the following way:\n\n$$g_2^1: (\\bigoplus_{[H]\\neq [Z(A_n)]} \\Z^2)\\oplus \\Z\\rightarrow (\\bigoplus_{[H]\\neq [Z(A_n)]} \\Z)\\oplus \\Z.$$\n\nHere the kernel of $g_{21[H]}^1$ is isomorphic to $\\Z$ for every $[H]\\neq [Z(A_n)]$.\nAs there is an infinite number of such commensurators, we conclude that the kernel should be also infinite, and then $H_2^{vc}(A_n,K_1 (\\Z[-]))=\\bigoplus_{\\aleph_0} \\Z$.\n\nObserve that, as $K_{-1}(\\Z)=0$ and $\\Z$ is regular, $N_0^{[H]}=T_i(K_0\\Z)=\\overline{C}(K_0\\Z)=0$.\nNow the values of $H_1^{vc}(A_n,K_0 (\\Z[-]))$ and $H_0^{vc}(A_n,K_0 (\\Z[-]))$ are easily deduced from items (4) and (5) of Theorem \\ref{Thm:Bredon}.\n\nNow take $q=1$. For $H_3^{vc}(A_n,K_1 (\\Z[-]))$ and $H_2^{vc}(A_n,K_1 (\\Z[-]))$ it is argued as in the previous case, taking into account that every subgroup of an $\\mathbb{F}_2$-vector space is again an $\\mathbb{F}_2$-vector space.\nNow observe that $N_1^{[H]}=K_0(\\Z)=\\Z$ by the Bass-Heller-Swan decomposition, and regularity and the fact that $K_0(\\Z)=\\Z$ is torsion-free imply that $T_1(K_1\\Z)=0$, $T_2(K_1\\Z)=0$ and $\\overline{C}(K_1\\Z)=\\Z$.\nThe values of $H_1^{vc}(A_n,K_1 (\\Z[-]))$ and $H_0^{vc}(A_n,K_1 (\\Z[-]))$ follow again from items (4) and (5) of Theorem \\ref{Thm:Bredon}.\n\nFinally, for $q=2$, the values of the homology are immediately implied by regularity and the fact that $K_2(\\Z)=K_1(\\Z)=\\Z \/{\\bf 2}$.\n\n\\end{ex}\n\n\n\\begin{ex}\n\nNow we will compute the groups $H_i^{vc}(A_n,K_q (\\mathbb{F}_2[-]))$, for $0\\leq q\\leq 3$. First, the following table (see \\cite{Qui73}) includes the algebraic $K$-groups that are necessary in our computations:\n\n\\begin{center}\\begin{tabular}{c|c|c|c|c|c}\n\\hline\n$q$ & $<0$ & 0 & 1 & 2 & 3 \\\\\n\\hline\n$K_q(\\mathbb{F}_2)$ & 0 & $\\Z$ & 0 & 0 & $\\Z\/{\\bf 3}$  \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\n\nAs $\\mathbb{F}_2$ is regular, the same considerations about the Nil-terms and the negative $K$-groups apply also in this case. Hence, again as a direct consequence of Theorem \\ref{Thm:Bredon} we have the following results. For $q=0$ and every $i\\in\\mathbb{N}$, $H_i^{vc}(A_n,K_0 (\\mathbb{F}_2[-]))=H_i^{vc}(A_n,K_0 (\\Z[-]))$, because $K_0(\\Z)=K_0(\\mathbb{F}_2)$ and then  $K_0 (\\mathbb{F}_2[-])=K_0 (\\Z[-])$ because both have the same Bass-Heller-Swan decomposition.\n\nFor $q=1$, we have $H_3^{vc}(A_n,K_1 (\\mathbb{F}_2[-]))=H_2^{vc}(A_n,K_1 (\\mathbb{F}_2[-]))=0$, as $K_1(\\mathbb{F}_2)=0$. On the other hand, as {$N_1^{[H]}=\\Z$} for every $H$, $H_1^{vc}(A_n,K_1 (\\mathbb{F}_2[-]))=H_0^{vc}(A_n,K_1 (\\mathbb{F}_2[-]))=\\bigoplus_{\\aleph_0}\\Z$.\n\n\nFor $q=2$, the triviality of $K_2(\\mathbb{F}_2)$ and $K_1(\\mathbb{F}_2)$ implies the triviality of $H_i^{vc}(A_n,K_2 (\\mathbb{F}_2[-]))$ for every $i$.\n\nFinally, for $q=3$, we have $H_2^{vc}(A_n,K_3 (\\mathbb{F}_2[-]))\\simeq H_3^{vc}(A_n,K_3 (\\mathbb{F}_2[-]))=\\bigoplus_{\\aleph_0}\\Z \/{\\bf 3}$.\nAs the groups $N_3^{[H]}$ are  trivial and $K_2(\\mathbb{F}_2)$ is so, we obtain that  $H_1^{vc}(A_n,K_3 (\\mathbb{F}_2[-]))=\\Z \/{\\bf 3}$ is trivial, and  $H_0^{vc}(A_n,K_3 (\\mathbb{F}_2[-]))=\\Z \/{\\bf 3}$.\n\\end{ex}\n\n\n\nWe follow with some non-regular examples, namely the rings $\\Z [\\Z \/{\\bf 2}]$, $\\Z [\\Z \/{\\bf 2}\\times \\Z \/{\\bf 2}]$ and $\\Z [\\Z \/{\\bf 4}]$, which we will respectively denote by $R_1$, $R_2$ and $R_3$.\nWe will compute the groups $H_i^{vc}(A_n,K_q (R_j[-]))$, for $0\\leq i\\leq 3$, $q=0,1$ and $1\\leq j\\leq 3$.\nIn order to do this we will need the values of their lower algebraic $K$-theory groups, as well as the Nil groups.\nAll these groups are displayed in the following table:\n\n\\begin{center}\\begin{tabular}{c|c|c|c|c|c}\n\\hline\n& $K_1$ & $K_0$ & $K_{-1}$ & $NK_0$ & $NK_1$ \\\\\n\\hline\n$\\Z [\\Z \/{\\bf 2}]$  & $(\\Z \/{\\bf 2})^2$ & $\\Z$ & 0 & 0 & 0 \\\\\n\\hline\n$\\Z [\\Z \/{\\bf 2}\\times \\Z \/{\\bf 2}]$  & $(\\Z \/{\\bf 2})^3$ & $\\Z$ & $\\Z^r$ & $\\bigoplus_{\\aleph_0} \\Z\/{\\bf 2}$ & $\\bigoplus_{\\aleph_0} \\Z\/{\\bf 2} $\\\\\n\\hline\n$\\Z [\\Z \/{\\bf 4}]$  & $\\Z \/{\\bf 2}\\times \\Z \/{\\bf 4}$ & $\\Z$ & $\\Z^s$ & $\\bigoplus_{\\aleph_0} \\Z\/{\\bf 2}$&$ \\bigoplus_{\\aleph_0} \\Z\/{\\bf 2}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\nLet us briefly explain the values of the table.  By work of Oliver \\cite[Theorem 14.1-2]{Oli88}, the kernels $SK_1(R_j)$ of the determinant maps are trivial, and hence (see for example \\cite{Ste78}) $K_1(R_j)$ is the group of units of $R_j$. Now the values of $K_1$ follow from a theorem of Higman (\\cite{Hig40}, see also \\cite[II.4.1]{She78}). In turn, by \\cite[Proposition 6]{Cas73}, the reduced $K_0$ of these three rings is trivial, and then \\cite[Lemma 2.18]{Luc21} implies that $K_0(R_j)=\\Z$ for every $j$. The values of the third column follow from work of Carter \\cite[Theorem 1]{Car80}, being the figures $r$ and $s$ positive integers that depend on the Schur indexes. Finally, the Nil-terms of the two last columns were computed by Weibel in \\cite{Wei09}.\n\nWe are now ready to describe the homology of the dihedral Artin groups with respect to the family of virtually cyclic groups, referred to the $K$-theory $K_q$ of these group rings, $q=0,1$. As before, our main tool is Theorem \\ref{Thm:Bredon}.\n\n\n\n\n\\begin{ex}\nWe start with $R_1=\\Z [\\Z \/{\\bf 2}]$.\nWhen $q=0$, $H^{vc}_3(A_n,K_0(R_1[-]))=H^{vc}_2(A_n,K_0(R_1[-]))=\\bigoplus_{\\aleph_0} \\Z $.\nTaking into account that $N_0^{[H]}=0$ for every $H$ (because $K_{-1}(R_1)$ is trivial, and also the Nil-terms), we obtain that $H^{vc}_1(A_n,K_0(R_1[-]))=0$.\nOn the other hand, $H^{vc}_0(A_n,K_0(R_1[-]))=\\Z$ too.\n\nIf $q=1$, $H^{vc}_3(A_n,K_1(R_1[-]))=H^{vc}_2(A_n,K_1(R_1[-]))=\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2} $.\nMoreover, as $\\Z$ is torsion-free and then the correspondent $T_1(K_1(R_1))$ is trivial, we have $H^{vc}_1(A_n,K_1(R_1[-]))=\\bigoplus_{\\aleph_0}\\Z$.\nFinally, taking into account that $C(K_1(R_1))=\\Z$ in this case, $H^{vc}_0(A_n,K_1(R_1[-]))=\\Z \/{\\bf 2}\\oplus \\Z \/{\\bf 2}\\oplus(\\bigoplus_{\\aleph_0} \\Z)$.\n\n\\end{ex}\n\n\\begin{ex}\nWe continue by considering $R_2=\\Z [\\Z \/{\\bf 2}\\times \\Z \/{\\bf 2}]$.\nNow $H^{vc}_3(A_n,K_0(R_2[-]))=H^{vc}_2(A_n,K_0(R_2[-]))=\\bigoplus_{\\aleph_0} \\Z $, exactly as in the previous example. We have $N_0^{[H]}=K_{-1}(R_2)\\oplus NK_0(R_2)\\oplus NK_0(R_2)=\\Z^r\\oplus (\\bigoplus_{\\aleph_0}\\Z \/{\\bf 2})$ and then $H^{vc}_1(A_n,K_0(R_2[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})$.\n Observe that the extra term $T_2(K_0(R_2))$ is an $\\mathbb{F}_2$-vector space of at most countable dimension, and then it is included in the previous direct sum.\n Finally, $H^{vc}_0(A_n,K_0(R_2[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})$ as in the previous case, taking into account that $\\overline{C}(K_0(R_2))$ is a direct sum of free abelian groups and an $\\mathbb{F}_2$-vector space, both of at most countable dimension.\n\nWhen $q=1$, $H^{vc}_3(A_n,K_1(R_2[-]))=H^{vc}_2(A_n,K_0(R_2[-]))=\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2}$, because $K_1(R_2)=(\\Z \/{\\bf 2})^3$.\nNow $N_q^{[H]}=\\Z\\oplus \\bigoplus_{\\aleph_0}\\Z \/{\\bf 2}$, and then by similar reasons to the previous case, $H^{vc}_1(A_n,K_1(R_2[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})$.\nSimilarly $H^{vc}_0(A_n,K_1(R_2[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})$.\n\n\\end{ex}\n\n\\begin{ex}\nTo conclude, we consider $R_2=\\Z [\\Z \/{\\bf 4}]$.\nAs the groups $K_0$, $NK_0$ and $NK_1$ are all isomorphic as abelian groups to their counterparts in the previous example\nand $K_{-1}$ is also free abelian, $H^{vc}_3(A_n,K_0(R_3[-]))=H^{vc}_2(A_n,K_0(R_3[-]))=\\bigoplus_{\\aleph_0} \\Z $, $H^{vc}_1(A_n,K_0(R_3[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})$, and $H^{vc}_0(A_n,K_0(R_3[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})$.\n\nFor $q=1$, it is clear that\n$H^{vc}_3(A_n,K_1(R_3[-]))=(\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 4})$.\nNow, the analysis of $g_2^1$ in Section \\ref{g21} guarantees that $H^{vc}_2(A_n,K_1(R_3[-]))$ should contain a subgroup isomorphic to $(\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 4})$, and then be isomorphic to it, as the source of $g_2^1$ is also isomorphic to $(\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 4})$.\nWe have $N_1^{[H]}=K_{0}(R_3)\\oplus NK_0(R_2)\\oplus NK_0(R_2)=\\Z\\oplus (\\bigoplus_{\\aleph_0}\\Z \/{\\bf 2})$.\nBy an analogous reasoning to the case of $R_2$, $H^{vc}_1(A_n,K_1(R_3[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})$.\nFinally, $H^{vc}_0(A_n,K_1(R_3[-]))=(\\bigoplus_{\\aleph_0} \\Z)\\oplus (\\bigoplus_{\\aleph_0} \\Z \/{\\bf 2})\\oplus \\Z \/{\\bf 4}$, corresponding the extra term to $K_1(R_3)$.\n\n\\end{ex}\n\n\\begin{rem}\nObserve that the proposed method permits computations of the Bredon homology (with respect to the family of virtually cyclic groups) of $A_n$ with respect to the $K$-theory of any ring for which there is some knowledge of the algebraic $K$-groups.\nWe remark that the computation of the lower algebraic $K$-theory groups is a hot topic nowadays (see for example \\cite{LaOr07}, \\cite{HiJu2021} or \\cite{GJM18}), so it seems possible that good knowledge about the $E^2$-term of the Atiyah-Hirzebruch spectral sequence is achieved in these cases, even for more general families of Artin groups.\n\n\\end{rem}\n\n\n\n\n\nWhen the ring $R$ is regular, it is possible to take a shortcut in order to compute the left-hand side of the Farrell-Jones conjecture. Observe that Theorem 0.1 in \\cite{LS16}, that establishes a splitting $$H^G({\\underline{\\underline{E}}} G,\\mathbf{K} (R))\\simeq H^G(\\underline{E} G,\\mathbf{K} (R))\\oplus H^G({\\underline{\\underline{E}}} G, \\underline{E} G, \\mathbf{K} (R)),$$ for every group $G$ and ring $R$.\nNow if $R$ is regular and $G$ is torsion-free, the second term of the direct sum vanishes \\cite[Proposition 2.6]{LR05}. Likewise, when $G$ is torsion-free $\\underline{E}  G=EG$, the classical universal space for principal $G$-bundles, and then $H^G(\\underline{E}  G,\\mathbf{K}(R))=H(BG, \\mathbf{K}(R))$, being the latter ordinary homology. Hence, in the case $G=A_n$, it is enough to compute $H^(BA_n, \\mathbf{K}(R))$ to obtain the left-hand side of the Farrell-Jones conjecture for these groups. When the coefficients $\\mathbf{K}(R)$ are known, and taking into account that the Artin groups of dihedral type are one-relator, Lemma 16.21 in \\cite{Luc21} provides an accurate description of these homology groups, corresponding case i) of the lemma to $n$ even and case ii) to $n$ odd. For example, when $K=\\mathbb{F}_p$ for $p$ prime, the classical results of \\cite{Qui73} provide a complete knowledge of the groups $H^(BA_n, \\mathbf{K}(R))$ , while for $K=\\mathbb{Z}$ much information is available (\\cite[page 2]{Wei05}). When $R$ is not regular, however, this strategy does not work, as in this case groups in the left-hand side of Farrell-Jones. In this context, we expect that our results on Bredon homology of $A_n$ (znd in particular the last examples of this section) can bring some light over the groups $H^{A_n}({\\underline{\\underline{E}}} A_n,\\mathbf{K} (R))$, via appropriate computations in the equivariant Atiyah-Hirzebruch spectral sequence. Observe that, according to Theorem \\ref{Thm:Bredon}, this spectral sequence has four columns. This fact indicates that the sequence should collapse at most in the page $E_4$, and hence this page should provide the knowledge of Farrell-Jones groups. The analysis of the differentials, as well as the subsequent description of the $E_3$ and $E_4$-pages, seems a difficult and interesting future line of research.\n\n\n\n\\noindent{\\textbf{{Acknowledgments.}}}\n\nWe warmly thank D. Juan-Pineda, W. L\\\"{u}ck, N. Petrosyan, L.J. S\\'anchez-Salda\\~{n}a, V. Srinivasan and J. Stienstra and an anonymous referee for their useful comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nEntanglement entropy is a non-local observable which measures entanglement between two subsystems of a quantum system.\nIt has many applications in studies of phenomena in quantum gravity, quantum information, condensed matter and high energy physics.\nParticularly, entanglement entropy in the context of the gauge\/gravity duality is aimed to shed some light on understanding of quantum gravity into the bulk \\cite{Rangamani:2016dms}.\n\nFollowing the holographic prescription, the  entanglement entropy between a subsystem (region) $A \\in\\mathbb{R}^{d}$ that has a  $d-1$-dimensional boundary $\\partial A$ and a remaining part $B$  can be calculated by Ryu-Takayanagi formula \\cite{Ryu:2006bv,Ryu:2006ef,Hubeny:2007xt}\n\\begin{eqnarray}\\label{1.1}\nS =  \\frac{\\text{Area}(\\gamma_{A})}{4G^{d+2}_{N}},\n\\end{eqnarray}\nwhere $\\gamma_{A}$ is the minimal $d$-dimensional surface in $AdS_{d+2}$ space whose boundary coincides with the boundary of the region $A$ ($\\partial A = \\partial \\gamma_{A}$), $G^{d+2}_{N}$ is $d +2$-dimensional Newton constant.\n\nFor the classic  case with $AdS$ on the gravity side, which geometry is not supported by any scalar field,\n and conformal theory of QFT side the area of the surface $\\gamma_{A}$ is defined through the induced metric  by the relation\n\\begin{equation}\\label{1.1a}\nA = \\int d^{d}\\sigma \\sqrt{|\\det{G_{\\alpha\\beta}}|},\n\\end{equation}\nwhere  $G_{\\alpha\\beta}=g_{MN}\\partial_{\\alpha}X^{M}\\partial_{\\beta}X^{N}$ is the induced metric of $\\gamma_{A}$ and $g_{MN}$ is the metric of the background. Important examples of $AdS$ spacetimes include near-horizon geometries of $p$-branes.\nIn the form (\\ref{1.1a}) it can be applied to studies of entanglement entropy for non-dilatonic branes, namely D3, M2 and   M5 branes \\cite{Quijada:2017zif}.\n\nThe  generalization of the entangled functional (\\ref{1.1}) with (\\ref{1.1a}) for branes with non-conformal boundaries reads\n\\begin{eqnarray}\\label{RTncbr}\nS = \\int d^{8}\\sigma \\frac{1}{4G^{10}_{N}} \\sqrt{|\\det G_{ind}|}e^{-2\\phi},\n\\end{eqnarray}\nwhere $\\phi$ is the dilaton.\nThe holographic entanglement entropy for configurations on D2 and NS five-branes was calculated in the original work  \\cite{Ryu:2006ef},\non D3 and  D4 branes in \\cite{Klebanov:2007km,Pakman:2008ui,Arean:2008az}, on D1-D5 brane intersection in \\cite{Asplund:2011cq}.\n The dilaton destroys the scale symmetry, \nbut  we still can detect  a certain field theory on the boundaries of the branes and discuss a holographic picture.\nFor example, \nfor NS5 brane in the long distances of the theory is governed by  the (2, 0) SCFT for IIA theory and the IR free SYM with sixteen supercharges for IIB, while\nthe short distance behavior leads to a linear dilaton geometry   \\cite{Aharony:1998ub} that can be described through the so called Little String Theory  $\\mathcal{N}=(2,0)$ and $\\mathcal{N}=(1,1)$ on the Type IIA and Type IIB NS5 branes respectively.\n\n\n\n\nIn this work we aim at studying T-duality aspects of entanglement entropy for field theories living on NS five branes, including the exotic branes $5_2^r$ with $r=0,1,2,3,4$. For the NS5 brane the decoupling limit is known to be LST, which is a 6-dimensional theory describing dynamics of string-like degrees of freedom which do not have gravitational modes in their spectrum. In all other respects they exhibit essentially stringy behaviour, such as Hagedorn temperature and T-duality of spectrum \\cite{Losev:1997hx, Kutasov:2001uf, Aharony:1999ks}. This is due to the fact that in contrast to D-branes the decoupling limit for the NS branes does not involve taking $\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon'\\to0$. This preserves stringy properties on the world-volume. The origin of T-duality in LST is the simple observation that a compactified NS5 brane transform into itself under T-duality along a world-volume direction. One the language of LST this transform into T-duality symmetry of the 6d theory with one compact direction, the direct analogue of that of the 10d string theory.\n\nIn addition however one may wonder what are the properties of the theory under T-duality transformations in the transverse directions, i.e. those which change the brane, say from NS5 brane to the KK5-monopole. By simple counting of degrees of freedom one concludes that the theory does not change under that. I.e. the Type IIA\/B NS5 brane carries the same world-volume theory as the Type IIB\/A Kaluza-Klein monopole \\cite{Sen:1997js}. Continuing this logic one concludes that the theory should not change along the whole T-duality orbit.\n\\begin{equation}\n\\begin{aligned}\n5_2^0(A\/B) && \\longleftrightarrow && 5_2^1(B\/A)\\longleftrightarrow && 5_2^2(A\/B)\\longleftrightarrow && 5_2^3(B\/A)\\longleftrightarrow && 5_2^4(A\/B)\n\\end{aligned}\n\\end{equation}\nHere we use the notations for the branes of \\cite{Obers:1998fb} (see also \\cite{deBoer:2012ma} for more on that), and the last three are exotic. The fact that the corresponding world-volume field theories do not change under T-duality trivially follows from the T-duality invariant world-volume effective action for these branes presented in \\cite{Blair:2017hhy}. This is a single action for the whole orbit, which drops into actions for a representative upon removing half of the scalar fields living on the brane (geometric or dual coordinates). Since from the world-volume point of view these are just scalar fields moving in a dynamical background, replacement one by its dual does not change anything for it. \n\n\nHowever, applying the Ryu-Takayanagi prescription for geometric entropy to the background of say Kaluza-Klein monopole one gets the answer which is different from the one for the NS5 brane, which clearly breaks the T-duality invariance. In this paper we show that the reason for that is that in its geometric and straightforward form this prescription does not take into account dependence on the winding direction of the localized Kaluza-Klein monopole. Indeed, in \\cite{Tong:2002rq, Harvey:2005ab,Kimura:2013fda,Kimura:2018hph} it has been shown that instanton corrections coming from the 2d sigma-model describing the KK5 background, change the geometry such that field start depending on a winding mode. This correct the throat behaviour of the KK5-monopole to make it the same as that of the NS5 brane. In \\cite{Jensen:2011jna, Berman:2014jsa, Bakhmatov:2016kfn}  it has been shown that this has simple explanation in terms of Double Field Theory, that is to do a T-duality transformation in a direction $z$ one replaces $z$ by it dual $\\tilde{z}$ in all expressions. The same is true for producing exotic backgrounds, and the corresponding instanton interpretation has been presented in \\cite{Kimura:2013zva}. In this work we consider the invariant action of \\cite{Blair:2017hhy} and propose an algorithm to calculate entanglement entropy for theories living on branes with non-trivial dynamics in doubled space.\n\nThis paper is structures as follows. In Section \\ref{geom} we present a short technical review of how the geometric entanglement entropy is calculated and explicitly show that the RT formula gives different results when applying to NS five-brane backgrounds belonging to the same T-duality orbit. In Section \\ref{dft} we turn to invariant dynamics governed by the action of \\cite{Blair:2017hhy}, shortly review how one obtains different action from the invariant one, and describe the algorithm which produces an invariant answer for entanglement entropy. In addition we comment on the geometric meaning of the expression, which is an important and subtle point due to lack of the notions of integration, distance and area in doubled geometry.\n\n\n\\section{Geometric entanglement entropy}\n\\label{geom}\n\n\n\n\nThe usual choice of areas which carry entangled states, which significantly simplifies calculations, is the infinite strip set-up. For that one considers a surface in the space transverse to a brane one which the field theory lives (shaded on Fig. \\ref{embed}). This surface is the boundary for the minimal surface, which tends to curve closer to the brane due to the transverse geometry. For D-branes this surface is identified with the AdS conformal boundary. The geometric formula of Ryu and Takayanagi gives entanglement entropy of states in the region A and B on the picture.\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=10cm]{embed.png}\n\\caption{Configuration of the embedding}\n\\label{embed}\n\\begin{tikzpicture}[overlay]\n\\node at (-0.1,9.5) (r) {$r$};\n\\node at (-1.8,3.8) (ra) {$r_a$};\n\\node at (-4,2) (X25) {$X^{2,\\dots,5}$};\n\\node at (5,3.8) (X1) {$X^1$};\n\\node at (1.5,8.3)  (A) {$A$};\n\\node at (3.7,8)  (B1) {$B$};\n\\node at (-0.8,8.5)  (B2) {$B$};\n\\node[rotate=-11] at (1,3) (br) {brane};\n\\end{tikzpicture}\n\\end{figure}\n\n\nIn this section the standard formula for calculation of geometric entanglement entropy is applied to the standard NS5 brane and to the KK5-monopole and exotic $5_2^2$ brane. Due to the special circle already for the KK5 background one gets expressions very different from that for the NS5 background. From this we conclude that one must develop a different algorithm and a different understanding of the RT expression to properly capture transformations along the NS five-brane T-duality orbit.\n\n\n\\subsection{Non-conformal theories: geometric five-branes}\n\\label{direct_KK}\n\nWhen turning to NS five branes one encounters 6d theories which describe string-like degrees of freedom which do not have gravitational excitation in their spectrum, the so-called Little String Theory. In the field theory limit these drop to non-conformal field theories since the corresponding brane backgrounds contain non-trivial dilaton and are not asymptotically AdS. However, for these one also can define entanglement entropy using the Ruy-Takayanagi conjecture (\\ref{RTncbr}) and write\n\\begin{equation}\nS=\\int_{\\S} d^5 \\s e^{-2\\f} \\sqrt{\\det G_{\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon\\b}},\n\\end{equation}\nwhere $\\{\\s^\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon\\}$ wuth $\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon=1,\\dots,5$ are coordinates on the space-like surface $\\S$  and $G_{\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon\\b}=\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon X^\\m \\dt_\\b X^\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho G_{\\m\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho}$ is the induced metric on the surface. For our purposes we choose the simplest shape for the surface $\\S$  generated by an infinite stripe. Theory for which the entanglement entropy is calculated lives on a surface parallel to the NS5 brane placed at some $r=r_b$. Entanglement is assumed for the states defined in the interior $A$ and exterior $B$ regions of the grey surface on the Fig.\\ref{embed}. According to the conjecture this is equal to area of the minimal surface $\\S$ whose boundary satisfies $\\dt \\S=\\dt A$.\n\nEmbedding of the brane and of the surface is given by\n\\begin{equation}\n\\begin{aligned}\n&      &&  0 && 1 && 2 && 3 && 4 && 5 && r && \\q_1 && \\q_2 && \\f \\\\\n\\hline\n& NS5  &&\\times &&\\times&&\\times&&\\times&&\\times&&\\times&&  \\\\\n& \\S   &&    &&\\bullet&& L && L && L && L && \\bullet\n\\end{aligned}\n\\end{equation}\nwhere $\\times$ denote the world-volume directions. The surface $\\S$ extends from $-L$ to $L$ in the directions denoted by $L$ above, while it is somehow curved in the directions denoted by bullets. I.e. one can choose coordinates and embedding functions for the surface as follows\n\\begin{equation}\n\\begin{aligned}\nX^1&=X(r),\\\\\nX^{2,\\dots,5}&=\\s^{2,\\dots,5},\\\\\nr&=\\s^1.\n\\end{aligned}\n\\end{equation}\n\nBackground for the NS5 brane is given by\n\\begin{equation}\n\\begin{aligned}\nds^2&=\\h_{rs}dx^rdx^s+H(dr^2+r^2\\d\\W_{3}^2),\\\\\n\\mathcal{H}&=dB,\\\\\ne^{-2(\\f-\\f_0)}&=H(r)^{-1},\n\\end{aligned}\n\\end{equation}\nwith the harmonic function $H(r)=1+h\/r^2$. Hence one writes for the entropy\n\\begin{equation}\n\\begin{aligned}\\label{RT-ns5}\nS_{NS5}\n16L^4 \\int dr H(r)^{-1}\\sqrt{H(r)+X'(r)^2}.\n\\end{aligned}\n\\end{equation}\nThe usual minimisation procedure implies that the embedding function $X(r)$ should satisfy\n\\begin{equation}\n\\label{Xemb}\nX_{NS5}'(r)=\\pm\\frac} \\def\\dfr{\\dfrac} \\def\\dt{\\partial{H(r_a)^{1\/2}H(r)}{\\sqrt{H(r_a)^2-H(r)^2}},\n\\end{equation}\nwhere we used the condition that $X'(r_a)=0$, which basically means that $r_a$ is the turning point for the surface $\\S$. Note, that one has to set $r>r_a$ to keep the expression in the square root positive, which means that the turning point is closer to the brane than the surface $r=r_b$ on which the  field theory is defined. This is the usual configuration for the AdS\/CFT correspondence and hence the initial setup and the Fig.\\ref{embed}.\n\nOne can apply the same procedure to the worldvolume theory of the KK-monopole which for the Type IIA(B) monopole is the same as for the Type IIB(A) NS5 brane. Background geometry is given by the following configuration\n\\begin{equation}\n\\begin{aligned}\nds^2&=\\h_{rs}dx^rdx^s+H^{-1}(d\\tilde{z}+A_idy^i)^2+H\\d_{ij}dy^i dy^j,\\\\\nB&=0,\\\\\ne^{-2(\\f-\\f_0)}&=1.\n\\end{aligned}\n\\end{equation}\nHere $\\tilde{z}$ is the normal geometric coordinate used to measure distances in space-time, however it is dual to the coordinate $z$ of the corresponding NS5 background. Note that the harmonic function is smeared $H=1+h\/r$.\n\nRepeating the same calculation as above one gets for the entropy and for the embedding function $X_{KK5}(r)$\n\\begin{equation}\n\\begin{aligned}\nS_{KK5}&=16L^4 \\int dr \\sqrt{H(r)+X'(r)^2},\\\\\nX_{KK5}'(r)&=\\pm C\\sqrt{H(r)}, \\quad C=\\mbox{const}.\n\\end{aligned}\n\\end{equation}\nOne first notices that the crucial difference with the previous case, that is $dr\/dX=0$ at $r=0$, i.e. on the brane itself, while for the NS5 brane background the turning point is at some $r_a\\neq 0$. This can be understood in terms of the short distance behaviour of NS5 branes and KK5 monopole. As it has been shown in \\cite{Tong:2002rq,Gauntlett:1992nn} the former is the  version of the H-monopole (which is the proper T-dual of the KK monopole) localized due to instanton corrections. However, the localization breaks isometry along the compact circle of H-monopole and one observes a throat behaviour at short distances. \n\nTo cure the near-brane behaviour of the KK5-monopole background one also considers instanton corrections \\cite{Harvey:2005ab}. Only in this case one may expect result for the entropy which reproduce those for the NS5 brane. Such corrections however deform the background by introducing a non-trivial dependence on string winding coordinates, which requires double field theory to consistently address the issue, as in \\cite{Jensen:2011jna,Bakhmatov:2016kfn}. \n\n\n\n\n\n\n\\subsection{Exotic five-branes}\n\\label{direct_exotic}\n\nHence, the answer for the entropy which one obtains for the theory living on the KK monopole is different from that for the NS5 brane. The important point here is that although T-duality exchanges IIA and IIB branes the theories living on the NS5A(B) and the KK5B(A) are the same and the entropy should not change. When going further along the T-duality orbit towards exotic branes the situation does not get better. Smearing the KK5 background along $y_3$ and T-dualizing one arrives at the exotic $5_2^2$-brane with background given by \\cite{deBoer:2012ma}\n\\begin{eqnarray}\nds^{2} &=& \\eta_{rs}dx^{r}dx^{s} + HK^{-1}\\left(d\\tilde{z}^{2} + d\\tilde{y}^{2}_{3}\\right) + H \\delta_{\\alpha\\beta}dy^{\\alpha}dy^{\\beta}, \\\\\nB& = & h\\theta K^{-1} d\\tilde{z}\\wedge d\\tilde{y}_{3}, \\\\\ne^{-2(\\phi - \\phi_{0})} &=& HK^{-1},\\\\\nK &= & H^{2} + (h\\theta)^2.\n\\end{eqnarray}\nHere the harmonic function is further smeared $H(r)=1+h \\log r$ and does not behave well at space infinity\n\nThis background is globally well-defined only up to a monodromy around the brane, hence the non-geometric properties of the background. Naively applying the above procedure one obtains\n\\begin{equation}\n\\begin{aligned}\nS_{5_2^2}&=16L^4\\int d r \\frac} \\def\\dfr{\\dfrac} \\def\\dt{\\partial{H(r)\\sqrt{H(r)+X'(r)^2}}{H(r)^2+h^2 \\q^2},\\\\\nX_{5_2^2}'&=\\pm\\frac} \\def\\dfr{\\dfrac} \\def\\dt{\\partial{C \\sqrt{H(r)}(H(r)^2+h^2 \\q^2)}{\\sqrt{H(r)^2-C^2 (H(r)^2+h^2 \\q^2)^2}}.\n\\end{aligned}\n\\end{equation} \nThe embedding function $X_{5_2^2}(r)$ delivering extremum to $S_{5_2^2}$ is apparently not well-defined and moreover it explicitly depends on $\\q$. Hence, the entropy also depends explicitly on the coordinate $\\q$ respecting the monodromy property of the background.\n\nOn the other hand, the worldvolume theory on $5_2^2$-brane should not differ from that of the KK-monopole or NS5 brane (with proper replacement of Type IIA with Type IIB). To perform calculation of entanglement entropy for such theories which respect T-duality we use the T-duality covariant action of \\cite{Blair:2017hhy} for the 5-brane orbit. It suggests that the worldvolume theory is the same irrespective of the choice of the brane (equivalently, the section constraint or orientation in the doubled space) upon the proper exchange of the worldvolume scalars $X^\\m$ with their duals $\\tilde{X}_{\\m}$. \n\n\n\n\\section{Entanglement entropy in  DFT}\n\\label{dft}\n\nDouble Field Theory being a T-duality covariant formulation of supergavity (string theory) allows to consider the whole T-duality orbit instead of a single representative. In this section we propose a deformation of the geometric prescription for entanglement entropy and embed the expression for entropy itself into the DFT framework. Let us start with brief description of how NS five-branes are embedded into doubled space.\n\n\\subsection{Embedding of NS five-branes in doubled space}\n\nIn \\cite{Blair:2017hhy} it was shown in details how one can construct a T-duality covariant action for NS five-branes. The covariancy here is understood in the following way: one has a single expression which is written in terms of DFT (covariant) fields and which reproduces the effective action for the NS5B-brane, KK5A monopole and exotic branes $5_2^2$B, $5_2^3$A, $5_2^4$B. The full action smartly chooses these frames depending on which symmetries of the doubled spaces are eventually realized on the world-volume. Let us briefly describe the process focusing only on the NS-NS sector and only on the DBI part of the action, which is given by\n\\begin{equation}\n\\label{full_5}\nS_{NS,DBI}[Y(\\x)]=\\int_V d^6 \\x e^{-2d}\\sqrt{\\det h_{ab}}\\sqrt{\\displaystyle -\\det\\Big(g_{\\m\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon{X^\\m} \\dt_\\b X^\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho+ \\mathcal{H}_{MN}\\hat{D}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^M \\hat{D}_\\b Y^N\\Big)},\n\\end{equation}\nwhere we introduce\n\\begin{equation}\n\\begin{aligned}\nh_{ab}&=k_a^M k_b^N \\mathcal{H}_{MN},\\\\\n\\hat{D}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^M&=\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^M+\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon X^\\m A_\\m{}^M,\\\\\n\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^M&=\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^M-(h^{-1})^{ab}k_a^M k_b^N\\mathcal{H}_{NP}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^P,\\\\\n\\mathcal{H}_{MN}&=\n\\begin{bmatrix}\nG_{mn}-B_{m}{}^k B_{kn} & B_{n}{}^q\\\\\nB_m{}^p & G^{pq}\n\\end{bmatrix}.\n\\end{aligned}\n\\end{equation}\nHere the full space-time is split into the part parallel to the five-brane, labelled by the indices $\\m,\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho=\\{0,5\\}$, and the part transverse to the branes, which is doubled and labelled by $M,N,P,Q=\\{6,7,8,9,\\tilde{6},\\tilde{7},\\tilde{8},\\tilde{9}\\}$. The vector fields $A_\\m{}^M$ result from the Kaluza-Klein decomposition of the full 10D theory \n\\begin{equation}\nA_\\m{}^M=\n\\begin{bmatrix}\nA_\\m{}^m \\\\\n-B_{\\m m}\n\\end{bmatrix}.\n\\end{equation}\nThe integration is performed over world-volume of the brane which is parametrized by six coordinates $\\{\\x^\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon\\}$. The hatted derivative $\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon$  contains a projector part and is designed in such a way as to always remove half of the fields $Y^M$ from the action. Upon adding the action for DFT fields this results in field configurations which do not depend on half of DFT coordinates and hence is a worldvolume realization of the section constraint. Finally, the choice of the section frame and hence a representative of the T-duality orbit is done by choosing the particular form of the vectors $k_a{}^M$, which must satisfy the following algebraic section constrain\n\\begin{equation}\nk_a^Mk_b^N\\h_{MN}=0,\n\\end{equation}\nwhere $\\h_{MN}$ is the usual O(4,4) invariant tensor \n\\begin{equation}\n\\h_{MN}=\\begin{bmatrix}\n0 & 1 \\\\\n1 & 0\n\\end{bmatrix}\n\\end{equation}\nand the indices $a,b={1,4}$ enumerate the Killing vectors. The reason why we call these vectors Killing will be clear in a moment.\n\nFor the O(4,4) configuration there exist five inequivalent solutions of the algebraic section constrain, each of which corresponds to the branes $5_2^r$ with $r=0,1,2,3,4$ showing the number of quadratic direction in the mass of the corresponding 3D BPS state (see \\cite{deBoer:2012ma} for more detailed description of these notations). Here we list five representative solutions\n\\begin{equation}\n\\label{Kill}\n\\begin{aligned}\nNS5=5_2^0: && k_a^M&=(0,0,0,0;\\tilde{k}_{a 1}, \\tilde{k}_{a 2},\\tilde{k}_{a 3},\\tilde{k}_{a 4}),\\\\\nKK5=5_2^1: && k_a^M&=(0,0,0,k_{a}^{4}; \\tilde{k}_{a 1},\\tilde{k}_{a 2},\\tilde{k}_{a 3}, 0),\\\\\nQ=5_2^2: && k_a^M&=(0,0,k_{a}^{3},k_{a}^{4};\\tilde{k}_{a 1},\\tilde{k}_{a 2},0,0),\\\\\nR=5_2^3: && k_a^M&=(0,k_{a}^{2},k_{a}^{3},k_{a}^{4};\\tilde{k}_{a 1},0,0,0),\\\\\nR'=5_2^4: && k_a^M&=(k_{a}^1,k_{a}^{2},k_{a}^{3},k_{a}^{4};0,0,0,0).\n\\end{aligned}\n\\end{equation}\nFor example, for the NS5 brane case, which is the first line above, one chooses all vectors $k_a^M$ to be along the dual coordinates. Substituting this back into the action one checks that all fields $Y_m$ drop from the expression rendering field configurations independent on the corresponding DFT coordinates. This is due to\n\\begin{equation}\n\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y_m=B_{mn}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^n,\n\\end{equation}\nwhere $B_{mn}$ is the usual Kalb-Ramond two-form gauge field. The same is true for all other configurations up to the R'-brane which is a co-dimension-0 object from the point of view of the conventional supergravity.\n\nTo obtain explicit expression for the background fields for a fixed choice of the Killing vectors, one considers the full action with the embedding given by Dirac delta functions $\\d^{(8)}(\\mathbb{X}^M-Y^M(\\x))$, where $\\mathbb{X}^M=(x^m,\\tilde{x}_m)$ are the coordinates of DFT. The reparametrization invariance of the world-volume is fixed as usual as\n\\begin{equation}\n\\label{gauge_fix0}\n\\begin{aligned}\nX^\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon&=\\x^\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon.\n\\end{aligned}\n\\end{equation}\nConsider for example the KK-monopole, which is the second line above, where the fields $Y_{1,2,3}$ and $Y^4$ drop from the action meaning that the field configurations as functions are of the form $H=H(x^1,x^2,x^3,\\tilde{x}_4)$. This is interpreted as a functional dependence of the background on three geometric coordinates $x^{1,2,3}$ and one non-geometric (dual or winding) coordinate $\\tilde{x}_4$. This is due to an additional piece of information fixed in the DFT action, where one always understands $\\mathbb{X}^m=x^m$ as geometric coordinates, i.e. those used to measure space distances, and $\\mathbb{X}_m=\\tilde{x}_m$ as their non-geometric duals. Without this fixing one will just count each brane four more times obtaining the same backgrounds but with different names for the same physical coordinates.\n\nSuch dependence of exotic backgrounds (starting from the KK monopole) on dual coordinates has been shown for the DFT monopole in \\cite{Bakhmatov:2016kfn} and will be important for our further discussion.\n\n\n\n\\subsection{Invariant entropy and minimal surface}\n\nThe main feature of the effective action \\eqref{full_5} is that it does not depend on the choice of the T-duality frame and describes dynamics of all five-branes dual to NS5 brane. Since the world-volume theory does not change when switching from (Type IIB) NS5 brane to (Type IIA) KK5-monopole, the corresponding entanglement entropy should not change as well. One can conjectures the following deformation of the Ryu-Takayanagi formula which provides such invariant description:\n\\begin{equation}\n\\label{inv_entr}\nS_5=\\int_\\S d^5 \\s e^{-2d}\\sqrt{\\det h_{ab}}\\sqrt{\\displaystyle -\\det\\Big(g_{\\m\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon{X^\\m} \\dt_\\b X^\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho+ \\mathcal{H}_{MN}\\hat{D}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^M \\hat{D}_\\b Y^N\\Big)},\n\\end{equation}\nwhere the notations are the same as before. In a moment we will explicitly show that this expression gives the usual RT formula whose minimization gives the geometric entanglement entropy for the NS5 brane case. For other representatives of the orbit one gets a deformation of the formula, however the integral itself does not distinguish between the allowed choices of the duality frame.\n\nBefore that it is important to discuss the meaning of the integration and of the surface $\\S$ here. Going back to the effective action \\eqref{full_5} one notes that the integration there is performed over the world-volume $V$ parametrized by $\\s^\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon$, which is a usual geometric manifold with properly defined integration measure. On this manifold one defines $6+(4+4)$ fields $\\{X^\\m(\\x),Y^M(\\x)\\}$, which are identified with coordinates in the space-time and the doubled coordinates of the O(4,4) DFT. The crucial point here is that without such identification, these fields do not carry the meaning of coordinates on a doubled space and hence one is not actually doing doubled geometry, and rather works with a number of fields. For more discussion on this see \\cite{Blair:2017hhy}.\n\nAlthough the expressions \\eqref{full_5} and \\eqref{inv_entr} look almost the same, there is fundamental difference between them. While in action one varies with respect to the background fields keeping the embedding fixes, for the entropy the background is fixed by our choice of the brane and variation goes with respect to the embedding. The latter is defined by identification of the surface coordinates $\\{\\s^\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon\\}$ with the fields $X^\\m,Y^M$, which define the (doubled) space-time dependence of the background. Since, the harmonic function depends only on a singlet combination $r$, a natural choice of the embedding is\n\\begin{equation}\n\\label{gauge_fix}\n\\begin{aligned}\nX^{2,3,4,5}&=\\s^{2,3,4,5},\\\\\n\\s^1&=r,\n\\end{aligned}\n\\end{equation}\nand the remaining field $X^1=X(\\s^1)$ is a function delivering minimum to the expression. Here the particular form of the field $r$ depends on the choice of the background and reads\n\\begin{equation}\n\\begin{aligned}\nNS5=5_2^0: &&  r^2&=(Y^1)^2+(Y^2)^2+(Y^3)^2+(Y^4)^2,\\\\\nKK5=5_2^1: && r^2&=(Y^1)^2+(Y^2)^2+(Y^3)^2+(\\tilde{Y}_4)^2,\\\\\nQ=5_2^2: && r^2&=(Y^1)^2+(Y^2)^2+(\\tilde{Y}_3)^2+(\\tilde{Y}_4)^2,\\\\\nR=5_2^3: && r^2&=(Y^1)^2+(\\tilde{Y}_2)^2+(\\tilde{Y}_3)^2+(\\tilde{Y}_4)^2,\\\\\nR'=5_2^4: && r^2&=(\\tilde{Y}_1)^2+(\\tilde{Y}_2)^2+(\\tilde{Y}_3)^2+(\\tilde{Y}_4)^2.\n\\end{aligned}\n\\end{equation}\nThese follow from solutions of the equations of motion for the full action $S_{DFT}+S_{brane}$ which boil down to Poisson equation with delta source whose solution is the harmonic function $H=H(r)$ with $r$ given by the above expression. The number of dual coordinates entering the dependence of the fields is equal to the number of special circles.\n\nThe gauge fixing conditions  \\eqref{gauge_fix} can be understood as a proper embedding of the surface $\\S$ in the doubled 5+(4+4)-dimensional space. This is similar to the way how the magnetic charge for these branes has been calculated in \\cite{Bakhmatov:2016kfn}, however now the integration remains proper integration over a conventional manifold with conventional measure. The structure of the doubled space shows up only at the level of the Killing vectors and of the interaction between the effective action and the full DFT action. Before that, the integration does not distinguish between $Y^m$ and $\\tilde{Y}_m$, as it should be since the corresponding world-volume theories do not feel this as well. The integration is then performed in $\\s^{2,3,4,5}\\in [-L,L]$ for some large $L$ and from the points $X'=0$ in the $\\s^1$ direction. This is what is usually called the rectangular strip area, which is the simplest to perform calculations. In principle, one may choose a different embedding which will correspond to a different area inside the world-volume theory.\n\nLet us postpone the discussion, of how this process is seen from the point of view of the world-volume theory, to the Discussion section and now move to explicit examples to show invariance of the expression.\n\n\n\\subsection{Explicit examples}\n\n\nLet us start with the T-duality frame which corresponds to NS5 brane, which fixes the Killing vectors to be\n\\begin{equation}\nk_a^M=(0;\\tilde{k}_{am}).\n\\end{equation}\nThen the matrix $h_{ab}$ becomes $h_{ab}=\\tilde{k}_{am}\\tilde{k}_{bn}g^{mn}$ and one has \n\\begin{equation}\n\\det{h_{ab}}=|\\tilde{k}|^2 g^{-1},\n\\end{equation}\nwhere $|\\tilde{k}|=\\det\\tilde{k}_{am}$ and $g=\\det G_{mn}$. The inverse of the matrix $h_{ab}$ is then\n\\begin{equation}\n(h^{-1})^{ab}=(\\tilde{k}^{-1})^{am}(\\tilde{k}^{-1})^{bn}G_{mn},\n\\end{equation}\nwhere $(\\tilde{k}^{-1})$ is the inverse of $\\tilde{k}_{am}$ understood simply as a $4\\times 4$ matrix. Hence, for derivatives of the fields $Y^M$ we have\n\\begin{equation}\n\\begin{aligned}\n\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^m&=\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^m,\\\\\n\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon \\tilde{Y}_m&=\\dt \\tilde{Y}_m-(h^{-1})^{ab}\\tilde{k}_{am}\\tilde{k}_{an}\\mathcal{H}^n{}_P\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^p=B_{mn}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^n.\n\\end{aligned}\n\\end{equation}\nWith this in hands it is easy to show that\n\\begin{equation}\n\\mathcal{H}_{MN}\\hat{D}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^M \\hat{D}_\\b Y^N= G_{mn}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^m \\dt_\\b Y^n,\n\\end{equation}\nwhere we used the fact that $A_{\\m}{}^M=0$ for the chosen embedding of the brane. \n\nFinally, substituting all this into the expression for the entropy \\eqref{inv_entr} one obtains\n\\begin{equation}\n\\label{Einv_NS5}\n\\begin{aligned}\nS_{NS5}&=\\int_\\S d^5 \\s e^{-2\\f}\\sqrt{G}|\\tilde{k}|\\frac} \\def\\dfr{\\dfrac} \\def\\dt{\\partial{1}{\\sqrt{G}}\\sqrt{-\\displaystyle \\Big(g_{\\m\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon X^\\m \\dt_\\b X^\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho+G_{mn}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^m \\dt_\\b Y^n\\Big)}\\\\\n&=|\\tilde{k}|\\int_\\S d^5 \\s e^{-2\\f}\\sqrt{-\\displaystyle \\Big(g_{\\m\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon X^\\m \\dt_\\b X^\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho+G_{mn}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^m \\dt_\\b Y^n\\Big)},\n\\end{aligned}\n\\end{equation}\nwhich is the conventional expression for the geometric entanglement entropy of Ryu and Takayanagi (for the chosen embedding, i.e. $g_{\\m m}=0$).\n\nThe same calculation can be repeated for the KK5-monopole. One starts with the following Killing vectors\n\\begin{equation}\nk_a{}^M=(0,k_4^m;\\tilde{k}_{e m}),\n\\end{equation}\nwhere $e,f,g,h=1,2,3$. And the direction $4$ is identified with the Taub-NUT direction (the special circle of the monopole). For further convenience it is natural to choose such basis for the vectors $k_a^M$ where $\\tilde{k}_{e 4}=0$ and $k_4^i$=0. Then the matrix $h_{ab}=k_a{}^Mk_b{}^N\\mathcal{H}_{MN}$ becomes\n\\begin{equation}\n\\begin{aligned}\nh_{ef}&=\\tilde{k}_{e m}\\tilde{k}_{fn}g^{mn}=\\tilde{k}_{e i}\\tilde{k}_{f j}G^{ij},\\\\\nh_{e4}&=0,\\\\\nh_{44}&=k_4^4k_4^4 G_{44},\n\\end{aligned}\n\\end{equation}\nand $\\det h_{ab}=|\\tilde{k}|^2 (k_4^4)^2 g^{-1} G_{44}$, where $g=\\det g_{ij}$ is determinant of the 3-dimensional part of the metric $G_{mn}$ defined as\n\\begin{equation}\n\\begin{aligned}\nG_{ij}&=g_{ij}+A_iA_j G_{44}, && & G_{i4}&=A_i G_{44},\\\\\nG^{ij}&=g^{ij}, && & G^{i4}&=-A_{i4}G_{44},\\\\\nG_{44}&=H^{-1},  && & G^{44}&=\\frac} \\def\\dfr{\\dfrac} \\def\\dt{\\partial{1}{G_{44}}+A_i A_i G_{44}.\n\\end{aligned}\n\\end{equation}\nFollowing the same procedure as before it is straightforward to obtain the following expression for derivatives of the fields $Y^M$\n\\begin{equation}\n\\begin{aligned}\n\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^i&= \\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^i, && &\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^4&=-A_i \\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^i\\\\\n\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon \\tilde{Y}_i&=A_i \\dt_a \\tilde{Y}_4, && &\\hat{\\dt} \\tilde{Y}_4&=\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon \\tilde{Y}_4.\n\\end{aligned}\n\\end{equation}\nThe crucial difference between NS5 brane and KK5-monopole here is that for the former one is left only with the fields $Y^i$, which upon embedding into the full DFT action are identified with the usual geometric coordinates. In contrast, for KK5-monopole after projection one has the fields $\\{Y^i,\\tilde{Y}_4\\}$ which results in dependence of the background fields on the corresponding dual (winding) coordinate $\\tilde{x}_4$. This behaviour has been observed in \\cite{Harvey:2005ab,Jensen:2011jna,Bakhmatov:2016kfn} for KK5 and in \\cite{Kimura:2013zva} for the exotic $5_2^2$-brane.\n\nFinally, collecting all these pieces together one arrives at the following expression for entanglement entropy of the world-volume theory on (localized) Kaluza-Klein monopole\n\\begin{equation}\n\\label{Einv_KK55}\n\\begin{aligned}\nS_{KK5}&=|\\tilde{k}||k_4^4|\\int_\\S d^5 \\s e^{-2\\f} G_{44}\\times\\\\\n&\\sqrt{-\\displaystyle \\det\\Big[g_{\\m\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon X^\\m \\dt_\\b X^\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho + \\big(G_{ij}-G_{44}A_iA_j\\big)\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^i \\dt_\\b Y^j +\\big(G^{44}-G^{ij}A_iA_j\\big)\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon \\tilde{Y}_4 \\dt_\\b \\tilde{Y}_4\\Big]}\\\\\n&=|\\tilde{k}||k_4^4|\\int_\\S d^5 \\s e^{-2\\f}G_{44}\\sqrt{-\\displaystyle \\det\\Big[g_{\\m\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon X^\\m \\dt_\\b X^\\nu} \\def\\x{\\xi} \\def\\p{\\pi} \\def\\vp{\\varpi} \\def\\r{\\rho + H\\big(\\d_{ij}\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon Y^i \\dt_\\b Y^j +\\dt_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon \\tilde{Y}_4 \\dt_\\b \\tilde{Y}_4\\big)\\Big]}.\n\\end{aligned}\n\\end{equation}\nWhere the last line is obtained by substituting the explicit background of KK5-monopole inside the square root. Taking into account that for the monopole one has $e^{-2\\f}=1$ and $G_{44}=H^{-1}$ the second line reproduces precisely the expression \\eqref{Einv_NS5} up to replacement $Y^4 \\to \\tilde{Y}_4$. Note however, that talking about world-volume dynamics and field theories on the branes one does not distinguish between fields $Y^m$ and their duals $\\tilde{Y}_m$. The only difference is that the latter see the background T-dual to the background seen by the former. This is a trivial consequence of the above considerations.\n\nNow, for exotic branes $5_2^r$ with $r=2,3,4$ the story is precisely the same and the algorithm is the following: fix the Killing vectors as in \\eqref{Kill}, calculate $h_{ab}$ and hatted derivatives $\\hat{\\dt}_\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon$, substitute everything in \\eqref{inv_entr}. The result will always be \\eqref{Einv_NS5} with the corresponding replacement of the fields $Y^m$ by their duals. Hence the name ``invariant entropy''. We postpone speculations on the physical and geometrical meaning of this procedure to the next section.\n\n\n\n\n\\section{Discussion}\n\nIn this letter we propose a T-duality invariant generalization of the Ryu-Takayanagi formula for geometric entanglement entropy for the case of NS  five-branes $5_2^r$ ($r=0,\\dots,4$), with tension proportional to $g_s^{-2}$. The result is the expression \\eqref{inv_entr} which is based on the same ideas as the effective action \\eqref{full_5} for these branes. In particular, to choose a representative brane from the orbit one must specify Killing vectors, which satisfy the so-called algebraic section constraint. The choice which gives the effective action of NS5B-brane also reproduces the RT-formula for entanglement entropy of $\\mathcal{N}=(1,1)$ Little String Theory living on this brane.\n\nWe check, that the same expression gives always the same result irrespective of which representative is chosen. This is in consistency with the fact, that e.g. the world-volume theory for the KK5A-brane is also $\\mathcal{N}=(1,1)$ LST and hence the entropy should be the same. As we show in Section \\ref{direct_KK} this is in contrast with the direct application of the Ryu-Takayanagi formula, which gives different results. \n\nOn the level of world-volume scalar fields $Y^M$ transition between orbit representatives (say NS5B and KK5A) is just replacement of a field $Y^m$ by its duality partner $\\tilde{Y}_m$. Although this has crucial impact on DFT and supergravity solutions changing the background, the world-volume theory has no way to see that, and hence it is always the same. For this reason, as the carrier of the $\\mathcal{N}=(1,1)$ LST in Type IIA string theory one should consider the localized Kaluza-Klein monopole rather than the smeared one \\cite{Harvey:2005ab}. The former is a deformation of the latter by instanton corrections, and is already exotic since its harmonic function depends on one dual coordinate \\cite{Jensen:2011jna,Bakhmatov:2016kfn}. The same is true for other exotic branes, which should also be localized.\n\nThe apparent issue that needs clarification is the following. For a theory on a Dp-brane one has apparent geometric picture, where the theory lives on a timelike surface at some $r\\neq 0$ in the transverse space. For AdS\/CFT correspondence one literally takes the conformal boundary of the anti-de-Sitter space. To calculate entanglement entropy geometrically one chooses a region $A$ on this surface and a surface $\\S$ in the transverse space of the brane such that $\\dt \\S =\\dt A$, and calculates its area in the given background. \n\nIn the case in question one cannot develop such simple geometric picture, and moreover one cannot do this already for the NS5 brane. Indeed, the corresponding geometry does not drop into AdS and the corresponding field theory is not conformal. However, the $6D$ field theory associated with the brane can be as well put at any $\\r$ in the transverse space, and the choice corresponds to the RG flow and one can still calculate area properly. The procedure described here suggests the following:\n\\begin{itemize}\n\\item start with a $5_2^r$-brane with any $r\\in\\{0,1,2,3,4\\}$ and its world-volume theory described by the doubled amount of scalar fields $\\Phi} \\def\\C{\\Chi} \\def\\Y{\\Psi} \\def\\W{\\Omega^M=(\\Phi} \\def\\C{\\Chi} \\def\\Y{\\Psi} \\def\\W{\\Omega^m,\\tilde{\\Phi} \\def\\C{\\Chi} \\def\\Y{\\Psi} \\def\\W{\\Omega}_m)$ half of which is projected out by the algebraic section constraint;\n\\item choose a region $A$ in the space of the theory with boundary $\\dt A$;\n\\item consider a surface $\\S$ with boundary $\\dt\\S$ parametrized by some coordinates $\\s^\\alpha} \\def\\b{\\beta} \\def\\g{\\gamma} \\def\\d{\\delta} \\def\\e{\\epsilon$;\n\\item this surface carries a doubled amount of scalar fields $\\{Y^M\\}$ with boundary conditions $Y^M\\big|_{\\dt\\S}=\\Phi} \\def\\C{\\Chi} \\def\\Y{\\Psi} \\def\\W{\\Omega^a\\big|_{\\dt A}$;\n\\item minimize the functional \\eqref{inv_entr}.\n\\end{itemize}\nThe theory in the first item here just descents from the full invariant effective action \\eqref{full_5}. The boundary condition is needed to identify the scalar fields living on the artificial surface $\\S$ with the actual fields of the theory. For the conventional geometric picture this is done automatically by the embedding functions, where both the theory and the surface live in a single geometric background. Apparently, this procedure trivially reproduces the conventional geometric calculation, and the only messages here are the following:\n\\begin{itemize}\n\\item to calculate entanglement entropy for the $\\mathcal{N}=(1,1)$ and $\\mathcal{N}=(2,0)$ $6D$ theories one may use equivalently any of the representative of the T-duality orbit;\n\\item to get the correct result one must take into account proper localization of the backgrounds in the dual space.\n\\end{itemize}\n\n\n\nAn interesting  further direction of research is to generalize the expression to the case of M5-brane which belongs to the same orbit as the $5^3$-brane under U-duality group. One then still works with Little String Theory and 6D, however the invariant expression will be different. One can also consider D-branes in DFT, which can also be non-geometric, i.e. localized in the dual space. The corresponding effective action will be presented in the forthcoming paper \\cite{axel-eric-fabio} and investigation of the corresponding world-volume theories and their entanglement entropy we reserve for future work.\n\n\n\n\n\\section*{Acknowledgements}  The authors are grateful to the Istanbul center of mathematical sciences and Bogazici University for hospitality during initial stages of this project. ETM would like to thank for hospitality Bogolyubov laboratory,  JINR, Dubna. The work of ETM was supported by the Russian state grant Goszadanie 3.9904.2017\/8.9 and by the Alexander von Humboldt return fellowship and partially by the program of competitive growth of Kazan Federal University. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Abstract}\n\nWe present the discovery of only the second radio-selected, z $\\sim$ 6 quasar.  We identified SDSS J222843.54+011032.2 (z=5.95) by matching the optical detections of the deep Sloan Digital Sky Survey (SDSS) Stripe 82 with their radio counterparts in the Stripe82 VLA Survey.  We also matched the Canadian-France-Hawaiian Telescope Legacy Survey Wide (CFHTLS Wide) with the Faint Images of the Radio Sky at Twenty-cm (FIRST) survey but have yet to find any z $\\sim$ 6 quasars in this survey area.  The discovered quasar is optically-faint, $z = 22.3$ and M$_{1450}$ $\\sim$ -24.5, but radio-bright, with a flux density of f$_{1.4GHz, peak}$ = 0.31mJy and a radio-loudness of R $\\sim$ 1100 (where R $\\equiv$ $f_{5GHz}\/f_{2500}$).  The $i-z$ color of the discovered quasar places it outside the color selection criteria for existing optical surveys.  We conclude by discussing the need for deeper wide-area radio surveys in the context of high-redshift quasars.    \n \n\\section{Introduction}\n\\label{sec:intro}\n\nHigh-redshift quasars (z $\\sim$ 6) which are powered by supermassive black holes (SMBH's) provide a window into the early universe, and, unsurprisingly, the study of these rare objects has grown in the decade since their discovery (\\citealp{2000AJ....120.1167F}; \\citealp{2001AJ....122.2833F}).  These distant SMBH's are important in the study of galaxy evolution and the epoch of re-ionization.   The strong correlation observed between the velocity dispersion of the stars in a galaxy and the mass of its central SMBH indicates a linkage between the mass of the SMBH and the evolution of the galaxy \\citep{2000ApJ...539L...9F}.  This relation may be explained by a ``quasar phase\" of the SMBH, where significant accretion is coupled with feedback into the stellar formation processes of the host galaxy \\citep{ 2005ApJ...625L..71H}.  Since nearly every galaxy is thought to have a central SMBH, active or not, high-redshift quasars are an important piece of the puzzle in understanding early galaxy evolution.  \n\nThe spectra of distant quasars probe the intergalactic medium (IGM) and can be used to study the re-ionization epoch along the line of sight (\\citealp{2001AJ....122.2850B}; \\citealp{2003AJ....126....1W}; \\citealp{2006AJ....132..117F}).  The presence of the Gunn-Peterson trough \\citep{1965ApJ...142.1633G} can indicate the neutral hydrogen fraction and act as a flag marking the re-ionization epoch.  There is evidence that the fraction of neutral hydrogen by volume increases an order of magnitude from z $=$ 5.7 to z $=$ 6.4 (\\citealp{2006AJ....132..117F}; \\citealp{2010ApJ...714..834C}).  This increase may mark the end of the re-ionization epoch, placing a high importance on any new discoveries of z $>$ 5.7 quasars, especially those which are radio-loud.  Radio-loud quasars are valuable as probes of the early IGM because the coming generation of radio telescopes (EVLA, ALMA, SKA, LOFAR, etc.) will use them to measure the properties of neutral hydrogen through 21 cm absorption studies (\\citealp{2004NewAR..48.1029C}; \\citealp{2010HiA....15..312K}).\n\nDespite large area surveys designed to find high-redshift quasars, only about 60 ($z > 5.7$) quasars have been found so far (\\citealp{2000AJ....120.1167F}; \\citealp{2007AJ....134.2435W}; \\citealp{2008AJ....135.1057J}).  These surveys all find quasars using similar methods that rely on red optical colors ($(i - z)_{AB}$ $>$ 2) and blue near-IR colors ($(z - J)_{AB}$ $<$ 1).  The red optical color of high-z quasars is a consequence of the Ly$\\alpha$ break moving through and out of the blueward optical bands.  The blue near-IR color cut is used to distinguish high-redshift quasars from the largest contaminant in these surveys: cool dwarf stars.  An alternative method to separate high-redshift quasars from cool dwarf stars is to require a radio detection, as very few stars are radio-bright (\\citealp{kimball}).  This method only finds $\\sim$5$\\%$ of quasars (3 of $\\sim$60 z $\\sim$ 6 quasars have been detected at $>$1mJy in the radio), but it can be used to select quasars with colors that fail the usual optical\/near-IR selection.  Previously, one radio-selected z $\\sim$ 6 quasar was found by \\citet{2006ApJ...652..157M} in a mere 4 ${deg}^2$ search area in the NOAO Deep Wide-Field Survey (NDWFS).  This quasar was bright enough to be discovered by other surveys; however, its red near-IR colors placed it in the color space occupied by cool dwarf stars, and hence it was missed by typical color-color selection methods.  \n\nThis work presents the discovery of the second radio-selected, z $\\sim$ 6 quasar.  We used two different survey combinations to search for radio-selected z $>$ 5.7 quasars: the Canadian-France-Hawaiian Telescope Legacy Survey Wide (CFHTLS Wide \\footnote[6]{http:\/\/www.cfht.hawaii.edu\/Science\/CFHTLS\/}) matched to the Faint Images of the Radio Sky at Twenty-cm (FIRST - \\citealp{1995ApJ...450..559B}) survey, and the deep Sloan Digital Sky Survey (SDSS Stripe 82 - \\citealp{2009ApJS..182..543A}) matched with the Stripe82 VLA Survey (Hodge et al. 2011).  We discovered the quasar in Stripe 82 with the deeper radio data of the Stripe82 VLA Survey.  \n\n In \\S 3 we describe our candidate selection methods and expected number of discoveries.  In \\S 4, we discuss the observations of SDSS J222843.53+011032.0 (SDSS J2228+0110; z $=$ 5.95). In \\S 5 we discuss the implications of further studies of high redshift radio-selected quasars.  All magnitudes are AB unless stated otherwise. This paper assumes a flat cosmological model with $\\Omega_{m} = 0.28$, $\\Omega_{\\Lambda} = 0.72$, and $H_{0} = 70$ km s$^{-1}$  Mpc$^{-1}$. \n\n\\section{Candidate Selection}\n\n\\subsection{CFHTLS Wide}\n\nCFHTLS Wide is an intermediate depth and area survey covering 171 deg$^2$ with $\\sim$130 deg$^2$ publicly available at the time of this work (release T0005).  The typical integration times are 4300s in $i$ and 3600s in $z$, reaching an average depth of 24.5 in i and 23.8 in z at the 80$\\%$ completeness limit determined through simulation\\footnote[7]{http:\/\/terapix.iap.fr\/cplt\/oldSite\/Descart\/CFHTLS-T0005-Release.pdf}. \n\n\\subsection{FIRST}\n\nFIRST is a 20 cm survey over $\\sim$10,000 deg$^2$.  With a sensitivity threshold of 1 mJy, it achieves a source density of $\\sim$90 deg$^{-2}$.  The FIRST survey covers the entirety of the CFHTLS Wide survey, and the typical astrometric accuracy between the two surveys is $<$ 0.5''.       \n\n\\subsection{Stripe 82}\n\nDuring the months when the primary SDSS area was not observable, SDSS repeatedly observed a strip of sky along the Galactic Equator known as Stripe 82.  This patch of sky is 300 deg$^2$ and spans roughly 20$^{h}$ $<$ RA $<$ 4$^{h}$ and -1.5$^{\\circ}$ $<$ Dec $<$ 1.5$^{\\circ}$.  The resulting co-additions (\\citealp{2009ApJS..182..543A}) go two magnitudes deeper than the typical SDSS images and reach 23.3 in $i$ and 22.5 in $z$ at the 95$\\%$ repeatable detection limit.  \n\n\\subsection{Stripe 82 VLA Survey}\n\nThis 1.4 GHz survey was conducted with the Very Large Array (VLA) in A-configuration and has an angular resolution of 1.8'' (Program ID AR646 and AR685).  It achieves a median rms noise of 52 $\\mu$Jy beam$^{-1}$ over 92 deg$^{2}$ (\\citealp{hodge2011}), making it the deepest 1.4 GHz survey to cover that much sky.  A catalog of 17,969 isolated radio components, for an overall source density of $\\sim$195 sources deg$^{-2}$, is publicly available.  The astrometric accuracy of the data is excellent, with an rms scatter of 0.25'' in both right ascension and declination when matched to SDSS's Stripe 82.\n\n\\subsection{Selection Method}\n\nHigh-redshift quasars are typically selected based on their very red optical colors: $(i - z)_{AB} > 1.5$.  Cool dwarf stars also have red optical colors, so in an effort to reduce contamination, surveys such as SDSS require a blue near-IR color (see Figure 1).  The optical color probes the Ly$\\alpha$ break while the near-IR color probes the quasar continuum as well as strong emission features.  Dust reddening can affect the color of the quasar continuum and displace a fraction of the high-z quasars outside of the typical selection criterion.  An alternative to a blue near-IR color cut is radio-selection, which reduces the contamination from cool dwarf stars to nearly zero but will only be able to recover $\\sim$5\\% of high-z quasars.   \n\nWe combined optical and radio data through catalog matching.  Counterparts between catalogs were defined using a matching radius of 0.6'' for Stripe 82 and 1'' for CFHTLS Wide, which was a  compromise between completeness and reliability.  We used a method similar to that of \\citet{2002ApJS..143....1M} to estimate completeness and reliability of the matches as a function of radius.  A matching radius of 1'' for FIRST-CFHTLS wide results in a completeness of 83\\% and a reliability of 94\\%.  A matching radius of 0.6'' for Stripe 82-Stripe 82 VLA results in a completeness of 91\\% and a reliability of 99\\%.   If there were multiple optical sources for a single radio source, then only the closest match was used.   After matching the two catalogs, two photometric cuts were applied to select the initial $z > 6$ quasar candidates:  $(i -z)_{AB} > 1.5$ and $z_{AB} <  23.8$ for CFHTLS Wide, and $(i -z)_{AB} > 1.7$ \\footnote[8]{The color conversion between SDSS and CFHT for typical z $\\sim$ 6 quasars is $(i-z)_{SDSS} > (i-z)_{CFHT} + 0.20$;  http:\/\/www.cadc.hia.nrc.gc.ca\/megapipe\/docs\/filters.html} and $z_{AB} <  22.5$ for Stripe 82.  The candidates went through another series of cuts which required that the \\textit{u}, \\textit{g}, and \\textit{r} band fluxes be below the 3$\\sigma$ detection limit.  Also, a visual inspection of the \\textit{i} and \\textit{z} band images was conducted to ensure there were no cosmic rays or bad pixels contaminating the photometry.  The remaining candidates were then checked against known sources as to not repeat observations.  One of the candidates was a previously found z$=$6.21 quasar by \\citet{2010AJ....139..906W}, CFHQS J1429+5447, and is the third radio-loud, z $\\sim$ 6 quasar to be identified.  CFHQS J1429+5447 is the radio-brightest z $\\sim$ 6 quasar yet to be found with f$_{1.4GHz, peak}$ = 2.93 mJy and has the highest radio-loudness value of R $\\sim$ 3200, where R $\\equiv$ $f_{5 GHz} \/ f_{2500\\AA}$.  After all of the cuts, there were 29 remaining candidates in CFHTLS Wide, i.e. $\\sim$ 0.2 per deg$^2$, and 27 in Stripe 82, giving $\\sim$ 0.3 per deg$^2$.  \n\nOur search area was previously mined by other high-z quasar searches and thus our candidate list has some overlap with those selection methods.  In the case of CFHTLS Wide, 3 of our 29 candidates satisfy the criteria set forth by \\citet{2009AJ....137.3541W}, $(i -z)_{AB} > 2.0$ and $z_{AB} < 23.0$ or 10$\\sigma$ $z_{AB}$ limit for the field (private communication).  Only 3 of the 27 candidates in Stripe 82 would have been selected by \\citet{2009AJ....138..305J} while the other 24 candidates were either too blue, $(i -z)_{AB} < 2.2$, or too faint, $z_{AB} > 21.8$ for their completeness-limited selection.      \n\nTo properly estimate how many quasars our radio-selection should find, we had to first estimate the completeness of our method as a function of redshift and rest-frame absolute magnitude, $M_{1450}$\\footnote[9]{$M_{1450}$ was calculated using the measured z-band magnitudes and assuming the \\citet{2001AJ....122..549V} spectrum corrected for the effective Gunn-Peterson optical depth due to Ly$\\alpha$ and Ly$\\beta$ absorption (\\citealp{2006AJ....132..117F})}.  The completeness of our optical selection was estimated through simulation. A relation between redshift and $i-z$ color was calculated using the median color track of z $\\sim$ 3 quasars from SDSS redshifted from z$=$5.5 to z$=$6.7 and corrected for the effective Gunn-Peterson optical depth due to Ly$\\alpha$ and Ly$\\beta$ absorption (\\citealp{2006AJ....132..117F}).  This track is very similar to that calculated by \\citet{2009AJ....137.3541W} as shown in Figure 1.  A ``measured'' $i-z$ color was drawn randomly from a gaussian distribution with a mean $i-z$ color taken from the median quasar color track and a sigma calculated using both the standard deviation of the median quasar color track and the median error as a function of the $i$ and $z$ magnitudes.  This was done hundreds of times for a grid of (z, $M_{1450}$), and the completeness was estimated by the fraction recovered by our selection method.  An example of our completeness as a function of redshift and absolute magnitude is shown in Figure 2.  The simulation of completeness only took into account the optical selection method and not the radio.  The radio-loud ($>$1mJy) fraction of quasars at 0 $<$ z $<$ 5 in SDSS is $\\sim$10\\%, but is strongly dependent on redshift and optical luminosity (\\citealp{jiang2007}).  We adopt a radio-loud fraction of 5\\% for z $\\sim$ 6 and apply this fraction to estimate the number of radio-selected quasars expected.                \n\nUsing the luminosity function for z $=$ 6 quasars calculated by \\citet{2010AJ....139..906W}, we are able to estimate the expected number of quasars in our search, N, \\begin{equation}\nN = A * \\iint \\ \\Phi(z,M_{1450}) \\, V_{c}(z)  \\,\\, p(z,M_{1450})   \\, \\mathrm{d}z \\, \\mathrm{d}M_{1450}.  \n\\end{equation}\nThe comoving volume, $V_{c}(z)$,  is corrected for the completeness, $p(z,M_{1450})$, to form an effective volume.  The luminosity function, $\\Phi(z,M_{1450})$, is in the form of a double power law, and we used the best fit parameters from \\citet{2010AJ....139..906W}.  Our search area is represented by A, in units of steradians.  From the calculation,  we expect 0.7 radio-selected quasars in Stripe 82 and 3.2 in CFHTLS Wide.  We find more candidates than the expected number of quasars for several reasons.  Firstly, some of our candidates are expected to be lower redshift, radio-loud AGN that share a red $i-z$ color with high-z quasars due to a strong Balmer break or dust extincted continuum.  Secondly, a few of our candidates may be spurious matches where the optical source is not really associated with the radio emission.  Thirdly, the use of a constant radio-loud fraction may not be appropriate as it has been observed to be a function of optical luminosity which could change our number estimates as much as a factor of 2 (\\citealp{jiang2007}).    Lastly, the best fit parameters of the luminosity function from \\citet{2010AJ....139..906W} have large errors and can change the expected number of quasars by a factor of 3 or 4.           \n\n\\section{Observations}\n\n\\subsection{Optical and Radio}\n\nThe quasar, SDSS J2228+0110, was selected for followup using the point spread function (PSF) magnitudes from the  co-added imaging catalog of Stripe 82 (\\citealp{2009ApJS..182..543A}).  It was found near our $z$-band magnitude limit at $z$ $=$ 22.28 and near our $i-z$ color limit at $i-z$ $=$ 1.81 which would have been too faint and too blue to be selected by \\citet{2009AJ....138..305J}.  It is one of the faintest quasars found to date, with M$_{1450}$ $=$ -24.53.  The quasar overlaps the UKIRT Infrared Deep Sky Survey (UKIDSS) sky coverage, but is undetected in the $J$-band placing an upper limit at $z-J \\le 1.4$\\footnote[10]{UKIDSS Large Area Survey detection limit is $J_{AB} = 20.9$, http:\/\/www.ukidss.org\/surveys\/surveys.html}.\n\nThe quasar was detected in the radio in the Stripe82 VLA Survey, which has a detection limit of 0.30 mJy.  SDSS J2228+0110 has a measured peak flux density of 0.31 mJy, just above the detection limit, and it is only the fourth z $\\sim$ 6 quasar discovered with a flux density f$_{1.4GHz} >$  0.3 mJy.  The faint optical luminosity and relatively high radio luminosity, $L_{5 GHz} = $ 2.55 $\\times$ 10$^{32}$, makes this one of the most radio-loud z $\\sim$ 6 quasars ever found, with a radio-loudness of R $\\sim$ 1100.  There is no discernible morphology from the optical or radio images given that the discovered quasar was at the limits of detection in both wavelengths (see Figure 3).\n\n\\subsection{Quasar Spectrum}\n\nFrom thirty-five candidates observed in June and December 2010 (sixteen from CFHTLS Wide and nineteen from Stripe 82 of which most remain unidentified with featureless continua), we discovered one quasar at z=5.95, SDSS J2228+0110 (see Table 1).  The discovery spectrum of SDSS J2228+0110 was taken using the Keck I telescope with the Low Resolution Imaging Spectrometer (LRIS; \\citealp{1995PASP..107..375O}).  The spectrum included four exposures of 900 seconds each and a 1\" long slit.  It was taken with a 600\/10000 grating on the red camera, resulting in a dispersion of $\\sim$0.8$\\AA$ per pixel.  The conditions were fair when the spectrum was taken, with $\\sim$1'' seeing at a high air mass of 1.5.  \n\nThe reduction of the spectrum was done in a standard way.  Bias frames were obtained and combined with the overscan bias, then subtracted off the science image.  A flat field correction was applied using flat field frames taken during the observing run.  A specialized routine was used for cosmic ray rejection which recognized and rejected cosmic rays based on their shape.  The spectrum of our quasar was extracted using optimal variance weighting through the IRAF task apall.  The wavelength was calibrated using arc line lamps.  The standard star used for flux calibration was Feige 34\\footnote[11]{The calibration flux was obtained from the Space Telescope standard star flux catalog} in combination with a custom Mauna Kea extinction curve as a function of wavelength.  A lower resolution spectrum, $\\sim$8$\\AA$ per pixel, was created using inverse sky-variance weighting.  The spectrum clearly identifies the source as a z=5.95 quasar with a large continuum break blueward of a strong emission line marked as Ly$\\alpha$ (see Figure 4).\n\nThe only strong emission feature in the discovery spectrum is Ly$\\alpha$.  The noise from sky emission lines resulted in a low signal to noise continuum, making it difficult to claim the detection of other, weaker emission lines.  However, the redshift calculated from Ly$\\alpha$ does seem consistent with possible detections of other common emission features such as NV and Si IV.  The wavelength coverage of the spectrum was not large enough to detect Ly$\\beta$ or OVI.    \n\nWe measured the rest-frame equivalent width (EW) and full width at half maximum (FWHM) of Ly$\\alpha$ for SDSS J2228+0110.  The  wavelength coverage for this spectrum was too small to fit the continuum slope; instead, we assumed it to be a power law with slope $\\alpha$ = -0.5 (f$_{\\nu} \\propto {\\nu}^{\\alpha}$).  The normalization of the power law was fit through a chi-squared minimization to the continuum redward of 1280$\\AA$.  We fit the Ly$\\alpha$ profile with a Gaussian on its red side, and we assumed the line to be symmetric to account for the Ly$\\alpha$ forest on the blue side.  We find a rest-frame FWHM of 7.68 $\\AA$, which is equivalent to a velocity of 1,890 km\/s, making it a narrow Ly$\\alpha$ but not abnormal.  The rest-frame EW is 21.9 $\\AA$ and is at the low end, but well within the range of Ly$\\alpha$ strengths of other z $\\sim$ 6 quasars.\n\n\\section{Discussion}\n\nHigh-redshift quasar searches have been popular over the last decade.  Large scale efforts have revealed that these objects are quite rare and quite difficult to find (\\citealp{2000AJ....120.1167F}; \\citealp{2007AJ....134.2435W}; \\citealp{2008AJ....135.1057J}).  Many of these surveys use optical data to select candidates and rely on very red optical colors.  Unfortunately, high-z quasars share the same red optical color space with cool dwarf stars.  In an effort to reduce contamination, a blue near-IR cut is used in the selection method to help separate the two populations.  This technique successfully increases efficiency; however, it reduces the completeness of the quasar sample by an unknown amount.   Our incompleteness was estimated by redshifting low-z quasar spectra to high-z and tracking their colors.  This track indicates that a large ``blue'' quasar population exists at high-z, and that optical surveys are selecting a significant sample of the population.  However, it is an open question as to whether low-z quasars are truly like their high-z counterparts.  \n\n\\citet{2006ApJ...652..157M} discovered, at the time, the highest redshift radio-loud quasar (FIRST J1427+3312) using radio selection in a search area of only 4 deg$^2$.  The quasar was bright enough in the optical to be detected in other surveys, but its red near-IR color would have prevented its selection.  This rare discovery in such a small area hints that there may be a larger population of ``red'' quasars than predicted.  If this is true, current estimates for the quasar number density at z $\\sim$ 6 are too low.  \n \nRadio-selection offers an alternative method for the selection of high-z quasars.  Requiring a radio detection is just as efficient in removing contaminants as a blue near-IR color cut; however, radio-selection also allows for the detection of ``red'' quasars, possibly from dust reddening.  A likely cause of dust reddening is the circumnuclear cocoon that is thought to encase relatively young quasars (\\citealp{ 2005ApJ...625L..71H}).  As the quasar ages, the dust cocoon may be blown away by energy released from accretion at near-Eddington luminosities.  The universe is less than one billion years old at z $\\sim$ 6, and it would not be unreasonable that the fraction of quasars in this dust cocoon phase is higher at higher redshift.  \n \nOf the seven z $\\sim$ 6 quasars with $z-J >$ 0.8, three of them are radio-loud.  It is not unusual that radio-loud quasars are redder than their radio-quiet counterparts, as this effect is also seen at lower redshift (\\citealp{2003AJ....126..706W}, \\citealp{2009AJ....138.1925M}).  However, when quasars at lower redshift, z $\\sim$ 3, are redshifted to z $=$ 6 and placed in the same colorspace as high-z quasars, at higher redshift radio-loud quasars tend to have redder $z-J$ colors than at lower redshift (see Figure 5).  This is a very intriguing result albeit in the small number regime, as it might suggest that radio-loud quasars are intrinsically redder at higher redshift or have higher quantities of dust.  The result should also be taken with caution as there is a substantial selection effect for z $\\sim$ 3 quasars in SDSS (\\citealp{richards}).  The selection efficiency at z $\\sim$ 3 is $\\sim$50$\\%$ and biased towards redder $u-g$ colors; however, radio-selected z $\\sim$ 3 quasars don't seem to show the same color bias (\\citealp{worseck}).  This comparison of radio-loud\/radio-selected z $\\sim$ 3 quasars redshifted to z $=$ 6 with observed radio-loud z $\\sim$ 6 quasars should be absent of a color selection-effect and serve as a viable comparison in cosmic time.    \n\nAlthough dust reddening of the quasar continuum may contribute to a significant ``red'' quasar population, it would also cause high levels of extinction in the detection bands.  This extinction can be greater than three magnitudes for E(B-V) = 0.1 using a typical SMC  reddening law (\\citealp{prevot}), placing even some of the brightest z $\\sim$ 6 quasars below the detection limit of wide-area surveys like CFHT and SDSS.  A more likely cause of a significant ``red'' quasar population is the strength of Ly$\\alpha$.  A strong Ly$\\alpha$ line leads to a ``bluer'' population of high-z quasars in the $z-J$ color while a weak Ly$\\alpha$ line leads to a ``redder'' population of z $\\sim$ 6 quasars.  A second factor that can lead to a ``red'' quasar population is the absorption blueward of Ly$\\alpha$ from neutral hydrogen in the intergalactic medium, which can lead to a ``red'' $i-z$ color for stronger absorption (see Figure 6).\n\nThis is still an ongoing search for radio-selected quasars.  We plan to observe more candidates in future observations and obtain a deeper optical spectrum of SDSS J2228+0110.  In an effort to measure the $z-J$ color of radio-selected z $\\sim$ 6 quasars, we also plan to follow up on our current discovery and any new discoveries with near-IR imaging.  It would be interesting to use the discovered z $\\sim$ 6 quasar to constrain the luminosity function of high-z quasars outside of the typical optical selection criteria; however, our survey is not complete.   \n \nThe greatest limitation to radio-selection is the depth of the radio data.  Only $\\sim$5\\% of z $\\sim$ 6 quasars are detected in FIRST.  Even a medium depth radio survey like the Stripe82 VLA Survey does not reach the median quasar radio luminosity.  With a deep and wide-area radio survey, questions about the existence of a large ``red'' high-z quasar population, as well as the mechanism that might cause the reddening, could finally be answered.                  \n\n{\\bf Acknowledgements}:  Some of the data presented herein were obtained at the W.M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W.M. Keck Foundation.\nBased on observations obtained with MegaPrime\/MegaCam, a joint project of CFHT and CEA\/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS. \nThe authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community.  We are most fortunate to have the opportunity to conduct observations from this mountain.  The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.\nGRZ acknowledges NRAO Grant GSSP 09-0010.  The work by RHB was partly performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#1} \\vspace{0mm}}\n\\newcommand{\\SubSection}[1]{\\vspace{-1mm} \\subsection{#1} \\vspace{-0mm}}\n\\newcommand{\\SubSubSection}[1]{\\vspace{-1mm} \\subsubsection{#1} \\vspace{-1mm}}\n\n\\newcommand\\Mark[1]{\\textsuperscript#1}\n\n\\iccvfinalcopy\n\n\\def\\iccvPaperID{10568}\n\\def\\mbox{\\tt\\raisebox{-.5ex}{\\symbol{126}}}{\\mbox{\\tt\\raisebox{-.5ex}{\\symbol{126}}}}\n\n\n\\begin{document}\n\t\n\\begin{textblock*}{\\textwidth}(0cm,0cm)\n\t\\large\\noindent{\\copyright~2021 IEEE.  Personal use of this material is permitted.  Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting\/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.}\n\\end{textblock*}\n\\thispagestyle{empty}\t\n\n\n\\title{\\textbf{Full-Velocity Radar Returns by Radar-Camera Fusion}}\n\n\\author{\nYunfei Long\\Mark{1}, Daniel Morris\\Mark{1},  Xiaoming Liu\\Mark{1}, \\\\\nMarcos Castro\\Mark{2},\nPunarjay Chakravarty\\Mark{2},\nand Praveen Narayanan\\Mark{2} \\\\ \n\\Mark{1}Michigan State University, \\Mark{2}Ford Motor Company \\\\\n{\\tt\\small \\{longyunf,dmorris,liuxm\\}@msu.edu},\n{\\tt\\small \\{mgerard8,pchakra5,pnaray11\\}@ford.com}\n}\n\\date{}\n\n\n\\maketitle\n\\pagenumbering{arabic}\n\n\\begin{abstract}\nA distinctive feature of Doppler radar is the measurement of velocity in the radial direction for radar points. \nHowever, the missing tangential velocity component hampers object velocity estimation as well as temporal integration of radar sweeps in dynamic scenes.  \nRecognizing that fusing camera with radar provides complementary information to radar, in this paper we present a closed-form solution for the point-wise, full-velocity estimate of Doppler returns using the corresponding optical flow from camera images.  \nAdditionally, we address the association problem between radar returns and camera images with a neural network that is trained to estimate radar-camera correspondences. \nExperimental results on the nuScenes dataset verify the validity of the method and show significant improvements over the state-of-the-art in velocity estimation and accumulation of radar points.   \n\\end{abstract}\n\n\\section{Introduction}\nRadar is a mainstream automotive 3D sensor, and along with LiDAR and camera, is used in perception systems for driving assistance and autonomous driving~\\cite{stanislas2015characterisation, li2020lidar,m3d-rpn-monocular-3d-region-proposal-network-for-object-detection}. \nUnlike LiDAR, radar has been widely installed on existing vehicles due to its relatively low cost and small sensor size, which makes it an easy fit into various vehicles without changing their appearance. Thus, advances in radar vision systems have potential to make immediate impact on vehicle safety. Recently, with the release of a couple of autonomous driving datasets with radar data included,~\\emph{e.g.}, Oxford Radar RobotCar~\\cite{barnes2020oxford} and nuScenes~\\cite{caesar2020nuscenes}, there is great interest in the community to explore how to leverage radar data in various vision tasks such as object detection~\\cite{nabati2021centerfusion, yang2020radarnet}.\n\n\n\\begin{figure}[t!]\n    \\captionsetup{font=small}\n\t\\centering\n\t\\begin{subfigure}[b]{\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.9\\textwidth]{.\/figures\/indeterminacy.pdf}\n\t\t\\vspace{-2mm}\n\t\t\\caption{$ $}\n\t\t\\label{fig:y equals x}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[b]{0.2\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{.\/figures\/flow1}\n\t\t\\caption{$ $}\n\t\t\\label{fig:three sin x}\n\t\\end{subfigure}\n\t\\hfill\n\t\\begin{subfigure}[b]{0.22\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{.\/figures\/ex_motion.pdf}\n\t\t\\caption{$ $}\n\t\t\\label{fig:five over x}\n\t\\end{subfigure}\n\t\\vspace{-2mm}\n\t\\caption{\\small (a) Full motion cannot be determined with a single sensor: all motions ending on the blue dashed line (\\emph{i.e.},~blue dashed arrows) map to the same optical flow and all motions terminated on the red dashed line (\\emph{i.e.},~red dashed arrows) fit the same radial motion. However, with a radar-camera pair, the full motion can be uniquely decided: only the motion drawn in black satisfies both optical flow and radial motion. (b) Optical flow in the camera-image and (c) a bird's-eye view of the observed vehicle. This shows measured radar points with radial velocity (red), our predicted point-wise, full velocity (black), and ground truth full velocity of the vehicle (green).}\n\t\\vspace{-3mm}\n\t\\label{Figure:indeterminacy}\n\\end{figure}\n\nIn addition to measuring 3D positions, radar has the special capability of obtaining radial velocity of returned points based on the Doppler effect. \nThis extra capability is a significant advantage over other 3D sensors like LiDAR, enabling, for instance, instantaneous moving object detection. \nHowever, due to the inherently ambiguous mapping from radial velocity to full velocity, using radial velocity directly to account for the real movement of radar points is inadequate and sometimes misleading. \nHere, the full velocity denotes the actual velocity of radar points in 2D or 3D space. \nWhile radial velocity can well approximate full velocity when a point is moving away from or towards the radar, these two can be very different when the point is moving in the non-radial directions. \nAn extreme case occurs for objects moving tangentially as these will have zero radial velocity regardless of target speed. Therefore, acquiring point-wise full velocity instead of radial velocity is crucial to reliably sense the motion of surrounding objects.\n\nApart from measuring the velocity of objects, another important application of point-wise velocity is the accumulation of radar points. \nRadar returns from a single frame are much sparser than LiDAR in both azimuth and elevation,~\\emph{e.g.}, typically LiDAR has an azimuth resolution $10\\times$ higher than radar~\\cite{yang2020radarnet}.\nThus, it is often essential to accumulate multiple prior radar frames to acquire sufficiently dense point clouds for downstream tasks,~\\emph{e.g.}, object detection~\\cite{nobis2019deep, chadwick2019distant, chang2020spatial}. \nTo align radar frames, in addition to compensating egomotion, we shall consider the motion of moving points in consecutive frames, which can be estimated by point-wise velocity and time of movement. \nAs the radial velocity does not reflect the true motion, it is desirable to have point-wise full velocity for point accumulation.\n\n\nTo solve the aforementioned dilemma of radial velocity, we propose to estimate point-wise full velocity of radar returns by fusing radar with a RGB camera. \nSpecifically, we derive a closed-form solution to infer point-wise full velocity from radial velocity as well as associated projected image motion obtained from optical flow. As shown in Fig.~\\ref{Figure:indeterminacy}, constraints imposed by optical flow resolve the ambiguities of radial-full velocity mapping and lead to a unique and closed-form solution for full velocity. \nOur method can be considered as a way to enhance raw radar measurement by upgrading point-wise radial velocity to full velocity, laying the groundwork for improving radar-related tasks, ~\\emph{e.g.}, velocity estimation, point accumulation and object detection. \n\nMoreover, a prerequisite for our closed-form solution is the association between moving radar points and image pixels. \nTo enable a reliable association, we train a neural network to predict radar-camera correspondences as well as discerning occluded radar points. \nExperimental results demonstrate that the proposed method improves point-wise velocity estimates and their use for object velocity estimation and radar point accumulation.\n\nIn summary, the main contributions of this work are:\n\\begin{itemize}\n    \\item We define a novel research task for radar-camera perception systems,~\\emph{i.e.}, estimating point-wise full velocity of radar returns by fusing radar and camera.\n    \n    \\item We propose a novel closed-form solution to infer full radar-return velocity by leveraging the radial velocity of radar points, optical flow of images, and the learned association between radar points and image pixels.\n    \n    \\item We demonstrate state-of-the-art (SoTA) performance in object velocity estimation, radar point accumulation, and 3D object localization.\n    \n\\end{itemize}\n\n\n\\section{Related Works}\n\n\\Paragraph{Application of Radar in Vision}\nRadar data differs from LiDAR data in various aspects~\\cite{brodeski2019deep}. \nIn addition to the popular point representation (also named radar target~\\cite{palffy2020cnn}), an analogy to LiDAR points, there are other radar data representations containing more raw measurements, {\\it e.g.}, range-azimuth image and spectrograms, which have been applied in tasks such as activity classification~\\cite{seyfiouglu2018deep}, detection~\\cite{lim2019radar}, and pose estimation~\\cite{roos2016reliable}.  \nOur method is based on radar points, with the format available in the  nuScenes dataset~\\cite{caesar2020nuscenes}. \n\nThe characteristics of radar have been explored to complement other sensors. \nThe Doppler velocity of radar points is used to distinguish moving targets. \nFor example, RSS-Net~\\cite{kaul2020rss} uses radial velocity as a motion cue for image semantic segmentation. Chadwick~\\emph{et al.}~\\cite{chadwick2019distant} use radial velocity to detect distant moving vehicles---difficult to detect with only images. \nFritsche~\\emph{et al.}~\\cite{fritsche2017fusion} combine radar with LiDAR for measurement under poor visibility. \nWith a longer detection range than LiDAR, radar is also deployed with LiDAR to better detect far objects~\\cite{yang2020radarnet}.\n\nThe {\\it sparsity} of radar makes it difficult to directly apply well-developed techniques for LiDAR on radar~\\cite{lim2019radar, nabati2021centerfusion}. \nFor example, Danzer~\\emph{et al.}~\\cite{danzer20192d} adopt PointNets~\\cite{qi2017pointnet} on radar points for 2D car detection, while sparsity limits it to large objects like cars. \nSimilar to LiDAR-camera depth completion~\\cite{depth-completion-with-twin-surface-extrapolation-at-occlusion-boundaries,depth-coefficients-for-depth-completion}, Long~\\emph{et al.}~\\cite{long2021radar} develop radar-camera depth completion by learning a probabilistic mapping from radar returns to images.\nTo obtain denser radar points, Lombacher~\\emph{et al.}~\\cite{lombacher2016potential} use occupancy grid~\\cite{elfes1989using} to accumulate radar frames.\nYet, the method assumes a static scene and cannot cope with moving objects. \nRadar points are projected on images and represented as regions near projected points, such as vertical bars~\\cite{nobis2019deep} and circles~\\cite{chadwick2019distant, chang2020spatial}, to account for uncertainty of projection due to measurement error. \nWhile accumulating radar frames is desirable, without reliably compensating object motion, these methods need to carefully decide the number of frames to trade off between the gain in accumulation and loss in accuracy due to delay~\\cite{nobis2019deep}. \nOur estimated point-wise velocity can compensate object motion and realize more accurate accumulation.\n\n\n\\Paragraph{Velocity Estimation in Perception Systems}\nResearchers have used monocular videos~\\cite{kinematic-3d-object-detection-in-monocular-video} or radial velocity of radar points to estimate {\\it object-wise} velocity. With only radar data of a single frame, Kellner~\\emph{et al.}~\\cite{kellner2013instantaneous, kellner2014instantaneous} compute full velocity of moving vehicles from radial velocities and azimuth angles of at least two radar hits. \nHowever, for a robust solution, the method requires that 1) radar captures more radar hits on each object, 2) radar points have significantly different azimuth angles and 3) object points are clustered before velocity estimation~\\cite{kellner2013instantaneous, schlichenmaier2019clustering, scheiner2019multi}. \nObviously due to sparsity of radar in a single frame, it is difficult to obtain at least two radar hits on distant vehicles, let alone objects of smaller sizes. \nAlso, it is common that radar points on the same object, {\\it e.g.}, a distant or small object, have similar azimuth. \n\n\\begin{figure*}[t!]\n    \\captionsetup{font=small}\n\t\\begin{center}\n\t\t\\includegraphics[width=\\linewidth]{figures\/diagram.pdf}\n\t\\end{center}\n\t\\vspace{-6mm}\n\t\\caption{\\small \\textbf{Full velocity estimation and learning to associate radar points to camera pixels}. (a) A 3D point, $\\bm{p}$, is observed by a camera at $B$.  A short interval, $\\Delta t$, later, the point has moved by $\\dot{\\bm{m}}\\Delta t$ to $\\bm{q}$ while the camera has moved by $\\dot{\\bm{c}}\\Delta t$ to $A$. At the same time, the radar measures both the position of $\\bm{q}$ and the radial speed $\\dot{r}$, which is the radial component of $\\dot{\\bm{m}}$. Using radial speed $\\dot{r}$ and the associated  optical flow of $\\bm{q}$ in images, we derive a closed-form equation (denoted as $\\bm{f}()$) to estimate $\\bm{q}$'s full velocity $\\dot{\\bm{m}}$. (b) As the closed-form solution requires point-wise association of two sensors, we train a Radar-2-Pixel (R2P) network to take a multi-channel input and predict the association probabilities for pixels within a neighborhood of the raw projection (white dot) obtained via known pose $\\prescript{A}{R}\\T$. A pixel with the highest probability (yellow arrow) is deemed as the associated pixel of a radar point. To obtain labels for training R2P, our label generation module uses $\\bm{f}()$ to compute velocities of all neighboring pixels, then calculates velocity error $E_m$ by using the ground truth velocity $\\dot{\\bm{m}}_{GT}$, and finally obtains association probabilities of these neighbors based on $E_m$.}\n\t\\label{fig:diagram_association}\n\t\t\\vspace{-3mm}\n\\end{figure*}\n\n\nRecognizing the density and accuracy limitation of radar, researchers fuse radar with other sensors, {\\it e.g.}, LiDAR and camera, for object-wise velocity estimation. \nSpecifically, existing techniques~\\cite{zhao2019object,wu2020deep,li2020deep} for images or LiDAR are employed to obtain preliminary detections. \nRadar data, including radial velocity, once associated with the initial detections, are used as additional cues to predict full velocities of objects. \nFor instance, in RadarNet~\\cite{yang2020radarnet} \ntemporal point clouds of radar and LiDAR, modeled as voxels, are used to acquire initial detections and their motions. \nObject motion direction is used to resolve the ambiguities in radar-point association by\nback-projecting their radial velocities on the motion direction. \nYet, a sequence of LiDAR frames is required to obtain the initial detection and motion estimation.\n\nCenterFusion~\\cite{nabati2021centerfusion} integrates radar with camera for object-wise velocity estimation. \nWell-developed image-based detector is applied to extract preliminary boxes.\nAfter associating radar points with detections, the method combines radar data, radial velocity and depth, with image features within detected regions to regress a full velocity per detection. \nHowever, without a closed-form solution, the mapping from radial to full velocity needs to be learned from a great number of labeled data. \nIn contrast, we present a point-wise {\\it closed-form solution} for full-velocity estimation of radar points, without performing object detection. \nTo our knowledge, there is no prior method able to perform point-wise full-velocity estimation for radar returns.\n\n\n\\section{Proposed Method}\n\nWe consider the case of a camera and radar rigidly attached to a moving platform, {\\it e.g.}, a vehicle, observing moving objects in the environment.  In this section we develop equations relating optical flow measurements in the camera to position and velocity measurements made by the radar.  \n\n\\subsection{Physical Configuration and Notation}\n\n\nThe physical configuration of our camera and radar measurements is illustrated in Fig.~\\ref{fig:diagram_association}(a). Three coordinate systems are shown: $A$ and $B$ specifying camera poses and $R$ specifying a radar pose.  The camera at $B$ observes a 3D point $\\bm{p}$.  A short interval later, $\\Delta t$, the point has moved to $\\bm{q}$, the camera to $A$ and the radar to $R$, and both the camera and radar observe the target point $\\bm{q}$.  These 3D points are specified by $4$-dim homogeneous vectors, and when needed, a left-superscript specifies the coordinate system in which it is specified, {\\it e.g.}, $\\prescript{A}{}\\bm{q}$ indicates a point relative to a coordinate system $A$.  The target velocity, $\\dot{\\bm{m}}$, and camera velocity $\\dot{\\bm{c}}$ are specified by $3$-dim vectors, again optionally with a left superscript to specify a coordinate system.    \n\nCoordinate transformations, containing both a rotation and translation, are specified by $4\\times 4$ matrices, such as $\\prescript{B}{A}{\\bm{T}}$, which transforms points from the left-subscript coordinate system to the left-superscript coordinate system.  In this case we transform a point from $A$ to $B$ with:\n\\begin{equation}\n    \\prescript{B}{}\\bm{q} = \\prescript{B}{A}{\\bm{T}} \\: \\prescript{A}{}\\bm{q}.\n\\end{equation}\nOnly the rotational component of these transformations is needed to transform velocities.  For example, $\\prescript{A}{}\\dot{\\bm{m}}$ is transformed to $\\prescript{B}{}\\dot{\\bm{m}}$ by the $3\\times 3$ rotation matrix $\\prescript{B}{A}{\\bm{R}}$:\n\\begin{equation}\n    \\prescript{B}{}\\dot{\\bm{m}} = \\prescript{B}{A}{\\bm{R}} \\: \\prescript{A}{}\\dot{\\bm{m}}.\n\\end{equation}\n\nA vector with a right subscript, {\\it e.g.}, ${\\bm{p}_i}$, indicates the $i$'th element of ${\\bm{p}}$, while a right subscript of ``1:3'' puts the first $3$ elements in a $3$-dim vector.  \nFor a matrix, the right subscript indicates the row. Thus $\\prescript{B}{A}{\\bm{R}_i}$ is a $1\\times 3$ row vector containing its $i$-th row.  A right superscript ``${}^\\mathsf{T}$'' is a matrix transpose.\n\nThe projections of points $\\bm{p}$ and $\\bm{q}$ are specified in either undistorted raw pixel coordinates, {\\it e.g.}, $(x_q,y_q)$ or their normalized image coordinates $(u_q,v_q)$ given by:\n\\begin{equation}\n    u_q = (x_q-c_x) \/ f_x, \\hspace{0.5cm} v_q = (y_q-c_y) \/ f_y.\n\\end{equation}\nHere $c_x,c_y,f_x,f_y$ are intrinsic camera parameters, while the right subscript of the pixel refers to the point being projected.  Vectors for 3D points can be expressed in terms of the normalized image coordinates:\n\\begin{align}\n\\prescript{A}{}\\qv = \\begin{pmatrix} u_qd_q\\\\ v_qd_q\\\\ d_q\\\\ 1 \\end{pmatrix} \n\\;\\; \\text{and} \\;\\;\n\\prescript{B}{}\\pv = \\begin{pmatrix} u_pd_p\\\\ v_pd_p\\\\ d_p\\\\ 1 \\end{pmatrix}.\n\\label{eq:qa}\n\\end{align}\nHere $d_q$ and $d_p$ are depths of points $\\prescript{A}{}\\qv$ and $\\prescript{B}{}\\pv$ respectively. \n\nWe assume dense optical flow is available that maps target pixel coordinates observed in $A$ to $B$ as follows:\n\\begin{equation}\n    \\text{Flow}\\left( (u_q,v_q) \\right) \\rightarrow (u_p,v_p).\n    \\label{eq:flow}\n\\end{equation}\nFurther, we assume the following are known: camera motion, $\\prescript{B}{A}\\T$, relative radar pose, $\\prescript{A}{R}\\T$, and intrinsic parameters.\n\n\\subsection{Full-Velocity Radar Returns}\n\nThe Doppler velocity measured by a radar is just one component of the three-component, full-velocity vector of an object point.  Here our goal is to leverage optical flow from a synchronized camera to augment radar and estimate this full-velocity vector for each radar return.  \n\n\\subsubsection{Relationship of Full Velocity to Radial Velocity}\n\nThe target motion from $\\bm{p}$ to $\\bm{q}$ is modeled as constant velocity, $\\dot{\\bm{m}}$, over time $\\Delta t$, such that \n\\begin{equation}\n    \\dot{\\bm{m}}=\\frac{\\bm{q}_{1:3}-\\bm{p}_{1:3}}{\\Delta t}.  \n    \\label{eq:velocity}\n\\end{equation}\nOur goal is to estimate the full target velocity, $\\dot{\\bm{m}}$.  Radar provides an estimate of the target position, $\\bm{q}$, but not the previous target location $\\bm{p}$.  Radar also provides the signed radial speed, $\\dot{r}$, which is one component of $\\dot{\\bm{m}}$.  In the nuScenes dataset $\\dot{r}$ is given by:\n\\begin{equation}\n    \\dot{r} = \\hat{\\rv}^\\mathsf{T}\\dot{\\bm{m}}.\n    \\label{eq:radial}\n\\end{equation}\nHere $\\hat{\\rv}$ is the unit-norm vector along the direction to the target $\\prescript{R}{}\\qv$.   Note that this equation is coordinate-invariant, and could  be equally written in $A$ using $\\prescript{A}{}\\hat{\\rv}$ and $\\prescript{A}{}\\mvdot$.  Now Eq.~\\eqref{eq:radial} is actually the egomotion-corrected Doppler speed.  The raw Doppler speed, $\\dot{r}_{raw}$, is the radial component of the \\emph{relative} velocity between target and sensor, $\\dot{\\bm{m}}-\\dot{\\bm{c}}$, and this constraint is given by:\n\\begin{equation}\n    \\dot{r}_{raw} = \\hat{\\rv}^\\mathsf{T}(\\dot{\\bm{m}}-\\dot{\\bm{c}}),\n    \\label{eq:radial_relative}\n\\end{equation}\nwhere $\\dot{\\bm{c}}$ is the known ego-velocity. Either Eq.~\\eqref{eq:radial} or \\eqref{eq:radial_relative} can be used in our formulation, depending on whether $\\dot{r}$ or $\\dot{r}_{raw}$ is available from the radar.\n\n\n\\subsubsection{Relationship of Full Velocity to Optical Flow}\n\nIn solving the velocity constraints, we first identify the known variables.  The radar measures $\\prescript{R}{}\\qv$, and transforming this we obtain $\\prescript{A}{}\\qv=\\prescript{A}{R}\\bm{T}\\:\\prescript{R}{}\\qv$ which contains $d_q$ as the third component.  Image coordinates $(u_q,v_q)$ are obtained by projection, and using optical flow in Eq.~\\eqref{eq:flow}, we can also obtain the $(u_p,v_p)$ components of $\\prescript{B}{}\\pv$.  The key parameter we do not know from this is the depth, $d_p$, in $B$.\n\nNext we eliminate this unknown depth from our constraints. Eq.~\\eqref{eq:velocity} can be rearranged and each component expressed in frame $B$:\n\\begin{equation}\n    \\prescript{B}{}\\pv_{1:3} = \\prescript{B}{}\\qv_{1:3} - \\prescript{B}{A}\\R\\: \\prescript{A}{}\\mvdot\\Delta t,\n    \\label{eq:pvB2}\n\\end{equation}\nwhere the second term on the right is the transformation of the target motion into $B$ coordinates.  The third row of this equation is an expression for $d_p$:\n\\begin{equation}\n    d_p = \\prescript{B}{}\\qv_3 - \\prescript{B}{A}\\R_3\\: \\prescript{A}{}\\mvdot\\Delta t.\n    \\label{eq:dp1}\n\\end{equation}\nSubstituting this for $d_p$, and the components of $\\prescript{B}{}\\pv$ from  Eq.~\\eqref{eq:qa}, into the first two rows of Eq.~\\eqref{eq:pvB2}, we obtain\n\\begin{eqnarray}\n\\begin{bmatrix}\nu_p(\\prescript{B}{}\\qv_3 - \\prescript{B}{A}\\R_3\\: \\prescript{A}{}\\mvdot\\Delta t) \\\\\nv_p(\\prescript{B}{}\\qv_3 - \\prescript{B}{A}\\R_3\\: \\prescript{A}{}\\mvdot\\Delta t) \\\\\n\\end{bmatrix}\n=\n\\begin{bmatrix}\n\\prescript{B}{}\\qv_1 - \\prescript{B}{A}\\R_1\\: \\prescript{A}{}\\mvdot\\Delta t \\\\\n\\prescript{B}{}\\qv_2 - \\prescript{B}{A}\\R_2\\: \\prescript{A}{}\\mvdot\\Delta t \\\\\n\\end{bmatrix},\n\\label{eq:dp2}\n\\end{eqnarray}\nand rearrange to give two constraints on the full velocity:\n\\begin{eqnarray}\n\\begin{bmatrix}\n\\prescript{B}{A}\\R_1 - u_p\\prescript{B}{A}\\R_3 \\\\\n\\prescript{B}{A}\\R_2 - v_p\\prescript{B}{A}\\R_3 \\\\\n\\end{bmatrix}\n\\prescript{A}{}\\mvdot\n=\n\\begin{bmatrix}\n\\left(\\prescript{B}{}\\qv_1 - u_p\\prescript{B}{}\\qv_3\\right)\/\\Delta t  \\\\\n\\left(\\prescript{B}{}\\qv_2 - v_p\\prescript{B}{}\\qv_3\\right)\/\\Delta t \\\\\n\\end{bmatrix}.\n\\label{eq:dp3}\n\\end{eqnarray}\n\n\\subsubsection{Full-Velocity Solution}\n\nWe obtain three constraints on the full velocity, $\\prescript{A}{}\\mvdot$, from Eq.~\\eqref{eq:dp3} and by converting Eq.~\\eqref{eq:radial}  to $A$ coordinates. Combining these we obtain: \n\\begin{eqnarray}\n\\begin{bmatrix}\n\\prescript{B}{A}\\R_1 - u_p\\prescript{B}{A}\\R_3 \\\\\n\\prescript{B}{A}\\R_2 - v_p\\prescript{B}{A}\\R_3 \\\\\n\\prescript{A}{}\\hat{\\rv}^\\mathsf{T} \\\\\n\\end{bmatrix}\n\\prescript{A}{}\\mvdot\n=\n\\begin{bmatrix}\n\\left(\\prescript{B}{}\\qv_1 - u_p\\prescript{B}{}\\qv_3\\right)\/\\Delta t  \\\\\n\\left(\\prescript{B}{}\\qv_2 - v_p\\prescript{B}{}\\qv_3\\right)\/\\Delta t \\\\\n\\dot{r} \\\\\n\\end{bmatrix}.\n\\label{eq:dp4}\n\\end{eqnarray}\nThen inverting the $3\\times 3$ coefficient of $\\prescript{A}{}\\mvdot$ gives a closed form solution for the full velocity:\n\\begin{eqnarray}\n\\prescript{A}{}\\mvdot =\n\\begin{bmatrix}\n\\prescript{B}{A}{\\bm{R}_1} - u_p \\prescript{B}{A}{\\bm{R}_3}\\\\\n\\prescript{B}{A}{\\bm{R}_2} - v_p \\prescript{B}{A}{\\bm{R}_3}\\\\\n\\prescript{A}{}\\hat{\\rv}^\\mathsf{T}\\\\ \n\\end{bmatrix}^{-1}\n\\begin{bmatrix}\n\\left(\\prescript{B}{}{\\bm{q}}_1 - u_p \\prescript{B}{}{\\bm{q}_3} \\right) \/ \\Delta t\\\\\n\\left(\\prescript{B}{}{\\bm{q}}_2 - v_p \\prescript{B}{}{\\bm{q}_3} \\right) \/ \\Delta t\\\\\n\\dot{r} \\\\ \n\\end{bmatrix}\n\\label{eq:full_v}.\n\\end{eqnarray}\n\n\n\\begin{figure}[t!]\n    \\captionsetup{font=small}\n\t\\centering\n\t\\scalebox{1}{\n\t\t\\begin{tabular}{@{}c@{}c@{}c@{}c@{}}\n\t\t\t\\includegraphics[width=1.4 in]{.\/figures\/flow2}  &\n\t\t\t\\includegraphics[width=1.4 in]{.\/figures\/bev1.pdf} \\vspace{-1mm}\\\\\n\t\t\t\\footnotesize{ (a) } &  \\footnotesize{ (b) } \\\\\n\t\t\t\\includegraphics[width=1.4 in]{.\/figures\/04_ev}  &\n\t\t\t\\includegraphics[width=1.4 in]{.\/figures\/02_ev} \\vspace{-1mm}\\\\\n\t\t\t \\footnotesize{ (c) } & \\footnotesize{ (d) } \\\\\n\t\\end{tabular}   }\n\t\\vspace{-2mm}\n\t\\caption{\\small (a) Optical flow; (b) Bird's-eye view of GT bounding box, radial velocity (red) and GT velocity (green); (c) and (d) show $E_m$, computed by using Eq.~(\\ref{eq:e_v}), for two radar projections (white square) over $41\\times41$ pixel regions, respectively. For radar hits reflected from the vehicle, $E_m$ is small for neighboring pixels on the car and large on the background.}\n\t\\label{Figure:label}\\vspace{-3mm}\n\\end{figure}\n\n\n\nRecall in Fig.~\\ref{Figure:indeterminacy}(a) the red\/blue dashed lines show the velocity constraints from radar\/flow.  The solution of Eq.~\\eqref{eq:full_v} is the full velocity that is consistent with both constraints. \nWe note that this can handle moving sensors, although Fig.~\\ref{Figure:indeterminacy}(a) shows the case of a stationary camera for simplicity.  Further, if we set $\\Delta t<0$, Eq.~\\eqref{eq:full_v} also applies to the case that the point shifts from $\\bm{q}$ to $\\bm{p}$ as the camera moves from $A$ to $B$.  And one limitation is that Eq.~\\eqref{eq:full_v} cannot estimate full velocity for radar points occluded in the camera view, although we can typically identify those occlusions.\n\n\n\\SubSection{Image Pixels and Radar Points Association}\n\nOur solution for point-wise velocity in Eq.~(\\ref{eq:full_v}) assumes that we know the pixel coordinates $(u_q,v_q)$ of the radar-detected point, $\\prescript{R}{}\\bm{q}$.\nIt appears straightforward to obtain this pixel correspondence by projecting a radar point onto the image using the known radar-image coordinate transformation, $\\prescript{A}{R}\\bm{T}$. \nWe refer to this corresponding pixel as ``raw projection''. \nHowever, there are a number of reasons why raw projection of radar points into an image is inaccurate. Radar beam-width typically subtends a few degrees and is large relative to a pixel, resulting in low resolution target location in both azimuth and elevation.  Also, a radar displaced from a camera can often see behind an object, as viewed by the camera, and when these returns are projected onto an image they incorrectly appear to correspond to the foreground occluding object. Using flow from an occluder or an incorrectly associated object pixel may result in incorrect full-velocity estimation.  To address these issues with raw projection, we train a neural network model, termed Radar-2-Pixel (R2P) network, to estimate associated radar pixels in the neighborhood of raw projection and identify occluded radar points.\nSimilar models have been applied to image segmentation~\\cite{kampffmeyer2018connnet} and radar depth enhancement~\\cite{long2021radar}.\n\n\n\\SubSubSection{Model Structure}\nOur method estimates association probabilities (ranging from $0$ to $1$) between a moving radar point and a set of pixels in the neighborhood of its raw projection. \nThe R2P network is an encoder-decoder structure with inputs and outputs of image resolution. \nStored in $8$ channels, the input data include image, radar depth map (with depth on raw projections) and optical flow. \nThe output has $N$ channels, representing predicted association probability for $N$ pixel neighbors. \nThe association between the radar point, $\\prescript{A}{}\\qv$, and the $k$-th neighbor of raw projection $(x,y)$ is stored in $A(x,y,k)$, where $k=1,2,...,N$.\n\n\n\\SubSubSection{Ground Truth Velocity of Moving Radar Points}\n\\label{sec:GT_velocity}\nThe nuScenes~\\cite{caesar2020nuscenes} provides the GT (ground truth) velocity of object bounding boxes. \nWe associate radar hits on an object to its labeled bounding box, and assign the velocity of the box to its associated radar points.\nThe association is determined based on two criteria:\n1) in radar coordinates, the distance between radar points and associated box is smaller than a threshold $T_d$; \nand 2) the percentage error between the radial velocity of a radar point and the radial component of the velocity of associated box is smaller than a threshold $T_p$.\n\n\\SubSubSection{Generating Association Labels}\nWe can project a radar point expressed in corresponding camera coordinates, $\\prescript{A}{}\\qv$, to pixel coordinates $(u_q,v_q)$, but as mentioned before, often this image pixel does not correspond to the radar return.  Our proposed solution is to search in a neighboring region around $(u_q,v_q)$ for a pixel whose motion is consistent with the radar return.  This neighborhood search is shown in Fig.~\\ref{fig:diagram_association}.   If a pixel is found, then we correct the 3D radar location $\\prescript{A}{}\\qv$ to be consistent with this pixel, otherwise we mark this radar return as occluded.\n\nWe learn this radar-to-pixel association and correction by training the R2P network.  \nWe generate true association score between a radar point and a pixel according to the compatibility between the true velocity and the optical flow at that pixel: high compatibility indicates high association. \nTo quantify the compatibility, assuming a pixel is associated with a radar point, we compute a hypothetical full velocity for the radar point by using the optical flow of that pixel according to Eq.~\\eqref{eq:full_v}. \nThe flow is considered compatible\nif the hypothetical velocity is close to the GT velocity. \nSpecifically, the  hypothetical velocity can be computed as\n\\begin{equation}\n\\resizebox{.88\\hsize}{!}{\n$\\prescript{A}{}\\mvdot_{est}(x,y,k) = \\bm{f}\\left(\\breve{u}_q,\\breve{v}_q,\\breve{u}_p,\\breve{v}_p, d_q, \\dot{r}, \\prescript{B}{A}\\T, \\prescript{A}{R}\\T \\right),$\n}\n\\label{eq:fvel}\n\\end{equation}\nwhere $k=1,\\cdots,N$, $\\bm{f}(\\cdot)$ is the function to solve full velocity via Eq.~\\eqref{eq:full_v}, and $(x,y)$ is the raw projection of the radar point.\nNote that  $\\breve{u}_q=u_q\\left[x+\\Delta x(k), y+\\Delta y(k)\\right]$, $\\breve{v}_q$ is defined similarly, \nand $[\\Delta x(k),\\Delta y(k)]$ is the coordinate offset from raw projection to the $k$-th neighbor.  Using flow, Eq.~\\eqref{eq:flow}, we obtain $(\\breve{u}_p,\\breve{v}_p)$ from $(\\breve{u}_q,\\breve{v}_q)$.\n\nSecond, we calculate the $L_2$ norm of errors between $\\prescript{A}{}\\mvdot_{est}(x,y,k)$ and ground truth velocity $\\prescript{A}{}\\mvdot_{GT}(x,y)$ by\n\\begin{equation}\nE_m(x,y,k)= \\lVert \\prescript{A}{}\\mvdot_{est}(x,y,k) - \\prescript{A}{}\\mvdot_{GT}(x,y) \\rVert_2\n\\label{eq:e_v}.\n\\end{equation}\nFig.~\\ref{Figure:label} shows examples of $E_m$ for two radar hits on a car.\n\nFinally, we transform $E_m$ to an association score with\n\\begin{equation}\nL(x,y,k)= e^{-\\frac{E^2_m(x,y,k)}{c}}\n\\label{eq:prob},\n\\end{equation}\nwhere $L$ is used as a label for association probability between a radar and its $k$-th neighbor. Note that $L$ increases with decreasing $E_v$, and $c$ is a parameter adjusting the tolerance of velocity errors when converting errors to association. \nWe use the cross entropy loss to train the model.\n\n\n\n\n\\SubSubSection{Estimate Association and Identify Occlusion}\nWith a trained model, we can estimate association probability between radar points and $N$ pixels around their raw projections $(x,y)$, {\\it i.e.}, $A(x,y,k)$.\nAmong the $N$ neighbors, the radar return velocity may be compatible with a number of pixels, and we select the pixel with the maximum association, $A_{max}$, as the neighbor ID $k_{max}$:\n\\begin{equation}\nk_{max}= \\underset{k}{\\arg\\max}[A(x,y,k)]\n\\label{eq:k_max}.\n\\end{equation}\n If $A_{max}$ is equal or larger than a threshold $T_a$, we estimate the associated pixel as $\\left[x+\\Delta x(k_{max}), y+\\Delta y(k_{max})\\right]$.  \nOtherwise there is no associated pixels in the neighborhood, and an occlusion is identified.\n\n\\section{Experimental Results}\n\\subsection{Comparison of Point-wise Full Velocity}\n\nTo the best of our knowledge, there is no existing method estimating {\\it point-wise} full velocity for radar returns. \nThus, we use point-wise radial velocity from raw radar returns as the baseline to compare with our estimation. \nWe extract data from the nuScenes Object Detection Dataset~\\cite{caesar2020nuscenes}, with $6432$, $632$, and $2041$ samples in training, validation and testing set, respectively. \nEach sample consists of a radar scan and two images for optical flow computation, {\\it i.e.}, one image synchronizing with the radar and the other is a neighboring image frame. \nThe optical flow is computed by the RAFT model~\\cite{teed2020raft} pre-trained on KITTI~\\cite{geiger2013vision}. \nThe R2P network is an U-Net~\\cite{ronneberger2015u, morris2018pyramid} with five levels of resolutions and $64$ channels for intermediate filters. \nThe neighborhood skips every other pixel, and its size (in pixels) is (left: $4$, right: $4$, top: $10$, bottom: $4$) and an example of the neighborhood is illustrated in Fig.~\\ref{fig:diagram_association}(b). \nThe threshold of association scores $T_a$ is $0.3$. Parameters associating radar points with GT bounding box are set as $T_d=0.5$m and $T_p=20\\%$. \nParameter $c$ in Eq.~\\eqref{eq:prob} is  $0.36$.\nTo obtain GT point-wise velocity, based on the criteria in Sec.~\\ref{sec:GT_velocity}, we first associate moving radar points to GT detection boxes, whose GT velocity is assigned to associated points as their GT velocity. The GT velocity of bounding boxes is estimated from GT center positions in neighboring frames with timestamps. \n\nTab.~\\ref{tab:baseline} shows the average velocity error for moving points. The proposed method achieves substantially more accurate velocity estimation than the baseline. \nFor instance, the error of our tangential component is only $21\\%$ of that of the baseline.\nWe also have much smaller standard deviation, indicating more stable estimates. In addition, we list in Tab.~\\ref{tab:baseline} velocity error of our method using raw radar projection for radar-camera association. Results show that, compared with using raw projection, using R2P network achieves higher estimation accuracy.\nFig.~\\ref{Figure:pv} illustrates qualitative results of our point-wise velocity estimation.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\vspace{-2mm} \n\t\\scalebox{1.02}{\n\t\t\\begin{tabular}{@{}c@{}c@{}c@{}c@{}}\n\t\t\t\\includegraphics[height=1.09in]{.\/figures\/fig_v\/Figure_1}     &\n\t\t\t\\includegraphics[height=1.16in]{.\/figures\/fig_v\/01_v_flow} &\n\t\t\t\\includegraphics[height=1.1in]{.\/figures\/fig_v\/01_v_aff}   &\n\t\t\t\\includegraphics[height=1.1in]{.\/figures\/fig_v\/01_v_bev}  \\vspace{-2mm} \\\\\n\t\t\t\\includegraphics[height=1.1in]{.\/figures\/fig_v\/Figure_2}     &\n\t\t\t\\includegraphics[height=1.12in]{.\/figures\/fig_v\/02_v_flow} &\n\t\t\t\\includegraphics[height=1.1in]{.\/figures\/fig_v\/02_v_aff}   &\n\t\t\t\\includegraphics[height=1.1in]{.\/figures\/fig_v\/02_v_bev}  \\vspace{-3mm} \\\\\n\t\t\t\\includegraphics[height=1.1in]{.\/figures\/fig_v\/Figure_3}     &\n\t\t\t\\includegraphics[height=1.15in]{.\/figures\/fig_v\/03_v_flow} &\n\t\t\t\\includegraphics[height=1.1in]{.\/figures\/fig_v\/03_v_aff}   &\n\t\t\t\\includegraphics[height=1.1in]{.\/figures\/fig_v\/03_v_bev}  \\vspace{-2mm} \\\\\n\t\t\t\\footnotesize{(a)} &  \\footnotesize{ (b)} & \\footnotesize{ (c)} & \\footnotesize{ (d)} \\vspace{-2mm}\\\\\n\t    \\end{tabular} }\n\t\\caption{\\small Visualization of point-wise velocity estimation: (a) depth of all measured radar returns as well as flow, (b) optical flow in the white box region, (c) association scores around the selected radar projections as well as predicted mapping from raw radar projections to image pixels (yellow arrow) and (d) radial velocity (red), estimated full velocity (black) and GT velocity (green) in bird's-eye view.}\n\t\\label{Figure:pv}\n\\end{figure*}\n\n\n\\begin{table}[t!]\n\t\\captionsetup{font=small}\n\t\\begin{center}\t\n\n\t\t\\scalebox{0.75}{\n\t\t\t\\begin{tabular}{|c|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\tMean Error (STD)  & Ours & Ours & Baseline \\\\ \n\t\t\t\t(m\/s) & (R2P Network) & (Raw Projection) & \\\\\n\t\t\t\t\\hline\n\t\t\t\tFull Velocity  \t      \t& $\\mathbf{0.433}\\; (\\mathbf{0.608})$ & $0.577\\; (1.010)$ & $1.599\\; (2.054)$ \\\\\n\t\t\t\t\\hline\n\t\t\t\tTangential Comp.    & $\\mathbf{0.322}\\; (\\mathbf{0.610})$ & $0.472\\; (1.024)$ & $1.536\\; (2.083)$ \\\\\n\t\t\t\t\\hline\n\t\t\t\tRadial Comp.    \t& $0.205\\; (0.196)$ & $0.205\\; (0.196)$ & $0.205\\; (0.196)$ \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{center}\n\t\\vspace{-5mm}\n\t\\caption{\\small Comparison of point-wise velocity error of our methods and the baseline (raw radial velocity).}\n\t\\label{tab:baseline}\n\t\\vspace{-2mm}\n\\end{table}\n\n\n\\subsection{Comparison of Object-wise Velocity}\nAlthough there are no existing methods for point-wise velocity estimation for radar, a related work, CenterFusion~\\cite{nabati2021centerfusion}, estimates {\\it object-wise} full velocity  via object detection with image and radar inputs. \nTo fairly compare with CenterFusion, we convert our point-wise velocity to object-wise velocity. \nSpecifically, we use the average velocity of radar points associated with the same detected box as our estimate of object velocity. \nPoints are associated with detected boxes according to distance. \nNote the point-wise velocity to object-wise velocity conversion is straightforward for comparison purposes, and there would be more advanced approaches to integrate point-wise full velocities in a detection network, which is beyond the scope of this work. \nTab.~\\ref{tab:det} shows that with our estimated full velocity, the velocity estimation for objects is significantly improved.\n\n\\begin{table}[t!]\n\t\\captionsetup{font=small}\n\t\\begin{center}\t\n\t\t\\scalebox{0.9}{\n\t\t\t\\begin{tabular}{|c|c|}\n\t\t\t\t\\hline\n\t\t\t\tMethods & Error (m\/s) \\\\ \n\t\t\t\t\\hline\n\t\t\t\tOurs \t        & $\\mathbf{0.451}$ \\\\\n\t\t\t\tCenterFusion~\\cite{nabati2021centerfusion}    & $0.826$  \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{center}\n\t\\vspace{-5mm}\n\t\\caption{\\small Comparison of object-wise velocity errors. For a fair comparison we inherit the same set of detected objects from~\\cite{nabati2021centerfusion}.}\n\t\\label{tab:det}\t\\vspace{-3mm}\n\\end{table}\n\n\n\\subsection{Radar Point Accumulation}\nAccumulating radar points over time can overcome the sparsity of radar hits acquired in a single sweep, achieving dense point cloud for objects and thus allowing techniques designed for processing LiDAR points to be applicable for radar. \nThe point-wise velocity estimate makes it possible to compensate the motion of dynamic objects appearing in a temporal sequence of measurements for accumulation. \nSpecifically, for a moving radar point (with estimated velocity $\\dot{\\bm{m}}$) in a previous frame $i$ captured at time $t_i$, its motion from $t_i$ to the time at the current frame, $t_0$, can be compensated by,\n\\begin{equation}\n\\bm{p_0}= \\bm{p_i} + \\dot{\\bm{m}}(t_0 - t_i),\n\\label{eq:motion_compensate}\n\\end{equation}\nwhere $\\bm{p_i}$ and $\\bm{p_0}$ are the radar point coordinates at $t_i$ and $t_0$ in radar coordinates of $t_i$. \nThen $\\bm{p_0}$ is transformed to current radar coordinates by known egomotion from $t_i$ to $t_0$.\n\n\\begin{figure}[t!]\n    \\captionsetup{font=small}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.8\\linewidth]{figures\/error_curve}\n\t\\end{center}\n\t\\vspace{-5mm}\n\t\\caption{\\small Error comparison when accumulating radar points from increasing number of frames. The lines represent mean error and shaded area $\\pm0.1\\times$ STD. Our  full velocity based accumulation outperforms the ones with radial velocity, or no compensation.}\n\t\\label{fig:curve}\\vspace{-3mm}\n\\end{figure}\n\n\\Paragraph{Qualitative results}\nFig.~\\ref{Figure:acc} shows accumulated points of moving vehicles in radar coordinates. \nFor comparison, we show accumulated radar points compensated by our estimated full velocity, compensated with radial velocity (baseline) and without motion compensation. Compared with the baseline and no motion compensation, our accumulated points are more consistent with the GT bounding boxes.\n\n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\scalebox{1}{\n\t\t\\begin{tabular}{@{}c@{}c@{}c@{}c@{}c@{}}\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/02_acc_im} &\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/02_acc_one}&\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/02_acc_no} &\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/02_acc_old}&\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/02_acc_our}\\vspace{-2mm}\\\\\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/03_acc_im} &\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/03_acc_one}&\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/03_acc_no} &\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/03_acc_old}&\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/03_acc_our}\\vspace{-2mm}\\\\\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/04_acc_im} &\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/04_acc_one}&\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/04_acc_no} &\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/04_acc_old}&\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/04_acc_our}\\vspace{-2mm}\\\\\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/05_acc_im} &\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/05_acc_one}&\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/05_acc_no} &\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/05_acc_old}&\n\t\t\t\\includegraphics[width=1.2in]{.\/figures\/fig_acc\/05_acc_our}\\vspace{-2mm}\\\\\n\t\t\t\\footnotesize{(a)} &  \\footnotesize{ (b)} & \\footnotesize{ (c)} & \\footnotesize{ (d)} & \\footnotesize{ (e)}\\\\\n\t\\end{tabular} }\n\t\\vspace{-2mm}\n\t\\caption{\\small Moving radar points are plotted with point-wise radial (red) and full (black) velocity, including image with bounding box (a), single-frame radar points in bird's-eye view (b), accumulated radar points from $20$ frames without motion compensation (c), with radial velocity based compensation (d), and with our full-velocity based compensation (e).\n\t Our accumulated points are tightly surrounding the bounding box, which will benefit downstream tasks such as pose estimation and object detection.}\n\t\\label{Figure:acc}\t\\vspace{-1mm}\n\\end{figure*}\n\n\n\n\\Paragraph{Quantitative results}\nTo quantitatively evaluate the accuracy of radar point accumulation, we use the mean distance from accumulated points (of up to $25$ frames) to their corresponding GT boxes as the accumulation error. \nThis distance for points inside the box is zero, and outside it is the distance from the radar point to the closest point on the box's boundary. \nIn Fig.~\\ref{fig:curve}, we compare the accumulation for our method, the baseline and accumulation without motion compensation. \nWhile error increases with the number of frames for all methods, our method has the lowest rate of error escalation. \n\n\n\\begin{table}[t!]\n    \\captionsetup{font=small}\n\t\\begin{center}\t\n\t\t\\scalebox{0.85}{\n\t\t\t\\begin{tabular}{|c|c|c|}\n\t\t\t\t\\hline\n\t\t\t\t  \t\t                    Metric\t& Ours &  Baseline \\\\ \n\t\t\t\t\\hline\t\t\t\t\n\t\t\t\t\tCenter Error (m) $\\downarrow$            & $\\mathbf{0.834}$ & $0.997$  \\\\\t\t\t\n\t\t\t\t\tOrientation Error (degree) $\\downarrow$  & $\\mathbf{6.873}$ & $7.517$ \\\\\n\t\t\t\t\tIoU $\\uparrow$ \t\t\t\t\t\t& $\\mathbf{0.546}$ & $0.462$ \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{center}\n\t\\vspace{-4mm}\n\t\\caption{\\small Comparison of pose estimation performance: average error in center and orientation as well as Intersection over Union (IoU), by using BoxNet~\\cite{nezhadarya2019boxnet} on radar points accumulated using our velocity and the radial velocity as a baseline.}\n\t\\label{tab:iou}\t\\vspace{-3mm}\n\\end{table}\n\n\\Paragraph{Application of pose estimation}\nTo demonstrate the utility of accumulated radar points for downstream applications, we apply a pose estimation method, {\\it i.e.}, BoxNet~\\cite{nezhadarya2019boxnet}, on the accumulated 2D radar points via our full velocity and radial velocity (baseline), respectively. BoxNet takes pre-segmented 2D point clouds of an object as input and predicts a 2D bounding box with parameters as center position, length, width and orientation. \nWe use accumulated radar points of $5702$, $559$ and $2001$ moving vehicles with corresponding GT bounding boxes as training, validation and testing data, respectively. Tab.~\\ref{tab:iou} shows our accumulated radar achieves higher accuracy than the baseline.\n\n\n\\Section{Conclusion}\nA drawback of Doppler radar has been that it provides only the radial component of velocity, which limits its utility in object velocity estimation, motion prediction and radar return accumulation. This paper addresses this drawback by presenting a closed-form solution to the full velocity of radar returns.  It leverages optical flow constraints to upgrade radial velocity into full velocity.  As part of this work, we use GT bounding-box velocities to supervise a  network that predicts association corrections for the raw radar projections.  We experimentally verify the effectiveness of our method and demonstrate its application on motion compensation for integrating radar sweeps over time.\n\nThis method developed here may apply to additional modalities such as full-velocity estimation from Doppler LiDAR and cameras.  \n\n\\vspace{1mm}\n\\noindent\\textbf{Acknowledgement}\nThis work was supported by the Ford-MSU Alliance.\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nTopology freezing or fixing are important issues in quantum field theory, in particular in QCD. For example, when simulating chirally symmetric overlap quarks, the corresponding algorithms do not allow transitions between different topological sectors, i.e.\\ topological charge is fixed (cf.\\ e.g.\\ \\cite{Aoki:2008tq,Aoki:2012pma}). Also when using other quark discretizations, e.g.\\ Wilson fermions, topology freezing is expected at lattice spacings $a \\lesssim 0.05 \\, \\textrm{fm}$, which are nowadays still fine, but realistic \\cite{Luscher:2011kk,Schaefer:2012tq}. There are also applications, where one might fix topology on purpose. For example, when using a mixed action setup with light overlap valence and Wilson sea quarks, approximate zero modes in the valence sector are not compensated by the sea. The consequence is an ill-behaved continuum limit \\cite{Cichy:2010ta,Cichy:2012vg}. A possible solution to overcome this problem is to restrict computations to a single topological sector, either by sorting the generated gauge link configurations with respect to their topological charge or by directly employing so-called topology fixing actions (cf.\\ e.g.\\ \\cite{Fukaya:2005cw,Bietenholz:2005rd,Bruckmann:2009cv}).\n\nIn view of these issues it is important to develop methods, which allow to obtain physically meaningful results (i.e.\\ results corresponding to unfixed topology) from fixed topology simulations. The starting point for our work are calculations from the seminal papers \\cite{Brower:2003yx,Aoki:2007ka}. We extend these calculations by including all terms proportional to $1\/V^2$ and $1\/V^3$. We apply the resulting equations to a quantum mechanical particle on a circle, to the Schwinger model and to SU(2) Yang-Mills theory and determine ``hadron masses'' at unfixed topology from fixed topology computations and simulations (for related exploratory studies in the Schwinger model and the $O(2)$ and $O(3)$ non-linear Sigma model cf.\\ \\cite{Bietenholz:2011ey,Bietenholz:2012sh,Bautista:2014tba}).\n\nPart of this work has already been published \\cite{Dromard:2013wja,Dromard:2014wja,Czaban:2013haa}.\n\n\n\n\n\\section{\\label{SEC1}Hadron masses from fixed topology simulations}\n\n\n\\subsection{\\label{SEC11}Two-point correlation functions at fixed topology}\n\nThe partition function and the two-point correlation function of a hadron creation operator $O$ at fixed topological charge $Q$ and finite spacetime volume $V$ are given by\n{\\small\n\\begin{equation}\n\\begin{aligned}\n & Z_{Q,V} \\equiv \\int DA \\, D\\psi \\, D\\bar{\\psi}\\, \\delta_{Q,Q[A]} e^{-S_E[A,\\bar{\\psi},\\psi]} \\\\\n & C_{Q,V}(t) \\equiv \\frac{1}{Z_{Q,V}} \\int DA \\, D\\psi \\, D\\bar{\\psi} \\, \\delta_{Q,Q[A]} O^\\dagger(t) O(0) e^{-S_E[A,\\bar{\\psi},\\psi]} .\n\\end{aligned}\n\\end{equation}\n}Using a saddle point approximation the correlation function has been expanded in \\cite{Brower:2003yx} according to\n{\\small\n\\begin{equation}\n\\label{EQN673} C_{Q,V}(t) = \\alpha(0) \\exp\\bigg(-M_H(0) t - \\frac{M^{(2)}_H(0) t}{2\\mathcal{E}_2 V} \\bigg(1- \\frac{ Q^2}{\\mathcal{E}_2 V}\\bigg)\\bigg) + \\mathcal{O}\\bigg(\\frac{1}{ V^2} \\bigg) ,\n\\end{equation}\n}where $\\alpha(0)$ is a constant, $M_H(\\theta)$ the hadron mass at vacuum angle $\\theta$, $\\mathcal{E}_k \\equiv e_0^{(k)}(\\theta)|_{\\theta=0}$ ($\\mathcal{E}_2 = \\chi_t$, the topological susceptibility) and $e_0$ is the vacuum energy density. In \\cite{Dromard:2014wja} we have extended this calculation by including all terms proportional to $1\/V^2$ and $1\/V^3$,\n{\\small{\n\\begin{equation}\n\\label{EQN674} \\begin{aligned}\n & C_{Q,V}(t) = \\alpha(0) \\exp\\bigg(-M_H(0) t - \\frac{x_2}{2\\mathcal{E}_2 V} - \\bigg(\\frac{x_4 - 2 (\\mathcal{E}_4\/\\mathcal{E}_2) x_2 - 2 x_2^2-4x_2Q^2}{8(\\mathcal{E}_2 V)^2} \\bigg) \\\\\n & \\hspace{0.6cm} - \\bigg(\\frac{16 (\\mathcal{E}_4\/\\mathcal{E}_2)^2 x_2 + x_6 - 3 (\\mathcal{E}_6\/\\mathcal{E}_2) x_2 - 8 (\\mathcal{E}_4\/\\mathcal{E}_2) x_4 - 12 x_2 x_4 + 18 (\\mathcal{E}_4\/\\mathcal{E}_2) x_2^2 + 8 x_2^3}{48(\\mathcal{E}_2 V)^3} \\\\\n & \\hspace{1.2cm} - \\frac{x_4 - 3 (\\mathcal{E}_4\/\\mathcal{E}_2) x_2 - 2 x_2^2}{4(\\mathcal{E}_2 V)^3} Q^2\\bigg)\\bigg) + \\mathcal{O}\\bigg(\\frac{1}{(\\mathcal{E}_2 V)^4} \\, , \\, \\frac{1}{(\\mathcal{E}_2 V)^4} Q^2 \\, , \\, \\frac{1}{(\\mathcal{E}_2 V)^4} Q^4\\bigg) ,\n\\end{aligned}\n\\end{equation}\n}where $x_n \\equiv M^{(n)}_H (0) t+ \\beta^{(n)}(0)$ (for the definition of $\\beta^{(n)}$ cf.\\ \\cite{Dromard:2014wja}). The expansions (\\ref{EQN673}) and (\\ref{EQN674}) are rather accurate approximations, if the following conditions are fulfilled: \\vspace{0.1cm}\n\\\\\\textbf{(C1)} $\\phantom{xxx} 1 \/ \\mathcal{E}_2 V \\ll 1 \\quad , \\quad |Q| \/ \\mathcal{E}_2 V \\ll 1$. \\vspace{0.1cm}\n\\\\\\textbf{(C2)} $\\phantom{xxx} |x_2| = |M_H^{(2)}(0) t + \\beta^{(2)}(0)| \\lesssim 1$. \\vspace{0.1cm}\n\\\\\\textbf{(C3)} $\\phantom{xxx} m_\\pi(\\theta) L \\gtrsim 3 \\ldots 5 \\gg 1$ $\\ \\ $ ($m_\\pi$: pion mass, $L$: periodic spatial extension). \\vspace{0.1cm}\n\\\\\\textbf{(C4)} $\\phantom{xxx} (M_H^\\ast(\\theta) - M_H(\\theta)) t \\gg 1 \\quad , \\quad M_H(\\theta) (T-2 t) \\gg 1$. \\vspace{0.1cm}\n\nNote that the effective mass at fixed topology, defined in the usual way,\n{\\small\n\\begin{equation}\n\\label{eq:MQ} M^\\textrm{eff}_{Q,V}(t) \\equiv -\\frac{1}{C_{Q,V}(t)}\\frac{d C_{Q,V}(t)}{dt} ,\n\\end{equation}\n}exhibits severe deviations from a constant behavior at large temporal separations $t$ \\cite{Dromard:2014wja}, which is in contrast to ordinary quantum field theory at unfixed topology.\n\n\n\\subsection{\\label{SEC13}Extracting hadron masses}\n\nA straightforward method to determine physical hadron masses (i.e.\\ hadron masses at unfixed topology) from fixed topology simulations is to fit either (\\ref{EQN673}) or (\\ref{EQN674}) to two-point correlation functions computed at fixed topology. Among the results of the fit are then the hadron mass at unfixed topology $M_H(0)$ and the topological susceptibility $\\mathcal{E}_2 = \\chi_t$. A similar method is to first determine hadron masses $M_{Q,V}$ at fixed topological charge $Q$ and spacetime volume $V$ and then use equations based on (\\ref{EQN673}) or (\\ref{EQN674}) to determine $M_H(0)$ and $\\mathcal{E}_2 = \\chi_t$. For a detailed discussion cf.\\ \\cite{Dromard:2014wja}}.\n\n\n\n\n\\section{\\label{SEC2}A quantum mechanical particle on a circle at fixed topology}\n\nFor a first test of the methods mentioned in section~\\ref{SEC13} we decided for a simple toy model, a quantum mechanical particle on a circle in a square well potential. This model shares some important features with QCD, e.g.\\ the existence of topological charge and the symmetry $+\\theta \\leftrightarrow -\\theta$. Moreover, it can be solved numerically up to arbitrary precision. We determine $M_H(0)$ (which is the energy difference between the ground state and the first excitation) and $\\chi_t$ from fixed topology two-point correlation functions as outlined in section~\\ref{SEC13}. We compare the $1\/V$ expansion from \\cite{Brower:2003yx} (eq.\\ (\\ref{EQN673})) and our $1\/V^3$ version (eq.\\ (\\ref{EQN674})). We find rather accurate results for $M_H(0)$ and $\\chi_t$ (cf.\\ Table~\\ref{TAB001}). Note that the relative errors for both $M_H(0)$ and $\\chi_t$ are smaller, when using the $1\/V^3$ version (\\ref{EQN674}). For details cf.\\ \\cite{Dromard:2013wja,Dromard:2014wja}.\n\n\\begin{table}[htb]\n\\begin{center}\n\\small{\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline \n & expansion  & $\\hat{M}_H(0)$ & error& $\\hat{\\chi}_t$ & error \\tabularnewline\n\\hline \n\\hline \n\\multirow{2}{*}{$\\frac{|Q|}{\\chi_{t}V} \\leq 0.5$} & (\\ref{EQN674}), hep-lat\/0302005 & $0.40702$ & $0.029\\%$& $0.00629$ & $2.5\\%$\\tabularnewline\n\\cline{2-6} \n & (\\ref{EQN674}) &  $0.40706$ & $0.019\\%$ & $0.00633$ & $1.9\\%$ \\tabularnewline\n\\hline \n\\end{tabular}\n}\n\n\\caption{\\label{TAB001}$M_H(0)$ and $\\chi_t$ from fixed topology two-point correlation functions; ``error'' denotes relative differences to the exact results $\\hat{M}_{H} = 0.40714$ and $\\hat{\\chi}_t = 0.00645$ at unfixed topology.}\n\\end{center}\n\\end{table}\n\n\n\n\n\\section{The Schwinger model at fixed topology}\n\n\nThe Schwinger model, defined by the Lagrangian\n{\\small\n\\begin{equation}\n\\mathcal{L}(\\psi,\\bar{\\psi},A_{\\mu}) \\equiv \\bar{\\psi} (\\gamma_\\mu (\\partial_\\mu + i g A_\\mu) + m) \\psi + \\frac{1}{2} F_{\\mu \\nu} F_{\\mu \\nu} ,\n\\end{equation}\n}also shares certain features with QCD, most prominently confinement. Furthermore, simulations are computationally inexpensive, because there are only $2$ spacetime dimensions.\n\nWe have studied the ``pion'' mass $m_\\pi$ and the static quark-antiquark potential ${\\mathcal V}_{q\\bar{q}}$ for various separations. Results are summarized in Table~\\ref{TAB002}. In the first line (``fixed top.'') results obtained from two-point correlation functions at fixed topology (as outlined in section~\\ref{SEC13}) are listed. In the second line (``unfixed top.'') they are compared to results from standard lattice simulations, where gauge link configurations from all topological sectors are taken into account. One can observe agreement demonstrating that one can obtain correct and accurate physical results from fixed topology simulations. For details cf.\\ \\cite{Czaban:2013haa}.\n\n\\begin{table}[htb]\n\\begin{center}\n{\\small\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n & $m_\\pi a$ & ${\\mathcal V}_{q\\bar{q}}(1a) a$ & ${\\mathcal V}_{q\\bar{q}}(2a) a$ & ${\\mathcal V}_{q\\bar{q}}(3a) a$ & ${\\mathcal V}_{q\\bar{q}}(4a) a$ \\\\ \\hline\\hline\nfixed top.\\ &  0.2747(2) & 0.12551(4) & 0.2247(2) & 0.3005(3) & 0.3581(7)  \\\\ \\hline\nunfixed top.\\ & 0.2743(3) & 0.12551(4) & 0.2247(2) & 0.3008(4) & 0.3577(9) \\\\ \\hline\n\\end{tabular}\n}\n\n\\caption{\\label{TAB002}Comparison of results obtained from computations at fixed and at unfixed topology.}\n\\end{center}\n\\end{table}\n\n\n\n\n\\section{SU(2) Yang-Mills theory at fixed topology}\n\nCurrently we perform fixed topology studies of SU(2) Yang-Mills theory,\n{\\small\n\\begin{equation}\n\\mathcal{L}(A_\\mu) \\equiv \\frac{1}{4} F_{\\mu \\nu}^a F_{\\mu \\nu}^a ,\n\\end{equation}\n}which is expected to be rather similar to QCD. Again we explore the static quark-antiquark potential for various separations.\n\nThe left plot in Fig.~\\ref{FIG342} shows that there is a significant discrepancy between the potential from computations restricted to a single topological sector and corresponding results obtained at unfixed topology. The plot, therefore, underlines the necessity of a method to extract physical results from fixed topology computations.\n\nIn the right plot of Fig.~\\ref{FIG342} we compare the static potential obtained from Wilson loops at fixed topology (as outlined in section~\\ref{SEC13}) and from standard lattice simulations, where gauge link configurations from all topological sectors are taken into account. As for the Schwinger model, one can observe excellent agreement demonstrating again that one can obtain correct and accurate physical results from fixed topology simulations.\n\nDetails regarding our study of Yang-Mills theory at fixed topology will be published in the near future.\n\n\\begin{figure}[htb]\n\\input{FIGSU001.pstex_t}\n\\caption{\\label{FIG342} \\textbf{(left)} ${\\mathcal V}_{q\\bar{q}}(6a)$ for different topological sectors $Q = 0, 1, 2, 3$ for spacetime volume $V\/a^4 = 16^4$. \\textbf{(right)} Comparison of potential results obtained from computations at fixed and at unfixed topology.}\n\n\\end{figure}\n\n\n\n\n\\section{Conclusions and outlook}\n\nWe have extended relations from the literature \\cite{Brower:2003yx,Aoki:2007ka} relating two-point correlation functions at fixed topology to physical hadron masses (i.e.\\ hadron masses at unfixed topology). We have successfully applied our resulting equations to various models. We plan to test the same methods for QCD in the near future, where hadron masses obtained from different topological sectors also exhibit clear differences (for an example cf.\\ \\cite{Galletly:2006hq}, where the pion mass has been computed in various topological charge sectors).\n\n\n\n\n\\section*{Acknowledgments}\n\nWe thank Wolfgang Bietenholz, Krzysztof Cichy, Dennis Dietrich, Gregorio Herdoiza, Karl Jansen and Andreas Wipf for discussions. We acknowledge support by the Emmy Noether Programme of the DFG (German Research Foundation), grant WA 3000\/1-1. This work was supported in part by the Helmholtz International Center for FAIR within the framework of the LOEWE program launched by the State of Hesse.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Abstract.}%\n      \\else\n        \\small\n        \\quotation\n\t\\noindent{\\bfseriesAbstract.}%\n      \\fi}\n      {\\if@twocolumn\\else\\endquotation\\fi}\n\\makeatother\n\\setcounter{secnumdepth}{2}\n\n\\title{\\Large \\bf Placing Green Bridges Optimally, with a Multivariate~Analysis}\n\\author{Till Fluschnik\\footnote{Supported by DFG, project TORE (NI\/369-18).} \\and Leon Kellerhals}\n\\date{\\small Technische Universit\u00e4t Berlin, Faculty~IV,\\\\ Algorithmics and Computational Complexity, Germany.\\\\\\texttt{\\{till.fluschnik,leon.kellerhals\\}@tu-berlin.de}}\n\n\\begin{document}\n\n\\maketitle\n\n\\begin{abstract}\nWe study the problem of placing wildlife crossings, \nsuch as green bridges,\nover human-made obstacles to challenge habitat fragmentation.\nThe main task\nherein is,\ngiven\na graph describing habitats or routes of wildlife animals and possibilities of building green bridges,\nto find a low-cost placement of green bridges that connects the habitats.\nWe develop different problem models for this task\nand study them from a computational complexity and parameterized algorithmics perspective.\n\n\\medskip\n\\noindent\n\\emph{Keywords.}\nwildlife crossings,\ncomputational complexity,\nNP-hardness,\nparameterized algorithmics,\nconnected subgraphs.\n\\end{abstract}\n\n\\section{Introduction}\n\\label{sec:intro}\n\nSustainability \nis\nan enormous concern \nimpacting today's\npolitics,\neconomy,\nand industry.\nAccordingly,\nsustainability sciences are well-established by now.\nYet,\nthe \\emph{interdisciplinary} scientific field ``computational sustainability''~\\cite{Gomes09a},\nwhich combines practical and theoretical computer science with sustainability sciences,\nis quite young.\nFor instance,\nthe Institute for Computational Sustainability \nat Cornell University was founded in 2008,\nthe 1st International Conference on Computational Sustainability \n(CompSust'09) \ntook place in 2009,\nand\nspecial tracks on\ncomputational sustainability and AI\nwere established\nin 2011 (AAAI) and 2013 (IJCAI).\nThis work\ncontributes to\ncomputational sustainability:\nWe model problems of elaborately placing wildlife crossings\nand give complexity-theoretical and algorithmic analysis for each.\nWildlife crossings are constructions (mostly bridges or tunnels) that allow wildlife animals to safely cross human-made transportation lines (mostly roads). \nWe will refer to wildlife crossings as~\\emph{green bridges}.\n\nHuijser~\\textsl{et~al.}~\\cite{HuijserMHKCSA08}\ngive an extensive report on wildlife-vehicle collisions.\nThey identify several endangered animal species suffering from high road mortality\nand estimate the annual cost associated with wildlife-vehicle collisions with about 8~billion US dollars.\nWildlife fencing with wildlife crossings\ncan reduce collisions by over 80\\%~\\cite{HuijserMHKCSA08},\nenables populations to sustain~\\cite{SawayaKC14},\nand are thereby among the most cost-effective~\\cite{HuijserDCAM09}. \nThe implementation,\nthough,\nis an important problem:\n\\begin{quote}\nThe location, type, and dimensions of wildlife\ncrossing structures must be carefully planned with regard to the species and surrounding\nlandscape.\\ \n[...]\\ \nIn addition, different species use different habitats, influencing their\nmovements and where they want to cross the road.\n~\\cite[p.\\;\\!16]{HuijserMHKCSA08}\n\\end{quote}\nIn addition,\nit is pointed out that data about wildlife habitats \nis basic for mitigation plans,\nyet challenging to obtain~\\cite{ZhengLY19}.\nIn this work,\nour main problem is placing green bridges\nat low cost and under several variants of habitat-connectivity requirements,\nthereby inherently modeling different availabilities of data on habitats.\nThe problem is hence the following:\n\\textsl{\nGiven a graph describing habitats of wildlife animals and possibilities of building green bridges,\nfind a low-cost placement of green bridges that sufficiently connects habitats.\n}\nIn particular,\nwe comparatively \nstudy in terms of computational complexity and parameterized algorithmics\nthe following three different (families of) decision problems.%\n\\footnote{The ($d$-th power) graph $G^d$\nof~$G$ \ncontains edge~$\\{v,w\\}$ if and only if~$\\dist_G(v,w)\\leq d$.}\n\\decprobX{$\\Pi$ Green Bridges Placement{} ($\\Pi$ \\prob{GBP}{})}\n{An undirected graph~$G=(V,E)$,\na set~$\\mathcal{H}=\\{V_1,\\dots,V_r\\}$ of habitats where~$V_i\\subseteq V$ for all~$i\\in\\set{r}$, \nand~$k\\in\\mathbb{N}_0$.}\n{Is there an edge set~$F\\subseteq E$ with~$|F|\\leq k$ such that for every~$i\\in\\set{r}$, \nit holds that\n\n\\smallskip\n\\noindent\n{\n\\setlength{\\tabcolsep}{7pt}\n\\begin{tabular}{@{\\hspace{1em}}rlll@{}}\n $\\Pi\\equiv{}$\\prob{$d$-Reach}: & $G[F]^d[V_i]$ is connected? & (\\cref{prob:rgbp}) & (\\cref{sec:rgbp}) \\\\\n $\\Pi\\equiv{}$\\prob{$d$-Closed}: & $G[F]^d[V_i]$ is a clique? & (\\cref{prob:cgbp}) &  (\\cref{sec:cgbp})\\\\\n $\\Pi\\equiv{}$\\prob{$d$-Diam(eter)}: & $\\diam(G[F][V_i])\\leq d$? & (\\cref{prob:dgbp}) & (\\cref{sec:dgbp})\n\\end{tabular}\n}}\n\n\\noindent\nIn words:\n\\RgbpAcr{} seeks to connect each habitat \nsuch\nthat \nevery patch has some other patch at short distance.\n\\CgbpAcr{} seeks to connect each habitat such that any two habitat's patches are\nat short distance.\nFinally,\n\\DgbpAcr{} seeks to connect each habitat such that the habitat forms a connected component of low diameter.\n\\cref{fig:relationship}\ndepicts a relationship between the problems in terms of \nKarp reductions.\n\\begin{figure}[t]\\centering\n  \\begin{tikzpicture}\n    \\def1.1{1}\n    \\def0.9{1}\n    \\node (con) at (0,-1.25*0.9)[]{\\hyperref[prob:congbp]{\\prob{Connect \\GBP}}};\n    \\node (r) at (-3*1.1,-1*0.9)[]{\\hyperref[prob:rgbp]{\\prob{Reach \\prob{GBP}{}}}};\n    \\node (c) at (3*1.1,-1*0.9)[]{\\hyperref[prob:cgbp]{\\prob{Closed \\prob{GBP}{}}}};\n    \n    \\newcommandx{\\AredB}[3][1=$\\geq_P$]{\n      \\draw[draw=none] (#2) to node[midway,sloped]{#1}(#3);\n    }\n    \\AredB{con}{r};\n    \\AredB[$\\leq_P$]{con}{c};\n    \n    \\node (r1) at (-4*1.1,-2*0.9)[]{\\RgbpAcr[1]{}};\n    \\node (c1) at (4*1.1,-2*0.9)[]{\\CgbpAcr[1]{}};\n    \n    \\AredB[$\\leq_P$]{r1}{r};\n    \\AredB{c1}{c};\n    \n    \\node (d) at (-1.25*1.1,-1.75*0.9)[]{\\hyperref[prob:dgbp]{\\prob{Diam \\prob{GBP}{}}}};\n    \\node (d1) at (1.25*1.1,-2*0.9)[]{\\DgbpAcr[1]{}};\n    \n    \\AredB[$\\leq_P$]{r1}{d}\n    \\AredB{d1}{d}\n    \\AredB[$\\equiv_P$]{d1}{c1}\n  \\end{tikzpicture} \n  \\caption{Polynomial-time many-one reducibility directly derived from problem definitions.}\n  \\label{fig:relationship}\n\\end{figure}\n\n\\paragraph{Our contributions.}\nOur results are summarized in \\cref{tab:results}.\n\\begin{table}[t]\n \\caption{Overview of our results. \n \\cocl{NP}-c., P, and K \n stand for \n \\cocl{NP}-complete,\n ``polynomial-size'',\n and ``problem kernel'',\n respectively.\n \\tss{a}(even on planar graphs or if~$\\Delta=4$)\n \\tss{b}(even on bipartite graphs with~$\\Delta=4$ or graphs of diameter four.)\n \\tss{c}(even if~$r=1$ or if~$r=2$ and~$\\Delta=4$)\n \\tss{d}(even on bipartite graphs of diameter three and~$r=1$, \n \\emph{but} linear-time solvable when~$r+\\Delta$ is constant)\n \\tss{e}(admits a linear-size problem kernel if~$\\Delta$ is constant)\n \\tss{f}(linear-time solvable when~$r+\\Delta$ is constant) \n \\tss{$\\dagger$}(no polynomial problem kernel unless~\\ensuremath{\\NP\\subseteq \\coNP\/\\poly}, but an~$\\O(k^3)$-vertex kernel on planar graphs)\n }\n \\label{tab:results}\n \\centering\n \\setlength{\\tabcolsep}{3.25pt}\n \\begin{tabular}{@{}p{0.12\\textwidth}l|p{0.13\\textwidth}|p{0.17\\textwidth}p{0.11\\textwidth}p{0.19\\textwidth}|p{0.1\\textwidth}@{}}\\toprule\n  Problem &  & Comput. & \\multicolumn{3}{p{0.46\\textwidth}|}{\n  Parameterized Algorithmics} & Ref.\\\\\n  ($\\Pi$~\\prob{GBP}) &  & Complex. & $k$ & $r$ & $k+r$ &\n  \\\\ \\midrule\\midrule\n  \\multirow{3}{*}{\\makecell{\\prob{$d$-Reach} \\\\ \\tref{sec:rgbp}}} & $d=1$ & \\cocl{NP}-c.~\\tss{a} & $O(k4^{k})$ K & \\emph{open} & $O(rk+k^2)$ PK & \\tref{ssec:1rgbp}\\\\\n                              & $d=2$ & \\cocl{NP}-c.\\tss{b}  & $O(2^{4k})$ K\\tss{$\\dagger$} & p-\\cocl{NP}-h. \\tss{c} & FPT\\tss{$\\dagger$} & \\tref{ssec:2rgbp} \\\\\n                              & $d\\geq 3$ & \\cocl{NP}-c. & \\cocl{XP}, \\W{1}-h. & p-\\cocl{NP}-h. & \\cocl{XP}, \\W{1}-h. & \\tref{ssec:3rgbp}\\\\\n  \\midrule\n  \\multirow{3}{*}{\\makecell{\\prob{$d$-Closed} \\\\ \\tref{sec:cgbp}}} & $d=1$ & Lin.~time & --- & --- & --- & \\tref{sec:cgbp} \\\\\n                              & $d=2$ & \\cocl{NP}-c. \\tss{d} & $O(2^{4k})$ K\\tss{$\\dagger$} & p-\\cocl{NP}-h.\\tss{e}  & FPT\\tss{$\\dagger$} & \\tref{ssec:2cgbp}\\\\\n                              & $d\\geq 3$ & \\cocl{NP}-c. & \\cocl{XP}, \\W{1}-h. & p-\\cocl{NP}-h.\\tss{e} & \\cocl{XP}, \\W{1}-h. & \\tref{ssec:3cgbp}\\\\\n  \\midrule\n  \\multirow{3}{*}{\\makecell{\\prob{$d$-Diam} \\\\ \\tref{sec:dgbp}}} & $d=1$ & Lin.~time & --- & --- & --- & \\tref{sec:dgbp}\\\\\n                              & $d=2$     & \\cocl{NP}-c. \\tss{f} & $2k$-vertex K & p-\\cocl{NP}-h. & $O(rk+k^2)$ PK & \\tref{ssec:2dgbp}\\\\\n                              & $d\\geq 3$ & \\cocl{NP}-c. & $2k$-vertex K & p-\\cocl{NP}-h. & $O(rk+k^2)$ PK & \\tref{ssec:3dgbp}\n  \\\\\\arrayrulecolor{black} \\bottomrule\n \\end{tabular}\n\\end{table}\nWe settle the classic complexity and parameterized complexity (regarding the number~$k$ of green bridges and the number~$r$ of habitats)\nof the three problems.\nWhile~\\RgbpAcr{} is \n(surprisingly) \nalready \\cocl{NP}-hard for~$d=1$ on planar or maximum degree~$\\Delta=4$ graphs,\n\\CgbpAcr{} and \\DgbpAcr{} become~\\cocl{NP}-hard for~$d\\geq 2$,\nbut admit an $(r+\\Delta)^{\\O(1)}$-sized problem kernel \nand thus are linear time solvable if~$r+\\Delta$ is constant.\nExcept for~\\RgbpAcr[1]{},\nwe proved all variants to be para-\\cocl{NP}-hard regarding~$r$.\n\\RgbpAcr{} and \\CgbpAcr{}\nare  fixed-parameter tractable regarding~$k$ when~$d\\leq 2$,\nbut become~\\W{1}-hard (yet~\\cocl{XP}) regarding~$k$ and~$k+r$ when~$d>2$.\nAdditionally,\nwe prove that\n\\RgbpAcr{} admits an $rd$-approximation in~$\\O(mn+rnd)$ time.\n\n\\paragraph{Further related work.}\nOur problems deal with\nfinding (small) spanning connected subgraphs\nobeying some (connectivity) constraints,\nand thus can be seen as \nnetwork design problems~\\cite{KerivinM05}.\nMost related to our problems are \nSteiner multigraph problems~\\cite{RicheyP86}\nwith an algorithmic study~\\cite{Gassner10,LaiGSMCM11}\n(also in the context of wildlife corridor construction).\nRequiring small diameter \nappears also in the context of spanning trees~\\cite{RaviSMRR96}\nand Steiner forests~\\cite{DingQ20}.\nA weighted version of\n\\DgbpAcr[4]{}\nis proven to be \\cocl{NP}-hard with two different weights~\\cite{Plesnik81}.\nAs to wildlife crossing placement,\nmodels and approaches different to ours are studied~\\cite{LoraammD16,DownsHLAKO14}.\n\n\\decprob{\\prob{Connected \\gbp}{} (\\prob{Connect \\GBP}{})}{congbp}\n{An undirected graph~$G=(V,E)$,\na set~$\\mathcal{H}=\\{V_1,\\dots,V_r\\}$ of habitats \nwhere~$V_i\\subseteq V$ for all~$i\\in\\set{r}$, \nand an integer~$k\\in\\mathbb{N}_0$.}\n{Is there a subset~$F\\subseteq E$ with~$|F|\\leq k$ such that\nfor every~$i\\in\\set{r}$\nit holds that in~$G[F]$ exists a connected component containing~$V_i$?\n}\n\n\\noindent\n\\prob{Connect \\GBP}{}\nwith edge costs\nis also known as \\prob{Steiner Forest}~\\cite{Gassner10}\nand generalizes the well-known \\cocl{NP}-hard \\prob{Steiner Tree} problem. \nGassner~\\cite{Gassner10} proved \\prob{Steiner Forest} to be \\cocl{NP}-hard \neven if every so-called terminal net contains two vertices,\nif the graph is planar and has treewidth three,\nand if there are two differrent edge costs,\neach being upper-bounded linearly in the instance size.\nIt follows that \\prob{Connect \\GBP}{} is also \\cocl{NP}-hard in this case.\nBateni~\\textsl{et~al.}~\\cite{BateniHM11} proved that \\prob{Steiner Forest} is polynomial-time solvable on treewidth-two graphs \nand admits approximation schemes on planar and bounded-treewidth graphs.\n\nFrom a modeling perspective,\nsolutions are valid in which any habitat\nis scattered and \na habitat's patch is far away from all the others;\nthus\nanimals may need to take long walks through areas outside of their habitats.\nWith our models we avoid solutions with this property.\n\n\\section{Preliminaries}\n\\label{sec:prelims}\n\nLet~$\\mathbb{N}$ and~$\\mathbb{N}_0$ be the natural numbers with and without zero,\nrespectively.\nWe use basic definitions from graph theory~\\cite{Diestel} and\nparameterized algorithmics~\\cite{cygan2015parameterized}.\n\n\\paragraph*{Graph Theory.}\n\nLet~$G=(V,E)$ be an undirected graph with vertex set~$V$ and edge set~$E\\subseteq \\binom{V}{2}$.\nWe also denote by~$V(G)$ and~$E(G)$ the vertices and edges of~$G$,\nrespectively.\nFor~$F\\subseteq E$ let~$V(F)\\ensuremath{\\coloneqq} \\{v\\in V\\mid \\exists e\\in F :\\: v\\in e\\}$ and\n$G[F]\\ensuremath{\\coloneqq} (V(F),F)$.\nA path~$P$ is a graph with~$V(P) \\ensuremath{\\coloneqq} \\{v_1, \\ldots, v_n\\}$ and~$E(P) \\ensuremath{\\coloneqq} \\{\\{v_i,v_{i+1}\\} \\mid 1 \\le i < n \\}$.\nThe length of the path~$P$ is $|E(P)|$.\nThe distance~$\\dist_G(v,w)$ between vertices~$v,w \\in V(G)$ is the length of the shortest path between~$v$ and~$w$ in~$G$.\nThe diameter~$\\diam(G)$ is the length of longest shortest path over all vertex pairs.\nFor~$p\\in \\mathbb{N}$,\nthe graph~$G^p$ is the $p$-th power of~$G$ \ncontaining the vertex set~$V$ and edge set~$\\{\\{v,w\\}\\in \\binom{V}{2}\\mid \\dist_G(v,w)\\leq p\\}$.\nLet~$N_G(v)\\ensuremath{\\coloneqq} \\{w\\in V\\mid \\{v,w\\}\\in E\\}$ be the (open) neighborhood of~$v$,\nand $N_G[v]\\ensuremath{\\coloneqq} N_G(v)\\cup\\{v\\}$ be the closed neighborhood of~$v$.\nFor~$p\\in\\mathbb{N}$,\nlet~$N_G^p(v)\\ensuremath{\\coloneqq} \\{w\\in V\\mid \\{v,w\\}\\in E(G^p)\\}$ be the (open) $p$-neighborhood of~$v$,\nand $N_G^p[v]\\ensuremath{\\coloneqq} N_G^p(v)\\cup\\{v\\}$ be the closed $p$-neighborhood of~$v$.\nTwo vertices~$v,w\\in V$ are called twins if~$N_G(v)=N_G(w)$.\nThe (vertex) degree~$\\deg_G(v)\\ensuremath{\\coloneqq} |N_G(v)|$ of~$v$ is the number if its neighbors.\nThe maximum degree~$\\Delta(G)\\ensuremath{\\coloneqq} \\max_{v\\in V}\\deg_G(v)$ is the maximum over all (vertex) degrees.\n\n\\section{Connecting Habitats with a Patch at Short Reach}\n\\label{sec:rgbp}\n\nThe following problem ensures that any habitat patch can reach the other patches via patches of the same habitat and short strolls over ``foreign'' ground.\n\n\\decprob{\\RgbpTsc{} (\\RgbpAcr{})}{rgbp}\n{An undirected graph~$G=(V,E)$,\na set~$\\mathcal{H}=\\{V_1,\\dots,V_r\\}$ of habitats where~$V_i\\subseteq V$ for all~$i\\in\\set{r}$, \nand an integer~$k\\in\\mathbb{N}_0$.}\n{Is there a subset~$F\\subseteq E$ with~$|F|\\leq k$ such that\nfor every~$i\\in\\set{r}$\nit holds that~$G[F]^d[V_i]$ is connected?\n}\n\n\\noindent\nNote that if~$d$ is part of the input,\nthen~$\\prob{Connect \\GBP}{} \\leq \\prob{Reach \\prob{GBP}{}}$.\nWe will prove the following.\n\n\\begin{theorem}\n\\label{thm:rgbp}\n \\RgbpAcr{} is\n \\begin{compactenum}[(i)]\n  \\item if~$d=1$, \\cocl{NP}-hard even on planar graphs or graphs with maximum degree four;\n  \\item if~$d=2$, \\cocl{NP}-hard even on graphs with maximum degree four and~$r=2$ or graphs with diameter four and~$r=1$, and in~\\cocl{FPT}{} regarding~$k$;\n  \\item if~$d\\geq 3$, \\cocl{NP}-hard and~\\W{1}-hard regarding~$k+r$.\n \\end{compactenum}\n Moreover, \\RgbpAcr{} admits an~$rd$-approximation of the minimum number of green bridges in~$\\O(mn+rnd)$ time.\n\\end{theorem}\n\n\\subsection{Approximation}\nThe approximation algorithm computes for every habitat~$V_i$ a spanning tree in~$G^d[V_i]$,\nand adds the edges of the corresponding paths to the solution~$F$.\nEach of the spanning trees then is a~$d$-approximation for just the one habitat,\nhence the union of the spanning trees is an~$rd$-approximation for all habitats.\n\n\\begin{lemma}\n \\label{lem:d-approx}\n For~$r=1$,\n \\RgbpAcr{} admits a~$d$-approximation \n of the minimum number of green bridges\n in~$\\O(mn)$ time.\n\\end{lemma}\n\n\\begin{proof}\n We start off by computing in~$\\O(mn)$ time the graph~$H \\ensuremath{\\coloneqq} G^d$ as well as for every edge~$e = \\{u, v\\} \\in E(H)$ the corresponding path~$P_e$ from~$u$ to~$v$ of length at most~$d$ in~$G$.\n If~$H[V_1]$ is not connected, then return~\\textnormal{\\texttt{no}}{}.\n If not,\n then compute a minimum spanning tree~$T \\subseteq H[V_1]$ in~$\\O(n \\log n)$ time.\n For each edge~$e = \\{u, v\\} \\in E(T)$ compute in~$\\O(m)$ time the corresponding path~$P_e \\subseteq G$ from~$u$ to~$v$ of length at most~$d$.\n Finally, return the set~$F \\ensuremath{\\coloneqq} \\bigcup_{e \\in E(T)} E(P_e)$, computable in~$\\O(m)$ time.\n Clearly, $G[F]^d[V_1]$ is connected.\n As a minimum solution~$F^*$ has at least~$|V_1|-1$ edges,\n and every of the paths~$P_e$ consists of at most~$d$ edges,\n \\[\n  |F| = |\\bigcup_{e\\in E(T)} E(P_e)| \\le \\sum_{e \\in E(T)} E(P_e) \\le (|V_1|-1)\\cdot d \\le d|F^*|.\\qedhere\n \\]\n\\end{proof}\n\nWith the~$d$-approximation algorithm for a habitat at hand, we present our~$rd$-approximation algorithm.\n\n\\begin{proposition}\n\t\\RgbpAcr{} admits an~$rd$-approximation \n\tof the minimum number of green bridges \n\tin $\\O(mn + rnd)$ time.\n\\end{proposition}\n\\begin{proof}\n\tWe initially compute the shortest paths between all vertex pairs in~$G$ in~$O(mn)$ time.\n\tWe obtain the graph~$H \\ensuremath{\\coloneqq} G^d$ as a byproduct.\n\tIf for some~$i \\in \\set{r}$, $H[V_i]$ is not connected,\n\tthen return~\\textnormal{\\texttt{no}}{}.\n\tIf not, then \n\tcompute for each~$i \\in \\set{r}$ a spanning tree~$T_i$ of~$H[V_i]$, or return~\\textnormal{\\texttt{no}}{} if~$H[V_i]$ is not connected.\n\tLet~$F_i \\subseteq E(G)$ be the edge set corresponding to~$T_i$ as in the proof of \\cref{lem:d-approx}.\n\tAs~$G[F_i]^d[V_i]$ is connected,\n\t$F \\ensuremath{\\coloneqq} \\bigcup_{i=1}^r F_i$ is a solution.\n\n\tNote that each of the~$r$ spanning trees~$T_i$ contain at most~$n$ edges,\n\tand for each of these edges~$e \\in F_i$ we can determine the corresponding paths~$P_e \\subseteq G$\n\tof length at most~$d$ in~$\\O(d)$ time.\n\tWe obtain an overall running time of~$\\O(mn + rnd)$.\n\n\tAs for the approximation ratio, let~$F^*$ be a minimum solution, and for every~$i \\in \\set{r}$ let~$F^*_i \\in E(G)$ be a minimum-size edge set such that~$G[F^*_i]^d[V_i]$ is connected.\n\tAs~$|F^*| \\ge \\max_{i\\in\\set{r}} |F^*_i|$,\n\twe have\n\t\\[\n\t\t|F| \\le \\sum_{i=1}^r |F_i| \\le \\sum_{i=1}^r d|F^*_i| \\le r \\cdot d|F^*|. \\qedhere\n\t\\]\n\\end{proof}\n\n\\subsection{When a next habitat is directly reachable (\\texorpdfstring{\\boldmath{$d=1$}}{d=1})}\n\\label{ssec:1rgbp}\n\nSetting~$d=1$ may reflect perfect knowledge about the habitats.\nIn this case,\nwe want that in~$G[F]$, \neach habitat~$V_i$ forms a connected component.\n\n\\begin{proposition}\n \\label{prop:1rgbp}\n \\RgbpAcr[1]{} is \\cocl{NP}-hard even on graphs of maximum degree four\n and on series-parallel graphs.\n\\end{proposition}\nWe remark that series-parallel graphs are also planar.\nWe first present the construction for showing hardness for graphs of degree at most four.\n\n\\begin{construction}\n \\label{constr:1rgbp}\n For an instance~$\\mathcal{I}=(G, k)$ of \\prob{3-Regular Vertex Cover} with~$G=(V, E)$,\n $V=\\set{n}$,\n and~$E=\\{e_1,\\dots,e_m\\}$,\n construct an instance~$\\mathcal{I}'\\ensuremath{\\coloneqq} (G',\\mathcal{H},k')$\n where\n $\\mathcal{H}\\ensuremath{\\coloneqq} 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\\x\/\\y in {1\/xnode,2\/xnodex,3\/xnode,4\/xnodex,5\/xnode,6\/xnodex,7\/xnode}{\n          \\node (a\\x) at (\\x*0.75*1.1-5*1.1,0)[\\y]{};\n          \\node (b\\x) at (\\x*0.75*1.1-5*1.1,-0.9*1.5*\\yr)[\\y]{};\n      }\n      \\draw[xedge,white] (a1) to node[midway]{\\color{cyan!50!green}$e_1$}(b1);\n      \\draw[xedge,white] (a3) to node[midway]{\\color{cyan!50!green}$e_s$}(b3);\n      \\draw[xedge,white] (a5) to node[midway]{\\color{cyan!50!green}$e_t$}(b5);\n      \\draw[xedge,white] (a7) to node[midway]{\\color{cyan!50!green}$e_m$}(b7);\n\n      \\foreach \\x\/\\y in {1\/xnodef,2\/xnodex,3\/xnodef,4\/xnodex,5\/xnodef,6\/xnodex,7\/xnodef}{\n          \\node (x\\x) at (9*0.75*1.1+\\x*0.75*1.1-5*1.1,0)[\\y]{};\n          \\node (y\\x) at (9*0.75*1.1+\\x*0.75*1.1-5*1.1,-0.9*1.5*\\yr)[\\y]{};\n          \\draw[xedge] (x\\x) to (y\\x);\n      }\n      \\draw[xedge] (x1) to node[midway,left]{$1$}(y1);\n      \\draw[xedge] (x3) to node[midway,left]{$i$}(y3);\n      \\draw[xedge] (x5) to node[midway,left]{$j$}(y5);\n      \\draw[xedge] (x7) to node[midway,left]{$n$}(y7);\n\n      \\foreach \\x in {2,4,6}{\n        \\node at (a\\x.north east)[anchor=south,rotate=25]{$\\cdots$};\n        \\node at (b\\x.south east)[anchor=north,rotate=-25]{$\\cdots$};\n      }\n      \\foreach \\x in {4}{\n        \\node at (x\\x.north west)[anchor=south,rotate=-25]{$\\cdots$};\n        \\node at (y\\x.south west)[anchor=north,rotate=25]{$\\cdots$};\n      }\n      \\foreach \\x\/\\y in {1\/7,1\/6,7\/2,7\/4}{\n        \\draw[xedgem,color=red!50!black] (a\\x) to [out=25,in=155](x\\y);\n        \\draw[xedgem,color=red!50!black] (b\\x) to [out=-25,in=-155](y\\y);\n      }\n      \\foreach \\x\/\\y in {3\/3,3\/5,5\/1,5\/3}{\n        \\draw[xedgef,color=red!50!black] (a\\x) to [out=25,in=155](x\\y);\n        \\draw[xedgef,color=red!50!black] (b\\x) to [out=-25,in=-155](y\\y);\n      }\n      \n      \\newcommand{\\ltb}[3]{\n     \\contourlength{0.09em}\n      \\node at (#1)[label={[align=center,font=\\scriptsize,color=black]\\contour*{white}{#2}},label={[align=center,font=\\scriptsize,color=black]-90:\\contour*{white}{#3}}]{};\n      }\n      \\ltb{a5}{~~~$\\in V_1,V_i,Z_t$}{};\n      \\ltb{b5}{}{~~~$\\in W_1,W_i,Z_t$}{};\n      \n      \\ltb{a3}{$\\in V_i,V_j,Z_s$~~~}{};\n      \\ltb{b3}{}{$\\in W_i,W_j,Z_s$~~~}{};\n      \n      \\ltb{x3}{$\\in V_i,Z_s,Z_t$~~~}{};\n      \\ltb{y3}{}{$\\in W_i,Z_s,Z_t$~~~}{};\n      \\ltb{x5}{$\\in V_j,Z_s$}{};\n      \\ltb{y5}{}{$\\in W_j,Z_s$}{};\n    \n    \\end{tikzpicture}\n  \\caption{Illustration to~\\cref{constr:1rgbp} for~\\RgbpAcr[1]{}. \n  Here,\n  e.g.,~$e_s=\\{i,j\\}$ and~$e_t=\\{1,i\\}$.\n  Every solution (if existent) contains all red-colored edges~(\\Cref{obs:1rgbp}).}\n  \\label{fig:1rgbp}\n \\end{figure}\n Construct vertex sets~$V_E\\ensuremath{\\coloneqq} \\{x_i,y_i\\mid e_i\\in E\\}$ \n and~$V_G\\ensuremath{\\coloneqq} \\{v_i,w_i\\mid i\\in V\\}$.\n Next,\n construct edge sets~$E^*\\ensuremath{\\coloneqq} \\bigcup_{i\\in V} \\{\\{v_i,x_j\\},\\{w_i,y_j\\}\\mid i\\in e_j\\}$\n and~$E'\\ensuremath{\\coloneqq} \\{\\{v_i,w_i\\}\\mid i\\in V\\}\\cup E^*$.\n Finally, \n construct habitats~$V_i\\ensuremath{\\coloneqq} \\{v_i\\}\\cup\\bigcup_{i\\in e_j} \\{x_j\\}$ and\n $W_i\\ensuremath{\\coloneqq} \\{w_i\\}\\cup\\bigcup_{i\\in e_j} \\{y_j\\}$ \n for every~$i\\in\\set{n}$,\n and~$Z_j\\ensuremath{\\coloneqq} \\{x_j,y_j\\}\\cup \\bigcup_{i\\in e_j} \\{v_i,w_i\\}$ for every~$j\\in\\set{m}$.\n\\end{construction}\n\n\\begin{observation}\n \\label{obs:1rgbp}\n Let~$\\mathcal{I}'$ be a \\textnormal{\\texttt{yes}}-instance.\n Then\n every solution~$F$ contains all edges in~$E^*$.\n\\end{observation}\n\n\\begin{proof}\n  Observe that by construction,\n  for every~$S\\in\\mathcal{V}\\cup\\mathcal{W}$,\n  $G[S]$ is a star with center in~$V_G$.\n  Hence,\n  all edges in~$G[S]$ must be contained in every solution.\n  Since~$E^* = \\bigcup_{S\\in\\mathcal{V}\\cup\\mathcal{W}} E(G[S])$,\n  the claim follows.\n\\end{proof}\n\n\\begin{lemma}\n \\label{lem:1rgbp}\n Let~$\\mathcal{I}'$ be the instance obtained from an instance~$\\mathcal{I}$ using~\\cref{constr:1rgbp}.\n Then,\n $\\mathcal{I}$ is a \\textnormal{\\texttt{yes}}-instance if and only if~$\\mathcal{I}'$ is a \\textnormal{\\texttt{yes}}-instance.\n\\end{lemma}\n\n\\begin{proof}\n  $(\\Rightarrow)\\quad${}\n  Let~$S\\subseteq V$ be a vertex cover of~$G$ of size~$k$.\n  We claim that~$F\\ensuremath{\\coloneqq} E^*\\cup \\bigcup_{i\\in S}\\{\\{x_i,y_i\\}\\}$\n  is a solution to~$\\mathcal{I}'$.\n  Note that~$|F|=4m+k$.\n  Observe that~$G[F][T]$ is connected for every~$T\\in\\mathcal{V}\\cup\\mathcal{W}$.\n  Suppose that there is~$Z_\\ell$ such that~$G[F][Z_\\ell]$\n  is not connected.\n  Let~$e_\\ell=\\{i,j\\}$.\n  Since~$E^*\\subseteq F$,\n  none of~$\\{v_i,w_i\\}$ and~$\\{v_j,w_j\\}$ is contained in~$F$.\n  It follows that~$\\{i,j\\}\\cap S=\\emptyset$,\n  contradicting the fact that~$S$ is a vertex cover.\n  \n  $(\\Leftarrow)\\quad${}\n  Let~$F$ be a solution to~$\\mathcal{I}'$.\n  We know that~$E^*\\subseteq V$.\n  We claim that~$S\\ensuremath{\\coloneqq} \\{i\\in V\\mid \\{x_i,y_i\\}\\in F\\}$\n  is a vertex cover of~$G$.\n  Note that~$|S|\\leq k$.\n  Suppose not,\n  that is,\n  there is an~$e_\\ell=\\{i,j\\}$ with~$\\{i,j\\}\\cap S=\\emptyset$.\n  Then,\n  $G[F][Z_\\ell]$ is not connected,\n  a contradiction.\n\\end{proof}\n\nThe construction for showing hardness for series-parallel graphs is very similar:\nWe replace the edges in~$E^*$ by two stars.\n\n\\begin{construction}\n \\label{constr:1rgbp-planar}\n For an instance~$\\mathcal{I}=(G,k)$ of~\\prob{3-Regular Vertex Cover}\n with~$G=(V,E)$, and~$V=\\set{n}$ and~$E=\\{e_1,\\dots,e_m\\}$,\n construct an instance~$\\mathcal{I}'\\ensuremath{\\coloneqq} (G',\\mathcal{H},k')$ with\n $\\mathcal{H}=\\{S, T, Z_1,\\dots,Z_m\\}$\n and~$k'\\ensuremath{\\coloneqq} 2m+2n+k$ as follows\n (see~\\cref{fig:1rgbp} for an illustration).\n\\begin{figure}[t]\n  \\centering\n    \\begin{tikzpicture}\n     \\contourlength{0.09em}\n\n      \\def1.1{1}\n      \\def0.9{1}\n      \\def0.75{0.75}\n      \\def1.5*\\yr{1}\n      \\def25{25}\n      \\def155{155}\n      \\tikzstyle{xnode}=[circle,fill,scale=0.6,draw,color=cyan!50!green]\n      \\tikzstyle{xnodex}=[label={[xshift=-0.55*1.1 cm]0:$\\cdots$},color=cyan!50!green]\n      \\tikzstyle{xnodef}=[circle,scale=0.6,draw,color=cyan!50!green,fill=white]\n      \\tikzstyle{xedge}=[thick,-,color=cyan!50!green]\n\n      \\foreach \\x\/\\y in {1\/xnode,2\/xnodex,3\/xnode,4\/xnodex,5\/xnode,6\/xnodex,7\/xnode}{\n          \\node (a\\x) at (\\x*0.75*1.1,0)[\\y]{};\n          \\node (b\\x) at (\\x*0.75*1.1,-0.9*1.5*\\yr)[\\y]{};\n      }\n      \\draw[xedge,white] (a1) to node[midway]{\\color{cyan!50!green}$e_1$}(b1);\n      \\draw[xedge,white] (a3) to node[midway]{\\color{cyan!50!green}$e_p$}(b3);\n      \\draw[xedge,white] (a5) to node[midway]{\\color{cyan!50!green}$e_q$}(b5);\n      \\draw[xedge,white] (a7) to node[midway]{\\color{cyan!50!green}$e_m$}(b7);\n\n      \\foreach \\x\/\\y in {1\/xnodef,2\/xnodex,3\/xnodef,4\/xnodex,5\/xnodef,6\/xnodex,7\/xnodef}{\n          \\node (x\\x) at (9*0.75*1.1+\\x*0.75*1.1,0)[\\y]{};\n          \\node (y\\x) at (9*0.75*1.1+\\x*0.75*1.1,-0.9*1.5*\\yr)[\\y]{};\n          \\draw[xedge] (x\\x) to (y\\x);\n      }\n      \\draw[xedge] (x1) to node[midway,left]{$1$}(y1);\n      \\draw[xedge] (x3) to node[midway,left]{$i$}(y3);\n      \\draw[xedge] (x5) to node[midway,left]{$j$}(y5);\n      \\draw[xedge] (x7) to node[midway,left]{$n$}(y7);\n\n      \n      \\foreach \\x in {2,4,6}{\n        \\node at (a\\x.north east)[anchor=south,rotate=25]{$\\cdots$};\n        \\node at (b\\x.south east)[anchor=north,rotate=-25]{$\\cdots$};\n      }\n      \\foreach \\x in {4}{\n        \\node at (x\\x.north west)[anchor=south,rotate=-25]{$\\cdots$};\n        \\node at (y\\x.south west)[anchor=north,rotate=25]{$\\cdots$};\n      }\n      \\node (a) at (8.5*0.75*1.1,1*0.9)[xnode]{};\n      \\node (b) at (8.5*0.75*1.1,-2*0.9)[xnode]{};\n      \\foreach \\x\/\\y in {1,...,7}{\n        \\draw[xedge,color=red!50!black] (a\\x) to (a);\n        \\draw[xedge,color=red!50!black] (x\\x) to (a);\n        \\draw[xedge,color=red!50!black] (b\\x) to (b);\n        \\draw[xedge,color=red!50!black] (y\\x) to (b);\n      }\n      \n      \\newcommand{\\ltb}[3]{\n      \\node at (#1)[label={[align=center,font=\\scriptsize,color=black]#2},label={[align=center,font=\\scriptsize,color=black]-90:#3}]{};\n      }\n      \\ltb{a3}{\\contour*{white}{$\\in Z_p,S$~~~}}{};\n      \\ltb{b3}{}{\\contour*{white}{$\\in Z_p,T$~~~}}{};\n      \n      \\ltb{a5}{\\contour*{white}{~~~$\\in Z_q,S$}}{};\n      \\ltb{b5}{}{\\contour*{white}{~~~$\\in Z_q,T$}}{};\n      \n      \\ltb{x3}{\\contour*{white}{$\\in Z_p,Z_q,S$}}{};\n      \\ltb{y3}{}{\\contour*{white}{$\\in Z_p,Z_q,T$}}{};\n      \\ltb{x5}{\\contour*{white}{$\\in Z_q,S$}}{};\n      \\ltb{y5}{}{\\contour*{white}{$\\in Z_q,T$}}{};\n      \n      \\ltb{a}{\\contour*{white}{$\\in S,Z_1,\\dots,Z_m$}}{$s$};\n      \\ltb{b}{$t$}{\\contour*{white}{$\\in T,Z_1,\\dots,Z_m$}}{};\n    \n    \\end{tikzpicture}\n  \\label{fig:1rgbp-planar}\n  \\caption{Illustration to~\\cref{constr:1rgbp-planar} for \\RgbpAcr[1]{} on planar series-parallel graphs. \n  In this example,\n  there are e.g.~$e_p=\\{1,i\\}$ and~$e_q=\\{i,j\\}$.\n  In case of a \\textnormal{\\texttt{yes}}-instance,\n  the red-colored edges are in every solution~(\\cref{obs:1rgbp-planar}).\n  ($k=2n+2m+k$)}\n \\end{figure}\n \n Add to~$G'$ the vertex sets~$V_E\\ensuremath{\\coloneqq} \\{x_j,y_j\\mid e_j\\in E\\}$,\n $V_G\\ensuremath{\\coloneqq} \\{v_i,w_i\\mid i\\in V\\}$,\n and~$V_C\\ensuremath{\\coloneqq} \\{s, t\\}$,\n and the edge sets\n $E_V\\ensuremath{\\coloneqq} \\{\\{v_i,w_i\\}\\mid i\\in V\\}$\n and~$E^*\\ensuremath{\\coloneqq} \\{\\{s, v_i\\}, \\{t, w_i\\} \\mid i \\in V\\} \\cup \\{\\{s, x_j\\}, \\{t, y_j\\} \\mid e_j \\in E\\}$.\n Finally,\n let~$S\\ensuremath{\\coloneqq} \\{s\\}\\cup\\bigcup_{i \\in V} \\{v_i\\} \\cup \\bigcup_{e_j\\in E} \\{x_j\\}$,\n let~$T\\ensuremath{\\coloneqq} \\{t\\}\\cup\\bigcup_{i \\in V} \\{w_i\\} \\cup \\bigcup_{e_j\\in E} \\{y_j\\}$,\n and for~$e_j \\in E$ let~$Z_j\\ensuremath{\\coloneqq}\\{x_j,y_j\\}\\cup\\bigcup_{i\\in e_j}\\{v_i, w_i\\}\\cup\\{s,t\\}$.\n\\end{construction}\n\n\\begin{observation}\n\tThe graph~$G'$ constructed in \\cref{constr:1rgbp-planar} is planar and series-parallel.\n\\end{observation}\n\n\\begin{observation}\n \\label{obs:1rgbp-planar}\n Let~$\\mathcal{I}'$ be a \\textnormal{\\texttt{yes}}-instance.\n Then\n every solution~$F$ contains all edges in~$E^*$.\n\\end{observation}\n\n\\begin{proof}\n  Observe that,\n  by construction,\n  $G[S]$ is a star with center~$s$\n  and~$G[T]$ is a star with center~$t$.\n  Hence,\n  all edges in~$G[S]$ and in~$G[T]$ are contained in every solution.\n  Since~$E^* = E(G[S]) \\cup E(G[T])$,\n  the claim follows.\n\\end{proof}\n\n\n\\begin{lemma}\n \\label{lem:1rgbp-planar}\n Let~$\\mathcal{I}'$ be the instance obtained from an instance~$\\mathcal{I}$ using~\\cref{constr:1rgbp-planar}.\n Then,\n $\\mathcal{I}$ is a \\textnormal{\\texttt{yes}}-instance if and only if~$\\mathcal{I}'$ is a \\textnormal{\\texttt{yes}}-instance.\n\\end{lemma}\n\n\\begin{proof}\n  $(\\Rightarrow)\\quad${}\n  Let~$S\\subseteq V$ be a vertex cover of~$G$ of size~$k$.\n  We claim that~$F\\ensuremath{\\coloneqq} E^*\\cup \\bigcup_{i\\in S}\\{\\{x_i,y_i\\}\\}$\n  is a solution to~$\\mathcal{I}'$.\n  Note that~$|F|=2m+2n+k$.\n  Observe that~$G[F][S]$ and~$G[F][T]$ are connected.\n  Suppose that there is~$Z_\\ell$ such that~$G[F][Z_\\ell]$\n  is not connected.\n  Let~$e_\\ell=\\{i,j\\}$.\n  Since~$E^*\\subseteq F$,\n  none of~$\\{v_i,w_i\\}$ and~$\\{v_j,w_j\\}$ is contained in~$F$.\n  It follows that~$\\{i,j\\}\\cap S=\\emptyset$,\n  contradicting the fact that~$S$ is a vertex cover.\n  \n  $(\\Leftarrow)\\quad${}\n  Let~$F$ be a solution to~$\\mathcal{I}'$.\n  We know that~$E^*\\subseteq V$.\n  We claim that~$S\\ensuremath{\\coloneqq} \\{i\\in V\\mid \\{x_i,y_i\\}\\in F\\}$\n  is a vertex cover of~$G$.\n  Note that~$|S|\\leq k$.\n  Suppose not,\n  that is,\n  there is an~$e_\\ell=\\{i,j\\}$ with~$\\{i,j\\}\\cap S=\\emptyset$.\n  Then,\n  $G[F][Z_\\ell]$ is not connected,\n  a contradiction.\n\\end{proof}\n\n\\subsection{One hop between habitat patches (\\texorpdfstring{\\boldmath{$d=2$}}{d=2})}\n\\label{ssec:2rgbp}\n\nWe prove that~\\RgbpAcr[2]{}\nis already~\\cocl{NP}-complete even if there are two habitats and the graph has maximum degree four,\nor if there is only one habitat.\n\n\\begin{proposition}\n \\label{prop:2rgbp}\n \\RgbpAcr{}  with~$d\\geq 2$ is \\cocl{NP}-complete \n even if\n (i) $r=2$ and~$\\Delta\\leq 4$\n or\n (ii) $r=1$ and the input graph has diameter~$2d$.\n\\end{proposition}\n\n\\noindent\nFor the sake of presentation,\nwe prove~\\cref{prop:2rgbp}(i) for~$d=2$.\nAfterwards,\nwe briefly explain how to adapt the proof for~$d>2$ and for \\cref{prop:2rgbp}(ii).\n\n\\begin{construction}\n \\label{constr:2rgbp}\n For an instance~$\\mathcal{I}=(G, k)$ of \\prob{3-Regular Vertex Cover} with~$G=(V, E)$ and~$V=\\set{n}$\n construct an instance of~\\RgbpAcr[2]{}\n with graph~$G'=(V',E')$, \n habitat sets~$V_1$ and~$V_2$,\n and integer~$k'\\ensuremath{\\coloneqq} |E|+(n-1)+k$ as follows \n (see~\\cref{fig:2rgbp}(a) for an illustration).\n \\begin{figure}[t]\n  \\centering\n  \\begin{tikzpicture}\n\n    \\def1.1{0.67}\n    \\def0.9{0.7}\n    \\def1.5*\\yr{1.5*0.9}\n    \\def0.125{0.125}\n    \\tikzstyle{xnode}=[circle,fill,scale=0.6,draw,color=cyan!50!green]\n    \\tikzstyle{xnodef}=[circle,scale=0.6,draw,color=cyan!50!green,fill=white]\n    \\tikzstyle{xnodex}=[label={[xshift=-0.64*1.1 cm]0:$\\cdots$},color=cyan!50!green]\n    \\tikzstyle{xedge}=[thick,-,color=cyan!50!green]\n\n    \\node at (-4.25*1.1,1*0.9)[]{(a)};\n    \n    \\foreach \\x\/\\y in {1\/xnode,2\/xnodex,3\/xnode,4\/xnodex,5\/xnode,6\/xnodex,7\/xnode,8\/xnodex,9\/xnode}{\n    \\node (e\\x) at (\\x*1.1-5*1.1,0)[\\y]{};\n    }\n    \\draw[dotted,rounded corners,draw] ($(e1.south west)-(0.125,0.125)$) rectangle ($(e9.north east)+(0.125,0.125)$);\n    \\foreach \\x\/\\y in {1\/xnodef,2\/xnodex,3\/xnodef,4\/xnodex,5\/xnodef,6\/xnodex,7\/xnodef}{\n    \\node (v\\x) at (\\x*1.1-4*1.1,-1*1.5*\\yr)[\\y]{};\n    }\n    \\draw[dotted,rounded corners,draw] ($(v1.south west)-(0.125,0.125)$) rectangle ($(v7.north east)+(0.125,0.125)$);\n    \\foreach \\x\/\\y in {1\/xnode,2\/xnodex,3\/xnode,4\/xnodex,5\/xnode,6\/xnodex,7\/xnode}{\n    \\node (z\\x) at (\\x*1.1-4*1.1,-2*1.5*\\yr)[\\y]{};\n    }\n    \\draw[dotted,rounded corners,draw] ($(z1.south west)-(0.125,0.125)$) rectangle ($(z7.north east)+(0.125,0.125)$);\n    \\foreach \\x\/\\y in {3\/3,3\/5,5\/3,5\/2,7\/5,7\/6}{\n      \\draw[xedge] (e\\x) to (v\\y);\n    }\n    \\foreach \\x\/\\y in {2\/1,8\/7}{\n      \\draw[draw=none] (e\\x) to node[midway]{$\\vdots$}(v\\y);\n    }\n    \\foreach \\x in {1,...,7}{\n      \\draw[xedge] (z\\x) to (v\\x);\n    }\n    \\foreach \\x\/\\y in {1\/2,2\/3,3\/4,4\/5,5\/6,6\/7}{\n      \\draw[xedge] (z\\x) to (z\\y);\n    }\n    \n    \\contourlength{0.09em}\n    \\newcommand{\\ltb}[3]{\n      \\node at (#1)[label={[align=center,font=\\footnotesize,color=black]#2},label={[align=center,font=\\scriptsize,color=black]-90:\\contour*{white}{#3}}]{};\n    }\n    \n    \\ltb{e3}{\\footnotesize$e=$\\\\\\footnotesize$\\{i,j\\}$}{$\\in V_1$};\n    \\ltb{e5}{\\footnotesize$e'=$\\\\\\footnotesize$\\{i,j'\\}$}{$\\in V_1$};\n    \\ltb{e7}{\\footnotesize$e''=$\\\\\\footnotesize$\\{i',j\\}$}{$\\in V_1$};\n    \n    \\ltb{z3}{\\contour*{white}{$x_i$}}{$\\in V_1,V_2$};\n    \\ltb{z5}{\\contour*{white}{$x_j$}}{$\\in V_1,V_2$};\n  \n    \\ltb{v3}{$v_i$}{};\n    \\ltb{v5}{$v_j$}{};\n  \\end{tikzpicture}\n  \\hfill\n  \\begin{tikzpicture}\n\n    \\def1.1{0.67}\n    \\def0.9{0.7}\n    \\def1.5*\\yr{1.5*0.9}\n    \\def0.125{0.125}\n    \\tikzstyle{xnode}=[circle,fill,scale=0.6,draw,color=cyan!50!green]\n    \\tikzstyle{xnodef}=[circle,scale=0.6,draw,color=cyan!50!green,fill=white]\n    \\tikzstyle{xnodex}=[label={[xshift=-0.64*1.1 cm]0:$\\cdots$},color=cyan!50!green]\n    \\tikzstyle{xedge}=[thick,-,color=cyan!50!green]\n\n    \\node at (-4.25*1.1,1*0.9)[]{(b)};\n    \n    \\foreach \\x\/\\y in {1\/xnode,2\/xnodex,3\/xnode,4\/xnodex,5\/xnode,6\/xnodex,7\/xnode,8\/xnodex,9\/xnode}{\n    \\node (e\\x) at (\\x*1.1-5*1.1,0)[\\y]{};\n    }\n    \\draw[dotted,rounded corners,draw] ($(e1.south west)-(0.125,0.125)$) rectangle ($(e9.north east)+(0.125,0.125)$);\n    \\foreach \\x\/\\y in {1\/xnodef,2\/xnodex,3\/xnodef,4\/xnodex,5\/xnodef,6\/xnodex,7\/xnodef}{\n    \\node (v\\x) at (\\x*1.1-4*1.1,-1*1.5*\\yr)[\\y]{};\n    }\n    \\draw[dotted,rounded corners,draw] ($(v1.south west)-(0.125,0.125)$) rectangle ($(v7.north east)+(0.125,0.125)$);\n    \\node (x) at (0*1.1,-2*1.5*\\yr)[xnodef]{};\n\n\n    \\foreach \\x\/\\y in {3\/3,3\/5,5\/3,5\/2,7\/5,7\/6}{\n      \\draw[xedge] (e\\x) to (v\\y);\n    }\n    \\foreach \\x\/\\y in {2\/1,8\/7}{\n      \\draw[draw=none] (e\\x) to node[midway]{$\\vdots$}(v\\y);\n    }\n    \\foreach \\x in {1,...,7}{\n      \\draw[xedge] (x) to (v\\x);\n    }\n    \n    \\contourlength{0.09em}\n    \\newcommand{\\ltb}[3]{\n      \\node at (#1)[label={[align=center,font=\\footnotesize,color=black]#2},label={[align=center,font=\\scriptsize,color=black]-90:\\contour*{white}{#3}}]{};\n    }\n    \n    \\ltb{e3}{\\footnotesize$e=$\\\\\\footnotesize$\\{i,j\\}$}{$\\in V_1$};\n    \\ltb{e5}{\\footnotesize$e'=$\\\\\\footnotesize$\\{i,j'\\}$}{$\\in V_1$};\n    \\ltb{e7}{\\footnotesize$e''=$\\\\\\footnotesize$\\{i',j\\}$}{$\\in V_1$};\n    \n    \\ltb{x}{\\contour*{white}{$x$}}{$\\in V_1$};\n  \n    \\ltb{v3}{$v_i$}{};\n    \\ltb{v5}{$v_j$}{};\n  \\end{tikzpicture}\n  \\caption{Illustration for~\\RgbpAcr[2]{} with \n  (a) $r=2$ and~$\\Delta=4$ ($k'=m+(n-1)+k$) and\n  (b) $r=1$ ($k'=m+k$).}\n  \\label{fig:2rgbp}\n\\end{figure}\n\n Add the vertex set~$V_E\\ensuremath{\\coloneqq} \\{v_e\\mid e\\in E\\}$\n and add~$v_e$ with $e=\\{i,j\\}\\in E$ to habitat~$V_1$.\n Next,\n add the vertex sets~$V_G=\\{v_i\\mid i\\in V\\}$,\n and connect each~$v_i$ with all edge-vertices corresponding to an edge incident with~$i$,\n i.e.,\n add the edge set~$E_G\\ensuremath{\\coloneqq} \\bigcup_{i\\in V}\\{\\{v_i,v_e\\}\\mid i\\in e\\}$.\n Next,\n add the vertex set~$V_X\\ensuremath{\\coloneqq} \\{x_i\\mid i\\in V\\}$,\n connect each~$x_i$ with~$v_i$,\n and add~$x_i$ to~$V_1,V_2$.\n Finally,\n add the edge set~$\\{\\{x_i,x_{i+1}\\}\\mid i\\in\\set{n-1}\\}$.\n\\end{construction}\n\n\\begin{observation}\n \\label{obs:2rgbp}\n Let~$\\mathcal{I}=(G,k)$ be an instance of \\prob{3-Regular Vertex Cover} and let~$\\mathcal{I}'=(G',\\{V_1,V_2\\},k')$ be the instance obtained from~$\\mathcal{I}$ using~\\cref{constr:2rgbp}.\n If~$\\mathcal{I}'$ is a \\textnormal{\\texttt{yes}}-instance,\n then every solution contains all edges in~$G[V_X]$.\n\\end{observation}\n\n\\begin{proof}\n Suppose not,\n and let~$F$ be a solution without some edge~$\\{x_i,x_{i+1}\\}$.\n Note that in~$G-\\{\\{x_i, x_{i+1}\\}\\}$, the distance between~$x_i$ and~$x_{i+1}$ is at least four;\n thus~$G[F]^2[V_X] = G[F]^2[V_2]$ is not be connected.\n A contradiction.\n\\end{proof}\n\n\\begin{lemma}\n \\label{lem:2rgbp:edinc}\n Let~$\\mathcal{I}=(G,k)$ be an instance of \\prob{3-Regular Vertex Cover} and let~$\\mathcal{I}'=(G',f,k')$ be the instance obtained from~$\\mathcal{I}$ using~\\cref{constr:2rgbp}.\n If~$\\mathcal{I}'$ is a \\textnormal{\\texttt{yes}}-instance,\n then there is a solution~$F\\subseteq E(G')$ such that~$\\deg_{G'[F]}(v_e)=1$ for all~$e\\in E(G)$.\n\\end{lemma}\n\n\\begin{proof}\n Clearly, in every solution,\n we have~$\\deg_{G'[F]}(v_e)\\geq 1$.\n Let~$F$ be a minimum solution with a minimum number of edges incident to vertices in~$\\{v_e \\mid e\\in E\\}$.\n Suppose that there is at least one~$e=\\{i,j\\}\\in E$ such that~$\\deg_{G'[F]}(v_e)=2$, that is, $\\{v_e, v_i\\}, \\{v_e, v_j\\} \\in F$.\n Since~$F$ is a solution, there is a path~$P$ in~$G'[F]$ from~$v_e$ to some~$x_i$.\n Let~$\\{v_e,v_i\\}$ be the first edge on this path.\n Let~$F'\\ensuremath{\\coloneqq} (F \\setminus\\{v_e,v_j\\})\\cup\\{v_j,x_j\\}$.\n We claim that~$F'$ is a solution,\n yielding a contradiction to the fact that~$F$ is a solution with a minimum number of edges incident with vertices in~$V_E$.\n\n Only a vertex~$v_{e'}$ can be disconnected from any~$V_X$ by removing~$\\{v_e,v_j\\}$ from~$F$.\n This vertex cannot be on the path~$P$,\n and hence is connected to~$v_e$ via edge~$\\{v_e,v_j\\}$.\n Since now edge~$\\{v_j,x_j\\}$ is present,\n $v_{e'}$ is again connected to~$V_X$.\n\\end{proof}\n\n\\begin{lemma}\n \\label{lem:2rgbp:cor}\n Let~$\\mathcal{I}=(G,k)$ be an instance of \\prob{3-Regular Vertex Cover} and let~$\\mathcal{I}'=(G',\\{V_1,V_2\\},k')$ be the instance obtained from~$\\mathcal{I}$ using~\\cref{constr:2rgbp}.\n Then~$\\mathcal{I}$ is a \\textnormal{\\texttt{yes}}-instance if and only if~$\\mathcal{I}'$ is a \\textnormal{\\texttt{yes}}-instance.\n\\end{lemma}\n\n\\begin{proof}\n $(\\Rightarrow)\\quad${}\n Let~$S\\subseteq V$ be a vertex cover of size~$k$ in~$G$.\n We construct a solution~$F\\subseteq E'$ as follows.\n Let~$F_X=\\bigcup_{i=1}^{n-1} \\{\\{x_i,x_{i+1}\\}\\}$\n and~$F_V\\ensuremath{\\coloneqq} \\{\\{v_i,x_i\\}\\mid i\\in S\\}$.\n We define the auxiliary function~$g\\colon E\\to V'$ with~$g(\\{i,j\\})=v_{\\min(\\{i,j\\}\\cap S)}$.\n Let~$F_E\\ensuremath{\\coloneqq} \\bigcup_{e=\\{i,j\\}\\in E} \\{v_e,g(e)\\}$.\n Let~$F\\ensuremath{\\coloneqq} F_X\\cup F_V\\cup F_E$. \n Note that~$|F|=|F_X|+|F_V|+|F_E|\\leq |E|+(n-1)+k = k'$.\n Moreover, \n every~$v_e\\in V_E$ is connected to~$x_i$ via a path~$(v_e,v_i,x_i)$, \n where~$i\\in (e\\cap S)$.\n Finally,\n observe that~$G'[F][V_X]$ is connected.\n \n $(\\Leftarrow)\\quad${}\n Let~$\\mathcal{I}'$ be a \\textnormal{\\texttt{yes}}-instance.\n Due to \\cref{lem:2rgbp:edinc} there\n is a solution~$F\\subseteq E'$ such that $\\deg_{G'[F]}(v_e)=1$ for all~$e\\in E$.\n Due to the observation,\n we know that~$\\bigcup_{i=1}^{n-1} \\{\\{x_i,x_{i+1}\\}\\}\\subseteq F$.\n Let~$S\\ensuremath{\\coloneqq} \\{i\\in V\\mid \\{v_i,x_i\\}\\in F\\}$.\n We claim that~$S$ is a vertex cover.\n Suppose not,\n that is,\n there is an edge~$e\\in E$ such that~$e\\cap S=\\emptyset$.\n That means that the unique neighbor of~$v_e$,\n say~$v_i$,\n is not adjacent with~$x_i$ in~$G'[F]$.\n Since $\\deg_{G'[F]}(v_e)=1$ for all~$e\\in E$,\n $N_{G'[F]}[v_i]\\neq G'[F]$ forms a connected component in~$G'[F]^2$.\n Hence,\n $F$ is not a solution.\n A contradiction.\n\\end{proof}\n\n\\begin{remark}\n  \\begin{inparaenum}[(i)]\n  \\item To make the reduction work for~$d\\geq 3$,\n  it is enough to subdivide each edge~$\\{v_e,v_i\\}$\n  $(d-2)$ times and set~$k'\\ensuremath{\\coloneqq} (d-1)m+(n-1)+k$.\n  \\item If we contract all~$x_i$,\n  set~$V_2=\\emptyset$ \n  (i.e., only one habitat remains),\n  and set~$k'\\ensuremath{\\coloneqq} (d-1)m+k$,\n  then the reduction is still valid\n  (see~\\cref{fig:2rgbp}(b) for an illustration).\n  \\end{inparaenum}\n  Thus,\n  \\cref{prop:2rgbp}(ii) follows.\n\\end{remark}\n\n\\noindent\n\\cref{prop:2rgbp}\nleaves $k$ unbounded.\nThis leads to the following.\n\n\\subsubsection{Parameterizing with~\\texorpdfstring{\\boldmath{$k$}}{k}.}\n\nWe show that \\RgbpAcr[2]{} admits a problem kernel of size exponential in~$k$.\n\\begin{proposition}\n \\label{prop:rgbp:kernel}\n \\RgbpAcr[2]{} admits a problem kernel with at most $2k+\\binom{2k}{k}$ vertices,\n at most $\\binom{2k}{2}+k\\binom{2k}{k}$ edges,\n and at most~$2^{2k}$ habitats.\n\\end{proposition}\n\n\\noindent\nLet~$\\bar{V}\\ensuremath{\\coloneqq} V\\setminus \\bigcup_{V'\\in\\mathcal{H}} V'$ for graph~$G=(V,E)$ and habitat set~$\\mathcal{H}=\\{V_1,\\dots,V_r\\}$.\nThe following reduction rules are immediate.\n\n\\begin{rrule}\n \\label{rr:immediate}\n \\begin{inparaenum}[(i)]\n  \\item If~$|V_i|=1$ some~$i$,\n  delete~$V_i$.\n  \\item If a vertex in~$\\bar{V}$ is of degree at most one,\n  delete it.\n  \\item If there is an~$i\\in\\set{r}$ with~$|V_i|>1$ and an~$v\\in V_i$ of degree zero,\n  return a trivial \\textnormal{\\texttt{no}}-instance.\n  \\item If there is~$v\\in V\\setminus\\bar{V}$ of degree at most one,\n  delete it (also from~$V_1,\\dots,V_r$),\n  and set~$k\\ensuremath{\\coloneqq} k-1$.\n \\end{inparaenum}\n\\end{rrule}\n\n\\noindent\nClearly, \n$k$ edges can connect at most~$2k$ vertices;\nthus we obtain the following.\n\n\\begin{rrule}\n \\label{rr:few-habitat-vertices}\n If~$|V\\setminus\\bar{V}|>2k$,\n then return a trivial \\textnormal{\\texttt{no}}{}-instance.\n\\end{rrule}\n\n\\noindent\nSo we have at most~$2k$ vertices in habitats.\nNext, we upper-bound the number of non-habitat vertices.\nNo minimal solution has edges between two such vertices.\n\n\\begin{rrule}\n \\label{rr:no-bar-edges}\n If there is an edge~$e \\in E$ with $e\\subseteq \\bar{V}$,\n then delete~$e$.\n\\end{rrule}\n\n\\noindent\nMoreover,\nno minimum solution connects through non-habitat twins.\n\n\\begin{rrule}\n \\label{rr:no-twins}\n If~$N(v)\\subseteq N(w)$ for distinct~$v,w\\in \\bar{V}$,\n then delete~$v$.\n\\end{rrule}\n\n\\noindent\nWe still need to bound the number of vertices in~$\\bar V$.\nFor an $n$-element set~$S$ let~$\\mathcal F \\subseteq 2^{S}$ be a family of subsets such that\nfor every~$A, B \\in \\mathcal F$ we have~$A \\not\\subseteq B$.\nThen~$|\\mathcal F| \\le \\binom{n}{\\lfloor n\/2 \\rfloor}$ by Sperner's Theorem.\nHence,\nafter \napplying the reduction rules,\nwe get an instance with at most~$2k+\\binom{2k}{k}$ vertices \nand~$\\binom{2k}{2}+2k\\binom{2k}{k}$ edges.\n\n\\begin{rrule}\n\\label{rr:habitats}\nIf $V_i=V_j$ for distinct~$i,j\\in\\set{r}$,\nthen delete~$V_j$.\n\\end{rrule}\n\n\\noindent\nIt follows that we can safely assume that~$r\\leq 2^{2k}$.\nThus,\n\\cref{prop:rgbp:kernel} follows.\nUnfortunately,\nimproving the problem kernel \nto polynomial-size appears unlikely.\n\n\\begin{proposition}\n \\label{prop:rgbp:nopk}\n Unless~\\ensuremath{\\NP\\subseteq \\coNP\/\\poly},\n \\RgbpAcr{} for~$d\\geq 2$ admits no problem kernel of size~$k^{\\O(1)}$,\n even if~$r\\geq 1$ is constant.\n\\end{proposition}\n\n\\noindent\nWe will give a linear parameteric transformation from the following problem:\n\n\\decprob{Set Cover (SC)}{setcover}\n{A universe~$U$, a set~$\\mathcal{F}\\subseteq 2^U$ of subsets of~$U$, and an integer~$k$.}\n{Is there~$\\mathcal{F}'\\subset\\mathcal{F}$ with~$|\\mathcal{F}'|\\leq k$ such that~$\\bigcup_{F\\in\\mathcal{F}'} F=U$?}\n\n\\noindent\nThe construction is basically the same as for \\cref{prop:2rgbp}(ii).\nNote that \\prob{Set Cover} admits no problem kernel of size polynomial in~$|U|+k$,\nunless~$\\ensuremath{\\NP\\subseteq \\coNP\/\\poly}$~\\cite{DomLS14}.\n\n\n\\begin{proof}\n Let~$\\mathcal{I}=(U,\\mathcal{F},k)$ be an instance of~\\prob{Set Cover},\n with~$U=\\{u_1,\\dots,u_n\\}$.\n Construct an instance~$\\mathcal{I}'\\ensuremath{\\coloneqq} (G,V_1,k')$ of \\RgbpAcr[2]{} with~$k'=|U|+k$ as follows \n (see~\\cref{fig:2rgbp:nopk}).\n \\begin{figure}[t]\n  \\centering\n  \\begin{tikzpicture}\n\n    \\def1.1{1.1}\n    \\def0.9{1}\n    \\def1.5*\\yr{1.5*0.9}\n    \\def0.125{0.125}\n    \\tikzstyle{xnode}=[circle,fill,scale=0.6,draw,color=cyan!50!green]\n    \\tikzstyle{xnodef}=[circle,scale=0.6,draw,color=cyan!50!green,fill=white]\n    \\tikzstyle{xnodex}=[label={[xshift=-0.4*1.1 cm]0:$\\cdots$},color=cyan!50!green]\n    \\tikzstyle{xedge}=[thick,-,color=cyan!50!green]\n\n    \\foreach \\x\/\\y in {1\/xnode,2\/xnodex,3\/xnode,4\/xnodex,5\/xnode,6\/xnodex,7\/xnode,8\/xnodex,9\/xnode}{\n    \\node (e\\x) at (\\x*1.1-5*1.1,-1*1.5*\\yr)[\\y]{};\n    }\n    \\draw[dotted,rounded corners,draw] ($(e1.south west)-(0.125,0.125)$) rectangle ($(e9.north east)+(0.125,0.125)$);\n    \\foreach \\x\/\\y in {1\/xnodef,2\/xnodex,3\/xnodef,4\/xnodex,5\/xnodef,6\/xnodex,7\/xnodef}{\n    \\node (v\\x) at (\\x*1.1-4*1.1,0*1.5*\\yr)[\\y]{};\n    }\n    \\draw[dotted,rounded corners,draw] ($(v1.south west)-(0.125,0.125)$) rectangle ($(v7.north east)+(0.125,0.125)$);\n    \\node (x) at (0*1.1,-2*1.5*\\yr)[xnodef]{};\n\n\n    \\foreach \\x\/\\y in {3\/2,3\/3,3\/5,5\/1,5\/3,5\/5,5\/7,7\/4,7\/5,7\/6}{\n      \\draw[xedge] (e\\x) to (v\\y);\n    }\n    \\foreach \\x\/\\y in {2\/1,8\/7}{\n      \\draw[draw=none] (e\\x) to node[midway]{$\\vdots$}(v\\y);\n    }\n    \\foreach \\x in {1,...,9}{\n      \\draw[xedge] (x) to (e\\x);\n    }\n    \n    \\newcommand{\\ltb}[3]{\n      \\contourlength{0.09em}\n      \\node at (#1)[label={[align=center,font=\\footnotesize,color=black]\\contour*{white}{#2}},label={[align=center,font=\\scriptsize,color=black]-90:\\contour*{white}{#3}}]{};\n    }\n    \n    \\ltb{e3}{$v_{F'}$}{};\n    \\ltb{e5}{$v_F$}{};\n    \\ltb{e7}{$v_{F''}$}{};\n    \n    \\ltb{x}{$x$}{$\\in V_1$};\n  \n    \\ltb{v1}{$u_1$}{$\\in V_1$};\n    \\ltb{v7}{$u_n$}{$\\in V_1$};\n    \\ltb{v3}{$u_i$}{$\\in V_1$};\n    \\ltb{v5}{$u_j$}{$\\in V_1$};\n    \n    \\node[right =of v7,xshift=-1.1*0.5 cm]{$V_U$};\n    \\node[right =of e9,xshift=-1.1*0.5 cm]{$V_\\mathcal{F}$};\n  \\end{tikzpicture}\n  \\caption{Illustration for the construction in the proof of~\\cref{prop:rgbp:nopk} for~\\RgbpAcr[2]{} with~$r=1$. \n  In this example, \n  $U=\\{u_1,\\dots,u_n\\}$ and\n  we have $\\{u_1,u_i,u_j,u_n\\}= F\\in\\mathcal{F}$.}\n  \\label{fig:2rgbp:nopk}\n  \\end{figure}\n Let~$G$ be initially empty.\n Add the vertex set~$V_U\\ensuremath{\\coloneqq} U$,\n the vertex set~$V_\\mathcal{F}\\ensuremath{\\coloneqq} \\{v_F\\mid F\\in\\mathcal{F}\\}$,\n and the vertex~$x$.\n Set~$V_1\\ensuremath{\\coloneqq} V_U\\cup\\{x\\}$.\n Make each vertex in~$V_\\mathcal{F}$ adjacent with~$x$.\n Finally,\n for each~$F\\in\\mathcal{F}$,\n add the edge set~$\\{\\{v_i,v_F\\}\\mid u_i\\in F\\}$.\n \n The proof that~$\\mathcal{I}$ is a \\textnormal{\\texttt{yes}}-instance if and only if~$\\mathcal{I}'$ is a \\textnormal{\\texttt{yes}}-instance \n is analogous with the correctness proof for \\cref{prop:2rgbp}(ii).\n \n Since \\prob{Set Cover} admits no problem kernel of size polynomial in~$|U|+k$,\n unless~$\\ensuremath{\\NP\\subseteq \\coNP\/\\poly}$~\\cite{DomLS14},\n neither does~\\RgbpAcr[2]{} when parameterized by~$k'=|U|+k$.\n\\end{proof}\n\n\\noindent\n\\cref{prop:rgbp:nopk} holds for general graphs.\nIn fact, \nfor planar graphs,\nthe above reduction rules allow for \nan~$\\O(k^3)$-vertex kernel.\n\n\\begin{proposition}\n\t\\label{prop:rgbp:planar}\t\n\t\\RgbpAcr[2]{} on planar graphs admits a problem kernel with~$\\O(k^3)$ vertices and edges and at most~$2^{2k}$ habitats.\n\\end{proposition}\n\n\\begin{observation}\n  \\label{obs:2rgbpplanar}\n\tSuppose all reduction rules were applied exhaustively.\n\tThen\n\t\\begin{inparaenum}[(i)]\n\t\\item there are at most~$k$ vertices of degree two in~$\\bar V$, and\n\t\\item there are at most~$3\\binom{2k}{3}$ vertices of dergee at least three in~$\\bar V$.\n\t\\end{inparaenum}\n\\end{observation}\n\n\\begin{proof}\n\t\\begin{inparaenum}[\\itshape (i)]\n\t\\item By \\cref{rr:no-bar-edges,rr:few-habitat-vertices,rr:no-twins},\n\t\tevery degree-two vertex in~$\\bar V$ has a pairwise different pair of neighbors in~$V \\setminus \\bar V$.\n\t\tIf there are more than~$2\\binom{2k}{2}$ degree-two vertices in~$\\bar V$,\n\t\tthen one of the reduction rules was not applied exhaustively.\n\n\t\\item Any three vertices~$u,v,w\\in V$ of a planar graph share at most two neighbors,\n\t\tthat is,\n\t\t$|N(u)\\cap N(v)\\cap N(w)| \\le 2$.\n\t\tSuppose there are more than~$3\\binom{2k}{3}$ vertices in~$\\bar V$ of degree at least three.\n\t\tThen,\n\t\tby \\cref{rr:no-bar-edges,rr:few-habitat-vertices,rr:no-twins},\n\t\tthere are three vertices~$u,v,w\\in\\bar V$ such that~$|N(u)\\cap N(v)\\cap N(w)| \\ge 3$,\n\t\ta contradiction to~$G$ being planar.\n\t\\end{inparaenum}\n\\end{proof}\n\n\\noindent\nAs~$|V\\setminus \\bar V| \\le 2k$ and we deleted all degree-one vertices,\n\\cref{prop:rgbp:planar} follows.\n\n\\subsection{At least two hops between habitat patches (\\texorpdfstring{\\boldmath{$d\\ge 3$}}{d\u2a7e3})}\n\\label{ssec:3rgbp}\n\nIf the data is more sparse,\nthat is,\nthe observed habitats to connect are rather scattered,\nthen the problem becomes significantly harder to solve from the parameterized complexity point of view.\n\n\\begin{proposition}\n \\label{prop:3rgbp}\n \\RgbpAcr{}  with~$d\\geq 3$ is \\cocl{NP}-complete and \n \\W{1}-hard when parameterized by~$k+r$.\n\\end{proposition}\n\n\\noindent\nWe give the construction for~$d$ being odd.\nAfterwards,\nwe explain how to adapt the reduction to~$d$ being even.\n\n\\begin{construction}\n \\label{constr:3rgbp}\n Let~$(G)$ with~$G=(U^1,\\dots,U^k,E)$ be an instance of \\prob{Multicolored Clique}\n where~$G[U^i]$ forms an independent set for every~$i\\in\\set{k}$.\n Assume without loss of generality that~$U^i=\\{u^i_1,\\dots,u^i_{|V^i|}\\}$.\n Construct the instance~$(G',V_1,\\dots,V_{\\binom{k}{2}},k')$ with~$k\\ensuremath{\\coloneqq} \\frac{(d-1)}{2}k+\\binom{k}{2}$ as follows\n (see~\\cref{fig:3rgbp} for an illustration).\n \\begin{figure}[t]\n  \\centering\n    \\begin{tikzpicture}\n\n      \\usetikzlibrary{calc}\n      \\def1.1{1}\n      \\def0.9{0.775}\n      \\def0.125{0.175}\n      \\tikzstyle{xnode}=[circle,fill,scale=0.6,draw,color=cyan!50!green]\n      \\tikzstyle{xedge}=[thick,-,color=cyan!50!green]\n      \\tikzstyle{xnodef}=[circle,scale=0.6,draw,color=cyan!50!green,fill=white]\n\n      \\newcommand{\\xpart}[6]{\n        \\begin{scope}[xshift=#5*1.1 cm,yshift=#6*0.9 cm,rotate around={#4:(0,0)}]\n          \\node (u#1)  at (0,2*0.9)[xnode]{};\n          \\node (v#11)  at (-1.5*1.1,0*0.9)[xnodef]{};\n          \\node (v#12)  at (-0.75*1.1,0*0.9)[rotate=#4]{$\\cdots$};\n          \\node (v#13)  at (0*1.1,0*0.9)[xnodef]{};\n          \\node (v#14)  at (0.75*1.1,0*0.9)[rotate=#4]{$\\cdots$};\n          \\node (v#15)  at (1.5*1.1,0*0.9)[xnodef]{};\n    \n\n          \\foreach \\x in {1,...,5}{\n            \\draw[xedge] (u#1) to (v#1\\x);\n            \\draw[dash pattern=on \\pgflinewidth off 8pt, fill=white,thick, line width=3pt,color=cyan!50!green] (u#1) to (v#1\\x);\n          }\n          \\draw[densely dotted,rounded corners] ($(v#11)-(0.125,0.125)$) rectangle ($(v#15)+(0.125,0.125)$);\n        \\end{scope}\n        \\node at (v#15)[label={[]#3}]{};\n        \\node at (u#1)[label={[]#2}]{};\n      }\n      \n      \\xpart{1}{90:$\\in V_\\ell$ if~$i\\in g^{-1}(\\ell)$}{90:$U^i$}{45}{-3}{2};\n      \\xpart{2}{-90:$\\in\\bigcap_{\\ell: j\\in g^{-1}(\\ell)} V_\\ell$}{180:$U^j$}{135}{-3}{-2};\n      \\xpart{3}{-90:$\\in\\bigcap_{\\ell: j'\\in g^{-1}(\\ell)} V_\\ell$}{-90:$U^{j'}$}{225}{3}{-2};\n      \\xpart{4}{90:$\\in\\bigcap_{\\ell: i'\\in g^{-1}(\\ell)} V_\\ell$}{0:$U^{i'}$}{315}{3}{2};\n\n      \\node (top) at (0,3.2*0.9)[scale=2]{$\\cdots$};\n      \\node (bot) at (0,-3.2*0.9)[scale=2]{$\\cdots$};\n      \\draw[xedge] (v13) to (v25) to (v33) to (v45) to (v13) to (v33);\n      \\draw[xedge] (v25) to (v45);\n      \\foreach \\x\/\\y in {1\/3,2\/5,3\/3,4\/5}{\n        \\draw[xedge] (v\\x\\y) to (top);\n        \\draw[xedge] (v\\x\\y) to (bot);\n      }\n      \\draw[draw=none] (v11) to node[midway,below,sloped,color=red]{$(d-1)\/2$ edges}(u1);\n\n      \\end{tikzpicture}\n  \\caption{Illustration to~\\cref{constr:3rgbp} for~\\RgbpAcr{} for~$d\\geq 3$.}\n  \\label{fig:3rgbp}\n \\end{figure}\n\n Let~$g\\colon \\binom{\\set{k}}{2}\\to \\set{\\binom{k}{2}}$ be a bijective function.\n Let~$G'$ be initially~$G$.\n For each~$i\\in\\set{k}$,\n add a vertex~$v_i$ to~$G'$,\n add~$v_i$ to each habitat~$V_\\ell$ with~$i\\in g^{-1}(\\ell)$,\n and connect~$v_i$ with~$u^i_j$ for each~$j\\in\\set{|u^i_{|U^i|}}$ via a path with~$\\frac{d-1}{2}$ edges,\n where~$v_i$ and~$u_i^j$ are the endpoints of the path.\n\\end{construction}\n\n\\begin{remark}\n For every even~$d\\geq 4$,\n we can adapt the reduction for~$d-1$:\n at the end of the construction,\n subdivide each edge between two vertices that are in the original graph~$G$.\n\\end{remark}\n\n\\begin{observation}\n \\label{obs:3rgbp:smallhabs}\n In the obtained instance,\n for every~$\\ell\\in\\set{\\binom{k}{2}}$,\n it holds that, \n $V_\\ell=\\{v_i,v_j\\}$ where~$\\{i,j\\}=g^{-1}(\\ell)$,\n and for every~$i,j\\in\\set{k}$,\n $i\\neq j$,\n it holds that~$\\{\\ell'\\mid \\{v_i,v_j\\}\\subseteq V_{\\ell'}\\}=\\{\\ell\\}$ with~$\\ell=g(\\{i,j\\})$.\n\\end{observation}\n\n\\begin{observation}\n \\label{obs:3rgbp:exone}\n If the obtained instance is a \\textnormal{\\texttt{yes}}-instance,\n then in every minimal solution~$F$,\n for every~$i\\in\\set{k}$ there is exactly one~$u^i_j$ in~$G[F]$.\n\\end{observation}\n\n\\begin{proof}\n Note that each~$v_i$ must be connected with at least one vertex from~$U^i$ in~$G[F]$.\n Thus,\n $|G[F]\\cap U^i|\\geq 1$.\n Moreover,\n from each~$i,j\\in\\set{k}$,\n $i\\neq j$,\n $F$~must contain an edge between~$U^i$ and~$U^j$,\n since~$\\dist_{G'}(v_i,u)+\\dist_{G'}(v_j,u')\\geq d-1$ for every~$u\\in U^i$, \n $u'\\in U^j$.\n Since additionally~$k= \\frac{(d-1)}{2}k+\\binom{k}{2}$,\n it follows that~$v_i$ can not be connected with two vertices from~$U^i$ in~$G[F][U^i\\cup \\{v_i\\}]$.\n Hence,\n if there are two vertices~$u,u'\\in U^i\\cap F$,\n with~$u$ being connected to~$v_i$ in~$G[F][U^i\\cup \\{v_i\\}]$,\n then~$u'$ is not part of an $v_a$-$v_b$~path in~$G[F]$ of length at most~$d$\n for every~$a,b\\in\\set{k}$.\n It follows that~$F$ is not minimal.\n\\end{proof}\n\n\n\\begin{lemma}\n \\label{lem:3rgbp}\n Let~$\\mathcal{I}=(G)$ with~$G=(U^1,\\dots,U^k,E)$ be an instance of \\prob{Multicolored Clique} and let~$\\mathcal{I}'=(G',\\mathcal{H},k')$ be the instance obtained from~$\\mathcal{I}$ using~\\cref{constr:3rgbp}.\n Then~$\\mathcal{I}$ is a \\textnormal{\\texttt{yes}}-instance if and only if~$\\mathcal{I}'$ is a \\textnormal{\\texttt{yes}}-instance.\n\\end{lemma}\n\n\\begin{proof}\n $(\\Rightarrow)\\quad${} \n Let~$W\\subseteq V(G)$ be a multicolored clique.\n Let~$F$ contain~$\\binom{W}{2}$ and all edges of a path from~$v_i$ to~$U^i\\cap W$.\n We claim that~$F$ is a solution.\n Note that~$|F|=\\binom{k}{2}+k\\frac{d-1}{2}$.\n Since~$V_\\ell$ is of size two for all~$\\ell\\in\\set{\\binom{k}{2}}$ (\\cref{obs:3rgbp:smallhabs}),\n we only need to show that~$v_i,v_j$ with~$\\{i,j\\}=g^{-1}(\\ell)$\n is connected by a path of length at most~$d$.\n We know that~$v_i$ is connected to some~$u^i_x$ by a path of length~$(d-1)\/2$,\n which is adjacent to some~$u^j_y$, \n which is connected to~$v_j$ by a path of length~$(d-1)\/2$.\n Thus, \n $v_i$ and~$v_j$ are of distance~$d$.\n \n $(\\Leftarrow)\\quad${}\n Let~$F$ be a solution.\n Note that~$|F|=\\binom{k}{2}+k\\frac{d-1}{2}$.\n We claim that~$W\\ensuremath{\\coloneqq} V(G'[F])\\cap V(G)$ is a multicolored clique.\n First, \n observe that~$|W|=k$ since for every~$v_i$ there is exactly one~$u^i_{\\ell_i}$ in~$G'[F]$ (\\cref{obs:3rgbp:exone}).\n Suppose that~$W$ is not a multicolored clique,\n that is,\n there are~$U^i$ and~$U^j$ such that there is no edge in~$F$ between them.\n Then~$v_i$ and~$v_j$ are of distance larger than~$d$ in~$G[F]$,\n contradicting that~$F$ is a solution.\n\\end{proof}\n\n\\section{Connecting Habitats at Short Pairwise Distance}\n\\label{sec:cgbp}\n\nIn the next problem, \nwe require short pairwise reachability.\n\n\\decprob{\\CgbpTsc{} (\\CgbpAcr{})}{cgbp}\n{An undirected graph~$G=(V,E)$,\na set~$\\mathcal H = \\{V_1, \\dots, V_r\\}$ of habitats where~$V_i\\subseteq V$ for all~$i\\in\\set{r}$,\nand~$k \\in \\mathbb{N}_0$.}\n{Is there a subset~$F\\subseteq E$ with~$|F|\\leq k$ such that\nfor every~$i\\in\\set{r}$\nit holds that~$G[F]^d[V_i]$ is a clique?\n}\n\n\\noindent\nNote that if $G[F]^d[V_i]$ is a clique, \nthen~$\\dist_{G[F]}(v,w) \\leq d$ for all~$v,w\\in V_i$.\nNote that~\\CgbpAcr[2]{} is an unweighted variant of the 2NET problem~\\cite{DahlJ04}.\n\n\\begin{theorem}\n \\CgbpAcr{} is,\n \\begin{compactenum}[(i)]\n  \\item if~$d=1$, linear-time solvable;\n  \\item if~$d=2$, \\cocl{NP}-hard even on bipartite graphs of diameter three and~$r=1$, \n  and in~\\cocl{FPT}{} regarding~$k$;\n  \\item if~$d\\geq 3$, \\cocl{NP}-hard and~\\W{1}-hard regarding~$k$ even if~$r=1$.\n \\end{compactenum}\n\\end{theorem}\n\n\n\\noindent\nFor~$d=1$,\nthe problem is solvable in linear time:\nCheck whether each habitat induces a clique.\nIf so,\ncheck if the union of the cliques is small enough.\n\n\\begin{observation}\n \\label{obs:1cgbp}\n \\CgbpAcr[1]{} is solvable in linear time.\n\\end{observation}\n\n\\begin{proof}\n  We employ the following algorithm:\n  For each~$i\\in\\set{r}$,\n  let~$G_i := G[V_i]$ and return~\\textnormal{\\texttt{no}}{} if~$G_i$ is not a clique.\n  Finally,\n  return~\\textnormal{\\texttt{yes}}{} if~$|\\bigcup_{i=1}^r E(G_i)|\\leq k$,\n  and~\\textnormal{\\texttt{no}}{} otherwise.\n  Clearly,\n  if the algorithm returns \\textnormal{\\texttt{yes}},\n  then~$\\mathcal{I}$ is \\textnormal{\\texttt{yes}}-instance.\n  Conversely,\n  let~$\\mathcal{I}$ be a \\textnormal{\\texttt{yes}}-instance\n  and let~$F'$ be a solution to~$\\mathcal{I}$.\n  We know that for every~$i\\in\\set{r}$,\n  and any two vertices~$v,w\\in V_i$,\n  edge~$\\{v,w\\}$ must be in~$F'$.\n  It follows that\n  $\\bigcup_{i=1}^r E(G_i)\\subseteq F'$.\n  Thus,\n  $|\\bigcup_{i=1}^r E(G_i)|\\leq |F'|\\leq k$ \n  and the algorithm correctly returns~\\textnormal{\\texttt{yes}}.\n\\end{proof}\n\n\\subsection{When each part is just two steps away (\\texorpdfstring{\\boldmath{$d=2$}}{d=2})}\n\\label{ssec:2cgbp}\n\nFor~$d=2$,\n\\CgbpAcr{}\nbecomes \\cocl{NP}-hard already on quite restrictive inputs.\n\n\\begin{proposition}\n \\label{prop:2cgbp}\n \\CgbpAcr[2]{} is \\cocl{NP}-complete,\n even\n if~$r=1$ and\n the input graph is bipartite and of diameter three.\n\\end{proposition}\n\n\\begin{construction}\n \\label{constr:2cgbp}\n Let~$\\mathcal{I}=(G,k)$ with~$G=(V,E)$ be an instance of \\prob{Vertex Cover},\n and assume without loss of generality that~$V=\\set{n}$.\n Construct an instance of~\\CgbpAcr[2]{}\n with graph~$G'=(V',E')$, \n habitat~$V_1$,\n and integer~$k'\\ensuremath{\\coloneqq} 2|E|+k+3$ as follows\n (see~\\cref{fig:2cgbp} for an illustration).\n \\begin{figure}[t]\n  \\centering\n  \\begin{tikzpicture}\n    \\def1.1{1.1}\n    \\def0.9{0.7}\n    \\def1.5*\\yr{1.5*0.9}\n    \\def0.125{0.125}\n    \\tikzstyle{xnode}=[circle,fill,scale=0.6,draw,color=cyan!50!green]\n    \\tikzstyle{xnodex}=[label={[xshift=-0.4*1.1 cm]0:$\\cdots$},color=cyan!50!green]\n    \\tikzstyle{xedge}=[thick,-,color=cyan!50!green]\n    \\tikzstyle{xnodef}=[circle,scale=0.6,draw,color=cyan!50!green,fill=white]\n    \n    \\node (xp) at (0,1*1.5*\\yr)[xnodef]{};\n    \\node (x) at (0*1.1,2*1.5*\\yr)[xnode]{};\n    \n    \\foreach \\x\/\\y in {1\/xnode,2\/xnodex,3\/xnode,4\/xnodex,5\/xnode,6\/xnodex,7\/xnode,8\/xnodex,9\/xnode}{\n      \\node (e\\x) at (\\x*1.1-5*1.1,0)[\\y]{};\n    }\n    \\draw[dotted,rounded corners,draw] ($(e1.south west)-(0.125,0.125)$) rectangle ($(e9.north east)+(0.125,0.125)$);\n    \\foreach \\x\/\\y in {1\/xnodef,2\/xnodex,3\/xnodef,4\/xnodex,5\/xnodef,6\/xnodex,7\/xnodef}{\n    \\node (v\\x) at (\\x*1.1-4*1.1,-1*1.5*\\yr)[\\y]{};\n    }\n    \\draw[dotted,rounded corners,draw] ($(v1.south west)-(0.125,0.125)$) rectangle ($(v7.north east)+(0.125,0.125)$);\n    \\node (z) at (0,-2*1.5*\\yr)[xnode]{};\n\n\n    \\foreach \\x\/\\y in {3\/3,3\/5,5\/3,5\/2,7\/5,7\/6}{\n    \\draw[xedge] (e\\x) to (v\\y);\n    }\n    \\foreach \\x\/\\y in {2\/1,8\/7}{\n    \\draw[draw=none] (e\\x) to node[midway]{$\\vdots$}(v\\y);\n    }\n\n    \\foreach \\x in {1,...,7}{\n    \\draw[xedge] (z) to (v\\x);\n    }\n    \n    \\foreach \\x in {1,...,9}{\n    \\draw[xedge] (e\\x) to (xp);\n    }\n    \\draw[xedge] (xp) to (x);\n    \n    \\node[right =of e9](y)[xnodef]{};\n    \\draw[xedge] (z) to [out=0,in=-90](y);\n    \\draw[xedge] (y) to [out=90,in=0](x);\n    \n    \\newcommand{\\dlabel}[3]{\n      \\node at (#1)[label={[align=center,font=\\footnotesize,color=black]#2},\n      label={[align=center,font=\\scriptsize,color=black]-90:#3}]{};\n    }\n    \\newcommand{\\ltb}[3]{\n      \\contourlength{0.09em}\n      \\node at (#1)[label={[align=center,font=\\footnotesize,color=black]\\contour*{white}{#2}},label={[align=center,font=\\scriptsize,color=black]-90:\\contour*{white}{#3}}]{};\n    }\n\n\n    \n    \\ltb{xp}{$y'$}{}\n    \\ltb{x}{$y$}{$\\in V_1$}\n    \n    \\ltb{e3}{$e=\\{i,j\\}$}{$\\in V_1$};\n    \\ltb{e5}{$e'=\\{i,j'\\}$}{$\\in V_1$};\n    \\ltb{e7}{$e''=\\{i',j\\}$}{$\\in V_1$};\n\n    \\ltb{v3}{$v_i$}{};\n    \\ltb{v5}{$v_j$}{};\n    \n    \\ltb{z}{$x$}{$\\in V_1$};\n    \\dlabel{y}{180:$z$}{};\n    \n  \\end{tikzpicture}\n  \\caption{Illustration to~\\cref{constr:2cgbp} for~\\CgbpAcr[2]{}.}\n  \\label{fig:2cgbp}\n \\end{figure}\n\n To construct~$G'$ and~$V_1$,\n add the vertex set~$V_E\\ensuremath{\\coloneqq} \\{v_e\\mid e\\in E\\}$ and add~$V_E$ to~$V_1$.\n Add two designated vertices~$y'$ and~$y$,\n add~$y$ to~$V_1$,\n and make~$y'$ adjacent with~$y$ and all vertices in~$V_E$.\n Add a designated vertex~$x$,\n add~$x$ to~$V_1$,\n and introduce a path of length two from~$x$ to~$y$ (call the inner vertex~$z$).\n Add the vertex set~$V_G\\ensuremath{\\coloneqq} \\{v_i\\mid i\\in V\\}$,\n and make each~$v_i$ adjacent with~$x$ \n and all edge-vertices corresponding to an edge incident with~$i$,\n i.e.,\n add the edge set~$E_G\\ensuremath{\\coloneqq} \\bigcup_{i\\in V}\\{\\{v_i,v_e\\}\\mid i\\in e\\}$.\n\\end{construction}\n\n\\begin{observation}\n \\label{obs:2cgbp}\n Let~$\\mathcal{I}=(G,k)$ be an instance of \\prob{Vertex Cover} and let~$\\mathcal{I}'=(G',\\{V_1\\},k')$ be the instance obtained from~$\\mathcal{I}$ using~\\cref{constr:2cgbp}.\n If~$\\mathcal{I}'$ is a \\textnormal{\\texttt{yes}}-instance,\n then for every solution~$F\\subseteq E(G')$ \n it holds that~$\\{\\{y,y'\\},\\{y,z\\},\\{z,x\\}\\}\\cup \\{\\{y',v_e\\}\\mid e\\in E(G)\\}\\subseteq F$.\n\\end{observation}\n\n\\begin{lemma}\n \\label{lem:2cgbp:edinc}\n Let~$\\mathcal{I}=(G,k)$ be an instance of \\prob{Vertex Cover} and let~$\\mathcal{I}'=(G',\\{V_1\\},k')$ be the instance obtained from~$\\mathcal{I}$ using~\\cref{constr:2cgbp}.\n If~$\\mathcal{I}'$ is a \\textnormal{\\texttt{yes}}-instance,\n then there is a solution~$F\\subseteq E(G')$ such that~$|N_{G'[F]}(v_e)\\cap V_G|=1$ for all~$e\\in E(G)$.\n\\end{lemma}\n\n\\begin{proof}\n Note that in every solution,\n clearly we have~$|N_{G'[F]}(v_e)\\cap V_G|\\geq 1$.\n Suppose there is a minimal solution~$F$ such that there is at least one~$e=\\{i,j\\}\\in E$ such that~$|N_{G'[F]}(v_e)\\cap V_G|=2$.\n Let~$F$ be a solution with a minimum number of edges incident to vertices in~$V_E$.\n \n Since~$\\dist_{G'[F]}(v_e,x)= 2$,\n at least one of the edges~$\\{v_i,x_i\\}$ or~$\\{v_j,x_j\\}$ are in~$F$.\n If both are present\n then we can remove one of the edges~$\\{v_e,v_i\\}$ or~$\\{v_e,v_j\\}$ to~$e$ to obtain a solution of smaller size.\n This yields a contradiction.\n \n Otherwise,\n assume there is exactly one edge, say~$\\{v_e,v_i\\}$, \n contained in~$F$.\n Then exchanging~$\\{v_e,v_j\\}$ with~$\\{v_j,x\\}$ yields a solution with a lower number of edges incident to vertices in~$V_E$.\n A contradiction.\n\\end{proof}\n\n\n\\begin{lemma}\n \\label{lem:2cgbp:cor}\n Let~$\\mathcal{I}=(G,k)$ be an instance of \\prob{Vertex Cover} and let~$\\mathcal{I}'=(G',\\{V_1\\},k')$ be the instance obtained from~$\\mathcal{I}$ using~\\cref{constr:2cgbp}.\n Then~$\\mathcal{I}$ is a \\textnormal{\\texttt{yes}}-instance if and only if~$\\mathcal{I}'$ is a \\textnormal{\\texttt{yes}}-instance.\n\\end{lemma}\n\n\\begin{proof}\n $(\\Rightarrow)\\quad${}\n Let~$W\\subseteq V$ be a vertex cover of size at most~$k$ in~$G$.\n We construct a solution~$F\\subseteq E'$ as follows.\n Let~$F'$ denote the set of all edges required due to~\\cref{obs:2cgbp}.\n Let~$F_V\\ensuremath{\\coloneqq} \\{\\{v_i,x\\}\\mid i\\in W\\}$.\n We define the auxiliary function~$g\\colon E\\to V'$ with~$g(\\{i,j\\})=v_{\\min(\\{i,j\\}\\cap W)}$.\n Let~$F_E\\ensuremath{\\coloneqq} \\bigcup_{e=\\{i,j\\}\\in E} \\{v_e,g(e)\\}$.\n Let~$F\\ensuremath{\\coloneqq} F'\\cup F_V\\cup F_E$. \n Note that~$|F|=|F'|+|F_V|+|F_E|\\leq |E|+3+|E|+k = k'$.\n Moreover, \n every~$v_e\\in V'$ is connected to~$x$ via a path~$(v_e,v_i,z)$, \n for some~$i\\in (e\\cap W)$,\n of length two.\n Thus all vertex pairs in~$V_1$ are at distance at most two.\n \n $(\\Leftarrow)\\quad${}\n Let~$\\mathcal{I}'$ be a \\textnormal{\\texttt{yes}}-instance.\n Due to~\\cref{lem:2cgbp:edinc},\n there\n is a solution~$F\\subseteq E'$ such that~$\\deg_{G'[F]}(v_e)=1$ for all~$e\\in E$.\n Let~$W\\ensuremath{\\coloneqq} \\{i\\in V\\mid \\{v_i,x\\}\\in F\\}$.\n We claim that~$W$ is a vertex cover.\n Suppose not,\n that is,\n there is an edge~$e\\in E$ such that~$e\\cap W=\\emptyset$.\n That means that the unique neighbor of~$v_e$,\n say~$v_i$,\n is not adjacent with~$x$ in~$G'[F]$.\n Then,\n $v_e$ is not connected with~$x$ in~$G'[F]^2$,\n and hence~$F$ is no solution,\n a contradiction.\n\\end{proof}\n\n\\subsubsection{Graphs of constant maximum degree.}\n\n\\RgbpAcr[2]{} is \\cocl{NP}-hard if the number~$r$ of habitats and the maximum degree~$\\Delta$ are constant \n(\\cref{prop:2rgbp}).\n\\CgbpAcr[2]{} is linear-time solvable in this~case:\n\n\\begin{proposition}\n \\label{prop:cgbpdelta}\n \\CgbpAcr{} admits an~$\\O(r\\Delta(\\Delta-1)^{3d\/2})$-sized problem kernel computable in~$\\O(r(n+m))$ time.\n\\end{proposition}\n\n\\begin{proof}\n\tLet~$\\mathcal{I} = (G, \\mathcal{H}, k)$ be an instance of \\CgbpAcr{}.\n\tFor every~$i\\in\\set{r}$, fix a vertex~$u_i\\in V_i$.\n\tWe assume that we have $V_i \\subseteq N_G^d[u_i]$ for all~$i\\in\\set{r}$, otherwise~$\\mathcal{I}$ is a \\textnormal{\\texttt{no}}-instance.\n\tNow let~$W_i = N_G^{\\lceil 3d\/2\\rceil}[u_i]$ and let~$G' \\ensuremath{\\coloneqq} G[\\bigcup_{i=1}^r W_i]$.\n\tNote that~$G'$ contains at most~$r\\Delta(\\Delta - 1)^{\\lceil 3d\/2\\rceil}$ vertices and can be computed by~$r$ breadth-first searches.\n\tWe claim that~$G'$ contains every path of length at most~$d$ between every two vertices~$v, w\\in V_i$, for every~$i \\in \\set{r}$.\n\tRecall that an edge set~$F \\subseteq E$ is a solution if and only if for every~$i\\in\\set{r}$ and for every~$v, w\\in V_i$,\n\tthe graph $G[F]$ contains a path of length at most~$d$ from~$v$ to~$w$.\n\tAs by our claim~$G'$ contains any such path,\n\tthis implies that~$\\mathcal{I}$ is a \\textnormal{\\texttt{yes}}-instance if and only if~$\\mathcal{I}'\\ensuremath{\\coloneqq}(G',\\mathcal{H},k)$ is a \\textnormal{\\texttt{yes}}-instance (note that~$V_i \\subseteq V(G')$ for every~$i \\in \\set{r}$).\n\n\tAssuming that~$V_i \\subseteq N_G^d[u_i]$,\n\t$G[W_i]$ contains all paths of length at most~$d$ between~$u_i$ and any~$v\\in V_i$.\n\tSo let~$v, w \\in V_i$ be two vertices, both distinct from~$u_i$.\n\tAs $v, w \\in N^d_G[u_i]$ and~$W_i=N^{\\lceil 3d\/2\\rceil}_G[u_i]$,\n\tthe subgraph $G[W_i]$ contains all vertices in~$N^{\\lceil d\/2\\rceil}_G[v]$ and~$N^{\\lceil d\/2\\rceil}_G[w]$.\n\tConsider now a path of length at most~$d$ between~$v$ and~$w$.\n\tSuppose it contains a vertex~$x \\in V(G) \\setminus (N^{\\lceil d\/2\\rceil}_G[v]\\cup N^{\\lceil d\/2\\rceil}_G[w])$.\n\tThen~$\\dist_G(v, x) + \\dist_G(w, x) > 2 \\lceil d\/2 \\rceil \\ge d$, a contradiction to~$x$ being on a path from~$v$ to~$w$ of length at most~$d$.\n\tThe claim follows.\n\\end{proof}\n\n\\subsubsection{Parameterizing with~\\texorpdfstring{\\boldmath{$k$}}{k}.}\n\nAll the reduction rules that worked for~\\RgbpAcr[2]{}\nalso work for~\\CgbpAcr[2]{}.\nIt thus follows that~\\CgbpAcr[2]{} admits a problem kernel of size exponentially in~$k$,\nand hence,\n\\CgbpAcr[2]{} is~\\cocl{FPT}.\nAs with~\\RgbpAcr[2]{},\nthe problem kernel presumably cannot be much improved.\nFor this, we combine the constructions for~\\cref{prop:rgbp:nopk,prop:2cgbp}\n(see \\cref{fig:2rgbp:nopk,fig:2cgbp}).\n\n\\begin{corollary}\n \\CgbpAcr[2]{} admits a problem kernel of size exponentially in~$k$\n and,\n unless~\\ensuremath{\\NP\\subseteq \\coNP\/\\poly},\n none of size polynomial in~$k$,\n even if~$r=1$.\n\\end{corollary}\n\n\\subsection{When reaching each part is a voyage (\\texorpdfstring{\\boldmath{$d=3$}}{d\u2a7e3})}\n\\label{ssec:3cgbp}\n\nFor~$d\\geq 3$,\nthe problem\nis\n\\W{1}-hard regarding the number~$k$ of green bridges,\neven for one habitat.\nThe reduction is similar to the one for \\cref{prop:3rgbp}.\n\n\\begin{proposition}\n \\label{prop:3cgbp}\n \\CgbpAcr{} with~$d\\geq 3$ is \\cocl{NP}-complete and~\\W{1}-hard when parameterized by the number~$k$,\n even if $r=1$.\n\\end{proposition}\n\n\\begin{proof}\n Let~$\\mathcal{I}=(G)$ with~$G=(U^1,\\dots,U^k,E)$ be an instance of \\prob{Multicolored Clique}.\n Apply\n \\cref{constr:3rgbp}\n to obtain instance~$\\mathcal{I}''=(G',\\{V_1,\\dots,V_{\\binom{k}{2}}\\},k')$\n (recall that~$k'=\\frac{d-1}{2}k+\\binom{k}{2}$).\n Let~$\\mathcal{I}'=(G',\\{V_1'\\},k')$ with~$V_1'\\ensuremath{\\coloneqq} \\bigcup_{i=1}^{\\binom{k}{2}} V_i=\\{v_1,\\dots,v_k\\}$ be the finally obtained instance of~\\CgbpAcr{}.\n We claim that~$\\mathcal{I}$ is a \\textnormal{\\texttt{yes}}-instance if and only if~$\\mathcal{I}'$ is a \\textnormal{\\texttt{yes}}-instance.\n \n $(\\Rightarrow)\\quad${}\n Let~$C$ be a multicolored clique in~$G$.\n Let~$z_i\\ensuremath{\\coloneqq} V(C)\\cap U^i$.\n We claim that~$F$,\n consisting of the edges of each shortest path from~$v_i$ to~$z_i$ and the edge set~$E(C)$,\n is a solution to~$\\mathcal{I}'$.\n Note that~$|F|=k'$.\n Moreover,\n for any two~$i,j\\in\\set{k}$,\n we have that~$v_i$ and~$v_j$ are of distance~$2\\frac{d-1}{2}+1=d$.\n Hence,\n $F$ is a solution.\n \n $(\\Leftarrow)\\quad${}\n Let~$F$ be a solution to~$\\mathcal{I}$.\n Since~$F$ must contain a path from~$v_i$ to some~$z_i\\in U^i$\n for every~$i\\in\\set{k}$,\n there are at most~$\\binom{k}{2}$ edges left to connect.\n Let~$Z\\ensuremath{\\coloneqq} \\{z_1,\\dots,z_k\\}$ be the vertices such that~$v_i$ is connected with~$z_i$ in~$G[F][U^i]$.\n Since~$d\\geq \\dist_{G'[F]}(v_i,v_j) = \\dist_{G'[F]}(v_i,z_i)+\\dist_{G'[F]}(z_i,z_j)+\\dist_{G'[F]}(z_j,v_j)$ and~$d-1=\\dist_{G'[F]}(v_i,z_i)+\\dist_{G'[F]}(z_j,v_j)$,\n it follows that~$\\dist_{G'[F]}(z_i,z_j)=1$.\n Thus,\n $G[Z]$ forms a mulicolored clique.\n\\end{proof}\n\n\\section{Connecting Habitats at Small Diameter}\n\\label{sec:dgbp}\n\nLastly,\nwe consider requiring short pairwise reachability\nin \\RgbpAcr[1]{}.\n\n\\decprob{\\DgbpTsc{} (\\DgbpAcr{})}{dgbp}\n{An undirected graph~$G=(V,E)$,\na set~$\\mathcal{H}=\\{V_1,\\dots,V_r\\}$ of habitats where~$V_i\\subseteq V$ for all~$i\\in\\set{r}$, \nand an integer~$k\\in\\mathbb{N}_0$.}\n{Is there a subset~$F\\subseteq E$ with~$|F|\\leq k$ such that\nfor every~$i\\in\\set{r}$\nit holds that~$G[F][V_i]$ has diameter~$d$?\n}\n\n\\noindent\nIn particular,\n$G[F][V_i]$ is connected.\nNote that\n\\RgbpAcr[1]{} reduces to~\\prob{Diam \\prob{GBP}} \n(where $d$ is part of the input and then set to the number of vertices in the input instance's graph).\nWe have the following.\n\n\\begin{theorem}\n \\label{thm:dgbp}\n \\DgbpAcr{} is\n \\begin{compactenum}[(i)]\n  \\item if~$d=1$, solvable in linear time;\\label{thm:dgbp:i}\n  \\item if~$d=2$, \\cocl{NP}-hard even if~$r=3$;\n  \\item if~$d=3$, \\cocl{NP}-hard even if~$r=2$.\n \\end{compactenum}\n Moreover,\n \\DgbpAcr{} admits a problem kernel with at most~$2k$ vertices and at most~$2^{2k}$ habitats.\n\\end{theorem}\n\n\\noindent\n\\DgbpAcr[1]{} is equivalent to \\CgbpAcr[1]{},\nwhich is linear-time solvable by \\cref{obs:1cgbp}.\nThus, \n\\cref{thm:dgbp}\\eqref{thm:dgbp:i} follows.\nApplying \\cref{rr:few-habitat-vertices,rr:habitats} and deleting all non-habitat vertices yields the problem kernel.\n\n\\subsection{Over at most one patch to every other (\\texorpdfstring{\\boldmath{$d=2$}}{d=2})}\n\\label{ssec:2dgbp}\n\n\\DgbpAcr[2]{} turns out to be \\cocl{NP}-hard even for three habitats.\n\n\n\n\\begin{proposition}\n \\label{prop:2dgbp}\n \\DgbpAcr[2]{} is \\cocl{NP}-hard even if~$r=3$.\n\\end{proposition}\n\n\\begin{construction}\n  \\label{constr:2dgbp}\n  Let~$\\mathcal{I}=(G,k)$ with~$G=(V,E)$ be an instance of~\\prob{Vertex Cover}\n  and assume without loss of generality that~$V=\\{1,\\dots,n\\}$ and~$E=\\{e_1,\\dots,e_m\\}$.\n  Construct an instance~$\\mathcal{I}'\\ensuremath{\\coloneqq} (G',\\{V_1,V_2,V_3\\},k')$\n  with~$k'=2m+2n+k+4$ as follows\n  (see~\\cref{fig:2dgbp} for an illustration).\n\\begin{figure}[t]\n  \\centering\n  \\begin{tikzpicture}\n\n    \\def1.1{1.1}\n    \\def0.9{0.9}\n    \\def1.5*\\yr{1.5*0.9}\n    \\def0.125{0.125}\n    \\tikzstyle{xnode}=[circle,fill,scale=0.6,draw,color=cyan!50!green]\n    \\tikzstyle{xnodef}=[circle,scale=0.6,draw,color=cyan!50!green,fill=white]\n    \\tikzstyle{xnodex}=[label={[xshift=-0.4*1.1 cm]0:$\\cdots$},color=cyan!50!green]\n    \\tikzstyle{xedge}=[thick,-,color=cyan!50!green]\n\n    \\foreach \\x\/\\y in {1\/xnode,2\/xnodex,3\/xnode,4\/xnodex,5\/xnode,6\/xnodex,7\/xnode,8\/xnodex,9\/xnode}{\n    \\node (e\\x) at (\\x*1.1-5*1.1,0)[\\y]{};\n    }\n    \\draw[dotted,rounded corners,draw] ($(e1.south west)-(0.125,0.125)$) rectangle ($(e9.north east)+(0.125,0.125)$);\n    \\foreach \\x\/\\y in {1\/xnodef,2\/xnodex,3\/xnodef,4\/xnodex,5\/xnodef,6\/xnodex,7\/xnodef}{\n    \\node (v\\x) at (\\x*1.1-4*1.1,-1*1.5*\\yr)[\\y]{};\n    }\n    \\draw[dotted,rounded corners,draw] ($(v1.south west)-(0.125,0.125)$) rectangle ($(v7.north east)+(0.125,0.125)$);\n    \\node (x) at (0*1.1,-2*1.5*\\yr)[xnodef]{};\n    \\node (z) at (5*1.1,-0.5*1.5*\\yr)[xnodef]{};\n    \\node (zp) at (6*1.1,-0.5*1.5*\\yr)[xnodef]{};\n    \\node (y) at (5*1.1,-1.5*1.5*\\yr)[xnodef]{};\n    \\node (yp) at (6*1.1,-1.5*1.5*\\yr)[xnodef]{};\n    \n\n    \\foreach \\x\/\\y in {3\/3,3\/5,5\/3,5\/2,7\/5,7\/6}{\n      \\draw[xedge] (e\\x) to (v\\y);\n    }\n    \\foreach \\x in {1,...,9}{\n        \\draw[xedge] (e\\x) to [out=-15,in=175](z);\n    }\n    \\foreach \\x\/\\y in {2\/1,8\/7}{\n      \\draw[draw=none] (e\\x) to node[midway]{$\\vdots$}(v\\y);\n    }\n    \\foreach \\x in {1,...,7}{\n        \\draw[xedge] (v\\x) to [out=15,in=185](z);\n        \\draw[xedge] (v\\x) to [out=-15,in=180](y);\n    }\n    \\foreach \\x in {1,...,7}{\n      \\draw[xedge] (x) to (v\\x);\n    }\n    \\draw[xedge] (x) to (y);\n    \\draw[xedge] (y) to (z);\n    \\draw[xedge] (y) to (yp);\n    \\draw[xedge] (z) to (zp);\n\n    \\newcommand{\\ltb}[3]{\n      \\contourlength{0.09em}\n      \\node at (#1)[label={[align=center,font=\\footnotesize,color=black]\\contour*{white}{#2}},label={[align=center,font=\\scriptsize,color=black]-90:\\contour*{white}{#3}}]{};\n    }\n    \n    \\ltb{e3}{$e=\\{i,j\\}$}{$\\in V_1,V_2$};\n    \\ltb{e5}{$e'=\\{i,j'\\}$}{$\\in V_1,V_2$};\n    \\ltb{e7}{$e''=\\{i',j\\}$}{$\\in V_1,V_2$};\n    \n    \\ltb{x}{$x$}{$\\in V_1, V_3$};\n    \\ltb{y}{$y$}{$\\in V_1,V_3$};\n    \\ltb{z}{$z$}{$\\in V_1,V_2,V_3$};\n    \\ltb{yp}{$y'$}{$\\in V_3$};\n    \\ltb{zp}{$z'$}{$\\in V_2$};\n  \n    \\ltb{v3}{$v_i$}{$\\in V_1,V_2,V_3$};\n    \\ltb{v5}{$v_j$}{$\\in V_1,V_2,V_3$};\n  \\end{tikzpicture}\n  \\caption{Illustration for~\\DgbpAcr[2]{} with~$r=3$. ($k'=2m+2n+k+4$)}\n  \\label{fig:2dgbp}\n\\end{figure}\n  Add the vertex sets~$V_E\\ensuremath{\\coloneqq} \\{v_e\\mid e\\in E\\}$ and~$V_G=\\{v_i\\mid i\\in V\\}$,\n  as well as the vertices~$x,y,y',z,z'$.\n  Add all vertices  except for~$y'$ and~$z'$ to~$V_1$.\n  Let~$V_2\\ensuremath{\\coloneqq} V_E\\cup V_G\\cup \\{z,z'\\}$\n  and\n  $V_3\\ensuremath{\\coloneqq} V_G\\cup \\{x,z,y,y'\\}$.\n  Next,\n  for each~$e=\\{i,j\\}\\in E$,\n  connect~$v_e$ with~$v_i$, \n  $v_j$, \n  and~$z$.\n  For each~$i\\in V$,\n  connect~$v_i$ with~$x$, \n  $y$, \n  and~$z$.\n  Lastly, \n  add the edge set~$E^*\\ensuremath{\\coloneqq}\\{\\{x,y\\},\\{y,y'\\},\\{z,z'\\},\\{z,y\\}\\}$ to~$E'$.\n  Let~$E_y \\ensuremath{\\coloneqq} \\{\\{y,v_i\\} \\mid i \\in V\\}$,\n  $E_{E}\\ensuremath{\\coloneqq}\\{\\{v_e,z\\} \\mid e \\in E\\}$,\n  and~$E_{V}\\ensuremath{\\coloneqq}\\{\\{v_i,z\\}\\mid i \\in V\\}$.\n\\end{construction}\n\n\\begin{observation}\n \\label{obs:2dgbp}\n Let~$\\mathcal{I}'$ be the instance obtained from some instance~$\\mathcal{I}$ using~\\cref{constr:2dgbp}.\n If~$\\mathcal{I}'$ is a \\textnormal{\\texttt{yes}}-instance,\n then every solution~$F$ for~$\\mathcal{I}'$ contains\n (i) $\\{\\{v,z\\}\\mid v\\in N_{G'}(z)\\}$ and \n (ii) $\\{\\{v,y\\}\\mid v\\in N_{G'}(y)\\}$.\n\\end{observation}\n\n\\begin{proof}\n Clearly,\n $\\{z,z'\\},\\{y,y'\\}\\in F$.\n Every path of length at most two from~$z'$ to any vertex in~$V_E\\cup V_G$ passes~$z$.\n Hence,\n each edge in~$\\{\\{v,z\\}\\mid v\\in V_G\\cup V_E\\}$ is contained in~$F$.\n This proves~(i).\n The proof for~(ii) is analogous.\n\\end{proof}\n\n\\noindent\nNote that~$|\\{\\{v,z\\}\\mid v\\in N_{G'}(z)\\}\\cup\\{\\{v,y\\}\\mid v\\in N_{G'}(y)\\}|=m+2n+4$.\n\n\\begin{lemma}\n \\label{lem:2dgbp}\n Let~$\\mathcal{I}'$ be the instance obtained from some instance~$\\mathcal{I}$ using~\\cref{constr:2dgbp}.\n Then,\n $\\mathcal{I}$ is a \\textnormal{\\texttt{yes}}-instance if and only if~$\\mathcal{I}'$ is a \\textnormal{\\texttt{yes}}-instance.\n\\end{lemma}\n\n\\begin{proof}\n $(\\Rightarrow)\\quad${}\n Let~$S\\subseteq V$ be a vertex cover of size~$k$.\n Let~$F'$ denote the set of all edges required to be in a solution due to \\cref{obs:2dgbp}.\n Let~$F_V\\ensuremath{\\coloneqq} \\{\\{v_i,x\\}\\mid i\\in S\\}$.\n We define the auxiliary function~$g\\colon E\\to V'$ with~$g(\\{i,j\\})=v_{\\min(\\{i,j\\}\\cap S)}$.\n Let~$F_E\\ensuremath{\\coloneqq} \\bigcup_{e\\in E} \\{\\{v_e,g(e)\\}\\}$.\n Let~$F\\ensuremath{\\coloneqq} F'\\cup F_V\\cup F_E$. \n Note that~$|F|=|F'|+|F_V|+|F_E|\\leq (m+2n+4)+k+m = k'$.\n Observe that~$G'[F][V_2]$ and~$G'[F][V_3]$ are connected and of diameter two.\n Next consider~$G'[F][V_1]$.\n We claim that for all~$e \\in E$, $\\dist_{G'[F][V_1]}(x,v_e)=2$.\n By construction,\n $\\dist_{G'[F][V_1]}(x,v_e)>1$.\n Suppose that there is~$v_e$ with~$e=\\{i,j\\}$ and~$\\dist_{G'[F][V_1]}(x,v_e)>2$.\n Then there is no path~$(x,v,v_e)$ with~$v\\in\\{v_i,v_j\\}$.\n Then~$\\{i,j\\}\\cap S=\\emptyset$,\n contradicting the fact that~$S$ is a vertex cover.\n \n $(\\Leftarrow)\\quad${} \n Let~$F$ be a solution to~$\\mathcal{I}'$.\n Let~$F'$ be the set of edges mentioned in \\cref{obs:2dgbp};\n so~$F' \\subseteq F$.\n Observe that in~$G'-V_G$,\n the distance of~$x$ to any~$v_e\\in V_E$ is larger than two.\n Hence,\n for each~$v_e$,\n there is a path~$(v_e,v,x)$ in~$G'[F][V_1]$ with~$v\\in V_G$.\n We claim that~$S\\ensuremath{\\coloneqq} \\{i\\in V\\mid \\{v_i,x\\}\\in F\\}$ is a vertex cover for~$G$ of size at most~$k$.\n Suppose not,\n that is,\n there is an edge~$e=\\{i,j\\}$ with~$e\\cap S=\\emptyset$.\n This contradict the fact that there is a path~$(v_e,v,x)$ in~$G'[F][V_1]$ with~$v\\in V_G$.\n It remains to show that~$|S|\\leq k$.\n As~$F$ contains an edge~$\\{v_e,v\\}$ with~$v \\in V_G$ for every~$e \\in E$,\n $|S| = |F\\cap \\{\\{v_i,x\\} \\mid i\\in V\\}| \\le k'-(|F'|+m) = k$,\n and the claim follows.\n\\end{proof}\n\n\\subsection{Over at most two patches to every other (\\texorpdfstring{\\boldmath{$d\\ge 3$}}{d\u2a7e3})}\n\\label{ssec:3dgbp}\n\n\\DgbpAcr[3]{} turns out to be \\cocl{NP}-hard even for two habitats.\n\n\\begin{proposition}\n \\label{prop:3dgbp}\n \\DgbpAcr[3]{} is \\cocl{NP}-hard even if~$r=2$.\n\\end{proposition}\n\n\\begin{construction}\n  \\label{constr:3dgbp}\n  Let~$\\mathcal{I}=(G,k)$ with~$G=(V,E)$ be an instance of~\\prob{Vertex Cover}.\n  Construct an instance~$\\mathcal{I}'\\ensuremath{\\coloneqq} (G',\\{V_1,V_2\\},k')$\n  with~$k'=2m+n+k+4$ as follows\n  (see~\\cref{fig:3dgbp} for an illustration).\n  \\begin{figure}[t]\n    \\centering\n    \\begin{tikzpicture}\n\n      \\def1.1{1.1}\n      \\def0.9{0.9}\n      \\def1.5*\\yr{1.5*0.9}\n      \\def0.125{0.125}\n      \\tikzstyle{xnode}=[circle,fill,scale=0.6,draw,color=cyan!50!green]\n      \\tikzstyle{xnodef}=[circle,scale=0.6,draw,color=cyan!50!green,fill=white]\n      \\tikzstyle{xnodex}=[label={[xshift=-0.4*1.1 cm]0:$\\cdots$},color=cyan!50!green]\n      \\tikzstyle{xedge}=[thick,-,color=cyan!50!green]\n\n      \\foreach \\x\/\\y in {1\/xnode,2\/xnodex,3\/xnode,4\/xnodex,5\/xnode,6\/xnodex,7\/xnode,8\/xnodex,9\/xnode}{\n      \\node (e\\x) at (\\x*1.1-5*1.1,0)[\\y]{};\n      }\n      \\draw[dotted,rounded corners,draw] ($(e1.south west)-(0.125,0.125)$) rectangle ($(e9.north east)+(0.125,0.125)$);\n      \\foreach \\x\/\\y in {1\/xnodef,2\/xnodex,3\/xnodef,4\/xnodex,5\/xnodef,6\/xnodex,7\/xnodef}{\n      \\node (v\\x) at (\\x*1.1-4*1.1,-1*1.5*\\yr)[\\y]{};\n      }\n      \\draw[dotted,rounded corners,draw] ($(v1.south west)-(0.125,0.125)$) rectangle ($(v7.north east)+(0.125,0.125)$);\n      \n      \\node (x) at (0*1.1,-2*1.5*\\yr)[xnodef]{};\n      \\node (xp) at (6*1.1,-2*1.5*\\yr)[xnodef]{};\n\n      \\node (y) at (5*1.1,-1.5*1.5*\\yr)[xnodef]{};\n      \\node (yp) at (6*1.1,-1*1.5*\\yr)[xnodef]{};\n      \\node (z) at (5*1.1,-0.5*1.5*\\yr)[xnodef]{};\n\n      \\foreach \\x\/\\y in {3\/3,3\/5,5\/3,5\/2,7\/5,7\/6}{\n        \\draw[xedge] (e\\x) to (v\\y);\n      }\n      \\foreach \\x in {1,...,9}{\n          \\draw[xedge] (e\\x) to [out=-15,in=180](z);\n      }\n      \\draw[xedge] (z) to (yp);\n      \\foreach \\x\/\\y in {2\/1,8\/7}{\n        \\draw[draw=none] (e\\x) to node[midway]{$\\vdots$}(v\\y);\n      }\n      \\foreach \\x in {1,...,7}{\n        \\draw[xedge] (v\\x) to [out=-15,in=180](y);\n      }\n      \\draw[xedge] (yp) to (y);\n      \\foreach \\x in {1,...,7}{\n        \\draw[xedge] (x) to (v\\x);\n      }\n      \\draw[xedge] (x) \n      to (xp) to (yp);\n      \n      \\contourlength{0.09em}\n      \\newcommand{\\ltb}[3]{\n        \\node at (#1)[label={[align=center,font=\\footnotesize,color=black]\\contour*{white}{#2}},label={[align=center,font=\\scriptsize,color=black]-90:\\contour*{white}{#3}}]{};\n      }\n      \n      \\ltb{e3}{$e=\\{i,j\\}$}{$\\in V_1$};\n      \\ltb{e5}{$e'=\\{i,j'\\}$}{$\\in V_1$};\n      \\ltb{e7}{$e''=\\{i',j\\}$}{$\\in V_1$};\n      \n      \\ltb{x}{$x$}{$\\in V_1$};\n      \\ltb{xp}{$x'$}{$\\in V_1$};\n      \\ltb{z}{$z$}{$\\in V_1$};\n      \\ltb{y}{$y$}{$\\in V_1,V_2$};\n      \\ltb{yp}{$y'$}{$\\in V_1,V_2$};\n    \n      \\ltb{v3}{$v_i$}{$\\in V_1,V_2$};\n      \\ltb{v5}{$v_j$}{$\\in V_1,V_2$};\n    \\end{tikzpicture}\n    \\caption{Illustration for~\\DgbpAcr[3]{} with~$r=2$. ($k'=2m+n+k+4$)}\n    \\label{fig:3dgbp}\n  \\end{figure}\n  Add the vertex sets~$V_E\\ensuremath{\\coloneqq} \\{v_e\\mid e\\in E\\}$,\n  $V_G\\ensuremath{\\coloneqq}\\{v_i\\mid i\\in V\\}$,\n  and~$V^*\\ensuremath{\\coloneqq} \\{x,x',y,y',z\\}$.\n  Add all vertices to~$V_1$,\n  and all vertices in~$V_G$ as well as~$y$ and~$y'$ to~$V_2$.\n  Next,\n  for each~$e=\\{i,j\\}\\in E$,\n  connect~$v_e$ with~$v_i$, $v_j$, and~$z$.\n  Denote by~$E_z$ the set of edges incident with~$z$.\n  For each~$i\\in V$,\n  connect~$v_i$ with~$x$ and~$y$.\n  Denote by~$E_y$ the set of edges incident with~$y$.\n  Finally,\n  add the edge set~$E^*\\ensuremath{\\coloneqq}\\{\\{x,x'\\},\\{y,y'\\},\\{z,y'\\},\\{y',x'\\}\\}$.\n\\end{construction}\n\n\\begin{observation}\n \\label{obs:3dgbp-aux}\n Let~$\\mathcal{I}'$ be the instance obtained from some instance~$\\mathcal{I}$ using~\\cref{constr:3dgbp}.\n Then every solution~$F$ for~$\\mathcal{I}'$ (if there is one) contains the set of edges~$F'\\ensuremath{\\coloneqq} E_y \\cup E_z \\cup E^*$.\n Further, $\\dist_{G[F']}(u, v) \\le 3$ for every~$u, v \\in V^*$ and for every~$u \\in V_G \\cup V_E$ and~$v \\in \\{y, y', z, x'\\}$.\n\\end{observation}\n\n\\begin{proof}\n Let~$\\mathcal{I}'$ be a \\textnormal{\\texttt{yes}}-instance and\n let~$F$ be a solution.\n Since~$G[V_2]$ is a tree and~$E_y = E(G[V_2])$,\n we have~$E_y \\subseteq F$.\n If for some~$e \\in E$, $\\{v_e, z\\} \\notin F$, \n then~$\\dist_{G[F]}(v_e, z) > 3$.\n If~$\\{z,y'\\}\\not\\in F$, \n then~$\\dist_{G[F]}(z, y') > 3$.\n Thus,\n $E_z \\subseteq F$.\n If~$\\{y',x'\\}\\not\\in F$, \n then~$\\dist_{G[F]}(z, x') > 3$.\n If~$\\{x,x'\\}\\not\\in F$, \n then~$\\dist_{G[F]}(x, x') > 3$.\n Thus,\n $E^* \\subseteq F$.\n The distances claimed for~$G[F']$ are immediate.\n\\end{proof}\n\n\\begin{lemma}\n Let~$\\mathcal{I}'$ be the instance obtained from an instance~$\\mathcal{I}$ using \\cref{constr:3dgbp}.\n Then~$\\mathcal{I}$ is a \\textnormal{\\texttt{yes}}-instance if and only if~$\\mathcal{I}'$ is a \\textnormal{\\texttt{yes}}-instance.\n\\end{lemma}\n\\begin{proof}\n $(\\Rightarrow)\\quad${}\n Let~$S \\subseteq V$ be a vertex cover of~$G$ of size~$k$.\n We construct a solution~$F \\subseteq E'$ as follows.\n Let~$g \\colon E \\to V'$ be an auxiliary function with~$g(\\{i, j\\}) = v_{\\min(\\{i, j\\}\\cap S)}$.\n Let~$F_V\\ensuremath{\\coloneqq}\\{\\{v_i, x\\} \\mid i \\in S\\}$\n and let~$F_E\\ensuremath{\\coloneqq}\\{\\{v_e, g(e)\\} \\mid e \\in E\\}$.\n Then we set~$F\\ensuremath{\\coloneqq} F' \\cup F_V \\cup F_E$, \n where~$F'$ is as defined in \\cref{obs:3dgbp-aux}.\n Note that~$|F| = n + m + 4 + k + m = k'$. \n Since~$S$ is a vertex cover, \n every vertex~$v_e \\in V_E$ is adjacent to a vertex~$v_i$, $i \\in S$, \n which in turn is adjacent to~$x$ in~$G[F]$ due to~$F_V$.\n Further, as every~$v_i$, $i \\in V \\setminus S$, \n has distance two to every vertex~$v_j$, $j \\in S$, \n we have~$\\dist_{G[F]}(v_i, v_e) = \\dist_{G[F]}(v_i, x) \\le 3$ for every~$v_e \\in V_E$.\n Hence, $F$ is a solution for~$\\mathcal{I}'$.\n \n $(\\Leftarrow)\\quad${}\n Let~$F \\subseteq E'$ be a solution.\n Observe that in~$G$ the only path of length at most three from a vertex in~$V_E$ to~$x$ is via a vertex in~$V_G$ (and thus of length exactly two).\n Hence, in~$G[F]$, for every~$v_e \\in V_E$ there exists a~$v_i \\in V_G$ such that~$\\{v_e, v_i\\}, \\{v_i, x\\} \\in F$.\n It follows that\n $G[F]$ contains at least~$m$ edges between~$V_E$ and~$V_I$ and hence at most~$k$ edges between~$V_I$ and~$x$.\n We claim that~$S\\ensuremath{\\coloneqq}\\{i \\in V \\mid \\{v_i, x\\} \\in G[F]\\}$ is a vertex cover in~$G$.\n First note that~$|S|\\leq |F|-(|F'|+m)=k$,\n where~$F' \\subseteq F$ is the set of edges from~\\cref{obs:3dgbp-aux}.\n Suppose not,\n that is,\n $S$ is no vertex cover and hence there is an edge~$e\\in E$ with~$e\\cap S=\\emptyset$.\n This contradicts the fact that~$\\dist_{G[F][V_1]}(v_e,x)\\leq 3$.\n\\end{proof}\n\n\\section{Conclusion, Discussion, and Outlook}\n\nWe modeled the problem of placing wildlife crossings\nwith three different problem (families):\n\\RgbpAcr{},\n\\CgbpAcr{},\nand~\\DgbpAcr{}.\nWe studied the practically desired cases~$d=1$ and $d=2$, \nas well as the cases~$d\\ge3$.\nFor all three problems,\nwe settled the classic as well as the parameterized complexity \n(regarding the number~$k$ of wildlife crossings and the number~$r$ of habitats),\nexcept for the parameterized complexity of \\RgbpAcr{} regarding~$r$.\n\n\\paragraph{Discussion.}\nWe derived an intriguing interrelation of\nconnection requirements,\ndata quality, \nand computational and parameterized complexity.\nWhile each problem admits its individual complexity fingerprint,\neach of them depends highly on the value of~$d$,\nthe level of the respective connectivity constraint.\nThis value can reflect the quality of the given data,\nsince naturally we assume that habitats are connected.\nThe worse the data, \nthe stronger are the relaxations according to the connectivity of habitats,\nand thus the larger is the value of~$d$.\nOur results show that having very small ($d=2$) data gaps already leads to the problems becoming \\cocl{NP}-hard,\nand that even larger gaps ($d\\geq 3$) yield \\W{1}-hardness (when parameterized by~$k$).\nHence, knowledge about habitats, connections, and data quality\ndecide which problem models can be applied,\nthus influencing the computation power required to determine an optimal placement of wildlife crossings.\nFor instance,\nfor larger networks,\nwe recommend to ensure data quality such that one of our proposed problems for~$d\\leq 2$ becomes applicable.\nThis\nin turn\nemphasizes the importance of careful habitat recognition.\n\nIn our models,\nwe neglected that different positions possibly lead to \\emph{different} costs of building bridges \n(i.e.,\\ edge~costs).\nThis neglect is justified when differentiating between types of bridges (and thus their costs)\nis not necessary \n(e.g., if the habitat's species share preferred types of green bridges,\nand the underlying human-made transportation lines are homogeneous).\nIn other scenarios,\nadditionally considering these costs may be beneficial for decision-making.\n\n\n\\paragraph{Outlook and open problems.}\nFor a final version,\nwe plan to continue our study with \napproximation and (refined) data reduction for our three problems,\nas well as planar input graphs,\nand to settle \\RgbpAcr[1]{}'s complexity regarding~$r$. \nNote that we obtained an~$\\O(rd)$-approximation for~\\RgbpAcr{},\nwhich does not directly transfer to the other two problems.\nFPT approximations may be lucrative.\nFor small~$d\\geq 2$,\nall problems allow for problems kernels where the number of vertices only depends on~$k$.\nIf more effective preprocessing is possible,\nthen data reduction on the habitats is required.\nIf the underlying street network is planar,\nthen the input graphs to our problems can be seen as their planar dual.\nThus, \ninput graphs may be planar in the applications.\n\nMoreover,\ninteresting directions for future work are,\nfor instance,\ndistinguishing types of green bridges to place,\ntaking into account possible movement directions within habitats (connectivity in directed graphs),\nidentifying real-world driven problem parameters leading to tractability,\nor the problem of maintaining and servicing green bridges over time under a possible seasonal change of wildlife habitats \n(temporal graph modeling could fit well).\n\n\\bigskip \n\n{\n  \\let\\clearpage\\relax\n  \\renewcommand{\\url}[1]{\\href{#1}{$\\ExternalLink$}}\n  ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe study of special knots in contact three manifolds provided great insight\ninto the geometry and topology of three manifolds.  In particular, the study\nof Legendrian knots (ones tangent to the contact planes) has been useful\nin distinguishing homotopic contact structures on $T^3$  \\cite{k} and\nhomology spheres \\cite{am}.  Moreover, Rudolph \\cite{r} has shown\nthat invariants of Legendrian knots can be useful in understanding\nslicing properties of knots.  The first example of the use of knot \ntheory in contact topology was in the work of Bennequin. In \\cite{be}\nBennequin used transversal knots (ones transversal to the contact\nplanes) to show that $\\hbox{$\\mathbb R$} ^3$ has exotic contact structures.  This was the\ngenesis of Eliashberg's insightful tight versus overtwisted dichotomy in three dimensional\ncontact geometry.\n\nIn addition to its importance in the understanding of contact geometry, the study\nof transversal and Legendrian knots is quite interesting in its own right.\nQuestions concerning transversal and Legendrian knots have most prominently appeared\nin \\cite{el:knots} and Kirby's problem list \\cite{kirby}.\nCurrently there are very few general theorems concerning the classification\nof these knots. In \\cite{el:knots}, Eliashberg classified transversal unknots\nin terms of their self-linking number.  In \\cite{ef}, Legendrian unknots\nwere similarly classified.  In this paper we will extend this classification \nto positive transversal torus knots\\footnote{By ``positive transversal torus knot'' we mean a positive (right handed) \n\ttorus\n\tknot that is transversal to a contact structure.}.  In particular we prove:\n\\begin{ttm}\n\tPositive transversal \n\ttorus knots\n\tare transversely isotopic if and only if they have the same topological knot\n\ttype and the same self-linking number.\n\\end{ttm}\nIn the process of proving this result we will examine {\\em transversal stabilization}.\nThis is a simple method for creating one transversal knot from another.  By \nshowing that all positive transversal torus knots whose self-linking number is less than \nmaximal come from this stabilization process we are able to reduce the above theorem\nto the classification of positive transversal torus knots with maximal self-linking number.\nStabilization also provides a general way to approach the classification problem for \nother knot types.  For example, we can reprove Eliashberg's classification\nof transversal unknots using stabilization ideas and basic contact topology.\n\nIt is widely believed that the\nself-linking number is not a complete invariant for transversal knots. However,\nas of the writing of this paper, there is no known knot type whose transversal realizations\nare not determined by their self-linking number. For Legendrian knots, in contrast,\nEliashberg and Hofer (currently unpublished)\nand Chekanov \\cite{chek} have produced examples of Legendrian knots that are not determined\nby their corresponding invariants.\n\nIn Section~\\ref{basic} we review some standard facts concerning contact geometry\non three manifolds.  In Section~\\ref{sec:main} we prove our main theorem modulo some\ndetails concerning the characteristic foliations on tori which are proved in\nSection~\\ref{tori} and some results on stabilizations proved in Section~\\ref{stab}.\nIn the last section we discuss some open questions.\n\n\\rk{Acknowledgments}\nThe author gratefully acknowledges the support of an NSF \nPost-Doctoral Fellowship (DMS--9705949) and Stanford University.\nConversations with Y Eliashberg and E Giroux were helpful\nin preparing this paper.\n\n\n\\section{Contact structures in three dimensions}\\label{basic}\n\nWe begin by recalling some basic facts from contact topology.  For a more detailed\nintroduction, see \\cite{a:etall, gi:survey}.\nRecall an orientable plane field $\\xi$ is a \\dfn{contact structure} on a three manifold\nif $\\xi=\\hbox{ker } \\alpha$ where $\\alpha$ is a nondegenerate 1--form for which\n$\\alpha\\wedge d\\alpha\\not=0$.  Note $d\\alpha$ induces an orientation on $\\xi$.\nTwo contact structures are called \\dfn{contactomorphic} if there is a diffeomorphism\ntaking one of the plane fields to the other.\nA contact structure $\\xi$ induces a singular\nfoliation on a surface $\\Sigma$  by integrating the singular line field\n$\\xi\\cap T\\Sigma$. This is called the \\dfn{characteristic foliation} and is denoted\n$\\Sigma_\\xi$. Generically, the singularities are elliptic (if local degree is 1)\nor hyperbolic (if the local degree is $-1$).  If $\\Sigma$ is oriented then the singularities\nalso have a sign.  A singularity is positive (respectively negative) if the orientations\non $\\xi$ and $T\\Sigma$ agree (respectively disagree) at the singularity. \n\n\\begin{lem}[Elimination Lemma \\cite{gi:convex}]\\label{lem:elimination}\n\tLet $\\Sigma$ be a surface in a contact $3$--manifold $(M,\\xi)$.  Assume\n\tthat $p$ is an elliptic and $q$ is a hyperbolic singular point in \n\t$\\Sigma_\\xi$, they both have the same sign and there is a leaf $\\gamma$\n\tin the characteristic foliation $\\Sigma_\\xi$ that connects $p$ to $q$.\n\tThen there is a $C^0$--small isotopy $\\phi\\colon\\thinspace \\Sigma\\times[0,1]\\to M$ such that\n\t$\\phi_0$ is the inclusion map, $\\phi_t$ is fixed on $\\gamma$ and outside any \n\t(arbitrarily small) pre-assigned\n\tneighborhood $U$ of $\\gamma$ and $\\Sigma'=\\phi_1(\\Sigma)$ has no singularities\n\tinside $U$.\n\\end{lem}\n\nIt is important to note that after the above cancellation there is a curve in the \ncharacteristic foliation on which the singularities had previously sat. \nIn the case of positive\nsingularities this curve will consist of the (closure of the) stable manifolds of the hyperbolic point\nand {\\em any} arc leaving the elliptic point (see \\cite{ef, e:dis}), and similarly for the negative singularity case.\nOne may also reverse this process and add a canceling pair of singularities along \na leaf in the characteristic foliation.  It is also important to note:\n\n\\begin{lem}\n\tThe germ of the contact structure $\\xi$ along a surface $\\Sigma$ is determined \n\tby $\\Sigma_\\xi$.\n\\end{lem}\n\nNow recall that a contact structure $\\xi$ on $M$ is called \\dfn{tight} if no disk embedded in \n$M$ contains a limit cycle in its characteristic foliation, otherwise it is called \n\\dfn{overtwisted}. The standard contact structure on $S^3$,\ninduced from the complex tangencies to $S^3=\\partial B^4$ where $B^4$ is the\nunit 4--ball in $\\hbox{$\\mathbb C$} ^2$, is tight.\n\nA closed curve $\\gamma\\colon\\thinspace S^{1}\\to M$ in a contact manifold $(M,\\xi)$ is called \n\\dfn{transversal} if $\\gamma'(t)$ is transverse to $\\xi_{\\gamma(t)}$ \nfor all $t\\in S^{1}$.  Notice a transversal curve can be \n\\dfn{positive} or \\dfn{negative} according as $\\gamma'(t)$ agrees with \nthe co-orientation of $\\xi$ or not. We will restrict our attention to \npositive transversal knots (thus in this paper ``transversal'' means \n``positive transversal''). It can be shown that any curve can be made \ntransversal by a $C^{0}$ small isotopy. It will be useful to note:\n\n\\begin{lem}[See \\cite{el:knots}]\\label{extend}\n\tIf $\\psi_t\\colon\\thinspace S^1\\to M$ is a transversal isotopy, then there\n\tis a contact isotopy $f_t\\colon\\thinspace M\\to M$ such that $f_t\\circ \\psi_0=\\psi_t$.\n\\end{lem}\n\nGiven a transverse knot $\\gamma$ in $(M,\\xi)$ that bounds a surface $\\Sigma$\nwe define the \\dfn{self-linking number}, $l(\\gamma)$, of $\\gamma$ as follows: \ntake a nonvanishing vector field $v$ in $\\xi\\vert_{\\gamma}$ that \nextends to a nonvanishing vector field in $\\xi\\vert_{\\Sigma}$ and let $\\gamma'$\nbe $\\gamma$ slightly pushed along $v$. Define\n$$l(\\gamma,\\Sigma)=I(\\gamma',\\Sigma),$$\nwhere $I(\\,\\cdot\\,,\\,\\cdot\\,)$ is the oriented intersection number.\nThere is a nice relationship between $l(\\gamma,\\Sigma)$ and the \nsingularities of the characteristic foliation of $\\Sigma$.  Let \n$d_{\\pm}=e_{\\pm}-h_{\\pm}$ where $e_{\\pm}$ and $h_{\\pm}$ are the \nnumber of $\\pm$ elliptic and hyperbolic points in the characteristic \nfoliation $\\Sigma_{\\xi}$ of $\\Sigma$, respectively.  In \\cite{be} it \nwas shown that\n\\begin{equation}\n\tl=d_{-}-d_{+}.\n\\end{equation}\nWhen $\\xi$ is a {\\em tight} contact structure and $\\Sigma$ is a \n{\\em disk},\nEliashberg \\cite{el:twenty} has shown, using the elimination lemma,  how to \neliminate all the positive hyperbolic and negative elliptic points \nfrom $\\Sigma_{\\xi}$. Thus in a \ntight contact structure when $\\gamma$ is an unknot $l(\\gamma,\\Sigma)$ is always negative.\nMore generally one can show (see \\cite{be, el:twenty}) that\n\\begin{equation}\\label{lbound}\n\tl(\\gamma)\\leq -\\chi(\\Sigma),\n\\end{equation}\nwhere $\\Sigma$ is a Seifert surface for $\\gamma$ and $\\chi(\\Sigma)$ is its Euler number.\n\nAny odd negative integer can be realized as the self-linking number for some \ntransversal unknot.\nThe first general result concerning the classification of transversal knots\nwas the following:\n\n\\begin{thm}[Eliashberg \\cite{el:knots}]\\label{unknots}\n\tTwo transversal unknots are transversely isotopic if and only if \n\tthey have the same self-linking number.\n\\end{thm}\n\nLet $\\mathcal{T}$ be the transversal isotopy classes of transversal knots in $S^3$ with its unique\ntight contact structure. Let $\\mathcal{K}$ be the isotopy classes of knots in $S^3$.\nGiven a transversal knot $\\gamma\\in \\mathcal{T}$ we have two pieces of information: its knot type \n$[\\gamma]\\in\\mathcal{K}$ and its self-linking number $l(\\gamma)\\in\\hbox{$\\mathbb Z$} $.  Define \n\\begin{equation}\\label{tmap}\n\t\\phi\\colon\\thinspace \\mathcal{T}\\to\\mathcal{K}\\times \\hbox{$\\mathbb Z$}  : \\gamma\\mapsto ([\\gamma],l(\\gamma)).\n\\end{equation}\nThe main questions concerning transversal knots can be phrased in terms of the image\nof this map and preimages of points.  In particular the above results say that $\\phi$\nis onto \n$$U=[\\hbox{unknot}]\\times\\{\\hbox{negative odd integers}\\}$$ and $\\phi$ is one-to-one\non $\\phi^{-1}(U)$.\n\nWe will also need to consider Legendrian knots.  A knot $\\gamma$ is a \\dfn{Legendrian knot} if\nit is tangent to $\\xi$.  The contact structure $\\xi$ defines a canonical framing on a Legendrian\nknot $\\gamma$. If $\\gamma$ is null homologous we may associate a number to this framing which\nwe call the \\dfn{Thurston--Bennequin invariant} of $\\gamma$ and denote it $\\hbox{tb}(\\gamma)$.\nIf we let $\\Sigma$ be the surface exhibiting the null homology of $\\gamma$ then we may trivialize\n$\\xi$ over $\\Sigma$ and use this trivialization to measure the rotation of $\\gamma'(t)$ around\n$\\gamma$.  This number $r(\\gamma)$ is called the \\dfn{rotation number} of $\\gamma$. Note that\nthe rotation number depends on an orientation on $\\gamma$. From an oriented \nLegendrian knot $\\gamma$ one can obtain canonical positive and negative transversal knots\n$\\gamma_\\pm$ by pushing $\\gamma$ by vector fields tangent to $\\xi$ but transverse to\n$\\gamma'(t)$.  One may compute\n\\begin{equation}\n\tl(\\gamma_\\pm)=\\hbox{tb}(\\gamma)\\mp r(\\gamma).\n\\end{equation}\nThis observation combined with \\eqn{lbound} implies\n\\begin{equation}\\label{tb-bound}\n\t\\hbox{tb}(\\gamma)+|r(\\gamma)|\\leq -\\chi(\\Sigma).\n\\end{equation}\nConsider an oriented (nonsingular) foliation $\\mathcal{F}$ on a torus $T$. The foliation is \nsaid to have a \\dfn{Reeb component} if two oppositely oriented periodic orbits cobound an\nannulus containing no other periodic orbits. \n\n\\begin{lem}\\label{curveinT}\n\tConsider a torus $T$ in a contact three manifold $(M,\\xi)$.  If the characteristic foliation\n\ton $T$ is nonsingular and contains no Reeb components then  \n\tany closed curve on $T$ may be isotoped to be transversal to $T_\\xi$ or into a leaf\n\tof $T_\\xi$. Moreover there is at most one homology class in $H_1(T)$ that can\n\tbe realized by a leaf of $T_\\xi$. \n\\end{lem}\n\nNow let $\\xi$ be a tight contact structure on a solid torus $S$ with nonsingular characteristic\nfoliation on it boundary $T=\\partial S$. It is easy to arrange for $T_\\xi$ to have no Reeb \ncomponents \\cite{ml}.\nSince $\\xi$ is tight the lemma above implies the meridian $\\mu$\ncan be made transversal to $T_\\xi$. We say $S$ has self-linking number $l$ if $l=l(\\mu)$\n(ie, the self-linking number of $S$ is the self-linking number of its meridian).\n\n\\begin{thm}[Makar--Limanov \\cite{ml}]\\label{solid_tori}\n\tAny two tight contact structures on $S$ which induce the same nonsingular\n\tfoliation on the boundary and have self-linking number $-1$ are contactomorphic.\n\\end{thm}\n\n\n\\section{Positive transversal torus knots}\\label{sec:main}\n\nLet $U$ be an unknot in a 3--manifold $M$, $D$ an embedded disk that it bounds\nand $V$ a tubular neighborhood of $U$.  The boundary $T$ of $V$ is an embedded torus in $M$,\nwe call such a torus a \\dfn{standardly embedded torus}. \nLet $\\mu$ be the unique curve on $T$ that bounds a disk in $V$ and\n$\\lambda=D\\cap V$.  Orient $\\mu$ arbitrarily and \nthen orient $\\lambda$ so that $\\mu, \\lambda$ form a positive basis for $H_1(T)$ where \n$T$ is oriented as the boundary of $V$. Up to homotopy any curve in \n$T$ can be written as $p\\mu + q\\lambda$, we shall denote this curve by $K_{(p,q)}$.   \nIf $p$ and $q$ are relatively prime\nthen $K_{(p,q)}$ is called a {\\em $(p,q)$--torus knot.}  If $pq>0$ we say $K(p,q)$ is a \n\\dfn{positive} torus knot otherwise we call it \\dfn{negative}. One\nmay easily compute that the Seifert surface of minimal genus for $K_{(p,q)}$\nhas Euler number $|p|+|q|-|pq|$.  Thus for a transversal torus knot Equation~\\ref{lbound} implies \n\\begin{equation}\n\tl(K_{(p,q)})\\leq -|p|-|q|+|pq|.\n\\end{equation}\nIn fact, if $\\overline{l}_{(p,q)}$ denotes the maximal self-linking number for a \ntransversal $K_{(p,q)}$ then one may easily check that\n\\begin{equation}\n\t\\overline{l}_{(p,q)}=-p-q+pq,\n\\end{equation}\nif $p,q>0$, ie, for a positive torus knot. (Note: for a positive transversal torus knot \nLemma~\\ref{basis}\nsays we have $p,q>0$ not just $pq>0$.) From the symmetries involved in the definition of\na torus knot we may assume that $p>q$, which we do throughout the rest of the paper.\nWe now state our main theorem.\n\n\\begin{thm}\\label{main}\n\tPositive transversal torus knots in a tight contact structure\n\tare determined up to transversal isotopy by \n\ttheir knot type and their self-linking number.\n\\end{thm}\n\n\\begin{rem}\n\t{\\em We may restate this theorem by saying\n\tthe map $\\phi$ defined in equation \\ref{tmap} is one-to-one when restricted to\n\t$$(\\hbox{pr}\\circ \\phi)^{-1}(\\hbox{positive torus knots})$$ \n\t(here $\\hbox{pr}\\colon\\thinspace \\mathcal{K}\\times \\hbox{$\\mathbb Z$} \\to \\mathcal{K}$ is projection). \n\tMoreover, the image of $\\phi$ restricted to the above set is \n\t$G=\\cup_{(p,q)} K_{(p,q)}\\times N(p,q)$ where the union\n\tis taken over relatively prime positive $p$ and $q$, and \n\t$N(p,q)$ is the set of odd integers less than or equal to $-p-q+pq$.}\n\\end{rem}\n\nWe first prove the following auxiliary result:\n\n\\begin{prop}\\label{aux}\n\tTwo positive transversal $(p,q)$--torus knots $K$ and $K'$ in a tight contact\n\tstructure with maximal self-linking number (ie, $l(K)=l(K')=\\overline{l}_{(p,q)}$) \n\tare transversally isotopic.\n\\end{prop}\n\n\\proof\nLet $T$ and $T'$ be tori standardly embedded in $M$ on which $K$ and $K'$,  \nrespectively, sit.\n\n\\begin{lem}\\label{nonsingular}\n\tIf the self-linking number of $K$ is maximal then $T$ may be\n\tisotoped relative to $K$ so that the characteristic\n\tfoliation on $T$ is nonsingular.\n\\end{lem}\n\nThis lemma and the next are proved in the following section.\n\n\\begin{lem}\\label{isotopic}\n\tTwo transversal knots on a torus $T$ with nonsingular characteristic foliation\n\tthat are homologous are transversally isotopic, except possibly when there is a closed leaf\n\tin the foliation isotopic to the transversal knots.\n\\end{lem}\n\nOur strategy is to isotop $T$ onto $T'$, keeping $K$ and $K'$ transverse to $\\xi$, \nso that $K$ and $K'$ are\nhomologous, and thus transversally isotopic.  We now show that $T$\ncan be isotoped into a standard form keeping $K$ transverse (and similarly for\n$K'$ and $T'$ without further mention).  Let $V$ be \nthe solid torus that $T$ bounds (recall we are choosing $V$ so that $p>q$). \nLet $D_\\mu$ and $D_\\lambda$ be the disk that $\\mu$ and $\\lambda$\nrespective bound. Now observe:\n\n\\begin{lem}\\label{basis}\n\tWe may take $\\mu$ and $\\lambda$ to be positive transversal curves and with this\n\torientation $\\mu,\\lambda$ form a positive basis for $T=\\partial V$.\n\\end{lem}\n\n\\proof\nClearly we may take $\\mu$ and $\\lambda$ to be positive transversal knots, for if we could\nnot then \\lm{curveinT} implies that we may isotop one of them to a closed leaf in $T_\\xi$ contradicting\nthe tightness of $\\xi$.  Thus we are left to see that \n$\\mu, \\lambda$ is a positive basis. Assume this is not the case.\nBy isotoping $T$ slightly we may assume that $T_\\xi$ has closed leaf\n(indeed if $T_\\xi$ does not already have a closed leaf then the isotopy will give an \nintervals worth  of rotation numbers, and hence\nsome rational rotation numbers, for the return map\ninduced on $\\mu$ by $T_\\xi$). Let $C$ be one of these closed leaves and let $n=\\lambda\\cdot C$\nand $m=\\mu\\cdot C$.  Note $n$ and $m$ are both positive since $\\mu$ and $\\lambda$ are\npositive transversal knots. Since $\\mu, \\lambda$ is not a positive basis $C$ is an $(n,m)$--torus knot. \nIn particular $C$ is a positive torus knot. Moreover, the framing on $C$ induced by $\\xi$ is\nthe same as the framing induced by $T$. Thus $\\hbox{tb}(C)=mn$ contradicting \\eqn{tb-bound}.\nSo $\\mu,\\lambda$ must be a positive basis for $T$.\n\\endproof\n\nNow let $m=l(\\mu)$ and $l=l(\\lambda)$ and recall $m,l\\leq -1$.\n\n\\begin{lem}\\label{l-formula}\n\tIf $\\gamma$ is a transversal $(p,q)$ knot on $T$ (with nonsingular \n\tcharacteristic foliation)  then \n\t\\begin{equation}\\label{eqn:l-formula}\n\t\tl(\\gamma)=pm+ql+pq.\n\t\\end{equation}\n\\end{lem} \n\n\\proof\nLet $v$ be a section of $\\xi$ over an open 3--ball containing $T$ and its meridional and longitudinal\ndisks. If $C$ is a curve on $T$ then define\n$f(C)$ to be the framing of $\\xi$ over $C$ induced by $v$ relative to the framing of $\\xi$  \nover $C$ induced by $T$.  Note $f$ descends to a map on $H_1(T)$ and $f(A+B)=f(A)+f(B)$\nwhere $A,B\\in H_1(T)$. One easily computes $f(\\mu)=m$ and $f(\\lambda)=l$. Thus \n$f(p\\mu+q\\lambda)=pm+ql$.  Now for a transversal curve $C$ on $T$ the normal bundle to $C$ can be\nidentified with $\\xi$ thus $f(C)$ differs\nfrom $l(C)$ by the framing induced on $C$ by $T$ relative to the framing induced\non $C$ by its Seifert surface.  So $l(C)=f(C)+pq=pm+ql+pq$. \n\\endproof\n\nThus since $K$ has maximal self-linking number we must have $m=l=-1$. \nNow by \\tm{solid_tori} we may find\na contactomorphism from $V$ to $S_f=\\{(r,\\theta,\\phi)\\in \\hbox{$\\mathbb R$} ^2\\times S^1|\nr\\leq f(\\theta,\\phi)\\}$ for some positive function $f\\colon\\thinspace T^2\\to \\hbox{$\\mathbb R$} $, with the \nstandard tight contact structure $\\hbox{ker}(d\\phi+r^2\\, d\\theta)$. \n\nClearly $T=\\partial S_f$ may be isotoped to $S_\\epsilon=\\{(r,\\theta,\\phi)\\in \\hbox{$\\mathbb R$} ^2\\times S^1|\nr<\\epsilon\\}$ for arbitrarily small $\\epsilon>0$. We now show this isotopy may be done\nkeeping our knot $K$ transverse to the characteristic foliation. To a foliation on $\\partial S_f$ we may\nassociate a real valued rotation number $r(S_f)$ for the return map on $\\mu$ induced\nby $(\\partial S_f)_\\xi$ (see \\cite{ml}). \nFor a standardly embedded torus this number must be negative since if not then some\nnearby torus would have a positive $(r,s)$ torus knot as a closed leaf in its characteristic\nfoliation violating the Bennequin inequality (as in the proof of Lemma~\\ref{basis}). \nSo as we isotop $\\partial S_f$ to\n$\\partial S_\\epsilon$ we may keep our positive torus knot transverse to the characteristic foliation\nby Lemma~\\ref{curveinT} (since closed leaves in $(\\partial S_f)_\\xi$ have slope $r(S_f)$ and\n$K$ has positive slope). \nThus we assume that the solid torus $V$ is contactomorphic to $S_\\epsilon$.\nIf $C$ is the core of $V (=S_\\epsilon)$ then it is a transversal unknot with self-linking \n$l(\\lambda)=-1$.\n\nFinally, let $V$ and $V'$ be the solid tori associated to the torus knots $K$ and $K'$ and\nlet $C$ and $C'$ be the cores of $V$ and $V'$.  Now since $C$ and $C'$ are unknots with the\nsame self-linking number they are transversely isotopic. Thus we may think of $V$ and $V'$ as\nneighborhoods of the same transverse curve $C=C'$. From above, $V$ and $V'$ may both be shrunk \nto be arbitrarily small neighborhoods of $C$ keeping $K$ and $K'$ transverse to $\\xi$. Hence\nwe may assume that $V$ and $V'$ both sit in a neighborhood of $C$ which is contactomorphic\nto, say, $S_c$ (using the notation from the previous paragraph). By shrinking $V$ and $V'$ \nfurther we may assume they are the tori $S_\\epsilon$ and $S_{\\epsilon'}$ inside $S_c$\nfor some $\\epsilon$ and $\\epsilon'$. Note that this is not immediately obvious but follows\nfrom the fact that a contactomorphism from the standard model $S_f$ for, say, $V$ to \n$V\\subset S_c$ may be constructed to take a neighborhood of the core of $S_f$ to a\nneighborhood of the core of $S_c$.\nThis allows us to finally conclude that we may isotop\n$V$ so that $V=V'$. Now since $K$ and $K'$ represent the same homology class on $\\partial V$ and\nthey are both transverse to the foliation we may use Lemma~\\ref{isotopic} to transversely \nisotop $K$ to $K'$.\n\\endproof\n\nA transversal knot $K$ is called a \\dfn{stabilization} of a transversal knot $C$ if\n$K=\\alpha\\cup A$, $C=\\alpha\\cup A'$ and $A\\cup A'$ cobound a disk\nwith only positive elliptic and negative hyperbolic singularities (eg Figure~\\ref{fig:stab}).  \n\\begin{figure}[ht]\n\t{\n\\epsfysize=2in\\centerline{\\relabelbox\\small\n\\epsfbox{stab.eps}\n\\relabel {A}{$A$}\n\\relabel {A'}{$A'$}\n\\adjustrelabel <-4pt,-1pt> {e}{$e_+$}\n\\relabel {h}{$h_-$}\n\\endrelabelbox}}\n\t\\caption{Stabilization disk}\n\t\\label{fig:stab}\n\\end{figure}\nWe say $K$ is obtained from $C$ by a \\dfn{single stabilization} if $K$ is a stabilization\nof $C$ and  $l(K)=l(C)-2$ (ie, the disk that $A\\cup A'$ cobound is the one shown in \nFigure~\\ref{fig:stab}).\nThe key observation concerning stabilizations is the following:\n\n\\begin{thm}\\label{tm-stab}\n\tIf the transversal knots $K$ and $K'$ are single stabilizations of transversal knots\n\t$C$ and $C'$ then $K$ is transversely isotopic to $K'$ if $C$ is transversely \n\tisotopic to $C'$.\n\\end{thm}\n\nThis theorem will be proved in Section~\\ref{stab}.  The proof of \\tm{main} is completed by an inductive\nargument using the following observation.\n\n\\begin{lem}\\label{canstab}\n\tIf $K$ is a positive transversal $(p,q)$--torus knot and $l(K)<\\overline{l}_{(p,q)}$ then $K$\n\tis a single stabilization of a $(p,q)$--torus knot with larger self-linking number.\n\\end{lem}\n\nThe proof of this lemma will be given in the next section following the proof of \\lm{nonsingular}.\n\n\\section{Characteristic foliations on tori}\\label{tori}\n\nIn this section we prove various results stated in Section~\\ref{sec:main} related to \nfoliations on tori.\nLet $T$ be a standardly embedded torus in $M^3$ and $K$ a positive $(p,q)$--torus\nknot on $T$ that is transverse to a tight contact structure $\\xi$. We are now ready to prove:\n\n\\proof[\\bf Lemma~\\ref{nonsingular}]\n{\\sl\n\tIf the self-linking number of $K$ is maximal then $T$ may be\n\tisotoped relative to $K$ so that the characteristic\n\tfoliation on $T$ is nonsingular.\n}\n\n\\proof\nBegin by isotoping $T$ relative to $K$ so that the number of singularities \nin $T_\\xi$ is minimal. Any singularities that are left must occur in pairs:\na positive (negative) hyperbolic $h$ and elliptic $e$ point connected by a stable\n(unstable) manifold $c$.  Moreover, since $h$ and $e$ cannot be canceled without\nmoving $K$ we must have $c\\cap K\\not=\\emptyset$.\n\nNow $T\\setminus K$ is an annulus $A$ with the characteristic foliation flowing\nout of one boundary component and flowing in the other. Let $c'$ be the component\nof $c$ connected to $h$ in $A$.  \nWe can have no periodic orbits in $A$ since such an orbit\nwould be a Legendrian $(p,q)$--torus knot with Thurston--Bennequin invariant\n$pq$ contradicting \\eqn{tb-bound}.\nThus the other stable (unstable) manifold $c''$ of $h$ will have to enter (exit) $A$ through the same\nboundary component. The manifolds $c'$ and $c''$ separate off a disk $D$ from $A$.\nWe may use $D\\subset T$ to push the arc $K\\cap D$ across $D$ to obtain another\ntransverse $(p,q)$--torus knot $K'$.  It is not hard to show that\n$K$ is a stabilization of $K'$. In particular $l(K')>l(K)$, contradicting\nthe maximality of $l(K)$.  Thus we could have not have had any singularities \nleft after our initial isotopy.\n\\endproof\n\nThe above proof provides some insight into \\lm{canstab}. Recall:\n\n\\proof[{\\bf \\lm{canstab}}]\n{\\sl\n\tIf $K$ is a positive transversal $(p,q)$--torus knot with and $l(K)<\\overline{l}_{(p,q)}$ then $K$\n\tis a single stabilization of a $(p,q)$--torus knot with larger self-linking number.\n}\n\n\\proof\nWe begin by noting that if $K$ is a stabilization of another transversal knot then it\nis also a single stabilization of some transversal knot. Thus we just demonstrate that \n$K$ is a stabilization of some transversal knot.\n\nFrom the above proof it is clear that if we cannot eliminate all the singularities in\nthe characteristic foliation of the torus $T$ on which $K$ sits then there is a disk on \nthe torus which exhibits $K$ as a stabilization. \n\nIf we can remove all the singularities from $T$ then by \\lm{l-formula} we know\nthat the self-linking number of, say, the meridian $\\mu$ is less than $-1$. Thus $\\mu$ bounds a \ndisk $D_\\mu$ containing only positive elliptic and at least one negative hyperbolic singularity.  \nTo form a positive transversal torus knot $K''$ we can take $p$ copies of the meridian $\\mu$ and \n$q$ copies of the longitude $\\lambda$\nand ``add'' them (ie, resolve all the intersection points keeping the curve \ntransverse to the characteristic foliation).  \nThis will produce a transversal knot on $T$ isotopic to $K$  thus transversely\nisotopic.  Moreover, we may use the graph of singularities on $D_\\mu$ to show that $K''$, and hence $K$, \nis a stabilization.  \n\\endproof\n\nWe end this section by establishing (a more general version of)\nLemma~\\ref{isotopic}.\n\\begin{lem}\n\tSuppose that $\\mathcal{F}$ is a nonsingular foliation on a torus\n\t$T$ and $\\gamma$ and $\\gamma'$ are two simple closed curves on $T.$\n\tIf $\\gamma$ and $\\gamma'$ are homologous and transverse to $\\mathcal{F}$\n\tthen they are isotopic through simple closed curves transverse to  \n\t$\\mathcal{F},$ except possibly if $\\mathcal{F}$ has a closed leaf \n\tisotopic to $\\gamma.$\n\\end{lem}\n\n\\proof\nWe first note that if $\\gamma$ and $\\gamma'$ are disjoint and there are not closed\nleaves isotopic to them then the annulus that they cobound will provide the desired transverse\nisotopy. Thus we are left to show that we can make $\\gamma$ and $\\gamma'$ disjoint.\nWe begin by isotoping them so they intersect transversely. Now assume we have transversely \nisotoped them so that the number of their intersection points is minimal.  We wish to show \nthis number is zero.  Suppose not, then there are an even number of intersection points\n(since homologically their intersection is zero).  \n\nUsing a standard innermost arc argument we may find a disk $D\\subset T$  such that\n$\\partial D$ consists of two arcs, one a subarc of $\\gamma$ the other a subarc of $\\gamma'$.\nWe can use the disk $D$ to guide a transverse isotopy of $\\gamma'$ that will decrease the\nnumber of intersections of $\\gamma$ and $\\gamma'$ contradicting our assumption of minimality.\nTo see this, note that the local orientability of the foliation implies that we can define\na winding number of $\\mathcal{F}$ around $\\partial D$. Moreover since $\\partial D$ is contractible\nand the foliation is nonsingular this winding number must be zero. Thus the foliation on \n$D$ must be diffeomorphic to the one shown in Figure~\\ref{fig:diskfol} where the desired isotopy is apparent.\n\\begin{figure}[ht]\n\t{\\epsfysize=2in\\centerline{\\epsfbox{diskfol.eps}}}\n\t\\caption{Foliation on $D$}\n\t\\label{fig:diskfol}\n\\end{figure}\n\\endproof\n\n\n\\section{Stabilizations of transversal knots}\\label{stab}\n\nThe main goal of this section is to prove \\tm{tm-stab}:\n\n\\proof[{\\bf \\tm{tm-stab}}]\n{\\sl\n\tIf the transversal knots $K$ and $K'$ are single stabilizations of transversal knots\n\t$C$ and $C'$ then $K$ is transversely isotopic to $K'$ if $C$ is transversely \n\tisotopic to $C'$.\n}\n\n\\proof\nSince $C$ and $C'$ are transversely isotopic we can assume that $C=C'$. \nLet $D$ and $D'$ be the disks that exhibit $K$ and $K'$ as stabilizations of\n$C$.  Let $e,h$ and $e',h'$ be the elliptic\/hyperbolic pairs on $D$ and $D'$.\nFinally, let $\\alpha$ and $\\alpha'$ be the Legendrian arcs formed by the (closure of the) \nunion of stable manifolds of $h$ and $h'$.  Using the characteristic foliation \non $D$ we may transversely isotop $K\\setminus C$ to lie arbitrarily close to $\\alpha$\n(and similarly for $K'$ and $\\alpha'$). We are thus done by the following simple lemmas.\n\n\\begin{lem}\n\tThere is a contact isotopy preserving $C$ taking $\\alpha\\cap C$ to \n\t$\\alpha'\\cap C$.\n\\end{lem} \n\nWorking in a standard model for a transverse curve\nthis lemma is quite simple to establish.\nThus we may assume that $\\alpha$ and $\\alpha'$ both touch $C$ at the same point.\n\\begin{lem}\n\tThere is a contact isotopy preserving $C$ taking $\\alpha$ to $\\alpha'$.\n\\end{lem}\nOnce again one can use a Darboux chart to check this lemma (for some details see \\cite{ef}).\n\\begin{lem}\n\tAny two single stabilizations of $C$ along a fixed Legendrian arc are transversely \n\tisotopic.\n\\end{lem}\nWith this lemma our proof of Theorem~\\ref{tm-stab} is complete.\n\\endproof\n\nWe now observe that using \\tm{tm-stab} we may reprove Eliashberg's result concerning \ntransversal unknots. The reader should note that this ``new proof'' is largely just\na reordering\/rewording of Eliashberg's  proof.\n\n\\begin{thm}\n\tTwo transversal unknots are transversally isotopic if and only if they\n\thave the same self-linking number.\n\\end{thm}\n\n\\proof\nUsing \\tm{tm-stab} we only need to prove that two transversal unknots with self-linking\nnumber $-1$ are transversally isotopic, since by looking at the characteristic foliation\non a Seifert disk it is clear that a transversal unknot with self-linking number less\nthan $-1$ is a single stabilization of another unknot.  But given a transversal unknot with self-linking\nnumber $-1$ we may find a disk that it bounds with precisely one positive elliptic \nsingularity in its characteristic foliation. Using the characteristic foliation on the disk \nthe unknot may be transversely isotoped into an arbitrarily small neighborhood of the elliptic \npoint. Thus given two such knots we may now find a contact isotopy of taking the elliptic point on\none of the Seifert disks to the elliptic point on the other. Since the Seifert disks are tangent\nat their respective elliptic points we may arrange that they agree in a neighborhood of the \nelliptic points. Now by shrinking the Seifert disks more we may assume that both unknots sit on the\nsame disk. It is now a simple matter to transversely isotop one unknot to the other.\n\\qed\n\n\\section{Concluding remarks and questions}\n\nWe would like to note that many of the techniques in this paper work for negative torus knots\nas well (though the proofs above do not always indicate this). There are two places where we cannot\nmake the above proofs work for negative torus knots, they are:\n\\begin{itemize}\n\\item From Equation~\\ref{eqn:l-formula} we cannot conclude that the self-linking numbers of\t\n\t$\\mu$ and $\\lambda$ are $-1$ when $l(K_{(p,q)})$ is maximal as we could for positive torus\n\tknots.\n\\item We cannot always conclude that a negative torus knot with self-linking less than maximal\n\tis a stabilization.\n\\end{itemize}\nDespite these difficulties we conjecture that negative torus knots are also determined by their\nself-linking number.\n\nLet $S=S^1\\times D^2$ and let $K$ be a $(p,q)$--curve on\nthe boundary of $S$. Now if $C$ is a null homologous knot in a three manifold $M$ then \nlet $f\\colon\\thinspace S\\to N$ be a diffeomorphism from $S$ to a neighborhood $N$ of $C$ in $M$ taking $S^1\\times\\{\\hbox{point}\\}$\nto a longitude for $C$.  We now define the \\dfn{$(p,q)$--cable of $C$} to be the knot $f(K)$. \n\\begin{quest}\\label{conj}\n\tIf $\\mathcal{C}$ is the class of topological knots whose transversal realizations\n\tare determined up to transversal isotopy by their self-linking number, then is\n\t$\\mathcal{C}$ closed under cablings?\n\\end{quest}\nEliashberg's \nTheorem~\\ref{unknots} says that the unknot $U$ is in $\\mathcal{C}$. Our main Theorem~\\ref{main} says\nthat any positive cable of the unknot is in $\\mathcal{C}$. \nThis provides the first bit of evidence that the answer to the question might be YES, at least for\n``suitably positive'' cablings.\n\nGiven a knot type one might hope,\nusing the observation on stabilizations in this paper, to prove that transversal knots in this\nknot type are determined by their self-linking number as follows:  First establishing that there is a \nunique transversal knot in this knot type with maximal self-linking number. Then showing that\nany transversal knot in this knot type that does not have maximal self-linking number is a stabilization.\nThe second part of this program is of independent interest so we ask the\nfollowing question:\n\\begin{quest}\n\tAre all transversal knots not realizing the maximal self-linking number of their knot type\n\tstabilizations of other transversal knots?\n\\end{quest}\nIt would be somewhat surprising if the answer to this question is YES in complete generality\nbut understanding when the answer is YES and when and why it is NO should provide insight\ninto the structure of transversal knots.\n\nWe end by mentioning that the techniques in this paper also seem to shed light on Legendrian\ntorus knots. It seems quite likely that their isotopy class may be determined by their \nThurston--Bennequin invariant\nand rotation number. We hope to return to this question in a future paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Interfacial Structures of Different Classical Water Force Fields}\nTo compare the interfacial structures of classical water force fields, we simulated the liquid-vapor interfaces of SPC\/E \\cite{Berendsen:1987gc}, TIP4P\/2005 \\cite{abascal2005general}, and TIP5P \\cite{Mahoney2000_SM} waters in NVT ensembles. Each system has 1944 water molecules in a slab geometry with dimensions of 5.0 nm $\\times$ 5.0 nm $\\times$ 3.0 nm. The system was equilibrated at $T=298 \\,\\text{K}$ and Particle Mesh Ewald was used to handle the long-range part of electrostatic interactions. The SHAKE and SETTLE algorithms were used to constrain the geometry of water. The simulations were performed using the LAMMPS package \\cite{Plimpton:1995fc} for SPC\/E and TIP4P and the GROMACS package \\cite{pronk2013gromacs} for TIP5P. Following the same procedure in Ref. \\onlinecite{Willard:2010da}, instantaneous liquid interface was constructed for each configuration of the generated statistics. Density profile, $\\rho(a)$, and orientational distribution, $P(\\cos\\theta_1,\\cos\\theta_2 | a)$, were computed according to Eqs. (7) and (11) in Ref. \\onlinecite{Willard:2010da}. Figure \\ref{fig:S1} shows the density profiles along with the reduced orientational distributions as defined in Eq. (5) of the main text. Both $\\rho(a)$ and $P(\\cos\\theta_\\text{OH}|a)$ exhibit the qualitatively same structural characteristics for three different force fields of water. \n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width = 4.0 in]{FigS1}\n\\caption{(a) Density profiles computed from the atomistic simulations of SPC\/E, TIP4P, and TIP5P. They are normalized by the bulk density, $\\rho_\\mathrm{b}$. (b) Orientational distributions, $P(\\cos\\theta_\\text{OH}|a)$, for SPC\/E (top), TIP4P (middle), and TIP5P (bottom). Color shading indicates the probability density.}\n\\label{fig:S1}\n\\end{figure}\n\nThe orientational polarization, $\\langle \\mu_{\\mathbf{\\hat{n}}}(a) \\rangle$, and polarizability, $\\langle (\\delta \\mu_{\\mathbf{\\hat{n}}}(a) )^2 \\rangle$, were computed according to Eqs.~(6) and (7) of the main text, and their interfacial profiles are plotted in Fig. \\ref{fig:S2}. The plots show that these interfacial properties also share the qualitatively same feature among three different force fields of water. It is notable that while there exists some quantitative difference in the scale of the polarization, the polarizability shows the almost identical trend across the force fields.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width = 6.4 in]{FigS2_v2}\n\\caption{Interfacial polarization, $\\langle \\mu_{\\mathbf{\\hat{n}}}(a) \\rangle$, and polarizability, $\\langle (\\delta \\mu_{\\mathbf{\\hat{n}}}(a) )^2 \\rangle$, computed from the atomistic simulations of SPC\/E, TIP4P, and TIP5P.}\n\\label{fig:S2}\n\\end{figure}\n\n\\section{Optimization for the Hydrogen Bond Energy Parameter}\nTo determine the optimal model parameter, $\\epsilon_\\text{w}$, we compute the Kullback-Leibler divergence \\cite{Kullback1951},\n\\begin{equation}\n\\Gamma(\\epsilon_\\text{w}) = \\int d a \\int d (\\cos\\theta_\\text{OH}) \\, P_\\text{ref} (\\cos\\theta_\\text{OH} | a) \\ln \\left[ \\frac{ P_\\text{ref} (\\cos\\theta_\\text{OH} | a)}{P (\\cos\\theta_\\text{OH} | a,\\epsilon_\\text{w})} \\right],\n\\tag{S.1}\n\\label{gamma}\n\\end{equation}\nwhere $P_\\text{ref}(\\cos\\theta_\\text{OH} | a)$ and $P (\\cos\\theta_\\text{OH} | a,\\epsilon_\\text{w})$ are the reduced orientational distributions obtained from atomistic simulation and our mean-field model, respectively. This quantity measures how far the probability distribution of our model deviates from the reference given $\\epsilon_\\text{w}$. As indicated above, it becomes a function of $\\epsilon_\\text{w}$ and thus we choose the parameter that minimizes the \\emph{fitness} function, \n\\begin{equation}\n\\epsilon_\\text{w}^* = \\argmin_{\\epsilon_\\text{w}}\\{\\Gamma(\\epsilon_\\text{w})\\},\n\\tag{S.2}\n\\end{equation}\nas the effective hydrogen bond energy.\n\n\n\\section{Fluctuations in hydrogen bond geometry}\nWe quantify the distortions in hydrogen bond geometry and the associated energetics by analyzing the atomistic simulation results as follows.\nNotably, here relatively simple algorithms for quantifying various aspects of molecular geometry translate into complicated mathematical expressions. \nLet $\\mathbf{v}_{k}^{(i)} = \\vec{r}_k^{(i)} - \\vec{r}_\\text{O}^{(i)}$, where $\\vec{r}_\\text{O}^{(i)}$ is the position of the oxygen of the $i$th water molecule and $\\vec{r}_k^{(i)}$ is the position of the $k$th bonding site on it ($k=1,2$ indicate the hydrogens and $k=3,4$ indicate the lone pairs for TIP5P water). \nThen $\\mathbf{v}_{k}^{(i)}$ represents the direction of ideal hydrogen bonding coordination through the $k$th site of the $i$th molecule.\nFor the $j$th molecule neighboring the $i$th, its deviation angle from the ideal coordination to the $i$th molecule is given by\n\\begin{equation}\n\\phi_j^{(i)} = \\min_k \\left\\{\\cos^{-1}\\!\\left( \\frac{\\mathbf{v}_{k}^{(i)} \\cdot \\mathbf{b}_j^{(i)} }{\\left|\\mathbf{v}_{k}^{(i)}\\right| \\left|\\mathbf{b}_j^{(i)} \\right|}\\right) \\right\\},\n\\label{angle_phi}\n\\tag{S.3}\n\\end{equation}\nwhere $\\mathbf{b}_j^{(i)} = \\vec{r}_\\text{O}^{(j)}- \\vec{r}_\\text{O}^{(i)}$ represents the hydrogen bond vector of the $i$th molecule to the $j$th. \nThe corresponding index,\n\\begin{equation}\ny_j^{(i)} = \\argmin_k \\left\\{\\cos^{-1}\\!\\left( \\frac{\\mathbf{v}_{k}^{(i)} \\cdot \\mathbf{b}_j^{(i)} }{\\left|\\mathbf{v}_{k}^{(i)}\\right| \\left|\\mathbf{b}_j^{(i)} \\right|} \\right) \\right\\},\n\\tag{S.4}\n\\label{b_label}\n\\end{equation}\nindicates which ideal boding direction $\\mathbf{b}_j^{(i)}$ is distorted from and thus whether it belongs to donor or acceptor hydrogen bond. \nThe inter-bond angle between the $j$th and $k$th molecules with respect to the $i$th is given by\n\\begin{equation}\n\\psi_{jk}^{(i)} = \\cos^{-1}\\!\\left( \\frac{\\mathbf{b}_j^{(i)} \\cdot \\mathbf{b}_k^{(i)} }{\\left|\\mathbf{b}_j^{(i)}\\right| \\left|\\mathbf{b}_k^{(i)}\\right|} \\right).\n\\tag{S.5}\n\\label{angle_psi}\n\\end{equation}\nThen the probability distribution of inter-bond angle at given distance $a$ is computed as,\n\\begin{equation}\nP_{\\alpha\\gamma}(\\psi|a) = \\frac{\\displaystyle \\left< \\sum_i \\sum_{j \\neq i} \\sum_{k \\neq i > j} \\delta(\\psi_{jk}^{(i)} - \\psi)\\delta(a^{(i)} - a) \\Theta_\\text{hyd}\\!\\left(\\mathbf{b}_j^{(i)} \\right) \\Theta_\\text{hyd}\\!\\left(\\mathbf{b}_k^{(i)} \\right) \\Phi_\\alpha \\!\\left(y_j^{(i)} \\right) \\Phi_\\gamma \\!\\left(y_k^{(i)} \\right) \\right>}{\\sin\\psi\\displaystyle\\left< \\sum_i \\sum_{j \\neq i} \\sum_{k \\neq i > j} \\delta(a^{(i)} - a)\\Theta_\\text{hyd}\\!\\left(\\mathbf{b}_j^{(i)} \\right) \\Theta_\\text{hyd}\\!\\left(\\mathbf{b}_k^{(i)} \\right) \\Phi_\\alpha \\!\\left(y_j^{(i)} \\right) \\Phi_\\gamma \\!\\left(y_k^{(i)} \\right)\\right>}~,\n\\tag{S.6}\n\\end{equation}\nwhere $a^{(i)}$ is the interfacial depth of the $i$th molecule,\n\\begin{equation}\n\\Theta_\\text{hyd}(\\vec{r}) = H(|\\vec{r}| - 2.4 \\,\\mathrm{\\AA})H(3.2 \\,\\mathrm{\\AA} - |\\vec{r}|)\n\\tag{S.7}\n\\end{equation} \nselects the molecules only in the first hydration shell of the $i$th molecule using the Heaviside step function, $H(x)$, and \n\\begin{equation}\n\\Phi_\\alpha \\left(x \\right) = \\left\\{ \\begin{array}{ll}\n\t\tH\\left(2.5 - x \\right), & \\quad   \\textrm{if $\\alpha$ is Donor}, \\\\\n          \tH\\left(x - 2.5 \\right), & \\quad \\textrm{if $\\alpha$ is Acceptor}, \\end{array} \\right.\n\\tag{S.8}\t\n\\end{equation}\nselects the neighboring molecule of specific bond type, $\\alpha$. Here the geometric factor, $\\sin\\psi$, corrects the bias coming from the variation of solid angle. \nFig. 3(b) of the main text shows the plots of $P_{\\alpha\\gamma}(\\psi|a)$ with $a = 10 \\,\\mathrm{\\AA}$ and $a = 0 \\,\\mathrm{\\AA}$ for the bulk and interface respectively ($0.1 \\,\\mathrm{\\AA}$ was used for the binning width of histogram).\nFor the plots in Fig. 4(a) of the main text, the distribution is integrated such that $P_{\\alpha\\gamma}(\\psi < 60^{\\circ}|a) = \\int_0^{60^{\\circ}} P_{\\alpha\\gamma}(\\psi|a)\\sin\\psi d\\psi$.\n\nComputing $P_\\text{sqz}(a)$ needs to specify the certain type of defect among the configurations of $\\psi < 60 \n\\,\\text{deg}$. There is the other type of defect than the squeezed triangular one, which is known as the intermediate of the water reorientation \\cite{Laage2006_SM}. \nThis type of defect has bifurcated hydrogen bonds through one site of the molecule such that $y_j^{(i)} = y_k^{(i)}$. \nBy excluding such cases, we can compute the probability to observe a squeezed configuration of two donor bonds as,\n\\begin{equation}\nP_\\text{sqz,DD}(a) = \\frac{\\displaystyle \\left< \\sum_i \\sum_{j \\neq i} \\sum_{k \\neq i > j} H\\!\\left(60^{\\circ} - \\psi_{jk}^{(i)} \\right)\\delta(a^{(i)} - a) \\Theta_\\text{hyd}\\!\\left(\\mathbf{b}_j^{(i)} \\right) \\Theta_\\text{hyd}\\!\\left(\\mathbf{b}_k^{(i)} \\right) \\delta \\!\\left(y_j^{(i)}y_k^{(i)} - 2 \\right) \\right>}{\\displaystyle\\left< \\sum_i \\sum_{j \\neq i} \\sum_{k \\neq i > j} \\delta(a^{(i)} - a)\\Theta_\\text{hyd}\\!\\left(\\mathbf{b}_j^{(i)} \\right) \\Theta_\\text{hyd}\\!\\left(\\mathbf{b}_k^{(i)} \\right) \\Phi_\\text{D} \\!\\left(y_j^{(i)} \\right) \\Phi_\\text{D} \\!\\left(y_k^{(i)} \\right)\\right>}~.\n\\tag{S.9}\n\\end{equation}\n\nThe average direct interaction energy for a hydrogen bond pair is computed in the bulk phase as a function of the deviation angle, $\\phi$, such that\n\\begin{equation}\nu_\\alpha (\\phi) = \\frac{\\displaystyle \\left< \\sum_i \\sum_{j \\neq i} U_{ij} \\,\\delta(\\phi_j^{(i)} - \\phi ) H\\!\\left(a^{(i)} - a_b\\right)\\Theta_\\text{hyd}\\!\\left(\\mathbf{b}_j^{(i)} \\right) \\Phi_\\alpha \\!\\left(y_j^{(i)} \\right) \\right>}{\\displaystyle \\left< \\sum_i \\sum_{j \\neq i} \\delta(\\phi_j^{(i)} - \\phi ) H\\!\\left(a^{(i)} - a_b\\right) \\Theta_\\text{hyd}\\!\\left(\\mathbf{b}_j^{(i)} \\right) \\Phi_\\alpha \\!\\left(y_j^{(i)} \\right) \\right>}~,\n\\tag{S.10}\n\\end{equation}\nwhere $U_{ij}$ is the pair potential energy between the $i$th and $j$th molecules and $a_b = 10 \\,\\mathrm{\\AA}$. The plots are given in Fig. 3(c) of the main text, normalized by the average bulk hydrogen bond energy where we used $E_\\text{HB} = 9.0 \\,k_B T$ (see the next section below for more details on this).\nSimilarly, the average direct interaction energy between two neighbors of a tagged molecule is computed as,\n\\begin{equation}\nv_{\\alpha\\gamma} (\\psi) = \\frac{\\displaystyle \\left< \\sum_i \\sum_{j \\neq i} \\sum_{k \\neq i > j} U_{jk} \\,\\delta( \\psi_{jk}^{(i)} - \\psi ) H\\!\\left(a^{(i)} - a_b\\right) \\Theta_\\text{hyd}\\!\\left(\\mathbf{b}_j^{(i)} \\right) \\Theta_\\text{hyd}\\!\\left(\\mathbf{b}_k^{(i)} \\right) \\Phi_\\alpha \\!\\left(y_j^{(i)} \\right) \\Phi_\\gamma \\!\\left(y_k^{(i)} \\right) \\right>}{\\displaystyle \\left< \\sum_i \\sum_{j \\neq i}\\sum_{k \\neq i > j}  \\delta (\\psi_{jk}^{(i)} - \\psi ) H\\!\\left(a^{(i)} - a_b\\right) \\Theta_\\text{hyd}\\!\\left(\\mathbf{b}_j^{(i)} \\right) \\Theta_\\text{hyd}\\!\\left(\\mathbf{b}_k^{(i)} \\right) \\Phi_\\alpha \\!\\left(y_j^{(i)} \\right) \\Phi_\\gamma \\!\\left(y_k^{(i)} \\right) \\right>}~.\n\\tag{S.11}\n\\end{equation}\n\n\\section{Implementation of the Three-body Fluctuation model} \nFollowing the notations used in the previous section, let $\\{\\mathbf{\\hat{v}}_1, \\mathbf{\\hat{v}}_2, \\mathbf{\\hat{v}}_3, \\mathbf{\\hat{v}}_4\\}$ be the unit vectors of ideal hydrogen bonding directions through the hydrogens and lone pairs of a probe water molecule of given orientation $\\vec{\\kappa}$.\nWe sample the hydrogen bond vectors, $\\{\\mathbf{b}_1, \\mathbf{b}_2, \\mathbf{b}_3, \\mathbf{b}_4 \\}$, each of which is within a certain solid angle around $\\mathbf{\\hat{v}}_i$ such that $\\mathbf{b}_i \\cdot \\mathbf{\\hat{v}}_i = |\\mathbf{b}_i| \\cos\\phi_i$ where the value for $\\cos\\phi_i$ is drawn from the uniform random distribution of [$\\cos 70^{\\circ}$, 1]. \nEach hydrogen bond vector is assigned a directionality of either donor or acceptor based on the proximity to the bonding sites (see Eq.~(\\ref{b_label})). \nThe six inter-bond angles, $\\psi_{ij}$, are calculated based on Eq.~(\\ref{angle_psi}). \nIf the bond vectors are too close with one another, \\emph{i.e.} $\\psi_{ij} < 44^{\\circ }$, they are not taken into account for computing $P(\\vec{\\kappa}|a)$ since there is almost no statistics below that in the atomistic simulation. \nHence $P(\\vec{\\kappa}|a)$ is computed as,\n\\begin{equation}\nP(\\vec{\\kappa} \\vert a) = \\int \\prod_{i=1}^{4} \\left[d\\mathbf{b}_i \\delta (|\\mathbf{b}_i| - d_\\text{HB})H\\!\\left(\\frac{\\mathbf{b}_i}{|\\mathbf{b}_i|} \\cdot \\mathbf{\\hat{v}}_i - \\cos 70^{\\circ} \\right)  \\prod_{j > i} H\\!\\left(\\cos44^{\\circ} - \\frac{\\mathbf{b}_i \\cdot \\mathbf{b}_j}{|\\mathbf{b}_i||\\mathbf{b}_j|} \\right)\\right]  \\frac{\\left \\langle e^{-\\beta E(\\vec{\\kappa},a,\\{n_k\\})} \\right\\rangle_\\mathrm{b}}{Z(a)},\n\\tag{S.12}\n\\label{eq:dist1}\n\\end{equation}\nwhere $E(\\vec{\\kappa},a,\\{n_k\\})$ follows the Eq.~(8) of the main text. \nHere we impose the same constraint on the length of hydrogen bond vectors as that in the rigid tetrahedral model. \nImplementing the fluctuations in $|\\mathbf{b}_i|$ provokes more details about the energetics, $\\tilde{u}_\\alpha$ and $\\tilde{v}_{\\alpha\\gamma}$, such as their dependence on both lengths and angles of the hydrogen bond vectors, which we have not detailed so far in this model. \n\nThe energy functions, $\\tilde{u}_\\alpha(\\phi)$ and $\\tilde{v}_{\\alpha\\gamma}(\\psi)$, are rescaled from the atomistic simulation data of $u_\\alpha(\\phi)$ and $v_{\\alpha\\gamma}(\\psi)$. \nAdditionally, we parametrize $\\tilde{u}_\\alpha(\\phi)$ by tuning the maximum value of $u_\\alpha(\\phi)$ such that\n\\begin{equation}\n\\tilde{u}_\\alpha(\\phi) = \\left\\{ \\begin{array}{ll}\n\t\t\\displaystyle\\left\\{ \\big[u_\\alpha(\\phi) - u_\\alpha(0) \\big]\\frac{u_\\alpha^* - u_\\alpha(0)}{u_{\\alpha,\\text{max}} - u_\\alpha(0)} + u_\\alpha(0) \\right\\}\\frac{|\\epsilon_\\text{w}|}{E_\\text{HB}}, & \\quad\\quad   \\textrm{if $\\phi \\le \\phi_c$},\\;\\;\\quad\\;\\;\\;\\; \\\\\\\\\n          \t\\displaystyle0, & \\quad\\quad \\textrm{if $\\phi > \\phi_c$}, \\end{array} \\right.\\\\\n\\tag{S.13}\n\\label{eq:u_alpha}\n\\end{equation}\nwhere $\\phi_c = 72^{\\circ}$, $u_{\\alpha,\\text{max}} = \\max_{\\phi < \\phi_c}\\left\\{u_\\alpha(\\phi)\\right\\}$, and $u_\\alpha^*$ is the parameter that sets the new maximum (in the original scale of $u_\\alpha$). \nHere the factor of $|\\epsilon_\\text{w}|\/E_\\text{HB}$ rescales the functions in units of the effective hydrogen bond energy of our model.\nWe observed that the behavior of $\\langle \\mu_{\\mathbf{\\hat{n}}}(a) \\rangle$ is largely sensitive to $\\tilde{u}_\\alpha(\\phi)$, and thus we optimized the parameters, $u_\\alpha^*$ and $\\epsilon_\\text{w}$, for the result expected from atomistic simulations.\nFor the results presented in Fig. 2 of the main text, we used  $\\epsilon_\\text{w} = -5.0 \\,k_B T$, $u_\\text{D}^* = +0.2 \\,k_B T$, and $u_\\text{A}^* = -3.7 \\,k_B T$. \nWe found that the optimized $\\tilde{u}_\\text{A}(\\phi)$ is quite different from $u_\\text{A}(\\phi)$ but more like the corresponding energy computed near the interface (see Fig. \\ref{fig:S3}). \n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width = 3.5 in]{FigS3_v2}\n\\caption{$\\tilde{u}_\\alpha(\\phi)$ compared with $u_\\alpha(\\phi)$ in the same scale. Solid lines are the simulation data of $u_\\alpha(\\phi)\/E_\\text{HB}$, as originally shown in Fig.~3(c) of the main text, and circles indicate the rescaled energy functions that are optimized for the accurate trend of $\\langle \\mu_{\\mathbf{\\hat{n}}}(a) \\rangle$. Gray and blue color correspond to $\\alpha = \\mathrm{D}$ and $\\alpha = \\mathrm{A}$, respectively. The blue dashed line corresponds to $u_\\text{A}(\\phi|a=1\\,\\mathrm{\\AA})$, \\emph{i.e.}, the average direct interaction energy for a hydrogen bond pair at $a = 1\\,\\mathrm{\\AA}$.}\n\\label{fig:S3}\n\\end{figure}\nFor $\\tilde{v}_{\\alpha\\gamma}(\\psi)$, we simply take the values from the atomistic simulation data and rescale them in units of $\\epsilon_\\text{w}$, that is,\n\\begin{equation}\n\\tilde{v}_{\\alpha\\gamma} (\\psi) = v_{\\alpha\\gamma}(\\psi)\\frac{|\\epsilon_\\text{w}|}{E_\\text{HB}}~.\n\\tag{S.14}\n\\end{equation}\n\nAs given in Eq.~(8) of the main text, $\\tilde{v}_{\\alpha\\gamma}(\\psi)$ is combined with the auxiliary function, $\\lambda(a_i,a_j,\\psi_{ij})$, in order to accounts for the interface-specific stability of hydrogen bond defects.\nThis auxiliary function represents the energetic cost for the defects to pay based on the hydrogen bonding status of the $i$th or $j$th hydrogen bond partner. \nAssuming that this penalty is imposed on the one that donates hydrogen, we describe this function in terms of the average number and energy of hydrogen bonds through donor sites, denoted by $N_\\text{D} (a)$ and $E_\\text{D} (a)$ respectively. \nSpecifically it is given by,\n\\begin{equation}\n\\lambda_{\\alpha\\gamma} (a_i,a_j,\\psi) = \\left\\{ \\begin{array}{ll}\n\t\t\\displaystyle \\frac{N_\\text{D} (\\bar{a}_{\\alpha\\gamma})}{2}E_\\text{D} (\\bar{a}_{\\alpha\\gamma})\\frac{|\\epsilon_\\text{w}|}{E_\\text{HB}}, & \\quad   \\textrm{if $\\psi \\le \\psi_{\\alpha\\gamma}$}, \\\\\\\\\n\t\t\\displaystyle \\left[ 1 - \\frac{ v_{\\alpha\\gamma}(\\psi) - v_{\\alpha\\gamma}( \\psi_{\\alpha\\gamma})}{v_{\\alpha\\gamma}(\\psi_c) - v_{\\alpha\\gamma}(\\psi_{\\alpha\\gamma}) } \\right] \\lambda_{\\alpha\\gamma}(a_i,a_j,\\psi_{\\alpha\\gamma}), & \\quad   \\textrm{if $\\psi_{\\alpha\\gamma} < \\psi < \\psi_c$}, \\\\\\\\\n          \t\\displaystyle 0, & \\quad \\textrm{if $\\psi \\ge \\psi_c$}, \\end{array} \\right.\n\\tag{S.15}\n\\end{equation}\nwhere $\\psi_{\\alpha\\gamma} = \\argmin_{\\psi}\\left\\{v_{\\alpha\\gamma}(\\psi)\\right\\}$, $\\psi_c = \\argmax_{\\psi}\\left\\{v_\\text{AA}(\\psi)\\right\\}$, and\n\\begin{equation}\n\\bar{a}_{\\alpha\\gamma} (a_i,a_j) = \\left\\{ \\begin{array}{ll}\n\t\t\\displaystyle \\min \\{a_i, a_j \\}, & \\quad   \\textrm{if $\\alpha = \\gamma$},\\\\\\\\\n          \t\\displaystyle \\Phi_\\text{D}\\!\\left(y_i \\right)a_i + \\Phi_\\text{D}\\!\\left(y_j \\right)a_j, & \\quad \\textrm{if $\\alpha \\ne \\gamma$}. \\end{array} \\right.\n\\tag{S.16}\n\\end{equation}\nHere we let the penalty taken by the hydrogen bond partner located closer to the interface, but we make the exception for donor-acceptor bond pairs based on the typical structure of cyclic water trimer \\cite{keutsch2003water}.\n$N_\\text{D}(a)$ and $E_\\text{D}(a)$ are computed from atomistic simulation as, \n\\begin{equation}\nN_\\text{D} (a) = \\frac{\\displaystyle \\left< \\sum_i \\sum_{j \\neq i} \\delta(a^{(i)} - a) H\\!\\left(30^{\\circ} - \\phi_j^{(i)} \\right) H\\!\\left(3.5 \\,\\mathrm{\\AA} - \\left\\vert \\mathbf{b}_j^{(i)}\\right\\vert \\right) \\Phi_\\text{D}\\!\\left(y_j^{(i)} \\right) \\right>}{\\displaystyle \\left< \\sum_i \\delta\\!\\left(a^{(i)} - a \\right) \\right>}~,\n\\tag{S.17}\n\\end{equation}\nand\n\\begin{equation}\nE_\\text{D} (a) = \\frac{\\displaystyle \\left< \\sum_i \\sum_{j \\neq i} U_{ij} \\, \\delta(a^{(i)} - a) H\\!\\left(30^{\\circ} - \\phi_j^{(i)} \\right) H\\!\\left(3.5 \\,\\mathrm{\\AA} - \\left\\vert \\mathbf{b}_j^{(i)}\\right\\vert \\right) \\Phi_\\text{D}\\!\\left(y_j^{(i)} \\right) \\right>}{\\displaystyle \\left< \\sum_i \\sum_{j \\neq i} \\delta(a^{(i)} - a) H\\!\\left(30^{\\circ} - \\phi_j^{(i)} \\right) H\\!\\left(3.5 \\,\\mathrm{\\AA} - \\left\\vert \\mathbf{b}_j^{(i)}\\right\\vert \\right) \\Phi_\\text{D}\\!\\left(y_j^{(i)} \\right) \\right>}~,\n\\tag{S.18}\n\\end{equation}\nwhere the definition of a good hydrogen bond follows the one by Luzar and Chandler \\cite{luzar1996effect}.\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width = 6.4 in]{FigS4_v2}\n\\caption{(a) Interfacial profiles of average number and energy of hydrogen bonds through donor sites, rendered in orange and blue lines respectively. (b) Illustration of a three-body interaction term implemented in our model. Solid lines show the different energetic preferences for highly distorted configurations depending on the interfacial depth of hydrogen bond partner. Dashed line corresponds to $\\tilde{v}_\\text{AA}$ without the attenuation by $\\lambda_\\text{AA}$.}\n\\label{fig:S4}\n\\end{figure}\nAs illustrated in Fig.~\\ref{fig:S4}, they change dramatically within the first $2\\,\\mathrm{\\AA}$ of the interfacial region such that the effect of three-body interaction also becomes significant in that region. \nHere we take the value of $E_\\text{HB}$ from $E_\\text{D}(a)$ by setting $E_\\text{HB} = |E_\\text{D}(a_b)| = 8.45 \\,k_B T = 20.9 \\,\\mathrm{kJ\/mol}$ at $T = 298 \\text{ K}$.\n\n\nIn order to evaluate $P(\\vec{\\kappa}|a)$, we obtain an approximate analytic expression for $\\left< e^{-\\beta E(\\vec{\\kappa},a,\\{n_k\\})} \\right>_\\text{b}$ in Eq.~(\\ref{eq:dist1}).\nHere we made the same assumption as that of the rigid tetrahedral model, such that $\\langle n_i n_j \\rangle \\approx \\langle n_i \\rangle \\langle n_j \\rangle $ for $i \\ne j$\n\\footnote{\nThis is definitely a rough approximation since it neglects the density correlation between hydrogen bond partners even in the squeezed configurations. However, more accurate treatment for the density correlation provokes again the distance dependence of the energy functions which we have not detailed so far in this model. \n}. \nWithin this approximation, we can write\n\\begin{align*}\n\\left< e^{-\\beta E(\\vec{\\kappa},a,\\{n_k\\})}\\right>_\\text{b} &= \\prod_{i=1}^4 \\langle n_i \\rangle_\\text{b} e^{-\\beta \\tilde{u}_\\alpha(\\phi_i)} \\prod_{j > i}^{4} e^{-\\beta  \\tilde{v}_{\\alpha\\gamma}(\\psi_{ij})} + \\sum_{k=1}^4 \\left[ 1 - \\langle n_k \\rangle_\\text{b}\\right] \\prod_{i \\neq k}^4 \\langle n_i \\rangle_\\text{b} e^{-\\beta \\tilde{u}_\\alpha(\\phi_i)} \\prod_{j\\neq k > i}^4 e^{-\\beta  \\tilde{v}_{\\alpha\\gamma}(\\psi_{ij})} \\nonumber\\\\ &\\quad+ \\sum_{i = 1}^4\\sum_{j > i}^4  \\langle n_i \\rangle_\\text{b}\\langle n_j \\rangle_\\text{b} \\left[1 - \\langle n_k \\rangle_\\text{b} \\right] \\left[1 - \\langle n_l \\rangle_\\text{b} \\right] e^{-\\beta \\left[\\tilde{u}_\\alpha(\\phi_i) + \\tilde{u}_\\alpha(\\phi_j) + \\tilde{v}_{\\alpha\\gamma}(\\psi_{ij}) \\right]} \\nonumber\\\\ &\\quad\\quad+ \\sum_{k=1}^4 \\langle n_k \\rangle_\\text{b} e^{-\\beta \\tilde{u}_\\alpha(\\phi_k)} \\prod_{i \\neq k}^4 \\left[1 - \\langle n_i \\rangle_\\text{b}\\right] + \\prod_{i=1}^4 \\left[1 - \\langle n_i \\rangle_\\text{b}\\right]~,\\nonumber\n\\tag{S.19}\n\\end{align*}\nwhere  $\\langle n_i \\rangle_\\text{b} = P_\\text{HB} (a_i) = \\rho(a_i)\/2\\rho_b$ and the dummy indices, $k$ and $l$, in the third term are the numbers among $\\{1,2,3,4\\}$ such that $i \\ne j \\ne k \\ne l$. \nThe resulting reduced probability distribution, $P(\\cos\\theta_\\text{OH}|a)$, is given in Fig.~\\ref{fig:S5}.\nAlthough its qualitative feature is still the same as the result from the rigid tetrahedral model, its details are closer to that from the atomistic simulation of TIP5P water (especially at $a < 3 \\,\\mathrm{\\AA}$).\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width = 3.4 in]{FigS5_v2}\n\\caption{Orientational distributions, $P(\\cos\\theta_\\text{OH}|a)$, computed from (a) the three-body fluctuation model and (b) the rigid tetrahedral model for the TIP5P force field. Color shading indicates the probability density.}\n\\label{fig:S5}\n\\end{figure}\n\n\\section{Application of mean-field models to SPC\/E force field}\nDespite sharing similar bulk hydrogen bonding structures, different classical water models can yield non-trivial differences in interfacial structure. This is highlighted to some extent in Figs.~\\ref{fig:S1} and \\ref{fig:S2}. As Fig.~\\ref{fig:S2} illustrates, different water models exhibit similar trends in $\\langle \\mu_{\\mathbf{\\hat{n}}}(a) \\rangle$ and $\\langle (\\delta \\mu_{\\mathbf{\\hat{n}}}(a) )^2 \\rangle$, but they differ in their quantitative characteristics. To evaluate the ability of our mean field model to capture these differences have also applied our model to the SPC\/E force field. To do this, we have followed the same procedure described herein but using the data from molecular dynamics simulations of SPC\/E water rather than TIP5P. For the SPC\/E-parameterized mean field model we have found similarly good agreement in reproducing $P(\\cos\\theta_\\text{OH}|a)$, as illustrated in Fig.~\\ref{fig:S6}, but as Fig.~\\ref{fig:S7}  illustrates, the agreement to $\\langle \\mu_{\\mathbf{\\hat{n}}}(a) \\rangle$ and $\\langle (\\delta \\mu_{\\mathbf{\\hat{n}}}(a) )^2 \\rangle$, is not as strong for SPC\/E as it is for TIP5P. \n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width = 3.0 in]{FigS6}\n\\caption{$P(\\cos\\theta_\\text{OH}|a)$ computed from (a) the atomistic simulation with SPC\/E water and (b) the rigid tetrahedral model optimized for the SPC\/E force field ($\\epsilon_\\text{w}^* = -1.8\\,k_B T = -4.46 \\text{ kJ\/mol}$).}\n\\label{fig:S6}\n\\end{figure}\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width = 5.0 in]{FigS7}\n\\caption{Interfacial polarization, $\\langle \\mu_{\\mathbf{\\hat{n}}}(a) \\rangle$, and polarizability, $\\langle (\\delta \\mu_{\\mathbf{\\hat{n}}}(a) )^2 \\rangle$, computed from the three-body fluctuation model parametrized for the SPC\/E force field, in comparison to the molecular dynamics simulation results.}\n\\label{fig:S7}\n\\end{figure}\n\nWe understand that the difference in the ability of our model to reproduce dipolar polarization\/polarizability between SPC\/E and TIP5P arises due to the differences in the geometric tendencies inherent to these force fields. The TIP5P force field is built upon a tetrahedral charge scaffold, so non-ideal hydrogen bond structures are more naturally described in terms of their deviations from this scaffold. On the other hand, SPC\/E is built upon a triangular charge scaffold (albeit with a tetrahedral bond angle) so non-ideal hydrogen bond structures are less well represented in term of deviation from a tetrahedral scaffold. We speculate that we could improve quantitative accuracy of our model in reproducing SPC\/E results by modifying the details of the underlying geometry of our mean field model.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:introduction}\n\nHeat exchanger network synthesis (HENS) \nminimizes cost and improves energy recovery in chemical processes \\citep{biegleretal:1997, smith:2000, eliaetal:2010, balibanetal:2012}.  \nHENS exploits excess heat by integrating process hot and cold streams and improves energy efficiency by reducing utility usage \\citep{floudas:1987, naess:1988, furman:2002, escobar-et-trierweiler:2013}. \n\\citet{floudas:2012} review the critical role of heat integration for energy systems producing liquid transportation fuels \\citep{niziolek:2015}.\nOther important applications of HENS include: refrigeration systems \\citep{shelton:1986}, batch semi-continuous processes \\citep{zhao:1998,castro:2015} and water utilization systems \\citep{bagakewicz:2002}.\n\n\n\nHeat exchanger network design is a mixed-integer nonlinear optimization (MINLP) problem \\citep{yee:1990, Ciric:1991, papalexandri-pistikopoulos:1994,hasan-etal:2010}. \\cite{mistry:2016} recently showed that expressions incorporating logarithmic mean temperature difference, i.e.\\ the nonlinear nature of heat exchange, may be reformulated to decrease the number of nonconvex nonlinear terms in the optimization problem. But HENS remains a difficult MINLP with many nonconvex nonlinearities. One way to generate good HENS solutions is to use the so-called \\emph{sequential method} \\citep{furman:2002}.\nThe sequential method decomposes the original HENS MINLP into three tasks: (i) minimizing utility cost, (ii) minimizing the number of matches, and (iii) minimizing the investment cost. \nThe method optimizes the three mathematical models sequentially with: (i) a linear program (LP) \\citep{cerdaetwesterberg:1983, papoulias:1983}, (ii) a mixed-integer linear program (MILP) \\citep{cerda:1983, papoulias:1983}, and (iii) a nonlinear program (NLP) \\citep{floudas:1986}.\nThe sequential method may not return the global solution of the original MINLP, but solutions generated with the sequential method are practically useful.\n\nThis paper investigates the \\emph{minimum number of matches} problem \\citep{floudas}, the computational bottleneck of the sequential method. \nThe minimum number of matches problem is a strongly $\\mathcal{NP}$-hard MILP \\citep{furman:2001}.\nMathematical symmetry in the problem structure combinatorially increases the possible stream configurations and deteriorates the performance of exact, tree-based algorithms \\citep{kouyialis:2016}. \n\nBecause state-of-the-art approaches cannot solve the minimum number of matches problem to global optimality for moderately-sized instances \\citep{chen:2015}, engineers develop experience-motivated heuristics \\citep{hindmarsh:1983,cerdaetwesterberg:1983}. {\\cite{hindmarsh:1983} highlight the importance of generating good solutions quickly: a design engineer may want to actively interact with a good minimum number of matches solution and consider changing the utility usage as a result of the MILP outcome.}\n\\citet{furman:2004} propose a collection of approximation algorithms, i.e.\\ heuristics with performance guarantees, for the minimum number of matches problem by exploiting the LP relaxation of an MILP formulation.\n\\citet{furman:2004} present a unified worst-case analysis of their algorithms' performance guarantees and show a non-constant approximation ratio scaling with the number of temperature intervals.\nThey also prove a constant performance guarantee for the single temperature interval problem.\n\nThe standard MILP formulations for the minimum number of matches contain big-M constraints, i.e.\\ the on\/off switches associated with weak continuous relaxations of MILP.\nBoth optimization-based heuristics and exact state-of-the-art methods for solving minimum number of matches problem are highly affected by the big-M parameter.\nTrivial methods for computing the big-M parameters are typically adopted, but \\citet{gundersen:1997} propose a tighter way of computing the big-M parameters.\n\nThis manuscript develops new heuristics and provably efficient approximation algorithms for the minimum number of matches problem. These methods have guaranteed solution quality and efficient run-time bounds. \nIn the sequential method, many possible stream configurations are required to evaluate the minimum overall cost \\citep{floudas}, so a complementary contribution of this work is a heuristic methodology for producing multiple solutions efficiently.\nWe classify the heuristics based on their algorithmic nature into three categories: (i) relaxation rounding, (ii) water filling, and (iii) greedy packing.\n\n{\nThe relaxation rounding heuristics we consider are (i) Fractional LP Rounding (FLPR), (ii) Lagrangian Relaxation Rounding (LRR), and (iii) Covering Relaxation Rounding (CRR).\nThe water-filling heuristics are (i) Water-Filling Greedy (WFG), and (ii) Water-Filling MILP (WFM). \nFinally, the greedy packing heuristics are (i) Largest Heat Match LP-based (LHM-LP), (ii) Largest Heat Match Greedy (LHM), (iii) Largest Fraction Match (LFM), and (iv) Shortest Stream (SS).\nMajor ingredients of these heuristics are adaptations of single temperature interval algorithms and maximum heat computations with match restrictions.\nWe propose (i) a novel MILP formulation, and (ii) an improved greedy approximation algorithm for the single temperature interval problem.\nFurthermore, we present (i) a greedy algorithm computing maximum heat between two streams and their corresponding big-M parameter, (ii) an LP computing the maximum heat in a single temperature interval using a subset of matches, and (iii) an extended maximum heat LP using a subset of matches on multiple temperature intervals.\n}\n\nThe manuscript proceeds as follows: \nSection \\ref{sec:preliminaries} formally defines the minimum number of matches problem and discusses mathematical models.\nSection \\ref{sec:heuristics_performance_guarantees} discusses computational complexity and introduces a new $\\mathcal{NP}$-hardness reduction of the minimum number of matches problem from bin packing. \nSection \\ref{sec:single_temperature_interval} \nfocusses on the single temperature interval problem.\nSection \\ref{sec:max_heat} explores computing the maximum heat exchanged between the streams with match restrictions.\nSections \\ref{sec:relaxation_rounding} - \\ref{sec:greedy_packing} present our heuristics for the minimum number of matches problem based on: (i) relaxation rounding, (ii) water filling, and (iii) greedy packing, respectively, as well as new theoretical performance guarantees.\nSection \\ref{sec:results} evaluates experimentally the heuristics and discusses numerical results.\nSections \\ref{sec:discussion} and \\ref{sec:conclusion} discuss the manuscript contributions and conclude the paper.\n\n\n\n\\begin{comment}\n\\begin{table}[h]\n\n\\scriptsize\n\\begin{center}\n\\begin{adjustbox}{center}\n\\begin{tabular}{ | c | c c c | c c | c c c c| } \n\\hline\n& \\multicolumn{3}{|c|}{\\textbf{Relaxation Rounding}} & \\multicolumn{2}{c}{\\textbf{Water Filling}} & \\multicolumn{4}{|c|}{\\textbf{Greedy Packing}} \\\\\n& FLPR & LRR & CRR & WFM & WFG & LHM & LFM & LHM-LP & SS \\\\\n& (\\ref{Subsection:FLPR}) & (\\ref{Subsection:LRR}) & (\\ref{Subsection:CRR}) & (\\ref{sec:water_filling}) & (\\ref{sec:water_filling}) & (\\ref{Subsection:Largest_Heat_Match}) & (\\ref{Subsection:LFM}) & (\\ref{Subsection:Largest_Heat_Match}) & (\\ref{Subsection:SS}) \\\\\n\\hline\n\\textbf{Single Temperature Interval Problem} & & & & & & & & & \\\\ \nMILP Model (\\ref{Sec:SingleTemperatureIntervalProblem-MILP}) & & & & \\checkmark & & & & & \\\\ \nApproximation Algorithm (\\ref{Sec:SingleTemperatureIntervalProblem-Approximation}) & & & & & \\checkmark & & & & \\\\ \n\\hline\n\\textbf{Maximum Heat Computations} & & & & & & & & & \\\\\nTwo Streams, Big-M Parameter (\\ref{Sec:MaxHeat_2Streams}) & \\checkmark & \\checkmark & \\checkmark & & & \\checkmark & \\checkmark & & \\checkmark \\\\ \nSingle Temperature Interval (\\ref{Sec:MaxHeat_SingleInterval}) & & & & \\checkmark & \\checkmark & & & & \\\\ \nMultiple Temperature Intervals (\\ref{Sec:MaxHeat_MultipleIntervals}) & & & \\checkmark & & & & & \\checkmark & \\\\ \n\\hline\n\\end{tabular}\n\\end{adjustbox}\n\\end{center}\n\\caption{Table indicating the single temperature interval problem and maximum heat with match restrictions components used by each heuristic. Each element is associated with a section number. This table may be used as a roadmap to the paper.}\n\\label{Table:Heuristic_Components}\n\\end{table}\t\n\\end{comment}\n\n\n \n\n\\section{Minimum Number of Matches for Heat Exchanger Network Synthesis}\n\\label{sec:preliminaries}\nThis section defines the minimum number of matches problem and presents the standard transportation and transshipment MILP models. \nTable \\ref{tbl:notation} contains the notation.\n\n\\singlespacing\n\\begin{longtable}{l l l}\n\\caption{Nomenclature}\\\\\n\\toprule\nName & Description \\\\\n\\midrule\n{\\bf Cardinalities} & \\\\\n$n$ & Number of hot streams \\\\\n$m$ & Number of cold streams \\\\\n$k$ & Number of temperature intervals\\\\\n$v$ & Number of matches (objective value) \\\\\n\\midrule\n{\\bf Indices} \\\\\n$i\\in H$ & Hot stream\\\\\n$j\\in C$ & Cold stream\\\\\n$s,t,u \\in T$ & Temperature interval\\\\\n$b\\in B$ & Bin (single temperature interval problem) \\\\\n\\midrule\n{\\bf Sets} & & \\\\\n$H$, $C$ & Hot, cold streams \\\\\n$T$ & Temperature intervals \\\\\n$M$ & Set of matches (subset of $H\\times C$) \\\\\n$C_i(M), H_j(M)$ & Cold, hot streams matched with $i\\in H$, $j\\in C$ in $M$ \\\\\n$B$ & Bins (single temperature interval problem) \\\\\n$A(M)$ & Set of valid quadruples $(i,s,j,t)$ with respect to a set $M$ of matches \\\\\n$A_u(M)$ & Set of quadruples $(i,s,j,t)\\in A(M)$ with $s\\leq u<t$ \\\\\n$V^H(M)$ & Set of pairs $(i,s)\\in H\\times T$ appearing in $A(M)$ (transportation vertices) \\\\\n$V^C(M)$ & Set of pairs $(j,t)\\in C\\times T$ appearing in $A(M)$ (transportation vertices) \\\\\n$V_{i,s}^C(M)$ & Set of pairs $(j,t)\\in V^C(M)$ such that $(i,s,j,t)$ belongs to $A(M)$ \\\\\n$V_{j,t}^H(M)$ & Set of pairs $(i,s)\\in V^H(M)$ such that $(i,s,j,t)$ belongs to $A(M)$ \\\\\n\\midrule\n{\\bf Parameters} & \\\\\n$h_i$ & Total heat supplied by hot stream $i$ ($h_i=\\sum_{s\\in T}\\sigma_{i,s}$) \\\\\n$h_{\\max}$ & Maximum heat among all hot streams ($h_{\\max}=\\max_{i\\in H}\\{h_i\\}$) \\\\\n$c_j$ & Total heat demanded by cold stream $j$ ($c_j=\\sum_{t\\in T}\\delta_{j,t}$) \\\\\n$\\sigma_{i,s}$ & Heat supply of hot stream $i$ in interval $s$ \\\\\n$\\delta_{j,t}$ & Heat demand of cold stream $j$ in interval $t$ \\\\\n$\\vec{\\sigma}, \\vec{\\delta}$ & Vectors of all heat supplies, demands \\\\ \n$\\vec{\\sigma}_t, \\vec{\\delta}_t$ & Vectors of all heat supplies, demands in temperature interval $t$ \\\\ \n$R_{t}$ & Residual heat exiting temperature interval $t$\\\\\n$U_{i,j}$ & Upper bound (big-M parameter) on the heat exchanged via match $(i,j)$ \\\\\n$\\lambda_{i,j}$ & Fractional cost approximation of match $(i,j)$ (Lagrangian relaxation) \\\\\n$\\vec{\\lambda}$ & Vector of all fractional cost approximations $\\lambda_{i,j}$ \\\\\n\\midrule\n{\\bf Variables} & \\\\\n$y_{i,j}$ & Binary variable indicating whether $i$ and $j$ are matched \\\\\n$q_{i,j,t}$ & Heat of hot stream $i$ received by cold stream $j$ in interval $t$ \\\\\n$q_{i,s,j,t}$ & Heat exported by hot stream $i$ in $s$ and received by cold stream $j$ in $t$ \\\\\n$\\vec{y}, \\vec{q}$ & Vectors of binary, continuous variables \\\\\n$r_{i,s}$ & Heat residual of heat of hot stream $i$ exiting $s$ \\\\\n$x_b$ & Binary variable indicating whether bin $b$ is used \\\\\n$w_{i,b}$ & Binary variable indicating whether hot stream $i$ is placed in bin $b$ \\\\\n$z_{j,b}$ & Binary variable indicating whether cold stream $j$ is placed in bin $b$ \\\\\n\\midrule\n{\\bf Other} & \\\\\n$N$ & Minimum cost flow network  \\\\\n$G$ & Solution graph (single temperature interval problem) \\\\\n$\\phi(M)$ & Filling ratio of a set $M$ of matches \\\\\n$\\vec{y}^f, \\vec{q}^f$ & Optimal fractional solution \\\\\n$\\alpha_i, \\beta_j$ & Number of matches of hot stream $i$, cold stream $j$ \\\\\n$L_{i,j}$ & Heat exchanged from hot stream $i$ to cold stream $j$ \\\\\n$I$ & Instance of the problem \\\\\n$r$ & Remaining heat of an algorithm \\\\\n\\bottomrule\n\\label{tbl:notation}\n\\end{longtable}\n\\onehalfspacing\n\n\\subsection{Problem Definition}\n\n{\n\nHeat exchanger network design involves a set $HS$ of \\emph{hot process streams} to be cooled and a set $CS$ of \\emph{cold process streams} to be heated.\nEach hot stream $i$ posits an initial temperature $T_{\\text{in},i}^{HS}$ and a target temperature $T_{\\text{out},i}^{HS}$ ($<T_{\\text{in},i}^{HS}$).\nAnalogously, each cold stream $j$ has an initial temperature $T_{\\text{in},j}^{CS}$ and a target temperature $T_{\\text{out},j}^{CS}$ ($>T_{\\text{in},j}^{CS}$).\nEvery hot stream $i$ and cold stream $j$ are associated flow rate heat capacities $FCp_i$ and $FCp_j$, respectively.\nMinimum heat recovery approach temperature $\\Delta T_{\\min}$ relates the hot and cold stream temperature axes.\nA hot utility $i$ in a set $HU$ and a cold utility $j$ in a set $CU$ may be purchased at a cost, e.g.\\ with unitary costs $\\kappa_i^{HU}$ and $\\kappa_j^{CU}$.\nLike the streams, the utilities have inlet and outlet temperatures $T_{\\text{in},i}^{HU}$, $T_{\\text{out},i}^{HU},T_{\\text{in},j}^{CU}$ and $T_{\\text{out},j}^{CU}$.\nThe first step in a sequential approach to HENS minimizes the utility cost and thereby specifies the heat each utility introduces in the network.\nThe next step minimizes the number of matches.\n\\ref{App:Minimum_Utility_Cost} discusses the transition from the minimizing utility cost to minimizing the number of matches.\nAfter this transition, each utility may, without loss of generality, be treated as a stream.\n}\n\nThe minimum number of matches problem posits a set of \\emph{hot process streams} to be cooled and a set of \\emph{cold process streams} to be heated.\nEach stream is associated with an initial and a target temperature. \nThis set of temperatures defines a collection of \\emph{temperature intervals}. \nEach hot stream exports (or supplies) heat in each temperature interval between its initial and target temperatures.\nSimilarly, each cold stream receives (or demands) heat in each temperature interval between its initial and target temperatures.\n\\ref{App:Minimum_Utility_Cost} formally defines the temperature range partitioning.\nHeat may flow from a hot to a cold stream in the same or a lower temperature interval, but not in a higher one.\nIn each temperature interval, the \\emph{residual heat} descends to lower temperature intervals.\nA zero heat residual is a \\emph{pinch point}.\nA pinch point restricts the maximum energy integration and divides the network into subnetworks. \n\nA problem instance consists of a set $H=\\{1,2,\\ldots,n\\}$ of hot streams, a set $C=\\{1,2,\\ldots,m\\}$ of cold streams, and a set $T=\\{1,2,\\ldots,k\\}$ of temperature intervals. \nHot stream $i\\in H$ has heat supply $\\sigma_{i,s}$ in temperature interval $s\\in T$ and cold stream $j\\in C$ has heat demand $\\delta_{j,t}$ in temperature interval $t\\in T$.\nHeat conservation is satisfied, i.e.\\ $\\sum_{i\\in H}\\sum_{s\\in T}\\sigma_{i,s} = \\sum_{j\\in C}\\sum_{t\\in T}\\delta_{j,t}$.\nWe denote by $h_i=\\sum_{s\\in T}\\sigma_{i,s}$ the total heat supply of hot stream $i\\in H$ and by $c_j=\\sum_{t\\in T}\\delta_{j,t}$ the total heat demand of cold stream $j\\in C$.\n\nA feasible solution specifies a way to transfer the hot streams' heat supply to the cold streams, i.e.\\ an amount $q_{i,s,j,t}$ of heat exchanged between hot stream $i\\in H$ in temperature interval $s\\in T$ and cold stream $j\\in C$ in temperature interval $t\\in T$.\nHeat may only flow to the same or a lower temperature interval, i.e.\\ $q_{i,s,j,t}=0$, for each $i\\in H$, $j\\in C$ and $s,t\\in T$ such that $s>t$.\nA hot stream $i\\in H$ and a cold stream $j\\in C$ are \\emph{matched}, if there is a positive amount of heat exchanged between them, i.e.\\ $\\sum_{s,t\\in T}q_{i,s,j,t}>0$.\nThe objective is to find a feasible solution minimizing the number of matches $(i,j)$.\n\n\n\n\\subsection{Mathematical Models}\nThe transportation and transshipment models formulate the minimum number of matches as a mixed-integer linear program (MILP).\n\n\\paragraph{Transportation Model \\citep{cerda:1983}} \nAs illustrated in Figure \\ref{Fig:transportation}, the transportation model represents heat as a commodity transported from supply nodes to destination nodes.\nFor each hot stream $i\\in H$, there is a set of supply nodes, one for each temperature interval $s\\in T$ with $\\sigma_{i,s}>0$.\nFor each cold stream $j\\in C$, there is a set of demand nodes, one for each temperature interval $t\\in T$ with $\\delta_{j,t}>0$.\nThere is an arc between the supply node $(i,s)$ and the destination node $(j,t)$ if $s\\leq t$, for each $i\\in H$, $j\\in C$ and $s,t\\in T$.\n\nIn the MILP formulation, variable $q_{i,s,j,t}$ specifies the heat transferred from hot stream $i\\in H$ in temperature interval $s\\in T$ to cold stream $j\\in C$ in temperature interval $t\\in T$. \nBinary variable $y_{i,j}$ if whether streams $i\\in H$ and $j\\in C$ are matched or not. \nParameter $U_{i,j}$ is a big-M parameter bounding the amount of heat exchanged between every pair of hot stream $i\\in H$ and cold stream $j\\in C$, e.g.\\ $U_{i,j}=\\min\\{h_i,c_j\\}$.\nThe problem is formulated:\n{\\allowdisplaybreaks\n\\begin{align}\n\\text{min} & \\sum_{i \\in H}\\sum_{j \\in C} y_{i,j} \\label{TransportationMIP_Eq:ObjMinMatches} \\\\ \n& \\sum_{j\\in C}\\sum_{t\\in T} q_{i,s,j,t} = \\sigma_{i,s} & i\\in H, s\\in T \\label{TransportationMIP_Eq:HotStreamConservation}\\\\\n& \\sum_{i\\in H}\\sum_{s\\in T} q_{i,s,j,t} = \\delta_{j,t} & j\\in C, t\\in T \\label{TransportationMIP_Eq:ColdStreamConservation}\\\\\n& \\sum_{s,t\\in T} q_{i,s,j,t}\\leq U_{i,j}\\cdot y_{i,j} & i\\in H, j\\in C \\label{TransportationMIP_Eq:BigM_Constraint}\\\\\n& q_{i,s,j,t} = 0 & i\\in H, j\\in C, s,t\\in T: s> t \\label{TransportationMIP_Eq:ThermoConstraint} \\\\\n& y_{i,j} \\in \\{0,\\,1\\},q_{i,s,j,t}\\geq 0 & i\\in H, j\\in C,\\; s,t\\in T \\label{TransportationMIP_Eq:IntegralityConstraints}\n\\end{align}\n}\nExpression (\\ref{TransportationMIP_Eq:ObjMinMatches}), the objective function, minimizes the number of matches.\nEquations (\\ref{TransportationMIP_Eq:HotStreamConservation}) and (\\ref{TransportationMIP_Eq:ColdStreamConservation}) ensure heat conservation.\nEquations (\\ref{TransportationMIP_Eq:BigM_Constraint}) enforce a match between a hot and a cold stream if they exchange a positive amount of heat.\nEquations (\\ref{TransportationMIP_Eq:BigM_Constraint}) are \\emph{big-M constraints}.\nEquations (\\ref{TransportationMIP_Eq:ThermoConstraint}) ensure that no heat flows to a hotter temperature.\n\n{\n\nThe transportation model may be reduced by removing redundant variables and constraints.\nSpecifically, a mathematically-equivalent \\emph{reduced transportation MILP model} removes: (i) all variables $q_{i,s,j,t}$ with $s> t$ and (ii) Equations (\\ref{TransportationMIP_Eq:ThermoConstraint}).\nBut modern commercial MILP solvers may detect redundant variables constrained to a fixed value and exploit this information to their benefit. Table \\ref{Table:Transportation_Models_Comparison} shows that the aggregate performance of CPLEX and Gurobi is unaffected by the redundant constraints and variables.\n\n}\n\n\\begin{figure*}[t!]\n\\centering\n\n\\begin{subfigure}[t]{0.45\\textwidth}\n\\centering\n\\includegraphics{transportation_graph.eps}\n\\vspace*{-1cm}\n\\caption{ Transportation Model}\n\\label{Fig:transportation}\n\\end{subfigure}\n\\begin{subfigure}[t]{0.45\\textwidth}\n\\centering\n\\includegraphics{transshipment_graph.eps}\n\\vspace*{-1cm}\n\\caption{Transshipment Model}\n\\label{Fig:transshipment}\n\\end{subfigure}\n\\vspace*{-0.5cm}\n\\caption{\nIn the transportation model \\citep{cerda:1983}, each hot stream $i$ supplies $\\sigma_{i,t}$ units of heat in temperature interval $t$ which can be received, in the same or a lower temperature interval, by a cold stream $j$ which demands $\\delta_{j,t}$ units of heat in $t$. \nIn the transshipment model \\citep{papoulias:1983}, there are also intermediate nodes transferring residual heat to a lower temperature interval.\nThis figure is adapted from \\citet{furman:2004}.\n}\n\\end{figure*}\n\n\n\\paragraph{Transshipment Model \\citep{papoulias:1983}} \nAs illustrated in Figure \\ref{Fig:transshipment}, the transshipment formulation transfers heat from hot streams to cold streams via intermediate transshipment nodes.\nIn each temperature interval, the heat entering a transshipment node either transfers to a cold stream in the same temperature interval or it descends to the transshipment node of the subsequent temperature interval as residual heat.\n\nBinary variable $y_{i,j}$ is 1 if hot stream $i\\in H$ is matched with cold stream $j\\in C$ and 0 otherwise.\nVariable $q_{i,j,t}$ represents the heat received by cold stream $j\\in C$ in temperature interval $t\\in T$ originally exported by hot stream $i\\in H$.\nVariable $r_{i,s}$ represents the residual heat of hot stream $i\\in H$ that descends from temperature interval $s$ to temperature interval $s+1$. \nParameter $U_{i,j}$ is a big-M parameter bounding the heat exchanged between hot stream $i\\in H$ and cold stream $j\\in C$, e.g.\\ $U_{i,j}=\\min\\{h_i,c_j\\}$.\nThe problem is formulated:\n{\\allowdisplaybreaks\n\\begin{align}\n\\text{min} & \\sum_{i\\in H}\\sum_{j \\in C} y_{i,j} \\label{TransshipmentMIP_Eq:ObjMinMatches}  \\\\ \n& \\sum_{j \\in C} q_{i,j,s} + r_{i,s} = \\sigma_{i,s} + r_{i,s-1} & i\\in H, s\\in T \\label{TransshipmentMIP_Eq:HotStreamConservation} \\\\\n& r_{i,k} = 0 & i\\in H \\label{TransshipmentMIP_Eq:HeatConservation} \\\\\n& \\sum_{i\\in H} q_{i,j,t}  = \\delta_{j,t} &  j\\in C, t \\in T \\label{TransshipmentMIP_Eq:ColdStreamConservation} \\\\ \n& \\sum_{t\\in T} q_{i,j,t}\\leq U_{i,j}\\cdot y_{i,j} & i\\in H, j\\in C \\label{TransshipmentMIP_Eq:BigM_Constraint} \\\\\n& y_{i,j}\\in \\{0,1\\},\\; q_{i,j,t}, r_{i,s}\\geq 0 & i\\in H, j\\in C, s,t\\in T\n\\end{align}\n}\nExpression (\\ref{TransshipmentMIP_Eq:ObjMinMatches}) minimizes the number of matches. \nEquations (\\ref{TransshipmentMIP_Eq:HotStreamConservation})-(\\ref{TransshipmentMIP_Eq:ColdStreamConservation}) enforce heat conservation.\nEquation (\\ref{TransshipmentMIP_Eq:BigM_Constraint}) allows positive heat exchange between hot stream $i\\in H$ and cold stream $j\\in C$ only if $(i,j)$ are matched.\n\n\n\\section{Heuristics with Performance Guarantees}\n\\label{sec:heuristics_performance_guarantees}\n\n\\subsection{Computational Complexity}\n\\label{sec:computational_complexity}\n\nWe briefly introduce $\\mathcal{NP}$-completeness and basic computational complexity classes \\citep{arora:2009,papadimitriou:1994}.\nA \\emph{polynomial algorithm} produces a solution for a computational problem with a running time polynomial to the size of the problem instance.\nThere exist problems which admit a polynomial-time algorithm and others which do not.\nThere is also the class of \\emph{$\\mathcal{NP}$-complete problems} for which we do not know whether they admit a polynomial algorithm or not.\nThe question of whether $\\mathcal{NP}$-complete problems admit a polynomial algorithm is known as the $\\mathcal{P}=\\mathcal{NP}$ question.\nIn general, it is conjectured that $\\mathcal{P}\\neq \\mathcal{NP}$, i.e.\\ $\\mathcal{NP}$-complete problems are not solvable in polynomial time.\nAn optimization problem is \\emph{$\\mathcal{NP}$-hard} if its decision version is $\\mathcal{NP}$-complete.\nA computational problem is \\emph{strongly $\\mathcal{NP}$-hard} if it remains $\\mathcal{NP}$-hard when all parameters are bounded by a polynomial to the size of the instance.\n\n\nThe minimum number of matches problem is known to be strongly $\\mathcal{NP}$-hard, even in the special case of a single temperature interval.\n\\citet{furman:2004} propose an $\\mathcal{NP}$-hardness reduction from the well-known 3-Partition problem, i.e.\\ they show that the minimum number of matches problem has difficulty equivalent to the 3-Partition problem.\n\\ref{App:NP_harndess} presents an alternative $\\mathcal{NP}$-hardness reduction from the bin packing problem. This alternative setting of the minimum number of matches problem gives new insight into the packing nature of the problem.\nA major contribution of this paper is to design efficient, greedy heuristics motivated by packing.\n\n\\begin{theorem}\n\\label{thm:NP_hardness}\nThere exists an $\\mathcal{NP}$-hardness reduction from bin packing to the minimum number of matches problem with a single temperature interval.\n\\end{theorem}\n\\begin{omitted_proof}\nSee \\ref{App:NP_harndess}.\n\\end{omitted_proof}\n\n\n\\subsection{Approximation Algorithms}\n\\label{sec:approximation_algorithms}\n\nA heuristic with a performance guarantee is usually called an \\emph{approximation algorithm} \\citep{vazirani:2001,williamson:2011}.\nUnless $\\mathcal{P}=\\mathcal{NP}$, there is no polynomial algorithm solving an $\\mathcal{NP}$-hard problem.\nAn approximation algorithm is a polynomial algorithm producing a near-optimal solution to an optimization problem.\nFormally, consider an an optimization problem, without loss of generality minimization, and a polynomial Algorithm $A$ for solving it (not necessarily to global optimality).\nFor each problem instance $I$, let $C_A(I)$ and $C_{OPT}(I)$ be the algorithm's objective value and the optimal objective value, respectively.\nAlgorithm $A$ is $\\rho$-approximate if, for every problem instance $I$, it holds that:\n\\begin{equation*}\nC_A(I)\\leq \\rho\\cdot C_{OPT}(I).\n\\end{equation*}\nThat is, a $\\rho$-approximation algorithm computes, in polynomial time, a solution with an objective value at most $\\rho$ times the optimal objective value.\nThe value $\\rho$ is the \\emph{approximation ratio} of Algorithm $A$. \nTo prove a $\\rho$-approximation ratio, we proceed as depicted in Figure \\ref{Fig:ApproximationAlgorithm}.\nFor each problem instance, we compute analytically a lower bound $C_{LB}(I)$ of the optimal objective value, i.e.\\ $C_{LB}(I) \\leq C_{OPT}(I)$, and we show that the algorithm's objective value is at most $\\rho$ times the lower bound, i.e.\\ $C_A(I)\\leq \\rho \\cdot C_{LB}(I)$.\nThe ratio of a $\\rho$-approximation algorithm is \\emph{tight} if the algorithm is not $\\rho-\\epsilon$ approximate for any $\\epsilon>0$. \nAn algorithm is $O(f(n))$-approximate and $\\Omega(f(n))$-approximate, where $f(n)$ is a function of an input parameter $n$, if the algorithm does not have an approximation ratio asymptotically higher and lower, respectively, than $f(n)$.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{approximation_algorithm.eps}\n\\caption{Analysis of an Approximation Algorithm}\n\\label{Fig:ApproximationAlgorithm}\n\\end{center}\n\\end{figure}\n\nApproximation algorithms have been developed for two problem classes relevant to process systems engineering: heat exchanger networks \\citep{furman:2004} and pooling \\citep{dey2015analysis}. Table \\ref{Table:Heuristics_Performance_Guarantees} lists performance guarantees for the minimum number of matches problem; most are new to this manuscript.\n\n\\begin{table}[t]\n\\small\n\\begin{adjustbox}{center} \n\\begin{tabular}{ | l l c c c | }\n\\hline\n\\textbf{Heuristic} & \\textbf{Abbrev.} & \\textbf{Section} & \\textbf{Performance Guarantee} & \\textbf{Running Time} \\\\ \n\\hline\n\\multicolumn{5}{|l|} {\\textbf{Single Temperature Interval Problem}} \\\\ \nSimple Greedy & SG & \\ref{Sec:SingleTemperatureIntervalProblem-Approximation} & 2$^{\\dagger}$ (tight) & $O(nm)$ \\\\\nImproved Greedy & IG & \\ref{Sec:SingleTemperatureIntervalProblem-Approximation} &  1.5 (tight) & $O(nm)$ \\\\\n\\hline\n\\multicolumn{5}{|l|} {\\textbf{Relaxation Rounding Heuristics}} \\\\ \nFractional LP Rounding & FLPR & \\ref{Subsection:FLPR} & $O(k)^{\\dagger}$, $O(U_{\\max})$, $\\Omega(n)$ & 1 LP \\\\\nLagrangian Relaxation Rounding & LRR & \\ref{Subsection:LRR} & & 2 LPs \\\\\nCovering Relaxation Rounding & CRR & \\ref{Subsection:CRR} & & $O(nm)$ ILPs \\\\\n\\hline\n\\multicolumn{5}{|l|} {\\textbf{Water Filling Heuristics}} \\\\ \nWater Filling MILP & WFM & \\ref{sec:water_filling} \\& \\ref{Sec:SingleTemperatureIntervalProblem-MILP} & $O(k)^{\\dagger}$, $\\Omega(k)$ & $O(k)$ MILPs \\\\ \nWater Filling Greedy & WFG & \\ref{sec:water_filling} \\& \\ref{Sec:SingleTemperatureIntervalProblem-Approximation} & $O(k)^{\\dagger}$, $\\Omega(k)$ & $O(n m k)$ \\\\\n\\hline\n\\multicolumn{5}{|l|} {\\textbf{Greedy Packing Heuristics}} \\\\ \nLargest Heat Match LP-based & LHM-LP & \\ref{Subsection:Largest_Heat_Match} & $O(\\log n + \\log (h_{\\max}\/\\epsilon))$ & $O(n^2m^2)$ LPs \\\\\nLargest Heat Match Greedy & LHM & \\ref{Subsection:Largest_Heat_Match} & & $O(n^2 m^2 k)$ \\\\\nLargest Fraction Match & LFM & \\ref{Subsection:LFM} & & $O(n^2 m^2 k)$  \\\\\nShortest Stream & SS & \\ref{Subsection:SS} & & $O(n m k)$  \\\\\n\\hline\n\\end{tabular}\n\\end{adjustbox} \n\\caption{Performance guarantees for the minimum number of matches problem. The performance guarantees marked ${\\dagger}$ are from \\cite{furman:2004}; all others are new to this manuscript.}\n\\label{Table:Heuristics_Performance_Guarantees}\n\\end{table}\n\n\n\n\\section{Single Temperature Interval Problem}\n\\label{sec:single_temperature_interval}\n\nThis section proposes efficient algorithms for the single temperature interval problem.\nUsing graph theoretic properties, we obtain: (i) a novel, efficiently solvable MILP formulation without big-M constraints and (ii) an improved 3\/2-approximation algorithm. \n{Of course, the single temperature interval problem is not immediately applicable to the minimum number of matches problem with multiple temperature intervals. But designing efficient approximation algorithms for the single temperature interval is the first, essential step before considering multiple temperature intervals. Additionally, the water filling heuristics introduced in Section \\ref{sec:water_filling} repeatedly solve the single temperature interval problem.}\n\nIn the single temperature interval problem, a feasible solution can be represented as a bipartite graph $G=(H\\cup C, M)$ in which there is a node for each hot stream $i\\in H$, a node for each cold stream $j\\in C$ and the set $M\\subseteq H\\times C$ specifies the matches. \n\\ref{App:Single_Temperature_Interval} shows the existence of an optimal solution whose graph $G$ does not contain any cycle. \nA connected graph without cycles is a \\emph{tree}, so $G$ is a forest consisting of trees.\n\\ref{App:Single_Temperature_Interval} also shows that the number $v$ of edges in $G$, i.e.\\ the number of matches, is related to the number $\\ell$ of trees with the equality $v=n+m-\\ell$.\nSince $n$ and $m$ are input parameters, minimizing the number of matches in a single temperature interval is equivalent to finding a solution whose graph consists of a maximal number $\\ell$ of trees.\n\n\\subsection{Novel MILP Formulation}\n\\label{Sec:SingleTemperatureIntervalProblem-MILP}\n\nWe propose a novel MILP formulation for the single temperature interval problem.\nIn an optimal solution without cycles, there can be at most $\\min\\{n,m\\}$ trees.\nFrom a packing perspective, we assume that there are $\\min\\{n,m\\}$ available bins and each stream is placed into exactly one bin.\nIf a bin is non-empty, then its content corresponds to a tree of the graph.\nThe objective is to find a feasible solution with a maximum number of bins. \n\nTo formulate the problem as an MILP, we define the set $B=\\{1,2,\\ldots,\\min\\{n,m\\}\\}$ of available bins.\nBinary variable $x_b$ is 0 if bin $b\\in B$ is empty and 1, otherwise. \nA binary variable $w_{i,b}$ indicates whether hot stream $i\\in H$ is placed into bin $b\\in B$.\nSimilarly, a binary variable $z_{j,b}$ specifies whether cold stream $j\\in C$ is placed into bin $b\\in B$. \nThen, the minimum number of matches problem can be formulated:\n{\\allowdisplaybreaks\n\\begin{align}\n\\text{max} & \\sum_{b\\in B} x_b \\label{Eq:SingleMIP_MaxBins}  \\\\\n& x_b \\leq \\sum_{i\\in H} w_{i,b} & b\\in B \\label{Eq:SingleMIP_HotBinUsage} \\\\ \n& x_b \\leq \\sum_{j\\in C} z_{j,b} & b\\in B \\label{Eq:SingleMIP_ColdBinUsage} \\\\ \n& \\sum_{b\\in B} w_{i,b} = 1 & i\\in H \\label{Eq:SingleMIP_HotAssignment} \\\\\n& \\sum_{b\\in B} z_{j,b} = 1 & j\\in C \\label{Eq:SingleMIP_ColdAssignment} \\\\\n& \\sum_{i\\in H} w_{i,b} \\cdot h_i = \\sum_{j\\in C} z_{j,b} \\cdot c_j & b\\in B \\label{Eq:SingleMIP_BinHeatConservation} \\\\\n& x_b, w_{i,b}, z_{j,b}\\in\\{0,1\\} & b\\in B, i\\in H, j\\in C \\label{Eq:SingleMIP_Integrality}\n\\end{align}\n}\nExpression (\\ref{Eq:SingleMIP_MaxBins}), the objective function, maximizes the number of bins. \nEquations (\\ref{Eq:SingleMIP_HotBinUsage}) and (\\ref{Eq:SingleMIP_ColdBinUsage}) ensure that a bin is used if there is at least one stream in it.\nEquations (\\ref{Eq:SingleMIP_HotAssignment}) and (\\ref{Eq:SingleMIP_ColdAssignment}) enforce that each stream is assigned to exactly one bin.\nFinally, Eqs.\\ (\\ref{Eq:SingleMIP_BinHeatConservation}) ensure the heat conservation of each bin. \nNote that, unlike the transportation and transshipment models, Eqs.\\ (\\ref{Eq:SingleMIP_MaxBins})-(\\ref{Eq:SingleMIP_BinHeatConservation}) do not use a big-M parameter. {\n\\ref{App:Water_Filling} formulates the single temperature interval problem \\emph{without} heat conservation. Eqs.\\ (\\ref{Eq:SingleMIPwithoutConservation_MaxBins})-(\\ref{Eq:SingleMIPwithoutConservation_Integrality}) are similar to Eqs.\\ (\\ref{Eq:SingleMIP_MaxBins})-(\\ref{Eq:SingleMIP_Integrality}) except (i) they drop constraints (\\ref{Eq:SingleMIP_HotBinUsage}) and (ii) equalities (\\ref{Eq:SingleMIP_HotAssignment}) \\& (\\ref{Eq:SingleMIP_BinHeatConservation}) become inequalities (\\ref{Eq:SingleMIPwithoutConservation_HotAssignment}) \\& (\\ref{Eq:SingleMIPwithoutConservation_BinHeatConservation}).\n}\n\n\\subsection{Improved Approximation Algorithm}\n\\label{Sec:SingleTemperatureIntervalProblem-Approximation}\n\n\\citet{furman:2004} propose a greedy 2-approxi\\-mation algorithm for the minimum number of matches problem in a single temperature interval.\nWe show that their analysis is tight. We also propose an improved, tight $1.5$-approximation algorithm by prioritizing matches with equal heat loads and exploiting graph theoretic properties.\n\nThe simple greedy (SG) algorithm considers the hot and the cold streams in non-increasing heat load order \\citep{furman:2004}.\nInitially, the first hot stream is matched to the first cold stream and an amount $\\min\\{h_1, c_1\\}$ of heat is transferred between them.\nWithout loss of generality $h_1 > c_1$, which implies that an amount $h_1 - c_1$ of heat load remains to be transferred from $h_1$ to the remaining cold streams.\nSubsequently, the algorithm matches $h_1$ to $c_2$, by transferring $\\min\\{h_1 - c_1, c_2\\}$ heat. \nThe same procedure repeats with the other streams until all remaining heat load is transferred.\n\n\n\n\\begin{algorithm}[t] \\nonumber\n\\caption[Simple Greedy (SG)]{Simple Greedy (SG), developed by \\cite{furman:2004}, is applicable to one temperature interval only.}\n\\begin{algorithmic}[1]\n\\State Sort the streams so that $h_1\\geq h_2\\geq \\ldots\\geq h_n$ and $c_1\\geq c_2\\geq \\ldots \\geq c_m$.\n\\State Set $i = 1$ and $j = 1$.\n\\While {there is remaining heat load to be transferred}\n\\State Transfer $q_{i,j}=\\min\\{h_i, c_j\\}$ \n\\State Set $h_i = h_i - q_{i,j}$ and $c_j = c_j - q_{i,j}$\n\\State \\textbf{if} $h_i = 0$, \\textbf{then} set $i = i+1$\n\\State \\textbf{if} $c_j = 0$, \\textbf{then} set $j = j+1$\n\\EndWhile\n\\end{algorithmic}\n\\label{Alg:SimpleGreedy}\n\\end{algorithm}\n\n\n\n\\citet{furman:2004} show that Algorithm SG is 2-approximate for one temperature interval.\nOur new result in Theorem \\ref{thm:greedy} shows that this ratio is tight.\n\n\\begin{theorem}\n\\label{thm:greedy}\nAlgorithm SG achieves an approximation ratio of 2 for the single temperature interval problem and it is tight.\n\\end{theorem}\n\\begin{omitted_proof}\nSee \\ref{App:Single_Temperature_Interval}.\n\\end{omitted_proof}\n\n\\medskip\n\nAlgorithm IG improves Algorithm SG by: (i) matching the pairs of hot and cold streams with equal heat loads and (ii) using the acyclic property in the graph representation of an optimal solution. {\nIn practice, hot and cold process streams are unlikely to have equal supplies and demands of heat, so discussing equal heat loads is largely a thought experiment. But the updated analysis allows us to claim an improved performance bound on Algorithm SG. Additionally, the notion of matching roughly equivalent supplies and demands inspires the Section \\ref{Subsection:LFM} \\emph{Largest Fraction Match First} heuristic.\n}\n\n\n\\begin{algorithm}[t] \\nonumber\n\\caption[Improved Greedy (IG)]{Improved Greedy (IG) is applicable to one temperature interval only.}\n\\label{alg:impgreedy}\n\\begin{algorithmic}[1]\n\\For {each pair of hot stream $i$ and cold stream $j$ s.t. $h_i=c_j$}\n\\State Transfer $h_i$ amount of heat load (also equal to $c_j$) between them and remove them.\n\\EndFor\n\\State Run Algorithm SG with respect to the remaining streams.\n\\end{algorithmic}\n\\label{Alg:ImprovedGreedy}\n\\end{algorithm}\n\n\n\n\\medskip\n\n\\begin{theorem}\\label{thm:impgreedy}\nAlgorithm IG achieves an approximation ratio of 1.5 for the single temperature interval problem and it is tight.\n\\end{theorem}\n\\begin{omitted_proof}\nSee \\ref{App:Single_Temperature_Interval}.\n\\end{omitted_proof}\n\n\\medskip\n\n\n\n\\section{Maximum Heat Computations with Match Restrictions}\n\\label{sec:max_heat}\nThis section discusses computing the maximum heat that can be feasibly exchanged in a minimum number of matches instance. Section \\ref{Sec:MaxHeat_2Streams} discusses the specific instance of two streams and thereby reduces the value of big-M parameter $U_{i,j}$. \nSections \\ref{Sec:MaxHeat_SingleInterval} \\& \\ref{Sec:MaxHeat_MultipleIntervals} generalize Section \\ref{Sec:MaxHeat_2Streams} from 2 streams to any number of the candidate matches. Section \\ref{Sec:MaxHeat_SingleInterval} is limited to a restricted subset of matches in a single temperature interval. Section \\ref{Sec:MaxHeat_MultipleIntervals} calculates the maximum heat that can be feasibly exchanged for the most general case of multiple temperature intervals.\nThese maximum heat computations are an essential ingredient of our heuristic methods and aim in using a match in the most profitable way. They also answer the feasibility of the minimum number of matches problem.\n\n\\subsection{Two Streams and Big-M Parameter Computation}\n\\label{Sec:MaxHeat_2Streams}\n\nA common way of computing the big-M parameters is setting $U_{i,j}=\\min\\{h_i,c_j\\}$ for each $i\\in H$ and $j\\in C$. \\citet{gundersen:1997} propose a better method for calculating the big-M parameter.\nOur novel Greedy Algorithm MHG (Maximum Heat Greedy) obtains tighter $U_{i,j}$ bounds than either the trivial bounds or the \\citet{gundersen:1997} bounds by exploiting the transshipment model structure.\n\nGiven hot stream $i$ and cold stream $j$, Algorithm MHG computes the maximum amount of heat that can be feasibly exchanged between $i$ and $j$ in any feasible solution.\nAlgorithm MHG is tight in the sense that there is always a feasible solution where streams $i$ and $j$ exchange exactly $U_{i,j}$ units of heat.\nNote that, in addition to $U_{i,j}$, the algorithm computes a value $q_{i,s,j,t}$ of the heat exchanged between each hot stream $i\\in H$ in temperature interval $s\\in T$ and each cold stream $j\\in C$ in temperature interval $t\\in T$, so that $\\sum_{s,t\\in T}q_{i,s,j,t}=U_{i,j}$.\nThese $q_{i,s,j,t}$ values are required by greedy packing heuristics in Section \\ref{sec:greedy_packing}.\n\nAlgorithm \\ref{Alg:MaximumHeat} is a pseudocode of Algorithm MHG.\nThe correctness, i.e.\\ the maximality of the heat exchanged between $i$ and $j$, is a corollary of the well known maximum flow - minimum cut theorem.\nInitially, the procedure transfers the maximum amount of heat across the same temperature interval; $q_{i,u,s,u}=\\min\\{\\sigma_{i,u},\\delta_{j,u}\\}$ for each $u\\in T$.\nThe remaining heat is transferred greedily in a top down manner, with respect to the temperature intervals, by accounting heat residual capacities.\nFor each temperature interval $u\\in T$, the heat residual capacity $R_u=\\sum_{i=1}^n\\sum_{s=1}^u\\sigma_{i,s} - \\sum_{j=1}^m\\sum_{t=1}^u\\delta_{j,t}$ imposes an upper bound on the amount of heat that may descend from temperature intervals $1,2,\\ldots,u$ to temperature intervals $u+1,u+2,\\ldots,k$.\n\n\\begin{algorithm}[t] \\nonumber\n\\caption{Maximum Heat Greedy (MHG)}\n\\textbf{Input:} Hot stream $i\\in H$ and cold stream $j\\in C$ \n\\begin{algorithmic}[1]\n\\State $\\vec{q} \\leftarrow \\vec{0}$\n\\For {{$u=1,2,\\ldots,k-1$}}\n\\State {$R_u=\\sum_{i=1}^n\\sum_{s=1}^u\\sigma_{i,s} - \\sum_{j=1}^m\\sum_{t=1}^u\\delta_{j,t}$}\n\\EndFor\n\\For {$u=1,2,\\ldots,k$}\n\\State $q_{i,u,j,u}\\leftarrow \\min\\{\\sigma_{i,u},\\delta_{j,u}\\}$\n\\State $\\sigma_{i,u}\\leftarrow \\sigma_{i,u}-q_{i,u,j,u}$\n\\State $\\delta_{j,u}\\leftarrow \\delta_{j,u}-q_{i,u,j,u}$\n\\EndFor\n\\For {$s=1,2,\\ldots,k-1$}\n\\For {$t=s+1,s+2,\\ldots,k$}\n\\State $q_{i,s,j,t}=\\min\\{\\sigma_{i,s},\\delta_{j,t},\\min_{s\\leq u\\leq t-1}\\{R_u\\}\\}$\n\\State $\\sigma_{i,s}\\leftarrow \\sigma_{i,s}-q_{i,s,j,t}$\n\\State $\\delta_{j,t}\\leftarrow \\delta_{j,t}-q_{i,s,j,t}$\n\\For {$u=s,s+1,s+2,\\ldots,t-1$}\n\\State $R_u\\leftarrow R_u-q_{i,s,j,t}$\n\\EndFor\n\\EndFor\n\\EndFor\n\\State Return $\\vec{q}$\n\\end{algorithmic}\n\\label{Alg:MaximumHeat}\n\\end{algorithm}\n\n\\subsection{Single Temperature Interval}\n\\label{Sec:MaxHeat_SingleInterval}\n\nGiven an instance of the single temperature interval problem and a subset $M$ of matches, the maximum amount of heat that can be feasibly exchanged between the streams using only the matches in $M$ can be computed by solving \\ref{EquationSet:SingleMaxHeatLP_initial_stattement}.\n{Like the single temperature interval algorithms of Section \\ref{sec:single_temperature_interval}, \\ref{EquationSet:SingleMaxHeatLP_initial_stattement} is not directly applicable to a minimum number of matches problem with multiple temperature intervals. But \\ref{EquationSet:SingleMaxHeatLP_initial_stattement} is an important part of our water filling heuristics.}\nFor simplicity, \\ref{EquationSet:SingleMaxHeatLP_initial_stattement} drops\ntemperature interval indices for variables $q_{i,j}$.\n{\\allowdisplaybreaks\n\\begin{align}\n\\tag{MaxHeatLP}\n\\label{EquationSet:SingleMaxHeatLP_initial_stattement}\n\\begin{aligned}\n\\text{max} & \\sum_{(i,j)\\in M} q_{i,j} \\\\\n& \\sum_{j\\in C} q_{i,j} \\leq h_i & i\\in H \\\\\n& \\sum_{i\\in H} q_{i,j} \\leq c_j & j\\in C \\\\\n& q_{i,j} \\geq 0 & i\\in H, j\\in C\n\\end{aligned}\n\\end{align}\n}\n\n\\subsection{Multiple Temperature Intervals}\n\\label{Sec:MaxHeat_MultipleIntervals}\n\nMaximizing the heat exchanged through a subset of matches across multiple temperature intervals can solved with an LP that generalizes \\ref{EquationSet:SingleMaxHeatLP_initial_stattement}. \nThe generalized LP must satisfy the additional requirement that, after removing a maximum heat exchange, the remaining instance is feasible.\nFeasibility is achieved using residual capacity constraints which are essential for the efficiency of greedy packing heuristics (see Section \\ref{Subsection:MonotonicGreedyHeuristics}).\n\nGiven a set $M$ of matches, let $A(M)$ be the set of quadruples $(i,s,j,t)$ such that a positive amount of heat can be feasibly transferred via the transportation arc with endpoints the nodes $(i,s)$ and $(j,t)$.\nThe set $A(M)$ does not contain any quadruple $(i,s,j,t)$ with: (i) $s>t$, (ii) $\\sigma_{i,s}=0$, (iii) $\\delta_{j,t}=0$, or (iv) $(i,j)\\not\\in M$. \nLet $V^H(M)$ and $V^C(M)$ be the set of transportation vertices $(i,s)$ and $(j,t)$, respectively, that appear in $A(M)$.\nSimilarly, given two fixed vertices $(i,s)\\in V^H(M)$ and $(j,t)\\in V^C(M)$, we define the sets $V_{i,s}^C(M)$ and $V_{j,t}^H(M)$ of their respective neighbors in $A(M)$.\n\nConsider a temperature interval $u\\in T$.\nWe define by $A_u(M)\\subseteq A(M)$ the subset of quadruples with $s\\leq u< t$, for $u\\in T$. \nThe total heat transferred via the arcs in $A_u(M)$ must be upper bounded by $R_u=\\sum_{i=1}^n\\sum_{s=1}^u\\sigma_{i,s} - \\sum_{j=1}^m\\sum_{t=1}^u\\delta_{j,t}$.\nFurthermore, $A(M)$ eliminates any quadruple $(i,s,j,t)$ with $R_u=0$, for some $s\\leq u<t$.  \nFinally, we denote by $T(M)$ the subset of temperature intervals affected by the matches in $M$, i.e.\\ if $u\\in T(M)$, then there exists a quadruple $(i,s,j,t)\\in A(M)$, with $s\\leq u<t$.\nThe procedure $MHLP(M)$ is based on solving the following LP:\n{\\allowdisplaybreaks\n\\begin{align}\n\\max & \\sum_{(i,s,j,t) \\in A(M)} q_{i,s,j,t} \\label{Eq:MaxHeatLP_Objective} \\\\ \n& \\sum_{(j,t)\\in V_{i,s}^C(M)} q_{i,s,j,t} \\leq \\sigma_{i,s} & (i,s)\\in V^H(M) \\label{Eq:MaxHeatLP_HotStreamSupply}\\\\\n& \\sum_{(i,s)\\in V_{j,t}^H(M)} q_{i,s,j,t} \\leq \\delta_{j,t} & (j,t)\\in V^C(M) \\label{Eq:MaxHeatLP_ColdStreamDemand}\\\\\n& \\sum_{(i,s,j,t)\\in A_u(M)} q_{i,s,j,t}\\leq R_u & u\\in T(M) \\label{Eq:MaxHeatLP_ResidualCapacity}\\\\\n& q_{i,s,j,t}\\geq 0 & (i,s,j,t) \\in A(M) \\label{Eq:MaxHeatLP_Positiveness}\n\\end{align}\n}\nExpression (\\ref{Eq:MaxHeatLP_Objective}) maximizes the total exchanged heat by using only the matches in $M$.\nConstraints (\\ref{Eq:MaxHeatLP_HotStreamSupply}) and (\\ref{Eq:MaxHeatLP_ColdStreamDemand}) ensure that each stream uses only part of its available heat.\nConstraints (\\ref{Eq:MaxHeatLP_ResidualCapacity}) enforce the heat residual capacities.\n\n\n\\section{Relaxation Rounding Heuristics}\n\\label{sec:relaxation_rounding}\nThis section investigates relaxation rounding heuristics for the minimum number of matches problem. \nThese heuristics begin by optimizing an efficiently-solvable relaxation of the original MILP. \nThe efficiently-solvable relaxation allows violation of certain constraints, so that the optimal solution(s) is (are) typically infeasible in the original MILP.\nThe resulting infeasible solutions are subsequently rounded to feasible solutions for the original MILP.\nWe consider 3 types of relaxations.\nSection \\ref{Subsection:FLPR} relaxes the integrality constraints and proposes fractional LP rounding.\nSection \\ref{Subsection:LRR} relaxes the big-M constraints, i.e.\\ Eqs.\\ (\\ref{TransportationMIP_Eq:BigM_Constraint}), \nand uses Lagrangian relaxation rounding.\nSection \\ref{Subsection:CRR} relaxes the heat conservation equations, i.e.\\ Eqs.\\ (\\ref{TransportationMIP_Eq:HotStreamConservation})-(\\ref{TransportationMIP_Eq:ColdStreamConservation}), \nand takes an approach based on covering relaxations.\n{Figure \\ref{Figure:RR_Ingredients} shows the main components of relaxation rounding heuristics.}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{relaxation_rounding_ingredients.eps}\n\\caption{\nThe main components of relaxation rounding heuristics are (i) a preprocessing step, (ii) a relaxation, and (iii) a rounding scheme.\nThe preprocessing step constructs the relaxation.\nFractional relaxation and covering relaxation require big-M parameter computations, while Lagrangian relaxation minimum cost LP requires cost calculations.\nFLPR and LRR compute a feasible heat exchange between all streams, i.e.\\ values to variables $q_{i,s,j,t}$, by solving their respective relaxations and round the relaxed solutions according to Algorithm \\ref{Alg:FLPR}.\nHeuristic CRR adds matches incrementally until it ends up with a feasible solution.\nFeasibility is determined using the maximum heat LP in Section \\ref{Sec:MaxHeat_MultipleIntervals}.}\n\\label{Figure:RR_Ingredients}\n\\end{center}\n\\end{figure}\n\n\\subsection{Fractional LP Rounding}\n\\label{Subsection:FLPR}\n\n\nThe LP rounding heuristic, originally proposed by \\citet{furman:2004},\ntransforms an optimal fractional solution for the transportation MILP to a feasible integral solution.\nWe show that the fractional LP can be solved efficiently via network flow techniques. \nWe observe that, in the worst case, the heuristic produces a weak solution if it starts with an arbitrary optimal solution of the fractional LP. \nWe derive a novel performance guarantee showing that the heuristic is efficient when \nthe heat of each chosen match $(i,j)$ is close to big-M parameter $U_{i,j}$, in the optimal fractional solution.\n\nConsider the fractional LP obtained by replacing the integrality constraints $y_{i,j}\\in\\{0,1\\}$ of the transportation MILP, i.e.\\ Eqs.\\ (\\ref{TransportationMIP_Eq:ObjMinMatches})-(\\ref{TransportationMIP_Eq:IntegralityConstraints}), with the constraints $0\\leq y_{i,j}\\leq 1$, for each $i\\in H$ and $j\\in C$:\n\\begin{align}\n\\tag{FracLP}\n\\label{EquationSet:FracLP}\n\\begin{aligned}\n\\min & \\sum_{i \\in H}\\sum_{j \\in C} y_{i,j} & \\\\\n& \\sum_{j\\in C}\\sum_{t\\in T} q_{i,s,j,t} = \\sigma_{i,s} & i\\in H, s\\in T \\\\\n& \\sum_{i\\in H}\\sum_{s\\in T} q_{i,s,j,t} = \\delta_{j,t} & j\\in C, t\\in T  \\\\\n& \\sum_{s,t\\in T} q_{i,s,j,t}\\leq U_{i,j}\\cdot y_{i,j} & i\\in H, j\\in C  \\\\\n& q_{i,s,j,t} = 0 & \\;\\; i\\in H, j\\in C, s,t\\in T: s\\leq t \\\\\n& 0\\leq y_{i,j}\\leq 1,\\; q_{i,s,j,t}\\geq 0 & i\\in H, j\\in C,\\; s,t\\in T\n\\end{aligned}\n\\end{align}\n\n\n\\ref{EquationSet:FracLP} can be solved via minimum cost flow methods.\nFigure \\ref{Fig:min_cost_flow_network} illustrates a network $N$, i.e.\\ a minimum cost flow problem instance, such that finding a minimum cost flow in $N$ is equivalent to optimizing the fractional LP.\nNetwork $N$ is a layered graph with six layers of nodes: (i) a source node $S$, (ii) a node for each hot stream $i\\in H$, (iii) a node for each pair $(i,s)$ of hot stream $i\\in H$ and temperature interval $s\\in T$, (iv) a node for each pair $(j,t)$ for each cold stream $j\\in C$ and temperature interval $t\\in T$, (v) a node for each cold stream $j\\in C$, and (vi) a destination node $D$.\nWe add: (i) the arc $(S,i)$ with capacity $h_i$ for each $i\\in H$, (ii) the arc $(i,(i,s))$ with capacity $\\sigma_{i,s}$ for each $i\\in H$ and $s\\in T$, (iii) the arc $((i,s),(j,t))$ with infinite capacity for each $i\\in H$, $j\\in C$ and $s,t\\in T$, (iv) the arc $((j,t),j)$ with capacity $\\delta_{j,t}$ for each $j\\in H$ and $t\\in T$, and (v) the arc $(j,D)$ with capacity $c_j$ for each $j\\in C$.  \nEach arc $((i,s),(j,t))$ has cost $1\/U_{i,j}$ for $i\\in H$, $j\\in C$ and $s,t\\in T$. \nEvery other arc has zero cost.\nAny flow of cost $\\sum_i h_i$ on network $N$ is equivalent to a feasible solution for \\ref{EquationSet:FracLP} with the same cost and vice versa.\n\n\\begin{figure}[t]\n\n\\begin{center}\n\\includegraphics{min_cost_flow_network.eps}\n\\caption{Minimum cost network flow formulation of \\ref{EquationSet:FracLP}.\nThe heat is modeled as flow transferred from a source node $S$ to a destination node $D$.\nAll finite capacities are labelled above the corresponding arcs.\nThe cost is incurred in each arc between node $(i,s)\\in H\\times T$ and node $(j,t)\\in C\\times T$ under the condition that heat flows to the same or a lower temperature interval.\n}\n\\label{Fig:min_cost_flow_network}\n\\end{center}\n\n\\end{figure}\n\n\\cite{furman:2004} observe that any feasible solution of \\ref{EquationSet:FracLP} can be rounded to a feasible solution of the original problem via Algorithm \\ref{Alg:FLPR}, a simple greedy procedure that we call FLPR. \nGiven a problem instance $I$, the procedure $FractionalLP(I)$ computes an optimal solution of \\ref{EquationSet:FracLP}.\nWe denote by $(\\vec{y}^f,\\vec{q}^f)$ the optimal fractional solution.\n\n\\begin{algorithm}[t] \\nonumber\n\\caption{Fractional LP Rounding (FLPR) \\citep{furman:2004}}\n\\begin{algorithmic}[1]\n\\State $(\\vec{y}^f,\\vec{q}^f)\\leftarrow FractionalLP(I)$\n\\Comment{{(\\ref{EquationSet:FracLP}) solving, Section \\ref{Subsection:FLPR}}}\n\\State $\\vec{q} \\leftarrow \\vec{q}^f$\n\\For {each $i\\in H$ and $j\\in C$}\n\\If {$\\sum_{s,t\\in T} q_{i,s,j,t}>0$}\n\\State $y_{i,j}\\leftarrow 1$\n\\Else\n\\State $y_{i,j}\\leftarrow 0$\n\\EndIf\n\\EndFor\n\\State Return $(\\vec{y},\\vec{q})$\n\\end{algorithmic}\n\\label{Alg:FLPR}\n\\end{algorithm}\n\n\nAn inherent drawback of the \\citet{furman:2004} approach is the existence of optimal fractional solutions with unnecessary matches.\nTheorem \\ref{Thm:FLPR_negative} shows that Algorithm FLPR performance is bad in the worst case, even for instances with a single temperature interval. \nThe proof, given in \\ref{App:Relaxation_Rounding}, can be extended so that unnecessary matches occur across multiple temperature intervals.\n\\begin{theorem}\n\\label{Thm:FLPR_negative}\nAlgorithm FLPR is $\\Omega(n)$-approximate.\n\\end{theorem}\n\\begin{omitted_proof}\nSee \\ref{App:Relaxation_Rounding}.\n\\end{omitted_proof}\n\n\\medskip\n\nConsider an optimal fractional solution to \\ref{EquationSet:FracLP} and suppose that $M\\subseteq H\\times C$ is the set of pairs of streams exchanging a positive amount of heat.\nFor each $(i,j)\\in M$, denote by $L_{i,j}$ the heat exchanged between hot stream $i$ and cold stream $j$.\nWe define: \n\\begin{equation*}\n\\phi(M) = \\min_{(i,j)\\in M} \\left\\{ \\frac{L_{i,j}}{U_{i,j}} \\right\\}\n\\end{equation*}\nas the \\emph{filling ratio}, which corresponds to the minimum portion of an upper bound $U_{i,j}$ filled with the heat $L_{i,j}$, for some match $(i,j)$.\nGiven an optimal fractional solution with filling ratio $\\phi(M)$, Theorem \\ref{Thm:FLPR_positive} obtains a $1\/\\phi(M)$-approximation ratio for FLPR.\n\n\\begin{theorem}\n\\label{Thm:FLPR_positive}\nGiven an optimal fractional solution with a set $M$ of matches and filling ratio $\\phi(M)$, FLPR produces a $\\left(1\/\\phi(M) \\right)$-approximate integral solution.\n\\end{theorem}\n\\begin{omitted_proof}\nSee \\ref{App:Relaxation_Rounding}.\n\\end{omitted_proof}\n\n\\medskip\n\nIn the case where all heat supplies and demands are integers, the integrality of the minimum cost flow polytope and Theorem \\ref{Thm:FLPR_positive} imply that FLPR is $U_{\\max}$-approximate, where $U_{\\max}=\\max_{(i,j)\\in H\\times C}\\{U_{i,j}\\}$ is the biggest big-M parameter. \n{A corollary of the $L_{i,j} \/ U_{i,j}$ ratio is that a fractional solution transferring heat $L_{i,j}$ close to capacity $U_{i,j}$ corresponds to a good integral solution. For example, if the optimal fractional solution satisfies $L_{i,j}>0.5 \\cdot U_{i,j}$, for every used match $(i,j)$ such that $L_{i,j} \\neq 0$, then FLPR gives a 2-approximate integral solution. \nFinally, branch-and-cut repeatedly solves the fractional problem, so our new bound proves the big-M parameter's relevance for exact methods.}\nBecause performance guarantee of FLPR scales with the big-M parameters $U_{i,j}$, we improve the heuristic performance by computing a small big-M parameter $U_{i,j}$ using Algorithm MHG in Section \\ref{Sec:MaxHeat_2Streams}.\n\n\\subsection{Lagrangian Relaxation Rounding}\n\\label{Subsection:LRR}\n\n\\cite{furman:2004} design efficient heuristics for the minimum number of matches problem by applying the method of Lagrangian relaxation and relaxing the big-M constraints.\nThis approach generalizes Algorithm FLPR by approximating the fractional cost of every possible match $(i,j)\\in H\\times C$ and solving an appropriate LP using these costs.\nWe present the LP and revisit different ways of approximating the fractional match costs.\n\nIn a feasible solution, the fractional cost $\\lambda_{i,j}$ of a match $(i,j)$ is the cost incurred per unit of heat transferred via $(i,j)$.\nIn particular, \n\\begin{equation*}\n\\lambda_{i,j}=\n\\left\\{\n\t\\begin{array}{ll}\n\t\t1\/L_{i,j},  & \\mbox{if $L_{i,j}>0$, and} \\\\\n\t\t0, & \\mbox{if $L_{i,j}=0$}\n\t\\end{array}\n\\right.\n\\end{equation*}\nwhere $L_{i,j}$ is the heat exchanged via $(i,j)$.\nThen, the number of matches can be expressed as $\\sum_{i,s,j,t}\\lambda_{i,j}\\cdot q_{i,s,j,t}$.\n\\citet{furman:2004} propose a collection of heuristics computing a single cost value for each match $(i,j)$ and constructing a minimum cost solution.\nThis solution is rounded to a feasible integral solution equivalently to FLPR.\n\nGiven a cost vector $\\vec{\\lambda}$ of the matches, a minimum cost solution is obtained by solving:\n\\begin{align}\n\\tag{CostLP}\n\\label{EquationSet:CostLP}\n\\begin{aligned}\n\\text{min} & \\sum_{i \\in H} \\sum_{j \\in C} \\sum_{s,t\\in T} \\lambda_{i,j} \\cdot q_{i,s,j,t} \\\\ \n& \\sum_{j\\in C}\\sum_{t\\in T} q_{i,s,j,t} = \\sigma_{i,s} & i\\in H, s\\in T \\\\\n& \\sum_{i\\in H}\\sum_{s\\in T} q_{i,s,j,t} = \\delta_{j,t} & j\\in C, t\\in T \\\\\n& q_{i,s,j,t}\\geq 0 & i\\in H, j\\in C,\\; s,t\\in T \n\\end{aligned}\n\\end{align}\n\nA challenge in Lagrangian relaxation rounding is computing a cost $\\lambda_{i,j}$ for each hot stream $i\\in H$ and cold stream $j\\in C$.\nWe revisit and generalize policies for selecting costs.\n\n\\paragraph{Cost Policy 1 (Maximum Heat)} \nMatches that exchange large amounts of heat incur low fractional cost.\nThis observation motivates selecting $\\lambda_{i,j}= 1\/U_{i,j}$, for each $(i,j)\\in H\\times C$, where $U_{i,j}$ is an upper bound on the heat that can be feasibly exchanged between $i$ and $j$.\nIn this case, Lagrangian relaxation rounding is equivalent to FLPR (Algorithm \\ref{Alg:FLPR}).\n\n\\paragraph{Cost Policy 2 (Bounds on the Number of Matches)}\nThis cost selection policy uses lower bounds $\\alpha_i$ and $\\beta_j$ on the number of matches of hot stream $i\\in H$ and cold stream $j\\in C$, respectively, in an optimal solution.\nGiven such lower bounds, at least $\\alpha_i$ cost is incurred for the $h_i$ heat units of $i$ and at least $\\beta_j$ cost is incurred for the $c_j$ units of $j$.\nOn average, each heat unit of $i$ is exchanged with cost at least ${\\alpha_i}\/{h_i}$ and each heat unit of $j$ is exchanged with cost at least ${\\beta_j}\/{c_j}$.\nSo, the fractional cost of each match $(i,j)\\in H\\times C$ can be approximated by setting $\\lambda_{i,j}={\\alpha_i}\/{h_i}$, $\\lambda_{i,j}={\\beta_j}\/{c_j}$ or $\\lambda_{i,j}=\\frac{1}{2}(\\frac{\\alpha_i}{h_i} + \\frac{\\beta_j}{c_j})$.\n\n\\citet{furman:2004} use lower bounds $\\alpha_i=1$ and $\\beta_j=1$, for each $i\\in H$ and $j\\in C$.\nWe show that, for any choice of lower bounds $\\alpha_i$ and $\\beta_j$, this cost policy for selecting $\\lambda_{i,j}$ is not effective. Even when $\\alpha_i$ and $\\beta_j$ are tighter than 1, all feasible solutions of \\ref{EquationSet:CostLP} attain the same cost. \nConsider any feasible solution $(\\vec{y},\\vec{q})$ and the fractional cost $\\lambda_{i,j}=\\alpha_i \/ h_i$ for each $(i,j)\\in H\\times C$.\nThen the cost of $(\\vec{y},\\vec{q})$ in \\ref{EquationSet:CostLP} is:\n\\begin{equation*}\n\\sum_{i\\in H} \\sum_{j\\in C} \\sum_{s,t\\in T} \\lambda_{i,j} \\cdot q_{i,s,j,t} = \n\\sum_{i\\in H} \\sum_{j\\in C} \\sum_{s,t\\in T} \\frac{\\alpha_i}{h_i} \\cdot q_{i,s,j,t} = \n\\sum_{i\\in H} \\alpha_i.\n\\end{equation*}\nSince every feasible solution in (\\ref{EquationSet:CostLP}) has cost $\\sum_{i\\in H} \\alpha_i$, Lagrangian relaxation rounding returns an arbitrary solution.\nSimilarly, if $\\lambda_{i,j}={\\beta_j}\/{c_j}$ for $(i,j)\\in H\\times C$, every feasible solution has cost $\\sum_{j\\in C}\\beta_j$.\nIf $\\lambda_{i,j}=\\frac{1}{2}(\\frac{\\alpha_i}{h_i} + \\frac{\\beta_j}{c_j})$, all feasible solutions have the same cost $1\/2 \\cdot (\\sum_{i\\in H} \\alpha_i + \\sum_{j\\in C}\\beta_j)$.\n\n\n\\paragraph{Cost Policy 3 (Existing Solution)}\nThis method of computing costs uses an existing solution.\nThe main idea is to use the actual fractional costs for the solution's matches and a non-zero cost for every unmatched streams pair. \nA minimum cost solution with respect to these costs may improve the initial solution.\nSuppose that $M$ is the set of matches in the initial solution and let $L_{i,j}$ be the heat exchanged via $(i,j) \\in M$.\nFurthermore, let $U_{i,j}$ be an upper bound on the heat exchanged between $i$ and $j$ in any feasible solution.\nThen, a possible selection of costs is $\\lambda_{i,j}= 1\/L_{i,j}$ if $(i,j)\\in M$, and $\\lambda_{i,j}=1\/U_{i,j}$ otherwise.\n\n\n\\subsection{Covering Relaxation Rounding}\n\\label{Subsection:CRR}\n\nThis section proposes a novel covering relaxation rounding heuristic for the minimum number of matches problem.\nThe efficiency of Algorithm FLPR depends on lower bounding the unitary cost of the heat transferred via each match. \nThe goal of the covering relaxation is to use these costs and lower bound the number of matches in a stream-to-stream to basis by relaxing heat conservation.\nThe heuristic constructs a feasible integral solution by solving successively instances of the covering relaxation.\n\nConsider a feasible MILP solution and suppose that $M$ is the set of matches.\nFor each hot stream $i\\in H$ and cold stream $j\\in C$, denote by $C_i(M)$ and $H_j(M)$ the subsets of cold and hot streams matched with $i$ and $j$, respectively, in $M$.\nMoreover, let $U_{i,j}$ be an upper bound on the heat that can be feasibly exchanged between $i\\in H$ and $j\\in C$.\nSince the solution is feasible, it must be true that $\\sum_{j\\in C_i(M)}U_{i,j}\\geq h_i$ and $\\sum_{i\\in H_j(M)}U_{i,j}\\geq c_j$.\nThese inequalities are necessary, though not sufficient, feasibility conditions. \nBy minimizing the number of matches while ensuring these conditions, we obtain a covering relaxation:\n\\begin{align}\n\\tag{CoverMILP}\n\\label{EquationSet:CoverMILP}\n\\begin{aligned}\n\\text{min} & \\sum_{i \\in H}\\sum_{j \\in C} y_{i,j} \\\\ \n& \\sum_{j\\in C} y_{i,j}\\cdot U_{i,j} \\geq h_i & i\\in H \\\\\n& \\sum_{i\\in H} y_{i,j}\\cdot U_{i,j} \\geq c_j & j\\in C \\\\\n& y_{i,j}\\in\\{0,1\\} & i\\in H, j\\in C\n\\end{aligned}\n\\end{align}\nIn certain cases, the matches of an optimal solution to \\ref{EquationSet:CoverMILP} overlap well with the matches in a near-optimal solution for the original problem. \nOur new Covering Relaxation Rounding (CRR) heuristic for the minimum number of matches problem successively solves instances of the covering relaxation \\ref{EquationSet:CoverMILP}.\nThe heuristic chooses new matches iteratively until it terminates with a feasible set $M$ of matches. \nIn the first iteration, Algorithm CRR constructs a feasible solution for the covering relaxation and adds the chosen matches in $M$.\nThen, Algorithm CRR computes the maximum heat that can be feasibly exchanged using the matches in $M$ and stores the computed heat exchanges in $\\vec{q}$.\nIn the second iteration, the heuristic performs same steps with respect to the smaller updated instance $(\\vec{\\sigma}',\\vec{\\delta}')$, where $\\sigma_{i,s}'=\\sigma_{i,s}-\\sum_{j,t}q_{i,s,j,t}$ and $\\delta_{j,t}'=\\delta_{j,t}-\\sum_{i,s}q_{i,s,j,t}$.\nThe heuristic terminates when all heat is exchanged.\n\nAlgorithm \\ref{Alg:CRR} is a pseudocode of heuristic CRR.\nProcedure $CoveringRelaxation(\\vec{\\sigma},\\vec{\\delta})$ produces an optimal subset of matches for the instance of the covering relaxation in which the heat supplies and demands are specified by the vectors $\\vec{\\sigma}$ and $\\vec{\\delta}$, respectively.\nProcedure $MHLP(\\vec{\\sigma},\\vec{\\delta},M)$ (LP-based Maximum Heat) computes the maximum amount of heat that can be feasibly exchanged by using only the matches in $M$ and is based on solving the LP in Section \\ref{Sec:MaxHeat_MultipleIntervals}.\n\n\\begin{algorithm}[t] \\nonumber\n\\caption{Covering Relaxation Rounding (CRR)}\n\\begin{algorithmic}[1]\n\\State $M\\leftarrow \\emptyset$\n\\State $\\vec{q} \\leftarrow \\vec{0}$\n\\State $r\\leftarrow \\sum_{i\\in H}h_i$\n\\While {$r>0$}\n\\State For each $i\\in H$ and $s\\in T$, set $\\sigma_{i,s}' \\leftarrow \\sigma_{i,s} - \\sum_{j\\in C} \\sum_{t\\in T} q_{i,s,j,t}$\n\\State For each $j\\in C$ and $t\\in T$, set $\\delta_{j,t}' \\leftarrow \\delta_{j,t} - \\sum_{i\\in H} \\sum_{s\\in T} q_{i,s,j,t}$\n\\State $M'\\leftarrow CoveringRelaxation(\\vec{\\sigma}',\\vec{\\delta}')$\n\\Comment{{(\\ref{EquationSet:CoverMILP}) solving, Section \\ref{Subsection:CRR}}}\n\\State $M\\leftarrow M\\cup M'$\n\\State $\\vec{q}\\leftarrow MHLP(\\vec{\\sigma},\\vec{\\delta},M')$\n{\\Comment{Equations (\\ref{Eq:MaxHeatLP_Objective}) - (\\ref{Eq:MaxHeatLP_Positiveness}) LP solving, Section \\ref{Sec:MaxHeat_MultipleIntervals}}}\n\\State $r\\leftarrow \\sum_{i\\in H}h_i-\\sum_{i\\in H}\\sum_{j\\in C}\\sum_{s,t\\in T} q_{i,s,j,t}$\n\\EndWhile\n\\end{algorithmic}\n\\label{Alg:CRR}\n\\end{algorithm}\n\n\n\\section{Water Filling Heuristics}\n\\label{sec:water_filling}\n\nThis section introduces \\emph{water filling heuristics} for the minimum number of matches problem.\nThese heuristics produce a solution iteratively by exchanging the heat in each temperature interval, in a top down manner. \nThe water filling heuristics use, in each iteration,\nan efficient algorithm for the single temperature interval problem \n(see Section \\ref{sec:single_temperature_interval}).\n\n\nFigure \\ref{Fig:Water_Filling} shows the main idea of a \\emph{water filling heuristic} for the minimum number of matches problem with multiple temperature intervals.\nThe problem is solved iteratively in a top-down manner, from the highest to the lowest temperature interval.\nEach iteration produces a solution for one temperature interval.\nThe main components of a water filling heuristic are: (i) a maximum heat procedure which reuses matches from previous iterations and (ii) an efficient single temperature interval algorithm. \n\n\\begin{figure*}[t!]\n    \\centering\n\n    \\begin{subfigure}[t]{0.5\\textwidth}\n    \\centering\n\t\\includegraphics{water_filling_all.eps}\n\t\\caption{Top Down Temperature Interval Structure}\n\t\\label{Fig:Water_Filling_Temperature_Intervals}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\\centering\n\n\t\\includegraphics{water_filling_one.eps}\n\n\t\\caption{Excess Heat Descending}\n\t\\label{Fig:Water_Filling_Excess_Heat}\n\t\\end{subfigure}\n\\caption{A water filling heuristic computes a solution by exploiting the top down temperature interval structure and moving from the higher to the lower temperature interval.\nIn each temperature interval $t$, the heuristic isolates the streams with positive heat at $t$, it matches them and descends the excess heat to the next interval which is sequentially solved. \n\\label{Fig:Water_Filling}}\n\n\\end{figure*}\n\n\nGiven a set $M$ of matches and an instance $(\\vec{\\sigma_t},\\vec{\\delta}_t)$ of the problem in the single temperature interval $t$, the procedure $MHS(\\vec{\\sigma}_t,\\vec{\\delta}_t,M)$ (Maximum Heat for Single temperature interval) computes the maximum heat that can be exchanged between the streams in $t$ using only the matches in $M$.\nAt a given temperature interval $t$, the $MHS$ procedure solves the LP in Section \\ref{Sec:MaxHeat_SingleInterval}.\nThe procedure $SingleTemperatureInterval(\\vec{\\sigma}_t,\\vec{\\delta}_t)$ produces an efficient solution for the single temperature interval problem with a minimum number of matches and total heat to satisfy one cold stream. $SingleTemperatureInterval(\\vec{\\sigma}_t,\\vec{\\delta}_t)$ either: (i) solves the MILP exactly (Water Filling MILP-based or WFM) or (ii) applies the improved greedy approximation Algorithm IG in Section \\ref{sec:single_temperature_interval} (Water Filling Greedy or WFG). \nBoth water filling heuristics solve instances of the single temperature interval problem in which there is no heat conservation, i.e.\\ the heat supplied by the hot streams is greater or equal than the heat demanded by the cold streams. \nThe exact WFM uses the MILP proposed in Eqs.\\ (\\ref{Eq:SingleMIPwithoutConservation_MaxBins}) -  (\\ref{Eq:SingleMIPwithoutConservation_Integrality}) of \\ref{App:Water_Filling}.\nThe greedy heuristic WFG adapts Algorithm IG by terminating when the entire heat demanded by the cold streams has been transferred.\nAfter addressing the single temperature interval, the excess heat descends to the next temperature interval.\nAlgorithm \\ref{Alg:Water_Filling} represents our water filling approach in pseudocode.\n{Figure \\ref{Figure:Water_Filling_Ingredients} shows the main components of water filling heuristics.}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{water_filling_ingredients.eps}\n\\caption{\nWater filling heuristics solve the temperature intervals serially in a top-down manner and keep composition feasible. \nThe main components are (i) a maximum heat computation re-using higher temperature interval matches, (ii) a single temperature interval problem algorithm, and (iii) excess heat descending between consecutive temperature intervals. \nHeuristic WFM uses the \\ref{App:Water_Filling} MILP formulation for solving the single temperature interval problem, while heuristic WFG uses the Section \\ref{Sec:SingleTemperatureIntervalProblem-Approximation} Algorithm IG.}\n\\label{Figure:Water_Filling_Ingredients}\n\\end{center}\n\\end{figure}\n\n\\begin{algorithm}[t] \\nonumber\n\\caption{Water Filling (WF)}\\label{alg:water_filling}\n\\begin{algorithmic}[1]\n\\State $M \\leftarrow \\emptyset$\n\\State $\\vec{q} \\leftarrow \\vec{0}$\n\\For {$t=1,2,\\ldots,k$}\n\\If {$t\\neq 1$}\n\\State $\\vec{q}\\;' \\leftarrow MHS(\\vec{\\sigma}_t,\\vec{\\delta}_t,M)$\n{\\Comment{(\\ref{EquationSet:SingleMaxHeatLP_initial_stattement}) solve, Section \\ref{Sec:MaxHeat_SingleInterval}}}\n\\State $\\vec{q} \\leftarrow \\vec{q} + \\vec{q}\\;'$\n\\State For each $i\\in H$, set $\\sigma_{i,t} \\leftarrow \\sigma_{i,t} - \\sum_{j\\in C} \\sum_{t\\in T} q_{i,j,t}'$\n\\State For each $j\\in C$, set $\\delta_{j,t} \\leftarrow \\delta_{j,t} - \\sum_{i\\in H} \\sum_{s\\in T} q_{i,j,t}'$\n\\EndIf\n\\State $(M',\\vec{q}\\;') \\leftarrow SingleTemperatureInterval(\\vec{\\sigma}_t,\\vec{\\delta}_t)$\n{\\Comment{Eqs (\\ref{Eq:SingleMIPwithoutConservation_MaxBins} - \\ref{Eq:SingleMIPwithoutConservation_Integrality}) or Alg IG, Sec \\ref{Sec:SingleTemperatureIntervalProblem-Approximation}}}\n\\State $M \\leftarrow M\\cup M'$\n\\State $\\vec{q} \\leftarrow \\vec{q} + \\vec{q}\\;'$\n\\If {$t\\neq k$}\n\\For {$i\\in H$}\n\\State $\\vec{\\sigma}_{i,t+1} \\leftarrow \\vec{\\sigma}_{i,t+1} + (\\vec{\\sigma}_{i,t} - \\sum_jq_{i,j,t})$ (excess heat descending)\n\\EndFor\n\\EndIf\n\\EndFor\n\\end{algorithmic}\n\\label{Alg:Water_Filling}\n\\end{algorithm}\n\n\n\n\\begin{theorem}\n\\label{Thm:WaterFillingRatio}\nAlgorithms WFG and WFM are $\\Omega(k)$-approximate.\n\\end{theorem}\n\\begin{omitted_proof}\nSee \\ref{App:Water_Filling}.\n\\end{omitted_proof}\n\n\n\n\\section{Greedy Packing Heuristics}\n\\label{sec:greedy_packing}\n\nThis section proposes greedy heuristics motivated by the packing nature of the minimum number of matches problem.\nEach greedy packing heuristic starts from an infeasible solution with zero heat transferred between the streams and iterates towards feasibility by greedily selecting matches.\nThe two main ingredients of such a heuristic are: (i) a match selection policy and (ii) a heat exchange policy for transferring heat via the matches.\nSection \\ref{Subsection:MonotonicGreedyHeuristics} observes that a greedy heuristic has a poor worst-case performance if heat residual capacities are not considered.\nSections \\ref{Subsection:Largest_Heat_Match} - \\ref{Subsection:SS} define formally the greedy heuristics: (i) Largest Heat Match First, (ii) Largest Fraction Match First, and (iii) Smallest Stream First. \n{Figure \\ref{Figure:Greedy_Packing_Ingredients} shows the main components of greedy packing heuristics.}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics{greedy_packing_ingredients.eps}\n\\caption{\nGreedy packing heuristics select matches iteratively one by one. \nThe main components of greedy packing heuristics are (i) a heat exchange policy, and (ii) a match selection policy.\nGreedy packing heuristics apply these policies with respect to all unmatched stream pairs, in each iteration.\nOptions for the heat exchange policy include \\emph{dynamic heat exchange}, which solves the Section \\ref{Sec:MaxHeat_MultipleIntervals} maximum heat LP, and \\emph{static heat exchange}, which uses the Section \\ref{Sec:MaxHeat_2Streams} greedy algorithm.\nOnce the heat exchange policy has been applied for every unmatched pair of streams, a match selection policy chooses the new match, e.g.\\\n(i) with the largest heat (LHM), (ii) with the largest fraction (LFM), or (iii) of the shortest stream (SS).}\n\\label{Figure:Greedy_Packing_Ingredients}\n\\end{center}\n\\end{figure}\n\n\\subsection{A Pathological Example and Heat Residual Capacities}\n\\label{Subsection:MonotonicGreedyHeuristics}\n\nA greedy match selection heuristic is efficient if it performs a small number of iterations and chooses matches exchanging large heat load in each iteration.\nOur greedy heuristics perform large moves towards feasibility by choosing good matches in terms of: (i) heat and (ii) stream fraction. \nAn efficient greedy heuristic should also be monotonic in the sense that every chosen match achieves a strictly positive increase on the covered instance size.\n\nThe Figure \\ref{Fig:non_monotonic_heuristic} example shows a pathological behavior of greedy non-monotonic heuristics.\nThe instance consists of 3 hot streams, 3 cold streams and 3 temperature intervals.\nHot stream $i\\in H$ has heat supply $\\sigma_{i,s}=1$ for $s=i$ and no supply in any other temperature interval.\nCold stream $j\\in C$ has heat demand $\\delta_{j,t}=1$ for $t=j$ and no demand in any other temperature interval.\nConsider the heuristic which selects a match that may exchange the maximum amount of heat in each iteration.\nThe matches $(h_1,c_2)$ and $(h_2,c_3)$ consist the initial selections.\nIn the subsequent iteration, no match increases the heat that can be feasibly exchanged between the streams and the heuristic chooses unnecessary matches.\n\n\\begin{figure}[t]\n\n\\begin{center}\n\\includegraphics{non_monotonic_heuristic.eps}\n\\caption{A bad example of a non monotonic heuristic. If a heuristic begins by matching $h_1$ with $c_2$ and $h_2$ with $c_3$, then many unnecessary matches might be required to end up with a feasible solution.}\n\\label{Fig:non_monotonic_heuristic}\n\\end{center}\n\n\\end{figure}\n\nA sufficient condition enforcing strictly monotonic behavior and avoiding the above pathology, is for each algorithm iteration to satisfy the heat residual capacities.\nAs depicted in Figure \\ref{Fig:greedy_decomposition}, a greedy heuristic maintains a set $M$ of selected matches together with a decomposition of the original instance $I$ into two instances $I^A$ and $I^B$.\nIf $I=(H,C,T,\\vec{\\sigma},\\vec{\\delta})$, then it holds that $I^A=(H,C,T,\\vec{\\sigma}^A,\\vec{\\delta}^A)$ and $I^B=(H,C,T,\\vec{\\sigma}^B,\\vec{\\delta}^B)$, where $\\mathbf{\\sigma}=\\vec{\\sigma}^A+\\vec{\\sigma}^B$ and $\\vec{\\delta}=\\vec{\\delta}^A+\\vec{\\delta}^B$. \nThe set $M$ corresponds to a feasible solution for $I^A$ and the instance $I^B$ remains to be solved.\nIn particular, $I^A$ is obtained by computing a maximal amount of heat exchanged by using the matches in $M$ and $I^B$ is the remaining part of $I$.\nInitially, $I^A$ is empty and $I^B$ is exactly the original instance $I$.\nA selection of a match increases the total heat exchanged in $I^A$ and reduces it in $I^B$.\n\\ref{App:Greedy_Packing} observes that a greedy heuristic is monotonic if $I^B$ is feasible in each iteration. \nFurthermore, $I^B$ is feasible if and only if $I^A$ satisfies the heat residual capacities $R_u = \\sum_{i\\in H}\\sum_{s=1}^u\\sigma_{i,s} - \\sum_{j\\in C}\\sum_{t=1}^u\\delta_{j,t}$, for $u\\in T$.\n\n\\begin{figure}[t]\n\n\\begin{center}\n\\includegraphics{greedy_packing_decomposition.eps}\n\\vspace{-20pt}\n\\caption{Decomposition of a greedy packing heuristic. The problem instance $I$ is the union of the instance $I_A$ already solved by the heuristic and the instance $I_B$ that remains to be solved.}\n\\label{Fig:greedy_decomposition}\n\\end{center}\n\n\\end{figure}\n\n\\subsection{Largest Heat Match First}\n\\label{Subsection:Largest_Heat_Match}\n\nOur Largest Heat Match First heuristics arise from the idea that the matches should individually carry large amounts of heat in a near optimal solution.\nSuppose that $Q_v$ is the maximum heat that may be transferred between the streams using only a number $v$ of matches.\nThen, minimizing the number of matches is expressed as $\\min\\{v:Q_v\\geq \\sum_{i=1}^nh_i\\}$.\nThis observation motivates the greedy packing heuristic which selects matches iteratively until it ends up with a feasible set $M$ of matches exchanging $\\sum_{i=1}^nh_i$ units of heat.\nIn each iteration, the heuristic chooses a match maximizing the additional heat exchanged. \nOur two variants of largest heat matches heuristics are: (i) LP-based Largest Heat Match (LHM-LP) and (ii) Greedy Largest Heat Match (LHM).\n\n\nHeuristic LHM-LP uses the $MHLP(M)$ (LP-based Maximum Heat) procedure to compute the maximum heat that can be transferred between the streams using only the matches in the set $M$.\nThis procedure is repeated $O(nm)$ times in each iteration, once for every candidate match, and solves an LP incorporating the proposed heat residual capacities.\nAlgorithm \\ref{Alg:LHM-LP} is an LHM-LP heuristic using the LP in Section \\ref{Sec:MaxHeat_MultipleIntervals}.\nThe algorithm maintains a set $M$ of chosen matches and selects a new match $(i',j')$ to maximize $MHLP(M\\cup(i',j'))$.\n\n\\begin{algorithm}[t] \\nonumber\n\\caption{Largest Heat Match First LP-based (LHM-LP)}\n\\begin{algorithmic}[1]\n\\State $M\\leftarrow\\emptyset$\n\\State $r\\leftarrow \\sum_{i\\in H}h_i$\n\\While {$r>0$}\n\\State $(i',j')\\leftarrow\\arg\\max_{(i,j)\\in H\\times C\\setminus M} \\{MHLP(M\\cup\\{(i,j)\\})\\}$\n{\\Comment{Eqs (\\ref{Eq:MaxHeatLP_Objective} - \\ref{Eq:MaxHeatLP_Positiveness}), Sec \\ref{Sec:MaxHeat_MultipleIntervals}}}\n\\State $M\\leftarrow M\\cup \\{(i',j')\\}$\n\\State $r\\leftarrow\\sum_{i\\in H}h_i-MHLP(M)$\n\\EndWhile\n\\State Return $M$\n\\end{algorithmic}\n\\label{Alg:LHM-LP}\n\\end{algorithm}\n\n\\begin{theorem}\n\\label{Thm:GreedyPackingRatio}\nAlgorithm LHM-LP is $O(\\log n + \\log \\frac{h_{\\max}}{\\epsilon})$-approximate, where $\\epsilon$ is the required precision.\n\\end{theorem}\n\\begin{omitted_proof}\nSee \\ref{App:Greedy_Packing}.\n\\end{omitted_proof}\n\n\\medskip\n\nLHM-LP heuristic is polynomial-time in the worst case.\nThe $i$-th iteration solves $nm-i+1$ LP instances which sums to solving a total of $\\sum_{i=1}^{nm}(nm-i+1)=O(n^2m^2)$ LP instances in the worst case.\nHowever, for large instances, the algorithm is time consuming because of this iterative LP solving.\nSo, we also propose an alternative, time-efficient greedy approach. \nThe new heuristic version builds a solution by selecting matches and deciding the heat exchanges, without modifying them in subsequent iterations.\n\nThe new approach for implementing the heuristic, that we call LHM, requires the $MHG(\\vec{\\sigma},\\vec{\\delta},i,j)$ procedure. \nGiven an instance $(\\vec{\\sigma},\\vec{\\delta})$ of the problem, it computes the maximum heat that can be feasibly exchanged between hot stream $i\\in H$ and cold stream $j\\in C$, as defined in Section \\ref{Sec:MaxHeat_2Streams}.\nThe procedure also computes a corresponding value $q_{i,s,j,t}$ of heat exchanged between $i\\in H$ in temperature interval $s\\in T$ and $j\\in C$ in temperature interval $t\\in T$.\nLHM maintains a set $M$ of currently chosen matches together with their respective vector $\\vec{q}$ of heat exchanges.\nIn each iteration, it selects the match $(i',j')$ and heat exchanges $q'$ between $i'$ and $j'$ so that the value $MHG(\\vec{\\sigma},\\vec{\\delta},i',j')$ is maximum.\nAlgorithm \\ref{Alg:LHM} is a pseudocode of this heuristic.\n\n\n\\begin{algorithm}[t] \\nonumber\n\\caption{Largest Heat Match First Greedy (LHM)}\n\\begin{algorithmic}[1]\n\\State $M \\leftarrow \\emptyset$\n\\State $\\vec{q} \\leftarrow \\vec{0}$\n\\State $r \\leftarrow \\sum_{i\\in H}h_i$\n\\While {$r > 0$}\n\\State $(i', j', \\vec{q}\\;') \\leftarrow \\arg \\max_{(i,j)\\in H\\times C\\setminus M} \\{MHG(\\vec{\\sigma},\\vec{\\delta},i,j)\\}$\n{\\Comment{Algorithm MHG, Section \\ref{Sec:MaxHeat_2Streams}}}\n\\State $M \\leftarrow M \\cup \\{(i',j')\\}$\n\\State $\\vec{q} \\leftarrow \\vec{q} + \\vec{q}\\;'$\n\\State For each $s\\in T$, set $\\sigma_{i',s} \\leftarrow \\sigma_{i',s} - \\sum_{t\\in T} q_{i',s,j',t}'$\n\\State For each $t\\in T$, set $\\delta_{j',t} \\leftarrow \\delta_{j',t} - \\sum_{s\\in T} q_{i',s,j',t}'$\n\\State $r \\leftarrow r - \\sum_{s,t\\in T}q_{i',s,j',t}'$\n\\EndWhile\n\\State Return $M$\n\\end{algorithmic}\n\\label{Alg:LHM}\n\\end{algorithm}\n\n\n\\subsection{Largest Fraction Match First}\n\\label{Subsection:LFM}\n\nThe heuristic \\emph{Largest Fraction Match First} (LFM) exploits the bipartite nature of the problem by employing matches which exchange large fractions of the stream heats.\nConsider a feasible solution with a set $M$ of matches.\nEvery match $(i,j)\\in M$ covers a fraction $\\sum_{s,t\\in T}\\frac{q_{i,s,j,t}}{h_i}$ of hot stream $i\\in H$ and a fraction $\\sum_{s,t\\in T}\\frac{q_{i,s,j,t}}{c_j}$ of cold stream $j\\in C$.\nThe total covered fraction of all streams is equal to $\\sum_{(i,j)\\in M} \\sum_{s,t\\in T} \\left( \\frac{q_{i,s,j,t}}{h_i} + \\frac{q_{i,s,j,t}}{c_j}\\right)=n+m$.\nSuppose that $F_v$ is the maximum amount of total stream fraction that can be covered using no more than $v$ matches.\nThen, minimizing the number of matches is expressed as $\\min\\{v:F_v\\geq n+m\\}$.\nBased on this observation, the main idea of LFM heuristic is to construct iteratively a feasible set of matches, by selecting the match covering the largest fraction of streams, in each iteration.\nThat is, LFM prioritizes proportional matches in a way that high heat hot streams are matched with high heat cold streams and low heat hot streams with low heat cold streams. \nIn this sense, it generalizes the idea of Algorithm IG for the single temperature interval problem (see Section \\ref{sec:single_temperature_interval}), according to which it is beneficial to match streams of (roughly) equal heat.\n\nAn alternative that would be similar to LHM-LP is an\nLFM heuristic \nwith an $MFLP(M)$ (LP-based Maximum Fraction) procedure computing the maximum fraction of streams that can be covered using only a given set $M$ of matches.\nLike the LHM-LP heuristic, this procedure would be based on solving an LP (see \\ref{App:Greedy_Packing}), except that the objective function maximizes the total stream fraction.\nThe LFM heuristic can be also modified to attain more efficient running times using Algorithm $MHG$, as defined in Section \\ref{Sec:MaxHeat_2Streams}. \nIn each iteration, the heuristic selects the match $(i,j)$ with the highest value $\\frac{U_{i,j}'}{h_i}+\\frac{U_{i,j}'}{c_j}$, where $U_{i,j}'$ is the maximum heat that can be feasibly exchanged between $i$ and $j$ in the remaining instance.\n\n\n\n\\subsection{Smallest Stream Heuristic}\n\\label{Subsection:SS}\n\nSubsequently, we propose \\emph{Smallest Stream First} (SS) heuristic based on greedy match selection, which also incorporates stream priorities so that a stream is involved in a small number of matches.\nLet $\\alpha_i$ and $\\beta_j$ be the number of matches of hot stream $i\\in H$ and cold stream $j\\in C$, respectively.\nMinimizing the number of matches problem is expressed as $\\min\\{\\sum_{i\\in H}\\alpha_i\\}$, or equivalently $\\min\\{\\sum_{j\\in C}\\beta_j\\}$.\nBased on this observation, we investigate heuristics that specify a certain order of the hot streams and match them one by one, using individually a small number of matches.\nSuch a heuristic requires: (i) a stream ordering strategy and (ii) a match selection strategy.\nTo reduce the number of matches of small hot streams, heuristic SS uses the order $h_1\\leq h_2\\leq \\ldots\\leq h_n$.  \n\nIn each iteration, the next stream is matched with a low number of cold streams using a greedy match selection strategy; we use greedy LHM heuristic.\nObserve that SS heuristic is more efficient in terms of running time compared to the other greedy packing heuristics, because it solves a subproblem with only one hot stream in each iteration. \nAlgorithm \\ref{Alg:ShortestStream} is a pseudocode of SS heuristic. \nNote that other variants of ordered stream heuristics may be obtained in a similar way.\nThe heuristic uses the $MHG$ algorithm in Section \\ref{Sec:MaxHeat_2Streams}.\n\n\\begin{algorithm}[t] \\nonumber\n\\caption{Smallest Steam First (SS)}\n\\begin{algorithmic}[1]\n\\State Sort the hot streams in non-decreasing order of their heat loads, i.e.\\ $h_1\\leq h_2\\leq\\ldots\\leq h_n$.\n\\State $M \\leftarrow \\emptyset$\n\\State $\\vec{q}\\leftarrow \\vec{0}$\n\\For {$i\\in H$}\n\\State $r\\leftarrow h_i$\n\\While {$r>0$}\n\\State $(i, j', \\vec{q}\\;') \\leftarrow \\arg \\max_{j\\in C} \\{MHG(\\vec{\\sigma},\\vec{\\delta},i,j)\\}$\n{\\Comment{Algorithm MHG, Section \\ref{Sec:MaxHeat_2Streams}}}\n\\State $M\\leftarrow M\\cup \\{(i,j')\\}$\n\\State $\\vec{q} \\leftarrow \\vec{q} + \\vec{q}\\;'$\n\\State For each $s\\in T$, set $\\sigma_{i,s} \\leftarrow \\sigma_{i,s} - \\sum_{t\\in T} q_{i,s,j',t}'$\n\\State For each $t\\in T$, set $\\delta_{j',t} \\leftarrow \\delta_{j',t} - \\sum_{s\\in T} q_{i,s,j',t}'$\n\\State $r \\leftarrow r - \\sum_{s,t\\in T}q_{i',s,j',t}'$\n\\EndWhile\n\\EndFor\n\\State Return $M$\n\\end{algorithmic}\n\\label{Alg:ShortestStream}\n\\end{algorithm}\n\n\n\n\\section{Numerical Results}\n\\label{sec:results}\n\nThis section evaluates the proposed heuristics on three test sets. \nSection \\ref{Section:Benchmark_Instances} provides information on system specifications and benchmark instances.\nSection \\ref{Section:Exact_Experiments} presents computational results of exact methods and shows that commercial, state-of-the-art approaches have difficult solving moderately-sized instances \nto global optimality.\nSection \\ref{Section:Heuristic_Experiments} evaluates experimentally the heuristic methods and compares the obtained results with those reported by \\citet{furman:2004}.\nAll result tables are provided in \\ref{App:Experimental_Results}.\n{\\cite{source_code} provide test cases and source code for the paper's computational experiments.}\n\n\\subsection{System Specification and Benchmark Instances}\n\\label{Section:Benchmark_Instances}\n\nAll computations are run on an Intel Core i7-4790 CPU 3.60GHz with 15.6 GB RAM running 64-bit Ubuntu 14.04.\nCPLEX 12.6.3 and Gurobi 6.5.2 solve the minimum number of matches problem exactly. \nThe mathematical optimization models and heuristics are implemented in Python 2.7.6 and Pyomo 4.4.1 \\citep{hart:2011, hart:2012}. \n\nWe use problem instances from two existing test sets \\citep{furman:2004, chen:2015}. {We also generate two collections of larger test cases. The smaller of the two sets uses work of \\citet{grossman:2017}. The larger of the two sets was created using our own random generation method.}\nAn instance of general heat exchanger network design consists of streams and utilities with inlet, outlet temperatures, flow rate heat capacities and other parameters.\n\\ref{App:Minimum_Utility_Cost} shows how a minimum number of matches instances arises from the original instance of general heat exchanger network design.\n \nThe \\emph{\\cite{furman:2000} test set} consists of test cases from the engineering literature. \nTable \\ref{Table:Problem_Sizes} reports bibliographic information on the origin of these test cases. \nWe manually digitize this data set and make it publicly available for the first time \\citep{source_code}.\nTable \\ref{Table:Problem_Sizes} lists the 26 problem instance names and information on their sizes.\nThe total number streams and temperature intervals varies from 6 to 38 and from 5 to 32, respectively. \nTable \\ref{Table:Problem_Sizes} also lists the number of binary and continuous variables as well as the number of constraints in the transshipment MILP formulation.\n\nThe \\emph{\\cite{minlp,chen:2015} test set} consists of 10 problem instances.\nThese instances are classified into two categories depending on whether they consist of balanced or unbalanced streams. \nTest cases with balanced streams have flowrate heat capacities in the same order of magnitude,\nwhile test cases with unbalanced streams have dissimilar flowrate heat capacities spanning several orders of magnitude.\nThe sizes of these instances range from 10 to 42 streams and from 12 to 35 temperature intervals. \nTable \\ref{Table:Problem_Sizes} reports more information on the size of each test case.\n\n\\emph{The \\cite{grossman:2017} test set} is generated randomly.\nThe inlet, outlet temperatures of these instances are fixed while the values of flowrate heat capacities are generated randomly with fixed seeds.\nThis test set contains 12 moderately challenging problems (see Table \\ref{Table:Problem_Sizes}) with a classification into balanced and unbalanced instances, similarly to the \\cite{minlp,chen:2015} test set. \nThe smallest problem involves 27 streams and 23 temperature intervals while the largest one consists of 43 streams and 37 temperature intervals.\n\n{\\emph{The Large Scale test set} is generated randomly. These instances have 80 hot streams, 80 cold streams, 1 hot utility and 1 cold utility.\nFor each hot stream $i\\in HS$, the inlet temperature $T_{\\text{in},i}^{HS}$ is chosen uniformly at random in the interval $(30,400]$. \nThen, the outlet temperature $T_{\\text{out},i}^{HS}$ is selected uniformly at random in the interval $[30, T_{\\text{in},i}^{HS})$.\nAnalogously, for each cold stream $j\\in CS$, the outlet temperature $T_{\\text{out},j}^{CS}$ is chosen uniformly at random in the interval $(20,400]$.\nNext, the inlet temperature $T_{\\text{in},j}^{CS}$ is chosen uniformly at random in the interval $[20,T_{\\text{out},j}^{CS})$.\nThe flow rate heat capacities $FCp_i$ and $FCp_j$ of hot stream $i$ and cold stream $j$ are chosen as floating numbers with two decimal digits in the interval $[0,15]$.\nThe hot utility has inlet temperature $T_{\\text{in}}^{HU}=500$, outlet temperature $T_{\\text{out}}^{HS}=499$, and cost $\\kappa^{HU}=80$.\nThe cold utility has inlet temperature $T_{\\text{in}}^{CU}=20$, outlet temperature $T_{\\text{out}}^{CU}=21$, and cost $\\kappa^{CU}=20$.\nThe minimum heat recovery approach temperature is $\\Delta T_{\\min}=10$.}\n\n\n\\subsection{Exact Methods} \n\\label{Section:Exact_Experiments}\n\nWe evaluate exact methods using state-of-the-art commercial approaches. \nFor each problem instance, CPLEX and Gurobi solve the Section \\ref{sec:preliminaries} transportation and transshipment models.\nBased on the difficulty of each test set, we set a time limit for each solver run as follows: (i) 1800 seconds for the \\cite{furman:2000} test set, (ii) 7200 seconds for the \\cite{minlp,chen:2015} test set, and (iii) 14400 seconds for the \\cite{grossman:2017} {and large scale} test sets. \nIn each solver run, we set absolute gap 0.99, relative gap $4\\%$, and maximum number of threads 1.\n\nTable \\ref{Table:Exact_Methods} reports the best found objective value, CPU time and relative gap, for each solver run.\nObserve that state-of-the-art approaches cannot, in general, solve moderately-sized problems with 30-40 streams to global optimality.\nFor example, none of the test cases in the \\cite{grossman:2017} {or large scale} test sets is solved to global optimality within the specified time limit. \nTable \\ref{Table:Comparisons} contains the results reported by \\citet{furman:2004} using CPLEX 7.0 with 7 hour time limit. \nCPLEX 7.0 fails to solve 4 instances to global optimality.\nInterestingly, CPLEX 12.6.3 still cannot solve 3 of these 4 instances with a 1.5 hour timeout.\n\nTheoretically, the transshipment MILP is better than the transportation MILP because the former has asymptotically fewer variables.\nThis observation is validated experimentally with the exception of very few instances, e.g.\\ \\texttt{balanced10}, in which the transportation model computes a better solution within the time limit.\nCPLEX and Gurobi are comparable and neither dominates the other.\nInstances with balanced streams are harder to solve, which highlights the difficulty introduced by symmetry, see \\cite{kouyialis:2016}. \n{\nThe preceding numerical analysis refers to the extended transportation MILP.\nTable \\ref{Table:Transportation_Models_Comparison} compares solver performance to the reduced transportation MILP, i.e.\\ a formulation removing redundant variables $q_{i,s,j,t}$ with $s> t$ and Equations (\\ref{TransportationMIP_Eq:ThermoConstraint}). Note that modern versions of CPLEX and Gurobi show effectively no difference between the two formulations.\n}\n  \n\\subsection{Heuristic Methods}  \n\\label{Section:Heuristic_Experiments}\n\n{\nWe implement the proposed heuristics using Python and develop the LP models with Pyomo \\citep{hart:2011,hart:2012}.\nWe use CPLEX 12.6.3 with default settings to solve all LP models within the heuristic methods. \\cite{source_code} make the source code available. The following discussion covers the 48 problems with 43 streams or fewer. Section \\ref{Section:Large_Scale} discusses the 3 examples with 160 streams each.}\n\nThe difficulty of solving the minimum number of matches problem to global optimality motivates the design of heuristic methods and approximation algorithms with proven performance guarantees.\nTables \\ref{Table:Heuristic_Upper_Bounds} and \\ref{Table:Heuristic_CPU_Times} contain the computed objective value and CPU times, respectively, of the heuristics for all test cases. \nFor the challenging \\cite{minlp,chen:2015} and \\cite{grossman:2017} test sets, heuristic LHM-LP always produces the best solution.\nThe LHM-LP running time is significantly higher compared to all heuristics due to the iterative LP solving, despite the fact that it is guaranteed to be polynomial in the worst case.\nAlternatively, heuristic SS produces the second best heuristic result with very efficient running times in the \\cite{minlp,chen:2015} and \\cite{grossman:2017} test sets. \nFigure \\ref{Fig:Boxplot_Performance_Ratios} depicts the performance ratio of the proposed heuristics using a box and whisker plot, where the computed objective value is normalized with the one found by CPLEX for the transshipment MILP.\nFigure \\ref{Fig:Boxplot_CPU_Times} shows a box and whisker plot of the CPU times of all heuristics in $\\log_{10}$ scale normalized by the minimum CPU time for each test case.\nFigure \\ref{Fig:Line_Chart} shows a line chart verifying that our greedy packing approach produces better solutions than the relaxation rounding and water filling ones. \n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[scale=0.6]{performance_ratios_boxplot.eps}\n\\end{center}\n\\vspace{-20pt}\n\\caption{Box and whisker diagram of {48} heuristic performance ratios, i.e.\\ computed solution \/ best known solution {for the problems with 43 streams or fewer.}}\n\\label{Fig:Boxplot_Performance_Ratios}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[scale=0.6]{elapsed_times_boxplot.eps}\n\\end{center}\n\\vspace{-20pt}\n\\caption{Box and whisker diagram of {48} CPU times ($\\log_{10}$ scale) normalized by the minimum CPU time for each test case.}\n\\label{Fig:Boxplot_CPU_Times}\n\\end{figure}\n\n\\begin{figure}[!ht] \n\\begin{center}\n\\includegraphics[scale=0.6]{heuristic_comparison_line_chart.eps}\n\\end{center}\n\\vspace{-20pt}\n\\caption{Line chart comparing the performance ratio, i.e.\\ computed solution \/ best known solution, of the best computed result by heuristic methods: relaxation rounding, water filling, and greedy packing. {This graph applies to the 48 problems with 43 streams or fewer.}}\n\\label{Fig:Line_Chart}\n\\end{figure}\n\nTable \\ref{Table:Comparisons} contains the heuristic results reported by \\citet{furman:2004} and the ones obtained with our improved version of the FLPR, LRR, and WFG heuristics of \\cite{furman:2004}.\nOur versions of FLPR, LRR, and WFG perform better for the \\cite{furman:2004} test set because of our new Algorithm MHG for tightening the big-M parameters.\nFor example, out of the 26 instances, our version of FLPR performs strictly better than the \\cite{furman:2004} version 20 times and worse only once (\\texttt{10sp1}). To further explore the effect of the big-M parameter, Table \\ref{Table:BigM_Comparisons} shows how different computations for the big-M parameter change the FLPR and LRR performance. Table \\ref{Table:BigM_Comparisons} also demonstrates the importance of the big-M parameter on the transportation MILP fractional relaxation quality. \n\nIn particular, Table \\ref{Table:BigM_Comparisons} compares the three big-M computation methods discussed in Section \\ref{Sec:MaxHeat_2Streams}: (i) the trivial bounds, (ii) the \\citet{gundersen:1997} method, and (iii) our greedy Algorithm MHG.\nOur greedy maximum heat algorithm dominates the other approaches for computing the big-M parameters. Algorithm MHG also outperforms the other two big-M computation methods by finding smaller feasible solutions via both Fractional LP Rounding and Lagrangian Relaxation Rounding.  In the 48 test cases, Algorithm MHG produces the best FLPR and LRR feasible solutions in 46 and 43 test cases, respectively. Algorithm MHG is strictly best for 33 FLPR and 32 LRR test cases. Finally, Algorithm MHG achieves the tightest fractional MILP relaxation for all test instances.\n\nFigure \\ref{Fig:Boxplot_Performance_Ratios} and Table \\ref{Table:Heuristic_Upper_Bounds} show that our\nnew CRR heuristic is competitive with the other relaxation rounding heuristics, performing as well or better than FLPR or LRR in 19 of the 48 test cases and strictly outperforming both FLPR and LRR in 8 test cases.\nAlthough CRR solves a sequence of MILPs, Figure \\ref{Fig:Boxplot_CPU_Times} and Table \\ref{Table:Heuristic_CPU_Times} show that its running time is efficient compared to the other relaxation rounding heuristics.\n\nOur water filling heuristics are equivalent to or better than \\citet{furman:2004} for 25 of their 26 test set instances (all except \\texttt{7sp2}).  \nIn particular, our Algorithm WFG is strictly better than their WFG in 18 of 26 instances and is worse in just one. This improvement stems from the new 1.5-approximation algorithm for the single temperature interval problem (see Section \\ref{Sec:SingleTemperatureIntervalProblem-Approximation}).\nThe novel Algorithm WFM is competitive with Algorithm WFG and produces equivalent or better feasible solutions for 37 of the 48 test cases. \nIn particular, WFM has a better performance ratio than WFG (see Figure \\ref{Fig:Boxplot_Performance_Ratios}) and WFM is strictly better than WFG in all but 1 of the \\citet{grossman:2017} instances.\nThe strength of WFM highlights the importance of our new MILP formulation in Eqs.\\ (\\ref{Eq:SingleMIP_MaxBins})-(\\ref{Eq:SingleMIP_Integrality}).\nAt each iteration, WFM solves an MILP without big-M constraints and therefore has a running time in the same order of magnitude as its greedy counterpart WFG (see Figure \\ref{Fig:Boxplot_CPU_Times}).\n\nIn summary, our heuristics obtained via the relaxation rounding and water filling methods improve the corresponding ones proposed by \\citet{furman:2004}.\nFurthermore, greedy packing heuristics achieve even better results in more than $90\\%$ of the test cases. \n\n{\n\n\\subsection{Larger Scale Instances}\n\\label{Section:Large_Scale}\n\nAlthough CPLEX and Gurobi do not converge to global optimality for many of the \n\\cite{furman:2000}, \\cite{minlp,chen:2015}, and \\cite{grossman:2017} instances,\nthe solvers produce the best heuristic solutions in all test cases.\nBut the literature instances are only moderately sized. We expect that the heuristic performance improves relative to the exact approaches as the problem sizes increase.\nTowards a more complete numerical analysis, we randomly generate 3 larger scale instances with 160 streams each.\n\nFor larger problems, the running time may be important to a design engineer \\citep{hindmarsh:1983}.\nWe apply the least time consuming heuristic of each type for solving the larger scale instances, i.e.\\ apply relaxation rounding heuristic FLPR, water filling heuristic WFG, and greedy packing heuristic SS.\nWe also solve the transshipment model using CPLEX 12.6.3 with a 4h timeout.  The results are in Table \\ref{Table:Large_Scale_Results}.\n\nFor these instances, greedy packing SS computes a better solution than the relaxation rounding FLPR heuristic or the water filling WFG heuristic, but SS has larger running time.\nIn instance \\texttt{large-scale1}, greedy packing SS computes 218, a better solution than the CPLEX value 219. Moreover, the CPLEX heuristic spent the first 1hr of computation time at solution 257 (18\\% worse than the solution SS obtains in 10 minutes) and the next 2hr of computation time at solution 235 (8\\% worse than the solution SS obtains in 10 minutes). Any design engineer wishing to interact with the results would be frustrated by these times.\n\nIn instance \\texttt{large-scale2}, CPLEX computes a slightly better solution (239) than the SS heuristic (242).\nBut the good CPLEX solution is computed slightly before the 4h timeout. For more than 3.5hr, the best CPLEX heuristic is 273 (13\\% worse than the solution SS obtains in 10 minutes).\nFinally, in instance \\texttt{large-scale0}, CPLEX computes a significantly better solution (175) than the SS heuristic (233).\nBut CPLEX computes the good solution after 2h and the incumbent is similar to the greedy packing SS solution for the first 2 hours.\nThese findings demonstrate that greedy packing approaches are particularly useful when transitioning to larger scale instances.\n\nNote that we could additionally study approaches to improve the heuristic performance of CPLEX, e.g.\\ by changing CPLEX parameters or using a parallel version of CPLEX. But the point of this paper is to develop a deep understanding of a very important problem that consistently arises in process systems engineering \\citep{floudas:2012}.\n}\n\n\\section{Discussion of Manuscript Contributions}\n\\label{sec:discussion}\n\nThis section reflects on this paper's contributions and situates the work with respect to existing literature. We begin in Section \\ref{sec:single_temperature_interval} by designing efficient heuristics for the minimum number of matches problem with the special case of a single temperature interval.\nInitially, we show that the 2 performance guarantee by \\citet{furman:2004} is tight.\nUsing graph theoretic properties, we propose a new MILP formulation for the single temperature interval problem which does not contain any big-M constraints. We also develop an improved, tight, greedy 1.5-approximation algorithm which prioritizes stream matches with equal heat loads. Apart from the its independent interest, solving the single temperature interval problem is a major ingredient of water filling heuristics.\n\n\nThe multiple temperature interval problem requires big-M parameters. We reduce these parameters in Section \\ref{sec:max_heat} by\ncomputing the maximum amount of heat transfer with match restrictions.\nInitially, we present a greedy algorithm for exchanging the maximum amount of heat between two streams.\nThis algorithm computes tighter big-M parameters than \\citet{gundersen:1997}.\nWe also propose LP-based ways for computing the maximum exchanged heat using only a subset of the available matches.\nMaximum heat computations are fundamental ingredients of our heuristic methods and detect the overall problem feasibility. This paper emphasizes how tighter big-M parameters improve heuristics with performance guarantees, but notice that improving the big-M parameters will also tend to improve exact methods.\n\nSection \\ref{sec:relaxation_rounding} further investigates the \\emph{relaxation rounding heuristics} of \\cite{furman:2004}. \n\\citet{furman:2004} propose a heuristic for the minimum number of matches problem based on rounding the LP relaxation of the transportation MILP formulation\n(\\emph{Fractional LP Rounding (FLPR)}).\nInitially, we formulate the LP relaxation as a minimum cost flow problem showing that it can be solved with network flow techniques which are more efficient than generic linear programming.\nWe derive a negative performance guarantee showing that FLPR has poor performance in the worst case.\nWe also prove a new positive performance guarantee for FLPR indicating that its worst-case performance may be improved with tighter big-M parameters.\nExperimental evaluation shows that the performance of FLPR improves with our tighter algorithm for computing big-M parameters.\nMotivated by the method of Lagrangian Relaxation, \\citet{furman:2004} proposed an approach generalizing FLPR by approximating the cost of the heat transferred via each match.\nWe revisit possible policies for approximating the cost of each match.\nInterestingly, we show that this approach can be used as a generic method for potentially improving a solution of the minimum number of matches problem.\nHeuristic \\emph{Lagrangian Relaxation Rounding (LRR)} aims to improve the solution of FLPR in this way.\nFinally, we propose a new heuristic, namely \\emph{Covering Relaxation Rounding (CRR)}, that successively solves instances of a new covering relaxation which also requires big-M parameters.\n\nSection \\ref{sec:water_filling} defines \\emph{water filling heuristics} as a class of heuristics solving the minimum number of matches problem in a top-down manner, i.e.\\ from highest to lowest temperature interval.\n\\citet{cerdaetwesterberg:1983} and \\citet{furman:2004} have solution methods based on water filling.\nWe improve these heuristics by developing novel, efficient ways for solving the single temperature interval problem.\nFor example, heuristics \\emph{MILP-based Water Filling (WFM)} and \\emph{Greedy Water Filling (WFG)} incorporate the new MILP formulation (Eqs.\\ \\ref{Eq:SingleMIP_MaxBins}-\\ref{Eq:SingleMIP_Integrality}) and greedy Algorithm IG, respectively.\nWith appropriate LP, we further improve water filling heuristics by reusing in each iteration matches selected in previous iterations.\n\\citet{furman:2004} showed a performance guarantee scaling with the number of temperature intervals. \nWe show that this performance guarantee is asymptotically tight for water filling heuristics. \n\nSection \\ref{sec:greedy_packing} develops a new \\emph{greedy packing approach} for designing efficient heuristics for the minimum the number of matches problem motivated by the packing nature of the problem.\nGreedy packing requires feasibility conditions which may be interpreted as a decomposition method analogous to pinch point decomposition, see\n\\cite{hindmarsh:1983}.\nSimilarly to \\cite{cerdaetwesterberg:1983}, stream ordering affects the efficiency of greedy packing heuristics.\nBased on the feasibility conditions, the LP in Eqs.\\ (\\ref{Eq:MaxHeatLP_Objective})-(\\ref{Eq:MaxHeatLP_Positiveness}) selects matches carrying a large amount of heat and incurring low unitary cost for exchanging heat.\nHeuristic \\emph{LP-based Largest Heat Match (LHM-LM)} selects matches greedily by solving instances of this LP.\nUsing a standard packing argument, we obtain a new logarithmic performance guarantee.\nLHM-LP has a polynomial worst-case running time but is experimentally time-consuming due to the repeated LP solving.\nWe propose three other greedy packing heuristic variants which improve the running time at the price of solution quality.\nThese other variants are based on different time-efficient strategies for selecting good matches.\nHeuristic \\emph{Largest Heat Match (LHM)} selects matches exchanging high heat in a pairwise manner.\nHeuristic \\emph{Largest Fraction Match (LFM)} is inspired by the idea of our greedy approximation algorithm for the single temperature interval problem which prioritizes roughly equal matches.\nHeuristic \\emph{Smallest Stream First (SS)} is inspired by the idea of the tick-off heuristic \\citep{hindmarsh:1983} and produces matches in a stream to stream basis, where a hot stream is ticked-off by being matched with a small number of cold streams. \n\nFinally, Section \\ref{sec:results} \nshows numerically that our new way of computing the big-M parameters, our improved algorithms for the single temperature interval, and the other enhancements improve the performance of relaxation rounding and water-filling heuristics.\nThe numerical results also show that our novel greedy packing heuristics {typically find better feasible solutions than} relaxation rounding and water-filling ones. {But the tradeoff is that the relaxation rounding and water filling algorithms achieve very efficient run times.}\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn his PhD thesis, Professor Floudas showed that, given a solution to the minimum number of matches problem, he could solve a nonlinear optimization problem designing effective heat recovery networks. But the sequential HENS method cannot guarantee that promising minimum number of matches solutions will be optimal (or even feasible!) to Professor Floudas' nonlinear optimization problem. Since the nonlinear optimization problem is relatively easy to solve, we propose generating many good candidate solutions to the minimum number of matches problem. This manuscript develops nine heuristics with performance guarantees to the minimum number of matches problem. Each of the nine heuristics is either novel or provably the best in its class. Beyond approximation algorithms, our work has interesting implications for solving the minimum number of matches problem exactly, e.g.\\ the analysis into reducing big-M parameters {or the possibility of quickly generating good primal feasible solutions}. \\\\[-4pt]\n\n\\noindent\n\\textbf{Acknowledgments} \\\\[2pt]\n\\noindent\nWe gratefully acknowledge support from EPSRC EP\/P008739\/1, an EPSRC DTP to G.K., and a Royal Academy of Engineering Research Fellowship to R.M.\n\n\n\\bibliographystyle{ijocv081}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe success of the manufacturing industry based on the revolutionary innovations in various technologies in hardware and software \\cite{8666558}. The conception of the Internet of Things (IoT) has already become a part of modern factories for automatizing of large-scale processes~\\cite{Microsoft2019}. \n\nManufacturing sector brings many different stakeholders together. This industry has mostly common fraud problems comparing to any sector. The significant rate (56\\%) of respondents reported the issues related to vendor or\nprocurement fraud~\\cite{kroll}. For some companies developing know-how products, intellectual property infringement could be fatal for the whole business. Not surprisingly, these issues make companies look for new ways to avoid them~\\cite{deloitte2014}.\n\nBlockchain technology, which stays behind Bitcoin, is nowadays a hype technology\\cite{8704309}. Its development could be revolutionized like the appearance of the Internet at the beginning of 90th's. Blockchain provides a secure mechanism for achieving an independent trusted between business partners, excluding intermediaries. \nThe rest of the paper is organized as follows. Section \\ref{sec:research:method} discusses information about research methods. In Section \\ref{sec:results} illustrates the results of searching and screening for relevant papers for this research, while section \\ref{sec:analysis} answers the research questions using these results. Section \\ref{sec:conlcusion} concludes the paper.\n\n\n\n\n\\subsection{Overview of Blockchain}\\label{sec:background}\nBlockchain is a new and successful combination of existing technologies. The technical concept of Blockchain describes how data are distributed saved across the user systems using cryptography algorithm.\nBlockchain organized logically centralized and organizationally decentralized. As a result, the Blockchain represents a distributed database that maintains an ever-expanding list of decentralized transaction, events, or records in hash form. The data is maintained in a distributed register (Distributed Ledger Technology) and all participants have a copy of the entire register. In this distributed approach, the data is grouped into individual blocks that are linked together to ensure the chronological order and immutable data integrity of the entire data set~\\cite{8715005}. \n\nThe innovation of Blockchain technology is that existing approaches have been successfully put together. Following approaches are the main components of Blockchain:\n\\begin{itemize}\n\\item \\textbf{Peer-to-peer network:} In this peer-to-peer network (P2P network), communication runs without a central point. All participants or nodes are connected to each other and communicate with each other at the same level. Since the nodes are equal to each other, or can use services and make them available at the same time, there is no classic client-server structure\\cite{8704309}.\n\\item \\textbf{Cryptography:} With the help of methods from cryptography, the distributed register is protected against manipulation and abuse. This enables traceability, data integrity and authentication of the data source\\cite{LI2018133}.\n\\item \\textbf{Consensus mechanism:} The consensus mechanism defines the criteria that provide evidence of permission to create new blocks (mining). To reach a consensus, various consensus algorithms have been developed\\cite{Innerbichler:2018:FBA:3211933.3211953}.\n\\end{itemize}\n\nDue to the different uses of Blockchain technology, there are different variations on how the Blockchain is constructed.\nIn a \\textit{public Blockchain}, there are no restrictions on who can see the public data and validate the transactions\\cite{LI2018133}. Furthermore, the Blockchain data may be encrypted and understandable only to the authorized user. In the case of a \\textit{private Blockchain}, the completed consortium of those who access the Blockchain and are allowed to validate transactions is predefined~\\cite{8560153}. In a \\textit{permissionless Blockchain}, there are no restrictions on the identity of the participants who are allowed to conduct the transactions. In a \\textit{permissioned Blockchain}, the user group that can execute the transactions and generate new blocks is predefined and known~\\cite{8678753}.\n\\subsection{Smart Contracts}\nIn addition to a decentralized database system for transactions, Blockchain technology is also a platform for the automation of processes, regulations and organizational principles.\nSmart contracts are new, intelligent forms of contracts. These are to be understood as peer-to-peer applications, which are distributed with the underlying Blockchain technology, such as Bitcoin, Ethereum or Hyperledger. Hyperledger also uses the term \"chaincode\"~\\cite{Wang:2019:SSB:3302505.3310086}.\n\nThe smart contracts enable the determination of the conditions that lead to certain decisions by the data provided; the automatic processing of contracts; the permanent and real-time monitoring of contract terms and the automatic enforcement of the rights of contractors. The smart contracts provide not only the information about the Blockchain network and the distribution of data, but also the business logic~\\cite{Baumung2018135}.\n\nThese P2P applications can be programmed, stored in the Blockchain and executed in there. Therefore, they have the same advantages as the Blockchain itself. In Bitcoin, the smart contracts are created in the form of scripts. In order to simplify the development of smart contracts for Ethereum platform, a new specific programming language \"Solidity\" has been developed~\\cite{Baumung2019456}. Compared to the scripting language in Bitcoin, where many program constructs, such as loops, are missing, Solidity is a higher-level, more abstract language that resembles the JavaScript language. The chain codes are written in various high-level languages, such as Java or Go, and during execution, access is made to the data stored in the Blockchain, or to read out the existing information and store new ones. These scripts are stored in the Blockchain at a particular address. This address is determined when the integration of  smart contract into the Blockchain is decided. When an event prescribed in the contract has occurred, a transaction is sent to this address. The distributed virtual machine executes the code of the script using the data sent with the transaction.\n\\subsection{State of Research on Blockchain}\nYli-Huumo et. al~\\cite{Yli_Huumo_2016} aim to understand the current research state of Blockchain technology, its technical challenges and limitations. This systematic review illustrates an sharply increasing number of publications each year beginning from 2012. It shows a growing interest in Blockchain technology. \nSwan~\\cite{swan2015blockchain} identified seven technical challenges of Blockchain for the future. Modern Blockchain implementations have to ensure security, throughput, size and bandwidth, performance, usability, data integrity and scalability. Being public Blockchains the \\textit{throughput} in the Bitcoin and Ethereum networks is from 10tps to 100tps (transactions per second).  For example, VISA Payment System proceeds 2,000tps. But a permissioned Blockchain Hyperledger Fabric overcomes these challenges~\\cite{Alharby2017}. In order to achieve adequate security in the Bitcoin network validation of transaction  takes roughly 10 minutes \\textit{(latency)}. In February 2016 the size of Bitcoin register was 50,000 MB. Current \\textit{size} of Bitcoin is 1 MB. This is the serious limitation of \\textit{bandwidths} for Blockchain, which should be solved to increase amount of transaction handled by register. The 51\\% attach on Blockchain network is still significant security issue. If the majority of the network  will be controlled by hackers, it will be possible to manipulate Blockchain. Issue of \\textit{waster resources} is caused  by Proof-of-Work effort in the mining process mainly in Bitcoin, which required huge amounts of energy.  But there are other consensus algorithms, like Proof-Of-Stake, which are energy friendly. \\textit{Usability} problems resulting  from difficulty of using Bitcoin API~\\cite{Yli_Huumo_2016}. \\textit{Versioning, hard forks, multiple chains} refers to a small chain with a small number of nodes, where a possibility of  51\\% attach is higher. Another issue become possible  when chains are split for administrative or versioning purposes.\n\\begin{figure*}[htbp]\n\\centerline{\\includegraphics[width=\\textwidth,height=4.5cm]{Mapping-Process.png}}\n\\caption{The Systematic Mapping Process~\\cite{Petersen:2008:SMS:2227115.2227123}.}\n\\label{fig:001}\n\\end{figure*}\n\\section{Research method}\\label{sec:research:method}\nA systematic mapping study was selected to identify and classify primary studies to provide a systematic overview on the topics of industrial manufacturing and Blockchain. Petersen et al.~\\cite{Petersen:2008:SMS:2227115.2227123} presented the guidelines for systematic mapping study, which we followed to conduct this study.\n\nThe process for the systematic mapping study falls into a  five-phase process as depicted in Figure \\ref{fig:001}: (1) Define research questions; (2) Search for primary studies; (3) Identify inclusion and exclusion criteria and screen primary studies based on these criteria; (4) Classify primary studies; (5) Mapping the data.\n\\subsection{Research questions}\nThe first step in systematic mapping study is the definition of the research questions. The purpose of this research is to classify current research and identify pertinent themes which relate directly to Blockchain technologies in manufacturing. This leads to the following research questions (RQs): \\\\\n\n\\textbf{RQ1:} \\textit{What are the problems between stakeholders in the manufacturing\nindustry? } \\\\\nRationale: The intention of this question is to identify current gaps in relationships of interested parties in manufacturing industry.\\\\\n\n\\textbf{RQ2:} \\textit{What are the data to secure in manufacturing process?} \\\\\nRationale: This question aims to identify the data, which should be insured during the whole process.\\\\\n\n\\textbf{RQ3:} \\textit{What are the use cases of Blockchain technology for manufacturing industry?} \\\\\nRationale: The intention of this question is to figure out the possible usage of Blockchain in manufacturing area.\\\\\n\n\\textbf{RQ4:} \\textit{What Blockchain frameworks are suitable for the scenario \"Assignment of production orders to an external manufacturer\"?} \\\\\nRationale: With this question, we investigate the existing Blockchain solutions to find appropriate framework. \\\\\n\\subsection{Search Strategy}\nThe search strategy is key to ensure a good starting point for the identification of studies and ultimately for the actual outcome of the study. An extensive and broad set of primary studies was needed to answer the research questions. The most popular academic databases in the domain of software engineering were selected to be used in this systematic mapping to search for potentially relevant papers:\n\\begin{itemize}\n\\item ACM Digital Library\\footnote{http:\/\/dl.acm.org}\n\\item IEEE Xplore Digital Library\\footnote{http:\/\/ieeexplore.ieee.org}\n\\item Scopus\\footnote{https:\/\/www.scopus.com}\n\\item Science Direct\\footnote{https:\/\/www.sciencedirect.com}\n\\end{itemize}\n\nFinding possibly relevant publications to answer the research questions requires creating an appropriate search clause. We chose the terms \"Blockchain\" and \"Manufacturing industry\" for this study as the main search keyword core, it focuses on Blockchain technology, manufacturing, production processes.\nThe final search strings were extended with alternative synonyms for main keywords. The term \"distributed ledger\" is a basic technology for \"blockchain\". We considered papers mentioning distributed manufacturing, manufacturing execution, programmable logic controller\" and included them into the search clause.\nRegarding the keywords for the search, after some exploratory searches using different combination of keywords, the researchers jointly established the final string to be used in the search for papers in the databases. Search terms with similar meanings were grouped in the same group and combined using the OR logical operator. To perform automatic searches in the selected digital libraries, the AND logical operator were used between combined terms of different groups, depicted in Table \\ref{tab:01}:\n\\begin{table}[htbp]\n\\centering\n\\label{tab:01}\n\\caption{Searches in databases.}\n\\begin{tabular}{|l|l|}\n\\hline\n\\rowcolor[HTML]{EFEFEF} \n\\textbf{Database} & \\textbf{Search}\\\\ \n\\hline\nACM               & \\begin{tabular}[c]{@{}l@{}}(+(\"Blockchain\" \"Distributed Ledger\") +\\\\(\"Manufacturing Execution\" \"Programmable Logic \\\\ Controller\" \"Manufacturing\" \"Distributed \\\\manufacturing\")\\end{tabular}                                                             \\\\ \\hline\nIEEE              & \\begin{tabular}[c]{@{}l@{}}('Blockchain' OR 'Distributed Ledger') AND \\\\ ('Industrial Control' OR 'Manufacturing Execution' \\\\ OR 'Programmable Logic Controller' OR \\\\'Manufacturing industry' OR 'Distributed \\\\manufacturing')\\end{tabular}           \\\\ \\hline\nScopus            & \\begin{tabular}[c]{@{}l@{}}ALL ((\"Blockchain\" OR \"Distributed Ledger\") AND\\\\ (\"Industrial Control\" OR \"Manufacturing Execution\" \\\\ OR \"Programmable Logic Controller\") OR \\\\ (\"Manufacturing industry\" OR \"Distributed\\\\ manufacturing\"))\\end{tabular} \\\\ \\hline\nScience Direct    & \\begin{tabular}[c]{@{}l@{}}(\"Blockchain\" OR \"Distributed Ledger\") AND \\\\ (\"Industrial Control\" OR \"Manufacturing Execution\" \\\\ OR \"Programmable Logic Controller\" OR \\\\ \"Manufacturing\" OR \"Distributed manufacturing\")\\end{tabular}                    \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nThe search string was applied to title, abstract, full-text and keywords, and limited to journal papers written in English. The search was performed at the beginning of 2016. A total of 258 papers were retrieved from the different databases, which are focusing on research regarding information technologies, as on 21st May 2019, 10.30 am (CEST) and displayed in Table \\ref{tab:studies}. \n\\begin{table}[htbp]\n\\label{tab:studies}\n\\caption{Number of studies per database.}\n\\centering\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n\\rowcolor[HTML]{EFEFEF} \n\\textbf{Database} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Search\\\\ results\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}Final\\\\ results\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}l@{}}\\% final papers\\\\ from search results\\end{tabular}}  \\\\ \\hline\nACM               & 24   & 4 & 1.6 \\%         \\\\ \\hline\nIEEE              & 61   & 12 & 4.7 \\%         \\\\ \\hline\nScopus            & 159  & 6 &  2.3 \\%       \\\\ \\hline\nScience Direct    & 14   & 6 & 2.3 \\%         \\\\ \\hline\n\\textbf{Total}    & \\textbf{258} &  \\textbf{28} &  \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\subsection{Screening for relevant papers}\nThis step is to exclude all research papers that are irrelevant to the research questions. To accomplish this step, We followed the screening approach described by Yli-Huumo et al.~\\cite{Yli_Huumo_2016} and defined inclusion\/exclusion criteria.\\\\\n\n\\textbf{Inclusion criteria} \n\nTo be considered for inclusion in the study, the research being evaluated had to originate from an academic source, such as a journal or conference, and clearly show its contribution was focused on applying Blockchain in manufacturing. Studies are accessible electronically. The paper title includes \"Blockchain\".\\\\\n\n\\textbf{Exclusion criteria}\n\nFor those publications that passed the inclusion criteria, two filters were applied to reduce the publications to only those that were deemed to be directly aligned with the focus of the study. These filters are described as follows:\n\\begin{itemize}\n\\item Study focuses on financial sector\n\\item Study contains \"Bitcoin\" or \"Cryptocurrency\"\n\\end{itemize}\nIt was decided to apply exclusion criteria on titles, keywords, abstracts and full-text. The final inclusion or exclusion could be decided based on the reading of publication's full text. Thus, this phase only eliminates publications clearly not within this study's scope and publications failing on formal requirements (such as duplicated papers). In detail, we identified 230 publications (about 90\\% of all papers) outside this study's scope: 20 duplicates, 156 papers accordingly with inclusion\/exclusion criteria and other based on abstract reading.\nThe search process is summarized in Figure \\ref{fig:sum}.\\\\\n\n\\begin{tikzpicture}\n[node distance=.6cm,\nstart chain=going below,]\n    \\node (step0) [punktchain ]  {Applying search on databases};\n        \\begin{scope}[start branch=venstre,\n            every join\/.style={->, thick, shorten <=1pt}, ]\n                \\node[punktchain, on chain=going right, join=by {->}]\n            (result0){Results = 258};\n        \\end{scope}\n    \\node (step1) [punktchain ]  {Remove duplicates};\n        \\begin{scope}[start branch=venstre,\n            every join\/.style={->, thick, shorten <=1pt}, ]\n                \\node[punktchain, on chain=going right, join=by {->}]\n            (result1){Results = 238};\n        \\end{scope}\n    \\node[punktchain] (step2) {Apply inclusion\/exclusion};\n        \\begin{scope}[start branch=hoejre1,]\n            every join\/.style={->, thick, shorten <=1pt}, ]\n                \\node [punktchain, on chain=going right, join=by {->}] \n            (result2) {Results = 94};\n        \\end{scope}\n    \\node[punktchain] (step3) {Exclusion based on abstract};\n        \\begin{scope}[start branch=hoejre1,]\n            every join\/.style={->, thick, shorten <=1pt}, ]\n                \\node [punktchain, on chain=going right, join=by {->}] \n            (result3) {Results = 36};\n        \\end{scope}\n    \\node[punktchain] (step4) {Exclusion based on full reading};\n        \\begin{scope}[start branch=hoejre1,]\n            every join\/.style={->, thick, shorten <=1pt}, ]\n                \\node [punktchain, on chain=going right, join=by {->}]\n            (result4) {Results = 32};\n        \\end{scope}\n    \\node[punktchain] (step5) {Final primary papers};\n        \\begin{scope}[start branch=hoejre1,]\n            every join\/.style={->, thick, shorten <=1pt}, ]\n                \\node [punktchain, on chain=going right, join=by {->}]\n            (result5) {Results = 28};\n        \\end{scope}\n  \\draw[|-,-|,->, thick,] (result0.south) |-+(0,-1em)-| (step1.north);\n  \\draw[|-,-|,->, thick,] (result1.south) |-+(0,-1em)-| (step2.north);\n  \\draw[|-,-|,->, thick,] (result2.south) |-+(0,-1em)-| (step3.north);\n  \\draw[|-,-|,->, thick,] (result3.south) |-+(0,-1em)-| (step4.north);\n  \\draw[|-,-|,->, thick,] (result4.south) |-+(0,-1em)-| (step5.north);\n  \\end{tikzpicture}\n  \\begin{figure}[htbp]\n\\caption{Search and Selection Process of the Papers.}\n\\label{fig:sum}\n\\end{figure}\n\\subsection{Key-wording using Abstracts}\nYli-Huumo et al.~\\cite{Yli_Huumo_2016} proposed the key-wording technique to classify all relevant research papers. We first went thought the abstract of each paper to identify the most important keywords and the key contribution. This purpose used to classify papers under different categories.  In some cases where it was difficult to classify a paper using\nits abstract, we looked through its introduction and conclusion. After classifying all papers, we read the papers and made changes to the classification when necessary.\n\\subsection{Data extraction and mapping process}\nThe bubble plot was designed to collect all the information needed to address the research questions of this mapping study. During the process of data extraction, we recorded major terms to Excel, which helped me to generate categories and proceed quality analysis. These data items embrace the main goals of papers. We used both qualitative and quantitative synthesis methods.\n\\\\\n\nThe final search process results 28 papers performed at the beginning of 2017. Their distribution over databases and percentage from all search results are illustrated in Table \\ref{tab:studies}.\\\\\n\\section{Study Results}\\label{sec:results}\nThis section demonstrates the findings from those the data was extracted regarding use cases of Blockchain in manufacturing industry, research type and attributes of Blockchain technology.\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[width=4.25cm,height=4.25cm]{Countries.png}}\n\\caption{Continent wise distribution of studies.}\n\\label{fig:country}\n\\end{figure} \n\n\\textbf{Top Countries and continents}\n\nGeographic distribution of the selected primary papers is shown in Fig. \\ref{fig:country}. The top continent was Europe with 14 studies being conducted there. Asia was second with 9 studies followed by America with 5 studies. China and Germany contributed towards 6 and 3 studies respectively. The rest of the countries had two or less papers published. It shows, that Blockchain technology has attracted attention worldwide.  \\\\\n\n\\textbf{Year Wise Distribution}\n\nFigure \\ref{year} depicts the distribution of the papers from 2017 to 2019 year. Literature related to Blockchain in manufacturing industry has increased enormously in the last 2 years. Due to increasing papers of applying Blockchain in industrial context it expected to save this trend for next years. \\\\\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[width=6cm,height=4cm]{Year.png}}\n\\caption{Publication year of the selected primary papers.}\n\\label{year}\n\\end{figure} \n\n\\textbf{Classification of research.}\n\nAll  of  the  publications  in  the  study were classified by considering the following\ncriteria: (1) Use case of Blockchain, (2) Research facet and (3) Blockchain facet.\n\\subsection{Use case of Blockchain:}\nIn order to answer RQ3, we classified  the  publications  in  the  study  under five dimensions. These dimensions describe different use cases of Blockchain in manufacturing industry on the current state of research in the area. The use cases of Blockchain in industrial manufacturing are follows: \\\\\n\n\\textbf{Secure transfer of order data.} This use case describes how the production orders can be assigned to an external manufacturer and securely transmitted between different systems. It enables mutual interaction between the producer and the customer~\\cite{8250199}.\\\\\n\n\\textbf{Product data storing.} The data can be intercepted during transmission from the user's computer to the cloud systems. In these use cases the focus is on secured storing of products data in Blockchain.\\\\\n\n\\textbf{Supply chain, Process traceability.} Creating and distributing of goods can span over multiple  locations, hundreds of stages etc. The use case aims to provide the ability to trace process in supply chain from procurement of raw materials to production~\\cite{8711819}.\\\\\n\n\\textbf{Prevention of fraud, Protection of Intellectual Property (IP).} The main point of this use case is to prove of products origin and intend for prevention of manipulation providing an indelible and traceable record of changes.\\\\\n\n\\textbf{Industrial IoT (IoT), Automation.} This use case illustrates how Blockchain can be used for integration with industrial IoT in automation context. \\\\\n\nFigure \\ref{trend} shows the amount of publications where each of use case above is described year wise. This figure illustrates, that the most researched topic is supply chain and process traceability (19 papers). Less researched (5 papers) is the scenario of how could the product data be stored in Blockchain. All use cases, except \"Secure transfer of order data\" show the trend of increasing interests. \n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[width=9cm,height=5cm]{Trend.png}}\n\\caption{Publication year of the identified use cases.}\n\\label{trend}\n\\end{figure} \n\n\\subsection{Research Facet:}\nThis facet is inspired by~\\cite{Petersen:2008:SMS:2227115.2227123} classify publications according to the type of research they contribute. (1) Review paper summarizes the current state of understanding on a topic. (2) Conceptual paper addresses a question that cannot be answered simply by getting more factual information. (3) Solution proposal includes an illustration or example of a solution to a particular problem. (4) Implementation research  provides a prototypical development of a solution. (5) Case Study provides an up-close, in-depth, and detailed examination of a subject of study.\n\n\\begin{figure*}[htbp]\n\\centerline{\\includegraphics[width=\\textwidth,height=11cm]{SM.png}}\n\\caption{Visualisation of a Systematic Map in the Form of a Bubble Plot}\n\\label{fig:007}\n\\end{figure*}\n\\subsection{Blockchain Facet:}\nTo answer the RQ4, this facet classified the publications along the Blockchain attributes like framework (Ethereum, Hyperledger or other frameworks), type of Blockchain (public. private, permissioned) and focus of research paper on  consensus algorithm.\n\nThe results of mapping process are summarized in Figure \\ref{fig:007} in form of the bubble plot to show the frequencies. This visualization gives a quick overview of publications for each category.\n\n\\section{Analysis}\\label{sec:analysis}\nThis section discusses the study results and answers the research questions that we defined in Section \\ref{sec:background}.\\\\ \n\n\\textbf{RQ1: What are the problems between stakeholders in manufacturing industry?}\\\\\\\\\nManufacturing industry is facing security incidents due to the competition~\\cite{Wang:2019:SSB:3302505.3310086}, data sharing with third companies~\\cite{8678753}, operational inefficiencies, losses and costs~\\cite{8626103}. Issue of limited trust is one of the complications in the industry~\\cite{Geiger:2019:PTD:3297280.3297546,Innerbichler:2018:FBA:3211933.3211953,8704309}. In~\\cite{Pinheiro2019331} authors illustrate the dependency of industrial companies on Trusted-Third-Party (TTP). It caused by the closed source code of programs, used in manufacturing industry~\\cite{8678753}. Moreover it is necessary to differentiating between original part or counterfeit products~\\cite{Holland2018,BANERJEE201869}. The outsourcing of production orders~\\cite{Baumung2019456} leads to limited flexibility  and considerable organizational effort.\n\\\\ \\\\\n\\textbf{RQ2: What are the data to secure in manufacturing process?} \\\\\\\\\nFor some businesses the data exchange is a key success factor, so we found several type of data that should be protected between stakeholders: \n\\begin{itemize}\n\\item Computer-Aided Design (CAD) file for design and technical documentation~\\cite{Holland2018},~\\cite{Papakostas2019}\n\\item Material specification~\\cite{Mondragon20181300,Papakostas2019}\n\\item Order details~\\cite{Baumung2019456} and product recipe~\\cite{WESTERKAMP2019}\n\\item Machine data~\\cite{Geiger:2019:PTD:3297280.3297546,ANGRISH20181180} (measurement data~\\cite{Wang:2019:SSB:3302505.3310086} and configuration\\cite{Mondragon20181300})\n\\item Process values~\\cite{Geiger:2019:PTD:3297280.3297546,MANDOLLA2019134} and process state~\\cite{LI2018133,8704309,8621042}\n\\end{itemize}\n\nSelection of data, which should be protected, depends on specific scenario and application area of Blockchain. \\\\ \n\\textbf{RQ3: What are the use cases of Blockchain technology for manufacturing industry?} \\\\ \\\\\nThere are various use cases of Blockchain technology in the industrial manufacturing. In this study we identified 5 papers (17\\% of all papers) describing use case \"Secure transfer of order data\" in form of solution proposal. All of this papers provide prototypical implementation. Only 4 papers (14\\% ) are relevant for storing of product data in distributed ledger. According to our findings, significant part of papers (around 43\\%) illustrate solution proposal for supply chain and process traceability and 9 of them (32\\%) demonstrate implementation examples. Fraud prevention and IP Protection are considered in 2 papers (around 7\\%) and both of these papers implement this use case. The last use case \"IoT, Automation\" are described in 5 papers and covers equally all types of research.\nThe application of Blockchain is not limited to the use cases above. \\\\ \\\\\n\\textbf{RQ4: What Blockchain frameworks are suitable for the scenario \"Assignment of production orders to an external manufacturer\"?} \\\\ \\\\\nThis scenario is implemented mostly on Ethereum Blockchain (in 4 of all papers)~\\cite{Baumung2018135,Baumung2019456,Papakostas2019,ANGRISH20181180}. In~\\cite{Baumung2019456} was illustrated an example of using Hyperledger Fabric and in~\\cite{LI2018133} was used Multichain. Public Blockchain was used in 2 papers~\\cite{Baumung2018135},~\\cite{Baumung2019456}, in 1 research work the authors used the private Blockchain~\\cite{Papakostas2019} and in 4 papers was chosen consortium or permissioned Blockchain~\\cite{LI2018133,Baumung2018135,Baumung2019456,ANGRISH20181180}. Some papers describe several types of Blockchain in the case use case. It means, that this use case is possible to implement based on different Blockchain networks and doesn't require a specific framework.  \n\\section{Conclusion}\\label{sec:conlcusion}\nIn the coming years it is expected, that the manufacturing sector will benefit from the use of Blockchain technology. In order to identify opportunities for integration of Blockchain in industrial processes, this research was made in form of systematic mapping study. After conducting the SMS and analysing the literature, a total of 28 primary papers were extracted from 4 different scientific databases, published mainly in journals and conference proceedings and classified into different facets. We have covered the time period of 2017-2019 and have classified the papers under different dimensions. We grouped these issues into five use cases, namely, secure transfer of order data, product data storing, supply chain and process traceability, fraud prevention and IP protection, IoT and automation. \n\nIn this study we found that the majority of papers describe the case \"supply chain and process traceability\" as solution proposal. There are essential less findings regarding assignment of production orders to an external manufacturer. It demonstrates the relative lack of research on this scenario, that requires effort lot more research. As found out, the last use case can be implemented using different frameworks. For example, this case could be implemented base on permissioned Blockchain using Hyperledger Fabric\\footnote{\\url{www.hyperledger.org}}. However, it is required to evaluate all frameworks to select the most suitable solution for the use case. \n\nThe result of this mapping study can be applied only on the selected research databases and may help the researchers to get an overview of the status of Blockchain in the manufacturing industry and highlight the research gaps. \n\nOur line of future research aims precisely to implement our own prototype of use case \"Assignment of production orders to an external manufacturer\" to demonstrate the benefits of using Blockchain in factory automation. Additionally, we are going to extend the literature survey to include other databases like SpringerLink and we will apply snowballing to assure that the search is as comprehensive as possible. \n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Introduction}\nIn recent years unstable periodic orbits have\nbeen shown to be an effective tool in the description of \ndeterministic dynamical systems of low intrinsic dimension\\cite{CHAOS92}, \nin diagnosing deterministic chaos in noisy biological \nsystems\\cite{Moss94}, and many other applications. \nThe theory has been successfully applied to low dimensional ordinary \ndifferential equations (deterministic chaos) and linear partial differential \nequations (semiclassical quantization). It is an open question whether\n the theory has anything to say about nonlinear partial differential \nequations (hydrodynamics, field theory).  \nIn this paper we show that \n the periodic orbit theory can be used to describe \nspatially extended systems by applying it to the Kuramoto-Sivashinsky\nequation\\cite{Kur,Siv}.\n\nIn what follows we shall refer to a periodic solution \n as a ``cycle'', and to the closure of the union of all \nperiodic solutions as the ``invariant set''.\nPeriodic solutions are important because they form the {skeleton}\nof the invariant set\\cite{cycprl,AACI}, with\ncycles ordered {hierarchically}; short cycles give good\napproximations to the invariant set, longer cycles refinements.\nErrors due to neglecting long cycles can be bounded, and for nice hyperbolic\nsystems they fall off exponentially or even superexponentially \nwith the cutoff cycle length\\cite{Rugh92,CRR93}.\nFurthermore, cycles are {structurally robust} as for smooth flows\neigenvalues of short cycles vary slowly with smooth parameter changes,\nshort cycles can be accurately extracted from experimental\nor numerical data, and\nglobal averages (such as \ncorrelation exponents, escape rates \nand other ``thermodynamic\" averages) can be efficiently computed from short\ncycles by means of {cycle expansions}.\n\nWhile the role of periodic solutions in elucidating the asymptotics of\nordinary differential equations was already appreciated by\nPoincar\\'e\\cite{poincare}, allegedly Hopf\\cite{Hopf42}\nand, more demonstrably, Spiegel and\ncollaborators\\cite{MS66,BMS71,EAS87} have argued that the asymptotics of\npartial differential equations should also\nbe described in terms of recurrent spatiotemporal patterns.\nPictorially, dynamics drives a given spatially extended system through a\nrepertoire of unstable patterns; as we watch  \na given ``turbulent'' system evolve, \nevery so often we catch a glimpse of a familiar pattern. \nFor any finite  spatial resolution,\nthe system follows approximately for a finite time \na pattern belonging to a finite \nalphabet of admissible patterns, and the long term dynamics can be thought\nof as a walk through the space of such patterns,\njust as chaotic dynamics with a  low dimensional\nattractor can be thought of as a succession of nearly periodic (but\nunstable) motions.\n\n\\section{Kuramoto-Sivashinsky system}\n\nWe offer here a modest implementation of the above program on\na prototype spatially extended dynamical system defined by\nthe Kuramoto-Sivashinsky equation\\cite{Kur,Siv}\n\\begin{equation}\nu_t=(u^2)_x-u_{xx}-\\nu u_{xxxx} \n\\label{ks}\n\\end{equation}\nwhich arises as a model amplitude equation for interfacial instabilities\nin a variety of contexts - see e.g. \\refref{KNS90}.\nHere $t \\geq 0$ is the time, $x \\in [0,2\\pi]$ is the space coordinate,\nand $\\nu$ is a fourth-order ``viscosity'' damping parameter.\nThe subscripts $x$ and $t$ denote the partial derivatives with respect to \n$x$ and $t$. \nWe take the Kuramoto-Sivashinsky system  because it is one of the\nsimplest physically interesting spatially extended nonlinear systems,\nbut in the present context the \ninterpretation of the equation, or the equation itself is not the most \nimportant ingredient;\nthe approach should be applicable to a wide class of \nspatially extended nonlinear systems. The salient feature of such \npartial differential equations is that\nfor any finite value of $\\nu$ their asymptotics is in principle \ndescribable by a\n{\\em finite} set of ``inertial manifold''\n ordinary differential equations\\cite{Foias88}.\n\nThe program of studying unstable solutions in this context\nwas initiated by Goren, Eckmann and Procaccia\\cite{GEP}\nwho have used a 2-unstable modes truncation of \nthe Kuramoto-Sivashinsky equation\nto study the dynamics connecting coexisting unstable \n{\\em temporally stationary} solutions.\nWe shall study here\nunstable {\\em  spatiotemporally  periodic} solutions of the {\\em full}\nKuramoto-Sivashinsky system.\nOur main result is that in the limit of weak turbulence or \n``spatiotemporal chaos'', we can determine hierarchically and exhaustively\ncycles of longer and longer periods, and apply this data \nto the evaluation of global averages.  \n\nThe function $u(x,t)=u(x+2\\pi,t)$ is assumed periodic on \nthe $x \\in [0,2\\pi]$ interval. \nAs $u(x,t)$ has compact support, the standard\nstrategy is to expand it in a discrete spatial Fourier series: \n\\begin{equation}\n u(x,t)= \\sum_{k=-\\infty}^{+ \\infty} b_k(t) \\e^{\\i k x}\n\\, . \n\\label{fseries}\n\\end{equation}\nSince $u(x,t)$ is real, $b_k=b_{-k}^*$.\nSubstituting (\\ref{fseries}) into (\\ref{ks}) yields \nthe infinite ladder of evolution equations for the Fourier coefficients $b_k$: \n\\begin{equation}\n\\dot{b}_k=(k^2-\\nu k^4)b_k +\\i k \\sum_{m=-\\infty}^{\\infty}\nb_m b_{k-m}\n\\,. \n\\label{expanfull}\n\\end{equation}\nAs  $\\dot{b}_0=0$, the average (the mean drift) of the solution is \nan integral of motion. In what follows we shall assume that this average is \nzero, $\\int \\d x \\, u(x,t) =0$.    \n\nThe coefficients $b_k$ are in general complex functions of time. \nWe can simplify the system (\\ref{expanfull}) further by assuming \nthat $b_k$ are pure imaginary, $b_k= \\i a_k$, \nwhere $a_k$ are real. \nAs we shall see below, this picks out the\nsubspace of odd solutions $u(x,t)=-u(-x,t)$, with \nthe evolution equations \n\\begin{equation}\n\\dot{a}_k=(k^2-\\nu k^4)a_k - k \\sum_{m=-\\infty}^{\\infty} a_m a_{k-m} \n\\,.\n\\label{expan}\n\\end{equation}\nWe shall determine \nthe periodic solutions in the space of Fourier coefficients,\nand then reconstitute from them the unstable spatiotemporally \nperiodic solutions of\n\\refeq{ks}. \n\nThe trivial solution $u(x,t)=0$ is  a fixed point of (\\ref{ks}). From \n(\\ref{expan}) it follows that the $|k|<1\/ \\sqrt{\\nu}$ \nlong wavelength modes of this fixed point \nare linearly unstable, and the \n$|k|>1\/ \\sqrt{\\nu}$ short wavelength modes are stable.  \nFor $\\nu > 1$,  $u(x,t)=0$ is the globally attractive stable fixed point;\nstarting with $\\nu =1$ the solutions go through a rich sequence of\nbifurcations, studied e.g. in \\refref{KNS90}. \nDetailed knowledge of the parameter dependence of bifurcations sequences\nis not needed for our\npurposes; we shall take $\\sqrt{\\nu}$ sufficiently small so that the\ndynamics can be spatiotemporally chaotic, but not so small that we would be\noverwhelmed by too many short wavelength modes needed in order to accurately\nrepresent the dynamics.\n\nThe growth of the unstable long wavelengths (low $|k|$) excites\nthe short wavelengths \nthrough the nonlinear term  in (\\ref{expan}). \nThe excitations thus transferred are dissipated by the strongly damped \nshort wavelengths, and  a sort of ``chaotic equilibrium'' \ncan emerge. The very short wavelengths $|k| \\gg  1 \/ \\sqrt{\\nu}$ \nwill remain small \nfor all times, but the intermediate wavelengths of order\n$|k| \\sim 1 \/ \\sqrt{\\nu}$ \nwill play an important role in maintaining the dynamical equilibrium. \nAs the damping parameter decreases, the solutions increasingly take on \n Burgers type shock front\ncharacter which is poorly represented by the Fourier basis, and many\nhigher harmonics need to be kept\\cite{KNS90,GEP} in truncations of\n(\\ref{expan}).\nHence, while one may truncate the high modes in the expansion (\\ref{expan}), \ncare has to be exercised to ensure that no\nmodes essential to the dynamics are  chopped away.\n\nBefore proceeding with the calculations, we take into account\nthe symmetries of the solutions and describe our criterion for reliable\ntruncations of the infinite ladder of \nordinary differential equations (\\ref{expan}).\n\n\\section{Symmetry decomposition}\nAs usual, the first step in analysis of such dynamical flows\nis to restrict the dynamics to a Poincar\\'e section. We\nshall fix the  Poincar\\'e section to be the hyperplane\n$a_1=0$. We integrate (\\ref{expan}) with the initial\n conditions  \n$a_1=0$, and arbitrary values of the coordinates  $a_2, \\ldots, a_N$, where \n$N$ is the truncation order.  When $a_1$ becomes \n$0$ the next time,  the coordinates  $a_2, \\ldots, a_N$ are mapped \ninto $(a_2', \\ldots a_N')=P(a_2, \\ldots, a_N)$, where $P$ is the  Poincar\\'e \nmap. $P$ defines a mapping of a $N-1$ dimensional hyperplane into itself. \nUnder successive iterations of  $P$, any trajectory\napproaches the attractor ${\\cal A}$, which itself is an invariant \nset under $P$. \n\nA trajectory of \n (\\ref{expan}) can cross the plane $a_1=0$ in two possible ways: \n either when   \n$\\dot{a_1}>0$ (``up'' intersection) \nor when  $\\dot{a_1}<0$ (``down'' intersection),\n with the ``down'' and ``up'' crossings \nalternating. \nIt then makes sense to define the  Poincar\\'e map $P$ as a transition between, \nsay, ``up'' and ``up'' crossing. \nWith  Poincar\\'e section defined as the ``up-up'' transition, \nit is natural to define a ``down-up'' transition map $\\Theta$. Since \n$\\Theta$ describes the transition from down to up (or up to down) state, \nthe map $\\Theta^2$ describes the transition  up-down-up, that is  \n$\\Theta^2=P$.\n\nConsider the spatial flip and\nshift symmetry operations $Ru(x)=u(-x)$, $Su(x)=u(x+\\pi)$. \n The latter symmetry reflects the invariance under\nthe shift $u(x,t) \\rightarrow u(x+ \\pi,t)$, and is a particular case of the   \ntranslational invariance of the Kuramoto-Sivashinsky equation (\\ref{ks}). \nIn the Fourier modes decomposition (\\ref{expan}) this \nsymmetry acts as\n$S: a_{2k} \\rightarrow a_{2k}, a_{2k+1} \\rightarrow -a_{2k+1}$.  \nRelations $R^2=S^2=1$\ninduce decomposition of the space of solutions into 4 invariant \nsubspaces\\cite{KNS90};\nthe above restriction to $b_k= \\i a_k$ amounts to specializing\nto a subspace of odd solutions $u(x,t)=-u(-x,t)$.\n\nNow, with the help of the symmetry $S$ \nthe  whole attractor ${\\cal A}_{tot}$ can be \ndecomposed into two  pieces: ${\\cal A}_{tot}={\\cal A}_0 \\cup S \n{\\cal A}_0 $ for some set ${\\cal A}_0$.\n It can happen that the set ${\\cal A}_0$\n (the symmetrically decomposed attractor)  \ncan be decomposed  even further by the action of the map $\\Theta$. In this \ncase the attractor will consist of four disjoint sets: \n ${\\cal A}_{tot}={\\cal A} \\cup S {\\cal A}  \\cup \\Theta {\\cal A} \n  \\cup \\Theta S {\\cal A} $. As we shall see, \nthis decomposition \nis not always possible, since sometimes $ {\\cal A}$ overlaps with \n$\\Theta S{\\cal A} $ (in this case $\\Theta  {\\cal A}$ will also  overlap with \n$S {\\cal A} $). \nWe shall carry out our calculations in the regime where \nthe decomposition into four disjoint pieces \nis possible. In this case  the set $ {\\cal A}$ can be taken as\nthe fundamental \ndomain of the Poincar{\\'e} map, with $S  {\\cal A} $, \n$\\Theta  {\\cal A} $ and $\\Theta S  {\\cal A} $ its images under the \n$S$ and $\\Theta$ mappings.\n\nThis reduction of the dynamics to the fundamental domain is particularly\nuseful in periodic orbit calculations, as it simplifies symbolic dynamics\nand improves the convergence of cycle expansions\\cite{CEsym}.\n\n\\section{Fourier modes truncations}\n \nWhen we simulate the equation (\\ref{expan}) on a computer, we  have \nto  truncate the ladder of equations to a finite length $N$, i.e., set \n $a_k=0$ for $k>N$. \n$N$ has to be sufficiently large that no harmonics \n$a_k$ important for the dynamics  with $k>N$ \nare truncated. On the other hand,\ncomputation time increases dramatically with the increase of $N$:\nsince we will be evaluating the stability matrices for the flow,\nthe computation time will grow at least as $N^2$. \n\n\nAdding an extra dimension to a truncation of the system (\\ref{expan})\nintroduces a small\nperturbation, and this can (and often will) \nthrow the system into a totally different asymptotic state. \nA  chaotic attractor for $N=15$ can become a period three \nwindow for $N=16$, and so on. \nIf we compute, for example, the Lyapunov exponent\n$\\lambda(\\nu,N)$ for the strange attractor of the \nsystem (\\ref{expan}), there is no reason to \nexpect $\\lambda(\\nu,N)$ to smoothly converge to the limit  \nvalue $\\lambda(\\nu,\\infty)$ as $N \\rightarrow \\infty$. \nThe situation is different in the periodic windows, \nwhere the system is structurally stable, and it makes sense to compute \n Lyapunov exponents, escape rates, etc. for the \n{\\em repeller}, i.e. the closure of the set of all \n{\\em unstable} periodic orbits. \nHere the power of cycle expansions comes in: \nto compute quantities on the repeller by direct averaging methods is \ngenerally more difficult, because the motion quickly collapses to the \nstable cycle. \n\n\\begin{figure}\n\\centerline{\\epsfig{file=feig16bm.ps,width=8cm}}\n\\caption[]{\nFeigenbaum tree for coordinate $a_6$, $N=16$ Fourier modes \ntruncation of (\\ref{expan}). The two upper arrows indicate \nthe values of damping parameter that we use in our \nnumerical investigations; $\\nu=0.029910$ \n(chaotic) and $\\nu=0.029924$ (period-3 window). The lower arrow indicates\nthe kink where the invariant set ${\\cal A}$ \nstarts to overlap with $\\Theta S {\\cal A}$.\nTruncation to $N=17$ modes yields a similar figure, with values for \nspecific bifurcation points shifted by $\\sim 10^{-5}$ with respect to the \n$N=16$ values. The choice of the coordinate $a_6$ is arbitrary;\nprojected down to any coordinate, the tree is qualitatively the same.\n}\n\\label{feig16}\n\\end{figure}\n\n\nWe have found that the minimum value of $N$ to get any chaotic behavior at all \nwas $N=9$. However, the dynamics for the $N=9$ truncated system is rather\ndifferent from the full system dynamics, and therefore we have performed our\nnumerical calculations for $N=15$, $N=16$ and $N=17$. \n\\refFig{feig16} is a representative plot of  the Feigenbaum \ntree for the Poincar\\'e map $P$. \nTo obtain this figure, we took a random \ninitial point, iterated it for a some  time to let it settle on the\nattractor and then plotted the $a_6$ coordinate of the next 1000 intersections\nwith the Poincar{\\'e} section.\nRepeating this for different values of the damping parameter $\\nu$, one can \nobtain a picture of the attractor as a function of $\\nu$.\nFor an intermediate range of values of\n$\\nu$, the dynamics exhibits a rich variety of behaviours, such\nas strange attractors, stable limit cycles, and so on.\nThe Feigenbaum trees for different values of $N$ resemble \neach other, but the precise \nvalues of $\\nu$ corresponding the various bifurcations \ndepend  on the order of truncation $N$. \n\nBased on  the observed numerical similarity\nbetween the Feigenbaum trees for $N=16$ and $N=17$ (cf. \\reffig{feig16}),\n we choose $N=16$ as a reasonable cutoff \nand will use only this truncation throughout the remainder of this \npaper. We will\nexamine  two values of the damping parameter: $\\nu=0.029910$, \nfor which\nthe system is chaotic, and $\\nu=0.029924$, for which  the system has a\nstable period-3 cycle.\nIn our numerical work we use both the pseudospectral\\cite{Laurette} and the \n$4$-th order variable-step Runge-Kutta integration routines\\cite{NAG};\ntheir results are in satisfactory  agreement. As will be seen below, the\ngood control of symbolic dynamics guarantees that we do not miss\nany short periodic orbits generated by the bifurcation sequence indicated\nby the  Feigenbaum tree of \\reffig{feig16}. However, even though\nwe are fairly sure that for this parameter value we have all\nshort periodic orbits, the possibility that\nother sets of periodic solutions exist somewhere else in the\nphase space has not been excluded.\n\n\n\\begin{figure}\n\\centerline{\\epsfig{file=solution123.ps,width=8cm}\n\t \\epsfig{file=solution124.ps,width=8cm}}\n\\caption[]{Projections of a typical 16-dimensional trajectory onto \n\t\tdifferent 3-dimensional subspaces, coordinates \n\t\t(a) $\\{a_1, a_2, a_3\\}$,\n\t\t(b) $\\{a_1, a_2, a_4\\}$. $N=16$ Fourier modes truncation with\n\t\t    $\\nu=0.029910$.}\n\\label{plot123}\n\\end{figure}\n\n\nThe problem with such high dimensional truncations of (\\ref{expan})\nis that the dynamics is difficult to visualize. We can \nexamine its projections onto any three axes\n$a_i,a_j,a_k$, as in \\reffig{plot123}\nor, alternatively, study \na return map for a given  coordinate\n$a_k \\rightarrow a_k' = P_k(a_2, \\ldots, a_{N})$\nas the one  plotted in \\reffig{returnmap}.\nThe full return map is $(N-1)$-dimensional\n${\\bf a} \\rightarrow {\\bf P}(a_2, \\dots, a_{N})={\\bf a}'$\nand single-valued, and for the values of $\\nu$ used here\nthe attractor is essentially 1-dimensional,\nbut its  projection into the $\\{a_k,P_k(a_2,\\ldots,a_{N})\\}$ plane \ncan be  multi-valued and self-intersecting.\nOne can  imagine a situation where no\n``good'' projection is possible,\nthat is, any projection onto any two-dimensional\nplane is a multiple-valued function.\nThe question is how to treat such a map?\n \n\n\\begin{figure}\n\\centerline{\\epsfig{file=return6.ps,width=8cm}}\n\\caption[]{The attractor of the system \\refeq{expan}, plotted as the $a_6$\ncomponent of the $a_1=0$  \nPoincar\\'e section return map, 10,000 Poincar\\'e section returns of\na typical trajectory. Indicated are the periodic points $\\overline{0}$, \n$\\overline{1}$ and $\\overline{01}$; as this is an arbitrary projection of\nthe invariant set, they exhibit no good spatial ordering.\n$N=16$ Fourier modes truncation with $\\nu=0.029910$. \n}\n\\label{returnmap}\n\\end{figure}\n\n\\section{One-dimensional visualization of  the dynamics} \n\nWe now describe an \napproach which simplifies matters a lot by reducing the \nmap to an approximate one-dimensional map. The multiple-valuedness in \n\\reffig{returnmap} arises from the fact that the return map \nis a 2-dimensional\nprojection of a convoluted \n1-dimensional curve embedded into a high-dimensional space.\nWe shall show that it is possible to find an {\\em intrinsic} \nparametrization $s$ along the unstable manifold, \nsuch that the map $s \\rightarrow f(s)$ induced by the\nfull $d$-dimensional flow is approximately {\\em   $1$-dimensional}. \nStrictly speaking, the attractor on \\reffig{returnmap} has a certain \nthickness transverse to it, but the contraction in the transverse\ndirections is so strong that the invariant set is effectively\n$1$-dimensional. \n\nSuppose we already have determined some of the shorter cycles for our\nsystem, i.e. \nthe fixed points of the Poincar\\'e map and its iterates. This is \naccomplished relatively easily \nby checking a trajectory of a random initial point for close returns\nand then using these as initial guesses for a cycle search algorithm.\nWe now assume that the invariant set can be approximated by a curve\npassing close to all periodic points,\nand determine the order of periodic points along such curve.\nThis is done in the following way: there exists a fixed point which\nis not connected to the attractor (the point \n$\\overline{0}$ in \\reffig{returnmap}) - we\nchoose this fixed point \nas the starting point and assign it number $1$. \nPoint number $2$ is the periodic point in the sample\nwhich is closest (in the full space) \nto this fixed point, and the $n$-th point is determined as the point \nwhich has the minimum distance from the point number $n-1$ among \nall the periodic points which have not yet been enumerated. \nProceeding this way, we order all the periodic points that we have found\nso far.\n\nSince all periodic points belong to cycles, \ntheir images are known and are simply the successive periodic points\nalong the cycle. We use this fact to recursively construct\na $1$-dimensional mapping $s_i \\rightarrow f(s_i)$. We approximate\nparametrization length $s$ along the invariant set \nby computing the Euclidean inter-distances between the\nsuccessive periodic points in the full dynamical\nspace,\n$s_1=0, s_2=\\| {\\bf a}_2-{\\bf a}_1 \\|, s_i-s_{i-1}=\\| \n{\\bf a}_i-{\\bf a}_{i-1} \\| $. \nThe $i$-th cycle point $s_i$ is mapped onto its image\n$s_{\\sigma{i}} = f(s_i)$, \nwhere $\\sigma i$ denotes the label of the next periodic point \nin the cycle.\nWe can now find longer \nperiodic orbits of the 1-dimensional map $f$  by standard methods  such\nas inverse iteration, and guess the location of \nthe corresponding points in the full $N$-dimensional\nspace by interpolating between the nearest known periodic points.\nThese will not be exact periodic orbits of the full system,\nbut are  very useful as good starting guesses in a search for the exact\nperiodic orbits. Iteratively, more and more periodic orbits can be computed. \nWhile it only pays to refine the 1-dimensional parametrization \nuntil the density\nof the periodic points become so high that the width of the attractor\nbecomes noticeable, the 1-dimensional map continues to provide good\ninitial guesses to longer periodic orbits.\nMore sophisticated methods are needed only if \nhigh accuracy around the folding region of $f(s)$ is required\nin order to distinguish between long cycles.\n\n\\begin{figure}\n\\centerline{\\epsfig{file=unfolded.ps,width=8cm}}\n\\caption[]{The return map $s_{n+1} = f(s_n)$ constructed\nfrom the images of periodic points. The diamonds were obtained by \nusing $34$ periodic points, the dots were obtained by using $240$ periodic \npoints. \nWe have indicated the periodic points $\\overline{0}$, \n$\\overline{1}$ and $\\overline{01}$. \nNote that the transverse fractal structure of the map shows when \nthe number of points is increased. \n$N=16$ Fourier modes truncation with $\\nu=0.029910$. }\n\\label{unfolded}\n\\end{figure}\nFor the values of $\\nu$ we are working with, the attractor consists \nof four disjoint sets, the fundamental domain  ${\\cal A}$ and \nits images under the maps $S$ and $\\Theta$. \nIn this case the approximate return map \n$s \\rightarrow f(s)$ is unimodal.\nThe corresponding map on the symmetric \npart of the attractor, $S \\Theta {\\cal A} $,  is likewise unimodal,\nand turned $180$ degrees around the origin. For the values of $\\nu$ we\nwork with the\ntwo maps do not interact and their domains are separate. \nHowever, if the value of the damping parameter $\\nu$ is decreased\nsufficiently, the domains of the\nmaps join and together they form a connected invariant set\ndescribed by a bimodal map\\cite{Glbtsky94}.\nThis joining of the fundamental domain ${\\cal A}$ and \nits symmetric image $\\Theta S {\\cal A} $ is visible \nin \\reffig{feig16} at\n$\\nu \\simeq 0.0299$, where the range of the $a_6$ coordinate \nincreases discontinuously.\n\n\\begin{figure}\n\\centerline{\\epsfig{file=timeceiling.ps,width=8cm}}\n\\caption[]{The return time $T(s)$ as a function of the parameter $s$,\nevaluated on the periodic points, as in \\reffig{unfolded}, with\nthe diamonds obtained by \n$34$ periodic points and the dots by $240$ periodic points. \nThe fine structure is due to the fractal structure of the attractor.\n}\n\\label{rettime}\n\\end{figure}\n\nWe use the unimodal map $s \\rightarrow f(s)$ to construct binary symbolic \ndynamics for the system in the usual way: assign the symbol '0' to points\nto the left of the maximum, '1' to the points to the right. \nIn the period-3 window with the stable cycle $\\overline{001}$, \nthe pruning rules are very easy: except for the stable $\\overline{001}$ cycle\nand the $\\overline{0}$ fixed point (both disjoint from the invariant set) \ntwo 0's in a row are forbidden. \nIn this case it is convenient to redefine the alphabet by\ndenoting the symbol pair $01$ by $a$ and the symbol $1$ by $b$.\nThis renders the symbolic dynamics of the points on the repeller\ncomplete binary: all sequences of\nthe letters $a$ and $b$ are admissible.\n\nA flow in $N$ dimensions \ncan be reduced to an $(N-1)$-dimensional \nreturn map by suspension on a Poincar\\'e section\nprovided that the Poincar\\'e return map is \nsupplemented by a ``time ceiling function''\\cite{bowen} which accounts\nfor a variation in the section return times. \nHence we also determine the return time $T(s_i)$ \nfor each periodic point $i$, and use those to construct recursively\nthe periodic orbit approximations to the time ceiling function,\n\\reffig{rettime}.\nThe mean Poincar\\'e section return time \nis of order $\\overline{T} \\approx .88$. \n\n\n\\subsection{Numerical results}\n\nWe have found all cycles \nup to topological length 10 (the redefined topological length in the case of\nthe period-3 window), 92 cycles in the chaotic regime and 228 \nin the  period-3 window,\nby using the 1-dimensional parametrization $f(s)$ to find initial guesses for\nperiodic points of the full $N=16$ Fourier modes truncation \nand then determining the cycles by\na multi-shooting Newton routine. It is worth\nnoting that the effectiveness of using the \n$1$-dimensional $f(s)$ approximation to the dynamics\nto determine initial guesses is\nsuch that for a typical cycle it takes only 2-3 Newton iterations to find\nthe cycle with an accuracy of $10^{-10}$.\n\n\\begin{table}\n\\caption[]{All cycles up to topological length\n5 for the $N=16$ Fourier modes truncation of the Kuramoto-Sivashinsky equation \n(\\ref{expan}),\ndamping parameter \n$ \\nu =0.029910$ (chaotic attractor) and $\\nu=0.029924$ (period-3\nwindow), their itineraries, periods, \nthe first four stability eigenvalues. For the chaotic attractor\npruning shows up at the \ntopological length $4$; $\\overline{0001}$ \nand $\\overline{0011}$ cycles are pruned. \nThe deviation from unity of $\\Lambda_2$, the eigenvalue along the flow, \nis an indication of the accuracy of the numerical integration.  \nFor the period-3 window we also give the itineraries in the redefined alphabet\nwhere $a=1$ and $b=10$.}\n\\begin{indented}\n{\\small \n\\lineup\n\\item[]\\begin{tabular}{@{}lllllll}\n\\br\n$p$ & & $T_p$ & $\\Lambda_1$ & $\\Lambda_2-1$ & $\\Lambda_3$ & $\\Lambda_4$ \\\\ \\mr\n\\multicolumn{7}{l}{Chaotic, $\\nu=0.029910$} \\\\ \\mr\n0 & & 0.897653 & \\03.298183 & 5$\\cdot10^{-12}$ \n& \\-2.793085$\\cdot 10^{-3}$ & \\-2.793085$\\cdot 10^{-3}$ \\\\\n1 & & 0.870729 & \\0\\-2.014326 & \\-5$\\cdot10^{-12}$ \n & 6.579608$\\cdot 10^{-3}$ & \\-3.653655$\\cdot 10^{-4}$ \\\\\n10 & & 1.751810 & \\0\\-3.801854 & 8$\\cdot10^{-12}$ &\n \\-3.892045$\\cdot 10^{-5}$ & 2.576621$\\cdot 10^{-7}$ \\\\\n100 & & 2.639954 &\\0\\-4.852486 & 1$\\cdot10^{-11}$\n & 3.044730$\\cdot 10^{-7}$ &\\-3.297996$\\cdot 10^{-10}$ \\\\\n110 & & 2.632544 & \\06.062332 & 2$\\cdot10^{-11}$ &\n \\-2.721273$\\cdot 10^{-7}$ & \\-1.961928$\\cdot 10^{-10}$ \\\\ \n1000 & & - & - & - & - & - \\\\\n1100 & & - & - & - & - & - \\\\\n1110 & &  3.497622 & \\-14.76756 & 2$\\cdot10^{-11}$ &\n \\-1.629532$\\cdot 10^{-9}$ & 6.041192$\\cdot 10^{-14}$ \\\\\n10100 & & 4.393973 & 19.64397 & 2$\\cdot10^{-11}$ & \n\\-1.083266$\\cdot 10^{-11}$ & 3.796396$\\cdot 10^{-15}$ \\\\\n11100 & & 4.391976 & \\-18.93979 & 2$\\cdot10^{-11}$ & \n 1.162713$\\cdot 10^{-11}$ &\\-1.247149$\\cdot 10^{-14}$ \\\\\n11010 & & 4.380100 & \\-26.11626 & 2$\\cdot10^{-11}$ & \n 1.005397$\\cdot 10^{-11}$ & 8.161650$\\cdot 10^{-15}$ \\\\\n11110 & & 4.370895 & 28.53133 & 2$\\cdot10^{-11}$ & \n1.706568$\\cdot 10^{-11}$ & 1.706568$\\cdot 10^{-14}$ \\\\ \\mr\n\\multicolumn{7}{l}{Period-3 window, $\\nu=0.029924$} \\\\ \\mr\n     0 & & 0.897809 &  \\03.185997 & 7$\\cdot10^{-13}$ & \n\\-2.772435$\\cdot10^{-3}$ & \\-2.772435$\\cdot10^{-3}$ \\\\ \n     1 & $a$ & 0.871737 & \\0\\-1.914257 & 5$\\cdot10^{-13}$ &\n 6.913449$\\cdot10^{-3}$ & \\-3.676167$\\cdot10^{-4}$ \\\\ \n    10 & $b$ & 1.752821 & \\0\\-3.250080 & 1$\\cdot10^{-12}$ &\n\\-4.563478$\\cdot10^{-5}$ &  2.468647$\\cdot10^{-7}$ \\\\ \n   100 & & 2.638794 & \\0\\-0.315134 & \\-4$\\cdot10^{-13}$ &\n 4.821809$\\cdot10^{-6}$ & \\-2.576341$\\cdot10^{-10}$ \\\\ \n   110 & $ab$ & 2.636903 &  \\02.263744 & 3$\\cdot10^{-12}$ &\n\\-6.923648$\\cdot10^{-7}$ & \\-2.251226$\\cdot10^{-10}$ \\\\ \n  1110 & $aab$ & 3.500743 & \\-10.87103 & 2$\\cdot10^{-12}$ &\n\\-2.198314$\\cdot10^{-9}$ &  3.302367$\\cdot10^{-14}$ \\\\ \n 11010 & $abb$ & 4.382927 & \\-15.84102 & 2$\\cdot10^{-12}$ &\n 1.656690$\\cdot10^{-11}$ &  1.388232$\\cdot10^{-14}$ \\\\ \n 11110 & $aaab$ & 4.375712 &  18.52766 & 3$\\cdot10^{-12}$ &\n\\-1.604898$\\cdot10^{-11}$ &  2.831886$\\cdot10^{-14}$ \\\\ \\br\n\\end{tabular} \n}\n\\end{indented}\n\\label{t_orbits}\n\\end{table}\n\n\\begin{figure}\n\\centerline{\\epsfig{file=eigenvalues.ps,width=8cm}}\n\\caption[]{Lyapunov exponents $\\lambda_k$ versus $k$ for the \nperiodic orbit $\\overline{1}$ compared with  the stability eigenvalues \nof the $u(x,t)=0$ stationary solution $k^2- \\nu k^4$.  \n$\\lambda_k$ for $k \\geq 8$ fall below the numerical accuracy of integration \nand are not meaningful. \n$N=16$ Fourier modes, $\\nu=0.029924$, chaotic regime.\n}\n\\label{eigenvalues}\n\\end{figure}\n\nIn \\reftab{t_orbits} we list the periodic orbits \nto topological length 5 found by our method.\nThe value of $\\Lambda_2$ serves as an indication of the \naccuracy of our numerics, as $\\Lambda_2$ corresponds to the marginal\neigenvalue along the periodic orbit, strictly equal to $1$.\nAll cycles seem to have \nreal eigenvalues (to within the numerical accuracy) except for the \n$\\overline{0}$-cycle \nwhich has a pair of complex eigenvalues, $\\Lambda_3$ and $\\Lambda_4$.\nWe therefore do not list the corresponding imaginary parts of the eigenvalues. \nTo illustrate the rapid contraction in the nonleading eigendirections\nwe plot all the eigenvalues of the \n$\\overline{1}$-cycle in \\reffig{eigenvalues}. \nAs the length of the orbit increases, the magnitude of\ncontracting eigenvalues falls quickly \nbellow the attainable numerical numerical \naccuracy $\\approx 10^{-16}$ and our numerical results for \n$\\Lambda_k$ are not meaningful for $ k \\geq 8$.\n\nHaving determined the periodic solutions $p$ in the Fourier modes space, \nwe now go back to the configuration space and plot the corresponding\nspatiotemporally periodic solutions $u_p(x,t)$: they are the\nrepertoire of the recurrent spatiotemporal patterns that Hopf wanted to\nsee in turbulent dynamics.\nDifferent spatiotemporally periodic solutions are qualitatively\nvery much alike but still different, as a closer inspection reveals. \nIn \\reffig{orbit0fig} we plot \n$u_0(x,t)$ corresponding to the Fourier space $\\overline{0}$-cycle. \nOther solutions, plotted in the configuration space, exhibit the same\noverall gross structure. For this reason we find it more informative\nto plot the difference $u_0(x,t'T_0)-u_p(x,t''T_p\/n_p)$\nrather than $u_p(x,t)$ itself. \nHere $p$  labels a given prime (non-repeating) cycle, \n$n_p$ is the topological cycle length, $T_p$ its period,\nand the time is rescaled to make this difference periodic in \ntime: $t'=t \/T_0$ and $t''=n_p t\/T_p$, so that $t''$ ranges from $0$ to $n_p$. \n$u_0(x,t'T_0)-u_1(x,t''T_1)$ is given in \\reffig{diff1fig}, and\n$u_0(x,t'T_0)-u_{01}(x,t''T_{01}\/2)$ in \\reffig{diff01fig}.\n\n\\begin{figure}\n\\centerline{\\epsfig{file=orbit0.ps,width=8cm}}\n\\caption[]{Spatiotemporally periodic solution $u_0(x,t)$.\nWe have divided $x$ by $\\pi$ and plotted only the  $x>0$ part, since we work in\nthe subspace of the odd solutions,  $u(x,t)=-u(-x,t)$. \n$N=16$ Fourier modes truncation with $\\nu=0.029910$.\n\t  }\n\\label{orbit0fig}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\epsfig{file=diff1.ps,width=8cm}}\n\\caption[]{The difference between the two shortest period\n\t   spatiotemporally periodic solutions \n\t   $u_0(x,t'T_0)$ and $u_1(x,t''T_1)$.\n\t  }\n\\label{diff1fig}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\epsfig{file=diff01.ps,width=8cm}}\n\\caption[]{The difference between solution  $u_0(x,t'T_0)$ repeated\n\t   twice and the $n_p=2$ period spatiotemporally periodic solution\n           $u_{01}(x,t''T_{01}\/2)$.\n          }\n\\label{diff01fig}\n\\end{figure}\n\n\\section{Global averaging: periodic orbits in action} \n\nThe above investigation of the Kuramoto-Sivashinsky\nsystem demonstrates that it is possible to\nconstruct recursively and exhaustively\na hierarchy of spatiotemporally periodic unstable solutions\nof a spatially extended nonlinear system.\n\nNow we turn to the central issue of this paper; qualitatively, these\nsolutions are indeed an implementation of Hopf's program, but how\nis this information to be used quantitatively? This is precisely\nwhat the periodic orbit theory is about;\nit offers machinery that puts together\nthe topological and the quantitative information about individual\nsolutions, such as their periods and stabilities, into predictions \nabout measurable global averages, such as the Lyapunov exponents,\ncorrelation functions, and so on. The proper tool for\ncomputing such global characterizations of the dynamics are the trace\nand determinant formulas of the periodic orbit theory.\n\nWe shall briefly summarize the aspects of the\nperiodic orbit theory relevant to the present application;\nfor a complete exposition of the theory \nthe reader is referred to \\refref{cycl_book}. The key idea is to\nreplace a time average $\\Phi^t (x)\/t$\nof an ``observable\"  $\\phi$ measured along\na dynamical trajectory $x(t) = f^t(x)$\n\\[\n\\Phi^t (x) = \\int_0^t \\d\\tau \\, \\phi(x(\\tau))\n\\]\nby the spatial average $\\left<\\e^{\\beta \\Phi^t}\\right>$\nof the quantity $\\e^{\\beta \\Phi^t (x)}$. Here $\\beta$ is\na dummy variable, used to recover the desired expectation value\n$\\left<\\phi\\right> = \\lim_{t\\to\\infty} \\left<\\Phi^t\/t \\right>$ \nby taking $\\frac{\\d}{\\d\\beta}$ derivatives\nof $\\left<\\e^{\\beta \\Phi^t}\\right>$ and then setting $\\beta=0$.\nFor large $t$ the average $\\left<\\e^{\\beta \\Phi^t}\\right>$ behaves as\nthe trace \n\\begin{equation}\n{\\rm tr}\\,  {\\cal L}^{t}\n     =   \\sum_{p} \\period{p} \\sum_{r=1}^\\infty\n                { \\e^{r \\beta \\Phi_p} \\over \\oneMinJ{r} }\n                \\prpgtr{t-r \\period{p}}\n\\ee{tr-L1}\nof the evolution operator\n\\[\n{\\cal L}^t (x,y) = \\delta(y-f^{t}(x))\\e^{\\beta \\Phi^t(x)} .\n\\]\nand is dominated by its largest eigenvalue $\\e^{ts(\\beta)}$.\n\nThe trace formula \\refeq{tr-L1} has an intuitive geometrical interpretation.\nThe sums in \\refeq{tr-L1}\nare over prime periodic orbits $p$ \nand their repeats $r$, $T_p$ are their periods,\nand ${\\bf J_p}$ are their stability matrices. \nPrime cycles partition the dynamical space into closed tubes of\nlength $\\period{p}$ and thickness\n$\\left|\\det({\\bf 1}-{\\bf J}_p)\\right|^{-1}$, \nand the trace picks up a periodic orbit contribution only when the\ntime $t$ equals a prime period or its repeat, hence the time\ndelta function $\\prpgtr{t- r\\period{p}}$. Finally,\n$\\e^{\\beta \\Phi_p }$\nis the mean value of\n$ \\e^{\\beta \\Phi^t(x)}$\nevaluated on this part of dynamical space, so the trace formula is \nthe average of $\\left< \\e^{\\beta \\Phi^t}\\right>$ expressed as a partition\nof the space of solutions into a repertoire of spatiotemporally\nperiodic solutions, each weighted by its stability, i.e. likelihood of its\noccurrence in a long time evolution of the system.\n\nIn applications of the periodic orbit theory \nthe related Fredholm determinant\n\\begin{equation}\nF(\\beta,s)=\n\\exp \\left ( \n{ \\displaystyle - \\sum_p \\sum_{r=1}^{\\infty} z ^{n_p r} \n\\frac{\\displaystyle \\e^{r( \\beta \\Phi_p -s T_p) } }\n{ \\displaystyle \nr \\left | \\det \\left ( {\\bf 1}- {\\bf J}_p^r \\right )  \\right | } } \\right )\n\\label{fredholm} \n\\end{equation}\nhas better convergence as a function of\nthe maximal cycle length truncation, so that is the function whose\nleading zero $F(\\beta,s(\\beta))=0$ we determine here in order\nto evaluate the leading eigenvalue $s(\\beta)$. \n\nThe dummy variable $z$ in \\refeq{fredholm} keeps track of \nthe topological lengths $n_p$ (number of the Poincar\\'e section crossings),\nand is used to expand $F$ as a series in\n$z$. If we know all cycles up to topological length $l$ we truncate $F$\nto $l$-th order polynomial:\n\\begin{equation}\nF(\\beta,s)=1-\\sum_{1}^{l} c_k z^k + (\\mbox{remainder})\n\\label{Fred_exp}\n\\end{equation}\nand set $z=1$. The general theory\\cite{Ruelle,Rugh92,CRR93} then \nguarantees that\nfor a hyperbolic dynamical system the coefficients $c_k$ fall\noff in magnitude exponentially or faster with increasing $k$.\nWe now calculate the leading eigenvalue $s(\\beta)$ \nby determining the smallest zero of $F(\\beta,s)$,\nand check the convergence of this estimate by studying it as a \nfunction of the maximal cycle length truncation $l$. \nIf the flow conserves all trajectories, the leading eigenvalue\nmust satisfy $s(0)=0$; if the invariant set is repelling, the\nleading eigenvalue yields $\\gamma= -s(0)$, the escape rate from the repeller.\nOnce the leading eigenvalue is determined we can\ncalculate the desired average $\\left<\\phi \\right>$ using formula\\cite{AACI}:\n\\begin{equation}\n\\left<\\phi \\right> =\n\\left. -\\frac{\\partial s}{\\partial \\beta}\\right|_{\\beta=0} =\n\\left. -\\frac{\\partial F}{\\partial \\beta} \n  \\left\/\n \\frac{ \\partial F}{\\partial s  }\n  \\right.  \\right|_{\\beta=0 \\atop s=s(0)}.\n\\label{cyc_aver}\n\\end{equation}\nFor example, if we take as our ``observable'' $\\log |\\Lambda_{1}^t(x)|$,\nthe largest eigenvalue of the linearized stability of the flow,\n$\\Phi_p$ will be\n$\\log |\\Lambda_{1,p}|$ where $\\Lambda_{1,p}$ is the largest eigenvalue\nof stability matrix of the cycle $p$, and the above \nformula yields the Lyapunov exponent $\\left<\\lambda\\right>$.\n \nBoth the numerator and the denominator in \\refeq{cyc_aver} have a cycle\nexpansion analogous to \\refeq{Fred_exp} (cf. \\refref{cycl_book}), \nand the same periodic orbit data suffices for their evaluation.\n\nConceptually the most\nimportant lesson of the periodic orbit theory \nis that the spatiotemporally periodic\nsolutions are {\\em not} to be thought of as eigenmodes to be used as a linear\nbasis for expressing solutions of the equations of motion - as the equations\nare nonlinear, the periodic solutions are in no sense additive.\nNevertheless, the trace formulas and determinants of the periodic\norbit theory give a precise prescription for how \nto systematically explore the repertoire of admissible spatiotemporal patterns,\nand how to put them together in order to predict measurable observables. \n\n\\subsection{Numerical results}\n\nOne of the objectives of a theory of turbulence is\nto predict measurable global averages over turbulent flows, such as\nvelocity-velocity correlations and transport coefficients. While in\nprinciple the periodic orbit averaging formulas should be applicable\nto such averages, with the present parameter values\nwe are far from any strongly turbulent regime, and here we shall\nrestrict ourselves to the simplest tests of chaotic dynamics: we shall\ntest the theory by evaluating Lyapunov exponents\nand escape rates.\n\n\\begin{figure}\n\\centerline{\\epsfig{file=coeff.ps,width=8cm}}\n\\caption[]{$\\log_{10}$ of the coefficients  $|c_k|$ in the cycle expansion \n\\refeq{Fred_exp} of $F(0,0)$ versus $k$ \nfor the period-$3$ window case (crosses) and the chaotic case\n(diamonds). $N=16$ Fourier modes truncation. }\n\\label{coeffval}\n\\end{figure}\n\nWe compute the periodic orbits, escape rates and  Lyapunov exponents \nboth for the period-$3$ window and a chaotic regime.\nIn the case of period-$3$ window the complete symbolics dynamics\nand grammar rules\nare known and  good convergence of cycle expansions is expected \nboth for the escape rate from the repeller\nand the Lyapunov exponent. Parenthetically, the stable\nperiod-3 orbit is separated from the rest of the invariant set\nby its immediate basin of attraction window, and its eigenvalues \nbear no immediate relation to the escape rate and the Lyapunov\nexponent of the repelling set.\n\nIn the case of a generic ``strange attractor'',\nthe convergence is not expected to be nearly as good, \nsince in this case there exist no finite description of the symbolic dynamics. \nFor closed systems (no escape) $\\gamma=0$ and $F(0,0)=0$. \nThe discrepancy of the value $F(0,0)$ from $0$ for a closed system \nallows us to estimate the accuracy of \nfinite cycle length approximations to the \nFredholm determinant. \n\nThe analytic properties of the Fredholm determinant are illustrated by \nthe decay rate of the coefficients $c_k$ as a function of $k$ in the expansion \n\\refeq{Fred_exp}.\nIf the complete symbolic dynamics is known and the system is hyperbolic, the \ndecay of $c_k$ should be superexponential\\cite{Rugh92}. \nThis is illustrated in \\reffig{coeffval}, where\nwe plot the coefficients $c_k$ for the $16$-dimensional \nsystem for the chaotic case and for the period-$3$ window. \nWe can clearly see the \nsuperexponential decay for the period-$3$ \nwindow case and at best exponential decay \nfor the chaotic case. \n\nOur results are presented in \\reftab{t_16_chaotic}. One \nobserves that when the symbolic dynamics is known \n(period-$3$ window), the convergence is much better than\nin the generic case, in accordance with the periodic orbit theory\nexpectations. \n\n\\begin{table}\n\\caption[]{\nThe escape rate $\\gamma$ and the leading Lyapunov exponent\nas a function of the cycle expansion truncation $n_{max}$ \nfor the $N=16$ Fourier modes truncation, chaotic regime \n$(\\nu=0.029910$) and period-3 window $(\\nu=0.029924)$. In the \nperiod-3 window the Fredholm determinant starts converging only\nfor $n_{max}>4$; for $n_{max}=4$ it has no real zero at all.\nA numerical simulation \nestimate for the Lyapunov exponent in the chaotic regime is given in the \nlast line; for this parameter value the escape rate, $\\gamma$,\nshould strictly equal zero.}\n\\begin{indented}\n{\\small \n\\lineup\n\\item[]\\begin{tabular}{lllll} \\br\n & \\multicolumn{2}{l}{chaotic} & \n    \\multicolumn{2}{l}{period-3 window} \\\\ \\mr\n$n_{max}$ & $\\gamma$ & $\\lambda_1$ & $\\gamma$ & $\\lambda_1$ \\\\ \\mr\n 1 & & & 0.428143 & 0.703010 \\\\ \\hline\n 2 & 0.441948 & 0.981267 &  \\-0.187882 & 0.430485 \\\\ \\hline\n 3 & 0.080117 & 0.765050 &  \\-0.049325 & 0.469350 \\\\ \\hline\n 4 & 0.148583 & 0.703072 & & \\\\ \\hline\n 5 & 0.068513 & 0.727498 &  1.072468 & 0.585506 \\\\ \\hline\n 6 & 0.027724 & 0.699907 &  0.078008 & 0.547005 \\\\ \\hline\n 7 & 0.035137 & 0.693852 &  0.088132 & 0.598977 \\\\ \\hline\n 8 & 0.007104 & 0.675529 &  0.090425 & 0.631551 \\\\ \\hline\n 9 & 0.021066 & 0.673144 &  0.090101 & 0.618160 \\\\ \\hline\n10 & 0.007367 & 0.646233 &  0.090065 & 0.621271 \\\\ \\hline\nnumer. &  & 0.629 & & \\\\  \\br\n\\end{tabular}\n}\n\\end{indented}\n\\label{table16_chaotic}\n\\label{t_16_chaotic}\n\\end{table}\n\n\\section{Summary}\n\nHopf's proposal for a theory of turbulence was, as we understand it, to think \nof turbulence as a sequence of near recurrences of a repertoire of unstable \nspatiotemporal patterns. Hopf's \nproposal is in its spirit very different from most ideas that animate\ncurrent turbulence research. \nIt is distinct from\nthe Landau quasiperiodic picture of turbulence as a sum of \ninfinite number of incommensurate frequencies, with dynamics taking place on a \nlarge-dimensional torus.\nIt is not the\nKolmogorov's 1941 homogeneous turbulence with no \ncoherent structures fixing the length scale, here all the action is \nin specific coherent structures.\nIt is emphatically {\\em not} universal; spatiotemporally periodic solutions \nare specific to the particular set of equations and boundary conditions.\nAnd it is {\\em not} probabilistic; everything is fixed by the deterministic\ndynamics with no probabilistic assumptions on the velocity distributions \nor external stochastic forcing. \n\nOur investigation of the Kuramoto-Sivashinsky system is a\nstep in the direction of implementing Hopf's program. \nWe have constructed \na complete and exhaustive hierarchy of spatiotemporally periodic solutions \nof spatially extended nonlinear system and applied the periodic orbit theory \nto evaluation of global averages for such system. Conceptually the most \nimportant lesson of this theory is that \nthe unstable spatiotemporally periodic\nsolutions serve to explore systematically the\nrepertoire of admissible spatiotemporal patterns, with the trace\nand determinant formulas and their cycle expansions being the proper tools for\nextraction of quantitative predictions from the periodic orbits data.\n\nWe have applied the\ntheory to a low dimensional attractor, not larger than the Lorenz's original \nstrange attractor\\cite{Lorenz}. As our aim was to solve the given equations \naccurately, we were forced to work with a high dimensional\nFourier modes truncations, and we succeeded in determining the\nperiodic orbits for flows of much higher dimension than in\nprevious applications of the periodic orbit theory. As something new, we \nhave developed an intrinsic parametrization of the invariant set that\nprovided the key to finding the periodic orbits. \n\nIn practice, the method \nof averaging by means of periodic orbits produced best results when \nthe complete symbolic dynamics was known. \nFor generic parameter values we cannot claim that the periodic\norbit approach is\ncomputationally superior to a direct numerical simulation. \nA program to find periodic orbits up to length 10 for one value of \nthe damping parameter $\\nu$ requires a day of CPU on a fast workstation, \nmuch longer than the time used in the direct numerical\nsimulations. \n\nThe parameter $\\nu$ values that we work with correspond to\nthe weakest nontrivial ``turbulence'', and it is an open \nquestion to what extent the approach remains implementable as the system goes \nmore turbulent. Our hope is that the unstable structures  captured so far \ncan be adiabatically tracked to the ``intermediate turbulence'' regime, \nand still remain sufficiently representative of the space of admissible \npatterns to allow meaningful estimates of global averages.  \nAs long as no effective coordinatization of the ``inertial\nmanifold'' exists and we rely on the spatial Fourier decomposition, the\npresent approach is \nbound to fail in the ``strong turbulence'' $\\nu \\rightarrow 0$ limit,\nwhere the dominant structures are Burgers-type shocks \nand truncations of the spatial Fourier modes \nexpansions are increasingly uncontrollable. \n\n\\ack\nWe are grateful to L. Tuckerman for patient instruction,\nE.A. Spiegel,\nG. Goren, \nR. Zeitek,\nand \nI. Procaccia\nfor inspiring conversations, P. Dahlqvist for a critical\nreading of an early version of the paper, and E. Bogomolny for the\ncatchy but all too ephemeral title for the paper.\n\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe connection between deep neural network and the ordinary differential equation has been studied and discussed in different recent works [1,2,3,4,5,6]. It has been shown that residual networks such as ResNet [7] and recurrent neural network decoders can be modeled as a discretization of a continuous ODE model. An ODE-based model and its relation to the residual network can be shown as follows:\n\\begin{equation} \\label{Eq1}\n\\begin{aligned}\n \\ \\ \\text L_{t_1} =\\text L_{t_0} + f(\\text L_{t_0},\\theta) \\ \\ \\ \\  \\ \\ \\ \\ \\ \\ \\ \\  \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (ResNet)\n\\end{aligned}\n\\end{equation}\n\\begin{equation} \\label{Eq2}\n\\begin{aligned}\n\\text L_{t_1} =\\text L_{t_0} + \\int_{t_0}^{t_1} f(\\text L_{t},\\theta)\\ dt \\ \\ \\ \\  \\ \\ \\ \\ \\ \\ \\ \\ (ODE)\n\\end{aligned}\n\\end{equation}\nwhere $\\text L_{t_0}, \\text L_{t_1}$ are the residual block input and output. $f$ represents the network-defined nonlinear operator which preserves the dimensionality of $\\text L_{t_0}$ and $\\theta$ represents the network weights. The defined ODE ($\\frac{d\\text L}{dt} = f(\\text L_{t},\\theta)$) is described in terms of its solution in $t=t_1$. \nThe forward step of ODE Euler discretization is as follows:\n\\begin{equation} \\label{Eq3}\n\\begin{aligned}\n \\ \\ \\text L_{t_{0}+h} =\\text L_{t_0} + hf(\\text L_{t_0},\\theta) \\ \\ \\ \\  \\ \\ \\ \\ \\ \\ \\ \\  \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\end{aligned}\n\\end{equation}\nIt can be observed that the single forward step of Eq~\\ref{Eq3}, can be considered as the equivalent to the formulation of the residual block. Therefore, The ODE discretization model can lead to different ODE-inspired network architectures. In this paper, we present an ODE-based deep network for MRI reconstruction which extends the conventional reconstruction framework to its data-adaptive variant using the ODE-based network and provides an end-to-end reconstruction scheme. We evaluate the reconstruction performance of our method to the reconstruction methods based on the standard UNet network [8] and Residual network. \n\\section{Method}\nThe discretized version of MR imaging model given by\t\t\n\\begin{equation} \\label{Eq4}\n\\begin{aligned}\n\\text d=\\text E \\text x + \\text n.\n\\end{aligned}\n\\end{equation}\nwhere $\\text x$ is the samples of unknown MR image, and $\\text d$ is the undersampled k-space data. $\\text E =\\text {FS}$ is an encoding matrix, and $\\text F$ is an undersampled Fourier operator. $\\text S$ is a matrix representing the sensitivity map of the coils, and $\\text n$ is noise. Assuming that the interchannel noise covariance has been whitened, the reconstruction relies on the least-square approach:\n\\begin{equation} \\label{Eq5}\n\\begin{aligned}\n\\hat{\\text x} =\\underset{\\text x}{ argmin} \\ \\|\\text d-\\text E\\text x\\|_{2}^{2}\n\\end{aligned}\n\\end{equation}\nThe ODE-based reconstruction framework we used for solving the Eq~\\ref{Eq5} is shown in Fig~\\ref{fig1}. \\\\\nFor a conventional neural network, we minimize the loss function ($l$) over a set of training pairs and we search for the weights ($\\theta$) that minimize that loss function:\n\\begin{equation} \\label{Eq6}\n\\begin{aligned}\n\\underset{\\theta}{ minimize} \\ \\frac{1}{M} \\sum\\limits_{i=1}^M l(L(\\theta;x_i,y_i)) + \\text R(\\theta)\n\\end{aligned}\n\\end{equation}\nwhere $(x_i,y_i)$ is the $i_{th}$ training pairs (input and ground truth). $R$ is a regularization operator and $M$ is the number of training pairs. The loss function depends implicitly on $\\theta$. This optimization is usually solved through Stochastic Gradient Descent (SGD) and backpropagation to compute the gradient of $L$ with respect to $\\theta$. In our ODE-based network, besides the $L$s, the network weights also change with respect to time as well. In this case, we need to solve the following constrained optimization problem :\n\\begin{equation} \\label{Eq7}\n\\begin{aligned}\n\\underset{p,\\theta}{ minimize} \\ \\frac{1}{M} \\sum\\limits_{i=1}^M l(L_{t_1};x_i,y_i) + \\text R(p,\\theta)\n\\end{aligned}\n\\end{equation}\n\\begin{equation} \\label{Eq8}\n\\begin{aligned}\n\\frac{d\\text L}{dt} = f(\\text L_{t},\\theta_{t})\n\\end{aligned}\n\\end{equation}\n\\begin{equation} \\label{Eq9}\n\\begin{aligned}\n\\frac{d\\theta}{dt} = w(\\theta_{t},p)\n\\end{aligned}\n\\end{equation}\nwhere $\\theta_t$ is parameterized by the learnable dynamics of\n\\begin{equation} \\label{Eq10}\n\\begin{aligned}\n\\theta_{t_1} =\\theta_{t_0} + \\int_{t_0}^{t_1} w(\\theta_{t},p)\\ dt\n\\end{aligned}\n\\end{equation}\nwhere $w$ is a nonlinear operator responsible for the network weights dynamics and $p$ is the parameters for $w$.\nWe also augment the learnable space and solve the ODE flow as follows so that the learned ODE representation won't preserve the input space topology [5].\n\\begin{equation} \\label{Eq11}\n\\begin{aligned}\n\\frac{d}{dt} \\begin{bmatrix}\n    \\text L\\\\\n    a\n\\end{bmatrix} = f(\\begin{bmatrix}\n    \\text L_{t}\\\\\n    a_{t}\n\\end{bmatrix},\\theta_{t})\n\\end{aligned}\n\\end{equation}\nwhere $a_{0}=0$. We use the discretize-then-optimize method [4,6] to calculate the gradients for backpropagating through ODE layers. Figure \\ref{fig2} shows the proposed ODE-based deep network. Five residual blocks have been used in our method (N=5).\n\\section{Results and Discussion}\nIn our experiments, we have tested our method with our MPRAGE brain datasets. The data on ten volunteers with a total of 750 brain images used as the training set. Images from fifteen different volunteers have used as the testing set. The sensitivity maps were computed from a block of size 24x24 using the ESPIRiT [9] method. Reconstruction results with the undersampling factor of 2x2 for different approaches are shown in Fig \\ref{fig3}. ResidualNet includes the same number of residual blocks as our proposed method (without ODE layers).\nTable \\ref{table1} shows\nthat our method consistently has higher Peak Signal-to-noise Ratios (PSNR) and structural similarities (SSIM) compared to the reconstructions using the other two networks. In conclusion, an ODE-based deep network for MRI reconstruction is proposed. It enables the rapid acquisition of MR images with improved image quality. The proposed ODE-based network can be easily adopted by unrolled optimization schemes for better MRI reconstruction accuracy.\n\n\n\\begin{figure}\n\\begin{floatrow}\n\\ffigbox{%\n  \\includegraphics[scale=0.37]{fig2.jpg\n  }{%\n  \\caption{The reconstruction framework}%\n  \\label{fig1}}\n\\capbtabbox{%\n\\begin{tabular}{rllll}\n\\hline\n\\multicolumn{3}{c} {Brain Dataset} \\\\\n\\cline{2-3} \n\\cline{4-5} \nMethod & PSNR  & SSIM   \\\\\n\\hline\nProposed & $54.5\\pm1.37$  & $0.99\\pm0.0063$   \\\\\nUNet & $52.4\\pm1.54$  & $0.98\\pm0.0075$   \\\\\nResidualNet & $50.1\\pm1.65$  & $0.978\\pm0.0097$   \\\\\n\\hline\n\\end{tabular}\n}\n{\n\\caption{PSNR and SSIM variations on Brain dataset}%\n\\label{table1}\n}\n\\end{floatrow}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\centerline{\\includegraphics[scale=0.5]{fig1.jpg}}\n  \\caption{The proposed ODE-based deep network.}\n  \\label{fig2}\n\\end{figure}\n\\begin{figure}\n  \\centering\n  \\centerline{\\includegraphics[scale=0.15]{fig3.jpg}}\n  \\caption{First row (left to right): Reference image using fully sampled data, ResidualNet reconstruction, UNet reconstruction, and our reconstruction all with undersampling factor of 2x2. Second row includes error maps correspond to each reconstruction results for comparison.}\n  \\label{fig3}\n\\end{figure}\n\\newpage\n\n\\subsubsection*{Acknowledgments}\n\nThis research was supported in part by NIH grants R01 NS079788, R01 EB019483, R01 DK100404, R44 MH086984, IDDRC U54 HD090255, and by a research grant from the Boston Children's Hospital Translational Research Program.\n\n\n\\section*{References}\n\n[1] Haber, E. and Ruthotto, L., \"Stable architectures for deep neural networks,\" Inverse Problems, 34(1), 014004 (2017).\n\n[2] Ruthotto,  L.  and  Haber,  E.,  \"Deep  neural  networks  motivated  by  partial  differential  equations,\" arXiv preprint arXiv:1804.04272 (2018).\n\n[3] Chen, T. Q., Rubanova, Y., Bettencourt, J., and Duvenaud, D. K., \"Neural ordinary differential equations,\" in [Advances in neural information processing systems], 6571\u20136583 (2018).\n\n[4] Gholami, A., Keutzer, K., and Biros, G., \"Anode:  Unconditionally accurate memory-efficient gradients for neural odes,\" arXiv preprint arXiv:1902.10298 (2019).\n\n[5] Dupont,  E.,  Doucet,  A.,  and  Teh,  Y.  W.,  \"Augmented  neural  odes,\" arXiv  preprint  arXiv:1904.01681 (2019).\n\n[6] Zhang,  T.,  Yao,  Z.,  Gholami,  A.,  Keutzer,  K.,  Gonzalez,  J.,  Biros,  G.,  and Mahoney,  M.,  \"Anodev2:  A coupled neural ode evolution framework,\" arXiv preprint arXiv:1906.04596 (2019).\n\n[7] He, K., Zhang, X., Ren, S., and Sun, J., \"Deep residual learning for image recognition,\" in [Proceedings of the IEEE conference on computer vision and pattern recognition], 770\u2013778 (2016).\n\n[8] Ronneberger,  O.,  Fischer,  P.,  and  Brox,  T., \"U-net:  Convolutional  networks  for  biomedical  image  segmentation,\" in [International Conference on Medical image computing and computer-assisted intervention], 234\u2013241, Springer (2015).\n\n[9] M. Uecker, P. Lai, M. J. Murphy, P. Virtue, M. Elad, J. M.Pauly, S. S. Vasanawala, and M. Lustig, \"Espirit an eigenvalue approach to autocalibrating parallel mri:  where sense meets grappa,\" Magnetic resonance in medicine, 71(3), 990\u20131001 (2014).\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe duration of particle collisions is an interesting and important aspect of\ngeneral scattering theory which is in a sense complementary  to the energy\nrepresentation ordinarily used. A collision is characterized in this case by\nthe time delay of particles in the region of interaction. Wigner \\cite{W-55}\nwas the first who considered the time on which a monochromatic particle with\ngiven\nangular momentum is delayed during its elastic scattering. He established\nthe connection of this time delay to the energy derivative of the scattering\nphase shift. The sharper the energy dependence of the phase shift is the\nlonger is the time delay.\n\nLater on Smith \\cite{S-60} extended the time delay concept on many channel\nproblems introducing the time delay matrix\n\\begin{equation}\\label{q-matr}\n Q^{ab}(E) = -i\\hbar\\left\\{\\frac{d}{d\\,\\varepsilon}\\,\n \\sum_{c}S^{ac}(E\\!+\\!\\frac{\\varepsilon}{2})\n S^{\\ast\\,cb}(E\\!-\\!\\frac{\\varepsilon}{2})\n \\right\\}_{\\varepsilon=0}\\,\\,,\n\\end{equation}\nin the channel space. Here $S$ is the scattering matrix and the summation\nindex $c$ runs over all the $M$ open scattering channels. The matrix $Q$ is\nhermitian; its diagonal element $Q^{cc}$ coincides with the mean duration of\ncollision (time delay) in the $c$-th entrance channel. Generally speaking,\nthe delays are different in different channels $c$. Taking the trace of the\nSmith matrix, one arrives at the simple weighted-mean characteristic\n\\begin{equation}\\label{q1}\nQ(E) = \\frac{1}{M}\\,\\sum_{c} Q^{cc}\n     = -\\frac{i}{M}\\,\\frac{d}{dE}\\,\\ln \\det S(E)\n\\end{equation}\nof the duration of collisions. Eq.  (\\ref{q1}) is just the many-channel\nversion of the well-known simple Wigner formula. (Here and below we set\n$\\hbar=1$.)\n\nThe time delay turns out to be an especially pertinent concept for the chaotic\nresonance scattering encountered in atomic, molecular and nuclear physics\n\\cite{L-77, LW-91}, as well as in the scattering of electromagnetic\nmicrowaves \\cite{Sr-91, GHLLRRSW-92, SS-92} in resonance billiard-like\ncavities. The quantity $Q(E)$, being closely connected to the complex energy\nspectrum of resonance states, shows in its energy dependence strong\nfluctuations around a smooth regular variation. The two kinds of variation on\ndifferent energy scales are naturally decomposed\n\\begin{equation}\\label{atd}\nQ(E)=\\langle Q(E)\\rangle+Q_{fl}(E)\\,\\,,\n\\end{equation}\nwith an energy or ensemble averaging. By this, the slow energy dependence of\ntime delay is revealed whereas the two-point autocorrelation function\n\\begin{equation}\\label{dcf}\nC_Q(E,\\varepsilon) =\n\\langle Q_{fl}(E\\!+\\!\\frac{\\varepsilon}{2})\nQ_{fl}(E\\!-\\!\\frac{\\varepsilon}{2})\\rangle=\n\\langle Q(E\\!+\\!\\frac{\\varepsilon}{2})\nQ(E\\!-\\!\\frac{\\varepsilon}{2})\\rangle -\n\\langle Q(E\\!+\\!\\frac{\\varepsilon}{2})\\rangle\n\\langle Q(E\\!-\\!\\frac{\\varepsilon}{2})\\rangle\n\\end{equation}\nis used to characterize the time delay fluctuations.\n\nTo the best of our knowledge, the first consideration of these fluctuations\nhas been made numerically as well as analytically in \\cite{WJ-89} and\n\\cite{SW-92} in the framework of rather peculiar model of resonance elastic\nquantum scattering on a leaky surface of constant negative curvature. The\nnoteworthy feature of this model is that the poles of the scattering\namplitude turn out to correspond to zeros of the famous Riemann\n$\\zeta$-function. The real parts of the poles are therefore supposed\n\\cite{Mo-73} to be chaotically distributed similar to the eigenvalues of the\nGaussian Unitary Ensemble whereas all their imaginary parts (the widths of\nresonances) are the same.  The latter very specific property partly deprives\nthe model of its interest since actual single-channel widths are known to\nexhibit quite large fluctuations \\cite{Po-65}.\n\nThe width fluctuations are suppressed when many channels are open. In this\ncase semiclassical approximation can be as a rule expected to be valid.  The\nsemiclassical analysis of the time delay in terms of closed periodic orbits\nis given in \\cite{Ec-93}. It is in particular emphasized there that only the\ntail of the correlation function (\\ref{dcf}) corresponding to the very large\nvalues of $\\varepsilon$ can immediately be related to the (short) periodic\norbits. Quite opposite, the central peak near the point $\\varepsilon=0$ is\nformed as a result of a strong interference of many orbits. Therefore, its\nwidth describing the long-time  asymptotic behaviour of the Fourier transform\nhas no direct connection to the classical escape rate and has rather to be\ncalculated on the pure quantum ground. This is in line with the results of\nthe analysis \\cite{GR-89} of distribution of the resonance widths in the\nthree discs scattering problem.\n\nIt is now generally acknowledged that the random matrix theory \\cite{Me-67}\nrepresents a suitable and reliable foundation for description of local\nproperties of dynamical quantum chaos \\cite{BG-84}. We therefore use below a\nrandom matrix model of chaotic scattering to calculate the time delay\nautocorrelation function. We suppose as usual that the number $N$ of\nresonances is asymptotically large and use the powerful supersymmetry\ntechnique \\cite{E-83}\nfirst applied to chaotic scattering problems in \\cite{VWZ-85}. The number $M$\nof the (statistically equivalent) scattering channels\ncan be small or large or can even scale with the number of resonance states.\nOne can treat the latter two cases \\cite{LW-91, HILSS-92, LSSS-94} as a\n\"semiclassical limit\" in the matrix model. We show here that the time-delay\nlocal fluctuations are governed, similar to those of the $S$-matrix\n\\cite{LSSS-94}, by the gap between the real axis and the upper edge of the\ndistribution of resonance energies in the complex energy plane. We compare\nthis result with that obtained in the framework of the periodic orbit\napproach.\n\nIn the next section our statistical matrix model is briefly presented.\nThe connections of average time delay with the resonance spectrum and S-matrix\nfluctuations are elucidated in sec.~3. After a short description in sec.~4\nof the supersymmetry method which we use the main analytical results for the\ntime delay correlation function are given and discussed in detail in sec.~5.\nSome numerical results shedding additional light upon properties of the time\ndelay correlations are gathered in sec. 6. We close with a brief summary\nin sec.~7.\n\n\\section{The Resonance Matrix Model}\nAccording to the general scattering theory, the evolution of the $N$-level\nunstable system formed on intermediate stage of a resonance collision is\ndescribed \\cite{MW-69, KNO-69, SZ-89} by the effective Hamiltonian\n\\begin{equation}\\label{hamil}\n{\\cal H} = H - i\\gamma\\, W,\\; \\; \\;\\;  W = VV^{T}\\,\\,.\n\\end{equation}\nThe Hamiltonian (\\ref{hamil}) acts within the intrinsic $N$-dimensional space\nbut acquires, due to the elimination of continuum variables, an antihermitian\npart. The hermitian matrix $H$ is the internal Hamiltonian with a discrete\nspectrum whereas the rectangular $N\\times M$ matrix $V$ consists of the\namplitudes $V_m^c$ of transitions between $N$ internal and $M$ channel\nstates. These amplitudes are real in T-invariant theory, so that the matrix\n$W$, similar to $H$, is real and symmetric. As usual, we neglect the smooth\nenergy dependence of $V$ and $W$.  The dimensionless parameter $\\gamma$\ncharacterizes the strength of the coupling of the internal motion to the\ncontinuum.\n\nThe poles of the resonance scattering matrix in the complex energy plane\nare those of the Green's function \\cite{MW-69, KNO-69, SZ-89}\n\\begin{equation}\\label{green}\n{\\cal G}(E) = (E-{\\cal H})^{-1}\\,\\,.\n\\end{equation}\nThey coincide with the eigenvalues ${\\cal E}_n=E_n-\\frac{i}{2}\\Gamma_n$ of\nthe effective Hamiltonian ${\\cal H}$ with $E_n$ and $\\Gamma_n$ being the\nenergy and width of $n$-th resonance state. It what follows, the properties\nof the spectrum of complex energies ${\\cal E}_n$ play the crucial role.\n\nThe intrinsic chaoticity of the internal motion of long-lived intermediate\nsystem manifests itself by chaotic fluctuations in resonance scattering and\ndemands a statistical consideration. Therefore the random matrix approach\nextending\nthe well-known \\cite{Po-65, Me-67} description of chaotic bounded systems\nhas been worked out in \\cite{W-84, VWZ-85, SZ-89}. It is usually\nassumed that the hermitian part $H$ of the effective Hamiltonian belongs to\nthe Gaussian Orthogonal Ensemble (GOE),\n\\begin{equation}\\label{goe}\n\\langle H_{nm} \\rangle = 0,\\ \\ \\ \\langle H_{nm}H_{n'm'} \\rangle =\n\\frac{\\lambda^2}{N}(\\delta_{nn'}\\delta_{mm'}+\\delta_{nm'}\\delta_{mn'})\\,\\,.\n\\end{equation}\nIn the limit $N\\rightarrow\\infty$ eigenvalues of $H$ are situated in the\ninterval $[-2\\lambda,2\\lambda]$ with the density given by Wigner's\nsemicircle law. Following \\cite{SZ-89}, we suggest the transition amplitudes\n$V_n^c$ also to be statistically independent Gaussian variables,\n\\begin{equation}\\label{rand}\n\\langle V^a_n \\rangle = 0,\\, \\, \\,\n\\langle V^a_nV^b_m \\rangle = \\frac{\\lambda}{N}\\delta^{ab}\\delta_{nm}\\,\\,.\n\\end{equation}\nWe will use below the ensemble (\\ref{goe},\\ref{rand}) to calculate the\naverage quantities defined in (\\ref{atd},\\ref{dcf}).\n\n\\section{Time Delay and Resonance Spectrum}\n\nSince we have neglected a smooth energy dependence of the effective\nHamiltonian (\\ref{hamil}),\nthe poles ${\\cal E}_n$ in the lower part of the complex energy\nplane are the only singularities of the resonance scattering matrix.  Due to\nthe unitarity condition their complex conjugates ${\\cal E}_n^*$ serve as\n$S$-matrix's zeros. These two conditions result in the representation\n\\begin{equation}\\label{det_s}\n\\det\\,S(E) = \\prod_{n} \\frac{E-{\\cal E}^{\\ast}_{n}}{E-{\\cal E}_{n}}\\,\\,.\n\\end{equation}\nSubstituting eq.(\\ref{det_s}) in eq.(\\ref{q1}), we come to the important\nconnection\n\\begin{equation}\\label{q2}\n Q(E) = -2\\,\\mbox{Im}\\,\\left\\{\\frac{1}{M}\\mbox{tr\\,} {\\cal G(E)}\\right\\}\n = \\frac{1}{M}\\,\\sum_n \\frac{\\Gamma_n}{(E-E_n)^2+\\frac{1}{4}\\Gamma_n^2}\n\\end{equation}\nbetween the time delay and the trace of the Green's function (\\ref{green}) of\nthe intermediate unstable system. The time delay is entirely determined by\nthe spectrum of complex energies of this system. The collision duration\ndirectly reflects the statistical properties of resonances. This is in\ncontrast to the scattering amplitudes $S^{cc'}$ which explicitly depend also\non the transition amplitudes $V_n^c$.\n\nThe ensemble averaging of eq.(\\ref{q2}) gives\n\\begin{equation}\\label{qav}\n \\langle Q(E) \\rangle = \\frac{2}{m\\lambda}\\,\\mbox{Re}\\,{\\sl g}(E)\n\\end{equation}\nwhere $m < 1$ is the ratio $M\/N$ and the function\n\\begin{equation}\\label{gav}\n {\\sl g}(E) = i\\lambda\\frac{1}{N}\\,\\langle \\mbox{tr\\,}\\,{\\cal G}(E)\\rangle\n\\end{equation}\nsatisfies the cubic equation \\cite{LSSS-94}\n\\begin{equation}\\label{qeq}\n {\\sl g}(E)-\\frac{1}{{\\sl g}(E)}+\\frac{m\\gamma}{1+\\gamma{\\sl g}(E)}-\n i\\frac{E}{\\lambda}=0\\,\\,.\n\\end{equation}\nThe (unique) solution with a positive real part has to be chosen.  It can be\nseen from the consideration given in \\cite{LSSS-94} that this real part is\nclose to $\\frac{\\lambda}{N}\\pi\\rho(E)$ with $\\rho(E)$ being the projection on\nthe real energy axis near the scattering energy $E$ of the density of\nresonance levels in the complex energy plane.\n\nOn the other hand, averaging eq.(\\ref{q-matr}) directly, we express\n$\\langle Q \\rangle$ in terms of the two-point $S$-matrix correlation function\n\\cite{VWZ-85, LSSS-94}. In the limit of a large number of statistically\nequivalent channels, $M\\gg 1$, scaling with the number of resonances $N$\n\\begin{equation}\\label{q-ss}\n\\langle Q \\rangle =\n-i\\frac{dC_S(\\varepsilon)}{d\\varepsilon}\\Bigg|_{\\varepsilon=0} +\ni\\frac{d{\\cal T}(\\varepsilon)}{d\\varepsilon}\\Bigg|_{\\varepsilon=0}\\,\\,.\n\\end{equation}\nHere \\cite{LSSS-94}\n\\begin{equation}\\label{scor}\n C_S(\\varepsilon) = \\frac{i\\Gamma(\\varepsilon)}{\\varepsilon +\n i\\Gamma(\\varepsilon)}\\,{\\cal T}(\\varepsilon) \\equiv\n K(\\varepsilon)\\,{\\cal T}(\\varepsilon)\n\\end{equation}\nwith the two smooth functions defined by\n\\begin{equation}\\label{G,T}\n \\Gamma(\\varepsilon) = \\frac{m}{2}\\lambda\\,\n \\frac{{\\cal T}(\\varepsilon)}{{\\sl g}(\\varepsilon\/2)}\\;\\;\\;,\n \\;\\;\\;{\\cal T}(\\varepsilon) = \\frac{4\\gamma{\\sl g}(\\varepsilon\/2)}\n {\\left[1+\\gamma{\\sl g}(\\varepsilon\/2)\\right]^2}\n\\end{equation}\nand we set $E=0$ for the sake of simplicity. The quantity\n\\begin{equation}\\label{trc}\nC_S(0) = {\\cal T}(0)\\equiv T\n\\end{equation}\ncoincides with the transmission coefficient $T=1-|\\langle S\\rangle|^2$. With\neq.(\\ref{scor}) taken into account we obtain from (\\ref{q-ss})\n\\begin{equation}\\label{qav2}\n\\langle Q \\rangle =\n-iT\\frac{dK(\\varepsilon)}{d\\varepsilon}\\Bigg|_{\\varepsilon=0}\n= \\frac{T}{\\Gamma_0}\\,\\,,\n\\end{equation}\nwhere we have designated $\\Gamma(0)$ as $\\Gamma_0$.\n\nAs long as the typical values of the quantity $\\Gamma(\\varepsilon)$ are small\nas compared to the parameter $\\lambda$ characterizing the scale of the smooth\n$\\varepsilon$-dependence of the function ${\\cal T}(\\varepsilon)$, the two\nfactors on the r.h.s. of eq.(\\ref{scor}) have quite different energy scales.\nOnly the first fast varying factor $K(\\varepsilon)$ describes the local\nfluctuations whereas the second one corresponds to the joint influence of all\nresonances giving rise to the processes with a very short duration. The\nlatter came out from eq.(\\ref{qav2}). The average time delay of a\nnon-monochromatic spatially small wave packet caused by the formation of a\nlong-lived intermediate state \\cite{L-77, DHM-92} is determined just by the\nfactor $K(\\varepsilon)$ \\cite{LSSS-94}\n\\begin{equation}\\label{dstd}\n\\langle \\tau \\rangle =\n-i\\frac{dK(\\varepsilon)}{d\\varepsilon}\\Bigg|_{\\varepsilon=0}\n= \\Gamma_0^{-1}\\,\\,.\n\\end{equation}\nThis implies the connection \\cite{DHM-92, LSSS-94}\n\\begin{equation}\\label{std}\n\\langle\\tau\\rangle = \\langle Q\\rangle\/T =\n\\frac{2N}{\\lambda MT}\\,{\\sl g}(0)\\approx\\frac{2\\pi\\rho}{MT}\\,\\,.\n\\end{equation}\n\n\\section{The Supersymmetry Method}\n\nNow we calculate the correlation function (\\ref{dcf}). Taking into account\nthe relation (\\ref{q2}), one can cast eq.(\\ref{dcf}) into the form\n\\begin{equation}\\label{qq2}\nC_Q(E,\\varepsilon)= \\frac{2}{M^2}\\,\\mbox{Re} \\left\\{\n\\langle \\mbox{tr\\,}{\\cal G}(E\\!+\\!\\frac{\\varepsilon}{2})\n\\mbox{tr\\,}{\\cal G}^{\\dagger}(E\\!-\\!\\frac{\\varepsilon}{2}) \\rangle\n- \\langle \\mbox{tr\\,}{\\cal G}(E\\!+\\!\\frac{\\varepsilon}{2}) \\rangle\n\\langle \\mbox{tr\\,}{\\cal G^{\\dagger}}(E\\!-\\!\\frac{\\varepsilon}{2})\n\\rangle \\right\\}\\,\\,.\n\\end{equation}\nWe also define the normalized quantity\n\\begin{equation}\\label{Nqq}\nK_Q(E,\\varepsilon)=\\frac{C_Q(E,\\varepsilon)}{{\\langle Q(E)\\rangle}^2}\\,\\,.\n\\end{equation}\nThe terms containing two Green's functions with poles at the same side from\nthe real energy axis are omitted in (\\ref{qq2}). We will briefly return to\nthis point later.\n\nIn the limit $\\gamma=0$, when the system gets closed, the correlation function\n(\\ref{qq2}) becomes proportional to the GOE density-density correlation which\nconsists \\cite{Me-67} of the singular term $\\delta(\\pi\\rho\\varepsilon)$ and\nDyson's smooth function $-Y_2(\\pi\\rho\\varepsilon)$. Coupling to the continuum\nleads to appearing of a new energy scale caused by the decay processes. This\nscale is defined \\cite{LSSS-94} by the quantity $\\Gamma(\\varepsilon)$ from\neq.(\\ref{G,T}).  One can anticipate a qualitative changing of the correlation\nfunction to occur on this scale.  For larger distances the influence of the\nantihermitian part should fade away and the asymptotics of $C_Q$ for\n$\\varepsilon\\rightarrow\\infty$ is expected to coincide with that of the\nDyson's function $-Y_2(\\pi\\rho\\varepsilon)$.\n\nTo perform the ensemble averaging in (\\ref{qq2}) we use the modification\nworked out in \\cite{LSSS-94} of the supersymmetry technique \\cite{VWZ-85}.\nUsing the integral representation of Green's function as a multivariate\nGaussian integral over commuting and anticommuting variables, one gains the\npossibility to accomplish the averaging exactly. With the help of the Fourier\ntransformation in the supermatrix space the integration over initial\nauxiliary supervectors is then carried out. Going along this line, one finally\narrives at\n\\begin{equation}\\label{grgr}\n \\langle \\mbox{tr\\,}{\\cal G}(E\\!+\\!\\frac{\\varepsilon}{2})\\,\n \\mbox{tr\\,}{\\cal G^{\\dagger}}(E\\!-\\!\\frac{\\varepsilon}{2}) \\rangle =\n -\\frac{N^2}{4}\\langle\\mbox{str\\,}(\\sigma \\eta_1)\\,\\mbox{str\\,}\n (\\sigma \\eta_2) \\rangle_{{\\cal L}}\\,\\,.\n\\end{equation}\nHere the shorthand $\\langle\\ldots\\rangle_{{\\cal L}}$ is used to denote\nthe integral\n\\begin{equation}\\label{shand}\n \\langle \\ldots \\rangle_{{\\cal L}} =\n \\int\\!d[\\sigma]\\,d[\\hat\\sigma]\\,\\exp \\{\n - N{\\cal L}(\\sigma,\\hat\\sigma) \\} (\\ldots)\n\\end{equation}\nover two $8\\times8$ supermatrices $\\sigma$ and $\\hat\\sigma$ with the\nmeasure defined by the Lagrangian \\cite{LSSS-94}\n\\begin{equation}\\label{Lg}\n {\\cal L}(\\sigma,\\hat\\sigma) = \\frac{1}{4}\\,\\mbox{str\\,}\\sigma^2\n - \\frac{i}{2}E\\,\\mbox{str\\,}\\sigma -\n \\frac{i}{2}\\mbox{str\\,}(\\sigma\\hat\\sigma) +\n \\frac{1}{2}\\mbox{str\\,}\\ln(\\hat\\sigma) + \\frac{m}{2}\\,\\mbox{str\\,} \\ln\n (1\\!+\\!\\gamma\\sigma \\eta) - \\frac{i}{4}\\varepsilon\\,\\mbox{str\\,}(\\sigma\n \\eta)\\,\\,.  \\end{equation} The  diagonal supermatrices appearing above are\nequal to \\[\\eta=\\mbox{diag}(1,1,-1,-1,1,1,-1,-1)\\]\n\\[\\eta_1=\\mbox{diag}(1,1,0,0,-1,-1,0,0)\\;\\;\\;\n\\eta_2=\\mbox{diag}(0,0,1,1,0,0,-1,-1)\\,\\,. \\]\nHere we have set the GOE parameter $\\lambda$ equal to one.\n\nThe supermatrix $\\sigma$ can be decomposed in the following way \\cite{VWZ-85}\n\\begin{equation}\n\\sigma=T_0\\,\\sigma_R\\,T_0^{-1}\n\\end{equation}\nwhere $T_0$ is a transformation from a non-compact manifold whereas the matrix\n$\\sigma_R$ is\ndiagonalized by transformations from a compact one. This implies a\ncorresponding decomposition of the integrals on the r.h.s. of (\\ref{shand})\n\\begin{equation}\\label{z4}\n \\langle \\ldots \\rangle_{{\\cal L}} =\n \\int\\!{\\cal F}(\\sigma_R)\\,d[\\sigma_R]\\,d[\\hat\\sigma]\n \\exp\\{-N{\\cal L}_R(\\sigma_R,\\hat\\sigma) \\}\n \\int\\!d\\mu\\exp\\{-N{\\cal L}_{\\mu}(\\sigma_R,T_0)\\}(\\ldots)\\,\\,.\n\\end{equation}\nThe Berezinian ${\\cal F}(\\sigma_R)$ depends only on the eigenvalues of\n$\\sigma_R$; $d\\mu$ is the invariant measure of the manifold of non-compact\ntransformations $T_0$.  At last, the Lagrangian (\\ref{Lg}) is splitted into\n two parts, ${\\cal L}_R$ and ${\\cal L}_{\\mu}$, given by\n\\begin{equation}\\label{dLg}\n\\begin{array}{l}\n {\\cal L}_R(\\sigma,\\hat\\sigma) = \\frac{1}{4}\\mbox{str\\,}\\sigma_R^2 -\n \\frac{i}{2}E\\,\\mbox{str\\,}\\sigma_R -\n \\frac{i}{2}\\mbox{str\\,}(\\sigma_R\\hat\\sigma) +\n \\frac{1}{2}\\mbox{str\\,}\\ln(\\hat\\sigma)\\,\\,,  \\\\  \\\\ {\\cal\n L}_{\\mu}(\\sigma_R,T_0) = -\\frac{i}{4}\\varepsilon\\,\\mbox{str\\,}(\\sigma_R\n T_0^{-1}\\eta T_0) + \\frac{m}{2}\\,\\mbox{str\\,}\\ln(1\\!+\\!\\gamma\\sigma_R\n T_0^{-1}\\eta T_0)\\,\\,.  \\end{array} \\end{equation} Only the second part\n${\\cal L}_{\\mu}$ depends on the non-compact variables. The first one\n${\\cal L}_R$ is invariant under a transformation by\n$T_0$ since it is fully absorbed by an appropriate transformation of\n$\\hat\\sigma$. One can easily verify that the corresponding Berezinian is\nequal to unity.\n\nSince the number of resonances $N\\rightarrow \\infty$, the integrations over\n$\\sigma_R$ and $\\hat\\sigma$ can be carried out in the saddle-point\napproximation. At the same time, one has to integrate exactly over\nnon-compact variables as long as the number of channels $M$ is finite\n($m=0$). The saddle-point approximation becomes valid for the latter\nintegration when the number $M$ also tends to infinity ($m$ is finite). We\nwill consider both cases mentioned. To simplify formulae we restrict\nour further consideration to the center of the GOE spectrum $E=0$.\n\n\\section{Time Delay Correlation Function}\n\nLet us first consider collisions with a fixed number of channels $M$. The\nlogarithmic term in ${\\cal L}_{\\mu}$ being proportional to the small ratio\n$m$ does not influence then the saddle-point equations in the\n$(\\sigma_R,\\hat\\sigma)$-sector. In particular, the term in (\\ref{qeq})\ncontaining this ratio has to be omitted. The saddle-point equations are\ntrivially solved in this case and at the point $E=0$\n\\begin{equation}\\label{sps1}\n\\hat\\sigma=-i\\sigma_R^{-1}\\;\\;,\\;\\;\\sigma_R=\\eta\\,\\,.\n\\end{equation}\nWith integrations over $\\sigma_R$ and $\\hat\\sigma$ being done, the\ncorrelation function (\\ref{dcf}) reduces to the integral\n\\begin{equation}\\label{Cqq}\n K_Q(\\varepsilon)=2\\,\\mbox{Re}\\int\\!d\\mu\\,\n \\mbox{str\\,}(\\kappa\\alpha_1)\\,\\mbox{str\\,}(\\kappa\\alpha_2)\\,\n \\exp\\Bigl\\{ \\frac{i}{2}\\pi\\rho\\varepsilon\\,\\mbox{str\\,}\\alpha_1\n-\\frac{M}{2}\\mbox{str\\,}\\ln(1\\!+\\!\\frac{1}{2}T\\alpha_1) \\Bigr\\}\n\\end{equation}\nover the invariant measure of the non-compact manifold of $T_0$-matrices.\nHere $\\alpha_{1,2}$ are the $4\\times 4$ supermatrices defined in\n\\cite{VWZ-85}, the supermatrix $\\kappa=\\mbox{diag}(1,1,-1,-1)$ and\n\\begin{equation}\\label{trc0}\nT=\\frac{4\\gamma}{(1+\\gamma)^2}\n\\end{equation}\nis the transmission coefficient (\\ref{trc}) calculated in the limit of $m=0$.\n\nThe further calculations go along the line described in details in\n\\cite{VWZ-85} and lead to the result\n\\[K_Q(\\varepsilon)=\\frac{1}{4}\\,\\int\\limits_0^1\\!d\\lambda_0\\!\n\\int\\limits_0^{\\infty}\\!d\\lambda_1\\!\\int\\limits_0^{\\infty}\\!d\\lambda_2\\,\n\\mu(\\lambda_0,\\lambda_1,\\lambda_2)(2\\lambda_0+\\!\\lambda_1\\!+\\!\\lambda_2\\!)^2\\,\n\\mbox{cos}\\{\\pi\\rho\\varepsilon(2\\lambda_0+\\!\\lambda_1\\!+\\!\\lambda_2\\!)\\}\\]\n\\begin{equation}\\label{K-f}\n\\times\\left[ \\frac{(1\\!-\\!T\\lambda_0)^2}\n{(1\\!+\\!T\\lambda_1)(1\\!+\\!T\\lambda_2) }\\right]^{M\/2}\n\\end{equation}\nwhere\n\\[\\mu(\\lambda_0,\\lambda_1,\\lambda_2)=\n\\frac{(1\\!-\\!\\lambda_0)\\lambda_0|\\lambda_1-\\lambda_2|}\n{[(1+\\lambda_1)\\lambda_1(1+\\lambda_2)\\lambda_2]^{1\/2}\n(\\lambda_0\\!+\\!\\lambda_1)^2(\\lambda_0\\!+\\!\\lambda_2)^2 }\\,\\,.\\]\n\nThe dependence of the function $K_Q$ on openness of the unstable system is\nfully contained in the last factor in (\\ref{K-f}). If at least one of the\nquantities $M$ or $T$ is equal to zero the threefold integral reduces to\nthe single one \\cite{E-83}\n\\begin{equation}\\label{d-d}\nK_Q^{(0)}(\\varepsilon)=\\int\\limits_0^2\\!dt\\,\nt\\left(1-\\frac{1}{2}\\ln(t\\!+\\!1)\\right)\n\\mbox{cos}(\\pi\\rho\\varepsilon t)+\n\\int\\limits_2^{\\infty}\\!dt\\,\n\\left(2-\\frac{t}{2}\\ln\\frac{t\\!+\\!1}{t\\!-\\!1}\\right)\n\\mbox{cos}(\\pi\\rho\\varepsilon t)\n\\end{equation}\n\\[=\\delta(\\pi\\rho\\varepsilon)-Y_2(\\pi\\rho\\varepsilon)\\]\nwhich is just the normalized GOE density-density correlation function.\n\nGenerally speaking, the threefold integral in (\\ref{K-f}) can be investigated\nfor arbitrary number of channels $M$ only numerically using the methods\ndeveloped in \\cite{V-86} (see the next section). However, this integral can\nbe simplified if $M$ becomes large enough. Let the number $M$ grow still\nkeeping the ratio $m=0$ and the product $MT=2\\pi\\rho\\Gamma_W$ (compare with\n(\\ref{std})) fixed. The quantity $\\Gamma_W$ is just the limiting value of\n$\\Gamma_0$ with $T$ and ${\\sl g}$ calculated in the limit $m=0$. It coincides\nwith the well-known semiclassical Weisskopf estimate \\cite{BW-79} of the\ncorrelation length of Ericson fluctuations.  Then\n\\begin{equation}\\label{t-exp}\n\\left[\\frac{(1\\!-\\!T\\lambda_0)^2}\n{(1\\!+\\!T\\lambda_1)(1\\!+\\!T\\lambda_2)}\\right]^{M\/2}\n\\rightarrow\\exp\\{-\\pi\\rho\\Gamma_W\\,\n(2\\lambda_0+\\!\\lambda_1\\!+\\!\\lambda_2\\!)\\},\n\\end{equation}\nand one obtains similar to eq.(\\ref{d-d})\n\\[K_Q(\\varepsilon)=\\int\\limits_0^2\\!dt\\,t e^{(-\\pi\\rho\\Gamma_W t)}\\,\n\\left(1-\\frac{1}{2}\\ln(t\\!+\\!1)\\right)\n\\mbox{cos}(\\pi\\rho\\varepsilon t)\\]\n\\begin{equation}\\label{lM}\n+\\int\\limits_2^{\\infty}\\!dt\\,e^{(-\\pi\\rho\\Gamma_W t)}\\,\n\\left(2-\\frac{t}{2}\\ln\\frac{t\\!+\\!1}{t\\!-\\!1}\\right)\n\\mbox{cos}(\\pi\\rho\\varepsilon t)\\,\\,.\n\\end{equation}\nThis is in close analogy with the consideration of the S-matrix correlation\nfunction made in \\cite{V-86}.\n\nA new convergency factor appeared in the integrals in (\\ref{lM}) as compared\nto (\\ref{d-d}) where only the oscillating cosine cuts the integral in the\nregion\nof asymptotically large $t$. This makes the function $K_Q$ finite for all\nvalues of $\\varepsilon$ including zero, so that the $\\delta$-function is now\nsmeared out. The behaviour of $K_Q(\\varepsilon)$ is quite different in the\nregions $\\varepsilon\\ll\\Gamma_W$ and $\\varepsilon\\gg \\Gamma_W$. In the first\none it is determined by decays and therefore is sensitive to the coupling to\nthe continuum. Quite opposite, for large $\\varepsilon$ the behaviour becomes\nuniversal since the GOE fluctuations described by the Dyson's function $Y_2$\nare restored. It is perfectly reasonable since an open system cannot be\ndistinguished from a closed one during a small time $t\\ll \\Gamma_W^{-1}$.\n\nThe first $\\gamma$-sensitive domain is widened when the width $\\Gamma_W$\ngrows.\nIn the case of small $\\rho\\Gamma_W\\ll 1$ (isolated resonances) it is natural\nto set aside the contribution of asymptotics of the integrand presenting\n(\\ref{lM}) in the form\n\\begin{equation}\\label{isol}\n K_Q(\\varepsilon) = \\frac{1}{\\pi\\rho}\\,\\frac{\\Gamma_W}\n{(\\varepsilon^2\\!+\\!\\Gamma_W^2)}\\, +\n\\end{equation}\n\\[\\int\\limits_0^2\\!dt\\,e^{-\\pi\\rho\\Gamma_W t}\\,\n\\left(t-\\frac{t}{2}\\ln(t\\!+\\!1)-1\\right)\n\\mbox{cos}(\\pi\\rho\\varepsilon t) +\n\\int\\limits_2^{\\infty}\\!dt\\,e^{-\\pi\\rho\\Gamma_W t}\\,\n\\left(1-\\frac{t}{2}\\ln\\frac{t\\!+\\!1}{t\\!-\\!1}\\right)\n\\mbox{cos}(\\pi\\rho\\varepsilon t)\\,\\,.\\]\nThe Lorentzian contribution with the width $\\Gamma_W$ directly traced to\nthe GOE $\\delta$-function dominates in the domain $\\varepsilon\\mathrel{\\mathpalette\\fun <}\\Gamma_W$.\nThe sum of the integrals in the second line is negative for all values of\n$\\varepsilon$ and approaches asymptotically the function $Y_2$ from above.\nWe thus come to the conclusion that the correlation function vanishes at some\nintermediate point $\\varepsilon_0$ which can be estimated as\n\\begin{equation}\\label{isoz}\n\\varepsilon_0\\simeq \\sqrt{\\frac{\\Gamma_W}{\\pi\\rho}}\n\\end{equation}\nusing the condition\n\\[\\frac{1}{\\pi\\rho}\\,\\frac{\\Gamma_W}{(\\varepsilon_0^2\\!+\\!\\Gamma_W^2)}\n\\sim |Y_2(\\rho\\varepsilon_0)|\\sim 1\\,\\,.\\]\n\nThe regime of strongly overlapping resonances, $\\rho\\Gamma_W\\gg 1$, is the\nmost interesting. In this case the\nmain contribution in $K_Q$ comes from the region of small $t$. Therefore, the\nsecond integral in (\\ref{lM}) can be neglected. Dropping then out the small\nlogarithmic term in the first integral and extending its upper limit to\ninfinity, we arrive at\n\\begin{equation}\\label{overl}\nK_Q(\\varepsilon)\\approx\\int\\limits_0^\\infty\\!dt\\,t e^{(-\\pi\\rho\\Gamma_W t)}\\,\n\\mbox{cos}(\\pi\\rho\\varepsilon t) =\n\\frac{1}{\\pi^2\\rho^2}\\frac{\\Gamma_W^2-\\varepsilon^2}\n{(\\varepsilon^2+\\Gamma_W^2)^2}\\,\\,.\n\\end{equation}\nCorrections to this result are of higher order with respect to the parameter\n$(\\rho\\Gamma_W)^{-1}$. The function (\\ref{overl}) is not a Lorentzian at all.\nDecreasing quadratically in a small vicinity of the point $\\varepsilon=0$, it\ndeviates subsequently from a Lorentzian, becomes zero at the point\n$\\varepsilon=\\Gamma_W$, reaches a negative minimum and approaches at last\nzero from below. Just the correlation function of such a form with $\\Gamma_W$\nsubstituted by the classical escape rate was conjectured in \\cite{Ec-93}\nas the limiting classical expression following from the periodic orbit\npicture. However, there is no room for the classical escape rate in the\nmatrix models considered here.  One can see that the found form has in fact\nquantum grounds.\n\nOne should return to the exact expressions (\\ref{z4},\\ref{dLg}) if the ratio\n$m$ is finite. The resonances strongly overlap in this case. The\nsaddle-point is now found to be\n\\begin{equation}\\label{sps2}\nT_0=1\\;\\,,\\;\\, \\hat\\sigma=-i\\sigma_R^{-1}\\;\\,,\n\\;\\, \\sigma_R={\\sl g}(\\varepsilon\/2)\\,\\eta\\,\\,,\n\\end{equation}\nwhere ${\\sl g}$ is the  solution chosen in sec. 5 of the cubic equation\n(\\ref{qeq}).  The sequential saddle-point integrations over\n$\\sigma_R,\\hat\\sigma$ and then over the non-compact manifold result in the\nexpression\n\\begin{equation}\\label{Cqqm}\nK_Q(\\varepsilon)=-\\frac{4}{M^2T^2}\\,\\mbox{Re}\\,\n\\frac{\\Gamma_0^2}{\\left[\\varepsilon+i\\Gamma(\\varepsilon)\\right]^2}\n\\end{equation}\nwhere the function $\\Gamma(\\varepsilon)$ defined in (\\ref{G,T}) is just the\none appearing when the $S$-matrix fluctuations are considered \\cite{LSSS-94}.\n\nThe explicit dependence on $\\varepsilon$ gives rise to a sharp variation of\nthe correlation function (\\ref{Cqqm}) in the vicinity of zero if the typical\nvalues $|\\Gamma(\\varepsilon)|\\ll 1$ (see eq.(\\ref{scor}) and the discussion\nbelow). As long as the ratio $m$ is small, the quantity $\\Gamma(\\varepsilon)$\nis small indeed and we can neglect its smooth $\\varepsilon$-dependence for\nall $\\varepsilon\\mathrel{\\mathpalette\\fun <}\\Gamma_0\\approx\\Gamma_W$.  Eq.(\\ref{Cqqm}) is equivalent\nto eq.~(\\ref{overl}) within this domain. The asymptotic behaviour for large\n$\\varepsilon$ also does not change since $\\Gamma(\\varepsilon)$ remains\nrestricted for all $\\varepsilon$. A small difference can appear only for\nintermediate values of $\\varepsilon$.\n\nHowever, for larger values of $m$ the deviation can become noticeable even\nnear the point $\\varepsilon=0$. In this case the next term in the power\nexpansion\n\\begin{equation}\\label{pex}\n\\Gamma(\\varepsilon)\\approx\\Gamma_0+\\Gamma_0'\\,\\varepsilon\n\\end{equation}\nwith respect to the smooth $\\varepsilon$-dependence should be taken into\naccount \\cite{LSSS-94}. Because of the smoothness, the derivative $\\Gamma_0'$\nis small. One can see from eq.(\\ref{qeq}) that this derivative is pure\nimaginary.  The form (\\ref{overl}) is now reproduced again for sufficiently\nsmall $\\varepsilon$ ,\n\\begin{equation}\\label{Cqqmap}\nK_Q(\\varepsilon) = \\frac{4\\Gamma_g^2}{M^2T^2}\\,\n\\frac{\\Gamma_g^2-\\varepsilon^2}{(\\varepsilon^2+\\Gamma_g^2)^2 }\\,\\,,\n\\end{equation}\nwith\n\\begin{equation}\\label{Gg}\n\\Gamma_g=\\frac{\\Gamma_0}{1+i\\Gamma_0'}\\,\\,.\n\\end{equation}\nIt has been proven in \\cite{LSSS-94} that $\\Gamma_g$, playing the role of the\ncorrelation length of the Ericson fluctuations, coincides with the gap between\nthe distribution of resonance energies in the complex energy plane and the\nreal energy axis. Therefore we come to the conclusion that the properties of\nfluctuations both of the $S$-matrix and time delay are described by the same\nquantity, the gap $\\Gamma_g$, rather than the classical escape rate.\n\nUntil now we neglected the \"one-sided\" contribution\n\\[\\widetilde{C}_Q(\\varepsilon)=\\langle Q\\rangle^2\\,\n\\widetilde{K}_Q(\\varepsilon)=\\]\n\\begin{equation}\\label{tqq2}\n-\\frac{2}{M^2}\\,\\mbox{Re}\\left\\{\\langle\\mbox{tr\\,}{\\cal G}(\\frac{\\varepsilon}\n{2})\\mbox{tr\\,}{\\cal G}(-\\frac{\\varepsilon}{2})\\rangle\n- \\langle \\mbox{tr\\,}{\\cal G}(\\frac{\\varepsilon}{2})\\rangle\n\\langle\\mbox{tr\\,}{\\cal G}(-\\frac{\\varepsilon}{2})\n\\rangle \\right\\}\n\\end{equation}\nto the correlation function (\\ref{dcf}). As long as $m=0$, this contribution\nis of higher order in the parameter $N^{-1}$. However, this is not the case\nwhen the ratio $M\/N$ is finite. So one has to calculate (\\ref{tqq2})\nexplicitly. The well-known replica method \\cite{EA-75} turns out to be\nsufficient for the latter purpose. Dropping here the corresponding rather\ncumbersome\nexpressions we only note that the function $\\widetilde{K}_Q(\\varepsilon)$ is\nentirely expressed in terms of the slowly varying ${\\sl\ng}(\\frac{\\varepsilon}{2})$ and varies slowly itself. It has got no pronounced\nresonance behaviour around the point $\\varepsilon=0$ and constitutes a\nsmooth background for the correlation function. Its value at the point\n$\\varepsilon=0$ is approximately equal to\n\\[\\widetilde{K}_Q(0)\\approx - \\frac{1}{8N^2}\\]\nso that\n\\[\\Bigg|\\widetilde K_Q(0)\/{K_Q(0)}\\Bigg|\n\\approx\\frac{1}{2}\\,\\left(\\frac{\\pi\\rho\\Gamma_0}{2N}\\right)^2\\,\\,.\\]\nThe ratio is small under the condition\n\\begin{equation}\\label{con}\n\\pi\\rho\\Gamma_0\\ll N \\quad\\mbox{or}\\quad \\Gamma_0\\ll 1\n\\end{equation}\nimplying a clear-cut distinction of the local and global scales\n\\cite{LSSS-94}. Such a scale separation is necessary for matrix models to\nbe valid so far as the fluctuations are concerned.\n\nThe obtained form of the $\\varepsilon$-dependence of the many-channel\ncorrelation function $C_Q$ is close to that found in \\cite{SW-92} for the\nGutzwiller's model of single-channel chaotic scattering on a space of\nnegative curvature. The same values of all resonance widths and the\noutcoming possibility  for resonances to overlap are two specific features\nof the model which are in fact in strong disagreement with properties of\nthe resonance spectra represented by matrix models. In particular, the\nsingle-channel resonances cannot overlap at all in the latter models\n\\cite{SZ-89} and their widths fluctuate strongly. That is why our result\nfor $M=1$ (see below) differs noticeably from the correlation function\nof ref.\\cite{SW-92}. The situation changes when the number of channels is\nlarge.  The width fluctuations diminish with the number $M$ of channels\ngrowing. Since the time delay depends, according to (\\ref{q2}), only on\nproperties of the complex energies of resonances and not on the number of\nchannels directly, the correlation functions become similar in the two\nquite different cases compared.\n\nIt is worthy to note that the resonances overlapping strongly suppress the\ntime delay fluctuations. Indeed, eq.(\\ref{isol}) gives for isolated\nresonances\n\\[K_Q(0)=\\frac{1}{\\pi\\rho\\Gamma_W}\\gg 1\\]\nwhereas\n\\[K_Q(0)=\\frac{1}{\\pi^2\\rho^2\\Gamma_W^2}\\ll 1\\]\nwhen they overlap. The duration of a collision thus becomes a good definite\nquantity in the \"quasiclassical\" limit.\n\n\\section{Numerical results}\nExcepting a few limiting cases considered above, further analytical study of\n(\\ref{K-f}) is not possible and one has to use numerical methods. However,\nthe threefold integral as it stands does not suit for numerical computation.\nA very convenient substitution of the integration variables has been proposed\nin \\cite{V-86} to overcome all difficulties appearing. Following this author\nwe reduce the expression (\\ref{K-f}) to the Fourier integral\n\\begin{equation}\\label{fourier}\n K_Q(\\varepsilon) =\n \\int\\limits_0^{\\infty}\\!dt\\, F(t)\\cos(\\pi\\rho\\varepsilon t)\n\\end{equation}\nwith the Fourier transform $F(t)$ given by a double integral of a smooth\nfunction quite convenient for the numerical work. The asymptotic behaviour\nof $F(t)$ can be easily found explicitly\n\\begin{equation}\\label{F-asymp}\n F(t) \\sim \\left\\{ \\begin{array}{ll} t\n &\\ \\ \\mbox{for}\\ t\\ll 1 \\\\ (1+Tt)^{-M\/2} &\\ \\\n \\mbox{for}\\ t\\gg 1 \\end{array} \\right.\\,\\,.\n\\end{equation}\n\nFor a closed system ($T=0$) the Fourier transform $F(t)$ tends to unity in\nthe large-$t$ asymptotics. This results in the $\\delta$-term in the GOE\ndensity-density correlation. A singularity still survives even for an open\nsystem with one or two decay channels. The asymptotics (\\ref{F-asymp})\nimplies square root or logarithmic divergences correspondingly at the\npoint $\\varepsilon=0$ in these two cases.\n\nIn Fig. 1 the function $K_Q(x)$ versus $x=\\rho\\varepsilon$ is plotted for the\ncase of a single open channel. The singular behaviour near zero as well as\nGOE-like asymptotics are shown. The dashed line represents the Dyson's\nfunction $-Y_2(\\pi x)$. The calculation was made for the value $\\gamma=1$;\nonly some small domain around zero is sensitive to the choice of $\\gamma$.\nThe correlation function Fig.1 has little in common with that found in\n\\cite{SW-92}. This discrepancy is due to the strong fluctuations of\nsingle-channel widths in our model in contrast to identical widths of all\nresonances in Gutzwiller's one.\n\nFor $M>2$ the quantity $K_Q(0)$ is finite and the correlation function\napproaches, as the number of channels grows, the asymptotics given by\n(\\ref{lM}). The Fig. 2 demonstrates this for the ratio\n$K_Q(\\varepsilon)\/K_Q(0)$ in the\ncase of overlapping resonances. In asymptotic regime (\\ref{overl}) such a\nratio is an universal function of the only variable $\\varepsilon\/\\Gamma_W$.\nOne can see how the exact result (\\ref{K-f}) gets more and more close to this\nuniversal behaviour.\n\nThe Lorentzian peak should dominate the ratio\n$K_Q(\\varepsilon)\/K_Q(0)$ in the domain\n$\\varepsilon\/\\Gamma_W\\mathrel{\\mathpalette\\fun <} (\\pi\\rho\\Gamma_W)^{-\\frac{1}{2}}\\gg 1$ when\nresonances are isolated (see (\\ref{isoz})). Fig. 3 demonstrates this for two\nvalues of coupling constant $\\gamma$.\n\nAs it has been mentioned above, the function $K_Q(\\varepsilon)$ vanishes at\nsome point $\\varepsilon_0$.  The position of this point as the function of\nthe number of channels $M$ at several fixed values of $\\gamma$ is shown in\nFig. 4 for three different values of the coupling constant $\\gamma$. It is\nclearly seen that the square root dependence for isolated resonances (see\n(\\ref{isoz})) is replaced by the linear one for overlapping ones.\n\n\n\n\\section{Summary.}\nIn this paper we have considered the fluctuations of the characteristic time\nof collisions in the framework of a random matrix model of resonance chaotic\nscattering.  These fluctuations are entirely due to the fluctuations of the\nspectrum of complex resonance energies. We calculate analytically the time\ndelay correlation function and investigate its properties analytically and\nnumerically for different values of the number of channels and the strength\nof the coupling to the continuum. For any values of these parameters this\nfunction is far from being a Lorentzian. In particular, it vanishes at some\npoint which plays the role of the characteristic correlation length of the\nfluctuations. In the \"quasiclassical\" limit of a large number of strongly\noverlapping resonances this length is given, similar to that of the S-matrix\nfluctuations, by the gap between the upper edge of the distribution of\ncomplex energies of resonances and the real energy axis. We do not expect\nthat this quantity may be connected to the escape rate appearing in the\nclassical theory of chaotic scattering. The latter has been conjectured in\n\\cite{Sm-91} to be the semiclassical limit for the correlation length in\nchaotic scattering.\n\n\\begin{center}\n{\\large\\bf Acknowledgements}\n\\end{center}\nWe are grateful to F.Izrailev for his permanent interest to this work.\nFinancial support by the Deutsche Forschungsgemeinschaft through the SFB 237\nis acknowledged. For two of us (V.V.S. and D.V.S.) the research described in\nthis publication was made possible in part by Grant No RB7000 from the\nInternational Science Foundation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nCrowd counting aims to count the number of people in a crowded scene where as density estimation aims to map an input crowd image to it's corresponding density map which indicates the number of people per pixel present in the image (as illustrated in Fig. \\ref{fig:task_illustration}) and the two problems have been jointly addressed by researchers. The problem of crowd counting and density estimation is of paramount importance  and it is essential for building higher level cognitive abilities  in crowded scenarios such as crowd monitoring \\cite{chan2008privacy} and scene understanding \\cite{shao2015deeply,zhou2012understanding}. Crowd analysis has attracted significant attention from researchers in the recent past due to a variety of reasons. Exponential growth in the world population and the resulting urbanization has led to an increased number of activities such as sporting events, political rallies, public demonstrations etc. (shown in Fig. \\ref{fig:crowd_scenes}), thereby resulting in more frequent crowd gatherings in the recent years. In such scenarios, it is essential to analyze crowd behavior for better management, safety and security. \n\nLike any other computer vision problem, crowd analysis comes with many challenges such as occlusions, high clutter, non-uniform distribution of people, non-uniform illumination, intra-scene and inter-scene variations in appearance, scale and perspective making the problem extremely difficult. Some of these challenges are illustrated in Fig. \\ref{fig:crowd_scenes}. The complexity of the problem together with the wide range of applications for crowd analysis has led to an increased focus by researchers in the recent past. \n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{minipage}{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig1}\n\\vskip-6pt\n\\captionof*{figure}{(a)}\n\\end{minipage}%\n\\hfill\n\\begin{minipage}{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig2}\n\\vskip-6pt\n\\captionof*{figure}{(b)}\n\\end{minipage}\n\\vskip-10pt\n\\captionof{figure}{Illustration of density map estimation. (a) Input image (b) Corresponding density map with count.}\n\\label{fig:task_illustration}\n\\end{center}\n\n\\end{figure}\n\nCrowd analysis is an inherently inter-disciplinary research topic with researchers from different communities (such as sociology \\cite{moussaid2010walking,blumer1951collective}, psychology \\cite{aveni1977not}, physics \\cite{castellano2009statistical,1971HendersonStatistics}, biology \\cite{parrish1999complexity,zhang2010collective}, computer vision and public safety) have addressed the issue from different viewpoints. Crowd analysis has a variety of critical applications of inter-disciplinarian nature:\n\n\\noindent \\textit{Safety monitoring}: The widespread usage of video surveillance cameras for security and safety purposes in places such as sports stadiums, tourist spots, shopping malls and airports has enabled easier monitoring of crowd in such scenarios. However, traditional surveillance algorithms may break down as they are unable to process high density crowds due to limitations in their design. In such scenarios, we can leverage the results of algorithms specially designed for crowd analysis related tasks such as behavior analysis \\cite{saxena2008crowd,ko2008survey}, congestion analysis \\cite{zhou2015learning,huang2015congestion}, anomaly detection \\cite{li2014anomaly,chaker2017social} and event detection \\cite{benabbas2010motion}. \n\n\n\\noindent \\textit{Disaster management}: Many scenarios involving crowd gatherings such as sports events, music concerts, public demonstrations and political rallies face the risk of crowd related disasters such as stampedes which can be life threatening. In such cases, crowd analysis can be used as an effective tool for early overcrowding detection and appropriate management of crowd, hence, eventual aversion of any disaster \\cite{abdelghany2014modeling,almeida2013crowd}. \n\n\\noindent \\textit{Design of public spaces}: Crowd analysis on existing public spots such as airport terminals, train stations, shopping malls and other public buildings \\cite{chow2008waiting,sime1995crowd} can reveal important design shortcomings from crowd safety and convenience point of view. These studies can be used for design of public spaces that are optimized for better safety and crowd movement \\cite{lu2016study,al2013crowd}. \n\n\\noindent \\textit{Intelligence gathering and analysis}: Crowd counting techniques can be used to gather intelligence for further analysis and inference. For instance, in retail sector, crowd counting can be used to gauge people's interest in a product in a store and this information can be used for appropriate product placement \\cite{lipton2015video,mongeon2015busyness}.  Similarly, crowd counting can be used to measure queue lengths to optimize staff numbers at different times of the day.  Furthermore, crowd counting can be used to analyze pedestrian flow at signals at different times of the day and this information can be used for optimizing signal-wait times \\cite{bernal2014system}. \n\n\\noindent \\textit{Virtual environments}: Crowd analysis methods can be used to understand the underlying phenomenon thereby enabling us to establish mathematical models that can provide accurate simulations. These mathematical models can be further used for simulation of crowd phenomena for various applications such as computer games, inserting visual effects in film scenes and designing evacuation plans \\cite{gustafson2016mure,perez2016task}.\n\n\\noindent \\textit{Forensic search}: Crowd analysis can be used to search for suspects and victims in events such as bombing, shooting or accidents in large gatherings.  Traditional face detection and recognition algorithms can be speeded up using crowd analysis techniques which are more adept at handling such scenarios \\cite{klontz2013case,barr2014effectiveness}.\n\n\n\n\nThese variety of applications has motivated researchers across various fields to develop sophisticated methods for crowd analysis and related tasks such as counting \\citep{chan2008privacy,chan2009bayesian,chen2012feature,idrees2013multi,chan2012counting,skaug2016end,ge2009marked,idrees2013multi}, density estimation \\citep{lempitsky2010learning,chen2013cumulative,zhang2016single,zhang2015cross,pham2015count,wang2016fast,boominathan2016crowdnet}, segmentation \\citep{kang2014fully,dong2007fast}, behaviour analysis \\citep{bandini2014towards,shao2014scene,cheng2014recognizing,zhou2012understanding,zhou2015learning,yi2016l0}, tracking  \\citep{rodriguez2011density,zhu2014crowd}, scene understanding \\citep{shao2015deeply,zhou2012understanding} and anomaly detection \\citep{mahadevan2010anomaly,li2014anomaly}. Among these, crowd counting and density estimation are a set of fundamental tasks and they form basic building blocks for various other applications discussed earlier. Additionally, methods developed for crowd counting can be easily extended to counting tasks in other fields such as cell microscopy \\cite{wang2016fast,walach2016learning,lempitsky2010learning,chen2012feature}, vehicle counting \\cite{onoro2016towards}, environmental survey \\cite{french2015convolutional,zhan2008crowd}, etc.\n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{minipage}{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig3}\n\\vskip-6pt \\captionof*{figure}{(a)}\n\\end{minipage}%\n\\hfill\n\\begin{minipage}{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig4}\n\\vskip-6pt \\captionof*{figure}{(b)}\n\\end{minipage}\n\\end{center}\n\n\\begin{center}\n\\begin{minipage}{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig5}\n\\vskip-6pt \\captionof*{figure}{(c)}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{0.49\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig6}\n\\vskip-6pt \\captionof*{figure}{(d)}\n\\end{minipage}\n\\vskip-10pt \\captionof{figure}{Illustration of various crowded scenes and the associated challenges. (a) Parade (b) Musical concert (c) Public demonstration (d) Sports stadium. High clutter, overlapping of subjects, variation in scale and perspective can be observed across images.}\n\\label{fig:crowd_scenes}\n\\end{center}\n\\end{figure}\n\nOver the last few years, researchers have attempted to address the issue of crowd counting and density estimation using a variety of approaches such as detection-based counting, clustering-based counting and regression-based counting \\cite{loy2013crowd}. The initial work on regression-based methods mainly use hand-crafted features and the more recent works use Convolutional Neural Network (CNN) based approaches. The CNN-based approaches have demonstrated significant improvements over previous hand-crafted feature-based methods, thus, motivating more researchers to explore CNN-based approaches further for related crowd analysis problems. In this paper, we review various single image crowd counting and density estimation methods with a specific focus on recent CNN-based approaches. \n\n\nResearchers have attempted to provide a comprehensive survey and evaluation of existing techniques for various aspects of crowd analysis \\citep{zhan2008crowd,ferryman2014performance,junior2010crowd,li2015crowded,zitouni2016advances}.  Zhan \\textit{et al. }  \\cite{zhan2008crowd} and Junior \\textit{et al. } \\cite{junior2010crowd} were among the first ones to study and review existing methods for general crowd analysis. Li \\textit{et al. }  \\cite{li2015crowded} surveyed different methods for crowded scene analysis  tasks such as crowd motion pattern learning, crowd behavior, activity analysis and anomaly detection in crowds. More recently, Zitouni \\textit{et al. } \\cite{zitouni2016advances} evaluated existing methods across different research disciplines by inferring key statistical evidence from existing literature and provided suggestions towards the general aspects of techniques rather than any specific algorithm. While these works focussed on the general aspects of crowd analysis, researchers have studied in detail crowd counting and density estimation methods specifically  \\cite{loy2013crowd,saleh2015recent,ryan2015evaluation}. Loy \\textit{et al. } \\cite{loy2013crowd} provided a detailed description and comparison of video imagery-based crowd counting and evaluation of different methods using the same protocol. They also analyzed each processing module to identify potential bottlenecks to provide new directions for further research. In another work, Ryan \\textit{et al. } \\cite{ryan2015evaluation} presented an evaluation of regression-based methods for crowd counting across multiple datasets and provided a detailed analysis of performance of various hand-crafted features. Recently, Saleh \\textit{et al. } \\cite{saleh2015recent} surveyed two main approaches which are direct approach (i.e., object based target detection) and indirect approach (e.g. pixel-based, texture-based, and corner points based analysis). \n\n\nThough existing surveys analyze various methods for crowd analysis and counting, they however cover only traditional methods that use hand-crafted features and do not take into account the recent advancements driven primarily by CNN-based approaches \\cite{shao2015deeply,hu2016dense,zhao2016crossing,boominathan2016crowdnet,skaug2016end,walach2016learning,arteta2016counting,wang2015deep,zhang2016single,zhang2015cross,onoro2016towards,shao2016slicing}\nand creation of new challenging crowd datasets \\cite{zhang2016data,zhang2015cross,zhang2016single}. While CNN-based approaches have achieved drastically lower error rates, the creation of new datasets has enabled learning of more generalized models. To keep up with the rapidly advancing research in crowd counting, we believe it is necessary to analyze these methods in detail in order to understand the trends.  Hence, in this paper, we provide a survey of recent state-of-the-art CNN-based approaches for crowd counting and density estimation for single images.   \n\n\n\nRest of the paper is organized as follows: Section \\ref{sec:review_traditional} briefly reviews the traditional crowd counting and density estimation approaches with an emphasis on the most recent methods. This is followed by a detailed survey on CNN-based methods along with a discussion on their merits and drawbacks in Section \\ref{sec:survey_cnn}. In Section \\ref{sec:datasets_and_results}, recently published challenging datasets for crowd counting are discussed in detail along with results of the state-of-the-art methods. We discuss several promising avenues for achieving further progress in Section \\ref{sec:future_research}.  Finally, concluding remarks are made in Section \\ref{sec:conclusion}.\n\n\\section{Review of traditional approaches}\n\\label{sec:review_traditional}\nVarious approaches have been proposed to tackle the problem of crowd counting in images \\cite{idrees2013multi,chen2013cumulative,lempitsky2010learning,zhang2015cross,zhang2016single} and videos \\cite{brostow2006unsupervised,ge2009marked,rodriguez2011density,chen2015person}. \nLoy \\textit{et al. } \\cite{loy2013crowd} broadly classified traditional crowd counting methods based on the approach into the following categories: (1) Detection-based approaches, (2) Regression-based approaches, and (3) Density estimation-based approaches. \n\nSince the focus of this work is on CNN-based approaches, in this section, we briefly review the detection and regression-based approaches using hand-crafted features for the sake of completeness. In addition, we present a review of the recent traditional methods \\cite{idrees2013multi,lempitsky2010learning,pham2015count,wang2016fast,xu2016crowd} that have not been analyzed in earlier surveys. \n\n\\subsection{Detection-based approaches}\nMost of the initial research was focussed on detection style framework, where a sliding window detector is used to detect people in the scene \\cite{dollar2012pedestrian} and this information is used to count the number of people \\cite{li2008estimating}. Detection is usually performed either in the monolithic style or parts-based detection. Monolithic detection approaches \\cite{dalal2005histograms,leibe2005pedestrian,tuzel2008pedestrian,enzweiler2009monocular} typically are traditional pedestrian detection methods which train a classifier using features (such as Haar wavelets \\cite{viola2004robust}, histogram oriented gradients \\cite{dalal2005histograms}, edgelet \\cite{wu2005detection} and shapelet \\cite{sabzmeydani2007detecting}) extracted from a full body. Various learning approaches such as Support Vector Machines, boosting \\cite{viola2005detecting} and random forest \\cite{gall2011hough} have been used with varying degree of success. Though successful in low density crowd scenes, these methods are adversely affected by the presence of high density crowds. Researchers have attempted to address this issue by adopting part-based detection methods \\cite{felzenszwalb2010object,lin2001estimation,wu2007detection}, where one constructs boosted classifiers for specific body parts such as\nthe head and shoulder to estimate the people counts in a designated area \\cite{li2008estimating}. In another approach using shape learning, Zhao et al. \\cite{zhao2008segmentation} modelled humans using 3D shapes composed of ellipsoids, and employed a stochastic process to estimate the number and shape configuration that best explains a given foreground mask in a scene. Ge and Collins \\cite{ge2009marked} further extended the idea by using flexible and practical shape models.\n\n\\subsection{Regression-based approaches}\nThough parts-based and shape-based detectors were used to mitigate the issues of occlusion, these methods were not successful in the presence of extremely dense crowds and\nhigh background clutter. To overcome these issues, researchers attempted to count by regression where they learn a mapping between features extracted from local image patches to their counts  \\cite{chan2009bayesian,ryan2009crowd,chen2012feature}. By counting using regression, these methods avoid dependency on learning detectors which is a relatively complex task. These methods have two major components: low-level feature extraction and regression modelling. A variety of features such as foreground features, edge features, texture and gradient features have been used for encoding low-level information. Foreground features are extracted from foreground segments in a video using standard background subtraction techniques. Blob-based holistic features such as area, perimeter, perimeter-area ration, etc. have demonstrated encouraging results \\cite{chan2008privacy,chen2012feature,ryan2009crowd}. While these methods capture global properties of the scene, local features such as edges and texture\/gradient features such as local binary pattern (LBP), histogram oriented gradients (HOG), gray level co-occurrence matrices (GLCM) have been \nused to further improve the results. Once these global and local features are extracted, different regression techniques such as linear regression \\cite{paragios2001mrf}, piecewise linear regression \\cite{chan2008privacy}, ridge regression \\cite{chen2012feature},  Gaussian process regression and neural network \\cite{marana1998efficacy} are used to learn a mapping from low-level feature to the crowd count.\n\nIn a recent approach, Idrees \\textit{et al. } \\cite{idrees2013multi} identified that no single feature or detection method is reliable enough to provide sufficient information for accurate counting in the presence of high density crowds due to various reasons such as low resolution, severe occlusion, foreshortening and perspective.  Additionally, they observed that there exists a spatial relationship that can be used to constrain the count estimates in neighboring local regions. With these observations in mind, they proposed to extract features using different methods that capture different information. By treating densely packed crowds of individuals as irregular and non-homogeneous texture, they employed Fourier analysis along with head detections and SIFT interest-point based counting in local neighborhoods. The count estimates from this localized multi-scale analysis are then aggregated subject to global consistency constraints. The  three sources, i.e., Fourier, interest points and head detection are then combined with their respective confidences and counts at localized patches are computed independently. These local counts are then globally constrained in a multi-scale Markov Random Field (MRF) framework to get an estimate of count for the entire image.  The authors also introduced an annotated dataset (UCF\\textunderscore CC\\textunderscore 50) of 50 images containing 64000 humans.\n\nChen \\textit{et al. } \\cite{chen2013cumulative} introduced a novel cumulative attribute concept for learning a regression model when only sparse and imbalanced data are available. Considering that the challenges of inconsistent features along with sparse  and imbalanced (encountered during learning a regression function) are  related, cumulative attribute-based representation for learning a regression model is proposed. Specifically, features extracted from sparse and imbalanced image samples are mapped onto a cumulative attribute space. The method is based on the notion of discriminative attributes used for addressing sparse training data. This method is inherently capable of handling imbalanced data.\n\n\\subsection{Density estimation-based approaches}\nWhile the earlier methods were successful in addressing the issues of occlusion and clutter, most of them ignored important spatial information as they were regressing on the global count. In contrast, Lempitsky \\textit{et al. } \\cite{lempitsky2010learning} proposed to learn a linear mapping between local patch features and corresponding object density maps, thereby incorporating spatial information in the learning process. In doing so, they avoided the hard task of learning to detect and localize individual object instances by introducing a new approach of estimating image density whose integral over any region in the density map gives the count of objects within that region. The problem of learning density maps is formulated as a minimization of a regularized risk quadratic cost function. A new loss function appropriate for learning density maps is introduced. The entire problem is posed as a convex optimization task which they solve using cutting-plane optimization.\n\nObserving that it is difficult to learn a linear mapping, Pham \\textit{et al. } \\cite{pham2015count} proposed to learn a non-linear mapping between  local patch features and density maps. They used random forest regression from multiple image patches to vote for densities of multiple target objects to learn a non-linear mapping. In addition, they tackled the problem of large variation in appearance and shape between crowded image patches and non-crowded ones by proposing a crowdedness prior and they trained two different forests corresponding to this prior.  Furthermore, they were able to successfully speed up the estimation process for real-time performance by proposing an effective forest reduction that uses permutation of decision trees. Apart from achieving real-time performance, another advantage of their method is that it requires relatively less memory to build and store the forest.\n\nSimilar to the above approach, Wang and Zou \\cite{wang2016fast} identified that though existing methods are effective, they were inefficient from computational complexity point of view. To this effect, they proposed a fast method for density estimation based on subspace learning. Instead of  learning a mapping between dense features and their corresponding density maps, they learned to compute the embedding of each subspace formed by image patches. Essentially, they exploited the relationship between images and their corresponding density maps in the respective feature spaces. The feature space of image patches are clustered and examples of each subspace are collected to learn its embedding. Their assumption that local image patches and their corresponding density maps share similar local geometry enables them to learn locally linear embedding using which  the density map of an image patch can be estimated by preserving the geometry. Since, implementing locally linear embedding (LLE) is time-consuming, they divided the feature spaces  of image patches and their counterpart density maps into subspaces, and computed the embedding of each subspace formed by image patches. The density map of input patch is then estimated by simple classification and mapping with the corresponding embedding matrix. \n\nIn a more recent approach, Xu and Qiu \\cite{xu2016crowd} observed that the existing crowd density estimation methods used a smaller set of features thereby limiting their ability to perform better.  Inspired by the ability of high-dimensional features in other domains such as face recognition, they proposed to boost the performances of crowd density estimation by using a much extensive and richer set of features. However, since the regression techniques used by earlier methods (based on Gaussian process\nregression or Ridge regression) are computationally complex and are unable to process very high-dimensional features,  they used random forest as the regression model whose tree structure is intrinsically fast and scalable. Unlike traditional approaches to random forest construction, they embedded random projection in the tree nodes to combat the curse of dimensionality and to introduce randomness in the tree construction.\n\n\\begin{figure}[t]\n\\begin{center}\n\\begin{minipage}{0.95\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig7}\n\\end{minipage}%\n\\vskip-10pt\n\\captionof{figure}{Categorization of existing CNN-based approaches.}\n\\label{fig:classification}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\section{CNN-based methods}\n\\label{sec:survey_cnn}\nThe success of CNNs in numerous computer vision tasks has inspired researchers to exploit their abilities for learning non-linear functions from crowd images to their corresponding density maps or corresponding counts. A variety of CNN-based methods have been proposed in the literature. We broadly categorize these methods based on property of the networks  and training approach as shown in Fig. \\ref{fig:classification}. Based on the property of the networks, we classify the approaches into the following categories:\n\\begin{itemize}[noitemsep]\n\\item \\textbf{Basic CNNs}: Approaches that involve basic CNN layers in their networks fall into this category. These methods are amongst initial deep learning approaches for crowd counting and density estimation.\n\\item \\textbf{Scale-aware models}: The basic CNN-based approaches evolved into more sophisticated models that were robust to variations in scale. This robustness is achieved through different techniques such as multi-column or multi-resolution architectures.\n\\item \\textbf{Context-aware models}: Another set of approaches attempted to incorporate local and global contextual information present in the image into the CNN framework for achieving lower estimation errors. \n\\item \\textbf{Multi-task frameworks}: Motivated by the success of multi-task learning for various computer vision tasks, various approaches have been developed to combine crowd counting and estimation along with other tasks such as foreground-background subtraction and crowd velocity estimation.\n\\end{itemize}\n\nIn an yet another categorization, we classify the CNN-based approaches based on the inference methodology into the following two categories:\n\\begin{itemize}[noitemsep]\n\\item \\textbf{Patch-based inference}: In this approach, the CNNs are trained using patches cropped from the input images. Different methods use different crop sizes. During the prediction phase, a sliding window is run over the test image and predictions are obtained for each window and finally aggregated to obtain total count in the image. \n\\item \\textbf{Whole image-based inference}: Methods in this category perform a whole-image based inference. These methods avoid computationally expensive sliding windows. \n\\end{itemize}\nTable~\\ref{tab:classification} presents a categorization of various CNN-based crowd counting methods based on their network property and inference process.  \n\n\n\\begin{table}[htp!]\n\\centering\n\\caption{Categorization of existing CNN-based approaches.}\n\\label{tab:classification}\n\\resizebox{0.99\\linewidth}{!}\n\\begin{tabular}{|l|l|l|}\n\\hline\n                                                 & \\multicolumn{2}{c|}{Category}                                                                                            \\\\ \\hline\nMethod                                           & \\begin{tabular}[c]{@{}l@{}}Network \\\\ property\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Inference\\\\ process\\end{tabular} \\\\ \\hline\nFu \\textit{et al. } \\cite{fu2015fast}                       & Basic                                                       & Patch-based                                                \\\\ \\hline\nWang \\textit{et al. } \\cite{wang2015deep}                   & Basic                                                       & Patch-based                                                 \\\\ \\hline\nZhang \\textit{et al. } \\cite{zhang2015cross}                & Multi-task                                                  & Patch-based                                                \\\\ \\hline\nBoominathan \\textit{et al. } \\cite{boominathan2016crowdnet} & Scale-aware                                                 & Patch-based                                                \\\\ \\hline\nZhang \\textit{et al. } \\cite{zhang2016single}               & Scale-aware                                                 & Whole image-based                                                 \\\\ \\hline\nWalach and Wolf \\cite{walach2016learning}        & Basic                                                       & Patch-based                                                \\\\ \\hline\nOnoro \\textit{et al. } \\cite{onoro2016towards}              & Scale-aware                                                 & Patch-based                                                \\\\ \\hline\nShang \\textit{et al. } \\cite{skaug2016end}                  & Context-aware                                               & Whole image-based                                                 \\\\ \\hline\nSheng \\textit{et al. } \\cite{sheng2016crowd}                & Context-aware                                               & Whole image-based                                                 \\\\ \\hline\nKumagai \\textit{et al. } \\cite{kumagai2017mixture}                & Scale-aware                                               & Patch-based                                                 \\\\ \\hline\nMarsden \\textit{et al. } \\cite{marsden2016fully}                & Scale-aware                                               & Whole image-based                                                 \\\\ \\hline\nMundhenk \\textit{et al. } \\cite{mundhenk2016large}          & Basic                                                       & Patch-based                                                \\\\ \\hline\nArtetta \\textit{et al. } \\cite{arteta2016counting}          & Multi-task                                                  & Patch-based                                                \\\\ \\hline\nZhao \\textit{et al. } \\cite{zhao2016crossing}               & Multi-task                                                  & Patch-based                                                \\\\ \\hline\nSindagi \\textit{et al. } \\cite{sindagi2017cnnbased}               & Multi-task                                                  & Whole image-based                                                \\\\ \\hline\nSam \\textit{et al. } \\cite{sam2017switching}               & Scale-aware & Patch-based                                                \\\\ \\hline\nKang \\textit{et al. } \\cite{zhao2016crossing}               & Basic & Patch-based                                                \\\\ \\hline\n\\end{tabular}\n}\n\\end{table}\n\n\\subsection{Survey of CNN-based methods}\nIn this section, we review various CNN-based crowd counting and density estimation methods along with their merits and drawbacks.\n\nWang \\textit{et al. } \\cite{wang2015deep} and Fu \\textit{et al. } \\cite{fu2015fast} were among the first ones to apply CNNs for the task of crowd density estimation. Wang \\textit{et al. } proposed an end-to-end deep CNN regression model for counting people from images in extremely dense crowds. They adopted AlexNet network \\cite{krizhevsky2012imagenet} in their architecture where the final fully connected layer of 4096 neurons is replaced with a single neuron layer for predicting the count. Besides, in order to reduce false responses background like buildings and trees in the images, training data is augmented with additional negative samples whose ground truth count is set as zero. In a different approach, Fu \\textit{et al. } proposed to classify the image into one of the five classes: very high density, high density, medium density, low density and very low density instead of estimating density maps. Multi-stage ConvNet from the works of Sermanet \\textit{et al. } \\cite{sermanet2012convolutional} was adopted for better shift, scale and distortion invariance. In addition, they used a cascade of two classifiers to achieve boosting in which the first one specifically samples misclassified images whereas the second one reclassifies rejected samples. \n\nZhang \\textit{et al. } \\cite{zhang2015cross} analyzed existing methods to identify that their performance reduces drastically when applied to a new scene that is different from the training dataset. To overcome this issue, they proposed to learn a mapping from images to crowd counts and to adapt this mapping to new target scenes for cross-scene counting. To achieve this, they first learned their network by alternatively training on two objective functions: crowd count and density estimation which are related objectives. By alternatively optimizing over these objective functions one is able to obtain better local optima. In order to adapt this network to a new scene, the network is  fine-tuned using training samples that are similar to the target scene. It is important to note that the network is adapted to new target scenes without any extra label information. The overview of their approach is shown in Fig. \\ref{fig:cross_scene}. Also, in contrast to earlier methods that use the sum of Gaussian kernels centered on the locations of objects, a new method for generating ground truth density map is proposed that incorporates perspective information. In doing so, the network is able to perform perspective normalization thereby achieving robustness to scale and perspective variations. Additionally, they introduced a new dataset for the purpose of evaluating cross-scene crowd counting. The network is evaluated for cross-scene crowd counting as well as single scene crowd counting and superior results are demonstrated for both scenarios.\n\n\n\\begin{figure}[htp!]\n\\begin{center}\n\\begin{minipage}{1\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig8}\n\\end{minipage}%\n\\vskip-8pt\n\\captionof{figure}{Overview of cross scene crowd counting proposed by Zhang \\textit{et al. } \\cite{zhang2015cross}.}\n\\label{fig:cross_scene}\n\\end{center}\n\\end{figure}\n\n\nInspired by the success of cross-scene crowd counting \\cite{zhang2015cross}, Walach and Wolf \\cite{walach2016learning} performed layered boosting and selective sampling. Layered boosting involves iteratively adding CNN layers to the model such that every new layer is trained to estimate the residual error of the earlier prediction. For instance, after the first CNN layer is trained, the second CNN layer is trained on the difference between the estimation and ground truth. This layered boosting approach is based on the notion of Gradient Boosting Machines (GBM) \\cite{friedman2001greedy} which are a subset of powerful ensemble techniques.  An overview of their boosting approach is presented in Fig. \\ref{fig:learning_to_count_arch}. The other contribution made by the authors is the use of sample selection algorithm to improve the training process by reducing the effect of low quality samples such as trivial samples or outliers. According to the authors, the samples that are correctly classified early on are trivial samples. Presenting such samples for training even after the networks have learned to classify them tends to introduce bias in the network for such samples, thereby affecting its generalization performance. Another source of training inefficiency is the presence of outliers such as mislabeled samples. Apart from affecting the network's performance, these samples increase the training time. To overcome this issue, such samples are eliminated out of the training process for a number of epochs. The authors demonstrated that their method reduces the count estimation error by 20\\% to 30\\%  over existing state-of-the-art methods at that time on different datasets. \n\\begin{figure}[t]\n\\begin{center}\n\\begin{minipage}{0.7\\linewidth}\n\\includegraphics[width=1\\linewidth]{Fig9}\n\\end{minipage}%\n\\vskip-8pt\n\\captionof{figure}{Overview of learning to count using boosting by Walach and Wolf \\cite{walach2016learning}.}\n\\label{fig:learning_to_count_arch}\n\\end{center}\n\\end{figure}\n\n\nIn contrast to the above methods that use patch-based training, Shang \\textit{et al. } \\cite{skaug2016end} proposed an end-to-end count estimation method using CNNs (Fig. \\ref{fig:end_to_end_arch}). Instead of dividing the image into patches, their method takes the entire image as input and directly outputs the final crowd count. As a result, computations on overlapping regions are shared by combining multiple stages of processing leading to a reduction of complexity. The network simultaneously learns to estimate local counts and can be viewed as learning a patch level counting model which enables faster training. By doing so, contextual information is incorporated into the network, enabling it to ignore background noises and achieve better performance. The network is composed of three parts: (1) Pre-trained GoogLeNet model \\cite{szegedy2015going}, (2) Long-short time memory (LSTM) decoders for local count, and (3) Fully connected layers for the final count. The network takes an image as input and computes high-dimensional  CNN feature maps using the GoogleNet network. Local blocks in these high-dimensional features are decoded into local count using a LSTM unit. A set of fully connected layers after the LSTM unit map the local counts into global count. The two counting objectives are jointly optimized during training.\n\n\\begin{figure}[htp!]\n\\begin{center}\n\\begin{minipage}{1\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig10}\n\\end{minipage}%\n\\vskip-8pt\n\\captionof{figure}{Overview of the end-to-end counting method proposed by Shang \\textit{et al. } \\cite{skaug2016end}.  GoogLeNet is used to compute high-dimensional features which are further decoded into local counts using LSTM units. }\n\\label{fig:end_to_end_arch}\n\\end{center}\n\\end{figure}\n\nIn an effort to capture semantic information in the image, Boominathan \\textit{et al. } \\cite{boominathan2016crowdnet} combined deep and shallow  fully convolutional networks to predict the density map for a given crowd image. The combination of two networks enables one to build a model robust to non-uniform scaling of crowd and variations in perspective. Furthermore, an extensive augmentation of the training dataset is performed in two ways. Patches from the multi-scale image representation are sampled to make the system robust to scale variations. Fig. \\ref{fig:boominathan_arch} shows overview of this method. \n\n\\begin{figure}[htp!]\n\\begin{center}\n\\begin{minipage}{1\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig11}\n\\end{minipage}%\n\\vskip-8pt\n\\captionof{figure}{Overview of counting method proposed by Boominathan \\textit{et al. } \\cite{boominathan2016crowdnet}.  A deep network is used in combination with a shallow network to address scale variations across images. }\n\\label{fig:boominathan_arch}\n\\end{center}\n\\end{figure}\n\nIn another approach, Zhang \\textit{et al. } \\cite{zhang2016single} proposed a  multi-column based architecture (MCNN) for images with arbitrary crowd density and arbitrary perspective. Inspired by the success of multi-column networks for image recognition \\cite{ciregan2012multi}, the proposed method ensures robustness to large variation in object scales by constructing a network that comprises of three columns corresponding to filters with receptive fields of different sizes (large, medium, small) as shown in Fig. \\ref{fig:single_image_arch}. These different columns are designed to cater to different object scales present in the images. Additionally, a new method for generating ground truth crowd density maps is proposed. In contrast to existing methods that either use sum of Gaussian kernels with a fixed variance or perspective maps, Zhang \\textit{et al. } proposed to take into account perspective distortion by estimating spread parameter of the Gaussian kernel based on the size of the head of each person within the image. However, it is impractical to estimate head sizes and their underlying relationship with density maps. Instead they used an important property observed in high density crowd images that the head size is related to distance between the centers of two neighboring persons. The spread parameter for each person is data-adaptively determined based on its average distance to its neighbors. Note that the ground truth density maps created using this technique incorporate distortion information without the use of perspective maps. Finally, considering that existing crowd counting datasets do not cater to all the challenging situations encountered in real world scenarios, a new ShanghaiTech crowd datasets is constructed. This new dataset includes 1198 images with about 330,000 annotated heads. \n\n\n\n\\begin{figure}[htp!]\n\\begin{center}\n\\begin{minipage}{1\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig12}\n\\end{minipage}%\n\\vskip-8pt\n\\captionof{figure}{Overview of single image crowd counting via multi-column network  by Zhang \\textit{et al. } \\cite{zhang2016single}.}\n\\label{fig:single_image_arch}\n\\end{center}\n\\end{figure}\n \n\n\nSimilar to the above approach, Onoro and Sastre \\citep{onoro2016towards} developed a scale aware counting model called Hydra CNN that is able to estimate object densities in a variety of crowded scenarios without any explicit geometric information of the scene. First, a deep fully-convolutional neural network (which they call as Counting CNN) with six  convolutional layers is employed.  Motivated by the observation of earlier work \\cite{zhang2015cross,loy2013crowd} that incorporating perspective information for geometric correction of the input features results in better accuracy, geometric information is incorporated into the Counting CNN (CCNN). To this effect, they developed Hydra CNN that learns a multi-scale non-linear regression model. As shown in Fig. \\ref{fig:hydra_cnn_arch} the network consists of 3 heads and a body with each head learning features for a particular scale.  Each head of the Hydra-CNN is constructed using the CCNN model whose outputs are concatenated and fed to the body. The body consists of a set of two fully-connected layers followed by a rectified linear unit (ReLu), a dropout layer and a final fully connected layer to estimate the object density map. While the different heads extract image descriptors at different scales, the body learns a high-dimensional representation that fuses the multi-scale information provided by the heads. This network design of Hydra CNN is inspired by the work of Li et al. \\cite{li2015visual}. Finally, the network is trained with  pyramid of image patches extracted at multiple scales. The authors demonstrated through their experiments that the Hydra CNN is able to perform successfully in scenarios and datasets with significant variations in the scene. \n\n\n\\begin{figure}[htp!]\n\\begin{center}\n\\begin{minipage}{1\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig13}\n\\includegraphics[width=\\linewidth]{Fig14}\n\\end{minipage}%\n\\vskip-10pt\n\\captionof{figure}{Overview of Hydra-CNN by Onoro \\textit{et al. }  \\citep{onoro2016towards}.}\n\\label{fig:hydra_cnn_arch}\n\\end{center}\n\\end{figure}\n\n\nInstead of training all regressors of a multi-column network \\cite{zhang2016single} on all the input patches, Sam \\textit{et al. } \\cite{sam2017switching} argue that better performance is obtained by training regressors with a particular set of training patches by leveraging variation of crowd density within an image. To this end, they proposed a switching CNN that cleverly selects an optimal regressor suited for a particular input patch. As shown in Fig. \\ref{fig:switching_cnn}, the proposed network consists of multiple independent regressors similar to multi-column network     \\cite{zhang2016single} with different receptive fields and a switch classifier. The switch classifier is trained to select the optimal regressor for a particular input patch. Independent CNN crowd density regressors are trained on patches sampled from a grid in a given crowd scene. The switch classifier and the independent regressors are alternatively trained. The authors describe multiple stages of training their network. First, the independent regressors are pretrained on image patches to minimize the Euclidean distance between the estimated density map and ground truth. This is followed by a differential training stage where, the count error is factored in to improve the counting performance by back-propagating a regressor with the minimum count error for a given training patch. After training the multiple regressors, a switch classifier based on VGG-16 architecture \\cite{simonyan2014very} is trained to select an optimal regressor for accurate counting. Finally, the switch classifier and CNN regressors are co-adapted in the coupled training stage.\n\n\n\\begin{figure}[htp!]\n\\begin{center}\n\\begin{minipage}{1\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig32}\n\\end{minipage}%\n\\vskip-8pt\n\\captionof{figure}{Overview of Switching CNN by Sam \\textit{et al. } \\cite{sam2017switching}.}\n\\label{fig:switching_cnn}\n\\end{center}\n\\end{figure}\n \n\nWhile the above methods concentrated on incorporating scale information in the network, Sheng \\textit{et al. } in \\cite{sheng2016crowd} proposed to integrate semantic information by learning locality-aware feature sets. Noting that earlier methods that use hand-crafted features ignored key semantic  and spatial information, the authors proposed a new image representation which incorporates semantic attributes as well as spatial cues to improve the discriminative power of feature representations. They defined semantic attributes at the pixel level and learned  semantic feature maps via deep CNN. The spatial information in the image is encoded using locality-aware features in the semantic attribute feature map space. The locality-aware features (LAF) are built on the idea of spatial pyramids on neighboring patches thereby encoding spatial context and local information.  The local descriptors from adjacent cells are then encoded into image representations using weighted VLAD encoding method. \n\n\n\n\nSimilar to \\cite{zhang2016single,onoro2016towards}, Kumagai \\textit{et al. } \\cite{kumagai2017mixture}, based on the observation that a single predictor is insufficient to appropriately predict the count in the presence of large appearance changes, proposed a Mixture of CNNs (MoCNN) that are specialized to a different scene appearances. As shown in Fig. \\ref{fig:mixture_cnn_arch}, the architecture consists of a mixture of expert CNNs and a gating CNN that adaptively selects the appropriate CNN among the experts according to the appearance of the input image. For prediction, the expert CNNs predict crowd count in the image while the gating CNN predicts appropriate probabilities for each of the expert CNNs. These probabilities are further used as weighting factors to compute the weighted average of the counts predicted by all the expert CNNs.\n\n\n\\begin{figure}[htp!]\n\\begin{center}\n\\begin{minipage}{.8\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig15}\n\\end{minipage}%\n\\vskip-8pt\n\\captionof{figure}{Overview of MoC (Mixture of CNN) for crowd counting by Kumagai \\textit{et al. } \\cite{kumagai2017mixture}.}\n\\label{fig:mixture_cnn_arch}\n\\end{center}\n\\end{figure}\n\n\n\n\nMotivated by the success of scale aware models \\cite{zhang2016single,onoro2016towards}, Marsden \\textit{et al. } \\cite{marsden2016fully} proposed to incorporate scale into the models with much less number of model parameters.  Observing that the earlier scale aware models \\cite{zhang2016single,onoro2016towards} are difficult to optimize and are computationally complex, Marsden \\textit{et al. } \\cite{marsden2016fully} proposed a single column fully convolutional network where the scale information is incorporated into the model using a simple yet effective multi-scale averaging step during prediction without any increase in the model parameters. The method addresses the issues of scale and perspective changes by feeding multiple scales of test image into the network during prediction phase. The crowd count is estimated for each scale and the final count is obtained by taking an average of all the estimates. Additionally,  a new training set augmentation scheme is developed to reduce redundancy among the training samples. In contrast to the earlier methods that use  randomly cropped patches with high degree of overlap, the training set in this work is constructed using the four image quadrants as well as their horizontal flips ensuring no overlap. This technique avoids potential overfit when the network is continuously exposed to the same set of pixels during training, thereby improving the generalization performance of the network. In addition, the generalization performance of the proposed method is studied by measuring cross dataset performance. \n\n\n\\begin{figure}[htp!]\n\\begin{center}\n\\begin{minipage}{0.8\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig16}\n\\end{minipage}%\n\\vskip-8pt\n\\captionof{figure}{Overview of Fully Convolutional Network  for crowd counting by Marsden \\textit{et al. } \\cite{marsden2016fully}.}\n\\label{fig:congested_arch}\n\\end{center}\n\\end{figure}\n\n\nInspired by the superior results achieved by simultaneous learning of related tasks \\cite{ranjan2016hyperface,yu2017iprivacy}, Sindagi \\textit{et al. } \\cite{sindagi2017cnnbased} and Marsden et al.  \\cite{marsden2017resnetcrowd} explored multi-task learning to boost individual task performance. Marsden et al. \\cite{marsden2017resnetcrowd} proposed a Resnet-18 \\cite{he2016deep} based architecture for simultaneous crowd counting, violent behaviour detection and crowd density level classification. The network consists of initial 5 convolutional layers of Resnet18 including batch normalisation layers and skip connections form the primary module. The convolutional layers are followed by a set of task specific layers. Finally, sum of all the losses corresponding to different tasks is minimized. Additionally, the authors constructed a new 100 image dataset specifically designed for multi-task learning of crowd count and behaviour. In a different approach, Sindagi \\textit{et al. } \\cite{sindagi2017cnnbased} proposed a cascaded CNN architecture to incorporate learning of a high-level prior to boost the density estimation performance. Inspired by \\cite{chen2016cascaded}, the proposed network simultaneously learns to classify the crowd count into various density levels and estimate density map (as shown in Fig. \\ref{fig:cascaded_mtcnn}). Classifying crowd count into various levels is equivalent to coarsely estimating the total count in the image thereby incorporating a high-level prior into the density estimation network. This enables the layers in the network to learn globally relevant discriminative features. Additionally, in contrast to most recent work, they make use of transposed convolutional layers to generate high resolution density maps.\n\n\n\\begin{figure}[htp!]\n\\begin{center}\n\\begin{minipage}{1\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig33}\n\\end{minipage}%\n\\vskip-8pt\n\\captionof{figure}{Overview of Cascaded Multi-task CNN by Sindagi \\textit{et al. } \\cite{sindagi2017cnnbased}.}\n\\label{fig:cascaded_mtcnn}\n\\end{center}\n\\end{figure}\n\n\n\nIn a recent work, Kang \\textit{et al. } \\cite{kang2017beyond} explored maps generated by density estimation methods for the purpose of various crowd analysis tasks such as counting, detection and tracking. They performed a detailed analysis of the effect of using full-resolution density maps on the performance of these tasks. They demonstrated through their experiments that full resolution density maps improved the performance of localization  tasks such as detection and tracking. Two different approaches are considered for generating full-resolution maps. In the first approach, a sliding window based CNN regressor is used for pixel-wise density prediction. In the second approach, Fully Convolutional Networks \\cite{long2015fully} along with skip connections are used to learning a non-linear mapping between input image and the corresponding density map.\n\n\n\n\nIn a slightly different application context of counting, Mundhenk \\textit{et al. } \\cite{mundhenk2016large} and Arteta \\textit{et al. } \\cite{arteta2016counting} proposed to count different types of objects such as cars and penguins respectively. Mundhenk \\textit{et al. } \\cite{mundhenk2016large} addressed the problem of automated counting of automobiles from satellite\/aerial platforms. Their primary contribution is the creation of a large diverse set of cars from overhead images. Along with the large dataset, they present a deep CNN-based network to recognize the number of cars in patches. The network is trained in a classification setting where the output of the network is a class that is indicative of the number of objects in the input image. Also, they incorporated contextual information by including additional regions around the cars in the training patches. Three different networks based on AlexNet \\cite{krizhevsky2012imagenet}, GoogLeNet \\cite{szegedy2015going} and ResNet \\cite{he2016deep} with Inception are evaluated. For a different application of counting penguins in images, Arteta \\textit{et al. } \\cite{arteta2016counting} proposed a deep multi-task architecture for accurate counting even in the presence of labeling errors. The network is trained in a multi-task setting where, the tasks of foreground-background subtraction and uncertainty estimation along with counting are jointly learned. The authors demonstrated that the joint learning especially helps in learning a counting model that is robust to labeling errors. Additionally, they exploited scale variations and count variability across the annotations  to incorporate scale information of the object and prediction of annotation difficulty respectively into the model. The network was evaluated on a newly created Penguin dataset. \n\n\nZhao \\textit{et al. } addressed a higher level cognitive task of counting people that cross a line in \\citep{zhao2016crossing}. Though the task is a video-based application, it comprises of a CNN-based model that is trained with pixel-level supervision maps similar to single image crowd density estimation methods, making it a relevant approach to be included in this article. Their method consists of a two-phase training scheme (as shown in Fig. \\ref{fig:crossing_line_arch}) that decomposes  original counting problem into two sub-problems: estimating crowd density map and crowd velocity map where  the two tasks share the initial set of layers enabling them to learn more effectively. The estimated crowd density and crowd velocity maps are then multiplied element-wise to generate the crowd counting maps. Additionally, they contributed a large-scale dataset for evaluating crossing-line crowd counting algorithms, which includes 5 different scenes, 3,100 annotated frames and 5,900 annotated pedestrians. \n\n\n\\begin{figure}[htp!]\n\\begin{center}\n\\begin{minipage}{1\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig17}\n\\end{minipage}%\n\\vskip-8pt\n\\captionof{figure}{Overview of the method proposed by Zhao \\textit{et al. } \\cite{zhao2016crossing} for counting people crossing a line.}\n\\label{fig:crossing_line_arch}\n\\end{center}\n\\end{figure}\n\n\\section{Discussion}\n\nWith a variety of methods discussed in Section \\ref{sec:survey_cnn}, we analyze various advantages and disadvantages of the broad approaches followed by these methods in this section. \n\nZhang \\textit{et al. } \\cite{zhang2015cross} were among the first ones to address the problem of adapting models to new unlabelled datasets using a simple and effective method based on finding similar patches across datasets. However, their method is heavily dependent on accurate perspective maps which may not be necessarily available for all the datasets. Additionally, the use of 72$\\times$72 sized patches for training and evaluation ignores global context which is necessary for accurate estimation of count. Walach \\textit{et al. } \\cite{walach2016learning} successfully addressed training inefficiencies in earlier methods using a layered boosting approach and a simple sample selection method. However, similar to Zhang \\textit{et al. } \\cite{zhang2015cross}, their method involves patch-based training and evaluation resulting in loss of global context information along with inefficiency during evaluation due to the use of a sliding window approach. Additionally, these methods tend to ignore scale variance among the dataset assuming that their models will implicitly learn the invariance.\n\nIn an effort to explicitly model scale invariance, several methods involving combination of networks were proposed (\\cite{zhang2016single,onoro2016towards,sam2017switching,kumagai2017mixture,boominathan2016crowdnet}). \nWhile these methods demonstrated significant improvements in the performance using multiple column networks and a combination of deep and shallow networks, the invariance achieved is limited by the number of columns present in the network and receptive field sizes which are chosen based on the scales present in the dataset. Additionally, these methods do not explicitly model global context information which is crucial for a task such as crowd counting. In a different approach, Marsden \\textit{et al. } \\cite{marsden2016fully} attempt to address the scale issue by performing a multi-scale averaging during the prediction phase. While being simple and effective, it results in an inefficient inference stage. Additionally, these methods do not explicitly encode global context present in an image which can be crucial for improving the count performance. To this end, few approaches model local and global context \\cite{sheng2016crowd,skaug2016end} by considering key spatial and semantic information present in the image. \n\nIn an entirely different approach, few methods \\cite{marsden2017resnetcrowd,sindagi2017cnnbased} take advantage of multi-task learning and incorporate high-level priors into the network. For instance, Sindagi \\textit{et al. } \\cite{sindagi2017cnnbased} simultaneously learn density estimation and a high-level prior in the form of crowd count classification. While they demonstrated high performance gain by learning an additional task of crowd density level classification, the number of density levels is dataset dependent and it needs to be carefully chosen based on the density levels present in the dataset. \n\n\n\n\n\n\n\n\\section{Datasets and results}\n\\label{sec:datasets_and_results}\nA variety of datasets have been created over the last few years driving researchers to create models with better generalization abilities. While the earlier datasets usually contain low density crowd images, the most recent ones focus on high density crowd thus posing numerous challenges such as scale variations, clutter and severe occlusion. The creation of these large scale datasets has motivated recent approaches to develop methods that cater to such challenges. In this section, we review five key datasets \\cite{chan2008privacy,chen2012feature,idrees2013multi,zhang2015cross,zhang2016single} followed by a discussion on the results of CNN-based approaches and recent traditional methods that were not included in the earlier surveys. \n\n\\begin{table*}[t!]\n\\caption{Summary of various datasets.}\n\\begin{center}\n\\vskip-15pt\\begin{tabular}{|l|c|c|c|c|c|c|}\n\\hline\nDataset & No. of images & Resolution & Min & Ave & Max & Total count \\\\\n\\hlin\nUCSD \\cite{chan2008privacy} & 2000 & 158x238 & 11 & 25 & 46 & 49,885\\\\\n\\hline\nMall \\cite{chen2012feature} & 2000 & 320x240 & 13 & - & 53 & 62,325\\\\\n\\hline\nUCF\\textunderscore CC\\textunderscore 50 \\cite{idrees2013multi} & 50 & Varied & 94 & 1279 & 4543 & 63,974\\\\\n\\hline\nWorldExpo '10 \\cite{zhang2016data,zhang2015cross} & 3980 & 576x720 & 1 & 50 & 253 & 199,923\\\\\n\\hline\nShanghaiTech Part A \\cite{zhang2016single} & 482 & Varied & 33 & 501 & 3139 & 241,677\\\\\n\\hline\nShanghaiTech Part B \\cite{zhang2016single}& 716 & 768x1024 & 9 & 123 & 578 & 88,488\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:datasetsummary}\n\\end{table*}\n\n\\begin{figure*}[ht!]\n\\begin{center}\n\\begin{minipage}{0.163\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig18}\n\\captionof*{figure}{(a)}\n\\end{minipage}%\n\\hfill\n\\begin{minipage}{0.163\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig19}\n\\captionof*{figure}{(b)}\n\\end{minipage}\n\\begin{minipage}{0.163\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig20}\n \\captionof*{figure}{(c)}\n\\end{minipage}\n\\begin{minipage}{0.163\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig21}\n\\captionof*{figure}{(d)}\n\\end{minipage}\n\\begin{minipage}{0.163\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig22}\n\\captionof*{figure}{(e)}\n\\end{minipage}\n\\begin{minipage}{0.163\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig23}\n\\captionof*{figure}{(f)}\n\\end{minipage}\n\\end{center}\n\\vskip-10pt \\captionof{figure}{Sample images from various datasets. (a) UCSD \\cite{chan2008privacy} (b) Mall \\cite{chen2012feature} (c)  UCF\\textunderscore CC\\textunderscore 50 \\cite{idrees2013multi} (d) WorldExpo '10 \\cite{zhang2015cross} (e) Shanghai Tech Part A \\cite{zhang2016single} (f) SHanghai Tech Part B \\cite{zhang2016single}. It can be observed that in the case of UCSD and Mall dataset , the images come from the same video sequence providing no variation in perspective across images.}\n\\label{fig:dataset}\n\\end{figure*}\n\n\n\\subsection{Datasets}\n\\textbf{UCSD dataset}: The UCSD dataset \\cite{chan2008privacy} was among the first datasets to be created for counting people. The dataset was collected from a video camera at a pedestrian walkway. The dataset consists of 2000 frames of size 238$\\times$158 from a video sequence along with ground truth annotations of each pedestrian in every fifth frame. For the rest of the frames, linear interpolation is used to create the annotations. A region-of-interest is also provided to ignore unnecessary moving objects such as trees. The dataset contains a total of 49,885 pedestrian instances and it is split into training and test set. While the training set contains frames with indices 600 to 1399, the test set contains the remaining 1200 images. This dataset has relatively low density crowd with an average of around 15 people in a frame and since the dataset was collected from a single location, there is no variation in the scene perspective across images. \n\\linebreak\n\n\n\\noindent \\textbf{Mall dataset}: Considering little variation in the scene type in the UCSD dataset, Chen \\textit{et al. } in \\cite{chen2012feature} collected a new Mall dataset with diverse illumination conditions and crowd densities. The dataset was collected using a surveillance camera installed in a shopping mall. Along with having various density levels, it also has different activity patterns (static and moving crowds). Additionally, the scene contained in the dataset has severe perspective distortion resulting in large variations in size and appearance of objects. The dataset also presents the challenge of severe occlusions caused by the scene objects, e.g.stall, indoor plants along the walking path. The video sequence in the dataset consists of 2000 frames of size 320$\\times$240 with 6000 instances of labelled pedestrians. The first 800 frames are used for training and the remaining 1200 frames are used for evaluation. In comparison to the UCSD dataset, the Mall dataset has relatively higher crowd density images. However, both the datasets do not have any variation in the scene perspective across images since they are a part of a single continuous video sequence. \n\\linebreak\n\n\\noindent \\textbf{UCF\\textunderscore CC\\textunderscore 50 dataset}: The UCF\\textunderscore CC\\textunderscore 50 \\cite{idrees2013multi} is the first truly challenging dataset constructed to include a wide range of densities and diverse scenes with varying perspective distortion. The dataset was created from publicly available web images. In order to capture diversity in the scene types, the authors collected images with different tags such as concerts, protests, stadiums and marathons. It contains a total of 50 images of varying resolutions with an average of 1280 individuals per image. A total of 63075 individuals were labelled in the entire dataset. The number of individuals varies from 94 to 4543 indicating a large variation across the images. The only drawback of this dataset is that only a limited number of images are available for training and evaluation. Considering the low number of images, the authors defined a cross-validation protocol for training and testing their approach where the dataset was divided into sets of 10 and a five fold cross-validation is performed. The challenges posed by this dataset are so enormous that even the results of recent CNN-based state-of-the-art approaches on this dataset are far from optimal.\n\\linebreak\n\n\\noindent \\textbf{WorldExpo '10 dataset}: Since some of the earlier approaches and datasets focussed primarily on single scene counting, Zhang \\textit{et al. } \\cite{zhang2015cross} introduced a dataset for the purpose of cross-scene crowd counting.  The authors attempted to perform a data-driven cross-scene crowd counting for which they collected a new large-scale dataset that includes 1132 annotated video sequences captured by 108 surveillance cameras, all from Shanghai 2010 WorldExpo event. Large diversity in the scene types is ensured by collecting videos from cameras having disjoint bird views. The dataset consists of a total of 3980 frames of size 576 $\\times$ 720 with 199923 labelled pedestrians. The dataset is split into two parts: training set consisting of 1,127 one-minute long video sequences from 103 scenes and test set consisting of 5 one-hour long video sequences from 5 different scenes. Each test scene consists of 120 labelled frames with the crowd count varying from 1 to 220. Though an attempt is made to capture diverse scenes with varying density levels, the diversity is limited to only 5 scenes in the test set and the maximum crowd count is limited to 220. Hence, the dataset is not sufficient enough for evaluating  approaches designed for extremely dense crowds in a variety of scenes. \n\\linebreak\n\n\n\\begin{table*}[htp!]\n\\centering\n\\caption{Comparison of results on various datasets. The CNN-based approaches provide significant improvements over traditional approaches that rely on hand-crafted representations. Further, among the CNN-based methods, scale aware and context aware approaches tend to achieve lower count error.}\n\\label{tab:results}\n\\resizebox{0.70\\textwidth}{!}{%\n\\begin{tabular}{|c|l|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n\\multicolumn{1}{|l|}{}                                                                        & Dataset                                                                                                   & \\multicolumn{2}{c|}{UCSD}                 & \\multicolumn{2}{c|}{Mall}                 & \\multicolumn{2}{c|}{UCF CC 50}              & \\multicolumn{2}{c|}{\\begin{tabular}[c]{@{}c@{}}WorldExpo\\\\  '10\\end{tabular}} & \\multicolumn{2}{c|}{\\begin{tabular}[c]{@{}c@{}}Shanghai\\\\ Tech-A\\end{tabular}} & \\multicolumn{2}{c|}{\\begin{tabular}[c]{@{}c@{}}Shanghai\\\\ Tech-B\\end{tabular}} \\\\ \\hline\n\\multicolumn{1}{|l|}{\\begin{tabular}[c]{@{}l@{}}Approach\\\\ type\\end{tabular}}                 & Method                                                                                                    & MAE                 & MSE                 & MAE                 & MSE                 & MAE                  & MSE                  & MAE                                           & MSE                           & MAE                                    & MSE                                   & MAE                                    & MSE                                   \\\\ \\hline\n\\multirow{5}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{4.5cm}{\\centering Traditional approaches}}} & \\begin{tabular}[c]{@{}l@{}}Multi-source multi-scale\\\\ Idrees \\textit{et al. } \\cite{idrees2013multi}\\end{tabular}    &                     &                     &                     &                     & 468.0                & 590.3                &                                               &                               &                                        &                                       &                                        &                                       \\\\ \\cline{2-14} \n                                                                                             \n                                                                                             & \\begin{tabular}[c]{@{}l@{}}Cumulative Attributes\\\\ Chen \\textit{et al. } \\cite{chen2013cumulative}\\end{tabular}   & 2.07                 &   6.86                  &  3.43                   &     17.07                &                 &                &                                               &                               &                                        &                                       &                                        &                                       \\\\ \\cline{2-14} \n                                                                                              & \\begin{tabular}[c]{@{}l@{}}Density learning\\\\ Lempitsky \\textit{et al. } \\cite{lempitsky2010learning}\\end{tabular}   & 1.7                 &                     &                     &                     & 493.4                & 487.1                &                                               &                               &                                        &                                       &                                        &                                       \\\\ \\cline{2-14} \n                                                                                              & \\begin{tabular}[c]{@{}l@{}}Count forest\\\\ Pham \\textit{et al. } \\cite{pham2015count}\\end{tabular}                    & 1.61                & 4.40                & 2.5                 & 10.0                &                      &                      &                                               &                               &                                        &                                       &                                        &                                       \\\\ \\cline{2-14} \n                                                                                              & \\begin{tabular}[c]{@{}l@{}}Exemplar density\\\\ Wang \\textit{et al. } \\cite{wang2016fast}\\end{tabular}                 & 1.98                & 1.82                & 2.74                & {\\underline \\textbf{2.10}} &                      &                      &                                               &                               &                                        &                                       &                                        &                                       \\\\ \\cline{2-14} \n                                                                                              & \\begin{tabular}[c]{@{}l@{}}Random projection forest\\\\ Xu \\textit{et al. } \\cite{xu2016crowd}\\end{tabular}            & 1.90                & 6.01                & 3.22                & 15.5                &                      &                      &                                               &                               &                                        &                                       &                                        &                                       \\\\ \\hline\n\\multirow{7}{*}{\\rotatebox[origin=c]{90}{\\parbox[c]{9.5cm}{\\centering CNN-based approaches}}}   & \\begin{tabular}[c]{@{}l@{}}Cross-scene\\\\ Zhang \\textit{et al. } \\cite{zhang2015cross}\\end{tabular}                   & 1.60                & 3.31                &                     &                     & 467.0                & 498.5                & 12.9                                          &                               & 181.8                                  & 277.7                                 & 32.0                                   & 49.8                                  \\\\ \\cline{2-14} \n                                                                                              & \\begin{tabular}[c]{@{}l@{}}Deep + shallow\\\\ Boominathan \\textit{et al. } \\cite{boominathan2016crowdnet}\\end{tabular} &                     &                     &                     &                     & 452.5                &                      &                                               &                               &                                        &                                       &                                        &                                       \\\\ \\cline{2-14} \n                                                                                              & \\begin{tabular}[c]{@{}l@{}}M-CNN\\\\ Zhang \\textit{et al. } \\cite{zhang2016single}\\end{tabular}                        & {\\underline \\textbf{1.07}} & {\\underline \\textbf{1.35}} &                     &                     & 377.6                & 509.1                & 11.6                           &                               & 110.2                  & 173.2                  & 26.4                  & 41.3                  \\\\ \\cline{2-14} \n                                                                                              & \\begin{tabular}[c]{@{}l@{}}CNN-boosting\\\\ Walach and Wolf \\cite{walach2016learning}\\end{tabular}          & 1.10                &                     & {\\underline \\textbf{2.01}} &                     & 364.4                &                      &                                               &                               &                                        &                                       &                                        &                                       \\\\ \\cline{2-14} \n                                                                                              & \\begin{tabular}[c]{@{}l@{}}Hydra-CNN\\\\ Onoro \\textit{et al. } \\cite{onoro2016towards}\\end{tabular}                   &                     &                     &                     &                     & 333.7                & 425.2 &                                               &                               &                                        &                                       &                                        &                                       \\\\ \\cline{2-14} \n                                                                                              & \\begin{tabular}[c]{@{}l@{}}Joint local \\& global count\\\\ Shang \\textit{et al. } \\cite{skaug2016end}\\end{tabular}     &                     &                     &                     &                     & {\\underline \\textbf{270.3}} &                      & 11.7                                          &                               &                                        &                                       &                                        &                                       \\\\ \\cline{2-14} \n                                                                                              \n                                                                                                                                                                                           \n                                                                                             & \\begin{tabular}[c]{@{}l@{}}MoCNN \\\\ Kumagai \\textit{et al. } \\cite{kumagai2017mixture}\\end{tabular}     &                     &                     &     2.75                &                    13.4 & 361.7 & 493.3                      &                                           &                               &                                        &                                       &                                        &  \n                                                                                                                              \\\\ \\cline{2-14} \n                                                                                                                              \n                                                                                                                                                                                                                           & \\begin{tabular}[c]{@{}l@{}}FCN \\\\ Marsden \\textit{et al. } \\cite{marsden2016fully}\\end{tabular}     &                     &                     &                     &                     & 338.6 & 424.5                      &                                           &                               & 126.5                                        &       173.5                                &       23.76                                 &  33.12\n                                                                                                                              \\\\ \\cline{2-14} \n                                                                                                                              \n                                                                                                                              & \\begin{tabular}[c]{@{}l@{}}CNN-pixel \\\\ Kang \\textit{et al. } \\cite{kang2017beyond}\\end{tabular}     &         1.12            &        2.06             &                     &                     & 406.2 & 404.0                      &                                          13.4 &                               &                                         &                                  &        & \n                                                                                                                              \\\\ \\cline{2-14} \n                                                                                                                              \n                                                                                                                              & \\begin{tabular}[c]{@{}l@{}}Weighted V-LAD\\\\ Sheng \\textit{et al. } \\cite{sheng2016crowd}\\end{tabular}     & 2.86                    &    13.0                &      2.41               &                     9.12 &  &   &                                            &                               &                                       &                                   &       & \n                                                                                                                              \\\\ \\cline{2-14}\n                                                                                                                              \n                                                                                                                                                                                                                                                            & \\begin{tabular}[c]{@{}l@{}}Cascaded-MTL\\\\ Sindagi \\textit{et al. } \\cite{sindagi2017cnnbased}\\end{tabular}     &                     &                    &                     &                     & 322.8 & {\\underline  \\textbf{341.4                    }}  &                                            &                               &    101.3                                     &                                 152.4 &      {\\underline  \\textbf{20.0}}  & {\\underline  \\textbf{31.1}}\n                                                                                                                              \\\\ \\cline{2-14}                                                                                                                              \n                                                                                                                                                                                                                                                            \n                                                                                             \n                                                                                             \n                                                                                              & \\begin{tabular}[c]{@{}l@{}}Switching-CNN\\\\ Sam \\textit{et al. } \\cite{sam2017switching}\\end{tabular}                & 1.62 & 2.10 &  &  &                     318.1 &        439.2              &                            {\\underline  \\textbf{9.4}}                   &                               &                         {\\underline  \\textbf{90.4}   }          &                                     {\\underline  \\textbf{ 135.0}} &   21.6                                     &                                   33.4 \\\\ \\hline\n                                                                                             \n                                                                                             \n                                                                                              \n                                                                                              \n\\end{tabular}%\n}\n\\end{table*}\n\n\n\\noindent \\textbf{Shanghai Tech dataset}: Zhang \\textit{et al. } \\cite{zhang2016single} introduced a new large-scale crowd counting dataset consisting of 1198 images with 330,165 annotated heads. The dataset is among the largest ones in terms of the number of annotated people and it contains two parts: Part A and Part B. Part A consists of 482 images that are randomly chosen from the Internet whereas Part B consists of images taken from the streets of metropolitan areas in Shanghai. Part A has considerably larger density images as compared to Part B. Both the parts are further divided into training and evaluation sets. The training and test of Part A has 300 and 182 images, respectively, whereas that of Part B has 400 and 316 images, respectively. The dataset successfully attempts to create a challenging dataset with diverse scene types and varying density levels. However, the number of images for various density levels are not uniform making the training and evaluation biased towards low density levels. Nevertheless, the complexities present in this dataset such as varying scales and perspective distortion has created new opportunities for more complex CNN network designs. \n\nSample images from the five datasets are shown in Fig. \\ref{fig:dataset}. The datasets are also summarized in Table \\ref{tab:datasetsummary}. It can be observed that the UCSD and the Mall dataset have relatively low density images and typically focus on single scene type. In contrast, the other datasets have significant variations in the density levels along with different perspectives across images. \n\n\\subsection{Discussion on results}\nResults of the recent traditional approaches along with CNN-based methods are tabulated in Table \\ref{tab:results}. The count estimation errors are reported directly from the respective original works. The following standard metrics are used to compare different methods:\n\\begin{equation}\nMAE = \\frac{1}{N}\\sum_{i=1}^{N}|y_i-y'_i|,\n\\end{equation}\n\\begin{equation}\nMSE = \\sqrt{\\frac{1}{N}\\sum_{i=1}^{N}|y_i-y'_i|^2},\n\\end{equation}\nwhere MAE is mean absolute error, MSE is mean squared error, $N$ is the number of test samples, $y_i$ is the ground truth count and $y'_i$ is the estimated count corresponding to the $i^{th}$ sample. We make the following observations regarding the results:\n\n\\begin{itemize}[noitemsep]\n\\item In general, CNN-based methods outperform the traditional approaches across all datasets. \n\\item While the CNN-based methods are especially effective in large density crowds with a diverse scene conditions, the traditional approaches suffer from high error rates in such scenarios. \n\\item Among the CNN-based methods, most performance improvement is achieved by scale-aware and context-aware models. It can be observed from Table \\ref{tab:results} that a  reduction in count error is largely driven by the increase in the complexity of CNN models (due to addition of context and scale information). \n\\item While the multi-column CNN architecture \\cite{zhang2016single} achieves the state-of-the-art results on 3 datasets: UCSD, WorldExpo '10 and ShanghaiTech, the CNN-boosting approach by \\cite{walach2016learning} achieves the best results on the Mall dataset. The best results on the UCF\\textunderscore CC\\textunderscore 50 dataset are achieved by joint local and global count approach \\cite{skaug2016end} and Hydra-CNN \\cite{onoro2016towards}. \n\\item The work in \\cite{walach2016learning} suggests that layered boosting can achieve performances that are comparable to scale aware models. \n\\item The improvements obtained by selective sampling in \\cite{wang2015deep} and \\cite{walach2016learning} suggests that it helps to obtain unbiased performance. \n\\item Whole image-based methods such as Zhang \\textit{et al. } \\cite{zhang2016single} and Shang \\textit{et al. } \\cite{skaug2016end} are less computationally complex from the prediction point of view and they have proved to achieve better results over patch-based techniques. \n\\item Finally, techniques such as layered boosting and selective sampling \\cite{onoro2016towards,wang2016fast} not only improve the estimation error but also reduce the training time significantly. \n\\end{itemize}\n\n\n\n\n\n\n\\begin{figure}[htp!]\n\\begin{center}\n\\begin{minipage}{0.33\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig24}\n\\end{minipage}%\n\\hfill\n\\begin{minipage}{0.33\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig25}\n\\end{minipage}\n\\begin{minipage}{0.33\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig26}\n\\end{minipage}\n\\end{center}\n\n\\begin{center}\n\\begin{minipage}{0.33\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig27}\n\\captionof*{figure}{(a)}\n\\end{minipage}%\n\\hfill\n\\begin{minipage}{0.33\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig28}\n\\captionof*{figure}{(b)}\n\\end{minipage}\n\\begin{minipage}{0.33\\linewidth}\n\\includegraphics[width=\\linewidth]{Fig29}\n\\captionof*{figure}{(c)}\n\\end{minipage}\n\\end{center}\n\\vskip-10pt \\captionof{figure}{Results of Zhang \\textit{et al. } \\cite{zhang2016single} on ShanghaiTech dataset. (a) Input image(b) Ground-truth density map (c) Estimated density maps. It can be observed that though the method is able to accurate estimation of crowd count, the estimated density maps are of poor quality.}\n\\label{fig:quality}\n\\end{figure}\n\n\n\\begin{figure}[htp!]\n\\begin{center}\n\\begin{minipage}{1\\linewidth}\n\\includegraphics[width=0.47\\linewidth]{Fig30}\n\\includegraphics[width=0.47\\linewidth]{Fig31}\n\\end{minipage}%\n\\vskip-8pt\n\\captionof{figure}{Distribution of crowd counts in ShanghaiTech dataset. It can be observed that the dataset is highly imbalanced. }\n\\label{fig:shanghaitech}\n\\end{center}\n\\end{figure}\n\n\\section{Future research directions}\n\\label{sec:future_research}\n\nBased on the analysis of various methods and results from Section \\ref{sec:survey_cnn} and \\ref{sec:datasets_and_results} and the trend of other developments in computer vision, we believe that CNN-based deeper architectures will dominate further research in the field of crowd counting and density estimation. We make the following observations regarding future trends in research on crowd counting:\n\n\\begin{enumerate}[noitemsep]\n\\item Given the requirement of large datasets for training deep networks, collection of large scale datasets (especially for extremely dense crowds) is essential. Though many datasets exist currently, only one of them (The UCF\\textunderscore CC\\textunderscore 50 \\cite{idrees2013multi}) caters to large density crowds. However, the size of the dataset is too small for training deeper networks. Though Shanghai Tech \\cite{zhang2016single}) attempts to capture large density crowds, the number of images per density level is non-uniform with a large number of images available for low density levels and very few samples for high density levels (as shown in Fig. \\ref{fig:shanghaitech}). \n\n\\item Considering the difficulty of training deep networks for new scenes, it would be important to explore how to leverage from models trained on existing sources. Most of the existing methods retrain their models on a new scene and it is impractical to do so in real world scenarios as it would be expensive to obtain annotations for every new scene. Zhang \\textit{et al. } \\cite{zhang2015cross} attempted to address this issue by performing a data driven training without the need of labelled data for new scenes. In an another approach, Liu \\textit{et al. } \\cite{liu2015bayesian} considered the problem of transfer learning for crowd counting. A model adaptation technique for Gaussian process counting model was introduced. Considering the source model as a prior and the target dataset as a set of observations, the components are combined into a predictive distribution that captures information in both the source and target datasets. However, the idea of transfer learning or domain adaptation \\cite{VMP_SPM_DA_2015} for crowd scenes is relatively unexplored and is a nascent area of research.\n\n\\item Most crowd counting and density estimation methods have been designed for and evaluated either only on single images or videos.  Combining the techniques developed separately for these methods is a non-trivial task. Development of low-latency methods that can operate in real-time for counting people in crowds from videos is another interesting problem to be addressed in future. \n\n\n\n\\item Another key issue ignored by earlier research is that the quality of estimated crowd density maps. Many existing CNN-based approaches have a number of max-pooling layers in their networks compelling them to regress on down-sampled density maps. Also, most methods optimize over traditional Euclidean loss which is known to have certain disadvantages \\cite{johnson2016perceptual1}. Regressing on down-sampled density maps using Euclidean loss results in low quality density maps. Fig. \\ref{fig:quality} demonstrates the results obtained using the state-of-the-art method \\cite{zhang2016single}. It can be observed that though accurate count estimates are obtained, the quality of the density maps is poor. As a result, these poor quality maps adversely affect other higher level cognition tasks which depend on them. Recent work on style-transfer \\cite{zhang17ICCV}, image de-raining \\cite{zhang2017image} and image-to-image translation \\cite{pix2pix2016} have demonstrated promising results from the use of additional loss functions such as adversarial loss and perceptual loss. In principle, density estimation can be considered as an image-to-image translation problem and it would be interesting to see the effect of these recent loss functions.  Generating high quality density maps along with low count estimation error would be another important issue to be addressed in the future.\n\n\n\n\n\n\\item Finally, considering advancements by scale-aware \\cite{zhang2016single,onoro2016towards}  and context-aware models \\cite{skaug2016end}, we believe designing networks to incorporate additional contextual and scale information will enable further progress. \n\n\n\\end{enumerate}\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nThis article presented an overview of recent advances in CNN-based methods for crowd counting and density estimation.   In particular, we summarized various methods for crowd counting into traditional approaches (that use hand-crafted features) and CNN-based approaches. The CNN-based approaches are further categorized based on the training process and the network property. Obviously all the literature on crowd counting cannot be covered, hence, we have chosen a representative subset of the latest approaches for a detailed analysis and review. We also reviewed the results demonstrated by various traditional and CNN-based approaches to conclude that CNN-based methods are more adept at handling large density crowds with variations in object scales and scene perspective. Additionally, we observed that incorporating scale and contextual information in the CNN-based methods drastically improves the estimation error. Finally, we identified some of the most compelling challenges and issues that confront research in crowd counting and density estimation using computer vision and machine leaning approaches.  \n\n\\vspace{5pt}\n\\noindent \\textbf{Acknowledgement}\nThis work was supported by US Office of Naval Research (ONR) Grant YIP N00014-16-1-3134.\n\n\n\\bibliographystyle{model2-names}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction to dibosons}\nProcesses involving multiple massive bosons probe right at the\ncore of the SM, i.e. its electro-weak symmetries and resulting\ngauge boson structure. Our current understanding requires some -- yet\nundetected --\nmechanism to regularize processes like $W_L W_L \\rightarrow W_L W_L$ at the\n$O(1)$ TeV scale in order to maintain unitarity \\cite{Chanowitz:2004gk}.\nThis hints at an experimental no-lose situation where we either will\nfind the scalar\nresponsible for the Higgs mechanism, or find new interactions\ninteracting strongly with the longitudinal gauge modes. If none of these cases\nare seen we will be obliged to go back to the sub TeV region and ask ourself\nwhat it is that we don't understand on the fundamental level.\n\nOur lack of knowledge concerning the underlying electro-weak symmetry\nbreaking (EWSB) mechanism of the SM desperately calls for experimental input to\nguide us. Clearly, as indicated from the discussion above diboson\nscattering is likely to provide the ultimate EWSB exploration tool.\nUnfortunately the cross-section for diboson scattering is expected to\nbe unobservable at the Tevatron compared to the LHC \\cite{Butterworth:2002tt}.\nAnother way to get a handle on the EWSB is to reconstruct massive triple gauge\nbosons, but also here is the expected Tevatron cross-section beyond reach.\nIt is primarily for inclusive dibosons that we currently are gaining\nsensitivity to anomalous gauge interactions where new phase-space regions\nare opening up with the increase of integrated Tevatron luminosity.\n\nLEP 2 puts stringent limits on\nthe anomalous triple gauge couplings \\cite{Bruneliere:2004ab} (ATGC) which\nare the lowest order general effective theory operator couplings related\nto the diboson final state. However, the LEP 2 energy scale\n$\\sqrt{\\hat{s}} \\sim 200$ GeV is sufficiently small compared to the new\nphysics scale $\\Lambda$ such that higher order operators can be neglected. At\nthe Tevatron this is no longer the case since $O(1)$ TeV resonances can\nbe produced on-shell and a form factor must be\nintroduced in order to maintain unitarity of the model \\cite{Hagiwara:1989mx}.\nThe form factor is equivalent to a infinite series of higher order operators\nand implies a non linear extension of the linear model commonly used\nat LEP. For this reason the quoted limits on the couplings refer to\nslightly different assumptions at the two accelerators. And one should be\naware of that Tevatron measurements are in many ways complementary to\nthose made at LEP, e.g. new unexpected\non-shell dynamics may kick-in at higher energies and with different\nproduction mechanisms.\n\n\\section{Tevatron measurements}\nAs can be seen from table \\ref{tab:diboxs} it is just recently that\nmassive diboson signals became statistically significant at hadron colliders.\n\\begin{table}[t]\n\\caption{New(**) and recent diboson measurements at the Tevatron. The\nphoton($\\gamma$) has a transverse momentum cut of 7(8) GeV for the\nCDF(D0) measurements. All cross-sections are within the theoretical\nexpectations.\n\\label{tab:diboxs}}\n\\vspace{0.4cm}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n& \\multicolumn{2}{|c|}{CDF} & \\multicolumn{2}{|c|}{D0} \\\\\n\\hline\nDiboson & Events & $\\sigma$[pb] & Events & $\\sigma$[pb] \\\\\n\\hline\n$W(l\\nu)\\gamma \\times BR(l\\nu)$ & 208(200\/pb)& $18.1 \\pm 3.1$ &\n141(162\/pb) & $14.8 \\pm 1.9$ \\\\\n$Z(ll)\\gamma \\times BR(ll)$ & 66 (200\/pb)& $4.6 \\pm 0.6$ &\n244(320\/pb) & $4.2 \\pm 0.5$ \\\\\n$W(l\\nu)W(l\\nu)$ & 12(200\/pb) &\n\\begin{minipage}{1in}\n$14.6^{+5.8}_{-5.1}$(stat) $^{+1.8}_{-3.0}$(sys) $\\pm 0.9$(lum)\n\\end{minipage}\n& 17(252\/pb) &\n\\begin{minipage}{1in}\n$13.8^{+4.3}_{-3.8}$(stat) $^{+1.2}_{-0.9}$(sys) $\\pm 0.9$(lum)\n\\end{minipage}\n\\\\\n$W(l\\nu)(W(jj)+Z(jj))$ & n.a.(350\/pb) & $<36$ (**) & & \\\\\n$W(l\\nu)Z(ll)$ & n.a.(825\/pb)& $<6.4$ (**) & n.a.(320\/pb) & $<13.3$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\nThe measured cross-sections decrease with the diboson mass and the first\nprocess that has been firmly established during the last two years is WW to\nleptons. This is very different from LEP 2 where thousands of WW pairs have\nbeen observed in many decay channels, both leptonic and pure hadronic\nmodes. However, constraints on new physics are dominated by high\nenergy scales and the inclusive cross-section is not the most efficient\nobservable at the Tevatron for anomalous gauge interactions. A closer\nlook at the anomalous cross-section contributions \\cite{Hagiwara:1989mx}\nreveal that they, apart from a boson angular dependence, scale with\nthe parton center-of-mass energy ($\\sqrt{\\hat{s}}$). Thus the very high energy\ntail of the boson transverse momentum turns out to be a extremely sensitive and\nrobust observable for ATGC.\n\nIn the following subsections I will report on two brand new measurements\nof WW and WZ dibosons from CDF. It is interesting to note that even though the\nprocesses themselves are just below the the threshold of observation they\nhave a significant constraining power for ATGC.\n\n\\subsection{The process $WZ \\rightarrow l\\nu ll$}\n\n\\begin{figure}\n\\begin{center}\n\\psfig{figure=feyn_wz.eps,width=12cm}\n\\caption{Tree level graphs for WZ.\n\\label{fig:wz}}\n\\end{center}\n\\end{figure}\n\nThe WZ process, see figure \\ref{fig:wz}, is particularly interesting among\nthe dibosons.\nSince it could not be be produced at tree level at LEP we have so far no\ndirect experimental measurement of the WWZ vertex. Furthermore,\nthe absence of interference with other triple gauge vertices at tree level\nmakes this channel unique since it can provide a unambiguous\nhandle on the WWZ couplings in case of any observed anomaly that needs to be\ndisentangled in e.g. the WW production.\n\nA new CDF search for WZ to leptons has been made using data\nequivalent to 825\/pb of integrated luminosity of proton-antiproton collisions\nat $\\sqrt{s}=1.96$ TeV. The events are triggered by a lepton (isolated\nelectron or muon) with at least 20 GeV transverse momentum. Two more leptons\nare then required with at least 10 GeV transverse momentum. A neutrino\nlike signature is selected by requiring at least 25 GeV, see figure\n\\ref{fig:wzmet}, of missing transverse\nenergy. One Z is selected by one opposite sign lepton pair within the\ndilepton mass $76 < M_{ll} < 106$ GeV, see figure \\ref{fig:wzmll},\nand ZZ events are vetoed by removing\nevents with tracks that falls within $76 < M_{trk,l} < 106$ GeV using the\nremaining unmatched lepton. Due to the specific multilepton signature the\nselected candidates have a relatively small expected background contamination.\nAn overview of measured and estimated events are shown in\ntable \\ref{tab:wz}. Only two candidates are observed and a limit is\nderived on the total cross-section\n$$\n\\sigma(p\\bar{p} \\rightarrow WZ)< 6.4 \\textrm{ pb (95\\% C.L.)}.\n$$\nThis is intriguingly close to the theoretical NLO value around 4 pb.\nLimits on the WWZ ATGC have not yet been extracted but we know\nfrom a recent and similar D0 measurement \\cite{Abazov:2005ys}\n(see also the corresponding entry in table 1)\nthat this analysis have the potential to provide strong constraints.\n\n\\begin{table}[t]\n\\caption{Measured and expected number of events for WZ to leptons\nat CDF using 825\/pb of data.\n\\label{tab:wz}}\n\\vspace{0.4cm}\n\\begin{center}\n\\begin{tabular}{|l|l|}\n\\hline\nProcess & Events \\\\\n\\hline\n$WZ$ & $3.72 \\pm 0.02$(stat) $\\pm 0.15$(sys) \\\\\n\\hline\n$ZZ$ &  $0.50 \\pm 0.01$(stat) $\\pm 0.05$(sys) \\\\\n$Z\\gamma$ &  $0.03 \\pm 0.01$(stat) $\\pm 0.01$(sys) \\\\\n$t\\bar{t}$ &  $0.05 \\pm 0.01$(stat) $\\pm 0.01$(sys) \\\\\n$Z+jets$ &  $0.34 \\pm 0.07$(stat) $^{+0.16}_{-0.10}$(sys) \\\\\n\\hline\nTotal background: &  $0.92 \\pm 0.07$(stat) $\\pm 0.15$(sys) \\\\\n\\hline\nTotal data: &  2 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[p]\n\\begin{center}\n\\psfig{figure=h_Met_noMetCut_wsf.eps,width=13cm}\n\\caption{Missing transverse energy in the WZ events. The cut for the signal\nregion is also shown.\n\\label{fig:wzmet}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[p]\n\\begin{center}\n\\psfig{figure=trileptons_Mll_Met.eps,width=13cm}\n\\caption{Scatter plot in the missing transverse energy and dilepton mass\nplane for WZ. Data is compared to expectation.\n\\label{fig:wzmll}}\n\\end{center}\n\\end{figure}\n\n\\subsection{The process $WW+WZ \\rightarrow l\\nu jj$}\n\nSemi-leptonic decay modes of WW events have yet not been observed at hadron\ncolliders, see figure \\ref{fig:ww} where one of the bosons decays into quarks\nand the other one into leptons.\n\\begin{figure}\n\\begin{center}\n\\psfig{figure=feyn_ww.eps,width=13cm}\n\\caption{Tree level and LO Higgs graphs for WW.\n\\label{fig:ww}}\n\\end{center}\n\\end{figure}\nThe reconstruction is challenging due\nto the limited dijet mass resolution of about 10\\% and the W+jets background\nwhich is $O(100)$ times larger than the signal after event preselection\nat the Tevatron. Note that events from diagrams in figure \\ref{fig:wz} (replace\nthe leptonic Z decay with quarks) cannot be excluded, hence they are included\nin the signal. Still, the hadronic decay modes are interesting since\nthey have larger event yield than the pure leptonic modes and they have\nthe important ability to reconstruct the W transverse momentum which we\nknow is a sensitive handle on the ATGC.\n\nIn order to constrain the ATGC and to verify the SM rate\na new CDF search for WW+WZ to leptons, missing transverse energy and jets\nhas been made using data equivalent to 350\/pb of integrated luminosity\nof proton-antiproton collisions at $\\sqrt{s}=1.96$ TeV. The events are\ntriggered by an isolated lepton with 25 GeV transverse energy (20 GeV\ntransverse momentum for muons). An inclusive set of leptonic W decays is\nselected by requiring the missing transverse energy to be at least 25 GeV.\nThe events selection then proceeds by requiring two jets with at least\n15 GeV transverse energy and with a dijet mass in the range $32 < M_{jj} <\n184$ GeV. An additional cut on the W transverse mass below 25 GeV is applied\nto reduce the multijet background. Also proximity cuts are applied among the\nlepton and jets. However care must be taken not to reject narrow jets since\nthe interesting high transverse momentum boson have narrow jets due to the\nboost. The expected number of events is shown in table \\ref{tab:ww}.\n\n\\begin{table}[t]\n\\caption{Measured and expected number of events for WW+WZ to leptons,\nmissing transverse energy and jets at CDF using 350\/pb of data.\n\\label{tab:ww}}\n\\vspace{0.4cm}\n\\begin{center}\n\\begin{tabular}{|l|c|r|}\n\\hline\nProcess & Uncertainty & Events \\\\\n\\hline\n$WW$ & 15\\% & 142.0 \\\\\n$WZ$ & 15\\% & 18.2 \\\\\n\\hline\n$W+jets$ & 20\\% & 6261.0 \\\\\n$Multijets$ & 40\\% & 263.4 \\\\\n$W(\\tau)+jets$ & 20\\% & 171.0 \\\\\n$Z+jets$ & 20\\% & 154.0 \\\\\n$t\\bar{t}$ & 25\\% & 171.6 \\\\\n$t(t-ch)$ & 25\\% & 14.4 \\\\\n$t(s-ch)$ & 25\\% & 8.2 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nSince the theoretical uncertainty on the W+jets background is huge we extract\nthe WW+WZ signal by assuming the SM ratio between WW\/WZ and fit the expected\nsignal and background dijet mass shapes to data, see figure \\ref{fig:wwxsfit}.\nThe fit gives 109 signal events and an statistical uncertainty of 110 events.\nSystematic uncertainties contributes with an additional 54 events. From this\nwe estimate the upper limit on the WW+WZ cross-section to be\n$$\n\\sigma(p\\bar{p} \\rightarrow WW+WZ) < 36 \\textrm{ pb}.\n$$\nThe SM expectation is 16 pb.\n\nThe ATGC are extracted from the signal region defined to be the\ndijet mass window $56<m_{jj}<112$ GeV. The fit and extra cuts used for\nestimating the ATGC background events are shown in figure \\ref{fig:wwbg}.\n\\begin{figure}[t]\n\\begin{center}\n\\psfig{figure=datafit_14_1.eps,width=9cm}\n\\caption{WW+WZ signal and background dijet mass spectrum shape fitted to the\nmeasured data. This is used for estimating the total cross-section.\n\\label{fig:wwxsfit}}\n\\end{center}\n\\end{figure}\n\\begin{figure}[t]\n\\begin{center}\n\\psfig{figure=datafit_16_4.eps,width=9cm}\n\\caption{WW+WZ signal and background dijet mass spectrum shape fit\nwith cuts used for estimating the ATGC background.\n\\label{fig:wwbg}}\n\\end{center}\n\\end{figure}\nThe actual ATGC limits are given by a binned likelihood fit to the\nW transverse momentum, see figure \\ref{fig:wwpt}. For completeness we also show\nthe transverse cluster mass which is a good estimator of $\\hat{s}$ in the event\nand a probe for unexpected resonances. Finally the 2D ATGC limits in the\n$(\\Delta\\kappa,\\lambda)$ plane is shown in figure \\ref{fig:ac}. The\ncorresponding 1D limits where one of the parameters are fixed to zero are\n$$\n-0.51 < \\Delta\\kappa < 0.44\n$$\n$$\n-0.28 < \\lambda < 0,28\n$$\nThis is with a form factor cut-off $\\Lambda=1.5$ TeV. Previous limits in a\ncomparable channel are from D0 and Run I \\cite{Abbott:1999aj}.\nThe quoted limits are\n$$\n-0.56 < \\Delta\\kappa < 0.75\n$$\n$$\n-0.42 < \\lambda < 0.44\n$$\n\n\\begin{figure}[t]\n\\begin{center}\n\\psfig{figure=eclum_stac.eps,width=11cm}\n\\psfig{figure=enupt_stac.eps,width=11cm}\n\\caption{Cluster transverse mass (upper plot) and W transerse momentum\n(lower plot).\n\\label{fig:wwpt}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{center}\n\\psfig{figure=acscan_mea.eps,width=13cm}\n\\caption{ATGC limits extracted from the W transverse momentum in\nWW+WZ to lepton + jets events. The 1D limit shows the case when one\nof the parameters is fixed to zero.\n\\label{fig:ac}}\n\\end{center}\n\\end{figure}\n\n\\section*{Acknowledgments}\nI want to thank the Moriond organizers for providing\nan exceptionally stimulating conference environment.\nAlso a big thanks to my colleagues and collaborators at the D0 and CDF\nexperiments.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe nonlinear Schr\\\"odinger (NLS) equation, as one of the universal\nequations that describe the evolution of slowly varying packets of\nquasi-monochromatic waves in weakly nonlinear dispersive media, has been\nvery successful in many applications such as nonlinear optics and water\nwaves \\cite{Kodamabook, Agrawalbook,Boydbook,Yarivbook}. The NLS equation is\nintegrable, which can be solved by the inverse scattering transform \\cit\n{Zakharov}.\n\nHowever, in the regime of ultra-short pulses where the width of optical\npulse is in the order of femtosecond ($10^{-15}$ s), the NLS equation\nbecomes less accurate \\cite{Rothenberg}. Description of ultra-short\nprocesses requires a modification of standard slow varying envelope models\nbased on the NLS equation. There are usually two approaches to meet this\nrequirement in the literature. The first one is to add several higher-order\ndispersive terms to get higher-order NLS equation \\cite{Agrawalbook}. The\nsecond one is to construct a suitable fit to the frequency-dependent\ndielectric constant $\\epsilon(\\omega)$ in the desired spectral range.\nSeveral models have been proposed by this approach including the short-pulse (SP) equation \\cite{SPE_Org,KimPRL,KimPRA,Bandelow}.\n\nRecently, Sch\\\"{a}fer and Wayne derived a so-called short pulse (SP)\nequation \\cite{SPE_Org}\n\\begin{equation}\nu_{xt}=u+ \\frac 16 \\left(u^{3}\\right)_{xx}\\,,  \\label{SPE}\n\\end{equation\nto describe the propagation of ultra-short optical pulses in nonlinear\nmedia. Here, $u=u(x,t)$ is a real-valued function, representing the\nmagnitude of the electric field, the subscripts $t$ and $x$ denote partial\ndifferentiation. Apart from the context of nonlinear optics, the SP equation\nhas also been derived as an integrable differential equation associated with\npseudospherical surfaces \\cite{Robelo}. The SP equation has been shown to be\ncompletely integrable \\cite{Robelo,Beals,Sakovich,Brunelli1,Brunelli2}. The\nperiodic and soliton solutions of the SP equation were found in \\cit\n{Sakovich2,Kuetche,Parkes}. The connection between the SP equation and the\nsine-Gordon equation through the hodograph transformation was clarified, and\nthen the $N$-soliton solutions including multi-loop and multi-breather ones\nwere given in \\cite{Matsuno_SPE,Matsuno_SPEreview} by using the Hirota\nbilinear method \\cite{Hirota}. The integrable discretization of the SP\nequation was studied in \\cite{SPE_discrete1}, the geometric interpretation\nof the SP equation, as well as its integrable discretization, was given in\n\\cite{SPE_discrete2}. The higher-order corrections to the SP equation was\nstudied in \\cite{Schafer13} most recently.\n\nSimilar to the case of the NLS equation \\cite{Manakov1974}, it is necessary\nto consider its two-component or multi-component generalizations of the SP\nequation for describing the effects of polarization or anisotropy. As a\nmatter of fact, several integrable coupled short pulse have been proposed in\nthe literature \\cit\n{PKB_CSPE,Sakovich3,Hoissen_CSPE,Matsuno_CSPE,Feng_CSPE,ZengYao_CSPE}. Most\nrecently, the bi-Hamiltonian structures for the above two-component SP\nequations were obtained by Brunelli \\cite{Brunelli_CSPE}.\n\n\nIn the present paper, we propose and study a complex short pulse (CSP)\nequation\n\\begin{equation}\nq_{xt}+q+ \\frac{1}{2} \\left(|q|^{2}q_x \\right)_{x}=0\\,,  \\label{CSP}\n\\end{equation\nand its two-component generalization\n\\begin{eqnarray}\n&& q_{1,xt}+q_1+\\frac{1}{2} \\left((|q_1|^2+|q_2|^2)q_{1,x}\\right)_{x}=0\\,,\n\\label{CCSP1} \\\\\n&& q_{2,xt}+q_2+\\frac{1}{2} \\left((|q_1|^2+|q_2|^2)q_{2,x}\\right)_{x}=0\\,.\n\\label{CCSP2}\n\\end{eqnarray}\nAs will be revealed in the present paper, both the CSP equation and its\ntwo-component generalization are integrable guaranteed by the existence of\nLax pairs and infinite number of conservation laws. They have $N$-soliton\nsolutions which can be constructed via Hirota's bilinear method.\n\n\nThe outline of the present paper is organized as follows. In section 2, we\nderive the CSP equation and coupled complex short pulse (CCSP) equation from\nthe physical context. In section 3, by providing the Lax pairs, the\nintegrability of the proposed two equations are confirmed, and further, the\nconservation laws, both local and nonlocal ones, are investigated. Then $N\n-soliton solutions to both the CSP and CCSP equations are constructed in\nterms of pfaffians by Hirota's bilinear method in section 4. In section 5,\nsoliton-interaction for coupled complex short pulse equation is investigated\nin details, which shows rich phenomena similar to the\ncoupled nonlinear Schr\\\"odinger equatoin. In particular, they may undergo\neither elastic or inelastic collision depending on the initial conditions.\nFor inelastic collisions, there is an energy exchange between solitons,\nwhich can allow the generation or vanishing of soliton. The dynamics is more\nricher in compared with the single component case. The paper is concluded by\ncomments and remarks in section 6.\n\n\\section{The derivation of the complex short pulse and coupled complex short\npulse equations}\n\nIn this section, following the procedure in \\cite{Agrawalbook,SPE_Org}, we\nderive the complex short pulse equation (\\ref{CSP}) and its two-component\ngeneralization that governs the propagation of ultra short pulse packet\nalong optical fibers.\n\n\\subsection{The complex short pulse equation}\n\nWe start with a wave equation for electric field\n\\begin{equation}  \\label{E-wave-equation}\n\\nabla^2 \\mathbf{E} -\\frac{1}{c^2} \\mathbf{E}_{tt} = \\mu_0 \\mathbf{P}_{tt}\\,,\n\\end{equation}\noriginated from the Maxwell equation. Here $\\mathbf{E} ( \\mathbf{r},t)$ and \n\\mathbf{P} ( \\mathbf{r},t)$ represent the electric field and the induced\npolarization, respectively, $\\mu_0$ is the vacuum permeability, $c$ is the\nspeed of light in vacuum. If we assume the local medium response and only\nthe third-order nonlinear effects governed by $\\chi^{(3)}$, the induced\npolarization consists of two parts, $\\mathbf{P} ( \\mathbf{r},t)=\\mathbf{P\n_{L} ( \\mathbf{r},t)+\\mathbf{P}_{NL} ( \\mathbf{r},t)$, where the linear part\n\\begin{equation}  \\label{P_L}\n\\mathbf{P}_{L} ( \\mathbf{r},t)=\\epsilon_0 \\int_{-\\infty}^{\\infty} \\chi^{(1)}\n(t-t^{\\prime })\\cdot \\mathbf{E} ( \\mathbf{r},t^{\\prime })\\,dt^{\\prime }\\,,\n\\end{equation}\nand the nonlinear part\n\\begin{equation}  \\label{P_NL}\n\\mathbf{P}_{NL}( \\mathbf{r},t)=\\epsilon_0 \\int_{-\\infty}^{\\infty} \\chi^{(3)}\n(t-t_1,t-t_2,t-t_3)\\times \\mathbf{E} ( \\mathbf{r},t_1) \\mathbf{E} ( \\mathbf{\n},t_2) \\mathbf{E} ( \\mathbf{r},t_3)\\,dt_1dt_2dt_3\\,.\n\\end{equation}\nHere $\\epsilon_0$ is the vacuum permittivity and $\\chi^{(j)}$ is the $j\nth-order susceptibility. Since the nonlinear effects are relatively small in\nsilica fibers, $\\mathbf{P}_{NL}$ can be treated as a small perturbation.\nTherefore, we first consider (\\ref{E-wave-equation}) with $\\mathbf{P}_{NL}=0\n. Furthermore, we restrict ourselves to the case that the optical pulse\nmaintains its polarization along the optical fiber, and the transverse\ndiffraction term $\\Delta_{\\perp} \\mathbf{E}$ can be neglected. In this case,\nthe electric field can be considered to be one-dimensional and expressed as\n\\begin{equation}  \\label{E-field}\n\\mathbf{E} = \\frac 12 \\mathbf{e_1} \\left(E(z,t)+c.c. \\right)\\,,\n\\end{equation}\nwhere $\\mathbf{e_1}$ is a unit vector in the direction of the polarization,\n$E(z,t)$ is the complex-valued function, and $c.c.$ stands for the complex\nconjugate. Conducting a Fourier transform on (\\ref{E-wave-equation}) leads\nto the Helmholtz equation\n\\begin{equation}  \\label{Waveequation-Fourier}\n\\tilde {E}_{zz} (z,\\omega) + \\epsilon(\\omega) \\frac{\\omega^2}{c^2} \\tilde {E}\n(z,\\omega)=0\\,,\n\\end{equation}\nwhere $\\tilde {E} (z,\\omega)$ is the Fourier transform of $E(z,t)$ defined\nas\n\\begin{equation}  \\label{E_Fourier}\n\\tilde E (z,\\omega)=\\int_{-\\infty}^{\\infty} {\\ E} (z,t) e^{\\mathrm{i} \\omega\nt}\\,dt\\,,\n\\end{equation}\n$\\epsilon(\\omega)$ is called the frequency-dependent dielectric constant\ndefined as\n\\begin{equation}  \\label{Dielectric}\n\\epsilon(\\omega)=1+ \\tilde \\chi^{(1)} (\\omega)\\,,\n\\end{equation}\nwhere $\\tilde \\chi^{(1)} (\\omega)$ is the Fourier transform of $\\chi^{(1)}\n(t)$\n\\begin{equation}  \\label{Chi_Fourier}\n\\tilde \\chi^{(1)} (\\omega)=\\int_{-\\infty}^{\\infty} \\chi^{(1)} (t) e^{\\mathrm\ni} \\omega t}\\,dt\\,.\n\\end{equation}\nNow we proceed to the consideration of the nonlinear effect. Assuming the\nnonlinear response is instantaneous so that $P_{NL}$ is given by \nP_{NL}(z,t)= \\epsilon_0 \\epsilon_{NL} E(z,t)$ \\cite{Agrawalbook} where the\nnonlinear contribution to the dielectric constant is defined as\n\\begin{equation}  \\label{Epsnl}\n\\epsilon_{NL}= \\frac 34 \\chi^{(3)}_{xxxx} |E(z,t)|^2\\,.\n\\end{equation}\nIn this case, the Helmholtz equation (\\ref{Waveequation-Fourier}) can be\nmodified as\n\\begin{equation}  \\label{Helmholtz}\n\\tilde {E}_{zz} (z,\\omega) + \\tilde \\epsilon(\\omega) \\frac{\\omega^2}{c^2}\n\\tilde {E} (z,\\omega)=0\\,,\n\\end{equation}\nwhere\n\\begin{equation}  \\label{Modified_Dielectric}\n\\tilde \\epsilon(\\omega)=1+ \\tilde \\chi^{(1)} (\\omega)+ \\epsilon_{NL}\\,.\n\\end{equation}\nAs pointed out in \\cite{SPE_Org,Boydbook,KimPRL}, the Fourier transform \n\\tilde \\chi^{(1)}$ can be well approximated by the relation $\\tilde\n\\chi^{(1)}=\\tilde \\chi_0^{(1)} - \\tilde \\chi_2^{(1)} \\lambda^2$ if we\nconsider the propagation of optical pulse with the wavelength between 1600\nnm and 3000 nm. It then follows that the linear equation (\\re\n{Waveequation-Fourier}) written in Fourier transformed form becomes\n\\begin{equation}  \\label{Wave_Fourier2}\n\\tilde {E}_{zz} + \\frac{1+\\tilde \\chi_0^{(1)}}{c^2} \\omega^2 \\tilde {E} -\n(2\\pi)^2 \\tilde \\chi_2^{(1)} \\tilde {E} + \\epsilon_{NL} \\frac{\\omega^2}{c^2}\n\\tilde {E} =0\\,.\n\\end{equation}\n\nApplying the inverse Fourier transform to (\\ref{Wave_Fourier2}) yields a\nsingle nonlinear wave equation\n\\begin{equation}  \\label{Wave_nonlinear1}\nE_{zz} - \\frac{1}{c_1^2} E_{tt} = \\frac{1}{c_2^2} E +\\frac 34\n\\chi^{(3)}_{xxxx} \\left(|E|^2 E\\right)_{tt} =0\\,.\n\\end{equation}\n\nSimilar to \\cite{SPE_Org}, we focus on only a right-moving wave packet and\nmake a multiple scales \\textrm{ansatz}\n\\begin{equation}  \\label{E_ansatz}\nE(z,t)=\\epsilon E_0(\\phi, z_1, z_2, \\cdots)+ \\epsilon^2 E_1(\\phi, z_1, z_2,\n\\cdots) + \\cdots\\,,\n\\end{equation}\nwhere $\\epsilon$ is a small parameter, $\\phi$ and $z_n$ are the scaled\nvariables defined by\n\\begin{equation}  \\label{Multiple_ansatz}\n\\phi= \\frac{t-\\frac{x}{c_1}}{\\epsilon}, \\quad z_n=\\epsilon^n z\\,.\n\\end{equation}\nSubstituting (\\ref{E_ansatz}) with (\\ref{Multiple_ansatz}) into (\\re\n{Wave_nonlinear1}), we obtain the following partial differential equation\nfor $E_0$ at the order $O(\\epsilon)$:\n\n\\begin{equation}  \\label{Wave_nonlinear2}\n- \\frac{2}{c_1} \\frac{\\partial^2 E_0}{\\partial{\\phi}\\partial{z_1}} = \\frac{\n}{c_2^2} E_0 +\\frac 34 \\chi^{(3)}_{xxxx} \\frac{\\partial}{\\partial{\\phi}}\n\\left(|E_0|^2 \\frac{\\partial E_0}{\\partial {\\phi}} \\right)\\,.\n\\end{equation}\n\n\nFinally, by a scale transformation\n\\begin{equation}  \\label{Scaling}\nx= \\frac{c_1}{2} \\phi, \\quad t={c_2} z_1, \\quad q = \\frac{c_1\\sqrt\n6c_2\\chi^{(3)}_{xxxx}}}{4} E_0\\,,\n\\end{equation}\nwe arrive at the normalized form of the complex short pulse equation (\\re\n{CSP}).\n\n\\subsection{Coupled complex short pulse equation}\n\nIn the previous subsection, a major simplification made in the derivation of\nthe complex short pulse equation is to assume that the polarization is\npreserved during its propagating inside an optical fiber. However, this is\nnot really the case in practice. For birefringent fibers, two orthogonally\npolarized modes have to be considered. Therefore, similar to the extension\nof coupled nonlinear Schr{\\\"o}dinger equation from the NLS equation, an\nextension to a two-component version of the complex short pulse equation \n\\ref{CSP}) is needed to describe the propagation of ultra-short pulse in\nbirefringent fibers. In fact, several generalizations have been proposed for\nthe short pulse equation \\cit\n{PKB_CSPE,Sakovich3,Hoissen_CSPE,Matsuno_CSPE,Feng_CSPE,ZengYao_CSPE}.\nParticularly, by taking into account the effects of anisotropy and\npolarization, Pietrzyk \\textit{et. al.} have derived a general two-component\nshort-pulse equation from the physical context \\cite{PKB_CSPE}. We follow\nthe approach by Pietrzyk \\textit{et. al.} to derive a two-component complex\nshort pulse equation. However, as shown in subsequent section, the\ntwo-component complex short pulse equation admits multi-soliton solutions\nwhich reveals richer dynamics in soliton interactions in compared with the\nreal SP equation.\n\nWe first consider the linear birefringent fiber such that the electric field\nwith an arbitrarily polarized optical fiber can be expressed as\n\\begin{equation}  \\label{E-field2}\n\\mathbf{E} = \\frac 12 \\left( \\mathbf{e_1} E_1(z,t)+ \\mathbf{e_2} E_2(z,t)\n\\right)+ c.c.\\,,\n\\end{equation}\nwhere $\\mathbf{e_1}$, $\\mathbf{e_2}$ are two unit vectors along positive $x\n- and $y$-direction in the transverse plane perpendicular to the optical\nfiber, respectively, $E_1$ and $E_2$ are the complex amplitudes of the\npolarization components correspondingly. Without the presence of nonlinear\npolarization ($P_{NL}=0$) and the transverse diffraction, the Fourier\ntransform converts (\\ref{E-wave-equation}) into a pair of Helmholtz\nequations\n\\begin{equation}  \\label{CCSPE_Helmholtz1}\n\\tilde {E}_{1,zz} (z,\\omega) + \\epsilon(\\omega) \\frac{\\omega^2}{c^2} \\tilde \nE_1} (z,\\omega)=0\\,,\n\\end{equation}\n\n\\begin{equation}  \\label{CCSPE_Helmholtz2}\n\\tilde {E}_{2,zz} (z,\\omega) + \\epsilon(\\omega) \\frac{\\omega^2}{c^2} \\tilde \nE_2} (z,\\omega)=0\\,.\n\\end{equation}\nSame as the scalar case, the frequency-dependent dielectric constant \n\\epsilon(\\omega)=1+ \\tilde \\chi^{(1)} (\\omega)$, where $\\tilde \\chi^{(1)}$\ncan be well approximated by the relation $\\tilde \\chi^{(1)}=\\tilde\n\\chi_0^{(1)} - \\tilde \\chi_2^{(1)} \\lambda^2$ for the propagation of optical\npulse with the wavelength between 1600 nm and 3000 nm.\n\nAs indicated in \\cite{Agrawalbook}, the nonlinear part of the induced\npolarization $\\mathbf{P}_{NL}$ can be written as\n\\begin{equation}\n\\mathbf{P}_{NL}=\\frac{1}{2}\\left( \\mathbf{e_{1}}P_{1}(z,t)+\\mathbf{e_{2}\nP_{2}(z,t)\\right) +c.c.\\,,  \\label{E-field22}\n\\end{equation\nwhere\n\\begin{equation}\nP_{1}=\\frac{3\\epsilon _{0}}{4}\\chi _{xxxx}^{(3)}\\left[ \\left( |E_{1}|^{2}\n\\frac{2}{3}|E_{2}|^{2}\\right) E_{1}+\\frac{1}{3}(E_{1}^{\\ast }E_{2})E_{2\n\\right] \\,,  \\label{Nonlinear_Polarization1}\n\\end{equation\n\\begin{equation}\nP_{2}=\\frac{3\\epsilon _{0}}{4}\\chi _{xxxx}^{(3)}\\left[ \\left( |E_{2}|^{2}\n\\frac{2}{3}|E_{1}|^{2}\\right) E_{2}+\\frac{1}{3}(E_{2}^{\\ast }E_{1})E_{1\n\\right] \\,.  \\label{Nonlinear_Polarization2}\n\\end{equation\nThe last term in Eqs. (\\ref{Nonlinear_Polarization1}) and (\\re\n{Nonlinear_Polarization2}) leads to the degenerate four-wave mixing. In\nhighly birefringent fibers, the four-wave-mixing term can often be\nneglected. In this case, we arrive at a coupled nonlinear wave equation\n\\begin{equation}  \\label{Coupled_equations1}\nE_{1,zz}-\\frac{1}{c_{1}^{2}}E_{1,tt}=\\frac{1}{c_{2}^{2}}E_{1}+\\frac{3}{4\n\\chi _{xxxx}^{(3)}\\left[ \\left( |E_{1}|^{2}+\\frac{2}{3}|E_{2}|^{2}\\right)\nE_{1}\\right] _{tt}\\,,\n\\end{equation}\n\n\\begin{equation}  \\label{Coupled_equations2}\nE_{2,zz}-\\frac{1}{c_{1}^{2}}E_{2,tt}=\\frac{1}{c_{2}^{2}}E_{2}+\\frac{3}{4\n\\chi _{xxxx}^{(3)}\\left[ \\left( |E_{2}|^{2}+\\frac{2}{3}|E_{1}|^{2}\\right)\nE_{2}\\right] _{tt}\\,.\n\\end{equation}\n\nSimilar to the scalar case, by a multiple scales expansion and an\nappropriate scaling transformation, a couple complex short pulse equation\ncan be obtained from (\\ref{Coupled_equations1})--(\\ref{Coupled_equations2})\n\\begin{equation}  \\label{CCSP_linear1}\nq_{1,xt}+q_{1}+\\frac{1}{2} \\left((|q_{1}|^{2}+\\frac{2}{3}|q_{2}|^{2})q_{1,x\n\\right) _{x}=0\\,,\n\\end{equation}\n\n\\begin{equation}  \\label{CCSP_linear2}\nq_{2,xt}+q_{2}+\\frac{1}{2} \\left((|q_{2}|^{2}+\\frac{2}{3}|q_{1}|^{2})q_{2,x\n\\right) _{x}=0\\,.\n\\end{equation}\n\nMore generally, we can consider the coupled short pulse equation for\nelliptically birefringent fibers. In this case, the electric field can be\nwritten as\n\\begin{equation}\n\\mathbf{E}=\\frac{1}{2}\\left( \\mathbf{e_{x}}E_{x}(z,t)+\\mathbf{e_{y}\nE_{y}(z,t)\\right) +c.c.\\,,  \\label{E-field3}\n\\end{equation\nwhere $\\mathbf{e_{x}}$ and $\\mathbf{e_{y}}$ are orthonormal polarization\neigenvectors\n\\begin{equation}\n\\mathbf{e_{x}}=\\frac{\\mathbf{e_{1}}+ir\\mathbf{e_{2}}}{\\sqrt{1+r^{2}}},\\quad\n\\mathbf{e_{y}}=\\frac{r\\mathbf{e_{1}}-i\\mathbf{e_{2}}}{\\sqrt{1+r^{2}}}\\,.\n\\label{Elliptical-unit}\n\\end{equation\nThe parameter $r$ represents the ellipticity. It is common to introduce the\nellipticity angle $\\theta $ as $r=\\tan (\\theta \/2)$. The case $\\theta =0$\nand $\\pi \/2$ correspond to linearly and circularly birefringent fibers,\nrespectively.\n\nFollowing a procedure similar to the case of linearly birefringent fibers,\none can drive the normalized form for the coupled complex short pulse\nequation\n\\begin{equation}  \\label{CCSP_general1}\nq_{1,xt}+ q_1+ \\frac 12 \\left((|q_1|^2+ B|q_2|^2) q_{1,x} \\right)_x = 0\\,,\n\\end{equation}\n\\begin{equation}  \\label{CCSP_general2}\nq_{2,xt}+ q_2+ \\frac 12  \\left((|q_2|^2+ B|q_1|^2) q_{2,x} \\right)_x = 0\\,.\n\\end{equation}\nwhere the parameter $B$ is related to the ellipticity angle $\\theta$ as\n\\begin{equation}  \\label{B_theta}\nB=\\frac{2+2\\sin^2 \\theta}{2+\\cos^2 \\theta}\\,.\n\\end{equation}\nFor a linearly birefringent fiber ($\\theta=0$), $B=\\frac 23$, and Eqs. (\\re\n{CCSP_general1})-- (\\ref{CCSP_general2}) reduces to Eqs. (\\ref{CCSP_linear1\n)-- (\\ref{CCSP_linear2}). For a circularly birefringent fiber ($\\theta=\\pi\/2\n), $B=2$. In general, the coupling parameter $B$ depends on the ellipticity\nangle $\\theta$ and can vary from $\\frac 23$ to $2$ for values of $\\theta$ in\nthe range from $0$ to $\\pi\/2$. Note that $B=1$ when $\\theta \\approx 35^\\circ\n. As discussed in the subsequent section, this case is of particular\ninterest because the coupled system is integrable and admits $N$-soliton\nsolution.\n\\section{Lax pairs and conservation laws for the complex and\ncoupled complex short pulse equations}\n\n\\subsection{Lax pairs and integrability}\nIn \\cite{PKB_CSPE}, a matrix generalization for the SP equation is given\nbased on zero-curvature representation, from which the Lax pairs for several\nintegrable two-component SP equations are explicitly provided. In this\nsubsection, we will show the integrability of the complex short pulse and\ncoupled complex short pulse equations by finding their Lax pairs constructed\nfrom another matrix generalization of the SP equation.\n\nThe Lax pair for the complex short pulse equation (\\ref{CSP}) can be\nexpressed as\n\\begin{equation}  \\label{comSPE_Laxa}\n\\Psi_x=U \\Psi, \\quad \\Psi_t=V\\Psi\\,,\n\\end{equation}\nwith\n\\begin{equation}  \\label{comSPE_Laxb}\n\\displaystyle U= \\lambda \\left\n\\begin{array}{cc}\n1 & q_x \\\\\nq^{*}_x & -\n\\end{array\n\\right), \\quad V= \\left\n\\begin{array}{cc}\n-\\frac {\\lambda}{2} |q|^2-\\frac{1}{4\\lambda} & -\\frac{\\lambda}{2\n|q|^2q_x+\\frac q2 \\\\\n-\\frac{\\lambda}{2}|q|^2q^{*}_x-\\frac {q^{*}}{2} & \\frac {\\lambda}{2}|q|^2+\\frac{1}{4\\lambda\n\\end{array\n\\right)\\,.\n\\end{equation}\nIt can be easily shown that the compatibility condition $U_t-V_x+[U,\\,V]=0$\ngives the complex short pulse equation (\\ref{CSP}).\n\nThe Lax pair for the coupled complex short pulse equation (\\ref{CCSP1})--\n\\ref{CCSP2}) is found to be of the form:\n\\begin{equation}\n\\Psi _{x}=U\\Psi ,\\quad \\Psi _{t}=V\\Psi \\,,  \\label{ccomSPE_Laxa}\n\\end{equation\nwith\n\\begin{equation}\nU=\\lambda \\left(\n\\begin{array}{cc}\nI_{2} & Q_{x} \\\\\nR_{x} & -I_{2\n\\end{array\n\\right) ,\\quad   V=\\left(\n\\begin{array}{cc}\n-\\frac{\\lambda }{2}QR-\\frac{1}{4\\lambda }I_{2} & -\\frac{\\lambda }{2}QRQ_{x}\n\\frac{1}{2}Q \\\\\n-\\frac{\\lambda }{2}RQR_{x}-\\frac{1}{2}R & \\frac{\\lambda }{2}QR+\\frac{1}\n4\\lambda }I_{2\n\\end{array\n\\right) \\,,  \\label{ccomSPE_Laxb}\n\\end{equation\nwhere $I_{2}$ is a $2\\times 2$ identity matrix, $Q$, $R$ are $2\\times 2$\nmatrices defined as\n\\begin{equation}\nQ=\\left(\n\\begin{array}{cc}\nq_{1} & q_{2} \\\\\n-q_{2}^{\\ast } & q_{1}^{\\ast \n\\end{array\n\\right) ,\\quad R=\\left(\n\\begin{array}{cc}\nq_{1}^{\\ast } & -q_{2} \\\\\nq_{2}^{\\ast } & q_{1\n\\end{array\n\\right) \\,.  \\label{ccomSPE_Laxc}\n\\end{equation\nNote that $R=Q^{\\dag }$, thus,\n\\begin{equation}\nQR=RQ=(|q_{1}|^{2}+|q_{2}|^{2})I_{2}\\,,\n\\end{equation\nthe compatibility condition $U_{t}-V_{x}+[U,\\,V]=0$ for (\\ref{ccomSPE_Laxa})\ngives the coupled complex short pulse equation (\\ref{CCSP1})--(\\ref{CCSP2}).\n\nAs a matter of fact, the coupled complex short pulse equation can be\ngeneralized into a multi-component, or a vector complex short pulse equation\n\\begin{equation}\nq_{i,xt}+q_{i}+\\frac{1}{2}\\left( |\\mathbf{q}|^{2}q_{i,x}\\right)\n_{x}=0\\,,\\quad i=1,\\cdots ,n,  \\label{NCSPE}\n\\end{equation\nwhere $\\mathbf{q}=(q_1,q_2, \\cdots, q_n)$. The integrability of Eq. (\\re\n{NCSPE})\ncan be guaranteed by the Lax pair constructed in a similar way as in \\cit\n{TsuchidaJPSJ}.\n\\begin{equation}\n\\Psi _{x}=U\\Psi ,\\quad \\Psi _{t}=V\\Psi \\,,  \\label{vcomSPE_Laxa}\n\\end{equation\nwith\n\\[\nU=\\lambda \\left(\n\\begin{array}{cc}\nI_{2^{n-1}} & Q_{x}^{(n)} \\\\\nR_{x}^{(n)} & -I_{2^{n-1}\n\\end{array\n\\right) ,\n\\\n\\[\nV=\\left(\n\\begin{array}{cc}\n-\\frac{1}{2}Q^{(n)}R^{(n)}-\\frac{1}{4\\lambda }I_{2^{n-1}} & -\\frac{\\lambda }{2\nQ^{(n)}R^{(n)}Q_{x}^{(n)}+\\frac{1}{2}Q^{(n)} \\\\\n-\\frac{\\lambda }{2}R^{(n)}Q^{(n)}R_{x}^{(n)}-\\frac{1}{2}R^{(n)} & \\frac{1}{2\nQ^{(n)}R^{(n)}+\\frac{1}{4\\lambda }I_{2^{n-1}\n\\end{array\n\\right) \\,,\n\\\nwhere $I_{2^{n-1}}$ is a $2^{n-1}\\times 2^{n-1}$ identity matrix, $Q^{(n)}$\nand $R^{(n)}$ are $2^{n-1}\\times 2^{n-1}$ matrices can be constructed\nrecursively as follows\n\\begin{equation}\nQ^{(1)}=q_{1},\\quad R^{(1)}=q_{1}^{\\ast }\\,,  \\label{NCSPE_Laxb1}\n\\end{equation\n\\begin{equation}\nQ^{(n+1)}=\\left(\n\\begin{array}{cc}\nQ^{(n)} & q_{n+1}I_{2^{n-1}} \\\\\n-q_{n+1}^{\\ast }I_{2^{n-1}} & R^{(n)\n\\end{array\n\\right) \\,,  \\label{NCSPE_Laxb2}\n\\end{equation}\n\n\\begin{equation}  \\label{NCSPE_Laxb3}\nR^{(n+1)}= \\left\n\\begin{array}{cc}\nR^{(n)} & -q_{n+1} I_{2^{n-1}} \\\\\nq^*_{n+1}I_{2^{n-1}} & Q^{(n)\n\\end{array\n\\right) \\,.\n\\end{equation}\nBy the above construction, we have $R^{(n+1)}=(Q^{(n+1)})^\\dag$, and further\n\\begin{equation}\nQ^{(n)} R^{(n)}=R^{(n)}Q^{(n)}=\\sum_{i=1}^n|q_i|^2I_{2^{n-1}}\\,.\n\\end{equation}\nTherefore, the zero curvature condition $U_t-V_x+[U,\\,V]=0$ gives the vector\ncomplex coupled short pulse equation (\\ref{NCSPE}).\n\n\\subsection{Local and nonlocal conservation laws}\n\nFollowing a systematic method developed by in \\cit\n{TsuchidaJPSJ,WadatiPTP75,WadatiJPSJ79,Zimerman}, we construct conservation\nlaws for the vector complex short pulse equation, the conservation laws for\nthe complex and coupled short pulse equations can be treated as special\ncases for $n=1,2$, respectively. To this end, let us rewrite the Lax pair\nfor the vector complex short pulse equation as follows:\n\\begin{equation}  \\label{Laxpair_vcspe1}\n\\left\n\\begin{array}{c}\n\\Psi_1 \\\\\n\\Psi_\n\\end{array}\n\\right)_x = \\left\n\\begin{array}{cc}\n\\lambda I & \\lambda Q_x \\\\\n\\lambda R_x & - \\lambda \n\\end{array\n\\right) \\left\n\\begin{array}{c}\n\\Psi_1 \\\\\n\\Psi_\n\\end{array}\n\\right)\\,,\n\\end{equation}\n\n\\begin{equation}  \\label{Laxpair_vcspe2}\n\\left\n\\begin{array}{c}\n\\Psi_1 \\\\\n\\Psi_\n\\end{array}\n\\right)_t = \\left(\\begin{array}{cc}\n-\\frac{\\lambda }{2}QR-\\frac{1}{4\\lambda }I_{2} & -\\frac{\\lambda }{2}QRQ_{x}\n\\frac{1}{2}Q \\\\\n-\\frac{\\lambda }{2}RQR_{x}-\\frac{1}{2}R & \\frac{\\lambda }{2}QR+\\frac{1}\n4\\lambda }I_{2\n\\end{array\n\\right)  \\left\n\\begin{array}{c}\n\\Psi_1 \\\\\n\\Psi_\n\\end{array}\n\\right)\\,.\n\\end{equation}\nHere the size of matrices in the entries of Eqs. (\\ref{Laxpair_vcspe1})--\n\\ref{Laxpair_vcspe2}) is of $2^{n-1} \\times 2^{n-1}$ and is omitted for\nbrevity. If we define\n\\begin{equation}  \\label{Cons_law1}\n\\Gamma \\equiv \\Psi_2 \\Psi_1^{-1}\\,\n\\end{equation}\nthen we have\n\\begin{equation}  \\label{Cons_law2}\n2 \\lambda Q_x \\Gamma = \\lambda Q_x R_x - Q_x ((Q_x)^{-1} \\cdot Q_x \\Gamma)_x\n- \\lambda (Q_x \\Gamma)^2\\,\n\\end{equation}\n\n\nExpanding $Q_x \\Gamma$ in terms of the spectral parameter $\\lambda$ as\nfollows\n\\begin{equation}  \\label{Cons_law4}\nQ_x \\Gamma = \\sum_{n=0}^\\infty F_n \\lambda^{-n}\\,,\n\\end{equation}\nand substituting into Eq. (\\ref{Cons_law2}), we obtain the following\nrelation\n\\begin{equation}  \\label{Cons_law5}\n2 \\lambda F_n = Q_x R_x \\delta_{n,0} - Q_x((Q_x)^{-1} F_{n-1})_x -\n\\sum_{l=0}^n F_l F_{n-l}.\n\\end{equation}\nThe first local conserved density turns out to be\n\\begin{equation}  \\label{Cons_law6}\nF_0=\\left(-1+\\sqrt{1+\\sum|q_{i,x}|^2}\\right)I\\,,\n\\end{equation}\n\nwhich is associated with a Hamiltonian of\n\\begin{equation}  \\label{Cons_law7a}\nH_0=\\int \\sqrt{1+|q_x|^2}\\, dx \\,,\n\\end{equation}\nfor the complex short pulse equation (\\ref{CSP}) and\n\\begin{equation}  \\label{Cons_law8a}\nH_0=\\int \\sqrt{1+|q_{1,x}|^2+|q_{2,x}|^2}\\, dx \\,,\n\\end{equation}\nfor the coupled complex short pulse equation (\\ref{CCSP1})--(\\ref{CCSP2}).\n\nFollowing the procedure in \\cite{Zimerman}, we can find the nonlocal\nconservation laws for vector complex short pulse equation. To this end, we\nexpand $Q_x \\Gamma$ as follows\n\\begin{equation}  \\label{Cons_law9}\nQ_x \\Gamma = \\sum_{n=1}^\\infty F_{-n} (2\\lambda)^{n}\\,.\n\\end{equation}\nThe first two orders in $\\lambda$ yield the following equations\n\\begin{equation}  \\label{Cons_law10}\n0 = Q_x R_x -Q_x \\left((Q_x)^{-1}F_{-1}\\right)_x\\,,\n\\end{equation}\n\\begin{equation}  \\label{Cons_law11}\n2F_{-1} = -Q_x \\left((Q_x)^{-1}F_{-2}\\right)_x\\,,\n\\end{equation}\nfrom which, the first two nonlocal conserved densities can be calculated as\n\\begin{equation}  \\label{Cons_law12}\nF_{-1}= \\frac 12 Q_x R\\,,\n\\end{equation}\n\\begin{equation}  \\label{Cons_law13}\nF_{-2}= \\frac 12 Q R\\ -\\frac 12 \\partial_x\\left(Q\\partial_x R \\right).\n\\end{equation}\nThe first one turns out to be a trivial one, the second one accounts for a\nHamiltonian\n\\begin{equation}  \\label{Cons_law7c}\nH_{-1}= \\frac 12 \\int |q|^2\\, dx \\,,\n\\end{equation}\nfor the complex short pulse equation (\\ref{CSP}) and\n\\begin{equation}  \\label{Cons_law8c}\nH_{-1}= \\frac 12\\int (|q_{1}|^2+|q_{2}|^2)\\, dx \\,,\n\\end{equation}\nfor the coupled complex short pulse equation (\\ref{CCSP1})--(\\ref{CCSP2}).\n\\section{Multi-soliton solutions by Hirota's bilinear method}\n\n\n\\subsection{Bilinear equations and $N$-soliton solution to the complex short\npulse equation}\n\n\\begin{proposition}\nThe complex short pulse equation is derived from the following bilinear\nequations.\n\\begin{equation}  \\label{CSPE_bilinear1}\nD_sD_y f \\cdot g =fg\\,,\n\\end{equation}\n\\begin{equation}  \\label{CSPE_bilinear2}\nD^2_s f \\cdot f =\\frac{1}{2} |g|^2\\,,\n\\end{equation}\nby dependent variable transformation\n\\begin{equation}  \\label{CSP_dtrf}\nq=\\frac{g}{f}\\,,\n\\end{equation}\nand hodograph transformation\n\\begin{equation}\nx = y -2(\\ln f)_s\\,, \\quad t=-s \\,,  \\label{CSP_hodograph}\n\\end{equation}\nwhere $D$ is called Hirota $D$-operator defined by\n\\[\nD_s^n D_y^m f\\cdot g=\\left(\\frac{\\partial}{\\partial s} -\\frac{\\partial}\n\\partial s^{\\prime }}\\right)^n \\left(\\frac{\\partial}{\\partial y} -\\frac\n\\partial}{\\partial y^{\\prime }}\\right)^m f(y,s)g(y^{\\prime },s^{\\prime\n})|_{y=y^{\\prime }, s=s^{\\prime }}\\,.\n\\]\n\\end{proposition}\n\n.\n\n\\begin{proof}\nDividing both sides by $f^2$, the bilinear equations (\\ref{CSPE_bilinear1\n)-- (\\ref{CSPE_bilinear2}) can be cast into\n\\begin{equation}\n\\left\\\n\\begin{array}{l}\n\\displaystyle \\left(\\frac{g}{f} \\right)_{sy} + 2\\frac{g}{f} \\left( \\ln\nf\\right)_{sy} = \\frac{g}{f}\\,, \\\\[5pt]\n\\displaystyle \\left( \\ln f\\right)_{ss} =\\frac{1}{4} \\frac{|g|^2}{f^2}\\,\n\\end{array\n\\right.  \\label{CSP_BL2}\n\\end{equation}\nFrom the hodograph transformation and dependent variable transformation, we\nthen have\n\\[\n\\frac{\\partial x}{\\partial s} = -2(\\ln f)_{ss} = -\\frac 12 |q|^2\\,, \\qquad\n\\frac{\\partial x}{\\partial y} = 1-2(\\ln f)_{sy}\\,,\n\\]\nwhich implies\n\\begin{equation}  \\label{CSP_BL3}\n{\\partial_y} = \\rho^{-1} {\\partial_x}\\,, \\qquad {\\partial_s} = -{\\partial_t}\n- \\frac 12 |q|^2 {\\partial_x}\\,\n\\end{equation}\nby letting $1-2(\\ln f)_{sy} = \\rho^{-1}$.\n\nNotice that the first equation in (\\ref{CSP_BL2}) can be rewritten as\n\\begin{equation}\n\\left(\\frac{g}{f} \\right)_{sy} = \\left(1-2(\\ln f)_{sy} \\right) \\frac{g}{f}\\,,\n\\end{equation}\nor\n\\begin{equation}  \\label{CSP_BL4}\n\\rho \\left(\\frac{g}{f} \\right)_{sy} = \\frac{g}{f}\\,,\n\\end{equation}\nwhich is converted into\n\\begin{equation}  \\label{CSPE1}\n\\partial_x \\left(-\\partial_t - \\frac 12 |q|^2 \\partial_x \\right)q = q\\,,\n\\end{equation}\nby using (\\ref{CSP_BL3}). Eq. (\\ref{CSPE1}) is nothing but the complex short\npulse equation (\\ref{CSP}).\n\\end{proof}\n\n$N$-soliton solution to the bilinear equations (\\ref{CSPE_bilinear1})--(\\re\n{CSPE_bilinear2}) can be expressed by pfaffians similar to the ones for\ncoupled modified KdV equation \\cite{IwaoHirota}. To this end, we need to\ndefine two sets: $B_\\mu$ ($\\mu=1,2$): $B_1 = \\{b_1, b_2, \\cdots, b_{N} \\}$, \nB_2 = \\{b_{N+1}, b_2, \\cdots, b_{2N} \\}$, and an index function of $b_j$ by \nindex(b_j) = \\mu$ \\ if $b_j \\in B_\\mu$.\n\n\\begin{theorem}\nThe pfaffians\n\\begin{eqnarray}\nf &=& \\mathrm{Pf} (a_1, \\cdots, a_{2N}, b_1, \\cdots, b_{2N})\\,,\n\\label{CSP_sl1} \\\\\ng &=& \\mathrm{Pf} (d_0, \\beta_1, a_1, \\cdots, a_{2N}, b_1, \\cdots, b_{2N})\\,.\n\\label{CSP_sl2}\n\\end{eqnarray}\nsatisfy the bilinear equations (\\ref{CSPE_bilinear1})--(\\ref{CSPE_bilinear2\n) provided that the elements of the pfaffians are defined by\n\\begin{equation}  \\label{CSPE_pf1}\n\\mathrm{Pf}(a_j,a_k)= \\frac{p_j-p_k}{p_j+p_k} e^{\\eta_j+\\eta_k}\\,, \\quad\n\\mathrm{Pf}(a_j,b_k)=\\delta_{j,k}\\,,\n\\end{equation}\n\\begin{equation}  \\label{CSPE_pf2}\n\\mathrm{Pf}(b_j,b_k)=\\frac 14 \\frac{\\alpha_j \\alpha_k}{p^{-2}_j-p^{-2}_{k}}\n\\delta_{\\mu+1, \\nu}\\,, \\quad \\mathrm{Pf}(d_l,a_k)= p_k^{l} e^{\\eta_k}\\,,\n\\end{equation}\n\n\\begin{equation}  \\label{CSPE_pf4}\n\\mathrm{Pf}(b_j,\\beta_1)=\\alpha_j \\delta_{\\mu, 1}\\,, \\quad \\mathrm{Pf\n(d_0,b_j) =\\mathrm{Pf}(d_0,\\beta_1) = \\mathrm{Pf}(a_j,\\beta_1)=0\\,.\n\\end{equation}\nHere $\\mu=index(b_j)$, $\\nu=index(b_k)$, $\\eta_j=p_j y + p_j^{-1} s$ which\nsatisfying $p_{j+N}=\\bar{p}_j$, $\\alpha_{j+N}=\\bar{\\alpha}_{j}$, $\\bar{p_j}$\nand $\\bar{\\alpha}_{j}$ represent the complex conjugates of $p_j$ and ${\\alph\n}_{j}$, respectively. The same notation will be used hereafter.\n\\end{theorem}\n\nThe proof of the Theorem is given in Appendix. Combined with dependent and\nhodograph transformations (\\ref{CSP_dtrf})--(\\ref{CSP_hodograph}), the above\npfaffians (\\ref{CSP_sl1})--(\\ref{CSP_sl2}) give $N$-soliton solution to the\ncomplex short pulse equation (\\ref{CSP}) in parametric form.\n\n\\subsection{One- and two-soliton solutions for the complex short pulse\nequation}\n\nIn this subsection, we provide one- and two-soliton to the complex short\npulse equation (\\ref{CSP}) and give a detailed analysis for their properties.\n\n\\subsubsection{One-soliton solution}\n\nBased on (\\ref{CSP_sl1})--(\\ref{CSP_sl2}), the tau-functions for one-soliton\nsolution ($N=1$) are\n\\begin{eqnarray}\n&& f = -1-\\frac 14 \\frac {|\\alpha_1|^2(p_1\\bar{p}_1)^2}{(p_1+\\bar{p}_1)^2}\ne^{\\eta_1+\\bar{\\eta}_1} \\,,\n\\end{eqnarray}\n\\begin{equation}\ng = -\\alpha_1 e^{\\eta_1}\\,.\n\\end{equation}\n\nLet $p_1=p_{1R}+ \\mathrm{i}p_{1I}$, and we assume $p_{1R} >0$ without loss\nof generality, then the one-soliton solution can be expressed in the\nfollowing parametric form\n\\begin{equation}  \\label{CSP1solitona}\nq=\\frac{\\alpha_1}{|\\alpha_1|}\\frac{2p_{1R}}{|p_1|^2}e^{\\mathrm{i} \\eta_{1I}}\n\\mbox{sech} \\left(\\eta_{1R}+\\eta_{10}\\right) \\,,\n\\end{equation}\n\\begin{equation}  \\label{CSP1solitonb}\nx=y-\\frac{2p_{1R}}{|p_1|^2}\\left(\\tanh\n\\left(\\eta_{1R}+\\eta_{10}\\right)+1\\right)\\,, \\quad t=-s\\,,\n\\end{equation}\nwhere\n\\begin{equation}\n\\eta_{1R}=p_{1R}y+\\frac{p_{1R}}{|p_1|^2}s , \\quad \\eta_{1I}=p_{1I} y-\\frac\np_{1I}}{|p_1|^2}s \\,,\\quad \\eta_{10}=\\ln \\frac{|\\alpha_1||p_1|^2}{4p_{1R}}\\,.\n\\end{equation}\n\nEq. (\\ref{CSP1solitona}) represents an envelope soliton of amplitude \n2p_{1R}\/|p_1|^2$ and phase $\\eta_{1I}$. To analyze the property for the\none-soliton solution, we calculate out\n\\begin{equation}\n\\frac{\\partial x}{\\partial y} = 1- \\frac{2p^2_{1R}}{|p_1|^2} {\\mbox{sech}\n^2(\\eta_{1R}+\\eta_{10})\\,.\n\\end{equation}\nTherefore, $\\partial x\/\\partial y \\to 1$ as $y \\to \\pm \\infty$. Moreover, it\nattains a minimum value of $({p^2_{1I}-p^2_{1R}})\/({p^2_{1I}+p^2_{1R}})$\nat the peak point of envelope soliton where $\\eta_{1R}+\\eta_{10}=0$. Since $\n\\partial |q|}\/{\\partial x}=\\frac{\\partial |q|\/\\partial y}{\\partial\nx\/\\partial y}$, we can classify this one-soliton solution as follows:\n\n\\begin{itemize}\n\\item \\textbf{smooth soliton:} when $|p_{1R}| < |p_{1I}|$, ${\\partial x}\/\n\\partial y}$ is always positive, which leads to a smooth envelope soliton\nsimilar to the envelope soliton for the nonlinear Schr{\\\"o}dinger equation.\nAn example with $p_1=1+1.5\\mathrm{i}$ is illustrated in Fig. 1 (a).\n\n\\item \\textbf{loop soliton:} when $|p_{1R}| > |p_{1I}|$, the minimum value\nof ${\\partial x}\/{\\partial y}$ at the peak point of the soliton becomes\nnegative. In view of the fact that $\\partial x\/\\partial y \\to 1$ as $y \\to\n\\pm \\infty$, ${\\partial x}\/{\\partial y}$ has two zeros at both sides of the\npeak of the envelope soliton. Moreover, ${\\partial x}\/{\\partial y}< 0$\nbetween these two zeros. This leads to a loop soliton for the envelope of $q\n. An example is shown in Fig. (b) with $p_1=1+0.5\\mathrm{i}$.\n\n\\item \\textbf{cuspon soliton:} when $|p_{1R}| = |p_{1I}|$, ${\\partial x}\/\n\\partial y}$ has a minimum value of zero at $\\eta_{1R}+\\eta_{10}=0$, which\nmakes the derivative of the envelope $|q|$ with respect to $x$ going to\ninfinity at the peak point. Thus, we have a cusponed envelope soliton, which\nis illustrated in Fig. 1 (c) with $p_1=1+\\mathrm{i}$.\n\\end{itemize}\n\\begin{figure}[htbp]\n\\centerline{\n\\includegraphics[scale=0.35]{onesolitona1b1p5.eps}\\quad\n\\includegraphics[scale=0.35]{onesolitona1b0p5.eps}} \\kern-0.3\\textwidth\n\\hbox to\n\\textwidth{\\hss(a)\\kern16em\\hss(b)\\kern11em} \\kern+0.3\\textwidth\n\\centerline{\n\\includegraphics[scale=0.35]{onesolitona1b1.eps}} \\kern-0.3\\textwidth\n\\hbox to\n\\textwidth{\\hss(c)\\kern21em} \\kern+0.3\\textwidth\n\\caption{Envelope soliton for the complex short pulse equation (\\protect\\re\n{CSP}), solid line: $Re(q)$, dashed line: $|q|$; (a) smooth soliton with \np_1=1+1.5\\mathrm{i}$, (b) loop soliotn with $p_1=1+0.5\\mathrm{i}$, (c)\ncuspon soliton with $p_1=1+\\mathrm{i}$.}\n\\label{figure:cspe1soliton}\n\\end{figure}\n\n\\begin{remark}\nThe one-soliton solution to the short pulse equation (\\ref{SPE}) is of\nloop-type, which lacks physical meaning in the context of nonlinear optics.\nHowever, the one-soliton solution to the complex short pulse equation (\\re\n{CSP}) is of breather-type, which allows physical meaning for optical pulse.\n\\end{remark}\n\n\\begin{remark}\nWhen $|p_{1R}| <|p_{1I}|$, there is no singularity for one-soliton solution.\nMoreover, in view of $\\eta_{1R}$ associated with the width of envelope\nsoliton and $\\eta_{1I}$ associated with the phase, it is obvious that this\nnonsingular envelope soliton can only contain a few optical cycle. This\nproperty coincides with the fact that the complex short pulse equation is\nderived for the purpose of describing ultra-short pulse propagation. When \n|p_{1R}| =|p_{1I}|$, the soliton becomes cuspon-like one, which agrees with\nthe results in \\cite{Bandelow} derived from a bidirectional model.\n\\end{remark}\n\n\\subsubsection{Two-soliton solution}\n\nBased on the $N$-soliton solution of the complex short pulse equation from \n\\ref{CSP_sl1})--(\\ref{CSP_sl2}), the tau-functions for two-soliton solution\ncan be expanded for $N=2$\n\\begin{eqnarray}\n&&f=\\mathrm{Pf}(a_{1},a_{2},a_{3},a_{4},b_{1},b_{2},b_{3},b_{4})  \\nonumber\n\\\\\n&&\\quad =1+a_{1\\bar{1}}e^{\\eta _{1}+\\bar{\\eta}_{1}}+a_{1\\bar{2}}e^{\\eta _{1}\n\\bar{\\eta}_{2}}+a_{2\\bar{1}}e^{\\eta _{2}+\\bar{\\eta}_{1}}+a_{2\\bar{2}}e^{\\eta\n_{2}+\\bar{\\eta _{2}}}  \\nonumber \\\\\n&&\\qquad +|P_{12}|^{2}\\left( a_{1\\bar{1}}a_{2\\bar{2}}P_{1\\bar{2}}P_{2\\bar{1\n}-a_{1\\bar{2}}a_{2\\bar{1}}P_{1\\bar{1}}P_{2\\bar{2}}\\right) e^{\\eta _{1}+\\eta\n_{2}+\\bar{\\eta}_{1}+\\bar{\\eta}_{2}}\\,,\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&&g=\\mathrm{Pf}(d_{0},\\beta\n_{1},a_{1},a_{2},a_{3},a_{4},b_{1},b_{2},b_{3},b_{4})  \\nonumber \\\\\n&&\\quad =\\alpha _{1}e^{\\eta _{1}}+\\alpha _{2}e^{\\eta _{2}}+P_{12}\\left(\n\\alpha _{1}P_{1\\bar{1}}a_{2\\bar{1}}-\\alpha _{2}P_{2\\bar{1}}a_{1\\bar{1\n}\\right) e^{\\eta _{1}+\\eta _{2}+\\bar{\\eta}_{1}}  \\nonumber \\\\\n&&\\qquad +P_{12}\\left( \\alpha _{1}P_{1\\bar{2}}a_{2\\bar{2}}-\\alpha _{2}P_{\n\\bar{2}}a_{1\\bar{2}}\\right) e^{\\eta _{1}+\\eta _{2}+\\bar{\\eta}_{2}}\\,,\n\\end{eqnarray\nwhere\n\\begin{equation}\nP_{ij}=\\frac{p_{i}-p_{j}}{p_{i}+p_{j}}\\,,\\quad P_{i\\bar{j}}=\\frac{p_{i}-\\bar\np}_{j}}{p_{i}+\\bar{p}_{j}}\\,,\\quad a_{i\\bar{j}}=\\frac{\\alpha _{i}\\bar{\\alpha\n_{j}(p_{i}\\bar{p}_{j})^{2}}{4(p_{i}+\\bar{p}_{j})^{2}}\\,,\n\\end{equation\nand $\\eta _{j}=p_{j}y+p_{j}^{-1}s$, $\\bar{\\eta}_{j}=\\bar{p}_{j}y+\\bar{p\n_{j}^{-1}s$.\n\n\\begin{figure}[htbp]\n\\centerline{\n\\includegraphics[scale=0.35]{CSPE_2soliton_ct.eps}\\quad\n\\includegraphics[scale=0.35]{CSPE_2soliton_elastic.eps}} \\kern-0.31\n\\textwidth \\hbox to\n\\textwidth{\\hss(a)\\kern4em\\hss(b)\\kern2em} \\kern+0.315\\textwidth\n\\caption{Two-soliton solution to the complex short pulse equation (a)\ncontour plot; (b) profiles at $t=-80$, $80$.}\n\\label{f:1com2soliton}\n\\end{figure}\nTo avoid the singularity of the envelope solitons, the conditions $|p_{1R}|<\n|p_{1I}|$ and $|p_{2R}|< |p_{2I}|$ need to be satisfied.\nWhen two solitons stay apart, the amplitude of each soliton is of $2|p_{iR}|\/|p_{i}|^2$, and the velocity is of $-1\/|p_i|^2$ in the $ys$-coordinate system. Therefore, the soliton of larger velocity will catch up with and collide with the soliton of smaller velocity if it is initially located on the left. Furthermore, the\ncollision is elastic, and there is no change in shape and amplitude of\nsolitons except a phase shift. In Fig. 2, we illustrate\nthe contour plot for the collision of two solitons (a), as well as the\nprofiles (b) before and after the collision. The parameters are taken as \n\\alpha_1=\\alpha_2=1.0$, $p_1=1+1.2\\mathrm{i}$ and $p_2=1+2\\mathrm{i}$.\n\nSince the velocity of single envelope soliton is $-1\/|p_i|^2$ in the $ys\n-coordinate system, a bound state can be formed under the condition of \n|p_1|^2 = |p_2|^2$ if two solitons stay close enough and move with the same\nvelocity. Such a bound state is shown in Fig. 3 for\nparameters chosen as $\\alpha_1=\\alpha_2=1.0$, $p_1=1.3+1.8193\\mathrm{i}$, \np_2=1+2\\mathrm{i}$.\nIt is interesting that the envelope of the bound state oscillates\nperiodically as it moves along $x$-axis.\n\\begin{figure}[htbp]\n\\centerline{\n\\includegraphics[scale=0.35]{1com_bd3d.eps}\\quad\n\\includegraphics[scale=0.35]{1com_bdprofile.eps}} \\kern-0.315\\textwidth\n\\hbox to\n\\textwidth{\\hss(a)\\kern4em\\hss(b)\\kern2em} \\kern+0.315\\textwidth\n\\caption{Bound state to the complex short pulse equation: (a) 3D plot (b)\nprofiles at $t=-100$, $40$.}\n\\label{f:1comboundstate}\n\\end{figure}\n\\subsection{Bilinear equations and $N$-soliton solutions to the coupled\ncomplex short pulse equation}\n\n\\begin{proposition}\nThe coupled complex short pulse equation is derived from bilinear equations\n\\begin{equation}  \\label{CCSPE_bilinear1}\nD_sD_y f \\cdot g_i =fg_i, \\quad i=1,2 \\,,\n\\end{equation}\n\\begin{equation}  \\label{CCSPE_bilinear2}\nD^2_s f \\cdot f =\\frac{1}{2} \\left(|g_1|^2+|g_2|^2\\right)\\,,\n\\end{equation}\nby dependent variable transformation\n\\begin{equation}  \\label{CCSPE_vartrf}\nq_1=\\frac{g_1}{f}, \\quad q_2=\\frac{g_2}{f}\\,,\n\\end{equation}\nand hodograph transformation\n\\begin{equation}\nx = y -2(\\ln f)_s\\,, \\quad t=-s \\,,  \\label{CCSP_hodograph}\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nDividing both sides of Eqs. (\\ref{CCSPE_bilinear1})--(\\ref{CCSPE_bilinear2})\nby $f^2$, we have\n\\begin{equation}\n\\left(\\frac{g_i}{f} \\right)_{sy} + 2\\frac{g_i}{f} \\left( \\ln f\\right)_{sy} =\n\\frac{g_i}{f}\\,,  \\label{CCSP_BL2a}\n\\end{equation}\n\\begin{equation}\n\\left( \\ln f\\right)_{ss} =\\frac{1}{4} \\left( \\frac{|g_1|^2}{f^2}+\\frac\n|g_2|^2}{f^2} \\right)\\,.  \\label{CCSP_BL2b}\n\\end{equation}\n\nFrom dependent variable and hodograph transformations (\\ref{CCSPE_vartrf})--\n\\ref{CCSP_hodograph}), we obtain\n\\[\n\\frac{\\partial x}{\\partial s} = -2(\\ln f)_{ss} = -\\frac 12\n\\left(|q_1|^2+|q_2|^2 \\right)\\,, \\qquad \\frac{\\partial x}{\\partial y} =\n1-2(\\ln f)_{sy}\\,,\n\\]\nwhich implies\n\\begin{equation}  \\label{CCSP_BL3}\n{\\partial_y} = \\rho^{-1} {\\partial_x}\\,, \\qquad {\\partial_s} = -{\\partial_t}\n- \\frac 12 \\left(|q_1|^2+|q_2|^2 \\right) {\\partial_x}\\,\n\\end{equation}\nby letting $1-2(\\ln f)_{sy} = \\rho^{-1}$.\n\nWith the use of (\\ref{CCSP_BL3}), Eq. (\\ref{CCSP_BL2a}) can be recast into\n\\begin{equation}  \\label{CCSP_BL4}\n\\rho \\left(\\frac{g_i}{f} \\right)_{sy} = \\frac{g_i}{f}\\,, \\quad i=1,2\\,,\n\\end{equation}\nwhich can be further converted into\n\\begin{equation}  \\label{CCSPE_alt}\n\\partial_x \\left(-\\partial_t - \\frac 12 (|q_1|^2+|q_2|^2) \\partial_x\n\\right)q_i = q_i\\,, \\quad i=1,2\\,.\n\\end{equation}\nEq. (\\ref{CCSPE_alt}) is, obviously, equivalent to the coupled complex short\npulse equation (\\ref{CCSP1})--(\\ref{CCSP2}).\n\\end{proof}\n\n\n$N$-soliton solution for the coupled complex short pulse equation is given\nin a similar way as the complex short pulse equation by the following\ntheorem.\n\n\n\\begin{theorem}\nThe coupled complex short pulse equation admits the following $N$-soliton\nsolution\n\\[\nq_i=\\frac{g_i}{f}, \\quad x = y -2(\\ln f)_s\\,, \\quad t=-s \\,,\n\\]\nwhere $f$, $g_i$ are pfaffians defined as\n\\begin{eqnarray}  \\label{CCSPE_Nsoliton1}\nf &=& \\mathrm{Pf} (a_1, \\cdots, a_{2N}, b_1, \\cdots, b_{2N})\\,, \\\\\ng_i &=& \\mathrm{Pf} (d_0, \\beta_{i}, a_1, \\cdots, a_{2N}, b_1, \\cdots,\nb_{2N})\\,,  \\label{CCSPE_Nsoliton2}\n\\end{eqnarray}\nand the elements of the pfaffians are determined as\n\\begin{equation}  \\label{NCSPE_pf1}\n\\mathrm{Pf}(a_j,a_k)= \\frac{p_j-p_k}{p_j+p_k} e^{\\eta_j+\\eta_k}\\,, \\quad\n\\mathrm{Pf}(a_j,b_k)=\\delta_{j,k}\\,,\n\\end{equation}\n\\begin{equation}  \\label{NCSPE_pf2}\n\\mathrm{Pf}(b_j,b_k)=\\frac 14 \\frac{\\sum^2_{i=1} \\alpha^{(i)}_j\n\\alpha^{(i)}_k }{p^{-2}_j-p^{-2}_{k}} \\delta_{\\mu+1, \\nu}\\,, \\quad \\mathrm{P\n}(d_l,a_k)= p_k^{l} e^{\\eta_k}\\,,\n\\end{equation}\n\n\\begin{equation}  \\label{NCSPE_pf4}\n\\mathrm{Pf}(b_j,\\beta_i)=\\alpha^{(i)}_j \\delta_{\\mu, i}\\,,\\quad \\mathrm{Pf\n(d_0,b_j) =\\mathrm{Pf}(d_0,\\beta_i) = \\mathrm{Pf}(a_j,\\beta_i)=0\\,.\n\\end{equation}\nHere $\\mu=index(b_j)$, $\\nu=index(b_k)$, $\\eta_j=p_j y + p_j^{-1} s +\n\\eta_{j,0}$ which satisfying $p_{j+N}=\\bar{p}_j$, $\\alpha_{j+N}=\\bar{\\alpha\n_{j}$.\n\\end{theorem}\n\nThe proof of the Theorem is given in the Appendix. In the subsequent\nsection, based on the $N$-soliton solution of coupled complex short pulse\nequation, we will investigate the dynamics of one- and two-solitons in\ndetails.\n\n\\begin{remark}\nThrough the transformations\n\\begin{equation}  \\label{NCSPE_trfs}\nx = y -2(\\ln f)_s\\,, \\quad t=-s \\,, \\quad q_i=\\frac{g_i}{f}\\,,\n\\end{equation}\nthe vector complex short pulse equation (\\ref{NCSPE}) can be decomposed into\nthe following bilinear equations\n\\begin{equation}  \\label{NCSPE_bilinear1}\nD_sD_y f \\cdot g_i =fg_i, \\quad i=1, \\cdots, n \\,,\n\\end{equation}\n\\begin{equation}  \\label{NCSPE_bilinear2}\nD^2_s f \\cdot f =\\frac{1}{2} \\left(\\sum^n_{i=1}|g_i|^2\\right)\\,.\n\\end{equation}\nThe parametric form of $N$-soliton solution in terms of pfaffians to the\nvector complex short pulse equation (\\ref{NCSPE}) can be given in a very\nsimilar from as to to the coupled complex short pulse equation. Here, we\nomit the details and will report the results later on.\n\\end{remark}\n\n\\section{Dynamics of solitons to the coupled complex short pulse equation}\n\\subsection{One-soliton solution}\nThe tau-functions for one-soliton solution to the coupled complex short\npulse equation are obtained from (\\ref{CCSPE_Nsoliton1})--(\\re\n{CCSPE_Nsoliton2}) for $N=1$\n\\begin{equation}\nf = -1-\\frac 14 \\frac {\\sum_{i=1}^2|\\alpha^{(i)}_1|^2(p_1\\bar{p}_1)^2}{(p_1\n\\bar{p}_1)^2} e^{\\eta_1+\\bar{\\eta}_1} \\,,\n\\end{equation}\n\\begin{equation}\ng_1 = -\\alpha^{(1)}_1 e^{\\eta_1}\\,, \\quad g_2 = -\\alpha^{(2)}_1 e^{\\eta_1}\\,.\n\\end{equation}\n\nLet $p_{1}=p_{1R}+\\mathrm{i}p_{1I}$, the one-soliton solution can be\nexpressed in the following parametric form\n\\begin{equation}\n\\left(\n\\begin{array}{c}\nq_{1} \\\\\nq_{2\n\\end{array\n\\right) =\\left(\n\\begin{array}{c}\nA_{1} \\\\\nA_{2\n\\end{array\n\\right) \\frac{2p_{1R}}{|p_{1}|^{2}}e^{\\mathrm{i}\\eta _{1I}}{\\mbox{sech}\n\\left( \\eta _{1R}+\\eta _{10}\\right) \\,,  \\label{1soliton_ay}\n\\end{equation\n\\begin{equation}\nx=y-\\frac{2p_{1R}}{|p_{1}|^{2}}\\left( \\tanh (\\eta _{1R}+\\eta _{10})+1\\right)\n\\,,\\quad t=-s\\,,  \\label{CCSP1solitonb}\n\\end{equation\nwhere\n\\begin{equation}\n\\eta _{1R}=p_{1R}\\left( y+\\frac{1}{|p_{1}|^{2}}s\\right) ,\\quad \\eta\n_{1I}=p_{1I}\\left( y-\\frac{1}{|p_{1}|^{2}}s\\right) \\,,\n\\end{equation\n\\begin{equation}\nA_{i}=\\frac{\\alpha _{1}^{(i)}}{\\sqrt{\\sum_{i=1}^{2}|\\alpha _{1}^{(i)}|^2}\n\\,,\\quad \\eta_{10}=\\ln \\frac{\\sqrt{\\sum_{i=1}^{2}|\\alpha _{1}^{(i)}|^2\n|p_{1}|^{2}}{4|p_{1R}|}\\,.\n\\end{equation\nThe amplitudes of the single soliton in each component are ${2|A_{1}|p_{1R}}\n{|p_{1}|^{2}}$ and ${2|A_{2}|p_{1R}}\/{|p_{1}|^{2}}$, respectively. Note that\n$|A_{1}|^{2}+|A_{2}|^{2}=1$. Same as the analysis for one-soliton solution\nof complex short pulse equation, if $|p_{1R}|<|p_{1I}|$, the envelope for\none-soliton in each of the component is smooth, whereas, if \n|p_{1R}|>|p_{1I}| $, it becomes a loop (multi-valued) soliton, if \n|p_{1R}|=|p_{1I}|$, it is a cuspon.\n\n\\subsection{Soliton interactions}\nTwo-soliton solution for coupled complex short pulse equation is obtained\nfrom (\\ref{CCSPE_Nsoliton1})--(\\ref{CCSPE_Nsoliton2}) for $N=2$. By\nexpanding the pfaffians, the tau-functions for two-soliton solution are\nexpressed by\n\\begin{eqnarray}\n&&f=1+e^{\\eta _{1}+\\bar{\\eta}_{1}+r_{1\\bar{1}}}+e^{\\eta _{1}+\\bar{\\eta\n_{2}+r_{1\\bar{2}}}+e^{\\eta _{2}+\\bar{\\eta}_{1}+r_{2\\bar{1}}}+e^{\\eta _{2}\n\\bar{\\eta}_{2}+r_{2\\bar{2}}}  \\nonumber \\\\\n&&\\qquad +|P_{12}|^{2}|P_{1\\bar{2}}|^{2}P_{1\\bar{1}}P_{2\\bar{2}}\\left( B_{\n\\bar{1}}B_{2\\bar{2}}-B_{2\\bar{1}}B_{1\\bar{2}}\\right) e^{\\eta _{1}+\\eta _{2}\n\\bar{\\eta}_{1}+\\bar{\\eta}_{2}}\\,,\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&& g_1= \\alpha^{(1)}_{1} e^{\\eta_1} + \\alpha^{(1)}_2 e^{\\eta_2} +P_{12} P_{\n\\bar{1}} P_{2\\bar{1}} \\left( \\alpha^{(1)}_2 B_{1\\bar{1}} - \\alpha^{(1)}_1\nB_{2\\bar{1}} \\right) e^{\\eta_1+\\eta_2+\\bar{\\eta}_1}  \\nonumber \\\\\n&& \\qquad +P_{12} P_{1\\bar{2}} P_{2\\bar{2}} \\left( \\alpha^{(1)}_2 B_{1\\bar{2\n} - \\alpha^{(1)}_1 B_{2\\bar{2}} \\right) e^{\\eta_1+\\eta_2+\\bar{\\eta}_2} \\,,\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&&g_{2}=\\alpha _{1}^{(2)}e^{\\eta _{1}}+\\alpha _{2}^{(2)}e^{\\eta\n_{2}}+P_{12}P_{1\\bar{1}}P_{2\\bar{1}}\\left( \\alpha _{2}^{(2)}B_{1\\bar{1\n}-\\alpha _{1}^{(2)}B_{2\\bar{1}}\\right) e^{\\eta _{1}+\\eta _{2}+\\bar{\\eta}_{1}}\n\\nonumber \\\\\n&&\\qquad +P_{12}P_{1\\bar{2}}P_{2\\bar{2}}\\left( \\alpha _{2}^{(2)}B_{1\\bar{2\n}-\\alpha _{1}^{(2)}B_{2\\bar{2}}\\right) e^{\\eta _{1}+\\eta _{2}+\\bar{\\eta\n_{2}}\\,,\n\\end{eqnarray\nwhere\n\\[\nP_{ij}=\\frac{p_{i}-p_{j}}{p_{i}+p_{j}}\\,,\\quad P_{i\\bar{j}}=\\frac{p_{i}-\\bar\np}_{j}}{p_{i}+\\bar{p}_{j}}\\,,\n\\\n\\[\nB_{i\\bar{j}}=\\frac{\\alpha _{i}^{(1)}\\bar{\\alpha}_{j}^{(1)}+\\alpha _{i}^{(2)\n\\bar{\\alpha}_{j}^{(2)}}{4(p_{i}^{-2}-\\bar{p}_{j}^{-2})}\\,,\\quad e^{r_{i\\bar{\n}}}=\\frac{\\alpha _{i}^{(1)}\\bar{\\alpha}_{j}^{(1)}+\\alpha _{i}^{(2)}\\bar\n\\alpha}_{j}^{(2)}}{4(p_{i}^{-1}+\\bar{p}_{j}^{-1})^2}\\,.\n\\\nand $\\eta _{j}=p_{j}y+p_{j}^{-1}s$, $p_{3}=\\bar{p}_{1}$, $p_{4}=\\bar{p}_{2}\n, thus, $\\eta _{3}=\\bar{\\eta}_{1}$, $\\eta _{4}=\\bar{\\eta}_{2}$.\n\nNext, we investigate the asymptotic behavior of two-soliton solution. To\nthis end, we assume $p_{1R}>p_{2R}>0$, $p_{1R}\/|p_{1}|^{2}>p_{2R}\/|p_{2}|^{2}\n$ without loss of generality. For the above choice of parameters, we have\n(i) $\\eta _{1R}\\approx 0$, $\\eta _{2R}\\rightarrow \\mp \\infty $ as \nt\\rightarrow \\mp \\infty $ for soliton 1 and (ii) $\\eta _{2R}\\approx 0$, \n\\eta _{2R}\\rightarrow \\pm \\infty $ as $t\\rightarrow \\mp \\infty $ for soliton\n2. This leads to the following asymptotic forms for two-soliton solution.\n\\newline\n(i) Before collision ($t\\rightarrow -\\infty $) \\newline\nSoliton 1 ($\\eta _{1R}\\approx 0$, $\\eta _{2R}\\rightarrow -\\infty $):\n\\begin{eqnarray}\n\\left(\n\\begin{array}{c}\nq_{1} \\\\\nq_{2\n\\end{array\n\\right)  &\\rightarrow &\\left(\n\\begin{array}{c}\n\\alpha _{1}^{(1)} \\\\\n\\alpha _{1}^{(2)\n\\end{array\n\\right) \\frac{e^{\\eta _{1}}}{1+e^{\\eta _{1}+\\bar{\\eta}_{1}+r_{1\\bar{1}}}}\\,,\n\\nonumber  \\label{soliton1_aybf} \\\\\n&\\rightarrow &\\left(\n\\begin{array}{c}\nA_{1}^{1-} \\\\\nA_{2}^{1-\n\\end{array\n\\right) \\frac{2p_{1R}}{|p_{1}|^{2}}e^{i\\eta _{1I}}{\\mbox{sech}}\\left( \\eta\n_{1R}+\\frac{r_{1\\bar{1}}}{2}\\right) \\,,\n\\end{eqnarray\nwhere\n\\begin{equation}\n\\left(\n\\begin{array}{c}\nA_{1}^{1-} \\\\\nA_{2}^{1-\n\\end{array\n\\right) =\\left(\n\\begin{array}{c}\n\\alpha _{1}^{(1)} \\\\\n\\alpha _{1}^{(2)\n\\end{array\n\\right) \\frac{1}{\\sqrt{|\\alpha _{1}^{(1)}|^{2}+|\\alpha _{1}^{(2)}|^{2}}}\\,.\n\\end{equation}\n\nSoliton 2 ($\\eta_{2R} \\approx 0$, $\\eta_{1R} \\to \\infty$):\n\\begin{equation}  \\label{soliton2_aybf}\n\\left\n\\begin{array}{c}\nq_1 \\\\\nq_\n\\end{array\n\\right) \\to \\left\n\\begin{array}{c}\nA^{2-}_1 \\\\\nA^{2-}_\n\\end{array\n\\right)\\frac{2p_{2R}}{|p_{2}|^2} e^{i\\eta_{2I}} {\\mbox{sech}}\n\\left(\\eta_{2R}+\\frac {r_{1\\bar{1}2\\bar{2}}-r_{1\\bar{1}}}{2} \\right)\\,,\n\\end{equation}\nwhere\n\\begin{equation}\n\\left\n\\begin{array}{c}\nA^{2-}_1 \\\\\nA^{2-}_\n\\end{array\n\\right) = \\left\n\\begin{array}{c}\ne^{r^{(1)}_{1\\bar{1}2}} \\\\\ne^{r^{(2)}_{1\\bar{1}2}\n\\end{array\n\\right) \\frac{e^{-(r_{1\\bar{1}2\\bar{2}}+r_{1\\bar{1}}-r_{2\\bar{2}})\/{2}}} \n\\sqrt{|\\alpha^{(1)}_2|^2+|\\alpha^{(2)}_2|^2}} \\,,\n\\end{equation}\nwith\n\\begin{equation}\ne^{r^{(i)}_{1\\bar{1}2}}= P_{12} P_{1\\bar{1}} P_{2\\bar{1}} \\left(\n\\alpha^{(i)}_2 B_{1\\bar{1}} - \\alpha^{(i)}_1 B_{2\\bar{1}} \\right)\\,, \\quad\n(i=1,2)\n\\end{equation}\n\\begin{equation}\ne^{r_{1\\bar{1}2\\bar{2}}}=|P_{12}|^2 |P_{1\\bar{2}}|^2 P_{1\\bar{1}} P_{2\\bar{2\n} \\left(B_{1\\bar{1}} B_{2\\bar{2}} -B_{2\\bar{1}}B_{1\\bar{2}}\\right)\\,.\n\\end{equation}\n\\newline\nAfter collision ($t \\to \\infty$) \\newline\nSoliton 1 ($\\eta_{1R} \\approx 0$, $\\eta_{2R} \\to \\infty$):\n\n\\begin{equation}\n\\left(\n\\begin{array}{c}\nq_{1} \\\\\nq_{2\n\\end{array\n\\right) \\rightarrow \\left(\n\\begin{array}{c}\nA_{1}^{1+} \\\\\nA_{2}^{1+\n\\end{array\n\\right) \\frac{2p_{1R}}{|p_{1}|^{2}}e^{i\\eta _{1I}}{\\mbox{sech}}\\left( \\eta\n_{2R}+\\frac{r_{1\\bar{1}2\\bar{2}}-r_{2\\bar{2}}}{2}\\right) \\,,\n\\label{soliton2_ayafter}\n\\end{equation\nwhere\n\\begin{equation}\n\\left(\n\\begin{array}{c}\nA_{1}^{1+} \\\\\nA_{2}^{1+\n\\end{array\n\\right) =\\left(\n\\begin{array}{c}\ne^{r_{12\\bar{1}}^{(1)}} \\\\\ne^{r_{12\\bar{1}}^{(2)}\n\\end{array\n\\right) \\frac{e^{-(r_{1\\bar{1}2\\bar{2}}-r_{1\\bar{1}}+r_{2\\bar{2}})\/{2}}}\n\\sqrt{|\\alpha _{1}^{(1)}|^{2}+|\\alpha _{1}^{(2)}|^{2}}}\\,,\n\\end{equation\nwith\n\\begin{equation}\ne^{r_{12\\bar{1}}^{(i)}}=P_{12}P_{1\\bar{2}}P_{2\\bar{2}}\\left( \\alpha\n_{2}^{(i)}B_{1\\bar{2}}-\\alpha _{1}^{(i)}B_{2\\bar{2}}\\right) \\,,\\quad\n(i=1,2)\\,.\n\\end{equation}\n\nSoliton 2 ($\\eta _{2R}\\approx 0$, $\\eta _{1R}\\rightarrow -\\infty $):\n\\begin{equation}\n\\left(\n\\begin{array}{c}\nq_{1} \\\\\nq_{2\n\\end{array\n\\right) \\rightarrow \\left(\n\\begin{array}{c}\nA_{1}^{2+} \\\\\nA_{2}^{2+\n\\end{array\n\\right) \\frac{2p_{2R}}{|p_{2}|^{2}}e^{i\\eta _{2I}}{\\mbox{sech}}\\left( \\eta\n_{2R}+\\frac{r_{2\\bar{2}}}{2}\\right) \\,,  \\label{soliton2_ayaf}\n\\end{equation\nwhere\n\\begin{equation}\n\\left(\n\\begin{array}{c}\nA_{1}^{2+} \\\\\nA_{2}^{2+\n\\end{array\n\\right) =\\left(\n\\begin{array}{c}\n\\alpha _{2}^{(1)} \\\\\n\\alpha _{2}^{(2)\n\\end{array\n\\right) \\frac{1}{\\sqrt{|\\alpha _{2}^{(1)}|^{2}+|\\alpha _{2}^{(2)}|^{2}}}\\,.\n\\end{equation}\n\nSimilar to the analysis for the CNLS equations \\cit\n{Lakshmanan1997,Lakshmanan2001,Lakshmanan2003}, the change in the amplitude\nof each of the solitons in each component can be obtained by introducing the\ntransition matrix $T^k_j$ by $A_j^{k+}= T_j^k A_j^{k-}$, $j,k=1,2$. The\nelements of transition matrix is obtained from the above asymptotic analysis\nas\n\\begin{equation}  \\label{trasition_matrix1}\nT_j^1=\\left(\\frac{P_{12}P_{1\\bar{2}}}{\\bar{P}_{12}\\bar{P}_{1\\bar{2}}}\n\\right)^{1\/2} \\frac{1}{\\sqrt{1-\\lambda_1 \\lambda_2}} \\left(1-\\lambda_2\\frac\n\\alpha_2^{(j)}}{\\alpha_1^{(j)}} \\right)\\,, \\quad j=1,2\\,,\n\\end{equation}\n\\begin{equation}  \\label{trasition_matrix2}\nT_j^2=\\left(\\frac{\\bar{P}_{12}P_{1\\bar{2}}}{P_{12}\\bar{P}_{1\\bar{2}}}\n\\right)^{1\/2} \\sqrt{1-\\lambda_1 \\lambda_2} \\left(1-\\lambda_1\\frac\n\\alpha_1^{(j)}}{\\alpha_2^{(j)}} \\right)^{-1}\\,, \\quad j=1,2\\,,\n\\end{equation}\nwhere $\\lambda_1=B_{2\\bar{1}}\/B_{1\\bar{1}}$, $\\lambda_2=B_{1\\bar{2}}\/B_{\n\\bar{2}}$.\n\nTherefore, in general, there is an exchange of energies between two components of two solitons\nafter the collision.\nAn example is shown in Fig. 4 for the\nparameters taken as follows\n$p_{1}=1+1.2\\mathrm{i}$, $p_{2}=1+2\\mathrm{i}$, $\\alpha^{(1)}_{1}\n\\alpha^{(2)}_{1}=1.0$, $\\alpha^{(1)} _{2}=2.0$, $\\alpha^{(2)}_{2}=1.0$.\n\\begin{figure}[tbph]\n\\centerline{\n\\includegraphics[scale=0.35]{Inelastic_ct_q1.eps}\\quad\n\\includegraphics[scale=0.35]{Inelastic_ct_q2.eps}} \\kern-0.315\\textwidth\n\\hbox to\n\\textwidth{\\hss(a)\\kern6.5em\\hss(b)\\kern-1.5em} \\kern+0.315\\textwidth\n\\centerline{\n\\includegraphics[scale=0.35]{Inelastic_q1.eps}\\quad\n\\includegraphics[scale=0.35]{Inelastic_q2.eps}} \\kern-0.315\\textwidth\n\\hbox to\n\\textwidth{\\hss(c)\\kern6.5em\\hss(d)\\kern-1.5em} \\kern+0.315\\textwidth\n\\caption{Inelastic collision in coupled complex short\npulse equation. (a)-(b): contour plot; (c)-(d): profiles before and after the collision.}\n\\label{f:inelastic1}\n\\end{figure}\nHowever, only for the special case\n\\begin{equation}\n\\frac{\\alpha _{1}^{(1)}}{\\alpha _{2}^{(1)}}=\\frac{\\alpha _{1}^{(2)}}{\\alpha\n_{2}^{(2)}}\\,,\n\\end{equation\nthere is no energy exchange between two compoents of solitons\nafter the collision.  An example is shown in Fig. 5 for the\nparameters\n$p_{1}=1+1.2\\mathrm{i}$,\\ $p_{2}=1+2\\mathrm{i}$,\\ $\\alpha^{(1)}_{1}\n\\alpha^{(2)}_{1}=1.0$, \\ $\\alpha^{(1)}_{2}=\\alpha^{(2)}_{2}=1.0$.\n\\begin{figure}[htbp]\n\\centerline{\n\\includegraphics[scale=0.35]{Elastic_ct_q1.eps}\\quad\n\\includegraphics[scale=0.35]{Elastic_ct_q2.eps}} \\kern-0.315\\textwidth\n\\hbox to\n\\textwidth{\\hss(a)\\kern6.5em\\hss(b)\\kern-1.5em} \\kern+0.315\\textwidth\n\\centerline{\n\\includegraphics[scale=0.35]{Elastic_q1.eps}\\quad\n\\includegraphics[scale=0.35]{Elastic_q2.eps}} \\kern-0.315\\textwidth\n\\hbox to\n\\textwidth{\\hss(c)\\kern6.5em\\hss(d)\\kern-1.5em} \\kern+0.315\\textwidth\n\\caption{Elastic collision in coupled complex short pulse equation.}\n\\label{f:elastic}\n\\end{figure}\n\nIt is interesting to note that if we just change the parameters in previous\ntwo examples as $\\alpha^{(1)}_{2}=0$, $\\alpha^{(2)}_{2}=1.0$,\nthe energy of one soliton is concentrated in component $q_2$ before the collision. However, component $q_1$ gains some energy after the collision.\nSuch an example is shown in Fig. 6.\n\n\\begin{figure}[htbp]\n\\centerline{\n\\includegraphics[scale=0.35]{Inelastic2_ct_q1.eps}\\quad\n\\includegraphics[scale=0.35]{Inelastic2_ct_q2.eps}} \\kern-0.315\\textwidth\n\\hbox to\n\\textwidth{\\hss(a)\\kern6.5em\\hss(b)\\kern-1.5em} \\kern+0.315\\textwidth\n\\centerline{\n\\includegraphics[scale=0.35]{Inelastic2_q1.eps}\\quad\n\\includegraphics[scale=0.35]{Inelastic2_q2.eps}} \\kern-0.315\\textwidth\n\\hbox to\n\\textwidth{\\hss(c)\\kern6.5em\\hss(d)\\kern-1.5em} \\kern+0.315\\textwidth\n\\caption{Inelastic collision in coupled\ncomplex short pulse equation for $p_{1}=1+1.2{\\rm i}$, $p_{2}=1+2{\\rm i}$, $\\alpha^{(1)}_{1}=\\alpha^{(2)}_{1}=1.0$, $\\alpha^{(1)}_{2}=0$, $\\alpha^{(2)}_{2}=1.0$. (a)-(b): contour plot; (c)-(d): profiles before and after the collision.}\n\\label{f:inelastic2}\n\\end{figure}\nOn the other hand, if we change the parameters as $\\alpha^{(1)}_{2}=1.0$, \n\\alpha^{(2)}_{2}=0$, then the energy of one soliton, which are distributed between two components  before the\ncollision is concentrated into one component $q_2$ after the collision.\n The example is\nshown in Fig. 7.\n\\begin{figure}[htbp]\n\\centerline{\n\\includegraphics[scale=0.35]{Inelastic3_ct_q1.eps}\\quad\n\\includegraphics[scale=0.35]{Inelastic3_ct_q2.eps}} \\kern-0.315\\textwidth\n\\hbox to\n\\textwidth{\\hss(a)\\kern6.5em\\hss(b)\\kern-1.5em} \\kern+0.315\\textwidth\n\\centerline{\n\\includegraphics[scale=0.35]{Inelastic3_q1.eps}\\quad\n\\includegraphics[scale=0.35]{Inelastic3_q2.eps}} \\kern-0.315\\textwidth\n\\hbox to\n\\textwidth{\\hss(c)\\kern6.5em\\hss(d)\\kern-1.5em} \\kern+0.315\\textwidth\n\\caption{Inelastic collision in coupled\ncomplex short pulse equation for $p_{1}=1+1.2\\mathrm{i}$, $p_{2}=1+2\\mathrm{\n}$, $\\protect\\alpha^{(1)}_{1}=\\alpha^{(2)}_{1}=1.0$, \\ $\\alpha^{(1)}_{2}=1.0$, $\\alpha^{(2)}_{2}=0$. (a)-(b): contour plot; (c)-(d): profiles before and after the collision.}\n\\label{f:inelastic3}\n\\end{figure}\n\\section{Concluding Remarks}\nIn this paper, we proposed a complex short pulse equation and its\ntwo-component generalization. Both of the equations can be used to model\nthe propagation of ultra-short pulses in optical fibers. We have shown their\nintegrability by finding the Lax pairs and infinite numbers of conservation\nlaws. Furthermore, multi-soliton solutions are constructed via Hirota's\nbilinear method. In particular, one-soliton solution for the CSP equation is\nan envelope soliton with a few optical cycles under certain condition, which\nperfectly match the requirement for the ultra-short pulses.\nThe $N$-solution for complex short pulse equation and its\ntwo-component generalization is a benchmark for the study of soliton\ninteractions in ultra-short pulses propagation in optical fibers. It is expected that these analytical\nsolutions can be confirmed from experiments.\n\nSimilar to our previous results for the integrable discretizations of the short pulse equation \\cite{SPE_discrete1}, how to construct integrable discretizations of the CSP and coupled CSP equations and how to\napply them for the numerical simulations is also an interesting topic to be\nstudied. It is obviously beyond the scope of the present paper, we\nare to report the results on this aspect in a forthcoming paper.\n\n\n\n\n\n\n\\apptitle\n\\section{}\n\\appeqn\n\\textbf{Proof of Theorem 4.2}\n\n\\begin{proof}\nFirst we define\n\\[\n(b_j, \\bar{\\beta}_1)= \\bar{\\alpha}_j\\delta_{\\mu,1} \\,, \\quad (b_j, \\bar{\\bet\n}_2)= \\bar{\\alpha}_j\\delta_{\\mu,2}\\,,\n\\]\nwhere $index(b_j)=\\mu$ , then from the fact\n\\[\n\\mathrm{Pf}(\\bar{a}_j,a_k)= \\mathrm{Pf} (a_{N+j},a_{N+k})\\,, \\mathrm{Pf}\n\\bar{b}_j,b_k)= \\mathrm{Pf} (b_{N+j},b_{N+k})\\,,\n\\]\nwe obtain\n\\[\n\\bar{f}=f\\,, \\quad \\bar{g}= \\mathrm{Pf} (d_0, \\bar{\\beta}_1, a_1, \\cdots,\na_{2N}, b_1, \\cdots, b_{2N})\\,.\n\\]\nSince\n\\[\n\\frac{\\partial} {\\partial y} \\mathrm{Pf} (a_j,a_k)= (p_j -\np_k)e^{\\eta_j+\\eta_k} = \\mathrm{Pf} (d_0, d_1, a_j,a_k)\\,,\n\\]\n\n\\[\n\\frac{\\partial} {\\partial s} \\mathrm{Pf} (a_j,a_k)= (p^{-1}_k - p^{-1}_j)\ne^{\\eta_j+\\eta_k} = \\mathrm{Pf} (d_{-1}, d_0, a_j,a_k)\\,,\n\\]\n\n\\[\n\\frac{\\partial^2} {\\partial s^2} \\mathrm{Pf} (a_j,a_k)= (p^{-2}_k -\np^{-2}_j) e^{\\eta_j+\\eta_k} = \\mathrm{Pf} (d_{-2}, d_0, a_j,a_k)\\,,\n\\]\n\\[\n\\frac{\\partial^2} {\\partial y \\partial s}\\mathrm{Pf} (a_j,a_k)= (p_jp^{-1}_k\n- p_k p^{-1}_j) e^{\\eta_j+\\eta_k} = \\mathrm{Pf} (d_{-1}, d_1, a_i,a_j)\\,,\n\\]\nwe then have\n\\[\n\\frac{\\partial f} {\\partial y} = \\mathrm{Pf} (d_0, d_1, \\cdots)\\,,\n\\]\n\n\\[\n\\frac{\\partial f} {\\partial s} = \\mathrm{Pf} (d_{-1}, d_0, \\cdots)\\,,\n\\]\n\n\\[\n\\frac{\\partial^2 f} {\\partial s^2} = \\mathrm{Pf} (d_{-2}, d_0, \\cdots)\\,,\n\\]\n\n\\[\n\\frac{\\partial^2 f} {\\partial y \\partial s} = \\mathrm{Pf} (d_{-1}, d_1,\n\\cdots)\\,.\n\\]\nHere $\\mathrm{Pf} (d_0, d_1, a_1, \\cdots, a_{2N}, b_1, \\cdots, b_{2N})$ is\nabbreviated by $\\mathrm{Pf} (d_0, d_1, \\cdots)$, so as other similar\npfaffians.\n\nFurthermore, it can be shown\n\\begin{eqnarray*}\n&& \\frac{\\partial g} {\\partial y} = \\frac{\\partial} {\\partial y} \\left\n\\sum_{j=1}^{2N} (-1)^{j} \\mathrm{Pf} (d_0, a_j) \\mathrm{Pf} (\\beta_1, \\cdots\n,\\hat{a}_j, \\cdots)\\right] \\\\\n&& =\\sum_{j=1}^{2N} (-1)^{j} \\left[ \\left( {\\partial_y} \\mathrm{Pf} (d_0,\na_j) \\right) \\mathrm{Pf} (\\beta_1, \\cdots ,\\hat{a}_j, \\cdots) + \\mathrm{Pf}\n(d_0, a_j) {\\partial_y} \\mathrm{Pf} (\\beta_1, \\cdots ,\\hat{a}_j, \\cdots)\n\\right] \\\\\n&& =\\sum_{j=1}^{2N} (-1)^{j} \\left[ \\mathrm{Pf} (d_1, a_j) \\mathrm{Pf}\n(\\beta_1, \\cdots ,\\hat{a}_j, \\cdots) + \\mathrm{Pf} (d_0, a_j) \\mathrm{Pf}\n(\\beta_1, d_0, d_1, \\cdots ,\\hat{a}_j, \\cdots) \\right] \\\\\n&& = \\mathrm{Pf} ( d_1, \\beta_1, \\cdots)+ \\mathrm{Pf} ( d_0, \\beta_1, d_0,\nd_1, \\cdots) \\\\\n&& = \\mathrm{Pf} (d_1, \\beta_1, \\cdots)\\,.\n\\end{eqnarray*}\n\nHere $\\hat{a}_j$ means that the index $j$ is omitted. Similarly, we can show\n\\[\n\\frac{\\partial g} {\\partial s} = \\mathrm{Pf} (d_{-1}, \\beta_1, \\cdots)\\,,\n\\]\n\n\\begin{eqnarray*}\n&& \\frac{\\partial^2 g} {\\partial y \\partial s} = \\frac{\\partial} {\\partial y}\n\\left[\\sum_{j=1}^{2N} (-1)^{j} \\mathrm{Pf} (d_{-1}, a_j) \\mathrm{Pf}\n(\\beta_1, \\cdots ,\\hat{a}_j, \\cdots)\\right] \\\\\n&& =\\sum_{j=1}^{2N} (-1)^{j} \\left[ \\left( {\\partial_y} \\mathrm{Pf} (d_{-1},\na_j) \\right) \\mathrm{Pf} (\\beta_1, \\cdots ,\\hat{a}_j, \\cdots) + \\mathrm{Pf}\n(d_{-1}, a_j) {\\partial_y} \\mathrm{Pf} (\\beta_1, \\cdots ,\\hat{a}_j, \\cdots)\n\\right] \\\\\n&& =\\sum_{j=1}^{2N} (-1)^{j} \\left[ \\mathrm{Pf} (d_0, a_j) \\mathrm{Pf}\n(\\beta_1, \\cdots ,\\hat{a}_j, \\cdots) + \\mathrm{Pf} (d_{-1}, a_j) \\mathrm{Pf}\n(\\beta_1, d_0, d_1, \\cdots ,\\hat{a}_j, \\cdots) \\right] \\\\\n&& = \\mathrm{Pf} (d_0, \\beta_1, \\cdots)+ \\mathrm{Pf} (d_{-1}, \\beta_1, d_0,\nd_1, \\cdots)\\,.\n\\end{eqnarray*}\n\nAn algebraic identity of pfaffian \\cite{Hirota}\n\\begin{eqnarray*}\n&& \\mathrm{Pf} (d_{-1}, \\beta_1, d_0, d_1, \\cdots) \\mathrm{Pf} (\\cdots)=\n\\mathrm{Pf} (d_{-1}, d_0, \\cdots) \\mathrm{Pf} (d_1, \\beta_1, \\cdots) \\\\\n&& \\quad - \\mathrm{Pf} (d_{-1}, d_1, \\cdots) \\mathrm{Pf} (d_0, \\beta_1,\n\\cdots) + \\mathrm{Pf} (d_{-1}, \\beta_1, \\cdots) \\mathrm{Pf} (d_0, d_1,\n\\cdots)\\,,\n\\end{eqnarray*}\nimplies\n\\[\n( {\\partial_s} {\\partial_y} g-g) \\times f = {\\partial_s} f \\times {\\partial_\n} g - {\\partial_s} {\\partial_y} f \\times g + {\\partial_s} g \\times \n\\partial_y} f \\,.\n\\]\nTherefore, the first bilinear equation is approved.\n\nThe second bilinear equation can be proved in the same way by Iwao and\nHirota \\cite{IwaoHirota}.\n\\begin{eqnarray}\n&& \\frac{\\partial^2 f} {\\partial s^2} \\times 0 - \\frac{\\partial f} {\\partial\ns} \\frac{\\partial f} {\\partial s}  \\nonumber \\\\\n&& = \\mathrm{Pf} (d_{-2}, d_0, \\cdots) \\mathrm{Pf} (d_{0}, d_0, \\cdots) -\n\\mathrm{Pf} (d_{-1}, d_0, \\cdots) \\mathrm{Pf} (d_{-1}, d_0, \\cdots)\n\\nonumber \\\\\n&& = \\sum_{i=1}^{2N} (-1)^i \\mathrm{Pf} (d_{-2}, a_i) \\mathrm{Pf} (d_0,\n\\cdots, \\hat{a}_i, \\cdots) \\sum_{j=1}^{2N} (-1)^j \\mathrm{Pf} (d_{0}, a_j)\n\\mathrm{Pf} (d_0, \\cdots, \\hat{a}_j, \\cdots)  \\nonumber \\\\\n&& -\\sum_{i=1}^{2N} (-1)^i \\mathrm{Pf} (d_{-1}, a_i) \\mathrm{Pf} (d_0,\n\\cdots, \\hat{a}_i, \\cdots) \\sum_{j=1}^{2N} (-1)^j \\mathrm{Pf} (d_{-1}, a_j)\n\\mathrm{Pf} (d_0, \\cdots, \\hat{a}_j, \\cdots)  \\nonumber \\\\\n&& =\\sum_{i,j=1}^{2N} (-1)^{i+j} \\left[ \\mathrm{Pf} (d_{-2}, a_i) \\mathrm{Pf}\n(d_{0}, a_j) -\\mathrm{Pf} (d_{-1}, a_i) \\mathrm{Pf} (d_{-1}, a_j) \\right]\n\\nonumber \\\\\n&& \\quad \\times \\mathrm{Pf} (d_0, \\cdots, \\hat{a}_i, \\cdots) \\mathrm{Pf}\n(d_0, \\cdots, \\hat{a}_j, \\cdots)  \\nonumber \\\\\n&&=\\sum_{i,j=1}^{2N} (-1)^{i+j+1} \\left[p_i^{-2} + p_i^{-1}p_j^{-1} \\right]\n\\mathrm{Pf} (a_i, a_j) \\mathrm{Pf} (d_0, \\cdots, \\hat{a}_i, \\cdots) \\mathrm\nPf} (d_0, \\cdots, \\hat{a}_j, \\cdots)  \\nonumber\n\\end{eqnarray}\n\nThe summation over the second term within the bracket vanishes due to the\nfact that\n\\begin{eqnarray*}\n&& \\sum_{i,j=1}^{2N} (-1)^{i+j+1} p_i^{-1}p_j^{-1} \\mathrm{Pf} (a_i, a_j)\n\\mathrm{Pf} (d_0, \\cdots, \\hat{a}_i, \\cdots) \\mathrm{Pf} (d_0, \\cdots, \\hat{\n}_j, \\cdots) \\\\\n&& = \\sum_{j,i=1}^{2N} (-1)^{j+i+1} p_j^{-1}p_i^{-1} \\mathrm{Pf} (a_j, a_i)\n\\mathrm{Pf} (d_0, \\cdots, \\hat{a}_j, \\cdots) \\mathrm{Pf} (d_0, \\cdots, \\hat{\n}_i, \\cdots) \\\\\n&& = -\\sum_{i,j=1}^{2N} (-1)^{i+j+1} p_i^{-1}p_j^{-1} \\mathrm{Pf} (a_i, a_j)\n\\mathrm{Pf} (d_0, \\cdots, \\hat{a}_i, \\cdots) \\mathrm{Pf} (d_0, \\cdots, \\hat{\n}_j, \\cdots)\\,.\n\\end{eqnarray*}\nTherefore,\n\\begin{eqnarray}\n&& - \\frac{\\partial f} {\\partial s} \\frac{\\partial f} {\\partial s}\n=\\sum_{i,j=1}^{2N} (-1)^{i+j+1} p_i^{-2} \\mathrm{Pf} (a_i, a_j) \\mathrm{Pf}\n(d_0, \\cdots, \\hat{a}_i, \\cdots) \\mathrm{Pf} (d_0, \\cdots, \\hat{a}_j, \\cdots)\n\\nonumber \\\\\n&&=\\sum_{i=1}^{2N} (-1)^{i+1} p_i^{-2} \\mathrm{Pf} (d_0, \\cdots, \\hat{a}_i,\n\\cdots) \\left[ \\sum_{j=1}^{2N} (-1)^{j} \\mathrm{Pf} (a_i, a_j) \\mathrm{Pf}\n(d_0, \\cdots, \\hat{a}_j, \\cdots) \\right]  \\nonumber  \\label{CSP1_proof5}\n\\end{eqnarray}\nFurther, we note that the following identity can be substituted into the\nterm within bracket\n\\begin{eqnarray*}\n&& \\sum_{j=1}^{2N} (-1)^{j} \\mathrm{Pf} (a_i, a_j) \\mathrm{Pf} (d_0, \\cdots,\n\\hat{a}_j, \\cdots) \\\\\n&& = \\mathrm{Pf} (d_{0}, a_i) \\mathrm{Pf} (\\cdots) + (-1)^{i+1} \\mathrm{Pf}\n(d_0, \\cdots, \\hat{b}_i, \\cdots)\\,\n\\end{eqnarray*}\nwhich is obtained from the expansion of the following vanishing pfaffian \n\\mathrm{Pf} (a_i, d_0, \\cdots)$ on $a_i$. Consequently, we have\n\\begin{eqnarray}\n&& - \\frac{\\partial f} {\\partial s} \\frac{\\partial f} {\\partial s} =\n\\nonumber \\\\\n&& \\sum_{i=1}^{2N} (-1)^{i+1} p_i^{-2} \\mathrm{Pf} (d_0, \\cdots, \\hat{a}_i,\n\\cdots) \\left[\\mathrm{Pf} (d_{0}, a_i) \\mathrm{Pf} (\\cdots) + (-1)^{i+1}\n\\mathrm{Pf} (d_0, \\cdots, \\hat{b}_i, \\cdots)\\right]\\,,  \\nonumber \\\\\n&& = -\\mathrm{Pf} (\\cdots) \\mathrm{Pf} (d_{-2}, d_0, \\cdots)+\n\\sum_{i=1}^{2N} p_i^{-2} \\mathrm{Pf} (d_0, \\cdots, \\hat{a}_i, \\cdots)\n\\mathrm{Pf} (d_0, \\cdots, \\hat{b}_i, \\cdots)\\,,\n\\end{eqnarray}\nwhich can be rewritten as\n\\begin{equation}\n\\frac{\\partial^2 f} {\\partial s^2} f- \\frac{\\partial f} {\\partial s} \\frac\n\\partial f} {\\partial s}= \\sum_{i=1}^{2N} p_i^{-2} \\mathrm{Pf} (d_0, \\cdots,\n\\hat{a}_i, \\cdots) \\mathrm{Pf} (d_0, \\cdots, \\hat{b}_i, \\cdots)\\,.\n\\end{equation}\n\nNow, we work on the r.h.s of the second bilinear equation.\n\\begin{eqnarray}  \\label{CSP1_proof1}\n&& \\frac 12 |g|^2 = \\frac 12 \\mathrm{Pf} (d_0,\\beta_1, \\cdots) \\mathrm{Pf}\n(d_0, \\bar{\\beta}_1, \\cdots)  \\nonumber \\\\\n&& = \\frac 12 \\sum_{i,j}^{2N} (-1)^{i+j}\\mathrm{Pf} (b_i, \\beta_1) \\mathrm{P\n} (d_0, \\cdots,\\hat{b}_i, \\cdots) \\mathrm{Pf} (b_j,\\bar{\\beta}_1) \\mathrm{Pf}\n(d_0, \\cdots,\\hat{b}_j, \\cdots)  \\nonumber \\\\\n&& = \\frac 14 \\sum_{i,j}^{2N} (-1)^{i+j} (\\alpha_i \\bar{\\alpha}_j) \\mathrm{P\n} (d_0, \\cdots,\\hat{b}_i, \\cdots) \\mathrm{Pf} (d_0, \\cdots,\\hat{b}_j, \\cdots)\n\\nonumber \\\\\n&& = \\sum_{i,j}^{2N} (-1)^{i+j} \\left(p_i^{-2}-p_{j}^{-2}\\right) \\mathrm{Pf}\n(b_i,b_j)\\mathrm{Pf}(d_0, \\cdots,\\hat{b}_i, \\cdots) \\mathrm{Pf} (d_0, \\cdots\n\\hat{b}_j, \\cdots)  \\nonumber \\\\\n\\end{eqnarray}\nNext, the expansion of the vanishing pfaffian $\\mathrm{Pf} (b_i, d_0,\n\\cdots) $ on $b_i$ yields\n\\begin{equation}\n\\sum_{j=1}^{2N} (-1)^{i+j}\\mathrm{Pf} (b_i, b_j) \\mathrm{Pf} (d_0, \\cdots,\n\\hat{b}_j, \\cdots) = \\mathrm{Pf} (d_0, \\cdots,\\hat{a}_i, \\cdots)\\,,\n\\end{equation}\nwhich subsequently leads to\n\\begin{eqnarray}\n&& \\sum_{i,j}^{2N} (-1)^{i+j} p_i^{-2} \\mathrm{Pf} (b_i,b_j)\\mathrm{Pf}(d_0,\n\\cdots,\\hat{b}_i, \\cdots) \\mathrm{Pf} (d_0, \\cdots,\\hat{b}_j, \\cdots)\n\\nonumber \\\\\n&& = \\sum_{i}^{2N} p_i^{-2} \\mathrm{Pf} (d_0, \\cdots,\\hat{a}_i, \\cdots)\n\\mathrm{Pf} (d_0, \\cdots,\\hat{b}_i, \\cdots)\\,.  \\label{CSP1_proof2}\n\\end{eqnarray}\nSimilarly, we can show that\n\\begin{eqnarray}\n&& -\\sum_{i,j}^{2N} (-1)^{i+j} p_j^{-2} \\mathrm{Pf} (b_i,b_j)\\mathrm{Pf\n(d_0, \\cdots,\\hat{b}_i, \\cdots) \\mathrm{Pf} (d_0, \\cdots,\\hat{b}_j, \\cdots)\n\\nonumber \\\\\n&& = \\sum_{j}^{2N} p_j^{-2} \\mathrm{Pf} (d_0, \\cdots,\\hat{a}_j, \\cdots)\n\\mathrm{Pf} (d_0, \\cdots,\\hat{b}_j, \\cdots)\\,.  \\label{CSP1_proof3}\n\\end{eqnarray}\nSubstituting Eqs. (\\ref{CSP1_proof2})--(\\ref{CSP1_proof2}) into Eq. (\\re\n{CSP1_proof1}), we arrive at\n\\begin{equation}  \\label{CSP1_proof4}\n\\frac 12 |g|^2= 2\\sum_{i}^{2N} p_i^{-2} \\mathrm{Pf} (d_0, \\cdots,\\hat{a}_i,\n\\cdots) \\mathrm{Pf} (d_0, \\cdots,\\hat{b}_i, \\cdots)\\,.\n\\end{equation}\nConsequently we have\n\\begin{equation}\n2\\frac{\\partial^2 f} {\\partial s^2} f- 2\\frac{\\partial f} {\\partial s} \\frac\n\\partial f} {\\partial s}= \\frac 12 |g|^2\\,,\n\\end{equation}\nwhich is nothing but the second bilinear equation. Therefore, the proof is\ncomplete.\n\\end{proof}\n\n\\textbf{The proof of Theorem 4.6}\n\n\\begin{proof}\nThe proof of the first bilinear equation can be done exactly in the same way\nas for the complex short pulse equation. In what follows, we prove the\nsecond equation by starting from the r.h.s of this equation. Because\n\\[\n\\bar{g}_1= \\mathrm{Pf} (d_0, \\bar{\\beta}_1, a_1, \\cdots, a_{2N}, b_1,\n\\cdots, b_{2N})\\,,\n\\]\n\\[\n\\bar{g}_2= \\mathrm{Pf} (d_0, \\bar{\\beta}_2, a_1, \\cdots, a_{2N}, b_1,\n\\cdots, b_{2N})\\,,\n\\]\nthe r.h.s of the bilinear equation turns out to be\n\\begin{eqnarray}  \\label{CCSP1_proof1}\n&& \\frac 12 \\left( g_{1} \\bar{g}_{1} + g_{2} \\bar{g}_{2} \\right)  \\nonumber\n\\\\\n&& = \\frac 12 \\sum^2_{k=1} \\sum_{i,j}^{2N} (-1)^{i+j}\\mathrm{Pf} (b_i,\n\\beta_k) \\mathrm{Pf} (d_0, \\cdots,\\hat{b}_i, \\cdots) \\mathrm{Pf} (b_j,\\bar\n\\beta}_k) \\mathrm{Pf} (d_0, \\cdots,\\hat{b}_j, \\cdots)  \\nonumber \\\\\n&& = \\frac 14 \\sum_{i,j}^{2N} (-1)^{i+j} \\sum^2_{k=1}(\\alpha^{(k)}_i \\bar\n\\alpha}^{(k)}_j) \\mathrm{Pf} (d_0, \\cdots,\\hat{b}_i, \\cdots) \\mathrm{Pf}\n(d_0, \\cdots,\\hat{b}_j, \\cdots)  \\nonumber \\\\\n&& = \\sum_{i,j}^{2N} (-1)^{i+j} \\left(p_i^{-2}-p_{j}^{-2}\\right) \\mathrm{Pf}\n(b_i,b_j)\\mathrm{Pf}(d_0, \\cdots,\\hat{b}_i, \\cdots) \\mathrm{Pf} (d_0, \\cdots\n\\hat{b}_j, \\cdots)  \\nonumber \\\\\n\\end{eqnarray}\nSimilar to the complex short pulse equation, we can show\n\n\\begin{equation}  \\label{CCSP1_proof4}\n\\frac 12 \\left(|g_{1}|^2 + |g_{2}|^2 \\right) = 2\\sum_{i}^{2N} p_i^{-2}\n\\mathrm{Pf} (d_0, \\cdots,\\hat{a}_i, \\cdots) \\mathrm{Pf} (d_0, \\cdots,\\hat{b\n_i, \\cdots)\\,.\n\\end{equation}\nRegarding the r.h.s of the bilinear equation, exactly the same as the proof\nof the Theorem 4.2, we have\n\\begin{equation}  \\label{CCSP1_proof5}\n\\frac{\\partial^2 f} {\\partial s^2} f- \\frac{\\partial f} {\\partial s} \\frac\n\\partial f} {\\partial s} =\\sum_{i}^{2N} p_i^{-2} \\mathrm{Pf} (d_0, \\cdots\n\\hat{a}_i, \\cdots) \\mathrm{Pf} (d_0, \\cdots,\\hat{b}_i, \\cdots)\\,.\n\\end{equation}\nTherefore the second bilinear equation is proved.\n\\end{proof}\n\n\\thank\n\\section{}\nThe author is grateful for the useful discussions with Dr. Yasuhiro Ohta\n(Kobe University) and Dr. Kenichi Maruno at Waseda University. This work is partially supported by the National Natural Science Foundation of China (No. 11428102).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWith the exponential growth of Internet usage, online users massively publish textual content on online media. For instance, a micro-blogging website, Twitter, allows users to post their content in 140-characters length. A popular social media like Facebook allows users to interact and share content in their communities, as known as ``Friends''. An electronic commercial website, Amazon, allows users to ask questions on their interested items and give reviews on their purchased products. While these textual data have been broadly studied in various research areas (e.g. automatic text summarization, information retrieval, information extraction, etc.), online debate domain, which recently becomes popular among Internet users, has not yet largely explored. For this reason, there are no sufficient resources of annotated debate data available for conducting research in this genre. This motivates us to explore online debate data. \n\nIn this paper, we collected and annotated debate data for an automatic summarization task. There are 11 debate topics collected. Each topic consists of different number of debate comments. In total, there are 341 debate comments collected, accounting for 2518 sentences. In order to annotate online debate data, we developed a web-based system which simply runs on web browsers. We designed the user interface for non-technical users. When participants logged into the system, a debate topic and a comment which is split to a list of consecutive sentences were shown at a time. The annotators were asked to select salient sentences from each comment which summarize it. The number of salient sentences chosen from each comment is controlled by a compression rate of 20\\% which is automatically calculated by the web-based system. For instance, Table \\ref{table_annotation} shows a debate comment to be annotated by an annotator. Based on the compression rate of 20\\%, the annotator needs to choose 1 sentence that summarizes the comment. This compression rate was also used in \\cite{Neto2002ATS} and \\cite{Morris199217}. In total, we obtained 5 sets of annotated debate data. Each set of data consists of 341 comments with total 519 annotated salient sentences. \n\n\nInter-annotator agreement in terms of Cohen's Kappa and Krippendorff's alpha are 0.28 and 0.27 respectively. For social media data such low agreements have been also reported by related work. For instance, \\cite{Mitrat} reports Kappa scores between 0.20 and 0.50 for human constructed newswire summaries. \\cite{Liu:2008:CRH:1557690.1557747} reports again Kappa scores between 0.10 and 0.35 for the conversation transcripts. Our agreement scores are based on strict conditions where agreement is achieved when annotators have selected exact the same sentences. However, such condition does not consider syntactically different sentences bearing the same semantic meaning. Thus we also experimented with a more relaxed version that is based on semantic similarity between sentences. We regard two sentences as identical when their semantic similarity is above a threshold. Our results revealed that after applying such an approach the averaged Cohen's Kappa and Krippendorff's alpha increase to 35.71\\% and 48.15\\% respectively. \n\n\n\n\nFinally we report our results of automatic debate data summarization. We implemented an extractive text summarization system that extracts salience sentences from user comments. Among the features the most contributing ones are sentence position, debate titles, and cosine similarity of the debate title words and sentences. \n\n\nThe paper is structured as follows. First we describe the nature of our online debate data. In Section \\ref{data_annotation} we discuss the procedures of data annotation and discuss our experiments with semantic similarity applied on inter-annotator agreement computation. In Section \\ref{experiment_salient}, we present our first results on automatically performing debate data summarization. We conclude in Section \\ref{conclusion}.\n\n\n\n\n\\begin{table}[ht]\n\\begin{flushleft}\n\\begin{framed}\n\\noindent\\textbf{Task 02: Is global warming fictitious?}\\\\\n\\emph{$[1]$} I do not think global warming is fictitious.\\\\\n\\emph{$[2]$} I understand a lot of people do not trust every source and they need solid proof.\\\\\n\\emph{$[3]$} However, if you look around us the proof is everywhere.\\\\\n\\emph{$[4]$} It began when the seasons started getting harsh and the water levels were rising.\\\\\n\\emph{$[5]$} I do not need to go and see the ice caps melting to know the water levels are rising and the weather is changing.\\\\\n\\emph{$[6]$} I believe global warming is true, and we should try and preserve as much of the Earth as possible.\n\\end{framed}\n\\end{flushleft}\n\\caption{Examples of the debate data to be annotated.}\\label{table_annotation} \n\\end{table}\n\n\n\n\n\n\\begin{table}[ht]\n\\begin{flushleft}\n\\begin{framed}\n\\textbf{Example 1: Propositions from the proponents}  \n\\\\ - Global warming is real.\n\\\\ - Global warming is an undisputed scientific fact.     \n\\\\ - Global warming is most definitely not a figment of anyone's imagination, because the proof is all around us.\n\\\\ - I believe that global warming is not fictitious, based on the observational and comparative evidence that is currently presented to us.\n\\vskip 0.2in\n\\textbf{Example 2: Propositions from the opponents}  \n\\\\ - Global warming is bull crap.\n\\\\ - Global Warming isn't a problem at all.\n\\\\ - Just a way for the government to tax people on more things by saying their trying to save energy.\n\\\\ - Yes, global warming is a myth, because they have not really proven the science behind it. \n\\\\  \n\\end{framed}\n\\end{flushleft}\n\\caption{Examples of Paraphrased Arguments.}\\label{table_pargument} \n\\end{table}\n\\FloatBarrier\n\n\n\\section{Online Debate Data and Their Nature} \\label{nature_debate}\n\nThe nature of online debate is different from other domains. It gives opportunities to users to discuss ideological debates in which users can choose a stance of a debate, express their opinions to support their stance, and oppose other stances. To conduct our experiments we collected debate data from the Debate discussion forum.\\footnote{http:\/\/www.debate.org} The data are related to an issue of the existence of global warming. In the data, there are two main opposing sides of the arguments. A side of proponents believes in the existence of global warming and the other side, the opponents, says that global warming is not true. When the proponents and the opponents express their sentiments, opinions, and evidences to support their propositions, the arguments between them arise. Moreover, when the arguments are referred across the conversation in the forum, they are frequently paraphrased. Table \\ref{table_pargument} illustrates examples of the arguments being paraphrased. Sentences expressing related meaning are written in different context. \n\n\n\\section{Annotation Procedures} \\label{data_annotation}\nIn this paper, we collected and annotated debate data for an automatic summarization task. There are 11 debate topics collected. Each topic consists of a different number of debate comments as shown in Table \\ref{table_stats_corpus}. The annotation was guided through a web-based application. The application was designed for non-technical users. When participants logged in to the system, a debate topic and a comment which is split to a list of sentences were shown at a time. The annotators were given a guideline to read and select salient sentences that summarize the comments. From each comment we allowed the participants to select only 20\\% of the comment sentences. These 20\\% of the sentences are treated as the summary of the shown comment. In the annotation task, all comments in the 11 debate topics were annotated. We recruited 22 participants: 10 males and 12 participants to annotate salient sentences. The participants' backgrounds were those who are fluent in English and aged above 18 years old. We aimed to have 5 annotations sets for each debate topic. Due to a limited number of annotators and a long list of comments to be annotated in each debate topic, 11 participants were asked to complete more than one debate topic, but were not allowed to annotate the same debate topics in which they had done before. In total, 55 annotation sets were derived: 11 debate topics and each with 5 annotation sets. Each annotation set consists of 341 comments with total 519 annotated salient sentences.\\footnote{This dataset can be downloaded at https:\/\/goo.gl\/3aicDN.}\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{@{}clccc@{}}\n\\toprule\n\\textbf{Topic ID} & \\multicolumn{1}{c}{\\textbf{Debate Topics}} & \\textbf{Comments} & \\textbf{Sentences} & \\textbf{Words} \\\\ \\midrule\n01 & Is global warming a myth? & 18 & 128 & 2701 \\\\\n02 & Is global warming fictitious? & 28 & 173 & 3346 \\\\\n03 & Is the global climate change man made? & 10 & 47 & 1112 \\\\\n04 & Is global climate change man-made? & 103 & 665 & 12054 \\\\\n05 & Is climate change man-made? & 9 & 46 & 773 \\\\\n06 & Do you believe in global warming? & 21 & 224 & 3538 \\\\\n07 & Does global warming exist? & 68 & 534 & 9178 \\\\\n08 & \\begin{tabular}[c]{@{}l@{}}Can someone prove that climate \\\\ change is real (yes) or fake (no)?\\end{tabular} & 8 & 49 & 1127 \\\\\n09 & Is global warming real? & 51 & 434 & 6749 \\\\\n10 & Is global warming true? & 5 & 26 & 375 \\\\\n11 & \\begin{tabular}[c]{@{}l@{}}Is global warming real (yes) or just a bunch \\\\ of scientist going to extremes (no)?\\end{tabular} & 20 & 192 & 2988 \\\\\\midrule\n\\textbf{} & \\multicolumn{1}{r}{\\textbf{Average}} & \\textbf{31} & \\textbf{229} & \\textbf{3995} \\\\\n\\textbf{} & \\multicolumn{1}{r}{\\textbf{Total}} & \\textbf{341} & \\textbf{2518} & \\textbf{43941} \\\\ \\bottomrule\n\\end{tabular}\n\\caption{Statistical information of the online debate corpus.}\n\\label{table_stats_corpus}\n\\end{table}\n\n\n\n\\subsection{Inter-Annotator Agreement}\n\nIn order to compute inter-annotator agreement between the annotators we calculated the averaged Cohen's Kappa and Krippendorff's alpha with a distant metric, Measuring Agreement on Set-valued Items metric (MASI). The scores of averaged Cohen's Kappa and Krippendorff's alpha are 0.28 and 0.27 respectively. According to the scale of \\cite{krippendorff-2004}, our alpha did neither accomplish the reliability scale of 0.80, nor the marginal scales between 0.667 and 0.80. Likewise, our Cohen's Kappa only achieved the agreement level of \\emph{fair agreement}, as defined by \\cite{Landis77}. However, such low agreement scores are also reported by others who aimed creating gold standard summaries from news texts or conversational data \\cite{Mitrat}  \\cite{Liu:2008:CRH:1557690.1557747} .\n\n\nOur analysis shows that the low agreement is caused by different preferences of annotators in the selection of salient sentences. As shown in Table \\ref{table_pargument} the sentences are syntactically different but bear the same semantic meaning. In a summarization task with a compression threshold, such situation causes the annotators to select one of the sentences but not all. Depending on each annotator's preference the selection leads to different set of salient sentences. To address this we relaxed the agreement computation by treating sentences equal when they are semantically similar. We outline details in the following section.\n\n\n\n\n\\subsection{Relaxed Inter-Annotator Agreement}\n\nWhen an annotator selects a sentence, other annotators might select other sentences expressing similar meaning. In this experiment, we aim to detect sentences that are semantically similar by applying Doc2Vec from the Gensim package \\cite{rehurek_lrec}. Doc2Vec model simultaneously learns the representation of words in sentences and the labels of the sentences. The labels are numbers or chunks of text which are used to uniquely identify each sentence. We used the debate data and a richer collections of sentences related to climate change to train the Doc2Vec model. In total, there are 10,920 sentences used as the training set. \n\nTo measure how two sentences are semantically referring to the same content, we used a function provided in the package to calculate cosine similarity scores among sentences. A cosine similarity score of 1 means that the two sentences are semantically equal and 0 is when it is opposite the case. In the experiment, we manually investigated pairs of sentences at different threshold values and found that the approach is stable at the threshold level above 0.44. The example below shows a pair of sentences obtained at 0.44 level. \\\\\n\n\n\\indent \\textbf{S1: }\\emph{Humans are emitting carbon from our cars, planes and factories, which is a heat trapping particle.}\\\\\n\\indent \\textbf{S2: }\\emph{So there is no doubt that carbon is a heat trapping particle, there is no doubt that our actions are emitting carbon into the air, and there is no doubt that the amount of carbon is increasing.}\\\\\n\n\nIn the pair, the two sentences mention the same topic (i.e. \\emph{carbon emission}) and express the idea in the same context. We used the threshold 0.44 to re-compute the agreement scores. By applying the semantic approach, the inter-annotator agreement scores of Cohen's Kappa and Krippendorff's alpha increase from 0.28 to 35.71\\% and from 0.27 to 48.15\\% respectively. The inter-annotator agreement results are illustrated in Table \\ref{iaa}. Note that, in the calculation of the agreement, we incremented the threshold by 0.02. Only particular thresholds are shown in the table due to the limited space.\n\n\n\n\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{@{}lcccc@{}}\n\\toprule\n\\multicolumn{1}{c}{\\textbf{Trial}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Threshold\\\\ ($\\ge$)\\end{tabular}} & \\textbf{$\\kappa$} & &\\textbf{$\\alpha$} \\\\ \\midrule\n\\multicolumn{1}{c}{Before} &  & 0.28  && 0.27 \\\\\\midrule\n\\multicolumn{1}{c}{After} & 0.00 & 0.81  && 0.83 \\\\ \n & 0.10 & 0.62 & & 0.65 \\\\\n & 0.20 & 0.46 & & 0.50 \\\\\n & 0.30 & 0.40 & & 0.43 \\\\\n & 0.40 & 0.39 & & 0.41 \\\\\n & 0.42 & 0.38 & & 0.41 \\\\\n & \\textbf{0.44} & \\textbf{0.38} & & \\textbf{0.40} \\\\\n & 0.46 & 0.38 & & 0.40 \\\\\n & 0.48 & 0.38 & & 0.40 \\\\\n & 0.50 & 0.38 & & 0.40 \\\\\n & 0.60 & 0.38 & & 0.40 \\\\\n & 0.70 & 0.38 & & 0.40 \\\\\n & 0.80 & 0.38 & & 0.40 \\\\\n & 0.90 & 0.38 & & 0.40 \\\\\n & 1.00 & 0.38 & & 0.40 \\\\\\bottomrule\n\\end{tabular}\n\\caption{Inter-Annotator Agreement before and after applying the semantic similarity approach.}\n\\label{iaa}\n\\end{table}\n\n\n\n\n\n\\section{Automatic Salient Sentence Selection} \\label{experiment_salient} \n\\subsection{Support Vector Regression Model}\n\n In this experiment, we work on extractive summarization problem and aim to select sentences that are deemed important or that summarize the information mentioned in debate comments. Additionally, we aim to investigate the keys features which play the important roles in the summarization of the debate data. We view this salient sentence selection as a regression task. A regression score for each sentence is ranged between 1 to 5. It is derived by the number annotators selected that sentence divided by the number of all annotators. In this experiment, a popular machine learning package which is available in Python, called Scikit-learn \\cite{scikitLearn} is used to build a support vector regression model. We defined 8 different features and the support vector regression model combines the features for scoring sentences in each debate comment. From each comment, sentences with the highest regression scores are considered the most salient ones. \n\n\\subsection{Feature Definition}\n\\begin{enumerate}\n    \\item \\textbf{Sentence Position (SP).}\nSentence position correlates with the important information in text \\cite{Baxendale,EdmundsonRatingSummary,Goldstein}. In general, humans are likely to mention the first topic in the earlier sentence and they express more information about it in the later sentences. We prove this claim by conducting a small experiment to investigate which sentence positions frequently contain salient sentences. From our annotated data, the majority votes of the sentences are significantly at the first three positions (approximately 60\\%), shaping the assumption that the first three sentences are considered as containing salient pieces of information. Equation \\ref{eq_sentence_position} shows the calculation of the score obtained by the sentence position feature.  \\\\\n    \\begin{equation}\t \\label{eq_sentence_position}\n      SP=\\left\\{\n      \\begin{array}{@{}ll@{}}\n        \\frac{1}{sentence \\; position}, & \\text{if}\\ position <4 \\\\\n        0, & \\text{otherwise}\n      \\end{array}\\right.\n     \\end{equation} \n    \n    \n\\item \\textbf{Debate Titles (TT).}\nIn writing, a writer tends to repeat the title words in a document. For this reason, a sentence containing title words is likely to contain important information. We collected 11 debate titles as shown in Table \\ref{table_stats_corpus}. In our experiment, a sentence is considered as important when it contains mutual words as in debate titles. Equation \\ref{eq_titleword} shows the calculation of the score by this feature. \\\\\n    \\begin{equation} \\label{eq_titleword}\n    TT  = \\frac{\\; number \\; of \\; title \\; words \\; in \\; sentence}{number \\; of \\; words \\;in \\;debate \\;titles}\n    \\end{equation}\n     \n     \n    \\item \\textbf{Sentence Length (SL).} \nSentence length also indicates the importance of sentence based on the assumption that either very short or very long sentences are unlikely to be included in the summary. Equation \\ref{eq_sentencelength} is used in the process of extracting salient sentences from debate comments. \\\\\n    \\begin{equation} \\label{eq_sentencelength}\n    SL  = \\frac{\\; number \\; of \\; words \\; in \\; a \\; sentence}{number \\; of \\; words\\; in \\; the \\; longest \\; sentence}\n    \\end{equation}\t\n\n\n    \\item \\textbf{Conjunctive Adverbs (CJ).}\nOne possible feature that helps identify salient sentence is to determine conjunctive adverbs in sentences. Conjunctive adverbs were proved that they support cohesive structure of writing. For instance, ``the conjunctive adverb \\emph{moreover} has been used mostly in the essays which lead to a conclusion that it is one of the best accepted linkers in the academic writing process.\" \\cite{januliene2015use}. The NLTK POS Tagger\\footnote{http:\/\/www.nltk.org\/api\/nltk.tag.html} was used to determine conjunctive adverbs in our data. \\\\\n\n    \\item \\textbf{Cosine Similarity.}\nCosine similarity has been used extensively in Information Retrieval, especially in the vector space model. Documents will be ranked according to the similarity of the given query. Equation \\ref{cosinesim} illustrates the equation of cosine similarity, where: \\emph{q} and \\emph{d} are n-dimensional vectors \\cite{Manning:1999:FSN:311445}. Cosine similarity is one of our features that is used to find similarity between two textual units. The following features are computed by applying cosine similarity. \n\n    \\begin{equation} \\label{cosinesim}\t\n    cos(q,d) = \\frac{\\sum\\limits_{i=1}^n q_{i} d_{i}}{\\sqrt{\\sum\\limits_{i=1}^n q^2_{i}}\\sqrt{\\sum\\limits_{i=1}^n d^2_{i}}} \n    \\end{equation}\n\n        \\begin{enumerate}\n            \\item \\textbf{Cosine similarity of debate title words and sentences (COS\\_TTS).} For each sentence in debate comments we compute its cosine similarity score with the title words. This is based on the assumption that a sentence containing title words is deemed as important. \\\\\n            \n            \\item \\textbf{Cosine similarity of climate change terms and sentences (COS\\_CCTS)}. The climate change terms were collected from news media about climate change. We calculate cosine similarity between the terms and sentences. In total, there are 300 most frequent terms relating to location, person, organization, and chemical compounds.\\\\\n            \n             \\item \\textbf{Cosine similarity of topic signatures and sentences (COS\\_TPS).} Topic signatures play an important role in automatic text summarization and information retrieval. It helps identify the presence of complex concepts or the importance in text. In a process of determining topic signatures, words appearing occasionally in the input text but rarely in other text are considered as topic signatures. They are determined by an automatic predefined threshold which indicates descriptive information. Topic signatures are generated by comparing with pre-classified text on the same topic using a concept of likelihood ratio \\cite{nenkova-mckeown-2011,Lin:2000:AAT:990820.990892}, $\\lambda$ presented by \\cite{Dunning1993}. It is a statistical approach which calculates a likelihood of a word. For each word in the input, the likelihood of word occurrence is calculated in pre-classified text collection. Another likelihood values of the same word is calculated and compared in another out-of-topic collection. The word, on the topic-text collection that has higher likelihood value than the out-of-topic collection, is regarded as topic signature of a topic. Otherwise the word is ignored. \\\\        \n        \\end{enumerate}\n    \\item \\textbf{Semantic Similarity of Sentence and Debate Titles (COS\\_STT).} Since the aforementioned features do not semantically capture the meaning of context, we create this feature for such purpose. We compare each sentence to the list of debate titles based on the assumption that forum users are likely to repeat debate titles in their comments. Thus, we compare each sentence to the titles and then calculate the semantic similarity score by using Doc2Vec \\cite{rehurek_lrec}.   \n\\end{enumerate}\n\n\n\n\n\n\n\\begin{table}[ht]\n\\centering\n\\scalebox{0.85}{\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n\n\n\\textbf{ROUGE-N} & \\textbf{CB} & \\textbf{CJ} & \\textbf{COS\\_CCT} & \\textbf{COS\\_TTS} & \\textbf{COS\\_TPS} & \\textbf{SL} & \\textbf{SP} & \\textbf{COS\\_STT} & \\textbf{TT} \\\\ \\hline\n\\textbf{R-1} & 0.4773 & 0.4988 & 0.3389 & 0.5630 & 0.3907 & 0.4307 & \\textbf{0.6124} & 0.4304 & 0.5407 \\\\ \\hline\n\\textbf{R-2} & 0.3981 & 0.4346 & 0.2558 & 0.5076 & 0.2986 & 0.3550 & \\textbf{0.5375} & 0.3561 & 0.4693 \\\\ \\hline\n\\textbf{R-SU4} & 0.3783 & 0.4147 & 0.2340 & 0.4780 & 0.2699 & 0.3335 & \\textbf{0.4871} & 0.3340 & 0.4303 \\\\ \\hline\n\\end{tabular}}\n\\caption{ROUGE scores after applying Doc2Vec to the salient sentence selection.}\\label{table_rouge_scores}\n\\end{table}\n\n\n\n\n\n\n\n\n\\begin{table}[ht]\n\\centering\n\\scalebox{0.85}{\n\\begin{tabular}{lcccccc}\n\\hline\n\\multicolumn{1}{|c|}{\\multirow{2}{*}{\\textbf{Comparison Pairs}}} & \\multicolumn{2}{c|}{\\textbf{ROUGE-1}} & \\multicolumn{2}{c|}{\\textbf{ROUGE-2}} & \\multicolumn{2}{c|}{\\textbf{ROUGE SU4}} \\\\ \\cline{2-7} \n\\multicolumn{1}{|c|}{} & \\multicolumn{1}{c|}{\\textbf{Z}} & \\multicolumn{1}{c|}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Asymp. Sig.\\\\ (2-tailed)\\end{tabular}}} & \\multicolumn{1}{c|}{\\textbf{Z}} & \\multicolumn{1}{c|}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Asymp. Sig.\\\\ (2-tailed)\\end{tabular}}} & \\multicolumn{1}{c|}{\\textbf{Z}} & \\multicolumn{1}{c|}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Asymp. Sig.\\\\ (2-tailed)\\end{tabular}}} \\\\ \\hline\n\\multicolumn{1}{|l|}{SP VS CB} & \\multicolumn{1}{c|}{$-4.246^b$} & \\multicolumn{1}{c|}{0*} & \\multicolumn{1}{c|}{$-3.962^b$} & \\multicolumn{1}{c|}{0*} & \\multicolumn{1}{c|}{$-3.044^b$} & \\multicolumn{1}{c|}{0.002} \\\\ \\hline\n\\multicolumn{1}{|l|}{SP VS CJ} & \\multicolumn{1}{c|}{$-3.570^b$} & \\multicolumn{1}{c|}{0*} & \\multicolumn{1}{c|}{$-3.090^b$} & \\multicolumn{1}{c|}{0.002} & \\multicolumn{1}{c|}{$-2.192^b$} & \\multicolumn{1}{c|}{0.028} \\\\ \\hline\n\\multicolumn{1}{|l|}{SP VS COS\\_CCTS} & \\multicolumn{1}{c|}{$-6.792^b$} & \\multicolumn{1}{c|}{0*} & \\multicolumn{1}{c|}{$-6.511^b$} & \\multicolumn{1}{c|}{0*} & \\multicolumn{1}{c|}{$-6.117^b$} & \\multicolumn{1}{c|}{0*} \\\\ \\hline\n\\multicolumn{1}{|l|}{SP VS COS\\_TTS} & \\multicolumn{1}{c|}{$-1.307^b$} & \\multicolumn{1}{c|}{0.191} & \\multicolumn{1}{c|}{$-.789^b$} & \\multicolumn{1}{c|}{0.43} & \\multicolumn{1}{c|}{$-.215^b$} & \\multicolumn{1}{c|}{0.83} \\\\ \\hline\n\\multicolumn{1}{|l|}{SP VS COS\\_TPS} & \\multicolumn{1}{c|}{$-6.728^b$} & \\multicolumn{1}{c|}{0*} & \\multicolumn{1}{c|}{$-6.663^b$} & \\multicolumn{1}{c|}{0*} & \\multicolumn{1}{c|}{$-6.384^b$} & \\multicolumn{1}{c|}{0*} \\\\ \\hline\n\\multicolumn{1}{|l|}{SP VS SL} & \\multicolumn{1}{c|}{$-4.958^b$} & \\multicolumn{1}{c|}{0*} & \\multicolumn{1}{c|}{$-4.789^b$} & \\multicolumn{1}{c|}{0*} & \\multicolumn{1}{c|}{$-4.110^b$} & \\multicolumn{1}{c|}{0*} \\\\ \\hline\n\\multicolumn{1}{|l|}{SP VS COS\\_STT} & \\multicolumn{1}{c|}{$-4.546^c$} & \\multicolumn{1}{c|}{0*} & \\multicolumn{1}{c|}{$-4.322^c$} & \\multicolumn{1}{c|}{0*} & \\multicolumn{1}{c|}{$-3.671^c$} & \\multicolumn{1}{c|}{0*} \\\\ \\hline\n\\multicolumn{1}{|l|}{SP VS TT} & \\multicolumn{1}{c|}{$-3.360^c$} & \\multicolumn{1}{c|}{0.001*} & \\multicolumn{1}{c|}{$-2.744^c$} & \\multicolumn{1}{c|}{0.006} & \\multicolumn{1}{c|}{$-2.641^c$} & \\multicolumn{1}{c|}{0.008} \\\\ \\hline\n\\multicolumn{7}{l}{a) Wilcoxon Signed Ranks Test.} \\\\\n\\multicolumn{7}{l}{b) Based on negative ranks.} \\\\\n\\multicolumn{7}{l}{c) Based on positive ranks.}\n\\end{tabular}}\n\\caption{The statistical information of comparing sentence position and other features after applying Doc2Vec.}\n\\label{table_sig_position}\n\\end{table}\n\n\n\n\n\\subsection{Results} \\label{results}\nIn order to evaluate the system summaries against the reference summaries, we apply ROUGE-N evaluation metrics. We report ROUGE-1 (unigram), ROUGE-2 (bi-grams) and ROUGE-SU4 (skip-bigram with maximum gap length of 4). The ROUGE scores as shown in Table \\ref{table_rouge_scores} indicate that sentence position feature outperforms other features. The least performing feature is the cosine similarity of climate change terms and sentences feature.\n\n\nTo measure the statistical significance of the ROUGE scores generated by the features, we calculated a pairwise Wilcoxon signed-rank test with Bonferroni correction. We report the significance p = .0013 level of significance after the correct is applied. Our results indicate that there is statistically significance among the features. Table \\ref{table_sig_position} illustrates the statistical information of comparing sentence position and other features. The star indicates that there is a statistical significance difference between each comparison pair. \n\n\n\n\n\n\n\\section{Conclusion} \\label{conclusion}\nIn this paper we worked on an annotation task for a new annotated dataset, online debate data. We have manually collected reference summaries for comments given to global warming topics. The data consists of 341 comments with total 519 annotated salient sentences. We have performed five annotation sets on this data so that in total we have 5 X 519 annotated salient sentences. We also implemented an extractive text summarization system on this debate data. Our results revealed that the key feature that plays the most important role in the selection salient sentences is sentence position. Other useful features are debate title words feature, and cosine similarity of debate title words and sentences feature. \n\nIn future work, we aim to investigate further features for the summarization purposes. We also plan to integrate stance information so that summaries with pro-contra sides can be generated.\n\n\\section*{Acknowledgments}\nThis work was partially supported by the UK EPSRC Grant No. EP\/I004327\/1, the European Union under Grant Agreements No. 611233 PHEME, and the authors would like to thank Bankok University of their support. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDeep learning methods are quite successful in many fields such as image analytics and natural language processing.\nDeep learning uses several stacked layers of neural networks, which are optimised using loss functions such as cross entropy with stochastic gradient descent.\nAn example is presented in eq.\\ref{eq:empirical-risk}, where\n$\\alpha$ represents the possible configurations of the learning machine, $z_{i}$ is a set of examples $i = 1, ..., l$ and $\\mathcal{Q}$ is a loss function.\nMinimisation of the empirical risk equals minimising eq.~\\ref{eq:empirical-risk}.\n\n\\begin{equation}\nR_{emp}(\\alpha) = \\frac{1}{l} \\sum_{i=1}^l \\mathcal{Q}(z_{i},\\alpha)\n\\label{eq:empirical-risk}\n\\end{equation}\n\nWe could consider several ways in which to reduce the risk and several advances have been done in improving stochastic gradient descent~\\cite{kingma2014adam}.\nNeural networks have a large number of neurons, which implies that they have a large capacity able of modeling a large set of problems and several sets of network weight values could be identified during learning that have minimal empirical risk, which might have different generalisation capabilities.\nTheir capacity in terms of VC-dimension is large due to the large number of neurons, even though it is finite.\n\nAs indicated in equation~\\ref{eq:risk-generalization}, the bound on the generalisation performance of a trained model depends on the performance on the training set $R_{emp}(\\alpha_{l})$, the VC-dimension $h$ of the classifier and the size of the number of examples used as training data.\n\n\\begin{equation}\nR(\\alpha_{l}) \\leq R_{emp}(\\alpha_{l}) + \\dfrac{B \\mathcal{E} (l) }{2} (1+ \\sqrt{1 + \\dfrac{4 R_{emp} (\\alpha_{l}) }{B \\mathcal{E} (l)}}) \n\\label{eq:risk-generalization}\n\\end{equation}\n\n\\begin{equation}\n    \\mathcal{E} (l) = 4 \\dfrac{h (ln \\dfrac{2l}{h}+1)- ln \\dfrac{\\eta}{4}}{l}\n\\end{equation}\n\nAssuming that the empirical risk on the training set is the same for several functions, in order to improve the predictive performance, the function with a lower VC-dimension or the one trained with more data should have a lower risk on the performance on an unseen test set.\n\nFor controlling the VC-dimension of a function, constraining the effective VC-dimension has been proposed~\\cite{vapnik1998statistical}.\nAmong existing work, regularisation is a way to constrain the search for functions that follow certain properties.\nLinear models have relied on Tikhonov regularisation~\\citep{tikhonov1977solutions}.\nNeural networks are made of a set of non-linearities, thus regularisation such as $L_2$ have been applied~\\cite{elsayed2018large}.\nThere have been several proposals, which include recent work to the models using knowledge~\\citep{roychowdhury2021regularizing}.\n\nThe structural risk minimisation framework~\\cite{vapnik1998statistical} intends to control parameters that minimise the VC-dimension and offers certain guarantees about the performance of the trained model.\nA good example of learning algorithm that implements the structural risk minimisation is support vector machines (SVM)~\\citep{vapnik2013nature}.\nSVM is a large margin classifier that aims separating the classes defining it as a constraint problem, whose solution builds on the Karush\u2013Kuhn\u2013Tucker approach that generalises the Lagrange multipliers.\nThe vectors from the training set that are on the margin are the named the support vectors.\nIf we have a training set defined by pairs $\\{x_i, y_i\\}$ where $x_i$ is a vector in $R^n$ and $y_i = \\{1,-1\\}$ defines the class of the instance where $i=1,...,l$.\nWhere the margin is defined by the $1\/w \\in R^n$ and $b \\in R^1$ is the bias term, the constraint optimisation problem for SVMs is defined as:\n\n\\begin{equation}\n\\begin{aligned}\nmin & & \\frac{1}{2}w^2 \\\\\n\\\\\ns.t. & & y_i (w x_i + b) \\geq 1\n\\end{aligned}\n\\end{equation}\n\nUsing Lagrange multipliers $\\alpha = \\{ \\alpha_1,..., \\alpha_l \\}$ with $\\alpha \\geq 0$ and $\\sum_{i=1}^l \\alpha_i y_i = 0$, the separating hyperplane has the formulation:\n\n\\begin{equation}\n    f(x,\\alpha) = \\sum_{i=1}^l y_i \\alpha_i (x_i*x) + b\n\\label{eq:svm-hyperplane}\n\\end{equation}\n\nEq.~\\ref{eq:svm-hyperplane} depends on the inner product of $x_i$ and $x$. The elements of the training set on the margin are named the support vectors.\n\nIf the data is not linearly separably in input space, a mapping into a feature space in which the data is linearly separable and an inner product exists would be a solution. For instance, in eq.~\\ref{eq:mapping-svm} function $z$ maps the input space into a feature space.\nAs we can see, what is important is the calculation of the inner product and not the dimensionality of the space.\n\n\\begin{equation}\n    f(x,\\alpha) = \\sum_{i=1}^l y_i \\alpha_i (z(x_i)*z(x)) + b\n\\label{eq:mapping-svm}\n\\end{equation}\n\nFrom the formulation of SVMs, the inner product between vectors is what is needed to define the solution to the large margin classifier.\nUsing Mercer's theorem allows using kernels to calculate the inner product in a Hilbert space.\nEq.~\\ref{eq:kernel-svm} shows how the SVM formulation would be defined using kernel $K$.\n\n\\begin{equation}\n    f(x,\\alpha) = \\sum_{i=1}^l y_i \\alpha_i K(x_i,x) + b\n\\label{eq:kernel-svm}\n\\end{equation}\n\nUsing kernels in this way allows working in high-dimensional feature spaces without having to work in the high-dimensional space explicitly, this is known as the \\textit{kernel trick}.\nSpecialised kernels have been developed to use SVMs in several problem types.\n\n\nIn this paper, we explore using non-linear functions similar to deep neural networks as mapping functions from input space to feature space.\nWe show examples of the expected performance based on structural risk minimisation principles.\nWe evaluate the proposed method on several data sets and baseline methods.\nWhen the training data is small, the proposed method largely improves over the baseline methods.\nAs expected, this improvement is reduced when more training data is provided.\n\nThe code used in this experiments is available from the following GitHub repository:\\\\\\url{https:\/\/github.com\/ajjimeno\/nn-hyperplane-bounds}\n\n\\section{Methods}\n\nIn this section, we define a large margin linear classifier and we introduce the structural risk minimisation principle.\nThen, we provide a way to define a large margin classifier using a feature space defined by a set of non-linear functions.\nThese non-linear functions are the equivalent of a deep neural network.\n\n\\subsection{Large margin classifier}\n\n\n\nThere are several kernels that have been developed over time that turn the input space into a feature space that captures relations among the features, from image analytics (e.g.~\\cite{szeliski2010computer,camps2006composite}) to text analytics (e.g. string kernel~\\cite{lodhi2002text}).\nThe kernel trick mentioned above allows working in high dimensional feature spaces without the cost of working directly in the feature space.\nThis is achieved by using the kernel to calculate the dot product in feature space without having to map the instances into the feature space.\nOn the other hand, it is difficult to design a kernel that will work well with all sorts of data when compared to the recent success of deep learning.\n\n\nDeep neural network seem to be effective at approximating many different functions, thus it is interesting to map our input feature space into a feature space in which an optimal hyperplane could be identified.\nThis means, using a neural network $z$ as the mapping function between the input space and the feature space, so the optimisation problem should consider now $z(x_i) \\in R^m$ instead of $x_i$ and now the constraints look like $y_i (w z(x_i) + b) \\geq 1$. and $w \\in R^m$.\n\nOne problem is that current neural networks have a large number of parameters, which are needed to be effective in the current tasks where they are successful.\nThis implies as well that they have a large capacity or VC-dimension.\nIn the next sections, we explore how to search for the mapping that has better generalisation guarantees. \n\n\n\n\n\n\n\n\\subsection{Hyperplane bounds}\n\n\nIn this section, we explore properties of the separating hyperplane and what constrains are needed to identify a configuration of the neural network used as mapping function that has better generalisation properties.\n\n\nIf we consider a training set $\\{Y,X\\}$, as defined above, the following inequality holds true for vector $w_0$ and $\\rho_0$,\n\n\\begin{equation}\n    min_{(y,x) \\in \\{Y,X\\}} \\frac{y(w_0 x)}{|w_0|} \\geq \\rho_0\n\\end{equation}\n\nwhich assumes that the training set is separable by a hyperplane with margin $\\rho_0$, then the following theorem holds true.\n\n\n\\paragraph{Novikoff theorem}\n\nGiven and infinite sequence of training examples $u_i$ with elements satisfying the inequality $|x_i| < D$, there is a hyperplane with coefficients $w_0$ that separates the training examples currently and satisfies conditions as above.\nUsing an iterative procedure, e.g. stochastic gradient descent, to construct such a hyperplane takes at most\n\n\\begin{equation}\n    M = [\\frac{D^2}{\\rho^2_0}]\n\\end{equation}\n\nAs a follow up theorem, we accept that for the algorithm constructing hyperplanes in the regime that separates training data without error, the following bound on error rate is valid\n\n\\begin{equation}\n    ER(w_l) \\leq \\frac{E[\\frac{D^2_l}{\\rho^2_l}]}{l+1}\n\\end{equation}\n\nwhere $[\\frac{D^2_l}{\\rho^2_l}]$ is estimated from training data.\n\nThese two theorems provide already a bound on the error of the separating hyperplane, which relies on parameters that can be estimated from the training data.\nIn the following two sections, we show theorems for the bounds on the VC-dimension of the separating hyperplane and properties of the optimal separating hyperplane that will be used to define the optimisation problem for the neural network mapping function.  \n\n\\subsection{Bounds on the VC-dimension for \\texorpdfstring{$\\Delta$-margin}{TEXT} separating hyperplanes}\n\nIn this section, we explore theorems that define bounds on the VC-dimension of the separating hyperplane.\n\n\\paragraph{Theorem} Let vectors $x \\in X$ belong to a sphere of radius $R$, the set of $\\Delta$-margin separating hyperplanes has the following VC-dimension h bounded by the inequality\n\n\\begin{equation}\n h \\leq min([\\frac{R^2}{\\Delta^2}], n)+1   \n\\end{equation}\n\n\\paragraph{Corollary} With probability $1-\\eta$ the probability that a test example will not be separated correctly by the $\\Delta$-margin hyperplane has the bound\n\n\\begin{equation}\n    P_{error}\\leq \\frac{m}{l} + \\frac{\\xi}{2}(1 + \\sqrt{1+\\frac{4m}{l\\xi}})\n\\end{equation}\n\nwhere \n\n\\begin{equation}\n    \\xi = 4 \\frac{h(ln \\frac{2l}{h} + 1) -ln \\frac{\\eta}{4}}{l}\n\\end{equation}\n\nwhere $m$ is the number of examples not separated correctly by this $\\Delta$-margin hyperplane, $h$ is the bound in the VC-dimension, where a good generalisation is dependent on $\\Delta$.\n\nSo, we have already a bound on the VC-dimension of the separating hyperplane.\nIn the next section, we present the idea of identifying the optimal hyperplane, which links the hyperplane optimisation problem with structural risk minimisation.\n\n\\subsection{Optimal hyperplane properties}\n\nThe optimal hyperplane is the one that separates the training examples from classes $y={1,-1}$ with the maximal margin.\nIt has been previously shown that the optimal hyperplane is unique~\\cite{vapnik1998statistical}.\nThe optimal hyperplane has some interesting properties that are relevant to our work.\nOne of them is that the generalisation ability of the optimal hyperplane is better than the general bounds obtained for methods that minimise the empirical risk.\nLet's define the set $X = {x_1, ..., x_l}$ in space $R^n$, where we have our training examples. Within the elements of $X$ that have the following property:\n\n\\begin{equation}\n    \\inf_{x \\in X} | w x + b | = 1\n\\end{equation}\n\nThese elements $x \\in X$ are the essential support vectors and are on the margin.\nHaving defined the essential support vectors, we define the number of essential support vectors $K_l$\n\n\\begin{equation}\n    K_l = K((x_1, y_1), ... (x_l, y_l))\n\\end{equation}\n\nAnd the maximum norm $D_l$ of the essential support vectors.\n\n\\begin{equation}\nD_l = D((x_1, y_1), ... (x_l, y_l)) = \\max_i |x_i|\n\\end{equation}\n\nBased on the definitions above for the essential support vectors, the following properties have been proved for the optimal hyperplane.\nThe following inequality $K_l \\leq n$ holds true, which implies that the number of essential support vectors is smaller than the dimensionality of the elements of $X$ and $w$.\nLet $ER(\\alpha_l)$ be defined as the expectation of the probability of error of the optimal hyperplane defined using the training data, then the following inequalities hold for the optimal hyperplane considering the values estimated on the essential support vectors. \n\n\n\n\\begin{equation}\n    ER(\\alpha_l) \\leq \\frac{EK_{l+1}}{l+1}\n\\end{equation}\n\n\nAdditionally, considering an optimal hyperplane passing through the origin:\n\n\\begin{equation}\n    ER(\\alpha_l) \\leq \\frac{E(\\frac{D_{l+1}}{\\rho_{l+1}})^2}{l+1}\n    \\label{eq:novikoff-error-expectation}\n\\end{equation}\n\nCombining the two previous inequalities, we obtain a bound on the expectation of the probability of error, which is bounded by the number of examples in the training data but as well by the number of essential support vectors and the relation between the ball in which the support vectors are and the margin, as shown below\n\n\\begin{equation}\n    ER(\\alpha_l) \\leq \\frac{E \\min(K_l, (\\frac{D_{l+1}}{\\rho_{l+1}})^2)}{l+1}\n\\end{equation}\n\nLeave-one-out error has been used as an unbiased estimator to prove the bounds on the optimal hyperplane~\\citep{luntz1969estimation}.\n\n\\begin{equation}\n    E\\frac{\\mathcal{L}(z_1,...,z_{l+1})}{l+1}=ER(\\alpha_l)\n\\end{equation}\n\nFirst, the number of errors by leave-one-out does not exceed the number of support vectors~\\citep{vapnik2000bounds}.\nIf a vector $x_i$ is not an essential support vector, then there is an expansion of the vector $\\phi_0$ that defines the optimal hyperplane that does not contain the vector $x_i$.\nThe optimal hyperplane is unique, so removing this vector from the training set does not change it.\nLeave-one-out recognizes correctly all the vectors that are not in the set of essential support vectors.\nThe number $\\mathcal{L}(z_1,...,z_{l+1})$ of errors in leave-one-out does not exceed $\\mathcal{K}_{l+1}$, which implies\n\n\n\\begin{equation}\n    ER(\\alpha_l) = \\frac{E \\mathcal{L}(z_1,...,z_{l+1})}{l+1} \\leq \\frac{EK_{l+1}}{l+1}\n\\end{equation}\n\nTo prove~eq.~\\ref{eq:novikoff-error-expectation}, the number of errors in leave-one-out does not exceed the number of corrections M to find the optimal hyperplane, as defined by Novikoff's theorem presented above.\n\n\n\n\n\n\n\n\n\n\\subsection{Mapping into feature space using neural networks}\n\nUp to this point, the formulations rely on an input space defined by the training data instances $x_1, ..., x_l$ in a given space $R^n$.\nIf a hyperplane separating the data instances into classes $y=\\{1,-1\\}$ does not exist in input space, the instances are mapped into a feature space $R^m$.\n\n\\begin{equation}\n    z(x) : R^n \\mapsto R^m\n\\end{equation}\n\nIn the case of support vector machines, the~\\textit{kernel trick} is used to calculate the dot product using a kernel in feature space without having to do the explicit mapping, which allows working with feature spaces of larger dimensions, even infinite dimensions.\nKernels have been designed to map the input space into separable feature spaces.\n\nOnce the kernel is designed, there is a point in the feature space for each training instance.\nThe properties described for the optimal hyperplane mentioned in the previous sections would still apply but in this case, these properties would be applied to the generated feature space.\nSpecific formulations are applied to profit from the kernel trick mentioned above.\n\nDeveloping the best kernel for a specific problem has shown to be a difficult task and multiple kernels have appeared for different tasks.\nRecently, neural networks have shown that they can learn a classifier with relatively less effort, despite the increase in computational power required to train these classifiers.\nNeural networks in a sense map the input space to another space using a concatenation of linear operators followed by a non-linearity (e.g. sigmoid, RELU, ...), as shown in equation~\\ref{eq:neural-network}.\n\n\\begin{equation}\nz(x) = \\sigma f_{k}(\\sigma f_{k-1}(... \\sigma f_1(x))))\n\\label{eq:neural-network}\n\\end{equation}\n\nEach function $f \\in {f_1, ..., f_k}$ will be a derivative of $f(x) = Wx+B$, where $W$ and $B$ are matrices of weights and biases.\nThese are parameters that need to be optimised for each function.\nStochastic gradient descent is typically used to optimize such functions, thus we prepare the optimisation to use stochastic gradient descent.\n\nRevisiting the classification that we intend to optimise and considering the mapping function $z$, we obtain eq.~\\ref{eq:neural-network-svm}.\nThe parameters to optimize are the vector $w$, the bias $b$ and the weights and biases of the neural network $z$.\nThe dimension of the weight vector $w$ is defined by $z$.\n\n\\begin{equation}\ny_i(w z(x_i) + b) \\geq 1\n\\label{eq:neural-network-svm}\n\\end{equation}\n\n\n\n\n\n\n\n\n\n\n\nIn a support vector machine in feature space, a way to compare different kernels is to measure the maximum radius $D_l$ in which the support vectors fit in and multiply it by $\\lVert w_{l} \\rVert ^2$ as in $[ D_l^2  \\lVert W_{l} \\rVert^2 ]$.\nConsidering that $D_l^2 \\lVert W_{l} \\rVert^2 = \\frac{D_l^2}{\\rho_l^2}$, the lower is this ration, the lower is the probability of error, which is considered as a mean to control the VC-dimension of the separating hyperplane for the setup defined in this work.\n\n\n\n\n\n\\subsection{Optimisation}\n\nTo find the values of the parameters of the system, we use stochastic gradient descent.\nOther approaches, such as Lagrange multipliers are not applicable to neural networks.\nOn the other hand, using stochastic gradient descent guarantees finding a local optima, which hopefully is an approximation with reasonable generalisation performance.\n\nFor optimisation, we use AdamW~\\citep{loshchilov2017decoupled} with eps=1e-08 and weight decay set to 0.1 (which already implies using an $L_{2}$ regularisation) and betas=(0.9, 0.999).\n\nWe have adapted~\\citep{zhang2004solving} using large margin loss.\nWe use the modified Huber loss, which has nicer properties, even though other large margin losses could be explored.\n\nModified Huber loss~\\citep{zhang2004solving}\n\\begin{equation}\n    l(y)=\n    \\begin{cases}\n    -4*h*y   & \\text{if }h*y <= -1\\\\\n    (1-h*y)^2  & \\text{if } -1 < h*y <= 1\\\\\n    0          & \\text{if } h*y > 1\n    \\end{cases}\n\\end{equation}\n\nGradient of the modified Huber loss:\n\n\\begin{equation}\n    \\frac{\\partial l}{\\partial w_i}=\n    \\begin{cases}\n    -4*h*x_i   & \\text{if }h*y <= -1\\\\\n    -2* (1-h*y)*h*x_i  & \\text{if } -1 < h*y <= 1\\\\\n    0          & \\text{if } h*y > 1\n    \\end{cases}\n\\end{equation}\n\nWe need to estimate $w$ and $z$ subject to the constraints above.\n$z$ will be defined using a neural network and the size of the feature space derived from $z$ will define the size of the vector $w$.\n$\\lVert w \\rVert$ will be minimised and so will be the $\\lVert z(x) \\rVert$ of the vectors.\nThe loss function of the optimisation problem is defined as follows\n\n\\begin{equation}\n\\mathcal{L}(x_1, ..., x_l)_{inv}= \\mathcal{L} (x_1, ..., x_l) + \\alpha \\lVert w \\rVert^2 + \\beta \\sum^l_{i=1} \\lVert z(x_i) \\rVert^2\n\\end{equation}\n\nThe development above is prepared for binary classification.\nIn a multi class setting, as many classifiers $y \\in \\{1, -1\\}$ as classes are trained and during prediction, the classifier with the maximum value is returned as prediction as shown in equation \\ref{eq:argmax-multi-class}.\n\n\\begin{equation}\n    \\arg \\max_{c \\in 1..n} \\{ f_{1}(x), ..., f_{n}(x) \\}\n\\label{eq:argmax-multi-class}\n\\end{equation}\n\nThe proposed method works in the multi class setting, but it has the advantage that it might be able of deciding when an instance does not belong to any class if all the classifier functions predict the -1 class.\n\n\n\n\n\n\\section{Results}\n\nWe have evaluated the proposed method using the MNIST and CIFAR 10 data sets.\nThe different methods have been evaluated using several algorithms.\n$\\alpha$ and $\\beta$ of the loss function have been set to the same value which is the best set up during the experiments.\nWe evaluate the both losses and the performance of dropout~\\citep{srivastava2014dropout} and augmentation based on affine transformations.\n\n\\subsection{MNIST}\n\nMNIST is a collection of hand written digits from 0 to 9.\nThe training set has a total of 60k examples while the test set contains a total of 10k examples.\nAll images are 28x28 with black background pixels and different white level pixels to define the numbers using just one channel.\nImages were normalised using a mean of 0.1307 and standard deviation of 0.3081.\n\nWe have used LeNet as base neural network, which has been adapted to be used in our approach.\nIn order to use LeNet in our approach, we have considered the last layer as the margin weights $w$, which defines each each one of the 10 MNIST classes functions.\nThe rest of the network has been considered as the function $z(x_i)$.\nThis means that both the vector $w$ and $z(x_i)$ belong to $R^{84}$.\nIf the parameters $\\alpha$ and $\\beta$ are set to zero, we are effectively using LeNet.\n\nTable~\\ref{tab:mnist-results-percentage} shows the results of different experimental setups, which includes dropout $p=0.5$, augmentation and a combination of several configurations.\nAugmentation was done using random affine transformations with a maximum rotation of 20 degrees, translation maximum of 0.1 on the x and y axis and scale between 0.9 and 1.1, as well brightness and contrast were randomly altered with a maximum change of 0.2.\nWe have used several partitions of the training set to simulate training the different configurations with data sets of several sizes, which evaluates as well the impact of the number of training examples in addition to the bounded VC-dimension of the separating hyperplane.\nExperiments have been run 5 times per configuration and results show the average and the standard deviation.\n\n\n\n\\begin{sidewaystable}\n\\begin{tabular}{l|c|c|c|c|c|c|c|c}\n\\hline\nMethod&1&5&10&20&40&60&80&100 \\\\\n\\hline\nce&91.52$\\pm$0.28&97.20$\\pm$0.16&98.07$\\pm$0.10&98.58$\\pm$0.10&99.04$\\pm$0.10&99.20$\\pm$0.06&99.25$\\pm$0.05&99.20$\\pm$0.03 \\\\\nce+do&94.36$\\pm$0.22&97.78$\\pm$0.09&98.54$\\pm$0.07&98.81$\\pm$0.07&99.17$\\pm$0.03&99.22$\\pm$0.03&99.32$\\pm$0.03&99.37$\\pm$0.04 \\\\\nce+lm-0.001&94.89$\\pm$0.23&97.95$\\pm$0.10&98.42$\\pm$0.09&98.61$\\pm$0.08&99.05$\\pm$0.10&99.01$\\pm$0.03&99.22$\\pm$0.07&99.25$\\pm$0.08 \\\\\nce+lm-0.001+do&95.12$\\pm$0.17&97.84$\\pm$0.15&98.53$\\pm$0.08&98.77$\\pm$0.04&98.98$\\pm$0.06&99.13$\\pm$0.02&99.22$\\pm$0.05&99.19$\\pm$0.04 \\\\\n\\hline\nce+aug&95.82$\\pm$0.17&98.37$\\pm$0.09&98.94$\\pm$0.09&99.23$\\pm$0.08&99.36$\\pm$0.04&99.40$\\pm$0.05&99.52$\\pm$0.04&99.50$\\pm$0.02 \\\\\nce+aug+do&95.18$\\pm$0.25&97.96$\\pm$0.10&98.56$\\pm$0.07&98.89$\\pm$0.11&99.11$\\pm$0.07&99.09$\\pm$0.06&99.16$\\pm$0.05&99.22$\\pm$0.06 \\\\\nce+aug+lm-0.001&96.82$\\pm$0.17&98.68$\\pm$0.12&99.08$\\pm$0.09&99.24$\\pm$0.06&99.37$\\pm$0.05&99.41$\\pm$0.04&99.47$\\pm$0.06&99.53$\\pm$0.01 \\\\\nce+aug+lm-0.001+do&95.57$\\pm$0.36&97.91$\\pm$0.09&98.55$\\pm$0.08&98.78$\\pm$0.06&98.95$\\pm$0.06&98.99$\\pm$0.04&99.05$\\pm$0.06&99.16$\\pm$0.05 \\\\\n\\hline\n\\hline\nmh&91.66$\\pm$0.48&97.22$\\pm$0.22&98.03$\\pm$0.15&98.49$\\pm$0.08&98.99$\\pm$0.08&99.13$\\pm$0.03&99.16$\\pm$0.03&99.27$\\pm$0.06 \\\\\nmh+do&94.75$\\pm$0.32&97.99$\\pm$0.10&98.51$\\pm$0.04&98.82$\\pm$0.05&99.16$\\pm$0.05&99.22$\\pm$0.08&99.33$\\pm$0.04&99.36$\\pm$0.08 \\\\\nmh+lm-0.001&94.17$\\pm$0.25&97.53$\\pm$0.12&98.26$\\pm$0.13&98.57$\\pm$0.10&98.89$\\pm$0.04&98.97$\\pm$0.07&99.05$\\pm$0.05&99.23$\\pm$0.04 \\\\\nmh+lm-0.001+do&95.03$\\pm$0.31&97.95$\\pm$0.13&98.50$\\pm$0.13&98.79$\\pm$0.07&99.01$\\pm$0.09&99.14$\\pm$0.03&99.14$\\pm$0.07&99.12$\\pm$0.08 \\\\\n\\hline\nmh+aug&96.20$\\pm$0.18&98.45$\\pm$0.05&98.96$\\pm$0.03&99.22$\\pm$0.04&99.40$\\pm$0.06&99.44$\\pm$0.02&99.53$\\pm$0.04&99.51$\\pm$0.04 \\\\\nmh+aug+do&95.25$\\pm$0.45&98.08$\\pm$0.13&98.60$\\pm$0.07&98.76$\\pm$0.07&99.01$\\pm$0.04&99.11$\\pm$0.06&99.19$\\pm$0.03&99.22$\\pm$0.04 \\\\\nmh+aug+lm-0.001&96.53$\\pm$0.12&98.62$\\pm$0.13&99.05$\\pm$0.08&99.19$\\pm$0.06&99.41$\\pm$0.02&99.46$\\pm$0.04&99.46$\\pm$0.06&99.50$\\pm$0.01 \\\\\nmh+aug+lm-0.001+do&95.14$\\pm$0.43&98.03$\\pm$0.06&98.52$\\pm$0.09&98.78$\\pm$0.12&98.90$\\pm$0.09&99.00$\\pm$0.06&99.09$\\pm$0.09&98.99$\\pm$0.06 \\\\\n\\hline\n\\end{tabular}\n\\caption{MNIST results using LeNet using cross-entropy (ce) vs Modified Huber Loss (ml), dropout (do), hyperplane bound loss factor (lm) and augmented (aug)}\n\\label{tab:mnist-results-percentage}\n\\end{sidewaystable}\n\n\\subsection{CIFAR10}\n\nCIFAR10 is a data set of 32x32 images with 10 classes of objects with 3 channels. There a total of 60k training images and 10k testing images, with an equal split between images.\nEach image channel was normalised using a mean of 0.5 and standard deviation of 0.5.\n\nFor CIFAR10, we have considered the vgg19~\\citep{simonyan14vgg19} network.\nThe last layer is considered the margin weights $w$ and the output of the rest of the network as the function $z(x_i)$, which belong to $R^{4096}$. \n\n\n\\begin{sidewaystable}\n\\begin{tabular}{l|c|c|c|c|c|c|c|c}\n\\hline\nMethod&1&5&10&20&40&60&80&100 \\\\\n\\hline\nce&28.77$\\pm$1.74&43.60$\\pm$0.76&51.51$\\pm$0.90&60.98$\\pm$0.61&70.99$\\pm$0.64&76.23$\\pm$0.49&78.98$\\pm$0.60&81.16$\\pm$0.33 \\\\\nce+do&29.92$\\pm$1.22&43.54$\\pm$0.94&52.05$\\pm$0.75&61.52$\\pm$0.81&71.11$\\pm$0.68&75.99$\\pm$0.68&79.15$\\pm$0.56&80.96$\\pm$0.37 \\\\\nce+lm-1e-05&29.73$\\pm$1.57&44.01$\\pm$0.80&51.54$\\pm$0.98&61.11$\\pm$0.91&70.92$\\pm$0.89&76.34$\\pm$0.28&79.68$\\pm$0.48&81.23$\\pm$0.21 \\\\\nce+lm-1e-05+do&30.55$\\pm$1.46&43.21$\\pm$1.36&51.37$\\pm$1.28&61.15$\\pm$0.73&71.16$\\pm$0.24&76.20$\\pm$0.55&79.64$\\pm$0.58&81.45$\\pm$0.56 \\\\\n\\hline\nce+aug&32.50$\\pm$1.44&49.81$\\pm$1.00&61.55$\\pm$0.65&70.58$\\pm$0.59&79.18$\\pm$0.32&83.03$\\pm$0.19&85.47$\\pm$0.16&87.09$\\pm$0.16 \\\\\nce+aug+do&32.77$\\pm$1.53&51.53$\\pm$0.92&61.38$\\pm$0.59&71.00$\\pm$0.48&79.22$\\pm$0.32&82.91$\\pm$0.29&85.34$\\pm$0.27&87.22$\\pm$0.19 \\\\\nce+aug+lm-1e-05&32.53$\\pm$1.56&50.03$\\pm$1.02&60.85$\\pm$0.73&70.81$\\pm$0.27&79.38$\\pm$0.38&83.21$\\pm$0.31&85.25$\\pm$0.22&87.12$\\pm$0.14 \\\\\nce+aug+lm-1e-05+do&33.31$\\pm$1.27&50.94$\\pm$1.93&61.29$\\pm$0.63&71.07$\\pm$0.64&79.50$\\pm$0.39&83.14$\\pm$0.32&85.66$\\pm$0.29&87.38$\\pm$0.14 \\\\\n\\hline\n\\hline\nmh&30.26$\\pm$1.59&45.11$\\pm$1.40&54.14$\\pm$0.93&63.39$\\pm$0.49&71.93$\\pm$0.71&76.83$\\pm$0.64&79.49$\\pm$0.48&81.58$\\pm$0.20 \\\\\nmh+do&30.63$\\pm$1.43&46.16$\\pm$1.01&54.66$\\pm$0.99&63.70$\\pm$0.94&71.99$\\pm$0.22&76.51$\\pm$0.54&79.13$\\pm$0.46&81.54$\\pm$0.32 \\\\\nmh+lm-1e-05&30.72$\\pm$1.35&45.22$\\pm$0.96&53.80$\\pm$1.19&63.46$\\pm$0.80&71.89$\\pm$0.59&76.78$\\pm$0.52&79.35$\\pm$0.56&81.14$\\pm$0.37 \\\\\nmh+lm-1e-05+do&31.47$\\pm$1.22&45.71$\\pm$0.96&53.98$\\pm$0.77&62.86$\\pm$0.82&71.56$\\pm$0.67&76.43$\\pm$0.50&79.29$\\pm$0.27&81.22$\\pm$0.51 \\\\\n\\hline\nmh+aug&37.25$\\pm$0.84&54.38$\\pm$0.42&63.54$\\pm$0.57&71.64$\\pm$0.45&79.66$\\pm$0.43&82.98$\\pm$0.40&85.25$\\pm$0.27&87.10$\\pm$0.36 \\\\\nmh+aug+do&36.37$\\pm$0.35&54.66$\\pm$0.66&63.11$\\pm$0.69&72.06$\\pm$0.53&79.46$\\pm$0.29&83.06$\\pm$0.40&85.32$\\pm$0.18&87.03$\\pm$0.21 \\\\\nmh+aug+lm-1e-05&36.91$\\pm$0.98&54.54$\\pm$0.76&63.53$\\pm$1.00&71.35$\\pm$0.51&79.49$\\pm$0.41&82.90$\\pm$0.37&85.36$\\pm$0.21&86.94$\\pm$0.23 \\\\\nmh+aug+lm-1e-05+do&36.30$\\pm$1.34&54.61$\\pm$0.70&63.34$\\pm$0.60&72.08$\\pm$0.37&79.44$\\pm$0.40&83.14$\\pm$0.21&85.45$\\pm$0.34&87.09$\\pm$0.16 \\\\\n\\hline\n\\end{tabular}\n\\caption{CIFAR10 results using vgg19 cross-entropy (ce) vs Modified Huber Loss (ml), dropout (do), hyperplane bound loss factor (lm) and augmented (aug).}\n\\label{tab:cifar-results-percentage}\n\\end{sidewaystable}\n\n\n\\section{Discussion}\n\nWe have evaluated the proposed approach using two well-known image classification data sets and compared againt several baselines.\nThe results are compared against several baselines, which includes dropout and augmentation using affine transformations.\nIn addition to the modified Huber loss, we compare as well the performance of cross entropy loss, which is typically used in deep learning.\n\nWe have mentioned at the beginning that there are two factors that are relevant for the risk of generalisation of machine learning algorithms.\nOne is the VC-dimension, which we have tried to influence in this work.\nThe second one is additional training data.\nProviding additional training data has been simulated in two ways.\nThe first one is by selecting portions of the training data available from the training sets.\nThe second one has been using affine transformations, which uses the examples in the portion of the data set used as training data.\n\nWe observe that the proposed method shows a strong performance improvement when less training data is being used.\nWe observe as well that by providing additional training data effectively affects performance on the test set positively, which can be combined with the proposed method.\nDropout performed well when no modification or data augmentation were used.\n\nWhen considering the values of $D_l^2$ and $\\lVert w_l \\rVert^2$ in the loss function, the changes in size on the training set seems to have limited impact. The values for the margin are very similar, while the radius $D_l^2$ for the support vectors are around 1.\nConsidering these variables in the loss function, their values become more stable between training runs.\nWhen not using these terms as part of the loss function, the values tend to change significantly between experiments.\n\n\\section{Related work}\n\nDeep learning has achieved tremendous success in many practical tasks but\nthe neural networks used in deep learning have a high number of parameters, which implies a high VC-dimension.\nThese networks are optimised to reduce the empirical risk and having large data sets helps reducing the generalisation risk linked to this.\n\nNeural networks are trained using stochastic gradient descent, which guarantees reaching an optimum even if it is a local one.\nThere is recent work in which the neural networks is studied within the kernel theory, which has shown interesting understanding about the global optimisation of neural networks~\\citep{du2019gradient} and the convergence of wide neural networks through\nNeural Tangent Kernels~\\citep{jacot2018neural}.\nThis work complements what we have studied in this work and provides as well directions for further research.\nOn the other hand, this previous research could be considered to define directions for better defining approximation functions that would use well know properties of kernels and the adaptability of neural networks. \n\nCompared as well to what we propose in this work from the point of view of regularisation, there have been several means of regularisation of neural networks that include $L^2$ regularisation and several variants where different layers are regularised independently.\nTransfer learning is a way to pre-train the neural networks using similar data to the training data, which has been quite successful in recent proposed systems to improve the capability of existing systems.\nIn addition, knowledge has been proposed as means of regularisation~\\citep{borghesi2020improving,roychowdhury2021regularizing}, which would reuse existing resources and reduce the need for training data.\n\nIn this work, we have used hyperplane bounds to study how this contribute to find a better hyperplane using neural networks as feature space mapper.\nThere are several recent directions that would provide directions to improve the definition of functions that have better guarantees in terms of optimisation.\n\n\\section{Conclusions and future work}\n\nImprove over the baseline, which is most significant when a small portion of the training data is used.\nWhen more training data is made available, the improvement is reduced, which is an expected behaviour as shown by the bounds of the VC-dimension for $\\Delta$-margin separating hyperplanes and formulation of the empirical risk.\n\nWe have considered well known networks for the experiments, based on convolution neural networks, it might be interesting to explore additional network configurations to understand the impact of the network architecture with the proposed approach.\nThere has been as well recent work to better understand the optimisation and behaviour of neural networks that we would like to explore as future work.\n\n\\section{Acknowledgements}\n\nThis research was undertaken using the LIEF HPC-GPGPU Facility hosted at the University of Melbourne. This Facility was established with the assistance of LIEF Grant LE170100200.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\\label{sec0}\nThe Lieb--Thirring inequalities~\\cite{LT} give estimates for $\\gamma$-moments\nof the negative eigenvalues of the Schr\\\"odinger operator $-\\Delta-V$ in $L_2(\\mathbb{R}^d)$,\nwhere  $V=V(x)\\ge0$:\n\\begin{equation}\\label{LT}\n\\sum_{\\lambda_i\\le0}|\\lambda_i|^\\gamma\\le\\mathrm{L}_{\\gamma,d}\n \\int_{\\mathbb{R}^d} V(x)^{\\gamma+ d\/2}dx.\n\\end{equation}\nIn the case $\\gamma=1$ estimate~\\eqref{LT} is equivalent to the dual\ninequality\n\\begin{equation}\\label{orth}\n \\int_{\\mathbb{R}^d} \\rho(x)^{1+2\/d}dx\\le\\mathrm{k}_d\\sum_{j=1}^N\\|\\nabla\\psi_j\\|^2,\n\\end{equation}\nwhere $\\rho(x)$ is as in \\eqref{rho}, and $\\{\\psi_j\\}_{j=1}^N\\in H^1(\\mathbb{R}^d)$\nis an arbitrary orthonormal system. Furthermore the (sharp) constants $\\mathrm{k}_d$ and\n $\\mathrm{L}_{1,d}$ satisfy\n\\begin{equation}\\label{kL}\n \\mathrm{k}_d=(2\/d)(1+d\/2)^{1+2\/d}\\mathrm{L}_{1,d}^{2\/d}.\n\\end{equation}\n\nSharp constants in  \\eqref{LT} were found in~\\cite{Lap-Weid} for $\\gamma\\ge 3\/2$,\nwhile for a long time  the best available estimates for  $1\\le\\gamma<3\/2$ were those  found in\n\\cite{D-L-L}. Very recently an important improvement in the area was made\nin~\\cite{Frank-Nam}, where the original idea of~\\cite{Rumin2}\nwas developed and extended in a substantial way.\n\n\nInequality \\eqref{orth} plays an important role in the theory of the\n Navier--Stokes equations\n\\cite{Lieb, B-V,  T}, where the constant $ \\mathrm{k}_2$ enters the estimates\nof the fractal dimension of the global attractors\nof the Navier--Stokes system in various two-dimensional formulations.\n(In the three-dimensional case the corresponding results are  of a conditional character.)\n\nAlong with the  problem in a bounded domain\n$\\Omega\\subset\\mathbb{R}^2$ with Dirichlet boundary conditions\nthe Navier--Stokes system is also studied with periodic boundary conditions,\nthat is, on a two-dimensional torus. In this case for the system to be dissipative\none has to impose the zero mean condition on the components of the velocity vector\nover the torus.\n\nAnother physically relevant model  is the Navier--Stokes system on the sphere.\nIn this case the system is dissipative without extra orthogonality conditions.\nHowever, if we want to study the system in the form of the scalar vorticity equation, then the\nscalar stream function of a divergence free vector field is defined up\nto an additive constant, and without loss of generality we can (and always) assume that\nthe integral of the stream function over the sphere vanishes.\n\n\nWe can formulate our main result as follows.\n\n\n\n\n\\begin{theorem}\\label{Th:1}\nLet ${M}$ denote either $\\mathbb{S}^2$ or $\\mathbb{T}^2$,\nand let $\\dot H^1({M})$ be the Sobolev space of functions with\nmean value zero.\nLet $\\{\\psi_j\\}_{j=1}^N \\in\\dot H^1({M})$ be an orthonormal family\nin $L_2({M})$.  Then\n\\begin{equation}\\label{rho}\n\\rho(x):=\\sum_{j=1}^N|\\psi_j(x)|^2\n\\end{equation}\nsatisfies the inequality\n\\begin{equation}\\label{M}\n\\int_{{M}}\\rho(x)^2d{M}\\le\\mathrm{k}\\sum_{j=1}^N\\|\\nabla\\psi_j\\|^2,\n\\end{equation}\nwhere\n$$\n\\mathrm{k}\\le\\frac{3\\pi}{32}=0.2945\\dots\\,.\n$$\n\\end{theorem}\n\n\\begin{corollary} Setting  $N=1$ and $\\psi=\\varphi\/\\|\\varphi\\|$ we obtain\nthe interpolation inequality which is often called the Ladyzhenskaya\ninequality (in the context of the Navier--Stokes equations) or the\nKeller--Lieb--Thirring  one-bound-state inequality (in the context of the spectral theory):\n$$\n\\|\\varphi\\|_{L_4}^4\\le \\mathrm{k}_{\\mathrm{Lad}}\\|\\varphi\\|^2\\|\\nabla\\varphi\\|^2,\n\\qquad \\mathrm{k}_{\\mathrm{Lad}}\\le\\mathrm{k}_{\\mathrm{LT}}.\n$$\n\\end{corollary}\n\\begin{remark}\n{\\rm\nThe previous estimate of the Lieb--Thirring constant\non $\\mathbb{T}^2$ and $\\mathbb{S}^2$ obtained in~\\cite{Zel-Il-Lap2019} and \\cite{I-L-AA} by\nmeans of the discrete version of the method of \\cite{Rumin2} was:\n$$\n\\mathrm{k}\\le\\frac{3}{2\\pi}=0.477\\,.\n$$\n}\n\\end{remark}\n\\begin{remark}\n{\\rm\nIn all cases $M=\\mathbb{S}^2, \\mathbb{T}^2$, or $\\mathbb{R}^2$ the Lieb--Thirring constant\nsatisfies the (semiclassical) lower bound\n$$\n0.1591\\dots=\\frac1{2\\pi}\\le\\mathrm{k}_{\\mathrm{LT}}.\\\n$$\nIn $\\mathbb{R}^2$ the sharp value of $\\mathrm{k}_{\\mathrm{Lad}}$ was found in\n\\cite{Weinstein} by the numerical solution of the corresponding Euler--Lagrange equation\n$$\n\\mathrm{k}_{\\mathrm{Lad}}=\\frac1{\\pi\\cdot 1.8622\\dots}=0.1709\\dots\\,,\n$$\nwhile the best to date  closed form estimate for this constant  was obtained in\n\\cite{Nasibov}\n$$\n\\mathrm{k}_{\\mathrm{Lad}}\\le\\frac{16}{27\\pi}=0.188\\dots,\n$$\nsee also \\cite[Theorem 8.5]{Lieb--Loss} where the  equivalent result is obtained for the inequality\nin the additive form.\n}\n\\end{remark}\n\n\n\n\n\n\n\\setcounter{equation}{0}\n\\section{Lieb--Thirring inequalities on  $\\mathbb{S}^2$ }\\label{sec2}\n\nWe  begin with the case of a sphere and first consider the scalar\ncase. We  recall the basic facts concerning the spectrum of the\nscalar Laplace operator $\\Delta=\\div\\nabla$ on the sphere\n$\\mathbb{S}^{2}$:\n\\begin{equation}\\label{harmonics}\n-\\Delta Y_n^k=n(n+1) Y_n^k,\\quad\nk=1,\\dots,2n+1,\\quad n=0,1,2,\\dots.\n\\end{equation}\nHere the $Y_n^k$ are the orthonormal real-valued spherical\nharmonics and each eigenvalue $\\Lambda_n:=n(n+1)$ has multiplicity $2n+1$.\n\n\nThe following identity is essential in what\nfollows \\cite{S-W}: for any $s\\in\\mathbb{S}^{2}$\n\\begin{equation}\\label{identity}\n\\sum_{k=1}^{2n+1}Y_n^k(s)^2=\\frac{2n+1}{4\\pi}.\n\\end{equation}\n\n\n\n\n\n\n\n\\begin{theorem}\\label{Th:S2}\nLet $\\{\\psi_j\\}_{j=1}^N\\in H^1(\\mathbb{S}^2)$ be an orthonormal family of scalar functions\nwith zero average: $\\int_{\\mathbb{S}^2}\\psi_j(s)dS=0$. Then $\\rho(s):=\\sum_{j=1}^N|\\psi_j(s)|^2$\nsatisfies the inequality\n\\begin{equation}\\label{LTS2}\n\\int_{\\mathbb{S}^2}\\rho(s)^2dS\\le\\frac{3\\pi}{32}\\sum_{j=1}^N\\|\\nabla\\psi_j\\|^2.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nWe use the discrete version of the recent important far-going improvement \\cite{Frank-Nam}\nof the approach of \\cite{Rumin2}.\n\nLet $f$ be a smooth non-negative function on $\\mathbb{R}^+$ with\n\\begin{equation}\\label{f}\n\\int_0^\\infty f(t)^2dt=1,\n\\end{equation}\nand therefore for any $a>0$\n\\begin{equation}\\label{fa}\na=\\int_0^\\infty f(E\/a)^2dE.\n\\end{equation}\nExpanding a function $\\psi$ with $\\int_{\\mathbb{S}^2}\\psi(s)dS=0$  in spherical harmonics\n$$\n\\psi(s)=\\sum_{n=1}^\\infty\\sum_{k=1}^{2n+1}\\psi_n^kY_n^k(s),\\qquad\n\\psi_n^k=\\int_{\\mathbb{S}^2}\\psi(s)Y_n^k(s)dS=(\\psi,Y_n^k)\n$$\nand observing that the summation starts with $n=1$ we see using\n\\eqref{fa} that\n\\begin{equation}\\label{chain}\n\\aligned\n\\|\\nabla\\psi\\|^2=\\int_{\\mathbb{S}^2}|\\nabla\\psi(s)|^2dS=\n\\sum_{n=1}^\\infty n(n+1)\\sum_{k=1}^{2n+1}|\\psi_n^k|^2=\\\\=\n\\int_0^\\infty \\sum_{n=1}^\\infty f\\biggl(\\frac E{ n(n+1)}\\biggr)^2\\,\\sum_{k=1}^{2n+1}|\\psi_n^k|^2dE=\\\\=\n\\int_0^\\infty\\int_{\\mathbb{S}^2}|\\psi^E(s)|^2dSdE=\n\\int_{\\mathbb{S}^2}\\int_0^\\infty|\\psi^E(s)|^2dEdS,\n\\endaligned\n\\end{equation}\nwhere\n$$\n\\psi^E(s)=\\sum_{n=1}^\\infty\\sum_{k=1}^{2n+1}f\\biggl(\\frac E{ n(n+1)}\\biggr)\n\\psi_n^k\\,Y_n^k(s).\n$$\nReturning to the family $\\{\\psi_j\\}_{j=1}^N$ we have for any $\\varepsilon>0$\n$$\n\\aligned\n\\rho(s)&=\\sum_{j=1}^N|\\psi_j(s)|^2=\\\\&=\\sum_{j=1}^N|\\psi^E_j(s)|^2\n+2\\sum_{j=1}^N\\psi^E_j(s)(\\psi_j(s)-\\psi_j^E(s))+\\sum_{j=1}^N|\\psi_j(s)-\\psi^E_j(s)|^2\\le\\\\&\\le\n(1+\\varepsilon)\\sum_{j=1}^N|\\psi^E_j(s)|^2+\n(1+\\varepsilon^{-1})\\sum_{j=1}^N|\\psi_j(s)-\\psi^E_j(s)|^2.\n\\endaligned\n$$\nFor each term in the second sum we have\n$$\n\\psi(s)-\\psi^E(s)=\\sum_{n=1}^\\infty\\sum_{k=1}^{2n+1}\\psi_n^k\\biggl(1-f\\biggl(\\frac E{ n(n+1)}\\biggr)\\biggr)\n\\,Y_n^k(s)=\\bigl(\\psi(\\cdot),\\chi^E(\\cdot,s)\\bigr),\n$$\nwhere\n$$\n\\chi^E(s',s)=\\sum_{n=1}^\\infty\\sum_{k=1}^{2n+1}\\biggl(1-f\\biggl(\\frac E{ n(n+1)}\\biggr)\\biggr)\n\\,Y_n^k(s')Y_n^k(s).\n$$\nSince the $\\psi_j$'s are orthonormal, we have by Bessel's inequality\n$$\n\\sum_{j=1}^N|\\psi_j(s)-\\psi^E_j(s)|^2=\\sum_{j=1}^N\\bigl(\\psi_j(\\cdot),\\chi^E(\\cdot,s)\\bigr)^2\n\\le\\|\\chi^E(\\cdot,s)\\|^2,\n$$\nwhere in view of \\eqref{identity} $\\|\\chi^E(\\cdot,s)\\|^2$, in fact, is independent of $s$:\n\\begin{equation}\\label{indeps}\n\\aligned\n\\|\\chi^E(\\cdot,s)\\|^2=\n\\sum_{n=1}^\\infty\\sum_{k=1}^{2n+1}\\biggl(1-f\\biggl(\\frac E{ n(n+1)}\\biggr)\\biggr)^2\nY_n^k(s)^2=\\\\=\\frac1{4\\pi}\\sum_{n=1}^\\infty(2n+1)\\biggl(1-f\\biggl(\\frac E{ n(n+1)}\\biggr)\\biggr)^2.\n\\endaligned\n\\end{equation}\n\nWe now specify the choice of $f$ by setting (see~\\cite{Frank-Nam}, \\cite{lthbook})\n\\begin{equation}\\label{choice}\nf(t)=\\frac1{1+\\mu t^2},\\qquad\\\n\\mu=\\frac{\\pi^2}{16}\\,.\n\\end{equation}\nThe function $f$ so chosen solves the minimization problem\n$$\n\\aligned\n\\int_{\\mathbb{R}^2}\\left(1-f(1\/|\\xi|^2)\\right)^2d\\xi=&\\pi\\int_0^\\infty(1-f(t))^2t^{-2}dt\\to\\min\\\\\n\\text{under condition}\\quad&\\int_0^\\infty f(t)^2dt=1,\n\\endaligned\n$$\nand the above integral over $\\mathbb{R}^2$ corresponds to the series\non the right-hand side in \\eqref{indeps} (see also~\\eqref{tor}).\n\nWe first observe that \\eqref{f} is satisfied and secondly,\nin view of the estimate for the series in the Appendix\n\\begin{equation}\\label{series1}\n\\aligned\n\\|\\chi^E(\\cdot,s)\\|^2=\\frac1{4\\pi}\\sum_{n=1}^\\infty\\frac\n{(2n+1)}{\\biggl({1+\\left(\\frac1{\\sqrt{\\mu}E}n(n+1)\\right)^2}\\biggr)^2}<\\\\<\n\\frac1{4\\pi}\\sqrt{\\mu}E\\int_0^\\infty\\frac{dt}{(1+t^2)^2}=\n\\frac1{4\\pi}\\sqrt{\\mu}E\\frac\\pi4=\\frac\\pi{64}E=:AE\\,.\n\\endaligned\n\\end{equation}\nHence\n\\begin{equation}\\label{epseps}\n\\rho(s)\\le(1+\\varepsilon)\\sum_{j=1}^N|\\psi^E_j(s)|^2+\n(1+\\varepsilon^{-1})AE.\n\\end{equation}\nOptimizing with respect to $\\varepsilon$ we obtain\n$$\n\\rho(s)\\le\\left(\\sqrt{\\sum_{j=1}^N|\\psi^E_j(s)|^2}+\\sqrt{AE}\\right)^2,\n$$\nwhich gives that\n$$\n\\sum_{j=1}^N|\\psi^E_j(s)|^2\\ge\\left(\\sqrt{\\rho(s)}-\\sqrt{AE}\\right)^2_+.\n$$\nSumming equalities \\eqref{chain} from $j=1$ to $N$ we\nobtain\n$$\n\\aligned\n&\\sum_{j=1}^N\\|\\nabla\\psi_j\\|^2=\\int_{\\mathbb{S}^2}\\int_0^\\infty\n\\sum_{j=1}^N|\\psi_j^E(s)|^2dEdS\\ge\\\\&\n\\int_{\\mathbb{S}^2}\\int_0^\\infty\\left(\\sqrt{\\rho(s)}-\\sqrt{AE}\\right)^2_+dEdS=\n\\frac1{6A}\\int_{\\mathbb{S}^2}\\rho(s)^2dS=\\frac{32}{3\\pi}\\int_{\\mathbb{S}^2}\\rho(s)^2dS.\n\\endaligned\n$$\nThe proof is complete.\n\\end{proof}\n\\begin{remark}\\label{R:semi}\n{\\rm\nThe constant $\\mathrm{k}$ in the theorem satisfies the (semiclassical) lower bound\n\\begin{equation}\\label{lb}\nk\\ge\\frac1{2\\pi}\\,,\n\\end{equation}\nwhich can easily be proved in our particular case of $\\mathbb{S}^2$. In fact, we\ntake for the orthonormal family the eigenfunctions $Y_n^k$ with $n=1,\\dots,N-1$,\nand $k=1,\\dots,2n+1$, so that\n$$\n\\sum_{n=1}^{N-1}(2n+1)=N^2-1\\ \\ \\text{and}\\ \\  \\sum_{n=1}^{N-1}(2n+1)n(n+1)=\\frac12N^2(N^2-1),\n$$\nthen \\eqref{M} and the Cauchy inequality give \\eqref{lb}, since\n$$\n(N^2-1)^2=\\left(\\int_{\\mathbb{S}^2}\\rho(s)dS\\right)^2\\le\n4\\pi\\|\\rho\\|^2\\le\n2\\pi \\mathrm{k}N^2(N^2-1).\n$$\n\n}\n\\end{remark}\n\n\n\\subsection{The vector case}\n\nThe vector case is similar, and the key identity~\\eqref{identity} is replaced by\nvector analogue\n(see \\cite{I93}): for any $s\\in\\mathbb{S}^{2}$\n\\begin{equation}\\label{identity-vec}\n\\sum_{k=1}^{2n+1}|\\nabla Y_n^k(s)|^2=n(n+1)\\frac{2n+1}{4\\pi}.\n\\end{equation}\n\n\n\n\nIn the vector case by the  Laplace operator\nacting on (tangent) vector\nfields on $\\mathbb{S}^2$ we mean  the Laplace--de Rham\noperator $-d\\delta-\\delta d$ identifying $1$-forms and\nvectors. Then for a two-dimensional manifold\n(not necessarily $\\mathbb{S}^2$) we have\n\\cite{I93}\n\\begin{equation}\\label{vecLap}\n\\mathbf{\\Delta} u=\\nabla\\div u-\\mathop\\mathrm{rot}\\rot u,\n\\end{equation}\nwhere the operators $\\nabla=\\mathop\\mathrm{grad}$ and $\\div$ have the\nconventional meaning. The operator $\\mathop\\mathrm{rot}$ of a vector $u$ is a\nscalar  and for a scalar $\\psi$,\n$\\mathop\\mathrm{rot}\\psi$ is a vector:\n\\begin{equation}\\label{divrot}\n\\mathop\\mathrm{rot} u:=\\div(u^\\perp),\\qquad\n\\mathop\\mathrm{rot}\\psi:=\\nabla^\\perp\\psi,\n\\end{equation}\nwhere  in the local frame $u^\\perp=(u_2,-u_1)$.\n\n Integrating by parts\nwe obtain\n\\begin{equation}\\label{byparts}\n(-\\mathbf{\\Delta} u,u)=\\|\\mathop\\mathrm{rot} u\\|^2+\\|\\div u\\|^2.\n\\end{equation}\n\n\nThe vector Laplacian has a complete in $L_2(T\\mathbb{S}^2)$ orthonormal basis\nof vector eigenfunctions:  corresponding\nto the eigenvalue\n$\\Lambda_n=n(n+1)$, where $n=1,2,\\dots$, there are two families of $2n+1$\northonormal vector-valued  eigenfunctions  $w_n^k(s)$ and $v_n^k(s)$\n\\begin{equation}\\label{bases}\n\\aligned\nw_n^k(s)&=(n(n+1))^{-1\/2}\\,\\nabla^\\perp Y_n^k(s),\\ -\\mathbf{\\Delta}w_n^k=n(n+1)w_n^k,\\\n\\div w_n^k=0;\\\\\nv_n^k(s)&=(n(n+1))^{-1\/2}\\,\\nabla Y_n^k(s),\\ \\ -\\mathbf{\\Delta}v_n^k=n(n+1)v_n^k,\\ \\mathop\\mathrm{rot} v_n^k=0,\n\\endaligned\n\\end{equation}\nwhere\n$k=1,\\dots,2n+1$,  and~(\\ref{identity-vec}) gives the\nfollowing important identities: for any $s\\in\\mathbb{S}^2$\n\\begin{equation}\\label{id-vec}\n\\sum_{k=1}^{2n+1}|w_n^k(s)|^2=\\frac{2n+1}{4\\pi},\\qquad\n\\sum_{k=1}^{2n+1}|v_n^k(s)|^2=\\frac{2n+1}{4\\pi}.\n\\end{equation}\nWe finally observe that  $-\\mathbf{\\Delta}$ is strictly\npositive $-\\mathbf{\\Delta}\\ge \\Lambda_1I=2I.$\n\n\n\n\\begin{theorem}\\label{Th:LT-vec}\nLet $\\{u_j\\}_{j=1}^N\\in H^1(T\\mathbb{S}^2)$\nbe an  orthonormal family  of vector fields in $L^2(T\\mathbb{S}^2)$.  Then\n\\begin{equation}\\label{orthvec}\n\\int_{\\mathbb{S}^2}\\rho(s)^2dS\\le\n \\frac{3\\pi}{16}\\sum_{j=1}^N(\\|\\mathop\\mathrm{rot} u_j\\|^2\n+\\|\\div u_j\\|^2),\n\\end{equation}\nwhere $\\rho(s)=\\sum_{j=1}^N|u_j(s)|^2$.\nIf, in addition, $\\div u_j=0$ $($or $\\mathop\\mathrm{rot} u_j=0$$)$,\nthen\n\\begin{equation}\\label{orthvecsol}\n\\int_{\\mathbb{S}^2}\\rho(s)^2dS\\le\\frac{3\\pi}{32}\\cdot\n\\begin{cases}\\displaystyle\n\\sum_{j=1}^N\\|\\mathop\\mathrm{rot} u_j\\|^2,\n\\quad \\ \\div u_j=0,\n\\\\\\displaystyle\n\\sum_{j=1}^N\\|\\div u_j\\|^2,\n\\quad \\mathop\\mathrm{rot} u_j=0.\n\\end{cases}\n\\end{equation}\n\\end{theorem}\n\\begin{proof} We prove the first inequality in~\\eqref{orthvecsol},\nthe proof of the second is  similar. Expanding a vector function\n$u$ with $\\div u=0$ in the basis $w_n^k$\n$$\nu(s)=\\sum_{n=1}^\\infty\\sum_{k=1}^{2n+1} u_n^kw_n^k(s),\\qquad\nu_n^k=(u,w_n^k),\n$$\nwe have instead of \\eqref{chain}\n\\begin{equation}\\label{chain1}\n\\aligned\n\\|\\mathop\\mathrm{rot} u\\|^2&=\n\\sum_{n=1}^\\infty n(n+1)\\sum_{k=1}^{2n+1}|u_n^k|^2=\\\\&=\n\\int_0^\\infty \\sum_{n=1}^\\infty f\\biggl(\\frac E{ n(n+1)}\\biggr)^2\\,\\sum_{k=1}^{2n+1}|u_n^k|^2dE=\n\\int_{\\mathbb{S}^2}\\int_0^\\infty|u^E(s)|^2dEdS,\n\\endaligned\n\\end{equation}\nwhere\n$$\nu^E(s)=\\sum_{n=1}^\\infty\\sum_{k=1}^{2n+1}f\\biggl(\\frac E{ n(n+1)}\\biggr)\nu_n^k\\,w_n^k(s).\n$$\nAs before\n$$\n\\aligned\n\\rho(s)\\le\n(1+\\varepsilon)\\sum_{j=1}^N|u^E_j(s)|^2+\n(1+\\varepsilon^{-1})\\sum_{j=1}^N|u_j(s)-u^E_j(s)|^2.\n\\endaligned\n$$\nWe now imbed $\\mathbb{S}^2$ into $\\mathbb{R}^3$\nin the natural way and use the standard basis $\\{e_1,e_2,e_3\\}$\nand the scalar product $\\langle \\cdot,\\cdot\\rangle$ in $\\mathbb{R}^3$.\nThen we see that\n$$\n\\aligned\n\\langle u(s)-u^E(s),e_1\\rangle=\\\\\\sum_{n=1}^\\infty\\sum_{k=1}^{2n+1}u_n^k\\biggl(1-f\\biggl(\\frac E{ n(n+1)}\\biggr)\\biggr)\n\\,\\langle w_n^k(s),e_1\\rangle=\\bigl(u(\\cdot),\\chi^E_1(\\cdot,s)\\bigr),\n\\endaligned\n$$\nwhere the vector function\n$$\n\\chi^E_1(s',s)=\\sum_{n=1}^\\infty\\sum_{k=1}^{2n+1}\\biggl(1-f\\biggl(\\frac E{ n(n+1)}\\biggr)\\biggr)\n\\,w_n^k(s')\\langle w_n^k(s),e_1\\rangle.\n$$\nBy orthonormality and Bessel's inequality\n$$\n\\aligned\n\\sum_{j=1}^N|u_j(s)-u^E_j(s)|^2=\\sum_{j=1}^N\\sum_{l=1}^3|\\langle u_j(s)-u^E_j(s),e_l\\rangle|^2=\\\\\n=\\sum_{l=1}^3\\sum_{j=1}^N\\bigl(u_j(\\cdot),\\chi^E_l(\\cdot,s)\\bigr)^2\n\\le\\sum_{l=1}^3\\|\\chi_l^E(\\cdot,s)\\|^2.\n\\endaligned\n$$\nHowever, in view of~\\eqref{id-vec}, the right hand side is again independent of $s$\n$$\n\\aligned\n\\sum_{l=1}^3\\|\\chi^E_l(\\cdot,s)\\|^2&=\n\\sum_{n=1}^\\infty\\biggl(1-f\\biggl(\\frac E{ n(n+1)}\\biggr)\\biggr)^2\\sum_{k=1}^{2n+1}\\sum_{l=1}^3\n|\\langle w_n^k(s),e_l\\rangle|^2=\\\\&=\n\\sum_{n=1}^\\infty\\biggl(1-f\\biggl(\\frac E{ n(n+1)}\\biggr)\\biggr)^2\\sum_{k=1}^{2n+1}\n| w_n^k(s)|^2=\\\\&=\\frac1{4\\pi}\\sum_{n=1}^\\infty(2n+1)\\biggl(1-f\\biggl(\\frac E{ n(n+1)}\\biggr)\\biggr)^2,\n\\endaligned\n$$\nand we complete the proof in exactly the same way as we have\ndone in the proof of Theorem~\\ref{Th:S2} after \\eqref{indeps}.\nFinally, in the proof of inequality~\\eqref{orthvec} both families\nof vector eigenfunctions \\eqref{bases} play equal roles,\nand the constant is increased by the factor of two.\n\\end{proof}\n\nThis, however, does not happen for a single vector function.\n\n\\begin{corollary}\\label{C:vec}\nLet $u\\in H^1(T\\mathbb{S}^2)$. Then\n\\begin{equation}\\label{vecLad}\n\\|u\\|^4_{L_4}\\le \\vec{\\mathrm{k}}_\\mathrm{Lad}\n\\|u\\|^2\\left(\\|\\mathop\\mathrm{rot} u\\|^2+\\|\\div u\\|^2\\right),\\qquad\n\\vec{\\mathrm{k}}_\\mathrm{Lad}\\le\\frac{3\\pi}{32}\\,.\n\\end{equation}\n\\end{corollary}\n\\begin{proof}\nThe proof is based on the equivalence\n\\eqref{LT}$_{\\gamma=1}\\Leftrightarrow$\\eqref{orth} with equality\nfor the constants~\\eqref{kL} and the fact that the eigenvalues of\nthe vector Schr\\\"odinger operator on $\\mathbb{S}^2$\n\\begin{equation}\\label{vecSchr}\nAv=-\\mathbf{\\Delta} v-Vv\n\\end{equation}\nhave even multiplicities as the following equality implies\n(see~\\eqref{vecLap}, \\eqref{divrot})\n$$\n\\mathbf{\\Delta}(v^\\perp)=(\\mathbf{\\Delta} v)^\\perp.\n$$\nNow let $u$ in \\eqref{vecLad} be normalized, $\\|u\\|=1$,  let\n$V(s)=\\alpha|u(s)|^2$, $\\alpha>0$, and let $E$ be the lowest\neigenvalue of~\\eqref{vecSchr}.  If $E<0$, then since $E$ is counted\nat least twice in the sum $\\sum_{\\lambda_j\\le0}\\lambda_j$, it\nfollows that\n\\begin{equation}\\label{E}\nE\\ge \\frac12\\sum_{\\lambda_j\\le0}\\lambda_j\\ge-\\frac12\\mathrm{L_1}\\int_{\\mathbb{S}^2}V(s)^2dS=\n-\\alpha^2\\frac12\\mathrm{L_1}\\|u\\|_{L_4}^4,\n\\end{equation}\nwhere the second inequality is~\\eqref{LT} with\n$$\n\\mathrm{L_1}\\le \\frac14\\cdot\\frac{3\\pi}{16},\n$$\nin view of \\eqref{kL} and \\eqref{orthvec}. If $E\\ge0$, then\n\\eqref{E} also formally holds.\n\nNext, by the variational principle\n\\begin{equation}\\label{Eless}\n\\aligned\nE\\le(Au,u)=\\|\\mathop\\mathrm{rot} u\\|^2+\\|\\div u\\|^2-\\int_{\\mathbb{S}^2}V(s)|u(s)|^2dS=\\\\\n\\|\\mathop\\mathrm{rot} u\\|^2+\\|\\div u\\|^2-\\alpha\\|u\\|^4_{L_4}.\n\\endaligned\n\\end{equation}\nCombining \\eqref{E} and \\eqref{Eless} and setting optimal\n$\\alpha=1\/\\mathrm{L}_1$ we finally obtain\n$$\n\\|u\\|_{L_4}^4\\le 2\\mathrm{L}_1\\left(\\|\\mathop\\mathrm{rot} u\\|^2+\\|\\div u\\|^2\\right)\\le\n\\frac{3\\pi}{32}\\left(\\|\\mathop\\mathrm{rot} u\\|^2+\\|\\div u\\|^2\\right).\n$$\n\\end{proof}\n\n\n\\begin{remark}\n{\\rm\nIt is worth pointing out that\n and for any domain on the sphere\n$\\Omega\\subseteq\\mathbb{S} ^2$ and an orthonormal family\n$\\{u_j\\}_{j=1}^N\\in H^1_0(\\Omega,T\\mathbb{S}^2)$ extension by zero\nshows that the corresponding Lieb--Thirring constants are uniformly\nbounded by the constants on the whole sphere whose estimates were\nfound  in Theorem~\\ref{Th:LT-vec} and Corollary~\\ref{C:vec}. }\n\\end{remark}\n\n\n\\setcounter{equation}{0}\n\\section{Lieb--Thirring inequalities on  $\\mathbb{T}^2$ }\\label{sec3}\nWe now prove Theorem~\\ref{Th:1} for the 2D torus. We first consider the torus  with equal periods\nand without loss of generality\nwe set $\\mathbb{T}^2=[0,2\\pi]^2$.\n\\begin{proof}[Proof of Theorem~\\ref{Th:1} for $\\mathbb{T}^2$]\nWe use the Fourier series\n$$\n\\psi(x)=\\frac1{2\\pi}\\sum_{k\\in\\mathbb{Z}^2_0}\\psi_k e^{ik\\cdot x},\\qquad\n\\psi_k=\\frac1{2\\pi}\\int_{\\mathbb{T}^2}\\psi(x)e^{-ik\\cdot x}dx,\\quad\n\\mathbb{Z}^2_0=\\mathbb{Z}^2\\setminus\\{0,0\\},\n$$\nso that\n$$\n\\|\\psi\\|^2=\\sum_{k\\in\\mathbb{Z}^2_0}|\\psi_k|^2,\n\\qquad\n\\|\\nabla\\psi\\|^2=\\sum_{k\\in\\mathbb{Z}^2_0}|k|^2|\\psi_k|^2.\n$$\nThen as before we have\n$$\n\\|\\nabla\\psi\\|^2=\n\\int_0^\\infty \\sum_{k\\in\\mathbb{Z}^2_0} f\\biggl(\\frac E{|k|^2}\\biggr)^2|\\psi_k|^2dE=\n\\int_{\\mathbb{T}^2}\\int_0^\\infty|\\psi^E(x)|^2dEdx,\n$$\nwhere\n$$\n\\psi^E(x)=\\frac1{2\\pi}\\sum_{k\\in\\mathbb{Z}^2_0} f\\biggl(\\frac E{|k|^2}\\biggr)\\psi_ke^{ik\\cdot x},\n$$\nand therefore\n$$\n\\psi(x)-\\psi^E(x)=\n\\frac1{2\\pi}\\sum_{k\\in\\mathbb{Z}^2_0}\n\\left(1-f\\biggl(\\frac E{|k|^2}\\biggr)\\right)\\psi_ke^{ik\\cdot x}=\n(\\psi(\\cdot), \\chi^E(\\cdot,x)),\n$$\nwhere\n$$\n\\chi^E(x',x)=\\frac1{2\\pi}\n\\sum_{k\\in\\mathbb{Z}^2_0}\n\\left(1-f\\biggl(\\frac E{|k|^2}\\biggr)\\right)e^{ik\\cdot x'}e^{-ik\\cdot x}.\n$$\nWith the  choice of $f$ given in \\eqref{choice} and setting $a=\\sqrt{\\mu}E$ below, we have\n\\begin{equation}\\label{tor}\n\\aligned\n\\|\\chi^E(\\cdot,x)\\|^2=\\frac1{4\\pi^2}\\sum_{k\\in\\mathbb{Z}^2_0}\n\\left(1-f\\biggl(\\frac E{|k|^2}\\biggr)\\right)^2=\\\\=\n\\frac1{4\\pi^2}\\sum_{k\\in\\mathbb{Z}^2_0}\\frac1{\\left(\\left(\\frac{|k|}{\\sqrt{a}}\\right)^4+1\\right)^2}\n<\\frac a{16}=\\frac\\pi{64}E=:AE,\n\\endaligned\n\\end{equation}\nwhere the key inequality for series is proved in the Appendix.\n\nAt this point we can complete the proof as in Theorem~\\ref{Th:S2}.\n\\end{proof}\n\n\\subsection{Elongated torus.}\nWe now briefly discuss the Lieb--Thirring constant on a 2D torus\nwith aspect ratio $\\alpha$. Since the Lieb--Thirring constant\ndepends only on $\\alpha$, we consider the torus $\\mathbb{T}^2_\\alpha=[0,2\\pi\/\\alpha]\\times\n[0,2\\pi]$. Furthermore, it suffices to consider the case $\\alpha\\le1$, since otherwise\nwe merely interchange the periods.\n\n\\begin{theorem}\\label{Th:alpha}\nThe Lieb--Thirring constant on the elongated torus $\\mathbb{T}^2_\\alpha$\nsatisfies the bound\n\\begin{equation}\\label{alpha}\n\\mathrm{k}_\\mathrm{LT}(\\mathbb{T}^2_\\alpha)\\le\\frac1\\alpha\\frac{3\\pi}{32}\\\n\\text{as}\\ \\alpha\\to0.\n\\end{equation}\n\\end{theorem}\n\\begin{proof} We shall prove \\eqref{alpha} under an additional technical assumption that\n$k=1\/\\alpha\\in\\mathbb{N}$. Given the orthonormal family\n$\\{\\psi_j\\}_{j=1}^N \\in\\dot H^1(\\mathbb{T}^2_\\alpha)$, we extend each $\\psi_j$\n by periodicity in the $x_2$ direction $k$ times, multiply the result by $\\sqrt{\\alpha}$\n and denote the resulting function defined on the square\n torus $\\mathbb{T}^2=[0,2\\pi k]^2$ by $\\widetilde\\psi_j$. Then the family\n $\\{\\widetilde\\psi_j\\}_{j=1}^N$ is orthonormal in $L_2(\\mathbb{T}^2)$ and\n for $\\rho_{\\widetilde\\psi}(x)=\\sum_{j=1}^N|\\widetilde\\psi_j(x)|^2$\n and $\\rho_{\\psi}(x)=\\sum_{j=1}^N|\\psi_j(x)|^2$  it holds\n $$\n \\int_{\\mathbb{T}^2}\\rho_{\\widetilde\\psi}(x)^2dx=\n \\alpha\\int_{\\mathbb{T}^2_\\alpha}\\rho_{\\psi}(x)^2dx,\\qquad\n \\int_{\\mathbb{T}^2}|\\nabla\\widetilde\\psi_j(x)|^2dx=\n \\int_{\\mathbb{T}^2_\\alpha}|\\nabla\\psi_j(x)|^2dx,\n $$\n which gives \\eqref{alpha}.\n \\end{proof}\n \\begin{remark}\n{\\rm\nThe rate of growth $1\/\\alpha$ of the Lieb--Thirring constant is sharp\nas $\\alpha\\to0$. To see this we set $N=1$ and consider a function\non $\\mathbb{T}^2_\\alpha$ depending on the long coordinate $x_1$ only.\nFor example, let $\\psi(x_1,x_2)=\\sin(2\\pi\\alpha x_1)$.\nThen $\\|\\psi\\|^4_{L_4}\\sim 1\/\\alpha$ $(=\\frac{3\\pi^2}{2\\alpha})$,\n$\\|\\psi\\|^2_{L_2}\\sim 1\/\\alpha$ $(=\\frac{2\\pi^2}{\\alpha})$,\n$\\|\\nabla\\psi\\|^2_{L_2}\\sim \\alpha$ $(=2\\pi^2\\alpha)$.\nTherefore $\\mathrm{k}_\\mathrm{LT}(\\mathbb{T}^2_\\alpha)\\succeq 1\/\\alpha$\n$(\\ge\\frac1\\alpha\\frac3{8\\pi^2})$.\n}\n\\end{remark}\n\\begin{remark}\n{\\rm\nThe orthogonal complement to the  subspace of functions depending only on the long coordinate\n$x_1$ consists of functions $\\psi(x_1,x_2)$ with mean value\nzero with respect to the short coordinate $x_2$:\n\\begin{equation}\\label{T2cond}\n\\int_0^{2\\pi}\\psi(x_1,x_2)dx_2=0\\quad\\forall x_1\\in[0,2\\pi\/\\alpha].\n\\end{equation}\nThe Lieb--Thirring constant on this subspace is bounded uniformly\nwith respect to $\\alpha$ as $\\alpha\\to0$. The similar\nresult holds for the multidimensional torus with different periods.\nSee \\cite{I-L-MS} for the details.\n}\n\\end{remark}\n\n\\begin{remark}\n{\\rm The lifting argument of \\cite{Lap-Weid} was used in\n\\cite{I-L-MS} to derive the Lieb--Thirring inequalities on the\nmultidimensional with pointwise orthogonality condition of the type\n\\eqref{T2cond}.   It is not clear how to use the lifting argument\nin the case of a global (and weaker) orthogonality contition\n$\\int_{\\mathbb{T}^d}\\psi(x)dx=0$.\n\nFinally, we do not know whether the lifting argument\ncan in some form be used for the Lieb--Thirring inequalities\non the sphere, say, when going over from $\\mathbb{S}^{d-1}$ to\n$\\mathbb{S}^{d}$. }\n\\end{remark}\n\n\n\n\n\\setcounter{equation}{0}\n\\section{Appendix. Estimates of the series}\\label{sec4}\n\n\\subsection*{Estimate for the sphere.} The series estimated in \\eqref{series1} is precisely of the type\n\\begin{equation}\\label{G}\nG(\\nu):=\\sum_{n=1}^\\infty(2n+1)g\\left(\\nu\\, n(n+1)\\right),\n\\end{equation}\nwhere $g$ is sufficiently smooth and sufficiently fast decays at\ninfinity. We need to find the asymptotic behavior of $G(\\nu)$ as\n$\\nu\\to0$. This has been done in~\\cite{IZ} where the following\nresult was proved.\n\\begin{lemma}\\label{L:E-M}\nThe following asymptotic expansion holds as $\\nu\\to0$:\n\\begin{equation}\\label{as2}\nG(\\nu)=\\frac1{\\nu}\\int_0^\\infty g(t)dt-\\frac23g(0)-\n\\frac1{15}\\nu g'(0)+\\frac4{315}\\nu^2g''(0)+\nO(\\nu^3).\n\\end{equation}\n\\end{lemma}\n\nThe series in \\eqref{series1} is of the form~\\eqref{G}\nwith\n$$\ng(t)=\\frac1{(1+t^2)^2}, \\qquad \\nu=\\frac1{\\sqrt{\\mu}E},\n$$\nso that $g(0)=1$, $g'(0)=0$, $g''(0)=-4$ and $\\int_0^\\infty g(t)dt=\\pi\/4$. Therefore\n\\eqref{as2} gives\n$$\n\\sum_{n=1}^\\infty\\frac\n{(2n+1)}{\\biggl({1+\\left(\\frac{n(n+1)}a\\right)^2}\\biggr)^2}=\n    a\\frac\\pi 4-\\frac23-\\frac{16}{315}a^{-2}+O(a^{-3}),\\ \\text{as}\\ a\\to\\infty,\n$$\nwhich shows that\ninequality  \\eqref{series1}\nclearly holds for large energies $E>E_0$. The proof of inequality \\eqref{series1}  for\nall $E\\in[0,\\infty)$ amounts to showing that the inequality\n$$\nH_{\\mathbb{S}^2}(a):=\\frac4\\pi\\,a^3\\sum_{n=1}^\\infty\\frac{2n+1}{\\bigl(\\bigl(n(n+1)\\bigr)^2+a^2\\bigr)^2}\n\\,<\\,1, \\quad a=\\sqrt{\\mu}E=\\frac\\pi 4E\n$$\nholds on a \\emph{finite} interval $ a\\in[0,a_0]$.\nThe value of $a_0$ (say, $20$) can be specified similarly to \\cite{I12JST}.\nFurthermore,  the sum of the series\ncan be found in an explicit form in terms of the (digamma) $\\psi$-function.\nThe function $H_{\\mathbb{S}^2}(a)$ and the third-order remainder term are shown in Fig.~\\ref{fig:S2}.\n\\begin{figure}[htb]\n\\centerline{\\psfig{file=S2.eps,width=7.5cm,height=6cm,angle=0}\n\\psfig{file=remainderS2.eps,width=7.5cm,height=6cm,angle=0}}\n\\caption{The graph of $H_{\\mathbb{S}^2}(a)$ is on the left;\nthe remainder term $\\left(H_{\\mathbb{S}^2}(a)-1-\\frac8{3\\pi a}\\right)\\cdot a^3$\nis shown on the right, the horizontal red line is  $-64\/(315\\pi)=-0.064$.}\n\\label{fig:S2}\n\\end{figure}\n\\subsection*{Estimate for the torus.}\n\n\\begin{lemma}\\label{L:Poisson}\nThe following asymptotic expansion holds as $a\\to\\infty$:\n\\begin{equation}\\label{at2}\n\\sum_{k\\in\\mathbb{Z}^2_0}\\frac1{\\left(\\left(\\frac{|k|}{\\sqrt{a}}\\right)^4+1\\right)^2}\n=\\frac{\\pi^2}4 a-1+O(e^{-C\\sqrt{a}}).\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWe use the Poisson summation formula\n(see, e.\\,g., \\cite{S-W}):\n$$\n\\sum_{m\\in\\mathbb{Z}^d}f(m\/\\nu)=\n(2\\pi)^{d\/2}\\nu^d\n\\sum_{m\\in\\mathbb{Z}^d}\\widehat{f}(2\\pi m \\nu),\n$$\nwhere\n$\\widehat{f}(\\xi)=(2\\pi)^{-d\/2}\\int_{\\mathbb{R}^d}\nf(x)e^{-i\\xi x}dx$.\n\nTaking into account that the term with $k=(0,0)$ is missing in\n\\eqref{at2}, the Poisson summation formula gives\n\\begin{equation}\\label{expsmall}\n\\aligned\n\\sum_{k\\in\\mathbb{Z}^2_0}\\frac1{\\left(\\left(\\frac{|k|}{\\sqrt{a}}\\right)^4+1\\right)^2}+1=\\\\=\na\\int_{\\mathbb{R}^2}\\frac{dxdy}{((x^2+y^2)^2+1)^2}+2\\pi a\\sum_{k\\in\\mathbb{Z}^2_0}\n\\widehat h(2\\pi\\sqrt{a}|k|)=\\frac{\\pi^2}4a+O(e^{-C\\sqrt{a}}),\n\\endaligned\n\\end{equation}\nwhere $h(x,y)=\\frac1{((x^2+y^2)^2+1)^2}$ is analytic\nand therefore its Fourier transform\n\\begin{equation}\\label{Fourier}\n\\widehat h(\\xi)=\\frac1{2\\pi}\\int_{\\mathbb{R}^2}\ne^{-ix\\xi_1-iy\\xi_2}h(x,y)dxdy\n\\end{equation}\nis exponentially decaying.\n\\end{proof}\n\nThe graph of the function\n\\begin{equation}\\label{lessthen1}\nH_{\\mathbb{T}^2}(a):=\\frac4{\\pi^2}\\,a^3\\sum_{k\\in\\mathbb{Z}^2_0}\\frac{1}{\\bigl(|k|^4+a^2\\bigr)^2}\n\\,<\\,1,\n\\end{equation}\nand the exponentially small remainder term\n$$\nR(a)=2\\pi a\\sum_{k\\in\\mathbb{Z}^2_0}\n\\widehat h(2\\pi\\sqrt{a}|k|)\n$$\nare shown in Fig.~\\ref{fig:T2}\n\\begin{figure}[htb]\n\\centerline{\\psfig{file=T2.eps,width=7.5cm,height=6cm,angle=0}\n\\psfig{file=remainderT2.eps,width=7.5cm,height=6cm,angle=0}}\n\\caption{The function $H_{\\mathbb{T}^2}(a)$ is shown on the left;\nthe exponentially small term $R(a)$ is shown on the right.}\n\\label{fig:T2}\n\\end{figure}\n\nWe now give an explicit estimate for the exponentially small remainder term in~\\eqref{expsmall}.\nThe function $h(z)$ is analytic in the domain $\\Omega\\subset\\mathbb{C}^2$:\n$\\Omega=\\{z\\in \\mathbb{C}^2,\\ |\\mathrm{Im} z_1|<\\frac12,\\ |\\mathrm{Im} z_2|<\\frac12\\}$.\nIn fact, the equation\n$$\n(x+i\\alpha)^2+(y+i\\alpha)^2=\\pm i\n$$\nhas real solutions $x$ and $y$ only for $\\alpha\\ge\\frac12$.\n\nFor $F(x,y,\\alpha)=((x+i\\alpha)^2+(y+i\\alpha)^2)^2+1$ we have\n$$\n\\aligned\n&\\mathrm{Re}\\,F=\n(x^2+y^2)^2-8\\alpha^2(x^2+y^2+xy)+4\\alpha^4+1\\ge t^2-12\\alpha^2t+4\\alpha^4+1,\\\\\n&\\mathrm{Im}\\,F=\n4a(x+y)(x^2+y^2-2\\alpha^2),\\quad|\\mathrm{Im}\\,F|\\le4\\sqrt{2}\\alpha t^{1\/2}|t-2\\alpha^2|,\n\\endaligned\n$$\nwhere $t:=x^2+y^2$. Next, by a direct inspection we verify that\nfor $t\\ge0$\n$$\n\\aligned\n|F^2|\\ge\n (\\mathrm{Re}\\,F)^2-(\\mathrm{Im}\\,F)^2=\\\\\n(t^2-12\\alpha^2t+4\\alpha^4+1)^2-32\\alpha^2t(t-2a^2)^2>\\frac1b(t^4+1),\n\\endaligned\n$$\nwhere $\\alpha=4.6^{-1}$ and $b=4.75$.\nThis gives that for $|\\mathrm{Im} z_1|\\le\\alpha,\\ |\\mathrm{Im} z_2|\\le\\alpha$\n$$\n|h(x+i\\alpha,y+i\\alpha)|\\le\\frac b{(x^2+y^2)^4+1}\\,.\n$$\nBy the Cauchy integral theorem we can shift the $x$ and $y$ integration in \\eqref{Fourier}\nin the complex plane by $\\pm i\\alpha$ (depending on the sign of $\\xi_1$ and $\\xi_2$)  and find that\n$$\n|\\widehat h(\\xi)|\\!\\le\\!\\frac b{2\\pi}e^{-(|\\xi_1|+|\\xi_2|)\\alpha}\\!\\int_{\\mathbb{R}^2}\n\\frac{dxdy}{(x^2+y^2)^4+1}\\!=e^{-(|\\xi_1|+|\\xi_2|)\\alpha}\\frac{b\\pi\\sqrt{2}}8\\le\ne^{-\\alpha|\\xi|}\\frac{b\\pi\\sqrt{2}}8\\,.\n$$\n\nWe write  the numbers $|k|^2$ over the lattice $\\mathbb{Z}^2_0$\nin non-decreasing order and denote them by $\\{\\lambda_j\\}^\\infty_{j=1}$.\nUsing that $\\lambda_j\\ge j\/4$ (see \\cite{I-L-AA}) and setting\n$L:=\\frac{\\alpha\\pi\\sqrt{a}}2$ and $A:=\\frac{\\pi^2\\sqrt{2}ab}4$ we\nestimate the series in \\eqref{expsmall} as follows\n$$\n\\aligned\n|R(a)|\\le\n2\\pi a\\sum_{k\\in\\mathbb{Z}^2_0}\n|\\widehat h(2\\pi\\sqrt{a}|k|)|=2\\pi a\\sum_{j=1}^\\infty\n|\\widehat h(2\\pi\\sqrt{a}\\lambda_j^{1\/2})|\\le\\\\\nA\\sum_{j=1}^\\infty\ne^{-2\\pi\\alpha\\sqrt{a}\\lambda_j^{1\/2}}\\le\nA\\sum_{j=1}^\\infty\ne^{-2Lj^{1\/2}}=Ae^{-L}\\sum_{j=1}^\\infty e^{-L(2j^{1\/2}-1)}\\le\\\\\nAe^{-L}\\sum_{j=1}^\\infty e^{-Lj^{1\/2}}<\nAe^{-L}\\int_0^\\infty e^{-L\\sqrt{x}}dx=\nAe^{-L}\\frac2{L^2}=\\frac{2^{3\/2}b}{\\alpha^2}e^{-\\frac{\\alpha\\pi\\sqrt{a}}2}\\,.\n\\endaligned\n$$\nInequality~\\eqref{lessthen1} holds if $R(a)<1$. The above estimates show that\n$|R(a)|<1$ for\n$$\na>\\left[\\frac2{\\alpha\\pi}\\log\\left(\\frac{2^{3\/2}b}{\\alpha^2}\\right)\\right]^2=\n273.8\\,.\n$$\n\nA more optimistic estimate follows from the fact that $h(x)$ is radial and therefore\nso is its Fourier transform\n $$\\widehat h(\\xi)=\\int_0^\\infty J_0(|\\xi|r)h(r)rdr,\n$$\nwhere $J_0$ is the Bessel function. The latter integral is expressed in terms of the\nMeijer G-function  and satisfies $|\\widehat h(\\xi)|<e^{-|\\xi|\/2}$. Similar estimates\ngive that $|R(a)|<1$ for\n$$\na>\\left[\\frac4\\pi\\log\\frac{64}\\pi\\right]^2=14.73\\,.\n$$\n\n\n\\subsection*{Acknowledgements}\\label{SS:Acknow}\nThe work of A.\\,I. and S.\\,Z. is supported in part by the  Russian\nScience Foundation grant 19-71-30004 (sections 1,2). Research of A.\\,L. is\nsupported by the Russian Science Foundation grant 19-71-30002 (sections 3,4).\n\n\n\n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Introduction}\n\nPartial least squares (PLS) is a regularized regression technique developed by \\citet{Wol84} to deal with collinearities in the regressor matrix. It is an iterative algorithm where the covariance between response and regressor is maximized at each step, see \\citet{Hel88} for a detailed description. Regularization in the PLS algorithm is obtained by stopping the iteration process early.\n\nSeveral studies showed that partial least squares algorithm is competitive with other regression methods such as ridge regression and principal component regression and it needs generally fewer iterations than the latter to achieve comparable estimation and prediction, see, e.g., \\citet{FrankFriedman} and \\citet{Krae07b}. For an overview of further properties of PLS we refer to \\citet{Ros06}.\n\nReproducing kernel Hilbert spaces (RKHS) have a long history in probability and statistics \\citep[see e.g.][]{Berlinet}. \nHere we focus on the supervised kernel based learning approach for the solution of non-parametric regression problems. RKHS methods are both computationally and theoretically attractive, due to the kernel trick \\citep{Sch98} and the representer theorem \\citep{Wah99} as well as its generalization \\citep{Sch01}. Within the reproducing kernel Hilbert space framework one can adapt linear regularized regression techniques like ridge regression and principal component regression to a non-parametric setting, see \\citet{Sau98} and \\citet{Ros00a}, respectively. We refer to \\citet{bSch} for more details on the kernel based learning approach.\n\nKernel PLS was introduced in \\citet{Ros01} who reformulated the algorithm presented in \\citet{Lin93}. The relationship to kernel conjugate gradient (KCG) methods was highlighted in \\citet{Blan10a}. It can be seen in \\citet{Hanke} that conjugate gradient methods are well suited for handling ill-posed problems, as they arise in kernel learning, see, e.g., \\citet{Vit06}.\n\n\\citet{Ros03} investigated the performance of kernel partial least squares (KPLS) for non-linear discriminant analysis.\n\\citet{Blan10a} proved the consistency of KPLS when the algorithm is stopped early without giving convergence rates.  \n\n\\citet{Cap07} showed that kernel ridge regression (KRR) attains optimal probabilistic rates of convergence for independent and identically distributed data, using a source and a polynomial effective dimensionality condition. A generalization of these results to a wider class of effective dimensionality conditions and extension to kernel principal component regression can be found in \\citet{Dicker17}.\n\nFor a variant of KCG \\citet{Blan10b} obtained probabilistic convergence rates for independent identically distributed data. The pointed explicitly out that their approach and results are not directly applicable to KPLS.\n\nWe study of the convergence of the kernel partial least squares estimator to the true regression function when the algorithm is stopped early. \nSimilar to \\citet{Blan10b} we derive explicit probabilistic convergence rates. In contrast to previously cited works on kernel regression our input data are not independent and identically distributed but rather stationary time series. We derive probabilistic convergence results that can be applied for arbitrary temporal dependence structures, given that certain concentration inequalities for these data hold.\nThe derived convergence rates depend not only on the complexity of the target function and of the data mapped into the kernel space, but also on the persistence of the dependence in the data. In the stationary setting we prove that the short range dependence still leads to optimal rates, but if the dependence is more persistent, the rates become slower.\n\n\\section{Kernel Partial Least Squares}\n\\label{sec:problem}\nConsider the non-parametric regression problem  \n\\begin{equation}\n\\label{eq:model}\n\ty_t = f^\\ast(X_t) + \\varepsilon_t,~~t \\in {\\mathbb Z}.\n\\end{equation}\nHere $\\{X_t\\}_{t \\in {\\mathbb Z}}$ is a $d$-dimensional, $d \\in {\\mathbb N}$, stationary time series on a probability space $(\\Omega,\\mathcal A,\\mathrm{P})$ and $\\{\\varepsilon_t\\}_{t \\in {\\mathbb Z}}$ is an independent and identically distributed sequence of real valued random variables with expectation zero and variance $\\sigma^2 > 0$ that is independent of $\\{X_t\\}_{t \\in {\\mathbb Z}}$. Let $X$ be a random vector that is independent of $\\{X_t\\}_{t \\in {\\mathbb Z}}$ and $\\{\\varepsilon_t\\}_{t \\in {\\mathbb Z}}$ with the same distribution as $X_0$. The target function we seek to estimate is $f^\\ast \\in \\mathcal L^2\\left(\\Prob^{X} \\right)$.\n\nFor the purpose of supervised learning assume that we have a training sample $\\{(X_t,y_t)\\}_{t=1}^n$ for some $n \\in {\\mathbb N}$. In the following we introduce some basic notation for the kernel based learning approach.\n\nDefine with $(\\mathcal{H},\\langle \\cdot,\\cdot\\rangle_\\mathcal H)$ the RKHS of functions on ${\\mathbb R}^d$ with reproducing kernel $k:{\\mathbb R}^d \\times {\\mathbb R}^d \\rightarrow {\\mathbb R}$, i.e., it holds\n\\begin{equation}\n\\label{eq:rep.property}\n\tg(x) = \\langle g, k(\\cdot,x) \\rangle_\\mathcal H, ~~x \\in {\\mathbb R}^d, g\\in \\mathcal H.\n\\end{equation} \n\nThe corresponding inner product and norm in $\\mathcal H$ is denoted by $\\langle \\cdot,\\cdot \\rangle_\\mathcal H$ and $\\|\\cdot\\|_\\mathcal H$, respectively. We refer to \\citet{Berlinet} for examples of Hilbert spaces and their reproducing kernels. In the following we deal with reproducing kernel Hilbert spaces which fulfill the following, rather standard, conditions:\n\\begin{enumerate}[label={(K\\arabic*})]\n\\item \\label{con:k1}\n$\\mathcal H$ is separable,\n\\item \\label{con:k2}\nThere exists a $\\kappa>0$ such that $|k(x,y)| \\leq \\kappa$ for all $x,y \\in {\\mathbb R}^d$ and $k$ is measurable.\n\\end{enumerate}\nUnder \\ref{con:k1} the Hilbert-Schmidt norm $\\|\\cdot\\|_{\\mathrm{HS}}$ for operators mapping from $\\mathcal H$ to $\\mathcal H$ is well defined.\nIf condition \\ref{con:k2} holds, all functions in $\\mathcal H$ are bounded, see \\citet{Berlinet}, chapter 2. \nThe conditions are satisfied for a variety of popular kernels, e.g., Gaussian or triangular. \n\nThe main principle of RKHS methods is the mapping of the data $X_t$ into $\\mathcal H$ via the feature maps $\\phi_t = k(\\cdot,X_t)$, $t=1,\\dots,n$. This mapping can be done implicitly by using the kernel trick $\\langle \\phi_t,\\phi_s \\rangle_\\mathcal H = k(X_t,X_s)$ and thus only the $n \\times n$ dimensional kernel matrix $K_n = n^{-1}[k(X_t,X_s)]_{t,s=1}^n$ is needed in the computations. Then the task for RKHS methods is to find coefficients $\\alpha_1,\\dots,\\alpha_n$ such that $f_\\alpha = \\sum_{t=1}^n \\alpha_t \\phi_t$ is an adequate approximation of $f^\\ast$ in $\\mathcal H$, measured in the $\\mathcal L^2\\left(\\Prob^{X} \\right)$ norm $\\|\\cdot\\|_2$.\n\nThere are a variety of different approaches to estimate the coefficients $\\alpha_1,\\dots,\\alpha_n$, including kernel ridge regression, kernel principal component regression and, of course, kernel partial least squares. The latter method was introduced by \\citet{Ros01} and is the focus of the current work.\n\nIt was shown by \\citet{Krae07b} that the KPLS algorithm solves\n\\begin{equation}\n\\label{eq:kpls.optim}\n\t\\widehat{\\alpha}_i = \\arg\\min\\limits_{v \\in \\mathcal K_i(K_n,y)} \\|y - K_n v\\|^2,~~i=1,\\dots,n,\n\\end{equation}\nwith $y=(y_1,\\dots,y_n)^{ \\mathrm{\\scriptscriptstyle T} }$. Here $\\mathcal K_i(K_n,y) = \\mathrm{span}\\left\\{\n\ty,K_n y, K_n^2y,\\dots,K_n^{i-1}y\n\\right\\}$, $i=1,\\dots,n$, is the $i$th order Krylov space with respect to $K_n$ and $y$ and $\\| \\cdot\\|$ denotes the Euclidean norm rescaled by $n^{-1}$. The dimension $i$ of the Krylov space is the regularization parameter for KPLS.\n\nWe will introduce several operators that will be crucial for our further analysis. Fist define two integral operators: the kernel integral operator $T^\\ast:\\mathcal L^2\\left(\\Prob^{X} \\right) \\rightarrow \\mathcal H, g \\mapsto\\mathrm{E}\\{k(\\cdot,X) g(X)\\}$ and the change of space operator $T:\\mathcal H \\rightarrow \\mathcal L^2\\left(\\Prob^{X} \\right), g \\mapsto g$, which is well defined if \\ref{con:k2} holds.\nIt is easy to see that $T, T^\\ast$ are adjoint, i.e., for $u \\in \\mathcal H$ and $v \\in \\mathcal L^2\\left(\\Prob^{X} \\right)$ it holds $\\langle T^\\ast v, u \\rangle_\\mathcal H = \\langle v, T u \\rangle_2$ with $\\langle\\cdot,\\cdot \\rangle_2$ being the inner product in $\\mathcal L^2\\left(\\Prob^{X} \\right)$. \n\nThe sample analogues of $T,T^\\ast$ are $T_n:\\mathcal H \\rightarrow {\\mathbb R}^n, g \\mapsto \\{g(X_1),\\dots,g(X_n)\\}^{ \\mathrm{\\scriptscriptstyle T} }$ and $T_n^\\ast:{\\mathbb R}^n \\rightarrow \\mathcal H, (v_1,\\dots,v_n)^{ \\mathrm{\\scriptscriptstyle T} } \\mapsto n^{-1} \\sum_{t=1}^n v_t k(\\cdot,X_t)$, respectively. Both operators are adjoint with respect to \nthe rescaled Euclidean product $\\langle u,v\\rangle = n^{-1} u^{ \\mathrm{\\scriptscriptstyle T} } v$, $u,v \\in {\\mathbb R}^d$\n\nFinally, we define the sample kernel covariance operator $S_n = T^\\ast_n T_n:\\mathcal H \\rightarrow \\mathcal H$ and the population kernel covariance operator $S = T^\\ast T:\\mathcal H \\rightarrow \\mathcal H$. Note that it holds $K_n = T_n T_n^\\ast$. Under \\ref{con:k1} and \\ref{con:k2} $S$ is a self-adjoint compact operator with operator norm $\\|S\\|_{\\mathcal L} \\leq \\kappa$, see \\citet{Cap07}.\n\nWith this notation we can restate (\\ref{eq:kpls.optim}) for the function $f_\\alpha$ \n\\begin{equation}\n\\label{eq:func.representation}\nf_{\\widehat{\\alpha}_i} = \\arg \\min_{g \\in \\mathcal K_i(S_n,T_n^\\ast y )} \\|y- \\{g(X_1),\\dots,g(X_n)\\}^{ \\mathrm{\\scriptscriptstyle T} }\\|^2=\\arg \\min_{g \\in \\mathcal K_i(S_n,T_n^\\ast y )}\\|y-T_ng\\|^2.\n\\end{equation}\nHence, we are looking for functions that minimize the squared distance to $y$ constrained to a sequence of Krylov spaces. \n\nIn the literature of ill-posed problems it is well known that without further conditions on the target function $f^\\ast$ the convergence rate of the conjugate gradient algorithm can be arbitrarily slow, see \\citet{Hanke}, chapter 3.2. One common a-priori assumption on the regression function $f^\\ast$ is \na source condition:\n\\begin{enumerate}[label={(S)}]\n\\item\n\\label{eq:source}\nThere exist $r \\geq 0$, $R>0$ and $u \\in \\mathcal L^2\\left(\\Prob^{X} \\right)$ such that $f^\\ast = (T T^\\ast)^{r} u$ and $\\|u\\|_2 \\leq R$.\n\\end{enumerate}\n\nIf $r \\geq 1\/2$, then the target function $f^\\ast \\in \\mathcal L^2\\left(\\Prob^{X} \\right)$ coincides almost surely with a function $f \\in \\mathcal H$ and we can write $f^\\ast = T f$, see \\citet{Cuc02}.\nWith this the kernel partial least squares estimator $f_{\\widehat{\\alpha}_i}$ estimates the correct target function, not only its best approximation in $\\mathcal{H}$. This case is known as the inner case. \n\nThe situation with $r<1\/2$ is referred to as the outer case. Under additional assumptions, e.g., the availability of additional unlabeled data, it is still possible that an estimator of $f^\\ast$ converges to the true target function in $\\mathcal L^2\\left(\\Prob^{X} \\right)$ norm with optimal rates (with respect to the number $n$ of labeled data points). See \\citet{Vit06} for a detailed description of this semi-supervised approach for kernel ridge regression in the independent and identically distributed case. We do not treat the case $r<1\/2$ in this work.\n\nA source conditions is often interpreted as an abstract smootheness condition. This can be seen as follows.\nLet $\\eta_1 \\geq \\eta_2 \\geq \\dots$ be the eigenvalues and $\\psi_1,\\psi_2,\\dots$ the corresponding eigenfunctions of the compact operator $S$. \nThen it is easy to see that the source condition \\ref{eq:source} is equivalent to $f = \\sum_{j=1}^\\infty b_j \\psi_j \\in \\mathcal L^2\\left(\\Prob^{X} \\right)$ with $b_j$ such that $\\sum_{j=1}^\\infty \\eta_j^{-2(r+1\/2)} b_j^2 < \\infty$. Hence, the higher $r$ is chosen the faster the sequence $\\{b_j\\}_{j=1}^\\infty$ must converge to zero. Therefore, the sets of functions for which source conditions hold are nested, i.e., the larger $r$ is the smaller the corresponding set will be. The set with $r=1\/2$ is the largest one and corresponds to a zero smoothness condition, i.e., $\\sum_{j=1}^\\infty \\eta_j^{-2} b_j^2 < \\infty$, which is equivalent to $f \\in \\mathcal H$. For more details we refer to \\citet{Dicker17}.\n\n\\section{Consistency of Kernel Partial Least Squares}\n\\label{sec:kpls.convergence}\nThe KCG algorithm as described by \\citet{Blan10b} is consistent when stopped early and convergence rates can be obtained when a source condition \\ref{eq:source} holds. Here we will proof the same property for KPLS. Early stopping in this context means that we stop the algorithm at some $a=a(n) \\leq n$ and consider the estimator $f_{\\widehat{\\alpha}_a}$ for $f^\\ast$.\n\nThe difference between KCG and KPLS is the norm which is optimized. The kernel conjugate gradient algorithm studied in \\citet{Blan10b} estimates the coefficients $\\alpha \\in {\\mathbb R}^n$ of $f_\\alpha$ via $\\widehat{\\alpha}_i^{CG} = \\arg\\min_{v \\in \\mathcal K_i(K_n,y)} \\langle y- K_n v, K_n(y- K_n v)\\rangle$.\nIt is easy to see that this optimization problem can be rewritten for the function $f_\\alpha$ as  \n\\[\n\\min_{g \\in \\mathcal K_i(S_n,T_n^\\ast y )} \\|T_n^\\ast y-S_ng\\|_\\mathcal H^2=\t\\min_{g \\in \\mathcal K_i(S_n,T_n^\\ast y )} \\|T_n^\\ast\\left(y-T_ng\\right)\\|_\\mathcal H^2,\n\\] \ncompared to (\\ref{eq:func.representation}) for KPLS. Thus, KCG obtains the least squares approximation $g$ in the $\\mathcal H$-norm for the normal equation $T_n^\\ast y = T^\\ast _nT_n g$ and KPLS finds a function that minimizes the residual sum of squares. In both methods the solutions are restricted to functions $g \\in \\mathcal K_i(S_n,T_n^\\ast y )$.\n\nAn advantage of the kernel conjugate gradient estimator is that concentration inequalities can be established for both $T_n^\\ast y$ and $S_n$ and applied directly as the optimization function contains both quantities. The stopping index for the regularization can be chosen by a discrepancy principle as ${a^\\ast} = \\min\\{1\\leq i \\leq n: \\|S_n f_{\\widehat{\\alpha}_i^{CG}} - T_n^\\ast y\\| \\leq \\Lambda_n\\}$ with $\\Lambda_n$ being a threshold sequence that goes to zero as $n$ increases.\n\nOn the other hand, the function to be optimized for KPLS contains only $y$ and $T_n g = \\{g(X_1),\\dots,g(X_n)\\}^{ \\mathrm{\\scriptscriptstyle T} }$ for which statistical properties are not readily available. Thus, we need to find a way to apply the concentration inequalities for $T_n^\\ast y$ and $S_n$ to this slightly different problem. This leads to complications in the proof of consistency and a rather different and more technical stopping rule for choosing the optimal regularization parameter $a^\\ast$ is used, as can be seen in Theorem \\ref{th:kpls}. This stopping rule has its origin in \\citet{Hanke}.\\\\\n\n In the following $\\|\\cdot\\|_{\\cal{L}}$ denotes the  operator norm and $\\|\\cdot\\|_{HS}$ is the Hilbert-Schmidt norm.\n\n\\begin{theorem}\n\\label{th:kpls}\nAssume that conditions \\ref{con:k1}, \\ref{con:k2}, \\ref{eq:source} hold with $r \\geq 3\/2$ and there are constants $C_\\delta(\\nu),C_\\epsilon(\\nu)>0$ and a sequence $\\{\\gamma_n\\}_{n \\in {\\mathbb N}} \\subset [0,\\infty)$, $\\gamma_n \\rightarrow 0$, such that we have for $\\nu \\in (0,1]$\n\\begin{align*}\n\t\\mathrm{P}\\left(\n\t\\|S_n-S\\|_{\\mathcal L} \\leq C_\\delta(\\nu) \\gamma_n\n\t\\right) &\\geq 1 - \\nu\/2,\\\\\n\t\\mathrm{P}\\left(\n\t\\|T_n^\\ast y - S f\\|_\\mathcal H \\leq C_\\epsilon(\\nu) \\gamma_n\n\t\\right) &\\geq 1- \\nu\/2.\n\\end{align*}\nDefine the stopping index $a^\\ast$ by\n\\begin{equation}\n\\label{eq:stopping}\na^\\ast = \\min\\left\\{1 \\leq a \\leq n: \\sum_{i=0}^a \\|S_n f_{\\widehat{\\alpha}_i} - T_n^\\ast y\\|^{-2}_\\mathcal H \\geq (C \\gamma_n)^{-2}\n\t\\right\\},\n\\end{equation}\nwith $C = C_\\epsilon(\\nu) + \\kappa^{r-1\/2}(r+1\/2) R \\{1 + C_\\delta(\\nu)\\}$.\n\nThen it holds with probability at least $1-\\nu$ that\n\\begin{align*}\n\t\\|f_{\\widehat{\\alpha}_{a^\\ast}} - f^\\ast\\|_2 &= O\\left\\{\\gamma_n^{2r\/(2r+1)}\\right\\},\\\\\n\t\\|f_{\\widehat{\\alpha}_{a^\\ast}} - f\\|_\\mathcal H &= O\\left\\{\\gamma_n^{(2r-1)\/(2r+1)}\\right\\},\n\\end{align*}\nwith $f^\\ast = T f$.\n\\end{theorem}\n\n\nIt can be shown that the stopping rule (\\ref{eq:stopping}) always determines a finite index, i.e., the set the minimum is taken over is not empty, see \\citet{Hanke}, chapter 4.3.\n\nThe theorem yields two convergence results, one in the $\\mathcal H$-norm and one in the $\\mathcal L^2\\left(\\Prob^{X} \\right)$-norm. It holds that $\\|v\\|_2 = \\|S^{1\/2}v\\|_\\mathcal H$. These are the endpoints of a continuum of norms $\\|v\\|_\\beta = \\|S^\\beta v\\|_\\mathcal H$, $\\beta \\in [0,1\/2]$ that were considered in \\citet{Nemirovskii86} for the derivation of convergence rates for KCG algorithms in a deterministic setting. \n\nThe convergence rate of the kernel partial least squares estimator depends crucially on the sequence $\\gamma_n$ and the source parameter $r$. If $\\gamma_n = O(n^{-1\/2})$, this yields the same convergence rate as Theorem 2.1 of \\citet{Blan10b} for kernel conjugate gradient or \\citet{Vito05} for kernel ridge regression with independent and identically distributed data. For stationary Gaussian time series we will derive concentration inequalities in the next section and obtain convergence rates depending on the source parameter $r$ and the range of dependence. Note that Theorem \\ref{th:kpls} is rather general and it can be applied to any kind of dependence structure, as long as the necessary concentration inequalities can be established.\n\nThe next theorem  derives faster convergence rates under assumptions on the effective dimensionality of operator $S$, which is defined as $d_\\lambda = \\mathrm{tr}\\{(S+\\lambda)^{-1} S\\}$. The concept of effective dimensionality was introduced in \\citet{Zho02} to get sharp error bounds for general learning problems considered there. If $\\mathcal H$ is a finite dimensional space it was shown in \\citet{Zho02} that $d_\\lambda \\leq \\mathrm{dim}(\\mathcal H)$. For infinite dimensional spaces it describes the complexity of the interactions between data and reproducing kernel. \n\nIf $d_\\lambda = O(\\lambda^{-s})$ for some $s \\in (0,1]$, \\citet{Cap07} showed that the order optimal convergence rates $n^{-r\/(2r+s)}$ are attained for KRR with independent and identically distributed data.\n\nThe effective dimensionality clearly depends on the behaviour of eigenvalues of $S$. If these converge sufficiently fast to zero, nearly parametric rates of convergence can be achieved for reproducing kernel Hilbert space methods, see, e.g., \\citet{Dicker17}. In particular, the behaviour of $d_\\lambda$ around zero is of interest, since it determines how ill-conditioned the operator $(S+\\lambda)^{-1}$ becomes. In the following theorem we set $\\lambda = \\lambda_n$ for a sequence $\\{\\lambda_n\\}_{n\\in {\\mathbb N}} \\subset (0,\\infty)$ that converges to zero. \n\n\\begin{theorem}\n\\label{th:kpls2}\nAssume that conditions \\ref{con:k1}, \\ref{con:k2}, \\ref{eq:source} hold with $r \\geq 1\/2$ and that the effective dimensionality $d_\\lambda$ is known. Additionally, there are constants $C_\\delta(\\nu),C_\\epsilon(\\nu),C_\\psi\n>0$ and a sequence $\\{\\gamma_n\\}_{n \\in {\\mathbb N}} \\subset [0,\\infty)$, $\\gamma_n \\rightarrow 0$, such that for $\\nu \\in (0,1]$ and $n$ sufficiently large\n\\begin{align*}\n\t\\mathrm{P}\\left\\{\n\t\\|S_n-S\\|_{\\mathcal L} \\leq C_\\delta(\\nu) \\gamma_n\n\t\\right\\} &\\geq 1 - \\nu\/3,\\\\\n\t\\mathrm{P}\\left\\{\n\t\\|(S+\\lambda_n)^{-1\/2}(T_n^\\ast y - S f)\\|_\\mathcal H \\leq C_\\epsilon(\\nu) \\sqrt{d_{\\lambda_n}}\\gamma_n\n\t\\right\\} &\\geq 1- \\nu\/3,\\\\\n\t\\mathrm{P}\\left\\{\n\t\\|(S+\\lambda_n)^{1\/2}(S_n+\\lambda_n)^{-1\/2}\\|_{\\mathcal L} \\leq C_\\psi\n\t\\right\\} &\\geq 1 - \\nu\/3,\n\\end{align*}\nHere $\\{\\lambda_n\\}_{n \\in {\\mathbb N}} \\subset (0,\\infty)$ is a sequence converging to zero such that for $n$ large enough \n\\begin{equation}\n\t\\label{eq:lambda.inequ}\n\t\\gamma_n \\leq \\lambda_n^{r-1\/2}.\n\\end{equation}\n\nTake $\\zeta_n = \\max\\{\\sqrt{\\lambda_n d_{\\lambda_n}} \\gamma_n, \\lambda_n^{r+1\/2}\\}$ \nDefine the stopping index $a^\\ast$ by\n\\begin{equation}\n\\label{eq:stopping2}\na^\\ast = \\min\\left\\{1 \\leq a \\leq n: \\sum_{i=0}^a \\|S_n f_{\\widehat{\\alpha}_i} - T_n^\\ast y\\|^{-2}_\\mathcal H \\geq (C \\zeta_n)^{-2}\\right\\},\n\\end{equation}\nwith $C=4 R \\max\\{1, C_\\psi^2,(r-1\/2)\\kappa^{r-3\/2} C_\\delta(\\nu),2^{-1\/2}R^{-1} C_\\psi C_\\epsilon(\\nu)\\}$.\n\nThen it holds with probability at least $1-\\nu$ that\n\\begin{align*}\n\t\\|f_{\\widehat{\\alpha}_{a^\\ast}} - f^\\ast\\|_2 &= O\\left\\{\\lambda_n^{-1\/2}\\zeta_n\\right\\},\\\\\n\\|f_{\\widehat{\\alpha}_{a^\\ast}} - f\\|_\\mathcal H &= O\\left\\{\\lambda_n^{-1}\\zeta_n\\right\\},\n\\end{align*}\nwith $f^\\ast = T f$.\n\\end{theorem}\nThe condition (\\ref{eq:lambda.inequ}) holds trivially for $r=1\/2$ as $\\gamma_n$ converges to zero. For $r >1\/2$ the sequence $\\lambda_n$ must not converge to zero arbitrarily fast. \n\nIn its general form Theorem \\ref{th:kpls2} does not give immediate insight in the probabilistic convergence rates of the kernel partial least squares estimator. Therefore, we state two corollaries, where the function $d_\\lambda$ is specified. In both corollaries we explicitly state the choice of the sequence $\\lambda_n$ that yield the corresponding rates.\n\n\\begin{corollary}\n\\label{cor:pol.ed}\nAssume that there exists $s \\in (0,1]$ such that $\n\td_\\lambda = O(\\lambda^{-s})\n$ for $\\lambda \\rightarrow 0$.\nThen under conditions of Theorem \\ref{th:kpls2} with $\\lambda_n = \\gamma_n^{2\/(2r + s)}$  it holds with probability at least $1-\\nu$ that\n\\begin{align*}\n\t\\|f_{\\widehat{\\alpha}_{a^\\ast}} - f^\\ast\\|_2 &= O\\left\\{\\gamma_n^{2r\/(2r+s)}\\right\\}.\n\\end{align*}\n\\end{corollary}\nPolynomial decay of the effective dimensionality $d_\\lambda = \\mathrm{tr}\\{(S+\\lambda)^{-1}S\\}$ occurs if the eigenvalues of $S$ also decay polynomially fast, that is, $\\mu_i = c_s i^{-1\/s}$ for $s \\in (0,1]$, since in this case $d_\\lambda = \\sum\\limits_{i=1}^\\infty \\{1+\\lambda\/c_s i^{1\/s}\\}^{-1} = O(\\lambda^{-s})$. This holds, for example, for the Sobolev kernel $k(x,y) = \\min(x,y)$, $x,y \\in [0,1]$ and data that are uniformly distributed on $[0,1]$, see \\citet{Ras14}.\n\nIf $\\gamma_n = n^{-1\/2}$, then the KPLS estimator converges in the $\\mathcal L^2\\left(\\Prob^{X} \\right)$-norm with a rate of $n^{-r\/(2r+s)}$. This rate is shown to be optimal in \\citet{Cap07} for KRR with independent identically distributed data.\n\nNote that the rate obtained in Theorem \\ref{th:kpls} corresponds to $\\gamma_n^{-2r\/(2r+s)}$ with $s=1$, i.e., the worst case rate with respect to the parameter $s \\in (0,1]$.\n\nIn the next corollary to Theorem \\ref{th:kpls2} we assume that the effective dimensionality behaves in a logarithmic fashion.\n\n\\begin{corollary}\n\\label{cor:log.ed}\nLet $d_\\lambda = O\\{\\log(1+a\/\\lambda)\\}$ for $\\lambda \\rightarrow 0$ and  $a>0$. Then under the conditions of Theorem \\ref{th:kpls2} with $\\lambda_n = \\gamma_n^2 \\log\\{ \\gamma_n^{-2}\\}$ and $r=1\/2$ it holds with probability at least $1-\\nu$ that\n\\begin{align*}\n\t\\|f_{\\widehat{\\alpha}_{a^\\ast}} - f^\\ast\\|_2 &= O\\left\\{\\gamma_n \\log(1\/2 \\gamma_n^{-2})\\right\\}.\n\\end{align*}\n\\end{corollary}\nThe effective dimensionality takes the special form considered in this corollary, for example, when the eigenvalues of $S$ decay exponentially fast. This holds, for example, if the data are Gaussian and the Gaussian kernel is used, see Section \\ref{sec:gauss.kern}.\nIf $\\gamma_n =O(n^{-1\/2})$, then the convergence rate is of order $O\\{n^{-1}\\log(n)\\}$, which are nearly parametric. It is noteworthy that the source condition only impacts the choice of the sequence $\\lambda_n$, not the convergence rates of the estimator in the $\\mathcal L^2\\left(\\Prob^{X} \\right)$-norm. Therefore, we stated the corollary for $r=1\/2$, which is a minimal smoothness condition on $f^\\ast$, i.e., that $f^\\ast = T f$ almost surely for an $f \\in \\mathcal H$.\n\nThe rates obtained in Corollaries \\ref{cor:pol.ed} and \\ref{cor:log.ed} were derived in \\citet{Dicker17} for kernel ridge regression and kernel principal component regression under the assumption of independent and identically distributed data.\n\n\\section{Concentration Inequalities for Gaussian Time Series}\n\\label{sec:concentration}\nCrucial assumptions of Theorem \\ref{th:kpls} and \\ref{th:kpls2} are the concentration inequalities\nfor $S_n$ and $T_n^\\ast y$ and convergence of the sequence $\\{\\gamma_n\\}_{n \\in {\\mathbb N}}$. Here we establish such inequalities in a Gaussian setting for stationary time series. At the end of this section we will state explicit convergence rates for $f_{\\widehat{\\alpha}_{a^\\ast}}$ that depend not only on the source parameter $r \\geq 1\/2$ and the effective dimensionality $d_\\lambda$, but also on the persistence of the  dependence in the data.\n\nThe Gaussian setting is summarized in the following assumptions\n\\begin{enumerate}[label={(D\\arabic*})]\n\\item\n\\label{D1}\n$(X_h,X_0)^{ \\mathrm{\\scriptscriptstyle T} } \\sim \\mathcal{N}_{2d}(0,\\Sigma_h)$, $h=1,\\dots,n-1$, with \n\\[\n      \\Sigma_h = \\left[\n\\begin{matrix}\n\t\\tau_0 & \\tau_h\\\\\n\t\\tau_h & \\tau_0\n\\end{matrix}\n\\right] \\otimes \\Sigma.\n\\]\nHere $\\Sigma \\in {\\mathbb R}^{d \\times d}$ and $V=[\\tau_{|i-j|}]_{i,j=1}^{n} \\in {\\mathbb R}^{n \\times n}$ are positive definite, symmetric matrices and $\\otimes$ denotes the Kronecker product between matrices. Furthermore $X_0 \\sim \\mathcal{N}_d(0,\\tau_0 \\Sigma)$.\n\\item\n\\label{D2}\nFor the autocorrelation function $\\rho_h = \\tau^{-1}_0\\tau_h$ there exists a $q>0$ such that $|\\rho_h| \\leq  (h+1)^{-q}$ for $h =0,\\dots,n-1$.\n\\end{enumerate}\nCondition \\ref{D1} is a separability condition for the covariance matrices $\\Sigma_h$, $h = 0,\\dots,n-1$. Due to \\ref{D1} the effects (on the covariance) over time and between the different variables can be treated separately. Under condition \\ref{D2} it is easy to see that from $q>1$ follows the absolute summability of the autocorrelation function $\\rho$ and thus $\\{X_t\\}_{t \\in {\\mathbb Z}}$ is a short memory process. Stationary short memory processes keep many of the properties of independent and identically distributed data, see, e.g., \\citet{bBrock}.\n\nOn the other hand $q \\in (0,1]$ yields a long memory process, see, e.g., Definition 3.1.2 in \\citet{Giraitis}. Examples of long memory processes are the fractional Gaussian noise with an autocorrelation function that behaves like $(h+1)^{-2(1-H)}$, with $H \\in [0,1)$ being the Hurst coefficient. Stationary long memory processes exhibit dependencies between observations that are more persistent and many statistical results that hold for independent and identically distributed data turn out to be false, see \\citet{Samoro} for details.\n\nThe next theorem gives concentration inequalities for both estimators $S_n$ and $T_n^\\ast y$ in a Gaussian setting with convergence rates depending on the parameter $q>0$. These inequalities are the ones needed in Theorem \\ref{th:kpls} and Theorem \\ref{th:kpls2}. Recall that $d_\\lambda = \\mathrm{tr}\\{(S+\\lambda)^{-1}S\\}$ denotes the effective dimensionality of $S$.\n\n\\begin{theorem}\n\\label{th:conc.equality}\n(i) Define $\\mathrm{d}\\mu_h(x,y) = \\mathrm{d}\\mathrm{P}^{X_h,X_0}(x,y) - \\mathrm{d} \\mathrm{P}^{X_0}(x)\\mathrm{d} \\mathrm{P}^{X_0}(y)$.\nUnder Assumptions \\ref{con:k1} and \\ref{con:k2} it holds for $\\nu \\in (0,1]$ with probability at least $1-\\nu$ that\n\\begin{align*}\n\\|S_n - S\\|^2_{\\mathcal L} &\\leq\n\\frac{2 \\nu^{-1}}{n^2}\\sum\\limits_{h=1}^{n-1} (n-h) \\int\\limits_{{\\mathbb R}^{2d}} k^2(x,y) \\mathrm{d}\\mu_h(x,y) + \\frac{\\nu^{-1}}{n} \\left\\{\n\t\\mathrm{E} k^2(X_0,X_0) - \\|S\\|^2_{\\mathrm{HS}}\n\\right\\},\\\\\n\\|T_n^\\ast y - S f\\|^2_\\mathcal H &\\leq\n\\frac{2\\nu^{-1}}{n^2}\\sum\\limits_{h=1}^{n-1} (n-h) \\int\\limits_{{\\mathbb R}^{2d}} k(x,y)f(x)f(y)\\mathrm{d}\\mu_h(x,y)\\\\\n&+ \\frac{\\nu^{-1}}{n} \\left[\n\t\\mathrm{E} \\left\\{k(X_0,X_0) f^2(X_0)\\right\\} - \\| S f\\|^2_\\mathcal H + \\sigma^2 \\mathrm{E}\\{ k(X_0,X_0)\\}\n\\right].\n\\end{align*}\n\n\n(ii) Assume that additionally to \\ref{con:k1}, \\ref{con:k2} also \\ref{D1}, \\ref{D2} for $q >0$ are fulfilled. Denote $M=\\sup_{x \\in {\\mathbb R}^d} |f(x)|$.\n\nThen there exists a constant $C(q)>0$ such that \n\\begin{align*}\n\\|S_n - S\\|_\\mathcal L\n\t&\\leq \n\t \\nu^{-1\/2} \\{\\gamma_n^2(q) \\kappa C_\\gamma\t\n\t\t+ n^{-1}(\\kappa^2-\\|S\\|_\\mathrm{HS}^2)\\}^{1\/2},\\\\\n \\|T_n^\\ast y - S f\\|_\\mathcal H\n\t&\\leq \n\t\\nu^{-1\/2} \\left[\\gamma_n^2(q) M C_\\gamma \n\t\t+ n^{-1}\\left\\{\n\t\t\\kappa (M + \\sigma^2) - \\| S f\\|^2_\\mathcal H\t\\right\\}\\right]^{1\/2}\n\t\t,\n\\end{align*}\nfor $C_\\gamma = C(q)\\{(2\\pi)^d \\mathrm{det}(\\Sigma)\\}^{-1\/2} \\kappa d^{1\/2}(1-4^{-q})^{-1\/4(d+2)}$. The function $\\gamma_n(q)$, $q>0$, is defined as\n\\[\n\t\\gamma_n(q) = \\left\\{\n\t\t\t\t\\begin{array}{clc}\n\t\t\t n^{-1\/2} &,& q>1\\\\\n\t\tn^{-1\/2} \\log(1\/2n)\t&,& q=1\\\\\n\t\t\tn^{-q\/2} \n\t\t\t&,& q \\in (0,1).\n\t\t\\end{array}\n\t\\right.\n\\]\n\n(iii) Let \\ref{con:k1}, \\ref{con:k2} and \\ref{eq:source} hold. Let $\\gamma_n(q)$ be the function as defined in (ii). Then there exists a constant $\\tilde{C}_\\epsilon>0$ such that it holds with probability at least $1-\\nu$ for $\\lambda>0$ that\n\\[\n \\|(S+\\lambda)^{-1\/2}(T_n^\\ast y - S_n f\\|_\\mathcal H \\leq  \\nu^{-1\/2}  \\tilde{C}_\\epsilon \\sigma \\sqrt{d_\\lambda}\\gamma_n(q).\n\\] \n\n(iv) Let \\ref{con:k1}, \\ref{con:k2}, \\ref{eq:source}, \\ref{D1} and \\ref{D2} hold. Let $\\lambda_n^{-1\/2}d_{\\lambda_n}^{1\/2} \\gamma_n(q) \\rightarrow 0$ for a sequence $\\lambda_n \\rightarrow 0$ and $\\gamma_n(q)$ the function defined in (ii). Then there exists an $n_0 = n_0(\\nu,q) \\in {\\mathbb N}$ such that with probability at least $1-\\nu$ we have for all $n \\geq n_0$\n\\[\\|(S+\\lambda_n)^{1\/2}(S_n+\\lambda_n)^{-1\/2}\\|_\\mathcal L \\leq \\sqrt{2}.\n\\]\n\\end{theorem}\n\nThe first part of the theorem is general and can be used to derive concentration inequalities not only in the Gaussian setting and is of interest in itself. The convergence rate is controlled by the sums appearing on the right hand side.\n If these sums are of $O(n)$ then the mean squared error of both $S_n$ and $T_n^\\ast y$ will converge to zero with a rate of $n^{-1}$. On the other hand, if the sums are of order $O(n^{2-q})$ for some $q\\in (0,1)$, the mean squared errors will converge with the reduced rate $n^{-q}$.\n\nThe second part derives explicit concentration inequalities in the Gaussian setting described by \\ref{D1} and \\ref{D2} with rates depending on the range of the dependence measured by $q>0$. These inequalities appear in Theorem \\ref{th:kpls}.\n\nParts (iii) and (iv) give the additional probabilistic bounds needed to apply Theorem \\ref{th:kpls2}. The condition $\\lambda_n^{-1\/2} d^{1\/2}_{\\lambda_n} \\gamma_n(q) \\rightarrow 0$ in Theorem \\ref{th:conc.equality} (iv) is fulfilled in the settings of Corollary \\ref{cor:pol.ed} and Corollary \\ref{cor:log.ed}.\n\n Theorem \\ref{th:kpls}, Corollary \\ref{cor:pol.ed}, Corollary \\ref{cor:log.ed} and Theorem \\ref{th:conc.equality} together imply\n\\begin{corollary}\n\\label{cor:convergence}\nLet the conditions of Theorem \\ref{th:kpls2} and \\ref{D1}, \\ref{D2} hold.\n\n(i) Assume that there exists $s\\in (0,1]$ such that $d_\\lambda = O(\\lambda^{-s})$ for $\\lambda \\rightarrow 0$. Then with probability at least $1-\\nu$\n\t\\[\n\t\t\\|f_{\\widehat{\\alpha}_{a^\\ast}} - f^\\ast\\|_2 = \\left\\{\n\t\t\t\\begin{array}{cc}\n\t\t\t\tO\\{n^{-r\/(2r+s)}\\}, & q>1,\\\\\n\t\t\t\tO\\{n^{-q r\/(2r+s)}\\}, & q \\in (0,1).\n\t\t\t\\end{array}\n\t\t\t\\right.\n\t\\]\nIf instead of conditions of Theorem \\ref{th:kpls2}, conditions of Theorem \\ref{th:kpls} are assumed, then the convergence rates  above have $s=1$. \n\t\n(ii) Assume that there exists $a>0$ such that $d_\\lambda = O\\{\\log(1+a\/\\lambda)\\}$ for $\\lambda \\rightarrow 0$ and $r=1\/2$. Then with probability at least $1-\\nu$\n\t\\[\n\t\t\\|f_{\\widehat{\\alpha}_{a^\\ast}} - f^\\ast\\|_2 = \\left\\{\n\t\t\t\\begin{array}{cc}\n\t\t\t\tO\\{n^{-1\/2}\\log(1\/2 n)\\}, & q>1,\\\\\n\t\t\t\tO\\{n^{-q\/2} \\log(1\/2 n^q)\\}, & q \\in (0,1).\n\t\t\t\\end{array}\n\t\t\t\\right.\n\t\\]\n\\end{corollary}\nHence, for $q>1$ the kernel partial least squares algorithm achieves the same rates as if the data were independent and identically distributed. For $q \\in (0,1)$ the convergence rates become substantially slower, highlighting that dependence structures that persist over a long time can influence the convergence rates of the algorithm.\n\n\\section{Source condition and effective dimensionality for Gaussian kernels}\n\\label{sec:gauss.kern}\nThe source condition \\ref{eq:source} and the effective dimensionality $d_\\lambda$ are of great importance in the  convergence rates derived in previous sections. Here we investigate these conditions for the reproducing kernel Hilbert space corresponding to the Gaussian kernel $k(x,y) = \\exp(-l\\|x-y\\|^2)$, $x,y \\in {\\mathbb R}^d$, $l>0$, for $d=1$. Hence, the space $\\mathcal H$ is the space of all analytic functions that decay exponentially fast, see \\citet{Steinwart05anexplicit}.\n\nWe also impose the normality conditions \\ref{D1} and \\ref{D2} on $\\{X_t\\}_{t\\in {\\mathbb Z}}$, where now $\\sigma^2_x =\\Sigma \\in {\\mathbb R}$ due to $d=1$. The following proposition derives a more explicit representation for $f \\in \\mathcal H$.\n\n\\begin{proposition}\n\\label{prop:source}\nAssume that \\ref{con:k1},\\ref{con:k2} and \\ref{eq:source} hold for $r \\geq 1\/2$. Let $d=1$, $X_0 \\sim \\mathcal{N}(0,\\sigma^2_x), \\sigma^2_x > 0$ and consider the Gaussian kernel $k(x,y) = \\exp\\{-l (x-y)^2\\}$ for $x,y \\in {\\mathbb R}$, $l>0$. Then $f$ can be expressed for $\\mu = r - 1\/2 \\in {\\mathbb N}$\n  via $f(x) = \\sum_{i=1}^\\infty c_i L_\\mu(x,z_i)$ for fixed $\\{z_i\\}_{i=1}^\\infty,\\{c_i\\}_{i=1}^\\infty \\subset {\\mathbb R}$ such that $\\sum_{i,j=1}^\\infty c_i c_j k(z_i,z_j) \\leq R^2$, $R>0$. Here we have for $x,z \\in {\\mathbb R}$ \n\t\\begin{align*}\n\tL_\\mu(x,z) &=  \\exp\n\t\t\\left[\n\t\t\t-1\/2 \\left\\{\n\t\t\t\t\\frac{\\det(\\Lambda)(x^2+z^2)-2 l^{\\mu+1} x z}{\\det(\\Lambda_{1:\\mu})}\n\t\t\t\\right\\}\n\t\t\\right],\n\t\t\\end{align*}\n\twith $\\Lambda \\in {\\mathbb R}^{(\\mu+1)\\times (\\mu+1)}$ being a tridiagonal matrix with elements \n\t\\[\n\t\t\\Lambda_{i,j} = \\left\\{\n\t\\begin{array}{cll}\n\t\t\\sigma^{-2}_x + 2 l &, & i=j<\\mu+1\\\\\n\t\tl &, & i=j=\\mu+1\\\\\n\t\t-l &, & |i-j| = 1\\\\\n\t\t0&, & else\n\t\\end{array}\\right.\n\t\\]\n\tfor $i,j=1,\\dots,\\mu+1$ and $\\Lambda_{1:\\mu}$ is the $\\mu \\times \\mu$-dimensional sub-matrix of $\\Lambda$ including the fist $\\mu$ columns and rows.\n\t\n\tConversely any function $f^\\ast = T f$ with $f$ of the above form fulfills a source condition $\\ref{eq:source}$ with $r = \\mu + 1\/2$, $\\mu \\in {\\mathbb N}$.\n\\end{proposition}\nHence if we fix an $r \\geq 1\/2$ with $r-1\/2 \\in {\\mathbb N}$ this theorem gives us a way to construct functions $f \\in \\mathcal H$ with $f^\\ast = Tf$ that fulfill \\ref{eq:source}.\n\nThe next proposition derives the effective dimensionality $d_\\lambda$ in this setting:\n\\begin{proposition}\n\\label{prop:ed}\nLet $d=1$, $X_0 \\sim \\mathcal{N}(0,\\sigma^2_x)$ for some $\\sigma^2_x>0$ and consider the Gaussian kernel $k(x,y) = \\exp\\{-l(x-y)^2\\}$, $x,y \\in {\\mathbb R}$, $l>0$. \n\nThen there is a constant $D>0$ such that it holds for any $\\lambda \\in (0,1]$\n\\[\n\td_\\lambda = \\mathrm{tr}\\{(S+\\lambda)^{-1} S\\} \\leq \tD\\log(1+a\/\\lambda),\n\\]\nwith $a =\\sqrt{2}(1+\\beta+\\sqrt{1+\\beta})^{-1\/2} $, $\\beta = 4 l \\sigma^2_x$.\n\\end{proposition}\n\nWith the latter result Corollary \\ref{cor:log.ed} is applicable and we expect convergence rates for the kernel partial least squares algorithm of order $O\\{\\gamma_n \\log(1\/2 \\gamma_n^{-2})\\}$ for a sequence $\\{\\gamma_n\\}_n$ as in Theorem \\ref{th:kpls2}.\n\n\n\\section{Simulations}\n\\label{sec:simulations}\nTo validate the theoretical results of the previous sections we conducted a simulation study. The reproducing kernel Hilbert space is chosen to correspond to the Gaussian kernel $k(x,y) = \\exp(-l\\|x-y\\|^2)$, $x,y \\in {\\mathbb R}^d$, $l=2$, for $d=1$.\n\nThe source parameter is taken $r=4.5$ and we consider the function \n\\[\n\tf(x) = 4.37^{-1}\\{3 {L}_4(x,-4) - 2 {L}_4(x,3)+ 1.5 {L}_4(x,9)\\}, ~~ x \\in {\\mathbb R}.\n\t\\] \n\tThe normalization constant is chosen such that $f$ takes values in $[-0.35,0.65]$ and $L_4$ is the exponential function given in Proposition \\ref{prop:source}. The function $f$ is shown in Figure \\ref{fig:func}.\n\\begin{figure}\n   \\begin{center}\n   \t\\includegraphics[width=.5\\linewidth]{f_plot}\n   \t\\caption{\\label{fig:func}\n\t\tThe function $f$ evaluated on $[-7.5,7.5]$ (black) and one realisation of the noisy data $y = f(x) + \\varepsilon$ (grey).   \t\t\n   \t\t   }\n   \\end{center}\n\\end{figure}\n\nIn condition \\ref{D1} we set $\\sigma^2_x =\\Sigma = 4$. For the matrix $V^2 = [\\tau_{|i-j|}]_{i,j=1}^n \\in {\\mathbb R}^{n \\times n}$ we choose three different structures  for $n\\in\\{200,400,1000\\}$. In the first setting $\\tau_h = \\mathbb{I}(h=0)$, which corresponds to independent data. The second setting with $\\tau_h = 0.9^{-h}$ implies an autoregressive process of order one. Finally, the third setting with $\\tau_h = (1+h)^{-q}$, $q=1\/4$, $h =0,\\dots,n-1$ leads to the long range dependent case. \n\nIn a Monte Carlo simulation with $M=1000$ repetitions the time series $\\{X_t^{(j)}\\}_{t=1}^n$ are generated via $X^{(j)} = V N^{(j)}$ with $N^{(j)} \\sim \\mathcal{N}_n(0,\\sigma^2 I_n)$, $j=1,\\dots,M$, where $I_n$ is the $n \\times n$-dimensional identity matrix. \n\nThe residuals $\\varepsilon_1^{(j)},\\dots,\\varepsilon_n^{(j)}$ are generated as independent standard normally distributed random variables and independent of $\\{X_t^{(j)}\\}_{t=1}^n$ . The response is defined as $y_t^{(j)} = f(X_t^{(j)}) + \\eta\\, \\varepsilon_t^{(j)}$, $t=1,\\dots,n$, $j=1,\\dots,M$, with $\\eta = 1\/16$.\n\nThe kernel partial least squares and kernel conjugate gradient algorithms are run for each sample $\\{(X_t^{(j)},y_t^{(j)})^{ \\mathrm{\\scriptscriptstyle T} }\\}_{t=1}^n$, $j=1,\\dots,M$, with a maximum of $40$ iteration steps. We denote the estimated coefficients with $\\widehat{\\alpha}_1^{(j,m)},\\dots,\\widehat{\\alpha}_{40}^{(j,m)}$, $j=1,\\dots,M$, with $m=CG$ meaning that the kernel conjugate gradient algorithm was employed and $m=PLS$ that kernel partial least squares was used to estimate $\\alpha_1,\\dots,\\alpha_n$. \n\nThe squared error in the $\\mathcal L^2\\left(\\Prob^{X} \\right)$-norm is calculated via\n\\[\n\t\\widehat{e}_{n,\\tau}^{(j,m)} =  \\min\\limits_{a=1,\\dots,40} \\left[ \\frac{1}{\\sqrt{2\\pi\\sigma_x^2}}\\int\\limits_{-\\infty}^\\infty \\left\\{f_{\\widehat{\\alpha}_a^{(j,m)}}(x) - f(x) \\right\\}^2 \\exp\n\t\\left(-\\frac{1}{2\\sigma_x^2} x^2 \\right) \\mathrm{d} x \\right],\n\\]\nfor $j=1,\\dots,M$, $n = 200,400,\\dots,1000$ and $m \\in \\{CG, PLS\\}$.\n\nThe results of the Monte-Carlo simulations are depicted in the boxplots of Figure \\ref{fig:box}.\n\\begin{figure}\n   \\begin{center}\n\t \t\\begin{minipage}{.33\\textwidth}\n\t  \t\t\\centering\n\t \t\t\\includegraphics[width=.98\\linewidth]{MSE_box_1}\n\t\t\\end{minipage}%\n\t\t\\begin{minipage}{.33\\textwidth}\n\t  \t\t\\centering\n\t \t\t\\includegraphics[width=.98\\linewidth]{MSE_box_2}\n\t\t\\end{minipage}%\n\t\t\\begin{minipage}{.33\\textwidth}\n\t  \t\t\\centering\n\t \t\t\\includegraphics[width=.98\\linewidth]{MSE_box_3}\n\t\t\\end{minipage}\n\t\t\\caption{\\label{fig:box}\n\t\t\tBoxplots of the $\\mathcal L^2\\left(\\Prob^{X} \\right)$-errors $\\{\\widehat{e}_{n,\\tau}^{(j,m)}\\}_{j=1}^{M}$ of kernel partial least squares (left side of each panel) and kernel conjugate gradient (right side of each panel) for different autocovariance functions $\\tau$ and $n = 200,400,1000$. On the left is $\\tau_h = \\mathbb{I}(h=0)$, in the middle $\\tau_h = 0.9^{-h}$ and on \n\t\t\tthe right $\\tau_h = (h+1)^{-1\/4}$.\n\t\t}\n\t\\end{center}\n\\end{figure}\nFor kernel partial least squares (left panels) one observes that independent and autoregressive dependent data have roughly the same convergence rates, although the latter have a somewhat higher error. In contrast, the long range dependent data show slower convergence with the larger interquartile range, supporting the theoretical results of Corollary \\ref{cor:convergence}.\n\nThe $\\mathcal L^2\\left(\\Prob^{X} \\right)$-error of kernel conjugate gradient estimators is generally slightly higher than that of kernel partial least squares. Nonetheless, both of them have a similar behaviour.\n\nWe also investigated the the stopping indices $a=1,\\dots,40$ for which the errors $\\widehat{e}_{n,\\tau}^{(j,m)}$ were attained. These are shown in Figure \\ref{fig:a} for independent and identically distributed data.\n\\begin{figure}\n   \\begin{center}\n   \t\\begin{minipage}{.45\\textwidth}\n\t\\center   \t\n   \t\\includegraphics[width=.98\\linewidth]{MSE_sim_a_200}\n   \t\\end{minipage}\n   \t\\begin{minipage}{.45\\textwidth}\n\t\\center   \t\n   \t\\includegraphics[width=.98\\linewidth]{MSE_sim_a_1000}\n   \t\\end{minipage}\n   \t\\caption{\\label{fig:a}\n   \t\tBoxplots of the optimal indices $a\\in\\{1,\\dots,40\\}$ for which the $\\mathcal L^2\\left(\\Prob^{X} \\right)$-errors $\\{\\widehat{e}_{n,\\tau}^{(j,m)}\\}_{j=1}^{M}$ were attained. Kernel partial least squares is on the left of each panel and kernel conjugate gradient on the right. On the left is $n = 200$, on the right $n=1000$. The data were assumed to be independent and identically distributed.\n   \t}\n   \\end{center}\n\\end{figure}\nIt can be seen that the optimal indices for both algorithms have a rather similar behaviour. Kernel conjugate gradient stops slightly later, but overall the differences seem negligible.\n\nFigure \\ref{fig:lines} shows the mean (over $j$) of the estimated $\\mathcal L^2\\left(\\Prob^{X} \\right)$ errors $\\{\\widehat{e}_{n,\\tau}^{(j,m)}\\}_{j=1}^{M}$ for different $n$, $\\tau$ and $m \\in \\{CG,PLS\\}$. The errors were multiplied by $n\/\\log(n)$ to illustrate the convergence rates. According to Proposition \\ref{prop:ed} and Corollary \\ref{cor:convergence} (ii) we expect the rates for the independent and autoregressive cases to be $n^{-1}\\log(n)$, which is verified by the fact that the solid black and grey lines are roughly constant.\nFor the long range dependent case we expect worse convergence rates which are also illustrated by the divergence of the dashed black line.\n\\begin{figure}\n   \\begin{center}\n   \t\\begin{minipage}{.45\\textwidth}\n\t\\center   \t\n   \t\\includegraphics[width=.98\\linewidth]{MSE_sim_pls_mean}\n   \t\\end{minipage}\n   \t\\begin{minipage}{.45\\textwidth}\n\t\\center   \t\n   \t\\includegraphics[width=.98\\linewidth]{MSE_sim_cg_mean}\n   \t\\end{minipage}\n   \t\\caption{\\label{fig:lines}\n   \t\tMean of the $\\mathcal L^2\\left(\\Prob^{X} \\right)$-errors $\\{\\widehat{e}_{n,\\tau}^{(j,m)}\\}_{j=1}^{M}$ of kernel partial least squares (left) and kernel conjugate gradient (right) for $n = 200,400,\\dots,1000$ multiplied by $n\/\\log(n)$. The solid black line is for $\\tau_h = \\mathbb{I}(h=0)$, the grey line for $\\tau_h = 0.9^{-h}$ and the dashed black line for $\\tau_h = (h+1)^{-1\/4}$.\n   \t}\n   \\end{center}\n\\end{figure}\n\n\\section{Application to Molecular Dynamics Simulations}\n\\label{sec:protein}\nThe collective motions of protein atoms are responsible for its biological function and molecular dynamics simulations is a popular tool to explore this \\citep{Henz07}. \n\nTypically, the $p \\in {\\mathbb N}$ backbone atoms of a protein are considered for the analysis with the relevant dynamics happening in time frames of nanoseconds. Although the dynamics are available exactly, the high dimensionality of the data and large number of observations can be cumbersome for regression analysis, e.g., due to the high collinearity in the columns of the covariates matrix. Many function-dynamic relationships are also non-linear \\citep{Hub09}.\nA further complication is the fact that the motions of different backbone atoms are highly correlated, making additive non-parametric models for the target function $f^\\ast$ less suitable. \n\nWe consider T4 Lysozyme (T4L) of the bacteriophage T4, a protein responsible for the hydrolisis of 1,4-beta-linkages in peptidoglycans and chitodextrins from bacterial cell walls.\nThe number of available observations is $n=4601$ and T4L consists of $p=486$ backbone atoms.\n\nDenote with $A_{t,i} \\in {\\mathbb R}^3$ the $i$-th atom, $i=1,\\dots,p$, at time $t=1,\\dots,n$ and $c_i \\in {\\mathbb R}^3$ the $i$-th atom in the (apo) crystal structure of T4L. A usual representation of the protein in a regression setting is the Cartesian one, i.e., we take as the covariate $X_t = (A_{1,t}^{ \\mathrm{\\scriptscriptstyle T} },\\dots,A^{ \\mathrm{\\scriptscriptstyle T} }_{p,t})^{ \\mathrm{\\scriptscriptstyle T} }$, $t=1,\\dots,n$, see \\citet{Bro83}.\nThe functional quantity to predict is the root mean square deviation of the protein configuration $X_t$ at time $t = 1,\\dots,n$ from the (apo) crystal structure $C = (c_1^{ \\mathrm{\\scriptscriptstyle T} },\\dots,c_d^{ \\mathrm{\\scriptscriptstyle T} })^{ \\mathrm{\\scriptscriptstyle T} }$, i.e.,\n\\[\n\ty_t = \\left\\{\n\t\tp^{-1} \\sum\\limits_{i=1}^{p} \\|X_{i,t}-C_i\\|^2\n\t\\right\\}^{1\/2}.\n\\] \nThis nonlinear function was previously considered in \\citet{Hub09}, where it was established that linear models are insufficient for the prediction.\n\nFigure \\ref{fig:sample.series} shows the time series corresponding to $X_{t,1}$ (i.e., the first coordinate of the first atom of T4L) on the left and the functional quantity $y_t$ on the right. These plots reveal certain persistent dependence over time.\n\n\\begin{figure}\n   \\begin{center}\n\t \t\\begin{minipage}{.48\\textwidth}\n\t  \t\t\\centering\n\t \t\t\\includegraphics[width=.98\\linewidth]{T4L_X1}\n\t\t\\end{minipage}%\n\t\t\\begin{minipage}{.48\\textwidth}\n\t  \t\t\\centering\n\t \t\t\\includegraphics[width=.98\\linewidth]{T4L_Y}\n\t\t\\end{minipage}\n\t\t\\caption{\\label{fig:sample.series}\n\t\t\tTime series of $X_{t,1}$, i.e., the first coordinate of the first atom T4L consists of (left) and the root mean squared deviation $y_t$ between the protein configuration at time $t$ and the (apo) crystal structure.\n\t\t}\n\t\\end{center}\n\\end{figure}\n\n\nFitting autoregressive moving average models of order $(3,2)$ ($ARMA(3,2)$) to $y_t$ and $ARMA(5,2)$ to $X_{t,1}$ shows that the smallest root of their respective characteristic polynomial is close to one ($1.009$ for $y_t$ and $1.003$ for $X_{t,1}$), highlighting that we are on the border of stationarity, see, e.g., \\citet{bBrock}.\n\nFigure \\ref{fig:acf} depicts the autocorrelation functions of $X_{t,1}$ and $y_t$, the theoretical autocorrelation function of the corresponding autoregressive moving average process and $\\rho_h \\propto (h+1)^{-q}$ for $q=0.134$ for $X_{t,1}$ and $q=0.066$ for $y_t$. The latter, as highlighted in Section \\ref{sec:concentration}, is an autocorrelation function for a stationary long range dependent process.\n\\begin{figure}\n   \\begin{center}\n\t \t\\begin{minipage}{.48\\textwidth}\n\t  \t\t\\centering\n\t \t\t\\includegraphics[width=.98\\linewidth]{T4L_acf_X}\n\t\t\\end{minipage}%\n\t\t\\begin{minipage}{.48\\textwidth}\n\t  \t\t\\centering\n\t \t\t\\includegraphics[width=.98\\linewidth]{T4L_acf_y}\n\t\t\\end{minipage}\n\t\t\\caption{\\label{fig:acf}\n\t\t\tAutocorrelation  \n\t\t\tplots of $X_{t,1}$ (left) and $y_t$ (right). The estimated autocorrelation function is grey, the theoretical one of a fitted $ARMA(3,2)$ process is solid black and $\\rho_h \\propto (h+1)^{-q}$ for a suitable choice of $q>0$ is dashed black.\n\t\t}\n\t\\end{center}\n\\end{figure}\nThese plots suggest that $X_{t,1}$ and $y_t$ follow some long-range stationary process.\n\nWe apply kernel partial least squares to this data set with the Gaussian kernel $k(x,y) = \\exp(-l \\|x-y\\|^2)$, $x,y \\in {\\mathbb R}^{3p}$, $l>0$. The function $f$ we aim to estimate is a distance between protein configurations, so using a distance based kernel seems reasonable. Moreover, we also investigated the impact of other bounded kernels such as triangular and Epanechnikov and obtained similar results. The first $25\\%$ of the data form a training set to calculate the kernel partial least squares estimator and the remaining data are used for testing.\n\nThe parameter $l>0$ is calculated via cross validation on the training set. \nIn our evaluation we obtained $l = 0.0189$.\n\nFigure \\ref{fig:prot} compares the observed response in the test set with the prediction on the test set obtained by kernel partial least squares, kernel principal component regression and linear partial least squares.\n\\begin{figure}[t!]\n   \\begin{center}\n\t \t\\begin{minipage}{.48\\textwidth}\n\t  \t\t\\centering\n\t \t\t\\includegraphics[width=.98\\linewidth]{T4L_pred_cor}\n\t\t\\end{minipage}%\n\t\t\\begin{minipage}{.48\\textwidth}\n\t  \t\t\\centering\n\t \t\t\\includegraphics[width=.98\\linewidth]{T4L_pred_rss}\n\t\t\\end{minipage}\n\t\t\\caption{\\label{fig:prot}\n\t\t\tCorrelation (left) and residual sum of squares (right) between predicted values and the observed response on the test set depending on the number of used components for kernel partial least squares (solid black), partial least squares (grey) and kernel principal component regression (dashed black).\n\t\t}\n\t\\end{center}\n\\end{figure}\nApparently, kernel partial least squares show the best performance and the kernel principal components algorithm is able to achieve comparable prediction with more components only. Obviously, linear partial least squares can not cope with the non-linearity of the problem. \n\nThis application highlights that kernel partial least squares still delivers a robust prediction even when the dependence in the data is more persistent, if enough observations are available.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{intro}\nEvery atom heavier than lithium has been processed by stars.  \nElements of the $\\alpha$ process, iron peak, and \n{\\em r} process have different formation sites, \nand therefore, understanding the distribution of \nthese elements in nearby stars can lead to a \nbetter understanding of the Galaxy's \nchemical enrichment history.\nThe {\\em r}-process\\ site is the least well understood.\nEuropium is our choice for an {\\em r}-process\\ investigation\nfor two reasons:\\ 96\\% of Galactic europium is \nformed through the {\\em r}-process\\ \\citep{burris_2000}, \nand it has several strong lines in the \nvisible portion of the electromagnetic spectrum.\nIn order to provide insight into the Galactic\nenrichment history, however, europium measurements\nin very large stellar samples will be needed.\n\nThe study of {\\em r}-process\\ elements is well established in metal-poor \nstars \\citep[e.g.,][]{sneden_2008,lai_2008,frebel_2007,\nsimmerer_2004,johnson_2001},\nwhere {\\em r}-process\\ abundances probe early Galactic history.  \nA handful of studies have measured europium in \nsolar-metallicity stars \n\\citep[e.g.,][]{reddy_2006,bensby_2005,koch_2002,woolf_1995}, \nbut have done so in samples of 50--200 stars.\nWe aim to extend such work to a substantial sample of \n1000 stars at solar metallicity, using spectra\ncollected by the California and Caltech Planet Search\n(CCPS).\nSuch a large sample will require automated analysis.\n\nOur goal in this paper is to establish a new, \nautomated abundance fitting method based on Spectroscopy Made Easy\n\\citep[SME;][]{valenti_1996}, an LTE spectral synthesis code \ndiscussed in further detail in \\S\\,\\ref{abund}. \nThe analysis builds on the framework of the \n\\citealt{valenti_2005} (hereafter VF05) Spectroscopic \nProperties of Cool Stars (SPOCS) catalog.\nOur method will yield consistent results across large stellar \nsamples, generating smaller errors than previous analyses.\n\nWe begin here with 41\\ stars that\nhave been examined in the literature, and we include \nthree europium lines:\\ 4129\\,\\AA, 4205\\,\\AA, and 6645\\,\\AA.\nOur stellar observations are detailed in \\S\\,\\ref{obs}.\nWe describe the details of our europium \nabundance measurement algorithm in \\S\\,\\ref{abund}, \ncompare our values with existing europium literature\nin \\S\\,\\ref{comp},\nand summarize the results of the study in \n\\S\\,\\ref{summary}.\n\n\\section{Observations}\n\\label{obs}\n\nThe spectra of the 41\\ stars included in this study were taken with the HIRES echelle spectrograph \\citep{vogt_1994} on the 10-m Keck I telescope in Hawaii.  \nThe spectra date from January 2004 to September 2008 and have resolution $R \\sim 50\\,000$ and signal-to-noise ratio (S\/N) of $\\sim$\\,$160$ near \\mbox{4200\\,\\AA}, where the two strongest Eu~{\\sc ii}\\ lines included in this study are located.  \nThe spectra have the same resolution but S\/N~$\\sim 340$ at \\mbox{6645\\,\\AA}, where a third, weaker Eu~{\\sc ii}\\ line is located. \n\nThe spectra were originally obtained by the CCPS with the intention of detecting exoplanets. \nThe CCPS uses the same spectrometer alignment each night and employs the HIRES exposure meter \\citep{kibrick_2006}, so the stellar observations are extremely consistent, even across years of data collection. \nFor a more complete description of the CCPS and its goals, see \\citealt{marcy_2008}.\nIt should be noted that the iodine cell Doppler technique \\citep{marcy_1992} imprints molecular iodine lines only between the wavelengths of 5000 and 6400\\,\\AA, leaving the regions of interest for this study iodine free.\n\n\\subsection{Stellar Sample}\n\\label{stellsamp}\nFor this study we choose to focus on a subset of \nstars that have been analyzed in other abundance studies.  \nWe compare our measurements with those from \n\\citealt{woolf_1995,simmerer_2004,bensby_2005}; and \n\\citealt{delpeloso_2005a}. We select CCPS target stars that \nhave temperature, gravity, mass, and metallicity measurements \nin the SPOCS catalog (VF05). \n\nOur initial stellar sample consisted of 44 objects that were \nmembers both of the aforementioned studies and the SPOCS catalog, \nand that were observed by the CCPS on the Keck I telescope.\nThree stars that otherwise fit our criteria, but for which we\ncould only measure the europium abundance in one line, were \nremoved from the sample.\nThey were HD\\,64090, HD\\,188510, and HIP\\,89215. \nWhile stellar europium abundances may be successfully \nmeasured from a single line, \nthe goal of this study is to establish the robustness \nof our method, \nand so we deem it necessary to determine the \neuropium abundance in at least two lines to \ninclude a star in our analysis here.\nWe describe our criteria for rejecting a fit in \n\\S\\,\\ref{stellar_abund}.\n\nThe 41\\ stars included in this study have \n$3.4 < V < 10.0$,\n$0.50 < B-V < 0.92$, and\n$5 < d < 75\\unit{pc}$ ({\\em Hipparcos}; \\citealt{hipparcos}).\nVF05 determine stellar properties using SME,\nso it is reasonable to adopt their values for our SME europium\nanalysis.\nBased on the VF05 analysis, our stars have\nmetallicity $-1.4 < \\textrm{[M\/H]} < 0.4$,\neffective temperature $4940 < T_{\\rm eff} < 6230$, and\ngravity $3.8 < \\log{g} < 4.8$.\nIt should be noted that throughout this study we use the \n[M\/H]\\ parameter for a star's metallicity, rather than \nthe iron abundance [Fe\/H]. \nOur [M\/H]\\ value is taken from VF05, where it \nis an independent model parameter that adjusts the\nrelative abundance of all metals together.\nIt is not an average of individual metal \nabundances.\nThe full list of stellar properties, from the {\\em Hipparcos} and\nSPOCS catalogs, appears in Table \\ref{starsinfo_table}.\n\n\\begin{deluxetable*}{rrrrrrrrrc}\n\\tablecaption{Stellar Data.\\label{starsinfo_table}}\n\\tablewidth{0pt}\n\\tablehead{\n  \\multicolumn{3}{c}{{Identification}}\n  & \\colhead{$V$\\tablenotemark{a}}\n  & \\colhead{$B-V$\\tablenotemark{a}}\n  & \\colhead{$d$\\tablenotemark{a}}\n  & \\colhead{$T_{\\rm eff}$\\tablenotemark{b}}\n  & \\multirow{2}{*}{[M\/H]\\tablenotemark{b}}\n  & \\colhead{$\\log{g}$\\tablenotemark{b}}\n  & \\multirow{2}{*}{Ref.\\tablenotemark{c}} \\\\\n  \\colhead{HD}\n  & \\colhead{HR}\n  & \\colhead{HIP}\n  & \\colhead{(mag)}\n  & \\colhead{(mag)}\n  & \\colhead{(pc)}\n  & \\colhead{(K)}\n  & \n  & \\colhead{(cgs)}\n  & \n}\n\\startdata \n    3795 &     173 &    3185 &  6.14 &  0.72 &  28.6 &  5369 & $ -0.41$ &  4.16 &       1 \\\\\n    4614 &     219 &    3821 &  3.46 &  0.59 &   6.0 &  5941 & $ -0.17$ &  4.44 &       4 \\\\\n    6734 &  \\ldots &    5315 &  6.44 &  0.85 &  46.4 &  5067 & $ -0.28$ &  3.81 &       1 \\\\\n    9562 &     448 &    7276 &  5.75 &  0.64 &  29.7 &  5939 & $  0.19$ &  4.13 & 1, 2, 4 \\\\\n    9826 &     458 &    7513 &  4.10 &  0.54 &  13.5 &  6213 & $  0.12$ &  4.25 &       4 \\\\\n   14412 &     683 &   10798 &  6.33 &  0.72 &  12.7 &  5374 & $ -0.45$ &  4.69 &       1 \\\\\n   15335 &     720 &   11548 &  5.89 &  0.59 &  30.8 &  5891 & $ -0.20$ &  4.07 &       4 \\\\\n   16397 &  \\ldots &   12306 &  7.36 &  0.58 &  35.9 &  5788 & $ -0.35$ &  4.50 &       1 \\\\\n   22879 &  \\ldots &   17147 &  6.68 &  0.55 &  24.3 &  5688 & $ -0.76$ &  4.41 & 1, 2, 4 \\\\\n   23249 &    1136 &   17378 &  3.52 &  0.92 &   9.0 &  5095 & $  0.03$ &  3.98 &       1 \\\\\n   23439 &  \\ldots &   17666 &  7.67 &  0.80 &  24.5 &  5070 & $ -0.73$ &  4.71 &       3 \\\\\n   30649 &  \\ldots &   22596 &  6.94 &  0.59 &  29.9 &  5778 & $ -0.33$ &  4.44 &       4 \\\\\n   34411 &    1729 &   24813 &  4.69 &  0.63 &  12.6 &  5911 & $  0.09$ &  4.37 &       4 \\\\\n   43947 &  \\ldots &   30067 &  6.61 &  0.56 &  27.5 &  5933 & $ -0.28$ &  4.37 &       2 \\\\\n   45184 &    2318 &   30503 &  6.37 &  0.63 &  22.0 &  5810 & $  0.03$ &  4.37 &       1 \\\\\n   48938 &    2493 &   32322 &  6.43 &  0.55 &  26.6 &  5937 & $ -0.39$ &  4.31 &       4 \\\\\n   84737 &    3881 &   48113 &  5.08 &  0.62 &  18.4 &  5960 & $  0.14$ &  4.24 &       4 \\\\\n   86728 &    3951 &   49081 &  5.37 &  0.68 &  14.9 &  5700 & $  0.11$ &  4.29 &       4 \\\\\n  102365 &    4523 &   57443 &  4.89 &  0.66 &   9.2 &  5630 & $ -0.26$ &  4.57 &       2 \\\\\n  \\ldots &  \\ldots &   57450 &  9.91 &  0.58 &  73.5 &  5272 & $ -1.42$ &  4.30 &       3 \\\\\n  103095 &    4550 &   57939 &  6.42 &  0.75 &   9.2 &  4950 & $ -1.16$ &  4.65 &       3 \\\\\n  109358 &    4785 &   61317 &  4.24 &  0.59 &   8.4 &  5930 & $ -0.10$ &  4.44 &       4 \\\\\n  115617 &    5019 &   64924 &  4.74 &  0.71 &   8.5 &  5571 & $  0.09$ &  4.47 &       4 \\\\\n  131117 &    5542 &   72772 &  6.30 &  0.61 &  40.0 &  5973 & $  0.10$ &  4.06 &       2 \\\\\n  144585 &  \\ldots &   78955 &  6.32 &  0.66 &  28.9 &  5854 & $  0.25$ &  4.33 &       1 \\\\\n  156365 &  \\ldots &   84636 &  6.59 &  0.65 &  47.2 &  5856 & $  0.24$ &  4.09 &       1 \\\\\n  157214 &    6458 &   84862 &  5.38 &  0.62 &  14.4 &  5697 & $ -0.15$ &  4.50 &       4 \\\\\n  157347 &    6465 &   85042 &  6.28 &  0.68 &  19.5 &  5714 & $  0.03$ &  4.50 &       1 \\\\\n  166435 &  \\ldots &   88945 &  6.84 &  0.63 &  25.2 &  5843 & $  0.01$ &  4.44 &       1 \\\\\n  169830 &    6907 &   90485 &  5.90 &  0.52 &  36.3 &  6221 & $  0.08$ &  4.06 &       1 \\\\\n  172051 &    6998 &   91438 &  5.85 &  0.67 &  13.0 &  5564 & $ -0.24$ &  4.50 &       1 \\\\\n  176377 &  \\ldots &   93185 &  6.80 &  0.61 &  23.4 &  5788 & $ -0.23$ &  4.40 &       1 \\\\\n  179949 &    7291 &   94645 &  6.25 &  0.55 &  27.0 &  6168 & $  0.11$ &  4.34 &       1 \\\\\n  182572 &    7373 &   95447 &  5.17 &  0.76 &  15.1 &  5656 & $  0.36$ &  4.32 &       1 \\\\\n  190360 &    7670 &   98767 &  5.73 &  0.75 &  15.9 &  5552 & $  0.19$ &  4.38 &       1 \\\\\n  193901 &  \\ldots &  100568 &  8.65 &  0.55 &  43.7 &  5408 & $ -1.19$ &  4.14 &       3 \\\\\n  199960 &    8041 &  103682 &  6.21 &  0.64 &  26.5 &  5962 & $  0.24$ &  4.31 &       1 \\\\\n  210277 &  \\ldots &  109378 &  6.54 &  0.77 &  21.3 &  5555 & $  0.20$ &  4.49 &       1 \\\\\n  217107 &    8734 &  113421 &  6.17 &  0.74 &  19.7 &  5704 & $  0.27$ &  4.54 &       1 \\\\\n  222368 &    8969 &  116771 &  4.13 &  0.51 &  13.8 &  6204 & $ -0.08$ &  4.18 &       4 \\\\\n  224383 &  \\ldots &  118115 &  7.89 &  0.64 &  47.7 &  5754 & $ -0.06$ &  4.31 &       1 \\\\\n\\enddata\n\\tablenotetext{a}{$V$-magnitude, color index, and parallax-based distance \nfrom the {\\em Hipparcos} catalogue \\citep{hipparcos}.}\n\\tablenotetext{b}{Stellar parameters previously published in VF05.}\n\\tablenotetext{c}{Star included for comparison to the following \nworks:~1---\\citealt{bensby_2005};\n2---\\citealt{delpeloso_2005a};\n3---\\citealt{simmerer_2004};\n4---\\citealt{woolf_1995}}\n\\end{deluxetable*}\n\n\n\\subsection{Co-adding Data}\n\\label{data}\nThe nature of the radial velocity planet search dictates that most stars \nwill have multiple, and in some cases, dozens of, observations.  \nTo take advantage of the multiple exposures, we carefully \nco-add the reduced echelle spectra where possible. \n\nThe co-adding procedure is as follows:\\ a 2000-pixel region \n(approximately half the full order width) near \nthe middle of each order is cross-correlated, order by order, \nwith one observation arbitrarily designated as standard. \nThe pixel shifts are then examined and a linear trend as a function of \norder number is fit to the pixel shifts.  \nFor any order whose pixel shift falls more than 0.4 pixels from the linear \ntrend, the value predicted by the linear trend is substituted.  \nThis method corrects outlying pixel shift values, which often proved to\nbe one of a handful of problematic orders where the echelle blaze\nfunction shape created a false cross-correlation peak.\n\nEach spectral order is adjusted by its appropriate fractional pixel\nshift before all the newly aligned spectra are added together.\nIn order to accurately add spectra that have been shifted by \nnon-integer pixel amounts, we use a linear interpolation between \npixel values to artificially (and temporarily) increase \nthe sampling density by a factor of 20.\nAfter the co-adding, the resultant high-S\/N \nmulti-observation spectrum is sampled back down to its \noriginal spacing. \n\nThe number of observations used per star is recorded in the $N_{obs}$ \ncolumn of Table \\ref{starsvals_table}. \nThe co-adding proved particularly beneficial when fitting the relatively \nweak Eu~{\\sc ii}\\ line at 6645\\,\\AA.  \nTwo sample stellar spectra, from a star with 1 observation\nand from a star with 17 observations, are given in Figure \n\\ref{coadding_works} as evidence of the S\/N advantages \nof this procedure.\n\n\\begin{deluxetable}{rrrrrrr}\n\\tablecaption{Stellar abundance values.\\label{starsvals_table}}\n\\normalsize\n\\tablehead{\n  \\multirow{2}{*}{Name\\tablenotemark{a}} \n  & \\multirow{2}{*}{$N_{obs}$}\n  & \\colhead{4129 \\AA}\n  & \\colhead{4205 \\AA}\n  & \\colhead{6645 \\AA}\n  & \\colhead{Weighted} \\\\\n & \n  & \\colhead{[Eu\/H]\\tablenotemark{b}}\n  & \\colhead{[Eu\/H]\\tablenotemark{b}}\n  & \\colhead{[Eu\/H]\\tablenotemark{b}}\n  & \\colhead{Average\\tablenotemark{c}}\n}\n \n\\startdata\n        3795 &    2 &  $0.07$ &  $0.06$ &      $0.13$ &$0.07 \\pm    0.03$ \\\\\n        4614 &    2 & $-0.17$ & $-0.19$ &     $-0.24$ &$-0.18 \\pm   0.02$ \\\\\n        6734 &    6 & $-0.02$ & $-0.06$ &      $0.01$ &$-0.03 \\pm   0.03$ \\\\\n        9562 &    3 &  $0.11$ &  $0.14$ &      $0.08$ &$0.12 \\pm    0.02$ \\\\\n        9826 &    1 &  $0.10$ &  $0.13$ &      $0.09$ &$0.12 \\pm    0.02$ \\\\\n       14412 &   70 & $-0.28$ & $-0.15$ &     $-0.16$ &$-0.24 \\pm   0.06$ \\\\\n       15335 &    2 & $-0.13$ & $-0.10$ &     $-0.08$ &$-0.12 \\pm   0.03$ \\\\\n       16397 &    1 & $-0.21$ & $-0.20$ &     $-0.22$ &$-0.21 \\pm   0.02$ \\\\\n       22879 &   27 & $-0.55$ & $-0.50$ &     $-0.57$ &$-0.54 \\pm   0.03$ \\\\\n       23249 &   34 &  $0.05$ &  $0.26$ &      $0.24$ &$0.17 \\pm    0.11$ \\\\\n       23439 &   24 & $-0.43$ & $-0.36$ &     $-0.47$ &$-0.38 \\pm   0.03$ \\\\\n       30649 &    3 & $-0.13$ & $-0.11$ &     $-0.13$ &$-0.12 \\pm   0.02$ \\\\\n       34411 &   70 &  $0.12$ &  $0.13$ &      $0.11$ &$0.13 \\pm    0.02$ \\\\\n       43947 &    6 & $-0.21$ & $-0.19$ &     $-0.21$ &$-0.20 \\pm   0.02$ \\\\\n       45184 &   95 &  $0.01$ & $-0.03$ &      $0.02$ &$0.00 \\pm    0.02$ \\\\\n       48938 &    2 & $-0.30$ & $-0.28$ &     $-0.41$ &$-0.30 \\pm   0.03$ \\\\\n       84737 &    7 &  $0.14$ &  $0.14$ &      $0.14$ &$0.14 \\pm    0.02$ \\\\\n       86728 &   46 &  $0.06$ &  $0.11$ &      $0.10$ &$0.09 \\pm    0.03$ \\\\\n      102365 &   12 & $-0.13$ & $-0.08$ &     $-0.14$ &$-0.11 \\pm   0.03$ \\\\\n    HIP57450 &    1 & $-1.10$ & $-1.03$ &$\\leq -0.95$ &$-1.06 \\pm   0.04$ \\\\\n      103095 &    9 &  \\ldots & $-0.37$ &     $-0.38$ &$-0.37 \\pm   0.02$ \\\\\n      109358 &   47 & $-0.12$ & $-0.16$ &     $-0.12$ &$-0.14 \\pm   0.03$ \\\\\n      115617 &  165 &  $0.05$ &  $0.02$ &      $0.09$ &$0.04 \\pm    0.03$ \\\\\n      131117 &    2 &  $0.10$ &  $0.09$ &      $0.12$ &$0.10 \\pm    0.02$ \\\\\n      144585 &   17 &  $0.22$ &  $0.25$ &      $0.22$ &$0.23 \\pm    0.03$ \\\\\n      156365 &    1 &  $0.11$ &  $0.20$ &      $0.14$ &$0.13 \\pm    0.04$ \\\\\n      157214 &   24 &  $0.05$ &  $0.06$ &      $0.07$ &$0.06 \\pm    0.02$ \\\\\n      157347 &   77 &  $0.09$ &  $0.12$ &      $0.13$ &$0.10 \\pm    0.02$ \\\\\n      166435 &    1 & $-0.05$ & $-0.17$ &      $0.04$ &$-0.07 \\pm   0.06$ \\\\\n      169830 &    8 &  $0.02$ &  $0.05$ &      $0.05$ &$0.03 \\pm    0.02$ \\\\\n      172051 &   37 & $-0.19$ & $-0.15$ &     $-0.12$ &$-0.17 \\pm   0.03$ \\\\\n      176377 &   56 & $-0.20$ & $-0.21$ &     $-0.23$ &$-0.20 \\pm   0.02$ \\\\\n      179949 &    1 &  $0.05$ &  $0.02$ &      $0.05$ &$0.04 \\pm    0.03$ \\\\\n      182572 &   56 &  $0.25$ &  $0.41$ &      $0.36$ &$0.31 \\pm    0.08$ \\\\\n      190360 &   73 &  $0.17$ &  $0.29$ &      $0.30$ &$0.24 \\pm    0.07$ \\\\\n      193901 &    2 & $-0.92$ & $-0.91$ &     $-0.98$ &$-0.91 \\pm   0.02$ \\\\\n      199960 &    3 &  $0.14$ &  $0.19$ &      $0.19$ &$0.16 \\pm    0.03$ \\\\\n      210277 &   74 &  $0.20$ &  $0.34$ &      $0.32$ &$0.27 \\pm    0.07$ \\\\\n      217107 &   37 &  $0.20$ &  $0.43$ &      $0.33$ &$0.29 \\pm    0.11$ \\\\\n      222368 &    2 &  $0.08$ &  $0.09$ &      $0.08$ &$0.08 \\pm    0.02$ \\\\\n      224383 &    2 &  $0.00$ & $-0.01$ &      $0.04$ &$0.00 \\pm    0.02$ \\\\\n       Vesta &    3 & $-0.04$ &  $0.00$ &      $0.00$ &$-0.03 \\pm   0.03$ \\\\\n\\enddata\n\\tablenotetext{a}{All names are HD numbers unless otherwise indicated.}\n\\tablenotetext{b}{For a given star, [Eu\/H] = $\\log{\\epsilon\\left(\\textrm{Eu}\\right)} - \\log{\\epsilon\\left(\\textrm{Eu}\\right)_{\\odot}}$, where \\mbox{$\\log{\\epsilon\\left(X\\right)} = \\log_{10}{\\left(N_{X}\/N_{H}\\right)}$.}}\n\\tablenotetext{c}{The weight of each line in the average is based on 50 Monte Carlo trials in each Eu~{\\sc ii}~line. We have adopted an error floor of $0.02\\unit{dex}$, added in quadrature to the errors determined by our Monte Carlo procedure. See \\S\\ref{errors} for a more complete description of the weighted average and associated uncertainty.}\n\\end{deluxetable}\n\n\\begin{figure} \n\\includegraphics[width=\\columnwidth]{f1.eps}\n\\figcaption[Co-added Stellar Spectra.]{The three Eu~{\\sc ii}\\ lines considered in this paper, \n  plotted for two different stars to demonstrate the advantage to be gained\n  from co-adding multiple observations of the same star. \n  Note that the rightmost plots have a different ordinate axis scaling than \n  the other four panels.\n  The top three panels, showing HD\\,156365, include 1 spectrum, while the bottom \n  three panels, showing HD\\,144585, include 17. \n  The two stars have approximately the same metallicity and\n  effective temperature. \n  The advantage from co-adding is most profound\n  in the weak 6645-\\AA\\ line.\n  \\label{coadding_works}}\n\\end{figure}\n\n\\section{Abundance Measurements}\n\\label{abund}\n\nWe use the SME suite of routines for our spectral synthesis, both for fine-tuning line lists based on the solar spectrum (\\S\\,\\ref{linelists}) and for measuring europium in each star (\\S\\,\\ref{stellar_abund}).\nSME is an LTE spectral synthesis code based on the \\citealt{kurucz_1992} grid of stellar atmosphere models. \nIn brief, to produce a synthetic spectrum, SME interpolates between the atmosphere models, calculates the continuous opacity, computes the radiative transfer, and then applies line broadening, which is governed by macroturbulence, stellar rotation, and instrumental profile. \nConsult \\citealt{valenti_1996} and VF05 for a more in-depth description of SME's inner workings.\n\nThroughout this study we use SME only to compute synthetic spectra.\nAll fitting is done in specialized routines of our own design, external to SME.\n\nIn this section we outline our technique for measuring europium abundances in the set of 41\\ stars included in this work. \nIn broad strokes, we first determine the atomic parameters of our spectral lines by fitting the solar spectrum (\\S\\,\\ref{linelists}). \nThen, we use that line list to measure the europium abundance in three transitions (4129\\,\\AA, 4205\\,\\AA, and 6645\\,\\AA); a weighted average of the three transitions determines a star's final europium value (\\S\\,\\ref{stellar_abund}). Finally, we estimate our uncertainties by adding artificial noise to our data in a series of Monte Carlo trials (\\S\\,\\ref{errors}).\nWe also here include notes on individual lines (\\S\\,\\ref{individual_lines}).\n\n\\subsection{Line Lists}\n\\label{linelists}\nWe use relatively broad spectral segments in our europium \nanalysis.\nThe regions centered on the Eu~{\\sc ii}\\ 4129\\,\\AA\\ and \n6645\\,\\AA\\ lines are 5\\,\\AA\\ wide, and the region centered\non the Eu~{\\sc ii}\\ 4205\\,\\AA\\ line is 8\\,\\AA\\ wide.\nWe find it necessary to use such broad spectral segments\nin order to fit a robust and consistent continuum in \nthe crowded blue regions.\n\nLine lists are initially drawn from the Vienna Astrophysics Line \nDatabase (VALD; \\citealt{piskunov_1995,kupka_1999}). \nThe VALD line lists, in the regions surrounding all three \nEu~{\\sc ii}\\ transitions, make extensive use of Kurucz line lists. \n\nWe apply the original VALD line list to an observed solar spectrum in\norder to determine the list's completeness and to adjust line parameters\nas needed.\nIn the blue, we use the disk-center solar spectrum from \n\\citealt{wallace_1998} with the following global parameters: \n$T_{\\rm eff}$~$ = 5770\\unit{K}$, \n$\\log{g} = 4.44$, \n[M\/H]~$ = 0$, \nmicroturbulence $v_{mic} = 1.0\\unit{km s$^{-1}$}$, \nmacroturbulence $v_{mac} = 3.6\\unit{km s$^{-1}$}$,\nrotational velocity $v\\sin{i} = 0\\unit{km s$^{-1}$}$, \nand radial velocity $v_{rad} = 0\\unit{km s$^{-1}$}$. \nThese are the same solar parameters adopted in VF05.\nIn the red, we find the \\citealt{wallace_1998} solar atlas \nto have insufficient S\/N to accurately determine\nthe atomic parameters.\nAt 6645\\,\\AA, therefore, we instead compare our \nline list to the disk-integrated NSO solar spectrum \n\\citep{kurucz_1984}, \nadjusting $v\\sin{i}$ to $1.6\\unit{km s$^{-1}$}$ because\nthe full solar disk has more substantial rotational \nbroadening.\n\nWe find that adjustments to the oscillator strength \n($\\log{\\textrm{\\em gf}}$) and van der Waals broadening ($\\Gamma_{6}$) parameters \nare required for the strongest lines in a given \nwavelength segment, even far from the Eu~{\\sc ii}\\ line of \ninterest.\nFor example, the Fe~{\\sc i}\\ line at 4202\\,\\AA\\ has an equivalent \nwidth $W=326\\unit{m\\AA}$ in the Sun \nand significantly affects \nthe continuum of the 4205-\\AA\\ region.\nThe $\\log{\\textrm{\\em gf}}$\\ parameter controls the line depth while the\n$\\Gamma_{6}$\\ parameter controls the line shape, so in general\nthe two parameters are orthogonal.\n\nWe used the Kurucz $\\log{\\textrm{\\em gf}}$\\ values provided \nby VALD where possible, but adjustments\nwere necessary where line depths were poorly fit.\nFor $\\Gamma_{6}$, VALD returns the \\citealt{barklem_2000} \nparameters for beryllium through barium ($Z$ of 4--56),\nbut has no $\\Gamma_{6}$\\ values above $Z=56$.\nWe therefore find it necessary to fit $\\Gamma_{6}$\\ in \nspecies heavier than barium and in deep features\nnot fit well by VALD values.\nWe take particular care to determine the appropriate \n$\\log{\\textrm{\\em gf}}$\\ and $\\Gamma_{6}$\\ parameters for all lines \nadjacent to the Eu~{\\sc ii}\\ line of interest.  \nWe find the best value for these parameters with the SME \nsynthesizer by performing a $\\chi^2$\\ minimization against \nthe solar spectrum on each parameter.\n\nWhere the VALD line list is insufficient, we add line data \nfrom \\citealt{moore_1966}, \nNIST (whose lists are based on a variety of sources) and \nC.~Sneden (private communication).\nWe also add CH and CN molecular lines based on values\nobtained from the Kurucz molecular line list web \nsite.\\footnote{\\tt http:\/\/kurucz.harvard.edu\/LINELISTS\/LINESMOL\/}\nIn the 4129-\\AA\\ region, we find it necessary to add \nthree artificial iron lines in order to match the solar spectrum.\nWe follow here the precedent of \\citealt{delpeloso_2005a},\nthough we find our fit requires the lines to have \nslightly different wavelengths and $\\log{\\textrm{\\em gf}}$\\ values.\nThe complete line list for Eu~{\\sc ii}\\ at 4129\\,\\AA\\ appears in\nTable \\ref{4129_nso_table}, \nfor 4205\\,\\AA\\ in Table \\ref{4205_nso_table}, and\nfor 6645\\,\\AA\\ in Table \\ref{6645_nso_table}.\nThe corresponding plots of the regions in these tables appear in\nFigures \\ref{eu4129_nso}, \\ref{eu4205_nso}, and \\ref{eu6645_nso}.\n\n\\begin{figure} \n\\includegraphics[width=\\columnwidth]{f2.eps}\n\\figcaption[Eu 4129\\,\\AA\\ in NSO.]{The 4129-\\AA\\ Eu~{\\sc ii}\\ line in the solar spectrum. \n  Individual lines are annotated and are listed in Table \\ref{4129_nso_table}.\n  The hyperfine components (see \\S\\,\\ref{linelists}) appear in the inset plot, \n  which is aligned with the wavelength scale of the main plot.\n  The relative strengths of the 32 hyperfine components \\citep{ivans_2006}\n  are plotted on a linear scale in the inset; the top half of the inset \n  contains the components from the $^{151}$Eu isotope while the\n  $^{153}$Eu isotope components appear on the bottom.\n  The gray box indicates the portion of the spectrum used to calculate \n  $\\chi^2$\\ during the abundance fitting step (see \\S\\,\\ref{stellar_abund}).\n  The cross-hatched region indicates a portion of the spectrum \n  used to fit a continuum. \n  This plot represents a subset of the spectral region used in our \n  analysis; the full region is 5\\,\\AA\\ wide and contains three additional \n  continuum fitting regions.\n  \\label{eu4129_nso}}\n\\end{figure}\n\n\\begin{figure} \n\\includegraphics[width=\\columnwidth]{f3.eps}\n\\figcaption[Eu 4205\\,\\AA\\ in solar atlas.]{The 4205-\\AA\\ Eu~{\\sc ii}\\ line in the solar spectrum. \n  Individual lines are annotated and are listed in Table \\ref{4205_nso_table}. \n  The hyperfine components (see \\S\\,\\ref{linelists}) appear in the inset plot, \n  which is aligned with the wavelength scale of the main plot.\n  The relative strengths of the 30 hyperfine components \\citep{ivans_2006}\n  are plotted on a linear scale in the inset; the top half of the inset \n  contains the components from the $^{151}$Eu isotope while the\n  $^{153}$Eu isotope components appear on the bottom.\n  The gray box indicates the portion of the spectrum used to calculate \n  $\\chi^2$\\ during the abundance fitting step (see \\S\\,\\ref{stellar_abund}).  \n  The cross-hatched region indicates a portion of the spectrum \n  used to fit a continuum. This plot represents a subset of the \n  spectral region used in our analysis; the full region is 8\\,\\AA\\ wide \n  and contains six additional continuum fitting regions.\n  \\label{eu4205_nso}}\n\\end{figure}\n \n\\begin{deluxetable*}{ccccccc}[!htp]\n\\tablecaption{Line list for the region near Eu~{\\sc ii}\\ at 4129\\,\\AA.\\label{4129_nso_table}}\n\\tablehead{\n  \\colhead{$\\lambda$} \n  & \\multirow{2}{*}{Element}\n  & \\colhead{Lower Level}\n  & \\multicolumn{2}{c}{{$\\log{\\textrm{\\em gf}}$}} \n  & \\multicolumn{2}{c}{{$\\Gamma_{6}$}}\\\\\n  \\colhead{(\\AA)}\n  & \n  & \\colhead{(eV)}\n  & \\colhead{solar fit} \n  & \\colhead{VALD\\tablenotemark{a}}\n  & \\colhead{solar fit}\n  & \\colhead{VALD\\tablenotemark{b}}\n}\n\n\\startdata\n   4129.147 &  Pr {\\sc ii} &    1.039 & $-0.100$ & $-0.100$ & $-7.454$ &   \\ldots \\\\\n   4129.159 &   Cr {\\sc i} &    3.013 & $-1.948$ & $-1.948$ & $-6.964$ & $-7.362$ \\\\\n   4129.159 &  Ti {\\sc ii} &    1.893 & $-2.300$ & $-1.730$ & $-6.900$ & $-7.908$ \\\\\n   4129.166 &   Ti {\\sc i} &    2.318 & $-0.200$ & $-0.231$ & $-6.900$ & $-7.572$ \\\\\n   4129.174 &  Ce {\\sc ii} &    0.740 & $-3.000$ & $-0.901$ & $-7.493$ &   \\ldots \\\\\n   4129.182\\tablenotemark{c} &\n                Cr {\\sc i} &    2.914 & $-0.100$ &   \\ldots & $-8.300$ &   \\ldots \\\\\n   4129.220 &   Fe {\\sc i} &    3.417 & $-3.500$ & $-2.030$ & $-6.857$ & $-7.255$ \\\\\n   4129.220 &  Sm {\\sc ii} &    0.248 & $-1.123$ & $-1.123$ & $-7.536$ &   \\ldots \\\\\n   4129.425 &  Dy {\\sc ii} &    0.538 & $-0.522$ & $-0.522$ & $-7.554$ &   \\ldots \\\\\n   4129.426 &   Nb {\\sc i} &    0.086 & $-0.780$ & $-0.780$ & $-7.462$ &   \\ldots \\\\\n   4129.458 &   Fe {\\sc i} &    3.396 & $-1.950$ & $-1.970$ & $-6.863$ & $-7.206$ \\\\\n   4129.522\\tablenotemark{d} &\n                Fe {\\sc i} &    3.140 & $-3.497$ &   \\ldots & $-6.873$ &   \\ldots \\\\\n   4129.643 &   Ti {\\sc i} &    2.239 & $-1.987$ & $-1.987$ & $-7.529$ & $-7.529$ \\\\\n   4129.705 &  Eu {\\sc ii} &    0.000 & $+0.260$ & $+0.173$ & $-7.174$ &   \\ldots \\\\\n   4129.817 &   Co {\\sc i} &    3.812 & $-1.808$ & $-1.808$ & $-6.099$ & $-7.782$ \\\\\n   4129.837 &  Nd {\\sc ii} &    2.024 & $-0.543$ & $-0.543$ & $-6.237$ &   \\ldots \\\\\n   4129.959\\tablenotemark{d} &\n                Fe {\\sc i} &    2.670 & $-3.139$ &   \\ldots & $-7.322$ &   \\ldots \\\\\n   4130.035 &   Fe {\\sc i} &    1.557 & $-3.900$ & $-4.345$ & $-7.885$ & $-7.826$ \\\\\n   4130.036 &   Fe {\\sc i} &    3.111 & $-2.350$ & $-2.636$ & $-8.026$ & $-7.857$ \\\\\n   4130.073 &   Cr {\\sc i} &    2.914 & $-1.971$ & $-1.971$ & $-6.929$ & $-7.349$ \\\\\n   4130.122\\tablenotemark{e} &\n                V  {\\sc i} &    1.218 & $-1.000$ & $-3.142$ & $-7.060$ & $-7.800$ \\\\\n   4130.233 &   Mn {\\sc i} &    2.920 & $-2.400$ & $-3.309$ & $-7.900$ & $-7.784$ \\\\\n   4130.315 &   V  {\\sc i} &    2.269 & $-0.300$ & $-0.607$ & $-7.187$ & $-7.585$ \\\\\n   4130.364 &  Gd {\\sc ii} &    0.731 & $+0.177$ & $-0.090$ & $-6.608$ &   \\ldots \\\\\n   4130.452 &   Cr {\\sc i} &    2.914 & $-1.099$ & $-2.751$ & $-6.805$ & $-7.348$ \\\\\n\\enddata\n\\tablenotetext{a}{All line data except \\gamsix\\ from Kurucz databases via VALD (unless otherwise noted).}\n\\tablenotetext{b}{Van der Waals parameters where available from \\citealt{barklem_2000} via VALD (unless otherwise noted).}\n\\tablenotetext{c}{Identification from \\citealt{moore_1966}, \\loggf\\ and \\gamsix\\ from \\chisq\\ minimization.}\n\\tablenotetext{d}{Artificial iron lines included after \\citealt{delpeloso_2005a}.}\n\\tablenotetext{e}{Identification and parameters from Kurucz line lists hosted by the University of Hannover:\\newline \\texttt{http:\/\/www.pmp.uni-hannover.de\/cgi-bin\/ssi\/test\/kurucz\/sekur.html}.}\n\\end{deluxetable*}\n\n\\begin{figure} \n\\includegraphics[width=\\columnwidth]{f4.eps}\n\\figcaption[Eu 6645\\,\\AA\\ in solar atlas.]{The 6645-\\AA\\ Eu~{\\sc ii}\\ line in the solar \n  spectrum. Note that the ordinate axis is scaled differently than in Figures \n  \\ref{eu4129_nso} and \\ref{eu4205_nso}.\n  Individual lines are annotated and are listed in Table \\ref{6645_nso_table}.\n  The hyperfine components (see \\S\\,\\ref{linelists}) appear in the inset plot, \n  which is aligned with the wavelength scale of the main plot.\n  The relative strengths of the 30 hyperfine components \\citep{ivans_2006}\n  are plotted on a linear scale in the inset; the top half of the inset \n  contains the components from the $^{151}$Eu isotope while the\n  $^{153}$Eu isotope components appear on the bottom.\n  The gray box indicates the portion of the spectrum used to calculate \n  $\\chi^2$\\ during the abundance fitting step (see \\S\\,\\ref{stellar_abund}).  \n  The two cross-hatched regions indicate the portion of the spectrum used \n  to fit a continuum.\n  This plot represents a subset of the spectral region used in our analysis; \n  the full region is 5\\,\\AA\\ wide and contains four additional continuum fitting regions.\n  \\label{eu6645_nso}}\n\\end{figure}\n\nHyperfine splitting is the dominant broadening mechanism for the\neuropium spectral lines. \nThe interaction between the nuclear spin and the atom's \nangular momentum vector causes energy level splitting\nin atoms with odd atomic numbers (europium is $Z=63$). \nThe effect is particularly pronounced in rare earth \nelements.\nThe 4129-\\AA\\ and 4205-\\AA\\ lines, for example, have FWHMs \nof 1.5\\,\\AA, due largely to hyperfine structure\n(but not isotope splitting---see insets of Figures \n\\ref{eu4129_nso}--\\ref{eu6645_nso}).\nSince the relative strengths of the hyperfine\ncomponents are constant without regard to\ntemperature, pressure, or magnetic field \\citep{abt_1952},\nthe components measured in laboratory settings\ncan be applied to stellar spectra.\n\nLike other spectral fitting packages,\nSME has no built-in treatment of hyperfine \nstructure. \nWe therefore convert a single europium line into \nits constituent hyperfine components and include them \nas separate entries in the star's line list.  \nThe relative strengths and wavelength offsets of the \nhyperfine components come from \\citealt{ivans_2006}, \nwhich bases the values on an FTS laboratory analysis. \nFollowing the procedure of \\citealt{ivans_2006}, \nwe assume a solar system \ncomposition for the relative abundance of the two \neuropium isotopes ($^{151}$Eu at 47.8\\% and \n$^{153}$Eu at 52.2\\%, from \\citealt{rosman_1998}).\nWe divide the Eu~{\\sc ii}\\ $\\log{\\textrm{\\em gf}}$\\ values listed in \nTables \\ref{4129_nso_table}, \\ref{4205_nso_table}, \nand \\ref{6645_nso_table} amongst the \nhyperfine components according to their relative strengths.  \nAll other attributes of the Eu~{\\sc ii}\\ lines remain the same in \nthe creation of the hyperfine lines.  \nThe relative strengths of the hyperfine components are displayed \nin the inset plots in the solar spectrum Figures\n\\ref{eu4129_nso}, \\ref{eu4205_nso}, and \\ref{eu6645_nso}.\n\n\\begin{deluxetable*}{ccccccc}\n\\tablecolumns{7}\n\\tablecaption{Line list for the region near Eu~{\\sc ii}\\ at 4205\\,\\AA.\\label{4205_nso_table}}\n\\tablewidth{0pt}\n\\scriptsize\n\\tablehead{\n  \\colhead{$\\lambda$} \n  & \\multirow{2}{*}{Element}\n  & \\colhead{Lower Level}\n  & \\multicolumn{2}{c}{$\\log{\\textrm{\\em gf}}$}\n  & \\multicolumn{2}{c}{$\\Gamma_{6}$} \\\\\n  \\colhead{(\\AA)}\n  & \n  & \\colhead{(eV)}\n  & \\colhead{solar fit} \n  & \\colhead{VALD\\tablenotemark{a}} \n  & \\colhead{solar fit}\n  & \\colhead{VALD\\tablenotemark{b}} \n}\n\n\\startdata\n   4204.695 &  Y  {\\sc ii} &    0.000 &  $-1.800$ &  $-1.760$ &  $-7.700$ &  \\ldots \\\\\n   4204.717 &  Ce {\\sc ii} &    0.792 &  $-0.963$ &  $-0.963$ &  $-7.000$ &  \\ldots \\\\\n   4204.759\\tablenotemark{c} &\n                        CH &    0.519 &  $-1.140$ &  $-1.158$ &  $-8.900$ &  \\ldots \\\\\n   4204.771\\tablenotemark{c} &\n                        CH &    0.520 &  $-1.900$ &  $-1.135$ &  $-8.500$ &  \\ldots \\\\\n   4204.801 &  Sm {\\sc ii} &    0.378 &  $-1.771$ &  $-1.771$ &  $-6.738$ &  \\ldots \\\\\n   4204.831\\tablenotemark{c} &\n                        CH &    0.520 &  $-3.360$ &  $-3.360$ &  $-7.699$ &  \\ldots \\\\\n   4204.858 &  Gd {\\sc ii} &    0.522 &  $-0.668$ &  $-0.668$ &  $-6.787$ &  \\ldots \\\\\n   4204.990 &   Cr {\\sc i} &    4.616 &  $-1.457$ &  $-1.457$ &  $-6.658$ &  $-7.852$ \\\\\n   4205.000 &   Fe {\\sc i} &    4.220 &  $-1.900$ &  $-2.150$ &  $-7.000$ &  $-7.548$ \\\\\n   4205.038 &  V  {\\sc ii} &    1.686 &  $-1.850$ &  $-1.875$ &  $-6.800$ &  $-7.913$ \\\\\n   4205.042 &  Eu {\\sc ii} &    0.000 &  $+0.250$ &  $+0.120$ &  $-6.800$ &  \\ldots \\\\\n   4205.084 &  V  {\\sc ii} &    2.036 &  $-1.100$ &  $-1.300$ &  $-6.900$ &  $-7.956$ \\\\\n   4205.098 &   Fe {\\sc i} &    2.559 &  $-4.900$ &  $-4.900$ &  $-6.671$ &  $-7.865$ \\\\\n   4205.107 &   Cr {\\sc i} &    4.532 &  $-1.160$ &  $-1.160$ &  $-6.582$ &  $-7.776$ \\\\\n   4205.163 &  Ce {\\sc ii} &    1.212 &  $-0.653$ &  $-0.653$ &  $-6.672$ &  \\ldots \\\\\n   4205.253 &  Nd {\\sc ii} &    0.680 &  $-0.992$ &  $-0.992$ &  $-6.699$ &  \\ldots \\\\\n   4205.303 &   Nb {\\sc i} &    0.049 &  $-0.850$ &  $-0.850$ &  $-6.677$ &  \\ldots \\\\\n   4205.381 &  Mn {\\sc ii} &    1.809 &  $-3.300$ &  $-3.376$ &  $-6.800$ &  $-8.001$ \\\\\n   4205.402\\tablenotemark{c} &\n                        CH &    0.488 &  $-2.300$ &  $-3.960$ &  $-8.000$ &  \\ldots \\\\\n   4205.427\\tablenotemark{c} &\n                        CH &    1.019 &  $-2.300$ &  $-1.130$ &  $-8.000$ &  \\ldots \\\\\n   4205.491\\tablenotemark{c} &\n                        CH &    1.019 &  $-3.463$ &  $-3.463$ &  $-8.000$ &  \\ldots \\\\\n   4205.533\\tablenotemark{c} &\n                        CH &    1.019 &  $-1.800$ &  $-1.149$ &  $-8.000$ &  \\ldots \\\\\n   4205.538 &   Fe {\\sc i} &    3.417 &  $-1.100$ &  $-1.435$ &  $-7.800$ &  $-7.224$ \\\\\n\\enddata\n\\tablenotetext{a}{All line data except \\gamsix\\ from Kurucz databases via VALD (unless otherwise noted).}\n\\tablenotetext{b}{Van der Waals parameters where available from \\citealt{barklem_2000} via VALD (unless otherwise noted).}\n\\tablenotetext{c}{All molecular line data (except \\gamsix, from \\chisq\\ minimization) from Kurucz web site:\\ \\texttt{http:\/\/kurucz.harvard.edu\/LINELISTS\/LINESMOL\/}.}\n\\end{deluxetable*}\n\n\\begin{deluxetable*}{ccccccc}\n\\tablecolumns{7}\n\\tablecaption{Line list for the region near Eu~{\\sc ii}\\ at 6645\\,\\AA.\\label{6645_nso_table}}\n\\tablewidth{0pt}\n\\scriptsize\n\\tablehead{\n  \\colhead{$\\lambda$} \n  & \\multirow{2}{*}{Element}\n  & \\colhead{Lower Level}\n  & \\multicolumn{2}{c}{$\\log{\\textrm{\\em gf}}$}\n  & \\multicolumn{2}{c}{$\\Gamma_{6}$} \\\\\n  \\colhead{(\\AA)}\n  & \n  & \\colhead{(eV)}\n  & \\colhead{solar fit} \n  & \\colhead{VALD\\tablenotemark{a}} \n  & \\colhead{solar fit}\n  & \\colhead{VALD\\tablenotemark{b}} \n}\n\n\\startdata\n   6644.320\\tablenotemark{c} &\n                         CN &    0.805 & $-1.456$ & $-2.258$ &  $-7.695$ &  \\ldots \\\\\n   6644.415\\tablenotemark{d} &\n                 La {\\sc i} &    0.131 & $-1.330$ & $-2.070$ &  $-8.000$ &  \\ldots \\\\\n   6645.111 &  Eu {\\sc ii} &    1.380 & $+0.219$ & $+0.205$ &  $-7.218$ &  \\ldots \\\\\n   6645.210 &   Si {\\sc i} &    6.083 & $-2.510$ & $-2.120$ &  $-7.118$ &  \\ldots \\\\\n   6645.372 &   Fe {\\sc i} &    4.386 & $-2.759$ & $-3.536$ &  $-6.780$ &  $-7.808$ \\\\\n\\enddata\n\\tablenotetext{a}{All line data except \\gamsix\\ from Kurucz databases via VALD (unless otherwise noted).}\n\\tablenotetext{b}{Van der Waals parameters where available from \\citealt{barklem_2000} via VALD (unless otherwise noted).}\n\\tablenotetext{c}{All molecular line data (except \\gamsix, from \\chisq\\ minimization) from Kurucz web site:\\ \\texttt{http:\/\/kurucz.harvard.edu\/LINELISTS\/LINESMOL\/}.}\n\\tablenotetext{d}{Identification and parameters from Kurucz line lists hosted by the University of Hannover:\\newline \\texttt{http:\/\/www.pmp.uni-hannover.de\/cgi-bin\/ssi\/test\/kurucz\/sekur.html}.}\n\\end{deluxetable*}\n\n\\subsection{Europium Abundances}\n\\label{stellar_abund}\nIn order to measure the europium abundances in our selected stars, \nwe begin with the co-added spectra described in \\S\\,\\ref{data}. \nWe do a preliminary continuum fit using the SME routines,\nwhich follow the VF05 procedure:\\ deep features are filled \nin with a median value from neighboring spectral orders, \nthen a sixth-order polynomial is fit to the region of interest.\nThe built-in procedure creates a flat, normalized\ncontinuum, though we find it necessary to fine-tune the \ncontinuum normalization in the course of our europium \nfitting.\n\nFrom the VF05 SPOCS catalog we take \n$T_{\\rm eff}$, $\\log{g}$, [M\/H], $v\\sin{i}$, $v_{mac}$, and $v_{mic}$\n(fixed at 0.85\\unit{km s$^{-1}$}) \nfor each star we consider. \nIn general, the global parameters from VF05 agree very well with\nthe values adopted in the studies we compare to here. \n(Most of the literature values fall within the 2-$\\sigma$ errors\nquoted in VF05.) The VF05 catalog is one of the largest and most \nreliable sources of stellar properties determined to date; \n\\citealt{haywood_2006}, for example, finds the VF05 $T_{\\rm eff}$\\ and [M\/H]\\\nto be in good agreement with other reliable measurements.\n\nFor most element abundances we use a scaled solar\nsystem composition, shifting the \n\\citealt{grevesse_1998} solar abundances by \nthe star's [M\/H].\nThe exceptions to this rule are \nsodium, silicon, titanium, iron, and nickel,\nwhich VF05 measured individually. \nThose $\\alpha$ and iron-peak elements are\ntherefore treated independently of the na\\\"ive scaled-solar\nadjustment.\nIt is possible that a more explicit treatment of iron-peak\nand $\\alpha$ elements would improve our europium measurement\naccuracy. \nFor the present study, however, we deem individual \nabundance analysis (apart from europium) unnecessary.\nWe hold fixed the abundances of all elements other than \neuropium in the subsequent analysis.\n\nWe fit for the europium abundance by iterating three \n$\\chi^2$-minimization routines that solve for the \nwavelength alignment, spectrum continuum, and \neuropium abundance.\nA summary of each routine follows:\n\\begin{enumerate}\n\\item Wavelength. \nThe pixel scale is pre-determined from\nthe thorium lamp calibration taken each night.\nA first estimate of the rest frame wavelengths comes from\na cross-correlation of the full spectral segment with the \nsolar spectrum, a built-in functionality of SME.\nWe then use a 2-\\AA\\ region immediately surrounding \nthe Eu~{\\sc ii}\\ line of interest to perform a \n$\\chi^2$\\ minimization between the modeled stellar \natmosphere and the spectral data, thus solving for\nthe wavelength scale alignment as precisely as \npossible in the Eu~{\\sc ii}\\ region.\n\\item Continuum.\nWe fit a quadratic function across the points designated \nin the solar spectrum as continuum-fitting points \n(the cross-hatch regions in \nFigures \\ref{eu4129_nso}, \\ref{eu4205_nso},\nand \\ref{eu6645_nso}).\nWe adjust the quadratic continuum function vertically\nto require that 1--2\\% of spectral points \nin the full spectral segment are above unity, \nthereby ensuring that all spectra are scaled identically.\n\\item Abundance.\nWe perform a $\\chi^2$\\ minimization \nadjusting only the abundance of europium.\nWe begin with the solar abundance value scaled\nby the star's metallicity, then search \n$1.0\\unit{dex}$\nof europium abundance space to find the \nbest-fit value.\nIn minimizing the $\\chi^2$\\ statistic, \nwe calculate the residuals between the data \nand the fit in a limited region around the \nEu~{\\sc ii}\\ line (the gray regions in Figures \n\\ref{eu4129_nso}, \\ref{eu4205_nso}, and \n\\ref{eu6645_nso}).\n\\end{enumerate}\n\\noindent The wavelength alignment, spectrum continuum, and europium abundance \nfitting routines are run in that order and iterated \nuntil a stable solution is reached.\nIn most cases a stable solution requires only one or \ntwo iterations.\nThe abundances for each line in each star as \ndetermined by this process are listed in Table \n\\ref{starsvals_table}.\n\nAfter running our automatic europium fitting algorithm \non all 41\\ stars,\nwe examine each line in each star by eye to \nconfirm that the fit is successful. \nIn a few of the metal-poor 4129-\\AA\\ fits, the blended lines\nthat encroach on Eu~{\\sc ii}\\ were poorly enough fit with the\nVF05 SPOCS values that the europium value was \nunconvincing. \nIn those cases, the 4129-\\AA\\ value is\nomitted from Table \\ref{starsvals_table} and does not\ncontribute to the average.\nSee Figure \\ref{keep} for a comparison of a rejected\n4129-\\AA\\ feature and a robust 4129-\\AA\\ fit.\nSimilarly, in a few metal-poor stars the\n6645-\\AA\\ line is too weak to be seen in the noise, \nand hence the output of the fitting routine serves only\nas an upper limit on the europium abundance.\nSee Figure \\ref{keep} for a comparison of a measureable \n6645-\\AA\\ feature and a feature that provides an \nupper limit.\nIn the cases where the 6645-\\AA\\ line is an upper limit\nonly, it is listed as such in Table \\ref{starsvals_table}\nand does not contribute to the average. \n\n\\begin{figure} \n\\includegraphics[width=\\columnwidth]{f5.eps}\n\\figcaption[Bad fits vs.\\ good fits in comparable stars.]{Bad fits (top) \n  versus good fits (bottom) in comparable stars. \n  In the upper left panel, the SPOCS stellar properties fit the lines adjacent to\n  Eu~{\\sc ii}\\ poorly enough that [Eu\/H]\\ is unreliable. That line is removed from further \n  analysis.\n  In the upper right panel, the 6645-\\AA\\ line is buried in the noise, meaning\n  the fit represents an upper limit to the europium abundance. The abundance\n  upper limit is noted in Table \\ref{starsvals_table}, but does not contribute\n  to the overall [Eu\/H]\\ measurement in the star.\n  The lower two panels include fits to stars with similar characteristics \n  to the stars in the upper panels, but where the 4129-\\AA\\ and 6645-\\AA\\\n  fits were more successful.\n  \\label{keep}}\n\\end{figure}\n\nIf both 4129\\,\\AA\\ and 6645\\,\\AA\\ proved problematic we removed the\nstar from our analysis entirely. \nThe three stars for which this was the case (listed in\n\\S\\,\\ref{stellsamp}) are omitted from \nTables \\ref{starsinfo_table} and \\ref{starsvals_table}.\nSince all of the rejected stars were\nmetal poor, we conclude that our \nfitting routine is most robust at solar metallicity, \nand becomes less reliable at [M\/H]~$< -1$. \nTemperature may also play a role, as one of the\nrejected stars, HD\\,64090, has $T_{\\rm eff} = 7300\\unit{K}$; \nVF05 determined SME to be reliable between 4800 and \n$6500\\unit{K}$.\n\nFor each star we calculate a weighted average europium \nabundance value based on the three (or, in some \ncases, two) spectral lines.\nWeighting the average is important because for stars with\nrelatively few observations (e.g., HD\\,156365 in Figure \n\\ref{coadding_works}), the weak line at 6645\\,\\AA\\ should \nbe weighted significantly less than the more robust blue \nlines. \nStars with a larger number of observations \nand higher S\/N spectra\n(e.g., HD\\,144585 in Figure \\ref{coadding_works})\nshould have the 6645-\\AA\\ line weighted more strongly.\nIn order to determine the relative weights of the three \nspectral lines, we tested the robustness of our fit by \nadding artificial noise to the spectra. \nWe describe that process in \\S\\,\\ref{errors}.\n\n\\subsection{Error Analysis}\n\\label{errors}\nWe begin our error analysis by adding to the data \nGaussian-distributed random noise with a standard \ndeviation set by the photon noise at each pixel.\nWe then fit the europium line again, using the \nsame iterative $\\chi^2$-minimization process described \nin \\S\\,\\ref{stellar_abund}, and repeat the process \n50 times.\nThe standard deviation of the 50 Monte Carlo \ntrials determines the relative weights of the \nlines in the average listed in \nTable \\ref{starsvals_table}, with lower \nstandard deviation lines (corresponding \nto a more robust fit) weighted more \nstrongly.\n\nAs expected, the results of the Monte Carlo trials show \nthat the larger the photon noise in an observation, the \nless that line is to be weighted. \nWe derive a linear relation between \nthe Poisson uncertainty of an observation\nand its relative weight, based on the Monte Carlo results.\nBecause the Monte Carlo trials are CPU intensive, we plan\nto use that relation to determine \nrelative weights in the future.\n\nWe estimate the uncertainty of our europium\nabundances to be the sum of the squares of the \nresiduals for all three lines, where \nwe assign the residuals the same relative\nweights we applied when calculating the average.\nWe also impose an error floor of $0.02\\unit{dex}$,\nadded to the uncertainty in quadrature.\n\nThe error floor comes from a comparison of\nabundance values from the individual lines\n(Figure \\ref{line_compare}, discussed in more \ndetail in \\S\\,\\ref{compare_lines});\nerror bars of $0.03\\unit{dex}$ on each measurement\nwould make the reduced $\\sqrt{\\chi^2}$~$=1$. \nA minimum error of $0.03\\unit{dex}$ on each \neuropium measurement\ntranslates into an uncertainty of $0.02\\unit{dex}$ \nfor the final averaged value.\nThe $0.02\\unit{dex}$ error floor is included in the \nuncertainty quoted in Table \\ref{starsvals_table}.\n\n\\begin{figure} \n\\includegraphics[width=\\columnwidth]{f6.eps}\n\\figcaption[Comparing three Eu~{\\sc ii}\\ lines.]{A comparison of the Eu~{\\sc ii}\\ values \n  measured in the 41\\ stars of this study, as listed in Table \n  \\ref{starsvals_table}. \n  Vesta is marked with a plus symbol.\n  The solid line represents a 1:1 correlation; the dashed line is a \n  best fit to the data.\n  Points are omitted where the fit was poor in one of the Eu~{\\sc ii}\\ lines. \n  Upper limits for the 6645-\\AA\\ line are marked with arrows.\n  See \\S\\,\\ref{compare_lines} for a discussion of the quality of the fit.\n  \\label{line_compare}}\n\\end{figure}\n\nAs an initial test of the robustness of our fitting routine,\nwe measure the europium abundance in a spectrum of Vesta,\nwhich serves as a solar proxy.\nThe values are listed in the last row of\nTable \\ref{starsvals_table}.\nThe three Vesta europium abundances have a standard\ndeviation of $0.022\\unit{dex}$, indicating that \nsystematic errors ($\\sim$\\,$0.03\\unit{dex}$ for Vesta,\nits offset from the solar value) may be a significant\nportion of the error budget.\n\nFor the moment we absorb any systematic error with \nour random error estimates.\nThe full sample of 1000 stars, the analysis \nof which will follow this work, will\nallow a far more thorough investigation of \nthe dependence of our results on various model \nparameters ($T_{\\rm eff}$, $\\log{g}$, etc.)\\ than can be \naccomplished here.\nTherefore, we delay a substantive discussion of \nsystematics until \nwe have more europium abundance measurements in hand,\nthough we touch on it again in \\S\\,\\ref{compare_lines}\nand \\S\\,\\ref{compare_lit}.\nIt is important to note that since most of the CCPS stars \nare similar to the Sun, and we are treating each star\nidentically, our results will be internally consistent.\n\n\\subsection{Notes on Individual Lines}\n\\label{individual_lines}\n\n\\subsubsection{Europium 4129\\,\\AA}\n\\label{abund:eu4129}\nThe europium line at 4129\\,\\AA\\ is the result of a \nresonance transition of Eu~{\\sc ii}, and is a strong, \nrelatively clean line.  \nIt provides the most reliable measurement of \neuropium abundance in a star.  \nOur fit to the Eu~{\\sc ii}\\ line at 4129\\,\\AA\\ in the \nsolar spectrum (described in \\S\\,\\ref{linelists}) \nappears as Figure \\ref{eu4129_nso}, with the 32\nhyperfine components (16 each from $^{151}$Eu \nand $^{153}$Eu; \\citealt{ivans_2006}) \nrepresented in the Figure \n\\ref{eu4129_nso} inset.\n\nIn the course of stellar fitting, the europium \nabundance is determined from the 4129\\,\\AA\\ line \nin all 41\\ stars except HD\\,103095. \nThat star is quite metal poor ([M\/H]~$= -1.16$) and cool \n($T_{\\rm eff} = 4950$), factors that likely contribute to the \npoor fit in the lines adjacent to the Eu~{\\sc ii}\\ line of \ninterest (see Figure \\ref{keep}).\n\n\\subsubsection{Europium 4205\\,\\AA}\n\\label{abund:eu4205}\nThe Eu~{\\sc ii}\\ line at 4205\\,\\AA\\ is the other fine-structure\ncomponent of the resonance transition responsible\nfor the line at 4129\\,\\AA.\nThough it is as strong as the Eu~{\\sc ii}\\ line at 4129\\,\\AA, \ncontamination from embedded lines (see Figure \\ref{eu4205_nso})\nhas the potential make the 4205-\\AA\\ line less reliable.  \nHowever, it is useful as a comparison line for the results\nfrom the 4129-\\AA\\ Eu~{\\sc ii}\\ line.\nOur solar spectrum fit at 4205\\,\\AA\\ appears in \nFigure \\ref{eu4205_nso}, with the  \n30 hyperfine components (15 each from $^{151}$Eu and $^{153}$Eu;\n\\citealt{ivans_2006}) \nin the inset.\nDespite its blended nature, the fit appears sound\nin all 41\\ stars.\n\n\\subsubsection{Europium 6645\\,\\AA}\n\\label{abund:eu6645}\nThe Eu~{\\sc ii}\\ line at 6645\\,\\AA\\ is weaker than the lines in the blue, but\nit is also relatively unblended, making it worthwhile to fit wherever \npossible.  \nOur solar spectrum fit at 6645\\,\\AA\\ appears in \nFigure \\ref{eu6645_nso},\nwith the inset plot showing the 30 hyperfine components (15\neach from $^{151}$Eu and $^{153}$Eu; \\citealt{ivans_2006}). \nIn HIP\\,57450, which is metal poor ([M\/H]~$= -1.42$) and has only\none observation, the 6645-\\AA\\ line was lost in the noise\n(see Figure \\ref{keep}), and\nonly the 4129-\\AA\\ and 4205-\\AA\\ lines contribute to the final\neuropium abundance; the 6645-\\AA\\ line provides an upper limit only.\n\n\n\\section{Results}\n\\label{comp}\n\\subsection{Comparison of Individual Lines}\n\\label{compare_lines}\nWe compare our results from the three Eu~{\\sc ii}\\ lines in\nFigure \\ref{line_compare}.\nThe Vesta abundances, included as a solar \nproxy, are consistent with the stellar \nabundances to within $0.03\\unit{dex}$.\nBy assigning a measurement uncertainty\nof $0.03\\unit{dex}$ to each line, the \nreduced $\\sqrt{\\chi^2}$\\ for each of the three plots is unity.\nThis measurement uncertainty is the source of the\nerror floor discussed in \\S\\,\\ref{errors}.\n\n\\begin{figure*} \n\\centering\n\\includegraphics[width=0.80\\textwidth]{f7.eps}\n\\figcaption[Comparing our Eu~{\\sc ii}\\ values to others'.]{A comparison of the \n  final Eu~{\\sc ii}\\ values from this work with literature measurements. \n  The solid line represents a 1:1 correlation. \n  The horizontal error bars represent the uncertainties as quoted in \n  the source literature. \n  The vertical error bars come from our analysis as described in \\S\\,\\ref{errors}. \n  Two stars, HD\\,9562 and HD\\,22879 (with [Eu\/H]\\ of $+0.12$ and $-0.54$, \n  respectively), were included in the \\citealt{bensby_2005}, \n  \\citealt{delpeloso_2005a}, and \\citealt{woolf_1995} analyses, so those \n  two stars are represented by three points along the abscissa at the \n  same ordinate value. \n  See Table \\ref{starsinfo_table} for the full list of the europium \n  abundances plotted here, and see \\S\\,\\ref{compare_lit} for a discussion\n  of the quality of our fit based on this plot. \n  \\label{me_vs_them}}\n\\end{figure*}\n\nCalculating the best-fit line for each \ncomparison plot (the dashed lines in\nFigure \\ref{line_compare}) lends insight\ninto possible systematic trends in our analysis.\nIn the blue, the two Eu~{\\sc ii}\\ lines (4129\\,\\AA\\ and\n4205\\,\\AA) have no apparent linear systematic \ntrend:\\ the slope of the best-fit line is 1.03.\nThe red Eu~{\\sc ii}\\ line (6645\\,\\AA), however, exhibits\na minor systematic trend:\\ the best-fit line\nin the red has a slope of 1.10 relative to the blue,\ni.e., a 10\\% stretching of the blue abundance values \nabout [Eu\/H]~$= 0$ roughly reproduces the red \nabundance values.\nSince this systematic trend is absorbed by the\n$0.03\\unit{dex}$ error bars assigned to each point \n(necessary to make reduced $\\sqrt{\\chi^2}$\\ unity even when\ncomparing the non-systematic blue lines),\nwe make no attempt to correct this systematic\ntrend here.\n\nBetween the relatively low measurement uncertainty\nneeded to achieve reduced $\\sqrt{\\chi^2}$\\ of unity and the minor\nsystematic trend that only appears in the red, \nwe conclude that the europium abundance values \nderived from the Eu~{\\sc ii}\\ lines at 4129\\,\\AA, \n4205\\,\\AA, and 6645\\,\\AA\\ are consistent with one \nanother.\n\n\\subsection{Comparison with Literature Europium Measurements}\n\\label{compare_lit}\nIn Figure \\ref{me_vs_them}, we compare our \nfinal europium abundance measurements to the\nliterature values.\nThe agreement is quite good.\nThe literature values, plotted with the error\nbars quoted in the original studies, appear\nas the abscissa.\nOur europium values, plotted with the error\nbars calculated in \\S\\,\\ref{errors} and listed\nin Table \\ref{starsvals_table}, appear\nas the ordinate.\nThe solid line represents a 1:1 correlation.\nComparing the points to the 1:1 line,\nthe reduced $\\sqrt{\\chi^2}$~$=\\rchisqtot$.\nThat value is dominated by\nHD\\,103095, the \\citealt{simmerer_2004} data \npoint with the smallest error bars, and \nomitting it makes\nthe reduced $\\sqrt{\\chi^2}$~$=\\rchisqsm$. \n\nAdopting the global values ($T_{\\rm eff}$, [M\/H], $\\log{g}$)\nused by the comparison studies (instead of the VF05 \nvalues) drops the reduced $\\sqrt{\\chi^2}$\\ to \\rchisqlit, indicating\nthat some of the scatter in Figure \\ref{me_vs_them}\nis from the choice of stellar parameters.\nWe emphasize that the VF05 stellar parameters are \nreliable; we calculate reduced $\\sqrt{\\chi^2}$\\ using\nthe comparison studies' values in an attempt to \nseparate how much of the disagreement in Figure \n\\ref{me_vs_them} is from the europium abundance\ntechnique and how much is from the parameters adopted.\n\nOmitting the outlier mentioned at the beginning\nof this section, we examine \nthe data in Figure \\ref{me_vs_them} to \nsearch for systematics in our results relative \nto the literature values. \nWe consider the effects of a global offset in\nour europium values and a linear trend\nwith europium abundance, finding that systematic\noffsets of $\\sim$\\,$0.1\\unit{dex}$ and linear trends\nof $\\sim$\\,$40\\%$ are needed\nto make reduced $\\sqrt{\\chi^2}$\\ climb to 2.\nWe conclude that our error bars have sufficiently\ncharacterized our uncertainties.\n\nOverall, we find reduced $\\sqrt{\\chi^2}$\\ to be dominated by the \npoints at [Eu\/H]~$< 0.5$, consistent with\nour conclusion in \\S\\,\\ref{stellar_abund} that\nour results are most reliable near solar \nmetallicities, although\nthe larger error bars on the \\citealt{woolf_1995}\npoints make the correlation very forgiving near\n[Eu\/H]~$=0$. \nOur spectra have very high S\/N:\\ $\\sim$\\,160 \nat 4200\\,\\AA\\ \nin a single observation, enhanced substantially\nby our co-adding procedure (\\S\\,\\ref{data}).\nThe automated\nSME synthesis treats all stars consistently,\nespecially important for line blends in the \ncrowded blue regions.\nFor these reasons we believe that our europium abundance\ntechnique is accurate and robust and\nthat our smaller error bars are warranted.\n\nWe conclude that\nour abundance measurements are consistent\nwith previous studies, not surprising\nsince most abundance techniques rely on the same\nKurucz stellar atmosphere models.\nBecause the majority of the points in\nFigures \\ref{me_vs_them} fall near the\n1:1 correlation, we also conclude that\nnear [Eu\/H]~$=0$ our errors are \n$\\sim$\\,$0.03\\unit{dex}$, though at\n[Eu\/H]~$< -0.5$, the errors may be as high as \n$0.1\\unit{dex}$.\n\n\n\\section{Summary}\n\\label{summary}\nWe have established that our method for measuring europium in \nsolar-metallicity stars using SME is sound.\nThe resolution and S\/N of the Keck HIRES spectra are \nsufficiently high to fit the Eu~{\\sc ii}\\ lines in question. \nThe values obtained from the \nthree europium lines are self-consistent, and our final \naveraged europium value for each of the 41\\ stars in this study\nare consistent with the literature values for those stars.\n\nBy employing SME to calculate our synthetic spectra,\nwe are self-consistently modeling all the lines in the regions\nof interest. \nAny blending from neighboring lines is treated consistently\nfrom star to star, adding robustness to our europium determination.\nUsing SME has the added benefit of allowing us to \nadopt the stellar parameters from the SPOCS catalog.\nOur automated procedure ensures all stars are treated consistently.\n\nHaving established a new method for measuring stellar \neuropium abundances, we intend to apply our technique to \n1000 F, G, and K stars from the Keck CCPS survey.\nOur analysis of europium in these stars will represent the \nlargest and most consistent set of europium measurements in\nsolar-metallicity stars to date, and will provide\ninsight into the question of the {\\em r}-process\\ formation\nsite and the enrichment history of the Galaxy.\n\n\n\\acknowledgements\nThe author is indebted to\nGeoffrey W.~Marcy, \nChristopher Sneden,\nDebra A.~Fischer, \nJeff A.~Valenti, \nAnna~Frebel,\nJames W.~Truran,\nand Taft E.~Armandroff\nfor productive and enlightening conversations about the\nprogress of this work.\nParticular thanks are extended to\nChristopher Sneden,\nGeoffrey W.~Marcy, and\nLouis-Benoit Desroches\nfor their thoughtful comments on this paper.\nThe author is also grateful to her fellow observers\nwho collected the Keck HIRES data used\nhere:\\ Geoffrey W.~Marcy, \nDebra A.~Fischer,\nJason T.~Wright, \nJohn Asher Johnson,\nAndrew W.~Howard, \nChris McCarthy,\nSuneet Upadhyay,\nR.~Paul Butler,\nSteven S.~Vogt,\nEugenio Rivera,\nand Joshua Winn.\nWe gratefully acknowledge the dedication of the\nstaff at Keck Observatory, particularly Grant Hill\nand Scott Dahm for their HIRES support.\nThis research has made use of \nthe SIMBAD database, operated at CDS, Strasbourg, France; \nthe Vienna Atomic Line Database; \nthe Kurucz Atomic and Molecular Line Databases; \nthe NIST Atomic Spectra Database;\nand NASA's Astrophysics Data System Bibliographic Services.\nThe author extends thanks to those of Hawaiian \nancestry on whose sacred mountain of Mauna Kea we\nare privileged to be guests. \nWithout their generous hospitality, the Keck observations\npresented here would not have been possible.\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{0pt}{8pt plus 2pt minus 1pt}{6pt plus 2pt minus 1pt}\n\n\\usepackage{balance}\n\n\\usepackage[scaled]{beramono}\n\\usepackage{listings}\n\n\\lstset{\n  language=Python,\n  showstringspaces=false,\n  formfeed=\\newpage,\n  tabsize=4,\n  commentstyle=\\itshape,\n  basicstyle=\\ttfamily,\n  breaklines=true,\n  morekeywords={models, lambda, forms}\n}\n\n\\usepackage{color}\n\\definecolor{Orange}{rgb}{0.9,0.5,0}\n\\definecolor{NavyBlue}{rgb}{0.1, 0.4, 0.8}\n\\definecolor{Magenta}{rgb}{0.8, 0.1, 0.6}\n\\definecolor{Red}{rgb}{1, 0, 0}\n\\newcommand{\\marco}[1]{\\textcolor{Magenta}{\\textbf{[By Marco: #1]}}}\n\\newcommand{\\gloria}[1]{\\textcolor{blue}{\\textbf{[By Gloria: #1]}}}\n\\newcommand{\\bruno}[1]{\\textcolor{Red}{\\textbf{[By Bruno: #1]}}}\n\\newcommand{\\yahui}[1]{\\textcolor{Orange}{\\textbf{[By Yahui: #1]}}}\n\n\\newcommand{${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}}{${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}}\n\\newcommand{$\\pmb{\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}}{$\\pmb{\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}}\n\n\n\n\n\n\\usepackage[belowskip=-0.3em,aboveskip=2pt]{caption}\n\\begin{document}\n\\fancyhead{}\n\\title{Gesture-to-Gesture Translation in the Wild via Category-Independent Conditional Maps}\n\n\\author{Yahui Liu}\n\\email{yahui.liu@unitn.it}\n\\affiliation[obeypunctuation=true]{\\institution{University of Trento}, \\city{Trento}, \\country{Italy}}\n\n\\author{Marco De Nadai}\n\\email{denadai@fbk.eu}\n\\orcid{0000-0001-8466-3933}\n\\affiliation[obeypunctuation=true]{\\institution{FBK}, \\city{Trento}, \\country{Italy}}\n\n\\author{Gloria Zen}\n\\email{gloria.zen@unitn.it}\n\\affiliation[obeypunctuation=true]{\\institution{University of Trento}, \\city{Trento}, \\country{Italy}}\n\n\\author{Nicu Sebe}\n\\email{niculae.sebe@unitn.it}\n\\affiliation[obeypunctuation=true]{\\institution{University of Trento}, \\city{Trento}, \\country{Italy}}\n\n\\author{Bruno Lepri}\n\\email{lepri@fbk.eu}\n\\affiliation[obeypunctuation=true]{\\institution{FBK}, \\city{Trento}, \\country{Italy}}\n\n\n\n\n\\renewcommand{\\shortauthors}{Liu, et al.}\n\\renewcommand{\\shorttitle}{Gesture-to-Gesture Translation in the Wild}\n\n\n\\begin{abstract}\n\\begin{sloppypar}\nRecent works have shown Generative Adversarial Networks (GANs) to be particularly effective in image-to-image translations.\nHowever, in tasks such as body pose and hand gesture translation, existing methods usually require precise annotations, e.g. key-points or skeletons, which are time-consuming to draw. \nIn this work, we propose a novel GAN architecture that decouples the required annotations into a category label - that specifies the gesture type - and a simple-to-draw category-independent conditional map - that expresses the location, rotation and size of the hand gesture.\nOur architecture synthesizes the target gesture while preserving the background context, thus effectively dealing with gesture translation \\textit{in the wild}.\nTo this aim, we use an attention module and a rolling guidance approach, which loops the generated images back into the network and produces higher quality images compared to competing works.\nThus, our GAN learns to generate new images from simple annotations without requiring key-points or skeleton labels.\nResults on two public datasets show that our method outperforms state of the art approaches both quantitatively and qualitatively.\nTo the best of our knowledge, no work so far has addressed the gesture-to-gesture translation \\emph{in the wild} by requiring user-friendly annotations.\n\n\\end{sloppypar}\n\n\\end{abstract}\n\n\\begin{CCSXML}\n<ccs2012>\n<concept>\n<concept_id>10010147.10010178.10010224<\/concept_id>\n<concept_desc>Computing methodologies~Computer vision<\/concept_desc>\n<concept_significance>500<\/concept_significance>\n<\/concept>\n<concept>\n<concept_id>10010147.10010257<\/concept_id>\n<concept_desc>Computing methodologies~Machine learning<\/concept_desc>\n<concept_significance>300<\/concept_significance>\n<\/concept>\n<\/ccs2012>\n\\end{CCSXML}\n\n\\ccsdesc[500]{Computing methodologies~Computer vision}\n\\ccsdesc[300]{Computing methodologies~Machine learning}\n\n\\keywords{GANs, image translation, hand gesture}\n\n\n\n\n\\maketitle\n\n\\begin{figure}[t]\n    \\includegraphics[width=1\\columnwidth]{figures\/teaser2.pdf}\n \n    \\caption\n    Our proposal decouples the \\emph{category label} that specifies the gesture type (e.g., gesture \"5\" or \"7\") from the \\emph{conditional map} (i.e. triangle) that controls the location, orientation and size of the target gesture.\n    Existing works require a detailed conditional map (e.g. skeleton) that is gesture-dependent. \n    In this example, we show that our method significantly lowers the drawing effort and expertise required by users. Our method can generate multiple output images with the same map for multiple gesture categories. \n   \n    }\n    \\label{fig:teaser}\n    \\vspace{-0.3em}\n\\end{figure}\n\n\\section{Introduction}\n\\begin{sloppypar}\nPhoto-editing software, fashion, and retail markets would enormously benefit from the possibility of modifying an image through a simple user input describing the changes to make (e.g. change the hand gesture of the person in the picture from ``open hand'' to ``ok'').\nHowever, despite the significant advances of Generative Adversarial Networks (GANs)~\\cite{goodfellow2014generative,radford2015unsupervised,arjovsky2017wasserstein,gulrajani2017improved}, the generation of images \\emph{in the wild} without precise annotations (e.g. hand skeleton) is still an open problem.\nPrevious literature on image-to-image translation has relied either on pixel-to-pixel mappings~\\cite{isola2017image,zhu2017unpaired,yang2018crossing} or precise annotations to localize the instance to be manipulated, such as segmentation masks~\\cite{mo2018instagan}, key-points~\\cite{siarohin2018deformable}, skeletons~\\cite{tang2018gesturegan} and facial landmarks~\\cite{sanchez2018triple}. \nHowever, obtaining such annotations is not trivial.\nOn one hand, automatic methods for key-points extraction~\\cite{cao2017realtime,simon2017hand} may fail or the reference gesture image\/video~\\cite{siarohin2018animating} may be not available. On the other, drawing such annotations is complicated, time-consuming, and their quality directly affects the performance of the network. \n\nMoreover, existing methods often focus on foreground content, e.g., the target gesture or facial expression, generating blurred and imprecise backgrounds~\\cite{siarohin2018deformable,ma2017pose,reed2016learning}.\nThese methods are well suited in the cases where\nshare fixed or similar spatial structures, such as facial expression datasets~\\cite{liu2015deep,fabian2016emotionet}.\nInstead, in image-to-image translations \\textit{in the wild}, both the foreground and background between the source image and target image can vary a lot~\\cite{tang2018gesturegan}. \n\nIn this paper, we propose a novel method, named as ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}, that requires a simple-to-draw annotation, such as a triangle, and focuses on the challenging task of hand gesture-to-gesture translation \\emph{in the wild}.\nIn general, annotations such as key-points or skeletons are category-dependent since they provide four types of information at the same time, namely \\textit{category} (e.g., gesture ``5\"), \\textit{location}, \\textit{scale} and \\textit{orientation} of the hand gesture.\nInstead, our ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}~decouples the \\textit{category} from the \\textit{location}-\\textit{scale}-\\textit{orientation} information.\nUsing a category-independent conditional map significantly lowers the annotation cost, allowing to generate multiple target images while requiring users to draw only a single map. \nIn this work, we refer to ``annotations'' as the user effort to draw the desired target gesture also at deploy time, besides the effort needed for generating the training data for our model. \nThe intuition of our approach is depicted in Figure~\\ref{fig:teaser}.\nFurthermore, we propose a novel architecture that uses an attention module and a rolling guidance approach to perform gesture-to-gesture translation \\textit{in the wild}.\nOur research yields three main contributions:\n\\begin{itemize}\n    \\item \\emph{Decoupled conditional map and category.} We designed a general architecture for gesture-to-gesture translations that separately encodes the category label (e.g., gesture ``5\") and the category-independent conditional map. \n    This allows to perform several image translations with the same conditional map.\n    \\item \\emph{Rolling guidance and attention mask.} We propose a novel rolling guidance approach that allows generating higher quality output images by feeding the generated image back to the input as an additional condition to be considered. \n   \n   \n    Also, ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}~learns unsupervisedly an attention mask that allows to preserve the details shared between the input and target image.\n   \n   \n    \\item \\emph{Simple-to-draw conditional maps.} \n    We propose a triangle conditional map as the simplest and minimal necessary user provided condition for gesture-to-gesture translation.\n    To the best of our knowledge, no work so far has addressed the gesture-to-gesture translation task \\emph{in the wild} by requiring user-friendly annotations. \n    Furthermore, we assess the performance of our method with different shapes, such as boundary and skeleton maps.\n   \n   \n    Finally, we enrich two public datasets with different conditional maps for each gesture image, specifically based on triangles and boundaries.\n\\end{itemize}\n\\end{sloppypar}\n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}[ht!]\n    \\centering\n    \\includegraphics[width=0.95\\linewidth]{figures\/diagram.pdf}\n    \\vspace{-1em}\n    \\caption{Overview of $\\pmb{\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}, that allows to translate hand gestures \\emph{in the wild} by separately encoding the input image and the target gesture attributes including the category label (e.g., gesture 3) and the category-independent conditional map. We include an unsupervised attention mask to preserve the details shared between the input image and the target image. Specially, we feed the first reconstructed image back to the input conditions encoding module to improve quality of output image.}\n    \\label{fig:ourGAN}\n \n\\end{figure*}\n\n\n\\section{Related work}\nRecently, there have been a lot of works on Generative Adversarial Networks (GANs) and, particularly, on conditional GANs for the task of image-to-image translation~\\cite{isola2017image,zhu2017unpaired,zhu2017toward,choi2018stargan,pumarola2018ganimation}. \nA significant line of these works have focused on translation tasks where the the input and target images are spatially aligned, as in the case of style transfer~\\cite{zhu2017unpaired,johnson2016perceptual,isola2017image}, emotional content transfer~\\cite{emotionGAN18} or \nimage inpaiting~\\cite{zhang2018semantic}.\nIn general, these works aim to preserve the main image content while presenting them in various styles.\nAnother line of works have tackled the more challenging task of image translation where the target object is spatially unaligned with respect to the original location, shape or size of the input original object.\nThis is the case of image-to-image translation tasks like facial expression~\\cite{sanchez2018triple,geng20193d}, \nhuman pose~\\cite{ma2017pose,siarohin2018deformable} or \nhand gesture translation~\\cite{tang2018gesturegan}.\nTo this aim, methods usually require geometry information as guidance to guarantee where and how to edit visual patterns corresponding to the image attributes,\nsuch as key-points~\\cite{ma2017pose,siarohin2018animating,ma2018disentangled}, skeletons~\\cite{tang2018gesturegan}, object segmentation masks~\\cite{mo2018instagan}, facial landmarks~\\cite{sanchez2018triple}, action units~\\cite{pumarola2018ganimation} or 3D models~\\cite{geng20193d}. \n\n\n GANimation~\\cite{pumarola2018ganimation} learns an attention mask to preserve the background content of the source image for the spatially aligned translation of facial expressions. \n InstaGAN~\\cite{mo2018instagan} performs multi-instance domain-to-domain image translation by requiring as input precise segmentation masks for the target objects.\n Pose Guided Person Generation Network (PG$^2$)~\\cite{ma2017pose} proposes a two stages generating framework to refine the output image given a reference image and a target pose.\n MonkeyNet~\\cite{siarohin2018animating} generates a conditioned video transferring body movements from a target driving video to a reference appearance image. The target key-points are obtained automatically using state-of-the-art detectors.\n\nRelatively few works have considered the challenging task of image translation \\textit{in the wild}, where both the foreground content and the background context undergoes significant variation~\\cite{tang2018gesturegan}. \nIn these cases, not only the networks learn to synthesize the target object or attribute, but they also have to correctly locate it in the image while the pixels depicting the content are preserved from the original image.\nGestureGAN~\\cite{tang2018gesturegan} proposes a novel color loss to improve the output image quality and to produce sharper results. However, this approach does not aim to separately encode the foreground content and the background context.\n\nFurthermore, the large majority of image-to-image translations works focus on one to one domain mapping. \nRecently, efficient solutions have been proposed to address the multi-domain translations~\\cite{choi2018stargan,geng20193d}. In particular, StarGAN~\\cite{choi2018stargan} proposes the use of multiple mask vectors to generalize on different datasets and domain labels. 3DMM~\\cite{geng20193d} decomposes an image into shape and texture spaces, and relies on an identity and target coefficients to generate images in multiple domains. They, however, focus on multi-domain face attributes where neither the background quality or the misalignment are considered.\n\nOur work falls within the category of multi-domain and image-to-image translation \\emph{in the wild}. Still, rather than asking expensive annotation to the user, we propose a novel user-friendly annotation strategy.\nTo the best of our knowledge, no works so far have investigated other approaches for gesture translation that also allow to reduce the annotation effort.\n\n\n\\section{Our approach}\nThe goal of this work is to perform gesture translation \\emph{in the wild}, conditioned on the user provided gesture attributes, such as the desired category, location, scale and orientation. \nSpecifically, we seek to estimate the mapping $\\mathcal{M}$: $(\\mathbf{I}_X, \\mathbf{S}_Y, C_Y) \\rightarrow \\mathbf{I}_Y$ that translates an input image $\\mathbf{I}_X$ into an output image $\\mathbf{I}_Y$, conditioned to a hand gesture category $C_Y$ and a simple-to-draw conditional map $\\mathbf{S}_Y$, which encodes the desired location, scale and orientation of the gesture.\nThe generated image is discriminated through a Conditional Critic discriminator~\\cite{pumarola2018ganimation} that judges the photo-realism of the generated image and assess the category of the generated gesture (e.g. gesture \"5\"). \nFurthermore, in order to efficiently deal with the gesture translation \\emph{in the wild}, we propose a rolling guidance approach and an attention mask module. \nFigure~\\ref{fig:ourGAN} depicts the architecture of our approach. In this Section we further explain the details of the architecture and the loss functions used to train our framework.\n\n\\subsection{Network Architecture}\n\\label{sec:network}\n\n\n\\noindent \\textbf{Generator.}  \nOur Generator $G$ takes as input the conditional image $\\textbf{I}_X$, the category label $C_Y$ and the conditional map $\\textbf{S}_Y$\nand it outputs the translated image $\\textbf{I}_Y$.\nThe original image and the target gesture conditions are encoded\nrespectively through the encoder $E_1$ and $E_2$, and then concatenated and provided to $F_{res}$. Then, the output features of $E_2$ and $F_{res}$ are concatenated and provided to $F_{dec}$. \nSimilarly to~\\cite{pumarola2018ganimation}, in the decoder we learn to predict in an unsupervised manner the approximated image $\\widetilde{\\textbf{I}}_Y$ and an \\textit{Attention Mask} $\\mathbf{A} \\in [0,1.0]^{{H\\times W}}$. \nSince both the background and foreground between the input and target images can vary a lot, the learned attention mask $\\mathbf{A}$ tends to preserve the shared part between $\\mathbf{I}_X$ and $\\mathbf{I}_Y$. \nPixels in $\\mathbf{A}$ with higher value indicate to preserve the corresponding pixels from $\\mathbf{I}_X$, otherwise from $\\widetilde{\\mathbf{I}}_Y$.\nThe final generated image is thus obtained as:\n\\begin{equation}\n\\widehat{\\mathbf{I}}_Y = \\mathbf{A}*\\mathbf{I}_X + (1-\\mathbf{A})*\\widetilde{\\mathbf{I}}_Y\n\\label{eq:attention-mask}\n\\end{equation}\nFurthermore, the generated image $\\mathbf{I}_Y$ is rolled back as an input condition and concatenated together with the category label $C_Y$ and the conditional map $\\textbf{S}_Y$. More details are provided in Section~\\ref{Sec:rolling-guidance}.  \n\n\\medskip\n\\noindent \\textbf{Discriminator.} Our Discriminator $D$ takes as input the generated image $\\widehat{\\textbf{I}}_Y$ and its conditional map $\\textbf{S}_Y$. \nThe outputs of $D$ consist of two parts: $D_{cat}$ predicts the category label \nand $D_{prob}$ classifies whether local image patches are real or fake. \nAs in~\\cite{pumarola2018ganimation}, D$_{prob}$ is based on PatchGANs~\\cite{isola2017image}.\nThe conditional map $\\textbf{S}_Y$ is needed by $D$ to verify that the generated gesture has also the right location, scale and orientation.\n\nWe refer to the Supplementary Material for additional details in the architecture.\n\n\\subsection{Objective Formulation}\n\\label{sec:loss}\n\nThe loss function of ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ is composed of four main components, namely \\emph{GAN Loss} that pushes the generated image distribution to the distribution of source images; \\emph{Reconstruction loss} that forces the generator to reconstruct the source and target image; \\emph{Category Label loss} that allows to properly classify the generated image into hand gesture classes; and \\emph{Total Variation loss} that indirectly learns the attention mask in an unsupervised fashion.\n\n\\medskip\n\\noindent \\textbf{GAN Loss.} To generate images with the same distribution of the source images,\nwe adopt an adversarial loss: \n\\begin{equation}\n\\begin{aligned}\n\\mathcal{L}_{GAN} = & \\mathbb{E}_{\\mathbf{I}_Y, \\mathbf{S}_Y\\sim \\mathbb{P}}[\\log D_{prob}(\\mathbf{I}_Y, \\mathbf{S}_Y)] + \\\\ & \\mathbb{E}_{\\mathbf{I}_X,\\mathbf{S}_Y, C_Y\\sim \\mathbb{P}}[\\log (1- D_{prob}(G(\\mathbf{I}_X, \\mathbf{S}_Y, C_Y), \\mathbf{S}_Y))] \\\\\n\\end{aligned}\n\\end{equation}\nwhere $\\mathbb{P}$ is the data distribution of the hand gesture images in the dataset, and\nwhere $G$ generates an image $G(\\mathbf{I}_X, \\mathbf{S}_Y, C_Y)$ conditioned on both the input image $\\mathbf{I}_X$, the conditional map $\\mathbf{S}_Y$, and the target category $C_Y$, while $D$ tries to distinguish between real and fake images.  In this paper, we refer to the term $D_{prob}(\\mathbf{I}, \\mathbf{S})$ as a probability distribution over sources given by $D$.\n\n\\medskip\n\\noindent \\textbf{Reconstruction Loss.} The adversarial loss does not guarantee that the generated image is consistent with both the target conditional map $\\mathbf{S}$ and category $C$.\nThus, we first apply a \\emph{forward reconstruction loss} that ties together the target image $\\mathbf{I}_Y$ with its target conditional map $\\mathbf{S}_Y$ and category $C_Y$:\n\\begin{equation}\n    \\mathcal{L}_{rec} = \\|G(\\mathbf{I}_X, \\mathbf{S}_Y, C_Y) - \\mathbf{I}_Y\\|_1\n\\end{equation}\nThen, instead of using perceptual features (e.g. extracted from VGG~\\cite{very2015simonyan} networks) to force the model to reconstruct the source image, we propose a simplified  \\emph{self-reconstruction (identity) loss}:\n\\begin{equation}\n    \\mathcal{L}_{idt} = \\|G(\\mathbf{I}_X, \\mathbf{S}_X, C_X) - \\mathbf{I}_X \\|_1\n\\end{equation}\nwhere $\\mathbf{S}_X$ is the conditional map of the source image and $C_X$ the category label of the source image.\nFinally, we apply the \\emph{cycle consistency loss}~\\cite{zhu2017unpaired,kim2017learning} to reconstruct the original image from the generated one:\n\\begin{equation}\n    \\mathcal{L}_{cyc}=\\|G(G(\\mathbf{I}_X, \\mathbf{S}_Y, C_Y), \\mathbf{S}_X, C_X) - \\mathbf{I}_X\\|_1\n\\end{equation}\nNote that we apply the cycle reconstruction loss only in one direction to reduce computation, i.e., A-to-B-to-A, since a translation pair based on two images A and B may be sampled either as A-to-B or as B-to-A\nduring the training.\n\n\\medskip\n\\noindent \\textbf{Category Label loss.} We enforce the generator to render realistic samples that have to be correctly classified to the hand gesture expressed by the input category label.\nSimilarly to StarGAN~\\cite{choi2018stargan}, we split the \\emph{Category Label loss} in two terms: a gesture classification loss of the real image $\\mathbf{I}_Y$ used to optimize $D$, and a gesture classification loss of the generated image $\\hat{\\mathbf{I}}_Y$, used to optimize $G$.\nSpecifically:\n\\begin{equation}\n    \\mathcal{L}_{cls} = \\mathbb{E}_{\\mathbf{I}_Y, C_Y}[- \\log D_{cat}(C_Y | \\mathbf{I}_Y, \\mathbf{S}_Y)]\n\\end{equation}\n\\begin{equation}\n\\mathcal{L}_{\\hat{cls}} = \\mathbb{E}_{\\hat{\\mathbf{I}}_Y, C_Y}[- \\log D_{cat}(C_Y | \\hat{\\mathbf{I}}_Y, \\mathbf{S}_Y)]\n\\end{equation}\nwhere $D_{cat}(C_Y | \\mathbf{I}_Y, \\mathbf{S}_Y)$ and $D_{cat}(C_Y | \\hat{\\mathbf{I}}_Y,\\mathbf{S}_Y)$ represent a probability distribution over the categories of hand gestures respectively in the real and generated images.\nIn other words, these losses allow to generate images that can be correctly classified as the target hand gesture category.\n\n\\medskip\n\\noindent \\textbf{Total Variation loss}. \nTo prevent the final generated image having artifacts, we use a Total Variation Regularization, $f_{tv}$,\nas in GANimation~\\cite{pumarola2018ganimation}.\nHowever, differently from them, we calculate $f_{tv}$ over the approximated image $\\widetilde{\\mathbf{I}}_Y$ instead of the attention mask $\\mathbf{A}$, \nthus allowing to freely explore the shared pixels between the source and target images. \nThe total variation loss is applied both to the \\textit{forward reconstruction} and \\textit{self-reconstruction} and is formulated as:\n\\begin{equation}\n    \\mathcal{L}_{tv} = f_{tv}(G_C(\\mathbf{I}_X, \\mathbf{S}_X, C_X)+f_{tv}(G_C(\\mathbf{I}_X, \\mathbf{S}_Y, C_Y))\n\\end{equation}\nThe total variation regularization $f_{tv}$ is defined as:\n\\begin{dmath}\n    f_{tv}(\\mathbf{I})=\n    \\mathbb{E}_{\\mathbf{I}} \\left[\\sum_{i,j}^{H-1,W-1}[(\\mathbf{I}_{i+1, j} - \\mathbf{I}_{i,j})^2 + (\\mathbf{I}_{i, j+1} - \\mathbf{I}_{i,j})^2]\\right]\n\\end{dmath}\nwhere $\\mathbf{I}_{i,j}$ is the entry $i,j$ of the image matrix $\\mathbf{I}$.\n\n\n\\medskip\n\\noindent \\textbf{Total loss}. The final objective function used to optimize $G$ and $D$ is formulated as follows:\n\\begin{dmath}\n\\mathcal{L}_{D} = \\lambda_{D}\\mathcal{L}_{GAN}  + \\lambda_{cls}\\mathcal{L}_{cls}\n\\label{eq:loss_D}\n\\end{dmath}\n\\begin{dmath}\n\\mathcal{L}_{G} = \\lambda_{G}\\mathcal{L}_{GAN} + \\lambda_{rec}\\mathcal{L}_{rec} + \\lambda_{idt}\\mathcal{L}_{idt} + \\lambda_{cyc}\\mathcal{L}_{cyc} + \\lambda_{cls}\\mathcal{L}_{\\hat{cls}} + \\lambda_{tv}\\mathcal{L}_{tv}\n\\label{eq:loss_G}\n\\end{dmath}\nwhere $\\lambda_{D}$, $\\lambda_{G}$, $\\lambda_{rec}$, $\\lambda_{idt}$, $\\lambda_{cyc}$, $\\lambda_{cls}$, and $\\lambda_{tv}$ are hyper-parameters that control the relative importance of each loss term.\n\n\n\\subsection{Rolling Guidance}\n\\label{Sec:rolling-guidance}\n\\begin{sloppypar}\nWhile the total variation loss $\\mathcal{L}_{tv}$ also enforces the approximated images $\\widetilde{\\mathbf{I}}_Y$ to be smooth, the source and target images might contain edges and details that have to be preserved. \nMoreover, the $E_1$ and $E_2$ encoders mostly focus on the gesture, failing to learn important details of the context, which might result in blurred images.\nInspired by previous works~\\cite{mosinska2018beyond,ma2017pose, zhang2014rolling}, we propose a Rolling Guidance approach to refine the generated image in a two-stage process. \nFirst, the network generates an initial version $\\widehat{\\mathbf{I}}_Y$ from input ($\\mathbf{I}_X$, $\\mathbf{S}_Y$, $C_Y$). \nThen, $\\widehat{\\mathbf{I}}_Y$ is fed back to $E_2$. Thus, the network generates a refined version of $\\widehat{\\mathbf{I}}_Y$ from input ($\\mathbf{I}_X$, $\\mathbf{S}_Y$, $C_Y$, $\\widehat{\\mathbf{I}}_Y$).\nNote that there exists some approaches, like PG$^2$~\\cite{ma2017pose}, feed the initial generated image back and concatenated to the source input to learn difference map to refine the results. However, images with a considerable variation in both the foreground and background between source and target images might result in an ill-posed problem. Gesture-to-gesture translation in the wild shows such an issue. For this reason, in ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}, we feedback the generated image to $E_2$ to refine the generated image and to learn the condition related features of the target gesture at the same time. This results in better generalization and significantly improves the results as well.\n\\end{sloppypar}\n\n\\section{Experiments}\nWe compare our model with the state of the art techniques on two hand gesture datasets.\nFirst, we evaluate our model quantitatively through various widely used metrics that compare the generated image with the ground truth. \nThen, we also evaluate our results qualitatively through a perceptual user study.\nWe released the resulted dataset and annotations, source code and trained models are available at: \\url{https:\/\/github.com\/yhlleo\/TriangleGAN}.\n\n\\subsection{Datasets}\nThe NTU Hand Gesture dataset~\\cite{ren2013robust} is a collection of 1,000 RGB-D images recorded with a Kinect sensor. It includes 10 gestures repeated 10 times by 10 subjects under a cluttered background. Image resolution is 640x480. The Creative Senz3D dataset~\\cite{memo2018head} is a collection of 1,320 images, where 11 gestures are repeated 30 times by 4 subjects. \nImage resolution is 640x480.\n\n\\medskip\n\\noindent \\textbf{Experimental setting.}\nWe consider two dataset setups in our experimental evaluation.\n\n\\medskip\n\\noindent \\textit{Normal} uses the same setup as GestureGAN~\\cite{tang2018gesturegan}, in order to directly compare with the state of the art.\nGestureGAN authors used only a subset of the datasets (acquired by OpenPose~\\cite{cao2018openpose}): 647 images out of 1,000 for NTU Hand Gesture and 494 out of 1,320 for Creative Senz3d. It shows that the state-of-the-art detector OpenPose~\\cite{cao2018openpose} fails at detecting key-points for about 50\\% of the images.\nThe resulting number of training and test data pairs obtained are respectively the following:\n21,153 and 8,087 for NTU Hand Gesture; \n30,704 and 11,234 for Creative Senz3D.\nThese numbers are different from those reported in~\\cite{tang2018gesturegan} since we here report only the unique pairs without considering flipping and A-to-B reverse ordering.\n\n\n\\medskip\n\\noindent \\textit{Challenging} pushes the limits of our model by ensuring that all the translation pairs \"A-to-B\" to a specific image \"B\" are included either in the train or in the test, a condition not ensured by the \\emph{normal} setting and, thus, by state of the art.\nAs a consequence, the model here generates multi-domain images without previous knowledge about it.\nWe randomly select for the training and test data the following number of pairs: \n22,050 and 13,500 for NTU Hand Gesture;\n138,864 and 16,500 for the Creative Senz3D. \n\n\n\n\n\\medskip\n\\noindent \\textbf{Conditional Maps.}\nWe consider three possible shapes of the conditional map to prove the effectiveness and generality of our method. Sample images are reported in Figure~\\ref{fig:maps}.\n\n\\medskip\n\\noindent \\textit{Triangle Map.}\nIn this type of annotation, the user has to provide an oriented triangle which outlines the size, base and orientation of the hand palm.\nThis conditional map is easy to draw, as it is possible to provide a simple interface where users can draw a triangle simply specifying the three delimiting points of the triangle, plus its base.\nMoreover, the triangle conditional map is category independent, as it does not contain any information about the gesture.\nWe annotated all images for both datasets with the corresponding triangle conditional maps.\n\n\\medskip\n\\noindent \\textit{Boundary Map.}\nIn the boundary map annotation, the user has to draw the contour of the desired target gesture. This type of annotation is weakly category dependent, since from the conditional map (see Figure~\\ref{fig:maps} in the center) it may be possible to infer the target gesture category. However, as this shape is easy to draw, it may be a valid alternative to the skeleton and triangle maps.\nWe annotated all 1,320 images for the NTU Hand dataset with the corresponding boundary annotations.\n\n\\medskip\n\\noindent \\textit{Skeleton Map.}\nIn the skeleton map, the user is required to draw either a complicated skeleton of the hand gesture or the exact position of the hand gesture key-points.\nHowever, when the target gesture image is available, it is sometimes possible to draw them automatically. \nAs in~\\cite{tang2018gesturegan}, we obtain the skeleton conditional maps by connecting the key-points obtained through OpenPose, a state of the art hand key-points detector~\\cite{simon2017hand, cao2018openpose}.\nIn the case of \\textit{normal} experimental setting, the hand pose is detected for all the 647 and 494 images of the two datasets. Instead, in the case of \\textit{challenging} experimental setting, the key-points could not be obtained for over half of the image set. \nTo this reason, the skeleton map is considered only in the \\textit{normal} experimental setting.\nThis conditional map is hard to draw, and strongly dependent on the category of hand gesture.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.16\\columnwidth]{figures\/maps\/S2.jpg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.16\\columnwidth]{figures\/maps\/S.png}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.16\\columnwidth]{figures\/maps\/B2.jpg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.16\\columnwidth]{figures\/maps\/B.png}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.16\\columnwidth]{figures\/maps\/T2.jpg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.16\\columnwidth]{figures\/maps\/T.png}}\n\t\\caption{The three considered shapes of the conditional map, sorted by user drawing effort: from (left) the most difficult to (right) the easiest to draw.}\n\t\\label{fig:maps}\n\\end{figure}\n\n\n\\subsection{Evaluation}\n\\label{Sec:evaluation}\n\n\\textbf{Baseline models.}\nAs our baseline models we select the state of the art for hand gesture translation \\emph{in the wild} GestureGAN~\\cite{tang2018gesturegan}. \nAlso, we adopt StarGAN~\\cite{choi2018stargan}, GANimation~\\cite{pumarola2018ganimation}, and PG$^2$~\\cite{ma2017pose} as they showed impressive results on multi-domain image-to-image translation. \nBoth StarGAN and GANimation learn to use attribute vectors to transfer facial images from one expression to another one. GestureGAN learns to transfer hand gestures via category-dependent skeleton maps.\n\n\\medskip\n\\noindent \\textbf{Evaluation metrics.}\nWe quantitatively evaluate our method performance using two metrics that measure the quality of generated images, namely Peak Signal-to-Noise Ratio (PSNR) and Fr\u00e9chet Inception Distance (FID)~\\cite{NIPS2017_7240}, and the F1-score, which measures whether the generated images depict a consistent category label. \nMoreover, to be comparable with GestureGAN~\\cite{tang2018gesturegan}, we employ the Mean Squared Error (MSE) between pixels and the Inception Score (IS)~\\cite{gulrajani2017improved}.\nHowever, these two metrics were indicated as highly unstable~\\cite{barratt2018note,borji2019pros}\nand they are not strictly related to the quality of generated images. To this reason, we report their results on a separate table and do not further discuss them. \n\n\\medskip\n\\noindent \\textit{PSNR.} It compares two images through their MSE and the maximal pixel intensity ($MAX_I = 255$). It is defined as: $PSNR = 20 \\log_{10}(\\frac{MAX_I}{\\sqrt{MSE}})$.\n\n\n\\medskip\n\\noindent \\textit{FID.} It is defined as the distance between two Gaussian with mean and covariance $(\\mu_x, \\Sigma_x)$ and $(\\mu_y, \\Sigma_y)$: $FID(x,y) = || \\mu_x- \\mu_y ||^2_2 + Tr(\\Sigma_x + \\Sigma_y - 2(\\Sigma_x\\Sigma_y)^{1\/2})$, where the two Gaussians are defined in the feature space of the Inception model.\n\n\\medskip\n\\noindent \\textit{F1.} The F1-score for binary classifiers is defined as $F_1 = (2 p r)\/(p+r)$ where $p$ and $r$ are the precision and recall. For multi-class classifiers it can be defined as the sum of \nthe F1-scores of each class, weighted by the percentage of labels belonging to the class.\nThe resulting measure ranges from 0 to 1, where 1 means that all the classes are correctly classified and 0 the opposite. Thus, we use the F1-score to evaluate the category consistence of our generated images. To compute the F1-score, we train a network to recognize hand gestures. Further details are provided in the Implementation Details.\n\n\n\\medskip\n\\noindent \\textit{Perceptual User study.} \nWe run a ``fake'' vs ``real'' perceptual user study following the same protocol as~\\cite{tang2018gesturegan,yang2018crossing}. Users are shown a pair of images, the original and the generated image, and they are asked to select the fake one. The images are displayed for only 1 second before the user can provide his or her choice. The image pairs are randomly shuffled in order to avoid introducing bias in the results. \nOverall, we collected perceptual annotations from 12 users and each user was asked to vote for 98 image comparisons. Specifically, 12 image translation pairs were selected for each dataset and experimental setting. \n\n\n\n\\begin{table}[t]\n\t\\centering\n\t\\caption{Quantitative comparison for the gesture-to-gesture translation task in \\textit{normal} experimental setting, using the same evaluation metrics as GestureGAN~\\cite{tang2018gesturegan}.}\n\n\t\\resizebox{0.99\\linewidth}{!}{ \n\t\\begin{tabular}{@{}l rrr rrr@{}}\n\t\\toprule\n\t\t\\textbf{Model}  & \n\t\t\\multicolumn{3}{c}{\\textbf{NTU Hand Gesture~\\cite{ren2013robust}}} & \\multicolumn{3}{c}{\\textbf{Creative Senz3D~\\cite{memo2018head}}} \\\\\n\t\t\\cmidrule(r{4pt}){2-4} \\cmidrule(l{4pt}){5-7}\n\t\t& MSE & PSNR & IS & MSE & PSNR & IS\n\t\t\\\\\n\t\t\\midrule\n\t\tPG$^2$~\\cite{ma2017pose}  & 116.10 & 28.24 & 2.42 & 199.44 & 26.51 & 3.37\n\t\t\\\\\n\t\tYan \\emph{et al.}~\\cite{yan2017skeleton}  & 118.12 & 28.02 & 2.49 & 175.86 & 26.95 & 3.33\n\t\t\\\\\n\t\tMa \\emph{et al.}~\\cite{ma2018disentangled}  & 113.78 & 30.65 & 2.45 & 183.65 & 26.95 & 3.38\n\t\t\\\\\n\t\tPoseGAN \\emph{et al.}~\\cite{siarohin2018deformable}  & 113.65 & 29.55 & 2.40 & 176.35 & 27.30 & 3.21\n\t\t\\\\\n\t\tGestureGAN~\\cite{tang2018gesturegan}  &  105.73 & 32.61 & \\textbf{2.55} & 169.92 & 27.97 & \\textbf{3.41}\n\t\t\\\\\n\t\t${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}~- no rolling &\n\t\t31.80 & 33.57 & 1.92 &\n\t\t46.79 & 31.98 & 2.31\n\t\t\\\\\n\t\t${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN} &\n\t\t\\textbf{15.76} & \\textbf{36.51} & 2.00 &\n\t\t\\textbf{21.73} & \\textbf{35.39} & 2.34\n\t\t\\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t}\n\t\\label{tab:QuantitativeResults_normal}\n\\end{table}\n\n\n\n\n\n\n\\begin{table*}[t]\n\t\\centering\n\t\\caption{Quantitative results for the gesture-to-gesture translation task for the two experimental settings.}\n\n\t\\begin{tabularx}{\\textwidth}{@{}Xccrrrrrrrrrrrrrrrrrrr@{}}\n\t\\toprule\n\t\t\\textbf{Model}  & \\textbf{Experimental setting} & \\textbf{Easy to draw map} & \\phantom{a} &\n\t\t\\multicolumn{3}{c}{\\textbf{NTU Hand Gesture~\\cite{ren2013robust}}} & \\phantom{a} & \\multicolumn{3}{c}{\\textbf{Creative Senz3D~\\cite{memo2018head}}} \\\\\n\t\t\\cmidrule{5-7} \\cmidrule{9-11}\n\t\t& & && PSNR & FID & F1 && PSNR & FID & F1\n\t\t\\\\\n\t\t\\midrule\n\t\tGANimation~\\cite{pumarola2018ganimation}  &  \\multirow{6}{*}{\\textit{Normal}}  & \\multirow{6}{*}{$\\times$} &&  10.03 & 440.45 & 0.08 && 11.11 & 402.99 & 0.23 \n\t\t\\\\\n\t\tStarGAN~\\cite{choi2018stargan}  &&&&  17.79  & 98.84 & 0.09 && 11.44 & 137.88 & 0.07 \n\t\t\\\\\n\t\tPG$^2$~\\cite{siarohin2018deformable}  &&&& 21.71 & 66.83 & 0.15 && 21.78 & 122.08 & 0.68\n\t\t\\\\\n\t\tGestureGAN~\\cite{tang2018gesturegan}  &&&&  34.24 & 15.06 & \\textbf{0.93} &  & 28.65 &  54.00 & 0.96\n\t\t\\\\\n\t\t${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ - no rolling &&&&\n\t\t34.07 & 20.29 & 0.90 &&\n\t\t31.98 & 58.28 & 0.88\n\t\t\\\\\n\t\t${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN} &&&&\n\t\t\\textbf{36.51} & \\textbf{13.73} & \\textbf{0.93} &&\n\t\t\\textbf{35.39} & \\textbf{32.04} & \\textbf{0.99}\n\t\t\\\\\n\t\t\\midrule\n\t\t\\midrule\n\t\tPG$^2$~\\cite{ma2017pose}  & \\multirow{5}{*}{\\textit{Challenging}}  & \\multirow{5}{*}{\\checkmark} && 21.94 & 135.32 & 0.10 && 18.30 & 265.73 & 0.10\n\t\t\\\\\n\t\tGestureGAN~\\cite{tang2018gesturegan}  &&  && 25.60 & 94.10 & 0.38 & & 18.69 & 254.37 & 0.34\n\t\t\\\\\n\t\tGestureGAN$^\\dag$  &&  && 27.32 & 59.79 & 0.43 & & 22.24 & 192.77 & 0.38\n\t\t\\\\\n\t\t${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ - no rolling     &&& &    27.14 & 61.14 & 0.43  && 22.77 & 134.54 & 0.33\n\t\t\\\\\n\t\t${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}     &&&&    \\textbf{28.11} & \\textbf{47.88} & \\textbf{0.61} && \\textbf{23.11} & \\textbf{101.22} & \\textbf{0.58}\n\t\t\\\\\n\t\t\\bottomrule\n\t\\end{tabularx}\n\t\\label{tab:QuantitativeResults_both}\n\\end{table*}\n\\begin{sloppypar}\n\n\n\n\\medskip\n\\noindent \\textbf{Implementation Details.}\nInspired by previous methods~\\cite{zhu2017unpaired,choi2018stargan}, both $E_1$ and $E_2$ are composed of two convolutional layers with the stride size of two for downsampling, $F_{res}$ refers to six residual blocks~\\cite{he2016deep}, and $F_{dec}$ is composed of two transposed convolutional layers with the stride size of two for upsampling. \nWe train our model using Adam~\\cite{kingma2014adam} with $\\beta_1=0.5$ and $\\beta_2=0.999$ and batch size 4. We use an $n$-dimensional one-hot vector to represent the category label ($n=10$ for NTU dataset and $n=11$ for Senz3D Dataset). For data augmentation we flip the images horizontally with a probability of 0.5 and we reverse the ``A-to-B\" direction with a probability of 0.5. The initial learning rate is set to $0.0002$. We train for 20 epochs and linearly decay the rate to zero over the last 10 epochs. To reduce model oscillation~\\cite{goodfellow2016nips}, we follow previous works~\\cite{shrivastava2017learning,zhu2017unpaired} and update the discriminators using a history of generated images rather than the ones produced by the latest generators. \nWe use instance normalization~\\cite{ulyanov2016instance} for the generator $G$.\nFor all the experiments, the weight coefficients for the loss term in Eq.~\\ref{eq:loss_D} and Eq.~\\ref{eq:loss_G} are set to $\\lambda_{D} = 1$, $\\lambda_{G} = 2$,\n$\\lambda_{cls} = 1$, $\\lambda_{rec}=100$, $\\lambda_{idt}=10$, $\\lambda_{cyc}=10$ and $\\lambda_{tv}=1e-5$.\nBaseline models are optimized using the same settings described in the respective articles.\nWe used the source code released by the authors for all competing works, except for GestureGAN, which was implemented from scratch following the description of the original article~\\cite{tang2018gesturegan}.\n${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}~is implemented using the deep learning framework PyTorch.\n\nTo compute the F1-score, we train a network on hand gesture recognition using Inception v3~\\cite{szegedy2016rethinking} network fine tuned on the NTU Hand Gesture and Creative Senz3D datasets. \nThe network achieves F1-score 0.93 and 0.99 on Creative Senz3D and NTU Hand Gesture test sets, respectively. Additional details on the training can be found in the supplementary materials.\n\n\n\\end{sloppypar}\n\n\n\n\n\n\n\n\n\n\\begin{figure*}[ht]\n\t\\centering\n\t\\subfloat[Source]{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/P1-G1-3-AB-P1-G5-6_real_A.jpeg}}\\hfill\n\t\\subfloat[Target skeleton]{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/P1-G1-3-AB-P1-G5-6_cond_B.jpeg}}\\hfill\n\t\\subfloat[Ground truth]{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/P1-G1-3-AB-P1-G5-6_real_B.jpeg}}\\hfill\n\t\\subfloat[GANimation~\\cite{pumarola2018ganimation}]{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/P1-G1-3-AB-P1-G5-6_fake_B_masked.jpeg}}\\hfill\n\t\\subfloat[StarGAN~\\cite{choi2018stargan}]{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/P1-G1-3-AB-P1-G5-6_fake_B.jpeg}}\\hfill\n\t\\subfloat[PG$^2$~\\cite{ma2017pose}]{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/P1-G1-3-AB-P1-G5-6_fake_B2.jpeg}}\\hfill\n\t\\subfloat[GestureGAN~\\cite{tang2018gesturegan}]{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/P1-G1-3-AB-P1-G5-6_fake_B_gest.jpeg}}\\hfill\n\t\\subfloat[$\\pmb{\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN} ]{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/P1-G1-3-AB-P1-G5-6_fake_B2_masked_roll.jpeg}}\n\n\t%\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n    \\\\\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/S1-G10-6-color-AB-S1-G7-18-color_real_A.jpeg}\n\t\t}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/S1-G10-6-color-AB-S1-G7-18-color_cond_B.jpeg}}\\hfill\n\t\\subfloat{%\n\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/S1-G10-6-color-AB-S1-G7-18-color_real_B.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/S1-G10-6-color-AB-S1-G7-18-color_fake_B_masked.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/S1-G10-6-color-AB-S1-G7-18-color_fake_B.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/S1-G10-6-color-AB-S1-G7-18-color_fake_B2_pg2.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/S1-G10-6-color-AB-S1-G7-18-color_fake_B_gesture.jpeg}}\\hfill\n\t\\subfloat{\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/S1-G10-6-color-AB-S1-G7-18-color_fake_B2_masked_rolling.jpeg}}\n\t\\\\\n\t\\subfloat\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/S2-G4-9-color-AB-S2-G8-4-color_real_A.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/S2-G4-9-color-AB-S2-G8-4-color_cond_B.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/S2-G4-9-color-AB-S2-G8-4-color_real_B.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/S2-G4-9-color-AB-S2-G8-4-color_fake_B_masked.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/S2-G4-9-color-AB-S2-G8-4-color_fake_B.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/S2-G4-9-color-AB-S2-G8-4-color_fake_B2.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/S2-G4-9-color-AB-S2-G8-4-color_fake_B_gesture.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.12\\textwidth]{figures\/qualitative\/S2-G4-9-color-AB-S2-G8-4-color_fake_B2_masked_roll.jpeg}}\n\n\t\\caption{Qualitative comparison between ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}~and competing works in the \\textit{normal} experimental setting. NTU Hand Gesture dataset (top two rows) and Creative Senz3D (bottom two rows).}\n\t\\label{fig:res_qual_normal}\n\\end{figure*}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\subfloat[Source]{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/P1-G2-6-AB-P1-G9-8_real_A.jpeg}}\\hfill\n\t\\subfloat[Target]{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/P1-G2-6-AB-P1-G9-8_real_B.jpeg}}\\hfill\n\t\\subfloat[PG$^2$]{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/P1-G2-6-AB-P1-G9-8_diff_map.jpeg}}\\hfill\n\t\\subfloat[GANimation]{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/S1-G1-16-color-AB-S1-G4-26-color_fake_B_mask.png}}\\hfill\n\t\\subfloat[$\\pmb{\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}]{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/P1-G2-6-AB-P1-G9-8_fake_B2_mask.jpeg}}\n\t\\\\\n\t%\n\n\n\n\n\n\n\n\n\n\n\n\t%\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/S1-G1-16-color-AB-S1-G4-26-color_real_A.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/S1-G1-16-color-AB-S1-G4-26-color_real_B.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/S1-G1-16-color-AB-S1-G4-26-color_diff_map.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/S1-G1-16-color-AB-S1-G4-26-color_fake_B_mask.png}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/S1-G1-16-color-AB-S1-G4-26-color_fake_B2_mask.jpeg}}\n\n\t%\n\n\n\n\n\n\n\n\n\n\n\n\t\\caption{Masks computed by the various state of the art methods. Specifically, PG$^2$ computes a difference map that is noisy, GANimation fails at computing the attention mask, while $\\pmb{\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}~computes the attention of the pixels that stay constant from source to target images.}\n\t\\label{fig:attention}\n\\end{figure}\n\n\n\n\n\n\n\n\\begin{figure*}[ht]\n\t\\centering\n\t\\begin{minipage}{.88\\textwidth}\n\t\\centering\n\t\\subfloat[Source]{%\n\t\t\\includegraphics[width=0.137\\textwidth]{figures\/qualitative\/P1-G2-6-AB-P1-G9-9_real_A.jpeg}}\\hfill\n\t\\subfloat[Target triangle]{%\n\t\t\\includegraphics[width=0.137\\textwidth]{figures\/qualitative\/P1-G2-6-AB-P1-G9-9_cond_B.jpeg}}\\hfill\n\t\\subfloat[Category label]{%\n\t\t\\squarecat{\"9\"}}\\hfill\n\t\\subfloat[Ground truth]{%\n\t\t\\includegraphics[width=0.137\\textwidth]{figures\/qualitative\/P1-G2-6-AB-P1-G9-9_real_B.jpeg}}\\hfill\n\t\\subfloat[PG$^2$]{%\n\t\t\\includegraphics[width=0.137\\textwidth]{figures\/qualitative\/P1-G2-6-AB-P1-G9-9_fake_B1.jpeg}}\\hfill\n\t\\subfloat[GestureGAN]{%\n\t\t\\includegraphics[width=0.137\\textwidth]{figures\/qualitative\/P1-G2-6-AB-P1-G9-9_fake_B.jpeg}}\\hfill\n\t\\subfloat[$\\pmb{\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}]{%\n\t\t\\includegraphics[width=0.137\\textwidth]{figures\/qualitative\/P1-G2-6-AB-P1-G9-9_fake_B2_masked.jpeg}} \n\t\t\\\\\n   \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.137\\textwidth]{figures\/qualitative\/tria\/S1-G1-16-color-AB-S1-G4-26-color_real_A.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.137\\textwidth]{figures\/qualitative\/tria\/S1-G1-16-color-AB-S1-G4-26-color_cond_B.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\squarecat{\"4\"}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.137\\textwidth]{figures\/qualitative\/tria\/S1-G1-16-color-AB-S1-G4-26-color_real_B.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.137\\textwidth]{figures\/qualitative\/tria\/S1-G1-16-color-AB-S1-G4-26-color_fake_B2.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.137\\textwidth]{figures\/qualitative\/tria\/S1-G1-16-color-AB-S1-G4-26-color_fake_B.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.137\\textwidth]{figures\/qualitative\/tria\/S1-G1-16-color-AB-S1-G4-26-color_fake_B2_masked.jpeg}}\n\t\\end{minipage}\n\n\t\\caption{Qualitative comparison between ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}~and competing works in the \\textit{challenging} experimental setting. NTU Hand Gesture dataset (top row) and Creative Senz3D (bottom row).\n\t}\n\t\\label{fig:res_qual_challenging}\n\t\n\\end{figure*}\n\n\n\n\n\n\\section{Results}\n\\noindent \\textbf{Quantitative Results.} \nWe begin by directly comparing ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ with the same experimental protocol and metrics used by our most similar competitor, GestureGAN~\\cite{tang2018gesturegan}.\nTable~\\ref{tab:QuantitativeResults_normal} shows that ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ performs better than all competing works in terms of MSE and PSNR, especially when the rolling guidance is employed. In terms of IS GestureGAN performs better.\nHowever, the MSE and the IS are not directly related to the quality of the generated images. The MSE is indeed a metric of pixel difference, while the low reliability of the Inception Score is well known~\\cite{barratt2018note}.\n\nFor this reason, we compare our method with competing works using the PSNR, F1-score and the FID score, both in \\textit{normal} and \\textit{challenging} experimental settings. These metrics compare the diversity, quality and hand-gesture consistency of the generated images.\nTable~\\ref{tab:QuantitativeResults_both} shows that ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ outperforms all competing works, in both the experimental settings and for all the metrics.\nIn the \\emph{normal} setting, compared to GestureGAN, we achieve higher PSNR (36.51 vs 34.24 and 35.39 vs 28.65), F1-score (0.99 vs 0.96) and lower FID (13.73 vs 15.06 and 32.04 vs 54.00) in both datasets. \nOther methods perform particularly poor in terms of F1-score. \nFor example, GANimation and StarGAN perform near randomness in the F1-score (random baseline $\\sim 0.09$), while ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}~ achieves near-perfect performance in Creative Senz3D ($0.99$)  and NTU Hand Gesture ($0.93$).\nThese outcomes might be related to the fact that GANimation, StarGAN and PG$^2$ are not designed for image translation \\emph{in the wild}.\n\n${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ significantly outperforms the competing methods in the \\emph{challenging} setting, where we enforce a stricter division of training and test gestures, and we use the 100\\% of the data, differently from GestureGAN's settings.\nIn particular, the FID score reduces by 49\\% and 60\\% from GestureGAN in NTU Hand Gesture and Creative Senz3D, respectively. \nIn terms of F1-score, the result improves by 61\\% and 71\\% in NTU Hand Gesture and Creative Senz3D, respectively. \nThe PSNR improves by 10\\% and 24\\% in NTU Hand Gesture and Creative Senz3D, respectively. We note that the rolling guidance applied to GestureGAN (denoted in Table~\\ref{tab:QuantitativeResults_both} as GestureGAN$^\\dag$) improves the original FID results by 36\\% and 24\\% in NTU Hand Gesture and Creative Senz3D, respectively. \n\n\nAltogether, the quantitative results show that ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ outperforms the state of the art, both in the \\emph{normal} and \\emph{challenging} setting. \n\n\\begin{sloppypar}\n\\medskip\n\\noindent \\textbf{Qualitative Results.} Figure~\\ref{fig:res_qual_normal} shows some randomly selected gesture-to-gesture translations in the \\textit{normal} experimental setting. \nBoth GestureGAN~\\cite{tang2018gesturegan} and ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ produce sharper output results while the output gestures from PG$^2$~\\cite{ma2017pose} are very blurry. ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ also produces a better define sharper background than GestureGAN. \nStarGAN and GANimation, however, fail to produce gestures from the provided conditional map.\n\nWe further inspect the reason behind the poor results of PG$^2$ and GANimation. \nThese methods focus on multi-domain translation and are specifically tailored to the task of facial expression translation and to the cases where the input and target objects are aligned. \nFigure~\\ref{fig:attention} depicts two sample cases of the difference and attention masks generated by these methods. It can be seen that PG$^2$ fails at finding the difference mask, especially in the NTU Hand Gesture dataset (top row). Similarly, GANimation generates an empty attention mask, which might also be the cause of its poor results in both Figure~\\ref{fig:res_qual_normal} and Table~\\ref{tab:QuantitativeResults_both}. \n${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}, instead, learns to focus on the differences between the source and target images.\n\nFigure~\\ref{fig:res_qual_challenging} shows the results in the \\textit{challenging} setting. \n${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ generates sharper images with recognizable hand gestures, while other methods such as GestureGAN fails at it. \nThis result is in line with the F1-scores reported in Table~\\ref{tab:QuantitativeResults_both} (bottom), which range within 0.10 and 0.38 for the competing works, and within 0.58 and 0.61 in case of ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}. \nBoth qualitative and quantitative results confirm that state-of-the-art methods are not adapted to perform gesture translation in the challenging settings, i.e. where a user-friendly category independent conditional map is provided, and where the network is asked to translate to an unseen gesture for a given user.\nWe refer to the Supplementary Material for additional qualitative Figures.\n\\end{sloppypar}\n\n\n\n\n\\begin{figure}[!htp]\n\t\\centering\n\t\\subfloat[Source]{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/diversity\/S2-G6-4-color-S2-G9-5-color-1_real_A.jpeg}}\\hfill\n\t\\subfloat[Target map]{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/diversity\/S2-G6-4-color-S2-G9-5-color-1_cond_B.jpeg}}\\hfill\n\t\\subfloat[\"7\"]{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/diversity\/S2-G6-4-color-S2-G9-5-color-7_fake_B2_masked.jpeg}}\\hfill\n\t\\subfloat[\"9\"]{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/diversity\/S2-G6-4-color-S2-G9-5-color-9_fake_B2_masked.jpeg}}\\hfill\n\t\\subfloat[\"11\"]{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/diversity\/S2-G6-4-color-S2-G9-5-color-11_fake_B2_masked.jpeg}}\n\t\\\\ \n\t%\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/diversity\/BOS2-G6-4-color-S2-G9-5-color-1_real_A.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/diversity\/BOS2-G6-4-color-S2-G9-5-color-1_cond_B.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/diversity\/BOS2-G6-4-color-S2-G9-5-color-7_fake_B2_masked.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/diversity\/BOS2-G6-4-color-S2-G9-5-color-9_fake_B2_masked.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/diversity\/BOS2-G6-4-color-S2-G9-5-color-7_fake_B2_masked.jpeg}}\n\t\\\\ \n\t\\subfloat{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/diversity\/SKS2-G6-4-color-S2-G9-5-color-1_real_A.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/diversity\/SKS2-G6-4-color-S2-G9-5-color-1_cond_B.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/diversity\/SKS2-G6-4-color-S2-G9-5-color-7_fake_B2_masked.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/diversity\/SKS2-G6-4-color-S2-G9-5-color-9_fake_B2_masked.jpeg}}\\hfill\n\t\\subfloat{%\n\t\t\\includegraphics[width=0.192\\columnwidth]{figures\/qualitative\/diversity\/SKS2-G6-4-color-S2-G9-5-color-11_fake_B2_masked.jpeg}}\n\n\t\\caption{${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}~decouples the conditional map from the category label of the hand gesture. The same conditional map can be used with different category labels to generate multiple images. \n\t}\n\t\\label{fig:diversity}\n\n\\end{figure}\n\\begin{figure}[!ht]\n\t\\centering\n\t\\renewcommand{\\tabcolsep}{1pt}\n\t\\subfloat{\\footnotesize\\begin{tabular}{>{\\centering\\arraybackslash}p{0.39\\columnwidth}>{\\centering\\arraybackslash}p{0.58\\columnwidth}}\n         \\textbf{Source} & \\textbf{Conditional maps and generated images} \\\\ \n    \\end{tabular}}\\\\\n\t%\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n    \\subfloat{%\n\t\t\\includegraphics[width=0.39\\columnwidth,height=0.396\\columnwidth]{figures\/qualitative\/shift\/S1-G1-21-color-S1-G7-8-color-7-a_real_A.jpeg}}\\hfill\n\t\\subfloat{\\begin{minipage}{0.192\\columnwidth}%\n\t\t\t\\setlength{\\lineskip}{3pt}%\n\t\t\t\\includegraphics[width=\\columnwidth]{figures\/qualitative\/shift\/S1-G1-21-color-S1-G7-8-color-7-13_cond_B.jpeg}\\\\\n\t\t\t\\includegraphics[width=\\columnwidth]{figures\/qualitative\/shift\/S1-G1-21-color-S1-G7-8-color-7-13_fake_B2_masked.jpeg}\n\t\\end{minipage}}\\hfill\n\t\\subfloat{\\begin{minipage}{0.192\\columnwidth}%\n\t\t\t\\setlength{\\lineskip}{3pt}%\n\t\t\t\\includegraphics[width=\\columnwidth]{figures\/qualitative\/shift\/SKS1-G1-21-color-S1-G7-8-color-7-6_cond_B.jpeg}\\\\\n\t\t\t\\includegraphics[width=\\columnwidth]{figures\/qualitative\/shift\/SKS1-G1-21-color-S1-G7-8-color-7-6_fake_B2_masked.jpeg}\n\t\\end{minipage}}\\hfill\n\t\\subfloat{\\begin{minipage}{0.192\\columnwidth}%\n\t\t\t\\setlength{\\lineskip}{3pt}%\n\t\t\t\\includegraphics[width=\\columnwidth]{figures\/qualitative\/shift\/SKS1-G1-21-color-S1-G7-8-color-7-7_cond_B.jpeg}\\\\\n\t\t\t\\includegraphics[width=\\columnwidth]{figures\/qualitative\/shift\/SKS1-G1-21-color-S1-G7-8-color-7-7_fake_B2_masked.jpeg}\n\t\\end{minipage}}\n\n\t\\caption\n\t${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}~generates images with the same conditional map that is rotated, shifted and resized.\n\t}\n\t\\label{fig:diversity2}\n\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\\medskip\n\\noindent \\textbf{Diversity Study.} ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ decouples the desired hand gesture category, specified through a class number, from its location, scale and orientation, which are specified by a conditional map.\nThis means that users can use the same conditional map with different hand gesture categories to generate multiple images.\nFigure~\\ref{fig:diversity} shows that ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ can generate three different distinguishable hand gestures by using the same \\emph{triangle} conditional map and different category numbers.\nInstead, when using non category-independent shapes are used (e.g. boundary or skeleton), ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ fails to synthesize several hand gestures categories from the same conditional map. \nAs mentioned before, \\emph{Boundary} maps are weakly category dependent, as their shape might suggest the type of gesture, while \\emph{Skeleton} and \\emph{Key-points} maps are category dependent. \n\nFurthermore, we test the performance of ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ with the same source image and category, but with different conditional maps. \nFor this test, we manually draw three triangle conditional maps with arbitrary size, location and orientation.\nFigure~\\ref{fig:diversity2} shows that our method faithfully synthesizes the target gestures in all cases.\nAltogether, we show that users can generate hand gestures \\emph{in the wild} with much less effort than state-of-the-art models, that require complex annotations that are dependent on the specific hand gesture users want to generate. \n\n\n\n\n\\begin{table}[t]\n    \\caption{Perceptual user study. Percentage of times, on average, when the translated images are selected as ``real'' by users, in the ``fake'' vs. ``real'' comparison.}\n   \n    \\centering\n    \\small\n    \\begin{tabularx}{\\columnwidth}{@{}X rr rrr@{}}\n    \\toprule \n        \\textbf{Model} & \\multicolumn{2}{c}{\\textbf{NTU Hand Gesture~\\cite{ren2013robust}}} & \n        \\multicolumn{2}{c}{\\textbf{Creative Senz3D~\\cite{memo2018head}}}  \\\\\n        \\cmidrule(r{4pt}){2-3} \\cmidrule(l{4pt}){4-5}\n        & \\textit{Normal} & \\textit{Challenging} & \\textit{Normal} & \\textit{Challenging} \\\\\n         \\midrule\n          GestureGAN & 36.54\\% & 3.21\\% & 3.85\\% & 0.64\\% \\\\\n          ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN} & \\textbf{44.32\\%} & \\textbf{7.05\\%} & \\textbf{16.03\\%} & \\textbf{3.85\\%} \\\\\n          \\bottomrule\n    \\end{tabularx}\n    \\label{tab:userstudy}\n\\end{table}\n\n\\medskip\n\\noindent \\textbf{Perception User Study.}\nTable~\\ref{tab:userstudy} shows the outcome of the perceptual user study, where we report the percentage of times when the translated image wins against the original target image in the real vs fake comparison, i.e when the original image is selected as fake. It can be seen that ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ outperforms GestureGAN~\\cite{tang2018gesturegan} in both experimental settings. \nThis means that ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ generates higher quality images that can be mistaken with the real ones at higher rates than competitors.\n\n\n\\begin{table}[t]\n\t\\centering\n\t\\small\n\t\\caption{Performance degradation of $\\pmb{\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ by removing the rolling guidance approach.\n\t}\n\n\t\\begin{tabularx}{\\columnwidth}{@{}X c rrr@{}}\n\t\\toprule\n\t\t\\textbf{Conditional map}   & \\textbf{Category} & \n\t\t\\multicolumn{3}{c}{\\textbf{Creative Senz3D~\\cite{memo2018head}}} \\\\ \n\t\t\\cmidrule{3-5} \n\t\t& \\textbf{independent} & PSNR & FID & F1 \n\t\t\\\\\n\t\t\\midrule\n\t\t\\emph{Triangle} & \\checkmark & 1.53\\% \\textcolor{red}{\\textdownarrow}  & 24.77\\% \\textcolor{red}{\\textuparrow} & 75.76\\% \\textcolor{red}{\\textdownarrow}\n\t\t\\\\\n\t\t\\emph{Boundary} & $\\sim$ &\n\t\t 1.84\\% \\textcolor{red}{\\textdownarrow}  & 35.37\\% \\textcolor{red}{\\textuparrow} & 58.06\\% \\textcolor{red}{\\textdownarrow}\n\t\t\\\\\n\t\t\\emph{Skeleton} & $\\times$ &\n\t\t 10.66\\% \\textcolor{red}{\\textdownarrow}  & 81.90\\% \\textcolor{red}{\\textuparrow} & 12.50\\% \\textcolor{red}{\\textdownarrow}\n\t\t\\\\\n\t\t\\bottomrule\n\t\\end{tabularx}\n\t\\label{tab:ablation}\n\t\\vspace{-0.3cm}\n\\end{table}\n\n\n\\medskip\n\\noindent \\textbf{Ablation study.} We further investigate the effect of the rolling guidance approach by removing this component from ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}. \nIn the quantitative results, Table~\\ref{tab:QuantitativeResults_both} it can be seen that the rolling guidance allows to significantly improves the quality of the generated images.\nIn Table~\\ref{tab:ablation} we report the degradation of our method performance without rolling guidance, on Creative Senz3D dataset, for the three types of annotation maps. \nSpecifically, we observe 24.77\\%, 35.37\\% and 81.90\\% worse (increased) FID score for the \\emph{Triangle}, \\emph{Boundary} and \\emph{Skeleton} maps, respectively. \nWhile the F1-score decreases by 75.76\\% and 58.06\\% on the \\emph{Triangle} and \\emph{Boundary} maps respectively, for \\emph{Skeleton} maps it decreases by 12.50\\%. In terms of PSNR the degradation is less significant for the \\emph{Triangle} and \\emph{Boundary}, but higher (10.66\\%) for \\emph{Skeleton} maps.\nFinally, we observed that a single rolling is sufficient to improve the generated results, while ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ does not benefit from additional rolling iterations.\n\n\n\n\n\n\n\n\\section{Conclusion}\nWe have presented a novel GAN architecture for gesture-to-gesture translation \\textit{in the wild}. Our model decouples the conditional input into a category label and an easy-to-draw conditional map.\nThe proposed attention module and rolling guidance approach allow generating sharper images \\textit{in the wild}, faithfully generating the target gesture while preserving the background context.  \nExperiments on two public datasets have shown that ${\\mathlarger{\\mathlarger{\\bigtriangleup}}}$-{GAN}\\ outperforms state of the art both quantitatively and qualitatively.\nMoreover, it allows the use of simple-to-draw and category-independent conditional maps, such as triangles.\nThis significantly reduces the annotation effort both at drawing time and allowing to use a single map for multiple gesture translations. The proposed framework is not limited to category labels but can also be used through embedding, learned from the data, that can express the gesture characteristics. In future work users could easily provide a reference image instead of a label, and translate the original image to the target gesture expressed by the reference image.\n\nOur approach is especially important when the target image, and thus the target conditional map, is not available, which is the typical scenario in photo-editing software. \n\n\n\n\n\n\\begin{acks}\nWe gratefully acknowledge NVIDIA Corporation for the donation of the Titan X GPUs and Fondazione Caritro for supporting the SMARTourism project.\n\\end{acks}\n\n\n\\bibliographystyle{ACM-Reference-Format}\n\\balance\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nStudying regions close to the Galactic plane in the optical is difficult due both due to dust obscuration and source confusion.  It was only recently Feast et al. (2014) reported the discovery of five classical Cepheid variables at distances of 13 - 22 kpc from the Galactic center, towards the Galactic bulge, that may be associated with the flared atomic hydrogen disk of our Galaxy.  Two classical Cepheid variables at 11 kpc close to the plane of the Milky Way have been recently uncovered from VVV data (Dekany et al. 2015), that indicates an underlying young star cluster.  Searches for dwarf galaxies in the optical have primarily targeted high latitudes (McConnachie 2012).  The Sagittarius (Sgr) dwarf galaxy is the closest known dwarf galaxy to the plane, at a latitude of $b = -14^\\circ$ (Ibata et al. 1994).  The dearth of Milky Way satellites at low latitudes (Mateo 1998; McConnachie 2012) is underscored by simulations that suggest that there may be massive, nearly dark satellites that have not yet been discovered (Boylan-Kolchin et al. 2011).  Not only dwarf galaxies, but even bright spiral galaxies are not easily seen if they are hidden behind the obscuring column of dust and gas of the Galactic disk (Kraan-Korteweg et al. 1994).\n\nMining data from deep infrared surveys of the Galactic plane may well uncover new dwarf galaxies and halo sub-structure.  This would alleviate several outstanding problems in near-field cosmology.  The \"missing satellites problem\", or the overabundance of dwarf galaxies in cosmological simulations relative to the number of observed dwarf galaxies in and around the Local Group (Klypin et al. 1999), and the \"too big to fail problem\", wherein there are too few massive satellites in the Milky Way relative to cosmological simulations (Boylan-Kolchin et al. 2011) are two such outstanding problems.   Yet another is the ostensibly anisotropic distribution of the Milky Way satellites (Kroupa et al. 2005).  These discepancies may be resolved by a more complete inventory of the structure of our Galaxy at low latitudes.  \n\nWe have searched for distant stars close to the Galactic plane using near-infrared data from the ESO Public survey VISTA Variables of the Via Lactea (VVV) (Minniti et al. 2011; S12), targeting the VVV disk area, which covers Galactic longitudes $-65.3^\\circ < l <\n-10^\\circ$ within Galactic latitudes $-2.25^\\circ < b < +2.25^\\circ$.    The VVV survey is an ongoing 5-band photometric survey in the Z (0.87$~\\mu\\rm m$), Y (1.02$~\\mu\\rm m$), \nJ (1.25 $~\\mu\\rm m$), H (1.64 $~\\mu\\rm m$) and $K_{s}$ (2.14 $~\\mu\\rm m$) bands (S12),\nand is multi-epoch in the $K_{s}$ band, with approximately 30-40 epochs per star across the VVV disk area at the time of writing.  \nIn \\S \\ref{sec:results}, we review the methods we used to identify Cepheid variables, and present the distance and extinction values.  We discuss possible interpretations and conclude in \\S \\ref{sec:conclusion}.\n\n\\section{Results \\& Analysis}\n\\label{sec:results}\n\nThe infrared photometry is from the VVV survey, which is based on aperture photometry\ncomputed on the individual tile images (S12).\nEach of the sources was observed with a median exposure time of $16$ s\nper pixel, depending on the position in the tile (each exposure is $8$ s\nlong, and most of the area in a tile is a combination of two\npointings).  The limiting magnitude of the VVV data using aperture photometry\nis $K_{s} \\sim 18.0$ mag in most fields (S12).  \nA particular pointing is called a ``tile'', covers $\\sim1.64$ square\ndegree, and contains approximately $10^{6}$ stars.  \nAs a preliminary search, we examined the disk area of the VVV survey by applying color cuts that correspond to distant ($D > 60~\\rm kpc$) red-clump\nstars.  Red-clump stars have been shown to be good distance indicators (Alves 2000; Paczynski \\& Stanek 1998).\nGiven the mean values of intrinsic near-infrared colors for red-clump stars in the Milky Way disk and the Cardelli et al. (1989) extinction law, we used the distance modulus noted in Minniti et al. (2011), which gives a color cut of $1.5<(J-K_{\\rm s})<1.8$ and\n$K_{s} > 17.6$ (which corresponds to distances in excess of $\\sim$ 60 kpc).  Using this color cut, we saw an excess of distant red-clump stars \nat $l \\sim -27^{\\circ}$.   We defer a detailed analysis of the red-clump stars and other stellar populations to a future paper.\n\nWe  carried out a search for variable stars, restricting our search to faint variables, with mean $K_{s} > 15$ mag, and periods greater than three days.  We examined the variability data in five tiles close to Galactic longitude $l \\sim -27^\\circ$ and searched six comparison tiles at other locations in the VVV disk area.  We found four Cepheid variables at $l \\sim -27^\\circ$ at an average distance of $\\sim$ 90 kpc,\nand none in the other tiles.  The survey strategy ensures that the tiles in the VVV disk area have similar number of observations and limiting magnitude (S12).  While the control on the cadence is limited (Saito et al. 2013), we have checked that there is no significant difference in the cadence for the $l \\sim -27^\\circ$ tiles relative to the rest of the disk area,  i.e., the region at $l \\sim -27^\\circ$ is not unique in terms of the way it was observed.\n\nIn identifying Cepheids, we employed\nseveral successive tests.  The first two tests are based on the statistical significance of the highest peak in the Lomb-Scargle periodogram, and the uncertainty of the period, respectively.  These two tests ensure that we have identified sources of a given pulsation period, and that there is a small uncertainty in the period that we derive.  For the final cut, we quantitatively assess the shapes of the light curves by calculating the Fourier parameters of the sources, as well as the skewness and acuteness parameters, and visually inspect the light curves.  \n\nA given tile is searched using the Lomb-Scargle algorithm (Lomb 1976; Scargle 1982), and periodograms\nare constructed for every source.   The statistical significance of the amplitude of the largest peak in the periodogram (Scargle 1982)\ncorresponds to a false alarm probability $p_{0}$.  Claiming the detection of the signal if the amplitude\nexceeds the threshold value, one can expect to be wrong a fraction $p_{0}$ of the time.  Alternately, the statistical significance\nlevel of the detection is $1-p_{0}$, and this quantity is listed in Table 1.\nFor the first test, we require $\\bar{K_{s}} >$ 15 mag\nto search for faint variables, and set the minimum and maximum period range in our variability search\nbetween 3 - 50 days.  To pass the first test, sources have to satisfy the following conditions:\n1) the period corresponding to the maximum in the Lomb-Scargle periodogram is greater\nthan three days, 2) the maximum in the Lomb-Scargle periodogram exceeds 90-th pericentile \nfor the significance level, 3) if there are other maxima in the periodogram that are at 90-th percentile\nor higher, the periods corresponding to these maxima must differ by a factor of two or less.  \nThe last condition amounts to requiring a clean periodogram without spuriously large multiple peaks.\n\nIn the second test, we assess the quality of the light curves of the variables that pass the tests above with a\nparametric bootstrap.  Assuming a Gaussian distribution of errors, we sample the distribution one thousand times\nto derive the distribution of periods for each source, which is similar to prior work (Klein et al. 2012; Klein et al. 2014)\non RR Lyrae stars.\nIf the mean of the period distribution agrees to within 20 \\% of the period calculated from the raw data, and \nif the mean of the period distribution $\\pm$ the standard deviation still exceeds three days, we consider the\nperiod distribution to be sufficiently well constrained.   The goal of this second cut is to select sources that have a small uncertainty in the derived period, given the photometric errors.  \nThe Lomb-Scargle algorithm allows us to derive the period and its statistical signficance, but not the uncertainty in the derived period.   \nIf the width of the histogram that gives the distribution of periods from the bootstrap calculation is narrow, the uncertainty in the derived period is low.  For the sources that pass the above tests, we fit the light curves with a Fourier series (Kovacs \\& Kupi 2007).\n The Fourier parameters are similar to the light curves of classical Cepheids for $P \\sim 3 - 15$ days observed in the K-band (Persson et al. 2004; Bhardwaj et al. 2015).  The light curves of Cepheid variables in the optical are different in shape and amplitude from the light curves of Cepheid variables in the K-band (Matsunaga et al. 2013; Bhardwaj et al. 2015), and our comparison here is to the observed light curves of classical Cepheid variables in the K-band.  The Cepheids we list here pass all of the automated and visual checks.  Because of this multi-tiered, conservative approach, the light curve analysis is time consuming, but allows us to derive accurate distances.  It is worth noting that in addition to lower extinctions in the infrared relative to the optical, another advantage of infrared photometry of Cepheids is that it is minimally affected by metallicity variations (Bono et al. 2010; Freedman et al. 2010).\n\nThe tiles close to longitude $l \\sim -27^\\circ$ produce a significantly larger number of variables\nthat pass the first of our tests than the other six tiles we examined (at $l = -15^{\\circ},-29^{\\circ}, -35^{\\circ}, -40^{\\circ}, -50^{\\circ}, -65^{\\circ}$).  Figure 2 of S12 depicts the VVV survey area.  The number of sources in tiles d027, d065, d103, and d141 \nthat are centered at $l \\sim -27 ^\\circ$ and extend upwards in latitude (S12), produce $\\sim 100-200$ sources\nthat pass our first cut.  In contrast, the average number that pass the first cut from the comparison tiles is $\\sim$ 60.  If we consider \nthis background number to be the mean of a Poisson distribution, and randomly sample a Poisson distribution with this mean value, values in excess of 100 are above 5-$\\sigma$, i.e., they are statistically extremely unlikely to occur by chance.  Figure \\ref{f:comb} shows the number of sources that pass the first of our tests as a function of longitude (the value at $l \\sim -27 ^\\circ$ is an average over latitude), as well as a function of the total number of variable stars in the tile.  While the number of sources that pass the first of our tests has some correlation with the number of variable stars (which is not unexpected), the region at $l \\sim -27 ^\\circ$ is a clear outlier. In some tiles centered at $l \\sim -27 ^\\circ$, there were a significant number that passed our automated and visual analysis of the light curves, but did not pass our visual inspection of the images, due to the possibility of spikes or blending.  \nThe number of Cepheid variables that we report here from the final cut is very likely an underestimate.  \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{comb.png}\n\\caption{The number of sources that pass the first of our tests (requiring statistical significance greater than 90-th percentile, $P >$ 3 days, $K_{s} > 15~\\rm mag$) is shown as a function of longitude in the top panel, and as a function of the total number of variable stars in the tile (normalized to one million) in the bottom panel.  The error bars in the top-panel is from Poisson noise. \n\\label{f:comb}}\n\\end{center}\n\\end{figure}\n\nThe phase folded light curves of the Cepheid variables, which show a clear resemblance to each other, and corresponding images are shown in Figure 1.   Table 1 summarizes the derived distances and other parameters for the Cepheids.  To estimate the dust extinction from the excess color, we use the quasi-simultaneous\nsingle epoch VVV measurements in the $J$, $H$ and $K_{s}$ bands ($\\sim$ 190 s between each band).  The near-infrared amplitudes of \nclassical Cepheids are relatively small (Persson et al. 2004) and as such an estimate of the\ndust extinction from the single epoch measurements of the colour should be sufficient.\nThe average extinction-corrected $(J-K_{s})$ colours of the Cepheids is $\\sim$ 0.4, which is\nconsistent with the colors of short-period classical Cepheids (Persson et al. 2004) in the LMC.\n\n\\begin{table*}\n\\centering\n        \\caption{Data For Individual Cepheids and Derived Parameters}\n\n          \\begin{tabular}{@{}lccccccc@{}}\n          \\hline\n\nVVV ID  &       $l~\\rm (deg)$      &    $b~\\rm (deg)$        &    D (kpc)  &  P (day)       &  $\\bar{K_{s}}$       &  Significance Level  \\\\\n\\hline\n\nVVVJ162559.36-522234.0     &   -27.5971       &  -2.23686           &  92         &  3.42        & 16.04            &   91 \\%           \\\\\nVVV J162328.18-513230.4        &   -27.2729    &  -1.37557          &  100      &  4.19      &   16.12    &         93 \\% \\\\\nVVVJ162119.39-520233.3      &  -27.8621       & -1.49382            &  71        &   5.69        & 15.1              &    97 \\%           \\\\\nVVV J161542.47-494439.0       &  -26.8882    &     0.768427         &  93        &  13.9        & 15.6      &         98 \\%    \\\\\n\n\n\n\\hline\n\n\\end{tabular} \n\n\\small {VVV ID, Galactic longitude ($l$) and latitude ($b$), $D$ is the distance from the sun, $P$ is the pulsation period, $\\bar{K_{s}}$ is the mean $K_{s}$-band magnitude, the last column is the significance level of the highest peak in the Lomb-Scargle periodogram.  }\n\n\n\n\\end{table*} \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.65]{Figure1Cep.png}\n\\caption{$JHK_{\\rm s}$ false color\nimage of the Cepheid variables, with phase folded $K_{\\rm s}$-band\nlight curves. All fields are $30\"\\times30\"$, oriented in Galactic\ncoordinates. The VVV ID and period are also listed in Table 1, along\nwith the Galactic latitude, longitude, distances and average $K_{\\rm\ns}$-band magnitude.  The four light curves have a clear resemblance to each other, and a quantitative assessment of their shapes shows they are similar to those of $K_{s}$-band light curves of classical Cepheids.\n\\label{f:cep}}\n\\end{center}\n\\end{figure}\n\nWe adopt the period-luminosity relations of classical Cepheids in the LMC (Matsunaga et al. 2011), with a LMC distance modulus\nof 18.5 mag and interstellar extinction value of $A_{K_{s}} = 0.02$ mag for the LMC\ndirection.  This gives the distance modulus $\\mu$ for a Cepheid with pulsation\nperiod $P$ (Feast et al. 2014):\n\n\\begin{equation}\n\\mu= K_{s} - A_{K_{s}} + 3.284~\\rm log(P) + 2.383 \\; ,\n\\end{equation}\nwhere $A_{Ks}$ is the extinction in the $K_{s}$ band, which we can express in terms\nof the colour excess:\n\n\\begin{equation}\nA_{Ks} = 0.6822 E(J - K_{s}) \\; ,\n\\end{equation}\nwhere $E(J-K_{s}) = (M_{J} - M_{K_{s}})_{\\rm obs} - (M_{J} - M_{K_{s}})_{\\rm int} $ is the difference between the observed and intrinsic colors,\nand we adopt the period-luminosity relations of classical Cepheids in the LMC (Matsunaga et al. 2011) and the Cardelli et al. (1989) extinction law.  The single-epoch colors and extinctions in the $K_{s}$ and $J$ bands,\nalong with the extinction corrected colors are listed in Table 2. Using extinction values from dust maps derived\nfrom far-IR colors (Schlegel et al. 1998) leads to slightly larger values close to the plane of the Galaxy along these lines of sight, as well as recent work that is based on spectra from the Sloan Digital Sky Survey (Schlafly \\& Finkbeiner 2011).  If we consider the standard deviation of these three values to be the uncertainty in the dust extinction, and include the photometric errors and uncertainty in the period distribution to derive the uncertainties in the distance, on average this gives a distance uncertainty of $\\sim$ 20 \\%, where the dominant term is the uncertainty in the extinction.  The dust extinction in this area (Schlegel et al. 1998) is not unusual, i.e., this area is neither a high or low region of dust extinction.  \n\n\\begin{table*}\n\\centering\n        \\caption{Photometry Data And Extinction}\n\n\n        \\begin{tabular}{@{}lccccccc@{}}\n          \\hline\n\nVVV ID                                   &   J            &     H          & $K_{s}$    & $A_{K_{s}} $   &   $A_{J}$      &  $(J-K_{s})_{\\rm corr}$    \\\\ \n\\hline\n\n\nVVVJ162559.36-522234.0    & 17.078   &  16.429  & 16.175    & 0.348          & 0.83          & 0.42  \\\\\nVVV J162328.18-513230.4     &    17.88  &   17.09      & 16.7         &   0.53          &   1.25          &       0.44        \\\\\nVVVJ162119.39-520233.3    & 16.416  & 15.414   & 14.96     & 0.7         &  1.68          &  0.48    \\\\\nVVV J161542.47-494439.0    &    18.71   &  16.69      &    15.45     &  1.89            &   4.52            &   0.6              \\\\\n\n\\hline\n\n\n\n\\end{tabular}\n\\small {\\center{Single epoch VVV photometry in the $J, H$ and $K_{s}$ bands, with extinction values derived from the color excess assuming the Cardelli et al. (1989) extinction law, along with the extinction corrected $(J-K_{s})$ color.}\n}\n\n\\end{table*}\n\nShort-period ($\\sim$ day) close contact eclipsing binaries like W Ursa Majoris stars can mimic the sinusoidal light curves of RR Lyrae stars, which have periods of $\\sim$ day (Rucinski 1993), but is less of a concern for long-period variables.  To ensure that the shapes of the light curves are quantitatively similar to classical Cepheids, we compute their Fourier parameters, as well as the skewness and acuteness parameter, and visually inspect all the light curves.\nWe have computed the Fourier parameters of our sources (Figure \\ref{f:fourier}), following Kovacs \\& Kupi (2007).  Out to fourth-order the Fourier series is expressed as:\n\n\\begin{equation}\nm(t) = A_{0} + \\sum_{i=1}^{4} A_{i} \\rm cos\\left(2\\pi i t\/P + \\phi_{i}  \\right)\\; ,\n\\end{equation}\nwhere $m(t)$ is the light curve, $P$ is the period, and $A_{i}$ and $\\phi_{i}$ are the amplitudes and phases respectively.\nThe top panel shows $R_{21} = A_{2}\/A_{1}$ and $\\phi_{21} = \\phi_{2} - 2 \\phi_{1}$, and the bottom panel shows $R_{31} = A_{3}\/A_{1}$ and $\\phi_{31} = \\phi_{3} - 3 \\phi_{1}$.  Eclipsing binaries have $\\phi_{21}$ and $\\phi_{31}$ values close to 2$\\pi$ or zero, which reflects their symmetric variations (Matsunaga et al. 2013).  Thus, the Fourier parameters of our sources indicate that they are not eclipsing binaries.   Bhardwaj et al. (2015) provides a compilation of the Fourier parameters of a large number of Galactic and LMC Type I Cepheids across a range of wavelengths.  This work shows the differences in the shape of the light curve in the K-band relative to the I-band, as well as differences between the K-band light curves of Cepheids in the Galaxy and in the LMC.   We have overplotted in Figure \\ref{f:fourier} the Fourier parameters of Type II Cepheids in the Milky Way (Matsunaga et al. 2013).   These Type II Cepheids tend to have lower $\\phi_{31}$ values than classical, Type I Cepheids, but there is scatter in the Fourier parameters derived from K-band light curves.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.35]{fourierall.png}\n\\caption{The Fourier parameters (defined by Eqn 3 and corresponding text) are plotted vs period.  We have plotted here the Fourier parameters derived from K-band light curves of Type I, classical Cepheids in the Milky Way (marked \"MW\"), Type I classical Cepheids from the LMC (marked \"LMC\") (Bhardwaj et al. 2015), the Cepheid variables discovered at $\\sim$ 90 kpc (marked \"this paper\"), eclipsing binaries (Matsunaga et al. 2013) (marked \"EB\"), and Type II Cepheids  (Matsunaga et al. 2013) (marked \"Type II\").\n\\label{f:fourier}}\n\\end{center}\n\\end{figure}\n\nThe shape of the light curve can be further quantified by the skewness ($S_{k}$) and acuteness ($A_{c}$) parameters:\n\\begin{equation}\nS_{k}=\\frac{1}{\\phi_{\\rm rb}} -1, ~~~~ \\phi_{\\rm rb} = \\phi_{\\rm max}-\\phi_{\\rm min},~~~~ A_{c} = \\frac{1} {\\phi_{\\rm fw}} -1 \\; ,\n\\end{equation}\nwhere $\\phi_{\\rm max}$ and $\\phi_{\\rm min}$ are the phases corresponding to the minimum and maximum of the rising branch, and $\\phi_{\\rm rb}$ is therefore the phase duration of the rising branch, and $\\phi_{\\rm fw}$ is the full width at half maximum of the light curve.  Bhardwaj et al. (2015) demonstrated that the skewness parameter derived from I-band light curves is significantly higher than K-band light curves.  The average skewness parameter of our sources is $\\sim$ 0.63 and the average acuteness parameter is $\\sim$ 0.8, which is comparable to classical Cepheids observed in the K-band (Bhardwaj et al. 2015). \n\n\\section{Discussion \\& Conclusion}\n\\label{sec:conclusion}\n\nBy employing a series of successive tests to determine the periods of variable stars, the uncertainty in their periods, and a quantitative assessment of the light curve shape, we have found four Cepheid variables within an angular extent of one degree centered at Galactic longitude of $l = -27.4^\\circ$ and Galactic latitude of $b = -1.08 ^\\circ$, at an average distance of 90 kpc.  These successive tests are not satisfied at any of the other locations where we searched for Cepheid variables.  Spectroscopic observations would be useful to confirm the spectral type and determine a radial velocity.  Type II Cepheids that are part of the Galactic halo are not expected to be clustered within a degree, which is what we see here.  \nType II Cepheids that are part of a dwarf galaxy can be clustered. There are many more Type I, classical Cepheids than Type II Cepheids; the OGLE survey has detected 3361 Type I, classical Cepheids in the LMC, and 197 Type II Cepheids (Soszynski et al. 2008; Soszynki et al. 2008a).  Unless this object is as massive and extended as the LMC, one would expect that these sources are more likely to be Type I rather than Type II Cepheids.  If they are Type II Cepheids, they would be at an average distance of $\\sim$ 50 kpc (Matsunaga et al. 2013), and such a concentration of Type II Cepheids (which are very rare) is unexpected beyond the edge of the Galactic disk.   Therefore, on the basis of the Fourier parameters, skewness and acuteness parameters, and their angular concentration, we conclude these sources are Type I Cepheids.\n\nEarlier work (Chakrabarti \\& Blitz 2009) predicted that the observed perturbations in the atomic hydrogen disk of our Galaxy (Levine, Blitz \\& Heiles 2006)\nare due to a recent (300 Myr years ago) interaction with a dwarf satellite galaxy that is one-hundredth the mass of our Galaxy,\ncurrently at a distance of 90 kpc from the Galactic center, close to the plane, and within\nGalactic longitudes of $-50^{\\circ} < l < -10^{\\circ}$ (Chakrabarti \\& Blitz 2011).  \nThis methodology was applied to spiral galaxies with known, tidally dominant optical companions to provide a proof of principle of the method (Chakrabarti et al. 2011).   \n\nThere are no known dwarf galaxies that have\ntidal debris at this location.  \nThe tidal debris of the Sgr dwarf does not not extend to within $\\sim$\ntwenty-five degrees of Galactic longitude of $l=-27^\\circ$ (Carlin et al. 2012), and the Magellanic stream does not extend to within $\\sim$ 40 degrees of this region (Putman et al. 2003).  \nThe Canis Major overdensity was identified as an excess of M-giant stars from the Two Micron All Sky Survey (2MASS)\nat $(l,b) = (-120^\\circ,-8^\\circ)$, at a distance of $\\sim$ 7 kpc from the Sun (Martin et al. 2004).\nIts proximity to the Milky Way indicates that this overdensity is also unlikely to be associated with the\nCepheids we report here.\n\nThese are the most distant Cepheid variables close to the plane of our Galaxy discovered to date.  The fact that the Cepheids that we detect are at an average distance of 90 kpc, highly clustered in angle (within one degree) and in distance (within 20 \\% of the mean value of 90 kpc), is difficult to explain without invoking the hypothesis of these stars being associated with a dwarf galaxy, which may be more extended in latitude than can be determined from the VVV survey alone.  Constraining the structure of this object should be possible with future deeper observations.\n\n\\bigskip\n\\bigskip\n\n\\acknowledgements\nWe gratefully acknowledge use of data from the ESO Public Survey\nprogramme ID 179.B-2002 taken with the VISTA Telescope, and data\nproducts from the Cambridge Astronomical Survey Unit.  R.K.S.\nacknowledges support from CNPq\/Brazil through projects 310636\/2013-2\nand 481468\/2013-7.  F.G. acknowledges support from CONICYT-PCHA Mag\\'{i}ster National 2014-22141509.\nS.C. thanks M. Feast, B. Madore, C.F. McKee, J. Scargle, and G. Kovacs for helpful discussions.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n{The  presence of renewable generations in power systems, especially solar generations, has increased rapidly in recent decades. Reference \\cite{SolarMart} reports that the global capacity of photovoltaic (PV) installment reached 634 GW in 2019.  The solar capacity in 2019 has grown nearly 400 times since 2000. California Independent System Operator (ISO) estimates that the renewable energy generations will contribute 50\\% power supplies by 2030 in California \\cite{CAISO}.  \n\nThe wide deployment of renewable generations\ndecreases greenhouse gas emissions, however, but also brings great challenges to the reliability and resiliency of existing power systems. For example, at the substation level, the measurements of power consumptions are the net loads that contain different types of loads.  The solar generation is invisible to the power system operator and thus  is behind-the-meter (BTM). {Because of the stochastic nature and high volatility of renewable generations, the accurate estimation of generated energy is challenging.}\nEnergy disaggregation at the substation level (EDS)\naims to disaggregate each individual load\\footnote{{Generation is considered as a negative load in this paper.}}  from aggregate measurement. The accurate information for load consumption are crucial for power system planning and operations,  such as hosting capacity evaluation \\cite{AMZ18,WDW19}, demand response and load dispatching \\cite{MB20,XGL20} and load forecasting \\cite{WZCK18,SZWS19}.\n \n \n {Energy disaggregation problem at the household level (EDH) has been extensively studied, see, e.g., \n\\cite{KBNY10,KDM16,CZWHFJ19,HSLS16},\nalso under the terminology \\textit{non-intrusive load monitoring (NILM)} \\cite{KBNY10,KDM16,CZWHFJ19,HSLS16,H92,GAM15,MSW16}}. The electric appliances are typically single-state or multi-state devices and patterns of their power consumptions usually are repeatable. The general procedure for EDH methods is to first collect  historical power consumption for each individual appliance and learn patterns from these well-annotated data. Then EDH methods disaggregate power consumptions for each appliance from the aggregate data based on these patterns. In comparison, obtaining historical power consumption for each individual load at the substation level is more difficult, as the measurements at the substation level are highly aggregated from different types of loads. Even though the operator has information about load types attached to a substation,  whether a   certain load is consuming\/generating energy or not in a certain time interval is not already clear. One example is the BTM solar generation.\n{Thus, the measurements at the substation  usually   contain multiple loads and are partially labeled. \n It is more challenging to learn distinctive load profiles\nunder this situation than learning from measurements on individual loads. Moreover, the volatility of load and renewable generation often lead to significant estimation errors. \nHowever, to the best of our knowledge, there is no work that provides a confidence measure of the energy disaggregation results.\n\n{ This paper summarizes our recent results\nfor solving these two challenges. Given partially labeled training data,  our work~\\cite{LYWW20} proposes a  deterministic dictionary learning-based method to learn load patterns and disaggregate the aggregate measurements into individual loads in real-time.\n\nNote that \\cite{LYWW20} is a deterministic approach and therefore is unable to provide the confidence measure of the estimation results. To estimate the reliability of the disaggregation results,\nin \\cite{YW21}, we propose a probabilistic energy disaggregation approach based on Bayesian dictionary learning.  }\n\n\n{The contributions of this paper are three folds: 1. We summarize our works~\\cite{LYWW20}  and  \\cite{YW21} for solving the ``partial label'' problem and modeling the uncertainty. 2. We compare  these two methods and other two existing works in the experiment. (3) We provide  more testing cases for these two methods in this paper. }\n\n\n{The remainder of this paper is organized as follows. Section II explains our partial label formulation. Section\nIII discusses our proposed deterministic approach to solve the issue of  partial labels and introduces our proposed Bayesian method for modeling the uncertainty of disaggregation results. Section IV summarizes this paper.}\n\n\n\n\n\n\n\n\n\n\n\\section{Problem Formulation}\n\n{A substation is connected to  $C$ ($C>1$) types of loads in total.\nLet $\\bm{x}\\in \\mathbb{R}^P$ denote the aggregate measurement with window length $P$. Let a binary vector $\\bm{y}= [y^1, y^2,..., y^C] \\subseteq \\{0,1\\}^C$ denote the  load existence in $\\bm{x}$.  For example,\nwhen $C=3$, $\\bm{y}=[0,0,1]^T$ means that only load 3 exists in $\\bm{x}$.}\n\n{In our paper  \\cite{LYWW20}, we propose a ``partial label formulation'' where the operator only knows partial entries in  $\\bm{y}$. The partial labels can be obtained by designing a load detector for each load separately \\cite{HCQ19,ZG16} or from engineering experience. As described in \\cite{LYWW20}, annotating partial labels  has a lower cost for manpower or communication burdens than annotating all the labels. Moreover,  if a detector fails to identify some loads \\cite{HD18}, we can only obtain partial labels.  \n\n\nLet $\\bar{\\bm{X}}=[\\bar{\\bm{x}}_1,\\bar{\\bm{x}}_2,...,\\bar{\\bm{x}}_N ] \\in \\mathbb{R}^{P \\times N}$ denote  $N$ measurements.  $\\bar{\\bm{x}}_i$ denotes the data at the $i$th time window.  $\\bm{y}_i \\in \\{0,1\\}^C$ denotes the labels in $\\bar{\\bm{x}}_i$. Let label matrix $\\bm{Y}=[\\bm{y}_1, \\bm{y}_2,...,\\bm{y}_N]$ denote all the labels in $\\bar{\\bm{X}}$. Let $\\Omega$ denote the indices of known entries in $\\bm{Y}$. $\\bm{Y}_\\Omega$ denotes all the known partial labels. In the above example, if one only knows $\\bar{\\bm{x}}$ contains load 3 and does not know whether the other two loads exist or not, then the corresponding $\\bm{Y}_\\Omega$ is $[?, ?, 1]^T$ where $?$ denotes one does not know the corresponding load exists or not.\n\nFig. 1 illustrates our partial label formulation. The aggregate data are aggregated from  two industrial loads and one solar generation. Each subfigure shows patterns of aggregate data and the corresponding individual loads at the same time interval. {In all these four cases, the label is $[?, ?, 1]^T$, indicating that load 3 always exists, while the existence of loads 1 and 2 is unknown. \n\n Given training dataset $\\bar{\\bm{X}}$, the corresponding partial label matrix $\\bm{Y}_\\Omega$  and an aggregate measurement $\\hat{\\bm{x}}\\in \\mathbb{R}^P$, the objective of this paper is to: (1)   learn distinctive patterns of individual loads from $\\bar{\\bm{X}}$ and disaggregate\n $\\hat{\\bm{x}}$, and (2)   characterize the uncertainty of the disaggregation results.\n\n}\n\n\n    \n    \n    \n  \n\n    \\begin{figure}[!ht]  \n     \\centering\n    \t\\includegraphics[width=0.45\\textwidth]{.\/Figures\/PartialLabel.pdf}  \n    \t\\caption{{An example of partial label formulation. There are three load types in the aggregate data. All four aggregate data have the same partial label  $[?,?,1]^T$.   The aggregate data contain (a) all three loads; (b) load 1 and 3; (c) load 2 and 3; (d) only load 3.} \\cite{LYWW20}}  \\label{fig1}\n    \\end{figure}\n    \n        \n   \n    \n\\section{Methodology}\n\n{In this section, we present our two model-free approaches based on deterministic dictionary learning \\cite{LYWW20} and Bayesian dictionary learning \\cite{YW21}, respectively.\n\\subsection{Deterministic Energy Disaggregation}\n\nTo learn patterns of each individual load from the given training data $\\bar{X}$, we formulate\na deterministic dictionary learning problem,\n \\begin{align} \n\\min_{ A, D }    f(A,D) &=\\lVert \\bar{X}- \\Sigma_{i = 1}^C D_i A_i \\rVert_F^2+\\Sigma_{i = 1}^C \\lambda_i \\Sigma_{j: i \\notin y_j }  \\lVert A_i^{j} \\rVert   \\nonumber\\\\\n &   + \\lambda_D \\text{Tr}(D\\Theta D^\\intercal)    \\label{eqn:dictionary} \\\\\n\\text{s.t. }&   \\lVert d^{m } \\rVert_2 \\leq 1,   d^{m} \\geq 0, m=1, \\cdots, K     \\label{cons} \\\\\n& c_i A_i   \\geq 0, \\forall i   \\label{eqn:positive}  \n\\end{align} \nwhere $D_i\\in \\mathbb{R}^{P\\times K_i}$ denotes the dictionary for  load $i$, and $A_i   \\in \\mathbb{R}^{K_i \\times N}$ denotes the corresponding coefficients   of load $i$.  $A_i^j$ is the $j$th column in $A_i$.   $A=[A_1; A_2; \\cdots; A_C]\\in \\mathbb{R}^{K \\times N}$ is the matrix that contains all coefficients. $D = \\begin{bmatrix}\nD_1, D_2, \\cdots, D_C\n\\end{bmatrix}  \\in \\mathbb{R}^{P \\times K }$    is the matrix that contains all dictionaries. $d^m$ is the $m$th column in $D$.  $K = \\Sigma_{i=1}^C K_i $. $\\text{Tr}(\\cdot)$  is the trace operator, and $D^\\intercal$ represents the transpose matrix of $D$.  $\\lambda_i$ and $\\lambda_D$ are pre-defined hyper-parameters.     }\n\n   \\begin{figure}[!ht]  \n     \\centering\n    \t\\includegraphics[width=0.48\\textwidth]{.\/Figures\/Dictionary1.pdf}  \n    \t\\caption{ {The dictionary representation in Fig. \\ref{fig1}.\n     The coefficients\t$A_1$ and $A_2$ are column-sparse.  \\cite{LYWW20}}}\n    \t\\label{diction}\n    \\end{figure}\n\n        \\begin{figure*}[!ht]  \n     \\centering\n    \t\\includegraphics[width=0.8\\textwidth]{.\/Figures\/figure1.pdf}  \n \n  \\caption{{An illustrative framework of the proposed Bayesian method. In the training stage, our method learns the posterior distribution of atom labels, coefficients and load labels from the training data. In the testing stage, the method samples learned distributions of the dictionary and learns the posterior distribution for atom labels, coefficients and load labels, respectively, for test data. \\cite{YW21}}}\n    \t\\label{fig3}\n  \n  \\label{fig_example}\n    \\end{figure*}\n    \n{The first term $\\lVert \\bar{X}- \\Sigma_{i = 1}^C D_i A_i \\rVert_F^2$ is the standard reconstruction error in dictionary learning. It measures the reconstruction error between the original data and the learned dictionaries and coefficients. $\\sum_{j: i \\notin y_j}  \\lVert A_i^{j} \\rVert$ is the column sparsity constraint. The motivation of using this regularization is illustrated in Fig. \\ref{diction}, which shows the dictionary representation of Fig. \\ref{fig1}.\nBecause the training data only have partial label 3, load 1 and load 2 may not exist in the training data. Therefore, we impose the column sparsity on $A_1$ and $A_2$ to promote the group sparsity of $A_1$ and $A_2$. \n    \nThe incoherence term $\\text{Tr} (D\\Theta D^\\intercal) $ is defined as\n\n\\begin{align}\\label{eqn:trace}\n\\text{Tr} (D\\Theta D^\\intercal) & = \\Sigma_{m = 1}^{K} \\Sigma_{p =1}^{K } \\theta_{mp}  (d^m)^\\intercal   d^{p}.\n\\end{align} \nThe $(m,p)$th entry $\\theta_{m p}$ in the weight matrix $\\Theta \\in R^{K \\times K}$ is $0$ if  $d^m$ and $d^p$ are in the same dictionary and $1$ otherwise. The incoherence term promotes a discriminative dictionary such that $D_i$ and $D_j$ are as different as possible. The discriminative dictionaries are able to enhance the disaggregation performance.}\n\n\n\n\n \n {Given an aggregate test data $\\hat{x}$, we aim to disaggregate the aggregate measurement into individual load $c$, denoted by $\\hat{x}^c$. The objective function in the testing stage can be written as\n\\begin{align} \\label{w} \n     \\min_{w \\in \\mathbb{R}^q }  &  \\lVert \\hat{x} - \\hat{D} \\tilde{A} w \\rVert_2       + \\mu  \\lVert w \\rVert_1,  \n     \\end{align} \nwhere   we select a submatrix $\\tilde{A}=[\\tilde{A}_1;\\cdots; \\tilde{A}_n] \\in \\mathbb{R}^{K \\times q}$ from $\\hat{A}$. $\\hat{D}$, $\\hat{A}$ is the solution by solving the objective function \\eqref{eqn:dictionary}. $\\mu$ is a pre-defined hyper-parameter. The intuition is that some load  combinations are repetitive in the training data. We can select some representative combinations and disaggregate the aggregate measurement with respect to these combinations  to improve the disaggregation accuracy. Let $\\hat{w}$ be the solution to \\eqref{w}, then the estimated load  for load $c$ is $\\hat{x}^c= \\hat{D}_i \\tilde{A}_i \\hat{w}$. }\n\n\n\n \n \\subsection{Bayesian Energy Disaggregation}\n \n{ In \\cite{YW21}, we  propose a   Bayesian method to deal with partial label data and provide the confidence measure of our disaggregation results. An overall framework is shown in Fig. \\ref{fig3}. Given the training data $\\bar{X}$ and   partial labels $\\bm{Y}_\\Omega$,  the proposed Bayesian method learns the posterior distribution of dictionaries and coefficients in the training stage. At the testing stage, the method learns the distributions of coefficients based  on the learned distributions of dictionaries. The distribution of $\\hat{x}^c$ is then computed, where  $\\hat{x}^c$ is the estimated power consumption of load $c$.  The mean of the distribution of $\\hat{x}^c$ is used as the estimation of the load $c$ and the covariance is computed to measure the uncertainty.  \n\n\nThe proposed method is based on a hierarchical probabilistic model. The prior distribution of the aggregate data ${x}_i$ can be written as \n\n \\begin{equation} \\label{equation1}\n\\bar{x}_i = \\sum_{c=1}^{C}{D}_{c} \\bm{\\omega}_i^c+\\bm{\\epsilon}_i \n\\end{equation}\n\\begin{equation}\\label{equation2}\n\\bm{\\omega}_i^c = (\\bm{z}_i^c \\odot \\bm{s}_i^c)y_i^c\n\\end{equation}\nfor all  $i=1,2,3,...,N$, $c=1,2,3,...,C$, \nwhere $\\bm{\\omega}_i^c \\in \\mathbb{R}^{K_c}$ is the coefficients for $D_c$, and $\\bm{\\epsilon}_i$ is the measurement noise. {In \\eqref{equation2}, $\\odot$ represents the element-wise product.} Let $\\bm{d}^c_k$ denote the $k$th column in the dictionary ${D}_{c}$. $\\bm{d}^c_k$ is sampled from a multivariate Gaussian distribution $\\mathcal{N}(\\bold{0},\\frac{1}{\\lambda_d} \\bm{I}_P)$, where $\\lambda_d$ is a pre-defined scalar, and $\\bm{I}_P$ is an  identity matrix with size $P\\times P$.  The  noise $\\bm{\\epsilon}_i$ is sampled from Gaussian $\\mathcal{N}(\\bold{0},\\frac{1}{\\gamma_\\epsilon} \\bm{I}_P)$.   One can see from (\\ref{equation2}) that $\\bm{\\omega}_i^c$ is the element-wise product of $\\bm{z}_i^c$ and $\\bm{s}_i^c$ and then multiplied by $y_i^c$. $y_i^c$ is a binary variable sampled from a Bernoulli distribution and $y_i^c=1$ indicates that load $c$ exists in ${x}_i$. $\\bm{z}_i^c$ is a binary vector. Let $\\bm{z}_{ik}^c$  denote the $k$th entry of  $\\bm{z}_i^c$. $\\bm{z}_{ik}^c=1$  indicates $d_k^c$ is used to represent ${x}_i$ and 0 otherwise. $\\bm{z}_{ik}^c$  is sampled from the Bernoulli distribution. Note that the Bayesian method is able to infer the actual dictionary size $K_c$ by gradually pruning the dictionary size based on  $\\bm{z}_i^c$ in the training stage. Therefore, the Bayesian method is not sensitive to the selection of initial dictionary size.\n{$\\bm{s}_i^c$ is sampled from   $\\mathcal{N}(\\bold{0},\\frac{1}{\\gamma_s^c} \\bm{I}_{Kc})$.   We put Gamma priors on $\\gamma_s^c$ and $\\gamma_\\epsilon$, respectively. { The Gamma priors are conjugate priors of the Gaussian distribution. If conjugate priors are selected, we can derive the analytical solution of the posterior distribution in the variational inference, which simplifies the updating process.}   Note that our model selects conjugate priors to simplify the updating process.} \n\n Let $\\bm{\\Theta}$ denote all the latent variables. Given   $\\bar{X}$ and partial labels $\\bm{Y}_\\Omega$, the objective is to obtain the posterior $P(\\bm{\\Theta}, \\bm{Y}_{\\bar{\\Omega}} |\\bm{X}, \\bm{Y}_\\Omega)$. From the Bayes theorem, \n\n\\begin{equation}\\label{eqn:bayes}\n  P(\\bm{\\Theta}, \\bm{Y}_{\\bar{\\Omega}} |\\bar{X}, \\bm{Y}_\\Omega)=\\frac{P(\\bm{\\Theta}, \\bar{X}, \\bm{Y})}{P(\\bar{X},\\bm{Y}_{\\Omega})}\n    \\end{equation}\nBecause computing \\eqref{eqn:bayes} directly is intractable, we use Gibbs sampling \\cite{I12} to compute the posterior distribution. Gibbs sampling  sequentially samples from the conditional probability of one variable in $\\bm{\\Theta}$ and $\\bm{Y}_{\\bar{\\Omega}}$ while keeping all other variables fixed. These conditional distributions have closed-form expressions because of the conjugate priors, which leads to an efficient updating process. \n\n\nIn the testing stage, given the aggregate test data $\\hat{\\bm{x}}$, the goal of our approach is to estimate  $\\hat{{x}}^c$. A similar probabilistic model for  $\\hat{\\bm{x}}$ and $\\hat{{x}}^c$ is described as: \n\n\n\\begin{equation} \\label{xhat}\n  \\hat{{x}}  = \\sum_{c=1}^{C}{D}_{c} (\\bm{\\hat{z}^c} \\odot \\bm{\\hat{s}^c})\\hat{y}^c+\\hat{\\bm{\\epsilon}}\\\\\n\\end{equation}\n \n\\begin{equation}\\label{eqn:hatxc}\n {\\hat{{x}}^c} = {D}_c (\\bm{\\hat{z}^c} \\odot \\bm{\\hat{s}^c})\\hat{y}^c+\\frac{\\hat{\\bm{\\epsilon}}}{C}.\n \\end{equation}\nfor all $c=1,...,C$, $k=1,...,K_c$.\n\nThe dictionary atom $d_k^c$ is sampled from learned distribution  $p({d_k^c}|\\bm{X},\\bm{Y}_\\Omega)$ in the training stage. We also assume that $\\hat{y}^c$ and $\\hat{\\bm{z}}^c$ are sampled from Bernoulli distributions. $\\hat{\\bm{s}}^c$ is sampled from $\\mathcal{N}(\\bold{0},\\frac{1}{\\gamma_s^c} \\bm{I}_{K_c})$ and  $\\hat{\\bm{\\epsilon}}$ is sampled from $\\mathcal{N}(\\bold{0},\\frac{1}{\\hat{\\gamma}_{\\epsilon}} \\bm{I}_P)$. Gibbs sampling is also employed for computing probabilistic distributions of $\\hat{y}^c$, $\\hat{\\bm{z}}^c$, $\\hat{\\bm{s}}^c$, and $\\hat{\\gamma}_\\epsilon$. \n\n\n{The per-iteration computational complexity of the Bayesian offline training is $\\mathcal{O}(CK_cPN)$. The per-iteration computational complexity of the online testing   is $\\mathcal{O}(CK_cP)$. Thus, the computational complexity scales linearly with respect to the number of loads. }\n\n \\subsection{Uncertainty Modeling}\n Equipped with all learned posterior distributions, we then estimate the distribution of  $\\hat{x}^c$. However, it is intractable to obtain the explicit expression for the distribution of  $\\hat{x}^c$.   Monte-Carlo integration \\cite{PBJ12} is employed to approximately compute the predictive mean and predictive variance.\n\n \n \nDefine\n\\begin{equation}\nf (\\bm{\\Psi})  = {D}_c (\\hat{\\bm{z}}^c \\odot \\hat{\\bm{s}}^c)\\hat{y}^c\n\\end{equation}\nwhere $\\bm{\\Psi}=\\{{D}_c, \\hat{\\bm{z}}^c, \\hat{\\bm{s}}^c, {\\hat{y}^c}, \\hat{\\gamma}_\\epsilon\\}$.\nThe predictive mean of ${\\hat{{x}}^c}$ is computed by\n\\begin{equation}  \\label{predictmean}\n\\begin{split}\nE[{\\hat{{x}}^c}]\n                     &\\approx\\frac{1}{L}\\sum_{l=1}^{l=L}  f (\\bm{\\Psi}^l)\n\\end{split}\n\\end{equation}\nwhere $L$ is the number of Monte-Carlo samples. {More Monte-Carlo samples increase the  estimation  accuracy, at the cost of higher computational burden. Our experiments show that $50$ Monte-Carlo samples suffice to provide accurate estimations of the predictive mean and the predictive variance.} $\\bm{\\Psi}^l$ is sampled  from the learned distributions of  variables in $\\bm{\\Psi}$.  $E[{\\hat{{x}}^c}]$ is then used as the estimation of the power consumption of load C.   \n\nThe predictive covariance is approximated by\n\n\\begin{equation} \\label{predictvariance}\n\\begin{split}\n \\textrm{Var}[{\\hat{{x}}^c}] \n =&E[{\\hat{x}^c}{\\hat{x}^c}{}^T]-{E[{\\hat{x}^c}]}{E[{\\hat{x}^c}]}^T \\\\\n \\approx &\\frac{\\bm{I}_P}{LC}\\sum_{l=1}^{l=L} \\frac{1}{\\hat{\\gamma}_{\\epsilon}^{l}} +\\frac{1}{L}\\sum_{l=1}^{l=L}  f (\\bm{\\Psi}_l){ f (\\bm{\\Psi}^l)}^T  \\\\ -&(\\frac{1}{L}\\sum_{l=1}^{l=L}  f (\\bm{\\Psi}^l))(\\frac{1}{L}\\sum_{l=1}^{l=L}  {f (\\bm{\\Psi}^l)}^T) \n\\end{split}\n\\end{equation}\n\n\n\\noindent Let  $\\sigma_i$ ($i=1,..., P$)  denote all the singular values  of $\\textrm{Var}[{\\hat{{x}}^c}]$.\n\nThe uncertainty index $U_c$   for individual load $c$ and the uncertainty index $U_{\\textrm{all}}$ for total estimated loads are computed  as\n\\begin{equation}\\label{Uncertaintyindex1}\nU_c = \\Sigma_{i = 1}^P \\sigma_i\n\\end{equation}\n\\begin{equation}\\label{Uncertaintyindex2}\nU_{\\textrm{all}}  = \\Sigma_{c = 1}^C U_c\n\\end{equation}\nThe intuition is that a large variance indicates higher uncertainty of the estimation. The uncertainty index is able to characterize the confidence level of disaggregation results. \n\n }\n\n    \n\\section{Numerical Experiment}\n{ The performance of the proposed methods is evaluated on a partially labeled dataset. The dataset contains two industry loads and one solar generation. $N=360$ training samples and $M=300$ testing samples are generated. Even though the generated training samples contain up to three loads, each sample is annotated with only one label. The testing samples also contain up to three loads and have no label. In the following experiments, $\\gamma$ represents the percentage of the training data that measure individual loads. For example, $\\gamma=50\\%$ denotes that $50\\%$ training data labeled as load $c$  contain pure load $c$ and the remaining $50\\%$ data contain other loads.   \n\n} \n\n\n{  \\subsubsection{Error Metrics}  Several metrics are employed to compute the disaggregation error.  The standard Root Mean Square Error (RMSE)   \\cite{RQFKT17, PGWRA12} is defined as, \n\\begin{align}\n\\text{RMSE}_c & =  \\sqrt{\\frac{\\Sigma_{i=1}^{M}  \\lVert \\hat{x}^c_i -x^c_i \\rVert_2^2}{P \\times M}.  } \n\\end{align}\nwhere $\\hat{x}^c_i, x^c_i  \\in \\mathbb{R}^P$ are the estimated   and the ground-truth load $c$ in  the $i$th testing sample, respectively.\n \nA new Total Error Rate (TER) is proposed to compute the disaggregation error of all the loads as follows,\n\\begin{align}\n\\text{TER}   = &  \\frac{  \\Sigma_{i = 1}^{M} \\Sigma_{c =1}^C  \\min(   \\lVert  \\hat{x}^c_i - x_i^c \\rVert_1 ,  \\lVert  {x}^c_i \\rVert_1 )  }{   \\Sigma_{i= 1}^{M} \\Sigma_{c = 1}^C \\lVert x^c_i \\rVert_1 }   \n\\end{align} \n\n \n \n \n\nThe Weighted Root Mean Square Error (WRMSE) is proposed to take the uncertainty index into account.  The weighted average disaggregation error is computed as, \n\n\n\\begin{equation}\\label{eqn:WRMSE}\n{\\text{WRMSE}_c   =  \\sqrt{\\frac{\\Sigma_{i=1}^{M}    \\frac{\\lVert \\hat{x}^c_i -{x}^c_i \\rVert_2^2}{U_c(\\hat{x}^c_i)} }{P\\Sigma_{i=1}^{M} \\frac{1}{U_c(\\hat{x}^c_i)} }  } }\n\\end{equation}\n\n\\noindent where   $U_c(\\hat{x}^c_i)$ denotes the uncertainty index of $\\hat{x}^c_i$.   A larger  $U_c(\\hat{x}^c_i)$  represents a less reliable estimation. If the estimated loads with higher disaggregation errors are accompanied by larger uncertainty indices, the RMSE$_c$ could be much larger than WRMSE$_c$. The scenario that RMSE$_c$ is much larger than WRMSE$_c$ indicates that the unreliable estimation results are correctly flagged by higher uncertainty indices.\n \n  \n\\subsubsection{Methods}  Our deterministic EDS method in \\cite{LYWW20} is abbreviated as ``D-EDS.'' Our Bayesian EDS method in \\cite{YW21} is abbreviated as ``B-EDS.''  Two other existing methods are employed for comparison. The work in \\cite{KBNY10} that is based on discriminative sparse coding is abbreviated as ``DDSC,'' and the work \\cite{RQFKT17} based on sum-to-k matrix factorization  is abbreviated as ``sum-to-k''. Because we set the Monte-Carlo samples $L=50$ in our method B-EDS, then D-EDS, DDSC and sum-to-k are averaged over 50 runs for\na fair comparison.  The comparisons  of disaggregation performance of B-EDS, D-EDS, DDSC and sum-to-k are shown in Table \\ref{table1}. $\\gamma=70\\%$. Note that all the existing works such as DDSC and sum-to-k methods require fully labeled data to obtain accurate estimation.\nDirectly applying the existing methods\nto partially labeled data leads to a low disaggregation accuracy.  The proposed two approaches B-EDS and D-EDS are designed for  partially labeled data and can achieve state-of-the-art disaggregation performance.\nBetween these two methods, the disaggregation accuracy of B-EDS is slightly better. Moreover, one can see from Table I that the WRMSE$_c$ is much smaller than the corresponding RMSE$_c$. As we discussed above, this means that those estimations with larger disaggregation errors also have large uncertainty indices. This validates the effectiveness of applying the proposed uncertainty index to measure the reliability of the disaggregation results. \n\n\nThe major advantage of B-EDS over D-EDS is that B-EDS is able to measure the confidence level of disaggregation results from the uncertainty index. We provide five case studies to verify the performance of uncertainty modeling   of B-EDS.\n\n\n\\begin{itemize}\n\\item Case 1: we select  test data from the testing datasets in Table I and this test data contains three types of loads.\n\n\\item Case 2:  the test data is as same as the data in Case 1 with an additional Gaussian noise $\\mathcal{N}(0,4^2)$ in each entry. \n\n\\item Case 3: the test data is as same as the data in Case 1 with an additional Gaussian noise $\\mathcal{N}(0,6^2)$ in each entry. \n\n\\item Case 4: the test data only contains one solar generation, but the pattern of solar generation is different from the solar patterns in the training data.\n\n\\item Case 5: the test data contains the same load 1 and 2 as those in Case 1,  and as well as a solar generation with a pattern different from the solar patterns in the training data. \n\\end{itemize}\n\n\nFigs.~\\ref{fig_4} and ~\\ref{fig_5} show the disaggregation performance of D-EDS on Case1 and Case 4. \nThe aggregate measurement  is shown in   (a), and the disaggregation results are shown in (b)-(d) in both figures. In Case 1, the disaggregation results by D-EDS follow the actual load pattern. In Case 4, the disaggregation result of the solar generation does not follow the actual solar pattern because it is different from the learned pattern from the training data. In both cases, the disaggregation results contain some errors. That motivates using the Bayesian approach to compute a probabilistic  distribution of load consumption rather than computing one deterministic estimation. \n\n\n{Fig.~\\ref{fig_uncertainty} shows the \n  disaggregation performance of B-EDS on these five cases, and Table II shows the corresponding uncertainty index.\n  Each subfigure in Fig.~\\ref{fig_uncertainty} plots the ground-truth load, the estimated load and the $99.7\\%$ confidence interval of the estimated load. One can see\n  that in Cases 1-3, although there are some errors in the disaggregation results, the ground-truth loads are within the confidence interval. Moreover, the estimation errors increase slightly when the noise levels increase.  Correspondingly, Table II shows that the total uncertainty indices in Case 1-3 also increase as the noise level increases, which indicates the effectiveness of using the uncertainty index to characterize  the uncertainty in the estimation. In Case 4 and Case 5, because the patterns of solar generation are far from the patterns in the training data, the ground-truth load consumption may not fall inside the confidence interval (especially Case 5).\n  The uncertainty indices in Table II also increase significantly, indicating that the estimated results are less reliable in these cases.\n  Therefore,\n  the users can use the uncertainty index to evaluate the accuracy of the disaggregation results.\nTable II also compares the TER of B-EDS and D-EDS, and B-EDS has  a smaller disaggregation error of D-EDS.}\n\n\\begin{figure*}[!ht]  \n     \\centering\n     \\subfigure[]\t{\\includegraphics[width=0.24\\textwidth]{.\/Figures\/netFig1.png}}\n    \\subfigure[]{\t\\includegraphics[width=0.24\\textwidth]{.\/Figures\/Case21Fig1.png}}\n \n    \\subfigure[]{\t\\includegraphics[width=0.24\\textwidth]{.\/Figures\/Case21Fig2.png}}\n\\subfigure[]{\t\\includegraphics[width=0.24\\textwidth]{.\/Figures\/Case21Fig3.png}}\n    \t\\caption{ {Disaggregation performance of D-EDS on Case 1.(a) Net load. (b) The ground-truth  and disaggregated load 1.(c) The ground-truth   and disaggregated load 2. (d) The ground-truth   and disaggregated solar.} } \\label{fig_4}\n    \\end{figure*}\n    \n\\begin{figure*}[!ht]  \n     \\centering\n     \\subfigure[]\t{\\includegraphics[width=0.24\\textwidth]{.\/Figures\/nettFig1.png}}\n    \\subfigure[]{\t\\includegraphics[width=0.24\\textwidth]{.\/Figures\/Case23Fig1.png}}\n \n    \\subfigure[]{\t\\includegraphics[width=0.24\\textwidth]{.\/Figures\/Case23Fig2.png}}\n\\subfigure[]{\t\\includegraphics[width=0.24\\textwidth]{.\/Figures\/Case23Fig3.png}}\n    \t\\caption{ {Disaggregation performance of D-EDS on Case 4.(a) Net load. (b) The ground-truth   and disaggregated load 1.(c) The ground-truth   and disaggregated load 2. (d) The ground-truth  and disaggregated solar.} } \\label{fig_5}\n    \\end{figure*}\n    \n    \n    \\begin{table}[]\n\\centering\n\\caption{Comparison between B-EDS, D-EDS, sum-to-k and DDSC methods on disaggregation accuracy}\n\\label{table1}\n\\begin{tabular}{lllll}\n \\hline          & B-EDS    & D-EDS  & sum-to-k  & DDSC    \\\\\n \\hline RMSE$_1$  & 6.20 & 6.62  & 13.17 & 22.77   \\\\\n \\hline RMSE$_2$  & 5.19  & 6.34  & 11.35 & 23.86   \\\\\n \\hline RMSE$_3$  & 5.82 & 4.65 & 10.70 & 13.49   \\\\\n \\hline WRMSE$_1$ & 0.16 &  -      &  -       &     -    \\\\\n \\hline WRMSE$_2$ & 0.13 &   -     &  -       &   -      \\\\\n \\hline WRMSE$_3$ & 0.13 &    -    &  -       &   -      \\\\\n \\hline TER      & 8.97\\%   & 9.95\\% & 20.61\\% & 37.12\\% \\\\\n  \\hline \n\\end{tabular}\n\\end{table}\n\n\n\n} \n\n\\begin{figure*} [h!]\n    \\centering\n    \\subfigure[]{\\includegraphics[width=0.18\\textwidth]{Figures\/Case4Fig1.png}} \n    \\addtocounter{subfigure}{+2}\n   \\subfigure[]{\\includegraphics[width=0.18\\textwidth]{.\/Figures\/Case5Fig1.png}} \n    \\addtocounter{subfigure}{+2}\n    \\subfigure[]{\\includegraphics[width=0.18\\textwidth]{.\/Figures\/Case3Fig1.png}}\n    \\addtocounter{subfigure}{+2}\n    \\subfigure[]{\\includegraphics[width=0.18\\textwidth]{.\/Figures\/Case10Fig1.png}}\n    \\addtocounter{subfigure}{+2}\n     \\subfigure[]{\\includegraphics[width=0.184\\textwidth]{.\/Figures\/Case13Fig1.png}}\n\n     \\addtocounter{subfigure}{-12}\n     \\subfigure[]{\\includegraphics[width=0.18\\textwidth]{Figures\/Case4Fig2.png}} \n      \\addtocounter{subfigure}{+2}\n    \\subfigure[]{\\includegraphics[width=0.18\\textwidth]{.\/Figures\/Case5Fig2.png}} \n          \\addtocounter{subfigure}{+2}\n    \\subfigure[]{\\includegraphics[width=0.18\\textwidth]{.\/Figures\/Case3Fig2.png}}\n          \\addtocounter{subfigure}{+2}\n    \\subfigure[]{\\includegraphics[width=0.18\\textwidth]{.\/Figures\/Case10Fig2.png}}\n          \\addtocounter{subfigure}{+2}\n     \\subfigure[]{\\includegraphics[width=0.184\\textwidth]{.\/Figures\/Case13Fig2.png}}\n         \\addtocounter{subfigure}{-12}\n     \\subfigure[]{\\includegraphics[width=0.18\\textwidth]{Figures\/Case4Fig3.png}} \n      \\addtocounter{subfigure}{+2}\n    \\subfigure[]{\\includegraphics[width=0.18\\textwidth]{.\/Figures\/Case5Fig3.png}} \n          \\addtocounter{subfigure}{+2}\n    \\subfigure[]{\\includegraphics[width=0.18\\textwidth]{.\/Figures\/Case3Fig3.png}}\n          \\addtocounter{subfigure}{+2}\n    \\subfigure[]{\\includegraphics[width=0.18\\textwidth]{.\/Figures\/Case10Fig3.png}}\n          \\addtocounter{subfigure}{+2}\n     \\subfigure[]{\\includegraphics[width=0.184\\textwidth]{.\/Figures\/Case13Fig3.png}}\n    \t\\caption{The disaggregation performance  of B-EDS on Cases 1-5. Each subfigure shows the ground-truth load, disaggregated load and the corresponding confidence interval. (a)-(c): the disaggregation results for Case 1.  (d)-(f): the disaggregation results for Case 2 where the test data contains Gaussian noise $\\mathcal{N}(0,4^2)$. (g)-(i):  the disaggregation results for Case 2 where the test data contains Gaussian noise  $\\mathcal{N}(0,6^2)$. (j)-(l):  the disaggregation results for Case 4 where the test data is a solar generation and its pattern  is different from the training data.  (m)-(o) the disaggregation results for Case 5 where the test data contains three loads, and the pattern of solar generation is different from the training data. }\\label{fig_uncertainty}\n    \n\\end{figure*}\n\n\n\\begin{table}[ht!]\n\\centering\n\\caption{The uncertainty indices and disaggregation accuracy on five testing cases}\n\\label{table_uncertainty}\n\\begin{tabular*}{\\hsize}{@{}@{\\extracolsep{\\fill}}c|ccccc@{}}\n\\hline     & Case 1 & Case 2 & Case 3 & Case 4 & Case 5  \\\\\n\\hline U$_1$ &\t243.72\t&\t280.35\t&\t201.52\t&\t0.058\t&\t160.89\t\\\\\n\\hline U$_2$&\t116.07\t&\t101.24\t&\t215.23\t&\t0.060\t&\t394.41\t\\\\\n\\hline U$_3$ &\t249.89\t&\t257.78\t&\t287.23\t&\t703.48\t&440.26\t\\\\\n\\hline U$_{\\textrm{all}}$ &609.69\t&\t639.37\t&\t703.98\t&\t703.60\t&788.44\t\\\\\n\\hline B-EDS TER &\t4.77\\%\t&\t5.10\\%\t&\t7.00\\%\t&\t6.77\\%\t&\t12.97\\%\t\\\\\n\\hline D-EDS TER &\t7.19\\%\t&\t8.86\\%\t&\t11.60\\%\t&\t11.01\\%\t&\t16.45\\%\t\\\\\n\\hline \n\\end{tabular*}\n\\end{table}\n\n\n\n\nThe Bayesian method B-EDS has slightly better disaggregation performance than\nthe deterministic approach D-EDS.   The major advantage of the Bayesian approach is to measure the confidence level of the disaggregation results. However, the deterministic approach\nis much  more computationally efficient than the Bayesian method. {In Table I, the B-EDS requires around 50 seconds for the offline training  and 4 seconds for each testing sample. In comparison, the D-EDS requires around 15 seconds for the offline training, and 0.9 seconds for each testing sample. } If users want to know the reliability of the estimation results, the Bayesian method should be selected. In contrast, if users need to disaggregate the aggregate measurement with limited computational resources in real-time, the deterministic approach is a better option. \n\n\n\\section{Conclusion}\n{\n\n\n\n\nEnergy disaggregation at substations with BTM solar generations has drawn increasing attention.\nAccurate energy disaggregation results are crucial for  power system planning and operations.  However, collecting training data with full labels at the substation level is challenging. Therefore, \nwe propose the concept of partially labeled data which is   applicable in practice and significantly reduces the burden of annotating data. This paper summarizes two new  load disaggregation approaches.\nBoth the deterministic approach and the Bayesian approach can achieve   accurate disaggregation results on partially labeled data. Moreover,\nan uncertainty index is proposed to measure the reliability of the disaggregation results.\nTo the best of our knowledge, this is the first work to provide the uncertainty measure for the energy disaggregation problem.}\n\n\n\n\n\\section*{Acknowledgment}\n\n\nThis work was supported in part by the NSF grant \\# 1932196, AFOSR FA9550-20-1-0122, and ARO W911NF-21-1-0255.\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran} \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nIn quantum computers, quantum bits (qubits) \\cite{nielsen2011quantum, scherer2019mathematics} are the elementary information carriers.  In such a computer, quantum gates \\cite{nielsen2011quantum, scherer2019mathematics, deutsch1989quantum} can manipulate arbitrary multi-partite quantum states \\cite{raussendorf2001one} including arbitrary superposition of the computational basis states, which are frequently also entangled. Thus the logic gates of quantum computation are considerably more varied than the logic gates of classical computation. In addition, a quantum computer can solve problems exponentially faster than any classical computer \\cite{deutsch1992rapid} because by exploiting superposition principle and entanglement allows the computer to manipulate and store more bits of information than a classical computer.\n\nIn this paper we present a theoretical approach of realizing Hadamard and controlled-NOT (C-NOT) quantum logic gates which form a universal set for quantum computation \\cite{barenco1995elementary, shi2002both, boykin2000new}.\nThe important discovery and proof of a conserved excitation number operator of the AJC Hamiltonian \\cite{omolo2017conserved} now means that dynamics generated by the AJC Hamiltonian is exactly solvable, as demonstrated in the polariton and anti-polariton qubit (photospin qubit) models in \\cite{omolo2017polariton, omolo2019photospins}. The reformulation developed in  \\cite{omolo2017conserved, omolo2017polariton, omolo2019photospins}, drastically simplifies exact solutions of the AJC model which we shall apply in the present work.\n\nWe define the quantum C-NOT gate as that which affects the unitary operation on two qubits which in a chosen orthonormal basis in $\\mathbb{C}^2$ gives the C-NOT operation obtained as\n\n\\begin{equation}\n|a\\rangle|b\\rangle\\rightarrow|a\\rangle|a\\oplus|b\\rangle\n\\label{eq:theqcnot}\n\\end{equation}\nwhere $|a\\rangle$ is the control qubit, $|b\\rangle$ is the target qubit and $\\oplus$ indicates addition modulo 2 \\cite{barenco1995conditional, scherer2019mathematics, nielsen2011quantum}. The C-NOT gate transforms superposition into entanglement thus acts as a measurement gate \\cite{nielsen2011quantum, scherer2019mathematics,barenco1995conditional} fundamental in performing algorithms in quantum computers \\cite{knill2002introduction}. Transformation to a separable state (product state) is realized by applying the C-NOT gate again. In this case, it is used to implement Bell measurement on the two qubits \\cite{braunstein1992maximal}. \n\nWe note here that the JC model has been applied extensively in implementing C-NOT and Hadamard gate operations. Domokos \\textit{et al} (1995) \\cite{domokos1995simple} showed that using induced transitions between dressed states, it is possible to implement a C-NOT gate in which a cavity containing at most one photon is the control qubit and the atom is the target qubit. Later, Vitali, D. \\textit{et al} (2001)\\cite{vitali2001quantum} proposed a scheme of implementing a C-NOT gate between two distinct but identical cavities, acting as control and target qubits respectively. By passing an atom prepared initially in ground state consecutively between the two cavities a C-NOT ($cavity\\rightarrow{atom}$) and a C-NOT ($atom\\rightarrow{cavity}$) is realised with the respective classical fields S. Saif, F. \\textit{et al} (2001) \\cite{saif2007engineering} presented a study of quantum computing by engineering non-local quantum universal gates based on interaction of a two-level atom with two modes of electromagnetic field in high Q superconducting cavity. The two-level atom acted as the control qubit and the two-mode electromagnetic field served as the the target qubit. In this letter, we apply an approach similar to that in \\cite{saif2007engineering} where we implement a quantum C-NOT gate operation between two cavities defined in a two-dimensional Hilbert space spanned by the state vectors $|\\mu_1\\rangle=|1_A,0_B\\rangle$ and $|\\mu_2\\rangle=|0_A,1_B\\rangle$ as target qubits. Here $|\\mu_1\\rangle$ expresses the presence of one photon in mode A, when there is no photon in mode B, and $|\\mu_2\\rangle$ indicates that mode A is in vacuum state and one photon is present in mode B.  The control qubit in this respect is a two-level atom. The important difference with the approach used in \\cite{saif2007engineering} is the model, i.e, while the initial absolute atom-field ground state $|g,0\\rangle$  in the AJC interaction is affected by atom-cavity coupling, the  ground state $|g,0\\rangle$ in the JC model \\cite{saif2007engineering} is not affected by atom-cavity coupling. A similar result was determined independently in \\cite{omolo2019photospins}. Further, with precise choice of interaction time in the AJC qubit state transition operations defined in the AJC qubit sub-space spanned by normalised but non-orthogonal basics qubit state vectors \\cite{omolo2017polariton, omolo2019photospins}, C-NOT gate operations are realized between the two cavities.\n\nThe Hadamard gate also known as the Walsh-Hadamard gate is a single qubit gate \\cite{scherer2019mathematics, nielsen2011quantum}. The Hadamard transformation is defined as \n \n\\begin{equation}\n\\hat{H}=\\frac{\\hat{\\sigma}_\\textit{x}+\\hat{\\sigma}_\\textit{z}}{\\sqrt{2}}\n\\label{eq:mathhadamard}\n\\end{equation}\nwhere it transforms atomic computational basis states $|e\\rangle(|0\\rangle)$, $|g\\rangle(|1\\rangle)$ into diagonal basis states according to\n\\begin{eqnarray}\n\\hat{H}|e\\rangle\\rightarrow\\frac{|e\\rangle+|g\\rangle}{\\sqrt{2}}\\quad;\\quad\\hat{H}|g\\rangle\\rightarrow\\frac{|e\\rangle-|g\\rangle}{\\sqrt{2}}\\nonumber\\\\\n\\hat{H}|0\\rangle\\rightarrow\\frac{|0\\rangle+|1\\rangle}{\\sqrt{2}}\\quad;\\quad\\hat{H}|1\\rangle\\rightarrow\\frac{|0\\rangle-|1\\rangle}{\\sqrt{2}}\n\\label{eq:mathhadamard1}\n\\end{eqnarray}\n Vitali, D. \\textit{et al} (2001)\\cite{vitali2001quantum} showed that one qubit operations can be implemented on qubits represented by two internal atomic states because it amounts to applying suitable Rabi pulses. He demonstrated that the most practical solution on implementing one qubit operations on two Fock states is sending the atoms through the cavity. If the atom inside the cavity undergoes a $\\frac{\\pi}{2}$ pulse one realizes a Hadamard-phase gate. Saif, F. \\textit{et al} (2001) \\cite{saif2007engineering} also showed that it is possible to realise Hadamard operation by a controlled interaction between a two-mode high Q electromagnetic cavity field and a two-level atom. In his approach, the two-level atom is the control qubit, whereas the target qubit is made up of two modes of cavity field. Precision of the gate operations is realised by precise selection of interaction times of the two-level atom with the cavity mode. In this paper, we show that Hadamard operation in the AJC interaction is possible for a specified initial atomic state by setting a specific sum frequency and photon number in the anti-Jaynes-Cummings qubit state transition operation \\cite{omolo2017polariton, omolo2019photospins}, noting that the interaction components of the anti-Jaynes-Cummings Hamiltonian generates state transitions.\n\nThe content of this paper is therefore summarised as follows. Section \\ref{sec:model} presents an overview of the theoretical model. In sections \\ref{sec:qcgate} and \\ref{sec:hadgatelogic} respectively, implementation of a quantum C-NOT and Hadamard gates in the AJC interaction are presented. Finally section \\ref{sec:conclusion} contains the conclusion. \n\\section{The model}\n\\label{sec:model}\nThe quantum Rabi model of a quantized electromagnetic field mode interacting with a two-level atom is generated by the Hamiltonian \\cite{omolo2017conserved} \n\n\\begin{equation}\n\\hat{H}_R=\\frac{1}{2}\\hbar\\omega(\\hat{a}^{\\dagger}\\hat{a}+\\hat{a}\\hat{a}^{\\dagger})+\\hbar\\omega_0\\hat{s}_z + \\hbar\\lambda(\\hat{a}+\\hat{a}^{\\dagger})(\\hat{s}_++\\hat{s}_-)\n\\label{eq:rabi1}\n\\end{equation}\nnoting that the free field mode Hamiltonian is expressed in normal and anti-normal order form $\\frac{1}{2}\\hbar\\omega(\\hat{a}^{\\dagger}\\hat{a}+\\hat{a}\\hat{a}^{\\dagger})$.\nHere, $\\omega\\hspace{1mm},\\hspace{1mm}\\hat{a}\\hspace{1mm},\\hspace{1mm}\\hat{a}^{\\dagger}$ are quantized field mode angular frequency, annihilation and creation operators, while $\\omega_0\\hspace{1mm},\\hspace{1mm}\\hat{s}_z\\hspace{1mm},\\hspace{1mm}\\hat{s}_+\\hspace{1mm},\\hspace{1mm}\\hat{s}_-$ are atomic state transition angular frequency and operators. The Rabi Hamiltonian in eq.~\\eqref{eq:rabi1} is expressed in a symmetrized two-component form \\cite{omolo2017conserved, omolo2017polariton, omolo2019photospins}\n\n\\begin{equation}\n\\hat{H}_R=\\frac{1}{2}(\\hat{H}+\\hat{\\overline{H}})\n\\label{eq:rabi2}\n\\end{equation}\nwhere $\\hat{H}$ is the standard JC Hamiltonian interpreted as a polariton qubit Hamiltonian expressed in the form \\cite{omolo2017conserved}\n\n\\begin{eqnarray}\n\\hat{H}&=&\\hbar\\omega\\hat{N}+2\\hbar\\lambda\\hat{A}-\\frac{1}{2}\\hbar\\omega\\quad;\\quad\\hat{N}=\\hat{a}^{\\dagger}\\hat{a}+\\hat{s}_+\\hat{s}_- \\nonumber\\\\\n\\hat{A}&=&\\alpha\\hat{s}_z+\\hat{a}\\hat{s}_++\\hat{a}^{\\dagger}\\hat{s}_-\\quad;\\quad\\alpha=\\frac{\\omega_0-\\omega}{2\\lambda}\n\\label{eq:pham1}\n\\end{eqnarray}\nwhile $\\hat{\\overline{H}}$ is the AJC Hamiltonian interpreted as an anti-polariton qubit Hamiltonian in the form \\cite{omolo2017conserved}\n\n\\begin{eqnarray}\n\\hat{\\overline{H}}&=&\\hbar\\omega\\hat{\\overline{N}}+2\\hbar\\lambda\\hat{\\overline{A}}-\\frac{1}{2}\\hbar\\omega\\quad;\\quad\\hat{\\overline{N}}=\\hat{a}\\hat{a}^{\\dagger}+\\hat{s}_-\\hat{s}_+\\nonumber\\\\\\hat{\\overline{A}}&=&\\overline{\\alpha}\\hat{s}_z+\\hat{a}\\hat{s}_-+\\hat{a}^{\\dagger}\\hat{s}_+\\quad;\\quad\\overline{\\alpha}=\\frac{\\omega_0+\\omega}{2\\lambda}\n\\label{eq:antpham1}\n\\end{eqnarray}\nIn eqs.~\\eqref{eq:pham1} and \\eqref{eq:antpham1}, $\\hat{N}$, $\\hat{\\overline{N}}$ and $\\hat{A}$, $\\hat{\\overline{A}}$ are the respective polariton and anti-polariton qubit conserved excitation numbers and state transition operators.\n\nFollowing the physical property established in  \\cite{omolo2019photospins}, that for the field mode in an initial vacuum state only an atom entering the cavity in an initial excited state $|e\\rangle$ couples to the rotating positive frequency field  component in the JC interaction mechanism, while only an atom entering the cavity in an initial ground state $|g\\rangle$ couples to the anti-rotating negative frequency field  component in an AJC interaction mechanism, we generally take the atom to be in an initial excited state $|e\\rangle$ in the JC model and in an initial ground state $|g\\rangle$ in the AJC model. \n\nConsidering the AJC dynamics, applying the state transition operator $\\hat{\\overline{A}}$ from eq.~\\eqref{eq:antpham1} to the initial atom-field \\textit{n}-photon ground state vector $|g,n\\rangle$, the basic qubit state vectors $|\\psi_{gn}\\rangle$ and $|\\overline{\\phi}_{gn}\\rangle$ are determined in the form (\\textit{n}=0,1,2,....) \\cite{omolo2019photospins}\n\n\\begin{equation}\n|\\psi_{gn}\\rangle=|g,n\\rangle\\quad;\\quad|\\overline{\\phi}_{gn}\\rangle=-\\overline{c}_{gn}|g,n\\rangle+\\overline{s}_{gn}|e,n+1\\rangle\n\\label{eq:entsuptate}\n\\end{equation}\nwith dimensionless interaction parameters $\\overline{c}_{gn}$, $\\overline{s}_{gn}$ and Rabi frequency $\\overline{R}_{gn}$ defined as\n\n\\begin{eqnarray}\n\\overline{c}_{gn}&=&\\frac{\\overline{\\delta}}{2\\overline{R}_{gn}}\\quad;\\quad\\overline{s}_{gn}=\\frac{2\\lambda\\sqrt{n+1}}{\\overline{R}_{gn}}\\quad;\\quad\\overline{R}_{gn}=2\\lambda{\\overline{A}_{gn}}\\nonumber\\\\\n\\overline{A}_{gn}&=&\\sqrt{(n+1)+\\frac{\\overline{\\delta}^2}{16\\lambda^2}}\\quad;\\quad\\overline{\\delta}=\\omega_0+\\omega\n\\label{eq:parameters}\n\\end{eqnarray}\nwhere we have introduced sum frequency $\\overline{\\delta}=\\omega_0+\\omega$ to redefine $\\overline{\\alpha}$ in eq.~\\eqref{eq:antpham1}.\n\nThe qubit state vectors in eq.~\\eqref{eq:entsuptate} satisfy the qubit state transition algebraic operations\n\n\\begin{equation}\n\\hat{\\overline{A}}|\\psi_{gn}\\rangle=\\overline{A}_{gn}|\\overline{\\phi}_{gn}\\rangle\\quad;\\quad\\hat{\\overline{A}}|\\overline{\\phi}_{gn}\\rangle=\\overline{A}_{gn}|\\psi_{gn}\\rangle\n\\label{eq:traans}\n\\end{equation}\nIn the AJC qubit subspace spanned by normalized but non-orthogonal basic qubit state vectors  $|\\psi_{gn}\\rangle$\\hspace{1mm},\\hspace{1mm} $|\\overline{\\phi}_{gn}\\rangle$  the basic qubit state transition operator $\\hat{\\overline{\\varepsilon}}_g$ and identity operator $\\hat{\\overline{I}}_g$ are introduced according to the definitions \\cite{omolo2019photospins}\n\n\\begin{equation}\n\\hat{\\overline{\\varepsilon}}_g=\\frac{\\hat{\\overline{A}}}{\\overline{A}_{gn}}\\quad;\\quad\\hat{\\overline{I}}_g=\\frac{\\hat{\\overline{A}}^2}{\\overline{A}_{gn}^2}\\quad\\Rightarrow\\quad\\hat{\\overline{I}}_g=\\hat{\\overline{\\varepsilon}}_g^2\n\\label{eq:anttransop1}\n\\end{equation}\nwhich on substituting into eq.~\\eqref{eq:traans} generates the basic qubit state transition algebraic operations\n\n\\begin{eqnarray}\n\\hat{\\overline{\\varepsilon}}_g|\\psi_{gn}\\rangle&=&|\\overline{\\phi}_{gn}\\rangle\\quad;\\quad\\hat{\\overline{\\varepsilon}}_g|\\overline{\\phi}_{gn}\\rangle=|\\psi_{gn}\\rangle\\nonumber\\\\\\hat{\\overline{I}}_g|\\psi_{gn}\\rangle&=&|\\psi_{gn}\\rangle\\quad;\\quad\\hat{\\overline{I}}_g|\\overline{\\phi}_g\\rangle=|\\overline{\\phi}_g\\rangle\n\\label{eq:algop11}\n\\end{eqnarray}\nThe algebraic properties \\hspace{0.5mm} $\\hat{\\overline{\\varepsilon}}^{2k}=\\hat{\\overline{I}}_g$ \\hspace{0.5mm} and \\hspace{0.5mm} $\\hat{\\overline{\\varepsilon}}^{2k+1}=\\hat{\\overline{\\varepsilon}}_g$ \\hspace{0.5mm}easily gives the final property  \\cite{omolo2019photospins}\n\n\\begin{equation}\ne^{-i\\theta\\hat{\\overline{\\varepsilon}}_g}=\\cos(\\theta)\\hat{\\overline{I}}_g-i\\sin(\\theta)\\hat{\\overline{\\varepsilon}}_g\n\\label{eq:antialgprop}\n\\end{equation}\nwhich is useful in evaluating time-evolution operators.\n\nThe AJC qubit Hamiltonian defined within the qubit subspace spanned by the basic qubit state vectors $|\\psi_{gn}\\rangle$ , $|\\overline{\\phi}_{gn}\\rangle$ is then expressed in terms of the basic qubit states transition operators $\\hat{\\overline{\\varepsilon}}_g$, $\\hat{\\overline{I}}_g$ in the form \\cite{omolo2019photospins}\n\n\\begin{equation}\n\\hat{\\overline{H}}_g=\\hbar\\omega(n+\\frac{3}{2})\\hat{\\overline{I}}_g+\\hbar\\overline{R}_{gn}\\hat{\\overline{\\varepsilon}}_g\n\\label{eq:antijch2}\n\\end{equation}\n\\section{Quantum c-not gate operations}\n\\label{sec:qcgate}\n\nIn order to realise a C-NOT quantum gate operation in this context, we take a two-level atom as the control qubit, which is defined in a two dimensional Hilbert space with $|e\\rangle$ and $|g\\rangle$  as basis vectors, where $|e\\rangle$  expresses the excited state of the two-level atom and $|g\\rangle$ indicates the ground state. Two non-degenerate and orthogonal polarized cavity modes $C_A$ and $C_B$ make the target qubit. The target qubit is defined in two-dimensional Hilbert space spanned by the state vector $|\\mu_1\\rangle=|1_A,0_B\\rangle$, which expresses the presence of one photon in mode A, when there is no photon in mode B, and the state vector $|\\mu_2\\rangle=|0_A,1_B\\rangle$, which indicates that mode A is in the vacuum state and one photon is present in mode B.\n\nWith reference to the AJC qubit state transition operator in eq.\\eqref{eq:antialgprop}, lets first consider when an atom in ground state $|g\\rangle$ enters an electromagnetic cavity with mode A in vacuum state and a single photon in mode B. The atom couples to the anti-rotating negative frequency component of the field mode undergoing an AJC qubit state transition. After the atom interacts with mode A for a time $t=\\frac{\\pi}{\\overline{R}_{g0}}$, equal to half Rabi oscillation time, the driving field is modulated such that $\\theta=\\overline{R}_{g0}t=2\\lambda\\overline{A}_{g0}t=\\pi$. Redefining \\cite{omolo2019photospins}\n\n\\begin{equation}\n\\overline{\\alpha}=\\frac{\\overline{\\delta}}{2\\lambda}=\\frac{\\omega_0-\\omega+2\\omega}{2\\lambda}=\\frac{\\delta}{2\\lambda}+\\frac{\\omega}{\\lambda}=\\alpha+\\frac{\\omega}{\\lambda}\n\\end{equation}\nand considering a resonance case where $\\delta=\\omega_0-\\omega=0$ with $\\lambda\\gg\\omega$,            $\\overline{\\alpha}$   becomes very small  thus $\\theta=\\lambda{t}=\\frac{\\pi}{2}$, since  $\\overline{A}_{g0}=1$ determined from eq.~\\eqref{eq:parameters}. The evolution of this interaction determined by applying the AJC qubit state transition operation in eq.~\\eqref{eq:antialgprop} noting the definition of $\\hat{\\overline{I}}_g$ and $\\hat{\\overline{\\varepsilon}}_g$  \\citep{omolo2019photospins} in eq.~\\eqref{eq:anttransop1} is of the form\n\n\\begin{equation}\ne^{-i\\theta\\hat{\\overline{\\varepsilon}}_g}|g,0_A\\rangle=\\cos(\\theta)|g,0_A\\rangle-i\\sin(\\theta)|e,1_A\\rangle\n\\label{eq:modeA}\n\\end{equation}\nwhich reduces to\n\n\\begin{equation}\n|g,0_A\\rangle\\rightarrow-i|e,1_A\\rangle\n\\label{eq:flip1}\n\\end{equation}\nWe observe that the atom interacted with mode A and completed half of the Rabi oscillation, as a result, it contributed a photon to mode A and evolved to excited state $|e\\rangle$. Now, after the interaction time, it enters mode B containing a single photon, interacting with the cavity mode as follows\n\n\\begin{equation}\n-ie^{i\\theta\\hat{\\overline{\\varepsilon}}_e}|e,1_B\\rangle=-i\\cos(\\theta)|e,1_B\\rangle+\\sin(\\theta)|g,0_B\\rangle\n\\label{eq:modeB}\n\\end{equation}\nAfter an interaction with mode B for a time $t_1=2t$ such that $t_1=\\frac{\\pi(\\overline{R}_{g0}+\\overline{R}_{e1})}{\\overline{R}_{g0}\\overline{R}_{e1}}$, the driving field is modulated such that $\\theta=\\left(\\frac{\\overline{R}_{g0}\\overline{R}_{e1}}{\\overline{R}_{g0}+\\overline{R}_{e1}}\\right)t=\\frac{\\pi}{2}$ with $\\overline{R}_{g0}=2\\lambda{\\overline{A}_{g0}}=2\\lambda$ since $\\overline{A}_{g0}=1$ and $\\overline{R}_{e1}=2\\lambda{\\overline{A}_{e1}}=2\\lambda$ since $\\overline{A}_{e1}=1$. Therefore, $\\lambda{t}=\\frac{\\pi}{2}$. The form of eq.~\\eqref{eq:modeB} results into the evolution\n\n\\begin{equation}\n-i|e,1_B\\rangle\\rightarrow|g,0_B\\rangle\n\\label{eq:flip2}\n\\end{equation}\nThe results in  eq.~\\eqref{eq:flip2} shows that the atom evolves to ground state and absorbs a photon initially in mode B. Therefore the atom clearly performs a swapping of the electromagnetic field between the two field modes by controlled interaction. \n\nWhen the atom in ground state $|g\\rangle$, enters the electromagnetic cavity containing a single photon in mode A and mode B in vacuum state, the atom and the field interacts as follows\n\n\\begin{equation}\ne^{-i\\theta\\hat{\\overline{\\varepsilon}}_g}|g,0_B\\rangle=\\cos(\\theta)|g,0_B\\rangle-\\sin(\\theta)|e,1_B\\rangle\n\\label{eq:modeB2}\n\\end{equation} \nAfter an interaction with field mode B for a time $t=\\frac{\\pi}{\\overline{R}_{g0}}$ equal to half Rabi oscillation time, the driving field is modulated such that $\\theta=\\overline{R}_{g0}t=\\pi$, with $\\overline{R}_{g0}=2\\lambda{\\overline{A}}_{g0}=2\\lambda$ since $\\overline{A}_{g0}=1$. Therefore $\\theta=\\lambda{t}=\\frac{\\pi}{2}$. The form of eq.~\\eqref{eq:modeB2} results into the evolution\n\n\\begin{equation}\n|g,0_B\\rangle\\rightarrow-|e,1_B\\rangle\n\\label{eq:flip3}\n\\end{equation}\nThe atom then enters mode A containing one photon and interacts as follows\n\n\\begin{equation}\n-e^{i\\theta\\hat{\\overline{\\varepsilon}}_e}|e,1_A\\rangle=-\\cos(\\theta)|e,1_A\\rangle-i\\sin(\\theta)|g,0_A\\rangle\n\\label{eq:modeA2}\n\\end{equation}\nAfter an interaction with the cavity mode for a time $t_1=2t$ such that $t_1=\\frac{\\pi(\\overline{R}_{e1}+\\overline{R}_{g0})}{\\overline{R}_{e1}\\overline{R}_{g0}}$ we obtain a driving field modulation $\\theta=\\left(\\frac{\\overline{R}_{e1}\\overline{R}_{g0}}{\\overline{R}_{e1}+\\overline{R}_{g0}}\\right)t=\\frac{\\pi}{2}$, with $\\overline{R}_{e1}=2\\lambda{\\overline{A}_{e1}}=2\\lambda$ since $\\overline{A}_{e1}=1$ and $\\overline{R}_{g0}=2\\lambda{\\overline{A}_{g0}}=2\\lambda$ since $\\overline{A}_{g0}=1$. Therefore $\\theta=\\lambda{t}=\\frac{\\pi}{2}$. The form of eq.~\\eqref{eq:modeA2} results into the evolution\n\n\\begin{equation}\n|e,1_A\\rangle\\rightarrow{i}|g,0_A\\rangle\n\\label{eq:flip4}\n\\end{equation}\nThis shows that the atom evolves to ground state and performs a field swapping by absorbing a photon in mode A.\n\nWhen the atom in excited state $|e\\rangle$ enters mode A in vacuum state,that is target qubit $|\\mu_2\\rangle$, the atom propagates as a free wave without coupling to the field mode in vacuum state $|0\\rangle$ \\cite{omolo2019photospins}, leaving the cavity without altering the state of the cavity-field mode. A similar observation is made when the atom enters cavity B in vacuum state for the case of target qubit  $|\\mu_1\\rangle$.\n\nFrom the results obtained, we conclude that the target qubit made up of the electromagnetic field remains unchanged if the control qubit, that is, the two-level atom, is initially in the excited state $|e\\rangle$, while when the atom is in ground state $|g\\rangle$, the cavity states $|0\\rangle$ and $|1\\rangle$ flip. We shall refer to this gate as the AJC C-NOT $(atom\\rightarrow{cavity})$\n\n\\subsection{Probability of success of the C-NOT gate} \n\\label{sec:cnotgate}\nSuccess probability for the C-NOT gate is given by\n\n\\begin{equation}\nP_s=1-(\\sin^2(\\theta_A)+\\cos^2(\\theta_A)\\sin^2(\\theta_B))\n\\label{eq:success1}\n\\end{equation}\nIn terms of the Rabi frequency we write eq.~\\eqref{eq:success1} as\n\n\\begin{equation}\nP_s=1-(\\sin^2(\\overline{R}_A\\Delta{t_A})+\\cos^2(\\overline{R}_A\\Delta{t_A})\\sin^2(\\overline{R}_B\\Delta{t_B}))\n\\label{eq:success2}\n\\end{equation}\nFor the case in which the atom is in the ground state $|g\\rangle$ and enters an electromagnetic cavity with mode A in vacuum state and a single photon of the field mode B\n\n\\begin{equation*}\n\\overline{R}_A=\\overline{R}_{{g}{0}}=2\\lambda{\\overline{A}_{{g}{0}}}=2\\lambda\n\\end{equation*}\n\n\\begin{equation*}\n\\Delta{t_A}=\\frac{\\pi}{\\overline{R}_A}=\\frac{\\pi}{\\overline{R}_{{g}{0}}}=\\frac{\\pi}{2\\lambda}\n\\end{equation*}\n\n\\begin{equation*}\n\\overline{R}_B=\\overline{R}_{{e}{1}}=2\\lambda{\\overline{A}_{{e}{1}}}=2\\lambda\n\\end{equation*}\n\n\\begin{equation}\n\\Delta{t_B}=\\frac{\\pi}{2}\\frac{(\\overline{R}_A+\\overline{R}_B)}{\\overline{R}_A\\overline{R}_B}=\\frac{\\pi}{2}\\frac{(\\overline{R}_{{g}{0}}+\\overline{R}_{{e}{1}})}{\\overline{R}_{{g}{0}}\\overline{R}_{{e}{1}}}=\\frac{\\pi}{2\\lambda}\n\\label{eq:first}\n\\end{equation}\nsubstituting eq.~\\eqref{eq:first} into eq.~\\eqref{eq:success2} we obtain\n\n\n\\begin{equation}\nP_s=1-(\\sin^2(\\pi)+\\cos^2(\\pi)\\sin^2(\\pi))=1\n\\label{eq:success3}\n\\end{equation}\na unit probability of success.\n\nWhen the atom in ground state $|g\\rangle$ enters an electromagnetic cavity containing a single photon in mode A, and mode B in the vacuum state\n\n\\begin{equation*}\n\\overline{R}_A=\\overline{R}_{{e}{1}}=2\\lambda{\\overline{A}_{{e}{1}}}=2\\lambda\n\\end{equation*}\n\n\\begin{equation*}\n\\Delta{t_A}=\\frac{\\pi}{2}\\frac{(\\overline{R}_A+\\overline{R}_B)}{\\overline{R}_A\\overline{R}_B}=\\frac{\\pi}{2}\\frac{(\\overline{R}_{{e}{1}}+\\overline{R}_{{g}{0}})}{\\overline{R}_{{e}{1}}\\overline{R}_{{g}{0}}}=\\frac{\\pi}{2\\lambda}\n\\end{equation*}\n\n\\begin{equation*}\n\\overline{R}_B=\\overline{R}_{{g}{0}}=2\\lambda{\\overline{A}_{{g}{0}}}=2\\lambda\n\\end{equation*}\n\n\\begin{equation}\n\\Delta{t_B}=\\frac{\\pi}{\\overline{R}_B}=\\frac{\\pi}{\\overline{R}_{{g}{0}}}=\\frac{\\pi}{2\\lambda}\n\\label{eq:second}\n\\end{equation}\nsubstituting eq.~\\eqref{eq:second} into eq.~\\eqref{eq:success2} we obtain\n\n\\begin{equation}\nP_s=1-(\\sin^2(\\pi)+\\cos^2(\\pi)\\sin^2(\\pi))=1\n\\label{eq:success4}\n\\end{equation}\na unit probability of success.\n\nWe observe that success probabilities depend mainly upon the precise selection of the interaction times of the two level atom with the successive cavity modes.\n\n\\section{Hadamard logic gate}\n\\label{sec:hadgatelogic}\n\nTo realise Hadamard operation in the AJC interaction, we apply the qubit state transition operation in eq.~\\eqref{eq:anttransop1} and the general form in \\cite{omolo2017polariton, omolo2019photospins}. In this respect, we define the Hadamard operation at sum frequency $\\overline{\\delta}=4\\lambda$ and $n=0$ specified for an initial atomic state $|g\\rangle$ as \n\n\\begin{equation}\n\\hat{\\overline{\\varepsilon}}_g=\\frac{1}{\\overline{A}_{g0}}\\left(2\\hat{s}_z+\\hat{a}\\hat{s}_-+\\hat{a}^\\dagger\\hat{s}_+\\right)\\hspace{5mm};\\hspace{5mm}\\overline{A}_{g0}=\\sqrt{2}\n\\end{equation}\nThe initial atomic state $|g\\rangle$ is rotated to\n\n\\begin{equation}\n|g\\rangle\\rightarrow\\frac{1}{\\sqrt{2}}(|e\\rangle-|g\\rangle)\n\\label{eq:hadop1}\n\\end{equation}\nSimilarly, the Hadamard operation at sum frequency $\\overline{\\delta}=4\\lambda$ and $n=1$ specified for an initial atomic state $|e\\rangle$  is defined as \\citep{omolo2019photospins}\n\\begin{equation}\n\\hat{\\overline{\\varepsilon}}_e=\\frac{1}{\\overline{A}_{e1}}\\left(2\\hat{s}_z+\\hat{a}\\hat{s}_-+\\hat{a}^\\dagger\\hat{s}_+\\right)\\hspace{5mm};\\hspace{5mm}\\overline{A}_{e1}=\\sqrt{2}\n\\end{equation}\nThe initial atomic state $|e\\rangle$ is rotated to\n\n\n\\begin{equation}\n|e\\rangle\\rightarrow\\frac{1}{\\sqrt{2}}(|e\\rangle+|g\\rangle)\n\\label{eq:hadop2}\n\\end{equation}\nThe Hadamard transformations in Eqs.~\\eqref{eq:hadop1} and \\eqref{eq:hadop2} realised in the AJC interaction process (AJC model) agree precisely with the standard definition in Eq.~\\eqref{eq:mathhadamard1}.\n\\section{ Conclusion}\n\\label{sec:conclusion}\nIn this paper we have shown how to implement quantum C-NOT and Hadamard gates in the anti-Jaynes-Cummings interaction mechanism. We obtained ideal unit probabilities of success due to precise selection of interaction times during the C-NOT gate operations. We also realised efficient Hadamard operations through application of respective AJC qubit state transitions. \n\\section*{Acknowledgment}\n\nWe thank Maseno University, Department of Physics and Materials Science for providing a conducive environment to do this work.\n\n \\bibliographystyle{apsrev}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{introduction}\n\nThe quantum information theory in the relativistic framework has\nreceived considerable attention due to its theoretical importance\nand practical application \\cite{Peres,Boschi,Pan}. Especially, more\nand more efforts have been expended on the study of quantum\nentanglement in a relativistic setting because people consider the\nentanglement to be a major resource for quantum information tasks\nsuch as quantum teleportation, quantum computation and so on\n\\cite{Bouwmeester}. With the intention of studying the entanglement\nbetween accelerated observers, the fidelity of teleportation between\ntwo parties in relative uniform acceleration was discussed by Alsing\n\\emph{et al.} \\cite{Alsing-Milburn,Alsing-McMahon-Milburn}. Xian-Hui\nGe \\emph{et al.} extended the gravitational field of the\nteleportation to the four and higher dimensional spacetimes, and\neven explicitly discussed what effects the shape of the cavity in\nwhich particles are confined has on the teleportation in a black\nhole spacetime \\cite{Ge-Shen,Ge-Kim}. In order to further\ninvestigate the observer-dependent character of the entanglement,\nFuentes-Schuller \\emph{et al.} analyzed the entanglement between two\nmodes of a non-interacting  massless scalar field when one of the\nobservers describing the state is uniformly accelerated\n\\cite{Schuller-Mann}. And then Alsing \\emph{et al.} calculated the\nentanglement between two modes of a free Dirac field described by\nrelatively accelerated parties in a flat spacetime\n\\cite{Alsing-Mann}. Their results \\cite{Schuller-Mann,Alsing-Mann}\nalso showed that the different type of field will have a\nqualitatively different effect on the degradation of entanglement\nproduced by the Unruh effect \\cite{Davies,unruh}. More recently, Ahn\n\\emph{et al.} extended the investigation to the entanglement of a\ntwo-mode squeezed state in Riemannian spacetime \\cite{Ahn-Kim}, Yi\nLing \\emph{et al.} discussed the entanglement of electromagnetic\nfield in noninertial reference frames \\cite{Ling}, and Adesso\n\\emph{et al.} investigated the distribution of entanglement between\nmodes of a free scalar field from the perspective of observers in\nuniform acceleration \\cite{Adesso}.\n\nAs a further step along this line, we will provide an analysis of\nthe entanglement for the scalar field in the spacetime of a most\ngeneral, static and asymptotically flat black hole with spherical\nsymmetry. It seems to be an interesting study to consider the\ninfluences of the Hawking effect\n\\cite{Hawking-1,Hawking-2,Hawking-3} on the quantum entangled states\nand show how the Hawking temperature will change the properties of\nthe entanglement and teleportation. Choosing a generically entangled\nstate as the initially entangled state for two observers in the flat\nregion of this black hole, we will also try to see what effects the\nuncertain entangled state will have on the degradation of\nentanglement in our scheme due to the presence of an arbitrary state\nparameter. Our scheme proposes that the two observers, Alice and\nBob, share an initially entangled state at the same initial point in\nflat Minkowski spacetime before the black hole is formed. After the\ncoincidence of Alice and Bob, Alice stays stationary at the\nasymptotically flat region, while Bob falls in toward the mass and\nthen hovers outside of it. Once Bob is safely hovering outside of\nthe object at some constant acceleration, let it collapse to form a\nblack hole. By Birkhoff's theorem \\cite{Birkhoff} this won't change\nthe metric outside of the black hole and therefore won't change\nBob's acceleration. Thus, Bob's detector registers only thermally\nexcited particles due to the Hawking effect \\cite{unruh-1,unruh-2}.\nIn order to investigate the teleportation between two modes of a\nscalar field as detected by the two observers, we assume that Alice\nand Bob each hold an optical cavity which is small and perfect for\nthe teleportation in the black hole spacetime. Just as suggested by\nRefs. \\cite{Alsing-Milburn,Alsing-McMahon-Milburn}, we further\nsuppose that each cavity supports two orthogonal modes, with the\nsame frequency, which are each excited to a single photon Fock state\nat the coincidence point for Alice and Bob. Different from the\nstandard teleportation protocol, our scheme assumes that Bob hovers\noutside of the object before it collapses, and turns on his detector\nafter the formation of the black hole. Then, Bob can check to see\nwhether any thermal photons have been excited in his local cavity\nusing the non-absorbing detector.\n\nThe organization of this paper is as follows. In Sec. 2 we discuss\nthe vacuum structure of the background spacetime and the Hawking\neffect for the scalar particles as experienced by the observer\noutside the black hole. In Sec. 3 we analyze the effects of the\nHawking temperature on the entanglement between the modes for the\ndifferent state parameter. In Sec. 4 we describe the process of the\nteleportation between Alice and Bob, and calculate the fidelity of\nteleportation. We summarize and discuss our conclusions in the last\nsection.\n\n\\section{Vacuum structure and Hawking Radiation of scalar field}\n\nIt is well known that the spherically symmetric line element of a\nstatic and asymptotically flat black hole such as Schwarzschild\nblack hole, Reissner-Nordstr\\\"{o}m black hole \\cite{Chandrasekhar},\nGarfinkle-Horowitz-Strominger dilaton black hole \\cite{Horowitz},\nCasadio-Fabbri-Mazzacurati (CFM) brane black hole \\cite{Casadio} and\nso on can be written in the form\n\\begin{eqnarray}\\label{metric}\nds^2=f(r)dt^{2}-\\frac{1}{h(r)}dr^{2}-R^{2}(r)(d\\theta^{2}+\\sin\\theta^{2}d\\varphi^{2}),\n\\end{eqnarray}\nwhere the functions $f(r)$ and $h(r)$ vanish at the event horizon\n$r=r_{+}$ of the black hole. Throughout this paper we use\n$G=c=\\hbar=\\kappa_{B}=1$. It is obvious that the surface gravity of\nthe event horizon is determined by\n$\\kappa=\\sqrt{f'(r_{+})h'(r_{+})}\/2$. Defining the tortoise\ncoordinates $r_{*}$ as $dr_{*}=dr\/\\sqrt{f(r)h(r)}$, we can rewrite\nthe metric (\\ref{metric}) as\n\\begin{eqnarray}\\label{new metric}\nds^2=f(r)(dt^{2}-dr_{*}^{2})-R^{2}(r)(d\\theta^{2}+\\sin\\theta^{2}d\\varphi^{2}).\n\\end{eqnarray}\n\nThe massless scalar field $\\psi$ satisfies the Klein-Gordon equation\n\\begin{eqnarray}\n\\label{K-G Equation}\\frac{1}{\\sqrt{-g}}\\frac{{\\partial}}{\\partial\nx^{\\mu}} \\left(\\sqrt{-g}g^{\\mu\\nu}\\frac{\\partial\\psi}{\\partial\nx^{\\nu}}\\right)=0.\n\\end{eqnarray}\nAfter expressing the normal mode solution as \\cite{unruh,D-R}\n\\begin{eqnarray}\n\\psi_{\\omega lm}=\\frac{1}{R(r)}\\chi_{\\omega\nl}(r)Y_{lm}(\\theta,\\varphi)e^{-i\\omega t},\n\\end{eqnarray}\nwe can easily get the radial equation\n\\begin{eqnarray}\\label{radial equation}\n\\frac{d^{2}\\chi_{\\omega\nl}}{dr_{*}^{2}}+[\\omega^{2}-V(r)]\\chi_{\\omega l}=0,\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nV(r)=\\frac{\\sqrt{f(r)h(r)}}{R(r)}\\frac{d}{dr}\\left[\\sqrt{f(r)h(r)}\\frac{d\nR(r)}{dr}\\right]+\\frac{l(l+1)f(r)}{R^{2}(r)},\n\\end{eqnarray}\nwhere $Y_{lm}(\\theta,\\varphi)$ is a scalar spherical harmonic on the\nunit twosphere. Solving Eq. (\\ref{radial equation}) near the event\nhorizon, we obtain the incoming wave function which is analytic\neverywhere in the spacetime manifold \\cite{D-R}\n\\begin{eqnarray}\n\\psi_{in,\\omega lm}=e^{-i\\omega v}Y_{lm}(\\theta,\\varphi),\n\\end{eqnarray}\nand the outgoing wave functions for the inside and outside region of\nthe event horizon\n\\begin{eqnarray}\\label{inside mode}\n\\psi_{out,\\omega lm}(r<r_{+})=e^{i\\omega u}Y_{lm}(\\theta,\\varphi),\n\\end{eqnarray}\n\\begin{eqnarray}\\label{outside mode}\n\\psi_{out,\\omega lm}(r>r_{+})=e^{-i\\omega u}Y_{lm}(\\theta,\\varphi),\n\\end{eqnarray}\nwhere $v=t+r_{*}$ and $u=t-r_{*}$. Eqs. (\\ref{inside mode}) and\n(\\ref{outside mode}) are analytic inside and outside the event\nhorizon respectively, so they form a complete orthogonal family. In\nsecond-quantizing the field $\\Phi_{out}$ in the exterior of the\nblack hole we can expand it as follows \\cite{unruh}\n\\begin{eqnarray}\\label{First expand}\n&&\\Phi_{out}=\\sum_{lm}\\int d\\omega[b_{in,\\omega lm}\\psi_{out,\\omega\nlm}(r<r_{+})+b^{\\dag}_{in,\\omega lm}\\psi^{*}_{out,\\omega\nlm}(r<r_{+})\\nonumber\\\\\n&& \\quad \\quad \\quad  \\quad \\quad \\quad +b_{out,\\omega\nlm}\\psi_{out,\\omega lm}(r>r_{+})+b^{\\dag}_{out,\\omega\nlm}\\psi^{*}_{out,\\omega lm}(r>r_{+})],\n\\end{eqnarray}\nwhere $b_{in,\\omega lm}$ and $b^{\\dag}_{in,\\omega lm}$ are the\nannihilation and creation operators acting on the vacuum of the\ninterior region of the black hole, and $b_{out,\\omega lm}$ and\n$b^{\\dag}_{out,\\omega lm}$ are the annihilation and creation\noperators acting on the vacuum of the exterior region respectively.\nThus, the Fock vacuum state can be defined as\n\\begin{eqnarray}\\label{dilaton vacuum}\nb_{in,\\omega lm}|0\\rangle_{in}=b_{out,\\omega lm}|0\\rangle_{out}=0.\n\\end{eqnarray}\n\nIntroducing the generalized light-like Kruskal coordinates\n\\cite{D-R,Sannan,Zhao,Birrell}\n\\begin{eqnarray}\n&&U=-\\frac{1}{\\kappa}e^{-\\kappa u},\\quad V=\\frac{1}{\\kappa}e^{\\kappa\nv},\\quad {\\rm if\\quad r>r_{+}};\\nonumber\\\\\n&&U=\\frac{1}{\\kappa}e^{-\\kappa u},\\quad V=\\frac{1}{\\kappa}e^{\\kappa\nv}, \\quad {\\rm if\\quad r<r_{+}},\n\\end{eqnarray}\nand noticing that near the event horizon\n\\begin{eqnarray}\nr_{*}\\simeq\\frac{1}{\\sqrt{f'(r_{+})h'(r_{+})}}\\ln(r-r_{+}),\n\\end{eqnarray}\nwe can obtain a complete basis of the outgoing modes according to\nthe suggestion of Damour-Ruffini \\cite{D-R}\n\\begin{eqnarray}\n\\psi_{I,\\omega lm}=e^{\\frac{\\pi\\omega}{2\\kappa}}\\psi_{out,\\omega\nlm}(r>r_{+})+e^{-\\frac{\\pi\\omega}{2\\kappa}}\\psi^{*}_{out,\\omega\nlm}(r<r_{+}),\n\\end{eqnarray}\n\\begin{eqnarray}\n\\psi_{II,\\omega\nlm}=e^{-\\frac{\\pi\\omega}{2\\kappa}}\\psi^{*}_{out,\\omega\nlm}(r>r_{+})+e^{\\frac{\\pi\\omega}{2\\kappa}}\\psi_{out,\\omega\nlm}(r<r_{+}).\n\\end{eqnarray}\nThus, we can also quantize the quantum field $\\Phi_{out}$ in terms\nof $\\psi_{I,\\omega lm}$ and $\\psi_{II,\\omega lm}$ in the Kruskal\nspacetime as\n\\begin{eqnarray}\\label{Second expand}\n\\Phi_{out}=\\sum_{lm}\\int\nd\\omega&[2&\\sinh(\\pi\\omega\/\\kappa)]^{-1\/2}[a_{out,\\omega\nlm}\\psi_{I,\\omega lm}+a^{\\dag}_{out,\\omega lm}\\psi^{*}_{I,\\omega\nlm}\\nonumber \\\\ &+&a_{in,\\omega lm}\\psi_{II,\\omega\nlm}+a^{\\dag}_{in,\\omega lm}\\psi^{*}_{II,\\omega lm}] ,\n\\end{eqnarray}\nwhere the annihilation operator $a_{out,\\omega lm}$ can be used to\ndefine the Kruskal vacuum outside the event horizon\n\\begin{eqnarray}\\label{Kruskal vacuum}\na_{out,\\omega lm}|0\\rangle_{K}=0.\n\\end{eqnarray}\n\nAccording to Eqs. (\\ref{First expand}) and (\\ref{Second expand}), we\nobtain the Bogoliubov transformations \\cite{Birrell,Barnett} for the\nparticle creation and annihilation operators in the black hole and\nKruskal spacetimes\n\\begin{eqnarray}\n&&a_{out,\\omega lm}=\\frac{b_{out,\\omega\nlm}}{\\sqrt{1-e^{-2\\pi\\omega\/\\kappa}}}-\\frac{b^{\\dag}_{in,\\omega lm}}{\\sqrt{e^{2\\pi\\omega\/\\kappa}-1}}, \\nonumber\\\\\n&&a^{\\dag}_{out,\\omega lm}=\\frac{b^{\\dag}_{out,\\omega\nlm}}{\\sqrt{1-e^{-2\\pi\\omega\/\\kappa}}}-\\frac{b_{in,\\omega\nlm}}{\\sqrt{e^{2\\pi\\omega\/\\kappa}-1}}.\n\\end{eqnarray}\nWe assume that the Kruskal vacuum $|0\\rangle_{K}$ is related to the\nvacuum of the black hole $|0\\rangle_{in}\\otimes|0\\rangle_{out}$ by\n\\begin{eqnarray}\\label{two vacuum}\n|0\\rangle_{K}=\\Upsilon(b_{in,\\omega lm},b^{\\dag}_{in,\\omega\nlm},b_{out,\\omega lm},b^{\\dag}_{out,\\omega\nlm})|0\\rangle_{in}\\otimes|0\\rangle_{out}.\n\\end{eqnarray}\nFrom $[b_{in,\\omega lm},b^{\\dag}_{in,\\omega lm}]=[b_{out,\\omega\nlm},b^{\\dag}_{out,\\omega lm}]=1$ and Eq. (\\ref{Kruskal vacuum}), we\nget \\cite{unruh,Ahn}\n\\begin{eqnarray}\n\\Upsilon\\propto \\exp({b^{\\dag}_{out,\\omega lm}b^{\\dag}_{in,\\omega\nlm}}~e^{-\\pi\\omega\/\\kappa}).\n\\end{eqnarray}\nAfter properly normalizing the state vector, we obtain the Kruskal\nvacuum which is a maximally entangled two-mode squeezed state\n\\cite{Barnett,Ahn}\n\\begin{eqnarray}\\label{Scalar-vacuum}\n|0\\rangle_{K}=\\sqrt{1-e^{-2\\pi\\omega\/\\kappa}}\n\\sum_{n=0}^{\\infty}e^{-n\\pi\\omega\/\\kappa}|n\\rangle_{in}\\otimes|n\\rangle_{out},\n\\end{eqnarray}\nand the first excited state\n\\begin{eqnarray}\\label{Scalar-excited}\n&&|1\\rangle_{K}=a^{\\dag}_{out,\\omega lm}|0\\rangle_{K}\\nonumber\\\\\n&&=(1-e^{-2\\pi\\omega\/\\kappa})\\sum_{n=0}^{\\infty}\n\\sqrt{n+1}~e^{-n\\pi\\omega\/\\kappa}|n\\rangle_{in}\\otimes|n+1\\rangle_{out},\n\\end{eqnarray}\nwhere $\\{|n\\rangle_{in}\\}$ and $\\{|n\\rangle_{out}\\}$ are the\northonormal bases for the inside and outside region of the event\nhorizon respectively. For the observer outside the black hole, he\nneeds to trace over the modes in the interior region since he has no\naccess to the information in this causally disconnected region.\nTherefore, when an outside observer travels through the Kruskal\nparticle vacuum $|0\\rangle_{K}$ of mode $\\omega$ his detector\nregisters a number of particles given by\n\\begin{eqnarray}\\label{Hawking}\n_{K}\\langle0|b^{\\dag}_{out,\\omega lm}b_{out,\\omega\nlm}|0\\rangle_{K}=\\frac{1}{e^{2\\pi\\omega\/\\kappa}-1}=\\frac{1}{e^{\\omega\/T}-1},\n\\end{eqnarray}\nwhere we have defined the Hawking temperature as\n\\cite{Kerner-Mann,Jiang-Wu-Cai}\n\\begin{eqnarray}\\label{Hawking temperature}\nT=\\frac{\\kappa}{2\\pi}=\\frac{\\sqrt{f'(r_{+})h'(r_{+})}}{4\\pi}.\n\\end{eqnarray}\nEq. (\\ref{Hawking}) is well known as the Hawking effect\n\\cite{Hawking-1,Hawking-2,Hawking-3}, which shows that the observer\nin the exterior of the black hole detects a thermal Bose-Einstein\ndistribution of particles as he traverses the Kruskal vacuum.\n\n\\section{Quantum Entanglement in background of a black hole}\n\nNow we assume that Alice has a detector which only detects mode\n$|n\\rangle_{A}$ and Bob has a detector sensitive only to mode\n$|n\\rangle_{B}$, and they share a generically entangled state at the\nsame initial point in flat Minkowski spacetime before the black hole\nis formed\n\\begin{eqnarray}\\label{initial}\n|\\Psi\\rangle=\\alpha|0\\rangle_{A}|0\\rangle_{B}\n+\\sqrt{1-\\alpha^{2}}|1\\rangle_{A}|1\\rangle_{B},\n\\end{eqnarray}\nwhere $\\alpha$ is some real number which satisfies\n$|\\alpha|\\in(0,1)$, $\\alpha$ and $\\sqrt{1-\\alpha^{2}}$ are the\nso-called ``normalized partners\". After the coincidence of Alice and\nBob, Alice remains at the asymptotically flat region but Bob freely\nfalls in toward the mass with his detector and then hovers outside\nof it before it collapses to form a black hole. Obviously, there\nwill be some thermal effects due to changes in Bob's acceleration,\nbut there will be no Hawking radiation. Note that such effects can\nbe negligible, or at least will disperse after some time. Then, once\nBob is safely hovering outside of the object at some constant\nacceleration, let it collapse to form a black hole. By Birkhoff's\ntheorem \\cite{Birkhoff} this won't change the metric outside of the\nblack hole and therefore won't change Bob's acceleration. Thus,\nBob's detector registers only thermally excited particles due to the\nHawking effect \\cite{unruh-1,unruh-2}. The states corresponding to\nmode $|n\\rangle_{B}$ must be specified in the coordinates of the\nblack hole in order to describe what Bob sees in this curved\nspacetime. Thus, using Eqs. (\\ref{Scalar-vacuum}) and\n(\\ref{Scalar-excited}), we can rewrite Eq. (\\ref{initial}) in terms\nof Minkowski modes for Alice and black hole modes for Bob. Since Bob\nis causally disconnected from the interior region of the black hole,\nwe will take the trace over the states in this region and obtain the\nmixed density matrix between Alice and Bob in exterior region\n\\begin{eqnarray}\\label{Bos-density}\n&&\\rho_{AB}=(1-e^{-\\omega\/T})\\sum_{n=0}^{\\infty}\\rho_{n}~e^{-n\\omega\/T},\n\\nonumber\\\\&& \\rho_{n}=\\alpha^{2}|0n\\rangle\\langle0n|\n+(n+1)(1-\\alpha^{2})(1-e^{-\\omega\/T})|1(n+1)\\rangle\\langle1(n+1)|\n\\nonumber\\\\&&\\qquad+\\alpha\\sqrt{(n+1)(1-\\alpha^{2})\n(1-e^{-\\omega\/T})}|0n\\rangle\\langle1(n+1)|\\nonumber\\\\&&\\qquad+\n\\alpha\\sqrt{(n+1)(1-\\alpha^{2})(1-e^{-\\omega\/T})}\n|1(n+1)\\rangle\\langle0n|,\n\\end{eqnarray}\nwhere $|nm\\rangle=|n\\rangle_{A}|m\\rangle_{B,out}$.\n\nIt should be noted that we do not trace over the states located\ninside the event horizon for Alice, even though she now is also\ncausally disconnected from the interior region of the black hole. As\na matter of fact, the causal structure of spacetime keeps every\nobserver exterior from the black hole disconnected from its\ninterior. Why do we not trace over the degree of freedom of Alice in\nthe region inaccessible to her? We now present the reasons as\nfollows. On the one hand, we can justify what we are doing by\ntheory. For a Schwarzschild black hole, the mass of which is assumed\nto be of the order of a solar mass ($M_{bh}\\sim M_{\\odot}$), the\nmagnitude of acceleration near this black hole that Bob needs is\nabout $10^{13}m\/s^{2}$, which is much larger than that of Alice\nneeds (almost equals to zero) in the asymptotical region. Thus, we\nargue that Alice's acceleration effects can be neglected, whereas\nBob's can't. On the other hand, we can think about it from\nexperiment. Though we do not observe the black hole directly,\nimpressive progress in optical, radio and $X$-ray astronomy greatly\nbolster the evidence for supermassive black holes in the centers of\ngalaxies \\cite{Frolov-Novikov}. Thus, the Earth can be argued to be\nan asymptotical region far from black holes, as far as we know. The\nstandard quantum field theory works fine for the earthbound\nexperiments, so we have at least some circumstantial empirical\nevidence that tracing over the black hole interior can be neglected\nin asymptotical regions.\n\nIt is clear that the partial transpose criterion provides a\nsufficient condition for the existence of entanglement in this case\n\\cite{peres}: if at least one eigenvalue of the partial transpose of\nthe density matrix is negative, the density matrix is entangled; but\na state with positive partial transpose can still be entangled. It\nis well-known bound or nondistillable entanglement\n\\cite{Vidal,Plenio}. Interchanging Alice's qubits\n$(|mn\\rangle\\langle pq| \\to |pn\\rangle\\langle mq|)$, we get the\nmatrix representation of the partial transpose in the ($n$,~$n+1$)\nblock {\\small\n\\begin{eqnarray}\n(\\rho_{AB}^{T_{A}})_{n,n+1}&=&e^{-n\\omega\/T}(1-e^{-\\omega\/T})\n\\nonumber \\\\ && \\times \\left(\n\\begin{array}{ll}\n n(1-\\alpha^{2})(e^{\\omega\/T}-1)\n & \\alpha\\sqrt{(n+1)(1-\\alpha^{2})(1-e^{-\\omega\/T})} \\\\ \\\\\n \\alpha\\sqrt{(n+1)(1-\\alpha^{2})(1-e^{-\\omega\/T})}\n & ~~~~~~~~\\alpha^{2}e^{-\\omega\/T}\\\\\n\\end{array}\\right),\n\\end{eqnarray}}\nand its eigenvalues\n\\begin{eqnarray}\n\\lambda _{\\pm }^{n}=\\frac{e^{-n\\omega\/T}(1-e^{-\\omega\/T})}{2}\n\\left[\\zeta_{n}\\pm\\sqrt{\\zeta_{n}^{2}\n+4\\alpha^{2}(1-\\alpha^{2})(1-e^{-\\omega\/T})}~\\right],\n\\end{eqnarray}\nwhere\n$\\zeta_{n}=\\alpha^{2}e^{-\\omega\/T}+n(1-\\alpha^{2})(e^{\\omega\/T}-1)$.\nObviously the eigenvalue $\\lambda_{-}^{n}$ is always negative for\nfinite value of the Hawking temperature. Hence, this mixed state is\nalways entangled for any finite value of $T$. It should be noted\nthat in the limit $T\\rightarrow\\infty$, the negative eigenvalue will\ngo to zero. In order to discuss this further, we will use the\nlogarithmic negativity which serves as an upper bound on the\nentanglement of distillation \\cite{Vidal,Plenio}. This entanglement\nmonotone is defined as $N(\\rho_{AB})=\\log\n_{2}||\\rho_{AB}^{T_{A}}||$, where $||\\rho_{AB}^{T_{A}}||$ is the\ntrace norm of the partial transpose $\\rho_{AB}^{T_{A}}$. Thus, we\nobtain the logarithmic negativity for this case\n\\begin{eqnarray}\n&&N(\\rho_{AB})=\\log_{2}\\left[\\alpha^{2}(1-e^{-\\omega\/T})\n+\\sum_{n=0}^{\\infty}e^{-n\\omega\/T}(1-e^{-\\omega\/T})\n\\sqrt{\\zeta_{n}^{2} +4\\alpha^{2}\n(1-\\alpha^{2})(1-e^{-\\omega\/T})}\\right].\\nonumber \\\\\n\\end{eqnarray}\nThe trajectories of the logarithmic negativity $N(\\rho_{AB})$ versus\n$T$ for different $\\alpha$ in Fig. \\ref{NegBos} just show how the\nHawking temperature $T$ would change the properties of the\nentanglement.\n\n\\begin{figure}[ht]\n\\includegraphics[scale=0.8]{NegBos}\n\\caption{\\label{NegBos}The logarithmic negativity as a function of\nthe Hawking temperature $T$ with the fixed $\\omega$ for different\n$\\alpha$.}\n\\end{figure}\n\nFor the Hawking temperature of zero, corresponding to the case of a\nsupermasive or an almost extreme black hole,\n$N(\\rho_{AB})=\\log_{2}(1+2|\\alpha|\\sqrt{1-\\alpha^{2}})$. In the\nrange $0<|\\alpha|\\leq1\/\\sqrt{2}$ the larger $\\alpha$, the stronger\nthe ``initial entanglement\"; but in the range\n$1\/\\sqrt{2}\\leq|\\alpha|<1$ the larger $\\alpha$, the weaker the\n``initial entanglement\". For finite Hawking temperature, the\nmonotonous decrease of $N(\\rho_{AB})$ with increasing $T$ for five\ndifferent $\\alpha$ means that the ``initial entanglement\" is lost to\nthe thermal fields generated by the Hawking effect. This result\nagrees well with the Hawking's original argument\n\\cite{Hawking-1,Hawking-2,Hawking-3}, which says that smaller black\nholes are at a higher temperature and so radiate more violently than\nmassive black holes. Fig. \\ref{NegBos} also shows that when the\n``initial entanglement\" is stronger, we lose it more rapidly. But it\nis surprisingly found that the same ``initial entanglement\" for\n$\\alpha$ and its ``normalized partner\" $\\sqrt{1-\\alpha^{2}}$ will be\ndegraded along two different curves except for the maximally\nentangled state, i.e., $|\\alpha|=1\/\\sqrt{2}$. This phenomenon, due\nto the coupling of $\\alpha$ and the exponential functions related to\n$T$, just shows the inequivalence of the quantization for a scalar\nfield in the black hole and Kruskal spacetimes. The logarithmic\nnegativity is exactly zero for any $\\alpha$ in the limit\n$T\\rightarrow\\infty$, which indicates that the state has no longer\ndistillable entanglement for the arbitrary values of $\\alpha$ when\nthe black hole evaporates completely.\n\nIn order to estimate the total amount of correlations between Alice\nand Bob, we will analyze the mutual information which is defined as\n\\cite{RAM}\n\\begin{eqnarray}\nI(\\rho_{AB})=S(\\rho_{A})+S(\\rho_{B})-S(\\rho_{AB}),\n\\end{eqnarray}\nwhere $S(\\rho )=-\\text{Tr}(\\rho \\log_{2}\\rho)$ is the entropy of the\ndensity matrix $\\rho$. From Eq. (\\ref{Bos-density}), we can give the\nentropy of this joint state\n\\begin{eqnarray}\n&&S(\\rho _{AB})=-\\sum_{n=0}^{\\infty }\ne^{-n\\omega\/T}(1-e^{-\\omega\/T})\\left[\\alpha^{2}+(n+1)(1-\\alpha^{2})(1-e^{-\\omega\/T})\\right]\n\\nonumber\\\\\n&&\\qquad\\qquad~\\times\n\\log_{2}e^{-n\\omega\/T}(1-e^{-\\omega\/T})\\left[\\alpha^{2}+(n+1)(1-\\alpha^{2})(1-e^{-\\omega\/T})\\right].\n\\end{eqnarray}\nTracing over Alice's states for the density matrix $\\rho_{AB}$, we\nget Bob's density matrix in exterior region of the event horizon\n\\begin{eqnarray}\n&&\\rho_{B}=(1-e^{-\\omega\/T})\\sum_{n=0}^{\\infty}e^{-n\\omega\/T}\\left[\\alpha^{2}|n\\rangle\\langle\nn|+(n+1)(1-\\alpha^{2})(1-e^{-\\omega\/T})|n+1\\rangle\\langle\nn+1|\\right],\n\\end{eqnarray}\nand its entropy\n\\begin{eqnarray}\n&&S(\\rho_{B})=-\\sum_{n=0}^{\\infty\n}e^{-n\\omega\/T}(1-e^{-\\omega\/T})\\left[\\alpha^{2}+n(1-\\alpha^{2})(e^{\\omega\/T}-1)\\right]\n\\nonumber\\\\\n&&\\qquad\\qquad\\times\n\\log_{2}e^{-n\\omega\/T}(1-e^{-\\omega\/T})\\left[\\alpha^{2}+n(1-\\alpha^{2})(e^{\\omega\/T}-1)\\right].\n\\end{eqnarray}\nWe can also obtain Alice's density matrix by tracing over Bob's\nstates\n\\begin{eqnarray}\\label{Alice-density}\n&&\\rho_{A}=\\alpha^{2}|0\\rangle\\langle0|\n+(1-\\alpha^{2})|1\\rangle\\langle1|,\n\\end{eqnarray}\nwhose entropy can be expressed as\n\\begin{eqnarray}\\label{Alice-entropy}\n&&S(\\rho_{A})=-[\\alpha^{2}\\log_{2}\\alpha^{2}\n+(1-\\alpha^{2})\\log_{2}(1-\\alpha^{2})].\n\\end{eqnarray}\nThus, we draw the behaviors of the mutual information $I(\\rho_{AB})$\nas a function of the Hawking temperature $T$ for different values of\nthe state parameter $\\alpha$ in Fig. \\ref{MuInBos}.\n\n\\begin{figure}[ht]\n\\includegraphics[scale=0.8]{MuInBos}\n\\caption{\\label{MuInBos}The mutual information as a function of the\nHawking temperature $T$ with the fixed $\\omega$ for different\n$\\alpha$.}\n\\end{figure}\n\nFig. \\ref{MuInBos} shows that for the Hawking temperature of zero,\nthe ``initially mutual information\" equals to\n\\begin{eqnarray}\\label{Boc-MuIn-Initial}\nI_{i}(\\rho _{AB})=-2[\\alpha^{2}\\log_{2}\\alpha^{2}\n+(1-\\alpha^{2})\\log_{2}(1-\\alpha^{2})].\n\\end{eqnarray}\nIn the range $0<|\\alpha|\\leq1\/\\sqrt{2}$ the larger $\\alpha$, the\nstronger $I_{i}(\\rho _{AB})$; but in the range\n$1\/\\sqrt{2}\\leq|\\alpha|<1$ the larger $\\alpha$, the weaker\n$I_{i}(\\rho _{AB})$. As the Hawking temperature increases, the\nmutual information becomes smaller. It is interesting to note that\nexcept for the maximally entangled state, the same ``initially\nmutual information\"  for $\\alpha$ and $\\sqrt{1-\\alpha^{2}}$ will be\ndegraded along two different trajectories. However, in the infinite\nHawking temperature limit $T\\rightarrow\\infty$, i.e., the black hole\nevaporates completely, the mutual information converges to the same\nvalue again\n\\begin{eqnarray}\\label{Boc-MuIn-Final}\nI_{f}(\\rho _{AB})=-[\\alpha^{2}\\log_{2}\\alpha^{2}\n+(1-\\alpha^{2})\\log_{2}(1-\\alpha^{2})],\n\\end{eqnarray}\nwhich equals to just half of $I_{i}(\\rho _{AB})$. Thus, we conclude\nthat\n\\begin{eqnarray}\n&&I_{f}(\\rho _{AB})=\\frac{1}{2}I_{i}(\\rho _{AB}),\n\\end{eqnarray}\nwhich is independent of the state parameter $\\alpha$. Obviously if\n$I_{i}(\\rho _{AB})$ is higher, it is degraded to a higher degree in\nthis limit. Since the distillable entanglement in the infinite\nHawking temperature limit is zero, we are safe to say that the total\ncorrelations consist of classical correlations plus bound\nentanglement in this limit.\n\n\\section{Quantum Teleportation in background of a black hole}\n\nIn this section we will concentrate on a particular quantum\ninformation task: quantum teleportation. We assume that Alice and\nBob each hold an optical cavity, at rest in their local frame. Each\ncavity supports two orthogonal modes (labeled by $A_{i}$ and $B_{i}$\nwith $i=1,~2$), with the same frequency, which are each excited to a\nsingle photon Fock state at the coincidence point for Alice and Bob.\nWe ignore the polarization of these modes and model the photons by\nthe massless modes of a scalar field as suggested by Refs.\n\\cite{Alsing-Milburn,Alsing-McMahon-Milburn}. Considering the\ntextbook teleportation protocol \\cite{Nielsen}, we let Alice and Bob\nshare a maximally entangled state, i.e., an entangled Bell state in\nflat Minkowski spacetime\n\\begin{eqnarray}\n|\\Psi\\rangle=\\frac{1}{\\sqrt{2}}\\left(|\\mathbf{0}\\rangle_{A}|\\mathbf{0}\\rangle_{B}\n+|\\mathbf{1}\\rangle_{A}|\\mathbf{1}\\rangle_{B}\\right),\n\\end{eqnarray}\nwhere the logical states $|\\mathbf{0}\\rangle_{A}$ and\n$|\\mathbf{1}\\rangle_{A}$ are defined in terms of the physical Fock\nstates for Alice's cavity by the dual-rail basis states\n\\cite{Alsing-Milburn,Alsing-McMahon-Milburn}\n\\begin{eqnarray}\n|\\mathbf{0}\\rangle_{A}=|1\\rangle_{A_{1}}|0\\rangle_{A_{2}},~~\n|\\mathbf{1}\\rangle_{A}=|0\\rangle_{A_{1}}|1\\rangle_{A_{2}},\n\\end{eqnarray}\nwith similar expressions for Bob's cavity. It should be noted that\n$|1\\rangle_{A_{1}}$ and $|1\\rangle_{A_{2}}$ are single photon\nexcitations of the Minkowski vacuum states in Alice's cavity. Our\nconstruction implicitly assumes that we have chosen a modal\ndecomposition of the Minkowski vacuum based on intra-cavity and\nextra-cavity modes, which is a legitimate alternative to the usual\nway of quantizing the vacuum in terms of plane wave modes\n\\cite{Dalton-Knight-1,Dalton-Knight-2}. Once the cavities are loaded\nwith a photon, we also assume each cavity is perfect and cannot emit\nthe photon.\n\nRecalling the usual teleportation protocol with the unknown state\n\\cite{Nielsen}\n\\begin{eqnarray}\\label{teleportation}\n|\\varphi\\rangle=a|\\mathbf{0}\\rangle+b|\\mathbf{1}\\rangle,\n\\end{eqnarray}\nwe assume that Alice has an additional cavity which contains this\nsingle qubit (\\ref{teleportation}) with dual-rail encoding by a\nphoton excitation of a two-mode Minkowski vacuum state. This will\nallow Alice to make a joint measurement on the two orthogonal modes\nof each cavity. For the usual teleportation protocol between two\nMinkowski observers Alice and Bob, after Alice's measurement, Bob's\nstate will be projected according to the measurement outcome. We can\ngive the final state received by Bob\n\\begin{eqnarray}\n|\\varphi_{ij}\\rangle=x_{ij}|\\mathbf{0}\\rangle+y_{ij}|\\mathbf{1}\\rangle,\n\\end{eqnarray}\nwith four possible conditional state amplitudes\n$(x_{00},y_{00})=(a,b)$, $(x_{01},y_{01})=(b,a)$,\n$(x_{10},y_{10})=(a,-b)$ and $(x_{11},y_{11})=(-b,a)$. Once\nreceiving the classical information of the result of Alice's\nmeasurement, Bob can apply a unitary transformation to verify the\nprotocol. Obviously the fidelity of the teleported state is unity in\nthis idealized situation.\n\nAlice now wishes to perform the same teleportation protocol with the\nnoninertial observer Bob. We assume that prior to their coincidence,\nAlice and Bob ensure that all photons are removed from their\ncavities. When Alice and Bob instantaneously share a maximally\nentangled state at the asymptotically flat region, we suppose that\nthe two cavities overlap and simultaneously a four photon source\nexcites a two photon state in each cavity. Then Alice remains there\nbut Bob falls in toward the mass and then hovers outside of it. Once\nBob is safely hovering outside of the object at some constant\nacceleration, let it collapse to form a black hole. Then, Bob turns\non his detector after the formation of the black hole. Bob can check\nto see whether any thermal photons have been excited in his local\ncavity using the non-absorbing detector. It should be noted that the\ncommon frequency of both Alice's and Bob's cavity is just the\nfrequency $\\omega$ of Eqs. (\\ref{Scalar-vacuum}) and\n(\\ref{Scalar-excited}) \\cite{Alsing-Milburn,Alsing-McMahon-Milburn}.\nFor Bob, the observer locates near the event horizon of a black\nhole, he needs to trace over the modes in the interior region since\nhe is causally disconnected from this region. Thus, when Alice sends\nthe result of her measurement to Bob, Bob's state can be projected\ninto\n\\begin{eqnarray}\\label{Bob teleport}\n&&\\rho_{ij}=\\sum_{k=0}^{\\infty}\\sum_{l=0}^{\\infty}~_{in}\\langle\nk,l|\\varphi_{ij}\\rangle\\langle\\varphi_{ij}|k,l\\rangle_{in}\\nonumber\\\\\n&&~~~~=(1-e^{-\\omega\/T})^{3}\\sum^{\\infty}_{n=0}\\sum^{n}_{m=0}\\{e^{-(n-1)\\omega\/T}[(n-m)|x_{ij}|^{2}+m\n|y_{ij}|^{2}]|m,n-m \\rangle_{out}\\langle m,n-m|\n\\nonumber\\\\&&\\qquad~~+[\n\\sqrt{(m+1)(n-m+1)}~x_{ij}y^{*}_{ij}e^{-n\\omega\/T}|m,n-m+1\\rangle_{out}\\langle\nm+1,n-m|+{\\rm H.c.}]\\},\\nonumber \\\\\n\\end{eqnarray}\nwhere $|m,n-m \\rangle_{out}=|m\\rangle_{B_{1}}\\otimes|n-m\n\\rangle_{B_{2}}$ is a state of $n$ total excitations in the exterior\nregion product state, with $0\\leq m\\leq n$ excitations in the\nleftmost mode. Eq. (\\ref{Bob teleport}) can be rewritten as\n\\begin{eqnarray}\n&&\\rho_{ij}=\\sum^{\\infty}_{n=0}p_{n}\\rho_{ij,n},\\quad {\\rm\nwith}\\quad p_{0}=0,\\quad\np_{n}=(1-e^{-\\omega\/T})^{3}e^{-(n-1)\\omega\/T}\\quad{\\rm for}\\quad\nn\\geq1.\n\\end{eqnarray}\nSince what we concern about is to which extent\n$|\\varphi_{ij}\\rangle$ might deviate from unitarity, so upon\nreceiving the result $(i,j)$ of Alice's measurement, Bob can apply\nthe rotation operators (a unitary transformation in his local frame)\nrestricted to the one-excitation sector of his state spanned by\n$\\{|\\mathbf{0}\\rangle_{out},|\\mathbf{1}\\rangle_{out}\\}=\\{|0,1\\rangle_{out},|1,0\\rangle_{out}\\}$\nto turn this portion of his density matrix into the exterior region\nanalogue of the state in Eq. (\\ref{teleportation})\n\\cite{Alsing-Milburn,Alsing-McMahon-Milburn,Ge-Shen,Ge-Kim}\n\\begin{eqnarray}\n|\\varphi\\rangle_{out}=a|\\mathbf{0}\\rangle_{out}+b|\\mathbf{1}\\rangle_{out}.\n\\end{eqnarray}\nThus, we can obtain the fidelity of Bob's final state with\n$|\\varphi\\rangle_{out}$\n\\begin{eqnarray}\nF\\equiv{}~_{out}\\langle\\varphi|\\rho_{ij}|\\varphi_{ij}\\rangle_{out}\n=(1-e^{-\\omega\/T})^{3}.\n\\end{eqnarray}\n\nFrom Fig. \\ref{FBosw}, we can see that the fidelity of teleportation\ndepends on the Hawking temperature $T$. It has been found that the\nfidelity decreases as the Hawking temperature increases, which just\nindicates the entanglement degradation obtained in previous section\nbecause the state fidelity in conventional teleportation protocol is\nrelated to the entanglement.\n\n\\begin{figure}[ht]\n\\includegraphics[scale=0.8]{FBosw}\n\\caption{\\label{FBosw}The fidelity of teleportation as a function of\nthe Hawking temperature $T$ with the fixed $\\omega$ for a maximally\nentangled state.}\n\\end{figure}\n\n\\section{summary}\n\nWe have analytically discussed the effect of the Hawking temperature\non the entanglement between two modes of a scalar field as detected\nby Alice who stays stationary at an asymptotically flat region and\nBob who locates near the event horizon in the background of a most\ngeneral, static and asymptotically flat black hole with spherical\nsymmetry. It is shown that the entanglement is degraded by the\nHawking effect with increasing Hawking temperature. It is found that\nthe stronger the ``initial entanglement\" which corresponds to the\nHawking temperature of zero, i.e., the case of a supermasive or an\nalmost extreme black hole, the faster it loses. It is found that the\nsame ``initial entanglement\" for the state parameter $\\alpha$ and\nits ``normalized partners\" $\\sqrt{1-\\alpha^{2}}$ will be degraded\nalong two different trajectories as the Hawking temperature\nincreases except for the maximally entangled state\n$\\alpha=1\/\\sqrt{2}$, which just shows the inequivalence of the\nquantization for a scalar field in the black hole and Kruskal\nspacetimes. In the infinite Hawking temperature limit\n$T\\rightarrow\\infty$, corresponding to the case of the black hole\nevaporating completely, the state has no longer distillable\nentanglement for the arbitrary values of $\\alpha$. Further analysis\nshows that the mutual information is degraded to a nonvanishing\nminimum value which is dependent of $\\alpha$ with increasing Hawking\ntemperature. However, it is interesting to note that the mutual\ninformation in the infinite Hawking temperature limit equals to just\nhalf of the ``initially mutual information\", which is independent of\n$\\alpha$. We have also investigated the scheme of teleportation in\nthis black hole spacetime. It has been demonstrated that the\nfidelity of teleportation decreases as the Hawking temperature\nincreases, which just indicates the entanglement degradation because\nthe state fidelity in conventional teleportation protocol is related\nto the entanglement.\n\n\\begin{acknowledgments}\nThe authors would like to thank the anonymous referee for his\/her\ninsightful and constructive criticisms and suggestions, which\nallowed us to improve the manuscript significantly. This work was\nsupported by the National Natural Science Foundation of China under\nGrant No. 10675045; the FANEDD under Grant No. 200317; and the Hunan\nProvincial Natural Science Foundation of China under Grant No.\n08JJ3010.\n\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThere is no established complete theory of 4-dimensional quantum gravity as yet, and several approaches are being pursued to achieve this goal; among these, spin foam models \\cite{review, alexreview} represent a new promising one in that they seem to be a point of convergence of quite a few lines of research, including loop quantum gravity and simplicial approaches. Among the various models proposed, the Barrett-Crane model \\cite{bc} is the one on which most of the work has focused so far; the basic idea of this model is to describe and quantize gravity, in a simplicial setting, as a constrained topological field theory, where the topological field theory one starts with is the so-called BF theory. The result is a purely combinatorial and algebraic version of a sum-over-histories formulation of the theory in which spacetime is replaced by a combinatorial 2-complex and spacetime geometry is encoded in algebraic data from the representation theory of the Lorentz group. These data are of two types: representation parameters labelling (unitary irreducible) simple, i.e. class I, representations of the group (in the principal series), and vectors in the homogeneous space ${\\cal H}^3 = \\mathrm{SO}(3,1)\/\\mathrm{SO}(3)$; both have a geometric interpretation within the simplicial spacetime one reconstructs from the spin foam (see \\cite{us} and references therein): the first are interpreted as areas of the triangles they label, while the second are thought of as normals to the tetrahedra of the simplicial complex. In this simplicial context, the spacetime behind spin foam models is made out of fundamental building blocks, \\lq\\lq atoms of a quantum spacetime\\rq\\rq, being the 4-simplices of the triangulation, or equivalently, the vertices of the spin foam, and the spin foam model describe how these building blocks interact, how they are coupled, with this coupling determining the details of the overall spacetime geometry. In the specific case of the Barrett-Crane model the coupling between 4-simplices is limited to the first set of algebraic data, i.e. to the representation labels only, in that triangles have the same area in all the 4-simplices they are shared by; on the other hand, there two normal variables for each tetrahedron in the manifold, one for each 4-simplex sharing it, and these two variables are completely un-correlated, i.e. un-coupled. This is not necessarily a problem and in fact it has a nice geometric interpretation (see \\cite{us} and references therein), but it represents a sort of \\lq\\lq ultra-locality\\rq\\rq of the model that one may want to relax (on the ultra-locality of the Barrett-Crane model seen from the point of view of group field theories, see \\cite{generalised}). One reason for this is that one expects this coupling to be affected by and somehow flow under renormalisation of the quantum amplitudes, and one would like to be able to tune the model to different degrees of locality, so to allow for propagating degrees of freedom; in fact it is not clear whether or not these are to be expected in a ultra-local model.\nTherefore, one would like to introduce an additional coupling between 4-simplices with respect to the Barrett-Crane model, and possibly in a tunable way, e.g. introducing an extra parameter in the theory that can be set, when wanted, to a value that gives back the un-coupled Barrett-Crane model, and on whose value one can play to obtain the various levels of locality. This is exactly what we achieve in this paper. The goal goes however beyond the construction of a coupled model: one would like to study the renormalisation of the coupling, as a step towards the difficult task of understanding the renormalisation of spin foam models in full generality, an issue made tricky by the absence of any background structure, as it should be the case for any sensible candidate to a theory of quantum gravity. We believe the model presented here may be of help regarding this more ambitious goal.\nIn this article we work in the framework and language of group field theory \\cite{review, alexreview,laurentgft}. Group field theories are known to be behind any spin foam model we know so far \\cite{carlomike} and to give the most complete formulation of them, in that they allow to remove the dependence on any specific discretization of spacetime obtaining thus a fully background independent model. Moreover, group field theories allow for an easy control over the degrees of freedom and the variables being dealt with in that it provides the basic ingredients of the spin foam model without being tied to any choice of underlying 2-complex, because the provide directly the basic building blocks of it, in the form of propagators and vertex amplitudes forming it as a Feynman diagram of the theory; it is then easier to work in the group field theory context when looking for generalisations of the current models because one sees clearly the structures being generalised even before one sees how the resulting spin foam amplitudes end up being modified. The use of the group field theory formalism descends however also, independently on the specific result one wants to achieve, from the general point of view that group field theories represent not only the most complete formulation of spin foam models, as said, but the most {\\it fundamental} definition of them, i.e. they are {\\it the} theory we are talking about when discussing spin foam models, whose content and significance goes far beyond that of spin foam models alone. This point of view is not that outrageous given that indeed spin foam models arise as Feynman diagrams amplitudes for the corresponding group field theory, and we know that there is much  more in a quantum field theory than its perturbative expansion in Feynman diagrams. However, at present there are too many aspects of group field theories that are not been studied and too many properties of spin foam models whose group field theory origin is not understood for being able to consider this point of view as {\\it the correct one} and not just a possible one. We content ourselves in this paper to show one instance in which working directly at the group field theory level proves advantageous over working at the level of its Feynman amplitudes. Moreover, the results we will present in the end of this work, concerning a new type of group field theory suggest that the group field theory formalism may contain more than usually suspected also with respect to the issue of renormalisation of spin foam models. \n\nThe paper is organised as follows: we start by showing how the {\\it usual} Barrett-Crane model can be obtained by a group field theory in for a field with {\\it five} arguments living in the group manifold, with the extra (with respect to the usual formulations) fifth variable playing the role, in the resulting amplitudes, of the normal to the tetrahedra of the triangulation dual to the Feynman 2-complex, but with the corresponding degrees of freedom not being propagated from vertex to vertex (from 4-simplex to 4-simplex); then in section \\ref{sec:genmodel} we construct a generalisation of this model that provides a non-trivial coupling of these variables across different 4-simplices; we write down the model in full generality, i.e. regardless of any specific choice of coupling function, but we then exhibit a particularly interesting example, with the coupling function being the heat kernel on the group, and thus carrying a tunable coupling parameter, and leading to a model interpolating between the Barrett-Crane model and a strongly coupled flat model, as we are going to discuss; in section \\ref{sec:partBC} we discuss how the mathematical structures involved in this generalised model are relevant also for the insertion of particles in the Barrett-Crane model; finally, in section \\ref{sec:gengft} we propose a further generalisation of the group field theory formalism in which the coupling parameter of the coupled model is promoted to a variable on the same footing as group variables, and in section \\ref{sec: renorm} we discuss at length the issue of renormalisation of spin foam model from the novel perspective provided by the results of this work.             \n\n\\section{Generalized Barrett-Crane model and the coupling of spacetime\natoms} \\label{sec:genBC}\n\\subsection{Cosmetics on the Barrett-Crane model} \\label{sec:cosm}\nBefore we show how it is possible to generalise the Barrett-Crane model to a new model that presents a non-trivial coupling between 4-simplices, we want to first show how the {\\it usual} Barrett-Crane model can be derived from a group field theory action defined for a field living on {\\it five} copies of the Lorentz group, instead of four, with the fifth group element having the interpretation of the normal to the tetrahedron of which the field represents the (second) quantization.\n\nWe start with a complex field $\\varphi(g_i,G)$\\footnote{Here and in the following we use a simplified notation labelling by a single group element $g_i$, indexed by $i$, the four group elements $g_1,g_2,g_3,g_4$ variables of the field} on $\\mathrm{SO}(3,1)^{\\times 5}$; this is our dynamical object\\footnote{The calculations and results of this paper are valid for both $\\mathrm{SO}(3,1)$ and $\\mathrm{SO}(4)$ in the same way, with the non-compactness of the Lorentz group causing only a higher degree of formal complications but no difference in substance; therefore in what follows we always refer explicitly only to $\\mathrm{SO}(3,1)$ because it is physically more interesting, but we use the notation that is customary in the $\\mathrm{SO}(4)$ case instead because it is easier to follow and less cumbersome.}. We define its projection on $(\\mathrm{SO}(3,1)\/\\mathrm{SO}(3))^{\\times 4}\\times \\mathrm{SO}(3,1)$ and expand it into Fourier modes:\n$$\nP_h\\varphi(g_i,G)=\\sum_{J_i A_i}\\varphi^{J_i}_{A_i,B_i}(G)D^{J_i}_{A_iB_i}(g_i)W^{J_i}_{B_i},\n$$\nwhere $\\varphi^{J_i}_{A_i}(G)$ are the Fourier modes of the field and $W^{J_i}_{A_i}$ the $\\mathrm{SO}(3)$-invariant vector in the $J_i$ (simple) representation.\nThe gauge invariant field is defined as:\n\\begin{eqnarray}\n\\phi(g_i,G)\\equiv P_gP_h\\varphi(g_i,G)\n=\\int d\\Lambda\\, P_h\\varphi(g_i\\Lambda,G\\Lambda) \\\\ = \\int d\\Lambda\\, \\sum_{J_i A_i}\\varphi^{J_i}_{A_i}(G\\Lambda)D^{J_i}_{A_iB_i}(g_i\\Lambda)W^{J_i}_{B_i} = \\label{eq:field1} \\\\\n=\\int d\\Lambda\\, \\sum_{J_i A_i}\\varphi^{J_i}_{A_i}(\\Lambda)D^{J_i}_{A_iB_i}(g_iG^{-1}\\Lambda)W^{J_i}_{B_i}.\n\\end{eqnarray}\nThe group variable $\\Lambda$ is then naturally projected down to the hyperboloid ${\\cal H}=\\mathrm{SO}(3,1)\/\\mathrm{SO}(3)$.\nThe field is a function of five variables, either five group elements or five representations of the Lorentz group, in the complete mode expansion, so it represents a generalisation of the field on which the Barrett-Crane model is based; this is also clear if one thinks that it decomposes into combinations of 5-valent intertwiners between simple representations. In a sense it can be said that it \\lq\\lq unlocks\\rq\\rq the structure of a quantum tetrahedron, by relaxing the closure constraint and allowing the four simple representations corresponding to the 4 triangles of a tetrahedron to be mapped to a non-trivial one. However, whether and how this is realised depends on what action one defines for this extended field, i.e. on whether or not one chooses to trivialise the dependence on the extra variables. Moreover, we see that for each given value of the $G$ variables, the extra fifth variables of the field are given by the $\\Lambda$s entering the amplitudes as normals to the tetrahedra, as anticipated. The presence of these variables, however, allows for $\\Lambda$s to be coupled in different 4-simplices, as we will see.  \n\nIn fact, using this definition of the field, one can define the action:\n\\begin{eqnarray} \nS[\\varphi]&=& \\frac{1}{2}\\prod_i\\int dg_i \\int dG\\int dG'\\phi(g_i,G)\\phi(g_i,G') \\\\\n&+&\\frac{\\lambda}{5!} \\prod_{i=1}^{10} \\int dg_i\\,\\int dG_A ..dG_E\n\\phi(g_1,g_2,g_3,g_4,G_A)\\phi(g_4,g_5,g_6,g_7,G_B)\\phi(g_7,..,G_C)\n\\phi(g_9,..,G_D)\\phi(g_{10},..,G_E), \\nonumber\n\\end{eqnarray}\nwith capital letters labeling the five group elements associated to the five tetrahedra of a 4-simplex.\nThis is a trivial extension of the group field theory action from which the usual Barrett-Crane model is derived, in the formulation of \\cite{DP-F-K-R}, in the sense that the above action reproduces {\\it exactly} the same amplitudes of the usual Barrett-Crane model, as it is easy to verify. For example, we can show how this is realised for the propagator of the theory. The kinetic term is decomposed in modes by standard methods, starting from expression \\Ref{eq:field1} as:\n\\begin{equation}\n\\prod_i\\int dg_i \\int dG  dG' \\phi(g_i,G)\\phi(g_i,G')= \\int d\\Lambda d\\Lambda' \\int dG dG' \\varphi^{J_i,J}_{A_i,m,n} \\varphi^{J_i,J'}_{A_i,k,l} \\prod_i D^{J_i}_{00}(\\Lambda \\Lambda'^{-1})\\, D^{J}_{mn}(G\\Lambda)\\,D^{J'}_{kl}(G'\\Lambda'),\n\\end{equation}\nwith implicit summation over all indices $J_i, A_i, J,J',m,n,k,l$. It is easy to see that, upon integration of the $G$ variables, the dependence on the variables $\\Lambda$ (those imposing the gauge invariance of the fields) is reduced and the fifth representations $J$,$J'$ are forced to be the singlet, and the usual kinetic term \\cite{DP-F-K-R} is obtained:\n\\begin{equation}\n\\prod_i\\int dg_i \\int dG \\int dG' \\phi(g_i,G)\\phi(g_i,G')= \\sum_{J_i,A_i} \\varphi^{J_i}_{A_i}\\varphi^{J_i}_{A_i}\\int d\\Lambda d\\Lambda' \\prod_i D^{J_i}_{00}(\\Lambda \\Lambda'^{-1}). \n\\end{equation}\nThe same can be verified for the interaction term, that gives the usual Barrett-Crane amplitude for a 4-simplex. In the end the Barrett-Crane model is obtained also in this 5-arguments formalism, with the (once) fifth variable entering in the model as the normal to the tetrahedra in each of the 4-simplices to which they belong \\cite{us}, but being completely decoupled in the different 4-simplices. Note that, had we imposed the projection $P_h$ only in the interaction term and not in the kinetic one, we would have obtained back the Barrett-Crane model in the version given in \\cite{P-R}; so the same relation between the various versions of the BC model is present in this five argument extension as in the usual four argument formulation of the group field theory. In the following we will discuss in parallel what happens when one does or does not impose the simplicity constraints in the kinetic term.\nThe usefulness of this extension of the model lies in the fact that, the dependence on the normal variables having been made explicit in the field theory, it is possible to modify the field theory itself in a very simple way, and thus to introduce a coupling between these variables. This is what we are now going to show.\n\n\\subsection{A generalised model with a non-trivial coupling between 4-simplices} \\label{sec:genmodel}\nWe modify the kinetic term, the one producing the propagator of the theory and thus the edge amplitudes of the spin foam model, in the above extended action by inserting an arbitrary coupling for the new extra variables $G$:\n\\begin{equation}\nS_{\\rm kin}[\\varphi]\\equiv\n\\int dg_i \\int dG \\, dG'\\,\nf(G,G')\\phi(g_i,G)\\phi(g_i,G').\n\\end{equation}\nThus the function $f(G,G')$ encodes in configuration space a non-trivial coupling between the fifth variables of the field, that, as we have seen in the previous section, enter in each 4-simplex (vertex) amplitude as the normals to the tetrahedra in the reference frame associated to each of them. It represents a non-trivial amplitude for the curvature, in the sense that it does not weight equally all the possible values of them, as it does the trivial choice $f(G,G')=1$. The coupling $f$ is naturally of the form $f(G,G')\\equiv F(G'G^{-1})$ where $F$ is a function over $\\mathrm{SO}(3,1)$.\nMoreover, we would like to have a Lorentz invariant coupling such that the correlations be invariant under $\\Lambda,\\Lambda'\\rightarrow g\\Lambda,g\\Lambda'$ for all $g\\in\\mathrm{SO}(3,1)$. This amounts to assuming the coupling function $F$ to be invariant under conjugation: $F(G)=F(g^{-1}Gg), \\,\\forall g$.\n\nWe want now to see what expression this coupling takes in momentum (representation) space; this is necessary when the simplicity constraints are imposed, i.e. in trying to construct a \\lq\\lq coupled extension\\rq\\rq of the version of the Barrett-Crane model presented in \\cite{DP-F-K-R}, because when this is done taking the inverse of the kinetic operator in configuration space is particularly cumbersome, and much simpler in momentum space. Expanding into Fourier modes, we obtain:\n\\begin{equation}\nS_{\\rm kin}[\\varphi]=\n\\int d\\Lambda\\, d\\Lambda'\\, \\sum_{J_i,A_i}\\frac{1}{\\Delta_{J_i}}\n\\varphi^{J_i}_{A_i}(\\Lambda)\\varphi^{J_i}_{A_i}(\\Lambda')\n\\,\\left[\n\\int dG\\,dG'\\, f(G,G') \\prod_{i=1}^4 K^{J_i}(\\Lambda'^{-1}G'G^{-1}\\Lambda)\n\\right],\n\\end{equation}\nwhere $K^{J}(g)=\\bar{W}^J_AD^J_{AB}(g)W^J_B=D^{J}_{00}(g)$ is the homogeneous kernel, the Hadamard propagator of 'mass' $J$ (more precisely, $m^2=J(J+1)$) on the hyperboloid ${\\cal H}$ \\cite{feynman}.\nThen the quadratic coupling in the action is given by a kind of Fourier transform of $F$ where the modes are the \\lq\\lq eye diagram evaluations\\rq\\rq:\n\\begin{equation}\n{\\cal C}^{J_i}(\\Lambda,\\Lambda')\\equiv\n\\int dG\\, F(G) \\prod_{i=1}^4 K^{J_i}(\\Lambda'^{-1}G\\Lambda).\n\\label{transform}\n\\end{equation}\nBecause $F$ is invariant under conjugation, ${\\cal C}^{J_i}$ is simply\na function of $\\Lambda'^{-1}\\Lambda$, so we will denote it more concisely\n${\\cal C}^{J_i}(\\Lambda,\\Lambda')\\equiv {\\cal\n  C}^{J_i}(\\Lambda'^{-1}\\Lambda)$, with:\n$$\n{\\cal C}^{J_i}(g)=\\int dG\\, F(G) \\prod_{i=1}^4 K^{J_i}(Gg).\n$$\n${\\cal C}^{J_i}$ is thus a zonal function, i.e. invariant under both\nthe left and right actions of $\\mathrm{SO}(3)$. Then, under the assumption\nthat it is a $L^2$ function over the hyperboloid, it can be decomposed\ninto Fourier modes:\n$$\n{\\cal C}^{J_i}(g)=\\sum_{J}\\Delta_J{\\cal C}^{J_i}_J K_J(g),\n$$\nwhere ${\\cal C}^{J_i}_J$ are its Fourier components.\n\nLet's point out that the modes of the transform \\Ref{transform} that\nwe introduced above $\\prod_{i=1}^4 K^{J_i}(\\Lambda'^{-1}G\\Lambda)$\nwere already discussed in \\cite{us}. Mathematically, it is the\nevaluation of the eye diagram labeled by the four representations\n$J_i$ on the group element $G$. It can be interpreted as the\nprobability amplitude of the quantum tetrahedron, with triangles\ndefined by the four representations $J_i$, carrying a curvature\ndefined by the parallel transport $G$ between the two 4-simplices\nsharing that tetrahedron.\n\nThe propagator of the theory will be the inverse kernel\n$P(\\Lambda,\\Lambda')$ such that:\n$$\n\\int_{{\\cal H}} d\\Lambda'\\,\n{\\cal P}^{J_i}(\\Lambda,\\Lambda'){\\cal\n  C}^{J_i}(\\Lambda',\\Lambda'')=\\delta_{{\\cal\n    H}}(\\Lambda\\Lambda''^{-1}),\n$$\nor equivalently\n\\begin{equation}\n\\int_{\\mathrm{SO}(3,1)} dg\\,\n{\\cal P}^{J_i}(Gg^{-1}){\\cal C}^{J_i}(g)=\\delta(G).\n\\end{equation}\nThen considering that $\\delta(G)=\\sum_J\\Delta_JK^J(G)$, it is straightforward to check that:\n$$\n{\\cal P}^{J_i}(g)=\\sum_{J}\\Delta_J\\frac{1}{{\\cal C}^{J_i}_J} K_J(g).\n$$\n\nIt is easy to see what changes in the above results if we do not impose the simplicity constraints in the kinetic term of our generalised group field theory action; of course there will still be a non-trivial coupling for the fifth variables of the field, resulting from a non-trivial coupling insertion $f(G,G')$ in configuration space, and the corresponding propagator in configuration space will be given by a product of delta functions for the first four arguments of the field, just as in \\cite{P-R} times a propagator for the fifth pair of variable given simply by $f^{-1}(G,G')$. The four deltas and the extra propagator will be intertwined by a global action of $\\mathrm{SO}(3,1)$; in momentum space this will lead for the kinetic term to an analogue of the operator $\\mathcal{C}^{J_{i}}$, given by:\n\\begin{equation}\n{\\cal C}_{B_i C_i}^{J_i}(g)=\\int dG\\, F(G) \\prod_{i=1}^4 D_{B_i C_i}^{J_i}(Gg).\n\\end{equation}\nwith extra indices $B_i,C_i$ (appearing also in the field modes, resulting from the fact that we do not project anymore on the invariant vectors in the relevant representation spaces. The rest of the construction proceeds analogously, leading to a propagator that carries also these extra indices (of course all the variables are in this case in the group and not in the homogeneous space).   \n\nWhat we have done up to now is completely general, in that it holds for any choice of coupling function $f$, and it gives a generalisation of the Barrett-Crane model with a non-trivial coupling between 4-simplices with respect to the normal vectors variables. \nHowever, we can do more than this: we can exhibit a specific choice of the coupling function satisfying all the properties required, thus providing us with a specific model with many interesting properties, that we are going to highlight. Moreover, this model leads naturally to a further generalisation of the Barrett-Crane model, to be discussed below, that can be the starting point for a new approach to the issue of renormalisation of spin foam models.\n  \nConsider the case in which the simplicity projections $P_h$ are not imposed in the kinetic term, as we assume from now on. The specific model we mentioned arise from choosing the (truncated) heat kernel function as the coupling function $f(G,G')$ in configuration space:\n\\begin{equation}\nf(G,G')=F(g)={\\cal K}^{-\\beta,L}(g)=\\sum_{J\\le L} \\Delta_J e^{\\beta C_J} K^J(g),\n\\end{equation}\nwhere $C_J$ is the quadratic Casimir of the Lorentz group; then the propagator is the inverse heat kernel ${\\cal K}^{\\beta,L}$, i.e. the (truncated) heat kernel function for the opposite value of the coupling $\\beta$, as it is easy to verify.\nConsidering (formally) the heat kernel ${\\cal K}^{-\\beta}(g)$ (untruncated) as coupling\nfunction $f(G,G')$, two limits are particularly interesting for the corresponding propagator ${\\cal K}^{\\beta}(g)$:\n\\begin{itemize}\n\\item $\\beta\\rightarrow \\infty$: the coupling becomes trivial, i.e.\n\\begin{equation}\n{\\cal K}^{\\beta}(g)\\rightarrow Id\n\\qquad \\Rightarrow \\qquad\nC^{J_i}(\\Lambda, \\Lambda')=1,\n\\end{equation}\nand we lose the\n  dependence on $\\Lambda$ and $\\Lambda'$. We recover the usual expression for the Barrett-Crane model in the expansion in Feynman graphs; in this limit the 4-simplices are decoupled as far as the $\\Lambda$ variables are concerned, meaning that the connection, that maps the normal vector to the tetrahedron as seen from the reference frame associated to one 4-simplex to that corresponding to the reference frame associated to the other 4-simplex, is completely arbitrary, and the only coupling is given by the representations assigned to the triangles of the tetrahedron, which are the same in both 4-simplices that share it.\n\\item $\\beta\\rightarrow 0$: the coupling becomes rigid, i.e. ${\\cal K}^{\\beta}(g)=\\delta(g)$ so we are in\n  a strong coupling limit. The propagator constraints\n  $G=G'$, where $G$ and $G'$ are the fifth variables in the two fields corresponding to the same tetrahedron and interacting in the two vertices, being propagated by ${\\cal K}^\\beta(g)$. We call this model the flat model since it\n  amounts to imposing a trivial parallel transport between\n  4-simplices, i.e. constraining the (boost part of the) connection to be flat. \n  Here we stress that we constrain the fifth group variables $G, G'$ to be equal and not the normals $\\Lambda, \\Lambda'$, so the coupling is simply the eye diagram evaluation:\n$$\nC^{J_i}(\\Lambda, \\Lambda')=\\prod_{i=1}^4K^{J_i}(\\Lambda'^{-1}\\Lambda).\n$$\nit would be interesting to understand the explicit relation between this limit of the model and BF theories. On the one hand it would seem that this limit indeed corresponds to a BF theory, since we constrain the curvature to be flat. On the other hand, it is not the BF theory based on the Lorentz group (the Crane-Yetter model), however it could well be a BF theory on the homogeneous space $\\mathrm{SL}(2,\\mathbb{C})\/SU(2)$.\n\\end{itemize}\nBoth these limits can be easily verified both in configuration space and in momentum space, by a straightforward calculation (in this context without simplicity projectors).\n\nNow let us have a look at the interaction term of the group field\ntheory, which defines the vertex amplitude of a spin foam or\nequivalently the 4-simplex amplitude. Let us consider the following\nintegral:\n\\begin{equation}\nI(G_A,..,G_E)\\equiv\n\\int \\prod_{i=1}^{10} dg_i\\,\n\\phi(g_1,g_2,g_3,g_4,G_A)\\phi(g_4,g_5,g_6,g_7,G_B)\\phi(g_7,g_3,g_8,g_9,G_C)\n\\phi(g_9,..,G_D)\\phi(g_{10},..,G_E),\n\\end{equation}\nwhere $i=1,..10$ refers to the triangles of the 4-simplex and  the letters\n$A,..,E$ to its tetrahedra.\nThen it is straightforward to check that:\n\\begin{equation}\nI(G_A,..,G_E)=\n\\int d\\Lambda_A\\, ..\\, d\\Lambda_E\\,\n\\sum_{J_i, A_i} \\frac{1}{\\Delta_{J_i}}\n\\varphi^{J_1..J_4}_{A_1..A_4}(\\Lambda_A)..\\varphi^{J_{10}..J_1}_{A_{10}..A_1}(\\Lambda_E)\n\\,\\prod_{i=1}^{10} K^{J_i}(\\Lambda_{t(i)}^{-1}G_{t(i)}G_{s(i)}^{-1}\\Lambda_{s(i)}),\n\\end{equation}\nwhere we label $s(i)$ and $t(i)$ the source and target vertices of the edge $i$ of the 4-simplex spin network \\cite{F-K}, or equivalently, the two tetrahedra sharing the triangle $i$ of the 4-simplex.\n\n\nIf we take as interaction term $\\int dG_A\\, ..\\,dG_E I(G_A,..,G_E)$,\nthen we erase all information on the normals $\\Lambda_A,..\\Lambda_E$\nand we recover the usual Barrett-Crane evaluation of a 4-simplex, as we have discussed in the previous section. As\nwe would like to keep a non-trivial dependence  of the $\\Lambda$'s, we\nsimply choose:\n\\begin{equation}\nS_{\\rm int}[\\varphi]=I(G_A={\\rm Id},..,G_E={\\rm Id}).\n\\end{equation}\n\nThis way, we keep an extended dependence on the variables $\\Lambda$, that can be coupled by the propagator in each Feynman graph, leading to a spin foam model which differs from the Barrett-Crane model. Note that choosing ${\\rm Id}$ is not essential. Fixing the $G_v$ variables to another value would amount to changing the origin -the reference point- of the hyperboloid for each variable $\\Lambda_v$.\n\nFinally, our group field theory for the field $\\varphi$ is defined by the action\n$S_{\\rm GFT}[\\varphi]\\equiv S_{\\rm kin}[\\varphi]+S_{\\rm int}[\\varphi]$, i.e. by:\n\\begin{eqnarray}\nS_{\\rm GFT}[\\varphi]&\\equiv& \\int dg_i \\int dG \\, dG'\\,\nf(G,G')\\phi(g_i,G)\\phi(g_i,G')  \\nonumber\\\\\n&&+\\frac{\\lambda}{5!}  \\int \\prod_{i=1}^{10} dg_i\\,\n\\phi(g_1,g_2,g_3,g_4,Id)\\phi(g_4,g_5,g_6,g_7,Id)\\phi(g_7,g_3,g_8,g_9,Id)\n\\phi(g_9,..,Id)\\phi(g_{10},..,Id),\n\\end{eqnarray}\nin configuration space, or\n\\begin{eqnarray}\nS_{\\rm GFT}[\\varphi]&\\equiv& \\int d\\Lambda d\\Lambda' \\sum_{J_i,A_i}\\frac{1}{\\Delta_{J_i}}\n\\varphi^{J_i}_{A_i B_i}(\\Lambda)\\varphi^{J_i}_{A_i C_i}(\\Lambda'){\\cal C}_{B_i C_i}^{J_i}(\\Lambda,\\Lambda') \\nonumber\\\\ \n&&+\\frac{\\lambda}{5!} \\int d\\Lambda_A ..\\Lambda_E \\sum_{J_i,A_i} \\varphi^{J_1..J_4}_{A_1..A_4}(\\Lambda_A)...\\varphi^{J_{10}..J_1}_{A_{10}..A_1}(\\Lambda_E) \\prod_i D^{J_i}_{00}(\\Lambda_{s(i)}\\Lambda_{t(i)}^{-1}) \\\\ &=& \\sum_{J_i,A_i}\n\\varphi^{J_i,J}_{A_i,B_i,k,l}\\varphi^{J_i,J'}_{A_i,C_i,m,n}\\left(\\int d\\Lambda d\\Lambda' D^{J}_{kl}(\\Lambda)D^{J'}_{mn}(\\Lambda')\\frac{1}{\\Delta_{J_i}} {\\cal C}_{B_i C_i}^{J_i}(\\Lambda,\\Lambda')\\right) \\nonumber\\\\ &&+\\frac{\\lambda}{5!}  \\sum_{J_i,A_i} \\varphi^{J_1..J_4,J_A}_{A_1..A_4,m_A,n_A}...\\varphi^{J_{10}..J_1,J_E}_{A_{10}..A_1,m_E,n_E} \\left(\\int d\\Lambda_A ..\\Lambda_E D^{J_A}_{m_A n_A}(\\Lambda_A)...D^{J_E}_{m_E n_E}(\\Lambda_E) \\prod_i D^{J_i}_{00}(\\Lambda_{s(i)}\\Lambda_{t(i)}^{-1})\\right)\\nonumber,\n\\end{eqnarray}\nin momentum space, from which it is immediate to read out the vertex amplitude for our Feynman diagrams encoding the interaction of our field $\\varphi$, given in configuration space by:\n\\begin{equation}\n\\mathcal{I}_v (g_{\\mid v}, \\tilde{g}_{f\\mid v}, \\Lambda_{e \\mid v}) = \\prod_{f\\mid v} \\delta(g_{f\\mid v}\\Lambda_{e1}^{-1}\\Lambda_{e2}\\tilde{g}_{f\\mid v}^{-1}) \\label{eq:vertex}\n\\end{equation}\n\nwhile the propagator is given by the expression: \n\n\\begin{equation}\n{\\cal P}(g_i,G; \\tilde{g}_i, G') = \\int d\\Lambda\\int d\\Lambda' \\prod_i \\delta(g_i \\Lambda^{-1},\\tilde{g}_i\\Lambda'^{-1}) \\, f^{-1}(G\\Lambda^{-1},G'\\Lambda'^{-1})\n\\end{equation} \nin configuration space, and by:\n\\begin{equation}\n{\\cal P}(J_i,A_i, B_i,J,mn; \\tilde{J}_i,\\tilde{A}_i, C_i,J',kl)=\\prod_i\\delta_{J_i,\\tilde{J}_i}\\prod_i \\delta_{A_i,\\tilde{A}_i} \\int dGdG' D^{J}_{mn}(G)D^{J'}_{kl}(G') {\\cal P}^{J_i}_{B_i C_i}(G,G')\n\\end{equation}\nin momentum space, with ${\\cal P}^{J_i}$ having been defined above. This is for the general case; for the specific choice of coupling function proposed above, i.e. the heat kernel on the group, the analogous expressions are easily written down: the interaction term is unaltered, while the propagator takes the form:\n\\begin{equation}\n{\\cal P}^\\beta(g_i,G; \\tilde{g}_i, G') = \\int d\\Lambda\\int d\\Lambda' \\prod_i \\delta(g_i \\Lambda^{-1},\\tilde{g}_i\\Lambda'^{-1}) \\, {\\cal K}^{\\beta}(G\\Lambda^{-1},G'\\Lambda'^{-1})\n\\end{equation}\nthat reduces as we had discussed to the usual Barrett-Crane propagator for $\\beta\\rightarrow \\infty$ and gives instead a BF-type product of five deltas if the limit $\\beta\\rightarrow 0$ is taken instead. \n\nAnalogously, one can impose the simplicity projectors in the kinetic term and work fully in the homogeneous space; the construction is not altered substantially, but checking the two limiting properties in $\\beta$ for the resulting Feynman diagrams is much less straightforward, especially in configuration space.   \n\nThe partition function is then defined in terms of its Feynman expansion, as usual, and it is given by:\n\\begin{equation}\nZ(\\lambda,\\beta) = \\int \\mathcal{D}\\varphi\\; e^{- S[\\varphi]} = \\sum_{\\Gamma} \\frac{\\lambda^N}{sym(\\Gamma)} \\,Z_{\\Gamma}[\\beta]\n\\end{equation}\nwhere $N$ is the number of vertices in the graph and $sym$ its order of automorphisms, and with the amplitude for each graph being given by the appropriate convolution of vertex amplitudes and propagators as:\n\n\\begin{equation}\nZ_{\\Gamma}[\\beta]= \\left(\\prod_e \\prod_{f\\mid e} \\int dg_{f\\mid e} \\int d\\tilde{g}_{f\\mid e}\\right) \\left( \\prod_e \\int dG_e \\int dG'_e\\right) \\prod_e {\\cal P}^{\\beta}(g_{f\\mid e},\\tilde{g}_{f\\mid e}, G_e,G'_e) \\prod_v \\mathcal{I}_v(g_{f\\mid v}, \\tilde{g}_{f\\mid v}, G_{e\\mid v})\n\\end{equation}\n \nIt is now easier to understand the structure of the resulting models; in particular, we see how indeed, in the $\\beta\\rightarrow 0$ limit of the coupled model, the fifth variables of the fields, having the interpretation of normals to the tetrahedra in the reference frames defined by each 4-simplex, are forced to be mapped trivially from one 4-simplex to the next. It is indeed a flat model, with a rigid coupling between 4-simplices and similar to a BF theory in that allows {\\it only} such a flat configuration at least for what concerns the extra variables. It is not however a GFT formulation of a true BF theory for a 5-dimensional spacetime, and differs of course also from a true BF theory in 4-dimensions, in that: 1) the first four variables of the field are projected down to the homogeneous space $\\mathrm{SO}(3,1)\/\\mathrm{SO}(3)$ while maintaining the gauge invariance under the full Lorentz group; 2) the fifth variables in the interaction term, i.e. in each 4-simplex, are completely decoupled.\n\n\nIt is important to stress that not only the new model gives a simple way of interpolating between the Barrett-Crane model and the flat model by moving in parameter space, i.e. by following a renormalisation group flow, but most importantly perhaps it allows for perturbation analysis around these limiting configurations, i.e. it permits to study the physics of the model near the \\lq decoupled phase', i.e. the ordinary Barrett-Crane model, and near the \\lq strongly coupled phase' instead. Interestingly, one would expect to see propagating \\lq gravitons', i.e. propagating perturbations in the curvature, in the almost flat case, thus in the strongly coupled phase, and \\lq confined gravitons' in the decoupled phase, when perturbing around the usual Barrett-Crane configurations. In the almost flat case it would correspond to a definition of a gravity theory in terms of a perturbative expansion around a flat connection configuration; in both cases it would amount to a fully covariant perturbation expansion, analogous to the one proposed in \\cite{laurent-artem}, and indeed the exact relationship with the model in \\cite{laurent-artem} should be analysed and we leave it for further study. In a renormalisation group interpretation, according to which the $\\beta$-dependent model would possess two physically very different phases, the Barrett-Crane or decoupled phase and the strongly coupled BF-type one, this perturbation expansion would then be an expansion around the two possible fixed points of the renormalisation group flow. \nMoreover, we are going to show in the following that the $\\beta$-dependent model bases on the heat kernel coupling lend itself to a further generalisation which could be the basis for further progress concerning the renormalisation of spin foam models. But before discussing this further generalised model, we would like to show how the five argument formalism for the field theory described above and the associated mathematical structures arise naturally when particles are inserted in the spin foam model.\n\n\n\\subsection{About particle insertions in the Barrett-Crane model} \\label{sec:partBC}\nIn flat spacetime, a basis of one-particle states is given by momentum\neigenstates $\\psi_{p,\\sigma}$, where $p$ is the four-momentum and\n$\\sigma$ the other degrees of freedom - the spin. A Lorentz\ntransformation $U(\\Lambda)$ will map a state of momentum $p$ to a\nstate of momentum $\\Lambda p$. Then we can choose a reference momentum\n$k^\\mu$ and define the states $\\psi_{p,\\sigma}$ from the action of Lorentz\nboosts on the reference state:\n\n$$\n\\psi_{p,\\sigma}=N(p)U(s(p))\\psi_{k,\\sigma},\n$$\nwhere $N(p)$ is a normalization factor and $s(p)$ is a Lorentz\ntransformation mapping $k$ to $p$.\nThe usual normalization is $N(p)=\\sqrt{k_0\/p_0}$ so that the scalar product\nis normalized\n$\\langle\\psi_{p,\\sigma},\\psi_{p,\\sigma}\\rangle=\n\\delta_{\\sigma\\sigma'}\\delta^{(2)}(\\vec{p}-\\vec{p'})$.\nFor massive particles, we usually take $k=(1,0,0,0)$ and the little\ngroup of Lorentz transformations $W$ leaving $k$ invariant is the\nrotation group $\\mathrm{SU}(2)$. Then the action of a Lorentz transformation\nreads:\n$$\nU(\\Lambda)\\psi_{p,\\sigma}=N(p)U(s(\\Lambda p))U(W(\\Lambda,p))\\psi_{k,\\sigma},\n$$\nwith $W(\\Lambda,p)=s(\\Lambda p)^{-1}\\Lambda s(p)$ living in the little group.\nFrom this, we see that it is enough to postulate the action of the little group:\n$$\nU(W)\\psi_{k,\\sigma}=\\sum_{\\sigma'}D_{\\sigma',\\sigma}(W)\\psi_{k,\\sigma'},\n$$\nin order to get the action of an arbitrary Lorentz transformations:\n$$\nU(\\Lambda)\\psi_{p,\\sigma}=\n\\left(\\frac{N(p)}{N(\\Lambda p)}\\right)\n\\sum_{\\sigma'}D_{\\sigma',\\sigma}(W(\\Lambda,p))\\psi_{\\Lambda p,\\sigma'}.\n$$\nThe {\\it spin} $s$ is the choice of a irreducible representation of the little group $\\mathrm{SU}(2)$,\nwhich defines the coefficient $D_{\\sigma',\\sigma}$, and $\\sigma$ is the angular momentum. Then we have induced a representation of the Lorentz group from a $\\mathrm{SU}(2)$ representation.\n\n\\medskip\n\nTo include particles in the Barrett-Crane model, we first look at the\nparticle insertion on a 4-simplex. The one-4-simplex amplitude is\nsimply the evaluation of its boundary spin network. Thus we need to\ninclude the particles in a spin network wave function. Inserting\nparticles at all vertices of the spin network graph, the wave function\nis then a function of $E$ holonomies $U_e\\in\\mathrm{SL}(2,\\mathbb{C})$, describing the\nparallel transport between vertices and carrying the gravitational\ndegrees of freedom, and $V$ group elements\n$\\Lambda_v\\in\\mathrm{SL}(2,\\mathbb{C})\/\\mathrm{SU}(2)$ in the hyperboloid ${\\cal H}$, defining the\nstate of the particles, where $E$ and $V$ are respectively the number\nof edges and vertices of the graph. Considering the transformations of\nholonomies and momenta under gauge transformations, we are looking at\nthe following gauge invariant functions:\n\\begin{equation}\n\\psi(U_1,..,U_E,\\Lambda_1,..,\\Lambda_V)=\n\\psi(g_{s(1)}^{-1}U_1g_{t(1)},..,g_{s(E)}^{-1}U_Eg_{t(E)},g_1\\Lambda_1,..,g_V\\Lambda_V),\n\\qquad\n\\forall g_v\\in(\\mathrm{SL}(2,\\mathbb{C}))^{\\otimes V}.\n\\end{equation}\nSuch wave functions have already been considered in the context of\n(covariant) loop quantum gravity, and in \\cite{proj} a basis for\nthe Hilbert space of such wave functions ($L^2$ invariant functions\nfor the Haar measure on the Lorentz group) is defined in terms of {\\it projected spin\n  networks}. Roughly, these are spin networks labeled with $\\mathrm{SL}(2,\\mathbb{C})$\nrepresentations on every edge, and with $\\mathrm{SU}(2)$ representations on\nevery edge and $\\mathrm{SU}(2)$ intertwiners at every vertex. In the case that\nwe choose the $\\mathrm{SU}(2)$ to be all the trivial scalar representation, we\nfind back the simple spin networks of the standard Barrett-Crane model\n\\cite{proj,us}:\n$$\n\\psi_{J_e}(U_1,..,U_E,\\Lambda_1,..,\\Lambda_V)=\\prod_e K^{J_e}(\\Lambda_{s(e)}^{-1}U_e\\Lambda_{t(e)}),\n$$\nwhere we note $s(e)$ and $t(e)$ the source and target vertices of the edge $e$.\nTo each vertex of the simple spin network is associated the\n$\\Lambda_v$ dependent $\\mathrm{SU}(2)$ intertwiner between the $\\mathrm{SL}(2,\\mathbb{C})$\nrepresentations $R^{J_i}$:\n\\begin{equation}\n\\bigotimes_i \\langle J_i,\\Lambda_v,j=m=0|\\,:\\bigotimes_i R^{J_i}\\,\\rightarrow\\,\\mathbb{C},\n\\label{simple}\n\\end{equation}\nwhere $i$ runs over all the edges meeting at the vertex\n$v$. $|J_i,\\Lambda,j=m=0\\rangle=D^{J_i}(\\Lambda)W_{J_i}$ is the vector in\n$R_{J_i}$ which is invariant under the $\\mathrm{SU}(2)$ subgroup of\n$\\mathrm{SL}(2,\\mathbb{C})$ leaving invariant the direction $\\Lambda\\in{\\cal H}$.\n\nWe recognize in the above structure the basis of states in which we decompose the 4-valent field of the usual group field theory formulation of the Barrett-Crane model, or, equivalently, the basis states arising in the trivial 5-valent extension of the same group field theory, discussed at the beginning. The field is indeed decomposed in 4-valent simple intertwiners of the Lorentz group, i.e. carrying simple representations of the Lorentz group on every link, and trivial $SU(2)$ representations on the same links, and a $\\Lambda$ dependent $SU(2)$ intertwiner at each vertex, with the various $\\Lambda$'s being integrated over at the end when evaluating the spin network amplitude. \n\nThere is another way of building a basis of the space of such\ninvariant wave functions, which makes the link with the particle\ninsertions clearer. In fact the construction is very close to the way spinning particles are coupled in 3d quantum gravity (see \\cite{pr1,pr3,3d} for the construction of coupled spin foam amplitudes, and \\cite{particle, dj} for the group field theory formalism). To each $n$-valent\nvertex of the spin network graph, we associate a $(n+1)$ valent intertwiner ${\\cal I}$\nwhich intertwines the $n$ edges meeting at the vertex and an extra\nedge describing the particle insertion. The particle is then\ncharacterized by a choice of a Lorentz representation $J$ and a vector\n${\\cal V}^J$ in the corresponding Hilbert space $R^J$. At each vertex, the\nintertwiner ${\\cal I}_v$ is a $\\mathrm{SL}(2,\\mathbb{C})$ invariant tensor in $\\otimes_i\nR^{J_i}\\,\\otimes R^J$. The wave functions is then the evaluation of\nthe holonomies on the spin network with the following tensors at each\nvertex:\n\\begin{equation}\n{\\cal I}_v(\\,\\cdot\\, ,D^J(\\Lambda){\\cal V}^J)\n\\,:\\bigotimes_i R^{J_i}\\,\\rightarrow\\,\\mathbb{C}.\n\\label{particle}\n\\end{equation}\nTo describe a particle of spin $s$, we simply choose the vector\n${\\cal V}^J$ to lie in the subspace of $R^{J}$ corresponding to the\n$\\mathrm{SU}(2)$ representation of spin $j\\equiv s$. Therefore, to describe a\nspinless particle, we choose the $\\mathrm{SU}(2)$ invariant vector\n${\\cal V}^J\\equiv W^J$.\n\nWe recognize in this structure the basis of states arising in the mode expansion of our 5-valent field, in the non-trivial extension of the group field theory presented above.\n\n\nThese two basis, projected spin networks and particle insertions,\ncorrespond thus to two choices of Fourier modes for the decomposition of\nthe $\\phi(g_i,G)$ in our group field theory context. It is easy to see\nthat the simple intertwiner \\Ref{simple} corresponds to the sum over\nall possible particle insertions \\Ref{particle} for fixed\nrepresentations $J_i$, i.e. sum over all possible $J$ and\n$(n+1)$-valent intertwiners.\nIndeed:\n\\begin{equation}\n\\bigotimes_i |J_i,\\Lambda, j=m=0\\rangle\n=\\prod_i D^{J_i}_{A_iB_i}(\\Lambda)W^{J_i}_{B_i}\n=\\sum_J\\sum_{{\\cal I}}{\\cal I}^{J_1..J_n,J}_{A_1..A_n,A}D^J_{AB}(\\Lambda)W^J_B,\n\\end{equation}\nwhere we are summing over an orthonormal basis of $(n+1)$ valent intertwiners between the representations $J_1,..,J_n,J$.\nNevertheless, more generally, if we want to include a particle of some\ngiven fixed spin $s$, we will have to go beyond simple spin networks\nand allow arbitrary $\\mathrm{SU}(2)$ representations in the projected spin\nnetwork basis. Here we will not write in details the explicit change\nof basis between the two choices of modes. \nInstead we would like to\nstress again that the generalized group field theory introduced in the\nprevious section with an extra group variable for each field (or\nquantized tetrahedron) would allow to compute transition amplitudes\nfor gravity plus particles in the spin foam formalism.\nThis also means that, on the one hand, as we anticipated, this kind of mathematical structures would probably play a role in any group field theoretic formulation of 4d quantum gravity coupled to point particles, and, on the other hand, that the insertion of point particle in the Barrett-Crane model, or a similar one, would provide the model with a non-trivial coupling between spacetime atoms, i.e. between 4-simplices, possibly of the type we have presented here. \n\n\\section{A new type of group field theory: turning the coupling into a variable} \\label{sec:gengft}\nWe have just seen that the choice of the heat kernel\n${\\cal K}^{\\beta}(g,g')$ for the propagator term, relating the two normals\ncorresponding to the same tetrahedron in two neighboring 4-simplices,\ngenerates a model that interpolates nicely between the usual\nBarrett-Crane model and a strongly coupled model where the only\ngeometric configuration allowed is flat space, with all the normals\nbeing identified. This very same choice leads naturally to a\nfurther generalization of the group field theory and of spin foam\nmodels, that we believe has even more interesting properties and may\nbe a good starting point for further investigations.\n\nConsider any given spin foam (i.e. Feynman graph) amplitude for the\nabove model, with the heat kernel with parameter $\\beta$ on each dual\nedge connecting the two group variables corresponding to the fifth\nvariables of the field, i.e. to the tetrahedron normals. Now\nre-interpret the $\\beta$ as a {\\it variable} of the model, as opposed\nto a freely specified {\\it parameter}. This implies that there is an\nintegral over it, with range from zero to infinity in the definition\nof the amplitudes; for the same reason, the two limits mentioned for\nthis interpolating model correspond now to two possible approximations\none can make for this integral, considering only the region of\nintegration near zero or the asymptotic region at infinity; this last\none describes perturbations around the Barrett-Crane configurations,\nwhile the first corresponds to a strong coupling region. \n\nEven more interesting is the group field theory derivation of this\nextended model, with $\\beta$ treated as a variable. First of all\nnotice that, if $\\beta$ has to be considered as a variable, the\ndefinition of the field has to be extended to a function \n$$ \\phi(g_1,...,g_4;G,\\beta) : SO(3,1)^5\\times \\mathbb{R} \\rightarrow\n\\mathbb{C}, $$ with the global invariance under $SO(3,1)$\ntransformations and the invariance under $SO(3)$ transformations of\nthe first four arguments to be then imposed in the action. \n\nNow, considering just the Feynman amplitudes in configuration space, i.e. not performing any mode expansion, we have now a propagator of the form:\n\\begin{equation}\n{\\cal P}(g_i,g'_i,G,G',\\beta)=\\left(\\prod_i\\delta(g_i,g'_i)\\right) {\\cal K}(G,G';\\beta),\n\\end{equation}\nwhere we switched to a different notation for the heat kernel to\nhighlight the fact that now $\\beta$ has the interpretation of a \\lq\\lq\nEuclidean time \\rq\\rq for a particle living on the homogeneous space\nto which $G$, and $G'$ belong, and the heat kernel function indeed\npropagates it from the point $G'$ at time 0 to the point $G$ at (Euclidean) time\n$\\beta$. We assume that ${\\cal K}(G,G';\\beta)$ is zero for $\\beta<0$, in order to have a well-defined mode expansion for the propagator, so we effectively work with ${\\cal K}^+(G,G';\\beta)=\\theta(\\beta){\\cal K}(G,G';\\beta)$, where $\\theta$ is the Heaviside function. \n\nIt is easy to identify the kinetic term from which this\npropagator comes. Indeed the heat kernel with \\lq\\lq time\\rq\\rq variable $\\beta$ is\nsimply the propagator for the heat equation \n\\begin{equation}\n\\left( -\\frac{\\partial}{\\partial \\beta} + \\nabla \\right) \\psi(g,\\beta) = 0,\n\\end{equation}\nwith $\\nabla$ being the Laplace-Beltrami operator on the group manifold.\nTherefore the appropriate kinetic term of the group field theory action is identified with:\n\n\\begin{equation}\nS_{kin}[\\phi] = \\prod_i \\int dg_i \\int dG\\int d\\beta \\;\\phi(g_1,...,g_4;G,\\beta)\\left[\\left( -\\partial_\\beta + \\nabla_G \\right) \\phi\\right](g_1,...,g_4;G,\\beta).\n\\end{equation}\nand the kinetic operator with:\n\\begin{equation}\nK(g_i,G,\\beta; \\tilde{g}_i,G',\\beta')= \\left( \\prod_i \\delta(g_i,\\tilde{g}_i) \\right) \\left( -\\partial_\\beta + \\nabla_G\\right)\\delta(G,G')\\delta(\\beta,\\beta').\n\\end{equation}\n\nIn the following we will actually use a symmetrized version of the propagator above, so to erase any asymmetry between the two possible sign of the $\\beta$ variable entering in it, so we will work with a propagator of the form:\n\\begin{equation}\n{\\cal P}_H(g_i,g'_i,G,G',\\beta)=\\left(\\prod_i\\delta(g_i,g'_i)\\right) \\left[ \\theta(\\beta){\\cal K}(G,G';\\beta) + \\theta(-\\beta) {\\cal K}(G,G',-\\beta)\\right].\n\\end{equation}\nThis has an interpretation in terms of different transition amplitudes one can define for quantum gravity in a group field theory context \\cite{generalised}.\n\nWe can now consider the interaction term for our generalized group field theory. A first choice is to define the dependence on the extra variable $\\beta$ in each field to be trivialized, i.e. a different $\\beta$ enters in the various fields in the interaction term, but no relation is assumed between them; the interaction is then of the form:\n\n\\begin{eqnarray}\n\\lefteqn{S_{int}^{(1)}[\\phi]=\\int d\\beta_A ...d\\beta_E\\,\\int d\\Lambda_A ...d\\Lambda_E \\int \\prod_{i=1}^{10} dg_i\\prod_i \\int dh_i d\\tilde{h}_i\\,\n\\phi(g_1 h_1\\Lambda_A,g_2 h_2\\Lambda_A,g_3 h_3\\Lambda_A,g_4 h_4\\Lambda_A,\\Lambda_A, \\beta_A)} \\nonumber \\\\ &&\\phi(g_4 \\tilde{h}_4 \\Lambda_B,g_5 h_5\\Lambda_B,g_6 h_6 \\Lambda_B,g_7 h_7 \\Lambda_B,\\Lambda_B, \\beta_B)\\; \\phi(g_7\\tilde{h}_7\\Lambda_C,g_3 \\tilde{h}_3\\Lambda_C,g_8 h_8 \\Lambda_C,g_9 h_9 \\Lambda_C,\\Lambda_C, \\beta_C) \\nonumber\n\\\\ &&\\phi(g_9 \\tilde{h}_9 \\Lambda_D,g_6\\tilde{h}_6\\Lambda_D,g_2\\tilde{h}_2\\Lambda_D,g_{10}h_{10}\\Lambda_D,\\Lambda_D, \\beta_D)\\; \\phi(g_{10}\\tilde{h}_{10}\\Lambda_E,g_8\\tilde{h}_8\\Lambda_E,g_5\\tilde{h}_5\\Lambda_E,g_1\\tilde{h}_1\\Lambda_E,\\Lambda_E,\\beta_E).\n\\end{eqnarray}\n\nThis is an admissible choice for the interaction term; however, the resulting dependent on the $\\beta$ variables is somehow unsatisfactory: there are two $\\beta$ variables associated with each tetrahedron in the dual formulation of the associated Feynman graphs, i.e. two $\\beta$ variables for each edge of the Feynman graph; the amplitudes depend only on the difference between them, but this dependence is \\lq\\lq localised\\rq\\rq along the edge of the Feynman graph only, i.e. the variables appearing there do not appear anywhere else in the amplitude, so the integral can actually be performed separately in each edge, leading to a modified but $\\beta$-independent propagator. Explicitly, this modified $\\beta$-independent propagator of this model is given by:\n\\begin{equation}\n{\\cal P}_H(g_i,\\tilde{g}_i,G,G')= \\prod_i \\delta(g_i,\\tilde{g}_i)\\,\\int d\\beta \\int d\\tilde{\\beta}\\;\\; {\\cal P}_H(G,G';\\beta-\\tilde{\\beta}), \n\\end{equation}\nwhere $\\beta$ and $\\tilde{\\beta}$ are the two \\lq\\lq Euclidean time\\rq\\rq variables associated to the give edge. The integral can be performed easily if one uses the mode expansion of the heat kernel, assuming that one can exchange the sum over representation and the integral over the $\\beta$ variables, leading to:\n\\begin{eqnarray}\n\\lefteqn{{\\cal P}_H(g_i,\\tilde{g}_i,G,G')=\\,2\\prod_i \\delta(g_i,\\tilde{g}_i)\\,\\int_0^\\infty d(\\beta -\\tilde{\\beta}) \\;\\;{\\cal K}(G,G';\\beta-\\tilde{\\beta})=} \\nonumber \\\\ &=& 2\\prod_i \\delta(g_i,\\tilde{g}_i)\\,\\int_0^\\infty d(\\beta -\\tilde{\\beta}) \\sum_J \\Delta_J e^{-(\\beta - \\tilde{\\beta}) C_J} \\chi^{J}(G G'^{-1}) = 2\\prod_i \\delta(g_i,\\tilde{g}_i)\\,\\sum_J \\frac{\\Delta_J}{C_J}\\, \\chi^{J}(G G'^{-1}),\n\\end{eqnarray}  \nwhere $C_J$ is the non-zero quadratic Casimir for simple representations of the Lorentz group\n(with an appropriate analytic continuation needed to regularise the integral), to which the sum refers, $\\chi^J$ is the character of the representation, and we have discarded an infinite factor coming from the integral over $\\beta + \\tilde{\\beta}$ and that is clearly pure gauge, the amplitude being completely independent of such a variable.\n\nOne can still consider the different regimes of integration over $\\beta - \\tilde{\\beta}$, i.e. the regime in which the difference is large and the model then approximates the usual Barrett-Crane model, and the one in which the difference is very small, and the model gives a BF-type behavior; however, such a localized dependence on the variables and the triviality of the amplitudes with respect to this dependence, to the point that the $\\beta$ variables can be quite simply eliminated from the amplitudes by integrating them out, make this choice of interaction term somewhat unpleasant, and we are led to look for a different and less trivial choice.\n\nIf we want a relationship between the various $\\beta$ variables in the fields in the same vertex term, then the simplest choice is to choose them to be all equal; the interaction term in the action would then be:\n\n\\begin{eqnarray}\n\\lefteqn{S_{int}^{(2)}[\\phi]=\\int d\\beta \\; \\int d\\Lambda_A ...d\\Lambda_E \\int \\prod_{i=1}^{10} dg_i\\prod_i \\int dh_i d\\tilde{h}_i\\,\n\\phi(g_1 h_1\\Lambda_A,g_2 h_2\\Lambda_A,g_3 h_3\\Lambda_A,g_4 h_4\\Lambda_A,\\Lambda_A, \\beta)} \\nonumber \\\\ &&\\phi(g_4 \\tilde{h}_4 \\Lambda_B,g_5 h_5\\Lambda_B,g_6 h_6 \\Lambda_B,g_7 h_7 \\Lambda_B,\\Lambda_B, \\beta)\\; \\phi(g_7\\tilde{h}_7\\Lambda_C,g_3 \\tilde{h}_3\\Lambda_C,g_8 h_8 \\Lambda_C,g_9 h_9 \\Lambda_C,\\Lambda_C,\\beta) \\nonumber\n\\\\ &&\\phi(g_9 \\tilde{h}_9 \\Lambda_D,g_6\\tilde{h}_6\\Lambda_D,g_2\\tilde{h}_2\\Lambda_D,g_{10}h_{10}\\Lambda_D,\\Lambda_D, \\beta)\\; \\phi(g_{10}\\tilde{h}_{10}\\Lambda_E,g_8\\tilde{h}_8\\Lambda_E,g_5\\tilde{h}_5\\Lambda_E,g_1\\tilde{h}_1\\Lambda_E,\\Lambda_E,\\beta)\n\\end{eqnarray}     \n\nThis choice is more satisfactory for several reasons; first of all it avoids the problems mentioned above for $S_{int}^{(1)}$, in that the dependence on $\\beta$ is much less trivial and not localized along each edge; second, with the interpretation of $\\beta$ as a interaction coupling parameter in each spin foam amplitude, this choice corresponds to having a different coupling parameter in each spacetime building block, i.e. 4-simplex, with the amplitude for the interaction between these building blocks depending only on the difference between the parameters associated to them; third, if one keeps in mind instead the interpretation of the $\\beta$ variable as a kind of \\lq\\lq Euclidean time\\rq\\rq, then the choice of $S_{int}^{(2)}$ made above is actually the most sensible one, as it corresponds to an interaction between the five fields corresponding to the five tetrahedra in 4-simplex that is {\\it local} in this time variable.     \nNote that also in this case we have a different coupling of spacetime atoms, i.e. 4-simplices, for each edge of the spin foam, but with a non-trivial relation among those referring to the same 4-simplex.\nThe vertex amplitude in configuration space corresponding to the interaction term $S_{int}^{(2)}[\\phi]$ is therefore still given by \\Ref{eq:vertex}:\n\\begin{equation}\n\\mathcal{I}_v(g_f,\\tilde{g}_f, \\Lambda_e, \\beta)= \\prod_{f\\mid v} \\delta(g_{f\\mid v}\\Lambda_{e1}^{-1}\\Lambda_{e2}\\tilde{g}_{f\\mid v}^{-1}),\n\\end{equation}\nwhere the dependence on $\\beta$ is limited to the fact that the same $\\beta$ should appear in any mode expansion of the fields with respect to this variable, that we do not perform explicitly, and in that the same $\\beta$ has to appear in any propagator connecting this vertex to any other one in the Feynman expansion. This is given of course by:\n\n\\begin{equation}\nZ(\\lambda) = \\int \\mathcal{D}\\phi\\; e^{- S[\\phi]} = \\int \\mathcal{D}\\phi\\; e^{- S_{kin}[\\phi] - \\frac{\\lambda}{5!} S_{int}^{(2)}[\\phi]} = \\sum_{\\Gamma} \\frac{\\lambda^N}{sym(\\Gamma)} \\,Z_{\\Gamma}, \\label{eq:Zvar}\n\\end{equation}\nwith:\n\\begin{equation}\nZ_{\\Gamma}= \\left(\\prod_e \\prod_{f\\mid e} \\int dg_{f\\mid e} \\int d\\tilde{g}_{f\\mid e}\\right) \\left( \\prod_e \\int dG_e \\int dG'_e\\right) \\prod_v \\int d\\beta_v \\prod_e {\\cal P}_H(g_{f\\mid e},\\tilde{g}_{f\\mid e}, G_e,G'_e, \\beta_{e}, \\tilde{\\beta}_e) \\prod_v \\mathcal{I}_v(g_{f\\mid v}, \\tilde{g}_{f\\mid v}, G_{e\\mid v}, \\beta_v).\n\\end{equation}\n\nWe think this new type of group field theory model possesses many interesting features, and the resulting spin foam amplitudes should be analysed in more detail, but we leave this analysis for future work; now we prefer to simply highlight some of its general aspects, its possible interpretation and uses. Note for example that in this new model the coupling relative to different edges of the spin foam, i.e. too different tetrahedra, may be different so that it is possible to study models that are almost flat, but not quite, and with small {\\it different} curvatures in different regions of space.\n\nFirst of all the model presented above can be seen at the same time as an extension and as a special case of the new type of group field theories studied and to be presented in \\cite{generalised}; there a \\lq\\lq proper time\\rq\\rq parametrisation of group field theories is studied in order to achieve a generalized formulation of them from which both orientation dependent and orientation independent spin foam models \\cite{us,feynman} can be obtained; the field is defined to be of the general form:\n\\begin{equation}\n\\varphi(g_1, s_1; g_2, s_2; g_3, s_3; g_4, s_4) : \\left( \\mathrm{SO}(3,1) \\times \\mathbb{R}\\right)^{\\times 4}\n\\end{equation}  \nand on this field both a simplicity projector onto $\\mathrm{SO}(3,1)\/\\mathrm{SO}(3)$ for the four arguments and a global \\lq\\lq gauge invariance\\rq\\rq projector, under $\\mathrm{SO}(3,1)\\times \\mathbb{R}$ transformations acting simultaneously (diagonally) on the four arguments, are applied at the level of the action; with respect to a model based on this type of field, the one we have outlined above represents a restriction, because the extension of the domain of the first four arguments of the field from $\\mathrm{SO}(3,1)$ to $\\mathrm{SO}(3,1)\\times \\mathbb{R}$, with an extra \\lq\\lq proper time\\rq\\rq variable, is dropped; but it is also a generalization, in the same sense as the model presented in section \\Ref{sec:genBC} is a generalization of the Barrett-Crane model, because it extends the field to a five argument one, again by basically turning the variables enforcing the gauge invariance of the four argument field into true extra variables of it, and extending appropriately the gauge invariance to be imposed.   \nIndeed we expect a direct generalisation along these lines of the type of models studied in \\cite{generalised} to be straightforward and to lead to a model of the same type as the one defined by the partition function \\Ref{eq:Zvar}.  The main difference is however that what appears in the model defined by \\Ref{eq:Zvar} is a kind of \\lq\\lq Euclidean time\\rq\\rq, not a Minkowskian one, so that the corresponding differential operator appearing in the action is of heat- or diffusion-type, and not of Schr\\\"oedinger type; this is needed for maintaining the property that the model reduces to the Barrett-Crane one in the large $\\beta-\\tilde{\\beta}$ limit. \n\nIt is interesting at this point to discuss briefly the classical equations of motion for the field $\\phi$  coming from the action $S[\\phi] = S_{kin}[\\phi] + S_{int}^{(2)}[\\phi]$. These are:\n\\begin{eqnarray}\n\\lefteqn{\\left(-\\partial_\\beta + \\nabla_G\\right)\\phi(g_1,g_2,g_3,g_4,G,\\beta) +} \\nonumber \\\\  &+&\\frac{\\lambda}{5!} \\int dg_5...dg_{10}\\;\\varphi(g_1,g_5,g_6,g_7,G,\\beta)\\phi(g_2,g_5,g_8,g_9,G,\\beta)\\varphi(g_3,g_6,g_8,g_{10},G,\\beta)\\varphi(g_4,g_7,g_9,g_{10},G,\\beta)= 0,\n\\end{eqnarray} \nwhich is indeed the form of a diffusion or heat transfer equation in Euclidean time $\\beta$, with an additional time dependent \\lq\\lq driving potential\\rq\\rq, for what concerns the fifth extra arguments; the dynamics defined by this equation for the other arguments of the field is basically the usual one, modulo the modified gauge invariance requirement.  However, the physical interpretation of such an equation is not straightforward from a quantum gravity perspective, mainly because, while the field $\\phi$ itself has the (quantum) geometric interpretation of a (second) quantized tetrahedron, the geometric interpretation of $\\beta$ is unclear at the present stage. On the other hand, recalling the interpretation of $\\beta$ as a coupling parameter, the above equation can be interpreted as a kind of renormalisation group equation for our coupled model and it is quite striking, in our opinion, that what is a renormalisation group equation from the point of view of the spin foam amplitudes as presented in the previous section is just an equation of motion from the point of view of the generalised group field theory described above. Indeed, this new type of group field theory can be a useful starting point, we think, for tackling the issue of renormalisation of spin foam models, or, better, of the group field theories generating them as Feynman graphs and amplitudes. Therefore we conclude now by discussing how this issue can be tackled from the starting point provided in this paper. \n\n\n\\section{Outlook: renormalisation of spin foam models} \\label{sec: renorm}\n\nThe final goal is to understand the semi-classical behavior of the spin foam quantum gravity, recover the standard effective physics on flat or curved space-time and compute the corrections due to the quantum fluctuations of the gravitational field. This is a long-term program, and a first step in this direction would be to understand the coarse-graining and renormalisation of the spin foam models such as the Barrett-Crane model. We could expect a semi-classical space-time to be built out of a large number of elementary Planck-size blocks (the 4-simplices) and the macroscopic effective spin foam model to have a structure similar to that of the microscopic theory, but with a slightly different form of the amplitudes and a different value of the couplings: the Barrett-Crane model would then be only be a particular model along the renormalisation flow. We propose to use our generalized Barrett-Crane with arbitrary couplings between 4-simplices as an ansatz to study explicitly this renormalisation flow.\n \n\\subsection{About the renormalisation of the coupling}\n\nThe Barrett-Crane amplitude for a 4-simplex is a function of the 10 representation labels $\\rho_f\\in\\mathbb{R}_+$ living on its faces and of 5 variables $\\Lambda_T\\in\\mathrm{SO}(3,1)\/\\mathrm{SO}(3)$ attached to each tetrahedron. We note it ${\\cal A}(\\rho_f,\\Lambda_T)$. The $\\rho_f$'s have the interpretation of the area of the corresponding triangle while the $\\Lambda_T$'s are thought the (time-like) normal to the corresponding tetrahedron. Then the Barrett-Crane $\\{10\\rho\\}$ symbol can be considered as a (quite peculiar) quantum amplitude for a single quantum 4-simplex described in terms of 1st order Regge calculus\\cite{us,barrett,barrettruth}. A full spin foam is constructed by gluing 4-simplices together. As explained above, the representation label attached to a triangle will be the same for all 4-simplices who share that triangle. However, the normal variables $\\Lambda_T$ are attached to the corresponding tetrahedron in the reference frame of a given 4-simplex: they are not shared between 4-simplices and we will not denote them $\\Lambda_{T,\\sigma}$ referring to both the tetrahedron $T$ and the 4-simplex $\\sigma$ to which it belongs. Then the Barrett-Crane spin foam amplitude is the product of the 4-simplex amplitudes summed over all assignments $\\{\\rho_f,\\Lambda_{T,\\sigma}\\}$.\n\nLet us consider a particular tetrahedron $T_0$ in a fixed spin foam. It belongs to two 4-simplices, $\\sigma_1$ and $\\sigma_2$. The two normal variables for $T_0$, $\\Lambda_{T_0,\\sigma_1}$ and $\\Lambda_{T_0,\\sigma_2}$, are  a priori independent. They are indeed both integrated out independently to get the Barrett-Crane amplitude. Nevertheless, if we compute the spin foam amplitude summing and integrating over all representation labels $\\rho_f$ and all other normal variables $\\Lambda_T$, $T\\ne T_0$, we will obtain a probability amplitude on  the space of these two variables, $\\Lambda_1$ and $\\Lambda_2$ in short; this amplitude will not be factorised in general, which implies that they will not be treated as fully independent by the model after this sort of \\lq\\lq renormalisation procedure\\rq\\rq involving the mentioned summation over representations and normal variables. Therefore, with the aim of understanding the renormalisation of the Barrett-Crane model, it is natural to work in a generalized formulation allowing for generic coupling between the two normals associated to each tetrahedron. This should be intended as an effective way for encoding the coupling between these variables that would result from renormalisation, before the actual renormalisation procedure is carried out. \nAnother way to state the same point is the following. The Lorentz transformation mapping $\\Lambda_{T_0,\\sigma_1}$ to $\\Lambda_{T_0,\\sigma_2}$ can naturally be interpreted as the parallel transport between the two reference frames associated to the two 4-simplices $\\sigma_1$ and $\\sigma_2$. In the \"fundamental\" model, this parallel transport is arbitrary, and therefore $\\Lambda_1$ and $\\Lambda_2$ are integrated over independently. Nevertheless, under renormalisation of that fundamental model, we expect that the effective probability amplitude for this parallel transport evolves to a non-trivial correlation ${\\cal C}(\\Lambda_1,\\Lambda_2)$ between $\\Lambda_1$ and $\\Lambda_2$.\n\n\\medskip\n\nFrom this perspective, it would be interesting to look for the existence of a fixed point of the renormalisation of the generalized coupled Barrett-Crane model, which we described above. We think that the group field theory provides us with the right framework to address this issue. A first result would be to check that the uncoupled spin foam model is not a fixed point (or at least not a stable fixed point) and then whether or not the rigidly coupled model is a fixed point. This rigid coupling, or strong coupling limit, ${\\cal C}(\\Lambda_1,\\Lambda_2)\\sim \\delta (\\Lambda_1^{-1}\\Lambda_2)$ would corresponds to a flat model, as we said: the 4-simplices would be glued with a trivial parallel transport, so that the spin foam should be describing a flat space-time \\cite{us}.\n\nThe recent study \\cite{wade} of the Lorentzian Barrett-Crane partition function in terms of the normal variables $\\Lambda$'s should be particularly relevant to this project: the author looks at the properties of the amplitudes obtained by integrating first the representation labels $\\rho_f$ and uses them to present a new proof of the finiteness theorems for the Barrett-Crane amplitudes. It would actually be rather natural to include arbitrary couplings between the $\\Lambda$'s in such a reformulation of the model.\n\n\nWe also proposed a more specific group field theory for a restricted class of coupling given by the heat kernel. It should be simpler to study the coarse-graining properties of this model for which the couplings are described by a single real parameter. This model is close to a very interesting model proposed and studied by Oeckl \\cite{robert}.\nHe introduces a new model replacing the $\\delta$-functions constituting the edge and vertex amplitudes (in configuration space) in the spin foam quantization of the topological BF theory and the various Barrett-Crane-like models by heat kernels, and discusses the stability properties of the weak and strong coupling limits under coarse-graining moves generalizing the standard Pachner moves, i.e. under renormalisation. Our proposal is in the same spirit. However, while Oeckl's model corresponds to using the usual definition of the field and just changing the coupling between its arguments, the areas of triangles in momentum space, we first introduce a generalisation of the same field to five arguments, and then use explicitly the $\\Lambda$ variables, representing the normal to the tetrahedra.\nWe feel that using the $\\Lambda$ allows us to introduce the coupling between 4-simplices in a transparent way, easier to interpret physically. Moreover, Oeckl's model interpolates between the Barrett-Crane model and BF theory, while as we have discussed our $\\beta\\rightarrow 0$ limit does not give a true quantum BF theory but a similar theory in which the only configuration allowed for the (boost part of the) connection is the flat one. We believe it would be very interesting to analyse in more detail the similaritites and differences between these two modifications of the Barrett-Crane model, and in particular check whether the respective fixed points in the renormalisation flow in $\\beta$ correspond to the same type of theory. \n\nThe \\lq\\lq parametrised \\rq\\rq group field theory we have introduced last, in which the heat kernel coupling is treated as a variable of the theory, suggests (actually, forces) a different way of dealing with the same issues. In fact, as we said, the renormalisation flow of the coupling is encoded there in the {\\it classical} equations of motion coming from the group field theory action; therefore, if we are to study the presence and nature of fixed points, we should check whether particular regimes for the coupling are favored by the classical equations of motion.\n\n\\medskip\n\nWe conclude this discussion by a speculation that the study of the renormalisation flow of the spin foam model could \nshed light on the issue of the diffeomorphism invariance of the spin foam amplitudes: diffeomorphisms may appear as renormalisation group transformations in the group field theory formulation since they seem to be related to Pachner moves in the state sum formulation of spin foams and to the Ward identities in the group field theory formulation. This was suggested in \\cite{laurentgft,instantons}, following the results of \\cite{laurentdiffeo}. Indeed Ward identities in quantum field theory relate the amplitudes of different Feynman diagrams. In the group field theory, they would then relate the amplitudes of different spin foams. It is likely that looking more precisely to this point would lead to a deeper understand of both the renormalisation of spin foams and the action of diffeomorphisms in this discrete quantum setting.  In the 2d spin foam case, the group field theory reduces to matrix models, for which the explicit link between the Ward identities of the path integral and the conformal symmetry is well-understood (see for example the review \\cite{matrix}), so methods and ideas from matrix models can be of help in analysing this same issue for group field theories. Finally, it would be interesting to compare the resulting Ward identities relating different spin foam amplitudes to the background independent coarse graining scheme proposed by Markopoulou \\cite{fotini}, using the Hopf algebra structure of 2-complexes in a way very similar to the Connes-Kreimer approach to the renormalisation of quantum field theory \\cite{connes}.\n\n\n\\subsection{A numerical simulation project}\n\nThe coarse-graining of the coupling can well be studied numerically. First, we would need to look at the single 4-simplex amplitude ${\\cal A}(\\rho_f,\\Lambda_T)$. It would be interesting to look at the probability amplitude of the $\\Lambda$ variables. We could restrict the study to regular 4-simplices where all the representation labels are identical, $\\rho_f =\\rho$. Fixing $\\rho$ and integrating over three of the $\\Lambda$ variables, we would get a probability amplitude on the remaining two $\\Lambda$ variables, which would describe the correlations between two tetrahedra within the considered 4-simplex. Actually, due to the Lorentz gauge invariance of the 4-simplex, we would obtain a simple distribution on the angle between the two normals. A first issue is whether this distribution is peaked or not at a given angle. A second issue is whether this peak moves or not when rescaling the representation label $\\rho$: this would show that the coupling would definitely be sensitive to the size of the 4-simplices (measured in Planck units). \n\nStill looking at a single 4-simplices, we could consider almost identical assignments of representation labels $\\rho_f$. More precisely, singling out two tetrahedra, and keeping fixed the 7 representation labels attached to these tetrahedra, we could look at the evolution of the coupling between these tetrahedra when we change the 3 remaining representation labels. This would show how the geometry of a tetrahedron does not solely depend on the representation labels assigned to it but on the representation labels assigned to the 4-simplex to which it belongs: the intrinsic geometry of a tetrahedron is determined by the space-time geometry of the 4-simplex.\n\nLet us point out that we could carry on this analysis unfolding the Barrett-Crane intertwiner at each tetrahedron in terms of $\\mathrm{SO}(3,1)$ simple representations $\\rho_{int}$ instead of the normal variable $\\Lambda$. $\\rho_{int}$ would be interpreted as describing the internal space geometry of the tetrahedron while the $\\Lambda$ seem to describe its embedding in the surrounding spacetime. A first computation would be to study the correlations between the internal representation label $\\rho_{int}$ of two tetrahedra within the same 4-simplex and see whether the maximal probability corresponds to small\/large-small\/large configurations. One might then hope that such correlations propagate within the spinfoam helped by the coupling between 4-simplices.\n\nThe next step would be to consider two 4-simplices or more and the coupling between two tetrahedra belonging to different 4-simplices. The coupling between the 4-simplices would then play a role, and therefore the new coupled model we have proposed in this paper can be of direct use. We could study how long-range correlations between spacetime points (tetrahedra) would depend on the coupling appearing between two glued 4-simplices. In this framework, we could either keep the representation labels fixed and look at the correlations induced solely by the coupling imposed by hand, or we could sum over internal representation labels (attached to the internal faces i.e not on the spin foam boundary) and we would look at the full effective correlations. \n\n\n\\section{Conclusions}\nLet us summarize what we have achieved in this work. Our main result has been to construct a group field theory formalism that generalizes the Barrett-Crane model to allow a non-trivial coupling between the two normals to the tetrahedra in the two 4-simplices that share them; the way we have done it is simple: we have shown first how the usual Barrett-Crane model can be seen as the result of a quantization of a field theory on five copies of a group manifold, with no coupling among the fifth arguments of the field, and then we have modified this field theory to introduce a coupling. We have exhibited a specific choice of coupling function and therefore constructed a specific coupled model; this model interpolates nicely between the usual Barrett-Crane model, with the parallel transport between 4-simplices being completely randomized, and a flat or BF-like model, in which the only allowed connection is the flat one, according to the value that a new additional parameter of the theory, not present in the usual Barrett-Crane model, takes. The tuning of the coupling parameter therefore corresponds to a tuning of the degree of \\lq\\lq locality\\rq\\rq of the resulting spin foam amplitudes. This new coupled model is amenable for further study of correlations between simplices, both analytical and numerical, that can shed light on the presence or absence of local propagating degrees of freedom in spin foams, and can play an important role in studying renormalisation of the spin foam amplitudes under coarse graining. Moreover, it can be the basis for developing fully covariant perturbation expansions in the coupling parameter that do not make any use of background structures and therefore remain fully background independent from the quantum spacetime point of view. We have also discussed how the mathematical structures on which the coupled model is based are going to be relevant for studies of particle and matter insertions in the Barrett-Crane or similar spin foam models. Finally, we have gone further in this process of generalization, by showing how a new type of group field theory can be constructed in which the coupling parameter is promoted to a variable of the theory, on the same footing as group variables, leading to the presence of derivatives in the action and therefore to a different type of classical equations of motion for the field. We concluded by discussing the issue of renormalisation of spin foam models from the new perspective that the results of this work suggest, and how the new coupled model can be a useful new starting point for the application of renormalisation group techniques, and therefore for understanding the semi-classical properties of spin foam models.  \n\n\n\n\n\n\n\\section*{Acknowledgements}\nWe would like to thank, for hospitality and beverages, Caffe' Nero in Cambridge and Cafe' 1842 in Waterloo, where part of this work was done. D.O. thanks also the Perimeter Institute for hospitality during the early stages of this project, and the Kalahari Meerkat Project for hospitality in the Kuruman River Reserve, South Africa, during its completion.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{0pt}{0.4em plus 0.2em minus 0.2em}{0.4em plus 0.2em minus 0.2em}\n\n\\title{Policy Evaluation with Latent Confounders via Optimal Balance}\n\n\n\\author{%\n  Andrew~Bennett\\thanks{Alphabetical order.} \\\\\n  Cornell University \\\\\n  \\texttt{awb222@cornell.edu} \n \n  \\and\n  Nathan~Kallus\\samethanks\\\\\n  Cornell University\\\\\n  \\texttt{kallus@cornell.edu} \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n}\n\\date{}\n\n\\begin{document}\n\n\\maketitle\n\n\\begin{abstract}\nEvaluating novel contextual bandit policies using logged data is crucial in applications where exploration is costly, such as medicine. But it usually relies on the assumption of no unobserved confounders, which is bound to fail in practice. We study the question of policy evaluation when we instead have proxies for the latent confounders and develop an importance weighting method that avoids fitting a latent outcome regression model. We show that unlike the unconfounded case no single set of weights can give unbiased evaluation for all outcome models, yet we propose a new algorithm that can still provably guarantee consistency by instead minimizing an adversarial balance objective. We further develop tractable algorithms for optimizing this objective and demonstrate empirically the power of our method when confounders are latent.\n\\end{abstract}\n\n\\input{src\/intro.tex}\n\\input{src\/problem.tex}\n\\input{src\/theory.tex}\n\\input{src\/algorithms.tex}\n\\input{src\/experiments.tex}\n\\input{src\/conclusion.tex}\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nPersonalized intervention policies are of increasing importance in education \\citep{mandel2014offline}, healthcare \\citep{bertsimas2017personalized}, and public policy \\citep{kube2019allocating}.\nIn many of these domains exploration is costly or otherwise prohibitive, and so it is crucial to evaluate new policies using existing observational data.\nUsually, this relies on an assumption of no unobserved confounding (aka unconfoundedness or ignorability): that conditioned on observables, interventions are independent of idiosyncrasies that affect outcomes, so that counterfactuals can be reliably and correctly predicted. In particular, this enables the use of inverse propensity score (IPS) estimators of policy value \\citep{beygelzimer2009offset,li2011unbiased,kallus2018policy} that eschew the need to actually fit outcome prediction models and doubly robust estimators that work even if such models are misspecified \\citep{dudik2011doubly}.\n\nIn practice, however, it may be unlikely that we observe confounders exactly. Nonetheless, if we observe very many features they may serve as good proxies for the true confounders, which can enable an alternative route to identification \\citep{louizos2017causal,kallus2018causal}. \nIn particular, noisy observations of true confounders can serve as valid proxies.\nFor example, if intelligence is latent but affects both selection and outcome, we can instead use many noisy observations of intelligence such as school grades, IQ test, etc. Similarly, many medical measurements taken together can serve as proxies for underlying healthfulness.\n\n\n\n\n\nIn this paper, we study the problem of policy evaluation from observational data where we observe proxies instead of true confounders and we develop new weighting estimators based on optimizing balance in the latent confounders. Unlike the unconfounded setting where IPS weights ensure balance regardless of outcome model, we show that in this new setting there cannot exist any single of weights that ensure such unbiasedness regardless of outcome model. Instead, we develop an adversarial objective that bounds the conditional mean square error (CMSE) of any weighted estimator and, by appealing to game theoretic and empirical process arguments, we show that this objective can actually be driven to zero by a single set of weights. We therefore propose a novel policy evaluation method that minimizes this objective, thus provably ensuring consistent estimation in the face of latent confounders. We develop tractable algorithms for this optimization problem. Finally, we provide empirical evidence demonstrating our method's consistent evaluation compared to standard evaluation methods and its improved performance compared to using fitted latent outcome models.\n\n\n\n\n\n\n\\section{Conclusion}\n\nWe presented theory for how to do optimal balancing for policy evaluation when we only have proxies for the true confounders, given an already identified model for the confounders, treatment, and proxies, but not for the outcomes. We provided an adversarial objective for selecting optimal weights given some class of mean outcome functions to be balanced, and proved that under mild conditions these optimal weights result in consistent policy evaluation. In addition, we presented a tractable algorithm for minimizing this objective when our function class is an RKHS, and we conducted a series of experiments to demonstrate that our method can achieve consistent evaluation even under sufficient levels of confounding where standard approaches fail.\n\nFor future work we note that the adversarial objective and theory presented here is fairly general, and could be used to develop new algorithms for balancing different function classes such as neural networks. An alternative direction would be to study how best to apply this methodology when an identified model is not already given.\n\n\\section{Weight-Balancing Objectives}\n\\label{sec:method}\n\n\\subsection{Infeasibility of IPS-Style Unbiased Weighting}\n\nIf we had unconfoundedness given $X$ (\\ie, $Y(t)\\indep T\\mid X$), the IPS weights $\\pi_T(X)\/\\eta_T(X)$ are immediately gotten as the solution to making every term in the weighted sum \\cref{eq:weighted} unbiased:\n\\begin{equation}\n\\label{eq:ips}\n\\mathbb{E}[W(X, T) \\delta_{T_i t} Y(t)] = \\mathbb{E}[\\pi_t(X) Y(t)].\n\\end{equation}%\nNotably the IPS weights do not depend on the outcome function.\nHowever, without unconfoundedness given $X$ and given only \\cref{asm:ignorability,asm:overlap,asm:variance}, this approach fails.\n\\begin{theorem}\n\\label{thm:ips}\nIf $W(x,t)$ satisfies \\cref{eq:ips} then for any $t\\in\\{1,\\dots,m\\}$\n\\begin{equation}\\label{eq:generalips} W(X,t) = \\pi_t(X) \\frac{\\sum_{t' \\in \\mathcal{\\tau}} \\eta_{t'}(X) \\nu_t(X,t') + \\Omega_t(X)}{\\eta_t(X) \\nu_t(X,t)}, \\end{equation}\nfor some $\\Omega_t(x)$ such that $\\E\\Omega_t(X)=0\\,\\forall t$.\n\\end{theorem}\nThe proof of \\cref{thm:ips} is given in \\cref{apx:ips}. \n\nNote that \\emph{if} we had unconfoundedness given $X$ then $\\nu_t(x,t')=\\nu_t(x,t)=\\rho_t(x)$ so that\nchoosing $\\Omega_t(X) = 0$ would recover the standard IPS weights. However, in our setting we generally have $\\nu_t(x,t')\\neq\\nu_t(x,t)$, and so \\cref{thm:ips} tells us that we cannot do unbiased IPS-style weighted evaluation without knowing the mean outcome functions $\\nu_t(x,t')$. In particular, there exists no single weight function that is simultaneously unbiased for all outcome functions.\n\n\nOn the other hand, \\cref{thm:ips} tells us that there do exist some weights that give unbiased and consistent policy evaluation via \\cref{eq:weighted} or \\cref{eq:dr}: we just may not be able to calculate them. \nThe existence of such weights motivates our subsequent approach, which seeks weights that mimic these weights for a wide class of possible outcome functions.\n\n\n\\subsection{Adversarial Error Objective}\n\nOver all weights that are functions of $X_{1:n},T_{1:n}$, the \\emph{optimal} choice of weights for estimating $\\tau^{\\pi}$ via \\cref{eq:weighted} would minimize the (unknown) conditional MSE (CMSE):\n\\begin{equation}\\label{eq:cmsetau}\\mathbb{E}[ (\\hat{\\tau}^{\\pi}_W - \\tau^{\\pi})^2 \\mid X_{1:n},T_{1:n} ].\\end{equation} \nIn particular, the weights in \\cref{eq:generalips} achieve $O_p(1\/n)$ control on this CMSE for many outcome functions, as long as the denominator is well behaved, which can be seen by applying concentration inequalities to \\cref{eq:cmsetau}. However, as discussed above the outcome function is unknown and these weights are therefore practically infeasible.\nOur aim is to find weights with similar near-optimal behavior but that do not depend on the particular unknown outcome function. To do this, we will find an upper bound for \\cref{eq:cmsetau} that we can actually compute.\n\nLet $f_{it} = W_i \\delta_{T_i t} - \\pi_t(X_i)$ and\n$$\nJ(W,\\mu) =  \\left( \\frac{1}{n} \\sum_{i=1}^n \\sum_{t=1}^m f_{it} \\nu_t(X_i,T_i) \\right)^2 + \\frac{2\\sigma^2}{n^2} \\|W\\|_2^2,\n$$\nwhere we embedded the dependence on $\\mu$ inside $\\nu_t(x,t')=\\Eb{\\mu_t(Z)\\mid X=x,T=t'}$.\n\n\\begin{theorem}\n\\label{thm:jbound}\n$\\displaystyle\\mathbb{E}[ (\\hat{\\tau}^{\\pi}_W - \\tau^{\\pi})^2 \\mid X_{1:n},T_{1:n} ]\n\\leq 2J(W,\\mu)+O_p(1\/n)\n$.\n\\end{theorem}\nWe prove this in \\cref{apx:jbound}.\n\nTherefore, if we find weights that obtain $O_p(1\/n)$ control on $J(W,\\mu)$, we can ensure that we also have $O_p(1\/n)$ control on $\\mathbb{E}[ (\\hat{\\tau}^{\\pi}_W - \\tau^{\\pi})^2 \\mid X_{1:n},T_{1:n} ]$. Combined with the following result, which follows from \\citep[Lemma~31]{kallus2016generalized}, this would give root-n consistent estimation.\n\\begin{lemma}\n\\label{lem:control}\nIf $\\mathbb{E}[ (\\hat{\\tau}^{\\pi}_W - \\tau^{\\pi})^2 \\mid X_{1:n},T_{1:n} ]=O_p(1\/n)$ then\n$\\hat{\\tau}^{\\pi}_W=\\tau^{\\pi}+O_p(1\/\\sqrt{n})$.\n\\end{lemma}\n\n\n\n\n\n\n\n\nIt remains to find weights that control $J(W,\\mu)$. The key obstacle for this is that $\\mu$ is unknown. Instead, we show how we can obtain weights that control $J(W,\\mu)$ over a whole class of given functions $\\mu$.\n\nSuppose we are given a set $\\mathcal{F}$ of functions mapping $\\mathcal{Z}$ to $\\mathbb{R}^m$, where each $\\mu \\in \\mathcal{F}$ corresponds to a vector of mean outcome functions $\\mu = (\\mu_1, \\ldots, \\mu_m)$. Then, motivated by \\cref{thm:jbound,lem:control}, we define our adversarial optimization problem as\n\n\\begin{equation}\n\\label{eq:optim}\n\\mathcal{W}^* = \\argmin_{W \\in \\mathcal{W}} \\sup_{\\mu \\in \\mathcal{F}} J(W, \\mu).\n\\end{equation}\n\nOne question the reader might ask at this point is why not solve the above optimization problem by ignoring the hidden confounders and directly balancing the conditional mean outcome functions $\\nu_t(x,t')$. The problem is that this would be impossible to do over any kind of generic flexible function space, since we have no data corresponding to terms in the form $\\nu_t(x,t')$ when $t \\neq t'$, so this is akin to an overlap problem. Conversely, if we were to ignore the conditioning on $t$ and balance against functions of the form $\\nu_t(x) = \\nu_t(x,t)$ this would be inadequate, as we couldn't hope for such a space to cover the true $\\mu$ since we don't assume ignorability given $Z$.\n\nIn light of these limitations, we can view what we are doing in optimizing \\cref{eq:optim} using an identified model $\\varphi(z;x,t)$ as implicitly balancing some controlled space of functions $\\nu_t(x,t')$ that do not have this overlap issue between different $t$ values. The following lemma makes this explicit, as it implies that that the terms $\\nu_t(x,t')$ are all mutually bounded by each other for fixed $x$ and $t$:\n\n\\begin{lemma}\n\\label{lem:gbound}\nAssuming $\\|\\mu_t\\|_\\infty\\leq b$, under \\cref{asm:overlap}, for all $x \\in \\mathcal{X}$, and $t,t',t'' \\in \\{1,\\ldots,m\\}$ we have\n\\[ |\\nu_t(x,t'')| \\leq \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} \\sqrt{8b\n\\Eb{e^{-2}_t(Z) \\mid X=x, T=t'} |\\nu_t(x,t')|}. \\]\n\\end{lemma}\n\nWe prove this in \\cref{apx:gbound}. \n\n\n\\subsection{Consistency of Adversarially Optimal Estimator}\n\nNow we analyze the consistency of our weighted estimator based on \\cref{eq:optim}. Given \\cref{lem:control}, all we need to justify to prove consistency is that $\\mu \\in \\mathcal{F}$ and that $\\inf_W \\sup_{\\mu \\in \\mathcal{F}} J(W,\\mu) \\in O_p(\\frac{1}{n})$. Define $\\mathcal{F}_t$ as the space of all functions for treatment level $t$ allowed by $\\mathcal{F}$. That is $\\mathcal{F}_t = \\{\\mu_t : \\exists (\\mu_1', \\ldots, \\mu_m') \\in \\mathcal{F} \\ \\text{with} \\ \\mu_t' = \\mu_t\\}$. We will use the following assumptions about $\\mathcal{F}$ to prove control of $J$:\n\n\\begin{assumption}[Normed]\n\\label{asm:norm}\nFor each $t \\in \\{1,\\ldots,m\\}$ there exists a norm $\\|\\cdot\\|_t$ on $\\op{span}(\\mathcal{F}_t)$, and there exists a norm $\\|\\cdot\\|$ on $\\op{span}(\\mathcal{F})$ which is defined given some $\\mathbb{R}^m$ norm as $\\|\\mu\\|~=~\\|(\\|\\mu_1\\|_1,\\ldots,\\|\\mu_m\\|_m)\\|$.\n\\end{assumption}\n\n\\begin{assumption}[Absolutely Star Shaped]\n\\label{asm:symmetry}\nFor every $\\mu \\in \\mathcal{F}$ and $\\abs{\\lambda} \\leq 1$, we have $\\lambda \\mu \\in \\mathcal{F}$.\n\\end{assumption}\n\n\\begin{assumption}[Convex Compact]\n\\label{asm:compact}\n$\\mathcal{F}$ is convex and compact \n\\end{assumption}\n\n\\begin{assumption}[Square Integrable]\n\\label{asm:ltwo}\nFor each $t \\in \\{1,\\ldots,m\\}$ the space $\\mathcal F_t$ is a subset of $\\ltwo(\\mathcal Z)$, and its norm dominates the $\\ltwo$ norm (i.e., $\\inf_{\\mu_t \\in \\mathcal F_t} \\|\\mu_t\\| \/ \\|\\mu_t\\|_{\\ltwo} > 0$).\n\\end{assumption}\n\n\\begin{assumption}[Nondegeneracy]\n\\label{asm:nondegen}\nDefine $\\mathcal B(\\gamma) = \\{\\mu \\in \\op{span}(\\mathcal F) : \\| \\mu \\| \\leq \\gamma\\}$. Then we have $\\mathcal B(\\gamma) \\subseteq \\mathcal F$ for some $\\gamma > 0$.\n\\end{assumption}\n\n\\begin{assumption}[Boundedness]\n\\label{asm:bounded}\n$\\sup_{\\mu \\in \\mathcal F} \\|\\mu\\|_{\\infty} < \\infty$.\n\\end{assumption}\n\n\\begin{definition}[Rademacher Complexity]\n$\\mathcal R_n(\\mathcal F) = \\mathbb{E}[\\sup_{f \\in \\mathcal F} \\frac{1}{n} \\sum_{i=1}^n \\epsilon_i f(Z_i)]$, where $\\epsilon_i$ are iid Rademacher random variables.\n\\end{definition}\n\n\\begin{assumption}[Complexity]\n\\label{asm:complexity}\nFor each $t \\in \\{1\\ldots,m\\}$ we have $\\mathcal R_n(\\mathcal F_t) = o(1)$.\n\\end{assumption}\n\nThese assumptions are satisfied for many commonly-used families of functions, such as RKHS spaces and families of neural networks. We shall prove this claim for RKHS spaces in \\cref{sec:algo}.\n\nIn order to justify that we can control $J$, first we will show that these assumptions allow us to reverse the order of minimization and maximization in our optimization problem. This means we can reduce problem to finding weights to control any particular $\\mu$ rather than controlling all of $\\mathcal{F}$.\n\n\n\\begin{lemma}\n\\label{lem:minimax}\nLet $B(W, \\mu) = \\frac{1}{n} \\sum_{i=1}^n \\sum_{t=1}^m f_{it} \\nu_t(X_i,T_i)$. Then under \\cref{asm:symmetry,asm:compact,asm:ltwo} for every $M>0$ we have the bound \n\\[ \\min_W \\sup_{\\mu \\in \\mathcal{F}} J(W, \\mu) \\leq \\sup_{\\mu \\in \\mathcal F} \\min_{\\|W\\|_2 \\leq M} B(W, \\mu)^2 + \\frac{\\sigma^2}{n^2}M^2. \\]\n\\end{lemma}\n\nWe prove this in \\cref{apx:minimax}.\n\nNext, we note that \\cref{lem:minimax} means that we can choose of weights given $\\mu$ to set $B(W,\\mu) = 0$, and therefore we have our desired control as long as we can justify that these weights have controlled euclidean norm. Using this strategy and optimizing for the weights of this kind with minimum euclidean norm, we are able to prove the following:\n\n\\begin{lemma}\n\\label{lem:jconvergence}\nUnder \\cref{asm:norm,asm:symmetry,asm:compact,asm:ltwo,asm:nondegen,asm:bounded,asm:complexity} we have $\\inf_W \\sup_{\\mu \\in \\mathcal{F}} J(W, \\mu) = O_p(1\/n)$.\n\\end{lemma}\n\nWe prove this in \\cref{apx:jconvergence}. This is the key lemma in proving our main consistency theorem:\n\n\\begin{theorem}\n\\label{thm:consistency}\nUnder \\cref{asm:norm,asm:symmetry,asm:compact,asm:ltwo,asm:nondegen,asm:bounded,asm:complexity} and assuming that $\\mu \\in \\mathcal F$ we have $\\hat{\\tau}^{\\pi}_{W^*} = \\tau^{\\pi} + O_p(1\/\\sqrt{n})$.\n\\end{theorem}\n\nThis theorem follows immediately from our previous results, since $\\mu \\in \\mathcal F$ and \\cref{lem:jconvergence} imply that $J(W^*, \\mu) = O_p(1\/n)$. This combined with \\cref{thm:jbound} imply that $\\mathbb{E}[ (\\hat{\\tau}^{\\pi}_{W^*} - \\tau^{\\pi})^2 \\mid X_{1:n},T_{1:n} ] = O_p(1\/n)$, which in turn combined with \\cref{lem:control} gives us our result.\n\n\n\n\n\n \n\n\n\\section{Algorithms for Optimal Kernel Balancing}\n\\label{sec:algo}\n\n\\subsection{Kernel Function Class}\n\nWe now provide an algorithm for optimal balancing when our function class consists of vectors of RKHS functions. Formally, given a kernel $K$ and corresponding RKHS norm $\\|\\cdot\\|_K$, we define the space $\\fk$ as follows:\n\n\\begin{definition}[Kernel Class]\n\\label{def:kernel-class}\n$\\fk = \\{\\mu : ||\\mu|| \\leq 1\\}$, where $||(\\mu_1,\\ldots,\\mu_m)|| = \\sqrt{\\sum_{t=1}^m ||\\mu_t||_K^2}$.\n\\end{definition}\n\n\\begin{theorem}\n\\label{thm:fk}\nAssuming $K$ is a Mercer kernel \\citep{zhou2002covering} and is bounded, $\\fk$ satisfies \\cref{asm:norm,asm:symmetry,asm:compact,asm:ltwo,asm:nondegen,asm:bounded,asm:complexity}.\n\\end{theorem}\n\nWe prove this in \\cref{apx:fk}.\n\nWe can remark that the commonly used Gaussian kernel is both Mercer and bounded, so it satisfies the conditions of \\cref{thm:fk}. Given this, and assuming that $\\fk$ covers the real mean outcome function $\\mu$, we can apply \\cref{thm:consistency} to see that solving \\cref{eq:optim} using $\\fk$ gives consistent evaluation.\n\nWe can note that the $\\fk$ having maximum norm 1 is without loss of generality, because if we wanted the maximum norm to instead be $\\gamma$ we could replace the $\\Sigma$ matrix by $\\Gamma = \\frac{1}{\\gamma} \\Sigma$ in our objective function, resulting in an equivalent re-scaled optimization problem. To make this explicit, we will replace the $\\Sigma$ matrix in the objective with $\\Gamma$ in this section, where it is assumed that $\\Gamma$ is freely chosen as a hyperparameter.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Kernel Balancing Algorithm}\n\\label{sec:qpalgo}\n\nIn order to optimize \\cref{eq:optim} over a class of kernel functions as defined by \\cref{def:kernel-class}, we can observe that the definition of $J(W, \\mu)$ looks very similar to the adversarial objective of \\citet{kallus2018balanced}, except that we have $\\nu_t(X_i,T_i)$ terms instead of $\\mu_t(X_i)$ terms. This motivates the idea that, given our identified posterior model $\\varphi(z;x,t)$, we may be able to employ a similar quadratic programming (QP)-based approach. The following theorem makes this explicit, by defining a QP objective for $W$ that we can approximate by sampling from $\\varphi$:\n\n\\begin{theorem}\n\\label{thm:continuous-algorithm}\nDefine $Q_{ij} = \\mathbb{E}[K(Z_i, Z_j')]$, $G_{ij} = \\frac{1}{n^2} (Q_{ij} \\delta_{T_i T_j} + \\Gamma_{ij})$, and $a_i = \\frac{2}{n^2} \\sum_{j=1}^n Q_{ij} \\pi_{T_j}(X_i)$, where for each i $Z_i$ and $Z_i'$ are iid shadow variables. Then for some $c$ that is constant in $W$ we have the identity\n\\[ \\sup_{\\mu \\in \\fk} J(W, \\mu) = W^T G W - a^T W + c.\\]\n\\end{theorem}\n\nWe prove this in \\cref{apx:continuous-algorithm}.\n\nGiven this our balancing algorithm is natural and straightforward, and is summarized by \\cref{algo:continuous}. Note that we provide an optional weight space constraint $\\mathcal{W}$ in this algorithm, since standard weighted estimator approaches for policy evaluation regularize by forcing constraints such as $W \\in n \\Delta^n$. Under this kind of constraint our unconstrained QP becomes a LCQP. However that our theory does not support this constraint, and that we find it hurts performance in practice, especially when $\\Gamma$ is large, so we do not use this constraint in our experiments.\n\n\n\n\\begin{algorithm}\n  \\caption{Optimal Kernel Balancing}\\label{algo:continuous}\n  \\begin{algorithmic}[1]\n  \\Require \\parbox[t]{0.9\\textwidth}{\n     Data $(X_{1:n}, T_{1:n})$, policy $\\pi$, kernel function $K$, posterior density $\\varphi$, regularization matrix $\\Gamma$, number samples $B$, optional weight space $\\mathcal{W}$ (defaults to $\\mathbb{R}^n$ if not provided)\n  }\n  \\Ensure Optimal balancing weights $W_{1:n}$\n      \\For {$i \\in \\{1, \\ldots, n\\}$}\n        \\State \\parbox[t]{0.9\\textwidth}{\\textbf{Sample Data.} Draw $B$ data points $Z_i^b$ from the posterior $\\varphi(\\cdot \\ ;X_i,T_i)$}\n      \\EndFor\n      \\State \\textbf{Estimate Q.} Calculate $Q_{ij} = \\frac{1}{B^2} \\sum_{b=1}^B \\sum_{c=1}^B K(Z_i^b, Z_i^c)$\n      \\State \\textbf{Calculate QP Inputs.} Calculate $G_{ij} = Q_{ij} \\delta_{T_i T_j} + \\Gamma_{ij}$, and $a_i = 2 \\sum_{j=1}^n Q_{ij} \\pi_{T_j}(X_i)$,\n      \\State \\textbf{Solve Quadratic Program.} Calculate $W = \\argmin_{W \\in \\mathcal{W}} W^T G W - a^T W$\n  \\end{algorithmic}\n\\end{algorithm}\n\n\n\\section{Problem}\n\n\\subsection{Setting and Assumptions}\n\\label{sec:setting}\n\nWe consider a contextual decision making setting with $m$ possible treatments (aka actions or interventions).\nEach unit is associated with \na set of potential outcomes $Y(1),\\dots,Y(m)\\in\\Rl$ corresponding to the reward\/loss for each treatment, an observed treatment $T\\in\\{1,\\dots,m\\}$, an observed outcome $Y=Y(T)$, true but latent confounders $Z\\in\\mathcal Z \\subseteq \\mathbb{R}^p$, and observed covariates $X\\in\\mathcal X \\subseteq \\mathbb{R}^q$.\nOur data consists of iid observations $X_{i},T_{i},Y_{i}$ of $X,T,Y$.\nBoth the latent confounders and potential outcomes of unassigned treatments are unobserved.\nNote that $Y_i=Y_i(T_i)$ encapsulates the assumptions of {consistency} between observed and potential outcomes and non-interference between units.\n\nA \\emph{policy} is a rule for assigning the probability of each treatment option given the observed covariates $X$. Given a policy $\\pi$, we use the notation $\\pi_t(x)$ to indicate the probability of assigning treatment $t$ when observed covariates are $x$. We define the \\emph{value} of a policy, $\\tau^{\\pi}$, as the expected outcome that would be obtained from following the policy in the population. Formally:%\n\\begin{definition}[Policy Value]\n$\\tau^{\\pi} = \\mathbb{E}[\\sum_{t=1}^m \\pi_t(X) Y(t)]$.\n\n\\end{definition}\n\\begin{wrapfigure}{r}{0.225\\textwidth}\n\\vspace{-2.75\\baselineskip}\n\\begin{minipage}{0.225\\textwidth}\\small\\centering%\n\\begin{tikzpicture}[%\n    >=latex',\n    circ\/.style={draw, shape=circle, node distance=1cm, line width=1pt}]\n    \\node[circ] (X) at (-0.5,0.8) {$X$};\n    \\node[circ][draw,dashed] (U) at (1,0.8) {$Z$};\n    \\node[circ] (Y) at (2,-.3) {$Y$};\n    \\node[circ] (T) at (0,-.3) {$T$};\n    \\draw[line width=1pt,->] (T) -- (Y);\n    \\draw[line width=1pt,->] (U) -- (T);\n    \\draw[line width=1pt,->] (U) -- (Y);\n    \\draw[line width=1pt,->] (U) -- (X);\n    \\draw[line width=1pt,->] (X) -- (T);\n\\end{tikzpicture}\n\\end{minipage}%\n\\captionof{figure}{DAG representation of problem.}\\label{fig:proxies}\n\\vspace{-\\baselineskip}\n\\end{wrapfigure}\n\nWe encapsulate the assumption that $Z$ are sufficient for unconfoundedness and that $X$ is a proxy for $Z$ in the following assumption. \\Cref{fig:proxies} provides a representation of this setting using a causal DAG \\citep{pearl2000causality}.\nNote importantly that we do \\emph{not} assume ignorability given $X$. \n\\begin{assumption}[$Z$ are true confounders]\n\\label{asm:ignorability}\nFor every $t\\in\\{1,\\dots,m\\}$, $Y(t)$ is independant of $(X,T)$, given $Z$.\n\\end{assumption}\n\nWe next define the average mean outcome given $Z$ and its conditional expectations given observables:\n\\begin{align*}\n\\mu_t(z) &= \\mathbb{E}[Y(t) \\mid Z=z],\\\\\n\\nu_t(x,t') &= \\mathbb{E}[\\mu_t(Z) \\mid X=x, T=t']= \\mathbb{E}[Y(t) \\mid X=x, T=t'], \\\\\n\\rho_t(x) &= \\mathbb{E}[\\mu_t(Z) \\mid X=x]= \\mathbb{E}[Y(t) \\mid X=x].\n\\end{align*}\nWe further define the propensity function and its conditional expectation given observables:\n\\begin{align*}\ne_t(z) &= \\Prb{T=t\\mid Z=z},\\\\\n\\eta_t(x) &= P(T=t \\mid X=x)=\\Eb{e_t(Z)\\mid X=x}.\n\\end{align*}\n\nFinally, we denote by $\\varphi(z; x,t)$ the conditional density of $Z$ given $X=x,T=t$. This density represents the latent variable model underlying the observables. For example, this can be a Gaussian mixture model, a PCA-type model as in \\citep{kallus2018causal}, or a deep variational autoencoder as in \\citep{louizos2017causal}. Because we focus on how one might use such a latent model rather than the estimation of this model, we just assume we have some oracle for calculating its values. (Note that for fair comparison, in experiments in \\cref{sec:experiments}, we similarly let the outcome regression methods use this oracle.)\n\n\n\n\n\n\n\n\n\nWe further make the following regularity assumptions:\n\\begin{assumption}[Weak Overlap]\n\\label{asm:overlap}\n$\\Eb{e^{-2}_t(Z)}<\\infty$\n\\end{assumption}\n\n\\begin{remark}\n\\label{rmk:overlap}\nGiven \\cref{asm:overlap} it trivially follows that for every $t \\in \\{1,\\ldots,m\\}$, $x \\in \\mathcal X$, $z \\in \\mathcal Z$ that $e_t(z) > 0$ and $\\eta_t(x) > 0$.\n\\end{remark}\n\n\\begin{assumption}[Bounded Variance]\n\\label{asm:variance}\nThe conditional variance of our potential outcomes given $X,T$ is bounded: $\\mathbb{V}[Y(t) \\mid X, T]\\leq\\sigma^2$.\n\\end{assumption}\n\n\\subsection{The Policy Evaluation Task}\n\nThe problem we consider is to estimate the policy value $\\tau^\\pi$ given a policy $\\pi$ and data $X_{1:n},T_{1:n},Y_{1:n}$. One standard approach to this is the \\emph{direct method} \\citep{qian2011performance}, which given an estimate $\\hat \\rho_t$ of $\\rho_t$ predicts the policy value as\n\\begin{equation}\n\\label{eq:direct}\n\\hat{\\tau}^\\pi_{\\hat\\rho} = \\frac{1}{n} \\sum_{i=1}^n \\sum_{t=1}^m \\pi_t(X_i) \\hat{\\rho}_t(X_i).\n\\end{equation}\n\nHowever this method is known to be biased and doesn't generalize well \\citep{beygelzimer2009offset}. Furthermore given \\cref{asm:ignorability} $\\hat \\rho$ is not straightforward to estimate, since the mean value of $Y$ observed in our logged data given $X=x$ and $T=t$ is  $\\nu_t(x,t)$ not $\\rho_t(x)$, so fitting $\\hat \\rho$ would require controlling for the effects of the unobserved $Z$.\n\nAn alternative to this is to come up with weights $W_{1:n}=f_W(X_{1:n},T_{1:n})$ according to some function $f_W$ of the observed covariates and treatments, in order to re-weight the outcomes to look more like those that would be observed under $\\pi$. Using these weights we can define the \\emph{weighted} estimator\n\\begin{equation}\\label{eq:weighted}\n\\hat{\\tau}^{\\pi}_W = \\frac{1}{n} \\sum_{i=1}^n W_i Y_i.\n\\end{equation}\n\nThis weighted estimator has the advantage that it does not require modeling the outcome distributions. Furthermore we could combine the weights $W_{1:n}$ with an outcome model $\\hat \\rho_t$ to calculate the \\emph{doubly robust} estimator \\citep{dudik2011doubly}, which is defined as\n\\begin{equation}\n\\label{eq:dr}\n\\hat{\\tau}^{\\pi}_{W,\\hat{\\rho}} = \\frac{1}{n} \\sum_{i=1}^n \\sum_{t=1}^m \\pi_t(X_i) \\hat{\\rho}_t(X_i) + \\frac{1}{n} \\sum_{i=1}^n W_i (Y_i - \\hat{\\rho}_{T_i}(X_i)).\n\\end{equation}\nThe doubly robust estimator is known to be consistent when either the weighted or direct estimator is consistent and can attain local efficiency \\citep{chernozhukov2016double,robins2000robust,robins1994estimation,scharfstein1999adjusting}. \n\nVarious approaches exist for coming up with weights for either the weighted or doubly robust estimators, which we discuss below. However none of these methods are applicable given \\cref{asm:ignorability}, and so we develop a theory for weighting using proxy variables in \\cref{sec:method}.\n\n\n\\subsection{Related Work}\n\n\nOne of the most standard approaches for policy evaluation is using the weighted or doubly robust estimator defined in \\cref{eq:weighted,eq:dr}, using inverse propensity score (IPS) weights. These are given by $W_i = \\pi_{T_i}(X_i) \/ e_{T_i}(Z_i)$ \\citep{bottou2013counterfactual}, where $e_t$ are known or estimated logging probabilities. \nSince these weights can be extreme, both normalization \\citep{austin2015moving,lunceford2004stratification,swaminathan2015self} and clipping \\citep{elliott2008model,ionides2008truncated,swaminathan2015counterfactual} are often employed. In addition some other approaches include recursive partitioning \\citep{kallus2016recursive}.\nNone of these methods are applicable to our setting however, since we do not know the true confounders $Z_{1:n}$.\n\n\n\nAn alternative to approaches based on fixed formulae for computing the importance weights is to compute weights that optimize an imbalance objective function \\citep{athey2018approximate,kallus2017framework,kallus2018optimal}. For policy evaluation, \\citet{kallus2018balanced} propose to choose weights that adversarially minimize the conditional mean squared error of policy evaluation in the worst case of possible mean outcome functions in some reproducing kernel Hilbert space (RKHS) class, by solving a linearly constraint quadratic program (LCQP). Our work follows a very similar style to this, however instead of using the true confounders we only assume access to proxies, and we prove our theory for more general families of functions. \n\nFinally there has been a long history of work in causal inference using proxies for true confounders \\citep{wickens1972note,frost1979proxy}. As in our problem setup, much of this work is based on the model of using an identified latent variable model for the proxies \\citep{wooldridge2009estimating,pearl2012measurement,kuroki2014measurement,miao2016identifying,edwards2015all}. Some recent work on this problem involves using techniques such as matrix completion \\citep{kallus2018balanced} or variational autoencoders \\citep{louizos2017causal} to infer confounders from the proxies. In addition, there is a variety of work that studies sufficient conditions for the identifiability of latent confounder models \\citep{cai2008identifying,pearl2012measurement,miao2016identifying}. Our work is complementary to this line of research in that we assume access to an identified latent confounder model, but do not study how to identify such models. Furthermore our work is novel in combining proxy variable models with optimal balancing and applying it to finding importance weights for policy evaluation.\n\n\n\n\n\n\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\\subsection{Experimental Setup}\n\\label{sec:exp-setup}\n\nWe now present a brief set of experiments to explore our methodology. The aim of these experiments is to be a proof of concept of our theory. We seek to show that given an identified posterior model $\\varphi$ policy as discussed in \\cref{sec:setting}, evaluation using the weights defined by \\cref{eq:optim} can give unbiased policy evaluation even in the face of sufficiently strong confounding where standard benchmark approaches that rely on ignorability given $X$ fail. We experiment with the following generalized linear model-style scenario:\n\n\\begin{center}\n\\begin{tabular}{lll}\n$Z \\sim \\mathcal{N}(0,1)$ \\qquad & \\qquad $X \\sim \\mathcal{N}(\\alpha^T Z + \\alpha_0, \\sigma^2_X)$ \\qquad & \\qquad $P_T = \\beta^T Z + \\beta_0$ \\\\\n$T \\sim \\op{softmax}(P_T)$ \\qquad & \\qquad $W(t) \\sim \\mathcal{N}(\\zeta(t)^T Z + \\zeta_0(t), \\sigma^2_Y)$ \\qquad & \\qquad $Y(t) = g(W(t))$ \n\\end{tabular}\n\\end{center}\n\nIn our experiments $Z$ is $1$-dimensional, $X$ is $10$-dimensional, and we have two possible treatment levels ($m=2$). We experiment with a parametric policy $\\pi$ and multiple link functions $g$ as follows:\n$$\\pi_t(X) = \\frac{\\exp(\\psi_t^T X)}{\\exp(\\psi_1^T X) + \\exp(\\psi_2^T X)}$$\n$$\n\\text{\\textbf{step}: } g(w) = 3 \\mathbbm{1}_{\\{w \\geq 0\\}} - 6 \\qquad\n\\text{\\textbf{exp}: } g(w) = \\exp(w) \\qquad\n\\text{\\textbf{cubic}: } g(w) = w^3 \\qquad\n\\text{\\textbf{linear}: } g(w) = w \n$$\n\nWe experiment with the following methods in this evaluation:\n\\begin{enumerate}\n\\item \\textbf{OptZ} Our method, using $\\Gamma = \\gamma \\op{Identity}(n)$ for $\\gamma \\in \\{0.001, 0.2, 1.0, 5.0\\}$.\n\\item \\textbf{IPS} IPS weights based on $X$ using estimated $\\hat\\eta_t$.\n\\item \\textbf{OptX} The optimal weighting method of \\citet{kallus2018balanced} with same values of $\\Gamma$ as our method.\n\\item \\textbf{DirX} Direct method by fitting $\\hat \\rho_t(x)$ incorrectly assuming ignorability given $X$.\n\\item \\textbf{DirZ} Direct method by first fitting $\\hat\\mu_t$ using posterior samples from $\\varphi$, then using the estimate $\\hat \\rho_t(x) = (1\/D) \\sum_{i=1}^D \\hat\\mu_t(z_i')$, where $z_i'$ are sampled from $\\varphi(\\cdot;x,t)$.\n\\item \\textbf{D:W} Doubly robust estimation using direct estimator $\\textbf{D}$ and weighted estimator $\\textbf{W}$.\n\\end{enumerate}\n\nFinally we detail all choices for scenario parameters in \\cref{apx:exp-scenario}, and provide implementation details of our methods in \\cref{apx:exp-benchmark}.\\footnote{Code available online at \\url{https:\/\/github.com\/CausalML\/LatentConfounderBalancing}.}\n\n\n\n\\subsection{Results}\nWe display results for our experiments using the \\textbf{step} link function in \\cref{tab:results-ours,tab:results-benchmarks}. For each of $n \\in \\{200, 500, 1000, 2000\\}$ we estimate the RMSE of policy evaluation using each method, as well as doubly robust evaluation using our best performing weights, by averaging over 64 runs. In addition, in \\cref{tab:results-ours-bias,tab:results-benchmarks-bias} we display the estimated bias from the evaluations. It is clear that the naive methods that assume ignorability given $X$ all hit a performance ceiling, where bias converges to some non-zero value. In particular for $\\text{IPS}$ we separately ran it on up to one million data points and found that the bias converged to $0.418\\pm0.001$. One the other hand, for our method it appears like we have consistency. This is particularly evident when we look at \\cref{tab:results-ours-bias}, as bias seems to be approximately converging to zero with vanishing variance. We can also observe that doubly robust estimation using either direct method does not appear to improve performance.\n\nIt is noteworthy that the \\textbf{DirZ} benchmark method fails horribly, despite being a correctly specified regression estimate. From our experience we observed that it is difficult to train the $\\mu_t$ functions accurately if there is a high amount of overlap in the $\\varphi(\\cdot;x,t)$ posteriors for fixed $t$. Therefore we postulate that in highly confounded settings this benchmark inherently difficult to train using a finite number of samples from $\\varphi(\\cdot;x,t)$, and the result seems to collapse to degenerate solutions.\n\nFinally we note that we observed similar trends to this using our other link functions, and other doubly robust estimators. We present more extensive tables of results in \\cref{apx:exp-results}. In addition we present some results there on the negative impact on our method's performance using the constraint $W \\in n\\Delta^n$, as mentioned in \\cref{sec:qpalgo}.\n\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$ & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.001}$\\\\\n\\hline\n200 & $.39\\pm.07$ & $.24\\pm.02$ & $.36\\pm.02$ & $.81\\pm.02$ & $.57\\pm.06$ & $.41\\pm.07$\\\\\n500 & $.19\\pm.02$ & $.18\\pm.02$ & $.23\\pm.02$ & $.49\\pm.02$ & $.55\\pm.03$ & $.20\\pm.02$\\\\\n1000 & $.11\\pm.01$ & $.11\\pm.01$ & $.13\\pm.01$ & $.27\\pm.01$ & $.49\\pm.02$ & $.11\\pm.01$\\\\\n2000 & $.08\\pm.01$ & $.08\\pm.01$ & $.09\\pm.01$ & $.17\\pm.01$ & $.48\\pm.01$ & $.08\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for for policy evaluation using our weights.}\n\\label{tab:results-ours}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccccccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.47\\pm.03$ & $2.0\\pm.03$ & $2.1\\pm.03$ & $2.3\\pm.02$ & $2.5\\pm.02$ & $.52\\pm.02$ & $2.6\\pm.02$\\\\\n500 & $.48\\pm.03$ & $2.0\\pm.02$ & $2.1\\pm.02$ & $2.3\\pm.02$ & $2.6\\pm.02$ & $.48\\pm.02$ & $2.6\\pm.01$\\\\\n1000 & $.39\\pm.02$ & $2.0\\pm.01$ & $2.1\\pm.01$ & $2.3\\pm.01$ & $2.5\\pm.01$ & $.48\\pm.02$ & $2.6\\pm.01$\\\\\n2000 & $.40\\pm.01$ & $2.0\\pm.01$ & $2.1\\pm.01$ & $2.3\\pm.01$ & $2.5\\pm.01$ & $.45\\pm.02$ & $2.6\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for benchmark methods.}\n\\label{tab:results-benchmarks}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$ & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.001}$\\\\\n\\hline\n200 & $.03\\pm.39$ & $.11\\pm.21$ & $.29\\pm.21$ & $.78\\pm.18$ & $.43\\pm.38$ & $.05\\pm.40$\\\\\n500 & $.09\\pm.17$ & $.10\\pm.15$ & $.17\\pm.16$ & $.47\\pm.15$ & $.51\\pm.19$ & $.10\\pm.18$\\\\\n1000 & $.02\\pm.11$ & $.05\\pm.09$ & $.08\\pm.09$ & $.25\\pm.09$ & $.47\\pm.13$ & $.04\\pm.11$\\\\\n2000 & $.03\\pm.07$ & $.05\\pm.06$ & $.07\\pm.07$ & $.16\\pm.07$ & $.47\\pm.09$ & $.03\\pm.07$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for policy evaluation using our weights.}\n\\label{tab:results-ours-bias}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccccccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.40\\pm.25$ & $1.9\\pm.21$ & $2.1\\pm.20$ & $2.3\\pm.19$ & $2.5\\pm.18$ & $.49\\pm.18$ & $2.6\\pm.14$\\\\\n500 & $.43\\pm.21$ & $2.0\\pm.16$ & $2.1\\pm.15$ & $2.3\\pm.14$ & $2.6\\pm.13$ & $.45\\pm.16$ & $2.6\\pm.12$\\\\\n1000 & $.37\\pm.12$ & $2.0\\pm.10$ & $2.1\\pm.09$ & $2.3\\pm.09$ & $2.5\\pm.08$ & $.46\\pm.15$ & $2.6\\pm.11$\\\\\n2000 & $.39\\pm.10$ & $2.0\\pm.08$ & $2.1\\pm.07$ & $2.3\\pm.07$ & $2.5\\pm.07$ & $.42\\pm.17$ & $2.6\\pm.11$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for benchmark methods.}\n\\label{tab:results-benchmarks-bias}\n\\end{table}\n\n\n\n\n\\section{Omitted Proofs}\n\n\n\\subsection{Proof of \\cref{thm:ips}}\n\\label{apx:ips}\n\n\nFirst we note that $W(X,T)Y - \\sum_t \\pi_t(X) Y(t)] = \\sum_t (W(X,t) \\delta_{Tt} - \\pi_t(X)) Y(t)$. Then analyzing each summand separately, we can obtain:\n\\begin{align}\n\\mathbb{E}&[W(X,t) \\delta_{Tt} Y(t) - \\pi_t(X) Y(t)] \\nonumber \\\\\n&= \\mathbb{E}[ \\mathbb{E}[ W(X,t) \\delta_{Tt} Y(t) - \\pi_t(X) Y(t) \\mid X] ] \\nonumber \\\\\n&= \\mathbb{E}[ W(X,t) \\mathbb{E}[\\delta_{Tt} Y(t) \\mid X] - \\pi_t(X) \\mathbb{E}[Y(t) \\mid X] ] \\nonumber \\\\\n&= \\mathbb{E}[ W(X,t) \\mathbb{E}[\\mathbb{E}[\\delta_{Tt} Y(t) \\mid X,T] \\mid X] - \\pi_t(X) \\mathbb{E}[\\mathbb{E}[Y(t) \\mid X,Z] \\mid X] ] \\nonumber \\\\\n&= \\mathbb{E}[ W(X,t) \\mathbb{E}[\\delta_{Tt} \\mathbb{E}[Y(t) \\mid X,T=t] \\mid X] - \\pi_t(X) \\mathbb{E}[\\mu_t(Z) \\mid X] ] \\nonumber \\\\ \n&= \\mathbb{E}[ W(X,t) \\mathbb{E}[\\delta_{Tt} \\mid X] \\mathbb{E}[\\mu_t(Z) \\mid X,T=t] - \\pi_t(X) \\mathbb{E}[\\mu_t(Z) \\mid X] ] \\nonumber \\\\ \n&= \\mathbb{E}[ W(X,t) \\eta_t(X) \\mathbb{E}[\\mu_t(Z) \\mid X,T=t] - \\pi_t(X) \\mathbb{E}[\\mu_t(Z) \\mid X] ] \\nonumber \\\\ \n&= \\mathbb{E}\\left[ W(X,t) \\eta_t(X) \\nu_t(X,t) - \\pi_t(X) \\sum_{t'} \\eta_{t'}(X) \\nu_t(X,t') \\right] \\nonumber \n\\end{align}\n\nTherefore the solution to the equation $\\mathbb{E}[W(X,t) \\delta_{Tt} Y(t) - \\pi_t(X) Y(t)] = 0$ is given by:\n\\begin{equation*}\nW(X,t) \\eta_t(X) \\nu_t(X,t) - \\pi_t(X) \\sum_{t'} \\eta_{t'}(X) \\nu_t(X,t') = \\Omega_t(X)\n\\end{equation*}\nwhere $\\Omega_t(X)$ is any arbitrary function of $X$ with mean zero. Solving this for $W$ gives\n\\begin{equation*}\nW(X,t) = \\frac{\\pi_t(X) \\sum_{t'} \\eta_{t'}(X) \\nu_t(X,t') + \\Omega_t(X)}{\\eta_t(X) \\nu_t(X,t)},\n\\end{equation*}\nand finally replacing $X$ with $x$ gives the required solution.\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{thm:jbound}}\n\\label{apx:jbound}\nDefine \n$$\\tau^{\\pi}_{SAPE} = \\frac{1}{n} \\sum_i \\sum_t \\pi_t(X_i) \\mu_t(Z_i).\n$$\nUsing $(x+y)^2\\leq2x^2+2y^2$, we have\n\\begin{align*}\n\\mathbb{E}[ (\\hat{\\tau}^{\\pi}_W - \\tau^{\\pi})^2 \\mid X_{1:n},T_{1:n} ]\n\\leq 2\\mathbb{E}[ (\\hat{\\tau}^{\\pi}_W - \\tau^{\\pi}_{SAPE})^2 \\mid X_{1:n},T_{1:n} ]+2\\mathbb{E}[ (\\tau^{\\pi}_{SAPE} - \\tau^{\\pi})^2 \\mid X_{1:n},T_{1:n} ].\n\\end{align*}\nNoting that $\\tau^{\\pi} = \\mathbb{E}[\\tau^{\\pi}_{SAPE}]$, \\Cref{asm:variance} implies that \n$\\mathbb{E}[ (\\tau^{\\pi}_{SAPE} - \\tau^{\\pi})^2]=O(1\/n)$.\nMarkov's inequality yields $\\mathbb{E}[ (\\tau^{\\pi}_{SAPE} - \\tau^{\\pi})^2 \\mid X_{1:n},T_{1:n} ]=O_p(1\/n)$.\n\nLet $CMSE(W,\\mu)=\\mathbb{E}[ (\\hat{\\tau}^{\\pi}_W - \\tau^{\\pi}_{SAPE})^2 \\mid X_{1:n},T_{1:n} ]$. We proceed to bound $CMSE$.\nBy iterating expectations we can obtain:\n\\begin{align}\nCMSE(W, \\mu) &= \\mathbb{E}[ ( \\hat{\\tau}^{\\pi}_W - \\tau^{\\pi}_{SAPE} )^2 \\mid X, T ] \\nonumber \\\\\n&= \\mathbb{E}[ \\mathbb{E}[ ( \\hat{\\tau}^{\\pi}_W - \\tau^{\\pi}_{SAPE} )^2 \\mid X, T, Z \\mid X, T ] \\nonumber \\\\\n&= \\mathbb{E}[ \\mathbb{E}[ \\hat{\\tau}^{\\pi}_W - \\tau^{\\pi}_{SAPE} \\mid X, T, Z]^2  \\mid X, T ] + \\mathbb{E}[ \\mathbb{V}[ \\hat{\\tau}^{\\pi}_W - \\tau^{\\pi}_{SAPE} \\mid X, T, Z] \\mid X, T] \\nonumber \\\\\n&= \\mathbb{E}[ ( \\frac{1}{n} \\sum_{i,t} f_{it} \\mu_t(Z_i) )^2 \\mid X, T ] + \\mathbb{E}[ \\mathbb{V}[ \\frac{1}{n} \\sum_i W_i Y(T_i)  \\mid X, T, Z] \\mid X, T] \\nonumber \\\\\n&\\leq \\mathbb{E}[ ( \\frac{1}{n} \\sum_{i,t} f_{it} \\mu_t(Z_i) )^2 \\mid X, T ] + \\frac{\\sigma^2}{n^2} \\|W\\|_2^2 \\nonumber\n\\end{align}\nwhere $\\sigma$ is the bound defined in \\cref{asm:variance}.\n\nNext observe that for any (possibly correlated) random variables $A_1,\\ldots,A_n$ and numbers $p_1,\\ldots,p_n \\in \\R n$ such that $\\sum_ip_i=1$, we have $\\mathbb{V}[\\sum_i p_i A_i] \\leq \\max_i \\mathbb{V}[A_i]$. Given this, we can simplify the first term above further, as follows:\n\\begin{align}\n\\mathbb{E}[ ( \\frac{1}{n} \\sum_{i,t} f_{it} \\mu_t(Z_i) )^2 \\mid X, T ] &= (\\frac{1}{n} \\sum_{i,t} f_{it} \\mathbb{E}[\\mu_t(Z_i) \\mid X_i, T_i])^2 + \\mathbb{V}[\\frac{1}{n} \\sum_{i,t} f_{it} \\mu_t(Z_i) \\mid X, T] \\nonumber \\\\\n&= (\\frac{1}{n} \\sum_{i,t} f_{it} \\nu_t(X_i,T_i))^2 + \\frac{1}{n^2} \\sum_i \\mathbb{V}[ \\sum_t f_{it} \\mu_t(Z_i) \\mid X, T] \\nonumber \\\\\n&\\leq (\\frac{1}{n} \\sum_{i,t} f_{it} \\nu_t(X_i,T_i))^2 + \\frac{1}{n^2} \\sum_i \\max_t f_{it}^2 \\mathbb{V}[\\mu_t(Z_i) \\mid X, T] \\nonumber \\\\\n&\\leq (\\frac{1}{n} \\sum_{i,t} f_{it} \\nu_t(X_i,T_i))^2 + \\frac{2\\sigma^2}{n^2} \\sum_i \\max_t W_i^2 \\delta_{T_i t} + \\pi_t(X_i)^2 \\nonumber \\\\\n&\\leq (\\frac{1}{n} \\sum_{i,t} f_{it} \\nu_t(X_i,T_i))^2 + \\frac{2\\sigma^2}{n^2} \\sum_i (W_i^2 + 1) \\nonumber \\\\\n&= (\\frac{1}{n} \\sum_{i,t} f_{it} \\nu_t(X_i,T_i))^2 + \\frac{2\\sigma^2}{n^2} \\|W\\|_2^2 + \\frac{2 \\sigma^2}{n} \\nonumber\n\n\\end{align}\nwhere for the second inequality we used the fact that $(x+y)^2 \\leq 2x^2 + 2y^2$. This gives us $CMSE(W, \\mu) \\leq J(W, \\mu) + O_p(1\/n)$, and combining this with the above gives $\\mathbb{E}[(\\hat{\\tau}^{\\pi}_W~-~\\tau^{\\pi})^2~\\mid~X_{1:N},~T_{1:N}] \\leq 2J(W, \\mu) + O_p(1\/n)$ as required.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{lem:gbound}}\n\\label{apx:gbound}\n\nFirst, we will use the notation $f(z;x,t)$ for the conditional measure of $Z$ given $X=x$ and $T=t$, and observe that according to Bayes rule we have:\n\\begin{equation}\n\\frac{f(z;x,t'')}{f(z;x,t')} = \\frac{e_{t''}(z)}{e_{t'}(z)} \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} \\nonumber\n\\end{equation}\n\nDefine $\\mathbb{E}_{xt}$ and $\\mathbb{P}_{xt}$ as shorthand for expectation and probability given $X=x,T=t$ respectively. Then given the above, for any $M > 0$ we can bound\n\\begin{align*}\n\\nu_t(x,t'') &= \\frac{\\eta_{t''}(x)}{\\eta_{t'}(x)} \\mathbb{E}[\\mu_t(Z) \\mid X=x, T=t''] \\\\\n&= \\int_{\\mathcal{Z}} f(z;x,t'') \\mu_t(z) dz \\\\\n&= \\frac{\\eta_{t''}(x)}{\\eta_{t'}(x)} \\int_{\\mathcal{Z}} \\frac{e_{t''}(z)}{e_{t'}(z)} \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} f(z;x,t') \\mu_t(z) dz \\\\\n&= \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} \\int_{\\mathcal{Z}} \\frac{e_{t''}(z)}{e_{t'}(z)} f(z;x,t') \\mu_t(z)  dz \\\\\n&\\leq \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} \\left( M \\mathbb{E}_{xt'}[\\indicator{\\frac{e_{t''}(z)}{e_{t'}(z)} \\leq M} \\mu_t(Z)] + \\mathbb{E}_{xt'}[\\indicator{\\frac{e_{t''}(z)}{e_{t'}(z)} > M} \\frac{e_{t''}(z)}{e_{t'}(z)} \\mu_t(Z)] \\right) \\\\\n\\end{align*}\n\nNow we can use the fact that $\\mu_t$ is $b$-bounded to bound the first term by\n\\begin{align*}\nM \\mathbb{E}_{xt'}[\\indicator{\\frac{e_{t''}(z)}{e_{t'}(z)} \\leq M} \\mu_t(Z)] &= M \\nu_t(x,t') - M\\mathbb{E}_{xt'}[\\indicator{\\frac{e_{t''}(z)}{e_{t'}(z)} > M} \\mu_t(Z)] \\\\\n&\\leq M \\nu_t(x,t') + Mb\\mathbb{P}_{xt'}[\\frac{e_{t''}(z)}{e_{t'}(z)} > M],\n\\end{align*}\nand in addition applying Cauchy Schwartz we can bound the second term by\n\\begin{align*}\n\\mathbb{E}_{xt'}[\\indicator{\\frac{e_{t''}(z)}{e_{t'}(z)} > M} \\frac{e_{t''}(z)}{e_{t'}(z)} \\mu_t(Z)]  &\\leq \\sqrt{\\mathbb{E}_{xt'}[\\indicator{\\frac{e_{t''}(z)}{e_{t'}(z)} > M} \\mu_t(Z)^2] \\mathbb{E}_{xt'}[\\left(\\frac{e_{t''}(z)}{e_{t'}(z)}\\right)^2]} \\\\\n&\\leq \\sqrt{b^2 \\mathbb{P}_{xt'}[\\frac{e_{t''}(z)}{e_{t'}(z)} > M] \\mathbb{E}_{xt'}[\\left(\\frac{e_{t''}(z)}{e_{t'}(z)}\\right)^2]} \\\\\n&\\leq b \\sqrt{\\mathbb{P}_{xt'}[\\frac{1}{e_{t'}(z)} > M] \\mathbb{E}_{xt'}[\\left(\\frac{1}{e_{t'}(z)}\\right)^2]}.\n\\end{align*}\n\nNow define $g(x,t) = \\mathbb{E}_{xt}[\\left(\\frac{1}{e_t(z)}\\right)^2]$. By \\cref{asm:overlap} we know $g(x,t)$ is finite for every $x$ and $t$. Also by Markov's inequality we know that $\\mathbb{P}_{xt}[\\frac{1}{e_t(z)} > M] \\leq \\frac{g(x,t)}{M^2}$. Therefore putting all of the above together we can obtain\n\\begin{align}\n\\nu_t(x,t'') &\\leq \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} \\left(M \\nu_t(x,t') + \\frac{2bg(x,t')}{M} \\right) \\nonumber \\\\\n&\\leq \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} \\left(M |\\nu_t(x,t')| + \\frac{2bg(x,t')}{M} \\right) \\nonumber \n\\end{align}\nThis inequality is valid every $M$, so we can pick $M$ to make it as tight as possible. Choosing $M = \\sqrt{\\frac{2bg(x,t')}{\\nu_t(x,t')}}$ gives us:\n\\begin{equation}\n\\nu_t(x,t'') \\leq \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} \\sqrt{8bg(x,t')|\\nu_t(x,t')|} \\nonumber \n\\end{equation}\n\nFinally note that, since by symmetry $\\mathbb{E}[\\mu(Z) \\mid X,T] = -\\mathbb{E}[-\\mu(Z) \\mid X,T]$, we can strengthen this inequality to the following\n\\begin{equation*}\n|\\nu_t(x,t'')| \\leq \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} \\sqrt{8bg(x,t')|\\nu_t(x,t')|},\n\\end{equation*}\nand noting that $g(x,t') = \\mathbb{E}[e^{-2}_t(Z) \\mid X=x, T=t']$ gives us our final result.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{lem:minimax}}\n\\label{apx:minimax}\n\n\nFirst note that by \\cref{asm:compact} $\\mathcal{F}$ is compact. Also $J(W,\\cdot)$ is continuous for every $W$, since by \\cref{asm:ltwo} we know that the norm on each $\\mathcal F_t$ dominates the norm on $\\ltwo(\\mathcal Z)$ and this continuity result would be trivial if $\\mathcal F_t = \\ltwo(\\mathcal Z)$. This means that by the Extreme Value theorem we can replace the supremum over $\\mu$ with a maximum over $\\mu$ in the quantity we are bounding. Given this, we will proceed by bounding $\\min_W \\max_{\\mu \\in \\mathcal{F}} B(W, \\mu)$ using von Neumann's minimax theorem to swap the minimum and the maximum, and then use this to establish the overall bound for $J(W,\\mu)$. \n\nNext we can observe that $B(W, \\mu)$ is linear, and therefore both convex and concave, for each of $W$ and $\\mu$. Next, by \\cref{asm:compact} $\\mathcal{F}$ is convex and compact, and following the same argument as above $B(W,\\cdot)$ is continuous for every $W$. In addition, $B(W, \\mu)$ is also clearly continuous in $W$ for fixed $\\mu$, and the set $\\{W : \\|W\\|_2 \\leq M\\}$ is obviously compact and convex for any constant $M$. Thus by von Neumann's minimax theorem we have the following for every finite $M$:\n\\begin{equation}\n\\label{eq:minimax}\n\\min_{\\|W\\|_2 \\leq M} \\max_{\\mu \\in \\mathcal{F}} B(W, \\mu) = \\max_{\\mu \\in \\mathcal{F}} \\min_{\\|W\\|_2 \\leq M} B(W, \\mu)\n\\end{equation}\n\nGiven this, we can bound $\\min_W \\max_{\\mu \\in \\mathcal{F}} \\hat{J}(W, \\mu)$ as follows, which is valid for any $M$:\n\\begin{align}\n\\min_{W} \\max_{\\mu \\in \\mathcal{F}} \\hat{J}(W, \\mu) &\\leq \\min_{W} \\max_{\\mu \\in \\mathcal{F}} B(W, \\mu)^2 + \\frac{1}{n^2} W^T \\Sigma W \\nonumber \\\\\n&\\leq \\min_{W} \\max_{\\mu \\in \\mathcal{F}} B(W, \\mu)^2 + \\frac{\\sigma^2}{n^2} \\|W\\|^2_2 \\nonumber \\\\\n&\\leq \\min_{\\|W\\|_2 \\leq M} \\max_{\\mu \\in \\mathcal{F}} B(W, \\mu)^2 + \\frac{\\sigma^2}{n^2}\\|W\\|^2_2 \\nonumber \\\\\n&\\leq (\\min_{\\|W\\|_2 \\leq M} \\max_{\\mu \\in \\mathcal{F}} B(W, \\mu))^2 + \\frac{\\sigma^2}{n^2}M^2 \\nonumber \\\\\n&= (\\max_{\\mu \\in \\mathcal{F}} \\min_{\\|W\\|_2 \\leq M} B(W, \\mu))^2 + \\frac{\\sigma^2}{n^2}M^2 \\nonumber \\\\\n&= \\max_{\\mu \\in \\mathcal{F}} \\min_{\\|W\\|_2 \\leq M} B(W, \\mu)^2 + \\frac{\\sigma^2}{n^2}M^2 \\nonumber\n\\end{align}\n\nIn these inequalities we use the fact that $\\min_W \\max_{\\mu} B(W, \\mu)^2 = (\\min_W \\max_{\\mu} B(W, \\mu))^2$ and $\\max_{\\mu} \\min_W B(W, \\mu)^2 = (\\max_{\\mu} \\min_W B(W, \\mu))^2$ due to the symmetry of $\\mu$ in $\\mathcal{F}$ implied by \\cref{asm:symmetry}.\n\n\n\n\n\n\n\\subsection{Proof of \\cref{lem:jconvergence}}\n\\label{apx:jconvergence}\n\nLet $\\prod$ denote Cartesian product. First we note that without loss of generality we can prove this lemma in the case that $\\mathcal F = \\prod_t \\mathcal F_t$, since in general $F \\subseteq \\prod_t \\mathcal F_t$ so $\\sup_{\\mu \\in \\mathcal F} \\inf_W J(W,\\mu) \\leq \\sup_{\\mu \\in \\prod_t \\mathcal F_t} \\inf_W J(W,\\mu)$, and it is easy to verify that all of our assumptions would still hold on the larger set $\\prod_t \\mathcal F_t$. \n\nNow define the set $\\mathcal{H}^0 = \\{\\mu \\in \\prod_t \\ltwo(\\mathcal Z) : \\mathbb{E}[\\nu_T(X,T)^2] = 0\\}$. Each coordinate $\\mathcal H_t^0$ of $\\mathcal H^0$ is a subspace of $\\ltwo(\\mathcal Z)$, so we can also define its orthogonal complement $\\mathcal{H}_t^+$. Also, we have separability since from \\cref{sec:setting} we know that $\\mathcal Z \\subseteq \\mathbb{R}^q$, so any function $f \\in \\ltwo(\\mathcal Z)$ can be uniquely represented as $f = f^0 + f^+$ where $f^0 \\in \\mathcal{H}_t^0$ and $f^+ \\in \\mathcal{H}_t^+$. This means that for each $\\mu_t \\in \\mathcal F_t$, we can similarly uniquely represent $\\mu_t = \\mu_t^0 + \\mu_t^+$, and we can easily extend this to a unique representation of the vector $\\mu = \\mu^0 + \\mu^+$. Now in the case that $\\mathbb{E}[\\nu_T(X,T)^2] = 0$ we have $\\nu_{T_i}(X_i,T_i) = 0$ almost surely for all $i$, and it follows from \\cref{lem:gbound} that $\\nu_{T_i}(X_i,t) = 0$ almost surely also for all $i$ and $t$. Therefore any component of $\\mu$ in $\\mathcal{H}^0$ has no effect on the function $J(W,\\mu)$ which we are bounding, so without loss of generality we can restrict our attention to the following space:\n\\begin{equation*}\n\\mathcal{F}^+ = \\prod_t (\\mathcal{H}_t^+ \\cap \\mathcal F_t).\n\\end{equation*}\n\nBy construction the only function in $\\mathcal{F}^+$ such that $\\mathbb{E}[\\nu_T(X,T)^2] = 0\\}$ is the zero function, which we can also ignore in our bounds below, since when $\\mu = 0$ we can easily obtain $J(W,\\mu) = 0$ by choosing $W = 0$. Furthermore, by \\cref{asm:ltwo} we know that for each $t$ the $\\mathcal F_t$ norm dominates the $\\ltwo(\\mathcal Z)$ norm, so it must be the case that that each space $\\mathcal{F}_t^+$ is closed, since $\\mathcal{H}_t^+$ is a closed subspace of $\\ltwo(\\mathcal Z)$ due to it being an orthogonal complement. Thus it follows easily from \\cref{asm:norm} that $\\mathcal{F}^+$ is closed, given that its norm is an $\\mathbb{R}^m$ norm on top of the corresponding $\\mathcal{F}^+_t$ norms and $m$ is finite.\n\nNow, based on \\cref{lem:minimax}, it is sufficient to pick weights in response to $\\mu$ that control for a single mean outcome function. Instead of actually constructing a particular set of weights, we take the approach of viewing this as a convex optimization problem. Specifically, given $\\mu$, we calculate the minimum euclidean norm of all weights that set the bias term $B(W,\\mu)$ to zero exactly. This can be formulated as the following convex optimization program\n\\begin{align*}\n&\\min_W \\sum_i W_i^2 \\\\\n&\\text{s.t.} \\sum_i W_i \\nu_{T_i}(X_i, T_i) = \\sum_{i,t} \\pi_t(X_i) \\nu_t(X_i, T_i).\n\\end{align*}\n\nGiven the program only has linear constraints with equality, it satisfies Slater's condition, and therefore satisfies strong duality, which we will use to find the optimal value of this program. First we calculate the Lagrangian as\n\\begin{equation*}\n\\mathcal L_n(W, \\lambda) = \\sum_i W_i^2 + \\lambda \\left( \\sum_i W_i \\nu_{T_i}(X_i, T_i) - \\sum_{i,t} \\pi_t(X_i) \\nu_t(X_i, T_i) \\right).\n\\end{equation*}\n\nIt can easily be verified by taking derivatives that for any $\\lambda \\in \\mathbb{R}$ this function is minimized by setting $W_i = -\\frac{\\lambda}{2} \\nu_{T_i}(X_i,T_i)$, and plugging this value in gives the dual formulation of the program as\n\\begin{align*}\n&\\max_{\\lambda} -\\frac{\\lambda^2}{4} \\sum_i \\nu_{T_i}(X_i,T_i)^2 - \\lambda \\sum_{i,t} \\pi_t(X_i) \\nu_t(X_i,T_i),\n\\end{align*}\nwhich is unconstrained. Again by taking derivatives we can maximize this function, and we find the maximum value is given by\n\\begin{equation*}\n\\lambda = -2\\frac{\\sum_i \\pi_t(X_i) \\nu_t(X_i, T_i)}{\\sum_i \\nu_{T_i}(X_i, T_i)^2},\n\\end{equation*}\nand finally plugging this value into the dual objective function we see that the euclidean norm of the weights $W^*$ solving the convex program above is given by\n\\begin{equation*}\n\\|W^*\\|_2^2 = \\frac{\\left(\\sum_i \\pi_t(X_i) \\nu_t(X_i, T_i)\\right)^2}{\\sum_i \\nu_{T_i}(X_i, T_i)^2}.\n\\end{equation*}\n\nNow define $\\mathbb{E}_n$ as the mean with respect to the empirical distribution of the logged data. Then this objective value can be reformulated as\n\\begin{equation*}\n\\|W^*\\|_2^2 = \\frac{n\\mathbb{E}_n[\\sum_t \\nu_t(X,T)]^2}{\\mathbb{E}_n[\\nu_{T}(X,T)^2]}.\n\\end{equation*}\n\nTherefore choosing $M = \\|W^*\\|$, combining this result with \\cref{lem:minimax} gives us\n\\begin{align*}\n\\min_W \\sup_{\\mu \\in \\mathcal{F}^+} J(W, \\mu) &\\leq \\sup_{\\mu \\in \\mathcal{F}^+} \\frac{1}{n} \\left( \\frac{\\sigma^2\\mathbb{E}_n[\\sum_t \\nu_t(X,T)]^2}{\\mathbb{E}_n[\\nu_{T}(X,T)^2]} \\right) \\\\\n&= \\sup_{\\mu \\in \\mathcal{F}^+} \\frac{1}{n} \\left( \\frac{\\sigma^2\\mathbb{E}_n[\\sum_t \\nu_t(X,T)]^2}{\\mathbb{E}[\\nu_{T}(X,T)^2] + (\\mathbb{E}_n[\\nu_{T}(X,T)^2] - \\mathbb{E}[\\nu_{T}(X,T)^2])} \\right).\n\\end{align*}\n\nGiven this we will proceed by arguing that we can bound the denominator away from zero. We can note that $\\mu$ appears in both the numerator and denominator on the same scale, so without loss of generality we can further restrict our attention to $\\mu$ with fixed norm. By \\cref{asm:nondegen} we know that we can rescale every $\\mu \\in \\mathcal F$ to have norm $\\gamma$ for some $\\gamma > 0$. Given this we will restrict ourselves to the set $\\mathcal{F}^+_{\\gamma} = \\{\\mu \\in \\mathcal{F}^+ : \\|\\mu\\| = \\gamma\\}$. Since $\\mathcal{F}^+_{\\gamma}$ is the intersection of two closed sets it must be closed. Furthermore by \\cref{asm:compact} it is also compact, so it satisfies the conditions for the extreme value theorem. By construction $\\mathbb{E}[\\nu_T(X,T)^2] > 0$ for every $\\mu \\in \\mathcal{F}^+_{\\gamma}$, so putting the above together we have $\\inf_{\\mu \\in \\mathcal{F}^+_{\\gamma}} \\mathbb{E}[\\nu_T(X,T)^2] > 0$. We will define this value to be $\\alpha$. \n\nNow the numerator in the above bound is clearly bounded above by some $\\beta > 0$ uniformly over $\\mu \\in \\mathcal F$, since by \\cref{asm:bounded} we know that every $\\mu_t \\in \\mathcal F_t$ is uniformly bounded by some global constant, and therefore all $\\nu$ terms are bounded by some constant $b$. Given this all that remains to be shown is that $\\sup_{\\mu \\in \\mathcal F} |\\mathbb{E}_n[\\nu_{T}(X,T)^2] - \\mathbb{E}[\\nu_{T}(X,T)^2]|$ converges in probability to zero. In order to show this we will define the following terms:\n\\begin{align*}\nD_n &= \\mathbb{E}_n[\\nu_T(X,T)^2] \\\\\nE_n &= \\sup_{\\mu \\in \\mathcal{F}} |D_n - \\mathbb{E}[D_n]|\n\\end{align*}\n\nWe need to show that $E_n$ converges uniformly to zero. Define $D_n'$ as an arbitrary recalculation of $D_n$ replacing $(X_{1:n}, T_{1:n})$ with $(X'_{1:n}, T'_{1:n})$, which differ from the originals at most in a single coordinate $i$, and define $E_n' = \\sup_{\\mu \\in \\mathcal{F}} |D_n' - \\mathbb{E}[D_n']|$. Furthermore as argued above all $\\nu$ terms are bounded above by some constant $b$, so each $\\nu_{T_i}(X_i,T_i)^2$ is bounded by $b^2$. Given this we can obtain\n\\begin{align*}\n|E_n - E_n'| &= |\\sup_{\\mu \\in \\mathcal{F}} |D_n - \\mathbb{E}[D_n]| - \\sup_{\\mu \\in \\mathcal{F}} |D_n' - \\mathbb{E}[D_n']| | \\\\\n&\\leq \\sup_{\\mu \\in \\mathcal{F}} |(D_n - \\mathbb{E}[D_n]) - (D_n' - \\mathbb{E}[D_n'])| \\\\\n&= \\sup_{\\mu \\in \\mathcal{F}} |D_n - D_n'| \\\\\n&= \\frac{1}{n} \\sup_{\\mu \\in \\mathcal{F}} |\\nu_{T_i}(X_i,T_i)^2 - \\nu_{T_i'}(X_i',T_i')^2| \\\\\n&\\leq \\frac{2b^2}{n}\n\\end{align*}\nGiven this we can apply McDiarmid's inequality to obtain the following bound:\n\\begin{equation*}\n\\label{eq:mcdiarmid}\nP(|E_n - \\mathbb{E}[E_n]| \\leq \\epsilon) \\leq 2\\exp \\left( -\\frac{n\\epsilon^2}{2b^4} \\right) \n\\end{equation*}\n\nThis implies that $E_n - \\mathbb{E}[E_n] = o_p(1)$. Next we show that $\\mathbb{E}[E_n] = o_p(1)$ also. We do this using a symmetrization argument as follows, where $D'_n$ is defined identically to $D_n$ using iid shadow variables $(X'_i,T'_i)$ in place of $(X_i,T_i)$ for each $i$, and $\\epsilon_i$ are iid Rademacher random variables:\n\\begin{align*}\n\\mathbb{E}[E_n] &= \\mathbb{E}\\left[ \\sup_{\\mu \\in \\mathcal{F}} \\left| D_n - \\mathbb{E}[D_n] \\right| \\right] \\\\\n&= \\mathbb{E} \\left[ \\sup_{\\mu \\in \\mathcal{F}} \\left| \\frac{1}{n} \\sum_i \\nu_{T_i}(X_i,T_i) - \\mathbb{E}[\\nu_{T_i'}(X_i',T_i')] \\right| \\right] \\\\\n&\\leq 2 \\mathbb{E} \\left[ \\sup_{\\mu \\in \\mathcal{F}} \\left| \\frac{1}{n} \\sum_i \\epsilon_i \\nu_{T_i}(X_i,T_i)  \\right| \\right]  \\\\\n&\\leq 2 \\sum_t \\mathbb{E} \\left[ \\sup_{\\mu \\in \\mathcal{F}} \\left| \\frac{1}{n} \\sum_i \\epsilon_i \\delta_{T_i t} \\nu_{t}(X_i,T_i)  \\right| \\right] \\\\\n&\\leq 4 \\sum_t \\mathbb{E} \\left[ \\sup_{\\mu \\in \\mathcal{F}} \\left| \\frac{1}{n} \\sum_i \\epsilon_i \\nu_{t}(X_i,T_i)  \\right| \\right] \\\\\n&\\leq 4 \\sum_t \\mathbb{E} \\left[ \\sup_{\\mu \\in \\mathcal{F}} \\left| \\frac{1}{n} \\sum_i \\epsilon_i \\mu_t(Z_i)  \\right| \\right] \\\\\n&\\leq 4 \\sum_t \\mathcal R_n(\\mathcal F_t) \\\\\n\\end{align*}\nwhere in the third inequality we appeal to the Rademacher comparison lemma \\citep[Thm.~4.12]{ledoux2013probability}. Thus since from \\cref{asm:complexity} we know that the Rademacher complexity of each set $\\mathcal R_n(\\mathcal F_n)$ vanishes, it follows that $\\mathbb{E}[E_n] = o_p(1)$. Putting everything from above together we get\n\\begin{equation*}\n\\min_W \\sup_{\\mu \\in \\mathcal{F}} J(W, \\mu) \\leq \\frac{1}{n} \\frac{\\beta}{\\alpha + o_p(1)},\n\\end{equation*}\nso we have $\\min_W \\sup_{\\mu \\in \\mathcal{F}} J(W, \\mu) \\leq O_p(1\/n)$ as required.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{thm:fk}}\n\\label{apx:fk}\n\nFirst, \\cref{asm:norm} follows trivially from the definition of $\\fk$. Next, \\cref{asm:symmetry} and $\\cref{asm:nondegen}$ follow from the fact that $\\fk$ consists of all functions in $\\op{span}(\\mathcal{F}_t)$ with norm at most 1, as does the fact that it is a closed space. Given that $K$ is a Mercer kernel, balls in the corresponding RKHS have finite covering number \\citep{zhou2002covering}, and it follows easily from this that $\\fk$ has finite covering numbers, as its covering number must be bounded above by the sum of the covering numbers of the spaces $\\fkt$. So $\\fk$ is closed and totally bounded with respect to its norm, and therefore compact, which gives us \\cref{asm:compact}. Clearly each $\\mathcal F_t$ is contained in $\\ltwo(\\mathcal Z)$ since RKHS spaces are square integrable, and the fact that the $K$ norm dominates the $\\ltwo$ follows from Mercer's Theorem, which implies that $\\|f\\|_K^2 = \\sum_{i=1}^{\\infty} f_i^2 \/ \\sigma_i$, where $f_i$ is the $i$th eigenvalue of $f$ for some orthonormal basis of $\\ltwo(\\mathcal Z)$, and $\\sigma_i \\geq 0$ converges to zero. This gives us \\cref{asm:ltwo}. Next, by construction each $\\fkt$ consists of all functions in the RKHS up to norm 1. Therefore assuming the kernel $K$ is bounded, it is trivial to verify that function application must be globally bounded, since for any function $f \\in \\fkt$ we have $f(x) \\leq <f, K_x> \\leq \\|f\\| \\sqrt{K(x,x)} \\leq \\sqrt{K(x,x)}$, which gives us \\cref{asm:bounded}. Finally, given this characterization of $\\fkt$ as the 1-ball of the RKHS, it has vanishing Rademacher complexity \\citep[Thm.~2.1]{mendelson2003performance}, so we have \\cref{asm:complexity}.\n\n\n\n\n\n\\subsection{Proof of \\cref{thm:continuous-algorithm}}\n\\label{apx:continuous-algorithm}\n\nFirst we will find a closed form expression for $\\sup_{\\mu \\in \\fk} (\\frac{1}{n} \\sum_{i,t} f_{it} \\nu_{t}(X_i,T_i))^2$. In this derivation we will use the shorthand $\\varphi_{ti}$ for the conditional density of $\\mu_t$ given $X_i$ and $T_i$, and $T_K$ for the kernel intergral operator defined according to $T_K f = \\int_{\\mathcal Z} K(\\cdot,z) f(z) dz$. In this derivation we will make use of the fact that $\\ltwoprod{f}{g} = \\rkhsprod{f}{T_K g}$ for any square integrable $f$ and $g$. Given all this we can obtain:\n\\begin{align*}\n\\sup_{\\mu \\in \\fk} \\left( \\frac{1}{n} \\sum_{i,t} f_{it} \\nu_{t}(X_i,T_i) \\right)^2 &= \\sum_t \\sup_{\\mu_t \\in \\fkt} \\left( \\frac{1}{n} \\sum_i f_{it} \\ltwoprod{\\mu_t}{\\varphi_i} \\right)^2 \\\\\n&= \\sum_t \\sup_{\\mu_t \\in \\fkt} \\ltwoprod{\\mu_t}{\\frac{1}{n} \\sum_i f_{it} \\varphi_i}^2 \\\\\n&= \\sum_t \\sup_{\\mu_t \\in \\fkt} \\rkhsprod{\\mu_t}{T_K \\frac{1}{n} \\sum_i f_{it} \\varphi_i}^2 \\\\\n&= \\sum_t \\frac{\\rkhsprod{T_K \\frac{1}{n} \\sum_i f_{it} \\varphi_i}{T_K \\frac{1}{n} \\sum_i f_{it} \\varphi_i}^2}{\\|T_K \\frac{1}{n} \\sum_i f_{it} \\varphi_i\\|_K} \\\\\n&= \\sum_t \\rkhsprod{T_K \\frac{1}{n} \\sum_i f_{it} \\varphi_i}{T_K \\frac{1}{n} \\sum_i f_{it} \\varphi_i} \\\\\n&= \\sum_t \\ltwoprod{\\frac{1}{n} \\sum_i f_{it} \\varphi_i}{T_K \\frac{1}{n} \\sum_i f_{it} \\varphi_i} \\\\\n&= \\frac{1}{n^2} \\sum_{i,j,t} f_{it} f_{jt} \\ltwoprod{\\varphi_i}{T_K \\varphi_j} \\\\\n&= \\frac{1}{n^2} \\sum_{i,j,t} f_{it} f_{jt} \\int_{\\mathcal Z} \\varphi_i(z) (\\int_{\\mathcal Z'} K(z,z') \\varphi_j(z') dz') dz \\\\\n&= \\frac{1}{n^2} \\sum_{i,j,t} f_{it} f_{jt} \\int_{\\mathcal Z} \\int_{\\mathcal Z'} \\varphi_i(z) \\varphi_j(z') K(z,z') dz' dz \\\\\n&= \\frac{1}{n^2} \\sum_{i,j,t} f_{it} f_{jt} \\mathbb{E}[K(Z_i,Z_j')] \n\\end{align*}\n\nGiven this, and recalling that $f_{it} = W_i \\delta_{T_i t} - \\pi_t(X_i)$ we can derive a closed form for our minimization objective, as follows:\n\\begin{align*}\n\\sup_{\\mu \\in \\fk} J(W, \\mu) &= \\frac{1}{n^2} \\sum_{i,j,t}  f_{it} f_{jt} Q_{ij} + \\frac{1}{n^2} \\sum_{i,j} W_i W_j \\Gamma_{ij} \\\\\n&= \\frac{1}{n^2} \\sum_{i,j,t}  Q_{ij} (W_i W_j \\delta_{T_i t} \\delta_{T_j t} - 2 W_j \\delta_{T_j t} \\pi_t(X_i) + \\pi_t(X_i) \\pi_t(X_j)) \\\\\n&\\qquad\\qquad + \\frac{1}{n^2} \\sum_{i,j} W_i W_j \\Gamma_{ij} \\\\\n&= \\frac{1}{n^2} \\sum_{i,j} W_i W_j (Q_{ij} \\delta_{T_i T_j} + \\Gamma_{ij}) - \\frac{2}{n^2} \\sum_j W_j (\\sum_i Q_{ij} \\pi_{T_j}(X_i)) \\\\\n&\\qquad\\qquad + \\frac{1}{n^2} \\sum_{i,j,t} Q_{ij} \\pi_t(X_i) \\pi_t(X_j)\n\\end{align*}\n\nFinally we can conclude by noting that this corresponds to the quadratic program formulation given in the question with $c = \\frac{1}{n^2} \\sum_{i,j,t} Q_{ij} \\pi_t(X_i) \\pi_t(X_j)$.\n\n\n\n\n\n\n\n\\section{Additional Experimentation Details}\n\n\\subsection{Experiment Scenario}\n\\label{apx:exp-scenario}\n\nAll our experiments were conducted using the setup described in \\cref{sec:exp-setup}. We used the following parameter values for our data-generating distribution:\n\\begin{align*}\n\\alpha &= [1.0, -2.0, -1.0, 2.0, 4.0, 0.0, -2.0, -1.0, -3.0, 1.0] \\\\\n\\alpha_0 &= 0 \\\\\n\\sigma^2_X &= 4.0 \\\\\n\\beta &= [0.5, -0.5] \\\\\n\\beta_0 &= 0 \\\\\n\\zeta(0) &= 1.0 \\\\\n\\zeta(0)_0 &= 0 \\\\\n\\zeta(1) &= -0.5 \\\\\n\\zeta(1)_0 &= 0 \\\\\n\\sigma^2_Y &= 0.01 \n\\end{align*}\n\nIn addiiton, the policy $\\pi$ we are evaluating takes the form as described in \\cref{sec:exp-setup}, and we used the following parameter values for this policy:\n\\begin{align*}\n\\psi_0 &= [-0.1, 0.2, 0.2, -0.1, -0.1, -0.1, 0.1, 0.1, 0.1, -0.1] \\\\\n\\psi_1 &= - \\psi_0 \n\\end{align*}\n\n\n\\subsection{Method Implementation Details}\n\\label{apx:exp-benchmark}\n\nIn all methods where we sampled from the posterior $\\varphi(\\cdot;x,t)$, this sampling was done using STAN \\citep{carpenter2017stan}, solving QPs and LCQPs was done using the Python package \\texttt{quadprog},\\footnote{https:\/\/pypi.org\/project\/quadprog\/} all stochastic gradient descent (SGD) learning was performed using the Adam \\citep{kingma2014adam} optimizer with a learning rate of 0.001.\n\n\\paragraph{\\textbf{OptZ}} We ran \\cref{algo:continuous} with $B = 50$.\n\n\\paragraph{\\textbf{IPS}} Since the propensity scores $\\eta_t(x)$ are not not tractable to compute analytically, we trained a neural network $\\hat \\eta$ to estimate this function. This was done by sampling batches of $(Z,X)$ pairs from the data model, and training the network using SGD to predict the vector of probabilities $P_T = \\beta^T Z + \\beta_0$ from $X$, using cross-entropy loss. We used a neural network with two hidden layers of size $200$ for $\\hat \\eta$, and trained for $2000$ iterations with a batch size of 32. We found in practice this training was very stable and gave accurate results. \n\n\\paragraph{\\textbf{DirX}} For each $t$ we trained a neural network $\\hat \\rho_t$ to predict $\\nu_t(x,t)$ by taking the set of $(X,T,Y)$ triplets in our training data where $T=t$, and training the network using SGD to predict $Y$ from $X$ using MSE loss. Based on pilot experiments we used a network architecture with a single hidden layer of size $100$, and trained using a batch size of 128. We used $80\\%$ of our data for training, and used the remaining $20\\%$ for the purpose of early stopping. We trained for a maximum of $500$ epochs, or until we made no progress on development data for $20$ epochs. \n\n\\paragraph{\\textbf{DirZ}} For each $t$ we trained a neural network $\\hat \\mu_t$ to predict $\\mu_t$. This was done by taking the set of $(X,T,Y)$ triplets in our training data, and for each sampling $200$ $Z$ values from the posterior using our identified model given $X$ and $T$. This gives us a set of $(Z,T,Y)$ triplets $200$ times as large as our original training set. We then trained each $\\hat \\mu_t$ network by taking the set of these triplets where $T=t$, and optimized the network using SGD on this data predicting $Y$ from $Z$. We used the same settings for this optimization as with the \\emph{direct-naive} method, except we allowed up to $1000$ epochs. Note that for both training and inference we limited ourselves to sampling $200$ $Z$ values per data point due to computational limitations.\n\n\n\\section{Additional Experiment Results}\n\\label{apx:exp-results}\n\nWe present here our additional experiment results. In these results \\textbf{SimplexOptZ} refers to our method using the simplex constraints discussed in \\cref{sec:qpalgo}.\n\n\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$\\\\\n\\hline\n200 & $.39\\pm.07$ & $.24\\pm.02$ & $.36\\pm.02$ & $.81\\pm.02$\\\\\n500 & $.19\\pm.02$ & $.18\\pm.02$ & $.23\\pm.02$ & $.49\\pm.02$\\\\\n1000 & $.11\\pm.01$ & $.11\\pm.01$ & $.13\\pm.01$ & $.27\\pm.01$\\\\\n2000 & $.08\\pm.01$ & $.08\\pm.01$ & $.09\\pm.01$ & $.17\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for weighted estimator using our weights, with \\textbf{step} link}\n\\label{tab:step-rmse}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirX:OptZ}_{0.2}$ & $\\textbf{DirX:OptZ}_{1.0}$ & $\\textbf{DirX:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.57\\pm.06$ & $.42\\pm.03$ & $.39\\pm.03$ & $.43\\pm.03$\\\\\n500 & $.55\\pm.02$ & $.46\\pm.02$ & $.39\\pm.02$ & $.37\\pm.02$\\\\\n1000 & $.49\\pm.02$ & $.45\\pm.01$ & $.39\\pm.01$ & $.32\\pm.01$\\\\\n2000 & $.48\\pm.01$ & $.47\\pm.01$ & $.42\\pm.01$ & $.34\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for doubly robust estimator using our weights and $\\textbf{DirX}$, with \\textbf{step} link}\n\\label{tab:step-rmse-drx}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirZ:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.2}$ & $\\textbf{DirZ:OptZ}_{1.0}$ & $\\textbf{DirZ:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.41\\pm.07$ & $.29\\pm.02$ & $.50\\pm.02$ & $1.1\\pm.03$\\\\\n500 & $.20\\pm.02$ & $.21\\pm.02$ & $.31\\pm.02$ & $.70\\pm.02$\\\\\n1000 & $.11\\pm.01$ & $.13\\pm.01$ & $.18\\pm.01$ & $.42\\pm.01$\\\\\n2000 & $.08\\pm.01$ & $.09\\pm.01$ & $.13\\pm.01$ & $.26\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for doubly robust estimator using our weights and $\\textbf{DirZ}$, with \\textbf{step} link}\n\\label{tab:step-rmse-drz}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{SimplexOptZ}_{0.001}$ & $\\textbf{SimplexOptZ}_{0.2}$ & $\\textbf{SimplexOptZ}_{1.0}$ & $\\textbf{SimplexOptZ}_{5.0}$\\\\\n\\hline\n200 & $.30\\pm.02$ & $.25\\pm.02$ & $.38\\pm.02$ & $.91\\pm.02$\\\\\n500 & $.18\\pm.02$ & $.19\\pm.02$ & $.24\\pm.02$ & $.54\\pm.02$\\\\\n1000 & $.12\\pm.01$ & $.11\\pm.01$ & $.13\\pm.01$ & $.29\\pm.01$\\\\\n2000 & $.07\\pm.01$ & $.08\\pm.01$ & $.10\\pm.01$ & $.18\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for weighted estimator using our weights and constraining $W \\in n \\Delta^n$, with \\textbf{step} link}\n\\label{tab:step-rmse-simplex}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{cccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.47\\pm.03$ & $2.0\\pm.03$ & $2.1\\pm.03$ & $2.3\\pm.02$ & $2.5\\pm.02$ & $.52\\pm.02$ & $2.6\\pm.02$\\\\\n500 & $.48\\pm.03$ & $2.0\\pm.02$ & $2.1\\pm.02$ & $2.3\\pm.02$ & $2.6\\pm.02$ & $.48\\pm.02$ & $2.6\\pm.01$\\\\\n1000 & $.39\\pm.02$ & $2.0\\pm.01$ & $2.1\\pm.01$ & $2.3\\pm.01$ & $2.5\\pm.01$ & $.48\\pm.02$ & $2.6\\pm.01$\\\\\n2000 & $.40\\pm.01$ & $2.0\\pm.01$ & $2.1\\pm.01$ & $2.3\\pm.01$ & $2.5\\pm.01$ & $.45\\pm.02$ & $2.6\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for benchmark methods, with \\textbf{step} link}\n\\label{tab:step-rmse-bench}\n\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$\\\\\n\\hline\n200 & $.03\\pm.39$ & $.11\\pm.21$ & $.29\\pm.21$ & $.78\\pm.18$\\\\\n500 & $.09\\pm.17$ & $.10\\pm.15$ & $.17\\pm.16$ & $.47\\pm.15$\\\\\n1000 & $.02\\pm.11$ & $.05\\pm.09$ & $.08\\pm.09$ & $.25\\pm.09$\\\\\n2000 & $.03\\pm.07$ & $.05\\pm.06$ & $.07\\pm.07$ & $.16\\pm.07$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for weighted estimator using our weights, with \\textbf{step} link}\n\\label{tab:step-bias}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirX:OptZ}_{0.2}$ & $\\textbf{DirX:OptZ}_{1.0}$ & $\\textbf{DirX:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.43\\pm.38$ & $.35\\pm.24$ & $.31\\pm.24$ & $.37\\pm.22$\\\\\n500 & $.51\\pm.19$ & $.42\\pm.18$ & $.35\\pm.18$ & $.33\\pm.17$\\\\\n1000 & $.47\\pm.13$ & $.44\\pm.11$ & $.37\\pm.10$ & $.30\\pm.11$\\\\\n2000 & $.47\\pm.09$ & $.46\\pm.08$ & $.41\\pm.08$ & $.33\\pm.08$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for doubly robust estimator using our weights and $\\textbf{DirX}$, with \\textbf{step} link}\n\\label{tab:step-bias-drx}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirZ:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.2}$ & $\\textbf{DirZ:OptZ}_{1.0}$ & $\\textbf{DirZ:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.05\\pm.40$ & $.19\\pm.22$ & $.45\\pm.22$ & $1.1\\pm.21$\\\\\n500 & $.10\\pm.18$ & $.14\\pm.16$ & $.26\\pm.16$ & $.68\\pm.17$\\\\\n1000 & $.04\\pm.11$ & $.09\\pm.10$ & $.15\\pm.10$ & $.41\\pm.10$\\\\\n2000 & $.03\\pm.07$ & $.06\\pm.07$ & $.10\\pm.07$ & $.25\\pm.07$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for doubly robust estimator using our weights and $\\textbf{DirZ}$, with \\textbf{step} link}\n\\label{tab:step-bias-drz}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{SimplexOptZ}_{0.001}$ & $\\textbf{SimplexOptZ}_{0.2}$ & $\\textbf{SimplexOptZ}_{1.0}$ & $\\textbf{SimplexOptZ}_{5.0}$\\\\\n\\hline\n200 & $.04\\pm.30$ & $.12\\pm.21$ & $.31\\pm.21$ & $.89\\pm.20$\\\\\n500 & $.08\\pm.15$ & $.10\\pm.15$ & $.18\\pm.16$ & $.51\\pm.16$\\\\\n1000 & $.01\\pm.12$ & $.06\\pm.09$ & $.09\\pm.09$ & $.27\\pm.10$\\\\\n2000 & $.03\\pm.07$ & $.05\\pm.06$ & $.07\\pm.07$ & $.17\\pm.07$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for weighted estimator using our weights and constraining $W \\in n \\Delta^n$, with \\textbf{step} link}\n\\label{tab:step-bias-simplex}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{cccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.40\\pm.25$ & $1.9\\pm.21$ & $2.1\\pm.20$ & $2.3\\pm.19$ & $2.5\\pm.18$ & $.49\\pm.18$ & $2.6\\pm.14$\\\\\n500 & $.43\\pm.21$ & $2.0\\pm.16$ & $2.1\\pm.15$ & $2.3\\pm.14$ & $2.6\\pm.13$ & $.45\\pm.16$ & $2.6\\pm.12$\\\\\n1000 & $.37\\pm.12$ & $2.0\\pm.10$ & $2.1\\pm.09$ & $2.3\\pm.09$ & $2.5\\pm.08$ & $.46\\pm.15$ & $2.6\\pm.11$\\\\\n2000 & $.39\\pm.10$ & $2.0\\pm.08$ & $2.1\\pm.07$ & $2.3\\pm.07$ & $2.5\\pm.07$ & $.42\\pm.17$ & $2.6\\pm.11$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for benchmark methods, with \\textbf{step} link}\n\\label{tab:step-bias-bench}\n\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$\\\\\n\\hline\n200 & $.07\\pm.01$ & $.04\\pm.00$ & $.04\\pm.00$ & $.07\\pm.00$\\\\\n500 & $.04\\pm.00$ & $.03\\pm.00$ & $.03\\pm.00$ & $.04\\pm.00$\\\\\n1000 & $.02\\pm.00$ & $.02\\pm.00$ & $.02\\pm.00$ & $.02\\pm.00$\\\\\n2000 & $.01\\pm.00$ & $.01\\pm.00$ & $.01\\pm.00$ & $.01\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for weighted estimator using our weights, with \\textbf{exp} link}\n\\label{tab:exp-rmse}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirX:OptZ}_{0.2}$ & $\\textbf{DirX:OptZ}_{1.0}$ & $\\textbf{DirX:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.13\\pm.01$ & $.10\\pm.01$ & $.12\\pm.01$ & $.11\\pm.01$\\\\\n500 & $.10\\pm.01$ & $.09\\pm.01$ & $.10\\pm.01$ & $.12\\pm.01$\\\\\n1000 & $.08\\pm.00$ & $.08\\pm.00$ & $.08\\pm.00$ & $.10\\pm.00$\\\\\n2000 & $.07\\pm.00$ & $.07\\pm.00$ & $.08\\pm.00$ & $.09\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for doubly robust estimator using our weights and $\\textbf{DirX}$, with \\textbf{exp} link}\n\\label{tab:exp-rmse-drx}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirZ:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.2}$ & $\\textbf{DirZ:OptZ}_{1.0}$ & $\\textbf{DirZ:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.15\\pm.02$ & $.15\\pm.01$ & $.25\\pm.01$ & $.44\\pm.01$\\\\\n500 & $.10\\pm.01$ & $.11\\pm.01$ & $.18\\pm.01$ & $.32\\pm.01$\\\\\n1000 & $.07\\pm.01$ & $.08\\pm.01$ & $.12\\pm.01$ & $.23\\pm.01$\\\\\n2000 & $.04\\pm.00$ & $.05\\pm.00$ & $.08\\pm.00$ & $.16\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for doubly robust estimator using our weights and $\\textbf{DirZ}$, with \\textbf{exp} link}\n\\label{tab:exp-rmse-drz}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{SimplexOptZ}_{0.001}$ & $\\textbf{SimplexOptZ}_{0.2}$ & $\\textbf{SimplexOptZ}_{1.0}$ & $\\textbf{SimplexOptZ}_{5.0}$\\\\\n\\hline\n200 & $.05\\pm.00$ & $.11\\pm.01$ & $.22\\pm.01$ & $.39\\pm.02$\\\\\n500 & $.04\\pm.00$ & $.08\\pm.01$ & $.15\\pm.01$ & $.28\\pm.01$\\\\\n1000 & $.02\\pm.00$ & $.05\\pm.00$ & $.09\\pm.00$ & $.19\\pm.00$\\\\\n2000 & $.02\\pm.00$ & $.03\\pm.00$ & $.07\\pm.00$ & $.14\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for weighted estimator using our weights and constraining $W \\in n \\Delta^n$, with \\textbf{exp} link}\n\\label{tab:exp-rmse-simplex}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{cccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.12\\pm.01$ & $.76\\pm.02$ & $.81\\pm.02$ & $.92\\pm.02$ & $1.0\\pm.02$ & $.10\\pm.01$ & $1.0\\pm.01$\\\\\n500 & $.11\\pm.00$ & $.76\\pm.01$ & $.82\\pm.01$ & $.92\\pm.01$ & $1.0\\pm.01$ & $.10\\pm.01$ & $1.0\\pm.01$\\\\\n1000 & $.10\\pm.00$ & $.74\\pm.01$ & $.79\\pm.01$ & $.90\\pm.01$ & $1.0\\pm.01$ & $.09\\pm.01$ & $1.1\\pm.01$\\\\\n2000 & $.10\\pm.00$ & $.73\\pm.00$ & $.78\\pm.00$ & $.88\\pm.00$ & $.99\\pm.01$ & $.10\\pm.01$ & $1.0\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for benchmark methods, with \\textbf{exp} link}\n\\label{tab:exp-rmse-bench}\n\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$\\\\\n\\hline\n200 & $.01\\pm.06$ & $.00\\pm.04$ & $-0.00\\pm.04$ & $-0.05\\pm.04$\\\\\n500 & $.01\\pm.04$ & $.01\\pm.03$ & $.00\\pm.03$ & $-0.03\\pm.03$\\\\\n1000 & $.00\\pm.02$ & $.01\\pm.02$ & $-0.00\\pm.02$ & $-0.01\\pm.02$\\\\\n2000 & $.01\\pm.01$ & $.01\\pm.01$ & $.00\\pm.01$ & $.00\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for weighted estimator using our weights, with \\textbf{exp} link}\n\\label{tab:exp-bias}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirX:OptZ}_{0.2}$ & $\\textbf{DirX:OptZ}_{1.0}$ & $\\textbf{DirX:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.07\\pm.10$ & $.08\\pm.06$ & $.10\\pm.05$ & $.11\\pm.04$\\\\\n500 & $.07\\pm.07$ & $.07\\pm.05$ & $.09\\pm.05$ & $.11\\pm.04$\\\\\n1000 & $.06\\pm.05$ & $.07\\pm.03$ & $.07\\pm.03$ & $.09\\pm.02$\\\\\n2000 & $.06\\pm.04$ & $.06\\pm.03$ & $.07\\pm.03$ & $.09\\pm.03$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for doubly robust estimator using our weights and $\\textbf{DirX}$, with \\textbf{exp} link}\n\\label{tab:exp-bias-drx}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirZ:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.2}$ & $\\textbf{DirZ:OptZ}_{1.0}$ & $\\textbf{DirZ:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.05\\pm.14$ & $.12\\pm.10$ & $.23\\pm.09$ & $.43\\pm.09$\\\\\n500 & $.03\\pm.10$ & $.09\\pm.07$ & $.16\\pm.07$ & $.32\\pm.07$\\\\\n1000 & $.02\\pm.06$ & $.06\\pm.05$ & $.10\\pm.05$ & $.23\\pm.06$\\\\\n2000 & $.01\\pm.04$ & $.03\\pm.03$ & $.07\\pm.03$ & $.16\\pm.04$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for doubly robust estimator using our weights and $\\textbf{DirZ}$, with \\textbf{exp} link}\n\\label{tab:exp-bias-drz}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{SimplexOptZ}_{0.001}$ & $\\textbf{SimplexOptZ}_{0.2}$ & $\\textbf{SimplexOptZ}_{1.0}$ & $\\textbf{SimplexOptZ}_{5.0}$\\\\\n\\hline\n200 & $.02\\pm.05$ & $.10\\pm.06$ & $.20\\pm.08$ & $.38\\pm.11$\\\\\n500 & $.02\\pm.04$ & $.07\\pm.04$ & $.14\\pm.05$ & $.27\\pm.06$\\\\\n1000 & $.01\\pm.02$ & $.04\\pm.02$ & $.08\\pm.03$ & $.19\\pm.04$\\\\\n2000 & $.01\\pm.01$ & $.03\\pm.02$ & $.06\\pm.02$ & $.14\\pm.02$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for weighted estimator using our weights and constraining $W \\in n \\Delta^n$, with \\textbf{exp} link}\n\\label{tab:exp-bias-simplex}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{cccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.10\\pm.06$ & $.74\\pm.13$ & $.80\\pm.13$ & $.91\\pm.15$ & $1.0\\pm.16$ & $.07\\pm.07$ & $1.0\\pm.08$\\\\\n500 & $.11\\pm.03$ & $.76\\pm.07$ & $.81\\pm.08$ & $.92\\pm.09$ & $1.0\\pm.10$ & $.07\\pm.08$ & $1.0\\pm.09$\\\\\n1000 & $.10\\pm.03$ & $.74\\pm.05$ & $.79\\pm.05$ & $.90\\pm.06$ & $1.0\\pm.06$ & $.06\\pm.07$ & $1.0\\pm.10$\\\\\n2000 & $.10\\pm.02$ & $.73\\pm.03$ & $.78\\pm.03$ & $.88\\pm.04$ & $.99\\pm.04$ & $.07\\pm.07$ & $1.0\\pm.09$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for benchmark methods, with \\textbf{exp} link}\n\\label{tab:exp-bias-bench}\n\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$\\\\\n\\hline\n200 & $.47\\pm.04$ & $.35\\pm.02$ & $.39\\pm.02$ & $.48\\pm.02$\\\\\n500 & $.36\\pm.05$ & $.27\\pm.02$ & $.30\\pm.02$ & $.40\\pm.02$\\\\\n1000 & $.25\\pm.02$ & $.22\\pm.01$ & $.25\\pm.01$ & $.37\\pm.01$\\\\\n2000 & $.14\\pm.01$ & $.14\\pm.01$ & $.17\\pm.01$ & $.27\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for weighted estimator using our weights, with \\textbf{cubic} link}\n\\label{tab:cubic-rmse}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirX:OptZ}_{0.2}$ & $\\textbf{DirX:OptZ}_{1.0}$ & $\\textbf{DirX:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.58\\pm.07$ & $.41\\pm.03$ & $.42\\pm.02$ & $.38\\pm.02$\\\\\n500 & $.37\\pm.04$ & $.33\\pm.02$ & $.35\\pm.02$ & $.37\\pm.02$\\\\\n1000 & $.31\\pm.02$ & $.31\\pm.02$ & $.33\\pm.02$ & $.39\\pm.01$\\\\\n2000 & $.21\\pm.02$ & $.23\\pm.02$ & $.26\\pm.01$ & $.32\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for doubly robust estimator using our weights and $\\textbf{DirX}$, with \\textbf{cubic} link}\n\\label{tab:cubic-rmse-drx}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirZ:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.2}$ & $\\textbf{DirZ:OptZ}_{1.0}$ & $\\textbf{DirZ:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.49\\pm.04$ & $.42\\pm.03$ & $.54\\pm.03$ & $.76\\pm.02$\\\\\n500 & $.38\\pm.05$ & $.30\\pm.02$ & $.38\\pm.02$ & $.59\\pm.02$\\\\\n1000 & $.27\\pm.02$ & $.25\\pm.02$ & $.32\\pm.02$ & $.52\\pm.02$\\\\\n2000 & $.16\\pm.01$ & $.16\\pm.01$ & $.22\\pm.01$ & $.39\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for doubly robust estimator using our weights and $\\textbf{DirZ}$, with \\textbf{cubic} link}\n\\label{tab:cubic-rmse-drz}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{SimplexOptZ}_{0.001}$ & $\\textbf{SimplexOptZ}_{0.2}$ & $\\textbf{SimplexOptZ}_{1.0}$ & $\\textbf{SimplexOptZ}_{5.0}$\\\\\n\\hline\n200 & $.45\\pm.04$ & $.41\\pm.03$ & $.52\\pm.03$ & $.70\\pm.03$\\\\\n500 & $.37\\pm.05$ & $.30\\pm.02$ & $.37\\pm.02$ & $.55\\pm.02$\\\\\n1000 & $.26\\pm.02$ & $.24\\pm.02$ & $.31\\pm.02$ & $.50\\pm.02$\\\\\n2000 & $.14\\pm.01$ & $.15\\pm.01$ & $.20\\pm.01$ & $.35\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for weighted estimator using our weights and constraining $W \\in n \\Delta^n$, with \\textbf{cubic} link}\n\\label{tab:cubic-rmse-simplex}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{cccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.46\\pm.04$ & $1.1\\pm.02$ & $1.1\\pm.03$ & $1.3\\pm.03$ & $1.4\\pm.03$ & $.36\\pm.02$ & $1.4\\pm.01$\\\\\n500 & $.38\\pm.02$ & $1.1\\pm.01$ & $1.2\\pm.01$ & $1.3\\pm.01$ & $1.4\\pm.01$ & $.34\\pm.02$ & $1.4\\pm.01$\\\\\n1000 & $.39\\pm.01$ & $1.1\\pm.01$ & $1.2\\pm.01$ & $1.3\\pm.01$ & $1.4\\pm.01$ & $.35\\pm.02$ & $1.4\\pm.01$\\\\\n2000 & $.35\\pm.01$ & $1.1\\pm.01$ & $1.2\\pm.01$ & $1.3\\pm.01$ & $1.4\\pm.01$ & $.39\\pm.02$ & $1.4\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for benchmark methods, with \\textbf{cubic} link}\n\\label{tab:cubic-rmse-bench}\n\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$\\\\\n\\hline\n200 & $.01\\pm.47$ & $.16\\pm.31$ & $.29\\pm.27$ & $.45\\pm.16$\\\\\n500 & $-0.01\\pm.36$ & $.12\\pm.24$ & $.22\\pm.21$ & $.37\\pm.15$\\\\\n1000 & $.03\\pm.25$ & $.12\\pm.18$ & $.20\\pm.15$ & $.35\\pm.12$\\\\\n2000 & $.02\\pm.14$ & $.07\\pm.12$ & $.13\\pm.11$ & $.26\\pm.08$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for weighted estimator using our weights, with \\textbf{cubic} link}\n\\label{tab:cubic-bias}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirX:OptZ}_{0.2}$ & $\\textbf{DirX:OptZ}_{1.0}$ & $\\textbf{DirX:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.13\\pm.57$ & $.26\\pm.32$ & $.33\\pm.27$ & $.34\\pm.17$\\\\\n500 & $.12\\pm.35$ & $.22\\pm.25$ & $.28\\pm.21$ & $.34\\pm.15$\\\\\n1000 & $.18\\pm.26$ & $.24\\pm.19$ & $.29\\pm.15$ & $.37\\pm.11$\\\\\n2000 & $.14\\pm.16$ & $.18\\pm.14$ & $.22\\pm.12$ & $.31\\pm.09$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for doubly robust estimator using our weights and $\\textbf{DirX}$, with \\textbf{cubic} link}\n\\label{tab:cubic-bias-drx}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirZ:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.2}$ & $\\textbf{DirZ:OptZ}_{1.0}$ & $\\textbf{DirZ:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.04\\pm.49$ & $.25\\pm.34$ & $.45\\pm.29$ & $.74\\pm.20$\\\\\n500 & $.03\\pm.38$ & $.17\\pm.25$ & $.32\\pm.21$ & $.57\\pm.16$\\\\\n1000 & $.04\\pm.26$ & $.16\\pm.19$ & $.28\\pm.16$ & $.50\\pm.13$\\\\\n2000 & $.03\\pm.16$ & $.10\\pm.13$ & $.19\\pm.12$ & $.37\\pm.10$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for doubly robust estimator using our weights and $\\textbf{DirZ}$, with \\textbf{cubic} link}\n\\label{tab:cubic-bias-drz}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{SimplexOptZ}_{0.001}$ & $\\textbf{SimplexOptZ}_{0.2}$ & $\\textbf{SimplexOptZ}_{1.0}$ & $\\textbf{SimplexOptZ}_{5.0}$\\\\\n\\hline\n200 & $.02\\pm.45$ & $.22\\pm.35$ & $.40\\pm.34$ & $.65\\pm.28$\\\\\n500 & $.01\\pm.37$ & $.15\\pm.26$ & $.29\\pm.23$ & $.52\\pm.18$\\\\\n1000 & $.04\\pm.25$ & $.15\\pm.19$ & $.27\\pm.17$ & $.47\\pm.14$\\\\\n2000 & $.02\\pm.14$ & $.08\\pm.13$ & $.17\\pm.11$ & $.34\\pm.09$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for weighted estimator using our weights and constraining $W \\in n \\Delta^n$, with \\textbf{cubic} link}\n\\label{tab:cubic-bias-simplex}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{cccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.26\\pm.38$ & $1.1\\pm.20$ & $1.1\\pm.20$ & $1.2\\pm.21$ & $1.4\\pm.22$ & $.32\\pm.16$ & $1.4\\pm.12$\\\\\n500 & $.31\\pm.22$ & $1.1\\pm.10$ & $1.2\\pm.10$ & $1.3\\pm.10$ & $1.4\\pm.11$ & $.29\\pm.18$ & $1.4\\pm.10$\\\\\n1000 & $.37\\pm.14$ & $1.1\\pm.07$ & $1.2\\pm.07$ & $1.3\\pm.08$ & $1.4\\pm.08$ & $.32\\pm.16$ & $1.4\\pm.10$\\\\\n2000 & $.34\\pm.09$ & $1.1\\pm.05$ & $1.2\\pm.06$ & $1.3\\pm.06$ & $1.4\\pm.06$ & $.34\\pm.18$ & $1.4\\pm.11$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for benchmark methods, with \\textbf{cubic} link}\n\\label{tab:cubic-bias-bench}\n\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$\\\\\n\\hline\n200 & $.09\\pm.01$ & $.08\\pm.01$ & $.13\\pm.01$ & $.23\\pm.01$\\\\\n500 & $.06\\pm.01$ & $.06\\pm.00$ & $.09\\pm.00$ & $.16\\pm.00$\\\\\n1000 & $.04\\pm.01$ & $.04\\pm.00$ & $.06\\pm.00$ & $.12\\pm.00$\\\\\n2000 & $.02\\pm.00$ & $.03\\pm.00$ & $.04\\pm.00$ & $.08\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for weighted estimator using our weights, with \\textbf{linear} link}\n\\label{tab:linear-rmse}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirX:OptZ}_{0.2}$ & $\\textbf{DirX:OptZ}_{1.0}$ & $\\textbf{DirX:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.15\\pm.01$ & $.13\\pm.01$ & $.13\\pm.01$ & $.13\\pm.01$\\\\\n500 & $.14\\pm.01$ & $.13\\pm.01$ & $.13\\pm.01$ & $.13\\pm.00$\\\\\n1000 & $.13\\pm.01$ & $.13\\pm.00$ & $.13\\pm.00$ & $.13\\pm.00$\\\\\n2000 & $.12\\pm.00$ & $.12\\pm.00$ & $.12\\pm.00$ & $.12\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for doubly robust estimator using our weights and $\\textbf{DirX}$, with \\textbf{linear} link}\n\\label{tab:linear-rmse-drx}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirZ:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.2}$ & $\\textbf{DirZ:OptZ}_{1.0}$ & $\\textbf{DirZ:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.11\\pm.01$ & $.11\\pm.01$ & $.18\\pm.01$ & $.35\\pm.01$\\\\\n500 & $.06\\pm.01$ & $.08\\pm.01$ & $.13\\pm.01$ & $.24\\pm.01$\\\\\n1000 & $.05\\pm.01$ & $.06\\pm.00$ & $.09\\pm.00$ & $.18\\pm.00$\\\\\n2000 & $.03\\pm.00$ & $.04\\pm.00$ & $.06\\pm.00$ & $.12\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for doubly robust estimator using our weights and $\\textbf{DirZ}$, with \\textbf{linear} link}\n\\label{tab:linear-rmse-drz}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{SimplexOptZ}_{0.001}$ & $\\textbf{SimplexOptZ}_{0.2}$ & $\\textbf{SimplexOptZ}_{1.0}$ & $\\textbf{SimplexOptZ}_{5.0}$\\\\\n\\hline\n200 & $.09\\pm.01$ & $.09\\pm.01$ & $.15\\pm.01$ & $.29\\pm.01$\\\\\n500 & $.06\\pm.01$ & $.07\\pm.01$ & $.10\\pm.01$ & $.19\\pm.01$\\\\\n1000 & $.04\\pm.01$ & $.04\\pm.00$ & $.07\\pm.00$ & $.14\\pm.00$\\\\\n2000 & $.02\\pm.00$ & $.03\\pm.00$ & $.05\\pm.00$ & $.09\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for weighted estimator using our weights and constraining $W \\in n \\Delta^n$, with \\textbf{linear} link}\n\\label{tab:linear-rmse-simplex}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{cccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.15\\pm.01$ & $.57\\pm.01$ & $.60\\pm.01$ & $.66\\pm.01$ & $.72\\pm.01$ & $.13\\pm.00$ & $.76\\pm.00$\\\\\n500 & $.15\\pm.01$ & $.57\\pm.00$ & $.60\\pm.00$ & $.66\\pm.00$ & $.72\\pm.00$ & $.13\\pm.00$ & $.76\\pm.00$\\\\\n1000 & $.14\\pm.00$ & $.57\\pm.00$ & $.60\\pm.00$ & $.66\\pm.00$ & $.72\\pm.00$ & $.13\\pm.00$ & $.76\\pm.00$\\\\\n2000 & $.14\\pm.00$ & $.57\\pm.00$ & $.60\\pm.00$ & $.66\\pm.00$ & $.72\\pm.00$ & $.13\\pm.00$ & $.76\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for benchmark methods, with \\textbf{linear} link}\n\\label{tab:linear-rmse-bench}\n\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$\\\\\n\\hline\n200 & $.03\\pm.09$ & $.06\\pm.06$ & $.11\\pm.05$ & $.23\\pm.04$\\\\\n500 & $.02\\pm.05$ & $.04\\pm.05$ & $.08\\pm.04$ & $.15\\pm.04$\\\\\n1000 & $.01\\pm.04$ & $.03\\pm.03$ & $.05\\pm.03$ & $.11\\pm.03$\\\\\n2000 & $.01\\pm.02$ & $.02\\pm.02$ & $.04\\pm.02$ & $.08\\pm.02$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for weighted estimator using our weights, with \\textbf{linear} link}\n\\label{tab:linear-bias}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirX:OptZ}_{0.2}$ & $\\textbf{DirX:OptZ}_{1.0}$ & $\\textbf{DirX:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.12\\pm.09$ & $.12\\pm.06$ & $.12\\pm.05$ & $.13\\pm.04$\\\\\n500 & $.13\\pm.06$ & $.12\\pm.05$ & $.12\\pm.04$ & $.12\\pm.04$\\\\\n1000 & $.12\\pm.04$ & $.12\\pm.03$ & $.12\\pm.03$ & $.12\\pm.03$\\\\\n2000 & $.12\\pm.03$ & $.12\\pm.03$ & $.12\\pm.02$ & $.12\\pm.02$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for doubly robust estimator using our weights and $\\textbf{DirX}$, with \\textbf{linear} link}\n\\label{tab:linear-bias-drx}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirZ:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.2}$ & $\\textbf{DirZ:OptZ}_{1.0}$ & $\\textbf{DirZ:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.04\\pm.10$ & $.09\\pm.07$ & $.17\\pm.06$ & $.34\\pm.05$\\\\\n500 & $.02\\pm.06$ & $.06\\pm.05$ & $.12\\pm.05$ & $.24\\pm.05$\\\\\n1000 & $.02\\pm.04$ & $.05\\pm.04$ & $.08\\pm.04$ & $.18\\pm.04$\\\\\n2000 & $.01\\pm.03$ & $.03\\pm.03$ & $.06\\pm.03$ & $.12\\pm.02$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for doubly robust estimator using our weights and $\\textbf{DirZ}$, with \\textbf{linear} link}\n\\label{tab:linear-bias-drz}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{SimplexOptZ}_{0.001}$ & $\\textbf{SimplexOptZ}_{0.2}$ & $\\textbf{SimplexOptZ}_{1.0}$ & $\\textbf{SimplexOptZ}_{5.0}$\\\\\n\\hline\n200 & $.02\\pm.08$ & $.07\\pm.06$ & $.14\\pm.06$ & $.29\\pm.06$\\\\\n500 & $.02\\pm.05$ & $.05\\pm.05$ & $.09\\pm.05$ & $.19\\pm.04$\\\\\n1000 & $.01\\pm.04$ & $.03\\pm.03$ & $.06\\pm.03$ & $.14\\pm.03$\\\\\n2000 & $.01\\pm.02$ & $.02\\pm.02$ & $.04\\pm.02$ & $.09\\pm.02$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for weighted estimator using our weights and constraining $W \\in n \\Delta^n$, with \\textbf{linear} link}\n\\label{tab:linear-bias-simplex}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{cccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.13\\pm.08$ & $.57\\pm.05$ & $.60\\pm.05$ & $.66\\pm.05$ & $.72\\pm.05$ & $.13\\pm.04$ & $.76\\pm.04$\\\\\n500 & $.14\\pm.05$ & $.57\\pm.04$ & $.60\\pm.03$ & $.66\\pm.03$ & $.72\\pm.03$ & $.12\\pm.04$ & $.76\\pm.02$\\\\\n1000 & $.14\\pm.04$ & $.57\\pm.02$ & $.60\\pm.02$ & $.66\\pm.02$ & $.72\\pm.02$ & $.13\\pm.03$ & $.76\\pm.03$\\\\\n2000 & $.13\\pm.03$ & $.57\\pm.02$ & $.60\\pm.02$ & $.66\\pm.02$ & $.72\\pm.02$ & $.13\\pm.03$ & $.76\\pm.02$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for benchmark methods, with \\textbf{linear} link}\n\\label{tab:linear-bias-bench}\n\n\\end{table}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMachine learning algorithms have been increasingly used as diagnostic tools in biomedical research\\cite{gulshan2016,esteva2017,golden2017}. The widespread availability of smartphones and other health tracking devices generates high volumes of sensor data, and makes machine learning uniquely well posed to impact clinical research using digital health tools. In clinical applications, gender, age, and other demographic characteristics of the study participants often play the role of confounders. Confounding is particularly prevalent in mobile health studies run under uncontrolled conditions outside clinical and laboratory settings, where we have little control over the demographic and clinical characteristics of the cohort of participants that self-select to participate in a study.\n\nIn the context of predictive modeling, we define a confounder as a variable that causes spurious associations between the features and response variable. In machine learning applications, the presence of confounding can lead to ambiguous inference and poor generalizability of models. Confounding is usually present when the joint probability distribution of the confounder and response variables is different in the data available to develop the learner (which we from now on denote as the \"development dataset\") relative to the population where the learner will be applied (denoted as the \"target population\")\\cite{rao2017}. For example, consider a diagnostic application where most cases are old aged while most controls are young, but where age is not associated with disease status in the target population (e.g., the target population is composed of older patients only). If the classifier can more efficiently detect age-related signals than disease-related signals, then it will likely perform poorly when deployed in the target population.\n\nConfounding adjustment is an active area of research in machine learning. The goal is to prevent an algorithm from learning the confounding signal. Since any variable that confounds the feature-response relationship has to be associated with both the features and the response, most of the methods proposed in the literature can be divided into two approaches: (i) methods  that remove the association between the confounder and the response; or (ii) methods that remove the association between the confounder and the features. A standard example of the first approach is to match subjects from the development data in order to obtain a subsample that more closely resembles the target population. This strategy, however, results in a smaller number of participants to train and evaluate the machine learning algorithm, and, in highly unbalanced situations, might lead to the exclusion of most of the participants from the analyses. Alternative methods that make more efficient use of the data include inverse probability weighting approaches\\cite{linn2016,rao2017}, which weight the training samples in order to make model training better tailored to the target population. A canonical example of the second approach is to separately regress each feature on the confounders, and use the residuals as the predictors in the machine learning algorithm. Other approaches that do not fall into categories (i) or (ii) include penalized learners\\cite{li2011} and backdoor adjustment\\cite{landeiro2016}.\n\nIn this paper, we present statistical methods to detect and quantify the influence of observed confounders,  and to estimate the actual (i.e., unconfounded) predictive performance of a learner. We use a large Parkinsons digital health study cohort to illustrate how our methods can be used to evaluate the effectiveness of standard confounding adjustment methods.\n\n\n\\section{Methods}\n\nWe adopt restricted permutations\\cite{good2000,rao2017} to isolate the contribution of the confounder from the predictive performance of a learner. The key idea is to shuffle the response data within the levels of a categorical\/ordinal confounder (as illustrated in the Supplementary Figure \\ref{fig:stratifiedperm}) in order to destroy the direct association between the response and the features while still preserving the indirect association due to the confounder. Algorithm \\ref{alg:stratifiedShuffling} describes the procedure for an arbitrary performance metric, $m$ (such as the area under the receiver operating characteristic curve, AUC, or root mean square error).\n\n\\begin{algorithm}\n\\caption{Restricted Monte Carlo permutation null distribution for performance metric $m$}\\label{alg:stratifiedShuffling}\n\\begin{algorithmic}[1]\n\\State \\textbf{Input}: Number of permutations, $b$. Development data set feature matrix, response vector, and confounder vector, $\\boldsymbol{X}$, $\\boldsymbol{y}$, $\\boldsymbol{c}$. Training and test set indexes, $i_{train}$, $i_{test}$\n\\State Split $\\boldsymbol{X}$, $\\boldsymbol{y}$ and $\\boldsymbol{c}$ into training and test sets\n\\For{$i = 1, 2, \\ldots, b$}\n  \\State $\\boldsymbol{y}^{\\ast}_{train} \\leftarrow \\mbox{RestrictedShuffle}(\\boldsymbol{y}_{train}, \\boldsymbol{c}_{train})$, and $\\boldsymbol{y}^{\\ast}_{test} \\leftarrow \\mbox{RestrictedShuffle}(\\boldsymbol{y}_{test}, \\boldsymbol{c}_{test})$\n  \\State Train a machine learning algorithm on the $\\boldsymbol{X}_{train}$ and $\\boldsymbol{y}_{train}^{\\ast}$ data\n  \\State Evaluate the algorithm on the $\\boldsymbol{X}_{test}$ and $\\boldsymbol{y}_{test}^{\\ast}$ data\n  \\State Record the value of the performance metric, $m^\\ast_i$, on the shuffled data\n\\EndFor\n\\State \\textbf{Output}: $m^\\ast_1$, $m^\\ast_2$, \\ldots, $m^\\ast_b$\n\\end{algorithmic}\n\\end{algorithm}\n\nBuilding upon the restricted permutation null distribution, we developed two statistical tools to deal with confounding. First, we estimate the ``unconfounded\" predictive performance of a learner by building a mapping from the restricted permutation null to the standard permutation null (where the standard permutation null distribution is generated by shuffling the labels in the usual unconstrained manner). As fully described in the Supplement, for any performance metric that can be expressed as a (generalized) U-statistic\\cite{hoeffding1948,lehmann1951,serfling1980,delong1988} (e.g., AUC), or expressed as a simple average (e.g., mean square error, mean absolute error, and classification accuracy), we have that an asymptotic estimate of the unconfounded performance metric is given by,\n\\begin{equation}\n\\hat{m}_u \\, = \\, (m_o - a_{\\hat{\\pi}^{\\ast}}) (s_{\\hat{\\pi}^{\\ast\\ast}}\/s_{\\hat{\\pi}^{\\ast}}) + a_{\\hat{\\pi}^{\\ast\\ast}}~,\n\\label{eq:correctedmetric}\n\\end{equation}\nwhere $m_o$ represents the uncorrected metric value; $a_{\\hat{\\pi}^{\\ast}}$ and $s^2_{\\hat{\\pi}^{\\ast}}$ represent the sample average and variance of the restricted permutation null; and $a_{\\hat{\\pi}^{\\ast\\ast}}$ and $s^2_{\\hat{\\pi}^{\\ast\\ast}}$ represent the analogous quantities for the standard permutation null. It is important to point out that we don't view the unconfounded metric estimation as an adjustment method (in the sense that it does not prevent an algorithm from learning the confounding signal in the first place). It simply quantifies the amount of response signal learned by the algorithm, after the algorithm has had a chance to learn both confounding and response signals.\n\n\nSecond, by noticing that the location of the restricted permutation null provides a natural measure of the amount of confounding signal learned by the algorithm, we adopt the average of the restricted permutation null as a test statistic, and develop a statistical test to compare the hypotheses,\n\\begin{align}\nH_0^c &: \\mbox{the algorithm has not learned the confounding signal~,} \\\\\nH_1^c &: \\mbox{the algorithm has learned the confounding signal~,} \\nonumber\n\\end{align}\nand detect confounding learning per se. In the Supplement we show that, under $H_0^c$, the average of the restricted permutation null distribution is asymptotically distributed as a $N(a_{\\hat{\\pi}^{\\ast\\ast}}, s^2_{\\hat{\\pi}^{\\ast\\ast}}\/n)$ distribution, where $n$ represents the test set sample size.\n\nFor the AUC metric, additional analytical results are available. It is well known\\cite{bamber1975,mason2002} that the standard permutation null can be approximated by,\n\\begin{equation}\nN(0.5 \\, , \\, (n_n + n_p + 1)\/(12 \\, n_n \\, n_p))~,\n\\label{eq:standnullapprox}\n\\end{equation}\nwhere $n_n$ and $n_p$ represent the number of negative and positive labels in the test set. Hence, $a_{\\hat{\\pi}^{\\ast\\ast}} \\approx 0.5$ and $s^2_{\\hat{\\pi}^{\\ast\\ast}} \\approx (n_n + n_p + 1)\/(12 \\, n_n \\, n_p)$ and the estimator in (\\ref{eq:correctedmetric}) becomes,\n\\begin{equation}\nauc_u = (auc_o -  a_{\\hat{\\pi}^{\\ast}})(n_n + n_p + 1)\/(12 \\, n_n \\, n_p \\, s_{\\hat{\\pi}^\\ast}) + 0.5~,\n\\label{eq:auccorrectionformula}\n\\end{equation}\nthe null distribution under $H_0^c$ reduces to $N(0.5, (n_n + n_p + 1)\/(12 \\, n_n \\, n_p \\, n))$, and,\n\\begin{equation}\np = 1 - \\Phi\\big((a_{\\hat{\\pi}^{\\ast}} - 0.5)\/[(n_n + n_p + 1)\/(12 \\, n_n \\, n_p \\, n)]^{1\/2}\\big)~,\n\\label{eq:confoundingpval}\n\\end{equation}\ncorresponds to the confounding p-value, where $\\Phi(.)$ represents the c.d.f. of standard normal variable.\n\n\\section{Real data illustrations}\n\nA key practical application of our tools is to evaluate if an adjustment method is working as expected. This is important in practice since most of these methods rely on assumptions, and it is generally unclear how robust they are to violations of these assumptions. Here we illustrate the application of our tools to two confounding adjustment methods: sample matching, and approximate inverse probability weighting (IPW) based on the propensity score\\cite{propensityscore1983}.\n\nOur development data was collected in a digital health study on Parkinsons disease\\cite{bot2016,trister2016} and consist of features generated from 30 second inertial sensor readings captured during walking. We focused on walking, as walking patterns are influenced by age and gender\\cite{ko2011} in addition to Parkinson's disease. The development data was split into training and test sets with similar joint distributions for the age, gender, and disease status (Supplementary Figure \\ref{sfig:agegenderassociations}). We apply the adjustment methods to both training and test sets, and the analyses are based on a combined gender\/discretized age\\footnote{While, in theory, we can only perform restricted permutations using categorical\/ordinal confounders, in practice we can discretize and evaluate continuous confounders as well. Clearly, if the discretization is too coarse the discretized confounder might not be able to fully capture the association between the confounder and the response, and we might end up underestimating the amount of confounding learned by the algorithm. In practice, one should experiment with distinct discretizations, as illustrated in Supplementary Figure \\ref{sfig:agediscretization}.} confounder with levels: young male, young female, middle age male, middle age female, senior male, and senior female.\n\nFigure \\ref{fig:adjustmentcomparison} shows the results based on logistic regression (top panels) and random forest (bottom panels) classifiers. In all panels, the blue histograms represent the restricted permutation null distributions generated by Algorithm 1, the red curves represent the normal approximation for the standard permutation null distribution presented in equation 3, the orange line shows the unconfounded estimate of AUC computed using equation \\ref{eq:auccorrectionformula}, and  the cyan line represents the observed AUC.\n\nFor the sake of comparison, panels a and d report the results when no confounding adjustment is performed. Both logistic regression and random forest classifiers are clearly learning confounding signal since the restricted permutation nulls are centered around 0.7, and the confounding test p-values (equation \\ref{eq:confoundingpval}) are highly significant ($p<10^{-16}$). Hence, the high AUC scores (cyan lines above 0.81) reflect the classifiers' ability to detect both disease and confounding signals, while the unconfounded estimates (orange scores around 0.66) are considerably more modest.\n\nPanels b and e show the results based on a matched subset of participants. The fact that the restricted permutation nulls are centered around 0.5, and closely match the standard permutation null density (red curve), suggests that matching effectively prevented the classifier from learning the confounding signal ($p<0.51$ and $p<0.58$, respectively) and that the classifiers are only learning the disease signal. As expected, the observed and unconfounded AUC scores match each other closely in this situation. Finally, note that the much larger spread of the null distributions (in comparison to panels a and d) is due to the smaller test set available after matching.\n\nPanels c and f report the results for the approximate IPW approach. This method makes use of the entire data set and attempts to prevent confounding learning by weighting the samples according to the inverse of their estimated propensity scores (i.e., the conditional probability that a participant has the disease given its gender and age). While several approaches have been proposed in the literature for the estimation of propensity scores\\cite{lee2010,pirracchio2014}, here, we adopt the most commonly used method based on logistic regression. The panels show that the approximate IPW approach managed to reduce the amount of confounding (the blue histograms are closer to 0.5 compared to panels a and d). However, it didn't remove it completely ($p < 10^{-16}$). This suggests that the estimated inverse probability weights did not generate a well balanced augmented data set. (Supplementary Figure \\ref{sfig:failuretobalance} confirms this is indeed the case.) Most likely, the reason for this suboptimal performance is that propensity score estimation using logistic regression makes the strong assumption that the observed associations between the confounders and disease labels can be well described by the logistic function. This example illustrates how the violation of a parametric modeling assumption can lead to an inefficient confounding adjustment.\n\n\n\\begin{figure}[h]\n  \\centering\n  \\centerline{\\includegraphics[width=\\linewidth, bb = 0 20 720 210]{adjustment_checks_glm_and_rf3.pdf}}\n  \\caption{Comparison of confounding adjustment methods.}\n  \\label{fig:adjustmentcomparison}\n\\end{figure}\n\n\n\\section{Final remarks}\n\nDigital health enabled diagnostic systems have the potential to provide low cost remote diagnostic tools to underserved communities that lack easy access to medical care. However this opportunity cannot be fully realized without (i) efficient approaches to combat confounding (without which we run the risk of making spurious inferences from the data) and (ii) rigorous methods to evaluate these adjustment methods. The tools proposed in this paper address the second need.\n\nTo the best of our knowledge, the use of restricted permutations in the context of predictive modeling has only been leveraged by\\cite{rao2017}. These authors, however, use restricted permutations to test if a machine learning algorithm has learned the response signal in the presence (or absence) of confounders, but not to detect and quantify confounding learning per se, as proposed in this paper.\n\nFor the sake of clarity, our illustrations focused on the case where confounder and response are associated in the development set but not in the target population. We can, however, still apply our methodology when response and confounder are known to be associated in the target population but have a different joint probability distribution compared to the development data. Section 9 of the Supplement provides an illustrative example based on synthetic data.\n\nWe also conducted a simulation study to evaluate the statistical properties of the confounding test (Section 10 of the Supplement). Our simulations show reasonable statistical power for the range of parameters investigated in our experiments, and well-controlled type I error rates.\n\nFinally, we point out that while this paper has focused on digital health applications, the proposed tools can be more generally applied to any other areas impacted by confounders.\n\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThis is a continuation of papers \\cite{toric} and \\cite{toricII}.\nWe have studied the effect of each elementary birational map in the minimal model program to the derived categories \nin the case of toric pairs.\nIn this paper we consider problems of more global nature.\nWe add some cases to the list of affirmative answers to the \nderived McKay correspondence conjecture and its generalization, \nthe \\lq\\lq $K$ implies $D$ conjecture''.\n\nFirst we remark that the results in \\cite{toric} and \\cite{toricII} are \neasily extended to the relative situation (Theorem~\\ref{relative}).\nThen we remark that the derived McKay correspondence holds for any abelian quotient singularity \nin the following sense; \nthe derived category defined by a finite abelian group contained in a general linear group \nis semi-orthogonally decomposed \ninto derived categories of a relative minimal model and subvarieties of the quotient space \n(Theorem~\\ref{abelian}).  \nAs an application, we prove that such derived McKay correspondence also holds for arbitrary quotient singularities in dimension $2$ (Theorem~\\ref{GL2}).\n\nNext we prove that any proper birational morphism between toric pairs which is a $K$-equivalence\nis decomposed into a sequence of flops (Theorem~\\ref{decomposition}).\nThis is a generalization, in the toric case, of a theorem in \\cite{flop}, which states that \nany birational morphism between minimal models is decomposed into a sequence of flops.\nWe note that the latter case is easier because the log canonical divisor stays \nat the bottom and cannot be decreased when the models are minimal, \nwhile it is more difficult to preserve the level of the log canonical divisor in the former case. \n\nNow we state the theorems in details.\nWe refer the terminology to the next section.\n\nThe following is the reformulation of the results of \\cite{toric} and \\cite{toricII} to the relative case: \n\n\\begin{Thm}\\label{relative}\nLet $(X,B)$ and $(Y,C)$ be $\\mathbf{Q}$-factorial toric pairs whose coefficients \nbelong to the standard set \n$\\{1 - 1\/m \\mid m \\in \\mathbf{N}\\}$, \nlet $\\tilde X$ and $\\tilde Y$ be smooth Deligne-Mumford stacks associated to \nthe pairs $(X,B)$ and $(Y,C)$, respectively, and\nlet $f: X \\dashrightarrow Y$ be a toric rational map.\nAssume one of the following conditions:\n\n(a) $f$ is the identity morphism, and $B \\ge C$.\n\n(b) $f$ is a flip, $C=f_*B$, and $K_X+B \\ge K_Y+C$.\n\n(c) $f$ is a divisorial contraction, $C=f_*B$, and $K_X+B \\ge K_Y+C$.\n\n(d) $f$ is a divisorial contraction, $C=f_*B$, and $K_X+B \\le K_Y+C$.\n\n(e) $f$ is a Mori fiber space and $C$ is determined by $(X,B)$ and $f$ as explained in \\cite{toric}. \n\nThen the following hold:\n\n(a), (b), (c): There are toric closed subvarieties $Z_i$ ($1 \\le i \\le l$) of $Y$ such that $Z_i \\ne Y$ \nfor some $l \\ge 0$,  \nand fully faithful functors $\\Phi: D^b(\\text{coh}(\\tilde Y)) \\to D^b(\\text{coh}(\\tilde X))$\nand $\\Psi_i: D^b(\\text{coh}(\\tilde Z_i)) \\to D^b(\\text{coh}(\\tilde X))$\nfor the smooth Deligne-Mumford stacks $\\tilde Z_i$ associated to the $Z_i$ \nsuch that there is a semi-orthogonal decomposition of triangulated categories \n\\[\nD^b(\\text{coh}(\\tilde X)) = \\langle \\Psi_1(D^b(\\text{coh}(\\tilde Z_1))), \\dots,\n\\Psi_l(D^b(\\text{coh}(\\tilde Z_l))), \\Phi(D^b(\\text{coh}(\\tilde Y))) \\rangle\n\\]\n\n(d): There are $Z_i \\subset Y$ ($1 \\le i \\le l$) as in (c), \nand fully faithful functors $\\Phi: D^b(\\text{coh}(\\tilde X)) \\to D^b(\\text{coh}(\\tilde Y))$\nand $\\Psi_i: D^b(\\text{coh}(\\tilde Z_i)) \\to D^b(\\text{coh}(\\tilde Y))$\nsuch that there is a semi-orthogonal decomposition\n\\[\nD^b(\\text{coh}(\\tilde Y)) = \\langle \\Psi_1(D^b(\\text{coh}(\\tilde Z_1))), \\dots,\n\\Psi_l(D^b(\\text{coh}(\\tilde Z_l))), \\Phi(D^b(\\text{coh}(\\tilde X))) \\rangle\n\\]\n\n(e):  There are fully faithful functors $\\Phi_j: D^b(\\text{coh}(\\tilde Y)) \\to D^b(\\text{coh}(\\tilde X))$ \n($1 \\le j \\le m$) for some $m \\ge 2$ and a semi-orthogonal decomposition\n\\[\nD^b(\\text{coh}(\\tilde X)) = \\langle \\Phi_1(D^b(\\text{coh}(\\tilde Y))), \\dots, \n\\Phi_m(D^b(\\text{coh}(\\tilde Y))) \\rangle\n\\]\n\nMoreover if $K_X+B = K_Y+C$ in the cases (b) or (c), then $\\Phi$ is an equivalence.\n\\end{Thm} \n\nWe note that, when $f$ is a divisorial contraction, \nthe direction of the inclusion of the derived categories \nis unrelated to the direction of the morphism, \nbut is the same as the direction of the inequality of the canonical divisors.\n\nIf $X$ and $Y$ are smooth and $f$ is a blowing up of a smooth center in the case (b), or if  \n$X$ and $Y$ are smooth and $f$ is a standard flip in the case (c), then\nit is a theorem by Bondal-Orlov (\\cite{BO}).\nThe derived equivalence in the case of flops (case (c) with $K_X+B = K_Y+C$) was \nproved for general (non-toric) case under the additional assumptions as follows:\nif $\\dim X = 3$ and $X$ is smooth by Bridgeland (\\cite{Bridgeland}),\nif $\\dim X = 3$ and $X$ has only Gorenstein terminal singularities by Chen (\\cite{Chen}) \nand Van den Bergh (\\cite{VdBergh}), \nif $\\dim X = 3$ and $X$ has only terminal singularities by \\cite{DK}, \nand if $X$ is a holomorphic simplectic manifold by Kaledin (\\cite{Kaledin}).\nThere are also interesting results concerning the last case by Cautis (\\cite{Cautis}) by using the \ncategorification of linear group actions, \nand by Donovan-Segal (\\cite{DS}), Ballard-Favero-Katzarkov (\\cite{BFK}) and\nHalpern-Leistner (\\cite{HL}) by using the theory of variations of GIT quotients.\n\nAs a corollary we obtain\n\n\\begin{Cor}\\label{combine}\nLet $(X,B)$ be a $\\mathbf{Q}$-factorial toric pair whose coefficients belong to the standard set \n$\\{1 - 1\/m \\mid m \\in \\mathbf{N}\\}$.\nAssume that there is a projective toric morphism $f: X \\to Z$ to another  \n$\\mathbf{Q}$-factorial toric variety.\nThen there exist toric closed subvarieties $Z_i$ ($1 \\le i \\le m$) of $Z$ \nfor some $m \\ge 1$ such that there are fully faithful functors \n$\\Phi_i: D^b(\\text{coh}(\\tilde Z_i)) \\to D^b(\\text{coh}(\\tilde X))$ with\na semi-orthogonal decomposition \n\\[\nD^b(\\text{coh}(\\tilde X)) \\cong \\langle \\Phi_1(D^b(\\text{coh}(\\tilde Z_1))), \\dots, \n\\Phi_m(D^b(\\text{coh}(\\tilde Z_m))) \\rangle\n\\]\nwhere $\\tilde X$ and the $\\tilde Z_i$ are smooth Deligne-Mumford stacks associated \nto $(X,B)$ and the $Z_i$. \n\\end{Cor}\n\nIn other words, one can say that the derived category \n$D^b(\\text{coh}\\tilde X)$ is generated by a {\\em relative exceptional collection}.\nThe proofs of the above statements are the same as in \\cite{log crepant}, \n\\cite{toric} and \\cite{toricII} except Theorem~\\ref{relative}~(a).\n\nThe following is the derived McKay correspondence for finite abelian groups:\n\n\\begin{Thm}\\label{abelian}\nLet $G \\subset GL(n,\\mathbf{C})$ be a finite abelian subgroup acting naturally on an affine \nspace $\\mathbf{A} = \\mathbf{C}^n$, and let $X=\\mathbf{A}\/G$ be the quotient space.\nLet $f: Y \\to X$ be a $\\mathbf{Q}$-factorial terminal relative minimal model of $X$.\nThen there exist toric closed subvarieties $Z_i$ ($1 \\le i \\le m$) of $X$ with $Z_i \\ne X$ \nfor some $m \\ge 0$, with possible repetitions, \nsuch that there is a semi-orthogonal decomposition \n\\[\nD^b(\\text{coh}([\\mathbf{A}\/G])) \\cong \\langle D^b(\\text{coh}(\\tilde Z_1)), \\dots, \nD^b(\\text{coh}(\\tilde Z_m)), D^b(\\text{coh}(\\tilde Y)) \\rangle\n\\]\nwhere $\\tilde Y$ and the $\\tilde Z_i$ are smooth Deligne-Mumford \nstacks associated to $Y$ and the $Z_i$, respectively. \nMoreover if $G \\subset SL(n,\\mathbf{C})$, then $m=0$.\n\\end{Thm}\n\nIn the case of dimension $2$, we do not need the assumption that the group is abelian:\n\n\\begin{Thm}\\label{GL2}\nLet $G \\subset GL(2,\\mathbf{C})$ be a finite subgroup acting naturally on an affine \nspace $\\mathbf{A} = \\mathbf{C}^2$, let $X = \\mathbf{A}\/G$, and\nlet $f: Y \\to X$ be the minimal resolution.\nThen there exist smooth closed subvarieties $Z_i$ ($1 \\le i \\le m$) of $X$ with $Z_i \\ne X$ \nfor some $m \\ge 0$, with possible repetitions, \nsuch that there is a semi-orthogonal decomposition \n\\[\nD^b(\\text{coh}([\\mathbf{A}\/G])) \\cong \\langle D^b(\\text{coh}(Z_1)), \\dots, D^b(\\text{coh}(Z_m)),\nD^b(\\text{coh}(Y)) \\rangle.\n\\]\nMoreover if there are no quasi-reflections in $G$, the elements whose invariant \nsubspaces are of codimension $1$, then $\\dim Z_i = 0$ for all $i$.\nThus the semi-orthogonal complement of $D^b(\\text{coh}(Y))$\nin $D^b(\\text{coh}([\\mathbf{A}\/G]))$ is generated by an exceptional collection in this case.\n\\end{Thm}\n\nThe derived McKay correspondence was already proved for finite subgroups of \n$SL(2,\\mathbf{C})$, $SL(3,\\mathbf{C})$ (Bridgeland-King-Reid \\cite{BKR}), \nand $Sp(2n,\\mathbf{C})$ (Bezrukavnikov-Kaledin \\cite{BK}).\n\nFinally, we prove the following decomposition theorem for $K$-equivalent toric birational map:\n\n\\begin{Thm}\\label{decomposition}\nLet $f: X \\dashrightarrow Y$ be a toric birational map between\nprojective $\\mathbf{Q}$-factorial toric varieties which is an isomorphism in codimension $1$, and\nlet $B$ be a toric $\\mathbf{R}$-divisor on $X$\nwhose coefficients belong to the interval $(0,1)$, and let $f_*B=C$.\nAssume that $K_X+B = K_Y+C$.\nThen the birational map $f: (X,B) \\dashrightarrow (Y,C)$ is decomposed into a \nsequence of flops.\n\\end{Thm}\n\nAs a corollary of Theorems~\\ref{relative} and \\ref{decomposition}, \nwe obtain an affirmative answer to \n\\lq\\lq $K$ implies $D$ conjecture'' in the toric case:\n\n\\begin{Cor}\nAssume additionally that the coefficients of $B$ belong to a set\n$\\{1 - 1\/m \\mid m \\in \\mathbf{Z}_{>0}\\}$.\nLet $\\tilde X$ and $\\tilde Y$ be the smooth Deligne-Mumford staks \nassociated to the pairs $(X,B)$ and $(Y,C)$, respectively.\nThen there is an equivalence of triangulated categories\n$D^b(\\text{coh}(\\tilde X)) \\cong D^b(\\text{coh}(\\tilde Y))$.\n\\end{Cor}\n\n\\section{Preliminaries}\n\nWe fix the terminology in this section.\nA pair $(X,B)$ consisting of a normal algebraic variety \nand an $\\mathbf{R}$-divisor\nis said to be {\\em KLT} (resp. {\\em terminal}) \nif there is a projective birational morphism \n$p: Z \\to X$ from a smooth varity with an $\\mathbf{R}$-divisor $D$ \nwhose support is a normal crossing divisor such that \nan equality $p^*(K_X+B)=K_Z+D$ holds \nand all the coefficients of $D$ are smaller than $1$\n(resp. all the coefficients of $D - p^{-1}_*B$ are smaller than $0$),\nwhere the canonical divisors $K_X$ and $K_Z$ are defined by using the \nsame rational differential forms.\nIt is called {\\em $\\mathbf{Q}$-factorial} if any prime divisor on $X$ is a \n$\\mathbf{Q}$-Cartier divisor.\n \nA birational map $h: X \\dashrightarrow Y$ between varieties is said to be {\\em proper}\nif there exists a third variety $Z$ with proper birational morphisms $f: Z \\to X$ and \n$g: Z \\to Y$ such that $g = h \\circ f$. \n\nLet $f: (X,B) \\dashrightarrow (Y,C)$ be a proper birational map between KLT pairs.\nWe say that there is an {\\em equality of log canonical divisors}\n$K_X+B = K_Y+C$\n(resp. {\\em an inequality} $K_X+B \\ge K_Y+C$), \nif $p^*(K_X+B) = q^*(K_Y+C)$\n(resp. $p^*(K_X+B) \\ge q^*(K_Y+C)$) \nas an ineqality (resp. an equality) of $\\mathbf{R}$-divisors for some $p,q$ as above.\nWe note that such equality or inequality can be defined only \nwhen the birational map \n$f$ is fixed, and the definition is independent of the choice of $p,q$.\nSuch an equality (resp. an inequality) is called a {\\em $K$-equivalence} \n(resp. {\\em $K$-inequality}).\n$f$ is also said to {\\em crepant} if it is a $K$-equivalence.\n\nA proper birational map $f: X \\dashrightarrow Y$ is said to be \n{\\em isomorphic in codimension $1$}\n(resp. {\\em surjective in codimension $1$}) if it induces a bijection\n(resp. surjection) between the sets of prime divisors.\n\nA {\\em flop} (resp. {\\em flip}) is a proper birational map\n$f: (X,B) \\dashrightarrow (Y,C)$ between \n$\\mathbf{Q}$-factorial KLT pairs satisfying the following conditions: \n$f_*B=C$, $K_X+B=K_Y+C$ (resp. $K_X+B > K_Y+C$), and \nthere exist projective birational morphisms \n$s: X \\to W$ and $t: Y \\to W$ to a normal variety \nwhich are isomorphisms in codimension $1$ and the relative Picard numbers \n$\\rho(X\/W)$ and $\\rho(Y\/W)$ are equal to $1$.\n\nA {\\em divisorial contraction} is a projective \nbirational morphism $f: (X,B) \\to (Y,C)$ between \n$\\mathbf{Q}$-factorial KLT pairs satisfying the following conditions: \n$f_*B=C$, and the exceptional locus of $f$ consists of a single prime divisor.\nWe have always $\\rho(X\/Y)=1$, and \nwe have either $K_X+B=K_Y+C$, $K_X+B > K_Y+C$, or $K_X+B < K_Y+C$.\nThe inverse birational map $f^{-1}: (Y,C) \\dashrightarrow (X,B)$ of a \ndivisorial contraction is called a {\\em divisorial extraction}.\n\nA {\\em Mori fiber space } is a projective morphism $f: (X,B) \\to Y$ from a  \n$\\mathbf{Q}$-factorial KLT pair to a normal variety satisfying the following conditions: \n$\\rho(X\/Y)=1$, $-(K_X+B)$ is ample, and $\\dim X > \\dim Y$.\n\nFor a $\\mathbf{Q}$-factorial KLT pair $(X,B)$, there exists a sequence of \ncrepat divisorial contractions \n\\[\n(Y,C)=(X_m,B_m) \\to \\dots \\to (X_1,B_1) \\to (X_0,B_0)=(X,B)\n\\]\nsuch that $(Y,C)$ is terminal and $\\mathbf{Q}$-factorial (\\cite{BCHM}).\nThe composition $f: (Y,C) \\to (X,B)$ is called a {\\em $\\mathbf{Q}$-factorial crepant terminalization} \nof the pair $(X,B)$.\nWe note that even if $B$ has standard coefficients, $C$ may not be so.\n\nLet $f: (Y,C) \\to (X,B)$ be a projective birational morphism\nfrom a $\\mathbf{Q}$-factorial terminal pair to a KLT pair. \nIt is called a {\\em $\\mathbf{Q}$-factorial terminal relative minimal model} \nif $C=f_*^{-1}B$ and $K_Y+C$ is relatively nef over $X$.\nWe note that $f$ is not necessarily crepant.\nThe existence of such a model is also proved in \\cite{BCHM}.\n\nA rational number is said to be {\\em standard} if it is of the form  \n$1-1\/m$ for a positive integer $m$.\nA KLT pair $(X,B)$ is said to be of {\\em quotient type}, if \nthere is a quasi-finite surjective morphism $\\pi': X' \\to X$ from a \nsmooth variety $X'$, which is not necessarily irreducible, such that \n$K_{X'}=(\\pi')^*(K_X+B)$.\nIn this case $B$ has only standard coefficients, and \n$X$ is automatically $\\mathbf{Q}$-factorial.\nIf the pair $(X,B)$ is toric, $X$ is $\\mathbf{Q}$-factorial, and if \n$B$ has standard coefficients, then it is of quotient type.\n\nThere is a {\\em smooth Deligne-Mumford stack} $\\tilde X$ \nassociated to the pair $(X,B)$ of quotient type in such a way that\na sheaf $\\mathcal{F}$ on $\\tilde X$ is nothing but a sheaf $\\mathcal{F}'$ on $X'$  \nwith an isomorphism $p_1^*\\mathcal{F}' \\cong p_2^*\\mathcal{F}'$ satisfying the cocycle condition, \nwhere $p_i: X' \\times_X X' \\to X'$ ($i=1,2$) are projections.\nIn this case, there is a finite birational morphism \n$\\pi: \\tilde X \\to X$ such that $\\pi^*(K_X+B)=K_{\\tilde X}$.\n\nFor an algebraic variety or a Deligne-Mumford stack $X$, \nlet $D^b(\\text{coh}(X))$ \ndenotes the bounded derived category of coherent sheaves on $X$.\nThere are variants of this notation; if a finite group $G$ acts on $X$, then\n$D^b(\\text{coh}^G(X))$ \ndenotes the bounded derived category of $G$-equivariant coherent \nsheaves on $X$.\nThus $D^b(\\text{coh}^G(X)) \\cong D^b(\\text{coh}([X\/G]))$.\n\nA triangulated category $\\mathcal{T}$ is said to have a {\\em semi-orthogonal decomposition}\n$\\mathcal{T} = \\langle \\mathcal{T}_1, \\dots, \\mathcal{T}_m \\rangle$ by \ntriangulated full subcategories $\\mathcal{T}_i$ if the \nfollowing conditions are satisfied:\n(a) $\\text{Hom}(a,b) = 0$ for $a \\in \\mathcal{T}_i$ and $b \\in \\mathcal{T}_j$ if $i > j$, and \n(b) for any object $a \\in \\mathcal{T}$ there is a sequence of objects $a_i \\in \\mathcal{T}$ \nand $b_i \\in \\mathcal{T}_i$ for $1 \\le i \\le m$ such that \n$a=a_1$ and there are distinguished triangles $a_{i+1} \\to a_i \\to b_i$, where\n$a_{m+1}=0$.\nIn particular, if the $\\mathcal{T}_i$ are equivalent to $D^b(\\text{Spec }k)$ for all $i$, then \n$\\mathcal{T}$ is said to have a {\\em full exceptional collection}.\n\nA {\\em toric pair} $(X,B)$ consists of a toric variety and an $\\mathbf{R}$-divisor \nwhose irreducible components are invariant under the torus action.\n$X$ is $\\mathbf{Q}$-factorial if and only if the corresponding fan is simplicial.\nIn this case it has only abelian quotient singularities.\nConversely, the quotient of an affine space by any finite abelian group is a \n$\\mathbf{Q}$-factorial toric variety. \nAny $\\mathbf{Q}$-factorial crepant terminalization and \n$\\mathbf{Q}$-factorial terminal relative minimal model of a toric pair is toric.\n\nFor a pair of quotient type, we can define a derived category whose homological dimension is finite.\nTherefore we would like to ask the following:\n\n\\begin{Que}\\label{quotient-flop}\nLet $f: (X,B) \\dashrightarrow (Y,C)$ be a flop or a flip.\nIf $(X,B)$ is of quotient type, then is $(Y,C)$ too?\n\\end{Que}\n\nWe note that a divisorial contraction of a smooth $3$-fold may produce a \nsingularity of non-quotient type even if it is crepant.\nWe note also that even if $X$ is smooth and $f$ is a flop, $Y$ is not necessarily smooth, but we have\nstill the derived equivalence in some examples when we consider the associated \nDeligne-Mumford stacks (cf. \\cite{Francia}).\n\nThe following lemma is standard:\n\n\\begin{Lem}\\label{codimension}\nLet $f: (X,B) \\dashrightarrow (Y,C)$ be a proper birational map between \n$\\mathbf{Q}$-factorial terminal pairs, and assume that\n$K_X+B \\ge K_Y+C$. \nThen $f$ is surjective in codimension $1$, and $f_*B \\ge C$.\nMoreover, if $K_X+B = K_Y+C$, then $f$ is an isomorphism in \ncodimension $1$.\n\\end{Lem}\n\n\\begin{proof}\nSuppose that there is a prime divisor $C_1$ on $Y$ \nwhich is not the strict transform of a prime\ndivisor on $X$.\nLet $c_1 \\ge 0$ be the coefficient of $C_1$ in $C$.\nLet $p: Z \\to X$ and $q: Z \\to Y$ be projective birational morphisms \nfrom a smooth variety, and let $D_1$ be the strict transform of $C_1$.\nSince we have an equality $p^*(K_X+B) \\ge q^*(K_Y+C)$, \nthe coefficient of $D_1$ in $p^*(K_X+B)-K_Z$ \nshould be non-negative, a contradiction to the fact that $(X,B)$ is terminal.\nTherefore $f$ is surjective in codimension $1$.\nIf we apply the first assertion to $f$ and $f^{-1}$, then we obtain the second assertion.\n\\end{proof}\n\n\\section{Decomposition theorem for toric pairs}\n\nWe prove Theorem~\\ref{decomposition} in this section.\n\nFirst we prove a condition for the decomposition theorem that is also valid for the non-toric case.\nLet $f: (X,B) \\dashrightarrow (Y,C)$ be a birational map between projective $\\mathbf{Q}$-factorial \nKLT pairs which is an isomorphic in codimension 1 and such that $K_X+B=K_Y+C$. \nLet $p:Z \\to X$ and $q: Z \\to Y$ be projective birational morphisms\nsuch that $f \\circ p = q$.\nLet $\\{l_{\\lambda}\\}$ be a set of curves on $X$ whose classes generate the \ncone of curves $\\overline{NE}(X)$ as a closed convex cone.\nLet $L_Y$ be an ample and effective divisor on $Y$ \nsuch that $L_Y-(K_Y+C)$ is also ample, and \nlet $L_X = f_*^{-1}L_Y$ be its strict transform.\n\n\\begin{Lem}\\label{condition}\nAssume that, if $(L_X,l_{\\lambda}) < 0$ for some $\\lambda$, \nthen $((K_X+B),l_{\\lambda}) = 0$.\nThen $f$ is decomposed into a sequence of flops.  \n\\end{Lem}\n\n\\begin{proof}\nWe shall find suitable extremal rays\nin order to find a path from $(X,B)$ to $(Y,C)$.\n\nAssume first that $L_X$ is nef.\nWe claim that $L_X-(K_X+B)$ is also nef.\nOtherwise, there exists a curve $l_{\\lambda}$ such that \n$((L_X-(K_X+B)),l_{\\lambda})<0$.\nBy the assumption, it follows that $((K_X+B),l_{\\lambda})=0$.\nTherefore $(L_X,l_{\\lambda})<0$, a contradiction.\n\nBy the base point free theorem (\\cite{KMM}), $L_X$ is semiample.\nSince $f$ is an isomorphism in codimension $1$, \nthe associated morphism coincides with $f$.\nSince both $X$ and $Y$ are $\\mathbf{Q}$-factorial, we conclude that $f$ is an\nisomorphism.\n\nNext assume that $L_X$ is not nef.\nWe take a small positive number $\\epsilon$ such that \nthe pair $(X,B+\\epsilon L_X)$ is KLT.\nWe would like to run a minimal model program for this pair in order to\nmove from the pair $(X,B)$ to the pair $(Y,C)$.\nHere we have to note that there may be extremal rays of $(X,B+\\epsilon L_X)$ \nwhich do not come from the negativity of $L_X$.\n\nSince $L_X$ is not nef, \nthere exists a curve $l_{\\lambda}$ such that $(L_X,l_{\\lambda}) < 0$.\nBy the assumption, $((K_X+B),l_{\\lambda})=0$ for such curve.\nTherefore $K_X+B+\\epsilon L_X$ is not nef.\n\nThe part of the closed cone of curves $\\overline{NE}(X)$ \nwhere $L_X$ is negative is contained in the part where $K_X+B+\\epsilon L_X$ \nis negative.\nThus there exists an extremal ray $R$ of $\\overline{NE}(X)$ on which \n$L_X$ is negative.\nSince $K_X+B$ is numerically trivial on $R$.\nthe flip of $R$ for the pair $(X,B+\\epsilon L_X)$ is a flop \nfor $(X,B)$.\nSince $B+\\epsilon L_X$ is big, this process terminates by \\cite{BCHM}, and \n$f$ is decomposed into the flops.\n\\end{proof}\n\nIt is important note that $\\{l_{\\lambda}\\}_{\\lambda \\in \\Lambda}$ is not necessarily the \nset of all curves.\n\nNow we consider the toric case.\nWe take $p: Z \\to X$, $q: Z \\to Y$ and $L_Y$ to be also toric.\nLet $\\Delta_X$ and $\\Delta_Y$ be the simplicial fans corresponding to \n$X$ and $Y$, respectively, in the same vector space $N_{\\mathbf{R}}$.\nBy assumption, the sets of primitive vectors on the edges of \n$\\Delta_X$ and $\\Delta_Y$ coincide.\nSince $\\overline{NE}(X)$ is generated by toric curves, \nTheorem~\\ref{decomposition} is a consequence of the following:\n\n\\begin{Lem}\n(1) Let $w$ be a wall in $\\Delta_X$ and let $l_w$ be the corresponding curve \non $X$.\nAssume that $(L_X,l_w) \\le 0$.\nThen $w$ is not a wall in $\\Delta_Y$.\n\n(2) Let $w$ be a wall in $\\Delta_X$ which does not belong to $\\Delta_Y$.\nThen $((K_X+B),l_w)=0$.\n\\end{Lem}\n\n\\begin{proof}\n(1) Since $f$ is an isomorphism in codimension $1$, \nthe set of primitive vectors $\\{v_i\\}$ on the edges of $\\Delta_X$ and $\\Delta_Y$ coincide.\nLet $g_X$ and $g_Y$ be functions on $N_{\\mathbf{R}}$ corresponding to the divisors\n$L_X$ and $L_Y$, respectively.\n$g_X$ and $g_Y$ have the same values at the $v_i$, \nand are linear inside each simplex \nbelonging to $\\Delta_X$ and $\\Delta_Y$, respectively.\nSince $L_Y$ is ample, $g_Y$ is convex along walls of $\\Delta_Y$.\n\nAssuming that $w$ is also a wall in $\\Delta_Y$, \nwe shall derive a contradiction.\nUnder this assumption, the values of $g_X$ and $g_Y$ are equal on $w$.\n\nLet $v_1,v_2$ be two vertexes adjacent to $w$ in $\\Delta_X$.\nWe take a general point $P$ on $w$ such that the line segments $I_i$ \nconnecting $P$ to $v_i$ for $i=1,2$ pass only the interiors of simplexes and \nwalls of $\\Delta_Y$.\nThen the function $g_Y$ is convex along the $I_i$.\n\nSince $g_X$ is linear on the $I_i$ and coincides with $g_Y$ at the end points,\nwe have an inequality $g_X \\ge g_Y$ on $I_1 \\cup I_2$.\nOn the other hand, $g_X = g_Y$ on $w$.\nSince $g_Y$ is convex along $w$, we conclude that $g_X$ is also convex along \n$w$, hence $(L_X,l_w) > 0$, a contradiction.\n\n(2) Let $D_i$ be the the prime divisors on $X$ corresponding the $v_i$.\nWe write $B = \\sum d_iD_i$, and let $h$ be the function on the vector space \n$N_{\\mathbf{R}}$ which takes values $1-d_i$ at the $v_i$ and linear \ninside each simplex of $\\Delta_X$.\nSince $K_X+B=K_Y+C$, $h$ is also linear inside each simplex of $\\Delta_Y$.\nThere is a point inside $w$ which is an interior point of a simplex in \n$\\Delta_Y$.\nThen $h$ is linear across $w$.\nTherefore $((K_X+B),l_w)=0$.\n\\end{proof}\n\n\\begin{Rem}\nThe relative version is similarly proved.\nWe need to assume that $X$ and $Y$ are projective and toric over a toric variety $S$ and \n$f$ is a toric birational map over $S$.\nWe only consider curves relative over $S$, i.e., those mapped to single points on $S$.\n\\end{Rem}\n\nIf $\\dim X = 3$, then the condition of Lemma~\\ref{condition} is easily verified\neven in the non-toric case:\n\n\\begin{Prop}\\label{dim3}\nAssume that $\\dim X = 3$.\nThen any $K$-equivalent birational map between projective $\\mathbf{Q}$-factorial  KLT pairs\n$f: (X,B) \\dashrightarrow (Y,C)$ which is an isomorphism in codimension $1$\nis decomposed into a sequence of flops.\n\\end{Prop}\n\n\\begin{proof}\nAssume that $(L_X,l) \\le 0$ for a curve $l$.\nSince $\\dim X = 3$, there are only finitely many such curves $l$.\nThen $l$ is the image of a fiber of $q$ by $p$.\nTherefore we have $((K_X+B),l)=0$.\n\\end{proof}\n\n\n\\section{Derived McKay correspondence}\n\nWe prove Theorems~\\ref{abelian} and \\ref{GL2} as well as Theorem~\\ref{relative}~(a) \nin this section.\n\n\\begin{proof}[proof of Theorem~\\ref{abelian}]\nLet $B$ be a $\\mathbf{Q}$-divisor on $X$ such that $p^*(K_X+B)=K_{\\mathbf{A}}$.\nSince the action of $G$ is diagonalizable, the pair $(X,B)$ is toric.\nWe have $D^b(\\text{coh}([\\mathbf{A}\/G]) \\cong D^b(\\text{coh}(\\tilde X))$ for the \nsmooth Deligne-Mumford stack $\\tilde X$ associated to the pair $(X,B)$.\n\nThere exists a toric relative minimal model of $X$.\nAny other relative minimal model is obtained by a sequence of flops from the toric \nrelative minimal model by \\cite{flop}. \nSince the extremal rays are toric, the sequence is automatically toric, \nhence any relative minimal model is again toric.\nBy Theorem~\\ref{relative}, it is sufficient to prove the theorem for a particular relative minimal model.\n\nSince $X$ is $\\mathbf{Q}$-factorial and $(X,B)$ is KLT, we can construct a sequence \ntoric birational morphisms\n\\[\n(X_t,B_t) \\to (X_{t_1},B_{t-1}) \\to \\dots \\to (X_0,B_0)=(X,B)\n\\]\nwhich satisfy the following conditions:\n\n(1) Each step $f_i: (X_i,B_i) \\to (X_{i-1},B_{i-1})$ is a divisorial contraction such that \n$B_i = f_{i*}^{-1}B_{i-1}$ and $K_{X_i}+B_i \\le K_{X_{i-1}}+B_{i-1}$ for $1 \\le i \\le t$.\n\n(2) The pair $(X_t,B_t)$ is terminal.\n\nNext we run a toric minimal model program for $X_t$ over $X$ \nto obtain a sequence of birational maps\n\\[\nX_t=Y_0 \\dashrightarrow Y_1 \\dashrightarrow \\dots \\dashrightarrow Y_s=Y\n\\]  \nconsisting of divisorial contractions and flips such that $K_Y$ is nef over $X$. \nWe have $K_{Y_{j-1}} > K_{Y_j}$ for $1 \\le j \\le s$.\n\nWe apply Theorem~\\ref{relative} and Corollary~\\ref{combine} \nto each step of the above sequences to obtain our result. \n\\end{proof}\n\n\\begin{proof}[proof of  Theorem~\\ref{GL2}]\nLet $H$ be the normal subgroup of $G$ defined by $H = G \\cap SL(2,\\mathbf{C})$.\nThe residue group is a cyclic group $\\mathbf{Z}_r=G\/H$ for $r=[G:H]$.\nBy \\cite{BKR}, \na component of the Hilbert scheme of $H$-invariant subschemes\nof $\\mathbf{A}$ gives a crepant resolution $f: M \\to \\mathbf{A}\/H$, and \nthere is a derived equivalence $\\Phi:D^b(\\text{coh}(M)) \\to D^b(\\text{coh}^H(\\mathbf{A}))\n= D^b(\\text{coh}([\\mathbf{A}\/H]))$.\nMore precisely, $M$ is the moduli space of closed subschemes\n$P \\subset \\mathbf{A}$ which is $H$-invariant\nand such that $\\text{length }\\mathcal{O}_P = \\#H$.\nFor a generic point $[P] \\in M$, $P$ is a reduced subscheme, \nand corresponds to a generic orbit of $H$ in $\\mathbf{A}$.\nThere exists a universal subscheme \n$W \\subset M\\times \\mathbf{A}$ with projections \n$p: W \\to M$ and $q: W \\to \\mathbf{A}$ such that the \nfunctor $\\Phi=q_*p^*$ gives the equivalence.\n\nFor $[P] \\in M$ and $g \\in G$, $gP$ is also $H$-invariant, since $H$ is a normal subgroup.\nHence $[gP] \\in M$, and $G$ acts on $M$ such that its subgroup $H$ acts trivially.\nThe equivariant derived category $D^b(\\text{coh}^{\\mathbf{Z}_r}(M))$ is identified with \nthe category where the objects are $G$-equivariant complexes on which $H$ acts \ntrivially, and the morphisms are $G$-equivariant morphisms.\n\n$G$ acts on the product $M \\times \\mathbf{A}$ diagonally and preserves the subscheme $W$.\nWe claim that the functor\n\\[\n\\Phi'= q_*p^*: D^b(\\text{coh}^{\\mathbf{Z}_r}(M)) \\to D^b(\\text{coh}^G(\\mathbf{A})).\n\\]\nis an equivalence.\nIndeed if $a,b$ are $\\mathbf{Z}_r$-equivariant objects, then\nthe isomorphism \n\\[\n\\text{Hom}_M(a,b) \\cong \\text{Hom}_{\\mathbf{A}}(\\Phi(a),\\Phi(b))^H\n\\]\nimplies an isomorphism\n\\[\n\\text{Hom}_M(a,b)^{\\mathbf{Z}_r} \\cong \n(\\text{Hom}_{\\mathbf{A}}(\\Phi(a),\\Phi(b))^H)^{\\mathbf{Z}_r}\n= \\text{Hom}_{\\mathbf{A}}(\\Phi(a),\\Phi(b))^G. \n\\]\n\nLet $g: M \\to M\/\\mathbf{Z}_r$ be the map to the quotient space.\nWe define a boundary divisor $B_{M\/\\mathbf{Z}_r}$ on $M\/\\mathbf{Z}_r$ by \n$g^*(K_{M\/\\mathbf{Z}_r}+B_{M\/\\mathbf{Z}_r})=K_M$. \nSince $M$ is smooth, $M\/\\mathbf{Z}_r$ has only cyclic quotient singularities.\nLet $h: Y' \\to M\/\\mathbf{Z}_r$ be the minimal resolution, and \nlet $k: Y' \\to Y$ be the contraction morphism to the minimal resolution of $\\mathbf{A}\/G$.\nThe former is a toroidal morphism, and decomposed into toroidal divisorial contractions.\nThe latter is a composition of divisorial \ncontractions of $(-1)$-curves, which are toroidal too.\n\nThe canonical divisors satisfy the following inequalities:\n\\[\nK_Y \\le K_{Y'} \\le K_{M\/\\mathbf{Z}_r} \\le K_{M\/\\mathbf{Z}_r}+B_{M\/\\mathbf{Z}_r}\n=K_{\\mathbf{A}\/G}.\n\\]\nBy Theorem~\\ref{relative}, we obtain fully faithful embeddings\n\\[\nD^b(\\text{coh}(\\tilde Y)) \\to D^b(\\text{coh}(\\tilde Y')) \\to D^b(\\text{coh}([M\/\\mathbf{Z}_r])) \n= D^b(\\text{coh}([\\mathbf{A}\/G]))\n\\]\nwhose semi-orthogonal complements are generated by the derived categories of subvarieties of \n$X=\\mathbf{A}\/G$.\n\\end{proof}\n\n\\begin{proof}[proof of Theorem~\\ref{relative}~(a)]\nIt is sufficient to consider the following situation:\n$X=Y$ is an affine toric variety corresponding to a simplicial cone \n$\\sigma \\subset N_{\\mathbf{R}}$ generated by primitive vectors $v_1,\\dots,v_n \\in N$.\nLet $B_i$ be prime divisors on $X$ corresponding to the $v_i$, and let \n$B = \\sum (1-1\/r_i)B_i$ and $C=\\sum (1-1\/s_i)B_i$, where $r_1 > s_1$ and\n$r_i=s_i$ for $i=2,\\dots,n$.\nLet $t_1=\\text{LCM}(r_1,s_1)$ and $t_i=r_i$ for $i=2,\\dots,n$.\n\nThe stacks $\\tilde X$,  $\\tilde Y$ and $\\tilde W$ are defined by the sublattices \n$N_X$, $N_Y$ and $N_W$ \nof $N$ generated by the $r_iv_i$, $s_iv_i$ and $t_iv_i$, respectively. \nLet $\\tilde B_i^X$, $\\tilde B_i^Y$ and $\\tilde B_i^W$ be prime divisors \non $\\tilde X$,  $\\tilde Y$ and $\\tilde W$ over $B_i$, respectively.\nThey correspond primitive vectors $r_iv_i$, $s_iv_i$ and $t_iv_i$, respectively.\nLet $p: \\tilde W \\to \\tilde X$ and $q: \\tilde W \\to \\tilde Y$ be the natural morphisms.\nThen the fully faithful functor $\\Phi: D^b(\\text{coh}(\\tilde Y)) \\to D^b(\\text{coh}(\\tilde X))$ is \nconstructed as $\\Phi=p_*q^*$ in \\cite{log crepant}~Theorem~4.2 (1).\nWe have \n\\[\n\\Phi(\\mathcal{O}_{\\tilde Y}(k\\tilde B_1^Y)) = \n\\mathcal{O}_{\\tilde X}(\\llcorner kr_1\/s_1 \\lrcorner \\tilde B_1^X)\n\\]\nfor $k = 1,\\dots, s_1-1$.\n\nWe consider the semi-orthogonal complement.\nIt is generated by the sheaves\n$\\mathcal{O}_{\\tilde B_1^X}(l\\tilde B_1^X)$ for $1 \\le l \\le r_1-1$ such that\n$l \\ne \\llcorner kr_1\/s_1 \\lrcorner$ for any $k$.\nWe have \n\\[\nR\\text{Hom}_{\\tilde X}(\\mathcal{O}_{\\tilde X}(\\llcorner kr_1\/s_1 \\lrcorner \\tilde B_1^X),\n\\mathcal{O}_{\\tilde B_1^X}(l\\tilde B_1^X))=0\n\\]\nfor such $k,l$, and \n\\[\nR\\text{Hom}_{\\tilde X}(\\mathcal{O}_{\\tilde B_1^X}(l\\tilde B_1^X), \n\\mathcal{O}_{\\tilde B_1^X}(l'\\tilde B_1^X))=0\n\\]\nfor such $l,l'$ with $l < l'$.\n\nWe define a toric boundary $D$ on $B_1$ \nfrom the pair $(X,B)$ as follows (cf. \\cite{toric}~\\S5).\nLet $\\bar N = N\/\\mathbf{Z}v_1$, and write $v_i + \\mathbf{Z}v_1 = d_i \\bar v_i \\in \\bar N$ for \nprimitive vectors $\\bar v_i$ ($i=2,\\dots,n$).\nThen define \n$D= \\sum_{i=2}^n (1-1\/d_ir_i)D_i$ for $D_i=B_1 \\cap B_i$. \nLet $\\tilde B_1$ be the associated smooth Deligne-Mumford stack above $(B_1,D)$.\nLet $\\tilde B_1'$ be the normalization of the fiber product $\\tilde B_1 \\times_X \\tilde X$ with \nnatural morphisms \n$p_1: \\tilde B_1' \\to \\tilde B_1$ and $p_2: \\tilde B_1' \\to \\tilde X$.\nThen as in \\cite{toric}, the semi-orthogonal complement is generated by \nsubcategories \n\\[\np_{2*}p_1^*D^b(\\text{coh}(\\tilde B_1)) \\otimes \\mathcal{O}_{\\tilde X}(l\\tilde B_1^X)\n\\]\n$1 \\le l \\le r_1-1$ such that $l \\ne \\llcorner kr_1\/s_1 \\lrcorner$ for any $k$.\nMoreover, there is no morphism between objects belonging to different $k$'s.\nTherefore we conclude the proof.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nGravitational lensing is recognized as a unique tool to conduct many important investigations in modern astrophysics \\cite{Liebes:1964,Schneider-Ehlers-Falco:1992,Schneider-etal:2006}. Since the beginning of the 21st century, it is used to study the distribution of matter in stelar structures, to probe the dark matter distribution in the universe, even to search for exoplanets orbiting distant stars \\cite{Refsdal:1964,Blandford-Narayan:1992,Wambsganss:1998,Gaudi:2012}.\n\nDue to the nonlinear nature of the equations involved, most relevant efforts were constrained to monopole lenses, where the gravitational field is taken to be that of a structureless point source \\cite{Herlt-Stephani:1976,Deguchi-Watson:1986,Narayan-Bartelmann:1996}. Nevertheless, there have been attempts to model extended lenses, including quadrupole and general shear distortions of the lensing potential \\cite{Kovner:1987,Schneider-Ehlers-Falco:1992,Erdl-Schneider:1993,Gould:2001} as well as to describe binaries \\cite{Congdon-Keeton-book:2018}.  It was recognized that such deviations from spherical symmetry lead to the formation of caustics, which complicates image formation \\cite{Ohanian:1983,Blandford-Kovner:1988,Nambu:2013,Chu:2016}. Most of these investigations were conducted using the geometric optics approximation, which is known to be of limited utility when it comes to describing light amplification, especially in the presence of caustics \\cite{Gaudi-Petters:2001,Gaudi-Petters:2002}, where such results have singularities. As caustics appear naturally in the point-spread function (PSF) characterizing the optical properties of an extended gravitational lens, there is a need to address these shortcomings.  In particular, it was recognized that a wave-optical treatment of gravitational lensing is needed \\cite{Nakamura-Deguchi:1999,Nambu:2012}. Until recently, such a description of an extended lens was not available.\n\nMeanwhile, the solar gravitational lens (SGL) gained attention as a possible means to obtain resolved images of exoplanets \\cite{Turyshev:2017,Turyshev-Toth:2017,Turyshev-Toth:2020-im-extend,Toth-Turyshev:2020}. From the beginning, our efforts to describe the SGL were conducted within the Mie theory \\cite{Mie:1908,Born-Wolf:1999}, aiming to describe diffraction of electromagnetic (EM) waves by a gravitational field. Such an approach solves a Schr\\\"odinger-like wave equation for Debye potentials, yielding a wave-optical description for the lens\\footnote{In Ref. \\cite{Turyshev-Toth:2021-multipoles} we show that although similar results may be obtained within a scalar theory by using a general Fresnel--Kirchhoff diffraction formula or the path integral formalism of quantum field theory, the Mie-inspired solution covers a method to treat vector fields. Such an approach is advantageous from a practical standpoint as it allows us to deal with directly observable quantities and evaluate detection sensitivities (i.e., signal-to-noise ratio \\cite{Turyshev-Toth:2020-im-extend,Toth-Turyshev:2020}) while preserving the vectorial nature of the EM field in a weak gravitational field.}. At first, the lens was modeled as a gravitational monopole. This established a good foundation on which increasingly refined models could be constructed. These refinements were needed to capture the fact that the Sun is not a perfect sphere: its rotation and the resulting oblateness result in small axisymmetric perturbations of its otherwise spherically symmetric gravitational field in the form of the quadrupole and, to a lesser extent, higher-order zonal harmonics.\n\nAlthough our initial objective was to capture only the dominant quadrupole perturbation captured by the $J_2$ zonal harmonics, we were able to do much more. We developed a wave-optical treatment that we call the angular eikonal method, which can be used to describe gravitational lensing by any axisymmetric gravitational field that is dominated by the monopole potential, but perturbed by an infinite set of zonal harmonics \\cite{Turyshev-Toth:2021-multipoles}.\nThe resulting wave-optical treatment of gravitational lensing focuses on evaluating the eikonal phase shift that an EM wave acquires as it travels from the source to the image plane. This phase shift now may be evaluated for any gravitational potential that can be modeled as a perturbed monopole gravitational field. The new method may in fact be used to recover the multipole moments that characterize the mass distribution of the lens, and thus recover its basic geometry and structure.  This wave-optical approach is especially useful to describe imaging with realistic axisymmetric astrophysical lenses \\cite{Turyshev-Toth:2021-imaging}.\n\nOur prior work on gravitational lensing focused on the strong interference region of the lens that exists in the vicinity of its primary optical axis (see Fig.~\\ref{fig:regions}). As we move further away from that axis,  we enter the weak interference region and then the region of geometric optics (see description in \\cite{Turyshev-Toth:2017}.) Clearly, the farther we are from the optical axis, the less is the impact of perturbations to the monopole gravitational field. At some distance from the optical axis, the behavior of an extended gravitational lens becomes indistinguishable from that of a monopole lens. In any case, a complete description of the transition process between various regions is needed to fully understand the behavior of the PSF of the lens. The images seen by a telescope at different distances from the optical axis are also of interest.  A similar discussion in the context of a monopole lens was presented in \\cite{Turyshev-Toth:2019-extend,Turyshev-Toth:2020-im-extend}. We can now extend these results to the case of a generic lens that can be described as a perturbed gravitational monopole.\n\nIn this paper, we apply our new approach beyond the strong interference region, describing gravitational lensing in all lensing regimes.  This paper is organized as follows:\nIn Section~\\ref{sec:EM-field} we summarize the wave-optical solution we call \\emph{the angular eikonal method} \\cite{Turyshev-Toth:2021-multipoles} that allows us to determine the EM field in all regions behind the extended axisymmetric SGL, including the regions of strong and weak interference as well as the region of geometric optics.\nIn Section~\\ref{sec:v-large-disp} we address imaging with the SGL of the extended Sun, where we describe the signal received at the focal plane of an imaging telescope which moves in the image plane.\nIn Section~\\ref{sec:sims} we demonstrate the power of our formalism by presenting results that show the view of a point source, projected by the SGL and observed at various distances from the optical axis by an imaging telescope. We also present simulations of an extended source modeling light from a distant resolved star.\nIn Section~\\ref{sec:end} we present our conclusions and identify next steps.\nIn Appendix~\\ref{sec:lim-cases} we consider some limiting cases and demonstrate agreement between previously obtained results and the results in the present paper.\n\n\n\\section{The EM field all regions behind the lens}\n\\label{sec:EM-field}\n\nIn Ref.~\\cite{Turyshev-Toth:2021-multipoles}, we studied diffraction of EM waves in the presence of gravity. For this we considered a Mie problem with the electromagnetic field propagating in the vicinity of an extended gravitational lens in the first post-Newtonian approximation of the general theory of relativity. We were able to reduce the problem to a Schr\\\"odinger-like equation describing the Debye potential and then derived a complete solution for the EM field on an image plane positioned behind the lens. The resulting EM field was used to compute the energy flux in various regions behind that lens by calculating the Poynting vector -- the quantity that is needed to study the optical properties of the lens\\footnote{To simplify the material and keep the focus of this paper on its broader objectives, here we only summarize the solution, inviting the reader to consult \\cite{Turyshev-Toth:2021-multipoles} for technical details if needed.}.\n\n\\subsection{Summary of the solution}\n\n\\begin{figure}\n\\includegraphics[scale=0.27]{regions}\n\\caption{\\label{fig:regions}The different optical regions of the SGL\n(adapted from \\cite{Turyshev-Toth:2019-extend}).\n}\n\\end{figure}\n\nWe use a heliocentric spherical coordinate system $(r,\\theta,\\phi)$ and consider a source positioned at a distance $r_0$ from a lens. In \\cite{Turyshev-Toth:2021-multipoles}, we studied propagation of a light ray with impact parameter $b$ with respect to the lens and determined the components of the EM field that would be observed on an image plane at distance $r$ from the lens. For a high-frequency EM wave (i.e., neglecting terms $\\propto(kr)^{-1}$) and for $r\\gg r_g$ (with $r_g=2GM\/c^2$ being the Schwarzschild radius of the lens), we derived the EM field  that is needed to estimate the flux through the image plane. Following the logic of solving the Mie problem \\cite{Born-Wolf:1999,Turyshev-Toth:2017}, this field can be given to the required order in the following form \\cite{Turyshev-Toth:2021-multipoles}:\n{}\n\\begin{eqnarray}\n    \\left( \\begin{aligned}\n{D}_\\theta& \\\\\n{B}_\\theta& \\\\\n  \\end{aligned} \\right) =    \\left( \\begin{aligned}\n{B}_\\phi& \\\\\n-{D}_\\phi& \\\\\n  \\end{aligned} \\right)&=&\n \\left( \\begin{aligned}\n \\cos\\phi& \\\\\n \\sin\\phi& \\\\\n  \\end{aligned} \\right)\\,e^{-i\\omega t}\\gamma(r, \\theta)+{\\cal O}(r_g^2, \\theta^2, b\/r_0),\n  \\label{eq:DB-sol-rho_go}\n\\end{eqnarray}\nwith the term $\\gamma(r, \\theta)$ given to ${\\cal O}\\big({r_g}\/{r},r_g^2\\big)$ as\n{}\n\\begin{eqnarray}\n \\gamma(r,\\theta,\\phi) &=&  \\frac{E_0}{r_0}\\frac{ue^{ik(r+r_0+r_g\\ln 4k^2rr_0)}}{ikr}\\sum_{\\ell=kR^\\star_\\odot}^\\infty\\frac{\\ell+{\\textstyle\\frac{1}{2}}}{\\ell(\\ell+1)}e^{i\\big(2\\sigma_\\ell+\\frac{\\ell(\\ell+1)}{2k\\tilde r}+2\\xi_b\n\\big)}\n\\Big\\{\\frac{\\partial P^{(1)}_\\ell(\\cos\\theta)}\n{\\partial \\theta} +\\frac{P^{(1)}_\\ell(\\cos\\theta)}{\\sin\\theta}\n \\Big\\},\n  \\label{eq:beta*1*}\n\\end{eqnarray}\nwhere $1\/\\tilde r=1\/r+1\/r_0$ (as discussed in \\cite{Turyshev-Toth:2019-extend}) and $\\sigma_\\ell$ is the Coulomb phase shift (see details in \\cite{Turyshev-Toth:2017}). The summation is conducted over the partial momenta $\\ell$ that, in a semiclassical analogy, is related to the impact parameter $b$ as $\\ell=kb$. Also, the sum in (\\ref{eq:beta*1*}) starts at $\\ell=kR^\\star_\\odot$, that corresponds to applying a fully-absolving boundary condition, capturing the fact that light rays with impact parameters $0\\leq b < R^\\star_\\odot=R_\\odot+r_g$ are completely absorbed by the opaque Sun.\nAs usual, $P^{(1)}_\\ell(\\cos\\theta)$ are Legendre polynomials of the first kind \\cite{Abramovitz-Stegun:1965}. The radial components of the EM wave behave as $({D}_r, {B}_r)\\sim {\\cal O}({\\rho}\/{z},b\/r_0)$; thus they are negligibly small compared to the other two components (\\ref{eq:DB-sol-rho_go}).\n\nThe quantity $\\xi_b$  in the phase of (\\ref{eq:beta*1*}) is the eikonal phase shift that is acquired by an EM wave as it travels in the vicinity of an extended axisymmetric gravitational lens (such as our Sun). To establish the form of this quantity,  in \\cite{Turyshev-Toth:2021-multipoles} we used a heliocentric coordinate system with its $z$-axis aligned with the wave vector $\\vec k$ of the incident wave, so that $\\vec k=(0,0,1)$, and introduce the vector of the impact parameter, $\\vec b=b \\vec n_\\xi$. Using $z$ to denote the heliocentric distance of the image plane, we define  $\\vec x$ to mark a position in the image plane. Lastly, we introduce a unit vector in the direction of the solar rotation axis, $\\vec s$. These quantities are given as:\n{}\n\\begin{eqnarray}\n{\\vec b}&=&b(\\cos\\phi_\\xi,\\sin \\phi_\\xi,0),\n\\label{eq:note-b}\\\\\n{\\vec x}&=&\\rho(\\cos\\phi,\\sin \\phi,0),\n\\label{eq:note-x}\\\\\n{\\vec s}&=&(\\sin\\beta_s\\cos\\phi_s,\\sin\\beta_s\\sin\\phi_s,\\cos\\beta_s).\n\\label{eq:note}\n\\end{eqnarray}\nWith this parametrization, the additional eikonal phase shift $\\xi_b$ induced by an extended, axisymmetric and rotating gravitational lens characterized in terms of zonal harmonics was determined \\cite{Turyshev-Toth:2021-multipoles} to have the form\n{}\n\\begin{eqnarray}\n\\xi_b=-kr_g\\sum_{n=2}^\\infty\\frac{J_n}{n}\\Big(\\frac{R_\\odot}{b}\\Big)^n \\sin^n\\beta_s\\cos[n(\\phi_\\xi-\\phi_s)],\n  \\label{eq:xi_b}\n\\end{eqnarray}\nwhere $J_n$ are the zonal harmonic coefficients of the gravitational field of the lens, such as the SGL.\n\nIn \\cite{Turyshev-Toth:2021-multipoles}, we considered solution (\\ref{eq:DB-sol-rho_go})--(\\ref{eq:xi_b}) only in the strong interference region that lies in the proximity of the primary optical axis where $\\theta\\simeq \\sqrt{2r_g\/r}$.  Our objective for this paper is to use the solution above and derive results  that will be applicable in all the gravitational lensing regions that are formed behind the else that also include the weak interference region and that of the geometric optics.\n\n\\subsection{Eikonal correction for the azimuthal term}\n\nTo evaluate expression (\\ref{eq:beta*1*}), following \\cite{Turyshev-Toth:2021-multipoles},  we use  the asymptotic representation for $P_\\ell(\\cos\\theta)$ and $\\ell\\gg1$ from\\footnote{For an improved, explicit, uniformly valid two-term asymptotic form of this expression, see \\cite{Bakaleinikov:2020}} \\cite{Bateman-Erdelyi:1953,Korn-Korn:1968,Kerker-book:1969,Abramovitz-Stegun:1965}:\n{}\n\\begin{eqnarray}\nP_\\ell(\\cos\\theta)&=& \\sqrt{\\frac{\\theta}{\\sin\\theta}} J_0\\big(\\ell \\theta\\big)+{\\cal O}(\\theta^2).\n\\label{eq:Bess0}\n\\end{eqnarray}\n\nNext, we use expression\n{}\n\\begin{eqnarray}\nP^{(1)}_\\ell(\\cos\\theta)=-\\frac{\\partial P_\\ell(\\cos\\theta)}{\\partial\\theta}=\\ell J_1(\\ell\\theta)+\n{\\textstyle\\frac{1}{6}}\\theta J_0(\\ell\\theta) +{\\cal O}(\\theta^2),\n\\label{eq:Bess}\n\\end{eqnarray}\nalongside with the recurrence relations for the Bessel functions \\cite{Abramovitz-Stegun:1965}\n{}\n\\begin{eqnarray}\n\\frac{2n}{x}J_n(x)=J_{n-1}(x)+J_{n+1}(x),\n\\label{eq:Bess-rec}\n\\end{eqnarray}\nand derive the following two well-known \\cite{Born-Wolf:1999} and  useful relations\n{}\n\\begin{eqnarray}\n\\frac{P^{(1)}_\\ell(\\cos\\theta)}{\\sin\\theta}&=& {\\textstyle\\frac{1}{2}}\\ell^2\\Big(J_0(\\ell \\theta)+J_2(\\ell \\theta)\\Big),\n\\label{eq:pi-l=}\n\\qquad\n\\frac{dP^{(1)}_\\ell(\\cos\\theta)}{d\\theta}= {\\textstyle\\frac{1}{2}}\\ell^2\\Big(J_0(\\ell \\theta)-J_2(\\ell \\theta)\\Big).\n\\label{eq:tau-l=}\n\\end{eqnarray}\n\nSubstituting (\\ref{eq:pi-l=}) in expression (\\ref{eq:beta*1*}), and, following the approach that we presented in \\cite{Turyshev-Toth:2019-extend}, we consider the case of the large partial momenta, $\\ell\\gg1$, which is certainly valid here, as the integration is done from $\\ell=kR_\\odot\\gg1$ to infinity. In this case,  the term $\\gamma(r, \\theta,\\phi)$ is determined from the following integral:\n{}\n\\begin{eqnarray}\n\\gamma(r,\\theta,\\phi) =\n\\frac{E_0}{r_0}\\frac{ue^{ik(r+r_0+r_g\\ln 4k^2rr_0)}}{ikr}\\int_{\\ell=kR^\\star_\\odot}^\\infty\n\\ell d\\ell e^{i\\big(2\\sigma_\\ell+\\frac{\\ell^2}{2k\\tilde r}+\n2\\xi_b\\big)}\\Big(\nJ_0(\\ell\\theta) +{\\cal O}\\big(\\theta^2,\\frac{r_g}{r},r_g^2\\big)\\Big).\n  \\label{eq:gamma**1*}\n\\end{eqnarray}\n\nTo evaluate this integral we used the \\emph{angular eikonal method} presented in \\cite{Turyshev-Toth:2021-multipoles}. For that, we first recognize that in the case of a point mass (i.e, when only monopole is present), the resulting gravitational field is spherically symmetric \\cite{Turyshev-Toth:2017,Turyshev-Toth:2019}. However, once we include the field from the gravitational multipoles, that symmetry is broken as the eikonal phase shift acquires an azimuthal term, namely $\\xi_b=\\xi_b(b, \\theta,\\phi)$. However, we found a way to develop the treatment of the problem even in this generic case.  First, we recall that the eikonal phase shift $\\xi_b$ was obtained through an iterative process involving the eikonal approximation that originates from the field of high-energy particle physics but is also applicable in the optics domain.\nNext, we recognize that for a spherically symmetric field (which is used as the starting point of our iterative process),  the Bessel function $J_0(\\ell \\theta)$ can be used in its integral form\\footnote{Note that we can use the same representation of this function with the positive sign in the phase, but the result is identical as it will be integrated over the entire range of the azimuthal angle $\\phi_\\xi$. }\n{}\n\\begin{eqnarray}\nJ_0(\\ell\\theta)&=& \\frac{1}{2\\pi}\\int_0^{2\\pi} d\\phi_\\xi e^{-i\\ell\\theta\\cos(\\phi_\\xi-\\phi)}.\n\\label{eq:J0}\n\\end{eqnarray}\nThis is the natural step that captures the spherical symmetry of the field of a gravitational monopole. So, the iterative process used to derive the eikonal phase is conducted under this integral over all the azimuthal angles, $\\phi_\\xi$. However, in the case of the multipoles the azimuthal symmetry is broken. The presence of this integral over $d\\phi_\\xi$ allows us to account for this azimuthal dependence within the angular eikonal approximation; hence the name of the method.\n\nWe now substitute (\\ref{eq:J0}) into  (\\ref{eq:gamma**1*}) and see that  expression (\\ref{eq:gamma**1*}), to the order of ${\\cal O}\\big(\\theta^2,{r_g}\/{r},r_g^2\\big)$, transforms as\n{}\n\\begin{eqnarray}\n\\gamma(r,\\theta,\\phi) &=&\n\\frac{E_0}{r_0} \\frac{ue^{ik(r+r_0+r_g\\ln 4k^2rr_0)}}{ikr} \\frac{1}{2\\pi}\\int_0^{2\\pi} d\\phi_\\xi \\int_{\\ell=kR^\\star_\\odot}^\\infty\n\\ell d\\ell e^{i\\big(2\\sigma_\\ell+\\frac{\\ell^2}{2k\\tilde r}+2\\xi_b-\\ell\\theta\\cos(\\phi_\\xi-\\phi)\\big)}.\n  \\label{eq:beta*3}\n\\end{eqnarray}\n\nIn this form, the integral over $d\\phi_\\xi$ properly acts not only on the monopole term represented by the term $2\\sigma_\\ell+\\frac{\\ell^2}{2k\\tilde r}-\\ell\\theta\\cos(\\phi_\\xi-\\phi)$ in the phase of the expression \\ref{eq:beta*3}, but on the entire phase  $2\\sigma_\\ell+\\frac{\\ell^2}{2k\\tilde r}+2\\xi_b-\\ell\\theta\\cos(\\phi_\\xi-\\phi)$, which now includes contributions from nonspherical  parts of the gravitational potential via the eikonal phase term, $2\\xi_b$. This process constitutes the \\emph{angular eikonal method},  valid for weak gravitational fields, which allows us to study the scattering of light on nonspherical potentials under the eikonal approximation.\n\n\\subsection{Taking the integral over $b$ with the method of stationary phase}\n\nTo develop a solution for (\\ref{eq:beta*3}), and for convenience, we use (\\ref{eq:xi_b}) and introduce quantity $\\psi(\\vec b)$, as\n{}\n\\begin{eqnarray}\n\\xi_b(\\vec b)=-kr_g\\psi(\\vec b),\\, \\qquad\n\\psi(\\vec b)=\\sum_{n=2}^\\infty\\frac{J_n}{n}\\Big(\\frac{R_\\odot}{b}\\Big)^n \\sin^n\\beta_s\\cos[n(\\phi_\\xi-\\phi_s)].\n  \\label{eq:psi}\n\\end{eqnarray}\n\n\nFurthermore, for $\\ell\\gg kr_g$, evaluate $\\sigma_\\ell$ as \\cite{Turyshev-Toth:2018-grav-shadow}:\n{}\n\\begin{eqnarray}\n\\sigma_\\ell&=& -kr_g\\ln \\ell.\n\\label{eq:sig-l*}\n\\end{eqnarray}\nThis form agrees with the other known forms of $\\sigma_\\ell$ \\cite{Cody-Hillstrom:1970,Barata:2009ma} that are approximated for large $\\ell$ (see discussion in \\cite{Turyshev-Toth:2017,Turyshev-Toth:2019-image}).\n\nWe rely on the semiclassical approximation (see relevant discussion in \\cite{Turyshev-Toth:2017,Turyshev-Toth:2019}) that connects the partial momenta, $\\ell$, to the impact parameter $b$:\n{}\n\\begin{equation}\n\\ell\\simeq kb,\n\\label{eq:S-l-pri-p-g}\n\\end{equation}\nwhich is applicable for small angles $\\theta$ (or, large distances from the Sun, $R_\\odot\/r<b\/r\\ll 1$),  (see \\cite{Turyshev-Toth:2017} for details) and present the phase in (\\ref{eq:beta*3}) as\n{}\n\\begin{eqnarray}\n\\varphi(\\vec b)&=&2\\sigma_\\ell+\\frac{\\ell^2}{2k\\tilde r}+2\\xi_b-\\ell\\theta\\cos(\\phi_\\xi-\\phi)=\nk\\Big\\{\\frac{b^2}{2\\tilde r} -b\\theta\\cos(\\phi_\\xi-\\phi)-2r_g\\big(\\ln kb+\\psi(\\vec b)\\big)\\Big\\}.\n  \\label{eq:beta*3+4}\n\\end{eqnarray}\n\nTo establish the nature of this quantity, we recognize from (\\ref{eq:note-b}) that the vector of the impact parameter has the form given by (\\ref{eq:note-b}) as ${\\vec b}=b(\\cos\\phi_\\xi,\\sin\\phi_\\xi,0)$. With this, we define the vector ${\\vec \\theta}$ to a point on the image plane with coordinates $(r, \\theta,\\phi)$ that has the form ${\\vec \\theta}=\\theta(\\cos\\phi,\\sin\\phi,0)$, developed from (\\ref{eq:note-x}) with $\\theta=\\rho\/r$. With these definitions, we see that $b\\theta\\cos(\\phi_\\xi-\\phi)=({\\vec b}\\cdot{\\vec \\theta})$, and, thus (\\ref{eq:beta*3+4}) takes the form\n{}\n\\begin{eqnarray}\n\\varphi(\\vec b)&=&\nk\\Big\\{\\frac{1}{2\\tilde r}\\big({\\vec b} -\\tilde r \\vec \\theta\\big)^2-2r_g\\big(\\ln kb+\\psi(\\vec b)\\big)\\Big\\}+{\\cal O}(\\theta^2).\n  \\label{eq:beta*3+}\n\\end{eqnarray}\nThus, $\\varphi(\\vec b)$ represents the Fermat potential that governs the gravitational lensing phenomena \\cite{Liebes:1964,Refsdal:1964,Schneider-Ehlers-Falco:1992}. Note that the same form of the expression (\\ref{eq:beta*3+}) may be obtained within the path integral approach (see \\cite{Turyshev-Toth:2021-multipoles}).\n\nUsing (\\ref{eq:beta*3+4}), we can present  the $\\gamma(r,\\theta,\\phi) $ factor from (\\ref{eq:beta*3}) as\n{}\n\\begin{eqnarray}\n\\gamma(r,\\theta,\\phi) &=&\n\\frac{E_0}{r_0}ue^{ik(r+r_0+r_g\\ln 4k^2rr_0)-i{\\textstyle\\frac{\\pi}{2}}}\n\\frac{k}{2\\pi r}\\int_0^{2\\pi} d\\phi_\\xi \\int_{b=R^\\star_\\odot}^\\infty\nb db \\, e^{ik\\big(\\frac{b^2}{2\\tilde r} -b\\theta\\cos(\\phi_\\xi-\\phi)-2r_g\\big(\\ln kb+\\psi(\\vec b)\\big)\\big)}.\n  \\label{eq:gamma0}\n\\end{eqnarray}\n\nWe recognize that this is the double integral is with respect to the impact parameter, $\\vec b$, specifically, $d^2{\\vec b}=d\\phi_\\xi bdb$. We may try to take this integral with using the 2-dimensional method of stationary phase. However, as even the presence a quadrupole leads to appearance of caustics such a solution will not be precise. This problem can be mitigated if we take only one of the two integrals in (\\ref{eq:gamma0}) may be taken with the method of stationary phase. When the multipoles represent a small distortion of the gravitational field, contributions to the eikonal phase shift due to multipoles (\\ref{eq:psi}) are much smaller than that of the monopole (given by the $\\ln kb$ term in (\\ref{eq:beta*3+})). In this case, we may take the integral over $db$ using the method of stationary phase, leaving the integral over $d\\phi_\\xi$ unevaluated.\n\nIn case of gravitation, the monopole term is responsible for a long-range gravitational field that effects rays of light over very large distances (similar to the effect of a Coulomb potential  in the time-independent Schr\\\"odinger equation). In that case, the behavior of the light ray is well-understood. The impact of the monopole is affects the light ray's trajectory along its entire path from emission to reception.  Within the eikonal approximation, any multipole distortion leads to a phase shift in addition to that produced by the monopole. This makes it possible ot evaluate the radial integral using the method of stationary phase. It can be shown that the error incurred by doing so is of $\\sim{\\cal O}\\big((kr_g)^{-1}\\big)$, which is negligible in practice. This justifies our approach in case of a weak gravitational field and its long-range behavior.\n\n\\subsubsection{Solving for the impact parameter for the stationary phase}\n\nAs was done in \\cite{Turyshev-Toth:2019-extend}, we evaluate this integral using the method of stationary phase. To do that, we note that the relevant $b$-dependent part of the phase in (\\ref{eq:gamma0})  is of the form (\\ref{eq:beta*3+4}).  The phase is stationary when $d\\varphi (\\vec b)\/db=0$, which implies\n{}\n\\begin{equation}\n\\frac{b}{\\tilde r}-\\theta\\cos(\\phi_\\xi-\\phi)-\\frac{2r_g}{b}\\Big(1-\\sum_{n=2}^\\infty J_n\\Big(\\frac{R_\\odot}{b}\\Big)^n \\sin^n\\beta_s\\cos[n(\\phi_\\xi-\\phi_s)]\\Big)={\\cal O}\\big(r_g^2, \\theta^2\\big).\n\\label{eq:S-l-pri=}\n\\end{equation}\nWe solve this equation iteratively, using a trial solution $b=b_{[0]}+b_{[1]}+{\\cal O}(J_n^2)$, which allows us to form two equations:\n{}\n\\begin{eqnarray}\nb_{[0]}^2- b_{[0]}\\tilde r \\theta \\cos(\\phi_\\xi-\\phi)-2r_g\\tilde r&=&{\\cal O}\\big(r_g^2, \\theta^2\\big),\n\\label{eq:b0}\\\\\nb_{[1]}\\Big(2b_{[0]}-\\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\Big)+\n2r_g\\tilde r\\sum_{n=2}^\\infty J_n\\Big(\\frac{R_\\odot}{b_{[0]}}\\Big)^n \\sin^n\\beta_s\\cos[n(\\phi_\\xi-\\phi_s)]&=&{\\cal O}\\big(r_g^2, \\theta^2\\big).\n\\label{eq:b1}\n\\end{eqnarray}\n\nThe quadratic equation (\\ref{eq:b0}) yields the following two solutions:\n{}\n\\begin{eqnarray}\nb_{[0]}^{\\pm}= {\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\pm\\sqrt{\\big({\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\big)^2+2r_g \\tilde r}+{\\cal O}(\\theta^3,r_g^2,J_n).\n\\label{eq:S-l-pri+0}\n\\end{eqnarray}\nWe require the impact parameter to be positive. This condition is be satisfied only for the positive sign in (\\ref{eq:S-l-pri+0}). Thus, the  impact parameter $b^{[0]}$ has the form\n{}\n\\begin{eqnarray}\nb_{[0]}= \\sqrt{\\big({\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\big)^2+2r_g \\tilde r}+{\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)+{\\cal O}(\\theta^3,r_g^2,J_n).\n\\label{eq:S-l-pri+}\n\\end{eqnarray}\n\nAs this solution has dependence on the azimuthal angle $\\phi$, in the case of a monopole, (\\ref{eq:S-l-pri+}) actually represents two families of impact parameters when $\\phi_\\xi-\\phi=0$ and $\\phi_\\xi-\\phi=\\pi$, yielding\n{}\n\\begin{eqnarray}\nb^{[0]}_{\\tt in}=\\sqrt{({\\textstyle\\frac{1}{2}} \\tilde r \\theta)^2+2r_g \\tilde r}+{\\textstyle\\frac{1}{2}} \\tilde r \\theta, \\qquad\nb^{[0]}_{\\tt sc}=\\sqrt{({\\textstyle\\frac{1}{2}} \\tilde r \\theta)^2+2r_g \\tilde r}-{\\textstyle\\frac{1}{2}} \\tilde r \\theta,\n\\label{eq:S-l-pri0d+}\n\\end{eqnarray}\nwhere $b^{[0]}_{\\tt in}$ and $b^{[0]}_{\\tt sc}$ are the two impact parameters describing incident and scattered EM waves, corresponding to light rays passing by the near side and the far side of the Sun (with respect to the location of the telescope), correspondingly. After it is diffracted by a point-source gravitational lens, a wavefront is described as the sum of a gravity-modified plane wave (the incident wave) and a spherical wave centered on the gravitational lensing source (the scattered wave);\nsee, for instance, Fig.~2 of \\cite{Turyshev-Toth:2017}.  The impact parameters $b^{[0]}_{\\tt in}$ and $b^{[0]}_{\\tt sc}$ correspond to images that appear close to the Einstein ring on opposite sides of the lens; the ``scattered'' image, denoted by ``${\\tt sc}$'', on the far side relative to the telescope (called the minor image) always appears inside the Einstein ring, and the ``incident'' image, denoted by ``${\\tt in}$'' on the near side always appears outside (major image, see \\cite{Schneider-Ehlers-Falco:1992} for details).\n\nFor $\\sqrt{2r_g\/\\tilde r}\\ll \\theta$, this result is equivalent to the two solutions derived  in Sec.~IV of  \\cite{Turyshev-Toth:2019-extend}. However, the form (\\ref{eq:S-l-pri+}) allows us to study the behavior  of the EM wave in the transition between the two solutions in the region where angle $\\theta$ is of the same order as the Einstein deflection angle $\\theta\\sim \\sqrt{2r_g\/\\tilde r}$.\n\nIt is convenient to use a shorthand notation $\\theta\\cos(\\phi_\\xi-\\phi)=({\\vec n}_\\xi\\cdot{\\vec \\theta})$, where ${\\vec \\theta}=\\theta(\\cos\\phi,\\sin\\phi,0)$. Then, by dividing the solution (\\ref{eq:S-l-pri+}) by $\\tilde r$, we may present the two solutions in term of the angles $\\theta^{[0]}_+=b^{[0]}_{\\tt in}\/\\tilde r$, for $\\phi_\\xi-\\phi=0$, and $\\theta^{[0]}_-=b^{[0]}_{\\tt in}\/\\tilde r$, for $\\phi_\\xi-\\phi=\\pi$, to ${\\cal O}(\\theta^3,r_g^2,J_n)$, we have\n{}\n\\begin{equation}\n\\theta^{[0]}= {\\textstyle\\frac{1}{2}} \\Big(\\sqrt{({\\vec n}_\\xi\\cdot{\\vec \\theta})^2+4\\theta_E^2}+({\\vec n}_\\xi\\cdot{\\vec \\theta})\\Big)\\quad\\rightarrow \\quad \\theta^{[0]}_+= {\\textstyle\\frac{1}{2}} \\Big(\\sqrt{ \\theta^2+4\\theta_E^2}+ \\theta\\Big) \\qquad {\\rm and} \\qquad\n\\theta^{[0]}_-= {\\textstyle\\frac{1}{2}} \\Big(\\sqrt{ \\theta^2+4\\theta_E^2}-\\theta\\Big),\n\\label{eq:S-l-pri-mic}\n\\end{equation}\nwhere $\\theta_E=\\sqrt{{2r_g}\/{\\tilde r}}$ is the Einstein deflection angle.  This establishes the correspondence of our analysis in this section to the well-known modeling of microlensing \\cite{Liebes:1964,Refsdal:1964,Schneider-Ehlers-Falco:1992}.\n\nExpressions (\\ref{eq:S-l-pri-mic})  lead to the familiar expression to describe the image magnification of $A=(u^2+2)\/(u\\sqrt{u^2+4})$, where $u= \\theta\/\\theta_E$.  Our description allows us to develop the vectorial description of the microlensing phenomena and, besides magnification, it also allows us to describe light amplification. Furthermore, our approach allows further improvement -- it allows to describe deflection on multipoles, where the motion is not loner in one plane, but is a function of all three coordinates $(\\tilde r, \\theta, \\phi)$.\n\nTo demonstrate this, we continue with the solution of (\\ref{eq:b1}). To solve for $b^{[1]}$, we use $b^{[0]}$ from  (\\ref{eq:b0}) and substitute it in  (\\ref{eq:b1}) to derive $b^{[1]}$ to ${\\cal O}\\big(r_g^2, \\theta^3\\big)$:\n{}\n\\begin{eqnarray}\nb_{[1]}=-\\frac{\nr_g\\tilde r}{\\sqrt{\\big({\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\big)^2+2r_g \\tilde r}}\\sum_{n=2}^\\infty J_n\\frac{R^n_\\odot \\sin^n\\beta_s\\cos[n(\\phi_\\xi-\\phi_s)]}{\\Big(\\sqrt{\\big({\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\big)^2+2r_g \\tilde r}+ {\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^n}.\n\\label{eq:b2}\n\\end{eqnarray}\n\nAs a result, using expression (\\ref{eq:S-l-pri+}) and (\\ref{eq:b2}) in the solution to (\\ref{eq:S-l-pri=}) takes the form valid to the order of ${\\cal O}\\big(r_g^2, \\theta^3\\big)$:\n\\begin{eqnarray}\nb&=&\\sqrt{\\big({\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\big)^2+2r_g \\tilde r}+{\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)-\\nonumber\\\\\n&-&\\frac{\nr_g\\tilde r}{\\sqrt{\\big({\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\big)^2+2r_g \\tilde r}}\\sum_{n=2}^\\infty J_n\\frac{R^n_\\odot \\sin^n\\beta_s\\cos[n(\\phi_\\xi-\\phi_s)]}{\\Big(\\sqrt{\\big({\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\big)^2+2r_g \\tilde r}+ {\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^n}.\n\\label{eq:b12}\n\\end{eqnarray}\n\nWith this result for the impact parameter $b$, we may now proceed with forming the stationary phase solution.\n\n\\subsubsection{Computing expressions needed the stationary phase}\n\nTo establish the solution with the method of stationary phase, we also need to compute the second derivative of the phase $\\varphi ({\\vec b})$ from (\\ref{eq:beta*3+4}). With respect to $b$, we have\n{}\n\\begin{eqnarray}\n\\frac{d^2\\varphi}{db^2}=k\\Big(\\frac{1}{\\tilde r} +\\frac{2r_g}{b^2}+{\\cal O}(J_n)\\Big).\n\\label{eq:S-l2+p}\n\\end{eqnarray}\nNote that we need $\\varphi''$ only to the order of ${\\cal O}(J_n)$ as in the eikonal approximation we may neglect the influence of the short-range potential  (that depends on the mass multipoles) on the amplitude of the EM wave.\n\nNow, using expression for $b\\equiv b_0$ from (\\ref{eq:b12}), we have\n{}\n\\begin{eqnarray}\n\\varphi''(b_0)&=&\\dfrac{2k}{\\tilde r}\\frac{\\sqrt{\\big({\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\big)^2+2r_g \\tilde r}}{\\sqrt{\\big({\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\big)^2+2r_g \\tilde r}+ {\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)}+\n{\\cal O}(r_g^2, \\theta^3, J_n),\n\\label{eq:S-l220-g=}\n\\end{eqnarray}\nwhich is always positive. Next, we compute $\\sqrt{{2\\pi}\/{\\varphi''(b_0)}}$ as\n{}\n\\begin{eqnarray}\n\\sqrt{\\frac{2\\pi}{\\varphi''(b_0)}}&=&\\sqrt{\\frac{\\pi \\tilde r}{k}}\\Bigg[1+ \\frac{{\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)}{\\sqrt{\\big({\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\big)^2+2r_g \\tilde r}}\\Bigg]^{1\/2}\n+{\\cal O}(r_g^2, \\theta^3, J_n).\n\\label{eq:S-l220-g}\n\\end{eqnarray}\n\nAs a result, the amplitude of the integrand in (\\ref{eq:gamma0}), for $b$ from (\\ref{eq:b12}), is taking the form\n{}\n\\begin{eqnarray}\nA(b_0)\\sqrt{\\frac{2\\pi}{\\varphi''(b_0)}}\n&=&\\frac{k}{rr_0}b\\sqrt{\\frac{2\\pi}{\\varphi''}}=\\frac{\\sqrt{\\pi k\\tilde r}}{r+r_0}\\Bigg[\\frac{\\Big(\\sqrt{\\Big({\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^2+\\frac{2r_g}{ \\tilde r}}+{\\textstyle\\frac{1}{2}} \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^3}{\\sqrt{\\Big({\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^2+\\frac{2r_g}{ \\tilde r}}}\\Bigg]^{1\/2}.~~~~~\n\\label{eq:apm}\n\\end{eqnarray}\n\nWe are now ready to assemble the stationary phase solution for (\\ref{eq:gamma0}), treating the radial part of the double integral.\n\n\\subsubsection{Summary of the stationary phase solution}\n\nFinally, we need to compute the stationary phase. For this, we substitute the solution for the impact parameter, $ b$, from  (\\ref{eq:b12}) into the expression for the phase, $\\varphi(\\vec b)$ given by (\\ref{eq:beta*3+4}):\n{}\n\\begin{eqnarray}\n\\varphi(b_{\\tt in\/sc})&=&\nk\\bigg\\{-{\\textstyle\\frac{1}{2}}  \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\Big(\\sqrt{\\Big({\\textstyle\\frac{1}{2}} \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^2+\\frac{2r_g}{\\tilde r}}+{\\textstyle\\frac{1}{2}} \\theta\\cos(\\phi_\\xi-\\phi)\\Big)-\\nonumber\\\\\n&&-\\,2r_g\\ln \\Big(\\sqrt{\\Big({\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^2+\\frac{2r_g}{\\tilde r}}+ {\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi) \\Big)+ r_g-2r_g\\ln k\\tilde r-\\nonumber\\\\\n&&-\\,2r_g\\sum_{n=2}^\\infty \\frac{J_n}{n}\\frac{R^n_\\odot \\sin^n\\beta_s\\cos[n(\\phi_\\xi-\\phi_s)]}{\\Big(\\sqrt{\\big({\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\big)^2+2r_g \\tilde r}+ {\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^n}\\bigg\\}.\n  \\label{eq:stat-ph}\n\\end{eqnarray}\n\nAs a result, the factor $\\gamma(\\tilde r,\\theta,\\phi)$ from (\\ref{eq:gamma0}) corresponding to the incident EM wave moving towards the interference region is given in the following form:\n{}\n\\begin{eqnarray}\n\\gamma(\\tilde r,\\theta,\\phi)&=&\n\\frac{E_0}{r+r_0}\\, e^{ik\\big(r+r_0+r_g\\ln 4k^2rr_0+r_g-2r_g\\ln k\\tilde r\\big)-i{\\textstyle\\frac{\\pi}{4}}} \\frac{1}{2\\pi}\\int_0^{2\\pi} d\\phi_\\xi \\,\na(\\tilde r,\\theta,\\phi)\ne^{i\\varphi_{\\tt }(\\tilde r,\\theta,\\phi)}+{\\cal O}(\\theta^2, \\frac{r_g}{r}\\theta^2),\n\\label{eq:Pi_s_exp4+1pp}\\\\\na(\\tilde r,\\theta,\\phi)&=&\n\\sqrt{\\pi k\\tilde r}\\Bigg[\\frac{\\Big(\\sqrt{\\Big({\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^2+\\frac{2r_g}{ \\tilde r}}+{\\textstyle\\frac{1}{2}} \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^3}{\\sqrt{\\Big({\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^2+\\frac{2r_g}{ \\tilde r}}}\\Bigg]^{1\/2},\n\\label{eq:Pi_s_exp4+1pp2}\\\\\n\\varphi(\\tilde r,\\theta,\\phi)\n&=&-k\\bigg\\{{\\textstyle\\frac{1}{2}}   \\tilde r\\theta\\cos(\\phi_\\xi-\\phi)\\Big(\\sqrt{\\Big({\\textstyle\\frac{1}{2}} \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^2+\\frac{2r_g}{\\tilde r}}+{\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi)\\Big)+\\nonumber\\\\\n&&\\hskip 20pt +\\,2r_g\\ln \\Big(\\sqrt{\\Big({\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^2+\\frac{2r_g}{\\tilde r}}+ {\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi) \\Big)+\\nonumber\\\\\n&&\\hskip 20pt +\\,2r_g\\sum_{n=2}^\\infty \\frac{J_n}{n}\\frac{R^n_\\odot \\sin^n\\beta_s\\cos[n(\\phi_\\xi-\\phi_s)]}{\\Big(\\sqrt{\\big({\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\big)^2+2r_g \\tilde r}+ {\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^n}\\bigg\\}.\n\\label{eq:Pi_s_exp4+1pp3}\n\\end{eqnarray}\n\nResults (\\ref{eq:Pi_s_exp4+1pp})--(\\ref{eq:Pi_s_exp4+1pp3}) provide all the necessary information for us to compute the components of the EM field in all the regions behind the lens in the case of the weak gravitational field.\n\n\\subsection{Deriving the EM field on the image plane}\n\nWith the expressions developed above, as a result, the components of the incident and scattered EM field from (\\ref{eq:DB-sol-rho_go}) to the order of ${\\cal O}\\big(r_g^2, \\theta^2, b\/z_0\\big)$ take the form\n{}\n\\begin{eqnarray}\n    \\left( \\begin{aligned}\n{D}_\\theta& \\\\\n{B}_\\theta& \\\\\n  \\end{aligned} \\right)_{\\tt \\hskip -2pt }\n=    \\left( \\begin{aligned}\n{B}_\\phi& \\\\\n-{D}_\\phi& \\\\\n  \\end{aligned} \\right)_{\\tt \\hskip -2pt }&=&\n \\frac{E_0}{r+r_0}e^{i\\Omega(t)}\n   \\frac{1}{2\\pi}\\int_0^{2\\pi} d\\phi_\\xi \\,\n  { A}_{\\tt } (\\tilde r,\\theta,\\phi)\n   \\left( \\begin{aligned}\n \\cos\\phi& \\\\\n \\sin\\phi& \\\\\n  \\end{aligned} \\right),~~~~\n  \\label{eq:DB-sol-in}\n\\end{eqnarray}\nwhere the phase $\\Omega(t)$ is given as\n{}\n\\begin{eqnarray}\n\\Omega(t)=k\\big(r+r_0+r_g\\ln 4k^2rr_0+r_g-2r_g\\ln k\\tilde r\\big)-{\\textstyle\\frac{\\pi}{4}}-\\omega t,~~~~~\n\\label{eq:omega}\n\\end{eqnarray}\nand the complex amplitude ${ A}_{\\tt }={ A}_{\\tt }  \\big(\\vec x,\\phi_\\xi\\big)$ is given as\n{}\n\\begin{eqnarray}\n{ A}_{\\tt }(\\vec x,\\phi_\\xi)&=&a_{\\tt }(\\vec x,\\phi_\\xi)e^{i \\varphi_{\\tt }(\\vec x,\\phi_\\xi)}=\\nonumber\\\\\n&=&\n\\sqrt{\\pi k\\tilde r}\\Bigg[\\frac{\\Big(\\sqrt{\\Big({\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^2+\\frac{2r_g}{ \\tilde r}}+{\\textstyle\\frac{1}{2}} \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^3}{\\sqrt{\\Big({\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^2+\\frac{2r_g}{ \\tilde r}}}\\Bigg]^{1\/2}\\times\\nonumber\\\\\n&&\\hskip 20pt\\times \\exp\\bigg[-ik\\bigg\\{{\\textstyle\\frac{1}{2}}  \\tilde r\\theta\\cos(\\phi_\\xi-\\phi)\\Big(\\sqrt{\\Big({\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^2+\\frac{2r_g}{\\tilde r}}+{\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi)\\Big)+\\nonumber\\\\\n&&\\hskip 54pt +\\,2r_g\\ln \\Big(\\sqrt{\\Big({\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^2+\\frac{2r_g}{\\tilde r}}+ {\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi) \\Big)+\\nonumber\\\\\n&&\\hskip 54pt +\\,2r_g\\sum_{n=2}^\\infty \\frac{J_n}{n}\\frac{R^n_\\odot \\sin^n\\beta_s\\cos[n(\\phi_\\xi-\\phi_s)]}{\\Big(\\sqrt{\\big({\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\big)^2+2r_g \\tilde r}+ {\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^n}\\bigg\\}\\bigg],\n  \\label{eq:DB-sol-inA}\n\\end{eqnarray}\nwhere the radial components of the EM waves behave as $({E}_r, {H}_r)_{\\tt \\hskip 0pt in\/sc} \\sim {\\cal O}({\\rho}\/{r},b\/r_0)$ and, thus, are negligible for any practical purposes. Note that if $ \\theta\\gg \\sqrt{2r_g\/\\tilde r}$, results are identical to those reported in \\cite{Turyshev-Toth:2019-extend}.\n\nAs our interest is the EM field on the image plane, it is convenient to transform these solutions to cylindrical coordinates $(\\rho,\\phi,z)$, as was done in  \\cite{Turyshev-Toth:2017,Turyshev-Toth:2019-extend}. Transforming (\\ref{eq:DB-sol-in}),  yields the components of both solutions, to ${\\cal O}(r_g^2, \\theta^2, b\/r_0)$, in the form\n{}\n\\begin{eqnarray}\n    \\left( \\begin{aligned}\n{E}_\\rho& \\\\\n{H}_\\rho& \\\\\n  \\end{aligned} \\right)_{\\tt \\hskip -2pt  } =    \\left( \\begin{aligned}\n{H}_\\phi& \\\\\n-{E}_\\phi& \\\\\n  \\end{aligned} \\right)_{\\tt \\hskip -2pt  }&=&\n\\frac{E_0}{r+r_0}e^{i\\Omega(t)}  B \\big(\\vec x\\big)\n \\left( \\begin{aligned}\n \\cos\\overline \\phi& \\\\\n \\sin\\overline \\phi& \\\\\n  \\end{aligned} \\right),\n \n  \\label{eq:DB-sol-in-cc8=}\n\\end{eqnarray}\nwhere the $z$-components of the EM waves behave as $({E}_z, {H}_z)_{\\tt \\hskip 0pt } \\sim {\\cal O}({\\rho}\/{z},\\sqrt{2r_gz}\/z_0)$, and where $\\overline\\phi$ is the angle that corresponds to the rotated $\\overline z$ coordinate axis described in \\cite{Turyshev-Toth:2019-extend}. The quantity $ B \\big(\\vec x\\big) $ is  the complex amplitude of the EM field has the following form:\n{}\n\\begin{eqnarray}\n B \\big(\\vec x\\big) =\n \\frac{1}{2\\pi}\\int_0^{2\\pi} d\\phi_\\xi \\,  { A}_{\\tt }  \\big(\\vec x,\\phi_\\xi\\big).~~~~\n  \\label{eq:DB-sol-in-cc=}\n\\end{eqnarray}\nWith ${ A}_{\\tt }  \\big(\\vec x,\\phi_\\xi\\big)$ given by (\\ref{eq:DB-sol-inA}), the complex amplitude takes the form\n\\begin{eqnarray}\n B \\big(\\vec x\\big) &=&\n\\sqrt{\\pi k\\tilde r}  \\,\n \\frac{1}{2\\pi}\\int_0^{2\\pi} d\\phi_\\xi \\,\n\\Bigg[\\frac{\\Big(\\sqrt{\\Big({\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^2+\\frac{2r_g}{ \\tilde r}}+{\\textstyle\\frac{1}{2}} \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^3}{\\sqrt{\\Big({\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^2+\\frac{2r_g}{ \\tilde r}}}\\Bigg]^{1\/2}\\times\\nonumber\\\\\n&&\\hskip 20pt\\times \\exp\\bigg[-ik\\bigg\\{{\\textstyle\\frac{1}{2}}  \\tilde r\\theta\\cos(\\phi_\\xi-\\phi)\\Big(\\sqrt{\\Big({\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^2+\\frac{2r_g}{\\tilde r}}+{\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi)\\Big)+\\nonumber\\\\\n&&\\hskip 80pt +\\,2r_g\\ln \\Big(\\sqrt{\\Big({\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^2+\\frac{2r_g}{\\tilde r}}+ {\\textstyle\\frac{1}{2}}  \\theta\\cos(\\phi_\\xi-\\phi) \\Big)+\\nonumber\\\\\n&&\\hskip 100pt +\\,2r_g\\sum_{n=2}^\\infty \\frac{J_n}{n}\\frac{R^n_\\odot \\sin^n\\beta_s\\cos[n(\\phi_\\xi-\\phi_s)]}{\\Big(\\sqrt{\\big({\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\big)^2+2r_g \\tilde r}+ {\\textstyle\\frac{1}{2}} \\tilde r \\theta\\cos(\\phi_\\xi-\\phi)\\Big)^n}\\bigg\\}\\bigg].~~~~~\n  \\label{eq:DB-sol-in-cc+}\n\\end{eqnarray}\n\nThis universal expression for the complex amplitude is valid in all regions of an extended axisymmetric lens, including the geometric optics region, weak and strong interference regions. It represents a powerful result that is now applicable in all these diverse regions with very different gravitational lensing behavior.\n\nWe may now evaluate the optical performance of the SGL of the extended Sun by computing its PSF. The PSF characterizes the impulse response of the optical system: it maps light from a point source into the image plane. We can follow the approach used in \\cite{Turyshev-Toth:2021-multipoles}, using the result (\\ref{eq:DB-sol-in-cc8=})--(\\ref{eq:DB-sol-in-cc+}) to compute the energy flux in the image region of the lens.\nWith overline and brackets denoting time averaging and ensemble averaging, the relevant components of the time-averaged Poynting vector for the EM field in the image volume may be given in the following form (see \\cite{Turyshev-Toth:2017,Turyshev-Toth:2019,Turyshev-Toth:2019-extend} for details):\n {}\n\\begin{eqnarray}\nS_z({\\vec x})=\\frac{c}{4\\pi}\\big<\\overline{[{\\rm Re}{\\vec E}\\times{\\rm Re}{\\vec H}]}_z\\big>=\\frac{c}{4\\pi}\\frac{E_0^2}{(r+r_0)^2}\n\\big<\\overline{\\big({\\rm Re}\\big[{ B}({\\vec x})e^{i\\Omega (t)}\\big]\\big)^2}\\big>=\\frac{c}{8\\pi}\\frac{E_0^2}{(r+r_0)^2}\n|B({\\vec x})|^2,\n  \\label{eq:S_z*6z}\n\\end{eqnarray}\nwhere $|B({\\vec x})|^2=B({\\vec x})B^*({\\vec x})$, with $B^*(\\vec x)$ being the complex conjugate of $B({\\vec x})$. Note that ${\\bar S}_\\rho= {\\bar S}_\\phi=0$ for all practical purposes. Defining light amplification as usual \\cite{Turyshev-Toth:2017,Turyshev-Toth:2019,Turyshev-Toth:2019-extend}, $\\mu_z({\\vec x})=S_z({\\vec x})\/|\\vec S_0({\\vec x})|$, where $\\vec S_0({\\vec x})=(c\/8\\pi){E_0^2}\/{(r+r_0)^2}\\, \\vec k$ being the Poynting vector carried by a plane wave in the vacuum in flat spacetime, we have the light amplification factor of the lens that, for short wavelengths (i.e., $kr_g\\gg1$) is given by\n {}\n\\begin{eqnarray}\n\\mu_z({\\vec x})=|B({\\vec x})|^2.\n  \\label{eq:S_mu}\n\\end{eqnarray}\n We recognize that the quantity $\\mu_z({\\vec x})$ is the PSF of the SGL that is scaled by the amplification factor and it describes all lensing regimes with this extended lens. In Appendix~\\ref{sec:small-dev} we show that, in some cases, the amplification factor explicitly multiplies the PSF, but, in general, the PSF (\\ref{eq:S_mu}) is implicitly scaled by the amplification factor via (\\ref{eq:DB-sol-in-cc+}).\n\nIn Appendix~\\ref{sec:lim-cases} we consider limiting cases of  $B\\big(\\vec x) $ from  (\\ref{eq:DB-sol-in-cc+}). Those cases include very small deviations $\\rho$ from the optical axis, namely $\\rho\/r\\equiv \\theta\\ll \\sqrt{2r_g\/r}$; very large deviations $\\rho\/r\\equiv \\theta\\gg \\sqrt{2r_g\/r}$; and those in-between. We show that far from the optical axis, the PSF that is constructed with the help of $B({\\vec x})$ from (\\ref{eq:DB-sol-in-cc+}) exhibits the behavior of the monopole PSF \\cite{Turyshev-Toth:2017}, but as we come closer to the optical axis, the effect of multipoles becomes more pronounced, ultimately bringing us to the caustic region, discussed in  \\cite{Turyshev-Toth:2021-multipoles}.\n\n\\section{Imaging with the  SGL of the extended Sun}\n\\label{sec:v-large-disp}\n\nThe complex amplitude (\\ref{eq:DB-sol-in-cc+}) developed in the previous section describes the EM field in the image plane. This field, however, is not what is usually observed. Rather, observations are made with an imaging telescope looking back in the direction of the lens. Our formalism also grants us the ability to accurately describe the image that forms in the focal plane of such a telescope: i.e., the actual observable.\n\n \\subsection{Description of the imaging geometry}\n\\label{sec:geom}\n\n With the knowledge of the EM field in the image plane behind an extended gravitational lens  (\\ref{eq:DB-sol-in-cc8=})--(\\ref{eq:DB-sol-in-cc+}) and following the approach developed in \\cite{Turyshev-Toth:2019-image,Turyshev-Toth:2020-im-extend,Turyshev-Toth:2021-imaging}, we can now describe what an imaging telescope would detect on its focal plane.  Such telescopic capability is important as it characterizes the measured optical signal \\cite{Turyshev-Toth:2021-imaging,Turyshev-Toth:2021-quartic}.\n\nSimilarly to \\cite{Turyshev-Toth:2019-image,Turyshev-Toth:2021-imaging}, we describe the geometry of the observation using ${\\vec x}$ to represent the current position of an optical telescope in the SGL's image plane, ${\\vec x}'$, denoting any point in the same plane, and ${\\vec x}_i$, representing a point on the focal plane of the optical telescope. These positions are given as\n{}\n\\begin{eqnarray}\n\\{{\\vec x}\\}&\\equiv& (x,y,0)=\\rho\\,\\big(\\cos\\phi,\\sin\\phi,0\\big)=\\rho{\\vec n}, \\label{eq:coord'}\\\\\n\\{{\\vec x'}\\}&\\equiv& (x',y',0)=\\rho'\\big(\\cos\\phi',\\sin\\phi',0\\big)=\\rho'{\\vec n'},\n\\label{eq:x-im} \\\\\n \\{{\\vec x}_i\\}&\\equiv& (x_i,y_i,0)=\\rho_i\\big(\\cos\\phi_i,\\sin\\phi_i,0\\big)=\\rho_i{\\vec n}_i.\n  \\label{eq:coord}\n\\end{eqnarray}\n\nTo convolve the PSF of the SGL with that of a thin lens that represents an aperture of a telescope, we first need to establish an appropriate form of the PSF for point sources. Examining (\\ref{eq:DB-sol-inA}), we see that it contains the expression $\\rho\\cos(\\phi_\\xi-\\phi)$, which may be transformed as\n{}\n\\begin{eqnarray}\n\\rho\\cos(\\phi_\\xi-\\phi)=({\\vec n}_\\xi\\cdot{\\vec x}).\n\\label{eq:rn}\n\\end{eqnarray}\n\nWe now transition from the current position $\\vec x$ of the telescope to an arbitrary location within the telescope's aperture by the substitution\n{}\n\\begin{eqnarray}\n\\vec x \\qquad \\Rightarrow \\qquad {\\vec x}+\\vec x'.\n\\label{eq:rn0}\n\\end{eqnarray}\nTherefore, we may write\n{}\n\\begin{eqnarray}\n({\\vec n}_\\xi\\cdot{\\vec x}) \\qquad \\rightarrow \\qquad\n({\\vec n}_\\xi\\cdot{\\vec x}) +({\\vec n}_\\xi\\cdot{\\vec x}').\n\\label{eq:rn2}\n\\end{eqnarray}\nWe note that $\\vec x'$ varies only with the aperture, whereas  $\\vec x$ can be anywhere in the SGL image plane. In much of the image plane, we have  $\\rho' \\ll \\rho$. This allows us to expand (\\ref{eq:DB-sol-inA}) in terms of the small parameter  $\\rho'\/\\rho$, keeping only terms of the first order in $\\rho'\/\\rho$. In addition, we recognize that the vector $\\vec \\theta=\\theta(\\cos\\phi,\\sin\\phi,0)=\\vec x\/r$, with $r$ being the distance to the image plane,  may be transformed as\n{}\n\\begin{eqnarray}\n({\\vec n}_\\xi\\cdot{\\vec \\theta})=({\\vec n}_\\xi\\cdot{\\vec x})\/r \\qquad \\rightarrow \\qquad\n({\\vec n}_\\xi\\cdot{\\vec x})\/r +({\\vec n}_\\xi\\cdot{\\vec x}')\/r=\\frac{\\rho}{r}\\cos(\\phi_\\xi-\\phi)+\\frac{\\rho'}{r}\\cos(\\phi_\\xi-\\phi').\n\\label{eq:the}\n\\end{eqnarray}\n\nThis approximation yields the following result for the complex amplitude, ${ A}_{\\tt }({\\vec x})$, from (\\ref{eq:DB-sol-inA}), but given with the shifted argument according to (\\ref{eq:rn0}):\n{}\n\\begin{eqnarray}\n{ A}_{\\tt }({\\vec x},{\\vec x}')&=&\na_{\\tt }(\\vec x,\\vec n_\\xi)\n\\exp\\Big[i\\Big(\\delta\\varphi_{\\tt }(\\vec x,{\\vec n}_\\xi)-\\nu{\\tt }({\\vec n}_\\xi\\cdot{\\vec x'})\\Big)\\Big],\n  \\label{eq:amp-Ain}\n\\end{eqnarray}\nwith the amplitude factor $a_{\\tt }(\\vec x)$ and phase $\\delta\\varphi_{\\tt }(\\vec x) $ given as\n{}\n  \\begin{eqnarray}\na_{\\tt }(\\vec x,\\vec n_\\xi) &=& \\sqrt{\\pi k\\tilde r}\\Bigg[\\frac{\\Big(\\sqrt{\\big({\\textstyle\\frac{1}{2}} ({\\vec n}_\\xi\\cdot{\\vec x})\/r \\big)^2+\\frac{2r_g}{ \\tilde r}}+{\\textstyle\\frac{1}{2}}  ({\\vec n}_\\xi\\cdot{\\vec x})\/r \\Big)^3}{\\sqrt{\\big({\\textstyle\\frac{1}{2}}  ({\\vec n}_\\xi\\cdot{\\vec x})\/r \\big)^2+\\frac{2r_g}{ \\tilde r}}}\\Bigg]^{1\/2}+{\\cal O}(\\vec x\/r),\n    \\label{eq:q_insc}\\\\\n    \\delta\\varphi_{\\tt }\n(\\vec x,\\vec n_\\xi) &=& -k\\bigg\\{{\\textstyle\\frac{1}{2}}  ({\\vec n}_\\xi\\cdot{\\vec x}) \\Big( \\sqrt{\\big({\\textstyle\\frac{1}{2}}  ({\\vec n}_\\xi\\cdot{\\vec x})\/r \\big)^2+\\frac{2r_g}{\\tilde r}}+{\\textstyle\\frac{1}{2}}  ({\\vec n}_\\xi\\cdot{\\vec x})\/r\\Big)+\n\\nonumber\\\\\n&+&\n2r_g\\bigg(\\ln \\Big(\\sqrt{\\big({\\textstyle\\frac{1}{2}}  ({\\vec n}_\\xi\\cdot{\\vec x})\/r\\big)^2+\\frac{2r_g}{\\tilde r}}+ {\\textstyle\\frac{1}{2}}  ({\\vec n}_\\xi\\cdot{\\vec x})\/r\\Big)+\n\\sum_{n=2}^\\infty  \\frac{J_n}{n}\\frac{R^n_\\odot \\sin^n\\beta_s\\cos[n(\\phi_\\xi-\\phi_s)]}{\\Big(\\sqrt{\\big({\\textstyle\\frac{1}{2}} ({\\vec n}_\\xi\\cdot{\\vec x})\\big)^2+2r_g \\tilde r}+ {\\textstyle\\frac{1}{2}} ({\\vec n}_\\xi\\cdot{\\vec x})\\Big)^n}\\bigg)\\bigg\\}.~~~~~\n    \\label{eq:Ain-d_ph}\n\\end{eqnarray}\nWe note that when the angle $\\theta$ is large,   $\\theta \\gg \\sqrt{2r_g\/\\tilde r}$ and thus, $\\rho\\gg \\sqrt{2r_g\\tilde r}$, and we get back the PSF of a monopole. Thus, the integral (\\ref{eq:beta*3}) may be taken using the method of stationary phase applied to the double integral. In that case, the factors $a_{\\tt in\/sc}$ in (\\ref{eq:q_insc})  take their known values (see \\cite{Turyshev-Toth:2019-extend} for details), namely $a^2_{\\tt in}(\\rho,\\tilde r)=1+{\\cal O}(r_g\\theta^2,r_g^2)$ and $a^2_{\\tt sc}(\\rho,\\tilde r) =({2r_g\\tilde r}\/{\\rho^2})^2 +{\\cal O}(r_g\\theta^2,r_g^2)$. However, our new expressions (\\ref{eq:q_insc}) allow studying the cases when  $\\rho\\simeq \\sqrt{2r_g\\tilde r}$ anywhere in the image plane.\nThe last quantity present in  (\\ref{eq:amp-Ain})  is the  spatial frequency $\\nu=\\nu_{\\tt }(\\vec x,\\vec n_\\xi)$, defined as\n{}\n  \\begin{eqnarray}\n\\nu_{\\tt }(\\vec x,\\vec n_\\xi)&=&k\\Big(\\sqrt{\\big({\\textstyle\\frac{1}{2}}  ({\\vec n}_\\xi\\cdot{\\vec x})\/r\\big)^2+\\frac{2r_g}{\\tilde r}}+ {\\textstyle\\frac{1}{2}}  ({\\vec n}_\\xi\\cdot{\\vec x})\/r\\Big).\n  \\label{eq:betapm}\n  \\end{eqnarray}\n\nThe quantities (\\ref{eq:amp-Ain})--(\\ref{eq:betapm}) describe the complex amplitude of the EM field, ${B}({\\vec x},{\\vec x}')$ from (\\ref{eq:DB-sol-in-cc=}), as measured in the focal plane of an imaging telescope.\n\n \\subsection{The EM field in the telescope's focal plane}\n\\label{sec:im-tele}\n\nThe focal plane of the optical telescope is located at the focal distance $f$ of the lens, centered on ${\\vec x}'$. Using the Fresnel--Kirchhoff diffraction formula, the amplitude of the image field in the optical telescope's focal plane at a location ${\\vec x}_i=(x_i,y_i)$ is derived from (\\ref{eq:DB-sol-in-cc=}) and is given by  \\cite{Wolf-Gabor:1959,Richards-Wolf:1959,Born-Wolf:1999}:\n{}\n\\begin{eqnarray}\n{B}({\\vec x},{\\vec x}_i)=\\frac{i}{\\lambda}\\iint \\displaylimits_{|{\\vec x}'|^2\\leq (d\/2)^2} \\hskip -7pt  B({\\vec x},{\\vec x}')e^{-i\\frac{k}{2f}|{\\vec x}'|^2}\\frac{e^{iks'}}{s'}d^2{\\vec x}'.\n  \\label{eq:amp-w-f0}\n\\end{eqnarray}\n\nThe function $\\exp[-i\\frac{k}{2f}|{\\vec x}'|^2]=\\exp[-i\\frac{k}{2f}(x'^2+y'^2)]$ in (\\ref{eq:amp-w-f0}) represents the action of the convex lens that transforms incident plane waves to spherical waves, focusing at the focal point. Assuming that the focal length is sufficiently greater than the radius of the lens, we may approximate the optical path $s'$ as $s'=\\sqrt{(x'-x_i)^2+(y'-y_i)^2+f^2}\\sim f+\\big((x'-x_i)^2+(y'-y_i)^2\\big)\/2f$. This allows us to present (\\ref{eq:amp-w-f0}) as\n{}\n\\begin{eqnarray}\n{B}({\\vec x},{\\vec x}_i)&=&\n- \\frac{e^{ikf(1+{{\\vec x}_i^2}\/{2f^2})}}{i\\lambda f}\\iint\\displaylimits_{|{\\vec x}'|^2\\leq (\\frac{1}{2}d)^2} d^2{\\vec x}'\n B({\\vec x},{\\vec x}') e^{-i\\frac{k}{f}({\\vec x}'\\cdot{\\vec x}_i)}.\n  \\label{eq:amp-w-f+}\n\\end{eqnarray}\n\nExpressions (\\ref{eq:amp-Ain})--(\\ref{eq:betapm})  allow us to consider imaging of point sources with the SGL, now treated as that produced by a gravitating body that is axisymmetric and rotating, thus admitting characterization of its external gravitational field by zonal harmonics. To accomplish this, following \\cite{Turyshev-Toth:2019-image,Turyshev-Toth:2020-im-extend}, we use the expression for $A({\\vec x},{\\vec x}')$ from (\\ref{eq:amp-Ain}) and present the Fresnel--Kirchhoff diffraction formula as\n{}\n\\begin{eqnarray}\n{\\cal A}_{\\tt }({\\vec x},{\\vec x}_i)&=&\n- \\frac{e^{ikf(1+{{\\vec x}_i^2}\/{2f^2})}}{i\\lambda f}\\iint\\displaylimits_{|{\\vec x}'|^2\\leq (\\frac{1}{2}d)^2} d^2{\\vec x'}\\,\n{ A}_{\\tt }({\\vec x},{\\vec x}') e^{-i\\eta_i({\\vec n}_i\\cdot{\\vec x}')}=\\nonumber\\\\\n&=&- \\frac{e^{ikf(1+{{\\vec x}_i^2}\/{2f^2})}}{i\\lambda f}\n a_{\\tt}(\\vec x,\\vec n_\\xi)\ne^{i\\delta\\varphi_{\\tt }(\\vec x,\\vec n_\\xi)}\\,\n\\iint\\displaylimits_{|{\\vec x}'|^2\\leq (d\/2)^2}\\hskip -8pt\n  d^2{\\vec x}'\\,\ne^{i\\big(-\\nu_{\\tt }({\\vec n}_\\xi\\cdot{\\vec x}')-\\eta_i({\\vec n}_i\\cdot{\\vec x}')\\big)},\n  \\label{eq:amp-w-f}\n\\end{eqnarray}\nwhere the spatial frequency $\\nu=\\nu_{\\tt }(\\vec x,\\vec n_\\xi)$ is given by (\\ref{eq:betapm}).  Also, for a telescope with focal length of $f$ and for a radial pixel position $\\rho_i$, the factor $ \\eta_i$ has the form \\cite{Turyshev-Toth:2019-image,Turyshev-Toth:2020-im-extend,Turyshev-Toth:2021-imaging}\n{}\n\\begin{eqnarray}\n \\eta_i=k\\frac{\\rho_i}{f}.\n\\label{eq:zerJ}\n\\end{eqnarray}\n\nTherefore, to derive  the amplitudes of the EM field in the focal plane of the optical telescope, corresponding to  (\\ref{eq:amp-Ain}), we need to evaluate an integral of the type\n{}\n\\begin{eqnarray}\n\\iint\\displaylimits_{|{\\vec x}'|^2\\leq (d\/2)^2}\\hskip -8pt\n  d^2{\\vec x}'\\,e^{i\\big(- \\nu_{\\tt }({\\vec n}_\\xi\\cdot{\\vec x}')-\\eta_i({\\vec n}_i\\cdot{\\vec x}')\\big)}.\n  \\label{eq:amp-int}\n\\end{eqnarray}\nTo evaluate this integral, we present the phase  in (\\ref{eq:amp-int}) as\n{}\n\\begin{eqnarray}\n- \\nu_{\\tt }({\\vec n}_\\xi\\cdot{\\vec x}')-\\eta_i({\\vec n}_i\\cdot{\\vec x}')=-u\\,\\rho' \\cos\\big(\\phi'-\\sigma\\big)+{\\cal O}(\\rho^2),\n  \\label{eq:ph4s}\n\\end{eqnarray}\nwhere, for convenience, we defined\n{}\n\\begin{eqnarray}\nu&=&\\sqrt{\\nu^2+2\\nu\\eta_i\\cos\\big(\\phi_\\xi-\\phi_i\\big)+\\eta_i^2},\n\\qquad\n\\cos\\sigma=\n\\frac{\\nu  \\cos\\phi_\\xi+\\eta_i\\cos\\phi_i}{u},\n\\qquad\n\\sin\\sigma=\\frac{\\nu \\sin\\phi_\\xi+\\eta_i\\sin\\phi_i}{u}.\n\\label{eq:vpms}\n\\end{eqnarray}\n\nWith these definitions, and using the parameterization given in (\\ref{eq:x-im}), the integral (\\ref{eq:amp-int}) may be evaluated as\n{}\n\\begin{eqnarray}\n  \\int_0^{2\\pi} \\hskip -4pt d\\phi' \\int_0^{d\/2} \\hskip -4pt \\rho' d\\rho'\\,\n  e^{-iu\\rho'\\cos(\\phi'-\\sigma)}=\\pi\\Big(\\frac{d}{2}\\Big)^2\\, \\frac{\n2J_1(u\\frac{1}{2}d)}{u\\frac{1}{2}d}.\n  \\label{eq:amp-int*}\n\\end{eqnarray}\nAs a result,  using (\\ref{eq:amp-Ain}) in  (\\ref{eq:amp-w-f})  leads to the following amplitude of the  EM wave on the optical telescope's image plane:\n{}\n\\begin{eqnarray}\n{\\cal A}_{\\tt }({\\vec x},{\\vec x}_i,\\vec n_\\xi)&=&\n\\Big(\\frac{kd^2}{8f}\\Big)\\, \\Big\\{\na_{\\tt }\n\\Big(\\frac{\n2J_1(u\\frac{1}{2}d)}{u\\frac{1}{2}d}\\Big)e^{i\\big(kf(1+{{\\vec x}_i^2}\/{2f^2})+\\delta\\varphi_{\\tt }\n(\\vec x,\\vec n_\\xi) +\\frac{\\pi}{2}\\big)}+{\\cal O}(r_g^2)\\Big\\}.\n  \\label{eq:amp-Aind}\n\\end{eqnarray}\n\nTherefore, the Fourier-transformed complex amplitude (\\ref{eq:amp-w-f0}) takes the from\n{}\n\\begin{eqnarray}\n&&  \\frac{1}{2\\pi}\\int_0^{2\\pi} d\\phi_\\xi \\, {\\cal A}_{\\tt }({\\vec x},{\\vec x}_i,\\vec n_\\xi)=\n \\Big(\\frac{kd^2}{8f}\\Big)e^{i\\big(kf(1+{{\\vec x}_i^2}\/{2f^2}) +\\frac{\\pi}{2}\\big)}\n {\\cal B} \\big(\\vec x,\\vec x_i\\big),~~~~~\n  \\label{eq:FtB}\n\\end{eqnarray}\nwhere ${\\cal B} \\big(\\vec x,\\vec x_i\\big) $ is given as\n{}\n\\begin{eqnarray}\n {\\cal B} \\big(\\vec x,\\vec x_i\\big) &=&\n \\frac{1}{2\\pi}\\int_0^{2\\pi} d\\phi_\\xi \\,  \\Big\\{\n  a_{\\tt }(\\vec x,\\vec n_\\xi)\n  \\Big(\\frac{\n2J_1(u({\\vec x}_i,\\vec x,\\vec n_\\xi)\\frac{1}{2}d)}{u({\\vec x}_i,\\vec x,\\vec n_\\xi) \\frac{1}{2}d}\\Big)\ne^{i\\delta\\varphi_{\\tt }(\\vec x,\\vec n_\\xi)}\\Big\\},~~~~~\n  \\label{eq:Binsc}\n\\end{eqnarray}\nwhere $a_{\\tt }(\\vec x,\\vec n_\\xi)$,  $\\delta\\varphi_{\\tt }(\\vec x,\\vec n_\\xi)$, and $u({\\vec x}_i,\\vec x,\\vec n_\\xi)$ are given by (\\ref{eq:q_insc}), (\\ref{eq:Ain-d_ph}), and (\\ref{eq:vpms}), correspondingly.\n\nUsing this result together with (\\ref{eq:DB-sol-in-cc8=}), we obtain the EM field on the detector that is given as below\n{}\n\\begin{eqnarray}\n    \\left( \\begin{aligned}\n{E}_\\rho& \\\\\n{H}_\\rho& \\\\\n  \\end{aligned} \\right)_{\\tt} =    \\left( \\begin{aligned}\n{H}_\\phi& \\\\\n-{E}_\\phi& \\\\\n  \\end{aligned} \\right)_{\\tt}&=&\n\\frac{E_0}{r+r_0}\n e^{i\\big(\\Omega(t)+\\frac{\\pi}{2}+kf(1+{{\\vec x}_i^2}\/{2f^2})\\big)}\n  \\Big(\\frac{kd^2}{8f}\\Big)\n {\\cal B} \\big(\\vec x,\\vec x_i\\big)\n \\bigg( \\begin{aligned}\n\\cos\\overline \\phi& \\\\\n\\sin\\overline \\phi& \\\\\n  \\end{aligned} \\bigg).\n  \\label{eq:EM-det}\n\\end{eqnarray}\n\nAfter time averaging, we derive the Poynting vector of the EM wave in the focal plane of the imaging telescope:\n {}\n\\begin{eqnarray}\nS_{\\tt }({\\vec x},{\\vec x}_i)&=&\\frac{c}{8\\pi}\n\\frac{E_0^2}{(r+r_0)^2}\n \\Big(\\frac{kd^2}{8f}\\Big)^2\n  {\\cal B}^2 \\big(\\vec x,\\vec x_i\\big) .\n  \\label{eq:PV}\n\\end{eqnarray}\n\nAs a result, the  intensity on the focal plane, ${\\cal I}({\\vec x},{\\vec x}_i)$,  of the system that includes the SGL and a thin lens is given in the form as below:\n {}\n\\begin{eqnarray}\n{\\cal I}({\\vec x},{\\vec x}_i)&=&\n  {\\cal B}^2 \\big(\\vec x,\\vec x_i\\big),\n  \\label{eq:PFT}\n\\end{eqnarray}\nwhere the Fourier-transformed complex amplitude $ {\\cal B}\\big(\\vec x,\\vec x_i\\big)$ from (\\ref{eq:Binsc}).\nWe emphasize that $\\mu_z({\\vec x})$ from (\\ref{eq:S_mu}) is the PSF of the extended SGL. It describes the image of a point source projected on the image plane at the SGL's focal region. At the same time, the quantity ${\\cal I}({\\vec x},{\\vec x}_i)$ from (\\ref{eq:PFT}) is the intensity of light received on the focal plane of an imaging telescope. This is a directly observable quantity that is accessible to an optical telescope. As such, it is of most importance for any practical applications of  the SGL.\nThe resulted expression for the intensity on the focal plane allows considering imaging of various sources with the SGL of an extended Sun. We will do that next.\n\n\\section{Application of results}\n\\label{sec:sims}\n\n\\begin{figure}\n\\includegraphics{psf4m}\n\\caption{\\label{fig:astroid}An illustrative example of the SGL PSF, appearing as the astroid caustic projected into the image plane by the SGL, with its recognizable cusps (vertices) and folds. As an imaging telescope enters this region in the image plane, its view of a distant source transitions from a pair of images (the primary and secondary image) into some variation of an Einstein cross or Einstein ring, depending on the size of the astroid, the imaging wavelength, and the size of the light source. Adapted from \\cite{Turyshev-Toth:2021-multipoles}.}\n\\end{figure}\n\nThe formalism developed in the preceding section opens the route to simulate the effects of the SGL beyond the immediate vicinity of the optical axis in its strong interference region (see Fig.~\\ref{fig:regions}). There is, however, first our obstacle: evaluation of the remaining integral in our final expression (\\ref{eq:Binsc}).\n\n\\subsection{Evaluation method}\n\nEquation~(\\ref{eq:Binsc}) describes the view seen by an imaging telescope of a distant source, both near and far from the optical axis of the gravitational lens. To use this equation, it is necessary to evaluate the remaining integral in the regions of interest. Examining it more closely, we note that the integral has finite integration limits, which makes numerical evaluation easier. However, it is still an oscillatory integral. Moreover, at large distances from the optical axis, the oscillations become very rapid. This makes direct numerical evaluation challenging.\n\nOn the other hand, a rapidly oscillating integral implies the possible use of the method of stationary phase once again. This is precisely what we have accomplished in \\cite{Turyshev-Toth:2021-quartic}, for the case when $J_4$ and higher order zonal harmonics can be safely neglected, thus leaving only the astroid caustic due to $J_2$. The result, expressed through the roots of a quartic equation, works reliably everywhere in the region of strong interference, only showing occasional rounding errors in the immediate vicinity of the caustic boundary of the projected astroid pattern of a quadrupole lens (Fig.~\\ref{fig:astroid} \\cite{Turyshev-Toth:2021-multipoles}).\n\nBeyond the region of strong interference, the contribution of the zonal harmonics is negligible and we can use previously developed monopole solutions for efficient evaluation.\n\nUsing this combination of methods, we are now in the position to evaluate (\\ref{eq:Binsc}) everywhere in the image plane, constructing simulated views of point sources as seen by an imaging telescope through the SGL.\n\n\\subsection{Simulated approach to the optical axis}\n\n\\begin{figure}\n\\includegraphics{ptearly-new}\n\\caption{\\label{fig:appr-pre}View of a distant compact source by a telescope approaching the SGL optical axis associated with that source. The telescope is positioned at $3\\times 10^5$~km, $1\\times 10^5$~km and $4\\times 10^3$~km from the optical axis. Note that at $3\\times 10^5$~km, the secondary image of the source is still obscured by the solar disk (shown as a yellow circle). By the time we reach $4\\times 10^3$~km, the images become indistinguishable, even as light amplification increases nearly hundredfold. For the full animation, see \\protect\\url{https:\/\/www.vttoth.com\/CMS\/physics-notes\/361}.}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics{ptfinal-new}\n\\caption{\\label{fig:approach1}View of a distant point source by a telescope near the optical axis, at 2~m, 1~m and positioned on the axis. The optical axis is at 5.74$^\\circ$ from the solar axis of rotation, a direction chosen because it representatively shows the development of an Einstein cross during this final approach. The view is that of a telescope with a 1~m aperture; light amplification is of ${\\cal O}(10^8)$.\nFor the full animation, see \\protect\\url{https:\/\/www.vttoth.com\/CMS\/physics-notes\/361}.}\n\\end{figure}\n\n\nTo demonstrate the power of the approach captured by the expression (\\ref{eq:Binsc}), we chose to simulate the view of a distant point source, as seen by an imaging telescope that is approaches the optical axis of that star with respect to the SGL.\n\nWe were able to assemble a series of still images, ultimately in the form of animations\\footnote{See \\protect\\url{https:\/\/www.vttoth.com\/CMS\/physics-notes\/361} for a full set of animations.}, which show how an imaging telescope would see the distant source as it was approaches the optical axis that corresponds to that source. Select frames from this animation are presented in this section.\n\nWe began the simulation with the imaging telescope located at $10^6$ km from the optical axis, looking in the direction of the Sun (see Fig.~\\ref{fig:appr-pre}). This distance was chosen because it is comparable in magnitude to the solar radius, thus placing the imaging telescope firmly in the region of geometric optics.\n\nAt the beginning, the source's ``primary image'' is outside the telescope's field of view, and no noticeable ``secondary image'' forms yet on the opposite side of the Sun. At $6\\times 10^5$~km from the optical axis, a faint secondary image emerges, or rather, would emerge if the Sun were transparent. In reality, light from that secondary image is yet blocked by the opaque disk of the Sun. When the telescope is at $3\\times 10^5$~km from the optical axis (less than half the solar radius) the primary image becomes clearly visible within the imaging area. This is the unobstructed view of the distant source, already amplified by the SGL, so its peak central brightness is $\\sim$1.8 times the brightness of the unamplified image. The secondary image, now less faint, is still obscured by the solar disk.\n\nWhen the telescope is only $\\sim1\\times 10^5$~km from the optical axis, the secondary image emerges from behind the Sun. Light amplification is becoming significant: the primary image's peak brightness is now more than four times as bright as the unamplified star. When the telescope approaches within $\\sim 2\\times 10^4$~km of the optical axis, the primary and secondary images are already nearly identical in appearance, at symmetric positions, settling at a distance from the solar limb that corresponds to the radius of a yet-to-form Einstein ring. Light amplification is substantial: the peak brightness that the imaging telescope sees is nearly 20 times the intensity of light from the unamplified star. Even so, the images remain point-like in appearance: This is dictated by the diffraction-limited resolution of the imaging telescope itself.\n\nAt this stage, the position of the two images of the point source is final. As the telescope continues to approach the optical axis, however, light amplification increases across several orders of magnitude.\n\nFor the purposes of this simulation, we chose to place the optical axis very near the solar axis of rotation, in order to keep the contribution of the $J_2$ zonal harmonic small.  Figure \\ref{fig:approach1} shows the telescope's final approach to an optical axis that is at $5.74^\\circ$ from the solar axis of rotation, which corresponds to $\\sin\\beta_s=0.1$. This yields an astroid PSF that is relatively small, convenient for visual presentation.\n\nOnce the telescope is within a distance comparable to the size of the astroid caustic (in this case, within 10 meters), the secondary image begins to widen into an arc. Even closer to the optical axis, the arc splits into three distinct spots of light. As the telescope settles on the optical axis, these spots migrate to their final positions on the circumference of the Einstein ring, resulting in a fully formed Einstein cross. (This simulation assumed that the telescope approaches from one of the principal directions of the astroid caustic, i.e., one of the cusps. To see what happens when the telescope approaches from a different angle, see, e.g., \\cite{Turyshev-Toth:2021-quartic}.)\n\nIt is remarkable that all these animation frames are simply surface density plots of the integral expression given by Eq.~(\\ref{eq:Binsc}), which accurately describes an axisymmetric gravitational lens dominated by a spherically symmetric gravitational potential in all regions, both near and far the optical axis. We can generate with equal ease images seen through a telescope that is positioned as far as a million kilometers or more from the optical axis or a telescope that is at the optical axis or its immediate vicinity.\n\n\\subsection{Viewing an extended object}\n\n\\begin{figure}\n\\includegraphics{hoststar-new}\n\\caption{\\label{fig:hoststar-new}View of a distant star by a telescope approaching the SGL optical axis, at distances of 1,000~km, $\\sim$200~km and $\\sim$150~km. The geometric projection of the start to the image plane would yield a disk with a radius of 200~km. As the imaging telescope approaches this distance, a full Einstein-ring forms; subsequently, the ring brightens and becomes uniform as the telescope settles on the optical axis.}\n\\end{figure}\n\nThe PSF of a lens represents its impulse response: the image that forms when the light source is a point source. An extended object can, of course, be considered as a collection of point sources. The most straightforward method (though computationally inefficient) of convolving an extended source with the PSF of the lens is by dividing the source into point sources and iterating through them.\n\nTo demonstrate this, we considered an extended source in the form of a uniformly illuminated disk, which could represent a host star. We chose a disk that would be geometrically projected to an image with a 200~km radius in the image plane. With the image plane at 650~AU, this would correspond to a Sun-sized star at $\\sim$36 light-years.\n\nFor computational efficiency, we modeled the extended source using a simple adaptive mesh implementation, refining the resolution for regions that are projected close to the telescope's location in the image plane. This approach was sufficient to create a series of animation frames\\footnote{For the full animation, see \\url{https:\/\/www.vttoth.com\/CMS\/physics-notes\/360}.}, several of which are shown in Fig.~\\ref{fig:hoststar-new}.\n\nWhen the telescope looking at such an extended object is far from the optical axis, the telescopic image appears similar to that produced by a point source (see Fig.~\\ref{fig:appr-pre}). However, when the telescope begins to approach the projected image area corresponding to the extended source, a very different picture emerges. Instead of developing into an Einstein cross, the view of the telescope shows a fully formed Einstein ring. We may think of this Einstein ring as a collection of a large number of overlapping Einstein crosses at various orientations, corresponding to the point sources constituting the extended source. Thus, instead of being dominated by light from a single point-like region in the source, the Einstein ring now contains a mix of light from many different regions of the extended source.\n\n\\section{Discussion and Conclusions}\n\\label{sec:end}\n\nWe studied the optical properties of an extended axisymmetric gravitational lens. The gravitational potential for such a lens can be described using an infinite series of zonal harmonics. We extended the description of the SGL optical properties from the strong interference region to all lensing regimes. The new results can now also describe lensing in the weak interference region and that in the geometric optics region.\n\nThe expressions that we obtained can be used to describe the light field that is created by the SGL in its focal region. It can also be convolved with a representation of an optical telescope (modeled as a thin lens telescope) to show the view seen by such a telescope. The results are ``actionable'' in the sense that they are reduced to a single integral expression that can be evaluated in many cases using direct numerical methods.\n\nMoreover, when used in conjunction with our earlier work \\cite{Turyshev-Toth:2021-caustics} in which we obtained a closed form expression of the SGL PSF monopole and quadrupole contribution (ignoring higher-order zonal harmonics that contribute little) the new formalism allows us to compute the light field of the SGL or the view seen by a model telescope without resorting to numerical integration, and thus not hindered by the properties of rapidly oscillating integrals.\n\nWe put these results to use, in particular, by creating a series of multiframe animations that show the view of a pont source through a telescope that is approaching the SGL optical axis from afar. The strength of our formalism is powerfully demonstrated when we consider that the same expression can model the (essentially unamplified) view of a distant object when the telescope is still far from the SGL optical axis; the emergence of a secondary image from behind the solar limb; and the eventual widening of these images into arcs and their transition to form an Einstein cross around the Sun. We can also simulate light from extended objects, showing how, even in the presence of multipole moments, such objects still form an Einstein ring around the Sun.\n\nFinally, we note that although our focus remains the SGL that can be represented elegantly using zonal harmonics, our approach can be readily extended to other gravitational lenses that can be represented using symmetric trace-free (STF) tensors \\cite{Turyshev-Toth:2021-multipoles}. The resulting formalism covers every gravitational lens that can be described by small deviations from the spherically symmetric gravitational field of a mass monopole. Our approach, therefore, is the most comprehensive wave-theoretical treatment of gravitational lensing in a weak gravitational field to date.\n\nConcluding, we emphasize that the analytical expressions derived in this paper are presented in terms of physically observable quantities and, as such, they are directly suitable for realistic data analysis. To that extent, we can use them to process, e.g., time series brightness data available from the OGLE\\footnote{\\url{https:\/\/en.wikipedia.org\/wiki\/Optical_Gravitational_Lensing_Experiment}} and MACHO\\footnote{\\url{https:\/\/en.wikipedia.org\/wiki\/MACHO_Project}}\nprojects, the upcoming Roman Space Telescope\\footnote{\\url{https:\/\/roman.gsfc.nasa.gov\/}}, or other microlensing projects that may benefit from the improved modeling.  In addition, the results presented in this paper offer a solution for establishing a local reference frame that can be used to achieve the required navigational precision for future missions to the SGL's focal region for high-resolution exoplanet imaging \\cite{Turyshev-etal:2020-PhaseII}. The corresponding efforts are under way; results, when available, will be published elsewhere.\n\n\\begin{acknowledgments}\nThis work in part was performed at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.\nVTT acknowledges the generous support of Plamen Vasilev and other Patreon patrons.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nAlgebraic classification (up to isomorphism) of algebras of small dimension from a certain variety defined by a family of polynomial identities is a classic problem in the theory of non-associative algebras. There are many results related to algebraic classification of small dimensional algebras in varieties of Jordan, Lie, Leibniz, Zinbiel and other algebras.\nAnother interesting approach of studying algebras of a fixed dimension is to study them from a geometric point of view (that is, to study degenerations and deformations of these algebras). The results in which the complete information about degenerations of a certain variety is obtained are generally referred to as the geometric classification of the algebras of these variety. There are many results related to geometric classification of Jordan, Lie, Leibniz, Zinbiel and other algebras \\cite{ack, kv17,ikv17, kv16}.\n\nIn 1972, Kantor introduced the notion a conservative algebra as a generalization of Jordan algebras \\cite{Kantor72}. Unlike other classes of non-associative algebras, this class is not defined by a set of identities. To introduce the notion of a conservative algebra, we need some notation. \n Let $\\mathbb V$ be a vector space, let $A$ be a linear operator on $\\mathbb V,$ and let $B$ and $C$ be bilinear operators on $\\mathbb V.$ For all $x,y,z \\in \\mathbb V,$ put\n\\begin{gather*}\n[B,x](y)=B(x,y),\\\\\n[A,B](x,y)= A(B(x,y))- B(A(x),y)-B(x,A(y)),\\\\\n[B,C](x,y,z)=B(C(x,y),z) + B(x,C(y,z))+ B(y,C(x,z))\\\\\n-C(B(x,y),z)- C(x,B(y,z))-C(y,B(x,z)).\n\\end{gather*}\n \nConsider an algebra as a vector space $\\mathbb V$ over a field $\\mathbb{C}$, together with an\n element $\\mu$ of $\\operatorname{Hom}(\\mathbb V \\otimes \\mathbb V,  \\mathbb V),$ so that $a \\cdot b =\\mu(a \\otimes b).$ \nFor an algebra $(\\mathbb V, \\mathcal P)$ with a multiplication $\\mathcal P$ and $x\\in \\mathbb V$ we denote by $L_x^{\\mathcal P}$ the operator of left multiplication by $x.$\nThus, Kantor defines conservative algebras as follows:\n\n\\begin{definition}\nAn algebra $(\\mathbb V, \\mathcal P),$ where $\\mathbb V$ is the vector space and $\\mathcal P$ is the multiplication, is called a conservative algebra if there is a new multiplication $\\mathcal P^*: \\mathbb V\\times \\mathbb V\\rightarrow \\mathbb V$ such that \n\\begin{equation}\\label{uno}\n[L_b^{\\mathcal P},[L_a^{\\mathcal P}, {\\mathcal P}]]=-[L_{{\\mathcal P^*}(a,b)}^{\\mathcal P},{\\mathcal P}] \n\\textrm{, for all $a, b \\in \\mathbb V.$}\n\\end{equation}\nSimple calculations take us to the following identity with an additional multiplication $\\mathcal P^*,$ which must  hold for all $a, b, x, y\\in \\mathbb V$:\n\\begin{equation}\\label{dos}\n\\begin{split}\nb(a(xy)-(ax)y-x(ay)) - a((bx)y) + (a(bx))y+(bx)(ay) -a(x(by))+(ax)(by)+x(a(by))= \\\\\n= - \\mathcal P^*(a,b)(xy)+(\\mathcal P^*(a,b)x)y + x(\\mathcal P^*(a,b)y).\n\\end{split}\n\\end{equation}\n\\end{definition}\n\nThe class of conservative algebras is very vast. It includes\n all associative algebras, \n all quasi-associative algebras,\n all Jordan algebras, \n all Lie algebras, \n all (left) Leibniz algebras,\n all (left) Zinbiel algebras, \n and many other classes of algebras.\nOn the other side, all conservative algebras are \"rigid\" (by Cantarini and Kac) algebras  \\cite{kacan}.\n\n\nHowever, this class is very hard to study and for now even the basic general questions about it remain unanswered, so it is a good idea to study its subclasses which are sufficiently wide but easier to deal with. In \\cite{Kantor89} Kantor, studying the generalized TKK functor, introduced the class of terminal algebras which is a subclass of the class of conservative algebras. \n\n\\begin{definition}\nAn algebra $(\\mathbb V, \\mathcal P),$ where $\\mathbb V$ is a vector space and $\\mathcal P$ is a multiplication, is called a terminal algebra if for all $a\\in \\mathbb V$ we have\n\\begin{equation}\\label{tress}\n[[[{\\mathcal P},a],{\\mathcal P}],{\\mathcal P}]=0.\n\\end{equation}\n\n\\end{definition}\n\nNote that we can expand the relation (\\ref{tress}), obtaining an identity of degree 4. Therefore, the class of terminal algebras is a variety.\nAlso, about terminal and conservative algebras see, for example \\cite{Kantor90, KPP18, cfk19}.\n \nThe following remark is obtained by straightforward calculations.\n\n\\begin{remark}\nA commutative algebra satisfying (\\ref{tress}) is a Jordan algebra. \n\\end{remark}\n\nAside from Jordan algebras, the class of terminal algebras includes all Lie algebras, all (left) Leibniz algebras and some other types of algebras.\n\nThe following characterization of terminal algebras, proved by Kantor \\cite[Theorem 2]{Kantor89}, provides a description of this class as a subclass of the class of conservative algebras.\n\n\\begin{remark}\nAn algebra $(\\mathbb{V}, {\\bf P})$ is terminal if and only if it is conservative\n and the multiplication in the associated superalgebra ${\\bf P}^*$ can be defined by\n \\begin{equation}\n\\label{terminal_associated} \n     {\\bf P}^*(x,y)=\\frac{2}{3}{\\bf P}(x,y)+\\frac{1}{3}{\\bf P}(y,x).\n \\end{equation}\n\\end{remark}\n\n\nOur method of classification of nilpotent terminal algebras is based on the calculation of central extensions of smaller nilpotent algebras from the same variety.  The algebraic study of central extensions of Lie and non-Lie algebras has a very big story \\cite{omirov,hac16,ss78}.\nSkjelbred and Sund used central extensions of Lie algebras for a classification of nilpotent Lie algebras  \\cite{ss78}. After that, using the method described by Skjelbred and Sund were described  all non-Lie central extensions of  all $4$-dimensional Malcev algebras \\cite{hac16}, all non-associative central extensions of $3$-dimensional Jordan algebras, all anticommutative central extensions of $3$-dimensional anticommutative algebras,\nall central extensions of $2$-dimensional algebras \\cite{cfk18}.\n\nThe algebraic classification of nilpotent algebras will be achieved by the calculation of central extensions of algebras from the same variety which have a smaller dimension.\nCentral extensions of algebras from various varieties were studied, for example, in \\cite{ss78,omirov}.\nSkjelbred and Sund \\cite{ss78} used central extensions of Lie algebras to classify nilpotent Lie algebras.\nUsing the same method,  \nall non-Lie central extensions of  all $4$-dimensional Malcev algebras \\cite{hac16},\nall non-associative central extensions of all $3$-dimensional Jordan algebras,\nall anticommutative central extensions of $3$-dimensional anticommutative algebras,\nall central extensions of $2$-dimensional algebras \\cite{cfk18}\nand some others were described.\nOne can also look at the classification of\n$3$-dimensional nilpotent algebras \\cite{fkkv},\n$4$-dimensional nilpotent associative algebras \\cite{degr1},\n$4$-dimensional nilpotent Novikov algebras,\n$4$-dimensional nilpotent bicommutative algebras,\n$4$-dimensional nilpotent commutative algebras in \\cite{fkkv},\n$5$-dimensional nilpotent restricted Lie agebras \\cite{usefi1},\n$5$-dimensional nilpotent Jordan algebras,\n$5$-dimensional nilpotent anticommutative algebras \\cite{fkkv},\n$6$-dimensional nilpotent Lie algebras \\cite{degr3, degr2},\n$6$-dimensional nilpotent Malcev algebras \\cite{hac18},\n$6$-dimensional nilpotent Tortkara algebras,\n$6$-dimensional nilpotent binary Lie algebras \\cite{ack}.\n\n\nIn the present paper, we classify nilpotent terminal algebras of dimension less than or equal to 4, and obtain the complete description of degenerations of these algebras.\n\n\n\\section{The algebraic classification of nilpotent terminal algebras}\n\\subsection{Method of classification of nilpotent algebras}\nThroughout this paper, we  use the notation and methods described in \\cite{hac16,cfk18}\nand adapted for the terminal case with some modifications (see also \\cite{Jac} for a discussion of extensions of algebras in an arbitrary nonassociative variety). Therefore, all statements in this subsection are given without proofs, which can be found in the papers cited above.\n\nLet ${\\bf A}$ be a terminal algebra over $\\mathbb C$ and $\\mathbb V$ a vector space of dimension $s$ over the same base field. Then the space ${\\rm Z_T^2}(\\bf A,\\mathbb V )$ is defined as the set of all  maps $\\theta :{\\bf A} \\times {\\bf A} \\to {\\mathbb V}$ such that \n\\[\\theta(b,a(xy) - (ax)y - x(ay)) - \\theta(a,(bx)y) + \\theta(a(bx),y) + \\theta(bx,ay)\\]\n\\[- \\theta(a,x(by)) + \\theta(ax,(by)) + \\theta(x,(ab)y) = -\\theta({\\bf P}^*(a,b),xy) + \\theta({\\bf P}^*(a,b)x,y) + \\theta(x,{\\bf P}^*(a,b)y),\\]\nwhere ${\\bf P}^*$ is given by (\\ref{terminal_associated}). Its elements will be called \\textit{cocycles}. For a linear map $f: \\bf A \\to \\mathbb V$ define $\\delta f: {\\bf A} \\times\n{\\bf A} \\to {\\mathbb V}$ by $\\delta f (x,y ) =f(xy).$ One can check that $\\delta f\\in {\\rm Z_T^2}({\\bf A},{\\mathbb V} ).$ Therefore, ${\\rm B^2}({\\bf A},{\\mathbb V}) =\\left\\{ \\theta =\\delta f\\ :f\\in \\operatorname{Hom}({\\bf A},{\\mathbb V}) \\right\\}$ is a subspace of ${\\rm Z_T^2}({\\bf A},{\\mathbb V})$ whose elements are called \\textit{coboundaries}. We define the \\textit{second cohomology space} ${\\rm H_T^2}({\\bf A},{\\mathbb V})$ as the quotient space ${\\rm Z_T^2}({\\bf A},{\\mathbb V}) \\big\/{\\rm B^2}({\\bf A},{\\mathbb V}).$\nThe equivalence class of $\\theta \\in {\\rm Z^2}({\\bf A},{\\mathbb V})$  in ${\\rm H^2}({\\bf A},{\\mathbb V})$ will be denoted by $[\\theta].$\n\n\\bigskip\n\nFor a bilinear map $\\theta :{\\bf A} \\times {\\bf A} \\to {\\mathbb V}$ define on the linear space ${\\bf A}_{\\theta }:={\\bf A}\\oplus {\\mathbb V}$ a bilinear product $[-,-]_{{\\theta}}$ by \n\\[[x+x^{\\prime },y+y^{\\prime }]_{{\\theta}}= xy +\\theta (x,y) \\mbox{ for all } x,y\\in {\\bf A},x^{\\prime },y^{\\prime }\\in {\\mathbb V}.\\]\nThen ${\\bf A}_{\\theta }$ is an algebra called an $s${\\it{-dimensional central extension}} of ${\\bf A}$ by ${\\mathbb V}.$ The following statement can be verified directly:\n\n\\begin{lemma}\nThe algebra ${\\bf A_{\\theta}}$ is terminal \\textit{if and only if} $\\theta \\in {\\rm Z_T^2}({\\bf A}, {\\mathbb V}).$\n\\end{lemma}\n\nRecall that the {\\it{annihilator}} of ${\\bf A}$ is defined as the ideal $\\operatorname{Ann}({\\bf A} ) =\\{ x\\in {\\bf A}:  x{\\bf A}+{\\bf A}x=0\\}.$ Given $\\theta \\in {\\rm Z_T^2}({\\bf A}, {\\mathbb V}),$ we call the set $\\operatorname{Ann}(\\theta)=\\left\\{ x\\in {\\bf A}:\\theta (x, {\\bf A} )+\\theta ({\\bf A},x ) = 0 \\right\\}$ the {\\it{annihilator}} of $\\theta.$ \n\n\\begin{lemma}\n\\label{ann_of_ext}\n$\\operatorname{Ann}({\\bf A}_{\\theta }) = (\\operatorname{Ann}(\\theta) \\cap \\operatorname{Ann}({\\bf A}))\n \\oplus {\\mathbb V}.$\n\\end{lemma}\n\n\\bigskip\n\nTherefore, $0 \\neq \\mathbb{V} \\subseteq \\operatorname{Ann}({\\bf A}_{\\theta}).$ The following lemma shows that every algebra with a nonzero annihilator can be obtained in the way described above:\n\n\\begin{lemma}\nLet ${\\bf A}$ be an $n$-dimensional terminal algebra such that $\\dim(\\operatorname{Ann}({\\bf A}))=m\\neq0.$ Then there exists, up to isomorphism, a unique $(n-m)$-dimensional terminal algebra ${\\bf A}^{\\prime }$ and a bilinear map $\\theta \\in {\\rm Z_T^2}({\\bf A}, {\\mathbb V})$ with $\\operatorname{Ann}({\\bf A}')\\cap \\operatorname{Ann}(\\theta)=0,$ where $\\mathbb V$ is a vector space of dimension $m,$ such that ${\\bf A}\\cong {\\bf A}^{\\prime }_{\\theta}$ and ${\\bf A}\/\\operatorname{Ann}({\\bf A})\\cong {\\bf A}^{\\prime }.$\n\\end{lemma}\n\nIn particular, any finite-dimensional nilpotent algebra is a central extension of another nilpotent algebra of strictly smaller dimension. Thus, to classify all nilpotent terminal algebras of a fixed dimension, we need to classify cocycles of nilpotent terminal algebras ${\\bf A}'$ of smaller dimension (with an additional condition $\\operatorname{Ann}({\\bf A}')\\cap \\operatorname{Ann}(\\theta)=0$) and central extensions that arise from them.\n\n\\bigskip\n\nWe can reduce the class of extensions that we need to consider.\n\n\n\\begin{definition}\nLet ${\\bf A}$ be an algebra and $I$ be a subspace of ${\\rm Ann}({\\bf A})$. If ${\\bf A}={\\bf A}_0 \\oplus I$\nthen $I$ is called an {\\it annihilator component} of ${\\bf A}$.\n A central extension of an algebra $\\bf A$ without an annihilator component is called a non-split central extension.\n\\end{definition} \n\nClearly, we are only interested in non-split extensions (in the contrary case we can cut off annihilator components lying in $\\mathbb{V}$ until we obtain a non-split extension). \n\nLet us fix a basis $e_{1},\\ldots,e_{s}$ of ${\\mathbb V},$ and $\\theta \\in {\\rm Z_T^2}({\\bf A},{\\mathbb V}).$ Then $\\theta$ can be uniquely written as $\\theta (x,y) =\\theta (x,y) =\\sum_{i=1}^s\\theta_{i}(x,y) e_{i},$ where $\\theta_{i}\\in {\\rm Z_T^2}({\\bf A},\\mathbb C).$ Moreover, $\\operatorname{Ann}(\\theta)=\\operatorname{Ann}(\\theta_{1})\\cap \\operatorname{Ann}(\\theta_{2})\\cap\\cdots \\cap \\operatorname{Ann}(\\theta_{s}).$ Further, $\\theta \\in {\\rm B^2}({\\bf A},{\\mathbb V}) $\\ if and only if all $\\theta_{i}\\in {\\rm B^2}({\\bf A},\\mathbb C).$ Using this presentation, one can determine whether the extension corresponding to a cocycle $\\theta$ is split:\n\n\\begin{lemma} \\cite[Lemma 13]{hac16}\n\\label{split_lindep}\nLet $\\theta (x,y) =\\sum_{i=1}^s\\theta_{i}(x,y) e_{i}\\in {\\rm Z_T^2}({\\bf A},{\\mathbb V})$ be such that $\\operatorname{Ann}(\\theta)\\cap \\operatorname{Ann}({\\bf A}) = 0.$ Then ${\\bf A}_{\\theta }$ has an annihilator component if and only if $[\\theta_{1}],[\\theta_{2}],\\ldots,[\\theta_{s}]$ are linearly dependent in ${\\rm H_T^2}({\\bf A},\\mathbb C).$\n\\end{lemma}  \n\n\\bigskip\n\nSome cocycles give rise to isomorphic extensions:\n\n\\begin{lemma}\nLet $\\theta, \\vartheta \\in {\\rm Z_T^2}({\\bf A}, {\\mathbb V})$ be such that $[\\theta] = [\\vartheta].$ Then ${\\bf A}_\\theta \\cong {\\bf A}_\\vartheta.$  \n\\end{lemma}\n\n\n\nBy above, the isomorphism classes of extensions correspond to certain equivalence classes on ${\\rm H_T^2}({\\bf A},{\\mathbb V}).$ These classes can be given in terms of actions of certain groups on this space. In particular, let $\\operatorname{Aut}({\\bf A})$ be the automorphism group of ${\\bf A},$ let $\\phi \\in \\operatorname{Aut}({\\bf A}),$ and let $\\psi \\in \\operatorname{GL}(\\mathbb V).$ For $\\theta \\in {\\rm Z_T^2}({\\bf A},{\\mathbb V})$ define \n\\[\\phi \\theta(x,y) =\\theta (\\phi(x), \\phi(y)), ~ \\psi\\theta (x,y) = \\psi(\\theta(x,y)).\\] \nThen $\\phi\\theta, \\psi\\theta \\in {\\rm Z_T^2}({\\bf A},{\\mathbb V}).$ Hence, $\\operatorname{Aut}({\\bf A})$ and $\\operatorname{GL}(\\mathbb V)$ act on ${\\rm Z_T^2}({\\bf A},{\\mathbb V}).$ It is easy to verify that ${\\rm B^2}({\\bf A},{\\mathbb V})$ is invariant under both actions. Therefore, we have induced actions on ${\\rm H_T^2}({\\bf A},{\\mathbb V}).$\n\n\\begin{lemma}\nLet $\\theta, \\vartheta \\in {\\rm Z_T^2}({\\bf A},{\\mathbb V})$ be such that $\\operatorname{Ann}({\\bf A}_\\theta) = \\operatorname{Ann}({\\bf A}_\\vartheta) = \\mathbb{V}.$ Then ${\\bf A}_\\theta \\cong {\\bf A}_\\vartheta$ if and only if there exist a $\\phi \\in \\operatorname{Aut}({\\bf A}), \\psi \\in \\operatorname{GL}(\\mathbb V)$ such that $[\\phi\\theta] = [\\psi\\vartheta].$\n\\end{lemma}\n\n\\bigskip\n\nNow we rewrite the above lemma in a form more suitable for computations.\n\nLet ${\\mathbb U}$ be a finite-dimensional vector space over $\\mathbb C.$ The {\\it{Grassmannian}} $G_{k}({\\mathbb U})$ is the set of all $k$-dimensional linear subspaces of ${\\mathbb V}.$ Let $G_{s}({\\rm H_T^2}({\\bf A},\\mathbb C) )$ be the Grassmannian of subspaces of dimension $s$ in ${\\rm H_T^2}({\\bf A},\\mathbb C).$ There is a natural action of $\\operatorname{Aut} ({\\bf A})$ on $G_{s}({\\rm H_T^2}({\\bf A},\\mathbb C)):$ for $\\phi \\in \\operatorname{Aut} ({\\bf A}), W=\\langle [\\theta_{1}],[\\theta_{2}],\\dots,[\\theta_{s}] \\rangle \\in G_{s}({\\rm H_T^2}({\\bf A},\\mathbb C)),$ define \n\\[\\phi W=\\langle [\\phi \\theta_{1}], [\\phi \\theta_{2}],\\dots,[\\phi \\theta_{s}]\\rangle.\\]\nNote that this action is compatible with the action of $\\operatorname{Aut} ({\\bf A})$ on ${\\rm H_T^2}({\\bf A},{\\mathbb V})$ and the above presentation of a cocycle as a collection of $s$ elements of ${\\rm H_T^2}({\\bf A},\\mathbb C).$ Denote the orbit of $W$ under the action of $\\operatorname{Aut} ({\\bf A})$ by $\\mathrm{Orb}(W).$ It is easy to check that given two bases of a subspace \n\\begin{equation*}\nW=\\langle [\\theta_{1}],[\\theta_{2}],\\dots,[\\theta_{s}] \\rangle =\\langle [\\vartheta_{1}],[\\vartheta_{2}],\\dots,[\\vartheta_{s}]\n\\rangle \\in G_{s}({\\rm H^2}({\\bf A},\\mathbb C)),\n\\end{equation*}\nwe have $\\cap_{i=1}^s\\operatorname{Ann}(\\theta_{i})\\cap \\operatorname{Ann}({\\bf A}) =\\cap_{i=1}^s\\operatorname{Ann}(\\vartheta_{i})\\cap \\operatorname{Ann}({\\bf A}).$ Therefore, we can introduce the set\n\n\\begin{equation*}\nT_{s}({\\bf A}) =\\left\\{ W=\\left\\langle [\\theta_{1}],%\n[\\theta_{2}],\\dots,[\\theta_{s}] \\right\\rangle \\in\nG_{s}({\\rm H^2}({\\bf A},\\mathbb C) ) : \\cap_{i=1}^s \\operatorname{Ann}(\\theta_{i})\\cap \\operatorname{Ann}({\\bf A}) =0\\right\\},\n\\end{equation*}\nwhich is stable under the action of $\\operatorname{Aut}({\\bf A}).$\n\n\\medskip\n\nLet us denote by $E({\\bf A},{\\mathbb V})$ the set of all {\\it non-split} central extensions of ${\\bf A}$ by ${\\mathbb V}.$ By Lemmas \\ref{ann_of_ext} and \\ref{split_lindep}, we can write\n\\begin{equation*}\nE({\\bf A},{\\mathbb V}) =\\left\\{ {\\bf A}_{\\theta }:\\theta (x,y) =\\sum_{i=1}^s\\theta_{i}(x,y) e_{i}\\mbox{ and }\\langle [\\theta_{1}],[\\theta_{2}],\\dots,\n[\\theta_{s}] \\rangle \\in T_{s}({\\bf A})\\right\\}.\n\\end{equation*}\nAlso, we have the next result, which can be proved as \\cite[Lemma 17]{hac16}.\n\n\\begin{lemma}\n Let $\\theta(x,y) =\\sum_{i=1}^s \\theta_{i}(x,y) e_{i}$ and $\\vartheta(x,y) = \\sum_{i=1}^s\\vartheta_{i}(x,y) e_{i}$ be such that ${\\bf A}_{\\theta},{\\bf A}_{\\vartheta }\\in E({\\bf A},{\\mathbb V}).$  Then the algebras ${\\bf A}_{\\theta }$ and ${\\bf A}_{\\vartheta }$ are isomorphic if and only if \n\\[\\mathrm{Orb}\\langle [\\theta_{1}], [\\theta_{2}],\\dots,[\\theta_{s}] \\rangle = \\mathrm{Orb}\\langle [\\vartheta_{1}],[\\vartheta\n_{2}],\\dots,[\\vartheta_{s}] \\rangle. \\]\n\\end{lemma}\n\nThus, there exists a one-to-one correspondence between the set of $\\operatorname{Aut}({\\bf A}) $-orbits on $T_{s}({\\bf A})$ and the set of isomorphism classes of $E({\\bf A},{\\mathbb V}).$ Consequently, we have a procedure that allows us, given a terminal algebra ${\\bf A}^{\\prime }$ of dimension $n,$ to construct all non-split central extensions of ${\\bf A}^{\\prime }.$ This procedure is as follows:\n\n\\medskip\n\n{\\centerline{\\it Procedure}}\n\n\\begin{enumerate}\n\\item For a given (nilpotent) terminal algebra $\\bf{A}^{\\prime }$\nof dimension $n-s,$ determine ${T}_{s}(\\bf{A}^{\\prime })$\nand $\\operatorname{Aut}(\\bf{A}^{\\prime }).$\n\n\\item Determine the set of $\\operatorname{Aut}(\\bf{A}^{\\prime })$-orbits on $%\n{T}_{s}(\\bf{A}^{\\prime }).$\n\n\\item For each orbit, construct the terminal algebra corresponding to one of its\nrepresentatives.\n\\end{enumerate}\n\nThe above described method gives all (Leibniz and non-Leibniz) terminal algebras. But we are interested in developing this method in such a way that it only gives non-Leibniz terminal algebras, because the classification of all Leibniz algebras is given in \\cite{demir}. Clearly, any central extension of a non-Leibniz terminal  algebra is non-Leibniz. But a Leibniz algebra may have extensions which are not Leibniz algebras. More precisely, let ${\\bf  L}$ be a Leibniz algebra and $\\theta \\in {\\rm Z_T^2}({\\bf  L}, {\\mathbb C}).$ Then ${\\bf  L}_{\\theta }$ is a Leibniz algebra if and only if \n\\begin{equation*}\n\\theta ( x, yz ) = \\theta (xy, z )+ \\theta (y, xz  ) \n\\end{equation*}\nfor all $x,y,z\\in {\\bf  L}.$ Define the subspace ${\\rm Z_L^2}({\\bf  L},{\\mathbb C})$ of ${\\rm Z_T^2}({\\bf  L},{\\mathbb C})$ by\n\\begin{equation*}\n{\\rm Z_L^2}({\\bf  L},{\\mathbb C}) =\\left\\{\\begin{array}{c} \\theta \\in {\\rm Z_T^2}({\\bf  L},{\\mathbb C}) : \\theta ( x, yz ) = \\theta (xy, z ) + \\theta (y, xz  ) \\text{ for all } x, y,z\\in {\\bf  L}\\end{array}\\right\\}.\n\\end{equation*}\nObserve that ${\\rm B^2}({\\bf  L},{\\mathbb C})\\subseteq{\\rm Z_L^2}({\\bf  L},{\\mathbb C}).$\nLet ${\\rm H_L^2}({\\bf  L},{\\mathbb C}) =%\n{\\rm Z_L^2}({\\bf  L},{\\mathbb C}) \\big\/{\\rm B^2}({\\bf  L},{\\mathbb C}).$ Then ${\\rm H_L^2}({\\bf  L},{\\mathbb C})$ is a subspace of $%\n{\\rm H_T^2}({\\bf  L},{\\mathbb C}).$ Define \n\\begin{eqnarray*}\n{\\bf R}_{s}({\\bf  L})  &=&\\left\\{ {\\bf W}\\in {T}_{s}({\\bf  L}) :{\\bf W}\\in G_{s}({\\rm H_L^2}({\\bf  L},{\\mathbb C}) ) \\right\\}, \\\\\n{\\bf U}_{s}({\\bf  L})  &=&\\left\\{ {\\bf W}\\in {T}_{s}({\\bf  L}) :{\\bf W}\\notin G_{s}({\\rm H_L^2}({\\bf  L},{\\mathbb C}) ) \\right\\}.\n\\end{eqnarray*}\nThen ${T}_{s}({\\bf  L}) ={\\bf R}_{s}(\n{\\bf  L})$ $\\mathbin{\\mathaccent\\cdot\\cup}$ ${\\bf U}_{s}(\n{\\bf  L}).$ The sets ${\\bf R}_{s}({\\bf  L}) $\nand ${\\bf U}_{s}({\\bf  L})$ are stable under the action\nof $\\operatorname{Aut}({\\bf  L}).$ Thus, the terminal  algebras\ncorresponding to the representatives of $\\operatorname{Aut}({\\bf  L}) $%\n-orbits on ${\\bf R}_{s}({\\bf  L})$ are Leibniz  algebras,\nwhile those corresponding to the representatives of $\\operatorname{Aut}({\\bf  L}%\n) $-orbits on ${\\bf U}_{s}({\\bf  L})$ are not\nLeibniz algebras. Hence, we may construct all non-split non-Leibniz terminal algebras $%\n\\bf{A}$ of dimension $n$ with $s$-dimensional annihilator \nfrom a given terminal algebra $\\bf{A}%\n^{\\prime }$ of dimension $n-s$ in the following way:\n\n\\begin{enumerate}\n\\item If $\\bf{A}^{\\prime }$ is non-Leibniz, then apply the Procedure.\n\n\\item Otherwise, do the following:\n\n\\begin{enumerate}\n\\item Determine ${\\bf U}_{s}(\\bf{A}^{\\prime })$ and $%\n\\operatorname{Aut}(\\bf{A}^{\\prime }).$\n\n\\item Determine the set of $\\operatorname{Aut}(\\bf{A}^{\\prime })$-orbits on ${\\bf U%\n}_{s}(\\bf{A}^{\\prime }).$\n\n\\item For each orbit, construct the terminal  algebra corresponding to one of its\nrepresentatives.\n\\end{enumerate}\n\\end{enumerate}\n\n\n\n\\medskip\n\n\\subsection{Notations}\nLet us introduce the following notations. Let ${\\bf A}$ be a terminal algebra with\na basis $e_{1},e_{2}, \\ldots, e_{n}.$ Then by $\\Delta_{ij}$\\ we will denote the\nbilinear form\n$\\Delta_{ij}:{\\bf A}\\times {\\bf A}\\longrightarrow \\mathbb C$\nwith $\\Delta_{ij}(e_{l},e_{m}) = \\delta_{il}\\delta_{jm}.$\nThe set $\\left\\{ \\Delta_{ij}:1\\leq i, j\\leq n\\right\\}$ is a basis for the linear space of \nbilinear forms on ${\\bf A},$ so every $\\theta \\in\n{\\rm Z^2}({\\bf A},\\bf \\mathbb V )$ can be uniquely written as $%\n\\theta ={\\sum }_{i,j=1}^nc_{ij}\\Delta_{ij},$ where $%\nc_{ij}\\in \\mathbb C.$\nWe also denote by\n\n\n$$\\begin{array}{lll}\n\\T {i*}{j}& \\mbox{the }j\\mbox{th }i\\mbox{-dimensional nilpotent Leibniz  algebra}, \\\\\n\\T i{j}& \\mbox{the }j\\mbox{th }i\\mbox{-dimensional nilpotent non-Leibniz terminal algebra}, \\\\\n{\\mathfrak{N}}_i& \\mbox{the }i\\mbox{-dimensional algebra with zero product}, \\\\\n({\\bf A})_{i,j} & j\\mbox{th }i\\mbox{-dimensional central extension of }\\bf A. \\\\\n\\end{array}$$\n\n\nAlso, it is easy to see that every central extension of $\\mathfrak{N}_i$ is a Leibniz algebra.\n\n\n\n\\subsection{The algebraic classification of 3-dimensional nilpotent terminal algebras}\n\nObserve that a 2-dimensional nilpotent algebra is either $\\mathfrak{N}_2,$ or is isomorphic to \n\\begin{align*\n    \\begin{array}{llllll}\n        \\T {2*}{01} &:& e_1 e_1 = e_2. \n    \\end{array}\n\\end{align*}\nBoth of these algebras are nilpotent of index less than 4, and hence are terminal. All central extensions of 2-dimensional nilpotent algebras were described in~\\cite{cfk18}. By a direct verification, we have the following list of all 3-dimensional nilpotent terminal algebras with annihilator of codimension 1 or 2\n\\begin{align}\\label{3-dim-term}\n    \\begin{array}{lllllllll}\n        \\T {3*}{01}&:&\\T {2*}{01}\\oplus\\mathfrak{N}_1 &:& e_1 e_1 = e_2;  \\\\\n        \\T {3*}{02}&:&(\\mathfrak{N}_2)_{3,1} &:& e_1 e_1 = e_3, & e_2 e_2=e_3; \\\\\n        \\T {3*}{03}&:& (\\mathfrak{N}_2)_{3,2} &:& e_1 e_2=e_3, & e_2 e_1=-e_3; \\\\\n        \\T {3*}{04}(\\lambda)&:&(\\mathfrak{N}_2)_{3,3} &:& e_1 e_1 = \\lambda e_3, &  e_2 e_1=e_3,  & e_2 e_2=e_3; \\\\\n        \\T {3*}{05}&:&(\\T {2*}{01})_{3,1} &:& e_1 e_1 = e_2, & e_1 e_2=e_3; \\\\\n        \\T 3{01}(\\lambda)&:&(\\T {2*}{01})_{3,2} &:& e_1 e_1 = e_2, & e_1 e_2=\\lambda e_3, & e_2 e_1=e_3. \n    \\end{array}\n\\end{align}\nNotice that only $\\T 3{01}(\\lambda)$ from list \\cref{3-dim-term} is non-Leibniz. \n\n\\subsection{The algebraic classification of $4$-dimensional nilpotent terminal algebras}\n\n\nAnalyzing the list of all the 4-dimensional nilpotent  algebras with annihilator of codimension 2, we have only one terminal non-Leibniz algebra:\n$$\n\\begin{array}{lllllllll}\n   \n   \n   \n   \n   \n   \n    \\T 4{02}&:&(\\T {2*}{01} )_{4,1} &:& e_1 e_1 = e_2, & e_1 e_2=e_4, & e_2 e_1=e_3.\n\\end{array}\n$$\nTo complete the algebraic classification of 4-dimensional nilpotent terminal algebras, we need to describe all the 1-dimensional terminal non-Leibniz extensions of the algebras from the table \\cref{3-dim-term}. This is done in Subsections~\\ref{aut-and-H^2}--\\ref{ext-T_05^3*} and summarized in the main theorem of the first part of the paper.\n\n\\begin{theorem}\\label{main-alg}\nLet $\\mathbf A$ be a 4-dimensional nilpotent non-Leibniz terminal algebra over $\\mathbb C$. Then $\\mathbf A$ is isomorphic to one of the algebras $\\T 4{01}-\\T 4{44}$ or $\\D 4{01}-\\D 4{40}$ found below.\n\\end{theorem}\n\\begin{Proof}\n The proof is split into several steps presented in Subsections~\\ref{aut-and-H^2}--\\ref{ext-T_05^3*} below.\n\\end{Proof}\n\n\\subsubsection{Automorphism and cohomology groups of $3$-dimensional nilpotent terminal algebras}\\label{aut-and-H^2}\n\n\\[\n\\tiny\n\\begin{tabular}{|c|c|c|c|}\n\\hline \n$\\mathbf{A}$ & $\\aut\\mathbf{A}$ & ${\\rm Z_T^2}(\\mathbf{A})$ & ${\\rm H_T^2}(\\mathbf{A})$\\\\\n\\hline\n\n$\\T {3*}{01}$ \n&\n$\\begin{pmatrix}\nx &    0  &  0\\\\\ny &  x^2  &  u\\\\\nz &   0  &  v\n\\end{pmatrix}$ \n& \n$\\Big\\langle\n\\begin{array}{l}\n     \\Dt 11, \\Dt 12, \\Dt 13, \\Dt 21,\\\\\n     \\Dt 23, \\Dt 31, \\Dt 32, \\Dt 33\n\\end{array}\n\\Big\\rangle$ \n&\n$\\Big\\langle\n\\begin{array}{l}\n     \\Dl 12, \\Dl 13, \\Dl 21, \\Dl 23\\\\\n     \\Dl 31, \\Dl 32, \\Dl 33\n\\end{array}\n\\Big\\rangle$\n\\\\\n\\hline\n\n$\\T {3*}{02}$ \n&\n$\\begin{pmatrix}\nx &    y  &  0\\\\\n(-1)^{n+1} y &  (-1)^n x  &  0\\\\\nz &   u  &  x^2+y^2\n\\end{pmatrix}$ \n&\n$\\Big\\langle\n\\begin{array}{l}\n     \\Dt 11, \\Dt 12, \\Dt 13, \\Dt 21\\\\\n     \\Dt 22, \\Dt 23, \\Dt 31, \\Dt 32\n\\end{array}\n\\Big\\rangle$ \n& \n$\\Big\\langle\n\\begin{array}{l}\n     \\Dl 11, \\Dl 12, \\Dl 13, \\Dl 21\\\\\n     \\Dl 23, \\Dl 31, \\Dl 32\n\\end{array}\n\\Big\\rangle$ \n\\\\\n\\hline\n\n$\\T {3*}{03}$ \n&\n$\\begin{pmatrix}\nx &    y  &  0\\\\\nz &    u  &  0\\\\\nv &    w  &  xu-yz\n\\end{pmatrix}$ \n&\n$\\Big\\langle\n\\begin{array}{l}\n     \\Dt 11, \\Dt 12, \\Dt 13, \\Dt 21\\\\\n     \\Dt 22, \\Dt 23, \\Dt 31, \\Dt 32\n\\end{array}\n\\Big\\rangle$ \n&\n$\\Big\\langle\n\\begin{array}{l}\n     \\Dl 11, \\Dl 12, \\Dl 13, \\Dl 22\\\\\n     \\Dl 23, \\Dl 31, \\Dl 32\n\\end{array}\n\\Big\\rangle$\n\\\\\n\\hline\n\n$\\T {3*}{04}$ \n&\n    $\\begin{pmatrix}\n    x &      y    &  0\\\\\n    -\\lambda y    &    x-y   &  0\\\\\n    z &      u      &  x^2-xy+\\lambda y^2\n    \\end{pmatrix}$\n&\n$\\Big\\langle\n\\begin{array}{l}\n     \\Dt 11, \\Dt 12, \\Dt 13, \\Dt 21\\\\\n     \\Dt 22, \\Dt 23, \\Dt 31, \\Dt 32\n\\end{array}\n\\Big\\rangle$ \n&\n$\\Big\\langle\n\\begin{array}{l}\n     \\Dl 11, \\Dl 12, \\Dl 13, \\Dl 21\\\\\n     \\Dl 23, \\Dl 31, \\Dl 32\n\\end{array}\n\\Big\\rangle$\n\\\\\n\\hline\n\n$\\T {3*}{05}$ \n&\n$\\begin{pmatrix}\nx &    0  &  0\\\\\ny &  x^2  &  0\\\\\nz &   xy  &  x^3\n\\end{pmatrix}$ \n& \n$\\Big\\langle\n\\begin{array}{l}\n     \\Dt 11, \\Dt 12, \\Dt 13,\\\\\n     \\Dt 21, \\Dt 22 - 3\\Dt 31\n\\end{array}\n\\Big\\rangle$ \n&\n$\\Big\\langle\n\\begin{array}{l}\n     \\Dl 13, \\Dl 21,\\\\\n     \\Dl 22 - 3\\Dl 31\n\\end{array}\n\\Big\\rangle$\n\\\\\n\\hline\n\n$\\T 3{01}$ \n&\n$\\begin{pmatrix}\nx &               0  &  0\\\\\ny &             x^2  &  0\\\\\nz &   (\\lambda+1)xy  &  x^3\n\\end{pmatrix}$ \n& \n$\\begin{cases}\n\\Big\\langle\n\\begin{array}{l}\n     \\Dt 11, \\Dt 12, -\\Dt 13 + 3\\Dt 31,\\\\\n     \\Dt 21,  {\\color{black}\\Dt 13 + \\Dt 22}, \\Dt 23  \n\\end{array}\n\\Big\\rangle & \\lambda=0\\\\\n\\Big\\langle\n\\begin{array}{l}\n     \\Dt 11, \\Dt 12, (\\lambda-1)\\Dt 13 + 3\\Dt 31,\\\\\n     \\Dt 21, \\color{black}{\\Dt 13 + \\Dt 22}\n\\end{array}\n\\Big\\rangle & \\lambda\\ne 0\n\\end{cases}$\n&\n\n$\\begin{cases}\n\\Big\\langle\n\\begin{array}{l}\n     \\Dl 12, -\\Dl 13 + 3\\Dl 31,\\\\\n     {\\color{black}\\Dl 13 + \\Dl 22}, \\Dl 23  \n\\end{array}\n\\Big\\rangle & \\lambda=0\\\\\n\\Big\\langle\n\\begin{array}{l}\n     \\Dl 12, (\\lambda-1)\\Dl 13 + 3\\Dl 31,\\\\\n     \\color{black}{\\Dl 13 + \\Dl 22}\n\\end{array}\n\\Big\\rangle & \\lambda\\ne 0\n\\end{cases}$\n\\\\\n\\hline\n\n\\end{tabular}\n\\] \n\n\\vskip0.5cm\n\nSince the algebras $\\T {3*}{01}, \\T {3*}{02},\\T {3*}{03},\\T {3*}{04}$ and $\\T {3*}{05}$ are Leibniz, it is natural to find the relation between the Leibniz and terminal cohomologies of these algebras in order to exclude those cocycles which give Leibniz algebras. \n\n\\vskip0.5cm\n\n$$\\begin{tabular}{|c|c|c|}\n\t\\hline \n\t$\\mathbf{A}$ & ${\\rm H_L^2}(\\mathbf{A})$ & ${\\rm H_T^2}(\\mathbf{A})$\\\\\n\t\\hline\n\t\n\t$\\T {3*}{01}$ \n\t&\n\t$\\langle \\Dl 12, \\Dl 13, \\Dl 31, \\Dl 33 \\rangle$\n\t&\n\t${\\rm H_L^2}(\\mathbf{A})\\oplus \\langle \\Dl 21, \\Dl 23, \\Dl 32 \\rangle$\n\t\\\\\n\t\\hline\n\t\n\t$\\T {3*}{02}$ \n\t&\n\t$\\langle \\Dl 11, \\Dl 12, \\Dl 21 \\rangle$ \n\t& \n\t${\\rm H_L^2}(\\mathbf{A})\\oplus\\langle \\Dl 13, \\Dl 23, \\Dl 31, \\Dl 32\\rangle$ \n\t\\\\\n\t\\hline\n\t\n\t$\\T {3*}{03}$ \n\t&\n\t$\\Big\\langle\n\t\\begin{array}{l}\n\t\\Dl 11, \\Dl 12, \\Dl 13 - \\Dl 31,\\\\\n\t\\Dl 22, \\Dl 23 - \\Dl 32\n\t\\end{array}\n\t\\Big\\rangle$ \n\t&\n\t${\\rm H_L^2}(\\mathbf{A})\\oplus\\langle \\Dl 31, \\Dl 32 \\rangle$\n\t\\\\\n\t\\hline\n\t\n\t$\\T {3*}{04}$ \n\t&\n\t$\\begin{cases}\n\t\\langle \\Dl 11, \\Dl 12, \\Dl 21 \\rangle,   &  \\lambda\\ne 0\\\\\n\t\\langle\\Dl 11, \\Dl 12, \\Dl 21, \\Dl 23\\rangle, &  \\lambda=0\n\t\\end{cases}$\n\t&\n\t$\n\t\\begin{cases}\n\t{\\rm H_L^2}(\\mathbf{A})\\oplus \\langle \\Dl 13, \\Dl 23, \\Dl 31, \\Dl 32 \\rangle,   &  \\lambda\\ne 0\\\\\n\t{\\rm H_L^2}(\\mathbf{A})\\oplus \\langle \\Dl 13, \\Dl 31, \\Dl 32 \\rangle,           &  \\lambda=0\n\t\\end{cases}$\n\t\\\\\n\t\\hline\n\t\n\t$\\T {3*}{05}$ \n\t&\n\t$\\langle \\Dl 13\\rangle$\n\t&\n\t${\\rm H_L^2}(\\mathbf{A})\\oplus \\langle \\Dl 21,\\Dl 22 - 3\\Dl 31 \\rangle$\n\t\\\\\n\t\\hline\n\t\n\t\\end{tabular}$$\n\n\n\\subsubsection{$1$-dimensional  central extensions of $\\T {3*}{01}$}\n\nLet us use the following notations: \n\\begin{align*}\n    \\nb 1 = \\Dl 12, \\nb 2 = \\Dl 13, \\nb 3 = \\Dl 31, \\nb 4 = \\Dl 33, \\nb 5 = \\Dl 21, \\nb 6 = \\Dl 23, \\nb 7 = \\Dl 32.    \n\\end{align*}\nTake $\\theta=\\sum_{i=1}^7\\alpha_i\\nb i\\in {\\rm H_T^2}(\\T {3*}{01}).$\nIf \n$$\n\\phi=\n\\begin{pmatrix}\nx &    0  &  0\\\\\ny &  x^2  &  u\\\\\nz &   0  &  v\n\\end{pmatrix}\\in\\aut{\\T {3*}{01}},\n$$\nthen\n$$\n\\phi^T\\begin{pmatrix}\n0& \\alpha_1& \\alpha_2\\\\\n\\alpha_5& 0& \\alpha_6\\\\\n\\alpha_3& \\alpha_7& \\alpha_4\n\\end{pmatrix} \\phi=\n\\begin{pmatrix}\n\\alpha^*& \\alpha_1^*& \\alpha_2^*\\\\\n\\alpha_5^*& \\alpha^{**}& \\alpha_6^*\\\\\n\\alpha_3^*& \\alpha_7^*& \\alpha_4^*\n\\end{pmatrix},\n$$\nwhere\n\\begin{align*}\n    \\alpha_1^* &= x^2(\\alpha_1x + \\alpha_7z),\\\\\n    \\alpha_2^* &= u(\\alpha_1x + \\alpha_7z) + v(\\alpha_2x + \\alpha_6y  + \\alpha_4z),\\\\\n    \\alpha_3^* &= u(\\alpha_5x + \\alpha_6z) + v(\\alpha_3x + \\alpha_7y + \\alpha_4z),\\\\\n    \\alpha_4^* &= v((\\alpha_6 + \\alpha_7)u + \\alpha_4v),\\\\\n    \\alpha_5^* &= x^2(\\alpha_5x + \\alpha_6z),\\\\\n    \\alpha_6^* &= \\alpha_6x^2v,\\\\\n    \\alpha_7^* &= \\alpha_7x^2v.\n\\end{align*}\nHence, $\\phi\\langle\\theta\\rangle=\\langle\\theta^*\\rangle,$ where $\\theta^*=\\sum\\limits_{i=1}^7 \\alpha_i^*  \\nb i.$\n \n We are interested in $\\theta$ with $(\\alpha_5,\\alpha_6,\\alpha_7) \\neq (0,0,0).$ Moreover, the condition $\\theta \\in \\mathbf{T}_1 (\\T{3*}{01})$ gives us $(\\alpha_2, \\alpha_3, \\alpha_4, \\alpha_6, \\alpha_7) \\neq (0,0,0,0,0)$ and \n $(\\alpha_1 + \\gamma\\alpha_2, \\alpha_5 + \\gamma\\alpha_3, \\alpha_6, \\alpha_7, \\alpha_6 + \\gamma\\alpha_4, \\alpha_7 + \\gamma\\alpha_4) \\neq (0,0,0,0,0,0)$ for all $\\gamma \\in \\mathbb{C}.$\n \n\\begin{enumerate}\n\n   \n\\item Let $\\alpha_6 \\neq 0, \\alpha_7 \\neq 0$ and $\\alpha_1 \\alpha_6 - \\alpha_5 \\alpha_7=0.$\n    Taking $v =\\frac{1}{x^2 \\alpha_7},$ \n    $z=-\\frac{x \\alpha_1}{\\alpha_7}$ and $y=\\frac{x (\\alpha_1 \\alpha_4 - \\alpha_3 \\alpha_7)}{\\alpha_7^2},$\n     we get the family of representatives \n     $$ \n     \\langle  \\alpha_2^\\star \\nabla_2 + \\alpha_4^\\star \\nabla_4 +  \\alpha_6^\\star \\nabla_6+  \\nabla_7\\rangle,\n     $$\n    where \n\n\\begin{align*}\n    \\as 2 &= \\frac 1{\\alpha_7^3x}(\\alpha_1\\alpha_4\\alpha_6 - \\alpha_1\\alpha_4\\alpha_7 - \\alpha_3\\alpha_6\\alpha_7 + \\alpha_2\\alpha_7^2),\\\\\n    \\as 4 &= \\frac 1{\\alpha_7^2x^4}(ux^2\\alpha_7(\\alpha_6 + \\alpha_7) + \\alpha_4),\\\\\n    \\alpha_6^\\star &= \\frac{\\alpha_6}{\\alpha_7}.\n\\end{align*}\n\n  \\begin{enumerate}\n      \\item $\\alpha_6\\neq -\\alpha_7$\n      and $\\alpha_1\\alpha_4\\alpha_6 - \\alpha_1\\alpha_4\\alpha_7 - \\alpha_3\\alpha_6\\alpha_7 + \\alpha_2\\alpha_7^2\\neq 0$.\nThen choosing $u =-\\frac{\\alpha_4}{x^2\\alpha_7(\\alpha_6 + \\alpha_7)}$, where $x$ is such that $\\as 2=1$, we have the family of representatives \n     $ \\langle \\nabla_2 + \\alpha  \\nabla_6+  \\nabla_7\\rangle_{\\alpha \\neq -1,0}.$ \n      \n      \\item $\\alpha_6\\neq -\\alpha_7$\n      and $\\alpha_1\\alpha_4\\alpha_6 - \\alpha_1\\alpha_4\\alpha_7 - \\alpha_3\\alpha_6\\alpha_7 + \\alpha_2\\alpha_7^2 = 0$. Then\n      choosing $u = -\\frac{\\alpha_4}{x^2 \\alpha_7 (\\alpha_6 + \\alpha_7)},$ we have the family of representatives \n     $ \\langle \\alpha  \\nabla_6+  \\nabla_7\\rangle_{\\alpha \\neq -1,0}.$ \n\n\n\n      \\item $\\alpha_6=- \\alpha_7,$ $\\alpha_4=0$\n      and $\\alpha_2 \\neq - \\alpha_3$. Then choosing $x = \\frac{\\alpha_2 + \\alpha_3}{\\alpha_7},$ we have the representative \n     $ \\langle \\nabla_2 -  \\nabla_6+  \\nabla_7\\rangle,$ which will be joined with the family $ \\langle \\nabla_2 + \\alpha  \\nabla_6+  \\nabla_7\\rangle_{\\alpha \\neq -1,0}.$ \n\n\n\n      \\item $\\alpha_6=- \\alpha_7,$ $\\alpha_4=0$\n      and $\\alpha_2 = - \\alpha_3$. Then\n       we have the representative \n     $ \\langle   -  \\nabla_6+  \\nabla_7\\rangle,$\n     which will be joined with the family         $ \\langle    \\alpha  \\nabla_6+  \\nabla_7\\rangle_{\\alpha \\neq -1,0}.$ \n\n      \\item $\\alpha_6=- \\alpha_7$ and $\\alpha_4\\neq 0$.\n      Then choosing $x = \\sqrt[4]{\\frac{\\alpha_4}{\\alpha_7^2}},$ we have the family of representatives \n     $ \\langle \\alpha \\nabla_2 +\\nabla_4  -  \\nabla_6+  \\nabla_7\\rangle.$\n    It gives only two distinct orbits with representatives \n         $ \\langle  \\nabla_2 +\\nabla_4  -  \\nabla_6+  \\nabla_7\\rangle$ and \n              $ \\langle \\nabla_4  -  \\nabla_6+  \\nabla_7\\rangle.$\n\n\\end{enumerate}\n\n\\item Let $\\alpha_6 \\neq 0, \\alpha_7 \\neq 0$ and $\\alpha_1 \\alpha_6 - \\alpha_5 \\alpha_7\\ne 0.$\n    Taking \n    $z=-\\frac{\\alpha_5x}{\\alpha_6},$ \n    $v = \\frac x{\\alpha_6\\alpha_7}(\\alpha_1\\alpha_6 - \\alpha_5\\alpha_7),$\n    $y = \\frac x{\\alpha_6\\alpha_7}(\\alpha_4\\alpha_5 - \\alpha_3\\alpha_6),$ \n    and \n    $u = \\frac x{\\alpha_6\\alpha_7^2}(\\alpha_3\\alpha_6^2 - \\alpha_2\\alpha_6\\alpha_7 + \\alpha_4\\alpha_5(\\alpha_7 -\\alpha_6)),$\n     we get the family of representatives \n     $$ \\langle   \\nabla_1  +  \\alpha_4^\\star \\nabla_4 +  \\alpha_6^\\star \\nabla_6+  \\nabla_7\\rangle,$$\n    where \n\n\\begin{align*}\n    \\alpha_4^\\star &=  \\frac 1{\\alpha_7^3x}(\\alpha_3\\alpha_6^2 + \\alpha_1\\alpha_4\\alpha_7 + \\alpha_3\\alpha_6\\alpha_7 - \\alpha_2\\alpha_6\\alpha_7  - \\alpha_2\\alpha_7^2 - \\alpha_4\\alpha_5\\alpha_6), \\\\\n    \\alpha_6^\\star &= \\frac{\\alpha_6}{\\alpha_7}.\n\\end{align*}\nIt gives two families of representatives of distinct orbits \n  $  \\langle   \\nabla_1  + \\alpha  \\nabla_6+  \\nabla_7\\rangle_{\\alpha\\neq 0} $\n  and \n  $  \\langle   \\nabla_1  +  \\nabla_4 +  \\alpha   \\nabla_6+  \\nabla_7\\rangle_{\\alpha\\neq 0}$ \n  depending on whether $\\alpha_3\\alpha_6^2 + \\alpha_1\\alpha_4\\alpha_7 + \\alpha_3\\alpha_6\\alpha_7 - \\alpha_2\\alpha_6\\alpha_7  - \\alpha_2\\alpha_7^2 - \\alpha_4\\alpha_5\\alpha_6=0$ or not.\n   \n \n \n    \n    \n    \n    \n    \n    \n    \n    \\item Let $\\alpha_6 = 0, \\alpha_7 \\neq 0.$\n    Taking \n    $y = \\frac x{\\alpha_7^2} (\\alpha_1 \\alpha_4 + \\alpha_4 \\alpha_5 - \\alpha_3 \\alpha_7),$ $z = -\\frac{x \\alpha_1}{\\alpha_7},$ \n    and $u = -\\frac{\\alpha_4v}{\\alpha_7} $\n    we get  a family of representatives \n     $$ \\langle \\alpha_2^\\star \\nabla_2 + \\alpha_5^\\star \\nabla_5 + \\nabla_7\\rangle,$$\nwhere \n\\begin{align*}\n  \\alpha_2^\\star &= \\frac 1{x \\alpha_7^2}( \\alpha_2 \\alpha_7-\\alpha_1 \\alpha_4), \\\\ \n\\alpha_5^\\star &=  \\frac{\\alpha_5x}{\\alpha_7v}.  \n\\end{align*}\n\\begin{enumerate}\n        \\item $\\alpha_5 = 0$ and $\\alpha_1\\alpha_4 - \\alpha_2\\alpha_7=0$. Then \n        we have the representative $\\langle\\nabla_7\\rangle,$\n        which will be joined with the family    $ \\langle \\alpha  \\nabla_6+  \\nabla_7\\rangle_{\\alpha \\neq 0}.$ \n\n\n        \\item $\\alpha_5 = 0$ and $\\alpha_1\\alpha_4 - \\alpha_2\\alpha_7\\ne 0$. Then \n        choosing $x= \\frac 1{\\alpha_7^2}(\\alpha_2 \\alpha_7-\\alpha_1 \\alpha_4),$ \n        we have the representative $\\langle\\nabla_2 + \\nabla_7\\rangle.$\n        which will be joined with the family    $ \\langle \\nabla_2 + \\alpha  \\nabla_6+  \\nabla_7\\rangle_{\\alpha \\neq 0}$, \n\n        \n        \\item  $\\alpha_5 \\neq 0$ and $\\alpha_1\\alpha_4 - \\alpha_2\\alpha_7\\ne 0$. Then  \n        choosing $x= \\frac 1{\\alpha_7^2}(\\alpha_2 \\alpha_7-\\alpha_1 \\alpha_4)$ and\n       $v=\\frac{\\alpha_5\\alpha_7x^2}{\\alpha_2\\alpha_7-\\alpha_1\\alpha_4}$\n        we get the representative $\\langle \\nabla_2 + \\nabla_5 + \\nabla_7 \\rangle.$\n        \n        \n        \\item  $\\alpha_5 \\neq 0$ and $\\alpha_1\\alpha_4 - \\alpha_2\\alpha_7=0$. Then  \n        choosing \n        $v = \\frac {\\alpha_5x}{\\alpha_7}$\n        we get the representative $\\langle  \\nabla_5 + \\nabla_7 \\rangle.$\n    \\end{enumerate}\n    \n    \n  \n    \n    \\item $\\alpha_6 \\neq 0$ and $\\alpha_7 = 0.$\n    Taking \n    $y = \\frac x{\\alpha_6^2} (\\alpha_1 \\alpha_4 + \\alpha_4 \\alpha_5 - \\alpha_2 \\alpha_6),$ \n    $z = -\\frac{x \\alpha_5}{\\alpha_6},$   and $u = -\\frac{\\alpha_4v}{\\alpha_6}$\n    we get the family of representatives \n     $$ \\langle \\alpha_1^\\star \\nabla_1  + \\alpha_3^\\star \\nabla_3 + \\nabla_6\\rangle,$$\nwhere \n\\begin{align*}\n   \\alpha_1^\\star &=  \\frac{\\alpha_1x}{\\alpha_6v}, \\\\ \n\\alpha_3^\\star &=  \\frac 1{x \\alpha_6^2}(\\alpha_3 \\alpha_6-\\alpha_4 \\alpha_5). \n\\end{align*}\n  \n   \\begin{enumerate}\n       \\item $\\alpha_1\\neq 0$ and $\\alpha_3 \\alpha_6- \\alpha_4 \\alpha_5\\ne 0$. Then\n       we have the representative \n            $  \\langle \\nabla_1  +  \\nabla_3 + \\nabla_6\\rangle.$\n\n       \\item $\\alpha_1 = 0$ and $\\alpha_3 \\alpha_6- \\alpha_4 \\alpha_5\\ne 0$. Then\n       we have the representative $  \\langle  \\nabla_3 + \\nabla_6\\rangle.$\n\n       \\item $\\alpha_1 \\neq 0$ and $\\alpha_3 \\alpha_6 - \\alpha_4 \\alpha_5=0$. Then\n       we have the representative $  \\langle  \\nabla_1 + \\nabla_6\\rangle.$\n       \n       \n       \\item $\\alpha_1 = 0$ and $\\alpha_3 \\alpha_6 - \\alpha_4 \\alpha_5=0$. Then\n       we have the representative $  \\langle  \\nabla_6\\rangle.$\n\n   \\end{enumerate}\n   \\item $\\alpha_5 \\neq 0$, $\\alpha_6=0$ and $\\alpha_7=0.$ \n    Taking \n    $u =\\frac{-v x \\alpha_3 - v z \\alpha_4}{x \\alpha_5}$ and $x = \\frac{1}{\\sqrt[3]{\\alpha_5}},$\n    we get a family of representatives \n    $$ \\langle \\alpha_1^\\star \\nabla_1 + \\alpha_2^\\star \\nabla_2 + \\alpha_4^\\star \\nabla_4 + \\nabla_5\\rangle,$$\nwhere \n\\begin{align*}\n   \\alpha_1^\\star &= \\frac{\\alpha_1}{\\alpha_5}, \\\\ \n   \\alpha_2^\\star &= \\frac v{\\alpha_5^2x^3}((\\alpha_2\\alpha_5 -\\alpha_1\\alpha_3)x + \\alpha_4(\\alpha_5 - \\alpha_1)z), \\\\\n   \\alpha_4^\\star &= \\frac{\\alpha_4v^2}{\\alpha_5x^3}. \n\\end{align*}\n\n    \\begin{enumerate}\n        \\item $\\alpha_4 \\neq 0$ and $\\alpha_1 - \\alpha_5\\ne 0$. Then \n        choosing $z= -\\frac{(\\alpha_2\\alpha_5 -\\alpha_1\\alpha_3)x}{\\alpha_4 (\\alpha_1 -\\alpha_5)}$ and $v = \\sqrt{\\frac{\\alpha_5x^3}{\\alpha_4}}$\n        we have the family of representatives of distinct orbits\n        $ \\langle \\alpha \\nabla_1 + \\nabla_4 + \\nabla_5\\rangle_{\\alpha \\neq 1}.$ \n\n        \\item $\\alpha_4 = 0$, $\\alpha_1 - \\alpha_5\\ne 0$ and $\\alpha_2\\alpha_5-\\alpha_1\\alpha_3=0$. Then \n        we have the family of representatives of distinct orbits\n        $ \\langle \\alpha \\nabla_1 + \\nabla_5\\rangle_{\\alpha \\neq 1}$. The corresponding extensions are split. \n        \n        \\item $\\alpha_4 = 0$, $\\alpha_1 - \\alpha_5\\ne 0$ and $\\alpha_2\\alpha_5-\\alpha_1\\alpha_3=0$. Then \n        we have the family of representatives of distinct orbits\n        $ \\langle \\alpha \\nabla_1 +\\nabla_2 +\\nabla_5\\rangle_{\\alpha \\neq 1}.$ \n\n       \\item $\\alpha_4 \\neq 0$, $\\alpha_1 - \\alpha_5=0$ and $\\alpha_2 - \\alpha_3\\ne 0$. Then \n        choosing $x = \\frac 1{\\alpha_4\\alpha_5}(\\alpha_2-\\alpha_3)^2$,\n        $v=\\frac 1{\\alpha_4^2\\alpha_5}(\\alpha_2- \\alpha_3)^3$,\n        $u = -\\frac {\\alpha_2 - \\alpha_3}{\\alpha_4^2\\alpha_5^2}(\\alpha_4^2\\alpha_5z + (\\alpha_2 - \\alpha_3)^2\\alpha_3)$\n        we have the representative \n             $ \\langle  \\nabla_1 + \\nabla_2 +\\nabla_4+ \\nabla_5\\rangle.$ \n            \n\n       \\item $\\alpha_4 \\neq 0$, $\\alpha_1 - \\alpha_5=0$ and $\\alpha_2 - \\alpha_3= 0$. Then \n        choosing $v=\\sqrt {\\frac{\\alpha_5x^3}{\\alpha_4}}$ and $u=-\\frac v{\\alpha_5x}(\\alpha_3x + \\alpha_4z)$\n        we have the representative \n             $ \\langle  \\nabla_1 + \\nabla_4+ \\nabla_5\\rangle,$\n              which will be joined with the family $\\langle \\alpha \\nabla_1 +  \\nabla_4+ \\nabla_5 \\rangle_{\\alpha \\neq 1}$.\n\n       \\item $\\alpha_4 = 0$, $\\alpha_1 - \\alpha_5=0$ and $\\alpha_2 - \\alpha_3\\ne 0$. Then \n        choosing $u=-\\frac{\\alpha_3v}{\\alpha_5}$ and $v=\\frac{\\alpha_5x^2}{\\alpha_2 - \\alpha_3}$\n        we have the representative \n             $ \\langle  \\nabla_1 + \\nabla_2+ \\nabla_5\\rangle,$\n              which will be joined with the family $\\langle \\alpha \\nabla_1 +  \\nabla_2+ \\nabla_5 \\rangle_{\\alpha \\neq 1}$.\n\n       \\item $\\alpha_4 = 0$, $\\alpha_1 - \\alpha_5=0$ and $\\alpha_2 - \\alpha_3= 0$. Then choosing $u=-\\frac{\\alpha_3v}{\\alpha_5}$\n        we have the representative \n             $ \\langle  \\nabla_1 +  \\nabla_5\\rangle,$\n             which gives a split extension. \n\n\n\n    \\end{enumerate}\n     \\end{enumerate}\n\n \\      \n      \n        \n \n  \n  \n \nSummarizing, we have the following representatives of distinct orbits: \n\n$   \n\\begin{array}{l}\n \\langle   \\nabla_1  +  \\nabla_4 +  \\alpha   \\nabla_6+  \\nabla_7\\rangle_{\\alpha\\neq 0},\\\\\n \\langle   \\nabla_1  + \\alpha  \\nabla_6+  \\nabla_7\\rangle_{\\alpha\\neq 0},\\\\\n \\langle  \\nabla_2 +\\nabla_4  -  \\nabla_6+  \\nabla_7\\rangle,\\\\\n \\langle \\nabla_2 + \\nabla_5 + \\nabla_7 \\rangle,\n\\end{array}\n\\begin{array}{l}\n    \\langle \\nabla_2 + \\alpha  \\nabla_6+  \\nabla_7\\rangle,\\\\\n    \\langle \\nabla_4  -  \\nabla_6+  \\nabla_7\\rangle,\\\\\n    \\langle  \\nabla_5 + \\nabla_7 \\rangle,\\\\\n    \\langle \\alpha  \\nabla_6+  \\nabla_7\\rangle, \n\\end{array}\n\\begin{array}{l}\n  \\langle  \\nabla_1 + \\nabla_6\\rangle,\\\\\n  \\langle \\nabla_1  +  \\nabla_3 + \\nabla_6\\rangle,\\\\  \n  \\langle  \\nabla_3 + \\nabla_6\\rangle,\\\\\n  \\langle  \\nabla_6\\rangle,\n\\end{array}\n\\begin{array}{l}\n     \\langle \\alpha \\nabla_1 +\\nabla_2 +\\nabla_5\\rangle,\\\\ \n    \\langle  \\nabla_1 + \\nabla_2 +\\nabla_4+ \\nabla_5\\rangle,\\\\\n    \\langle \\alpha \\nabla_1 + \\nabla_4 + \\nabla_5\\rangle.\n\\end{array}\n$\n\n\nThe corresponding algebras are:\n\\[\\begin{array}{lllllllllll}\n\\T {4}{03}(\\alpha)_{\\alpha\\neq 0}&:& e_1e_1 = e_2,& e_1e_2=e_4,& e_2e_3=\\alpha e_4,& e_3e_2=e_4,& e_3e_3=e_4; \\\\\n\\T {4}{04}(\\alpha)_{\\alpha\\neq 0}&:& e_1e_1 = e_2,& e_1e_2=e_4,& e_2e_3=\\alpha e_4,& e_3e_2=e_4; \\\\\n\\T {4}{05} &:& e_1e_1 = e_2,& e_1e_3=e_4,& e_2e_3=- e_4,& e_3e_2=e_4,& e_3e_3=e_4; \\\\\n\\T {4}{06}  &:& e_1e_1 = e_2,& e_1e_3=e_4,& e_2e_1= e_4,& e_3e_2=e_4; \\\\\n\\T {4}{07}(\\alpha) &:& e_1e_1 = e_2,& e_1e_3=e_4,& e_2e_3=\\alpha e_4,& e_3e_2=e_4; \\\\\n\\T {4}{08}  &:& e_1e_1 = e_2,&  e_2e_3=- e_4,& e_3e_2=e_4,& e_3e_3=e_4; \\\\\n\\T {4}{09}  &:& e_1e_1 = e_2,&  e_2e_1= e_4,& e_3e_2=e_4;  \\\\\n\\T {4}{10}(\\alpha) &:& e_1e_1 = e_2,&   e_2e_3=\\alpha e_4,& e_3e_2=e_4;  \\\\\n\\T {4}{11} &:& e_1e_1 = e_2,& e_1e_2=e_4,& e_2e_3=  e_4; \\\\\n\\T {4}{12} &:& e_1e_1 = e_2,& e_1e_2=e_4,& e_2e_3=  e_4,& e_3e_1=e_4; \\\\\n\\T {4}{13} &:& e_1e_1 = e_2,&  e_2e_3=  e_4,& e_3e_1=e_4; \\\\\n\\T {4}{14} &:& e_1e_1 = e_2,&   e_2e_3=  e_4;  \\\\\n\\T {4}{15}(\\alpha) &:& e_1e_1 = e_2,& e_1e_2= \\alpha e_4,& e_1e_3=  e_4,& e_2e_1=e_4; \\\\\n\\T {4}{16} &:& e_1e_1 = e_2,& e_1e_2=  e_4,& e_1e_3=  e_4,& e_2e_1=e_4,& e_3e_3=e_4; \\\\\n\\T {4}{17}(\\alpha) &:& e_1e_1 = e_2,& e_1e_2= \\alpha e_4,& e_2e_1=e_4,& e_3e_3=  e_4.\n\n\\end{array}\\]\n\n\n\\subsubsection{$1$-dimensional  central extensions of $\\T {3*}{02}$}\nLet us use the following notations: \n\\begin{align*}\n    \\nb 1 = \\Dl 11, \\nb 2 = \\Dl 12, \\nb 3 = \\Dl 21, \\nb 4 = \\Dl 13, \\nb 5 = \\Dl 23, \\nb 6 = \\Dl 31, \\nb 7 = \\Dl 32.    \n\\end{align*}\nTake $\\theta=\\sum_{i=1}^7\\alpha_i\\nb i\\in {\\rm H_T^2}(\\T {3*}{02}).$\nIf \n$$\n\\phi=\n\\begin{pmatrix}\nx          &       y& 0 \\\\\n(-1)^{n+1}y& (-1)^nx& 0\\\\\nz          & u      & x^2+y^2\n\\end{pmatrix}\\in\\aut{\\T {3*}{02}},\n$$\nthen\n$$\n\\phi^T\\begin{pmatrix}\n\\alpha_1& \\alpha_2& \\alpha_4\\\\\n\\alpha_3& 0& \\alpha_5\\\\\n\\alpha_6& \\alpha_7& 0\n\\end{pmatrix} \\phi=\n\\begin{pmatrix}\n\\alpha_1^*+\\alpha^*& \\alpha_2^*& \\alpha_4^*\\\\\n\\alpha_3^*& \\alpha^*& \\alpha_5^*\\\\\n\\alpha_6^*& \\alpha_7^*& \\alpha^{**}\n\\end{pmatrix},\n$$\nwhere\n\\begin{align*}\n    \\alpha_1^* &= \\alpha_1(x^2 - y^2) + 2(-1)^{n+1}(\\alpha_2 + \\alpha_3)xy+(\\alpha_4x+(-1)^{n+1}\\alpha_5y+\\alpha_6x+ (-1)^{n+1}\\alpha_7y)z\\\\\n\t&\\quad - (\\alpha_4y +(-1)^n\\alpha_5x+\\alpha_6y+(-1)^n\\alpha_7x)u,\\\\\n\t\\alpha_2^* &= (-1)^n\\alpha_2x^2 + \\alpha_1xy + (-1)^{n+1}\\alpha_3y^2 + (\\alpha_4x + (-1)^{n+1}\\alpha_5y)u + (\\alpha_6y  + (-1)^n\\alpha_7x)z,\\\\\n\t\\alpha_3^* &= (-1)^n\\alpha_3x^2 + \\alpha_1xy + (-1)^{n+1}\\alpha_2y^2 + (\\alpha_4y + (-1)^n\\alpha_5x)z + (\\alpha_6x + (-1)^{n+1}\\alpha_7y)u,\\\\\n\t\\alpha_4^* &= (\\alpha_4x+(-1)^{n+1}\\alpha_5y)(x^2+y^2),\\\\\n\t\\alpha_5^* &= (\\alpha_4y+(-1)^n\\alpha_5x)(x^2 + y^2),\\\\\n\t\\alpha_6^* &= (\\alpha_6x+(-1)^{n+1}\\alpha_7y)(x^2+y^2),\\\\\n\t\\alpha_7^* &= (\\alpha_6y + (-1)^n\\alpha_7x)(x^2+y^2).\n\\end{align*}\nHence, $\\phi\\langle\\theta\\rangle=\\langle\\theta^*\\rangle,$ where $\\theta^*=\\sum\\limits_{i=1}^7 \\alpha_i^*  \\nb i.$\n \n We are interested in $\\theta$ with $(\\alpha_4,\\alpha_5,\\alpha_6,\\alpha_7) \\neq (0,0,0,0).$  The condition $\\theta \\in \\mathbf{T}_1 (\\T{3*}{02})$ does not give us any new restrictions on parameters.\n\n\\begin{enumerate}\n    \\item $\\alpha_6=0$ and $\\alpha_7=0.$ Then $\\alpha_4\\ne 0$ or $\\alpha_5\\ne 0$. If $\\alpha_4\\ne 0$ and $\\alpha_5=0$, then choosing $x=y=1$, we have $\\alpha^*_4\\ne 0$ and $\\alpha^*_5\\ne 0$. The same holds for $\\alpha_4=0$ and $\\alpha_5\\ne 0$. Thus, we shall assume from the very beginning that $\\alpha_4\\ne 0$ and $\\alpha_5 \\neq 0$. \n    \\begin{enumerate}\n        \\item $\\alpha^2_4+\\alpha^2_5\\ne 0$. Then choosing $x=(-1)^{n+1}\\frac{\\alpha_4y}{\\alpha_5},\n                    z = \\frac{(-1)^ny}{\\alpha_5(\\alpha_4^2 + \\alpha_5^2)}(\\alpha_1\\alpha_4^2 + 2(\\alpha_2 + \\alpha_3)\\alpha_4\\alpha_5 - \\alpha_1\\alpha_5^2)$ and $u = \\frac y{\\alpha_5(\\alpha_4^2 + \\alpha_5^2)}(\\alpha_2\\alpha_4^2 -\\alpha_3\\alpha_5^2 - \\alpha_1\\alpha_4\\alpha_5)$ we obtain the representative $\\langle\\as 3\\nb 3+\\as 4\\nb 4\\rangle$, where\n                    \\begin{align*}\n                        \\as 3 &= \\frac{(-1)^ny^2}{\\alpha_5^2}(\\alpha_3\\alpha_4^2 - \\alpha_1\\alpha_4\\alpha_5 - \\alpha_2\\alpha_5^2),\\\\\n                        \\as 4 &= \\frac{(-1)^{n+1}y^3}{\\alpha_5^3}(\\alpha_4^2 + \\alpha_5^2)^2.\n                    \\end{align*}\n                    Thus, we have two representatives $\\langle \\nabla_4 \\rangle $ and $\\langle \\nabla_3 + \\nabla_4 \\rangle $ depending on whether $\\alpha_3\\alpha_4^2 - \\alpha_1\\alpha_4\\alpha_5 - \\alpha_2\\alpha_5^2=0$ or not.\n        \\item  $\\alpha^2_4+\\alpha^2_5=0$. If $\\alpha_4=-i\\alpha_5$, then choosing $x=0$ and $y=n=1$, we obtain $\\alpha^*_4=i\\alpha^*_5$, so we shall assume from the very beginning that $\\alpha_4= i \\alpha_5.$ \n        Taking $n=0$,\n        $u=\\frac{i(\\alpha_2x^2 + \\alpha_1xy - \\alpha_3y^2)}{\\alpha_5(x + iy)}$ and \n        $z=-\\frac{\\alpha_3x^2 + \\alpha_1xy - \\alpha_2y^2}{\\alpha_5(x + iy)}$\n        we have the representative $\\langle \\alpha_1^{\\star} \\nabla_1 + i  \\nabla_4 + \\nabla_5 \\rangle$, where\n        \\begin{align*}\n            \\as 1 &= \\frac{x - iy}{\\alpha_5(x + iy)^2}(\\alpha_1 - i\\alpha_2 - i\\alpha_3).\n        \\end{align*}\nThus, we have two representatives $\\langle i  \\nabla_4 +  \\nabla_5 \\rangle$ and  $\\langle \\nabla_1 + i  \\nabla_4 +  \\nabla_5 \\rangle$ depending on whether $\\alpha_1 - i\\alpha_2 - i\\alpha_3=0$ or not.\n\n        \n        \\end{enumerate}\n        \n\\item $\\alpha_6\\neq 0$ or $\\alpha_7\\neq0$, and $\\alpha_6^2+\\alpha_7^2\\ne 0$. \n Then we may make $\\alpha_6\\ne 0$ and $\\alpha_7\\neq 0$. After that, if we choose $x=(-1)^{n+1}\\frac{\\alpha_6y}{\\alpha_7}\\ne 0,$ then $x^2+y^2=\\frac {y^2}{\\alpha^2_7}(\\alpha_6^2+\\alpha_7^2)\\ne 0$, $\\alpha^*_6=\\frac{(-1)^{n+1}y^3}{\\alpha_7^3}(\\alpha_6^2 + \\alpha_7^2)^2\\ne 0$ and $\\alpha_7^* = 0$. Thus, we shall assume that $\\alpha_6\\ne 0$ and $\\alpha_7=0$ from the very beginning. Taking $y = 0,$ we get $\\alpha^*_7=0$ and\n \\begin{align*}\n    \\alpha_1^* &= x (x \\alpha_1 + (-1)^{n + 1} u \\alpha_5 + z (\\alpha_4 + \\alpha_6)),\\\\\n\t\\alpha_2^* &= x ((-1)^n x \\alpha_2 + u \\alpha_4),\\\\\n\t\\alpha_3^* &= x ((-1)^n x \\alpha_3 + (-1)^n z \\alpha_5 + u \\alpha_6),\\\\\n\t\\alpha_4^* &= x^3 \\alpha_4,\\\\\n\t\\alpha_5^* &= (-1)^n x^3 \\alpha_5,\\\\\n\t\\alpha_6^* &= x^3 \\alpha_6.\n\\end{align*}\nConsider $\\alpha_2^* = 0, \\alpha_3^* = 0$ as a system of linear equations on $z$ and $u.$ Its determinant is $(-1)^{n+1}x^2\\alpha_4\\alpha_5,$ so we have the following cases (each of which defines an $\\operatorname{Aut}{\\T {3*}{02}}$-invariant subset):\n\\begin{enumerate}\n    \\item $\\alpha_4 \\neq 0$ and $\\alpha_5 \\neq 0$. Taking $u = \\frac{(-1)^{n+1} x \\alpha_2}{\\alpha_4}, z = \\frac{x (\\alpha_2 \\alpha_6-\\alpha_3 \\alpha_4)}{\\alpha_4 \\alpha_5},$ we get $\\alpha_2^* = 0, \\alpha_3^* = 0$ and \n    \\begin{align*}\n    \\alpha_1^* &= \\frac{x^2}{\\alpha_4\\alpha_5}(\\alpha_2\\alpha_5^2 - \\alpha_3\\alpha_4^2 + \\alpha_1\\alpha_4\\alpha_5 + \\alpha_2\\alpha_4\\alpha_6 - \\alpha_3\\alpha_4\\alpha_6 +\\alpha_2\\alpha_6^2),\\\\\n\t\\alpha_4^* &= x^3 \\alpha_4,\\\\\n\t\\alpha_5^* &= (-1)^n x^3 \\alpha_5,\\\\\n\t\\alpha_6^* &= x^3 \\alpha_6.\n\\end{align*}\nThus, we obtain two families of representatives of distinct orbits $\\langle \\nabla_1 +\\alpha  \\nabla_4+\\beta  \\nabla_5+\\nabla_6 \\rangle_{\\alpha\\beta\\neq 0, \\beta \\in \\mathbb C_{\\geq 0}}$ and $\\langle \\alpha  \\nabla_4+\\beta \\nabla_5+\\nabla_6 \\rangle_{\\alpha\\beta\\neq 0, \\beta \\in \\mathbb C_{\\geq 0}}. $\n    \\item $\\alpha_4 \\neq 0$ and $\\alpha_5 = 0$. Taking $u = \\frac{(-1)^{n+1} x \\alpha_2}{\\alpha_4}$, we get $\\alpha_2^* = 0$ and    \n    \\begin{align*}\n    \\alpha_1^* &= x (x \\alpha_1 + z (\\alpha_4 + \\alpha_6)),\\\\\n    \\alpha_3^* &= \\frac{(-1)^nx^2}{\\alpha_4}(\\alpha_3\\alpha_4 - \\alpha_2\\alpha_6),\\\\\n\t\\alpha_4^* &= x^3 \\alpha_4,\\\\\n\t\\alpha_6^* &= x^3 \\alpha_6.\n\\end{align*}\n    \\begin{enumerate}\n        \\item $\\alpha_4 + \\alpha_6 \\neq 0$. Taking $z = -\\frac{x\\alpha_1}{\\alpha_4 + \\alpha_6},$ we get two series of representatives of distinct orbits $\\langle \\nabla_3 +\\alpha \\nabla_4 +\\nabla_6\\rangle_{\\alpha\\neq -1,0}$ and $\\langle \\alpha \\nabla_4 +\\nabla_6\\rangle_{\\alpha\\neq -1,0}.$\n        \\item $\\alpha_4 + \\alpha_6 = 0.$ In this case we get the series of representatives of distinct orbits $\\langle \\nabla_1 +\\alpha  \\nabla_3 - \\nabla_4+\\nabla_6 \\rangle_{\\alpha \\in \\mathbb C_{\\geq 0}}$ and two distinct representatives $\\langle \\nabla_3 - \\nabla_4+\\nabla_6 \\rangle$ and $\\langle - \\nabla_4+\\nabla_6 \\rangle$, which will be joined with the families $\\langle \\nabla_3 +\\alpha \\nabla_4 +\\nabla_6\\rangle_{\\alpha\\neq -1,0}$ and $\\langle \\alpha \\nabla_4 +\\nabla_6\\rangle_{\\alpha\\neq -1,0}$ found above.\n    \\end{enumerate}\n    \\item $\\alpha_4 = 0$ and $\\alpha_5 \\neq 0.$ Consider $\\alpha_1^* = 0, \\alpha_3^* = 0$ as a system of linear equations on $z,u.$ Its determinant is $x^2(\\alpha_5^2 + \\alpha_6^2).$ \n    \\begin{enumerate}\n        \\item $\\alpha_5^2 + \\alpha_6^2 \\neq 0.$ Taking $u = \\frac{(-1)^n x (\\alpha_1 \\alpha_5 - \\alpha_3 \\alpha_6)}{ \\alpha_5^2 +  \\alpha_6^2}, z = -\\frac{x ( \\alpha_3 \\alpha_5 + \\alpha_1 \\alpha_6)}{ \\alpha_5^2 + \\alpha_6^2},$ we get $\\alpha^*_1 = \\alpha^*_3 = 0$ and\n        \\begin{align*}\n    \\alpha_2^* &= (-1)^n x^2 \\alpha_2,\\\\\n\t\\alpha_5^* &= (-1)^n x^3 \\alpha_5,\\\\\n\t\\alpha_6^* &= x^3 \\alpha_6.\n\\end{align*}\nTherefore, we have two series of representatives of distinct orbits $\\langle \\nabla_2 + \\alpha \\nabla_5+\\nabla_6 \\rangle_{\\alpha \\in \\mathbb C_{\\geq 0}, \\alpha \\neq 0, i}$ and $\\langle \\alpha \\nabla_5+\\nabla_6 \\rangle_{\\alpha \\in \\mathbb C_{\\geq 0}, \\alpha \\neq 0, i}.$\n    \\item $\\alpha_5^2 + \\alpha_6^2 = 0.$ We may assume that $\\alpha_5 = i \\alpha_6$ (if $\\alpha_5=-i\\alpha_6$, then choose $n=-1$). Taking $n=0$, $u = -\\frac{x\\alpha_3}{\\alpha_6}$and $z = 0,$ we get $\\alpha_3 = 0$ and \n    \\begin{align*}\n    \\alpha_1^* &= x^2 (\\alpha_1+i\\alpha_3),\\\\\n    \\alpha_2^* &= x^2 \\alpha_2,\\\\\n\t\\alpha_5^* &= ix^3 \\alpha_6,\\\\\n\t\\alpha_6^* &= x^3 \\alpha_6.\n\\end{align*}\nTherefore, we get the family of representatives of distinct orbits $\\langle \\nabla_1 + \\alpha \\nabla_2 + i \\nabla_5 + \\nabla_6 \\rangle$ and two representatives $\\langle \\nabla_2 + i \\nabla_5 + \\nabla_6 \\rangle, \\langle i \\nabla_5 + \\nabla_6 \\rangle$, which will be joined with the families $\\langle \\nabla_2 + \\alpha \\nabla_5+\\nabla_6 \\rangle_{\\alpha \\in \\mathbb C_{\\geq 0}, \\alpha \\neq 0, i}$ and $\\langle \\alpha \\nabla_5+\\nabla_6 \\rangle_{\\alpha \\in \\mathbb C_{\\geq 0}, \\alpha \\neq 0, i}$ found above. \n    \\end{enumerate}\n    \\item $\\alpha_4 = 0$ and $\\alpha_5 = 0.$ Taking $z = -\\frac{x\\alpha_1}{\\alpha_6}$ and $u = \\frac{(-1)^{n+1}x\\alpha_3}{\\alpha_6},$ we get $\\alpha_1^* = \\alpha_3^* = 0$ and \n     \\begin{align*}\n    \\alpha_2^* &= (-1)^nx^2 \\alpha_2,\\\\\n\t\\alpha_6^* &= x^3 \\alpha_6.\n\\end{align*}\nTherefore, we get two representatives $\\langle \\nabla_2 + \\nabla_6 \\rangle$ and $\\langle \\nabla_6 \\rangle$, which will be joined with the families $\\langle \\nabla_2 + \\alpha \\nabla_5+\\nabla_6 \\rangle_{\\alpha \\in \\mathbb C_{\\geq 0}, \\alpha \\neq 0, i}$ and $\\langle \\alpha \\nabla_5+\\nabla_6 \\rangle_{\\alpha \\in \\mathbb C_{\\geq 0}, \\alpha \\neq 0, i}$ found above. Note that representatives $\\langle \\nabla_2 + \\nabla_6 \\rangle$ and $\\langle \\nabla_6 \\rangle$ define the same orbit.\n\\end{enumerate}\n\n\n\n\n\n\n\n    \n\n\n\n\n\n\n\n\n\n\n\n        \n            \n\n        \n        \n\n\\item $\\alpha_6\\neq 0$ or $\\alpha_7\\neq0$, and $\\alpha_6^2+\\alpha_7^2=0$. Then we may assume that $\\alpha_6= i \\alpha_7\\ne 0.$\n\n\n\\begin{enumerate}\n    \\item $\\alpha_4 \\neq \\pm i \\alpha_5.$ We may assume that $\\alpha_4 \\neq 0$ (otherwise $\\alpha_5\\ne 0$, so we may take $x=y=1$ to make $\\alpha^*_4\\ne 0$). Then choosing $n=0$ and $y=-\\frac{x\\alpha_5}{\\alpha_4}x,$ we get $\\alpha^*_4=\\frac{x^3}{\\alpha_4^3}(\\alpha_4^2 + \\alpha_5^2)^2\\ne 0$ and $\\alpha_5^* = 0$, so we shall assume that $\\alpha_4\\ne 0$ and $\\alpha_5=0$ from the very beginning. Now, choosing $n=y=0$, $u = \\frac{i x \\alpha_3}{\\alpha_7}, z = -\\frac{x (i \\alpha_3 \\alpha_4 + \\alpha_2 \\alpha_7)}{\\alpha_7^2},$ we get $\\alpha_2^* = \\alpha_3^* = \\alpha^*_5=0$ and \n    \\begin{align*}\n    \\alpha_1^* &= \\frac{x^2}{\\alpha_7^2}(\\alpha_3\\alpha_4\\alpha_7 + \\alpha_1\\alpha_7^2 - \\alpha_2\\alpha_4\\alpha_7 -i\\alpha_3\\alpha_4^2 - i\\alpha_2\\alpha_7^2 -i\\alpha_3\\alpha_7^2),\\\\\n    \\alpha_4^* &= x^3 \\alpha_4,\\\\\n\t\\alpha_6^* &= ix^3 \\alpha_7,\\\\\n\t\\alpha_7^* &= x^3 \\alpha_7.\n\\end{align*}\nThus, we get two series of representatives of distinct orbits $\\langle \\nabla_1 + \\alpha \\nabla_4 + i \\nabla_6 + \\nabla_7 \\rangle_{\\alpha \\neq 0}$ and $\\langle \\alpha \\nabla_4 + i \\nabla_6 + \\nabla_7 \\rangle_{\\alpha \\neq 0}.$\n    \n   \n    \n  \n \n \n\n \n    \n    \\item $\\alpha_4= i \\alpha_5$. Then choosing $n=0$ we have \n    \\begin{align*}\n    \\alpha_1^* &= x^2 \\alpha_1 - x (2 y (\\alpha_2 + \\alpha_3) + (u - i z) (\\alpha_5 + \\alpha_7)) -  y (y \\alpha_1 + (i u + z) (\\alpha_5 + \\alpha_7)),\\\\\n    \\alpha_2^* &= x^2 \\alpha_2 - y (y \\alpha_3 + u \\alpha_5 -i z \\alpha_7) +  x (y \\alpha_1 + i u \\alpha_5 + z \\alpha_7),\\\\\n    \\alpha_3^* &= (x + i y) z \\alpha_5 + x (y \\alpha_1 + x \\alpha_3 + i u \\alpha_7) -  y (y \\alpha_2 + u \\alpha_7),\\\\\n    \\alpha_4^* &= i (x + i y) (x^2 + y^2) \\alpha_5,\\\\\n    \\alpha_5^* &= (x + i y) (x^2 + y^2) \\alpha_5,\\\\\n\t\\alpha_6^* &= i (x + i y) (x^2 + y^2) \\alpha_7,\\\\\n\t\\alpha_7^* &= (x + i y) (x^2 + y^2) \\alpha_7.\n\\end{align*}\n    Consider $\\alpha_1^* = 0, \\alpha_3^* = 0$ as a system of linear equations in $u,z.$ Its determinant is $i(x+iy)^2(\\alpha_7^2 - \\alpha_5^2).$\n    \\begin{enumerate}\n        \\item $\\alpha_5^2 - \\alpha_7^2 \\neq 0$. Then by choosing the appropriate values of $u$ and $z$ we get $\\alpha_1^* = 0, \\alpha_3^* = 0$, $\\alpha_2^* = i(i\\alpha_1 + \\alpha_2 + \\alpha_3)(x - iy)^2$ and $\\alpha^*_4,\\alpha^*_5,\\alpha^*_6,\\alpha^*_7$ as above. Therefore, we have two series of representatives of distinct orbits $\\langle \\nabla_2 + i \\alpha \\nabla_4 + \\alpha \\nabla_5 + i \\nabla_6 + \\nabla_7 \\rangle_{\\alpha \\neq \\pm 1}$ and $\\langle  i \\alpha \\nabla_4 + \\alpha \\nabla_5 + i \\nabla_6 + \\nabla_7 \\rangle_{\\alpha \\neq \\pm 1}$.\n    \\item $\\alpha_5 = \\alpha_7$. Then taking $n=0$, $z = \\frac{-x y \\alpha_1 + y^2 \\alpha_2 - x^2 \\alpha_3 - i u x \\alpha_7 +  u y \\alpha_7}{(x + i y)\\alpha_7}$ we get $\\alpha_3^* = 0$ and\n    \\begin{align*}\n    \\alpha_1^* &= (x - i y) (x (\\alpha_1- 2i\\alpha_3) - y (i \\alpha_1 + 2 \\alpha_2)),\\\\\n    \\alpha_2^* &= (x^2 + y^2) (\\alpha_2-\\alpha_3),\\\\\n    \\alpha_4^* &= i (x + i y) (x^2 + y^2) \\alpha_7,\\\\\n    \\alpha_5^* &= (x + i y) (x^2 + y^2) \\alpha_7,\\\\\n\t\\alpha_6^* &= i (x + i y) (x^2 + y^2) \\alpha_7,\\\\\n\t\\alpha_7^* &= (x + i y) (x^2 + y^2) \\alpha_7.\n\\end{align*}\n\\begin{enumerate}\n    \\item $i\\alpha_1 + \\alpha_2 + \\alpha_3\\ne 0$, $\\alpha_2 - \\alpha_3\\ne 0$ and $i\\alpha_1 + 2\\alpha_2\\ne 0$. Then taking $y = \\frac{(\\alpha_1 - 2i\\alpha_3)x}{i\\alpha_1 + 2\\alpha_2}$ we get the representative $\\langle\\as 2\\nb 2+i \\nabla_4 + \\nabla_5 + i \\nabla_6 + \\nabla_7\\rangle$, where\n        \\begin{align*}\n            \\as 2 &= \\frac{(i\\alpha_1 + 2\\alpha_2)(\\alpha_2 - \\alpha_3)}{2\\alpha_7 x(i\\alpha_1 + \\alpha_2 + \\alpha_3)}. \n        \\end{align*}\nSo, choosing the appropriate value of $x$, we get the representative $\\langle \\nabla_2 + i \\nabla_4 + \\nabla_5 + i \\nabla_6 + \\nabla_7\\rangle.$\n    \\item $i\\alpha_1 + \\alpha_2 + \\alpha_3 = 0$. Then we have \n         \\begin{align*}\n    \\alpha_1^* &= i(x^2+y^2)(\\alpha_2-\\alpha_3),\\\\\n    \\alpha_2^* &= (x^2+y^2)(\\alpha_2-\\alpha_3),\n\\end{align*}\nTherefore, we get two representatives $\\langle i\\nabla_1 + \\nabla_2 + i \\nabla_4 + \\nabla_5 + i \\nabla_6 + \\nabla_7\\rangle$ and $\\langle i \\nabla_4 + \\nabla_5 + i \\nabla_6 + \\nabla_7\\rangle.$\n    \\item $\\alpha_2-\\alpha_3 = 0$. Then $\\alpha_1^* =  (x-iy)^2(\\alpha_1 - 2i\\alpha_3)$, and we have only one new representative $\\langle \\nabla_1 + i \\nabla_4 + \\nabla_5 + i \\nabla_6 + \\nabla_7\\rangle.$\n    \\item $i\\alpha_1 + 2\\alpha_2=0$. Then\n    $\n        \\alpha_1^* = x(x - i y) (\\alpha_1- 2i\\alpha_3),\n    $\n    so choosing $x=0$ we have two representatives $\\langle \\nabla_2 + i \\nabla_4 + \\nabla_5 + i \\nabla_6 + \\nabla_7\\rangle$ and $\\langle i \\nabla_4 + \\nabla_5 + i \\nabla_6 + \\nabla_7\\rangle$ found above.\n\\end{enumerate}\n\\item $\\alpha_5 = -\\alpha_7$. Then taking $n=0$ and $z = \\frac{x y \\alpha_1 - y^2 \\alpha_2 + x^2 \\alpha_3 + i u x \\alpha_7 -  u y \\alpha_7}{(x + i y) \\alpha_7}$ we get $\\alpha_3^*=0$ and\n     \\begin{align*}\n    \\alpha_1^* &= (x^2 - y^2)\\alpha_1  - 2 x y (\\alpha_2+\\alpha_3),\\\\\n    \\alpha_2^* &= (x^2 - y^2)(\\alpha_2+\\alpha_3) +2 x y \\alpha_1,\\\\\n    \\alpha_4^* &= -i(x + i y) (x^2 + y^2) \\alpha_7,\\\\\n    \\alpha_5^* &= -(x + i y) (x^2 + y^2) \\alpha_7,\\\\\n\t\\alpha_6^* &= i (x + i y) (x^2 + y^2) \\alpha_7,\\\\\n\t\\alpha_7^* &= (x + i y) (x^2 + y^2) \\alpha_7.\n\\end{align*}\n    \\begin{enumerate}\n        \\item $(\\alpha_1,\\alpha_2+\\alpha_3) \\neq (0,0)$ and $\\alpha_1^2 + (\\alpha_2+\\alpha_3)^2 \\neq 0$. Then we may assume that $\\alpha_1\\ne 0$ and $\\alpha_2+\\alpha_3\\ne 0$. In this case the equality $\\alpha^*_1=0$ has two distinct roots $y_1=\\mu_1x$ and $y_2=\\mu_2x$, where at least one of $\\mu_1,\\mu_2$ is different from $\\pm i$ (otherwise $\\alpha_2+\\alpha_3=0$). Let $\\mu_1\\ne\\pm i$. Then choosing $y = \\mu_1x$ we get $\\alpha_1^* = 0$. Observe that in this case $x^2 + y^2\\ne 0$ and $\\alpha_1 = \\frac{2(\\alpha_2 + \\alpha_3)\\mu_1}{1 - \\mu_1^2}$. Substituting this into $\\alpha^*_2$, we obtain $\\alpha_2^* = \\frac{x^2(1 + \\mu_1^2)^2(\\alpha_2 + \\alpha_3)}{1-\\mu_1^2}\\ne 0.$ Therefore, we have the representative $\\langle \\nabla_2 - i \\nabla_4 - \\nabla_5 + i \\nabla_6 + \\nabla_7\\rangle.$\n        \\item $\\alpha_2+\\alpha_3 \\neq 0$ and $\\alpha_1 = \\pm i (\\alpha_2+\\alpha_3)\\ne 0$. Then we get \n        \\begin{align*}\n            \\alpha_1^* &= \\pm i (x \\pm i y)^2 (\\alpha_2+\\alpha_3),\\\\\n            \\alpha_2^* &= (x \\pm i y)^2 (\\alpha_2+\\alpha_3),\n        \\end{align*}\n        so we obtain two representatives $\\langle i\\nabla_1 + \\nabla_2 - i \\nabla_4 - \\nabla_5 + i \\nabla_6 + \\nabla_7\\rangle$ and $\\langle -i\\nabla_1 + \\nabla_2 - i \\nabla_4 - \\nabla_5 + i \\nabla_6 + \\nabla_7\\rangle.$\n        \\item $(\\alpha_1,\\alpha_2+\\alpha_3) = (0,0)$. Then we get the representative $\\langle - i \\nabla_4 - \\nabla_5 + i \\nabla_6 + \\nabla_7\\rangle.$\n    \\end{enumerate}\n   \n\n\n\n    \n\n    \\end{enumerate}\n    \\item $\\alpha_4 = -i\\alpha_5$. Choosing $n=0$, we have \n     \\begin{align*}\n    \\alpha_1^* &= x^2 \\alpha_1 -  x (2 y (\\alpha_2 + \\alpha_3) + i z (\\alpha_5 - \\alpha_7) + u (\\alpha_5 + \\alpha_7)) - y (y \\alpha_1 - i u (\\alpha_5 - \\alpha_7) + z (\\alpha_5 + \\alpha_7)),\\\\\n    \\alpha_2^* &= x^2 \\alpha_2 - y (y \\alpha_3 + u \\alpha_5 - i z \\alpha_7) + x (y \\alpha_1 - i u \\alpha_5 + z \\alpha_7),\\\\\n    \\alpha_3^* &= (x - i y) z \\alpha_5 + x (y \\alpha_1 + x \\alpha_3 + i u \\alpha_7) - y (y \\alpha_2 + u \\alpha_7),\\\\\n    \\alpha_4^* &= -i(x - iy) (x^2 + y^2) \\alpha_5,\\\\\n    \\alpha_5^* &= (x - i y) (x^2 + y^2) \\alpha_5,\\\\\n\t\\alpha_6^* &= i (x + i y) (x^2 + y^2) \\alpha_7,\\\\\n\t\\alpha_7^* &= (x + i y) (x^2 + y^2) \\alpha_7.\n\\end{align*}\nConsider $\\alpha_2^* = 0, \\alpha_3^* = 0$ as a linear system in $u, z.$ Its determinant is $-i((x - i y)^2 \\alpha_5^2 + (x + i y)^2 \\alpha_7^2).$ So, we may choose $x,y$ such that $x^2 + y^2 \\neq 0, -i((x - i y)^2 \\alpha_5^2 + (x + i y)^2 \\alpha_7^2) \\neq 0$ and $u$ and $z$ to make $\\alpha_2^* = \\alpha_3^*=0$. Observe that this does not change the conditions on $\\alpha_4,\\alpha_5,\\alpha_6,\\alpha_7$. So, we may assume that $\\alpha_2=\\alpha_3=0$ from the very beginning. \nWe may also suppose that $\\alpha_5 + \\alpha_7 \\neq 0.$         \n\\begin{enumerate}\n    \\item $\\alpha_5 \\neq 0$. Then taking $y = \\frac{\\alpha_5 - \\alpha_7}{i(\\alpha_5+\\alpha_7)}x$ we may make $\\alpha_5^* = \\alpha_7^*$, so we shall assume $\\alpha_5=\\alpha_7$. Now, choosing $n=y = u=z=0,$ we get \n     \\begin{align*}\n    \\alpha_1^* &= x^2 \\alpha_1,\\\\\n    \\alpha_4^* &= -ix^3 \\alpha_7,\\\\\n    \\alpha_5^* &= x^3 \\alpha_7,\\\\\n\t\\alpha_6^* &= i x^3 \\alpha_7,\\\\\n\t\\alpha_7^* &= x^3 \\alpha_7.\n\\end{align*}\nTherefore, we get two representatives $\\langle \\nabla_1 - i \\nabla_4 + \\nabla_5 + i\\nabla_6 + \\nabla_7 \\rangle$ and $\\langle - i \\nabla_4 + \\nabla_5 + i\\nabla_6 + \\nabla_7 \\rangle.$\n    \\item $\\alpha_5 = 0$. Then $\\alpha_4 = 0$, so that $\\alpha^*_4=\\alpha^*_5=0$. Choosing $n=0$ and the appropriate values of $u$ and $z$ we have $\\alpha^*_2=\\alpha^*_3=0$ and\n        \\begin{align*}\n    \\alpha_1^* &= (x - i y)^2 \\alpha_1,\\\\\n\t\\alpha_6^* &= i (x + i y) (x^2 + y^2) \\alpha_7,\\\\\n\t\\alpha_7^* &= (x + i y) (x^2 + y^2) \\alpha_7.\n\\end{align*}\nTherefore, we get two representatives $\\langle \\nabla_1 + i\\nabla_6 + \\nabla_7 \\rangle$ and $\\langle i\\nabla_6 + \\nabla_7 \\rangle,$ which will be joined with the series $\\langle \\nabla_1 + \\alpha \\nabla_4 + i \\nabla_6 + \\nabla_7 \\rangle_{\\alpha \\neq 0}$ and $\\langle \\alpha \\nabla_4 + i \\nabla_6 + \\nabla_7 \\rangle_{\\alpha \\neq 0}.$ Note that by above representatives $\\langle \\nabla_1 + i\\nabla_6 + \\nabla_7 \\rangle$ and $\\langle \\nabla_2 + i\\nabla_6 + \\nabla_7 \\rangle$ define the same orbit.\n\\end{enumerate}\n    \n\\end{enumerate}\n\n\\end{enumerate}\n\nSummarizing, \nwe have the following distinct orbits\n\n\\[ \\langle \\nabla_4\\rangle,\n\\langle \\nabla_3+ \\nabla_4\\rangle,\n\\langle \\nabla_1+i\\nabla_4+\\nabla_5\\rangle,\n\\langle i\\nabla_4 +\\nabla_5\\rangle,\\]\n\n\\[ \\langle \\alpha \\nabla_4+\\beta\\nabla_5+\\nabla_6 \\rangle_{ \\beta \\in \\mathbb C_{\\geq 0}}, \\langle \\nabla_1+ \\alpha \\nabla_4+\\beta\\nabla_5+\\nabla_6 \\rangle_{\\alpha\\beta \\neq 0, \\beta \\in \\mathbb C_{\\geq 0}},\\]\n\\[  \\langle  \\nabla_3+\\alpha\\nabla_4+\\nabla_6 \\rangle_{\\alpha \\neq 0}, \\langle  \\nabla_1 +\\alpha\\nabla_3 - \\nabla_4 +\\nabla_6 \\rangle_{\\alpha \\in \\mathbb C_{\\geq 0}}, \\langle \\nabla_1 + \\alpha \\nabla_2 + i \\nabla_5 + \\nabla_6 \\rangle, \\langle  \\nabla_2 +\\alpha\\nabla_5 +\\nabla_6 \\rangle_{\\alpha \\in \\mathbb C_{\\geq 0}}\\]\n\n\\[\\langle  \\nabla_1+\\alpha\\nabla_4+i\\nabla_6 + \\nabla_7 \\rangle, \\langle  \\alpha\\nabla_4+i\\nabla_6 + \\nabla_7 \\rangle,\\]\n\\[\\langle  \\nabla_2+i\\alpha\\nabla_4 + \\alpha\\nabla_5+i\\nabla_6 + \\nabla_7 \\rangle_{\\alpha \\neq 0}, \\langle  i\\alpha\\nabla_4 + \\alpha\\nabla_5+i\\nabla_6 + \\nabla_7 \\rangle_{\\alpha \\neq 0},\\]\n\\[\\langle i \\nabla_1 + \\nabla_2 + i \\nabla_4 + \\nabla_5 + i\\nabla_6 + \\nabla_7\\rangle,  \\langle \\pm i \\nabla_1 + \\nabla_2 - i \\nabla_4 - \\nabla_5 + i\\nabla_6 + \\nabla_7\\rangle\\]\n\nThe corresponding algebras are\n\n$$    \\begin{array}{lllllllllll}\n        \\T {4}{18}&:& e_1 e_1 = e_3, & e_1e_3=e_4, & e_2 e_2=e_3; \\\\\n        \\T {4}{19}&:& e_1 e_1 = e_3, & e_1e_3=e_4,& e_2e_1=e_4, & e_2 e_2=e_3; \\\\\n        \\T {4}{20}&:& e_1 e_1 = e_3+e_4, & e_1e_3=ie_4, & e_2 e_2=e_3, & e_2e_3=e_4; \\\\\n        \\T {4}{21}&:& e_1 e_1 = e_3, & e_1e_3=ie_4, & e_2 e_2=e_3,& e_2e_3=e_4; \\\\\n        \\T {4}{22}(\\alpha,\\beta)_{\\beta\\in \\mathbb C_{\\geq0}}&:& e_1 e_1 = e_3, & e_1e_3=\\alpha e_4,&  e_2 e_2=e_3, &e_2e_3=\\beta e_4,& e_3e_1=e_4; \\\\\n        \\T {4}{23}(\\alpha,\\beta)_\n        \\beta \\in \\mathbb C_{\\geq0}}&:& e_1 e_1 = e_3+e_4, & e_1e_3=\\alpha e_4,&  e_2 e_2=e_3, &e_2e_3=\\beta e_4,& e_3e_1=e_4; \\\\\n        \\T {4}{24}(\\alpha\n        &:& e_1 e_1 = e_3,& e_1e_3 = \\alpha e_4, & e_2e_1 = e_4, & e_2e_2 = e_3, & e_3e_1 = e_4; \\\\\n        \\T {4}{25}(\\alpha)_{\\alpha \\in \\mathbb C_{\\geq0}}&:& e_1 e_1 = e_3+e_4, & e_1e_3=- e_4,& e_2e_1 = \\alpha e_4, & e_2e_2 = e_3, & e_3e_1 = e_4; \\\\\n        \\T {4}{26}(\\alpha)&:& e_1 e_1 = e_3+e_4, & e_1e_2=\\alpha e_4, & e_2 e_2=e_3, &e_2e_3=i e_4,& e_3e_1=e_4; \\\\\n        \\T {4}{27}(\\alpha)_{\\alpha \\in \\mathbb C_{\\geq0}}&:& e_1 e_1 = e_3, & e_1e_2= e_4, & e_2 e_2=e_3,& e_2e_3=\\alpha e_4,& e_3e_1=e_4; \\\\\n        \\T {4}{28}(\\alpha)&:& e_1 e_1 = e_3 + e_4, & e_1e_3=\\alpha e_4, & e_2 e_2=e_3,& e_3e_1=ie_4,& e_3e_2=e_4; \\\\\n        \\T {4}{29}(\\alpha)&:& e_1 e_1 = e_3, & e_1e_3=\\alpha e_4, & e_2 e_2=e_3,& e_3e_1=ie_4,& e_3e_2=e_4; \\\\\n        \\T {4}{30}(\\alpha\n        &:& e_1 e_1 = e_3, & e_1 e_2 = e_4, & e_1e_3=i\\alpha e_4, & e_2 e_2=e_3,& e_2e_3=\\alpha e_4, & e_3e_1=ie_4,& e_3e_2=e_4; \\\\\n        \\T {4}{31}(\\alpha\n        &:& e_1 e_1 = e_3, & e_1e_3=i\\alpha e_4, & e_2 e_2=e_3,& e_2e_3=\\alpha e_4, & e_3e_1=ie_4,& e_3e_2=e_4; \\\\\n        \\T {4}{32}&:& e_1 e_1 = e_3 + ie_4, & e_1 e_2 = e_4, & e_1e_3=i e_4, & e_2 e_2=e_3,& e_2e_3= e_4, & e_3e_1=ie_4,& e_3e_2=e_4; \\\\\n        \\T {4}{33}&:& e_1 e_1 = e_3 + ie_4, & e_1 e_2 = e_4, & e_1e_3= -ie_4, & e_2 e_2=e_3,& e_2e_3= -e_4, & e_3e_1=ie_4,& e_3e_2=e_4; \\\\\n        \\T {4}{34}&:& e_1 e_1 = e_3 -ie_4, & e_1 e_2 = e_4, & e_1e_3= -ie_4, & e_2 e_2=e_3,& e_2e_3= -e_4, & e_3e_1=ie_4,& e_3e_2=e_4.\n    \\end{array}$$\n\n    \nThe algebras above are pairwise non-isomorphic, except $\\T {4}{23}(\\alpha,0) \\cong \\T {4}{22}(\\alpha,0)$ for $\\alpha \\neq -1,$ $\\T {4}{23}(-1,0) \\cong \\T {4}{25}(0),$ $\\T {4}{23}(0,\\beta) \\cong \\T {4}{22}(0,\\beta)$ for $\\beta \\neq i,$ $\\T {4}{23}(0,i) \\cong \\T {4}{26}(0), \\T {4}{24}(0) \\cong \\T {4}{22}(0,0), \\T {4}{30}(0) \\cong \\T {4}{28}(0), \\T {4}{31}(0) \\cong \\T {4}{29}(0).$ \n\n\\newpage \n\\subsubsection{$1$-dimensional  central extensions of $\\T {3*}{03}$}\nLet us use the following notations \n\\begin{align*}\n    \\nb 1 = \\Dl 11, \\nb 2 = \\Dl 12, \\nb 3 = \\Dl 13 - \\Dl 31, \\nb 4 = \\Dl 22, \\nb 5 = \\Dl 23 - \\Dl 32, \\nb 6 = \\Dl 31, \\nb 7 = \\Dl 32.    \n\\end{align*}\nTake $\\theta=\\sum_{i=1}^7\\alpha_i\\nb i\\in {\\rm H_T^2}(\\T {3*}{03}).$\nIf \n$$\n\\phi=\n\\begin{pmatrix}\nx &    y  &  0\\\\\nz &    u  &  0\\\\\nv &    w  &  xu-yz\n\\end{pmatrix}\\in\\aut{\\T {3*}{03}},\n$$\nthen\n$$\n\\phi^T\\begin{pmatrix}\n\\alpha_1& \\alpha_2& \\alpha_3\\\\\n 0& \\alpha_4& \\alpha_5\\\\\n\\alpha_6-\\alpha_3& \\alpha_7-\\alpha_5& 0\n\\end{pmatrix} \\phi=\n\\begin{pmatrix}\n\\alpha_1^*& \\alpha_2^*-\\alpha^*& \\alpha_3^*\\\\\n\\alpha^*& \\alpha_4^*& \\alpha_5^*\\\\\n\\alpha_6^*-\\alpha_3^*& \\alpha_7^*-\\alpha_5^*& \\alpha^{**}\n\\end{pmatrix},\n$$\nwhere\n\\begin{align*}\n\\alpha^*_1 &= \\alpha_1x^2 + \\alpha_2xz + \\alpha_4z^2 + v(\\alpha_6x + \\alpha_7z),\\\\\n\\alpha^*_2 &= x(2\\alpha_1y + \\alpha_2u) + z(\\alpha_2y + 2\\alpha_4u) + w(\\alpha_6x +\\alpha_7z) + v(\\alpha_6y + \\alpha_7u),\\\\\n\\alpha^*_3 &= (\\alpha_3x+\\alpha_5z)(xu - yz),\\\\\n\\alpha^*_4 &= y(\\alpha_1y + \\alpha_2u) + \\alpha_4u^2 + w(\\alpha_6y + \\alpha_7u),\\\\\n\\alpha^*_5 &= (\\alpha_3y+\\alpha_5u)(xu - yz),\\\\\n\\alpha^*_6 &= (\\alpha_6x+\\alpha_7z)(xu - yz),\\\\\n\\alpha^*_7 &= (\\alpha_6y+\\alpha_7u)(xu - yz).\n\\end{align*}\nHence, $\\phi\\langle\\theta\\rangle=\\langle\\theta^*\\rangle,$ where $\\theta^*=\\sum\\limits_{i=1}^7 \\alpha_i^*  \\nb i.$\n\n We are interested in $\\theta$ with $(\\alpha_6,\\alpha_7) \\neq (0,0).$ Moreover, the condition $\\theta \\in \\mathbf{T}_1 (\\T{3*}{03})$ gives us $(\\alpha_3,\\alpha_5, \\alpha_6 -\\alpha_3, \\alpha_7 - \\alpha_5) \\neq (0,0,0,0).$\n\n    If $\\alpha_7\\neq 0$, then taking $u=-\\frac{\\alpha_6y}{\\alpha_7}$ we have $\\alpha^*_7=0$, so we shall assume that $\\alpha^*_6\\ne 0$ and $\\alpha^*_7=0$ from the very beginning. Choosing $y=0$, $z=-\\frac{\\alpha_3x}{\\alpha_5}$, $u= \\frac{\\alpha_6x}{\\alpha_5}$, $v = \\frac x{\\alpha_5^2\\alpha_6}(\\alpha_2\\alpha_3\\alpha_5-\\alpha_3^2\\alpha_4 - \\alpha_1\\alpha_5^2)$, $w = \\frac x{\\alpha_5^2}(2\\alpha_3\\alpha_4 - \\alpha_2\\alpha_5)$,  we get  the family of representatives \n     $ \\langle  \\frac{\\alpha_4}{\\alpha_5 x}\\nabla_4 +  \\nabla_5+  \\nabla_6\\rangle$. It gives two distinct representatives $ \\langle  \\nabla_5+  \\nabla_6\\rangle$ and $ \\langle  \\nabla_4 +  \\nabla_5+  \\nabla_6\\rangle$ depending on whether $\\alpha_4=0$ or not.\n   \n   The algebras corresponding to $ \\langle  \\nabla_4 +  \\nabla_5+  \\nabla_6\\rangle$ and $ \\langle  \\nabla_5+  \\nabla_6\\rangle$ are:\n   \n  \n \n \n \n \n \n \n \n\n\n\n\n\n\n    \n    \n\n\n\n \n\n\n\n\n\n\n$$   \n\\begin{array}{lllllllll}\n\\T {4}{35}        &:&  e_1 e_2=e_3, & e_2 e_1=-e_3,& e_2e_2=e_4,& e_2e_3=e_4,& e_3e_1=e_4,&  e_3e_2=-e_4;\\\\\n\\T {4}{36}        &:&  e_1 e_2=e_3, & e_2 e_1=-e_3,&  e_2e_3=e_4,& e_3e_1=e_4,&  e_3e_2=-e_4.\n \\end{array} \n $$\n\n\n\n\n\n\\subsubsection{$1$-dimensional  central extensions of $\\T {3*}{04}$}\nLet us use the following notations \n\\begin{align*}\n    \\nb 1 = \\Dl 11, \\nb 2 = \\Dl 12, \\nb 3 = \\Dl 13, \\nb 4 = \\Dl 21,\n\t\\nb 5 = \\Dl 23, \\nb 6 = \\Dl 31, \\nb 7 = \\Dl 32.    \n\\end{align*}\nTake $\\theta=\\sum_{i=1}^7\\alpha_i\\nb i\\in {\\rm H_T^2}(\\T {3*}{04}).$\nIf \n$$\n\\phi=\n \\begin{pmatrix}\n    x &      y    &  0\\\\\n    -\\lambda y    &    x-y   &  0\\\\\n    z &      u      &  x^2-xy+\\lambda y^2\n    \\end{pmatrix}\\in\\aut{\\T 3{04}},\n$$\nthen\n$$\n\\phi^T\\begin{pmatrix}\n\t\\alpha_1            &  \\alpha_2        & \\alpha_3\\\\\n\t\\alpha_4            &      0        & \\alpha_5\\\\\n\t\\alpha_6            &  \\alpha_7        & 0\n\t\\end{pmatrix} \\phi=\n\t\\begin{pmatrix}\n\t\\alpha_1^*+\\lambda\\alpha^*      &  \\alpha_2^*  & \\alpha_3^*\\\\\n\t\\alpha_4^*+\\alpha^*         &  \\alpha^*    & \\alpha_5^*\\\\\n\t\\alpha_6^*               &  \\alpha_7^*  & 0\n\t\\end{pmatrix},\n$$\nwhere\n\\begin{align*}\n\t\\alpha^*_1 &= \\alpha_1x^2 + \\lambda(-\\alpha_1 + \\alpha_2 + \\alpha_4) y^2  -2\\lambda(\\alpha_2 + \\alpha_4) xy + (\\alpha_3 + \\alpha_6)xz - (\\alpha_5 + \\alpha_7) yz\\\\\n\t&\\quad- \\lambda(\\alpha_5 + \\alpha_7) ux + \\lambda(- \\alpha_3 + \\alpha_5 - \\alpha_6 + \\alpha_7) uy,\\\\\n    \\alpha^*_2 &= \\alpha_2x^2 - \\lambda\\alpha_4 y^2 + (\\alpha_1 - \\alpha_2)xy + \\alpha_7xz + (\\alpha_6 - \\alpha_7)yz -\\alpha_5\\lambda uy  + \\alpha_3ux,\\\\\n    \\alpha^*_3 &= (x^2 - xy + \\lambda y^2)(\\alpha_3x -\\lambda\\alpha_5 y),\\\\\n    \\alpha^*_4 &= \\alpha_4x^2 + (- \\alpha_1 + (1-\\lambda)\\alpha_2 + \\alpha_4)y^2 + (\\alpha_1 - \\alpha_2 - 2\\alpha_4)xy + (\\alpha_3 - \\alpha_5)yz + \\alpha_5xz\\\\\n    &\\quad - \\alpha_7\\lambda uy + (- \\alpha_5 + \\alpha_6 - \\alpha_7)ux  + (- \\alpha_3 + \\alpha_5 - \\alpha_6 + \\alpha_7)uy,\\\\\n    \\alpha^*_5 &= (x^2 - xy + \\lambda y^2)(\\alpha_5x + (\\alpha_3 - \\alpha_5)y),\\\\\n    \\alpha^*_6 &= (x^2 - xy + \\lambda y^2)(\\alpha_6x -\\lambda\\alpha_7 y),\\\\\n    \\alpha^*_7 &= (x^2 - xy + \\lambda y^2)(\\alpha_7x + (\\alpha_6-\\alpha_7)y).\n\t\\end{align*}\nHence, $\\phi\\langle\\theta\\rangle=\\langle\\theta^*\\rangle,$ where $\\theta^*=\\sum\\limits_{i=1}^7 \\alpha_i^*  \\nb i.$\n\n We are interested in $\\theta$ with $(\\alpha_3,\\alpha_5,\\alpha_6,\\alpha_7) \\neq (0,0,0,0)$ (if $\\lambda \\neq 0$) and $(\\alpha_3,\\alpha_6,\\alpha_7) \\neq (0,0,0)$ (if $\\lambda = 0$). Moreover, the condition $\\theta \\in \\mathbf{T}_1 (\\T{3*}{04})$ gives us $(\\alpha_4,\\alpha_5,\\alpha_6,\\alpha_7) \\neq (0,0,0,0).$\n \n    \\begin{enumerate}\n    \n        \\item $\\alpha_7\\ne 0$. We have the following subcases.\n\t\\begin{enumerate}\n\t    \\item $\\alpha_6^2 - \\alpha_6\\alpha_7 + \\lambda\\alpha_7^2\\ne 0$. Choosing $x=-\\frac{(\\alpha_6-\\alpha_7)y}{\\alpha_7}$, we have $\\alpha^*_7=0$. Since $(\\alpha^*_6)^2 - \\alpha^*_6\\alpha^*_7 + \\lambda(\\alpha^*_7)^2=(\\alpha_6^2 - \\alpha_6\\alpha_7 + \\lambda\\alpha_7^2)(x^2 - xy + \\lambda y^2)^3$, the condition $\\alpha_6^2 - \\alpha_6\\alpha_7 + \\lambda\\alpha_7^2\\ne 0$ is invariant under the action of the automorphism group. Thus, we may assume that $\\alpha_7=0$ and $\\alpha_6\\ne 0$ from the very beginning. Then choosing $y=0$ we obtain $\\alpha^*_7=0$ and\n        \\begin{align*}\n\t    \\alpha^*_1 &= x(\\alpha_1x  + (\\alpha_3 + \\alpha_6)z - \\lambda\\alpha_5u),\\\\\n        \\alpha^*_2 &= x(\\alpha_2x  + \\alpha_3u),\\\\\n        \\alpha^*_3 &= \\alpha_3 x^3,\\\\\n        \\alpha^*_4 &= x(\\alpha_4x + \\alpha_5z + (- \\alpha_5 + \\alpha_6)u),\\\\\n        \\alpha^*_5 &= \\alpha_5 x^3,\\\\\n        \\alpha^*_6 &= \\alpha_6 x^3.\n\t    \\end{align*}\n\t    \\begin{enumerate}\n\t        \\item $\\alpha_5\\ne 0$ and $\\alpha_3\\ne 0$. Then we choose $u=-\\frac{\\alpha_2x}{\\alpha_3}$ and $z=-\\frac{\\alpha_4x+(- \\alpha_5 + \\alpha_6)u}{\\alpha_5}$ and obtain the family of representatives $\\langle\\alpha\\nb 1+\\beta\\nb 3+\\gamma\\nb 5+\\nb 6\\rangle_{\\beta,\\gamma\\ne 0}$. It gives two families of representatives of distinct orbits: $\\langle\\nb 1+\\alpha\\nb 3+\\beta\\nb 5+\\nb 6\\rangle_{\\alpha,\\beta\\ne 0}$ and $\\langle\\alpha\\nb 3+\\beta\\nb 5+\\nb 6\\rangle_{\\alpha,\\beta\\ne 0}$.\n\t        \n\t        \\item $\\alpha_5\\ne 0$ and $\\alpha_3=0$. Then $\\alpha^*_3=0$ and $\\alpha^*_4$, $\\alpha^*_5$, $\\alpha^*_6$ are as above and\n\t        \\begin{align*}\n\t            \\alpha^*_1 &= x(\\alpha_1x  + \\alpha_6z - \\lambda\\alpha_5u),\\\\\n                \\alpha^*_2 &= \\alpha_2x^2.\n\t        \\end{align*}\n\t        Choosing $z=\\frac{\\lambda\\alpha_5u-\\alpha_1x}{\\alpha_6}$, we have $\\alpha^*_1=0$. Now we have the following subcases:\n\t        \\begin{enumerate}\n\t            \\item $\\alpha_6^2 - \\alpha_5\\alpha_6 + \\lambda\\alpha_5^2\\ne 0$. Then choosing $u=\\frac{(\\alpha_1\\alpha_5 - \\alpha_4\\alpha_6)x}{\\alpha_6^2 - \\alpha_5\\alpha_6 + \\lambda\\alpha_5^2}$ we have $\\alpha^*_4=0$, so we obtain the family of representatives $\\langle\\alpha\\nb 2+\\beta\\nb 5+\\nb 6\\rangle_{1-\\beta+\\lambda\\beta^2\\ne 0}$. It determines two families of representatives of distinct orbits: $\\langle\\nb 2+\\alpha\\nb 5+\\nb 6\\rangle_{\\alpha \\neq 0, 1-\\alpha+\\lambda\\alpha^2\\ne 0}$ and $\\langle\\alpha\\nb 5+\\nb 6\\rangle_{\\alpha \\neq 0, 1-\\alpha+\\lambda\\alpha^2\\ne 0}$.\n\t            \\item $\\alpha_6^2 - \\alpha_5\\alpha_6 + \\lambda\\alpha_5^2=0$. \n\t            \n\t            If $\\lambda\\not\\in\\{0,\\frac 14\\}$, then we have two families of representatives of distinct orbits\n\t            $\\Big\\langle \\lambda\\nb 2+\\lambda\\alpha\\nb 4+\\frac{1+\\sqrt{1-4\\lambda}}{2}\\nb 5+\\lambda\\nb 6\\Big\\rangle, \\Big\\langle\\nb 2+\\alpha\\nb 4+\\frac{2}{1+\\sqrt{1-4\\lambda}}\\nb 5+\\nb 6\\Big\\rangle$ and 4 separate representatives\n\t           $\\Big\\langle\\lambda \\nb 4+\\frac{1+\\sqrt{1-4\\lambda}}{2}\\nb 5+\\lambda \\nb 6\\Big\\rangle, \\Big\\langle\\nb 4+\\frac{2}{1+\\sqrt{1-4\\lambda}}\\nb 5+\\nb 6\\Big\\rangle,$ $\\Big\\langle\\frac{1\\pm\\sqrt{1-4\\lambda}}{2\\lambda}\\nb 5+\\nb 6\\Big\\rangle,$ the last two belonging to the family above.\n\t            \n\t            If $\\lambda=\\frac 14$, then we have the family of representatives of distinct orbits $\\langle\\nb 2+\\alpha\\nb 4+2\\nb 5+\\nb 6\\rangle$ and two separate representatives $\\langle\\nb 4+2\\nb 5+\\nb 6\\rangle$ and $\\langle 2\\nb 5+\\nb 6\\rangle$. \n\t            \n\t            If $\\lambda=0$, then we have $\\alpha_5=\\alpha_6$, so we obtain the family of representatives of distinct orbits $\\langle\\nb 2+\\alpha\\nb 4+\\nb 5+\\nb 6\\rangle$ and two separate representatives $\\langle\\nb 4+\\nb 5+\\nb 6\\rangle$ and $\\langle\\nb 5+\\nb 6\\rangle$.\n\t        \n\t        \\end{enumerate}\n\t        \n\t        \\item $\\alpha_3\\ne 0$ and $\\alpha_5=0$. Then $\\alpha^*_5=0$, $\\alpha^*_2$, $\\alpha^*_3$ and $\\alpha^*_6$ are as above and\n\t        \\begin{align*}\n\t            \\alpha^*_1 &= x(\\alpha_1x  + (\\alpha_3 + \\alpha_6)z),\\\\\n                \\alpha^*_4 &= x(\\alpha_4x + \\alpha_6u).\n\t        \\end{align*}\n\t        Choosing $u=-\\frac{\\alpha_4x}{\\alpha_6}$, we get $\\alpha^*_4=0$. Now we have two subcases:\n\t        \\begin{enumerate}\n\t            \\item $\\alpha_3 + \\alpha_6\\ne 0$. Then choosing $z=-\\frac{\\alpha_1x}{\\alpha_3 + \\alpha_6}$, we obtain two families of representatives of distinct orbits $\\langle \\nb 2+\\alpha\\nb 3+\\nb 6\\rangle_{\\alpha\\not\\in \\{0,-1\\}}$ and $\\langle \\alpha\\nb 3+\\nb 6\\rangle_{\\alpha\\not\\in \\{0,-1\\}}$.\n\t            \\item $\\alpha_3 + \\alpha_6=0$. Then choosing $z=0$, we have the family of representatives of distinct orbits $\\langle\\nb 1+\\alpha\\nb 2-\\nb 3+\\nb 6\\rangle$ and two separate representatives $\\langle\\nb 2-\\nb 3+\\nb 6\\rangle$ and $\\langle-\\nb 3+\\nb 6\\rangle$ which will be joined with the families $\\langle \\nb 2+\\alpha\\nb 3+\\nb 6\\rangle_{\\alpha\\not\\in \\{0,-1\\}}$ and $\\langle \\alpha\\nb 3+\\nb 6\\rangle_{\\alpha\\not\\in \\{0,-1\\}}$ found above.\n\t        \\end{enumerate}\n\t        \n\t        \\item $\\alpha_3=\\alpha_5=0$. Then $\\alpha^*_3=\\alpha^*_5=0$, $\\alpha^*_6$ is as above and \n\t        \\begin{align*}\n\t            \\alpha^*_1 &= x(\\alpha_1x + \\alpha_6z),\\\\\n                \\alpha^*_2 &= \\alpha_2x^2,\\\\\n                \\alpha^*_4 &= x(\\alpha_4x + \\alpha_6u).\n\t        \\end{align*}\n\t        Thus, choosing $z=-\\frac{\\alpha_1x}{\\alpha_6}$ and $u=-\\frac{\\alpha_4x}{\\alpha_6}$, we have two representatives depending on whether $\\alpha_2=0$ or not: $\\langle\\nb 2+\\nb 6\\rangle$ and $\\langle\\nb 6\\rangle$. They will be joined with the families $\\langle\\nb 2+\\alpha\\nb 5+\\nb 6\\rangle_{\\alpha \\neq 0, 1-\\alpha+\\lambda\\alpha^2\\ne 0}$ and $\\langle\\alpha\\nb 5+\\nb 6\\rangle_{\\alpha \\neq 0, 1-\\alpha+\\lambda\\alpha^2\\ne 0}$ found above.\n\t    \\end{enumerate}\n\t    \n\t    \\item $\\alpha_6^2 - \\alpha_6\\alpha_7 + \\lambda\\alpha_7^2=0$ and $\\alpha_5\\ne 0$ and $\\alpha_3^2 - \\alpha_3\\alpha_5 + \\lambda\\alpha_5^2\\ne 0$. Then choosing $x=-\\frac{(\\alpha_3 - \\alpha_5)y}{\\alpha_5}$, we get $\\alpha^*_5=0$. Since $(\\alpha^*_3)^2 - \\alpha^*_3\\alpha^*_5 + \\lambda(\\alpha^*_5)^2=(\\alpha_3^2 - \\alpha_3\\alpha_5 + \\lambda\\alpha_5^2)(x^2-xy+\\lambda y^2)^3\\ne 0$, we may assume that $\\alpha_5=0$ and $\\alpha_3\\ne 0$ from the very beginning. Then choosing $y=0$ and $z=-\\frac{\\alpha_2x+ \\alpha_3u}{\\alpha_7}$ we have $\\alpha^*_2=\\alpha^*_5=0$ and \n\t    \\begin{align*}\n\t        \\alpha^*_1 &= \\frac x{\\alpha_7}((\\alpha_1\\alpha_7- \\alpha_2\\alpha_3 - \\alpha_2\\alpha_6)x - (\\alpha_3^2  + \\alpha_3\\alpha_6 - \\alpha_6^2 + \\alpha_6\\alpha_7)u),\\\\\n            \\alpha^*_3 &= \\alpha_3x^3,\\\\\n            \\alpha^*_4 &= x(\\alpha_4x - (\\alpha_6 - \\alpha_7)u),\\\\\n            \\alpha^*_6 &= \\alpha_6x^3,\\\\\n            \\alpha^*_7 &= \\alpha_7x^3.\n\t    \\end{align*}\n\t    \\begin{enumerate}\n\t        \\item $\\alpha_6 - \\alpha_7\\ne 0$. Then choosing $u=\\frac{\\alpha_4x}{\\alpha_6 - \\alpha_7}$ we have $\\alpha^*_4=0$.\n\t        \n\t        If $\\lambda\\not\\in\\{0,\\frac 14\\}$, then we have four families of representatives of distinct orbits $\\Big\\langle \\nb 1+\\alpha\\nb 3+ \\frac{1\\pm\\sqrt{1-4\\lambda}}{2}\\nb 6+\\nb 7\\Big\\rangle_{\\alpha\\ne 0}$, $\\Big\\langle \\alpha\\nb 3+ \\frac{1\\pm\\sqrt{1-4\\lambda}}{2}\\nb 6+\\nb 7\\Big\\rangle_{\\alpha\\ne 0}$.\n\t        \n\t       \tIf $\\lambda=\\frac 14$, then $\\alpha_6^2 - \\alpha_6\\alpha_7 + \\lambda\\alpha_7^2=0$ implies $\\alpha_6=\\frac 12\\alpha_7$, so $\\alpha_6 - \\alpha_7\\ne 0$ is satisfied. Thus, we have two families of representatives of distinct orbits $\\langle\\nb 1+\\alpha\\nb 3+\\frac 12\\nb 6+\\nb 7\\rangle_{\\alpha\\ne 0}$ and $\\langle\\alpha\\nb 3+\\frac 12\\nb 6+\\nb 7\\rangle_{\\alpha\\ne 0}$.\n\t        \t        \n\t        If $\\lambda = 0$, then $\\alpha_6^2 - \\alpha_6\\alpha_7 + \\lambda\\alpha_7^2=0$ and $\\alpha_6 - \\alpha_7\\ne 0$ imply that $\\alpha_6=0$. Hence, we have two families of representatives of distinct orbits $\\langle\\nb 1+\\alpha\\nb 3+\\nb 7\\rangle_{\\alpha\\ne 0}$ and $\\langle\\alpha\\nb 3+\\nb 7\\rangle_{\\alpha\\ne 0}$.\n\t        \n\t        \\item $\\alpha_6 - \\alpha_7=0$. Then $\\alpha_6^2 - \\alpha_6\\alpha_7 + \\lambda\\alpha_7^2=0$ and $\\alpha_7\\ne 0$ imply that $\\lambda = 0$. Now,\n\t        \\begin{align*}\n\t        \\alpha^*_1 &= \\frac x{\\alpha_7}((\\alpha_1\\alpha_7- \\alpha_2\\alpha_3 - \\alpha_2\\alpha_7)x - \\alpha_3(\\alpha_3  + \\alpha_7)u),\\\\\n            \\alpha^*_3 &= \\alpha_3x^3,\\\\\n            \\alpha^*_4 &= \\alpha_4x^2,\\\\\n            \\alpha^*_6 &= \\alpha_7x^3,\\\\\n            \\alpha^*_7 &= \\alpha_7x^3.\n\t    \\end{align*}\n\t        \\begin{enumerate}\n\t            \\item $\\alpha_3  + \\alpha_7\\ne 0$. Then choosing $u=\\frac{(\\alpha_1\\alpha_7- \\alpha_2\\alpha_3 - \\alpha_2\\alpha_7)x}{\\alpha_3(\\alpha_3  + \\alpha_7)}$, we have $\\alpha^*_1=0$. Thus, we obtain two families of representatives of distinct orbits $\\langle\\alpha\\nb 3+\\nb 4+\\nb 6+\\nb 7\\rangle_{\\alpha\\ne 0,-1}$ and $\\langle\\alpha\\nb 3+\\nb 6+\\nb 7\\rangle_{\\alpha\\ne 0,-1}$.\n\t            \\item $\\alpha_3  + \\alpha_7=0$. Then we obtain the family of representatives of distinct orbits $\\langle\\nb 1-\\nb 3+\\alpha\\nb 4+\\nb 6+\\nb 7\\rangle$ and two separate representatives $\\langle-\\nb 3+\\nb 4+\\nb 6+\\nb 7\\rangle$ and  $\\langle-\\nb 3+\\nb 6+\\nb 7\\rangle,$ which will be joined with the families $\\langle\\alpha\\nb 3+\\nb 4+\\nb 6+\\nb 7\\rangle_{\\alpha\\ne 0,-1}$ and $\\langle\\alpha\\nb 3+\\nb 6+\\nb 7\\rangle_{\\alpha\\ne 0,-1}$ found above.\n\t        \\end{enumerate}\n\t    \\end{enumerate}\n\t    \\item $\\alpha_6^2 - \\alpha_6\\alpha_7 + \\lambda\\alpha_7^2=0$ and $\\alpha_5\\ne 0$ and $\\alpha_3^2 - \\alpha_3\\alpha_5 + \\lambda\\alpha_5^2=0$. Choosing $x,y$ such that $(x^2 - xy + \\lambda y^2)(\\alpha_7x + (\\alpha_6-\\alpha_7)y)=1$, we have $(\\alpha_3 - \\alpha_5)y + \\alpha_5x\\ne 0$, since otherwise $x^2 - xy + \\lambda y^2=(\\alpha_3^2 - \\alpha_3\\alpha_5 + \\lambda\\alpha_5^2)\\frac{y^2}{\\alpha_5^2}=0$. Now, the suitable value of $z$ gives $\\alpha^*_4=0$, so we shall assume $\\alpha_7=1$ and $\\alpha_4=0$ from the very beginning. The equality $\\alpha_6^2 - \\alpha_6\\alpha_7 + \\lambda\\alpha_7^2=0$ takes the form $\\alpha_6^2 - \\alpha_6 + \\lambda=0$, whence $\\lambda=\\alpha_6-\\alpha_6^2$. On the other hand, $\\alpha_3^2 - \\alpha_3\\alpha_5 + \\lambda\\alpha_5^2=0$ implies $\\lambda=\\frac{\\alpha_3\\alpha_5-\\alpha_3^2}{\\alpha_5^2}$. Therefore, $\\alpha_6-\\alpha_6^2=\\frac{\\alpha_3\\alpha_5-\\alpha_3^2}{\\alpha_5^2}$, i.e. $\\alpha_3^2-\\alpha_3\\alpha_5+\\alpha_5^2(\\alpha_6-\\alpha_6^2)=0$. This equation has two solutions in $\\alpha_3$, namely, $\\alpha_3=\\alpha_5\\alpha_6$ and $\\alpha_3=\\alpha_5(1-\\alpha_6)$.\n\t    \n\t    \\begin{enumerate}\n\t    \t\\item $\\alpha_3=\\alpha_5\\alpha_6$. Then\n\t    \t\\begin{align*}\n\t    \t\t\\alpha^*_2 &= (x + (\\alpha_6-1)y)z + \\alpha_5\\alpha_6(x + (\\alpha_6-1)y)u  + \\alpha_2x^2 + \\alpha_1xy - \\alpha_2xy,\\\\\n\t    \t\t\\alpha^*_4 &= \\alpha_5(x + (\\alpha_6-1)y)z + ((\\alpha_6 - \\alpha_5 - 1)x + (\\alpha_6^2  - \\alpha_5\\alpha_6  + \\alpha_5 - 2\\alpha_6 + 1)y)u\\\\\n\t    \t\t&\\quad + (\\alpha_1-\\alpha_2)xy  + (\\alpha_2 + \\alpha_2\\alpha_6^2 - \\alpha_1 - \\alpha_2\\alpha_6)y^2. \n\t    \t\\end{align*}\n\t    \tWe consider $\\alpha^*_2=\\alpha^*_4=0$ as a system of linear equations in $z$ and $u$. Its determinant is $(\\alpha_5 + 1)(\\alpha_6 - \\alpha_5\\alpha_6 - 1)( x + (\\alpha_6-1)y)^2$. So, we have the following cases:\n\t    \t\\begin{enumerate}\n\t    \t\t\\item $\\alpha_5 + 1\\ne 0$ and $\\alpha_6 - \\alpha_5\\alpha_6 - 1\\ne 0$. Then choosing $x,y$ such that $x + (\\alpha_6-1)y\\ne 0$, $x - \\alpha_6y\\ne 0$ and $z,u$ such that $\\alpha^*_2=\\alpha^*_4=0$ we obtain \n\t    \t\t\\begin{align*}\n\t    \t\t\t\\alpha^*_1 &= (\\alpha_1 - \\alpha_2\\alpha_6)(x - \\alpha_6y)^2,\\\\\n\t    \t\t\t\\alpha^*_3 &= \\alpha_5\\alpha_6(x - \\alpha_6y)(x + (\\alpha_6-1)y)^2,\\\\\n\t    \t\t\t\\alpha^*_5 &= \\alpha_5(x - \\alpha_6y)(x + (\\alpha_6-1)y)^2,\\\\\n\t    \t\t\t\\alpha^*_6 &= \\alpha_6(x - \\alpha_6y)(x + (\\alpha_6-1)y)^2,\\\\\n\t    \t\t\t\\alpha^*_7 &= (x - \\alpha_6y)(x + (\\alpha_6-1)y)^2.\n\t    \t\t\\end{align*}\n\t    \t\t\\begin{itemize}\n\t    \t\t    \\item Let $\\alpha_1 - \\alpha_2\\alpha_6\\ne 0$. Taking $y=0$ and $x=\\alpha_1 - \\alpha_2\\alpha_6$ we obtain the family of representatives $\\langle\\nb 1+\\alpha\\beta\\nb 3+\\alpha\\nb 5+\\beta\\nb 6+\\nb 7\\rangle$, where $\\alpha\\not\\in \\{0,-1\\}$, $\\beta-\\alpha\\beta-1\\ne 0$ and $\\beta^2 - \\beta + \\lambda=0$.\n\t    \t\t\n\t    \t\tIf $\\lambda\\not\\in\\{0,\\frac 14\\}$, then $\\beta=\\frac{1\\pm\\sqrt{1-4\\lambda}}2$. Observe that $\\beta\\ne 0$, so the condition $\\beta-\\alpha\\beta-1\\ne 0$ is equivalent to $\\alpha\\ne 1-\\frac 1\\beta=\\frac{\\sqrt{1-4\\lambda}\\mp 1}{\\sqrt{1-4\\lambda}\\pm 1}$. Thus, we obtain two families $\\Big\\langle\\nb 1+\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\alpha\\nb 3+\\alpha\\nb 5+\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle_{\\alpha\\not\\in\\left\\{0,-1,\\frac{\\sqrt{1-4\\lambda}\\mp 1}{\\sqrt{1-4\\lambda}\\pm 1}\\right\\}}$ of representatives of distinct orbits.\n\t    \t\t\n\t    \t\tIf $\\lambda=\\frac 14$, then $\\beta=\\frac 12$, and $\\alpha\\ne 1-\\frac 1\\beta$ becomes $\\alpha\\ne -1$. Thus, we obtain the family $\\langle\\nb 1+\\frac 12\\alpha\\nb 3+\\alpha\\nb 5+\\frac 12\\nb 6+\\nb 7\\rangle_{\\alpha\\not\\in\\{0,-1\\}}$ of representatives of distinct orbits.\n\t    \t\t\n\t    \t\tIf $\\lambda=0$, then $\\beta=0$ or $\\beta=1$. If $\\beta=0$, then the condition $\\beta-\\alpha\\beta-1\\ne 0$ becomes $-1\\ne 0$; and if $\\beta=1$, then it is $\\alpha\\ne 0$. Thus, we obtain two families $\\langle\\nb 1+\\alpha\\nb 5+\\nb 7\\rangle_{\\alpha\\not\\in\\{0,-1\\}}$ and $\\langle\\nb 1+\\alpha\\nb 3+\\alpha\\nb 5+\\nb 6+\\nb 7\\rangle_{\\alpha\\not\\in\\{0,-1\\}}$ of representatives of distinct orbits.\n\t    \t\t    \\item Let  $\\alpha_1 - \\alpha_2\\alpha_6=0$. Then we obtain the same families as in the previous case, but without $\\nb 1$.\n\t    \t\t\\end{itemize}\n\t   \n\t    \t\t\\item $\\alpha_6 - \\alpha_5\\alpha_6 - 1=0$. Then clearly $\\alpha_6\\ne 0$ and $\\alpha_5=1-\\frac 1{\\alpha_6}$, so $\\alpha_3=\\alpha_6-1$. Moreover, $\\alpha_6\\ne 1$ since $\\alpha_5\\ne 0$. In particular, $\\lambda\\ne 0$. Choosing $z$ such that $\\alpha^*_4=0$, we obtain\n\t    \t\t\\begin{align*}\n\t    \t\t\t\\alpha^*_1 &= \\frac{x - \\alpha_6y}{\\alpha_6 - 1}(\\alpha_1(\\alpha_6-1)x + \\alpha_6(\\alpha_2(2\\alpha_6^2 + 1) - \\alpha_6(\\alpha_1 + 2\\alpha_2))y),\\\\\n\t    \t\t\t\\alpha^*_2 &= \\frac{x - \\alpha_6y}{\\alpha_6 - 1}(\\alpha_2(\\alpha_6-1)x  + (\\alpha_2\\alpha_6(\\alpha_6-1) - \\alpha_1 + \\alpha_2)y),\\\\\n\t    \t\t\t\\alpha^*_3 &= (\\alpha_6 - 1)(x - \\alpha_6y)(x + (\\alpha_6-1)y)^2,\\\\\n\t    \t\t\t\\alpha^*_5 &= \\left(1 - \\frac 1{\\alpha_6}\\right)(x - \\alpha_6y)(x + (\\alpha_6-1)y)^2,\\\\\n\t    \t\t\t\\alpha^*_6 &= \\alpha_6(x - \\alpha_6y)(x + (\\alpha_6-1)y)^2,\\\\\n\t    \t\t\t\\alpha^*_7 &= (x - \\alpha_6y)(x + (\\alpha_6-1)y)^2.\n\t    \t\t\\end{align*}\n\t    \t\tLet $(\\alpha_1,\\alpha_2) =(0,0)$. Then we get the representative $\\Big\\langle(\\alpha-1)\\nb 3+\\left(1-\\frac 1{\\alpha}\\right)\\nb 5+\\alpha\\nb 6+\\nb 7\\Big\\rangle$, where $\\alpha^2 - \\alpha + \\lambda=0$.\n\t\t\t\t\n\t\t\t\tIf $\\lambda\\not\\in\\{0,\\frac 14\\}$, then we obtain two representatives $\\Big\\langle\\frac{-1\\pm\\sqrt{1-4\\lambda}}2\\nb 3+\\frac{\\sqrt{1-4\\lambda}\\mp 1}{\\sqrt{1-4\\lambda}\\pm 1}\\nb 5+\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7 \\Big\\rangle$ which will be joined with the families $\\Big\\langle\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\alpha\\nb 3+\\alpha\\nb 5+\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle_{\\alpha\\not\\in\\left\\{0,-1,\\frac{\\sqrt{1-4\\lambda}\\mp 1}{\\sqrt{1-4\\lambda}\\pm 1}\\right\\}}$.\n\t\t\t\t\n\t\t\t\tIf $\\lambda=\\frac 14$, then we obtain the representative $\\langle -\\frac 12\\nb 3-\\nb 5+\\frac 12\\nb 6+\\nb 7\\rangle$ which will be joined with the family $\\langle\\frac 12\\alpha\\nb 3+\\alpha\\nb 5+\\frac 12\\nb 6+\\nb 7\\rangle_{\\alpha\\not\\in\\{0,-1\\}}.$\n\t\t\t\n\t\t\t\t\n\t\t\t\tLet $(\\alpha_1,\\alpha_2) \\neq (0,0).$ If $\\alpha_2 =0,$ then  \n\t\t\t\t\\begin{align*}\n\t\t\t\t\\alpha^*_1 &= \\frac{x - \\alpha_6y}{\\alpha_6 - 1}((\\alpha_6-1)x - \\alpha_6^2y)\\alpha_1,\\\\\n\t\t\t\t\\alpha^*_2 &= -\\frac{x - \\alpha_6y}{\\alpha_6 - 1}y\\alpha_1,\n\t\t\t\t\\end{align*}\n\t\t\t\tso choosing $y \\neq 0$ we may suppose for the rest of the case that $\\alpha_2 \\neq 0.$\n\\begin{itemize}\n    \\item Let $\\alpha_1-\\alpha_2\\alpha_6\\ne 0$ and $\\alpha_1 - \\alpha_2 - 2\\alpha_2\\alpha_6^2 + 2\\alpha_2\\alpha_6\\ne 0$. Then choosing $z=\\frac{\\alpha_6(\\alpha_2\\alpha_6(\\alpha_6-1) - \\alpha_1 + \\alpha_2)y}{\\alpha_6 - 1}-(\\alpha_6-1)u$ and $x=\\frac{(\\alpha_1 - \\alpha_2 -\\alpha_2\\alpha_6(\\alpha_6-1))y}{\\alpha_2(\\alpha_6 - 1)}$ we obtain $\\alpha^*_2=\\alpha^*_4=0$ and \n\t    \t\t\\begin{align*}\n\t    \t\t\t\\alpha^*_1 &= (\\alpha_1 - \\alpha_2 - 2\\alpha_2\\alpha_6^2 + 2\\alpha_2\\alpha_6)^2(\\alpha_1 - \\alpha_2\\alpha_6)\\frac{y^2}{\\alpha_2^2(\\alpha_6 - 1)^2},\\\\\n\t    \t\t\t\\alpha^*_3 &= (\\alpha_1 - \\alpha_2 - 2\\alpha_2\\alpha_6^2 + 2\\alpha_2\\alpha_6)(\\alpha_1 - \\alpha_2\\alpha_6)^2\\frac{y^3}{\\alpha_2^3(\\alpha_6 - 1)^2},\\\\\n\t    \t\t\t\\alpha^*_5 &= (\\alpha_1 - \\alpha_2 - 2\\alpha_2\\alpha_6^2 + 2\\alpha_2\\alpha_6)(\\alpha_1 - \\alpha_2\\alpha_6)^2\\frac{y^3}{\\alpha_2^3\\alpha_6(\\alpha_6 - 1)^2},\\\\\n\t    \t\t\t\\alpha^*_6 &= (\\alpha_1 - \\alpha_2 - 2\\alpha_2\\alpha_6^2 + 2\\alpha_2\\alpha_6)(\\alpha_1 - \\alpha_2\\alpha_6)^2\\frac{\\alpha_6y^3}{\\alpha_2^3(\\alpha_6 - 1)^3},\\\\\n\t    \t\t\t\\alpha^*_7 &= (\\alpha_1 - \\alpha_2 - 2\\alpha_2\\alpha_6^2 + 2\\alpha_2\\alpha_6)(\\alpha_1 - \\alpha_2\\alpha_6)^2\\frac{y^3}{\\alpha_2^3(\\alpha_6 - 1)^3}.\n\t    \t\t\\end{align*}\n\t    \t\tChoosing $y=\\frac{\\alpha_2(\\alpha_6 - 1)(\\alpha_1 - \\alpha_2 - 2\\alpha_2\\alpha_6^2 + 2\\alpha_2\\alpha_6)}{\\alpha_1 - \\alpha_2\\alpha_6}$ we obtain the  representative $\\Big\\langle\\nb 1+(\\alpha-1)\\nb 3+\\left(1-\\frac 1{\\alpha}\\right)\\nb 5+\\alpha\\nb 6+\\nb 7\\Big\\rangle$, where $\\alpha^2 - \\alpha + \\lambda=0$.\n\t    \t\t\n\t    \t\tIf $\\lambda\\not\\in\\{0,\\frac 14\\}$, then $\\alpha=\\frac{1\\pm\\sqrt{1-4\\lambda}}2$, so $\\alpha-1=\\frac{-1\\pm\\sqrt{1-4\\lambda}}2$ and $1-\\frac 1{\\alpha}=\\frac{\\sqrt{1-4\\lambda}\\mp 1}{\\sqrt{1-4\\lambda}\\pm 1}$, and we obtain the following two representatives $\\Big\\langle \\nb 1+\\frac{-1\\pm\\sqrt{1-4\\lambda}}2\\nb 3+\\frac{\\sqrt{1-4\\lambda}\\mp 1}{\\sqrt{1-4\\lambda}\\pm 1}\\nb 5+\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7 \\Big\\rangle$ which will be joined with the families $\\Big\\langle\\nb 1+\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\alpha\\nb 3+\\alpha\\nb 5+\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle_{\\alpha\\not\\in\\left\\{0,-1,\\frac{\\sqrt{1-4\\lambda}\\mp 1}{\\sqrt{1-4\\lambda}\\pm 1}\\right\\}}$ found above.\n\t    \t\t\n\t    \t\tIf $\\lambda=\\frac 14$, then $\\alpha=\\frac 12$, so $\\alpha-1=-\\frac 12$ and $1-\\frac 1{\\alpha}=-1$, and we obtain the representative $\\langle\\nb 1-\\frac 12\\nb 3-\\nb 5+\\frac 12\\nb 6+\\nb 7\\rangle$ which will be joined with the family $\\langle\\nb 1+\\frac 12\\alpha\\nb 3+\\alpha\\nb 5+\\frac 12\\nb 6+\\nb 7\\rangle_{\\alpha\\not\\in\\{0,-1\\}}$ found above.\n    \\item Let $\\alpha_1 - \\alpha_2 - 2\\alpha_2\\alpha_6^2 + 2\\alpha_2\\alpha_6=0$. Then $\\alpha_1=(2\\alpha_6^2 - 2\\alpha_6 + 1)\\alpha_2$.\n\t   \n\t   \n\t \n\t\n\t\n\n\n\t\n\t\n\t    \t\t\n\t\n\t    \t\t\n\t\n\t   \n\t  \n\tIn this case $\\alpha_3^*, \\alpha_5^*, \\alpha_6^*, \\alpha_7^*$ are as above and\n\t\t\\begin{align*}\n\t    \t\t\\alpha^*_1 &= \\alpha_2 (x - y \\alpha_6)^2 (2\\alpha_6^2 - 2\\alpha_6 + 1)\\\\\n\t    \t\t\\alpha^*_2 &=\\alpha_2 (x - y \\alpha_6)^2.\n\t    \t\t\\end{align*}\n\tTherefore, we obtain the representative $\\Big\\langle (2 \\alpha^2-2\\alpha + 1)\\nabla_1+\\nabla_2+ (\\alpha-1)\\nabla_3 + (1-\\frac{1}{\\alpha})\\nabla_5 +\\alpha\\nabla_6+\\nabla_7\\Big\\rangle$, where $\\alpha^2 - \\alpha + \\lambda=0$. Note that $2 \\alpha^2-2\\alpha + 1 = 1- 2\\lambda.$\n\t\n\tIf $\\lambda\\not\\in\\{0,\\frac 14\\},$ then we obtain two representatives $\\Big\\langle (1-2\\lambda)\\nabla_1 + \\nabla_2 +\\frac{-1+\\sqrt{1-4\\lambda}}2\\nb 3+\\frac{\\sqrt{1-4\\lambda}- 1}{\\sqrt{1-4\\lambda}+ 1}\\nb 5+\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7 \\Big\\rangle, \\Big\\langle (1-2\\lambda)\\nabla_1 + \\nabla_2 +\\frac{-1-\\sqrt{1-4\\lambda}}2\\nb 3+\\frac{\\sqrt{1-4\\lambda}+ 1}{\\sqrt{1-4\\lambda}- 1}\\nb 5+\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7 \\Big\\rangle = \\Big\\langle (1-2\\lambda)\\lambda\\nabla_1 + \\lambda\\nabla_2 -\\lambda\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 3-\\Big(\\frac{\\sqrt{1-4\\lambda}+ 1}{2}\\Big)^2\\nb 5+\\lambda\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 6+\\lambda\\nb 7 \\Big\\rangle$.\n\t\n\t\tIf $\\lambda=\\frac 14$, then we obtain the representative $\\langle \\frac 12\\nb 1 + \\nabla_2 -\\frac 12\\nb 3-\\nb 5+\\frac 12\\nb 6+\\nb 7\\rangle$.\n\t\n    \\item \tLet $\\alpha_1-\\alpha_2\\alpha_6=0$. We may assume that $\\alpha_6\\ne\\frac 12$ (and hence $\\lambda\\ne\\frac 14$), since otherwise $\\alpha_1=(2\\alpha_6^2 - 2\\alpha_6 + 1)\\alpha_2$, which has already been considered.\n    \n   \n\n\n\n\n\n\n\n\t\t\t\t\n\n\t\t\t\t\n\n\t\t\t\t\n\n\t\t\t\t\n\tIn this case $\\alpha_3^*, \\alpha_5^*, \\alpha_6^*, \\alpha_7^*$ are as above and\n\t\t\\begin{align*}\n\t    \t\t\\alpha^*_1 &= \\alpha_2\\alpha_6 (x - y \\alpha_6)(x+(\\alpha_6 - 1)y)\\\\\n\t    \t\t\\alpha^*_2 &=\\alpha_2 (x - y \\alpha_6)(x+(\\alpha_6 - 1)y).\n\t    \t\t\\end{align*}\n\tTherefore, we obtain the representative $\\Big\\langle \\alpha\\nabla_1+\\nabla_2+ (\\alpha-1)\\nabla_3 + (1-\\frac{1}{\\alpha})\\nabla_5 +\\alpha\\nabla_6+\\nabla_7\\Big\\rangle$, where $\\alpha^2 - \\alpha + \\lambda=0$. \n\t\n\t\tIf $\\lambda\\not\\in\\{0,\\frac 14\\},$ then we obtain two representatives $\\Big\\langle \\frac{1+\\sqrt{1-4\\lambda}}2\\nabla_1 + \\nabla_2 +\\frac{-1+\\sqrt{1-4\\lambda}}2\\nb 3+\\frac{\\sqrt{1-4\\lambda}- 1}{\\sqrt{1-4\\lambda}+1}\\nb 5+\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7 \\Big\\rangle, \\Big\\langle \\frac{1-\\sqrt{1-4\\lambda}}2\\nabla_1 + \\nabla_2 -\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 3+\\frac{\\sqrt{1-4\\lambda}+ 1}{\\sqrt{1-4\\lambda}- 1}\\nb 5+\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7 \\Big\\rangle = \\Big\\langle \\lambda\\frac{1-\\sqrt{1-4\\lambda}}2\\nabla_1 + \\lambda\\nabla_2 -\\lambda\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 3-\\Big(\\frac{\\sqrt{1-4\\lambda}+ 1}{2}\\Big)^2\\nb 5+\\lambda\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 6+\\lambda\\nb 7 \\Big\\rangle$\n\t\n\t    \n\t   \n\t\n\t\n\n\\end{itemize}\n\t\t\t\t\\item $\\alpha_5 + 1=0$. Then $\\alpha_3=-\\alpha_6$. Observe that we may assume that $\\alpha_6\\ne\\frac 12$ (and hence $\\lambda\\ne\\frac 14$), since otherwise $\\alpha_6 - \\alpha_5\\alpha_6 - 1=0,$ which was considered above. Choosing $z$ such that $\\alpha^*_4=0$, we have\n\t\t\t\t\\begin{align*}\n\t\t\t\t\t\\alpha^*_1 &= \\alpha_1x^2 + 2\\alpha_2\\alpha_6(\\alpha_6 - 1)xy + \\alpha_6(\\alpha_6-1)(\\alpha_1 - \\alpha_2)y^2,\\\\\n\t\t\t\t\t\\alpha^*_2 &= \\alpha_2x^2 + 2(\\alpha_1 - \\alpha_2)xy + (\\alpha_2\\alpha_6(\\alpha_6-1) - \\alpha_1 + \\alpha_2)y^2,\\\\\n\t\t\t\t\t\\alpha^*_3 &= -\\alpha_6(x - \\alpha_6y)(x + (\\alpha_6-1)y)^2,\\\\\n\t\t\t\t\t\\alpha^*_5 &= -(x - \\alpha_6y)(x + (\\alpha_6-1)y)^2,\\\\\n\t\t\t\t\t\\alpha^*_6 &= \\alpha_6(x - \\alpha_6y)(x + (\\alpha_6-1)y)^2,\\\\\n\t\t\t\t\t\\alpha^*_7 &= (x - \\alpha_6y)(x + (\\alpha_6-1)y)^2.\n\t\t\t\t\\end{align*}\n\t\t\t\t\n\t\t\t\tIf $(\\alpha_1,\\alpha_2)=(0,0)$, then we obtain the representative $\\langle-\\beta\\nb 3-\\nb 5+\\beta\\nb 6+\\nb 7\\rangle$, where $\\beta^2 - \\beta + \\lambda=0$.\n\t\t\t\t\n\t\t\t\tIf $\\lambda\\not\\in\\{0,\\frac 14\\}$, then we have two representatives $\\Big\\langle\\frac{-1\\mp\\sqrt{1-4\\lambda}}2\\nb 3-\\nb 5+\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle$ which will be joined with the families $\\Big\\langle\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\alpha\\nb 3+\\alpha\\nb 5+\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle_{\\alpha\\ne 0,-1}$ found above.\n\t\t\t\t\n\t\t\t\t\n\t\t\t\tIf $\\lambda=0$, then we have two representatives $\\langle-\\nb 3-\\nb 5+\\nb 6+\\nb 7\\rangle$ and $\\langle-\\nb 5+\\nb 7\\rangle$ which will be joined the families $\\langle\\alpha\\nb 3+\\alpha\\nb 5+\\nb 6+\\nb 7\\rangle_{\\alpha\\ne 0,-1}$ and $\\langle\\alpha\\nb 5+\\nb 7\\rangle_{\\alpha\\ne 0,-1}$ found above.\n\t\t\t\t\n\t\t\t\tLet $(\\alpha_1,\\alpha_2) \\neq (0,0).$ Suppose that $\\alpha_2=0$. Then we have $\\alpha^*_2 =  \\alpha_1 y (2x - y).$ So, whenever $(\\alpha_1,\\alpha_2) \\neq (0,0)$, we may find $x,y$ such that $\\alpha^*_2\\neq 0$ and $\\alpha^*_7=1$. In the rest of the case we will suppose that $\\alpha_2 \\neq 0.$\n\t\t\t\t\n\t\t\t\tThe determinant of the equation $\\alpha^*_2=0$ is $4(\\alpha_2\\alpha_6 + \\alpha_1 - \\alpha_2)(\\alpha_1-\\alpha_2\\alpha_6)$.\n\t\t\t\t\n\t\t\t\t\\begin{itemize}\n\t\t\t\t    \\item Let $(\\alpha_2\\alpha_6 + \\alpha_1 - \\alpha_2)(\\alpha_1-\\alpha_2\\alpha_6)\\ne 0.$ \n\t\t\t\t    \n\t\t\t\t    Suppose that $\\alpha_2 = 2\\alpha_1.$ Observe that in this case $(\\alpha_2\\alpha_6 + \\alpha_1 - \\alpha_2)(\\alpha_2\\alpha_6 - \\alpha_1)\\ne 0$ is equivalent to $\\alpha_1^2(2\\alpha_6 - 1)^2\\ne 0$. Then $\\alpha_2^* - 2\\alpha_1^* = -2\\alpha_1(2x-y)y(2\\alpha_6-1)^2,$ so choosing appropriate $x,y$ we may suppose that $\\alpha_2\\ne 2\\alpha_1.$\n\t\t\t\t    \n\t\t\t\t    The equation $\\alpha^*_2=0$ has two  solutions $x_1=\\mu_1 y$ and $x_2=\\mu_2 y$, where $\\mu_1,\\mu_2\\in{\\mathbb C}$, $\\mu_1\\ne\\mu_2$. As in the case 1(a) if $\\mu_1^2 - \\mu_1 + \\lambda=\\mu_2^2 - \\mu_2 + \\lambda=0$, then $\\mu_1+\\mu_2=1$. But $\\mu_1+\\mu_2=-\\frac{2(\\alpha_1 - \\alpha_2)}{\\alpha_2}$, whence $2(\\alpha_1 - \\alpha_2)=-\\alpha_2$, which contradicts the assumption that $\\alpha_2\\ne 2\\alpha_1$. Thus, we may choose $x=\\mu_iy$ to make $\\alpha^*_2=0$. Since in this case the condition $(\\alpha_1,\\alpha_2,\\alpha_4)\\ne (0,0,0)$ is invariant under automorphisms, we have $\\alpha^*_1\\ne 0$ for such a choice of $x$. Thus, we obtain the family of representatives $\\langle\\alpha\\nb 1-\\beta\\nb 3-\\nb 5+\\beta\\nb 6+\\nb 7\\rangle$, where $\\beta^2 - \\beta + \\lambda=0$ and $\\alpha\\ne 0$. Then taking $y=0$ and $x=\\alpha$, we obtain the representative $\\langle\\nb 1-\\beta\\nb 3-\\nb 5+\\beta\\nb 6+\\nb 7\\rangle.$  It will be joined with the family $\\langle\\nb 1+\\alpha\\beta\\nb 3+\\alpha\\nb 5+\\beta\\nb 6+\\nb 7\\rangle_{\\alpha\\not\\in \\{0,-1\\}}.$\n\t\t\t\t\\item \tLet $\\alpha_1-\\alpha_2\\alpha_6=0$. Then\n\t\t\t\t\\begin{align*}\n\t\t\t\t\\alpha^*_1 =\\alpha_2\\alpha_6(x + (\\alpha_6-1)y)^2,\n\t\t\t\t\\alpha^*_2 =\\alpha_2(x + (\\alpha_6-1)y)^2.\n\t\t\t\t\\end{align*}\n\t\t\t \tSo, taking $x=\\alpha_6y+\\alpha_2$, we obtain the representative $\\langle\\beta\\nb 1+\\nb 2-\\beta\\nb 3-\\nb 5+\\beta\\nb 6+\\nb 7\\rangle$, where $\\beta^2 - \\beta + \\lambda=0$. If $\\lambda\\not\\in\\{0,\\frac 14\\}$, then we have two representatives $\\Big\\langle\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\nb 1+\\nb 2-\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\nb 3-\\nb 5+\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle$.\n\t\t\t \tIf $\\lambda=0$, then we have two representatives $\\langle\\nb 1+\\nb 2-\\nb 3-\\nb 5+\\nb 6+\\nb 7\\rangle$ and $\\langle\\nb 2-\\nb 5+\\nb 7\\rangle$.\n\t\t\t \t\\item \t\tLet $\\alpha_2\\alpha_6 + \\alpha_1 - \\alpha_2=0$. Then\n\t\t\t \t\\begin{align*}\n\t\t\t \t\\alpha^*_1 =\\alpha_2(1-\\alpha_6)(x - \\alpha_6y)^2,\n\t\t\t \t\\alpha^*_2 =\\alpha_2(x - \\alpha_6y)^2.\n\t\t\t \t\\end{align*}\n\t\t\t \tTaking $x,y$ such that $(x + (\\alpha_6-1)y)^2=\\alpha_2(x - \\alpha_6y)\\ne 0$, we obtain the representative $\\langle(1-\\beta)\\nb 1+\\nb 2-\\beta\\nb 3-\\nb 5+\\beta\\nb 6+\\nb 7\\rangle$, where $\\beta^2 - \\beta + \\lambda=0$. If $\\lambda\\not\\in\\{0,\\frac 14\\}$, then we have two representatives $\\Big\\langle\\frac{1\\mp\\sqrt{1-4\\lambda}}2\\nb 1+\\nb 2-\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\nb 3-\\nb 5+\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle$. \n\t\t\t \n\t\t\t \tIf $\\lambda=0$, then we have two representatives $\\langle\\nb 2-\\nb 3-\\nb 5+\\nb 6+\\nb 7\\rangle$ and $\\langle\\nb 1+\\nb 2-\\nb 5+\\nb 7\\rangle$.\n\t\t\t\t\\end{itemize}\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\t\t\n\t\t\n\t\t\t \t\n\t    \t\\end{enumerate}\n    \t\n    \t\t\\item $\\alpha_3=\\alpha_5(1-\\alpha_6)$. Observe that we may assume that $\\alpha_6\\ne\\frac 12$ (and hence $\\lambda\\ne\\frac 14$), since otherwise $\\alpha_3=\\alpha_5\\alpha_6$, which has already been considered.\n    \t\t\n  \t\tConsider $\\alpha^*_2=\\alpha^*_4=0$ as a system of linear equations in $z$ and $u$. Then its determinant is the following polynomial in $x$ and $y$:\n    \t\t\\begin{align*}\n    \t\t\tD(x,y)&=(\\alpha_5^2\\alpha_6 - \\alpha_5^2 - \\alpha_5 + \\alpha_6 - 1)x^2\\\\\n    \t\t\t&\\quad-(2\\alpha_5^2\\alpha_6^2 - 2\\alpha_5^2\\alpha_6 - 2\\alpha_6^2 - \\alpha_5 + 4\\alpha_6 - 2)xy\\\\\n    \t\t\t&\\quad+(\\alpha_5^2\\alpha_6^2 + \\alpha_5\\alpha_6 + \\alpha_6^2 - 2\\alpha_6 + 1)(\\alpha_6 - 1)y^2.\n    \t\t\\end{align*}\n    \t\t\n    \t\tSuppose that $\\alpha_5^2\\alpha_6 - \\alpha_5^2 - \\alpha_5 + \\alpha_6 - 1 = 0.$ In this case $\\alpha_5^2+1\\ne 0$, since otherwise $(\\alpha_5^2+1)\\alpha_6 = \\alpha_5^2 + \\alpha_5 + 1$ would imply $\\alpha_5=0$, whence $\\alpha_5^2+1=1$, a contradiction. Thus, $\\alpha_6=\\frac{\\alpha_5^2 + \\alpha_5 + 1}{\\alpha_5^2+1}$ and hence $\\alpha_3=-\\frac{\\alpha_5^2}{\\alpha_5^2+1}$. Then $D(x,y) = \\frac{y \\alpha_5 (1 + \\alpha_5)^3 ((1-\\alpha_5)x + \\alpha_5y )}{(1 + \\alpha_5^2)^2}.$ If $\\alpha_5+1=0,$ then $\\alpha_5=-1$, $\\alpha_6=\\frac 12$, $\\alpha_3=-\\frac 12=\\alpha_5\\alpha_6$. This case has been considered above. Therefore, we may suppose that $D(x,y)$ is not identically zero, and hence we may solve $\\alpha^*_2=\\alpha^*_4=0$ in $z$ and $u$. Choosing these values of $z$ and $u$, we may suppose that $\\alpha_2 = \\alpha_4 = 0$ from the very beginning.\n    \t\t\nThen taking the appropriate values of $z$ and $u$ we have \t\n\\begin{align*}  \t       \n                    \\alpha^*_1&=\\frac{\\alpha_1(\\alpha_5^2\\alpha_6 - \\alpha_5^2 - \\alpha_5 + \\alpha_6 - 1)(x + y(\\alpha_6-1))^2(x-y\\alpha_6)^2}{D(x,y)},\\\\\t\n                    \\alpha^*_3&=\\alpha_5 (1 - \\alpha_6) (x - y \\alpha_6)^2 (x + y(\\alpha_6-1)),\\\\\n    \t\t\t\t\\alpha^*_5&=\\alpha_5 (x - y \\alpha_6)^2 (x + y(\\alpha_6-1)),\\\\\n    \t\t\t\t\\alpha^*_6&=\\alpha_6 (x - y \\alpha_6) (x + y(\\alpha_6-1))^2,\\\\\n    \t\t\t\t\\alpha^*_7&=(x - y \\alpha_6) (x + y(\\alpha_6-1))^2.\n    \t\t\t\\end{align*}\n    \t\t\t\n    \t\tObserve that for $x,y$ such that $\\alpha_7^* = 1$ and $D(x,y)\\ne 0$ we have $(\\alpha_5^*)^2\\alpha_6^* - (\\alpha_5^*)^2 - \\alpha_5^* + \\alpha_6^* - 1 = \\frac{D(x,y)}{(x + y(\\alpha_6-1))^2}\\ne 0,$ so we may assume that $\\alpha_5^2\\alpha_6 - \\alpha_5^2 - \\alpha_5 + \\alpha_6 - 1 \\neq 0$ from the very beginning.\n    \t\t\n    \t\tLet $\\alpha_5 \\neq -1$.  Taking $x = \\frac{y (\\alpha_6(\\alpha_5-1)+1)}{\\alpha_5+1},$ we have $ D(x,y) = \\frac{y^2 \\alpha_5^2 (2 \\alpha_6-1)^2}{(\\alpha_5+1)^2}\\ne 0$, $x^2 - xy + \\lambda y^2 = -\\frac{y^2 \\alpha_5 (2 \\alpha_6-1)^2}{(\\alpha_5+1)^2}\\ne 0$, $\\alpha_5^* = -\\alpha_7^*$, so we get the representative $\\langle \\alpha \\nabla_1 + (\\beta-1)\\nabla_3 - \\nabla_5 + \\beta \\nabla_6 + \\nabla_7 \\rangle_{\\beta^2 - \\beta + \\lambda=0},$ where\n            \\[\n            \\alpha = -\\frac{\\alpha_1 (\\alpha_5+1) (\\alpha_5^2\\alpha_6 - \\alpha_5^2 - \\alpha_5 + \\alpha_6 - 1)}{y \\alpha_5^2 (2 \\alpha_6-1)^2}\n            \\]\n            \n    \\begin{itemize}\n        \\item  If $\\alpha_1 \\neq 0,$ then we take $y = -\\frac{\\alpha_1 (\\alpha_5+1) (\\alpha_5^2\\alpha_6 - \\alpha_5^2 - \\alpha_5 + \\alpha_6 - 1)}{\\alpha_5^2 (2 \\alpha_6-1)^2} \\neq 0,$ so that $x^2 - xy + \\lambda y^2 \\neq 0$, $D(x,y) \\neq 0$, and we get the representative $\\langle \\nabla_1 + (\\alpha-1)\\nabla_3 - \\nabla_5 + \\alpha \\nabla_6 + \\nabla_7 \\rangle_{\\alpha^2 - \\alpha + \\lambda=0}.$\n        \n            If $\\lambda \\neq 0$, then we get two distinct representatives $\\Big\\langle \\nb 1 - \\frac{1\\mp\\sqrt{1-4\\lambda}}2\\nb 3-\\nb 5+\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle.$\n            \n            If $\\lambda = 0$, then we get two distinct representatives $\\Big\\langle \\nb 1 - \\nb 3-\\nb 5 +\\nb 7\\Big\\rangle$ and $\\Big\\langle \\nb 1 -\\nb 5+ \\nb 6+\\nb 7\\Big\\rangle.$\n            \n            \\item If $\\alpha_1 = 0,$ then we get the same representatives but without $\\nabla_1.$\n    \\end{itemize}       \n    \t\t\n\n\n               \n \n    \t\t\t\n  \n    \t\t\t\n   \n  \n            \n\n  \n            \n   \n            \n      \n            \n     \n            \n   \n\n            \n      \n            \n      \n      \n          \n   \n      \n            \n \n \n  \n  \n  \n\n \n           \n\n\n\n\n    \t\t\t\n \n    \t\t\t\n    \t\t\t\n  \n    \t\t\t\n \n    \t\t\t\n    \t\n    \t\n   \n    \t\n\n    \t\t\n\n    \t\t\n \n    \t\t\n \n   \n   \n   \n    \t        \n    \n    \n    \t\n   \n    \t\t\n   \n    \t\t\n \n\n\n \n   \n   \n    \t\t\n   \n   \n   \n   \n   \n    \t\t\n    \t\t\t\n    \t\t\n    \n    \t\n    \t\t\t\n    \n    \t\t\t\n    \n    \t\t\t\n    \t\n    \t\t\t\n    \t\t\n    \n    \t\t\t\n    \n    \t\t\t\n    \t\n    \t\t\t\n    \t\n    \t\t\t\n    \n     \n    \t   \\end{enumerate}\n\t    \n\t    \\item $\\alpha_6^2 - \\alpha_6\\alpha_7 + \\lambda\\alpha_7^2=0$ and $\\alpha_5=0$. We may assume that $\\alpha_3=0$, since the case $\\alpha_5=0$ and $\\alpha_3\\ne 0$ was considered in 1(b). Then $\\alpha^*_3=\\alpha^*_5=0$. A suitable choice of $x,y$ and $z$ gives $\\alpha^*_7=1$ and $\\alpha^*_2=0$, so we shall assume that $\\alpha_7=1$ and $\\alpha_2=0$. As above, $\\lambda=\\alpha_6-\\alpha_6^2$. Now, choosing $z=-\\frac{\\alpha_1xy + (\\alpha_4\\alpha_6^2 - \\alpha_4\\alpha_6)y^2}{x+(\\alpha_6 - 1)y}$ we obtain $\\alpha^*_2=0$.\n\t    \n\t    \\begin{enumerate}\n\t    \t\\item $\\alpha_6\\ne 1$. Then choosing $u=-\\frac{\\alpha_4x^2 + (\\alpha_1 - 2\\alpha_4)xy - (\\alpha_1 - \\alpha_4)y^2}{(\\alpha_6 - 1)(x + (\\alpha_6 - 1)y)}$ we have $\\alpha^*_4=0$ and\n\t    \t\\begin{align*}\n\t    \t\t\\alpha^*_1&=(\\alpha_1-\\alpha_4\\alpha_6)(x-\\alpha_6y)^2,\\\\\n\t    \t\t\\alpha^*_6 &= \\alpha_6(x - \\alpha_6y)(x + (\\alpha_6-1)y)^2,\\\\\n\t    \t\t\\alpha^*_7 &= (x - \\alpha_6y)(x + (\\alpha_6-1)y)^2.\n\t    \t\\end{align*}\n\t    \t\\begin{enumerate}\n\t    \t\t\\item $\\alpha_1-\\alpha_4\\alpha_6\\ne 0$. Then choosing $y=0$ and $x=\\alpha_1-\\alpha_4\\alpha_6$ we obtain the representative $\\langle\\nb 1+\\alpha\\nb 6+\\nb 7\\rangle$, where $\\alpha^2 - \\alpha + \\lambda=0$. If $\\lambda\\not\\in\\{0,\\frac 14\\}$, then we have two representatives $\\Big\\langle\\nb 1+\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle$. We will join them with the families $\\Big\\langle\\nb 1+\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\alpha\\nb 3+\\alpha\\nb 5+\\frac{1\\pm\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle_{\\alpha\\ne 0}$found above. If $\\lambda=\\frac 14$, then we have the representative $\\langle\\nb 1+\\frac 12\\nb 6+\\nb 7\\rangle$, which will be joined with the family $\\langle\\nb 1+\\frac 12\\alpha\\nb 3+\\alpha\\nb 5+\\frac 12\\nb 6+\\nb 7\\rangle_{\\alpha\\ne 0}$ found above. If $\\lambda=0$, then $\\alpha_6=0$, since $\\alpha_6\\ne 1$ by assumption. So, we have the representative $\\langle\\nb 1+\\nb 7\\rangle$. It will be joined with $\\langle\\nb 1+\\alpha\\nb 5+\\nb 7\\rangle_{\\alpha\\ne 0}$.\n\t    \t\t\n\t    \t\t\\item $\\alpha_1-\\alpha_4\\alpha_6=0$. Then we have the same representatives as above, but without $\\nb 1$. We join them with the families found above. \n\t    \t\\end{enumerate}\n    \t\n    \t\t\\item $\\alpha_6=1$, so that $\\lambda =0$. Then \n    \t\t\\begin{align*}\n    \t\t\t\\alpha^*_1 &=\\alpha_1x(x - y),\\\\\n    \t\t\t\\alpha^*_4 &=(\\alpha_4(x-y) + \\alpha_1y)(x - y),\\\\\n    \t\t\t\\alpha^*_6 &= x^2(x - y),\\\\\n    \t\t\t\\alpha^*_7 &= x^2(x - y).\n    \t\t\\end{align*}\n    \t\t\\begin{enumerate}\n    \t\t\t\\item $\\alpha_1\\ne 0$ and $\\alpha_1-\\alpha_4\\ne 0$. Then choosing $x=\\alpha_1$ and $y=-\\frac{\\alpha_1\\alpha_4}{\\alpha_1-\\alpha_4}$ we obtain the representative $\\langle\\nb 1+\\nb 6+\\nb 7\\rangle$ which will be joined with the family $\\langle\\nb 1+\\alpha\\nb 3+\\alpha\\nb 5+\\nb 6+\\nb 7\\rangle_{\\alpha\\ne 0}$ found above.\n    \t\t\t\n    \t\t\t\\item $\\alpha_1=0$. Then we have two representatives $\\langle\\nb 4+\\nb 6+\\nb 7\\rangle$ and $\\langle\\nb 6+\\nb 7\\rangle$ depending on whether $\\alpha_4=0$ or not. Both of them belong to the families found above.\n    \t\t\t\n    \t\t\t\\item $\\alpha_1-\\alpha_4=0$. Then\n    \t\t\t\\begin{align*}\n    \t\t\t\\alpha^*_1 &=\\alpha_1x(x - y),\\\\\n    \t\t\t\\alpha^*_4 &=\\alpha_1x(x - y),\\\\\n    \t\t\t\\alpha^*_6 &= x^2(x - y),\\\\\n    \t\t\t\\alpha^*_7 &= x^2(x - y).\n    \t\t\t\\end{align*}\n    \t\t\tThus, we have two representatives $\\langle\\nb 1+\\nb 4+\\nb 6+\\nb 7\\rangle$ and $\\langle\\nb 6+\\nb 7\\rangle$. The second one was found above.\n    \t\t\\end{enumerate}\n\t    \\end{enumerate}\n\t\\end{enumerate}\n\t\n        \\item $\\alpha_7=0$. We may assume that $\\alpha_6=0$, since the case $\\alpha_7=0$ and $\\alpha_6\\ne 0$ was considered in 1(a).\n        \\begin{enumerate}\n            \\item Let $\\alpha_3^2 - \\alpha_3\\alpha_5 + \\lambda\\alpha_5^2 \\neq 0.$ If $\\alpha_5 \\neq 0,$ we may take $x = \\frac{(\\alpha_5 - \\alpha_3)y}{\\alpha_5}$ and get $\\alpha^*_5=0$. Observe that $(\\alpha^*_3)^2 - \\alpha^*_3\\alpha^*_5 + \\lambda(\\alpha^*_5)^2=(\\alpha_3^2 - \\alpha_3\\alpha_5 + \\lambda\\alpha_5^2)(x^2 - xy + \\lambda y^2)^3$, so the condition $\\alpha_3^2 - \\alpha_3\\alpha_5 + \\lambda\\alpha_5^2\\ne 0$ is invariant under the automorphisms. Thus, we shall assume that $\\alpha_5=0$ and $\\alpha_3\\ne 0$ from the very beginning. Choosing $y=0$, we have\n            \\begin{align*}\n\\alpha^*_1 &= x(\\alpha_1x + \\alpha_3z),\\\\\n\\alpha^*_2 &= x(\\alpha_2x + \\alpha_3u),\\\\\n\\alpha^*_3 &= \\alpha_3x^3,\\\\\n\\alpha^*_4 &= \\alpha_4x^2.\n\\end{align*}\nTaking $z = -\\frac{\\alpha_1x}{\\alpha_3}, u =  -\\frac{\\alpha_2x}{\\alpha_3},$ we get two representatives $\\langle \\nabla_3 + \\nabla_4 \\rangle$ and $\\langle \\nabla_3 \\rangle$ depending on whether $\\alpha_4=0$ or not. \n    \\item Let $\\alpha_3^2 - \\alpha_3\\alpha_5 + \\lambda\\alpha_5^2 = 0, \\alpha_5 \\neq 0.$ Taking $x,y$ such that $(x^2-xy+\\lambda y^2)(\\alpha_5x + (\\alpha_3-\\alpha_5)y) = 1,$ we may suppose that $\\alpha_5^* = 1$ and $\\lambda = \\alpha_3 - \\alpha_3^2.$ Consider $\\alpha_2^* = 0, \\alpha_4^* = 0$ as a linear system in $u, z.$ Its determinant is $\\alpha_3(x + (\\alpha_3-1)y)^2.$ If $\\alpha_3 = 0,$ then $\\lambda = 0$ and we get a Leibniz cocycle. Therefore, we may suppose that this determinant is nonzero and the system $\\alpha_2^* = 0, \\alpha_4^* = 0$ has a unique solution. We get the family of representatives $\\langle \\alpha_1\\nabla_1 + \\alpha_3\\nabla_3 + \\nabla_5 \\rangle,$ where $\\alpha_3^2 - \\alpha_3 + \\lambda = 0,$ except for $(\\lambda,\\alpha_3) = (0,0)$ which gives a Leibniz cocycle. Therefore, we may suppose that $\\alpha_2 = \\alpha_4 = 0$ from the very beginning and\n    \\begin{align*}\n    \\alpha^*_1 &= \\alpha_1(x-\\alpha_3y)^2,\\\\\n    \\alpha^*_3 &= \\alpha_3(x-\\alpha_3y)(x+(\\alpha_3-1)y)^2,\\\\\n    \\alpha^*_5 &= (x-\\alpha_3y)(x+(\\alpha_3-1)y)^2.\n    \\end{align*}\n    If $\\lambda\\not\\in\\{0,\\frac 14\\}$, then we have the following representatives: $\\Big\\langle \\nabla_1 +  \\frac{1\\pm\\sqrt{1-4\\lambda}}{2}\\nabla_3 + \\nabla_5 \\Big\\rangle$ and $\\Big\\langle \\frac{1\\pm\\sqrt{1-4\\lambda}}{2}\\nabla_3 + \\nabla_5 \\Big\\rangle$. If $\\lambda=\\frac 14$, then we have the representatives $\\langle\\nb 1+\\frac 12\\nb 3+\\nb 5\\rangle$ and $\\langle\\frac 12\\nb 3+\\nb 5\\rangle$.  If $\\lambda=0$, then we have the representatives $\\langle \\nabla_1 + \\nabla_3 + \\nabla_5 \\rangle$ and  $\\langle \\nabla_3 + \\nabla_5 \\rangle$. \n        \\end{enumerate}\n    \\end{enumerate}\n\n\t\\begin{landscape}\n\t\\tiny{\n\t\\begin{tabular}{|c|c|c|}\n\t    \\hline\n\t    $\\lambda=0$                      & $\\lambda=\\frac 14$                & $\\lambda\\not\\in\\{0,\\frac 14\\}$\\\\\n\t    \\hline\n\t    $\\begin{array}{l}\n\t       \\langle\\nb 1+\\alpha\\nb 3+\\beta\\nb 5+\\nb 6\\rangle_{\\alpha,\\beta\\ne 0},\\\\\n\t       \\langle\\alpha\\nb 3+\\beta\\nb 5+\\nb 6\\rangle_{\\alpha,\\beta\\ne 0},\\\\\n\t       \\langle\\nb 2+\\alpha\\nb 5+\\nb 6\\rangle_{\\alpha\\ne 1},\\\\\n\t       \\langle\\alpha\\nb 5+\\nb 6\\rangle,\\\\\n\t       \\langle\\nb 2+\\alpha\\nb 4+\\nb 5+\\nb 6\\rangle,\\\\\n\t       \\langle\\nb 4+\\nb 5+\\nb 6\\rangle,\\\\\n\t       \\langle \\nb 2+\\alpha\\nb 3+\\nb 6\\rangle_{\\alpha\\ne 0},\\\\\n\t       \\langle \\alpha\\nb 3+\\nb 6\\rangle_{\\alpha\\ne 0},\\\\\n\t       \\langle\\nb 1+\\alpha\\nb 2-\\nb 3+\\nb 6\\rangle,\\\\\n\t       \\langle\\nb 1+\\alpha\\nb 3+\\nb 7\\rangle_{\\alpha\\ne 0},\\\\\n\t       \\langle\\alpha\\nb 3+\\nb 7\\rangle_{\\alpha\\ne 0},\\\\\n\t       \\langle\\alpha\\nb 3+\\nb 4+\\nb 6+\\nb 7\\rangle,\\\\\n\t       \\langle\\alpha\\nb 3+\\nb 6+\\nb 7\\rangle,\\\\\n\t       \\langle\\nb 1-\\nb 3+\\alpha\\nb 4+\\nb 6+\\nb 7\\rangle,\\\\\n\t       \\langle\\nb 1+\\alpha\\nb 5+\\nb 7\\rangle,\\\\\n\t       \\langle\\nb 1+\\alpha\\nb 3+\\alpha\\nb 5+\\nb 6+\\nb 7\\rangle,\\\\\n\t       \\langle\\alpha\\nb 5+\\nb 7\\rangle,\\\\\n\t       \\langle\\alpha\\nb 3+\\alpha\\nb 5+\\nb 6+\\nb 7\\rangle_{\\alpha\\ne 0},\\\\\n\t       \\langle\\nb 1+\\nb 2-\\nb 3-\\nb 5+\\nb 6+\\nb 7\\rangle,\\\\\n\t       \\langle\\nb 2-\\nb 5+\\nb 7\\rangle,\\\\\n\t       \\langle\\nb 2-\\nb 3-\\nb 5+\\nb 6+\\nb 7\\rangle,\\\\\n\t       \\langle\\nb 1+\\nb 2-\\nb 5+\\nb 7\\rangle,\\\\\n\t       \\langle\\nb 1-\\nb 3-\\nb 5+\\nb 7\\rangle,\\\\\n\t       \\langle\\nb 1-\\nb 5+\\nb 6+\\nb 7\\rangle,\\\\\n\t       \\langle-\\nb 3-\\nb 5+\\nb 7\\rangle,\\\\\n\t       \\langle-\\nb 5+\\nb 6+\\nb 7\\rangle,\\\\\n\t       \\langle\\nb 1+\\nb 4+\\nb 6+\\nb 7\\rangle,\\\\\n\t       \\langle \\nb 3 + \\nb 4 \\rangle,\\\\\n\t       \\langle \\nb 3 \\rangle,\\\\\n\t       \\langle \\nb 1 + \\nb 3 + \\nb 5 \\rangle,\\\\\n\t       \\langle \\nb 3 + \\nb 5 \\rangle\n\t    \\end{array}$\n\t    &\n\t    $\\begin{array}{l}\n\t         \\langle\\nb 1+\\alpha\\nb 3+\\beta\\nb 5+\\nb 6\\rangle_{\\alpha,\\beta\\ne 0},\\\\\n\t         \\langle\\alpha\\nb 3+\\beta\\nb 5+\\nb 6\\rangle_{\\alpha,\\beta\\ne 0},\\\\\n\t         \\langle\\nb 2+\\alpha\\nb 5+\\nb 6\\rangle_{\\alpha\\ne 2},\\\\\n\t         \\langle\\alpha\\nb 5+\\nb 6\\rangle,\\\\\n\t         \\langle\\nb 2+\\alpha\\nb 4+2\\nb 5+\\nb 6\\rangle,\\\\\n\t         \\langle\\nb 4+2\\nb 5+\\nb 6\\rangle,\\\\\n\t         \\langle \\nb 2+\\alpha\\nb 3+\\nb 6\\rangle_{\\alpha\\ne 0},\\\\\n\t         \\langle \\alpha\\nb 3+\\nb 6\\rangle_{\\alpha\\ne 0},\\\\\n\t         \\langle\\nb 1+\\alpha\\nb 2-\\nb 3+\\nb 6\\rangle,\\\\\n\t         \\langle\\nb 1+\\alpha\\nb 3+\\frac 12\\nb 6+\\nb 7\\rangle_{\\alpha\\ne 0},\\\\\n\t         \\langle\\alpha\\nb 3+\\frac 12\\nb 6+\\nb 7\\rangle_{\\alpha\\ne 0},\\\\\n\t         \\langle\\nb 1+\\frac 12\\alpha\\nb 3+\\alpha\\nb 5+\\frac 12\\nb 6+\\nb 7\\rangle,\\\\\n\t         \\langle\\frac 12\\alpha\\nb 3+\\alpha\\nb 5+\\frac 12\\nb 6+\\nb 7\\rangle,\\\\\n\t         \\langle \\frac 12\\nb 1+ \\nb 2 -\\frac 12\\nb 3-\\nb 5+\\frac 12\\nb 6+\\nb 7\\rangle,\\\\\n\t       \n\t         \\langle \\nb 3 + \\nb 4 \\rangle,\\\\\n\t         \\langle \\nb 3 \\rangle,\\\\\n\t         \\langle\\nb 1+\\frac 12\\nb 3+\\nb 5\\rangle,\\\\\n\t         \\langle\\frac 12\\nb 3+\\nb 5\\rangle\n\t    \\end{array}$\n\t    &\n\t   $\\begin{array}{l}\n\t       \\langle\\nb 1+\\alpha\\nb 3+\\beta\\nb 5+\\nb 6\\rangle_{\\alpha,\\beta\\ne 0},\\\\\n\t       \\langle\\alpha\\nb 3+\\beta\\nb 5+\\nb 6\\rangle_{\\alpha,\\beta\\ne 0},\\\\\n\t       \\langle\\nb 2+\\alpha\\nb 5+\\nb 6\\rangle_{\\alpha\\ne\\frac{1\\pm\\sqrt{1-4\\lambda}}{2\\lambda}},\\\\\n\t       \\langle\\alpha\\nb 5+\\nb 6\\rangle,\\\\\n\t       \\Big\\langle \\lambda\\nb 2+\\lambda\\alpha\\nb 4+\\frac{1+\\sqrt{1-4\\lambda}}{2}\\nb 5+\\lambda\\nb 6\\Big\\rangle,\\\\\n\t      \n\t       \\Big\\langle\\nb 2+\\alpha\\nb 4+\\frac{2}{1+\\sqrt{1-4\\lambda}}\\nb 5+\\nb 6\\Big\\rangle,\\\\\n\t      \n\t       \\Big\\langle\\lambda \\nb 4+\\frac{1+\\sqrt{1-4\\lambda}}{2}\\nb 5+\\lambda \\nb 6\\Big\\rangle,\\\\\n\t       \\Big\\langle\\nb 4+\\frac{2}{1+\\sqrt{1-4\\lambda}}\\nb 5+\\nb 6\\Big\\rangle,\\\\\n\t       \\langle \\nb 2+\\alpha\\nb 3+\\nb 6\\rangle_{\\alpha\\ne 0},\\\\\n\t       \\langle \\alpha\\nb 3+\\nb 6\\rangle_{\\alpha\\ne 0},\\\\\n\t       \\langle\\nb 1+\\alpha\\nb 2-\\nb 3+\\nb 6\\rangle,\\\\\n\t       \\Big\\langle \\nb 1+\\alpha\\nb 3+ \\frac{1+\\sqrt{1-4\\lambda}}{2}\\nb 6+\\nb 7\\Big\\rangle_{\\alpha\\ne 0},\\\\\n\t       \\Big\\langle \\nb 1+\\alpha\\nb 3+ \\frac{1-\\sqrt{1-4\\lambda}}{2}\\nb 6+\\nb 7\\Big\\rangle_{\\alpha\\ne 0},\\\\\n\t       \\Big\\langle \\alpha\\nb 3+ \\frac{1+\\sqrt{1-4\\lambda}}{2}\\nb 6+\\nb 7\\Big\\rangle_{\\alpha\\ne 0},\\\\\n\t       \\Big\\langle \\alpha\\nb 3+ \\frac{1-\\sqrt{1-4\\lambda}}{2}\\nb 6+\\nb 7\\Big\\rangle_{\\alpha\\ne 0},\\\\\t       \n\t       \\Big\\langle\\nb 1+\\frac{1+\\sqrt{1-4\\lambda}}2\\alpha\\nb 3+\\alpha\\nb 5+\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle,\\\\\n\t       \\Big\\langle\\nb 1+\\frac{1-\\sqrt{1-4\\lambda}}2\\alpha\\nb 3+\\alpha\\nb 5+\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle,\\\\\n\t       \\Big\\langle\\frac{1+\\sqrt{1-4\\lambda}}2\\alpha\\nb 3+\\alpha\\nb 5+\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle,\\\\\n\t       \\Big\\langle\\frac{1-\\sqrt{1-4\\lambda}}2\\alpha\\nb 3+\\alpha\\nb 5+\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle,\\\\\n\t       \\Big\\langle (1-2\\lambda)\\nb 1+ \\nb 2 + \\frac{-1+\\sqrt{1-4\\lambda}}2\\nb 3+\\frac{\\sqrt{1-4\\lambda}- 1}{\\sqrt{1-4\\lambda}+ 1}\\nb 5+\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7 \\Big\\rangle,\\\\\n\t       \\Big\\langle (1-2\\lambda)\\lambda\\nabla_1 + \\lambda\\nabla_2 -\\lambda\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 3-\\Big(\\frac{\\sqrt{1-4\\lambda}+ 1}{2}\\Big)^2\\nb 5+\\lambda\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 6+\\lambda\\nb 7 \\Big\\rangle\\\\\n\t      \n\t       \\Big\\langle \\frac{1+\\sqrt{1-4\\lambda}}2\\nb 1+ \\nb 2 + \\frac{-1+\\sqrt{1-4\\lambda}}2\\nb 3+\\frac{\\sqrt{1-4\\lambda}- 1}{\\sqrt{1-4\\lambda}+ 1}\\nb 5+\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7 \\Big\\rangle,\\\\\n\t       \\Big\\langle \\lambda\\frac{1-\\sqrt{1-4\\lambda}}2\\nabla_1 + \\lambda\\nabla_2 -\\lambda\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 3-\\Big(\\frac{\\sqrt{1-4\\lambda}+ 1}{2}\\Big)^2\\nb 5+\\lambda\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 6+\\lambda\\nb 7 \\Big\\rangle\\\\\n\t      \n\t       \\Big\\langle\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 1+\\nb 2-\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 3-\\nb 5+\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle,\\\\\n\t       \\Big\\langle\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 1+\\nb 2-\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 3-\\nb 5+\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle,\\\\\n\t       \\Big\\langle\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 1+\\nb 2-\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 3-\\nb 5+\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle,\\\\\n\t       \\Big\\langle\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 1+\\nb 2-\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 3-\\nb 5+\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle,\\\\\n\t       \\Big\\langle\\nb 1-\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 3-\\nb 5+\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle,\\\\\n\t       \\Big\\langle\\nb 1-\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 3-\\nb 5+\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle,\\\\\n\t       \\Big\\langle-\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 3-\\nb 5+\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle,\\\\\n\t       \\Big\\langle-\\frac{1+\\sqrt{1-4\\lambda}}2\\nb 3-\\nb 5+\\frac{1-\\sqrt{1-4\\lambda}}2\\nb 6+\\nb 7\\Big\\rangle,\\\\\n\t       \\langle \\nb 3 + \\nb 4 \\rangle,\\\\\n\t       \\langle \\nb 3 \\rangle,\\\\\n\t       \\Big\\langle \\nb 1 + \\frac{1+\\sqrt{1-4\\lambda}}{2}\\nb 3 + \\nb 5 \\Big\\rangle,\\\\\n\t       \\Big\\langle \\nb 1 + \\frac{1-\\sqrt{1-4\\lambda}}{2}\\nb 3 + \\nb 5 \\Big\\rangle,\\\\\n\t       \\Big\\langle \\frac{1+\\sqrt{1-4\\lambda}}{2}\\nb 3 + \\nb 5 \\Big\\rangle,\\\\\n\t       \\Big\\langle \\frac{1-\\sqrt{1-4\\lambda}}{2}\\nb 3 + \\nb 5 \\Big\\rangle.\n\t       \\end{array}$\n\t    \\\\\n\t    \\hline\n\t\\end{tabular}\n\t}\n \n\n\t\\end{landscape}\n\n\n\\newpage \n\n\n\nDenote $\\Theta=\\frac{1+\\sqrt{1-4\\lambda}}{2}.$ The orbits above correspond to the following algebras:\t\n\n{\\tiny\n\\[\\begin{array}{lllllllllll}\n\n\n\n        \\D{4}{01}(\\lambda,\\alpha,\\beta)&:& e_1 e_1 = \\lambda e_3 + e_4, & e_1 e_3 = \\alpha e_4, & e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = \\beta e_4, & e_3e_1 = e_4; \\\\\n\t    \\D{4}{02}(\\lambda,\\alpha,\\beta)&:& e_1 e_1 = \\lambda e_3, & e_1 e_3 = \\alpha e_4, & e_2 e_1=e_3 & e_2 e_2 = e_3, & e_2 e_3 = \\beta e_4, & e_3e_1 = e_4; \\\\\n\t    \\D{4}{03}(\\lambda,\\alpha)&:& e_1 e_1 = \\lambda e_3, & e_1 e_2 = e_4, & e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = \\alpha e_4, & e_3e_1 = e_4; \\\\\n\t    \\D{4}{04}(\\lambda,\\alpha)&:& e_1 e_1 = \\lambda e_3, & e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = \\alpha e_4, & e_3e_1 = e_4; \\\\\n\t    \\D{4}{05}(\\lambda,\\alpha)&:& e_1 e_1 = \\lambda e_3, & e_1 e_2 = \\lambda e_4, & e_2 e_1=e_3 + \\lambda\\alpha e_4, & e_2 e_2 = e_3, & e_2 e_3 = \t\\Theta e_4, & e_3e_1 = \\lambda e_4; \\\\\n\t   \n\t    \\D{4}{06}(\\lambda,\\alpha)&:& e_1 e_1 = \\lambda e_3, & e_1 e_2 = e_4, & e_2 e_1=e_3 + \\alpha e_4, & e_2 e_2 = e_3, & e_2 e_3 = \t\\Theta^{-1} e_4, & e_3e_1 = e_4; \\\\\n\t   \n\t    \\D{4}{07}(\\lambda)&:& e_1 e_1 = \\lambda e_3, & e_2 e_1=e_3 + \\lambda e_4, & e_2 e_2 = e_3, & e_2 e_3 = \t\\Theta e_4, & e_3e_1 = \\lambda e_4; \\\\\n\t   \n\t    \\D{4}{08}(\\lambda)&:& e_1 e_1 = \\lambda e_3, &  e_2 e_1=e_3 + e_4, & e_2 e_2 = e_3, & e_2 e_3 = \t\\Theta^{-1} e_4, & e_3e_1 = e_4; \\\\\n\t   \n\t    \\D{4}{09}(\\lambda,\\alpha)&:& e_1 e_1 = \\lambda e_3, & e_1 e_2 = e_4,& e_1 e_3 = \\alpha e_4, & e_2 e_1=e_3, & e_2 e_2 = e_3, & e_3e_1 = e_4; \\\\\n\t    \\D{4}{10}(\\lambda,\\alpha)&:& e_1 e_1 = \\lambda e_3,& e_1 e_3 = \\alpha e_4, &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_3e_1 = e_4; \\\\\n        \\D{4}{11}(\\lambda,\\alpha)&:& e_1 e_1 = \\lambda e_3 + e_4,& e_1e_2 = \\alpha e_4, & e_1 e_3 = -e_4, &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_3e_1 = e_4; \\\\\n        \\D{4}{12}(\\lambda,\\alpha)&:& e_1 e_1 = \\lambda e_3 + e_4,& e_1e_3 = \\alpha e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_3e_1 = \\Theta e_4, & e_3e_2 = e_4; \\\\\n        \\D{4}{13}(\\lambda,\\alpha)&:& e_1 e_1 = \\lambda e_3 + e_4,& e_1e_3 = \\alpha e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_3e_1 = (1-\\Theta)e_4, & e_3e_2 = e_4; \\\\\n        \\D{4}{14}(\\lambda,\\alpha)&:& e_1 e_1 = \\lambda e_3,& e_1e_3 = \\alpha e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_3e_1 = \\Theta e_4, & e_3e_2 = e_4; \\\\\n        \\D{4}{15}(\\lambda,\\alpha)&:& e_1 e_1 = \\lambda e_3,& e_1e_3 = \\alpha e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_3e_1 = (1-\\Theta)e_4, & e_3e_2 = e_4; \\\\\n        \\D{4}{16}(\\alpha)&:& e_1e_3 = \\alpha e_4,  &  e_2 e_1=e_3 + e_4, & e_2 e_2 = e_3, & e_3e_1 = e_4, & e_3e_2 = e_4; \\\\\n        \\D{4}{17}(\\alpha)&:& e_1 e_1 = e_4, & e_1e_3 = -e_4,  &  e_2 e_1=e_3 + \\alpha e_4, & e_2 e_2 = e_3, & e_3e_1 = e_4, & e_3e_2 = e_4; \\\\\n        \\D{4}{18}(\\lambda,\\alpha)&:& e_1 e_1 = \\lambda e_3 + e_4,& e_1e_3 = \\Theta\\alpha e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = \\alpha e_4, & e_3e_1 = \\Theta e_4,\\\\\n        && e_3e_2 = e_4; \\\\\n        \\D{4}{19}(\\lambda,\\alpha)&:& e_1 e_1 = \\lambda e_3 + e_4,& e_1e_3 = (1-\\Theta)\\alpha e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = \\alpha e_4, & e_3e_1 = (1-\\Theta)e_4,\n        \\\\&& e_3e_2 = e_4; \\\\\n        \\D{4}{20}(\\lambda,\\alpha)&:& e_1 e_1 = \\lambda e_3,& e_1e_3 = \\Theta\\alpha e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = \\alpha e_4, & e_3e_1 = \\Theta e_4,\n        \\\\&& e_3e_2 = e_4; \\\\\n        \\D{4}{21}(\\lambda,\\alpha)&:& e_1 e_1 = \\lambda e_3,& e_1e_3 = (1-\\Theta)\\alpha e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = \\alpha e_4, & e_3e_1 = (1-\\Theta)e_4, \\\\&& e_3e_2 = e_4; \\\\\n        \\D{4}{22}(\\lambda)&:& e_1 e_1 = \\lambda e_3 + (1-2\\lambda)e_4,& e_1 e_2 = e_4,& e_1e_3 = (\\Theta - 1) e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = (1-\\Theta^{-1}) e_4,\n        \\\\&& e_3e_1 = \\Theta e_4, & e_3e_2 = e_4; \\\\\n        \\D{4}{23}(\\lambda)&:& e_1 e_1 = \\lambda e_3 + \\lambda(1-2\\lambda) e_4& e_1 e_2 = \\lambda e_4& e_1e_3 = -\\lambda\\Theta e_4 &  e_2 e_1=e_3 & e_2 e_2 = e_3 & e_2 e_3 = -\\Theta^2 e_4, \n        \\\\&& e_3e_1 = \\lambda(1-\\Theta)e_4 & e_3e_2 = \\lambda e_4; \\\\\n        \\D{4}{24}(\\lambda)&:& e_1 e_1 = \\lambda e_3 + \\Theta e_4,& e_1 e_2 = e_4,& e_1e_3 = (\\Theta - 1) e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = (1-\\Theta^{-1}) e_4, \n        \\\\&& e_3e_1 = \\Theta e_4, & e_3e_2 = e_4; \\\\\n        \\D{4}{25}(\\lambda)&:& e_1 e_1 = \\lambda e_3 + \\lambda(1-\\Theta)e_4,& e_1 e_2 = \\lambda e_4,& e_1e_3 = -\\lambda\\Theta e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = -\\Theta^2 e_4, \n        \\\\&& e_3e_1 = \\lambda (1-\\Theta)e_4, & e_3e_2 = \\lambda e_4; \\\\\n        \\D{4}{26}(\\lambda)&:& e_1 e_1 = \\lambda e_3 + \\Theta e_4,& e_1 e_2 = e_4,& e_1e_3 = -\\Theta e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = -e_4,\n        \\\\&& e_3e_1 = \\Theta e_4, & e_3e_2 = e_4; \\\\\n        \\D{4}{27}(\\lambda)&:& e_1 e_1 = \\lambda e_3 + (1-\\Theta)e_4,& e_1 e_2 = e_4,& e_1e_3 = (\\Theta-1) e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = -e_4,\n        \\\\&& e_3e_1 = (1-\\Theta)e_4, & e_3e_2 = e_4; \\\\\n        \\D{4}{28}(\\lambda)&:& e_1 e_1 = \\lambda e_3 + (1-\\Theta)e_4,& e_1 e_2 = e_4,& e_1e_3 = -\\Theta e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = -e_4, \n        \\\\&& e_3e_1 = \\Theta e_4, & e_3e_2 = e_4; \\\\\n        \\D{4}{29}(\\lambda)&:& e_1 e_1 = \\lambda e_3 + \\Theta e_4,& e_1 e_2 = e_4,& e_1e_3 = (\\Theta-1) e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = -e_4, \n        \\\\&& e_3e_1 = (1-\\Theta)e_4, & e_3e_2 = e_4; \\\\\n        \\D{4}{30}(\\lambda)&:& e_1 e_1 = \\lambda e_3 + e_4, & e_1e_3 = (\\Theta-1) e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = -e_4, & e_3e_1 = \\Theta e_4,\n        \\\\&& e_3e_2 = e_4; \\\\\n        \\D{4}{31}(\\lambda)&:& e_1 e_1 = \\lambda e_3 + e_4, & e_1e_3 = -\\Theta e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = -e_4, & e_3e_1 = (1-\\Theta)e_4,\n        \\\\&& e_3e_2 = e_4; \\\\\n        \\D{4}{32}(\\lambda)&:& e_1 e_1 = \\lambda e_3, & e_1e_3 = (\\Theta-1) e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = -e_4, & e_3e_1 = \\Theta e_4, \n        \\\\&& e_3e_2 = e_4; \\\\\n        \\D{4}{33}(\\lambda)&:& e_1 e_1 = \\lambda e_3, & e_1e_3 = -\\Theta e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3, & e_2 e_3 = -e_4, & e_3e_1 = (1-\\Theta)e_4, \n        \\\\&& e_3e_2 = e_4; \\\\\n       \\D{4}{34}&:& e_1 e_1 = e_4, &  e_2 e_1=e_3 + e_4, & e_2 e_2 = e_3, & e_3 e_1 = e_4,& e_3 e_2 = e_4; \\\\\n        \\D{4}{35}(\\lambda)&:& e_1 e_1 = \\lambda e_3, & e_1e_3 = e_4,  &  e_2 e_1=e_3 + e_4, & e_2 e_2 = e_3; \\\\\n        \\D{4}{36}(\\lambda)&:& e_1 e_1 = \\lambda e_3, & e_1e_3 = e_4,  &  e_2 e_1=e_3, & e_2 e_2 = e_3; \\\\\n        \\D{4}{37}(\\lambda)&:& e_1 e_1 = \\lambda e_3 + e_4, & e_1e_3 = \\Theta e_4, &  e_2 e_1=e_3, & e_2 e_2 = e_3,&  e_2 e_3 = e_4; \\\\\n        \\D{4}{38}(\\lambda)&:& e_1 e_1 = \\lambda e_3 + e_4, & e_1e_3 = (1-\\Theta)e_4, &  e_2 e_1=e_3, & e_2 e_2 = e_3,&  e_2 e_3 = e_4; \\\\\n        \\D{4}{39}(\\lambda)&:& e_1 e_1 = \\lambda e_3, & e_1e_3 = \\Theta e_4, &  e_2 e_1=e_3, & e_2 e_2 = e_3,&  e_2 e_3 = e_4; \\\\\n        \\D{4}{40}(\\lambda)&:& e_1 e_1 = \\lambda e_3, & e_1e_3 = (1-\\Theta)e_4, &  e_2 e_1=e_3, & e_2 e_2 = e_3,&  e_2 e_3 = e_4.\n\\end{array}\\]}\n\n\nThe algebras above are pairwise non-isomorphic, except for {\\tiny \\begin{gather*}\n\\D{4}{01}(\\lambda,0,\\beta) \\cong \\D{4}{02}(\\lambda,0,\\beta) \\cong \\D{4}{04}(\\lambda,\\beta),\\quad \\D{4}{01}(\\lambda,\\alpha,0)_{\\alpha \\neq -1} \\cong \\D{4}{02}(\\lambda,\\alpha,0) \\cong \\D{4}{10}(\\lambda,\\alpha),\\quad \\D{4}{01}(\\lambda,-1,0) \\cong \\D{4}{11}(\\lambda,0),\\quad\\\\\n\\D{4}{03}(\\lambda,0) \\cong \\D{4}{09}(\\lambda,0),\\quad \\D{4}{03}\\left(\\lambda,(1-\\Theta)^{-1}\\right)_{\\lambda \\neq 0} \\cong \\D{4}{05}(\\lambda,0)_{\\lambda \\neq 0}, \\D{4}{03}\\left(\\lambda,\\Theta^{-1}\\right)\\cong \\D{4}{06}(\\lambda,0),\\quad \\D{4}{04}(\\lambda,0) \\cong \\D{4}{10}(\\lambda,0),\\quad\\\\\n\\D{4}{05}(1\/4,\\alpha) \\cong \\D{4}{06}(1\/4,\\alpha),\\quad \\D{4}{07}(1\/4) \\cong \\D{4}{08}(1\/4),\\quad\\\\\n\\D{4}{05}(0,\\alpha) \\cong \\D{4}{07}(0) \\cong \\D{4}{23}(0) \\cong  \\D{4}{25}(0) \\cong  \\D{4}{40}(0),\\quad\\\\\n\\D{4}{12}(\\lambda,0) \\cong \\D{4}{18}(\\lambda,0),\\quad  \\D{4}{12}(1\/4,\\alpha) \\cong \\D{4}{13}(1\/4,\\alpha),\\quad \\D{4}{12}(0,\\alpha)_{\\alpha \\neq -1} \\cong \\D{4}{14}(0,\\alpha),\\quad \\D{4}{12}(0,-1) \\cong \\D{4}{17}(0),\\quad\\\\\n\\D{4}{13}(\\lambda,0) \\cong \\D{4}{19}(\\lambda,0),\\quad \\D{4}{14}(\\lambda,0) \\cong \\D{4}{20}(\\lambda,0),\\quad \\D{4}{14}(1\/4,\\alpha) \\cong \\D{4}{15}(1\/4,\\alpha),\\quad \\D{4}{15}(\\lambda,0) \\cong \\D{4}{21}(\\lambda,0),\\quad\\\\\n\\D{4}{18}(1\/4,\\alpha) \\cong \\D{4}{19}(1\/4,\\alpha),\\quad \\D{4}{18}(0,0) \\cong \\D{4}{22}(0) \\cong  \\D{4}{24}(0),\\quad \\D{4}{18}(1\/4,-1) \\cong \\D{4}{19}(1\/4,-1) \\cong \\D{4}{30}(1\/4) \\cong \\D{4}{31}(1\/4),\\quad\\\\\n\\D{4}{20}(1\/4,\\alpha) \\cong \\D{4}{21}(1\/4,\\alpha),\\quad \\D{4}{20}(1\/4,-1) \\cong \\D{4}{21}(1\/4,-1) \\cong \\D{4}{32}(1\/4) \\cong \\D{4}{33}(1\/4),\\quad\\\\\n\\D{4}{22}(1\/4) \\cong \\D{4}{23}(1\/4) \\cong \\D{4}{24}(1\/4) \\cong \\D{4}{25}(1\/4) \\cong \\D{4}{26}(1\/4) \\cong \\D{4}{27}(1\/4) \\cong \\D{4}{28}(1\/4) \\cong \\D{4}{29}(1\/4),\\quad\\\\\n \\D{4}{37}(1\/4) \\cong \\D{4}{38}(1\/4),\\quad \\D{4}{39}(1\/4) \\cong \\D{4}{40}(1\/4).\n\\end{gather*}}\n\nMoreover, the algebras $\\D{4}{05}(0,\\alpha) \\cong \\D{4}{07}(0) \\cong \\D{4}{23}(0) \\cong  \\D{4}{25}(0) \\cong  \\D{4}{40}(0), \\D{4}{38}(0)$ are Leibniz.\n\t\n\n\n\n\n\\newpage \n\n\n\\subsubsection{$1$-dimensional  central extensions of $\\T {3*}{05}$}\\label{ext-T_05^3*}\n\tLet us use the following notations \n\t\\begin{align*}\n\t\\nb 1 = \\Dl 13, \\nb 2 = \\Dl 21, \\nb 3 = \\Dl 22 - 3\\Dl 31.    \n\t\\end{align*}\n\tTake $\\theta=\\sum_{i=1}^3\\alpha_i\\nb i\\in {\\rm H_T^2}(\\T {3*}{05}).$\n\tIf \n\t$$\n\t\\phi=\n\t\\begin{pmatrix}\n\tx &    0  &  0\\\\\n\ty &  x^2  &  0\\\\\n\tz &   xy  &  x^3\n\t\\end{pmatrix}\\in\\aut{\\T {3*}{05}},\n\t$$\n\tthen\n\t$$\n\t\\phi^T\\begin{pmatrix}\n\t0      &  0    & \\alpha_1\\\\\n\t\\alpha_2  & \\alpha_3 & 0\\\\\n\t-3\\alpha_3&  0    & 0\n\t\\end{pmatrix} \\phi=\n\t\\begin{pmatrix}\n\t\\alpha^*      &  \\alpha^{**}    & \\alpha^*_1\\\\\n\t\\alpha^*_2    & \\alpha^*_3      & 0\\\\\n\t-3\\alpha^*_3  &  0           & 0\n\t\\end{pmatrix},\n\t$$\n\twhere\n\t\\begin{align*}\n\t\\alpha^*_1 &= \\alpha_1x^4,\\\\\n\t\\alpha^*_2 &= x^2(\\alpha_2x - 2\\alpha_3y),\\\\\n\t\\alpha^*_3 &= \\alpha_3x^4.\n\t\\end{align*}\n\tHence, $\\phi\\langle\\theta\\rangle=\\langle\\theta^*\\rangle,$ where $\\theta^*=\\sum\\limits_{i=1}^3 \\alpha_i^*  \\nb i.$\n\t\n\tWe are interested in $\\theta$ with $(\\alpha_2,\\alpha_3) \\neq (0,0).$ Moreover, the condition $\\theta \\in \\mathbf{T}_1 (\\T{3}{05})$ gives us $(\\alpha_1, \\alpha_3) \\neq (0,0).$\n\t\n\tLet $\\theta$ be as above. Consider two mutually exclusive cases:\n\t\n\t\\begin{enumerate}\n\t    \\item $\\alpha_3 = 0.$ The conditions above imply that $\\alpha_1 \\neq 0, \\alpha_2 \\neq 0.$\n\t    Then choosing $x=\\frac{\\alpha_2}{\\alpha_1},$ we have the representative $\\langle \\nabla_1+\\nabla_2 \\rangle.$      \n\n        \\item $\\alpha_3 \\neq 0.$ Choosing\n        $x = \\frac{1}{\\sqrt[4]{\\alpha_3}}$ and $y=\\frac{\\alpha_2}{ 2 \\sqrt[4]{\\alpha_3^5}}$ we have the family of representatives  $\\langle\\alpha\\nabla_1 + \\nabla_3 \\rangle.$ \n\\end{enumerate}\n\nSummarizing, we have the following distinct orbits: \n   \n\\[ \\langle \\nabla_1+\\nabla_2 \\rangle, \\ \\langle\\alpha\\nabla_1 + \\nabla_3 \\rangle.\\]\n\nThey correspond to the following algebras:\t        \n    \n\\[\\begin{array}{lllllllllll}\n\n\n\n        \\T{4}{37}&:& e_1e_1 = e_2,& e_1e_2 = e_3,& e_1e_3 = e_4,& e_2e_1 = e_4; \\\\\n\t    \\T{4}{38}(\\alpha)&:& e_1e_1 = e_2,& e_1e_2 = e_3,& e_1e_3 = \\alpha e_4,& e_2e_2 = e_4,& e_3e_1 = -3e_4.\n\t\n\\end{array}\\]\n\n\\subsubsection{$1$-dimensional  central extensions of $\\T {3}{01}(\\lambda)$}\n\n\n\n\\begin{enumerate}\n    \\item \t$\\lambda \\neq 0.$\n\n    \n    Let us use the following notations \n\t\\begin{align*}\n\t\\nb 1 = \\Dl 12, \\nb 2 = (\\lambda-1)\\Dl 13 + 3 \\Dl 31, \\nb 3 = \\color{black}{\\Dl 13 + \\Dl 22}.    \n\t\\end{align*}\n\tTake $\\theta=\\sum_{i=1}^3\\alpha_i\\nb i\\in {\\rm H_T^2}(\\T {3}{01}).$\n\tIf \n\t$$\n\t\\phi=\n\t\\begin{pmatrix}\nx &               0  &  0\\\\\ny &             x^2  &  0\\\\\nz &   (\\lambda+1)xy  &  x^3\n\\end{pmatrix}\n\\in\\aut{\\T {3}{01}},\n\t$$\n\tthen\n\t$$\n\t\\phi^T\\begin{pmatrix}\n\t0      &  \\alpha_1    & (\\lambda-1)\\alpha_2\\color{black}{+\\alpha_3}\\\\\n\t0  & \\color{black}{\\alpha_3} & 0\\\\\n\t\\color{black}{3\\alpha_2}&  0    & 0\n\t\\end{pmatrix} \\phi=\n\t\\begin{pmatrix}\n\t\\alpha^*      &  \\alpha_1^* + \\lambda\\alpha^{**}    & (\\lambda-1)\\alpha_2^*\\color{black}{+\\alpha_3^*}\\\\\n\t\\alpha^{**}  & \\color{black}{\\alpha_3^*} & 0\\\\\n\t\\color{black}{3\\alpha_2^*}&  0    & 0\n\t\\end{pmatrix},\n\t$$\n\twhere\n\t\\begin{align*}\n\n\n\t\\alpha^*_1 &= x^2(\\alpha_1x+({\\color{black} 2}\\alpha_3 - (\\lambda + 1)(2\\lambda + 1)\\alpha_2)y),\\\\\n\t\\alpha^*_2 &= \\alpha_2x^4,\\\\\n\t\\alpha^*_3 &= \\alpha_3x^4.\n\t\\end{align*}\n\tHence, $\\phi\\langle\\theta\\rangle=\\langle\\theta^*\\rangle,$ where $\\theta^*=\\sum\\limits_{i=1}^3 \\alpha_i^*  \\nb i.$\n\t\n\tThe condition $\\theta \\in \\mathbf{T}_1 (\\T{3}{01}(\\lambda))$ gives us ${\\color{black}(\\alpha_2, \\alpha_3)} \\neq (0,0).$\n\t\n\t\t\\begin{enumerate}\n\t\t\\item $(\\lambda + 1)(2\\lambda + 1)\\ne 0$.\n\t\t\\begin{enumerate}\n\t\t\t\\item $\\alpha_3\\ne 0$.\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item $\\alpha_2 \\ne \\frac{\\color{black} 2}{(\\lambda + 1)(2\\lambda + 1)}\\alpha_3$. Then choosing $x=\\frac 1{\\sqrt[4]{\\alpha_3}}$ and $y=-\\frac{\\alpha_1x}{{\\color{black} 2}\\alpha_3 - (\\lambda + 1)(2\\lambda + 1)\\alpha_2}$, we obtain the family of representatives of distinct orbits\n\t\t\t\t$\\langle\\alpha\\nb 2+\\nb 3\\rangle$, where $\\alpha\\ne \\frac{\\color{black} 2}{(\\lambda + 1)(2\\lambda + 1)}$.\n\t\t\t\t\\item $\\alpha_2=\\frac{\\color{black} 2}{(\\lambda + 1)(2\\lambda + 1)}\\alpha_3$. Then we have two representatives \n\t\t\t\t$\\Big\\langle\\frac{\\color{black} 2}{(\\lambda + 1)(2\\lambda + 1)}\\nb 2+\\nb 3\\Big\\rangle$ and $\\Big\\langle\\nb 1+\\frac{\\color{black} 2}{(\\lambda + 1)(2\\lambda + 1)}\\nb 2+\\nb 3\\Big\\rangle$ depending on whether $\\alpha_1=0$ or not. For convenience, we shall join the representative $\\Big\\langle\\frac{\\color{black} 2}{(\\lambda + 1)(2\\lambda + 1)}\\nb 2+\\nb 3\\Big\\rangle$ with the family $\\langle\\alpha\\nb 2+\\nb 3\\rangle_{\\alpha\\ne \\frac{\\color{black} 2}{(\\lambda + 1)(2\\lambda + 1)}}$ found above.  \n\t\t\t\\end{enumerate}\n\t\t\t\\item $\\alpha_3=0$. Then $\\alpha_2\\ne 0$. Choosing $y=\\frac{\\alpha_1x}{(\\lambda + 1)(2\\lambda + 1)\\alpha_2}$, we obtain the representative $\\langle\\nb 2\\rangle$.\n\t\t\\end{enumerate}\n\n\t\\item  $(\\lambda + 1)(2\\lambda + 1)=0$. \n\t\t\\begin{enumerate}\n\t\t\t\\item $\\alpha_3\\ne 0$. Then choosing $x=\\frac 1{\\sqrt[4]{\\alpha_3}}$ and $y=-\\frac{\\alpha_1x}{{\\color{black} 2}\\alpha_3}$, we obtain the family of representatives of distinct orbits $\\langle\\alpha\\nb 2+\\nb 3\\rangle$.\n\t\t\t\\item $\\alpha_3=0$. Then $\\alpha_2\\ne 0$. So, we obtain two representatives $\\langle\\nb 2\\rangle$ and $\\langle\\nb 1+\\nb 2\\rangle$ depending on whether $\\alpha_1=0$ or not.\n\t\t\\end{enumerate}\n\n\t\\end{enumerate}\n\n\n\nSummarizing, in the case (1) we have the following distinct orbits: \n$$\n     \\langle\\nb 1+\\nb 2\\rangle_{\\lambda\\in\\{-1,-\\frac 12\\}},\n    \\left\\langle\\nb 1+\\frac{\\color{black} 2}{(\\lambda + 1)(2\\lambda + 1)}\\nb 2+\\nb 3\\right\\rangle_{\\lambda\\not\\in\\{-1,-\\frac 12,0\\}},\n     \\langle\\nb 2\\rangle_{\\lambda\\ne 0},\n    \\langle\\alpha\\nb 2+\\nb 3\\rangle_{\\lambda\\ne 0}.\n$$\n\n\n\n\n\n\n  \n\n\n\t\\item $\\lambda=0.$ \n\tLet us use the following notations \n\t\\begin{align*}\n\t\\nb 1 = \\Dl 12, \\nb 2 = -\\Dl 13 + 3 \\Dl 31, \\nb 3 = {\\color{black}\\Dl 13 + \\Dl 22}, \\nb 4 = \\Dl 23.    \n\t\\end{align*}\n\tTake $\\theta=\\sum_{i=1}^4\\alpha_i\\nb i\\in {\\rm H_T^2}(\\T {3}{01}(0)).$\n\tIf \n\t$$\n\t\\phi=\n\t\\begin{pmatrix}\n    x &               0  &  0\\\\\n    y &             x^2  &  0\\\\\n    z &   xy  &  x^3\n    \\end{pmatrix}\n\\in\\aut{\\T {3}{01}(0)},\n\t$$\n\tthen\n\t$$\n\t\\phi^T\\begin{pmatrix}\n\t0      &  \\alpha_1    & -\\alpha_2{\\color{black}+\\alpha_3}\\\\\n\t0  & \\color{black}{\\alpha_3} & \\alpha_4\\\\\n\t{\\color{black} 3\\alpha_2}&  0    & 0\n\t\\end{pmatrix} \\phi=\n\t\\begin{pmatrix}\n\t\\alpha^*      &  \\alpha^*_1    & -\\alpha^*_2{\\color{black}+\\alpha_3^*}\\\\\n\t\\alpha^{**}  & {\\color{black}\\alpha^*_3} & \\alpha^*_4\\\\\n\t{\\color{black} 3\\alpha^*_2}&  0    & 0\n\t\\end{pmatrix},\n\t$$\n\twhere\n\t\\begin{align*}\n\t\\alpha^*_1 &= x (x^2 \\alpha_1 - x y (\\alpha_2 {\\color{black}-2}\\alpha_3) + y^2 \\alpha_4),\\\\\n    \\alpha^*_2 &= {\\color{black} \\alpha_2 x^4},\\\\\n    \\alpha^*_3 &= x^3 (x \\alpha_3 {\\color{black}+} y \\alpha_4),\\\\\n\t\\alpha^*_4 &= \\alpha_4x^5.\n\t\\end{align*}\n\tHence, $\\phi\\langle\\theta\\rangle=\\langle\\theta^*\\rangle,$ where $\\theta^*=\\sum\\limits_{i=1}^4 \\alpha_i^*  \\nb i.$\n\t\n\tThe condition $\\theta \\in \\mathbf{T}_1 (\\T{3}{01}(0))$ gives us $(\\alpha_2, {\\color{black}\\alpha_3},\\alpha_4) \\neq (0,0,0).$\n\t\n\t\n\t\\begin{enumerate}\n\t    \\item $\\alpha_4\\neq0$. Then choosing $y={\\color{black}-}\\frac{\\alpha_3x}{\\alpha_4}$ we have the family of representatives \n\t     $\\langle \\alpha_1^{\\star} \\nabla_1 + {\\color{black}\\alpha_2^*} \\nabla_2+ \\alpha^*_4\\nabla_4 \\rangle$,\n\t     where\n\t     \\begin{align*}\n\t        \\alpha^\\star_1=& \\frac{x^3}{\\alpha_4}(\\alpha_1\\alpha_4{\\color{black}+}\\alpha_2\\alpha_3{\\color{black}-\\alpha_3^2}).\n\t     \\end{align*}\n\t     \n\t     \\begin{enumerate}\n\t         \\item ${\\color{black}\\alpha_2}\\ne 0$. Then we have the family of representatives of distinct orbits $\\langle  \\alpha \\nabla_1+ \\nabla_2+ \\nabla_4 \\rangle$.\n\t         \n\t         \\item ${\\color{black}\\alpha_2}=0$. Then we have two representatives  $\\langle   \\nabla_4 \\rangle$ and $\\langle \\nabla_1+ \\nabla_4 \\rangle$ depending on whether $\\alpha_1\\alpha_4{\\color{black}+}\\alpha_2\\alpha_3{\\color{black}-\\alpha_3^2}=0$ or not.\n    \n\t     \\end{enumerate}\n\n\t    \\item $\\alpha_4=0.$ \n\t    \\begin{enumerate}\n\t        \\item $\\alpha_2 {\\color{black}-2}\\alpha_3\\ne 0$. Then choosing $y = \\frac{\\alpha_1x}{\\alpha_2 {\\color{black}-2}\\alpha_3}$ we have the representative $\\langle\\nb 2\\rangle$ and the family of representatives of distinct orbits $\\langle\\alpha\\nb 2+\\nb 3\\rangle_{\\alpha\\ne{\\color{black} 2}}$ depending on whether $\\alpha_3=0$ or not.\n\t        \\item $\\alpha_2 {\\color{black}-2}\\alpha_3=0$. Then we have two representatives $\\langle{\\color{black} 2}\\nb 2+\\nb 3\\rangle$ and $\\langle\\nb 1{\\color{black} +2}\\nb 2+\\nb 3\\rangle$ depending on whether $\\alpha_1=0$ or not. The representative $\\langle{\\color{black} 2}\\nb 2+\\nb 3\\rangle$ will be joined with the family $\\langle\\alpha\\nb 2+\\nb 3\\rangle_{\\alpha\\ne{\\color{black} 2}}$.\n\t    \\end{enumerate}\n\t    \\end{enumerate}\n\t    \nSummarizing, in the case (2)  we have the following distinct orbits: \n$$ \n\\langle\\nb 1{\\color{black} +2}\\nb 2+\\nb 3\\rangle_{\\lambda=0}, \\langle  \\alpha \\nabla_1+ \\nabla_2+ \\nabla_4 \\rangle_{\\lambda=0}, \\langle \\nabla_1+ \\nabla_4 \\rangle_{\\lambda=0}, \\langle\\nb 2\\rangle_{\\lambda=0}, \\langle\\alpha\\nb 2+\\nb 3\\rangle_{\\lambda=0}, \\langle   \\nabla_4 \\rangle_{\\lambda=0}.\n$$\n\\end{enumerate}\nNow, taking into account the both cases (1) and (2), we have the following distinct orbits:\n\n$\n\\begin{array}{l}\n     \\langle\\nb 1+\\nb 2\\rangle_{\\lambda\\in\\{-1,-\\frac 12\\}},\\\\ \n     \\left\\langle\\nb 1+\\frac{\\color{black} 2}{(\\lambda + 1)(2\\lambda + 1)}\\nb 2+\\nb 3\\right\\rangle_{\\lambda\\not\\in\\{-1,-\\frac 12\\}},\n\\end{array}\n\\begin{array}{l}\n    \\langle  \\alpha \\nabla_1+ \\nabla_2+ \\nabla_4 \\rangle_{\\lambda=0},\\\\\n    \\langle \\nabla_1+ \\nabla_4 \\rangle_{\\lambda=0},\\\\\n\\end{array}\n\\begin{array}{l}\n    \\langle\\nb 2\\rangle,\\\\\n    \\langle\\alpha\\nb 2+\\nb 3\\rangle,\n\\end{array}\n\\begin{array}{l}\n    \\langle \\nabla_4 \\rangle_{\\lambda=0}.\n\\end{array}\n$\n\nObserve that $\\left\\langle\\nb 1+\\frac{\\color{black} 2}{(\\lambda + 1)(2\\lambda + 1)}\\nb 2+\\nb 3\\right\\rangle=\\langle(\\lambda + 1)(2\\lambda + 1)\\nb 1+{\\color{black} 2}\\nb 2+(\\lambda + 1)(2\\lambda + 1)\\nb 3\\rangle$, the latter being $\\langle\\nb 2\\rangle$, when $\\lambda\\in\\{-1,-\\frac 12\\}$. Now, if $\\lambda\\not\\in\\{-1,-\\frac 12\\}$, then $\\mathrm{Orb}\\langle\\nb 2\\rangle=\\mathrm{Orb}\\langle\\nb 1+\\nb 2\\rangle$. Thus, we may reorganize our orbits as follows:\n\n$\n\\begin{array}{l}\n     \\langle\\nb 1+\\nb 2\\rangle,\\\\ \n     \\langle(\\lambda + 1)(2\\lambda + 1)\\nb 1+{\\color{black} 2}\\nb 2+(\\lambda + 1)(2\\lambda + 1)\\nb 3\\rangle,\n\\end{array}\n\\begin{array}{l}\n    \\langle  \\alpha \\nabla_1+ \\nabla_2+ \\nabla_4 \\rangle_{\\lambda=0},\\\\\n    \\langle \\nabla_1+ \\nabla_4 \\rangle_{\\lambda=0},\n\\end{array}\n\\begin{array}{l}\n     \\langle\\alpha\\nb 2+\\nb 3\\rangle,\\\\\n     \\langle\\nb 4 \\rangle_{\\lambda=0}.\n\\end{array}\n$\n\nThe corresponding algebras are:\t    \n\n\n$$\n\\begin{array}{lllllllll}\n     \\T 4{39}(\\lambda)   &:& e_1 e_1 = e_2, & e_1 e_2=\\lambda e_3+e_4, & e_1e_3 = (\\lambda-1)e_4,\\\\\n     && e_2 e_1=e_3, & e_3e_1 = 3e_4;\\\\ \n     \n     \\T 4{40}(\\lambda)   &:& e_1 e_1 = e_2, & e_1 e_2=\\lambda e_3+(\\lambda + 1)(2\\lambda + 1)e_4, & e_1e_3 = {\\color{black}(2\\lambda^2  + 5\\lambda  - 1)}e_4,\\\\\n     && e_2 e_1=e_3, & e_2e_2={\\color{black}(\\lambda + 1)(2\\lambda + 1)}e_4, & e_3e_1 = {\\color{black} 6}e_4;\\\\ \n     \n     \\T 4{41}(\\alpha)   &:& e_1 e_1 = e_2, & e_1 e_2=\\alpha e_4, & e_1e_3=-e_4,\\\\\n     && e_2 e_1=e_3, & e_2e_3 = e_4, & e_3e_1=3e_4;\\\\\n     \n     \\T 4{42}   &:& e_1 e_1 = e_2, & e_1 e_2=e_4, & e_2 e_1=e_3, & e_2e_3 = e_4;\\\\\n     \n     \\T 4{43}(\\lambda,\\alpha)   &:& e_1 e_1 = e_2, & e_1 e_2=\\lambda e_3, & e_1e_3 = (\\alpha(\\lambda-1){\\color{black}+1})e_4,\\\\\n     && e_2 e_1=e_3, & e_2e_2={\\color{black} e_4}, & e_3e_1 = 3{\\color{black}\\alpha} e_4;\\\\ \n     \n     \\T 4{44}   &:& e_1 e_1 = e_2, & e_2 e_1=e_3,    & e_2e_3 = e_4.\n\\end{array}\n$$\n\n\n \\section{The geometric classification of nilpotent terminal algebras}\n\n\\subsection{Degenerations of algebras}\nGiven an $n$-dimensional vector space ${\\bf V}$, the set ${\\rm Hom}({\\bf V} \\otimes {\\bf V},{\\bf V}) \\cong {\\bf V}^* \\otimes {\\bf V}^* \\otimes {\\bf V}$ \nis a vector space of dimension $n^3$. This space inherits the structure of the affine variety $\\mathbb{C}^{n^3}.$ \nIndeed, let us fix a basis $e_1,\\dots,e_n$ of ${\\bf V}$. Then any $\\mu\\in {\\rm Hom}({\\bf V} \\otimes {\\bf V},{\\bf V})$ is determined by $n^3$ structure constants $c_{i,j}^k\\in\\mathbb{C}$ such that\n$\\mu(e_i\\otimes e_j)=\\sum_{k=1}^nc_{i,j}^ke_k$. A subset of ${\\rm Hom}({\\bf V} \\otimes {\\bf V},{\\bf V})$ is {\\it Zariski-closed} if it can be defined by a set of polynomial equations in the variables $c_{i,j}^k$ ($1\\le i,j,k\\le n$).\n\n\nThe general linear group ${\\rm GL}({\\bf V})$ acts by conjugation on the variety ${\\rm Hom}({\\bf V} \\otimes {\\bf V},{\\bf V})$ of all algebra structures on ${\\bf V}$:\n$$ (g * \\mu )(x\\otimes y) = g\\mu(g^{-1}x\\otimes g^{-1}y),$$ \nfor $x,y\\in {\\bf V}$, $\\mu\\in {\\rm Hom}({\\bf V} \\otimes {\\bf V},{\\bf V})$ and $g\\in {\\rm GL}({\\bf V})$. Clearly, the ${\\rm GL}({\\bf V})$-orbits correspond to the isomorphism classes of algebras structures on ${\\bf V}$. Let $T$ be a set of polynomial identities which is invariant under isomorphism. Then the subset $\\mathbb{L}(T)\\subset {\\rm Hom}({\\bf V} \\otimes {\\bf V},{\\bf V})$ of the algebra structures on ${\\bf V}$ which satisfy the identities in $T$ is ${\\rm GL}({\\bf V})$-invariant and Zariski-closed. It follows that $\\mathbb{L}(T)$ decomposes into ${\\rm GL}({\\bf V})$-orbits. The ${\\rm GL}({\\bf V})$-orbit of $\\mu\\in\\mathbb{L}(T)$ is denoted by $O(\\mu)$ and its Zariski closure by $\\overline{O(\\mu)}$.\n\n\n\nLet ${\\bf A}$ and ${\\bf B}$ be two $n$-dimensional algebras satisfying the identities from $T$ and $\\mu,\\lambda \\in \\mathbb{L}(T)$ represent ${\\bf A}$ and ${\\bf B}$ respectively.\nWe say that ${\\bf A}$ {\\it degenerates} to ${\\bf B}$ and write ${\\bf A}\\to {\\bf B}$ if $\\lambda\\in\\overline{O(\\mu)}$.\nNote that in this case we have $\\overline{O(\\lambda)}\\subset\\overline{O(\\mu)}$. Hence, the definition of a degeneration does not depend on the choice of $\\mu$ and $\\lambda$. It is easy to see that any algebra degenerates to the algebra with zero multiplication. If ${\\bf A}\\to {\\bf B}$ and ${\\bf A}\\not\\cong {\\bf B}$, then ${\\bf A}\\to {\\bf B}$ is called a {\\it proper degeneration}. We write ${\\bf A}\\not\\to {\\bf B}$ if $\\lambda\\not\\in\\overline{O(\\mu)}$ and call this a {\\it non-degeneration}. Observe that the dimension of the subvariety $\\overline{O(\\mu)}$ equals $n^2-\\dim\\mathfrak{Der}({\\bf A})$. Thus if ${\\bf A}\\to {\\bf B}$ is a proper degeneration, then we must have $\\dim\\mathfrak{Der}({\\bf A})>\\dim\\mathfrak{Der}({\\bf B})$.\n\nLet ${\\bf A}$ be represented by $\\mu\\in\\mathbb{L}(T)$. Then  ${\\bf A}$ is  {\\it rigid} in $\\mathbb{L}(T)$ if $O(\\mu)$ is an open subset of $\\mathbb{L}(T)$.\nRecall that a subset of a variety is called {\\it irreducible} if it cannot be represented as a union of two non-trivial closed subsets. A maximal irreducible closed subset of a variety is called an {\\it irreducible component}.\nIt is well known that any affine variety can be represented as a finite union of its irreducible components in a unique way.\nThe algebra ${\\bf A}$ is rigid in $\\mathbb{L}(T)$ if and only if $\\overline{O(\\mu)}$ is an irreducible component of $\\mathbb{L}(T)$. \n\n\n\n\n\n\n\nIn the present work we use the methods applied to Lie algebras in \\cite{GRH,GRH2}.\nTo prove\ndegenerations, we will construct families of matrices parametrized by $t$. Namely, let ${\\bf A}$ and ${\\bf B}$ be two algebras represented by the structures $\\mu$ and $\\lambda$ from $\\mathbb{L}(T)$, respectively. Let $e_1,\\dots, e_n$ be a basis of ${\\bf V}$ and $c_{i,j}^k$ ($1\\le i,j,k\\le n$) be the structure constants of $\\lambda$ in this basis. If there exist $a_i^j(t)\\in\\mathbb{C}$ ($1\\le i,j\\le n$, $t\\in\\mathbb{C}^*$) such that the elements $E_i^t=\\sum_{j=1}^na_i^j(t)e_j$ ($1\\le i\\le n$) form a basis of ${\\bf V}$ for any $t\\in\\mathbb{C}^*$, and the structure constants $c_{i,j}^k(t)$ of $\\mu$ in the basis $E_1^t,\\dots, E_n^t$ satisfy $\\lim\\limits_{t\\to 0}c_{i,j}^k(t)=c_{i,j}^k$, then ${\\bf A}\\to {\\bf B}$. In this case  $E_1^t,\\dots, E_n^t$ is called a {\\it parametric basis} for ${\\bf A}\\to {\\bf B}$.\n\n\nTo prove a non-degeneration ${\\bf A}\\not\\to {\\bf B}$ we will use the following lemma (see \\cite{GRH}).\n\n\\begin{lemma}\\label{main}\nLet $\\mathcal{B}$ be a Borel subgroup of ${\\rm GL}({\\bf V})$ and $\\mathcal{R}\\subset \\mathbb{L}(T)$ be a $\\mathcal{B}$-stable closed subset.\nIf ${\\bf A} \\to {\\bf B}$ and ${\\bf A}$ can be represented by $\\mu\\in\\mathcal{R}$ then there is $\\lambda\\in \\mathcal{R}$ that represents ${\\bf B}$.\n\\end{lemma}\n\n\nIn particular, it follows from Lemma \\ref{main} that ${\\bf A}\\not\\to {\\bf B}$, whenever $\\dim({\\bf A}^2)<\\dim({\\bf B}^2)$.\n\n\nWhen the number of orbits under the action of ${\\rm GL}({\\bf V})$ on  $\\mathbb{L}(T)$ is finite, the graph of primary degenerations gives the whole picture. In particular, the description of rigid algebras and irreducible components can be easily obtained. Since the variety of $4$-dimensional nilpotent terminal algebras contains infinitely many non-isomorphic algebras, we have to fulfill some additional work. Let ${\\bf A}(*):=\\{{\\bf A}(\\alpha)\\}_{\\alpha\\in I}$ be a family of algebras and ${\\bf B}$ be another algebra. Suppose that, for $\\alpha\\in I$, ${\\bf A}(\\alpha)$ is represented by a structure $\\mu(\\alpha)\\in\\mathbb{L}(T)$ and ${\\bf B}$ is represented by a structure $\\lambda\\in\\mathbb{L}(T)$. Then by ${\\bf A}(*)\\to {\\bf B}$ we mean $\\lambda\\in\\overline{\\cup\\{O(\\mu(\\alpha))\\}_{\\alpha\\in I}}$, and by ${\\bf A}(*)\\not\\to {\\bf B}$ we mean $\\lambda\\not\\in\\overline{\\cup\\{O(\\mu(\\alpha))\\}_{\\alpha\\in I}}$.\n\nLet ${\\bf A}(*)$, ${\\bf B}$, $\\mu(\\alpha)$ ($\\alpha\\in I$) and $\\lambda$ be as above. To prove ${\\bf A}(*)\\to {\\bf B}$ it is enough to construct a family of pairs $(f(t), g(t))$ parametrized by $t\\in\\mathbb{C}^*$, where $f(t)\\in I$ and $g(t)=\\left(a_i^j(t)\\right)_{i,j}\\in {\\rm GL}({\\bf V})$. Namely, let $e_1,\\dots, e_n$ be a basis of ${\\bf V}$ and $c_{i,j}^k$ ($1\\le i,j,k\\le n$) be the structure constants of $\\lambda$ in this basis. If we construct $a_i^j:\\mathbb{C}^*\\to \\mathbb{C}$ ($1\\le i,j\\le n$) and $f: \\mathbb{C}^* \\to I$ such that $E_i^t=\\sum_{j=1}^na_i^j(t)e_j$ ($1\\le i\\le n$) form a basis of ${\\bf V}$ for any  $t\\in\\mathbb{C}^*$, and the structure constants $c_{i,j}^k(t)$ of $\\mu\\big(f(t)\\big)$ in the basis $E_1^t,\\dots, E_n^t$ satisfy $\\lim\\limits_{t\\to 0}c_{i,j}^k(t)=c_{i,j}^k$, then ${\\bf A}(*)\\to {\\bf B}$. In this case, $E_1^t,\\dots, E_n^t$ and $f(t)$ are called a {\\it parametric basis} and a {\\it parametric index} for ${\\bf A}(*)\\to {\\bf B}$, respectively. In the construction of degenerations of this sort, we will write $\\mu\\big(f(t)\\big)\\to \\lambda$, emphasizing that we are proving the assertion $\\mu(*)\\to\\lambda$ using the parametric index $f(t)$.\n\n\nThrough a series of degenerations summarized in the table below by the corresponding parametric bases and indices, we obtain the main result of the second part of the paper.\n\n\n\\begin{theorem}\\label{main-geo}\nThe variety of $4$-dimensional complex nilpotent terminal algebras has 3 irreducible components: one of dimension $17$ determined by the family of algebras $\\D 4{01}(\\lambda,\\alpha,\\beta)$ and two of dimension 15 determined by the families of algebras $\\T 4{41}(\\alpha)$ and $\\T 4{43}(\\lambda,\\alpha)$. \n\\end{theorem}\n\n\n\n\\begin{proof}[{\\bf Proof}]\nThanks to \\cite{kppv} the algebras \n$\\mathfrak{L}_2,$ $\\mathfrak{L}_5,$ $\\mathfrak{L}_{11}$ and $\\mathfrak{N}_3(\\alpha)$  \n define irreducible components in the variety of $4$-dimensional nilpotent Leibniz  algebras. \n Note that in \\cite{kppv} right Leibniz algebras were considered, and here we use their opposite versions which are left Leibniz algebras (and hence are terminal):\n$$ \n    \\begin{array}{lllllllll}\n    \n\\mathfrak{N}_3(\\alpha) & :&  e_1e_1 = e_4 &  e_1e_2 = -\\alpha e_4 & e_2e_1 = \\alpha e_4 & e_2e_2 = e_4&  e_3e_3 = e_4  \\\\\n\n\\mathfrak{L}_2  &: &  e_1e_1 = e_2 & e_1e_2 = e_3 &  e_1e_3 = e_4 \\\\  \n\n\\mathfrak{L}_5  &:&    e_1e_1 = e_3&   e_1e_2 = e_3& e_2e_2 = e_4&  e_1e_3=e_4 \\\\  \n\n\\mathfrak{L}_{11}  &:&  e_1e_1 = e_4& e_1e_2 = e_3 &  e_1e_3=e_4 & e_2e_1=-e_3 &   e_2e_2=e_4 & e_3e_1=-e_4.    \\\\ \n\\end{array}\n$$\n The list of all $4$-dimensional nilpotent non-Leibniz terminal  algebras was found in Theorem~\\ref{main-alg}. \n All these algebras  degenerate from one of the families: $\\D 4{01}(\\lambda,\\alpha,\\beta)$, $\\T 4{41}(\\alpha)$ or $\\T 4{43}(\\lambda,\\alpha)$, as it is shown in the table below. By a straightforward computation we have $\\dim\\mathfrak{Der}(\\D 4{01}(\\lambda,\\alpha,\\beta))=2$ for $\\alpha,\\beta\\ne 0$, and $\\dim\\mathfrak{Der}(\\D 4{01}(\\lambda,\\alpha,\\beta))>2$ otherwise. Hence, the closure of the orbit of $\\D 4{01}(\\lambda,\\alpha,\\beta)$ has dimension $4^2-2+3=17$. A similar argument yields that the dimensions of the orbit closures of $\\T 4{41}(\\alpha)$ and $\\T 4{43}(\\lambda,\\alpha)$ are both equal to $15<17$. In particular, this shows that $\\D 4{01}(\\lambda,\\alpha,\\beta)$ cannot degenerate from $\\T 4{41}(\\alpha)$ or $\\T 4{43}(\\lambda,\\alpha)$. Moreover, $\\T 4{41}(\\alpha)$ and $\\T 4{43}(\\lambda,\\alpha)$\\ do not degenerate from $\\D 4{01}(\\lambda,\\alpha,\\beta)$, since the squares of $\\T 4{41}(\\alpha)$ and $\\T 4{43}(\\lambda,\\alpha)$ have dimension 3, while the square of $\\D 4{01}(\\lambda,\\alpha,\\beta)$ has dimension $2<3$. This completes the proof of the theorem.\n\n\\end{proof}\n\n\n\n\n\n\n{\\tiny \n\n\n\\begin{longtable}{|lcl|ll|}\n\\multicolumn{5}{c}{  {\\bf Table.} {\\it  Degenerations of $4$-dimensional nilpotent terminal algebras.}}\\\\ \n\n\\hline\n$\\T {4}{03}\\left(\\frac{1}{1 - t}\\right)$ &$\\to$&  $\\mathfrak{N}_3(\\alpha)$&\n$E^t_1= \\frac{2 \\alpha t}{\\sqrt{t-1}} e_1 -  \\alpha t \\sqrt{t-1} e_2 + \\alpha t \\sqrt{t-1} e_3$ &\n$E^t_3= t e_3 + \\frac{t}{4}  \\left(1 + \\frac{1 - t}{\\alpha^2} - t\\right) e_4$\\\\\n\n&&&\n$E^t_2= -t \\sqrt{t-1} e_2 + t \\sqrt{t-1}  e_3$ &\n$E^t_4= t^2 e_4$ \\\\\n\n\n\\hline\n$\\T {4}{38}\\left(t^{-1}\\right)$ &$\\to$& $\\mathfrak{L}_2$&\n\n$E^t_1= e_1$ &\n$E^t_3= e_3$  \\\\\n&&& \n$E^t_2= e_2$&\n$E^t_4= t^{-1}  e_4$\\\\\n\n\\hline\n$\\D {4}{01}\\left( 2t^{-1},  t^{-1}, t \\right)$ &$\\to$& $\\mathfrak{L}_5$&\n\n$E^t_1= 2 e_1 - 2 e_2 - 2 e_3$ &\n$E^t_3= 4 t e_3 - 4t^{-1} e_4$\\\\\n&&& \n$E^t_2= -2 t e_2 - 2 e_3$ &\n$E^t_4= 8 e_4$\\\\\n\n\\hline\n$\\D {4}{01}\\left( -\\frac 12, 1, \\frac{1 + t^2}{4}\\right)$ &$\\to$& $\\mathfrak{L}_{11}$&\n$E^t_1= 4t^{-1} e_1 - 2t^{-1} e_2$ &\n$E^t_3= -8t^{-1} e_3 + 32t^{-3}  e_4$\\\\\n&&&\n$E^t_2= 4  e_2 - 8t^{-2} e_3$ &\n$E^t_4= 32t^{-2} e_4$\\\\\n\n\\hline\n\n\n\\color{black} $\\T {4}{43} \\left(0,0\\right)$   &\\color{black} $\\to$& \\color{black} $\\T {4}{01}$ & \n\n\\color{black} $E^t_1= te_1$ &\n\\color{black} $E^t_3= e_3$  \\\\\n&&& \n\\color{black} $E^t_2= t^2e_2$&\n\\color{black} $E^t_4= e_4$\\\\\n\\hline\n\n\n\\color{black}$\\T {4}{43} \\left(0, 0\\right)$   &\\color{black}$\\to$& \\color{black}$\\T {4}{02}$ & \n\n\\color{black} $E^t_1= te_1 + \\frac 12 t^{-2}e_2$ &\n\\color{black} $E^t_3= t^3e_3 + \\frac 12 e_4$ \\\\ \n&&& \n\\color{black} $E^t_2= t^2e_2 + \\frac 12 t^{-1}e_3 + \\frac 14 t^{-4}e_4$ &\n\\color{black} $E^t_4= e_4$\\\\\n\\hline\n\n$\\D {4}{01} \\left(0, t + \\alpha^{-1},t\\right)$   &$\\to$& $\\T {4}{03} (\\alpha)$ & \n$E^t_1= \\alpha^2 t e_2$ &\n$E^t_3= \\alpha^3 t^2 (e_1 - e_2)$\\\\\n&&&\n$E^t_2= \\alpha^4 t^2 e_3$ &\n$E^t_4= \\alpha^6 t^4 e_4$\n \\\\\n\\hline\n\n$\\T {4}{03} \\left(\\alpha \\right)$   &$\\to$& $\\T {4}{04} (\\alpha)$ & \n$E^t_1= t^{-1} e_1$ &\n$E^t_3= t^{-1} e_3$\\\\\n&&&\n$E^t_2= t^{-2} e_2$ &\n$E^t_4= t^{-3} e_4$\n \\\\\n\\hline\n\n$\\T {4}{07} \\left(\\frac{1}{t-1} \\right)$   &$\\to$& $\\T {4}{05}$  & \n\n$E^t_1= \\frac{(t-1)^2}{1 - 3 t + t^2} e_1  + \\frac{(t-1)^3}{(1 - 3 t + t^2)^2} e_2+  t e_3$ &\n\n$E^t_2= \\frac{(t-1)^4}{ (1 - 3 t + t^2)^2} e_2 + \\frac{(t-1)^4 t}{ (1 - 3 t + t^2)^2} e_4$ \\\\\n&&&\n$E^t_3=   - \\frac{(t-1)^4}{ t (1 - 3 t + t^2)^2} e_2+e_3$ &\n$E^t_4= -\\frac{(t-1)^3}{ (1 - 3 t + t^2)^2} e_4$\n\n \\\\\n\\hline\n\n$\\T {4}{03} \\left(t \\right)$   &$\\to$& $\\T {4}{06}$  & \n\n$E^t_1= \\frac{t - 1}{1 + t^2} e_1 + \\frac{t^2 - 1}{(1 + t^2)^2} e_2 + \\frac{1 - t}{1 + t^2} e_3$ &\n$E^t_3= \\frac{(1 - t) t}{1 + t^2} e_3$ \\\\\n&&& \n$E^t_2= \\frac{(t - 1)^2}{(1 + t^2)^2} e_2 +  \\frac{(1 - t)^3}{(1 + t^2)^3} e_4$ &\n$E^t_4= \\frac{(1 - t)^3 t}{(1 + t^2)^3} e_4$\\\\\n\\hline\n\n$\\T {4}{03} \\left( \\alpha \\right)$   &$\\to$& $\\T {4}{07} (\\alpha)$  & \n$E^t_1= -\\frac{1}{ 1 +\\alpha} e_1 - \\frac{1}{ (1 +\\alpha)^2} e_2 + \\frac{1}{ 1 +\\alpha} e_3$ &\n$E^t_2= \\frac{1}{(1 +\\alpha)^2} e_2 + \\frac{1}{ (1 +\\alpha)^3} e_4$\\\\\n&&&\n$E^t_3=   t^{-1} (e_2-(1 +\\alpha) e_3)$ &\n$E^t_4= -\\frac{1}{ t(1 +\\alpha)} e_4$ \\\\\n\n\\hline\n\n\n$\\T {4}{03} \\left(-1 \\right)$   &$\\to$& $\\T {4}{08}$  & \n\n$E^t_1= t^{-1} e_1$ & $E^t_3= t^{-2} e_3$ \\\\\n&&&\n$E^t_2= t^{-2} e_2$ &\n$E^t_4= t^{-4} e_4$\\\\\n\n\n\\hline\n\n$\\T {4}{03} \\left(t+t^4 \\right)$   &$\\to$& $\\T {4}{09}$  & \n$E^t_1= \\frac{t-1}{t^2 + t^5} e_1 + \\frac{t-1}{t^3 (1 + t^3)^2}e_2 + \\frac{1 - t}{t^2 + t^5} e_3$ &\n$E^t_3= -\\frac{(t-1 ) (-1 + t - t^2 - t^3 + 2 t^4 + t^7)}{ t (1 + t^3)^2 (-1 + t + t^4)} e_3$ \\\\\n&&& \n$E^t_2= \\frac{(t-1)^2}{t^4 (1 + t^3)^2} e_2 + \\frac{t-1}{(1 + t^3)^2} e_3$& \n$E^t_4= -\\frac{(t-1)^2 (-1 + t - t^2 - t^3 + 2 t^4 + t^7)}{ t^5 (1 + t^3)^4} e_4$\\\\ \n\\hline\n  \n$\\T {4}{03} (\\alpha )$   &$\\to$& $\\T {4}{10}(\\alpha )$  & \n\n$E^t_1= t^{-2} e_1$ & $E^t_3= t^{-3} e_3$\\\\\n&&&\n$E^t_2= t^{-4} e_2$  & $E^t_4= t^{-7} e_4$\\\\\n\\hline\n  \n$\\T {4}{03} (1+t^{-2})$   &$\\to$& $\\T {4}{11}$   & \n$E^t_1= t^3 e_1$ & $E^t_3= t^5 e_3$ \\\\\n\n&&&\n$E^t_2= t^6 e_2 - t^6 e_3$ &\n$E^t_4= t^9 e_4$\\\\\n\\hline\n\n$\\T {4}{41} (t^{-1})$    &$\\to$& $\\T {4}{12}$   & \n\n$E^t_1= 3 e_1 + 3 e_2$ &\n$E^t_3= 3t^{-1} e_3 + 9t^{-1} e_4$ \\\\\n&&& \n$E^t_2= 9 e_2 + 9 e_3 + 9t^{-1} e_4$ &\n$E^t_4= 27t^{-1} e_4$\\\\\n\\hline\n\n\n\n$\\T {4}{12}$    &$\\to$& $\\T {4}{13}$   & \n\n$E^t_1=  e_1$ & $E^t_3= t^{-1}  e_3$ \\\\\n&&& $E^t_2=  e_2$  & $E^t_4= t^{-1}  e_4$\\\\\n\\hline\n\n\n$\\T {4}{03} (1\/t )$   &$\\to$& $\\T {4}{14}$   & \n\n$E^t_1= t^{-1}   e_1$ & $E^t_3= t^{-2}  e_3$ \\\\\n&&& $E^t_2= t^{-2}  e_2$  & $E^t_4= t^{-5}  e_4$\\\\\n\\hline\n\n\n$\\T {4}{03} (\\frac{1 - t}{\\alpha} )$   &$\\to$& $\\T {4}{15}(\\alpha)$   & \n\n$E^t_1= e_1 + \\frac{\\alpha}{ t ( \\alpha -1+ t)} e_2 -  \\frac{\\alpha}{t (  \\alpha-1 + t)} e_3$ &\n$E^t_3= \\frac{1 - t}{\\alpha-1 + t} e_3 + \\frac{\\alpha (\\alpha t -1+ t^2)}{  t^3 ( \\alpha-1 + t)^2} e_4$ \\\\\n&&& \n$E^t_2= e_2 + \\frac{(t-1) t}{\\alpha-1 + t} e_3$ &\n$E^t_4= \\frac{-1 + t}{t ( \\alpha-1 + t)} e_4$\\\\\n\\hline\n\n\n$\\T {4}{03} (1 + t )$   &$\\to$& $\\T {4}{16}$   & \n\n\n$E^t_1= \\frac{t}{(1 + t) X^2} e_1 + \\frac{1 - t}{(1 + t) X^3} e_2 +  \\frac{1}{(1 + t) X^2}e_3$ &\n$E^t_3= \\frac{t}{(1 + t) X^3} e_3$ \\\\\n\\multicolumn{3}{|l|}{$X=-1 + t + t^2$}&\n$E^t_2= \\frac{t^2}{(1 + t)^2 X^4} e_2 + \\frac{ 1 + t - t^2}{(1 + t)^2 X^5} e_4$ &\n$E^t_4= \\frac{t^2}{(1 + t)^2 X^6} e_4$\\\\\n\\hline\n\n\n\n$\\T {4}{03} (\\frac{1}{1 - t^2} )$   &$\\to$& $\\T {4}{17}(\\alpha)$   & \n\n$E^t_1= t^2 X e_1 + t^2(-1 + t^2) X^2 e_2 + (t^2 - t^4) X^2 e_3$ &\n$E^t_3=  t^3 X^2 e_3$ \\\\\n\n\\multicolumn{3}{|l|}{$X=(-1 + \\alpha + t^2)^{-1}$}&\n$E^t_2= t^4  X^2 e_2 + t^4 ( t^2-1)(\\alpha + t^2) X^4 e_4$ &\n$E^t_4=  t^6 X^4 e_4$\\\\\n\\hline\n\n\n\n$\\T 4{19}$ &$\\to$& $\\T 4{18}$  & \n\n$E_1^t = \\sqrt{1-t^2}e_1 + te_2 - te_3$ &  $E_3^t = e_3$\\\\\n&&& $E_2^t = e_2$ & $E_4^t = \\sqrt{1-t^2}e_4$\\\\\n\\hline\n\n$\\T 4{23}(t^{-1},-t-1)$ &$\\to$& $\\T 4{19}$  & \n\n$E_1^t = t^2e_1 - \\frac{t^3}{t + 1}e_3$ & $E_3^t = t^4e_3$ \\\\\n&&& $E_2^t = t^2e_2$ & $E_4^t = t^5e_4$\\\\\n\\hline\n\n$\\T 4{23}(it^{-1},t^{-1})$ &$\\to$& $\\T 4{20}$  & \n\n$E_1^t = te_1$ & $E_3^t = t^2e_3$ \\\\\n&&& $E_2^t = te_2$ & $E_4^t = t^2e_4$\\\\\n\\hline\n\n\n$\\T 4{20}$ &$\\to$& $\\T 4{21}$  & \n\n$E_1^t = t^{-1}e_1$ &  $E_3^t = t^{-2}e_3$\\\\\n&&& $E_2^t = t^{-1}e_2$ & $E_4^t = t^{-3}e_4$\\\\\n\\hline\n\n$\\T 4{23}(\\alpha,\\beta)$ &$\\to$& $\\T 4{22}(\\alpha,\\beta)$  & \n\n$E_1^t = t^{-1}e_1$ & $E_3^t = t^{-2}e_3$ \\\\\n&&& $E_2^t = t^{-1}e_2$ & $E_4^t = t^{-3}e_4$\\\\\n\\hline\n\n$\\D 4{01}\\left(\\beta^2t^{-2},t+\\alpha,t\\right)$ &$\\to$& $\\T 4{23}(\\alpha,\\beta)$  & \n\n$E_1^t = t^2\\beta^{-2}(e_1 - e_2)$ & $E_3^t = t^2\\beta^{-2}e_3$ \\\\\n&&& $E_2^t = t\\beta^{-1}e_2$ & $E_4^t = t^4\\beta^{-4}e_4$\\\\\n\\hline\n\n$\\T 4{23}(\\alpha,-(\\alpha + 1)t)$ &$\\to$& $\\T 4{24}(\\alpha)$  & \n\n$E_1^t = te_1 - t(\\alpha + 1)^{-1}e_3$ & $E_3^t = t^2e_3$ \\\\\n&&& $E_2^t = te_2$ & $E_4^t = t^3e_4$\\\\\n\\hline\n\n$\\T 4{23}(-1,t)$ &$\\to$& $\\T 4{25}(\\alpha)$  & \n\n$E_1^t = e_1 + \\alpha t^{-1}e_3$ & $E_3^t = e_3$ \\\\\n&&& $E_2^t = e_2$ & $E_4^t = e_4$\\\\\n\\hline\n\n$\\T 4{23}(t,i)$ &$\\to$& $\\T 4{26}(\\alpha)$  & \n\n$E_1^t = \\frac i{\\alpha + i}e_1 -\\frac{\\alpha i}{(\\alpha + i)^2t}e_3$ & $E_3^t = -\\frac 1{(\\alpha + i)^2}e_3 + \\frac{\\alpha}{(\\alpha + i)^3t}e_4$ \\\\\n&&& $E_2^t = \\frac i{\\alpha + i}e_2 - \\frac{\\alpha}{(\\alpha + i)^2t}e_3$ & $E_4^t = -\\frac i{(\\alpha + i)^3}e_4$\\\\\n\\hline\n\n$\\T 4{23}\\left(\\frac{(\\alpha^2 + 1)t}{\\alpha - t},\\alpha\\right)$ &$\\to$& $\\T 4{27}(\\alpha)$  & \n\n$E_1^t =  te_1 +\\frac{(t-\\alpha)t}{\\alpha(\\alpha^2 + 1)}e_3$ & $E_3^t = t^2e_3 - \\frac{\\alpha(t-\\alpha)t^2}{\\alpha^2 + 1}e_4$ \\\\\n&&& $E_2^t = te_2 - \\frac{(t-\\alpha)t}{\\alpha^2 + 1}e_3$ & $E_4^t = t^3e_4$\\\\\n\\hline\n\n$\\T 4{23}\\left(\\frac{i\\alpha(1-t)}t,\\frac{\\alpha\\sqrt{1-t}}t\\right)$ &$\\to$& $\\T 4{28}(\\alpha)$  & \n\n$E_1^t =  \\frac{\\alpha^2(t - 1)}t\\left(-i e_1 + \\frac 1{\\sqrt{1-t}} e_2 - (\\alpha - i) e_3\\right)$ & $E_3^t = -\\frac{\\alpha^4(t - 1)}t e_3$ \\\\\n&&& $E_2^t = \\frac{\\alpha^2(t - 1)}t(-e_1 + i\\sqrt{1-t} e_2 +  e_3)$ & $E_4^t = \\frac{\\alpha^6(t - 1)^2}{t^2}e_4$\\\\\n\\hline\n\n$\\T 4{28}(\\alpha)$ &$\\to$& $\\T 4{29}(\\alpha)$  & \n\n$E_1^t = t^{-1}e_1$ & $E_3^t = t^{-2}e_3$ \\\\\n&&& $E_2^t = t^{-1}e_2$ & $E_4^t = t^{-3}e_4$\\\\\n\\hline\n\n$\\T 4{23}\\left(1,(1 - \\alpha)\\sqrt{t - 1}\\right)$ &$\\to$& $\\T 4{30}(\\alpha)$  & \n \\multicolumn{2}{l|}{$E_1^t = \\frac{(\\alpha - 1)t(t - 1)}D\\left(-e_1+\\frac 1{\\sqrt{t - 1}} e_2-\\frac{(\\alpha - 2)t - \\alpha  + 1}D e_3\\right)$}\\\\\n \\multicolumn{3}{|l|}{$D=\\alpha^2t^2 - 2\\alpha^2t - 2\\alpha t^2 + \\alpha^2 + 2\\alpha t + t^2 + 2t - 1$}\n& \\multicolumn{2}{l|}{$E_2^t = \\frac{i(\\alpha - 1)t(t - 1)}D\\left(e_1+\\sqrt{t - 1}e_2-\\frac{(\\alpha-1)t^2 - (2\\alpha-3)t + \\alpha - 1}D e_3\\right)$}\\\\\n&&& \\multicolumn{2}{l|}{$E_3^t = \\frac{(\\alpha - 1)^2(t - 1)t^3}{D^2}\\left(e_3+\\frac{((\\alpha-1) t - \\alpha)(\\alpha - 1)(t - 1)}De_4\\right)$} \\\\\n&&& \\multicolumn{2}{l|}{$E_4^t = \\frac{i(\\alpha - 1)^3(t - 1)^2t^4}{D^3}e_4$}\\\\\n\\hline\n\n$\\T 4{30}(\\alpha)$ &$\\to$& $\\T 4{31}(\\alpha)$ & \n\n$E_1^t = -\\frac{i(\\alpha^2 - 1)}{\\alpha t}e_1 - \\frac i{\\alpha t}e_3$ & $E_3^t = -\\frac{(\\alpha^2 - 1)^2}{\\alpha^2t^2}e_3 - \\frac{i(\\alpha + 1)^2(\\alpha - 1)}{\\alpha t^2}e_4$\\\\\n&&& $E_2^t = -\\frac{i(\\alpha^2 - 1)}{\\alpha t}e_2 + \\frac 1t e_3$ & $E_4^t = i\\frac{(\\alpha^2 - 1)^3}{\\alpha^3 t^3}e_4$\\\\\n\\hline\n\n$\\T 4{23}(\\sqrt{t-1} + 1,1)$ &$\\to$& $\\T 4{32}$  & \n\n$E_1^t = \\frac 1{(\\sqrt{t-1} + 1)^2}\\left(e_1 - \\sqrt{t-1}e_2 -\\frac{\\sqrt{t-1}}{(\\sqrt{t-1} + 1)^2}e_3\\right)$ & $E_3^t = \\frac t{(\\sqrt{t-1} + 1)^4}e_3$ \\\\\n&&& $E_2^t = \\frac i{(\\sqrt{t-1} + 1)^2}\\left(-e_1 - \\frac 1{\\sqrt{t-1}}e_2 + \\frac {\\sqrt{t - 1}}{(\\sqrt{t - 1} + 1)^2}e_3\\right)$ & $E_4^t = -\\frac {it}{(\\sqrt{t - 1} + 1)^6}e_4$\\\\\n\n\\hline\n$\\T 4{23}(\\sqrt{t-1} - 1,1)$ &$\\to$& $\\T 4{33}$  & \n\n$E_1^t = \\frac 1t\\left(e_1 - \\sqrt{t - 1}e_2 -\\frac{\\sqrt{t - 1}}t e_3\\right)$ & $E_3^t = \\frac 1t e_3$ \\\\\n&&& $E_2^t = \\frac it\\left(-e_1 -\\frac 1{\\sqrt{t - 1}}e_2 + \\frac{\\sqrt{t - 1}}t e_3\\right)$ & $E_4^t = -\\frac i{t^2}e_4$\\\\\n\\hline\n\n$\\T 4{23}(\\sqrt{t-1} - 1,1)$ &$\\to$& $\\T 4{34}$  & \n\n\\multicolumn{2}{l|}{$E_1^t = \\frac{\\sqrt{t-1} - 1}{t\\sqrt{t-1} - t + 2}\\left(e_1 -\\sqrt{t-1}e_2 -\\frac{\\sqrt{t-1}(t\\sqrt{t-1} - t + 4)}{t(t\\sqrt{t-1} - t + 2)}e_3\\right)$}\\\\\n&&& \n\\multicolumn{2}{l|}{$E_2^t = \\frac{i(\\sqrt{t-1}-1)}{t\\sqrt{t-1} - t + 2}\\left(-e_1 -\\frac 1{\\sqrt{t-1}}e_2 +\\frac{t^2 - t\\sqrt{t-1} - 3t + 4\\sqrt{t-1}}{t(t\\sqrt{t-1} - t + 2)}e_3\\right)$}\\\\\n&&& \n\\multicolumn{2}{l|}{$E_3^t = \\frac{(\\sqrt{t-1} - 1)^2}{(t\\sqrt{t-1} - t + 2)^2}\\left(te_3 +\\frac{2\\sqrt{t-1}(t - \\sqrt{t-1})}{t\\sqrt{t-1} - t + 2}e_4\\right)$}\\\\\n&&& \n\\multicolumn{2}{l|}{$E_4^t = -\\frac{it(\\sqrt{t-1} - 1)^3}{(t\\sqrt{t-1} - t + 2)^3}e_4$}\\\\\n\\hline\n\n$\\D 4{01}\\left(\\frac 14,\\frac 1{2t},\\frac 1t\\right)$ &$\\to$& $\\T 4{35}$  & \n\n$E_1^t = -t^{-1}(2e_1 - e_2 - 2e_3)$ & $E_3^t = -2t^{-1}e_3$ \\\\\n&&& $E_2^t = t^{-1}(2e_1 - (2t + 1)e_2 - 4e_3)$ & $E_4^t = 4t^{-2}e_4$\\\\\n\\hline\n\n$\\T 4{35}$ &$\\to$& $\\T 4{36}$  & \n\n$E_1^t = t^{-1}e_1$ & $E_3^t = t^{-2}e_3$ \\\\\n&&& $E_2^t = t^{-1}e_2$ & $E_4^t = t^{-3}e_4$\\\\\n\\hline\n\n$\\T 4{38}\\left(t^{-1}\\right)$ &$\\to$& $\\T 4{37}$  & \n\n$E_1^t = e_1 - \\frac 1{2t}e_2$ & $E_3^t = e_3 - \\frac 1{2t}(1 + \\frac 1t)e_4$ \\\\\n&&& $E_2^t = e_2 - \\frac 1{2t}e_3 + \\frac 1{4t^2}e_4$ & $E_4^t = \\frac 1te_4$\\\\\n\\hline\n\n\\color{black} $\\T 4{43}\\left((1-\\alpha)(t + 1)t^{-1}, t(\\alpha - 1)^{-1}\\right)$ &\\color{black} $\\to$& \\color{black} $\\T 4{38}(\\alpha)$  & \n\n\\color{black} $E^t_1= e_1$ & \n\\color{black} $E^t_3= (1-\\alpha)t^{-1}e_3$ \\\\\n&&& \n\\color{black} $E^t_2= e_2$ &\n\\color{black} $E^t_4= e_4$\\\\\n\\hline\n\n\n\\color{black} $\\T 4{43}\\left(\\lambda+t, -t^{-1}\\right)$ &\\color{black} $\\to$& \\color{black} $\\T 4{39}(\\lambda)$  & \n\n\\multicolumn{2}{l|}{\\color{black} $E_1^t = e_1 - De_2$}\\\\\n \\multicolumn{3}{|l|}{\\color{black} $D=(2t^2 + (4\\lambda +5)t + 2\\lambda^2 + 3\\lambda + 1)^{-1}$}\n& \\multicolumn{2}{l|}{\\color{black} $E_2^t = e_2 - (t + \\lambda + 1)De_3 + D^2 e_4$}\\\\\n&&& \\multicolumn{2}{l|}{\\color{black} $E_3^t = e_3 + t^{-1}(2t + 3\\lambda + 3)De_4$} \\\\\n&&& \\multicolumn{2}{l|}{\\color{black} $E_4^t = -t^{-1}e_4$}\\\\\n\\hline\n\\color{black}\n$\\T 4{43}\\left(\\lambda  + t, \\frac 2{(\\lambda + 1)(2\\lambda + 1)}\\right)$ &\\color{black}$\\to$& \\color{black}$\\T 4{40}(\\lambda)$ & \n\n \\multicolumn{2}{l|}{\\color{black} $E_1^t = e_1 - \\frac 12(\\lambda + 1)(2\\lambda + 1)D e_2$}\\\\\n \\multicolumn{3}{|l|}{\\color{black} $D=(t(2t + 4\\lambda + 3))^{-1}$}\n& \\multicolumn{2}{l|}{\\color{black} $E_2^t = e_2 - \\frac 12(\\lambda + 1)(2\\lambda + 1)(t + \\lambda + 1)De_3 + \\frac 14(\\lambda + 1)^2(2\\lambda + 1)^2D^2e_4$}\\\\\n&&& \\multicolumn{2}{l|}{\\color{black} $E_3^t = e_3 - \\frac 12(6t + 2\\lambda^2 + 9\\lambda + 7)De_4$} \\\\\n&&& \\multicolumn{2}{l|}{\\color{black} $E_4^t =  \\frac 1{(\\lambda + 1)(2\\lambda + 1)}e_4$}\\\\\n\\hline\n\n$\\T 4{41}(t^{-2})$ &$\\to$& $\\T 4{42}$  & \n\n$E_1^t = t^{-1}e_1 + t^{-1}e_2$ &  $E_3^t = t^{-3}e_3 + 3t^{-3}e_4$\\\\\n&&& $E_2^t = t^{-2}e_2 + t^{-2}e_3 + t^{-4}e_4$ & $E_4^t = t^{-5}e_4$\\\\\n\\hline\n\n$\\T 4{41}(0)$ &$\\to$& $\\T 4{44}$  & \n\n$E_1^t = t^{-1}e_1 + t^{-1}e_2$ & $E_3^t = t^{-3}e_3 + 3t^{-3}e_4$ \\\\\n&&& $E_2^t = t^{-2}e_2 + t^{-2}e_3$ & $E_4^t = t^{-5}e_4$\\\\\n\\hline\n\n\n$\\D 4{01}\\left(\\lambda, \\alpha,\\beta\\right)$ &$\\to$& $\\D 4{02}\\left(\\lambda, \\alpha,\\beta\\right)$ \n& $E_1^t = t^{-1}e_1$  & $E_3^t = t^{-2}e_3$\\\\\n &&& $E_2^t = t^{-1}e_2$ & $E_4^t = t^{-3}e_4$\\\\\n\\hline\n\n$\\D 4{01}\\left( \\frac{\\lambda - t}{(t-1)^2}, t - \\alpha  t ,\\alpha  - \\alpha  t \\right)$ &$\\to$& $\\D 4{03}\\left(\\lambda, \\alpha \\right)$ \n& \n\n\\multicolumn{2}{l|}{$E^t_1= \\frac{(1 - \\alpha ) \\alpha  (-1 + t)^2 t}{1 + \\alpha ^2 \\lambda + t - \\alpha  (1 + t)} e_1 + \n\\frac{(-1 + \\alpha ) \\alpha  (-1 + t) t^2}{1 + \\alpha ^2 \\lambda + t - \\alpha  (1 + t)} e_2 - \n\\frac{(-1 + \\alpha )^2 \\alpha  (-1 + t)^2 t}{ (1 + \\alpha ^2 \\lambda + t - \\alpha  (1 + t))^2} e_3$} \\\\\n\n&&& \\multicolumn{2}{l|}{$E^t_2= \\frac{(-1 + \\alpha ) \\alpha  (-1 + t) t}{ 1 + \\alpha ^2 \\lambda + t - \\alpha  (1 + t)}e_2 - \n\\frac{(-1 + \\alpha ) \\alpha ^2 (-1 + t)^2 t}{ (1 + \\alpha ^2 \\lambda + t - \\alpha  (1 + t))^2} e_3$ }\\\\\n\n&&&\n\\multicolumn{2}{l|}{$E^t_3= \\frac{(-1 + \\alpha )^2 \\alpha ^2 (-1 + t)^2 t^2}{(1 + \\alpha ^2 \\lambda + t - \\alpha  (1 + t))^2} e_3 + \n\\frac{ (-1 + \\alpha )^2 \\alpha ^4 (-1 + t)^4 t^2}{  (1 + \\alpha ^2 \\lambda + t - \\alpha  (1 + t))^3} e_4$}\\\\\n\n\n&&&\\multicolumn{2}{l|}{$E^t_4= -\\frac{ (-1 + \\alpha )^3 \\alpha ^3 (-1 + t)^4 t^3}{  (1 + \\alpha ^2 \\lambda + t - \\alpha  (1 + t))^3} e_4$ }\\\\\n    \n    \n\\hline\n\n\n$\\D 4{01}\\left(\\lambda, 0,\\alpha \\right)$ &$\\to$& $\\D 4{04}\\left(\\lambda, \\alpha\\right)$ \n\n\n& $E_1^t = t^{-1}e_1$  & $E_3^t = t^{-2}e_3$\\\\\n &&& $E_2^t = t^{-1}e_2$ & $E_4^t = t^{-3}e_4$\\\\\n \\hline\n \n \n$\\D 4{01}\\left(\\lambda, t , \\frac{1 + \\sqrt{1 - 4 \\lambda}}{2 \\lambda }\\right)$ &$\\to$& $\\D 4{05}\\left(\\lambda, \\alpha\\right)$ \n&\n\n$E^t_1= -\\frac{1 + \\sqrt{1 - 4 \\lambda}}{D} e_1 + \n\\frac{(1 + \\sqrt{1 - 4 \\lambda}) (D-2 \\lambda \\alpha )}{ t D^2} e_3$ &\n  \n$E^t_3= \\frac{(1 + \\sqrt{1 - 4 \\lambda})^2}{  D^2} e_3 - \\frac{(1 + \\sqrt{1 - 4 \\lambda})^4}{ 2 \\lambda t D^3} e_4$ \\\\\n\n\\multicolumn{3}{|l|}{$D=1 + \\sqrt{1 - 4 \\lambda} + 2 \\lambda (-1 + \\alpha + \\alpha t)$}&\n$E^t_2= -\\frac{1 + \\sqrt{1 - 4 \\lambda}}{  D} e_2 + \\frac{(1 + \\sqrt{1 - 4 \\lambda})^2}{ t D^2} e_3$ &\n\n$E^t_4= -\\frac{(1 + \\sqrt{1 - 4 \\lambda})^3}{  D^3} e_4$\n    \\\\\n\\hline\n\n$\\D 4{01}\\left(\\lambda, t , \\frac{1 -\\sqrt{1 - 4 \\lambda}}{2 \\lambda }\\right)$ &$\\to$& $\\D 4{06}\\left(\\lambda, \\alpha\\right)$ \n&\n\n$E^t_1= -\\frac{1 - \\sqrt{1 - 4 \\lambda}}{D} e_1 + \n\\frac{(1 - \\sqrt{1 - 4 \\lambda}) (D-2 \\lambda \\alpha )}{ t D^2} e_3$ &\n  \n\n$E^t_3= \\frac{(1 - \\sqrt{1 - 4 \\lambda})^2}{  D^2} e_3 - \\frac{(1 - \\sqrt{1 - 4 \\lambda})^4}{ 2 \\lambda t D^3} e_4$ \\\\\n\n\\multicolumn{3}{|l|}{$D=1 - \\sqrt{1 - 4 \\lambda} + 2 \\lambda (-1 + \\alpha + \\alpha t)$}&\n$E^t_2= -\\frac{1 - \\sqrt{1 - 4 \\lambda}}{  D} e_2 + \\frac{(1 - \\sqrt{1 - 4 \\lambda})^2}{ t D^2} e_3$ &\n\n\n$E^t_4= -\\frac{(1 - \\sqrt{1 - 4 \\lambda})^3}{  D^3} e_4$   \n    \\\\\n\\hline\n\n$\\D 4{01}\\left(\\lambda, t , \\frac{1 + \\sqrt{1 - 4 \\lambda}}{2 \\lambda }\\right)$ &$\\to$& $\\D 4{07}\\left(\\lambda\\right)$ \n&\n\n\\multicolumn{2}{l|}{$E^t_1=-\\frac{D  - (1 + \\sqrt{1 - 4 \\lambda}) \\lambda}{D \\lambda (1 + t)} e_1 + \n\\frac{2 (D -(1 + \\sqrt{1 - 4 \\lambda}) \\lambda) (D  -2 (1 + \\sqrt{1 - 4 \\lambda})\\lambda + 2 \\lambda^2)}{D^3 \\lambda (1 + t)^2} e_3$} \\\\\n  \n\\multicolumn{3}{|l|}{$D=1 + \\sqrt{1 - 4 \\lambda} - 2 \\lambda$}&\n\\multicolumn{2}{l|}{$E^t_2= -\\frac{D  - (1 + \\sqrt{1 - 4 \\lambda}) \\lambda}{D \\lambda (1 +t)}e_2$}  \\\\\n\n&&&\n$E^t_3= \\frac{(D  - (1 + \\sqrt{1 - 4\\lambda}) \\lambda)^2}{D^2 \\lambda^2 (1 + t)^2} e_3$ &\n$E^t_4= \\frac{(D - (1 + \\sqrt{1 - 4 \\lambda}) \\lambda)^3}{ D^3 \\lambda^3 (1 + t)^3} e_4$\\\\\n\\hline\n \n$\\D 4{01}\\left(\\lambda, t , \\frac{1 - \\sqrt{1 - 4 \\lambda}}{2 \\lambda }\\right)$ &$\\to$& $\\D 4{08}\\left(\\lambda\\right)$ \n&\n\n\\multicolumn{2}{l|}{$E^t_1=-\\frac{D  - (1 - \\sqrt{1 - 4\\lambda}) \\lambda}{D\\lambda (1 + t)} e_1 + \n\\frac{2 (D - (1 - \\sqrt{1 - 4\\lambda}) \\lambda) (D  - 2 (1 - \\sqrt{1 - 4\\lambda}) \\lambda + 2\\lambda^2)}{D^3\\lambda (1 + t)^2} e_3$} \\\\\n  \n\\multicolumn{3}{|l|}{$D=1 - \\sqrt{1 - 4\\lambda} - 2\\lambda$}&\n\\multicolumn{2}{l|}{$E^t_2= -\\frac{D - (1- \\sqrt{1 - 4\\lambda})\\lambda}{D\\lambda (1 +t)}e_2$}  \\\\\n\n&&&\n$E^t_3= \\frac{(D  - (1 - \\sqrt{1 - 4\\lambda})\\lambda)^2}{D^2\\lambda^2 (1 + t)^2} e_3$ &\n$E^t_4=- \\frac{(D - (1 - \\sqrt{1 - 4\\lambda})\\lambda)^3}{ D^3\\lambda^3 (1 + t)^3} e_4$\\\\\n\\hline\n    \n$\\D 4{01}\\left( \\frac{\\lambda  - t}{(t-1)^2}, \\alpha  - t^2,  t - t^2 \\right)$ &$\\to$& $\\D 4{09}\\left(\\lambda, \\alpha \\right)$ \n&\n\n\\multicolumn{2}{l|}{$E^t_1=(1-t)^2 t ( t^2-\\alpha) D  e_1  + (1 -t) t^2 ( t^2-\\alpha) D  e_2  + ( t-1 )^3\n   t (\\alpha  - t^2) D^2 e_3$  } \\\\\n  \n\\multicolumn{3}{|l|}{$D=(-1 + \\alpha  (-1 + t) + t - \\lambda  t^2)^{-1}$}&\n\\multicolumn{2}{l|}{$E^t_2= (1 - t) t (-\\alpha  + t^2) D e_2  - (-1 + t)^2 t^2 (-\\alpha  + t^2) D^2 e_3$ }  \\\\\n\n&&&\n\\multicolumn{2}{l|}{$E^t_3=(1 - t)^2 t^2 (\\alpha  - t^2)^2 D^2 e_3 + (1 - t)^4 t^4 (\\alpha  - t^2)^2 D^3 e_4$ } \\\\\\\n\n\n&&&\n\\multicolumn{2}{l|}{$E^t_4=( 1-t )^4 ( t^3-\\alpha  t)^3 D^3 e_4$}\\\\\n\\hline\n    \n\n$\\D 4{09}\\left(\\lambda, \\alpha \\right)$ &$\\to$& $\\D 4{10}\\left(\\lambda, \\alpha \\right)$ \n& $E_1^t = t^{-1}e_1$  & $E_3^t = t^{-2}e_3$\\\\\n &&& $E_2^t = t^{-1}e_2$ & $E_4^t = t^{-3}e_4$\\\\\n\\hline\n\n\n\n$\\D 4{01}\\left( \\frac{\\lambda  - t}{(t-1)^2},  t^2-1,  t^2 - t\\right)$ &$\\to$& $\\D 4{11}\\left(\\lambda, \\alpha \\right)$ \n&\n\n\\multicolumn{2}{l|}{$E^t_1=(1 - t)^3 (1 + t) D  e_1 + (1 - t)^2 t (1 + t) D e_2 - \\alpha (1 - t)^3 (1 + t)^2 D^2t^{-1}  e_3$ } \\\\\n  \n\\multicolumn{3}{|l|}{$D=(1 + \\alpha \\lambda t - t^2)^{-1}$}&\n\\multicolumn{2}{l|}{$E^t_2= (1 - t)^2 (1 + t) D  e_2 -  \\alpha (1 - t)^3 (1 + t) D^2 e_3$  }  \\\\\n\n&&&\n\\multicolumn{2}{l|}{$E^t_3= (1 - t)^4 (1 + t)^2 D^2 e_3 +  \\alpha (1 - t)^6 t (1 + t)^2 D^3 e_4 $} \\\\\\\n\n\n&&&\n\\multicolumn{2}{l|}{$E^t_4=(1 - t)^7 (1 + t)^3 D^3 e_4$}\\\\\n\\hline\n\n\n$\\D 4{01}\\left(\\lambda\\left(1-\\frac {\\alpha t}{\\Theta^2}\\right), \\frac 1\\Theta\\left(\\alpha+\\frac{\\lambda}{t}\\right), \\frac 1t\\right)$ &$\\to$& $\\D 4{12}\\left(\\lambda,\\alpha\\right)$\n& $E_1^t = -\\alpha\\Theta t^{-1}e_1 + \\alpha\\lambda t^{-1}e_2 + \\alpha(\\Theta - \\alpha)t^{-1}e_3$  & $E_3^t = -\\alpha^3 t^{-1}e_3$\\\\\n &&& $E_2^t = -\\alpha t^{-1}e_1 + \\alpha(\\alpha t + \\lambda)(\\Theta t)^{-1}e_2 + \\alpha t^{-1}e_3$ & $E_4^t = \\alpha^4 t^{-2}e_4$\\\\\n\\hline\n\n$\\D 4{01}\\left(\\lambda\\left(1-\\frac {\\alpha t}{\\Psi^2}\\right), \\frac 1{\\Psi}\\left(\\alpha+\\frac{\\lambda}{t}\\right), \\frac 1t\\right)$ &$\\to$& $\\D 4{13}\\left(\\lambda,\\alpha\\right)$ \n& $E_1^t = -\\alpha\\Psi t^{-1}e_1 + \\alpha\\lambda t^{-1}e_2 + \\alpha(\\Psi - \\alpha)t^{-1}e_3$  & $E_3^t = -\\alpha^3 t^{-1}e_3$\\\\\n\\multicolumn{3}{|l|}{$\\Psi=1-\\Theta$}&\n $E_2^t = -\\alpha t^{-1}e_1 + \\alpha(\\alpha t + \\lambda)(\\Psi t)^{-1}e_2 + \\alpha t^{-1}e_3$ & $E_4^t = \\alpha^4 t^{-2}e_4$\\\\\n\\hline\n\n$\\D 4{12}\\left(\\lambda,\\alpha\\right)$ &$\\to$& $\\D 4{14}\\left(\\lambda,\\alpha\\right)$ \n& $E_1^t = t^{-1}e_1$  & $E_3^t = t^{-2}e_3$\\\\\n &&& $E_2^t = t^{-1}e_2$ & $E_4^t = t^{-3}e_4$\\\\\n\\hline\n\n$\\D 4{13}\\left(\\lambda,\\alpha\\right)$ &$\\to$& $\\D 4{15}\\left(\\lambda,\\alpha\\right)$ \n& $E_1^t = t^{-1}e_1$  & $E_3^t = t^{-2}e_3$\\\\\n &&& $E_2^t = t^{-1}e_2$ & $E_4^t = t^{-3}e_4$\\\\\n\\hline\n\n$\\D 4{01}\\left(t^2, \\alpha , t^{-1}\\right)$ &$\\to$& $\\D 4{16}\\left(\\alpha\\right)$ \n&\n\n$E^t_1= -\\alpha D e_1 + \\alpha(\\alpha t - 1) D^2 e_3$ &\n \n$E^t_3= -\\alpha^3 t D^2 e_3 -\\alpha^3 D^3 e_4$ \\\\\n\n\\multicolumn{3}{|l|}{$D=((\\alpha^2 + \\alpha + 1)t - \\alpha  - 1)^{-1}$}&\n$E^t_2= -\\alpha D e_1 + \\alpha^2t D e_2 + \\alpha(\\alpha t - 1) D^2 e_3$ &\n\n$E^t_4= \\alpha^4 t D^3 e_4$\\\\\n\\hline\n\n$\\D 4{01}\\left(t^2, -1 , t^{-1}\\right)$ &$\\to$& $\\D 4{17}\\left(\\alpha\\right)$ \n&\n\n$E^t_1= D e_1 + ((\\alpha - 2) t + \\alpha)D^2 e_3$ &\n \n$E^t_3= tD^2 e_3 + \\alpha D^3 e_4$ \\\\\n\n\\multicolumn{3}{|l|}{$D=((\\alpha + 1)t)^{-1}$}&\n$E^t_2= D e_1 + tD e_2 + ((\\alpha - 1)t + \\alpha )D^2 e_3$ &\n\n$E^t_4= tD^3 e_4$\\\\\n\\hline\n\n$\\D 4{01}\\left(\\lambda\\left(1+\\frac t{\\Theta^2}\\right), \\alpha + \\Psi, 1\\right)$ &$\\to$& $\\D 4{18}\\left(\\lambda,\\alpha\\right)$ \n&\n\\multicolumn{2}{l|}{$E^t_1= t\\Theta^3D(\\alpha + \\Psi)\\left( e_1 - \\Psi e_2 - \\Theta(\\Theta^2\\alpha + \\alpha t + \\lambda) D e_3\\right)$}\\\\\n\n\\multicolumn{3}{|l|}{$D=(\\alpha t^2+\\Theta\\alpha(\\alpha + \\Theta) t+\\Theta^2(\\alpha  + 1)(\\Theta\\alpha + \\Psi))^{-1}$}&\n\n\\multicolumn{2}{l|}{$E^t_2= t\\Theta(\\alpha + \\Psi)D\\left(\\Theta e_1+(t-\\lambda)e_2 - (t^2+(\\alpha  - \\Psi)\\Theta t+\\Theta(\\Theta\\alpha + \\Psi)\\Theta^2) D e_3\\right)$}\\\\\n\n\\multicolumn{3}{|l|}{$\\Psi=1-\\Theta$}&\n\n\\multicolumn{2}{l|}{$E^t_3= t^3\\Theta^4(\\alpha + \\Psi)^2D^2\\left(e_3 -\\Theta(\\Theta\\alpha  + t) D e_4\\right)$} \\\\\n\n&&&\n\n\\multicolumn{2}{l|}{$E^t_4= t^4\\Theta^6(\\alpha + \\Psi)^3D^3e_4$}\\\\\n\\hline\n\n$\\D 4{01}\\left(\\lambda\\left(1+\\frac t{\\Psi^2}\\right), \\alpha + \\Theta, 1\\right)$ &$\\to$& $\\D 4{19}\\left(\\lambda,\\alpha\\right)$ \n&\n\\multicolumn{2}{l|}{$E^t_1= t\\Psi^3D(\\alpha + \\Theta)\\left( e_1 - \\Theta e_2 - \\Psi(\\Psi^2\\alpha + \\alpha t + \\lambda) D e_3\\right)$}\\\\\n\n\\multicolumn{3}{|l|}{$D=(\\alpha t^2+\\Psi\\alpha(\\alpha + \\Psi) t+\\Psi^2(\\alpha  + 1)(\\Psi\\alpha + \\Theta))^{-1}$}&\n\n\\multicolumn{2}{l|}{$E^t_2= t\\Psi(\\alpha + \\Theta)D\\left(\\Psi e_1+(t-\\lambda)e_2 - (t^2+(\\alpha - \\Theta)\\Psi t+\\Psi(\\Psi\\alpha + \\Theta)\\Psi^2) D e_3\\right)$}\\\\\n\n\\multicolumn{3}{|l|}{$\\Psi=1-\\Theta$}&\n\n\\multicolumn{2}{l|}{$E^t_3= t^3\\Psi^4(\\alpha + \\Theta)^2D^2\\left(e_3 -\\Psi(\\Psi\\alpha  + t) D e_4\\right)$} \\\\\n\n&&&\n\n\\multicolumn{2}{l|}{$E^t_4= t^4\\Psi^6(\\alpha + \\Theta)^3D^3e_4$}\\\\\n\\hline\n\n$\\D 4{18}\\left(\\lambda,\\alpha\\right)$ &$\\to$& $\\D 4{20}\\left(\\lambda,\\alpha\\right)$ \n& $E_1^t = t^{-1}e_1$  & $E_3^t = t^{-2}e_3$\\\\\n &&& $E_2^t = t^{-1}e_2$ & $E_4^t = t^{-3}e_4$\\\\\n\\hline\n\n$\\D 4{19}\\left(\\lambda,\\alpha\\right)$ &$\\to$& $\\D 4{21}\\left(\\lambda,\\alpha\\right)$ \n& $E_1^t = t^{-1}e_1$  & $E_3^t = t^{-2}e_3$\\\\\n &&& $E_2^t = t^{-1}e_2$ & $E_4^t = t^{-3}e_4$\\\\\n\\hline\n\n$\\D 4{01}\\left(\\lambda\\left(1+\\frac t{\\Theta^2}\\right), -\\frac{\\lambda^2}{\\Theta^3}, 1\\right)$ &$\\to$& $\\D 4{22}\\left(\\lambda\\right)$ \n&\n\\multicolumn{2}{l|}{$E^t_1= \\Theta^2D(\\Theta e_1 - \\lambda e_2 - \\Theta\\lambda D e_3)$}\\\\\n\n\\multicolumn{3}{|l|}{$D=(\\Psi(t-2\\lambda+3\\Theta-1))^{-1}$}&\n\n\\multicolumn{2}{l|}{$E^t_2= \\Theta D\\left(\\Theta e_1 + (t-\\lambda)e_2 - \\Theta^3\\lambda^{-1}((2\\Theta - 1)t - \\Theta (2\\lambda + 1)  + 1)  D e_3\\right)$}\\\\\n\n\\multicolumn{3}{|l|}{$\\Psi=1-\\Theta$}&\n\n\\multicolumn{2}{l|}{$E^t_3= \\Theta^4D^2\\left(t e_3 - \\Theta^3\\lambda^{-1}(2\\Theta - 1)(t - \\Psi) D e_4\\right)$} \\\\\n\n&&&\n\n\\multicolumn{2}{l|}{$E^t_4= t\\Theta^6 D^3 e_4$}\n   \n    \\\\\n\\hline\n\n$\\D 4{01}\\left(\\lambda\\left(1+\\frac t{\\Psi^2}\\right), -\\frac{\\lambda^2}{\\Psi^3}, 1\\right)$ &$\\to$& $\\D 4{23}\\left(\\lambda\\right)$ \n&\n\\multicolumn{2}{l|}{$E^t_1= \\Psi^2D(\\Psi e_1 - \\lambda e_2 - \\Psi\\lambda D e_3)$}\\\\\n\n\\multicolumn{3}{|l|}{$D=(\\Theta(t-2\\lambda+3\\Psi-1))^{-1}$}&\n\n\\multicolumn{2}{l|}{$E^t_2= \\Psi D\\left(\\Psi e_1 + (t-\\lambda)e_2 - \\Psi^3\\lambda^{-1}((2\\Psi - 1)t - \\Psi (2\\lambda + 1)  + 1)  D e_3\\right)$}\\\\\n\n\\multicolumn{3}{|l|}{$\\Psi=1-\\Theta$}&\n\n\\multicolumn{2}{l|}{$E^t_3= \\Psi^4D^2\\left(t e_3 - \\Psi^3\\lambda^{-1}(2\\Psi - 1)(t - \\Theta) D e_4\\right)$} \\\\\n\n&&&\n\n\\multicolumn{2}{l|}{$E^t_4= t\\Psi^6 \\lambda^{-1} D^3 e_4$}\n   \n    \\\\\n \n\\hline\n\n$\\D 4{01}\\left(\\lambda\\left(1-\\frac t{\\Theta^2}\\right), -\\frac{\\lambda^2}{\\Theta^3}, 1\\right)$ &$\\to$& $\\D 4{24}\\left(\\lambda\\right)$ \n&\n\n$E^t_1=  -\\Theta D\\left(\\Theta e_1 - \\lambda e_2 + \\Theta^2 D e_3\\right)$ &\n \n$E^t_3= -t\\Theta^2 D^2 e_3$ \\\\\n\n\\multicolumn{3}{|l|}{$D=(t - 2\\Theta + 1)^{-1}$}&\n$E^t_2= -D\\left(\\Theta e_1 - (t+\\lambda) e_2 + \\Theta^2 D e_3\\right)$ &\n\n$E^t_4= t\\Theta^3 D^3 e_4$ \\\\\n\\hline\n\n$\\D 4{01}\\left(\\lambda\\left(1-\\frac t{\\Psi^2}\\right), -\\frac{\\lambda^2}{\\Psi^3}, 1\\right)$ &$\\to$& $\\D 4{25}\\left(\\lambda\\right)$ \n&\n\n$E^t_1=  -\\Psi D\\left(\\Psi e_1 - \\lambda e_2 + \\Psi^2 D e_3\\right)$ &\n  \n$E^t_3= -t\\Psi^2 D^2 e_3$ \\\\\n\n\\multicolumn{3}{|l|}{$D=(t - 2\\Psi + 1)^{-1},\\ \\Psi=1-\\Theta$}&\n$E^t_2= -D\\left(\\Psi e_1 - (t+\\lambda) e_2 + \\Psi^2 D e_3\\right)$ &\n\n$E^t_4= t\\Psi^3\\lambda^{-1} D^3 e_4$\\\\\n\\hline\n\n$\\D 4{01}\\left(\\lambda\\left(1+\\frac t{\\Theta^2}\\right), -\\Theta, 1\\right)$ &$\\to$& $\\D 4{26}\\left(\\lambda\\right)$ \n& $E_1^t = \\Theta t^{-1}\\left(\\Theta e_1 - \\lambda e_2 - \\Theta t^{-1} e_3\\right)$  & $E_3^t = \\Theta^2 t^{-1} e_3$\\\\\n &&& $E_2^t = t^{-1}\\left(\\Theta e_1 + (t-\\lambda) e_2 - \\Theta^2 t^{-1} e_3\\right)$ & $E_4^t = \\Theta^3 t^{-2} e_4$\\\\\n\\hline\n\n$\\D 4{01}\\left(\\lambda\\left(1+\\frac t{\\Psi^2}\\right), -\\Psi, 1\\right)$ &$\\to$& $\\D 4{27}\\left(\\lambda\\right)$\n& $E_1^t = \\Psi t^{-1}\\left(\\Psi e_1 - \\lambda e_2 - \\Psi t^{-1} e_3\\right)$  & $E_3^t = \\Psi^2 t^{-1} e_3$\\\\\n &&& $E_2^t = t^{-1}\\left(\\Psi e_1 + (t-\\lambda) e_2 - \\Psi^2 t^{-1} e_3\\right)$ & $E_4^t = \\Psi^3 t^{-2} e_4$\\\\\n\\hline\n\n$\\D 4{01}\\left(\\lambda\\left(1+\\frac t{\\Theta^2}\\right), -\\Theta, 1\\right)$ &$\\to$& $\\D 4{28}\\left(\\lambda\\right)$ \n&\n\n\\multicolumn{2}{l|}{$E^t_1=  \\Theta^3D\\left(\\Theta e_1 - \\lambda e_2 - t^{-1}\\Theta^2(\\Psi^2t + 4\\Theta\\lambda - \\Theta) D e_3\\right)$}\\\\\n  \n\\multicolumn{3}{|l|}{$D=(\\Psi t-2\\lambda+\\Theta)^{-1},\\ \\Psi=1-\\Theta$}&\n\n\\multicolumn{2}{l|}{$E^t_2= \\Theta^2D\\left(\\Theta e_1 + (t-\\lambda) e_2 + t^{-1}\\Theta((1 - 2\\Theta)t^2 + \\Theta(3\\lambda-2\\Psi)t -\\Theta^2(4\\lambda-1)) D e_3\\right)$}\\\\\n\n&&&\n\n\\multicolumn{2}{l|}{$E^t_3=  \\Theta^6 D^2\\left(t e_3 + (2\\lambda-\\Theta)(t - \\Theta) D e_4\\right)$} \\\\\n\n&&&\n\n\\multicolumn{2}{l|}{$E^t_4= t\\Theta^9 D^3 e_4$}\\\\\n\\hline\n\n$\\D 4{01}\\left(\\lambda\\left(1+\\frac t{\\Psi^2}\\right), -\\Psi, 1\\right)$ &$\\to$& $\\D 4{29}\\left(\\lambda\\right)$ \n&\n\n\\multicolumn{2}{l|}{$E^t_1=  \\Psi^3D\\left(\\Psi e_1 - \\lambda e_2 - t^{-1}\\Psi^2(\\Theta^2t + 4\\Psi\\lambda - \\Psi) D e_3\\right)$}\\\\\n  \n\\multicolumn{3}{|l|}{$D=(\\Theta t-2\\lambda+\\Psi)^{-1},\\ \\Psi=1-\\Theta$}&\n\n\\multicolumn{2}{l|}{$E^t_2= \\Psi^2D\\left(\\Psi e_1 + (t-\\lambda) e_2 + t^{-1}\\Psi((1 - 2\\Psi)t^2 + \\Psi(3\\lambda-2\\Theta)t -\\Psi^2(4\\lambda-1)) D e_3\\right)$}\\\\\n\n&&&\n\n\\multicolumn{2}{l|}{$E^t_3=  \\Psi^6 D^2\\left(t e_3 + (2\\lambda-\\Psi)(t - \\Psi) D e_4\\right)$} \\\\\n\n&&&\n\n\\multicolumn{2}{l|}{$E^t_4= t\\Psi^9 D^3 e_4$}\\\\\n\\hline\n\n$\\D 4{01}\\left(\\lambda\\left(1-\\left(2+\\frac{2\\lambda-1}{\\Theta^2}\\right)t\\right), \\frac\\lambda{\\Theta^2}\\left(1-\\frac\\Theta t\\right), \\frac 1t\\right)$ &$\\to$& $\\D 4{30}\\left(\\lambda\\right)$ \n&\n\n\\multicolumn{2}{l|}{$E^t_1= -D\\left(\\Theta e_1 - \\lambda e_2 + t\\Psi^{-1}D e_3\\right)$}\\\\\n  \n\\multicolumn{3}{|l|}{$D=\\frac{\\lambda(t-\\Theta)}{t\\left((1-2\\Theta)t+\\Theta^2\\right)},\\ \\Psi=1-\\Theta$}&\n\n\\multicolumn{2}{l|}{$E^t_2= -D\\left(e_1 - \\Theta^{-1}((2\\Theta-1)t + \\lambda) e_2 + t\\Psi^{-1}(2t-\\Theta)(t-\\Theta)^{-1} D e_3\\right)$}\\\\\n\n&&&\n\n\\multicolumn{2}{l|}{$E^t_3= tD^2\\left((1-2\\Theta)e_3 + \\Theta^2\\lambda^{-1}(t-\\Theta)^{-1}D e_4\\right)$} \\\\\n\n&&&\n\n\\multicolumn{2}{l|}{$E^t_4= t D^3(2\\Theta - 1) e_4$}\n\n    \\\\\n\\hline\n\n$\\D 4{01}\\left(\\lambda\\left(1-\\left(2+\\frac{2\\lambda-1}{\\Psi^2}\\right)t\\right), \\frac\\lambda{\\Psi^2}\\left(1-\\frac\\Psi t\\right), \\frac 1t\\right)$ &$\\to$& $\\D 4{31}\\left(\\lambda\\right)$ \n&\n\n\\multicolumn{2}{l|}{$E^t_1= -D\\left(\\Psi e_1 - \\lambda e_2 + t\\Theta^{-1}D e_3\\right)$}\\\\\n  \n\\multicolumn{3}{|l|}{$D=\\frac{\\lambda(t-\\Psi)}{t\\left((1-2\\Psi)t+\\Psi^2\\right)},\\ \\Psi=1-\\Theta$}&\n\n\\multicolumn{2}{l|}{$E^t_2= -D\\left(e_1 - \\Psi^{-1}((2\\Psi-1)t + \\lambda) e_2 + t\\Theta^{-1}(2t-\\Psi)(t-\\Psi)^{-1} D e_3\\right)$}\\\\\n\n&&&\n\n\\multicolumn{2}{l|}{$E^t_3= tD^2\\left((1-2\\Psi)e_3 + \\Psi^2\\lambda^{-1}(t-\\Psi)^{-1}D e_4\\right)$} \\\\\n\n&&&\n\n\\multicolumn{2}{l|}{$E^t_4= t D^3(2\\Psi - 1) e_4$}\n\n    \\\\\n\\hline\n\n$\\D 4{30}\\left(\\lambda\\right)$ &$\\to$& $\\D 4{32}\\left(\\lambda\\right)$ \n& $E_1^t = t^{-1}e_1$  & $E_3^t = t^{-2}e_3$\\\\\n &&& $E_2^t = t^{-1}e_2$ & $E_4^t = t^{-3}e_4$\\\\\n\\hline\n\n$\\D 4{31}\\left(\\lambda\\right)$ &$\\to$& $\\D 4{33}\\left(\\lambda\\right)$ \n& $E_1^t = t^{-1}e_1$  & $E_3^t = t^{-2}e_3$\\\\\n &&& $E_2^t = t^{-1}e_2$ & $E_4^t = t^{-3}e_4$\\\\\n\\hline\n\n$\\D 4{01}\\left(\\frac t{(t + 1)2}, 1 , 1 + \\frac 1t\\right)$ &$\\to$& $\\D 4{34}$\n&\n\n\\multicolumn{2}{l|}{$E^t_1= -(t + 1)^2 D e_1 + (t + 1)t D e_2 + (t + 1)^2(t - 1)D^2 e_3$}\\\\\n  \n\\multicolumn{3}{|l|}{$D=(t^2 + t - 1)^{-1}$}&\n\n\\multicolumn{2}{l|}{$E^t_2= -(t + 1)^2 D e_1 + (t + 1)^2t D e_2 + (t + 1)^2 D e_3$}\\\\\n\n&&&\n\\multicolumn{2}{l|}{$E^t_3=  -(t + 1)^2t^2 D^2 e_3 - (t + 1)^4t D^3 e_4$} \\\\\n\n&&&\n\n\\multicolumn{2}{l|}{$E^t_4= (t + 1)^4t^2 D^3 e_4$}\n    \\\\\n\\hline\n\n$\\D 4{01}\\left(\\lambda, t^{-1}, 1\\right)$ &$\\to$& $\\D 4{35}\\left(\\lambda\\right)$ \n& $E_1^t = t^{-1}e_1$  & $E_3^t = t^{-2}e_3 + (\\lambda t^2)^{-1}e_4$\\\\\n &&& $E_2^t = t^{-1}e_2 + (\\lambda t)^{-1}e_3$ & $E_4^t = t^{-4}e_4$\\\\\n\\hline\n\n$\\D 4{35}\\left(\\lambda\\right)$ &$\\to$& $\\D 4{36}\\left(\\lambda\\right)$ \n& $E_1^t = t^{-1}e_1$  & $E_3^t = t^{-2}e_3$\\\\\n &&& $E_2^t = t^{-1}e_2$ & $E_4^t = t^{-3}e_4$\\\\\n\\hline\n\n$\\D 4{01}\\left(\\lambda,\\Theta t^{-1},t^{-1}\\right)$ &$\\to$& $\\D 4{37}\\left(\\lambda\\right)$ \n& $E_1^t = te_1$  & $E_3^t = t^2e_3$\\\\\n &&& $E_2^t = te_2$ & $E_4^t = t^2e_4$\\\\\n\\hline\n\n$\\D 4{01}\\left(\\lambda,(1-\\Theta) t^{-1},t^{-1}\\right)$ &$\\to$& $\\D 4{38}\\left(\\lambda\\right)$ \n& $E_1^t = te_1$  & $E_3^t = t^2e_3$\\\\\n &&& $E_2^t = te_2$ & $E_4^t = t^2e_4$\\\\\n\\hline\n\n$\\D 4{37}\\left(\\lambda\\right)$ &$\\to$& $\\D 4{39}\\left(\\lambda\\right)$ \n& $E_1^t = t^{-1}e_1$  & $E_3^t = t^{-2}e_3$\\\\\n &&& $E_2^t = t^{-1}e_2$ & $E_4^t = t^{-3}e_4$\\\\\n\\hline\n\n$\\D 4{38}\\left(\\lambda\\right)$ &$\\to$& $\\D 4{40}\\left(\\lambda\\right)$ \n& $E_1^t = t^{-1}e_1$  & $E_3^t = t^{-2}e_3$\\\\\n &&& $E_2^t = t^{-1}e_2$ & $E_4^t = t^{-3}e_4$\\\\\n\\hline\n\n\\end{longtable}\n}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Perfect matchings in the operator formalism}\n\\label{sec:perf-match-oper}\n\n\nIn Section~\\ref{sec:loop-expans-ferm}, we found the loop states to be\ncombinations of two perfect matchings. It would be interesting to\nrecover the same result in terms of the operator formalism of\nSection~\\ref{sec:loops-as-fermionic}. To do so, we start by\ndistinguishing between even and odd plaquettes. We will say that a\nplaquette with coordinates $\\mathbf{z} = (z, w) $ is \\emph{even}\n(resp. \\emph{odd}) if $z + w $ is even (resp. odd). We use the\ndefinition of the height function from the superposition of two\nperfect matchings $\\mathrm{PM}_1-\\mathrm{PM}_2$ given in Section\n\\ref{sec:preliminaries}. If an odd plaquette contains either an upward\narrow on the left side, a downward arrow on the right side, a right\npointing arrow on the bottom side or a left pointing arrow on the\nupper side, it belongs to the first perfect matching $\\mathrm{PM}_1$,\nin the converse case, it belongs to $\\mathrm{PM}_2$. For an even\nplaquette, the arrows are reversed. The strategy to construct an\noperator which only preserves the operators corresponding to one of\nthe perfect matchings, say $\\mathrm{PM}_1$, when acting on a state\n$\\ket{\\Psi}$ is simple. Consider \\emph{e.g.} all the black nodes. For each of\nthem add all the annihilators corresponding to the arrows belonging to\n$\\mathrm{PM}_2$. If $\\ket{\\Psi}$ is an allowed state, only one\noperator is paired for each node which gives a $1$ as a result of the\nanticommutation, while all the others will just annihilate the\nstate. This results in only the operators belonging to $\\mathrm{PM}_1$\nsurviving and so yielding the result we are looking for. We can define\nthe operators ${PM}_1$ and ${PM}_2$ explicitly as follows:\n\\begin{subequations}\n  \\label{eq:loop-splitting}\n  \\begin{align}\n    {PM}_1 &= \\prod_{z = 1}^{N} \\prod_{w = 1}^{M} \\frac{1 + \\left( -1 \\right)^{z + w}}{2} \\left[ a(z,w) + d(z, w) + b(z, w - 1) + c(z-1,w) \\right] \\, ,\\\\\n    {PM}_2 &= \\prod_{z = 1}^{z} \\prod_{w = 1}^{w} \\frac{1 +\n      \\left( -1 \\right)^{z + w}}{2} \\left[ b(z,w) + c(z,\n      w) + a(z, w - 1) + d(z-1,w) \\right] \\, .\n  \\end{align}\n\\end{subequations}\nA state $\\ket{\\Psi} = \\kket{m_1} \\otimes \\kket{m_2}$ is decomposed\ninto its two constituent perfect matchings by their action:\n\\begin{equation}\n  PM_i \\ket{\\Psi} = \\kket{m_i} \\, , \\hspace{1em} i = 1,2 \\, .\n\\end{equation}\nNow we are able to give a consistency condition for a state to\ncorrespond to a loop appearing in the expansion of the fermionic\naction. The idea is that the state should be decomposable into two\nmatchings, which both touch each node once. Let us therefore define\n\\begin{subequations}\n  \\begin{align}\n    P_1 (z,w) &= \\textstyle{\\frac{1 + \\left( -1 \\right)^{z + w}}{2}}\n    \\left[ N_a(z,w) + N_d(z, w) + N_b(z, w - 1) +\n      N_c(z-1,w)\n      - 1\\right] \\, , \\\\\n    P_2 (z,w) &= \\textstyle{\\frac{1 + \\left( -1 \\right)^{z + w}}{2}}\n    \\left[ N_b(z,w) + N_c(z, w) + N_a(z, w - 1) +\n      N_d(z-1,w) - 1\\right] \\, .\n  \\end{align}\n\\end{subequations}\nAn allowed state $\\ket{\\Psi}$ must satisfy the local conditions\n\\begin{equation}\n  \\label{eq:state-consistency}\n  P_1 (\\mathbf{z}) \\ket{\\Psi} = P_2 (\\mathbf{z}) \\ket{\\Psi} = 0 \\, , \\hspace{1em} \\forall \\mathbf{z} \\, .  \n\\end{equation}\n\n\n\nThe action of the height operators in Eq.~\\eqref{eq:global-height} is\nstill defined on the perfect matching states $\\kket{\\Psi}$. To be\nconsistent with the constructions in Section \\ref{sec:preliminaries},\nwe introduce a sign to distinguish between $\\kket{m_1}$ and\n$\\kket{m_2}$:\n\\begin{subequations}\n\\label{eq:matching-charges}\n  \\begin{align}\n    H_z \\kket{m_i} &= \\left(-1 \\right)^{i+1} \\left[ \\sum_{\\zeta =\n        1}^{N} N_a ( \\zeta, \\bar w) - N_b ( \\zeta, \\bar w)\n    \\right] \\kket{m_i} \\, , \\\\\n    H_w \\kket{m_i} &= \\left(-1 \\right)^{i+1} \\left[ \\sum_{\\omega =\n        1}^{M} N_c ( \\bar z, \\omega) - N_d ( \\bar z, \\omega)\n    \\right] \\kket{m_i} \\, .\n  \\end{align}\n\\end{subequations}\nWe find that each loop can be identified by four charges,\ncorresponding to the eigenvalues of the winding operators acting on\nthe two perfect matchings. The relation between the winding of a loop\nand the weights of the constituent perfect matchings is\n\\begin{subequations}\n  \\begin{align}\n    H_z \\ket{\\Psi} &= H_z \\kket{m_1} - H_z \\kket{m_2} \\, ,\\\\\n    H_w \\ket{\\Psi} &= H_w \\kket{m_1} - H_w \\kket{m_2} \\, .\n  \\end{align}\n\\end{subequations}\nExtending the definition of the height function to perfect matchings\nrequires some care, since the equality in\nEq.~\\eqref{eq:height-equality} does not hold for perfect matchings. A\npossible way out consists in symmetrizing the sum of the two\nexpressions and define\n\\begin{equation}\n  h (z, w ) \\kket{m_i} = \\frac{\\left(-1 \\right)^{i+1}}{2} \\left[ \\sum_{\\zeta = 0}^{z} N_a ( \\zeta, w) - N_b ( \\zeta, w) + \\sum_{\\omega = 0}^{w} N_c ( z, \\omega) - N_d ( z, \\omega) \\right] \\kket{m_i} \\, . \n\\end{equation}\nIn this way, one recovers once more\n\\begin{equation}\n  h (\\mathbf{z}) \\ket{\\Psi} = h (\\mathbf{z}) \\kket{m_1} - h (\\mathbf{z}) \\kket{m_2} \\, .\n\\end{equation}\n\n\n\n\n\n\n\\section{Generating functions}\n\\label{sec:generating-functions}\n\nIn this section, we give the generating function which contains the full\ninformation of the fermion loop gas, \\emph{i.e.} all the\nloop states. The dimer model partition function and the Witten index\ncan be recovered from it.\n\n\n\\subsection{Generating function for the fermion loops on the cylinder}\n\\label{sec:generating-function}\n\nWe compute the generating functions\nfor the loops defined as\n\\begin{equation}\n  G (q, z, y) = \\Tr \\left[ q^H z^K y^F \\right] \\, ,  \n\\end{equation}\nwhich keeps track of all the information. Expanding the trace one finds\nan explicit expression depending on the matching degeneracies $N_k$:\n\\begin{equation}\n  \\label{eq:generating-cylinder}\n  G( q, z, y ) = \\sum_{k=0}^N N_k\\, y^{2k} z^k + \\left( 1 + y \\right) \\left\\{ \\frac{1}{2}\\sum_{k=0}^N N_k \\left( N_k - 1 \\right) y^{2k} z^k + \\sum_{l=1}^N q^l \\sum_{k=l}^N N_k\\, N_{k-l}\\, y^{2k} z^k \\right\\} \\, . \n\\end{equation}\n\nAll the physical information about the system is contained in the\ngenerating function $G$. For example:\n\\begin{itemize}\n\\item the Euler number for the complex in Eq.~\\eqref{eq:SQM-complex}\n  is $\\,\\chi = \\Tr \\left[ \\left( -1 \\right)^F \\right] = G(1,1,-1)$ ;\n\\item the Poincar\\'e polynomial for the complex is $\\,\\chi (z) = \\Tr \\left[ \\left( -1\n    \\right)^F z^k \\right] = G(1,z,-1)$\\,;\n\\item the total number of perfect matchings is $N = \\Tr \\left[ \\left(\n      -1 \\right)^{K+F} \\right] = G(1,-1,-1)$~;\n\\item the multiplicity of weight-$k$ matchings is $N_k = \\frac{1}{k!}\n  \\left. \\frac{\\del^k}{\\del z^k } G(q,\\frac{z}{y},y)\\right|_{q=1,z=0,y=-1}$~;\n\\item the dimension of the $k$-th chain group is $\\,\\dim C_k =\n  \\frac{1}{k!}  \\left. \\frac{\\del^k}{\\del y^k}\n    G(q,z,y)\\right|_{q=1,z=1,y=0}$~;\n\\item the number of loops with winding number $h$ is $L_h =\n  \\frac{1}{h!}  \\left. \\frac{\\del^h}{\\del q^h}\n    G(q,z,y)\\right|_{q=0,z=1,y=1}$~.\n\\end{itemize}\n\n\n\\subsection{Generating function for the fermion loops on the torus}\n\\label{sec:generating-function-1}\n\nLet us now extend the definition of generating function for the loops\nto graphs embedded on the torus. Following the formula for the\ncylinder in Eq.~\\eqref{eq:generating-cylinder} we define:\n\\begin{equation}\n  G (\\mathbf{q}, \\mathbf{z}, y) = \\Tr \\left[ \\mathbf{q}^{\\mathbf{H}} \\mathbf{z}^{\\mathbf{K}} y^F \\right] \\, ,\n\\end{equation}\nwhere the notation $\\mathbf{q}^{\\mathbf{H}} $ is a shortcut for\n$\\mathbf{q}^{\\mathbf{H}} \\doteq q_z^{H_z} q_w^{H_w}$ and\n$\\mathbf{z}^{\\mathbf{k}} \\doteq z^{k_z} w^{k_w}$. Expanding in the loop basis we find:\n\\begin{multline}\n  G ( \\mathbf{q}, \\mathbf{z}, y ) %\n\n  = \\sum_{\\mathbf{k}=(0,0)}^{\\mathbf{N}} N_{\\mathbf{k}}\n  y^{2 k_z} \\mathbf{z}^\\mathbf{k} + \\frac{ 1 + y}{2} \\left\\{\n     \\sum_{\\mathbf{k} = (0,0)}^{\\mathbf{N}}\n     N_{\\mathbf{k}} \\left( N_{\\mathbf{k}} - 1 \\right) y^{2k_z} \\mathbf{z}^{\\mathbf{k}} +  \\right. \\\\\n   \\left. + \\sideset{}{'}\\sum_{\\mathbf{l} = (0,0)}^{\\mathbf{N}} \\mathbf{q}^{\\mathbf{l}} \\sum_{\\mathbf{k}\n       = \\mathbf{l}}^{\\mathbf{N}} \\left( N_{\\mathbf{k}}\n       N_{\\mathbf{k}-\\mathbf{l}} + N_{k_z (k_w-l_w)} N_{(k_z-l_z)k_w}\n     \\right)  y^{2 k_z} \\mathbf{z}^{\\mathbf{k}} \\right\\} \\, ,\n\\end{multline}\nwhere the primed sum $\\displaystyle{\\sideset{}{'}\\sum_{\\mathbf{l}}}$\nmeans that the value $\\mathbf{l}=(0,0)$ is to be omitted. All the\nphysics of the system is encoded into $G ( \\mathbf{q}, \\mathbf{z}, y\n)$. In particular:\n\\begin{itemize}\n\\item the Euler number for the complex in Eq.~\\eqref{eq:torus-complex}\n  is $\\,\\chi(w)=\\Tr \\left[ (-1)^F w^{K_w} \\right]$ \\\\ $= G((1,1),(1,w),-1)$~;\n\\item the Poincar\\'e characteristic for the complex is $ \\chi (z,w) = \\Tr \\left[ (-1)^F\n    \\mathbf{z}^{\\mathbf{k}} \\right]$\\\\  $= G((1,1),\\mathbf{z},-1)$~;\n\\item the number of perfect matchings with height $\\mathbf{k}$ is\n  $N_{\\mathbf{k}} = \\frac{1}{\\mathbf{k}!}\n  \\left. \\frac{\\del^{\\abs{\\mathbf{k}}}}{\\del \\mathbf{z}^{\\mathbf{k}}}\n    G \\right|_{\\mathbf{q}= (1,1), \\mathbf{z} = (0,0), y = -1}$~, where\n  we introduced the notation $\\mathbf{k}! \\doteq k_z! k_w !$~,\n  $\\abs{\\mathbf{k}} \\doteq k_z + k_w $~, and\n  $\\frac{\\del^{\\abs{\\mathbf{k}}}}{\\del \\mathbf{z}^{\\mathbf{k}}} \\doteq\n  \\frac{\\del^{k_z + k_w}}{\\del z^{k_w} \\del w^{k_w}}$~;\n\\item the dimension of the $C_{\\mathbf{k}}$ chain group is $\\dim\n  C_{\\mathbf{k}} = \\frac{1}{\\mathbf{k!}}\n  \\left. \\frac{\\del^{\\abs{\\mathbf{k}}}}{\\del y^{k_z} \\del w^{k_w}} G\n  \\right|_{\\mathbf{q}= (1,1), \\mathbf{z} = (0,1), y = 1}$~;\n\\item the number of loops with winding $\\mathbf{h}$ is $L_{\\mathbf{h}}\n  = \\frac{1}{\\mathbf{h}!} \\left. \\frac{\\del^{\\abs{\\mathbf{h}}}}{\\del\n      \\mathbf{q}^{\\mathbf{h}}} G \\right|_{\\mathbf{q}= (0,0),\n    \\mathbf{z} = (1,1), y = 1}$~.\n\\end{itemize}\n\n\n\n\n\\section{Example: One square on the torus}\n\\label{sec:ex_torus}\n\n\nTo illustrate the construction given in Section~\\ref{sec:sqm_torus}, we now present the smallest possible example\non the torus, consisting only of one square.  The eight possible\nmatchings and their quantum numbers are shown in Figure\n\\ref{fig:ex_matchings}.\n\\begin{figure}[h!]\n  \\begin{center}\n    \\includegraphics[width=110mm]{ex_matchings-T}\n    \\caption{Matchings for the square on the torus}\n    \\label{fig:ex_matchings-T}\n  \\end{center}\n\\end{figure}\nThey combine into 64 loop configurations, which are, sorted into their\nco--chain groups according to total winding number, depicted in Table~\\ref{chaingroupsT}.\n\nThe partition function of the dimer model is the one given in (\\ref{eq:partition_square}),\nthe generating function for this example reads\n\\begin{eqnarray}\n  G( \\mathbf{q}, \\mathbf{z}, y ) &=& w + y^2 z + 4 w y^2 z + w^2 y^2 z + w y^4 z^2 +\\nonumber \\\\\n && +\\left( 1 + y \\right) \\left[ \\left( 6 w y^2 z + 4 q_z + 4 q_w +  q_zq_w \\right) w y^2 z + \\left( 4 q_w + q_w^2 + q_zq_w \\right) w^2 y^2 z + \\right. \\nonumber\\\\\n && \\left. + \\left( 4 q_z + q_z^2 + q_zq_w \\right) w y^4 z^2 + q_zq_w w^2 y^4 z^2 \\right].\n\\end{eqnarray}\n\n\\afterpage{%\n  \\begin{center}\n    \\begin{longtable}{ccccccc}\n      \\caption[Co--chain groups for the square on the torus]{Co--chain groups\n        for the square on the torus}\n      \\label{chaingroupsT}\\\\\n      \n      \\toprule {$\\,K_w\\,$}& $C^0$ &{$C^1$} \n      &{$C^2$}& {$C^3$} & {$C^4$} & {$C^5$}  \\cr \\midrule\n      \\endfirsthead\n\n      \\multicolumn{7}{l}\n      {\\tablename\\ \\thetable{} -- continued from previous page} \\\\\n      \\toprule\n      $\\,K_w\\,$& $C^0$ &{$C^1$} & {$C^2$}& {$C^3$} & {$C^4$} & {$C^5$} \\cr\n      \\midrule\n      \\endhead\n      \n      \\midrule \\multicolumn{7}{r}{{Continued on next page}} \\\\ \\bottomrule\n      \\endfoot\n      \n      \\bottomrule\n      \\endlastfoot\n      \n      \\rowcolor[gray]{.95} 2& -& - & \\pb{$\\ket{1,2,1;0,1,1}_+$  \\includegraphics[height=10mm]{121211}} & \\pb{$\\ket{1,2,1;0,1,1}_{-}$  \\includegraphics[height=10mm]{211121}} & - & - \\cr\n      2 & -& - & \\pb{$\\ket{1,2,1;1,0,1}_+$ \\includegraphics[height=10mm]{121101}}&\\pb{$\\ket{1,2,1;1,0,1}_{-}$ \\includegraphics[height=10mm]{101121}} & - & - \\cr\n      \\rowcolor[gray]{.95} 2 & -& - &  \\pb{$\\ket{1,2,1;1,1,4}_+$ \\includegraphics[height=10mm]{121114}} &\\pb{$\\ket{1,2,1;1,1,4}_{-}$ \\includegraphics[height=10mm]{114121}} & - & - \\cr\n      2 & -& - &  \\pb{$\\ket{1,2,1;1,1,3}_+$ \\includegraphics[height=10mm]{121113}}&\\pb{$\\ket{1,2,1;1,1,3}_{-}$ \\includegraphics[height=10mm]{113121}} & - & - \\cr\n      \\rowcolor[gray]{.95} 2 & -& - &  \\pb{$\\ket{1,2,1;1,1,2}_+$  \\includegraphics[height=10mm]{121112}}&\\pb{$\\ket{1,2,1;1,1,2}_{-}$ \\includegraphics[height=10mm]{112121}} & - & - \\cr\n      2 & -& - &  \\pb{$\\ket{1,2,1;1,1,1}_+$ \\includegraphics[height=10mm]{121111}} &\\pb{$\\ket{1,2,1;1,1,1}_{-}$ \\includegraphics[height=10mm]{111121}} & - & - \\cr\n      \\rowcolor[gray]{.95} 2 & -& - &  \\pb{$\\ket{1,2,1;1,2,1}$ \\includegraphics[height=9mm]{121121}}&-&\\pb{$\\ket{2,1,1;1,2,1}_+$ \\includegraphics[height=10mm]{011121}} &\\pb{$\\ket{2,1,1;1,2,1}_-$ \\includegraphics[height=10mm]{121011}} \\cr\n      \\midrule\n      1 & -& - & \\pb{$\\ket{1,1,4;0,1,1}_+$ \\includegraphics[height=10mm]{114211}} & \\pb{$\\ket{1,1,4;0,1,1}_-$ \\includegraphics[height=10mm]{211114}} & - &-\\cr\n      \\rowcolor[gray]{.95} 1 & -& - & \\pb{$\\ket{1,1,3;0,1,1}_+$ \\includegraphics[height=10mm]{113211}}& \\pb{$\\ket{1,1,3;0,1,1}_-$ \\includegraphics[height=10mm]{211113}}&  -&-\\cr\n      1 & -& - & \\pb{$\\ket{1,1,2;0,1,1}_+$ \\includegraphics[height=10mm]{112211}} & \\pb{$\\ket{1,1,2;0,1,1}_-$ \\includegraphics[height=10mm]{211112}} &  -&-\\cr\n      \\rowcolor[gray]{.95} 1 & -& - & \\pb{$\\ket{1,1,1;0,1,1}_+$ \\includegraphics[height=10mm]{111211}}& \\pb{$\\ket{1,1,1;0,1,1}_-$ \\includegraphics[height=10mm]{211111}}&  -&-\\cr\n      1 & -& - & \\pb{$\\ket{1,0,1;0,1,1}_+$ \\includegraphics[height=10mm]{101211}}& \\pb{$\\ket{1,0,1;0,1,1}_-$ \\includegraphics[height=10mm]{211101}}&  -&-\\cr\n      \\rowcolor[gray]{.95} 1 & -& - & \\pb{$\\ket{1,1,4;1,0,1}_+$ \\includegraphics[height=10mm]{101114}}& \\pb{$\\ket{1,1,4;1,0,1}_-$ \\includegraphics[height=10mm]{114101}}&  -&- \\cr\n      1 & -&  - & \\pb{$\\ket{1,1,3;1,0,1}_+$ \\includegraphics[height=10mm]{113101}}& \\pb{$\\ket{1,1,3;1,0,1}_-$ \\includegraphics[height=10mm]{101113}}&  -&-\\cr\n      \\rowcolor[gray]{.95} 1 & -& - & \\pb{$\\ket{1,1,2;1,0,1}_+$ \\includegraphics[height=10mm]{112101}} & \\pb{$\\ket{1,1,2;1,0,1}_-$ \\includegraphics[height=10mm]{101112}}&  -&-\\cr\n      \\rowcolor[gray]{.95} 1 & -& - & \\pb{$\\ket{1,1,1;1,0,1}_+$ \\includegraphics[height=10mm]{101111}} & \\pb{$\\ket{1,1,1;1,0,1}_-$ \\includegraphics[height=10mm]{111101}}&  -&-\\cr\n      1 & -& - & \\pb{$\\ket{1,1,4;1,1,3}_+$ \\includegraphics[height=10mm]{113114}}& \\pb{$\\ket{1,1,4;1,1,3}_-$ \\includegraphics[height=10mm]{114113}}&  -&-\\cr\n      \\rowcolor[gray]{.95} 1 & -& - & \\pb{$\\ket{1,1,4;1,1,2}_+$ \\includegraphics[height=10mm]{114112}}& \\pb{$\\ket{1,1,4;1,1,2}_-$ \\includegraphics[height=10mm]{112114}}&  -&-\\cr\n      1 & -& - & \\pb{$\\ket{1,1,4;1,1,1}_+$ \\includegraphics[height=10mm]{114111}} & \\pb{$\\ket{1,1,4;1,1,1}_-$ \\includegraphics[height=10mm]{111114}}&  -&-\\cr\n      \\rowcolor[gray]{.95} 1 & -& - & \\pb{$\\ket{1,1,3;1,1,2}_+$ \\includegraphics[height=10mm]{113112}}& \\pb{$\\ket{1,1,3;1,1,2}_-$ \\includegraphics[height=10mm]{112113}} & \\pb{$\\ket{2,1,1;0,1,1}_+$ \\includegraphics[height=10mm]{011211}}  &\\pb{$\\ket{2,1,1;0,1,1}_-$ \\includegraphics[height=10mm]{211011}} \\cr\n      1 & -& - & \\pb{$\\ket{1,1,3;1,1,1}_+$ \\includegraphics[height=10mm]{113111}} & \\pb{$\\ket{1,1,3;1,1,1}_-$ \\includegraphics[height=10mm]{111113}} & \\pb{$\\ket{2,1,1;1,0,1}_+$ \\includegraphics[height=10mm]{011101}} & \\pb{$\\ket{2,1,1;1,0,1}_-$ \\includegraphics[height=10mm]{101011}} \\cr\n      \\rowcolor[gray]{.95} 1 & -& - & \\pb{$\\ket{1,1,2;1,1,1}_+$ \\includegraphics[height=10mm]{112111}} & \\pb{$\\ket{1,1,2;1,1,1}_-$ \\includegraphics[height=10mm]{111112}} &\\pb{$\\ket{2,1,1;1,1,4}_+$ \\includegraphics[height=10mm]{011114}}  &\\pb{$\\ket{2,1,1;1,1,4}_-$  \\includegraphics[height=10mm]{114011}} \\cr\n      1 & -& - & \\pb{$\\ket{1,1,4;1,1,4}$ \\includegraphics[height=10mm]{114114}} &- & \\pb{$\\ket{2,1,1;1,1,3}_+$ \\includegraphics[height=10mm]{011113}}  &\\pb{$\\ket{2,1,1;1,1,3}_-$ \\includegraphics[height=10mm]{113011}} \\cr\n      \\rowcolor[gray]{.95} 1 & -& - & \\pb{$\\ket{1,1,3;1,1,3}$ \\includegraphics[height=10mm]{113113}} &- & \\pb{$\\ket{2,1,1;1,1,2}_+$ \\includegraphics[height=10mm]{011112}} &\\pb{$\\ket{2,1,1;1,1,2}_-$ \\includegraphics[height=10mm]{112011}} \\cr\n      1 & -& - & \\pb{$\\ket{1,1,2;1,1,2}$ \\includegraphics[height=10mm]{112112}} &-& \\pb{$\\ket{2,1,1;1,1,1}_+$ \\includegraphics[height=10mm]{011111}} &\\pb{$\\ket{2,1,1;1,1,1}_-$ \\includegraphics[height=10mm]{111011}} \\cr\n      \\rowcolor[gray]{.95} 1 & \\pb{ $\\ket{0,1,1;0,1,1}$ \\includegraphics[height=10mm]{211211}}  & -& \\pb{$\\ket{1,1,1;1,1,1}$ \\includegraphics[height=10mm]{111111}}&-& {\\pb{$\\ket{2,1,1;2,1,1}$ \\includegraphics[height=10mm]{001001}}}&-\\cr\n      \\midrule\n      0 & -& -  & \\pb{$\\ket{1,0,1;1,0,1}$ \\includegraphics[height=10mm]{101101}} & - &-  &- \\cr\n    \\end{longtable}\n  \\end{center}\n}\n\n\n\\section{Categorification of the Newton polynomial}\n\\label{sec:categorification}\n\nWhat follows is inspired by the categorification programme for knot\npolynomials of Khovanov~\\cite{Khovanov1, Khovanov2, Khovanov3}. The\nbasic idea of his work is that for a knot $K$, a doubly graded\nhomology theory $H_{i, j}(K)$ can be constructed, whose graded Euler\ncharacteristic with respect to one of the gradings yields the knot\npolynomial in question, \\emph{e.g.} the Jones polynomial, or the\nAlexander polynomial.  The homology groups are constructed through a\ncategorification process which starts with a state sum representation\nof the invariant polynomial.  A group is constructed for each term in\nthe summation, and differential maps between these groups are\nappropriately defined.  Similar categorifications were subsequently\nperformed for other invariants with state sums, \\emph{e.g.} various\ngraph polynomials, such as the chromatic polynomial~\\cite{Helme1,\n  Helme2}, and the more general Tutte polynomial~\\cite{Rong}.\n\nBut exactly what is meant by the term categorification?  When we\ncategorify, we replace set theoretic concepts by category theoretic\nones. Sets become categories, functions become functors, and equations\nbetween functions become natural isomorphisms between functors\n(fulfilling certain relations). The opposite process,\ndecategorification, is much more familiar to us. When we decategorify,\nwe are basically throwing away extra information by taking all isomorphic\nobjects to be equal. When we say that all vector spaces of the same\ndimension are the same, we decategorify and are left with a set of\nisomorphism classes.  Obviously, the reverse process of inventing the\nmissing information is much more difficult.  The category\n$\\mathrm{FinSet}$ of all finite sets for example is a categorification\nof the set of natural numbers. For an easy to read introduction to the\nconcept of categorification, we refer the reader to~\\cite{Baez}.\n\n\\bigskip\n\nAfter this general digression, we return to our own system of\nfermion loops. Its partition function, the Newton polynomial, is, once\nmore, a state sum. From the point of view of the dimer model, its\ninteger coefficients and polynomial nature are obvious. From the point\nof view of the free fermion, this is much less so. Giving this\npartition function the interpretation of a graded Euler characteristic\nexplains its integer nature and gives it a geometric interpretation,\nwhile at the same time suggesting its relation to an index.\n\n\nThe task at hand is obvious: construct a bigraded co--chain complex from the\nfermion loop states with a differential map preserving the grading, such\nthat the Euler characteristic of this co--chain complex yields the Newton\npolynomial.\n\nThe fact that we can associate four different charges to each loop\nstate living on the torus (see Eq.~\\eqref{eq:matching-charges}) gives\nus a certain freedom in the choice of the gradings in the double\ncomplex. Our emphasis is placed on the matchings, so we choose to sort\nthe loop states into co--chain groups according to the maximum winding\nnumber of the two constituent matchings in one direction ($\\max_i H_p\n\\kket{m_i}$) and to define an internal grading based on the maximum\nweight in the other direction ($\\max_i H_q \\kket{m_i}$). This\nseemingly arbitrary choice just amounts to picking a horizontal and a\nvertical direction in the double complex. The Euler characteristic,\nfor example, is not sensitive to this choice.\n\n\nWhen studying the loop states, there is an obvious observation to\nmake: because of the two possible orientations of the loops, all\nstates are paired with the exception of the double line perfect\nmatchings. This very much smells of supersymmetric Quantum mechanics\n(\\textsc{sqm}), where the supersymmetric ground states are the only ones which\nneed not be paired. Indeed, if one takes the total weight of a\nconfiguration to be its energy, the perfect matchings, having zero\nwinding, become ground states.\n\nThe Witten index,\n\\begin{equation}\n  \\label{eq:wittenindex}\n  \\mathrm{dim}\\mathcal{H}^B_{(0)}-\\dim\\mathcal{H}^F_{(0)}=\\Tr(-1)^F,\n\\end{equation}\ncounts the number of bosonic minus fermionic ground states and is an\ninvariant of the system. This is, indeed, also what the Newton\npolynomial is doing! Once we re--interpret our fermion loop gas as a\n\\textsc{sqm} system and construct a co--chain complex such that its boundary\noperator can be identified with the $Q$ operator and states with\nopposite orientations become superpartners, the partition function\nbecomes the generalized Witten index of the\nsystem.\n\nThis will be done explicitly in the following sections.\n\n\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusions} \n\nIn this note, we relate a statistical mechanical system, the dimer\nmodel on a torus, to a system of lattice quantum field theory,\n\\emph{i.e.} the massless free fermion. The loop configurations\nobtained after a diagrammatic expansion of the fermion determinant\n(Sec.~\\ref{sec:loop-expans-ferm}) can be interpreted as states in\n\\textsc{sqm} after employing categorification techniques inspired by\nKhovanov's work (Sec.~\\ref{sec:categorification}). The Newton\npolynomial of the dimer model becomes the generalized Euler\ncharacteristic of a co--chain complex (Sec.~\\ref{sec:sqm_torus}).  The\nstates with vanishing overall winding become the supersymmetric ground\nstates of the system. Since all loop states except the double line\nperfect matchings are paired, taking the Witten index of this system\nagain counts the number of perfect matchings\n(Eq.~\\eqref{eq:Witten-torus}).\\\\\nIn the following, we make use of the dimer model--quiver gauge theory\ncorrespondence to give the loop configurations an interpretation in\nyet another picture. Interpreting the loops as maps from one perfect\nmatching to another (Sec.~\\ref{sec:loops-as-operators}), maps between\ninternal perfect matchings associated to the same point in the toric\ndiagram become changes of foundation of the associated helix and\nsequences of double Seiberg dualities in the acyclic quiver\n(Sec.~\\ref{sec:doubleseiberg}). When specializing to the honeycomb\ngraph, the plaquette moves which generate all $(0,0)$--maps correspond\ndirectly to the \\textsc{bps} black hole configurations which are parametrized\nby crystal melting configurations (Sec.~\\ref{sec:melting}).\n\nMapping between matchings associated to different internal points is a\nmore severe change and cannot be achieved as a sequence of Seiberg\ndualities. In the language of exceptional collections, we map from one\nhelix to another.\n\n\nIn Table~\\ref{table:dictionary}, we summarize the results of this\npaper by giving a dictionary between the loop states, the fermion\nstates, the gauge theory and the geometry.\n\n\n\\bigskip\n\nFrom this point, one can go to several different directions for\nfurther research.\n\\begin{itemize}\n\\item The dimer model admits a description as a free fermion model\n  based on a transfer matrix approach\n  (see~\\cite{lieb:2339,Sutherland:1968,alet:041124} and\n  also~\\cite{Okounkov:2003sp}) which, at first sight, is different\n  from the one we introduced in this note. It would be very\n  interesting to clarify the link between these two descriptions.\n\\item Furthermore, it would be interesting to use a more general\n  definition for the Dirac operator on a graph than the one in\n  Eq.~\\eqref{eq:Lattice-Dirac}, to prove the equivalence to the dimer\n  model completely independently of the type of graph and its\n  representation. Indeed, this could provide an elegant solution to\n  the problem of choosing a Dirac operator on a random graph that\n  depends only on the the adjacency.\n\\item The algebraic structure we introduce in\n  Sec.~\\ref{sec:categorification} also deserves further\n  investigation. A better understanding of the complex and the mapping\n  to \\textsc{sqm} might indeed cast a new light on the dual quiver\n  gauge theory.\n\\item A final word must be spent on the geometric interpretation in\n  Sec.~\\ref{sec:geometry}. To our knowledge, the meaning of the\n  boundary perfect matchings (and, correspondingly, of the\n  infinite-dimensional representations of the path algebras) in terms\n  of derived category is not yet clear. This is an obvious\n  prerequisite to a complete understanding of the geometrical meaning\n  of all the loop states that appear in the theory.\n\\end{itemize}\n\nThe study of the dimer model, which due to its combinatorial nature provides a relatively easy playground, has proven to be very fruitful thanks to its manifold connections to other physical systems.\nWe believe that these inter--relations have not yet been fully exploited and deserve further attention.\n\n\\begin{landscape}\n  \\begin{table}[]\n    \\begin{center}\n      \\begin{tabular}{cccp{4.5cm}p{4cm}}\n        \\toprule\n        & \\textbf{Loop state} & \\textbf{Gauge theory on $S$}&\n        \\begin{minipage}{4.4cm}\n          \\center \\textbf{Geometry}\n        \\end{minipage} &\n        \\begin{minipage}{3.9cm}\n          \\center \\textbf{Free fermion}\n        \\end{minipage}\n        \\\\ \\midrule\n        \\rowcolor[gray]{.95}\n        \\cellcolor[gray]{1} &  double line $\\mathrm{PM}$ & Identity &\n        \\begin{minipage}{4.4cm}\n          \\center Identity\n        \\end{minipage}\n        & time evolution with no local charge creation\\\\[.3cm]\n        \\cellcolor[gray]{1} $\\{\\mathrm{PM}^{int}\\} \\to \\{\\mathrm{PM}^{int}\\}$ &$(0,0)$&Seiberg duality  & change of foundation in helix & time evolution in which no net charge is created\\\\[0.3cm]\n        \\rowcolor[gray]{.95}\n        \\cellcolor[gray]{1} & $(p,q)\\neq (0,0)$ & general gauge duality &\n        general sequence of mutation, change of helix, birational\n        transformation\n        &\n        initial conditions with gradient $q$ and creation of net charge $p$\n        \\\\   \\midrule\n        $\\{\\mathrm{PM}^{int}\\} \\leftrightarrow  \\{\\mathrm{PM}^{bd}\\}$ &   no corner $ \\mathrm{PM}$ & ? & birational transformation & \\hspace{1.8cm}\" \\\\  \\midrule\n        \\rowcolor[gray]{.95}\n        \\cellcolor[gray]{1}$\\set{\\mathrm{PM}^{bd}} \\to \\set{\\mathrm{PM}^{bd}}$ &\n        \\begin{minipage}{3.3cm}\n          \\tabtop adjacent boundary matchings. Zig-zag path\n        \\end{minipage}\n        & \\begin{minipage}{3.5cm}\n          gauge-invariant multi--trace operator\n        \\end{minipage}\n        &  & \\hspace{1.8cm}\" \\\\[.7cm]\n        & no corner PM  & \n        & birational transformation & \\hspace{1.8cm}\" \\\\ \\bottomrule\n      \\end{tabular}\n      \\caption{Dictionary}\n      \\label{table:dictionary}\n    \\end{center}\n  \\end{table}\n\\end{landscape}\n\n\n\n\n\\section{Dimer Model Preliminaries}\n\\label{sec:preliminaries}\n\n\n\n\nTake a plane \\emph{bipartite} graph $\\gra$, \\emph{i.e.} one in which all\nvertices can be colored black and white, such that each black vertex\nis only connected by links to white vertices and vice versa. Let $M$\nbe a subset of the set $E$ of edges of $\\gra$. $M$ is called a\nmatching, if its elements are links and no two of them are\nadjacent. If every vertex of $\\gra$ is saturated under $M$, the\nmatching is called \\emph{perfect}. Such a link which joins a black and\na white vertex is also called a \\emph{dimer}. The dimer model describes the\nstatistical mechanics of a system of random perfect matchings.  In the\nsimplest case, we ask for the number of close packed dimer\nconfigurations, \\emph{i.e.} the number of perfect matchings.\n\nLet us label the vertices of the underlying graph $\\gra$ on which the\ndimer model lives consecutively.  The topology of $\\gra$ can be encoded\nin its adjacency matrix $A$, where $A_{x,y}=1$ if the vertices $x$ and\n$y$ are joined by an edge, and $A_{x,y}=0$ otherwise. The number of\nperfect matchings is given by the number of ways in which the vertices\ncan be partitioned into adjacent pairs.\n\nThe \\emph{Hafnian} of a symmetric matrix $A$, introduced by Caianiello\n\\cite{Caianiello}, is defined as\n\\begin{equation}\n  \\label{eq:hafnian}\n  \\Hf(A)=\\sum_{\\pi \\in \\mathcal{S}}A_{\\pi(1),\\pi(2)}A_{\\pi(3),\\pi(4)} \\dots A_{\\pi(n-1),\\pi(n)},\n\\end{equation}\nwhere $\\mathcal{S}$ is the set of all permutations $\\pi$ of $\\set{1, \\dots \n n}$ satisfying $\\pi(1)<\\pi(3)< \\dots <\\pi(n-1)$ and $\\pi(2i-1)<\\pi(2i)$\n for $1\\leq i \\leq n$, $n$ even.  If we take $A$ to be the adjacency\n matrix of ${\\gra}$, $\\Hf(A)$ counts the perfect matchings, since\n terms in (\\ref{eq:hafnian}) which contain non--adjacent pairs are\n zero.  The \\emph{permanent} of an $n\\times n$ matrix $A$ is, like the\n determinant, given by the sum over all permutations, except that the\n sign of the permutations is not taken into account:\n\\begin{equation}\n  \\label{eq:permanent}\n  \\perm(A)=\\sum_{\\pi\\in S_n}\\prod_{i=1}^n A_{i,\\pi(i)}.\n\\end{equation}\nWhile the determinant can be evaluated in polynomial time via Gaussian\nelimination, computing the permanent is $\\#P$--complete\n\\cite{Valiant}.  The Hafnian can be regarded as a generalization of\nthe permanent. In fact~\\cite{Kuperberg1},\n\\begin{equation}\n  \\Hf(A)=\\perm(B)\\ \\  \\text{for $A$ of the form}\\ \\ A= \\left( \\begin{array}{c|c}\n      0&B\\\\ \\hline\n      B^t&0\n    \\end{array} \\right).\n\\end{equation}\nThis identity accounts for the fact that sometimes, the number of\nperfect matchings is given as the permanent of the adjacency matrix in\nthe literature.  Like the permanent, also the Hafnian is difficult to\nevaluate and does not satisfy any useful identities.\n\nThe counterparts of the Hafnian and permanent are the \\emph{Pfaffian} and determinant, which take into account the sign of the permutations:\n\\begin{align}\n  \\label{eq:pfaffian}\n  \\Pf(A)&= \\sum_{\\pi \\in \\mathcal{S}}\\sign(\\pi)\\,A_{\\pi(1),\\pi(2)}A_{\\pi(3),\\pi(4)} \\dots A_{\\pi(n-1),\\pi(n)},\\\\\n  \\label{eq:det}\n  \\det(A)&=\\sum_{\\pi\\in S_n}\\sign(\\pi)\\prod_{i=1}^n A_{i,\\pi(i)},\n\\end{align}\nwhere the permutations contributing to the Pfaffian are subject to the same conditions as for the Hafnian.\nHere, we have\n\\begin{equation}\n  \\Pf(A')=(-1)^{\\binom{n}{2}}\\det(B)\\ \\  \\text{for $A'$ of the form}\\ \\ A'=\\left(\\begin{array}{c|c}\n      0 & B \\\\ \\hline\n      \\! \\! -B^t & 0 \\end{array} \\right).\n\\end{equation}\nMoreover, for every anti--symmetric matrix of even order, the relation\n\\begin{equation}\n  \\label{eq:rel}\n  \\det A= \\left( \\Pf A \\right)^2\n\\end{equation}\nholds.  Already in 1913, Polya suggested to change some signs in a\nmatrix $A$, such that the determinant of the new matrix $\\tilde A$\nwould be the permanent of $A$~\\cite{polya}.  Kasteleyn\n\\cite{Kasteleyn1, Kasteleyn2} actually does exactly this: he\nintroduces an orientation on \\gra, which leads to a signed adjacency\nmatrix $K$, now called the \\emph{Kasteleyn matrix}. The Pfaffian of $K$\ngives the number of perfect matchings.\n\nA \\emph{Kasteleyn orientation} fulfills the following\ncondition: the product of all edge weights around a face must equal $-1$\nif the number of edges around the face is $0 \\mod 4$.  If the number\nof edges equals $2 \\mod 4$, the product must equal $1$~\\cite{Kenyon2}. One\ncan choose an orientation by consistently assigning arrows to\nthe edges of the graph, as originally suggested by Kasteleyn\n\\cite{Kasteleyn1, Kasteleyn2}. For our purposes, it turns out to be\nmore convenient to allow the roots of 1 as (complex) edge weights.\n\nAlready Caianiello~\\cite{Caianiello} remarked that the expectation\nvalue of any product of free Fermi fields is a Pfaffian, while the\nexpectation value of any product of free Bose fields is a Hafnian. In\nthis sense Kasteleyn's method of attaching signs to the edges of a graph \\gra\\\ncould be seen as a fermionization of a bosonic system. Given the\ncorrespondence of the dimer model to the free fermion on \\gra, one\ncould look for its bosonic equivalent, but this goes beyond the scope\nof the present work.\n\n\nIn the following, we will consider graphs embedded on a cylinder or a\ntorus. The treatment can be straightforwardly generalized to any genus\n$g$ Riemann surface.  In essence, all of the above remains true. The\nonly change is that on the torus (cylinder), we have two (one)\nnon--trivial cycles, which we will denote by $z$ and $w$. In the case\nof the plane graph, the edge weights originated solely from the\nKasteleyn orientation. More generally, the Kuperberg flatness condition must be respected~\\cite{Kuperberg2}. We choose a positive direction on the dimers,\nsay $\\bullet\\to\\circ$. Now we assign the weight $z$ ($w$) to each edge\nwhich crosses the cycle $z$ ($w$) in positive direction and the weight\n$1\/z$ ($1\/w$) to each edge which crosses it in negative\ndirection. While the Pfaffian of the Kasteleyn matrix yielded a number\nin the case of the plane graph, it now becomes a polynomial in $z$ and\n$w$, the so--called characteristic polynomial or \\emph{Newton\npolynomial} of the graph. The coefficient of each monomial $z^pw^q$\ngives the number of matchings with \\emph{weight} $(z,w)=(p,q)$.\nThese are matchings with the number of dimers crossing $z$ in positive\ndirection minus the number of dimers crossing $z$ in negative\ndirection equal to $p$ (analogous for $q$). In the literature,\nwhat we call the weight is usually referred to as the slope of a height\nfunction defined on the composition of two matchings. The height\nfunction is defined as follows. Choose a reference matching $\\mathrm{PM}_0$. To\nfind the slope of a matching $\\mathrm{PM}$, compose it with the reference\nmatching, $\\mathrm{PM}-\\mathrm{PM}_0$, where the minus serves to change the orientation of\n$\\mathrm{PM}_0$ to $\\circ\\to\\bullet$. This results in closed loops (composition\ncycles) and double line dimers. The rule is that when an edge in $\\mathrm{PM}$\nbelonging to a closed loop is crossed such that the black node is to\nits left (right), the height changes by $+1$ $(-1)$. If an edge\nbelonging to $\\mathrm{PM}_0$ is crossed, the signs are reversed. This height\nfunction is defined up to the choice of the reference matching\n$\\mathrm{PM}_0$. Crossing the boundary of the fundamental region of the torus, this function\ncan jump. If the height function jumps by $p$ units crossing $z$, it\nis associated to the power $z^p$ in the Newton polynomial of the graph\n(and equivalently for $w$). Choosing a different reference matching\nresults in a common prefactor of $z^{p_0}w^{q_0}$ for all\nmonomials. Our method of assigning weights to a matching corresponds\nto choosing a reference matching of weight $(0,0)$ that does not\nintersect the $z$ or $w$ cycle. The matching shown in Figure\n\\ref{fig:example} has weight $(1,0)$, where $1=2-1$.\n\\begin{figure}[h!]\n  \\begin{center}\n    \\includegraphics[width=50mm]{example}\n    \\caption{Example of a square graph on the torus}\n    \\label{fig:example}\n  \\end{center}\n\\end{figure}\n\\\\\nThe characteristic, or Newton polynomial, of the dimer model on\nthe torus takes the form\n\\begin{equation}\n\\label{eq:torus-partition}\n  P(z,w)=\\sum_{n_z, n_w} N_{n_z,n_w}\\,(-1)^{n_z+n_w+n_zn_w}z^{n_z}w^{n_w},\n\\end{equation}\nwhere the $N_{n_z,n_w}$ count the number of matchings that have weight\n(height change) $(n_z,n_w)$.\nFurthermore, the partition function which counts the total number of matchings for the square graph on the torus is given by\n\\begin{equation}\\label{eq:totalnumber}\nZ=\\frac{1}{2}\\left(-P(1,1)+P(1,-1)+P(-1,1)+P(-1,-1)\\right),\n\\end{equation}\nwhere the first term is always zero. $P(z,w)$ evaluated in $z,w=\\pm1$ corresponds to the contributions of the four different spin structures.\n\n\n\n\n\n\\subsection*{Acknowledgements}\n\nWe would like to thank Luis \\'Alvarez-Gaum\\'e, David Cimasoni, Davide Forcella,\nEmanuel Scheidegger, and Alberto Zaffaroni for enlightening\ndiscussions. Furthermore, we would like to thank David Cimasonifor detailed comments on the manuscript.\n\nD.O. and S.R. would like to thank \\textsc{CERN} for hospitality, where\npart of this work was carried out.\n\nThe research of R.D. was supported by a NWO Spinoza grant and the FOM program \"String Theory and Quantum Gravity.\"\nD.O. is supported in part by \\textsc{INFN} and \\textsc{MIUR} under contract 2005-024045 and by \nthe European Community's Human Potential Program \\textsc{MRTN}-\\textsc{CT}-2004-005104.\nS.R. is supported by the EC's Marie Curie Research Training Network under the contract \\textsc{MRTN}-\\textsc{CT}-2004-512194 \"Superstrings\".\n\n\n\\section{The dimer model as a free fermion}\n\\label{sec:fermions-dimers}\n\nAs already mentioned in the introduction, many clues point towards the\nequivalence of the dimer model and the free fermion living on the\nsame lattice. On one hand, we know that the dimer model corresponds to\nthe two--dimensional Ising model. In the continuum limit, the critical\nIsing model corresponds in turn to the free Majorana fermion. Another clue is\nthat the partition function of the dimer model on a genus $g$ Riemann\nsurface is a linear combination of $2^{2g}$ Pfaffians, one for each of\nthe different boundary conditions. The one--to--one correspondence\nbetween the $2^{2g}$ equivalence classes of Kasteleyn orientations and\nthe spin structures was proved in~\\cite{cimasoni-2006}. \n\nNote that our construction given in the following is different from the transfer matrix approach used\nin~\\cite{lieb:2339,Sutherland:1968,alet:041124}.\n\n\n\\subsection{The Dirac operator and the Kasteleyn matrix}\n\\label{sec:dirac}\n\nIn this section, we show that the dimer model on a square lattice can be\nnaturally mapped to the dynamics of a free massless fermion on the\nsame lattice. \nMore precisely, we will see how introducing a Kasteleyn orientation on\nthe graph can be thought of as the projection of the $\\del$ operator\non $\\gamma $ matrices, see also~\\cite{KenyonDirac, Kenyon2}.\n\nLet us consider the theory for a free massless fermion on a bipartite\nsquare graph\\footnote{ A different definition for the Dirac operator,\n  such as the one given in~\\cite{Rabin:1981qj,Becher:1982ud} would be\n  needed to deal with a general random graph. This goes beyond the\n  scope of the present note.}. At this stage, the construction is\nindependent of the boundary conditions, \\emph{i.e.} we can think of\nthe model as living on the infinite plane. The graph being bipartite,\nit is natural to use staggered fermions~\\cite{PhysRevD.16.3031} and\nassociate a two--component real spinor to each pair of nodes:\n\\begin{equation}\\label{eq:twocomponent}\n  \\Psi =\n  \\begin{pmatrix}\n    \\chi_{\\bullet} \\\\ \\chi_{\\circ}\n  \\end{pmatrix} .\n\\end{equation}\nThe naive discretization of the free massless fermion action becomes then\n\\begin{equation}\n  S = \\int \\di x \\: \\bar \\Psi (x) \\slashed{\\del} \\Psi (x) = \\sum_{ \\braket{x,  y}} \\chi (x)\\, \\slashed{\\del}(x,y)\\, \\chi(y)  ,\n\\end{equation}\nwhere the sum runs over all neighbouring vertices $ x $ and $y$\non the lattice. The discretized Dirac operator\ncan be cast into the form\n\\begin{equation}\n\\label{eq:Lattice-Dirac}\n  \\slashed{\\del}(x,y) = \\gamma^\\mu \\frac{ x_\\mu - y_\\mu}{\\abs{x-y}} \\, ,  \n\\end{equation}\nwhere $\\gamma^\\mu$ are two matrices satisfying the Clifford algebra\n$\\set{ \\gamma^\\mu, \\gamma^\\nu} = 2\\, \\delta^{\\mu \\nu}$ in two\nEuclidean dimensions. On the square lattice, each node has\nfour neighbours, separated by unit vectors, so the action reads\n\\begin{equation}\n  S = \\sum_{x \\in E (\\gra)} \\sum_{k=1}^4 \\chi(x)\\, \\gamma^\\mu e\\ud{k}{\\mu}\\, \\chi (x + e^k) \\, ,\n\\end{equation}\nwhere $e^k$ is the vector $e^k= ( \\cos \\frac{2k \\pi}{4},\\, \\sin \\frac{2k\n  \\pi}{4})$.  If $x$ is a $\\bullet $ site, $x \\pm  e^k$ is a\n$\\circ$ site and vice versa. This means that we can split the action\ninto two pieces:\n\\begin{equation}\n  \\label{eq:Dirac-action}\n  S = \\left(\n    \\begin{array}{c|c}\n      \\vec{\\chi}_\\bullet & \\vec{\\chi}_\\circ \n    \\end{array} \\right) \\left(\n    \\begin{array}{c|c}\n      0 & \\slashed{\\del}_{\\bullet \\circ} \\\\ \\hline \n      \\slashed{\\del}_{\\circ \\bullet} & 0 \n    \\end{array} \\right) \\left(\n    \\begin{array}{c}\n      \\vec{\\chi}_\\bullet \\\\ \\hline\n      \\vec{\\chi}_\\circ \n    \\end{array} \\right) =  \\vec{\\chi}_\\bullet \\cdot \\slashed{\\del}_{\\bullet \\circ} \\cdot \\vec{\\chi}_\\circ + \\vec{\\chi}_\\circ \\cdot \\slashed{\\del}_{\\circ \\bullet} \\cdot \\vec{\\chi}_\\bullet \\, ,\n\\end{equation}\nwhere $\\vec{\\chi}_\\bullet$ is the vector of Grassmanian variables\nliving on the $\\bullet $ nodes.  Let us consider the first term. The\noperator $\\slashed{\\del}_{\\bullet \\circ}$ is non--vanishing if and only\nif $\\bullet $ and $\\circ$ are neighbours, and takes values\nproportional to the components of the $\\gamma $ matrices. More\nprecisely, if we think of $\\slashed{\\del}_{\\bullet \\circ}$ as a\nmatrix, each line will contain four non--vanishing entries $\\set{\n  \\gamma_{\\bullet \\circ}^1, \\gamma_{\\bullet \\circ}^2, -\n  \\gamma_{\\bullet \\circ}^1, - \\gamma_{\\bullet \\circ}^2}$. Choosing the\nrepresentation\n\\begin{align}\n  \\gamma^1 =\n  \\begin{pmatrix}\n    0 & 1 \\\\ 1 & 0\n  \\end{pmatrix} \\, ,&&\n  \\gamma^2 =\n  \\begin{pmatrix}\n    0 & \\imath \\\\ - \\imath & 0\n  \\end{pmatrix} \\, ,\n\\end{align}\nthese components read $\\set{1, \\imath, -1, - \\imath}$. These are the\nweights of the four links around each $\\bullet$ node. Since their\nproduct equals to $\\left( - 1 \\right)$ and on this graph the product of weights around a node is the same as the product around a plaquette, this is precisely a Kasteleyn\norientation (see Fig.~\\ref{fig:Dirac-Kasteleyn})\\footnote{This argument\n  can be easily generalized to any regular tessellation of the\n  hyperbolic plane (leading to higher--genus graphs). One can show that\n  for an $n$-gon, the product of the weights around a node is $\\left(\n    -1 \\right)$ if $n = 0 \\mod 4$ and $\\left(+1 \\right)$ if $n = 2\n  \\mod 4$. For the hexagonal lattice (genus one), the\n  weights around a $\\bullet$ site read $\\set{ 1, e^{2 \\pi \\imath \/3},\n    e^{-2\\pi \\imath \/3} }$ and their product is $\\left( + 1\n  \\right)$. It is worth to remark that for higher genus surfaces, there is no unique global choice of complex structure.}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics{Dirac-Kasteleyn}\n  \\caption{Dirac operator}\n  \\label{fig:Dirac-Kasteleyn}\n\\end{figure}\n\nThe same construction can of course be repeated for the second term in\nEq.~\\eqref{eq:Dirac-action}. In this case, we get for each line of the\noperator $\\slashed{\\del}_{\\circ \\bullet}$ the four non--vanishing elements\n$\\{ \\gamma_{\\circ \\bullet}^1, \\gamma_{\\circ \\bullet}^2,$ $ -\n  \\gamma_{\\circ \\bullet}^1, - \\gamma_{\\circ \\bullet}^2\\}$ which, in the\nsame representation for the $\\gamma$ matrices that we used above, read\n$\\set{ 1, - \\imath, - 1, \\imath} $, yielding again a Kasteleyn\norientation.\n\nAdjacency matrices with equivalent Kasteleyn orientations have the same determinant (up to an overall sign), therefore we have\n\\begin{equation}\n  \\det ( \\slashed{\\del}_{\\bullet \\circ} ) = \\det ( \\slashed{\\del}_{\\circ \\bullet}) = \\Pf ( K )\\, ,\n\\end{equation}\nwhere $K$ is a Kasteleyn matrix for the graph.\nWriting the partition function for the free massless fermion gives\n\\begin{equation}\n  \\label{eq:fermion-path-integral}\n  Z_{\\text{fermion}} = \\int \\di \\psi \\: e^{-S[\\psi]} = \\sqrt{ \\det (\\slashed{\\del})} = \\sqrt{ \\det (\\slashed{\\del}_{\\bullet \\circ} )  \\det ( \\slashed{\\del}_{\\circ \\bullet}) } = \\Pf (K) \\, ,\n\\end{equation}\nwhich is precisely the number of perfect matchings on the lattice for a given boundary condition.\n\nExplicit expressions for both the dimer model on a square graph and\nthe free fermion theory are known. \nTake a  triangular lattice describing a torus with\nperiodicity\n\\begin{align}\n  z = z + i N + j M e^{\\imath \\theta} \\, ,&& i,j \\in \\setZ,\\ z\\in\\setC,  \n\\end{align}\n\\emph{i.e.} with modular parameter $\\tau = \\tfrac{M}{N} e^{\\imath\n  \\theta}$.  In~\\cite{Nash:1996kn}, it is shown that the partition\nfunction for a vortex with winding $(a,b)$ on this lattice,\nwhich can be mapped to a free fermion, is given by the infinite product\n\\begin{multline}\n  Z_{\\text{vortex}}  \\oao{a}{b} (M,N, \\theta) =  \\prod_{k=-M\/2}^{M\/2} \\prod_{l=-N\/2}^{N\/2} 2 \\abs{ \\alpha \\cos \\left( 2 \\pi \\frac{k}{M} + \\pi \\frac{a}{M} \\right)  + \\beta  \\cos \\left( 2 \\pi \\frac{l}{N} + \\pi  \\frac{b}{N} \\right) + \\right. \\\\\n  \\left. + \\gamma \\cos \\left( 2 \\pi \\left( \\frac{k}{M} - \\frac{l}{N} \\right) + \\pi \\left( \\frac{a}{M} - \\frac{b}{N} \\right) \\right) - \\sigma}, \n\\end{multline}\nwhere\n\\begin{align}\n  \\alpha = \\beta = \\frac{1 - \\cos \\theta}{\\sin^2 \\theta}, && \\gamma = \\frac{\\cos \\theta}{\\sin^2 \\theta}, && \\sigma = \\frac{2 - \\cos \\theta}{\\sin^2 \\theta} . \n\\end{align}\nSpecializing this formula for a square torus with $\\theta = \\pi \/2 $ such that\n$\\alpha = \\beta = 1, \\gamma = 0, \\ \\sigma =2 $, we find:\n\\begin{equation}\n  Z_{\\text{vortex}}  \\oao{a}{b} (M,N, \\tfrac{\\pi}{2}) =  \\prod_{k=-M\/2}^{M\/2} \\prod_{l=-N\/2}^{N\/2} 2 \\abs{ \\cos \\left( \\frac{2k \\pi}{M} + a \\frac{\\pi}{M} \\right) + \\cos \\left( \\frac{2 l \\pi}{N} + b \\frac{\\pi}{N} \\right) - 2},\n\\end{equation}\nwhich is precisely the partition function of the dimer model.\n\n\n\n\n\\subsection{The scaling limit}\n\\label{sec:scaling-limit-theta}\n\nHaving shown that the dimer model on a graph is equivalent to a free\nMajorana fermion theory, we expect the torus partition function to be\nmodular invariant in the thermodynamical limit, and to consist of\ntheta functions.\n\nSince this is a universal property, we specialize to a\nparticular lattice, for example the $M \\times N$ square graph. The\npartition function for this model was discussed in\n\\cite{Kasteleyn2,Kenyon1} and takes the form\n\\begin{equation}\n  Z_{MN} = \\tfrac{1}{2} \\sum_{a,b=0}^1 \\left( -1 \\right)^{a+b+ab} Z_{MN} \\oao{a}{b} \\, ,\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq:partition-square-mn}\n  Z_{MN} \\oao{a}{b} = \\prod_{k=-M\/2}^{M\/2} \\prod_{l=-N\/2}^{N\/2} 2 \\abs{ \\cos \\left( \\frac{2k \\pi}{M} + a \\frac{\\pi}{M} \\right)  +  \\cos \\left( \\frac{2 l \\pi}{N} + b \\frac{\\pi}{N} \\right) - 2 } \\, .\n\\end{equation}\nIf we regard the $M \\times N$ lattice as a discretization of a square\ntorus with modular parameter $\\tau = \\imath t$ into $M \\times N$\nsquares, the scaling limit is obtained by taking\n\\begin{align}\n  M, N \\to \\infty && \\frac{M}{N} = t , \\quad \\text{$t\\,$ fixed}.  \n\\end{align}\nThis corresponds to sending the lattice size $\\frac{1}{M}$ to zero.\n\nAs we have previously pointed out, this is precisely the same\nexpression found for a soliton on a triangular lattice~\\cite{Nash:1995ba}. After a\ncareful analysis, the authors prove that in the continuum limit, this\nreproduces the Ray--Singer result for the $\\bar \\del$--torsion of the\ntorus~\\cite{Ray:1973sb} (see also~\\cite{Ferdinand1,Ferdinand2}):\n\\begin{equation}\n  Z_{MN} \\oao{a}{b} \\xrightarrow[\\genfrac{}{}{0pt}{}{M,N \\to\n    \\infty}{M\/N = - \\imath \\tau }]{} \\left[\n    \\frac{\\vartheta \\oao{a}{b} (\\tau)}{\\eta(\\tau)} \\right]^2 \\, .\n\\end{equation}\n\n\nInstead of reproducing the proof,\nwe will limit ourselves here to a heuristic argument. In the infrared\nlimit, the cosine can be approximated by a parabola and the functions\nabove behave as follows:\n\\begin{multline}\n  Z_{MN} \\oao{a}{b} = \\prod_{k=-M\/2}^{M\/2} \\prod_{l=-N\/2}^{N\/2} 2\n  \\abs{\\, \\cos \\left( \\frac{2k \\pi}{M} + a \\frac{\\pi}{M} \\right) +\n    \\cos \\left( \\frac{2 l \\pi}{N} + b \\frac{\\pi}{N} \\right) - 2\\, }\n  \\sim \\\\\n  \\sim \\prod_{k=-M\/2}^{M\/2} \\prod_{l=-N\/2}^{N\/2} \\frac{8 \\pi^2}{M^2}\n  \\abs{ \\left(k+ \\frac{a}{2} \\right)^2 + \\left(l+ \\frac{b}{2}\n    \\right)^2 t^2} \\, .\n\\end{multline}\nIn the $\\zeta$--function regularization, one recognizes the usual product\nexpansion for the theta functions. More precisely, introducing $\\tau =\n\\imath t$ we find %\n\\begin{comment}\n\n\n\nIt is a known fact that $\\vartheta \\oao{a}{b} (\\tau)\/ \\eta(\\tau)$ can be expanded as a double product as follows:\n\\begin{equation}\n  \\frac{\\vartheta \\oao{a}{b} (\\tau)}{\\eta(\\tau)} = \\prod_{k \\in \\setZ} \\prod_{l \\in \\setZ} \\abs{ k + \\frac{a}{2} + \\left( l + \\frac{b}{2} \\right) \\tau}  .\n\\end{equation}\nIf we decompose\n\\begin{equation}\n  \\abs{ \\left(k+ \\frac{a}{2} \\right)^2 + \\left(l+ \\frac{b}{2} \\right)^2 t^2} = \\abs{ \\left(k+ \\frac{a}{2} \\right) + \\imath \\left(l+ \\frac{b}{2} \\right) t } \\abs{ \\left(k+ \\frac{a}{2} \\right) -  \\imath \\left(l+ \\frac{b}{2} \\right) t } ,\n\\end{equation}\nit is immediate to see that in $\\zeta$-function regularization, given\nthat\n\\begin{align}\n  \\prod_{k=1}^\\infty a = \\sqrt{a}, && \\prod_{k = -\\infty}^\\infty a =\n  a^{2 \\zeta(0) + 1} = 1,\n\\end{align}\nwe find:\n\\end{comment}\n\\begin{gather}\n  Z_{MN} \\oao{a}{b} \\xrightarrow[\\genfrac{}{}{0pt}{}{M,N \\to\n    \\infty}{M\/N = - \\imath \\tau }]{} \\left[ \\prod_{k \\in \\setZ}\n    \\prod_{l \\in \\setZ} \\abs{ k + \\frac{a}{2} + \\left( l + \\frac{b}{2}\n      \\right) \\tau} \\right]^2= \\left[\n      \\frac{\\vartheta \\oao{a}{b} (\\tau)}{\\eta(\\tau)} \\right]^2 \\, ,  \\\\\n    Z_{MN} = \\tfrac{1}{2} \\sum_{a,b=0}^1 \\left( -1 \\right)^{a+b+ab}\n    Z_{MN} \\oao{a}{b} \\xrightarrow[\\genfrac{}{}{0pt}{}{m,n \\to\n      \\infty}{m\/n = - \\imath \\tau }]{} \\tfrac{1}{2} \\sum_{a,b=0}^1\n    \\left( -1 \\right)^{a+b+ab} \\left[ \\frac{\\vartheta \\oao{a}{b}\n        (\\tau)}{\\eta(\\tau)} \\right]^2 = Z_{\\text{Dirac}} (\\tau).\n\\end{gather}\nThe final result is that in the scaling limit, the partition function\nis the one of a Dirac fermion, or equivalently, for two Majorana\nfermions. It is not surprising that  we encounter a fermion doubling effect~\\cite{Nielsen:1981hk, Nielsen:1980rz} since we have started with a periodic\ndispersion relation.\n\n\n\n\\section{Gauge theory interpretation: Preliminaries}\n\\label{sec:geometry}\n\n\nAs mentioned in the introduction, there exists a correspondence\nbetween quiver gauge theories describing $D$--branes probing singular\ntoric surfaces and the dimer\nmodel~\\cite{Franco:2005rj,Franco:2005sm}. We will make use of it to\nre--interpret the loops in terms of the quiver gauge theory and the\ntoric geometry.  We will quickly summarize the necessary knowledge of\nquiver gauge theories and this correspondence in the following.\n\n\\subsection{The quiver gauge theory}\n\nConsider $D3$--branes in type IIB string theory probing a Calabi--Yau\n$X$. They correspond to BPS $B$--type branes given by points on\n$X$. If the $D$--branes are placed at a singularity, they are expected\nto decay into a collection of so--called \\emph{fractional} branes.\nThe resulting world--volume gauge theory can be summarized in a\n\\emph{quiver} graph $Q_X$ as follows. Each constituent brane appearing\nwith multiplicity $N_i$ corresponds to a $U(N_i)$ gauge group which is\nrepresented as a node in the quiver graph. The massless open strings\nstretching between the fractional branes correspond to chiral fields\nin the $(\\overline{N}_i, N_j)$ representation and are depicted as\narrows pointing from the $U(N_i)$--node to the $U(N_j)$--node in the\ndiagram. For the theory to be anomaly free, the number of arrows\n$a_{ik}$ going from node $N_i$ to node $N_k$ (in our convention\n$a_{ki} = - a_{ik}$) must fulfill\n\\begin{equation}\n  \\label{eq:anomaly}\n  \\sum_i N_i a_{ik} = 0 \\, , \\hspace{2em} \\forall k \\, .\n\\end{equation}\nIf the ranks of all gauge groups are equal, this reduces to the number\nof incoming and outgoing arrows being equal at each node.\n\nWe will be concentrating on the case of the Calabi--Yau $X$ being the cone over a singular toric surface $S$.\n\nTo fully capture the physics of the $D$--branes, they must be described in terms of the (bounded) derived category of coherent sheaves $D^\\flat(X)$. A full explanation of the machinery of the derived category is beyond the scope of this paper, the reader is therefore referred to~\\cite{Aspinwall:2004jr}. Luckily, we can avoid working directly in $D^\\flat(X)$ by using so--called \\emph{exceptional collections} of sheaves supported on a partial resolution of the singular surface $S$. In our case, these sheaves can be mostly thought of as line bundles. The exceptional collections form a convenient basis for the fractional branes and it is possible to construct the quiver from a given exceptional collection and vice versa~\\cite{Aspinwall:2004vm}. The whole treatment is based on the fact that the derived category of coherent sheaves is equivalent to the derived category of quiver representations.\n\n\nThe open strings in the topological sector between two $B$--branes $A$\nand $B$ are parametrized by $\\oplus_p \\mathrm{Ext}^p(A,B)$. The mass\nof such an open string between two branes with the same grade is given\nin string units by\n\\begin{equation}\\label{eq:mass}\nm^2=\\tfrac{1}{2}(p-1).\n\\end{equation}\nFrom this, we learn that the massless open strings (\\emph{i.e.} the arrows in\nthe quiver) are counted by $\\mathrm{Ext}^1$, while $\\mathrm{Ext}^0$\ncounts the tachyons. For $p>1$, the strings are very massive and are\nnot seen by the quiver gauge theory.\n\n\nIf oriented loops appear in a quiver (which is always the case for our\nquiver gauge theories because of the anomaly--freedom condition in\nEq.~\\eqref{eq:anomaly}), the quiver representations become infinite\ndimensional, causing the methods employed in~\\cite{Aspinwall:2004vm}\nto fail. Fortunately, this problem can be evaded by considering\nacyclic subquivers. Some of the arrows of the gauge theory quiver\n$Q_X$ are linked to the properties of the surface $S$ itself, while\nthe others parametrize the embedding of $S$ in the Calabi--Yau $X$.\nDeleting these latter arrows removes all oriented loops. The resulting\nquiver, $Q_S$, allows an ordering relation and is also called the\nBeilinson or Bondal quiver in the literature.\n\nThe derived category of coherent sheaves on $S$, $D^\\flat(S)$, is\nequivalent to the derived category of quiver representations for\n$Q_S$, denoted by $D^\\flat(A-{\\mathrm{mod}})$ (where $A$ is the path\nalgebra of $Q_S$). The basic representations $L_i$ of the path algebra\n$A$ are associated to the fractional branes at the nodes of the\nquiver. The arrows in the quiver are thus associated with the\n$\\mathrm{Ext}^1(L_i, L_j)$. The $\\mathrm{Ext}^2(L_i, L_j)$ correspond\nto relations in the quiver which the paths must obey. There is another\nquiver representation given by the projective objects $P_i$, which are\nthe subspaces of $A$ generated by all paths starting at node $i$. The\nsheaves in the exceptional collections we consider are the projective\nobjects $P_i$, which are in some sense dual to the fractional branes\n$L_i$.\n\nLet us illuminate a bit the connection between the quivers $Q_S$ and\n$Q_X$ and its geometrical meaning. Consider the embedding of the\nsurface $S$ into $X$ via $i:\\,S\\hookrightarrow X$. A $D$--brane in $S$\ncan obviously also be regarded as a $D$--brane in $X$, \\emph{i.e.} we can map\nobjects in $D^\\flat(S)$ injectively to objects in $D^\\flat(X)$ using\n$i_*$. \nSince there might be more morphisms in $D^\\flat(X)$ than $D^\\flat(S)$,\nsome of the open strings between two branes in $S$ may live in $X$ but\noutside of $S$. This fact is captured by the relation\n\\begin{equation}\\label{eq:Ext_decomposition}\n\\mathrm{Ext}^1_X(i_*L_i,i_*L_j)=\\mathrm{Ext}^1_S(L_i,L_j)\\oplus\\mathrm{Ext}^2_S(L_j,L_i).\n\\end{equation}\nThus, the decomposition of the arrows of the full quiver $Q_X$ into $\\mathrm{Ext}^1$ and $\\mathrm{Ext}^2$ on $S$ accounts for the embedding of $S$ in $X$. \n\n\n\n\\subsection{Exceptional collections and helices}\n\nIn this section, we collect the definitions on exceptional collections\nnecessary for later chapters.  An (ordered) collection of sheaves\n$\\{\\mathcal{F}_0, \\dots , \\mathcal{F}_{n-1}\\}$ on $S$ is called\n\\emph{exceptional} if \n\\begin{subequations}\n  \\label{eq:exceptional}\n  \\begin{align}\n    \\mathrm{Ext}^0_S(\\mathcal{F}_i, \\mathcal{F}_i)&=\\setC \\\\\n    \\mathrm{Ext}^p_S(\\mathcal{F}_i, \\mathcal{F}_i)&=0, \\quad p\\geq1 \\\\\n    \\mathrm{Ext}^p_S(\\mathcal{F}_i, \\mathcal{F}_j)&=0, \\quad i>j.\n  \\end{align}\n\\end{subequations}\nThe collection is called \\emph{strongly} exceptional, if\n$\\mathrm{Ext}^p_S(\\mathcal{F}_i, \\mathcal{F}_j)=0$ for $p\\neq0$ and\n\\emph{complete} if it generates $D^\\flat(S)$. Since the $D$--branes on\n$S$ can wrap $0-$, $2-$ and $4-$cycles, a collection must contain $n$\nsheaves to be complete, where $n=\\,$sum over all Betti numbers\n($=\\chi(S)$). For physics, the collection being strong means that the\nbasis of fractional branes it generates does not contain ghost matter.\n\nExceptional collections can be transformed into new exceptional\ncollections by left and right \\emph{mutations}, which represent an\naction of the braid group on the set of possible collections.  On a\nneighboring pair of sheaves in an exceptional collection, the\nmutations act as\n\\begin{align}\n  \\label{eq:mutation}\n  L_i &: (E_i, E_{i+1})\\mapsto(L_{E_i}E_{i+1}, E_i),\\\\\n  R_i &: (E_i, E_{i+1})\\mapsto(E_{i+1}, R_{E_{i+1}}E_i).\n\\end{align}\nThe sheaves $L_{E_i}E_{i+1}$ and $R_{E_{i+1}}E_i$ are defined via\nshort exact sequences, see~\\cite{Rudakov,Zaslow:1994nk}. Left and\nright mutations can be seen as braiding and unbraiding.  A\n\\emph{helix} $\\mathcal{H}$ is an infinite collection of coherent\nsheaves $\\{\\mathcal{F}_i\\}_{i\\in\\setZ}$ such that\n\\begin{itemize}\n\\item[(a)] for any $i\\in\\setZ$,\n  $\\{\\mathcal{F}_{i+1},\\dots,\\mathcal{F}_{i+n}\\}$ is an exceptional\n  collection.\n\\item[(b)] $R^{n-1}\\mathcal{F}_i=\\mathcal{F}_{i+n}$.\n\\end{itemize}\nThe name stems from the fact that after moving $n$ steps to the right,\none is back at the original place up to a translation. $n$ is called\nthe \\emph{period} $n$ of $\\mathcal{H}$.  Each collection of the form\n$\\{\\mathcal{F}_{i+1}, \\dots,\\mathcal{F}_{i+n}\\}$ is called a\n\\emph{foundation} of the helix and $\\mathcal{H}$ is uniquely determined by\neach of its foundations.\n\n\n\n \n\n\n\n\n\\subsection{The correspondence}\n\nHere we quickly summarize the correspondence worked out in\n\\cite{Franco:2005rj, Franco:2005sm}.\n\n\\begin{enumerate}\n\\item Start with a toric quiver gauge theory, given by a quiver graph\n  and its tree--level superpotential.\n\\item Draw the quiver diagram on a torus, such that each of the terms\n  of the superpotential corresponds to a plaquette. The resulting\n  graph is called the \\emph{periodic quiver}.\n\\item Take the graph dual of this graph (i.e. vertices become faces,\n  faces become vertices and edges remain edges). Colour the vertices\n  coming from the faces associated to a superpotential term with\n  negative sign black; the ones coming from a positive term colour\n  white. The resulting graph is bipartite and also lives on a torus.\n\\item Solve the dimer model on this graph by taking the Pfaffian of\n  the Kasteleyn matrix. This yields the Newton polynomial. The\n  associated \\emph{Newton polygon} is given by taking the exponents of\n  the monomials in $zw$ to be coordinates in the $(z,w)$--plane. The\n  resulting polygon is exactly the toric diagram of the surface the\n  quiver gauge theory comes from!\n\\end{enumerate}\n\nThe singular surface $S$ is specified by the corners of the toric diagram. These corner vertices each correspond to Weil divisors and therefore to line bundles. The fully resolved surface is given by adding all the points lying in the intersection of the convex hull of the polygon and the integer $\\setZ^2$--lattice to the toric diagram. Each such vertex corresponds to an exceptional divisor which is blown up to resolve the singularity. The surface $S$ can also be resolved partially by blowing up only part of the exceptional divisors. Passing from one partial resolution $S'$ corresponding to the exceptional divisor $E'$ being blown up to another partial resolution $S''$ with $E''$ being blown up corresponds to a sequence of birational transformations (blow down $E'$ and blow up $E''$ instead).\n \nThe vertices of the toric diagram can be labeled with the multiplicities of the perfect matchings with the corresponding weight. There are certain patterns in the multiplicities which are worth pointing out:\n\\begin{itemize}\n\\item The vertices at the corners of the toric diagram always have multiplicity one, since they correspond to the highest or lowest weight states in either $z$ or $w$, which are unique.\n\\item The multiplicities on the edges of the graph follow the rule of Pascal's triangle (i.e. $1,\\,2,\\,1$ or $1,\\,3,\\,3,\\,1$, etc.). Indeed, the polynomial consisting only of the monomials associated to the vertices of an edge has the form $(z+w)^n$, or can be brought to this form by multiplication with a prefactor $z^{n_0}w^{m_0}$ or by redefinition of the variables.\n\\end{itemize}\nFor the multiplicities of the vertices in the interior of the toric diagram, a pattern is much less obvious.\n\nWe illustrate the above by the example of one square on the torus, which corresponds to the quiver gauge theory on $\\mathbb{F}_0$ in Figure \\ref{table:corr}.\n\\begin{comment}\n  \\begin{table}[h!]\n    \\begin{center}\n      \\begin{tabular}{ccc}\n        \\includegraphics[width=28mm]{torusquiver} &\n        \\begin{minipage}{6em}\n          \\vspace{-5em} ${\\text{graph dual}}$ \\\\ \\vspace{5em}\n        \\end{minipage}\n        &  \\includegraphics[width=25mm]{one_square} \\\\\n        $\\updownarrow$ &  & $\\downarrow$ \\\\\n        \\includegraphics[width=24mm]{quiver} & \\begin{minipage}{6em}\n          \\vspace{-3em} $\\longleftrightarrow$  \\\\ \\vspace{3em}\n        \\end{minipage} & \\includegraphics[width=32mm]{F0}\n      \\end{tabular}\n      \\caption{Web of correspondences for one square on the torus\/\n        quiver gauge theory on $\\mathbb{F}_0$}\n      \\label{table:corr}\n    \\end{center}\n  \\end{table}\n\\end{comment}\n\n\n\\newarrow{Corresponds}<--->\n\\begin{figure}\n  \\centering\n  \\begin{diagram}\n    \\begin{minipage}{29mm}\n      \\includegraphics[width=28mm]{torusquiver}\n    \\end{minipage} & \\rCorresponds^{\\text{\\hspace{1em}graph dual}} &\n    \\begin{minipage}{26mm}\n      \\includegraphics[width=25mm]{one_square}\n    \\end{minipage} \\\\\n    \\\\\n    \\uTo^{\\text{on the torus}} & & \\dTo^{\\det(K)} \\\\\n    \\\\\n    \\begin{minipage}{23mm}\n      \\includegraphics[width=22mm]{quiver}\n    \\end{minipage} & \\rCorresponds &\n    \\begin{minipage}{33mm}\n      \\includegraphics[width=32mm]{F0}\n    \\end{minipage}\n  \\end{diagram}\n\\caption{Web of correspondences for one square on the torus\/ quiver gauge theory on $\\mathbb{F}_0$}\n    \\label{table:corr}\n\\end{figure}\nThe tree-level superpotential is given by\n\\begin{equation}\\label{eq:supo}\nW=-2\\,X_{DC}X_{CA}X_{AB}X_{BD}+2\\,X_{CA}X_{AB}X_{BD}X_{DC},\n\\end{equation}\nthe Newton polynomial is \n\\begin{equation}\n  \\label{eq:partition_square}\n  P (z,w) = zw \\left(- \\frac{1}{z}-\\frac{1}{w} + 4 - z -w \\right) \\, .\n\\end{equation}\n\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nThe dimer model is concerned with the statistical mechanics of close\npacked dimer arrangements on a bipartite graph.  The physical system\nwhich can be thought of as a real world representation of the dimer\nmodel is the adsorption of diatomic molecules on a crystal surface. \n\nIn the 1960s, the question of how many perfect matchings exist on a plane\ngraph was solved independently by Kasteleyn~\\cite{Kasteleyn1,\nKasteleyn2}, and Temperley and Fisher~\\cite{Temperley, Fisher}: the total number is given by the Pfaffian of a signed, weighted adjacency matrix of the graph (the Kasteleyn matrix).\nMuch of\nthe original interest in the dimer model arose because it provides a simple\nand elegant solution for the 2--dimensional Ising model~\\cite{Hurst}.\n\nThe problem of enumerating perfect matchings is of course a classical problem in graph theory and combinatorics, see \\emph{e.g.}~\\cite{Kenyon2}. It can also be phrased in terms of domino tilings~\\cite{Kenyon1}.\nDuring the last years, the interest in the dimer model was revived\nthanks its manifold connections to other branches of\nmathematics and physics. The dimer model is related to\n\\begin{itemize}\n\\item configurations of a melting crystal corner and the\ntopological string A--model \\cite{Okounkov:2003sp, Iqbal:2003ds},\n\\item real algebraic geometry~\\cite{okounkov1,okounkov2}, \n\\item BPS black holes from $D$--branes wrapping collapsed cycles~\\cite{Heckman:2006sk}.\n\\end{itemize}\nFurthermore, a correspondence between the dimer model and quiver\ngauge theories arising from $D3$--branes probing a singular toric\nsurface was discovered and worked out in great detail\n\\cite{Hanany:2005ve, Franco:2005rj, Franco:2005sm, Hanany:2005ss,\nHanany:2006nm}. An explanation of this\ncorrespondence via mirror symmetry was given in~\\cite{Feng:2005gw}.\n\n\\bigskip\nThat the dimer model must be related to the free massless fermion on\nthe same lattice has been known for a long time already. Its relation\nto the Ising model, which in turn corresponds to a free fermion model\nclearly indicates this.  Moreover, Kasteleyn also showed that the\npartition function of the dimer model living on a graph embeddable on\na genus $g$ Riemann surface is given by a linear combination of\n$2^{2g}$ Pfaffians. This is, of course, reminiscent of the number of\nspin structures on a Riemann surface. In fact, a one-to-one\ncorrespondence between equivalence classes of Kasteleyn orientations\nand spin structures was proved in~\\cite{cimasoni-2006}. \n\nIn the following, we will make the connection between the dimer model and the\nfree fermion precise by identifying the Dirac operator with the Kasteleyn matrix and relating the expansion of the determinant\nappearing in the free fermion path integral to loop configurations\ncomposed of two perfect matchings. This picture of free fermions as a\nloop gas already appeared in a different context, for example in\n\\cite{Fermionloops}. \n\nThe perfect matchings form a\nbasis for the space of fermion loop configurations.  If the loop gas lives on a surface of non--trivial homotopy, non--contractible loops of non--zero winding\nnumber can occur, and the Pfaffian of the Kasteleyn matrix becomes a\npolynomial. Drawing inspiration from the categorification programme of\nKhovanov~\\cite{Khovanov1, Khovanov2, Khovanov3}, we \ninterpret this polynomial as the Euler characteristic of a co--chain\ncomplex. \nThe loop configurations of the free fermion model can\nbe naturally classified by the\nweight of the constituent matchings of the loop state and used to generate Abelian groups. %\nFor the Euler\ncharacteristic to coincide with the Newton polynomial, a differential\noperator must be constructed accordingly.  In our fermion loop gas,\nall loop states are paired since they come in two opposite\norientations, except for states consisting only of double line perfect\nmatchings. This is reminiscent of the supersymmetric ground states in\nsupersymmetric quantum mechanics. Indeed, it is possible to\nre--interpret the fermion loop gas as \\textsc{sqm} and to construct a differential\noperator which can be identified with the $Q$ operator. Like this, the\nNewton polynomial becomes the generalized Witten index of the system.\nSince the fermion loop states are bilinear in the perfect matchings, they can also be interpreted as maps from one matching to another. \n\nThanks to the correspondence between the dimer model and quiver gauge theory, \nwe know that the perfect matchings, \\emph{i.e.} our zero modes, parametrize the toric surface being probed by the $D$--branes. \nIn the following, we make use of this correspondence to give the loops states an interpretation in terms of the quiver gauge theory. Loops of vanishing net winding can be related to sequences of double Seiberg dualities on the associated acyclic quivers. In the special case of the hexagon graph, they correspond to certain BPS black hole configurations which are parametrized by melting crystal configurations~\\cite{Heckman:2006sk}. \n\n\\bigskip The plan of this paper is as follows.  In Section\n\\ref{sec:preliminaries}, we review the necessary background and\ndefinitions of the dimer model.  In Section \\ref{sec:dirac},\nwe relate the Kasteleyn matrix to the discretized Dirac operator\nof the free fermion. In Section~\\ref{sec:scaling-limit-theta}, we go to the limit of vanishing lattice\nspacing in which we recover the partition function of the Dirac\nfermion, which in the light of the interpretation of the dimer model\nas a free fermion model is no longer surprising. In Section~\\ref{sec:loop-expans-ferm}, we\nexpand the determinant arising in the free fermion path integral into\ncyclic permutations, which correspond to closed loops on the graph. Furthermore, we give an interpretation of the loops as fermionic states in Subsection~\\ref{sec:loops-as-fermionic}.\n\nIn Section \\ref{sec:categorification}, the basic idea of\ncategorification is explained, and in Section \\ref{sec:sqm}, the\nfermion loop system is re--interpreted as supersymmetric quantum\nmechanics. The simpler case of a square graph on the cylinder\nis worked out first, together with a simple explicit example in Section~\\ref{sec:example:-one-square}.\nIn Section~\\ref{sec:loops-as-operators}, we interpret the loops as operators mapping from one perfect matching to another. Finally, Section~\\ref{sec:sqm_torus} gives the full construction on the torus.\n\nIn Section \\ref{sec:geometry}, the basics of quiver gauge theories and the\ncorrespondence to the dimer model is reviewed. The interpretation of\nthe loop states in terms of the physics of $D$--branes probing a singular toric CY is attempted in Section \\ref{sec:results}. In Section~\\ref{sec:acyclic}, the relation between perfect matchings and corresponding acyclic quiver graphs is clarified. In Section~\\ref{sec:doubleseiberg}, we show that sequences of double Seiberg dualities correspond to cyclic permutations of the nodes of the acyclic quiver and to isomorphisms of the full gauge quiver. With these results, we can interpret the loops with vanishing overall winding in Section~\\ref{sec:0-0-loops}. The loops with non--trivial winding are discussed in Section~\\ref{sec:winding_loops}.\n\nThe results of this paper are summarized in a dictionary given in\nSection~\\ref{sec:conclusions}, where we close with some concluding\nremarks.\n\n\n\n\n\n\n\\subsection{The loop expansion of the fermion determinant}\n\\label{sec:loop-expans-ferm}\n\nAfter having shown the equivalence of the discretized Dirac operator\nand the Kasteleyn orientations and the appearance of the Dirac fermion\npartition function in the scaling limit, we will now directly relate\nthe dimer model to the free fermion via a diagrammatic expansion of\nthe fermion determinant.\n\nConsider a Grassmanian field $\\chi$ living on the nodes $V(\\gra)$ of a\ngraph $\\gra $ described by a quadratic action:\n\\begin{equation}\n  S[\\Psi] = \\sum_{x,y \\in V(\\gra)} \\chi_x A_{x,y} \\chi_y  . \n\\end{equation}\nAs we have seen in Eq.~(\\ref{eq:twocomponent}), we can associate a\ntwo--component Majorana fermion $\\Psi ={\\chi_\\bullet \\choose\n  \\chi_\\circ}$ to each black--white pair in a bipartite graph. The\npath integral for such an action is easily computed and is given by\nthe Pfaffian of the matrix $A$:\n\\begin{equation}\n  Z_{\\text{fermion}} = \\int \\left[  \\prod_{x \\in V(\\gra )} \\di \\chi_x \\right] e^{\\frac{1}{2} \\vec{\\chi} A \\vec{\\chi}} = \\Pf (A),\n\\end{equation}\nwhere $\\vec{\\chi}$ is the vector $\\vec{\\chi} = \\set{ \\chi_1, \\chi_2,\n  \\dots, \\chi_{\\abs{V(\\gra)}} }$. In fact, expanding the exponential\n(and using the fact that $\\chi_x$ are Grassmanian variables), we find:\n\\begin{equation}\n  Z_{\\text{fermion}} = \\int \\left[ \\prod_{x \\in V(\\gra )} \\di \\chi_x \\right]\n  \\prod_{x<y} \\left( 1 - \\chi_x A_{x,y} \\chi_y \\right).\n\\end{equation}\nAll the factors in the product commute, therefore we can order them by\nincreasing $x$. Expanding, the terms that survive the integration are\nthose which contain each variable exactly once, hence they are products\nof the $\\abs{V(\\gra)}\/2$ matrix elements $A_{x,y}$. There is one such\nproduct for each of the permutations of the nodes. Taking into account\nthe signs, we recover the expression for the Pfaffian in\nEq.~\\eqref{eq:pfaffian}.\n\n\nIn the following, we give a diagrammatic expansion for the\npartition function in terms of loops on the graph. Instead of the\nPfaffian, we can thanks to relation \\eqref{eq:rel} expand\nthe determinant under the square root. By definition, the determinant\nis a sum over all possible permutations $\\pi$ of the lattice points,\nsee Eq.~\\eqref{eq:det}.\nThe signature $\\sign(\\pi)$ of a permutation is easily found by\nexpressing $\\pi$ as the composition of cyclic permutations:\n\\begin{equation}\n  \\pi = \\pi_s \\circ \\pi_{s-1} \\circ \\dots \\circ \\pi_l \\circ \\dots \\circ \\pi_1  ,\n\\end{equation}\nwhere $\\pi_l$ is a cyclic permutation of $l_r$ elements. We define\n\\begin{equation}\n  \\sign (\\pi_l) = \\left( -  1\\right)^{l_r + 1}  \n\\end{equation}\nand\n\\begin{equation}\n  \\sign (\\pi) = \\prod_{l=1}^s \\sign (\\pi_l)  .\n\\end{equation}\n\nEach cyclic permutation $\\pi_l$ can be represented by a closed loop in\nthe graph by joining the vertex $x$ by a directed line to the point\n$\\pi_l(x)$, see \\emph{e.g.}~\\cite{Fermionloops}. Each vertex in $V(\\gra)$ can be:\n\\begin{itemize}\n\\item isolated if the permutation does not act on it;\n\\item joined to another one by a double line if the permutation\n  interchanges the two;\n\\item joined to two others as part of a cyclic chain.  \n\\end{itemize}\nIn particular, a permutation of two neighbouring points is a dimer\n(with a double line). \n\nA general permutation $\\pi$ is represented by a graph obtained as the\ndisjoint union of such loops, and the determinant is written as the\nweighted sum of all possible such graphs.  Note that two points can be\njoined by a line even if they are not neighbouring on the\nlattice. If we do not impose further\nconstraints on the form of the matrix $A$, we have a sum over all the\n$\\abs{V(\\gra)} !$ possible permutations and the general term\n\\emph{will not} be a subgraph of $\\gra$ (see\nFig.~\\ref{fig:example-loop}).\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=.25\\textwidth]{loop-01}\n  \\caption{The diagrammatic representation for the permutation $\\pi =\n    \\pi_3 \\circ \\pi_2 \\circ \\pi_1$ with $\\pi_1 : (17, 18) \\mapsto\n    (18,17)$, $\\pi_2 : (8,9,10,13) \\mapsto (9,10,13,8) $ and $\\pi_3 :\n    (1,3,11,6) \\mapsto (3,11,6,1)$.}\n  \\label{fig:example-loop}\n\\end{figure}\n\n\\bigskip\n\nA given graph only contributes to the determinant if the product of\nthe $A_{x \\pi(x)}$ along its edges is non-vanishing. This means that\nimposing constraints on the form of $A$ reduces the number of terms\nneeded.\n\\begin{itemize}\n\\item $A$ is a \\emph{local} operator if $A_{x,y} = 0 $ when $x $ and\n  $y$ are not neighbouring. In this case, the only loops contributing\n  to the determinant are those constructed as unions of edges in\n  $E(\\gra)$, and we can limit ourselves to summing over subgraphs of\n  $\\gra$.\n\\item If $A$ does not allow self--interactions, \\emph{i.e.}\n  $A_{x,x} = 0, \\ \\forall x \\in V(\\gra)$, we can consider only\n  permutations such that $\\pi(x) \\neq x,\\ \\forall x \\in\n  V(\\gra)$. Diagrammatically, this means that we only consider\n  subgraphs without isolated points.\n\\end{itemize}\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=.25\\textwidth]{loop-02}\n  \\caption{The diagrammatic representation for a permutation allowed\n    by the action for a free massless fermion. Orientations are not represented.}\n  \\label{fig:local-loop}\n\\end{figure}\nThe Dirac operator on the graph in Eq.~\\eqref{eq:Lattice-Dirac} enjoys\nboth properties.  In other words, when representing the path integral\nfor massless free fermions diagrammatically, we consider all subgraphs\ncontaining all the vertices, composed of closed loops or dimers, and\nwithout any isolated points. An example of such a configuration is\ndepicted in Fig.~\\ref{fig:local-loop}.\n\n\\bigskip \n\nOn a bipartite graph, each of these contributing loop configurations\ncan be seen as the composition of two perfect matchings.  This\ndecomposition is indeed unique.  Let $\\kket{m_i}$ and $\\kket{m_j}$ be\ntwo perfect matchings. We define the combination $\\ket{\\Psi} =\n\\kket{m_i} \\otimes \\kket{m_j}$ as the collection of all dimers\nbelonging to either of the matchings, taken with the usual $\\bullet\n\\to \\circ$ orientation if the edge is in $\\kket{m_i}$ and with the\nopposite one if it is in $\\kket{m_j}$. Any subgraph $\\ket{\\Psi}$\ncontributing to the determinant can always be written in this way. In\nfact:\n\\begin{itemize}\n\\item if $\\ket{\\Psi}$ does not contain any loops, its edges are in\n  one--to--one correspondence with those of a perfect matching\n  $\\kket{m}$ and we define $\\ket{\\Psi} \\doteq \\ket{m}= \\kket{m} \\otimes \\kket{m}$;\n\\item if $\\ket{\\Psi}$ contains one loop, this loop consists of an even\n  number of edges since the graph is bipartite. Therefore we can\n  decompose it into two disjoint sets of edges that join all the\n  vertices. It follows that we can always find two matchings\n  $\\kket{m_i}$ and $\\kket{m_j}$ that only differ by a shift by one\n  dimer on the loop and are equal on all the isolated dimers in\n  $\\ket{\\Psi}$. Note that the state $\\ket{\\tilde\\Psi}$ containing the\n  same loop with reversed orientation corresponds simply to\n  $\\ket{\\tilde\\Psi} = \\kket{m_j} \\otimes \\kket{m_i}$;\n\\item if $\\ket{\\Psi}$ contains $n$ loops (which we regard here as unoriented), it can be constructed as the\n  composition of any of the perfect matchings containing the disjoint\n  sets of dimers making up each loop. It is easy to convince oneself\n  that there are $2^{n}$ such combinations, corresponding to the\n  $2^n$ possible orientations of the loops (see\n  Fig.~\\ref{fig:loop-composition}).\n\\end{itemize}\n\\begin{figure}\n  \\centering\n  \\begin{tabular}[c]{ccccc}\n    \\includegraphics[width=.25\\linewidth]{loop-03} &\n    \\begin{minipage}{1em}\n      \\vspace{-1cm}$=$ \\vspace{1cm}\n    \\end{minipage}\n    & \\includegraphics[width=.25\\linewidth]{loop-03-01} & \\begin{minipage}{1em}\n      \\vspace{-1cm}$\\otimes$ \\vspace{1cm}\n    \\end{minipage} & \\includegraphics[width=.25\\linewidth]{loop-03-02} \\\\\n    $\\ket{\\Psi}$ && $\\kket{m_1}$ && $\\kket{m_2}$\n  \\end{tabular}\n  \\caption{Decomposition of a subgraph $\\ket{\\Psi}$ containing two\n    loops as the combination $\\ket{\\Psi} = \\kket{m_1} \\otimes\n    \\kket{m_2}$. Orientations are not represented.}\n  \\label{fig:loop-composition}\n\\end{figure}\n\nNote that the composition of two perfect matchings always generates an\nallowed configuration. In fact, in each of the matchings, every point\nin $V(\\gra)$ is touched by an edge and in the resulting subgraph, each\npoint is either reached by two distinct lines (and then is part of a\nloop) or by a double one (and is part of a dimer).\n\nIf on a graph $\\gra$, there are $k$ perfect\nmatchings, the determinant is written as the sum over $k^2$ subgraphs.\n\n\n  \n\\subsection{Loops as fermionic states}\n\\label{sec:loops-as-fermionic}\n\nAs we have seen in the last section, the partition function for the free fermion can be expanded as a sum\nover loop configurations on the lattice. Here we show how these\nconfigurations can be understood in terms of fermionic states.\n\nConsider a square graph on the torus with $N$ horizontal and $M$\nvertical cells, each identified by a vector $\\mathbf{z} = \\left( z,\n  w \\right) = z\\, \\mathbf{e}_1 + w\\, \\mathbf{e}_2$. Let $h\n(\\mathbf{z})$ be the height function defined on a loop\nconfiguration. The function can be labelled by two integers $p,q$,\ncorresponding to the following periodicity conditions:\n\\begin{equation}\n  h_{p,q} ( \\mathbf{z} + n_1 N\\, \\mathbf{e}_1  + n_2 M\\, \\mathbf{e}_2  ) = h_{p,q} (\\mathbf{z}) + n_1 p + n_2 q \\, ,\\hspace{1em} \\forall n_i \\in \\setZ \\, .\n\\end{equation}\nThe integers $(p,q)$ are clearly the\nwinding numbers for the loop configuration.\\footnote{One can define a\nconsistent wavefunction for a $\\setZ_k$ parafermion as\n\\begin{equation}\n  \\psi^{(k)}_{p,q} ( \\mathbf{z} ) = e^{\\frac{2 \\pi \\imath}{k} h (\\mathbf{z})}  \\, .\n\\end{equation}\nHere, we concentrate on the $k=2$ fermionic case, but the formalism can\nbe extended easily.}\n\nWe would like to consider one of the two coordinates (say $z$) as\nEuclidean time and hence see the system as the description of a\nfermion in $1+1$ Euclidean dimensions. To each plaquette we associate\nthe operators $a (\\mathbf{z})$, $a^\\dagger (\\mathbf{z})$,\n$b(\\mathbf{z})$, $b^\\dagger (\\mathbf{z})$ living on the left edge,\nwith the following interpretations. The operator\n$a^\\dagger(\\mathbf{z})$ creates a particle at point $w$ for all\ntimes $z^\\prime \\ge z$ and raises the height function by $1$ on\nthe right of point $\\mathbf{z}$, while the operator $b^\\dagger\n(\\mathbf{z})$ creates an antiparticle at point $\\mathbf{z}$ and lowers\nthe height function by $1$ on the right. We impose the usual\nanticommutation relations\n\\begin{align}\n\\label{eq:anticomm-ab}\n  \\set{a (\\mathbf{z}), a^\\dagger (\\mathbf{z}^\\prime )} =\n  \\delta_{\\mathbf{z},\\mathbf{z}^\\prime} \\, , && \\set{b (\\mathbf{z}),\n    b^\\dagger (\\mathbf{z}^\\prime )} =\n  \\delta_{\\mathbf{z},\\mathbf{z}^\\prime} \\, ,\n\\end{align}\nall the other anticommutators vanishing. In particular, we recover the\nconstraint $\\left( a^\\dagger \\right)^2 = \\left( b^\\dagger \\right)^2 =\n0 $, which can be translated to the fact that the height function\ncannot jump by more than one unit when passing to an adjacent\nplaquette.  As usual, one can construct a highest--weight\nrepresentation for the algebra by introducing a state $\\ket{0}$ which\nis annihilated by all the $a$ and $b$\n\\begin{equation}\n  a (\\mathbf{z} ) \\ket{0} = b (\\mathbf{z}) \\ket{0} = 0,  \n\\end{equation}\nand defining a general state as the result of the action of a sequence\nof $a^\\dagger $ and $b^\\dagger$ on $\\ket{0}$:\n\\begin{equation}\n  \\ket{\\Psi} = b^\\dagger (\\mathbf{z}_1 ) \\dots b^\\dagger(\\mathbf{z}_{N_b}) a^\\dagger (\\mathbf{z}_{N_b+1}) \\dots a^\\dagger (\\mathbf{z}_{N_b+N_a}) \\ket{0} \\, .\n\\end{equation}\nIn this operator formalism, the height function is written as\n\\begin{equation}\n  h ( z, w ) = \\sum_{\\zeta = 0}^{z} a^\\dagger (\\zeta, w) a ( \\zeta, w) - b^\\dagger (\\zeta, w) b ( \\zeta, w) = \\sum_{\\zeta = 0}^{z} N_a (\\zeta, w) - N_b ( \\zeta, w) \\, ,\n\\end{equation}\nwhere we introduced the number operators $N_a (\\mathbf{z}) =\na^\\dagger(\\mathbf{z}) a(\\mathbf{z})$.\\footnote{In a typical Fermi\n  system, $h(z,w)$ is the total charge at point $w$ in\n  space and time $z$.}\n\nEven though one can reproduce any height function by just using the\n$a$ and $b$ operators, the relations in Eq.~\\eqref{eq:anticomm-ab} are\nnot yet enough to describe all the states. This is because we have\narbitrarily chosen a time direction. One can show that the\nanticommutation constraints should be supplemented by a set of\nnon--local constraints in order to ensure the consistency of the\nheight function. Yet it is more convenient to introduce two other sets\nof operators $c, c^\\dagger$ and $d, d^\\dagger$ which when acting on a\npoint $(z, w)$ create a particle (respectively an antiparticle) at\nfixed time $z$ for each $w^\\prime \\ge w$. They also satisfy\nanticommutation relations of the type\n\\begin{align}\n  \\set{c (\\mathbf{z}), c^\\dagger (\\mathbf{z}^\\prime )} =\n  \\delta_{\\mathbf{z},\\mathbf{z}^\\prime} \\, , && \\set{d (\\mathbf{z}),\n    d^\\dagger (\\mathbf{z}^\\prime )} =\n  \\delta_{\\mathbf{z},\\mathbf{z}^\\prime} \\, ,\n\\end{align}\nand anticommute with all the $a$s and $b$s. In this way, we\nagain obtain $\\left( c^\\dagger \\right)^2 = \\left( d^\\dagger \\right)^2\n= 0$, which means that the height function cannot jump by more than one unit\nwhen passing from a plaquette to the one on top or below. Redefining the vacuum\n$\\ket{0}$ as the state annihilated by $a (\\mathbf{z})$,\n$b(\\mathbf{z})$, $c (\\mathbf{z})$ and $d(\\mathbf{z})$, we finally find\nthat a general state is written as\n\\begin{multline}\n  \\ket{\\Psi} = d^\\dagger (\\mathbf{z}_1 ) \\dots d^\\dagger\n  (\\mathbf{z}_{N_d}) c^\\dagger (\\mathbf{z}_{N_d+1}) \\dots c^\\dagger\n  (\\mathbf{z}_{N_d+N_c}) b^\\dagger (\\mathbf{z}_{N_d+N_c+1}) \\dots\n  b^\\dagger (\\mathbf{z}_{N_d+N_c+N_b}) \\\\\n  a^\\dagger (\\mathbf{z}_{N_d+N_c+N_b+1}) \\dots a^\\dagger\n  (\\mathbf{z}_{N_d+N_c+N_b+N_a}) \\ket{0} \\, .\n\\end{multline}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=.3\\textwidth]{HeightFunction-crop}\n  \\caption{Height function for a loop state. }\n  \\label{fig:height-function-loop}\n\\end{figure}\n\nSome consistency conditions must be satisfied. In particular, it should\nbe verified that the actions of $b^\\dagger(z + 1, w)\\,\na^\\dagger(z,w)$ and of $d^\\dagger(z,w + 1 )\\,\nc^\\dagger(z,w)$ change the height function in the same way, namely\nthey raise $h(z,w)$ by one unit (see Figure\n\\ref{fig:height-function-loop}). It is easy to see that this is\nequivalent to requiring that on any allowed state $\\ket{\\Psi}$:\n\\begin{equation}\n  \\label{eq:height-equality}\n  h (z, w ) \\ket{\\Psi} = \\left[ \\sum_{\\zeta = 0}^{z} N_a ( \\zeta, w) - N_b ( \\zeta, w) \\right] \\ket{\\Psi} =   \\left[ \\sum_{\\omega = 0}^{w} N_c ( z, \\omega) - N_d ( z, \\omega) \\right] \\ket{\\Psi} \\, ,\n\\end{equation}\nwhere by allowed we mean that it contributes to the Berezin integral.\\\\\nNote that $h (\\mathbf{z}) \\ket{0} = 0$, and since the operator\n$a^\\dagger (\\mathbf{z})\\, b^\\dagger (\\mathbf{z})$ anticommutes with $h\n(\\mathbf{z})$, one can easily see that an allowed state $\\ket{\\Psi}$\ncorresponds to a perfect matching if and only if it is annihilated by\n$h(\\mathbf{z})$:\n\\begin{equation}\n  h (\\mathbf{z}) \\ket{PM} = 0 \\, .  \n\\end{equation}\n\n\nOther interesting operators can be defined:\n\\begin{subequations}\n  \\label{eq:global-height}\n  \\begin{align}\n    H_z \\ket{\\Psi}&= \\left[ h (N, \\bar w) - h (0, \\bar w)\n    \\right] \\ket{\\Psi} = \\left[\\sum_{\\zeta = 1}^{N} N_a ( \\zeta,\n      \\bar w) - N_b ( \\zeta, \\bar w) \\right] \\ket{\\Psi} \\, , \\\\\n    H_w \\ket{\\Psi}&= \\left[h (\\bar z, M) - h (\\bar z, 0)\\right]\n    \\ket{\\Psi} = \\left[\\sum_{\\omega = 1}^{M} N_c ( \\bar z, \\omega) -\n      N_d ( \\bar z, \\omega) \\right] \\ket{\\Psi}\\, ,\n  \\end{align}\n\\end{subequations}\nwhere $\\bar w$ (resp. $\\bar z$) is an arbitrary value. Note that\nboth operators are global since they are defined for the whole graph\nand do not depend on the choice of $\\bar z$ or $\\bar w$. We can consider $h\n(\\mathbf{z})$ as their local counterpart. $H_z,\\,H_w$ count the winding\nnumbers of a state $\\ket{\\Psi}$ in the $z$ and $w $ directions:\n\\begin{subequations}\n  \\begin{align}\n    H_z \\ket{\\Psi_{p,q}} &= p \\ket{\\Psi_{p,q}} \\, , \\\\\n    H_w \\ket{\\Psi_{p,q}} &= q \\ket{\\Psi_{p,q}} \\, .\n  \\end{align}\n\\end{subequations}\nIn particular, states with vanishing winding numbers will be\nannihilated by both $H_z$ and $H_w$. Another operator is the one that\ninterchanges each $a^\\dagger $ with a $b^\\dagger$, each $c^\\dagger$\nwith a $d^\\dagger$ and viceversa:\n\\begin{equation}\n  \\label{eq:Q1-creator}\n  Q_1 \\ket{\\Psi} = \\prod_{\\mathbf{z}} \\left[ a^\\dagger (\\mathbf{z} ) b (\\mathbf{z} )  + b^\\dagger (\\mathbf{z} ) a (\\mathbf{z} ) \\right] \\left[ c^\\dagger (\\mathbf{z} ) d (\\mathbf{z} )  + d^\\dagger (\\mathbf{z} ) c (\\mathbf{z} ) \\right] \\ket{\\Psi} \\, .\n\\end{equation}\nOne can easily verify that $\\set{Q_1, h (\\mathbf{z})} = 0$. It follows\nthat $Q_1$ leaves the perfect matchings invariant and pairs all the\nother states such that $Q_1 \\ket{\\Psi_{p,q} } =\n\\ket{\\tilde{\\Psi}_{p,q}} = \\ket{\\Psi_{-p,-q} }$.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=.3\\textwidth]{operators-crop}\n  \\caption{Diagrammatic representation for the four creation operators\n    around a plaquette with coordinates $(z,w)$.}\n  \\label{fig:operators}\n\\end{figure}\n\nA diagrammatic representation for a general state $\\ket{\\Psi} $ is\neasily obtained as follows. For each operator $a^\\dagger (\\mathbf{z})$,\none adds a vertical upward arrow on the left edge of the\nplaquette at point $\\mathbf{z}$, for each $b^\\dagger (\\mathbf{z})$ a\ndownward arrow on the left edge of the plaquette, for each\n$c^\\dagger(\\mathbf{z})$ a left pointing arrow on the bottom of the\nplaquette and for each $d^\\dagger(\\mathbf{z})$ a right pointing arrow\non the bottom of the plaquette (see Fig.~\\ref{fig:operators}).\n\nIn Sec.~\\ref{sec:loop-expans-ferm}, we found the loop states to be\ncombinations of two perfect matchings. We can indeed recover the same\nresult in terms of this operator formalism, see\nAppendix~\\ref{sec:perf-match-oper} for the details, where we also give\na necessary and sufficient condition for a state $\\ket{\\Psi}$ to be\nallowed.\n\n\\bigskip\n\nLet us summarize the contents of this rather technical section. The\nterms contributing to the expansion of the path integral in\nEq.~\\eqref{eq:fermion-path-integral} can be represented as resulting\nfrom the action of a set of operators $a^\\dagger, b^\\dagger, c^\\dagger\n$ and $d^\\dagger$, defined on the lattice, on a vacuum $\\ket{0}$. This\nset is redundant in the sense that the knowledge of the $a^\\dagger $\nand $b^\\dagger$ operators, plus the consistency conditions in\nEq.~\\eqref{eq:state-consistency} uniquely identifies the $c^\\dagger$\nand $d^\\dagger$.\n\nWe interpret the two directions in the graph as a Euclidean time ($z$)\nand a space coordinate ($w$). The operator $a^\\dagger (z,w)$\ncreates a particle at time $z$ and position $w$ that propagates at\nthe same position for all times $z \\le z^\\prime \\le N$. In the\nsame way, $b^\\dagger (z, w)$ creates an antiparticle. The height\nfunction $h(z,w)$ computes the local charge ($+1$ for particles\nand $-1$ for antiparticles) at point $w$ and time $z$. The charge\ngradient $H_w$ between the points $w = 0 $ and $w = M$ at fixed\ntime $\\bar z$ is conserved during the time evolution. \\emph{A loop\n  configuration with winding numbers $\\left( H_z, H_w \\right)$\n  represents the time evolution of a system with initial charge\n  gradient $H_w$, whose net charge in any fixed point in space $\\bar\n  w$ jumps by $H_z$ between $z=0$ and $z=N$}. In particular, a\n$\\left(0 , H_w \\right)$ loop describes a time evolution where no net\ncharge is created globally, while a perfect matching corresponds to a\nsystem where no net charge is present at the initial time nor is\ncreated locally.\n\n\n\n\n\n\n\\section{Gauge theory interpretation: Results}\n\\label{sec:results}\n\nIn Section~\\ref{sec:loops-as-operators}, we have seen that our fermion loop configurations can be interpreted as maps from one perfect matching to another.\nGeometrically, a marked difference exists between the inner vertices of a toric diagram and those on the boundary. The inner vertices correspond to exceptional divisors and are compact. The vertices on the boundary must be subdivided into the ones at the corners of the diagram and the others. The latter are also exceptional divisors and are of the form $\\mathbb{P}^1\\times \\mathbb{C}$ with possible blow--ups, \\emph{i.e.} have a compact and a non--compact factor. The corner vertices are non--compact and have the form $\\mathbb{C}\\times\\mathbb{C}$ plus possible blow--ups. It is therefore not surprising that it makes a difference whether we only map between internal matchings or if boundary matchings appear on one side of the map. When the size of the graph is very large, \\emph{i.e.} $N,\\,M\\to\\infty$, maps between internal matchings indeed are the generic case.\n\nIt was shown in~\\cite{Hanany:2006nm} that for each internal matching\n$\\mathrm{PM}_i\\in\\{\\mathrm{PM}^{int}$\\}, an acyclic quiver\n$Q_{\\mathrm{PM}_i}$ is obtained by deleting all edges which are part\nof $\\mathrm{PM}_{i}$ from the dual quiver $Q_X$. We can construct an\nexceptional collection of sheaves for each such\n$Q_{\\mathrm{PM}_i}$. We associate $Q_{\\mathrm{PM}_i}$ to the quiver\n$Q_{S_i}$ corresponding to the partial resolution $S_i$ of $S$\nobtained by blowing up the divisor $E_i$ to which $\\mathrm{PM}_{i}$\nbelongs. The acyclic quiver $Q_S$ represents the system as seen by an\nobserver living only on $S$, while from the point of view of the whole\nCalabi--Yau $X$, the low--energy theory remains unchanged, regardless\nof the resolution. The different $Q_{S_i}$ associated to perfect\nmatchings belonging to the same internal point can be seen as\ndifferent ways the gauge theory is embedded in the surface $S_i\\subset\nX$. The acyclic quivers associated to different resolutions $S_i$ and\n$S_j$ also correspond to different decompositions of the arrows in\n$Q_X$ into $\\mathrm{Ext}^1_S$ and ${Ext}^2_S$, see\nEq.~(\\ref{eq:Ext_decomposition}), but this difference should rather be\nseen as due to the different topologies of the partially resolved\nsurfaces $S_i,\\,S_j$ the exceptional collections are supported on.\n\nOn the quiver side, a map between internal matchings corresponds thus\nto a map $f_{ji}:Q_{S_i} \\mapsto Q_{S_j}$, while on the geometry side,\nit corresponds to mapping from one exceptional collection to another.\nThese maps are \\emph{always graph isomorphisms of the full quiver}\n$Q_X$, in the sense that the adjacency remains unchanged and only the\nlabels of the nodes are permuted. This means that the physical gauge\nquiver $Q_X$ remains unchanged by the loop maps, \\emph{i.e.} the\nquiver gauge theory is too coarse to see their effect.  At the level\nof the acyclic quivers $Q_{S_i}$, the maps correspond to gauge\ndualities. Geometrically, the decomposition of the arrows of $Q_X$\ninto $\\mathrm{Ext}^1_S$ and $\\mathrm{Ext}^2_S$ changes, which\ncorresponds to a change of embedding.\n\nIn the case of a boundary perfect matching, cycles will remain in the\nreduced quiver $Q_{\\mathrm{PM}_{bd}}$ which causes the quiver\nrepresentation to be infinite dimensional and the construction of the\nexceptional collection to fail. This failure of the known methods to\nextract useful information for these cases prevents us from giving a\nsatisfactory interpretation of maps involving boundary matchings.\n\nThere is one special case in which the loop states admit a direct\ninterpretation in the quiver gauge theory. It is the case of loops\narising from the composition of two adjacent boundary matchings. It\nwas noticed in~\\cite{Hanany:2005ss} that this results in a so--called\n\\emph{zig--zag path}.  A zig--zag path is a path which turns\nalternatingly maximally left and maximally right at the vertices. This\nmaximal turning condition results in also the edges in the dual graph\nforming a closed path, which is not the case for a general path. A\nclosed zig--zag path on the dimer model graph $\\gra$ therefore\nresults in a closed zig--zag path in the dual quiver, which\ncorresponds to a gauge invariant trace operator.\n\n\n\n\n\n\\subsection{Matchings and acyclic quivers}\n\\label{sec:acyclic}\n\nThe problem of finding a minimal set of arcs in a directed graph upon\nthe deletion of which the graph becomes acyclic is well studied in\nmathematics and computer science~\\cite{Karp:1972}. Such a set is\ncalled a minimal \\emph{feedback arc set} (\\textsc{fas}). There exists\na precise relation between the problem of finding all \\textsc{fas} of\na digraph and the problem of identifying all the perfect matchings in\nits dual\ngraph.\\\\\nConsider a bipartite plane graph $\\gra$ with $N$ nodes and its graph\ndual $\\gra^\\prime$. The one--cycles in $\\gra^\\prime$ are generated by\nthe plaquettes $\\set{p_j}_{j=1}^N$ which correspond to the vertices of\n$\\gra$. Removing the edge $e_{ij}$ shared by the cycles $p_i $ and\n$p_j$ breaks both cycles. A minimal \\textsc{fas} is obtained as the\ncollection of $N\/2$ edges $\\set{e_{i_k j_k}}_{k=1}^{N\/2}$ shared by\ndisjoint pairs of plaquettes $p_{i_j}$ and $p_{i_k}$. In the dual\ngraph $\\gra$, this corresponds to selecting a set of edges joining all\nthe nodes and touching all of them only once. In other words, the\n\\textsc{fas} in $\\gra^\\prime $ is a perfect matching in $\\gra$. The\nsituation is different when $\\gra $ is embedded on a Riemann surface\nof genus $g>0$, because in addition to the one--cycles generated by\nthe plaquettes, there are $2g$ equivalence classes of cycles of\nnon--trivial holonomy. In this case, the winding cycles in\n$\\gra^\\prime$ are generated by the zig--zag paths. It follows that\nbeing a perfect matching in $\\gra$ is only a necessary condition for a\nset of edges to be a \\textsc{fas} in $\\gra^\\prime$. It was shown\nin~\\cite{Hanany:2006nm} that a perfect matching corresponding to an\ninternal point in the toric diagram is always a \\textsc{fas}. Removing\na boundary matching, on the other hand, always preserves at least one\nzig--zag path.\n\n\\bigskip\n\nAs mentioned before, the gauge quiver $Q_X$ is the graph dual of the\ngraph $\\gra$ on which the dimer model lives.  The acyclic quiver\n$Q_{S_n}$ is obtained from $Q_X$ by deleting all the edges contained\nin the $n$th perfect matching\n$\\mathrm{PM}_n\\in\\{\\mathrm{PM}^{int}\\}$. The deleted arrows are\nrepresented by dashed lines and correspond to relations in the quiver,\n\\emph{i.e.} $\\mathrm{Ext}^2$s. Such an acyclic quiver allows an\nordering of its nodes such that there are no arrows between the nodes\n$i$ and $j$ if $i<j$. If $Q_{S}$ has more than one starting or end\npoint, an ordering ambiguity arises. Indeed, $Q_{S}$ is in general a\npartially ordered set which requires two ordering relations, say left\nto right and bottom to top.\n\nThe maps $f^{(0,0)}_{m,n}$ corresponding to $(0,0)$--loops turn out to\nhave a particularly simple structure: \\emph{they always correspond to\n  a cyclic permutation of the nodes in one of the two ordering\n  relations} (\\emph{i.e.} up to re--ordering in the vertical\ndirection). Maps of non--trivial weight correspond to a general\npermutation of the nodes. In particular, they do not preserve either\nof the ordering relations. In the following, we will attempt to give\nthese maps an interpretation as gauge dualities of the acyclic quivers\n$Q_{S_i}$.\n\n\\begin{figure}\n  \\centering\n  \\subfigure[Perfect matching $\\mathrm{PM}_1$ (weight $(0,0)$) and quiver $Q_{{\\mathrm{PM}}_1}$]{\\includegraphics[height=40mm]{ex_qui0}} \\hfill\n  \\subfigure[Perfect matching $\\mathrm{PM}_2$ (weight $(0,0)$) and quiver $Q_{{\\mathrm{PM}}_2}$]{\\includegraphics[height=40mm]{ex_qui1}}  \\vfill\n  \\subfigure[Perfect matching $\\mathrm{PM}_3$ (weight $(1,0)$) and quiver $Q_{{\\mathrm{PM}}_3}$]{\\includegraphics[height=34mm]{ex_qui2}} \\hfill  \\caption{Example: Three inner perfect matchings and associated acyclic quivers}\n  \\label{fig:iso}\n\\end{figure}\n\n\nAn example of the above maps is given in Figure \\ref{fig:iso}. This\nexample is big enough to have more than one inner point for the toric\ndiagram. Fig.  \\ref{fig:iso}.(a) shows a perfect matching of weight\n$(0,0)$ and the corresponding dual quiver with the deleted edges\ndepicted by dashed arrows. The map from this quiver to the one shown\nin Fig. \\ref{fig:iso}.(b) obviously corresponds to the cyclic\npermutation $f_{2,1}^{(0,0)}:\\, (F,E,G,A,C,B,D,H)\\mapsto\n(A,C,B,D,H,F,E,G)$. Graphically, $\\mathrm{PM}_1$ can be recovered from\n$\\mathrm{PM}_2$ by first moving $G$ in front of all nodes its arrows\nare attached to, then $E$, and finally $F$. In the process, dashed\narrows which change their direction from pointing left to pointing\nright become solid and solid arrows which change from pointing left to\nright become dashed.  Notice the ordering ambiguity in the vertical\ndirection. We could have ordered $Q_{{\\mathrm{PM}}_2}$ also as\n\\emph{e.g.} $(C,A,B,D,F,H,E,G)$. Now consider the map from\n$Q_{{\\mathrm{PM}}_1}$ to $Q_{{\\mathrm{PM}}_3}$, shown in\nFig. \\ref{fig:iso}.(c). It corresponds to the permutation\n$f_{3,1}^{(1,0)}:\\, (F,E,G,A,C,B,D,H)\\mapsto (C,D,H,E,A,B,F,G)$ which\nhas an obviously more complicated structure and does not preserve\neither ordering relation.\n\n\n\\subsection{Seiberg dualities and graph isomorphisms}\n\\label{sec:doubleseiberg}\n\nIn the following, we would like to connect the graph isomorphisms of\n$Q_X$ arising in the maps $f_{m,n}$ to known gauge dualities of the\nacyclic quivers. Seiberg dualities~\\cite{Seiberg:1994pq}, which are IR\ndualities between gauge theories, are obvious candidates. It was first\nremarked in~\\cite{Cachazo:2001sg} and later elaborated upon\nin~\\cite{Aspinwall:2004vm, Herzog:2004qw} that certain Seiberg\ndualities are realized as a sequence of exceptional mutations.  Here, we are\nonly interested in producing quivers which are subquivers of $Q_X$. In\ngeneral, this will not be the case for Seiberg dualities because the\nrank of the gauge groups can change, as in\nEq.~\\eqref{eq:Seiberg-ranks}.\n\nWe want to show here that the effect of performing two consecutive\nSeiberg dualities on the same node is a graph isomorphism on the\nphysical quiver $Q_X$ and a cyclic permutation on the nodes of the\nquiver $Q_S$ with respect to one of the ordering relations. Let $i_0$\nbe a starting node in the quiver, $A_{\\text{out}}$ the set of\nendpoints of the arrows starting in $i_0$, and $A_{\\text{in}}$ the set\nof starting points for the arrows ending in $i_0$. The effect of a\nSeiberg duality on $i_0$ is the following~\\cite{He:2004rn}: $i_0$ is\nmoved to the left of the nodes in $A_{\\text{out}}$, the directions of\nthe arrows coming to and going from $i_0$ are reversed, and the arrows\nbetween $A_{\\text{out}}$ and $A_{\\text{in}}$ are changed to preserve\nconsistency. More precisely, for $A \\in A_{\\text{out}}$ and $B \\in\nA_{\\text{in}}$,\n\\begin{equation}\n  \\begin{cases}\n    a_{A i_0}^\\prime = a_{i_0 A} \\, ,\\\\\n    a_{i_0 B}^\\prime = a_{B i_0} \\, ,\\\\\n    a_{A B}^\\prime = a_{AB} - a_{i_0 A} a_{B i_0} \\, ,\n  \\end{cases}\n\\end{equation}\nwhere $a_{ij}$ is the number of arrows going from node $i$ to node $j$\n(with the convention $a_{ji} = - a_{ij}$) in the initial quiver $Q_S$,\nand $a^\\prime_{ij}$ the corresponding in the quiver $Q_S^\\prime$\nobtained after the duality. The ranks of the gauge groups are\nchanged as follows:\n\\begin{equation}\n  \\label{eq:Seiberg-ranks}\n  \\begin{cases}\n    N^\\prime_j = N_j & \\text{if $j \\neq i_0$} \\, \\\\\n    N^\\prime_{i_0} = \\displaystyle{\\sum_{A \\in A_{\\text{out}}}} a_{i_0 A} N_A - N_{i_0} \\, . \n  \\end{cases}\n\\end{equation}\nIn general, $N^\\prime_{i_0} \\neq N_{i_0}$ and the transformation\nis not an isomorphism of $Q_X$. Let us now apply the \ntransformation again on the same node $i_0$. The arrows joining $i_0$ with\nthe other nodes have been reversed, so the role of $A_{\\text{in}}$ and\n$A_{\\text{out}}$ is interchanged:\n\\begin{align}\n  A^\\prime_{\\text{out}} = A_{\\text{in}} \\, ,&&\n  A^\\prime_{\\text{in}} = A_{\\text{out}} \\, .\n\\end{align}\nThe node $i_0$ is now moved to the left of $A^\\prime_{\\text{out}} =\nA_{\\text{in}}$ and the arrows are changed as follows:\n\\begin{equation}\n  \\begin{cases}\n    a^{\\prime\\prime}_{B i_0} = a^\\prime_{i_0 B} = a_{B i_0} \\, \\\\\n    a^{\\prime \\prime}_{i_0 A} = a^\\prime_{A i_0} = a_{i_0 A} \\, \\\\\n    a^{\\prime\\prime}_{BA} = a^\\prime_{BA} - a^\\prime_{i_0 B} a^\\prime{A i_0} = - \\left( a_{AB} - a_{i_0 A} a_{B i_0} \\right) -  a_{i_0 A} a_{B i_0}  = a_{BA} \\, .\n  \\end{cases}\n\\end{equation}\nThe ranks now read:\n\\begin{equation}\n  \\begin{cases}\n    N^{\\prime \\prime}_j = N^\\prime_j = N_j & \\text{for $j \\neq i_0$}\\\\\n    N^{\\prime \\prime}_{i_0} = \\displaystyle{ \\sum_{B \\in A^\\prime_{\\text{out}}} a^\\prime_{i_0 B} N^\\prime_B - N^\\prime_{i_0} = \\sum_{B \\in A_{\\text{in}}} a_{B i_0} N_B  - \\sum_{A \\in A_{\\text{out}}} a_{i_0 A} N_A + N_{i_0} = N_{i_0}},\n  \\end{cases}\n\\end{equation}\nwhere we used the anomaly--cancellation condition around $i_0$ in\n$Q_X$.\n\nAs expected, applying the Seiberg duality twice just amounts to an\nisomorphism of the gauge quiver $Q^{\\prime \\prime}_X \\sim Q_X$. On\nthe other hand, the node $i_0$ has been changed from a starting point\nin $Q_S$ to an end point in $Q_{S^{\\prime \\prime}}$, and continuous\nand dashed arrows from\/to $i_0$ have been interchanged (see\nFig.~\\ref{fig:Double-Seiberg}).\n\n\\begin{figure}\n  \\centering\n  \\subfigure[$Q_S$]{\\includegraphics[height=2.7cm]{DoubleSeiberg-1-crop}} \\hfill\n  \\subfigure[$Q_S^\\prime$]{\\includegraphics[height=2.7cm]{DoubleSeiberg-2-crop}} \\hfill\n  \\subfigure[$Q_S^{\\prime\\prime}$]{\\includegraphics[height=2.7cm]{DoubleSeiberg-3-crop}} \n  \\caption{Double Seiberg duality around the node $i_0$.}\n  \\label{fig:Double-Seiberg}\n\\end{figure}\n\n\\subsection{Interpretation of the $\\left(0,0\\right)$--loops}\n\\label{sec:0-0-loops}\n\n\\subsubsection{Plaquette moves}\n\\label{sec:plaquette-moves}\n\n\nWith the results of the last section, we can now attempt an\ninterpretation of the $(0,0)$--loops, which form the diagonal\ncomponent of the cohomology $H^{(k,k)}$ for the complex in\nEq.~\\eqref{eq:torus-complex}.\n\nConsider a loop configuration which contains the boundary of a basic plaquette and otherwise only consists of double--line perfect matchings.\nWe define a \\emph{plaquette move} to be a map that switches between the two perfect matchings making up the above plaquette loop while leaving the rest of the configuration invariant, see Figure \\ref{fig:plaquette}. This is obviously a $(0,0)$ map and corresponds to the minimal loop. It can be shown that the set of loops with windings $\\left(\\bar p, \\bar q \\right)$ fixed is connected, with the arcs\ngiven by the plaquette moves. It follows that all maps of weight $(0,0)$ are generated by plaquette moves, \\emph{i.e.} we can move between all perfect matchings of fixed weight $\\left(\\bar p, \\bar q \\right)$ by a sequence of plaquette moves.\n\\begin{figure}\n \\begin{center}\n  \\includegraphics[width=50mm]{Plaquette} \n  \\caption{Plaquette move in the square graph}\n  \\label{fig:plaquette}\n   \\end{center}\n\\end{figure}\n\nThe incoming and outgoing arrows of a node $i_0$ in $Q_X$ correspond to the edges\nof the plaquette $i_0$ in the dual dimer model graph \\gra. From this dual point of\nview, the double Seiberg duality described in the last section is precisely a\nplaquette move since the deleted edges around $i_0$ are interchanged while all\nother edges are left untouched~\\cite{Heckman:2006sk}. \nSince any $\\left( 0, 0 \\right)$--loop can be decomposed \ninto a product of plaquette moves, it can be interpreted as a sequence of double Seiberg dualities on the acyclic quiver.\n\nIt was shown in~\\cite{Herzog:2003zc} that a cyclic permutation of the nodes of $Q_X$ corresponds to choosing a different foundation of the same helix. So whenever we map between perfect matchings belonging to the same internal point of the toric diagram, we remain in the same helix.\n\nMoreover, as we already pointed out, a Beilinson quiver\n$Q_S \\subset Q_X$ corresponds to the embedding of a (partially\nresolved) singular surface $S$ in the Calabi--Yau three--fold $X$. We\ncan therefore read a $\\left( 0,0 \\right)$--loop as a map between two\npossible embeddings $i : S \\hookrightarrow X$.\n\n\n\n\n\n\n\\subsubsection{Relation to crystal melting and black holes}\n\\label{sec:melting}\n\n\nThe dimer configurations of the infinite hexagon graph are in one--to--one correspondence to the configurations of a melting crystal corner~\\cite{Okounkov:2003sp}. It was shown in~\\cite{Heckman:2006sk} that they also correspond to certain \\textsc{bps} black holes given by $D$--branes wrapping collapsed cycles in $\\setC^3\/\\IZ_n\\times\\IZ_n$ in the large $n$ limit. The black holes are generated by non--geometry changing flop transitions, which in the dimer picture are identified with so--called \\emph{local dimer moves}. The local dimer moves of~\\cite{Heckman:2006sk} correspond precisely to the plaquette moves of the hexagon graph. The $S_R^2$ transformations on the fractional branes to which their local dimer moves are related, correspond to the double Seiberg dualities described in Section \n\\ref{sec:doubleseiberg}. Thus, when specializing to the hexagon graph, the $(0,0)$--loops are the non--geometry changing flop transitions in $\\setC^3\/\\IZ_n\\times\\IZ_n$ which generate those \\textsc{bps} black hole charge configurations that are parametrized by the 3d crystal melting configurations. On a plane graph, everything is captured by the $(0,0)$--loops since no non--trivial winding states exist.\n\n\n\n\\subsection{Interpretation of the loops with non--trivial winding}\n\\label{sec:winding_loops}\n\nAs mentioned before, the maps corresponding to loops of non--trivial\nwinding result in a general permutation of the nodes of the gauge\nquiver $Q_X$. Geometrically, we map from one partial resolution of $S$\nto another, which is a birational map. The associated exceptional\ncollections produce equivalent derived categories.\n\nUnlike the $(0,0)$--maps, loops with non--trivial winding can in\ngeneral not be written as a sequence of Seiberg dualities. We have\nseen that plaquette moves cannot generate winding loops, therefore we\ncannot write the latter as a sequence of double Seiberg\ndualities. Single Seiberg dualities in general change the ranks of the\ngauge groups, see Eq.~\\eqref{eq:Seiberg-ranks}, and thus do not lead to\na graph isomorphism of the gauge quiver $Q_X$.\n\nSince the full quiver gauge theory remains unchanged, the action of\nthe weight changing maps $f^{(p,q)}_{m,n}$ on the acyclic quivers can\nstill be seen as a generalized gauge duality on the theory on the\n(partially resolved) surface $S$.\n\nWhat is the geometrical meaning of this? Morally speaking, an\nexceptional collection forms a basis for the physical branes. All\npartial resolutions lead to the same quiver gauge theory in $X$, so at\nthe level of the derived category, the associated exceptional\ncollections are equivalent. Passing from one to the other corresponds\nto a change of basis. We have seen that with the $(0,0)$--maps, we\nchange between foundations of a helix. This is a very particular\nchange of basis, which preserves the order of the sheaves in the\ncollection and only amounts to tensoring the canonical bundle a\ncertain number of times to all sheaves in the collection. We have seen\non the other hand, that the general permutation of nodes which takes\nplace in a weight--changing map does not preserve the order of the\nsheaves in the collection. This means that we no longer remain in the\nsame helix. At the level of the sheaves, this map is realized by a\ngeneral sequence of mutations.\n\n\n\n\n\n\n\n\n\n\n\\subsection{The dimer model on the cylinder and supersymmetric QM}\n\\label{sec:sqm}\n\nIn this section, we show how to construct a co--chain complex out of the\nloop configurations of a free fermion on a lattice, and how this\nsystem can be mapped to a \\textsc{sqm} system in which the loop\nconfigurations correspond to pure states.  For simplicity, we first\nconfine ourselves to a square graph embedded on a cylinder where only\none non--trivial cycle is present. A typical example of such a loop\nconfiguration is represented in Figure~\\ref{fig:artistic-cylinder}.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=.4\\textwidth]{cilindro}\n  \\caption{A loop configuration on the cylinder. In red: closed loops; in blue: double lines. This configuration has winding number $k-l = +1$}\n  \\label{fig:artistic-cylinder}\n\\end{figure}\n\nWe have seen in Section~\\ref{sec:loop-expans-ferm} that a loop\nconfiguration can always be decomposed into a superposition of two\nmatchings. Let us introduce the following notation:\n\\begin{itemize}\n\\item a matching with weight $k$ is written as $\\kket{k,a}$, where $a=1,\n  \\dots, N_k$ labels the matchings of the same weight;\n\\item we label a loop as $\\kket{k,a; l,b}_\\epsilon$ with $k \\geq l$,\n  $\\epsilon = \\pm 1$. This corresponds to one of the following\n  combinations of matchings:\n  \\begin{equation}\n    \\begin{array}{ccll}\n      \\ket{k,a; l,b}_{+} &=& \\kket{k,a} \\otimes \\kket{l,b} & \\text{if $k>l$,} \\\\ \n      \\ket{k,a; l,b}_{-} &=& \\kket{l,b} \\otimes \\kket{k,a} & \\text{if $k>l$,} \\\\ \n      \\ket{k,a; k,b}_{+} &=& \\kket{k,a} \\otimes \\kket{k,b} & \\text{if $a>b$,} \\\\ \n      \\ket{k,a; k,b}_{-} &=& \\kket{k,b} \\otimes \\kket{k,a} & \\text{if $a>b$,} \\\\ \n      \\ket{k,a; k,a}_{+} &=& \\kket{k,a} \\otimes \\kket{k,a} \\, ,\\\\ \n      \\ket{k,a; k,a}_{-} &=& 0 \\, .\n    \\end{array}\n  \\end{equation}\n  Opposite polarizations $\\epsilon$ correspond to opposite orientations\n  of the same loop configuration. States of the type $\\ket{k,a;\n    k,a}_+$ (which are in one--to--one correspondence with the perfect\n  matchings) are singled out because they are not paired.\n\\end{itemize}\n\n\n\n\\begin{comment}\nOur system of loops on a graph can be thought of as an example of\nsupersymmetric quantum mechanics. Then the partition function for the\nperfect matchings becomes a generalization of the Witten index.\n\nA generic loop state is characterized by five quantum numbers:\n\\begin{align}\n  \\ket{\\text{loop}} = \\ket{k,a; l,b}_\\epsilon , && l \\le k, \\, a = 0,\n  \\dots, N_k - 1, \\, b = 0, \\dots, N_l - 1, \\, \\epsilon = \\pm 1,\n\\end{align}\nwhere $k$ and $l$ are the weights of the matchings that make up the\nloop, $N_k$ is the degeneracy of matchings with weight $k$, and\n$\\epsilon $ is the polarization. Opposite polarizations correspond to\ntwo loops with reversed orientations. We define a loop with\ncounterclockwise orientation to have positive polarization,\ni.e. $\\epsilon=1$.\n\\end{comment}\n\n\nSince each loop configuration with a given orientation can be uniquely\ndecomposed into matchings, these quantum numbers can be assigned to a\ngiven configuration by inspection. The weight of a matching is given\nby the number of dimers crossing the boundary of the fundamental\nregion in positive direction minus the number of dimers crossing the\nboundary of the fundamental region in negative direction (where we\nchoose $\\bullet\\to\\circ$ as orientation).  $\\epsilon \\left( k-l\n\\right)$ is the total winding number of the loop configuration (which\ncoincides with the slope of the height function as defined in\nSec.~\\ref{sec:preliminaries}). States of the type $\\ket{k,a ; k,a}$\nare in one--to--one correspondence with the perfect matchings. All other\nstates are paired (i.e. there are two polarizations). From now on, we\nwill omit to specify the degeneracy quantum numbers and write a loop\nas $\\ket{k ; l}_\\epsilon$ (note the semicolon).\n\n\nThe quantum numbers can be seen as eigenvalues for the following operators:\n\\begin{align}\n  H \\ket{k;l}_\\epsilon &= \\left( k - l \\right) \\ket{k; l}_\\epsilon \\, ,\\\\\n  K \\ket{k;l}_\\epsilon &= k \\ket{k;l}_\\epsilon \\, ,\\\\\n  \\Pi \\ket{k; l}_\\epsilon &= \\epsilon \\ket{k;l}_\\epsilon \\, .\n\\end{align}\nThey commute on the above loop configurations by definition. It\nfollows that $H$ (which we want to identify with the energy) commutes\nalso with the operator\n\\begin{equation}\n  F = 2 K - \\frac{1}{2} \\left( \\Pi - 1 \\right) \\, ,\n\\end{equation}\nwhich we identify with the fermion number operator.\n\nIn terms of the loops, the above operators have the following\ninterpretation. $H$ counts the (unsigned) total winding number of the\nstate. $K$ counts the maximum attainable total monodromy of a loop\nconfiguration. The state $\\ket{k; l}_\\epsilon$ allows for maximally\n$k$ overall (i.e. positively minus negatively oriented) homologically\nnon--trivial windings, of which $k-l$ are actually realized. If $H$ is\nidentified with the energy, $k$ gives the maximum possible\nenergy. Each loop with non--trivial winding which is added to the\nconfiguration raises the energy of the state by one unit. Ground\nstates do not contain non--contractible loops. The matchings, in\nparticular, contain no loops at all. %\n\\begin{comment}\n  In the simple case of a product of spin chains on the cylinder, the\n  states $\\ket{k; l}_\\epsilon,\\ \\, l\\leq k$ can be viewed as a system\n  with $k$ boxes, into which $k-l$ particles (non--trivial loops) are\n  put.\n\\end{comment}\nThe eigenvalue of the polarization operator $\\Pi$ is the sign of the\nwinding number. The fermion number operator $F$ is constructed such,\nthat positive polarization states $\\ket{k; l}_+$ have even fermion\nnumber $2k$, while the fermion number of the corresponding negative\npolarization states $\\ket{k; l}_-$ is raised by one, \\emph{i.e.} is\n$2k+1$. The operator $\\left(-1\\right)^F$ gives a $\\IZ_2$ grading\naccording to the fermion parity. States with positive polarization\nhave even parity, while states negative polarization have odd parity.\n\n\\bigskip\n\nTo obtain the structure of a \\textsc{sqm}, we need to construct two\nconserved supercharges $Q$ and $Q^\\dag$ which satisfy the following\nalgebra (see \\emph{e.g.} Chapter 10 of~\\cite{Hori}):\n\\begin{align}\n  \\acomm{Q, Q} &= 0 & \\acomm{Q^\\dag, Q^\\dag} &= 0 \\\\\n  \\acomm{Q, Q^\\dag} &= 2 H \\\\\n  \\comm{F, Q} &= Q & \\comm{F, Q^\\dag} &= - Q^\\dag .\n\\end{align}\nA possible choice is the following:\n\\begin{align}\n\\label{eq:Q-cylinder}\n  \\begin{cases}\n    Q \\ket{k; l}_+ = \\sqrt{2 \\left( k - l \\right)} \\ket{k; l}_- ,\\\\\n    Q \\ket{k; l}_- = 0 .\n  \\end{cases} &&\n  \\begin{cases}\n    Q^\\dag \\ket{k; l}_+ = 0 ,\\\\\n    Q^\\dag \\ket{k; l}_- = \\sqrt{2 \\left( k - l \\right)} \\ket{k; l}_+ .\n  \\end{cases}\n\\end{align}\nIt is also useful to define the operator $Q_1$ as in\nEq.~\\eqref{eq:Q1-creator}:\n\\begin{equation}\n  Q_1 = Q + Q^\\dag.\n\\end{equation}\nIt reverses the polarization of a state,\n\\begin{equation}\n  Q_1 \\ket{k ; l}_\\epsilon = \\sqrt{2 \\left(k -l \\right)} \\ket{k; l}_{-\\epsilon} , \n\\end{equation}\nand squares to the energy:\n\\begin{equation}\n  Q_1^2 = 2 H .  \n\\end{equation}\n\n\nThe space of states can be decomposed into subspaces according to the fermion number. In the following, we will concentrate our attention on $Q$.\n\\begin{subequations}\n  \\begin{align}\n    C^p &= \\Sp \\set{\\ket{k,a; l,b}_\\epsilon | F \\ket{k,a;\n        l,b}_\\epsilon = p\n      \\ket{k,a; l,b}_\\epsilon } \\, , \\\\\n    C^{2k} &= \\Sp \\set{ \\ket{k,a; l, b}_+ } \\, ,\\\\\n    C^{2k+1} &= \\Sp \\set{\\ket{k,a ; l, b}_- } \\, .\n  \\end{align}\n\\end{subequations}\nWe also define\n\\begin{subequations}\n  \\begin{align}\n    C^+ &= \\bigoplus_{k} C^{2k} \\, , \\\\\n    C^- &= \\bigoplus_{k} C^{2k + 1} \\, , \\\\\n    C^* &= C^+ \\oplus C^- = \\bigoplus_{p} C^{p} \\, .\n  \\end{align}\n\\end{subequations}\nOne verifies easily that\n\\begin{equation}\n  Q : C^p \\to C^{p+1}  \n\\end{equation}\n(and conversely $Q^\\dag : C^p \\to C^{p-1}$). We can now construct the co--chain complex\n\\begin{equation}\n\\label{eq:SQM-complex}\n  0 \\hookrightarrow  C^0 \\xrightarrow{Q} C^1 \\xrightarrow{Q} \\dots \\xrightarrow{Q} C^N \\xrightarrow{Q} 0 \\, .  \n\\end{equation}\n\nLet us now study the cohomology of such a complex. The $0$--energy states\n$\\ket{k,a; k,b}_\\epsilon$ and the negative polarization states\n$\\ket{k,a;l,b}_-$ are $Q$--closed. On the other hand,\nnegative polarization states (with $k\\neq l$) are also $Q$--exact, such that the\ncohomology coincides with the set of supersymmetric ground states:\n\\begin{equation}\n  \\bigoplus_p H^p(Q) = \\Sp \\set{ \\ket{k, a; l, b}_\\epsilon| H \\ket{k, a; l, b}_\\epsilon = 0} = \\Sp \\set{\\ket{k,a;k,b}_\\epsilon} .\n\\end{equation}\nMoreover, all the states with $a \\neq b$ are paired (opposite parities). This means that when taking the Euler number, only the $\\ket{k,a;k,a}$ states (perfect matchings) contribute:\n\\begin{equation}\n  \\chi = \\Tr \\left( - 1 \\right)^F = \\sum_p \\left( - 1\\right)^p \\dim \\left[ H^p (Q) \\right] = \\sum_p N_p .  \n\\end{equation}\nAs we remarked above, diagrammatically, ground states are\nconfigurations with vanishing total winding number. The ones\ncontributing to the Euler number are only those that do not contain\nany loops.\n\nA different invariant can be constructed by taking the trace of the $\\left(\n  - 1\\right)^F z^K$ operator (which is still a conserved quantity since\nit is a function of conserved operators). It is easy to see that it is\ngiven by\n\\begin{equation}\n  \\chi (z)= \\Tr \\left[ \\left( - 1\\right)^F z^K \\right] = \\sum_{k,\\epsilon} \\left( - 1\\right)^p z^k \\dim \\left[ H^p (Q) \\right] = \\sum_k z^k N_k,\n\\end{equation}\nwhere $p = 2k - \\tfrac{1}{2}(\\epsilon -1)$. This is the Poincar\\'e\npolynomial for the sequence in Eq.~(\\ref{eq:SQM-complex}) and in\nparticular, $\\chi(1) = \\chi$. Moreover, $\\chi(-z)$ is by construction\nthe partition function for the free Majorana fermion on the lattice.\n\nIt is worth stressing that even if, as it is often the case, the\nzero--modes alone are enough to describe important features of the\nphysics of the system such as the partition function, we have complete\ncontrol over all the states in the theory. In Appendix~\\ref{sec:generating-functions}, the generating functions capturing all loop states are given.\n\n\n\\subsection{Example: One square on the cylinder}\n\\label{sec:example:-one-square}\n\nTo illustrate the above, we now present the smallest possible example\non the cylinder, consisting only of one square.  The five possible\nmatchings and their quantum numbers are shown in Figure\n\\ref{fig:ex_matchings}.\n\\begin{figure}[h!]\n  \\begin{center}\n    \\includegraphics[width=110mm]{ex_matchings}\n    \\caption{Matchings for the square on the cylinder}\n    \\label{fig:ex_matchings}\n  \\end{center}\n\\end{figure}\nThey combine into 25 loop configurations, which are, sorted into their\nco--chain groups according to total winding number, depicted in Table\n\\ref{table:chaingroups}.\n\nThe partition function can easily be found by inspection and reads:\n\\begin{equation}\n  P (z) = z \\left(- \\frac{1}{z} + 3 - z \\right) \\, .\n\\end{equation}\nThe generating function for this example reads\n\\begin{equation}\n  G( \\mathbf{q}, \\mathbf{z}, y ) = 1 + 3 y^2 z + y^4 z^2 + \\left( 1 + y \\right)  \\left[ 3 y^2 z + 3 q \\left( y^2 z + y^4 z^2 \\right) + q^2 y^4 z^2 \\right] \\, .\n\\end{equation}\n\n\n\n\\begin{table}[h!]\n  \\begin{center}\n    \\begin{tabular}{ccccccc}\n      \\toprule\n      \\boldmath $k - l$& $C^0$ & $C^1$ & $C^2$ & $C^3$ & $C^4$ & $C^5$ \\unboldmath \\cr \\midrule\n      \\rowcolor[gray]{.95}  2 & - & -& - & -& \\pb{$\\ket{2,1;0,1}_+$  \\includegraphics[width=13mm]{2101plus}} &\\pb{$\\ket{2,1;0,1}_{-}$ \\includegraphics[width=13mm]{2101minus}} \\cr \n      \\midrule\n      1 & - & - & \\pb{$\\ket{1,3;0,1}_+$ \\includegraphics[width=13mm]{1301plus}} & \\pb{$\\ket{1,3;0,1}_-$ \\includegraphics[width=13mm]{1301minus}}& \\pb{$\\ket{2,1;1,3}_+$ \\includegraphics[width=13mm]{2113plus}}&\\pb{$\\ket{2,1;1,3}_{-}$ \\includegraphics[width=13mm]{2113minus}} \\cr\n      \\rowcolor[gray]{.95}  1 & -  & -& \\pb{$\\ket{1,2;0,1}_+$ \\includegraphics[width=13mm]{1201plus} }& \\pb{$\\ket{1,2;0,1}_-$ \\includegraphics[width=13mm]{1201minus}}& \\pb{$\\ket{2,1;1,2}_+$ \\includegraphics[width=13mm]{2112plus}}&\\pb{$\\ket{2,1;1,2}_{-}$ \\includegraphics[width=13mm]{2112minus}}\\cr\n      1 & - & - & \\pb{$\\ket{1,1;0,1}_+$ \\includegraphics[width=13mm]{1101plus}} & \\pb{$\\ket{1,1;0,1}_-$ \\includegraphics[width=13mm]{1101minus}}& \\pb{$\\ket{2,1;1,1}_+$ \\includegraphics[width=13mm]{2111plus}}&\\pb{$\\ket{2,1;1,1}_{-}$ \\includegraphics[width=13mm]{2111minus}} \\cr\n      \\midrule\n      \\rowcolor[gray]{.95}  0 & -  & - & \\pb{$\\ket{1,3;1,2}_+$ \\includegraphics[width=13mm]{1312plus}} & \\pb{$\\ket{1,3;1,2}_-$ \\includegraphics[width=13mm]{1312minus}} &- &- \\cr\n      0 & -  & - & \\pb{$\\ket{1,3;1,1}_+$ \\includegraphics[width=13mm]{1311plus}} & \\pb{$\\ket{1,3;1,1}_-$ \\includegraphics[width=13mm]{1311minus}} &- &- \\cr\n      \\rowcolor[gray]{.95}  0 & -   & -& \\pb{$\\ket{1,2;1,1}_+$ \\includegraphics[width=13mm]{1211plus}} & \\pb{$\\ket{1,2;1,1}_-$ \\includegraphics[width=13mm]{1211minus}} &- &- \\cr\n      0 & -  & - & \\pb{$\\ket{1,3;1,3}$ \\includegraphics[width=13mm]{1313}} & - &- &- \\cr\n      \\rowcolor[gray]{.95}  0 & -  & - & \\pb{$\\ket{1,2;1,2}$ \\includegraphics[width=13mm]{1212}} & - &- &- \\cr\n      0 & \\pb{$\\ket{0,1;0,1}$ \\includegraphics[width=13mm]{00}}  & - & \\pb{$\\ket{1,1;1,1}$  \\includegraphics[width=13mm]{1111}} & -& {\\pb{$\\ket{2,1;2,1}$ \\includegraphics[width=13mm]{2121}} } &- \\cr \\bottomrule\n    \\end{tabular}\n    \\caption{Co--chain groups for the square on the cylinder}\n    \\label{table:chaingroups}\n  \\end{center}\n\\end{table}\n\n\n\n\n\n\\subsection{Loops as operators}\n\\label{sec:loops-as-operators}\n\nIn this section, we introduce a different notation for the loops that\nwill allow us to make contact with the geometric and gauge theory descriptions in\nSec.~\\ref{sec:geometry} and \\ref{sec:results}. Each perfect matching can be seen as a pure\nstate in a quantum system obeying a normalization condition\n$\\braket{k,a|l,b} = \\delta_{kl} \\delta_{ab}$. The general (quantum)\nconfiguration for the system is obtained as a linear combination\n\\begin{equation}\n  \\ket{\\Psi} = \\sum_k \\sum_a \\lambda_{k,a} \\ket{k,a} \\, ,\n\\end{equation}\nwhere $\\lambda_{k,a}$ are complex parameters normalized as usual\n$\\sum_{k,a} \\abs{\\lambda_{k,a}}^2 =1 $.\n\nUsing this point of view, a loop can be identified with a map going\nfrom a state $\\ket{k,a}$ to a state $\\ket{l,b}$ and as such is\nrepresented in bracket notation by $\\ketbra{l,b}{k,a}$ acting as\n\\begin{equation}\n  \\left(\\ketbra{l,b}{k,a}  \\right) \\ket{k',a'} = \\delta_{k k'} \\delta_{a a'} \\ket{l,b} \\, .\n\\end{equation}\nThe linear combination of two loops is still well defined, but a new\noperation is naturally defined by the product of two loops:\n\\begin{equation}\n  \\left( \\ketbra{l,b}{k,a} \\right) \\times \\left( \\ketbra{l',b'}{k',a'} \\right) \\mapsto \\ketbra{l,b}{k',a'} \\delta_{k l'} \\delta_{ab'} \\, .\n\\end{equation}\nNote that this is the usual path algebra on a complete graph whose\nnodes are labelled by $\\ket{k,a}$. It is useful to introduce a new\nrepresentation for our states. Consider a graph with $N$ nodes\narranged on the lattice $\\setN^2$ at coordinates $\\left( k, a \\right)$,\nand represent the loop $\\ketbra{l,b}{k,a}$ as the arrow from\n$\\left(k,a \\right)$ to $\\left(l,b \\right)$. Our system of loops\ncorresponds to a complete graph with double lines pointing in\nopposite directions plus a loop on top of each vertex (this is a state graph,\nsee Fig.~\\ref{fig:state-graph-cylinder}(a)).  The co--chain groups\n(Abelian, freely generated by the loops) receive the following\ninterpretation:\n\\begin{itemize}\n\\item $C^{2p}$ is the group generated by the loops mapping from a\n  state with winding $p$ to a state with winding smaller than $p$\n  (arrows pointing left), from a state with winding $p$ to another one\n  with a smaller degeneracy number (arrows pointing down), or from a\n  state to itself (loops). Fig.~\\ref{fig:state-graph-cylinder}(b);\n\\item $C^{2p+1}$ is the group generated by the loops mapping from a\n  state with winding smaller than $p$ to one with winding $p$ (arrows\n  pointing right), or from a state with winding $p$ to another with\n  the same winding and higher degeneracy number (arrows pointing\n  up). Fig.~\\ref{fig:state-graph-cylinder}(c).\n\\end{itemize}\nIn terms of the state graph, this means that the differential map\n$Q_{2p} : C^{2p} \\to C^{2p+1}$ maps every arrow with a non--vanishing\nleft component to its opposite and annihilates all the others:\n\\begin{gather}\n  Q_{2p}: \\ketbra{l,b}{p,a} \\mapsto \\ketbra{p,a}{l,b} \\, , \\\\\n  Q_{2p}: \\ketbra{p,b}{p,a} \\mapsto 0 \\, .\n\\end{gather}\nOn the other hand, $Q_{2p+1} : C^{2p+1} \\to C^{2p+2}$ annihilates every map\n\\begin{equation}\n    Q_{2p + 1}: \\ketbra{p,a}{l,b} \\mapsto 0 \\, .  \n\\end{equation}\nIt is immediate to see that when considering  the cohomology one finds:\n\\begin{itemize}\n\\item $H^{2p} \\subset C^{2p}$ is generated by downwards pointing arrows and\n  loops. Fig.~\\ref{fig:state-graph-cylinder}(d);\n\\item $H^{2p+1} \\subset C^{2p+1}$ is generated by upwards pointing\n  arrows. Fig.~\\ref{fig:state-graph-cylinder}(e).\n\\end{itemize}\nThis implies in turn that taking the Euler character, only the loops\nsurvive since upward and downward pointing arrows are paired and\ncounted with opposite signs \n(Fig.~\\ref{fig:state-graph-cylinder}(f)). They are in one--to--one\ncorrespondence with the nodes and all counted with $\\left(+ \\right)$\nsign, such that\n\\begin{equation}\n  \\chi = \\sum_p \\left( - 1\\right)^p \\dim (H^p) = \\sum_p N_p = N \\, .  \n\\end{equation}\n\n\\begin{figure}\n  \\centering\n  \\subfigure[Complete state graph $C^\\ast$]{\\includegraphics[width=.4\\textwidth]{StateGraph-131-crop}} \\hfill\n  \\subfigure[$C^+ = \\bigoplus_{p} C^{2p}$]{\\includegraphics[width=.4\\textwidth]{StateGraph-131-Cp-crop}}  \\vfill\n  \\subfigure[$C^- = \\bigoplus_{p} C^{2p+1}$]{\\includegraphics[width=.35\\textwidth]{StateGraph-131-Cm-crop}} \\hfill\n  \\subfigure[$H^+ = \\bigoplus_{p} H^{2p}$]{\\includegraphics[width=.4\\textwidth]{StateGraph-131-Hp-crop}}  \\vfill\n  \\subfigure[$H^- = \\bigoplus_{p} H^{2p+1}$]{\\includegraphics[width=.35\\textwidth]{StateGraph-131-Hm-crop}} \\hfill\n  \\subfigure[Loops contributing to the Euler number]{\\includegraphics[width=.4\\textwidth]{StateGraph-131-Eu-crop}}\n  \\caption{State graph for the dimer model in\n    Sec.~\\ref{sec:example:-one-square}}\n  \\label{fig:state-graph-cylinder}\n\\end{figure}\n\n\n \n\n\n\n\n\\subsection{The dimer model on the torus and supersymmetric QM}\n\\label{sec:sqm_torus}\n\nWe are now ready to pass to the actual case of interest, \\emph{i.e.}\nto extend the categorification construction of Sec.~\\ref{sec:sqm} to\ngraphs embedded on a torus. Now we have two non--trivial cycles, so\neach matching has two weights, $k_z$ and $k_w$, as explained in\nSec.~\\ref{sec:preliminaries}. These weights will be treated on a\ndifferent footing, but the final result will restore the expected\nsymmetry.\n\nThe notation is modified as follows:\n\\begin{itemize}\n\\item a perfect matching is labelled as $\\kket{\\mathbf{k},a}$, where\n  $\\mathbf{k}$ is the vector $\\mathbf{k} = \\left( k_z, k_w \\right) $\n  and $a = 1, \\dots, N_{\\mathbf{k}}$;\n\\item a loop is the combination of two matchings and is labelled by\n  $\\ket{\\mathbf{k},a;\\mathbf{l},b}_\\epsilon$, where $k_z \\geq l_z$,\n  $\\epsilon = \\pm 1$.\n  \\begin{equation}\n    \\begin{array}{ccll}\n      \\ket{\\mathbf{k},a; \\mathbf{l},b}_{+} &=& \\kket{\\mathbf{k},a} \\otimes \\kket{\\mathbf{l},b} & \\text{if $\\mathbf{k} \\succ \\mathbf{l}$,} \\\\ \n      \\ket{\\mathbf{k},a; \\mathbf{l},b}_{-} &=& \\kket{\\mathbf{l},b} \\otimes \\kket{\\mathbf{k},a} & \\text{if $\\mathbf{k} \\succ \\mathbf{l}$,} \\\\ \n      \\ket{\\mathbf{k},a; \\mathbf{k},b}_{+} &=& \\kket{\\mathbf{k},a} \\otimes \\kket{\\mathbf{k},b} & \\text{if $a>b$,} \\\\ \n      \\ket{\\mathbf{k},a; \\mathbf{k},b}_{-} &=& \\kket{\\mathbf{k},b} \\otimes \\kket{\\mathbf{k},a} & \\text{if $a>b$,} \\\\ \n      \\ket{\\mathbf{k},a; \\mathbf{k},a}_{+} &=& \\kket{\\mathbf{k},a} \\otimes \\kket{\\mathbf{k},a} \\, ,\\\\ \n      \\ket{\\mathbf{k},a; \\mathbf{k},a}_{-} &=& 0 \\, ,\n    \\end{array}\n  \\end{equation}\n  where $\\mathbf{k} \\succ \\mathbf{l} $ if $k_z > l_z $ or if $\\set{ k_z = l_z, \\,k_w>l_w}$. This corresponds to lexicographic order.\n\\end{itemize}\nAgain, the loops can be seen as eigenstates for the following operators:\n\\begin{align}\n  H \\ket{\\mathbf{k};\\mathbf{l}}_\\epsilon &= \\left( k_z - l_z \\right)\\ket{\\mathbf{k}; \\mathbf{l}}_\\epsilon \\, ,\\\\\n  K_z \\ket{\\mathbf{k};\\mathbf{l}}_\\epsilon &= k_z \\ket{\\mathbf{k};\\mathbf{l}}_\\epsilon \\, ,\\\\\n  K_w \\ket{\\mathbf{k};\\mathbf{l}}_\\epsilon &= \\max (k_w, l_w) \\ket{\\mathbf{k};\\mathbf{l}}_\\epsilon \\, ,\\\\\n  \\Pi \\ket{\\mathbf{k}; \\mathbf{l}}_\\epsilon &= \\epsilon \\ket{\\mathbf{k};\\mathbf{l}}_\\epsilon \\, .\n\\end{align}\nAs opposed to the example on the cylinder, a loop state now comes with\ntwo natural signs, $\\epsilon_z=\\sign{(k_z - l_z)}$, and\n$\\epsilon_w=\\sign{(k_w- l_w)}$. For the grading by the fermion number,\nwe assign only one sign $\\epsilon$ to each loop state according to\nTable \\ref{table:signs}.\n\\begin{table}[h!]\n  \\begin{center}\n    \\begin{tabular}{ccc}\n      \\toprule\n      \\boldmath $\\epsilon_z$&$\\epsilon_w$&$\\epsilon$ \\unboldmath \\cr \\midrule\n      \\rowcolor[gray]{.95} $+$&$+$&$+$\\\\\n      $-$&$-$&$-$\\\\\n      \\rowcolor[gray]{.95} $+$&$-$&$+$\\\\\n      $-$&$+$&$-$\\\\\n      \\bottomrule\n    \\end{tabular}\n    \\caption{Sign table for torus loop states}\n    \\label{table:signs}\n  \\end{center}\n\\end{table}\nThis is achieved by \n\\begin{equation}\n  \\label{eq:signdef}\n  \\epsilon=\\epsilon_z+(1-\\epsilon_z^2)\\,\\epsilon_w.\n\\end{equation}\nBy choosing the right order on the perfect matchings of the same\nweight and choosing a sign for some loop configurations of winding\nnumber 0, the signs of the overall winding numbers around the $z$ and\n$w$ cycles can be brought to correspondence with the\nsigns $\\epsilon_z,\\,\\epsilon_w$ defined above.  We assign a sign to an oriented loop\nby choosing a right handed (counterclockwise) system to have positive\nsign and a left handed (clockwise) to have negative sign. The signs of\na loop configuration are determined by the overall winding number\naround the cycles. Only in cases with an equal number of loops winding\na cycle in opposite directions, a sign has to be chosen arbitrarily.\n\nThe fermion number operator only depends on $K_z$ and is defined by\n\\begin{equation}\n  F = 2\\, K_z - \\frac{1}{2} \\left( \\Pi - 1 \\right) \\, .\n\\end{equation}\n\nStates can be again collected into subspaces according to the\neigenvalues of $F$, but in this case, we have an extra grading given by $k_w$. Since $\\comm{F, K_w} = 0$, we obtain a direct sum structure, and define:\n\\begin{gather}\n  C^p = \\bigoplus_{q=0}^M C^{p,\\,q}  \\, , \\\\\n  C^{p,\\,q} = \\Sp \\set{\\ket{\\mathbf{k};\\mathbf{l}}_\\epsilon | F \\ket{\\mathbf{k};\\mathbf{l}}_\\epsilon = p, \\, K_w \\ket{\\mathbf{k};\\mathbf{l}}_\\epsilon = q} \\, .\n\\end{gather}\nThe differential operators (or supercharges) are defined precisely as\nin Eq.~(\\ref{eq:Q-cylinder})\n\\begin{align}\n  \\begin{cases}\n    Q \\ket{\\mathbf{k}; \\mathbf{l}}_+ = \\sqrt{2 \\left( k_z - l_z \\right)} \\ket{\\mathbf{k}; \\mathbf{l}}_- ,\\\\\n    Q \\ket{\\mathbf{k}; \\mathbf{l}}_- = 0 .\n  \\end{cases} &&\n  \\begin{cases}\n    Q^\\dag \\ket{\\mathbf{k}; \\mathbf{l}}_+ = 0 ,\\\\\n    Q^\\dag \\ket{\\mathbf{k}; \\mathbf{l}}_- = \\sqrt{2 \\left( k_z - l_z \\right)} \\ket{\\mathbf{k}; \\mathbf{l}}_+ ,\n  \\end{cases}\n\\end{align}\nand it is immediate to verify that they respect the grading since\n$\\comm{K_w, Q} = \\comm{K_w, Q^\\dag} = 0$. Again we concentrate on $Q$. It can be decomposed into the sum\n\\begin{align}\n  Q_p &= \\sum_{q=0}^M Q_{p,\\,q} \\, , \\\\\n  \\intertext{where}\n  Q_p &: C^{p,\\,q} \\to C^{p+1,\\,q} \\, ,\\hspace{2em} q = 0, \\dots, M \\, . \n\\end{align}\nAs a result, the $C^\\ast$ complex is decomposed into $C_{\\ast,q}$\ncomplexes:\n\\begin{equation}\n  \\label{eq:torus-complex}\n  \\begin{array}{rccccc}\n    0 &\\hookrightarrow  C^{0,M} &\\xrightarrow{Q_{0,M}} C^{1,M} &\\xrightarrow{Q_{1,M}} \\dots &\\xrightarrow{Q_{N-1,M}} C^{N,M} &\\xrightarrow{Q_{N,M}} 0 \\, , \\\\\n    & & & \\dots \\\\\n    0 &\\hookrightarrow  C^{0,q} &\\xrightarrow{Q_{0,q}} C^{1,q} &\\xrightarrow{Q_{1,q}} \\dots &\\xrightarrow{Q_{N-1,q}} C^{N,q} &\\xrightarrow{Q_{N,q}} 0 \\, , \\\\\n    & & &\\dots \\\\\n    0 &\\hookrightarrow  C^{0,0} &\\xrightarrow{Q_{0,0}} C^{1,0} &\\xrightarrow{Q_{1,0}} \\dots &\\xrightarrow{Q_{N-1,0}} C^{N,0} &\\xrightarrow{Q_{N,0}} 0 \\, .\n  \\end{array}\n\\end{equation}\n\nThe analysis of the cohomology is the same as in the case of the cylinder and one can easily convince oneself that it consists of states of zero energy:\n\\begin{equation}\n  \\bigoplus_{p,q} H^{p,q} = \\Sp \\set{ \\ket{\\mathbf{k},a; \\mathbf{l},b}_\\epsilon | k_z = l_z} .  \n\\end{equation}\nOnce more, only the states that are in one--to--one correspondence with the perfect\nmatchings (\\emph{i.e.} the ones that do not contain non--trivial closed loops) are not paired\nand thus survive the projection by the $(-1)^F$ operator:\n\\begin{equation}\n  \\chi(w)=\\Tr \\left[ (-1)^F w^{K_w} \\right] = \\sum_{p,q}  w^q N_{p,q} \\,,\n\\end{equation}\nwhere we introduced the variable $w$ to keep track of the internal\ngrading. The partition function described in\nEq.~(\\ref{eq:torus-partition}) is obtained as the trace of the\noperator $(-1)^{F + K_z K_w} \\left( - z \\right)^{K_z} \\left( - w \\right)^{K_w}$:\n\\begin{equation}\n\\label{eq:Witten-torus}\n  P (z,w) = \\Tr \\left[ (-1)^{F + K_z  K_w} \\left( - z \\right)^{K_z} \\left( - w \\right)^{K_w} \\right] = \\sum_{p,q} (-1)^{p + q + pq}\\, z^p w^q\\, N_{p,q} \\, .\n\\end{equation}\nNote the presence of the factor $(-1)^{p+q+p\\,q}$, which coincides with\nthe signature of the spin structure. Because of this, the expression\ndiffers from the Poincar\\'e polynomial for the graded complex\n\\begin{equation}\n  \\chi (z,w) =   \\sum_{p,q}\\, z^p\\, w^q\\, N_{p,q} \\, .\n\\end{equation}\nThe two polynomials, on the other hand, contain the same information\nand one can pass from one to the other using the identity\n\\begin{equation}\n  2 \\,\\chi (z,w) = - P(z,w) + P(-z,w) + P(z,-w) + P(-z,-w) \\, .  \n\\end{equation}\nIn particular, $\\chi(w) = \\chi(-1,w) $ and $\\chi (1,1)$ is the overall\nnumber of perfect matchings on the graph.  As already remarked at the\nbeginning of this section, it is worth emphasizing that the choice of\nthe cycles is arbitrary. This means in particular that one could have\nused $K_w$ to construct the Fermion number operator and $K_z$ as an\ninternal grading without affecting the final result.\n\nThe generalization to graphs embedded on higher--genus Riemann surfaces\nis straightforward and leads to a $\\left( 2g -1 \\right)$--graded complex.\n\n\nA simple example to illustrate the above is given in\nAppendix~\\ref{sec:ex_torus}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe description of physical systems is standardly given in terms of coordinates as defined by reference frames. Thanks to the principle of covariance, stating the equivalence of all descriptions regardless of the choice of the reference frame, it is possible to choose the reference frame where the relevant dynamical quantities can be most conveniently described. For example, it is typically easier to describe the dynamics of a system from the point of view of its rest frame, because only internal degrees of freedom contribute to the dynamics in the rest frame. \n\nWhen the external degrees of freedom (momentum) of the system are in a quantum superposition from the perspective of the laboratory, no classical reference frame transformation can map the description of physics from the laboratory to the rest frame. However, this can be achieved via a quantum reference frame (QRF) transformation between two frames moving in a superposition of velocites relative to one another. In order to achieve such change of quantum reference frame in the nonrelativistic regime, a formalism was introduced in Ref.~\\cite{QRF} to change the description to a refence frame which is in a quantum relationship with the initial one. This QRF transformation only depends on relational quantities, and it has also been derived starting from a gravity inspired symmetry principle in a perspective neutral model \\cite{perspective1, perspective2} . An immediate consequence of the formalism is that entanglement and superposition are QRF-dependent features. This formalism naturally leads to the possibility of identifying the rest frame of a quantum system in an operational way. \n\nHere, we further develop this approach in the case of a relativistic quantum particle with spin, with the goal of finding an operational description of the spin in a special-relativistic setting. Spin is operationally defined in the rest frame of a particle (or, to a good approximation, for slow velocities) via the Stern-Gerlach experiment. When the particle has relativistic velocities, the spin degree of freedom transforms in a momentum-dependent way. If a standard Stern-Gerlach measurement is performed on a particle in a pure quantum state moving in a superposition of relativistic velocities, the operational identification of the spin fails, because no orientation of the Stern-Gerlach apparatus returns an outcome with unit probability. This happens because, as shown in Ref.~\\cite{peres_first}, the reduced density matrix of the spin degree of freedom is mixed when a Lorentz boost is performed and the momentum is traced out. The question arises whether it is possible to find `covariant measurements' of the spin and possibly momentum, which predict invariant probabilities in different Lorentzian reference frames also for the case of a quantum relativistic particle moving in a superposition of velocities. In this case, it would be possible to map the unambiguous description of spin in the rest frame of the particle to the frame of the laboratory, and therefore derive the corresponding observables to be measured in the laboratory frame to verify spin with probability one.\n\nThe question of finding such covariant measurements is motivated by the ubiquitous applications where the spin degree of freedom is used as a qubit, to encode and transmit quantum information. Such protocols are no longer valid in a relativistic context, thus limiting the range of applicability of techniques involving spin as a quantum information carrier. It is then important to explore possible alternative methods which could overcome this limitation. In the context of relativistic quantum information, this question has been extensively discussed \\cite{peres_first, peresterno_review, alsingmilburn, gingrichadami, leeyoung_entanglementboost, bartlettterno_encoding, vedral_paradox, esteban, friis_relativisticentanglement, caban_reducedspin, vedral_operational, palmer_sterngerlach, markusphilipp_lorentz, terno_localisation} in relation to Wigner rotations \\cite{wigner1939, weinberg, sexlurbantke} and has been related to the problem of identifying a covariant spin operator. The problem of identifying such covariant spin operator has arisen long before the birth of relativistic quantum information, and dates back to the early times of quantum mechanics \\cite{thomas_precession, frenkel_spin, bargmannOnFrenkel}. Since then, a multitude of relativistic spin operators have been proposed \\cite{bauke_relativisticspin}, such as the Frenkel \\cite{frenkel_spin}, the Pauli-Luba\\'{n}ski \\cite{lubanski, ryder_paulilubanski, terno_tworoles}, the Pryce \\cite{pryce_spin}, the Foldy-Wouthuysen \\cite{foldywouthuysen, caban_spinobservable}, the Czachor \\cite{czachor_spin} the Fleming \\cite{fleming_spin} the Chakrabarti \\cite{Chakrabarti_spin}, and the Fradkin-Good \\cite{fradkingood_spin} spin operators. A comparative description of spin observables can be found in Ref.~\\cite{terno_localisation}.\n\nHere, we introduce `superposition of Lorentz boosts' which allow us to ``jump'' into the rest frame of a relativistic quantum particle even if the particle is {\\it not} in a momentum eigenstate. In the rest frame, the spin observables fulfill the spin $\\mathfrak{su}(2)$ algebra (the algebra of a qubit) and are operationally defined through the Stern-Gerlach experiment. We transform the set of spin observables in the rest frame back to an isomorphic set of observables in the laboratory frame. The transformed observables are in general entangled in the spin and momentum degrees of freedom. The set fulfills the $\\mathfrak{su}(2)$ algebra and is operationally defined through a `relativistic Stern-Gerlach experiment': we construct the interaction and the measurement between the spin-momentum degrees of freedom and the electromagnetic field in the laboratory frame which gives the same probabilities as the Stern-Gerlach experiment in the rest frame. This set of observables in the laboratory frame allows us to partition the total Hilbert space into two (highly degenerate) subspaces corresponding to the two outcomes ``spin up'' and ``spin down''. Hence, with QRFs techniques the relativistic spin can effectively be described as a qubit in an operationally well-defined way.\n\n\\section{A relativistic Stern-Gerlach experiment}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.3]{spinQRF}\n\t\t\\caption{(a) The state of a Dirac particle A with spin $\\tilde{A}$ as seen from the laboratory perspective (C). When the state is in a superposition of relativistic velocities $-v_1$ and $-v_2$, the spin degree of freedom and the momentum degree of freedom are no longer separable. (b) The state of the spin $\\tilde{A}$ and of the laboratory C as seen in the rest frame of the quantum particle A. In this quantum reference frame, the spin is operationally defined by means of the Stern-Gerlach experiment.}\n\t\t\\label{fig:spinQRF}\n\t\\end{center}\n\\end{figure}\n\nIn the following, we build a QRF transformation between the reference frame of a laboratory C of mass $m_C$ and the rest frame of the external degrees of freedom A of a relativistic quantum particle of mass $m_A>0$ with spin degrees of freedom $\\tilde{A}$, as illustrated in Fig.~\\ref{fig:spinQRF}. We allow the particle to have any quantum state, and in particular to move in a superposition of momenta. This implies that there is a non-classical relationship between the initial and the final reference frame, i.e., that the rest frame A and the laboratory frame C are not related by a standard boost transformation. We show in this section how to generalise the boost transformation to this case. Formally, the situation we consider can be described by taking the one-particle sector of the positive-energy solutions of the Dirac equation\\footnote{For simplicity, we only consider spin-$1\/2$ particles, but the method can be straightfowardly applied to arbitrary spin.} in the Foldy-Wouthuysen representation \\cite{foldywouthuysen}. \n\nFollowing Ref.~\\cite{QRF} (see Supplemental Information for a review of the original formalism), when we ``stand'' in the rest frame of a particle, we describe all the systems external to the particle, but not the external degrees of freedom (i.e., the momentum degrees of freedom) of the particle itself. Hence, the quantum state describes the relational information in a given reference frame. In the reference frame in which A is at rest, the quantum state is assigned to the internal degrees of freedom $\\tilde{A}$ and the laboratory  C. For simplicity, we consider that the particle and the laboratory are moving with constant, yet not necessarily well-defined, relative velocity and define the $x$ axis along the direction of the relative motion. The total state of the spin and the laboratory is assumed to be\n\\begin{equation} \\label{eq:stateA}\n\t\\left| \\Psi \\right>_{\\tilde{A}C}^{(A)} = \\left|\\vec{\\sigma} \\right>_{\\tilde{A}}\\left| \\psi \\right>_{C},\n\\end{equation}\nwhere $\\left|\\vec{\\sigma} \\right>_{\\tilde{A}}$ is any vector representing the state of the spin in the rest frame A. In the rest frame, the spin state can in principle be tomographically verified by performing a series of standard Stern-Gerlach measurements. The state of the laboratory has a momentum-basis representation along the $x$ direction (at this stage, we neglect the quantum state in the $y$ and $z$ direction) $\\left| \\psi \\right>_{C} = \\int d\\mu_C(\\pi_C) \\psi (\\pi_C) \\left| \\pi_C \\right>_{C}$, where $d\\mu_C(\\pi_C) =  \\frac{d\\pi_C}{(2\\pi)^{1\/2}\\sqrt{2(m_C^2 c^2 + \\pi_C^2)}}$ is the Lorentz-covariant integration measure. \n\nWe now construct the transformation corresponding to the ``superposition of Lorentz boosts'' to the QRF of the laboratory. The unitary operator to boost to the QRF C is\n\\begin{equation} \\label{eq:QRFTransf}\n\t\\hat{S}_L = \\mathcal{P}_{CA}^{(v)} U_{\\tilde{A}}(\\hat{\\pi}_C),\n\\end{equation}\n where $U_{\\tilde{A}}(\\hat{\\pi}_C)$ is a unitary transformation acting on the total Hilbert space $\\mathcal{H}_{\\tilde{A}}\\otimes \\mathcal{H}_C$ (notice that $\\hat{\\pi}_C$ is an operator), and $\\mathcal{P}_{AC}^{(v)}$ is the `generalised parity operator' introduced in Ref.~ \\cite{QRF}, whose explicit expression is $\\mathcal{P}_{CA}^{(v)}= P_{AC} \\exp \\left(\\frac{i}{\\hbar}\\log \\sqrt{\\frac{m_A}{m_C}}(\\hat{q}_C \\hat{\\pi}_C + \\hat{\\pi}_C \\hat{q}_C)\\right)$, where $P_{AC}$ is the parity-swap operator mapping $\\hat{x}_A \\rightarrow -\\hat{q}_C$ and $\\hat{p}_A \\rightarrow -\\hat{\\pi}_C$ (and viceversa), where $\\hat{q}_C, \\,\\hat{\\pi}_C$ are canonically-conjugated one-particle operators of C in the reference frame of A and $\\hat{x}_A, \\,\\hat{p}_A$ are canonically-conjugated one-particle operators of A in the reference frame of C. Additionally to the action of $P_{AC}$, the operator $\\mathcal{P}_{AC}^{(v)}$ rescales the momentum of A by the ratio of the masses of A and C, i.e., $\\mathcal{P}_{AC}^{(v)} \\hat{p}_A \\mathcal{P}_{AC}^{(v)\\dagger} = - \\frac{m_A}{m_C} \\hat{\\pi}_C$. This enforces the physical condition that the velocity of A is mapped to the opposite of the velocity of C via the transformation\\footnote{For a relativistic particle the relation between the $i$-th velocity component and the momentum is $v_i = \\frac{p_i}{m_i} \\left( 1+ \\frac{|\\vec{p}|^2}{m_i^2 c^2} \\right)^{-1\/2}$, where $|\\vec{p}|^2$ is the norm of the spatial momentum. Therefore, only the ratio between momentum and mass determines the velocity.}. The operator $\\hat{S}_L$ can be defined via its action on a basis of the total Hilbert space of the spin and the laboratory $\\hat{S}_L |\\vec{\\sigma} \\rangle_{\\tilde{A}}|\\pi\\rangle_C = | -\\frac{m_A}{m_C}\\pi; \\Sigma_{\\pi}\\rangle_{A\\tilde{A}}$, where the state $|p; \\Sigma_{p}\\rangle_{A\\tilde{A}}$ is defined via a standard Lorentz boost $\\hat{U}(L_p)$ from the rest frame as $|p; \\Sigma_{p}\\rangle_{A\\tilde{A}}=\\hat{U}(L_p)| k; \\vec{\\sigma}\\rangle_{A\\tilde{A}}$ and $k= (mc, \\vec{0})$ is the momentum in the rest frame. \n\n  In Supplemental Information we derive the transformation $\\hat{S}_L$ in terms of standard Lorentz boosts connecting two relativistic reference frames where the parameter of the boost transformation is promoted to an operator. \n \n The state of $A$ and $\\tilde{A}$ expressed in the laboratory frame is $\\left| \\Psi \\right>_{A\\tilde{A}}^{(C)} = \\hat{S}_L \\left| \\Psi \\right>_{\\tilde{A}C}^{(A)}$, and is explicitly written as\n\\begin{equation} \\label{eq:StateTrans}\n\t\\left| \\Psi \\right>_{A\\tilde{A}}^{(C)} = \\int d\\mu_A(p_A) \\psi\\left(-\\frac{m_C}{m_A}p_A\\right) \\left| p_A; \\Sigma_{p_A} \\right>_{A\\tilde{A}},\n\\end{equation}\nwhere $d\\mu_A(p_A) =  \\frac{d p_A}{(2\\pi)^{1\/2}\\sqrt{2(m_A^2 c^2 + p_A^2)}}$ and the spin degree of freedom cannot be separated anymore from the momentum degree of freedom, which means that the state is not a product state in the laboratory frame. Notice that the effect of the $\\hat{S}_L$ transformation is to apply the usual boost transformation conditional on C's momentum degree of freedom. In the laboratory frame C, unless particle A is in a sharp momentum state, no spin measurement in a standard Stern-Gerlach experiment would give a result with probability one, because of two reasons: the spin and momentum are no longer separable, and the relation between the laboratory and the rest frame is not a standard (classical) reference frame transformation. Our goal is to devise a different measurement in the laboratory reference frame, possibly involving both the spin and momentum degrees of freedom, which gives the same probability distribution as a standard Stern-Gerlach would give, if performed in the rest frame.\n\nIn order to devise such measurement we note that, in the laboratory frame, it is possible to define the observables corresponding to the spin operators in the rest frame by transforming the spin, as defined in the rest frame, with a QRF transformation\n\\begin{equation}\n\t\\hat{\\Xi}_i = \\hat{S}_L ( \\hat{\\sigma}_i \\otimes \\mathbb{1}_C) \\hat{S}_L^\\dagger, \\qquad i=x,y,z.\n\\end{equation}\nIn terms of the momenta and of the manifestly covariant Pauli-Luba\\'{n}ski operator $\\hat{\\Sigma}_{\\hat{p}_A} = (\\hat{\\Sigma}^0_{\\hat{p}_A}, \\vec{\\hat{\\Sigma}}_{\\hat{p}_A})$, the operators $\\hat{\\Xi}_i$ are expressed as (see Supplemental Information) $\\vec{\\hat{\\Xi}} = \\vec{\\hat{\\Sigma}}_{\\hat{p}_A} - \\frac{\\hat{\\gamma}_A}{\\hat{\\gamma}_A +1}\\left(\\vec{\\hat{\\Sigma}}_{\\hat{p}_A} \\cdot \\vec{\\hat{\\beta}}_A  \\right) \\vec{\\hat{\\beta}}_A$, where $\\hat{\\gamma}_A = \\sqrt{1 + \\frac{\\hat{p}_A^2}{m_A^2 c^2}}$ and $\\vec{\\hat{\\beta}}_A = \\left( \\hat{\\beta}_A^x, \\hat{\\beta}_A^y, \\hat{\\beta}_A^z \\right)$, where each component is $\\hat{\\beta}_A^i = \\frac{\\hat{p}_A^i}{\\sqrt{m_A^2 c^2 + \\vec{\\hat{p}}_A^2}}$ with $i= x, y, z$. The operators $\\hat{\\Xi}_i$ are equivalent to the Foldy-Wouthuysen \\cite{foldywouthuysen} or Pryce spin operator \\cite{pryce_spin}. By definition, these operators satisfy the $\\mathfrak{su}(2)$ algebra $\\left[ \\hat{\\Xi}_i, \\hat{\\Xi}_j \\right] = i \\epsilon_{ijk} \\hat{\\Xi}_k$, and have the same eigenvalues as the Pauli operators $\\hat{\\sigma}_i$, $i=x,y,z$. This last property can be easily checked by choosing an eigenvector $| \\lambda_i \\rangle$ of the operator $\\hat{\\sigma}_i$ in the rest frame A, such that $\\hat{\\sigma}_i | \\lambda_i \\rangle = \\lambda_i | \\lambda_i \\rangle$ and by noting that $\\hat{\\Xi}_i \\hat{S}_L | \\lambda_i \\rangle_{\\tilde{A}}| \\psi \\rangle_{C} = \\lambda_i \\hat{S}_L | \\lambda_i \\rangle_{\\tilde{A}}| \\psi \\rangle_{C}$. Hence, it is possible to partition the total Hilbert space $\\mathcal{H}_A \\otimes \\mathcal{H}_{\\tilde{A}}$ into two equivalence classes, defined as\n\\begin{subequations}\n\t\\begin{equation}\n\t\t\\mathcal{H}_0 = \\left\\lbrace | \\Psi \\rangle_{A\\tilde{A}} \\in \\mathcal{H}_A \\otimes \\mathcal{H}_{\\tilde{A}}\\,\\, \\text{s.t.}\\,\\, | \\Psi \\rangle_{A\\tilde{A}} \\sim \\hat{S}_L | 0 \\rangle_{\\tilde{A}}| \\psi \\rangle_{C}, \\forall \\, | \\psi \\rangle_{C} \\in \\mathcal{H}_C \\right\\rbrace,\n\t\\end{equation}\n\t\\begin{equation}\n\t\t\\mathcal{H}_1 = \\left\\lbrace | \\Phi \\rangle_{A\\tilde{A}} \\in \\mathcal{H}_A \\otimes \\mathcal{H}_{\\tilde{A}}\\,\\, \\text{s.t.}\\,\\, | \\Phi \\rangle_{A\\tilde{A}} \\sim \\hat{S}_L | 1 \\rangle_{\\tilde{A}}| \\phi \\rangle_{C}, \\forall \\, | \\phi \\rangle_{C} \\in \\mathcal{H}_C \\right\\rbrace,\n\t\\end{equation}\n\\end{subequations}\nwhere $| 0 \\rangle_{\\tilde{A}}$ and $| 1 \\rangle_{\\tilde{A}}$ are the eigenvectors of $\\hat{\\sigma}_z$\\footnote{Notice that we could have chosen any other Pauli operator to define this partition.} and two states are said to be equivalent, i.e., $|\\Psi \\rangle_{A\\tilde{A}} \\sim \\hat{S}_L | i \\rangle_{\\tilde{A}}| \\psi \\rangle_{C}$, with $i=0,1$, if they are both eigenvectors of the $\\hat{\\Xi}_z$ operator with the same eigenvalue. We can then build a partition of the Hilbert space into two highly degenerate subspaces, one corresponding to the ``spin up'' and the other to the ``spin down'' eigenvalue, and on which it is possible to define a set of operators satisfying the $\\mathfrak{su}(2)$ algebra, which can be used to encode or decode information of a single qubit.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[scale=0.3]{SternGerlachnew}\n\t\t\\caption{The relativistic Stern-Gerlach experiment as seen from the QRF A (above) and from the QRF C (below). In the rest frame of particle A, the spin is operationally defined via the Stern-Gerlach experiment. To measure spin along direction $\\vec{n}$ the spin (Pauli operator) $\\vec{\\sigma}$ is coupled to an inhomogeneous magnetic field oriented along $\\vec{n}$. The particle is then deflected towards the direction $\\vec{n}$ and $-\\vec{n}$ corresponding to outcome ``spin up'' and ``spin down'' respectively. When transforming to the laboratory frame C, the magnetic field and the spin transform with a superposition of Lorentz boosts for $v_1$ and $v_2$. The interaction Hamiltonian is also transformed, giving rise to a coupling between the transformed vector $\\vec{\\mathcal{S}}_\\Lambda(\\vec{B}^{(A)}) = \\hat{\\gamma}_A \\left[ \\vec{B}^{(C)} - \\frac{\\hat{\\gamma}_A}{\\hat{\\gamma}_A+1}\\left(\\vec{\\hat{\\beta}}_A \\cdot \\vec{B}^{(C)}\\right)\\vec{\\hat{\\beta}}_A + \\left(\\vec{\\hat{\\beta}}_A \\times \\vec{E}^{(C)}\\right) \\right]$ aligned in the same direction $\\vec{n}$ as the magnetic field in the rest frame, and the transformed spin operator $\\vec{\\Xi}$. The particle is again deflected either to $\\vec{n}$ or $-\\vec{n}$ corresponding to the outcome ``spin up'' and ``spin down'' respectively. The probability of detecting the outcomes ``spin up'' and ``spin down'' is preserved under change of QRF.}\n\t\t\\label{fig:SternGerlach}\n\t\\end{center}\n\\end{figure}\n\nThe operators $\\vec{\\hat{\\Xi}}$ in general act on both the external and the internal degrees of freedom of the particle. Operationally, they can be defined via a ``relativistic Stern-Gerlach experiment,'' illustrated in Fig.~\\ref{fig:SternGerlach}. Traditionally, in a Stern-Gerlach experiment, the spin measurement is performed by applying a magnetic field, which interacts with the spin as $\\vec{B} \\cdot \\vec{\\sigma}$ and is inhomogeneous along the direction of its orientation, i.e.,  $\\vec{B} = B(\\vec{r}\\cdot \\vec{n})\\vec{n}$, where $\\vec{n}$ gives the direction and $\\vec{r} = (x, y, z)$. If the magnetic field is aligned precisely in the direction in which the spin state is prepared, the outcome is obtained with certainty. However, if the particle carrying the spin is moving in a superposition of relativistic velocities, no measurement of the spin alone in the laboratory frame will return the result with probability one in general. To treat such a case we set up a hypothetical Stern-Gerlach experiment in the rest frame of the particle, where the interaction Hamiltonian is $H_{int}^{(A)} = \\mu \\vec{B}^{(A)} \\cdot \\vec{\\sigma}$ and $\\mu$ is a coupling constant. We assume that the direction in which the magnetic field is aligned $\\vec{n}$ is orthogonal to the direction of the boost $x$. Formally, this geometric configuration requires to enlarge the Hilbert space of the laboratory to the $z$ direction, which we identify with the direction $\\vec{n}$ of deflection, and modify our previous definition of the state in Eq.~\\eqref{eq:stateA} as $|\\psi \\rangle_C = |\\psi_x \\rangle_C |\\psi_z \\rangle_C$, where $|\\psi_x \\rangle_C$ transforms with $\\hat{S}_L$ and $|\\psi_z \\rangle_C$ is left invariant by the transformation $\\hat{S}_L$, except for the fact that the label is changed from C to A, i.e., $\\hat{S}_L|\\psi_z \\rangle_C = |\\psi_z \\rangle_A$. Additionally, we assume that the motion in the $z$ direction is non relativistic. We then transform the Hamiltonian to the laboratory frame via the QRF transformation $\\hat{S}_L$. Knowing that the magnetic field transforms under superposition of Lorentz boosts as $\\vec{\\hat{\\mathcal{S}}}_\\Lambda(\\vec{B}^{(A)}) = \\hat{\\gamma}_A \\left[ \\vec{B}^{(C)} - \\frac{\\hat{\\gamma}_A}{\\hat{\\gamma}_A+1}\\left(\\vec{\\hat{\\beta}}_A \\cdot \\vec{B}^{(C)}\\right)\\vec{\\hat{\\beta}}_A + \\left(\\vec{\\hat{\\beta}}_A \\times \\vec{E}^{(C)}\\right) \\right]$, we find that the interaction Hamiltonian $H_{int}^{(A)}$ is transformed to\n\\begin{equation} \\label{eq:SternGerlachHamiltonian}\n\n\tH_{int}^{(C)} = \\mu \\hat{\\gamma}_A^{-1} \\vec{\\hat{\\mathcal{S}}}_\\Lambda(\\vec{B}^{(A)}) \\cdot \\vec{\\hat{\\Xi}}.\n\\end{equation}\n\nIt is straightforward to check that the direction of $\\vec{\\hat{\\mathcal{S}}}_\\Lambda (\\vec{B}^{(A)})$ is also $\\vec{n}$, therefore the deflection of the particle in the laboratory frame happens in the same direction as in the rest frame. Notice that, since both the quantum state and the observables transform unitarily, probabilities are automatically conserved after the change of QRF. In particular, if in the rest frame of the particle A the Stern Gerlach measurement detects that the spin is ``up'' with probability one, the ``relativistic Stern-Gerlach'' experiment in the laboratory frame with the interaction Hamiltonian of Eq.~\\eqref{eq:SternGerlachHamiltonian} will also detect ``spin up'' with probability one. Note that the specific form of the electromagnetic field in Eq.~\\eqref{eq:SternGerlachHamiltonian} is not crucial to our result, but we can design the coupling between the particle and the electromagnetic field according to our experimental capabilities in each reference frame. However, it is crucial that the electromagnetic field couples to the operator $\\vec{\\hat{\\Xi}}$, unlike in the standard Stern-Gerlach experiment. In Supplemental Information, we set up a different experiment, where we couple an ihnomogenous magnetic field in the laboratory frame to give an explicit analysis of a relativistic Stern-Gerlach experiment.\n\nIt is worth noting that the interaction Hamiltonian of Eq.~\\eqref{eq:SternGerlachHamiltonian} is covariant, because the quantity $H^{0} : =\\hat{\\gamma}_A H_{int}^{(C)}$ transforms like the zero-component of a $4$-vector. Therefore, the Schr\\\"{o}dinger equation in the reference frame of A, $i \\hbar \\frac{d}{dt_A} \\left| \\psi \\right>_{\\tilde{A}C}^{(A)} = H_{int}^{(A)}\\left| \\psi \\right>_{\\tilde{A}C}^{(A)}$, where $t_A$ is the proper time in the rest frame of A, is mapped to $i \\hbar \\frac{d}{dt_C} \\left| \\psi \\right>_{\\tilde{A}A}^{(C)} = H_{int}^{(C)}\\left| \\psi \\right>_{\\tilde{A}A}^{(C)}$, where $t_C$ is the proper time in the rest frame of C and the relation $t_C = \\hat{\\gamma}_A t_A$ holds. The general, manifestly covariant expression of  $H^0$ is\n\\begin{equation}\n\tH^{0} = \\frac{1}{2}\\eta^{0 \\rho}\\epsilon_{\\rho \\mu \\nu \\lambda} \\hat{\\Sigma}_{p_A}^{\\mu}F^{\\nu \\lambda},\n\\end{equation} \nwhere $\\eta^{\\mu \\nu} = \\text{diag}(1, -1, -1, -1)$ is the Minkowski metric, $F^{\\nu \\lambda}$ is the electromagnetic tensor and $\\epsilon_{\\rho \\mu \\nu \\lambda}$ is the totally antisymmetric tensor such that $\\epsilon_{0123}= 1$. \n\nIn order to complete the measurement, we now have to project the position of the particle along the $z$ direction. Formally, this is achieved by defining the two operators $\\hat{\\Pi}_{+}^{(A)} = \\int^{+\\infty}_{0} dz_c | z_C \\rangle_C \\langle z_C |$ and $\\hat{\\Pi}_{-}^{(A)} = \\int_{-\\infty}^{0} dz_c | z_C \\rangle_C \\langle z_C |$, distinguishing whether the particle is respectively  deflected upwards or downwards. For a thorough analysis of a concrete detection of spin via the ``relativistic Stern-Gerlach'' proposed here and more details on the measurement, see Supplemental Information.\n\nThe QRF transformation provides the description of the same experiment from the point of view of two different QRFs, which move in a superposition of velocities relative to each other. This treatment of the relativistic Stern-Gerlach experiment makes it possible to associate an operational meaning to the spin of a relativistic quantum particle, thus solving the problem of encoding quantum information in a particle with spin degrees of freedom as in a qubit.\n\n\\section{Conclusions}\n\nIn this paper, we have provided an operational description of the spin of a special-relativistic quantum particle. Such operational description is hard to obtain with standard methods due to the combined effect of special relativity, which makes the spin and momentum not separable, and quantum mechanics, which makes it impossible to jump to the rest frame with a standard reference frame transformation. We have introduced the `superposition of Lorentz boosts' transformation to the rest frame of a quantum particle, moving in a superposition of relativistic velocities from the point of view of the laboratory. We have found how the state transforms under such quantum reference frame transformation and identified a set of observables in the laboratory frame which satisfies the $\\mathfrak{su}(2)$ algebra and has the same eigenvalues as the spin in its rest frame. In addition, this set complies with the desiderata for a relativistic spin operator in Ref. \\cite{bauke_relativisticspin}: it commutes with the free Dirac Hamiltonian, it satisfies the $\\mathfrak{su}(2)$ algebra, and it has the same eigenvalues as the spin in its rest frame. In addition, it has the correct nonrelativistic limit. It can be easily shown, in fact, that our operator $\\vec{\\hat{\\Xi}}$ coincides with the Foldy-Wouthuysen spin operator \\cite{foldywouthuysen, caban_spinobservable}. Thanks to the unitarity of the transformation, probabilities are the same in the rest frame and in the laboratory frame. Finally, we have generalised the Stern-Gerlach to the special-relativistic regime by means of a transformation of the interaction Hamiltonian from the rest frame to the laboratory frame. Such generalisation opens up the possibility of performing quantum information protocols with spin in the special-relativistic regime. \n\n\\acknowledgements{We would like to thank Carlos Pineda for helpful discussions. We acknowledge support from the research platform \"Testing Quantum and Gravity Interface with Single Photons\" (TURIS), the Austrian Science Fund (FWF) through the project I-2526-N27 and I-2906, the \\\"{O}AW Innovationsfonds-Projekt ``Quantum Regime of Gravitational Source Masses'', and the doctoral program \"Complex Quantum Systems\" (CoQuS) under Project W1210-N25. We also acknowledge financial support from the EU Collaborative Project TEQ (Grant Agreement 766900). This work was funded by a grant from the Foundational Questions Institute (FQXi) Fund. This publication was made possible through the support of a grant from the John Templeton Foundation  (Project 60609). The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe computational analysis of mIF histopathological images, by enabling simultaneous and quantitative analysis of multiple markers, is becoming key for the development of novel therapeutic drugs. Because most deep-learning based segmentation and detection methods require large datasets of pixel-precise annotation to yield accurate and robust results, methods to transfer annotations or models from one domain to another are of increasing interest \\cite{brieu2019}. However, the standard CycleGAN approach \\cite{zhu2017} for the underlying unsupervised and unpaired domain translation have limitations in preservation of pathology structures, which led to the introduction of segmentation-based guidance \\cite{mahapatra2020}. We propose in this study a novel and simple guidance scheme tailored to the translation from and to the IF domain, based on the conversion from the RGB to Haematoxylin-DAB (HD) colorspace \\cite{ruifrok2001}. As shown qualitatively and quantitatively, our approach yields improved translation results.\n\n\\vspace{-3mm}\n\\section{Methods}\nWe incorporate two losses $\\mathcal{L}^{IF}$ and $\\mathcal{L}^{IHC}$ to the CycleGAN training (cf. Fig.~1). These respectively guide the translation from the IHC domain to the IF domain by generator $G_{AB}$ and the translation from the IF domain to the IHC domain by generator $G_{BA}$. \nWith $x_{IHC}$ an IHC patch input to $\\mathcal{G}_{AB}$ and $x_{IF}$ an IF patch input to $\\mathcal{G}_{BA}$, we define the stain-isolation guidance loss $\\mathcal{L}_g^{IF}$ as the $\\mathcal{L}_1$ norm between the DAB channel of the generated IF image $\\mathcal{G}_{AB}(x_{IHC})$ and the DAB channel of the pseudo-IF image $\\mathcal{F}(x_{IHC})$. The later is derived from the conversion of $x_{IHC}$ to the Haematoxylin-DAB (HD) colorspace followed by its rescaling\\footnote{https:\/\/scikit-image.org\/docs\/stable\/auto\\_examples\/color\\_exposure\/plot\\_ihc\\_color\\_separation.html} using a set of representative IHC images as reference. The inaccuracy of the estimated Haematoxylin channel leads us to consider only the DAB channel into the loss:\n\\vspace{-1mm}\n\\begin{equation}\n\\mathcal{L}_g^{IF} = \\left\\|\\mathcal{G}_{AB}(x_{IHC}).{DAB} - \\mathcal{F}(x_{IHC}).{DAB}\\right\\|_1\n\\end{equation}\n\\vspace{-1mm}\nWe similarly define the pseudo-IHC loss $\\mathcal{L}_g^{IHC}$ as the $\\mathcal{L}_1$ norm between the generated IHC image $\\mathcal{G}_{BA}(x_{IF})$ and the image $\\mathcal{F}^{-1}(x_{IF})$ obtained by color conversion from HD to RGB:\n\\vspace{-1mm}\n\\begin{equation}\n\\mathcal{L}_g^{IHC} = \\left\\|\\mathcal{G}_{BA}(x_{IF}) - \\mathcal{F}^{-1}(x_{IF})\\right\\|_1\n\\end{equation}\n\\vspace{-1mm}\nThe stain isolation and pseudo-IHC losses are added to the following CycleGAN losses: a cycle consistency loss $\\mathcal{L}_{cycle}$, two adversarial losses on the output of the generators $\\mathcal{G}_{AB}$ and $\\mathcal{G}_{BA}$, an identity loss $\\mathcal{L}_{id}$ as well as an embedding loss $\\mathcal{L}_{emb}$. The overall loss reads as:\n\\vspace{-1mm}\n\\begin{equation}\n\\mathcal{L} = \\mathcal{L}_{GAN}^{AB} + \\mathcal{L}_{GAN}^{BA} + \\lambda_1 \\mathcal{L}_{cycle} + \\lambda_2 \\mathcal{L}_{id} + \\lambda_3 \\mathcal{L}_{emb} + \\lambda_4 \\mathcal{L}_g^{IHC} + \\lambda_5 \\mathcal{L}_g^{IF}\n\\label{eq:loss}\n\\end{equation}\n\\vspace{-1mm}\nwith the following fixed weighting parameters: $\\lambda_1=10$, $\\lambda_2=2$ and $\\lambda_3=10$. In the remaining of this study, the two parameters $\\lambda_4$ and $\\lambda_5$ are either switched off or set to 10.\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=0.95\\linewidth]{methods.eps}\n\t\\caption{$\\mathcal{L}_g^{IF}$ and $\\mathcal{L}_g^{IHC}$ guide the training of $\\mathcal{G}_{AB}$ and $\\mathcal{G}_{BA}$. Note the improved DAPI channel of $\\mathcal{G}_{AB}(x_{IHC})$ vs. $\\mathcal{F}(x_{IHC})$ and the more realistic color of $\\mathcal{G}_{BA}(x_{IF})$ vs. ${F}^{-1}(x_{IF})$.\\vspace{-2mm}}\n\t\\label{fig:methods}\n\\end{figure}\n\n\\vspace{-2mm}\n\\section{Results}\nThe unpaired IF and IHC datasets respectively contains 11K and 160K patches of $256\\times256$px with $0.5\\mu$m\/px resolution. On the IF images, the PDL1 and DAPI channels are selected out of the seven initially available channels. The batch size  is 1, the learning rates 0.0001 for the generators and 0.0005 for the discriminators. Adam optimizer ($\\beta_1=0.5$, $\\beta_2=0.999$) is used to minimize the loss (cf. Eq.~\\ref{eq:loss}) for 100K iterations. We qualitatively study the following configurations: (a) no guidance ($\\lambda_4=0$, $\\lambda_5=0$), (b) a virtual-IHC guidance ($\\lambda_4=10$, $\\lambda_5=0$), (c) a stain-isolation guidance ($\\lambda_4=0$, $\\lambda_5=10$) and (d) a combine guidance ($\\lambda_4=10$, $\\lambda_5=10$).\nFig.~2A shows the difficulty of baseline (a) to translate highly saturated regions: dark-brown pixels in IHC are wrongly translated into high DAPI signal in IF while some high DAPI signal in IF is wrongly translated into brown pixels in IHC. While these are partially corrected by the virtual-IHC loss (b), the stain-isolation loss (c) yields better translation. Combining both losses (d) does not improve over (c). This is quantitatively confirmed in Fig.~2B: proposed guidance schemes (b)-(d) prevent the generation of high DAPI signal in membrane regions, otherwise observed in (a).\n\n\\vspace{-2mm}\n\\section{Discussion}\nWe propose two novel loss functions based on color conversion in order to provide guidance at training time for the CycleGAN-based translation between the IF and IHC stain domains. Doing so, we are able to prevent cycle-consistent but non structure preserving translation errors. Future work include the inclusion of the proposed guidance losses to the second translation steps of each cycle as well as the downstream task of transferring segmentation and detection models from the IHC domain to the IF domain.\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=0.99\\linewidth]{results2.eps}\n\t\\caption{A - Examples of $\\mathcal{G}_{AB}(x_{IHC})$ and $\\mathcal{G}_{BA}(x_{IF})$ generated using: (a) the baseline CycleGAN, (b) the virtual-IHC guidance, (c) the stain-isolation guidance, and (d) combined guidance; B - Respective histograms of the DAPI values of $\\mathcal{G}_{AB}(x_{IHC})$ measured on saturated ($S>0.90$) pixels of annotated membrane regions of 222 unseen IHC images $x_{IHC}$. \\vspace{-2mm}}\n\\end{figure}\n\n\\vspace{-3mm}\n\\midlacknowledgments{\\vspace{-1mm}We thank the mIF and IHC-HER2 teams of AstraZeneca Translation Medicine - Oncology R\\&D, for their support in generating the image datasets used in this study.}\n\n\\vspace{-2mm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIn his 1984 AMS Memoir, George Andrews \n\\cite{AndMem} \ndefined two families of combinatorial objects known as {\\it generalized Frobenius partitions}.  These are generalizations of the two--rowed arrays, often known as Frobenius symbols, which arise from considering the rows and columns of the Ferrers graph of an ordinary partition once the Durfee square has been ``removed''.  In the process, Andrews defined two families of functions, $\\phi_k(n)$ and $c\\phi_k(n),$ as the number of generalized Frobenius partitions of weight $n$ in these two families of objects, respectively.  In \n\\cite{AndMem}, Andrews studied these functions $\\phi_k(n)$ and $c\\phi_k(n)$ from several perspectives, including proving a number of Ramanujan--like congruences satisfied by these functions.  This, in turn, led a number of others to extend Andrews' congruence results.  \n\nWhile there exists an extensive literature on the subject of congruences satisfied by generalized Frobenius partition functions, our focus in this note will be on parity results.  We highlight here that a number of authors have proven congruence results with even moduli for these functions; see, for example, the work of \nAndrews \n\\cite[Theorem 10.2]{AndMem}, \nBaruah and Sarmah \n\\cite{BarSar1, BarSar2},   \nChan, Wang, and Yang \n\\cite{ChanWangYang}, \nCui and Gu, \n\\cite{CuiGu}, \nCui, Gu, and Huang \n\\cite{CuiGuHuang},  \nand \nJameson and Wieczorek \n\\cite{JamWie}\nwhere specific congruence results with even moduli are proved.  Several additional papers involving congruence results for generalized Frobenius partitions, but with odd moduli, also appear in the literature.  \n\nWhat is striking about many of the works cited above is that the authors focus specifically on a particular value of the parameter $k$ in order to manipulate the generating function in question to prove their results.  One exception to this rule is Andrews' Theorem 10.2 in \n\\cite[Theorem 10.2]{AndMem}:  \n\n\\begin{theorem}\nLet $p$ be prime and and let $r$ be an integer such that $0<r<p.$ For all $n\\geq 0,$ $$c\\phi_p(pn+r) \\equiv 0\\pmod{p^2}.$$ \n\\end{theorem}\n\nAnother exception to focusing on a particular value of the subscript $k$ appears in the work of Garvan and Sellers \n\\cite[Theorem 2.2]{GarSel}\nwhere the authors prove the following theorem:  \n\n\\begin{theorem}\n\\label{GarSelmainthm}\nLet $p$ be prime and let $r$ be an integer such that $0<r<p.$  If \n$$\nc\\phi_k(pn+r) \\equiv 0\\pmod{p}\n$$ \nfor all $n\\geq 0,$ then \n$$\nc\\phi_{pN+k}(pn+r) \\equiv 0\\pmod{p}\n$$ \nfor all $N\\geq 0$ and $n\\geq 0.$\n\\end{theorem}\nOur goal in this note is to follow a path similar to the above theorem of Garvan and Sellers, where an infinite family of values of $k$ is identified while the value of the modulus is fixed.  \n\nIt is clear, when one reviews the literature on the subject of congruences satisfied by generalized Frobenius partition functions, that the functions $c\\phi_k(n)$ satisfy many more congruences than their counterpart functions $\\phi_k(n).$  One might argue that this is true ``combinatorially'' given the structure of the objects being counted by these functions (and symmetries that are inherent in the two--rowed arrays counted by $c\\phi_k(n)$). It is also true that, although the generating functions for each of these two families of functions are extremely similar, the presence of certain powers of roots of unity in the generating function for $\\phi_k(n),$ and the corresponding absence of such roots of unity in the generating function for $c\\phi_k(n),$ may contribute to the relative lack of congruences satisfied by $\\phi_k(n).$  Whatever the case, our primary goal in this note is to alter this narrative by proving the following surprising result:  \n\n\\begin{theorem}\n\\label{mainresult} \nLet $\\ell$ be a positive integer,  $p\\geq 5$ be prime, and let  $r,$ $ 0< r < p,$ be an integer such that $24r+1$ is a quadratic nonresidue modulo $p.$ For all $n\\geq 0,$\n $$\n \\phi_{p\\ell-1}(pn+r) \\equiv 0 \\pmod{2}.\n $$\n\\end{theorem}\n\n\n\\section{Proof of Theorem \\ref{mainresult}}\n\nIn order to prove Theorem \\ref{mainresult}, we need a few preliminary facts.  First, we remind the reader of the $q$--Pochhammer symbol which is defined as follows:  \n$$(A;q)_\\infty = (1-A)(1-Aq)\\cdots(1-Aq^{n})\\cdots$$\n\nWe will also need the following two well--known results:  \n\\begin{theorem}\n\\label{eulerPNT}\n\\begin{equation*}\n    (q;q)_\\infty = \\sum\\limits_{k = -\\infty}^\\infty (-1)^k q^{\\frac{3k^2-k}{2}}\n\\end{equation*}\n\\end{theorem}\n\\begin{proof}\nSee Hirschhorn \n\\cite[(1.6.1)]{Hir}.  \n\\end{proof}\n\n\\begin{theorem}\n\\label{jacobi_cube} \n\\begin{equation*}\n    (q;q)_\\infty^3 = \\sum\\limits_{k=0}^\\infty (-1)^k(2k+1)q^{\\frac{k^2+k}{2}}.\n\\end{equation*}\n\\end{theorem}\n\\begin{proof}\nSee Hirschhorn \n\\cite[(1.7.1)]{Hir}.  \n\\end{proof}\n\nNext, we prove an extremely important fact about the generating function for $\\phi_k(n)$ for all $k\\geq 1$ using a key result that appears in Andrews \n\\cite{AndMem}.\n\n\\begin{theorem}\n\\label{Mod2Lemma} \nLet $$\n\\Phi_k(q) = \\sum_{n=0}^\\infty \\phi_k(n)q^n\n$$\nbe the generating function for the generalized Frobenius partition function $\\phi_k(n).$  Then \n$$\n\\Phi_k(q) \\equiv \\frac{(q;q)_\\infty}{(q^{k+1};q^{k+1})_\\infty} \\pmod{2}.  \n$$\n\\end{theorem}\n\\begin{proof}\nFrom Andrews' Memoir \n\\cite[Theorem 7.1]{AndMem},\nwe know \n\\begin{equation*}\n    \\Phi_k(q) = \\frac{1}{ (q;q)_\\infty^2(q^{k+1};q^{k+1})_\\infty}\\sum\\limits_{j,r = -\\infty \\atop r \\geq (k+1)|j|}^{\\infty}(-1)^{r+kj}q^{\\binom{r+1}{2}-\\binom{k+1}{2}j^2}.\n\\end{equation*}\nTherefore, we have \n\\begin{eqnarray*}\n\\Phi_k(q) \n&=& \n\\frac{1}{ (q;q)_\\infty^2(q^{k+1};q^{k+1})_\\infty}\\sum\\limits_{j,r = -\\infty \\atop r \\geq (k+1)|j|}^{\\infty}(-1)^{r+kj}q^{\\binom{r+1}{2}-\\binom{k+1}{2}j^2}\\\\\n&= & \n\\frac{(q;q)_\\infty}{ (q;q)_\\infty^3(q^{k+1};q^{k+1})_\\infty}\\sum\\limits_{r =0}^\\infty q^{\\binom{r+1}{2}} \\sum\\limits_{|j|\\leq r\/(k+1)}(-1)^{r+kj}q^{-\\binom{k+1}{2}j^2}  \\\\\n&\\equiv & \n\\frac{(q;q)_\\infty}{ (q;q)_\\infty^3(q^{k+1};q^{k+1})_\\infty}\\sum\\limits_{r =0}^\\infty q^{\\binom{r+1}{2}} \\sum\\limits_{|j|\\leq r\/(k+1)}q^{-\\binom{k+1}{2}j^2} \\pmod{2} \\\\\n&=& \n\\frac{(q;q)_\\infty}{ (q;q)_\\infty^3(q^{k+1};q^{k+1})_\\infty}\\sum\\limits_{r =0}^\\infty q^{\\binom{r+1}{2}}\\left(1+2 \\sum\\limits_{1\\leq j \\leq r\/(k+1)}q^{-\\binom{k+1}{2}j^2} \\right)   \\\\\n&\\equiv & \n\\frac{(q;q)_\\infty}{ (q;q)_\\infty^3(q^{k+1};q^{k+1})_\\infty}\\sum\\limits_{r =0}^\\infty q^{\\binom{r+1}{2}} \\pmod{2} \\\\\n&\\equiv & \n\\frac{(q;q)_\\infty(q;q)_\\infty^3}{ (q;q)_\\infty^3(q^{k+1};q^{k+1})_\\infty} \\pmod{2} \\\\\n\\end{eqnarray*}\nthanks to Theorem \\ref{jacobi_cube}.  The result immediately follows.  \n\\end{proof} \n\n\nWe are now in an excellent position to prove Theorem \\ref{mainresult}.  \n\n\\begin{proof}(of Theorem \\ref{mainresult}) \nThanks to Theorem \\ref{Mod2Lemma}, we know that the generating function for $\\phi_{p\\ell-1}$ satisfies \n\\begin{eqnarray*}\n\\sum_{n=0}^\\infty \\phi_{p\\ell-1}(n)q^n \n&\\equiv &\n\\frac{(q;q)_\\infty}{(q^{p\\ell};q^{p\\ell})_\\infty} \\pmod{2} \\\\\n&\\equiv & \n\\frac{1}{(q^{p\\ell};q^{p\\ell})_\\infty} \\sum\\limits_{k = -\\infty}^\\infty q^{\\frac{3k^2-k}{2}} \\pmod{2}  \n\\end{eqnarray*}\nwhere the last statement follows from Theorem \\ref{eulerPNT}.  \nSince $(q^{p\\ell};q^{p\\ell})_\\infty$ is a function of $q^p,$ and since we are interested in the parity of the values $\\phi_{p\\ell-1}(pn+r)$ where $0<r<p,$ we simply need to determine when $$pn+r = \\frac{3k^2-k}{2}$$ for some integer $k.$  After completing the square, this is equivalent to asking whether\n\\begin{align*}\n    24r+1 & \\equiv 36k^2 -12k + 1 \\pmod{p}\\\\\n    & = (6k-1)^2.\n\\end{align*}\nHowever, we assumed that $24r+1$ is a quadratic nonresidue modulo $p$ in the statement of this theorem.   Therefore, $pn+r$ can never be represented as $\\frac{3}{2}k^2-\\frac{1}{2}k$ for any integer $k$. This implies that \n$$\\phi_{p\\ell-1}(pn+r) \\equiv 0 \\pmod{p}.$$\n\\end{proof}\n\n\\section{Concluding Remarks}  \nIt should be noted that a companion result to Theorem \\ref{mainresult}  exists for the functions $c\\phi_k(n)$ as well (which is not all that surprising).  \n\n\\begin{theorem} \n\\label{cphi_mod2} \nFor all $k\\geq 1$ and all $n\\geq 0,$\n $$\n c\\phi_{2k}(2n+1) \\equiv 0 \\pmod{2}.\n $$\n\\end{theorem}\n\nIn a real sense, the proof of this result already appears in the literature; see, for example, Garvan and Sellers \n\\cite{GarSel}\nas well as Baruah and Sarmah \n\\cite{BarSar2}. \nFor completeness' sake, we provide a proof here.  \n\n\\begin{proof} \nThe generating function for $c\\phi_{2k}(n),$ as provided by Andrews \n\\cite{AndMem},\nis the constant term in \n$$\nCG_{2k}(z) = \\prod_{n=0}^\\infty(1+zq^{n+1})^{2k}(1+z^{-1}q^n)^{2k}.\n$$\nThanks to the Binomial Theorem, we have \n\\begin{align*}\n    CG_{2k}(z) \n    &= \\prod_{n=0}^\\infty(1+zq^{n+1})^{2k}(1+z^{-1}q^n)^{2k}\\\\\n    & \\equiv \\prod_{n=0}^\\infty(1+z^2q^{2n+2})^{k}(1+z^{-2}q^{2n})^{k} \\pmod{2}.\n\\end{align*}\nThe theorem immediately follows because the last product above is a function of $q^2.$\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe goal of this survey paper is to give an overview of results for the uniqueness of bounded solutions\nof the heat equation with continuous time parameter, aka stochastic completeness, on infinite weighted graphs.\nWe first discuss some equivalent formulations of stochastic completeness and how\nwe have to go beyond the realm of bounded operators on graphs\nin order for this property to be of interest.  \nFurthermore, we discuss how the combinatorial graph distance is not an appropriate choice\nof metric for the purpose of finding results analogous to those found in the setting of Riemannian manifolds.\nThis leads to the use of so-called intrinsic metrics which give an appropriate notion of\nvolume growth. Finally, we discuss how some recently developed\nversions of curvature on graphs can be used to give conditions for stochastic completeness.\n\n\n\\subsection{Stochastic completeness, uniqueness class and volume growth}\nIn 1986, Alexander Grigor$\\cprime$yan published an optimal volume growth condition\nfor the stochastic completeness of a geodesically complete Riemannian manifold. More specifically, if\na geodesically complete manifold $M$ satisfies\n$$\\int^\\infty \\frac{r}{\\log V(r)}dr =\\infty$$\nwhere $V(r)$ denotes the volume of a ball of radius $r$ with an arbitrary center, then \n$M$ is stochastically complete \\cite{Gri86}. In particular, we note\nthat any Riemannian manifold with volume growth satisfying $V(r) \\leq Ce^{r^2}$ for $C>0$\nwill be stochastically complete. This result improved all other known volume growth\ncriteria at the time \\cite{Gaf59, KLi} and is sharp in the sense that there exist stochastically incomplete manifolds\nwith volume growth of order $e^{r^{2+\\epsilon}}$ for any $\\epsilon>0$. In particular, the class of model manifolds\nalready provides such examples, see the survey article of Grigor$'$yan for this and \nmany other results \\cite{Gri99}.\n\nGrigor$'$yan's volume growth criterion was proven via a uniqueness class result for solutions\nof the heat equation. More specifically, if $u$ is a solution of the heat equation on $M \\times (0,T)$\nwith initial condition $0$ and for all $r$ large $u$ satisfies\n$$\\int_0^T \\int_{B_r} u^2(x,t)\\  d\\mu \\ dt \\leq e^{f(r)}$$\nwhere $f$ is a monotone increasing function on $(0,\\infty)$ which satisfies\n$$\\int^\\infty \\frac{r}{f(r)}dr=\\infty,$$\nthen $u=0$ on $M \\times (0,T)$.\nIt follows that bounded solutions of the heat equation are unique by taking the difference\nof two solutions and letting $f(r)=\\log(C^2TV(r))$ where $C$ is a bound on the solutions.\nSee \\cite{Gaf59, KLi, Dav92, Tak89} for other techniques for proving volume\ngrowth criteria for stochastic completeness in the manifold setting.\n\nIn the setting of graphs, an explicit study of geometric conditions for the \nuniqueness of bounded solutions of the heat equation\nwith continuous time parameter can be found in\nwork of J{\\'o}zef Dodziuk and Varghese Mathai \\cite{DM06} as well as that of Dodziuk \\cite{Dod06}\nand subsequently taken up in the author's Ph.D. thesis \\cite{Woj08} and independently\nin work of Andreas Weber \n\\cite{Web10}. In particular,\nDodziuk\/Mathai show that whenever the Laplacian on a graph with standard weights is a bounded\noperator, then the graph is stochastically complete. Dodziuk then extends this result\nto allow weights on edges. The technique used to establish these results is that \nof a minimum principle for the heat equation.\n\nIn the thesis \\cite{Woj08} and follow-up paper \\cite{Woj09}\nwe rather use an equivalent formulation of stochastic completeness in terms\nof bounded $\\lambda$-harmonic functions to derive criteria for stochastic completeness which allow for\nunbounded operators. Furthermore, we give a full characterization of stochastic completeness\nin the case of trees which enjoy a certain symmetry. This characterization\nalready shows a disparity between the graph and manifold\nsettings in that there  exist stochastically incomplete graphs with factorial volume growth, that is, \nif $V(r)$ denotes the  \nnumber of vertices within $r$ steps of a center vertex, then the tree is stochastically\nincomplete and $V(r)$ grows factorially with $r$.\nHowever, a more striking disparity appeared in a subsequent paper \nwhich introduced a class of graphs called anti-trees. \nThese graphs can be stochastically incomplete and have polynomial volume growth \\cite{Woj11}. \nIn particular, there exist stochastically incomplete anti-trees with volume growth like \n$$V(r)\\sim r^{3+\\epsilon}$$\nfor any $\\epsilon>0$. This provides a very strong contrast with the borderline for the manifold case\ngiven by Grigor$'$yan's result.\n\nThe volume growth in these examples involves taking balls via the usual combinatorial graph metric, that is,\ntaking the least number of edges in a path connecting two vertices. This notion reflects only the global connectedness properties of the graph. However, it is natural to expect that a metric should\nalso reflect the local geometry of a graph, i.e., the valence or degree of vertices. Furthermore, if the\ngraph has weights on both edges and a measure on vertices, then the metric should \ninteract with both the edge weights and the vertex measure.\nFor Riemannian manifolds, there exists the notion of an intrinsic metric which naturally arises\nfrom the energy form as well as from\nthe geometry of the manifold. This notion of an intrinsic metric for the energy form was then extended to strongly \nlocal Dirichlet forms by Karl-Theodor Sturm \\cite{Stu94}. Now, graphs which have both a weight on \nedges and a measure on vertices can be put into a one-to-one correspondence\nwith regular Dirichlet forms on discrete measure spaces as discussed in work of Matthias Keller and Daniel Lenz \\cite{KL12}. \nHowever, the Dirichlet forms that arise from graphs are not strongly local. Thus, the notion\nof an intrinsic metric from strongly local Dirichlet froms has to be extended to the non-local setting. \nThis was done systematically in a paper by Rupert Frank, Daniel Lenz and Daniel Wingert \\cite{FLW14}. \n\nThe notion of intrinsic metrics for non-local Dirichlet forms was quickly put to use by the graph theory community. \nA first example of a concrete intrinsic metric for weighted graphs already appers in the \nPh.D. thesis of Xueping Huang \\cite{Hua11b} and can also be found in the work of Matthew Folz \non heat kernel estimates around the same time \\cite{Fol11}.\nHowever, even given the tool of intrinsic metrics, there are still difficulties in proving an analogue\nto Grigor$'$yan's criterion for graphs. In particular, Huang gives an example of a graph for which there\nexists a non-zero bounded solution of the heat equation $u$ with zero initial condition which satisfies\n$$\\int_0^T \\int_{B_r} u^2(x,t)\\  d\\mu \\ dt \\leq e^{f(r)}$$\nfor $f(r)=C r \\log r$ for some constant $C$, see \\cite{Hua11b, Hua12}. \nHence, as $f$ in this case clearly satisfies $\\int^\\infty {r}\/{f(r)} \\ dr=\\infty$, \nwe see that even when using intrinsic metrics, a direct analogue\nto Grigor$'$yan's proof is not possible for all graphs. \n\nA recent breakthrough in resolving this issue can be found in the work of Xueping Huang, Matthias Keller\nand Marcel Schmidt \\cite{HKS}. In this paper, the authors first prove a uniqueness class result\nwhich is valid for a certain class of graphs called globally local. They can then\nreduce the study of stochastic completeness of general graphs to that of globally local graphs using\nthe technique of refinements first found in \\cite{HS14}. With these two results, they are able to establish\nan exact analogue to the volume growth criterion of Grigor$'$yan which is valid for all graphs. That is, letting\n$V_\\varrho(r)$ denote the measure of a ball with respect to an intrinsic metric and letting \n$\\log^{\\#}(x)=\\max\\{\\log(x), 1\\}$ if\n$$\\int^\\infty \\frac{r}{\\log^{\\#} V_\\varrho(r)} dr = \\infty,$$\nthen the graph is stochastically complete. We note that taking the minimum with 1 \nis only necessary to cover the case of when the entire graph has small measure;\nthe actual value of the constant 1 is not relevant.\n\nLet us mention that the volume growth criterion for stochastic completeness of graphs involving\nintrinsic metrics was first proven under \nsome additional assumptions by Folz \\cite{Fol14}. \nThe proof technique of Folz, however, is different from that of Grigor$'$yan.\nMore specifically, Folz bypasses Grigor$'$yan's uniqueness class technique via a\nprobabilistic approach involving synchronizing the random walk on the graph with a random walk\non an associated quantum graph and then applying a generalization of Grigor$'$yan's result for manifolds\nto strongly local Dirichlet forms found in work of Sturm \\cite{Stu94}. A similar proof involving quantum graphs\nbut using analytic techniques can also be found in a paper by Huang \\cite{Hua14b}.  \n\n\nWe would also like to highlight earlier work focused on a volume growth criterion \nby Alexander Grigor$'$yan, Xueping Huang and Jun Masamune \\cite{GHM12} \nusing a technique from \\cite{Dav92}. While this did not yield the optimal volume growth condition when using\nintrinsic metrics, it did yield an optimal volume growth condition for the combinatorial graph metric in that \n$$V(r) \\leq Cr^3$$\nimplies stochastic completeness where $V(r)$ is the volume defined with respect to the combinatorial graph metric.\nThus, we see that the anti-tree examples found in \\cite{Woj11} are the smallest stochastically incomplete\ngraphs in the combinatorial graph distance. \n\n\n\\subsection{Curvature and stochastic completeness}\nLet us now turn to curvature. For Riemannian manifolds, in a paper from 1974, \nRobert Azencott gave both a curvature criterion for stochastic completeness and the first examples of stochastically incomplete manifolds \\cite{Az74}.\nIn Azencott's example, the curvature decays to negative infinity rapidly, thus it is natural to expect that\nlower curvature bounds are necessary for stochastic completeness.\nAn optimal result in this direction involving Ricci curvature\nwas established by Nicholas Varopoulos \\cite{Var83} and Pei Hsu\n\\cite{Hsu89}. It can be formulated as follows: let $M$ be a geodesically complete Riemannian manifold\nand suppose that $\\kappa$ is a positive increasing continuous function on $(0,\\infty)$ such that for all points\naway from the cut locus on the sphere of radius $r$ we have $\\mathrm{Ric}(x)\\geq -C \\kappa^2(r)$ for all $r$ large\nand $C>0$. \nIf\n$$\\int^\\infty \\frac{1}{\\kappa(r)}dr=\\infty,$$\nthen $M$ is stochastically complete. This improved the previously known results which gave that Ricci curvature\nuniformly bounded from below implied stochastic completeness as proven by Shing-Tung Yau \\cite{Yau78}, \nsee also the work of Dodziuk \\cite{Dod83}. However, due to the connection between\nRicci curvature and volume growth, this result is already implied by Grigor$'$yan's volume growth result.\nThere is also a number of comparison results for stochastic completeness involving curvature, see\n\\cite{Ichi82} or Section 15 in the survey of Grigor$'$yan \\cite{Gri99}.\n\nIn recent years, there has been a tremendous interest in notions of curvature\non graphs. We focus here on two formulation. One definition of curvature \noriginates in work of Dominique Bakry\nand Michele \\'{E}mery on hypercontractive semigroups \\cite{BE85}. Thus, we refer to it as Bakry--\\'{E}mery curvature.\nA second formulation comes from the work of Yann Ollivier on Markov chains on metric spaces in \\cite{Oll07, Oll09}.\nThis was later modified to give an infinitesimal version by Yong Lin, Linyuan Lu and Shing-Tung Yau \\cite{LLY11}\nand then extended to the case of possibly unbounded operators on graphs by Florentin M{\\\"u}nch and the author \\cite{MW19}. In any case, we refer to this as Ollivier Ricci curvature.\nFor Bakry--\\'{E}mery curvature, Bobo Hua and Yong Lin proved that a uniform lower bound implies stochastic completeness\nin \\cite{HL17}.\nOn the other hand, in \\cite{MW19} we prove that for Ollivier Ricci curvature, if\n$$\\kappa(r) \\geq -C\\log r$$\nfor $C>0$ and all large $r$ where $\\kappa(r)$ denotes the spherical curvature on a sphere\nof radius $r$, then the graph is stochastically complete.\nThis is optimal in the sense that for any $\\epsilon>0$ there exist stochastically incomplete\ngraphs with $\\kappa(r)$ decaying like $-(\\log r)^{1+\\epsilon}$. Thus, there is still a disparity in this condition for graphs and\nfor the Ricci curvature condition for manifolds as presented above. However, this disparity cannot\nbe resolved by using the notion of intrinsic metrics as was the case for stochastic completeness and volume growth.\n \n\\subsection{The structure of this paper}\nWe now briefly discuss the structure of this paper. Although we do not give full proofs of results,\nwe also do not assume any particular background of the reader and thus try to make the presentation\nas self-contained as possible in terms of concepts and definitions. We also give specific references\nfor all results that we do not prove completely.\n\nIn Section~\\ref{s:heat} we introduce the setting of weighted graphs and discuss the heat\nequation. In particular, we outline an elementary construction of bounded solutions of the \nheat equation using exhaustion techniques. In Section~\\ref{s:sc} we present some equivalent formulations for \nstochastic completeness. In particular, stochastic completeness is equivalent to the uniqueness of this bounded solution\nof the heat equation. In Section~\\ref{s:bounded} we discuss boundedness of the Laplacian and\nhow boundedness is related to stochastic completeness. In Section~\\ref{s:symmetry} we then introduce\nthe class of weakly spherically symmetric graphs and present the examples of anti-trees which show the\ndisparity between the continuous and discrete settings in the case of the combinatorial graph metric.\nIn Section~\\ref{s:intrinsic} we introduce intrinsic metrics and discuss how they can differ from the\ncombinatorial graph metric and how this is related to stochastic completeness. \nFinally, in Sections~\\ref{s:volume}~and~\\ref{s:curvature} we present the criteria\nfor stochastic completeness in terms of volume growth and curvature mentioned above.\n\nWe also mention here a recent survey article by Bobo Hua and Xueping Huang which has some contact\npoints with our article but also discusses heat kernel estimates, ancient solutions of the heat\nequation and upper escape rates \\cite{HH}.\n\n\n\\section{The heat equation on graphs}\\label{s:heat}\n\\subsection{Weighted graphs}\nWe start by introducing our setting following \\cite{KL12}. We note that this setting is very general\nin that we allow for arbitrary weights on both edges and vertices. We also do not assume local finiteness,\ni.e., that every vertex has only finitely many edges coming out of the vertex.\n\n\n\\begin{definition}[Weighted graphs] \nWe let $X$ be a countably infinite set whose elements we refer to as \n\\emph{vertices}. We then let $m:X \\longrightarrow (0,\\infty)$ denote a \\emph{measure} on the vertex set which can\nbe extended to all subsets by additivity. Finally, we let $b:X \\times X \\longrightarrow [0,\\infty)$ denote a function\ncalled the \\emph{edge weight} which satisfies\n\\begin{enumerate}\n\\item[(b1)] \\quad $b(x,x)=0$ for all $x \\in X$\n\\item[(b2)] \\quad $b(x,y)=b(y,x)$ for all $x, y \\in X$\n\\item[(b3)] \\quad $\\sum_{y \\in X} b(x,y) <\\infty$ for all $x \\in X$.\n\\end{enumerate}\nWhenever $b(x,y)>0$, we think of the vertices $x$ and $y$ as being \\emph{connected} by an edge with \nweight $b(x,y)$, call $x$ and $y$ \\emph{neighbors} and write $x \\sim y$. \nThus, (b1) gives that there are no loops,\n(b2) that edge weights are symmetric and (b3) that the total sum of the edge weights is finite.\nWe call the triple $G=(X, b, m)$ a \\emph{weighted graph} or just \\emph{graph} for short.\n\\end{definition}\n\nWe note, in particular, that condition (b3) above allows for a vertex to have infinitely many neighbors.\nWhenever, each vertex has only finitely neighbors, we call the graph \\emph{locally finite}. We call the quantity\n$${\\mathrm{Deg}}(x)= \\frac{1}{m(x)}\\sum_{y \\in X} b(x,y)$$\nthe \\emph{weighted vertex degree} of $x \\in X$ or just \\emph{degree} for short. We will see\nthat this function plays a significant role in what follows.\n\n\\begin{example}\nWe now present some standard choices for $b$ and $m$ to help orient the reader.\nIn particular, we discuss the case of standard edge weights, counting and degree measures.\n\\begin{enumerate}\n\\item Whenever $b(x,y) \\in \\{0,1\\}$ for all $x, y \\in X$, we say that the graph has \\emph{standard edge weights}.\nIn this case, it is clear that condition (b3) in the definition of the edge weights \nimplies that the graph must be locally finite.\n\\item One choice of vertex measure is the counting measure, that is, $m(x)=1$\nfor all $x \\in X$. In this case, \n$m(K)=\\# K$ is just the cardinality of any finite subset $K$. In the case of standard edge weights\nand counting measure, we then\nobtain\n$${\\mathrm{Deg}}(x) =\\# \\{y \\ | \\ y \\sim x\\}$$\nso that the weighted vertex degree is just the number of neighbors of $x$, that is, the valence\nor degree of a vertex.\n\\item Another choice for the vertex measure is \n$$m(x) = \\sum_{y \\in X}b(x,y)$$\nfor $x \\in X$. \nIn the case of standard edge weights, \nit then follows that $m(x) = \\# \\{ y \\ | \\ y \\sim x \\}$ is the number of neighbors of $x$.\nIn any case, with this choice of measure, it is clear that\n$${\\mathrm{Deg}}(x)=1$$ \nfor all $x \\in X$. \n\\end{enumerate}\n\\end{example}\n\nOften we will assume that graphs are \\emph{connected} in the usual geometric sense, namely,\nfor any two vertices $x, y \\in X$, there exists a sequence of vertices $(x_k)_{k=0}^n$ with $x_0=x$,\n$x_n =y$ and $x_k \\sim x_{k+1}$ for $k=0, 1, \\ldots n-1$. We note that we include the case of $x=y$\nwhen a vertex can be connected to itself via a path consisting of a single vertex and thus no edges.\nSuch a sequence is called a \\emph{path}\nconnecting $x$ and $y$. We then let \n$$d:X \\times X \\longrightarrow [0,\\infty)$$\ndenote the \\emph{combinatorial graph distance} on $X$,\nthat is, $d(x,y)$ equals the least number of edges in a path connecting $x$ and $y$.\nWe note that this metric only considers the combinatorial properties of the graph encoded in $b$ but not\nthe actual value of $b(x,y)$ nor the vertex measure $m$. We will have more to say about this later.\n\n\\subsection{Laplacians and forms}\nWe now denote the set of all functions on $X$ by $C(X)$, that is,\n$$C(X) = \\{ f: X \\longrightarrow {\\mathbb R}\\}$$\nand the subset of finitely supported functions by $C_c(X)$.  The Hilbert space\nthat we will be interested in at various points is $\\ell^2(X,m)$, the space of square summable functions\non $X$ with respect to the measure $m$. That is,\n$$\\ell^2(X,m) = \\{ f \\in C(X) \\ | \\ \\sum_{x \\in X}f^2(x)m(x)<\\infty\\}$$\nwith inner product\n$\\langle f, g \\rangle = \\sum_{x \\in X}f(x)g(x)m(x)$. \n\nIn order to introduce a formal Laplacian, we first have to restrict to a certain class of functions\nas we do not assume local finiteness so that summability becomes an issue. \n\\begin{definition}[Formal Laplacian and energy form]\nWe let\n$$\\mathcal{F}=\\{ f\\in C(X) \\ | \\ \\sum_{y \\in X} b(x,y) |f(y)| <\\infty \\textup{ for all } x \\in X \\}$$\nand for $f \\in \\mathcal{F}$, we let \n$$\\mathcal{L} f(x) = \\frac{1}{m(x)} \\sum_{y \\in X} b(x,y) (f(x)-f(y))$$\nfor $x \\in X$. The operator $\\mathcal{L}$ is then called the \\emph{formal Laplacian} associated to $G$. We furthermore let\n$$\\mathcal{D}=\\{ f\\in C(X) \\ | \\  \\sum_{x, y \\in X} b(x,y) (f(x)-f(y))^2 <\\infty\\}$$\ndenote the space of \\emph{functions of finite energy}. For $f, g \\in \\mathcal{D}$, we let\n$$\\mathcal{Q}(f,g) = \\frac{1}{2}\\sum_{x,y \\in X} b(x,y) (f(x)-f(y))(g(x)-g(y)) $$\ndenote the \\emph{energy form} associated to $G$.\n\\end{definition}\n\nWe denote the restriction of $\\mathcal{Q}$ to $C_c(X) \\times C_c(X)$ by $Q_c$. It then follows that a \nversion of Green's formula holds for $Q_c$:\n$$Q_c(\\varphi,\\psi) = \\sum_{x \\in X} \\mathcal{L} \\varphi(x)\\psi(x) m(x) = \\sum_{x \\in X} \\varphi(x) \\mathcal{L} \\psi(x) m(x) $$\nfor all $\\varphi, \\psi \\in C_c(X)\\subseteq \\ell^2(X,m)$, see, for example \\cite{HK11}.\nThe form $Q_c$ is closable and thus there exists a \nunique self-adjoint operator $L$ with domain \n$D(L) \\subseteq \\ell^2(X,m)$ associated to the closure of $Q_c$ denoted by $Q$. \nFor a discussion of the closure of a form and the construction of the \nassociated operator in a general Hilbert space, see Theorem~5.37 in \\cite{Weid80}.\nWe refer to $L$ as the \\emph{Laplacian} associated to the graph $G$. We note that with our sign\nconvention, we have\n$$\\langle Lf, f \\rangle = Q(f,f) \\geq 0$$\nfor all $f \\in D(L)$ so that $L$ is a positive operator.\n\n\n\n\\subsection{A word about essential self-adjointness and $\\ell^2$ theory}\nAlthough not a main concern of this article as we mostly deal with bounded solutions, \nwe want to mention another approach to the construction of the Laplacian $L$.\nIn this viewpoint, one starts by restricting $\\mathcal{L}$ to $C_c(X)$ and denoting the resulting operator by $L_c$,\nthat is, $D(L_c)=C_c(X)$ and $L_c$ acts as $\\mathcal{L}$. \n\nHowever, due to the lack of local finiteness,\n$\\mathcal{L}$ does not necessarily map $C_c(X)$ into $\\ell^2(X,m)$. Thus, whenever we want to consider $L_c$ as an operator on $\\ell^2(X,m)$, we have to assume\nthat $\\mathcal{L}$ maps $C_c(X)$ into $\\ell^2(X,m)$.\nUnder this additional assumption, it is easy to see that $L_c$ is a symmetric operator on $\\ell^2(X,m)$, i.e.,\n$$\\langle L_c \\varphi, \\psi \\rangle = \\langle \\varphi, L_c \\psi \\rangle$$\nfor all $\\varphi, \\psi \\in C_c(X)$ and that the Green's formula reads as\n$$Q_c(\\varphi,\\psi) = \\langle L_c \\varphi, \\psi \\rangle$$\nfor $\\varphi, \\psi \\in D(L_c)$. In this case, the self-adjoint operator associated to the closure of $Q_c$, which is just \nthe Laplacian $L$,\nis called the \\emph{Friedrichs extension} of $L_c$, see\nTheorem~5.38 in \\cite{Weid80} for further details on the construction of this extension for general Hilbert\nspaces. \n\nWe note that $\\mathcal{L}$ maps $C_c(X)$ into $\\ell^2(X,m)$ whenever \n$\\mathcal{L}1_x \\in \\ell^2(X,m)$ for all $x \\in X$ where $1_x$\ndenotes the characteristic function of the singleton set $\\{x\\}$. It is a direct calculation that \n$\\mathcal{L}1_x \\in \\ell^2(X,m)$ for all $x \\in X$ if and only if \n$$\\sum_{y \\in X} \\frac{b^2(x,y)}{m(y)} <\\infty$$\nfor all $x \\in X$. In particular, all locally finite graphs or, more generally, all graphs\nwith $\\inf_{y \\sim x} m(y)>0$ automatically satisfy this assumption.\nFurthermore, the condition $\\mathcal{L}1_x \\in \\ell^2(X,m)$ for all $x \\in X$ is equivalent\nto a variety of other conditions, for example, that $C_c(X) \\subseteq D(L)$, for more details,\nsee \\cite{Kel15}.\n\n\nWe further note that, in general, $L_c$ may have many self-adjoint extensions and\nthat processes associated to these different extensions may have different stochastic properties.\nWhen $L_c$ has a unique self-adjoint extension, $L_c$ is called \\emph{essentially self-adjoint}.\nIt was first shown as Theorem~1.3.1 in \\cite{Woj08} that $L_c$ is essentially self-adjoint in the case \nof standard edge weights and counting measure. This was then extended to allow for general edge \nweights and any measure such that the measure of infinite paths is infinite as Theorem~6 in \\cite{KL12}. \nThis criterion was further improved and generalized in \\cite{Gol14, Schm20} \nwhich consider more general operators on graphs.\nFor further discussion of essential self-adjointness, see \\cite{HKLW12, HKMW13, Schm20} and reference therein.\nWe will also discuss the connection between essential self-adjointness and metric completeness in Subsection~\\ref{ss:metric_esa} below.\n\n\\subsection{The heat equation: existence of solutions}\nWe now introduce a continuous time heat equation on $G$. We let $\\ell^\\infty(X)$ denote the set of bounded\nfunctions on $X$, that is,\n$$\\ell^\\infty(X) = \\{ f \\in C(X) \\ | \\ \\sup_{x \\in X} |f(x)| <\\infty \\}.$$\nWe now make precise the requirements\nfor summability, differentiability and boundedness of a solution.\n\n\\begin{definition}[Bounded solution of the heat equation]\nLet $u_0 \\in \\ell^\\infty(X).$\nBy a \\emph{bounded solution\nof the heat equation with initial condition} $u_0$ we mean a bounded function\n$$u:X \\times [0,\\infty) \\longrightarrow {\\mathbb R}$$\nsuch that $u(x,\\cdot)$ is continuous for every $t\\geq0$, differentiable\nfor $t>0$ and all $x \\in X$ and\n$$(\\mathcal{L} + \\partial_t)u(x,t) = 0$$\nfor all $x \\in X$ and $t>0$ with $u(x,0)=u_0(x)$.\n\\end{definition}\n\nWe note, in particular, that as $u(\\cdot, t) \\in \\ell^\\infty(X)$ for every $t \\geq 0$, we obtain\nthat $u(\\cdot, t) \\in \\mathcal{F}$. Thus, we may apply the formal Laplacian to $u$ at every time $t \\geq 0$.\n\nThe definition of bounded solutions raises two immediate questions: the existence and uniqueness of solutions. \nWe will first address existence by showing that there always\nexists a bounded solution which is minimal in a certain sense. On the other hand, uniqueness \nis one of the various formulations of stochastic completeness as we will discuss in the next section.\n\nWe note that although we have a self-adjoint operator $L$ on $\\ell^2(X,m)$ so that we may apply the spectral\ntheorem and \nfunctional calculus to obtain a heat semigroup $e^{-tL}$ for $t \\geq 0$, this semigroup acts on $\\ell^2(X,m)$ and we\nare actually interested in bounded solutions, i.e., solutions in $\\ell^\\infty(X)$. There is a number\nof ways around this. One approach taken in \\cite{KL12} is to extend the heat semigroup on $\\ell^2(X,m)$\nto all $\\ell^p(X,m)$ spaces for $p \\in [1,\\infty]$ via monotone limits. Another approach is via the general\ntheory of Dirichlet forms and interpolation between $\\ell^p(X,m)$ spaces, see \\cite{Dav90, FOT94}.\nWe highlight a slightly different approach\nin that we rather exhaust the graph via finite subgraphs, apply the spectral theorem to each operator\non the finite subgraph in order to get a solution and then take the limit. This rather elementary approach\nhas its roots in \\cite{Dod83} which gave the first construction of the heat kernel\non a general Riemannian manifold without any geodesic completeness assumptions.\n\nA basic tool behind the construction is the following minimum principle.\nWe call a subset $K \\subseteq X$ \\emph{connected} if any two vertices in $K$ can be connected via a\npath that remains within $K$.\n\\begin{lemma}[Minimum principle for the heat equation]\\label{l:minimum_principle}\nLet $G$ be a connected weighted graph and let $K \\subset X$ be a finite connected subset.  \nLet $ T\\ge0 $ and let $ u:X\\times[0,T] \\longrightarrow {\\mathbb R} $ be such that $t \\mapsto u(x,t) $ is continuously \ndifferentiable on $(0,T)$ for every $ x\\in K $ and $u(\\cdot, t) \\in \\mathcal{F}$ for all $t\\in(0,T]$. \nAssume that $u$ satisfies\n\\begin{itemize}\n\\item[(A1)] \\quad $ (\\mathcal{L}+\\partial_t) u\\ge0 $ on $ K\\times (0, T) $\n\\item[(A2)] \\quad $ u\\ge0 $ on $ \\left(X\\setminus K \\times (0,T] \\right) \\cup (K\\times \\{0\\}). $\n\\end{itemize}\nThen, $ u\\ge0 $ on $ K\\times [0,T] $.\n\\end{lemma}\n\\begin{proof}\nSuppose to the contrary that there exists $(x_0,t_0) \\in K \\times [0,T]$ such that $u(x_0,t_0) <0$.\nBy continuity, we can assume that $(x_0,t_0)$ is a minimum for $u$ on $K \\times [0,T]$. \nBy assumption (A2), it follows\nthat $t_0 >0$ so that $\\partial_t u(x_0,t_0) \\leq 0$. Furthermore, by the definition of $\\mathcal{L}$, at a minimum we have\n$\\mathcal{L} u(x_0,t_0) \\leq 0$. Therefore, $(\\mathcal{L} + \\partial_t)u(x_0,t_0) \\leq 0$ and assumption (A1) \ngives $(\\mathcal{L}+\\partial_t) u(x_0,t_0)=0 $ from\nwhich \n$$\\mathcal{L} u(x_0,t_0)= \\frac{1}{m(x_0)}\\sum_{y \\in X} b(x_0,y)(u(x_0,t_0)-u(y,t_0))=0$$ \nfollows. Therefore, since we are at a minimum, we now obtain that\n$$u(y,t_0)=u(x_0,t_0)<0$$\nfor all $y \\sim x_0$. Iterating this argument and using the connectedness of $K$ now gives a contradiction\nto (A2) as $K \\neq X$ and we assume that $G$ is connected.\n\\end{proof}\n\n\\begin{remark}\nWe note that the finiteness of $K$ is not necessary. It suffices to assume that there is at least\none vertex outside of $K$ and that the negative part of $u$\nattains a minimum on $K \\times [0,T]$. The minimum principle then follows with basically the same proof,\nsee, for example, Lemma~3.5 in \\cite{KLW13}. However, assuming the finiteness of $K$ is sufficient\nfor our purposes. For a much more elaborate discrete integrated minimum principle for solutions of the heat \nequation, see Lemma~1.1 in \\cite{Hua12}.\n\\end{remark}\n\nWe now sketch the construction of the minimal bounded solution of the heat equation. \nWe note that if $G$ is not connected, we work on each connected component of $G$ separately.\nThus, for the construction, we can assume without loss of generality that $G$ is connected.\nWe let $(K_n)_{n=0}^\\infty$\nbe an \\emph{exhaustion sequence} of the graph $G$ by which we mean that each $K_n$ is finite and connected,\n$K_n \\subseteq K_{n+1}$ and $X=\\bigcup_n K_n$. For each $n$, we let $L_n$ denote the restriction of $\\mathcal{L}$ to\n$C(K_n)=\\ell^2(K_n,m)$. More precisely, for a function $f \\in C(K_n)$, we extend $f$ by 0 to be defined on all of $X$ and let\n$L_n f(x) = \\mathcal{L} f(x)$ for $x \\in K_n$. Then, $L_n$ is an operator on a finite dimensional Hilbert space and we can define\n$$e^{-tL_n}=\\sum_{k=0}^\\infty \\frac{(-t)^k}{k!}L_n^k$$\nfor $t \\geq 0$. We then define the \\emph{restricted heat kernels} $p_t^n(x,y)$ for $t \\geq 0$ and $x, y \\in K_n$ \nvia\n$$p_t^n(x,y)=e^{-tL_n}\\hat{1}_y(x)$$ \nwhere $\\hat{1}_y = 1_y\/m(y)$.\nIt is immediate that  \n$$u_n(x,t)=e^{-tL_n}u_0(x)=\\sum_{y\\in K_n}p_t^n(x,y)u_0(y)m(y)$$\nsatisfies the heat equation\non $K_n \\times [0,\\infty)$ with initial condition $u_0$. \n\nFurthermore, applying Lemma~\\ref{l:minimum_principle}, gives $0 \\leq p_t^n(x,y)m(y) \\leq 1$ and\n$p_t^n(x,y) \\leq p_t^{n+1}(x,y)$ for all $x,y \\in K_n, t \\geq 0$ and $n \\in {\\mathbb N}$. \nThus, we may take the limit\n$$p_t^n(x,y) \\to p_t(x,y)$$ \nas $n \\to \\infty$  to define $p_t(x,y)$ which is called the \n\\emph{heat kernel} on $G$. \nThen, by applying Dini's theorem and monotone convergence, we can show that\n$$u(x,t)=\\sum_{y \\in X}p_t(x,y)u_0(y)m(y)$$ \nis a bounded solution of the heat equation with initial condition $u_0$ on $G$. \nFor further details and proofs, see Section~2 in \\cite{Woj08} for the case of standard edge weights\nand counting measure. An alternative approach for general graphs involving resolvents is given\nin Section 2 of \\cite{KL12}, in particular, Proposition 2.7. \n\n\\begin{remark}\nThe approach via resolvents in \\cite{KL12}\nis equivalent to the heat kernel approach above via the Laplace transform formulas, that is,\n$$e^{-tL_n}= \\lim_{k\\to \\infty}\\left(\\frac{k}{t} \\left(L_n+\\frac{k}{t}\\right)^{-1}\\right)^k$$\nfor all $t>0$ and\n$$(L_n + \\alpha)^{-1} = \\int_0^\\infty e^{-t\\alpha}{e^{-tL_n}} dt$$\nfor all $\\alpha>0$. Both of these formulas also hold for the Laplacian $L$ defined \non the entire $\\ell^2(X,m)$ space.\nWe further note that $L_n$ is a positive definite operator on $\\ell^2(K_n,m)$\nas can be seen by direct calculation which gives\n$$\\langle L_n f, f \\rangle = \\frac{1}{2}\\sum_{x, y \\in K_n} b(x,y)(f(x)-f(y))^2 \n+ \\sum_{x \\in K_n}f^2(x) \\sum_{y \\not \\in K_n} b(x,y)$$\nfor all $f \\in \\ell^2(K_n,m)$.\n\\end{remark}\n\nWe mention two further properties that follow from the construction and Lemma~\\ref{l:minimum_principle}\nabove. First, the solution $u$ is minimal in the following sense: if $u_0\\geq0$ and $w \\geq 0$ is any\nsolution of the heat equation with initial condition $u_0$, then $u \\leq w$. Secondly, as $0 \\leq \\sum_{y \\in K_n} p_t^n(x,y)m(y) \\leq 1$ for all $n$, we\nget\n$$0 \\leq \\sum_{y \\in X} p_t(x,y)m(y) \\leq 1$$\nby taking the limit $n \\to \\infty$. \nWe will return to the second inequality in the following section.\n\nWe note that the approach above also gives that\nif $f \\in \\ell^\\infty(X)$ with $0 \\leq f \\leq 1$, then\n$$0 \\leq \\sum_{y \\in X} p_t(x,y)f(y)m(y) \\leq 1.$$ \nThis property is referred to by saying that the heat semigroup is \\emph{Markov}. The fact\nthat the semigroup is Markov will\nbe used later in our discussion of curvature on graphs.\n\n\n\n\n\\section{Formulations of stochastic completeness}\\label{s:sc}\n\\subsection{Stochastic completeness and uniqueness of solutions}\nWe have seen that given any bounded function, we can construct a bounded\nsolution of the heat equation with the given function as an initial condition. \nWe now address the uniqueness of this solution.\nIn fact, we will see that the uniqueness is equivalent to the following property.\n\n\\begin{definition}[Stochastic completness]\nLet $G$ be a weighted graph. If for all $x \\in X$ and all $t \\geq 0$, \n$$\\sum_{y \\in X} p_t(x,y)m(y)=1,$$ then \n$G$ is called \\emph{stochastically complete.} Otherwise, $G$ is called \\emph{stochastically incomplete}.\n\\end{definition}\n\n\\begin{remark}\nThere is a short way to state the definition above which we will have recourse to later in our\ndiscussion of curvature. Namely, letting $P_t = e^{-tL}$ denote the heat semigroup\non $\\ell^2(X,m)$ for $t \\geq 0$ defined via the spectral theorem, it follows that $P_t$ can be extended\nto $\\ell^\\infty(X)$, the space of bounded functions, via monotone limits, see Section~6 in \\cite{KL12} where\nthis is actually shown for all $\\ell^p(X,m)$ spaces with $p \\in [1,\\infty]$. In particular, letting $1 \\in \\ell^\\infty(X)$\ndenote the function which is constantly 1 on all vertices, we then have\n$$P_t 1(x) = \\sum_{y \\in X}p_t(x,y)m(y)$$\nfor $x \\in X$. Thus, stochastic completeness can also be written as $P_t 1 =1$ for all $t \\geq 0$.\n\\end{remark}\n\nThe goal of this section is to give a variety of characterizations for this property. We do not aim\nto be exhaustive but rather highlight the characterizations that will be useful later in our presentation.\nIf $v \\in \\mathcal{F}$ satisfies $\\mathcal{L} v = \\lambda v$ for $\\lambda \\in {\\mathbb R}$, then $v$ is called a $\\lambda$-\\emph{harmonic} function.\nIn particular, the theorem below characterizes stochastic completeness in terms of non-existence\nof $\\lambda$-harmonic bounded functions for $\\lambda<0$. This will be used in several places in what follows.\n\n\\begin{theorem}[Characterizations of stochastic completeness]\\label{t:sc_characterizations}\nLet $G$ be a weighted graph. The following statements are equivalent:\n\\begin{itemize}\n\\item[(i)] $G$ is stochastically complete.\n\\item[(ii)] Bounded solutions of the heat equations are uniquely determined by initial conditions.\n\\item[(ii$'$)] The only bounded solution of the heat equation with initial condition $u_0=0$ is $u=0$.\n\\item[(iii)] The only bounded solution to $\\mathcal{L} v = \\lambda v$ for some\/all $\\lambda <0$\nis $v=0$.\n\\item[(iii$'$)] The only non-negative bounded solution to $\\mathcal{L} v \\leq \\lambda v$ \nfor some\/all $\\lambda<0$ is $v=0$\n\\item[(iv)] Every bounded function $v$ with $v^* = \\sup v>0$ satisfies $\\sup_{\\Omega_\\alpha} \\mathcal{L} v \\geq 0$\nfor every $\\alpha < v^*$ where $\\Omega_\\alpha = \\{ x\\in X \\ | \\ v(x) > v^*- \\alpha\\}$.\n\\end{itemize}\n\\end{theorem}\n\\begin{remark}[History and intuition]\nWe give a partial history with references for the equivalences above in various settings. The equivalence of (i),\n(ii), and (iii) for diffusion processes on Euclidean spaces goes back to \\cite{Fel54, Has60}.\nFor manifolds, see Theorem~6.2~and~Corollary~6.3 in \\cite{Gri99} which also gives further\nhistorical references. For Markov processes on discrete spaces, these equivalence go back to \\cite{Fel57, Reu57}. \nFor a proof in the case of graphs with\nstandard edge weights and counting measure, see Theorem~3.1.3 in \\cite{Woj08}. For an extension\nto weighted graphs see Theorem~1~in~\\cite{KL12} which deals also with a more general phenomenon\ncalled stochastic completeness at infinity. This allows for a discussion of these properties\nin the case of operators of the type Laplacian plus a positive potential, see also \\cite{MS} for a \ndiscussion of this property in the case of manifolds.\nCondition (iv) is referred to as a weak Omori-Yau maximum principle after the original\nwork in \\cite{Om67, Yau75}. The equivalence\nof (iv) and stochastic completeness was shown for manifolds in \\cite{PRS03} and for graphs as\nTheorem 2.2 in \\cite{Hua11a}.\n\nWe would also like to mention some of the intuition behind the equivalences. Roughly speaking, as mentioned\nin the introduction and discussed further below,\na large volume growth or curvature decay is required for\nstochastic completeness to fail. Let us discuss how this large volume growth can cause the failure of the\nother properties listed in the theorem above. First, failure of (i) means that\nthe total probability of the process determined by the Laplacian to remain in the graph when\nstarting at a vertex $x$ is less than 1 at some time.\nHence, under a large enough volume growth (or curvature decay) the process can be swept off the graph\nto infinity in a finite time. \nSecond, failure of (ii) means that there exists a non-zero bounded solution of the heat equation with zero\ninitial condition. In other words, a large volume growth can create something out of nothing.\nThird, by looking at the equation $\\mathcal{L} v = \\lambda v$ for $v>0$ and $\\lambda<0$,\nwe see that $v$ must increase at some neighbor of each vertex. Hence, stochastic incompleteness\nmeans that there is a sufficient amount of space in the graph to accommodate this growth while keeping $v$ bounded.\nThis gives the intuition for the failure of condition (iii).\nFinally, $\\mathcal{L} v \\leq -C< 0$ on a set of vertices where $v$ is near its supremum means that there is always more \nroom for $v$ to grow in the graph. This gives the intuition behind the failure of (iv). \n\nTo summarize, in order for any of the four conditions above to fail requires a large amount of\nspace in the graph. Conversely, if there is no large growth, then the process remains in the graph\nand the graph cannot accommodate non-zero bounded solutions to various equations.\nThus, stochastic completeness is also referred to as \\emph{conservativeness}\nor \\emph{non-explosion}.\n\\end{remark}\n\n\\begin{proof}[Sketch of the proof of Theorem~\\ref{t:sc_characterizations}] \nWe now sketch a proof. For full details, please see the references\ngiven in the remark directly above.\nTo show the equivalence between (i) and (ii), observe that both the constant function \n1 and $u(x,t) = \\sum_{y \\in X}p_t(x,y)m(y)$ are bounded solutions of the heat equation with initial condition 1.\nThe equivalence between (ii) and (ii$'$) is shown by taking the difference of two solutions of the heat equation\nwith initial conditions $u_0$. \nThe equivalence between (ii$'$) and (iii) can be established via the fact that if $u$ is a\nbounded solution of the heat equation with initial condition 0, then\n$v(x) = \\int_0^\\infty e^{t\\lambda}u(x,t) dt$ is a bounded $\\lambda$-harmonic function for $\\lambda<0$.\nTo show the equivalence between (iii) and (iii$'$) one can use exhaustion and minimum principle arguments.\nFinally, if (iii) fails and $v$ is a non-trivial bounded $\\lambda$-harmonic function for $\\lambda<0$, then\nletting $\\alpha = \\sup v\/2$, it can be shown that $\\mathcal{L} v \\leq -C < 0$ on $\\Omega_\\alpha$ so that (iv)\nfails.\nConversely, if (iv) fails, then there exists a bounded function $v$ such that $\\mathcal{L}v \\leq -C$ for $C>0$\non $\\Omega_\\alpha$ for some $0< \\alpha<v^*$. \nThen, $w= (v+\\alpha -v^*)_+$ is a non-negative non-trivial bounded function with\n$\\mathcal{L} w \\leq \\lambda w$ for $\\lambda = -C\/\\alpha$. Thus, (iii$'$) fails.\n\\end{proof}\n\n\\begin{remark}\nWe note that connectedness of the graph and the\nsemigroup property can be used to show that if $\\sum_{y \\in X}p_t(x,y)m(y)=1$\nholds for one $x \\in X$ and one $t>0$, then it holds for all $x \\in X$ and all $t > 0$. However, we do not\nrequire connectedness for the equivalence of the properties above as later we will need to consider\na possibly unconnected scenario. Thus, in the definition, we assume that the sum is 1\nfor all $x \\in X$ and all $t \\geq 0$.\n\\end{remark}\n\n\\subsection{A Khas$'$minskii criterion}\nWe will also need another property which implies stochastic completeness. This is sometimes referred to \nas a Khas$'$minskii-type criterion after \\cite{Has60}. The formulation below is Theorem~3.3~in~\\cite{Hua11a}, the\nproof given there uses the weak Omori-Yau maximum principle, that is, condition (iv) in \nTheorem~\\ref{t:sc_characterizations}.\n\\begin{theorem}\\label{t:Khas}\nLet $G$ be a weighted graph. If there exists $v \\in \\mathcal{F}$ which satisfies $v \\geq 0$, $v(x_n) \\to \\infty$\nfor all sequences of vertices with ${\\mathrm{Deg}}(x_n) \\to \\infty$ and\n$$\\mathcal{L} v + f(v) \\geq 0$$\non $X \\setminus K$ where $K\\subseteq X$ is a set such that  ${\\mathrm{Deg}}$ is a bounded function on $K$ and \n$f:[0,\\infty) \\longrightarrow (0,\\infty)$ is an increasing continuously differentiable function with \n$$\\int^\\infty \\frac{1}{f(r)}dr = \\infty,$$\nthen $G$ is stochastically complete.\n\\end{theorem}\n\n\\begin{remark}\nThis formulation of a Khas$'$minskii-type criterion is very precise\nas it involves the weighted degree function\nas well as the function $f$. \nA more general formulation is that the existence of\na function $v$ which satisfies $\\mathcal{L} v \\geq \\lambda v$ outside of a compact set and which goes to infinity \nin all directions implies stochastic\ncompleteness, see Corollary~6.6 in \\cite{Gri99} for the manifold case and Proposition~5.5 in \\cite{KLW13} for \nweighted graphs.\nFurthermore, we note that, in the manifold setting, the equivalence of this formulation and stochastic completeness\nwas shown as Theorem~1.2 in \\cite{MV13}.\n\\end{remark}\n\n\n\n\\section{Boundedness of geometry and of Laplacians}\\label{s:bounded}\n\\subsection{Boundedness of the Laplacian}\nWe now discuss the boundedness of the Laplacian which turns out to be equivalent to boundedness of the\nweighted vertex degree. Furthermore, it turns out that boundedness always implies stochastic completeness.\n\nWe start by a simple observation. We recall that $L$ is the self-adjoint operator on $\\ell^2(X,m)$\nwhich is obtained from the closure of the form $Q_c$ acting on $C_c(X) \\times C_c(X)$ and that \n${\\mathrm{Deg}}(x) = 1\/m(x) \\sum_{y \\in X}b(x,y)$ for $x \\in X$ is the weighted degree of a vertex $x$. \nWe first characterize\nthe boundedness of this operator in terms of the boundedness of the weighted degree function. This\nfact is certainly well-known, see, for example Theorem~11 in \\cite{KL10}.\n\n\\begin{theorem}[Boundedness of $L$]\\label{t:boundedness_l2}\nLet $G$ be a weighted graph. The Laplacian $L$ is a bounded operator on $\\ell^2(X,m)$\nif and only if ${\\mathrm{Deg}}$ is bounded on $X$.\n\\end{theorem}\n\\begin{proof}\nA direct calculation gives \n$$\\langle L1_x, 1_x \\rangle = {\\mathrm{Deg}}(x)m(x)$$\nwhere $1_x$ is the characteristic function of the set $\\{x\\}$ for $x \\in X$.\nNow, the result follows from the general theory of self-adjoint operators on Hilbert space, see for \nexample Theorem~4.4 in \\cite{Weid80}, by noting that $\\{1_x\/\\sqrt{m(x)} \\ | \\ x\\in X\\}$ forms an\northonormal basis for $\\ell^2(X,m)$.\n\\end{proof}\n\nWeighted graphs which satisfy the condition that ${\\mathrm{Deg}}$ is a bounded function are sometimes referred\nto as having \\emph{bounded geometry}. We now have a look at this in the cases most commonly appearing\nin the graph theory literature.\n\n\\begin{example} Let $G$ be a weighted graph.\n\n\\begin{enumerate}\n\\item If $m(x)=\\sum_{y \\in X} b(x,y)$ is the sum of the edge weights, then ${\\mathrm{Deg}}(x)=1$\nfor all $x \\in X$. Thus, in this case, $L$ is always a bounded operator.\n\\item If $G$ has standard edge weights, i.e., $b(x,y)\\in \\{0,1\\}$ for all $x, y \\in X$ and $m$ is the counting\nmeasure, i.e.,  $m(x)=1$ for all $x \\in X$, then\n$${\\mathrm{Deg}}(x) = \\# \\{y \\ | \\ y \\sim x \\}$$\nfor all $x \\in X$. Thus, ${\\mathrm{Deg}}$ is just the usual vertex degree which counts the number of\nneighbors of $x$. We see that $L$ is bounded in this case if and only if there is a uniform upper \nbound on this quantity.\n\\end{enumerate}\n\\end{example}\n\n\\subsection{Boundedness and stochastic completeness}\nWe now discuss the connection between boundedness and stochastic completeness.\nIn particular, we show that if ${\\mathrm{Deg}}$ is bounded on $X$, then the graph is stochastically complete.\nThis follows from a more general result which allows for some growth of the weighted \nvertex degree which we state below.\n\n\\begin{theorem}[Boundedness implies stochastic completeness]\\label{t:bounded_SC}\nLet $G$ be a weighted graph. If for every infinite path $(x_n)_{n=0}^\\infty$ \n$$\\sum_{n=0}^\\infty \\frac{1}{{\\mathrm{Deg}}(x_n)} = \\infty,$$ \nthen $G$ is stochastically complete.\nIn particular, if\n${\\mathrm{Deg}}$ is bounded on $X$, then $G$ is stochastically complete.\n\\end{theorem}\n\\begin{proof}\nBy Theorem~\\ref{t:sc_characterizations}~(iii), it suffices to show that any non-trivial $v \\in \\mathcal{F}$ with \n$\\mathcal{L} v = \\lambda v$ for $\\lambda <0$ is not bounded. Suppose that $v(x_0)>0$ for some $x_0 \\in X$. The equation\n$\\mathcal{L} v(x_0) = \\lambda v(x_0)$ can be rewritten as\n$$\\frac{1}{m(x_0)} \\sum_{y \\in X} b(x_0,y)v(y) = \\left({\\mathrm{Deg}}(x_0)-\\lambda \\right)v(x_0).$$\nHence, there exists $x_1 \\sim x_0$ such that\n$$v(x_1) \\geq \\left( 1 - \\frac{\\lambda}{{\\mathrm{Deg}}(x_0)} \\right) v(x_0).$$\nNow, we iterate this argument to get a sequence of vertices $x_0 \\sim x_1 \\sim x_2 \\ldots$\nsuch that\n\\begin{align*}\nv(x_{n+1}) &\\geq \\left(1 - \\frac{\\lambda}{{\\mathrm{Deg}}(x_n)} \\right)v(x_n) \\\\\n&\\geq \\prod_{k=0}^n \\left(1 - \\frac{\\lambda}{{\\mathrm{Deg}}(x_k)}\\right) v(x_0).\n\\end{align*}\nAs $\\sum_{k=0}^\\infty 1\/{\\mathrm{Deg}}(x_k) = \\infty$ if and only if $\\prod_{k=0}^\\infty (1 - \\lambda\/{\\mathrm{Deg}}(x_k))=\\infty$,\nit follows that $v$ cannot be bounded.\n\\end{proof}\n\n\\begin{remark} \nFor an even shorter proof of the boundedness portion using the Omori-Yau maximum principle, see\nLemma~2.3 in \\cite{Hua11a}. This is then extended to a boundendess of a notion of a global weighted degree\nin Theorem~2.9 in \\cite{Hua11a}. For a more precise result which only considers the maximal outward degree\non spheres in the case of standard edge weights and counting measure, see Theorem~4.2 in \\cite{Woj11}\nand Theorem~5.5 in \\cite{Hua11a}.\n\nWhen $b$ is the standard edge weight and $m$ is the counting measure, the boundendess result\nwas first shown via a minimum principle in \\cite{DM06}. This proof was then extended to the case\nof arbitrary edge weights and counting measure in \\cite{Dod06}.\n\nA more structural proof of the boundedness portion of Theorem~\\ref{t:bounded_SC} can be \nfound as Corollary~27 in \\cite{KL10} and can be described as follows.  It turns out that \nthe boundedness of $L$ on $\\ell^2(X,m)$ also implies\nboundedness of $\\mathcal{L}$ acting on $\\ell^\\infty(X)$. In fact, the \nboundendess of $\\mathcal{L}$ restricted to $\\ell^p(X,m)$ for one $p \\in [1,\\infty]$ implies the boundedness of the restriction\nof $\\mathcal{L}$ to $\\ell^p(X,m)$ for all $p \\in [1,\\infty]$. This was shown via the Riesz-Thorin interpolation\ntheorem as Theorem~9.3 in \\cite{HKLW12} following earlier work presented as Theorem~11 in \\cite{KL10}. Now, \nif $\\mathcal{L}$ gives a bounded operator on $\\ell^\\infty(X)$,\nthen it is clear that the equation $\\mathcal{L} v = \\lambda v$ cannot have a non-zero bounded solution for all $\\lambda <0$. \nThus, by Theorem~\\ref{t:sc_characterizations}~(iii), $G$ is stochastically complete.\n\nWe also note that the lower bound for the $\\lambda$-harmonic function appearing in the proof above \ncan also be used to establish the essential self-adjointness\nof the restriction of $\\mathcal{L}$ to $C_c(X)$. See Proposition~2.2 in \\cite{Gol14} \nor, more generally, Theorem~11.5.2 in \\cite{Schm20}.\n\\end{remark}\n\n\\section{Weakly spherically symmetric graphs, trees and anti-trees}\\label{s:symmetry}\n\\subsection{Weakly spherically symmetric graphs}\nWe now discuss a class of graphs for which we will give a full characterization of stochastic completeness.\nThese are weakly spherically symmetric graphs. They are an analogue to model manifolds which are\nextensively discussed in \\cite{Gri99}. The definition we give here was first presented in \\cite{KLW13}\nand later generalized in \\cite{BG15}.\n\nWe start with some definitions. \nWe assume that $G$ is connected and\nrecall that $d(x,y)$ denotes the combinatorial graph distance between\nvertices $x$ and $y$, that is, the least number of edges in a path connecting $x$ and $y$. For a vertex\n$x_0 \\in X$ and $r \\in {\\mathbb N}_0$, we let $S_r(x_0)$ and $B_r(x_0)$ denote the sphere and ball of radius $r$\nabout $x_0$, that is,\n$$S_r(x_0) = \\{ x \\in X \\ | \\ d(x,x_0)=r \\}$$\nand $ B_r(x_0) = \\bigcup_{k=0}^r S_k(x_0) = \\{ x \\in X \\ | \\ d(x,x_0) \\leq r \\}.$\nTo ensure that these are finite sets, we now assume that $G$ is locally finite.\n\nWe will generally suppress the dependence on $x_0$ and just write $S_r$ and $B_r$.\nWe can then define the \\emph{outer} and \\emph{inner degrees} of a vertex $x \\in S_r$ as\n$${\\mathrm{Deg}}_{\\pm}(x)  =\\frac{1}{m(x)} \\sum_{y \\in S_{r \\pm 1}} b(x,y).$$\nThat is, ${\\mathrm{Deg}}_+(x)$ gives the total edge weight of edges going ``away'' from $x_0$ divided by the vertex\nmeasure while ${\\mathrm{Deg}}_-(x)$ of those going ``back'' towards $x_0$.\n\n\\begin{definition}[Weakly spherically symmetric graphs]\nA locally finite connected weighted graph $G$ is called \\emph{weakly spherically symmetric} \nif there exists a vertex $x_0 \\in X$ such\nthat the functions ${\\mathrm{Deg}}_\\pm$ depend only on the distance to $x_0$. In this case, we will write\n${\\mathrm{Deg}}_{\\pm}(r)$ for ${\\mathrm{Deg}}_{\\pm}(x)$ when $x \\in S_r(x_0)$.\n\\end{definition}\nAgain, although all of the concepts above depend on the choice of $x_0$, we will suppress this dependence\nin our notation.  We note that this notion of symmetry is weak in the sense that we do not assume\nanything about the edge weights between vertices on the same sphere nor do we assume anything about\nthe structure of the connections between vertices on successive spheres.\n\nWe will now state a full characterization for the stochastic completeness of such graphs. In order to do so, we introduce\nthe notion of \\emph{boundary growth} of a ball as\n$$\\partial B(r) = \\sum_{x \\in S_r}\\sum_{y \\in S_{r+1}} b(x,y).$$\nWe note that $\\partial B(r)$ reflects the total edge weight of edges leaving the ball $B_r$. Furthermore, \nfor weakly spherically symmetric graphs, this can be written as\n$$\\partial B(r) = {\\mathrm{Deg}}_+(r)m(S_r)= {\\mathrm{Deg}}_{-}(r+1)m(S_{r+1})$$\nas follows directly from the definitions. \nIn particular, we note that\n$$\\partial B(r)= \\frac{\\partial B(r-1) {\\mathrm{Deg}}_+(r)}{{\\mathrm{Deg}}_-(r)}.$$\nIn what follows, we also let\n$$V(r) = m(B_r)=\\sum_{k=0}^r m(S_k)$$\ndenote the measure of a combinatorial ball of radius $r$.\n\n\\begin{theorem}[Stochastic completeness of weakly spherically symmetric graphs]\\label{t:sc_symmetry}\nIf $G$ is a weakly spherically symmetric graph, then $G$ is stochastically complete if and only if\n$$\\sum_{r=0}^\\infty \\frac{V(r)}{\\partial B(r)} =  \\infty.$$\n\\end{theorem}\nWe give a sketch of the proof. For further details, see the proof of Theorem~5 in \\cite{KLW13}.\nFor standard edge weights and counting measure, this was first shown as Theorem~4.8 in \\cite{Woj11},\nsee also Theorem~5.10 in \\cite{Hua11a} for an alternative proof in this case using the weak Omori-Yau maximum\nprinciple.\n\\begin{proof}\nBy Theorem~\\ref{t:sc_characterizations}~(iii)\nit suffices to show that any bounded solution to $\\mathcal{L} v = \\lambda v$ for $\\lambda<0$ is zero if and only if \n$\\sum_{r=0}^\\infty \\frac{V(r)}{\\partial B(r)} = \\infty.$ By applying the characterization in \nterms of non-negative subsolutions in Theorem~\\ref{t:sc_characterizations}~(iii$'$) and \nthe Khas$'$minskii criterion from Theorem~\\ref{t:Khas}, it suffices to consider only non-negative solutions $v$.\nFinally, by averaging a solution over spheres, it suffices to consider only solutions depending on the\ndistance to $x_0$. \n\nThus, we may write $v(r)$ for $v(x)$ for all $x \\in S_r$ and note that stochastic completeness\nis equivalent to the triviality of $v$ if $v$ is bounded. Now, by induction on $r \\in {\\mathbb N}_0$, it can\nbe shown by using the formulas above that  \n$\\mathcal{L} v(r) = \\lambda v(r)$ if and only if\n$$v(r+1)-v(r) = \\frac{-\\lambda}{\\partial B(r)} \\sum_{k=0}^r v(k)m(S_k).$$\nIn particular, if $v(0)>0$, then $v$ is strictly increasing with respect to $r$. Therefore, we estimate\n$$ -\\frac{\\lambda V(r)}{\\partial B(r)} v(0) \\leq v(r+1)-v(r) \\leq -\\frac{\\lambda V(r)}{\\partial B(r)} v(r)$$\nso that \n$$ v(r) - \\frac{\\lambda V(r)}{\\partial B(r)} v(0) \\leq v(r+1) \\leq \\left(1 - \\frac{\\lambda V(r)}{\\partial B(r)} \\right) v(r).$$\nIterating this down to $r=0$, gives \n$$ -\\lambda \\sum_{k=0}^r \\frac{V(k)}{\\partial B(k)}  v(0) \\leq v(r+1) \\leq \n\\prod_{k=0}^r \\left(1 - \\frac{\\lambda V(k)}{\\partial B(k)} \\right) v(0).$$\nHence, if $v$ is bounded, then $\\sum_{k=0}^\\infty \\frac{V(k)}{\\partial B(k)} <\\infty$.\nOn the other hand, if $v$ is not bounded, then\n$ \\prod_{k=0}^\\infty \\left(1 - \\frac{\\lambda V(k)}{\\partial B(k)} \\right) =\\infty$ which is\nequivalent to\n$\\sum_{k=0}^\\infty \\frac{V(k)}{\\partial B(k)} =\\infty$. This completes the proof.\n\\end{proof}\n\n\\subsection{Trees and anti-trees} \nWe now illustrate the theorem above linking stochastic completeness of weakly spherically symmetric graphs \nand the ratio of the growth of the ball and the boundary of the ball with several examples. \nIn particular, we introduce the class of spherically symmetric trees and anti-trees.\n\nWe start with spherically symmetric trees. \nFor this, we first take standard edge weights and counting measure.\nSuch a graph $G$ is then called a \\emph{spherically symmetric tree} if $G$ contains no cycles\nand there exists a vertex $x_0 \\in X$ such for all $x \\in S_r$\n$${\\mathrm{Deg}}_+(x) = \\# \\{ y \\ | \\  y \\sim x, y \\in S_{r+1} \\}$$ \nonly depends on $r$. Thus, we may write ${\\mathrm{Deg}}_+(x)={\\mathrm{Deg}}_+(r)$ for\nall $x \\in S_r$.  Note that the lack of cycles implies that ${\\mathrm{Deg}}_-(r)=1$ for all\n$r \\in {\\mathbb N}$ so that the number of edges leading back to $x_0$ is minimal in order to \nhave a connected graph.\n\nWe note that for spherically symmetric trees, we have\n$$m(S_r) = \\prod_{k=0}^{r-1} {\\mathrm{Deg}}_+(k)$$ \nand $\\partial B(r) = m(S_{r+1})$ as follows by direct calculations.\nWe now apply our characterization of stochastic completeness of weakly spherically\nsymmetric graphs to the case of such trees.\n\n\\begin{corollary}[Stochastic completeness and spherically symmetric trees]\nIf $G$ is a spherically symmetric tree, then $G$ is stochastically complete\nif and only if\n$$\\sum_{r=0}^\\infty \\frac{1}{{\\mathrm{Deg}}_+(r)} = \\infty.$$\n\\end{corollary}\n\\begin{proof}\nFrom the remarks directly above we obtain\n$$\\sum_{r=0}^\\infty  \\frac{V(r)}{\\partial B(r)} = \\sum_{r=0}^\\infty \\frac{1+\\sum_{k=1}^r \\prod_{j=0}^{k-1} {\\mathrm{Deg}}_+(j)}{ \\prod_{k=0}^{r} {\\mathrm{Deg}}_+(k)}.$$\nBy the limit comparison test, it then follows that the divergence of the series above is equivalent to divergence\nof the series $\\sum_{r=0}^\\infty 1\/{\\mathrm{Deg}}_+(r)$. Thus, the conclusions follows by Theorem~\\ref{t:sc_symmetry}.\n\\end{proof}\n\nThe result above was first presented as Theorem~3.2.1 in \\cite{Woj08}.   \nIt establishes the sharpness of the condition given for stochastic completeness in terms of the weighted\nvertex degree on paths presented in Theorem~\\ref{t:bounded_SC} in the previous section.\n\nWe note that the case of\nspherically symmetric trees already provides a contrast with the manifold case as if we take\n$V(r) = m(B_r)$ to be the counterpart of the volume growth in the Riemannian setting, then there\nexist stochastically incomplete trees with factorial volume growth.\nHowever, a much more striking example is that of anti-trees which we define next.\nThe basic idea is that we choose an arbitrary sequence of natural numbers for the number of \nvertices on the sphere and then connect all vertices between successive spheres. Thus, these\nare the antithesis of trees in the sense that for trees the removal of a single edge between spheres\ncreates a disconnected graphs while for anti-trees one must remove all of the edges between spheres.\n\n\\begin{definition}[Anti-trees]\nLet $(a_r)$ be a sequence with $a_r \\in {\\mathbb N}$ for $r \\in {\\mathbb N}$ and $a_0=1$.\nA graph $G$ is called an \\emph{anti-tree} with sphere growth $(a_r)$ \nif $G$ has standard edge weights and counting measure and the vertex set $X$ can be written\nas a disjoint union $X= \\bigcup_r A_r$ where $m(A_r)=a_r$ and $b(x,y)= b(y,x) = 1$ for all $x \\in A_r, y \\in A_{r+1}$\nfor $r \\in {\\mathbb N}_0$ and zero otherwise.\n\\end{definition}\n\nThus, by the definition of the edge weight, an anti-tree with sphere growth $(a_r)$  satisfies\n$m(S_r)=a_r$ and is weakly spherically symmetric with ${\\mathrm{Deg}}_{\\pm}(r)=a_{r \\pm 1}$. \nFurthermore,\n$\\partial B(r) = a_r a_{r+1}$ as each vertex in the sphere $S_r$ is connected to\nall vertices in the sphere $S_{r+1}$. Therefore, we obtain the following characterization of stochastic completeness\nin the case of anti-trees.\n\\begin{corollary}[Stochastic completeness and anti-trees]\\label{c:antitree_sc}\nIf $G$ is an anti-tree with sphere growth $(a_r)$, then $G$ is stochastically complete\nif and only if\n$$\\sum_{r=0}^\\infty \\frac{\\sum_{k=0}^r a_k}{a_r a_{r+1}} = \\infty.$$\n\\end{corollary}\n\\begin{proof}\nThis follows directly from the definition of an anti-tree and Theorem~\\ref{t:sc_symmetry}.\n\\end{proof}\n\nWe note that if $a_r$ grows like $r^{2+\\epsilon}$ for any $\\epsilon>0$, then the corresponding anti-tree is\nstochastically incomplete. Furthermore, $V(r)$ grows like $r^{3+\\epsilon}$ in this case. Thus, unlike in the case\nof manifolds, for the combinatorial graph metric, there exists stochastically incomplete graphs\nwith polynomial volume growth. We will also see later that these are the smallest such examples.\nThis motivates the move to different graph metrics which take into\naccount not only the combinatorial graph structure but also the vertex degree. \nThese are the so-called intrinsic metrics which we introduce in the next section.\n\nThe result on stochastic incompleteness of anti-trees presented above originally appeared as \nExample~4.11 in \\cite{Woj11}. To the best of our knowledge, the first example of \nan anti-tree in the special case of sphere growth $a_r=r+1$ appears as Example~2.5 in \\cite{DK88}.\nThis anti-tree is a transient graph with the bottom of the spectrum at 0. The same graph\nappears in \\cite{Web10} as an example of a stochastically complete graph with unbounded\nvertex degree.\n\n\n\\section{Intrinsic metrics}\\label{s:intrinsic}\n\\subsection{A brief historical overview}\nAs we have seen, in order to hope for a counterpart for Grigor$'$yan's volume growth result\nfor graphs, we must go beyond the combinatorial graph distance when defining volume growth. In this section we introduce\nthe notion of an intrinsic metric for a weighted graph. This concept arises from Dirichlet form theory.\nAlthough beyond the scope of this article, we mention that the form associated to the Laplacian,\nwhich is a restriction of the graph energy form,\nis a regular Dirichlet form which is not strongly local. For background on Dirichlet forms see \\cite{FOT94},\nfor the connection between graphs and non-local regular Dirichlet forms see \\cite{KL12}.\nFurthermore, let us caution that the notion of an intrinsic metric for a Dirichlet form is distinct from the notion\nof an intrinsic metric in the sense of length spaces as discussed, for example, in \\cite{BBI01}.\n\n\nThe concept of an intrinsic metric for strongly local Dirichlet forms was brought into full fruition\nin \\cite{Stu94}. This allowed for the extension of a variety of results for Riemannian manifolds,\nincluding Grigor$'$yan's volume growth result, to the setting of strongly local Dirichlet forms.\nIn particular, this covers the Riemannian setting as \nthe Riemannian geodesic distance is an intrinsic metric for the strongly local\nDirichlet form arising in the manifold setting.\nHowever, as mentioned above, the energy form of a graph is not strongly local so that the notions of \\cite{Stu94}\ndo not cover the graph setting.\n\nFor non-local Dirichlet forms, such as particular restrictions of the energy form of a graph, the concept of an intrinsic\nmetric was discussed in full generality in \\cite{FLW14}, see also \\cite{MU11} as well as\n\\cite{Fol11, Fol14, GHM12} for the related notion of an adapted metric. \nHowever, as noted in \\cite{FLW14}, the concept\nof an intrinsic metric for a non-local form is more complicated than in the local setting\nas the maximum of two intrinsic metrics is not necessarily an intrinsic metric. This can already be seen\nin an easy example of a graph with three vertices, see Example~6.3 in \\cite{FLW14}. The fact that\nthe maximum of two intrinsic metrics is an intrinsic metric for strongly local Dirichlet forms is essential\nto establish the existence of a maximal intrinsic metric.\n\nThus, there does not exist a maximal intrinsic metric for graphs.\nHowever, for proving statements in graph theory which are analogous to the strongly local setting, \nthe tool of an intrinsic metric is \nquite useful, see the survey article \\cite{Kel15} for an overview of results in this direction and further\nhistorical notes and also \\cite{HH} for some further recent applications.\n\n\\subsection{Intrinsic metrics, combinatorial graph distance and boundedness}\nAfter this brief discussion of the history of intrinsic metrics, we now present the definition for graphs.\nWe call a function mapping pairs of vertices to non-negative real numbers \na \\emph{pseudo metric} if the map is symmetric, vanishes\non the diagonal and satisfies the triangle inequality. In other words, a pseudo metric is a metric\nexcept for the fact that it might be zero for pairs of distinct vertices.\nIn general, intrinsic metrics are only assumed to be pseudo metrics. However, we will follow\nconvention and refer to them as metrics in any case.\n\n\\begin{definition}[Intrinsic metrics, jump size]\nA pseudo metric $\\varrho:X \\times X \\longrightarrow [0,\\infty)$ is called an \\emph{intrinsic metric}\nif\n$$\\sum_{y \\in X} b(x,y) \\varrho^2(x,y) \\leq m(x)$$\nfor all $x \\in X$.\nThe quantity $j = \\sup_{x \\sim y} \\varrho(x,y)$ is called the \\emph{jump size} of $\\varrho$. \nWhen $j < \\infty$, we say that $\\varrho$ has \\emph{finite jump size}.\n\\end{definition}\n\nThe use of intrinsic metrics often lies in a scenario when we want to estimate the energy\nof a cut-off function defined with respect to an intrinsic metric via the measure. In the easiest example,\nlet $\\varrho$ be an intrinsic metric and let $\\rho(x) = \\varrho(x,x_0)$ be the distance with respect to $\\varrho$\nto a fixed vertex $x_0$. If $K \\subseteq X$, then\n\\begin{align*}\n\\sum_{x,y \\in K} b(x,y)(\\rho(x)-\\rho(y))^2 &= \\sum_{x,y \\in K} b(x,y)(\\varrho(x,x_0)-\\varrho(y,x_0))^2\\\\\n&\\leq \\sum_{x \\in K} \\sum_{y \\in K} b(x,y)\\varrho^2(x,y)\\\\\n&\\leq \\sum_{x \\in K} m(x) = m(K).\n\\end{align*}\nIn particular, we see that if $K$ is a set with finite measure, then the energy of $\\rho$ on $K$ is finite.\nThe jump size becomes relevant whenever we have a cut-off function which is supported on $K$\nand we need to control how far outside of the set $K$ the sum above reaches.\n\nAfter this brief discussion, let us mention some examples. We recall that $d$ denotes the combinatorial graph \ndistance, that is, the least number of edges in a path connecting two vertices. The case of when the\ncombinatorial graph distance is equivalent to an intrinsic metric can be characterized in terms\nof the boundedness of the weighted vertex degree.\n\\begin{proposition}\\label{p:intrinsic}\nLet $G$ be a connected weighted graph. The combinatorial graph distance $d$ is equivalent to an \nintrinsic metric if and only if ${\\mathrm{Deg}}$ is a bounded function on $X$.\n\\end{proposition}\n\\begin{proof}\nWe note that $d(x,y)=1$ for all $x \\sim y$. Thus,\n$$\\sum_{y \\in X} b(x,y)d^2(x,y) = \\sum_{y \\in X} b(x,y) = {\\mathrm{Deg}}(x)m(x).$$\nThe conclusion now follows directly.\n\\end{proof}\nRecall that by Theorem~\\ref{t:boundedness_l2} this is the case exactly when the Laplacian is a bounded\noperator and by Theorem~\\ref{t:sc_symmetry}, the graph is stochastically complete in this case. \n\nThus, we see that the combinatorial graph distance may or may not be intrinsic. We now look\nat some further examples. In particular, the first example below gives a case when the combinatorial\ngraph distance is in fact intrinsic and the second gives a pseudo metric which is intrinsic for any given graph.\n\\begin{example}[Intrinsic metrics]\\label{ex:intrinsic}\nWe now give two examples.\n\\begin{enumerate}\n\\item If $m(x)=\\sum_{y\\in X}b(x,y)$, then ${\\mathrm{Deg}}(x)=1$ for all $x \\in X$ and thus the combinatorial\ngraph distance $d$ is equivalent to an intrinsic metric by Proposition~\\ref{p:intrinsic} directly above.\n\\item For a pair of neighboring vertices $x \\sim y$ we let \n$$\\sigma(x,y) = \\left( \\max \\{ {\\mathrm{Deg}}(x), {\\mathrm{Deg}}(y) \\} \\right)^{-1\/2}$$\ndenote the length of the edge connecting $x$ and $y$. Now, we can extend from the length of an edge to the\nlength of a path in a natural way, that is, if $(x_k)=(x_k)_{k=0}^n$ is a path, we let\n$$l_\\sigma((x_k)) = \\sum_{k=0}^{n-1} \\sigma(x_k, x_{k+1})$$\ndenote the length of the path.\nFinally, we define a pseudo metric via\n$$\\varrho_\\sigma(x,y) = \\inf \\{ l_\\sigma((x_k)) \\ | \\ (x_k) \\textup{ is a path connecting } x \\textup{ and } y\\}.$$\nAs $\\varrho_\\sigma(x,y) \\leq \\sigma(x,y)$ for $x \\sim y$ it is then clear that $\\varrho_\\sigma$ is an intrinsic metric as\n$$\\sum_{y \\in X} b(x,y) \\varrho_\\sigma^2(x,y) \\leq \\frac{1}{{\\mathrm{Deg}}(x)} \\sum_{y \\in X} b(x,y) =m(x).$$\nThis metric was first introduced in \\cite{Hua11b}, see also \\cite{Fol11}. \n\nWe note that if the graph is not\nlocally finite, then this intrinsic metric may be only a pseudo metric; however, in the locally finite case,\npath metrics are metrics and give the discrete topology, see for example Lemma~A.3 in \\cite{HKMW13}.\n\\end{enumerate}\n\\end{example}\n\nThe metric $\\varrho_\\sigma$ introduced in the second example above shows that there always exists an intrinsic metric on a weighted\ngraph. This metric, which utilizes the weighted vertex degree function, makes sense in the context\nof the process determined by the heat kernel. Namely, if the Markov process with transition probabilities\ngiven by the heat kernel is a vertex $x$, then at the next jump time it moves with probability $b(x,y)\/\\sum_{z}b(x,z)$\nto a neighbor $y$ of $x$. Furthermore, the wait time at the vertex $x$ is an exponentially distributed\nrandom variable with parameter given by the weighted vertex degree, that is, the probability that \nthe random walker is still at a vertex $x$ after time $t$ without having jumped is given by $e^{-{\\mathrm{Deg}}(x)t}$. \nSee Section~7 in \\cite{KL10} for a further discussion of the connection between the heat semigroup and Markov\nprocesses.\nTherefore, at vertices with a large vertex degree, the process accelerates and thus will more quickly explore\nneighboring vertices. Hence, for the process, neighbors of vertices of large vertex degree are close\nwhich is consistent with the values of $\\varrho_\\sigma$.\n\n\nWe note that given an intrinsic metric $\\varrho$, we can always obtain an intrinsic metric of small\njump size merely by cutting from above. That is, if $\\varrho$ is an intrinsic metric and $C>0$, then \n$$\\varrho_C(x,y) = \\min \\{ \\varrho(x,y), C\\}$$\nis also intrinsic with jump size at most $C$. On the other hand, having a uniform lower bound from below\non the distance between neighbors is equivalent to bounded geometry as we now show.\n\\begin{proposition}\\label{p:intrinsic_2}\nLet $G$ be a weighted graph. There exists an intrinsic metric $\\varrho$ such that $\\varrho(x,y) \\geq C>0$\nfor all $x \\sim y$ if and only if ${\\mathrm{Deg}}$ is a bounded function on $X$.\n\\end{proposition}\n\\begin{proof}\nIf $\\varrho$ is an intrinsic metric with $\\varrho(x,y) \\geq C>0$ for all $x\\sim y$, then\n$$C^2 {\\mathrm{Deg}}(x)m(x) \\leq \\sum_{y \\in X} b(x,y)\\varrho^2(x,y) \\leq m(x)$$\nso that ${\\mathrm{Deg}}$ is bounded. \nConversely, if ${\\mathrm{Deg}}$ is bounded, then by Proposition~\\ref{p:intrinsic},\nit follows that the combinatorial graph distance is equivalent to an intrinsic metric $\\varrho$. In particular, there\nexists a constant $C>0$ such that $C d(x,y) \\leq \\varrho(x,y)$ for an intrinsic metric $\\varrho$. As $d(x,y)=1$\nfor all $x \\sim y$, the conclusion follows.\n\\end{proof}\n\nThus, we obtain two conditions involving the existence of intrinsic metrics which imply stochastic completeness.\n\\begin{corollary}[Stochastic completeness and intrinsic metrics]\nLet $G$ be a connected weighed graph. If either the combinatorial graph distance is equivalent\nto an intrinsic metric or if there exists an intrinsic metric which is uniformly bounded below\non neighbors, then $G$ is stochastically complete.\n\\end{corollary}\n\\begin{proof}\nThis follows immediately by combining Propositions~\\ref{p:intrinsic}~and~\\ref{p:intrinsic_2}\nwith Theorem~\\ref{t:bounded_SC}. \n\\end{proof}\n\n\\subsection{A word about essential self-adjointness and metric completeness}\\label{ss:metric_esa}\nWe briefly mention here some additional facts about metrics, geometry and analysis.\nIn Riemannian geometry, there is the famous Hopf-Rinow theorem, which gives a connection between\nmetric completeness, geodesic completeness and compactness of balls defined with respect to the geodesic metric,\nsee \\cite{doCar92} for example.\nFor locally finite graphs, a counterpart is shown in \\cite{HKMW13}, \nsee also \\cite{KM19} for a recent extension to a more general class of graphs. More specifically,\nTheorem A.1 in \\cite{HKMW13} shows that\nfor locally finite graphs and path metrics, the notions of metric completeness, geodesic completeness\nin the sense that all infinite geodesics have infinite length, and finiteness of balls are equivalent.\nFurthermore, if an intrinsic path metric on a locally finite graph\nsatisfies any of these equivalent conditions, then the restriction of the formal\nLaplacian to the finitely supported functions is essentially self-adjoint, see Theorem~2 in \\cite{HKMW13}. \nThus, metric completeness with respect to an intrinsic metric\nimplies that there exists a unique Laplacian, at least for locally finite graphs. \nThis corresponds to results known for Riemannian manifolds, see \\cite{Che73, Stri83}.\nFor a more general and thorough discussion which includes this question for magnetic Schr{\\\"o}dinger operators on graphs\nsee \\cite{Schm20}.\n\nSubsequently, the assumption that there exists an intrinsic metric for which balls are finite has\noften been used as a substitute for geodesic completeness in the graph setting. In particular, we will\nsee this assumption appearing in our criteria for stochastic completeness in the following sections.\n\n\n\\section{Uniqueness class, stochastic completeness and volume growth}\\label{s:volume}\nIn this section, we will discuss the connections between uniqueness class results for the heat\nequation, stochastic completeness and volume growth. As we have seen, using the combinatorial\ngraph metric gives a very different volume growth borderline for stochastic completeness compared to the manifold\nsetting. We will see that the results in these two settings can ultimately be reconciled via the use of intrinsic metrics.\n\n\\subsection{Uniqueness class}\nWe start by recalling Grigor$'$yan's uniqueness\nclass result on Riemannian manifolds. \nSpecifically, if $u$ is a solution of the heat equation on a Riemannian manifold with zero\ninitial condition and if there\nexists a monotone increasing function $f$ on $(0,\\infty)$ such that $\\int^\\infty {r}\/{f(r)}dr=\\infty$\nwhich dominates the growth of $u$ in the sense that\nfor all $r$ large\n$$\\int_0^T \\int_{B_r} u^2(x,t)\\  d\\mu \\ dt \\leq e^{f(r)},$$\nthen $u=0$, see Theorem~9.2 in \\cite{Gri99} or Theorem~11.9 in \\cite{Gri09} for a proof. \nBy Theorem~\\ref{t:sc_characterizations} above\nthis immediately implies stochastic completeness under a suitable volume growth restriction.\nMore precisely, if $u$ is a bounded solution of the heat equation\nwith bound given by $C$, then letting $f(r)=\\log(C^2 T V(r))$ shows that $u$ must be zero\nwhenever\n$$\\int^\\infty \\frac{r}{\\log V(r)}dr=\\infty.$$\nThus, the only bounded solution of the heat equation with trivial initial condition is trivial.\n\nHowever, in \\cite{Hua11b, Hua12}, there is already a counterexample to an analogue of this \nuniqueness class result when using intrinsic metrics. Namely, for the graph with\n$X={\\mathbb Z}$, \n$$b(x,y)=\\begin{cases}\n1 & \\textup{ if } |x-y|=1 \\\\\n0 & \\textup{ otherwise}\n\\end{cases}$$\nand counting measure $m=1$, there exists an explicit function $u$ which is non-zero, satisfies the\nheat equation with initial condition $0$ as well as the estimate\n$$\\int_0^T \\sum_{x \\in B_r}u^2(x,t) dt \\leq  e^{f(r)}$$\nfor all large $r$ with $f(r)=\\log T + C r \\log r$ for some constant $C$. \nIn particular, it is clear that\n$$\\int^\\infty \\frac{r}{f(r)}dr=\\infty.$$\nThus, no analogue to Grigor$'$yan's uniqueness class result can hold for all graphs, even when using\nintrinsic metrics.\n\nWe note that for this graph ${\\mathrm{Deg}}(x)=2$ for all $x \\in X$\nand thus this graph is stochastically complete by Theorem~\\ref{t:bounded_SC}. Therefore, this non-zero solution\ncannot be bounded by Theorem~\\ref{t:sc_characterizations}. Furthermore, we note by Proposition~\\ref{p:intrinsic}\nthat the combinatorial graph metric is equivalent to an intrinsic metric in this case and that the volume\ngrowth with respect to this metric is only quadratic. Finally, the constant $C$ appearing in the definition of $f$ \nin the example above is crucial\nas for $C<1\/2$, the uniqueness class result holds for all graphs, see Theorem~0.8 in \\cite{Hua12}. \n\nThe difference between these results in the discrete and continuous settings \nwas ultimately resolved in \\cite{HKS} by introducing a class of graphs for which a uniqueness\nclass result analogous to Grigor$'$yan's does hold.\nThese are the globally local\ngraphs which we introduce next. The idea for these graphs in the context of\na uniqueness class result is that the growth of the solution of the heat equation is balanced\nby a decay in the jump size of an intrinsic metric as we go further out in the graph.\nWe note that in the counter example to the uniqueness class result above\nwe use the combinatorial graph distance whose jump size is always one and thus does not decay.\n\n\\begin{definition}[Globally local graphs]\nLet $G$ be a weighted graph with a pseudo metric $\\rho$. Let $B_r$ denote the ball of radius\n$r$ with respect to $\\rho$ and let $j_r$ denote the jump size of $\\rho$ outside of $B_r$, that is,\n$$j_r = \\sup \\{\\rho(x,y) \\ | \\ x \\sim y, \\ x,y \\not \\in B_r\\}.$$\n$G$ is said to be \\emph{globally local} in $\\rho$ with respect to a monotone increasing function $f:(0,\\infty) \\longrightarrow (0,\\infty)$\nif $G$ has finite jump size, i.e., $j_0<\\infty$ and if there exists a constant $A>1$ such that\n$$\\limsup_{r \\to \\infty} \\frac{j_r f(Ar)}{r} < \\infty.$$\n\\end{definition}\n\nThus, globally local graphs not only have finite jump size but provided that $f$ has a certain growth,\nthe jump size must decay outside of balls. In the borderline case for the uniqueness class result, \n$f$ is of the order $r \\log r$ so that\n$j_r$ must take care of the growth of $\\log r$. We note, in particular, that the example mentioned above,\nthat the combinatorial metric used there as an intrinsic metric will not be globally local\nwith respect to $f(r)=r \\log r$.\n\nIn any case, with this notion of globally local graphs, Theorem~1.3 in \\cite{HKS} presents the following result.\n\\begin{theorem}[Uniqueness class for globally local graphs]\\label{t:uniqueness_class}\nLet $G$ be a weighted graph. Let $\\varrho$ be an intrinsic metric with finite balls $B_r^\\varrho$ and assume\nthat $G$ is globally local in $\\varrho$ with respect to a monotone increasing function $f:(0,\\infty) \\longrightarrow (0,\\infty)$\nsuch that\n$$\\int^\\infty \\frac{r}{f(r)}dr=\\infty.$$\nIf $u:X \\times [0,T] \\longrightarrow {\\mathbb R}$ is a solution of the heat equation with initial condition $0$ and\n$$\\int_0^T \\sum_{x \\in B_r^\\varrho} u^2(x,t) m(x) dt \\leq e^{f(r)}$$\nfor all $r >0$, then $u=0$.\n\\end{theorem}\nThe proof of Theorem~\\ref{t:uniqueness_class} can be found in Section~2 of \\cite{HKS}. Though rather\nlong and technical, the main\nidea is to estimate the size of a solution of the heat equation over a small ball at some time via the size of the\nsolution over a larger ball at an earlier time. Then one iterates this estimate down to time zero to show\nthat the solution must be trivial. Along the way, the use of cut-off functions involving intrinsic metrics\nis crucial. This is, in part, because of the fact that in the discrete setting there is no good substitute\nfor the chain rule which is used throughout the proof of Grigor$'$yan's uniqueness class\nresult for manifolds.\n\n\\subsection{Stochastic completeness and volume growth in intrinsic metrics}\nWe want to use the uniqueness class result above to establish stochastic completeness under a volume\ngrowth restriction which is valid for all graphs which allow for an intrinsic metric with finite distance balls. \nHowever, we note that the uniqueness class result above involves the additional assumption of being\nglobally local. Thus, some additional considerations are required in order\nto reduce from general graphs to the class of globally local ones.\n\nAs a first step, it turns out that one can reduce to the case of an intrinsic metric with finite jump size \nvia the notion of truncating the edge weights which\nis already contained in \\cite{GHM12}, see also \\cite{MU11}. \nMore specifically, if $\\varrho$ is an intrinsic metric for $G=(X,b,m)$, we define\nnew edge weights on $X$ via \n$$b_s(x,y) = \\begin{cases}\nb(x,y) & \\textup{ if } \\varrho(x,y) \\leq s \\\\\n0 & \\textup{ otherwise.}\n\\end{cases}$$\nThat is, we remove edges for which the distance between the adjacent vertices is large.\nIt follows that $\\varrho$ is also intrinsic for $G_s=(X,b_s,m)$ and $\\varrho$ now has finite jump size \nof at most $s$ on $G_s$.\nFurthermore, Lemma~3.4 in \\cite{HKS} gives that if $G_s$ is stochastically complete, then $G$ is stochastically\ncomplete. We note that $G_s$ is not necessarily connected even if we start with a connected graph; \nhowever, the equivalent notions of stochastic \ncompleteness presented in Theorem~\\ref{t:sc_characterizations} do not require connectedness of the graph.\nAs an alternate viewpoint, one may apply them on connected components of the graph. \nIn particular, Lemma~3.4 in \\cite{HKS}\nuses the weak Omori-Yau characterization of stochastic completeness, that is, condition (iv) in \nTheorem~\\ref{t:sc_characterizations} above to establish the result. \nSee also Theorem~2.2 in \\cite{GHM12} for a more general\nstatement involving Dirichlet forms associated to general jump processes. \n\nThus, without loss of generality, we may assume that the intrinsic metric has finite jump size. \nThe assumption of finite jump size along with finiteness of balls is easily seen to imply\nlocal finiteness of the graph, see, for example Lemma~3.5 in \\cite{Kel15}. Thus, we have reduced\nto the case of locally finite graphs with finite jump size and finite distance balls.\n\nFinally to reduce to the case of globally local graphs, the authors of \\cite{HKS} use the notion of refinements\nfor locally finite graphs found in \\cite{HS14}. The idea is to insert additional vertices within edges and extend the definitions\nof the edge weights, vertex measure and the intrinsic metric in such a way that both the finiteness of balls\nis preserved and that the measure of balls is only rescaled by a constant. Furthermore, as the inserted vertices\nare now closer together with respect to the new intrinsic metric, it follows by Lemma~3.3 in \\cite{HKS} \nthat this can be done in such a way that\nthe refined graph is globally local with respect to an arbitrarily chosen function. Finally, Theorem~1.5 in \\cite{HKS}\nshows that stochastic completeness is preserved during the process of refining the graph via the use\nof the weak Omori-Yau maximum principle.\n\nPutting everything together, we get the following analogue to Grigor$'$yan's volume growth result which\ncan be found as Theorem~1.1 in \\cite{HKS}.\n\\begin{theorem}[Volume growth and stochastic completeness]\\label{t:sc_volume1}\nLet $G$ be a weighted graph with an intrinsic metric $\\varrho$ with finite distance balls $B_r^\\varrho$.\nLet $V_\\varrho(r) = m(B_r^\\varrho)$ and let $\\log^{\\#}(x) = \\max\\{\\log(x), 1\\}$. If\n$$\\int^\\infty \\frac{r}{\\log^{\\#} V_\\varrho(r)}dr =\\infty,$$\nthen $G$ is stochastically complete.\n\\end{theorem}\n\\begin{proof}[Sketch of proof]\nFrom the discussion above, we can reduce to the case of finite jump size and finite balls and, thus, \nto locally finite graphs. In this case, the technique of refinements allows us to reduce to the case of graphs which are\nglobally local with respect to $f(r)=\\log^{\\#}V_\\varrho(r)$. Thus, given a bounded solution of the heat\nequation $u$ with initial condition 0 and bound $C$, we obtain\n$$\\int_0^T \\sum_{x \\in B_r^\\varrho}u^2(x,t) m(x) dt \\leq C^2T e^{f(r)} = e^{f(r)+C_1}$$\nfor some constant $C_1$.\nTherefore, by Theorem~\\ref{t:uniqueness_class}, we obtain that $u=0$ and by Theorem~\\ref{t:sc_characterizations}~(ii$'$)\nwe get that $G$ is stochastically complete.\n\\end{proof}\n\n\\begin{remark}\n\\begin{enumerate}\n\\item We note that the use of $\\log^{\\#}$ instead of just $\\log$ is to deal with the case when the measure\n$m$ is small. In particular, this covers the case when the entire vertex set has finite measure, that\nis, $m(X)<\\infty$.\nIn this case, stochastic completeness is actually equivalent to two other properties, namely, \nto recurrence and to form uniqueness, see Theorem~16\nin \\cite{Schm17} or Theorem~7.1 in \\cite{GHKLW15} for further details.  Thus, we see that in the case of finite measure,\nthe existence of an intrinsic metric which gives finite balls implies all three of these properties. A partial\nconverse to this result was recently proven in \\cite{Puch}. More specifically, if a graph is recurrent, then \nthere exists a finite measure and an intrinsic metric which has finite distance balls.\nFor a precise statement, see Theorem~11.6.15 in \\cite{Schm20}.\n\n\\item In the case of locally finite graphs with counting measure and an intrinsic metric with finite jump size\nand finite balls, \nTheorem~\\ref{t:sc_volume1} was first\nshown as Theorem~1.2 in \\cite{Fol14}. However, not only does the formulation have additional assumptions\non the graph,\nbut the proof is quite different from Grigor$'$yan's original proof on manifolds. Namely, the approach in \\cite{Fol14} is to synchronize the random\nwalk on the discrete graph with a random walk on a metric graph. Metric graphs are graphs where edges are \nintervals of real numbers. In particular, the energy form on a metric graph is a strongly local Dirichlet form. \nThus, the extension of Grigor$'$yan's result to strongly local Dirichlet forms shown in \\cite{Stu94} implies\nstochastic completeness given that the volume growth of the discrete graph is comparable to the volume\ngrowth of the metric graph. We note that the proof in \\cite{Stu94} also does not invoke the heat equation.\n\nAn analytic proof of the result in \\cite{Fol14} using the Omori-Yau maximum principle and \nwhich allows for an arbitrary vertex measure but still uses metric graphs and assumes local finiteness\nand finite jump size can be found in \\cite{Hua14b}. Thus,\nwhile an analogue of Grigor$'$yan's result was known for some classes of graphs since \\cite{Fol14}, the paper\n\\cite{HKS} contains the first proof which does not assume finite jump size nor local finiteness and does not\ninvoke metric graphs.\n\\end{enumerate}\n\\end{remark}\n\n\\subsection{Stochastic completeness and volume growth in the combinatorial graph metric} \nWe now briefly discuss how in the case of standard edge weights and counting measure the paper \\cite{GHM12}\nalready contains the optimal growth rate result for stochastic completeness when using the combinatorial\ngraph distance. More specifically, the authors of \\cite{GHM12} first use the method of \\cite{Dav85} from the\nstrongly local setting to establish that the volume growth condition\n$$\\liminf_{r \\to \\infty} \\frac{\\log V_{\\varrho_1}(r)}{r \\log r}<\\frac{1}{2}$$\nimplies stochastic completeness of general jump processes, see also \\cite{MUW12}\nwhere it is shown that the $1\/2$ on the right hand side can be replaced by $\\infty$. \nHere, the volume growth is defined with\nrespect to what the authors of \\cite{GHM12} call an \\emph{adapted} metric. \nThe idea for an adapted metric in \\cite{GHM12} is that the intrinsic condition\nneeds to be only satisfied for pairs of vertices that are close with respect to $\\varrho$, that is, \none truncates the metric before imposing the intrinsic condition.\nMore specifically, letting $\\varrho$ be a pseudo metric and $\\varrho_1 =\\min \\{ \\varrho, 1\\}$, then\n$\\varrho_1$ must satisfy\n$$\\sum_{y \\in X}b(x,y)\\varrho_1^2(x,y) \\leq m(x)$$\nfor all $x \\in X$ in order for $\\varrho$ to be called adapted.\n\nClearly any intrinsic metric is adapted. \nOn the other hand, we can also modify Example~\\ref{ex:intrinsic}~(2) in a natural way\nto get an adapted metric. \nMore specifically, by defining the length of an edge now by \n$$\\sigma_1(x,y) = \\left( {\\mathrm{Deg}}(x) \\vee {\\mathrm{Deg}}(y) \\right)^{-1\/2} \\wedge 1$$ \nwhere $a \\vee b = \\max \\{a, b \\}$ and $a \\wedge b = \\min \\{a,b\\}$ \nthen we can extend to paths to get an adapted metric $\\varrho_{\\sigma_1}$. Then, for locally finite graphs,\nCorollary~4.3 in \\cite{GHM12} gives that if $\\inf_{x \\in X} m(x)>0$ and\n$$V_{\\varrho_{\\sigma_1}}(r) \\leq e^{Cr}$$ \nfor $C>0$ and all large $r$, then $G$ is stochastically complete. \nWe note that the additional assumption that $\\inf_{x \\in X} m(x)>0$\nis used along with the volume growth assumption to show that balls defined with respect to $\\varrho_{\\sigma_1}$ are finite.\nThus, these assumptions are used as a replacement for the finiteness of balls assumption found \nin Theorem~\\ref{t:sc_volume1} above.\n\nAlthough clearly not optimal when compared with Theorem~\\ref{t:sc_volume1},\nit turns out that this volume growth result already gives an optimal volume growth condition in the case of standard edge\nweights, counting measure and combinatorial growth distance. More specifically, letting $B_r$ denote the ball\nof radius $r$ defined with respect to the combinatorial graph metric $d$ and $V(r)=m(B_r)$ which \nis the number of vertices in the ball in this case, \nTheorem~1.4 in \\cite{GHM12} gives the following result.\n\n\\begin{theorem}\\label{t:sc_volume_combinatorial}\nLet $G$ be a graph with standard edge weights and counting measure.\nIf \n$$V(r) \\leq C r^3$$\nfor some constant $C>0$ and all large $r$, then $G$ is stochastically complete.\n\\end{theorem}\n\\begin{proof}[Idea of proof]\nIt can be shown that under the assumption $V(r) \\leq C r^3$, there\nare sufficiently many vertices with small degree so that a ball defined with respect $\\varrho_{\\sigma_1}$ is contained\nin a ball of larger radius with respect to the combinatorial graph distance $d$. \nIn particular, the volume growth restriction on $V(r)$ can be used to get a volume growth restriction\non $V_{\\varrho_{\\sigma_1}}(r)$ and then stochastic completeness follows from the volume growth\ncriterion for $V_{\\varrho_{\\sigma_1}}(r)$ mentioned above.\nFor more details, see Section~4\nin \\cite{GHM12}.\n\\end{proof}\n\nWe note that the characterization of stochastic completeness of anti-trees provided in Corollary~\\ref{c:antitree_sc} above\ngives the sharpness of Theorem~\\ref{t:sc_volume_combinatorial} as for an anti-tree with sphere growth \n$a_r$ of the order $r^{1+\\epsilon}$ for any $\\epsilon>0$, we have that $V(r)$ grows like $r^{3+\\epsilon}$ and that the anti-tree is stochastically incomplete. Furthermore, in this case the weighted degree of vertices grows like\n$r^{2+\\epsilon}$ so that balls defined with respect to $\\varrho_{\\sigma_1}$ are not finite and thus the graph is not geodesically\ncomplete. Finally, we note that\nTheorem~\\ref{t:sc_volume_combinatorial} remains valid in the more general setting where the edge weights\nand vertex measure satisfy\n$$b(x,y) \\leq C m(x)m(y)$$\nfor all $x, y \\in X$ and some $C>0$, see Remark~4.4 in \\cite{GHM12}. This type of assumption is sometimes\ncalled an \\emph{ellipticity condition} on graphs, for more details and some applications of this condition\nin the context of curvature see \\cite{KM}.\n\n\\section{Stochastic completeness and curvature}\\label{s:curvature}\nIn recent years there has been a surge of interest in various notions of curvature in the discrete setting.\nWe do not even attempt to give a comprehensive overview of definitions nor of results. We rather confine\nourselves to two of the most prominent definitions of curvature and discuss results which relate curvature and stochastic \ncompleteness. The two notions of curvature that we will discuss are that of Bakry--{\\`E}mery \nwhich arises from the $\\Gamma$-calculus as outlined in \\cite{BE85}\nand Ollivier Ricci which arises from optimal transportation theory and is defined for general Markov\nchains in \\cite{Oll09}. \nWe will briefly introduce each\nand present the relevant results for stochastic completeness.\n\nWe note that, unlike in the manifold case where the Bishop--Gromov inequality relates lower bounds on Ricci curvature\nto volume growth, there is no analogous connection between lower curvature bounds and volume growth in\nfull generality in the discrete setting thus far. \nTherefore, we are not able to relate the \nvolume growth criteria for stochastic completeness presented in the previous section to the curvature\nconditions given in this section.\nHowever, for Bakry--{\\'E}mery curvature, see Theorem~1.8 in \\cite{HM} for some recent progress in connecting \ncurvature and volume growth for a specific class of graphs and Theorem~4.1 in \\cite{M} for\na connection between lower curvature bounds and the volume doubling property for finite\ngraphs. Furthermore, \\cite{AS} establishes comparisons between averaged inner and outer\ndegrees and volume growth and also gives an example of two graphs \nwith equal Olliver Ricci curvatures but different volume growths. \n\n\\subsection{Bakry--{\\'E}mery curvature and stochastic completeness}\nWe start with Bakry--{\\'E}mery curvature. This notion has origins in work on hypercontractive\nsemigroups found in \\cite{BE85}. For early manifestations in the graph setting, see \\cite{LY10,Schm96}. \nWe caution the reader at the outset that in the curvature on graphs community, one usually takes the Laplacian\nwith the opposite sign of ours.\n\nWe first introduce the $\\Gamma$-calculus. In order to take care of convergence of sums, we now assume\nthat all graphs are locally finite. In this case, we note that the domain of the formal Laplacian is the set of all functions\non $X$, that is, $\\mathcal{F} = C(X)$. For $f, g \\in C(X)$ and $x \\in X$, we let\n$$\\Gamma(f,g)(x) = -\\frac{1}{2} \\left(\\mathcal{L}(fg)-f \\mathcal{L} g - g \\mathcal{L} f\\right)(x).$$\nWe will follow convention and write $\\Gamma(f)$ for $\\Gamma(f,f)$. \nBy a direct calculation we then obtain\n$$\\Gamma(f)(x)=\\frac{1}{2m(x)}\\sum_{y \\in X}b(x,y)(f(x)-f(y))^2.$$\nIn particular, if $\\Gamma(f)=0$, then $f$ is constant on any connected component\nof the graph. In some sense, $\\Gamma$ can be thought of as an analogue to the norm squared of the gradient\nfrom the continuous setting.\n\nWe then define\n$$\\Gamma_2(f)= -\\frac{1}{2}\\mathcal{L} \\Gamma(f) + \\Gamma(f, \\mathcal{L} f).$$\nWith these notations $G$ is said to satisfy\n$CD(K,\\infty)$ at $x \\in X$ for $K \\in {\\mathbb R}$ if\n$$\\Gamma_2(f)(x) \\geq K \\Gamma(f)(x)$$\nfor all $f \\in C(X)$. The\n$CD(K,\\infty)$ condition on $G$ is then just the fact\nthat $G$ satisfies the conditions at all $x \\in X$.\n\n\\begin{definition}\nA locally finite weighted graph $G$ is said to satisfy\n$CD(K,\\infty)$ for $K \\in {\\mathbb R}$ if\n$$\\Gamma_2(f) \\geq K \\Gamma(f)$$\nfor all $f \\in C(X)$.\n\\end{definition}\n\nThe idea of the definition is to mimic the inequality obtained via Bochner's formula and a lower\nRicci curvature bound in the Riemannian manifold setting. The number $K$ is then thought\nto be a lower curvature bound on the graph.\n\nWe now work towards giving criteria for stochastic completeness involving Bakry--{\\'E}mery curvature\nconditions. We start with the main result found as Theorem~1.2 in \\cite{HL17} which gives stochastic completeness\nunder the condition $CD(K,\\infty)$, finiteness of distance balls and a uniform lower bound on the measure. \nWe recall that we denote the heat semigroup by \n$$P_t=e^{-tL}$$\nfor $t \\geq 0$. This semigroup is originally defined on $\\ell^2(X,m)$ via the spectral theorem and can then be extended to\nall $\\ell^p(X,m)$ spaces for $p \\in [1,\\infty]$ via monotone limits. \nThe heat semigroup is also strongly continuous, that is,\n$P_t f \\to f$ as $t \\to 0^+$ for all $f \\in \\ell^\\infty(X)$.\nFurthermore, $P_t$ is Markov, specifically,\nfor $0 \\leq f \\leq 1$, we have $0 \\leq P_t f \\leq 1$. Stochastic completeness is then equivalent to the \nfact that $P_t 1 =1$ for all $t \\geq 0$. \n\n\\begin{theorem}[Stochastic completeness and Bakry--{\\'E}mery curvature]\\label{t:sc_BE}\nLet $G$ be a locally finite graph connected with $\\inf_x m(x) >0$ \nand an intrinsic metric $\\varrho$ with finite distance balls.\nIf $G$ satisfies $CD(K,\\infty)$, then $G$ is stochastically complete.\n\\end{theorem}\n\\begin{proof}[Idea of proof]\nThe bulk of the work is in showing that $CD(K,\\infty)$ is actually equivalent to the \nfollowing gradient estimate on the heat semigroup\n$$ \\Gamma (P_t \\varphi) \\leq e^{-2Kt}P_t(\\Gamma(\\varphi))$$\nfor all $\\varphi \\in C_c(X)$ and $t \\geq 0$, see Theorem~4.1 in \\cite{HL17}. \nFurthermore, the finiteness of balls\nwith respect to an intrinsic metric allows for the construction of a sequence $\\varphi_n \\in C_c(X)$ such that\n$0 \\leq \\varphi_n \\leq 1$, $\\varphi_n(x) \\to 1$ as $n \\to \\infty$ for every $x \\in X$ and such that\n$$\\Gamma(\\varphi_n) \\leq \\frac{1}{n}$$\nfor all $n \\in {\\mathbb N}$.\nIn particular, one lets\n$$\\varphi_n(x) = \\left(\\frac{2n-\\varrho(x, x_0)}{n}\\right) \\vee 0 \\wedge 1$$\nwhere $x_0 \\in X$ is an arbitrary vertex, $a\\vee b = \\max \\{a,b\\}$ and $a\\wedge b = \\min\\{a, b\\}$.\nIt is then clear from the definition that $0 \\leq \\varphi_n \\leq 1$, $\\varphi_n =1$ on $B_n$, $\\varphi_n$ is supported\non $B_{2n}$, and thus $\\varphi_n \\in C_c(X)$ by the assumption of finite distance balls,\nand using the fact that $\\varrho$ is intrinsic, a direct calculation gives $\\Gamma(\\varphi_n) \\leq \\frac{1}{n}$.\n\nAs the semigroup is Markov on $C_c(X)$, we then obtain \n$$P_t (\\Gamma(\\varphi_n)) \\leq \\frac{1}{n}.$$\nGiven these ingredients, the proof is now straightforward as\n$P_t \\varphi_n(x) \\to P_t 1(x)$ for all $x \\in X$ by the monotone convergence theorem and thus\n\\begin{align*}\n\\Gamma(P_t 1)(x) &= \\lim_{n \\to \\infty} \\Gamma(P_t \\varphi_n)(x) \\leq \\liminf_{n \\to \\infty} e^{-2Kt}P_t(\\Gamma(\\varphi_n))(x) \\\\\n&\\leq \\liminf_{n \\to \\infty} e^{-2Kt} \\frac{1}{n} = 0\n\\end{align*}\nfor all $x \\in X$. \nThus, $P_t 1$ is constant for every $t \\geq 0$. \nFrom the heat equation we obtain\n$$\\partial_t P_t 1 = - \\mathcal{L} P_t 1 =0$$\nfor all $t >0$.\nAs $P_0 1 =1$ and the semigroup is strongly continuous\nwe obtain that $P_t 1 = 1$ for all $t \\geq 0$.\n\\end{proof}\n\n\\begin{remark}\nWe note that Theorem~\\ref{t:sc_BE} includes not only the assumption of finiteness of balls but also\nthe additional assumption that $\\inf_x m(x)>0$. Both of\nthese assumptions appear in the context of essential self-adjointness.\nIn particular, $\\inf_x m(x)>0$ implies both that $\\mathcal{L}$ maps $C_c(X)$ \ninto $\\ell^2(X,m)$ and that the restriction of $\\mathcal{L}$ to $C_c(X)$ is\nessentially self-adjoint, see Theorem~6 in \\cite{KL12}. In the context of the proof of \nTheorem~\\ref{t:sc_BE} in \\cite{HL17}, this assumption is used to establish the convergence of sums in the \nproof of the estimate $ \\Gamma (P_t \\varphi) \\leq e^{-2Kt}P_t(\\Gamma(\\varphi))$. However, it is not\nclear whether this assumption is really necessary for the ultimate result of stochastic completeness.\n\nOn the other hand, the result in \\cite{HL17} actually assumes a seemingly weaker assumption than finiteness\nof balls found in the formulation of Theorem~\\ref{t:sc_BE} above. Namely, Theorem~1.2 in \\cite{HL17} only\nassumes the existence of a sequence of finitely supported functions $(\\varphi_n)$ \nsuch that $0 \\leq \\varphi_n \\leq 1,\n\\varphi_n(x) \\to 1$ and $\\Gamma(\\varphi_n) \\leq 1\/n$ for all $n \\in {\\mathbb N}$. \nThis is sometimes referred to as a \\emph{completeness} assumption\non the graph as, in the manifold setting, the existence of such a sequence \nis known to be equivalent to geodesic completeness, see \\cite{BGL14,\nStri83}. It is clear in the proof above that the existence of an intrinsic metric with finite distance balls\nimplies the existence of such a sequence. On the other hand,\nMarcel Schmidt recently communicated to us that the converse is also true, that is, that\nthe existence of an intrinsic metric with finite distance balls is actually equivalent to completeness as defined above.\nFor a proof of this fact, see Appendix~A in the updated version of \\cite{LSW}. Furthermore, for graphs satisfying\nthe ellipticity condition $b(x,y) \\leq C m(x)m(y)$ for all $x, y \\in X$ and which are\nstochastically complete and satisfy an additional condition called the Feller property, see \\cite{PS12, Woj17} \nfor more details, $CD(0,\\infty)$ implies that the graph is complete, see Theorem~6.1 in \\cite{KM}.\n\\end{remark}\n\nTheorem~\\ref{t:sc_BE} gives stochastic completeness in the case of a uniform lower Bakry--{\\'E}mery curvature \nbound in the spirit of \\cite{Yau78}.\nHowever, looking at the optimal curvature results from the Riemannian setting mentioned in the introduction,\nwe would expect to allow for some decay of Ricci curvature as in \\cite{Var83, Hsu89}. One improvement of\nTheorem~\\ref{t:sc_BE} in this direction is contained for a special class of graphs in \\cite{HM}. More specifically,\nif $G$ is a graph with $X = {\\mathbb N}_0$, $x \\sim y$ if and only if $|x-y|=1$, $m$ is the counting measure\nand $\\varrho$ is an intrinsic metric on $G$, then letting\n$$\\kappa(x) = \\sup \\{K \\in {\\mathbb R} \\ | \\ G \\textup{ satisfies } CD(K,\\infty) \\textup{ at } x \\}$$\nif $\\kappa(x)$ decays like $-\\varrho^2(0,x)$, then $G$ is stochastically complete,\nsee Theorem~1.6 in \\cite{HM} for a more precise statement and proof.\n\n\n\n\n\\subsection{Ollivier Ricci curvature and stochastic completeness}\nWe now discuss a second commonly appearing manifestation of curvature in the discrete setting. \nThis formulation comes from optimal transport theory, see \\cite{Vill03} for a general background, and was defined for\nMarkov chains in \\cite{Oll07, Oll09}. For graphs, the basic idea is to transport a mass, which is given\nby the transition probability of a simple random walker starting at a vertex, to that of a mass at another vertex \nwith minimal effort. This definition\nwas then modified to give an infinitesimal version for the case of bounded degree in \\cite{LLY11} and extended\nto graphs with general measure and edge weights in \\cite{MW19}. \nWe note that, in contrast to the Bakry--{\\'E}mery formulation which\nis defined at a vertex, this curvature is defined for pairs of vertices.\n\nWe start with some basic definitions. As in the previous subsection, we assume that all graphs are locally finite.\nWe will further assume that all graphs are connected.\nFor a vertex $x \\in X$ and $\\epsilon>0$ small, one defines a transition probability distribution $\\mu_x^\\epsilon$ on $X$ via\n$$\\mu_x^\\epsilon(y)= \n\\begin{cases}\n1 - \\epsilon {\\mathrm{Deg}}(x) & \\textup{ if } y =x \\\\\n\\epsilon b(x,y)\/m(x)& \\textup{ otherwise}\n\\end{cases}\n$$\nwhere ${\\mathrm{Deg}}(x)$ is the weighted degree of $x$.\nThis is a probability distribution provided that $\\epsilon \\leq 1\/{\\mathrm{Deg}}(x)$. We note that a connection to the\nLaplacian is given by\n$$\\mu_x^\\epsilon(y)= 1_y(x) -\\epsilon L 1_y(x)$$\nas follows by a direct calculation.\n\nNow, for two vertices $x_1, x_2 \\in X$, we define the Wasserstein distance between $\\mu_{x_1}^\\epsilon$\nand $\\mu_{x_2}^\\epsilon$ via\n$$W(\\mu_{x_1}^\\epsilon, \\mu_{x_2}^\\epsilon) = \\inf_{\\pi} \\sum_{x,y \\in X}\\pi(x,y) d(x,y)$$\nwhere the infimum is taken over all $\\pi:X \\times X \\longrightarrow [0,1]$ with\n$\\sum_{y \\in X}\\pi(x,y) = \\mu_{x_1}^\\epsilon(x)$ and $\\sum_{x \\in X}\\pi(x,y) = \\mu_{x_2}^\\epsilon(y)$\nand $d$ is the combinatorial graph distance. The idea behind this is that $\\pi$ transports the mass\ndistribution from $\\mu_{x_1}^\\epsilon$ to $\\mu_{x_2}^\\epsilon$ and thus $W(\\mu_{x_1}^\\epsilon, \\mu_{x_2}^\\epsilon)$\nminimizes the effort required to carry out this transport.\n\nWe let $Lip(1)$ denote the set of functions with Lipshitz constant 1 with respect to the combinatorial\ngraph distance, that is,\n$$Lip(1) = \\{ f \\in C(X) \\ | \\ |f(x)-f(y)| \\leq d(x,y) \\textup{ for all } x, y \\in X \\}$$\nand let $\\ell^\\infty(X)$ denote the set of bounded functions.\nBy Kantorovich duality, see Theorem~1.14 in \\cite{Vill03}, we have\n$$W(\\mu_{x_1}^\\epsilon, \\mu_{x_2}^\\epsilon) = \n\\sup_{f \\in Lip(1) \\cap \\ell^\\infty(X)} \\sum_{x \\in X}f(x)(\\mu_{x_1}^\\epsilon(x)-\\mu_{x_2}^\\epsilon(x)).$$\nFinally, following \\cite{Oll09, LLY11}, in \\cite{MW19} we define the Ollivier Ricci curvature\nbetween two vertices as follows.\n\n\\begin{definition}[Ollivier Ricci curvature]\nFor vertices $x, y \\in X$ with $x \\neq y$, we let \n$$\\kappa^\\epsilon(x,y)=1- \\frac{W(\\mu_x^\\epsilon, \\mu_y^\\epsilon)}{d(x,y)}$$\nand define the \\emph{Ollivier Ricci curvature} as\n$$\\kappa(x,y)= \\lim_{\\epsilon \\to 0^+} \\frac{\\kappa^\\epsilon(x,y)}{\\epsilon}.$$\n\\end{definition}\n\n\\begin{remark}\nWe note that \\cite{Oll09} often considers the case when $m(x)=\\sum_{y \\in X}b(x,y)$\nand $\\epsilon=1$. In this case, ${\\mathrm{Deg}}(x)=1$ so that\n$$\\mu_x^1(y) = \\frac{b(x,y)}{\\sum_{z \\in X} b(x,z)}$$\nis just the one step transition probability at $x$ of the simple random walk on $G$.\nMore generally, when $\\epsilon<1$, we get that\n$$\\mu_x^\\epsilon(y) = \\begin{cases}\n1 - \\epsilon & \\textup{ if } y =x \\\\\n {\\epsilon b(x,y)}\/{\\sum_{z \\in X} b(x,z)} & \\textup{ otherwise}.\n\\end{cases}$$\nIn this case, the simple random walk is given a positive probability to remain at the vertex $x$. As such,\nthe constant $1- \\epsilon$ is sometimes referred to as the idleness parameter in this setting,\nsee \\cite{BCLMP18}.\n\nThe idea of letting $\\epsilon \\to 0^+$ in the case when $m(x) = \\sum_{y \\in X}b(x,y)$ then\nappears in \\cite{LLY11}. In particular, the concavity of the function $\\kappa^\\epsilon$ is used\nto establish the existence of the limit. However, as noted previously, the assumption\n$m(x) = \\sum_{y \\in X}b(x,y)$ gives that ${\\mathrm{Deg}}(x)=1$ and thus all such graphs are stochastically\ncomplete by Theorem~\\ref{t:bounded_SC}. \nThe contribution of \\cite{MW19} is to allow for this definition in the case of\npossibly unbounded vertex degree which makes the question of stochastic completeness\ninteresting. The existence of the limit as $\\epsilon \\to 0^+$ follows analogously to the argument\nin \\cite{LLY11}, see also \\cite{BCLMP18}.\n\\end{remark}\n\nWe note that the Ollivier Ricci curvature as defined above can be explicitly calculated in many cases.\nFor example, if the vertices $x\\sim y$ are not contained in any 3-, 4-, or 5-cycles, then\n$$\\kappa(x,y) = 2b(x,y)\\left( \\frac{1}{m(x)}+\\frac{1}{m(y)}\\right) - {\\mathrm{Deg}}(x) - {\\mathrm{Deg}}(y)$$\nsee Example~2.3 in \\cite{MW19}. As a concrete illustration, in the case of standard edge weights\nand counting measure, the curvature of an edge in a $k$-regular tree is \n$\\kappa(x,y)=4-2k$. This confirms the notion that regular trees of degree greater than 2 are the analogues\nof hyperbolic space as they have constant negative curvature.\n\nAs a second example, let $X={\\mathbb N}_0$ and let $x\\sim y$ if and only if $|x-y|=1$.\nSuch graphs will be referred to as \\emph{birth-death chains}.\nWe note that they automatically fall into the framework of weakly spherically symmetric graphs as \ndiscussed in Section~\\ref{s:symmetry}.\nLet $j, k \\in {\\mathbb N}_0$ with $k > j$. The Ollivier Ricci curvature on a birth-death chain can be calculated directly as\n$$\\kappa(j, k) = \\frac{1}{k-j} \\left( \\frac{b(j, j+1)-b(j,j-1)}{m(j)} - \\frac{b(k,k+1)-b(k,k-1)}{m(k)}\\right)$$\nsee Theorem~2.10 in \\cite{MW19}.  We note that for $j \\sim k$, this reduces to the formula above.\n\nBoth of the examples above are easily derived from the following\nformula which allows us to compute the curvature in terms of the Laplacian. \nTo state it, we let\n$$\\nabla_{xy}f= \\frac{f(x)-f(y)}{d(x,y)}$$\nfor $x \\neq y$.\nFor a proof of the following formula see Theorem~2.1 in \\cite{MW19}.\n\\begin{theorem}[Ollivier Ricci curvature and Laplacian]\\label{t:curvature_Lap}\nLet $G$ be a locally finite connected weighted graph. For vertices $x \\neq y$, we have\n$$\\kappa(x,y) =\\inf_{\\substack{f \\in Lip(1) \\\\ \\nabla_{xy}f=1}} \\nabla_{xy}\\mathcal{L} f\n =\\inf_{\\substack{f \\in Lip(1) \\cap C_c(X) \\\\ \\nabla_{xy}f=1}} \\nabla_{xy}\\mathcal{L} f.$$\n\\end{theorem}\n\nWe now show how Theorem~\\ref{t:curvature_Lap} allows us to prove a Laplacian\ncomparison result. In this result, we compare the Laplacian applied to a distance function\non a general graph to the Laplacian applied to a distance function on a birth-death chain.\nWe first define the notion of sphere curvature which will be involved. We recall that for a vertex $x_0 \\in X$, $S_r$ denotes\nthe sphere of radius $r$ around $x_0$ with respect to the combinatorial graph distance.\nWe then let\n$$\\kappa(r) = \\min_{y \\in S_r} \\max_{\\substack{x \\in S_{r-1} \\\\ x \\sim y}}\\kappa(x,y)$$\nfor $r\\in {\\mathbb N}$ with $\\kappa(0)=0$ and call $\\kappa$ the \\emph{sphere curvature}.\n\n\\begin{theorem}[Laplacian comparison]\\label{t:Lap_comp}\nLet $G$ be a locally finite connected weighted graph. If $x_0 \\in X$, $\\rho(x)=d(x,x_0)$ and $\\kappa$ denotes\nthe sphere curvature, then\n$$\\mathcal{L} \\rho(x) \\geq \\sum_{j=1}^{\\rho(x)} \\kappa(j) - {\\mathrm{Deg}}(x_0)$$\nfor all $x \\in X$.\n\\end{theorem}\n\\begin{proof}\nWe note that in general\n$$\\mathcal{L} \\rho(x) = {\\mathrm{Deg}}_-(x)-{\\mathrm{Deg}}_+(x)$$\nwhere ${\\mathrm{Deg}}_\\pm(x)$ are the outer and inner degrees as defined in Section~\\ref{s:symmetry}. \nIn particular, $\\mathcal{L} \\rho(x_0)=-{\\mathrm{Deg}}(x_0)$ so taking the\nsum to be zero in this case gives the statement for $x=x_0$.\n\nThe proof is now by induction on $r=\\rho(x)$ for $x \\in S_r$. Assume that the statement is true for $r-1$. \nLet $y \\in S_r$ and let $x\\in S_{r-1}$\nbe such that $\\kappa(x,y) \\geq \\kappa(x,z)$ for all $z \\sim x$ with $z \\in S_r$. We note that\n$\\kappa(r) \\leq \\kappa(x,y)$, $\\nabla_{yx}\\rho=1$ and $\\rho \\in Lip(1)$ \nso that Theorem~\\ref{t:curvature_Lap} gives \n\\begin{align*}\n\\kappa(r) \\leq \\kappa(x,y) &\\leq \\nabla_{yx}\\mathcal{L} \\rho =\\mathcal{L} \\rho(y) - \\mathcal{L} \\rho(x).\n\\end{align*}\nTherefore, by the inductive hypothesis,\n$$\\sum_{j=1}^{r} \\kappa(j) - {\\mathrm{Deg}}(x_0) = \\sum_{j=1}^{r-1} \\kappa(j) - {\\mathrm{Deg}}(x_0) + \\kappa(r) \\leq \\mathcal{L} \\rho(x) + \\kappa(r) \\leq \\mathcal{L} \\rho(y)$$\nwhich completes the proof.\n\\end{proof}\n\nWe note that the Laplacian comparison result is sharp on birth-death chains as\ncan be seen by a direct calculation. That is, for birth-death chains, we get\n$$\\mathcal{L} \\rho(x) = \\sum_{j=1}^{\\rho(x)} \\kappa(j) - {\\mathrm{Deg}}(x_0).$$\nThus, Theorem~\\ref{t:Lap_comp} compares the Laplacian of a distance function\non a general graph to that of a birth-death chain.\n\nWe now put the various pieces together to obtain a stochastic completeness result for Ollivier Ricci\ncurvature. This can be found as Theorem~4.11 in \\cite{MW19}.\n\\begin{theorem}\\label{t:SC_Oll_curv}\nLet $G$ be a locally finite connected weighted graph. If\n$$\\kappa(r) \\geq -C \\log r$$\nfor some constant $C>0$ and all large $r$, then $G$ is stochastically complete.\n\\end{theorem}\n\\begin{proof}\nLet $\\rho(x)=d(x,x_0)$. It follows from Theorem~\\ref{t:Lap_comp} that\n$$\\mathcal{L} \\rho(x) \\geq \\sum_{j=1}^{\\rho(x)} \\kappa(j) - {\\mathrm{Deg}}(x_0).$$\nNow, from the assumption that $\\kappa(r) \\geq -C \\log r$ we may choose an\nincreasing continuously differentiable function $f:[0,\\infty) \\longrightarrow (0,\\infty)$ such that\n$$\\mathcal{L} \\rho + f(\\rho) \\geq 0$$\nand \n$$\\int^\\infty \\frac{1}{f(r)}dr=\\infty.$$\nAs $\\rho(x_n) \\to \\infty$ along any sequence of vertices $(x_n)$ with\n${\\mathrm{Deg}}(x_n) \\to \\infty$, $G$ is stochastically complete by Theorem~\\ref{t:Khas}.\n\\end{proof}\n\nIt turns out that Theorem~\\ref{t:SC_Oll_curv} is sharp in the sense that for any $\\epsilon>0$,\nthere exists a stochastically incomplete graph with $\\kappa(r) \\geq -(\\log r)^{1+\\epsilon}$. This can already\nbe seen from the case of birth-death chains for which stochastic completeness is equivalent to\n$$\\sum_{r=0}^\\infty \\frac{r+1}{b(r,r+1)}=\\infty$$\nby Theorem~\\ref{t:sc_symmetry} above. For further details, see Theorem~4.11 in \\cite{MW19}.\n\nWe note that the optimal curvature criterion for stochastic completeness in terms of curvature in the \nmanifold setting gives the borderline for stochastic completeness around the curvature decay of order\n$-r^2$, see \\cite{Var83, Hsu89}.\nOne might be tempted to try and reconcile the difference between the manifold and graph setting\nby using intrinsic metrics as was successfully carried out in the case of volume growth, however, for Ollivier Ricci\ncurvature this approach turns out to not work, see Example~4.13 in \\cite{MW19}. \n\nWe note that\nthis is not the only difference between the continuous and discrete settings when it comes to curvature.\nAs another example,\nthere exist infinite graphs which have uniformly positive Ollivier Ricci curvature, see Example~4.18 in \\cite{MW19}.\nThis is known to be impossible in the manifold setting by the Bonnet--Myers theorem. Here,\nanti-trees also prove to be a source of counterexamples as they provide examples of infinite \ngraphs satisfying and $CD(K,\\infty)$ for $K>0$ and well as having uniformly positive Ollivier Ricci curvature, \nsee \\cite{CLMP20} in \\cite{KLW} for the Bakry--{\\'E}mery and Ollivier\nRicci curvature of anti-trees. However, as soon as one either imposes upper bounds on the vertex degree or\nlower bounds the measure, then a graph with uniformly positive lower curvature bounds \nmust be finite, see \\cite{LMP18} for the case of Bakry--{\\'E}mery\nand \\cite{MW19} for the case of Ollivier Ricci.\n\n\n\n\\subsection*{Acknowledgements} I would like to thank J{\\'o}zef Dodziuk\nfor suggesting such a fruitful area of study and for sustaining me\nover the years.  I am also happy to acknowledge the inspiration and support offered by\nIsaac Chavel and Leon Karp.\nFurthermore, I would like to thank Alexander Grigor$'$yan for encouragement and support,\nearly in my career up until the present moment. And also many thanks to my coauthors who contributed \nto this story and whom I also consider to be good friends. In alphabetical order:\nSebastian Haeseler, Bobo Hua, Xueping Huang, Matthias Keller, Daniel Lenz, Jun Masamune, \nFlorentin M{\\\"u}nch and Marcel Schmidt. \nFinally, I would like to thank Isaac Pesenson for the invitation to contribute\nthis article.\n\n\n\n\n\n\n\n\n\n\n\n\\begin{bibdiv}\n\\begin{biblist}\n\n\\bib{AS}{article}{\n   author={Adriani, Andrea},\n   author={Setti, Alberto G.},\n   title={Curvatures and volume of graphs},\n   eprint={arXiv:2009.12814 [math.DG]}\n}  \n\n\n\\bib{Az74}{article}{\n   author={Azencott, Robert},\n   title={Behavior of diffusion semi-groups at infinity},\n   journal={Bull. Soc. Math. 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+{"text":"\\section{Introduction}\n\n\nThe Pipe nebula\\ is an ideal test site to study early cloud formation. From this massive, nearby, elongated cloud (\\tenpow{4}~$M_{\\odot}$; 145~pc away; 18~pc$\\times$3~pc in size) arises $^{12}$CO~1--0 emission that shows two filamentary structures: one elongated along the N--S direction at a $v_{\\rm LSR}$ range of 6--7~km~s$^{-1}$; and the other one along the E--W direction at 2--4~km~s$^{-1}$\\ \\citep{onishi99}.\nThe magnetic field is strikingly uniform along the N--S direction with optical polarization degrees up to an outstanding 15\\% value  (\\citealp{alves08}, hereafter AFG08; \\citealp{alves14}). Overall, the cloud seems to be magnetically dominated and turbulence seems to be sub-Alfv\\'enic \\citep{franco10}. Besides the simple structure, the cloud is at a very early stage of evolution with little to none star-formation feedback. Only 14 young stars have been detected, mostly in the B59 region at the west end, delivering a very low $\\sim$0.06\\%\\ star formation efficiency \\citep{forbrich09}. In addition, the cloud harbors more than 150 starless cores with very early chemistry that seem to be gravitationally unbound and pressure confined \\citep{lada08,rathborne08,roman09,roman10,frau10,frau12}. \n\n\n\n\n\n\n\n\n\n\n\n   \\begin{figure*}[t]\n   \\centering\n   \\includegraphics[width=\\textwidth,angle=0]{veloPol3.pdf}\n   \\caption{Velocity and polarization distribution of the Pipe nebula.\n   {\\it Color code}: blue and red for field groups NS and S, respectively.\n      {\\bf Central maps}: \n        {\\it Grayscale maps}: top and bottom panels show the regions with $^{13}$CO\\ $v_{\\rm m1}$\\ in the $>$4~km~s$^{-1}$\\ and $<$4~km~s$^{-1}$, respectively \\citep[fitted to the map by][]{onishi99}. The N--S filament is evident in $^{12}$CO emission \\citep{onishi99}.\n       {\\it Contours}: visual extinction map \\citep{lombardi06}. Levels are 5.5 and 20~mag.\n   {\\it Segments}: average polarization segment on each observed field (AFG08; \\citealp{franco10}). Length is proportional to $\\langle$P$_{\\rm deg}\\rangle$. \n {\\it Circles}: positions of the embedded dense cores with extinction peaks $>10$~mag \\citep{roman09,roman10}. \n   {\\bf Left-hand side panels}: data averaged over the fields observed by AFG08. {\\it Top panel}: $\\langle\\Delta$P$_{\\rm ang}\\rangle$\\ versus $\\langle$P$_{\\rm deg}\\rangle$\\ as in AFG08. {\\it Central panels}: $\\langle\\Delta$P$_{\\rm ang}\\rangle$\\ and $\\langle$P$_{\\rm deg}\\rangle$\\ versus $\\langle v_{\\rm m1}\\rangle$, where m1 stands for the first-order moment of the $^{13}$CO\\ emission.\n   {\\it Bottom panel}: average linewidth of the $^{13}$CO\\ velocity components as a function of $v_{i}$ for the different cloud regions. $v_{i}$ is sampled in 0.25~km~s$^{-1}$--wide bins. Error bars indicate 1-$\\sigma$ dispersion.\n   {\\bf Right-hand side panels}: data for every pixel separated by groups. {\\it Top panel}: P$_{\\rm ang}$\\ versus P$_{\\rm deg}$. {\\it Bottom panels}: P$_{\\rm ang}$\\ and P$_{\\rm deg}$\\ versus $v_{\\rm m1}$. \n   }\n   \\label{fig-vel}\n    \\end{figure*}\n\n\n\n\n\n\n\n\n\\section{Datasets and data process\\label{sec-data}}\n\nThe three datasets used in this work are published in different articles. We limit ourselves to provide the basic information and we refer the reader to the original publications.\n\nThe dust extinction maps were derived by \\citet{roman09,roman10} through the NICER infrared color excess technique \\citep{lombardi01}. The angular resolution is 20\\arcsec\\ and the typical rms is $<$1~mag. \n\nThe $^{13}$CO~1--0 map was presented by \\citet{onishi99} and has spectral and angular resolution of $\\sim$0.1~km~s$^{-1}$\\ and 4\\arcmin, respectively. The $^{13}$CO\\ emission is enclosed in the 5~mag contour of the $A_{v}$\\ map (Fig.~\\ref{fig-vel}). Hence, $^{13}$CO\\ is a good tracer of the gas kinematics (Section~~\\ref{ssec-resolution}).\n\nThe optical linear polarization data were published by AFG08 and \\citet{franco10}. They observed 46 fields of 12\\arcmin$\\times$12\\arcmin\\ in the $R$-band, covering most of the Pipe nebula. We used the $\\sim$6,600 stars detected with $P\/\\sigma P\\geq10$.\n\nOur data processing followed two steps. First, the $^{13}$CO\\ datacube was converted to spectra on a pixel-by-pixel basis. A semi-automatic algorithm detected the number of peaks in each spectrum and carried out a Gaussian fit to derive the peak velocity, $v_{i}$, and the linewidth, FWHM$_{i}$, of the different components. For each pixel, we also derived the first-order moment of the $^{13}$CO\\ emission ($v_{\\rm m1}$). Second, we paired the position of each $R$-band star to its pixel in the $A_{v}$\\ and $^{13}$CO\\ map. Upon positional match, we composed a set of properties for each pixel: $A_{v}$, $v_{\\rm m1}$, $v_{i}$, FWHM$_{i}$, and polarization angle (P$_{\\rm ang}$) and degree (P$_{\\rm deg}$).  The effects of the different angular resolutions are discussed in Section~\\ref{ssec-resolution}.\n\n\n\n\n\n   \\begin{figure}\n   \\centering\n   \\includegraphics[width=\\columnwidth,angle=0]{stemOscillationsPaper.pdf}\n   \\vskip .1cm\n   \\includegraphics[width=.95\\columnwidth,angle=0]{pipe2.png}\n \\caption{\n {\\bf Top panels}: projections of the position-position-velocity cube in two perpendicular position-velocity planes. The orientation of the projection planes is indicated as blue lines in the $A_{v}$\\ map subpanels. {\\it Bottom panel}: velocity as a function of position. The projection plane is at an angle of 124$^\\circ$\\ (shock direction: projection on the plane of the sky of the velocity difference of the filaments). {\\it Top panels}: $v_{i}$ and FWHM$_{i}$ as a function of position. The projection plane is at an angle of 34$^\\circ$\\ (perpendicular to the shock direction). {\\it Circles}: dense cores as in Fig.~\\ref{fig-vel}.\n{\\bf Cartoon}: proposed scenario. \n The antenna represents the observer. Blue and red cylinders represent the E--W and N--S filaments, respectively. The LOS velocities of the filaments are shown as arrows and the stem and bowl regions are labeled. The translucent plane shows the shock direction on the plane of the sky extended along the line-of-sight direction (blue line on bottom panel above). The velocities of the filaments along the shock are shown as arrows. Green arrows represent the tentative direction of the magnetic field (Section~\\ref{ssec-mag-field}). }\n   \\label{fig-ppv}\n    \\end{figure}\n\n\n\n\n\n\n\n\\section{Results and analysis\\label{sec-results}}\n\n\n\\subsection{Dual large-scale velocity structure}\n\nFigure~\\ref{fig-vel} outlines the main velocity structures of the Pipe nebula\\ derived from the $^{13}$CO\\ spectra. The cloud can be separated in two different filaments with narrow velocity ranges: one oriented E--W at $<$4~km~s$^{-1}$\\ and another one oriented N--S at $>$4~km~s$^{-1}$\\ \\citep[this filament is better seen in the $^{12}$CO\\ map: see][]{onishi99}. The spectra from the E--W filament typically have one velocity component, two in some cases. Instead, the spectra from the N--S filament show a richer velocity structure with up to four components. Both structures overlap along the line-of-sight (LOS) at the bowl (Fig.~\\ref{fig-vel}). The velocities at the overlapping region cover the entire range between those of the two filaments. This transition is illustrated in Fig.~\\ref{fig-ppv} through two orthogonal projections of the position-position-velocity cube \\citep[see][]{hacar13} that show distinct patterns (Section~\\ref{ssec-collision}).\n\nWe define two regions: the non-shocked gas --group NS-- that roughly includes B59 and the stem ($\\langle$P$_{\\rm deg}\\rangle$$\\la$5\\%) and the shocked gas --group S-- that roughly coincides with the bowl ($\\langle$P$_{\\rm deg}\\rangle$$\\ga$5\\%). These groups appear color-coded on Fig.~\\ref{fig-vel} and also have distinct properties in polarimetry, density, and turbulence (Section~\\ref{ssec-resPol}). \n\n\n\n\\subsection{Bimodal trends in velocity, polarization, and linewidth\\label{ssec-resPol}}\n\nThe top left panel on Fig.~\\ref{fig-vel} replicates Fig.~2 by \\citet{alves08}. They report a systematic anti-correlation between $\\langle\\Delta$P$_{\\rm ang}\\rangle$\\ and $\\langle$P$_{\\rm deg}\\rangle$. The color-coded NS and S groups appear clearly differentiated: S fields have high $\\langle$P$_{\\rm deg}\\rangle$\\ and low $\\langle\\Delta$P$_{\\rm ang}\\rangle$, oppositely to NS fields. This differentiation goes beyond field averages and holds at a single pixel level (top right panels). \n\nThe central left panels on Fig.~\\ref{fig-vel} show the distinct distribution of $\\langle\\Delta$P$_{\\rm ang}\\rangle$\\ and $\\langle$P$_{\\rm deg}\\rangle$\\ as a function of $\\langle v_{\\rm m1}\\rangle$. The bottom right panels of Fig.~\\ref{fig-vel} show similar plots but for all the pixels. These panels show that the clear polarization differences between group~S and group~NS are mostly independent of the velocity of the gas, thus suggesting different scenarios for the two regions. Group~S gas at 4--5~km~s$^{-1}$\\ shows the largest polarization degree.\n\nThe bottom left panel of Fig.~\\ref{fig-vel} and top panel of Fig.~\\ref{fig-ppv} show that the gas in the bowl has a larger linewidth than in B59 and the stem. Strikingly, the star-forming B59 complex has a smaller linewidth than group~S. This suggests that the Bowl gas is more disturbed than that of B59.\n\n\n\n\n\n\\section{Discussion\\label{sec-disc}}\n\n\n\\subsection{Data suitability and spatial resolution\\label{ssec-resolution}}\n\nThe polarization data trace material with $A_{v}$  $\\la$15~mag that includes 99.1\\% of the $A_{v}$\\ maps pixels, and \\citet{franco10} show that P$_{\\rm deg}$\\ and $A_{v}$\\ grow roughly proportionally. Hence, optical data are good tracers of the magnetic field of the cloud. The NICER $A_{v}$\\ maps are very accurate: a comparison to 1.2~mm emission maps of dense cores shows a perfect within-the-errors agreement for up to 45~mag \\citep{frauthesis}. The spatial resolution of the $A_{v}$\\ map is $\\sim$0.015~pc. This is significantly smaller than the typical 0.1~pc diameter of dense cores, and thus, the map resolution is more than enough for the large scales. $^{13}$CO\\ is a good tracer of the dense gas (Fig.~\\ref{fig-vel}). The $^{13}$CO\\ pixel size is $\\sim$0.17~pc, comparable to a dense core. With a mean $^{13}$CO\\ linewidth of 0.6$\\pm$0.3~km~s$^{-1}$, the pixel crossing time is $\\sim$0.28~Myr. The dense core formation timescale is 0.5--1~Myr \\citep{bergin07}, and thus, it seems unlikely that any velocity substructure would form. Therefore, the $^{13}$CO\\ map should suffice to reflect all significant variations in the gas velocity.\n\n\n\n\\subsection{Interaction of gas filaments and core formation\\label{ssec-collision}}\n\nThe two filaments overlap at the bowl and the gas velocities cover the entire range between the original velocities (Fig.~\\ref{fig-ppv}). This fact points to an interaction, possibly a collision, as reported toward denser and more massive IRDCs \\citep{duartecabral11,henshaw13,nakamura14}. The sound speed at 12~K is $c_{s}$=0.27~km~s$^{-1}$\\ and the pre-shock Alfv\\'en speed\\footnote{Assuming a magnetic field on the plane of the sky with $B_{\\rm stem}$=30~$\\mu$G, and a density of \\scitenpow{3}{3}~cm$^{-3}$\\  (AFG08).} is $v_{\\rm A}$=0.78~km~s$^{-1}$. The magnetosonic speed $v_{m} = \\sqrt{c_{s}^{2}+v_{A}^{2}}$ is 0.83~km~s$^{-1}$, smaller than the velocity separation of the filaments. Therefore, an MHD wave is slower than the velocity difference between the filaments and a shock would indeed form. Hence, in the Pipe nebula\\ it is possible to compare the evolution from non-shocked (stem) to shocked (bowl) gas.\n\nThe two orthogonal projections of the position-position-velocity cube show that LOS velocities are anisotropic (Fig.~\\ref{fig-ppv}). On the one hand, the bottom panel shows highly ordered velocities likely to be the velocity gradient generated along the direction of the collision. On the other hand, the middle panel shows a large velocity dispersion likely to be the projection across the shock. This identified direction for the plausible shock is the projection on the plane of the sky of the velocity difference between the filaments. In addition, for the collision to happen, the N--S filament must be closer to us because it is going away faster than the E--W filament. The total shock velocity would be a combination of two components: (a) the line-of-sight velocity and (b) the plane-of-the-sky velocity whose direction is deduced from the projections of the position-position-velocity cube. Figure~\\ref{fig-ppv} shows a cartoon of the proposed scenario. \n\nFigure~\\ref{fig-vel} shows that linewidths increase {\\it only} at the bowl {\\it and} for the velocities between the two filaments (roughly 4 to 6~km~s$^{-1}$). This points to a more turbulent region where the interaction takes place. The interaction would settle the gas at the observed intermediate velocities and result in sub- and trans-Alfv\\'enic gas motions \\citep{franco10}. The accumulation of material would cause the increase in density and would lead to the formation of the bowl. In fact, the average column density of 6.0$\\pm$1.8~mag at the bowl is twice as high as the average 3.0$\\pm$1.3~mag at the stem. This increase reflects an increase in volume density due to the plausible filament collision.\n\nThe bowl is the region with the largest column density and the most fertile in number of dense cores (Fig.~\\ref{fig-vel}). B59, the other region rich in dense cores, is also proposed to be formed through an external shock that triggered star formation \\citep{peretto12}. Cores have velocities in good agreement with gas velocities. At the bowl, cores cover the entire velocity range between the filaments, follow the velocity gradient along the shock, and tend to cluster at the larger velocities (bottom panel in Fig.~\\ref{fig-ppv}). This combined structure of cores and gas is the ``molecular ring'' reported by \\citet{muench07}. There are other clusters of cores, one in B59 and two in the stem (middle panel in Fig.~\\ref{fig-ppv}). The gas surrounding these clusters shows similar properties: a larger dispersion of velocities, and increases in linewidth, $^{13}$CO\\ brightness, and column density. These properties resemble those of the shocked bowl but with smaller values and at a smaller spatial scale. In general, it seems that in the slowed, shocked gas the higher density and sub-Alfv\\'enic motions enhance the creation of cores.\n\n\n\n\\subsection{Magnetic field compression and polarization increase\\label{ssec-mag-field}}\n\nThe high P$_{\\rm deg}$\\ values --close to the theoretical maximum \\citep{whittet03}-- suggest that the magnetic field direction is close to the plane of the sky AFG08. The higher P$_{\\rm deg}$\\ values are detected towards the bowl, where the two filaments interact. The shock can have reoriented the magnetic field placing it closer to the plane-of-the-sky (Fig.~\\ref{fig-ppv}). This scenario could explain the variation of P$_{\\rm deg}$. Moreover, the peak of P$_{\\rm deg}$\\ is at a velocity of $\\sim$4.5~km~s$^{-1}$, the intermediate velocity between the filaments where the shock would have the strongest effects.\n\nThe material traced by the optical polarization is dominated by magnetic field over gravity and turbulence \\citep{franco10}. This low-density gas is in flux-freezing conditions, this is, matter and magnetic field are tied. As expected in this regime, the large-scale gas compression that causes the column density to double also doubles the magnetic field strength: from 30~$\\mu$G at the stem to 65~$\\mu$G at the bowl (AFG08). This implies that the magnetic energy density is four times higher in the bowl than in the stem ($E_{\\rm mag}^{\\rm stem}$$\\simeq$\\scitenpow{3.6}{-11}~erg~cm$^{-3}$\\ and $E_{\\rm mag}^{\\rm bowl}$$\\simeq$\\scitenpow{1.7}{-10}~erg~cm$^{-3}$). Similarly, the larger line widths in the bowl than in the stem (0.6$\\pm$0.3 and 0.48$\\pm$0.19~km~s$^{-1}$, respectively) give a turbulent kinetic energy density about 40\\% higher in the bowl ($E_{\\rm kin}^{\\rm bowl}$$\\simeq$\\scitenpow{2.3}{-11}~erg~cm$^{-3}$). This generates an increase of the surface pressure on dense cores, reported to be $\\sim$60\\% higher in the bowl with respect to the stem \\citep{lada08}. The energy increase can be explained by the plausible shock scenario: assuming an average volume density of \\scitenpow{3}{3}~cm$^{-3}$\\ (AFG08) and that the shock velocity is the velocity difference between the two filaments, 2.5~km~s$^{-1}$, then the shock energy density is $E_{\\rm shock}$$\\simeq$\\scitenpow{3.7}{-10}~erg~cm$^{-3}$, twice the energy increase from the stem to the bowl. Consequently, the kinetic energy of the collision could increase the measured turbulence levels, and then drag and compress the magnetic field, transferring and storing some of the kinetic energy into magnetic field energy. \n\nThe Alfv\\'enic Mach number $M_{A}=v_{\\rm shock}\/v_{A}$ of the plausible shock is $\\sim$3.0. Simulations of low $M_{A}$ (highly magnetized) collisions predict the formation of shocks \\citep{lesaffre13}. Indeed, models with low Alfv\\'enic Mach numbers ($M_{A}$$\\sim$4--5) drive more energy into compressing the magnetic field \\citep{pon12}. In particular, for clouds of similar temperature and density to the Pipe nebula\\ colliding at 2--3~km~s$^{-1}$, the density and magnetic field strength increase by a factor similar to $M_{\\rm A}$. Moreover, 20--50\\% of the shock energy is stored in the magnetic field. These predictions match our results at the Pipe nebula, also if taking the factor of 2 error in the measured quantities into account. This provides a plausible theoretical framework that reproduces our observations. The majority of the remaining energy is expected to be radiated away through mid-J CO lines that can be used to further test this scenario.\n\n\n\n\\section{Conclusions\\label{sec-concl}}\n\nWe combined visual extinction, optical polarimetry, and $^{13}$CO\\ data of the Pipe nebula. Here, we summarize the conclusions.\n\n\\begin{enumerate}\n\n\\item The velocity patterns of the gas suggest that the N--S and E--W filaments are colliding along the NW-SE direction at the bowl, where turbulence increases.\n\n\\item There is a good correlation between polarization properties and velocity of the gas, suggesting that magnetization increases at the interaction site. The column density and magnetic field strength double with respect to the non-shocked gas, pointing to a highly magnetized, continuous collision under flux-freezing.\n\n\\item Cores in the bowl tend to cluster in space and tend to follow the $^{13}$CO\\ velocity gradient. Dense cores tend to form in regions similar to the bowl where the cloud gas has large velocity dispersion, linewidths, column density, and $^{13}$CO\\ emission. It seems that in the slowed, shocked gas the higher density and sub-Alfv\\'enic motions enhance the creation of cores.\n\n\\end{enumerate}\n\n\n\n\n\n\n\\begin{acknowledgements}\n\nWe thank Daniele Galli and Andy Pon for useful comments and suggestions that are greatly appreciated. P.F. is supported by the CONSOLIDER project CSD2009-00038, Spain. P.F. and J.M.G. are supported by the MINECO project AYA2011-30228-C03-02, Spain. G.A.P.F. is supported by the CNPq agency, Brazil. CRZ is supported by program CONACYT CB-2010 152160, M\\'exico.\n\n\\end{acknowledgements}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nParticles that carry a fraction of the electron charge, $Q=\\epsilon e$, also called millicharged particles (mCPs) are an elegant extension to the standard model (SM). The discovery of their simplest iteration, without the presence of a dark photon, would be a violation of the charge quantization hypothesis. Alternatively, mCPs could also arise as charges quantized under a dark force which kinematically mixes with the SM photon \\cite{Izaguirre:2015eya}. \n\nMillicharged particles have attracted significant interest and there have been many attempts to detect them.\nCollider and beam-dump experiments have been performed to search for mCPs and null results have lead to strong limits on their parameter space in the MeV-TeV mass range. These include LEP~\\cite{akers1995search}, the SLAC millicharge experiment~\\cite{Prinz:1998ua}, neutrino experiments~\\cite{Magill:2018tbb}, the Argoneut experiment~\\cite{Acciarri:2019jly}, MilliQan pathfinder experiment at the LHC~\\cite{Ball:2020dnx} and the BEBC beam dump experiment~\\cite{Marocco:2020dqu}. A plethora of experiments have been proposed to improve these tests and search for still smaller charges~\\cite{Ball:2016zrp,Berlin:2018bsc,Kelly:2018brz,Harnik:2019zee}. At lower masses, stringent limits arise from absence of anomalous emission in stellar environments~\\cite{Davidson:2000hf,Chang:2018rso} at masses below an MeV. The prospect of mCPs making up some or all of the dark matter (DM) has also received a lot of interest over the years. There are robust predictions for their relic density from the early universe~\\cite{Dvorkin:2019zdi,Creque-Sarbinowski:2019mcm}, as well as numerous ways to detect them~\\cite{Knapen:2017ekk,Blanco:2019lrf,Essig:2019kfe,Berlin:2019uco,Kurinsky:2019pgb,Barak:2020fql,Griffin:2020lgd} depending on their mass $m_Q$ and charge. Subcomponent millicharge DM (mCDM) has also been invoked to explain several recent experimental anomalies~\\cite{Barkana:2018qrx,Munoz:2018pzp,Liu:2019knx,Kurinsky:2020dpb,Bloch:2020uzh,Harnik:2020ugb}. \n\nSince mCPs interact with the SM particles through a massless mediator, their transfer cross-section can be large at small velocities. As a result, mCDM with charge large enough to scatter in the atmosphere and Earth overburden of direct detection experiments, rapidly loses its virial kinetic energy and thermalizes with the environment. When it eventually reaches a direct detection experiment, it does not possess enough energy to deposit in the detector and can not be observed at direct detection experiments~\\cite{Emken:2019tni}. However, these mCPs which are now cooled to the ambient temperature, get trapped due to Earth's gravity and build up for the duration of the Earth's existence. This can lead to mCP densities on Earth up to fourteen orders of magnitude larger than that of the virial population in the galaxy \\cite{Pospelov:2020ktu}. This slow, albeit dense population requires novel detection strategies, some of which include mCP particle-antiparticle annihilation in a large-volume detector~\\cite{Pospelov:2020ktu} and accelerating mCPs present in electrostatic accelerator tubes to higher energies, sufficient for subsequent direct detection~\\cite{Pospelov:2020ktu}. \nIf the charge is large enough, negatively charged mCPs can bind with large positively charged SM nuclei, thereby creating fractional charge for a macroscopic material. Searches for such fractional charges bound to a sample material include Millikan-like oil-drop experiments~\\cite{kim2007search}, as well as more recent levitation experiments with microspheres~\\cite{moore2014search,afek2020limits}. However, these limits hinge critically on the assumption that the negatively charged mCPs bind to matter, thereby restricting their validity to only large charges and masses. \n\nIn this work, we point out an alternate search strategy; the remarkably stable trapped ions developed for metrology and quantum information science are the ideal targets to detect an ambient thermalized mCP population. There is a long history of using trapped charged SM  particles for particle physics applications. Trapped ions have been used to measure the anomalous magnetic moment of the electron~\\cite{hanneke2011cavity}, the electron's electric dipole moment~\\cite{cairncross2017precision}, proton and antiproton magnetic moments \\cite{schneider2017double,smorra2017parts}, as well as time variation in fundamental constants which can be induced by ultralight bosonic dark matter~\\cite{Roberts_2020}. Ion traps are also the preeminent candidate for qubits to realize quantum computing \\cite{haffner2008quantum}. \n\nFor all of the applications listed above and especially for magnetic moment measurements and quantum computing, it is important to keep the ion trapped for sufficiently long duration without heating from the surroundings. In the last couple of decades, there has been remarkable progress in reduction of the measured heating rate of trapped ions \\cite{hite2012100}. Further progress is expected and is an active area of research in order to achieve scalability of multi-ion systems and increased sampling rate in precision measurements. We briefly explain why the above properties make ion traps the ideal candidate for detecting mCPs next. \n\n\n\n\\subsection{Summary of Findings}\n\\label{sec:exec}\n\nWe propose the use of ion traps as detectors for millicharged particle (mCP) dark matter.  While an individual ion constitutes a much smaller target mass than any other dark matter direct detection experiment, a trapped ion has significant advantages for detection of mCPs including isolation from the environment, a lower energy threshold for detection, and a larger scattering cross section with mCPs.  These advantages outweigh the small target mass, allowing ion traps to reach many orders of magnitude past other detection methods for mCPs. \n\nSignificant effort has been put into isolating trapped ions from their environments.\nThus trapped ions are now sensitive to small energy depositions down to $\\sim$ neV.  \nThe dense, thermal gas mCPs, if they exist on Earth, would permeate the detector and can scatter off the ion.  \nDepending on the nature of the experiment, it may be possible to measure individual scattering events or just an overall heating rate of the ion.\nThe mCPs are thermalized with the walls of the detector which is held at a temperature much higher than the temperature of the trapped ion. Thus, the higher-energy mCPs can transfer kinetic energy which can be detected either as a single jump of the trapped ion, or an accumulation of several scatters resulting in heating of the ion.  We will discuss both types of signals.\nThe high degree of isolation of the ion achievable in these traps makes them sensitive to scattering rates for mCDM over a wide range of parameter space.\n\nIon traps also make excellent mCDM detectors because of their low energy thresholds that are set by the energy-level spacings in the trap that can be as low as $\\sim \\text{neV}$.  This allows detection of single scattering events with this energy and also allows such low-energy scatters to contribute to the heating rate.\n\nFurthermore, because they are charged, ions make excellent targets for mCP scattering.  mCPs scatter with ions via Rutherford scattering which is greatly enhanced at small relative velocities and momentum transfers.  Since the mCPs are thermalized with the walls, they generally have much lower velocities than virialized dark matter. The corresponding boost to the scattering cross section combined with the large number density of the thermalized mCPs makes direct detection with single ions viable. \n\n\n\nThe rest of this paper is organized as follows.\nIn Sec.\\,\\ref{sec:traps} we provide a description of ion traps. In Sec.\\,\\ref{sec:mcpdyn}, we provide an overview of the mCP density on Earth, as well as their dynamics, including propagation through the trap. The detection signals are explained in Sec.\\,\\ref{sec:signals} and results and projections are presented in Sec.\\,\\ref{sec:results}. We conclude with a discussion in Sec.\\,\\ref{sec:discussion}. \n\n\\section{Ion traps}\n\\label{sec:traps}\nTraps for (single) charged particles belong to the basic tool-set of atomic, molecular, and optical physics \\cite{safronova2018search}. They have wide applications in determinations of atomic masses and fundamental constants,  in precision measurements to test fundamental symmetries, and in quantum information technology. \nTypical ingredients common to all trap experiments are shown in Fig$.\\,$\\ref{fig:Trap}. The core of the experiments are usually sets of electrodes supplied by AC and DC voltages, optionally placed in strong magnetic fields.  \n \\begin{figure}[h!]\n\\centering\n\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/fig1.pdf} \n\n\n    \\caption{Elements common to typical trap experiments. A set of electrodes supplied by AD and DC voltages is mounted in a vacuum chamber, optionally in a strong magnetic field. Cryogenic traps have in addition thermal shielding and cryogenic vacuum chambers at low temperatures in their surroundings. The particles are manipulated with laser beams, microwave and radio-frequency generators. Signals are read-out by detecting image currents or fluorescent light. }\n    \\label{fig:Trap}\n\\end{figure}\nThe electrodes are mounted in vacuum chambers, in some cryogenic trap experiments pressures on the level of 10$^{-18}\\,$mbar are achieved,  which provides ultra-long particle storage times \\cite{sellner2017improved} and enables non-destructive long-term studies at low background. Thermal shielding and an additional insulation vacuum chamber usually surround the inner vacuum chamber. The trapped particles are manipulated, cooled and excited via laser, microwave and radio-frequency-drives. The experimental signals for ultra-sensitive precision studies, frequency measurements, and monitoring of quantum information-processing protocols are either image-currents picked up by sensitive detection circuits, or fluorescence signals. The signals are acquired and processed with spectrum analyzers and charge coupled device (CCD) cameras, respectively.\n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{ ||c|c|c|c|c|c|c|c| }\n\\hline\n Experiment & Type & Ion & $V_z$  &  $T_{\\rm wall}$ & $\\omega_p$ [neV] & $T_{\\rm ion}$[neV] & Heating Rate (neV\/s) \\\\ \n \\hline \n Hite et al, 2012 \\cite{hite2012100}& Paul &$^9\\text{Be}^+$ & 0.1 V&300 K&$\\omega_z=14.8$ &$14.8$&640 \\\\\n Goodwin et al, 2016\\,\\cite{goodwin2016resolved} &Penning&$^{40}\\text{Ca}^+$ & 175\\,V  &300\\,K& $\\omega_z=1.24$ & $1.24$ & 0.37 \\\\\n Borchert et al, 2019\\,\\cite{borchert2019measurement}&Penning&$\\bar{p}$& 0.633\\,V& 5.6\\,K & $\\omega_+=77.4$&7240 & 0.13 \\\\\n &&& & & $\\omega_-=0.050$& &  \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{List of ion traps and the relevant experimental parameters used for setting limits in this paper. The ion used, $V_z$,  potential barrier in the axial direction and $T_{\\rm wall}$, the temperature of the walls of the trap and $\\omega_p$, the fundamental frequency of the trap in the relevant direction are listed. Also listed are $T_{\\rm ion}$, the temperature of the ion in the trap and the measured heating rate. }\n\\label{table:data}\n\\end{table*}\n\n\n\\subsection{Penning Traps}\nIn a Penning trap, there is a strong magnetic field $B_0$ superimposed with an electrostatic quadrupolar potential $\\Phi(z,\\rho)$, which is attractive along the magnetic field axis. The motion of a charged particle in such crossed static fields is composed of harmonic oscillator modes of three independent types. The modified cyclotron and the magnetron modes correspond to oscillations perpendicular to the axis, while the axial mode corresponds to oscillations along the magnetic field lines. Associated with the mode oscillations are the three trap eigenfrequencies $\\nu_+$, $\\nu_-$, and $\\nu_z$, respectively. Room-temperature traps such as that in Ref.\\,\\cite{goodwin2016resolved}, cool and trap ions such as $^{40}\\textrm{Ca}^+$ or $^{9}\\textrm{Be}^+$ to the axial ground state using optical sideband cooling~\\cite{mavadia2014optical}. A delay period after cooling is followed by subsequent spectroscopy which determines the final state of the ion, thus measuring the heating rate. The lowest heating rate achieved thus far at room temperature is reported in Ref.\\,\\cite{goodwin2016resolved} where the increase in the number of phonons of the axial mode was reported to be $\\dot{n}=0.3\/\\textrm{s}$ with the axial frequency $\\nu_z=0.3\\,\\textrm{MHz}$, see  Tab.\\,~\\ref{table:data}. \\\\\nParticularly interesting for the detection of mode energy changes are experiments dedicated to direct measurements of nuclear magnetic moments such as those of the proton \\cite{schneider2017300}, the antiproton \\cite{smorra2017parts}, or $^3$He$^{2+}$ \\cite{mooser2018new}. These experiments operate advanced cryogenic Penning-trap systems consisting of multi-trap assemblies, common to all of them is a so-called analysis trap with a strong superimposed magnetic inhomogeneity $B(z)=B_0+B_2 z^2$, where $B_2$ characterizes its strength \\cite{ulmer2011observation}. \nThe interaction of the magnetic bottle with the particle's magnetic moment $\\mu_z=\\mu_++\\mu_-+\\mu_s$ results in a magnetostatic axial energy $E_{B,z}=-\\mu_z B_z$, where $\\mu_+$ and $\\mu_-$ are the orbital angular magnetic moments associated with the modified cyclotron and the magnetron mode, while $\\mu_s$ is the spin magnetic moment. \nAs a result, the particle's axial frequency $\\nu_z=\\nu_{z,0}+\\Delta\\nu_z (n_+,n_-,m_s)$ becomes a function of the radial trap eigenstates $n_+$ and $n_-$, as well as the spin eigenstate $m_s$ with\n\\begin{align}\n\\Delta\\nu_z=\\frac{h\\nu_+}{4\\pi^2 m \\nu_z}\\frac{B_2}{B_0} \\left(\\left(n_++\\frac{1}{2}\\right)+\\frac{\\nu_-}{\\nu_+}  \\left(n_-+\\frac{1}{2}\\right)+\\frac{g}{2} m_s\\right).\n\\end{align}\nMeasurements of single-particle magnetic moments in Penning traps rely on the detection of axial frequency shifts $\\Delta\\nu_{z,\\text{SF}}$  induced by driven spin quantum transitions $\\Delta m_s=1$. Since nuclear magnetic moments are about three orders of magnitude smaller than the Bohr magneton, these experiments require highest sensitivity with respect to magnetic moments, which is usually achieved by the utilization of magnetic bottle strengths of the order $B_2\\approx100\\,$kT\/m$^2$ to $B_2\\approx400\\,$kT\/m$^2$. Combined with continuous measurements of the axial frequency, such strong magnetic bottles provide excellent resolution of the radial mode energies with \n\\begin{eqnarray}\n\\frac{(\\Delta\\nu_z)}{\\Delta E_\\rho}=\\frac{1}{4\\pi^2m \\nu_z}\\frac{B_2}{B_0}\\approx 1 \\frac{\\text{Hz}}{\\mu \\textrm{eV}}, \n\\end{eqnarray}\nwhile in axial frequency measurements resolutions of order  200$\\,$mHz$\\cdot\\sqrt{\\text{s}}\/\\sqrt{t_\\text{avg}}$ are achieved, $t_\\text{avg}$ being the averaging time which is typically on the order of several tens of seconds. Transition rates in the radial modes\n$(dn_{+,-})\/dt\\propto(n_{+,-}\/\\omega_{+,-} ) S_E (\\omega_{+,-})$ lead to random walks in radial energy space and to axial frequency diffusion. Here  $S_E (\\omega_{+,-})$ is the power spectral density of a noisy background drive, and $n_{+,-}$ is the principal quantum number of the modified cyclotron ($n_+$) \/ magnetron ($n_-$) oscillator. The scaling of the heating rate with $n_{+,-}$ is related to eigenstate-overlap of harmonic oscillator states \\cite{ulmer2011observation}. \\\\ \nBy analyzing time sequences of axial frequency measurements $\\nu_z (t)$, the average radial quantum transition rates are obtained. With a highly optimized trap setup with which the antiproton magnetic moment was measured with 1.5 parts per billion precision, the BASE experiment at CERN reports on the observation of absolute cyclotron transition rates of 6(1) quanta per hour~\\cite{borchert2019measurement}. Together with the determination of the $n_+$ state during the recorded measurement, this result is  consistent with a projected ground state heating rate of 0.1 cyclotron quantum transitions per hour, setting an upper limit which is by a factor of 1800 lower than the best reported Paul-trap heating rates, and by a factor of 230 lower than the best room-temperature Penning trap. These numbers are summarized in Tab.\\,\\ref{table:data}. Note that the antiproton experiments are conducted in a background vacuum of $\\approx 10^{-18}\\,$mbar \\cite{sellner2017improved}, constraining parasitic heating induced by collisions with  background-gas to a level of $4\\times10^{-9}$\/s. \n\n\n\\subsection{Paul Traps}\nPaul traps or radiofrequency traps utilize an oscillating voltage to confine in the perpendicular direction instead of the magnetic field used for the same purpose in the Penning trap. Paul traps have a rich history of being used as mass spectrometers and more recently in building quantum computers~\\cite{brownnutt2015ion}. \n\nThe effective potential in the presence of both DC and AC potentials can be written as\n\\begin{equation}\n    \\psi(x,y,z,t)=\\left(U_{\\rm DC}+V_{\\rm AC} \\cos \\Omega t\\right)\\frac{r^2+2 (z_0^2-z^2)}{r_0^2+2z_0^2} .\n\\end{equation}\nHere $z$ is the axial direction, $r$ the radial direction with the distance to the electrodes given by $r_0$ and $z_0$. The rapidly oscillating potential creates a pseudopotential for charges of both signs and this leads to approximately simple harmonic motion very close to the trap center.  After laser cooling to the ground state, the total heating rate can be measured via Raman sideband technique~\\cite{turchette2000heating}. There has been extensive study of the heating rate in Paul traps and its dependence on distance to electrodes, wall temperature, trap temperature~\\cite{brownnutt2015ion} as well as ion beam treatment of electrodes~\\cite{hite2012100}. Electric field noise from the electrodes has been identified as the dominant heating source, with the dependence on distance scaling as $d^{-2}$~\\cite{brownnutt2015ion} to $d^{-4}$~\\cite{daniilidis2011fabrication}. Although heating rates are lower for bigger traps~\\cite{poulsen2012efficient}, smaller sized traps employ shallower potential wells $\\approx0.1~\\textrm{V}$. As we shall see, this allows mCPs with large charge to reach the trap and hence provide complementary reach at large charge parameter space. Hence we reinterpret limits only from a microtrap~\\cite{hite2012100}.  The heating rates reported in ~\\cite{hite2012100} are $\\dot{n}=43\/\\textrm{s}$ for the axial frequency $\\nu_z=3.6~\\textrm{MHz}$. These numbers are tabulated in Tab.\\,~\\ref{table:data}.\n\n\n\n\\section{Millicharged Particle Dynamics}\n\\label{sec:mcpdyn}\n\\subsection{Terrestrial Accumulation} \nIf mCDM exists and is virialized in the galaxy, there is a non-zero flux of mCPs flowing through the Earth at all times.  This mCDM stops in the atmosphere or rock overburden for large enough charge, and can accumulate on Earth. This process was treated in detail in~\\cite{Emken:2019tni} and the subsequent accumulation in~\\cite{Pospelov:2020ktu}. We provide here a summary of the relevant results of these papers that are used in Section.\\,\\ref{sec:results} and refer the reader to ~\\cite{Pospelov:2020ktu} for details.\n\nFollowing~\\cite{Pospelov:2020ktu} we consider effectively millicharge particles mediated by a dark photon which kinematically mixes with the SM photon. The dark photon mass is taken to be large enough ($m_{A'} \\gtrsim 10^{-12}~\\textrm{eV}$) such that the effect of large-scale electric and magnetic fields can be ignored while considering mCP propagation\\footnote{We leave to upcoming work the calculation of the accumulation of millicharged dark matter in cases where the electric and magnetic fields are relevant \\cite{blprfuture}. Once such calculations are complete, our given limits on the number density in the lab can be translated to limits on dark matter fraction in theses cases.}.  In this limit, the mCPs with mass $m_Q\\ge 1$\\,GeV are stuck on the Earth after thermalization.\\footnote{Millicharged particles with masses below 1\\,GeV can accumulate on Earth temporarily before evaporating. We leave limits on these masses for future work.} The volume averaged DM number density on Earth, $\\langle n_Q \\rangle$ can be several orders of magnitude larger than the virial DM density. It is given by, \n\\begin{align}\n\\label{start}\n\\langle n_Q \\rangle&= \\frac{\\pi R_{\\oplus}^2 v_{\\rm vir}  t_{\\oplus}}{4\/3 \\pi R_{\\oplus}^3}n_{\\rm vir} \\nonumber \\\\\n&\\approx \\frac{3\\times 10^{15} }{\\textrm{cm}^3} \\frac{t_{\\oplus}}{10^{10}\\,\\textrm{y}} f_Q \\frac{\\rm GeV}{m_Q},\n\\end{align}\nwhere, $R_{\\oplus}$ and $t_{\\oplus}$ are the radius and age of the Earth and $n_{\\rm vir}$ and $v_{\\rm vir}$ are the galactic virial number density and velocity. However, the equilibrium density profile is peaked at the Earth core. This density $n_{\\rm static}$ was calculated by taking into account the Earth's temperature and density variations in \\cite{Neufeld:2018slx}. However, the sinking to the Earth's core is not immediate, and there is a dynamic population, the so-called ``traffic-jam\" density, $n_{\\rm tj}$, given by,\n\\begin{equation}\n    n_{\\rm tj}=n_{\\rm vir}\\frac{v_{\\rm vir}}{v_{\\rm term}} .\n\\end{equation}\nHere, $v_{\\rm term}$ is the terminal velocity in rock, given in \\cite{Pospelov:2020ktu}. Finally, the density in the laboratory, $n_{\\rm lab}$ is given by, \n\\begin{equation}\nn_{\\rm lab} = \\textrm{Max}\\left( n_{\\rm static}, \\textrm{Min}\\left( n_{\\rm tj} , \\langle n_Q \\rangle \\right)\\right)\n\\label{eq:regimes}\n\\end{equation}\n\nThese results are applicable only for $\\epsilon$ large enough, such that it stops in the corresponding overburden. According to \\cite{Pospelov:2020ktu} this is valid only for\n\\begin{align}\n\\epsilon&\\gtrsim 2\\times10^{-4} \\sqrt{\\frac{m_Q}{\\rm GeV}} \\quad &&\\textrm{surface,} \\nonumber \\\\\n&\\gtrsim 3\\times 10^{-6}\\sqrt{\\frac{m_Q}{\\rm GeV}} \\quad &&\\textrm{1 km mine.} \n\\label{eq:applicable}\n\\end{align}\n\nIt is also important to comment on the asymmetry of the mCP population. If the mCDM is a symmetric population with equal number of particles and antiparticles, accumulation on Earth can result in Sommerfeld-enhanced annihilations which prevent build up. We consider here the asymmetric case such that opposite charges are carried by different species just like the the SM proton and electron, such that annihilations are absent. In this scenario, for large enough $\\epsilon$, the negatively charged mCPs can form deep bound states with the large positively nuclei, whereas the positive mCPs can only bind with the less massive electrons which also have smaller charge compared to the heavy SM nuclei. In ref.~\\cite{Pospelov:2020ktu} it was pointed out that positive mCPs with $\\epsilon \\ge 0.042$  bind to electrons at 300 K (room temperature). We find that such bound states are temporary with electrons rapidly preferring to bind with ions which possess larger charge than the positive mCPs. Thus the terrestrial population of positive mCPs remains free of binding for $\\epsilon\\lesssim 1$ and is present as a locally thermalized population that is diffusing everywhere. Thus, the results we derive apply to all positive charges with $\\epsilon \\lesssim 1$ and negative charges which do not bind with nuclei, i.e \\cite{Pospelov:2020ktu} $\\epsilon < \\frac{m_e}{\\mu_{Q,N}}$ where $\\mu_{Q,N}$ is the reduced mass of the mCDM-nuclear system.  \n\n\nIt is important to emphasize that the limits put on the ambient number density are applicable to mCPs charged directly under the SM photon as well as to ones mediated by a dark photon as long as the dark photon mass is below the relevant momentum transfer for scattering with ions which is around 1 eV.   \n\n\\subsection{Passage Through Apparatus}\n\n\\label{trappassage}\nWe next turn to the trajectory of mCPs through the trap peripherals in order to reach the trapped ion.  We need to know the density and temperature of the mCPs reaching the ion.  In this subsection we explain the main factors entering the calculation, but as the calculation itself is somewhat involved we leave the details to Appendix \\ref{Appendix:passage}.  Our main result is in Eq.\\,\\eqref{eqn:qion} relating the number density of mCPs at the position of the ion, $n^Q_{\\rm ion}$, to the ambient number density on the Earth $n_{\\rm lab}$.  \nThis is the equation we use to set our limits on mCPs.\n\nAt equilibrium, mCPs are expected to have roughly uniform density near the Earth's surface, including permeating all materials.  However the conditions of the experiment can affect this naive expectation for several reasons.\n\nFirst, some of the ion traps we consider are cryogenic.  Over essentially all of our parameter space the mCPs have a short interaction length in material and so will rapidly thermalize to the cryostat temperature as they enter the experiment.  By itself this would lead to an increase in mCP density by a factor linear in mCP velocity ($\\propto \\sqrt{T}$) because the fluxes entering and leaving the cryostat must be equal in equilibrium.\n\nSecond, the ion traps are surrounded by metal which has a work function that can affect the passage of mCPs.  This is only relevant for mCPs of relatively large charge ($\\epsilon \\gtrsim 10^{-2}$) but for those it can be a significant effect.  The work function for mCPs, which we will call $\\epsilon \\phi$, is not simply $\\epsilon$ times the work function for electrons.  For electrons the work function arises from several contributions of varying signs including e.g.~the binding to the lattice of nuclei, the Fermi sea of other electrons, and surface effects such as the ``double layer\" or the image charge potential.  Several of these do not apply or are negligible for mCPs.  Recall we are only considering positively charged mCPs since the negative ones may be stuck deeply bound to some nucleus somewhere on the Earth.  Thus the main effects are repulsion by the double layer and possibly also repulsion from the nuclei.   We consider the work function for mCPs in more detail in  Appendix \\ref{Appendix:work function}.  Our conclusion is that the work function is repulsive for positively charged mCPs and thus every metal sheet provides a barrier for mCPs to cross. \nThe size of the potential barrier that has to be crossed is $\\phi \\sim \\text{few} \\, \\text{eV}$.\nNote that for an experiment at room temperature ($T \\sim 0.03 \\, \\text{eV}$) the metal barriers are then irrelevant for charges $\\epsilon \\lesssim 10^{-1}$ because the Boltzmann tail easily pushes a fast enough rate of mCPs over the barrier.  For a cryogenic experiment at $T \\sim 6 \\, \\text{K} \\sim 5 \\times 10^{-4} \\, \\text{eV}$, the metal barrier will be relevant for charges $\\epsilon \\gtrsim 10^{-3}$ and essentially insurmountable for charges $\\epsilon \\gtrsim 10^{-2}$.  As we show in Appendix \\ref{Appendix:passage}, the most important effect comes from an experiment encased in two different metals where the work function for mCPs rises from the outer metal to the inner metal.  We will take this difference to be $\\Delta \\phi = 3 \\, \\text{eV}$ for all experiments we consider since this will be a conservative estimate as we show in Appendix \\ref{Appendix:work function}.\n\n\nThird, the ions are always in a region of ultrahigh vacuum.  This means that pumps were used to remove the Standard Model (SM) particles.  Given a short interaction length of the mCP in materials, these pumps could remove the mCPs from the ion chamber as well.  In one of the experiments we consider (Goodwin et al.~\\cite{goodwin2016resolved}) this effect is not relevant because the trap is at room temperature and the region of sensitivity is at low enough $\\epsilon$ that the millicharges pass easily through the walls\\footnote{The sensitivity is limited to be below $10^{-3}$ charge because of the large trap potential as will be discussed in Section \\ref{sec:results}.}.  In the other two experiments we consider (Hite et al.~\\cite{hite2012100} and Borchert et al.~\\cite{borchert2019measurement}), the vacuum pumps are turned off well before the actual data taking is begun.  In the case of Borchert this is at least a year, while for Hite we conservatively assume it is only a day\\footnote{The final answer is only logarithmically sensitive to this timescale anyway.}.  And of course mCPs are always continually flowing in from the walls of the vacuum chamber.  This would rapidly refill the trap region and this effect would not be relevant, except for the largest charges where the refill can be slow because of the work function of the surrounding metal.  This does mean that, depending on the parameters of the mCP and of the ion trap, the number density in the trap may be either in the equilibrium regime or in the filling regime.  This is why Eq.\\,\\eqref{eqn:final trap density} has two different regimes.  Eq.\\,\\eqref{eqn:final trap density} relates the number density of mCPs inside the trap $n_{\\rm trap}$ to the ambient number density on the Earth $n_{\\rm lab}$.  There is then one remaining step to find the number density of mCPs at the position of the ion.\n\nFourth, the ion trap itself has applied electromagnetic fields to trap the ion.  These can affect the passage of the mCPs, though again this is only relevant for larger charges $\\epsilon \\gtrsim 10^{-3}$.\nTab.\\,\\ref{table:data} lists parameters of the various experiments we consider.  All the experiments use an electric DC potential to confine the ion in the axial direction.  The height of the potential barrier along the axial direction is listed as $V_z$.  The mCPs in the trap are thermalized to the temperature of the walls of the trap $T_{\\rm wall}$.  Starting from the number density of mCPs calculated inside the trap $n_{\\rm trap}$ in Eq.\\,\\eqref{eqn:final trap density}, we take a Boltzmann suppression on the number density which can make it up the axial barrier height.  Thus we take the final number density at the position of the ion to be given by Eq.\\,\\eqref{eqn:qion}.  For the experiments we consider this Boltzmann suppression is relevant for setting the ceiling (largest $\\epsilon$ values) of the Goodwin regions in Figure \\ref{fig:evsm}, but is irrelevant for the other regions. \nThe Penning traps use a magnetic field to confine the ions in the radial direction.  This B-field will cause mCPs with a large enough charge to circle around the axial B-field lines but they are still free to move along the axis.  The magnetic field does not change the phase-space density of the mCPs and so it does not significantly affect our signal.  The Paul traps use an RF potential to confine the ion in the radial direction.  This potential around the minimum is locally attractive for the positive mCP independent of the sign of the charge of the trapped ion.  An mCP coming from far outside the RF fields in a radial direction could give a barrier in principle.  However for mCPs approaching along the axial direction, the RF pseudopotential in the axial direction is very weak and negligible.  And then  such mCPs will actually be concentrated in the radial direction towards the ion at the center of the trap since the potential is locally attractive.  We conservatively ignore this possible (Sommerfeld-like) enhancement though it could be large. For large enough mCP masses, $m_Q \\gtrsim 10^{10} \\textrm{GeV}$, the free-fall under gravity \ncan generate velocities much larger than the thermal velocities assumed. This can increase the heating rate further. We conservatively ignore this effect and leave its consideration for future work.\n\n\\section{Observables for Millicharged Particles in Ion Traps}\n\\label{sec:signals}\nIn this section we consider the interaction of ambient mCPs with ion traps to identify observables for detection. As seen in Section.~\\ref{trappassage}, the mCPs enter the ion trap with effective temperature $T_Q \\ge T_{\\rm wall}$, where $T_{\\rm wall}$ is the wall temperature. $T_{\\rm wall}= 5.6$ K for the cryogenic trap we consider~\\cite{borchert2019measurement} and $T_{\\rm wall}=T_{\\rm room}\\approx 300$ K for room temperature traps. The ions in the trap are at a much colder temperature $T_{\\rm ion} \\ll T_{\\rm wall}$. With this hierarchy of temperatures, the mCPs can cause two types of signal.\n\nThe first signal involves individual scattering events that impart energy $E_{\\rm ion}$ to the ion thus leading to a change in its harmonic oscillator quantum number. This signal is very similar to dark matter scattering in a conventional dark matter detector. The rate of events has to be slower than the rate at which the ions are interrogated. $E_{\\rm ion}$ also needs to exceed the energy resolution $E_{\\rm res}$ for detection. \n\nThe second type of signal is the heating of the trapped ion due to collisions with multiple mCPs. In this case, the individual hits $E_{\\rm ion}$ can be smaller than $E_{\\rm res}$ and only the sum needs to exceed this resolution. \n\nFor both of these signal types, we only consider individual energy transfers $E_{\\rm ion}$ much larger than the typical energy spacing of the trap $\\omega$, for concreteness $E_{\\rm ion} \\ge 10\\omega$. \\footnote{In principle, energy transfers $E_{\\rm ion} =n\\times \\omega$ where all integers $n\\ge 1$ are allowed. We restrict $E_{\\rm ion} \\ge 10 \\times \\omega$, a conservative choice that helps avoid form-factor calculations.} In this limit, the trapped ion can be approximated as a free particle with initial energy $T_{\\rm ion}$ and final energy $T_{\\rm ion}+E_{\\rm ion}$. Equivalently, for energy transfers much larger than the spacing, the form-factor that incorporates wave-function overlap can be approximated to unity.  \nWe start quantifying both of these signals by the angular differential cross-section given by the Rutherford formula,\n\\begin{equation}\n    \\frac{d\\sigma}{d\\Omega}=\\frac{2\\pi\\alpha^2 \\epsilon^2}{\\mu^2 v_{\\rm rel}^4\\left(1-\\cos\\theta\\right)^2} ,\n\\end{equation}\nwhere $\\theta$ is the scattering angle. We next introduce kinematic variables that  simplify the computation of the scattering rate. The results are presented here, with the details of the derivation presented in Appendix.~\\ref{app1}. The incoming mCP and trapped ion velocities are assumed to be $\\mathbf{v_Q}$ and $\\mathbf{v_{\\rm ion}}$ respectively. \nThe center of mass (CM) velocity $\\mathbf{v_{CM}}$ is given by,\n\\begin{equation}\n    \\mathbf{v_{CM}}=\\frac{\\left(m_{\\rm ion} \\mathbf{v_{\\rm ion}}+m_Q \\mathbf{v_Q}\\right)}{m_{\\rm ion}+m_Q} .\n\\end{equation}\n\nThe change in velocity of the ion, $\\mathbf{\\Delta v_{\\rm ion}}$ is given by,\n\\begin{align}\n    \\mathbf{\\Delta v_{\\rm ion}}=\\frac{m_Q}{m_{\\rm ion}+m_Q}\n    \\left[\\left(\\cos \\theta-1 \\right)\n    \\left(\\mathbf{v_{\\rm ion}}-\\mathbf{v_Q}\\right)\n    \\right. \n    \\nonumber \\\\ \n    \\left.\n    +\\sin \\theta |\\mathbf{v_{\\rm ion}}-\\mathbf{v_Q}|\n    \\mathbf{n_{\\perp}}\n    \\right] .\n\\end{align}\nThe transferred energy $E_{\\rm ion}$ is given by,\n\\begin{equation}\n    E_{\\rm ion} = m_{\\rm ion} \\mathbf{v_{CM}}.\\mathbf{\\Delta v_{\\rm ion}} .\n\\end{equation}\n\nGiven a threshold $E_{\\rm thr}$, the single event rate $R_{\\rm single}$ with energy transfer $E_{\\rm ion}$ above this threshold is,\n\\begin{align}\n    R_{\\rm single}\\left(E_{\\rm ion}\\ge E_{\\rm thr}\\right)=n^Q_{\\rm ion} \\int d^3 \\mathbf{v_Q} g_Q  \\int d^3 \\mathbf{v_{\\rm ion}} g_{\\rm ion}  \\int d \\Omega  \\nonumber \\\\ |v_Q-v_{\\rm ion}| \\frac{d\\sigma} {d \\Omega} \\Theta \\left(|E_{\\rm ion}|-E_{\\rm thr}\\right) . \n\\end{align}\nHere $n^Q_{\\rm ion} $ is the number density of mCPs at the ion position and is given by Eq.\\,\\eqref{eqn:qion}, $g_{\\textrm{ion}(Q)}$ is the Maxwell--Boltzmann distribution for the ion\/mCP with temperatures $T_{\\rm ion}$ and $T_{\\rm wall}$, respectively. \n\n\nThe heating rate per ion, $\\dot{H}$ can be computed through,\n\\begin{align}\n    \\dot{H}&=n^Q_{\\rm ion}  \\int d^3 \\mathbf{v_Q} g_Q  \\int d^3 \\mathbf{v_{\\rm ion}} g_{\\rm ion}  \\int d \\Omega |v_Q-v_{\\rm ion}|  \\nonumber \\\\ &\\frac{d\\sigma} {d \\Omega}E_{\\rm ion} \\Theta \\left(|E_{\\rm ion}|-E_{\\rm thr}\\right) \\Theta\\left(E_{\\rm samp}-|E_{\\rm ion}|\\right) . \n\\end{align}\nThe Heaviside theta function ensures the inequality $E_{\\rm thr}\\le E_{\\rm ion} \\le E_{\\rm samp}$. Here $E_{\\rm samp}$ is defined so as to prevent the average heating rate from including contribution from extremely rare events. It is defined through,\n\\begin{equation}\nR_{\\rm single}\\left(E_{\\rm ion}\\ge E_{\\rm samp}\\right) t_{\\rm obs}=1 ,\n\\end{equation}\nwhere $t_{\\rm obs}$ is the total observation time. \n\n\n \\section{Results}\n \\label{sec:results}\n \n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}{0.48\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/comparetraps1e3.pdf} \n\n\\end{subfigure}\n\\begin{subfigure}{0.48\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/evsmmainall.pdf} \n \n\\end{subfigure}\n    \\caption{Compilation of new limits using existing heating measurements from various traps in Table \\ref{table:data}: a room temperature Paul trap, Hite et al.~\\cite{hite2012100}, a room temperature Penning trap, Goodwin et al.~\\cite{goodwin2016resolved} and a cryogenic penning trap, Borchert et al. \\cite{borchert2019measurement}. {\\bf (left)} Comparison between traps for an ambient density $n_{\\rm lab}=10^3 \\textrm{cm}^{-3}$; {\\bf (right)} Combined limits from the three traps for different  $n_{\\rm lab}$} \n    \\label{fig:evsm}\n\\end{figure*}\n In this section, we set constraints on mCPs and make projections for the future by using the expressions for the signal rates derived in Sec.\\,\\ref{sec:signals}. \n \n \\subsection{Limits from existing measurements}\n To obtain existing limits, we use the data presented in Tab.\\,\\ref{table:data}. All of the trap parameters for the $^9\\text{Be}^+$  trap are taken from \\cite{hite2012100} while the trap depth is conservatively taken to be $V_z=0.1$~V, an order of magnitude larger than the typical potential depths in microtraps\\footnote{see for e.g.~\\cite{warring2013techniques} from the same group where the potential depth is report to be $V_z$=5 mV}. While the rest of the parameters for the $^{40}\\textrm{Ca}^+$ experiment are provided in ~\\cite{goodwin2016resolved}, we obtained the potential depth $V_z=175$~V from the authors. Finally, for ref.\\,\\cite{borchert2019measurement}, we use the parameters $T_{\\rm wall}=5.6$~K and $V_z$=0.6~V. While the analysis in ~\\cite{borchert2019measurement} dealt with measuring the cyclotron mode ($\\omega_+)$, the observable heating rate is equivalently a limit on the magnetron mode ($\\omega_-$) also. While individual jumps in $\\omega_-$ are unobservable with existing precision, the frequency shift due to the heating rate accumulates. Since the Rutherford cross-section increases for smaller energy transfers, we use $\\omega_-$ to convert existing heating limits into limits on mCPs. \n\n We start by plotting existing limits in Fig.\\ref{fig:evsm} in the millicharge $\\epsilon$ vs mass $m_Q$ parameter space for contours of constant ambient density $n_{\\rm lab}$. As mentioned earlier, for mCDM, the lab density is expected to be orders of magnitude larger than the virial density, i.e. $n_{\\rm lab} \\gg n_{\\rm vir}$. In the left panel, limits arising from the antiproton trap \\cite{borchert2019measurement} in blue, $^{40}\\textrm{Ca}^+$~\\cite{goodwin2016resolved} in red and $^{9}\\textrm{Be}$ in green \\cite{hite2012100} for an ambient density of $n_{\\rm lab}=10^3\/\\textrm{cm}^3$. The dominant limits arise from the antiproton trap owing to its superior heating rate as seen in Tab.\\,\\ref{table:data}. However, since it is a cryogenic trap, the limits disappear at $\\epsilon \\approx 10^{-2}$ owing to the suppression arising from mCPs finding it increasingly difficult to penetrate metals due to their work function. As explained in Section.~\\ref{sec:mcpdyn} and Appendix.~\\ref{Appendix:passage}, this factor roughly scales as $e^{-\\frac{\\epsilon \\Delta \\phi}{T_{\\rm wall}}}$ where $\\Delta \\phi$ is the difference in work functions between two adjacent metals. Hence this suppression is ameliorated for room temperature traps like the $^9\\textrm{Be}^+$ trap \\cite{hite2012100} in green which extends to $\\epsilon\\approx 0.1$. While the $^{40}\\textrm{Ca}^+$ trap in red is also at room temperature, the trap is at a potential of 175~V and thus mCP charges above $\\epsilon \\approx 10^{-3}$ do not reach the ion. However, there is reach to higher mCP masses for this trap as a result of the ion being more massive. The right panel of Fig.~\\ref{fig:evsm}\n corresponds to combined limits from these three experiments for different $n_{\\rm lab}$.  \n \n \\begin{figure}\n\\centering\n\n    \\includegraphics[width=\\linewidth]{figs\/compwithexisting.pdf} \n\n    \\caption\n    Comparison of limits derived in this work from ~\\cite{hite2012100,goodwin2016resolved,borchert2019measurement} with existing limits from Oil Drop~\\cite{kim2007search}, levitation experiments~\\cite{moore2014search,afek2020limits} and LEP~\\cite{Marocco:2020dqu} for $m_Q=10$ GeV.}\n    \\label{fig:compexist}\n\\end{figure}\n \n \n \n As is clear from both figures, existing data for anomalous heating in traps sets exquisite bounds on mCPs thermalized locally. Bounds are applicable to orders of magnitude in the mCP mass $m_Q$ as well as many orders of magnitude in charge. Number densities as small as $n_{\\rm lab}=1\\, \\textrm{cm}^{-3}$ are ruled out around the $\\epsilon\\approx 10^{-3}$ and $m_Q\\approx 10\\, \\textrm{GeV}$ parameter point. \n \nIn order to compare these limits on the ambient mCP population to ones that already exist in literature, we fix the mCP mass $m_Q=10$\\,GeV and show limits in the $n_{\\rm lab}$ vs $\\epsilon $ plane in Fig.~\\ref{fig:compexist}. The same color coding as Fig.~\\ref{fig:evsm} is followed. In gray we show limits from LEP~\\cite{Marocco:2020dqu}, as well as limits on mCPs bound in matter arising from Oil drop experiments~\\cite{kim2007search}, and levitation experiments~\\cite{moore2014search,afek2020limits}. As noted earlier, the limits on mCPs bound in matter are applicable only to negative mCPs with large enough charge such that binding with SM nuclei is possible. Furthermore, if mCP-SM bound states exist, there is no guarantee for these bound states to be evenly distributed all over the Earth. However, for the positive mCPs none of these caveats apply and they thermalize and distribute themselves over the entire Earth volume. Regardless, as seen in Fig.~\\ref{fig:compexist} the limits obtained from ion traps are orders of magnitude stronger than the levitation experiments. For $\\epsilon\\approx 3\\times 10^{-3}$, lab densities as small as $n_{\\rm lab} \\gtrsim 1 \\textrm{cm}^{-3}$ are ruled out by the measured heating rate at the antiproton experiment \\cite{borchert2019measurement}. \n \\begin{figure}\n\\centering\n\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/dmplot.pdf} \n  \n\n    \\caption{Limits and projections on the virial DM fraction in mCPs as a function of $\\epsilon$ and $m_Q$. Since the current experiments were all conducted at the surface, the robust limits are only above the ``$>$1m Thermalization\" line where the colored contours are solid. For a hypothetical deep mine experiment, the dashed part of the contours are also accessible.}\n    \\label{fig:virial}\n\\end{figure}\n\nNext, in Fig.~\\ref{fig:virial} we convert limits on $n_{\\rm lab}$ into limits on the fraction of virial DM existing in mCPs, $f_Q= \\frac{\\rho_Q}{\\rho_{\\rm DM}}$. For this purpose we use $n_{\\rm lab}$ from Eq.\\,\\ref{eq:regimes}. Existing limits from colliders are shown in gray. The solid parts of the colored contours correspond to the region where the incoming virial DM gets thermalized within 1 meter and hence the robust current limits we put are restricted to this region i.e. above the top black line. The dashed lines show the reach for an identical heating rate experiment that is conducted in a deep mine at 1km depth. Virial DM fractions as small as $f_Q\\approx 10^{-12}$ are already ruled using existing heating data for DM masses in the 1-10 GeV mass range. For heavier masses, the terminal velocity is larger and hence the traffic jam densities are smaller. Nonetheless we set limits on DM fractions as small as $f_Q=10^{-3}$ for masses as large as $m_Q\\approx 10^5$~GeV.\n\nThe limits presented above were derived using data from existing experiments that measure the anomalous heating rate. We next make projections for the future to capture improvements in reducing the heating rate as well as to incorporate mCP detection specific modifications. \n\n\\subsection{Projections}\n\\begin{figure}\n\\centering\n    \\includegraphics[width=\\linewidth]{figs\/future1e3.pdf} \n    \n    \\caption{Event Rate limits and projections for $n_{\\rm lab}=10^3 \\textrm{cm}^{-3}$. Existing limits from heating of the $\\omega^-$ mode are shown (brown shaded) from Fig \\ref{fig:evsm}.  The projection for a search for single events in the BASE experiment \\cite{borchert2019measurement} with energy deposit above $10\\omega^+$ and rate 0.1 event\/hr are shown (dark blue). Here $\\omega_+=77.4$ neV and $\\omega_-= 0.050$ neV.  Next, projections are also shown assuming sensitivity to 1 Event\/year event rates. In pink, we show sensitivity from the existing setup of Ref.~\\cite{borchert2019measurement}. The light blue curve corresponds to swapping a single (anti)-proton with fully ionized Calcium or equivalently trapping 400 Calcium ions in a Coulomb crystal. We also show the reach for a futuristic experiment with energy thresholds of $E_{\\rm min}=10\\omega_{-}$ in green. Finally, reach from a hypothetical electron trap with trap frequency $\\omega=200$~neV and $E_{\\rm min}=2~\\mu$eV is shown in orange.\n    }\n    \\label{fig:futureproj}\n\\end{figure}\n\n\nLimits shown thus far arise from the cumulative heating rate measured. Another promising avenue, is the (non)-observation of individual event rates. In Fig.~\\ref{fig:futureproj}, we compare projections from a non-observation of single event with $E_{\\rm ion}\\ge 10\\omega_+$ with the same parameters as existing data in Ref.~\\cite{borchert2019measurement} in dark blue with the heating of the much smaller $\\omega_-$  in brown for $n_{\\rm lab}=10^3~\\textrm{cm}^{-3}$. \n\nWe find that this projection is near-identical to the heating limit in brown. Both the heating limits as well as the event rate sensitivity are expected to improve in the future. For e.g. heating rates are known to decrease with larger electrode distance or increasing the frequency of the trapped ion~\\cite{brownnutt2015ion}. Whereas the mCP search with the heating rate in the current BASE apparatus is already background limited, the event rate analysis is not. \n\nIt is unclear what the limiting background will be for events with $E_{\\min}=10\\omega$. The harmonic oscillator selection rules prevent excitation of $E_{\\rm ion}>\\omega$. Background gas particles at the existing pressure of $3\\times 10^{-18}$~mbar and $100\\textrm{\\AA}^2$ cross-section, correspond to 1 event every 5000 years. The limiting rate will perhaps be electric field noise, whose estimate is unknown. In Fig.~\\ref{fig:futureproj} we make projections for various experimental choices for an optimistic choice of $\\textrm{1 yr}^{-1}$ event rates for $n_{\\rm lab}=10^3 ~\\textrm{cm}^{-3}$. In pink, we show projections for a trapped proton keeping the existing energy threshold $E_{\\rm min}=10\\omega_+$ and other parameters in \\cite{borchert2019measurement}. \nNext, we explore the reach for highly charged ions as well as ions in a lattice (see for e.g. Ref.~\\cite{micke2020coherent} for recent heating limits from highly charged $^{40}\\textrm{Ar}^{13+}$ ions in a lattice). In light blue, we consider the same set up as \\cite{borchert2019measurement} but consider the reach for fully ionized Calcium, which enhances the Rutherford scattering cross-section due to large ionic charge. This limit is also  equivalent to 400 ions in a Coulomb crystal observed for 1 year.  In green, we consider the effect of a vast and futuristic improvement in sensitivity to the energy jump of a single event, $E_{\\rm min}=10\\omega_-$. Finally we consider a hypothetical electron trap, with trap frequency $\\omega=0.3$~GHz and $E_{\\rm min}$=3~GHz. Despite the large trap frequency required for electron trapping, it is competitive with ions in the small $m_Q$ regime. This is because the momentum transfer is much smaller for electron targets compared to ions for the same energy transfer. \n\n\n\\section{Discussion}\n\\label{sec:discussion}\nGeneric cosmologies should produce non-trivial abundances of well motivated stable particles that make up some or all of the observed DM density provided the reheat temperature is high enough. Such a cosmic density of millicharge particles, with large enough charge, can get stopped on Earth and accumulate through the planet's history, forming an overdense, locally thermalized population. In this work, we analyzed the utility of ion traps as detectors of such an mCP population. Ions trapped in harmonic potentials can detect energy deposits as small as neV with low intrinsic backgrounds. We have shown that the existing measurement of heating rates in Penning and Paul traps~\\cite{borchert2019measurement,goodwin2016resolved,hite2012100} sets strong limits on a wide range of mCP masses and charges as seen in Fig.\\,\\ref{fig:evsm}. These limits on the ambient thermalized population improve on existing limits from levitated spheres by several orders of magnitude with no assumptions about binding of mCPs in material. This can be seen in Fig.\\,\\ref{fig:compexist}.\n\nThese limits can in turn be interpreted as limits on the fraction of virial DM existing in mCPs, $f_Q= \\frac{\\rho_Q}{\\rho_{\\rm DM}}$. We find in Fig.\\,\\ref{fig:virial} that fractions as small as $f_Q\\gtrsim 10^{-12}$ are ruled out for masses around 1-100 GeV for millicharge around $10^{-3}$. Smaller charges can be probed with a similar setup conducted deep underground. \n\nTurning to future prospects, while modest improvements in the observed heating rates are expected, greater strides can be made with single event observation as seen in Fig.\\,\\ref{fig:futureproj}. With event\/year sensitivities, a single (anti)proton can improve upon the existing bounds from heating by one order of magnitude with energy thresholds of $\\approx 100$ neV. Reducing energy threshold can further increase the parameter space that can be probed.\n\nAnother viable direction is using a single fully ionized heavy ion ~\\cite{micke2020coherent} which increases the Rutherford cross-section with mCPs. Multiple ions that form a Coulomb crystal also result in increased sensitivity. While we made projections only for the single event rate for a Coulomb crystal, it is reasonable to expect better signal\/background discrimination by considering in detail the selection rules for collective excitations of the crystal. Heavier ions will however require improvements in energy resolution in order to be sensitive to the same quantum jump as an (anti)proton experiment. \n\nIon experiments have been shown to be transportable~\\cite{cao2016transportable,delehaye2018single,gellesch2020transportable} and can thus be conducted at different altitudes; deep underground in mines, at high altitudes or in space will have drastically different mCP induced heating rate due to the large densities at mines and virial densities in space. This can be used for sensitivity to smaller ambient mCP densities as well as a viable background discrimination tool.\n\n\n\nFinally, electrons can be stably trapped in deep potential wells. Due to its lower mass, an electron can extract $\\approx\\frac{m_p}{m_e}$ more energy than a proton for the same momentum transfer. For masses around 1\\,GeV and below, electron traps might be a promising alternative to probe low charges. It is important to emphasize the complementarity of different traps. Existing Traps of larger sizes have lower heating rates but feature deeper potentials. Hence, they are suited for probing small charges, whereas microtraps are ideal for larger charges. The choice of the trapped charged SM particle, a heavy ion, a proton or an electron can provide optimal sensitivity to different masses due to kinematic matching. For mCP detection purposes, a large trap with shallow trapping potentials will be optimal. \n\nIon trapping has myriad applications including the realization of qubits for a quantum computer. There are significant resources invested in this endeavor which should translate to longer stability, reduced heating rates, scaling to large numbers of ions, as well as the realization of long-term stability in electrons. It is exciting that these improvements translate directly into increased sensitivity of dark matter detection.  \n\\\\\n\\\\\n\\textbf{Note Added:} While this paper was in preparation, ideas related to detecting mCPs using ion traps appeared in Ref~\\cite{Carney:2021irt}. Event rate measurements were explored, which is a subset of the observables discussed in this work. While future projections were made in Ref.~\\cite{Carney:2021irt}, our work additionally provides existing limits from the heating rate, which is already measured in many traps. \nWhile we agree qualitatively with the broad conclusions of the Ref.~\\cite{Carney:2021irt} for the future, that ion traps are ideal mCP detectors, we disagree with the quantitative results.\nIn particular, for the future projections using their parameters, we find different answers for the reach.  The discrepancy may arise from the following differences.\nFirst, in Ref.~\\cite{Carney:2021irt} the scattering is treated  as a  Rutherford scatter on a free, stationary ion. Taking into account the initial energy of the ion makes a significant difference. Second, in Ref.~\\cite{Carney:2021irt}, the ambient number density is conservatively assumed to be the virial one, whereas, our work incorporates the orders of magnitude increase in local density. Third, we analyze the effect of work functions, trapping potentials and mean-free-paths which are very important at large charge. This  provides a maximum charge above which there is no reach. Fourth, we point out that experiments performed near the surface of the earth have access to only a constrained region in charge vs mass, which can be expanded if the experiment was to instead be performed in a deep mine. \nThe initial arxiv version of the publication \\cite{Carney:2021irt} takes into account only virialized velocity distributions, so it did not apply to the parameter space we consider where current ion traps have sensitivity.  The subsequent slowdown to room and cryogenic temperatures were incorporated in the published version which just appeared during the final stages of this work, though with the differences noted above.\n\n\n\\acknowledgments\nWe would like to thank Richard Thompson for useful discussions regarding the particulars of the experimental data in ~\\cite{goodwin2016resolved}.  The work of DB and FSK was supported by the Cluster of Excellence ``Precision Physics, Fundamental Interactions, and Structure of Matter'' (PRISMA+ EXC 2118\/1) funded by the German Research Foundation (DFG) within the German Excellence Strategy (Project ID 39083149), by the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (project Dark-OST, grant agreement No 695405), and by the DFG Reinhart Koselleck project. SU acknowleges support by RIKEN and the Max Planck, RIKEN, PTB Center for Time, Constants and Fundamental Symmetries. \nCS acknowledges support by the ERC (project STEP, grant agreement No 852818) and the Institute of Physics in Mainz. \nPWG and HR acknowledge support from the Simons Investigator Award 824870, DOE Grant DE-SC0012012, NSF Grant PHY-2014215, DOE HEP QuantISED award \\#100495, and the Gordon and Betty Moore Foundation Grant GBMF7946.  This work was also supported by the U.S.~Department of Energy, Office of Science, National Quantum Information (NQI) Science Research Centers through the Fermilab SQMS NQI Center.\n\n\n\\onecolumngrid\n\n\n\\section{Introduction}\nParticles that carry a fraction of the electron charge, $Q=\\epsilon e$, also called millicharged particles (mCPs) are an elegant extension to the standard model (SM). The discovery of their simplest iteration, without the presence of a dark photon, would be a violation of the charge quantization hypothesis. Alternatively, mCPs could also arise as charges quantized under a dark force which kinematically mixes with the SM photon \\cite{Izaguirre:2015eya}. \n\nMillicharged particles have attracted significant interest and there have been many attempts to detect them.\nCollider and beam-dump experiments have been performed to search for mCPs and null results have lead to strong limits on their parameter space in the MeV-TeV mass range. These include LEP~\\cite{akers1995search}, the SLAC millicharge experiment~\\cite{Prinz:1998ua}, neutrino experiments~\\cite{Magill:2018tbb}, the Argoneut experiment~\\cite{Acciarri:2019jly}, MilliQan pathfinder experiment at the LHC~\\cite{Ball:2020dnx} and the BEBC beam dump experiment~\\cite{Marocco:2020dqu}. A plethora of experiments have been proposed to improve these tests and search for still smaller charges~\\cite{Ball:2016zrp,Berlin:2018bsc,Kelly:2018brz,Harnik:2019zee}. At lower masses, stringent limits arise from absence of anomalous emission in stellar environments~\\cite{Davidson:2000hf,Chang:2018rso} at masses below an MeV. The prospect of mCPs making up some or all of the dark matter (DM) has also received a lot of interest over the years. There are robust predictions for their relic density from the early universe~\\cite{Dvorkin:2019zdi,Creque-Sarbinowski:2019mcm}, as well as numerous ways to detect them~\\cite{Knapen:2017ekk,Blanco:2019lrf,Essig:2019kfe,Berlin:2019uco,Kurinsky:2019pgb,Barak:2020fql,Griffin:2020lgd} depending on their mass $m_Q$ and charge. Subcomponent millicharge DM (mCDM) has also been invoked to explain several recent experimental anomalies~\\cite{Barkana:2018qrx,Munoz:2018pzp,Liu:2019knx,Kurinsky:2020dpb,Bloch:2020uzh,Harnik:2020ugb}. \n\nSince mCPs interact with the SM particles through a massless mediator, their transfer cross-section can be large at small velocities. As a result, mCDM with charge large enough to scatter in the atmosphere and Earth overburden of direct detection experiments, rapidly loses its virial kinetic energy and thermalizes with the environment. When it eventually reaches a direct detection experiment, it does not possess enough energy to deposit in the detector and can not be observed at direct detection experiments~\\cite{Emken:2019tni}. However, these mCPs which are now cooled to the ambient temperature, get trapped due to Earth's gravity and build up for the duration of the Earth's existence. This can lead to mCP densities on Earth up to fourteen orders of magnitude larger than that of the virial population in the galaxy \\cite{Pospelov:2020ktu}. This slow, albeit dense population requires novel detection strategies, some of which include mCP particle-antiparticle annihilation in a large-volume detector~\\cite{Pospelov:2020ktu} and accelerating mCPs present in electrostatic accelerator tubes to higher energies, sufficient for subsequent direct detection~\\cite{Pospelov:2020ktu}. \nIf the charge is large enough, negatively charged mCPs can bind with large positively charged SM nuclei, thereby creating fractional charge for a macroscopic material. Searches for such fractional charges bound to a sample material include Millikan-like oil-drop experiments~\\cite{kim2007search}, as well as more recent levitation experiments with microspheres~\\cite{moore2014search,afek2020limits}. However, these limits hinge critically on the assumption that the negatively charged mCPs bind to matter, thereby restricting their validity to only large charges and masses. \n\nIn this work, we point out an alternate search strategy; the remarkably stable trapped ions developed for metrology and quantum information science are the ideal targets to detect an ambient thermalized mCP population. There is a long history of using trapped charged SM  particles for particle physics applications. Trapped ions have been used to measure the anomalous magnetic moment of the electron~\\cite{hanneke2011cavity}, the electron's electric dipole moment~\\cite{cairncross2017precision}, proton and antiproton magnetic moments \\cite{schneider2017double,smorra2017parts}, as well as time variation in fundamental constants which can be induced by ultralight bosonic dark matter~\\cite{Roberts_2020}. Ion traps are also the preeminent candidate for qubits to realize quantum computing \\cite{haffner2008quantum}. \n\nFor all of the applications listed above and especially for magnetic moment measurements and quantum computing, it is important to keep the ion trapped for sufficiently long duration without heating from the surroundings. In the last couple of decades, there has been remarkable progress in reduction of the measured heating rate of trapped ions \\cite{hite2012100}. Further progress is expected and is an active area of research in order to achieve scalability of multi-ion systems and increased sampling rate in precision measurements. We briefly explain why the above properties make ion traps the ideal candidate for detecting mCPs next. \n\n\n\n\\subsection{Summary of Findings}\n\\label{sec:exec}\n\nWe propose the use of ion traps as detectors for millicharged particle (mCP) dark matter.  While an individual ion constitutes a much smaller target mass than any other dark matter direct detection experiment, a trapped ion has significant advantages for detection of mCPs including isolation from the environment, a lower energy threshold for detection, and a larger scattering cross section with mCPs.  These advantages outweigh the small target mass, allowing ion traps to reach many orders of magnitude past other detection methods for mCPs. \n\nSignificant effort has been put into isolating trapped ions from their environments.\nThus trapped ions are now sensitive to small energy depositions down to $\\sim$ neV.  \nThe dense, thermal gas mCPs, if they exist on Earth, would permeate the detector and can scatter off the ion.  \nDepending on the nature of the experiment, it may be possible to measure individual scattering events or just an overall heating rate of the ion.\nThe mCPs are thermalized with the walls of the detector which is held at a temperature much higher than the temperature of the trapped ion. Thus, the higher-energy mCPs can transfer kinetic energy which can be detected either as a single jump of the trapped ion, or an accumulation of several scatters resulting in heating of the ion.  We will discuss both types of signals.\nThe high degree of isolation of the ion achievable in these traps makes them sensitive to scattering rates for mCDM over a wide range of parameter space.\n\nIon traps also make excellent mCDM detectors because of their low energy thresholds that are set by the energy-level spacings in the trap that can be as low as $\\sim \\text{neV}$.  This allows detection of single scattering events with this energy and also allows such low-energy scatters to contribute to the heating rate.\n\nFurthermore, because they are charged, ions make excellent targets for mCP scattering.  mCPs scatter with ions via Rutherford scattering which is greatly enhanced at small relative velocities and momentum transfers.  Since the mCPs are thermalized with the walls, they generally have much lower velocities than virialized dark matter. The corresponding boost to the scattering cross section combined with the large number density of the thermalized mCPs makes direct detection with single ions viable. \n\n\n\nThe rest of this paper is organized as follows.\nIn Sec.\\,\\ref{sec:traps} we provide a description of ion traps. In Sec.\\,\\ref{sec:mcpdyn}, we provide an overview of the mCP density on Earth, as well as their dynamics, including propagation through the trap. The detection signals are explained in Sec.\\,\\ref{sec:signals} and results and projections are presented in Sec.\\,\\ref{sec:results}. We conclude with a discussion in Sec.\\,\\ref{sec:discussion}. \n\n\\section{Ion traps}\n\\label{sec:traps}\nTraps for (single) charged particles belong to the basic tool-set of atomic, molecular, and optical physics \\cite{safronova2018search}. They have wide applications in determinations of atomic masses and fundamental constants,  in precision measurements to test fundamental symmetries, and in quantum information technology. \nTypical ingredients common to all trap experiments are shown in Fig$.\\,$\\ref{fig:Trap}. The core of the experiments are usually sets of electrodes supplied by AC and DC voltages, optionally placed in strong magnetic fields.  \n \\begin{figure}[h!]\n\\centering\n\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/fig1.pdf} \n\n\n    \\caption{Elements common to typical trap experiments. A set of electrodes supplied by AD and DC voltages is mounted in a vacuum chamber, optionally in a strong magnetic field. Cryogenic traps have in addition thermal shielding and cryogenic vacuum chambers at low temperatures in their surroundings. The particles are manipulated with laser beams, microwave and radio-frequency generators. Signals are read-out by detecting image currents or fluorescent light. }\n    \\label{fig:Trap}\n\\end{figure}\nThe electrodes are mounted in vacuum chambers, in some cryogenic trap experiments pressures on the level of 10$^{-18}\\,$mbar are achieved,  which provides ultra-long particle storage times \\cite{sellner2017improved} and enables non-destructive long-term studies at low background. Thermal shielding and an additional insulation vacuum chamber usually surround the inner vacuum chamber. The trapped particles are manipulated, cooled and excited via laser, microwave and radio-frequency-drives. The experimental signals for ultra-sensitive precision studies, frequency measurements, and monitoring of quantum information-processing protocols are either image-currents picked up by sensitive detection circuits, or fluorescence signals. The signals are acquired and processed with spectrum analyzers and charge coupled device (CCD) cameras, respectively.\n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{ ||c|c|c|c|c|c|c|c| }\n\\hline\n Experiment & Type & Ion & $V_z$  &  $T_{\\rm wall}$ & $\\omega_p$ [neV] & $T_{\\rm ion}$[neV] & Heating Rate (neV\/s) \\\\ \n \\hline \n Hite et al, 2012 \\cite{hite2012100}& Paul &$^9\\text{Be}^+$ & 0.1 V&300 K&$\\omega_z=14.8$ &$14.8$&640 \\\\\n Goodwin et al, 2016\\,\\cite{goodwin2016resolved} &Penning&$^{40}\\text{Ca}^+$ & 175\\,V  &300\\,K& $\\omega_z=1.24$ & $1.24$ & 0.37 \\\\\n Borchert et al, 2019\\,\\cite{borchert2019measurement}&Penning&$\\bar{p}$& 0.633\\,V& 5.6\\,K & $\\omega_+=77.4$&7240 & 0.13 \\\\\n &&& & & $\\omega_-=0.050$& &  \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{List of ion traps and the relevant experimental parameters used for setting limits in this paper. The ion used, $V_z$,  potential barrier in the axial direction and $T_{\\rm wall}$, the temperature of the walls of the trap and $\\omega_p$, the fundamental frequency of the trap in the relevant direction are listed. Also listed are $T_{\\rm ion}$, the temperature of the ion in the trap and the measured heating rate. }\n\\label{table:data}\n\\end{table*}\n\n\n\\subsection{Penning Traps}\nIn a Penning trap, there is a strong magnetic field $B_0$ superimposed with an electrostatic quadrupolar potential $\\Phi(z,\\rho)$, which is attractive along the magnetic field axis. The motion of a charged particle in such crossed static fields is composed of harmonic oscillator modes of three independent types. The modified cyclotron and the magnetron modes correspond to oscillations perpendicular to the axis, while the axial mode corresponds to oscillations along the magnetic field lines. Associated with the mode oscillations are the three trap eigenfrequencies $\\nu_+$, $\\nu_-$, and $\\nu_z$, respectively. Room-temperature traps such as that in Ref.\\,\\cite{goodwin2016resolved}, cool and trap ions such as $^{40}\\textrm{Ca}^+$ or $^{9}\\textrm{Be}^+$ to the axial ground state using optical sideband cooling~\\cite{mavadia2014optical}. A delay period after cooling is followed by subsequent spectroscopy which determines the final state of the ion, thus measuring the heating rate. The lowest heating rate achieved thus far at room temperature is reported in Ref.\\,\\cite{goodwin2016resolved} where the increase in the number of phonons of the axial mode was reported to be $\\dot{n}=0.3\/\\textrm{s}$ with the axial frequency $\\nu_z=0.3\\,\\textrm{MHz}$, see  Tab.\\,~\\ref{table:data}. \\\\\nParticularly interesting for the detection of mode energy changes are experiments dedicated to direct measurements of nuclear magnetic moments such as those of the proton \\cite{schneider2017300}, the antiproton \\cite{smorra2017parts}, or $^3$He$^{2+}$ \\cite{mooser2018new}. These experiments operate advanced cryogenic Penning-trap systems consisting of multi-trap assemblies, common to all of them is a so-called analysis trap with a strong superimposed magnetic inhomogeneity $B(z)=B_0+B_2 z^2$, where $B_2$ characterizes its strength \\cite{ulmer2011observation}. \nThe interaction of the magnetic bottle with the particle's magnetic moment $\\mu_z=\\mu_++\\mu_-+\\mu_s$ results in a magnetostatic axial energy $E_{B,z}=-\\mu_z B_z$, where $\\mu_+$ and $\\mu_-$ are the orbital angular magnetic moments associated with the modified cyclotron and the magnetron mode, while $\\mu_s$ is the spin magnetic moment. \nAs a result, the particle's axial frequency $\\nu_z=\\nu_{z,0}+\\Delta\\nu_z (n_+,n_-,m_s)$ becomes a function of the radial trap eigenstates $n_+$ and $n_-$, as well as the spin eigenstate $m_s$ with\n\\begin{align}\n\\Delta\\nu_z=\\frac{h\\nu_+}{4\\pi^2 m \\nu_z}\\frac{B_2}{B_0} \\left(\\left(n_++\\frac{1}{2}\\right)+\\frac{\\nu_-}{\\nu_+}  \\left(n_-+\\frac{1}{2}\\right)+\\frac{g}{2} m_s\\right).\n\\end{align}\nMeasurements of single-particle magnetic moments in Penning traps rely on the detection of axial frequency shifts $\\Delta\\nu_{z,\\text{SF}}$  induced by driven spin quantum transitions $\\Delta m_s=1$. Since nuclear magnetic moments are about three orders of magnitude smaller than the Bohr magneton, these experiments require highest sensitivity with respect to magnetic moments, which is usually achieved by the utilization of magnetic bottle strengths of the order $B_2\\approx100\\,$kT\/m$^2$ to $B_2\\approx400\\,$kT\/m$^2$. Combined with continuous measurements of the axial frequency, such strong magnetic bottles provide excellent resolution of the radial mode energies with \n\\begin{eqnarray}\n\\frac{(\\Delta\\nu_z)}{\\Delta E_\\rho}=\\frac{1}{4\\pi^2m \\nu_z}\\frac{B_2}{B_0}\\approx 1 \\frac{\\text{Hz}}{\\mu \\textrm{eV}}, \n\\end{eqnarray}\nwhile in axial frequency measurements resolutions of order  200$\\,$mHz$\\cdot\\sqrt{\\text{s}}\/\\sqrt{t_\\text{avg}}$ are achieved, $t_\\text{avg}$ being the averaging time which is typically on the order of several tens of seconds. Transition rates in the radial modes\n$(dn_{+,-})\/dt\\propto(n_{+,-}\/\\omega_{+,-} ) S_E (\\omega_{+,-})$ lead to random walks in radial energy space and to axial frequency diffusion. Here  $S_E (\\omega_{+,-})$ is the power spectral density of a noisy background drive, and $n_{+,-}$ is the principal quantum number of the modified cyclotron ($n_+$) \/ magnetron ($n_-$) oscillator. The scaling of the heating rate with $n_{+,-}$ is related to eigenstate-overlap of harmonic oscillator states \\cite{ulmer2011observation}. \\\\ \nBy analyzing time sequences of axial frequency measurements $\\nu_z (t)$, the average radial quantum transition rates are obtained. With a highly optimized trap setup with which the antiproton magnetic moment was measured with 1.5 parts per billion precision, the BASE experiment at CERN reports on the observation of absolute cyclotron transition rates of 6(1) quanta per hour~\\cite{borchert2019measurement}. Together with the determination of the $n_+$ state during the recorded measurement, this result is  consistent with a projected ground state heating rate of 0.1 cyclotron quantum transitions per hour, setting an upper limit which is by a factor of 1800 lower than the best reported Paul-trap heating rates, and by a factor of 230 lower than the best room-temperature Penning trap. These numbers are summarized in Tab.\\,\\ref{table:data}. Note that the antiproton experiments are conducted in a background vacuum of $\\approx 10^{-18}\\,$mbar \\cite{sellner2017improved}, constraining parasitic heating induced by collisions with  background-gas to a level of $4\\times10^{-9}$\/s. \n\n\n\\subsection{Paul Traps}\nPaul traps or radiofrequency traps utilize an oscillating voltage to confine in the perpendicular direction instead of the magnetic field used for the same purpose in the Penning trap. Paul traps have a rich history of being used as mass spectrometers and more recently in building quantum computers~\\cite{brownnutt2015ion}. \n\nThe effective potential in the presence of both DC and AC potentials can be written as\n\\begin{equation}\n    \\psi(x,y,z,t)=\\left(U_{\\rm DC}+V_{\\rm AC} \\cos \\Omega t\\right)\\frac{r^2+2 (z_0^2-z^2)}{r_0^2+2z_0^2} .\n\\end{equation}\nHere $z$ is the axial direction, $r$ the radial direction with the distance to the electrodes given by $r_0$ and $z_0$. The rapidly oscillating potential creates a pseudopotential for charges of both signs and this leads to approximately simple harmonic motion very close to the trap center.  After laser cooling to the ground state, the total heating rate can be measured via Raman sideband technique~\\cite{turchette2000heating}. There has been extensive study of the heating rate in Paul traps and its dependence on distance to electrodes, wall temperature, trap temperature~\\cite{brownnutt2015ion} as well as ion beam treatment of electrodes~\\cite{hite2012100}. Electric field noise from the electrodes has been identified as the dominant heating source, with the dependence on distance scaling as $d^{-2}$~\\cite{brownnutt2015ion} to $d^{-4}$~\\cite{daniilidis2011fabrication}. Although heating rates are lower for bigger traps~\\cite{poulsen2012efficient}, smaller sized traps employ shallower potential wells $\\approx0.1~\\textrm{V}$. As we shall see, this allows mCPs with large charge to reach the trap and hence provide complementary reach at large charge parameter space. Hence we reinterpret limits only from a microtrap~\\cite{hite2012100}.  The heating rates reported in ~\\cite{hite2012100} are $\\dot{n}=43\/\\textrm{s}$ for the axial frequency $\\nu_z=3.6~\\textrm{MHz}$. These numbers are tabulated in Tab.\\,~\\ref{table:data}.\n\n\n\n\\section{Millicharged Particle Dynamics}\n\\label{sec:mcpdyn}\n\\subsection{Terrestrial Accumulation} \nIf mCDM exists and is virialized in the galaxy, there is a non-zero flux of mCPs flowing through the Earth at all times.  This mCDM stops in the atmosphere or rock overburden for large enough charge, and can accumulate on Earth. This process was treated in detail in~\\cite{Emken:2019tni} and the subsequent accumulation in~\\cite{Pospelov:2020ktu}. We provide here a summary of the relevant results of these papers that are used in Section.\\,\\ref{sec:results} and refer the reader to ~\\cite{Pospelov:2020ktu} for details.\n\nFollowing~\\cite{Pospelov:2020ktu} we consider effectively millicharge particles mediated by a dark photon which kinematically mixes with the SM photon. The dark photon mass is taken to be large enough ($m_{A'} \\gtrsim 10^{-12}~\\textrm{eV}$) such that the effect of large-scale electric and magnetic fields can be ignored while considering mCP propagation\\footnote{We leave to upcoming work the calculation of the accumulation of millicharged dark matter in cases where the electric and magnetic fields are relevant \\cite{blprfuture}. Once such calculations are complete, our given limits on the number density in the lab can be translated to limits on dark matter fraction in theses cases.}.  In this limit, the mCPs with mass $m_Q\\ge 1$\\,GeV are stuck on the Earth after thermalization.\\footnote{Millicharged particles with masses below 1\\,GeV can accumulate on Earth temporarily before evaporating. We leave limits on these masses for future work.} The volume averaged DM number density on Earth, $\\langle n_Q \\rangle$ can be several orders of magnitude larger than the virial DM density. It is given by, \n\\begin{align}\n\\label{start}\n\\langle n_Q \\rangle&= \\frac{\\pi R_{\\oplus}^2 v_{\\rm vir}  t_{\\oplus}}{4\/3 \\pi R_{\\oplus}^3}n_{\\rm vir} \\nonumber \\\\\n&\\approx \\frac{3\\times 10^{15} }{\\textrm{cm}^3} \\frac{t_{\\oplus}}{10^{10}\\,\\textrm{y}} f_Q \\frac{\\rm GeV}{m_Q},\n\\end{align}\nwhere, $R_{\\oplus}$ and $t_{\\oplus}$ are the radius and age of the Earth and $n_{\\rm vir}$ and $v_{\\rm vir}$ are the galactic virial number density and velocity. However, the equilibrium density profile is peaked at the Earth core. This density $n_{\\rm static}$ was calculated by taking into account the Earth's temperature and density variations in \\cite{Neufeld:2018slx}. However, the sinking to the Earth's core is not immediate, and there is a dynamic population, the so-called ``traffic-jam\" density, $n_{\\rm tj}$, given by,\n\\begin{equation}\n    n_{\\rm tj}=n_{\\rm vir}\\frac{v_{\\rm vir}}{v_{\\rm term}} .\n\\end{equation}\nHere, $v_{\\rm term}$ is the terminal velocity in rock, given in \\cite{Pospelov:2020ktu}. Finally, the density in the laboratory, $n_{\\rm lab}$ is given by, \n\\begin{equation}\nn_{\\rm lab} = \\textrm{Max}\\left( n_{\\rm static}, \\textrm{Min}\\left( n_{\\rm tj} , \\langle n_Q \\rangle \\right)\\right)\n\\label{eq:regimes}\n\\end{equation}\n\nThese results are applicable only for $\\epsilon$ large enough, such that it stops in the corresponding overburden. According to \\cite{Pospelov:2020ktu} this is valid only for\n\\begin{align}\n\\epsilon&\\gtrsim 2\\times10^{-4} \\sqrt{\\frac{m_Q}{\\rm GeV}} \\quad &&\\textrm{surface,} \\nonumber \\\\\n&\\gtrsim 3\\times 10^{-6}\\sqrt{\\frac{m_Q}{\\rm GeV}} \\quad &&\\textrm{1 km mine.} \n\\label{eq:applicable}\n\\end{align}\n\nIt is also important to comment on the asymmetry of the mCP population. If the mCDM is a symmetric population with equal number of particles and antiparticles, accumulation on Earth can result in Sommerfeld-enhanced annihilations which prevent build up. We consider here the asymmetric case such that opposite charges are carried by different species just like the the SM proton and electron, such that annihilations are absent. In this scenario, for large enough $\\epsilon$, the negatively charged mCPs can form deep bound states with the large positively nuclei, whereas the positive mCPs can only bind with the less massive electrons which also have smaller charge compared to the heavy SM nuclei. In ref.~\\cite{Pospelov:2020ktu} it was pointed out that positive mCPs with $\\epsilon \\ge 0.042$  bind to electrons at 300 K (room temperature). We find that such bound states are temporary with electrons rapidly preferring to bind with ions which possess larger charge than the positive mCPs. Thus the terrestrial population of positive mCPs remains free of binding for $\\epsilon\\lesssim 1$ and is present as a locally thermalized population that is diffusing everywhere. Thus, the results we derive apply to all positive charges with $\\epsilon \\lesssim 1$ and negative charges which do not bind with nuclei, i.e \\cite{Pospelov:2020ktu} $\\epsilon < \\frac{m_e}{\\mu_{Q,N}}$ where $\\mu_{Q,N}$ is the reduced mass of the mCDM-nuclear system.  \n\n\nIt is important to emphasize that the limits put on the ambient number density are applicable to mCPs charged directly under the SM photon as well as to ones mediated by a dark photon as long as the dark photon mass is below the relevant momentum transfer for scattering with ions which is around 1 eV.   \n\n\\subsection{Passage Through Apparatus}\n\n\\label{trappassage}\nWe next turn to the trajectory of mCPs through the trap peripherals in order to reach the trapped ion.  We need to know the density and temperature of the mCPs reaching the ion.  In this subsection we explain the main factors entering the calculation, but as the calculation itself is somewhat involved we leave the details to Appendix \\ref{Appendix:passage}.  Our main result is in Eq.\\,\\eqref{eqn:qion} relating the number density of mCPs at the position of the ion, $n^Q_{\\rm ion}$, to the ambient number density on the Earth $n_{\\rm lab}$.  \nThis is the equation we use to set our limits on mCPs.\n\nAt equilibrium, mCPs are expected to have roughly uniform density near the Earth's surface, including permeating all materials.  However the conditions of the experiment can affect this naive expectation for several reasons.\n\nFirst, some of the ion traps we consider are cryogenic.  Over essentially all of our parameter space the mCPs have a short interaction length in material and so will rapidly thermalize to the cryostat temperature as they enter the experiment.  By itself this would lead to an increase in mCP density by a factor linear in mCP velocity ($\\propto \\sqrt{T}$) because the fluxes entering and leaving the cryostat must be equal in equilibrium.\n\nSecond, the ion traps are surrounded by metal which has a work function that can affect the passage of mCPs.  This is only relevant for mCPs of relatively large charge ($\\epsilon \\gtrsim 10^{-2}$) but for those it can be a significant effect.  The work function for mCPs, which we will call $\\epsilon \\phi$, is not simply $\\epsilon$ times the work function for electrons.  For electrons the work function arises from several contributions of varying signs including e.g.~the binding to the lattice of nuclei, the Fermi sea of other electrons, and surface effects such as the ``double layer\" or the image charge potential.  Several of these do not apply or are negligible for mCPs.  Recall we are only considering positively charged mCPs since the negative ones may be stuck deeply bound to some nucleus somewhere on the Earth.  Thus the main effects are repulsion by the double layer and possibly also repulsion from the nuclei.   We consider the work function for mCPs in more detail in  Appendix \\ref{Appendix:work function}.  Our conclusion is that the work function is repulsive for positively charged mCPs and thus every metal sheet provides a barrier for mCPs to cross. \nThe size of the potential barrier that has to be crossed is $\\phi \\sim \\text{few} \\, \\text{eV}$.\nNote that for an experiment at room temperature ($T \\sim 0.03 \\, \\text{eV}$) the metal barriers are then irrelevant for charges $\\epsilon \\lesssim 10^{-1}$ because the Boltzmann tail easily pushes a fast enough rate of mCPs over the barrier.  For a cryogenic experiment at $T \\sim 6 \\, \\text{K} \\sim 5 \\times 10^{-4} \\, \\text{eV}$, the metal barrier will be relevant for charges $\\epsilon \\gtrsim 10^{-3}$ and essentially insurmountable for charges $\\epsilon \\gtrsim 10^{-2}$.  As we show in Appendix \\ref{Appendix:passage}, the most important effect comes from an experiment encased in two different metals where the work function for mCPs rises from the outer metal to the inner metal.  We will take this difference to be $\\Delta \\phi = 3 \\, \\text{eV}$ for all experiments we consider since this will be a conservative estimate as we show in Appendix \\ref{Appendix:work function}.\n\n\nThird, the ions are always in a region of ultrahigh vacuum.  This means that pumps were used to remove the Standard Model (SM) particles.  Given a short interaction length of the mCP in materials, these pumps could remove the mCPs from the ion chamber as well.  In one of the experiments we consider (Goodwin et al.~\\cite{goodwin2016resolved}) this effect is not relevant because the trap is at room temperature and the region of sensitivity is at low enough $\\epsilon$ that the millicharges pass easily through the walls\\footnote{The sensitivity is limited to be below $10^{-3}$ charge because of the large trap potential as will be discussed in Section \\ref{sec:results}.}.  In the other two experiments we consider (Hite et al.~\\cite{hite2012100} and Borchert et al.~\\cite{borchert2019measurement}), the vacuum pumps are turned off well before the actual data taking is begun.  In the case of Borchert this is at least a year, while for Hite we conservatively assume it is only a day\\footnote{The final answer is only logarithmically sensitive to this timescale anyway.}.  And of course mCPs are always continually flowing in from the walls of the vacuum chamber.  This would rapidly refill the trap region and this effect would not be relevant, except for the largest charges where the refill can be slow because of the work function of the surrounding metal.  This does mean that, depending on the parameters of the mCP and of the ion trap, the number density in the trap may be either in the equilibrium regime or in the filling regime.  This is why Eq.\\,\\eqref{eqn:final trap density} has two different regimes.  Eq.\\,\\eqref{eqn:final trap density} relates the number density of mCPs inside the trap $n_{\\rm trap}$ to the ambient number density on the Earth $n_{\\rm lab}$.  There is then one remaining step to find the number density of mCPs at the position of the ion.\n\nFourth, the ion trap itself has applied electromagnetic fields to trap the ion.  These can affect the passage of the mCPs, though again this is only relevant for larger charges $\\epsilon \\gtrsim 10^{-3}$.\nTab.\\,\\ref{table:data} lists parameters of the various experiments we consider.  All the experiments use an electric DC potential to confine the ion in the axial direction.  The height of the potential barrier along the axial direction is listed as $V_z$.  The mCPs in the trap are thermalized to the temperature of the walls of the trap $T_{\\rm wall}$.  Starting from the number density of mCPs calculated inside the trap $n_{\\rm trap}$ in Eq.\\,\\eqref{eqn:final trap density}, we take a Boltzmann suppression on the number density which can make it up the axial barrier height.  Thus we take the final number density at the position of the ion to be given by Eq.\\,\\eqref{eqn:qion}.  For the experiments we consider this Boltzmann suppression is relevant for setting the ceiling (largest $\\epsilon$ values) of the Goodwin regions in Figure \\ref{fig:evsm}, but is irrelevant for the other regions. \nThe Penning traps use a magnetic field to confine the ions in the radial direction.  This B-field will cause mCPs with a large enough charge to circle around the axial B-field lines but they are still free to move along the axis.  The magnetic field does not change the phase-space density of the mCPs and so it does not significantly affect our signal.  The Paul traps use an RF potential to confine the ion in the radial direction.  This potential around the minimum is locally attractive for the positive mCP independent of the sign of the charge of the trapped ion.  An mCP coming from far outside the RF fields in a radial direction could give a barrier in principle.  However for mCPs approaching along the axial direction, the RF pseudopotential in the axial direction is very weak and negligible.  And then  such mCPs will actually be concentrated in the radial direction towards the ion at the center of the trap since the potential is locally attractive.  We conservatively ignore this possible (Sommerfeld-like) enhancement though it could be large. For large enough mCP masses, $m_Q \\gtrsim 10^{10} \\textrm{GeV}$, the free-fall under gravity \ncan generate velocities much larger than the thermal velocities assumed. This can increase the heating rate further. We conservatively ignore this effect and leave its consideration for future work.\n\n\\section{Observables for Millicharged Particles in Ion Traps}\n\\label{sec:signals}\nIn this section we consider the interaction of ambient mCPs with ion traps to identify observables for detection. As seen in Section.~\\ref{trappassage}, the mCPs enter the ion trap with effective temperature $T_Q \\ge T_{\\rm wall}$, where $T_{\\rm wall}$ is the wall temperature. $T_{\\rm wall}= 5.6$ K for the cryogenic trap we consider~\\cite{borchert2019measurement} and $T_{\\rm wall}=T_{\\rm room}\\approx 300$ K for room temperature traps. The ions in the trap are at a much colder temperature $T_{\\rm ion} \\ll T_{\\rm wall}$. With this hierarchy of temperatures, the mCPs can cause two types of signal.\n\nThe first signal involves individual scattering events that impart energy $E_{\\rm ion}$ to the ion thus leading to a change in its harmonic oscillator quantum number. This signal is very similar to dark matter scattering in a conventional dark matter detector. The rate of events has to be slower than the rate at which the ions are interrogated. $E_{\\rm ion}$ also needs to exceed the energy resolution $E_{\\rm res}$ for detection. \n\nThe second type of signal is the heating of the trapped ion due to collisions with multiple mCPs. In this case, the individual hits $E_{\\rm ion}$ can be smaller than $E_{\\rm res}$ and only the sum needs to exceed this resolution. \n\nFor both of these signal types, we only consider individual energy transfers $E_{\\rm ion}$ much larger than the typical energy spacing of the trap $\\omega$, for concreteness $E_{\\rm ion} \\ge 10\\omega$. \\footnote{In principle, energy transfers $E_{\\rm ion} =n\\times \\omega$ where all integers $n\\ge 1$ are allowed. We restrict $E_{\\rm ion} \\ge 10 \\times \\omega$, a conservative choice that helps avoid form-factor calculations.} In this limit, the trapped ion can be approximated as a free particle with initial energy $T_{\\rm ion}$ and final energy $T_{\\rm ion}+E_{\\rm ion}$. Equivalently, for energy transfers much larger than the spacing, the form-factor that incorporates wave-function overlap can be approximated to unity.  \nWe start quantifying both of these signals by the angular differential cross-section given by the Rutherford formula,\n\\begin{equation}\n    \\frac{d\\sigma}{d\\Omega}=\\frac{2\\pi\\alpha^2 \\epsilon^2}{\\mu^2 v_{\\rm rel}^4\\left(1-\\cos\\theta\\right)^2} ,\n\\end{equation}\nwhere $\\theta$ is the scattering angle. We next introduce kinematic variables that  simplify the computation of the scattering rate. The results are presented here, with the details of the derivation presented in Appendix.~\\ref{app1}. The incoming mCP and trapped ion velocities are assumed to be $\\mathbf{v_Q}$ and $\\mathbf{v_{\\rm ion}}$ respectively. \nThe center of mass (CM) velocity $\\mathbf{v_{CM}}$ is given by,\n\\begin{equation}\n    \\mathbf{v_{CM}}=\\frac{\\left(m_{\\rm ion} \\mathbf{v_{\\rm ion}}+m_Q \\mathbf{v_Q}\\right)}{m_{\\rm ion}+m_Q} .\n\\end{equation}\n\nThe change in velocity of the ion, $\\mathbf{\\Delta v_{\\rm ion}}$ is given by,\n\\begin{align}\n    \\mathbf{\\Delta v_{\\rm ion}}=\\frac{m_Q}{m_{\\rm ion}+m_Q}\n    \\left[\\left(\\cos \\theta-1 \\right)\n    \\left(\\mathbf{v_{\\rm ion}}-\\mathbf{v_Q}\\right)\n    \\right. \n    \\nonumber \\\\ \n    \\left.\n    +\\sin \\theta |\\mathbf{v_{\\rm ion}}-\\mathbf{v_Q}|\n    \\mathbf{n_{\\perp}}\n    \\right] .\n\\end{align}\nThe transferred energy $E_{\\rm ion}$ is given by,\n\\begin{equation}\n    E_{\\rm ion} = m_{\\rm ion} \\mathbf{v_{CM}}.\\mathbf{\\Delta v_{\\rm ion}} .\n\\end{equation}\n\nGiven a threshold $E_{\\rm thr}$, the single event rate $R_{\\rm single}$ with energy transfer $E_{\\rm ion}$ above this threshold is,\n\\begin{align}\n    R_{\\rm single}\\left(E_{\\rm ion}\\ge E_{\\rm thr}\\right)=n^Q_{\\rm ion} \\int d^3 \\mathbf{v_Q} g_Q  \\int d^3 \\mathbf{v_{\\rm ion}} g_{\\rm ion}  \\int d \\Omega  \\nonumber \\\\ |v_Q-v_{\\rm ion}| \\frac{d\\sigma} {d \\Omega} \\Theta \\left(|E_{\\rm ion}|-E_{\\rm thr}\\right) . \n\\end{align}\nHere $n^Q_{\\rm ion} $ is the number density of mCPs at the ion position and is given by Eq.\\,\\eqref{eqn:qion}, $g_{\\textrm{ion}(Q)}$ is the Maxwell--Boltzmann distribution for the ion\/mCP with temperatures $T_{\\rm ion}$ and $T_{\\rm wall}$, respectively. \n\n\nThe heating rate per ion, $\\dot{H}$ can be computed through,\n\\begin{align}\n    \\dot{H}&=n^Q_{\\rm ion}  \\int d^3 \\mathbf{v_Q} g_Q  \\int d^3 \\mathbf{v_{\\rm ion}} g_{\\rm ion}  \\int d \\Omega |v_Q-v_{\\rm ion}|  \\nonumber \\\\ &\\frac{d\\sigma} {d \\Omega}E_{\\rm ion} \\Theta \\left(|E_{\\rm ion}|-E_{\\rm thr}\\right) \\Theta\\left(E_{\\rm samp}-|E_{\\rm ion}|\\right) . \n\\end{align}\nThe Heaviside theta function ensures the inequality $E_{\\rm thr}\\le E_{\\rm ion} \\le E_{\\rm samp}$. Here $E_{\\rm samp}$ is defined so as to prevent the average heating rate from including contribution from extremely rare events. It is defined through,\n\\begin{equation}\nR_{\\rm single}\\left(E_{\\rm ion}\\ge E_{\\rm samp}\\right) t_{\\rm obs}=1 ,\n\\end{equation}\nwhere $t_{\\rm obs}$ is the total observation time. \n\n\n \\section{Results}\n \\label{sec:results}\n \n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}{0.48\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/comparetraps1e3.pdf} \n\n\\end{subfigure}\n\\begin{subfigure}{0.48\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/evsmmainall.pdf} \n \n\\end{subfigure}\n    \\caption{Compilation of new limits using existing heating measurements from various traps in Table \\ref{table:data}: a room temperature Paul trap, Hite et al.~\\cite{hite2012100}, a room temperature Penning trap, Goodwin et al.~\\cite{goodwin2016resolved} and a cryogenic penning trap, Borchert et al. \\cite{borchert2019measurement}. {\\bf (left)} Comparison between traps for an ambient density $n_{\\rm lab}=10^3 \\textrm{cm}^{-3}$; {\\bf (right)} Combined limits from the three traps for different  $n_{\\rm lab}$} \n    \\label{fig:evsm}\n\\end{figure*}\n In this section, we set constraints on mCPs and make projections for the future by using the expressions for the signal rates derived in Sec.\\,\\ref{sec:signals}. \n \n \\subsection{Limits from existing measurements}\n To obtain existing limits, we use the data presented in Tab.\\,\\ref{table:data}. All of the trap parameters for the $^9\\text{Be}^+$  trap are taken from \\cite{hite2012100} while the trap depth is conservatively taken to be $V_z=0.1$~V, an order of magnitude larger than the typical potential depths in microtraps\\footnote{see for e.g.~\\cite{warring2013techniques} from the same group where the potential depth is report to be $V_z$=5 mV}. While the rest of the parameters for the $^{40}\\textrm{Ca}^+$ experiment are provided in ~\\cite{goodwin2016resolved}, we obtained the potential depth $V_z=175$~V from the authors. Finally, for ref.\\,\\cite{borchert2019measurement}, we use the parameters $T_{\\rm wall}=5.6$~K and $V_z$=0.6~V. While the analysis in ~\\cite{borchert2019measurement} dealt with measuring the cyclotron mode ($\\omega_+)$, the observable heating rate is equivalently a limit on the magnetron mode ($\\omega_-$) also. While individual jumps in $\\omega_-$ are unobservable with existing precision, the frequency shift due to the heating rate accumulates. Since the Rutherford cross-section increases for smaller energy transfers, we use $\\omega_-$ to convert existing heating limits into limits on mCPs. \n\n We start by plotting existing limits in Fig.\\ref{fig:evsm} in the millicharge $\\epsilon$ vs mass $m_Q$ parameter space for contours of constant ambient density $n_{\\rm lab}$. As mentioned earlier, for mCDM, the lab density is expected to be orders of magnitude larger than the virial density, i.e. $n_{\\rm lab} \\gg n_{\\rm vir}$. In the left panel, limits arising from the antiproton trap \\cite{borchert2019measurement} in blue, $^{40}\\textrm{Ca}^+$~\\cite{goodwin2016resolved} in red and $^{9}\\textrm{Be}$ in green \\cite{hite2012100} for an ambient density of $n_{\\rm lab}=10^3\/\\textrm{cm}^3$. The dominant limits arise from the antiproton trap owing to its superior heating rate as seen in Tab.\\,\\ref{table:data}. However, since it is a cryogenic trap, the limits disappear at $\\epsilon \\approx 10^{-2}$ owing to the suppression arising from mCPs finding it increasingly difficult to penetrate metals due to their work function. As explained in Section.~\\ref{sec:mcpdyn} and Appendix.~\\ref{Appendix:passage}, this factor roughly scales as $e^{-\\frac{\\epsilon \\Delta \\phi}{T_{\\rm wall}}}$ where $\\Delta \\phi$ is the difference in work functions between two adjacent metals. Hence this suppression is ameliorated for room temperature traps like the $^9\\textrm{Be}^+$ trap \\cite{hite2012100} in green which extends to $\\epsilon\\approx 0.1$. While the $^{40}\\textrm{Ca}^+$ trap in red is also at room temperature, the trap is at a potential of 175~V and thus mCP charges above $\\epsilon \\approx 10^{-3}$ do not reach the ion. However, there is reach to higher mCP masses for this trap as a result of the ion being more massive. The right panel of Fig.~\\ref{fig:evsm}\n corresponds to combined limits from these three experiments for different $n_{\\rm lab}$.  \n \n \\begin{figure}\n\\centering\n\n    \\includegraphics[width=\\linewidth]{figs\/compwithexisting.pdf} \n\n    \\caption\n    Comparison of limits derived in this work from ~\\cite{hite2012100,goodwin2016resolved,borchert2019measurement} with existing limits from Oil Drop~\\cite{kim2007search}, levitation experiments~\\cite{moore2014search,afek2020limits} and LEP~\\cite{Marocco:2020dqu} for $m_Q=10$ GeV.}\n    \\label{fig:compexist}\n\\end{figure}\n \n \n \n As is clear from both figures, existing data for anomalous heating in traps sets exquisite bounds on mCPs thermalized locally. Bounds are applicable to orders of magnitude in the mCP mass $m_Q$ as well as many orders of magnitude in charge. Number densities as small as $n_{\\rm lab}=1\\, \\textrm{cm}^{-3}$ are ruled out around the $\\epsilon\\approx 10^{-3}$ and $m_Q\\approx 10\\, \\textrm{GeV}$ parameter point. \n \nIn order to compare these limits on the ambient mCP population to ones that already exist in literature, we fix the mCP mass $m_Q=10$\\,GeV and show limits in the $n_{\\rm lab}$ vs $\\epsilon $ plane in Fig.~\\ref{fig:compexist}. The same color coding as Fig.~\\ref{fig:evsm} is followed. In gray we show limits from LEP~\\cite{Marocco:2020dqu}, as well as limits on mCPs bound in matter arising from Oil drop experiments~\\cite{kim2007search}, and levitation experiments~\\cite{moore2014search,afek2020limits}. As noted earlier, the limits on mCPs bound in matter are applicable only to negative mCPs with large enough charge such that binding with SM nuclei is possible. Furthermore, if mCP-SM bound states exist, there is no guarantee for these bound states to be evenly distributed all over the Earth. However, for the positive mCPs none of these caveats apply and they thermalize and distribute themselves over the entire Earth volume. Regardless, as seen in Fig.~\\ref{fig:compexist} the limits obtained from ion traps are orders of magnitude stronger than the levitation experiments. For $\\epsilon\\approx 3\\times 10^{-3}$, lab densities as small as $n_{\\rm lab} \\gtrsim 1 \\textrm{cm}^{-3}$ are ruled out by the measured heating rate at the antiproton experiment \\cite{borchert2019measurement}. \n \\begin{figure}\n\\centering\n\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/dmplot.pdf} \n  \n\n    \\caption{Limits and projections on the virial DM fraction in mCPs as a function of $\\epsilon$ and $m_Q$. Since the current experiments were all conducted at the surface, the robust limits are only above the ``$>$1m Thermalization\" line where the colored contours are solid. For a hypothetical deep mine experiment, the dashed part of the contours are also accessible.}\n    \\label{fig:virial}\n\\end{figure}\n\nNext, in Fig.~\\ref{fig:virial} we convert limits on $n_{\\rm lab}$ into limits on the fraction of virial DM existing in mCPs, $f_Q= \\frac{\\rho_Q}{\\rho_{\\rm DM}}$. For this purpose we use $n_{\\rm lab}$ from Eq.\\,\\ref{eq:regimes}. Existing limits from colliders are shown in gray. The solid parts of the colored contours correspond to the region where the incoming virial DM gets thermalized within 1 meter and hence the robust current limits we put are restricted to this region i.e. above the top black line. The dashed lines show the reach for an identical heating rate experiment that is conducted in a deep mine at 1km depth. Virial DM fractions as small as $f_Q\\approx 10^{-12}$ are already ruled using existing heating data for DM masses in the 1-10 GeV mass range. For heavier masses, the terminal velocity is larger and hence the traffic jam densities are smaller. Nonetheless we set limits on DM fractions as small as $f_Q=10^{-3}$ for masses as large as $m_Q\\approx 10^5$~GeV.\n\nThe limits presented above were derived using data from existing experiments that measure the anomalous heating rate. We next make projections for the future to capture improvements in reducing the heating rate as well as to incorporate mCP detection specific modifications. \n\n\\subsection{Projections}\n\\begin{figure}\n\\centering\n    \\includegraphics[width=\\linewidth]{figs\/future1e3.pdf} \n    \n    \\caption{Event Rate limits and projections for $n_{\\rm lab}=10^3 \\textrm{cm}^{-3}$. Existing limits from heating of the $\\omega^-$ mode are shown (brown shaded) from Fig \\ref{fig:evsm}.  The projection for a search for single events in the BASE experiment \\cite{borchert2019measurement} with energy deposit above $10\\omega^+$ and rate 0.1 event\/hr are shown (dark blue). Here $\\omega_+=77.4$ neV and $\\omega_-= 0.050$ neV.  Next, projections are also shown assuming sensitivity to 1 Event\/year event rates. In pink, we show sensitivity from the existing setup of Ref.~\\cite{borchert2019measurement}. The light blue curve corresponds to swapping a single (anti)-proton with fully ionized Calcium or equivalently trapping 400 Calcium ions in a Coulomb crystal. We also show the reach for a futuristic experiment with energy thresholds of $E_{\\rm min}=10\\omega_{-}$ in green. Finally, reach from a hypothetical electron trap with trap frequency $\\omega=200$~neV and $E_{\\rm min}=2~\\mu$eV is shown in orange.\n    }\n    \\label{fig:futureproj}\n\\end{figure}\n\n\nLimits shown thus far arise from the cumulative heating rate measured. Another promising avenue, is the (non)-observation of individual event rates. In Fig.~\\ref{fig:futureproj}, we compare projections from a non-observation of single event with $E_{\\rm ion}\\ge 10\\omega_+$ with the same parameters as existing data in Ref.~\\cite{borchert2019measurement} in dark blue with the heating of the much smaller $\\omega_-$  in brown for $n_{\\rm lab}=10^3~\\textrm{cm}^{-3}$. \n\nWe find that this projection is near-identical to the heating limit in brown. Both the heating limits as well as the event rate sensitivity are expected to improve in the future. For e.g. heating rates are known to decrease with larger electrode distance or increasing the frequency of the trapped ion~\\cite{brownnutt2015ion}. Whereas the mCP search with the heating rate in the current BASE apparatus is already background limited, the event rate analysis is not. \n\nIt is unclear what the limiting background will be for events with $E_{\\min}=10\\omega$. The harmonic oscillator selection rules prevent excitation of $E_{\\rm ion}>\\omega$. Background gas particles at the existing pressure of $3\\times 10^{-18}$~mbar and $100\\textrm{\\AA}^2$ cross-section, correspond to 1 event every 5000 years. The limiting rate will perhaps be electric field noise, whose estimate is unknown. In Fig.~\\ref{fig:futureproj} we make projections for various experimental choices for an optimistic choice of $\\textrm{1 yr}^{-1}$ event rates for $n_{\\rm lab}=10^3 ~\\textrm{cm}^{-3}$. In pink, we show projections for a trapped proton keeping the existing energy threshold $E_{\\rm min}=10\\omega_+$ and other parameters in \\cite{borchert2019measurement}. \nNext, we explore the reach for highly charged ions as well as ions in a lattice (see for e.g. Ref.~\\cite{micke2020coherent} for recent heating limits from highly charged $^{40}\\textrm{Ar}^{13+}$ ions in a lattice). In light blue, we consider the same set up as \\cite{borchert2019measurement} but consider the reach for fully ionized Calcium, which enhances the Rutherford scattering cross-section due to large ionic charge. This limit is also  equivalent to 400 ions in a Coulomb crystal observed for 1 year.  In green, we consider the effect of a vast and futuristic improvement in sensitivity to the energy jump of a single event, $E_{\\rm min}=10\\omega_-$. Finally we consider a hypothetical electron trap, with trap frequency $\\omega=0.3$~GHz and $E_{\\rm min}$=3~GHz. Despite the large trap frequency required for electron trapping, it is competitive with ions in the small $m_Q$ regime. This is because the momentum transfer is much smaller for electron targets compared to ions for the same energy transfer. \n\n\n\\section{Discussion}\n\\label{sec:discussion}\nGeneric cosmologies should produce non-trivial abundances of well motivated stable particles that make up some or all of the observed DM density provided the reheat temperature is high enough. Such a cosmic density of millicharge particles, with large enough charge, can get stopped on Earth and accumulate through the planet's history, forming an overdense, locally thermalized population. In this work, we analyzed the utility of ion traps as detectors of such an mCP population. Ions trapped in harmonic potentials can detect energy deposits as small as neV with low intrinsic backgrounds. We have shown that the existing measurement of heating rates in Penning and Paul traps~\\cite{borchert2019measurement,goodwin2016resolved,hite2012100} sets strong limits on a wide range of mCP masses and charges as seen in Fig.\\,\\ref{fig:evsm}. These limits on the ambient thermalized population improve on existing limits from levitated spheres by several orders of magnitude with no assumptions about binding of mCPs in material. This can be seen in Fig.\\,\\ref{fig:compexist}.\n\nThese limits can in turn be interpreted as limits on the fraction of virial DM existing in mCPs, $f_Q= \\frac{\\rho_Q}{\\rho_{\\rm DM}}$. We find in Fig.\\,\\ref{fig:virial} that fractions as small as $f_Q\\gtrsim 10^{-12}$ are ruled out for masses around 1-100 GeV for millicharge around $10^{-3}$. Smaller charges can be probed with a similar setup conducted deep underground. \n\nTurning to future prospects, while modest improvements in the observed heating rates are expected, greater strides can be made with single event observation as seen in Fig.\\,\\ref{fig:futureproj}. With event\/year sensitivities, a single (anti)proton can improve upon the existing bounds from heating by one order of magnitude with energy thresholds of $\\approx 100$ neV. Reducing energy threshold can further increase the parameter space that can be probed.\n\nAnother viable direction is using a single fully ionized heavy ion ~\\cite{micke2020coherent} which increases the Rutherford cross-section with mCPs. Multiple ions that form a Coulomb crystal also result in increased sensitivity. While we made projections only for the single event rate for a Coulomb crystal, it is reasonable to expect better signal\/background discrimination by considering in detail the selection rules for collective excitations of the crystal. Heavier ions will however require improvements in energy resolution in order to be sensitive to the same quantum jump as an (anti)proton experiment. \n\nIon experiments have been shown to be transportable~\\cite{cao2016transportable,delehaye2018single,gellesch2020transportable} and can thus be conducted at different altitudes; deep underground in mines, at high altitudes or in space will have drastically different mCP induced heating rate due to the large densities at mines and virial densities in space. This can be used for sensitivity to smaller ambient mCP densities as well as a viable background discrimination tool.\n\n\n\nFinally, electrons can be stably trapped in deep potential wells. Due to its lower mass, an electron can extract $\\approx\\frac{m_p}{m_e}$ more energy than a proton for the same momentum transfer. For masses around 1\\,GeV and below, electron traps might be a promising alternative to probe low charges. It is important to emphasize the complementarity of different traps. Existing Traps of larger sizes have lower heating rates but feature deeper potentials. Hence, they are suited for probing small charges, whereas microtraps are ideal for larger charges. The choice of the trapped charged SM particle, a heavy ion, a proton or an electron can provide optimal sensitivity to different masses due to kinematic matching. For mCP detection purposes, a large trap with shallow trapping potentials will be optimal. \n\nIon trapping has myriad applications including the realization of qubits for a quantum computer. There are significant resources invested in this endeavor which should translate to longer stability, reduced heating rates, scaling to large numbers of ions, as well as the realization of long-term stability in electrons. It is exciting that these improvements translate directly into increased sensitivity of dark matter detection.  \n\\\\\n\\\\\n\\textbf{Note Added:} While this paper was in preparation, ideas related to detecting mCPs using ion traps appeared in Ref~\\cite{Carney:2021irt}. Event rate measurements were explored, which is a subset of the observables discussed in this work. While future projections were made in Ref.~\\cite{Carney:2021irt}, our work additionally provides existing limits from the heating rate, which is already measured in many traps. \nWhile we agree qualitatively with the broad conclusions of the Ref.~\\cite{Carney:2021irt} for the future, that ion traps are ideal mCP detectors, we disagree with the quantitative results.\nIn particular, for the future projections using their parameters, we find different answers for the reach.  The discrepancy may arise from the following differences.\nFirst, in Ref.~\\cite{Carney:2021irt} the scattering is treated  as a  Rutherford scatter on a free, stationary ion. Taking into account the initial energy of the ion makes a significant difference. Second, in Ref.~\\cite{Carney:2021irt}, the ambient number density is conservatively assumed to be the virial one, whereas, our work incorporates the orders of magnitude increase in local density. Third, we analyze the effect of work functions, trapping potentials and mean-free-paths which are very important at large charge. This  provides a maximum charge above which there is no reach. Fourth, we point out that experiments performed near the surface of the earth have access to only a constrained region in charge vs mass, which can be expanded if the experiment was to instead be performed in a deep mine. \nThe initial arxiv version of the publication \\cite{Carney:2021irt} takes into account only virialized velocity distributions, so it did not apply to the parameter space we consider where current ion traps have sensitivity.  The subsequent slowdown to room and cryogenic temperatures were incorporated in the published version which just appeared during the final stages of this work, though with the differences noted above.\n\n\n\\acknowledgments\nWe would like to thank Richard Thompson for useful discussions regarding the particulars of the experimental data in ~\\cite{goodwin2016resolved}.  The work of DB and FSK was supported by the Cluster of Excellence ``Precision Physics, Fundamental Interactions, and Structure of Matter'' (PRISMA+ EXC 2118\/1) funded by the German Research Foundation (DFG) within the German Excellence Strategy (Project ID 39083149), by the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (project Dark-OST, grant agreement No 695405), and by the DFG Reinhart Koselleck project. SU acknowleges support by RIKEN and the Max Planck, RIKEN, PTB Center for Time, Constants and Fundamental Symmetries. \nCS acknowledges support by the ERC (project STEP, grant agreement No 852818) and the Institute of Physics in Mainz. \nPWG and HR acknowledge support from the Simons Investigator Award 824870, DOE Grant DE-SC0012012, NSF Grant PHY-2014215, DOE HEP QuantISED award \\#100495, and the Gordon and Betty Moore Foundation Grant GBMF7946.  This work was also supported by the U.S.~Department of Energy, Office of Science, National Quantum Information (NQI) Science Research Centers through the Fermilab SQMS NQI Center.\n\n\n\\onecolumngrid\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nHadronic scattering in the late stages of a heavy-ion collision may modify the chemical composition of the system up to the time of chemical freeze-out, defined as when all non-resonant particle ratios are fixed.  However, resonances may still be produced through inelastic hadronic scatterings, their production rates being dependent on the relatively poorly known hadronic cross-sections ($\\Xi^{-}+\\pi^{+}\\rightarrow\\Xi^{*}(1530)$, $\\Lambda+\\pi^{\\pm}\\rightarrow\\Sigma^{*}(1385)$, etc.) \\cite{Bass00,Rapp01,vanHees05,Adler02}.  Similarly, rescattering of the daughters of resonances that decay during the lifetime of the colliding system may destroy their correlations.  This prohibits the parent resonance reconstruction, leading to an underestimated primary yield.  The interplay of these two effects continues from the time of chemical freeze-out ($t_{chem}$) to the time of thermal freeze-out ($t_{therm}$, currently estimated to be $\\sim$8-10 fm\/$c$ \\cite{Retiere04}), the degree to which the primary resonance yields are modified depends on both the length of this time interval ($\\Delta t = t_{therm} - t_{chem}$) and the relative strengths of the two effects \\cite{Bleicher02, Bleicher03, Torrieri01, Rafelski01, Rafelski02}.  It has also been suggested that the \\textit{in vacuo} resonance masses and widths might be modified in the presence of the nuclear medium \\cite{STARRes}.\n\nThe STAR experiment \\cite{STAR} has measured several mesonic and baryonic resonance states \\cite{STARRes, STARK*, STARrho, STARphi}.  The variation in the composition and lifetimes of the measured states provides multiple constraints on $\\Delta t_{therm-chem}$ and the degree of late stage hadronic interactions.  The measurements presented herein add to the emerging picture of how the medium produced in heavy-ion collisions at RHIC affects resonance production.\n\n\\section{Analysis}\n\nThe results presented here come from an analysis of 7.6$\\times 10^{6}$, 5.4$\\times 10^{6}$, and 6.8$\\times 10^{6}$ Au+Au events in the 0-12\\%, 10-40\\%, and 40-80\\% centrality ranges respectively.  The greatest difficulty in performing an analysis of short-lived particles is acquiring an accurate description of the underlying combinatorial  background.  For this analysis, we use a narrow rotational method that is implemented as follows.  \n\nIn each event, the invariant mass is calculated for each topologically found $\\Xi^{-}$ candidate that passes a set of quality cuts combined with each $\\pi^{+}$ candidate that has passed a set of particle identification cuts.  These combinations comprise the ``signal''.  A second set of combinations is then made in which the transverse momentum ($\\mathrm p_{\\mathrm T}$) vectors of the $\\Xi^{-}$ candidates are rotated in the azimuthal plane by angles from 170$^{\\circ}$ to 188$^{\\circ}$ and the $\\pi^{+}$ candidates are rotated from -8$^{\\circ}$ to +8$^{\\circ}$, both in steps of 2$^{\\circ}$.  The invariant mass is then calculated for each iteration and each pair.  These combinations comprise the ``background''.  The process is repeated as a function of $\\mathrm p_{\\mathrm T}$ and the background is then subtracted from the signal in each bin.  Lastly, the yield is determined by summing the bin contents in a narrow region around the canonical mass.  Efficiency and acceptance corrections are applied to the results.  Lastly, a correction is applied to account for contribution to the overall yield coming from the neutral decay channel, which we do not measure here.  The corrected mid-rapidity spectra for three centralities are show in Figure \\ref{spectra}.\n\\begin{figure}\n \\centering\n \\includegraphics[bb= 0 0 567 547, scale=0.45]{figures\/figure1.eps}\n\n \\caption{Corrected $\\Xi^{0}(1530)$ $\\mathrm p_{\\mathrm T}$ spectra from Au+Au collisions at $\\sqrt{\\mathrm{s}_{\\mathrm NN}}=200$ GeV\/$c$ for three centrality bins.}\n \\label{spectra}\n\\end{figure}\n\n\\section{Results and Conclusions}\n\nInformation about medium effects on resonance production is gained by comparing the resonance yield to that of its non-resonant partner.  In Figure \\ref{ratios} we show this comparison, plotted as a ratio of the resonance to non-resonance yield, for the relevant species measured by STAR.  The mid-rapidity ($|y|<1.0$) $\\Xi^{0}(1530)$ $dN\/dy$ yields are compared with ($|y|<0.75$) $\\Xi$ $dN\/dy$ yields from reference \\cite{STARXi}.\n\\begin{figure*}[h]\n\\hspace*{-1.0cm}\n\\subfigure[Resonance to non-resonance ratios for several species in $p+p$, d+Au, and Au+Au collisions.]{\n\\hspace*{-1.0cm}\n \\label{ratios}\n \\includegraphics[bb= 6 5 561 542, scale=0.45]{figures\/figure2.eps}}\n\n\\hspace*{0.1cm}\n\\subfigure[Thermal model calculation.  Black dots are data, model fit and predictions are horizontal lines.  Data to the right side of the ``Fitted'' line were not included in the fit, but rather compared to predictions from the fit.]{\n \\label{thermus}\n \\includegraphics[bb= 0 0 567 547, scale=0.45]{figures\/figure3.eps}}\n\n\\end{figure*}\nThese comparisons are usually shown scaled by the $p+p$ value.  However, as there is currently no measured $p+p$ ratio for the $\\Xi^{0}(1530)$, the ratios have all been scaled by the most central Au+Au values.  This alternative scaling results in what is usually seen as a supression in central Au+Au appearing as an enhancement in peripheral (and $p+p$) collisions and an enhancement in central Au+Au appearing as a suppression in peripheral (and $p+p$) collisions.\n\nExamining the ratios we find the following: The shortest lived resonance measured by STAR, the K$^*$, shows a small suppression in central collisions relative to peripheral \\cite{STARK*ratio}, and the only slightly longer-lived $\\Sigma^{*}(1385)$ shows no measurable effect \\cite{STARRes}.  The $\\Xi^{0}(1530)$ ratios show an \\textit{enhancement} in the most central collisions relative to the peripheral collisions, the opposite of what is seen for the other reasonably long-lived baryonic resonance, the $\\Lambda^{*}(1520)$, which shows a strong suppression in central collisions relative to $p+p$.\n\nThe short-lived resonances would be expected to show suppression in more central events due to the rescattering of their decay products.  The fact that they are only weakly suppressed, if at all, suggests there must be significant regeneration contributing to the final state yields.  This conclusion is strengthened when one adds the $\\Xi^{0}(1530)$ measurement to the picture.  The $\\Xi^{0}(1530)$ should easily outlive the system (on average) once it's produced.  This means rescattering does not significantly affect the $\\Xi^{0}(1530)$ yield, (re-)generation plays a stronger role.  The observed $\\Xi^{0}(1530)$ enhancement in central collisions seems to indicate that significant interaction, and resonance production, occurs between $\\Xi$ and $\\pi$ particles in the (late) hadronic phase.  The $\\phi$ meson, like the $\\Xi^{0}(1530)$, will not on average decay before $t_{therm}$.  However, hadronic phase $\\phi$ production may be suppressed by the relatively low probability for K$^{+}$+K$^{-}$ scattering in the final stages of the collision \\cite{STARphi}.  The $\\Lambda^{*}(1520)$ is also reasonably long-lived.  However, alternative explanations have been put forward for the observed $\\Lambda^{*}(1520)$ suppression that do not involve rescattering and regeneration \\cite{Enyo,Kaskulov}.\n\nLastly, in Figure \\ref{thermus} we show the results of a thermal model fit to the published STAR data from central $\\sqrt{\\mathrm{s}_{\\mathrm NN}}=200$ Au+Au collisions.  The fit was performed using the THERMUS interface \\cite{thermus}.  The resulting fit parameters are T=0.169$\\pm$0.006 GeV, $\\mu_{\\mathrm B}$=0.04$\\pm$0.01 GeV, $\\mu_{\\mathrm S}$=0.016$\\pm$0.009 GeV, $\\mu_{\\mathrm Q}$=-0.01$\\pm$0.01 GeV, $\\gamma_{\\mathrm S}$=0.91$\\pm$0.06, and radius=7.5$\\pm$1.0 fm.  All of the resonance to non-resonance ratios are shown to the right of the ``Fitted'' line compared to their predicted values from the thermal fit.  The same pattern of suppression and enhancement as was seen in the ratios alone shows-up again.  The thermal model gives predictions for particle ratios only at $t_{chem}$, and therefore does not account for production that occurs during the hadronic phase, and so the observed $\\Xi^{0}(1530)$ enhancement could again be attributed to significant resonance production occurring during the hadronic phase.  This result strengthens the conclusion drawn from the resonance to non-resonance ratios alone.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction: Cantor Ternary Set.}\nThe Cantor  ternary set $C$ iis probably the best known example of a {\\em perfect nowhere-dense} set in the real line. It was constructed by George Cantor in 1883, see \\cite{cantor}.\\\\\n\n$C$ is obtained from the closed interval $[0,1]$ by a sequence of deletions of open intervals known as ''middle thirds\".\nWe begin with the interval $[0,1]$, let us call it $C_0$, and remove the middle third, leaving us with leaving us with the union of two closed intervals of length $1\/3$\n$$C_1= \\left[0,\\frac{1}{3}\\right]  \\cup \\left[\\frac{2}{3}, 1\\right] .$$ Now we remove the middle third from each of these intervals, leaving us with the union of four closed intervals of length $1\/9$\n\\begin{equation*}\nC_2=\\left[0,\\frac{1}{9}\\right] \\cup \\left[\\frac{2}{9},\\frac{1}{3}\\right] \\cup\\left[\\frac{2}{3},\\frac{7}{9}\\right] \\cup\\left[\\frac{8}{9},1\\right] .\n\\end{equation*}\nThen we remove the middle third of each of these intervals leaving us with eight intervals of length $1\/27$,\n\\begin{equation*}\nC_{3}= [0,\\frac{1}{27}]\\cup[\\frac{2}{27}, \\frac{1}{9}]\\cup[\\frac{2}{9}, \\frac{7}{27}]\\cup [ \\frac{8}{27},\\frac{1}{3}] \\cup [\\frac{2}{3},\\frac{19}{27}]\\cup[\\frac{20}{27},\\frac{7}{9}]\\cup[\\frac{8}{9},\\frac{25}{27}]\\cup[\\frac{26}{27},1].\n\\end{equation*}\n\nWe continue this process inductively, then for each $k=1,2, 3\\cdots $ we get a set $C_k$  which is the union of $2^k$ closed intervals of length $1\/3^k$. This iterative construction is illustrated in the following figure, for the first four steps:\n\\begin{center}\n\\includegraphics[width=3in]{outputs-figure0.pdf}\n\\end{center}\nFinally, we define the {\\em Cantor ternary set} $C$ as the intersection\n\\begin{equation}\nC= \\bigcap_{n=0}^\\infty C_n.\n\\end{equation}\n\nClearly $C \\neq \\emptyset $, since  trivially $0,1 \\in C$. Moreover  $C$ is a a closed set, being the countable intersection of closed sets, and  trivially bounded, since it is a subset of $[0,1]$. Therefore, by the Heine-Borel theorem $C$ is a {\\em compact set}. Moreover, observe by the construction that if $y$ is the end point of some closed subinterval of a given  $C_n$ then it is also the end point of some of the subintervals of $C_{n+1}$. Because at each stage, endpoints are never removed, it follows that $y \\in C_n$ for all $n$. Thus $C$ contains all the end points of all the intervals that make up each of the sets $C_n$ (or alternatively,  the endpoints to the intervals removed)  all of which are rational ternary numbers in $[0,1]$, i.e. numbers of the form $k\/3^n$. But $C$ contains much more than that; actually it is an uncountable set since it is a {\\em perfect set} \\footnote{A perfect set $P$ is a set that is closed and every point  $x \\in P$ is a limit point i.e there is a sequence $\\{x_n\\} \\subset P$, $x_n \\neq x$ and $x_n \\rightarrow x$.}. To prove that simply observe that every point of $C$ is approachable arbitrarily closely by the endpoints of the intervals removed (thus for any $x\\in C$ and for each $n \\in \\mathbb{N}$ there is an endpoint, let us call it $y_n \\in C_n$, such that $|x-y_n| < 1\/3^n$).\n\nThere is an alternative characterization of $C$, the {\\em ternary expansion characterization}.\nConsider the ternary representation for  $x \\in [0,1],$ \\footnote{Observe, for the ternary rational  numbers $k\/3^n$ there are two possible ternary expansions, since\n$$ \\frac{k}{3^n} = \\frac{k-1}{3^n} + \\frac{1}{3^n}  =  \\frac{k-1}{3^n} + \\sum_{k=n+1}^\\infty \\frac{2}{3^k}.$$ \nSimilarly, for the dyadic rational numbers $k\/2^n$ there are two possible dyadic expansions as\n$$ \\frac{k}{2^n} = \\frac{k-1}{2^n} + \\frac{1}{2^n}  =  \\frac{k-1}{2^n} + \\sum_{k=n+1}^\\infty \\frac{1}{2^k}.$$ \nThus for the uniqueness of the dyadic and the ternary representations we will take  the infinite expansions representations for the dyadic and ternary rational numbers.}\n \n\\begin{equation}\\label{ternaryexp}\n x = \\sum_{k=1}^\\infty \\frac{\\varepsilon_k(x)}{3^k}, \\quad \\varepsilon_k(x)=0, 1, 2 \\quad \\mbox{for all} \\,k = 1, 2, \\cdots \n\\end{equation}\n\nObserve that removing the elements where at least one of the \\(\\varepsilon_{k}\\) is equal to one is the same as removing the middle third in the iterative construction, thus \nthe Cantor ternary set is the set of numbers in $[0,1]$ that can be written in base 3 without using the digit 1, i.e.\n\\begin{equation}\\label{ternaryChar}\n C=\\left\\{x\\in [0,1] : x = \\sum_{k=1}^\\infty \\frac{\\varepsilon_k(x)}{3^k}, \\quad \\varepsilon_k(x)=0, 2 \\quad \\mbox{for all} \\; k = 1, 2, \\cdots\n \\right\\}.\n\\end{equation}\n\nUsing this characterization  of  $C$ we can get a direct proof that it is uncountable. Define \nthe mapping $f: C \\rightarrow [0,1]$ for $  x=\\sum_{k=1}^\\infty \\frac{\\varepsilon_k(x)}{3^k} \\in C$, as \n\\begin{equation}\\label{cantorfun}\nf(x) =    \\sum_{k=1}^\\infty \\frac{\\varepsilon_k(x)\/2}{2^k}= \\frac{1}{2}    \\sum_{k=1}^\\infty \\frac{\\varepsilon_k(x)}{2^k}.\n\\end{equation}\n\nIt is clear that $f$ is one-to-one correspondence from $C$ to $[0,1]$ (observe that as $\\varepsilon_k=0, 2$ then $\\varepsilon_k\/2=0, 1$). \nalso, the uncoantablity of $C$ can be obtained from the fact that $C$ is perfect, see Abbott \\cite{abot}, page 90. \\\\\n\n$C$ is a {\\em nowhere-dense} set, that is, there are no intervals included in $C$. One way to prove that is using the Taking two arbitrary points in $C$ we can always find a number between them that requires the digit 1 in its ternary representation, and therefore there are no intervals included in $C$, thus $C$ is a nowhere dense set. Alternatively, we can prove this simply by contradiction. Assuming that there is a interval $I=[a,b] \\subset C, \\; a <b$. Then $I=[a,b] \\subset C_n$ for all $n$ but as $|C_n| \\rightarrow 0$ as $n \\rightarrow \\infty$ then $|I| = b-a =0.$\\\\\n\n\n$C$ has measure zero, since\n$$ m(C) = m([0,1]) - m(C^c) = 1 - \\sum_{n=1}^\\infty \\frac{2^{n-1}}{3^n} = 1 -\\frac{1}{3}  \\sum_{n=0}^\\infty (\\frac{2}{3})^{n} = 1 -  \\frac{1\/3}{1- 2\/3}=1-1=0.$$\\\\\n\nSo far we have study the main properties of the Cantor set: $C$ is, non-empty (moreover uncountable) compact, perfect , nowhere dense set, with measure zero. Also we have seen that $C$ can then be obtained by any of the following constructions:\n\\begin{enumerate}\n\\item [i)] {\\em Proportional (fractral) construction}: by removing a fixed proportion (one third) of each subinterval in each of the iterative steps.\n\\item[ii)] {\\em Power construction}: by removing the length $1\/3^k$ from the center of each subinterval in the $k^\\text{th}-$step.\n\\item [iii)] {\\em Ternary expansion characterization} : by removing  the digit one in each position of the ternary expansion.\n\\end{enumerate}\n\nOf course these three constructions are equivalent. Nevertheless, the first construction, removing a fixed proportion of each subinterval in each of the iterative steps, give us another important property of the Cantor set, its self-similarity (fractal characteristic) across scales, this is illustrated in the following figure:\n\\begin{center}\n\\includegraphics[width=3in]{outputs-figure1.pdf}\n\\end{center}\n\nIn order to compute the Hausdorff dimension of $C$, S. Abbott in his book uses a nice little trick (avoiding the technicalities of the definition of Hausdoff dimension) by dilating $C_1$ by a the factor $3$ (see \\cite{abot} page 77) obtaining\n$$ [0,1] \\cup [2, 3].$$\nThus if we continue the iterations we will get two Cantor-like sets, then it can be concluded that\n$$3^d= 2, $$\nand therefore the Hausdorff dimension of $C$ is\n $$d= \\frac{\\ln 2}{\\ln 3}=0.630930.$$\n \n The Cantor ternary set $C$ was not the first perfect nowhere-dense set  in the real line to be constructed. The first construction was done by the a British mathematician Henry J. S. Smith in 1875, but not many mathematician where awarw of Smith's construction. Vito Volterra, still a graduate student in Italy, also showed how to construct such a set  in 1881, but he published his result in an italian journal not widely read.In 1883 Cantor rediscover this construction himself, and due to Cantor's prestige, then the Cantor ternary set was  (and still is) the typical example of a perfect nowhere-dense set. Following D. Bresoud \\cite{bres} we will refer as the Smith-Volterra-Cantor sets or $SVC(n)$ sets to the family of examples of perfect, nowhere-dense sets exemplified by the work of Smith, Voterra and Cantor, by removing in the $k$-iteration, an open interval of length $1\/n^k$ from the center of the remaining closed intervals. Observe that $C = SVC(3)$.\n \nAdditionally, the Cantor set can also be obtained using an Iterated Function System (IFS), see \\cite{hutch}. Consider the maps:\n\\begin{eqnarray}\n\\omega_0(x) = x\/3, \\quad \\omega_1(x) = x\/3 + 2\/3,\n\\end{eqnarray}\nthen\n$$ C = \\omega_0([0,1]) \\cup \\omega_1([0,1]),$$\nwhere this is a disjoint union. $C$ is the only set in the real line that satisfies this relation. This is a very interesting construction with important applications in the study of $C$, but we will not consider it in further details in this article.\n\n\\section{Variations of the Cantor ternary set:  Cantor-like sets.}\n\nIs there something special with the number $3$? The answer is actually yes and no!. As we have already mentioned  $C$ can then be obtained by removing a fixed proportion (one third) of each subinterval in each of the iterative steps, by removing the length $1\/3^k$ from each subinterval in the $k^\\text{th}-$step, or removing  the digit one of the ternary expansion. What happen if we choose $n=2$ instead of $3$?\nUsing the proportional construction, removing the open ``middle half\" from each component, we get at the end of the iterative process the set that we will denote as $C^{1\/2}$.  Here are the first 3 iterative steps,\n\n\\begin{eqnarray*}\nC^{1\/2}_{1}&=& [0,\\frac{1}{4}]\\cup [\\frac{3}{4},1],\\\\\nC^{1\/2}_{2}&=& [0,\\frac{1}{16}]\\cup[\\frac{3}{16}, \\frac{1}{4}]\\cup [\\frac{3}{4},\\frac{13}{16}]\\cup[\\frac{15}{16},1],\\\\\n\\end{eqnarray*}\n\\begin{eqnarray*}\nC^{1\/2}_{3}&=& [0,\\frac{1}{64}]\\cup[\\frac{3}{64}, \\frac{1}{16}]\\cup[\\frac{3}{16}, \\frac{13}{64}]\\cup [ \\frac{15}{64},\\frac{1}{4}] \\cup [\\frac{3}{4},\\frac{49}{64}]\\cup[\\frac{51}{64},\\frac{13}{16}]\\cup[\\frac{15}{16},\\frac{61}{64}]\\cup[\\frac{63}{64},1].\n\\end{eqnarray*}\nThen, continuing this process inductively , then for each $k=1,2, \\cdots $ we get a set $C^{1\/2}_k$  which is the union of $2^k$ closed intervals of length $1\/4^k$, and \n$$ C^{1\/2} = \\bigcap_{k=1}^\\infty C^{1\/2}_{k}.$$\n$C^{1\/2}$ share the same properties as $C$, i.e. it is a perfect, nowhere dense set in the real line. Also $C^{1\/2}$ has measure zero, as \n\\begin{eqnarray*}\n1- \\frac{1}{2} -2 \\frac{1}{8}-4\\frac{1}{32} +\\cdots &=& 1- \\frac{1}{2}[1+ \\frac{1}{2}+\\frac{1}{4} +\\cdots ]\\\\\n&=& 1- \\frac{1}{2}  \\sum_{k=0}^\\infty (\\frac{1}{2})^k =  1- \\frac{1}{2}  \\frac{1}{1-\\frac{1}{2}} =  1- \\frac{1}{2} \\frac{1}{\\frac{1}{2}} = 1- 1=0.\n\\end{eqnarray*}\nIn order to compute the Hausdorff dimension of this Cantor-like set, following the same argument as before, if we dilate $C_1^{1\/2}$ by $4$ we obtain\n$$ [0,1] \\cup [3, 4].$$\nThus, if we continue the iterations we will get at the end two Cantor-like sets, therefore\n$$4^d= 2,\\; \\mbox{so}\\; d= \\frac{\\ln 2}{\\ln 4}= \\frac{\\ln 2}{2\\ln 2} = \\frac{1}{2}= 0.5,$$\n so for this Cantor-like set the{\\em dust} obtained  is {\\em more sparse} than the one generated by $C$.\nAlso observe that, we can get a {\\em expansion characterization} of $C^{1\/2}$ but with base $n=4$, \n\\begin{equation*}\n C^{1\/2}=\\left\\{x\\in [0,1] : x = \\sum_{k=1}^\\infty \\frac{\\varepsilon_k(x)}{4^k}, \\quad \\varepsilon_k(x)=0, 3 \\quad \\mbox{for all} \\; k = 1, 2, \\cdots\n \\right\\}.\n\\end{equation*}\n\nOn the other hand, what happen if we use the power construction? i.e.what happen if we remove the length $1\/2^k$ from the center of each subinterval in the $k^\\text{th}-$step?.  The first step is the same as before, obtaining the set\n$$ [0,\\frac{1}{4}]\\cup [\\frac{3}{4},1],$$\nbut next we need to remove two open intervals of length $1\/2^2 = 1\/4$, so we get only four points,\n$$\\{0, 1\/4, 3\/4,1\\}$$\nso the process stops there, at $k=2$, since there are no more intervals to remove form. Clearly this is not the same as removing  open ``middle half\" from each component.\\\\\n\nWhat happen if we consider now $n=4$? In this case both the proportional and the power construction can be iterated an infinity number of times:\n\n\\begin{enumerate}\n\\item [i)] {\\em Proportional construction}: If we repeat the Cantor set's construction starting with the interval [0,1], removing the open ``middle fourth\" from the center of the remaining closed intervals, we get at the end of the iterative process a set that we will denote as $ C^{1\/4}$. Here are the first three iterations\n\\begin{eqnarray*}\nC^{1\/4}_1 &=& [0,\\frac{3}{8}] \\cup [\\frac{5}{8}, 1]\\\\\nC^{1\/4}_2 &=& [0,\\frac{9}{64}]\\cup [\\frac{15}{64}, \\frac{3}{8}]\\cup[\\frac{5}{8},\\frac{49}{64}]\\cup[\\frac{55}{64},1]\\\\\nC^{1\/4}_3 &=& [0,\\frac{27}{512}]\\cup [\\frac{45}{512}, \\frac{9}{64}]\\cup[\\frac{15}{64},\\frac{147}{512}]\\cup[\\frac{165}{512},\\frac{3}{8}]\\cup[\\frac{5}{8},\\frac{347}{512}]\\cup[\\frac{365}{512},\\frac{49}{64}]\\cup[\\frac{55}{64},\\frac{467}{512}]\\cup[\\frac{485}{512},1].\n\\end{eqnarray*}\nThen, continuing this process inductively , then for each $k=1,2, \\cdots $ we get a set $C^{1\/4}_k$  which is the union of $2^k$ closed intervals of length $(3\/8)^k$. \n$$ C^{1\/4} = \\bigcap_{k=1}^\\infty C^{1\/4}_{k}.$$\n\\begin{center}\n\\includegraphics[width=3in]{C025-proportions.pdf}\n\\end{center}\n$C^{1\/4}$ share the same properties as $C$, i.e. it is a perfect, nowhere dense set in the real line.\nAlso $C^{1\/4}$ has measure zero, as \n\\begin{eqnarray*}\n1- \\frac{1}{4} -2 \\frac{3}{32}-4\\frac{9}{256} +\\cdots &=& 1- \\frac{1}{4}[1+ \\frac{3}{4}-\\frac{9}{16} +\\cdots ]\\\\\n&=& 1- \\frac{1}{4} \\frac{1}{1-\\frac{3}{4}} =  1- \\frac{1}{4} \\frac{1}{\\frac{1}{4}} = 1-1 =0.\n\\end{eqnarray*}\n\nIn order to compute the Hausdorff dimension of this Cantor-like set, following the same argument as before, if we dilate $C^{1\/4}_1$ by $8\/3$ we obtain\n$$ [0,1] \\cup [\\frac{5}{3}, \\frac{8}{3}].$$\nThus if we continue the iterations we will get two Cantor-like sets, thus\n$$(\\frac{8}{3})^d= 2, \\mbox{i.e} \\quad d= \\frac{\\ln 2}{\\ln 8\/3} = \\frac{\\ln 2}{\\ln 8- \\ln3}= .706695.$$\nObserve that if we do the same argument in the second or third step we get the same result,\n$$(\\frac{64}{9})^d= 4, \\mbox{i.e} \\quad d= \\frac{\\ln 4}{\\ln 64\/9} = \\frac{\\ln 2}{\\ln 8- \\ln3}= .706695,$$\n$$(\\frac{512}{27})^d= 8, \\mbox{i.e} \\quad d= \\frac{\\ln 8}{\\ln 512\/27} = \\frac{\\ln 2}{\\ln 8- \\ln3}= .706695.$$\nThus, this Cantor-like  set is less sparse than $C$.\\\\\n\n\\item [ii)] {\\em Power construction}: if we repeat the Cantor construction starting with the interval [0,1], removing, in the $k$-iteration, an open interval of length $1\/4^k$ from the center of the remaining closed intervals, we get \n\\begin{eqnarray*}\nSVC(4)_1 &=& [0,\\frac{3}{8}] \\cup [\\frac{5}{8}, 1]\\\\\nSVC(4)_2 &=& [0,\\frac{5}{32}]\\cup [\\frac{7}{32}, \\frac{3}{8}]\\cup[\\frac{5}{8},\\frac{25}{32}]\\cup[\\frac{27}{32},1]\\\\\nSVC(4)_3 &=& [0,\\frac{9}{128}]\\cup [\\frac{11}{128}, \\frac{5}{32}]\\cup[\\frac{7}{32},\\frac{37}{128}]\\cup[\\frac{39}{128},\\frac{3}{8}]\\cup[\\frac{5}{8},\\frac{89}{128}]\\cup[\\frac{91}{128},\\frac{25}{32}]\\cup[\\frac{27}{32},\\frac{59}{64}]\\cup[\\frac{119}{128},1].\n\\end{eqnarray*}\nThen\n$$ SVC(4) = \\bigcap_{k=1}^\\infty SVC(4) _{k}.$$\n\\begin{center}\n\\includegraphics[width=3in]{SVC4-proportions.pdf}\n\\end{center}\n The total length of $SVC(4)$ can be obtained after removing all the open intervals in constructing this set,\n\\begin{eqnarray*}\n1- \\frac{1}{4} -2 \\frac{1}{16}-4\\frac{1}{64} +\\cdots &=& 1- \\frac{1}{4}[1+ \\frac{1}{2}-\\frac{1}{4} +\\cdots ]\\\\\n&=& 1- \\frac{1}{4} \\frac{1}{1-\\frac{1}{2}} =  1- \\frac{1}{4} \\frac{1}{\\frac{1}{2}} = 1-\\frac{1}{2} = \\frac{1}{2}.\n\\end{eqnarray*}\\\\\n\nThus the set $SVC(4)$ must have positive measure $1\/2$. Cantor-like sets with positive measure are called {\\em fat-Cantor sets}.\\footnote{Moreover, observe that the only $SCV(n)$ set that has measure zero is $C$, since the total length of what is remove is\n$$ 1 - \\frac{1}{n} -\\frac{2}{n^2} - \\frac{4}{n^3}- \\cdots = 1 - \\frac{1}{n}[1 -\\frac{2}{n} - \\frac{2^2}{n^2}- \\cdots ] =1 - \\frac{1}{n} \\frac{1}{1- 2\/n} = \\frac{n-3}{n-2}.$$}\\\\\n\nIn this case if we try to compute the Hausdorff dimension, since the sets $SVC(4)_k$ are not proportional,  a similar  argument as  before fails,  since if we dilate $SVC(4)_1$ by $8\/3$ we obtain\n$$ [0,1] \\cup [\\frac{5}{3}, \\frac{8}{3}].$$\nThus if we continue the iterations we will get two Cantor-like sets, thus\n$$(\\frac{8}{3})^d= 2, \\mbox{i.e.} \\quad d= \\frac{\\ln 2}{\\ln 8\/3} = \\frac{\\ln 2}{\\ln 8- \\ln3}= .706695.$$\nBut now if we do the same argument in the second or third step we get the different  results,\n$$(\\frac{32}{5})^d= 4, \\mbox{i.e.} \\quad d= \\frac{\\ln 4}{\\ln 32\/5} = \\frac{2\\ln 2}{\\ln 32- \\ln 5}= .746806,$$\n$$(\\frac{128}{9})^d= 8, \\mbox{i.e.} \\quad d= \\frac{\\ln 8}{\\ln 128\/9} = \\frac{3\\ln 2}{\\ln 128- \\ln 9}= .783274.$$\nSince $$\\lim_{k\\rightarrow \\infty} \\frac{k \\ln 2}{ \\ln (2 \\cdot 4^k) - \\ln (1+ 2^k)} = 1,$$ \nwe  can conclude that the Hausdorff dimension of $SVC(4)$  is actually one.\\\\\n$SVC(4)$ is called the {\\em Volterra set}. This is the set that was considered in 1881 by V.  Volterra to construct his famous counter-example of a function with bounded derivative that exists everywhere but the derivative is not Riemann integrable in any closed bounded interval i.e the Fundamental Theorem of Calculus fails!.\\\\\nObserve that in this case there is not a expansion characterization of the set $SVC(4)$.\\\\\n\\end{enumerate}\n\nNow, if we repeat the Cantor set's {\\em proportional construction} starting with the interval [0,1], removing the open ``middle three fourth\" from the center of the remaining closed intervals, then we get in the first three iterations,\n\\begin{eqnarray*}\nC^{3\/4}_1 &=& [0,\\frac{1}{8}] \\cup [\\frac{7}{8}, 1]\\\\\nC^{3\/4}_2 &=& [0,\\frac{1}{64}]\\cup [\\frac{7}{64}, \\frac{1}{8}]\\cup[\\frac{7}{8},\\frac{57}{64}]\\cup[\\frac{63}{64},1]\\\\\nC^{3\/4}_3 &=& [0,\\frac{1}{512}]\\cup [\\frac{7}{512}, \\frac{1}{64}]\\cup[\\frac{7}{64},\\frac{57}{512}]\\cup[\\frac{63}{512},\\frac{1}{8}]\\cup[\\frac{7}{8},\\frac{449}{512}]\\cup[\\frac{455}{512},\\frac{57}{64}]\\cup[\\frac{63}{64},\\frac{505}{512}]\\cup[\\frac{511}{512},1].\n\\end{eqnarray*}\nThen, continuing this process inductively , then for each $k=1,2, \\cdots $ we get a set $C^{3\/4}_k$  which is the union of $2^k$ closed intervals of length $(3\/8)^k$, and\n$$ C^{3\/4} = \\bigcap_{k=1}^\\infty C^{3\/4}_{k}.$$\n$C^{3\/4}$ share the same properties as $C$, i.e. it is a perfect, nowhere dense set in the real line.\nAlso $C^{3\/4}$ has measure zero, as \n\\begin{eqnarray*}\n1- \\frac{3}{4} -2 \\frac{3}{32}-4\\frac{3}{256} +\\cdots &=& 1- \\frac{3}{4}[1+ \\frac{1}{4}-\\frac{1}{16} +\\cdots ]\\\\\n&=& 1- \\frac{3}{4} \\frac{1}{1-\\frac{1}{4}} =  1- \\frac{3}{4} \\frac{1}{\\frac{3}{4}} = 1-1 =0.\n\\end{eqnarray*}\n\nIn order to compute the Hausdorff dimension of this Cantor-like set, following the same argument as before, if we dilate $C^{3\/4}_1$ by $8$ we obtain\n$$ [0,1] \\cup [7, 8].$$\nThus if we continue the iterations we will get two Cantor-like sets, thus\n$$8^d= 2, \\mbox{i.e} \\quad d= \\frac{\\ln 2}{\\ln 8} = \\frac{\\ln 2}{3\\ln 2}= \\frac{1}{3}= .666666.$$\n so this Cantor-like  set is less sparse than $C$. Also observe that in this case there is no {\\em power construction.}\n \nFinally, it is possible  an {\\em expansion characterization} of that $C^{3\/4}$ for $n=8$,\n\\begin{equation}\\label{Cant2}\n C^{3\/4}=\\left\\{x\\in [0,1] : x = \\sum_{k=1}^\\infty \\frac{\\varepsilon_k(x)}{8^k}, \\quad \\varepsilon_k(x)=0, 7 \\quad \\mbox{for all} \\; k = 1, 2, \\cdots\n \\right\\}.\n\\end{equation}\n \\\\\n\nOn the other hand, using the idea behind the ternary expansion characterization of the Cantor ternary set \\ref{ternaryChar}, we can get a general method to generalize $C$. For instance, take the base $n=5$, and \nconsider\n$$C^5= \\left\\{x\\in [0,1] : x = \\sum_{k=1}^\\infty \\frac{\\varepsilon_k(x)}{5^k}, \\quad \\varepsilon_k(x)=0, 2, 4 \\quad \\mbox{for all} \\; k = 1, 2, \\cdots\n \\right\\}.$$\n\nThis characterization is equivalent, since, in the step $k \\geq 1$ we are removing all the numbers in $[0,1]$ such that they have in its $5$-adic expansion the $k$-th digit is  $1$ or $3$, to the following proportional contraction:\nStart again with the unit interval $[0,1]$, partition the interval into $5$  intervals of equal length and remove the second and  the fourth open intervals of the partition obtaining $C^5_{1}$. \n$$C^5_{1}= [0,\\frac{1}{5}]\\cup [\\frac{2}{5},\\frac{3}{5}]\\cup [\\frac{4}{5},1]$$\nIn the second iteration, partition each of the remain $3$ intervals in $C^5_{1}$ into $5$ partitions of equal length and remove the same $2$ open intervals of the partition corresponding to the intervals removed in the first iteration, obtaining $C^k_{2}$, \n\\begin{eqnarray*}\nC^5_{2}&=& [0,\\frac{1}{25}]\\cup[\\frac{2}{25}, \\frac{3}{25}]\\cup [\\frac{4}{25},\\frac{1}{5}]\\cup[\\frac{2}{5},\\frac{11}{25}]\\cup[\\frac{12}{25},\\frac{13}{25}]\\cup[\\frac{14}{25},\\frac{3}{5}],\\\\\n&& \\quad \\cup [\\frac{4}{5},\\frac{21}{25}]\\cup[\\frac{22}{25}, \\frac{23}{25}]\\cup [\\frac{24}{25},1]\\\\\n\\end{eqnarray*}\nOne more iteration give us $C^5_{3}$,\n\\begin{eqnarray*}\nC^5_{3}&=& [0,\\frac{1}{125}]\\cup[\\frac{2}{125}, \\frac{3}{125}]\\cup [\\frac{4}{125},\\frac{1}{25}]\\cup[\\frac{2}{5},\\frac{11}{125}]\\cup[\\frac{12}{125},\\frac{13}{25}]\\cup[\\frac{14}{25},\\frac{3}{25}],\\\\\n&& \\quad \\cup [\\frac{4}{25},\\frac{21}{125}]\\cup[\\frac{22}{125}, \\frac{23}{125}]\\cup [\\frac{24}{125},\\frac{1}{5}]\\ \\cup [\\frac{2}{5},\\frac{51}{125}]\\cup[\\frac{52}{125}, \\frac{53}{125}]\\cup [\\frac{54}{125},\\frac{11}{25}]\\\\\n&& \\quad \\cup [\\frac{12}{25},\\frac{61}{125}]\\cup[\\frac{62}{125}, \\frac{63}{125}]\\cup [\\frac{64}{125},\\frac{13}{125}]\\ \\cup [\\frac{14}{25},\\frac{71}{125}]\\cup[\\frac{72}{125}, \\frac{73}{125}]\\cup [\\frac{74}{125},\\frac{3}{5}]\\\\\n&& \\quad \\cup [\\frac{4}{5},\\frac{101}{125}]\\cup[\\frac{102}{125}, \\frac{103}{125}]\\cup [\\frac{104}{125},\\frac{21}{25}]\\ \\cup [\\frac{22}{25},\\frac{111}{125}]\\cup[\\frac{112}{125}, \\frac{113}{125}]\\cup [\\frac{114}{125},\\frac{23}{52}]\\\\\n&& \\quad \\cup [\\frac{24}{25},\\frac{121}{125}]\\cup[\\frac{122}{125}, \\frac{123}{125}]\\cup [\\frac{124}{125},1].\n\\end{eqnarray*}\nContinue partitioning each subinterval in $5$ subinterval and removing $2$ of them in the same pattern.  Observe that for each $k$,  $C^{5}_{k}$ is the union of $3^{k-1}$ closed intervals of length $1\/5^k$. Then\n$$C^5=\\bigcap_{k=i}^{\\infty} C^5_{k}.$$\nBy construction $ C^{5} $ is also perfect nowhere set. The total length after removing all the open intervals in constructing this set is\n\\begin{eqnarray*}\n1- 2\\frac{1}{5} -6 \\frac{1}{25}-18\\frac{1}{125} +\\cdots &=& 1- \\frac{2}{5}[1+ \\frac{3}{5}+\\frac{9}{25} +\\cdots ]\\\\\n&=& 1- \\frac{2}{5}  \\sum_{k=0}^\\infty (\\frac{3}{5})^k = 1- \\frac{2}{5}  \\frac{1}{1-\\frac{3}{5}} \\\\\n&=&   1- \\frac{2}{5} \\frac{1}{\\frac{2}{5}} = 1- 1=0.\n\\end{eqnarray*}\nThus $ C^{5}$ must have measure $0$.\n\nIn order to compute the Hausdorff dimension of this Cantor-like set, following the same argument as before, if we dilate $C_1^{5}$ by $5$ we obtain\n$$ [0,1] \\cup [2, 3] \\cup [4, 5].$$\nThus if we continue the iterations we will get three Cantor-like sets, thus\n$$5^d= 3, \\; \\mbox{i.e} \\; d= \\frac{\\ln 3}{\\ln 5}=0.6826,$$\nso this Cantor-like  set is even less sparse than $C$.\\\\\n\nOf course this construction can be generalized for any $n \\in \\mathbb{N}$ with $n>2$, $1<p<n$, and start again with the unit interval $[0,1]$.  Partition the interval into $n$  intervals of equal length and remove $p$ open intervals of the partition making sure to leave the first and last intervals\\footnote{ if not, the set obtained would not be perfect since the construction would produce isolated points.} obtaining $C^n_{1}$. In the second iteration, partition each of the remain $n-p$ intervals in $C^n_{1}$ into $n$ partitions of equal length and remove the same $p$ open intervals of the partition corresponding to the intervals removed in the first iteration, obtaining $C^n_{2}$.  Continue partitioning each subinterval in $n$ subinterval and removing $p$ of them in the same pattern.  Then\n$$C^k=\\bigcap_{k=i}^{\\infty} C^k_{k}.$$\nObserve that in this construction in each step $p \\geq 1$ subintervals are removed from each closed subinterval in  $C^n_{k}$.\\\\\n   \nClearly, the $n$-adic Cantor  set $C^n$,  has the {\\em expansion characterization} \n\\begin{eqnarray*}\n&& C^n = \\{ x \\in [0,1] :  x=\\sum_{k=1}^{\\infty}\\frac{\\epsilon_{k}}{n^{k}}\\,  \\text{where }\\epsilon_{k} \\in \\{ \\eta_0, \\eta_1, \\cdots, \\eta_p\\} \\subseteq \\{0,1,\\cdots,n-1\\}, \\\\\n&& \\hspace{8cm} \\,  \\eta_0=0 \\text{ and } \\;\\eta_p=n-1\\} \n\\end{eqnarray*}\n\nObserve that in this case, using the dimension trick, we get\n$$ n^d = p,$$\nand therefore the Hausdorff dimension is\n$$ d =\\frac{\\ln p}{\\ln n}.$$\\\\\n\nSeveral other constructions are possible. For instance,  take $ 0< \\lambda \\leq 1$ and repeat the construction of the Cantor set such that at the $k$-th step, instead of taking out an open interval of length $1\/3^k$ from the center of each of the $2^{k-1}$ intervals of equal length, take out an open interval of length $\\lambda\/3^k$ to obtain $C_{\\lambda,k}$, then, the first 3 iterative steps $C_{\\lambda,1}$, $C_{\\lambda,2}$, $C_{\\lambda,3}$ are\n\n\\begin{eqnarray*}\nC_{\\lambda,1}&=& [0,\\frac{3-\\lambda}{6}]\\cup [\\frac{3+\\lambda}{6},1],\\\\\nC_{\\lambda,2}&=& [0,\\frac{9-5\\lambda}{36}]\\cup[\\frac{9-\\lambda}{36}, \\frac{3-\\lambda}{6}]\\cup [\\frac{3+\\lambda}{6},\\frac{27+\\lambda}{36}]\\cup[\\frac{27+5\\lambda}{36},1],\\\\\n\\end{eqnarray*}\n\\begin{eqnarray*}\nC_{\\lambda,3}&=& [0,\\frac{27-19\\lambda}{216}]\\cup[\\frac{27-11\\lambda}{216}, \\frac{9-5\\lambda}{36}]\\cup[\\frac{9-\\lambda}{36}, \\frac{81-25\\lambda}{216}]\\cup [ \\frac{81-17\\lambda}{216},\\frac{3-\\lambda}{6}]\\\\\n&& \\cup [\\frac{3+\\lambda}{6},\\frac{135+17\\lambda}{216}]\\cup[\\frac{135+25\\lambda}{216},\\frac{27+\\lambda}{36}]\\cup[\\frac{27+5\\lambda}{36},\\frac{189+11\\lambda}{216}]\\cup[\\frac{189+19\\lambda}{216},1].\n\\end{eqnarray*}\nThen,\n$$ C_{\\lambda} = \\bigcap_{k=1}^\\infty C_{\\lambda,k}$$\n$ C_{\\lambda}$ is, again, a perfect, nowhere dense set with positive measure, as\n\\begin{eqnarray*}\n1- \\frac{\\lambda}{3} -2 \\frac{\\lambda}{9}-4\\frac{\\lambda}{27} +\\cdots &=& 1- \\frac{\\lambda}{3}[1+ \\frac{2}{3}-\\frac{4}{9} +\\cdots ]\\\\\n&=& 1- \\frac{\\lambda}{3}  \\sum_{k=0}^\\infty (\\frac{2}{3})^k =  1- \\frac{\\lambda}{3}  \\frac{1}{1-\\frac{2}{3}} =  1- \\frac{\\lambda}{3} \\frac{1}{\\frac{1}{3}} = 1- \\lambda,\n\\end{eqnarray*}\nThus $C_{\\lambda}$ is a {\\em fat Cantor-like} set. In order to compute the Hausdorff dimension, if we dilate $C_{\\lambda,1}$ by $6\/(3-\\lambda)$ we obtain\n$$ [0,1] \\cup [\\frac{3+\\lambda}{3-\\lambda}, \\frac{6}{3-\\lambda}].$$\nThus if we continue the iterations we will get two Cantor-like sets, thus\n$$(\\frac{6}{3-\\lambda})^d= 2, \\,\\mbox{i.e.} \\; d= \\frac{\\ln 2}{\\ln [6\/(3-\\lambda)]} = \\frac{\\ln 2}{\\ln 6- \\ln(3-\\lambda)}.$$\n\nIt is worth noting that, unlike the other previously explored constructions, $C^n$ need not be symmetric with respect to 1\/2. As long as the first and last intervals of the {\\em expansion characterization} partition remain, in order to avoid isolated points, any choice of removed partitions can be made. Neither the measure nor the dimension of $C^n$ is affected by the choice to remove intervals asymmetrically.\\\\\n\nA more systematic study of the possible generalizations of the Cantor ternary set can be found in \\cite{dimartinourb}.\\\\\n\nFinally, a nice application of fat Cantor sets is the following. It is mention in almost anywhere that if we consider the space of Riemann integrable functions  it turns out that is not a complete space, this is one of the reasons why the Lebesgue integral is needed. Unfortunately it is hard to find a concrete counterexample for that statement. Using fat Cantor set it is easy to construct such a counterexample. First of all remember that a function is Riemann integrable if and only if the set of discontinuities has measure zero. Now let us take a fat Cantor set, let us take $SVC(4)$, the Volterra set, $ C^{1\/4} = \\bigcap_{k=1}^\\infty C^{1\/4}_{k}.$ We have proved that $SVC(4)$ has measure $1\/2>0$. If we consider \n$$K= [0,1] - SVC(4) = (\\frac{3}{8}, \\frac{5}{8}) \\cup ( \\frac{5}{32}, \\frac{7}{32})\\cup (\\frac{25}{32},  \\frac{27}{32})\\cup ( \\frac{9}{128}, \\frac{11}{128}) \\cup \\cdots =\\bigcup_{i=1}^\\infty E_i.$$\nLet $f_n$ be the indicator function of the set $\\bigcup_{i=1}^n E_i$, and $f$ be the indicator function of $K$. Then $f_n$ is Riemann integrable on $[0,1]$ since it is equal to one (i.e. continuous) for all but a finite number of points in [0,1]. Also trivially by construction $f_n (x)\\rightarrow f(x),$ for any $x \\in [0,1]$ and moreover,\n$$ \\int_0^1 |f_n(x) - f(x) | dx = \\int_{\\bigcup_{i=n+1}^\\infty E_i} dx = \\sum_{i=n+1}^\\infty |E_i| \\rightarrow 0, $$\nas $n \\rightarrow \\infty.$ But $f$ is not Riemann integrable on $[0,1]$ since its set of  discontinuities is precisely $SVC(4)$ that has positive measure.\\\\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction.} \n\nIn the book \\cite{FOOO}, the authors studied the moduli spaces \nof bordered stable \nmaps of genus 0 with Lagrangian boundary condition \nin a systematic way and constructed \nthe filtered $A_{\\infty}$-algebra associated to Lagrangian \nsubmanifolds.  \nSince our construction depends on various auxiliary choices, \nwe considered the {\\it canonical model} of filtered \n$A_{\\infty}$-algebras, which is unique up to filtered \n$A_{\\infty}$-isomorphisms.  \nThe aim of this note is to explain the construction of the canonical \nmodel and to apply such an argument to obtain the filtered \n$A_{\\infty}$-structure to the Morse complex on the Lagrangian \nsubmanifold.  \nThe resulting filtered $A_{\\infty}$-operations are described \nby the moduli \nspaces of certain configurations consisting of pseudo-holomorphic \ncurves and gradient flow lines.   \nNote that the first named author \\cite{Fuk94} studied \nthe quantization of Morse homotopy based on \nthe moduli spaces of certain configurations consisting \nof pseudo-holomorphic discs and gradient flow lines of {\\it multiple} \nMorse functions, \nsee \\cite{Oh2} for monotone case and Theorem A4.28 in \\S A 4 in \n\\cite{2000}.  \nSuch configurations are also studied in monotone case by \nBuhovsky \\cite{Buk} and Biran and Cornea \\cite{BC}.  \nWe follow Chapter 5 in \\cite{FOOO} to explain the algebraic aspect \nof canonical models and use the geometric construction in Chapter 7 \nin \\cite{FOOO}.  \n\nWe briefly review the background of our study.  \nFloer \\cite{Fl} invented a new theory, which is now called \nFloer (co)homology for Lagrangian intersections.  \nVery roughly speaking, it is an analog of Morse theory for \nthe action functional on the space of paths with end points \non Lagrangian submanifolds.  \nFor a transversal pair of Lagrangian submanifolds $L_0, L_1$, \nthe cochain complex is generated by the intersection points \nof $L_0$ and $L_1$.  The coboundary operator is defined by \ncounting connecting orbits joining the intersection points.  \nThe theory was extended by the second named author \\cite{Oh} \nto the class of \nmonotone Lagrangian submanifolds with the minimal Maslov number \nbeing at least 3.  \nIn general, however, there arise obstructions to constructing the Floer \ncochain complex caused by the bubbling-off of pseudo-holomorphic \ndiscs in the moduli space of connecting orbits.  \nWe started a systematic study of the moduli spaces of \npseudo-holomorphic discs with Lagrangian boundary condition \nand formulated the obstructions in terms of the Mauer-Cartan \nequation on the filtered $A_{\\infty}$-algebra associated to \nthe Lagrangian submanifold \\cite{FOOO}. \nIn order to give consistent orientations on the moduli spaces, \nwe introduced the notion of relative spin structure and \ncosidered relative spin Lagrangian submanifolds.  \nFor a relative spin pair $(L_0,L_1)$ of Lagrangian submanifolds, \nwe constructed a filtered $A_{\\infty}$-bimodule over \nthe $A_{\\infty}$-algbras associated to $L_0$ and $L_1$.  \nIf each $L_i$, $i=0,1$, admits a solution $b_i$ of \nthe Maurer-Cartan equation, we can rectify the Floer operator \nto obtain a coboundary operator $\\delta^{b_0,b_1}$.  \nHence the Floer complex \n$(CF^{\\bullet}(L_0,L_1), \\delta^{b_0,b_1})$ is obtained.  \nWe also considered the case that the Lagrangian submanifolds \nadmit solutions of the Maurer-Cartan equation modulo multiples of \nthe fundamental class $[L_i]$ ({\\it weak solution}).  \nFor a weak solution $b_i$, we assign the potential \n${\\mathfrak{PO}}(b_i)$. \nIf ${\\mathfrak{PO}}(b_0)={\\mathfrak{PO}}(b_1)$, we can construct \nthe Floer complex $(CF(L_0,L_1),\\delta^{b_0,b_1})$ deformed by \n$b_0, b_1$.  \nThis extension with the weak bounding cochains plays crutial role \nin our study of Floer theory on compact toric manifolds \\cite{FOOO2}.   \n\nWe firstly constucted the filtered $A_{\\infty}$-algebra \nmentioned above on suitable subcomplex of \nthe singular cochain complex of $L_i$ using systematic \nmulti-valued perturbation of Kuranshi maps describing \nthe moduli spaces.  \nWe briefly review these constructions in subsequent section.  \nThus the resulting filtered $A_{\\infty}$-algebra depends \non various choices, i.e., the choice of the subcomplex, \nthe choice of systematic multi-valued perturbation, etc.  \nIn order to make the construction canonical, we introduced \nthe notion of the {\\it canonical model}.  \nSince the structure constants of the filtered $A_{\\infty}$-algebra \ndepends on these choices, it is appropriate to work with \nthe canonical model when we make practical computation of \nthe structure constants.  \nWhen we consider ${\\mathfrak{PO}}(b)$ as a function \non the set of weak solutions of the Maurer-Cartan equation, \nwe call it the {\\it potential function}.  \nThe canonical model provides an appropriate domain of \nthe definition of the potential function.  \nThe canonical models also play a role in the convergence of a certain \nspectral sequence, see Chapter 6 in \\cite{FOOO}.  \n(In \\cite{2000}, we used another kind of \nfinitely generated complex to ensure the weak finiteness property \nof the filtered $A_{\\infty}$ algebras.)    \nIt may be also worth mentioning that we rely on canonical models in \nsome places in \\cite{FOOO}, since the degree of the ordinary \ncohomology is bounded, though the degree of the singular complex is \nnot bounded above.  \n\nWe also developed \nan algebraic theory for filtered $A_{\\infty}$-algebras, bimodules, \nin particular, \nthe homotopy theory of the filtered $A_{\\infty}$-algebras, \nbimodules and proved that the homotopy type \nof the resulting algebraic object does not depend on such choices.  \nWe can also reduce the filtered $A_{\\infty}$-structure to \nappropriate free subcomplexes of the original complex.  \nIn particular, if we work over the ground coefficient field, \nwe obtain the filtered $A_{\\infty}$-structure on the {\\it classical} \n(co)homology of the complex.  \nIn this note, we review the construction of the canonical models \nof filtered $A_{\\infty}$-algebras and filtered $A_{\\infty}$-bimodules \nand explain its implication in a geometric setting.  \n\nIn section 5, we induce the filtered $A_{\\infty}$-algebra structure \non Morse complex based on the argument in the construction of \ncanonical models.  \nWe choose a Morse function $f$ on $L$, which is adapted to \na triangulation of $L$ (see section 5).   \n\\smallskip\n\n\\noindent\n{\\bf Theorem \\ref{Morse}}\n{\\it \nLet $L$ be a relatively spin Lagrangian submanifold in a closed \nsymplectic manifold $(M,\\omega)$ and $f$ a Morse function on $L$ \nas above.  \nThen Morse complex $CM^*(f)\\otimes \\Lambda_{nov}$ \ncarries a structure of a filtered $A_{\\infty}$-algebra, \nwhich is homotopy equivalent to the filtered $A_{\\infty}$-algebra \nassociated to $L$ constructed in \\cite{FOOO}.  \n}\n\n\\smallskip\n\nThis note is based on the lecture at Yashafest and we thank  \nthe organizers for the invitation.  \n\n\\section{Flitered $A_{\\infty}$-algebras, bimodules} \nIn this subsection, we recall the definition of (filtered) \n$A_{\\infty}$-algebras, bimodules, homomorphisms \nand prepare necessary notations.  \nThen we explain the notion of homotopy between (filtered) \n$A_{\\infty}$-homomorphisms.  \nIn fact, the notion of homotopy between \n$A_{\\infty}$-homomorphisms can be found in the literature, \ne.g., \\cite{Smi}.  \nFor differential graded algebras, homotopy theory was studied \nin rational homotopy theory, see \\cite{Su}, \\cite{G-M}.  \nIn order to make clear the relation among such notions, \nwe introduced the notion of the model of \n$[0,1] \\times \\overline{C}$ ($[0,1] \\times C$) for \na (filtered) $A_{\\infty}$-algebra $C$ ($\\overline{C}$) \nand defined the notion of homotopy using such a model.  \n\n\\smallskip\n\\noindent\n{\\bf 2.1) Unfiltered $A_{\\infty}$-algebras, homomorphisms, bimodules, \nbimodule homomorphisms.}\n\nLet $R$ be a commutative ring, e.g., $\\Z$, $\\Q$.  \nLet $\\overline{C}^{\\bullet}$ be a cochain complex over $R$.  \nWe assume that $\\overline{C}^k = 0$ for $k < 0$.  \nDenote by $\\overline{\\mathfrak m}_1$ its differential.  \nSet $\\overline{C}[1]^k=\\overline{C}^{k+1}$ and denote the shifted degree by \n$\\deg' x = \\deg x -1$, where $\\deg$ is the original degree \nof $\\overline{C}^{\\bullet}$.  \nIn this section, we use only shifted degrees.  \nConsider a series of $k$-ary operations, $k=1,2, \\dots,$ \n\n\\[\n\\overline{\\mathfrak m}_k : \n(\\overline{C}[1]^{\\bullet})^{\\otimes k} \\to \\overline{C}[1]^{\\bullet}\n\\]\nof degree $1$ with respect to the shifted degrees.  \n\nBefore giving the definition of $A_{\\infty}$-algebras, we explain \nthe case of differential graded algebras.  \nLet $(\\overline{C}^{\\bullet}, d, \\cdot)$ be a differential \ngraded algebra.  \nDefine $\\overline{\\mathfrak m}_1(x)=(-1)^{\\deg x} da$ and \n$\\overline{\\mathfrak m}_2(x \\otimes y) = \n(-1)^{\\deg x \\cdot (\\deg y +1)} x \\cdot y$.  \nThen we find that \n\n\\begin{align}\n&\\overline{\\mathfrak m}_1 \\circ \\overline{\\mathfrak m}_1 (x)= 0,\n\\nonumber \\\\\n&\\overline{\\mathfrak m}_1 \\circ \\overline{\\mathfrak m}_2(x \\otimes y) \n+ \\overline{\\mathfrak m}_2 (\\overline{\\mathfrak m}_1(x_1) \\otimes x_2) \n+ (-1)^{\\deg' x_1} \\overline{\\mathfrak m}_2 (x_1 \\otimes \n\\overline{\\mathfrak m}_1(x_2))=0, \n\\nonumber \\\\\n&\\overline{\\mathfrak m}_2(\\overline{\\mathfrak m}_2(x_1 \\otimes x_2) \\otimes x_3) + (-1)^{\\deg' x_1} \\overline{\\mathfrak m}_2(x_1 \\otimes \n\\overline{\\mathfrak m}_2(x_2 \\otimes x_3))=0, \\nonumber\n\\end{align}\nwhich follow from the facts that $d$ is a differential, \nthe multiplication and the differential $d$ satisfies Leibniz' rule \nand the multiplication is associative.  \n\nThere are some geometric situations where multiplicative structures \nare defined but not exactly associative.  \nA typical example is the composition in based loop spaces.  \nIn fact, Stasheff \\cite{St} introduced the notion of \n$A_{\\infty}$-structure on topological spaces in order to characterize \nthe homotopy types of based loop spaces.  \nHe also defined the $A_{\\infty}$-structure in algebraic setting.  \nFor instance, a multiplicative structure is said to be associative \nup to homotopy, if there exists \n$\\overline{\\mathfrak m}_3: (\\overline{C}[1]^{\\bullet})^{\\otimes 3} \\to \n\\overline{C}[1]^{\\bullet}$ such that \n\n\\begin{align}\n& \\overline{\\mathfrak m}_2(\\overline{\\mathfrak m}_2(x_1 \\otimes x_2) \\otimes x_3) + (-1)^{\\deg' x_1} \\overline{\\mathfrak m}_2(x_1 \\otimes \n\\overline{\\mathfrak m}_2(x_2 \\otimes x_3)) \\nonumber \\\\ \n+ & \\overline{\\mathfrak m}_1 \\circ \\overline{\\mathfrak m}_3\n(x_1 \\otimes x_2 \\otimes x_3) + \n\\overline{\\mathfrak m}_3(\\overline{\\mathfrak m}_1(x_1) \n\\otimes x_2 \\otimes x_3) \\nonumber \\\\\n+ & (-1)^{\\deg' x_1} \\overline{\\mathfrak m}_3\n(x_1 \\otimes \\overline{\\mathfrak m}_1(x_2) \\otimes x_3)  \n+ (-1)^{\\deg' x_1 + \\deg' x_2} \\overline{\\mathfrak m}_3 \n(x_1 \\otimes x_2 \n\\otimes \\overline{\\mathfrak m}_1(x_3)) \\nonumber \\\\\n= & 0.  \\nonumber\n\\end{align}\n\nNote that it coincides with the relation corresponding to the \nassociativity, if $\\overline{\\mathfrak m}_3=0$.  \nWe can continue higher homotopies in a similar way:  \n$$\n\\sum_{k_1+k_2=k+1} \\sum_i (-1)^{\\sum_{j=1}^{i-1} \\deg' x_j} \n\\overline{\\mathfrak m}_{k_1}\n(x_1, \\dots, \\overline{\\mathfrak m}_{k_2}\n(x_i, \\dots, x_{i+k_2-1}), \\dots, x_k) = 0.\n$$  \nHere $k_1, k_2$ are positive integers.  \nFor a concise description of relations among higher homotopies, \nwe introduce the bar complex of $\\overline{C}^{\\bullet}$, which is \ndefined by \n\\[\nB(\\overline{C}[1]^{\\bullet})=\\bigoplus_{k=0}^{\\infty} B_k(\\overline{C}[1]^{\\bullet}), \\ \\ \nB_k(\\overline{C}[1]^{\\bullet})=\\bigoplus_{m_1,\\dots, m_k} \n\\overline{C}[1]^{m_1} \\otimes \\cdots \\otimes \\overline{C}[1]^{m_k}, \n\\]\nwhich we consider as a tensor coalgebra.  \nThe comultiplication is given by \n$$\n\\Delta (x_1 \\otimes \\dots \\otimes x_k) \n= \\sum_{i=0}^k (x_1 \\otimes \\dots \\otimes x_i) \\otimes (x_{i+1} \\otimes \n\\dots \\otimes x_k),\n$$ \nwhere $x_1 \\otimes \\dots \\otimes x_i, x_{i+1} \\otimes \\dots \\otimes \nx_k \\in B(\\overline{C}[1]^{\\bullet})$ and \nthe former with $i=0$ and the latter with $i=k$ are understood as \n$1 \\in B_0(\\overline{C}[1]^{\\bullet})$.  \nExtend $\\overline{\\mathfrak m}_k$ to the graded coderivation \n$\\widehat{\\overline{\\mathfrak m}_k}$ on $B(\\overline{C}[1]^{\\bullet})$. \nNamely, \n\n\\begin{align}\n& \\widehat{\\overline{\\mathfrak m}}_k(x_1 \\otimes \\dots \\otimes x_N) \n\\nonumber \\\\\n= & \\sum_{i=1}^{N-k+1} (-1)^{\\sum_{j=1}^{i-1} \\deg' x_j} \nx_1 \\otimes \\dots \\otimes x_{i-1} \\otimes \n\\overline{\\mathfrak m}_k (x_i \\otimes \\dots \\otimes x_{i+k-1}) \\otimes \nx_{i+k} \\otimes \\dots \\nonumber \\\\\n& \\hspace{0.5in} \\cdots \\otimes x_N. \\nonumber\n\\end{align}\n\nWe call $(\\overline{C}^{\\bullet}, \\{\\overline{\\mathfrak m}_k\\})$ \nan $A_{\\infty}$-algebra, if \n\\[\n\\widehat{\\overline{d}}=\\sum_k \\widehat{\\overline{\\mathfrak m}}_k:\nB(\\overline{C}[1]^{\\bullet}) \\to B(\\overline{C}[1]^{\\bullet})\n\\] \nsatisfies $\\widehat{\\overline{d}} \\circ \\widehat{\\overline{d}} = 0$.  \nIn the case that $\\overline{\\mathfrak m}_k=0$ for $k > 2$, \nthis condition \nis equivalent to the notion of differential graded algebras.  \n\nFor a collection $\\{\\overline{\\mathfrak f}_k:B_k(\\overline{C}[1]^{\\bullet}) \\to \n\\overline{C}'[1]^{\\bullet}\\}_{k=1}^{\\infty}$ of degree $0$, \nwe extend it to a homomorphism as tensor coalgebras\n\\[\n\\widehat{\\overline{f}}(x_1 \\otimes \\cdots \\otimes x_k)  \n= \n\\sum_{k_1 + \\cdots + k_n=k} \n\\overline{\\mathfrak f}_{k_1}(x_1 \\otimes \\cdots \\otimes x_{k_1}) \n\\otimes \\cdots \\otimes \n\\overline{\\mathfrak f}_{k_n}(x_{k+1-k_n} \\otimes \\cdots \\otimes x_k).\n\\]\nWe call $\\{\\overline{\\mathfrak f}_k\\}$ an $A_{\\infty}$-homomorphism, \nif $\\widehat{\\overline{f}}$ satisfies \n$\\widehat{\\overline{d}}_{\\overline{C}'} \\circ \\widehat{\\overline f} = \n\\widehat{\\overline f} \\circ \\widehat{\\overline{d}}_{\\overline{C}}.$  \n\nIn terms of the components $\\overline{\\mathfrak m}_k$'s and \n$\\overline{\\mathfrak f}_k$'s, \nthis is equivalent to \n\n\\begin{align} \n& \\sum_{i_1+ \\dots + i_k=n} \\overline{\\mathfrak m}_k(\n\\overline{\\mathfrak f}_{i_1}(x_1 \\otimes \\dots \\otimes x_{i_1}) \n\\otimes \\dots \n\\otimes \\overline{\\mathfrak f}_{i_k}(x_{i_1+ \\dots + i_{k-1} +1} \n\\otimes \n\\dots \\otimes x_{n})) \\nonumber \\\\\n= & \n\\sum_{j_1+ \\dots + j_{\\ell}=n} \\sum_{p=1}^{\\ell} \n(-1)^{\\sum_{i=1}^{j_1+ \\dots j_{p-1}} \\deg' x_i}\n\\overline{\\mathfrak f}_{j_1}(x_1 \\otimes \\dots \\otimes x_{j_1}) \\otimes \n\\dots \\otimes \\nonumber \\\\\n& \\overline{\\mathfrak m}_{j_p}(x_{j_1+ \\dots + j_{p-1}+1} \n\\otimes \\dots \\otimes \nx_{j_1+ \\dots + j_p}) \\otimes \\dots \\otimes \n\\overline{\\mathfrak f}_{j_{\\ell}}(x_{j_1+ \\dots j_{\\ell -1} +1} \n\\otimes \\dots \\otimes x_n) \\nonumber\n\\end{align}\n\nLet $(\\overline{C}^{\\bullet}_i, \\{\\overline{\\mathfrak m}_k^{(i)}\\})$, \n$i=0,1$, \nbe $A_{\\infty}$-algebras, $\\overline{D}^{\\bullet}$ a graded module and \n$\\overline{\\mathfrak n}_{k_1,k_0}:B_{k_1}(\\overline{C}_1[1]^{\\bullet}) \\otimes \n\\overline{D}[1]^{\\bullet} \\otimes B_{k_0}(\\overline{C}_0[1]^{\\bullet}) \\to \n\\overline{D}[1]^{\\bullet}$ homomorphisms of degree $1$.  \nWe call $(\\overline{D}^{\\bullet},\\{\\overline{\\mathfrak n}_{k_1,k_0}\\})$ \nan $A_{\\infty}$-bimodule, if \n\\[\n\\widehat{\\overline{d}}_{\\overline{\\mathfrak n}} \\circ \n\\widehat{\\overline{d}}_{\\overline{\\mathfrak n}} = 0, \n\\]\nwhere $\\widehat{\\overline{d}}_{\\overline{\\mathfrak n}}$ is \ndefined on \n$\nB(\\overline{C}_1[1]^{\\bullet}) \\otimes \\overline{D}[1]^{\\bullet} \\otimes \nB(\\overline{C}_0[1]^{\\bullet})\n$ as follows:  \n\n\\begin{align}\n& \\widehat{\\overline{d}}_{\\overline{\\mathfrak n}}\n(x_{1,1} \\otimes \\dots \\otimes x_{1,k_0} \\otimes y \\otimes \nx_{0,1} \\otimes \\dots \\otimes x_{0,k_0}) \\nonumber \\\\\n= & \\widehat{\\overline{d}}^{(1)}(x_{1,1} \\otimes \\dots \n\\otimes x_{1,k_1}) \n\\otimes y \\otimes x_{0,1} \\otimes \\dots \\otimes x_{0,k_0} \\nonumber \\\\\n& + \\sum_{k'_1 \\leq k_1, k'_0 \\leq k_0} \n(-1)^{\\sum_{i=1}^{k_1-k'_1} \\deg' x_i}\nx_{1,1} \\otimes \\dots x_{k_1-k'_1} \\nonumber \\\\\n& \\otimes {\\overline{\\mathfrak n}}_{k'_1,k'_0}(x_{1,k_1-k'_1+1} \n\\otimes \\dots x_{k_1} \\otimes y \\otimes x_{0,1} \n\\otimes \\dots \n\\otimes x_{0,k'_0}) \\otimes x_{0,k_0+1} \\otimes \\dots \\otimes x_{0,k_0} \n\\nonumber \\\\\n& + (-1)^{\\sum_{i=1}^{k_1} \\deg' x_i + \\deg' y}\nx_{1,1} \\otimes \\dots x_{1,k_1} \\otimes y \\otimes \\widehat{\\overline{d}}^{(0)}(x_{0,1} \\otimes \\dots \\otimes x_{0,k_0}). \\nonumber\n\\end{align}\n\nThe condition for $(\\overline{D},\\{\\overline{\\mathfrak n}_{k_1,k_0}\\})$ \nto be an $A_{\\infty}$-bimodules over $\\overline{C}_i$, $i=0,1$ is \nequivalent to the identity \n \n\\begin{align}\n& \\overline{\\mathfrak n}_{*,*}(\\widehat{\\overline{d}}^{(1)}(\nx_{1,1} \\otimes \\dots \\otimes x_{1,k_1}) \\otimes y \\otimes \nx_{0,1} \\otimes \\dots \\otimes x_{0,k_0}) \\nonumber \\\\\n+ & \\sum_{k'_1 \\leq k_1, k'_0 \\leq k_0} (-1)^{\\sum_{i=1}^{k_1-k'_1} \n\\deg' x_i} \n\\overline{\\mathfrak n}_{k_1-k'_1,k_0-k'_0}\n(x_{1,1} \\otimes \\dots \\otimes \nx_{k_1-k'_1} \\otimes \\nonumber \\\\\n& \\overline{\\mathfrak n}_{k'_1,k'_0}\n(x_{1,k_1-k'_1+1} \\otimes \\dots \\otimes x_{1,k_1} \\otimes y \n\\otimes x_{0,1} \\otimes x_{0,k'_0}) \\otimes x_{0,k'_0+1} \\otimes \n\\dots \\otimes x_{0,k_0}) \\nonumber \\\\\n+ & (-1)^{\\sum_{i=1}^{k_1} \\deg' x_i + \\deg' y} \n\\overline{\\mathfrak n}_{*,*} (x_{1,1} \\otimes \\dots \\otimes \nx_{1,k_1} \\otimes y \\otimes \\widehat{\\overline{d}}^{(0)} \n(x_{0,1} \\otimes \\dots \\otimes x_{0,k_0})) \\nonumber \\\\\n= & 0. \\nonumber\n\\end{align}\n\nHere $\\overline{\\mathfrak n}_{*,*}:B(\\overline{C}_1[1]^{\\bullet}) \n\\otimes \\overline{D}[1] \\otimes B(\\overline{C}_0[1]^{\\bullet}) \n\\to \\overline{D}[1]$ is defined to be \n$\\overline{\\mathfrak n}_{k_1,k_0}$ on \n$B_{k_1}(\\overline{C}_1[1]^{\\bullet}) \n\\otimes \\overline{D}[1] \\otimes B_{k_0}(\\overline{C}_0[1]^{\\bullet})$.  \n\nLet $\\{\\overline{\\mathfrak f}_k^{(i)}:\\overline{C}_i[1]^{\\bullet} \\to \n\\overline{C}_i'[1]^{\\bullet}\\}$, $i=0,1$, be $A_{\\infty}$-homomorphisms \nand $\\overline{D}^{\\bullet}$, resp. $\\overline{D'}^{\\bullet}$ \nan $A_{\\infty}$-bimodules over \n$\\overline{C}_i$, resp. $\\overline{C}_i'$, $i=0,1$.  \nFor a collection $\\{\\overline{\\phi}_{k_1,k_0}\\}:\nB_{k_1}(\\overline{C}_1[1]^{\\bullet}) \\otimes \\overline{D}[1]^{\\bullet} \n\\otimes \nB_{k_0}(\\overline{C}_0[1]^{\\bullet}) \n\\to \\overline{D}'[1]^{\\bullet}$ of degree $0$, we define \n\\[\n\\widehat{\\overline{\\phi}}:B(\\overline{C}_1[1]^{\\bullet}) \\otimes \\overline{D}[1]^{\\bullet} \n\\otimes B(\\overline{C}_0[1]^{\\bullet}) \\to \nB(\\overline{C}_1'[1]^{\\bullet}) \\otimes \\overline{D}'[1]^{\\bullet} \\otimes \nB(\\overline{C}_0'[1]^{\\bullet})\n\\]\nas the homomorphism determined by $\\{\\overline{\\mathfrak f}_k^{(i)}\\}$, \n$i=0,1$ and $\\{\\overline{\\phi}_{k_1,k_0}\\}$.  \nWe call $\\{\\overline{\\phi}_{k_1,k_0}\\}$ a homomorphism of \n$A_{\\infty}$-bimodules, if \n\\[\n\\widehat{\\overline{d}}_{\\overline{\\mathfrak n}'} \\circ \n\\widehat{\\overline{\\phi}} = \\widehat{\\overline{\\phi}} \\circ \n\\widehat{\\overline{d}}_{\\overline{\\mathfrak n}}.\n\\]\n \n\\smallskip\n\\noindent \n{\\bf 2.2) The universal Novikov ring and the energy filtration.}\n\nTo explain the notion of filtered $A_{\\infty}$-algebras,   \nwe introduce the universal Novikov ring.  \nLet $e$ and $T$ be formal variables of degree $2$ and $0$, \nrespectively.  \nSet  \n\\begin{align}\n \\Lambda_{nov} & = \\{ \\sum_i a_i e^{\\mu_i} T^{\\lambda_i} \\ \n\\vert \\ a_i \\in R, \n\\ \\mu_i \\in \\Z, \\ \\lambda_i \\in \\R, \\ \n\\lambda_i \\to +\\infty (i \\to +\\infty)\\} \\nonumber \\\\\n \\Lambda_{0,nov} & = \n\\{ \\sum_i a_i e^{\\mu_i} T^{\\lambda_i} \\in \\Lambda_{nov} \n\\ \\vert \\ \\lambda_i \\geq 0 \\}. \\nonumber \n\\end{align}\n\nSet \n$$C^{\\bullet}=\n\\{\\sum c_i e^{\\mu_i} T^{\\lambda_i} \\vert \nc_i \\in \\overline{C}^{\\bullet}, \\mu_i \\in \\Z, \\lambda_i \\in \\R, \n\\lambda_i \\to +\\infty (i \\to +\\infty)\\},\n$$\nwhich is the completion of the graded tensor product \n$\\overline{C}^{\\bullet} \\otimes_R \\Lambda_{0,nov}$ with respect to \nthe energy filtration given below. \nWe define the filtration defined by \n\\[\nF^{\\lambda}C^{\\bullet}= \\{ \\sum_i x_i e^{\\mu_i} T^{\\lambda_i} \\in C^{\\bullet}\n\\ \\vert \\ x_i \\in \\overline{C}^{\\bullet}, \\lambda_i \\geq \\lambda \\} \n\\]\non $C^{\\bullet}$ and denote by \n$\nF^{\\lambda}(C[1]^{m_1} \\otimes \\cdots \\otimes \nC[1]^{m_k})\n$ \nthe submodule of $\\C[1]^{m_1} \\otimes \\cdots \\otimes C[1]^{m_k}$ \nspanned by \n\\[\nF^{\\lambda_1}(C[1]^{m_1}) \\otimes \\cdots \\otimes \nF^{\\lambda_k}(C[1]^{m_k}), \\ \\ \n\\sum_{i=1}^k \\lambda_i = \\lambda.\n\\]  \nDefine the bar complex of $C[1]^{\\bullet}$ by \nthe completion with respect to the energy filtration and denote it by  \n$B_k(C[1]^{\\bullet})$.\n\n\\smallskip\n\\noindent\n{\\bf 2.3) Filtered case and $G$-gapped conditions.}\n\nConsider the $k$-ary operations, $k=0,1,2, \\dots$,  \n\\[\n{\\mathfrak m}_k:B_k(C[1]^{\\bullet}) \\to C[1]^{\\bullet}\n\\]\nsuch that \n\\[\n{\\mathfrak m}_k(F^{\\lambda_1}C[1]^{\\bullet} \\otimes \\cdots \n\\otimes F^{\\lambda_k} C[1]^{\\bullet}) \\subset F^{\\lambda_1 + \\cdots \n\\lambda_k} C[1]^{\\bullet}\n\\]\nand \n\\[\n{\\mathfrak m}_0(1) \\in F^{\\lambda'}C[1]^{\\bullet} \n\\text{ for some } \\lambda'>0.\n\\]\n\nWe used the induction on the energy level in various arguments \nin \\cite{FOOO}, see also section 3 in this note.  \nFor such purposes, we introduced the \n$G$-gapped condition, which we assume from now on, as follows.  \nNote that the $G$-gapped condition follows from Gromov's compactness \ntheorem in the case of symplectic Floer theory.  \nLet $G \\subset \\R_{\\geq 0} \\times 2\\Z$ be a monoid such that \n${\\rm pr}_1^{-1}([0,c])$ is finite for any $c \\geq 0$ and \n\\[\n{\\rm pr}_1^{-1} (0) = \\{{\\mathbf 0}=(0,0)\\}.\n\\]\nHere ${\\rm pr}_i$ is the projection to the $i$-th factor, $i=1,2$.  \nThe filtered $A_{\\infty}$-algebra is said to be $G$-gapped, if \nthere exist \n\\[\n{\\mathfrak m}_{k,\\beta_i}:B_k(\\overline{C}[1]^{\\bullet}) \\to \n\\overline{C}[1]^{\\bullet}\n\\]\nfor $\\beta_i=(\\lambda_i,\\mu_i) \\in G \\subset \\R_{\\geq 0} \\times 2\\Z$ \nsuch that \n${\\mathfrak m}_{0,{\\mathbf 0}}=0$ and \n\\[\n{\\mathfrak m}_k = \\sum_{i} \nT^{\\lambda_i} e^{\\mu_i\/2} {\\mathfrak m}_{k, \\beta_i}.  \n\\] \nExtend ${\\mathfrak m}_k$ to the graded coderivation \n$\\widehat{\\mathfrak m}_k$ on $B(C[1]^{\\bullet})$.  \nWe call $(C^{\\bullet},\\{{\\mathfrak m}_k\\})$ a filtered \n$A_{\\infty}$-algebra, if \n\\[\n\\widehat{d} = \\sum_k \\widehat{\\mathfrak m}_k:B(C[1]^{\\bullet}) \\to B(C[1]^{\\bullet})\n\\] \nsatisfies $\\widehat{d} \\circ \\widehat{d} = 0$.  \nIn other words, \n$$\n\\sum_{k_1+k_2=k+1} \\sum_i (-1)^{\\sum_{j=1}^{i-1} \\deg' x_j} \n{\\mathfrak m}_{k_1}\n(x_1, \\dots, {\\mathfrak m}_{k_2}\n(x_i, \\dots, x_{i+k_2-1}), \\dots, x_k) = 0.\n$$  \nHere $k_1$ is a positive integer and $k_2$ is a non-negative \nintegers.  \nWhen $k_2=0$, ${\\mathfrak m}_{k_2}(x_i, \\dots, x_{i+k_2-1})$ \nis understood as ${\\mathfrak m}_0(1)$.  \n\nFor a filtered $A_{\\infty}$-algebra \n$(C^{\\bullet},\\{{\\mathfrak m}_k\\})$,\nset $\\overline{\\mathfrak m}_k= {\\mathfrak m}_{k,{\\bf 0}}$, \n${\\bf 0}=(0,0) \\in \\R_{\\geq 0} \\times 2\\Z$.  \nThen $(\\overline{C}^{\\bullet},\\{\\overline{\\mathfrak m}_k\\})$ is an \n$A_{\\infty}$-algebra.  \nWe call $(C^{\\bullet},\\{{\\mathfrak m}_k\\})$ a deformation of \n$(\\overline{C}^{\\bullet},\\{\\overline{\\mathfrak m}_k\\})$.  \n\nNote that ${\\mathfrak m}_1 \\circ {\\mathfrak m}_1$ may not be zero \nand we have \n$$\n{\\mathfrak m}_1 \\circ {\\mathfrak m}_1 (x) + {\\mathfrak m}_2(\n{\\mathfrak m}_0(1), x) + (-1)^{\\deg' x} {\\mathfrak m}_2(x,\n{\\mathfrak m}_0(1)) = 0.\n$$\nWe set \n$$\ne^b=1+b+b\\otimes b + b \\otimes b \\otimes b + \\dots, \n$$\nfor $b \\in {\\mathcal F}^{\\lambda}(C[1]^{0})$ with $\\lambda > 0$ \nand consider the {\\it Maurer-Cartan} equation: \n$$\n\\widehat{d}(e^b) = 0,\n$$\nwhich is equivalent to \n$$\n{\\mathfrak m}_0(1)+{\\mathfrak m}_1(b)+{\\mathfrak m}_2(b,b)+ \n{\\mathfrak m}_3(b,b,b) + \\dots = 0.\n$$\nFor a given $b$, we define a coalgebra homomorphism\n$$\n\\Phi^b(x_1 \\otimes x_2 \\otimes \\dots \\otimes x_k) \n=e^b \\otimes x_1 \\otimes e^b \\otimes x_2 \\otimes e^b \n\\otimes \\dots \\otimes e^b \\otimes x_k \\otimes e^b.\n$$\nThen define \n$$\n{\\mathfrak m}_k^b(x_1 \\otimes \\dots \\otimes x_k)\n={\\mathfrak m}_* \\circ \\Phi^b(x_1 \\otimes \\dots \\otimes x_k),\n$$\nwhere \n${\\mathfrak m}_*:B(C[1]^{\\bullet}) \\to C[1]^{\\bullet}$ is defined \nby ${\\mathfrak m}_*\\vert_{B_k(C[1]^{\\bullet})}={\\mathfrak m}_k$.  \nThen, for a solution $b$ of the Maurer-Cartan equation, we find that \n${\\mathfrak m}_0^b(1)=0$, hence ${\\mathfrak m}_1^b \\circ \n{\\mathfrak m}_1^b=0$.  \nNamely, the original ${\\mathfrak m}_1$ is rectified to \na coboundary operator ${\\mathfrak m}_1^b$ using a solution of \nthe Maurer-Cartan equation, which we also call a bounding cochain.  \n\n\nFor a collection $\\{ {\\mathfrak f}_k:B_k(C[1]^{\\bullet}) \\to C'[1]^{\\bullet} \\}_{k=0}\n^{\\infty}$ of degree $0$, we define \n\\[\n\\widehat{\\mathfrak f}(x_1 \\otimes \\cdots \\otimes x_k) \n=  \n\\sum_{k_1 + \\cdots k_n=k} \n{\\mathfrak f}_{k_1}(x_1 \\otimes \\cdots \\otimes x_{k_1}) \n\\otimes \\cdots \\otimes \n{\\mathfrak f}_{k_n}(x_{k+1-k_n} \\otimes \\cdots \\otimes x_k), \n\\]\nfor $k>0$ and \n\\[\n\\widehat{\\mathfrak f}(1)=1 + {\\mathfrak f}_0(1) + \n{\\mathfrak f}_0(1) \\otimes {\\mathfrak f}_0(1) + \\cdots,\n\\]\nwhere $1 \\in \\Lambda_{0,nov}=B_0(C[1]^{\\bullet})$.  \nWe assume the $G$-gapped condition, i.e., \nthere exist \n\\[\n{\\mathfrak f}_{k,\\beta_i}:B_k(\\overline{C}[1]^{\\bullet}) \\to \n\\overline{C}'[1]^{\\bullet}\n\\] \nfor $\\beta_i=(\\lambda_i,\\mu_i) \\in G$ \nwith $\\lambda_i \\to +\\infty$ as \n$i \\to +\\infty$ \nsuch that \n\\[\n{\\mathfrak f}_k = \\sum_i T^{\\lambda_i} e^{\\mu_i\/2} \n{\\mathfrak f}_{k,\\beta_i}.\n\\]\nIn particular, $\\widehat{\\mathfrak f}$ preserves the energy \nfiltration.  Namely, \n\\[\n\\widehat{\\mathfrak f}(F^{\\lambda}B(C[1]^{\\bullet})) \\subset F^{\\lambda}C'[1]^{\\bullet},\n\\]\nwhere $\\{F^{\\lambda}B(C[1]^{\\bullet})\\}$ is the filtration derived from \nthe filtration $F^{\\lambda}$ on $C[1]^{\\bullet}$.  \nWe call $\\{ {\\mathfrak f}_k \\}$ a $G$-gapped \nfiltered $A_{\\infty}$-homomorphism, \nif \n$\\widehat{d}_{C}' \\circ \\widehat{\\mathfrak f} = \\widehat{\\mathfrak f} \n\\circ \\widehat{d}_{C}$.  \nWhen we do not specify the monoid $G$, we call gapped filtered $A_{\\infty}$-algebras, gapped filtered $A_{\\infty}$-homomorphisms, etc.  \n\n\\smallskip\n\\noindent\n{\\bf 2.4) Homotopy theory.}\n\nIn \\cite{FOOO}, we introduced the notion of models of \n$[0,1] \\times C^{\\bullet}$ (Definition 15.1) and gave two constructions. Using this notion, we developed \nthe homotopy theory of filtered\n$A_{\\infty}$-algebras and filtered-$A_{\\infty}$ bimodules.\nOur formulation has an advantage to clarify \nequivalence of various definitions of homotopy\nof $A_{\\infty}$ algebras appearing in the literature\neven for the unfiltered cases.\n\nLet $C$ be the completion of $\\overline{C} \\otimes \\Lambda_0$, which is \na filtered $A_{\\infty}$-algebra. \n\n\\begin{defn}\nLet ${\\mathfrak C}$ be the completion of $\\overline{\\mathfrak C} \\otimes \\Lambda_0$, which is \na filtered $A_{\\infty}$-algebra together with \nfiltered $A_{\\infty}$-homomorphisms.  \n\\[\n{\\rm Incl}:C \\to {\\mathfrak C}, \\ \n{\\rm Eval}_{s=i}:{\\mathfrak C} \\to C, i=0,1.\n\\]  \nWe call $\\mathfrak C$ a model of $[0,1] \\times C$, if \nthe following conditions are satisfied:  \n\n\\begin{itemize}\n\\item\n${\\rm Incl}_{k,\\beta}$ and ${\\rm Eval}_{s=i}, i=0,1$ are \nzero unless $(k,\\beta)=(1,\\beta_0)$.  \n\\item\n${\\rm Incl}_{1,\\beta_0}, ({\\rm Eval}_{s=0})_{1,\\beta_0}$ \nare cochain homotopy equivalences between \n$\\overline{C}$ and $\\overline{\\mathfrak C}$.  \n\\item\n${\\rm Eval}_{s=0} \\circ {\\rm Incl} = {\\rm Eval}_{s=1} \\circ {\\rm Incl} \n= {\\rm id.}$  \n\\item\n${\\rm Eval}_{s=0} \\oplus {\\rm Eval}_{s=1}:{\\mathfrak C} \\to C \\oplus C$ \nis surjective.  \n\\end{itemize}\n\\end{defn}\n\nWe quote here one of constructions of the models of $[0,1] \\times C$ \nfor reader's convenience.  \n\nSet $$C^{[0,1]}=C \\oplus C[-1] \\oplus C,$$\nand define ${\\mathfrak I}_0, {\\mathfrak I}_1:C \\to C^{[0,1]}$ of \ndegree 0 and \n${\\mathfrak I}:C \\to C^{[0,1]}$ of degree 1 by \n$$\n{\\mathfrak I}_0(x)=(x,0,0), {\\mathfrak I}_1(x)=(0,0,x), \n{\\mathfrak I}_1(x)=(0,x,0).\n$$\nWe extend ${\\mathfrak I}_0,{\\mathfrak I}_1$ to \n$B(C[1]) \\to B(C^{[0,1]}[1])$ and denote them by the same symbol.  \nDefine \n\n\\begin{align}\n({\\rm Eval}_{s=0})_1(x,y,z)=x, \\ ({\\rm Eval}_{s=1})_1(x,y,z)=z, \n\\nonumber \\\\\n({\\rm Incl})_1(x)={\\mathfrak I}_0(x)+{\\mathfrak I}_1(x)=(x,0,x). \n\\nonumber\n\\end{align}\n\nWe define the filtered $A_{\\infty}$-structure $\\{{\\mathfrak M}_k\\}$.  \n\nFor ${\\mathfrak M}_0$, ${\\mathfrak M}_1$, we set  \n\\begin{align}\n{\\mathfrak M}_0(1) & \n= ({\\rm Incl})_1({\\mathfrak m}_0(1)), \\nonumber \\\\\n{\\mathfrak M}_1({\\mathfrak I}_0(x)) & \n={\\mathfrak I}_0({\\mathfrak m}_1(x)) \n+ (-1)^{\\deg' x} {\\mathfrak I}(x), \\nonumber \\\\\n{\\mathfrak M}_1({\\mathfrak I}_1(x)) & \n={\\mathfrak I}_1({\\mathfrak m}_1(x)) - (-1)^{\\deg' x} {\\mathfrak I}(x), \n\\nonumber \\\\\n{\\mathfrak M}_1({\\mathfrak I}(x)) & \n={\\mathfrak I}({\\mathfrak m}_1(x)).\n\\nonumber\n\\end{align}\n\nWe define ${\\mathfrak M}_k$, $k \\geq 2$ as follows.  \nFor ${\\mathbf x} \\in B_k(C[1])$, $y \\in C$ and \n${\\mathbf z} \\in B_{\\ell}(C[1])$, we set \n$$\n{\\mathfrak M}_{k+\\ell + 1}({\\mathfrak I}_0({\\mathbf x}), {\\mathfrak I}(y), {\\mathfrak I}_1({\\mathbf z})) = \n(-1)^{\\deg' {\\mathbf z}} {\\mathfrak I}({\\mathfrak m}_{k + \\ell + 1} \n({\\mathbf x}, y, {\\mathbf z}),\n$$\nand \n$$\n{\\mathfrak M}_k({\\mathfrak I}_0({\\mathbf x}))=\n{\\mathfrak I}_0({\\mathfrak m}_k({\\mathbf x})), \n{\\mathfrak M}_{\\ell}({\\mathfrak I}_1({\\mathbf z}))=\n{\\mathfrak I}_1({\\mathfrak m}_{\\ell}({\\mathbf z})), \\ \\ \n{\\text{ for }} k, \\ell \\geq 2.\n$$  \nHere the order of \n${\\mathfrak I}_0({\\mathbf x}), {\\mathfrak I}(y), \n{\\mathfrak I}_1({\\mathbf z})$ is important.  \nWe define operators ${\\mathfrak M}_k$ on $C^{[0,1]}$ other than \nthose defined above to be zero.  \n\n\nModels of $[0,1] \\times C$ are not unique, but we proved the following:\n\n\\begin{thm}[Theorem 15.34 in \\cite{FOOO}]\nLet $C_1,C_2$ be gapped filtered $A_{\\infty}$-algebras and \n${\\mathfrak C}_1, {\\mathfrak C}_2$ any models for $[0,1] \\times C_1, \n[0,1] \\times C_2$, respectively.  \nLet ${\\mathfrak f}:C_1 \\to C_2$ be a gapped filtered \n$A_{\\infty}$-homomorphism.  \nThen there exists a gapped filtered $A_{\\infty}$-homomorphism \n${\\mathfrak F}:{\\mathfrak C}_1 \\to {\\mathfrak C}_2$ such that \n$$\n{\\rm Eval}_{s=s_0} \\circ {\\mathfrak F} = {\\mathfrak f} \\circ \n{\\rm Eval}_{s=s_0}, \\ s_0=0,1\n$$\nand \n$$\n{\\rm Incl} \\circ {\\mathfrak f} = {\\mathfrak F} \\circ {\\rm Incl}.\n$$\n\\end{thm}\n\nWe define two filtered $A_{\\infty}$-homomorphisms \n${\\mathfrak f}_i:C_1 \\to C_2, i=0,1$ are homotopic, if \nthere is a model ${\\mathfrak C}_2$ of $C_2$ and \na filtered $A_{\\infty}$-homomorphism \n${\\mathfrak F}:C_1 \\to {\\mathfrak C}_2$ such that \n${\\mathfrak f}_i={\\rm Eval}_{s=i} \\circ {\\mathfrak F}$.  \nAlthough the definition literally depends on the choice of the model \n${\\mathfrak C}_2$, \nwe can show that the notion of homotopy between ${\\mathfrak f}_i$ \ndoes not depend on the choice of the model and the homotopy \nis, in fact, an equivalence relation, see Chapter 4 in \\cite{FOOO}.  \n\nNote that the notion of homotopy between $A_{\\infty}$-homomorphisms \nin the unfiltered case appeared in literaturure, e.g., \\cite{Smi}.  \nBy taking a suitable model, we can find that \nour definition above coincides with such a definition.  \nIt also implies that various definitions which appear in the literature \nare equivalent to one another.  \nWe think that the notion of models clarifies arguments and is \nalso useful when we consider the gauge equivalence between \nsolutions of the Maurer-Cartan equation.  \nNamely, two solutions $b, b'$ of the Maurer-Cartan equation is \ngauge equivalent, if there is a model ${\\mathfrak C}$ of \n$[0,1] \\times C$ and a solution $\\widetilde{b}$ of the Maurer-Cartan \neqution on ${\\mathfrak C}$ such that \n${\\rm Eval}_{s=0}(\\widetilde{b})=b$ and \n${\\rm Eval}_{s=1}(\\widetilde{b})=b'$.  \nFor details, see section 16 in \\cite{FOOO}.  \n\nAmong other things, we proved the Whitehead type theorem as follows.  \nAn $A_{\\infty}$-homomorphism $\\{\\overline{\\mathfrak f}_k\\}$ from \n$\\overline{C}^{\\bullet}$ to $\\overline{C}^{' \\bullet}$ \nis called a weak homotopy equivalence, if \n$\\overline{f}_1:\\overline{C}^{\\bullet} \\to \\overline{C}^{' \\bullet}$ \nis a cochain homotopy equivalence \nbetween $\\overline{\\mathfrak m}_1$-complexes. \nA filtered $A_{\\infty}$-homomorphism $\\{{\\mathfrak f}_k\\}$ from \n$C[1]^{\\bullet}$ to $C'[1]^{\\bullet}$ is called \na weak homotopy equivalence, if \n$\\overline{\\mathfrak f}_1={\\mathfrak f}_{1,{\\bf 0}}$ is a cochain \nhomotopy equivalence between \n$\\overline{\\mathfrak m}_1={\\mathfrak m}_{1,{\\bf 0}}$-complexes.  \n\n\\begin{thm}[Theorem 15.45 in \\cite{FOOO}]\\label{whitehead}\n(1) A weak homotopy equivalence of $A_{\\infty}$-algebras is \na homotopy equivalence.  \n\n\\noindent\n(2) A gapped weak homotopy equivalence between gapped filtered \n$A_{\\infty}$-algebras is a homotopy equivalence.  \nThe homotopy inverse of a strict weak homotopy equivalence can be taken \nto be strict.  \n\\end{thm}\n\nNote that the above theorem does not hold in the realm of \ndifferential graded algebras.  The notion of $A_{\\infty}$-homomorphism \nis much wider than that of homomorphisms as differential graded \nalgebras.  \n\n\\smallskip\n\\noindent\n{\\bf 2.5) Filtered $A_{\\infty}$-bimodules.} \n\nLet $(C_i^{\\bullet},\\{{\\mathfrak m}_k^{(i)}\\})$, $i=0,1$, be filtered \n$A_{\\infty}$-algebras and $\\overline{D}^{\\bullet}$ a graded module.  \nWrite \n\\[\nD[1]^{\\bullet}=\\overline{D}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}\n\\]\nand \n\\[\n\\widetilde{D}[1]^{\\bullet}=\\overline{D}[1]^{\\bullet} \\otimes \\Lambda_{nov}.\n\\]\nLet \n${\\mathfrak n}_{k_1,k_0}:B_{k_1}(C_1[1]^{\\bullet}) \\otimes D[1]^{\\bullet} \\otimes \nB_{k_0}(C_0[1]^{\\bullet}) \\to D[1]^{\\bullet}$, $k_1,k_0=0,1,2,\\dots$, \nbe $\\Lambda_{0,nov}$-module homomorphisms of degree $1$.  \nWe also denote its extension to $\\widetilde{D}[1]^{\\bullet}$ by the same symbol \n${\\mathfrak n}_{k_1,k_0}$.  \nWe call $(D^{\\bullet},\\{{\\mathfrak n}_{k_1,k_0}\\})$ and \n$(\\widetilde{D}^{\\bullet},\\{{\\mathfrak n}_{k_1,k_0}\\})$ a filtered \n$A_{\\infty}$-bimodule over $(C_i^{\\bullet},\\{{\\mathfrak m}_k\\})$, if \n\\[\n\\widehat{d}_{\\mathfrak n} \\circ \\widehat{d}_{\\mathfrak n} = 0,\n\\]\nwhere $\\widehat{d}_{\\mathfrak n}$ is the coderivation \non $B(C_1[1]^{\\bullet}) \\otimes D[1]^{\\bullet} \\otimes B(C_0[1]^{\\bullet})$ determined by \n$\\{{\\mathfrak m}_k^{(i)}\\}$, $i=0,1$, and \n$\\{{\\mathfrak n}_{k_1,k_0}\\}$.  \nThe $G$-gapped condition is defined in a similar way to \nthe case of filtered $A_{\\infty}$-algebras: \n\\[\n{\\mathfrak n}_{k_1,k_0} = \\sum_{\\beta \\in G} T^{{\\rm pr}_1(\\beta)} \ne^{{\\rm pr}_2(\\beta)\/2} {\\mathfrak n}_{k_1,k_0,\\beta}, \n\\]\nwhere \n\\[\n{\\mathfrak n}_{k_1,k_0,\\beta}:B_{k_1}(\\overline{C}_1[1]^{\\bullet}) \\otimes \n\\overline{D}[1]^{\\bullet} \\otimes \nB_{k_0}(\\overline{C}_0[1]^{\\bullet}) \\to \\overline{D}[1]^{\\bullet}.  \n\\]\nFor $\\lambda \\in \\R$, we set \n\\[\n{\\mathcal F}^{\\lambda}\\widetilde{D}^{\\bullet}=T^{\\lambda} \\cdot \nD^{\\bullet}.\n\\]  \nFor $\\lambda \\geq 0$, \n${\\mathcal F}^{\\lambda}\\widetilde{D}^{\\bullet} \\subset D^{\\bullet}$, \nhence we obtain  \na filtration on $D^{\\bullet}$.  \nIt is clear that \n\\[\n\\widehat{d}_{\\mathfrak n}(F^{\\lambda_1}B(C_1[1]^{\\bullet}) \\otimes \n{\\mathcal F}^{\\lambda_2}\\widetilde{D}^{\\bullet} \n\\otimes F^{\\lambda_3}B(C_0[1]^{\\bullet})) \n\\subset {\\mathcal F}^{\\lambda_1+\\lambda_2+\\lambda_3}\\widetilde{D}[1]^{\\bullet}.  \n\\]\n\nNote that ${\\mathfrak n}_{0,0} \\circ {\\mathfrak n}_{0,0}$ may not \nbe zero and we have \n$$\n{\\mathfrak n}_{0,0} \\circ {\\mathfrak n}_{0,0}(y) \n+ {\\mathfrak n}_{1,0}({\\mathfrak m}_0^{(1)}(1), y) \n+(-1)^{\\deg' y} {\\mathfrak n}_{0,1}(y,{\\mathfrak m}_0^{(0)}(1)) = 0.\n$$\nFor $b_i \\in {\\mathfrak F}^{\\lambda^{(i)}}(C_i[1]^{\\bullet})$ \nwith $\\lambda^{(i)} > 0$, we define \n$$\n{\\mathfrak n}^{b_0,b_1}_{k_1,k_0}({\\mathbf x} \\otimes y \\otimes \n{\\mathbf z})={\\mathfrak n}_{*,*}(\\Phi^{b_1}({\\mathbf x}) \n\\otimes y \\otimes \\Phi^{b_0}({\\mathbf z})), \n$$\nfor ${\\mathbf x} \\in B_{k_1}(C_1[1]^{\\bullet})$ and \n${\\mathbf z} \\in B_{k_0}(C_0[1]^{\\bullet})$.  \nIn particular, \n$$\n{\\mathfrak n}_{0,0}^{b_0,b_1}(y)=\n{\\mathfrak n}_{*,*}(e^{b_1} \\otimes y \\otimes e^{b_0}).\n$$\nIf $b_0, b_1$ are solutions of the Maurer-Cartan equations \nin the filtered $A_{\\infty}$-algebras $C_0, C_1$, respectively, \nwe find that \n$${\\mathfrak n}_{0,0}^{b_0,b_1} \\circ \n{\\mathfrak n}_{0,0}^{b_0,b_1} = 0.$$  \nNamely, we can rectify the original ${\\mathfrak n}_{0,0}$ \nto a coboundary operator ${\\mathfrak n}_{0,0}^{b_0,b_1}$ on $D[1]$.  \n\nLet $\\{{\\mathfrak f}_k^{(i)}\\}$, $i=0,1$, be \nfiltered $A_{\\infty}$-homomorphisms from $C_i^{\\bullet}$ to \n$C_i^{' \\bullet}$ and \n$(\\widetilde{D}^{\\bullet},\\{{\\mathfrak n}_{k_1,k_0}\\})$, resp. \n$(\\widetilde{D}^{' \\bullet},\\{{\\mathfrak n}_{k_1,k_0}'\\})$, \nfiltered $A_{\\infty}$-bimodules over \n$(C_i^{\\bullet},\\{{\\mathfrak m}_k^{(i)}\\})$, resp. \n$(C_i^{' \\bullet},\\{{\\mathfrak m}_k{'(i)}\\})$.  \nSuppose that there exist a real number $c$ and \n$\\Lambda_{nov}$-homomorphisms \n${\\phi}_{k_1,k_0}:B_{k_1}(C_1[1]^{\\bullet}) \\otimes \\widetilde{D}[1]^{\\bullet} \n\\otimes B_{k_0}(C_0[1]^{\\bullet}) \n\\to \\widetilde{D}'[1]^{\\bullet}$, $k_1,k_0=0,1,2,\\dots$, \nsuch that \n\\[\n{\\phi}_{k_1,k_0}(F^{\\lambda_1}B_{k_1}(C_1[1]^{\\bullet}) \\otimes \n{\\mathcal F}^{\\lambda_2} \\widetilde{D}[1]^{\\bullet} \\otimes \nF^{\\lambda_3}B_{k_0}(C_0[1]^{\\bullet})) \\subset \n{\\mathcal F}^{\\lambda_1+\\lambda_2+\\lambda_3 -c}\\widetilde{D}'[1]^{\\bullet}.\n\\]\nWe call such $c$ the energy loss of $\\{{\\phi}_{k_1,k_0}\\}$.  \nFor such a collection $\\{{\\phi}_{k_1,k_0}\\}$, \nwe define \n\\[\n\\widehat{\\mathfrak \\phi}:B(C_1[1]^{\\bullet}) \\otimes \\widetilde{D}[1]^{\\bullet} \\otimes \nB(C_0[1]^{\\bullet}) \\to B(C_1'[1]^{\\bullet}) \\otimes \\widetilde{D}'[1]^{\\bullet} \\otimes B(C_0'[1]^{\\bullet}) \n\\]\nas the homomorphism determined by \n$\\{{\\mathfrak f}_k^{(i)}\\}$, $i=0,1$, and $\\{{\\phi}_{k_1,k_0}\\}$.  \nWe call $\\phi=\\{{\\phi}_{k_1,k_0}\\}$ a weakly filtered \n$A_{\\infty}$-homomorphism of filtered $A_{\\infty}$-bimodules, if \n\\[\n\\widehat{d}_{{\\mathfrak n}'} \\circ \\widehat{\\phi} = \n\\widehat{\\phi} \\circ \\widehat{d}_{\\mathfrak n}.  \n\\]\nWhen we can take $c=0$ , $\\phi=\\{{\\phi}_{k_1,k_0}\\}$ is called \na filtered $A_{\\infty}$-homomorphism.  \nSuppose that $C_i^{\\bullet},C_i^{' \\bullet}$ are $G$-gapped.  \nLet $G' \\subset \\R \\times 2\\Z$ be a $G$-set such that \n${\\rm pr}_1|_G^{-1} ((-\\infty,\\lambda])$ is finite \nfor any $\\lambda \\in \\R$ \nand ${\\rm pr}_1(G)$ is bounded from below.   \nWe say that $\\phi=\\{{\\phi}_{k_1,k_0}\\}$ is $G'$-gapped, if \n\\[\n\\phi_{k_1,k_0}=\\sum_{\\beta' \\in G'} T^{{\\rm pr}_1(\\beta')} \ne^{{\\rm pr}_2(\\beta')\/2} \\phi_{k_1,k_0,\\beta'},\n\\]\nwhere \n\\[\n\\phi_{k_1,k_0,\\beta'}:B_{k_1}(\\overline{C}_1[1]^{\\bullet}) \\otimes \n\\overline{D}[1]^{\\bullet} \\otimes B_{k_0}(\\overline{C}_0[1]^{\\bullet}) \n\\to \\overline{D}'[1]^{\\bullet}.\n\\]\n\nThe homotopy theory between filtered $A_{\\infty}$-homomorphisms of \nfiltered $A_{\\infty}$-bimodules is also developed in \\cite{FOOO}.  \n\nWe also proved the Whitehead theorem for (filtered) \n$A_{\\infty}$-bimodules.  \n\n\\begin{thm}[Theorem 21.35 \\cite{FOOO}]\nLet $\\phi:D^{\\bullet} \\to D^{' \\bullet}$ be a gapped \nfiltered $A_{\\infty}$-bimodule \nhomomorphism over $({\\mathfrak f}^{(1)},{\\mathfrak f}^{(0)})$, where \n${\\mathfrak f}^{(i)}:C_i^{\\bullet} \\to C_i^{' \\bullet}$ \nare homotopy equivalences.  \nSuppose that $\\phi_{(0,0,{\\mathbf 0})}$ is a chain homotopy equivalence.  \nThen $\\phi$ is a homotopy equivalence of filtered \n$A_{\\infty}$-bimodules.\n\\end{thm}\n\n\\begin{rem}\nHere we require $\\phi$ is a filtered $A_{\\infty}$-homomorphism.  \nSince a weakly filtered $A_{\\infty}$-homomorphism, \n$\\phi_{(0,0,{\\mathbf 0})}$ may not induce a chain map with respect to \n${\\mathfrak n}_{0,0,{\\mathbf 0}}$ and \n${\\mathfrak n}_{0,0,{\\mathbf 0}}'$.\n\\end{rem}\n\n\\smallskip\n\\noindent\n{\\bf 2.6) Filtered $A_{n,K}$ structure.}\n\nFor a later argument in section 4, \nwe recall the notion of filtered $A_{n,K}$-algebras.  \nLet $G \\subset \\R{\\geq 0} \\times 2\\Z$ be a monoid as above and \n$\\beta_0={\\mathbf 0} \\in G$.    \nFor $\\beta \\in G$, we define \n\n\\[\n\\parallel \\beta \\parallel = \\left\\{\n\\begin{array}{ll} \n\\sup \\{ n | \\exists \\beta_i \\in G \\setminus \\{\\beta_0\\}, \\ \n\\sum_{i=1}^n \\beta_i = \\beta \\} + [{\\rm pr}_1(\\beta)] -1 & \n\\text{ if } \\beta \\neq \\beta_0 \\\\\n-1 & \\text{ if } \\beta = \\beta_0,   \n\\end{array}\n\\right.\n\\]\n\nThen we introduce a partial order on $(G \\times \\Z_{\\geq 0})  \n\\setminus \\{(\\beta_0,0)\\}$ by \n$(\\beta_1,k_1) \\succ (\\beta_2,k_2)$ if and only if \neither \n\\[\n\\parallel \\beta_1 \\parallel + k_1 > \n\\parallel \\beta_2 \\parallel + k_2\n\\] or \n\\[\n\\parallel \\beta_1 \\parallel + k_1 = \n\\parallel \\beta_2 \\parallel + k_2, \\text{ and } \n\\parallel \\beta_1 \\parallel > \\parallel \\beta_2 \\parallel.\n\\]\nWe write $(\\beta_1,k_1) \\sim (\\beta_2,k_2)$, when \n\\[\n\\parallel \\beta_1 \\parallel + k_1 = \n\\parallel \\beta_2 \\parallel + k_2, \\text{ and } \n\\parallel \\beta_1 \\parallel = \\parallel \\beta_2 \\parallel.\n\\]\nWe define $(\\beta_1,k_1) \\succsim (\\beta_2,k_2)$ if \neither $(\\beta_1,k_1) \\succ (\\beta_2,k_2)$ or $(\\beta_1,k_1) \\sim \n(\\beta_2,k_2)$.  \n\nWe also write $(\\beta,k) \\prec (n',k')$, when \n$\\parallel \\beta \\parallel + k < n' + k'$ or \n$\\parallel \\beta \\parallel + k = n' + k'$ and \n$\\parallel \\beta \\parallel < n'$.  \n\nLet $\\overline{C}^{\\bullet}$ be a cochain complex over $R$ and \n$C^{\\bullet}=\\overline{C}^{\\bullet}\\otimes \\Lambda_{0,nov}$.  \nSuppose that there are \n\\[\n{\\mathfrak m}_{k,\\beta}:B(\\overline{C}[1]^{\\bullet}) \\to \\overline{C}[1]^{\\bullet}\n\\]\nfor $(\\beta,k) \\in (G \\times \\Z) \\setminus \\{(\\beta_0,0)\\}$ \nwith $(\\beta,k) \\prec (n,K)$.  \nWe also suppose that ${\\mathfrak m}_{1,\\beta_0}$ is \nthe boundary operator of the cochain complex $C^{\\bullet}$.  \nWe call $(C^{\\bullet},\\{{\\mathfrak m}_{k,\\beta}\\})$ \na $G$-gapped filtered $A_{n,K}$-algebra, if the following holds\n\\[\n\\sum_{\\beta_1+\\beta_2=\\beta, k_1+k_2=k+1} \\sum_i \n(-1)^{\\deg'{\\mathbf x}_i^{(1)}} {\\mathfrak m}_{k_2,\\beta_2}\n\\bigl({\\mathbf x}_i^{(1)}, \n{\\mathfrak m}_{k_1,\\beta_1}({\\mathbf x}_i^{(2)}), \n{\\mathbf x}_i^{(3)} \\bigr) = 0\n\\] \nfor all $(\\beta,k) \\prec  (n,K)$, \nwhere \n\\[\n\\Delta^2({\\mathbf x})=\\sum_i{\\mathbf x}_i^{(1)} \\otimes \n{\\mathbf x}_i^{(2)} \\otimes {\\mathbf x}_i^{(3)}.\n\\]\nHere $\\Delta$ is the coproduct of the tensor coalgebra.  \n\nWe also have the notion of filtered $A_{n,K}$-homomorphisms, \nfiltered $A_{n,K}$-homotopy equivalences in a natural way.  \nIn \\cite{FOOO}, we proved the following:\n\n\\begin{thm}[Theorem 30.72 in \\cite{FOOO}]\\label{ext(n,K)}\nLet $C_1^{\\bullet}$ be a filtered $A_{n,K}$-algebra and \n$C_2^{\\bullet}$ a filtered $A_{n',K'}$-algebra such that \n$(n,K) \\prec (n',K')$.  \nLet ${\\mathfrak h}:C_1^{\\bullet} \\to C_2^{\\bullet}$ \nbe a filtered $A_{n,K}$-homomorphism.  \nSuppose that ${\\mathfrak h}$ is a filtered $A_{n,K}$-homotopy \nequivalence.  \nThen there exist a filtered $A_{n',K'}$-algebra structure on \n$C_1^{\\bullet}$ \nextending the given filtered $A_{n,K}$-algebra structure \nand a filtered $A_{n',K'}$-homotopy equivalence \n$C_1^{\\bullet} \\to C_2^{\\bullet}$ extending the given \nfiltered $A_{n,K}$-homotopy equivalence ${\\mathfrak h}$.  \n\\end{thm}\n\n\n\\section{Canonical models}\nIn this section, we give the notion of canonical models and \nexplain their construction after section 23, Chapter 5 of \\cite{FOOO}.  \nThe unfiltered version of such a result goes back to Kadeishvili \n\\cite{Kad}.  \nThere are two methods to construct canonical models in unfiltered \ncase.  \nOne is based on obstruction theory due to Kadeishvili and \nthe other uses the summation over trees due to Kontsevich and \nSoibelman \\cite{KS}.  \nOur argument is an adaptation of the latter argument and we also \nconstructed the canonical models for filtered case.  \n\nWhen the ground coefficent ring $R$ is a field, we have the \nfollowing:  \n\n\\begin{thm}[Theorem 23.1, Theorem 23.2 in \\cite{FOOO}]\\label{canmodel}\n(1) Any unfiltered $A_{\\infty}$-algebra \n$(\\overline{C}^{\\bullet},\\{\\overline{\\mathfrak m}_k\\})$ is homotopy equivalent to an $A_{\\infty}$-algebra \n$(\\overline{C'}^{\\bullet},\\{\\overline{\\mathfrak m}'_k\\})$ with \n$\\overline{\\mathfrak m}'_1=0$.  \n\n\\noindent\n(2) Any gapped filtered $A_{\\infty}$-algebra \n$(C^{\\bullet},\\{{\\mathfrak m}_k\\})$ \nis homotopy equivalent to a gapped filtered $A_{\\infty}$-algebra \n$(C^{' \\bullet},\\{{\\mathfrak m}'_k\\})$ \nwith $\\overline{\\mathfrak m}'_1=0$.  \nMoreover, the homotopy equivalence can be taken as a gapped \n$A_{\\infty}$-homomorphism.  \n\\end{thm}\n \nAn $A_{\\infty}$-algebra is called {\\it canonical}, if \n$\\overline{\\mathfrak m}_1=0$.  \nA canonical model of an $A_{\\infty}$-algebra is \na canonical $A_{\\infty}$-algebra homotopy equivalent to \nthe original one.  \nThe statement (1) is Kadeishvili's theorem and implies that the \n$\\overline{\\mathfrak m}_1$-cohomology has a structure of an \n$A_{\\infty}$-algebra.  \nNote that, in general,  \nwe do not have ${\\mathfrak m}_1$-cohomology, \nsince ${\\mathfrak m}_1 \\circ {\\mathfrak m}_1$ may not be zero.  \nA filtered $A_{\\infty}$-algebra is called {\\it canonical}, if \n${\\mathfrak m}_{1,0}=\\overline{\\mathfrak m}_1=0$.  \nA canonical model of a filtered $A_{\\infty}$-algebra is \na canonical filtered $A_{\\infty}$-algebra homotopy equivalent to \nthe original one.  \n \nPick a submodule \n${\\mathcal H}^{\\bullet} \\stackrel{\\iota}{\\hookrightarrow} \n\\ker \\overline{\\mathfrak m}_1 \\cap \\overline{C}^{\\bullet}$ such that \n$\\iota_*: {\\mathcal H}^k \\cong \n{\\rm H}^k(\\overline{C}^{\\bullet},\\overline{\\mathfrak m}_1)$, \nand $\\Pi^k:\\overline{C}^k \\to {\\mathcal H}^k \\subset \\overline{C}^k$ \nsuch that $\\Pi^k \\circ \\Pi^k = \\Pi^k$ and \n$\\Pi^k \\circ \\overline{\\mathfrak m}_1 = 0$.  \nWe will construct a structure of a filtered $A_{\\infty}$-algebra on \n${\\mathcal H}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}$ and \na filtered $A_{\\infty}$-homomorphism from \n${\\mathcal H}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}$ \nto $C[1]^{\\bullet}$, which is a weak homotopy equivalence.  \nSince $R$ is a field, any cochain homomorphism inducing an isomorphism \non cohomologies is a weak homotopy equivalence.  \nFirstly, we observe the following:  \n\n\\begin{lem}\\label{H}\nThere exist $G^k:\\overline{C}^k \\to \\overline{C}^{k-1}$, \n$k=0,1,\\dots, n$, such that \n\\begin{align}\nid - \\Pi^k & =  -(\\overline{\\mathfrak m}_1 \\circ G^k + G^{k+1} \\circ \\overline{\\mathfrak m}_1), \\label{htpy} \\\\\nG^{k} \\circ G^{k+1} & =  0. \\label{sqzero} \n\\end{align}\n\\end{lem}\n\nFrom now on, let ${\\mathcal H}^{\\bullet} \n\\stackrel{\\iota}{\\hookrightarrow} \\overline{C}^{\\bullet}$ \nbe a subcomplex and $\\Pi:\\overline{C}^k \\to \n\\overline{C}^k$ be a projection to ${\\mathcal H}^k$ such that \nthere exist $G^k:\\overline{C}^k \\to \\overline{C}^{k-1}$ \nsatisfying (\\ref{htpy}), (\\ref{sqzero}) in Lemma \\ref{H}.  \nWe do not assume that $\\overline{\\mathfrak m}_1|_{\\mathcal H}=0$.  \nThus ${\\mathcal H}^{\\bullet}$ is not necessarily isomorphic \nto the cohomology $H^{\\bullet}(\\overline{C}^{\\bullet})$.  \nBut the condition (\\ref{htpy}) implies that \n$\\iota_*: {\\rm H}^{\\bullet}({\\mathcal H},\\overline{\\mathfrak m}_1|_{{\\mathcal H}}) \\cong \n{\\rm H}^{\\bullet}(\\overline{C}^{\\bullet},\\overline{\\mathfrak m}_1)$.  \nTheorem \\ref{canmodel} follows from the following:\n\n\\begin{thm}\\label{reduction}\n(1) There exists a structure \n$\\{\\overline{\\mathfrak m}_k'\\}_{k=1}^{\\infty}$ of \nan $A_{\\infty}$-algebra on ${\\mathcal H}$ with \n${\\mathfrak m}_1'={\\mathfrak m}_1|_{\\mathcal H}$.  \nThe inclusion $\\iota$ extends to an $A_{\\infty}$-homomorphism \n$\\{\\overline{\\mathfrak f}_k\\}_{k=1}^{\\infty}$ \nwith $\\overline{\\mathfrak f}_1 = \\iota$.  \n\n\\noindent\n(2) There exists a structure \n$\\{{\\mathfrak m}_k\\}_{k=0}^{\\infty}$ of a filtered $A_{\\infty}$-algebra \non ${\\mathcal H} \\otimes \\Lambda_{0,nov}$.  \nThe inclusion $\\iota$ extends to a filtered $A_{\\infty}$-homomorphism \n$\\{{\\mathfrak f}\\}_{k=0}^{\\infty}$ with \n${\\mathfrak f}_{1,0}=\\iota$.  \n\\end{thm}\n\n\nLet $G$ be a monoid as in section 2 and ${\\rm pr}_1(G)=\\{\\lambda_{(i)}\\}$ \nsuch that \n\\[\n0=\\lambda_{(0)} < \\lambda_{(1)} < \\lambda_{(2)} < \\cdots \\to +\\infty, \n\\]\nunless $G=\\{(0,0)\\}$.  \nWe write \n\\[\n{\\mathfrak m}_{k,i}=\n\\sum_{\\beta \\in G \\ \\ {\\rm pr}_1(\\beta)=\\lambda_{(i)}} \ne^{{\\rm pr}_2(\\beta)\/2} {\\mathfrak m}_{k,\\beta}\n\\]\nand\n\\[\n{\\mathfrak m}_{k,i}^{\\circ}=T^{\\lambda_{(i)}}{\\mathfrak m}_{k,i}.\n\\]\nThus ${\\mathfrak m}_k= \\sum_i {\\mathfrak m}_{k,i}^{\\circ}$. \nHere ${\\mathfrak m}_{k,i}$ is considered as \n\\[\n{\\mathfrak m}_{k,i}:B_k(\\overline{C}[1]^{\\bullet}) \\otimes R[e,e^{-1}] \n\\to \\overline{C}[1]^{\\bullet} \\otimes R[e,e^{-1}].\n\\]  \nBy an abuse of notation, we also denote by $\\Pi^k$, $G^k$ \nthe extensions thereof \nto $\\overline{C}^k \\otimes R[e,e^{-1}]$ as a $R[e,e^{-1}]$-module \nhomomorphism.  \n\nIn order to define a $G$-gapped filtered $A_{\\infty}$-structure \non ${\\mathcal H}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}$ and \na $G$-gapped $A_{\\infty}$-homomorphism from ${\\mathcal H}[1]^{\\bullet} \n\\otimes \\Lambda_{0,nov}$ to $C[1]^{\\bullet}$, we introduce some notation.  \n\nA decorated planar rooted tree is a quintet  \n$\\Gamma=(T,i,v_0,V_{tad}, \\eta)$, which consists of \n\\begin{itemize}\n\\item $T$ is a tree, \n\\item $i:T \\to D^2$ is an embedding, \n\\item $v_0$ is the root vertex, \n\\item $V_{tad}=\\{ \\text{vertices of valency } 1\\} \n\\setminus C^0_{ext}(T)$,\n\\item $\\eta:C^0_{int}(T)=C^0(T) \\setminus C^0_{ext}(T) \\to \\{0,1,2,\\dots\\}.$\n\\end{itemize}\nHere $C^0(T)$ is the set of vertices of the tree $T$, \n$C^0_{ext}(T)=i^{-1}(\\partial D^2)$ is the set of exterior vertices \nand $C^0_{int}(T)$ is the set of interior vertices.  \nNote that the root vertex $v_0$ is an exterior vertex and \n$V_{tad} \\subset C^0_{int}(T)$.  \nLet ${G}_k^+$ be the set of $\\Gamma=(T,i,v_0,V_{tad},\\eta)$ \nsuch that $\\# C^0_{ext} = k$ and $\\eta(v) > 0$ if $v \\in C^0_{int}(T)$ \nis a vertex of valency $1$ or $2$.  \nWe set $E(\\Gamma)=\\sum_{v \\in C^0_{int}(T)} \\lambda_{(\\eta(v))}$.  \n\nFor each $\\Gamma \\in {G}_{k+1}^+$, we construct \n\\[\n{\\mathfrak m}_{\\Gamma}:B_k({\\mathcal H}[1]^{\\bullet}) \\otimes R[e,e^{-1}] \\to \n{\\mathcal H}[1]^{\\bullet} \\otimes R[e,e^{-1}],\n\\] \nwhich is of degree $1$ and \n\\[\n{\\mathfrak f}_{\\Gamma}:B_k({\\mathcal H}[1]^{\\bullet}) \\otimes R[e,e^{-1}] \\to \n\\overline{C}[1]^{\\bullet} \\otimes R[e,e^{-1}],\n\\]\nwhich is of degree $0$.  \nThen we define \n\\[\n{\\mathfrak m}_k'=\\sum_{\\Gamma \\in {G}_{k+1}^+} \nT^{E(\\Gamma)} {\\mathfrak m}_{\\Gamma}:\nB_k({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov} \\to \n{\\mathcal H}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}\n\\]\nand \n\\[\n{\\mathfrak f}_k=\\sum_{\\Gamma \\in {G}_{k+1}^+}\nT^{E(\\Gamma)} {\\mathfrak f}_{\\Gamma}:\nB_k({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov} \\to \n\\overline{C}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}.\n\\]\nWe will show that \n$({\\mathcal H}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}, \\{{\\mathfrak m}_k'\\})$ \nis a $G$-gapped filtered $A_{\\infty}$-algebra and \n${\\mathfrak f}=\\{{\\mathfrak f}_k\\}$ is a $G$-gapped \n$A_{\\infty}$-homomorphism, which is a weak homotopy equivalence.  \nThen the Whitehead type theorem implies that ${\\mathfrak f}$ is \na homotopy equivalence.    \n\n\\smallskip \\noindent\n{\\bf Step 1.} \\ \\ The case that $\\# C^0_{int}(T)=0$.\n\nSuch a $T$ consists of two exterior vertices and an edge \njoining them.  \nTherefore, there is unique element $\\Gamma_0$, which belongs to \n$G_2^+$.  \nWe define \n\\[\n{\\mathfrak m}_{\\Gamma_0}=\n\\overline{\\mathfrak m}_1|_{{\\mathcal H}[1]^{\\bullet}}\n\\] \nand \n\\[\n{\\mathfrak f}_{\\Gamma_0}:{\\mathcal H}[1]^{\\bullet} \\otimes R[e,e^{-1}] \\to \n\\overline{C}[1]^{\\bullet} \\otimes R[e,e^{-1}]\n\\]\nto be the inclusion $\\iota$.  \n\n\\smallskip \\noindent\n{\\bf Step 2.} \\ \\ The case that $\\# C^0_{int}(T)=1$.\n\nFor any $k=0,1,2, \\dots$, \nthere is a unique planar tree with $\\#C^0_{ext}(T)=k+1$ and \n$\\#C^0_{int}(T)=1$.   \nLet $\\Gamma_{k+1} \\in G_{k+1}^+$ be a decorated planar tree with \none interior vertex $v$, see Figure \\ref{elmdectree}.  \n\n\\begin{figure}\n\\begin{center}\n\\include{elmdectree}\n\\end{center}\n\\caption{}\n\\label{elmdectree}\n\\end{figure}\n\nWe define \n\\[\n{\\mathfrak m}_{\\Gamma_{k+1}}=\\Pi \\circ {\\mathfrak m}_{k,\\eta(v)}:\nB_k({\\mathcal H}[1]^{\\bullet}) \\otimes R[e,e^{-1}] \\to \n{\\mathcal H}[1]^{\\bullet} \\otimes R[e,e^{-1}]\n\\]\nand \n\\[\n{\\mathfrak f}_{\\Gamma_{k+1}}=G \\circ {\\mathfrak m}_{k,\\eta(v)}: \nB_k({\\mathcal H}[1]^{\\bullet}) \\otimes R[e,e^{-1}] \\to \n\\overline{C}[1]^{\\bullet} \\otimes R[e,e^{-1}].\n\\]\nSince the degree of $\\Pi$, resp. $G$, is $0$, resp. $-1$, \n${\\mathfrak m}_{\\Gamma_{k+1}}$, resp. ${\\mathfrak f}_{\\Gamma_{k+1}}$, \nis of degree $1$, resp. $0$.  \n\n\\smallskip \\noindent\n{\\bf Step 3.} \\ \\ General case. \n\nLet $v_1$ is the vertex closest to the root vertex $v_0$.  \nCut the decorated planar tree at $v_1$, then $\\Gamma$ is decomposed \ninto decorated planar trees $\\Gamma^{(1)}, \\dots, \\Gamma^{({\\ell})}$ \nand an interval toward $v_0$ in counter-clockwise order, see \nFigure \\ref{gendectree}.  \n\n\\begin{figure}\n\\begin{center}\n\\include{gendectree}\n\\end{center}\n\\caption{}\n\\label{gendectree}\n\\end{figure}\n\nThen we define \n\\[\n{\\mathfrak m}_{\\Gamma}=\\Pi \\circ {\\mathfrak m}_{\\ell, \\eta(v_1)} \n\\circ ({\\mathfrak f}_{\\Gamma^{(1)}} \\otimes \\cdots \\otimes {\\mathfrak f}_{\\Gamma^{({\\ell})}})\n\\]\nand \n\\[\n{\\mathfrak f}_{\\Gamma}=G \\circ {\\mathfrak m}_{\\ell, \\eta(v_1)} \n\\circ ({\\mathfrak f}_{\\Gamma^{(1)}} \\otimes \\cdots \\otimes {\\mathfrak f}_{\\Gamma^{({\\ell})}}).\n\\]\n\n\\smallskip\n\nFinally we define \n\\[\n{\\mathfrak m}'_k=\\sum_{\\Gamma \\in G_{k+1}^+} \nT^{E(\\Gamma)} {\\mathfrak m}_{\\Gamma}\n\\] \nand \n\\[\n{\\mathfrak f}_k=\\sum_{\\Gamma \\in G_{k+1}^+} \nT^{E(\\Gamma)} {\\mathfrak f}_{\\Gamma}.\n\\] \nAs in \\S 2.1, we obtain a graded coderivation \n\\[\n\\widehat{d}'=\\sum_k \\widehat{\\mathfrak m}'_k:B({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov} \\to B({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov}\n\\] \nand a (formal) coalgebra homomorphism \n\\[\n\\widehat{\\mathfrak f}:B({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov} \\to \nB(C[1]^{\\bullet}).\n\\] \n\nWe will show the following:\n\n\\begin{prop}\\label{mainprop}\n\\[\n\\widehat{\\mathfrak f} \\circ \\widehat{d}' = \\widehat{d} \\circ \n\\widehat{\\mathfrak f},\n\\]\nwhere $\\widehat{d}=\\sum_k \\widehat{\\mathfrak m}_k:B(C[1]^{\\bullet}) \\to B(C[1]^{\\bullet})$. \n\\end{prop}\n\nSince $\\overline{\\mathfrak f}_1={\\mathfrak f}_{\\Gamma_0}$ is the \ninclusion, we find that $\\widehat{\\mathfrak f}$ is injective \nusing the energy filtration and the number filtration on the bar \ncomplex.  \nThen $\\widehat{d}' \\circ \\widehat{d}' = 0$ follows from \n$\\widehat{d} \\circ \\widehat{d} = 0$.  \nHence we obtain the following:\n\n\\begin{cor}\n(1) $({\\mathcal H}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}, \\{{\\mathfrak m}'_k\\})$ \nis a $G$-gapped filtered $A_{\\infty}$-algebra.  \n\n\\noindent\n(2) $\\widehat{\\mathfrak f}$ is a $G$-gapped $A_{\\infty}$-homomorphism \nfrom $({\\mathcal H}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}, \\{{\\mathfrak m}'_k\\})$ \nto $(C[1]^{\\bullet}, \\{{\\mathfrak m}_k\\})$.  \n\\end{cor}\n\nThe rest of this section is devoted to the proof of \nProposition \\ref{mainprop}, \nwhich is equivalent to that \n\\[\n{\\mathfrak f} \\circ \\widehat{d}' = {\\mathfrak m} \\circ \n\\widehat{\\mathfrak f}  \n\\]\nas maps $B({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov} \\to C[1]^{\\bullet}$, \nwhere \n\\[\n{\\mathfrak f}:B({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov} \n\\stackrel{\\widehat{\\mathfrak f}}{\\to} B(C[1]^{\\bullet}) \\stackrel{\\rm pr}{\\to} \nC[1]^{\\bullet},\n\\]\nand \n\\[\n{\\mathfrak m}:B(C[1]^{\\bullet}) \\stackrel{\\widehat{d}}{\\to} B(C[1]^{\\bullet}) \n\\stackrel{\\rm pr}{\\to} C[1]^{\\bullet}.\n\\]\nNamely, ${\\mathfrak f}\\vert_{B_k({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov}} \n= {\\mathfrak f}_k$, \n${\\mathfrak m}\\vert_{B_k({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov}} \n= {\\mathfrak m}_k$.  \n\nWe introduce an order on $\\{(k,i) \\vert k,i =0,1,2, \\dots \\}$ by \n$(k_1,i_1) \\prec (k_2,i_2)$ if \neither $i_1 < i_2$ or $i_1=i_2$ and $k_1 < k_2$.  \nWe show the following claim by the induction on $(k,i)$.  \n\n\\begin{claim}\n\\[\n{\\mathfrak f} \\circ \\widehat{d}' \\equiv {\\mathfrak m} \\circ \n\\widehat{\\mathfrak f}  \\mod T^{\\lambda_{(i+1)}}\\cdot C[1]^{\\bullet} \\text{ \\rm on } \nB_k({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov}.\n\\]\n\\end{claim}\n\nThe key ingredients in the proof are the following relations \npresented in Figures \\ref{green}, \\ref{ainfty0}. \n\n\\begin{figure}\n\\begin{center}\n\\include{green}\n\\end{center}\n\\caption{}\n\\label{green}\n\n\\begin{center}\n\\include{ainfty0}\n\\end{center}\n\\caption{}\n\\label{ainfty0}\n\\end{figure}\n\n\nFirstly, we consider the case that $i=0$.  \nClaim $(0,0)$ follows from the gapped condition.  \nBy the choice of ${\\mathcal H}$, Claim (1,0) holds clearly.  \nSuppose that Claim $(\\ell,0)$ holds for $\\ell < k$.  \nNote that \n\\[\n\\overline{\\mathfrak f}_k={\\mathfrak f}_{k,0}^{\\circ}  \n= \\bigl(\\sum_{1<\\ell\\leq k} G \\circ \\overline{\\mathfrak m}_{\\ell} \\circ \n\\widehat{\\overline{\\mathfrak f}} + \n\\delta_{k1} {\\mathfrak f}_{\\Gamma_0}\\bigr)|_{B_k({\\mathcal H}[1]^{\\bullet})}\n\\] \nand \n\\[\n\\overline{\\mathfrak m}'_k={\\mathfrak m}_{k,0}^{\\circ}\n= \\bigl(\\sum_{1<\\ell\\leq k} \\Pi \\circ \\overline{\\mathfrak m}_{\\ell} \n\\circ \\widehat{\\overline{\\mathfrak f}} + \n\\delta_{k1} {\\mathfrak m}_{\\Gamma_0}\\bigr)|_{B_k({\\mathcal H}[1]^{\\bullet})}.\n\\]\nHere $\\delta_{ij}$ is Kronecker's delta.  \nRecall that ${\\mathfrak m}_{\\Gamma_0}={\\mathfrak m}_{1,0}|_{\\mathcal H}$ and \n${\\mathfrak f}_{\\Gamma_0}$ is the inclusion.  \nNote also that the restriction of $\\widehat{\\overline{\\mathfrak f}}$ \nto $B_k({\\mathcal H}[1]^{\\bullet})$ in the right hand sides is determined by \n$\\overline{\\mathfrak f}_1, \\dots, \\overline{\\mathfrak f}_{k-1}$.  \n\nThus we have \n\n\\begin{align}\n\\overline{\\mathfrak m} \\circ \\widehat{\\overline{\\mathfrak f}}\n|_{B_k({\\mathcal H}[1]^{\\bullet})} \n& = \n\\bigl(\\overline{\\mathfrak m}_1 \\circ \\widehat{\\overline{\\mathfrak f}} + \n\\sum_{1< \\ell \\leq k} \\overline{\\mathfrak m}_{\\ell} \\circ \n\\widehat{\\overline{\\mathfrak f}}\\bigr)|_{B_k({\\mathcal H}[1]^{\\bullet})}  \\nonumber \\\\\n& =  \n\\bigl(\\sum_{1< \\ell \\leq k} \\overline{\\mathfrak m}_1 \\circ G \\circ \n\\overline{\\mathfrak m}_{\\ell} \\circ \\widehat{\\overline{\\mathfrak f}} + \n\\delta_{k1} \\overline{\\mathfrak m}_1 \\circ {\\mathfrak f}_{\\Gamma_0} \n+ \\sum_{1< \\ell \\leq k} \\overline{\\mathfrak m}_{\\ell} \n\\circ \\widehat{\\overline{\\mathfrak f}}\\bigr)|_{B_k({\\mathcal H}[1]^{\\bullet})}.  \n\\nonumber \\\\\n& = \n\\bigl(\\sum_{1< \\ell \\leq k} \\Pi \\circ \\overline{\\mathfrak m}_{\\ell} \n\\circ \\widehat{\\overline{\\mathfrak f}} - \n\\sum_{1< \\ell \\leq k} G \\circ \\overline{\\mathfrak m}_1 \\circ \n\\overline{\\mathfrak m}_{\\ell} \\circ \\widehat{\\overline{\\mathfrak f}}\n+\\delta_{k1} \\overline{\\mathfrak m}_1 \\circ {\\mathfrak f}_{\\Gamma_0} \n\\bigr)\n|_{B_k({\\mathcal H}[1]^{\\bullet})}\n\\nonumber \\\\\n& =  \n\\bigl({\\mathfrak f}_{\\Gamma_0} \\circ \\overline{\\mathfrak m}'_k - \n\\sum_{1< \\ell \\leq k} G \\circ \\overline{\\mathfrak m}_1 \\circ \n\\overline{\\mathfrak m}_{\\ell} \\circ \\widehat{\\overline{\\mathfrak f}}\n\\bigr)\n|_{B_k({\\mathcal H}[1]^{\\bullet})} \n\\nonumber \\\\\n& = \n\\bigl({\\mathfrak f}_{\\Gamma_0} \\circ \\overline{\\mathfrak m}'_k + \n\\sum_{1< \\ell' \\leq k} G \\circ \\overline{\\mathfrak m}_{\\ell'} \\circ \n\\widehat{\\overline{d}} \\circ \\widehat{\\overline{\\mathfrak f}}\\bigr)\n|_{B_k({\\mathcal H}[1]^{\\bullet})}. \n\\nonumber\n\\end{align}\n\nHere we used the fact that \n$\\overline{\\mathfrak m}_1 \\circ G + G \\circ \n\\overline{\\mathfrak m}_1 = \\Pi - id$ and the $A_{\\infty}$-relation \n$\\widehat{\\overline{d}} \\circ \\widehat{\\overline{d}} = 0$.   \nSince we assumed Claim $(\\ell,0)$ for $\\ell < k$, i.e., \n\\[\n\\overline{\\mathfrak m} \\circ \\widehat{\\overline{\\mathfrak f}} = \n\\overline{\\mathfrak f} \\circ \\widehat{\\overline{d}'} \n\\text{ on } B_{\\ell}({\\mathcal H}[1]^{\\bullet}),\n\\]  \nwe have \n\\[\n\\widehat{\\overline{d}} \\circ \\widehat{\\overline{\\mathfrak f}} \\equiv \n\\widehat{\\overline{\\mathfrak f}} \\circ \\widehat{\\overline{d}'} \n\\mod \\overline{C}[1]^{\\bullet}=B_1(\\overline{C}[1]^{\\bullet})\n\\text{ on } B_k({\\mathcal H}[1]^{\\bullet}).\n\\]\nTherefore we find that \n\\[\n(\\sum_{1< \\ell' \\leq k} G \\circ \\overline{\\mathfrak m}_{\\ell'} \\circ \n\\widehat{\\overline{d}} \\circ \\widehat{\\overline{\\mathfrak f}})\n|_{B_k({\\mathcal H}[1]^{\\bullet})}\n= \n(\\sum_{1< \\ell' \\leq k} G \\circ \\overline{\\mathfrak m}_{\\ell'} \\circ \n\\widehat{\\overline{\\mathfrak f}} \\circ \\widehat{\\overline{d}'})\n|_{B_k({\\mathcal H}[1]^{\\bullet})}.  \n\\]\nHence we showed Claim $(k,0)$, i.e.,   \n\\[\n\\overline{\\mathfrak m} \\circ \\widehat{\\overline{\\mathfrak f}} \n= {\\overline{\\mathfrak f}} \\circ \\widehat{\\overline{d}'}\n\\] \non $B_k({\\mathcal H}[1]^{\\bullet})$.  \n\nNext we assume that Claim $(k,i)$ holds for all $k=0,1,2, \\dots$.  \nWe prove Claim $(k,i+1)$ by the induction on $k$.  \nNote that Case 3-1 below does not occur in the case that $k=0$.  \n\nFirst of all, we recall from the definition of \n$G_{k+1}^+$ that \n\\[\n{\\mathfrak f}_k=\\sum_{\\Gamma \\in G_{k+1}^+} T^{E(\\Gamma)} \n{\\mathfrak f}_{\\Gamma} \n= \\sum_{(\\ell, j) \\neq (1,0)} G \\circ {\\mathfrak m}_{\\ell,j}^{\\circ} \n\\circ \\widehat{\\mathfrak f} |_{B_k({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov}} + \\delta_{k1} \n{\\mathfrak f}_{\\Gamma_0}.\n\\]\nThen we have \n\n\\begin{align}\n{\\mathfrak m} \\circ \\widehat{\\mathfrak f}\n|_{B_k({\\mathcal H}[1]^{\\bullet})} \n= &\n\\bigl({\\mathfrak m}_{1,0} \\circ \\widehat{\\mathfrak f} + \n\\sum_{(\\ell, j) \\neq (1,0)} {\\mathfrak m}_{\\ell,j}^{\\circ} \\circ \n\\widehat{\\mathfrak f}\\bigr)|_{B_k({\\mathcal H}[1]^{\\bullet})}  \\nonumber \\\\\n= & \n\\bigl(\\sum_{(\\ell,j) \\neq (1,0)} {\\mathfrak m}_{1,0} \\circ G \\circ \n{\\mathfrak m}_{\\ell,j}^{\\circ} \\circ \\widehat{\\mathfrak f} + \n\\delta_{k1} {\\mathfrak m}_{1,0} \\circ {\\mathfrak f}_{\\Gamma_0} \n\\nonumber \\\\\n& \n+ \\sum_{(\\ell,j) \\neq (1,0)} {\\mathfrak m}_{\\ell,j}^{\\circ} \n\\circ \\widehat{\\mathfrak f}\\bigr)|_{B_k({\\mathcal H}[1]^{\\bullet})}.  \n\\nonumber \\\\\n= & \n\\bigl(\\sum_{(\\ell,j) \\neq (1,0)} \\Pi \\circ \n{\\mathfrak m}_{\\ell,j}^{\\circ} \\circ \\widehat{\\mathfrak f} - \n\\sum_{(\\ell,j) \\neq (1,0)} G \\circ {\\mathfrak m}_{1,0} \\circ \n{\\mathfrak m}_{\\ell,j}^{\\circ} \\circ \\widehat{\\mathfrak f} \n\\nonumber \\\\\n& \n+\\delta_{k1} {\\mathfrak m}_{1,0} \\circ {\\mathfrak f}_{\\Gamma_0} \n\\bigr)|_{B_k({\\mathcal H}[1]^{\\bullet})}\n\\nonumber \\\\\n= & \n\\bigl({\\mathfrak f}_{\\Gamma_0} \\circ {\\mathfrak m}'_k - \n\\sum_{(\\ell,j) \\neq (1,0)} G \\circ {\\mathfrak m}_{1,0} \\circ \n{\\mathfrak m}_{\\ell,j}^{\\circ} \\circ \\widehat{\\mathfrak f}\n\\bigr)\n|_{B_k({\\mathcal H}[1]^{\\bullet})} \n\\nonumber \\\\\n= & \n\\bigl({\\mathfrak f}_{\\Gamma_0} \\circ {\\mathfrak m}'_k + \n\\sum_{(\\ell',j') \\neq (1,0)} G \\circ \n{\\mathfrak m}_{\\ell',j'}^{\\circ} \n\\circ \\widehat{d} \\circ \\widehat{\\mathfrak f}\\bigr)\n|_{B_k({\\mathcal H}[1]^{\\bullet})}. \n\\nonumber\n\\end{align}\nIn the third equality, we used the fact that \n${\\mathfrak m}_{1,0} \\circ G + G \\circ \n{\\mathfrak m}_{1,0} = \\Pi - id$.   \n\nWe will show that \n\\[\n\\sum_{(\\ell',j') \\neq (1,0)} G \\circ {\\mathfrak m}_{\\ell',j'}^{\\circ} \n\\circ \\widehat{d} \\circ \\widehat{\\mathfrak f} \\equiv \n\\sum_{(\\ell',j') \\neq (1,0)} G \\circ {\\mathfrak m}_{\\ell',j'}^{\\circ} \n\\circ \\widehat{\\mathfrak f} \\circ \\widehat{d'} \\mod \nT^{\\lambda_{(i+2)}},\n\\]\nwhich implies that \n\\[\n{\\mathfrak m} \\circ \\widehat{\\mathfrak f} \\equiv \n\\widehat{\\mathfrak f} \\circ \\widehat{d'} \\mod T^{\\lambda_{(i+2)}}.\n\\]\n\n\\smallskip \\noindent\n{\\bf Case 1: $\\ell'=0$.} \\ \\ \nNote that the $B_0(C[1]^{\\bullet})=\\Lambda_{0,nov}$-components of \n${\\mathrm{Im}} \\ \\widehat{d} \\circ \\widehat{\\mathfrak f}$ and \n${\\mathrm{Im}} \\ \\widehat{\\mathfrak f} \\circ \\widehat{d}'$ are zero.  \nHence we have\n\\[\n{\\mathfrak m}_{0,j'}^{\\circ} \n\\circ \\widehat{d} \\circ \\widehat{\\mathfrak f} =\n{\\mathfrak m}_{0,j'}^{\\circ} \n\\circ \\widehat{\\mathfrak f} \\circ \\widehat{d'} = 0.  \n\\]\n\n\\smallskip \\noindent\n{\\bf Case 2: $\\ell'=1$.} \\ \\ \nFor $j' \\neq 0$, ${\\mathfrak m}_{1,j'}^{\\circ} \\equiv 0 \n\\mod T^{\\lambda_{(1)}}$.  \nBy the induction hypothesis, we have \n\\[\n\\widehat{\\mathfrak f} \\circ \\widehat{d'} \\equiv \n\\widehat{d} \\circ \\widehat{\\mathfrak f} \\mod T^{\\lambda_{i+1}}.\n\\]\nSince $\\lambda_{(i+2)} \\leq \\lambda_{(i+1)} + \\lambda_{(1)}$, \nwe obtain \n\\[\n{\\mathfrak m}_{1,j'}^{\\circ} \\circ \\widehat{\\mathfrak f} \\circ \n\\widehat{d'} \\equiv \n{\\mathfrak m}_{1,j'}^{\\circ} \\circ \\widehat{d} \\circ \n\\widehat{\\mathfrak f} \\mod T^{\\lambda_{(i+2)}}.\n\\] \n\n\\smallskip \\noindent\n{\\bf Case 3: $\\ell' \\geq 2$.} \\ \\ \nLet ${\\mathbf x} \\in B_k({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov}$.  \nWrite \n\\[\n\\Delta^{\\ell'-1} {\\mathbf x} = \\sum_a {\\mathbf x}_{1,a} \\otimes \n\\cdots \\otimes {\\mathbf x}_{\\ell',a},\n\\]\nwhere $\\Delta$ is the coproduct and ${\\mathbf x}_{i,a} \\in \nB_{k_{i,a}}({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov}$.  \nThen we have \n\n\\begin{align}\n\\widehat{\\mathfrak f} \\circ \\widehat{d'} ({\\mathbf x}) \n= & \\sum_a \\sum_j (-1)^{\\deg' {\\mathbf x}_{1,a}+\\cdots + \n\\deg'{\\mathbf x}_{j-1,a}}\n{\\mathfrak f}_{k_{1,a}}({\\mathbf x}_{1,a}) \\otimes \\cdots \\otimes \n{\\mathfrak f}_{k_{j,a}}(\\widehat{d'}({\\mathbf x}_{j,a})) \\otimes \n\\cdots \\nonumber \\\\\n& \\hspace{0.3in} \\cdots \\otimes {\\mathfrak f}_{k_{\\ell',a}}({\\mathbf x}_{\\ell',a}). \n\\nonumber\n\\end{align}\n\n\\smallskip \\noindent\n{\\bf Case 3-1: $k_{j,a} < k$.} \nIn this case, we have \n\\[\n{\\mathfrak f}_{k_{j,a}}(\\widehat{d'}({\\mathbf x}_{j,a}) \\equiv \n{\\mathfrak m} \\circ \\widehat{\\mathfrak f}({\\mathbf x}_{j,a}) \n\\mod T^{\\lambda_{(i+2)}}\n\\]\nby the induction hypothesis.  \nHence \n\n\\begin{align}\n& \n{\\mathfrak f}_{k_{1,a}}({\\mathbf x}_{1,a}) \\otimes \\cdots \\otimes \n{\\mathfrak f}_{k_{j,a}}(\\widehat{d'}({\\mathbf x}_{j,a})) \\otimes \n\\cdots \\otimes {\\mathfrak f}_{k_{\\ell',a}}({\\mathbf x}_{\\ell',a}) \n\\nonumber \\\\ \n\\equiv & \n{\\mathfrak f}_{k_{1,a}}({\\mathbf x}_{1,a}) \\otimes \\cdots \\otimes \n{\\mathfrak m} \\circ \\widehat{\\mathfrak f}({\\mathbf x}_{j,a}) \\otimes \n\\cdots \\otimes {\\mathfrak f}_{k_{\\ell',a}}({\\mathbf x}_{\\ell',a}) \n \\ \n\\mod T^{\\lambda_{(i+2)}}. \\nonumber \n\\end{align}\n\n\\smallskip \\noindent\n{\\bf Case 3-2: $k_{j,a} = k$.} \nIn this case, $k_{j',a} = 0$ for $j' \\neq j$, i.e., \n${\\mathbf x}_{j',a} \\in B_0({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov}$.  \nWithout loss of generality, we may assume that ${\\mathbf x}_{j',a}=1$ \nfor $j' \\neq j$.  \n\nBy the induction hypothesis, we have \n\\[\n{\\mathfrak f}(\\widehat{d'}({\\mathbf x}_{j,a})) \\equiv \n{\\mathfrak m}(\\widehat{\\mathfrak f}({\\mathbf x}_{j,a})) \n\\mod T^{\\lambda_{(i+1)}},\n\\]\nwhich implies that \n\\[\n{\\mathfrak f}_0(1) \\otimes \\cdot \\otimes {\\mathfrak f}\n(\\widehat{d'}({\\mathbf x}_{j,a})) \\otimes \\cdots \\otimes \n{\\mathfrak f}_1(1)  \\equiv  \n{\\mathfrak f}_0(1) \\otimes \\cdot \\otimes {\\mathfrak m} \n(\\widehat{\\mathfrak f}({\\mathbf x}_{j,a})) \\otimes \\cdots \\otimes \n{\\mathfrak f}_1(1) \n\\mod T^{\\lambda_{(i+2)}}.\n\\]\nHere we used ${\\mathfrak f}_0(1) \\equiv 0 \\mod T^{\\lambda_{(1)}}$. \n\nIn sum, we obtain Claim $(k,i+1)$ for all $k$.  \n\nBy the construction, $\\overline{\\mathfrak f}_1$ is a chain homotopy \nequivalence ($\\Pi$ is a homotopy inverse).  \nTherefore, Theorem \\ref{whitehead} implies that \n$\\{{\\mathfrak f}_k\\}$ is a homotopy equivalence of \nfiltered $A_{\\infty}$-algebras.  \n\n\\section{Filtered $A_{\\infty}$-algebra associated to Lagrangian \nsubmanifolds}\n\nLet $(M,\\omega)$ be a closed symplectic manifold and \n$L$ a Lagrangian submanifold.  \nWe only consider the case that $L$ is an embedded compact \nLagrangian submanifold without boundary equipped with a \nrelative spin structure, see \\S 44 in \\cite{FOOO}.   \nWe constructed a filtered $A_{\\infty}$-algebra \nassociated to $L$ in $(M,\\omega)$.  \nAs we explained in section 2, the framework of (filtered) \n$A_{\\infty}$-algebras, bimodules, etc. is adequate to formulate \nthe condition under which Floer complex is obtained.  \n\nIn this section, we briefly recall the way of constructing \nfiltered $A_{\\infty}$-algebra associated to $L$.  \nAlthough the readers may find Proposition 4.1 below too technical, \nwe present it precisely so that we can explain how to modify it \nfor the purpose of section 5.  \n\nA naive idea of the construction is to use the moduli space \nof pseudo-holomorphic discs to {\\it deform} \nthe intersection products of chains in $L$ \nin a similar way to the quantum cohomology, where the intersection \nproduct on (co)homology is deformed by the moduli space \nof pseudo-holomorphic spheres, more precisely, stable maps of genus 0.  \nHere appears a difference: while the moduli spaces of stable maps of \ngenus 0 are (virtual) cycles, the moduli spaces of stable \nbordered stable maps are, in general, not (virtual) cycles, \nbut with codimension 1 boundary (in the sense of Kuranishi structure).  \nTherefore, we cannot restrict ourselves to cycles and forced to \nwork with chains.  \nHowever, the intersection product is not defined \nin chain level, e.g., the self intersection of chains.  \nWe start with a subcomplex of the singular \nchain complex such that the inclusion induces an isomorphism \non homology.   \nThen take {\\it perturbed} intersection product \nof generators of the subcomplex and add them to get a larger \nsubcomplex such that the inclusion induces an isomorphism on homology.  \nOnce we get such nested subcomplexes, we apply the argument \nin the proof of Theorem 3.3 to define the operation\n$\\overline{\\mathfrak m}_2$ on a fixed subcomplex.\nThis multiplicative structure is not associative, but associative \nup to homotopy.  \nSo we proceed to constructed other operations \n$\\overline{\\mathfrak m}_k$ in a similar way, see \nCorollary 30.89 in section 30.6, \\cite{FOOO} for a detailed argument.  \nIn this way, we obtain an $A_{\\infty}$-algebra.  \n\nFor the construction of the filtered $A_{\\infty}$-algebra, \nwe include the effect from the moduli space of bordered stable \nmaps.  \nWe need to take perturbations of the moduli spaces to define \nthe operations not only perturbation in the intersection product \nmentioned above.  \nOur strategy is to construct an $A_{n,K}$-algebra on \n$C_{(g)}(L)$, which is generated by $\\chi_{(g)}$ in \nProposition \\ref{transv}, for a sufficiently large $g$.  \nThen we use the obstruction theory to extend a filtered \n$A_{n,K}$-structure to a filtered $A_{n',K'}$-structure \n($(n,K) \\precsim (n',K')$).  \nThe resulting filtered $A_{\\infty}$-structure is unique up to \nhomotopy, see \\S 30 in Chapter 7, \\cite{FOOO}.  \n\nLet $\\mu_L \\in {\\rm H}^2(M,L;\\Z)$ be the Maslov class of the \nLagrangian submanifold $L$.  \nWe introduce an equivalence relation $\\sim$ on ${\\rm H}_2(M,L;\\Z)$ \nby $\\beta_1 \\sim \\beta_2$ if and only if \n$\\omega(\\beta_1) = \\omega(\\beta_2)$ and \n$\\mu_L(\\beta_1)=\\mu_L(\\beta_2)$.  \n\nPick an almost complex structure $J$ compatible with $\\omega$.  \nDenote by ${\\mathcal M}(\\beta;L,J)$ the moduli space \nof bordered stable maps $u:(\\Sigma, \\partial \\Sigma) \\to (M,L)$ \nof genus $0$ representing $\\beta$ and \nby ${\\mathcal M}_{k+1}(\\beta;L,J)$ be the moduli space of \nbordered stable maps in the class $\\beta$ of genus $0$ \nwith $k+1$ marked points \n$z_0,z_1, \\dots, z_k$ on the regular part of $\\partial \\Sigma$.  \nDenote by ${\\mathcal M}_{k+1}^{\\rm main}(\\beta;L,J)$ the component, \non which the marked points $z_0, z_1, \\dots, z_k$ respect \nthe {\\it counter-clockwise} cyclic order on the boundary of bordered \nsemi-stable curve of genus 0 with connected boundary.  \nLet ${\\mathfrak G}(L)$ be \nthe monoid contained in $\\Pi(M,L)$ generated by \n$\\beta$ with ${\\mathcal M}(\\beta;L,J) \\neq \\emptyset$.  \nWe write $\\beta_0=0 \\in {\\mathfrak G}(L)$.  \n\nOur basic idea is as follows.  \nFor singular simplices $P_1, \\dots, P_k$ in $L$, \nwe consider the fiber product in the sense of Kuranishi structure \n\\[\n{\\mathcal M}_{k+1}^{\\rm main}(\\beta;P_1, \\dots, P_k) ={\\mathcal M}_{k+1}\n^{\\rm main}(\\beta;L,J)_{\\mathbf{ev}} \\times_{L^k} (P_1 \\times \\dots \\times P_k), \n\\]\nwhere ${\\mathbf{ev}}=(ev_1, \\dots , ev_k)$ is the evaluation map at \n$z_1, \\dots , z_k$.  \n(For the orientation issue, see Chapter 9 \\cite{FOOO}.)  \nThen we would like to define \n\\[\n{\\mathfrak m}_{k,\\beta}(P_1, \\dots, P_k) = (ev_0: \n{\\mathcal M}_{k+1}^{\\rm main}(\\beta;P_1, \\dots, P_k) \\to L),\n\\]\nwhere $ev_0$ is the evaluation at $z_0$.  \n\nNote that ${\\mathcal M}_{k+1}^{\\rm main}(\\beta)$ is not necessarily \na manifold or an orbifold and that \n$ev_i$ are not necessarily submersions even if \n${\\mathcal M}_{k+1}^{\\rm main}(\\beta)$ is such a nice space.  \nIn order to deal with this issue, we introduced the notion of \nKuranishi structure \\cite {FO}, see also Appendix in \\cite{FOOO}.  \nHere is a digression on Kuranishi structure.  \n\nLet $X$ be a compact Hausdorff space.  \nA Kuranishi structure on $X$ consists of \na covering of $X$ by Kuranishi neighborhoods of the same \n{\\it virtual} dimension and coordinate changes among them.  \nA Kuranishi neighborhood around $p \\in X$ is a quintet  \n $(V_p,E_p,\\Gamma_p, s_p, \\psi_p)$, where \n\\begin{itemize}\n\\item\n$V_p$ is a smooth manifold of finite dimension,\n\\item\n$E_p$ is a real vector bundle over $V_p$ of finite rank, \n\\item\n$\\Gamma_p$ is a finite group acting smoothly and effectively \non $V_p$ and $E_p$ such that $E_p \\to V_p$ is a $\\Gamma_p$-equivariant \nvector bundle, \n\\item\n$s_p$ is a $\\Gamma_p$-equivariant section of $E_p \\to V_p$, \n\\item \n$\\psi_p$ is a homeomorphism from $s_p^{-1}(0)\/\\Gamma_p$ to \na neighborhood of $p$ in $X$.\n\\end{itemize}\nThe vector bundle $E_p \\to V_p$ is called the {\\it obstruction} bundle \nand the section $s_p$ the Kuranishi map.  \nWe have coordinate changes among Kuranishi neighborhoods, see \n\\cite{FO}, \\cite{FOOO}.  \nWe require that $\\dim V_p - {\\rm rank}~ E_p$ does not depend on \n$p \\in X$ and call it the {\\it virtual} dimension of the space $X$ \nequipped with Kuranishi structure.  \n\nThe moduli spaces of stable maps, bordered stable maps carry \nKuranishi structures, hence we can locally describe the moduli space \nas $s_p^{-1}(0)\/\\Gamma_p$ in the definition of Kuranishi neighborhoods. \nIf $s_p$ is transversal to the zero section, the moduli space is \nlocally an orbifold.  \nIn general, we cannot perturb $s_p$ to \na $\\Gamma_p$-equivariant section $s'_p$, \nwhich is transversal to the zero section.  \nInstead of single valued sections,  \nwe consider perturbations by \n$\\Gamma_p$-equivariant {\\it multi-valued} sections, \neach branch of which is transversal to the zero section.  \nThen we arrange them compatible under the coordinate change.  \nIn this way, we obtain {\\it perturbed} moduli spaces.  \n\nWe take a multi-valued perturbation $\\mathfrak s$ of \nKuranishi maps for \n${\\mathcal M}_{k+1}^{\\rm main}(\\beta;P_1, \\dots, P_k)$ such that \neach branch of $\\mathfrak s$ is transversal to the zero section.  \nAfter taking a triangulation of \nthe perturbed zero locus \n${\\mathcal M}_{k+1}^{\\rm main}(\\beta;P_1, \\dots, P_k)^{\\mathfrak s}$ \nof $\\mathfrak s$, we obtain a {\\it virtual} chain \n$$\nev_0: {\\mathcal M}_{k+1}^{\\rm main}(\\beta;P_1, \\dots, P_k)^{\\mathfrak s}\n\\to L.\n$$  \n\nTo make this argument rigorous, we build a sequence of \nsubcomplexes of the singular chain complex of $L$ and \na series of operations ${\\mathfrak m}_{k,\\beta}^{geo}$.    \nFor details, see Chapter 7 in \\cite{FOOO}.  \nHere we briefly recall a part of it, in particular, \nthe construction of a series of subcomplexes of singular \nchain complex of $L$.  \nIn section 5, we explain how to arrange this construction \nin relation with the Morse theory.  \n\nIn \\S 30 in \\cite{FOOO}, we constructed countable sets $\\chi_g(L)$ \nof singular $C^{\\infty}$-simplices on $L$.  \nFor a simplex $P \\in \\chi_g(L)$, we call $g$ the generation of $P$.   \nWrite \n\\[\n\\chi_{(g)}=\\bigcup_{g' \\leq g} \\chi_{g'}(L)\n\\] \nand denote by $C_{(g)}(L;R)$ the $R$-vector space generated by \n$\\chi_{(g)}(L)$.  \nLet $S(L;R)$ be the singular $C^{\\infty}$-chain complex of $L$ \nwith coefficients in $R$.  \n\n\n\\smallskip\n\\noindent\n{\\bf Condition 1.} \\ \\  Any face of $P \\in \\chi_g(L)$ belongs to \n$\\chi_{(g)}(L)$.  \n\n\\smallskip\n\\noindent\n{\\bf Condition 2.} \\ \\ The inclusion $C_{(g)}(L) \\to S(L;R)$ \ninduces an isomorphism on homology.  \n\n\\smallskip\n\nFor $\\beta \\in {\\mathfrak G}(L)$, we define \n\n\\[\n\\parallel \\beta \\parallel = \n\\left\\{ \n\\begin{array}{ll}\n\\sup\\{n|\\exists \\beta_1,\\dots,\\beta_n \\in {\\mathfrak G}(L) \\setminus \n\\{\\beta_0\\}, \\sum_{i=1}^n \\beta_i = \\beta \\} \n+[\\omega(\\beta)] -1 & \n\\text{ if } \\beta \\neq \\beta_0 \\\\\n-1 & \\text{ if } \\beta = \\beta_0\n\\end{array}\n\\right.\n\\]\nHere $[\\omega(\\beta)]$ is the largest integer not greater than \n$\\omega(\\beta)$.  \n\nBy Gromov's compactness, \nthe number of $\\beta \\in {\\mathfrak G}(L)$ with \n$\\parallel \\beta \\parallel \\leq C$ is finite for any $C$.  \n\nNext we introduce an additional data \n${\\mathfrak d}:\\{1, \\dots ,k\\} \\to \\Z_{\\geq 0}$, which is called \na decoration.  \nFor a pair $({\\mathfrak d},\\beta)$ such that \n${\\mathcal M}_{k+1}^{\\rm main}(\\beta) \\neq \\emptyset$, we define \n\n\\[\n\\parallel ({\\mathfrak d},\\beta) \\parallel = \n\\left\\{ \n\\begin{array}{ll} \n\\max_{i \\in \\{1, \\dots, k\\}} {\\mathfrak d}(i) + \n\\parallel \\beta \\parallel + k & \\text{ if } k \\neq 0 \\\\\n\\parallel \\beta \\parallel & \\text{ if } k = 0.  \n\\end{array}\n\\right.\n\\]\n\nWe will take the fiber product of \n${\\mathcal M}_{k+1}^{\\rm main}(\\beta)$ and singular simplices \n$P_i$ in $L$.  \nThe decoration $\\mathfrak d$ is introduced in order to \ninclude the generations of singular simplices $P_i$ into the data.  \nWhen we emphasize that the decoration $\\mathfrak d$ \nis equipped with the moduli space \n${\\mathcal M}_{k+1}^{\\rm main}(\\beta)$, we denote it by \n${\\mathcal M}_{k+1}^{{\\rm main}, {\\mathfrak d}}(\\beta)$.  \n\n\\begin{prop}[Proposition 30.35 in \\cite{FOOO}]\\label{transv}\nFor any $\\delta > 0$ and ${\\mathcal K} > 0$, there exist \n$\\chi_{(g)}(L)$, $g=0,\\dots,{\\mathcal K}$, and multisections \n${\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}$ for \n$\\parallel ({\\mathfrak d},\\beta) \\parallel \\leq {\\mathcal K}$ \nwith the following properties: \n\n\\begin{itemize}\n\\item $\\chi_{(g)}(L)$ satisfies Conditions 1 and 2 above.  \n\\item Let $P_i \\in \\chi_{{\\mathfrak d}(i)}(L), i=1,\\dots,k$.  \nWe put \n\\[\n{\\mathcal M}_{k+1}^{{\\rm main},{\\mathfrak d}}(\\beta;P_1,\\dots,P_k)\n= \n{\\mathcal M}_{k+1}^{{\\rm main},{\\mathfrak d}}(\\beta)\\times_{L^k} \n\\prod P_i\n\\]\nand define a multisection \n${\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}$ thereof.   \n${\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}$ is transversal to \nthe zero section.  \n\\item If $g=\\parallel ({\\mathfrak d}, \\beta) \\parallel$, then \n\\[\nev_{0*}\\bigl({\\mathcal M}_{k+1}^{{\\rm main},{\\mathfrak d}}\n(\\beta;P_1,\\dots,P_k)^{{\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}} \n\\bigr)\n\\]\nis decomposed into elements of $\\chi_{(g)}(L)$.  \nHere and henceforth we denote \n\\[\n{\\mathcal M}_{k+1}^{{\\rm main},{\\mathfrak d}}\n(\\beta;P_1,\\dots,P_k)^{{\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}}\n:= {\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}^{-1}(0).\n\\]\n\\item The multisections ${\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}$ \nsatisfy certain compatibility conditions.\n\\item The zero locus ${\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}^{-1}\n(0)$ is in a $\\delta$-neighborhood of the zero locus of the original \nKuranishi map.  \n\\end{itemize}\n\\end{prop}\n\nFor the compatibility conditions in the above statement, see \nConditions 30.38 and 30.44 in \\cite{FOOO}.  \n\nNow we explain the way of constructing the filtered \n$A_{\\infty}$-algebra associated to $L$.  \n\nWe put \n\\[\n{\\mathfrak m}_{k,\\beta}^{geo}(P_1,\\dots,P_k)\n=(ev_0:{\\mathcal M}_{k+1}^{{\\rm main},{\\mathfrak d}}(\\beta;P_1,\\dots,\nP_k)^{{\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}} \\to L),\n\\]\nwhen $P_i \\in \\chi_{({\\mathfrak d}(i))}$, $i=1,\\dots, k$.  \nThen ${\\mathfrak m}_{k,\\beta}^{geo}(P_1,\\dots,P_k)$ is decomposed \ninto elements of $\\chi_{(g)}$, \nwhere $g = \\parallel ({\\mathfrak d},\\beta) \\parallel$.  \nUsing the idea in section 3, we showed the following:  \n\n\\begin{prop}[Proposition 30.78 in \\cite{FOOO}]\\label{geo}\nFor any $g_0, n, K$, there exists $g_1 > g_0$ and a filtered \n$A_{n,K}$-structure ${\\mathfrak m}_{k,\\beta}$ on $C_{(g_1)}(L) \n\\otimes \\Lambda_{0,nov}$ such that \n\\[\n{\\mathfrak m}_{k,\\beta}(P_1,\\dots,P_k)\n={\\mathfrak m}_{k,\\beta}^{geo}(P_1,\\dots,P_k),\n\\]\nif $P_i \\in \\chi_{(g_0)}(L)$.  \n\\end{prop}\n\nCombining Theorem \\ref{ext(n,K)} and Proposition \\ref{geo}, \nwe can construct a filtered $A_{\\infty}$-algebra associated to \n$L$, for details see \\cite{FOOO}.  Hence we obtain \n\n\\begin{thm}[Theorem 10.11 in \\cite{FOOO}]\nLet $L$ be a relatively spin Lagrangian submanifold.  \nThen there exist a countably generated subcomplex $C(L)$ of \nthe singular chain complex and a filtered $A_{\\infty}$-algebra \nstructure on $C(L)\\otimes \\Lambda_{0,nov}$.  \n\\end{thm}\n\nWe also proved that the homotopy type of \nthe filtered $A_{\\infty}$-algebra is unique.  \n\nApplying the construction of canonical models in section 3, \nwe obtain a filtered $A_{\\infty}$-algebra structure on \n${\\rm H}(L) \\otimes \\Lambda_{0,nov}$.  \n\nLet $(L_0,L_1)$ be a relative spin pair of Lagrangian submanifolds.  \nAssume that $L_0$ and $L_1$ intersect transversely.  \nThen we have the following:\n\n\\begin{thm}\nLet $D^{\\bullet}$ be a free $\\Lambda_{0,nov}$-module \ngenerated by $L_0 \\cap L_1$.  \nThen there exists a filtered $A_{\\infty}$-bimodule structure \nover filtered $A_{\\infty}$-algebras associated to $L_i$, $i=0,1$.  \n\\end{thm}\n\n\\section{Canonical models and Morse complexes}\n\nIn this section, we apply Theorem \\ref{reduction} \nand reduce the filtered $A_{\\infty}$-structure \non $C^{\\bullet}(L) \\otimes \\Lambda_{0,nov}$ to the Morse complex \n$CM^{\\bullet}(f) \\otimes \\Lambda_{0,nov}$.   \n\n\n\nWe pick a specific Morse function \nas follows.  \nChoose and fix a triangulation ${\\mathfrak T}$ of $L$.  \nWe may assume that the triangulation is sufficiently fine \nby taking subdivision.  \nPick a Morse function $f:L \\to \\R$ with the following property.  \nCritical points of $f$ are in one-to-one correspondence with \nbarycenters of simplices.  \nMoreover, the Morse index of a critical point \nis equal to the dimension of the corresponding simplex.  \nThen we can take a gradient-like vector field $X$ such that \nthe unstable manifold $W^u(p)$ at each critical point $p$ is \nthe interior of the corresponding simplex.  \nDenote by $\\{\\rho_t\\}$ the flow generated by $X$.  \n(The function $f$ increases along the orbits of $\\{\\rho_t\\}$.)  \n\nNow we prove the following:\n\n\\begin{thm}\\label{Morse}\nLet $L$ be a relatively spin Lagrangian submanifold in a closed \nsymplectic manifold $(M,\\omega)$ and $f$ a Morse function on $L$ \nas above.  \nThen Morse complex $CM^*(f)\\otimes \\Lambda_{0,nov}$ \ncarries a structure of a filtered $A_{\\infty}$-algebra, \nwhich is homotopy equivalent to the filtered $A_{\\infty}$-algebra \nassociated to $L$ constructed in \\cite{FOOO}.  \n\\end{thm} \nThe proof occupies the rest of this section.  \nWe explain how to choose $\\chi_g(L)$ in section 4. \nFirstly, we choose and fix a linear order on the set of vertices \nin ${\\mathfrak T}$.  \nThen we regard each $T_i \\in {\\mathfrak T}$ as \na singular simplex by the affine parametrization \n$\\sigma_i:\\Delta_{k_i} \\to T_i$ \nrespecting the order of the vertices.  \nIn particular, all simplices are oriented, hence \nthe unstable manifolds $W^u(p)$.  \nFor our construction, we have to start with the following \nset of singular simplices.  \nSet $\\chi_{\\mathfrak T}(L)=\\{\\sigma_i\\}$ \nand identify the Morse complex $CM^{\\bullet}(f)$ \nwith $C_{\\mathfrak T}(L)$, which is a subcomplex of the singular \nchain complex of $L$ generated by $\\chi_{\\mathfrak T}(L)$.  \nNote that $\\chi_{\\mathfrak T}(L)$ satisfies Conditions 1 and 2 \ngiven in section 4.  \n\nWe define $\\chi_g(L) \\supset \\chi_{\\mathfrak T}(L)$ \nin an inductive way as follows.  \nFor $g=-1$, we set $\\chi_{-1}(L)=\\chi_{\\mathfrak T}(L)$.\nFor $g=0,1,\\dots$, suppose that \nwe constructed $\\chi_{g'}(L)$, $g' < g$.  \n\nWe can choose the perturbations \n${\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}$ in \nProposition \\ref{transv} with the following property.  \n\n\\smallskip\n\nEach face $\\tau$ of any simplex in the triangulation of \n\\[\nev_{0*}\\bigl({\\mathcal M}_{k+1}^{{\\rm main},{\\mathfrak d}}\n(\\beta;P_1,\\dots,P_k)^{{\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}} \n\\bigr)\n\\] \nwith $g=\\parallel ({\\mathfrak d}, \\beta) \\parallel$\nis transversal to the stable manifold $W^s(p)$ \nat any $p \\in {\\rm Crit}(f)$.  \nMoreover, for each $\\tau$ of dimension at most \n$\\dim L$, there exists at most one \n$p=p(\\tau) \\in {\\rm Crit}(f)$ such that the stable submanifold \n$W^s(p)$ is of complementary dimension to $\\tau$ and \n$W^s(p)$ and $\\tau$ intersect at a unique point.  \nDenote by $T(p) \\in {\\mathfrak T}$ the simplex containing $p$.  \nLet $\\chi_g^{\\circ}(L)$ be the set of these singular simplices \n$\\tau$.\n\n\\smallskip\n\nWe have to add $\\chi_g^{\\circ}(L)$ to previous \n$\\bigcup_{g' < q}\\chi_{g'}(L)$.  \nIn order to guarantee Condition 2, we further add the following \nsingular simplices to $\\chi_g^{\\circ}(L)$ and obtain \n$\\chi_g(L)$.  \nDenote by $\\sigma^{\\tau} \\in \\chi_{\\mathfrak T}$ \nthe singular simplex corresponding to $T(p(\\tau))$.  \nDefine $\\Pi(\\tau)=\\epsilon \\sigma^{\\tau}$, where \n$\\epsilon = \\pm 1$ is given by the following equation.  \n\\[\n\\tau \\cap W^s(p(\\tau)) = \\epsilon W^u(p(\\tau)) \\cap W^s(p(\\tau)),\n\\]\nif there exists a unique stable manifold $W^s(p(\\tau))$, \nwhich intersects $\\tau$ transversely at a unique point. \nOtherwise, we define $\\Pi(\\tau)=0$.  \nIn particular, if $\\tau > \\dim L$, $\\Pi(\\tau)=0$.  \nFor each $\\tau$ as above, we will find a singular chain $G(\\tau)$ \nsuch that \n$$\n\\Pi (\\tau) - \\tau = \\overline{\\mathfrak m}_1 G(\\tau) \n+ G (\\overline{\\mathfrak m}_1 \\tau),\n$$\nwhere $\\overline{\\mathfrak m}_1=(-1)^{\\dim L} \\partial$.  \nWe can find such $G(\\tau)$ by induction on dimension of $\\tau$.  \nIn our case, we construct $G(\\tau)$ using the gradient-like \nflow $\\{\\rho_t\\}$.  \nSet \n\\[\n{\\mathfrak r}({\\mathrm{Im}} \\tau)=\\bigcup_{t\\leq 0} \\rho_t(\n{\\mathrm{Im}} \\tau).\n\\]\nBy the choice of our perturbations above, \nthe closure of ${\\mathfrak r}({\\mathrm{Im}} \\tau)$ can be \ntriangulated in a compatible way with $\\tau$ and $\\Pi(\\tau)$.  \nPick such a triangulation and then define $G(\\tau)$ the corresponding \nsingular chain.  \nFor the chain $G(\\tau)$, we define $G(G(\\tau))=0$.   \n\n\nNote that $\\Pi:C_{(g)}(L) \\to C_{(0)}(L)$ and \n$G:C_{(g)}(L) \\to C_{(g)}(L)$ satisfy the conditions in Lemma \n3.2, hence $C_{(g)}(L)$ satisfies Condition 2.  \nTherefore we can apply Theorem \\ref{reduction} to reduce \nthe filtered $A_{\\infty}$-structure on $C^{\\bullet}(L; \n\\Lambda_{0,nov})$ to $CM^{\\bullet}(f) \\otimes \\Lambda_{0,nov}$ \nand obtain a filtered $A_{\\infty}$-algebra \n$(CM^{\\bullet}(f) \\otimes \\Lambda_{0,nov},\\{{\\mathfrak m}'_k\\})$, \nwhich is homotopy equivalent to $(C_{(g)}(L) \\otimes \\Lambda_{0,nov},\n\\{{\\mathfrak m}_k\\})$.  \nTheorem \\ref{Morse} is proved.   \n\nIn the proof of Theorem 3.3, we constructed the operator \n${\\mathfrak m}'_k$ from ${\\mathfrak m}_{\\Gamma}$, \n$\\Gamma \\in G^+_{k+1}$.  \nThe geometric meaning of ${\\mathfrak m}_{\\Gamma}$ \nis as follows.  \nRecall that $G(\\tau)$ assigns the closure of the union of \nflow lines arriving at $\\tau$.  \nWe assign $G$ to the interior edges.   \nThe interior vertices correspond to $J$-holomorphic discs, \nmore precisely, bordered stable maps of genus 0.  \nIn order to describe the operation ${\\mathfrak m}_{\\Gamma}$,  \nwe need only rigid configuration of $\\tau_i \\in \n\\chi_{\\mathfrak T}(L)$ (the barycenters of $\\tau_i$ are inputs), \n$J$-holomorphic discs, (broken) negative flow lines of $X$ and \n$W^s(q)$ ($q$ is the output).  \nWe choose the perturbation ${\\mathfrak s}$ generically so that \nthe moduli spaces of holomorphic discs and the flow $\\{\\rho_t\\}$ \nare in general position so that \nthe inner edges correspond to negative flow lines of $X$.  \nHence the ${\\mathfrak m}_{\\Gamma}$ is defined by using \nthe configuration of pseudo-holomorphic discs and \nMorse negative gradient trajectories according to the decorated tree \n$\\Gamma \\in \\cup_k G^+_{k+1}$.  \n\nFor a decorated tree $\\Gamma \\in G_{k+1}^+$, \neach edge is oriented in the direction \nfrom the $k$ input vertices to the root vertex.  \nWe denote by $v^{\\pm}(e)$ the vertices such that \n$e$ is an oriented edge from $v^-(e)$ to $v^+(e)$.  \nConsider \nthe moduli space ${\\mathcal M}_{\\Gamma}(h;p_1, \\dots. p_k, q)$ \nconsisting of the configuration of the following \n\n\\begin{itemize}\n\\item \nfor each interior vertex $v \\in \\Gamma$, a bordered \nstable map $u_v$ representing the class $\\beta_{\\eta(v)}$ with \n$\\ell (v)$ boundary marked points, \nwhere $\\ell(v)$ is the valency of $v$, \n(we denote by $p(e,v)$ the marked point corresponding to the  \nedge $e$ attached to $v$)\n\\item\nthe $i$-th input edge $e_i$ corresponds to a broken negative \ngradient flow line $\\gamma_i$ starting from the critical point $p_i$ \nto $u_{v^+(e_i)}(p(e_i,v^+(e_i)))$, \n\\item\nthe output edge corresponds to a broken negative gradient flow line \n$\\gamma_0$ from $u_{v^-(e_0)}(p(e_0,v^-(e_0)))$ \nending at the critical point $q$,\n\\item\nan interior edge $e$ corresponds to a broken negative gradient flow line $\\gamma_e$ from $u_{v^-(e)}(p(e,v^-(e)))$ to $u_{v^+(e)}(p(e,v^+(e)))$. \n\\end{itemize}\n\nCounting the weighted order of the moduli spaces of vitual \ndimension $0$, we get\n$$\n{\\mathfrak m}_{\\Gamma}(p_1 \\otimes \\dots \\otimes p_k) \n= \\sum_q  \\# {\\mathcal M}_{\\Gamma}(h;p_1, \\dots, p_k,q) \\cdot \ne^{\\sum_{v} \\mu(\\beta_{\\eta(v)})\/2}q\n$$\nand \n$${\\mathfrak m}_k= \\sum_{\\Gamma \\in G^+_{k+1}} \nT^{E(\\Gamma)} {\\mathfrak m}_{\\Gamma}.\n$$\n\n\nFor example, \nwe obtain the configuration as in Figure \\ref{configuration} associated \nto the decorated planar tree $\\Gamma$ \nwith inputs $T(p),T(p'),T(p'') \\in \n\\chi_{\\mathfrak T}(L)$  as in Figure \\ref{modelconfiguration}.\n\n\n\\begin{figure}\n\\begin{center}\n\\include{modelconfiguration}\n\\end{center}\n\\caption{}\n\\label{modelconfiguration}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\include{configuration}\n\\end{center}\n\\caption{}\n\\label{configuration}\n\\end{figure}\n\nThis is essentially the configuration introduced in \\cite{Fuk94}.  \nNote that the first named author \\cite{Fuk94} took multiple \nMorse functions to achieve transversality.  \nHere we use one Morse function and apply the argument in section 3 \nto squeeze the filtered \n$A_{\\infty}$-algebra structure to the Morse complex.   \nWe emphasize that this becomes possible only after working out \nthe chain level intersection theory in detail, which we explained \nin section 4.  \nTo find an appropriate perturbation of \n${\\mathcal M}_{\\Gamma}(h;p_1, \\dots, p_k,q)$ directly without using \nthe argument in section 4 (or section 30 in \\cite{FOOO}) seems \nextremely difficult.  \n\nThe use of multiple Morse functions enables to construct the \ntopological (or partial) filterd $A_{\\infty}$-category of \nMorse functions on $L$ in the case that ${\\mathfrak m}_0 = 0$.  \nNote that, in a topological (or partial) filtered $A_{\\infty}$-category \n${\\mathcal A}$, \nthe set $Ob_{\\mathcal A}$ of objects is a topological space and \nthe set $Mor_{\\mathcal A}(a,b)$ of morphisms is defined \nfor $(a,b)$ in an open dense subset of $Ob_{\\mathcal A} \\times \nOb_{\\mathcal A}$.  \nWhen ${\\mathcal A}$ is a filtered $A_{\\infty}$-category, each \nobject $a$ is equipped with the filtered $A_{\\infty}$-algebra \n$Mor_{\\mathcal A}(a,a)$.   \nIn our case, the filtered $A_{\\infty}$-algebra on Morse complex \n$CM^{\\bullet}(f)$ \ncorresponds to the filtered $A_{\\infty}$-algebra associated to \nthe object $f$.  \nNote that, in the construction of this paper and in Theorem 5.1, \nwe do not need to assume that ${\\mathfrak m}_0=0$ \nin our construction. \n\nFor a relative spin pair $(L_0,L_1)$ of Lagrangian submanifolds, \nwhich intersect transversely, we obtain \nthe filtered $A_{\\infty}$-bimodule over the filtered \n$A_{\\infty}$-algebras on $CM^{\\bullet}(f_i) \\otimes \\Lambda_{0,nov}$, \nwhere $f_i:L_i \\to \\R$, \n$i=0,1$, are Morse functions.  \nWhen $L_0$ and $L_1$ are of clean intersection, \nthere exists a certain local system $\\Theta$ on $L_0 \\cap L_1$ \nand we can reduce the filtered $A_{\\infty}$-bimodule structure on \n$C^{\\bullet}(L_0 \\cap L_1;\\Theta) \\otimes \\Lambda_{0,nov}$ to \n$CM^{\\bullet}(h;\\Theta) \n\\otimes \\Lambda_{0,nov}$ \nover the filtered $A_{\\infty}$-algebras on $CM^{\\bullet}(f_i) \\otimes \n\\Lambda_{0,nov}$.  \nHere $h$ is a Morse function on $L_0 \\cap L_1$, which may be \ndisconnected with various dimensions.  \nFor the canonical models of filtered $A_{\\infty}$-bimodules, \nsee \\cite{FOOO}.  \n\n\\smallskip \\noindent\n{\\bf Acknowledgement.}  We thank Otto van Koert for his kind \ninstruction of making figures in this article.  \n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nEccentric gaseous disks arise in a surprisingly wide variety of astrophysical\ncontexts. A number of mechanisms can explain the existence of eccentric disks,\nthe most commonly invoked one being external perturbation. In eccentric\nbinaries, secular gravitational interaction endows forced and free\neccentricities upon circumbinary and circumobject disks\n\\citep[e.g.,][]{2000ssd..book.....M}; in circular binaries, tidal forces couple\nto circumobject disks through the $3:1$ mean motion resonance and allow free\neccentricity to grow exponentially \\citep{1991ApJ...381..259L}. Another\npossibility is that disks become more eccentric over time. Viscous\noverstability \\citep{1978MNRAS.185..629K}, which amplifies small-scale\neccentric perturbations in isolated disks \\citep[e.g.,][]{1994MNRAS.266..583L,\n2001MNRAS.325..231O}, is often cited in this connection. A third option is for\ndisks to be born eccentric. Outgassing from planetesimals can create eccentric\ndisks \\citep{2021MNRAS.505L..21T}, and so can the tidal disruption of stars\n\\citep{2015ApJ...804...85S, 2015ApJ...806..164P, 2017MNRAS.467.1426S} and\nmolecular clouds \\citep[e.g.,][]{2008Sci...321.1060B} by supermassive black\nholes. On the phenomenological side, eccentric disks are sometimes invoked to\nexplain asymmetric lines in white dwarfs \\citep[e.g.,][]{2006Sci...314.1908G},\nas well as asymmetric broad emission lines in active galactic nuclei\n\\citep[e.g.,][]{1995ApJ...438..610E, 2021MNRAS.506.6014T} and \\acp{TDE}\n\\citep[e.g.,][]{2014ApJ...783...23G, 2017MNRAS.472L..99L}.\n\nBecause ideal \\acp{MHD} is supported even by low levels of ionization\n\\citep{1994ApJ...421..163B, 1996ApJ...457..355G}, we expect magnetic fields to\nplay a role in many of the eccentric disks enumerated above. The presence of\nmagnetic fields changes the way disks evolve because of the \\ac{MRI}\n\\citep{1991ApJ...376..214B, 1991ApJ...376..223H}. Simply put, in a disk whose\ninner parts rotate faster than the outer parts, differential rotation can latch\nonto horizontal bits of the magnetic field, stretch them out, and amplify them.\nThe gas connected to one of these bits on the inside is pulled back by magnetic\ntension, loses angular momentum, migrates inward, and picks up orbital speed.\nIn the meantime, the gas on the outside is dragged forward, gains angular\nmomentum, drifts outward, and slows down. The rising velocity difference across\nthe horizontal magnetic field in turn enhances its stretching, precipitating an\ninstability.\n\nAnalytic calculations for circular disks show that a perturbation can grow by\norders of magnitude per orbit in the linear stage \\citep{1991ApJ...376..214B},\nmaking the \\ac{MRI} among the most vigorous \\ac{MHD} instabilities. The\ninitially exponential amplification eventually enters the nonlinear stage and\nbreaks down into \\ac{MHD} turbulence. Orbital shear enforces a correlation\nbetween the radial and azimuthal components of the turbulent velocity, and\nbetween the same components of the magnetic field. Turbulent stresses transport\nangular momentum outward; gas robbed of angular momentum sinks to smaller\nradii, and the disk accretes.\n\nThe saturation process is amenable only to numerical investigation. Previous\nsimulations of circular disks, numbering in the hundreds, are divided into\nshearing-box simulations, which consider a small neighborhood of the disk as\nrepresentative of the whole \\citep[e.g.,][]{1992ApJ...400..595H,\n1995ApJ...440..742H, 1996ApJ...464..690H, 1995ApJ...446..741B,\n1996ApJ...463..656S}, and global simulations, which take in the entire disk\n\\citep[e.g.,][]{1991ApJ...376..223H, 1998ApJ...501L.189A, 1999ASSL..240..195M,\n2000ApJ...528..462H, 2003ApJ...592.1060D}. This large body of work converged on\nthe consensus that, irrespective of the circumstances simulated, \\ac{MHD}\nturbulence in circular disks saturates within 10 to 20 orbits, and the stresses\nat saturation correspond to a \\citet{1973A&A....24..337S} alpha parameter\nbetween \\numrange[range-phrase={ and }]{0.01}{1}. For circular disks around\nblack holes, the alpha parameter may change substantially near their inner\nedges at the \\ac{ISCO}. In disks with gas-dominated pressure, the alpha\nparameter can increase very rapidly as matter approaches and crosses the\n\\ac{ISCO} \\citep{2010ApJ...711..959N}; alternatively, in super-Eddington\nradiation-dominated disks, it may exhibit a sharp peak at a radius a short\ndistance outside the \\ac{ISCO} \\citep{2014ApJ...796..106J}.\n\nThere is no reason to expect the \\ac{MRI} and the associated \\ac{MHD} stresses\nto be absent from eccentric disks, though their character and the exact manner\nin which they approach saturation may differ from circular disks. Very little\nheed, however, has hitherto been paid to any aspect of the role of magnetic\nfields in eccentric disks. We were the first to establish that eccentric disks\nare susceptible to the \\ac{MRI} \\citep{2018ApJ...856...12C}. Compared to the\ncircular \\ac{MRI}, the growth rate of the eccentric \\ac{MRI} is smaller at the\norder-unity level and the range of unstable wavelengths is wider. That work,\nhowever, is incomplete because it examined only linear stability. It remains an\nopen question whether the robust growth of \\ac{MHD} stresses in the linear\nstage would, as the \\ac{MRI} turns nonlinear, translate to saturated stress\nlevels significant enough to affect disk evolution. There are, however, two\nchallenges to the numerical work requisite to follow the nonlinear process.\n\nThe first problem stems from the fact that existing Newtonian \\acp{MHD} codes\nare capable of handling only Cartesian, cylindrical, and spherical coordinate\nsystems. If a moderately eccentric disk were simulated in one of these\ncoordinate systems, the streamlines would be oblique to the grid. Numerical\nartifacts reflecting grid symmetry would creep in; at the same time, excessive\nnumerical dissipation would prevent the disk from maintaining its shape over a\nlong time. These drawbacks limited previous simulations of isolated eccentric\ndisks, all dealing with pure hydrodynamics, to eccentricities small enough to\nbe implementable as a $m=1$ perturbation to the initial velocity\n\\citep[e.g.,][]{2005A&A...432..757P}. Here we demonstrate that Newtonian\nsimulations can be performed in arbitrary coordinate systems with \\acp{GRMHD}\ncodes, provided that the metric is judiciously chosen. Employing a coordinate\nsystem molded to the shape of moderately eccentric disks significantly\nsuppresses the numerical errors arising from ordinary cylindrical coordinates.\n\nThe second difficulty is with the simulation setup. One may think that the\nsetting of localized perturbations in \\citet{2018ApJ...856...12C} lends itself\nnaturally to shearing-box simulations \\citep[e.g.,][]{2014MNRAS.445.2621O,\n2018MNRAS.477.4838W}. Drawing inspiration from circular disks, one may imagine\nshearing boxes in eccentric disks to have edges running along curves of\nconstant semilatus rectum and constant azimuth. However, the very notion of an\neccentric shearing box is suspect. Circular shearing boxes assume that the\ndisk, being homogeneous, is equivalent to a tiling of the shearing box; this\njustifies periodic azimuthal and shift-periodic radial boundary conditions. The\nassumption breaks down for eccentric disks. \\citet{2018ApJ...856...12C} showed\nthat a perturbation grows differently at different positions along the orbit,\ndepending on the local orbital shear. This means the conditions at the leading\nedge of an eccentric shearing box are different from the trailing edge, and\nthey also vary along each of the other two edges, so boundary conditions that\ndirectly copy one edge to the other would be inappropriate. We can avoid these\nquestions about the eccentric shearing box by performing global simulations\ninstead. It is worth noting that the first simulations of the circular \\ac{MRI}\nwere also global \\citep{1991ApJ...376..223H}.\n\nWe recount our simulation setup in \\cref{sec:methods}. The results from the\nsimulations are presented in \\cref{sec:results} (see\n\\href{https:\/\/youtube.com\/playlist?list=PLlhsZldWhMs6OIvfFxf5DZy9UbpimeGGt}{movies})\nand discussed in \\cref{sec:discussion}. Our concluding remarks are gathered in\n\\cref{sec:conclusions}.\n\n\\section{Methods}\n\\label{sec:methods}\n\nWe outline our simulation strategy in \\cref{sec:overall strategy}. We continue\nwith the details of the simulation setup in the subsequent subsections and in\nthe Appendix; readers uninterested in the technicalities may skip to the\nresults in \\cref{sec:results}.\n\n\\subsection{Overall strategy}\n\\label{sec:overall strategy}\n\nOur goal is to simulate the nonlinear evolution of the eccentric \\ac{MRI} in a\npurely Newtonian setting. The only reason we turn to a \\acp{GRMHD} code is\nbecause numerical issues force us to tailor the coordinate system to the\neccentric disk shape, but existing Newtonian codes lack the facility to deal\nwith bespoke coordinate systems. The \\acp{GRMHD} code we use for this purpose\nis Athena++ \\citep{2016ApJS..225...22W, 2020ApJS..249....4S}.\n\nThere can be drawbacks to solving Newtonian problems with a \\acp{GRMHD} code,\nprincipally the large truncation error potentially created by the smallness of\nthe typical kinetic and internal energies compared to the rest energy. This\nerror can, however, be mitigated by careful design of the simulation setup, as\ndescribed in later subsections.\n\nA major difference of \\acp{GRMHD} codes compared with Newtonian codes is that\ngravity enters not as explicit momentum and energy source terms, but through\nthe metric. Our choice of the metric in \\cref{sec:coordinates,sec:potential}\nensures that orbits are closed ellipses that do not apsidally precess, allowing\nour simulations to closely approximate Newtonian behavior.\n\nWe simulate a suite of disks. The initial hydrodynamic configuration of the\ndisks follow the common prescription in \\cref{sec:hydrodynamic initial\ncondition}: Gas is placed within a limited radial range, so that the inner and\nouter edges of the disk are well-separated from the inner and outer boundaries,\nrespectively, of the simulation domain. All disks are tracked for 15 orbits so\nturbulence may have enough time to reach saturation. The individual disks\ncomprising the suite are designed with contrast in mind. They are classified\nalong two orthogonal dimensions: circular versus eccentric, unmagnetized versus\nmagnetized.\n\nComparison between circular and eccentric disks gives us an idea whether the\nsaturation level of the \\ac{MRI} depends on eccentricity. The eccentric disks\nhave a moderate eccentricity of 0.5, so that the character of the \\ac{MRI}\nspecific to eccentric disks can reveal itself without being overwhelmed by\nhydrodynamic effects.\n\nComparison between unmagnetized and magnetized disks helps us disentangle\n\\ac{MHD} effects from hydrodynamic effects. Magnetized disks are seeded with an\ninitial magnetic field as described in \\cref{sec:magnetic initial condition}.\nTwo magnetic topologies are considered, vertical- and dipolar-field, because\ntopology can influence the saturated \\ac{MHD} turbulence\n\\citep[e.g.,][]{1995ApJ...440..742H, 1996ApJ...464..690H, 2004ApJ...605..321S,\n2013ApJ...767...30B}.\n\n\\subsection{Equations}\n\nWe employ natural units, which means the length and velocity units are the\ngravitational radius and the speed of light, respectively. The sign convention\nis $(-,+,+,+)$, Greek indices range over $\\{0,1,2,3\\}$, Latin indices range\nover $\\{1,2,3\\}$, and Einstein summation is implied. \n\nThe equations of \\acp{GRMHD} are\n\\begin{alignat}{3}\n& \\partial_t[(-g)^{1\/2}\\rho u^t\n  &]& +\\partial_j[(-g)^{1\/2}\\rho u^j\n  &]&= 0, \\\\\n& \\partial_t[(-g)^{1\/2}T^t_\\mu\n  &]& +\\partial_j[(-g)^{1\/2}T^j_\\mu\n  &]&= (-g)^{1\/2}T^\\nu_\\sigma\\Gamma^\\sigma_{\\mu\\nu}, \\\\\n& \\partial_t[(-g)^{1\/2}\\astF^{i0}\n  &]& +\\partial_j[(-g)^{1\/2}\\astF^{ij}\n  &]&= 0.\n\\end{alignat}\nHere $t$ is the coordinate time, $\\rho$ is the comoving mass density, $u^\\mu$\nis the velocity, $g$ is the determinant of the metric $g_{\\mu\\nu}$, and\n$\\Gamma^\\sigma_{\\mu\\nu}$ is the Christoffel symbol of the second kind. From the\nHodge dual of the electromagnetic tensor $\\astF^{\\mu\\nu}$ we obtain the\nmagnetic field $B^i=\\astF^{i0}$ and the projected magnetic field\n$b^\\mu=u_\\nu\\astF^{\\nu\\mu}$. Lastly, the stress--energy tensor is\n\\begin{equation}\nT^{\\mu\\nu}=\\biggl(p+\\frac12b_\\sigma b^\\sigma\\biggr)g^{\\mu\\nu}\n  +\\biggl(\\rho+\\frac\\gamma{\\gamma-1}p+b_\\sigma b^\\sigma\\biggr)u^\\mu u^\\nu\n  -b^\\mu b^\\nu,\n\\end{equation}\nwhere $p$ and $\\gamma$ are the gas pressure and adiabatic index, respectively.\n\n\\subsection{Orbital coordinate system}\n\\label{sec:coordinates}\n\nWe solve the \\acp{GRMHD} equations in an orbital coordinate system\n\\citep{2001MNRAS.325..231O}. It is similar to the cylindrical coordinate\nsystem, except that circular coordinate surfaces are replaced by axially\naligned elliptical ones, chosen such that their cross sections along the\nmidplane match the initial disk streamlines. Adapting the coordinate system to\nthe disk reduces numerical artifacts and dissipation in our simulations. The\neccentricities and orientations of the coordinate surfaces can in principle\nvary from one elliptical cylinder to the next, but here we specialize to the\ncase in which both are spatially uniform.\n\nLet $(t,R,\\varphi,z)$ be cylindrical coordinates. We work with gravity weak\nenough to be well-described by a quasi-Newtonian potential $\\Phi(R,z)$; the\nnonzero components of the metric in this limit are\n\\begin{align}\ng_{tt} &= -[1+2\\Phi(R,z)], \\\\\ng_{RR} &= 1, \\\\\ng_{\\varphi\\varphi} &= R^2, \\\\\ng_{zz} &= 1.\n\\end{align}\nLet $(t,\\log\\lambda,\\phi,z)$ be orbital coordinates specialized for use in our\nsimulations; they are related to cylindrical coordinates by\n\\begin{align}\nR &= \\lambda\/(1+e\\cos\\phi), \\\\\n\\varphi &= \\phi.\n\\end{align}\nHere $e$ is the eccentricity of our orbital coordinates, set to 0 for circular\ndisks and 0.5 for eccentric disks. Coordinate surfaces of constant $\\lambda$\nare elliptical cylinders of semilatus rectum $\\lambda$, or equivalently,\nsemimajor axis $a=\\lambda\/(1-e^2)$. We opt for $\\log\\lambda$ instead of\n$\\lambda$ in order to generate a logarithmic grid, but for ease of\nunderstanding we continue to label that coordinate by $\\lambda$ and express\nresults in terms of $\\lambda$. We reuse $t$ and $z$ without risk of ambiguity\nbecause these two coordinates do not participate in the coordinate\ntransformation from cylindrical to orbital. The metric and connection in both\ncoordinates are provided in \\cref{sec:metric}.\n\n\\subsection{Gravitational potential and orbits}\n\\label{sec:potential}\n\nWe ignore vertical gravity in these first simulations of the eccentric\n\\ac{MRI}, so the potential depends only on $R$. In this sense, our simulations\nresemble earlier simulations of unstratified circular disks\n\\citep[e.g.,][]{1991ApJ...376..223H, 1995ApJ...440..742H, 1996ApJ...464..690H,\n2004ApJ...605..321S, 2007A&A...476.1113F, 2007A&A...476.1123F,\n2007MNRAS.378.1471L, 2009ApJ...690..974S, 2009ApJ...694.1010G,\n2011ApJ...739...82B}. However, instead of the point-mass gravitational\npotential $\\Phi(R,z)=-1\/R$, we adopt\n\\begin{equation}\n\\Phi(R,z)=-1\/(R+2)\n\\end{equation}\nbecause, as proven in \\cref{sec:potential derivation}, this potential admits\nclosed eccentric orbits at all distances. The modification matters because\ngeneral-relativistic apsidal precession, albeit small at large distances,\naccumulates over the tens of orbits during which the \\ac{MRI} saturates.\n\nThe properties of orbits in our potential are also derived in\n\\cref{sec:potential derivation}; here we repeat the parts that support our\nexposition. For an orbit along a coordinate curve of semilatus rectum\n$\\lambda$ or semimajor axis $a=\\lambda\/(1-e^2)$, the orbital period is\n\\begin{equation}\\label{eq:orbital period}\nT=2\\pi a^{3\/2}(1+2\/a).\n\\end{equation}\nThe specific energy and angular momentum conserved with respect to our metric\nare\n\\begin{align}\n\\label{eq:orbit energy}\nE &= (1+1\/a)^{-1\/2}, \\\\\n\\label{eq:orbit angular momentum}\nL &= \\lambda^{1\/2}E.\n\\end{align}\nThe specific energy includes the rest energy; $E=1$ corresponds to marginally\nbound material, and $E\\to0$ as material becomes more bound. The nonzero\ncomponents of the orbital velocity are\n\\begin{align}\n\\label{eq:orbit velocity 0} u^t &= E\/[1+2\\Phi(R,z)], \\\\\n\\label{eq:orbit velocity 2} u^\\phi &= L\/R^2,\n\\end{align}\nso the physical velocity at pericenter is\n\\begin{equation}\nv_\\su p(\\lambda)=\\frac{Ru^\\phi}{u^t}=\n  \\frac{\\lambda^{1\/2}(1+e)}{\\lambda+2(1+e)}.\n\\end{equation}\n\nConsider a collection of such orbits nested within each other, all following\ncoordinate curves. The velocity field thus generated has finite divergence:\n\\begin{equation}\n(-g)^{-1\/2}\\partial_\\mu[(-g)^{1\/2}u^\\mu]=\n  \\frac{u^\\phi}{R+2}\\frac{e\\sin\\phi}{1+e\\cos\\phi}.\n\\end{equation}\nConsequently, if the motion of a gas without pressure is described by this\nvelocity field, the density along a streamline of constant $\\lambda$ cannot be\nuniform, but must instead vary with $\\phi$ in proportion to\n\\begin{equation}\nd(\\lambda,\\phi)\\eqdef\n  \\biggl[\\frac{\\lambda+2(1+e\\cos\\phi)}{\\lambda+2(1+e)}\\biggr]^{1\/2}.\n\\end{equation}\nSo as to guarantee an initial condition that is a genuine hydrodynamical steady\nstate, we take this modulation into account in \\cref{sec:hydrodynamic initial\ncondition} even though the modulation amplitude is tiny for our disk\nparameters.\n\n\\subsection{Hydrodynamic initial condition}\n\\label{sec:hydrodynamic initial condition}\n\nBecause we ignore vertical gravity, our initial disk is translationally\nsymmetric along the $z$\\nobreakdash-direction. It can be described as an\nelliptical annular cylinder, each shell of which orbits the coordinate axis\nalong a coordinate surface of constant $\\lambda$ with a velocity as given by\n\\cref{eq:orbit velocity 0,eq:orbit velocity 2}. The scale of the disk is\ncharacterized by its fiducial orbit, whose semilatus rectum is the geometric\nmean of the semilatera recta of its inner and outer edges.\n\nThe disk should be large enough that orbital velocities are non-relativistic,\nand small enough that severe truncation errors do not arise from the evolution\nof the total energy as a result of the smallness of the kinetic energy with\nrespect to the rest energy. In light of the fact that the linear growth rate of\nthe \\ac{MRI} is inversely proportional to the orbital period\n\\citep{2018ApJ...856...12C}, we additionally require that runs of different $e$\nhave the same semimajor axis and thus orbital period. We settle on a semilatus\nrectum of $\\lambda_*=200(1-e^2)$ for the fiducial orbit, and we report time in\nunits of the orbital period at this orbit.\n\nThe initial density profile is\n\\begin{equation}\\label{eq:initial density}\n\\rho(\\lambda,\\phi)=\\rho_*m(\\lambda,\\phi),\n\\end{equation}\nwith $\\rho_*$ the density at the pericenter of the fiducial orbit,\n$(\\lambda,\\phi)=(\\lambda_*,0)$. To give the disk edges that are not too sharp\nand to build in a numerical vacuum, the density is modulated spatially as\n\\begin{equation}\nm(\\lambda,\\phi)=\n  (1-f_\\su v)d(\\lambda,\\phi)h(l_\\su{gb},l_\\su{gt};2q^\\lambda)+f_\\su v,\n\\end{equation}\nwhere\n\\begin{equation}\nh(a,s;x)\\eqdef\n  \\frac14\\biggl(1+\\tanh\\frac{x+a}s\\biggr)\\biggl(1+\\tanh\\frac{a-x}s\\biggr)\n\\end{equation}\nis a smoothed top-hat function and\n\\begin{equation}\nq^\\lambda\\eqdef\n  \\frac{\\log\\lambda-\\log\\lambda_*}{\\max(\\log\\lambda)-\\min(\\log\\lambda)}\n\\end{equation}\nis the fractional $(\\log\\lambda)$\\nobreakdash-position within the simulation\ndomain. The modulation is governed by three parameters: $f_\\su v=0.01$ is the\nvacuum-to-disk density ratio, and $l_\\su{gb}=0.5$ and $l_\\su{gt}=0.1$ are the\nfractional $(\\log\\lambda)$\\nobreakdash-extents of the disk body and the\ndisk--vacuum transition, respectively.\n\nBecause the most unstable mode of the circular \\ac{MRI} is incompressible, the\nthermodynamic properties of the gas is expected to have little bearing on the\ngrowth of the eccentric \\ac{MRI}. Even so, we would like the pressure to be\nsmall enough initially that the configuration above is close to equilibrium,\nand the internal energy to be large enough at all times that catastrophic\ntruncation errors are avoided. We balance these competing desires by setting\nthe Mach number at the fiducial pericenter to $M_*=30$. Additionally, we adopt\nan adiabatic index of $\\gamma=1+10^{-5}$, corresponding to a nearly isothermal\ngas, so that the internal energy is larger at fixed pressure. The initial\npressure is therefore\n\\begin{equation}\np_*=\\rho_*[v_\\su p(\\lambda_*)\/M_*]^2\/\\gamma\n\\end{equation}\nat the fiducial pericenter and\n\\begin{equation}\np(\\lambda,\\phi)=p_*[m(\\lambda,\\phi)]^\\gamma\n\\end{equation}\neverywhere else. This initial condition is not strictly hydrostatic; transient\noutgoing waves are launched from the inner edge as the disk seeks force\nbalance. Moreover, our use of a soft equation of state means that density\nperturbations are stronger and pressure gradients have a lesser effect on disk\nevolution than if an adiabatic equation of state were used.\n\n\\subsection{Magnetic initial condition}\n\\label{sec:magnetic initial condition}\n\nThe magnetized runs are initialized with the two kinds of magnetic topologies\nillustrated in \\cref{fig:topology}. The vertical-field topology refers to a\nmagnetic field with one nonzero component\n\\begin{equation}\nB^z\\propto d(\\lambda,\\phi)h(l_\\su{mb},l_\\su{mt},2q^\\lambda),\n\\end{equation}\nwhere $l_\\su{mb}=0.4$ and $l_\\su{mt}=0.1$ are the fractional\n$(\\log\\lambda)$\\nobreakdash-extents of the magnetized disk body and the\ntransition from the magnetized disk body to the unmagnetized disk edges,\nrespectively. The net vertical magnetic flux persists throughout the\nsimulation. Thanks to its simplicity and its ability to generate the\nfastest-growing instabilities, the vertical-field topology was considered in\nthe first studies of the circular \\ac{MRI} \\citep{1991ApJ...376..214B,\n1991ApJ...376..223H} and in our analytic study of the eccentric \\ac{MRI}\n\\citep{2018ApJ...856...12C}.\n\n\\begin{figure}\n\\includegraphics{topology}\n\\caption{Poloidal slices showing with arrows the initial magnetic field for our\ntwo magnetic topologies. Background colors plot the initial density on the same\nscale as \\cref{fig:density}. The ordinate is more stretched than the abscissa,\nand the vectors are arbitrarily scaled.}\n\\label{fig:topology}\n\\end{figure}\n\nThe dipolar-field topology is derived from a magnetic potential with one\nnonzero\ncomponent\n\\begin{equation}\nA_\\phi\\propto\n  (-g)^{1\/2}\\cos^2(\\min(\\tfrac12,\\abs{q^\\lambda\/l_\\su{mb}})\\pi)\\cos^2(q^z\\pi),\n\\end{equation}\nwhere\n\\begin{equation}\nq^z\\eqdef\\frac z{\\max(z)-\\min(z)}\n\\end{equation}\nis the fractional $z$\\nobreakdash-position within the simulation domain and\n$l_\\su{mb}=0.4$. The inclusion of the metric determinant makes the\nmagnetic-field strength more uniform over azimuth. The dipolar-field topology\nsees frequent application in global simulations of the circular \\ac{MRI}\n\\citep[e.g.,][]{2000ApJ...528..462H}.\n\nFor both topologies, the initial plasma beta, defined as the initial volume\nintegral of gas pressure to that of magnetic pressure, is 100. The pressure due\nto the magnetic field is subtracted from the gas to preserve the total. The gas\npressure in magnetized regions is perturbed at the 0.01 level to seed the\n\\ac{MRI}.\n\n\\subsection{Simulation domain, boundary conditions, and other numerical\nconcerns}\n\nThe simulation domain spans\n$[\\exp(-2)\\lambda_*,\\exp(1)\\lambda_*]\\times[-\\pi,\\pi]\\times[-10,10]$ in\n$(\\lambda,\\phi,z)$. The lower end of the $\\lambda$\\nobreakdash-range ensures\nvelocities are never close to the speed of light, and the asymmetry of the\n$\\lambda$\\nobreakdash-range gives the infalling disk more room to evolve freely\nbefore hitting the inner boundary. The resolution is $240\\times240\\times60$ in\nthe three directions.\n\nPeriodic boundary conditions apply to the $\\phi$\\nobreakdash-direction. They\nare also employed in the $z$\\nobreakdash-direction, in accordance with our\nneglect of vertical gravity. Outflow boundary conditions are used in the\n$\\lambda$\\nobreakdash-direction: We copy all quantities to the ghost zone, zero\nthe $\\lambda$\\nobreakdash-component of the velocity if it points into the\nsimulation domain, and zero the $\\phi$- and $z$\\nobreakdash-components of the\nmagnetic field always. This last step reduces unphysical influences from the\nboundaries.\n\nTo prevent numerical issues, we require that the pressure in the simulation\ndomain always satisfies $(\\gamma p\/\\rho)^{1\/2}\\ge\\num{2e-4}$. In addition,\nwhenever the recovery of primitive variables fails, the primitive variables\nfrom the previous time step are carried forward.\n\nAs the simulation progresses, different parts of the disk may evolve\ndifferently in eccentricity and orientation, so the disk could eventually\nbecome misaligned with the grid, resulting in greater numerical dissipation.\nThe disk also occupies a wider range of semilatera recta due to pressure\ngradients and outward angular momentum transport, bringing the now differently\nshaped disk into contact with the boundaries. The inner boundary poses a lower\nlimit on the pericenter, but this restriction is arguably physical because\nthere are indeed radii an accretion flow cannot return from. If the central\nobject is a star, the disk cannot extend inside the star or its magnetosphere.\nIf the central object is a black hole, \\cref{fig:effective potential} tells us\nthat material with sufficiently low angular momentum and high energy can plunge\ndirectly into the black hole; low angular momentum and high energy are, of\ncourse, the hallmarks of an eccentric orbit. By contrast, the interaction with\nthe outer boundary is unphysical and can lead to numerical artifacts;\ntherefore, we restrict our attention to the first 15 orbits at $a=200$, before\nthe disk starts running into the outer boundary and numerical artifacts appear.\nSteady state appears to obtain at this time for the plasma beta and the alpha\nparameter, despite the continual evolution in disk shape.\n\n\\begin{figure}\n\\includegraphics{effpotl}\n\\caption{Effective potential $V_\\su{eff}=-1\/r+\\tfrac12(L^2\/r^2)(1-2\/r)$ in\nSchwarzschild spacetime as a function of spherical radius $r$ and specific\nangular momentum $L$. The specific energy $E$ must satisfy\n$\\tfrac12(E^2-1)>V_\\su{eff}$, and marginally bound trajectories have $E=0$.\nMaterial accretes by losing angular momentum; thus, its trajectory is described\nby potentials of decreasing $L$. In circular disks, trajectories have the\nlowest energy allowed by the stable potential well; such trajectories evolve\nalong the sequence of dots downward until they arrive at the smallest radius\nthat supports circular orbits, the \\ac{ISCO}. In eccentric disks, trajectories\nhave energies above the bottom of the potential well, which allows them to make\nradial excursions as suggested by the double-headed arrow. The smallest radius\na bound trajectory can reach without falling in is the marginally bound orbit;\nsuch a trajectory has $E=0$ and $L^2=16$. Because trajectories energetic enough\nto overcome the centrifugal barrier have energies greater than \\iac{ISCO}\norbit, material plunging into the black hole on these trajectories have less\nenergy available for radiation compared to material accreting on circular\ntrajectories.}\n\\label{fig:effective potential}\n\\end{figure}\n\n\\section{Results}\n\\label{sec:results}\n\n\\subsection{Overview}\n\n\\Cref{fig:density} tells us how much the disks have changed by the end of the\nsimulations. The most conspicuous contrast between unmagnetized and magnetized\ndisks, of whatever eccentricity, is that unmagnetized disks remain smooth while\nmagnetized disks develop large density fluctuations. This, of course, is due to\nthe \\ac{MRI} creating \\ac{MHD} turbulence in the magnetized disks. The\nfluctuations are larger in vertical-field disks than in dipolar-field disks.\n\nComparison can also be made between circular and eccentric disks regardless of\nmagnetic topology. Unlike the circular disks, which stay circular despite the\n\\ac{MHD} turbulence, the inner parts of eccentric disks grow more eccentric and\nprecess prograde. As the inner parts of eccentric disks shrink, their\npericenters move inside the inner boundary, and their material accretes across\nthe boundary while retaining its eccentricity. This loss of material from the\nmost eccentric orbits is the reason why there is a sparsely populated region\nbetween the inner parts and the inner boundary, visible in the late-time\neccentric disks in \\cref{fig:density}.\n\n\\begin{figure}\n\\includegraphics{density}\n\\caption{Midplane slices of density, in units of the fiducial density $\\rho_*$\nfrom \\cref{eq:initial density}. The top row shows two initial disks with\ndifferent eccentricities. The panels under each top-row panel are the outcomes\nof imposing various magnetic topologies on an initial disk and evolving it for\n15 orbits. The boundaries of the simulation domain are traced by thin ellipses\nin order to better distinguish low-density regions from regions not covered by\nthe simulations. Short lines from the origin indicate the approximate\norientations of the inner parts of the eccentric disks.}\n\\label{fig:density}\n\\end{figure}\n\nThere are also differences among the eccentric disks, concerning chiefly\neccentricity evolution and to a lesser degree precession. The unmagnetized and\nvertical-field disks in \\cref{fig:density} have largely preserved their initial\neccentricity, even though the outer parts of the unmagnetized disk have become\nsomewhat rounder, and the inner parts of the vertical-field disk have precessed\nslightly more. The dipolar-field disk features the steepest eccentricity\ngradient and a much higher degree of precession.\n\nThe eccentricity gradient can be quantified by computing the instantaneous\neccentricity $\\bar e$, which is the eccentricity of the orbit material would\nfollow given its instantaneous velocity $u^\\mu$ if only gravitational forces\nact; this orbit is also known as the osculating orbit. We calculate $\\bar e$\nfrom $u^\\mu$ using \\cref{eq:orbit velocity 0,eq:orbit velocity\n2,eq:eccentricity}. \\Cref{fig:instantaneous eccentricity} contains plots of the\nmass-weighted vertical average of $\\bar e$:\n\\begin{equation}\\label{eq:instantaneous eccentricity map}\n\\mean{\\bar e}_{z;\\rho}\\eqdef\\int dz\\,\\rho\\bar e\\bigg\/\\int dz\\,\\rho.\n\\end{equation}\nThe instantaneous eccentricity deviates little from its initial value in the\nunmagnetized and vertical-field disks, but it develops a clear gradient in the\ndipolar-field disk, with the eccentricity higher than its initial value in the\ninner parts and lower in the outer parts. The implications of the eccentricity\ngradient will be discussed in \\cref{sec:inner edge}. It is also apparent in\nthis figure that the prograde precession is largest in the dipolar-field disk.\n\n\\begin{figure}\n\\includegraphics{eccntrcy}\n\\caption{Mass-weighted vertical average of the instantaneous eccentricity, as\ndefined in \\cref{eq:instantaneous eccentricity map}. The top-left panel shows\nthe eccentric initial disk. The other panels are the outcomes of imposing\nvarious magnetic topologies on the initial disk and evolving it for 15 orbits.\nThe boundaries of the simulation domain are traced by thin ellipses.}\n\\label{fig:instantaneous eccentricity}\n\\end{figure}\n\n\\subsection{Plasma beta and the alpha parameter}\n\\label{sec:magnetic field}\n\nThe top half of \\cref{fig:magnetic field map} portrays the mass-weighted\nvertical average of the plasma beta at the end of the simulations, defined as\n\\begin{equation}\\label{eq:plasma beta map}\n\\mean\\beta_{z;\\rho}\\eqdef\n  \\int dz\\,\\rho\\frac{2p}{b_\\mu b^\\mu}\\bigg\/\\int dz\\,\\rho.\n\\end{equation}\nThe plasma beta is \\num{\\sim10} in vertical-field disks and \\num{\\sim100} in\ndipolar-field disks; the variation within a disk is about one order of\nmagnitude. Comparable levels of plasma beta are witnessed in circular and\neccentric disks with the same magnetic topology, suggesting that the \\ac{MRI}\nis unimpeded by eccentricity. The stronger magnetic fields in the\nvertical-field topology accord with simulations of circular disks in the\nliterature \\citep[e.g.,][]{1995ApJ...440..742H, 1996ApJ...464..690H,\n2004ApJ...605..321S, 2013ApJ...767...30B}.\n\n\\begin{figure}\n\\includegraphics{magmap}\n\\caption{Mass-weighted vertical averages of the plasma beta in the top half and\nMaxwell-only alpha parameter in the bottom half, as defined in \\cref{eq:plasma\nbeta map,eq:alpha parameter map}, respectively. The panels are the outcomes of\nimposing various magnetic topologies on an initial disk and evolving it for 15\norbits. Regions with negligible levels of magnetic field have exceedingly large\nvalues of plasma beta. The boundaries of the simulation domain are traced by\nthin ellipses.}\n\\label{fig:magnetic field map}\n\\end{figure}\n\nWe can also examine the plasma beta along a one-dimensional profile running\nfrom the inside of the disk to the outside at different times during the\nsimulations. Considering that the disk evolves in eccentricity and orientation,\nit makes little sense to look at profiles over semilatus rectum; instead, we\nconstruct profiles over cylindrical radius. The mass-weighted average plasma\nbeta at cylindrical radius $R$ is given by\n\\begin{equation}\\label{eq:plasma beta profile}\n\\mean\\beta_{t\\varphi z;\\rho}\\eqdef\n  \\int dS\\,\\rho\\frac{2p}{b_\\mu b^\\mu}\\bigg\/\\int dS\\,\\rho,\n\\end{equation}\nwhere the hypersurface element is\n\\begin{equation}\\label{eq:hypersurface element}\ndS\\eqdef(-g)^{1\/2}\\,dt\\,d(\\log\\lambda)\\,d\\phi\\,dz\n  \\mathop\\delta\\biggl(\\frac\\lambda{1+e\\cos\\phi}-R\\biggr).\n\\end{equation}\nTemporal and spatial averaging smooths out turbulent fluctuations. Temporal\naveraging is performed over two intervals, each lasting one-third of the\nsimulation duration; comparison between the intervals gives us an idea how\nclose the \\ac{MRI} is to saturation. Spatial averaging is limited to the\ncylindrical shell of radius $R$ picked out by the delta-function. The results\nare plotted in the top half of \\cref{fig:magnetic field profile}, and the\nlegend lists the intervals of temporal averaging. The relatively small\ndifference between the two intervals at radii $100\\lesssim R\\lesssim200$\nsuggests that steady state is achieved to some degree at those radii, despite\nthe relatively short simulation duration. We also reach similar conclusions as\nwe did with \\cref{fig:magnetic field map}: The plasma beta is quite uniform\nover the disk, it is not significantly modified by the introduction of\neccentricity, but it is one to two orders of magnitude lower in vertical-field\ndisks than in dipolar-field disks.\n\n\\begin{figure}\n\\includegraphics{magprof}\n\\caption{Mass-weighted radial profiles of the plasma beta in the top half and\nMaxwell-only alpha parameter in the bottom half, as defined in \\cref{eq:plasma\nbeta profile,eq:alpha parameter profile}, respectively. The profiles are\ntime-averaged over the two intervals in the legend. The two rows in the lower\nhalf have different vertical scales. The magnetic field strength varies weakly\nwith eccentricity, but is much stronger in vertical-field disks than in\ndipolar-field disks.}\n\\label{fig:magnetic field profile}\n\\end{figure}\n\nThe alpha parameter is conventionally taken to be the sum of Reynolds and\nMaxwell stresses divided by the gas pressure. However, it is difficult to\ndetermine the mean flow and departures from it in a disk whose inner and outer\nparts evolve differently in eccentricity and orientation. We therefore consider\nonly the Maxwell, not Reynolds, contribution to the alpha parameter, working\nunder the assumption that the Maxwell stress dominates the total, as is\nuniformly the case for circular disks \\citep[e.g.,][]{1995ApJ...440..742H}. The\nMaxwell stress is defined in terms of the projected magnetic field $b^\\mu$,\nsimilar to the stress--energy tensor. To make the stress more physically\ninterpretable, we measure $b^\\mu$ in a local orthonormal basis whose basis\nvectors are those of cylindrical coordinates, but with lengths normalized to\nunity:\n\\begin{align}\n\\label{eq:physical magnetic field 1}\nb^{\\hat R}\n  &= R\\biggl[b^\\lambda+b^\\phi\\frac{e\\sin\\phi}{1+e\\cos\\phi}\\biggr], \\\\\n\\label{eq:physical magnetic field 2}\nb^{\\hat\\varphi}\n  &= Rb^\\phi, \\\\\n\\label{eq:physical magnetic field 3}\nb^{\\hat z}\n  &= b^z.\n\\end{align}\nWe then define the Maxwell stress to be $b^{\\hat\\varphi}b^{\\hat R}$. A factor\nof $R$ is attached to $b^\\lambda$ in \\cref{eq:physical magnetic field 1}\nbecause our orbital coordinates use $\\log\\lambda$, not $\\lambda$.\nOrthonormality guarantees that\n\\begin{equation}\ng_{\\lambda\\lambda}b^\\lambda b^\\lambda+2g_{\\lambda\\phi}b^\\lambda b^\\phi\n  +g_{\\phi\\phi}b^\\phi b^\\phi=\n  b^{\\hat R}b^{\\hat R}+b^{\\hat\\varphi}b^{\\hat\\varphi}.\n\\end{equation}\n\nThe bottom half of \\cref{fig:magnetic field map} depicts the mass-weighted\nvertical average of the Maxwell-only alpha parameter at the end of the\nsimulations:\n\\begin{equation}\\label{eq:alpha parameter map}\n\\mean{\\alpha_\\su m}_{z;p}\\eqdef\n  -\\int dz\\,b^{\\hat\\varphi}b^{\\hat R}\\bigg\/\\int dz\\,p.\n\\end{equation}\nThe alpha parameter is positive almost everywhere in circular disks, as\nrequired for outward angular momentum transport. By contrast, the alpha\nparameter is uniformly positive in the lower half of the disk where material\nfalls to pericenter, but it is consistently \\textit{negative} in certain\nsectors of the upper half where material flies out to apocenter. In the\nvertical-field disk, positive sectors occupy significantly more area than\nnegative sectors. The magnitude of the alpha parameter, whether positive or\nnegative, varies from \\numrange{\\sim0.2}{5}; however, if we construct\narea-weighted histograms of the magnitude separately for positive and negative\nsectors, we find that the positive histogram is shifted by a factor of\n\\num{\\sim2} toward larger values relative to its negative counterpart. In the\ndipolar-field disk, the total area of positive sectors is only slightly larger\nthan that of negative sectors. In addition, the magnitude of the alpha\nparameter has a narrower distribution, \\numrange{\\sim0.3}{3}; the positive\nhistogram is again displaced by a factor of \\num{\\sim2} compared to the\nnegative one. We shall speculate about why the alpha parameter switches sign in\n\\cref{sec:stresses}.\n\nThe net effect of angular momentum transport is revealed by integrating the\nMaxwell stress over an orbit. When doing so by eye on the basis of\n\\cref{fig:magnetic field map}, it is important to take into account the\nrelative areas and alpha-parameter ranges of the positive and negative sectors.\nMore quantitatively, we construct in \\cref{fig:magnetic field profile}\nmass-weighted radial profiles of the alpha parameter at different times using\nthe prescription\n\\begin{equation}\\label{eq:alpha parameter profile}\n\\mean{\\alpha_\\su m}_{t\\varphi z;p}\\eqdef\n  -\\int dS\\,b^{\\hat\\varphi}b^{\\hat R}\\bigg\/\\int dS\\,p,\n\\end{equation}\nwith $dS$ the same hypersurface element from \\cref{eq:hypersurface element}.\nThe alpha parameter is comparable in circular and eccentric disks of the same\nmagnetic topology. In vertical-field disks, the alpha parameter is\n\\numrange{\\sim0.5}{1}; in dipolar-field disks, it is still positive, but only\n\\numrange{\\sim0.05}{0.15}. Stronger stress for vertical than dipolar magnetic\nfield agrees with previous simulations of circular disks\n\\citep[e.g.,][]{1995ApJ...440..742H, 1996ApJ...464..690H, 2004ApJ...605..321S,\n2013ApJ...767...30B}.\n\n\\subsection{Specific angular momentum and binding energy}\n\\label{sec:angular momentum and binding energy}\n\nTo investigate the effect of internal stresses, we examine how the specific\nangular momentum squared $L^2$ and binding energy $E_\\su b=1-E$ evolve. The\nmass-weighted vertical averages of the two quantities at the end of the\nsimulations are\n\\begin{align}\n\\label{eq:angular momentum squared map}\n\\mean{L^2}_{z;\\rho} &\\eqdef\n  \\int dz\\,\\rho(Ru^{\\hat\\varphi})^2\\bigg\/\\int dz\\,\\rho, \\\\\n\\label{eq:binding energy map}\n\\mean{E_\\su b}_{z;\\rho} &\\eqdef\n  \\int dz\\,\\rho(1+u_t)\\bigg\/\\int dz\\,\\rho.\n\\end{align}\nHere $u^{\\hat\\varphi}=Ru^\\phi$ is the velocity measured in the local\northonormal cylindrical basis, defined analogously to $b^{\\hat\\varphi}$ in\n\\cref{eq:physical magnetic field 2}, and the covariant velocity component\n$u_t=-E$ is a conserved quantity of our time-independent metric.\n\n\\Cref{fig:angular momentum squared and binding energy} displays the specific\nangular momentum squared and binding energy, normalized to their initial values\nat the inner edge. Our focus is on the eccentric disks. Because of pressure\ngradients, even the inner edge of the unmagnetized disk experiences a reduction\nin specific angular momentum squared by \\SI{\\sim25}{\\percent} from its initial\nvalue. The decrease is greater in magnetized disks where the Maxwell stress\nalso contributes: the dipolar-field disk reports a drop by\n\\SI{\\sim33}{\\percent}, and the vertical-field disk, which has a smaller plasma\nbeta and larger alpha parameter than the dipolar-field disk, records a\nsuppression by \\SI{\\sim50}{\\percent}. It should be noted that the range of\nspecific angular momentum is constrained by the geometry of the simulation\ndomain: Once material loses enough angular momentum that its pericenter recedes\ninside the inner boundary, it leaves the simulation domain.\n\nThe specific binding energy at the inner edge changes by rather less. It is\nalmost identical to its initial value in the unmagnetized disk,\n\\SI{\\sim30}{\\percent} higher in the vertical-field disk, and\n\\SI{\\sim10}{\\percent} \\textit{lower} in the dipolar-field disk. The smaller\nfractional changes suggest that the torques transporting angular momentum\noutward, regardless of whether they are hydrodynamic or magnetic in nature,\noccur preferentially near apocenter where the attendant work done is smaller.\n\n\\begin{figure*}\n\\includegraphics{angmenrg}\n\\caption{Mass-weighted vertical averages of the specific angular momentum\nsquared in the left half and specific binding energy in the right half, as\ndefined in \\cref{eq:angular momentum squared map,eq:binding energy map},\nrespectively. The top row shows two initial disks with different\neccentricities. The panels under each top-row panel are the outcomes of\nimposing various magnetic topologies on an initial disk and evolving it for 15\norbits. In all panels, the inner edge is arbitrarily defined to be where the\ndensity is $0.1\\,\\rho_*$, with $\\rho_*$ the fiducial density from\n\\cref{eq:initial density}, and regions less dense than that are left blank.\nFurthermore, the specific angular momentum squared or binding energy in each\npanel is normalized by its value at the inner edge of the corresponding initial\ndisk, which is why the inner edge appears yellow in the top row. The boundaries\nof the simulation domain are traced by thin ellipses. The fractional changes in\nthe specific angular momentum squared and binding energy together determine the\nchange in eccentricity.}\n\\label{fig:angular momentum squared and binding energy}\n\\end{figure*}\n\n\\subsection{Quality factors}\n\nWe close this section by examining how well our magnetized disks resolve the\n\\ac{MRI}. The figures of merit for circular disks are the quality factors,\ndefined as the ratio of the characteristic wavelength of the \\ac{MRI} to cell\nsizes in different directions \\citep{2011ApJ...738...84H}. We generalize their\nmass-weighted vertical averages to eccentric disks as\n\\begin{align}\n\\label{eq:quality 3}\n\\mean{Q_z}_{z;\\rho} &\\eqdef \\int dz\\,\\rho\\frac{\\abs{b^{\\hat z}}T}\n  {\\rho^{1\/2}\\Delta z}\\bigg\/\\int dz\\,\\rho, \\\\\n\\label{eq:quality 2}\n\\mean{Q_\\phi}_{z;\\rho} &\\eqdef \\int dz\\,\\rho\\frac{\\abs{b^{\\hat\\varphi}}T}\n  {\\rho^{1\/2}R\\Delta\\phi}\\bigg\/\\int dz\\,\\rho,\n\\end{align}\nwhere $T$ is the orbital period from \\cref{eq:orbital period}, and $\\Delta z$\nand $\\Delta\\phi$ are the cell sizes in the $z$- and\n$\\phi$\\nobreakdash-directions, respectively. In keeping with work on circular\ndisks, we base our quality factors on the physical, cylindrical components\n$b^{\\hat\\mu}$ of the projected magnetic field $b^\\mu$, given by\n\\cref{eq:physical magnetic field 2,eq:physical magnetic field 3}.\n\n\\Cref{fig:quality} showcases the quality factors at the end of the simulations.\nFor ease of comparison with the criteria that indicate adequate resolution for\ncircular disks, to wit, $\\mean{Q_z}\\gtrsim15$ and $\\mean{Q_\\phi}\\gtrsim20$\n\\citep{2013ApJ...772..102H}, the color scales of the figure are centered on\nthese values. In terms of $\\mean{Q_z}$, the vertical-field disks are extremely\nwell-resolved everywhere, but the same is true for the dipolar-field disks only\nfor a limited range of semilatera recta. In terms of $\\mean{Q_\\phi}$, all disks\nare marginally under-resolved. Interestingly, $\\mean{Q_\\phi}$ is noticeably\nhigher in the pericentric half of the disks; this is because the strong orbital\nshear near pericenter amplifies $b^{\\hat\\varphi}$, not $b^{\\hat z}$.\n\n\\begin{figure}\n\\includegraphics{quality}\n\\caption{Mass-weighted vertical averages of the vertical quality factor in the\ntop half and azimuthal quality factor in the bottom half, as defined in\n\\cref{eq:quality 3,eq:quality 2}, respectively. The panels are the outcomes of\nimposing various magnetic topologies on an initial disk and evolving it for 15\norbits. The boundaries of the simulation domain are traced by thin ellipses.}\n\\label{fig:quality}\n\\end{figure}\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\\subsection{Stresses in circular and eccentric disks}\n\\label{sec:stresses}\n\nOur earlier investigation into the linear stage of the \\ac{MRI} found that the\n\\ac{MRI} grows in both circular disks and eccentric disks; the growth rate in\neccentric disks is about half that in circular disks\n\\citep{2018ApJ...856...12C}. The present simulations suggest that the nonlinear\nstage of the \\ac{MRI} is also not so different in circular and eccentric disks,\nin that it saturates to comparable levels of plasma beta and alpha parameter in\nthe two kinds of disks. Thus, the \\ac{MRI} functions in much the same capacity\nin eccentric disks as in circular disks, mediating outward angular momentum\ntransport.\n\nThat being said, the \\ac{MRI} in eccentric disks exhibits two intriguing\nfeatures not witnessed in circular disks. The first one is the sign flip in the\nmagnetic-only alpha parameter. In circular disks, the alpha parameter is\npositive almost everywhere; by contrast, in eccentric disks, the alpha\nparameter can be consistently negative in some sectors of the disk even though\nthe azimuthal integral of the Maxwell stress remains positive.\n\nTo understand the basic principle underlying this surprising behavior, we break\ntemporarily from the general-relativistic treatment used in the rest of the\narticle and work in the Newtonian limit. We denote the Newtonian velocity and\nmagnetic field by $\\vec{\\tilde v}$ and $\\vec{\\tilde B}$, respectively, and\ntheir components measured against the local orthonormal cylindrical basis by\n$\\tilde v_i$ and $\\tilde B_i$. We conflate $\\tilde B^i$ with $b^{\\hat i}$\nelsewhere in the text, so the Maxwell stress is simply $-\\tilde B_R\\tilde\nB_\\varphi$. Our starting point is the induction equation:\n\\begin{equation}\n\\pd{\\vec{\\tilde B}}t=\\curl(\\vec{\\tilde v}\\cross\\vec{\\tilde B}).\n\\end{equation}\nTo track the time-evolution of the magnetic field at a point comoving with the\nflow, we consider\n\\begin{equation}\n\\od{\\vec{\\tilde B}}t\n  =\\pd{\\vec{\\tilde B}}t+(\\vec{\\tilde v}\\cdot\\grad)\\vec{\\tilde B}\n  =(\\vec{\\tilde B}\\cdot\\grad)\\vec{\\tilde v}\n  -\\vec{\\tilde B}(\\divg\\vec{\\tilde v}).\n\\end{equation}\nWe set $\\divg\\vec{\\tilde v}=0$ because the \\ac{MRI} is largely incompressible,\nand we take $\\pds{}z=0$ because we ignore vertical gravity in our simulations.\nConsequently,\n\\begin{equation}\\label{eq:Maxwell stress evolution}\n\\od{}t(-\\tilde B_R\\tilde B_\\varphi)=\n  -\\tilde B_R^2\\pd{\\tilde v_\\varphi}R\n  -\\tilde B_\\varphi^2\\biggl(\\frac1R\\pd{\\tilde v_R}{\\varphi}\\biggr)\n  -(-\\tilde B_R\\tilde B_\\varphi)\\frac{\\tilde v_R}R.\n\\end{equation}\n\nIn circular disks, the time- and azimuth-averaged $\\pds{\\tilde v_\\varphi}R$ is\nnegative, the averaged $(1\/R)(\\pds{\\tilde v_R}\\varphi)$ is zero, and the\naveraged $\\tilde v_R$ is also negative, albeit with a very small magnitude.\nBecause $-\\tilde B_R\\tilde B_\\varphi>0$ nearly everywhere, all terms on the\nright-hand side of \\cref{eq:Maxwell stress evolution} are therefore on average\nzero or positive, so $-\\tilde B_R\\tilde B_\\varphi$ grows over time until\nlimited by dissipation.\n\nIn eccentric disks, the time-averaged $\\pds{\\tilde v_\\varphi}R$ is also\nnegative everywhere. However, the averaged $(1\/R)(\\pds{\\tilde v_R}\\varphi)$ is\nno longer zero: it changes from negative to positive shortly before pericenter\nand back after pericenter. The averaged $\\tilde v_R$ also changes sign, from\nnegative to positive at pericenter and back at apocenter. The second and third\nterms therefore have azimuth-dependent signs. When orbits are significantly\neccentric, the averaged $(1\/R)(\\pds{\\tilde v_R}\\varphi)$ is comparable in\nmagnitude to the averaged $\\pds{\\tilde v_\\varphi}R$, and the averaged $\\tilde\nv_R$ is comparable to the averaged $\\tilde v_\\varphi$, so all three terms can\nbe important.\n\nConsider material starting from apocenter with positive Maxwell stress, that\nis, $-\\tilde B_R\\tilde B_\\varphi>0$. Initially all three terms work together to\nmake $-\\tilde B_R\\tilde B_\\varphi$ more positive. As the material swings toward\npericenter and its velocity becomes more azimuthal than radial,\n$(1\/R)(\\pds{\\tilde v_R}\\varphi)$ turns positive while $\\abs{\\tilde v_R}$ drops,\nflipping the sign of the second term and reducing the magnitude of the third\nterm. If $\\abs{\\tilde B_R}\\lesssim\\abs{\\tilde B_\\varphi}$, which is marginally\nsatisfied at these azimuths, the second term can dominate, pulling $-\\tilde\nB_R\\tilde B_\\varphi$ toward negative values. For some distance past pericenter,\nthe second term continues to be negative; at the same time, $\\tilde v_R>0$ and\n$\\ods{\\tilde v_R}t>0$, so the third term now tends to reduce the magnitude of\n$-\\tilde B_R\\tilde B_\\varphi$ at an increasing rate regardless of its sign. The\nsecond and third terms combined make $-\\tilde B_R\\tilde B_\\varphi<0$ at some\npoint near pericenter. Further beyond pericenter, the second term changes back\nto positive and the third term decreases in magnitude. By the time the material\nreturns to apocenter, the first and second terms have restored $-\\tilde\nB_R\\tilde B_\\varphi$ to positive, transporting angular momentum outward.\n\nBoth magnetic topologies share these qualitative features, but they differ in\nthe quantitative details. The dipolar-field topology yields a very close\nalignment between the local directions of velocity and magnetic field, whereas\nthe vertical-field topology produces merely a correlation between the two\ndirections that has much more scatter. In addition, as we can see in\n\\cref{fig:magnetic field map}, the sectors of the dipolar-field disk with\n$-\\tilde B_R\\tilde B_\\phi<0$ coincide with the region where $\\tilde v_R>0$,\nwhereas the vertical-field disk sectors with $-\\tilde B_R\\tilde B_\\phi<0$ take\nup only a small part of the $\\tilde v_R>0$ region, and their boundary is much\nmore irregular. We speculate that both of these contrasts are due to the larger\namplitude of turbulence the \\ac{MRI} drives in the presence of a vertical\nmagnetic field, an effect well-documented in circular disks\n\\citep[e.g.,][]{1995ApJ...440..742H, 1996ApJ...464..690H, 2004ApJ...605..321S,\n2013ApJ...767...30B}. Because the third term of \\cref{eq:Maxwell stress\nevolution} is sensitive to correlations between fluctuations in velocity and\nmagnetic field, large-amplitude turbulence may change the stress evolution.\nThis supposition is corroborated by the fact that the vertical root-mean-square\nof $v_z$, a good proxy for the magnitude of turbulent fluctuations in\nunstratified simulations, in the vertical-field disk is \\num{\\sim4} times that\nin the dipolar-field disk.\n\nThe other curiosity is that, whereas \\ac{MHD} stresses in circular disks\ntransport angular momentum and energy at such rates as to keep shrinking orbits\ncircular, the two rates can be independent of each other in eccentric disks.\nCircular disks have $E_\\su bL^2\\approx\\tfrac12$ at all radii, according to\n\\cref{eq:Newtonian eccentricity}; by contrast, $E_\\su bL^2$ varies spatially\nover our eccentric disks because $E_\\su b$ and $L^2$ change by different\nfractional amounts, and the mismatch dictates how the eccentricity evolves. The\ninner edge of the vertical-field disk is the least eccentric of all three\neccentric disks because the fractional changes in $E_\\su b$ and $L^2$ are\nnearly equal and opposite; conversely, the inner edge of the dipolar-field disk\nis the most eccentric because the two changes are in the same direction.\n\n\\subsection{Dynamics and energy dissipation at the inner edge}\n\\label{sec:inner edge}\n\nAs demonstrated in \\cref{sec:angular momentum and binding energy}, \\ac{MHD}\nstresses in eccentric disks are typically more effective at moving angular\nmomentum than energy. Because angular momentum constrains the size of the inner\nedge, and because the binding energy at the inner edge determines the amount of\nenergy available for radiation, the different transport rates of these two\nquantities could have observable effects on eccentric disks.\n\nTo explore these effects, let us first consider how disks behave around a star.\nFor a circular disk, the inner edge is located at the larger of the stellar or\nAlfv\\'en radius. The total energy a given amount of material dissipates in the\ndisk is its binding energy on a circular orbit at that radius. Additional\ndissipation happens in the boundary layer at the inner edge as the material\ncomes into corotation with the star or its magnetosphere. If the star-regulated\nrotation speed in this boundary layer is small compared to the orbital speed,\nthe additional dissipation is equal to the dissipation that took place in the\ndisk itself.\n\nFor an eccentric disk, the stellar or Alfv\\'en radius likewise defines the\ninner edge, but in this case, it is through a match to the pericenter of the\ninner edge. Because the corresponding semimajor axis is larger than the stellar\nor Alfv\\'en radius, material dissipates less energy in the disk proper. The\ntotal energy dissipated, however, is exactly the same as in the circular case\nbecause the eccentric orbit of the material ultimately transforms into a\ncircular one at the stellar or Alfv\\'en radius. What makes eccentric disks\ndifferent is that dissipation in the boundary layer accounts for a larger\nfraction of the total. Energy can be dissipated there in a variety of ways,\ndepending on how the inner edge interacts with the star or its magnetosphere.\nIn one extreme, the material immediately comes into corotation. The fast-moving\nmaterial dissipates large amounts of energy in a small region, potentially\nproducing hard radiation. In the other extreme, the material loses a tiny part\nof its kinetic energy each time it grazes past the star or its magnetosphere,\nand the material migrates inward gradually on orbits of decreasing semimajor\naxes and eccentricities. Dissipation in this case could be more spatially\ndistributed, happening both in the grinding encounters and on the outer parts\nof the circularizing orbits; if so, the eccentric disk may resemble a circular\none in appearance.\n\nWe now turn to disks around black holes. As shown in \\cref{fig:effective\npotential}, material on eccentric orbits can reach the event horizon even if it\nhas more angular momentum than that of \\iac{ISCO} orbit provided that it has\nenough energy to overcome the centrifugal barrier inside the \\ac{ISCO} radius;\nequivalently, it can do so if its eccentricity is high enough. Moreover, when\nthe angular momentum is at least that of \\iac{ISCO} orbit, the peak of the\ncentrifugal barrier is always above the energy of \\iac{ISCO} orbit.\nConsequently, the amount of energy available for such material to radiate is\nalways smaller than the binding energy at the \\ac{ISCO}, the conventional\nestimate for radiative efficiency in circular disks \\citep{1973blho.conf..343N,\n1974ApJ...191..499P}. The energy available to eccentric material is even\nsmaller when we take into account extra energy extraction by the Maxwell stress\nin circular disks \\citep{1974ApJ...191..507T, 1999ApJ...515L..73K,\n1999ApJ...522L..57G, 2002ApJ...573..754K, 2009ApJ...692..411N,\n2016ApJ...819...48S, 2016MNRAS.462..636A, 2021ApJ...922..270K}. Unlike\neccentric disks around stars, the energy retained by the plunging material\ncannot be recovered for radiation in some boundary layer.\n\nThe diminution of the radiative efficiency of eccentric disks around black\nholes could be particularly relevant to \\acp{TDE}, in which the energy radiated\nis considerably below that expected from the accretion of a reasonable fraction\nof a stellar mass of material onto a black hole through a conventional circular\ndisk \\citep{2015ApJ...806..164P}. \\Citet{2017MNRAS.467.1426S} suggested that\nthis \\textquote{inverse energy problem} could be resolved by internal stresses\nthat transport angular momentum preferentially and cause the most bound debris\nto plunge; our results provide quantitative support to that qualitative\nargument.\n\n\\subsection{Effects of vertical gravity}\n\nOur simulations of the nonlinear development of the \\ac{MRI} in eccentric disks\nignored vertical gravity. We did so for two reasons. The first is to isolate\nthe effects of orbital-plane motions, the chief driver of the \\ac{MRI}, when\nthey are azimuthally modulated in eccentric disks. The second is to produce a\nbaseline for comparison with future simulations that do include vertical\ngravity.\n\nBecause vertical gravity is a form of tidal gravity, it is much weaker than\nradial gravity as long as the disk is thin. For this reason, the modifications\nit introduces are likely minor and unable to quench the \\ac{MRI}. Here we\nexamine what these modifications may be.\n\nOne effect of vertical gravity that is discernible in circular disks carries\nover to eccentric disks: it regulates the buoyant eviction of accumulated\nmagnetic flux that may determine the saturation level of the \\ac{MRI}\n\\citep[e.g.,][]{1992MNRAS.259..604T, 2010ApJ...713...52D, 2011MNRAS.416..361B,\n2011ApJ...732L..30H, 2015ApJ...809..118B, 2018ApJ...861...24H}.\n\nThe variation of vertical gravity around an eccentric orbit causes the disk to\nexpand in height as it travels from pericenter to apocenter and to collapse on\nthe way back. This \\textquote{breathing} can have large amplitudes even for\nmildly eccentric disks \\citep{2014MNRAS.445.2621O, 2021MNRAS.500.4110L}. For\nthe thickest and most eccentric disks, breathing can even become non-adiabatic:\nextreme compression can create shocks that eject material vertically\n\\citep{2021ApJ...920..130R}.\n\nVertical gravity can also influence the eccentric \\ac{MRI} more subtly, without\nshocks. Vertical motion opens up more avenues of energy exchange between the\norbit and the magnetic field. Compression and expansion can alter the Alfv\\'en\nspeed, hence the critical wavelength for stability against the \\ac{MRI}, and it\ncan also modulate the wavelengths of advected perturbations. All these\nvariations are periodic, raising the possibility of instability through\nparametric resonance \\citep{2005A&A...432..743P}.\n\nIt bears repeating that the changes due to vertical gravity should be small for\nthin, moderately eccentric disks, so we expect our results here to remain\ngenerally valid.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn circular disks, the \\ac{MRI} stirs up correlated \\ac{MHD} turbulence,\nturbulent stresses transport angular momentum outward, and the disk accretes.\nOur simulations demonstrate that much the same process operates in eccentric\ndisks, and with comparable efficiency. Like circular disks, the quantitative\nlevel of Maxwell stress achieved in eccentric disks depends on the magnetic\ntopology, but eccentric and circular disks with the same magnetic topology\nreach comparable levels of plasma beta and alpha parameter.\n\nAlthough the mass-weighted, disk-averaged Maxwell stress in an eccentric disk\nproduces an outward angular momentum flux similar in magnitude to that in a\ncircular disk with the same magnetic topology, the Maxwell stress in an\neccentric disk can cause \\textit{inward} angular momentum transport in certain\ndisk sectors. This behavior is seen only in eccentric disks likely because the\nradial velocity and the azimuthal gradient of the radial velocity are both\nnonzero and have signs that vary over azimuth.\n\nBy removing proportionately more angular momentum than energy from the inner\nparts of the disk, \\ac{MHD} stresses can promote accretion of highly eccentric\nmaterial. This material has higher energy than a circular orbit with the same\nangular momentum, so the radiative efficiency of a disk around a black hole can\nbe suppressed relative to a circular disk. These findings corroborate earlier\nsuggestions about how the power output from eccentric disks may explain why the\ntotal observed radiated energy in many \\acp{TDE} is one to two orders of\nmagnitude lower than expected from circular accretion of a stellar mass of\nmaterial \\citep{2017MNRAS.467.1426S}.\n\n\\begin{acknowledgments}\nThe authors thank Scott Noble for useful discussions. CHC and TP were supported\nby ERC Advanced Grant \\textquote{TReX}. CHC was additionally supported by NSF\ngrant AST-1908042. JHK was partially supported by NSF grants AST-1715032 and\nAST-2009260. The simulations were performed on the Rockfish cluster at the\nMaryland Advanced Research Computing Center (MARCC).\n\\end{acknowledgments}\n\n\\software{Athena++ \\citep{2016ApJS..225...22W, 2020ApJS..249....4S}, NumPy\n\\citep{2020Natur.585..357H}, SymPy \\citep{10.7717\/peerj-cs.103}, Matplotlib\n\\citep{2007CSE.....9...90H}}\n\n\\begin{appendices}\n\n\\section{Metric components and Christoffel symbols}\n\\label{sec:metric}\n\nIn cylindrical coordinates $(t,R,\\varphi,z)$, the nonzero metric components are\n\\begin{align}\ng_{tt} &= -\\symsf P, \\\\\ng_{RR} &= 1, \\\\\ng_{\\varphi\\varphi} &= R^2, \\\\\ng_{zz} &= 1, \\\\\ng^{tt} &= -1\/\\symsf P, \\\\\ng^{RR} &= 1, \\\\\ng^{\\varphi\\varphi} &= 1\/R^2, \\\\\ng^{zz} &= 1,\n\\end{align}\nthe metric determinant is\n\\begin{equation}\ng=-R^2\\symsf P,\n\\end{equation}\nand the nonzero Christoffel symbols of the second kind are\n\\begin{align}\n\\Gamma^t_{tR}=\\Gamma^t_{Rt}\n  &= \\partial_R\\Phi\/\\symsf P, \\\\\n\\Gamma^t_{tz}=\\Gamma^t_{zt}\n  &= \\partial_z\\Phi\/\\symsf P, \\\\\n\\Gamma^R_{tt}\n  &= \\partial_R\\Phi, \\\\\n\\Gamma^R_{\\varphi\\varphi}\n  &= -R, \\\\\n\\Gamma^\\varphi_{R\\varphi}=\\Gamma^\\varphi_{\\varphi R}\n  &= 1\/R, \\\\\n\\Gamma^z_{tt}\n  &= \\partial_z\\Phi,\n\\end{align}\nwhere\n\\begin{equation}\n\\symsf P\\eqdef1+2\\Phi(R,z).\n\\end{equation}\n\nPerforming a coordinate transformation to orbital coordinates\n$(t,\\log\\lambda,\\phi,z)$, we find that the nonzero metric components are\n\\begin{align}\ng_{tt} &= -\\symsf P, \\\\\ng_{\\lambda\\lambda} &= R^2, \\\\\ng_{\\lambda\\phi}=g_{\\phi\\lambda} &= R^2\\symsf Q, \\\\\ng_{\\phi\\phi} &= R^2(1+\\symsf Q^2), \\\\\ng_{zz} &= 1, \\\\\ng^{tt} &= -1\/\\symsf P, \\\\\ng^{\\lambda\\lambda} &= (1+\\symsf Q^2)\/R^2, \\\\\ng^{\\lambda\\phi}=g^{\\phi\\lambda} &= -\\symsf Q\/R^2, \\\\\ng^{\\phi\\phi} &= 1\/R^2, \\\\\ng^{zz} &= 1,\n\\end{align}\nthe metric determinant is\n\\begin{equation}\ng=-R^4\\symsf P,\n\\end{equation}\nand the nonzero Christoffel symbols of the second kind are\n\\begin{align}\n\\Gamma^t_{t\\lambda}=\\Gamma^t_{\\lambda t}\n  &= (\\partial_R\\Phi)R\/\\symsf P, \\\\\n\\Gamma^t_{t\\phi}=\\Gamma^t_{\\phi t}\n  &= (\\partial_R\\Phi)R\\symsf Q\/\\symsf P, \\\\\n\\Gamma^t_{tz}=\\Gamma^t_{zt}\n  &= \\partial_z\\Phi\/\\symsf P, \\\\\n\\Gamma^\\lambda_{tt}\n  &= \\partial_R\\Phi\/R, \\\\\n\\Gamma^\\lambda_{\\lambda\\lambda}\n  &= 1, \\\\\n\\Gamma^\\lambda_{\\phi\\phi}\n  &= -R\/\\lambda, \\\\\n\\Gamma^\\phi_{\\lambda\\phi}=\\Gamma^\\phi_{\\phi\\lambda}\n  &= 1, \\\\\n\\Gamma^\\phi_{\\phi\\phi}\n  &= 2\\symsf Q, \\\\\n\\Gamma^z_{tt}\n  &= \\partial_z\\Phi,\n\\end{align}\nwhere\n\\begin{equation}\n\\symsf Q\\eqdef e\\sin\\phi\/(1+e\\cos\\phi).\n\\end{equation}\n\n\\section{Precession-free gravitational potential}\n\\label{sec:potential derivation}\n\nSchwarzschild spacetime in general relativity produces prograde apsidal\nprecession, while an extended gravitating mass in Newtonian mechanics causes\nretrograde apsidal precession. It is natural to ask if one can be made to\ncancel the other exactly.\n\nWe work in cylindrical coordinates $(t,R,\\varphi,z)$. The metric in the\nweak-gravity limit is given in \\cref{sec:metric}, but here we assume $\\Phi$ is\nnot yet decided. The velocity of a particle is $u^\\mu$; without loss of\ngenerality we restrict the particle to the midplane, so $z=0$ and $u^z=0$.\nBecause $t$ and $\\varphi$ are ignorable in the metric, we immediately have two\nintegrals of motion\n\\begin{align}\n\\label{eq:conserved energy}\nE &= u^t\\symsf P, \\\\\n\\label{eq:conserved angular momentum}\nL &= R^2u^\\varphi.\n\\end{align}\nVelocity normalization requires\n\\begin{equation}\\label{eq:velocity normalization}\n(u^R)^2=-1+E^2\/\\symsf P-L^2\/R^2.\n\\end{equation}\nDividing the equation by $R^4(u^\\varphi)^2=L^2$, changing variable to\n$\\xi\\eqdef1\/R$, and differentiating with respect to $\\xi$ yields\n\\begin{equation}\\label{eq:inverse radius equation}\n\\odd\\xi\\varphi+\\xi=-\\frac{E^2}{2L^2\\symsf P^2}\\od{\\symsf P}\\xi.\n\\end{equation}\nClosed eccentric orbits exist for all values of $E$ and $L$ if and only if\n\\begin{equation}\n-\\frac1{2\\symsf P^2}\\od{\\symsf P}\\xi=C_1,\n\\end{equation}\nwhich has the solution\n\\begin{equation}\n\\symsf P=1\/(2C_1\\xi+C_2),\n\\end{equation}\nwhere $C_1$ and $C_2$ are constants. We take $C_1=C_2=1$ so that $\\symsf\nP=1+2\\Phi\\approx1-2\\xi$ for $\\xi\\ll1$, as befitting point-mass gravity. The\npotential $\\Phi=-1\/(R+2)$ describes softened gravity, so there are no\ncoordinate or physical singularities.\n\nOur next step is to determine the velocity $u^\\mu$ at any point along an\neccentric orbit. The solution to \\cref{eq:inverse radius equation} is\n\\begin{equation}\nR=\\bar\\lambda\/[1+\\bar e\\cos(\\varphi+C_3)],\n\\end{equation}\nwhere $\\bar e$ and $C_3$ are constants and\n\\begin{equation}\\label{eq:semilatus rectum}\n\\bar\\lambda\\eqdef L^2\/E^2.\n\\end{equation}\nThe embellishments on $\\bar e$ and $\\bar\\lambda$ serve to distinguish them from\n$e$ and $\\lambda$ defining our orbital coordinates. \\Cref{eq:semilatus rectum}\ndescribes an ellipse of eccentricity $\\bar e$, semilatus rectum $\\bar\\lambda$,\nand semimajor axis $\\bar a=\\bar\\lambda\/(1-\\bar e^2)$. We pick $C_3=0$ so that\nthe pericenter of the ellipse is at $\\varphi=0$, and we fix $\\bar e$ by solving\n\\cref{eq:velocity normalization} at pericenter, where $u^R=0$:\n\\begin{equation}\\label{eq:eccentricity}\n\\bar e^2=1+(E^2-1)L^2\/E^4.\n\\end{equation}\nThis expression reduces to its Newtonian equivalent\n\\begin{equation}\\label{eq:Newtonian eccentricity}\n\\bar e^2\\approx1-2E_\\su bL^2\n\\end{equation}\nwhen $\\abs{E_\\su b}=\\abs{1-E}\\ll1$. Once we know $\\bar e$ and $\\bar\\lambda$ for\nan eccentric orbit, we can solve for $E$ and $L$ using \\cref{eq:semilatus\nrectum,eq:eccentricity}; the results are \\cref{eq:orbit energy,eq:orbit angular\nmomentum}. These can then be substituted into \\cref{eq:conserved\nenergy,eq:conserved angular momentum,eq:velocity normalization} to yield the\nvelocity $u^\\mu$ in cylindrical coordinates. The velocity $u^\\mu$ in orbital\ncoordinates, given in \\cref{eq:orbit velocity 0,eq:orbit velocity 2}, follows a\ncoordinate transformation.\n\nThe final missing piece is the analogue of Kepler's equation for our potential.\nWe introduce the eccentric anomaly $\\symcal E$, defined by\n\\begin{equation}\nR=\\bar a(1-\\bar e\\cos\\symcal E).\n\\end{equation}\nUsing a result from geometry,\n\\begin{equation}\n(1-\\bar e)^{1\/2}\\tan\\tfrac12\\varphi=\n  (1+\\bar e)^{1\/2}\\tan\\tfrac12\\symcal E,\n\\end{equation}\nwe can express $t$ along the orbit as a function of $\\varphi$:\n\\begin{equation}\\label{eq:time}\nt=\\int d\\symcal E\\,\\od\\varphi{\\symcal E}\\frac{u^t}{u^\\varphi}\n  =\\bar a^{3\/2}[(1+2\/\\bar a)\\symcal E-\\bar e\\sin\\symcal E].\n\\end{equation}\nThe integration constant is set to zero for simplicity. As $\\symcal E$\nincreases by $2\\pi$, $t$ also increases by an amount equal to the orbital\nperiod:\n\\begin{equation}\nT=2\\pi\\bar a^{3\/2}(1+2\/\\bar a).\n\\end{equation}\nMultiplying both sides of \\cref{eq:time} by $2\\pi\/T$ furnishes us with the\nanalogue of Kepler's equation:\n\\begin{equation}\n\\symcal M=\\symcal E-\\frac{\\bar e\\sin\\symcal E}{1+2\/\\bar a},\n\\end{equation}\nwhere $\\symcal M=2\\pi t\/T$ is the mean anomaly.\n\n\\end{appendices}\n\n\\ifapj","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Preliminaries}\\label{sec:intro}\n\nA {\\em polymatroid $(f,M)$} is a non-negative, monotone and submodular\nfunction $f$ defined on the collection of non-empty subsets of the finite\nset $M$. Here $M$ is the {\\em ground set}, and $f$ is the {\\em rank function}.\nThe polymatroid is {\\em integer} if all ranks are integer. An integer\npolymatroid is a {\\em matroid}, if the rank of singletons are either zero or\none. Matroids are combinatorial objects which generalize the properties of\nlinear dependence among a finite set of vectors. For an introduction to\nmatroids, see \\cite{oxley}; and about polymatroids consult \\cite{Lov,fmadhe}. The\nrank function $f$ can be identified with a $(2^{|M|}-1)$-dimensional real vector,\nwhere the indices are the non-empty subsets of $M$. \nIn this paper he {\\em distance} of two polymatroids $f$ and $g$ on the same ground\nset is measured as the usual Euclidean distance of the corresponding vectors, and is\ndenoted as $\\|f-g\\|$.\n\nFollowing the usual practice, ground sets and their subsets are denoted by\ncapital letters, their elements by lower case letters. The union sign\n$\\cup$ is frequently omitted as well as the curly brackets around singletons,\nthus $Aab$ denotes the set $A\\cup\\{a,b\\}$. For a function $f$\ndefined on the subsets of the finite set $M$ (such as the rank function of a\npolymatroid) the usual information-theoretical abbreviations are used. Here\n$I$, $J$, $K$ are disjoint subsets of the ground set:\n\\begin{align*}\n   f(I,J|K) &= f(IK)+f(JK)-f(IJK)-f(K),\\\\\n   f(I,J)   &= f(I,J|\\emptyset) = f(I)+f(J)-f(IJ)-f(\\emptyset),\\\\\n   f(I|K)   &= f(IK)-f(K).\n\\end{align*}\nWhen $f$ is a rank function, $f(\\emptyset)$ is considered to be zero.\nIn cases when the function $f$ is clear from the context, even $f$ is\nomitted. \nAdditionally, the {\\em Ingleton\nexpression} \\cite{ingleton} is abbreviated as\n$$\n   f[I,J,K,L] = -f(I,J)+f(I,J|K)+f(I,J|L)+f(K,L).\n$$\nObserve that it is symmetrical for swapping $I$ and\n$J$ as well as swapping $K$ and $L$. \n\nVectors corresponding to polymatroids on the ground set $M$ form the pointed polyhedral cone \n$\\Gamma_M$ \\cite{yeungbook}. \nIts facets are the hyperplanes determined by the\nbasic submodular inequalities $(i,j|K)\\ge 0$ with distinct $i,j\\in M{-}K$\nand $K\\subseteq M$ ($K$ can be empty), and the monotonicity\nrequirements $(i|M{-}i)\\ge 0$, see \\cite[Theorem 2]{fmadhe}. Much less is known\nabout the extremal rays of this cone. They have been computed for ground sets\nup to five elements \\cite{studeny-kocka}, without indicating any structural\nproperty.\n\n\\subsection{Entropic, linear, and modular polymatroids}\n\nAn important class of polymatroids describes the entropy structure of the\nmarginals of finitely many discrete random variables. Assume\n$\\{\\xi_i:i\\in M\\}$ is a collection of (jointly distributed) random\nvariables. For $A\\subseteq M$ let $\\mathbf{H}(\\xi_A)$ be the usual Shannon entropy \nof the marginal distribution $\\xi_A = \\{ \\xi_i: i\\in A\\}$. The function $f(A) =\n\\mathbf{H}(\\xi_A)$ is a polymatroid \\cite{fuji}. Such polymatroids are called {\\em\nentropic}, and the collection of entropic polymatroids is\n$\\Gamma^*_M\\subseteq \\Gamma_M$ \\cite{yeungbook}. The closure of $\\Gamma^*_M$ \n(in the usual\nEuclidean topology) is the collection of {\\em almost entropic} or\n{\\em aent} polymatroids.\nStudying\npolymatroids is motivated partly by the difficult task of understanding the\nentropic region as well as solving problems arising in secret sharing\n\\cite{padro1,seymour}, network coding \\cite{acy}, and other areas.\n\nAnother important subclass is the linear polymatroids. \n$(f,M)$ is {\\em linearly representable} if there is a vector\nspace $V$ over some finite field, linear subspaces $V_i\\subseteq V$ for\neach $i\\in M$, such that $f(A)$ is the dimension of the linear subspace\nspanned by the vectors in $\\bigcup\\{ V_i:i\\in A\\}$. Linearly\nrepresentable polymatroids are integer.\nA polymatroid is {\\em\nlinear} if it is in the conic hull of linearly representable polymatroids,\nnamely, it can be written as a non-negative linear combination of such\npolymatroids. Linear polymatroids are almost entropic, see\n\\cite{entreg,M.twocon,padro}.\n\nThe polymatroid $(f,M)$ is {\\em modular} if $f(I|J)=0$ for any two\ndisjoint non-empty subsets $I,J\\subset M$, or, equivalently, if \nfor all $A\\subseteq M$ we have\n$$\n    f(A) = {\\textstyle\\sum}\\, \\{ f(i)\\,:\\, i\\in A\\}.\n$$\nModular polymatroids are entropic and linear \\cite{fmadhe}.\n\nIt is well known that all polymatroids on two or three \nelements are linear, moreover a polymatroid $f$ on the four element set\n$abcd$ is linear if and only if it satisfies all six instances of the\nIngleton inequality::\n\\begin{align*}\n   f[a,b,c,d]\\ge 0, ~~~ f[a,c,b,d]\\ge 0, ~~~ f[a,d,b,c]\\ge 0,\\\\\n   f[b,c,a,d]\\ge 0, ~~~ f[b,d,a,c]\\ge 0, ~~~ f[c,d,a,b]\\ge 0,\n\\end{align*}\nsee \\cite{matus-studeny}.\nLinear polymatroids on a five element set can also be characterized by \nsome finite set of linear inequalities \\cite{dougherty5}. Polymatroids on\nground set of size at most five have the following {\\em simultaneous\napproximation property}, which will be used in Section \\ref{sec:sticky-2}.\n\\begin{proposition}\\label{prop:pm-approx}\nLet $|M|\\le 5$, and\nlet $f_1$ and $f_2$ be linear polymatroids on $M$.\nFor each large enough vector space $V$ and positive $\\varepsilon$ there is a\n$\\lambda>0$ and integer polymatroids $\\ell_1$ and $\\ell_2$ on $M$\nlinearly representable over $V$, such that $\\|f_i-\\lambda\\ell_i\\|<\\varepsilon$,\nadditionally $\\ell_1(A)=\\ell_2(A)$ whenever $f_1(A)=f_2(A)$ ($A\\subseteq\nM$).\n\\end{proposition}\n\\begin{proof}\nOn ground set $|M|\\le5$ linear polymatroids form a polyhedral cone.\nMoreover, for every large enough vector space $V$, extremal rays of this\ncone contain polymatroids linearly representable over $V$, see\n\\cite{dougherty5,matus-studeny}. On one hand, non-negative\nrational combinations of these polymatroids form a dense subset of\nlinear polymatroids. On the other, integer combinations of linearly\nrepresentable polymatroids over the same space are linearly\nrepresentable. From these facts the claim follows.\n\\end{proof}\n\n\n\\subsection{Amalgam and adhesive extension}\\label{subsec:adhesive}\n\nLet $M$, $X$, and $Y$ be disjoint sets. \nPolymatroids $f_X$ and $f_Y$ on the ground sets $M\\cup X$ and $M\\cup Y$,\nrespectively, with joint restriction on $M$,\nhave an {\\em amalgam}, or {\\em can be glued together}, if there is a\npolymatroid $f$ on $M\\cup X\\cup Y$ extending both $f_X$ and $f_Y$\n\\cite{oxley}. This\nextension is {\\em modular} if, in addition, $X$ and $Y$ are independent over\n$M$, that is, $f(X,Y|M)=0$. If $f_X$ and $f_Y$ have such a modular\nextension $f$, then $f_X$ and $f_Y$ are {\\em adhesive}, and $f$ is an {\\em\nadhesive extension}. Adhesive extensions were defined and studied by\nF.~Mat\\'u\\v s in \\cite{fmadhe}. The main observation is that restrictions of\nan almost entropic polymatroid are adhesive \\cite[Lemma 2]{fmadhe}. In this\npaper we investigate adhesive extensions on their own right.\n\nWhen speaking about amalgam, or adhesive extension, the polymatroids are\ntacitly assumed to have the same restriction on the intersection of their\nground sets.\n\n\\smallskip\n\nWe have defined the amalgam of $f_X$ and $f_Y$ as a {\\em polymatroid} extending\nboth $f_X$ and $f_Y$. The amalgam of two matroids is traditionally required to be a\nmatroid. It is an interesting problem to decide whether the two different\nnotions of amalgam coincide.\n\\begin{problem}\\label{problem:1}\\rm\nSuppose the matroids $f_X$ and $f_Y$ on $M\\cup X$ and $M\\cup Y$,\nrespectively, have a polymatroid amalgam on $M\\cup X\\cup Y$.\nIs it true that then they have a matroid amalgam as well?\n\\end{problem}\n\\noindent\nIt is easy to see that if the extension polymatroid is integer valued then\nit is a matroid; and if there is an extension at all, then there is one with\nrational values.\n\n\\smallskip\n\nWhether two matroids have an amalgam is a combinatorial question; the\nsame question about polymatroids is a {\\em geometrical} one. Polymatroids\n$f_X$ and $f_Y$ have an amalgam if and only if the point $(f_X,f_Y)$ (merged\nalong coordinates corresponding to subsets of $M$) is in the {\\em\ncoordinatewise projection} of the polymatroid cone $\\Gamma_{MXY}$ to the\nsubspace with coordinates $I\\subseteq MXY$ where $I\\subseteq MX$ or\n$I\\subseteq MY$.\nThe projection is a polyhedral cone whose bounding\nhyperplanes correspond to (homogeneous) linear inequalities on the projected\ncoordinates. Thus $f_X$ and $f_Y$ have an amalgam if and only if the\nvector $(f_X,f_Y)$ satisfies all of these inequalities. While\ntheoretically simple, in practice it is unclear how to calculate\nthe facets of the projection efficiently.\n\nThe same reasoning applies to the adhesive extensions. Such an extension\nsatisfies the additional constraint $f(X,Y|M)=0$, thus the modular extensions\nform a subcone of dimension one less: the intersection of\n$\\Gamma_{MXY}$ and the hyperplane $f(XM)+f(YM)-f(XYM)-f(Y)=0$. $f_X$ and\n$f_Y$ have an adhesive extension if an only if the point $(f_X,f_Y)$ is in\nthe projection of this restricted cone.\n\n\\smallskip\nThe polymatroid $h$ is {\\em sticky} if any two extensions\nof $h$ have an amalgam. Modular polymatroids are sticky, the proof in\n\\cite[Theorem 12.4.10]{oxley} works in the polymatroid case as well, see\nalso \\cite[Theorem 1]{fmadhe}. The ``sticky matroid conjecture'' asserts that all\nsticky matroids are modular \\cite{bonin}. The same conjecture is stated here\nfor polymatroids.\n\\begin{polymatroidconjecture}\nSticky polymatroids are modular.\n\\end{polymatroidconjecture}\nFactors of sticky polymatroids are sticky, and the collection of sticky\npolymatroids on a given ground set forms a closed cone, thus to settle the\nabove conjecture it is enough to consider polymatroidal extensions of a\nmatroid.\n Consequently, if the the answer to Problem \\ref{problem:1} is {\\em yes} and\nthe sticky matroid conjecture is true, then so is the\nsticky polymatroid conjecture.\n\n\\medskip\n\nTo state some of our results we need one more definitions.\nThe polymatroid $(h,M)$ is {\\em $k$-$\\ell$-sticky}, if any two of its extensions\n$(f_X,MX)$ and $(f_Y,MY)$ with $|X|\\le k$ and $|Y|\\le\\ell$ have an amalgam.\nA polymatroid is {\\em $k$-sticky}, if it is $k$-$k$-sticky.\nSticky polymatroids on small ground sets are discussed in Sections\n\\ref{sec:sticky-2} and \\ref{sec:sticky-3}.\n\n\n\\subsection{New polymatroids from old ones}\\label{subsec:constructions}\n\nEach polymatroid can be decomposed as a sum of a modular and a tight\npolymatroid as described in Lemmas \\ref{lemma:down} and \n\\ref{lemma:tightening}; it is\na generalization of \\cite[Lemma 2]{entreg}. Lemma \\ref{lemma:excess}\ndiscusses how one can extend a polymatroid adding a new element to the base set.\nThe method will be used in later sections to create several extensions.\n\n\\medskip\n\nFor a subset $A\\subset M$ define the function $\\r_A$ on (non-empty)\nsubsets of $M$ as follows:\n$$\n    \\r_A(I) = \\begin{cases} 1 & \\mbox{if $A\\cap I$ is not empty,}\\\\\n                            0 & \\mbox{otherwise}.\n              \\end{cases}\n$$\nClearly $(\\r_A,M)$ is a matroid. In Information Theory it corresponds \nto the atoms in the $I$-measure defined in \\cite{yeungbook}.\n\\begin{lemma}\\label{lemma:down}\nLet $(f,M)$ be a polymatroid and $A\\subset M$. Suppose the real number $\\lambda$ satisfies the\nfollowing conditions:\n$$\\begin{array}{r@{\\;\\le\\;}ll}\n\\lambda & f(x,y|B)    &\\mbox{for {\\em different} $x,y\\in A$ and all $B\\subseteq\nM{-}A$; and}\\\\[3pt]\n\\lambda & f(x|M{-}A)  &\\mbox{for every $x\\in A$.}\n\\end{array}$$\nThen $(f-\\lambda \\r_A,M)$ is a polymatroid.\n\\end{lemma}\n\\noindent\nObserve that if $A$ has a single member $a$, then the first condition\nholds vacuously, and the second condition simplifies to $\\lambda \\le\nf(a|M{-}a)$.\n\\begin{proof}\nThe claim clearly holds when $\\lambda\\le 0$, so assume $\\lambda>0$, and\nlet $f^*=f-\\lambda \\r_A$. If $I$ and $A$ are disjoint, then $f(I)-f^*(I)=0$,\nin the other cases this difference is $\\lambda$. One has to check the monotonicity\nfor the special case $f^*(Cx)-f^*(C)\\ge 0$, $Cx\\subseteq M$ only. This difference equals to\n$f(Cx)-f(C)$ except when $A$ and $C$ are disjoint and $x\\in A$. But then\n\\begin{align*}\n   f^*(Cx)-f^*(C)&=f(Cx)-f(C)-\\lambda \\\\\n               {}&=f(x|C)-\\lambda \\ge f(x|M{-}A)-\\lambda\\ge 0\n\\end{align*}\nby assumption.\n\nTo check submodularity, observe that $f^*(x,y|B)=f(x,y|B)$ except when $A$\nand $B$ are disjoint and both $x$ and $y$ are in $A$. In the latter case\n$f^*(x,y|B)=f(x,y|B)-\\lambda$, which is non-negative by the fist assumption.\n\\end{proof}\n\nLet $(f,M)$ be any polymatroid and $a\\in M$. Choosing $\\lambda$ to be the\nmaximal $f(a|M{-}a)$,\nthe polymatroid $f-\\lambda\\r_a$ is denoted by\n$f\\down a$, and called {\\em tightening of $f$ at} (or on) $a$. $f$ is\n{\\em tight at $a$} if $f=f\\down a$, that is, if $f(a|M{-}a)=0$.\nNote that\n$(f\\down a)\\down a = f\\down a$, thus $f\\down a$ is tight at $a$; moreover \n$(f\\down a)\\down b = (f\\down\nb)\\down a$. Thus one can define the {\\em tight part of $f$} at\n$A=\\{a_1,\\dots,a_k\\}$ as $f\\down a_1\\down\\cdots\\down a_k$. $f$ is {\\em tight\non $A$}, if $f=f\\down A$, and is {\\em tight} if $f=f\\down M$. The next\nlemma summarizes the properties of tightening used in this paper.\n\n\\begin{lemma}\\label{lemma:tightening}\nLet $(f,M)$ be a polymatroid and $A\\subseteq M$. \\begin{itemize}\n\\item $f$ is tight on $A$ if and only if it is tight on all elements of $A$.\n\\item $f\\down A$ is tight on $A$.\n\\item $f-f\\down A$ is a modular polymatroid.\n\\item $f\\down M$ is tight, and $f=f\\down M + (f-f\\down M)$ is the unique\ndecomposition of $f$ into the sum of a tight and modular part. \\qed\n\\end{itemize}\n\\end{lemma}\n\n\n\\medskip\n\nIn the last part of this section we investigate how to extend the\npolymatroid $(f,M)$ to the ground set $Mx$ using the {\\em excess\nfunction} $e(A)=f(xA)- f(A)$ defined for all subsets $A\\subseteq M$ including\nthe empty set. In agreement with the previous notation,\n$e(a,b|A)$ abbreviates $e(aA)+e(bA)-e(abA)-e(A)$, in particular,\n$e(a,b)=e(a,b|\\emptyset)=e(a)+e(b)-e(ab)-e(\\emptyset)$.\n\n\\begin{lemma}\\label{lemma:excess}\nSuppose $x$ is not in the ground set $M$ of the polymatroid $f$. Extend\n$f$ to the subsets of $Mx$ by $f_x(Ax) = f(A)+e(A)$. Then $f_x$ is a\npolymatroid on $Mx$ if and only if the following conditions hold:\n\\begin{enumerate}\n\\item[\\rm 1.] $e$ is non-negative and non-increasing: $e(A)\\ge e(B) \\ge 0$\nfor $A\\subseteq B\\subseteq M$;\n\\item[\\rm 2.] $e(a|M{-}a)+f(a|M{-}a)\\ge 0$ for all $a\\in M$;\n\\item[\\rm 3.] $e(a,b|A)+f(a,b|A)\\ge 0$ for all $abA\\subseteq M$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nAn easy case by case checking.\n\\end{proof}\n\\begin{example}\\label{example:excess-1}\nLet $f$ be a polymatroid on $M$ and $\\,0\\le u,t$. Define the excess function\n$e_x$ by\n$$\n   e_x(A) = \\left\\{\\begin{array}{ll}\n         u+t & \\mbox{ if $A=\\emptyset$,}\\\\\n         u & \\mbox{ otherwise}.\n          \\end{array}\\right.\n$$\nIf $t\\le f(a,b)$ for all pairs $a,b\\in M$, then $f_x$ is a polymatroid .\n\\end{example}\n\\begin{proof}\nConditions 1 and 2 of Lemma \\ref{lemma:excess} trivially hold. As for\nCondition 3,\n$e_x(a,b|A)$ is zero except when $A=\\emptyset$, and then $e_x(a,b)=-t$. Thus\nit also holds by the assumption on $t$.\n\\end{proof}\n\\noindent\nAn easy calculation shows that for this extension $f_x$, for all pairs\n$a,b\\in M$ and non-empty $A\\subseteq M{-}a$ we have $f_x(x,a|A)=0$, \nand $f_x(a,b|x)=f(a,b)-t$.\n\\begin{example}\\label{example:excess-2}\nLet $c\\in M$ and $\\,0\\le u,t$. Define the excess function $e_x$ by\n$$\n    e_x(A) =\\left\\{\\begin{array}{ll}\n         u+t & \\mbox{ if $A=\\emptyset$ or $A=\\{c\\}$,}\\\\\n         u & \\mbox{ otherwise}.\n          \\end{array}\\right.\n$$\nIf $t\\le f(a,b)$ and $t\\le f(a,b|c)$ for all pairs $a,b\\in M{-}c$,\nthen $f_x$ is a polymatroid.\n\\end{example}\n\\begin{proof}\nSimilar to the previous Example. Conditions 1 and 2 hold, moreover\n$e_x(a,b|A)$ is either zero or $-t$, and the latter case holds when $A=\\emptyset$\nand $a,b\\in M{-}c$, or when $A=\\{c\\}$. Thus in all cases Condition 3 holds\nas well.\n\\end{proof}\n\n\n\\section{Adhesivity versus amalgam}\\label{sec:adh-eq-amalgam}\n\nAs defined in Section \\ref{subsec:adhesive}, polymatroids $f_X$ and $f_Y$ on\nground set $MX$ and $MY,$ respectively, have an {\\em amalgam} if there is a\npolymatroid\non $MXY$ extending both $f_X$ and $f_Y$. The same polymatroids are\n{\\em adhesive} if, in addition, they have a modular extension.\nWhen $Y$ has a single element $y$, then the polymatroid on $My$ will be\ndenoted by $f_y$. In this special case adhesivity of $f_X$ and $f_y$ is\nequivalent to the existence of the amalgam of closely related\npolymatroids. Recall that $f_y$ is {\\em tight on $y$} if $f_y(y|M)=0$,\nand by tightening $f_y$ on $y$ one gets the (tight) polymatroid\n$$\n    f_y\\down y = f_y - f_y(y|M)\\cdot\\r_y.\n$$\n\n\\begin{theorem}\\label{thm:amalgam-Xy}\nPolymatroids $f_X$ and $f_y$ are adhesive if and only if $f_X$ and\n$f_y\\down y$ have an amalgam.\n\\end{theorem}\n\\begin{proof}\nFirst let $g$ be the modular extension of $f_X$ and $f_y$, that is\n$g(X,y|M)=0$. This equality rewrites to\n$$\n   g(y|MX) = g(XMy)-g(MX)=g(My)-g(M)=f_y(My)-f_y(M)=f_y(y|M).\n$$\nLet $g^*=g\\down y$. The above equality means that restricting $g^*$ to $My$\none gets $f_y\\down y$, and, as $g$ and $g^*$ on $MX$ are the same, \nrestricting $g^*$ to $MX$ one gets\n$f_X$. Consequently $g^*$ is the required amalgam of $f_X$ and $f_y\\down y$.\n\nConversely, let $g^*$ be an amalgam of $f_X$ and $f_y\\down y$. Then using\nthat $f_y\\down y$ is tight on $y$, \n$g^*(My)=f_y\\down y(My) = f_y\\down y(M) = g^*(M)$, thus\n$$\n  g^*(XMy)-g^*(XM)\\le g^*(My)-g^*(M)=0,\n$$\nwhich means that $g^*(X,y|M)=0$. Let\n$g=g^*+\\lambda\\r_y$ with $\\lambda=f_y(y|M)$. Then $g$ extends $f_X$ (as\n$g\\restr MX=g^*\\restr MX=f_X$), and $f_y$ (as $g\\restr My=\ng^*\\restr My +\\lambda\\r_y = (f_y-\\lambda\\r_y)+\\lambda\\r_y$). Finally,\n$g(X,y|M)=g^*(X,y|M)=0$, as required.\n\\end{proof}\n\nThe last step in the proof works in a more general setting.\n\n\\begin{proposition}\\label{prop:tight-XY}\nSuppose $f_X\\down X$ and $f_Y\\down Y$ have an amalgam. Then $f_X$ and $f_Y$\nhave an amalgam as well. \n\\end{proposition}\n\\begin{proof}\nIf $g$ is an amalgam of $f_X\\down X$ and $f_Y\\down Y$, then\n$g+(f_X-f_X\\down X) + (f_Y-f_Y\\down Y)$ is an amalgam of $f_X$ and $f_Y$.\n\\end{proof}\n\nIn particular, to show that $f$ is sticky, it is enough to consider\nextensions $f_X$ and $f_Y$ which are tight on $X$ and $Y$, respectively. The\ncondition stated in Proposition \\ref{prop:tight-XY} is sufficient but not \nnecessary. Polymatroids $f_x$ and $f_y$ in\nExample \\ref{example:ex1} have an amalgam but are not adhesive. Thus, by\nTheorem \\ref{thm:amalgam-Xy}, $f_x\\down x$ and $f_y\\down y$ have no\namalgam.\n\n\n\\section{One-element extensions}\\label{sec:sticky-2}\n\nThis section starts with an alternative proof for a result of F.~Mat\\'u\\v s \n\\cite{fmadhe} which claims, using our terminology, that polymatroids on two\nelement sets are 1-sticky. A similar proof to this one will be given for\nTheorem \\ref{thm:nonsticky-2}.\nTheorem \\ref{thm:amalgam3} gives a sufficient and necessary condition for\na pair of one-element extensions of a polymatroid on three elements \nto have an amalgam. \nUsing Theorem \\ref{thm:amalgam-Xy}, this\nis turned into sufficient and necessary conditions for such\npolymatroid pairs to be adhesive, which, in turn, yields new 5-variable \nnon-Shannon entropy inequalities stated in Corollary \\ref{corr:5ineq}.\n\nThe section concludes with several examples. The first one specifies two\nlinearly representable (entropic) polymatroids which have an amalgam, but\nare not adhesive. Put in other words, they have a polymatroid\nextension, but no almost entropic (or linear) extension. Finally, there are\ntwo general examples for 1-sticky and not 1-sticky polymatroids on three\nelements.\n\n\\def\\cite[Corollary 2]{fmadhe}{\\cite[Corollary 2]{fmadhe}}\n\\begin{theorem}[\\cite[Corollary 2]{fmadhe}]\\label{thm:1-1-sticky}\nAll Polymatroids $f_x$ and $f_y$ on the ground set $abx$ and $aby$ with\ncommon restriction to $ab$\nare adhesive. In particular, such polymatroids have \nan amalgam, thus every polymatroid on a two element set is 1-sticky.\n\\end{theorem}\n\\begin{proof}\nAs discussed in Section \\ref{subsec:adhesive}, adhesive polymatroid pairs\n$(f_x,f_y)$ form a polyhedral cone. Consequently, $(f_x,f_y)$ is adhesive\nif and only if $(\\lambda f_x,\\lambda f_y)$ is adhesive for some (or all)\npositive $\\lambda$. The adhesive cone is closed, thus to show that a\nparticular pair $(f_x,f_y)$ is\nadhesive, it is enough to find, for each positive $\\varepsilon$, some adhesive pair\n$(\\ell_x,\\ell_y)$ such that $\\|f_x-\\lambda\\ell_x\\|<\\varepsilon$, and\n$\\|f_y-\\lambda\\ell_y\\|<\\varepsilon$. In this particular case $\\ell_1$ and $\\ell_2$\nwill be the linearly representable polymatroids guaranteed by\nProposition \\ref{prop:pm-approx}. Thus $\\ell_1$ and $\\ell_2$ are represented\nover the same vector space $V$,\n$\\lambda\\ell_x$ and $\\lambda\\ell_y$ are \n$\\varepsilon$-close to $f_x$ and $f_y$, respectively, and the linear subspaces\nin both representations corresponding to\nsubsets of $\\{ab\\}$ have the same dimensions: $\\ell_x(a)=\\ell_y(a)$,\n$\\ell_x(b)=\\ell_y(b)$ and $\\ell_x(ab)=\\ell_y(ab)$ as these\nequalities are true for polymatroids $f_x$ and $f_y$. To conclude\nthe claim of the theorem it is enough to show that $(\\ell_x,\\ell_y)$ is an\nadhesive pair.\n\nThe dimensions of subspaces spanned by $V_a$, $V_b$, and $V_a\\cup V_b$ are\nthe same in both representations. Choose a base in the first representation\nwhich can be partitioned to $B^x_x\\cup B^x_a \\cup B^x_b\\cup B^x_{ab}$ such\nthat $\\ell_x(a)=|B^x_a\\cup B^x_{ab}|$, $\\ell_x(b)=|B^x_b\\cup B^x_{ab}|$, and\n$\\ell_x(ab)=|B^x_a\\cup B^x_b\\cup B^x_{ab}|$, and similarly for $\\ell_y$.\nIdentify $B^x_a$ and $B^y_a$, $B^x_b$ and $B^y_b$, $B^x_{ab}$ and\n$B^y_{ab}$, and take the vector space with base $B^x_x\\cup B_a\\cup B_b\\cup\nB_{ab}\\cup B^y_y$ (that is, glue the representations of $\\ell_x$ and\n$\\ell_y$ along their common part). It will be a linear representation of a\npolymatroid on $abxy$, where $x$ and $y$ are independent given $ab$.\nConsequently $\\ell_x$ and $\\ell_y$ have an adhesive extension, which\nconcludes the proof.\n\\end{proof}\n\nNow we turn to the case of one-point extensions of polymatroids on\nthree-element sets. If not mentioned otherwise, all polymatroids on this\nsection are extensions of a fixed polymatroid on $M=\\{a,b,c\\}$.\n\n\\begin{theorem}\\label{thm:amalgam3}\nPolymatroids $f_x$, $f_y$ on the ground sets $abcx$ and $abcy$ have an amalgam if \nand only if the following inequalities and their permutations\n(permuting $a,b,c$ and $x,y$) hold:\n\\begin{align}\\label{eq:amalgam3}\n  & ~~~ f_x(a,x|c)+f_x(a,b|x)+f_y(a,b|y)+f_y(c,y)+{}\\\\\n  &+\\twoline{f_x(b,x|ac)}{f_y(b,y|ac)} + \\twoline{f_x(c,x|ab)}{f_y(c,y|ab)} + \n    2\\twoline{f_x(x|abc)}{f_y(y|abc)} \\ge f_x(a,b).\\nonumber\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nIt is clear that all terms are defined over one of\nthe polymatroids $f_x$ and $f_y$. Also, these inequalities hold for\nany polymatroid with ground set $abcxy$, which can easily be checked\nusing an automated entropy checker, thus they {\\em must} hold when $f_x$ and\n$f_y$ have an amalgam. Actually, inequalities in (\\ref{eq:amalgam3}) rewrite to\n\\begin{align}\n   & ~~~ (a,b|xy)+(x,y|a)+(x,y|b)+(c,y|x)+(a,x|cy)+{}\\label{eq:amalgamproof}\\\\\n   &+\\twoline{(b,x|acy)}{(b,y|acx)}+\\twoline{(c,x|aby)}{(c,y|abx)} +\n    2\\twoline{(x|abcy)}{(y|abcx)} \\ge 0,\\nonumber\n\\end{align}\nwhich evidently holds for any polymatroid on five elements.\n\nThe sufficiency can be checked by the method indicated in Section\n\\ref{subsec:adhesive}. Polymatroids $f_x$ and $f_y$ determine 23 out of the\n31 coordinates of the polymatroid on $N=abcxy$. The missing \n8 variables are indexed by subsets of the form $Axy$ with $A\\subseteq\n\\{abc\\}$.\n\nThe facets of the polymatroid cone $\\Gamma_N$ are determined by the basic submodular\ninequalities $(i,j|K)\\ge 0$ and by the monotonicity requirements $(i|N{-}i)\\ge\n0$. The strong duality of linear programming says that the facet equations of the\nprojection are non-negative linear combinations of these inequalities in\nwhich the combined coefficients of the projected (dropped) variables are zero. Let $\\mathcal\nM_N$ denote the matrix whose columns are indexed by the non-empty subsets of\n$N$, and whose rows contain the coefficients of the bounding facets of\n$\\Gamma_N$. In each row there are two, three, or four non-zero entries only.\nWhen $\\mathcal M$ is restricted to the eight columns labeled by $xyA$,\n\\begin{table}[!htb]\n\\begin{center}\\begin{tabular}{|cccccccc|ll|}\n\\hline\n\\rule{0pt}{11pt}$xy$&$axy$&$bxy$&$cxy$&$abxy$&$acxy$&$bcxy$&$abcxy$&&\\\\\n\\hline\n\\rule{0pt}{12pt}\n-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & $(x,y)$&\\\\[2pt]\n1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & $(c,x|y)$&  $(c,y|x)$\\\\[2pt]\n-1 & 1 & 1 & 0 & -1 & 0 & 0 & 0 & $(a,b|xy)$&\\\\[2pt]\n0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & $(x,y|a)$&\\\\[2pt]\n0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & $(x,y|b)$&\\\\[2pt]\n0 & 0 & 0 & 1 & 0 & -1 & 0 & 0 & $(a,x|cy)$&  $(a,y|cx)$\\\\[2pt]\n0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & $(c,x|aby)$&  $(c,y|abx)$\\\\[2pt]\n0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & $(b,x|acy)$&  $(b,y|acx)$\\\\[2pt]\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & $(x|abcy)$& $(y|abcx)$\\\\[2pt]\n\\hline\n\\end{tabular}\\end{center}\\vskip -5pt\n\\caption{A submatrix of $\\mathcal M_{abcxy}$}\\label{table:3-facets}\n\\end{table}\n27 different non-zero rows remain. Table \\ref{table:3-facets} shows some of\nthem with the corresponding facet equations (one or two). There are 154\nextremal non-negative linear combinations of the 27 rows which give zero for\nall eight columns. One of them is the combination taking each but the\nfirst and last row from Table \\ref{table:3-facets} once, and take the last\nrow twice. Eight of the corresponding facet combinations give the\ninequalities in (\\ref{eq:amalgamproof}). The other $3\\cdot 8$ combinations,\nwhen one takes $(c,x|y)$ instead of $(c,y|x)$, or $(a,y|cx)$ instead of\n$(a,x|cy)$, or both, yield supporting hyperplanes to the projected cone, but\nnot facets as they are consequences of the basic inequalities for $abcx$ and\n$abcy$. In other words, these hyperplanes do not cut into the cones\n$\\Gamma_{abcx}$ and $\\Gamma_{abcy}$.\n\nThe extremal combinations of the 27 rows were generated by a computer\nprogram. From that list the possible facets of the projection were generated\nand checked whether they are really facets of the projection. This search\nresulted in the statement of the Theorem.\n\\end{proof}\n\n\\begin{corollary}\\label{corr:3-adhesive}\nPolymatroids $f_x$ and $f_y$ on the ground sets $abcx$ and $abcy$ are\nadhesive if and only if the following inequalities and their permutations\nhold:\n\\begin{align}\\label{eq:adhesive3}\n  & ~~~ f_x(a,x|c)+f_x(a,b|x)+f_y(a,b|y)+f_y(c,y) +{}\\\\\n  &+\\twoline{f_x(b,x|ac)}{f_y(b,y|ac)} + \\twoline{f_x(c,x|ab)}{f_y(c,y|ab)} \\ge\nf_x(a,b).\\nonumber\n\\end{align}\n\\end{corollary}\n\\begin{proof}\nBy Theorem \\ref{thm:amalgam-Xy}, $f_x$ and $f_y$ are adhesive if and only if\n$f_x\\down x$ and $f_y\\down y$ have an amalgam. All terms in\n(\\ref{eq:amalgam3})\nare the same for $f_x$ and $f_x\\down x$ ($f_y$ and\n$f_y\\down y$) except for $(f_x\\down x)(x|abc)=0$ and \n$(f_y\\down y)(y|abc)=0$.\n\\end{proof}\n\n\\begin{corollary}\\label{corr:5ineq}\nThe following are five-variable non-Shannon information inequalities, that\nis, they hold in every entropic polymatroid on  at least five elements:\n\\begin{align*}\n  & ~~~ (a,x|c)+(a,b|x)+(a,b|y)+(c,y) +{}\\\\\n  &+\\twoline{(b,x|ac)}{(b,y|ac)} + \\twoline{(c,x|ab)}{(c,y|ab)} \\ge\n(a,b).\n\\end{align*}\n\\end{corollary}\n\\begin{proof}\nAs observed in \\cite{fmadhe}, restrictions of an entropic polymatroid are\nadhesive, consequently the inequalities (\\ref{eq:adhesive3}) in Corollary\n\\ref{corr:3-adhesive} must hold.\n\\end{proof}\n\n\\subsection{Examples}\\label{subsec:examples}\n\n\\begin{example}\\label{example:ex1}\nThere are linearly representable polymatroids $f_x$ and $f_y$ on $abcx$ and\n$abcy$ which have an amalgam but are not adhesive.\n\\end{example}\n\\begin{proof}\nPolymatroids $f_x$ and $f_y$ will be extensions of the uniform polymatroid\n$$\n   f(A) = \\left\\{\\begin{array}{ll}\n            4 & \\mbox{ if $|A|=1$},\\\\\n            6 & \\mbox{ otherwise},\n          \\end{array}\\right. ~~~~ A\\subseteq \\{abc\\}.\n$$\nClearly $f(i,j)=f(i,j|k)=2$ for\nall distinct $i$, $j$, $k$. The excess functions defining $f_x$ and $f_y$ are\n$$\n e_x(A)=\\left\\{\\begin{array}{ll} \n            3 & \\mbox{ if $A=\\emptyset$, }\\\\\n            1 & \\mbox{ otherwise},\n        \\end{array}\\right.\n~~ \\mbox{ and } ~~\n e_y(A)=\\left\\{\\begin{array}{ll} \n            3 & \\mbox{ if $A=\\emptyset$ or $A=\\{c\\}$,}\\\\\n            1 & \\mbox{ otherwise}.\n        \\end{array}\\right.\n$$\n\\newcommand\\mf[1]{\\mathbf{#1}}%\nBy Examples \\ref{example:excess-1} and \\ref{example:excess-2} both\n$f_x$ and $f_y$ are polymatroids. They are {\\em not} adhesive, as all terms on the\nleft hand side of (\\ref{eq:adhesive3}) are zero, while $f_x(a,b)=f(a,b)=2$.\nTo show that they have an amalgam, one can check that all conditions of\nTheorem \\ref{thm:amalgam3} hold. The polymatroid $f_{xy}$ specified in \nTable \\ref{table:ex1} gives such an extension explicitly.\n\\begin{table}[!htb]\\begin{center}\\begin{tabular}{|ccc@{\\qquad}ccc@{\\qquad}ccc@{\\qquad}ccc|}\n\\hline\\multicolumn{3}{|c@{\\qquad}}{\\rule{0pt}{12pt}$A$}&\n      \\multicolumn{3}{@{}c@{\\qquad}}{$Ax$ }&\n      \\multicolumn{3}{@{}c@{\\qquad}}{$Ay$ }&\n      \\multicolumn{3}{@{}c|}{$Axy$ }\\\\[2pt]\n\\hline\\rule{0pt}{13pt}\n    & 6 &   &    & 7 &   &    & 7 &   &    & 7 &     \\\\[2pt]\n~ 6 & 6 & 6 &  7 & 7 & 7 &  7 & 7 & 7 &  7 & 7 & 7~  \\\\[2pt]\n~ 4 & 4 & 4 &  5 & 5 & 5 &  5 & 5 & 7 &  6 & 6 & 7~  \\\\[2pt]\n    & 0 &   &    & 3 &   &    & 3 &   &    & 5 &     \\\\[2pt]\n\\hline\n\\end{tabular}\\end{center}\\vskip -5pt\n\\caption{The polymatroid $f_{xy}$ for $A\\subseteq \\{abc\\}$}\\label{table:ex1}\n\\end{table}\nThe four groups contain the values for the subsets indicated at the top line\nwhere $A$ runs over all subsets of $abc$. The values are arranged\nin four lines (from bottom to top) for $A=\\emptyset$, one-element\nsubsets $a$, $b$, $c$, two-element subsets $ab$, $ac$, $bc$, and $abc$ at\nthe top.\n\nFinally, the polymatroids $f_x$ and $f_y$ are linearly representable over\nany field. Choose seven independent vectors $\\mf s_1$, $\\mf s_2$, $\\mf u_1$,\n$\\mf u_2$, $\\mf v_1$, $\\mf v_2$, and $\\mf r$. Subspaces assigned to the\nground elements are are the ones spanned by the vectors listed below:\n$$\\begin{array}{rl}\n\\multicolumn{2}{c}{\\mbox{$abcx$ is linear}}\\\\[3pt]\na: & \\mf s_1,\\mf s_2, \\mf u_1,\\mf u_2 \\\\[2pt]\nb: & \\mf s_1, \\mf s_2, \\mf v_1, \\mf v_2\\\\[2pt]\nc: & \\mf s_1,\\mf s_2,\\mf u_1+\\mf v_1, \\mf u_2+\\mf v_2 \\\\[2pt]\nx: & \\mf s_1,\\mf s_2,\\mf r\n\\end{array}\n\\qquad\n\\begin{array}{rl}\n\\multicolumn{2}{c}{\\mbox{$abcy$ is linear}}\\\\[3pt]\na: & \\mf s_1,\\mf s_2, \\mf u_1,\\mf u_2 \\\\[2pt]\nb: & \\mf s_1, \\mf s_2, \\mf v_1, \\mf v_2\\\\[2pt]\nc: & \\mf u_1,\\mf u_2,\\mf v_1, \\mf v_2 \\\\[2pt]\ny: & \\mf s_1,\\mf s_2,\\mf r\n\\end{array}\n$$\nIt is easy to check that all generated subspaces have the right dimension.\nNote that while the dimensions of the subspaces corresponding to subsets of \n$abc$ are the same, the subspace arrangements are {\\em not} isomorphic.\n\\end{proof}\n\nIt is easy to check that $f_x\\down x$ and $f_y\\down y$ are also linearly\nrepresentable. As $f_x$ and $f_y$ are not adhesive, according to Theorem\n\\ref{thm:amalgam-Xy}, $f_x\\down x$ and $f_y\\down y$ have no amalgam.\n\n\\medskip\n\nTheorem \\ref{thm:amalgam3} can be used to characterizes 1-sticky polymatroids on\nthree-element sets. The following examples show some particular cases.\n\n\\begin{example}\\label{example:nonsticky-1}\nLet $f$ be a polymatroid on $\\{abc\\}$. If $f(a,b)$,\n$(a,b|c)$ are positive, $(a,b)\\le (a,c)$, $(a,b)\\le (b,c)$,\nthen $f$ is not 1-sticky.\n\\end{example}\n\\begin{proof}\nWe specify two extensions $f_x$ and $f_y$ so that one of the\ninequalities in Theorem \\ref{thm:amalgam3} fails.\nLet $t=(a,b)>0$, and $u=\\min\\{ (a,b),~ (a,b|c) \\}>0$.\nDefine the excess functions $e_x$, $e_y$ by\n$$\ne_x(A) = \\left\\{\n\\begin{array}{rl}\n    t & \\mbox{ if $A=\\emptyset$,}\\\\\n    0 & \\mbox{ otherwise},\n\\end{array}\\right.\n~~~\\mbox{ and } ~~~\ne_y(A)=\\left\\{\n\\begin{array}{rl}\n    u & \\mbox{ if $A=\\emptyset$ or $A=\\{c\\}$,} \\\\\n    0 & \\mbox{ otherwise}.\n\\end{array}\\right.\n$$\nAccording to Examples \\ref{example:excess-1} and \\ref{example:excess-2},\n$f_x$ and $f_y$ are polymatroids. In this case\n$f_x(a,x|c)=f_x(b,x|ac)=f_x(c,x|ab)=0$, $f_x(x|abc)=0$, and\n$f_x(a,b|x)=f(a,b)-t=0$, see the remark following Example\n\\ref{example:excess-1}. Similarly, we have \n$f_y(a,b|y)=f(a,b)-u=t-u$, $f_y(c,y)=0$, thus the\nleft hand side of the top line in (\\ref{eq:amalgam3}) is\n\\begin{align*}\n  & f_x(a,x|c)+(f_x(a,b|x)+f_y(a,b|y)+f_y(c,y)+{}\\\\\n  +& f_x(b,x|ac)+f_x(c,x|ab)+2f_x(x|abc) = t-u,\n\\end{align*}\nwhile the right hand side is $f(a,b)=t$. Thus no amalgam of $f_x$ and $f_y$\nexists.\n\\end{proof}\n\n\\begin{example}\\label{example:is-sticky}\nSuppose $(a|bc)=(b|ac)=(c|ab)=0$, and at least one of $(a,b|c)$,\n$(a,c|b)$, $(b,c|a)$ is zero. Then $f$ is 1-sticky.\n\\end{example}\n\\begin{proof}\nLet $f_x$ and $f_y$ be two extensions of $f$. Our goal is to show that all\ninstances of the inequalities in Theorem \\ref{thm:amalgam3} hold. From the\nassumptions it follows that for $|A|\\ge 2$ we have $f(A)=f(abc)=t$; moreover\nat least one of $f(a)$, $f(b)$, $f(c)$ also equals $t$. Suppose\n$f_x$ and $f_y$ are specified by the excess functions $e_x$ and\n$e_y$. By Proposition \\ref{prop:tight-XY} we can assume that $f_x$ is tight\non $x$ and $f_y$ is tight on $y$, which gives $e_x(M)=e_y(M)=0$. where\n$M=\\{abc\\}$. In our case\n$f(i|M{-}i)=0$, thus we must also have $e_x(i|M{-}i)=e_y(i|M{-}i)=0$, thus\n$e_x(A)=e_y(A)=0$ for all two-element subsets of $M$. This means\n$$\n   f_x(i,x|M{-}i) = f_x(x|M)= 0, ~~~ f_y(i,y|M{-}i)=f_y(y|M)=0,\n$$\nthus all terms in the second line of (\\ref{eq:amalgam3}) are zero.\nConsequently we only need to\nshow that\n$$\n  f_x(a,x|c)+f_x(a,b|x)+f_y(a,b|y)+f_y(c,y)\\ge f(a,b),\n$$\nwhich rewrites to\n\\begin{equation}\\label{eq:is-sticky}\n  f(a,b)+e_x(a)+e_x(b)+e_x(c)-e(x) + e_y(a)+e_y(b)-e_y(c)\\ge 0.\n\\end{equation}\nThe condition that one of $f(a)$, $f(b)$, $f(c)$ equals $t$ was not used\nyet.\nIf $f(c)=t$, then $e_x(c)=e_y(c)=0$, and then (\\ref{eq:is-sticky})\nfollows from\n$$\n    f(a,b)+e_x(a,b)+e_y(a)+e_y(b) \\ge 0,\n$$\nwhich holds by Lemma \\ref{lemma:excess}, Condition 3. When $f(a)=t$ (or,\nsymmetrically, $f(b)=t$), then $e_x(a)=e_y(a)=0$,\n$f(a,b)=f(b)=f(b,c)+f(a,b|c)$, and (\\ref{eq:is-sticky}) rewrites to\n$$\n   f(b,c)+e_x(b,c)+f(a,b|c)+e_y(a,b|c)+e_y(b)\\ge 0,\n$$\nwhich, again, holds by Lemma \\ref{lemma:excess}.\n\\end{proof}\n\n\n\n\\section{Two-element extensions}\\label{sec:sticky-3}\n\nUsing similar techniques necessary and sufficient conditions for the\nexistence of an amalgam of polymatroids on larger sets can be obtained.\nTheorem \\ref{thm:sticky-2-1} is\nsuch an example. It is a consequence of \\cite[Remark 6]{fmadhe} and Theorem\n\\ref{thm:amalgam-Xy}; we sketch a direct proof. The result is used to\nget a characterization of 2-sticky polymatroids on two-element sets.\n\n\\begin{theorem}\\label{thm:sticky-2-1}\nPolymatroids $f_X$ and $f_y$ on $abx_1x_2y$ and $aby$, respectively, have an amalgam if and\nonly if the following inequalities and all of their permutations (permuting\n$a$ and $b$, and $x_1$ and $x_2$) hold:\n$$\n   \\twoline{f_X[a,b,x_1,x_2]}{f_X[a,x_1,b,x_2]}\n   + f_y(y,a|b)+f_y(y,b|a)+f_y(a,b|y)\n +3f_y(y|ab)\\ge 0.\n$$\n\\end{theorem}\n\\begin{proof}\nThe inequalities rewrite to\n\\begin{align*}\n&~~~\\twoline{(x_2,y|b)+(x_1,x_2|y)+(a,b|x_2y)+(a,y|x_1x_2)}\n            {(x_2,b|y)+(x_2,y|x_1)+(a,y|bx_2)+(a,x_1|x_2y)}+{}\\\\\n&+ (x_1,y|a)+(x_2,y|a)+(x_1,y|b)+(a,b|x_1y)+{}\\\\\n&+(y|abx_1)+(y|abx_2)+(y|ax_1x_2) \\ge 0,\n\\end{align*}\nthus if $f_X$ and $f_y$ have an amalgam, then the expressions must be\nnon-negative.\n\nThe sufficiency can be checked similarly as in Theorem \\ref{thm:amalgam3} by\ncomputing the facets of the projection of the cone $\\Gamma_{\\{abx_1x_2y\\}}$ \nto the coordinates which are subsets of $abx_1x_2$ and $aby$. There are 12\ndropped coordinates: $x_1y$, $x_2y$, $x_1x_2y$, \\dots, $abx_1y$, $abx_2y$,\n$abx_1x_2y$. Restricting the matrix\n$\\mathcal M_{\\{abx_1x_2y\\}}$ to these columns, one gets 48 different non-zero\nrows, some of them are shown in Table \\ref{table:proj-facets}.\n\\begin{table}[!htb]\n\\begin{center}\\begin{tabular}{|ccc|ccc|ccc|ccc|l|}\n\\hline\n\\rule{0pt}{11pt}$x_1y$&$x_2y$&$x_1x_2y$&\\multicolumn{3}{|c}{$a$}&\\multicolumn{3}{|c}{$b$}&\\multicolumn{3}{|c|}{$ab$}&\\\\\n\\hline\n\\rule{0pt}{12pt}\n  1&0&0& -1&0&0& 0&0&0& 0&0&0  &$(a,x_1|y)$, $(a,y|x_1)$\\\\[2pt]\n-1&0&0& 1&0&0& 1&0&0& -1&0&0   &$(a,b|x_1y)$\\\\[2pt]\n 1&1&-1&0&0&0& 0&0&0&  0&0&0   &$(x_1,x_2|y)$\\\\[2pt]\n 0&-1&0& 0&1&0& 0&1&0& 0&-1&0  &$(a,b|x_2y)$\\\\[2pt]\n 0&0&1&  0&0&-1&0&0&0& 0&0&0   &$(a,y|x_1x_2)$\\\\[2pt]\n 0&0&0&-1&0&0& 0&0&0& 0&0&0    &$(x_1,y|a)$\\\\[2pt]\n 0&0&0& 0&-1&0&0&0&0& 0&0&0    &$(x_2,y|a)$\\\\[2pt]\n 0&0&0& 0&0&1 &0&0&0& 0&0&-1   &$(b,y|ax_1x_2$\\\\[2pt]\n 0&0&0& 0&0&0&-1&0&0& 0&0&0    &$(x_1,y|b)$\\\\[2pt]\n 0&0&0& 0&0&0& 0&-1&0&0&0&0    &$(x_2,y|b)$\\\\[2pt]\n 0&0&0& 0&0&0& 0&0&0& 1&0&-1   &$(x_2,y|abx_1)$\\\\[2pt]\n 0&0&0& 0&0&0& 0&0&0& 0&1&-1   &$(x_1,y|abx_2)$\\\\[2pt]\n 0&0&0& 0&0&0& 0&0&0& 0&0&1    &$(y|abx_1x_2)$\\\\[2pt]\n\\hline\n\\end{tabular}\\end{center}\\vskip -5pt\n\\caption{A submatrix of $\\mathcal M_{\\{abuxy\\}}$}\\label{table:proj-facets}\n\\end{table}\nOne facet of the projection is generated by the linear combination taking\nall but the first and last rows once, and the last row three times. There are\n6938 extremal non-negative linear combinations \ngiving zeros for the 12 projected variables. From\nthis list the possible facets of the projection were generated and checked\nif they are really facets. This search confirmed the claim.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:nonsticky-2}\nThe polymatroid on the two-element set $ab$ is 2-sticky if and only if\none of the following cases hold: $(a,b)=0$ (it is modular); $(a|b)=0$, or\n$(b|a)=0$ (one of them determines the other).\n\\end{theorem}\n\\begin{proof}\nFirst we show that these polymatroids are 2-2-sticky. Modular polymatroids are sticky\nwithout any restriction, so suppose, e.g., that $f(a|b)=0$. Let $f_X$ be an\nextension on $abx_1x_2$.\nAll six\nIngleton expressions for $f_X$ are non-negative using the following\nequalities and their symmetric versions:\n\\begin{align*}\n   [a,b,x_1,x_2] &+(a|b)=(a,x_1|b)+(a,x_2|b)+(x_1,x_2|a)+(a|x_1x_2);\\\\\n   [a,x_1,b,x_2] &+(a|b)=(a,x_1|b)+(b,x_2|a)+(a,x_2|x_1)+(a|bx_2);\\\\\n   [b,x_1,a,x_2]\n   &+(a|b)=(a,x_1|b)+(a,x_2|b)+(a,x_1|x_1)+(b,x_1|ax_2)+(a|bx_1x_2);\\\\\n   [x_1,x_2,a,b]\n   &+(a|b)=(a,x_1|b)+(a,x_2|x_1)+(a,b|x_2)+(x_1,x_2|ab)+(a|bx_1x_2).\n\\end{align*}\nIt means that $f_X$ is linear, and the same is true for $f_Y$. As in the\nproof of Theorem \\ref{thm:1-1-sticky}, using Proposition \\ref{prop:pm-approx}\nwe may assume that $f_X$ and $f_Y$ are linearly representable over the same\nfield, and the\ndimensions of the subspaces corresponding to the common subsets $a$, $b$\nand $ab$ are the same in both representations. \nChoose maximal independent set of vectors in both representations\nwhich span these subspaces in an equivalent way. Extend this set to be a\nbase in both representations. Glue the \ntwo vector spaces together along the equivalent set of base vectors. \nThis gives an amalgam (even an adhesive extension) of $f_X$ and $f_Y$, as\nrequired.\n\nIn the other direction first we show that given $f$ with $f(a,b)>0$, $f(a|b)>0$,\n$f(b|a)>0$, it can be extended to a polymatroid $f_X$ on $abx_1x_2$\nso that $f_X[a,b,x_1,x_2]<0$. For the construction we recall the {\\em \nnatural coordinates} of polymatroids on four elements from \\cite{entreg}. \nThis coordinate system has the additional advantage\nthat points with natural coordinates in the non-negative orthant $\\R_{\\ge 0}^{15}$\nare polymatroids.\nLet us recall these coordinates below:\n\\begin{align*}\n   & -[a,b,x_1,x_2], \\\\\n   & (a,b|x_1),~ (a,b|x_2), ~ (a,x_1|b), ~ (b,x_1|a), ~ (a,x_2|b), ~ (b,x_2|a),\\\\\n   & (x_1,x_2|a), ~ (x_1,x_2|b), ~ (x_1,x_2), ~(a,b|x_1x_2), \\\\\n   & (a|bx_1x_2), ~ (b|ax_1x_2), ~ (x_1|abx_2), (x_2|abx_1).\n\\end{align*}\nFrom these coordinates the values $f_X(a)$, $f_X(b)$, and $f_X(ab)$ can be \nexpressed as\nfollows, where only the coefficients of the coordinates in the above order\nare shown:\n\\def\\sb#1{\\mathbf #1}\n\\begin{align*}\n   f_X(a) &= (\\sb2,~ 1,1,1,0,1,0,~ \\sb1,\\sb0,\\sb1,1,~ 1,0,0,0)\\\\\n   f_X(b) &= (\\sb2,~ 1,1,0,1,0,1,~ \\sb0,\\sb1,\\sb1,1,~ 0,1,0,0)\\\\\n  f_X(ab) &= (\\sb3,~ 1,1,1,1,1,1,~ \\sb1,\\sb1,\\sb1,2,~ 1,1,0,0).\n\\end{align*}\nChoose the coordinates first and from eighth to tenth (typeset in bold)\nto have the positive values $\\varepsilon$, $f(a|b)-\\varepsilon$, $f(b|a)-\\varepsilon$, and \n$f(a,b)-\\varepsilon$, respectively, for some small enough $\\varepsilon$; set all other\ncoordinates to zero. With this choice $f_X$ will be a polymatroid which\nextends the one given on $ab$ as, e.g.,\n$f_X(a)=2\\varepsilon+f(a,b)-\\varepsilon+f(a|b)-\\varepsilon=f(a)$,\nmoreover the Ingleton value $f_X[a,b,x_1,x_2]$, as given by the first coordinate, \nis $-\\varepsilon$, which is negative.\n\nDefine the other extension $f_y$ by the excess function\n$$\n    e_y(A) = \\left\\{\\begin{array}{ll}\n               f(a,b) & \\mbox{ if $A=\\emptyset$,}\\\\\n               0     & \\mbox{ otherwise}.\n             \\end{array}\\right.\n$$\nBy the remark after Example \\ref{example:excess-1}, $f_y$ is a polymatroid\nand $f_y(a,b|y)=f_y(a,y|b)=f_y(b,y|a)=0$ as well as $f_y(y|ab)=0$.\nAccording to Theorem \\ref{thm:sticky-2-1} if $f_X$ and $f_y$ have an amalgam, \nthey must satisfy\n$$\n   f_X[a,b,x_1,x_2] + f_y(a,b|y)+f_y(a,y|b)+f_y(b,y|a)+3f_y(y|ab)\\ge 0.\n$$\nThis value, however, is negative, which proves the theorem.\n\\end{proof}\n\n\n\n\\section*{Acknowledgment}\nThe author would like to thank the generous support of \nthe Institute of Information Theory and Automation of the CAS, Prague.\nThe research reported in this paper has been supported by GACR project\nnumber 19-04579S, and partially by the Lend\\\"ulet program of the HAS.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Discussion}\n\\label{sec:discussion}\n\nWe have presented a reanalysis of a cosmological simulation designed\nto study the intergalactic medium with the aims to both better understand the systematics of the methods used to identify absorbers and better understand the gas comprising the warm-hot intergalactic medium (WHIM; $10^5$~K$<T<10^7$~K) by analyzing the physical environments associated with the absorbers. We have devised three different methods for\nidentifying QSO absorption line systems in simulation data-- the cut\nmethod, the contour method, and the spectral\nmethod. The cut method slices the line of sight into regions of equal\nredshift. The contour method identifies continuous regions above a\nnumber density cutoff. The spectral method fits absorption spectra\ngenerated for each line of sight. Often these methods give different\nresults, so it is important to choose the method with care.\n\n\\subsection{Absorber Identification Method Comparison}\n\nCreating a one-to-one correspondence of absorbers between the two methods\nallows us to get a sense of how well the two methods agree on column\ndensity for an arbitrary absorber.  However, this method of matching is\nimperfect and does not always match two absorbers that correspond to the same\nphysical structure. One way we combat this problem is by considering both\nspectral absorber matching of individual line components as well as total absorbing\ncomplexes. We do this to account for the cases where the spectral method\nfinds substructure that the contour method is unable to identify. \n\nOur spectral fitting routine does a good job of finding\nabsorbers in a manner that is consistent with the contour method for both\nLy$\\alpha$\\ and \\ion{O}{6}. Some fraction of Ly$\\alpha$\\ lines with column densities in the\nneighborhood of $N_{HI}\\sim 10^{16}$~cm$^{-2}$ are difficult\ndifficult to fit accurately due to line saturation. \n\nResults from the $d{\\cal N}\/dz$\\ comparison for Ly$\\alpha$\\ absorbers show very good\nagreement outside the $N_{HI}=10^{16}$~cm$^{-2}$ range between the\nthree methods. \\ion{O}{6}\\ absorbers show considerable \ndifference with observations, however, the $d{\\cal N}\/dz$\\ values produced \nby the contour and spectral methods are quite consistent with each other. We hypothesize\nthat differences between simulated and observed \\ion{O}{6}\\ absorption line statistics \nare due to ionization and feedback choices made in the underlying simulation, \nrather than in the methods we use to create and determine the processes of \nsynthetic absorbers.\n\nAs the contour method does not take line of sight velocities into account\nwhen identifying absorbers\nwe do not expect the b-values found by the spectral method to\ninstrinsically correspond to the b-values from the contour method. Despite\nthis difference, the two methods find comparable b-value distributions\nto both each other and observations \\citep{tripp2008} for Ly$\\alpha$, especially\nat lower column densities ($N_{HI}\\leq 10^{15}$~cm$^{-2}$).  \n\nThere is a larger discrepancy between the contour method and the \nspectral method in \\ion{O}{6}\\ which may be an indication of line of sight \nvelocities having a larger effect or a systematic underfitting\nby the spectral method. The slight systematic increase in the contour\nmethod b-values at high column densities is consistent with observations \\citep{tripp2008}\nand some simulations \\citep{2012ApJ...753...17C}, but contrast others\nthat find a much flatter distribution \\citep{teppergarcia2011}. The b-value distribution \nof \\ion{O}{6}\\ is also much more sensitive to the method of feedback used in \nthe simulation indicating, so it is not instructive to read too much into\nsuch a comparison.\n\nOne interesting result that comes from our comparison of the two\nmethods is the linear size-column density relation for both \\ion{H}{1}\\ and\n\\ion{O}{6}.  We find (as can be seen in Figure~\\ref{fig:size_Var}) that in\nour calculations absorber size is a weak function of column density,\nwith $L \\propto N^{-0.12}$ for both \\ion{H}{1}\\ and \\ion{O}{6}.  This is in\ndisagreement with analytic predictions\n\\citep[e.g.,][]{2001ApJ...559..507S}, though does not appear to\ndisagree with observations.  We speculate that this may be due to\nsimulation resolution or choice of cooling model, but also note that\nthe analytic models make the assumption of hydrostatic equilibrium,\nwhich is not necessarily accurate.\n\n\\subsection{Understanding the WHIM}\n\nThe physical conditions of the absorbers showed roughly the\nsame results as found by \\citet{britton2011}. The same general trends held\nfor mean \\ion{H}{1}\\ number density, metallicity, and \\ion{O}{6}\\ fraction as a\nfunction of column density. \n\nWe also find a similar phase distribution of absorbers as \\cite{britton2011}\nwith $69\\%$ in the WHIM phase, $30\\%$ in the warm phase, and\n$1\\%$ in the condensed phase. We do not, however, see strong evidence for a \ndistinct bimodality in temperature of absorbers as \\cite{britton2011} found for the phase\ndistribution of the total gas; instead we see a smoothly varying distribution across the temperature range. This does not indicate that there cannot\nbe two distinct collisionally ionized and photoionized populations, only that they are not represented in these absorber statistics. Such a population could still exist in the very rarest, highest column absorbers. It is also possible that these populations exist as part of a multi-phase absorber that would in effect, hide such a bimodality from observations. Further analysis could be done to examine such a distribution of gas within a single absorbing feature.\n\nIn effort to distinguish the characteristics of absorbers found in the WHIM\nphase versus the warm phase, we segment the absorber population by temperature\ncuts, with a temperature greater than $10^5$~K indicating that the\nabsorber is in the WHIM phase. These absorbers have similar \ncolumn density distributions as the rest of the population, but b-values that are\nmuch higher on average. There are no \\ion{H}{1}\\ (\\ion{O}{6}) absorbers in the WHIM phase with\nb-values less than 40~km~s$^{-1}$ (10~km~s$^{-1}$) by definition, but above those \nlimits the fraction of total absorbers is strongly dominated by WHIM absorbers.\nThese high b-value, low column \\ion{H}{1}\\ absorbers correspond to the population \nof broad Ly$\\alpha$\\ (BLA) absorbers identified through COS \\citep{2010ApJ...710..613D}\nand STIS \\citep{2004ApJS..153..165R,2006A&A...451..767R}.\n\nThere has been substantial recent debate about the origin of \\ion{O}{6}\\ absorption line systems, with some work suggesting that it comes entirely from collisional ionization, and other work suggesting that it comes from a combination of collisional and photo ionization.  As part of this debate, there has been controversy about the temperature of the plasma where \\ion{O}{6}\\ absorption lines predominantly occur. In this work, we investigate the nature of \\ion{O}{6}\\ absorbers in the intergalactic medium, but take a slightly different approach: we break the absorber population into absorbers dominated by collisional ionization and those dominated by photo ionization, and compare their thermal phase-space properties and column density distributions.  We find that the two populations have similar ranges of column densities, but (perhaps unsurprisingly) have bimodal distributions in both temperature and b-value.\n\nTo investigate the correlation between \\ion{H}{1}\\ and \\ion{O}{6}\\ and the phases of gas\nthat they trace, we find the associated \\ion{O}{6}\\ column densities for regions\nidentified as \\ion{H}{1}\\ absorbers. This shows a marked distinction between the\ncolumn of associated \\ion{O}{6}\\ by column density and b-value of \\ion{H}{1}\\ for the\ntwo phases, WHIM and warm, defined as having temperatures greater than and less thatn $10^5$~K respectively. The WHIM phase has a much higher median column density of associated \\ion{O}{6}\\ in all regimes except the highest column \\ion{H}{1}\\ absorbers. This is \nconsistent with the idea that the WHIM is effectively traced by low column, high b-value Ly$\\alpha$\\ absorbers (BLAs), as well as by \\ion{O}{6}.\n\nFinally, in consideration of the product $(Z\/Z_\\odot)f_{\\mathrm{OVI}}$, we find a\nmuch lower average value than the literature standard, comparable with that\nfound by \\cite{shull2012}. This may significantly affect predictions of IGM metallicity that have been primarily based on \\ion{O}{6}\\ absorption.\n\n\\section{Introduction}\n\nIt is now well accepted that a large portion of all baryons at $z =\n0$, up to $\\sim60$\\%, exist in a phase that is extremely difficult to\ndetect.  Unlike at high redshift ($z \\gtrsim 4$) where nearly all\nbaryons are accountable through Ly$\\alpha$\\ absorption \n\\citep{1997ApJ...489....7R, 2009and..book..419P}, only $\\sim30$\\% are\nobserved to be associated with the cool, photoionized Lyman alpha\nforest today \\citep{1992MNRAS.258P..14P, 1994MNRAS.267...13B,\n1998ApJ...503..518F, 2004ApJ...616..643F, danforth2008, shull2012},\nwith only an additional $\\sim10$\\% in galaxies.  However, it has been\nshown by cosmological simulations \\citep[e.g.][]{1999ApJ...514....1C,\n1999ApJ...511..521D, 2001ApJ...552..473D, britton2011} that most of these ``missing\nbaryons'' still exist in the IGM, but have simply been heated through\na combination of accretion shocks from structure formation and\nfeedback from galaxies to temperatures too hot ($\\sim 10^{5-7}$ K) to\nbe easily traceable with neutral hydrogen.  Because of these high\ntemperatures, this so-called warm\/hot intergalactic medium (WHIM) can\nonly be traced through absorption lines of highly ionized metals, such\nas \\ion{O}{6}\\ and \\ion{Ne}{8}\\ at ultraviolet wavelengths, and \\ion{O}{7}\\ and\n\\ion{O}{8}\\ in the X-ray.  To date, the \\ion{O}{6}\\ doublet at\n$\\lambda\\lambda$1032, 1038~\\AA\\ has proven to be the most fruitful of\nthese detection methods, with numerous large samples having been\ncompiled \\citep{2005ApJ...624..555D, danforth2008,\n2006ApJ...640..716D, tripp2008, 2008ApJS..179...37T,\n2008ApJ...683...22T, tilton2012, 2014arXiv1402.2655D, 2014ApJS..212....8S}, providing an excellent benchmark for\nsimulations.\n\nNumerous simulation studies have succeeded at reproducing the observed\nnumber density per unit redshift $d{\\cal N}\/dz$\\ of \\ion{O}{6}\\ absorbers\n\\citep[][]{2006ApJ...650..573C, 2009MNRAS.395.1875O,\nbritton2011, 2011ApJ...731...11C, teppergarcia2011,\n2012MNRAS.420..829O,2012ApJ...753...17C}, despite making significantly different\npredictions as to the physical conditions of the associated gas.  For\nexample, \\citet{2009MNRAS.395.1875O} and \\citet{2012MNRAS.420..829O}\nfind that an overwhelming majority of \\ion{O}{6}\\ absorbers are associated\nwith cool, photoionized gas with a mean temperature of $\\sim15,000$ K.\nOn the other hand, \\citet{britton2011} and \\citet{shull2012} find that \\ion{O}{6}\\ absorbers come\nfrom gas in two distinct temperature regimes -- a photoionized\ncomponent at $T \\sim 30,000$ K and a hotter, collisionally ionized\ncomponent at $T \\sim 300,000$ K, comprising $\\sim35$\\% and $\\sim55$\\%\nof all \\ion{O}{6}\\ absorption, respectively (with the remaining 10\\%\ncontained in collapsed structures).  \\citet{teppergarcia2011} find a\nsimilar distribution of \\ion{O}{6}\\ as \\citet{britton2011} in their\nsimulated volume, but \\ion{O}{6}\\ absorption is\nbiased towards detecting the hotter, collisionally ionized phase.  Some of these\ndisagreements may arise from differences in the simulations, such as\nthe methods by which metals are injected and mixed into the\nintergalactic medium by the feedback scheme, but\nanother potential source of difference is the method in which\nsynthetic \\ion{O}{6}\\ absorbers are created and analyzed in these works.\n\\citet{2009MNRAS.395.1875O}, \\citet{teppergarcia2011}, and\n\\citet{2012MNRAS.420..829O} create synthetic absorption spectra which\nare then analyzed in a fashion similar to actual observations.\nHowever, \\citet{britton2011} create absorbers simply by summing the\ntotal column density along a small section of a randomly oriented line\nof sight in their simulation box.\n\nIn this paper we present a re-analysis of a simulation from\n\\citet{britton2011}, with the intent being to better understand the\ncorrespondence between absorption lines and the physical conditions\nwith which they are associated, and to better understand possible systematics\nbetween the usage of different means of extracting absorber data\nfrom simulations.  To accomplish this, we create a\nsophisticated pipeline for fitting absorption-line spectra that can be\nused for both synthetic and actual absorption spectra.  Our primary\ngoal is to understand the systematic differences between quantities\ninferred from absorption lines in synthetic spectra and quantities \nderived more directly from the simulation outputs.\nIn Section~\\ref{sec:sim}, we describe\nbriefly the simulation upon which our analysis is focused.  In\nSection~\\ref{sec:methods}, we detail three different methods of\nvarying sophistication that we use to create synthetic absorber\nsamples.  In Section~\\ref{sec:results}, we examine a range of\nproperties of the synthetic Ly$\\alpha$\\ and \\ion{O}{6}\\ absorbers created using\neach of these three methods, comparing each method both to the other\ntwo and to observations of the statistical properties of real quasar\nabsorption line systems.  We provide a discussion of our results in\nSection~\\ref{sec:discussion}, and summarize in\nSection~\\ref{sec:summary}.\n\n\\subsection{Choosing an Appropriate Contour Cut-off Density}\n\\label{sec:cutoff}\nAs discussed in Section~\\ref{sec:contour}, for each spatially\ncontiguous region of cells above the mean number density\nthe characteristic density is calculated as some fraction \nof the peak number density of the region. Absorbers are then \ndetermined by spatially contiguous cells above the characteristic\ndensity. Thus, lowering or raising the characteristic density \nfraction could affect the number of absorbers\nand the extent of individual absorbers. As the effects of a poor\nchoice in characteristic density fraction are not immediately obvious,\nappropriate care must be taken to select a reasonable value.\n\nTo check that the characteristic density approach is reasonable\nat all we examine Figure\n\\ref{fig:spatial}, where we average the number densities for a set\nof absorbers with low column densities ($N_{HI}=10^{12}-10^{13}$~cm$^{-2}$,\n$N_{OVI}=10^{13}-10^{13.5}$~cm$^{-2}$) over physical distance along\nthe line of sight from the peak\nnumber density for the region. A total of 5692 and 1319 absorbers were\nused for Ly$\\alpha$\\ and \\ion{O}{6}\\ respectively. The solid line indicates\nthe average number density, and the shaded region shows $1\\sigma$\ndeviations. The averages were taken by creating bins of physical\ndistance and, for every absorber, assigning the lixel with the most\nclosely corresponding physical distance from peak number density to\nthe appropriate bin. We then fit the resulting distribution with a\npower law at small radii and a constant at larger radii (i.e., we fit \n$n(r) = r^\\alpha + b$ for $r<d$ and $n(r)=n_o$ for $r>d$). The\nslope of the power law, $\\alpha$,  allows us to\nassess the extent to which an absorber will be dominated by its peak, and\ncorrespondingly, how much a change in characteristic\ndensity fraction could affect column\ndensity weighted properties for absorbers of a given species. The\nconstant fit for the tail of the distribution, $b$, reflects the average ion \nnumber density of the regions \\textit{between} absorbers. This number\ndensity is adopted as the mean number density of the species, but again \nthis mean refers to the mean of the background between absorbers, not\nthe mean number density of the entire volume or a given sightline. We note\nthat the constant background value is a very low \\ion{O}{6}\\ number\ndensity.  Examination of the raw simulation data shows that these values correspond to cells in the\nsimulation volume with baryon densities of $10^{-3} - 10^{-1}$ of the cosmic\nmean, metallicities of $10^{-4} - 10$~Z$_\\odot$, and temperatures of\n$10^5$~K or below (resulting in \\ion{O}{6}\\ fractions of $10^{-7}$ or\nbelow at the densities in question when the metagalactic ultraviolet\nbackground is taken into consideration).\n\n\\begin{figure*}\n\\begin{center}\n    \\includegraphics{spatial.pdf}\n    \\end{center}\n  \\caption{Average number density as a function of distance along the\n    line of sight from the center of an absorber.  The shaded region\n    shows $1\\sigma$ deviations. All absorbers with column density in\n    range ($10^{12}-10^{13}$~cm$^{-2}$) for \\ion{H}{1}\\ (left) and ($10^{12}-10^{14}$~cm$^{-2}$)for \\ion{O}{6}\\ (right) identified with the contour method. The center of\n    the absorber is defined as the lixel with the highest number\n    density. The dashed line indicates a power-law fit of the\n    average number density.  We adopt cutoff densities of\n    $10^{-12}$~cm$^{-3}$ (\\ion{H}{1}) and $10^{-13}$~cm$^{-3}$ (\\ion{O}{6}). }\n  \\label{fig:spatial}\n\\end{figure*}\n     \n\n\n\nWe recognize that due to the shallowness of the power law for \\ion{H}{1}\\\nnumber density ($\\alpha_{HI} = -2.4$), varying the characteristic density \nfraction may affect the properties of the absorbers, especially if a very low\ncharacteristic density fraction is chosen. In this case, contour-identified absorbers will fail to\nreflect actual structures, and quantities that should be dominated by peak values\nwill begin to deviate due to the large excess of sampled\nmaterial. \\ion{O}{6}\\ should be much less sensitive to the choice in cutoff\ngiven the steepness of the power law ($\\alpha_{OVI} = -6.3$).\n\nThis intuition is reflected in Figure \\ref{fig:char_frac_tests}, where we show\nnumber of absorbers, line of sight size, and $b$ as a function of \ncolumn density for a variety of characteristic density fractions.\nFor each cutoff, a corresponding set of absorbers is found independently. \nThis analysis is performed for a subset of the total light rays; however,\nthe results are representative of the total sample. We find as expected that\nmost quantities are fairly robust to changing characteristic \ndensity fraction, as displayed by \nthe consistency of the b-value results.  The linear size of absorbers, which is not weighted by number\ndensity, increases dramatically when lowering the characteristic density\nfraction.  We further investigate the size as a function of column density in \nsection \\ref{sec:size}. Given the later results we chose a\n  characteristic density that is 0.5 times the absorber's maximum baryon density.\n\n\\begin{figure*}[ht!]\n\\begin{center}\n    \\includegraphics[width=.9\\textwidth]{char_frac_tests.pdf}\n  \\caption{ Absorbers were identified using the contour method for a variety of characteristic density fractions (\\emph{colors}). For each set of identified absorbers number of absorbers (\\emph{Top}), average b-value (\\emph{Middle}), and average size (\\emph{Bottom}) were found as a function of column density. \\emph{Left:} \\ion{H}{1}\\ absorbers, \\emph{Right: } \\ion{O}{6}\\ absorbers.}\n  \\label{fig:char_frac_tests}\n\\end{center}\n\\end{figure*}\n\n\n\\subsubsection{Absorber Size}\n\\label{sec:size}\n\nAs absorber size was the most clearly varying parameter with characteristic \ndensity fraction, it is a useful metric to assess the choice of said \nfraction. Analytic estimates \\citep{2001ApJ...559..507S} give an expected \nsize-column density relation of $L\\propto N^{-1\/3}$ for \\ion{H}{1}\\ absorbers. \nIn Figure \\ref{fig:size_Var} we plot mean absorber size as a function of \ncolumn density for \\ion{H}{1}\\ and \\ion{O}{6}. We also overplot a line of best fit for \nthe size-column relation obtained through a least squares fit of the median \nvalue in each column density bin weighted by the number of absorbers in the \nbin. We find a best-fit power law of $-0.12$ for \\ion{H}{1}, which is substantially \nshallower than the expected slope of $-1\/3$.\nFor \\ion{O}{6}\\ we find a power law of $-0.12$.\n\n\\begin{figure*}\n\\begin{center}\n    \\includegraphics{size_Var.pdf}\n  \\caption{Median size of absorbers binned by column density using . Error bars \n    given by 1st and 4th quartiles. Black dashed line shows line of best \n    fit for median values weighted by number of absorbers in each bin, \n    with fit given in the legend. Grey dashed line in left hand panel shows \n    expected analytic relation, normalized to a 100~kpc absorber at \n    a column density $N_{\\mathrm{HI}}=10^{14}$~cm$^{-2}$. \n    Dotted line indicates the resolution limit of the simulation.\n    \\emph{Left:} \\ion{H}{1}\\ absorbers. \\emph{Right:} \\ion{O}{6}\\\n    absorbers.}\n  \\label{fig:size_Var}\n\\end{center}\n\\end{figure*}\n\n\\subsection{Comparison of Methods}\n\n\\subsubsection{Comparison of Contour and Cut Methods}\n\\label{sec:cutcontour}\nAlthough we do not expect \\textit{a priori} that the cut and contour methods \nwill give identical results,  we give a short comparison of these\nmethods. It is important to show comparison to the cut method despite\nits lack of physical motivation, as it was used\nfor a previous analysis of these simulations in \\citet{britton2011}. In\norder to correlate the absorbers, each absorber from the contour\nmethod is matched by redshift with an absorber from the cut method.\n\nMatching absorbers between methods is accomplished by finding an\nabsorber from the contour method within a given redshift ($\\delta z =\n10^{-6}$) of an absorber from the cut method. This process is then\nrepeated several times after increasing the tolerance by an order of\nmagnitude until $\\delta z = 10^{-3}$. As the redshift range we consider ($0\\leq\nz\\leq 0.4$) is quite small in comparison to the range of redshift windows we\nconsider, we do not bother to use a window of size $\\delta z\/(1+z)$. We have chosen to match the absorbers\nin this way to create a robust way of matching as many absorbers as\npossible, while still ensuring that the matches are as accurate as\npossible. Changing the start and end tolerances by an order of\nmagnitude has minimal effect on the final matches, as long as the\nmatching is accomplished using a series of monotonically increasing\nwindow sizes.\n\nThe column densities of each method for the appropriately matched\nand then binned by contour column density. Median contour and cut\ncolumn densities and first and fourth quartiles were found for\neach bin. This comparison, shown in the lower panels of Figure\n\\ref{fig:NVar_cutcontour}, indicates no obvious systematic difference \nat any column density. This is indicative of the absorbers being dominated\nby the sharp peaks in number density.  \n\nAdditionally, the total number of Ly$\\alpha$\\ absorbers found at low column\ndensities, shown in the upper panel of Figure \\ref{fig:NVar_cutcontour}, is\nslightly larger when using the cut method. This is evidence of\nthe splitting effect, as more absorbers of low column will be found\nwhen a single physical feature is inappropriately identified as two\nabsorbers. \n\n\nGiven that these results show a comparable column density result, the\ncontour method is clearly an adequate substitute in analyzing the\ndirect output. Furthermore, since the contour method finds fewer total\nabsorbers than the cut method (as it effectively combines multiple\nartificially-segmented absorbers into a single, physically meaningful object), as well as absorbers that directly\ncorrelate to the actual cosmological structure in the simulation\nvolume, it is straightforward to correlate absorbers found with the\ncontour method with those found using the spectral method. Thus, all\nof the analysis in the following sections of this paper will be\ncompleted using values found through the contour method.\n\n\\begin{figure*}\n\\begin{center}\n  \\includegraphics[width=.95\\textwidth]{NVar_cut_contour.pdf}\n  \\caption{\\emph{Top: } Total number of absorbers found in each column\ndensity range for contour method (\\emph{light blue}) and cut method\n(\\emph{dark blue}). \\emph{Bottom: } The contour absorbers were binned by\ncolumn density and the median column density of the matching cut\nabsorbers was plotted against the median column density of the\ncontour absorbers for that bin with vertical error bars showing\nfirst and fourth quartiles in cut column density. Dashed line\nshows $\\rm{N}_{\\rm{contour}}=\\rm{N}_{\\rm{cut}}$.  \\emph{Left: } \\ion{H}{1}\\\nabsorbers. \\emph{Right: } \\ion{O}{6}\\ absorbers.}\n  \\label{fig:NVar_cutcontour}\n\\end{center}\n\\end{figure*}\n\t\n\\subsubsection{Comparison of Contour and Spectral Methods}\n\\label{sec:NVar}\n\nWe perform an analysis similar to Section~\\ref{sec:cutcontour} in\ncomparing the contour and spectral methods. After matching absorbers\nfrom the contour method with the column density determined by\na single component line fit using the spectral method technique,\nthe contour absorbers were then binned by\ncolumn density. The median spectral and contour column densities for\nthe absorbers in each of these bins was then plotted with error\nbars showing the corresponding first and fourth quartiles for each\nbin. These matches were determined using the same\nincreasing redshift tolerance process as described in\nSection~\\ref{sec:cutcontour}. \n        \n\\begin{figure*}\n\\begin{center}\n    \\includegraphics[width=.95\\textwidth]{NVar_split.pdf}\n  \\caption{ The contour\n    absorbers were binned by column density and the median column\n    density  of the matching spectral absorbers was plotted against\n    the median column density of the contour absorbers for that bin\n    with vertical error bars showing first and fourth quartiles in spectral\n    column density and horizontal error bars showing first and fourth \n    quartiles in contour column density. Dashed line shows\n    $\\rm{N}_{\\rm{contour}}=\\rm{N}_{\\rm{spectral}}$. The spectral method \n    treats each component found in multi-component fits of\n    a line complex separately.  \\emph{Left: } \\ion{H}{1}\\\n    absorbers. \\emph{Right: } \\ion{O}{6}\\ absorbers. }\n  \\label{fig:NVar_split}\n\\end{center}\n\\end{figure*}\n        \nWe find that for Ly$\\alpha$\\ the pipeline has significant trouble\nappropriately matching corresponding absorbers at all column\ndensities and significantly underpredicts the results from the contour method.\nFor \\ion{O}{6}\\ the contour method finds higher column densities\nby approximately a factor of $2$ for low column densities\n($N_{OVI}=10^{12}-10^{14}$~cm$^{-2}$) and increasing to an order of\nmagnitude for higher column densities\n($N_{OVI}=10^{14}-10^{16}$~cm$^{-2}$).\n\nThis is expected as the spectral method is not biased against using\nmultiple component fits if the $\\chi^2$ error is lower in using more \ncomponents. In order to negate this effect, we sum together components\nidentified in the same complex. A complex is defined as a contiguous region \nwith a flux less than $F\/F_{continuum}=0.99$. The total column density \nof the region is thus $N_{total} = \\sum N_i$. The redshift of the\ncomplex is slightly more difficult to define, but we use a column\ndensity weighted average to assign a single value of redshift to \nthe entire complex. Performing the analysis in this way loses some\ninformation about physical structure from line of sight velocities \nthat the spectral method is able to identify, but allows us to check \nif the two methods track similar amounts of total material\n\n\nAs seen in Figure \\ref{fig:NVar_combine}, matching the contour method\nabsorbers to a line complex rather than a line component\ngives more comparable column density results for Ly$\\alpha$\\ and \\ion{O}{6}\\ at\nall column densities.\nThere is a larger scatter about the mean for absorbers with higher\ncolumn densities ($\\mathrm{N_{HI}}\\geq 10^{16}$~cm$^{-2}$,\n$\\mathrm{N_{OVI}}\\geq 10^{14.5}$~cm$^{-2}$). This may be indicative\nof the difficulty of fitting saturated lines. \n  We can now be reasonably certain that the spectral\nand contour methods identify the same absorbing structures, and we can\nmatch them reasonably effectively. The spectral method may find different underlying\nIGM substructure than the contour method due to line of sight velocity effects, but the two methods find roughly\nequivalent bulk material.\n\n        \n\\begin{figure*}  \n\\begin{center}\n    \\includegraphics[width=.95\\textwidth]{NVar_combine.pdf}\n  \\caption{The contour\n    absorbers were binned by column density and the median column\n    density  of the matching spectral absorbers was plotted against\n    the median column density of the contour absorbers for that bin\n    with vertical error bars showing first and fourth quartiles in spectral\n    column density. Dashed line shows\n    $\\rm{N}_{\\rm{contour}}=\\rm{N}_{\\rm{spectral}}$. The spectral method \n    combines each component found in multi-component fits of\n    a line complex into a single absorber whose column density is the sum of\n    the components' column densities.   \\emph{Left: } \\ion{H}{1}\\\n    absorbers. \\emph{Right: } \\ion{O}{6}\\ absorbers.} \n  \\label{fig:NVar_combine}\n\\end{center}\n\\end{figure*}\n\nIn Figure \\ref{fig:sliced_b_distribution} we compare the distribution of b-values for \\ion{H}{1}\\ and \\ion{O}{6}\\ absorbers over slices in column density. The slices were chosen to give a relative idea of of how the methods compare for absorbers of interest; as such, we do not show higher column density slices for \\ion{H}{1}, despite the fact that higher column absorbers are fit. We find that for \\ion{H}{1}, the distribution of b-values is shifted for the spectral method relative to the contour method at low column ($10^{13}\\leq N \\leq 10^{13.5}$~cm$^{-2}$) with the peak of the spectral distribution at 25~km~s$^{-1}$ compared to 20~km~s$^{-1}$ for the contour method. For increasing column density slices this distinction shifts with spectral the spectral distribution peaking around 30~km~s$^{-1}$ and the contour distribution peaking around 25~km~s$^{-1}$. The \\ion{O}{6}\\ distributions look quite distinct for all column density slices. It seems that compared to the contour method, the spectral method identifies an overabundance of low column absorbers with very low b-values as well as a relative lack of high column aborbers with higher b-values. \n        \n\\begin{figure*}  \n\\begin{center}\n    \\includegraphics[width=0.95\\textwidth]{sliced_b_distribution.pdf}\n  \\caption{ Histogram of b-values for absorbers with column densities in ranges $10^{13}<N<10^{13.5}$~cm$^{-2}$ (\\emph{Top row}), $10^{13.5}<N<10^{14}$~cm$^{-2}$ (\\emph{2nd row}), $10^{14}<N<10^{14.5}$~cm$^{-2}$ (\\emph{3rd row}), and  $10^{14.5}<N<10^{15}$~cm$^{-2}$ (\\emph{Bottom row}). Red bars indicate spectral absorbers, blue bars indicate contour absorbers, and purple indicates an overlap in the distributions. \\emph{Left: } \\ion{H}{1}\\ absorbers. \\emph{Right: } \\ion{O}{6}\\ absorbers.} \n  \\label{fig:sliced_b_distribution}\n\\end{center}\n\\end{figure*}\n\n\n        \n\n\\section{Approaches to Identifying IGM Absorbers}\n    \\label{sec:methods}\n\nUsing the simulations described in Section \\ref{sec:sim} and the\n\\texttt{yt} analysis toolkit\\footnote{\\url{http:\/\/yt-project.org}}\n\\citep{yt}, we generate 2000 pencil-beam\ndata arrays meant to be representative of QSO sight lines. Each line\nof sight extends from $z=0$ to $z=0.4$ and has a random starting\nposition in $(x,y,z)$ and random orientation in $(\\theta,\\phi)$. As\nthe physical distance corresponding to \n$\\Delta z = 0.4$ is much larger than the extent of the simulation box\nsize at low redshifts, the line of sight generation tool stacks rays from multiple\nepochs together. This is accomplished by pre-calculating the number of\ndata outputs and the corresponding position in redshift space needed\nto traverse the full length of the desired sample.  All pertinent\ninformation is recorded for each cell intersected by a line of sight,\nincluding baryon density, metallicity, ion densities, ion fractions,\nand temperature.  The pathlength $dl$ of the ray as it passes through\nthe cell is also recorded; this length is not constant given the\nrandom orientation of the sight line, but is on the order of the resolution of the simulation.\n\n The methods with which observers find structure in the universe is\nintrinsically different than the ways simulators\napproach the same problem. Even after generating lines of sight,\nsimulators still have access to all the information retained by the\npixels along the line of sight, or `lixels' as per \\citet{britton2011},\nfrom temperature to specific number densities. Observationally, we are\nlimited to interpreting these quantities through indirect methods such\nas fitting absorption lines, apparent optical depth, or pixel optical\ndepth. Each of these approaches is susceptible to its own systematic\neffects that may make it difficult to compare them. By performing both\nan observationally motivated analysis technique as well as a computational\none on the same lines of sight we can better correlate\nthe products of simulations with actual observations. Thus, in the\nfollowing sections we will detail several methods with which we\nanalyzed our synthetic lines of sight; one, the `spectral' method,\nthat uses only absorption line fits of synthetic spectra \nand two approaches, the `contour' and\n`cut' methods, which utilize cell-by-cell output. A sample of the\nresults of identification of absorbers using these methods can be seen\nin Figure~\\ref{fig:methods}. The relevant number densities ($\\mathrm{n_{HI}}$,\n$\\mathrm{n_{OVI}}$) are computed in both photo- and collisional ionization equilibrium, with the former assuming a uniform \\citet{2001cghr.confE..64H} radiation background. \n\n\nFor each method, after generating a full set of absorbers we filter by column density and only include \\ion{H}{1}\\ absorbers with column densities N (cm$^{-2}$) in the range $12.5\\leq \\log \\mathrm{N_{HI}} \\leq 17.2$ and \\ion{O}{6}\\ absorbers with column densities in the range $12.5\\leq \\log \\mathrm{N_{OVI}} \\leq 15.2$. We restrict our column density range on the higher end to reflect the absence of radiative transfer in our simulation. On the low end we restrict our column density range to reflect observable limits.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.45\\textwidth]{schematic_methods.pdf}\n\\caption{Schematic example of three different methods for\n  finding absorbers in mock QSO lines of sight. In this figure\n  we examine \\ion{H}{1}; this applies equally well to any\n  species. \\emph{Top row:} The ``spectral method.'' A synthetic spectrum is\n  generated using the region of number density as a function of\n  redshift  (with density fields shown in the bottom two rows). Normalized flux plotted against\n  redshift appears shifted in redshift relative to peaks in\n  number density due to line-of-sight\n  velocities. \\emph{Middle row:} The ``cut method.''  \\ion{H}{1}\\ number density as\n  a function of redshift along the line of sight. Each shaded\n  region of fixed redshift width is a separate absorber.\n  \\emph{Bottom row:}  The ``contour method.''  The same region of \\ion{H}{1}\\ number\n  density as a function of redshift as above, but regions are identified as spatially contiguous cells above the mean number density of the species. The mean number density is given by the solid line at $\\mathrm{n_{HI}}=10^{-14}$~cm$^{-3}$. The dashed line is the characteristic density for each region above the mean number density. The spatially contiguous cells identified with number densities above the characteristic densities (shaded in blue) are then identified as absorbers. }\n\\label{fig:methods}\n\\end{center}\n\\end{figure}\n        \n\\subsection{Spectral Method}\n\nIn order to mimic true observations with the data generated via\nsimulation, a synthetic QSO absorption spectrum was created and\nthen fitted for each line of sight. The fitted components then\ndetermined the absorber -- in particular, the absorber's column density and\ndoppler-value. This method will henceforth be referred to as the\n``spectral method.'' \n\n       \n\\subsubsection{Creating Absorption Spectra}\n\nFor each of the 2000 simulated QSO sight lines we create a corresponding\nabsorption spectrum by considering the effects of absorption for each \nlixel of gas. The optical depth for a cloud of gas with column density N of a\nwavelength $\\lambda$ is given by $\\tau_i(\\lambda) = C_i N a H(a,x(\\lambda))$,\nwhere $x=(\\lambda-\\lambda_o)\/\\lambda_o$, $a=\\Gamma_i\/(4\\pi \\Delta \\nu_D)$, and $C_i$\nis a transition-dependent constant. Here $\\Delta  \\nu_D = b_{th}\/(c\\lambda_i)$ and \n$\\Gamma_i$ is the reciprocal of the mean lifetime of the transition. The Voigt-Hjerting function, $H$, is given by \n\n\\begin{equation}\n  H(a,x) = \\frac{a}{\\pi}\\int_{-\\infty}^{\\infty}\\frac{e^{-y^2}}{(x-y)^2+a^2}dy\\;,\n  \\label{eq:voigt_long}\n\\end{equation}\n\nwhere $y=v\/b_{th}$. The central wavelength $\\lambda_o$ is given by \n\n\\begin{equation}\n  \\lambda_o = \\lambda_i(1+z)(1+v_{los}\/c) \\;,\n\\end{equation}\n\nwhere $\\lambda_i$ is the rest wavelength of the transition for species $i$,\n$z$ is the redshift of the lixel generating the profile, and $v_{los}$ is\nthe peculiar velocity of the lixel generating the profile. \n\nBy defining a Doppler parameter as in Equation \\eqref{eq:bth}, we\ncan define and $\\Delta \\lambda_D = (b_{th}\/c)\\lambda_i$ \nwhere $\\lambda_i$ is the resonant wavelength\nof the corresponding transition given by $\\lambda_i = hc\/E_i$. The\nthermal Doppler parameter $b_{th}$ quantifies the width of a\nsingle line:\n\n\\begin{equation}\n  b_{th}=\\sqrt{2 k_B T\/m_i} \\;.\n  \\label{eq:bth}\n\\end{equation}\n\nThe resulting normalized flux is then given by $F(\\tau) =\ne^{-\\tau}$. The absorption feature generated in this fashion is\nknown as the Voigt profile. One of these profiles is generated for\neach lixel using the algorithm detailed by \\citet{armstrong1967}. \n\nThe spectra were generated at a resolving power of\n$R=\\lambda\/\\Delta\\lambda=10^{5}$. This is substantially higher\nresolution than a typical IGM observation using Hubble's Cosmic\nOrigins Spectrograph (COS), which has resolving power of $R=2,000$ (G140L) and\n$18,000$ (G130M); however, the simulated spectra can easily be degraded\nto any desired lower resolution, and noise can be added.\n\nAs each lixel generates its own Voigt profile, large scale\nfeatures in the spectra are typically composed of contributions\nfrom multiple lixels and their corresponding column densities.\nThis effect can be seen in Figure~\\ref{fig:methods} where, despite\na wide range of \\ion{H}{1}\\ number densities over a number of cells, only four\nsignificant spectral features are visible. This causes a\ndegeneracy in our notion of b-value, since broadening can occur as a\nresult of disordered line-of-sight motion within a cloud of gas (as a\nresult of, e.g., turbulence) in addition to\nthe broadening due to thermal motion. Cells with different\nvelocities create Voigt profiles at slightly different wavelengths\ndue to the Doppler effect and, as a result, the overall line to\nlooks broader. This broadening is typically indistinguishable from the\nbroadening due to thermal motion.\n\nIn generation of the spectrum as many ions as desired\ncan be included. Fitting a single ion at a time removes any\nambiguity of what ion is being analyzed, so all further analysis\nwas completed in this manner. Our spectra were also fit without\nthe effects of noise as an initial test of our methods, though we\nnote that the method works with noisy spectra as well. \n\n\\subsubsection{Fitting Absorption Spectra}     \nDue to the challenge of fitting a large number of synthetic\nspectra, we developed an automated line-fitting pipeline. This\nsystem takes the synthetic spectra and fits a series of Voigt\nprofiles to each spectrum in a completely automated\nfashion. To automatically identify absorbers, each contiguous\nregion with normalized flux below an adjustable cut-off\n($F\/F_{cont}=0.99$) is identified. This cut-off was chosen\nprimarily to underestimate the observable limit due to noise\nlimitations, without over-fitting very shallow and hard-to-detect lines. Lowering this value by a few percent does not\nsignificantly affect results. Raising the value begins to\nsubstantially increase computational cost as regions become\nlarger and a typical region contains more lines; however, the\nresults are not typically affected as a region is generally\nconsidered successfully fit regardless of the inclusion of\nthe smaller fluctuations.\n\nExceptionally large complexes ($\\Delta \\lambda >50\\,\\r{A}$) with multiple lines\nare broken up at points of minimum absorption, if that minimum\npoint has a flux within a few percent of the adjustable\ncutoff. This is done to prevent very large regions from\ndramatically slowing down the fitting procedure due to the\ndifficulty of optimizing the fits of many lines\nsimultaneously.\n\nEach region is then taken in turn from minimum to maximum\nwavelength and fit by iteratively adding and adjusting Voigt\nprofile parameters (column density, broadening value, and\nredshift) using the least squares method until the total\n$\\chi^2$ error is smaller than an acceptable fractional error threshold\n($10^{-4}$) multiplied by the number of points in the region\nin units of normalized flux. We attempt to fit up to 8\nindividual components simultaneously. If this is\nunsuccessful or if the error begins to increase upon adding\nmore components, the fit is accepted if the total error\nis within two orders of magnitude of the desired error. Given\nthe stringent conditions used for our standard fitting procedure, these fits are\ntypically quite good, and would certainly be acceptable if noise\nwere included in our synthetic spectra (which we defer to a later paper).\nChanging the value of the acceptable fractional error threshold \nby an order of magnitude has little effect on the overall results.\n\nIn cases of ions that create more than one line (as in the\n\\ion{O}{6}\\ doublet), the lower wavelength line is fit to\nthe region as, in the case of Li-like ions (2s-2p), this is the stronger\nline. We then attempt to fit the higher wavelength line\nwith the parameters calculated for the lower wavelength\ncounterpart and if the resulting total fit has a low enough\nerror then it is accepted. This allows some amount of leeway\nfor blanketed line identification where a large line occurs in\nthe same wavelength space as another smaller line, effectively\nhiding the smaller line.\n\nAn example of this fitting procedure is shown in Figure~\\ref{fig:fitex}. \nOnce the region is identified, a single line\nis optimized to fit the whole region. Given the structure of\nthe region, a single line is insufficient to constrain the\nregion and the difference between the fit and the data is\nstill larger than allowed by the average error per point. Thus,\nanother line is added and the region is fit using two lines,\nwith three free parameters, N, $b$, and $z$. This fit is again not\ngood enough so this procedure is repeated now with three lines for\na total of nine free parameters. The three-line fit then satisfies the\naverage error per point bound and the region is considered\nsuccessfully fit.\n\n\n\\begin{figure}\n  \\includegraphics[width=0.45\\textwidth]{multifit.pdf}\n  \\caption{An example Ly$\\alpha$\\ fit for a region of wavelength space where\nthe best fit consists of multiple line components.  The x-axis\nindicates wavelength difference measured from an arbitrary reference\nwavelength, and the y-axis indicates normalized flux.  The solid blue\nline indicates the total normalized flux generated from a range of\n\\ion{H}{1}\\ number density along a line of sight, while the dashed lines\nindicate components found for the fit.}\n  \\label{fig:fitex}\n\\end{figure}\n        \nIn regions where the regular fitting tools fail to perform\nproperly and the line in consideration is Ly$\\alpha$, a more\nrobust approach is used. Absorbers with low temperatures\n($T\\leq10^4$ K) and relatively high column densities\n($N_{HI}\\sim 10^{18}$ cm$^{-2}$) generate Voigt profiles\nwith damping wings. Typical parameters that\nare appropriate for the large majority of absorbers often fail\nto converge to a good fit for absorbers with these properties.\nWhen such a region is identified, a separate set of widely\nvaried initial temperatures and column densities are tried,\nwhich allows for a more accurate fit at the expense of\nsubstantially increased computational cost. In\nthis paper we primarily examine \\ion{H}{1}\\ and \\ion{O}{6}\\ absorption. Most\nof the challenging fits are of \\ion{H}{1}\\ absorbers, due to their more\nvaried (and higher) column densities and environments. \n\n\n\n\\subsection{Cut Method}\n\\label{sec:cut}\nInstead of creating absorption spectra, simulators have the\nopportunity to examine exactly how number density and other\nphysical quantities vary with redshift along a line of sight.\nHowever, in order to compare with observation, it is necessary\nto break up this continuous variation into discrete absorbing\nstructures. A simple approach, henceforth referred to as the\n`cut method,' was employed by  \\citet{britton2011}\nin a previous analysis of the simulations examined in this\npaper. In this method, a given line of sight is cut into\npieces of constant length in redshift space where the cells\nbetween each cut contribute to that absorber. The sampling\nresolution ($\\lambda\/\\delta\\lambda$) is set at 5000; this\nallows smoothing of the region over multiple lixels while\nmaintaining a constant resolution in the range of\nobservation. A change of resolution by an order of magnitude \nin either direction does not significantly affect\nthe results. \n\nThe column density for ion x is given by the sum over cells\n$N_x=\\sum dl_i n_{x,i}$, where $dl_i$ is the pathlength and\n$n_{x,i}$ is the number density of ion x in a given absorber species in a given cell.\nThe b-value is comprised of a thermal and a non-thermal\ncomponent for a total b-value given by\n$b=\\sqrt{b_{th}^2+(\\sigma_{v})^2}$.  The thermal component is\nthe same as the spectral thermal Doppler parameter,\n$b_{th}=\\sqrt{2kT\/m_i}$. To approximate the non-thermal\ncomponent of broadening we find the variance in the line of\nsight velocities weighted by column density of each cell. All\nother relevant quantities such as ion fraction, metallicity,\nand temperature are found via column density-weighted average for all cells contributing to the\nabsorber. \n\nThis method is simple because it requires no additional\nanalysis beyond dividing the line of sight properties into\nequal redshift bins; however, it does not identify\nphysical structures at all. Large absorbers may happen to fall\nalong the boundary of two redshift bins as seen at $z\\sim0.085$\nin Figure~\\ref{fig:methods}. This may cause a single region to\nbe interpreted as multiple absorbers of lower column density\nand distort the inferred physical conditions associated with\nthe absorber. Although properties dominated by the sharp peaks\nin number density may still hold, this method cannot be\nexpected to give good intuition for the physical nature of a\nstructure because, quite simply, the method of defining\nabsorbers in not based on physics, but rather numerically\nconvenient quantities.\n\n        \n\\subsection{Contour Method}\n\\label{sec:contour}\nThe `contour method' of defining IGM absorbers was developed\nin response to the limitations of the cut method described in\nSection~\\ref{sec:cut}. Instead of cutting blindly by redshift,\nregions are identified where groups of spatially contiguous cells\nall have number densities above the mean number density of a given \nspecies. A discussion of identifying the mean number density of each\nspecies is given in Section~\\ref{sec:cutoff}, but the values adopted\nfor the remained of this work are $\\mathrm{n_{HI} = 10^{-14}~cm^{-3}}$ \nand $\\mathrm{n_{OVI} = 10^{-22}~cm^{-3}}$.\nFor each region, a peak species number density is \ndetermined. An absorber is then quantified as spatially contiguous \ncells within the original region  with number densities \nabove a characteristic density, set by some\nfraction of the peak number density. For most of the following analysis\nwe chose the characteristic density as 0.5 times the\nthe peak number density, but the effects of varying this fraction are \ninvestigated in section~\\ref{sec:cutoff}. This process is illustrated\nin Figure~\\ref{fig:schematic_contour}. Multiple absorbers can be identified \nin a single region as illustrated in Figure~\\ref{fig:methods} at \n$z \\simeq 0.086$. By setting a characteristic density for each region \nrather than a single number density cutoff for the whole species we \nensure that only the cells that maximally contribute to a given absorbing \nfeature as selected as part of the absorber, while still identifying low \ncolumn absorbers. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.45\\textwidth]{schematic_contour.pdf}\n\\caption{Schematic example of the ``contour method''. Initially, the light blue region is identified as a spatially contiguous group of cells above the mean number density (\\emph{Bottom Line}). Within this region the maximum number density is identified. The characteristic number density (\\emph{Top line}) of this region is then calculated as some fraction of the maximum number density. The absorber for this region is then identified as the spatially contguous subset of cells with number densities greater than the characteristic number density.}\n\\label{fig:schematic_contour}\n\\end{center}\n\\end{figure}\n\nAll properties of these absorbers are found using the \nsame methodology as was used for the cut method. The identification \nof the regions in this way isolates large structures in a \nphysically-motivated way that preserves overall structure. \n\n\n\\section{Results}\n\\label{sec:results}\n\n\\input{methods-compare}\n\\input{obvs-compare}\n\\input{physical-conditions}\n\n\\input{discussion}\n\\input{summary}\n\n\n\\bibliographystyle{apj}\n\n\\subsection{Comparison to Observations}\nWe now compare our synthetic absorber population with observed absorbers to assess any systematic differences, attributable to our simulation or methods. \n\n\\subsubsection{Differential and Integral dN\/dz}\n\nA common test of the accuracy of simulations is to recreate the\nobserved number density of \\ion{O}{6}\\ absorbers per unit redshift\n\\citep{tilton2012,danforth2008}, as well as the cumulative number\ndensity of \\ion{O}{6}\\ absorbers per unit redshift\n\\citep[e.g.,][]{fang2001,cen2006,oppenheimer2009,teppergarcia2011,britton2011}.\nIt is thus quite useful to understand how generating this statistic\nfrom simulations using an observationally-motivated method\nintrinsically differs from generating this statistic from absorbers\nfound using analysis of cell-by-cell output.\n\nFigure~\\ref{fig:dndz} shows this analysis for \\ion{H}{1}\\ and \\ion{O}{6}\\ absorbers\nfound in our simulation, along\nwith observational data from \\citet{tilton2012} and\n\\citet{danforth2008} for comparison for \\ion{H}{1}\\ and \\ion{O}{6}\\, respectively. We consider each component\nof a multi-component fit of a complex separately because, although\nsingle-component fits were favored in both observational cases,\ncomplexes with clear evidence of substructure were fit in a\nmulti-component fashion with each component listed separately in the\ncalculation of $d \\mathcal{N}\/dz$. In the top row of this Figure we\nshow the cumulative line number density of \\ion{H}{1}\\ and \\ion{O}{6}\\ shown in the\nstandard way -- i.e., we show the number of\nabsorbers above a given species column density at the mean redshift of\nthe simulation outputs, normalized by the total redshift interval\n$\\Delta z$ of the synthetic observations.  The bottom row displays the\ndifferential line number density normalized by the total redshift\ninterval $\\Delta z$, also known as the column density distribution function.\n\nThe two methods agree quite well at all column densities for both\nLy$\\alpha$\\ and \\ion{O}{6}\\ as seen in both the differential and integral forms.\nThere is a systematic under-prediction relative to the observed\nabsorber frequency using either simulated method in \\ion{O}{6}; however, this was also\nfound in the initial analysis presented by \\citet{britton2011}, and is\nlikely to be an intrinsic property of the simulation rather than a\nfeature of our method of determining absorber properties.\n\nIt should be noted that we consider absorbers above a column\ndensity of $10^{12.5}$~cm$^{-2}$ for the contour method, and fit below\nthe observable limit for both \\ion{H}{1}\\ and \\ion{O}{6}\\ with the spectral method using noiseless\nspectra. As a result, no significant completion correction is required\nfor low column absorbers in our simulated results, which may result in\nsystematic differences from the cited observational results.\n        \n\\begin{figure*}\n\\begin{center}\n  \\includegraphics[width=.95\\textwidth]{dndz.pdf}\n  \\caption{\\emph{Top: } Number of\n    absorbers ($\\mathcal{N}$) with a column density greater than N\n    per redshift z (i.e., cumulative line number density). \\emph{Bottom:} Number of absorbers ($\\mathcal{N}$) per column\n    density ($N$) per redshift ($z$) (i.e., line number density). \\emph{Left:} \\ion{H}{1}\\ column densities with\n    observational points and error bars from \\citet{tilton2012}. \\emph{Right:}\n    \\ion{O}{6}\\ column densities, with observational points and error bars\n    from \\citet{danforth2008}.  In all panels, the red line\n    corresponds to the spectral method, the blue line to the contour\n    method, and the black line to observations.}\n  \\label{fig:dndz}\n\\end{center}\n\\end{figure*}\n\n\n\\subsubsection{Broadening Value}\n\nThe other parameter that is found directly using the spectral method\nis the Doppler parameter, or `b-value.'  This parameter determines the width of a line and\ntypically has contributions from both thermal and non-thermal broadening.\n\nWhile observationally these components are very challenging to distinguish and\nrequire simultaneous fits of multiple species with different masses,\nin a simulation we have perfect knowledge\nof the thermal and kinematic behavior of the plasma everywhere in our\nvolume. To that end, a more detailed and precise study is\npossible. Figure~\\ref{fig:bVar} shows the median b-value plotted against\ncolumn density for both the contour and spectral methods, as well as\nthe squared fraction of total b-value due to thermal motion as a function of\ncolumn density for the contour method. We look at the squared fraction as\nthe thermal and non-thermal components of the total b-value are added in\nquadrature.\n\nThe Ly$\\alpha$\\ values agree reasonably well between methods at lower column\ndensities and but diverge slightly at high column\ndensities ($N_{\\mathrm{HI}}\\geq 10^{16}$~cm$^{-2}$). The values increase\nslightly with column to $N_{\\mathrm{HI}}\\sim10^{14.5}$~cm$^{-2}$. The\nb-values from the contour method then begin to slightly decrease again\nwhile the values derived from the spectral method remain fairly constant.\nObservational results from \\citet{tripp2008} show a similar pattern as \nthe contour method.\n\nFor \\ion{O}{6}, the median b-value increases stays roughly constant at \n$b\\sim 20 $~km s$^{-1}$ using the contour method. The spectral method \nstays roughly constant at  $b\\sim 15$~km s$^{-1}$. \nWe do not see the trend indicated by \\citet{tripp2008} of increasing\nb-value with column density for either method.\nWe note that the biggest divergence between the two\nmethods occurs where the thermal component of the b-value becomes less\ndominant; however, this does not cause a systematic under or over\nprediction of b-value by either method across both species. \n\nWe note that here we only consider the b-values of each individual\nspectral component in this analysis. Although combining complexes was\nuseful in an attempt to correlate spectral structures with contour\nstructures as in Sections \\ref{sec:NVar}, this method obscures the\nunderlying structure that can be determined with the aid of the\nb-values. We thus do not expect the contour and spectral b-value\ndistributions to be the same, as they simply do not correspond to the\nsame quantities; the contour method takes the variation in bulk motion\nof separate features and gives a higher non-thermal motion component of\nthe b-value, whereas the spectral method divides up the same region of\nphysical space into separate absorbing components. \n\n\n\\begin{figure*}\n\\begin{center}\n    \\includegraphics[width=0.95\\textwidth]{bVar.pdf}\n  \\caption{\\emph{Top: } Squared fraction (due to component b-value \n    addition in quadrature) of b-value due to thermal motion as a\n    function of column density for the contour method. \\emph{Bottom: }Mean b-value vs Column\n    density for contour (\\emph{blue}) and spectral (\\emph{red})\n    methods. Observational points (\\emph{black}) from\n    \\citet{tilton2012}. Error bars show $1\\sigma$\n    deviations. \\emph{Left:} \\ion{H}{1}\\ absorbers. \\emph{Right:} \\ion{O}{6}\\\n    absorbers.}\n  \\label{fig:bVar}\n\\end{center}\n\\end{figure*}\n\n\\subsection{Physical Conditions of Absorbers}\nIn order to extend our understanding of the WHIM and its relationship to observations, we examine the physical environment associated with the absorbers identified with the contour method.\n\n\n\\subsubsection{Median Quantities Of Absorber Systems}\n\nFigure~\\ref{fig:param} details the physical condition of the gas as a\nfunction of absorber column density. \nGas temperature appears to increase and then decrease with increasing column\ndensity for \\ion{H}{1}\\  absorbers, while\nthe temperature of \\ion{O}{6}\\ absorbers decreases with increasing\ncolumn. The temperature\nbehavior for \\ion{H}{1}\\  also typically has much less variance at a given column density. \nMetallicity, defined as the total mass in elements heavier than helium relative \nto the total gas mass normalized by the Solar metal fraction, increases \nsteadily with increasing \\ion{O}{6}\\ column density.\nThere is slight increase with increasing \\ion{H}{1}\\ column density, albeit \nwith an increasing variance at low $\\mathrm{N_{HI}}$, suggesting that \nlow $\\mathrm{N_{HI}}$ systems may trace a large variety of environments. \nClear trends in \\ion{O}{6}\\ fraction \ncan be seen as a function of \\ion{H}{1}\\ column density, with a peak at \nN$_{HI} \\simeq 10^{14}$~cm$^{-2}$ and lower fractions at larger and smaller columns.\nThe median number density of H increases steadily for both \\ion{H}{1}\\ \nand \\ion{O}{6}\\ \nabsorbers, although over a wider range with smaller variance for \\ion{H}{1}.  The fraction \nof \\ion{H}{1}\\ ionization steadily increases as a function of increasing \\ion{H}{1} column density. The \\ion{H}{1}\\\nfraction also seems to increase with \\ion{O}{6}\\ column density, but the variance in this\ntrend is far larger. These trends appear largely consistent with those initially\nprovided by \\citet{britton2011}. \n\n\\begin{figure*}\n\\begin{center}\n    \\includegraphics[width=.95\\textwidth]{pVar.pdf}  \n  \\caption{Median temperature, metallicity, \\ion{O}{6}\\ fraction, total H number density,\n    and \\ion{H}{1}\\ fraction plotted against column density for absorbers found using the contour method. Error bars show first and\n    fourth quartiles. \\emph{Left:} Ly$\\alpha$\\ absorbers. \\emph{Right:} \\ion{O}{6}\\\n    absorbers.}\n  \\label{fig:param}\n\\end{center}\n\\end{figure*}\n            \n\\subsubsection{Thermal State of \\ion{O}{6}\\ Absorbers}\nOne of the primary purposes of these simulations was to investigate the \nutility of \\ion{O}{6}\\ as a tracer of the WHIM. The initial analysis presented by\n\\citet{britton2011} showed a bimodality in the temperature distribution\nof \\ion{O}{6}\\ also seen in \\citet{teppergarcia2011}. The bimodality was centered\nat $T\\sim10^5$ K, with 57\\% of \\ion{O}{6}\\ absorbers found\naround temperatures of $10^{5.5}$ K in the WHIM phase and 37\\% found\nat temperatures of $10^{4.5}$ K in the warm phase (and the remaining 6\\%\nfound at higher densities in what \\citet{britton2011} defined as the\n`condensed' phase, with a baryon overdensity of $\\Delta_b \\geq 1000$). Such a\nbimodality suggests that both collisionally ionized and photo-ionized \\ion{O}{6}\\ are present in significant amounts in the IGM.\n\nWe then must ask if these statistics are proportionally recreated when\nlooking at absorbers found using the methods presented in this\npaper. \\citet{britton2011} found there to be no bias between the true phase distribution of \\ion{O}{6}\\ in the simulation and that inferred from absorbers created with the cut method,\nwhereas \\citet{teppergarcia2011} found \\ion{O}{6}\\ absorption to be biased towards higher temperatures. Figure~\\ref{fig:phase} shows the baryon overdensity and\ntemperature for each absorber found using the spectral method.  Baryon\nproperties are determined by using the mean overdensity and\ntemperature of the gas in the same absorber found with the contour method, \nas determined using column density-weighted averages of all cells in the absorber.\nThe distribution shows no evidence of a strong temperature bimodality, but \ninstead shows a roughly smooth distribution over the temperature range \n$4.5 \\leq \\log (T \/ K) \\leq 5.5$. If a bimodality\ndoes exist it is only present in the very highest column absorbers. There are no\nsignificant discrepancies between the absorbers identified here with\nthe contour method versus those identified initially with the cut\nmethod. We also find similar overall phase fractions, with $69\\%$ of\n\\ion{O}{6}\\ absorbers in the WHIM phase, $30\\%$ of absorbers in the warm\nphase, and $1\\%$ of absorbers in the condensed phase.\n            \n\\begin{figure*}\n  \\begin{center}\n  \\includegraphics[width=0.85\\textwidth]{phase.pdf}\n  \\caption{Thermal-state distribution of \\ion{O}{6}\\ absorbers.  Each absorber is plotted\n    as a function of its mean baryon overdensity\n    (defined as $\\rho\/\\rho_o$) and temperature,\n    with points colored according to absorber column density. The dashed line\n    indicates where collisional ionization equals photoionization (collisional\n    ionization dominates above the curve).\n    Histograms along the top and right side of the scatter plot show\n    overdensity and\n    temperature, respectively, for all \\ion{O}{6}\\ absorbers with column\n    densities greater than $10^{15}$~cm$^{-2}$\n    (light blue), 10$^{14.5}$~cm$^{-2}$ (dark blue), $10^{14}$~cm$^{-2}$ (dark green),\n     and $10^{13}$~cm$^{-2}$ (light green).}\n  \\label{fig:phase}\n  \\end{center}\n\\end{figure*}\n\n\\subsubsection{Cool vs. Warm IGM}\n\\label{sec:temp_bimodality}\n\nThe WHIM is often defined as gas with temperatures in the range \n$10^5-10^7$ K \\citep{1999ApJ...514....1C,2001ApJ...552..473D}. \nUsing temperatures derived via the contour method, we attempt to \ndetermine the characteristics of observables for absorbers in this \ntemperature range and assess any systematic differences between \nabsorbers with lower temperatures. For this analysis we henceforth\ndefine a WHIM asborber as an absorber with $\\mathrm{T>10^5~K}$ and\na warm absorber as one with $\\mathrm{T<10^5~K}$.\nFigure \\ref{fig:temperature_bimodality} \nshows column density and $b$-value histogrammed separately for \nWHIM and warm absorbers. The warm \\ion{H}{1}\\ absorbers have a roughly \nlinear distribution by column density in logspace, while the WHIM \n\\ion{H}{1}\\ absorbers fall off sharply after $\\mathrm{N_{HI}}\\sim 10^{14}$~cm$^{-2}$. \nThe warm and WHIM \\ion{O}{6}\\ absorbers show very similar distributions \nby column density, with the WHIM absorbers dominating slightly below \n$N_{OVI}\\sim 10^{14}$~cm$^{-2}$. The $b$-value histograms show \ntwo distinct distributions for the WHIM and warm absorbers \nfor both \\ion{H}{1}\\ and \\ion{O}{6}. \\ion{H}{1}\\ warm absorbers have a peak $b$-value \nof $~15$~km~s$^{-1}$ whereas WHIM absorbers peak at around \n45~km~s$^{-1}$. In \\ion{O}{6}, warm absorbers peak at 10~km~s$^{-1}$ while \nWHIM absorbers peak at 20~km~s$^{-1}$.\n\n\n\\begin{figure*}  \n\n\\begin{center}\n  \\includegraphics[width=0.95\\textwidth]{temperature_bimodality.pdf}\n  \\caption{  Absorbers binned by column density (\\emph{top}) and b-value (\\emph{bottom}). WHIM absorbers are in blue, warm absorbers are in purple.\\emph{Left: } \\ion{H}{1}\\ absorbers. \\emph{Right: } \\ion{O}{6}\\ absorbers. All absorbers have been identified using the contour method.}\n  \\label{fig:temperature_bimodality}\n\\end{center}\n\\end{figure*}\n\nAfter establishing the relative distributions of absorber observables \nin the warm and WHIM phases, we examine the relationship between the \nobservables. In Figure \\ref{fig:temperature_bVar} we show median \nb-value plotted over column density bins. We find a general decrease \nin the b-value of an absorber with column density for WHIM phase \\ion{H}{1}\\ \nabsorbers. A absorber with column density in the last bin appears to \nreverse the trend, but this point is not statistically significant. \nFor \\ion{H}{1}\\ absorbers in the warm phase, b-values increase with column density \nuntil roughly N$_{HI}=10^{15}$~cm$^{-2}$ and then begin to decrease again \nslightly. For \\ion{O}{6}\\ absorbers both the warm and WHIM absorbers have \nslightly increasing b-values with column density, but the median b-value \nof WHIM absorbers is typically 5-10~km~s$^{-1}$ higher than the median \nb-value of the warm absorbers for a given column density bin.\n\n\n\\begin{figure*}  \n\n\\begin{center}\n    \\includegraphics[width=.95\\textwidth]{temperature_bVar.pdf}\n  \\caption{  Median b-value vs column\n    density for WHIM (\\emph{blue}) and warm (\\emph{purple}) \n    absorbers. Error bars show 1st and 4th quartiles. \\emph{Left: } \\ion{H}{1}\\\n    absorbers. \\emph{Right: } \\ion{O}{6}\\ absorbers.} \n  \\label{fig:temperature_bVar}\n\\end{center}\n\\end{figure*}\n\n\nIn Figure \\ref{fig:temperature_2dhist} we show the fraction of \nWHIM absorbers out of total absorbers histogrammed two-dimensionally \nalong column density and $b$-value. We find that above $b$-values of \n40 and 15 km~s$^{-1}$ for \\ion{H}{1}\\ and \\ion{O}{6}, respectively, the fraction \nis dominated by WHIM absorbers. These sharp cutoffs are to be expected \nbecause for a temperature of at least $10^5$ K there is a minimum \n$b$-value given by Equation \\ref{eq:bth}. We also see that the higher \n$b$-values are associated with lower column densities, although this \ncorrelation is stronger for \\ion{H}{1}\\ absorbers.\n\n\\begin{figure*}\n\\begin{center}\n    \\includegraphics[width=.95\\textwidth]{temperature_2dhist.pdf}\n  \\caption{ Fraction of total absorbers that are WHIM absorbers placed into two-dimensional histograms of column density and b-value. The color shows the fraction of total absorbers where red indicates no absorbers in the WHIM and white indicates no absorbers of any phase. \\emph{Left: } \\ion{H}{1}\\ absorbers. \\emph{Right: } \\ion{O}{6}\\ absorbers.} \n  \\label{fig:temperature_2dhist}\n\\end{center}\n\\end{figure*}\n\n\n\n\\subsubsection{Collisionally Ionized vs. Photoionized \\ion{O}{6}}\n\nIn Figure \\ref{fig:phase} we plotted a dotted line indicating where \ncollisional ionization begins to dominate over photoionization for \\ion{O}{6}, given by \n\n\\begin{equation}\n  \\begin{split}\n    \\log \\rho \/ \\bar{\\rho}_b = \n    7.68  \\left( \\frac{1+z}{1.2} \\right)^{-3}  T^{-1\/2} \\left(1 + \\frac{T}{1.32 \\times 10^7}\\right)\\\\\n    \\times  e^{1.32 \\times 10^6 \/ T}\n  \\end{split}\n\\end{equation}\n\nderived in \\citet{shull2012}, where $\\bar{\\rho}_b$ is the\nuniversal mean\nbaryon density such that $\\rho\/\\bar{\\rho}_b = \\Delta_b$.  This\nrelation is valid in the density regime where the IGM is optically\nthin to the metagalactic ionizing background, which includes all of\nthe absorbers shown in Figure~\\ref{fig:phase}.  In this section we \nperform a similar analysis as was done in section \\ref{sec:temp_bimodality}, \nbut instead of making a temperature cut to distinguish between WHIM \nand warm absorbers, we differentiate between absorbers where collisional \nionization dominates and where photoionization dominates. As this \ndifferentiation is only appropriate for \\ion{O}{6}, we do not show results \nfor \\ion{H}{1}\\ absorbers here. \n\nFigure \\ref{fig:ionization_bimodality} shows column density, $b$-value, \nand temperature histogrammed for photoionization dominated and \ncollisional ionization dominated absorbers. The distribution of \ncolumn density shows no significant difference for the two absorber \npopulations. The $b$-value distributions both appear roughly exponential \nbut the photoionization dominated population's distribution peaks at \n$b$ $\\sim 15$~km~s$^{-1}$. while the collisional ionization dominated \npopulation peaks at $b\\sim 20$~km~s$^{-1}$. Similarly, both temperature \ndistributions appear roughly gaussian with photoionization dominated \npeaking at T$\\sim 10^5$~K and collisionally ionization dominated at \nT$\\sim 10^{5.5}$~K. \n\n\\begin{figure*}  \n\\begin{center}\n  \\includegraphics[width=0.95\\textwidth]{ionization_bimodality.pdf}\n  \\caption{\\ion{O}{6}\\ absorbers binned by column density (\\emph{left}), b-value (\\emph{center}), and temperature (\\emph{right}). Collisional ionization dominated absorbers are in blue, photoionization dominated absorbers are in purple. }\n  \\label{fig:ionization_bimodality}\n\\end{center}\n\\end{figure*}\n\n\nIn an approach similar to the one in Figure \\ref{fig:temperature_2dhist}, \nFigure \\ref{fig:ionization_2dhist} shows the fraction of collisional \nionization-dominated absorbers out of total absorbers in a two \ndimensional histogram of column density and $b$-value. The results are \nqualitatively similar to those seen in the \\ion{O}{6}\\ panel of Figure \n\\ref{fig:temperature_2dhist}, but with a slightly smoother transition \nalong the $b$-value axis.\n\n\\begin{figure}\n\\begin{center}\n    \\includegraphics[width=0.45\\textwidth]{ionization_2dhist.pdf}\n  \\caption{Fraction of total \\ion{O}{6}\\ absorbers that are collisional ionization dominated placed into two-dimensional histograms of column density and b-value. The color shows the fraction of total absorbers where red indicates no absorbers in the WHIM and white indicates no absorbers of any phase.  } \n  \\label{fig:ionization_2dhist}\n\\end{center}\n\\end{figure}\n\n\n\\subsubsection{Relating \\ion{H}{1}\\ and \\ion{O}{6}}\n\nOne advantage of using the contour method is that all information \nassociated with a single grid cell is also associated with a given \nlixel and consequently associated with a given absorber. This \nallows us to identify \\ion{H}{1}\\ absorbers and then automatically \nidentify the associated \\ion{O}{6}\\ content by summing the \\ion{O}{6}\\ number \ndensities along the lixels in the absorber. Thus, after identifying \n\\ion{H}{1}\\ absorbers using the contour method we find the column density \nof \\ion{O}{6}\\ associated with this \\ion{H}{1}\\ identified set of cells. We plot \nthis associated \\ion{O}{6}\\ column density as a function of the original \n\\ion{H}{1}\\ number density and b-value in Figure \\ref{fig:OVI_in_HI}. \n\nWe find that the associated \\ion{O}{6}\\ column shows very different \nbehavior for WHIM absorbers ($T>10^5~K$) versus warm absorbers. In \nnearly all cases but the highest column \\ion{H}{1}\\ absorbers, the \nassociated \\ion{O}{6}\\ column density is significantly greater for WHIM \nabsorbers than for warm absorbers. As a function of \\ion{H}{1}\\ column \ndensity the \\ion{O}{6}\\ column for WHIM absorbers increases and then \ndecreases, with the peak at around $10^{14}$~cm$^{-2}$. The \nmedian \\ion{O}{6}\\ column for warm absorbers increases fairly steadily \nas a function of \\ion{H}{1}\\ column density. \n\nThe median \\ion{O}{6}\\ column density increases steadily as a function \nof b-value starting with the minimum possible b-value for WHIM \nabsorbers. Warm absorbers show an initially sharp increase, but after \n$b\\sim 60$~km~s$^{-1}$ the points become statistically insignificant. \n\n\n\\begin{figure*}  \n\\begin{center}\n  \\includegraphics[width=0.95\\textwidth]{OVI_in_HI.pdf}\n  \\caption{  Median associated \\ion{O}{6}\\ column density versus \\ion{H}{1}\\ column density (\\emph{Left}) and \\ion{H}{1}\\ b-value (\\emph{Right}). Plotted for both WHIM (\\emph{blue}) and warm (\\emph{purple}) absorbers. Error bars show 1st and 4th quartiles. Dashed line indicates minimum b-value for a WHIM absorber, as given by Equation \\eqref{eq:bth}.}\n  \\label{fig:OVI_in_HI}\n\\end{center}\n\\end{figure*}\n\n\n\n\\subsubsection{Metallicity Ionization Fraction Product}\n\nThe product $(Z\/Z_\\odot)\\times f_{OVI}$, where $(Z\/Z_\\odot)$ is the gas\nmetallicity in solar units and $f_{OVI}$ is the fraction of oxygen\nin the \\ion{O}{6}~ionization state, is of key importance for constraining\nthe budget of baryons traced by \\ion{O}{6},  as given by \n\n\\begin{equation}\n  \\Omega_b^{\\mathrm{OVI}}=\n  \\left[\n  \\frac{\\mu_bH_o}{c\\rho_{cr}(O\/H)_\\odot}\\right] \\int_{N_{min}}^{N_{max}}\n  \\left(\\frac{\\partial^2\\mathcal{N}}{\\partial z\\partial N}\\right)\\frac{N}{Z_O(N)f_{\\mathrm{OVI}}(N)}dN\n  \\label{eq:baryonOVI}\n\\end{equation}\n\nwhere $\\mu_b$ is the mean baryon mass per hydrogen, $\\rho_{cr}$ is\nthe cosmic closure density, and $(O\/H)_\\odot$ is the solar oxygen abundance.\n\nPreviously the product $(Z\/Z_\\odot)\\times f_{OVI}$ has been\nassumed to be constant, with typical estimates of $Z\/Z_\\odot=0.1$ and\n$f_{OVI}=0.2$, giving a product of $0.02$. \\citet{teppergarcia2011} first\nexploited the advantage of simulations in calculating $\\Omega_{\\mathrm{OVI}}$\ntaking into account the ionisation fraction and metallicity of each\nindividual absorber, and \nprevious analysis of our simulations using the cut method \ncontinued this study to find that\nthis product varied in a power-law distribution\nproportional to $(N_{OVI})^\\gamma,\\, \\gamma={0.7}$. Using this variation with an\nobservational fit of $\\partial\\mathcal{N}\/\\partial z \\propto N^{-\\beta},\\, \\beta=2.0$ \\citep{danforth2008}, one\ncan constrain the baryon budget as follows: \n\\begin{equation}\n  \\Omega_b^{\\mathrm{OVI}}\\propto \\int_{N_{min}}^{N_{max}}\n  N^{1-\\beta-\\gamma}dN  \\; .\n\\end{equation}\n\nAs can be seen in Figure~\\ref{fig:zzf}, we find a similar power-law \ndistribution when using our more physically-motivated contour method\nto find absorbers.  The product of ($Z\/Z_{\\odot}$) and $f_{OVI}$ is\nproportional to $(N_{OVI})^{0.7}$. In an effort to make this result applicable\nto observational surveys, we limit the absorbers we consider for the\naverage and fit to those with \\ion{O}{6}\\ column densities between\n$10^{13}$~cm$^{-2}$ and $10^{15}$~cm$^{-2}$. We then weight \nthe product by $(N_{\\mathrm{OVI}})^{1-\\beta},\\, \\beta=2$ \\citep{danforth2008} to estimate the\naverage contribution to the estimate \n$(Z\/Z_\\odot)f_{\\mathrm{OVI}}=0.007$, which is once again a significantly lower value\nthan previous literature suggests.  As before, we only use absorbers within the\ncolumn density range $10^{13}$~cm$^{-2}\\leq N_{\\mathrm{OVI}}\\leq 10^{15}$~cm$^{-2}$ to mimic\nobservational results. For further analysis of the implications of a lower value of the product $(Z\/Z_\\odot)\\times f_{OVI}$ , we direct readers to \\citet{shull2012}.\n\n\\begin{figure}\n  \\includegraphics[width=0.45\\textwidth]{zzf.pdf}\n  \\caption{Values of the product $(Z\/Z_\\odot) \\times f_{OVI}$ over column\n    density.   The vertical gray dotted lines show limits of\n    points that were included in averages and fit, representative of\n    range of typical \\ion{O}{6}\\ surveys \\citep{danforth2008}. The\n    horizontal blue dashed\n    line shows the previously assumed value of 0.02\n    obtained by using the standard values of $Z\/Z_\\odot=0.1$ and\n    $f_{OVI}=0.2$. The horizontal red dashed line\n    shows the average of column-limited points\n    weighted by $(N_{OVI})^{-1}$. The horizontal green dashed line \n    shows value previously obtained in a similar $\\log(N_{OVI})$\n    weighted average by \\citet{shull2012}.  The solid yellow line\n    indicates a power law fit of the column-limited sample of\n    points given\n    by $(Z\/Z_\\odot)\\times f_{OVI} = 0.021\\times[N_{OVI}\/(10^{14}\n    \\mathrm{cm}^{-2})]^{0.715}$. }\n  \\label{fig:zzf}\n\\end{figure}\n\t\t\n\n\t\t\t\n\n\\section{Simulation}\n\\label{sec:sim}\n\nIn this work, we analyze a simulation performed with the open source\ncosmological adaptive mesh refinement + N-body code,\n\\texttt{Enzo}\\footnote{\\url{http:\/\/enzo-project.org}}.\n\\citep{2013arXiv1307.2265T, 2004astro.ph..3044O, 2007arXiv0705.1556N}.\nThe simulation, which is run 50\\_1024\\_2 from \\citet{britton2011}, has\na comoving box size of 50 Mpc\/$h$ with 1024$^{3}$ grid cells and dark\nmatter particles, corresponding to a comoving spatial resolution of 49\nkpc\/$h$ and a dark matter particle mass of $7\\times10^{6}\\ M_{\\odot}$.\nThe simulation includes the metallicity dependent radiative cooling\nmethod of \\citet{2008MNRAS.385.1443S}, modified to include a\nmetagalactic UV background and a modified version of the star\nformation and feedback method of \\citet{1992ApJ...399L.113C} that\ninjects stellar feedback into a $3\\times3\\times3$ cube of grid cells\ncentered on the star particle (referred to in \\citet{britton2011} as\nthe ``distributed'' method). Note that the distributed feedback method\ndoes not take into account a physical scale, it suffers from the same\noverproduction of stars and metals as its predecessor when used with higher \nresolution simulations. However,  \\citet{britton2011} have shown that this\nsimulation is able to reasonably reproduce both the observed global\nstar formation history \\citep{0004-637X-651-1-142} and the number density of \\ion{O}{6}\\ absorbers per\nunit redshift ($d{\\cal N}\/dz$) in the redshift range $0 \\le z \\le 0.4$.\n\\citet{britton2011} performed simulations using two different stellar\nfeedback models.  However, since the goal of this work is to\nunderstand how the statistics of synthetic absorbers are affected by\nthe method with which they are produced, we choose to focus on just the\nsingle simulation from \\citet{britton2011} that best matched\nobservations.\n\n\\section{Summary}\n\\label{sec:summary}\nIn this paper, we compare two primary methods for finding\nIGM absorbers along lines of sight cast through a simulation box.  One\nmethod (the ``spectral method'') uses synthetic absorption line\nspectra, and is meant to directly correspond to observational attempts\nto find structure through fitting Voigt profiles to variations in\nflux.  The other primary method (the ``contour method'') relies on\ndefining absorbers by associating contiguous regions along the line of\nsight based on a threshold number density of the species of interest.\nAfter comparing these methods to each other, we compare to observational data of \\ion{H}{1}\\ and \\ion{O}{6}\\ absorption\nline systems.  The key results of this paper are as follows:\n\n\\begin{enumerate}\n\n\\item The two methods give comparable column densities for a given\nabsorber, although there is some difficulty in creating a one-to-one\ncorrespondence of absorbers between the two methods. The primary issue\nfor comparing absorbers generated with the two methods appears to be how \none decides whether an absorber is a single coherent structure or a complex.\n\n\\item The number of \\ion{H}{1}\\ and \\ion{O}{6}\\  absorbers per column density per unit redshift, or $d{\\cal N}\/dz$, traced by the two\nmethods give similar results, indicating that the two methods find\nsimilar amounts of overall baryons in these ionization states regardless of the ability to match each\nindividual absorber. \\ion{H}{1}\\ $d{\\cal N}\/dz$\\ compares favorably to observation,\nwhile the \\ion{O}{6}\\ $d{\\cal N}\/dz$\\ in our simulation underpredicts the observational results.\nThis is likely a shortcoming of the simulation itself, as the abundance \nof OVI depends sensitively on the assumptions of star formation, feedback, and metal transport.\n\n\\item The distribution of Doppler parameters (or b-values) by column\ndensity in our simulations are similar using the contour and spectral\nmethods in both Ly$\\alpha$\\ and \\ion{O}{6}.\nBoth methods of extracting b-values from our simulation match\nobservations of Ly$\\alpha$\\ systems as a function of column density. Neither method\nprovides a particularly good fit for the Doppler parameters measured in observed \\ion{O}{6}\\ systems; we speculate that this \nis due to our choice stellar feedback algorithms.\n\n\\item The distribution of \\ion{O}{6}\\ absorbers over baryon overdensity and\ntemperature was found to be similar to a previous analysis of the same simulation data,\nbut we do not see evidence for a bimodality in absorber distribution by temperature.\nInstead, we see that \\ion{O}{6}\\ absorbers are distributed smoothly in temperature-space \nfrom $4.5 < \\log (T \/ K) < 5.5$.\n\n\\item Using the contour method, we find that the relationship between linear size and column density for a given absorber scales as L~$\\propto$~N$^{-0.12}$, rather than the analytically-predicted exponent $-1\/3$.  The reason for this is unclear, though we speculate that it may be related to simulation resolution or physics.\n\n\\item We examine the individual properties of warm vs. WHIM tracing \\ion{H}{1}\\ and \\ion{O}{6}\\ absorbers (warm vs. WHIM having temperatures greater and less than $10^5$~K respectively) as well as photo- vs. collisional ionization dominated OVI absorbers.  \\ion{H}{1}\\ warm\/WHIM-tracing absorbers show slightly different column density distributions with number of WHIM absorbers of a given column density falling off much more sharply than number of warm absorbers after $\\mathrm{N_{HI}}\\sim10^{14}$~cm$^{-2}$. \\ion{O}{6}\\ show similarly shaped column density distributions, albeit with different normalizations.  \\ion{H}{1}\\ and \\ion{O}{6}\\ absorbers associated with WHIM gas have systematically higher b-values.   \\ion{H}{1}\\ and \\ion{O}{6}\\ absorbers associated with warm gas have b-value distributions centered around 12 and 10~km~s$^{-1}$, respectively, while \\ion{H}{1}\\ and \\ion{O}{6}\\ absorbers associated with the WHIM have b-value distributions centered around 45 and 20~km~s$^{-1}$.  Dividing \\ion{O}{6}\\ absorbers into photoionized  and collisionally ionized populations shows a similar results to the warm\/WHIM division.\n\n\\item We investigate the association of warm and WHIM-tracing \\ion{H}{1}\\ absorbers with \\ion{O}{6}\\ absorbers.  For \\ion{H}{1}\\ absorbers emanating from warm gas, there is a positive correlation between \\ion{H}{1}\\ column density and column density of associated \\ion{O}{6}.  Higher column density warm \\ion{H}{1}\\ absorbers tend to be associated with higher column density \\ion{O}{6}\\ absorbers.  Comparatively, low column density WHIM-tracing \\ion{H}{1}\\ absorbers are associated with higher column density \\ion{O}{6}\\ absorbers than are the warm-tracing \\ion{H}{1}.  However, there does not exist such a trend of increasing \\ion{O}{6}\\ column density with increasing \\ion{H}{1}\\ column density.  Instead, the average associated \\ion{O}{6}\\ column density peaks at $\\sim10^{13}$~cm$^{-2}$ at an \\ion{H}{1}\\ column density of $\\sim 10^{14}$~cm$^{-2}$.\n\n\\item Finally, we study the relation between the column density of \\ion{O}{6}\\ absorbers and the value of  $(Z\/Z_\\odot)\\times\nf_{\\mathrm{OVI}}$, finding that $f_{\\mathrm{OVI}}\\,\\propto\\,\n[N_{\\mathrm{OVI}}\/(10^{14}\\,\\mathrm{cm}^{-2})]^{0.7}$.  Over the column density range $13\\leq \\log N_{OVI}\/ \\mathrm{cm}^{-2} \\leq 15$ this yields an average value of $(Z\/Z_\\odot)\\times\nf_{\\mathrm{OVI}} = 0.007$, in reasonable agreement with \\citet{shull2012}, but nearly a factor of three lower than earlier estimates.  This would imply that \\ion{O}{6}\\ traces roughly triple the number of baryons previously thought.\n\n\\end{enumerate}\n\nIn subsequent papers we hope to expand our analysis of the systematic\ndifferences in the contour and spectral methods, and use the nearly\none-to-one correlation between the two sets of absorbers to better\nunderstand the correspondence between properties of absorption line\nsystems to features in physical structure. We also intend to look at\nthe effects of noise, as well as line blanketing when fitting multiple\nions together.\n\n\\acknowledgments\n\nThis work was supported by NASA through grants NNX09AD80G,\nNNX12AC98G, NNX08AC14G, and NNX07AG77G,  and by the NSF through AST grant 0908819.  The simulations\npresented here were performed and analyzed on the NICS Kraken and\nNautilus supercomputing resources under XSEDE allocations TG-AST090040\nand TG-AST120009.  We thank Charles Danforth for helpful discussions\nduring the course of preparing this manuscript.  BWO was supported in\npart by the MSU Institute for Cyber-Enabled Research and by Hubble Theory Grant  HST-AR-13261.01-A.  HE was\nsupported in part by the MSU Hantel Fellowship and the MSU College of\nNatural Sciences.  \\texttt{Enzo} and \\texttt{yt} are developed by a\nlarge number of independent research from numerous institutions around\nthe world.  Their commitment to open science has helped make this\nwork possible.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nAutonomous vehicles have attracted significant attention in the last few years. These vehicles require a proper perception and understanding of the world to determine their localization. In these scenarios, Visual Odometry (VO) methods provide a solution by estimating the egomotion using only visual changes from the input images. These methods require the environment to have sufficient light, the objects to have texture, and the subsequent images to overlap. However, traditional VO methods still present severe issues in real-world environments due to sudden changes in the agent's speed, changes in the scene such as illumination, shadows, occlusions, and simultaneous motion of numerous objects~\\cite{geiger_vision_2013}. In recent years, Deep Learning (DL) appeared as a novel way to learn, directly from the data, the various nonlinear factors that influence scene generation and motion~\\cite{goodfellow_deep_2016}. DL methods commonly outperform either direct or indirect hand-crafted solutions and traditional learning methods that usually suffer from non-linearities~\\cite{ciarfuglia_evaluation_2014,guizilini_semi-parametric_2013,roberts_memory-based_2008}.\n\nHowever, deep learning methods for visual odometry make extensive use of convolutional neural networks (CNN), which add a substantial cost when dealing with high-resolution images. Further, more input data does not mean a better prediction; on the contrary, the network may have to learn how to filter out useless information. Therefore, the implementation of computationally efficient and lightweight architectures, especially for mobile devices, has attracted significant interest in approaching the problem from a new perspective. Though capturing only the necessary information is fundamental, learning where to look requires elaborating several cognitive concepts, such as attention. \n\n\\begin{figure}[t!]\n    \\centering\n    \\subfloat{{\\includegraphics[width=0.98\\linewidth]{images\/092.png} }}\n    \\vspace{-0.6em}\n    \\subfloat{{\\includegraphics[width=0.48\\linewidth]{images\/095.png} }}\n    \\subfloat{{\\includegraphics[width=0.48\\linewidth]{images\/100.png} }}\n\n    \\caption{Sequences of eight observations by frame. A learned policy defines the patch location (red square). The total input information corresponds to only 5.7\\% of the total available.}\n    \\label{fig:baseline_all_glimpses}\n    \\vspace{-1.5em}\n\\end{figure}\n\nIn this context, the Recurrent Attention Model (RAM) \\cite{mnih2014recurrent} have emerged as a novel architecture, which implements a recurrent attentional glimpse, incorporating the attention concept by incrementally selecting the essential pieces of information. One of the RAM's main advantages is employing Reinforcement Learning (RL) to guide the glimpse sensor through the image; the RL paradigm allows the model to learn a more robust and efficient policy by trial and error. However, RAM was introduced mainly as a concept proof, only implemented for classification tasks on the MNIST~\\cite{lecun1998mnist} dataset. RAM also uses the REINFORCE rule~\\cite{williams1992simple} to guide the glimpse sensor, but this algorithm presents convergence problems and slowness in challenging scenarios.\n\nTherefore, in this work, we propose a monocular end-to-end visual odometry architecture --- RAM-VO --- that employs the reinforcement learning paradigm to train an attentional glimpse sensor over time. The proposed RAM-VO architecture extends RAM by introducing spatial and temporal elements to enable the 6-degree-of-freedom (6-DoF) pose regression in real-world sequences. Furthermore, RAM-VO is more computationally efficient than similar VO methods due to the addition of attentional mechanisms and the use of Proximal Policy Optimization (PPO)~\\cite{schulman2017proximal} to learn a robust policy. To the best of our knowledge, this is the first architecture to perform visual odometry that implements reinforcement learning in part of the pipeline. \n\n\\subsection{Contributions}\n\nThis work provides the following contributions:\n\n\\begin{itemize}\n\n    \\item A lightweight VO method that selects the important input information via attentional mechanisms;\n    \n    \\item The first visual odometry architecture that implements reinforcement learning in part of the pipeline;\n    \n    \\item Several experiments on KITTI~\\cite{geiger_vision_2013} sequences demonstrating the validity and efficiency of RAM-VO.\n\n\\end{itemize}\n\n\n\n\\section{Related Work}\n\nThe visual odometry field has seen a massive increase in architectures and publications in recent years. In 2015, Konda et al.~\\cite{konda_learning_2015} proposed the first architecture in the field, implemented as an end-to-end CNN model to estimate direction and velocity from raw stereo images. In the same period, extracting the optical flow was a common practice to initialize the models~\\cite{muller_flowdometry_2017}. Subsequent architectures coupled LSTM layers to provide temporal representation~\\cite{mohanty_deepvo_2016}. In 2017, Wang et al.~\\cite{wang_deepvo_2017} proposed DeepVO, which is an end-to-end monocular architecture capable of extracting visual features directly from the input raw images with a CNN; and determining temporal relation with an LSTM. \n\nAfter that, several supervised learning methods appeared to tackle distinct issues in the field. Peretroukhin and Kelly~\\cite{peretroukhin_dpc-net_2018} proposed the DPC-Net architecture, which aims to integrate the representation capabilities of deep neural networks with the efficiency of geometric and probabilistic algorithms. DPC-Net implements a CNN-based architecture to learn the corrections for the pose estimator. Zhao et al.~\\cite{zhao_learning_2018} propose the L-VO architecture, which predicts the 6-DoF pose from 3D optical flow for monocular VO. Valada et al.~\\cite{valada_deep_2018} proposed the VLocNet architecture, which is capable of estimating the 6-DoF pose in a monocular setup. Their architecture fuses relative and global RCNN-based architectures to improve accuracy. Saputra et al.~\\cite{saputra_distilling_2019} proposed to distill knowledge from a pose regressor employing concepts like Knowledge Distillation (KD) to train two networks jointly. Saputra et al. also proposed the first Curriculum Learning (CL) architecture, called CL-VO~\\cite{saputra_learning_2019}, aiming to learn the scene geometry for monocular VO by gradually increasing the task's difficulty.\n\nSeveral methods started to use concepts of attention in their pipeline in recent years, mainly for unsupervised learning. In 2020, Damirchi et al.~\\cite{damirchi2020exploring} explored the concept of self-attention to extract meaningful features in complex scenarios, which usually have many moving objects and low textures.  Also, Kuo et al.~\\cite{kuo2020dynamic} proposed the DAVO architecture, which is a dynamic attention-based visual odometry composed of two attentional networks. Their first network can generate semantic masks for determining the weights that each piece of the input image should have, while the second network uses a squeeze-and-excite attentional block. \n\nAttentional concepts are also employed to support the salient feature extraction from the input images. Liang et al.~\\cite{liang_salientdso_2018} proposes the SalientDSO architecture, which applies attention in the Direct Sparse Odometry (DSO)~\\cite{engel_direct_2018} algorithm. The proposed method runs in two distinct modules, the first one detects the visual salience using SalGAN, and the second one performs visual odometry with DSO. The drift error has substantially decreased compared to the original DSO due to the improved sampling. Chen et al.~\\cite{chen_salient_2019} also proposed an end-to-end CNN+LSTM salient-feature attention and context-guided networks for robust visual odometry. Their architecture can be trained using only monocular images and aimed to decouple rotational and translational motion.\n\nDeep learning methods demand the adoption of large and representative datasets; conversely, lightweight and efficient methods are fundamental to the field. Reinforcement Learning (RL) and attention applied to visual odometry can provide a solution in such scenarios. The architecture becomes efficient by selecting only the necessary input data, and learning a robust policy can mitigate the drift error. However, we have not found any method that implements RL in any part of the visual odometry pipeline.\n\n\\subsection{The Recurrent Attention Model (RAM)}\n\nThe Recurrent Attention Model (RAM)\\cite{mnih2014recurrent} implements a hard attention mechanism similar to the biological visual system, which iteratively builds an informative vector through multiple observations in the input image. Hard attention forces the model to consider only the relevant elements, discarding the others entirely~\\cite{correia2021attention}.\nThe observations are iteratively stored in a latent space, providing the knowledge to perform the task. The location of each observation is determined by a policy learned through REINFORCE~\\cite{williams1992simple}. RAM is composed of four distinct networks (Figure~\\ref{fig:ram}).\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{images\/ram.pdf}\n    \\caption{The RAM architecture is composed of four distinct networks, each one encapsulating a specific behavior.}\n    \\label{fig:ram}\n\\end{figure}\n\n\\begin{figure*}[bp]\n    \\centering\n    \\includegraphics[width=0.99\\linewidth]{images\/ramvo_baseline.pdf}\n    \\caption{RAM-VO is composed of five subnetworks that perform specific functions. The Glimpse allows efficient data consumption by successively observing regions of interest $\\textbf{l}_t$ on the input images. The Core sequentially integrates these observations $\\textbf{g}_t$ in nested LSTMs, the Locator generates the next observation's location $\\textbf{l}_{t+1}$ by sampling a Gaussian distribution parametrized by the internal state $\\textbf{h}_t$, and the Regressor predicts the 6-DoF pose regression in the end.}\n    \\label{fig:ramvo_baseline}\n\\end{figure*}\n\nThe glimpse network $f_{g}$ represents the attentional system and comprises a glimpse sensor to extract meaningful patches from the input images, and fully connected layers to encode the visual information. First, the glimpse sensor receives an image $\\textbf{x}_{t}$ and a location $\\textbf{l}_ {t-1}$ as input. Then, several patches are extracted in different resolutions, centered at the location $\\textbf{l}_{t-1}$. This process builds a pyramidal-like structure $\\rho (\\textbf{x}_{t}, \\textbf{l}_{t-1})$ similar to biological vision, representing what was observed on the image. Finally, the glimpse network concatenates $\\rho_{t}$ and $\\textbf{l}_{t-1}$ to include the location where the information was extracted, resulting in the final vector $\\textbf{g}_{t}$.\n\nThe core network $f_{h}$ stores the multiple observations by receiving the glimpse feature vector $\\textbf{g}_{t}$ and the previous internal state $\\textbf{h}_{t-1}$ as input at every time step $t$. Through fully connected layers, the core network outputs the current internal state $\\textbf{h}_{t}$, which condenses all the sequential information provided by the glimpse network. The location network $f_{l}$ generates the location $\\textbf{l}_t$ for the subsequent observation by sampling a Gaussian distribution with two dimensions $(x,~y)$ and a fixed standard deviation. For each subsequent observation, a novel Gaussian distribution is generated by using the internal state $\\textbf{h}_{t}$ to parameterize the mean $\\mu_t$. After all observations, the action network $f_{a}$ consumes the internal state $\\textbf{h}_{t}$ to predict the class $a_{t}$, which is the ultimate goal.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.99\\linewidth]{images\/ramvo_baseline_glimpse.pdf}\n    \\caption{The Glimpse Network. The consecutive input images are concatenated and cropped at the location $\\textbf{l}_t$, generating three patches $\\textbf{P}_{1,2,3}$. The patches are processed by independent CNN pipelines and concatenated in an internal vector; further multiplied by the encoded location $\\textbf{l}_t$, providing the final glimpse vector $\\textbf{g}_t$. This design favors the learning of optical flow.}\n    \\label{fig:ramvo_glimpse_baseline}\n    \\vspace{-1em}\n\\end{figure*}\n\nThe hard attention mechanism requires reinforcement learning strategies to train the model. Therefore, the RL setup is an instance of a Partially Observable Markov Decision Process (POMDP), in which the true state of the environment is unobserved; hence the model needs to learn a stochastic policy  $\\pi \\left (\\textbf{l}_{t} | \\textbf{h}_{1:T};\\boldsymbol\\theta \\right )$ that maps the environment history $\\textbf{h}_{1:T} = \\{x_{1}, l_{1}, ..., x_{t}, l_{t}\\} $ to a distribution over the actions in time step $t$, restricted by the sensor. In this sense, the glimpse sensor is the agent, the whole image is the environment, and the rewards are defined according to the success in the classification. From the ground-truth values, the agent receives $r_{t} = 1$ if the class is classified correctly after $T$ time steps, and 0 otherwise. The goal is to maximize the return $G = \\sum_{t=1}^{T}r_{t}$, which is sparse and delayed. \n\nThe architecture parameters $\\boldsymbol\\theta$ are optimized by maximizing the return $G$ when the agent interacts with the environment. The agent's policy, in combination with the environment's dynamics, produces a distribution over the possible iteration sequences $\\textbf{h}_{1:N} $, and the goal is to maximize the return under that distribution via $J(\\boldsymbol\\theta) = \\mathbb{E}_{p(\\textbf{h}_{1:T};\\boldsymbol\\theta)}[G]$, where $p(\\textbf{h}_{1:T};\\boldsymbol\\theta)$ depends on the policy. Maximizing $J(\\boldsymbol\\theta)$ is not trivial because it involves an expectation about high-dimension iteration sequences, which may involve unknown environment dynamics. However, we can obtain an approximation of the gradient with the REINFORCE rule~\\cite{williams1992simple} as\n\n\\begin{equation}\n\\nabla J(\\boldsymbol\\theta) = \\frac{1}{M}\\sum_{i=1}^{M}\\sum_{t=1}^{T} \\nabla \\mathrm{log}\\pi\\left ( \\textbf{l}_{t}^{i} | \\textbf{h}_{1:t}^{i}; \\boldsymbol\\theta\\right )(G^{i}_{t} - b_{t}),\n\\end{equation}\n\n\\noindent where $\\textbf{h}^{i}_{1:t}$ are the sequences obtained by running the current agent policy $\\pi_{\\theta}$ for $i=1, ..., M$ episodes, $G^{i}_{t}$ is the accumulated reward obtained after executing action $\\textbf{l}_t^i$, and $b_{t}$ is the baseline value, which reduces the variance for the gradient updates. The baseline value $b_{t}$ depends on sequence $\\textbf{h}^{i}_{1:t}$ but not directly of the action $\\textbf{l}^{i}_{t}$. As a result, the algorithm increases the log-probability of actions that generate a high cumulative reward and diminishes the probability of actions that generates a low cumulative reward.\n\nRAM is a hybrid architecture in which the location network is trained by the REINFORCE rule~\\cite{williams1992simple}, while the other networks employ supervised learning. Although bringing innovative ideas from biology, RAM was proposed mainly as a proof of concept, lacking the necessary complexity to deal with high-resolution images and regression tasks.\n\n\\section{The RAM-VO Architecture}\n\nThe Recurrent Attentional Model for Visual Odometry (RAM-VO) was constructed by extending RAM~\\cite{mnih2014recurrent} and changing the architecture's goal from classification to regression. We propose several modifications to learn a robust policy, deal with complex visual information, and enable 6-DoF pose regressions --- the next sections detail each modification. RAM-VO is shown in Figures~\\ref{fig:ramvo_baseline} and \\ref{fig:ramvo_glimpse_baseline}.\n\n\\subsection{Glimpse Network}\n\n\\begin{figure}[b!]\n\n    \\centering\n     \\includegraphics[width=1\\linewidth]{images\/scales.pdf}\n\n    \\caption{The glimpse sensor extracts three patches from the input image at a given location (top row). Each patch has an increased size, forming a pyramidal-like structure. The first patch $\\textbf{P}_{1}$ is cropped with $32 \\times 32$ pixels; the others $\\textbf{P}_{2,3}$ are twice larger than the previous one (bottom row).}\n    \\label{fig:city_scales}\n\\end{figure}\n\nThe visual odometry task requires processing two consecutive frames in the glimpse network, allowing the detection of the same features in both images and their correspondence. In this sense, the glimpse network receives two temporally consecutive images $(\\textbf{x}_t, \\textbf{x}_{t+1})$ and a location of interest $\\textbf{l}_t$ as input. The images are stacked and cropped at the location provided, generating three patches of 32, 64, 128 pixels. An average pooling operation is performed in the larger patches to reduce their size to $32 \\times 32$ pixels. Then, the three final patches $\\textbf{P}_{1,2,3}$ are processed through convolutional layers, flattened, and concatenated into a temporary internal vector. The patches locations are encoded and multiplied by the internal vector, originating the final glimpse vector $\\textbf{g}_t$. In this sense, the glimpse network implements a top-down attentional mechanism by capturing small portions of visual information that we will fuse with the information locations to generate a final vector with the correspondence between the two input images. \n\nGenerating an efficient representation $\\textbf{g}_t$ for the input images is closely associated with learning the geometry presented in the scene. Therefore, based on the FlowNetS~\\cite{fischer_flownet_2015}, we proposed to apply the convolutional operations in both images simultaneously to learn the optical flow and considerably reducing the computational cost compared to extracting the features separately. In this sense, the glimpse network uses 6 CNN layers with 32 to 128 channels for the patch $\\textbf{P}_1$, and 4 CNN layers with 32 to 64 channels for the other two patches $\\textbf{P}_{2,3}$. We observed that convolutional operations with smaller kernels significantly improve the representation; therefore, we processed the patch $\\textbf{P}_1$ with a kernel size of $3 \\times 3$ pixels and the others with a $5 \\times 5$ kernel. Padding values are defined as zero to avoid reducing the input size across the CNN layers, and we remove pooling operations entirely. We perform dimensionality reduction by varying the kernel stride between 1 and 2.\n\n\\textbf{Glimpse Scales}. The extraction of the image patches is performed in three different scales, simulating the human visual system \\cite{mnih2014recurrent}. The first scale corresponds to a central, high-resolution region but with a smaller dimension, and the second and third scales present larger dimensions but lower resolutions (Figure~\\ref{fig:city_scales}). This pyramidal-like structure provides a trade-off between the amount of information and computational cost. Therefore, the agent can observe the environment's details on the center and also the most salient elements presented on the boundaries. Also, the peripheral information helps the agent to determine the next location of interest $\\textbf{l}_{t+1}$ for subsequent observations. \n\n\\subsection{Core Network}\n\nThe core network is responsible for integrating all observations from the input images and providing the internal state $\\textbf{h}_t$ for the pose regression and for generating the next location of interest $\\textbf{l}_{t+1}$. Therefore, the observation $\\textbf{g}_t$ received from the glimpse network is recurrently integrated, updating the internal state $\\textbf{h}_t$ for every step. This process is repeated accordingly to the number of steps previously defined. In this sense, to track long-term dependencies and better represent the internal state, we adapted the core network to include two stacked LSTM layers with 1024 hidden units. LSTMs also diminished the problem of vanishing gradients during training, stabilizing the model. The ability to generate efficient RL policies is directly dependent upon the representation of the internal state $\\textbf{h}_t$, especially during the first epochs when high exploration is desired; hence, we initialized the weights orthogonally for $\\textbf{h}_t$.\n\n\\subsection{Locator and Baseliner Network}\n\nThe locator network provides the next location of interest $\\textbf{l}_{t+1}$ by sampling a Gaussian distribution, whose mean and standard deviation are parametrized by the internal state $\\textbf{h}_t$. The first location is defined randomly, and the others are sampled accordingly to the learned policy. Unlike the original RAM, the policy's standard deviation is also learned during training, promoting exploration in the first epochs. The locator network is detached from the supervised graph and trained separately by the REINFORCE rule~\\cite{williams1992simple}. We jointly train a baseliner network to provide the state value $b_t$ for each step, reducing the variance between the returns. Both locator and baseliner networks are composed of fully connected layers with 256 to 32 hidden units.\n\n\\subsection{Regressor Network}\n\nThe regressor network generates the pose prediction after the last observation is integrated into the internal state $\\textbf{h}_t$. The predictions are decoupled in rotational and translational components, which comprehends the orientation $\\boldsymbol{\\varphi}$ with the Euler angles as roll $\\phi$, pitch $\\theta$, and yaw $\\psi$; and the position $\\boldsymbol{p}$, composed of the coordinates $x$, $y$, and $z$. The main goal is to regress the 6-DoF pose vector $[\\phi, \\theta, \\psi, x, y, z]^T$ for every pair of frames. In this sense, the regressor network comprises three fully connected layers from 256 to 32 hidden units; the last layers provide linear outputs for the prediction.\n\n\\subsection{Loss and Reward Function}\n\nRAM-VO is a hybrid architecture where the regressor, core, glimpse, and baseliner networks are trained in a supervised learning fashion, while the locator is trained by reinforcement learning. The supervised loss and the reward function are both defined in terms of the MSE to minimize the outliers as much as possible since only one poor prediction can harm the entire trajectory. Therefore, the supervised loss $L$ is defined as\n\n\\begin{equation}\nL = \\frac{1}{N}\\sum_{i=1}^{N} \\left\\| \\boldsymbol{\\hat p} - \\boldsymbol{p} \\right\\|_2^2 + k \\left\\| \\boldsymbol{\\hat \\varphi} - \\boldsymbol{\\varphi} \\right\\|^2_2,\n\\label{eq:loss_ramvo_supervised}\n\\end{equation}\n\n\\noindent where $\\boldsymbol{\\hat p}$ and $\\boldsymbol{\\hat \\varphi}$ are the position and orientation prediction, respectively; $\\boldsymbol{p}$ and $\\boldsymbol{\\varphi}$ are the ground-truth values; and $k$ is the constant factor weighting the two losses, favoring the rotational or the translational component; we kept the ground-truth values normalized and $k=1$. The reward function $R$ is defined as \n\n\\begin{equation}\nR = \\frac{1}{1 + L}.\n\\label{eq:loss_ramvo_reward}\n\\end{equation}\n\nWe prefer not to bias the RL agent towards a specific behavior; therefore, only the visual odometry error $L$ is employed in the reward function. \n\n\\subsection{Proximal Policy Optimization (PPO)}\n\nThe REINFORCE rule~\\cite{williams1992simple} is known for presenting converge issues and slowness; this occurs by sudden updates on the policy's parameters, which can harm the entire training by converging to suboptimal solutions. Proximal Policy Optimization (PPO)~\\cite{schulman2017proximal} aims to attenuate these problems by updating the policy inside trusted regions. Therefore, PPO's surrogate function determines that the current policy must be close to the last one, avoiding large parameters shift. In practice, the PPO implementation consisted of replacing the locator and baseliner network with a similar structure in terms of layers and hidden units. We also use memory replay to enable the policy refinement with already sampled data, improving the architecture's efficiency. The policy refinement proportion is 20:1 concerning the supervised network --- we want the best policy to control the input information flow.\n\n\\section{Materials and Methods}\n\n\\subsection{Dataset}\n\nIn this work, we used the KITTI dataset~\\cite{geiger_vision_2013}, one of the most popular datasets for evaluating visual odometry methods. The entire dataset consists of 22 sequences (39.2 km) of real-world traffic data captured by a car moving across urban and rural areas in Germany. However, only the first eleven (00-10) sequences have ground-truth information. Therefore, we used the grayscale images provided by the left camera, resized to $1200 \\times 360$ pixels in resolution. \n\nIn order to compare our results with other methods, we chose sequences 0, 2, 4, 5, 6, 8, 9 for training, sequences 10 for validation, and sequences 3, 7 for testing. We did not use sequence 1 due to the higher average variation for the translational component. The whole data used comprehends 18,990 images for training, 1,200 for validation, and 1,902 for testing. We did not use data augmentation.\n\nWe preprocessed each image by equalizing the pixel intensity histogram in small windows of $8 \\times 8$ pixels with the Contrast Limited Adaptive Histogram Equalization method. In this way, we can highlight the image's features without increasing noise. Also, we normalized the images with the z-score function before entering the glimpse network.\n\n\\begin{figure*}\n    \\centering\n    \\subfloat{{\\includegraphics[width=4.3cm]{images\/glimpse_0.pdf} }}\\hfill\n    \\subfloat{{\\includegraphics[width=4.3cm]{images\/glimpse_1.pdf} }}\\hfill\n    \\subfloat{{\\includegraphics[width=4.3cm]{images\/glimpse_2.pdf} }}\\hfill\n    \\subfloat{{\\includegraphics[width=4.3cm]{images\/glimpse_3.pdf} }}\\hfill\n    \\vspace{-0.6em}\n    \\subfloat{{\\includegraphics[width=4.3cm]{images\/glimpse_4.pdf} }}\\hfill\n    \\subfloat{{\\includegraphics[width=4.3cm]{images\/glimpse_5.pdf} }}\\hfill\n    \\subfloat{{\\includegraphics[width=4.3cm]{images\/glimpse_6.pdf} }}\\hfill\n    \\subfloat{{\\includegraphics[width=4.3cm]{images\/glimpse_7.pdf} }}\\hfill\n    \n    \\caption{The temporal sequence of eight observations on the first frame on sequence 2. The glimpse sensor exploits high-gradient regions in most captures, such as corners and edges.}\n    \\label{fig:baseline_glimpses}\n    \\vspace{-1.5em}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\subfloat{{\\includegraphics[width=4.3cm]{images\/heatmap_1.pdf} }}\n    \\subfloat{{\\includegraphics[width=4.3cm]{images\/heatmap_2.pdf} }}\n    \\subfloat{{\\includegraphics[width=4.3cm]{images\/heatmap_3.pdf} }}\n    \\subfloat{{\\includegraphics[width=4.3cm]{images\/heatmap_4.pdf} }}\n    \n    \\caption{Heatmap of observations for the entire sequence 2. The first observation is randomly defined, and the RL agent chooses the others based on the learned policy.}\n    \\label{fig:baseline_heatmap}\n    \\vspace{-1em}\n\\end{figure*}\n\n\\subsection{Evaluation Metrics}\n\nThe most common evaluation metrics for visual odometry compute the agent's absolute trajectory error (ATE) and the relative pose error (RPE). These metrics are commonly reported for the entire trajectory by computing the root mean square error (RMSE) for all frames.\n\n\\textbf{Absolute Trajectory Error (ATE)} is used to compute the method's global consistency. We compare the estimated pose with the ground-truth pose for each frame. However, the poses usually are specified in arbitrary coordinate frames and must be aligned to be compared. The absolute pose error at instant $i$ is given by $\\mathcal{E}_i = \\textbf{G}_i^{-1}\\textbf{A}\\textbf{H}_i$, where $\\textbf{G}_i$ is the ground-truth pose at instant $i$, $\\textbf{H}_i$ is the estimated trajectory pose at instant $i$, $\\textbf{A}$ is the best alignment transformation.\n\n\\textbf{Relative Pose Error (RPE)} computes the error using only a relative relationship between frames, which could solve the issues associated with a global frame comparison. In this sense, RPE measures the local consistency of the trajectory and is a reliable metric for the drift. The relative pose error at instant $i$ is given by $   \\mathcal{F}_i = \\frac{\\textbf{H}_i^{-1} \\textbf{H}_{i+k}}{\\textbf{G}_i^{-1} \\textbf{G}_{i+k}}$, where $k$ is a fixed time interval, which determines the trajectory consistency accuracy; for visual odometry, $k=1$ is usually used. We compute RPE by averaging all sub-sequences ranging from 100 to 800 meters in KITTI's sequences.\n\n\\subsection{Hardware and Hyperparameters}\n\nWe implemented this work in Python 3 with Pytorch. The hardware used for building and training the models was an Intel Core i7-10700KF @ 3.80GHz, Nvidia RTX 2060 with 6Gb, and Cuda v11.1. The model configuration consisted of 3 image patches of $32 \\times 32$ pixels, batch size of 128, supervised learning rate of \\num{1e-4}, and RL learning rate of \\num{1e-6}; we employed the Adam optimizer for both networks. We trained the models for 400 epochs without early stopping or learning rate decay. The average training time was around 13 hours. The inference time is 35ms for a pair of frames.\n\n\\section{Experimental Results}\n\nWe conducted several experiments (Table~\\ref{table:baseline_configurations}) to provide a better understanding of the RAM-VO behavior on complex sequences in the KITTI dataset~\\cite{geiger_vision_2013}. First, we validated the number of glimpses\/observations on building a usable internal state $\\textbf{h}_t$ for visual odometry. Thus, we varied the number of observations from 1 to 12 and also tested with random observations. Second, we replaced the REINFORCE algorithm with PPO to evaluate the impact of the policy on the generalization; finally, we varied the internal state $\\textbf{h}_t$ capacity from 1024 to 256 hidden units to evaluate the impact on the drift error.\n\n\\subsection{Number of Observations}\n\nThe first experiment consisted of making a single observation at the image's center. The error metrics indicate that the model overfits the train sequences and cannot generalize, possibly by learning the appearance instead of geometry. The information captured from only a central region facilitates the learning of the scale since it tends to be constant for most frames; however, some frames show different behaviors (e.g., obstructions, reflections) and quickly degrades the trajectory prediction. Also, only one capture in the same location reduces the data diversity necessary to learn complex behaviors; the model tends to memorize the single observation instead of learning a general dynamics. \n\nThe second experiment consisted of a single observation in a random location. Although a single random observation provides more data diversity during training, it still captures little and sparse information, affecting the scale learning, hampering robust predictions. Furthermore, good predictions with random observations require learning the general dynamics since the input space is ample and the model's capacity is limited. Finally, we highlight that these experiments did not use reinforcement learning since the location is already determined; and a single observation did not provide enough information for generalization.\n\nThe subsequent experiments aim to determine the impact of the observations, and consequently, the learned policy to select informative patches and the core network's ability to integrate them. Therefore, we set the number of observations to 4, 8, and 12; all locations are determined by the policy. The model achieved the best generalization results with 4 and 8 glimpses; 12 glimpses have not provided better results, as more data not necessarily mean better predictions. More observations demand more from the core and locator networks since a single poor observation can harm the entire internal state and delay, even more, the sparse reward. \n\nConsidering the experiment with 8 observations, the totality of input information is 5.7\\% of the total available. We conclude that the agent is retrieving high informative patches for most observations, which consisted of edges and corners (Figure~\\ref{fig:baseline_glimpses}). Also, the learned policy displayed the traditional Gaussian pattern indicating a preference for observing the right-center portion of the image (Figure~\\ref{fig:baseline_heatmap}).\n\n\\begin{table}[b!]\n\\centering\n\\caption{The impact of the number of observations on the average RPE and ATE for all sequences on train and test set. $\\mathrm{\\bar{t}_{rpe}}$ represents the average translational RMSE drift (\\%) on length of 100m to 800m. $\\mathrm{\\bar{r}_{rpe}}$ is the average rotational RMSE drift (\\degree\/100m) on length of 100m to 800m. $\\mathrm{\\overline{ATE}}$ represents the average absolute trajectory error.}\n\\label{table:baseline_configurations}\n\n\\begin{tabular}{p{2.15cm}p{0.6cm}p{0.5cm}p{0.7cm}p{0.6cm}p{0.5cm}p{0.6cm}}\n\\toprule\n                       & \\multicolumn{3}{c}{\\textbf{Train set}} & \\multicolumn{3}{c}{\\textbf{Test set}}                   \\\\ \\cline{2-7} \n\\textbf{Configuration} & $\\mathrm{\\bar{t}_{rpe}}$       & $\\mathrm{\\bar{r}_{rpe}}$         & $\\mathrm{\\overline{ATE}}$         & $\\mathrm{\\bar{t}_{rpe}}$ & $\\mathrm{\\bar{r}_{rpe}}$ &$\\mathrm{\\overline{ATE}}$  \\\\ \\hline\n1 glimpse at center    &   0.984 & 0.408 & 3.216 & 16.369 &\t7.666 & 36.104 \\\\ \\hline\n1 glimpse random       & 16.516 &\t4.940 &\t127.158 &  25.969\t& 7.939 &\t42.221   \\\\ \\hline\n4 glimpses             &  3.599 &\t1.608 &\t36.463  & 10.929 &\t\\textbf{3.985} &\t\\textbf{17.418}   \\\\ \\hline\n8 glimpses             & \\textbf{3.021} &\t\\textbf{1.393} &\t\\textbf{20.311} & \\textbf{10.888} &\t4.206 &\t19.806            \\\\ \\hline\n8 glimpses random      &  5.227 &\t2.126 &\t58.193  &  12.461 &\t4.127 &\t21.004    \\\\\n\\hline\n12 glimpses            & 3.335&\t1.504 &\t32.517 & 13.181 &\t5.771 &\t24.762  \\\\  \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsection{Experiments with PPO}\n\nThe following experiments (Table~\\ref{table:ppo_configurations}) evaluate the results achieved by replacing the REINFORCE algorithm with PPO. We also investigated the impact of reducing the internal state $\\textbf{h}_t$ capacity from 1024 to 256 hidden units. The core network corresponds to most of the model's parameters; therefore, knowing the minimum capacity required to achieve good results is crucial for delivering lightweight models. For all experiments, we captured 8 glimpses and computed the statistics for three distinct executions. \n\n\\begin{table}[h!]\n\\centering\n\n\\caption{The impact of PPO and the internal state capacity. For metrics explanation, see Table~\\ref{table:baseline_configurations}.}\n\\label{table:ppo_configurations}\n\n\\begin{tabular}{p{0.6cm}p{0.8cm}p{0.6cm}p{0.6cm}p{0.6cm}p{0.6cm}p{0.6cm}p{0.6cm}}\n\\toprule\n & & \\multicolumn{3}{c}{\\textbf{Train set}} & \\multicolumn{3}{c}{\\textbf{Test set}}                  \n \\\\ \\cline{3-8} \n\\textbf{Config.} & \\textbf{Param.} & $\\mathrm{\\bar{t}_{rpe}}$       & $\\mathrm{\\bar{r}_{rpe}}$         & $\\mathrm{\\overline{ATE}}$         & $\\mathrm{\\bar{t}_{rpe}}$ & $\\mathrm{\\bar{r}_{rpe}}$ &$\\mathrm{\\overline{ATE}}$  \\\\ \\hline\nPPO 1024   & \\vspace{-0.1em} 16.54M & $\\textbf{4.095} \\pm 0.27$ &\t$\\textbf{1.681} \\pm 0.17$ & $\\textbf{30.066} \\pm 9.89$ & $\\textbf{10.634}  \\pm 1.38$ &\t$\\textbf{4.369}  \\pm  0.43$ &\t$\\textbf{17.181}  \\pm  2.20$\\\\ \\hline\n\nPPO 512    & \\vspace{-0.1em} 5.84M &  $7.068\t\\pm 2.43$ & $2.735 \\pm 0.74$ &\t$54.587 \\pm 26.91$ & $15.486 \\pm 4.06$ &\t$7.084 \\pm 1.87$ &\t$24.810 \\pm 6.67$ \\\\ \\hline\n\nPPO 256     & \\vspace{-0.1em} 2.92M & $7.287 \\pm\t1.68$ & $2.791 \\pm 0.44$ &\t$52.211 \\pm 21.08$ & $15.499 \\pm 5.18$ &\t$6.685 \\pm 2.00$ &\t$25.624 \\pm 8.88$ \\\\  \\bottomrule\n\n\\end{tabular}\n\n\\end{table}\n\nPPO with 1024 hidden units provided the best generalization, mainly due to the increased capacity. We observed that decreasing the number of parameters increases the relative error during training, and the generalization is strongly affected on average. Although PPO 1024 had the best performance, the difference in the results' quality may not justify a three-fold increase in parameters. PPO algorithm is sensitive to the initialization; therefore, we selected the best models of three executions to predict the trajectories (Figure~\\ref{fig:baseline_trajectory}). PPO 256 has only 2.92 million parameters and provides results compatible with the PPO 1024 on the best execution. In conclusion, we asserted that the PPO algorithm provided a slightly better generalization capacity than the REINFORCE algorithm. PPO learned a more centered policy, although very similar to the one learned by the REINFORCE algorithm.\n\n\\begin{figure}[tb!]\n    \\centering\n    \\subfloat{{\\includegraphics[width=4.2cm]{images\/trajectory_0.pdf} }}\n    \\subfloat{{\\includegraphics[width=4.2cm]{images\/trajectory_3.pdf} }}\n    \\vspace{-1.3em}\n    \\subfloat{{\\includegraphics[width=4.2cm]{images\/trajectory_8.pdf} }}\n    \\subfloat{{\\includegraphics[width=4.2cm]{images\/trajectory_7.pdf} }}\n    \\vspace{-1.3em}\n    \\subfloat{{\\includegraphics[width=4.2cm]{images\/trajectory_9.pdf} }}\n    \\subfloat{{\\includegraphics[width=4.2cm]{images\/trajectory_10.pdf} }}\n    \\vspace{-1em}\n    \\caption{RAM-VO trajectories' predictions for training (left column) and testing (right column) sequences on the KITTI dataset using our best model.}\n    \\label{fig:baseline_trajectory}\n    \\vspace{-1em}\n\\end{figure}\n\n\n\\subsection{Comparison with Literature}\n\nOur best RAM-VO with PPO obtained competitive results (Table~\\ref{table:comparison_literature}) using less input information than similar methods, around 5.7\\% of the total available, considering 8 glimpses with the size of $32 \\times 32$ pixels. While RAM-VO uses top-down attention to capture regions of interest, methods like ORB-SLAM~\\cite{mur2015orb} need to analyze an entire image to detect keypoints. Besides, ORB-SLAM is a geometric method that depends on high-texture regions for an accurate keypoint match between frames. DeepVO~\\cite{wang_deepvo_2017} and ESP-VO~\\cite{wang_end--end_2018} are both based on the FlowNet~\\cite{fischer_flownet_2015} architecture and therefore have more convolutional layers and channels than RAM-VO. These architectures perform direct VO  by determining the frames' correspondence from the pixel values; consequently, they are more robust to outliers than ORB-SLAM. However, direct methods are costly, especially DeepVO and ESP-VO, which use the entire image as input. In visual odometry, the motion is present in the whole image; hence, more data does not necessarily bring novel information. \n\n\\begin{table}[b!]\n\\centering\n\n\\caption{RAM-VO results compared to other methods on test sequences. For metrics explanation, see Table~\\ref{table:baseline_configurations}.  }\n\\label{table:comparison_literature}\n\n\\begin{tabular}{p{2cm}p{0.4cm}p{0.5cm}p{0.4cm}p{0.4cm}p{0.5cm}p{0.5cm}p{0.5cm}}\n\\toprule\n& \\textbf{Data} & \\multicolumn{2}{c}{\\textbf{Seq. 03}}     \n& \\multicolumn{2}{c}{\\textbf{Seq. 07}}   \n& \\multicolumn{2}{c}{\\textbf{Seq. 10}} \n\\\\ \\cline{3-8} \n\\textbf{Method} & \\textbf{\\%} & $\\mathrm{t_{rpe}}$ & $\\mathrm{r_{rpe}}$ & $\\mathrm{t_{rpe}}$ & $\\mathrm{r_{rpe}}$ & $\\mathrm{t_{rpe}}$ & $\\mathrm{r_{rpe}}$ \n\\\\ \\hline\nORB-SLAM~\\cite{mur2015orb} & - & 21.07 & 18.36 & 24.53 & 38.90 & 86.51 & 98.90   \\\\  \\hline\nDeepVO~\\cite{wang_deepvo_2017} & 100 & 8.49 & 6.89 & 3.91 & 4.60 & \\textbf{8.11} & 8.83  \\\\ \\hline\nESP-VO~\\cite{wang_end--end_2018} & 100 & 6.72 & 6.46 & \\textbf{3.52} & 5.02 & 9.77 & 10.20  \\\\  \\hline\nRAM-VO 1024 & \\textbf{5.68} & \\textbf{5.72} & \\textbf{3.08} & 9.17 & 5.63 & 13.85 & \\textbf{3.24} \\\\ \\hline\nRAM-VO 256 & \\textbf{5.68} & 7.08 & 4.01 & 7.55 & \\textbf{4.30} & 15.02 & 5.12   \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\nThe model's capacity is also a relevant fact to be considered. Although DeepVO and ESP-VO may present similar results on average, RAM-VO with 256 hidden units in the core network achieves comparable results with only 2.7 million parameters --- which is much lower than the 17 million parameters reached by RAM-VO with 1024 hidden units. Concurrent methods regularly pass 32 million parameters, especially when they are extensions of architectures like AlexNet~\\cite{krizhevsky_imagenet_2012}, and FlowNet~\\cite{fischer_flownet_2015}. Also, these CNN networks are considerably deep and add a high cost in terms of trainable parameters, making the training slow and the deployment a complicated task for mobile devices.\n\nLearning-based methods tend to be slow and costly to run due to requiring high-end devices with large processing power, GPUs, and better batteries; these issues make adopting learning methods problematic. In the same way, the results presented by ORB-SLAM are worse compared to learning-based methods, but they are sufficiently fast to build online applications on mobile devices. In this context, RAM-VO represents an alternative method capable of providing results similar to large models but with a smaller cost in trainable parameters and data consumption.\n\n\\subsection{Limitations}\n\nRAM-VO has difficulty in obtaining the world scale and determining the translational motion with greater precision. This issue probably happens due to the small image patches that, depending on the vehicle's velocity, can prevent the overlap between frames, compromising the regression since it relies on the features' correspondence. We aim to assess these problems in future formulations.\n\n\\section{Conclusion}\n\nIn this work, we proposed the RAM-VO model for monocular end-to-end visual odometry. Our model was extended from RAM~\\cite{mnih2014recurrent} and therefore implemented attention and reinforcement learning to optimize the selection of visual information for regression tasks. RAM-VO innovated with the addition of 6-DoF pose regression, a robust glimpse network to learn optical flow, an improved core network to store temporal relation between observations, and the replacement of REINFORCE by the PPO algorithm to learn better policies. To the best of our knowledge, RAM-VO is the first architecture for visual odometry that implements reinforcement learning in part of the pipeline. The experimental results indicate that RAM-VO can predict 6-DoF poses in the real-world KITTI~\\cite{geiger_vision_2013} dataset with generalization for unseen sequences. The comparison with the literature indicated that RAM-VO could achieve competitive results using significantly less trainable parameters and input information. Similar learning methods consume the whole input image to determine the pose, while the RAM-VO uses a small fraction, around 5.7\\% of the total input data.\n\n\n\\section*{Acknowledgment}\n\nThe authors would like to thank the Brazilian National Council for Scientific and Technological Development (CNPq), grant 130834\/2019-0, and Bradesco Bank.\n\n\n\\bibliographystyle{Transactions-Bibliography\/IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the last years  the need of enhancing the speed of modern spin-electronic and \nmagneto-optic devices, and of further developing the magnetic storage technology, has \nstimulated many studies aimed at achieving a fundamental understanding of the \nmechanisms of magnetization dynamics and switching. In addition to ultrafast magnetic\nfield pulses which require complex devices for their generation, the spin dynamics has \nbeen induced by ultrafast optical laser pulses. Recent experiments have shown that \nsignificant demagnetization of magnetic compounds can be measured on the scale of a few \nhundred femtoseconds. \\cite{beaure,hohle,schol,koopm}   \nTypically the light is absorbed by the material, giving rise to a rapid increase of\ntemperature responsible for the change of magnetization \\cite{hubne,guido,ju} and spin \nreorientation. \\cite{kimel} The question concerning the exact speed of the initial \nsub-picosecond magnetization breakdown is still subject of debate and partly relates to \nthe question of how to interpret magneto-optical experiments. \n\\cite{koopm,koopm1,liebs} Furthermore, the cooling time could limit the repetition \nfrequency whose value is fundamental for actual applications. \\cite{hohle1}   \n\nRecently, non-thermal ultrafast optical control of magnetization has been achieved \nin canted antiferromagnet samples of dysprosium orthoferrites by using circularly \npolarized femtosecond pulses. \\cite{kimel1} Via the inverse Faraday effect, the light \nexcitation acts on the spins of the system as a magnetic field pulse directed along \nthe wave vector of the radiation and proportional to its intensity. \\cite{ziel,pershan} \nThe inverse Faraday effect does not rely on absorption and has its fingerprint in the \nfact that the helicity of the pump controls the sign of the photo-induced \nmagnetization.  This effect plays a role also in the femtosecond photomagnetic \nswitching of spins in ferromagnetic garnet films.\\cite{hanst}\n\nSince the manipulation of spins by means of circularly polarized laser \npulses represents an advance in the field of ultrafast magnetization dynamics, \nwe analyze thoroughly the experimental work by Kimel {\\it et al.} \\cite{kimel1}\nand discuss its pecularities. In this experiment the difference between the Faraday \nrotations induced by right- and left-handed polarized pulses has been studied in the \ntemperature range between 20 K and 175 K. A characteristic spin-wave mode, called \nquasi-antiferro mode, is excited by the light pulse along the z-direction.\nWhen the sample is heated, the frequency of the mode oscillations increases and \nthe amplitude decreases from the low-temperature maximum value of the order of $M_S\/16$,\nwhere $M_S$ is the saturation magnetization. We point out that there is no \ntheoretical explanation concerning the magnitude and the temperature behavior of the \noscillations. Furthermore, some aspects of the cited experiment deserve attention. \nActually, in the so called $\\Gamma_4$ phase, stable at temperatures higher than 50 K, \nthe oscillations have temperature-dependent frequencies in full agreement with \nthose measured by Raman experiments. \\cite{white,koshi} On the other hand,  \nat temperatures below 50 K, the frequency of the photoinduced magnetization \nstays constant in contrast with the results of the Raman spectra and \nother measurements which signal the discontinuous transition to another phase, \ncalled $\\Gamma_1$. Therefore, the experimental data obtained by laser pulses \nreflect the excitation of resonance modes characteristic of the $\\Gamma_4$ phase \neven at very low temperatures. Finally, the field pulse is also directed along the \nx-axis: another spin-wave mode, called quasi-ferro mode, is excited and \nis characterized by an extraordinarily, not well understood, small amplitude. \n\nIn order to obtain a deeper understanding of the non-thermal control of magnetization, \nin this paper we perform simulations related to the experiment by \nKimel {\\it et al.} \\cite{kimel1} We have studied the \nmagnetization dynamics employing a model for orthoferrites that was previously proposed \nfor the analysis of resonance and high-frequency susceptibility. \\cite{hermann} \nThe parameters of the free energy, such as the symmetric and antisymmetric exchange, and\nthe anisotropy constants, are determined by using the experimental Raman spectra of \nRef.17. The dynamical behavior is described by solving two coupled sublattice nonlinear\nLandau-Lifshitz-Gilbert equations through a fifth-order Runge-Kutta algorithm. \nThe damping term in the dynamical equations is related to magnon-magnon interaction \nand its temperature behavior is provided by a calculation of the scattering rate in \northoferrites. \\cite{tsang}  \nExploiting the inverse Faraday effect, we have analyzed the effect of Gaussian \nmagnetic field pulses whose time width is of the order of hundred femtoseconds. \nSince, in the regime considered in the experiments, the effective magnetic fields are \nnot large if compared with exchange fields, the solution of the linearized system \nrepresents a reasonable approximation to the numerical results. Therefore \nwe have studied the dynamics within the linear solution after the excitation by a pulse \nshaped as a delta function, since the magnetic field pulse takes place on a time scale \nshorter than the period of the resonance modes.\n\nOne result of this work is that, in the $\\Gamma_4$ phase, the quasi-antiferro mode of\nthe magnetization is excited by a field pulse along the z-axis and the \noscillations have amplitudes in agreement with experimental results. \nMoreover, the oscillations induced by pulses directed along the z and -z axis show the \ncharacteristic out-of-phase behavior.  \nWe point out that, even for the dynamics, the ratio between the antisymmetric and \nsymmetric exchange energies is important. Furthermore, we stress that in the \ncanted antiferromagnets, such as rare-earth orthoferrites, the largest amplitudes of the\noscillation are not obtained for the ferromagnetic sum vector of the sublattice \nmagnetizations but for the antiferromagnetic difference vector. The behavior of the \nmagnetization has been analyzed in the temperature range between 20 K and 175 K. The \ndamping process based on magnon-magnon scattering describes the \ndecay of the oscillations with results consistent with the experiment. \n\nThe case of the field pulse directed along the x-axis has been analyzed in the \n$\\Gamma_4$ phase. The magnetization along the x-axis oscillates with the frequency of \nthe quasi-ferro mode as found in the experiment. However, the calculated and the \nexperimental amplitudes of the oscillation are different if the magnetic field pulse \nalong the x-axis has the same intensity of that along the z-axis. \nTherefore, in the comparison with experimental data, we discuss the role\nof anisotropies between z and x directions as a source of discrepancy between \ntheory and experiment. \n\nFinally, we consider the actual stable phase in equilibrium at low temperatures, \nthe $\\Gamma_1$ phase, and its resonance modes. We point \nout that the difference in energy between the $\\Gamma_1$ and $\\Gamma_4$ can be very \nsmall, so that even a small laser-heating effect could be responsible for the \nstabilization of the $\\Gamma_4$ phase at very low temperatures on a picosecond time \nscale.\n\nThe outline of this paper is as follows. In the next section we discuss\nthe numerical approach for the system and the analytic \nsolution of the linearized equations for excitation by a delta function shaped \nmagnetic field pulse. Section III provides the numerical and \nanalytical results: in the first and second subsection the excitation due to the pulses \nalong the z-axis and x-axis, respectively, is analyzed when the system at equilibrium \nis in the $\\Gamma_4$ phase. In the final subsection the effect on the dynamics of a \n$\\Gamma_1$ phase stable at low temperatures is considered. \nSection IV provides a summary.\n\n\\section{Free energy and dynamical equations}\nRare-earth orthoferrites are represented by the formula $ReFeO_3$, \nwhere $Re$ stands for rare-earth. They have a perovskite-type structure with \nslight deformation from cubic to orthorombic. In many of them the spins of iron \nions are antiferromagnetically aligned through a strong super-exchange interaction \nwith a N\\`eel transition temperature of about 700K. Moreover, promoted by the \northorombic deformation, an antisymmetric exchange interaction acts between iron spins, \nresulting in spin canted magnetism with a feeble saturation moment. In the case of \ndysprosium orthoferrites, the temperature dependence of the ferromagnetic moment is \ncharacterized by a steep rise around 50 K in coincidence with the stabilization of the \n$\\Gamma_4$ phase. \\cite{wolf} Actually \nthese compounds are also known for their spin reorientation properties: continuous \nrotational-type (ferromagnetism present in the low-temperature phase) and, only for \ndysprosium, abrupt-type from $\\Gamma_1$ (antiferromagnetic) to $\\Gamma_4$ with \nincreasing temperature. \n\n\\begin{figure}[t!\n  \\begin{center}\n  \\includegraphics[width=6cm,height=7cm,angle=-90]{Fig1.eps}\n  \\end{center}\n  \\vskip-3mm\n\\caption{\nThe equilibrium positions of sublattice magnetization vectors for the high-temperature \n$\\Gamma_4$ (ferromagnetic) and low-temperature $\\Gamma_1$ (antiferromagnetic) phases. \nDue to the ratio between the antisymmetric $D$ and symmetric $E$ exchange fields, \nthe angle $\\beta_0$ is actually very small.\n}\\end{figure}\n\nWe use the free energy and the dynamical equations proposed in a previous work and \nfocus on the behavior of the magnetization in the $\\Gamma_4$ phase. \\cite{hermann} \nThe static and dynamical behavior is studied within the two-sublattice model for iron spins that takes into account the two active resonance modes. \nThis represents an excellent approximation since the remaining resonance modes\nare almost inactive, and their interaction with active modes is negligible. \nIn Fig.1 we show a schematic representation of equilibrium positions of two-sublattice \nmagnetization vectors for dysprosium orthoferrites. \\cite{koshi}\n\nThe normalized free energy $V=F\/M_0$, with $M_0$ the modulus of the sublattice \nmagnetization, is composed of a part $V_{exc}$ due to  exchange interactions and a part \n$V_{ani}$ due to the anisotropy:\n\\begin{equation}\nV= V_{exc}+V_{ani}.\n\\label{e0}\n\\end{equation}\nThe free energy is expanded as a power series in the magnetization components and, \nin order to properly describe the $\\Gamma_4$ phase, only quadratic terms are sufficient. \nThe exchange energy is written as the sum of a scalar and a pseudo-vector part\n\\begin{equation}\nV_{exc}=E {\\vec R}_1 \\cdot {\\vec R}_2 +D (X_1 Z_2-X_2 Z_1),\n\\label{e1a}\n\\end{equation}\nwhere $E$ and $D$ are the symmetric and antisymmetric exchange fields, respectively, \n${\\vec R}_1={\\vec M}_1\/M_0 \\equiv (X_1,Y_1,Z_1)$, and \n${\\vec R}_2={\\vec M}_2\/M_0 \\equiv (X_2,Y_2,Z_2)$.\nThe anisotropic energy is\n \\begin{equation}\nV_{ani}=-A_{xx}(X_1^{2}+X_2^2)-A_{zz}(Z_1^2+Z_2^2).\n\\label{e1b}\n\\end{equation}        \nThe exchange field $E$ is quite large compared to the other terms. It is \nrelated to the exchange spin-spin interaction $J$ via the equation $E=12 J S \/ g \\mu_B$, \nwith $S=5\/2$ the spin of iron ions, $g=2$, and $\\mu_B$ the Bohr magneton. Since $J$ is of\nthe order of $20$ $cm^{-1}$, $E$ is approximately $6.4 \\times 10^6$ Oe. \\cite{koshi} \nThe field $D$ is related to the antisymmetrical exchange energy $d$ through the relation\n$D=6 d S \/ g \\mu_B$ and is of the order of $1.4 \\times 10^5$ Oe ($d$ is about \n$0.88$ $cm^{-1}$). The anisotropy constants $A_{xx}$ and $A_{zz}$ depend on \ntemperature and are of the order of hundreds of Oersted. \n \nUsing the energy given in Eq.(\\ref{e0}), the equilibrium position can be derived for \nthe $\\Gamma_4$ phase. This is shown in Fig. 1, where the small canting angle $\\beta_0$ \nis determined by the equation\n\\begin{equation}\ntan(2 \\beta_0)=\\frac{D}{E+A_{xx}-A_{zz}} \\simeq \\frac{D}{E} =0.022\n\\label{e2}.\n\\end{equation}\nHence the magnetization  \n$M_S=| \\vec{M}_1+ \\vec{M}_2 |=2 M_0 \\sin (\\beta_0)\\sim 0.022 M_0$ is two orders of \nmagnitude less than $M_0$.\n\nThe equilibrium position of $\\vec{M}_1$ and $\\vec{M}_2$ represents a stationary \nsolution of the nonlinear Landau-Lifshitz-Gilbert equations \n\n\\begin{equation}\n\\frac{1}{\\gamma} \\frac{d \\vec R_1 }{dt}=-{\\vec\nR}_1 \\wedge \\left({\\vec H}(t)-\\vec{\\nabla}_1 V \\right)+\\alpha\n{\\vec R}_1\n\\wedge \\frac{d \\vec R_1}{dt} \n\\label{e3} \n\\end{equation}\n\\begin{equation}\n\\frac{1}{\\gamma} \\frac{d \\vec R_2}{dt}=-{\\vec\nR}_2 \\wedge \\left({\\vec H}(t)-\\vec{\\nabla}_2 V \\right)+\\alpha\n{\\vec R}_2 \\wedge \\frac{d \\vec R_2}{dt}, \\label{e4}\n\\end{equation}\nwhere $\\gamma= 17.6$ MHz\/Oe is the gyroscopic ratio, $\\vec{\\nabla}_1$ and \n$\\vec{\\nabla}_2$ are gradients with respect to $\\vec R_1$ and $\\vec R_2$, \nrespectively, and $V$ is the energy in Eq.(\\ref{e0}). \nClearly the dynamical equations satisfy the following constraints: \n$X_1^2+Y_1^2+Z_1^2=1$ and $X_2^2+Y_2^2+Z_2^2=1$.\n\n\nThe quantity ${\\vec H}(t)$ in Eqs.(\\ref{e3},\\ref{e4}) is the time-dependent magnetic \nfield simulating the effect of laser pulses due to the inverse Faraday effect. In the \nexperiment by Kimel {\\it et al.} \\cite{kimel1} the magnetic field pulse is of the \norder of fractions of Tesla with time width  of hundred femtoseconds. In the \nnumerical simulations we consider a pulse directed along the propagation \ndirection of the light with a Gaussian shape\n\\begin{equation}\n{\\vec H}(t)=\\hat{k} \\frac{F_0}{\\sqrt{\\pi} \\tau_p} \\exp\\left[-(t\/\\tau_p)^2\\right],\n\\label{e4a}\n\\end{equation}\nwhere $\\hat{k}$ defines the direction of the light wave-vector and $\\tau_p $ indicates \napproximately the duration of the pulse. \\cite{notarot}\n\nFinally, in Eqs.(\\ref{e3},\\ref{e4}) $\\alpha$ is the Gilbert constant. Actually, \n$\\alpha$ takes into account the damping of the oscillations due to the magnon-magnon \nscattering and to the interaction of magnons with dysprosium spins and phonons. \nWe notice that the scattering via dysprosium spins should be larger at very low \ntemperatures where dysprosium ions tend to order. Moreover, the phonon-magnon \nscattering should be effective only on the nanosecond time scale. \\cite{hanst,kimel2} \nIt is the magnon-magnon interaction that provides the preminent source of \nscattering on the picosecond time scale.  \n\n\n\\subsection{Solution of linearized system}\nIn this subsection we consider the solution determined by linearizing \nEqs.(\\ref{e3},\\ref{e4}) and study the excitation of this linear system due to a \nmagnetic field pulse shaped as a delta function. \nThis is reasonable since, in the regime considered in the experiments, the effective \nmagnetic fields are not large if compared with exchange fields, and the temporal width \nof the pulse is much smaller than the periods of the resonance modes.\n\nIn order to take into account small deviations from the equilibrium, the standard \napproach is to consider two separate coordinate systems, \n($S_1$,$T_1$,$Y_1$) and ($S_2$,$T_2$,$Y_2$), which describe the \ndynamics of  $\\vec{M}_1$ and $\\vec{M}_2$, respectively. \\cite{hermann}\nThe variables $S_1$ and $S_2$ are chosen in order to coincide with the equilibrium \npositions of $\\vec{M}_1$ and $\\vec{M}_2$, respectively, so that\n\\begin{eqnarray}\nS_1&=&\\sin(\\beta_0) Z_1+\\cos(\\beta_0) X_1, \\\\ \nT_1&=&- \\cos(\\beta_0) Z_1 + \\sin(\\beta_0) X_1,\n\\label{e4b}\n\\end{eqnarray}  \nand\n\\begin{eqnarray}\nS_2&=&\\sin(\\beta_0) Z_2-\\cos(\\beta_0) X_2, \\\\ \nT_1&=&\\cos(\\beta_0) Z_2 + \\sin(\\beta_0) X_2.\n\\label{e4c}\n\\end{eqnarray} \nBy linearizing the system, we obtain the frequencies of two modes, the quasi-antiferro \n$\\omega_{AFM}$ and the quasi-ferro $\\omega_{FM}$ modes, involving cooperative motions \nof spins of the two sublattices. The energy of the quasi-antiferro and quasi-ferro  \nmodes is of the order of several $cm^{-1}$.\n\nThe first mode is characterized by the frequency \n\\begin{equation}\n\\frac{\\omega^2_{AFM}}{\\gamma^2}=4 E A_{xx}+4A_{xx}(A_{xx}-A_{zz})+D^2.\n\\label{e5}\n\\end{equation}\nThe dynamic in this mode shows the following behavior: $\\Delta X_1(t)=- \\Delta X_2(t)$, \n$ \\Delta Y_1(t)=- \\Delta Y_2(t)$ and \n$ \\Delta Z_1(t)=\\Delta Z_2(t)$, where $ \\Delta W_i(t) = W_i(t)-W_i^{eq}$, with $W=X,Y,Z$,\n$i=1,2$, and $W_i^{eq}$ corresponding to the equilibrium position shown in Fig.1. \nThus, the only ferromagnetic component different from zero is the magnetization along \nthe z-axis with respect to the equilibrium,   $\\Delta {M}_Z = M_0 \\Delta \\tilde{M}_Z$, \nwhere $\\Delta \\tilde{M}_Z$ is defined by \n\\begin{equation}\n\\Delta \\tilde{M}_Z = \\Delta Z_1 (t)+\\Delta Z_2 (t) = Z_1(t)+Z_2(t)-(Z_1^{eq}+Z_2^{eq}).\n\\label{e14a}\n\\end{equation}\nSince the net spin does not reflect the motions of the sublattice spins, \nthis mode is called quasi-antiferromagnetic. \n\nThe second mode, with frequency\n\\begin{equation}\n\\frac{\\omega^2_{FM}}{\\gamma^2}=4E(A_{xx}-A_{zz})+4A_{xx}(A_{xx}-A_{zz}),\n\\label{e6}\n\\end{equation}\nis characterized by $\\Delta X_1(t)=\\Delta X_2(t)$, $\\Delta Y_1(t)=\\Delta Y_2(t)$ and \n$\\Delta Z_1(t)=-\\Delta  Z_2(t)$. Since the net spin executes the\nsame rocking behavior as the sublattice spins, this mode is called quasi-ferromagnetic.\n\n\nEqs.(\\ref{e5},\\ref{e6}) relate the mode frequencies to the model parameters. The \nexchange fields are assumed constant in temperature at the values given above, since \nthey represent the highest energy scales. Using Eqs.(\\ref{e5},\\ref{e6}) and the \nexperimentally measured Raman spectra of Ref.17, we derive the anisotropy parameters. \nThe quasi-antiferro mode increases linearly as a function of temperature in the \n$\\Gamma_4$ phase: from 150 GHz at 50 K to 450 GHz at about 200 K, while the quasi-ferro \nmode stays constant at about 375 GHz. Therefore the anisotropic terms change upon \nheating the sample: $A_{xx}$ varies from about -640 Oe at T=50 K to about 200 Oe at \nT=200 K, $A_{zz}$ from -1540 Oe to -700 Oe in the same temperature range, with \napproximately fixed $A_{xx}-A_{zz}=900$ Oe. The knowledge of the frequencies is \nimportant also to determine the value of the damping constant in the dynamical \nequations \n(\\ref{e3},\\ref{e4}). \nIndeed, if $\\omega_0$ is the temperature-dependent frequency of one of the modes, the \ndamping constant $\\alpha$ could be related to the damping rate $1\/\\tau_0$ by the \nrelation $1\/\\tau_0=\\omega_0 \\alpha$.   \nAt the picosecond scale, the dominant spin-wave damping is due to four-magnon \nscattering. By using many-body perturbation theory, the rate for spin-wave at zero \nwave-vector is estimated to be $1\/\\tau_0 \\sim 2.66 \\cdot 10^4$ $T^2$ $s^{-1}$, with $T$ \ntemperature in units of Kelvin degrees. \\cite{tsang} The scattering rate grows \nquadratically in temperature, while the frequency of the quasi-antiferro mode is an \nincreasing linear function of the temperature. \\cite{koshi} The quantity \n$\\alpha$ gets larger with temperature since the most important contribution is given by \n$1\/\\tau_0$. Actually, the values of $\\alpha$ corresponding to \nthe quasi-antiferro mode range from about $0.4 \\cdot 10^{-4}$ at T=50 K to \n$3 \\cdot 10^{-4}$ at T=200 K. The smallness of $\\alpha$ implies that the oscillations \nof the magnetization are not strongly damped.      \n\nIt is useful to consider a magnetic field pulse shaped as a delta function\n\\begin{equation}\n{\\vec H}(t)=\\hat{k} F_0 \\delta(t).\n\\label{e6a}\n\\end{equation}\nSince the pulse is instantaneous, it provides an initial condition to the dynamics \ndescribed by the linearized equations of motions. We have analyzed two cases: \nfield along z- and x-axis, since these are prominent for the experiment that we \nwant to discuss. \n\nFor the field along the z-axis, starting from the equilibrium position at $t=0^-$, we \nfind at $t=0^+$\n\\begin{eqnarray}\n\\Delta X_1(0^+)&=&-\\Delta X_2(0^+)=\\cos(\\beta_0) [\\cos(\\gamma F_0) -1] \\\\ \n\\Delta Y_1(0^+)&=&-\\Delta Y_2(0^+)= \\cos(\\beta_0) \\sin(\\gamma F_0) \\\\\n\\Delta Z_1(0^+)&=&\\Delta  Z_2(0^+)=0.\n\\end{eqnarray}\nTherefore this pulse will excite only the quasi-antiferro mode. \n\nThe field along the x-axis yields at $t=0^+$\n\n\\begin{eqnarray}\n\\Delta X_1(0^+) &=&\\Delta X_2(0^+) =0  \\\\ \n\\Delta Y_1(0^+) &=&\\Delta Y_2(0^+) =-\\sin(\\beta_0) \\sin(\\gamma F_0) \\\\\n\\Delta Z_1(0^+) &=&-\\Delta  Z_2(0^+)= \\sin(\\beta_0) [\\cos(\\gamma F_0) -1].\n\\end{eqnarray} \nThus the system will be excited in the quasi-ferro mode. \n\nAs the experimental field magnitude is lower than the exchange fields, we study the\nsubsequent dynamical evolution within the linear approximation. \\cite{hermann}\nWithin a relaxation time approximation, we include the damping time  $\\tau$ \ndescribing the decay towards the equilibrium position. \\cite{morrish}  \n\nFrom the solution of the linear system in the case of a pulse along the z-axis, we get \nthe evolution of the quasi-antiferro mode. The only ferromagnetic component, \ndefined in Eq.(\\ref{e14a}), is  \n\\begin{eqnarray}\n\\Delta \\tilde{M}_Z=&&e^{- t \/ \\tau} \\frac{\\cos(\\beta_0)}{2} \\sin(2 \\beta_0) \n\\sin^2(\\frac{\\gamma F_0}{2}) \\cos(\\omega_{AFM} t) \\nonumber \\\\ \n&&+2 e^{- t \/ \\tau}  R \\cos^2(\\beta_0) \\sin(\\gamma F_0)  \n\\sin(\\omega_{AFM} t) \\nonumber  \\\\\n&&+2 e^{- t \/ \\tau}   \\sin{\\beta_0} \\cos^2(\\beta_0) \\left[\\cos(\\gamma F_0)-1 \\right].\n\\label{e14}\n\\end{eqnarray}\nWe notice in Eq.(\\ref{e14}) that the even terms in the magnetic field are at least two \norders of magnitude less than the odd term. \n\nAlong the other two directions, there are the antiferromagnetic component along x-axis \n(with respect to the equilibrium), $\\Delta {A}_X = M_0 \\Delta \\tilde{A}_X$, with \n$\\Delta \\tilde{A}_X$ defined as\n\\begin{equation}\n\\Delta \\tilde{A}_X=\\Delta X_1(t)-\\Delta X_2(t)=X_1(t)-X_2(t)-(X_1^{eq}-X_2^{eq}), \n\\label{e15a}\n\\end{equation}\nand the antiferromagnetic component along y-axis (with respect to the equilibrium), \n$\\Delta {A}_Y = M_0 \\Delta \\tilde{A}_Y$, with  $\\Delta \\tilde{A}_Y$ \n\\begin{equation}\n\\Delta \\tilde{A}_Y=\\Delta Y_1(t)-\\Delta Y_2(t)=Y_1(t)-Y_2(t)-(Y_1^{eq}-Y_2^{eq}). \n\\label{e16a}\n\\end{equation}\nUsing again the linearized form of Eqs.(\\ref{e3},\\ref{e4}), we obtain \n\\begin{eqnarray}\n\\Delta \\tilde{A}_X=&&-e^{- t \/ \\tau} \\sin(2 \\beta_0) \\sin(\\beta_0) \n\\sin^2(\\frac{\\gamma F_0}{2}) \\cos(\\omega_{AFM} t) \\nonumber \\\\\n&&- e^{- t \/ \\tau}  R \\sin(2 \\beta_0) \\sin(\\gamma F_0)  \n\\sin(\\omega_{AFM} t) \\nonumber  \\\\\n&&+2 e^{- t \/ \\tau}   \\cos{\\beta_0} \\cos^2(\\beta_0) \\left[\\cos(\\gamma F_0)-1 \\right],\n\\label{e15}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n\\Delta \\tilde{A}_Y=&&-e^{- t \/ \\tau}  \\frac{1}{R}  \\sin(2 \\beta_0) \n\\sin^2(\\frac{\\gamma F_0}{2}) \\sin(\\omega_{AFM} t)  \\nonumber \\\\ \n&&+2 e^{- t \/ \\tau}  \\cos(\\beta_0) \\sin(\\gamma F_0) \\cos(\\omega_{AFM} t).\n\\label{e16}\n\\end{eqnarray}\nThe components specified in Eqs.(\\ref{e14},\\ref{e15},\\ref{e16}) are \ndetermined by the mode frequency $\\omega_{AFM}$, the angle $\\beta_0$, and  \nthe quantity $R$ which depends on the model parameters in the form \n\\begin{equation}\nR=\\sqrt{\\frac{2 A_{xx}+D^2\/2E}{2 E}}. \n\\label{e16d}\n\\end{equation}\nAt intermediate temperatures ($A_{xx}$ very small) one gets $R \\simeq D\/2E$.\n\nWhen the pulse is directed along the x-direction, the quasi-ferro mode is excited. \nThe components of the magnetization different from zero are those along x- and y-axis. \nIn particular we consider the magnetization along x-axis \n(with respect to equilibrium), $\\Delta {M}_X = M_0 \\Delta \\tilde{M}_X$, where\n$\\Delta \\tilde{M}_X$ is defined as\n\\begin{equation}\n\\Delta \\tilde{M}_X=\\Delta X_1(t)+\\Delta X_2(t).\n\\label{e16c}\n\\end{equation}\nFrom the linearized dynamical equations this quantity is calculated as\n\\begin{equation}\n\\Delta \\tilde{M}_X= e^{- t \/ \\tau}  \\sqrt{\\frac{4 E}{A_{xx}-A_{zz}}}  \n\\sin^2(\\beta_0) \\sin(\\gamma F_0) \\sin(\\omega_{FM} t).\n\\label{e17}\n\\end{equation}\n\n\n\\section{Numerical and analytical results}\nUsing the procedure outlined in the previous section, all the parameters appearing\nin the dynamical equations (\\ref{e3},\\ref{e4}) can be determined. These nonlinear \nequations are numerically integrated through a fifth-order Runge-Kutta algorithm. \n\\cite{differe} The experimental laser pulses along the z-axis are \nestimated to be equivalent to magnetic fields with an amplitude of 0.3 T and a \nfull-width at half-maximum of about $200$ fs. \\cite{kimel1} \nTherefore, we consider Gaussian fields of the form given in Eq.(\\ref{e4a}), with \n$F_0=\\sqrt{\\pi} \\tau_p Amp(H)$, $\\tau_p=200$ fs and $Amp(H)$ typically 3000 Oe. \nThis value of the effective field is small when compared with the exchange fields, so \nthat effects due to the nonlinearity of the equations are negligible. Since the time \nwidth of the pulse is small on the scale of the mode periods, the results are not \nstrongly dependent on $\\tau_p$. The linear solution with a delta pulse represents a good \napproximation, provides the right orders of magnitude  but does not reproduce \nexactly the numerical results. Therefore, in all the following figures \nwe plot the results obtained by the numerical integration.     \n\nIn the first and second subsections we discuss the effects of a Gaussian pulse along \nz and x directions, respectively, when the system at equilibrium is in the \n$\\Gamma_4$ phase. In the final one we briefly analyze the resonance modes and \nmagnetization dynamics if the $\\Gamma_1$ phase is stable at low temperatures.\n\n\n\\subsection{Pulse along z-direction}\nAs illustrated in Ref. 12, pulses along the z and -z directions excite the \nquasi-antiferro mode. Using the numerical procedure discussed in the preceding section,\nwe have calculated the oscillating behavior of the ferromagnetic vector\nalong z focusing on $\\Delta M_Z$ at $T=95$ K. \nAs shown in Fig.2(a), the magnetization has a sine-like behavior with weak damping on \na picosecond scale. In addition, opposite fields give rise to \nout-of-phase oscillations in agreement with the experimental data. \nWe notice that after 50 picoseconds the amplitude is about half of the initial one, a \nvalue that is compatible with results reported in Fig.1 of Ref.12. This suggests that\nthe damping term associated to magnon-magnon scattering provides a reasonable \ndescription of the reduction of the amplitude as a function of temperature. \n\nComparing the experimental data shown in Fig.1 of Ref.12 with the theoretical results \nshown in Fig.2(a) of this paper, we point out similarities and differences. \nIndeed, on the scale of the initial pulse width, there is a strong enhancement of the \nexperimental Faraday rotation, probably due to the interference between pump and probe \npulses. After this transient the oscillations induced by pulses with opposite \nhelicities show similar amplitudes in time. In contrast with \ntheoretical results, the equilibrium point of the Faraday rotations shows a decay \nand its behavior is different for left- and right-handed polarized pulses. Only later \nthe oscillations tend toward a common equilibrium point that is different, however, \nfrom that before the pulse excitation. \nIf the shift of the equilibrium positions is caused by a change of the anisotropy \nconstants due to an intrinsic photomagnetic effect, then the amplitude of the \noscillations should be different for the two helicities, as reported in a recent \nexperiment. \\cite{hanst} Since this is not the case in dysprosium orthoferrites, \nthe change in the orientation equilibrium is more probably associated with a small but \nunavoidable laser-heating effect. Actually, within the model proposed in this paper, a \nshift of the equilibrium configuration could in principle be caused by the magnetic \npulse during the excitation. \\cite{hermann}\nThis shift is negligible, however, since the amplitude of the effective magnetic field \nis much smaller than the antisymmetrical exchange field $D$. Only by enhancing the \namplitude of the pulse to values comparable to $D$, the shift becomes important, and, \nupon further increase, even a precessional switching can occur. These \namplitudes imply very high light intensities that are probably above the damage \nthreshold of the samples.\n\nIn Fig. 2(b) we focus on the related quantity $d (\\Delta M_Z)$, defined as\n\\begin{equation}\nd (\\Delta M_Z)=\\Delta M_Z (H+)-\\Delta M_Z (H-),\n\\label{e17a}\n\\end{equation}\nwith $H+$ and $H-$ indicating positive and negative amplitudes, respectively. \nAs reported in the experimental work, \n\\cite{kimel1} this quantity is less dependent on initial effects. \nJust after the transient induced by the pulse, this quantity has an amplitude of the \norder of $M_S\/16$, in close agreement with the calculated data reported in Fig. 2(b). \nIt is worthwhile understanding the order of magnitude of this amplitude by exploiting \nthe result of the previous section for the delta pulse. From Eq.(\\ref{e14}) we get \nfor short times ($t \\ll \\tau_2$)\n\\begin{eqnarray}\n\\Delta {M}_Z \\simeq \n2 M_0 R \\sin(\\gamma F_0) \\sin(\\omega_{AFM} t), \n\\label{e18}\n\\end{eqnarray}\nwith $R$ given in Eq.(\\ref{e16d}).\nThe amplitude is determined by the term $R$ depending on the model parameters and by \n$\\sin(\\gamma F_0)$ related to the pulse intensity. \nUsing the experimental data, the impulse $F_0$ is estimated to be of the order of \n$3000 \\times 400 $ Oe$\\cdot$fs. This implies that $\\gamma F_0$ is of the same order as \nthe angle $\\beta_0$ responsible for the canted antiferromagnetism:\n$ \\sin(\\gamma F_0) \\sim \\gamma F_0 \\sim 2 \\beta_0$. Thus, we have\n\\begin{eqnarray}\n\\Delta {M}_Z\/M_S \\simeq 2 R \\sin(\\omega_{AFM} t).\n\\label{e19}\n\\end{eqnarray}\nIf we consider $R \\simeq D\/2E$, we obtain the order of magnitude \n$16 d (\\Delta {M}_Z)\/M_S \\sim 16 \\times 2 \\times 0.02 =0.64 $  not so far from the\nexperiment. Hence the ratio $D\/E$ is fundamental not only for the equilibrium \nconfiguration, but also for the magnetization dynamics in the canted antiferromagnet. \n   \n \n\\begin{figure}[t!\n  \\begin{center}\n  \\includegraphics[width=8cm,height=12cm]{Fig2new.ps}\n  \\end{center}\n\\caption{\n(a) The variation $\\Delta {M}_Z$ of the magnetization along z-axis with respect to the \nequilibrium value in units of $M_S\/16$ as a function of time for two opposite \namplitudes.\n(b) The difference $ d (\\Delta {M}_Z)$ in units of $M_S\/16$ between the magnetizations \nobtained upon excitation by opposite fields along z-direction as function of time. The \ndashed line includes the effect of the anisotropy that will be introduced in the next \nsubsection. \n}\\end{figure}\n\nWithin the theoretical approach, the dynamics of the \nantiferromagnetic vectors can be easily obtained, but these quantities have not been \nmeasured in the experimental work. \\cite{kimel1} \nDue to the fact that the equilibrium position corresponds to a maximum of \n$A_X=M_0 (X_1 - X_2)$, the temporal evolution of \n$\\Delta A_X=M_0 \\Delta \\tilde{A}_X$, with $\\Delta \\tilde{A}_X$ defined in \nEq.(\\ref{e15a}), is characterized by a very small amplitude. From Eq.(\\ref{e15}) one can\ndeduce that the amplitude of $\\Delta A_X$ should be at least an order of magnitude \nsmaller than that of $\\Delta {M}_Z$.  \n\nIn Fig. 3 we show the results of the numerical calculation for the quantity \n$\\Delta A_Y=M_0 \\Delta \\tilde{A}_Y$, where $ \\Delta \\tilde{A}_Y$ is defined in \nEq.(\\ref{e16a}). We point out that now the amplitude is an \norder of magnitude larger than that of $\\Delta {M}_Z$ and the response is more sensitive\nto the pulse. Actually this is due to the fact that the dynamics of the components along \nthe y-axis is strongly influenced by the highest energy scale $E$. \\cite{hermann} \nFrom Eq.(\\ref{e16}) we obtain a rough estimate for short times  ($t \\ll \\tau_2$)  \n\\begin{equation}\n\\Delta {A}_Y \\simeq 2 M_0 \\sin(\\gamma F_0) \\cos(\\omega_{AFM} t).\n\\label{e20}\n\\end{equation}\nTherefore the amplitude is \n$2 \\sin(\\gamma F_0) M_0 \\sim 2 \\beta_0 M_0 \\sim 0.022 M_0=M_S$. Finally, we notice \nthat, after 50 picoseconds, the damping acts in the same way as for the ferromagnetic \nvector along the z-direction, causing a reduction of about half of the initial \namplitude.\n  \n      \n\\begin{figure}[t!\n  \\begin{center}\n  \\includegraphics[width=8cm,height=9cm,angle=-90]{Fig3new.ps}\n  \\end{center}\n   \\vskip-3mm\n\\caption{\nThe variation of the antiferromagnetic vector along y-axis $\\Delta {A}_Y$ with respect \nto $M_0$ as a function of time for two opposite amplitudes.\n}\\end{figure}\n\nIn order to make contact to the experimental measurements, it is interesting to analyze \nthe behavior of the magnetization dynamics at different temperatures. As reported in \nFig.3 of the paper by Kimel {\\it et al.},\\cite{kimel1} upon heating the sample, \nthe frequency of the oscillations increases and the amplitude decreases. The \noscillations have temperature-dependent frequencies that are very close to those \nmeasured by Raman experiments at temperatures higher than 50 K. \nIn Fig.4 of this paper the calculated $d (\\Delta M_Z)$ is shown. \nWe find agreement with experimental decay of amplitudes and behavior of the \nfrequency upon increasing the temperature for times after the transient. \nWithin the theoretical approach, the temperature dependence of the anisotropy constants \n$A_{xx}$ and $A_{zz}$ affects the frequency of the modes, but is not so important in the \nchange of the amplitude after the laser transient. The decrease of the magnetization \nseen at a fixed time (of the order of tens of picoseconds) for different temperatures \nis dominated by the increase of the damping constant $\\alpha$ in temperature. \nDue to the agreement with experiment, the estimate of the damping constant \nderived on the basis of the magnon-magnon scattering is reliable.\n\nThe results of Fig.4 show the magnetization normalized by the sublattice magnetization \nmodulus $M_0$. We point out that, in principle, $M_0$ can vary as function of \ntemperature. However, the inclusion of the temperature dependence of \n$M_0=M_z\/(2 \\sin(\\beta_0)$) is not easy, since only the magnetization $M_z$ is \ntypically experimentally available. \nIn any case, even if the angle $\\beta_0$ is assumed fixed, the change of $M_z$ is small\nin the considered temperature range. \\cite{wolf} \n\n\n\n\\begin{figure}[t!\n  \\begin{center}\n  \\includegraphics[width=8cm,height=9cm,angle=-90]{Fig4new.ps}\n  \\end{center}\n   \\vskip-3mm\n\\caption{The difference between the magnetizations along z for\nopposite fields as a function of temperature. Every new curve is shifted from the \nprevious one along the vertical axis over 1.\n}\\end{figure}\n\nBefore closing this section, we focus on the dependence of the magnetic response\non the amplitude of the pulse. As inferred by Eq.(\\ref{e18}), for the effective magnetic\npulses used in the experiments, the ferromagnetic vector is proportional to $F_0$, \n{\\it i.e.}, to the intensity of the light pulse used in the inverse Faraday effect. In \nFig.5 we plot the numerical results for the amplitude of the magnetization  \nat t=40 ps, {\\it i.e.}, after about 10 periods as in the experimental data. \n\\cite{kimelp} \nWe find the expected linear behavior as a function of the amplitude of the pulse. \nThe value obtained at 5 T for $T=60$ K is not far from unity.\nThus, the saturation of the magnetization along z is reached for this high intensity, \nin agreement with the extrapolation of the experimental data in the inset of Fig.2 of \nRef.12. \n\n\\begin{figure}[t!\n  \\begin{center}\n  \\includegraphics[width=7cm,height=8cm,angle=-90]{Fig5new.ps}\n  \\end{center}\n   \\vskip-3mm\n\\caption{Amplitude of the spin oscillations as a function of the\namplitude of the magnetic field pulse for different temperatures.\n}\\end{figure}\nSummarizing, in this subsection we have focused on the magnetization dynamics in the \n$\\Gamma_4$ phase when the field pulse is directed along the z-axis. We have analyzed \nthe amplitude and the decay of the oscillations as a function of temperature finding \nagreement with experimental data for times longer than the initial transient. Actually, \non the time scale of the initial pulse width, discrepancies with experiments appear. \nThese could be due to a small but unavoidable laser-heating effect that is not included\nin the present theoretical approach.\n\n\\subsection{Pulse along x-direction: role of anisotropy}\nIn this subsection we analyze the effects of the excitation due to the pulse along the \nx-axis. \n\nThe fields along the x-direction excite another mode. As shown\nin Eq.(\\ref{e17}), $\\Delta \\tilde{M}_X$ oscillates with the frequency of the \nquasi-ferro mode in agreement with experiment. On the other hand, the magnitude of the \nmagnetic response along x is of the same order as that obtained along the z-axis {\\it \nwith a pulse of equal amplitude}. \nEven if the torque exerted by the field along x is tiny, \nthe subsequent temporal evolution, in particular, that of the variables $T_1$ and \n$T_2$, is able to give a non-negligible amplitude to the response along $x$. \nThis is in discrepancy with the experimental results which show that \nthe {\\it Faraday rotation} along x is at least an order of magnitude smaller \nthan that along the z-axis. \\cite{kimel1}\nAlso, as shown in Fig.3 of Ref.12, it does not show any temperature dependence \nfollowing the behavior of the frequency of the quasi-ferro mode, in striking contrast \nwith the case of a pulse along the z-axis. \n\nIn order to properly compare the experimental and theoretical results, we should take \ninto account several effects in orthoferrites. \\cite{simon} \nFirst of all, the experimental estimate \nof the effective magnetic field pulse due to the inverse Faraday effect is available \nonly when the pulse is along the z-axis. Moreover, it is well known that in rare-earth \northoferrites optical birefringence is not negligible. \\cite{tabor,usachev} Therefore \nthe experimental Faraday rotations along the z- and x-axis can be different, even if \nthe oscillations of the magnetizations are of the same order. Since the Faraday \nrotation along the x direction is very small in comparison with that along the z-axis, \npart of the effect could be also associated with the anisotropy of the magneto-optical\nsusceptibility. Indeed, the effective magnetic field generated via the inverse Faraday \neffect can be different for the two orthogonal orientations. Finally, there is another \nsource of anisotropy: the field along the z-axis can show a renormalization of the \ncoupling to the system that is different from that along the x-axis.   \n\nIn order to elucidate this last point, we have included in our model the anisotropy \ninduced by the dysprosium ions. As a result, a modification of \nthe coupling of the magnetic field to the iron ions takes place. According to a simple \nscheme proposed long ago by Zvezdin and Matveev, \\cite{zvezdin} \nthe coupling to the field along the z-direction is reduced by the factor \n$1+\\eta_z \\simeq 0.74$ (see in Fig.2(b) the effect of this reduction). \nInstead, the reduction $1+\\eta_x$ of the coupling along the x-axis is temperature \ndependent and can be extracted by the measurements of magnetic properties. \n\\cite{zvezdin} With temperature decreasing toward 50 K, these coupling factors can \nbecome a non-negligible source of anisotropy. As shown in Fig. 6, the difference in \nthe amplitudes for the response along the z- and x-axis can become relevant at those \ntemperatures. However, as shown in the inset of Fig. 6, the ratio between the \namplitudes along x and z axis increases with temperature up to an inversion point. \nTherefore the source of the anisotropy associated only to dysprosium ions seems not \nto be effective for the interpretation of the experimental data.    \n\n\n\\begin{figure}[t!\n  \\begin{center}\n  \\includegraphics[width=8.0cm,height=9.0cm,angle=-90]{Fig6new.ps}\n  \\end{center}\n\\caption{\nComparison between amplitudes for excitations with fields along z and x axis as \nfunction of time at T=60 K. In the inset the ratio of these amplitudes as function of the\ntemperature for times just after the pulse.  \n}\\end{figure}\n\n\n\\subsection{Stable phase at low temperatures}\nIn this last subsection we focus on the $\\Gamma_1$ phase that is stable at low \ntemperatures. \n\nThe measurements reported in Ref.12 show the maximum value of the amplitude of the \nmagnetization between 20 and 50 K. Moreover, the frequency of the photoinduced \nmagnetization is constant in this temperature range at the value close to 50 K \nthat is characteristic of the $\\Gamma_4$ phase. Therefore, the laser pulse is inducing \nexcitation modes of the phase that should be unstable at low temperatures.\n\nIn order to stabilize the $\\Gamma_1$ phase, the free energy (\\ref{e0}) has to be \nsupplemented with a quartic term, \\cite{zvezdin} for instance, a term that decreases the\nenergy when the moments are antiferromagnetically aligned along the y axis (see Fig.1).\nWe assume $V \\rightarrow V^{\\prime}=V-A^{(2)}_{yy} (Y_1-Y_2)^4$ in order to describe \nthe system in simple terms.\nUpon linearization around the equilibrium configuration $Y_1^{eq}=1$ and $Y_2^{eq}=-1$, \nthe two modes characteristic of this phase can be derived. The first one, \nwith frequency \n\\begin{equation}\n\\frac{\\omega^2_{z}}{\\gamma^2}=2 E (32 A^{(2)}_{yy}-2 A_{xx})-D^2,\n\\label{e5f}\n\\end{equation}\ncorresponds to a dynamic with $\\Delta X_1=-\\Delta X_2$ and $\\Delta Z_1=\\Delta Z_2$, \nwhile the second mode, with frequency\n\\begin{equation}\n\\frac{\\omega^2_{x}}{\\gamma^2}=2 E (2 A_{zz}-32 A^{(2)}_{yy})-D^2,\n\\label{e6f}\n\\end{equation}\nis characterized by $\\Delta X_1=\\Delta X_2$ and $\\Delta Z_1=-\\Delta Z_2$. We follow the\nprocedure of Sec. II: the anisotropy constants are derived by Eqs.(\\ref{e5f},\\ref{e6f}) \nand the experimental Raman spectra \\cite{koshi} by imposing $A^{(2)}_{yy}=0$. At T=50 \nwe find  $A_{xx}(\\Gamma_1)= - 1040$ Oe that is close to $A_{xx}(\\Gamma_4)=-640$ Oe \nobtained if the $\\Gamma_4$ phase is stable at that temperature.\n\nThe values of $A^{(2)}_{yy}$ required to stabilize the $\\Gamma_1$ phase are small. \nThe free energy corresponding to the equilibrium position \nof the $\\Gamma_1$ phase is $V_1=-E-16 A^{(2)}_{yy}$, while that of the $\\Gamma_4$ phase \nis at first order in the canting angle \n$\\beta_0$ \n\\begin{equation}\nV_4= -E -D (2 \\beta_0) -2 A_{xx}= -E -\\frac{D^2}{E} -2 A_{xx}.  \n\\end{equation}\nThe condition $V_1<V_4$ implies that \n\\begin{equation}\nA^{(2)}_{yy}> \\frac{D^2}{16 E}+\\frac{ A_{xx} }{8}. \n\\end{equation}\nTaking into account the values of the exchange fields given in the previous section and \nthe anisotropy constant $A_{xx}(\\Gamma_1)$, we derive $A^{(2)}_{yy}> 60$ Oe. Therefore \nthe value of the constant  $A^{(2)}_{yy}$ is consistent with the \nexpansion of the free energy being only a fraction of the anisotropy energies \n$A_{xx}$ and $A_{zz}$. \n\nIf the parameter  $A^{(2)}_{yy}$ is of the order of hundreds of Oersted, the phases \n$\\Gamma_4$ and $\\Gamma_1$ are close in energy. Due to a small light absorption, the \nlaser pulse could affect the stability of the system by favoring the $\\Gamma_4$ phase \non a short time scale. \nTherefore the femtosecond pulse could induce a reorientational phase transition \nfrom $\\Gamma_1$ to $\\Gamma_4$ state in analogy with the \nantiferromagnetic-to-ferromagnetic phase transition induced by heating with a laser in \nFeRh films. \\cite{thiele} \nFinally, we point out that a static magnetic field gives rise to a \nspin reorentational transition in orthoferrites. \\cite{zvezdin} Hence the role of the \nmagnetic field pulse induced via the inverse Faraday effect could be investigated in \nrelation to the perturbation of the phase stability. This is left for future \ninvestigations. \n           \n\n\\section{Summary}\nStimulated by recent experimental results showing ultrafast non-thermal control of \nmagnetization by instantaneous photomagnetic pulses in dysprosium orthoferrites, \na theoretical study of magnetization dynamics has been presented in this paper.  \nWe have employed a general form of free energy suitable for dysprosium orthoferrites \nwhose parameters are derived from experimental measurements. \nWe have solved coupled sublattice Landau-Lifshitz-Gilbert equations whose damping \nparameter is determined by considering the scattering rate due to magnon-magnon \ninteraction.\nDue to the inverse Faraday effect, the magnetic fields perturbing the equilibrium \nconfiguration can be modeled as Gaussian pulses with amplitude proportional to the \nintensity of the light pulse and time width of the order of hundred femtoseconds. \nThe non-linear dynamical equations have been integrated through an optimized Runge-Kutta\nalgorithm and an analytical solution of the linearized system has been discussed in the \ncase when the magnetic field pulse is assumed to have the shape of a delta function. \nThis solution provides the right orders of magnitude allowing to interpret the \nexperimental results in simple terms. \n\nWe have found that the quasi-antiferro mode is excited by the pulse along the z-axis\nand the oscillations of the magnetization have amplitudes compatible with experiment. \nMagnetic fields in opposite directions give rise to out-of-phase oscillations \nshowing a behavior in agreement with experimental results for times longer than the \ninitial transient. \nWe have stressed that the magnetization dynamics is not only strongly influenced \nby the amplitude of the magnetic field pulse, but also by the parameters determining \nthe free energy, in particular the ratio between the antisymmetric and symmetric \nexchange energies. The temperature dependence of the magnetization dynamics has been \ndiscussed showing that the proposed damping mechanism based on magnon-magnon \nscattering can be effective on the picosecond scale. When the field pulse is \nalong the x-axis, the quasi-ferro mode is excited, but there are some discrepancies \nin the comparison between theory and data. We point out that the response along the \nx-axis can be strongly influenced in orthoferrites by several effects such as the \noptical birefringence and the anisotropy of the magneto-optical susceptibility.  \nFinally, the behavior of the magnetization has been analyzed in the low-temperature \nrange where, due to an unavoidable heating effect, the laser pulse could perturb \nthe stability between $\\Gamma_1$ and $\\Gamma_4$ state.\n\nWe notice that the model proposed in this work has neglected dipolar contributions \nbecause they are orders of magnitude smaller than exchange and anisotropy terms. \nMoreover, due to the fact that the effective magnetic field obtained through the \ninverse Faraday effect shows spatial variations negligible on the microscopic scale, \nonly the spin-wave modes at zero wave-vector are excited. Since the static \nmagnetization changes slowly in the investigated temperature range, the presence of \nspin-waves with wave-vectors different from zero should not provide sizable \ncontributions to the dynamic behavior. Therefore, the macrospin approximation employed \nin this paper can be considered reliable.       \n\nFinally, we point out that the approach employed for dysprosium orthoferrites can be \nalso generalized to describe the magnetization dynamics of other rare-earth \northoferrites, at least in the $\\Gamma_4$ phase. The anisotropy constants are the only \nquantities strongly dependent on the rare-earth ion, but these do not play a major role \nin affecting statics and dynamics. On the other hand, the most important values of the \nexchange fields are of the same order in several rare-earth orthoferrites. \\cite{koshi}\nClearly the approach proposed in this paper is suitable for magnetic dielectrics \nand not for metallic itinerant magnets. However, up to now, due to the unavoidable light \nabsorption, it has been impossible to ascertain the role played by the inverse Faraday \neffect on the magnetization dynamics of itinerant magnets.\n \n\\section*{Acknowledgments}\nWe like to thank A.V. Kimel and A. Bringer for critical reading of the manuscript, \nand A. Kirilyuk and S. Woodford for useful discussions.\nC.A. Perroni acknowledges financial support by the European RTN network DYNAMICS.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION} \\label{sec:Introduction}\n\nGamma-ray bursts (GRBs) are extremely energetic transient events in the Universe, signifying catastrophic events that involve core collapse in some massive stars and mergers of two compact objects \\citep[e.g.,][]{kumarzhang15}. These events power an energetic, relativistic jet, which has powerful $\\gamma$-ray emission with an isotropic equivalent energy $\\sim 10^{50}$--$10^{55}$ erg. Observationally, the prompt $\\gamma$-ray emission lasts from milliseconds to several thousand seconds, with most of the light curves showing rapid variability. Following the prompt $\\gamma$-ray emission, the blast wave interacts with the circumburst medium and produces an afterglow, which is in principle detectable at X-ray through radio wavelengths.\n\nThe most popular model for GRBs is the standard fireball model, in which the broadband afterglow is from external shocks \\citep[][]{meszarosrees97,sari98}, while the prompt $\\gamma$-ray emission is from internal shocks \\citep[][]{rees98,kobayashipiran97,daigne98}. Recent studies suggest that the observed afterglow data can be quite consistent with the predictions of the external shock model \\citep[e.g.,][]{wang15}. Although the prompt emission is observed much earlier, it is less understood than the afterglow \\citep{zhang11}. Alternative models have been proposed suggesting that the prompt emission may be from the photosphere \\citep[][]{paczynski86,goodman86,meszarosree00,meszaros02,peer08}, magnetic dissipation regions \\citep[][]{lyutikov03,zhangyan11}, or external shocks \\citep[][]{reesmesazros92,meszarosrees93,dermer99}.\n\nThe rapid variability poses a problem for the external shock model. However, observationally there do exist some GRBs that only have a single smooth peak, or just a few long and temporally separated peaks with a large single smooth peak. The simple smooth profiles could arise from an external shock \\citep[e.g.,][]{mcmahon04,dermer04,ramirez07,guidorzi07,golkhou14,golkhou15,burgess16}.\n\n\nGRB 120729A is an interesting case with a single smooth pulse of $\\gamma$-ray emission. Furthermore, the prompt $\\gamma$-ray emission detected by \\emph{Swift}\/BAT smoothly extends to X-rays. Our observations are presented in \\S 2. In \\S 3, we perform the temporal and spectral analyses, and suggest the same physical origin (an external shock) for both the prompt $\\gamma$-ray emission and afterglow. Then, in \\S 4, we model the light curves with an external shock model by considering energy injection and time-dependent microphysics. Our discussion and conclusions are given in \\S 5.  Temporal and spectral slopes are defined as $F\\propto t^{-\\alpha} \\nu^{-\\beta}$ throughout this paper.\n\n\\section{OBSERVATIONS}\n\nGRB 120729A was triggered by the \\emph{Swift}\/BAT on 2012 July 29 (UT dates are adopted) at $10:56:14$, with $T_{90} = 71.5$~s \\citep[][]{ukwatta12a, palmer12}. It was also detected by the {\\it Fermi}\/GBM with $T_{90} = 25$~s \\citep {rau12}. The discrepancy of the detected duration ($T_{\\rm 90}$) is caused by the difference in sensitivity of the BAT and GBM energy bands \\citep{qiny13}. The X-ray Telescope (XRT) and the UV-optical Telescope (UVOT) onboard \\emph{Swift} began observing the X-ray and optical afterglows of GRB 120729A at 68~s and 77~s after the BAT trigger, respectively \\citep{ukwatta12b, oates12}.\n\nWe downloaded the BAT data from the NASA {\\em Swift} website, and used a Python source package  {\\it gtBurst}\\footnote{https:\/\/github.com\/giacomov\/gtburst} to extract light curves and spectra. The {\\em Swift}\/XRT light curve and spectrum are taken from the {\\it Swift} Burst Analyzer (Evans et al. 2010)\\footnote{http:\/\/www.swift.ac.uk\/burst\\_analyser\/00599037\/}. We also downloaded the {\\it Fermi}\/GBM data of GRB\\,120729A from the {\\it Fermi} Archive website\\footnote{ftp:\/\/legacy.gsfc.nasa.gov\/Fermi\/data\/}. We extract the light curve and spectrum from the {\\it Fermi}\/GBM data with our Python code. Spectral fitting package {\\it Xspec} was used for our spectral analysis. Both the BAT and GBM light curves exhibit a single smooth pulse (Figure \\ref{Gam-opt}a,b), and the BAT data smoothly connect to the XRT data (Figure \\ref{Gam-opt}c).\n\nGRB 120729A was rapidly followed up by ground-based telescopes: robotic 2~m Faulkes Telescope North (FTN) \\citep{virgili12}, 2~m Liverpool Telescope (LT), 0.82~m Instituto de Astrofisica de Canarias IAC80 telescope (IAC), 3.6~m Telescopio Nazionale Galileo (TNG), 10.4~m Gran Telescopio Canarias (GTC), Wide Field Camera (WFCAM) on the United Kingdom Infrared Telescope (UKIRT) \\citep{im12}, 0.4~m Rapid Telescopes for Optical Response (RAPTOR) \\citep{wren12}, Virtual Telescope \\citep{masi12}, 1.5~m Russian-Turkish telescope (RTT150) \\citep{khamitov12}, 1.5~m Observatorio de Sierra Nevada (OSN) \\citep{gorosabel12}, Burst Optical Observer and Transient Exploring System \\citep{gorosabel12}, James Clerk Maxwell Telescope (JCMT) \\citep{smith12}, and 0.8~m Tsinghua University -- National Astronomical Observatory Telescope (TNT) \\citep{xin12}.\n\nThe 0.76~m Katzman Automatic Imaging Telescope (KAIT; \\citep[][]{filippenko01,liw03}) at Lick Observatory also responded automatically to the {\\it Swift}\/BAT trigger and began imaging the field 98.0~s later. In this paper, we report the KAIT follow-up observations, which were performed in the $B$, $V$, $R$, and $I$ filters. Data reduction was carried out following standard routines in IRAF\\footnote{IRAF is distributed by the National Optical Astronomy Observatory (NOAO), which is operated by AURA, Inc., under a cooperative agreement with the NSF.} package. The photometry is reported in Table \\ref{table:magnitude}. Figure \\ref{Gam-opt}(c) shows the optical light curves, which include the data from KAIT and Cano et al. (2014). The redshift of GRB 120729A was measured to be $z = 0.80$ \\citep{tanvir12}. Note that BAT detections start 3.08~s before the BAT trigger (which occurred near the peak); thus, we shift $T_0$ to be this time of initial detection.\n\n\\section{MULTIWAVELENGTH LIGHT-CURVE BEHAVIOR}\n\nAn empirical model with a broken power-law function was employed to fit the light curves for temporal analysis \\citep[e.g.,][]{liang07,lil12,wang15}:\n\\begin{equation} F=F_{\\rm 0}\\left [\n\\left (   \\frac{t}{t_{\\rm b}}\\right)^{\\omega\\alpha_{\\rm 1}}+\\left (\n\\frac{t}{t_{\\rm b}}\\right)^{\\omega\\alpha_{\\rm 2}}\\right]^{{\\rm -1}\/\\omega},\n\\end{equation}\nwhere $\\alpha_{\\rm 1}$ ($\\alpha_{\\rm 2}$) is the temporal slope before (after) the break time $t_{\\rm b}$ and $\\omega$ represents the sharpness of the break. We found that the multiwavelength light curves can be well decomposed into four phases, as shown in Figure \\ref{Gam-opt}(c). The BAT light curve smoothly onsets with a slope of $\\alpha_{\\rm BAT,I}=-0.86\\pm0.11$ (Phase I) and peaks at $\\sim T_0+3.5$~s. The flux smoothly connects to the X-ray band and decays as a power law from $T_0+3.5$~s to $T_0+1950$~s (Phase II), with the power-law index $\\alpha_{\\rm BAT,II}=\\alpha_{\\rm X,II}=1.27\\pm0.09$ for both the BAT and  X-ray bands. The $R$-band light curve also decays as a power law with $\\alpha_{\\rm R,II}=0.96\\pm0.02$ in Phase II. Subsequently, the flux remains almost constant in both the X-ray and $R$ bands from $T_0+1950$~s to $T_0+3153$~s (Phase III) with $\\alpha_{\\rm X,III}=0.18$ (fixed) and $\\alpha_{\\rm R,III}=0.18\\pm0.12$, and then transits to a post-jet break (Phase IV) with $\\alpha_{\\rm X,IV}=1.83$ (fixed) and $\\alpha_{\\rm R,IV}=1.70\\pm0.09$.\n\nTo get more information, the analysis of broadband spectral energy distributions (SEDs) needed to be done. The optical data were corrected for Galactic extinction based on the burst direction \\citep[][]{schlegel98}, with $A_V=0.444$~mag, $A_R=0.351$~mag, $A_I=0.244$~mag, $A_g=0.534$~mag, and $A_i=0.275$~mag. The Galactic hydrogen column density in the burst direction is $N_{\\rm H}=2.15 \\times 10^{\\rm 21}$ cm$^{\\rm -2}$ (Willingale et al. 2013). The extinction law of the host galaxy was taken to be that of the Small Magellanic Cloud (SMC; $R_V=2.93$). The line-of-sight value of $N_{\\rm H}$ in the host galaxy is $\\sim 1.0\\times 10^{\\rm 21}$~cm$^{\\rm -2}$, derived from the time-integrated X-ray afterglow spectrum and fixed at this value in our time-resolved spectral fits. The ``Xspec'' package was employed for the spectral analysis. We subdivided the broadband data into five temporal ranges (as marked in Figure \\ref{Gam-opt}). Slice 1 covers the single pulse of $\\gamma$-ray emission from BAT to GBM in the time interval $T_0$ + [0,~15]~s. We take Slices 2, 3, and 4 to cover both the $\\gamma$-ray (BAT) and early afterglow (X-ray and optical) emission in respective time intervals $T_0$ + [15,~95]~s, $T_0$ + [95,~135]~s, and $T_0$ + [185,~330]~s. Slice 5 covers the late afterglow in the time interval $T_0$ + [3900,~6050]~s.\n\nThe fitting results are presented in Figure \\ref{spectral_analysis} and Table \\ref{tab-SED}. The SED of Slice 1 shows the BAT spectrum and GBM spectrum fitted with a single power-law function, with a photon index $\\Gamma_{\\rm \\gamma}=1.36\\pm0.02$ (seen in Figure \\ref{spectral_analysis}a) and a quite large $\\chi^{2}=2.13$. Calibration inconsistencies among the different instruments (GBM, BAT) could produce the high $\\chi^{2}$. The SED of the joint $\\gamma$-ray, X-ray, and optical spectrum (Slices 2, 3, and 4; see Figure \\ref{spectral_analysis}b) can be well fitted with a single absorbed power-law function, without considering the host-galaxy extinction. Their corresponding photon indices are 1.65, 1.70, and 1.73 for (respectively) Slices 2, 3, and 4. The SED of the joint optical and X-ray spectrum at the late-time epoch (Slice 5, seen in Figure \\ref{spectral_analysis}b) can also be well fitted with a single absorbed power-law function, with photon index 1.83. We take $\\Gamma = 1.75$ as a rough average (and stable) value of the photon index during the afterglow phase.\n\nAssuming the GRB 120729A multiwavelength data are located in the same spectral regime ($\\nu_{\\rm m}<\\nu<\\nu_{\\rm c}$)  and the circumburst medium is just the interstellar medium (ISM), one has the $\\alpha-\\beta$ closure relation with $\\alpha= 3\\beta \/2$, $\\alpha= 3\\beta \/2+0.75$, and $\\alpha=(q-1)+(2+q)\\beta\/2$ for the normal decay phase, the shallow decay phase, and the post-jet-break phases (respectively). Here the parameter $q$ is the energy injection parameter, which represents the central engine with a power-law luminosity history $L(t)=L_{0}(t\/t_{0})^{-q}$ \\citep[][]{zhangmeszaros01,zhang06}.\n\nCombining the results of our temporal analysis and spectral analysis, we use the closure relation ($\\alpha-\\beta$) of the fireball external shock model to test the multiwavelength data. We can see that the rising slope of the smooth bump is $\\alpha_{\\rm BAT,I}=-0.86\\pm0.11$ (Phase I). After the fireball starts decelerating, it transitions to a normal decay phase with a theoretical value  $\\alpha= 3\\beta \/2=3\/2\\times0.75=1.13$. The average of the observational spectral indices after the deceleration time is $\\beta=\\Gamma - 1 \\approx 0.75$, without considering the uncertainties. The observed values, $\\alpha_{\\rm BAT,II}=\\alpha_{\\rm X,II}=1.27\\pm0.09$ and $\\alpha_{\\rm R,II}=0.96\\pm0.02$ in Phase II, are approximately equivalent to the theoretical ones. In Phase III, we obtained an energy injection parameter $q=\\frac{(2+2\\alpha-2\\beta)}{(2-\\beta)}\\approx 0.12$ for both the X-ray and optical afterglow with  $\\alpha_{\\rm X,III}=0.18$ (fixed) and $\\alpha_{\\rm R,III}=0.18\\pm0.12$, which is located in a reasonable range predicted by the external shock. The theoretical value of the post-jet break is $\\alpha= 3\\beta \/2+0.75=1.88$, which is also close to the observed one, $\\alpha_{\\rm X,IV}=1.83$ and $\\alpha_{\\rm R,IV}=1.70\\pm0.09$ (Phase IV).\n\n The results of the temporal and spectral analyses show that the multiwavelength observed light curves have an achromatic behavior, consistent with the predictions of the external forward shock models with energy injection in the thin-shell case \\citep{zhang06,lil12},\n suggesting that both the prompt $\\gamma$-ray emission and the afterglow have the same physical origin.\n The rising slope of this model for the forward-shock onset is typically steep \\citep{gao13}.\n On the other hand, the circumburst medium may be more complex earlier \\citep[e.g.,][]{dai03,liang10,liang13,yi13},\n with the possibility of a transition between a wind profile and a constant-density medium. Adopting a more general\n profile with power-law index $k$ of the circumburst medium [$n(r)\\propto r^{-k}$], the predicted temporal index\n is \\textbf{$\\alpha=3-[k(\\beta+3)\/2]$} \\citep{liang13}. The rising slope of the smooth bump of GRB 120729A ($\\alpha_{\\rm BAT,I}=-0.86\\pm0.11$)\n  is consistent with a moderate $k$ value.\n\n\\section{MODELING: EXTERNAL SHOCK ORIGIN FOR BOTH THE PROMPT GAMMA-RAY EMISSION AND AFTERGLOW}\n\nWe further investigate our data with the standard forward shock model with energy injection. The light curves of GRB 120729A indicate that the $\\gamma$-ray emission phase (Phase I) is the blast-wave deceleration phase.  It then transitions to the self-similar deceleration phase (Phase II), followed by the energy injection which may invoke either a long-lasting central engine \\citep{dai98,zhangmeszaros01} or a Lorentz-factor-stratified ejecta \\citep{rees98,sari00,uhm12} exhibiting a shallow decay (Phase III). A post-jet-break phase (Phase IV) is entered when the $1\/\\Gamma$ cone is no longer filled with emission owing to the edge effect, with a steepening light curve\\citep[e.g.,][]{panaitescu98}. The late-time afterglow of GRB 120729A (beyond $10^5$~s) has a contribution from a possible accompanying supernova \\citep{cano14} and is beyond the scope of this paper.\n\n We adopt the standard external shock model by Sari et al. (1998) and Huang et al. (1999). The spectra in both the $\\gamma$-ray and afterglow regions are denoted as $N_e\\propto \\gamma_e^{-p}$. The spectral regimes are assumed to be located in $\\nu_{\\rm m}<\\nu<\\nu_{\\rm c}$, and we fix $p=2\\beta+1=2.5$. Our empirical analysis also indicates that the features of GRB 120729A emission are consistent with expectations in the ISM scenario. We adopt a constant ISM density $n$. The free parameters of our model include the isotropic kinetic energy $E_{\\rm K,iso}$, the initial Lorentz factor $\\Gamma_{\\rm 0}$, the fraction of shock energy to electron energy $\\epsilon_{\\rm e}$, the fraction of shock energy to magnetic field energy $\\epsilon_{\\rm B}$, the jet opening angle $\\theta_{\\rm j}$, and the parameters of the energy injection $L_{\\rm 0}$, $q$, $t_{\\rm s}$ (energy injection starting time), and $t_{\\rm e}$ (energy injection ending time). We can see that they can well constrain the observational data when $T > T_{\\rm 0} + 157$~s (as shown in Figure \\ref{theory-fit}a). When $T > T_{\\rm 0} + 157$~s, the best-fitting parameters of the standard external forward shock model are $E_{\\rm K,iso} = 3.36\\times 10^{\\rm 54}$ erg, $\\Gamma_{\\rm 0}=760$, $n = 8$ cm$^{\\rm -3}$, $\\epsilon_{\\rm e} = 0.01$, $\\epsilon_{\\rm B} = 4.0\\times 10^{\\rm -6}$, $\\theta_{\\rm j} = 0.0238$ rad,  $L_{\\rm 0} = 5.0\\times10^{\\rm 50}$ erg s$^{-1}$, $q = -0.1$, $t_{\\rm s} = 0$~s, and $t_{\\rm e} = 3698$~s.\n\nIn the standard model, the energy goes into electrons ($\\epsilon_e$) and magnetic fields ($\\epsilon_B$). The microphysical parameters are typically assumed to be not varying, and in fact constant $\\epsilon_e$ and $\\epsilon_B$ are consistent with the observations of late-time afterglows (e.g., Yost et al. 2003; Wang et al. 2015). The mechanism of energy transfer from protons to electrons and magnetic fields in the relativistic shocks is complicated. Modifications of the standard afterglow model with time-dependent microphysics models have been proposed to solve difficulties encountered with observations, such as X-ray afterglow plateaus, chromatic breaks, and afterglow rebrightenings (Ioka et al. 2006, Panaitescu et al. 2006; Kong et al. 2010; van der Horst et al. 2014). The early-time emission is even more complicated.\n\nTo explain the early emission of GRB 120729A, we consider a time-dependent microphysics model with $\\epsilon_e\\propto t^{\\alpha_e}$ and\/or $\\epsilon_B\\propto t^{\\alpha_B}$ \\citep{06ioka}. In the standard forward shock model, $\\epsilon_e$ and $\\epsilon_B$ evolve with $\\epsilon_e \\propto t^{-2p\/(p+4)}$ and $\\epsilon_B \\propto t^{2p\/(2p-1)}$ (for $\\nu_{\\rm m}<\\nu<\\nu_{\\rm c}$ and $p>2$), respectively. Figure \\ref{theory-fit}a shows that the modeled light curves with constant microphysics are lower than the observed data at early time ($T < T_{\\rm 0} + 157$~s). We therefore assume that $\\epsilon_B$ evolves with time and $\\epsilon_e$ is still constant, while the rest of the parameters are the same as above. We found that modeling the light curves after modification is well consistent with the data for both the prompt $\\gamma$-ray emission and the afterglow light curve (also shown in the Figure \\ref{theory-fit}a). The value of $\\epsilon_B$ in the early-emission epoch ($T < T_{\\rm 0} + 157$~s) evolves from $2.0\\times 10^{\\rm -5}$ to $4.0\\times 10^{\\rm -6}$ (Figure \\ref{theory-fit}b). It can be fitted with a broken power-law function, with $\\alpha_{\\rm B,1} = 0.18 \\pm 0.04$, $\\alpha_{\\rm B,2} = 0.84 \\pm 0.04$, and $t_{\\rm \\alpha_B,b} = 8.0 \\pm 1.6$~s.\n\n\\section{Discussion and Conclusions}\\label{sec:Conclusion}\n GRB 120729A has a single smooth $\\gamma$-ray emission pulse detected by {\\it Swift}\/BAT and {\\it Fermi}\/GBM. Its broadband afterglow was detected by {\\it Swift}\/XRT, {\\it Swift}\/UVOT, and ground-based optical telescopes. We obtained well-sampled KAIT $BVRI$ light curves from 98.9~s to 1.4~hr after the {\\it Swift}\/BAT trigger. We found that the prompt $\\gamma$-ray emission from {\\it Swift}\/BAT extends smoothly to the X-ray emission. Extensive analysis and modeling of the multiwavelength light curves shows that an external shock model can explain both the prompt $\\gamma$-ray emission and the afterglow. The properties of GRB 120729A are summarized as follows.\n \\begin{enumerate}\n   \\item The temporal and spectral joint fits of the multiwavelength light curves of GRB 120729A reveal achromatic behavior consistent with the external shock model. The light curves from the prompt $\\gamma$-ray emission phase to the optical can be decomposed into four phases: onset bump (Phase I), normal decay (Phase II), shallow decay (Phase III), and post-jet break (Phase IV).\n   \\item There is no obvious evolution of the SED between the X-ray and optical afterglows, with an average value of the photon index $\\Gamma \\approx 1.75$. The SED exhibits slight evolution from the prompt $\\gamma$-ray emission to the afterglow, with $\\Gamma_{\\rm \\gamma}=1.36$ to $\\Gamma \\approx 1.83$.\n   \\item The multiwavelength light curves from the $\\gamma$-ray emission to the optical afterglow can be well fitted with the external shock model, by introducing energy injection and time-dependent microphysics $\\epsilon_B\\propto t^{\\beta_B}$.  The best parameters are $E_{\\rm K,iso} = 3.36\\times 10^{\\rm 54}$ erg, $\\Gamma_{\\rm 0}=760$, $n = 8$ cm$^{\\rm -3}$, $\\epsilon_{\\rm e} = 0.01$, $\\epsilon_{\\rm B} \\approx [9\\times 10^{\\rm -5}, 4\\times 10^{\\rm -6}]$, $\\theta_{\\rm j} = 0.0238$ rad, $L_{\\rm 0} = 5.0\\times10^{\\rm 50}$ erg s$^{\\rm -1}$, $q = -0.1$, $t_{\\rm s} = 0$~s, and $t_{\\rm e} = 3698$~s. There is obvious evolution of $\\epsilon_B$ from $9\\times 10^{\\rm -5}$ to $4\\times 10^{\\rm -6}$ in the early emission phase when $T < T_{\\rm 0} + 157$~s.\n \\end{enumerate}\n\n The theoretical model suggests that $\\epsilon_B$ evolves with time as  $\\epsilon_B \\propto t^{2p\/(2p-1)}$ for the ISM case with $\\nu_{\\rm m}<\\nu<\\nu_{\\rm c}$ and $p>2$ \\citep{06ioka}. For GRB 120729A, we can see that the photon index evolves from the prompt $\\gamma$-ray emission to the afterglow with $\\Gamma_{\\rm \\gamma}=1.36$ to $\\Gamma \\approx 1.75$. The $\\epsilon_B$ in the early emission ($T < T_{\\rm 0} + 157$~s) can be described as a broken power-law function, with $\\alpha_{\\rm B,1} = 0.18 \\pm 0.04$, $\\alpha_{\\rm B,2} = 0.84 \\pm 0.04$, and $t_{\\rm \\alpha_B,b} = 8.0 \\pm 1.6$~s.  The results of the time-dependent microphysics model are consistent with the observed data. We obtain a low value of $\\epsilon_B \\approx [9\\times 10^{\\rm -5}, 4\\times 10^{\\rm -6}]$. The value of $\\nu_c$ is very sensitive to $\\epsilon_B$, with $\\nu_c \\propto \\epsilon_B^{-3\/2}$. The extremely low value of $\\epsilon_B$ ensures that both the optical and X-ray emission are still in the regime $\\nu_{\\rm m}<\\nu<\\nu_{\\rm c}$. These results are also consistent with previous studies \\citep[e.g.,][]{kumar09,kumar10,santana14,duran14,wang15,zhong16}.\n\nIn general, the characteristics of external shock pulses (e.g., pulse width, peak energy in the GRB spectrum $E_p$, and spectral-index evolution with time) can be very different from those of internal dissipation pulses \\citep[e.g.,][]{fishman95,norris96,peng07}. In order to systematically search for possible external-shock-origin GRBs, we collected candidate GRBs with a clear single pulse observed with {\\it Swift}\/BAT up to December 2017. There are 34 GRBs showing a very smooth single pulse, 29 a smooth single pulse followed by a small pulse, and 15 a small pulse followed by a smooth single pulse. The other 923 GRBs exhibit more pulses and are highly variable. We found that 11 GRBs in the single-pulse sample (GRBs 050721A, 050801, 051109A, 051111, 060912A, 070318, 070531, 080805, 110503A, 121117A, and 151006A) show a smooth connection between the BAT and XRT data with a slope less than the typical value of 1.5. Among them, 6 GRBs (GRBs 050721, 060912A, 070531, 080805, 121117A, and 151006A) lack early-time optical detections. Inspecting the other 5 GRBs, we find that only GRB 051111 and GRB 070318 may originate from an external shock for both the prompt $\\gamma$-ray emission and the afterglow (as shown in Figures \\ref{GRB-051111} and \\ref{GRB-070318}). Similar to GRB 120729A, the BAT detections of GRB 05111 and GRB 070318 start 3.03~s and 0.28~s (respectively) before the BAT trigger; thus, we have  shifted $T_0$ to be the time of initial BAT detection.\n\nFor GRB 051111, the best-fit parameters are $E_{\\rm K,iso} = 4.06\\times 10^{\\rm 54}$ erg, $\\Gamma_{\\rm 0}=650$, $n = 12$ cm$^{\\rm -3}$, $\\epsilon_{\\rm e} = 0.01$, and $\\theta_{\\rm j} = 0.1$ rad. The $\\epsilon_B$ value evolves as $\\epsilon_B \\propto t^{\\alpha_B}$ at early times ($T < T_{\\rm 0} + 120$~s) from $9.0\\times 10^{\\rm -4}$ to $1.0\\times 10^{\\rm -5}$; it can be fitted with a broken power-law function (dashed line) with $\\alpha_{\\rm B,1} = 0.21 \\pm 0.08$, $\\alpha_{\\rm B,2} = 2.75 \\pm 0.32$, and $t_{\\rm \\alpha_B,b} = 20.6 \\pm 2.5$~s. Subsequently, $\\epsilon_B$ stays constant at $\\epsilon_{\\rm B} = 1.0\\times 10^{\\rm -5}$. For GRB 070318, the best-fit parameters are $E_{\\rm K,iso} = 3.86\\times 10^{\\rm 53}$ erg, $\\Gamma_{\\rm 0}=1000$, $n = 8$ cm$^{\\rm -3}$, $\\epsilon_{\\rm e} = 0.12$, and $\\theta_{\\rm j} = 0.2$. The value of $\\epsilon_B$ evolves as  $\\epsilon_B\\propto t^{\\alpha_B}$ at early times ($T < T_{\\rm 0} + 85$~s) from $9.5\\times 10^{-6}$ to $2.0\\times 10^{-6}$, with $\\alpha_{\\rm B,1} = 0.80 \\pm 0.36$, $\\alpha_{\\rm B,2} = 0.25 \\pm 0.08$, and $t_{\\rm \\alpha_B,b} = 2.2 \\pm 2.0$~s. Thereafter ($T = T_{\\rm 0} + 85$~s), $\\epsilon_B$ stays constant at $\\epsilon_{\\rm B} = 2.0\\times 10^{\\rm -6}$.\n\nNo other GRBs display a smooth connection between the BAT and XRT data. In principle, all GRBs should have an external shock onset component similar to the one seen in GRB 120729A. However, whether it will appear in $\\gamma$-rays depends on the shock parameters. Our results suggest that once internal dissipation occurs, the $\\gamma$-ray efficiency is much higher than that of the external shock emission, and the external shock component is likely masked by internal dissipation emission in most events. This is consistent with theoretical modeling \\citep{maxham09}.\nIt seems that only a small fraction of GRBs may originate from an external shock for both the prompt $\\gamma$-ray emission and the afterglow.\n\nIf the composition of the GRB jet is a matter-dominated fireball, the absence of a ``prompt emission'' signal caused by internal dissipation within the jet is puzzling. Even though the lack of the internal shock emission may be circumvented by assuming that there is no significant variability within the outflow, the photospheric emission from the fireball would nevertheless show up, which should be above the detection threshold owing to the high efficiency expected for photospheric emission \\citep{meszarosree00}. The absence of such bright emission before the external-shock-origin $\\gamma$-ray peak therefore suggests that the outflow is Poynting-flux dominated, so that the photospheric luminosity is suppressed by a factor of $(1+\\sigma_{\\rm ph})$, where $\\sigma_{\\rm ph}$ is the magnetization parameter at the photosphere \\citep{zhangpeer09,15gao}. Such an outflow may still keep a moderate $\\sigma$ at the deceleration radius, so that the reverse-shock emission is also suppressed \\citep{zhangk05,gao15b}. Incidentally, the three GRBs we are studying (GRBs 051111, 070318, and 120729A) all did not show a reverse-shock emission component in the optical band, suggesting a self-consistent picture. It is possible that a dissipationless Poynting-flux-dominated outflow is a requirement to produce external-shock-origin single $\\gamma$-ray pulses. The scarcity of such bursts is consistent with the hypothesis that GRBs have diverse jet compositions \\citep{zhang11}, and the pure fireballs and pure Poynting-flux-dominated flows reflect the two extreme regimes of the jet composition.\n\n\n\\acknowledgments\nWe thank an anonymous referee for helpful suggestions. This work is supported by the National Basic Research Program of China (973 Program, Grant No. 2014CB845800),\nthe National Natural Science Foundation of China (Grant No. 11673006, 11533003), and the Guangxi Science\n Foundation (Grant No. 2016GXNSFFA380006, AD17129006).  We also acknowledge the use of public data from the \\emph{Swift} and {\\it Fermi} data archives.\nA.V.F.'s supernova\/GRB group is grateful for financial\nassistance from NSF grant AST-1211916, the TABASGO Foundation,\nthe Christopher R. Redlich Fund, NASA\/{\\it Swift} grants NNX10AI21G\nand NNX12AD73G, and the Miller Institute for Basic\nResearch in Science (U.C. Berkeley).\nKAIT and its ongoing operation were made possible by\ndonations from Sun Microsystems, Inc., the Hewlett-Packard Company,\nAutoScope Corporation, Lick Observatory, the NSF, the University of\nCalifornia, the Sylvia \\& Jim Katzman Foundation, and the TABASGO\nFoundation.\nResearch at Lick Observatory is partially supported by a generous\ngift from Google.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction \\label{sec1}}\n\n\nCoupled quantum wells (CQWs) provide a unique platform  to study the rich quantum properties of excitons \\cite{Miller1985a, Le1987a,butov1999,larionov2000,butov2002,combescot2017} that  have found numerous applications in optoelectronics \\cite{Ahn1989a, Tokuda1990a, High2007a, High2008a}. Of particular interest is a CQW  under  an  electric field perpendicular to the well plane: electrons and holes, which  for high enough field are separated in the two adjacent wells, form spatially indirect excitons, in contrast to direct excitons that are formed from electrons and holes in the same well. \nThe charge separation generates a built-in electric dipole moment, that can be controlled by the applied electric field. This has been used to control exciton transport  in  electrostatic \\cite{Winbow2011a, schinner2013} or optical \\cite{alloing2013, hammack2006} ways, and to develop excitonic devices \\cite{andreakou2014}. \nSuch indirect excitons have a long lifetime ranging from 10 ns to 10 $\\mu$s \\cite{High2008a}, which exceeds that of  direct excitons by orders of magnitude due to the small overlap between electron and  hole wave functions  that strongly decreases their  radiative recombination \\cite{Kash1985a,Voros2005a, sivalertporn2012,sivalertporn2015, Beian2015a}. Thus, they can easily reach  thermal equilibrium and allow exploring collective quantum phenomena   such as  exotic many-body phases \\cite{anankine2017} and exciton Bose-Einstein condensation \\cite{Beian2017a}. \n \n\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=15cm]{fig1_scheme-centered}\n\\caption{Schematic representation of carrier energies (dashed lines) and wave functions (solid lines) for electrons (blue) and holes (red) in a coupled quantum well, without (a) or with (b,c) electric field  $F$. The ground level $\\varepsilon_0^{(i)}$ for $F=0$ (a)  transforms into two levels under increasing $F$ (b,c). \n}\n\\label{fig1}\n\\end{center}\n\\end{figure*}\n\n\n\nThe small overlap between electron and hole wave functions renders indirect excitons  quite long-lived, but it also makes them weakly coupled to light. A way to enhance the photon coupling to excitons is to embed the CQW  in a microcavity \\cite{cristofolini2012}. The resulting exciton-photon particles can be transformed from hybrid polaritons to dipolaritons by applying an external field.  \nIn contrast to hybrid  polariton made from cavity photons strongly coupled to hybrid carriers that belong to the two quantum wells, the dipolaritonic modes result  from the strong  coupling of  indirect excitons,  direct excitons, and  cavity photons. The  controllability of the resulting system has  inspired  new applications such as  quantum logic gates \\cite{kyriienko2016}, optical parametric oscillators \\cite{khadzhi2015}, tunable single-photon emission \\cite{kyriienko2014} and dipolariton Bose-Einstein condensation \\cite{shahnazaryan2015}. \n\n In this paper, we consider a CQW  in an optical cavity, and we study how the cavity photon lifetime and the carrier coherence time  affect the dipolariton eigenstates. \n We wish to recall that, for polaritonic modes to be formed, the  coherence time of the exciton wave vector must be long at the exciton-photon coupling scale; otherwise  the exciton wave vector $\\mathbf{Q}$ would change and the exciton would emit a different photon $\\mathbf{Q}' \\neq \\mathbf{Q}$. The initial $\\mathbf{Q}$ photon would thus be lost, and the polaritonic mode $\\mathbf{Q}$ cannot develop \\cite{dubin2005}. \n \n We here focus on the  two limiting cases: \\\\\n (1) In the absence of electric field (see Fig.~\\ref{fig1}(a)), the electron and hole states result from the hybridization of individual quantum well states, depending on the  barrier height and well thickness. The excitons formed from these hybridized states are coupled to photons. The resulting polariton branches then have a lifetime that  depends on the photon detuning and the photon and carrier lifetimes. \\\\\n (2) When the electric field is large (see Fig.~\\ref{fig1}(c)), the electron ground state with energy $\\varepsilon_0^{(e)}$ is mainly localized in one well, while the hole ground state with energy $\\varepsilon_0^{(h)}$ is mainly in the other well. The first excited states of these carriers, with energies $\\varepsilon_1^{(e)}$ and  $\\varepsilon_1^{(h)}$, are mainly localized the other way around, provided that  the electric field is not too large; otherwise the carrier first excited state could be located in the same well as the ground state \\cite{sivalertporn2016}. The  exciton ground state, made of carriers in the $(\\varepsilon_0^{(e)}, \\varepsilon_0^{(h)})$ levels, is an indirect exciton. The next higher excitonic states, made of carriers in the same well, are direct excitons. The  quantum nature of the particles  allows them  to penetrate  the barriers. This induces a  small but finite overlap between the electron and hole wave functions. \n  The indirect exciton thus has a small but direct coupling to photons, with no need of any tunneling process that could induce a coupling to photons through  direct excitons. \nThe fact that the photon is coupled to both direct and indirect excitons leads to  eigenstates,  named dipolaritons, whose lifetime depends not only on the photon detuning to direct and indirect excitons, but also on the lifetimes of  excitons and cavity photons. \n\n\n\n\nThe exciton binding energies and the resulting excitonic photoluminescence spectra in a CQW under a static electric field have been studied using  variational methods \\cite{Kamizato1989a, Dignam1991a, Soubusta1999a}, or  by numerically solving the Schr\\\"odinger equation  of the coupled carriers \\cite{sivalertporn2012,sivalertporn2015}. We here propose an analytical approach to study these energies, in which the particle lifetimes is handled through the non-hermitian Hamiltonian approach. \n\n\n\n The paper is organized as follows:\n \\\\\n$\\bullet$ In section  \\ref{sec2}, we explain how to analytically study cavity photons coupled to quantum well excitons without and with a bias voltage between the two wells. Particular attention is paid to the photon coupling with indirect excitons, that was previously understood \\cite{cristofolini2012} through  electron tunneling between the wells, instead of by just taking into account the existing leakage of the carrier wave functions. \nWe show how to derive the hybrid polariton and dipolariton eigenstates when the particle  lifetimes are finite. These lifetimes    bring imaginary parts to the particle energies, that mathematically lead  to non-hermitian Hamiltonians with different bra and ket eigenstates. The diagonal form of the Hamiltonian then appears in terms of  creation operators for ket eigenstates, and destruction operators for bra eigenstates. Using this non-hermitian formalism, it becomes easy to analytically derive the time evolution of  photons coupled to electronic excitations located in two quantum wells, without and with a bias voltage, and to see the transition from hybrid polariton to dipolariton  when the detuning and particle lifetimes change.\n \\\\\n $\\bullet$ The unbiased case (no field)  is presented in Sec.~\\ref{sec3}, with photons coupled to excitons made of electrons and holes in hybridized states. The Hamiltonian then depends on  two operators only, the creation operators for photon and  hybridized excitons. \n \\\\\n$\\bullet$  In  Sec.~\\ref{sec4}, we consider a bias voltage large enough so that the direct excitons, made of electrons and holes in the same quantum well, are  well separated in energy from the indirect excitons, made of electrons and holes in different wells. Two kinds of direct excitons and two kinds of indirect excitons \\textit{a priori} exist. While the two indirect excitons are well separated in energy so that the one with  the highest energy can be eliminated from the problem, the two direct excitons have close energies; they would be equal for identical carrier leakages. It is possible to show that, for equal coupling  between photon and   direct excitons, one combination of direct excitons is decoupled, while the other gains  a $\\sqrt{2}$ bosonic enhancement. This leads to a three-body Hamiltonian made of photon, direct and indirect excitons, that is, a rank-3 eigenvalue equation which can be analytically solved using the Cardan's trick (see Appendix A). \\\\\n $\\bullet$   The main equations derived in this work are summarized in Table \\ref{table1}, at the end of the manuscript.\n \n \\section{General formalism \\label{sec2}}\n \\subsection{Semiconductor Hamiltonian}\n We consider two quantum wells having  equal width $\\ell$ separated by a potential barrier with thickness $d$, as shown in Fig.~\\ref{fig1}. This CQW structure can be made of two layers of InGaAs materials embedded in a GaAs material. Since the GaAs gap is larger than the InGaAs gap, the electron and hole wave functions along the growth axis $z$ tend to localize inside the InGaAs wells. The carriers  move freely in the well planes in the absence of Coulomb interaction.\n \n \n The semiconductor Hamiltonian for one electron and one hole reduces to \n \\begin{equation}\\label{eq:Hsc}\n H_{eh} =H_e + H_h + V_{eh}\\,. \n \\end{equation}\nThe ($H_e, H_h$) parts consist of  kinetic energies and  quantum well potentials for the electron and the hole, with a   Coulomb interaction $V_{eh}$ between them. \n \n\\noindent\\textbf{(1) One-body parts}: \nin the presence of external electric field $F$, the Hamiltonian for the carrier $i=(e,h)$  reads, in first quantization, as \n \\begin{equation}\\label{eq:He}\n H_i = \\frac{\\mathbf{p}^2}{2 m_i} + U_i(z)\\mp |e|Fz    \\,,\n\\end{equation}\n where the minus and plus sign in front of the field energy refers  to  electron and  hole, respectively. The potential $U_e(z)$ is equal to the difference between the  conduction band bottoms of the two semiconductors composing the quantum well, with $U_h(z)$  for the valence band  tops. Specifically, \n \\begin{equation}\nU_i(z)= -U_i  \\,\\Theta(z) \\,,\n\\end{equation} \n  with $\\Theta(z)=1$ inside the wells,  $d\/2 < |z| < \\ell+d\/2$, and $\\Theta(z)=0$  outside (see Fig.~\\ref{fig1}).\n  \n  \nBy writing $\\mathbf{p}^2$ as $p_z^2+\\mathbf{p}_{\\parallel}^2$, we can separate the electron Hamiltonian $H_e$ into a  part $\\mathbf{p}_{\\parallel}^2\/2 m_e$ that describes the free motion in the well plane, and a part  that depends on $z$   \n \\begin{equation}\n h_{z,e}=\\frac{p_{z}^2}{2 m_e}-U_e  \\,\\Theta(z) - eFz\\,.\n \\end{equation}\nThe wave function of the $H_e$ eigenstate, solution of $\\left(H_e-\\varepsilon^{(e)}_{n_e,\\mathbf{k}_{\\parallel}}\\right)|n_e^{(e)},\\mathbf{k}_{\\parallel}\\rangle=0$, then splits as \n \\begin{equation} \n \\langle \\mathbf{r}|n_e^{(e)},\\mathbf{k}_{\\parallel}\\rangle=\\braket{z}{n_e^{(e)}} \\, \\braket{ \\br_\\parallel}{\\bk_\\parallel}=\\phi_{n_e}^{(e)}(z) \\,\\frac{e^{i \\bk_\\parallel \\cdot \\br_\\parallel}}{L}\\,,\n  \\end{equation}\n where $L^2$ is the well area, in the direction orthogonal to $z$, while $\\phi_{n_e}^{(e)}(z)$ is the  wave function of the  $h_{z,e}$ eigenstate $|n_e^{(e)}\\rangle$ \n  \\begin{equation} \n  0 = \\left(  h_{z,e} - \\varepsilon_{n_e}^{(e)} \\right)|n_e^{(e)}\\rangle\\,.\\label{hzeeigenstate}\n  \\end{equation}\nAnd similarly for  holes.\n \n \n \n\n Let $a^{\\dagger}_{n_e,\\bk_\\parallel}$ and $b^{\\dagger}_{n_h,\\bk_\\parallel}$ be the creation operators of the electron and hole states, \\begin{subeqnarray}\\label{4}\na^{\\dagger}_{n_e,\\bk_\\parallel}\\ket{v} &=&|n_e^{(e)},\\mathbf{k}_{\\parallel}\\rangle \\,,\\label{4_0}\\\\\nb^{\\dagger}_{n_h,\\bk_\\parallel}\\ket{v} &=&|n_h^{(h)},\\mathbf{k}_{\\parallel}\\rangle\\,,\\label{4_1}\n\\end{subeqnarray} \nwith $\\ket{v}$ denoting the vacuum state. \n The one-body part of the electron-hole Hamiltonian \\eqref{eq:Hsc} then reads in terms of electron and hole operators as  \n\\begin{eqnarray}\nH_e + H_h &\\simeq&\\sum_{n_e=(0,1)} \\sum_{\\bk_\\parallel} \\left(\\varepsilon_{n_e}^{(e)} + \\varepsilon_{\\bk_\\parallel}^{(e)}\\right) a_{n_e,\\bk_\\parallel}^{\\dagger} a_{n_e,\\bk_\\parallel} \\label{hehh} \\\\\n&&+ \\sum_{n_h=(0,1)} \\sum_{\\bk_\\parallel} \\left( \\varepsilon_{n_h}^{(h)} +\\varepsilon_{\\bk_\\parallel}^{(h)}\\right) b_{n_h,\\bk_\\parallel}^{\\dagger} b_{n_h,\\bk_\\parallel}\\,,\\nonumber\n\\end{eqnarray}\nwith $\\varepsilon_{\\bk_\\parallel}^{(e,h)} = \\hbar^2 \\bk_\\parallel^2\/2 m_{e,h}$. \n\nThe $\\varepsilon_{n_i}^{(i)}$ energies depend on the electric field $F$. The carrier ground level, $n_i^{(i)}=0$ for $F=0$ transforms under increasing $F$ into a ground and an excited level, denoted as  $|0^{(i)}\\rangle$ and $|1^{(i)}\\rangle$. For small $F$, the carrier wave functions  are hybridized over the two wells (see Fig.~\\ref{fig1}(b)), while for large $F$, they are mainly localized in a single well when the barrier potential is large enough (see Fig.~\\ref{fig1}(c)). \n For small or large $F$, direct electron-hole pairs  have a far larger wave function overlap than indirect pairs, which  makes them  more coupled to photons. As shown below, their large coupling has a significant consequence on the  effective coupling between photons and  indirect pairs. This is why  direct pairs have to be taken into account in the construction of the polaritonic states,  although their energy is higher than for  indirect pairs. By contrast, the other indirect pairs made of electron with energy $\\varepsilon_1^{(e)}$  and hole with energy $\\varepsilon_1^{(h)}$  have a much higher energy, and thus can be neglected.\n \n\n\n\n\\noindent \\textbf{(2) Two-body interaction}: \nthe Coulomb attraction between one electron and one hole reads  in first quantization as\n\\begin{equation}\nV_{eh} = -\\frac{e^2}{\\epsilon_{\\rm sc}|\\mathbf{r}_{e} - \\mathbf{r}_{h}|}\\,,\n\\end{equation}\nwhere $\\epsilon_{\\rm sc}$ is the semiconductor  dielectric constant.\nIn second quantization, $V_{eh}$ appears in terms of  electron and holes operators  as\n\\begin{eqnarray}\nV_{eh}  &\\simeq& - \\sum_{n'_e,n'_h}\\sum_{n_e,n_h}\\sum_{\\bk_\\parallel,\\bk_\\parallel'} \\sum_{\\mathbf{q}_\\parallel} \\mathcal{V}\\Big(_{n'_h,\\bk_\\parallel'-\\mathbf{q}_\\parallel\\:\\:\\: n_h,\\bk_\\parallel'}^{ n'_e,\\bk_\\parallel+\\mathbf{q}_\\parallel \\:\\:\\: n_e,\\bk_\\parallel}\\Big)\\nonumber\\\\\n&&\\times b^{\\dagger}_{n'_h,\\bk_\\parallel'-\\mathbf{q}_\\parallel} a^{\\dagger}_{n'_e, \\bk_\\parallel+\\mathbf{q}_\\parallel}a_{n_e,\\bk_\\parallel}   b_{n_h,\\bk_\\parallel'}\\,.\\label{eq:VcoulV}\n\\end{eqnarray}\nAccording to the second quantization procedure \\cite{CSbook}, the prefactor of this two-body operator is given by \n\\begin{eqnarray} \\label{eq:vq}\n\\lefteqn{\\mathcal{V}\\Big(_{n'_h,\\bk_\\parallel'-\\mathbf{q}_\\parallel\\:\\:\\: n_h,\\bk_\\parallel'}^{ n'_e,\\bk_\\parallel+\\mathbf{q}_\\parallel \\:\\:\\: n_e,\\bk_\\parallel}\\Big) = {\\int} d\\mathbf{r}_e {\\int} d\\mathbf{r}_h   \\frac{e^2}{\\epsilon_{\\rm sc}|\\mathbf{r}_e - \\mathbf{r}_h|} } \\nonumber \\\\\n&&\\times \\bigg( \\phi_{n'_e}^{(e)}(z_e) \\frac{e^{i (\\bk_\\parallel + \\mathbf{q}_\\parallel)\\cdot \\mathbf{r}_{e\\parallel}}}{L}\\bigg)^*\\!\n \\bigg( \\phi_{n'_h}^{(h)}(z_h) \\frac{e^{i (\\bk_\\parallel' - \\mathbf{q}_\\parallel)\\cdot \\mathbf{r}_{h\\parallel}}}{L}\\bigg)^* \\nonumber \\\\\n&&\\times \\bigg( \\phi_{n_e}^{(e)}(z_e) \\frac{e^{i \\bk_\\parallel \\cdot \\mathbf{r}_{e\\parallel}}}{L}\\bigg) \n  \\bigg( \\phi_{n_h}^{(h)}(z_h) \\frac{e^{i \\bk_\\parallel'\\cdot \\mathbf{r}_{h\\parallel}}}{L}\\bigg) \\,,\n\\end{eqnarray}\n with  $d\\mathbf{r}_i = dz_i \\, d\\mathbf{r}_{i\\parallel}$. After integrating over $(\\mathbf{r}_{e\\parallel},\\mathbf{r}_{h\\parallel})$, we find that this scattering does not depend on $(\\bk_\\parallel,\\bk_\\parallel')$; it  reduces to\n\\begin{eqnarray}\\label{eq:vq_reduced}\n\\mathcal{V}_{\\mathbf{q}_\\parallel}\\big(_{n'_h\\: n_h}^{n'_e \\: n_e}\\big)&&=\\frac{2\\pi e^2}{L^2 \\epsilon_{\\rm sc}q_\\parallel} {\\iint} dz_e dz_h   \\, e^{-q_\\parallel |z_e-z_h|}\\\\\n&&\\times \\Big(\\phi_{n'_e}^{(e)}(z_e) \\phi_{n'_h}^{(h)}(z_h)\\Big)^* \\Big(\\phi_{n_e}^{(e)}(z_e)\\phi_{n_h}^{(h)}(z_h)\\Big)\\,,\\nonumber\n\\end{eqnarray}\nwhich evidences its dependence  on the wave function overlap between the  $n_e$ and $n'_e$ electron states and  between the  $n_h$ and $n'_h$ hole states. The largest overlaps correspond to $n_e=n'_e$ and $n'_h=n_h$, that is, to Coulomb processes in which the carriers do not change well, which are the ones leading to direct excitons. \n\n\n\n\\noindent \\textbf{(3) Exciton states}: \n the last step is to introduce electron-hole pair operators for the physically relevant states. \n %\nThe creation operator for a free pair made of an electron in the $n_e$ level and a hole in  the $n_h$ level, with a center-of-mass wave vector $\\bQ_\\parallel$ and a relative-motion wave vector $\\bk_\\parallel$, reads  \n\\begin{equation}\nB^{\\dagger}_{n_e,n_h;\\bQ_\\parallel,\\bk_\\parallel} = a^{\\dagger}_{n_e, \\bk_\\parallel + \\mu_e \\bQ_\\parallel} b^{\\dagger}_{n_h, -\\bk_\\parallel + \\mu_h \\bQ_\\parallel}\\,,\n\\end{equation}\nwith  $\\mu_e = 1 - \\mu_h = m_e \/M_{_X}$ for $M_{_X} = m_e + m_h$.\nThe electron-hole attraction  transforms this  free pair into a correlated pair, that is, an exciton with a center-of-mass wave vector $\\bQ_\\parallel$ and a relative-motion index $\\nu$, its creation operator reading as\n\\begin{equation}\\label{corrPair}\nB^{\\dagger}_{\\bQ_\\parallel,\\nu} = \\sum_{n_e,n_h}\\sum_{\\bk_\\parallel}  B^{\\dagger}_{n_e,n_h;\\bQ_\\parallel,\\bk_\\parallel} \\braket{n_e,n_h;\\bk_\\parallel}{\\nu}\\,,\n\\end{equation}\nthe  exciton energy being equal to  $E_{\\bQ_\\parallel,\\nu} =  \\hbar^2 \\bQ_\\parallel^2\/2 M_{_X} +\\varepsilon_\\nu$. \nThe $\\braket{n_e,n_h;\\bk_\\parallel}{\\nu}$ prefactor is solution of the  Schr\\\"{o}dinger equation\n\\begin{eqnarray}\\label{15}\n0&=&\\left(\\varepsilon_{n_e}^{(e)} {+} \\varepsilon_{n_h}^{(h)}{+}\\frac{\\hbar^2 \\bk_\\parallel^2}{2\\mu_X}{-}\\varepsilon_\\nu \\right)\\braket{n_e, n_h;\\bk_\\parallel}{\\nu}\\\\\n&&-\\sum_{n'_e,n'_h}\\sum_{\\mathbf{q}_\\parallel}\\mathcal{V}_{\\mathbf{q}_\\parallel}\\big(_{n_h\\,\\: n'_h}^{n_e \\,\\: n'_e}\\big)\\braket{n'_e,n'_h;\\bk_\\parallel-\\mathbf{q}_\\parallel}{\\nu},\\nonumber\n\\end{eqnarray}\n where $\\mu_{_X}$ is the electron-hole reduced mass given by $\\mu_{_X}^{-1}=m_e^{-1}+m_h^{-1}$. The exciton relative-motion energy $ \\varepsilon_{\\nu}$  depends on the single particle energies, \n $\\varepsilon_{n_e}^{(e)}$ and $\\varepsilon_{n_h}^{(h)}$, and the Coulomb scattering $\\mathcal{V}_{\\mathbf{q}_\\parallel}\\big(_{n_h\\: n'_h}^{n_e \\: n'_e}\\big)$;  through them, it also depends on the CQW parameters, $U_e$, $U_h$, $\\ell$, and $d$.   \n \n\nFor problems involving one electron-hole pair only, as the present one, we can replace  the $H_{eh}$ Hamiltonian by its diagonal form \n\\begin{equation}\\label{Hsc_diagonal}\nH_{eh}\\simeq\\sum_{\\bQ_\\parallel}\\sum_{\\nu}  E_{\\bQ_\\parallel,\\nu}   B^{\\dagger}_{\\bQ_\\parallel,\\nu}  B_{\\bQ_\\parallel,\\nu} \\, ,\n\\end{equation}\nwhich is convenient to derive polaritons. \n\nIt is important to note that the exciton eigenstates are linear combinations of $n_e$ electrons and $n_h$ holes; the way these  particle wave functions are distributed along $z$  in the two wells depends on the electric field. In general, this distribution  is nontrivial  and can only be numerically obtained. Excitons that are  mainly made of electron-hole pairs located in the same well are called direct, while those mainly made of  electron-hole pairs located in different wells are called indirect. When the carrier wave functions are extended in the two wells, the excitons are called hybrid.   \n\n\n\n\n\\subsection{Photon-semiconductor coupling}\nIn this section, we consider a CQW located in a photon cavity. The photon coupling to  direct and indirect excitons is controlled by the electric field through the exciton wave functions. We  first consider photon coupling to a bulk exciton and then turn to a CQW exciton, to better catch the consequences of the well confinement. \n\n\\noindent\\textbf{(1) Bulk}: in 3D, a photon with wave vector $\\mathbf{Q}$ and creation operator $\\alpha^{\\dagger}_\\mathbf{Q}$ is coupled to  free pairs with center-of-mass wave vector $\\mathbf{Q}$ and creation operators \n\\begin{equation}\\label{freeB}\nB^{\\dagger}_{\\mathbf{Q},\\mathbf{k}} = a^{\\dagger}_{\\mathbf{k}+ \\mu_e \\mathbf{Q}} b^{\\dagger}_{-\\mathbf{k} + \\mu_h \\mathbf{Q}}\\,.\n\\end{equation} \nThe photon-semiconductor interaction  then reads  \n \\begin{equation}\\label{Hph-sc}\n V_{\\rm ph-sc} = \\sum_\\mathbf{Q} \\sum_\\mathbf{k} \\Omega_{\\mathbf{Q}, \\mathbf{k}} \\,B^{\\dagger}_{\\mathbf{Q}, \\mathbf{k}}\\, \\alpha_{\\mathbf{Q}} + \\mathrm{h.c.} \\,.\n \\end{equation}\nThe coupling $\\Omega_{\\mathbf{Q}, \\mathbf{k}} $ is essentially $\\mathbf{k}$-independent for semiconductors having conduction and valence bands with different parities, as the ones we here consider; so, $\\Omega_{\\mathbf{Q}, \\mathbf{k}} \\sim \\Omega_\\mathbf{Q} \\sim \\Omega_0$, since the photon wave vector $\\mathbf{Q}$ is very small compared to the relevant electron wave vectors. \n \n \n Bulk excitons $ B^{\\dagger}_{\\mathbf{Q}, \\nu}\\ket{v}$ are related to free pairs $ B^{\\dagger}_{\\mathbf{Q}, \\mathbf{k}} \\ket{v}$ through \n \\begin{subeqnarray}\\label{Bfree}\n B^{\\dagger}_{\\mathbf{Q}, \\nu} &=& \\sum_\\mathbf{k} B^{\\dagger}_{\\mathbf{Q}, \\mathbf{k}} \\braket{\\mathbf{k}}{\\nu}\\,,\\\\\n B^{\\dagger}_{\\mathbf{Q}, \\mathbf{k}} &=& \\sum_\\nu B^{\\dagger}_{\\mathbf{Q}, \\nu} \\braket{\\nu}{\\mathbf{k}}\\,, \n \\end{subeqnarray}\n where $\\braket{\\mathbf{k}}{\\nu}$ is the bulk exciton relative-motion wave function in momentum space. This gives  \n the $ V_{\\rm ph-sc}$ interaction  in terms of excitons as \n \\begin{equation}\n V_{\\rm ph-sc} = \\Omega_0 \\sum_\\mathbf{Q} \\sum_\\nu  B^{\\dagger}_{\\mathbf{Q}, \\nu} \\alpha_{\\mathbf{Q}} \\sum_\\mathbf{k} \\braket{\\nu}{\\mathbf{k}} + \\mathrm{h.c.}\\, . \n \\end{equation}\n %\n The  sum over $\\mathbf{k}$ can be calculated by using $\\braket{\\mathbf{k}}{\\mathbf{r}} = e^{-i \\mathbf{k} \\cdot \\mathbf{r}}  \/ L^{3\/2}$: if we multiply $\\sum_\\mathbf{k} \\braket{\\nu}{\\mathbf{k}}$ by $1$ written as $L^{3\/2} \\braket{\\mathbf{k}}{\\mathbf{r}=0}$ and use the $\\ket \\mathbf{k}$ state closure relation,  the sum over $\\mathbf{k}$  reduces to $L^{3\/2}\\braket{\\nu}{\\mathbf{r}=0}$. \nSo, the  photon-semiconductor interaction ultimately appears in terms of 3D exciton creation operators  as \n \\begin{equation}\n  V_{\\rm ph-sc} = \\sum_\\mathbf{Q} \\sum_\\nu \\Omega_\\nu \\, B^{\\dagger}_{\\mathbf{Q}, \\nu} \\, \\alpha_\\mathbf{Q} + \\mathrm{h.c.}\\,,\n \\end{equation}\n with $\\Omega_\\nu = \\Omega_0 L^{3\/2}\\braket{\\nu}{\\mathbf{r}=0}$. \nThis, in particular,  shows that photons have a larger coupling to low 3D exciton states, due to their smaller spatial extension.\n\n\\noindent\\textbf{(2) Coupled quantum wells}: the quantum well structure breaks the translational invariance of bulk systems in the $z$ direction. \n We can derive the photon coupling  from Eq.~(\\ref{Hph-sc}), by first writing $B^{\\dagger}_{\\mathbf{Q}, \\mathbf{k}}$ in terms of the electron and hole states relevant for CQW, that are defined in Eq.~(\\ref{4}). This is done by writing the free electron creation operator $a^{\\dagger}_\\mathbf{k}$ in terms of a complete set of  $|n_e^{(e)}\\rangle$ states along $z$ that are defined  in Eq.~(\\ref{hzeeigenstate}), namely \n  \\begin{equation}\n a^{\\dagger}_\\mathbf{k} \\equiv a^{\\dagger}_{k_z, \\bk_\\parallel} = \\sum_{n_e} a^{\\dagger}_{n_e, \\bk_\\parallel} \\braket{n_e^{(e)}}{k_z} \\,.\n \\end{equation} \nAnd similarly for the hole.  The free pair creation operator $ B^{\\dagger}_{\\mathbf{Q}, \\mathbf{k}}$ given in Eq.~(\\ref{freeB}) becomes, in terms of   $(a_{n_e,\\bk_\\parallel}^{\\dagger},b_{n_h,\\bk_\\parallel}^{\\dagger})$  operators,  \n \\begin{eqnarray}\\label{23}\n  \\lefteqn{B^{\\dagger}_{Q_z,\\bQ_\\parallel; k_z,\\bk_\\parallel} \n\\!\\!= a^{\\dagger}_{k_z + \\mu_e Q_z, \\bk_\\parallel+\\mu_e \\bQ_\\parallel} b^{\\dagger}_{-k_z + \\mu_h Q_z, - \\bk_\\parallel+ \\mu_h \\bQ_\\parallel} }\\hspace{1.8cm}\n  \\nonumber \\\\\n & = & \\sum_{n_e,n_h}\\braket{n_e^{(e)}}{k_z {+} \\mu_e Q_z} \\braket{n_h^{(h)}}{{-}k_z {+} \\mu_h Q_z} \\nonumber\\\\\n &&\\times a^{\\dagger}_{n_e, \\bk_\\parallel + \\mu_e \\bQ_\\parallel} b^{\\dagger}_{n_h,-\\bk_\\parallel {+} \\mu_h \\bQ_\\parallel}\\,, \n \\end{eqnarray}\n $ \\braket{k_z}{n_e^{(e)}}$ being the Fourier transform of the $\\braket{z}{n_e^{(e)}}$ electron wave function. \n %\n In the following, we will  restrict the sum over quantum-well levels to the  two lowest states,  $|0^{(i)}\\rangle$ and $|1^{(i)}\\rangle$. \n %\n \n The next step is to transform the free pair creation operator $a^{\\dagger}_{n_e, \\bk_\\parallel + \\mu_e \\bQ_\\parallel} b^{\\dagger}_{n_h,-\\bk_\\parallel {+} \\mu_h \\bQ_\\parallel}$  of Eq.~(\\ref{23}) into creation operators for excitons with a center-of-mass wave vector $ \\bQ_\\parallel$, as  in Eq.~(\\ref{corrPair}). We get \n\\begin{eqnarray}\\label{eq24}\nB^{\\dagger}_{Q_z,\\bQ_\\parallel;k_z, \\bk_\\parallel} &&=\\hspace{-0.5em}\\sum_{n_e,n_h} \\braket{n_e^{(e)}}{k_z {+} \\mu_e Q_z} \\braket{n_h^{(h)}}{{-} k_z {+} \\mu_h Q_z}\\nonumber \\\\\n&&\\times\\sum_{\\nu} B^{\\dagger}_{\\bQ_\\parallel, \\nu} \\braket{\\nu}{n_e,n_h;\\bk_\\parallel}\\,.\n \\end{eqnarray}\n \n \n The photon-semiconductor coupling given in Eq.~(\\ref{Hph-sc})  then reads, in terms of CQW excitons, as\n \\begin{equation}\\label{Hphsc-final}\n  V_{\\rm ph-sc} =  \\sum_{Q_z,\\bQ_\\parallel} \\sum_{\\nu} \\sum_{n_e, n_h}  \\Omega_{n_e, n_h; \\nu} \\, B^{\\dagger}_{\\bQ_\\parallel, \\nu}\\,  \\alpha_{Q_z, \\bQ_\\parallel}  + \\mathrm{h.c.}\\,,\n \\end{equation}\n with  the prefactor given by \n \\begin{equation}\\label{30}\n \\Omega_{n_e, n_h; \\nu} {=} \\Omega_0  L  \\braket{\\nu}{n_e, n_h;\\br_\\parallel{=}0} \\! \\int_L \\! dz \\braket{n_e^{(e)}}{z} \\braket{n_h^{(h)}}{z}\\,.\n \\end{equation}\n To obtain this prefactor, we first note that $L \\braket{\\nu}{n_e, n_h;\\br_\\parallel=0}$  is equal to $\\sum_{\\bk_\\parallel}\\braket{\\nu}{n_e, n_h;\\bk_\\parallel}$, as obtained by using $1 = L\\braket{\\bk_\\parallel}{\\br_\\parallel=0}$ and the  closure relation for 2D states $|\\bk_\\parallel\\rangle$. Next,  to obtain  the integral over $z$, we have used the fact that the photon wave vector $Q_z$ is far smaller than the relevant electronic wave vectors; so, the  sum over $k_z$ reduces to  \n \\begin{eqnarray}\n&& \\sum_{k_z} \\braket{n_e^{(e)}}{k_z} \\braket{n_h^{(h)}}{{-} k_z} = \\\\\n &&\\!\\! \\int_L dz_{e} \\int_L dz_{h} \\braket{n_e^{(e)}}{z_{e}} \\braket{n_h^{(h)}}{z_{h}} \n  \\sum_{k_z} \\braket{z_{e}}{k_{z}} \\braket{z_{h}}{{-}k_z}. \\nonumber\n \\end{eqnarray}\n As $\\braket{z_{h}}{-k_z}$\n $=\\braket{k_z}{z_{h}} $, \n the sum over $k_z$ is equal to $\\braket{z_{e}}{z_{h}}$; so, we end  with\n \\begin{equation}\n \\sum_{k_z} \\braket{n_e^{(e)}}{k_z} \\braket{n_h^{(h)}}{- k_z} = \\int_L dz  \\braket{n_e^{(e)}}{z} \\braket{n_h^{(h)}}{z}, \n \\end{equation}\n which   corresponds to the overlap of the  electron and hole wave functions inside the wells. \n \n \nFor indirect excitons, the overlap between the  electron and hole localized in different wells  is very small, but differs from zero  for finite barrier height and thickness, because the carrier wave functions  do penetrate into the  barrier. \n So, indirect excitons are not fully decoupled from photons. As a result, it is not necessary to invoke a tunneling effect between direct and indirect excitons  to explain the coupling between photon and indirect exciton. This tunneling   actually is  another, less direct, way to say that carriers leak out of the well. \n \n\\subsection{Cavity photons}\nDue to the carrier confinement in the $z$ direction, bulk photons are poorly coupled to CQW excitons. Confining photons in a cavity selects a set of discrete photon wave vectors along the $z$ direction. This is crucial to possibly form polaritons with CQW excitons. \nIndeed, as seen from the photon-semiconductor coupling given in Eq.~(\\ref{Hphsc-final}), a $\\mathbf{Q} = (Q_z, \\bQ_\\parallel)$ photon is coupled to a CQW exciton with the same $\\bQ_\\parallel$ wave vector, whereas the conservation of the $Q_z$ component  is lost due to the well confinement. \nSo, the exciton can recombine into a photon with different $Q_z$ having essentially the same energy, except if the cavity forces the  allowed $Q_z$ components to be different enough in energy. \n\nLet us call $Q_z^{(c)}$ the wave vector of the  cavity photon having the lowest energy. For a cavity  such that  the other possible photon energies are outside the relevant energy range, we can restrict the photon Hamiltonian to \n\\begin{equation}\\label{33}\nH_{\\rm ph} \\simeq \\sum_{\\bQ_\\parallel} \\omega_{ \\bQ_\\parallel} \\alpha^{\\dagger}_{ \\bQ_\\parallel} \\alpha_{ \\bQ_\\parallel}\\,,\n\\end{equation}\nwith $\\alpha^{\\dagger}_{ \\bQ_\\parallel}\\equiv \\alpha^{\\dagger}_{Q_z^{(c)} ,\\bQ_\\parallel}$ and  $\\omega_{\\bQ_\\parallel} = v_{ph} \\sqrt{ \\bQ_\\parallel^2 + (Q_z^{(c)})^2}$, where $v_{ph}$ is the photon velocity in the material at hand.\n\n\\subsection{System Hamiltonian relevant to CQW}\nBy collecting the above results, we can write down the  system Hamiltonian for cavity photons interacting with carriers in the CQW as   \n\\begin{equation}\nH = H_{\\rm ph} + H_{eh} + V_{\\rm ph-sc}\\,, \n\\end{equation}\nwith the photon Hamiltonian  given in Eq.~(\\ref{33}). The other two Hamiltonians depend on the electric field applied to the CQW.\n\n\n(i) \\textbf{In the absence of electric field ($F=0$)}, the  exciton ground state is mainly made of  electron-hole pairs in their ground level, $n_i^{(i)}=0$, with their wave functions evenly distributed in the two wells (Fig. \\ref{fig1}(a)). \n The electron-hole Hamiltonian (\\ref{Hsc_diagonal}) then reduces to \n\\begin{equation}\nH_{eh} \\simeq \\sum_{\\bQ_\\parallel}\\varepsilon_{0,0;\\bQ_\\parallel} B^{\\dagger}_{0,0;\\bQ_\\parallel} B_{0,0;\\bQ_\\parallel}\\,,\n\\end{equation}\nwith  $\\varepsilon_{0,0;\\bQ_\\parallel} =  \\hbar^2 \\bQ_\\parallel^2\/2 M_{_X}  + \\varepsilon_{\\nu_0}$ where $\\varepsilon_{\\nu_0}$ is the exciton ground-state energy obtained by restricting $(n_e, n_h)$ states in Eq.~(\\ref{15}) to $(0,0)$. \nThese  excitons are coupled to cavity photons having the same $\\bQ_\\parallel$ through\n\\begin{equation}\\label{eq:37}\nV_{\\rm ph-sc} \\simeq \\Omega_{0,0} \\sum_{\\bQ_\\parallel} B^{\\dagger}_{0,0;\\bQ_\\parallel} \\alpha_{\\bQ_\\parallel} + \\mathrm{h.c.}\n\\end{equation}\nwith $\\Omega_{0,0} \\equiv \\Omega_{0,0;\\nu_0}$ given in Eq.~(\\ref{30}).\n\n\nThe system Hamiltonian then splits as $H = \\sum_{\\bQ_\\parallel} h_{\\bQ_\\parallel}$ with $h_{\\bQ_\\parallel}$ given by \n\\begin{eqnarray}\nh_{\\bQ_\\parallel}&=&\\omega_{\\bQ_\\parallel} \\alpha_{\\bQ_\\parallel}^{\\dagger} \\alpha_{\\bQ_\\parallel} +\\varepsilon_{0,0;\\bQ_\\parallel} B^{\\dagger}_{0,0;\\bQ_\\parallel} B_{0,0;\\bQ_\\parallel} \\nonumber\\\\\n&&+ \\left( \\Omega_{0,0} B^{\\dagger}_{0,0;\\bQ_\\parallel} \\alpha_{\\bQ_\\parallel} + \\mathrm{h.c.} \\right)\\, .\n\\end{eqnarray}\nIn the following, we will drop the $\\bQ_\\parallel$ index to simplify the notation, with $h_{\\bQ_\\parallel}$ now written as  \n\\begin{equation} \\label{39_0}\nh=\\omega \\alpha^{\\dagger} \\alpha+  \\varepsilon_{0, 0} B^{\\dagger}_{0, 0 } B_{0, 0}+  \\big(\\Omega_{0, 0} B^{\\dagger}_{0, 0 }\\alpha + \\mathrm{h.c.}  \\big)\\,.\n\\end{equation}\n\n(ii) \\textbf{When  the electric field is large}, it is necessary to include both direct and indirect excitons into the problem. The Hamiltonian $H$ also splits as a sum of $h_{\\bQ_\\parallel}$ Hamiltonians which reads, if we again drop the $\\bQ_\\parallel$ index, as \n\\begin{eqnarray} \\label{39}\nh&=&\\omega \\alpha^{\\dagger} \\alpha+ \\sum_{(n_e,n_h) = (0,1)} \\varepsilon_{n_e, n_h} B^{\\dagger}_{n_e, n_h } B_{n_e, n_h}\\nonumber\\\\\n&&+\\sum_{(n_e,n_h) = (0,1)} \\big(\\Omega_{n_e, n_h} B^{\\dagger}_{n_e, n_h }\\alpha + \\mathrm{h.c.}  \\big)\\,.\n\\end{eqnarray}\n\nAs schematically shown in  Fig.~\\ref{fig1}(c),  the energies of the two direct excitons, $\\varepsilon_{0,1}$ and $\\varepsilon_{1,0}$, are very close, and higher than the  $\\varepsilon_{0,0}$ energy of the indirect exciton. By contrast,  the energy $\\varepsilon_{1,1}$ of the other  indirect exciton is  much higher than  the ones of the other three species of excitons. This is why in the following, we will drop this $(1,1)$ indirect exciton from the $h$ Hamiltonian. \nMoreover, although the carriers  leak out of the well not exactly in the same way,  we will for simplicity take $\\varepsilon_{0,1} \\simeq \\varepsilon_{1,0}$  and $\\Omega_{0,1} \\simeq\\Omega_{1,0}$, the photon coupling to  direct excitons being much larger than the  $\\Omega_{0,0}$ coupling to the indirect exciton having the lowest energy. \n\n\n\n\\subsection{Finite lifetimes}\n\n The main purpose of this work is to study the effect of finite lifetimes on the coupled photon-CQW system. A mathematically simple and physically intuitive way to include them into the problem is to add an imaginary part to the exciton energies, that is, to replace $\\varepsilon_{n_e, n_h}$ by $\\tilde{\\varepsilon}_{n_e, n_h}=\\varepsilon_{n_e, n_h} - i\\gamma_{_X}$,  where $ \\gamma_{_X}$ denotes the inverse lifetime induced by non-radiative mechanisms that weakly depend on the well level. Indeed, the physical processes responsible for the lifetime we here consider, better called `coherence time', are the ones that cause a change in the exciton center-of-mass wave vector, these processes being  the same in the two wells. \n We also include the  lifetime of the cavity photon induced by mirror imperfections, through replacing $\\omega$ by $\\tilde{\\omega}=\\omega-i\\gamma_{ph}$. \n \nThe imaginary parts representing these lifetimes render  the system Hamiltonian non-hermitian. \nIn the next sections, we show how this can be analytically handled. \n\n\\section{Zero electric field: Hybrid polariton\\label{sec3}}\nIn the absence of electric field (Fig.~\\ref{fig1}(a)),  the relevant excitons are  hybrid excitons with carriers lying in the two wells. Including their lifetimes transforms  the  Hamiltonian \\eqref{39_0} into \n \\begin{equation} \\label{h:zeroF_lf}\nh=\\tilde{\\omega}\\, \\alpha^{\\dagger} \\alpha +\\tilde{\\varepsilon}_{0,0} B_{0,0}^{\\dagger} B_{0,0} +  \\left( \\Omega_{0,0}B_{0,0}^{\\dagger} \\alpha  + \\mathrm{h.c.} \\right)\\, ,\n\\end{equation} \nwith \n\\begin{eqnarray} \n\\tilde{\\omega}&=&\\omega-i\\gamma_{ph}\\,,\\label{eq37}\\\\\n\\tilde{\\varepsilon}_{0,0} &=&  \\varepsilon_{0,0}-i\\gamma_{_X} \\label{eq38}\\,. \n\\end{eqnarray} \n\n\n\n\n\\noindent\\textbf{(1) Hybrid polariton state and energy}: \ndue to the particle lifetimes, $h$ differs from $ h^{\\dagger}$; so, these two operators have different eigenstates. \nMoreover, since $h$ is not hermitian, its eigenstates are not orthogonal, and their energies are not necessarily real.\n\n$\\bullet$ Let us  look for the $h$ eigenstates \n \\begin{equation} \\label{40}\n0=\\big(h-\\mathcal{E}\\big)\\ket{P} \n\\end{equation}\nas linear combination of exciton state $\\ket{X}=B_{0,0}^{\\dagger} \\ket{v}$ and photon state $\\ket{\\alpha}=\\alpha^{\\dagger} \\ket{v}$, namely $\\ket{P}=x\\ket{X}+y\\ket{\\alpha}$.\nThe  eigenstate equation (\\ref{40}) for hybrid polariton made of photon coupled to hybrid exciton lying in the two wells leads to \n\\begin{equation} \\label{1}\n0=\\Big[x(\\tilde{\\varepsilon}_{0,0}-\\mathcal{E})+y\\Omega_{0,0} \\Big]\\ket{X}+\\Big[y(\\tilde{\\omega}-\\mathcal{E})+x\\Omega_{0,0}^*  \\Big]\\ket{\\alpha}\\,.\n\\end{equation} \nIts projection over $\\bra{X}$ and $\\bra{\\alpha}$  gives two coupled equations for $(x,y)$, which have a nonzero solution provided that their determinant is equal to zero. \nFor \\begin{equation}\n\\label{40bis}\n\\mathcal{E} \\equiv \\mathcal{E}' + \\frac{\\tilde{\\omega} + \\tilde{\\varepsilon}_{0,0}}{2}\\, ,\n\\end{equation}\nthis determinant appears as \n\\begin{equation}\n0 = \\begin{vmatrix}\n-\\tilde{\\omega}' - \\mathcal{E}' &\\Omega_{0,0} \\\\ \n\\Omega_{0,0}^*& \\tilde{\\omega}' - \\mathcal{E}' \n\\end{vmatrix}\n=\\mathcal{E}'^2-   \\tilde{\\omega}'^2 - |\\Omega_{0,0}|^2 \\,.\\label{matrixE'mathcal}\n\\end{equation}\n for $\\tilde{\\omega} - \\mathcal{E} \\equiv \\tilde{\\omega}' - \\mathcal{E}'$. \nThe  hybrid polariton energies $\\mathcal{E}$ follow from the two solutions of this equation. Let us call $\\mathcal{E}_{_X}$ the solution that goes to $\\tilde{\\varepsilon}_{0,0}$ and $\\mathcal{E}_{ph}$ the solution that goes to $\\tilde{\\omega}$, when the coupling $\\Omega_{0,0}$ goes to zero. The associated (unnormalized) eigenstates are given by\n\\begin{subeqnarray}\\label{43}\n\\ket{P_{_X}}&=&\\ket{X}+\\frac{\\Omega_{0,0}^*}{\\mathcal{E}_{_X}-\\tilde{\\omega}}\\ket{\\alpha}\\, ,\\\\\n\\ket{P_{ph}}&=& \\ket{\\alpha} + \\frac{\\Omega_{0,0}}{\\mathcal{E}_{ph}-\\tilde{\\varepsilon}_{0,0}} \\ket{X}\\, .\n\\end{subeqnarray}\n\n$\\bullet$ Since $h$ is non-hermitian, $h\\not= h^\\dagger$, the $h^\\dagger$ eigenstates, that \n  differ from the $h$ eigenstates, also play a role in the problem. The same derivation  shows that the  $h^\\dagger$ eigenstates simply read as the $h$ eigenstates with   $(\\mathcal{E}, \\tilde{\\omega},\\tilde{\\varepsilon}_{0,0})$ replaced by  $(\\mathcal{E}^*, \\tilde{\\omega}^*,\\tilde{\\varepsilon}_{0,0}^*)$: they read \n\\begin{subeqnarray}\\label{ketQrmx1}\n\\ket{Q_{_X}}&=&\\ket{X}+\\frac{\\Omega_{0,0}^*}{\\mathcal{E}^*_{_X}-\\tilde{\\omega}^*}\\ket{\\alpha}\\, ,\\\\\n\\ket{Q_{ph}}&=& \\ket{\\alpha} + \\frac{\\Omega_{0,0}}{\\mathcal{E}^*_{ph}-\\tilde{\\varepsilon}_{0,0}^*} \\ket{X}\\, .\n\\end{subeqnarray}\nNote that  $\\braket{Q}{P}\\neq 0$ for $(\\gamma_{ph},\\gamma_{_X}) \\neq0$, while   $\\ket{Q}=\\ket{P}$  for  $(\\gamma_{ph},\\gamma_{_X}) = 0$, as expected. \n\n$\\bullet$ To go further, we introduce the polariton operators associated with  the $h$ and $h^\\dagger$ eigenstates, $\\ket{P_{_X}}=P^\\dagger_{_X}\\ket{v}$ and $\\ket{Q_{_X}}=Q^\\dagger_{_X}\\ket{v}$, and similarly for $P^\\dagger_{ph}$ and  $Q^\\dagger_{ph}$. Using Eqs.~(\\ref{43},\\ref{ketQrmx1}), these operators, that read\n\\begin{eqnarray}\\nonumber\n\\:\\:P^\\dagger_{_X}{=}B_{0,0}^\\dagger{+}\\frac{\\Omega_{0,0}^*}{\\mathcal{E}_{_X}{-}\\tilde{\\omega}} \\alpha^\\dagger,\n\\qquad P_{ph}^{\\dagger}{=} \\alpha^{\\dagger}{+} \\frac{\\Omega_{0,0}}{\\mathcal{E}_{ph}{-}\\tilde{\\varepsilon}_{0,0}} B_{0,0}^{\\dagger} ,\\quad\\\\\n\\:\\: Q^\\dagger_{_X}{=}B_{0,0}^\\dagger{+}\\frac{\\Omega_{0,0}^*}{\\mathcal{E}^*_{_X}{-}\\tilde{\\omega}^*}\\alpha^\\dagger,    \n\\qquad Q_{ph}^{\\dagger} {=}  \\alpha^{\\dagger}{+} \\frac{\\Omega_{0,0}}{\\mathcal{E}^*_{ph}{-}\\tilde{\\varepsilon}_{0,0}^*} B_{0,0}^{\\dagger} ,\\quad \\nonumber\n\\end{eqnarray}\n fulfill the commutation relations \n\\begin{equation}\n\\frac{\\big[Q_{_X}, P^\\dagger_{_X}\\big]_-}{\\braket{Q_{_X}}{P_{_X}}}=1=\\frac{\\big[Q_{ph}, P^\\dagger_{ph}\\big]_-}{\\braket{Q_{ph}}{P_{ph}}}\\,, \n\\end{equation}\nthe other commutators being equal to zero.\nThese relations lead us to write the closure relation for hybrid polaritons as\n\\begin{equation}\\label{51}\n\\mathbb{I}= \\frac{\\ket{P_{_X}}\\bra{Q_{_X}}}{\\braket{Q_{_X}}{P_{_X}}}+\\frac{\\ket{P_{ph}}\\bra{Q_{ph}}}{\\braket{Q_{ph}}{P_{ph}}}\\,. \n\\end{equation}\n The diagonal form of the $h$ Hamiltonian then reads \n\\begin{equation}\nh=\\mathcal{E}_{_X}\\frac{P^\\dagger_{_X} Q_{_X}}{\\braket{Q_{_X}}{P_{_X}}}+\\mathcal{E}_{ph}\\frac{P^\\dagger_{ph} Q_{ph}}{\\braket{Q_{ph}}{P_{ph}}}\\, ,\\label{hasoperatorPQ}\n\\end{equation} \nas necessary  to  satisfy  $0=\\big(h-\\mathcal{E}_{_X}\\big)\\ket{P_{_X}}$ and $0=\\big(h-\\mathcal{E}_{ph}\\big)\\ket{P_{ph}}$, which is  easy to check. \n The above equations also lead to\n\\begin{equation}\n\\big[h,P^\\dagger_{_X}\\big]_-=\\mathcal{E}_{_X}P^\\dagger_{_X}\\, ,\\quad  \\big[h,P^\\dagger_{ph}\\big]_-=\\mathcal{E}_{ph}P^\\dagger_{ph}.\n\\end{equation}\n\nEquations (\\ref{51},\\ref{hasoperatorPQ}) also  provide a compact form for the time evolution operator  in the $\\ket{\\alpha}\\otimes\\ket{X}$ subspace, namely  \n\\begin{equation}\n e^{-iht}=e^{-i\\mathcal{E}_{_X}t}\\,\\frac{\\ket{P_{_X}}\\bra{Q_{_X}} }{\\braket{Q_{_X}}{P_{_X}}}+e^{-i\\mathcal{E}_{ph}t}\\,\\frac{\\ket{P_{ph}}\\bra{Q_{ph}} }{\\braket{Q_{ph}}{P_{ph}}}\\, . \n\\end{equation}\n\n\n\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=8.1cm]{fig2}\n\\caption{\n\\textbf{Hybrid polariton}. Dependence of the hybrid polariton energies ($\\mathcal{E'}, \\mathcal{E}$) defined in Eq. (\\ref{40bis},\\ref{eq59}), on the ratio of the photon coupling to hybrid exciton $\\Omega_{0,0}$  (a) over the detuning $\\delta$ when the cavity photon lifetime is equal to the exciton coherence time, $0=\\gamma=\\gamma_{_X}- \\gamma_{ph}$, and  (b) over the lifetime difference $\\gamma$  at the photon-exciton resonance, $0 = \\delta = \\omega-\\varepsilon_{0,0}$. Real parts (blue) and imaginary parts (red) of ($\\mathcal{E'}, \\mathcal{E}$). \\label{fig2}}\n\\end{figure}\n\n\n\n\n\\noindent\\textbf{(2) Hybrid polariton lifetimes}: \nThe finite lifetimes of the exciton and cavity photon are inherited by the hybrid polariton through the imaginary part of its eigenvalue given in Eqs. (\\ref{40bis}, \\ref{matrixE'mathcal}). By writing  \n\\begin{equation}\n\\tilde{\\omega}-\\tilde{\\varepsilon}_{0,0}=  \\delta + i\\gamma\n\\end{equation} \nwhere $\\delta$ is the photon detuning and $\\gamma$ is the  difference between the exciton and photon inverse lifetimes, \n \\begin{equation}\n\\delta=\\omega-\\varepsilon_{0,0}, \\,\\,\\,\\,\\,\\qquad  \\gamma=\\gamma_{_X}-\\gamma_{ph}\\,, \\label{57}\n\\end{equation}\nwe can deduce how the photon-exciton interaction couples the two particle lifetimes to the detuning. Let us study this tricky coupling in more detail.\n\n\n\n$\\bullet$ \\textbf{In the absence of photon-exciton interaction} ($\\Omega_{0,0}=0$), Eqs. (\\ref{40bis}, \\ref{matrixE'mathcal}) give $\\mathcal{E}'=\\pm (\\tilde{\\omega}-\\tilde{\\varepsilon}_{0,0})\/2$, that is, $\\mathcal{E}_{_X} =\\tilde{ \\varepsilon}_{0,0}$ and $\\mathcal{E}_{ph} =\\tilde{ \\omega}$, as expected. \n\n$\\bullet$  \\textbf{When the photon and the exciton have the same lifetime} ($\\gamma =0$), the detuning modifies the energies of the two polariton branches, but their lifetimes  stay equal to the common particle lifetime. Indeed, Eqs. (\\ref{40bis}, \\ref{matrixE'mathcal}) give $\\mathcal{E}'=\\pm \\sqrt{\\delta^2 \/4+ |\\Omega_{0,0}|^2}$ which is  real; so, the two polariton branches have the  lifetime of the particle, regardless of the detuning. \nFor a detuning small compared to the photon-exciton coupling,  $\\mathcal{E}' \\simeq \\pm  |\\Omega_{0,0}|$, while for large detuning, $\\mathcal{E}' \\simeq \\pm \\delta\/2$  whatever the coupling, as shown in Fig.~\\ref{fig2}(a). \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=8.1cm]{fig3}\n\\caption{\\textbf{Hybrid polariton}. Same as Fig.~\\ref{fig2},  for  general  $\\mathcal{E}'$ and $\\mathcal{E}$ given in Eq.~(\\ref{eq59}), as a function of the photon-exciton coupling $\\Omega_{0,0}$  (a) over the detuning $\\delta$ for  $\\delta > \\gamma>0$  and (b) over the lifetime difference $\\gamma$ for $\\gamma>\\delta>0$. \\label{fig3}}\n\\end{figure}\n\n$\\bullet$ \\textbf{At the photon-exciton resonance} ($\\delta=0$), the energies and lifetimes of the two polariton branches depend on the photon-exciton coupling since  at resonance $\\mathcal{E}'$ is equal to $ \\pm \\sqrt{|\\Omega_{0,0} |^2-\\gamma^2 \/4}$. For small coupling, $|\\Omega_{0,0} | <| \\gamma|\/2$,  the two solutions $\\mathcal{E}' = \\pm i \\sqrt{\\gamma^2\/4 -  |\\Omega_{0,0} |^2}$ are purely imaginary; so, the energies of the two polariton branches  stay equal to the particle energy. Under a coupling increase, their initially different lifetimes converge to the same value when $|\\Omega_{0,0}|=| \\gamma|\/2 $ (see Fig.~\\ref{fig2}(b)).  \n This holds until the exceptional point \\cite{kato1966,ashida2020},  at which  the two eigenvalues coalesce, while for larger coupling, the coalescence is lifted.\nFor large coupling, $ |\\Omega_{0,0} | > |\\gamma|\/2 $, the two solutions, $\\mathcal{E}' = \\pm  \\sqrt{  |\\Omega_{0,0} |^2-\\gamma^2\/4}$, are real; so, the two polariton energies depend on the coupling but  their lifetimes stay equal to the lifetime obtained for $|\\Omega_{0,0} | = |\\gamma|\/2 $, that is,\n\\begin{equation}\\label{ave_lifetime}\n\\bar{\\gamma}=\\frac{\\gamma_{_X}+\\gamma_{ph}}{2}\\,.\n \\end{equation}\nThe fact that the two polariton branches have the same average lifetime indicates that polaritons  oscillate between photon and exciton faster then the relaxation of any particle.\n\n\n\n$\\bullet$ These limiting cases help us catch the evolution of the hybrid polariton energies and lifetimes as a function of the photon-exciton coupling. We now consider the general case, that is, photon and exciton having different energies and different lifetimes. To do it, we go back to Eq.~(\\ref{matrixE'mathcal}); its two solutions read\n\\begin{equation} \\label{eq59}\n\\mathcal{E}' = \\pm \\sqrt{ (\\delta + i \\gamma)^2\/4 + |\\Omega_{0,0} |^2 }\\,. \n\\end{equation}\nThese two $\\mathcal{E}'$ values  evolve from $\\pm(\\delta  +  i \\gamma)\/2$ when $|\\Omega_{0,0}|= 0$, to $\\pm |\\Omega_{0,0}|$ for $|\\Omega_{0,0}|$ large compared to $|\\delta|$ (see Fig.~\\ref{fig3}). The energies of the two polariton branches increase with $|\\Omega_{0,0}|$, while their lifetimes converge  to the average value given in Eq.~(\\ref{ave_lifetime}).\n We note the absence of exceptional point in this setting.\n \nAll this shows that the coupling between photon and exciton not only changes their energies but also their lifetimes in a tricky way, except when the cavity photon lifetime is equal to the exciton coherence time.\n\n\n\n\\section{Large electric  field: dipolariton \\label{sec4}}\n\nWhen the external electric field is large (see Fig.~\\ref{fig1}(c)), we have shown in Eq.~(\\ref{39}) that the relevant Hamiltonian consists of  cavity photon coupled to two direct excitons, $B_{0,1}^{\\dagger}$ and $B_{1,0}^{\\dagger}$ with energy $\\varepsilon_d\\equiv \\varepsilon_{0,1}\\simeq \\varepsilon_{1,0}$, and to  a  ground-state exciton with energy $\\varepsilon_{id}\\equiv \\varepsilon_{0,0}$, which is an indirect exciton, $B_{id}^{\\dagger}$, its electron and hole being in their ground level localized in different wells. We now add the particle lifetimes. \n  By introducing the two linear combinations of direct excitons, that are normalized and commute,\n \\begin{eqnarray}\\label{BD_transform}\n B_{d}^{\\dagger}=\\frac{B_{0,1}^{\\dagger}+B_{1,0}^{\\dagger}} {\\sqrt{2}}  \\, ,\\quad         D_{d}^{\\dagger}=\\frac{B_{0,1}^{\\dagger}-B_{1,0}^{\\dagger}} {\\sqrt{2} }\\,, \n  \\end{eqnarray}\n the $h$ Hamiltonian appears,   after\ndropping the (1,1) indirect exciton, as  \n \\begin{eqnarray} \\label{64}\nh&\\textcolor{black}{\\approx}&\\tilde{\\omega}\\, \\alpha^{\\dagger} \\alpha +\\tilde{\\varepsilon}_{d}\\, \\Big(B_{d}^{\\dagger} B_{d}+ D_{d}^{\\dagger} D_{d}\\Big)+\\tilde{\\varepsilon}_{id}\\, B_{id}^{\\dagger} B_{id}\\nonumber  \\\\ \n &&+  \\Big( \\big(\\sqrt{2}\\, \\Omega_{d} \\,B_{d}^{\\dagger}+\\Omega_{id} \\,B_{id}^{\\dagger} \\big)\\alpha + \\mathrm{h.c.} \\Big)\\, ,\n\\end{eqnarray}\nwith the couplings given by $ \\Omega_d\\equiv \\Omega_{0,1}\\simeq\\Omega_{1,0}$ and  $\\Omega_{id}\\equiv\\Omega_{0,0}$, while the energies are given by  \n\\begin{eqnarray}\n&\\tilde{\\varepsilon}_d  \\equiv\\varepsilon_{0,1}-i\\gamma_{_X} = \\tilde{\\varepsilon}_{0,1} \\simeq  \\tilde{\\varepsilon}_{1,0} , \\label{eq55} \\\\\n&\\tilde{\\varepsilon}_{id}\\equiv\\varepsilon_{0,0}-i\\gamma_{_X}.  \\label{eq56}\n\\end{eqnarray}\nNote the $ \\sqrt{2} $ enhancement factor that appears in the   photon coupling to the $B_d^{\\dagger}$ direct exciton combination. We also note that  the $D_d^{\\dagger}$ exciton is not coupled to the cavity photon; so, in the following, we will drop it from the $h$ Hamiltonian given in  Eq. \\eqref{64}. \n\n\n\n\n\\noindent\\textbf{(1) Dipolariton eigenstates}\n\n\n\n$\\bullet$ \nWe proceed as we did for zero electric field. We first look for the $h$ eigenstates, \n$\n0=\\big(h-\\mathcal{E}\\big)\\ket{P} \n$, as a linear combination of  indirect exciton $\\ket{\\textsc{ix}}=\\,B_{id}^{\\dagger} \\ket{v}$, direct exciton $\\ket{\\textsc{dx}}=\\,B_{d}^{\\dagger} \\ket{v}$ and  cavity photon $\\ket{\\alpha}=\\,\\alpha^{\\dagger} \\ket{v}$, namely \n\\begin{eqnarray} \n\\ket{P}=x_{id}\\ket{\\textsc{ix}}+x_{d}\\ket{\\textsc{dx}}+y\\ket{\\alpha}\\,. \\label{EQ57}\n\\end{eqnarray} \nThe equation for dipolariton eigenstates then reads \n\\begin{eqnarray} \n0&=&\\Big(x_{id}(\\tilde{\\varepsilon}_{id}{-}\\mathcal{E}){+}y\\Omega_{id} \\Big)\\ket{\\textsc{ix}} \\nonumber \\\\\n&+&\\Big(x_{d}(\\tilde{\\varepsilon}_{d}{-}\\mathcal{E}){+}\\sqrt{2}y\\Omega_{d} \\Big)\\ket{\\textsc{dx}}\n\\\\ \n&+&\\Big(y(\\tilde{\\omega}-\\mathcal{E})+x_{id}\\Omega_{id}^* +\\sqrt{2}x_{d}\\Omega_{d}^* \\Big)\\ket{\\alpha} \\,. \\nonumber\\label{64_1}\n\\end{eqnarray} \nIts projection over $\\bra{\\textsc{dx}}$, $\\bra{\\textsc{ix}}$ and $\\bra{\\alpha}$  gives three coupled equations for $(x_{id},x_{d},y)$, which have a nonzero solution provided that their determinant is equal to zero. In terms of \n\\begin{equation}\n\\label{eq58}\n\\mathcal{E} \\equiv \\mathcal{E}' + \\frac{\\tilde{\\omega} + \\tilde{\\varepsilon}_{id}+ \\tilde{\\varepsilon}_d}{3}\\, ,\n\\end{equation}\n this determinant appears as\n\\begin{eqnarray}\n0 = \\begin{vmatrix}\n\\tilde{\\omega}'_{id} - \\mathcal{E}' & 0 & \\Omega_{id}\\\\ \n0 &\\tilde{\\omega}'_{d} - \\mathcal{E}'  & \\sqrt{2}\\,\\Omega_{d} \\\\ \n\\Omega_{id}^* &\\sqrt{2}\\, \\Omega_{d}^* & \\tilde{\\omega}' - \\mathcal{E}'\n\\end{vmatrix}\\,,\\label{65}\n\\end{eqnarray}\nwith  $\\tilde{\\omega} - \\mathcal{E} \\equiv \\tilde{\\omega}' - \\mathcal{E}'$; and similarly for $( \\tilde{\\omega}'_{id}, \\tilde{\\omega}'_{d})$. \n\nLet us call $\\mathcal{E}_{ph}$ the eigenvalue that tends to $\\tilde{\\omega}$  when the photon-exciton couplings go to zero, and similarly for ($\\mathcal{E}_{id},\\mathcal{E}_{d}$). The associated (unnormalized) eigenstates are given by\n\\begin{eqnarray}\n&&\\ket{P_{id}}=\\ket{\\textsc{ix}} \\label{ketPrmx1}\\\\\n&&+\\frac{\\Omega^*_{id}}{(\\mathcal{E}_{id}{-}\\tilde{\\varepsilon}_{d})(\\mathcal{E}_{id}{-}\\tilde{\\omega}){-}2 |\\Omega_d|^2}  \\left[\\sqrt{2}\\Omega_d\\ket{\\textsc{dx}}{+}(\\mathcal{E}_{id}{-}\\tilde{\\varepsilon}_{d})\\ket{\\alpha}\\right]  \\nonumber\n\\end{eqnarray} \nfor the indirect-exciton-like branch,\n\\begin{eqnarray}\\label{67}\n&&\\ket{P_{d}}=\\ket{\\textsc{dx}} \\\\\n&&+\\frac{\\sqrt{2}\\Omega^*_{d}}{(\\mathcal{E}_{d}{-}\\tilde{\\varepsilon}_{id})(\\mathcal{E}_{d}{-}\\tilde{\\omega}){-}|\\Omega_{id}|^2}\n \\left[\\Omega_{id}\\ket{\\textsc{ix}}{+}(\\mathcal{E}_{d}{-}\\tilde{\\varepsilon}_{id}\\big)\\ket{\\alpha}\\right], \\nonumber\n\\end{eqnarray}\nfor the direct-exciton-like branch, and \n\\begin{equation} \\label{Pph}\n\\ket{P_{ph}}= \\ket{\\alpha}+\n\\frac{\\Omega_{id}}{\\mathcal{E}_{ph}-\\tilde{\\varepsilon}_{id}} \\ket{\\textsc{ix}} + \\frac{\\sqrt{2}\\Omega_{d}}{\\mathcal{E}_{ph}-\\tilde{\\varepsilon}_{d}} \\ket{\\textsc{dx}}\\,\n\\end{equation}\nfor the photon-like branch. \n\n\nTo get  the $h^\\dagger$ eigenstates, $\n0=(h^\\dagger- \\mathcal{E}^*)\\ket{Q}$, we again have to replace $\\mathcal{E}$ by $\\mathcal{E}^*$,  the $\\ket{Q}$ eigenstates  being obtained from the $\\ket{P}$ eigenstates by changing  $(\\tilde{\\omega},\\tilde{\\varepsilon}_{d},\\tilde{\\varepsilon}_{id})$ into  $(\\tilde{\\omega}^*,\\tilde{\\varepsilon}_{d}^*,\\tilde{\\varepsilon}_{id}^*)$.\n\n\n\n\n\n  \n\n\n\n$\\bullet$ \nTo go further, we introduce the creation operators for the $h$ and $h^\\dagger$ eigenstates $\\ket{P}=P^\\dagger \\ket{v}$ and $\\ket{Q}=Q^\\dagger \\ket{v}$. These operators fulfill\n\\begin{equation}\n1=\\frac{\\big[Q_{d}, P^\\dagger_{d}\\big]_-}{\\braket{Q_{d}}{P_{d}}}=\\frac{\\big[Q_{id}, P^\\dagger_{id}\\big]_-}{\\braket{Q_{id}}{P_{id}}}=\\frac{\\big[Q_{ph}, P^\\dagger_{ph}\\big]_-}{\\braket{Q_{ph}}{P_{ph}}}\\,,\n\\end{equation}\nthe other commutators being equal to zero. These relations lead to the following closure relation\n\\begin{equation}\\label{clos_dip}\n\\mathbb{I}= \\frac{\\ket{P_{d}}\\bra{Q_{d}}}{\\braket{Q_{d}}{P_{d}}}+\\frac{\\ket{P_{id}}\\bra{Q_{id}}}{\\braket{Q_{id}}{P_{id}}}+\\frac{\\ket{P_{ph}}\\bra{Q_{ph}}}{\\braket{Q_{ph}}{P_{ph}}}\\,.\n\\end{equation}\n \n\n\n\nThe $h$ Hamiltonian takes a diagonal form in terms of these operators,\n\\begin{equation}\nh=\\mathcal{E}_{d}\\frac{P^\\dagger_{d} Q_{d}}{\\braket{Q_{d}}{P_{d}}}+\\mathcal{E}_{id}\\frac{P^\\dagger_{id} Q_{id}}{\\braket{Q_{id}}{P_{id}}}+\\mathcal{E}_{ph}\\frac{P^\\dagger_{ph} Q_{ph}}{\\braket{Q_{ph}}{P_{ph}}}\\, ,\\label{69}\n\\end{equation}\nthat satisfies $0=(h-\\mathcal{E}_{d})\\ket{P_d}$,  $0=(h-\\mathcal{E}_{id})\\ket{P_{id}}$, and  $0=(h-\\mathcal{E}_{ph})\\ket{P_{ph}}$, as easy to check. Equations (\\ref{clos_dip},\\ref{69}) also give the time evolution operator  for one photon coupled to  direct and  indirect excitons in a compact form as   \n\\begin{eqnarray}\ne^{-iht}&=&e^{-i\\mathcal{E}_{d}t}\\frac{|P_{d} \\rangle \\bra{Q_{d}}}{\\braket{Q_{d}}{P_{d}}}+e^{-i\\mathcal{E}_{id}t}\\frac{|P_{id}\\rangle \\bra{Q_{id}}}{\\braket{Q_{id}}{P_{id}}} \\nonumber \\\\\n&+& e^{-i\\mathcal{E}_{ph}t}\\frac{|P_{ph} \\rangle\\bra{Q_{ph}}}{\\braket{Q_{ph}}{P_{ph}}}\\,.\\label{70}\n\\end{eqnarray}\n \n\n\n\\noindent \\textbf{(2) Dipolariton energies and lifetimes}: The eigenvalue equation that follows from the  determinant of Eq.~(\\ref{65}) reads\n\\begin{equation}\n 0 = \\mathcal{E}'^3 - z_1 \\mathcal{E}'^2 + z_2 \\mathcal{E}' - z_3 \\,,\\label{66}\n\\end{equation}\nwith $z_i$ given by\n\\begin{eqnarray}\nz_1 &=& \\tilde{\\omega}'_{id}+\\tilde{\\omega}'_{d}+\\tilde{\\omega}' = 0  \\,,\\nonumber \\\\\nz_2 &=&\\tilde{\\omega}'_{id}\\tilde{\\omega}'_{d}+\\tilde{\\omega}'_{d}\\tilde{\\omega}'+\\tilde{\\omega}'\\tilde{\\omega}'_{id}\n-  | \\Omega_{id}|^2- 2| \\Omega_{d}|^2\\,, \\label{z2}\n \\nonumber \\\\\nz_3 &=&\\tilde{\\omega}'_{id}\\tilde{\\omega}'_{d}\\tilde{\\omega}'- 2| \\Omega_d|^2 \\tilde{\\omega}'_{id}- | \\Omega_{id}|^2   \\tilde{\\omega}'_{d}\\, .  \\label{z3}\n\\end{eqnarray}\nThe  solution of this third-order equation can be analytically obtained by using the Cardan's trick \\cite{nickalls1993}  (see Appendix \\ref{app:A}). The three solutions read, since $z_1=0$, \n\\begin{equation}\\label{mathE'n}\n\\mathcal{E}^{'(n)}=\\sum_{s=\\pm} e^{i s  \\varphi_n }\\left[ \\frac{z_3}{2} + s\\sqrt{ \\left( \\frac{z_3}{2} \\right)^2 + \\left( \\frac{z_2}{3} \\right)^3}\\right]^{1\/3}  \\,,\n\\end{equation}\nwith $\\varphi_n=(0,\\pm2\\pi\/3)$.\n\n\n\\begin{figure}\n\\includegraphics[width=8.1cm]{fig4}\n\\vspace{-0.1cm}\n\\caption{\\textbf{Dipolariton}. Dependence of the dipolariton energies $\\mathcal{E}$  on the ratio of the photon coupling to \\textit{direct} exciton $\\Omega_d$ over the detuning with respect to the indirect exciton, $\\delta= \\omega - \\varepsilon_{id}$, when the cavity photon and the exciton have (a) the same lifetime, $\\gamma=0$, and (b) a  lifetime difference $\\gamma=\\delta\/6$. \nThe other parameters are taken as $|\\Omega_{id}|=\\delta\/3$, $\\varepsilon_{id}=2\\delta$, $\\gamma_{_X}=\\delta$, and $\\delta_{_X}=\\varepsilon_d - \\varepsilon_{id}= 2\\delta$. The dipolariton has three energy branches given in Eqs.~(\\ref{eq58},\\ref{z3},\\ref{mathE'n}). Their energy real parts are shown as solid lines, while  their inverse lifetimes are shown as dashed lines (inset in (b)). \\label{fig4}\n}\n\\end{figure}\n\n\n\n\n\\noindent\\textbf{(3) Physical understanding}: Let us now discuss how the couplings of the direct and indirect excitons to the \\textit{same} cavity photon modify the particle energies and lifetimes; in particular, how a strong photon coupling to the direct exciton modifies the photon coupling to the indirect exciton  when \n\\begin{equation}\n\\lambda\\equiv\\left| \\frac{\\Omega_{id}}{ \\sqrt{2}\\, \\Omega_{d}}\\right|<1\n\\end{equation}\nas obtained for a  poor overlap between the carriers making the indirect exciton. \n\nIn addition to the ratio $\\lambda$ of the photon couplings to indirect and direct excitons, the other two relevant parameters of the dipolariton problem are the photon detuning $\\delta$ relative to the indirect exciton ground state and the energy difference $\\delta_{_X}$  between direct and indirect excitons\n\\begin{equation} \\label{eq72}\n\\delta=\\omega-\\varepsilon_{id}\\,,\\,\\,\\,\\,\\qquad  \\delta_{_X}=\\varepsilon_{d}-\\varepsilon_{id}\\,.                                \n\\end{equation}\nThese three  parameters can be experimentally controlled either directly or indirectly. \nThe $\\tilde{\\omega}'$ parameters defined in Eq.~(\\ref{65}) then read, for $\\gamma=\\gamma_{_X} - \\gamma_{ph}$ as in Eq.~(\\ref{57}),\n\\begin{eqnarray}\n\\tilde{\\omega}'_{id}&=&(-\\delta_{_X}- \\delta  - i\\gamma)\/3 \\,,  \\nonumber \\\\\n\\tilde{\\omega}'_{d} &=& (2\\delta_{_X}- \\delta  - i\\gamma)\/3 \\,,\\label{78}\\\\\n\\tilde{\\omega}' &=& (2\\delta- \\delta_{_X} + 2i\\gamma)\/3 \\,. \\nonumber \n\\end{eqnarray}\n\n\n\n\n\\begin{figure}\n\\includegraphics[width=8.1cm]{fig5}\n\\vspace{-0.1cm}\n\\caption{\\textbf{Dipolariton}. Same as Fig. \\ref{fig4} with ($\\delta, \\delta_{_X}$) exchanged.  \n\\label{fig5}}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{fig6}\n\\caption{\\textbf{Dipolariton}. Dependence of the dipolariton energies $\\mathcal{E}$ on the ratio of the photon coupling to \\textit{indirect} exciton  $|\\Omega_{id}|$ over the lifetime difference $\\gamma = \\gamma_{_X} - \\gamma_{ph}$\nfor various $\\gamma$ and coupling ratios $\\lambda = |\\Omega_{id} \/ \\sqrt{2}\\, \\Omega_d|$. In (a,b,c), $\\gamma$ stays equal to $4 \\delta\/3$ while $\\lambda$ is equal to (0.1; 0.25; 0.5). In (d), $\\lambda$ is equal to 0.25 while $\\gamma=2\\delta\/3$. All these curves are calculated for $\\delta_{_X} = 2 \\delta$. The dipolariton energy real parts are shown as solid lines, while the dashed lines represent the  dipolariton inverse lifetimes. \\label{fig6}}\n\\end{figure*}\n\n\n\\noindent $\\bullet$ When  the exciton and photon lifetimes are equal, $\\gamma=0$, and when $0 < \\delta \\ll \\delta_{_X}$, that is, when the energy of the direct exciton is much higher than those of the photon and the indirect exciton, the large coupling between photon and direct exciton coupling, $\\Omega_d$, makes the upper branch of the dipolariton go up to $\\mathcal{E}_{d}\\simeq\\varepsilon_d+\\sqrt{2}\\,|\\Omega_d|$, and the middle branch go down to $\\mathcal{E}_{ph}\\simeq \\omega-\\sqrt{2}\\,|\\Omega_d|$, until this middle branch anticrosses with the lower branch; the minimum energy splitting between the two is $\\simeq 2|\\Omega_{id}|$, as  obtained by minimizing the energy difference of the two branches with respect to $|\\Omega_d|$. Such an anticrossing exists because the lifetimes of the middle and lower branches are equal. These behaviors are shown in Fig.~\\ref{fig4}(a)  for $\\delta_{_X}= 2\\delta>0$ and $|\\Omega_{id}|=\\delta\/3$: \n as  $|\\Omega_d|$ increases, an anti-crossing occurs between the middle and lower branches, while the middle-branch energy remains flat due to its weak dependence on $|\\Omega_{d}|$. So, we end with a hybrid polariton, with the middle branch coupled to the lower branch. \n\n\n\n\\noindent $\\bullet$ When the exciton and photon lifetimes are equal, $\\gamma=0$, and when $0<\\delta_{_X}\\ll \\delta$, that is, when the photon energy is much higher than the energies of direct and indirect excitons, the large photon-direct exciton coupling $\\Omega_d$ makes the upper branch go up to $\\mathcal{E}_{ph}\\simeq\\omega+\\sqrt{2}\\,|\\Omega_d|$, and the middle branch go down to $\\mathcal{E}_{d}\\simeq\\varepsilon_d-\\sqrt{2}\\,|\\Omega_d|$, until it anticrosses with the lower branch, the minimum energy splitting being  \n\\begin{equation}\n\\simeq \\frac{2|\\Omega_{id}|\\sqrt{2|\\Omega_{d}|^2+|\\Omega_{id}|^2}}{\\delta}\\,,\n\\end{equation}\nas obtained by minimizing the energy difference of the two branches with respect to $|\\Omega_d|$. So, we again end with a hybrid polariton with the middle branch coupled to the lower branch, but the effective coupling between the two branches is different from the one for the $0<\\delta\\ll \\delta_{_X}$ case. The minimum splitting at the anticrossing shown in  Fig.~\\ref{fig5}(a), is  smaller than $2|\\Omega_{id}|$ as seen from Fig.~\\ref{fig4}(a).\n\n\n\\noindent $\\bullet$ When the exciton and photon have different lifetimes, $\\gamma=\\gamma_{_X} - \\gamma_{ph}\\not=0$,   the  lifetimes of the three dipolariton branches depend on the photon-direct exciton coupling $|\\Omega_d|$, as seen from the differences between Figs.~\\ref{fig4}(a)-\\ref{fig5}(a) and Figs.~\\ref{fig4}(b)-\\ref{fig5}(b). Surprisingly, for $\\gamma=|\\Omega_{id}|\/2\\neq0$,  the inverse lifetimes of the three dipolariton branches go to a single value as $|\\Omega_{d}|\/\\delta$ increases (see Fig.~\\ref{fig4}(b)), while the middle and lower branches anticross with a splitting smaller than the one for $\\gamma=0$. \nBefore this point, the photonic branch has the largest lifetime, while  for larger $|\\Omega_d|$, this branch has the smallest lifetime,  the two inverse excitonic lifetimes converging to a smaller value in the large $|\\Omega_{d}|$ limit. \nIndeed, for $\\delta \\ll \\delta_{_X}$, the splitting at the anticrossing is   approximately equal to $2\\sqrt{(1+\\delta\/\\delta_{_X})|\\Omega_{id}|^2-\\gamma^2\/4}$. \nEqual inverse lifetimes mathematically mean that the three $\\mathcal{E}'$ solutions of Eq.~(\\ref{66})  are real. This provides a tool to mitigate the short non-radiative lifetime of the excitons. \n\nWhen the upper branch is photon-like, as shown in Fig.~\\ref{fig5}(b), the middle and lower exciton-like branches anticross; their inverse lifetimes  go  to a single value at $|\\Omega_d| = \\delta_{_X}$, while the inverse lifetime of the upper branch, that remains the smallest for all $|\\Omega_{d}|$ values, goes to the lower branch value for large $|\\Omega_{d}|$.  \n\n\n\n\n\n\n\n\\noindent $\\bullet$ We now turn to the effect of the photon couplings to indirect and direct excitons, through their ratio $\\lambda$. When $\\lambda$ is small, see Figs.~\\ref{fig6}(a,b), the energies of the middle and lower branches can cross because the lifetimes of these two branches are different whatever  $|\\Omega_{id}|$. The energy and lifetime of the branch that start from the indirect exciton (green curves) remain flat as $|\\Omega_{id}|$ increases, which is  the signature of  indirect exciton being weakly coupled to  photon and to direct exciton.  This is in contrast to Fig.~\\ref{fig4}(b) which exhibits an anticrossing: as  $\\lambda$ increases, the lower two branches anticross and the lifetimes of the three branches go to the same value, as shown in Fig.~\\ref{fig6}(c) for  $\\lambda=0.5$, which is reminiscent of the results in Fig.~\\ref{fig4}(b). \nNote that we can also obtain an anticrossing by decreasing the inverse lifetime difference $\\gamma$, as can be seen by comparing Fig.~\\ref{fig6}(b) and Fig.~\\ref{fig6}(d).\n\n\n\\begin{table*}\\centering\n\\ra{1.4}\n\\begin{tabular}{@{}l c c c  cc@{}}\n\\toprule[1.1pt]\n&$F=0$ &Eqs.  &\\phantom{a}&$F\\gg 0$ &Eqs. \\\\\n\\cmidrule[0.5pt]{2-3} \\cmidrule[0.5pt]{5-6}\nHamiltonian  $h$ &  \n$h=\\tilde{\\omega}\\, \\alpha^{\\dagger} \\alpha +\\tilde{\\varepsilon}_{0,0} B_{0,0}^{\\dagger} B_{0,0}$&& &\n$h\\approx\\tilde{\\omega}\\, \\alpha^{\\dagger} \\alpha +\\tilde{\\varepsilon}_{d}\\, \\big(B_{d}^{\\dagger} B_{d}+ D_{d}^{\\dagger} D_{d}\\big)+\\tilde{\\varepsilon}_{id}\\, B_{id}^{\\dagger} B_{id}$&\\\\ \n&  +  $\\big( \\Omega_{0,0}B_{0,0}^{\\dagger} \\alpha  + \\mathrm{h.c.} \\big)$&\\eqref{h:zeroF_lf} && $+  \\Big( \\big(\\sqrt{2}\\, \\Omega_{d} \\,B_{d}^{\\dagger}+\\Omega_{id} \\,B_{id}^{\\dagger} \\big)\\alpha + \\mathrm{h.c.} \\Big)$ &\\eqref{64}\\\\\nEigenvectors: &&&&\\\\\n $(h - \\mathcal{E}) \\ket{P}=0$ &  $\\ket{P_{_X}},\\: \\ket{P_{ph}}$ &\\eqref{43} &\n & \n $\\ket{P_{id}}, \\: \\ket{P_{d}},\\: \\ket{P_{ph}}$ & (\\ref{ketPrmx1}-\\ref{Pph})\\\\ \n$(h^{\\dagger} - \\mathcal{E}^* )\\ket{Q}=0$ &  $\\ket{Q_{_X}},\\: \\ket{Q_{ph}}$ &\\eqref{ketQrmx1}& &\n$\\ket{Q_{id}}, \\: \\ket{Q_{d}},\\: \\ket{Q_{ph}}$ \\\\ \n&&&&\\\\\nDiagonal form& \n $h=\\mathcal{E}_{_X}\\frac{P^\\dagger_{_X} Q_{_X}}{\\braket{Q_{_X}}{P_{_X}}}+\\mathcal{E}_{ph}\\frac{P^\\dagger_{ph} Q_{ph}}{\\braket{Q_{ph}}{P_{ph}}}$&\\eqref{hasoperatorPQ} && \n $h=\\mathcal{E}_{d}\\frac{P^\\dagger_{d} Q_{d}}{\\braket{Q_{d}}{P_{d}}}+\\mathcal{E}_{id}\\frac{P^\\dagger_{id} Q_{id}}{\\braket{Q_{id}}{P_{id}}}+\\mathcal{E}_{ph}\\frac{P^\\dagger_{ph} Q_{ph}}{\\braket{Q_{ph}}{P_{ph}}}$ &\\eqref{69}\\\\ \n&&&&\\\\\nEigenenergies $\\mathcal{E}$ &  $\\mathcal{E} \\equiv \\mathcal{E}' + \\frac{\\tilde{\\omega} + \\tilde{\\varepsilon}_{0,0}}{2}$ &\\eqref{40bis}&\n& $\\mathcal{E} \\equiv \\mathcal{E}' + \\frac{\\tilde{\\omega} + \\tilde{\\varepsilon}_{id}+ \\tilde{\\varepsilon}_d}{3}$& \\eqref{eq58}\\\\ \nwith $\\mathcal{E}'$ &  $\\mathcal{E}' = \\pm \\sqrt{ (\\delta + i \\gamma)^2\/4 + |\\Omega_{0,0} |^2 }$& \\eqref{eq59} && $\n\\mathcal{E}^{'(n)}=\\sum_{s=\\pm} e^{i s  \\varphi_n }\\left[ \\frac{z_3}{2} + s\\sqrt{ \\left( \\frac{z_3}{2} \\right)^2 + \\left( \\frac{z_2}{3} \\right)^3}\\right]^{1\/3} $ &\\eqref{mathE'n}\\\\ \nParameters&$\\tilde{\\omega} = \\omega - i \\gamma_{ph}; \\quad \\tilde{\\varepsilon}_{0,0} = \\varepsilon_{0,0} - i \\gamma_{_X}$&(\\ref{eq37}-\\ref{eq38})&\n&$\\tilde{\\omega} = \\omega - i \\gamma_{ph}; \\quad \\tilde{\\varepsilon}_{d} = \\varepsilon_{0,1} - i \\gamma_{_X}; \\quad \\tilde{\\varepsilon}_{id} = \\varepsilon_{0,0} - i \\gamma_{_X}$&(\\ref{eq55}-\\ref{eq56})\\\\\n&$\\delta = \\omega - \\varepsilon_{0,0}; \\quad \\gamma = \\gamma_{_X} - \\gamma_{ph}$&\\eqref{57}&\n&$\\delta = \\omega - \\varepsilon_{id};\\quad \\delta_{_X} - \\varepsilon_d  - \\varepsilon_{id}; \\quad  \\gamma = \\gamma_{_X} - \\gamma_{ph}$& \\eqref{eq72}\\\\\n&$\\tilde{\\omega}' = \\tilde{\\omega} - (\\tilde{\\omega}  + \\tilde{\\varepsilon}_{0,0})\/2$&&\n&$\\{\\tilde{\\omega}'_{id}, \\tilde{\\omega}'_d, \\tilde{\\omega}'\\} = \\{\\tilde{\\varepsilon}_{id}, \\tilde{\\varepsilon}_d, \\tilde{\\omega}\\} - (\\tilde{\\omega} + \\tilde{\\varepsilon}_{id} + \\tilde{\\varepsilon}_d)\/3 $&\\eqref{78}\\\\\n\\bottomrule[1.1pt]\n\\end{tabular}\n\\caption{Summary of the main equations and the parameters used in the manuscript. \\label{table1} }\n\\end{table*}\n\n\n\\section{Conclusion}\nIn this work, we studied how cavity photons couple to direct and indirect excitons in a coupled quantum well when the external electric field increases, with a particular focus on the effect of the particle lifetimes. \nTo do so, we first constructed the relevant system Hamiltonians step by step, from scratch. Their diagonal form allowed us to identify the various states that are relevant to the problem at hand. \nThe procedure to derive diagonal Hamiltonian operators can be easily extended to multilevel systems. \nThen, we showed how, at low electric field, the photon couples to a hybrid exciton made of carriers lying in the two quantum wells, to form a hybrid polariton. \nBy contrast, at large electric field, the photon couples to one indirect exciton and two direct excitons, then forming a dipolariton. \nBy adding the lifetime of the cavity photon and the coherence time of the exciton carriers, we have derived the consequences of the carrier relaxation processes and  precisely tracked  the time evolution of this open quantum system. \nWe have been able to do it  with the help of non-hermitian Hamiltonians. \nBy showing how the polariton energies are affected by these lifetimes, our work  provides physical insights \n to possible identify the parameter regime in which dipolariton can be created. \n\n\n\\begin{acknowledgements}\nWe wish to thank Fran\\c cois Dubin for initiating this work and for numerous  insights. \n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\n\nRandom Matrix Theory is known to enjoy a wide range of applications in the\nphysical sciences and beyond. This remains true when the \neigenvalues of the operator to be described move into the complex plane.\nHowever, the ensemble that is perhaps the most interesting of these, \nthe real Ginibre ensemble \\cite{Ginibre65} \ndealing with real-valued asymmetric matrix entries, \nhas turned out to be the most difficult.\nPossible applications of these ensembles include\nneural networks \\cite{Sompolinsky88}, directed Quantum Chaos \n\\cite{Efetov97}, Quantum Chromodynamics \\cite{HOV}, financial markets \n\\cite{KWAPIEN06}, and quantum information theory \\cite{Bruzda08}.\n\nThe mathematical difficulty in solving these ensembles is due to the fact that\nthey allow for combinations of both real and complex conjugate eigenvalue\npairs, with their characteristic equation having only real entries.\nApart from results on the spectral density \n\\cite{Sommers88,EKS94,Efetov97} \nan eigenvalue representation \\cite{Lehmann91,Edelman97} was derived as a\nstarting point for studying systematically \nhigher order eigenvalue correlation functions. \nOnly very recently was their Pfaffian structure explicitly revealed\n\\cite{Kanzieper05,Sinclair06}, and\nthe probability $p_{N,k}$ that an $N\\times N$ matrix has exactly $k$ real\neigenvalues \\cite{Kanzieper05,Akemann07} \nas well as all correlations for $k=0$ were computed \n\\cite{Akemann07}. Finally the complete solution for all\nreal and complex eigenvalue correlations was achieved independently by three\ndifferent groups\n\\cite{Sommers2007,Forrester07,Borodin07,Forrester08,Sommers2008}. \n\nIn this paper we will give a very simple derivation for the generating kernel \nof all complex eigenvalue correlations in the general so called elliptic case,\ndealing with partly symmetric matrices depending on an asymmetry parameter.\nWe also present new results for the chiral real Ginibre ensemble as a\ntwo-matrix model which has not yet been considered.\n\nIn the next section\nwe will explain the relation between the complex\neigenvalue density and characteristic polynomials. After defining the elliptic\nreal Ginibre ensemble and its new chiral extension\nwe consider their corresponding results \nin two separate sections \\ref{Gin1} and \\ref{chGin1}.\nOur conclusions are presented in section \\ref{concs}.\n\n\n\n\n\\section{The r\\^ole of characteristic polynomials generating the kernel}\n\\label{role}\n\nWe start with the simplest ensemble considered here, the real Ginibre ensemble\nat maximal asymmetry. It is just given by a Gaussian measure in the space\nof real, asymmetric  $N \\times N$ matrices \nwhich is invariant under orthogonal transformations:\n\\begin{equation}\n\\label{dmuA} \nd\\mu (A) \\equiv \\prod_{i,j=1}^{N} \n\\left( \\frac{dA_{ij}}{\\sqrt{2 \\pi}} \\right) \n\\exp \\left( - \\frac{1}{2}A_{ij}^2\\right)\\equiv  {\\rm D}A\\ {\\rm\n  e}^{-\\frac12\\Tr AA^T}. \n\\end{equation}\nThe eigenvalues $\\lambda_i$ obey the equation $\\det [\\lambda_i -A ] = 0$, and\nthus are real or occur in complex conjugate pairs. They enjoy the\nfollowing ordered joint probability density function (jpdf)\n\\cite{Lehmann91,Edelman97} \n\\begin{equation}\n\\label{dmuL}\nd\\mu(\\lambda_1,\\lambda_2, \\ldots, \\lambda_N) \n= C_N \\cdot  d\\lambda_1 \\ldots d\\lambda_N   \n\\cdot \\prod_{i<j}^N (\\lambda_i - \\lambda_j) \\cdot \\prod_k^N f(\\lambda_k)\\ ,   \n\\end{equation}\nwith some positive definite weight function $ f(\\lambda_k)=f(\\bar\\lambda_k)$ and\na normalisation constant $C_N$. Here the eigenvalues are ordered as follows \nif they are real: $\\lambda_1>\\lambda_2>\\ldots$ , if they are complex: \n${\\rm  Re}\\lambda_1={\\rm Re}\\lambda_2>{\\rm Re}\\lambda_3={\\rm Re}\\lambda_4>\\ldots,\n\\ {\\rm Im}\\lambda_1=-{\\rm Im}\\lambda_2>0, \n\\ {\\rm Im}\\lambda_3=-{\\rm  Im}\\lambda_4>0,\\ldots$ , \nand similarly if they are mixed (see also \\cite{Sommers2008}). \n\nThis implies that the spectral  density of \ncomplex eigenvalues of the $N+2$ dimensional\nensemble, which can be obtained by inserting a two-dimensional delta-function\nin the complex plane, is proportional to \n\\begin{equation}\n\\label{R_2^C}\nR_{N+2,\\,1}^C(\\lambda) \\propto  i(\\lambda-\\bar \\lambda) f(\\lambda)^2 \n\\langle \\det[\\lambda-A] \\det[\\bar\\lambda-A]  \\rangle _N \\ .\n\\end{equation} \nThe brackets mean the average over the ensemble (\\ref{dmuA})\nwith the partition function \n${\\cal Z}_N\\equiv \\int  {\\rm D}A\\ \\e^{-\\frac12\\Tr AA^T}$. \nThere is an additional contribution to the total spectral density from the\nreal eigenvalues, which is\nobtained by inserting a delta-function on the real axis, \nwhich we do not consider here. \nWe are therefore led to consider the following correlation \nof two characteristic polynomials of $A$ \n\\footnote{The Hermitian conjugate  \nis put here merely to stress the analogy to other ensembles discussed below, \nas for real matrices $\\det[ A^T]=\\det [A]$.}\n\\begin{equation}\n\\label{F}\n {\\cal F }_N (\\lambda, \\gamma)\n\\ \\equiv\\ \\langle \\det [\\lambda-A] \\det[\\gamma-A^\\dag] \\rangle _N\n\\equiv\\frac{{\\cal K }_N^{\\beta=1} (\\lambda ,\\gamma)}{\\lambda-\\gamma}\\ \\\n\\mbox{with}\\ \\ \\lambda\\neq\\gamma\\ .\n\\end{equation}\nIt determines the antisymmetric kernel\n${\\cal K }_N^{\\beta=1} (\\lambda ,\\gamma)$, \nfrom which all correlation functions of complex eigenvalues \nfollow for even $N$. While the result for odd $N$ is known in the real Ginibre\nensemble \\cite{Sommers2008}, \nvery recently a general technique has been proposed for obtaining the\nodd $N$ result from even $N$ by removing an eigenvalue\n\\cite{Forrester08II}. \nThe kernel in eq. (\\ref{F}) has been derived in \\cite{Sommers2007} \nusing the Edelman result\n\\cite{Edelman97} for the density of complex eigenvalues, and using\neq.(\\ref{R_2^C}). Here we will give an independent derivation which makes\nclear why the result is so simple, when the jpdf eq. (\\ref{dmuL}) is so\ncomplicated. \n\nThe relation eq. (\\ref{F}) is far more general. Not only does it hold for\nthe other real Ginibre ensembles to be introduced below, but it also holds for\nother symmetry classes with complex eigenvalues having \nunitary or symplectic invariance. For the\nquaternionic Ginibre ensembles at $\\beta=4$ an identical relation to \neq. (\\ref{F}) was shown in \\cite{ABa} to give the skew-symmetric kernel.\nIn the Ginibre ensembles with unitary symmetry $\\beta=2$ the kernel is\nsymmetric and the following modified, simpler relation is known to hold\n\\cite{AV} \n\\begin{equation}\n\\label{Fb2}\n\\langle \\det [\\lambda-A] \\det[\\gamma-A^\\dag] \\rangle _N\n={\\cal K }_N^{\\beta=2} (\\lambda ,\\gamma)\\ .\n\\end{equation}\n\n\nThe argument we just presented above for the real Ginibre ensemble\nat maximal asymmetry can easily be translated to the partially symmetric \ncase depending on an asymmetry parameter $\\tau$. \nHere in the large-$N$ limit\nthe complex eigenvalues lie inside an ellipse with axes $\\sim(1\\pm\\tau)$\n\\cite{Sommers88}. \nIt is known \\cite{Lehmann91} that \nits jpdf is related to eq. (\\ref{dmuL}) by a simple\nrescaling of the eigenvalues, and we readily obtain\n\\begin{equation}\n\\label{Ft}\n\\fl {\\cal F }_N (\\lambda, \\gamma;\\tau)\n\\ \\equiv\\ \\langle \\det [\\lambda-(S+vA)] \\det[\\gamma-(S+vA)^T] \\rangle _N\n\\equiv\\frac{{\\cal K }_N^{\\beta=1} (\\lambda ,\\gamma;\\tau)}{\\lambda-\\gamma}\\ ,\n\\end{equation}\nwith \n\\begin{equation} \n\\label{tvdef}\n\\tau\\in[0,1]\\ ,\\ \\ v^2=\\frac{1-\\tau}{1+\\tau}\\ .\n\\end{equation}\nThe average is with respect to the following partition function\n\\begin{equation}\n\\label{Zt}\n{\\cal Z}_N\\ \\equiv\\ \\int {\\rm D}S\\  {\\rm D}A\\ \n\\exp\\left[-\\frac{1}{2(1+\\tau)}\\Tr  (SS^T+AA^T)\\right]\\ ,\n\\end{equation}\nand we consider the eigenvalues of the partly symmetric matrix $J=S+vA$. Here \n$S$ and $A$ are $N\\times N$ matrices being symmetric and antisymmetric\nrespectively, with a particular choice of variance. \nThe limiting case $\\tau=0$ brings us back to the ensemble eq. (\\ref{dmuA})\nwhile setting $\\tau=1$ would lead to the Gaussian Orthogonal Ensemble. \nHowever, in that case the eigenvalues become real and this limit is\nsubtle. \n\n\nThe second ensemble we consider in this paper is the chiral counterpart \nof the real Ginibre ensemble. Following \\cite{HOV} and its extension to a\ntwo-matrix model \\cite{James} we define the following chiral real Ginibre\nensemble ($ch$) with a particular variance $n$, \n\\begin{equation}\n\\label{Zchmu}\n{\\cal Z}_N^{ch}\\ \\equiv\\ \\int {\\rm D}A\\  {\\rm D}B\\ \n\\exp\\left[-\\frac{n}{2}\\Tr (AA^T+BB^T)\\right]\\ .\n\\end{equation}\nAgain we compute the average of characteristic polynomials to obtain the\nkernel, \n\\begin{eqnarray}\n\\label{Kchdef}\n&&{\\cal F }_N^{ch} (\\lambda, \\gamma;\\mu)\n\\equiv \\langle \n\\det[\\lambda-M]\\det[\\gamma-M^T]\n\\rangle _N\\ \\equiv\\ \n\\frac{{\\cal K }_N^{ch,\\,\\beta=1} (\\lambda ,\\gamma;\\mu)}{\\lambda^2-\\gamma^2}\\ ,\n\\\\\n&&M\\equiv \\left(\n\\begin{array}{cc}\n0& A+\\mu B\\\\\nA^T-\\mu B^T & 0\\\\\n\\end{array}\n\\right) .\n\\label{Mdef}\n\\end{eqnarray}\nHere both $A$ and $B$ are rectangular $N\\times (N+\\nu )$ \nmatrices without further symmetry among the real matrix elements. They are \ndrawn independently from the ensemble (\\ref{dmuA}) extended to $\\nu\\geq0$.\nThe asymmetry parameter is given here by $\\mu\\in[0,1]$, where $\\mu=1$ denotes\nmaximal asymmetry, and $\\mu=0$ takes us back to the chiral Gaussian Orthogonal\nEnsemble. In \\cite{HOV} initially a one-matrix model was proposed, replacing\n$B$ by the identity. Whilst we expect that in the large-$N$ limit both lead to\nthe same universal result our choice allows for an eigenvalues basis, as in\nthe corresponding chiral extensions of Ginibre at \n$\\beta=2$ \\cite{James} and $\\beta=4$\n\\cite{A05}, having complex and quaternion real matrix elements, respectively.\nThe ensemble in eq. (\\ref{Zchmu})\nhas not been solved before and we will give a  completely new result\nbelow, depending parametrically on $\\nu$. \nFor maximal asymmetry it corresponds to class $2P$ in \\cite{Magnea08} for real\nelements. \n\n\nThe $2N+\\nu$ eigenvalues $\\lambda_i$ of the matrix $M$ defined in eq. (\\ref{Mdef})\nsatisfy\n\\begin{equation}\n\\label{chev}\n0\\ =\\ \\det[\\lambda-M]\\ =\\ \\lambda^\\nu\\det\\left[\\lambda^2-(A+\\mu B)(A^T-\\mu B^T)\n\\right]\\ .\n\\end{equation}\nIt will be shown elsewhere that the jpdf of these eigenvalues\nis again of the form in eq. \n(\\ref{dmuL}).  From that it follows that the kernel \nderived from ${\\cal F}_N^{ch} (\\lambda, \\gamma;\\mu)$ again determines all correlation \nfunctions of complex eigenvalues. \n\nA peculiarity of the chiral ensemble is the following: the non-zero\neigenvalues $\\lambda^2_i$ solving the second equation in (\\ref{chev}) are real but\nnot necessarily positive. Thus the eigenvalues of $M$ can have both real and\npurely imaginary eigenvalues as well as complex conjugate\neigenvalue pairs. \nMoreover, all non-zero eigenvalues come in $\\pm$ pairs due to the\nchirality of the matrix $M$. \n\n\n\n\n\n\\section{Characteristic polynomials for the real Ginibre ensemble}\n\\label{Gin1}\n\nFor pedagogical reasons we begin with the maximally asymmetric case\neq. (\\ref{F}). The partly symmetric case at $\\tau\\neq 0$ is given as a second\nexample below.\n\\begin{equation}\n\\label{F_1}\n {\\cal F }_N(\\lambda, \\gamma)= \\frac{1}{{\\cal Z}_N}\n\\int {\\rm D}A\\ {\\rm e}^{-\\frac12{\\rm Tr} A A^T}\n \\det[\\lambda-A]\\ \\det[\\gamma - A^T] \\ . \n\\end{equation}\nWriting the determinants in terms of two $N$-dimensional complex \nGrassmann vectors  $\\eta_i$ and $\\zeta_i$, with $i=1,\\ldots,N$, we obtain \n\\begin{eqnarray}\n\\label{F_A2}\n\\fl \n{\\cal F }_N (\\lambda, \\gamma)&=& \\frac{1}{{\\cal Z}_N}\n\\int {\\rm D}A \\int d\\zeta\\ d\\eta\\  \n\\exp\\left[-\\frac12A_{ij}A^T_{ji} -\\lambda\\zeta^*_i\\zeta_i - \\gamma\n\\eta^*_i\\eta_i+\\zeta^*_iA_{ij}\\zeta_j +\\eta^*_jA^T_{ji}\\eta_i\\right] \\nonumber\\\\ \n\\fl &=&   \\int d\\zeta\\ d\\eta\\  \n\\exp\\left[     -\\lambda\\zeta^*_i\\zeta_i - \\gamma \\eta^*_i\\eta_i+\n\\frac12(\\zeta_i^*\\zeta_j+\\eta_j^*\\eta_i)^2\\right] \\ ,\n\\end{eqnarray}\nafter integrating out the Gaussian matrix $A$. \nHere and in the following we will use summation conventions over double\nindices. \nThe last term in the exponent can be written as\n\\begin{equation}\n\\label{F_A4}\n\\frac12 (\\zeta_i^*\\zeta_j+\\eta_j^*\\eta_i)(\\zeta_i^*\\zeta_j+\\eta_j^*\\eta_i)\n=\\zeta_i^*\\eta_i\\ \\zeta_j\\eta_j^* \\ .\n\\end{equation}\nWith the help of a complex Hubbard-Stratonovich (HS) transformation \nwe can bilinearise and integrate out the Grassmann variables:\n\\begin{eqnarray}\n\\label{F_A5}\n\\fl \n{\\cal F }_N (\\lambda,\\gamma)&=&\\frac{1}{\\pi}\\int{ d^2z}  \n\\int d\\zeta\\ d\\eta\\ \n\\exp\\left[-|z|^2    -\\lambda\\zeta^*_i\\zeta_i - \\gamma \\eta^*_i\\eta_i\n+z\\zeta_i\\eta_i^* +\\bar z \\zeta_i^*\\eta_i \\right] \\nonumber\\\\ \n&=& \\frac1\\pi\\int d^2z\\ {\\rm e}^{-|z|^2}\\ (\\lambda\\gamma+|z|^2)^N\\ =\\ \nN!\\sum_{n=0}^N{(\\lambda\\gamma)^{n}\\over n!}    \\ . \n\\end{eqnarray}\nThis gives a polynomial with leading power $(\\lambda\\gamma)^N$ as expected. \nThus our first main result leads to the following antisymmetric kernel \n\\begin{equation}\n\\label{kernel1}\n{\\cal K }_N^{1} (\\lambda ,\\gamma)\\ =\\ (\\lambda-\\gamma)\\ \nN!\\sum_{n=0}^N{(\\lambda\\gamma)^{n}\\over n!} \\ ,\n\\end{equation}\nwhich is enough to derive all complex correlation functions. \nOn setting $\\gamma=\\bar\\lambda$ and multiplying by the weight $f(\\lambda)^2$, \nEdelman's complex density \\cite{EKS94} in terms of an incomplete exponential \nfollows. It is\nremarkable that it only depends on $|\\lambda|^2$ while the jpdf eq. \n(\\ref{dmuL}) is not isotropic. We note that Edelman derived his result using\nmethods from multivariate statistics, and not from the jpdf.\n\n\nWe now turn to the partly symmetric case with $\\tau\\in[0,1]$, where we can\nfollow the same path, \n\\begin{eqnarray}\n\\fl{\\cal F }_N (\\lambda, \\gamma;\\tau)\n&\\!\\!\\!=&\\!\\frac{1}{{\\cal Z}_N}\\int\\! {\\rm D}S\\,  {\\rm D}A\\ \n\\e^{-\\frac{1}{2(1+\\tau)}\\Tr  (SS^T+AA^T)}\n\\det [\\lambda-(S+vA)] \\det[\\gamma-(S+vA)^T] \\nonumber\\\\\n&\\!\\!\\!=&\\!\\frac{1}{{\\cal Z}_N}\n\\int\\! {\\rm D}S\\, {\\rm D}A\\int d\\zeta\\ d\\eta\\ \n\\exp\\Big[-\\frac{1}{2(1+\\tau)}(S_{ij}^2-A_{ij}^2) \n-\\lambda\\zeta^*_i\\zeta_i - \\gamma\\eta^*_i\\eta_i\\nonumber\\\\\n&&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n+\\zeta^*_i(S_{ij}+vA_{ij})\\zeta_j +\\eta^*_i(S_{ij}-vA_{ij})\\eta_j\\Big].\n\\end{eqnarray}\nAfter symmetrising and antisymmetrising the terms in the last line, e.g. \n$\\zeta^*_iS_{ij}\\zeta_j=\\frac12S_{ij}(\\zeta^*_i\\zeta_j+\\zeta^*_j\\zeta_i)$, we\ncan complete the squares in $S_{ij}$ and $A_{ij}$ respectively, and integrate\nthem out to obtain\n\\begin{eqnarray}\n\\label{FtnoAS}\n\\fl{\\cal F }_N (\\lambda, \\gamma;\\tau)\n&=&\\int d\\zeta\\ d\\eta\\ \\exp\\Big[-\\lambda\\zeta^*_i\\zeta_i - \\gamma\\eta^*_i\\eta_i\n-c_-^2((\\zeta^*_i\\zeta_i)(\\zeta^*_j\\zeta_j)\n+(\\eta^*_i\\eta_i)(\\eta^*_j\\eta_j)) \\nonumber\\\\\n&&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n-2c_+^2\\eta^*_i\\zeta^*_i\\eta_j\\zeta_j +2c_-^2\\eta^*_i\\zeta_i\\eta_j\\zeta_j^* \n\\Big]\\ .\n\\end{eqnarray}\nHere we have introduced the constants $c_\\pm^2\\equiv\\frac12(1+\\tau)(1\\pm\nv^2)$.\nThe quartic terms in the Grassmann variables can be rewritten using two real \nHS transformations for the first line of eq. (\\ref{FtnoAS}), and two complex\nones for the second line:\n\\begin{eqnarray}\n&&\\fl{\\cal F }_N (\\lambda, \\gamma;\\tau)\n=\\frac{1}{\\pi^3}\\int dx\\ dy \\int d^2z\\ d^2w\n\\int d\\zeta\\ d\\eta\\ \\exp\\Big[-x^2- y^2 \n-\\zeta^*_j(\\lambda+2ic_-x)\\zeta_j \n\\nonumber\\\\\n&&\\fl\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -\\eta^*_j(\\gamma+2ic_-y)\\eta_j -2|z|^2-2|w|^2\n+2c_+(\\bar z \\eta_j\\zeta_j-z\\eta_j^*\\zeta_j)\n-2c_-(\\bar w \\eta_j\\zeta_j^*+w\\eta_j^*\\zeta_j)\n\\Big]\\nonumber\\\\\n&&\\fl\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ =\\frac{1}{\\pi^3}\\int dx\\ dy \\int d^2z\\ d^2w\n\\ \\exp\\Big[-x^2- y^2 -2|z|^2-2|w|^2\\Big]\\nonumber\\\\\n&&\\fl\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\times\\Big[ \n(\\lambda+2ic_-x)(\\gamma+2ic_-y)+4c_+^2|z|^2+4c_-^2|w|^2\\Big]^N\\ .\n\\label{FtHS}\n\\end{eqnarray}\nExpanding the last factor twice into binomial series in powers of $|z|^2$ and \n$|w|^2$ we can apply the following integral representation of the Hermite\npolynomials: \n\\begin{equation}\n\\label{Hint}\n\\left( \\frac{\\tau}{2}\\right)^{\\frac{k}{2}}\nH_k\\left( \\frac{\\lambda}{\\sqrt{2\\tau}}\\right)=\\frac{1}{\\sqrt{\\pi}}\n\\int dx \\ \\e^{-x^2} \\left(\\lambda+i\\sqrt{2\\tau}x\\right)^k\\ ,\n\\end{equation}\nwhere using eq. (\\ref{tvdef}) we have $2c_-=\\sqrt{2\\tau}$. \nThis eliminates the two real integrations.\nAfter integrating out the two remaining complex variables $z$ and $w$ we\nfinally arrive at\n\\begin{equation}\n\\label{Ftfinal}\n{\\cal F }_N (\\lambda, \\gamma;\\tau)=N!\\sum_{l=0}^N\\tau^l\\sum_{k=0}^l\n\\frac{1}{k!\\ 2^k}H_k\\left( \\frac{\\lambda}{\\sqrt{2\\tau}}\\right)\nH_k\\left( \\frac{\\gamma}{\\sqrt{2\\tau}}\\right)\\ .\n\\end{equation}\nAs a check this is again a polynomial with leading order $(\\lambda\\gamma)^N$. \nAlthough our result eq. (\\ref{Ftfinal}) could be further simplified \nthis form is most useful for obtaining the antisymmetric kernel by applying the\nChristoffel-Darboux formula to the inner sum:\n\\begin{equation}\n\\fl {\\cal K }_N^{1} (\\lambda ,\\gamma;\\tau) \\ = \\ N!\\sum_{l=0}^N\\frac{1}{l!}\n\\left( \\frac{\\tau}{2}\\right)^{l+\\frac{1}{2}}\\left( \nH_{l+1}\\left( \\frac{\\gamma}{\\sqrt{2\\tau}}\\right)\nH_l\\left( \\frac{\\lambda}{\\sqrt{2\\tau}}\\right)\n-(\\gamma\\leftrightarrow\\lambda)\\right)\n\\ .\n\\end{equation}\nThis coincides precisely with the kernel of skew-orthogonal Hermite\npolynomials derived in \\cite{Forrester08} via the jpdf, which is much more\nelaborate. As was shown there independently, this kernel generates all\ncomplex eigenvalue correlation functions of the partly symmetric ensemble eq. \n(\\ref{Zt}) for even $N$, depending parametrically on $\\tau$.\n\nA similar kernel given in terms of orthogonal (for $\\beta=2$) \\cite{FKS98} \nor skew-orthogonal (for $\\beta=4$) \\cite{EK01}\nHermite polynomials is known for the partly symmetric Ginibre ensembles.\n\n \n\n\n\n\n\n\\section{Characteristic polynomials for the chiral real Ginibre ensemble}\n\\label{chGin1}\n\nIn this section we present the calculation only for the \npartly symmetric case of the chiral extension of the real Ginibre \nensemble, depending on asymmetry parameter $\\mu$.\nThe simpler result at\nmaximal asymmetry with $\\mu=1$ is given at the end of this section. \n\\begin{eqnarray}\n\\fl{\\cal F }_N^{ch} (\\lambda, \\gamma;\\mu)\n&=&\\frac{1}{{\\cal Z}_N^{ch}}\\int \\!\\!{\\rm D}A\\,  {\\rm D}B\\ \n\\e^{-\\frac{n}{2}\\Tr  (AA^T+BB^T)}\n\\det \\left[\n\\begin{array}{cc}\n\\lambda& -(A+\\mu B)\\\\\n-(A^T-\\mu B^T) & \\lambda\\\\\n\\end{array}\n\\right]\\nonumber\\\\ \n&&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\,\n\\times \\det\\left[\n\\begin{array}{cc}\n\\gamma          & -(A-\\mu B)\\\\\n-(A^T+\\mu B^T) & \\gamma\\\\\n\\end{array}\n\\right] \\nonumber\\\\\n&&\\fl\\!\\!\\!\\!\\!\\!\\!\\!\\!=\\frac{1}{{\\cal Z}_N^{ch}}\n\\int \\!\\!{\\rm D}A\\,  {\\rm D}B\\int \\!\\!d\\eta d\\psi d\\zeta d\\varphi\\ \n\\exp\\Big[-\\frac{n}{2}(A_{ia}^2+B_{ia}^2) \n-\\lambda(\\eta^*_i\\eta_i +\\psi_a^*\\psi_a)- \\gamma(\\zeta^*_i\\zeta_i+\\varphi_a^*\\varphi_a) \\nonumber\\\\\n&&\\fl\\!\\!\n+\\eta^*_i(A_{ia}+\\mu B_{ia})\\psi_a +\\psi_a^*(A_{ai}^T-\\mu B_{ai}^T)\\eta_i\n+\\zeta_i^*(A_{ia}-\\mu B_{ia})\\varphi_a +\\varphi_a^*(A_{ai}^T+\\mu B_{ai}^T)\\zeta_i\n\\Big].\\ \\ \n\\end{eqnarray}\nHere we have written each determinant of size $2N+\\nu$ in terms of {\\it two}\nGrassmann vectors, $\\eta_i$($\\zeta_i$) and $\\psi_a$($\\varphi_a$) \nof size $N$ and $N+\\nu$,\nrespectively. Our summation conventions imply for $i=1,\\ldots,N$, and \nfor $a=1,\\ldots,N+\\nu$. \nIn addition to $\\mu$ we have a parameter $n$ for the\nvariance. After completing the square and integrating out the matrices $A$ and\n$B$ we obtain\n\\begin{eqnarray}\n\\fl{\\cal F }_N^{ch} (\\lambda, \\gamma;\\mu)&=& \\int d\\eta d\\psi d\\zeta d\\varphi\\ \n\\exp\\Big[-\\lambda(\\eta^*_i\\eta_i +\\psi_a^*\\psi_a)\n-\\gamma(\\zeta^*_i\\zeta_i+\\varphi_a^*\\varphi_a) \\nonumber\\\\\n&&\n\\fl+\\frac{1}{2n}(\\eta_i^*\\psi_a-\\eta_i\\psi_a^*+\\zeta_i^*\\varphi_a-\\zeta_i\\varphi_a^*)^2\n+\\frac{\\mu^2}{2n}\n(\\eta_i^*\\psi_a+\\eta_i\\psi_a^*-\\zeta_i^*\\varphi_a-\\zeta_i\\varphi_a^*)^2\n\\Big]\\ .\n\\label{FchGrass}\n\\end{eqnarray}\nMultiplying out and collecting all nonzero terms we need 6 new complex \nintegrations to perform the HS transformations that bilinearise the Grassmann\nvariables. We only give the result obtained after performing all Grassmann\nintegrations:\n\\begin{eqnarray}\n\\label{FHS}\n\\fl{\\cal F }_N^{ch} (\\lambda, \\gamma;\\mu)&=& \\frac{1}{\\pi^6}\n\\int d^2u\\ d^2v\\ d^2w\\ d^2p\\ d^2q\\ d^2z\n\\ \\e^{-|u|^2 -|v|^2-|w|^2-|p|^2-|q|^2-|z|^2}\\\\\n&&\\fl\\!\\!\\!\\!\\!\\!\n\\times \\left(\n(\\lambda-i\\delta_-u)(\\gamma-i\\delta_-\\bar v)\n-\\delta_+^2wz+\\delta_-^2pq\\right)^N \n\\left( \n(\\lambda-i\\delta_-\\bar u)(\\gamma-i\\delta_-v)\n-\\delta_+^2\\bar w\\bar z\n+\\delta_-^2\\bar p\\bar q \\right)^{N+\\nu}\\!\\!,\\nonumber\n\\end{eqnarray}\nwhere we have used the following abbreviations\n\\begin{equation}\n\\delta_\\pm^2 \\equiv \\frac1n (1\\pm\\mu^2)\\ .\n\\label{deltadef}\n\\end{equation}\nExpanding the first factor in the second line of eq. (\\ref{FHS}) as \n\\begin{equation}\n\\fl(\\hat{\\lambda}\\hat{\\bar\\gamma}\\ -\\ \\delta_+^2wz\\ +\\ \\delta_-^2pq)^N=\\sum_{l=0}^N\n{N\\choose l}( -\\delta_+^2wz)^{N-l}\\sum_{k=0}^l{l \\choose k} \n(\\hat{\\lambda}\\hat{\\bar\\gamma})^k (\\delta_-^2pq)^{l-k}\\ , \n\\end{equation}\nwith $\\hat{\\lambda}\\hat{\\bar\\gamma}=(\\lambda-i\\delta_-u)(\\gamma-i\\delta_-\\bar v)$,\nand likewise the second factor, we can use the following orthogonality\nrelation\n\\begin{equation}\n\\frac1\\pi \\int d^2p\\ \\e^{-|p|^2} p^k {\\bar p}^{\\,l} \\ = \\ \\delta_{kl}k! \\ .\n\\end{equation}\nApplying this first to the integrations over variables \n$p$ and $q$, and then to $w$\nand $z$ we can reduce the four sums to two. As a final step we employ the\nfollowing complex integral representation for Laguerre polynomials\n\\begin{equation}\n\\frac1\\pi \\int d^2u\\ \\e^{-|u|^2} (\\lambda+iu)^k(\\lambda+i\\bar u)^{k+\\nu}\n\\ =\\ k!\\ (-)^k\\lambda^\\nu\nL_k^\\nu(\\lambda^2) \\ . \n\\label{Lknurep}\n\\end{equation}\nWhilst we did not find this representation in tables it can be easily verified\nfrom the standard representation of generalised Laguerre polynomials\n\\begin{equation}\nL_k^\\nu(x)\\ =\\ \\sum_{m=0}^k \\frac{(-1)^m(k+\\nu)!}{(k-m)!\\,(m+\\nu)!\\,m!}\\ x^m\\ .\n\\end{equation}\nUsing the integral representation eq. (\\ref{Lknurep}) as well as its complex\nconjugate we finally arrive at the following result:\n\\begin{equation}\n\\fl{\\cal F }_N^{ch} (\\lambda, \\gamma;\\mu)\\ =\\ N!\\,(N+\\nu)!\\,\\delta_+^{4N}(\\lambda\\gamma)^\\nu\n\\sum_{l=0}^N\\left( \\frac{\\delta_-}{\\delta_+}\\right)^{4l}\n\\sum_{k=0}^l \\frac{k!}{(k+\\nu)!} L_k^\\nu\\left( \\frac{\\lambda^2}{\\delta_-^2}\\right)\nL_k^\\nu\\left( \\frac{\\gamma^2}{\\delta_-^2}\\right).\n\\label{Fchfinal}\n\\end{equation}\nIt is a polynomial in $\\lambda$ and $\\gamma$ with the correct leading power \n$(\\lambda\\gamma)^{2N+\\nu}$. \nLooking back to the definition of the antisymmetric \nkernel in our chiral case, eq. \n(\\ref{Kchdef}), we can read off the following result, after \nusing the Christoffel-Darboux formula for Laguerre polynomials:\n\\begin{eqnarray}\n\\fl{\\cal K }_N^{ch,\\,1} (\\lambda ,\\gamma;\\mu)&=&\nN!\\,(N+\\nu)!\\ \\delta_+^{4N}(\\lambda\\gamma)^\\nu\\nonumber\\\\\n\\fl&&\\times\n\\sum_{l=0}^N\\left( \\frac{\\delta_-}{\\delta_+}\\right)^{4l}\n\\frac{(l+1)!}{(l+\\nu)!}\\ \\delta_-^2\n\\left(\\! \nL_{l+1}^\\nu\\Big( \\frac{\\gamma^2}{\\delta_-^2}\\Big)\nL_{l}^\\nu\\Big( \\frac{\\lambda^2}{\\delta_-^2}\\Big)\n-(\\gamma\\leftrightarrow\\lambda)\\!\n\\right)\\!.\n\\label{Kchfinal}\n\\end{eqnarray}\nThis gives our new kernel of the chiral real Ginibre ensemble, from which all\nits complex eigenvalue correlations follow. In particular for \n$\\gamma=\\bar\\lambda$ it is proportional to a new complex eigenvalue\ndensity as in eq. (\\ref{R_2^C}).\nIt is similar to the corresponding \nexpressions for the kernel at $\\beta=2$ \\cite{James} \nand $\\beta=4$ \\cite{A05}, also given \nin terms of Laguerre polynomials in the complex\nplane. \n \nAfter dealing with the general case we can go to maximal asymmetry, by\nsetting $\\mu=1$. In this limit only the leading power of the Laguerre\npolynomials contributes, and we obtain \n\\begin{equation}\n\\fl{\\cal F }_N^{ch} (\\lambda, \\gamma;\\mu=1)\\ =\\ N!\\,(N+\\nu)!\n\\left( \\frac2n\\right)^{2N}(\\lambda\\gamma)^\\nu\n\\sum_{k=0}^N\\frac{1}{k!\\,(k+\\nu)!}\\left(\\frac{n\\lambda\\gamma}{2}\\right)^{2k}\n\\label{Fchmax}\n\\end{equation}\nfor the characteristic polynomials, with $\\lim_{\\mu\\to1}\\delta_+^2=\\frac2n$. \nFor the corresponding kernel we have to properly rescale with $\\delta_-^2$ \nand we obtain\n\\begin{equation}\n\\fl\\lim_{\\mu\\to1}\\delta_-^2{\\cal K }_N^{ch,\\,1} (\\lambda ,\\gamma;\\mu)=\nN!\\,(N+\\nu)!\n\\left( \\frac2n\\right)^{2N}(\\lambda\\gamma)^\\nu (\\lambda^2-\\gamma^2)\n\\sum_{l=0}^N\\frac{1}{l!\\,(l+\\nu)!}\n\\left(\\frac{n\\lambda\\gamma}{2}\\right)^{2l}\\!.\n\\label{Kchmax}\n\\end{equation}\nWhen setting $\\gamma=\\bar\\lambda$ and comparing to eq. (\\ref{kernel1}) \nwe again find a dependence on the modulus only, despite the anisotropic jpdf.\nHere the incomplete exponential is replaced by an incomplete modified\n$I$-Bessel function of the first kind.  \n\n\n\n\n\\section{Conclusions}\n\\label{concs}\n\nWe have calculated the expectation value of the product \nof two characteristic polynomials \nwith respect the following two Gaussian random matrix models: \nthe partly symmetric real Ginibre ensemble, and its chiral counterpart, \na newly introduced two-matrix model. \nIn our calculation we have used the supersymmetric method, \nwithout the need to explicitly go to an eigenvalue basis. \nIn this simple way we can determine a skew-symmetric kernel which is  \nthe main building block for all \ncomplex eigenvalue correlation functions that can be written as\nPfaffians. One could calculate this kernel directly from\nthe joint eigenvalue distribution (jpdf), \nbut this turns out to be a very difficult task.\n\nThis kernel\nis given by a sum over Hermite polynomials for the real Ginibre case, \ndepending on the asymmetry parameter. Here we have recovered a known, very \nrecent result. In the chiral real Ginibre ensemble we find a new kernel given\nin terms of generalised \nLaguerre polynomials. In addition to the asymmetry $\\mu$ \nit depends on the parameter\n$\\nu$ labelling the number of exact zero eigenvalues.\nOur method offers an explanation of why the spectral density of complex\neigenvalues is so simple, i.e. being an incomplete exponential or $I$-Bessel\nfunction at maximal asymmetry, while the jpdf is so complicated. \n\nOne possible application of our new chiral result would be in field theory \nfor Dirac operators with a real representation. The reason complex\neigenvalues appear here is due to a chemical potential $\\mu$ of the quarks. \n\nIt is an open question for the chiral ensemble if for all $N$ the kernel \nalso determines the weight function $f(\\lambda)$, and if both\ningredients (i.e. kernel and weight) determine all correlation functions \nof real, complex and mixed eigenvalues. For the real Ginibre ensemble \nthis fact is known to hold, and the similarity in structure makes this very\nsuggestive. \n\\\\\n\n\n\nH.-J. S. acknowledges discussions with  D. Savin as well as the kind\nhospitality at Brunel University   with thanks. This work has been supported by\nEPSRC grant EP\/D031613\/1, European Network ENRAGE MRTN-CT-2004-005616 (G.A.), \nan EPSRC doctoral training grant (M.J.P.) and the \nSFB\/TR12 of the Deutsche Forschungsgemeinschaft (H.-J.S.). \n\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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{W^\\pm}}{\\mathswitchr {W^\\pm}}\n \n\\def\\mathswitch#1{\\relax\\ifmmode#1\\else$#1$\\fi}\n\\newcommand{\\mathswitch {m_\\Pf}}{\\mathswitch {m_\\mathswitch  f}}\n\\newcommand{\\mathswitch {m_\\Pl}}{\\mathswitch {m_\\Pl}}\n\\newcommand{\\mathswitch {m_\\Pq}}{\\mathswitch {m_\\Pq}}\n\\newcommand{\\mathswitch {M_\\PV}}{\\mathswitch {M_\\PV}}\n\\newcommand{\\mathswitch {M_\\PW}}{\\mathswitch {M_\\mathswitchr W}}\n\\newcommand{\\mathswitch {\\lambda}}{\\mathswitch {\\lambda}}\n\\newcommand{\\mathswitch {M_\\PZ}}{\\mathswitch {M_\\mathswitchr Z}}\n\\newcommand{\\mathswitch {M_\\PH}}{\\mathswitch {M_\\mathswitchr H}}\n\\newcommand{\\mathswitch {m_\\Pe}}{\\mathswitch {m_\\mathswitchr e}}\n\\newcommand{\\mathswitch {m_\\mu}}{\\mathswitch {m_\\mu}}\n\\newcommand{\\mathswitch {m_\\tau}}{\\mathswitch {m_\\tau}}\n\\newcommand{\\mathswitch {m_\\Pd}}{\\mathswitch {m_\\mathswitchr d}}\n\\newcommand{\\mathswitch {m_\\Pu}}{\\mathswitch {m_\\mathswitchr u}}\n\\newcommand{\\mathswitch {m_\\Ps}}{\\mathswitch {m_\\mathswitchr s}}\n\\newcommand{\\mathswitch {m_\\Pc}}{\\mathswitch {m_\\mathswitchr c}}\n\\newcommand{\\mathswitch {m_\\Pb}}{\\mathswitch {m_\\mathswitchr b}}\n\\newcommand{\\mathswitch {m_\\Pt}}{\\mathswitch {m_\\mathswitchr t}}\n \n\\newcommand{\\mathswitch {s_\\PW}}{\\mathswitch {s_\\mathswitchr W}}\n\\newcommand{\\mathswitch {c_\\PW}}{\\mathswitch {c_\\mathswitchr W}}\n\\newcommand{\\mathswitch {\\bar s_\\PW}}{\\mathswitch {\\bar s_\\mathswitchr W}}\n\\newcommand{\\mathswitch {\\bar c_\\PW}}{\\mathswitch {\\bar c_\\mathswitchr W}}\n\\newcommand{\\mathswitch {Q_\\Pf}}{\\mathswitch {Q_\\mathswitch  f}}\n\\newcommand{\\mathswitch {Q_\\Pl}}{\\mathswitch {Q_\\Pl}}\n\\newcommand{\\mathswitch {Q_\\Pq}}{\\mathswitch {Q_\\Pq}}\n\\newcommand{\\mathswitch {v_\\Pf}}{\\mathswitch {v_\\mathswitch  f}}\n\\newcommand{\\mathswitch {a_\\Pf}}{\\mathswitch {a_\\mathswitch  f}}\n\\newcommand{\\mathswitch {g_\\Pe}^{\\sigma}}{\\mathswitch {g_\\mathswitchr e}^{\\sigma}}\n\\newcommand{\\mathswitch {g_\\Pe}^-}{\\mathswitch {g_\\mathswitchr e}^-}\n\\newcommand{\\mathswitch {g_\\Pe}^+}{\\mathswitch {g_\\mathswitchr e}^+}\n\\newcommand{\\mathswitch {G_\\mu}}{\\mathswitch {G_\\mu}}\n \n\\def\\mathop{\\mathrm{Li}}\\nolimits{\\mathop{\\mathrm{Li}}\\nolimits}\n\\def\\mathop{\\mathrm{Re}}\\nolimits{\\mathop{\\mathrm{Re}}\\nolimits}\n\\defi.e.\\ {i.e.\\ }\n\\defe.g.\\ {e.g.\\ }\n\\defcf.\\ {cf.\\ }\n\\newcommand{self-energy}{self-energy}\n\\newcommand{self-energies}{self-energies}\n\\newcommand{counterterm}{counterterm}\n\\newcommand{counterterms}{counterterms}\n\\newcommand{cross-section}{cross-section}\n\\newcommand{cross-sections}{cross-sections}\n\n\\hyphenation{brems-strah-lung}\n \n\\newcommand{\\mpar}[1]{{\\marginpar{\\hbadness10000%\n                      \\sloppy\\hfuzz10pt\\boldmath\\bf#1}}%\n                      \\typeout{marginpar: #1}\\ignorespaces}\n\\marginparwidth 1.2cm\n\\marginparsep 0.2cm\n\n\\renewcommand{\\topfraction}{1.0}\n\\renewcommand{\\bottomfraction}{1.0}\n\\renewcommand{\\textfraction}{0.2}\n\\renewcommand{\\floatpagefraction}{0.7}\n\n\\newcommand{\\gamma\\gamma\\to\\PWp\\PWm}{\\gamma\\gamma\\to\\mathswitchr {W^+}\\mathswitchr {W^-}}\n\\newcommand{\\Pep\\Pem\\to f\\bar{f}}{\\mathswitchr {e^+}\\mathswitchr {e^-}\\to f\\bar{f}}\n\\newcommand{\\Pep\\Pem\\to\\PWp\\PWm}{\\mathswitchr {e^+}\\mathswitchr {e^-}\\to\\mathswitchr {W^+}\\mathswitchr {W^-}}\n\\newcommand{\\Pem\\gamma\\to\\PWm\\nu}{\\mathswitchr {e^-}\\gamma\\to\\mathswitchr {W^-}\\nu}\n\\newcommand{\\mathrm{Born}}{\\mathrm{Born}}\n\\newcommand{\\mathrm{SM}}{\\mathrm{SM}}\n\\newcommand{\\mathrm{GNLSM}}{\\mathrm{GNLSM}}\n\\newcommand{E_{\\mathrm{CMS}}}{E_{\\mathrm{CMS}}}\n\n\\newcommand{\\mathrm{Born}}{\\mathrm{Born}}\n\\newcommand{\\mathrm{self}}{\\mathrm{self}}\n\\newcommand{\\mathrm{corr}}{\\mathrm{corr}}\n\\newcommand{\\mathrm{weak}}{\\mathrm{weak}}\n\\newcommand{\\mathrm{bos}}{\\mathrm{bos}}\n\\newcommand{\\mathrm{ferm}}{\\mathrm{ferm}}\n\\newcommand{\\mathrm{Coul.}}{\\mathrm{Coul.}}\n\\newcommand{\\mathrm{counter}}{\\mathrm{counter}}\n\\newcommand{\\mathrm{cut}}{\\mathrm{cut}}\n\\newcommand{\\mathrm{SB}}{\\mathrm{SB}}\n\\newcommand{\\mathrm{unpol}}{\\mathrm{unpol}}\n\\newcommand{\\mathrm{gauge}}{\\mathrm{gauge}}\n\\newcommand{\\mathrm{CMS}}{\\mathrm{CMS}}\n\\newcommand{\\mathrm{NL}}{\\mathrm{NL}}\n\\newcommand{\\mathrm{tHF}}{\\mathrm{tHF}}\n\\renewcommand{\\min}{\\mathrm{min}}\n\\renewcommand{\\max}{\\mathrm{max}}\n\n\\begin{document}\n\n\\thispagestyle{empty}\n\\def\\arabic{footnote}{\\fnsymbol{footnote}}\n\\setcounter{footnote}{1}\n\\null\n\\strut\\hfill  BI-TP 95\/04 \\\\\n\\strut\\hfill WUE-ITP-95-002\\\\\n\\strut\\hfill hep-ph\/9503442\n\\vskip 0cm\n\\vfill\n\\begin{center}\n{\\Large \\bf \n\\boldmath{Radiative Corrections to $\\gamma\\gamma\\to\\mathswitchr {W^+}\\mathswitchr {W^-}$ \\\\\n    in the Electroweak Standard Model}\n\\par} \\vskip 2.5em\n{\\large\n{\\sc A.~Denner%\n\\footnote{On leave of absence from \nInstitut f\\\"ur Theoretische Physik, Universit\\\"at W\\\"urzburg,\nAm Hubland,\\\\ \\hspace*{.5cm} D-97074 W\\\"urzburg, Germany.}\n}\\\\[1ex]\n{\\normalsize \\it Institut f\\\"ur Theoretische Physik, Universit\\\"at Leipzig\\\\\nAugustusplatz 10, D-04109 Leipzig, Germany}\n\\\\[2ex]\n{\\sc S.~Dittmaier%\n\\footnote{Supported by the Bundesministerium f\\\"ur Forschung und\nTechnologie, Bonn, Germany.} }\\\\[1ex]\n{\\normalsize \\it Theoretische Physik, Universit\\\"at Bielefeld\\\\ \nPostfach 100131, D-33501 Bielefeld, Germany}\n\\\\[2ex]\n{\\sc R.~Schuster%\n\\footnote{Supported by the Deutsche Forschungsgemeinschaft.}\n} \\\\[1ex]\n{\\normalsize \\it Institut f\\\"ur Theoretische Physik, Universit\\\"at W\\\"urzburg\\\\\nAm Hubland, D-97074 W\\\"urzburg, Germany}\n}\n\\par \\vskip 1em\n\\end{center} \\par\n\\vskip 1cm \n\\vfill\n{\\bf Abstract:} \\par\nThe cross-section\\ for $\\gamma\\gamma\\to\\mathswitchr W^+\\mathswitchr W^-$ with arbitrary\npolarized photons and W bosons is calculated within the electroweak\nStandard Model including the complete virtual and soft-photonic \n${\\cal O}(\\alpha)$ corrections. We present a detailed numerical discussion\nof the complete radiative corrections and an analytical investigation of\nthe leading corrections. It turns out that in the on-shell\nrenormalization scheme for fixed $\\mathswitch {M_\\PW}$ no leading corrections \nassociated with the \nrunning of $\\alpha$ or heavy top-quark \nand Higgs-boson masses occur.\nThe corrections are typically of the order of 10\\%. \nThey reach, however, larger values where the lowest-order cross-sections\\ are suppressed.\n\\par\n\\vskip 1cm \n\\noindent BI-TP 95\/04 \\\\\nWUE-ITP-95-002\\par\n\\vskip .15mm\n\\noindent March 1995 \\par\n\\null\n\\setcounter{page}{0}\n\\clearpage\n\\def\\arabic{footnote}{\\arabic{footnote}}\n\\setcounter{footnote}{0}\n\n\\section{Introduction}\n\nThe $\\mathrm{SU}(2)\\times\\mathrm{U}(1)$ standard electroweak theory \nhas passed many\nprecision tests during the last years. In particular, measurements of\nthe muon decay constant $G_\\mu$, the gauge-boson masses \\mathswitch {M_\\PW}\\ and \\mathswitch {M_\\PZ},\nand the decay widths and asymmetries of the \\mathswitchr Z\\ boson at LEP have \nprovided stringent constraints which are successfully fulfilled by \nthe Standard Model (SM) evaluated at one-loop level. \nThe experimental data favor\na value for the top-quark mass which is in accordance with the\ndirect measurements \\cite{mt} of CDF, $\\mathswitch {m_\\Pt}=176\\pm16\\unskip\\,\\mathrm{GeV}$, and D\\O,\n$\\mathswitch {m_\\Pt}=199\\pm30\\unskip\\,\\mathrm{GeV}$. Nevertheless, further precision tests of the SM\nare required. Up to now, only weak direct experimental information exists on\nthe non-Abelian self-interaction of the gauge bosons \\cite{WWZ}.\nMoreover, no experimental evidence on the mechanism of spontaneous \nsymmetry breaking, which is responsible for mass generation and\npostulates the existence of the scalar Higgs boson, has been found yet. \nFor such\ninvestigations, energies of several hundred GeV or even few TeV are\nneeded, since the sensitivity to deviations from the SM gauge-boson\nself-interaction grows strongly with energy, and the existence of the\nHiggs particle can be proven only by direct production. To this end, \na ``Next Linear Collider'' (NLC) for $\\mathswitchr e\\Pe$,\n$\\mathswitchr e\\gamma$, and $\\gamma\\ga$ collisions was proposed \\cite{nlc} which offers\na unique environment for such precision experiments owing to the \ncomparably small background.\n\nA particularly interesting process in $\\gamma\\ga$ collisions is $\\gamma\\gamma\\to\\PWp\\PWm$.\nIts total cross-section\\ approaches a constant of about\n$80\\unskip\\,\\mathrm{pb}$ at high energies \ncorresponding to $8\\times10^6$ \\mathswitchr W~pairs for $10\\unskip\\,\\mathrm{fb}^{-1}$.\nThis large cross-section\\ is due to the massive $t$-channel exchange and is\ndrastically reduced by angular cuts. But even for $|\\!\\cos\\theta|< 0.8$\nthe cross-section\\ is still 15 and $4\\unskip\\,\\mathrm{pb}$ at a center-of-mass energy of 500 and\n$1000\\unskip\\,\\mathrm{GeV}$, respectively, and thus much larger as e.g.\\  the one for $\\Pep\\Pem\\to\\PWp\\PWm$.\nHence $\\gamma\\gamma\\to\\PWp\\PWm$ is very well-suited for precision investigations of the SM.\n\nSeveral features of $\\gamma\\gamma\\to\\PWp\\PWm$ have already been discussed in the literature.\nMost of the existing works concentrated on tree-level predictions\n\\cite{Pe73}, in particular on the influence of anomalous\nnon-Abelian gauge couplings \\cite{Ki73,Ye91,Be92}.\nThe process $\\gamma\\gamma\\to\\PWp\\PWm$  depends at tree level both on the triple\n$\\gamma\\mathswitchr W\\PW$ and the quartic $\\gamma\\ga\\mathswitchr W\\PW$ coupling, and\nno other vertices are involved in the unitary gauge at lowest order.\nThe sensitivity to the $\\gamma\\mathswitchr W\\PW$ coupling is comparable and \ncomplementary to the reactions $\\Pep\\Pem\\to\\PWp\\PWm$ and $\\Pem\\gamma\\to\\PWm\\nu$: the first\ninvolves a mixture of the $\\gamma\\mathswitchr W\\PW$ and the $\\mathswitchr Z\\mathswitchr W\\PW$ coupling, the\nsecond involves the $\\gamma\\mathswitchr W\\PW$ alone but is less sensitive \\cite{Ye91}.\nBecause the sensitivity to the $\\gamma\\ga\\mathswitchr W\\PW$ coupling is much larger \nthan the one in $\\mathswitchr {e^+}\\mathswitchr {e^-}$ processes, $\\gamma\\gamma\\to\\PWp\\PWm$ is the ideal process to study \nthis coupling \\cite{Be92}. \n\nThe one-loop diagrams involving a resonant Higgs boson have been\ncalculated in order to \nstudy the possible investigation of the Higgs boson via\n$\\gamma\\ga\\to\\mathswitchr H^*\\to\\mathswitchr {W^+}\\mathswitchr {W^-}$ \\cite{Va79,Bo92,Mo94,Ve94}.\nBased on our complete one-loop calculation, we have\nsupplemented these investigations by a discussion of the heavy-Higgs\neffects in \\citere{HH}. As a matter of fact, only the (suppressed)\nchannels of longitudinal \\mathswitchr W-boson production are sensitive to the Higgs\nmechanism, but the (dominant) channels of purely transverse \\mathswitchr W-boson \nproduction are extremely insensitive. \nThis insensitivity to the Higgs sector renders $\\gamma\\gamma\\to\\PWp\\PWm$ even more\nsuitable for the investigation of the non-Abelian self couplings.\n\nIn this paper, we focus on the complete SM one-loop corrections to $\\gamma\\gamma\\to\\PWp\\PWm$.\nOne reason why these have not\nbeen calculated so far is certainly their analytical\ncomplexity. We have calculated the numerous Feynman graphs (roughly\n300--550 depending on the gauge fixing) by using {\\it Mathematica\\\/}\n\\cite{math}. More precisely, we have generated and drawn the Feynman\ngraphs by {\\it FeynArts} \\cite{fa} and performed three different\ncalculations, one in 't~Hooft--Feynman gauge using {\\it FeynCalc}\n\\cite{fc} and two in a non-linear gauge with and without using\n{\\it FeynCalc}.\nAs the final result exhibits a very complicated and\nuntransparent analytical form, we refrain from writing\nit down in full detail. Instead, we indicate its general structure and\npresent a detailed discussion of\nthe numerical results for the $\\mathswitch{{\\cal{O}}(\\alpha)}$ (virtual and real\nsoft-photonic) corrections to the polarized as well as unpolarized\ncross-sections. We restrict the presentation of the analytical results\nto the lowest-order cross-sections\\ and the most important $\\mathswitch{{\\cal{O}}(\\alpha)}$ corrections.\nIn particular, we discuss the Higgs resonance, the heavy-Higgs effects,\nthe Coulomb singularity, and the leading\neffects from light fermions and a heavy top quark.\n\nThe paper is organized as follows: After fixing our notation and\nconventions in \\refse{se:notcon}, we consider the lowest-order cross-sections\\ for\nvarious polarizations in \\refse{se:born}. In \\refse{se:RC} we discuss \nthe evaluation and general features of the radiative corrections and in\n\\refse{se:num} the numerical results. \n\n\\section{Notation and conventions}\n\\label{se:notcon}\n\nWe consider the reaction\n$$ \\gamma(k_1,\\lambda_1) + \\gamma(k_2,\\lambda_2) \\rightarrow\n   {\\mathswitchr {W^+}}(k_3,\\lambda_3) + {\\mathswitchr {W^-}}(k_4,\\lambda_4) \\; ,\n$$\nwhere $\\lambda_{1,2} = \\pm 1$ and $\\lambda_{3,4} = 0,\\pm 1$\ndenote the helicities of the incoming photons and outgoing W bosons,\nrespectively.\n\nIn the center-of-mass system (CMS)\nthe momenta read \nin terms of the beam energy $E$ of the incoming photons and \nthe scattering angle $\\theta$\n\\begin{eqnarray}\nk^\\mu_1 &=& E(1,  0,  0,  -1) \\; ,\n\\nonumber \\\\\nk^\\mu_2 &=& E(1,  0,  0, 1) \\; ,\n\\nonumber \\\\\nk^\\mu_3 &=& E(1,  -\\beta  \\sin\\theta,\n                             0,  -\\beta \\cos\\theta) \\; ,\n\\nonumber \\\\\nk^\\mu_4 &=& E(1,  \\beta  \\sin\\theta,\n                             0,  \\beta \\cos\\theta) \\; ,\n\\end{eqnarray}\nwhere $\\beta = \\sqrt{1 -   \\mathswitch {M_\\PW}^2 \/ E^2}$\nis the velocity of the \\mathswitchr W\\ bosons in the CMS.\nWe define the Mandelstam variables\n\\begin{eqnarray}\ns &=& (k_1 + k_2)^2 = (k_3 + k_4)^2 = 4  E^2 \\; ,\n\\nonumber \\\\\nt &=& (k_1 - k_3)^2 = (k_2 - k_4)^2 = \\mathswitch {M_\\PW}^2 - \\frac{s}{2}\n                                      (1 - \\beta \\cos \\theta) \\; ,\n\\nonumber \\\\\nu &=& (k_1 - k_4)^2 = (k_2 - k_3)^2 = \\mathswitch {M_\\PW}^2 - \\frac{s}{2}\n                                      (1 + \\beta \\cos \\theta)  \\; .\n\\end{eqnarray}\n\nIn order to calculate polarized cross-sections, we introduce explicit polarization\nvectors for the photons and W bosons as follows\n\\begin{eqnarray}\n\\varepsilon_1^\\mu(k_1, \\lambda_1 =\\pm 1) &=& \\frac{-1}{\\sqrt{2}}  (0,   1, \\mp i, 0) \\; ,\n\\nonumber \\\\\n\\varepsilon_2^\\mu(k_2, \\lambda_2 =\\pm 1) &=& \\frac{1}{\\sqrt{2}}  (0,   1, \\pm i, 0) \\; ,\n\\nonumber \\\\\n{\\varepsilon_3^*}^\\mu(k_3, \\lambda_3 = \\pm 1) &=& \\frac{-1}{\\sqrt{2}} \n                   (0,  \\cos\\theta,  \\pm i,  -\\sin\\theta) \\; ,\n\\nonumber \\\\\n{\\varepsilon_4^*}^\\mu(k_4, \\lambda_4 =\\pm 1) &=& \\frac{1}{\\sqrt{2}} \n                   (0,  \\cos\\theta,  \\mp i, -\\sin\\theta) \\; ,\n\\nonumber \\\\\n{\\varepsilon_3^*}^\\mu(k_3, \\lambda_3 =0)\\;\\;\\; &=& \\frac{E}{\\mathswitch {M_\\PW}} \n                   (\\beta,   -\\sin\\theta,  0,  -\\cos\\theta) \\; ,\n\\nonumber \\\\\n{\\varepsilon_4^*}^\\mu(k_4, \\lambda_4 =0)\\;\\;\\; &=& \\frac{E}{\\mathswitch {M_\\PW}} \n                   (\\beta,   \\sin\\theta,  0,  \\cos\\theta) \\; .\n\\end{eqnarray}\n\nWe decompose the amplitude ${\\cal M}$ into invariant functions $F_{ijkl}$\nand standard matrix elements (SME) ${\\cal M}_{ijkl}$, which\ncontain the whole information about the boson polarizations.\nUsing the transversality condition for the polarization vectors\nand Schouten's identity,\nthe amplitude ${\\cal M}$ can be reduced to\n\\begin{eqnarray}\n{\\cal M}(\\lambda_1,\\lambda_2,\\lambda_3,\\lambda_4,s,t) &=& \n         \\sum_{i,j = \\{0,3,4\\}}\\sum_{k,l = \\{0,1,2\\}}\n          F_{ijkl}(s,t)  \n       {\\cal M}_{ijkl}(\\lambda_1,\\lambda_2,\\lambda_3,\\lambda_4,s,t) \n\\nonumber \\\\\n       & &{}+ F^{(t)}_{0000}(s,t) \n       {\\cal M}_{0000}^{(t)}(\\lambda_1,\\lambda_2,\\lambda_3,\\lambda_4,s,t) \n\\nonumber \\\\\n       & &{}+ F^{(u)}_{0000}(s,t) \n       {\\cal M}_{0000}^{(u)}(\\lambda_1,\\lambda_2,\\lambda_3,\\lambda_4,s,t)\n\\label{decom}\n\\end{eqnarray}\nwith\n($i,j = \\{3,4\\}$, $k,l = \\{1,2\\}$)\n\\begin{eqnarray}\n{\\cal M}_{ijkl} &=& (\\varepsilon_1 \\cdot  k_i)  (\\varepsilon_2 \\cdot  k_j) \n           (\\varepsilon_3^* \\cdot  k_k)  (\\varepsilon_4^* \\cdot  k_l) \/ s^2,\n\\label{5}\n\\\\[.3em]\n{\\cal M}_{0jkl} &=& (\\varepsilon_1 \\cdot  A)  (\\varepsilon_2 \\cdot  k_j) \n           (\\varepsilon_3^* \\cdot  k_k)  (\\varepsilon_4^* \\cdot  k_l) \/ s,\n\\nonumber \\\\\n{\\cal M}_{i0kl} &=& (\\varepsilon_1 \\cdot k_i)  (\\varepsilon_2 \\cdot  A) \n           (\\varepsilon_3^* \\cdot  k_k)  (\\varepsilon_4^* \\cdot  k_l) \/ s,\n\\nonumber \\\\\n{\\cal M}_{ij0l} &=& (\\varepsilon_1 \\cdot  k_i)  (\\varepsilon_2 \\cdot  k_j) \n           (\\varepsilon_3^* \\cdot  A)  (\\varepsilon_4^* \\cdot  k_l) \/ s,\n\\nonumber \\\\\n{\\cal M}_{ijk0} &=& (\\varepsilon_1 \\cdot  k_i)  (\\varepsilon_2 \\cdot  k_j) \n           (\\varepsilon_3^* \\cdot  k_k)  (\\varepsilon_4^* \\cdot  A) \/ s,\n\\\\[.3em]\n{\\cal M}_{00kl} &=& (\\varepsilon_1 \\cdot  \\varepsilon_2) \n           (\\varepsilon_3^* \\cdot  k_k)  (\\varepsilon_4^* \\cdot  k_l) \/ s,\n\\nonumber \\\\\n{\\cal M}_{0j0l} &=& (\\varepsilon_1 \\cdot  \\varepsilon_3^*) \n           (\\varepsilon_2 \\cdot  k_j)  (\\varepsilon_4^* \\cdot  k_l) \/ s,\n\\nonumber \\\\\n{\\cal M}_{0jk0} &=& (\\varepsilon_1 \\cdot  \\varepsilon_4^*) \n           (\\varepsilon_2 \\cdot  k_j)  (\\varepsilon_3^* \\cdot  k_k) \/ s,\n\\nonumber \\\\\n{\\cal M}_{i00l} &=& (\\varepsilon_2 \\cdot  \\varepsilon_3^*) \n           (\\varepsilon_1 \\cdot  k_i)  (\\varepsilon_4^* \\cdot  k_l) \/ s,\n\\nonumber \\\\\n{\\cal M}_{i0k0} &=& (\\varepsilon_2 \\cdot  \\varepsilon_4^*) \n           (\\varepsilon_1 \\cdot  k_i)  (\\varepsilon_3^* \\cdot  k_k) \/ s,\n\\nonumber \\\\\n{\\cal M}_{ij00} &=& (\\varepsilon_3^* \\cdot  \\varepsilon_4^*) \n           (\\varepsilon_1 \\cdot  k_i)  (\\varepsilon_2 \\cdot  k_j) \/ s,\n\\\\[.3em]\n{\\cal M}_{000l} &=& (\\varepsilon_1 \\cdot  \\varepsilon_2) \n           (\\varepsilon_3^* \\cdot  A)  (\\varepsilon_4^* \\cdot  k_l),\n\\nonumber \\\\\n{\\cal M}_{00k0} &=& (\\varepsilon_1 \\cdot  \\varepsilon_2) \n           (\\varepsilon_3^* \\cdot  k_k) (\\varepsilon_4^* \\cdot  A),\n\\nonumber \\\\\n{\\cal M}_{0j00} &=& (\\varepsilon_3^* \\cdot  \\varepsilon_4^*) \n           (\\varepsilon_1 \\cdot  A)  (\\varepsilon_2 \\cdot  k_j),\n\\nonumber \\\\\n{\\cal M}_{i000} &=& (\\varepsilon_3^* \\cdot  \\varepsilon_4^*) \n           (\\varepsilon_1 \\cdot  k_i)  (\\varepsilon_2 \\cdot  A),\n\\\\[.3em]\n{\\cal M}_{0000}  &=& (\\varepsilon_1 \\cdot  \\varepsilon_2) \n           (\\varepsilon_3^* \\cdot  \\varepsilon_4^*)  ,\n\\nonumber \\\\\n{\\cal M}^{(t)}_{0000}  &=& (\\varepsilon_1 \\cdot  \\varepsilon_3^*) \n           (\\varepsilon_2 \\cdot  \\varepsilon_4^*)  ,\n\\nonumber \\\\\n{\\cal M}^{(u)}_{0000}  &=& (\\varepsilon_1 \\cdot  \\varepsilon_4^*) \n           (\\varepsilon_2 \\cdot  \\varepsilon_3^*)  , \\label{9}\n\\end{eqnarray}\nand \n\\begin{equation}\nA_\\mu =  \\frac{i}{ut - \\mathswitch {M_\\PW}^4} \n       \\varepsilon_{\\mu\\nu\\rho\\sigma} k_1^\\nu k_2^\\rho k_3^\\sigma,\n\\qquad \\varepsilon_{0123} = -1 \\;.\n\\end{equation}\nOur choice of polarization vectors for the photons implies\n\\begin{eqnarray}\n\\varepsilon_i k_j = 0, \\qquad i,j = 1,2 \\;,\n\\end{eqnarray}\nand thus by virtue of momentum conservation\n\\begin{eqnarray}\n\\varepsilon_i k_3 = - \\varepsilon_i k_4,\n \\qquad i = 1,2 \\;.\n\\end{eqnarray}\nWe use this relation to eliminate all SME\ninvolving $\\varepsilon_1 k_4$ and $\\varepsilon_2 k_3$.\nThis reduces the 83 SME defined in (\\ref{5}) -- \n(\\ref{9}) to 38 for the process under consideration. \n\nAs a consequence of CP invariance and Bose symmetry\nonly the sum of each SME and the one \nwith  $(\\varepsilon_1,k_1,\\varepsilon_3,k_3)$ and\n$(\\varepsilon_2,k_2,\\varepsilon_4,k_4)$ interchanged occurs.\nFor instance, ${\\cal{M}}_{0401}$ only appears in the\ncombination ${\\cal{M}}_{0401}+{\\cal{M}}_{3020}$ in the expansion of ${\\cal{M}}$ in\n\\refeq{decom}. This leaves 22 independent SME. \n\nIn four dimensions, the matrix elements ${\\cal M}^{(t)}_{0000}$ and\n${\\cal M}^{(u)}_{0000}$ are not linearly independent from the set of all\n${\\cal{M}}_{ijkl}$ and \ncan be reduced to linear combinations of the other matrix\nelements using the identities \n\\begin{eqnarray}\n\\delta^{\\varepsilon_1\\varepsilon_3^*k_1k_2k_3}\n      _{\\varepsilon_2\\varepsilon_4^*k_1k_2k_3} =\n\\delta^{\\varepsilon_1\\varepsilon_4^*k_1k_2k_3}\n      _{\\varepsilon_2\\varepsilon_3^*k_1k_2k_3} = 0\n\\end{eqnarray}\ninvolving the Gram determinant\n\\begin{eqnarray}\n\\delta^{p_1 \\ldots p_n}_{q_1 \\ldots q_n} =\n\\left| \\matrix{p_1\\cdot q_1&\\ldots&p_1\\cdot q_n\\cr\n                   \\vdots&\\ddots&\\vdots\\cr\n                   p_n\\cdot q_1&\\ldots&p_n\\cdot q_n\\cr}\\right| \\;.\n\\end{eqnarray}\nNevertheless, we keep ${\\cal M}^{(t)}_{0000}$ and\n${\\cal M}^{(u)}_{0000}$ for convenience.\n\nBose symmetry implies that the amplitude ${\\cal{M}}$ is invariant under \nthe interchange $(k_1,\\varepsilon_1) \\leftrightarrow (k_2,\\varepsilon_2)$.\nSince many diagrams can be related to others by this transformation,\nit is useful to introduce a second set of SME which is obtained\nfrom \\refeqs{5}--\\refeqf{9} by this interchange.\nOf course, this second set of SME can be expressed by the original set.\n\nBesides Bose symmetry also CP is an exact symmetry,\nsince we use a unit quark-mixing \nmatrix.%\n\\footnote{For a non-trivial quark-mixing matrix, CP would be violated\nin the considered process first at two-loop level.} \nThe helicity amplitudes for fixed \npolarization configurations are related as follows\n\\begin{equation}\n\\begin{array}[b]{ll}\n{\\cal M}_{\\lambda_1 \\lambda_2 \\lambda_3 \\lambda_4}(s,t,u) = \n{\\cal M}_{\\lambda_2 \\lambda_1 \\lambda_3 \\lambda_4}(s,u,t) \n& \\hbox{(Bose)} \\\\\n{\\cal M}_{\\lambda_1 \\lambda_2 \\lambda_3 \\lambda_4}(s,t,u) = \n{\\cal M}_{-\\lambda_1 -\\lambda_2 -\\lambda_4 -\\lambda_3}(s,u,t) \n\\qquad & \\hbox{(CP)} \\\\\n{\\cal M}_{\\lambda_1 \\lambda_2 \\lambda_3 \\lambda_4}(s,t,u) =\n{\\cal M}_{-\\lambda_2 -\\lambda_1 -\\lambda_4 -\\lambda_3}(s,t,u)\n\\qquad & \\hbox{(Bose+CP)}.\n\\end{array}\n\\end{equation}\n\nIn the following, we only consider the sum of the two transverse \n\\mathswitchr W~polarizations.\nTherefore we indicate the polarizations of the external particles \nby four labels, the first pair corresponding to the photons, and the \nsecond pair to\nthe \\mathswitchr W~bosons. The labels $+$,$-$ represent right-handed and left-handed\nphotons, respectively, ${\\mathrm{L}}$ stands for longitudinal, and\n${\\mathrm{T}}$ for the sum of the two transverse \\mathswitchr W~polarizations.\n\nThe combination of Bose and CP symmetry leads to the following relations\nbetween the differential cross-sections\\ with equal photon helicities\n\\begin{eqnarray}\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{--{\\rm TT}} &=&\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{++{\\rm TT}}\\; ,\n\\nonumber \\\\\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{--{\\rm LL}} &=&\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{++{\\rm LL}}\\; ,\n\\nonumber \\\\\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{--{\\rm (LT+TL)}} &=&\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{++{\\rm (LT+TL)}} \\; .\n\\label{eq:CPBds}\n\\end{eqnarray}\nMoreover, Bose symmetry implies that all cross-sections\\ in (\\ref{eq:CPBds})\nare forward--backward symmetric. \nFor different photon helicities Bose symmetry leads to\n\\begin{eqnarray}\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{-+{\\rm TT}}(s,t,u) &=&\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{+-{\\rm TT}}(s,u,t)\\; ,\n\\nonumber \\\\\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{-+{\\rm LL}}(s,t,u) &=&\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{+-{\\rm LL}}(s,u,t)\\; ,\n\\nonumber \\\\\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{-+{\\rm (LT+TL)}}(s,t,u) &=&\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{+-{\\rm (LT+TL)}}(s,u,t) \\; ,\n\\label{eq:Bds}\n\\end{eqnarray}\nwhereas Bose+CP does not yield further relations.\n\nC and P symmetry are only violated by the fermionic loop corrections,\nbut hold in lowest order and for the bosonic loop corrections. We\nindicate these restricted symmetries by a modified equality sign\n\\begin{eqnarray}\n{\\cal M}_{\\lambda_1 \\lambda_2 \\lambda_3 \\lambda_4}(s,t,u) &\\Nequal{P}&\n{\\cal M}_{-\\lambda_1 -\\lambda_2 -\\lambda_3 -\\lambda_4}(s,t,u) \\; ,\n\\nonumber\\\\\n{\\cal M}_{\\lambda_1 \\lambda_2 \\lambda_3 \\lambda_4}(s,t,u) &\\Nequal{C}&\n{\\cal M}_{\\lambda_1 \\lambda_2 \\lambda_4 \\lambda_3}(s,u,t) \\; .\n\\end{eqnarray}\nP invariance then implies for the differential cross-sections\\\n\\begin{eqnarray}\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{-+{\\rm TT}} &\\Nequal{P}&\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{+-{\\rm TT}}\\; ,\n\\nonumber \\\\\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{-+{\\rm LL}} &\\Nequal{P}&\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{+-{\\rm LL}}\\; ,\n\\nonumber \\\\\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{-+{\\rm (LT+TL)}} &\\Nequal{P}&\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)_{+-{\\rm (LT+TL)}}\\; .\n\\label{eq:Pds}\n\\end{eqnarray}\nIn combination with (\\ref{eq:Bds}) this means that the forward-backward\nasymmetries of the differential cross-sections\\ for unequal photon helicities \nare entirely due to fermionic corrections.\n\nWe perform the calculation both in 't~Hooft--Feynman (tHF) gauge and in a\ngauge with the following non-linear (NL) gauge-fixing term \\cite{gavela}\n\\begin{eqnarray}\n{\\cal L}_{\\rm GF} &=& -\\left| \\partial^\\mu W^+_\\mu + i e\n                (A^\\mu - \\frac{\\mathswitch {c_\\PW}}{\\mathswitch {s_\\PW}} Z^\\mu) W^+_\\mu \n               -i \\mathswitch {M_\\PW} \\phi^+ \\right|^2\n\\nonumber \\\\\n            & &{}-  \\frac{1}{2} (\\partial^\\mu Z_\\mu - \\mathswitch {M_\\PZ} \\chi)^2\n              - \\frac{1}{2} (\\partial^\\mu A_\\mu)^2 \\;,\n\\label{nl}\n\\end{eqnarray}\nwith the conventions of \\citere{ad&mex} for the fields.\nIn particular, $\\phi^\\pm$ and $\\chi$ denote the charged and neutral \nwould-be-Goldstone fields, respectively.\nIn this NL gauge the $\\phi^\\pm W^\\mp A$ vertices\nvanish. This reduces the number of Feynman graphs in comparison to the \ntHF gauge considerably.\n\n\\section{Lowest-order cross-section}\n\\label{se:born}\nIn NL gauge, only the three diagrams of \\reffi{fi:borndia} \ncontribute to the lowest-order amplitude. \n\\begin{figure}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(160,35)\n\\put(-13,-154){\\special{psfile=aawwborn.nlgauge.ps hscale=90 vscale=90}}\n\\end{picture}\n\\caption{Lowest-order diagrams for $\\gamma\\gamma\\to\\PWp\\PWm$ in NL gauge}\n\\label{fi:borndia}\n\\end{figure}\nIn tHF gauge two additional diagrams exist which \ninvolve internal $\\phi$ fields.\nEvaluation of the tree diagrams in either gauge yields the Born amplitude\n\\begin{equation}\n{\\cal{M}}_{\\mathrm{Born}} = 8\\pi\\alpha\\Biggl\\{\n\\frac{s}{\\mathswitch {M_\\PW}^2-t}{\\cal{M}}_{0,t} + \\frac{s}{\\mathswitch {M_\\PW}^2-u}{\\cal{M}}_{0,u} - {\\cal{M}}_{0000} \n\\Biggr\\},\n\\end{equation}\nwhere\n\\begin{eqnarray}\n{\\cal{M}}_{0,t} &=& \n2 {\\cal{M}}_{0012} + 2 {\\cal{M}}_{3400} - 2 {\\cal{M}}_{0401} - 2 {\\cal{M}}_{3020} + 2 {\\cal{M}}_{0410} \n+ 2 {\\cal{M}}_{3002} + {\\cal{M}}_{0000}^{(t)},  \\nonumber\\\\\n{\\cal{M}}_{0,u} &=& \n2 {\\cal{M}}_{0021} + 2 {\\cal{M}}_{4300} - 2 {\\cal{M}}_{0310} - 2 {\\cal{M}}_{4002} + 2 {\\cal{M}}_{0301} \n+ 2 {\\cal{M}}_{4020} + {\\cal{M}}_{0000}^{(u)}.\n\\end{eqnarray}\nThe lowest-order matrix element vanishes for the helicities \n$(\\lambda_1,\\lambda_2,\\lambda_3,\\lambda_4) = \n(\\pm,\\pm,0,\\pm)$, $(\\pm,\\pm,\\pm,0)$, $(\\pm,\\pm,0,\\mp)$, \n$(\\pm,\\pm,\\mp,0)$, $(\\pm,\\pm,\\pm,\\mp)$, $(\\pm,\\pm,\\mp,\\pm)$.\n\nThe differential Born cross-section\\ \\cite{Ye91,Be92} is obtained as\n\\begin{eqnarray}\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)^{\\mathrm{Born}} &=&\n\\frac{\\beta}{64 \\pi^2 s}\n\\sum_{\\lambda_1 \\lambda_2 \\lambda_3 \\lambda_4}\n\\frac{1}{4} \\left(1+ \\lambda_1 P^\\gamma_1\\right)\n            \\left(1+ \\lambda_2 P^\\gamma_2\\right)\n\\left| {\\cal M}_{{\\mathrm{Born}}} \\right|^2 \\; ,\n\\end{eqnarray} \nwhere $P^\\gamma_{1,2}$ denote the degrees of photon-beam polarization and\nthe sum over $\\lambda_3$, $\\lambda_4$ include the desired W polarizations.\n\nWe list the differential cross-sections\\ for several helicity configurations:\n\\begin{eqnarray}\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)^{\\mathrm{Born}}_{\\pm\\pm{\\mathrm{T}}\\rT} &=& \n\\frac{\\alpha^2 \\beta s (2 \\mathswitch {M_\\PW}^4 -4 \\mathswitch {M_\\PW}^2 s +s^2)}{(\\mathswitch {M_\\PW}^2 -t)^2(\\mathswitch {M_\\PW}^2 -u)^2} \\; ,\n\\nonumber \\\\\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)^{\\mathrm{Born}}_{\\pm\\pm{\\mathrm{L}}\\rL} &=& \n \\frac{\\alpha^2 \\beta \\mathswitch {M_\\PW}^4 s}{(\\mathswitch {M_\\PW}^2 -t)^2(\\mathswitch {M_\\PW}^2 -u)^2} \\; ,\n\\nonumber \\\\\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)^{\\mathrm{Born}}_{\\pm\\pm({\\mathrm{L}}{\\mathrm{T}}+{\\mathrm{T}}{\\mathrm{L}})} \n&=& 0 \\; ,  \\nonumber\\\\\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)^{\\mathrm{Born}}_{\\pm\\mp{\\mathrm{T}}\\rT} &=& \n \\frac{\\alpha^2 \\beta s^3}{(\\mathswitch {M_\\PW}^2 -t)^2(\\mathswitch {M_\\PW}^2 -u)^2} \n       \\biggl\\{2 \\frac{(16 \\mathswitch {M_\\PW}^4 +s^2) (ut-\\mathswitch {M_\\PW}^4)^2}{s^6\\beta^4}\n           +\\frac{(t-u)^2}{s^2\\beta^2} \\biggr\\} \\; , \\nonumber\\\\ \n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)^{\\mathrm{Born}}_{\\pm\\mp{\\mathrm{L}}\\rL} &=& \n \\frac{\\alpha^2 (4 \\mathswitch {M_\\PW}^2+s)^2 (\\mathswitch {M_\\PW}^4 - t u)^2}\n      {\\beta^3 s^3  (\\mathswitch {M_\\PW}^2 -t)^2(\\mathswitch {M_\\PW}^2 -u)^2} \\; ,\n\\nonumber \\\\\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)^{\\mathrm{Born}}_{\\pm\\mp({\\mathrm{L}}{\\mathrm{T}}+{\\mathrm{T}}{\\mathrm{L}})} &=& \n \\frac{16 \\alpha^2 \\mathswitch {M_\\PW}^2 (2 \\mathswitch {M_\\PW}^4 -t^2-u^2) (\\mathswitch {M_\\PW}^4 - t u)}{\n      \\beta^3 s^2  (\\mathswitch {M_\\PW}^2 -t)^2(\\mathswitch {M_\\PW}^2 -u)^2}  \\; .\n\\end{eqnarray}\nThese results can be reconstructed from equation (5) in \\citere{Be92}\nor from equation (4.5) in \\citere{Ye91}.\n\nAdding up the single contributions,\nwe get for the unpolarized differential cross-section\n\\footnote{The second term in equation (6) of \\citere{Be92} should be \nmultiplied by 2.}\n\\begin{eqnarray}\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)^{\\mathrm{Born}}_{\\mathrm{unpol}} &=&\n \\frac{3 \\alpha^2 \\beta}{2s} \n\\Biggl\\{1 - \\frac{2s^2}{(\\mathswitch {M_\\PW}^2 -t)(\\mathswitch {M_\\PW}^2 -u)} \\left(\n                    \\frac{2}{3} + \\frac{\\mathswitch {M_\\PW}^2}{s} \\right) \n\\nonumber \\\\ \n& &{}+ \\frac{2s^4}{(\\mathswitch {M_\\PW}^2 -t)^2(\\mathswitch {M_\\PW}^2 -u)^2} \\left(\n                    \\frac{1}{3} + \\frac{\\mathswitch {M_\\PW}^4}{s^2} \\right)\n   \\Biggr\\} \\; .\n\\label{unpol}\n\\end{eqnarray}\n\nIntegration over $\\theta_{\\mathrm{cut}} \\leq \\theta \\leq \\pi - \\theta_{\\mathrm{cut}}$\nyields:\n\\begin{eqnarray}\n\\sigma^{\\mathrm{Born}}_{\\pm\\pm{\\mathrm{T}}\\rT} &=& \\frac{16 \\pi \\alpha^2}{s}\\, \n\\frac{s^2 - 4 \\mathswitch {M_\\PW}^2s + 2\\mathswitch {M_\\PW}^4}{s^2} \\left\\{\n\\log\\left(\\frac{1 + \\beta \\cos\\theta_{\\mathrm{cut}}}{1 - \\beta\\cos\\theta_{\\mathrm{cut}}}\\right)\n   + \\frac{2\\beta \\cos\\theta_{\\mathrm{cut}}}{1 - \\beta^2 \\cos^2\\theta_{\\mathrm{cut}}}\n       \\right\\} \\; ,\n\\nonumber \\\\\n\\sigma^{\\mathrm{Born}}_{\\pm\\pm{\\mathrm{L}}\\rL} &=& \\frac{16 \\pi \\alpha^2}{s}\\frac{\\mathswitch {M_\\PW}^4}{s^2} \n\\left\\{\n\\log\\left(\\frac{1 + \\beta\\cos\\theta_{\\mathrm{cut}}}{1 - \\beta\\cos\\theta_{\\mathrm{cut}}}\\right)\n   + \\frac{2\\beta \\cos\\theta_{\\mathrm{cut}}}{1 - \\beta^2 \\cos^2\\theta_{\\mathrm{cut}}}\n       \\right\\} \\; ,\n\\nonumber \\\\\n\\sigma^{\\mathrm{Born}}_{\\pm\\mp{\\mathrm{T}}\\rT} &=& \\frac{8 \\pi \\alpha^2 }{s \\beta^4} \\Biggl\\{\n \\frac{s^2 + 16\\mathswitch {M_\\PW}^4}{s^2} \\beta \\cos\\theta_{\\mathrm{cut}}\\nonumber\\\\\n& & {} -2\\frac{s^4 - 2 \\mathswitch {M_\\PW}^2 s^3 - 2 \\mathswitch {M_\\PW}^4 s^2 + 32 \\mathswitch {M_\\PW}^6 s - 32 \\mathswitch {M_\\PW}^8} {s^4}\n\\log\\left(\\frac{1 + \\beta \\cos\\theta_{\\mathrm{cut}}}{1 - \\beta\\cos\\theta_{\\mathrm{cut}}}\\right) \n\\nonumber\\\\\n& &{}+ 4 \\frac{s^4 - 4 \\mathswitch {M_\\PW}^2 s^3 + 2 \\mathswitch {M_\\PW}^4 s^2 + 32 \\mathswitch {M_\\PW}^8}{s^4}\\,\n   \\frac{\\beta \\cos\\theta_{{\\mathrm{cut}}}} {1 - \\beta^2 \\cos^2\\theta_{{\\mathrm{cut}}}} \n\\Biggr\\} \\; , \\nonumber\\\\\n\\sigma^{\\mathrm{Born}}_{\\pm\\mp{\\mathrm{L}}\\rL} &=& \\frac{4 \\pi \\alpha^2}{s\\beta^4}\n\\frac{ (4 \\mathswitch {M_\\PW}^2 + s)^2}{s^2}\n\\Biggl\\{\\beta \\cos\\theta_{{\\mathrm{cut}}}\n- 4 \\frac{\\mathswitch {M_\\PW}^2 (s-\\mathswitch {M_\\PW}^2)}{s^2}\n\\log\\left(\\frac{1 + \\beta\\cos\\theta_{\\mathrm{cut}}}{1 - \\beta\\cos\\theta_{\\mathrm{cut}}}\\right) \n\\nonumber\\\\\n& &{} + \\frac{8\\mathswitch {M_\\PW}^4}{s^2}\n\\frac{\\beta \\cos\\theta_{{\\mathrm{cut}}}}{1 - \\beta^2 \\cos^2\\theta_{{\\mathrm{cut}}}} \n       \\Biggr\\} \\; ,\n\\nonumber \\\\\n\\sigma^{\\mathrm{Born}}_{\\pm\\mp({\\mathrm{L}}{\\mathrm{T}}+{\\mathrm{T}}{\\mathrm{L}})} &=&  \\frac{128 \\pi \\alpha^2}{s\\beta^4}\n\\frac{\\mathswitch {M_\\PW}^2}{s}\n\\Biggl\\{\n- \\beta \\cos\\theta_{{\\mathrm{cut}}}\n+ \\frac{s^2 - 2 \\mathswitch {M_\\PW}^2s + 4\\mathswitch {M_\\PW}^4}{s^2} \n\\log\\left(\\frac{1 + \\beta\\cos\\theta_{{\\mathrm{cut}}}} {1 - \\beta\\cos\\theta_{{\\mathrm{cut}}}}\n\\right) \\nonumber\\\\\n& &{}\n- \\frac{4 \\mathswitch {M_\\PW}^2 (s - 2\\mathswitch {M_\\PW}^2)}{s^2} \\,\n\\frac{\\beta \\cos\\theta_{{\\mathrm{cut}}}}{1 - \\beta^2 \\cos^2\\theta_{\\mathrm{cut}}}  \n       \\Biggr\\} \\; ,\n\\end{eqnarray}\nand for the unpolarized cross-section\\\n\\begin{eqnarray}\n\\sigma^{\\mathrm{Born}}_{{\\mathrm{unpol}}} &=&  \\frac{6\\pi \\alpha^2 }{s}\n\\Biggl\\{\n\\beta \\cos\\theta_{{\\mathrm{cut}}} - \n4 \\frac{\\mathswitch {M_\\PW}^2}{s}\\left(1 - \\frac{2\\mathswitch {M_\\PW}^2}{s}\\right)\n\\log\\left(\\frac{1 + \\beta \\cos\\theta_{{\\mathrm{cut}}}}\n{1 - \\beta \\cos\\theta_{{\\mathrm{cut}}}}\\right)\n\\nonumber \\\\\n& &{}+ \\left(\\frac{1}{3} + \\frac{\\mathswitch {M_\\PW}^4}{s^2} \\right)\n\\frac{16 \\beta \\cos\\theta_{\\mathrm{cut}}}{1 - \\beta^2\\cos^2\\theta_{\\mathrm{cut}}} \\Biggr\\} \\;.\n\\end{eqnarray}\n\nIn \\reffis{fi:intcs10} and \\ref{fi:intcs20} \nwe show the lowest-order cross-sections\\ for various polarizations\nand two different angular cuts $\\theta_{\\mathrm{cut}} = 10^\\circ, 20^\\circ$.\nFor $\\theta_{{\\mathrm{cut}}} = 0$, the cross-sections\\ for\ntransverse W~bosons approach a constant at high energies, $s \\gg \\mathswitch {M_\\PW}^2$, \nowing to the massive $t$-channel exchange \n\\begin{equation}\n\\sigma^{\\mathrm{Born}}_{\\pm\\pm{\\mathrm{T}}\\rT}, \\sigma^{\\mathrm{Born}}_{\\pm\\mp{\\mathrm{T}}\\rT}\n\\;\\Nlim{s\\rightarrow\\infty}\\;\n\\frac{8 \\pi \\alpha^2}{\\mathswitch {M_\\PW}^2} = 80.8 \\unskip\\,\\mathrm{pb}.\n\\end{equation}\nFor a finite cut, $\\sigma^{\\mathrm{Born}}_{\\pm\\pm{\\mathrm{T}}\\rT}$  and\n$\\sigma^{\\mathrm{Born}}_{\\pm\\mp{\\mathrm{T}}\\rT}$ behave as $1\/s$ for large $s$.\nThe cross-sections\\ $\\sigma^{\\mathrm{Born}}_{\\pm\\mp{\\mathrm{L}}\\rL}$ and\n$\\sigma^{\\mathrm{Born}}_{\\pm\\mp({\\mathrm{T}}{\\mathrm{L}}+{\\mathrm{L}}{\\mathrm{T}})}$ are\nproportional to $1\/s$ and $1\/s^2$, respectively, independently\n of the cut-off.\nThe cross-section\\ $\\sigma^{\\mathrm{Born}}_{\\pm\\pm{\\mathrm{L}}\\rL}$ goes \nlike $1\/s^2$ at high energies for $\\theta_{\\mathrm{cut}} = 0$\nand like $1\/s^3$ for a finite cut-off. \nIt is suppressed by about a factor of $10^3$ at $E_{\\mathrm{CMS}} = 500$ GeV.\nNote that the latter cross-section\\ can be\nenhanced drastically by non-standard physics \\cite{HH}.\nAt high energies, the unpolarized cross-section\\ $\\sigma^{\\mathrm{Born}}_{\\mathrm{unpol}}$\nis dominated by transverse \\mathswitchr W~bosons,\nand all polarized cross-sections\\ involving two transverse \\mathswitchr W~bosons are of the\nsame order-of-magnitude.\nClose to threshold the differential and integrated \ncross-sections\\ for all polarization configurations vanish like $\\beta$.\nNumerical values for the lowest-order cross-sections\\ can be found in \n\\refta{table_born}.\n\\begin{table}\n\\footnotesize\n\\begin{center}\n\\arraycolsep 6pt\n$$\\begin{array}{|c|c||c|c|c|c|c|c|}\n\\hline\n\\sqrt{s}\/\\mathrm{GeV}\n & \\theta &\n{\\mathrm{unpol}} & {{\\pm\\pm}\\mathrm{TT}} & {{\\pm\\pm}\\mathrm{LL}} &\n{{\\pm\\mp}\\mathrm{TT}} & {{\\pm\\mp}\\mathrm{LL}} & {{\\pm\\mp}\\mathrm{(LT+TL)}} \\\\\n\\hline\\hline\n\\phantom{0}500 & \\phantom{0}0^\\circ<\\theta<180^\\circ & \n77.6 &  82.2 & 6.10\\times 10^{-2} & 70.2 & 9.99 \\times 10^{-1} & \n1.69 \\phantom{{}\\times 10^{-1}} \\\\\n\\cline{2-8}\n& 20^\\circ<\\theta<160^\\circ & \n36.7 & 42.7 & 3.17\\times 10^{-2}& 28.2 & 9.89\\times 10^{-1}& \n1.49\\phantom{{}\\times 10^{-1}} \\\\\n\\hline\\hline\n1000 & \\phantom{0}0^\\circ<\\theta<180^\\circ & \n80.1 & 82.8 & 3.54 \\times 10^{-3} & 76.9 & 2.52\\times 10^{-1} & \n1.70\\times 10^{-1}\\\\\n\\cline{2-8}\n& 20^\\circ<\\theta<160^\\circ & \n14.2 & 16.8 &7.18 \\times 10^{-4}& 11.2 & 2.44\\times 10^{-1} & \n1.21\\times 10^{-1}\\\\\n\\hline\\hline\n2000 & \\phantom{0}0^\\circ<\\theta<180^\\circ & \n80.6 & 81.6 & 2.14\\times 10^{-4}& 79.5 & 6.41\\times 10^{-2}& \n1.50\\times 10^{-2}\\\\\n\\cline{2-8}\n& 20^\\circ<\\theta<160^\\circ & \n4.07& 4.84 & 1.27\\times 10^{-5}& 3.23 & 6.11\\times 10^{-2}& \n8.26\\times 10^{-3}  \\\\\n\\hline\n\\end{array}$$\n\\caption{Lowest-order integrated cross-sections\\ in pb for several polarizations}\n\\label{table_born}\n\\end{center}\n\\end{table}\n\nFigures \\ref{fi:difcspp} and \\ref{fi:difcsmm} show \nthe angular distributions of the differential\n lowest-order cross-sections\\ for various polarizations at \n $E_{\\mathrm{CMS}} = 500$, $1000$ and $2000$ GeV.\nThe cross-sections\\ involving transverse W bosons are characterized by the \n$t$- and $u$-channel poles in the forward and backward directions,\nrespectively. With increasing energy they increase in the very forward and\nbackward direction proportional to $s$ but decrease in the central angular\nregion proportional to $1\/s$.\nThe respective behavior of \n$({{{\\mathrm{d}}}\\sigma}\/{{{\\mathrm{d}}}\\Omega})^{\\mathrm{Born}}_{\\pm\\pm{\\mathrm{L}}\\rL}$\nis $1\/s$ and $1\/s^3$.\nThe cross-sections\\ $({{{\\mathrm{d}}}\\sigma}\/{{{\\mathrm{d}}}\\Omega})^{\\mathrm{Born}}_{\\pm\\mp{{\\mathrm{L}}\\rL}}$\nand $({{{\\mathrm{d}}}\\sigma}\/{{{\\mathrm{d}}}\\Omega})^{\\mathrm{Born}}_{\\pm\\mp{({\\mathrm{L}}{\\mathrm{T}}+{\\mathrm{T}}{\\mathrm{L}})}}$\nvanish in the forward and backward direction. \nWhile $({{{\\mathrm{d}}}\\sigma}\/{{{\\mathrm{d}}}\\Omega})^{\\mathrm{Born}}_{\\pm\\mp{{\\mathrm{L}}\\rL}}$\nreach their maxima at $90^\\circ$ and decrease proportional to $1\/s$\nfor all angles,\n$({{{\\mathrm{d}}}\\sigma}\/{{{\\mathrm{d}}}\\Omega})^{\\mathrm{Born}}_{\\pm\\mp{({\\mathrm{L}}{\\mathrm{T}}+{\\mathrm{T}}{\\mathrm{L}})}}$\npossess maxima at $|\\!\\cos\\theta| = \\beta$ decreasing\nproportional to $1\/s$ and relative minima at\n$\\theta = 90^\\circ$ decreasing proportional to $1\/s^2$.\n   \n\\section{Radiative corrections}\n\\label{se:RC}\n\n\\subsection{Non-linear gauge fixing}\n\nWe have performed the calculation of the radiative corrections\nin tHF gauge\nand a NL gauge with the gauge-fixing term given in equation\n\\refeq{nl} applying the complete on-shell renormalization\nscheme in both cases \\cite{ad&mex}.\nAs pointed out in Sect.~2 the $\\phi^\\pm W^\\mp A$ vertices\nvanish in NL gauge. As a consequence \nthe $\\phi$ self-energy\\ and the $\\phi W$ mixing energy do not contribute, and \nthe number of vertex and box diagrams is reduced from 441 in \ntHF gauge to 268 in NL gauge (for one fermion generation).\n\nFurthermore, the analytical expressions for the\n$W^\\pm \\overline{u}^\\pm u^{A,Z}$\nvertices (with $\\bar u$, $u$ denoting the Fadeev--Popov ghost fields)\nare proportional to the W-boson momentum\nin NL gauge and thus vanish for on-shell W~bosons.\nFor this reason most of the box\nand vertex diagrams with internal ghost fields vanish.\nAs the corrections to the $AAA$ and $AAZ$ vertices vanish in both gauges,\nthe number of non-vanishing vertex and box diagrams reduces to 365\nin tHF gauge and to 168 in NL gauge.\nMoreover, many diagrams have a simpler structure in NL gauge.\n\nIn order to determine the counterterms\\ necessary for renormalization\none has to calculate the self-energies.  Here we list the differences\n$\\Delta\\Sigma = \\Sigma^{\\mathrm{NL}} -\\Sigma^{\\mathrm{tHF}}$ between the self-energies\\ in\nNL gauge and the ones in tHF gauge.\nThe transverse parts of the latter can be found e.g.~in Ref.~\\cite{ad&mex}.\nFor the transverse part of the W self-energy\\ we find\n\\begin{equation}\n\\Delta\\Sigma^{WW}_{{\\mathrm{T}}} = \n \\frac{\\alpha}{2 \\pi} \\bigl(k^2 - \\mathswitch {M_\\PW}^2\\bigr)\n       \\biggl[B_0\\bigl(k^2,0,\\mathswitch {M_\\PW}\\bigr) + \n             \\frac{\\mathswitch {c_\\PW}^2}{\\mathswitch {s_\\PW}^2} B_0\\bigl(k^2,\\mathswitch {M_\\PZ},\\mathswitch {M_\\PW}\\bigr)\n       \\biggr] \\; ,\n\\end{equation}\nand for its longitudinal part\n\\begin{eqnarray}\n\\Delta\\Sigma^{WW}_{{\\mathrm{L}}} &=& \n- \\frac{\\alpha}{4 \\pi} \\biggl\\{\n      \\frac{\\mathswitch {c_\\PW}^2}{\\mathswitch {s_\\PW}^2} \\bigl(5 k^2 + 5 \\mathswitch {M_\\PZ}^2 - 3 \\mathswitch {M_\\PW}^2\\bigr) \n                                 B_0\\bigl(k^2,\\mathswitch {M_\\PW},\\mathswitch {M_\\PZ}\\bigr) +\n          \\bigl(5 k^2 - 3 \\mathswitch {M_\\PW}^2\\bigr) B_0\\bigl(k^2,\\mathswitch {M_\\PW},0\\bigr) \n\\nonumber \\\\\n&&\\phantom{\\frac{- \\alpha}{4 \\pi} \\biggl\\{}\n+ \\frac{5}{\\mathswitch {s_\\PW}^2}\\mathswitch {M_\\PW}^2 \\bigl[\n             B_0\\bigl(0,0,\\mathswitch {M_\\PW}\\bigr) - B_0\\bigl(0,0,\\mathswitch {M_\\PZ}\\bigr)\\bigr]\n - \\frac{2}{\\mathswitch {s_\\PW}^2}k^2\n       \\biggr\\}\\;,\n\\end{eqnarray}\nwhere $B_0$ is the scalar one-loop two-point function \\cite{ad&mex,scalar}.\nThe differences for the self-energies\\ involving neutral gauge bosons can be given \nin a compact way\n\\begin{eqnarray}\n\\Delta\\Sigma^{BB'}_{{\\mathrm{T}}} &=& \n- \\frac{\\alpha}{2 \\pi} f^{ B B'} \n             \\bigl(2 k^2 - M^2_{ B} - M^2_{ B'}\\bigr)\n               B_0\\bigl(k^2,\\mathswitch {M_\\PW},\\mathswitch {M_\\PW}\\bigr) \\; ,\n\\\\\n\\Delta\\Sigma^{BB'}_{{\\mathrm{L}}} &=&\n\\frac{\\alpha}{2 \\pi} f^{B B'}\n             \\bigl(M^2_{B} + M^2_{B'}\\bigr)\n               B_0\\bigl(k^2,\\mathswitch {M_\\PW},\\mathswitch {M_\\PW}\\bigr) \\; ,\n\\end{eqnarray}\nwith ${B}^{(\\prime)} = A,Z$ and\n\\begin{equation}\nf^{A A} = 1, \\qquad\nf^{A Z} = -\\frac{\\mathswitch {c_\\PW}}{\\mathswitch {s_\\PW}}, \\qquad\nf^{ZZ} = \\frac{\\mathswitch {c_\\PW}^2}{\\mathswitch {s_\\PW}^2} \\;.\n\\end{equation}\n\nNote that the differences for the transverse parts of the W, Z, and $A$ \nself-energies\\ are proportional to $(k^2 - M_{\\mathswitchr W,\\mathswitchr Z}^2)$ and $k^2$,\nrespectively.\n\n\\subsection{Inventory of ${\\cal O}(\\alpha)$ corrections}\n\nIn the following we list the virtual corrections,\ni.e.\\ the contributions to $\\delta\\cal M$, in NL gauge.\nWe adopt the conventions of\nRef.~\\cite{ad&mex},\nwhere the necessary explicit results for the transverse parts of the \nself-energies\\ and the renormalization constants can be found.\nBecause of the length of our results we do not explicitly write down the\nanalytic expressions.\n\nOwing to our renormalization scheme,\nwe have to deal with self-energy\\ insertions only into the internal\nlines of the tree diagrams of Fig.{} 1.\nThese result in the following contribution to the invariant matrix\nelement\n\\begin{eqnarray}\n\\delta {{\\cal{M}}}_{\\mathrm{self}} &=& 4\\pi\\alpha\\Biggl\\{\n\\frac{2s}{(\\mathswitch {M_\\PW}^2-t)^2} \\Sigma^{WW}_{{\\mathrm{T}}}(t) {\\cal{M}}_{0,t} \n+ \\frac{2s}{(\\mathswitch {M_\\PW}^2-u)^2} \\Sigma^{WW}_{{\\mathrm{T}}}(u){\\cal{M}}_{0,u} \\\\\n&& \\phantom{4\\pi\\alpha\\biggl(}\n{}+\\frac{1}{t} \n\\Bigl(\\Sigma^{WW}_{{\\mathrm{T}}}(t) - \\Sigma^{WW}_{{\\mathrm{L}}}(t)\\Bigr){\\cal{M}}_{0000}^{(t)} \n+\\frac{1}{u} \n\\Bigl(\\Sigma^{WW}_{{\\mathrm{T}}}(u) - \\Sigma^{WW}_{{\\mathrm{L}}}(u)\\Bigr){\\cal{M}}_{0000}^{(u)} \n\\Biggr\\} .  \\nonumber\n\\end{eqnarray}\n\n\\begin{figure}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(160,168)\n\\put(-2,-55){\\special{psfile=aaww.aww.ps voffset=0 hoffset=0}}\n\\end{picture}\n\\caption{The $t$-channel diagrams for the upper $AWW^*$ vertex }\n\\label{fi:AWW}\n\\end{figure}\n\nFigure \\ref{fi:AWW} shows the $t$-channel graphs for the upper\n$A W W^*$ vertex (asterics denote off-shell fields);\nthe diagrams (h), (i), (k), (l), and (n) vanish for on-shell \nexternal photons and \\mathswitchr W~bosons.\nThe diagrams for the lower $A W W^*$ vertex can be\nconstructed in an analogous way, and the $u$-channel diagrams are obtained\nvia crossing, i.e.\\\nthe interchange of the two external photons.\n\nThe $AAA^*$ and $AAZ^*$ vertex corrections vanish according\nto Yangs theorem \\cite{ya49} and because the virtual $A$ and $Z$ are \ncoupled to a conserved current.\nThus, the only $s$-channel vertex corrections, which contribute to \n$\\delta{\\cal M}$, are the Higgs-resonant \n$AAH^*$-vertex graphs shown in \\reffi{fi:AAH}. For the graphs (a)--(d)\nalso crossed ones exist.  The $AAH^*$ corrections are discussed in the\nnext subsection.\n\nThe box diagrams are shown in \\reffis{fi:box1} and \\ref{fi:box2}.\nWhile to each diagram in \\reffi{fi:box1} a crossed partner diagram\ncorresponds, those in \\reffi{fi:box2}  are symmetric under crossing.\nFor on-shell external bosons the graphs (h) and (i) of \\reffi{fi:box1}\nand the graphs (e) and (f) of \\reffi{fi:box2} vanish.\n\\begin{figure}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(160,165)(0,0)\n\\put(-34,-113){\\special{psfile=aaww.box1.ps voffset=0 hoffset=0}}\n\\end{picture}\n\\caption{Non-crossing-symmetric box diagrams}\n\\label{fi:box1}\n\\end{figure}\n\\begin{figure}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(160,98)(0,0)\n\\put(+3,-179){\\special{psfile=aaww.box2.ps voffset=0 hoffset=0}}\n\\end{picture}\n\\caption{Crossing-symmetric box diagrams}\n\\label{fi:box2}\n\\end{figure}\n\nThe renormalization is performed in the on-shell renormalization\nscheme.  Evaluation of the counterterms\\ diagrams yields\n\\begin{eqnarray}\n\\delta{\\cal M}_{\\mathrm{counter}} &=&\n{\\cal M}_{\\mathrm{Born}} \\biggl(\n2 \\delta { Z}_{ e} + \\delta { Z}_{ W} +\n\\delta { Z}_{A A} - \\frac{\\mathswitch {c_\\PW}}{\\mathswitch {s_\\PW}} \\delta Z_{ZA} \\biggr)\n\\nonumber \\\\ && {}-  8\\pi\\alpha   \n\\left(\\frac{s\\delta \\mathswitch {M_\\PW}^2}{(\\mathswitch {M_\\PW}^2-t)^2}{\\cal{M}}_{0,t} \n+ \\frac{s\\delta \\mathswitch {M_\\PW}^2}{(\\mathswitch {M_\\PW}^2-u)^2}{\\cal{M}}_{0,u}  \\right) .\n\\end{eqnarray}\nIn this context we mention that the massive gauge-boson sector does not\nbreak electromagnetic gauge invariance if the NL gauge fixing\n\\refeq{nl} is applied. As a consequence on-shell photons do not mix with\n\\mathswitchr Z\\ bosons rendering the counterterm\\ $\\delta Z_{ZA}$ zero,\n\\begin{equation}\n\\delta Z_{ZA}=2\\frac{\\Sigma^{AZ}_{\\rm T}(0)}{\\mathswitch {M_\\PZ}^2}=0.\n\\end{equation}\nThe charge renormalization constant $\\delta Z_e$ is then given by\n\\cite{ad&mex}\n\\begin{equation}\n\\delta Z_e = -\\frac{1}{2}\\delta Z_{AA} =\n\\left.\\frac{1}{2}\\frac{\\partial\\Sigma^{AA}_{\\rm T}(k^2)}{\\partial k^2}\n\\right|_{k^2=0},\n\\label{eq:dze}\n\\end{equation}\nso that the complete counterterm\\ contribution to the matrix element ${\\cal{M}}$\nreduces to \n\\begin{eqnarray}\n\\delta{\\cal M}^{\\mathrm{NL}}_{\\mathrm{counter}} &=&\n{\\cal{M}}_{\\mathrm{Born}} \\delta { Z}_{ W} \n-  8\\pi\\alpha   \n\\left(\\frac{s\\delta \\mathswitch {M_\\PW}^2}{(\\mathswitch {M_\\PW}^2-t)^2}{\\cal{M}}_{0,t} \n+ \\frac{s\\delta \\mathswitch {M_\\PW}^2}{(\\mathswitch {M_\\PW}^2-u)^2}{\\cal{M}}_{0,u}  \\right) .\n\\end{eqnarray}\n\n\\subsection{Higgs resonance}\nThe Higgs resonance in $\\gamma\\gamma\\to\\PWp\\PWm$\nwas discussed extensively in the literature;\nsee e.g.\\ \\citeres{Mo94,Ve94}.\nSo we restrict ourselves to the listing of our results for the \nHiggs-resonant graphs.\n\nThe Higgs-resonant part of the process is caused by the graphs\nof \\reffi{fi:AAH}.\n\\begin{figure}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(160,63)(0,0)\n\\put(16,-133){\\special{psfile=aaww.aah.ps voffset=00 hoffset=0}}\n\\end{picture}\n\\caption{The $AAH^*$ vertex diagrams}\n\\label{fi:AAH}\n\\end{figure}\nThese yield a contribution of the form (compare \\citere{Ve94})\n\\begin{eqnarray}\n\\delta {\\cal{M}}_{AAH^*} = \\frac{F^H(s)}{s-\\mathswitch {M_\\PH}^2}\n{\\cal{M}}_{0000} \n\\label{eq_higgs}\n\\end{eqnarray}\nwith\n\\strut\n\\begin{eqnarray}\nF^H(s) &=& -\\frac{\\alpha^2}{\\mathswitch {s_\\PW}^2} \\biggl\\{\n  6 \\mathswitch {M_\\PW}^2 + \\mathswitch {M_\\PH}^2 + \\mathswitch {M_\\PW}^2  C^{(\\mathrm{gauge})}\nC_0\\bigl(s,0,0,\\mathswitch {M_\\PW}^2,\\mathswitch {M_\\PW}^2,\\mathswitch {M_\\PW}^2\\bigr)\n\\nonumber \\\\\n&&\\phantom{\\frac{e^2}{8 \\pi \\mathswitch {s_\\PW}^2} \\biggl\\{}\n-2 \\sum_{f} N_f^c Q_{f}^2m_f^2 \\left[ 2 + (4m_f^2 -s)\n                 C_0\\bigl(s,0,0,m_{f}^2,m_{f}^2,m_{f}^2\\bigr) \\right]  \n\\biggr\\}.\n\\label{eq:FH}\n\\end{eqnarray}\n\\strut\nThe sum in (\\ref{eq:FH}) extends over all massive fermions with charge\n\\pagebreak[2]\n$Q_f$ and color factor $N_f^c$. \nThe coefficient $C^{(\\mathrm{gauge})}$ is gauge-dependent and\nreads\n\\begin{eqnarray}\nC^{(\\mathrm{gauge})} = 12\\mathswitch {M_\\PW}^2 + 2\\mathswitch {M_\\PH}^2 - 8s\n\\end{eqnarray}\nin NL gauge and\n\\begin{eqnarray}\nC^{(\\mathrm{gauge})} = 12\\mathswitch {M_\\PW}^2 +  \\mathswitch {M_\\PH}^2 - 7s\n\\end{eqnarray}\nin tHF gauge.\nNote that $\\delta {\\cal{M}}_{AAH^*}$ vanishes for opposite helicities of the\nincoming photons or outgoing \\mathswitchr W~bosons together with ${\\cal{M}}_{0000}$. \nHence the Higgs resonance is only present for \nphotons and \\mathswitchr W~bosons with equal helicities.\n\nIn the literature \\cite{Bo92,Mo94,Ve94}, the Higgs-boson width has been\nintroduced na{\\\"\\i}vely by the replacement \n\\begin{equation}\n\\frac{F^H(s)}{s-\\mathswitch {M_\\PH}^2} \\rightarrow \\frac{F^H(s)}{s-\\mathswitch {M_\\PH}^2 + i \\mathswitch {M_\\PH}\\Gamma_{\\mathswitchr H}}\n\\end{equation}\nin \\refeq{eq_higgs}.\nOwing to the gauge dependence of $F^H(s)$, this treatment \ndestroys gauge invariance. The violation of gauge invariance occurs at\nthe level of the non-resonant $\\mathswitch{{\\cal{O}}(\\alpha)}$ corrections, which were neglected in\n\\citeres{Bo92,Mo94,Ve94}. Since our main concern are exactly these\ncorrections we have to take care of gauge invariance.\nTo this end we \ndecompose (\\ref{eq_higgs}) into a gauge-invariant resonant part and a \ngauge-dependent non-invariant part and introduce $\\Gamma_\\mathswitchr H$ only \nin the former. This results in the following replacement in (\\ref{eq_higgs})\n\\begin{equation}\n\\frac{F^H(s)}{s-\\mathswitch {M_\\PH}^2} \\rightarrow\n\\frac{F^H(\\mathswitch {M_\\PH}^2)}{s-\\mathswitch {M_\\PH}^2 + i \\mathswitch {M_\\PH}\\Gamma_{\\mathswitchr H}} \n+ \\frac{F^H(s) - F^H(\\mathswitch {M_\\PH}^2)}{s-\\mathswitch {M_\\PH}^2} \\;.\n\\label{eq_higgs2}\n\\end{equation}\nEquations (\\ref{eq_higgs}) and (\\ref{eq_higgs2}) yield a gauge-invariant\namplitude including the finite width in the resonant Higgs contributions.\n\nSince the resonant Higgs contributions are large for $s\\approx\\mathswitch {M_\\PH}^2$,\nwe take also the square of the resonant part of the\nmatrix element into account in the numerical analysis [compare\n\\refeq{eq:dsido}].\n\nFor a calculation with order $\\mathswitch{{\\cal{O}}(\\alpha)}$ accuracy also near $s=\\mathswitch {M_\\PH}^2$, \none should take into account the $\\mathswitch{{\\cal{O}}(\\alpha)}$ corrections to the \nHiggs-boson width \\cite{Fl81} and to $F^H(\\mathswitch {M_\\PH}^2)$ in the resonant\ncontribution. Since the\nHiggs resonance is not our main concern, we only take into account the\nlowest-order decay width determined from the imaginary part of the\none-loop Higgs-boson self-energy\\ and \\refeq{eq:FH} for $F^H(\\mathswitch {M_\\PH}^2)$. \n\n\\subsection{Leading corrections}\n\\label{leadrcs}\n\nThe electroweak radiative corrections typically involve leading\ncontributions of universal origin such as the leading-logarithmic QED\ncorrections, corrections arising from the running of $\\alpha$,\ncorrections associated with large top-quark or Higgs-boson \nmasses, and the Coulomb\nsingularity at threshold for the production of a pair of charged\nparticles.\n\nWe first discuss the leading weak corrections:\n\\begin{itemize}\n\\item\nFor $\\gamma\\gamma\\to\\PWp\\PWm$, the running of $\\alpha$ is not relevant, as the {\\it two} \nexternal\nphotons are on mass shell, i.e.\\  the relevant effective coupling is the\none at zero-momentum transfer. Technically, the large logarithms present\nin the renormalization constant $\\delta Z_e$ of the electron charge are \ncanceled by the corresponding logarithms in the wave-function \nrenormalization constant $\\delta Z_{AA}$ of the external photons, as can\nbe explicitly seen in \\refeq{eq:dze}.\n\\item\nThe Higgs-mass-dependent corrections have been discussed in detail in\n\\citere{HH}. \nIn the heavy-Higgs limit, $\\mathswitch {M_\\PH}\\gg\\sqrt{s}$, no corrections involving \n$\\log(\\mathswitch {M_\\PH}\/\\mathswitch {M_\\PW})$ or $\\mathswitch {M_\\PH}^2\/\\mathswitch {M_\\PW}^2$ arise. Consequently the Higgs-mass\ndependence is small. However, for $\\sqrt{s}\\gg\\mathswitch {M_\\PH}\\gg\\mathswitch {M_\\PW}$ corrections\nproportional to $\\mathswitch {M_\\PH}^2\/\\mathswitch {M_\\PW}^2$ appear for the cross-sections\\ involving\nlongitudinal gauge bosons as a remnant of the unitarity cancellations\n(compare \\citere{eeWWhe}). These give rise to large effects in\nparticular for $\\sigma_{\\pm\\pm{\\mathrm{L}}\\rL}$.\n\\item\nThe situation is similar for the top-dependent corrections.\nAs the lowest-order matrix element is independent of the weak mixing \nangle, no universal corrections proportional to $\\mathswitch {m_\\Pt}^2\/\\mathswitch {M_\\PW}^2$ arise from\nrenormalization. It can be easily derived by power counting that such\nterms do also not result from loop diagrams in the heavy top limit \n$\\mathswitch {m_\\Pt}\\gg\\sqrt{s}$. A more accurate analysis\nreveals that even no terms involving $\\log(\\mathswitch {m_\\Pt}\/\\mathswitch {M_\\PW})$ occur in this limit. \nOn the other hand, for $\\sqrt{s}\\gg\\mathswitch {m_\\Pt}\\gg\\mathswitch {M_\\PW}$ corrections\nproportional to $\\mathswitch {m_\\Pt}^2\/\\mathswitch {M_\\PW}^2$ appear for longitudinal gauge bosons \n(compare \\citere{eeWWhe}).\n\\end{itemize}\nAll these statements hold in the on-shell renormalization scheme with\n$\\alpha$, \\mathswitch {M_\\PW}\\ and \\mathswitch {M_\\PZ}\\ as input parameters. \nIf the \\mathswitch {M_\\PW}\\ mass is determined from \\mathswitch {G_\\mu}, \ncorrections involving logarithms of\nthe light fermion masses, \\mathswitch {m_\\Pt}, and \\mathswitch {M_\\PH}\\ occur together with the\nuniversal corrections proportional to $\\mathswitch {m_\\Pt}^2\/\\mathswitch {M_\\PW}^2$ associated with the\n$\\rho$ parameter.\n\nThe leading corrections of electromagnetic origin are independent of \nthe renormalization scheme and the input parameters:\n\\begin{itemize}\n\\item\nAs $\\gamma\\gamma\\to\\PWp\\PWm$ involves no light charged external particles,\nno large logarithmic corrections associated with collinear\nphotons show up apart from the region of very high energies, $s\\gg\\mathswitch {M_\\PW}^2$.\nAs a consequence, the photonic corrections are not enhanced with respect\nto the weak corrections.\n\\item\nClose to threshold, the Coulomb singularity gives rise to large effects \nas in any pair-production process of charged particles. \nThese effects can be extracted on general grounds or directly\nfrom the Feynman diagrams. To this end one has to consider all diagrams\nresulting from the lowest-order diagrams with an additional photon\nexchanged between the final state \\mathswitchr W~bosons (\\reffi{fi:coul}). \nIn the limit $\\beta\\ll 1$ one obtains:\n\\begin{equation}\n\\delta\\sigma^{\\mathrm{Coul.}} = \\frac{\\alpha\\pi}{2\\beta} \\sigma^{\\mathrm{Born}}.\n\\label{eq:coul}\n\\end{equation}\nThe $\\beta^{-1}$ correction factor in \\refeq{eq:coul} to the Born\ncross-section\\ near threshold is typical for the pair production of\nstable (on-shell) particles. The generalization to unstable (off-shell)\nparticles can be found in the literature \\cite{coul}.\n\\end{itemize}\n\n\\begin{figure}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(160,30)(0,0)\n\\put(14,-188){\\special{psfile=aaww.coulsing.ps voffset=0 hoffset=0}}\n\\end{picture}\n\\caption{The diagrams that contribute to the Coulomb singularity in\nNL gauge}\n\\label{fi:coul}\n\\end{figure}\n\nAt high energies, $s\\gg\\mathswitch {M_\\PW}^2$, the radiative corrections are dominated \nby terms like\n$(\\alpha\/\\pi)\\log^2(s\/\\mathswitch {M_\\PW}^2)$, which arise from vertex and box diagrams\n(comp.\\ \\citere{eeWWhe}).\nAt $1\\unskip\\,\\mathrm{TeV}$ these are about 10\\%, setting the scale for the (weak)\nradiative corrections at this energy.\n\n\\subsection{Structure of the final result}\n\nFor a consistent treatment of the virtual one-loop radiative corrections\nthe squared transition matrix element has to be expanded in a power series\nof the coupling constant $\\alpha$\n\\begin{equation}\n|{\\cal M}|^2 = |{\\cal M}_{\\mathrm{Born}}|^2 \n+ 2\\mathop{\\mathrm{Re}}\\nolimits\\{\\delta {\\cal M} {\\cal M}^*_{\\mathrm{Born}}\\} + \\hbox{higher orders}.\n\\end{equation}\nThe ${\\cal {O}}(\\alpha)$ correction $\\delta {\\cal M}$ to the matrix element\n${\\cal M}$ is decomposed as in (\\ref{decom}).\nWe do not consider those polarization configurations for which the\nlowest-order matrix element vanishes.\n\nThe invariant functions $F_{ijkl}$ are calculated in terms\nof standard tensor integrals, which are reduced to scalar integrals by the\nprocedure proposed in \\citere{pas79}. The scalar one-loop integrals are\nevaluated using the methods and general results of \\citere{scalar}. Whereas\nUV divergences are regularized dimensionally, we treat IR divergences\nby introducing an infinitesimal photon mass $\\lambda$. The\nartificial $\\lambda$ dependence drops out when soft-photon\nbremsstrahlung is added. \n\nThe cross-section\\ including full ${\\cal {O}}(\\alpha)$ corrections and the \nsquared Higgs-resonant $\\mathswitch{{\\cal{O}}(\\alpha)}$ contributions read\n\\begin{eqnarray}\n\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right) &=&\n\\frac{\\beta}{64\\pi^2s} \\sum_{\\lambda_1 \\lambda_2 \\lambda_3 \\lambda_4}\n\\biggl[\n\\left| {\\cal M}_{\\mathrm{Born}} \\right|^2 (1 + \\delta_{\\mathrm{SB}}) \n+ 2\\mathop{\\mathrm{Re}}\\nolimits\\{\\delta {\\cal M} {\\cal{M}}_{{\\mathrm{Born}}}^*\\}\n+ \\frac{|F^H(\\mathswitch {M_\\PH}^2){\\cal{M}}_{0000}|^2}{(s-\\mathswitch {M_\\PH}^2)^2 + \\Gamma_H^2 \\mathswitch {M_\\PH}^2}\\biggr]\n\\nonumber \\\\*\n&=& \\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right)^{\\mathrm{Born}}(1+\\delta),\n\\label{eq:dsido}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\\delta_{\\mathrm{SB}} &=& -\\frac{\\alpha}{\\pi}\n\\biggl\\{2 \\log \\frac{2 \\Delta E}{\\lambda} +\n\\frac{1}{\\beta} \\log\\biggl(\\frac{1-\\beta}{1+\\beta}\\biggr)\n  +      \\frac{s - 2 \\mathswitch {M_\\PW}^2}{s \\beta} \\biggl[2 \\log \\frac{2 \\Delta E}{\\lambda}\n         \\log\\biggl(\\frac{1-\\beta}{1+\\beta}\\biggr)\n\\nonumber \\\\\n\\phantom{-\\frac{\\alpha}{\\pi}\\biggl\\{}\n&& \n       - 2 {\\mathop{\\mathrm{Li}}\\nolimits}_2 \\biggl(\\frac{1-\\beta}{1+\\beta}\\biggr) + \\frac{1}{2}\n       \\log^2\\biggl(\\frac{1-\\beta}{1+\\beta}\\biggr) + \\frac{\\pi^2}{3}\n      - 2 \\log\\biggl(\\frac{1-\\beta}{1+\\beta}\\biggr)\n      \\log\\biggl(\\frac{2 \\beta}{1+\\beta}\\biggr)\\biggr]\\biggr\\}\n\\end{eqnarray}\ndenotes the soft-photon correction factor, \n$\\Delta E$ is the maximal energy of the emitted photon, \n$F^H$ is given in \\refeq{eq:FH},\nand $\\delta$ is the relative correction.\n\nFor the integrated cross-section\\ $\\sigma$, the relative correction is defined\nanalogously\n\\begin{equation}\n\\sigma = \\int_{\\theta_{\\min}}^{\\theta_{\\max}} {\\mathrm{d}}\\!\\cos\\theta \\int_0^{2\\pi}\n{\\mathrm{d}}\\phi\\,\\left(\\frac{{\\mathrm{d}}\\sigma}{{\\mathrm{d}}\\Omega}\\right) = \\sigma^{\\mathrm{Born}}(1+\\delta).\n\\end{equation}\n\nIn order to ensure the correctness of our results we have performed \nthree different calculations. The corrections were calculated with \n{\\it FeynCalc\\\/} \\cite{fc} both in tHF gauge and NL\ngauge \\refeq{nl}. A further calculation was performed independently with \n{\\it Mathematica\\\/} without using {\\it FeynCalc\\\/} in NL gauge.\nThe results of these various calculations agree numerically within\n8--9 digits for the corrected cross-section.\nMoreover, we have checked that all \nUV and IR singularities cancel, and that the symmetries discussed in\n\\refse{se:notcon}\nhold. Finally, the leading corrections discussed in \\refse{leadrcs} have\nbeen deduced analytically and checked numerically.\n \n\\section{Numerical results}\n\\label{se:num}\n\nFor the numerical evaluation we used the following set of parameters\n\\cite{PDG94}\n\\begin{equation}\n\\begin{array}[b]{lcllcllcl}\n\\alpha &=& 1\/137.0359895 &\n\\mathswitch {G_\\mu} & = & 1.166390 \\times 10^{-5} \\unskip\\,\\mathrm{GeV}^{-2} \\\\[.3em]\n\\mathswitch {M_\\PZ} & = & 91.187\\unskip\\,\\mathrm{GeV}, &\n\\mathswitch {M_\\PH} & = & 250\\unskip\\,\\mathrm{GeV}, &&& \\\\[.3em]\n\\mathswitch {m_\\Pe} & = & 0.51099906\\unskip\\,\\mathrm{MeV},  \\hspace{1.5em} &\nm_{\\mu} & = & 105.65839\\unskip\\,\\mathrm{MeV},  \\hspace{1.5em} &\nm_{\\tau} & = & 1.777\\;\\unskip\\,\\mathrm{GeV}, \\\\[.3em]\n\\mathswitch {m_\\Pu} & = & 46.0\\;\\unskip\\,\\mathrm{MeV}, &\n\\mathswitch {m_\\Pc} & = & 1.50\\;\\unskip\\,\\mathrm{GeV}, &\n\\mathswitch {m_\\Pt} & = & 170\\;\\unskip\\,\\mathrm{GeV}, \\\\[.3em]\n\\mathswitch {m_\\Pd} & = & 46.0\\;\\unskip\\,\\mathrm{MeV}, &\n\\mathswitch {m_\\Ps} & = & 150\\;\\unskip\\,\\mathrm{MeV}, &\n\\mathswitch {m_\\Pb} & = & 4.50\\;\\unskip\\,\\mathrm{GeV}.\n\\end{array}\n\\label{eq:par}\n\\end{equation}\nThe masses of the light quarks are adjusted such that the\nexperimentally measured hadronic vacuum polarization is reproduced\n\\cite{Ei95}. As discussed in the previous section, \nno large logarithms associated with fermion masses enter the\n$\\mathswitch{{\\cal{O}}(\\alpha)}$ corrections for $\\gamma\\gamma\\to\\PWp\\PWm$ in the on-shell renormalization scheme,\nand the fermion mass contributions are only of the order $\\alpha\nm_f^2\/\\mathswitch {M_\\PW}^2$.\nHowever, as the Fermi-constant $\\mathswitch {G_\\mu}$ is empirically much better\nknown than the \\mathswitchr W~mass, \\mathswitch {M_\\PW}\\ is usually calculated from all the other \nparameters using the muon decay width including radiative corrections. \nIn this calculation of \\mathswitch {M_\\PW}\\ all parameters given above enter sensibly.\nIf not stated otherwise, $\\mathswitch {M_\\PW}$ is determined in the following using\nformulae (2.56) and (2.57) of \\citere{wwrev}. The above set of\nparameters yields\n$$\\mathswitch {M_\\PW}=80.333\\unskip\\,\\mathrm{GeV}.$$\n\nAs discussed above, no leading collinear logarithms occur in \n$\\gamma\\gamma\\to\\PWp\\PWm$. Thus, the only source of enhanced photonic corrections are\nthe soft-photon-cut-off-dependent terms which yield the following\nrelative correction\n\\begin{equation}\n\\delta_{\\mathrm{cut}} = -\\frac{2\\alpha}{\\pi} \\log\\frac{\\Delta E}{E} \n\\left(1 - \\frac{s-2\\mathswitch {M_\\PW}^2}{s\\beta}\\log\\frac{1+\\beta}{1-\\beta}\\right).\n\\label{eq:cut}\n\\end{equation}\nWhile these cut-off-dependent terms are \ndefinitely of electromagnetic origin, the complete electroweak \\mathswitch{{\\cal{O}}(\\alpha)}\\ \ncorrections cannot be\nseparated on the basis of Feynman diagrams in a gauge-invariant way.\nSince we are mainly interested in the weak corrections we discard the \ncut-off-dependent terms \\refeq{eq:cut} \nand consider the rest as a suitable measure of\nthe weak corrections for the process at hand. The elimination of the \ncut-off-dependent terms can be achieved simply by setting the soft-photon\ncut-off energy equal to the beam energy. If not stated otherwise, the\ncorrection $\\delta$ stands in the following for\nthe complete soft-photonic and virtual electroweak corrections\nas defined in \\refeq{eq:dsido} for $\\Delta E = E$.\n\nFigure \\ref{fi:intcs10} shows the corrections to the total cross-sections\\ integrated \nover $10^\\circ \\leq \\theta \\leq 170^\\circ$ for different boson polarizations.\n\\begin{figure}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(160,210)(0,0)\n\\put(0,110){\\input{aaww0.tot10.tex}}\n\\put(0,05){\\input{aaww.tot10.tex}}\n\\end{picture}\n\\caption{Integrated lowest-order cross-sections\\ and corresponding relative \ncorrections for several polarizations with an angular cut \n         $10^\\circ \\leq \\theta \\leq 170^\\circ$}\n\\label{fi:intcs10}\n\\end{figure}\nThe dominating channels involving transverse \\mathswitchr W~bosons get \ncorrections which almost coincide with each other as well as\nthe unpolarized case and reach roughly $-20\\%$ at $\\sqrt{s}=2\\unskip\\,\\mathrm{TeV}$.\nFor $\\theta_{\\mathrm{cut}} = 10^\\circ$ the corrections to $\\sigma_{\\pm\\mp{\\mathrm{L}}\\rL}$ are \nsimilar, and those to $\\sigma_{\\pm\\mp({\\mathrm{L}}{\\mathrm{T}}+{\\mathrm{T}}{\\mathrm{L}})}$ are only slightly larger.\nThe corrections to $\\sigma_{\\pm\\pm{\\mathrm{L}}\\rL}$ are \ncompletely different. At low energies they are dominated by the Higgs\nresonance, at high energies \nby corrections proportional to $\\mathswitch {M_\\PH}^2\/\\mathswitch {M_\\PW}^2$ which are additionally\nenhanced owing to the suppression of the corresponding lowest-order cross-section.\nThis cross-section, which is also most sensitive to a very heavy Higgs boson, has been\ndiscussed in detail in \\citere{HH}. \nNote that owing to helicity conservation only the cross-sections\\ with\nequal photon and \\mathswitchr W~boson helicities are affected by the Higgs resonance.\n\nImposing a more stringent angular cut $20^\\circ <\\theta <160^\\circ$ \nto the phase-space integration, the corrections become\nlarger at high energies for all polarizations \ninvolving $t$- and $u$-channel poles\nand reach about $-35\\%$ at $\\sqrt{s}=2\\unskip\\,\\mathrm{TeV}$  (\\reffi{fi:intcs20}).\n\\begin{figure}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(160,210)(0,0)\n\\put(0,110){\\input{aaww0.tot20.tex}}\n\\put(0,05){\\input{aaww.tot20.tex}}\n\\end{picture}\n\\caption{Same as in \\protect\\reffi{fi:intcs10} but with an angular cut \n         $20^\\circ \\leq \\theta \\leq 160^\\circ$}\n\\label{fi:intcs20}\n\\end{figure}\nThis is due to the fact that after cutting off the dominant forward \nand backward peaks we are left\nwith a region in phase space where the influence of the radiative\ncorrections becomes more important.\nThe corrections to the other cross-sections, in particular to $\\sigma_{\\pm\\mp{\\mathrm{L}}\\rL}$,\nare hardly affected.\n \nIn \\reffis{fi:difcspp} and \\ref{fi:difcsmm} we\nshow the corrections to the differential cross-sections\\ for $\\sqrt{s} =\n0.5$, 1 and $2\\unskip\\,\\mathrm{TeV}$.\n\\begin{figure}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(160,210)(-1,0)\n\\put(26,168){\\makebox(0,0)[l]{{\\footnotesize unpol}}}\n\\put(26,78){\\makebox(0,0)[l]{{\\footnotesize unpol}}}\n\\put(77,168){\\makebox(0,0)[l]{{\\footnotesize $\\pm\\pm$TT}}}\n\\put(77,78){\\makebox(0,0)[l]{{\\footnotesize $\\pm\\pm$TT}}}\n\\put(130,168){\\makebox(0,0)[l]{{\\footnotesize $\\pm\\pm$LL}}}\n\\put(130,28){\\makebox(0,0)[l]{{\\footnotesize $\\pm\\pm$LL}}}\n\\put(-10,110){\\input{aaww0.unpol}\n\\hspace*{-1.7cm}\\input{aaww0.22TT}\n\\hspace*{-1.7cm}\\input{aaww0.22LL}}\n\\put(-10,0){\n\\input{aaww.unpol}\n\\hspace*{-1.7cm}\\input{aaww.22TT}\n\\hspace*{-1.7cm}\\input{aaww.22LL}}\n\\end{picture}\n\\caption{Differential lowest-order cross-sections\\ and relative \ncorrections for the unpolarized cross-section\\ and the cross-sections\\ with equal photon helicities}\n\\label{fi:difcspp}\n\\end{figure}%\n\\begin{figure}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(160,210)(-1,0)\n\\put(21,164){\\makebox(0,0)[l]{{\\footnotesize $\\pm$$\\mp$(LT+TL)}}}\n\\put(21,78){\\makebox(0,0)[l]{{\\footnotesize $+$$-$(LT+TL)}}}\n\\put(77,164){\\makebox(0,0)[l]{{\\footnotesize $\\pm\\mp$TT}}}\n\\put(77,78){\\makebox(0,0)[l]{{\\footnotesize $+$$-$TT}}}\n\\put(130,164){\\makebox(0,0)[l]{{\\footnotesize $\\pm\\mp$LL}}}\n\\put(130,78){\\makebox(0,0)[l]{{\\footnotesize $+$$-$LL}}}\n\\put(-10,110){\\input{aaww0.21_LT_TL}\n\\hspace*{-1.7cm}\\input{aaww0.21TT}\n\\hspace*{-1.7cm}\\input{aaww0.21LL}}\n\\put(-10,0){\n\\input{aaww.21_LT_TL}\n\\hspace*{-1.7cm}\\input{aaww.21TT}\n\\hspace*{-1.7cm}\\input{aaww.21LL}}\n\\end{picture}\n\\caption{Differential lowest-order cross-sections\\ and relative \ncorrections for the cross-sections\\ with opposite photon helicities}\n\\label{fi:difcsmm}\n\\end{figure}\nWhenever the differential cross-section\\ is sizable, $\\delta$ is of the\norder of $10\\%$. The corrections are in particular small in the forward\nand backward direction for all cross-sections\\ that involve $t$- and $u$-channel\npoles in lowest order. On the other hand, the corrections get very large\nwhen the lowest-order cross-section\\ is suppressed or tends to zero, in particular\nfor ${\\mathrm{d}}\\sigma_{\\pm\\pm{\\mathrm{L}}\\rL}\/{\\mathrm{d}}\\Omega$ \nat high energies and intermediate scattering angles.\nThe maximal corrections are usually reached for central values of the\nscattering angle.\nIn accordance with the discussion in \\refse{se:notcon}, the corrections are\nforward-backward symmetric for equal photon helicities.\nFor opposite photon helicities they\ninclude an asymmetric contribution originating from box diagrams and\n$AWW$ vertex corrections involving fermion loops.\nThe corrections for two negative helicity photons are equal to those for\ntwo positive helicity photons. As a consequence of Bose symmetry, the\ncorrections to ${\\mathrm{d}}\\sigma_{+-}\/{\\mathrm{d}}\\Omega$ are obtained from those to \n${\\mathrm{d}}\\sigma_{-+}\/{\\mathrm{d}}\\Omega$ upon \nexchanging $u$ and $t$, i.e.\\  $\\theta\\leftrightarrow 180^\\circ-\\theta$. \nThus, the unpolarized cross-section\\ is forward--backward-symmetric.\n \n\\begin{table}\n\\footnotesize\n\\newdimen\\digitwidth\n\\setbox0=\\hbox{0}\n\\digitwidth=\\wd0\n\\catcode`!=\\active \n\\def!{\\kern\\digitwidth}\n\\newdimen\\minuswidth\n\\setbox0=\\hbox{$-$}\n\\minuswidth=\\wd0\n\\catcode`?=\\active \n\\def?{\\kern\\minuswidth}\n\\begin{center}\n\\arraycolsep 6pt\n$$\\begin{array}{|c|c||c|c|c|c|c|c|}\n\\hline\n\\sqrt{s}\/\\mathrm{GeV} & \\theta & \\sigma^{\\mathrm{Born}}\/\\mathrm{pb} &\n\\delta_{\\Delta E = 0.1E} \/\\% & \n\\delta_{\\mathrm{cut}} \/\\% & \n\\delta_{\\Delta E = E} \/\\% & \n\\delta_{\\mathrm{bos}}\/\\% &\n\\delta_{\\mathrm{ferm}}\/\\% \\\\\n\\hline\\hline\n       & !5^\\circ &  98.13  & !?0.02 &  -2.79 & ?!2.81 & ?!1.49 &  ?1.32 \\\\\n \\cline{2-8}\n       & 20^\\circ &  26.04  & !-2.68 &  -2.79 & ?!0.11 & !-0.08 &  ?0.19 \\\\\n\\cline{2-8}\n  !500 & 90^\\circ &  0.724  & -10.79 &  -2.79 & !-8.00 & !-5.62 &  -2.38 \\\\\n\\cline{2-8}\n       & !0^\\circ< \\theta <180^\\circ &  77.55     & !-3.38 &   -2.79 \n& !-0.59 & !-0.65 &   ?0.06  \\\\\n\\cline{2-8}\n       & 10^\\circ< \\theta <170^\\circ &  60.74     & !-4.27 &   -2.79 &\n  !-1.48 & !-1.21 &   -0.27  \\\\\n\\cline{2-8}\n       & 20^\\circ< \\theta <160^\\circ &  36.67     & !-6.06 &  -2.79 \n& !-3.27 & !-2.39 &   -0.89 \\\\\n\\hline\\hline\n       & !5^\\circ &  291.9  & !-2.06 &  -4.31 & ?!2.25 & ?!1.04 &  ?1.21 \\\\\n\\cline{2-8}\n       & 20^\\circ &  15.61  & -11.90 &  -4.31 & !-7.59 & !-6.37 &  -1.22 \\\\\n\\cline{2-8}\n 1000  & 90^\\circ &  0.193  & -31.64 &  -4.31 & -27.33 & -21.93 &  -5.40 \\\\\n\\cline{2-8}\n       & !0^\\circ< \\theta <180^\\circ &  80.05    & !-7.08 &  -4.31 \n& !-2.77 & !-2.71 &   -0.06 \\\\ \n\\cline{2-8}\n       & 10^\\circ< \\theta <170^\\circ &  37.06    & -12.26 &  -4.31 \n& !-7.95 & !-6.65 &   -1.30 \\\\\n\\cline{2-8}\n       & 20^\\circ< \\theta <160^\\circ & 14.16     & -19.29 &  -4.31 \n& -14.98 & -12.20 &  -2.78 \\\\\n\\hline\\hline\n       & !5^\\circ &  418.8 & !-7.14 &  -5.80 & !-1.33 & !-1.59 &  ?0.25 \\\\\n\\cline{2-8}\n       & 20^\\circ &  5.163 & -30.31 &  -5.80 & -24.51 & -20.96 &  -3.55 \\\\\n\\cline{2-8}\n 2000  & 90^\\circ &  0.049 & -59.59 &  -5.80 & -53.78 & -45.47 &  -8.32 \\\\ \n\\cline{2-8}\n       & !0^\\circ< \\theta <180^\\circ &  80.59     & !-9.85 &  -5.80 \n& !-4.04 & !-3.95 &  -0.09 \\\\\n\\cline{2-8}\n       & 10^\\circ< \\theta <170^\\circ &  14.14     & -27.15 &  -5.80 \n& -21.35 & -18.34 &  -3.01 \\\\\n\\cline{2-8}\n       & 20^\\circ< \\theta <160^\\circ &  4.068     & -41.22 &  -5.80 \n& -35.41 & -30.12 &  -5.29 \\\\\n\\hline\n\\end{array}$$\n\\caption{Lowest-order cross-sections\\ and relative corrections for unpolarized\nparticles}\n\\label{ta:born}\n\\end{center}\n\\end{table}\nIn \\refta{ta:born} we list the unpolarized cross-section\\ and the corresponding \ncorrections for several energies and scattering angles. We include the\ncorrections for a soft-photon-energy cut-off $\\Delta E = 0.1E$, the \ncut-off-dependent corrections $\\delta_{\\mathrm{cut}}$ from \\refeq{eq:cut}, and the \nindividual (gauge-invariant) fermionic $\\delta_{\\mathrm{ferm}}$ \nand bosonic corrections  $\\delta_{\\mathrm{bos}}$.\nThe fermionic corrections consist of all loop diagrams and counterterm\\ \ncontributions involving fermion loops, all other contributions form the \nbosonic corrections.\nThe fermionic corrections stay below 5--10\\% even for high energies. \nOn the other hand, the bosonic contributions are responsible for the\nlarge corrections at high energies, in\nparticular in the central angular region. \n \nIn \\citere{Ye91} various observables have been investigated in view of\ntheir sensitivity to anomalous couplings, \ninvolving the total cross-section\\ and the following ratios%\n\\footnote{Note that we do not perform a convolution with a realistic\n\\label{fo:3}%\nphoton spectrum but consider the incoming photons as monochromatic.}\n\\begin{eqnarray}\nR_{\\mathrm{IO}}&=& \\frac{\\sigma(|\\!\\cos\\theta|<0.4)}{\\sigma(|\\!\\cos\\theta|<0.8)}, \\\\\nR_{\\mathrm{LT}}&=& \\frac{\\sigma_{{\\mathrm{L}}\\rL}}{\\sigma_{{\\mathrm{T}}\\rT}}, \\\\\nR_{\\mathrm{02}}&=& \\frac{\\sigma_{++}}{\\sigma_{+-}} .\n\\end{eqnarray}\nWe list the lowest-order predictions together with the $\\mathswitch{{\\cal{O}}(\\alpha)}$-corrected\nones and the relative corrections for these observables in\n\\refta{ta:obs} using $|\\!\\cos\\theta_{\\mathrm{cut}}|=0.8$.\n\\begin{table}\n\\newdimen\\digitwidth\n\\setbox0=\\hbox{0}\n\\digitwidth=\\wd0\n\\catcode`!=\\active \n\\def!{\\kern\\digitwidth}\n\\newdimen\\minuswidth\n\\setbox0=\\hbox{$-$}\n\\minuswidth=\\wd0\n\\catcode`?=\\active \n\\def?{\\kern\\minuswidth}\n\\begin{center}\n\\arraycolsep 6pt\n$$\\begin{array}{|c|c||c|c|c|c|}\n\\hline\n\\sqrt{s}\/\\mathrm{GeV} & & \\sigma\/\\mathrm{pb} &\nR_{\\mathrm{IO}} & R_{\\mathrm{LT}} & R_{02} \\\\\n\\hline\\hline\n       & \\mathrm{Born~level}  & ?15.74  & ?0.265  &  0.0308  & ?1.934 \\\\\n\\cline{2-6}\n !500  & \\mathrm{corrected}   & ?14.82  & ?0.259  &  0.0325  & ?1.950 \\\\\n\\cline{2-6}\n       & \\mathrm{corrections\/\\%} & !-5.83  & !-2.02  &  !5.43!  & ?0.78! \\\\\n\\hline\\hline\n       & \\mathrm{Born~level}  & ?4.659  & ?0.241  &  0.0235  & ?2.229 \\\\\n\\cline{2-6}\n 1000  & \\mathrm{corrected}   & ?3.617  & ?0.227  &  0.0276  & ?2.184 \\\\\n\\cline{2-6}\n       & \\mathrm{corrections\/\\%} & -22.36  & !-5.64  &  17.08!  & -2.05! \\\\\n\\hline\\hline\n       & \\mathrm{Born~level}  & ?1.218  & ?0.234  &  0.0220  & ?2.307 \\\\\n\\cline{2-6}\n 2000  & \\mathrm{corrected}   & ?0.647  & ?0.207  &  0.0321  & ?2.168 \\\\\n\\cline{2-6}\n       & \\mathrm{corrections\/\\%} & -46.86  &  -11.53  &  46.11!  & -6.02!\\\\\n\\hline\n\\end{array}$$\n\\caption{Lowest-order predictions and corresponding  corrections for various\nobservables and $|\\!\\cos\\theta_{\\mathrm{cut}}|=0.8$}\n\\label{ta:obs}\n\\end{center}\n\\end{table}\n\nIn \\reftas{ta:MTvar.mw} -- \\ref{ta:MHvar.gf} we show the variation of the \nSM corrections with the top-quark and Higgs-boson masses at\n$\\sqrt{s}=500\\unskip\\,\\mathrm{GeV}$ \nin per cent of the cross-section\\ for our standard set of parameters \\refeq{eq:par}.\nWe have determined this variation by searching the maximum and minimum\ncross-sections\\ in the range $130\\unskip\\,\\mathrm{GeV} < \\mathswitch {m_\\Pt} < 210\\unskip\\,\\mathrm{GeV}$ for the variation with\n$\\mathswitch {m_\\Pt}$ and in the ranges $60\\unskip\\,\\mathrm{GeV} < \\mathswitch {M_\\PH} < 400\\unskip\\,\\mathrm{GeV}$ and $600\\unskip\\,\\mathrm{GeV} < \\mathswitch {M_\\PH} <\n1000\\unskip\\,\\mathrm{GeV}$ for the variation with $\\mathswitch {M_\\PH}$. The range $400\\unskip\\,\\mathrm{GeV} < \\mathswitch {M_\\PH} <\n600\\unskip\\,\\mathrm{GeV}$ has been left out as there the Higgs-mass dependence is\ndominated by the Higgs resonance. Because the resonance dominates\n$\\sigma_{\\pm\\pm{\\mathrm{L}}\\rL}$ in an even wider range, we have omitted this\ncross-section\\ in the tables for the Higgs dependence.\n\n\\begin{table}\n$$ \\begin{array}{|c||*6{c|}}\n\\hline\n$\\mathswitch {M_\\PW}$ \\mathrm{~fixed}\n&{\\mathrm{U}}\\rU{\\mathrm{U}}\\rU & +{+}{\\mathrm{T}}\\rT & +{+}{\\mathrm{L}}\\rL & +{-}{\\mathrm{T}}\\rT & \n+{-}{\\mathrm{L}}\\rL & +{-}({\\mathrm{L}}{\\mathrm{T}}+{\\mathrm{T}}{\\mathrm{L}}) \\\\\n\\hline\n\\theta=20^\\circ & 0.15\\% & 0.18\\%  & 0.46\\%  & 0.14\\% &  0.53\\% &  0.06\\% \\\\\n\\theta=90^\\circ & 0.62\\% & 0.58\\%  & 3.52\\% & 0.44\\% & 1.50\\% &  0.29\\% \\\\\n\\mbox{integrated over} &  & & &  &  & \\\\\n10^\\circ<\\theta<170^\\circ\n&  0.22\\% & 0.26\\% & 1.06\\% & 0.16\\% & 1.23\\% &  0.25\\% \\\\\n20^\\circ<\\theta<160^\\circ \n&  0.29\\% & 0.33\\% & 1.48\\% & 0.21\\% & 1.23\\% &  0.26\\% \\\\\n\\hline\n\\end{array} $$\n\\caption{Variation of various polarized cross-sections\\ at $E_{\\mathrm{CMS}} = 500\\unskip\\,\\mathrm{GeV}$\nfor fixed $\\mathswitch {M_\\PW}$\nwith the top-quark mass in the range $130\\unskip\\,\\mathrm{GeV} < \\mathswitch {m_\\Pt} < 210\\unskip\\,\\mathrm{GeV}$ in per cent \nof the cross-section\\ for $\\mathswitch {m_\\Pt} = 174\\unskip\\,\\mathrm{GeV}$}\n\\label{ta:MTvar.mw}\n\\end{table}\n\\begin{table}\n$$ \\begin{array}{|c||*5{c|}}\n\\hline\n$\\mathswitch {M_\\PW}$ \\mathrm{~fixed}\n&{\\mathrm{U}}\\rU{\\mathrm{U}}\\rU & +{+}{\\mathrm{T}}\\rT & +{-}{\\mathrm{T}}\\rT & \n+{-}{\\mathrm{L}}\\rL & +{-}({\\mathrm{L}}{\\mathrm{T}}+{\\mathrm{T}}{\\mathrm{L}}) \\\\\n\\hline\n\\theta=20^\\circ & 0.16\\% & 0.18\\%  & 0.13\\%  & 0.37\\% &  0.81\\%  \\\\\n\\theta=90^\\circ & 0.44\\% & 0.21\\%  & 0.34\\% & 2.62\\% & 1.51\\%  \\\\\n\\mbox{integrated over} &  & & &  &  \\\\\n10^\\circ<\\theta<170^\\circ\n&  0.09\\% & 0.12\\% & 0.10\\% & 2.04\\% & 0.37\\% \\\\\n20^\\circ<\\theta<160^\\circ \n&  0.07\\% & 0.09\\% & 0.06\\% & 2.06\\% & 0.52\\% \\\\\n\\hline\n\\end{array} $$\n\\caption{Variation of various polarized cross-sections\\ at $E_{\\mathrm{CMS}} = 500\\unskip\\,\\mathrm{GeV}$\nfor fixed $\\mathswitch {M_\\PW}$\nwith the Higgs-boson mass\nin the ranges $60\\unskip\\,\\mathrm{GeV} < \\mathswitch {M_\\PH} < 400\\unskip\\,\\mathrm{GeV}$ and $600\\unskip\\,\\mathrm{GeV} < \\mathswitch {M_\\PH} <\n1000\\unskip\\,\\mathrm{GeV}$ in per cent\nof the cross-section\\ for $\\mathswitch {M_\\PH} = 250\\unskip\\,\\mathrm{GeV}$}\n\\label{ta:MHvar.mw}\n\\end{table}\nIn \\reftas{ta:MTvar.mw} and \\ref{ta:MHvar.mw} the $\\mathswitchr W$-boson mass is\nkept fixed at $\\mathswitch {M_\\PW}=80.22\\unskip\\,\\mathrm{GeV}$. Then,\nas argued in the previous section, the variation is \nsmall owing to the absence of large top- and Higgs-mass-dependent\ncorrections. The larger variation of the cross-sections\\ \ninvolving longitudinal \\mathswitchr W~bosons\nis due to terms proportional to $\\mathswitch {m_\\Pt}^2\/\\mathswitch {M_\\PW}^2$ or $\\mathswitch {M_\\PH}^2\/\\mathswitch {M_\\PW}^2$ \narising as a remnant of the unitarity cancellations for\n$\\sqrt{s}\\gg\\mathswitch {m_\\Pt},\\mathswitch {M_\\PH}$. \nThese terms induce a sizable variation of these cross-sections\\\nwith $\\mathswitch {m_\\Pt}$ and $\\mathswitch {M_\\PW}$ at higher energies.\n\n\\begin{table}\n$$ \\begin{array}{|c||*6{c|}}\n\\hline\n$\\mathswitch {G_\\mu}$ \\mathrm{~fixed}\n&{\\mathrm{U}}\\rU{\\mathrm{U}}\\rU & +{+}{\\mathrm{T}}\\rT & +{+}{\\mathrm{L}}\\rL & +{-}{\\mathrm{T}}\\rT & \n+{-}{\\mathrm{L}}\\rL & +{-}({\\mathrm{L}}{\\mathrm{T}}+{\\mathrm{T}}{\\mathrm{L}}) \\\\\n\\hline\n\\theta=20^\\circ & 1.20\\% & 1.24\\%  & 1.39\\%  & 1.17\\% &  0.45\\% &  0.05\\% \\\\\n\\theta=90^\\circ & 0.18\\% & 0.06\\%  & 3.23\\%  & 0.20\\% & 1.56\\% &  1.22\\% \\\\\n\\mbox{integrated over} &  & & &  &  & \\\\\n10^\\circ<\\theta<170^\\circ\n                & 0.98\\% & 0.94\\% & 1.62\\% & 1.12\\% & 1.17\\% &  0.84\\% \\\\\n20^\\circ<\\theta<160^\\circ \n                & 0.52\\% & 0.51\\% & 2.03\\% & 0.67\\% & 1.19\\% &  0.95\\% \\\\\n\\hline\n\\end{array} $$\n\\caption{Same as in \\protect\\refta{ta:MTvar.mw} but now for fixed $\\mathswitch {G_\\mu}$}\n\\label{ta:MTvar.gf}\n\\end{table}\n\\begin{table}\n$$ \\begin{array}{|c||*5{c|}}\n\\hline\n$\\mathswitch {G_\\mu}$ \\mathrm{~fixed}\n&{\\mathrm{U}}\\rU{\\mathrm{U}}\\rU & +{+}{\\mathrm{T}}\\rT & +{-}{\\mathrm{T}}\\rT & \n+{-}{\\mathrm{L}}\\rL & +{-}({\\mathrm{L}}{\\mathrm{T}}+{\\mathrm{T}}{\\mathrm{L}}) \\\\\n\\hline\n\\theta=20^\\circ & 0.33\\% & 0.33\\%  & 0.35\\% &  0.73\\% &  0.81\\% \\\\\n\\theta=90^\\circ & 0.56\\% & 0.36\\%  & 0.55\\% & 2.63\\% &  1.26\\% \\\\\n\\mbox{integrated over} &  & &  &  & \\\\\n10^\\circ<\\theta<170^\\circ\n&  0.36\\% & 0.34\\% & 0.38\\% & 2.09\\% & 0.25\\% \\\\\n20^\\circ<\\theta<160^\\circ \n&  0.31\\% & 0.27\\% & 0.32\\% & 2.10\\% & 0.37\\% \\\\\n\\hline\n\\end{array} $$\n\\caption{Same as in \\protect\\refta{ta:MHvar.mw} but now for fixed $\\mathswitch {G_\\mu}$}\n\\label{ta:MHvar.gf}\n\\end{table}\nThe variations of the corrections for fixed $\\mathswitch {G_\\mu}$ are shown in\n\\reftas{ta:MTvar.gf} and \\ref{ta:MHvar.gf}. \nIt is larger in particular for the cross-sections\\ for purely transverse\n\\mathswitchr W~bosons. This fact results\nfrom the dependence of $\\mathswitch {M_\\PW}$ on $\\mathswitch {m_\\Pt}$ and $\\mathswitch {M_\\PH}$ that involves\nlogarithmic top- and Higgs-mass-dependent terms and terms  proportional\nto $\\mathswitch {m_\\Pt}^2\/\\mathswitch {M_\\PW}^2$. \n\nTo visualize the Higgs resonance, we plot in \\reffi{fi:intcshiggs} the cross-section\\ \nincluding \\mathswitch{{\\cal{O}}(\\alpha)}\\ corrections\nintegrated over $20^\\circ <\\theta <160^\\circ$ for various values of the\nHiggs-boson mass. \n\\begin{figure}\n\\setlength{\\unitlength}{1mm}\n\\begin{picture}(160,110)(0,0)\n\\put(0,10){\\input{aaww.higgs.tex}}\n\\end{picture}\n\\caption{Integrated unpolarized cross-section\\ including $\\mathswitch{{\\cal{O}}(\\alpha)}$ corrections\nfor various Higgs-boson masses ($20^\\circ<\\theta<160^\\circ$)}\n\\label{fi:intcshiggs}\n\\end{figure}\nOur results agree qualitatively%\n\\footnote{See footnote \\ref{fo:3}.}\nwell with those of \\citere{Mo94}.\nWhile the resonance\nis comparably sharp at small energies, it is washed out by the large width \nof the Higgs boson at high energies. Already for $\\mathswitch {M_\\PH}=400\\unskip\\,\\mathrm{GeV}$ \nthe Higgs resonance is hardly visible in $\\gamma\\gamma\\to\\PWp\\PWm$.\n\n\\section{Summary}\n \nThe process $\\gamma\\gamma\\to\\PWp\\PWm$ will be one of the most interesting reactions at future\n$\\gamma\\ga$ colliders.\nIn particular, it is very useful to study triple and quartic \nnon-Abelian gauge couplings.\n \nWe have calculated the one-loop radiative corrections to $\\gamma\\gamma\\to\\PWp\\PWm$ within the\nelectroweak Standard Model in the soft-photon approximation for arbitrary\npolarizations of the photons and \\mathswitchr W~bosons. \nBy using a non-linear gauge-fixing term the number of contributing\ndiagrams can be reduced by roughly a factor of two.\nAn interesting peculiarity of $\\gamma\\gamma\\to\\PWp\\PWm$\nis the absence of most (universal) leading corrections,\nsuch as leading logarithms of light quark masses associated with the\nrunning of $\\alpha$ and leading logarithms associated with collinear\nbremsstrahlung. \nTherefore, the theoretical predictions are very clean.\n \nIn the heavy mass limit no leading $\\mathswitch {m_\\Pt}^2$- and \n$\\log\\mathswitch {m_\\Pt}$-terms and $\\log\\mathswitch {M_\\PH}$-terms exist.\nConsequently, \nthe variation of the cross-sections\\ with the top-quark and Higgs-boson masses\nis small if $\\mathswitch {M_\\PW}$ is kept fixed with the exception of the cross-sections\\\ninvolving longitudinal \\mathswitchr W~bosons at high energies. For fixed $\\mathswitch {G_\\mu}$\nthe variation arises mainly from the variation of $\\mathswitch {M_\\PW}$ with \n$\\mathswitch {m_\\Pt}$ and $\\mathswitch {M_\\PH}$ and is thus of similar origin like \nthe one of $\\Pep\\Pem\\to f\\bar{f}$ or $\\Pep\\Pem\\to\\PWp\\PWm$.\n\nWe have presented a detailed numerical discussion of the lowest-order\ncross-sections\\ and the virtual and soft-photonic corrections to $\\gamma\\gamma\\to\\PWp\\PWm$.\nThe soft-photon-cut-off-independent radiative corrections to the\ntotal cross-section\\ are of the order of 10\\%. They are \nincreased at high energies if the forward and backward regions are \nexcluded by an angular cut.\nThis is due to the fact that at high energies the radiative corrections\nreach several\n10\\% for intermediate scattering angles whereas they\nare at the level of several per cent in the forward and backward direction \nwhich dominate the total cross-section. \nThe large corrections are caused by bosonic loop diagrams whereas the\neffects of the fermionic diagrams are of the order of 5--10\\%.\n\n\\section*{Acknowledgement}\nWe are grateful to M.~B\\\"ohm for useful discussions.\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nX-ray ptychography\nhas a resurgence of interest in late 2000s due to the increase of both brilliance and coherence in modern synchrotron light sources \\cite{PfeifferNatPho17}. Inheriting the sophisticated techniques and advantages from both coherent diffraction imaging (CDI) and scanning transmission X-ray microscopy (STXM),\nit has become an essential tool in X-ray imaging at nanometer scale, and has the ability to produce high-resolution sample images while mitigating requirements for sample preparation, data analysis, optics, \\emph{a priori} knowledge of probe, etc \\cite{PfeifferNatPho17}. By scanning an X-ray probe over an extended specimen, one could measure the intensity of far-field transmission diffraction for each point, sometimes referred to as a ``view'', on a 2D scanning grid. However, phase information of the diffraction cannot be directly measured.  In order to reconstruct both the amplitude  and the phase for the light profile (probe) and specimen (object), \noverlapped scanning spots are necessary for providing sufficient information so that iterative algorithms can be used to retrieve the diffraction phase. \n\nThere are several phase-retrieval algorithms in X-ray ptychography, including the ptychographical iterative engine (PIE) \\cite{FaulknerPRL04,RodenburgAPL04} and its extension (ePIE) \\cite{MaidenUlt09}, the difference map (DM) algorithm \\cite{ThibaultSci08,ThibaultUlt09}, the non-linear optimization approach \\cite{Guizar-SicairosOE08}, the maximum-likelihood optimization \\cite{ThibaultNJP12}, etc.\nIn the present paper we focus on the DM algorithm, and implement it as part of the state-of-the-art multi-mode high-resolution tool suite at the Hard X-ray Nanoprobe (HXN) beamline at National Synchrotron Light Source II (NSLS-II) \\cite{YanNF18}.\n\n\nWithout any parallelization, a typical ptychographic reconstruction of an image can take hours to process on a single CPU core of a standard workstation. Furthermore, in order to reduce the data acquisition time, most of the measurements are conducted in the ``on-the-fly'' scan mode \\cite{PelzAPL14,DengOE15,HuangSR15} --- the sample is continuously moving relative to the probe. For accommodating the blurriness caused by continuous motion in the recorded diffraction data, multiple illumination modes have to be included in the reconstruction, which is introduced in \\cite{ThibaultNat13}. This multi-mode approach increases both  memory footprint and computation time for the ptychographic reconstruction, making it crucial to have a high-performance ptychography reconstruction software.  \n\nIn this paper, we report the steps we took to port the HXN ptychography reconstruction software to distributed GPUs and the resulting performance improvements. The new GPU version reduces the computation time significantly down to only tens of seconds for reconstructing a single image with one or more probe\/object modes, thereby fundamentally changing the workflow in the HXN beamline and providing real-time feedback to the facility users for arranging or adjusting the experimental setup.\n\n\n\n\\section{X-ray ptychography: difference map algorithm}\nThe DM algorithm is one of the most widely used and powerful phase-retrieval algorithms, which allows simultaneous reconstruction of the probe and the object, and our implementation follows closely  the original work \\cite{ElserACSA03,ThibaultSci08,ThibaultNat13}. \nAs far as we know this is the first reported GPU acceleration effort in the DM algorithm. Similar efforts utilizing GPUs include \\cite{NashedOE14,MandulaJAC16,MarchesiniJAC16,NashedPCS17}.\n\nFor the fly scans,\nthe key insight is that the resulting blurriness can be equivalently seen as caused by an incoherent illumination beam.\nAs a result, the imperfection can be compensated by allowing the presence of multiple probe and object modes in the reconstruction.\nFirst, in the far-field limit the recorded transmission intensity is written as an incoherent sum of multiple modes, labeled by $k$ for probe and $l$ for object, respectively:\n\\begin{equation}\n\tI_j(\\mathbf{q}) = \\sum_{k,l} \\left| \\mathcal{F} \\left[\\psi_j^{(k,l)}(\\mathbf{r})\\right]\\right|^2,\n\t\\label{eq: intensity mode}\n\\end{equation}\nwhere $\\mathbf{r}$ is the position vector in the specimen plane and $\\mathbf{q}$ the corresponding reciprocal vector, $\\psi_j$ is the complex exit wave produced at the $j$-th scanning point and $\\mathcal{F}$ stands for Fourier transform. In the following, it is assumed $j\\in[1, N]$, $k\\in [1, M_P]$, and $l\\in[1, M_O]$. \nNext, we require $\\psi$ to be expressed as a product of the $k$-th probe mode and $l$-th object mode,\n\\begin{equation}\n\\psi^{(k, l)}_j(\\mathbf{r}) = P^{(k)}(\\mathbf{r}-\\mathbf{r}_j) O^{(l)}(\\mathbf{r}), \n\\label{eq: field product mode}\n\\end{equation}\nwhere \n$\\mathbf{r}_j$ is the scan position. \nThe underlying assumption\nis that the probe variation along the propagation direction is negligible within the sample thickness.\nThe goal of ptychography is to reconstruct $P$ and $O$ iteratively subject to \\eqref{eq: intensity mode} and \\eqref{eq: field product mode}, \nwhich can be regarded as a constrained search in the hyperspace of complex pixels.\n\n\nIn the first step, one provides an initial guess of $P$ and $O$ so that the initial fields  can be constructed for each view $j$ based on \\eqref{eq: field product mode}.\nNext, the fields $\\{\\psi_j\\}$ are iteratively updated according to\n\\begin{equation}\n\\begin{aligned}\n \\psi_j^{(k,l)}(\\mathbf{r}) &\\leftarrow \\psi_j^{(k,l)}(\\mathbf{r}) + \\beta \\left\\{ \\mathcal{F}^{-1}\\circ\\mathcal{F}_c \\left[ 2P^{(k)}(\\mathbf{r}-\\mathbf{r}_j)O^{(l)}(\\mathbf{r})\\right.\\right.\\\\\n &\\left.\\left.\\quad- \\psi_j^{(k,l)}(\\mathbf{r}) \\right] - P^{(k)}(\\mathbf{r}-\\mathbf{r}_j)O^{(l)}(\\mathbf{r}) \\right\\},\n  \\label{eq: update field}\n \\end{aligned}\n\\end{equation}\n where $\\beta\\in[0, 1]$ is a free parameter empirically adjusted to speed up convergence (usually we set $\\beta=0.8$), and $\\mathcal{F}_c$ refers to ``constrained'' Fourier transform, in which the computed amplitude is replaced by the measured one while the computed phase is kept, \n \\begin{equation}\n \\mathcal{F}_c \\left[\\psi_j^{(k,l)}(\\mathbf{r})\\right] \\equiv \\sqrt{\\frac{I_j(\\mathbf{q})}{\\sum_{k,l}\\left|\\tilde{\\psi}_j^{(k,l)}(\\mathbf{q})\\right|^2}}\n \\tilde{\\psi}_j^{(k,l)}(\\mathbf{q})\n \\end{equation}\n with $\\tilde{\\psi} = \\mathcal{F} \\left[\\psi\\right]$.\n The probe and object are also iteratively updated as follows:\n \\begin{align}\n P^{(k)}(\\mathbf{r}) &\\leftarrow \\frac{\\sum_l\\sum_j \\left[ O^{(l)}(\\mathbf{r}+\\mathbf{r}_j) \\right]^*\\psi_j^{(k,l)}(\\mathbf{r}+\\mathbf{r}_j)} {\\sum_l\\sum_j |O^{(l)}(\\mathbf{r}+\\mathbf{r}_j)|^2},  \\label{eq: update probe}\\\\\n O^{(l)}(\\mathbf{r}) &\\leftarrow \\frac{\\sum_k\\sum_j \\left[ P^{(k)}(\\mathbf{r}-\\mathbf{r}_j) \\right]^*\\psi_j^{(k,l)}(\\mathbf{r})} {\\sum_k\\sum_j |P^{(k)}(\\mathbf{r}-\\mathbf{r}_j)|^2}. \\label{eq: update object}\n \\end{align}\n\n The convergence of all quantities can be monitored by calculating the relative 2-norm (L2-norm) distance between iterations, for example,\n \\begin{equation}\n \\varepsilon_P^{(k)} = \\sqrt{\\frac{\\int d^2\\mathbf{r}\\left| P^{(k)}_\\text{new}(\\mathbf{r}) -  P^{(k)}_\\text{old}(\\mathbf{r}) \\right|^2}{\\int d^2\\mathbf{r}\\left|P^{(k)}_\\text{new}(\\mathbf{r})\\right|^2}}.\n \\end{equation}\n In our experience, the DM algorithm does not need many iterations to converge and to reach satisfying resolution, although the number of required iterations is sample-dependent. \n\nIn practice, to determine the best choice  for the number of probe modes ($M_P$), we increase $M_P$ incrementally in a few trial runs and examine whether the contribution of each mode to the total intensity is larger than at least 1\\% (if not, then stop).\nSo far we are unaware of any situation in which $M_O>1$ is useful, but we keep this flexibility in the code for future extensions.\n\n\n\n\\section{Porting to GPUs: strategy and implementation details}\n\nWe implemented the multi-mode DM algorithm on both CPU and GPU.\nThe original CPU code was written in Python for easy integration with other data acquisition, analysis, and visualization tools provided at NSLS-II. Therefore, in our port to NVIDIA GPUs we use PyCUDA (wrapper for CUDA API) \\cite{PyCUDA}, scikit-cuda (for cuFFT and other CUDA libraries) \\cite{scikit-cuda} and MPI for Python (mpi4py) \\cite{mpi4py1,mpi4py2,mpi4py3} to accelerate the existing Python code.\nIn addition, most of the computation are rewritten as CUDA C kernels, which are called through the PyCUDA binding for the best performance. An example of such binding is shown  below:\n\\begin{lstlisting}\nimport pycuda.driver as cuda\nimport pycuda.gpuarray as gpuarray\nimport pycuda.autoinit\nimport numpy as np\n\n# assuming kernel \"fx1\" is implemented in sample.cu:\n# extern \"C\" { \/\/ avoid C++ name mangling\n#   __global__ void fx1(float* input) { \n#     \/* code *\/ \n#   } \n#   \/* other kernels *\/\n# }\n\n# load the CUDA binary (.cubin) generated by\n# nvcc -cubin -o sample.cubin sample.cu\ngpu_func_mod = cuda.module_from_file(\"sample.cubin\")\n\n# obtain a handle to the CUDA kernel\nkernel_fx1 = gpu_func_mod.get_function(\"fx1\")\n\n# copy an array to GPU (a bit slower than htod)\na = gpuarray.to_gpu(np.arange(100, dtype=float))\n\n# launch the kernel as if it were a Python function\n# block and grid mean the same as in CUDA C\nkernel_fx1(a, block=(100,1,1), grid=(1,1,1))\n\\end{lstlisting}\n\n\nWe first identified the hotspots in the code by coarse-grained performance profiling of the different function calls in the algorithm.  We found that in the serial implementation, the update of the views $\\{\\psi_j\\}$ consumes the most CPU time and was our first target for GPU acceleration. Once this part has been ported to the GPUs, we redid the profiling and identified subsequently the error estimation for $\\{\\psi_j\\}$, the update of the probe and object, and so on, as the next target for acceleration.\n\nBecause GPU memory is scarce (ranging from a few to 32 GB depending on the device model), the full computation with realistic data size is usually difficult to be carried out using one GPU alone, especially for the multi-mode case as discussed earlier. It is therefore necessary to distribute the computation to multiple GPUs, either on the same compute node or across multiple nodes.\nFurthermore, in order to carry out most of the computation in the device and avoid large data transfer, \nwe need to allocate extra GPU memory as buffer for FFT, product of $P$ and $O$, \nparallel reduction, etc.\nAs a result, assuming a single GPU can take a reconstruction of $N=40000$, probe size $200\\times200$, and object size $10^3\\times10^3$, the memory footprint is about $3$ times larger than the raw data. Another factor of roughly $3.7$ is needed if we have five probe modes ($M_P=5$). \nIn total, over $10$ times of GPU memory compared to the raw data size is required for multi-mode ptychographic reconstruction\\footnote{We note that cuFFT allocates additional memory that is not taken into account in our estimates.}, which \ncan certainly be alleviated with multiple GPUs.\n\n\n\\begin{figure}[b]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{ptycho_workflow}\n\t\\caption{The workflow of our implementation. The diffraction data associated with $N$ scanning points are distributed (almost) evenly to available GPUs, along with initial guess of probe $P$ and object $O$, so that most of the computation is done in the device. At each iteration, \n\n\ta few MPI collective communications (reduce and broadcast) are necessary to keep updated $P$ and $O$ visible across all GPUs.\n\tOther actions done in each iteration, such as error estimation, are not shown for clarity.\n\t}\n\t\\label{fig: workflow}\n\\end{figure}\n\n\n\nThe workflow of our Python code is shown in Fig.~\\ref{fig: workflow}. When the code starts, we divide (almost) equally the array indices of the dataset ($N$ intensity measurements)\nby the number of available GPUs, possibly on different physical machines as long as reachable by MPI, let each GPU  grab its assigned portion from the disk, send  initial guesses for the probe and object to all GPUs, and start the iteration loop. \nSince each view $j$ is updated independently [cf.~\\eqref{eq: update field}], this is a natural way to parallelize the computation,\nand the consequence of such workload distribution is that the summation over $j$ in \\eqref{eq: update probe} and \\eqref{eq: update object} is partitioned into each GPU.\nTherefore,\nwhile each view $\\psi_j$ is only updated in the GPU that it resides and no MPI collective communications are needed,\nin each iteration we need to collect the (partially summed) probe and object  from each GPU, perform an MPI reduce followed by an MPI broadcast to keep the updated $P$ and $O$ visible to all GPUs.\nWe stress that because all views are needed to compute the probe and the object at each iteration, such synchronization is necessary, and we do not find this synchronization to be a significant burden to the entire calculation. Moreover, our approach avoids the complication of image stitching, such as those reported in \\cite{NashedOE14}.  \n\nOne remark on the object reconstruction: Typically the probe dimension is much smaller than that of object. If we consider the functions of interest as matrices (since they live on a 2D plane labeled by pixel position $\\mathbf{r}$), then effectively the object update according to \\eqref{eq: update object} is an \\emph{embedding} procedure:  the Hadamard product of two smaller matrices ($P$ and $\\psi$) is embedded into a larger matrix ($O$). Such embedding is sketched as follows:\n\\begin{lstlisting}\n# For each point j, the array indices (x_start and\n# others) are calculated during initialization. Note \n# that the 1st dimension of prb and psi[j] is equal\n# to x_end-x_start, and similarly for the 2nd dim\nfor j, (x_start, x_end, y_start, y_end) in \\\n    enumerate(point_info):\n    obj_update[x_start:x_end, y_start:y_end] \\\n        += np.conjugate(prb) * psi[j]\n    # perform other computations\n\\end{lstlisting}\nCurrently we perform this embedding in series (\\textit{i.e.,} loop over scanning points $\\mathbf{r}_j$) because such parallelization requires determining and grouping non-overlapping points in runtime, which is not trivial, and we are exploring possibilities of parallelizing it. As a workaround for mitigating this issue, we find that if the points on each GPU are batch-processed, then swapping the order of the $P$ and $O$ updates and then overlapping the $\\psi$ update for batch $i+1$ with the $O$ update for batch $i$ can speed up by about 20\\%. On the other hand, the probe counterpart \\eqref{eq: update probe} does not have this problem as\nall pixels in $P$ are updated at once. \n\nFinally, we note that the initial guesses of $P$ and $O$ can be completely random or supplied by pre-existing results, and that initializing different probe modes with different degrees of  blurriness can help converge faster.\n\n\\section{Performance benchmark}\n\\begin{figure}[t\n\t\\centering\n\t\\includegraphics[width=1.0\\linewidth]{strong_scaling_rev2}\n\t\\caption{Strong scaling in the number of GPUs for completing 50 iterations for single-mode reconstruction, plotted in log-log scale. Real measurements of different $N$ and dimensions are used, and the raw data sizes are indicated. For some datasets (e.g.\\ \\#6) the data size is too large to fit in one GPU. Dashed lines are perfect scalings to guide the eyes. Tested on the HPC1 cluster in CSI, BNL. Each K20 compute node has Intel Xeon CPU E5-2670 @2.60GHz, 128GB physical memory, and one NVIDIA Tesla K20 GPU.\n\t}\n\t\\label{fig: strong scaling}\n\\end{figure}\n\n\nTo benchmark the performance of our implementation, we first show in Fig.~\\ref{fig: strong scaling} time-to-solution scaling as a function of the number of GPUs for realistic datasets with various sizes (different $N$ and small variation in probe dimension), a strong scaling in other words.\nThe speedup is most apparent for large datasets. In particular, for datasets of gigabytes order, the reconstruction can still be done on the order of a minute when using multiple GPUs. In fact, it can be seen that some datasets are too large to be fit in a single GPU (thus no data point on the plot), showing the necessity of using distributed GPUs. \n\n\n\nNext, we simultaneously increase both the total problem size (the number of views $N$) and the number of GPUs while keeping the workload distributed to each GPU fixed, which is known as a weak scaling.\nIn this case, we use instead synthesized data similar to the standard practice in the literature \\cite{MaidenUlt09,HuangOE14,NashedOE14}:\nwe take two arbitrary images to be the object's amplitude and phase and generate the hypothetical diffraction pattern through a zone plate, with variable number of scanning points such that roughly $5000$ points are assigned to each compute node. The result is shown in Fig.~\\ref{fig: weak scaling}, which shows a nearly constant scaling. We note that in this case due to the increasing communication overhead with more GPUs, the network infrastructure is critical to the scaling behavior. The measurement was done using the InfiniBand connection; with the TCP connection, we found that the scaling is less ideal but in general the elapsed time is still below 2 minutes for the highest $N$ (not shown).\n\n\\begin{figure}[b\n\t\\centering\n\t\\includegraphics[width=0.85\\linewidth]{weak_scaling_rev}\n\t\\caption{Weak scaling in the number of GPUs for completing 50 iterations for single-mode reconstruction, plotted in log-linear scale. Synthesized data of dimension $200\\times200$ are used, and each node is assigned about 5000 points. Tested on the Institutional Cluster in BNL. Each compute node has Intel Xeon CPU E5-2695 v4 @2.10GHz, 256GB physical memory, Mellanox EDR InfiniBand connection, and either two NVIDIA Tesla K80 or two Pascal P100 GPUs.\n\t}\n\t\\label{fig: weak scaling}\n\\end{figure}\n\n\n\nFinally, as a representative example, we present the results of a multi-mode reconstruction in Fig.~\\ref{fig: reconstruction}. The raw measurement data is about 2.63GB ($N=10000$, image size $188\\times188$, double precision), but since some scratch space is allocated by our code,\nfor this particular case more than 32 GB of memory is required if only one GPU were used, which is not possible even on the latest NVIDIA Tesla V100 product. Therefore, we use two and four V100 GPUs, and the corresponding timings for completing 50 iterations are 49.82s and 25.69s, respectively. Compared with the single-core CPU performance \n(8.8hr) on the same test machine,\nthe  speedup is about 1235x with 4 GPUs\\footnote{The beamline machines have no job scheduler and can be accessed by all internal users, so a certain degree of performance degradation due to resource competition is expected\\label{HXNnote}.}. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{ptycho_34802}\n\t\\caption{Multi-mode ptychographical reconstruction. (a) The first four dominant probe modes. (b) Reconstructed object phase, in which the field of view is $1\\times 1 \\mu$m. The sample is gold nano-crystals, prepared by annealing 20 nm thick gold film at 800$^{\\circ}$C for 8 hours. Test machine: xf03id-srv5 in HXN, which has Intel Xeon CPU E5-2630 v4 @2.20GHz, 256GB physical memory, and four NVIDIA Tesla V100 GPUs\\textsuperscript{\\ref{HXNnote}}.\n\t\n\t}\n\t\\label{fig: reconstruction}\n\\end{figure}\n\n\n\\section{Conclusion and outlook}\n\n\nIn summary, we present a GPU implementation of single- and multi- mode DM algorithm for X-ray ptychography, which is already deployed at the HXN beamline of NSLS-II. The significant reduction of computation time from hours to tens of seconds is a game changer for the beamline scientists and users, allowing real-time feedback and shortening the analysis workflow. We emphasize that the GPU runtime is much shorter than the data acquisition time in diffraction measurements, therefore  the reconstruction can be done effectively in real-time as the measurement progresses.\n\nWe are continuing tuning the performance to further reduce the computation time and\/or the memory footprint. For example, our preliminary tests indicate a slight improvement (about 10\\% to 30\\% speedup depending on the dataset size) on top of the achieved speedup by using page-locked host memory for data transfer, and we expect that this will benefit near-future experiments in which the object size is one order of magnitude larger in each dimension. Other possible  routes include using single-precision floating point arithmetic, adapting CUDA-aware MPI, overlapping kernel execution and data transfer, and parallelizing the object update as discussed earlier. \nWe are also  porting other ptychographic algorithms to GPU as part of a high-performance ptychographic toolbox,\nall of which will soon be open-sourced  on \\url{https:\/\/github.com\/NSLS-II\/}. \n\n\n\n\n\\bibliographystyle{IEEEtran.bst}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWe consider the \\emph{single-channel source separation} problem, in\nwhich we wish to separate a single aggregate signal into mixture of\nunobserved component signals. Traditionally, this problem has been\napproached in two ways: the \\emph{supervised} setting\n\\cite{kolter2010energy,roweis2001one,schmidt2006single},\nwhere we have access to training data with the true signal\nseparations and the \\emph{unsupervised} (or ``blind'')\nsetting \\cite{blumensath2005shift, davies2007source, lewicki2000learning,\n  schmidt2006nonnegative}, where we have only the\naggregate signal.  However,\nboth settings have potential drawbacks: for many problems, including\nenergy disaggregation---which looks to separate individual energy uses\nfrom a whole-home power signal \\cite{hart1992nonintrusive}---it can\nbe difficult to obtain training data with the true separated signals\nneeded for the supervised setting; in contrast, the unsupervised\nsetting is an ill-defined problem with arbitrarily many solutions, and\nthus algorithms are highly\ntask-dependent.\n\nIn this work, we propose an alternative approach that lies between\nthese two extremes. We propose a framework of \\emph{contextual\n  supervision}, whereby along with the input signal to be separated, we\nprovide contextual features correlated with the\nunobserved component signals. In practice, we find that this is often\nmuch easier than providing a fully supervised training set; yet it\nalso allows for a well-defined problem, unlike the unsupervised\nsetting. The approach is a natural fit for energy disaggregation, since\nwe have strong correlations between energy usage and easily observed\ncontext---air conditioning spikes in hot summer months,\nlighting increases when there is a lack of sunlight, etc. We\nformulate our model directly as an optimization problem in which we jointly\nestimate these correlations along with the most likely source\nseparation. Theoretically, we show that when the contextual features are\nrelatively uncorrelated between different groups, we can recover the\ncorrect separation with high probability. We demonstrate that our\nmodel recovers the correct separation on synthetic examples resembling\nthe energy disaggregation task and apply the\nmethod to the task of separating sources of energy usage for thousands of homes\nover 4 years, a scale much larger than previously published efforts.  The main\ncontributions of this\npaper are 1) the proposed contextually supervised setting and the\noptimization formulation; 2) the theoretical analysis showing that\naccurate separation only requires linear independence between features\nfor different signals; and 3) the application of this approach to the\nproblem of energy disaggregation, a task with significant potential to\nhelp inform consumers about their energy behavior, thereby increasing\nefficiency.\n\n\n\\section{Related work}\n\nAs mentioned above, work in single-channel source separation\nhas been separated along the lines of supervised and\nunsupervised algorithms, although several algorithms can be\napplied to either setting.  A common strategy is to\nseparate  the observed aggregate signal into a linear combination\nof several \\emph{bases}, where different bases correspond to different\ncomponents of the signal; algorithms such as Probabilistic Latent\nComponent Analysis (PLCA) \\cite{smaragdis2006probabilistic}, sparse coding\n\\cite{olshausen1997sparse}, and factorial hidden Markov models (FHMMs)\n\\cite{ghahramani1997factorial} all fall within this category, with\nthe differences concerning 1) how bases are represented and assigned\nto different signal components and 2) how the algorithm infers the\nactivation of the different bases given the aggregate signal.  For\nexample, PLCA typically uses pre-defined basis functions (commonly\nFourier or Wavelet bases), with a probabilistic model for how sources\ngenerate different bases; sparse coding learns bases tuned to data while\nencouraging sparse activations;\nand FHMMs use hidden Markov models to represent each source.  In the\nsupervised setting, one typically uses the individual signals to learn\nparameters for each set of bases (e.g., PLCA will learn which bases\nare typical for each signal), whereas unsupervised methods learn\nthrough an EM-like procedure or by maximizing some separation\ncriteria for the learned bases.  The method we propose here is\nconceptually similar, but the nature of these bases is rather different: instead\nof fixed bases with changing activations, we require features that\neffectively generate time-varying bases and learn\nactivations that are constant over time.\n\nOrthogonal to this research, there has also been a great deal of work in\n\\emph{multi-channel} blind source separation problems, where we observe multiple\nmixings of the same sources (typically, as many mixings as there are signals)\nrather than in isolation.\nThese methods can exploit significantly more structure and algorithms like Independent Component Analysis\n\\cite{comon1994independent,bell1995information}\ncan separate signals with virtually no supervised information.\nHowever, when applied to the single-channel problem (when this is even\npossible), they typically perform substantially worse than\nmethods which exploit structure in the problem, such as those described above.\n\nFrom the applied point of view, algorithms for energy disaggregation\nhave received growing interest in recently years\n\\cite{kolter2010energy,zeifman.11,kolter2012approximate,parson2012non}.\nThis is an important task since\nmany studies have shown that consumers naturally adopt energy\nconserving behaviors when presented with a breakdown of their energy\nusage \\cite{darby.06,neenan.09,ehrhardt2010advanced}. Algorithmic\napproaches to disaggregation are appealing as they allow for these\ntypes of breakdowns, but existing disaggregation approaches virtually all use\nhigh-frequency sampling of the whole-building power signal (e.g. per second)\nrequiring the installation of custom monitoring hardware for data collection.  In contrast,\nthis work focuses on disaggregation using data from ``smart meters'',\ncommunication-enabled power meters that are currently installed in\nmore than 32 million homes \\cite{iee.12}, but are limited to recording\nusage at low frequencies (every 15 minutes or hour), leading to a substantially\ndifferent set of challenges. Smart meters are relatively new and deployment is\nongoing, but due to the large amount of data available now and in the near\nfuture, successful disaggregation has the potential to have a profound impact on\nenergy efficiency.\n\n\\section{Optimization Formulation}\n\nWe begin by formulating the optimization problem for contextual source\nseparation. Formally, we assume there is some unknown matrix of $k$\ncomponent signals\n\\begin{equation}\nY \\in \\mathbb{R}^{T \\times k} = \\left [ \\begin{array}{cccc}\n\\mid & \\mid & & \\mid \\\\ y_1 & y_2 & \\cdots & y_k \\\\ \\mid & \\mid & &\n\\mid \\end{array} \\right ]\n\\end{equation}\nfrom which we observe the sum $\\bar{y} = \\sum_{i=1}^k y_i$.  For\nexample, in our disaggregation setting, $y_i \\in \\mathbb{R}^T$ could denote  a\npower trace (with $T$ total readings) for a single type of\nappliance, such as the air conditioning, lighting, or electronics, and\n$\\bar{y}$ denotes the sum of all these power signals, which we observe\nfrom a home's power meter.\n\nIn our proposed model, we represent each individual component signal\n$y_i$ as a linear function of some component-specific bases $X_i \\in\n\\mathbb{R}^{T \\times n_i}$\n\\begin{equation}\ny_i \\approx X_i \\theta_i\n\\end{equation}\nwhere $\\theta_i \\in \\mathbb{R}^{n_i}$ are the signal's\ncoefficients. The formal objective of our algorithm is: given the\naggregate signal $\\bar{y}$ and the component features $X_i$,\n$i=1,\\ldots,k$, estimate both the parameters $\\theta_i$ and the\nunknown source components $y_i$.  We cast this as an optimization problem\n\\begin{equation}\n\\label{eq-opt}\n\\begin{split}\n\\minimize_{Y,\\theta} \\;\\; & \\sum_{i=1}^k \\left \\{ \\ell_i(y_i, X_i\\theta_i) +\ng_i(y_i) + h_i(\\theta_i) \\right \\} \\\\\n\\subjectto \\;\\; & \\sum_{i=1}^k y_i = \\bar y \\\\\n\\end{split}\n\\end{equation}\nwhere $\\ell_i : \\mathbb{R}^{T} \\times \\mathbb{R}^{T} \\rightarrow\n\\mathbb{R}$ is a loss function penalizing differences between the\n$i$th reconstructed signal and its linear representation; $g_i$ is a\nregularization term encoding the ``likely'' form of the signal $y_i$,\nindependent of the features; and $h_i$ is a regularization penalty on\n$\\theta_i$. Choosing $\\ell_i$, $g_i$ and $h_i$ to be convex functions\nresults in a convex optimization problem.\n\nA natural choice of loss function $\\ell_i$ is a norm\npenalizing the difference between the reconstructed signal and its features\n$\\|y_i - X_i \\theta_i\\|$, but since our formulation enables loss functions that\ndepend simultaneously on all $T$ values of the signal, we allow for more complex\nchoices as well.  For example\nin the energy disaggregation problem, air conditioning is correlated with\nhigh temperature but does not respond to outside temperature changes instantaneously;\nthermal mass and the varying occupancy in buildings often results in air\nconditioning usage that correlates with high temperature over some\nwindow (for instance, if no one is in a room during a period of high\ntemperature, we may not use electricity then, but need to ``make up'' for this\nlater when someone does enter the room).  Thus, the loss function\n\\begin{equation}\n\\ell_i(y_i, X_i \\theta_i) = \\|(y_i - X_i \\theta_i)(I \\otimes\n1^T)\\|^2_2\n\\end{equation}\nwhich penalizes the aggregate difference of $y_i$ and $X_i\\theta_i$ over a\nsliding window, can be used to capture such dynamics. In many settings, it\nmay also make sense to use $\\ell_1$ or $\\ell_\\infty$ rather than $\\ell_2$ loss,\ndepending on the nature of the source signal.\n\nLikewise, since the objective term $g_i$ depends on all $T$ values of $y_i$, we\ncan use it to encode the likely dynamics of the source signal independent of\n$X_i\\theta_i$. For air conditioning and other single appliance types, we expect\ntransitions between on\/off\nstates which we can encode by penalizing the $\\ell_1$ norm of $Dy_i$ where\n$D$ is the linear difference operator subtracting $(y_i)_{t-1} - (y_i)_t$. For\nother types of energy consumption, for example groups of many electronic\nappliances, we expect the signal to have smoother\ndynamics and thus $\\ell_2$ loss is more appropriate.  Finally, we also include\n$h_i$ for statistical regularization purposes---but for problems where $T \\gg\nn_i$, such as the ones we consider in our experiments, the choice of $h_i$ is\nless important.\n\n\\begin{comment}\nwe find it most natural not\nto assume a particular distribution for $w_i$ but to instead capture\nthis correspondence through a\nloss function, $f_i(y_i, X_i\\theta_i)$; common choices\nbeing a norm $\\|y_i - X_i\\theta_i\\|$, or more complex alternatives such as\n$\\|(x - y)(I \\otimes 1^T)\\|$  which allows some amount of ``shift'' in the\ncorrespondence between the signal and the input features over time. We\ncombine these functions in an optimization problem that jointly estimates both\nthe model parameters and the source signals\n where the additional term in the objective, $g_i(y_i)$, encodes the model of\n how the source evolves over time. For example, the $\\ell_2$ norm of difference\noperator, $\\|Dy\\|_2$, smoothes the signal over time while the difference operator\nwith the $\\ell_1$ norm tends to encourage a piecewise constant signal. The\noptimization framework makes it straightforward to include additional\nconstraints such as the $y_i \\ge 0$ which satisfies the requirement\nthat energy usage be\nnonnegative. It would similarly be straightforward to add additional\npenalty terms on $\\theta_i$ for statistical regularization, but the energy\ndisaggregation task that we consider in this work exploits the fact that $T \\gg\nn$ and thus for this application, regularization is unnecessary.\n\\end{comment}\n\n\\section{Theoretical analysis}\n\nNext, we consider the ability of our model to recover the true source\nsignals as $T$ grows while $k$ and $n_i$ remain fixed. For the purposes\nof this section only, we restrict our attention to the choice of\n$\\ell_2$ loss, no $g_i$ or $h_i$ terms, and\nGaussian noise (the extension to the sub-Gaussian case is\nstraightforward).  We show that under this specialization of\nthe model, the optimization problem recovers the underlying signals\nat a rate dependent on the linear independence between blocks of input features\n$X_i$. In practice, the choice of $\\ell_i$, $g_i$ and\n$h_i$ is problem-specific, but $\\ell_2$ loss is a reasonable default\nand while simplifying the theoretical analysis dramatically, captures\nthe essential behavior of the model in the large $T$ regime.\n\nFormally, for this section we assume the source signals have Gaussian noise\n\\begin{equation}\n\\label{eq-model}\ny_i = X_i\\theta^\\star_i + w_i\n\\end{equation}\nfor some $\\theta^\\star_i \\in \\mathbb{R}^{n_i}$ and $w_i \\sim\n\\mathcal{N}(0, \\sigma_i^2 I)$.  Under the choice of $\\ell_2$ loss, our\noptimization problem simplifies to\n\\begin{equation}\n\\begin{split}\n\\minimize_{Y,\\theta} \\;\\; &\\|Y1 - X\\theta\\|_2^2 \\\\\n\\subjectto \\;\\; & Y1 = \\bar y\n\\end{split}\n\\end{equation}\nwhere $1 \\in \\mathbb{R}^k$ is the all-ones vector, $\\theta \\in\n\\mathbb{R}^{n}$ is a concatenation of all the $\\theta_i$'s, $X \\in\n\\mathbb{R}^{T \\times n}$ is a concatenation of all the\n$X_i$'s  and $n = \\sum_{i=1}^k n_i$ is the total number of features. In\nparticular, estimation of $\\theta$ is equivalent to\nleast-squares\n\\begin{equation}\n\\label{eq-theta-estimate}\n\\hat{\\theta} \\in \\mathbb{R}^{n} = \\arg \\min_{\\theta} \\|\\bar y - X\n\\theta\\|_2^2 = (X^T X)^{-1} X^T{\\bar y}.\n\\end{equation}\nSince each $y_i$ has it's own noise term, we can never expect to recover $y_i$\nexactly, but we can recover the true $\\theta^\\star$ with analysis that is the same\nas for standard linear regression. However, in the context of source separation,\nwe are interested in the recovery of the ``noiseless'' $y_i$, $X_i\n\\theta^\\star_i$,  and thus in our analysis we consider how the root mean\nsquared error\n\\begin{equation}\n\\label{eq-err}\n\\mathrm{RMSE}(X_i\\hat{\\theta}_i) = \\sqrt{\\frac{1}{T} \\left\\|X_i\n  \\hat{\\theta}_i - X_i \\theta_i^\\star \\right \\|_2^2}\n\\end{equation}\nvanishes for large $T$; indeed, a key feature of our analysis is to\nshow that we may recover the underlying signals faster than $\\theta^\\star_i$.\n\n\\begin{theorem}\n\\label{theorem-bounds}\nGiven data generated by the model \\eqref{eq-model}, and estimating\n$\\hat{\\theta}$ via \\eqref{eq-theta-estimate}, we have that\n\\begin{equation}\n\\label{eq-expected-value}\n\\mathbf{E}\\left[\\|X_i \\hat{\\theta}_i - X_i \\theta^\\star\\|_2^2 \\right ]\n= \\sigma^2 \\tr X_i^T X_i (X^T X)^{-1}_{ii} \\leq \\sigma^2 n_i\n\\rho_i\n\\end{equation}\nwhere $\\sigma^2 = \\sum_{i=1}^k \\sigma_i^2$ and $\\rho_i =\n\\lambda_{\\max}(X_i^T X_i (X^T X)^{-1}_{ii})$.  Furthermore, for $\\delta\n\\leq 0.1$, with probability greater than $1 - \\delta$\n\\begin{equation}\n\\label{eq-sample-complexity}\n\\mathrm{RMSE}(X_i\\hat{\\theta}_i) \\leq \\sqrt{\\frac{4 \\sigma^2 n_i\n    \\rho_i \\log (1\/\\delta)}{T}}.\n\\end{equation}\n\\end{theorem}\n\nA key quantity in this theorem is the matrix $X_i^T X_i (X^T\nX)^{-1}_{ii} \\in \\mathbb{R}^{n_i \\times n_i}$; $(X^T X)^{-1}_{ii}$\ndenotes the $i,i$ block of the full inverse $(X^TX)^{-1}$ (i.e., first\ninverting the joint covariance matrix of all the features, and then\ntaking the $i,i$ block), and this term provides a measure of the\nlinear independence between features corresponding to \\emph{different}\nsignals.   To see this, note that if features across different signals\nare orthogonal, $X^T X$ is block diagonal, and thus $X_i^T X_i (X^T\nX)^{-1}_{ii} = X_i^T X_i (X_i^T X_i)^{-1} = I$, so $\\rho_i = 1$.\nAlternatively, if two features provided for different signals\nare highly correlated, entries of $(X^TX)_{ii}^{-1}$ will have large\nmagnitude that is not canceled by $X_i^T X_i$ and $\\rho_i$ will be large.  This\nformalizes an intuitive notion for\ncontextually supervised source separation: for recovering the\nunderlying signals, it does not matter if two features for the\n\\emph{same} signal are highly correlated (this contrasts to the case\nof recovering $\\theta^\\star$ itself which depends on all correlations), but two\ncorrelated signals for different features make estimation difficult;\nintuitively, if two very similar features are provided for two different source\nsignals, attribution becomes difficult. A particularly useful property of these\nbounds  is that all terms can be computed using just $X_i$, so we can estimate\nrecovery rates when choosing our design matrix.\n\n\\subsection{Proofs}\n\nThe proof of Theorem \\ref{theorem-bounds} proceeds in two steps.\nFirst, using rules for linear transformations of Gaussian random\nvariables, we show that the quantity $X_i(\\hat{\\theta}_i -\n\\theta^\\star_i)$ is also (zero-mean) Gaussian, which immediately leads\nto \\eqref{eq-expected-value}.  Second, we derive a tail bound on the\nprobability that $X_i(\\hat{\\theta}_i - \\theta^\\star_i)$ exceeds some\nthreshold, which leads to the sample complexity bound\n\\eqref{eq-sample-complexity}; because this quantity has a singular\ncovariance matrix, this requires a slightly specialized probability\nbound, given by the following lemma.\n\n\\begin{lemma}\n\\label{lemma-tail}\nSuppose $x \\in \\mathbb{R}^p \\sim \\mathcal{N}(0,\\Sigma)$ with\n$\\mathrm{rank}(\\Sigma) = n$.  Then\n\\begin{equation}\nP\\left(\\|x\\|_2^2 \\geq t\\right) \\leq \\left (\\frac{t}{n \\lambda} \\right )^{n\/2}\n\\exp\\left \\{-\\frac{1}{2}(t\/\\lambda - n) \\right \\}\n\\label{eq-syn}\n\\end{equation}\nwhere $\\lambda$ is the largest eigenvalue of $\\Sigma$.\n\\end{lemma}\n\n\\begin{proof}\nBy Chernoff's bound\n\\begin{equation}\nP\\left(\\|x\\|_2^2 \\geq t\\right) \\leq\n\\frac{\\mathbf{E}\\left[e^{\\alpha \\|x\\|_2^2}\\right ]}{e^{\\alpha t}}.\n\\end{equation}\nfor any $\\alpha \\geq 0$.  For any $\\epsilon > 0$, $z \\sim\n\\mathcal{N}(0, \\Sigma + \\epsilon I)$,\n\\begin{equation}\n\\begin{split}\n\\mathbf{E}\\left[e^{\\alpha \\|z\\|_2^2}\\right] & = (2\\pi)^{-p\/2}|\\Sigma + \\epsilon\n  I|^{-1\/2}  \\int \\exp\\left \\{ - \\frac{1}{2} z^T (\\Sigma + \\epsilon I)^{-1}\n  z + \\alpha z^T z \\right \\} dz \\\\\n& = (2\\pi)^{-p\/2}|\\Sigma + \\epsilon\n  I|^{-1\/2}  \\int \\exp\\left \\{ - \\frac{1}{2} z^T (\\Sigma + \\epsilon\n  I)^{-1}(I - 2 \\alpha (\\Sigma + \\epsilon I)^{-1})\n  z \\right \\} dz \\\\\n& = (2\\pi)^{-p\/2}|\\Sigma + \\epsilon I|^{-1\/2} (2\\pi)^{p\/2}|\\Sigma +\n  \\epsilon I|^{1\/2} |I - 2 \\alpha (\\Sigma + \\epsilon I)|^{-1\/2}\n\\end{split}\n\\end{equation}\nso taking the limit $\\epsilon \\rightarrow 0$, we have that\n$\\mathbf{E}[e^{\\alpha \\|x\\|_2^2}] = |I - 2 \\alpha\n\\Sigma|^{-1\/2}$.  Since $\\Sigma$ has only $n$\nnonzero eigenvalues,\n\\begin{equation}\n|I - 2 \\alpha \\Sigma| = \\prod_{i=1}^n (1 - 2 \\alpha \\lambda_i)\n\\geq (1 - 2 \\alpha \\lambda)^n\n\\end{equation}\nand so\n\\begin{equation}\nP\\left(\\|x\\|_2^2 \\geq t\\right) \\leq \\frac{1}{(1 - 2 \\alpha\n  \\lambda)^{n\/2}e^{\\alpha t}}.\n\\end{equation}\nMinimizing this expression over $\\alpha$ gives $\\alpha = \\frac{t - n\n  \\lambda}{2t\\lambda}$ and substituting this into the equation\nabove gives the desired bound.\n\\end{proof}\n\n\nTo write the problem more compactly, we define the block matrix $W \\in\n\\mathbb{R}^{T \\times k}$ with columns $w_i$, and define the\n``block-diagonalization'' operator $B : \\mathbb{R}^{n} \\rightarrow\n\\mathbb{R}^{n \\times k}$ as\n\\begin{equation}\nB(\\theta) = \\left [ \\begin{array}{cccc}\n\\theta_1 & 0 & \\cdots & 0 \\\\\n0 & \\theta_2 & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & \\cdots & \\theta_k \\end{array} \\right ].\n\\end{equation}\n\n\\begin{proof}\n(of Theorem \\ref{theorem-bounds}) Since $Y = X B(\\theta^\\star) + W$,\n\\begin{equation}\n\\begin{split}\nX B(\\hat{\\theta}) & = X B\\left((X^T X)^{-1} X^T (X B(\\theta^\\star) +\nW)1 \\right) \\\\ & = X B(B(\\theta^\\star)) 1 + X B\\left((X^T X)^{-1} X^T W 1\n\\right) \\\\ & = X B(\\theta^\\star) + X B\\left((X^T X)^{-1} X^T W 1\n\\right)\n\\end{split}\n\\end{equation}\nFor simplicity of notation we also denote\n\\begin{equation}\nu \\in \\mathbb{R}^{n} \\equiv (X^T X)^{-1} X^T W 1.\n\\end{equation}\nThus\n\\begin{equation}\nX B(\\theta^\\star) - X B(\\hat{\\theta})  = X B(u).\n\\end{equation}\nNow, using rules for Gaussian random variables under linear transformations\n$W 1 \\sim \\mathcal{N}(0, \\sigma^2 I_T)$ and so $u \\sim \\mathcal{N}(0,\n\\sigma^2(X^T X)^{-1})$.  Finally, partioning $u_1, u_2, \\ldots,\nu_k$ conformally with $\\theta$,\n\\begin{equation}\nX_i \\theta^\\star - X_i \\hat{\\theta}_i = X_i u_i \\sim \\mathcal{N}(0,\n\\sigma^2 X_i(X^T X)^{-1}_{ii} X_i^T)\n\\end{equation}\nso\n\\begin{equation}\n\\mathbf{E} \\left [\\left \\|X_i \\theta^\\star - X_i \\hat{\\theta}_i \\right\n  \\|_2^2 \\right ] = \\sigma^2  \\tr X_i^T X_i(X^T X)^{-1}_{ii}.\n\\end{equation}\n\nSince $\\sigma^2 X_i(X^T X)^{-1}_{ii}X_i^T$ is a rank $n_i$ matrix\nwith maximum eigenvalue equal to $\\sigma^2 \\rho_i$, applying\nLemma \\ref{lemma-tail} above gives\n\\begin{equation}\nP\\left ( \\mathrm{RMSE}(X_i\\hat{\\theta}_i) \\geq \\epsilon \\right) =\nP\\left ( \\|X_iu_i\\|^2_2 \\geq T \\epsilon^2 \\right) \\leq \\left\n  (\\frac{T \\epsilon^2}{n_i \\sigma^2 \\rho_i} \\right )^{n_i\/2} \\exp\\left\n  \\{-\\frac{1}{2}\\left (\\frac{T \\epsilon^2}{\\sigma^2 \\rho_i} -\n  n_i\\right) \\right \\}.\n\\end{equation}\nSetting the right hand side equal to $\\delta$ and solving for\n$\\epsilon$ gives\n\\begin{equation}\n\\epsilon = \\sqrt{\\frac{-W(-\\delta^{2\/n}\/e) n_i \\rho_i \\sigma^2}{T}}\n\\end{equation}\nwhere $W$ denotes the Lambert $W$ function (the inverse of $f(x) = x\ne^x$).  The theorem follows by noting that $-W(-\\delta^{2\/n}\/e)\n\\leq 4 \\log \\frac{1}{\\delta}$ for all $n \\geq 1$ when $\\delta \\leq\n0.1$, with both quantities always positive in this range (note that\nleaving the $W$ term in the bound can be substantially tighter in some\ncases).\n\\end{proof}\n\n\n\\section{Experimental results}\n\n\\begin{figure}\n\\includegraphics{recovery_T}\n\\includegraphics{recovery_rho}\n\\includegraphics{recovery_comparison}\n\\caption{Comparing 100 random experiments with the theoretical\n  mean and 90\\% upper bound for increasing $T$ (left); fixing $T = 500$ and\n  varying $\\rho_i$ (center); and compared to the recovery of\n  $\\theta_1$ directly (right).}\n\\label{fig-syn1}\n\\end{figure}\n\nIn this section we evaluate contextual supervision on synthetic\ndata and apply in to disaggregate smart meter data collected from thousands of\nhomes. Since labeled data is unavailable, we design a synthetic\ndisaggregation task similar to energy disaggregation from smart meters in order\nto evaluate our performance on this task quantitatively. Here we explore the\nchoice of loss functions and demonstrate that contextual supervision\ndramatically outperforms the unsupervised approach.\n\n\\textbf{Rates for signal recovery}. We begin with a set of experiments examining\nthe ability of the model to recover the\nunderlying source signals as predicted by our theoretical analysis. In these\nexperiments, we consider the problem of separating two source signals with $X_i\n\\in \\mathbb{R}^{T \\times 16}$\nsampled independently from a zero-mean Normal with covariance $I + (1-\\mu)11^T$; we sample $\\theta_i^\\star$ uniformly from $[-1,1]$ and $Y \\sim\n\\mathcal{N}(X\\theta^\\star, I)$. Setting $\\mu = 0.01$ causes the\nfeatures for each signal to be highly correlated with each other but since $X_i$\nare sampled independently, not highly correlated across signals. In Figure \\ref{fig-syn1},\nwe see that MSE vanishes as $T$ grows; when comparing these experimental results to\nthe theoretical mean and 90\\% upper bound, we see that at least in these\nexperiments, the bound is somewhat loose for large values of $\\rho_i$. We are\nalso able to recover $\\theta^\\star$ (which is expected since the least-squares\nestimator $\\hat \\theta$ is consistent), but the rate is much slower due to\nhigh correlations in $X_i$.\n\n\\textbf{Disaggregation of synthetic data}. The next set of experiments considers\na synthetic generation process that more\nclosely mimics signals that we encounter in energy disaggregation. The process\ndescribed visually in Figure\n\\ref{fig-syn2} (top) begins with two signals, the first is smoothly varying over\ntime while the other is a repeating step function\n\\begin{equation}\nX_1(t) = \\sin(2\\pi t \/ \\tau_1) + 1,  \\quad X_2(t) = I(t \\bmod \\tau_2 <\n\\tau_2\/2)\n\\end{equation}\nwhere $I(\\cdot)$ is the indicator function and $\\tau_1$, $\\tau_2$ are the\nperiod of each signal. We also use\ntwo different noise models: for the smooth signal we sample Gaussian noise from\n$\\mathcal{N}(0, \\sigma^2)$ while for the step function,\nwe sample a distribution with a point mass at zero, uniform probability\nover $[1, 0) \\cup (0, 1]$ and correlate it across time by summing over a\nwindow of size $\\beta$. Finally, we constrain both noisy signals to be\nnonnegative and sum them to generate our input.\n\n\\begin{figure}\n\\includegraphics{synthetic_x}\n\\includegraphics{synthetic_noise}\n\\includegraphics{synthetic_y}\n\\includegraphics{synthetic_y1_comparison}\n\\includegraphics{synthetic_y2_comparison}\n\\caption{(top) Synthetic data generation process starting with two underlying\n  signals (left), corrupted by different noise models (center), and constrained\n  to be nonnegative\n  and sum'd to give the observed input (right); (bottom) comparison of the true noisy\n  signal and estimates.}\n\\label{fig-syn2}\n\\end{figure}\n\nWe generate data under this model for $T = 50000$ time points and consider\nincreasingly specialized optimization objectives while measuring the error in\nrecovering $Y^\\star = XD(\\theta^\\star) + W$, the underlying source signals\ncorrupted by noise. As can be seen in Table \\ref{tab-syn2-mse}, by using $\\ell_1$\nloss for $y_2$ and adding $g_i(y_i)$ terms\npenalizing $\\|Dy_1\\|_2^2$ and $\\|Dy_2\\|_1$, error decreases by 25\\% over\njust $\\ell_2$ loss alone; in Figure \\ref{fig-syn2}, we observe that our\nestimations recovers the true source signals closely with the $g_i$ terms\nhelping to capture the dynamics of the noise model for $w_2$.\n\nAs a baseline for this result, we compare to an unsupervised method, nonnegative\nsparse coding \\cite{hoyer2002non}. We apply sparse coding\nby segmenting the input signal into $1000$ examples of $50$ time points\n(1\/4 the period of the sine wave, $X_1(t)$) and fit a sparse\nmodel of 200 basis functions. We report the best possible source separation by\nassigning each basis function according to an oracle measuring\ncorrelation with the true source signal and using the best value over a grid of\nhyperparameters; however, performance\nis still significantly worse than the contextually supervised method which makes\nexplicit use of additional information.\n\\begin{table}\n\\label{tab-syn2-mse}\n\\centering\n\\caption{Comparison of models for source separation on synthetic data.}\n\\begin{tabular}{|l|r|}\n\\hline\n\\textbf{Model} & \\textbf{RMSE} \\\\\n\\hline\nNonnegative sparse coding & 0.4035 \\\\\n$\\ell_2$ loss for $y_1$ & 0.1640 \\\\\n$\\ell_2$ loss for $y_1$, $\\ell_1$ loss for $y_2$ & 0.1520 \\\\\n$\\ell_2$ loss for $y_1$, $\\ell_1$ loss for $y_2$ and $g_i$ penalizing $\\|Dy_i\\|$  & \\textbf{0.1217} \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\textbf{Energy disaggregation on smart meter data}. Next, we turn to the motivating problem for our model: disaggregating\nlarge-scale low resolution smart meter data into its component sources of\nconsumption. Our dataset comes from PG\\&E and was collected by the utility from\ncustomers in Northern California who had smart meters between 1\/2\/2008\nand 12\/31\/2011. According to estimations based on survey data, heating and cooling (air\nconditioning and refrigerators) comprise over 39\\% of total consumer\nelectricity usage \\cite{recs} and thus are dominant uses for\nconsumers. Clearly, we expect temperature to have a strong correlation with\nthese uses and thus we provide contextual supervision in the form of\ntemperature information. The PG\\&E data is anonymized, but the\nlocation of individual customers is identified at the census block\nlevel; we use this information to construct a parallel temperature\ndataset using data from Weather Underground\n(\\url{http:\/\/www.wunderground.com\/}).\n\nThe exact specification of our energy disaggregation model is given in Table\n\\ref{tab-dis-features}---we capture the non-linear dependence on temperature\nwith radial-basis functions (RBFs), include a ``Base'' category which models\nenergy used as a function of time of day, and featureless ``Other'' category\nrepresenting end-uses not explicitly modeled. For simplicity, we penalize each\ncategory's deviations from the model using $\\ell_1$ loss; but for heating and\ncooling we first multiply by a smoothing matrix $S_n$ ($1$'s on\nthe diagonal and $n$ super diagonals) capturing the thermal mass\ninherent in heating and cooling: we expect energy usage to correlate\nwith temperature over a window of time, not immediately. Finally, we use $g_i(y_i)$ and the\ndifference operator to encode our intuition of how energy consumption in each\ncategory evolves over time. The ``Base'' category represents an aggregation of\nmany sources of consumption and which we expect to evolve smoothly over time,\nwhile the on\/off behavior in other categories is best represented by the\n$\\ell_1$ penalty.\n\nWe present the result of our model at two time scales, starting with Figure 3\n(top), where we show aggregate energy consumption across all homes at the\nweek level to demonstrate basic trends in usage. Quantitatively, our model\nassigns 15.6\\% of energy consumption to ``Cooling'' and 7.7\\% to\n``Heating'', which is reasonably close to estimations based on survey\ndata \\cite{recs} (10.4\\% for air conditioning and 5.4\\% for space heating). We\nhave deliberately kept the model simple and thus our higher estimations are\nlikely due to conflating other temperature-related energy usages, such as refrigerators and\nwater heating. In Figure \\ref{fig-dis} (bottom), we present the\nresults of the model in disaggregating energy usage for a single hot summer\nweek where the majority of energy usage is estimated to be cooling due\nto the context provided by high temperature.\n\n\\begin{table}\n\\label{tab-dis-features}\n\\centering\n\\caption{Model specification for contextually supervised energy disaggregation.}\n\\begin{tabular}{|l|l|l|l|}\n\\hline\n\\textbf{Category} & \\textbf{Features} & \\textbf{$\\ell_i$} & \\textbf{$g_i$} \\\\\n\\hline\nBase    & Hour of day & $\\|y_1 - X_1\\theta_1 \\|_1 $ & $\\|Dy_1 \\|_2^2$   \\\\\nCooling & RBFs over temperatures $>70^\\circ\\mathrm{F}$ & $\\|S_2(y_2 - X_2\\theta_2) \\|_1$ &\n$0.1 \\times \\| Dy_2\\|_1$ \\\\\nHeating & RBFs over temperatures $<50^\\circ\\mathrm{F}$ & $\\|S_2(y_3 - X_3\\theta_3) \\|_1$ &\n$0.1 \\times \\| Dy_3 \\|_1$ \\\\\nOther   & None & $\\| y_4 \\|_1 $ & $0.05 \\times \\| Dy_4 \\|_1$ \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}\n\\includegraphics{homes}\n\\includegraphics{single_home_model}\n\\includegraphics{single_home}\n\\caption{Disaggregated energy usage for 1276 homes over nearly four years shown weekly\n  (top); and for a single home near Fresno, California over the week starting 6\/28\/2010 (bottom) with\n  estimated correlations $X_i\\hat{\\theta}_i$ (left) and estimated energy uses $\\hat{y}_i$ (right).}\n\\label{fig-dis}\n\\end{figure}\n\n\\section{Conclusion and discussion}\n\nWe believe the advances in this work, formalizing contextually supervised source\nseparation and theoretically analyzing the requirements for accurate source\nsignal recovery, will enable new applications of single-channel source separation\nin domains with large amounts of data but no access to explicit supervision. In\nenergy disaggregation, this approach has allowed us to reasonably separate\nsources of consumption from extremely low-frequency smart meter data. This a\nsignificant advancement with the potential to drive increases in energy\nefficiency through programs that expose this information to consumers and\nautomated systems for demand response. Developing algorithms that use this\ninformation to achieve these goals is an interesting direction for\nfuture work.\n\nAnother interesting direction is the explicit connection of our large-scale\nlow-resolution methods with the more sophisticated appliance models developed on\nsmaller supervised datasets with high-frequency measurements. As a few examples, a\nsmall amount of supervised information would enable us to calibrate\nour models automatically, while a semi-supervised approach would enable\nspreading some of the benefit of high-resolution load monitoring to the vast\nnumber of homes where only smart meter data is available.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTesting the predictions of the Cabibbo-Kobayashi-Maskawa (CKM) mechanism for violation of the combined charge-parity ($CP$) symmetry~\\cite{C,KM} is one of the major precision tests~\\cite{JPsiKs_theo,ccK_Belle,BigiSander2} of the flavor sector of the Standard-Model (SM). The Belle experiment at KEK significantly contributed to the validation of the CKM scheme and to constraining the unitarity triangle for $B$ decays to its current precision. Up-to-date, the SM describes the data impressively well, but there is still room for a deviation from unitarity, which clearly would point towards physics beyond the SM. \nThese proceedings give a summary of the experimental status of measurements related to two of the CKM angles defined from the CKM matrix elements as: $\\ensuremath{\\phi_{2}} \\equiv \\arg(V_{td}V^{*}_{tb})\/(-V_{ud}V^{*}_{ub})$ and $\\phi_3 \\equiv \\arg(V_{ud}V^*_{ub})\/(-V_{cd}V^*_{cb})$. All measurements presented here are based on Belle's final data set of $772 \\times 10^{6}$ $B\\bar{B}$ pairs. \n\n\n\\section{The CKM Angle \\ensuremath{\\phi_{2}}}\n\nThe CKM angle $\\phi_2$ can be determined by measuring the time-dependent asymmetry between \\ensuremath{B^{0}}\\ and \\ensuremath{\\bar{B}^{0}}\\ decays into a common $CP$ eigenstate~\\cite{CP} made out of unflavored quarks ($\\bar{b} \\rightarrow \\bar{u}u\\bar{d}$ quark transitions). Examples are the decays $\\ensuremath{B^{0}} \\rightarrow \\pi\\pi$, $\\rho\\pi$, $\\rho\\rho$ and $\\ensuremath{a_{1}(1260)}\\pi$~\\cite{pipi_Belle,pipi_BaBar,pipi_LHCb,rhoprhom_BaBar, rhoprhom_Belle1, rhoprhom_Belle2,belle_r0r0,r0r0Babar}. In the decay sequence, $\\ensuremath{\\Upsilon(4S)} \\rightarrow B_{CP}B_{\\rm Tag} \\rightarrow f_{CP}f_{\\rm Tag}$, where one of the $B$ mesons decays into a $CP$ eigenstate  $f_{CP}$ at a time $t_{CP}$ and the other decays into a flavor specific final state $f_{\\rm Tag}$ at a time $t_{\\rm Tag}$, the time-dependent decay rate is given by\n\\begin{equation}\n  {P}(\\Delta t, q) = \\frac{e^{-|\\ensuremath{\\Delta t}|\/\\tau_{B^0}}}{4\\tau_{B^0}} \\bigg[ 1+q(\\ensuremath{{A}_{CP}}\\cos\\Delta m_d \\ensuremath{\\Delta t} + \\ensuremath{{S}_{CP}}\\sin\\Delta m_d \\ensuremath{\\Delta t}) \\bigg],\n\\label{eq1}\n\\end{equation}\nwhere $\\ensuremath{\\Delta t} \\equiv t_{CP}- t_{\\rm Tag}$ is the lifetime difference between the two $B$ mesons, $\\Delta m_d$ is the mass difference between the mass eigenstates  $B_{H}$ and $B_{L}$ and $q = +1 (-1)$ for $B_{\\rm Tag} = \\ensuremath{B^{0}} (\\ensuremath{\\bar{B}^{0}})$. The $CP$ asymmetry is given by \n\\begin{equation}\n\\frac{N(\\bar{B}\\to f_{CP}) - N(B\\to f_{CP})}{N(\\bar{B}\\to f_{CP}) + N(B\\to f_{CP})},\n\\label{eq_asym}\n\\end{equation}\nwhere $ N(B(\\bar{B})\\to f_{CP})$ is the number of events of a $B(\\bar{B})$ decaying to $f_{CP}$, the asymmetry can be time-dependent.\n The parameters \\ensuremath{{A}_{CP}}\\ and \\ensuremath{{S}_{CP}}\\ describe direct and mixing-induced $CP$ violation, respectively~\\footnote{There exists an alternate notation where $C_{CP} = -\\ensuremath{{A}_{CP}}$.}. \\\\\n At tree level one expects $\\ensuremath{{A}_{CP}}=0$ and $\\ensuremath{{S}_{CP}}=\\sin2\\ensuremath{\\phi_{2}}$ for the above mentioned decays sensitive to $\\phi_2$. Possible penguin contributions can give rise of direct $CP$ violation, $\\ensuremath{{A}_{CP}}\\neq 0$ and also pollute the measurement of \\ensuremath{\\phi_{2}}, $\\ensuremath{{S}_{CP}}=\\sqrt{1-\\ensuremath{{A}_{CP}}^{2}}\\sin(2\\ensuremath{\\phi_{2}}^{eff})$ where the observed $\\ensuremath{\\phi_{2}}^{eff} \\equiv \\ensuremath{\\phi_{2}} - \\Delta \\ensuremath{\\phi_{2}}$ is shifted by $\\Delta \\phi_2$ due to different weak and strong phases from additional non-leading contributions.\n\\begin{figure}[htb]\n  \\centering\n  \\includegraphics[width=.45\\textwidth]{pipi_feyn.eps}\n  \\includegraphics[width=.45\\textwidth]{pippim_peng.eps}\n \\put(-380,1){\\scriptsize a)}\n \\put(-180,1){\\scriptsize b)}\\\\\n  \\caption{a) leading order and b) penguin feynman diagrams for color-allowed $b \\rightarrow u \\bar{u} d$ transitions.}\n  \\label{fig_pipi_feyn}\n\\end{figure}\nDespite this, it is possible to determine $\\Delta \\ensuremath{\\phi_{2}}$ in $\\ensuremath{B^{0}} \\rightarrow h^{+} h^{-}$ with an $SU(2)$ isospin analysis by considering the set of three $B \\rightarrow hh$ decays where the $hh$s are either two pions or two longitudinally polarized $\\rho s$, related via isospin symmetry~\\cite{iso}. The $B \\rightarrow h^{i} h^{j}$ amplitudes $A_{ij}$ obey the triangle relations,\n\\begin{equation}\n  A_{+0} = \\frac{1}{\\sqrt{2}}A_{+-} + A_{00}, \\;\\;\\;\\; \\bar{A}_{-0} = \\frac{1}{\\sqrt{2}}\\bar{A}_{+-} + \\bar{A}_{00}.\n  \\label{eq_iso}\n\\end{equation}\nIsospin arguments demonstrate that $\\ensuremath{B^{+}} \\rightarrow h^{+} h^{0}$ is a pure first-order mode in the limit of neglecting electroweak penguins, thus these triangles share the same base, $A_{+0}=\\bar{A}_{-0}$, see Fig.~\\ref{fig_iso} for an illustration. $\\Delta \\ensuremath{\\phi_{2}}$ can then be determined from the difference between the two triangles. This method has an inherent four-fold discrete ambiguity in the determination of $\\sin(2\\ensuremath{\\phi_{2}})$.\n\n\\begin{figure}[htb]\n  \\centering\n  \\includegraphics[width=.5\\columnwidth]{iso_anal2.eps}\n  \\caption{A sketch of the isospin triangle.}\n  \\label{fig_iso}\n\\end{figure}\n\n\\subsection{The Decay $B^0\\to \\rho^+\\rho^-$}\n Having a decay into two vector particles, an angular analysis is performed to separate the $CP$-even states from the  $CP$-odd states for the isopsin analysis. Longitudinal polarized states correspond to pure $CP$-even states and their fraction, $f_L$, is obtained from a fit to the cosine of helicity angles, $\\Theta_{\\rm H}^{\\pm}$, which are defined as sketched in Fig.~\\ref{fig_hel}. Previous measurements show that $f_L$ is consistent with one~\\cite{rhoprhom_BaBar, rhoprhom_Belle1, rhoprhom_Belle2}, consequently the isospin analysis is performed for longitudinal polarization only.\n\\begin{figure}[htb]\n  \\centering\n \\includegraphics[height=!,width=0.6\\columnwidth]{helicity.eps}\n \\caption{The helicity angles $\\cos\\Theta_{\\rm H}^{\\pm}$; each one is defined in its $\\rho$ rest frame.}\n  \\label{fig_hel}\n\\end{figure}\nIn addition to combinatorial background, the presence of multiple, largely unknown backgrounds with the same four-pion final state as $B^0\\to \\rho^+\\rho^-$ make this decay quite difficult to isolate and interference between the various $4\\pi$ modes needs to be considered. Besides updating to the full data set, the branching fraction, the fraction of longitudinally polarized $\\rho$ mesons on this decay and the $CP$ violating parameters are obtained simultaneously from the fit to the data. In the fit, Belle uses the missing energy $(\\Delta E \\equiv E_{B}^{*}-E_{\\rm beam}^*)$, the beam-constraint $B$ mass ($M_{\\rm bc}=\\sqrt{E_{\\rm beam}^{*2} - p_{B}^{*2}})$, the masses and helicity angles of the two reconstructed $\\rho^\\pm$ mesons, a fisher discriminant to separate the jet-like $e^+e^-\\to q\\bar{q},\\; (q=u,\\;d,\\;s,\\;c)$ background from the spherical $B\\bar{B}$ decays and the $\\Delta t$ distribution for the two flavors of $B_{\\rm Tag}$. \nThey obtain\n\\begin{itemize}\n\\item[]${\\cal B}(B^0\\to\\rho^+\\rho^-)=(28.3\\pm 1.5\\;(\\rm stat) \\pm 1.4\\;(\\rm syst))\\times 10^{-6},$\n\\item[]$f_L = 0.988\\pm\\; 0.012\\;(\\rm stat ) \\pm\\;0.023(\\rm syst),$\n\\item[]${S}_{CP} =-0.13\\pm0.15(\\rm stat ) \\pm0.05\\;(\\rm syst)$ and \n\\item[]${A}_{CP} =0.00\\pm0.10(\\rm stat ) \\pm0.06\\;(\\rm syst)$.\n\\end{itemize}\nThis is currently the most precise measurement of the branching fraction and polarization of $B\\to \\rho^+\\rho^-$ decays as well as the tightest constraint on $CP$ violation in this mode.\n Fig.~\\ref{p_r0r0} shows the projections onto $\\Delta E$, $\\cos\\Theta_{\\rm H}^{+}$ and onto $\\Delta t$ for the two flavor of $B_{\\rm Tag}$, each with the fit result on top.\n\\begin{figure}[htb]\n  \\centering\n\\includegraphics[height=!,width=0.32\\columnwidth]{sigReg_dE.eps}                                                              \n\\includegraphics[height=!,width=0.32\\columnwidth]{sigReg_H1.eps}                                                             \n\\includegraphics[height=!,width=0.32\\columnwidth]{sigReg_dt_asym.eps}                                                         \n\\put(-50,105){\\textcolor{blue}{\\footnotesize $q=+1$}}\\put(-50,95){\\textcolor{red}{\\footnotesize $q=-1$}} \n\\put(-195,110){\\footnotesize  \\textcolor{red}{signal}} \\put(-195,102){\\footnotesize  \\textcolor{cyan}{$B\\to 4\\pi$}}\\put(-195,94){\\footnotesize  \\textcolor{mygreen}{$B\\bar{B}$}}\\\\\n  \\caption{ Signal enhanced distributions of (a) $\\Delta E$, (b) $\\cos\\Theta_H$, and (c) $\\Delta t$ for the two flavors of $B_{\\rm Tag}$ ($B_{\\rm Tag} = B^0$ for $q=+1$) with the fit result on top. The shaded red area is the $B^0\\to\\rho^0\\rho^0$ contribution. Furthermore, all $B$ decays with a four pion final state are shown in cyan, the entire ($B\\bar{B}$) background in dashed (dash-dotted dark) green and the full PDF in blue.}\n  \\label{p_r0r0}\n\\end{figure}\nThe results from this measurement are used in an isospin analysis together with other Belle measurements~\\cite{belle_r0r0, rpr0_Belle} (longitudinal polarization only).  Fig.~\\ref{fig_phi2}~b) shows the \\ensuremath{\\phi_{2}}\\ scan from the isospin analysis, the constraint most consistent with other measurements of the CKM triangle is $\\ensuremath{\\phi_{2}} = (93.7 \\pm 10.6)^{\\circ}$ and the penguin pollution is consistent with zero: $\\Delta\\phi_2 = (0.0\\pm9.6)^{\\circ}$. In the $B\\to\\rho\\rho$ system, the relatively small amplitude of $B^0\\to\\rho^0\\rho^0$ makes the isospin triangles flat and therefore the isospin analysis has no ambiguity.\n\\begin{figure}[htb]\n  \\centering\n \\includegraphics[height=!,width=0.65\\columnwidth]{rho_phi2_CL_rprm2.eps}\n \\put(-15, 70){\\small \\textcolor{black}{ $68\\%$}}                                                                                                           \n    \\put(-68, 120){\\small \\textcolor{black}{\\large $\\rho\\rho_{LP}$}}\\\\          \n \\caption{Probability scan of $\\phi_2$ in the $B\\to\\rho\\rho$ system.}\n  \\label{fig_phi2}\n\\end{figure}\n\n\\subsection{The Decay $B^0\\to\\ensuremath{\\pi^{0}}\\piz$}\nThis decay is an important input for the isospin analysis in the $B\\to\\pi\\pi$ system. Being reconstructed from four $\\gamma$s makes this measurement experimentally quite challenging. A fit to \\ensuremath{\\Delta E}, \\ensuremath{M_{\\rm bc}}\\ and a fisher discriminant $T_C$ is performed and a preliminary branching fraction of \n\\begin{itemize}\n\\item[] ${\\cal B}(B\\to\\ensuremath{\\pi^{0}}\\piz) = (0.9\\pm 0.12\\;{\\rm stat}\\pm 0.10\\;{\\rm syst})\\times 10^{-6}$\n\\end{itemize}\n is obtained. Signal enhanced projections are shown in Fig.~\\ref{p_p0p0}. It is planned to supersede this currently most precise measurement of the branching fraction with one including the determination of the direct $CP$ violation in this mode.\\\\\nThe upcoming Belle 2 experiment will make a time-dependent analysis possible, as the accumulated data will provides enough data to use converted photons to determine the $B$ vertex. Including the mixing-induced $CP$-violation parameter of $B\\to\\ensuremath{\\pi^{0}}\\piz$ in the isospin analysis might remove the four-fold ambiguity from the isospin analysis being currently present in the $B\\to\\pi\\pi$ system.\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[height=!,width=0.9\\columnwidth]{pi0pi0_B.eps}\n \\caption{ Signal enhanced distributions of (a) $\\Delta E$, (b) $\\cos\\Theta_H$ and (c) $\\Delta t$ with the fit result on top. Contributions from signal, continuum, $\\rho\\pi$ and other rare B decays are shown by blue, green, red and cyan respectively.}\n  \\label{p_p0p0}\n\\end{figure}\n\n\\section{The Decay \\ensuremath{B^0 \\to D^0[K_S^0\\pip\\pim]K^{*0}}\\ and the CKM Angle $\\phi_3$}\n\nA model independent dalitz plot analysis~\\cite{phi3_theo, B2DK_Belle} of the decay \\ensuremath{B^0 \\to D^0[K_S^0\\pip\\pim]K^{*0}}\\ has been performed for the first time. The flavor-specific decay, $K^{*0}\\to K^+\\ensuremath{\\pi^{-}}$, allows to determine the flavor of the $B$ meson. The number of events in different bins of the dalitz plot of the $D$ meson coming from either a $B^0$ ($N_i^+$) or a $\\bar{B}^0$ ($N_i^-$) is given by\n\\begin{equation}\nN_i^\\pm = h_B[K_{\\pm i} + r_S^2K_{\\mp i} + 2k\\sqrt{K_iK_{-i}}(x_\\pm c_i \\pm y_\\pm s_i))],\n\\label{e_BDK}\n\\end{equation}\nwhere $h_B$ is a normalization constant, $K_{\\pm i}$ are the entries in the $D$ (+) or $\\bar{D}$ (-) dalitz plot, $k=0.95 \\pm 0.03$~\\cite{B2DK_Babar} accounts for interference effects in the $D$ decay, and $c_i$ and $s_i$ include information on the average of the phase variation within a dalitz plot bin. All information on the $D$ dalitz plot is provided by measurements from the CLEO collaboration~\\cite{Cleo_D}. The observables $x_\\pm\\equiv r_{s}\\cos(\\delta_S \\pm \\phi_3)$ and $y_\\pm \\equiv r_{s}\\sin(\\delta_S \\pm \\phi_3)$ from the interference term in Equ.~\\ref{e_BDK} allow an extraction of the CKM angle $\\phi_3$ in general, where the ratio between the cabbibo-allowed and double-cabbibo-suppressed amplitudes, $r_S\\equiv\\frac{\\bar{A}}{A}=\\frac{A(\\bar{B}\\to \\bar{D}^0\\bar{K}^{*0})}{A(\\bar{B}\\to D^0\\bar{K}^{*0})}$, indicates the sensitivity to $\\phi_3$. Belle obtains\n\\begin{eqnarray}\nx_+=+0.1^{+0.7+0.0}_{-0.4-0.1}\\pm0.1, &\\;\\;\\;\\;\\;\\;\\;y_+=+0.3^{+0.5+0.0}_{-0.8-0.1}\\pm0.1,\\nonumber \\\\\nx_-=+0.4^{+0.5+0.0}_{-0.8-0.1}\\pm0.0, &\\;\\;\\;\\;\\;\\;\\;y_-=-0.6^{+0.8+0.1}_{-1.0-0.0}\\pm0.1,\\nonumber\n\\end{eqnarray}\nand uses the result to obtain an upper limit of \n\\begin{eqnarray}\nr_S<0.87 \\nonumber \n\\end{eqnarray}\nat the one $\\sigma$ level. The (dalitz plot integrated) fit results are shown in Fig.~\\ref{p_DK} and the $r_S$ scan is shown in Fig.~\\ref{p_DK2}. This mode is still statistically limited but will give additional insights on $\\phi_3$ when the Belle 2 data will be available.\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[height=!,width=0.99\\columnwidth, bb=2 500 1300 830]{B2DKs_result.eps}\n \\caption{ Signal enhanced distributions of (a) $\\Delta E$, (b) $\\cos\\Theta_H$ and (c) $\\Delta t$ with the fit result on top. The signal contribution is shown in red.}\n  \\label{p_DK}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[height=!,width=0.7\\columnwidth, bb = 20 330 1150 900]{B2DKs_rS.eps}\n \\caption{Probability scan of $r_S$.}\n  \\label{p_DK2}\n\\end{figure}\n\n\\section{Summary}\nWe have presented three recent and preliminary measurements from Belle sensitive to the CKM phases \\ensuremath{\\phi_{2}}\\ and $\\phi_3$ using the full data set of $772$ million $B\\bar{B}$ pairs .\nMeasurements of the branching fraction, the polarization and the $CP$ asymmetries in $B \\rightarrow \\rho^+\\rho^-$ were used to update the $\\phi_2$ isospin constraint from Belle. The branching fraction measurement of of $B\\to\\pi^0\\pi^0$ has been presented and the importance for this mode the isospin analysis in the $B\\to \\pi\\pi$ system has been discussed. The current world averages of \\ensuremath{\\phi_{2}}\\ as computed by the CKMfitter~\\cite{CKMfitter} and UTfit~\\cite{UTfit} collaborations are $\\ensuremath{\\phi_{2}} = (87.6^{+3.5}_{-3.3})^{\\circ}$ and $\\ensuremath{\\phi_{2}} = (88\/6 \\pm 3.3 )^{\\circ}$, respectively. Furthermore we presented a preliminary measurement of \\ensuremath{B^0 \\to D^0[K_S^0\\pip\\pim]K^{*0}}\\ decays and discussed the sensitivity to $\\phi_3$. All shown results are in good agreement with other SM based constraints on the CKM triangle. With Belle 2 being built~\\cite{Belle2} and the LHCb operating, the next generation of $B$ physics experiments are expected to further reduce the uncertainty of the CKM observables and might reveal new phenomena.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{\\ensuremath{\\mathsf{LRSyn}}\\xspace: The Solution Outline}\n\nOur solution data-extraction system is parametrized by a number of components\nthat need to be instantiated for each domain.\n\\begin{compactitem}\n  \\item \\emph{Region extraction program synthesizer.}\n    The region extraction DSL $\\mathcal{L}_{rx}$ is equipped with a synthesizer that\n    takes as input examples of the form $(\\ensuremath{\\mathsf{doc}}, \\ensuremath{\\ell}) \\mapsto \\ensuremath{\\mathsf{R}}$, and\n    produces programs from $\\mathcal{L}_{rx}$.\n    %\n    Here, the example maps a document $\\ensuremath{\\mathsf{doc}}$ and a location $\\ensuremath{\\ell}$ within\n    $\\ensuremath{\\mathsf{doc}}$ to a region $\\ensuremath{\\mathsf{R}}$ of the document.\n    %\n    For instance, an example in the HTML domain will have an input HTML\n    document, an input location (DOM node), and an output region (set of\n    contiguous DOM nodes).\n    %\n  \\item \\emph{Value extraction program synthesizer.}\n    The value extraction DSL $\\mathcal{L}_{vx}$ is equipped with a synthesizer that\n    takes as input examples of the form $\\ensuremath{\\mathsf{R}} \\mapsto \\ensuremath{v}$, and produces\n    a program from $\\mathcal{L}_{vx}$.\n    %\n    Here, $\\ensuremath{\\mathsf{R}}$ is a region in a document and $\\ensuremath{v}$ is the field-value\n    for that document.\n    %\n    For instance, an example might have an input region given by the blue\n    rectangle in Figure~\\ref{fig:formed_documents}(a), and an output value\n    ``Friday, Apr 3 8:18 PM''.\n    %\n  \\item \\emph{Blueprinting and Locating functions.}\n    The blueprinting function $\\ensuremath{\\mathsf{BP}}$, locating function $\\ensuremath{\\mathsf{Locate}}$, and the\n    blueprint distance function $\\ensuremath{\\delta}$ (as described in\n    Section~\\ref{sec:problem}) need to be specified per domain.\n\\end{compactitem}\n\n\\begin{figure*}[t]\n  \\scalebox{0.7}{\n\\begin{tikzpicture}[scale=0.5, every node\/.style={inner sep=5}]\n  \\node[draw, rectangle] (trainset)\n      {\\tabular{c}Training Subset\\\\with Annotations $\\ensuremath{\\mathcal{A}}$ \\endtabular} ;\n  \\node[rectangle, below of=trainset, yshift=-5mm] (dataset) {Documents} ;\n  \\node[draw, rectangle, thick, fit=(trainset)(dataset)] (fulldataset) {};\n\n  \\node[draw, rectangle, right of=fulldataset, xshift=5cm, yshift=1cm, minimum width=3.7cm] (cl1)\n      {\\tabular{c}Cluster \\& Landmark\\\\ $(\\ensuremath{C}_1, \\ensuremath{\\mathsf{m}}_1)$ \\endtabular};\n  \\node[draw, rectangle, dashed, below of=cl1, yshift=-2mm, minimum width=3.7cm] (cl2)\n      {$(\\ensuremath{C}_2, \\ensuremath{\\mathsf{m}}_2)$};\n  \\node[rectangle, below of=cl2, minimum width=3.7cm, yshift=0.5cm] (cl3)\n      {$\\ldots$};\n  \\node[rectangle, below of=cl3, minimum width=3.7cm, yshift=0.5cm] (cl4)\n      {$\\ldots$};\n\n  \\draw[->] (fulldataset.east) -- ++(1.5cm, 0cm) |- node[above] {} (cl1);\n  \\draw[->, dashed] (fulldataset.east) ++(1.5cm, 0cm) |- node[above] {} (cl2);\n  \\draw[->, dashed] (fulldataset.east) ++(1.5cm, 0cm) |- node[above] {} (cl3);\n  \\draw[->, dashed] (fulldataset.east) ++(1.5cm, 0cm) |- node[above] {} (cl4);\n\n  \\node[draw, rectangle, right of=cl1, xshift=5cm, yshift=1cm, minimum width=5cm] (inferroi)\n    {\\tabular{c}For each $\\ensuremath{\\mathsf{doc}}_i \\in \\ensuremath{C}_1$ \\\\ Infer ROI $\\ensuremath{\\mathsf{R}}_i$ \\endtabular};\n\n  \\node[draw, rectangle, below of=inferroi, yshift=-0.5cm, minimum width=5cm] (regionsynth)\n    {\\tabular{c}Synthesize Region Program\\\\mathsf{Ex}: $(\\ensuremath{\\mathsf{doc}}_i, \\ensuremath{\\mathsf{m}}_i) \\mapsto \\ensuremath{\\mathsf{R}}_i$ \\endtabular} ;\n\n  \\node[draw, rectangle, below of=regionsynth, yshift=-0.5cm, minimum width=5cm] (valuesynth)\n    {\\tabular{c}Synthesize Value Program\\\\mathsf{Ex}: $\\ensuremath{\\mathsf{R}}_i \\mapsto \\ensuremath{\\mathcal{A}}(\\ensuremath{\\mathsf{doc}}_i)$ \\endtabular} ;\n\n  \\node[draw, rectangle, below of=valuesynth, yshift=-0.5cm, minimum width=5cm] (inferbp)\n    {\\tabular{c}Compute average \\\\ ROI blueprint \\endtabular} ;\n\n  \\node[draw, thick, fit=(inferroi)(regionsynth)(valuesynth)(inferbp)] (synth) {};\n  \\draw[->] (cl1.east) -- (cl1.east-|synth.west) ;\n\n  \\node[draw, rectangle, right of=regionsynth, xshift=6cm, yshift=1cm, minimum width=4cm] (rprog)\n    {Region Program $\\ensuremath{\\mathsf{RProg}}_1$} ;\n  \\node[draw, rectangle, below of=rprog, minimum width=4cm] (vprog)\n    {Value Program $\\ensuremath{\\mathsf{EProg}}_1$} ;\n  \\node[draw, rectangle, below of=vprog, minimum width=4cm] (bpvalue)\n    {ROI Blueprint $\\ensuremath{\\mathsf{b}}_1$} ;\n  \\node[rectangle, above of=rprog, minimum width=4cm,yshift=-0.3cm] (fullprog1)\n    {Extraction Program for $\\ensuremath{C}_1$} ;\n  \\node[draw, thick, fit=(rprog)(vprog)(bpvalue)(fullprog1)] {};\n\n  \\draw[->] (regionsynth.east) -- ++(1.00cm,0cm) |- (rprog.west) ;\n  \\draw[->] (valuesynth.east)  -- ++(1.25cm,0cm) |- (vprog.west) ;\n  \\draw[->] (inferbp.east)     -- ++(1.50cm,0cm) |- (bpvalue.west) ;\n\n  \\node[draw, dashed, below of=fullprog1, yshift=-3cm, inner sep=0.4cm] (fullprog2)\n    {Extaction Program for $C_2$};\n  \\node[below of=fullprog2] (fullprog3) {...};\n\n  \\draw [decorate, thick, decoration = {calligraphic brace}]\n    ($(cl1.east|-synth.south)+(0cm,-0.5cm)$)\n    -- node[below] {\\large Joint Cluster and Infer Landmarks}\n    ($(fulldataset.west|-synth.south)+(0cm,-0.5cm)$) ;\n\n  \\draw [decorate, thick, decoration = {calligraphic brace}]\n    ($(fullprog1.east|-synth.south)+(0cm,-0.5cm)$)\n    -- node[below] {\\large Synthesize Extraction Programs}\n    ($(synth.west|-synth.south)+(0cm,-0.5cm)$) ;\n\n\\end{tikzpicture}\n  }\n \\vspace{-2ex}\n \\caption{Outline of landmark-based robust synthesis \\ensuremath{\\mathsf{LRSyn}}\\xspace}\n \\label{fig:lrsynblocks}\n \\vspace{-3ex}\n\\end{figure*}\n\n\\begin{algorithm}\n\\small\n\\caption{Landmark-based robust synthesis $\\mathsf{LRSyn}$}\n\\label{algo:lrsyn}\n\\begin{algorithmic}[1]\n  \\Require Training set $\\ensuremath{\\dataset_\\train} \\subseteq \\ensuremath{D}$\n  \\Require Annotation $\\ensuremath{\\mathcal{A}}(\\ensuremath{\\mathsf{doc}})$ for each document $\\ensuremath{\\mathsf{doc}} \\in \\ensuremath{\\dataset_\\train}$\n \n  \\Require Region extraction DSL $\\mathcal{L}_{rx}$\n  \\Require Value extraction DSL $\\mathcal{L}_{vx}$\n  %\n  \\State $[ (\\ensuremath{C}, \\ensuremath{\\mathsf{m}}) ] \\gets \\text{\\textsc{InferLandmarksAndCluster}}(\\ensuremath{\\dataset_\\train}, \\ensuremath{\\mathcal{A}})$\n      \\label{line:joint-cluster-and-landmark}\n  %\n  \\For{Cluster and landmark $(\\ensuremath{C}_i, \\ensuremath{\\mathsf{m}}_i) \\ \\in [(\\ensuremath{C}, \\ensuremath{\\mathsf{m}})]$:} \n    %\n    \\State $(\\ensuremath{\\mathsf{RProg}}_i,\\ensuremath{\\mathsf{b}}_i,\\ensuremath{\\mathsf{EProg}}_i) \\gets$\n    \\Statex\\hspace{\\algorithmicindent}$\\hspace*{3mm}\\text{\\textsc{SynthesizeExtractionProgram}}(\\ensuremath{C}_i, \\ensuremath{\\mathsf{m}}_i, \\ensuremath{\\mathcal{A}}, \\mathcal{L}_{rx}, \\mathcal{L}_{vx})$\n  \\EndFor\n  %\n  \\State \\Return $\\mathsf{Extract}(\\{ (\\ensuremath{\\mathsf{m}}_i, \\ensuremath{\\mathsf{RProg}}_i, \\ensuremath{\\mathsf{b}}_i, \\ensuremath{\\mathsf{EProg}}_i) \\})$\n\\end{algorithmic}\n\\end{algorithm}\n\nAlgorithm~\\ref{algo:lrsyn} presents an outline of our landmark-based robust\nsynthesis algorithm.\nThe high-level components are illustrated in Figure~\\ref{fig:lrsynblocks}.\nGiven an annotated training set $\\ensuremath{\\dataset_\\train}$, the first task is to infer the clustering of\n$\\ensuremath{\\dataset_\\train}$ into $\\ensuremath{C}_0, \\ldots, \\ensuremath{C}_n$.\nIn Algorithm~\\ref{algo:lrsyn}, this step is combined with the inference of\nlandmarks.\nThe procedure $\\text{\\textsc{InferLandmarksAndCluster}}$\n(line~\\ref{line:joint-cluster-and-landmark}) produces a set of clusters $C_i$\neach associated with a landmark $\\ensuremath{\\mathsf{m}}_i$.\nThen, for each cluster $\\ensuremath{C}_i$, the algorithm calls the subroutine\n$\\text{\\textsc{SynthesizeExtractionProgram}}$ to synthesize a region extraction program $\\ensuremath{\\mathsf{RProg}}_i$, a\nregion blueprint $\\ensuremath{\\mathsf{b}}_i$, and a value extraction program $\\ensuremath{\\mathsf{EProg}}_i$.\n\\begin{comment}\n\\begin{compactitem}\n  \\item The region extraction program $\\ensuremath{\\mathsf{RProg}}_i$ takes $\\ensuremath{\\mathsf{doc}}$ and $\\ensuremath{\\mathsf{m}}$ as inputs and\n    produces the corresponding ROI $\\ensuremath{\\mathsf{R}}$ as the output.\n    %\n  \\item The region blueprint $\\ensuremath{\\mathsf{b}}_i$ represents the typical blueprint\n    of an ROI for a document in $\\ensuremath{C}_i$.\n  \\item The value extraction program $\\ensuremath{\\mathsf{EProg}}_i$ produces the output field value given\n    the ROI.\n\\end{compactitem}\n\\end{comment}\nThe algorithm combines these with the landmark to  output an extraction program in the Landmark-based DSL $\\mathcal{L}_{ld}$, which can be executed using semantics shown in Algorithm 1.\n\n\n\\subsection {Clustering  Documents and Inferring Landmarks}\n\\label{subsec:clustering-and-landmark-inference}\n\n\\leaveout{\nIn Algorithm~\\ref{algo:lrsyn}, the first step is to cluster the set of documents\n$\\ensuremath{D}$ into clusters $\\{ \\ensuremath{C}_0, \\ldots, \\ensuremath{C}_n \\}$ and identify a\nlandmark $\\ensuremath{\\mathsf{m}}_i$ for each cluster.\nIdeally, we want to cluster the documents not based on the global structure, but\nthe local structure of the ROI around the landmark and field values.\nHowever, the appropriate ROI cannot be identified without the\ncorresponding landmark, and in turn, landmarks can only be identified given the\nclusters as they are defined in terms of values common to all documents in a cluster.\nTo avoid this circular dependence, we first generate fine-grained clusters based\non the global structure, compute landmarks and regions of interest for these\nclusters, and then merge these fine-grained clusters based on the blueprints\nof the regions of interest.\n}\n\n\\begin{algorithm}\n\\small\n\\caption{Joint clustering and landmark inference \\\\\n \\textbf{Procedure} $\\text{\\textsc{InferLandmarksAndCluster}}(\\ensuremath{\\dataset_\\train},\\ensuremath{\\mathcal{A}})$ }\n\\begin{algorithmic}[1]\n\\Require Training dataset $\\ensuremath{\\dataset_\\train}$ along with annotations $\\ensuremath{\\mathcal{A}}$.\n\\Require Blueprint function $\\ensuremath{\\mathsf{BP}}$.\n\\Require Blueprint distance metric dataset $\\ensuremath{\\delta}$.\n\\State \\Comment{\\textbf{Initial clustering using whole document blueprints}}\n\\State $\\Delta_{\\mathsf{fine}}(\\ensuremath{\\mathsf{doc}}, \\ensuremath{\\mathsf{doc}}') \\gets\n        \\ensuremath{\\delta}(\\ensuremath{\\mathsf{BP}}(\\ensuremath{\\mathsf{doc}}), \\ensuremath{\\mathsf{BP}}(\\ensuremath{\\mathsf{doc}}')), \\forall \\ensuremath{\\mathsf{doc}}, \\ensuremath{\\mathsf{doc}}' \\in \\ensuremath{\\dataset_\\train}$\n        \\label{line:fine-grained-distance}\n\\State $ [ \\ensuremath{C} ] \\gets \\mathsf{Cluster}(\\ensuremath{\\dataset_\\train}, \\Delta_\\mathsf{fine})$\n        \\label{line:initial-clustering}\n\\State \\Comment{\\textbf{Compute landmark and blueprint candidates}}\n\\For{$\\ensuremath{C}_i \\in [\\ensuremath{C}]$} \n\\State $\\ensuremath{\\mathsf{M}}_i \\gets \\ensuremath{\\mathsf{LandmarkCandidates}}(\\ensuremath{C}_i, \\ensuremath{\\mathcal{A}})$\n    \\label{line:landmark-candidates}\n\\For{$\\ensuremath{\\mathsf{doc}} \\in \\ensuremath{\\dataset_\\train}$} \\label{line:compute-blueprints-start}\n\\State $\\ensuremath{\\mathsf{R}}_{\\ensuremath{\\mathsf{doc}},\\ensuremath{\\mathsf{m}}} \\gets \\ensuremath{\\mathsf{EncRgn}}(\\ensuremath{\\mathcal{A}}(\\ensuremath{\\mathsf{doc}}) \\cup\n          \\ensuremath{\\mathsf{Locate}}(\\ensuremath{\\mathsf{m}}, \\ensuremath{\\mathsf{doc}}))$\n\\Statex\\hspace{20em}$, \\forall \\ensuremath{\\mathsf{m}} \\in \\ensuremath{\\mathsf{M}}_i$\n\\State $\\mathsf{roi}[\\ensuremath{\\mathsf{doc}}] \\gets \\{ (\\ensuremath{\\mathsf{m}}, \\ensuremath{\\mathsf{BP}}(\\ensuremath{\\mathsf{R}}_{\\ensuremath{\\mathsf{doc}},\\ensuremath{\\mathsf{m}}})) \n    \\mid \\ensuremath{\\mathsf{m}} \\in \\ensuremath{\\mathsf{M}}_i \\}$\n    \\label{line:compute-blueprints-end}\n\\EndFor\n\\EndFor\n\\State \\Comment{\\textbf{Merge clusters}}\n\\State Define $\\Delta_{\\mathsf{c}}(\\ensuremath{\\mathsf{doc}}_1, \\ensuremath{\\mathsf{doc}}_2) \\gets$ \\par\n    $\\min(\\{ \\ensuremath{\\delta}(\\ensuremath{\\mathsf{b}}_1, \\ensuremath{\\mathsf{b}}_2)\n      \\mid (\\ensuremath{\\mathsf{m}}_{1,2}, \\ensuremath{\\mathsf{b}}_{1,2}) \\in \\mathsf{roi}[\\ensuremath{\\mathsf{doc}}_{1,2}] \\land\n         \n          \\ensuremath{\\mathsf{m}}_1 = \\ensuremath{\\mathsf{m}}_2 \\}$\n    \\label{line:coarse-grained-distance}\n\n\\While{No change in $[\\ensuremath{C}]$} \\label{line:merge-clusters-start}\n\\State Let $\\Delta(\\ensuremath{C}_1, \\ensuremath{C}_2) = \n          \\mathsf{Avg}(\\{ \\Delta_\\mathsf{c}(\\ensuremath{\\mathsf{doc}}_1, \\ensuremath{\\mathsf{doc}}_2) \\mid \\ensuremath{\\mathsf{doc}}_i \\in \\ensuremath{C}_i \\})$\n\\If{$\\exists \\ensuremath{C}_1, \\ensuremath{C}_2 \\in [\\ensuremath{C}]$ such that $\\Delta(\\ensuremath{C}_1, \\ensuremath{C}_2) \\leq \\mathsf{threshold}$}\n  \\State $[\\ensuremath{C}] \\gets ([\\ensuremath{C}] \\setminus \\{ \\ensuremath{C}_1, \\ensuremath{C}_2 \\})\n    \\cup \\{ \\ensuremath{C}_1 \\cup \\ensuremath{C}_2 \\}$\n    \\label{line:merge-clusters-end}\n\\EndIf\n\\EndWhile\n\n\\State \\Return $[\\ensuremath{C}, \\mathsf{TopLandmarkCandidate}(\\ensuremath{C}) ]$\n\\end{algorithmic}\n\\label{algo:joint-clustering-landmarks}\n\\end{algorithm}\n\nThe \\text{\\textsc{InferLandmarksAndCluster}}\\ procedure (Algorithm~\\ref{algo:joint-clustering-landmarks}) outlines how we jointly perform clustering and landmark detection, using the approach described in Section~\\ref{subsec:joint-infer-cluster}.\n\n\\subparagraph{Initial clustering}\nLines~\\ref{line:fine-grained-distance}-\\ref{line:initial-clustering} perform the initial clustering to obtain the initial fine-grained clusters.\nHere, the clustering is by the blueprint of the whole document, and\nhence, two documents will be in the same cluster only if they have more or less\nexactly the same format with little or no variations.\n\n\\subparagraph{Landmark and blueprint identification}\nThe procedure $\\ensuremath{\\mathsf{LandmarkCandidates}}$ identifies common values in the\ndocuments of $\\ensuremath{C}_i$ as landmark candidates and orders\nthem by a \\emph{scoring function} (line~\\ref{line:landmark-candidates}).\nThe scoring function is based on two features:\n\\begin{inparaenum}[(a)]\n  \\item the distance between the landmark candidate and the annotated values in\n    the document, and\n  \\item the size of the region that encloses the landmark candidates and\n    annotated values.\n\\end{inparaenum}\nThese features were determined after initial experiments on a small fraction of\nour evaluation datasets.\nThe procedure $\\ensuremath{\\mathsf{LandmarkCandidates}}$ only return candidates with a score\nover a certain threshold.\nThen, for each landmark candidate and document, we compute and store the\nblueprint of the ROI in lines 8-9.\n\n\\subparagraph{Coarse-grained clustering}\nNow, for each document, we have a number of landmark candidates along with their\nassociated ROIs.\nWith the ROIs, we can now define a coarse-grained distance over\ndocuments that is based only on the blueprints of the local structure of ROIs\n(line~\\ref{line:coarse-grained-distance}).\nWith this coarse-grained distance, we now repeatedly merge clusters based on\ntheir average document distance\n(line~\\ref{line:merge-clusters-start}-\\ref{line:merge-clusters-end}).\nSince the coarse-grained distances are based on the blueprints of ROI, we now\nhave clusters that are solely based on the local structure,\nwhich was our intention in the first place.\n\n\\subparagraph{Finalizing landmarks}\nFinally, the procedure returns each coarse-grained cluster along with its top\nlandmark candidate.\n\n\n\\subsection{Synthesizing Extraction Programs}\n\\label{subsec:synth}\n\n\n\\begin{algorithm}\n\\small\n\\caption{Synthesize Extraction Program\\\\\n  \\textbf{Proc.} $\\text{\\textsc{SynthesizeExtractionProgram}}(\n    \\ensuremath{C},\n    \\ensuremath{\\mathsf{m}},\n    \\ensuremath{\\mathcal{A}},\n    \\mathcal{L}_{rx},\n    \\mathcal{L}_{vx})$}\n  \\label{algo:synthesize}\n\\begin{algorithmic}[1]\n\\Require Cluster $\\ensuremath{C}$ with annotations $\\ensuremath{\\mathcal{A}}$\n\\Require Landmark value $\\ensuremath{\\mathsf{m}}$\n\\Require Region and value extraction DSLs: $\\mathcal{L}_{rx}$ and $\\mathcal{L}_{vx}$\n\\State \\textbf{for all}~$\\ensuremath{\\mathsf{doc}}_i \\in \\ensuremath{C}$~\\textbf{define}:\n\\label{line:region-compute-begin}\n\\State \\hspace{\\algorithmicindent}\n        $\\ensuremath{\\ell}_i \\gets \\ensuremath{\\mathsf{Locate}}(\\ensuremath{\\mathsf{m}}, \\ensuremath{\\mathsf{doc}}_i)$\n\\State \\hspace{\\algorithmicindent}\n        $(\\ensuremath{\\mathsf{Locs}}_i, \\ensuremath{\\mathsf{Agg}}_i) \\gets \\ensuremath{\\mathcal{A}}(\\ensuremath{\\mathsf{doc}}_i)$\n\\State \\hspace{\\algorithmicindent}\n        $\\ensuremath{\\mathsf{R}}_i \\gets \\ensuremath{\\mathsf{EncRgn}}(\\{ \\ensuremath{\\ell}_i \\} \\cup \\ensuremath{\\mathsf{Locs}}_i, \\ensuremath{\\mathsf{doc}}_i)$\n\\label{line:region-compute-end}\n\\State \\Comment{\\textbf{Synthesize region program}}\n\\State $\\ensuremath{\\mathsf{RegionSpec}} \\gets \\{ (\\ensuremath{\\mathsf{doc}}_i, \\ensuremath{\\ell}_i) \\mapsto \\ensuremath{\\mathsf{R}}_i\n           \\mid \\ensuremath{\\mathsf{doc}}_i \\in \\ensuremath{C}\n      \\}$\n  \\label{line:region-spec}\n\\State $\\ensuremath{\\mathsf{RProg}} \\gets \\mathsf{Synthesize}(\\ensuremath{\\mathsf{RegionSpec}}, \\mathcal{L}_{rx})$\n  \\label{line:region-synth}\n\\State \\Comment{\\textbf{Compute region blueprint}}\n\\State $\\ensuremath{\\mathsf{b}} \\gets\n    \\mathsf{Average}(\\{ \\ensuremath{\\mathsf{BP}}(\\ensuremath{\\mathsf{RegionSpec}}(\\ensuremath{\\mathsf{doc}}))) \\mid \\ensuremath{\\mathsf{doc}} \\in \\ensuremath{C} \\})$\n  \\label{line:fp-compute}\n\\State \\Comment{\\textbf{Synthesize extraction program}}\n\\State $\\ensuremath{\\mathsf{ValueSpec}} \\gets \\{ \\ensuremath{\\mathsf{R}}_i \\mapsto\n        \\ensuremath{\\mathsf{Agg}}_i(\\ensuremath{\\mathsf{Locs}}_i) \\mid \\ensuremath{\\mathsf{doc}}_i \\in \\ensuremath{C} \\}$\n  \\label{line:extraction-spec}\n\\State $\\ensuremath{\\mathsf{EProg}} \\gets \\mathsf{Synthesize}(\\ensuremath{\\mathsf{ValueSpec}}, \\mathcal{L}_{vx})$\n  \\label{line:extraction-synth}\n\\State \\Return $(\\ensuremath{\\mathsf{RProg}}, \\ensuremath{\\mathsf{b}}, \\ensuremath{\\mathsf{EProg}})$\n\\end{algorithmic}\n\\end{algorithm}\n\n\nThe $\\text{\\textsc{SynthesizeExtractionProgram}}$ procedure (Algorithm~\\ref{algo:synthesize}) outlines\nhow we process a cluster with a given landmark, and calculate a region extraction program, blueprint, and a value extraction program.\nThe algorithm takes as input:\n\\begin{inparaenum}[(a)]\n  \\item a cluster $\\ensuremath{C}$ and corresponding landmark $\\ensuremath{\\mathsf{m}}$ \n \n  \\item the annotations $\\ensuremath{\\mathcal{A}}$ for the documents in $\\ensuremath{C}$, and\n  \\item the DSLs for region programs and extraction programs.\n\\end{inparaenum}\n\nIn the first step, the algorithm computes the ROI $\\ensuremath{\\mathsf{R}}_i$\nfor each document $\\ensuremath{\\mathsf{doc}}_i$ from the landmark and the annotations\n(lines~\\ref{line:region-compute-begin}-\\ref{line:region-compute-end}).\nThen, we synthesize the region program $\\ensuremath{\\mathsf{RProg}}$ using a set of examples of the\nform $(\\ensuremath{\\ell}_i, \\ensuremath{\\mathsf{doc}}_i) \\mapsto \\ensuremath{\\mathsf{R}}_i$\n(lines~\\ref{line:region-spec} and~\\ref{line:region-synth}).\nWe also compute the average or typical blueprint $\\ensuremath{\\mathsf{b}}$ for all the\nROIs in the cluster (line~\\ref{line:fp-compute}).\nThe region extraction program $\\ensuremath{\\mathsf{RProg}}$ and filtering based on blueprint $\\ensuremath{\\mathsf{b}}$ (used in the execution semantics in Algorithm 1) together act as a robust system for detecting the\nROIs.\nNext, we synthesize a value extraction program $\\ensuremath{\\mathsf{EProg}}$ using examples where the\ninputs are the ROIs in the document, and the outputs are the\nexpected field values (line~\\ref{line:extraction-spec}\nand~\\ref{line:extraction-synth}).\nThe region and value extraction programs work not only on the documents in $\\ensuremath{C}$, but typically also on unseen documents and formats where the global structure changes, \nwithout changes in the ROI.\n\nThe algorithm finally returns $(\\ensuremath{\\mathsf{RProg}}, \\ensuremath{\\mathsf{b}}, \\ensuremath{\\mathsf{EProg}})$, which is combined\nwith the landmark value $\\ensuremath{\\mathsf{m}}$ to produce a complete extraction program in the landmark DSL $\\mathcal{L}_{ld}$ in\nAlgorithm~\\ref{algo:lrsyn}.\n\n\n\\section{Region program for HTML}\n\\begin{algorithm}\n\\small\n\\caption{$\\mathsf{LearnRegionProgram}$ for a HTML document}\n\\label{algo:regionhtml}\n\\begin{algorithmic}[1]\n\\Require Annotated HTML documents $D_{tr}$ from a single cluster \n\\Require Field values $f_T(d)$ \\spsays{Do we remove the field here as well?}\n\\Require $\\ell^{f_T}_d$\n\\State $\\mathsf{BluePrint} \\gets \\langle\\rangle$\n\\State $\\mathsf{Hops} \\gets \\langle\\rangle$\n\\For {$Document \\; d \\in D_{tr}$}\n\\State $\\mathsf{ParentHops}, \\mathsf{SiblingHops} \\gets 0, 0$\n\\State $\\mathsf{landmarkRegion} \\gets \\ensuremath{\\mathsf{Locate}}({\\ell^{f_T}_d})$\n\\State $\\mathsf{fieldRegions} \\gets f_T(d)$\n\\State $\\mathsf{region} \\gets \\mathsf{landmarkRegion}$\n\\Do\n    \\If{$\\forall \\; \\mathsf{fieldRegions} \\subseteq \\mathsf{region}$}\n    \\State break\n    \\EndIf\n    \\If{$\\forall \\; \\mathsf{fieldRegions}$ are k siblings from $\\mathsf{region}$}\n    \\State $\\mathsf{SiblingHops} \\gets k$\n    \\State break\n    \\EndIf\n    \\State $\\mathsf{ParentHops} \\gets \\mathsf{ParentHops} + 1$\n    \\State $\\mathsf{region} \\gets \\mathsf{Expand} \\; \\mathsf{region}$ by one parent\n\\doWhile {$\\mathsf{True}$}\n\\State $\\mathsf{Hops} \\gets \\mathsf{Hops} + \\langle \\mathsf{ParentHops}, \\mathsf{SiblingHops} \\rangle$\n\\EndFor\n\\State\n\\State ParentHops = max over all parenthops in Hops \\spsays{Write this formally}\n\\State SiblingHops = max over all siblinghops for parenthop == ParentHops found above\n\\State\n\\State $\\mathsf{BluePrint} \\gets \\ensuremath{\\mathsf{BPrint}}(P_{rg}(d))$ \\spsays{we take blueprints over all documents and maintain them as set, usually the size is <=3. Should we simplify this?}\n\\State\n\\State $P_{rg} \\gets \\mathsf{ParentHops}, \\mathsf{SiblingHops} , \\mathsf{BluePrint}$\n\\State \\Return $P_{rg}$\n\\end{algorithmic}\n\\end{algorithm}\n\nAlgorithm~\\ref{algo:regionhtml}~describes the region growing algorithm for a HTML document. The algorithm loops through the training documents $D_{tr}$ and for each document, finds the $\\mathsf{ParentHops}$ and $\\mathsf{SiblingHops}$ required to grow the region. The first step in the algorithm (Line 5) is to use the $\\ensuremath{\\mathsf{Locate}}$ function to find the region associated with the landmark value. In this case, $\\ensuremath{\\mathsf{Locate}}$ function translates to a simple \\emph{grep} on the document. We initialize the $region$ variable to encompass the landmark region (Line 7).\n\nThe algorithm proceeds as follows: We check if all the field regions are contained within the $region$ itself, in which case we are done. Else, we check if all the field regions are within $k$ siblings away from the region. We call this as the \\emph{sibling consistency} check. If this check passes, we set $\\mathsf{SiblingHops}$ to $k$ and break. Else, it implies that we need a larger region to encompass the landmark and field regions. We increment the $\\mathsf{SiblingHops}$ by $1$ and expand the region by one parent (Lines $14 \\to 15$) and continue with the iteration from Line $9$.\n\nAt the end of the loop, we have $\\mathsf{ParentHops}$ and $\\mathsf{SiblingHops}$ across all training documents $D_{tr}$. Since we require a common parameter which will work across documents during inference time, we reconcile these two parameters as follows: We take the maximum over $\\mathsf{ParentHops}$. Lets call this $M$. And for all pairs with $\\mathsf{ParentHops} == M$, we take the maximum over $\\mathsf{SiblingHops}$. Theorem~\\ref{}~proves that this operation always computes the required region correctly across all documents.\n\nIn addition to the hops, we also output a blueprint characterizing the region. For doing this, we again grow the regions for each document using the final $\\mathsf{ParentHops}$ and $\\mathsf{SiblingHops}$ and add the resulting region blueprints to a set and return them.\n\n\\begin{theorem}\nThe operation described above to reconcile $\\mathsf{ParentHops}$ and $\\mathsf{SiblingHops}$ always produces the correct region for the given input documents.\n\n\\spsays{Write this formally}\n\nExample: If we have (3, 0) and (2, 1), this will result in (3, 0). And (3, 0) will be a superset of the region and will contain (2, 1) as well.\n\nLikewise, if we have (3, 0) and (3, 1), if we take the region (3, 1), it will also contain (3, 0). \n\n\\end{theorem}\n\n{\\bf[MOHAMMAD COMMENT: ] It is not clear what \"correct\" here means. Do you mean a region that contains the desired field values and landmark values? If so, then that sounds more like a soundness result that many possible answers could satisfy rather than one \"correct\" answer. So may be good to reword the theorem to \"a correct region\". Also wonder if theorem could be made stronger - e.g. is it the \"smallest\" such region or something like that? }\n\n\n\n\n\\begin{figure}\n\\small\n\\begin{tabular}{r c l}\nProg & := & \\textsf{map}(\\(\\lambda\\) node .\\; \\textsf{FFProg}(node), \\textsf{Nodes}) \\\\\nNodes & := & \\textsf{AllNodes}(\\textsf{input}) | \\textsf{Descendants}(\\textsf{Nodes}) \\\\\n     &  &    | \\textsf{filter}(\\textsf{Selector}, \\textsf{Nodes}) | \\textsf{Children}(\\textsf{Nodes}) \\\\\nSelector & := & \\textbf{tag} = c | \\textbf{class} = c | \\textbf{id} = c | \\textbf{nth-child}(n) | \\ldots \\\\\nFFProg & := & \\textsf{Substring} | \\textsf{Concat}(\\textsf{SubString}, \\textsf{FFProg}) \\\\\nSubString & := & \\textsf{node.TextValue}  \\\\\n          &    & | \\textsf{Extract}(RegexPos, RegexPos, SubString) \\\\\nRegexPos & := & \\textsf{RegexSearch}(\\textit{regex}, k)\n\\end{tabular}\n\\vspace{-2ex}\n\\caption{Syntax of the HTML extraction language $\\mathcal{L}_{ex}$}\n\\vspace{-3ex}\n\\end{figure}\n\n\\begin{table*}[ht]\n\\small\n\\label{table:m2hexpmt1}\n\\resizebox{8cm}{!}{\n\\begin{tabular}{|c|l|c|c|c|c|c|c|c|c|}\n\\hline\n{Domain} & Fields & \\multicolumn{4}{|c|}{$\\ensuremath{\\mathsf{NDSyn}}\\xspace$} & \\multicolumn{4}{|c|}{$\\ensuremath{\\mathsf{LRSyn}}\\xspace$}\\\\\n\\cline{3-10}&  & Prgs & Pre. & Rec. & F1 & Prgs & Pre. & Rec. & F1\\\\\n\\hline\n\\hline\\multirow{8}{*}{ifly.alaskaair.com}\n\n& AIata & 2 & 0.99 & 0.47 & 0.64 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& ATime & 2 & 0.97 & 0.46 & 0.62 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DIata & 2 & 0.89 & 0.42 & 0.57 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DDate & 1 & 1.00 & 1.00 & 1.00 & 2 & 1.00 & 1.00 & 1.00 \\\\\n& DTime & 2 & 0.89 & 0.42 & 0.58 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& FNum  & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& Name  & 1 & 1.00 & 0.99 & 0.99 & 1 & 1.00 & 0.99 & 0.99 \\\\\n& RId   & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n\n\\hline\\multirow{9}{*}{delta.com}\n\n& AIata & 2 & 1.00 & 1.00 & 1.00 & 2 & 1.00 & 1.00 & 1.00 \\\\\n& ATime & 2 & 1.00 & 1.00 & 1.00 & 2 & 1.00 & 1.00 & 1.00 \\\\\n& DIata & 2 & 1.00 & 1.00 & 1.00 & 2 & 1.00 & 1.00 & 1.00 \\\\\n& DDate & 1 & 0.94 & 0.97 & 0.95 & 2 & 1.00 & 1.00 & 1.00 \\\\\n& DTime & 2 & 1.00 & 1.00 & 1.00 & 2 & 1.00 & 1.00 & 1.00 \\\\\n& FNo   & 2 & 1.00 & 1.00 & 1.00 & 2 & 1.00 & 1.00 & 1.00 \\\\\n& Name  & 2 & 0.95 & 0.88 & 0.91 & 2 & 0.95 & 1.00 & 0.97 \\\\\n& Pvdr  & 2 & 1.00 & 1.00 & 1.00 & 2 & 1.00 & 1.00 & 1.00 \\\\\n& RId   & 2 & 1.00 & 0.99 & 0.99 & 2 & 1.00 & 0.99 & 0.99 \\\\\n\n\\hline\\multirow{9}{*}{booking.airasia.com}\n\n& AIata & 1 & 0.67 & 1.00 & 0.67 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& ATime & 0 & 0.00 & 0.00 & 0.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DIata & 1 & 0.67 & 1.00 & 0.67 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DDate & 1 & 0.67 & 1.00 & 0.67 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DTime & 0 & 0.00 & 0.00 & 0.00 & 1 & 1.00 & 1.00 & 1.00\\\\\n& FNo   & 1 & 1.00 & 0.92 & 0.92 & 1 & 1.00 & 0.92 & 0.92 \\\\\n& Name  & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& Pvdr  & 1 & 1.00 & 0.92 & 0.92 & 1 & 1.00 & 0.92 & 0.92 \\\\\n& RId   & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n\n\\hline\\multirow{9}{*}{getthere.com}\n\n& AIata & 3 & 0.73 & 0.84 & 0.78 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& ATime & 2 & 0.94 & 0.91 & 0.92 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DIata & 3 & 0.94 & 0.95 & 0.94 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DDate & 3 & 0.98 & 0.94 & 0.96 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DTime & 2 & 0.86 & 0.85 & 0.85 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& FNo   & 1 & 0.94 & 0.96 & 0.95 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& Name  & 1 & 0.83 & 0.96 & 0.89 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& Pvdr  & 1 & 0.98 & 0.95 & 0.97 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& RId   & 1 & 1.00 & 0.88 & 0.94 & 1 & 1.00 & 1.00 & 1.00 \\\\\n\n\\hline\\multirow{9}{*}{t.delta.com}\n\n& AIata & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& ATime & 1 & 0.99 & 1.00 & 0.99 & 1 & 0.99 & 1.00 & 0.99 \\\\\n& DIata & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DDate & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DTime & 1 & 0.99 & 1.00 & 0.99 & 1 & 0.99 & 1.00 & 0.99 \\\\\n& FNo   & 2 & 1.00 & 1.00 & 1.00 & 2 & 1.00 & 1.00 & 1.00 \\\\\n& Name  & 1 & 0.99 & 0.99 & 0.99 & 1 & 0.99 & 0.99 & 0.99 \\\\\n& Pvdr  & 2 & 0.99 & 0.97 & 0.98 & 2 & 1.00 & 0.97 & 0.98 \\\\\n& RId   & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n\n\\hline\\multirow{9}{*}{philippineairlines.com}\n\n& AIata  & 3 & 0.91 & 1.00 & 0.96 & 3 & 1.00 & 1.00 & 1.00 \\\\\n& ATime  & 2 & 1.00 & 1.00 & 1.00 & 3 & 1.00 & 0.98 & 0.99 \\\\\n& DIata  & 2 & 0.99 & 0.99 & 0.99 & 2 & 0.99 & 0.99 & 0.99 \\\\\n& DeDate & 2 & 0.98 & 0.94 & 0.96 & 2 & 0.98 & 0.94 & 0.96 \\\\\n& DTime  & 2 & 1.00 & 1.00 & 1.00 & 3 & 1.00 & 0.98 & 0.99 \\\\\n& FNo    & 3 & 0.74 & 0.68 & 0.70 & 3 & 1.00 & 0.93 & 0.97 \\\\\n& Name   & 3 & 1.00 & 0.97 & 0.99 & 2 & 1.00 & 0.99 & 0.99 \\\\\n& Pvdr   & 2 & 1.00 & 1.00 & 1.00 & 2 & 1.00 & 1.00 & 1.00 \\\\\n& RId    & 2 & 1.00 & 1.00 & 1.00 & 2 & 1.00 & 1.00 & 1.00 \\\\\n\n\\hline\\multirow{9}{*}{itinerary.westjet.com}\n\n& AIata & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& ATime & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DIata & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DDate & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DTime & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& FNo   & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& Name  & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& Pvdr  & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& RId   & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n\n\\hline\\multirow{9}{*}{aeromexico.com}\n\n& AIata & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& ATime & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DIata & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DDate & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DTime & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& FNo   & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& Name  & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& Pvdr  & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& RId   & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n\n\\hline\\multirow{9}{*}{mytrips.amexgbt.com}\n& AIata & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& ATime & 1 & 0.99 & 0.99 & 0.99 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DIata & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DDate & 1 & 0.99 & 1.00 & 0.99 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DTime & 1 & 0.99 & 1.00 & 0.99 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& FNo   & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& Name  & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& Pvdr  & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& RId   & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n\n\\hline\\multirow{9}{*}{qatarairways.com.qa}\n\n& AIata & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& ATime & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DIata & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DDate & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& DTime & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& FNo   & 1 & 0.99 & 1.00 & 0.99 & 1 & 0.99 & 1.00 & 0.99 \\\\\n& Name  & 1 & 0.99 & 1.00 & 0.99 & 1 & 0.99 & 1.00 & 0.99 \\\\\n& Pvdr  & 1 & 1.00 & 1.00 & 1.00 & 1 & 1.00 & 1.00 & 1.00 \\\\\n& RId   & 2 & 1.00 & 0.99 & 0.99 & 2 & 1.00 & 1.00 & 1.00 \\\\\n\n\\hline\n\\end{tabular}\n}\n\\caption{No. Program, Precision, Recall and F1 numbers on HTML extraction scenario (M2H dataset) with NDSyn and ReSyn}\n\\end{table*}\n\n\\begin{table*}[t]\n\\small\n\\label{table:m2hexpmt2appendix}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n{Domain} & Fields & \\multicolumn{4}{|c|}{HDEF} & \\multicolumn{5}{|c|}{$\\ensuremath{\\mathsf{LRSyn}}\\xspace$}\\\\\n\\cline{3-11}&  & Programs & Pre. & Rec. & F1 & Clusters & Programs & Pre. & Rec. & F1\\\\\n\\hline\n\\hline\\multirow{9}{*}{delta.com}\n& ArrivalAirportIata&2&0.9967&0.9967&0.9967&2&2&0.9967&0.9967&0.9967\\\\\n& ArrivalTime&2&0.9962&0.9975&0.9968&2&3&0.9971&0.9975&0.9973\\\\\n& DepartureAirportIata&2&0.9934&0.9967&0.9950&2&2&0.9934&0.9967&0.9950\\\\\n& DepartureDate&2&0.932&0.9659&0.9486&2&2&0.9986&0.9977&0.9981\\\\\n& DepartureTime&2&1&0.9975&0.9987&2&2&1&0.9975&0.9987\\\\\n& FlightNumber&2&1&1&1.00&2&2&1&1&1.00\\\\\n& Name&2&0.9447&0.8795&0.9109&3&3&0.951&1&0.9749\\\\\n& Provider&2&1&1&1.00&2&2&1&1&1.00\\\\\n& ReservationId&2&1&0.9989&0.9994&3&3&1&0.9989&0.9994\\\\\n\\hline\\multirow{9}{*}{philippineairlines.com}\n& ArrivalAirportIata&3&1&1&1.00&2&3&1&1&1.00\\\\\n& ArrivalTime&2&1&1&1.00&2&3&1&1&1.00\\\\\n& DepartureAirportIata&2&0.9937&0.9937&0.9937&2&2&0.9935&0.9935&0.9935\\\\\n& DepartureDate&2&0.9815&0.9463&0.9636&2&2&0.9809&0.9449&0.9626\\\\\n& DepartureTime&2&1&1&1.00&2&3&1&1&1.00\\\\\n& FlightNumber&2&0.8983&1&0.9464&2&2&0.9012&1&0.9480\\\\\n& Name&3&1&0.9745&0.9871&2&2&1&0.9953&0.9976\\\\\n& Provider&2&1&1&1.00&2&2&1&1&1.00\\\\\n& ReservationId&2&1&1&1.00&2&2&1&1&1.00\\\\\n\\hline\\multirow{9}{*}{itinerary.westjet.com}\n& ArrivalAirportIata&1&1&1&1.00&1&1&1&1&1.00\\\\\n& ArrivalTime&1&1&1&1.00&1&1&1&1&1.00\\\\\n& DepartureAirportIata&1&1&1&1.00&1&1&1&1&1.00\\\\\n& DepartureDate&1&1&1&1.00&1&1&1&1&1.00\\\\\n& DepartureTime&1&1&1&1.00&1&1&1&1&1.00\\\\\n& FlightNumber&1&1&1&1.00&1&1&1&1&1.00\\\\\n& Name&1&0.9931&1&0.9965&1&2&1&0.9942&0.9971\\\\\n& Provider&1&1&1&1.00&1&1&1&1&1.00\\\\\n& ReservationId&1&1&1&1.00&1&1&1&1&1.00\\\\\n\\hline\\multirow{9}{*}{qatarairways.com.qa}\n& ArrivalAirportIata&1&1&1&1.00&1&1&1&1&1.00\\\\\n& ArrivalTime&1&1&1&1.00&1&1&1&1&1.00\\\\\n& DepartureAirportIata&1&1&1&1.00&1&1&1&1&1.00\\\\\n& DepartureDate&1&1&1&1.00&1&1&1&1&1.00\\\\\n& DepartureTime&1&1&1&1.00&1&1&1&1&1.00\\\\\n& FlightNumber&1&0.9982&1&0.9991&1&1&0.9982&1&0.9991\\\\\n& Name&1&0.9988&1&0.9994&1&1&0.9988&1&0.9994\\\\\n& Provider&1&1&1&1.00&1&1&1&1&1.00\\\\\n& ReservationId&2&1&1&1.00&2&2&1&1&1.00\\\\\n\\hline\n\\end{tabular}\n\\caption{Results after adding new equivalence classes}\n\\end{table*}\n\\begin{table*}[t]\n\\small\n\\label{table:m2hexpmt2-Add}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n{Domain} & \\multicolumn{2}{c}{Original} &\\multicolumn{2}{|c|}{Added} & \\multicolumn{2}{|c|}{Final} & {New Equi Classes}\\\\\n\\cline{2-7}&  Train & Test & Train & Test & Train & Test& \\\\\n\\hline\ndelta&102&927&8&8&110&935&1\\\\\nphilippines&74&90&5&4&79&94&4\\\\\nqatar&55&581&10&20&65&601&9\\\\\nwestjet&50&427&5&73&55&500&0\\\\\n\\hline\n\\end{tabular}\n\\caption{Additions made to the dataset to account for out of cluster entities}\n\\end{table*}\n\n\n\n\\begin{table*}\n\\small\n\\label{table:m2hexpmt2}\n\\resizebox{17cm}{!}{\n\\begin{tabular}{|l|c|l|}\n\\hline\nDomain : Field &  & Program\\\\\n\n\\hline\\multirow{3}{*}{ifly.alaskaair.com:ArrivalAirportIata}\n&HDEF&$TR:nth-child(1):nth-last-child(1) > [style*=\"width\\:48\\%\"]:nth-child(3) > TABLE[cellpadding=\"0\"][cellspacing=\"0\"][border=\"0\"][style*=\"width\\:100\\%\"]:nth-child(1):nth-last-child(1) > TBODY:nth-child(1):nth-last-child(1) > :nth-child(1)$\\\\\n& &$TR:nth-child(1):nth-last-child(1) > [style*=\"padding-bottom\\:25px\"] > TABLE[cellpadding=\"0\"][cellspacing=\"0\"][border=\"0\"][style*=\"width\\:100\\%\"]:nth-child(1):nth-last-child(1) > TBODY:nth-child(1):nth-last-child(1) > :nth-last-child(4)$\\\\\n& HiSyn &$TR:nth-child(1)$ \\\\\n\n\\hline\\multirow{4}{*}{ifly.alaskaair.com:DepartureTime}\n&HDEF&$TR:nth-child(1):nth-last-child(1) > [style*=\"width\\:48\\%\"]:nth-child(1) > TABLE[cellpadding=\"0\"][cellspacing=\"0\"][border=\"0\"][style*=\"width\\:100\\%\"]:nth-child(1):nth-last-child(1) > TBODY:nth-child(1):nth-last-child(1) > :nth-child(2)$\\\\\n& &$TR:nth-child(1):nth-last-child(1) > [style*=\"padding-bottom\\:25px\"] > TABLE[cellpadding=\"0\"][cellspacing=\"0\"][border=\"0\"][style*=\"width\\:100\\%\"]:nth-child(1):nth-last-child(1) > TBODY:nth-child(1):nth-last-child(1) > :nth-last-child(3)$ \\\\ \n& HiSyn &$:nth-child(2)$ \\\\\n\\hline\\multirow{2}{*}{ifly.alaskaair.com:ReservationID}\n&HDEF&$TR:nth-child(1):nth-last-child(1) > [style*=\"padding-bottom\\:20px\"] > [align=\"center\"] > TBODY:nth-child(1):nth-last-child(1) > :nth-child(1)$\\\\\n&HiSyn&$TR:nth-child(1)$\\\\\n\n\\hline\\multirow{2}{*}{delta.com:DepartureAirportIata}\n&HDEF&$TABLE:nth-child(1) > TBODY:nth-child(1):nth-last-child(1) > TR > [style*=\"font-family\\:Lucida Grande\\, Lucida Sans\\, Lucida Sans Unicode\\, Trebuchet MS\\, Verdana\\, Tahoma\\, sans-serif\"]:nth-last-child(4) > SPAN[style*=\"color\\:rgb(112, 112, 112)\"]:nth-child(1):nth-last-child(1)$\\\\\n&HiSyn&$TBODY > TR > :nth-child(3)$\\\\\n\n\\hline\\multirow{2}{*}{delta.com:DepartureAirportIata}\n&HDEF&$TABLE:nth-child(1) > TBODY:nth-child(1):nth-last-child(1) > TR > [style*=\"font-family\\:Lucida Grande\\, Lucida Sans\\, Lucida Sans Unicode\\, Trebuchet MS\\, Verdana\\, Tahoma\\, sans-serif\"]:nth-last-child(4) > SPAN[style*=\"color\\:rgb(112, 112, 112)\"]:nth-child(1):nth-last-child(1)$\\\\\n&HiSyn&$TBODY > TR > :nth-child(3)$ \\\\\n\n\\hline\\multirow{2}{*}{booking.airasia.com:DepartureTime} \n&HDEF &$ [style*=\"border-collapse\\:collapse\"][style*=\"font-family\\:Roboto\\, Arial\\, sans-serif\"] > TBODY:nth-child(1):nth-last-child(1) > :nth-last-child(1):nth-child(1) > [valign=\"middle\"]:nth-child(3) > :nth-child(3)$\\\\\n&HiSyn&$TD:nth-child(3) > :nth-child(3)$ \\\\\n\n\\hline\\multirow{2}{*}{getthere.com:Provider} & HDEF&\n$DIV:nth-last-child(13) > TABLE > TBODY:nth-child(1):nth-last-child(1) > :nth-child(1) > :nth-child(2)$\\\\&HiSyn&$:nth-child(2)$\\\\\n\\hline\\multirow{2}{*}{getthere.com:ReservationID} & HDEF &\n$:nth-last-child(17) > TABLE > TBODY:nth-child(1):nth-last-child(1) > :nth-child(2) > :nth-child(2)$\\\\&HiSyn&$:nth-child(2)$\\\\\n\\hline\\multirow{2}{*}{t.delta.com:DepartureAirportIata}\n&HDEF&$.mj-column-per-100 > TABLE[border=\"0\"][cellpadding=\"0\"][cellspacing=\"0\"][role=\"presentation\"]:nth-child(1):nth-last-child(1) > TBODY:nth-child(1):nth-last-child(1) > TR > :nth-child(2)$\\\\&HiSyn&$TBODY:nth-child(1):nth-last-child(1) > TR > :nth-child(2)$\\\\\n\\hline\\multirow{2}{*}{t.delta.com:DepartureTime}\n&HDEF&$.mj-column-per-100 > :nth-child(1):nth-last-child(1) > TBODY:nth-child(1):nth-last-child(1) > TR > :nth-child(2)$\\\\&HiSyn&$TBODY:nth-child(1):nth-last-child(1) > TR > :nth-child(2)$\\\\\n\n\\hline\\multirow{2}{*}{philippineairlines.com:Name}\n&HDEF &$.content > [cellpadding=\"2\"] > TBODY:nth-child(1):nth-last-child(1) > [valign=\"top\"] > :nth-last-child(4):nth-child(1)$\n\\\\&HiSyn&$TR:nth-last-child(29)$\\\\\n\n\\hline\\multirow{2}{*}{itinerary.westjet.com:DepartureAirportIata} \n&HDEF&$[style*=\"font-size\\:13px\"][width=\"100\\%\"] > TBODY:nth-child(1):nth-last-child(1) > :nth-last-child(3):nth-child(1) > .accent > [id*=\"airSegment-departure-city-\"]$\\\\&HiSyn&$.accent > SPAN[id*=\"airSegment-departure-city-\"][style*=\"text-transform\\:uppercase\"]:nth-child(1)$\\\\\n\n\\hline\\multirow{2}{*}{aeromexico.com:ArrivalTime} &HDEF &\n$[id*=\"itinerary-container\"] > TABLE[cellspacing=\"0\"][width=\"100\\%\"]:nth-child(1):nth-last-child(1) > TBODY:nth-child(1):nth-last-child(1) > [style*=\"vertical-align\\:top\"] > [id*=\"-arrival-time\"]$\\\\&HiSyn&$TBODY:nth-child(1):nth-last-child(1) > [style*=\"vertical-align\\:top\"] > [id*=\"-arrival-time\"]$\\\\\n\n\\hline\\multirow{2}{*}{mytrips.amexgbt.com:FlightNumber} \n&HDEF&$TBODY > :nth-child(1):nth-last-child(1) > :nth-child(2) > DIV.right_section:nth-child(1):nth-last-child(1) > :nth-child(7):nth-last-child(12)$\\\\&HiSyn&$P:nth-child(7):nth-last-child(12)$ \\\\\n\n\\hline\\multirow{2}{*}{mytrips.amexgbt.com:FlightNumber}&HDEF&\n$TBODY > :nth-child(1):nth-last-child(1) > :nth-child(2) > DIV.right_section:nth-child(1):nth-last-child(1) > :nth-child(7):nth-last-child(12)$ \\\\&HiSyn& $P:nth-child(7):nth-last-child(12)$ \\\\\n\n\\hline\\multirow{2}{*}{qatarairways.com.qa:DepartureDate} \n&HDEF&$[style*=\"margin-bottom\\:10px\"]:nth-child(2) >  TBODY:nth-child(1):nth-last-child(1) > TR:nth-child(1):nth-last-child(1) > :nth-child(4)$\\\\&HiSyn&$TABLE > TBODY:nth-child(1):nth-last-child(1) > TR:nth-child(1):nth-last-child(1) > :nth-child(4)$\\\\\n\n\\hline\n\\end{tabular}\n}\n\\caption{HDEF vs ReSyn Programs}\n\\end{table*}\n\n\n\\subsection{Hierarchical Landmarks}\n\nConsider again the email in Figure~\\ref{fig:formed_documents}b) and a variant where the term\n\\emph{Pick-up} has been replaced by \\emph{Depart}.\nNow, using the landmark \\emph{Depart} for extraction will unintentionally also extract the car trip departure time.\nIn Section~\\ref{sec:overview}, the human used a hierarchy of landmarks (i.e.,\n\\emph{AIR} followed by \\emph{Depart}) to obtain the correct results.\nAlgorithm~\\ref{algo:lrsyn} can similarly be extended to hierarchical\nextractions.\n\nSay we first synthesized a program $\\ensuremath{\\mathsf{Prog}}_0$ using\nAlgorithm~\\ref{algo:lrsyn} that uses the landmark \\emph{Depart}.\nWe run $\\ensuremath{\\mathsf{Prog}}_0$ on the training document and realize that a spurious\nlandmark location (car departure) is identified.\nIn this case, we use the correct landmark locations (i.e., the first and last\noccurrence of \\emph{Depart}) as a new annotation.\nRunning Algorithm~\\ref{algo:lrsyn} with this annotation will produce a program\n$\\ensuremath{\\mathsf{Prog}}_1$ that uses \\emph{AIR} as a landmark and extracts precisely the\nrelevant occurrences of \\emph{Depart}.\nAt inference time, we run $\\ensuremath{\\mathsf{Prog}}_1$ to identify only the correct occurrences\nof \\emph{Depart} and then run $\\ensuremath{\\mathsf{Prog}}_0$ starting with only those\noccurrences of \\emph{Depart}.\nIn our implementation in Section~\\ref{sec:evaluation}, we have implemented the\nfull hierarchical extraction algorithm for the HTML domain.\n\n\\begin{comment}\nThat is, if \\emph{AIR} was also ambiguous, we create a new annotation with only\nthe relevant occurrences of \\emph{AIR} and iterate.\n\nAlgorithm~\\ref{algo:hierarchy} depicts this iterative process in an informal\nway.\nFor now, we are assuming only one document in the training set to simplify\npresentation.\nIn the first step, the algorithm runs the standard $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ algorithm to obtain\n$\\ensuremath{\\mathsf{Prog}}$ and compares the results of running $\\ensuremath{\\mathsf{Prog}}$ against the\nannotations.\nIf they are equal, we return the learned program.\nIf not, we identify the landmark locations that lead to ``good'' extractions.\nThat is, we run $\\ensuremath{\\mathsf{Prog}}$ on $\\ensuremath{\\mathsf{doc}}$, but restrict ourselves to a single\nlandmark location after finding the landmark locations in $\\ensuremath{\\mathsf{Locate}}$.\nWe continue running the rest of the programs with this single location, and if\nthe results we obtain are a subset of the annotations, it is considered good.\nWe then use these good landmark locations as a new annotation and recursively\ncall the algorithm, and finally return a hierarchical program which combines\n$\\ensuremath{\\mathsf{Prog}}$ with the results of the recursive call.\n\n\\begin{algorithm}\n\\small\n  \\caption{Hierarchical Landmark based Synthesis}\n  \\label{algo:hierarchy}\n\\begin{algorithmic}[1]\n\\Require Training set $\\ensuremath{\\dataset_\\train} = \\{ \\ensuremath{\\mathsf{doc}} \\}$\n\\Require Annotation $\\ensuremath{\\mathcal{A}}$ \n\\Require Region program DSL $\\mathcal{L}_{rg}$\n\\Require Extraction program DSL $\\mathcal{L}_{ex}$\n\\State $\\ensuremath{\\mathsf{Prog}} \\gets \\ensuremath{\\mathsf{LRSyn}}\\xspace(\\ensuremath{\\dataset_\\train}, \\ensuremath{\\mathcal{A}}, \\mathcal{L}_{rg}, \\mathcal{L}_{ex})$\n\\State $\\mathsf{Results} \\gets \\ensuremath{\\mathsf{Prog}}(\\ensuremath{\\mathsf{doc}})$\n\\If{$\\mathsf{Results} = \\ensuremath{\\mathcal{A}}(\\ensuremath{\\mathsf{doc}})$}\n\\Return $\\langle \\ensuremath{\\mathsf{Prog}} \\rangle$\n\\EndIf\n\\State $\\ensuremath{\\mathsf{Locs}} \\gets \\ensuremath{\\mathsf{Locate}}(\\ensuremath{\\mathsf{Prog}}.\\mathsf{Landmark}, \\ensuremath{\\mathsf{doc}})$\n\\State $\\ensuremath{\\mathsf{Locs}}_\\mathsf{good} \\gets\n    \\{ \\ensuremath{\\ell} \\mid \\ensuremath{\\ell} \\in \\ensuremath{\\mathsf{Locs}} \\land \\ensuremath{\\mathsf{Prog}}(\\ensuremath{\\mathsf{doc}}, \\ensuremath{\\ell})\n    \\subseteq \\ensuremath{\\mathcal{A}}(\\ensuremath{\\mathsf{doc}}) \\}$\n\\State $\\ensuremath{\\mathcal{A}}' \\gets \\{ \\ensuremath{\\mathsf{doc}} \\mapsto \\ensuremath{\\mathsf{Locs}}_\\mathsf{good} \\}$\n\\State \\Return $\\langle \\ensuremath{\\mathsf{Prog}} \\rangle +\n    \\mathsf{HierarchicalLRSyn}(\\ensuremath{\\mathsf{doc}}, \\ensuremath{\\mathcal{A}}', \\mathcal{L}_{rg}, \\mathcal{L}_{ex})$\n\\end{algorithmic}\n\\end{algorithm}\n  \n\\end{comment}\n\n\\subsection{Robustness of \\ensuremath{\\mathsf{LRSyn}}\\xspace\\xspace}\n\n\\begin{comment}\nManually examining our datasets, we have seen $4$ common ways in which formats\nchange.\nExpressing these in terms of ROIs, we see changes to:\n\\begin{inparaenum}[(1)]\n  \\item the format outside the ROIs,\n  \\item the format inside the ROIs,\n  \\item the position of the ROIs inside the document, and\n  \\item the number of ROIs in the document.\n\\end{inparaenum}\nOf these, \\ensuremath{\\mathsf{LRSyn}}\\xspace\\xspace handles $3$ kinds of variations (1, 3, and 4) to a greater\nor lesser extent.\nFor changes inside the ROIs (2), we discuss how \\ensuremath{\\mathsf{LRSyn}}\\xspace\\xspace can be leveraged to\nease the task of maintaining and updating the system to handle the new formats.\n\nThe major advantage \\ensuremath{\\mathsf{LRSyn}}\\xspace\\xspace has over related techniques is the use of the\n$\\ensuremath{\\mathsf{Locate}}$ function to narrow down on the ROIs using data content rather than the\ndocument structure, making structural format changes outside the ROIs \nand changes to the position of ROIs inside the document mostly irrelevant.\nOne special case we need to discuss is when the format change adds a new\nlandmark location outside the ROIs.\nFor example, in Figure~\\ref{fig:xxx}, suppose the format changes adding the word\n\\emph{Depart}.\nWe have two separate cases:\n\\begin{compactitem}\n  \\item If the word \\emph{Depart} is added in a position where the local\n    structure is different, the region blueprint learned by \\ensuremath{\\mathsf{LRSyn}}\\xspace\\xspace will\n    filter out this new landmark location.\n    %\n    For example, if \\emph{Depart} is part of an advertisement \\emph{Depart today\n    for your dream destination!}, the local structure around the new occurrence\n    of the landmark is significantly different.\n    %\n  \\item On the other hand, the word may be added in a context which is similar\n    to the ROIs.\n    %\n    For example, as a part of a new \\emph{TRAIN} section that is similar to the\n    \\emph{AIR} section.\n    %\n    In this case, \\ensuremath{\\mathsf{LRSyn}}\\xspace\\xspace does not naturally handle the change in the\n    variation, unless appropriate hierarchical landmarks are present in the program.\n    %\n    In general, this kind of landmark addition is problematic for \\ensuremath{\\mathsf{LRSyn}}\\xspace\\xspace.\n\\end{compactitem}\n\\end{comment}\n\nBy design, \\ensuremath{\\mathsf{LRSyn}}\\xspace\\xspace is robust to format variations that change\n\\begin{inparaenum}[(a)]\n  \\item document structure outside the ROIs,\n  \\item position of ROIs, and\n  \\item the number of ROIs.\n\\end{inparaenum}\nHowever, there are $2$ clear limitations to the robustness of \\ensuremath{\\mathsf{LRSyn}}\\xspace:\n \n  If a format changes by adding a new part (e.g., car departure time) that\n    contains the landmark the \\ensuremath{\\mathsf{LRSyn}}\\xspace generated program uses, the program may\n    generate a spurious output.\n    %\n    This is only a problem if the new part added also has similar blueprint to\n    the existing ROIs.\n    %\n    For example, the program will not be mislead by an new banner advertisement\n    saying \\emph{Depart today for your dream destination!}\n    %\n \n  %\n  The second case is when the format inside the ROI changes.\n    %\n    In this case, the underlying assumption about invariant local structure\n    is violated and \\ensuremath{\\mathsf{LRSyn}}\\xspace is unlikely to cope with this variation.\n    %\n    One possible solution that we discuss in Section~\\ref{sec:conclusion} is to use a trained ML model to automatically re-synthesize as  in~\\cite{ij2019}.\n   \n   \n    %\n   \n   \n   \n   \n\n\\begin{comment}\nThe other variation that \\ensuremath{\\mathsf{LRSyn}}\\xspace\\xspace is not naturally robust to (2 in the\nlist above), sometimes the robustness of the underlying program synthesis\ntechnique is sufficient.\nFor example, if the text in the ROI in Figure~\\ref{fig:xxx} changed to \\emph{To\nDenver (DEN) at 8:18 PM on Friday, April 3}, the underlying text extraction\ntechnique (FlashFill) will naturally handle the variation.\nHowever, non-robustness to variations within ROIs is a natural limitation of\n\\ensuremath{\\mathsf{LRSyn}}\\xspace\\xspace.\nOne potential solution is to use an ML model to account for this variation as\nin~\\cite{hdef}.\nHowever, \\ensuremath{\\mathsf{LRSyn}}\\xspace\\xspace does provide a significant advantage here: the model can be\ntrained on significantly smaller inputs (i.e., the ROIs) compared to the\n\\cite{hdef} (which uses the whole document).\n\\end{comment}\n\n\\begin{comment}\n  \n\\subsection{Practical advantages}\n\nIn addition to robustness discussed above, \\ensuremath{\\mathsf{LRSyn}}\\xspace\\xspace provides several practical\nadvantages compared to similar techniques.\nFor one, we do not need an ML model to achieve similar results to other\ntechniques.\nThis greatly expands the kind of devices on which the technique can be deployed,\ni.e., on low power portable devices like inexpensive phones.\n\nSecondly, compared to~\\cite{hdef}, the number and size of programs that are\ngenerated by \\ensuremath{\\mathsf{LRSyn}}\\xspace\\xspace are significantly lower.\nThis makes manual examination of the programs generated feasible, allowing an\nextraction system developer to gain confidence in the program generated.\nFor example, in the program in Figure~\\ref{fig:yyy}, it is significantly easier\nto examine the correctness of the \\ensuremath{\\mathsf{LRSyn}}\\xspace program---{\\color{red} write one short\nsentence after the figure if finalized}\n\nAnother significant advantage is the extraction performance.\nThis is very relevant in industrial settings where an extraction system is\nexpected to process billions of documents each day ({\\color{red} put numbers\nfrom Amit?}).\nThe first step of executing an \\ensuremath{\\mathsf{LRSyn}}\\xspace program is using the $\\ensuremath{\\mathsf{Locate}}$ function to\nnarrow down on a small part of the input document.\nIn most cases, the $\\ensuremath{\\mathsf{Locate}}$ function is extremely inexpensive, involving just\nstring searches.\nHence, the rest of the program runs on a small sub-document and hence, is quite\nefficient as well.\nIn our experiments, programs learned by \\ensuremath{\\mathsf{LRSyn}}\\xspace outperformed those\nfrom~\\cite{hdef} by a factor of {\\color{red}$XXX$}.\n\n\\arsays{Training data required. This is not as easy to explain.}\n\\end{comment}\n\\subsection{HTML extraction}\n\\label{sec:html-eval}\nOur HTML document dataset, called the {\\em machine-to-human (M2H)} email\ndataset, consists of anonymized flight reservation emails.\nIt consists of $3503$ emails from $6$ different flight providers and is divided\ninto training and test sets of size $362$ and $3141$, respectively.\nFor each provider, we synthesize and test programs using Algorithm~\\ref{algo:lrsyn},\nas instantiated in Section~\\ref{sec:html}.\nWe compare against $2$ state-of-the-art techniques, namely\n$\\ensuremath{\\mathsf{NDSyn}}\\xspace$~\\cite{ij2019} and $\\textsf{ForgivingXPaths}$~\\cite{omari2017synthesis}.\n\n\\paragraph{Overall results}\nTable~\\ref{table:m2hoverall}~shows the average precision, recall and F1 scores\nacross various extraction tasks for $\\ensuremath{\\mathsf{ForgivingXPaths}}$, $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ and $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ for both\ncontemporary and longitudinal setting.\nAs seen in the table, $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ has near perfect precision and recall, with\n$\\ensuremath{\\mathsf{NDSyn}}\\xspace$ performing quite well with numbers $>0.9$.\nFurther, as expected, the gap between $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ and $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ is higher in the longitudinal dataset indicating that $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ can cope with format variations better than $\\ensuremath{\\mathsf{NDSyn}}\\xspace$.\n\nUnlike $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ or $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ which use a combination of structure and text\nprograms for extraction, $\\textsf{ForgivingXPaths}$ only outputs XPaths which\ncorrespond to the entire node, rather than the sub-text contained within that\nnode.\nConsequently, it has high recall and poor precision when the field value is a substring of the entire DOM node text.\nWe therefore omit it from the more detailed results below.\n\n\\begin{table}\n\\scriptsize\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{Contemporary} \\\\\n\\hline\nMetric & $\\ensuremath{\\mathsf{ForgivingXPaths}}$ & $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n\\hline\nAvg. Precision & 0.17 & 0.96 & 1.00\\\\ \\hline\nAvg. Recall & 0.99 & 0.91 & 1.00\\\\ \\hline\nAvg. F1 & 0.22 & 0.93 & 1.00 \\\\ \\hline\n\\end{tabular}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{Longitudinal} \\\\\n\\hline\nMetric & $\\ensuremath{\\mathsf{ForgivingXPaths}}$ & $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n\\hline\nAvg. Precision & 0.15 & 0.99 & 1.00\\\\ \\hline\nAvg. Recall & 0.98 & 0.89 & 1.00\\\\ \\hline\nAvg. F1 & 0.20 & 0.92 & 1.00 \\\\ \\hline\n\\end{tabular}\n\\caption{Overall scores of $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ and $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ on the M2H Contemporary and Longitudinal datasets}\n\\label{table:m2hoverall}\n\\end{table}\n\n\n\\paragraph{Detailed comparison}\nTable~\\ref{tab:m2hexpmt1} shows a more detailed drill-down of the F1 scores for\n$\\ensuremath{\\mathsf{NDSyn}}\\xspace$ and $\\ensuremath{\\mathsf{LRSyn}}\\xspace$, in the two settings.\nIn summary, in all cases, the hit to the F1-scores come from lower precision numbers\nrather than lower recall numbers.\n\n\\highlightbox{\n$\\ensuremath{\\mathsf{LRSyn}}\\xspace$ is very robust to variations in the longitudinal\nsetting, achieving $>95\\%$ F1 score in all $53$ out of $53$ fields, with a perfect F1 score of $1.00$ in $49$ cases.\nIn comparison, $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ achieves $>95\\%$ and perfect scores in $40$ and $33$ cases respectively.\n}\nIn many applications, having a score of $1.00$ is crucial, i.e., even a system\nwith $0.99$ precision cannot be deployed in practice.\nFor example, even a tiny imprecision in a system that automatically adds\ncalendar entries based on flight reservation emails is disastrous for millions\nof users.\nComparing the numbers precisely, $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ outperforms $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ in 19 and 20 out\nof the 53 fields in the contemporary and longitudinal setting, respectively.\nIn the remaining fields, the two approaches have comparable F1 scores.\n\nWe examined the domains \\emph{aeromexico} and \\emph{mytrips.amexgbt} where both\n$\\ensuremath{\\mathsf{LRSyn}}\\xspace$ and $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ achieved perfect scores. \nIn the \\emph{aeromexico} domain, each field has a unique dedicated ID attribute\nin the HTML domain which act as \\emph{implicit landmarks}, and both $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ and\n$\\ensuremath{\\mathsf{LRSyn}}\\xspace$ are able to latch on to this unique ID.\nFor example, the arrival time and departure city DOM nodes have id\n\\emph{arrival-city} and \\emph{departure-city}, and $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ produces a program\nthat searches for this ID across the whole document, emulating the landmark\nlocation step of a $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ program.\n\nIn the \\emph{mytrips.amexgbt} domain, the $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ program while perfectly\naccurate on all the variations in our dataset, is very fragile.\nIn the final CSS selector step of the web extraction component, it looks for the $10^{th}$ child of a DOM element corresponding to a flight details section.\nIncidentally, all new sections (car reservations, hotel reservations, etc) added\nin the new variations have at most $5$ children, and hence, they are\nautomatically ignored by the $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ program.\nAny new variation that would add a long enough section will break this program.\nIn contrast, the $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ program narrows down on the right region with a\nlandmark and is resistant to such variations.\n\n\\begin{table*}\n\\scriptsize\n\\begin{tabular}{|l|c|c|c||c|c|}\n\\hline\n & & \\multicolumn{2}{|c||}{ Contemporary} &  \\multicolumn{2}{|c|}{Longitudinal} \\\\\n {Fields} & Domain & $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ & $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ \\\\\n\\hline\n\\hline\n AIata & \\multirow[b]{4}{*}{ifly}   & 0.81 & \\textbf{1.00} & 0.64 & \\textbf{1.00} \\\\\n ATime &                            & 0.76 & \\textbf{1.00} & 0.62 & \\textbf{1.00} \\\\\n DIata &                            & 0.73 & \\textbf{1.00} & 0.55 & \\textbf{1.00} \\\\\n DDate &                            & 1.00 &         1.00  & 1.00 &         1.00  \\\\\n DTime &  alaska                    & 0.73 & \\textbf{1.00} & 0.55 & \\textbf{1.00} \\\\\n FNum  & \\multirow[t]{4}{*}{air}    & 1.00 &         1.00  & 1.00 &         1.00  \\\\\n Name  &                            & 1.00 &         1.00  & 0.99 &         0.99  \\\\\n Pvdr  &                            & --   &         --    & --   &         --    \\\\\n RId   &                            & 1.00 &         1.00  & 1.00 &         1.00  \\\\\n\n\\hline\n AIata & \\multirow{9}{*}{airasia} & 0.67 & \\textbf{1.00} & 0.67 & \\textbf{1.00} \\\\\n ATime &                          & NaN  & \\textbf{1.00} & NaN  & \\textbf{1.00} \\\\\n DIata &                          & 0.67 & \\textbf{1.00} & 0.67 & \\textbf{1.00} \\\\\n DDate &                          & 0.67 & \\textbf{1.00} & 0.67 & \\textbf{1.00} \\\\\n DTime &                          & NaN  & \\textbf{1.00} & NaN  & \\textbf{1.00} \\\\\n FNum  &                          & 1.00 &          1.00 & 0.96 & 0.96          \\\\\n Name  &                          & 1.00 &          1.00 & 1.00  & 1.00 \\\\\n Pvdr  &                          & 1.00 &          1.00 & 0.96 & 0.96          \\\\\n RId   &                          & 1.00 &          1.00 & 1.00 & 1.00          \\\\\n\n\\hline \n\\end{tabular}\n\\begin{tabular}{|c|c|c||c|c|}\n\\hline\n & \\multicolumn{2}{|c||}{ Contemporary} &  \\multicolumn{2}{|c|}{Longitudinal} \\\\\n {Domain} & $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ & $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ \\\\\n\\hline\n\\hline\\multirow{9}{*}{getthere}\n\n & 0.75 & \\textbf{1.00} & 0.74 & \\textbf{1.00} \\\\\n & 0.94 & \\textbf{1.00} & 0.91 & \\textbf{1.00} \\\\\n & 0.94 & \\textbf{1.00} & 0.95 & \\textbf{1.00} \\\\\n & 0.98 & \\textbf{1.00} & 0.95 & \\textbf{1.00} \\\\\n & 0.76 & \\textbf{1.00} & 0.78 & \\textbf{1.00} \\\\\n & 0.98 & \\textbf{1.00} & 0.98 & \\textbf{1.00} \\\\\n & 1.00 & 1.00 & 0.89 & \\textbf{1.00} \\\\\n & 0.98 & \\textbf{1.00} & 0.97 & \\textbf{1.00} \\\\\n & 0.93 & \\textbf{1.00} & 0.94 & \\textbf{1.00} \\\\\n\n\\hline\\multirow{9}{*}{delta}\n & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n & 0.94 & \\textbf{1.00} & 0.95 & \\textbf{1.00} \\\\\n & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n & 0.85 & \\textbf{0.97} & 0.91 & \\textbf{0.97} \\\\\n & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n\n\\hline \n\\end{tabular}\n\\begin{tabular}{|c|c|c||c|c|}\n\\hline\n & \\multicolumn{2}{|c||}{ Contemporary} &  \\multicolumn{2}{|c|}{Longitudinal} \\\\\n {Domain} & $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ & $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ \\\\\n\\hline\n\\hline\\multirow[b]{4}{*}{aero} & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                               & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                               & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                               & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n\\multirow[t]{4}{*}{mexico}     & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                               & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                               & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                               & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                               & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n\n\\hline\n\\multirow[b]{4}{*}{mytrips} & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                            & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                            & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                            & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n amex                          & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n \\multirow[t]{3}{*}{gbt}   & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                            & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                            & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                            & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n\n\\hline \n\\end{tabular}\n\\caption{F1 scores of \\ensuremath{\\mathsf{NDSyn}}\\xspace\\ and \\ensuremath{\\mathsf{LRSyn}}\\xspace\\ for M2H HTML dataset. The Pvdr field is not relevant for iflyalaskaair}\n\\label{tab:m2hexpmt1}\n\\vspace{-5ex}\n\\end{table*}\n\n\n\\begin{table}\n\\scriptsize\n\\begin{tabular}{|c|l|c|c|c|}\n\\hline\n{Domain} & Fields & $\\ensuremath{\\mathsf{AFR}}$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n\\hline\\multirow{9}{*}{AccountsInvoice}\n  & Amount & 0.99 & \\textbf{1.00} \\\\\n  & Chassis & 0.82 & \\textbf{0.99} \\\\\n  & CustAddr & 0.98 & \\emph{0.96} \\\\\n  & Date & 0.93 & \\textbf{0.98} \\\\\n  & Dnum & 0.96 & \\textbf{0.97} \\\\\n  & Engine & 0.82 & \\textbf{1.00} \\\\\n  & InvoiceAddress & 0.90 & \\textbf{0.95} \\\\\n  & Model & 0.75 & \\textbf{1.00} \\\\\n\n\\hline\\multirow{8}{*}{CashInvoice}\n  & Amount & 1.00 & 1.00 \\\\\n  & Chassis & 0.99 & 0.99 \\\\\n  & CustAddr & 0.99 & \\emph{0.97} \\\\\n  & Date & 0.99 & 0.99 \\\\\n  & Dnum & 0.96 & 0.96 \\\\\n  & Engine & 0.93 & \\textbf{0.95} \\\\\n  & InvoiceAddress & 0.99 & 0.99 \\\\\n  & Model & 0.99 & \\textbf{1.00} \\\\\n\n\\hline\\multirow{5}{*}{CreditNote}\n  & Amount & 1.00 & 1.00 \\\\\n  & CreditNoteAddress & 0.99 & \\textbf{1.00} \\\\\n  & CreditNoteNo & 0.94 & \\emph{0.93} \\\\\n  & CustRefNo & 1.00 & 1.00 \\\\\n  & Date & 1.00 & 1.00 \\\\\n  & RefNo & 1.00 & 1.00 \\\\\n\n\\hline\\multirow{6}{*}{SalesInvoice}\n  & Amount & 1.00 & 1.00 \\\\\n  & CustomerReferenceNo & 1.00 & 1.00 \\\\\n  & Date & 1.00 & 1.00 \\\\\n  & InvoiceAddress & 0.94 & \\textbf{0.99} \\\\\n  & RefNo & 0.99 & 0.99 \\\\\n  & SalesInvoiceNo & 0.99 & 0.99 \\\\\n\n\\hline\\multirow{6}{*}{SelfBilledCreditNote}\n  & Amount & 1.00 & 1.00 \\\\\n  & CustomerAddress & 1.00 & \\emph{0.99} \\\\\n  & CustomerReferenceNo & 0.99 & 0.99 \\\\\n  & Date & 1.00 & 1.00 \\\\\n  & DocumentNumber & 1.00 & 1.00 \\\\\n  & VatRegNo & 1.00 & 1.00 \\\\\n\n\\hline\n\\end{tabular}\n\\caption{F1 scores for Finance dataset}\n\\label{tab:financedataset}\n\\vspace{-3ex}\n\\end{table}\n\n\\begin{table}\n\\scriptsize\n\\begin{tabular}{|l|p{1.5cm}|c|c|}\n\\hline\nFields & {Domain} & $\\ensuremath{\\mathsf{AFR}}$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n \\hline\nAIata & \\multirow{9}{*}{aeromexico}        & 0.62 & \\textbf{0.65} \\\\\nATime &                                 & 0.69 & \\textbf{0.99} \\\\\nDIata &                                 & 0.36 & \\textbf{0.66} \\\\\nDDate &                                 & 0.71 & \\textbf{0.89} \\\\\nDTime &                                 & 0.65 & \\textbf{0.97} \\\\\nFNum  &                                 & 0.66 & \\textbf{0.83}  \\\\\nName  &                                 & 0.96 & \\textbf{0.98}  \\\\\nPvdr  &                                 & 0.69 & \\textbf{0.78}  \\\\\nRId   &                                 & 1.00 & 1.00            \\\\\n \n\\hline\n\\end{tabular}\n\\begin{tabular}{|p{1.5cm}|c|c|}\n\\hline\n{Domain} & $\\ensuremath{\\mathsf{AFR}}$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n\\hline\n\\multirow{9}{*}{getthere}  & 0.94 & \\textbf{1.00} \\\\\n                           & 0.87 & \\textbf{1.00} \\\\\n                           & 0.93 & \\textbf{1.00} \\\\\n                           & 0.96 & \\textbf{0.99} \\\\\n                           & 0.88 & \\textbf{1.00} \\\\\n                           & 0.94 & \\textbf{1.00} \\\\\n                           & 0.99 &         0.99  \\\\\n                           & 0.75 & \\textbf{1.00} \\\\\n                           & 0.89 & \\textbf{0.95} \\\\\n \n\\hline\n\\end{tabular}\n\n\\begin{tabular}{|l|p{1.5cm}|c|c|}\n\\hline\n{Fields} & Domain & $\\ensuremath{\\mathsf{AFR}}$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n\\hline\nAIata & \\multirow[b]{4}{*}{ifly.}     & 0.99 & \\textbf{1.00} \\\\\nATime &                                    & 0.95 &   \\textbf{1.00} \\\\\nDIata &                                    & 0.98 & \\textbf{1.00} \\\\\nDDate &                                    & 0.98 & \\emph{-} \\\\\nDTime &  \\multirow[t]{4}{*}{alaskaair}       & 0.95 & \\textbf{0.98} \\\\\nFNum  &                                    & 0.97 & \\textbf{1.00} \\\\\nName  &                                    & 0.98 & 0.98 \\\\\nPvdr  &                                    & 0.93 & \\textbf{0.99} \\\\\nRId   &                                    & 1.00 &         \\emph{0.86} \\\\\n \n\\hline\n\\end{tabular}\n\\begin{tabular}{|p{1.5cm}|c|c|}\n\\hline\n{Domain} & $\\ensuremath{\\mathsf{AFR}}$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n \\hline\n\\multirow[b]{4}{*}{mytrips}  & 0.85 & \\textbf{0.98} \\\\\n                             & 0.97 & \\textbf{1.00} \\\\\n                             & 0.96 & \\textbf{0.99} \\\\\n                             & 0.93 & \\textbf{1.00} \\\\\n\\multirow[t]{4}{*}{amexgbt}  & 0.99 & \\textbf{1.00} \\\\\n                             & 0.98 & \\textbf{1.00} \\\\\n                             & 0.98 & \\textbf{1.00} \\\\\n                             & 0.91 & \\textbf{0.99} \\\\\n                             & 0.61 & \\textbf{0.96} \\\\\n\n\\hline\n\\end{tabular}\n\\caption{F1 Score for M2H-Images Dataset}\n\\label{tab:m2himages}\n\\vspace{-2ex}\n\\end{table}\n\n\n\n\\subsection{Form Image Extraction}\n\\label{sec:forms-eval}\n\nWe consider two datasets of form images:\n\\begin{compactitem}\n\\item {\\em Finance} dataset:\nThis consists of ~850 images of receipts, purchase\norders, credit notes, sales invoice and similar such documents.\nHere, the training and test data are from the same time period and we only\nevaluate the contemporary setting.\n\\item {\\em M2H-Images} dataset:\nWe convert M2H emails from $4$ domains to images and extract the same fields as\nbefore.\nThis represents common scenarios in practice where HTML documents such\nas booking mails or receipts may be printed and then scanned again, say when\nexpense reports are filed.\nThe OCR service we use produced extremely poor\nresults on $2$ of the $6$ domains from the HTML experiments, and hence, we used only $4$ domains in this dataset (The same OCR service is used by our baseline as well, see below).\n\\end{compactitem}\nFor both datasets, we use a training data size of $10$ for each field, and compare\nagainst Azure Form Recognizer (\\ensuremath{\\mathsf{AFR}})~\\cite{AFR}, a cloud-based form data extraction service.\n\n\\paragraph{Overall Results}\nTable~\\ref{tab:imagesoverall}~shows the average precision, recall, F1 and\naccuracy scores for \\ensuremath{\\mathsf{AFR}}\\ and \\ensuremath{\\mathsf{NDSyn}}\\xspace\\, for both the Finance and M2H-image\ndatasets.\nAs we can see from the table, both $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ and $\\ensuremath{\\mathsf{AFR}}$ perform very well on the\nFinance dataset, with $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ performing marginally better.\nIn this dataset, the image formats do not vary much, resulting in these\nhigh-quality results.\n\\highlightbox{\n$\\ensuremath{\\mathsf{LRSyn}}\\xspace$ outperforms $\\ensuremath{\\mathsf{AFR}}$, a state-of-the-art industrial neural form extraction\nsystem with just $10$ training images per\nfield, having a precision of $0.97$ vs $0.90$ on the M2H-Images dataset.\n}\n\n\\begin{table}\n\\scriptsize\n\\parbox{.45\\linewidth}{\n\\centering\n\\begin{tabular}{|l|c|c|c|}\n\\hline\nMetric & $\\ensuremath{\\mathsf{AFR}} $ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n\\hline\nAvg. Pre.  & 0.98\t& 0.99 \\\\ \\hline\nAvg. Rec.  & 0.96\t& 0.99 \\\\ \\hline\nAvg. F1  & 0.97\t& 0.99 \\\\ \\hline\n\\end{tabular}\n\\caption*{Finance dataset}\n}\n\\hfill\n\\parbox{.45\\linewidth}{\n\\centering\n\\begin{tabular}{|l|c|c|c|}\n\\hline\nMetric & $\\ensuremath{\\mathsf{AFR}} $ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n\\hline\nAvg. Prec.  & 0.90\t& 0.97 \\\\ \\hline\nAvg. Rec.  & 0.93\t& 0.97 \\\\ \\hline\nAvg. F1  & 0.91\t& 0.97 \\\\ \\hline\n\\end{tabular}\n\\caption*{M2H-Images dataset}\n}\n\\vspace{-2ex}\n\\caption{Average precision, recall, F1 numbers on Finance and M2H-Images dataset (Ignoring DDate field in ifly.alaskaair)}\n\\label{tab:imagesoverall}\n\\vspace{-3ex}\n\\end{table}\n\n\\paragraph{Detailed comparison}\nTable~\\ref{tab:financedataset}~shows the results of $\\ensuremath{\\mathsf{AFR}}$ and $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ on the Finance dataset with respect to $34$ extraction tasks.\n$\\ensuremath{\\mathsf{LRSyn}}\\xspace$ performs better than $\\ensuremath{\\mathsf{AFR}}$ in $12$ out of the $34$ cases and is on par on the rest, with significant gains in some domains like \"AccountsInvoice\". \n\nThough the neural model in $\\ensuremath{\\mathsf{AFR}}$ is trained with thousands of invoices and\nreceipts, and further fine-tuned with our training data, we observe that it is\nsensitive to the region coordinates in a given document.\nIf these regions are translated, or if the document scan is tilted, $\\ensuremath{\\mathsf{AFR}}$\nproduces erroneous results.\nOn the other hand, $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ is partially robust to such changes as we use a text\nlandmark.\n$\\ensuremath{\\mathsf{AFR}}$ is marginally better than $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ in some extraction tasks.\nThese are cases where there is no clear bounding pattern for the field values.\nOn the other hand, $\\ensuremath{\\mathsf{AFR}}$'s semantic understanding of the data is not\naffected by boundary text patterns.\n\nTable~\\ref{tab:m2himages}~shows the results of $\\ensuremath{\\mathsf{AFR}}$ and $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ on the M2H\ndataset with respect to $45$ extraction tasks.\nThis dataset exhibits more variations at the visual level as compared to the\nFinance dataset, and hence, $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ performs better than $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ in $35$ out\nof the $45$ tasks and is on par on most of the remaining extraction tasks.\nThere is $1$ specific case where $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ fails altogether, producing no\nprograms.\nThese are cases where there is no local textual landmark geometrically near the\nfield value.\nHowever, the region around the field value may still be similar across documents.\nWe discuss the possibility of using visual landmarks as opposed to textual\nones in Section~\\ref{sec:conclusion}.\n\n\\paragraph{Summary of results}\nIn the HTML domain, the prior work \\ensuremath{\\mathsf{NDSyn}}\\xspace\\ is a high-performing system with F1 scores in the range of 0.9. \\ensuremath{\\mathsf{LRSyn}}\\xspace\\ is able to push the F1 scores to a perfect 1.0 in most cases. In longitudinal scenarios, \\ensuremath{\\mathsf{LRSyn}}\\xspace\\ improves \\ensuremath{\\mathsf{NDSyn}}\\xspace\\ in 20 out of 53 fields, with significant lift in F1 scores in many cases.\nIn the images domain, even with very little training data, \\ensuremath{\\mathsf{LRSyn}}\\xspace\\ matches AFR, which is a released product on contemporary settings, and outperforms AFR in longitudinal settings. In addition, \\ensuremath{\\mathsf{LRSyn}}\\xspace\\ produces simpler interpretable programs that match human intuition for extraction, and are much easier to maintain. Hence, we see a lot of promise in this approach.\n\n\\subsection{Nature of Synthesized Programs}\n\\label{sec:programs-eval}\n\nAdditionally, we performed secondary\nanalysis to understand the features of the $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ programs.\n\n\\paragraph{Program size}\nIn the HTML domain, we compared size of the $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ and $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ programs.\nSince the programs are naturally of different shapes, we only compare the web extraction part of the programs.\nThe final text extraction program is generally the same across both algorithms.\nNote that these numbers need to be taken in the context that $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ programs\nadditionally have a landmark and blueprint.\n\\highlightbox{\n  For the M2H dataset, the web extraction part of $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ programs have $2.95$\n  CSS selector components as compared to $8.51$ for $\\ensuremath{\\mathsf{NDSyn}}\\xspace$.\n}\n\n\\ignore{\nHere is an example region and text program synthesized for the \"Date\" field in \\emph{SalesInvoice} document, with respect to a landmark \"Date\". \\\\\nRegion program: $Extend(\"Date\", Relative(Right, EOL, false)$ \\\\\nText program: Extract a substring between \":\" and EOL\\\\\n\nThe region program starts from the landmark \"Date\" and extends right to collect all boxes until the end of line (EOL). And the corresponding text program extracts the text between a colon and EOL. We refer the readers to Table~\\ref{}~ in the Appendix for more examples. \\spsays{looks ugly, needs more work!!}\n\n\\paragraph{Synthesized programs}\nFigures~\\ref{lst:ndsynprogram}~and~\\ref{lst:lrsynprogram} show an example extraction program synthesized by $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ and $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ respectively. \nAs seen from these, extraction programs synthesized by $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ programs are way more complex as they process the entire HTML page, when compared to the programs synthesized by $\\ensuremath{\\mathsf{LRSyn}}\\xspace$, which only process the local region of interest.\nConsequently, if the formats change, the programs synthesized by $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ continue to function correctly.\nFurther, the shape and size of the $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ programs makes them easy to \nexamine by hand and gain confidence in their correctness.\n\\arsays{Do we have any numbers here?}\n\\highlightbox{\nThe $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ programs are significantly smaller compared to $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ programs.\nFor the web extraction component, $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ programs have $XXX$ components\nin their CSS selectors as compared to $YYY$ for $\\ensuremath{\\mathsf{NDSyn}}\\xspace$.\n}\n}\n\n\n\\paragraph{Quality of Inferred Landmarks}\nWe infer landmarks automatically using the techniques and scoring functions from Sections~\\ref{sec:algorithms} and~\\ref{sec:instantiations}.\nTo check the quality of landmark inference, we also asked data annotators to tag landmarks manually.\n\\highlightbox{\nIn $57$ out of $63$ clusters across all fields, the inferred landmarks are\nthe same as manually provided landmarks.\nIn $5$ of the remaining $6$ cases, the human annotator agreed that the inferred landmark \nwas of equal quality.\n}\nIn the remaining $1$ case, the algorithm chose the human annotated landmark as well, but \nin addition chose a disambiguating hierarchical landmark of low quality.\nIn particular, it disambiguated the term \\emph{Name} occurred in reference to both\nthe name of the passenger and the name in the  billing address.\nHere, the algorithm chose to disambiguate using the term \\emph{Meal}, i.e.,\npassengers have a meal preference while the billed person does not.\n\n\\subsection{Robustness of Experimental Results}\n\\label{sec:results-robustness}\n\n\n\\paragraph{Training set choice}\nIn our experiments, the training set is small compared to the\nfull dataset leading to a possibility of over-fitting, with different training\nsets potentially producing significantly differing results.\nHowever, our techniques are robust even with small training sets:\n\\begin{inparaenum}[(a)]\n  \\item Landmark identification can leverage the full dataset of both labeled\n    and unlabeled documents.\n  \\item $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ does not need to see all format variations that differ only\n    outside the ROIs, and a small set covering only the variations within the ROIs is sufficient.\n\\end{inparaenum}\nTo confirm this, we reran all our experiments on the M2H-HTML dataset with $4$\ndifferent randomly chosen training datasets.\nIn all runs, the F1 scores of the generated programs for each field and domain\nvaried by no more than $0.01$ from the results presented in\nTable~\\ref{tab:m2hexpmt1}, confirming our hypothesis that the results are robust\nto training set choice.\n\n\\paragraph{Landmark identification threshold}\nFor the landmark candidate score threshold, we picked threshold values that\nresulted in \\textasciitilde$10$ candidates for each case.\nTo study the robustness of the results to this choice, we reran all experiments\nwith a threshold value that returned $2X$ as many candidates.\nThe obtained results were exactly identical to the results presented in the previous sections. \nThis is expected as ``bad'' landmark candidates are eliminated in subsequent\nsteps, i.e., there is usually no program that extracts the required field value\nstarting from the landmark.\nHence, as long as the threshold is high enough to allow for some good landmark\ncandidates, it does not matter how many bad landmark candidates are included.\n\n\n\\ignore{\n\\section{Ablation studies}\nThough the training set is small, landmark identification uses both labeled\nand unlabeled documents. Hence, signal from landmarks is robust.\nAlso, extraction is unaffected if changes happen outside ROIs, helping with\nrobustness. Table~\\ref{tab:ablation_trainingsize}~shows F1 scores for M2H-HTML dataset in the longitudinal for $4$ different training sets. As we can see, $LRSyn$ is robust even with small training set sizes.\n\nThe procedure LandmarkCandidates only return candidates with a score>threshold.\nWe chose this threshold based on subset of training data so that we obtained\n~10 candidates. We believe that the results are not very sensitive to this\nthreshold as \"bad landmarks\" will be eliminated in subsequent steps. We are\nwilling to add an experiment to show this.\n\n\\begin{table}\n\\scriptsize\n\\begin{tabular}{|l|p{1.5cm}|c|c|c|c|}\n\\hline\nFields & {Domain} & Seed1 & Seed2 & Seed3 & Seed4\\\\\n \\hline\nAIata & \\multirow{9}{*}{aeromexico}     & 1.00 & 1.00 & 1.00 & 1.00\\\\\nATime &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nDIata &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nDDate &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nDTime &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nFNum  &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nName  &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nPvdr  &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nRId   &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|l|p{1.5cm}|c|c|c|c|}\n\\hline\nFields & {Domain} & Seed1 & Seed2 & Seed3 & Seed4\\\\\n\\hline\nAIata & \\multirow{9}{*}{getthere}       & 1.00 & 1.00 & 1.00 & 1.00\\\\\nATime &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nDIata &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nDDate &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nDTime &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nFNum  &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nName  &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nPvdr  &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nRId   &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|l|p{1.5cm}|c|c|c|c|}\n\\hline\nFields & {Domain} & Seed1 & Seed2 & Seed3 & Seed4\\\\\n \\hline\nAIata & \\multirow[b]{4}{*}{ifly.}       & 1.00 & 1.00 & 1.00 & 1.00\\\\\nATime &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nDIata &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nDDate &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nDTime & \\multirow[t]{4}{*}{alaskaair}   & 1.00 & 1.00 & 1.00 & 1.00\\\\\nFNum  &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nName  &                                 & 0.99 & 0.99 & 0.98 & 0.99\\\\\nRId   &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\n\\hline\n\\end{tabular}\n\\begin{tabular}{|l|p{1.5cm}|c|c|c|c|}\n\\hline\nFields & {Domain} & Seed1 & Seed2 & Seed3 & Seed4\\\\\n\\hline\nAIata & \\multirow[b]{4}{*}{mytrips}     & 1.00 & 1.00 & 1.00 & 1.00\\\\\nATime &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nDIata &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nDDate &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nDTime &  \\multirow[t]{4}{*}{amexgbt}    & 1.00 & 1.00 & 1.00 & 1.00\\\\\nFNum  &                                 & 1.00 & 0.98 & 1.00 & 0.98\\\\\nName  &                                 & 1.00 & 1.00 & 1.00 & 1.00\\\\\nPvdr  &                                 & 1.00 & 0.98 & 1.00 & 0.98\\\\\nRId   &                                 & 0.98 & 1.00 & 1.00 & 1.00\\\\\n\\hline\n\\end{tabular}\n\n\\caption{F1 Score for M2H-HTML Dataset in Longitudinal setting for $4$ different training sets}\n\\label{tab:ablation_trainingsize}\n\\vspace{-2ex}\n\\end{table}\n}\n\n\\subsection{HTML documents}\n\\label{sec:html}\n\n\n\\subparagraph{Landmarks and Landmark Candidates}\nWe use \\emph{$n$-grams} as landmarks ($n \\leq 5$), and the $\\ensuremath{\\mathsf{Locate}}$ function\nlists all the document DOM nodes and finds those containing the landmark\n$n$-gram.\nThe $\\ensuremath{\\mathsf{LandmarkCandidates}}$ procedure for identifying the top landmark\ncandidates lists all $n$-grams in the document, filters out those containing\nstop words, retains those $n$-grams common to all documents in the cluster, and\nthen scores them according to the criteria from Section~\\ref{sec:algorithms}.\nIn particular, the score for a landmark candidate $\\ensuremath{\\mathsf{m}}$ is given by a\nweighted sum of:\n\\begin{inparaenum}[(a)]\n  \\item the number of nodes in the path from the DOM nodes corresponding to\n    $\\ensuremath{\\mathsf{m}}$ and field value $\\ensuremath{v}$, \n  \\item the number of nodes in the smallest region enclosing both $\\ensuremath{\\mathsf{m}}$\n    and $\\ensuremath{v}$, and\n  \\item the Euclidean distance between $\\ensuremath{\\mathsf{m}}$ and $\\ensuremath{v}$ in the rendered\n    document.\n\\end{inparaenum}\n\n\\subparagraph{Blueprints}\nWe define the blueprint of a region to be the set of XPaths to the \\emph{common\nvalue} DOM nodes in the region, ignoring the DOM node order.\nFor example, the XPath \\texttt{body[1]\/table[4]\/tr[3]\/td[2]} is\nsimplified to \\texttt{body\/table\/tr\/td} before adding it to the blueprint set. \n\n\\subparagraph{Region Extraction DSL}\nA program in the region extraction DSL $\\mathcal{L}_{rx}$ is a pair of integers $(\\ensuremath{\\mathsf{parentHops}}, \\ensuremath{\\mathsf{siblingHops}})$\nof \\emph{parent hops} and \\emph{sibling hops}.\nGiven a landmark location $\\ensuremath{\\ell}$, the semantics of the $(\\ensuremath{\\mathsf{parentHops}},\n\\ensuremath{\\mathsf{siblingHops}})$ is as follows:\n\\begin{inparaenum}[(a)]\n  \\item from $\\ensuremath{\\ell}$ go up the DOM tree $\\ensuremath{\\mathsf{parentHops}}$ steps to get node $n_1$,\n  \\item from $n_1$ go $\\ensuremath{\\mathsf{siblingHops}}$ right to obtain node $n_2$, and\n  \\item the result is the set of all descendants of all sibling nodes between\n    $n_1$ and $n_2$ (inclusive).\n\\end{inparaenum} \nFor synthesizing program in $\\mathcal{L}_{rg}$ given the landmark location $\\ensuremath{\\ell}$\nand the annotated location $\\ensuremath{\\mathsf{Locs}}$, we first take the lowest common\nancestor (LCA) $n$ of $\\ensuremath{\\ell}$ and all nodes in $\\ensuremath{\\mathsf{Locs}}$.\nThe $\\ensuremath{\\mathsf{parentHops}}$ is given by the difference in depths of $n$ and $\\ensuremath{\\ell}$\nminus $1$, and $\\ensuremath{\\mathsf{siblingHops}}$ is given by the difference in index of the\nleft-most and right-most child of $n$ that have one of $\\ensuremath{\\ell}$ or\n$\\ensuremath{\\mathsf{Locs}}$ as a descendant.\n\n\\subparagraph{Value Extraction DSL}\nFor the value extraction DSL $\\mathcal{L}_{vx}$, we build upon the synthesis techniques\nfrom~\\cite{raza2017automated} and~\\cite{flashfill}, as in~\\cite{ij2019}.\nWe do not discuss the DSL and synthesis techniques in detail, but refer the\nreader to~\\cite{ij2019}.\nFrom a bird's eye view, a program in $\\mathcal{L}_{vx}$ consists of two parts: a web\nextraction program which extracts the particular DOM node which contains the\nfield value, and a text extraction program which extracts the field value from\nthe text present in the extracted DOM node.\nGiven an example $\\ensuremath{\\mathsf{R}} \\mapsto \\ensuremath{v}$, the synthesis procedure first finds\nthe DOM node $n$ which contains the text $\\ensuremath{v}$.\nThen, we use $\\ensuremath{\\mathsf{R}} \\mapsto n$ as the example to synthesize the web extraction\nprogram using techniques from~\\cite{raza2017automated}, and $\\mathsf{text}(n) \\mapsto\n\\ensuremath{v}$ as the example to synthesize the text extraction program using\ntechniques from~\\cite{flashfill}.\n\n\n\\begin{example}\n    Consider the task of extracting departure time from the email in Figure~\\ref{fig:formed_documents} (a). The synthesized region extraction, value extraction programs and blueprint are shown in Figure~\\ref{lst:lrsynprogram}.\n\\end{example}\n\n\n\\subsection{Form Images}\n\nThis domain concerns images that are obtained by scanning or photographing of\nphysical paper documents.\nThese images are first processed by an Optical Character Recognition\n(OCR) technique to obtain a list of text boxes along with their coordinates.\nThe form images domain is significantly more complex than the HTML domain as:\n\\begin{inparaenum}[(a)]\n    \\item The OCR output is generally very noisy, sometimes splitting up field\n      values into a varying number of different text boxes.\n    %\n   \n   \n   \n    %\n    \\item These documents do not come equipped with a hierarchical structure\n        that defines natural regions.\n    %\n   \n   \n\\end{inparaenum}\n\n\\subparagraph{Landmarks and Landmark Candidates}\nAs in the HTML case, we use $n$-grams as landmarks.\nThe $\\ensuremath{\\mathsf{Locate}}$ and $\\ensuremath{\\mathsf{LandmarkCandidates}}$ functions work similarly to the HTML\ncase with OCR output text boxes replacing DOM nodes.\nThe scoring function for $\\ensuremath{\\mathsf{LandmarkCandidates}}$ computes the score for a landmark\ncandidate $\\ensuremath{\\mathsf{m}}$ as a weighted sum of:\n\\begin{inparaenum}[(a)]\n  \\item the Euclidean distance between $\\ensuremath{\\mathsf{m}}$ and field value $\\ensuremath{v}$, and\n  \\item the area of the smallest rectangle that encloses both $\\ensuremath{\\mathsf{m}}$ and $\\ensuremath{v}$.\n\\end{inparaenum}\n\n\\subparagraph{Blueprints}\nRather than considering all common boxes for blueprinting as in the HTML case,\nwe instead use only the boxes containing the top $50\\%$ most frequent $n$-grams.\nThe blueprint of a region is defined to be the $\\ensuremath{\\mathsf{BoxSummary}}$ of each such box\ntaken in document order.\nThe $\\ensuremath{\\mathsf{BoxSummary}}$ of  $\\ensuremath{\\mathsf{box}}$ consists of $2$ parts:\n\\begin{inparaenum}[(a)]\n    \\item The frequent $n$-gram that is present in the box, and\n    \\item For each of the directions top, left, right, and bottom, the content\n        type in the text box that immediately neighbors $\\ensuremath{\\mathsf{box}}$ in the direction.\n        %\n        The content type of a box is either:\n        %\n        \\begin{inparaenum}[(1)]\n            \\item $\\bot$ if the box does not exist,\n            \\item the frequent $n$-gram in the text of the box if one exists, and\n            \\item $\\top$ if the box exists, but does not contain a frequent $n$-gram.\n        \\end{inparaenum}\n\\end{inparaenum}\n\n\\begin{example}\n    Consider the text box enclosing \\emph{Engine number} in the Accounts Invoice image in\n    Figure~\\ref{fig:formed_documents}(c).\n    %\n    The $n$-gram \\emph{Engine number} is frequent and hence, is included in the\n    blueprint.\n    %\n    The $\\ensuremath{\\mathsf{BoxSummary}}$ of the box is given by:\n    %\n    $\\langle\n        \\mathsf{ngram} \\mapsto \\text{\\emph{Engine number}},\n        \\mathsf{Top} \\mapsto \\bot,\n        \\mathsf{Left} \\mapsto \\text{\\emph{Chassis number}},$\n        $\\mathsf{Right} \\mapsto \\text{\\emph{Reg Date}}, \n        \\mathsf{Bot} \\mapsto \\top\n    \\rangle$\n    %\n    Here, \\emph{Engine number} and \\emph{Reg Date} are also a frequent $n$-grams, while the value of\n    the Engine number \\emph{4713872198212} is not.\n\\end{example}\n\n\\begin{figure}\n\\begin{alltt}\nRProg     := Disjunct(path, path, ...)\npath      := input | Expand(path, motion)\nmotion    := Absolute(dir, k)\n           | Relative(dir, pattern, inclusive)\ndir       := Top | Left | Right | Bottom\n\\end{alltt}\n\\vspace{-2ex}\n\\caption{The Form Images Region extraction DSL $\\mathcal{L}_{rx}$}\n\\label{fig:region_dsl}\n\\end{figure}\n\n\\subparagraph{Region Extraction DSL}\nFigure~\\ref{fig:region_dsl} depicts a novel region extraction DSL $\\mathcal{L}_{rx}$ for this domain.\n$\\mathcal{L}_{rx}$ has the following components:\n      The top operator is a disjunction of \\emph{path programs}:\n      operationally, these programs are executed in sequence and the first\n      non-null result is returned.\n      %\n      Due to the OCR noise and variations in form images, often a single\n      non-disjunctive path program is not sufficient.\n      %\n     \n     \n     \n      %\n      Each path program starts at the input landmark and repeatedly extends the\n      path in steps till the path's bounding box covers all the annotated\n      values.\n      %\n      Each extension step is specified by a direction and a motion.\n      %\n      The motion may be \\emph{absolute} (e.g., move right by $4$ boxes) or\n      \\emph{relative} (e.g., move down till you hit a text box that matches\n      the regex \\texttt{[0-9]\\{5\\}}).\n      %\n      The additional $\\mathtt{inclusive}$ parameter indicates whether the box\n      that matches the pattern should be included or excluded in the path.\n     \n     \n     \n\n\n\\begin{example}\n    \\label{ex:pathexample}\n    In Figure~\\ref{fig:formed_documents}(c), let us consider the landmark \\emph{Chassis number} and the\n    annotated value \\emph{WDX 28298 2L SHX 3 }.\n    %\n    The field value here is a variable-length string and the OCR splits the\n    value into $1-4$ separate boxes.\n   \n   \n    Consider the two region programs given below:\\\\\n    $\\bullet~\\mathsf{Ext(Ext(input, Abs(down, 1)), Rel(Right, [0-9]\\{13\\}, false))}$\\\\\n    $\\bullet~\\mathsf{Ext(Ext(input, Abs(down, 1)), Rel(Right, DATE, false))}$\\\\\n   \n    Both programs first move one step down from the landmark \\emph{Chassis\n    number}.\n    %\n    However, the first moves to the right till it hits a $13$ digit engine\n    number, while the other till it hits a date.\n    %\n    In case the engine number is present in a given form, the first program\n    produces a path which ends with the annotated value, while the second one\n    does so if the engine number is absent.\n    %\n    When combined disjunctively in the right order, they together cover both\n    cases.\n    %\n   \n   \n   \n   \n   \n   \n   \n    %\n   \n   \n   \n\\end{example}\n\nThe synthesis algorithm for $\\mathcal{L}_{rx}$ is split into two parts: generating path\nprograms and selecting path programs to construct a disjunction.\nFix a set of input documents with annotations.\nWe first synthesize path programs for small subsets (size $\\leq 3$) of input\ndocuments.\nFor synthesizing path programs, we use \\emph{enumerative\nsynthesis}~\\cite{transit,sygus} to generate numerous candidate programs and then\nfilter them by whether they cover the annotated values when starting from the\nlandmark.\nWe enumerate paths of up to $4$ motions, bounding $\\mathtt{k}$ to positive\nintegers $< 5$.\nFor $\\mathtt{pattern}$, we enumerate a finite set of regular expression patterns\ngenerated using a string profiling technique \\cite{flashprofile,confminer}\nover all the common and field text values present in the cluster.\nFor example, when given a cluster of documents of similar to\nFigure~\\ref{fig:formed_documents}c), one of the patterns returned is $\\mathtt{[0-9]\\{13\\}}$ as\nthe cluster contains many engine numbers of that form. \n\n\nAfter the enumeration step, we have a collection $\\{ P_1, \\ldots, P_n \\}$ of\npath programs that are each synthesized from a small subset of input examples.\nNow, we use the $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ algorithm from~\\cite{ij2019} to select a subset of these\nprograms to construct the disjunctive program. \nNow, for each program $P_i$, we define the set $\\mathsf{Ex}_i$ to be the subset of\n$\\mathsf{Examples}$ that $P_i$ is correct on.\nThe $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ algorithm selects a subset of $\\mathcal{P}$ of programs such that\n$\\bigcup_{P_i \\in \\mathcal{P}} \\mathsf{Ex}_i = \\mathsf{Examples}$, optimizing\nfor F1 score and program size~\\cite{ij2019}.\n\n\\begin{example}\n   Consider the two path programs from Example~\\ref{ex:pathexample}, along with the\n   additional program $\\mathsf{Ext(Ext(input, Abs(Down, 1)), Abs(Right, 2))}$.\n   %\n   Given a collection of such path programs, $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ builds the disjunctive\n   program using the two from Example~\\ref{ex:pathexample} as they cover a large\n   fraction of the documents in the cluster.\n   %\n   The additional program above will be ignored as it is only correct when the\n   chassis number field value is split into $2$ boxes by the OCR.\n\\end{example}\n\n\\subparagraph{Value Extraction DSL}\nFor the value extraction DSL, we use FlashFill~\\cite{flashfill}.\nThe input to the value extraction program is the concatenation of all the\ntext values in the boxes returned by the path program.\n\n\n\\begin{comment}\n\\spsays{Lincy removed the last k parameter, we just aggregate the text content and pass it to TText. But I think last k will be good for generality? }\nA program $(k, P_{FF})$ in the extraction DSL $\\mathcal{L}_{ex}$ is primarily a\nFlashFill~\\cite{flashfill} program $P_{FF}$ along with one auxiliary integer\nvalue $k$.\nGiven a path program, a program $(k, P_{FF})$ in $\\mathcal{L}_{ex}$ chooses the last $k$\nboxes that appear in the path, and returns the result of running $P_{FF}$ on the\nconcatenation of the text content of the last $k$ boxes appearing in the path.\nDuring synthesis, we first determine $k$ to be the smallest integer such that\nthe last $k$ boxes in the path of each example contains the corresponding\nannotated values in the document.\nFixing this $k$, w\n\\end{comment}\n\n\n\n\\section{Introduction}\n\\input{intro}\n\n\\section{Overview}\n\\label{sec:overview}\n\\input{overview}\n\n\\section{Formed Document Extraction}\n\\label{sec:problem}\n\\input{problem}\n\n\\section{Landmark and Region based Synthesis}\n\\label{sec:algorithms}\n\\input{algorithms}\n\n\n\\section{Instantiating \\ensuremath{\\mathsf{LRSyn}}\\xspace\\xspace}\n\\label{sec:instantiations}\n\\input{instantiations}\n\n\\section{Discussion}\n\\input{discussion}\n\n\\section{Evaluation}\n\\input{evaluation}\n\n\\section{Related work}\n\\input{related}\n\n\\vspace{-1ex}\n\\section{Conclusion}\n\\label{sec:conclusion}\n\\input{conclusion}\n\\balance\n\n\n\\section{Synthesis}\n\\begin{itemize}\n    \\item Provide dsls here\n\\end{itemize}\n\\spsays{TODO. Copied this segment from HDEF paper, revise it appropriately}\nMention cover loop\n\n\\paragraph{The HTML Extraction DSL}\nThe DSL $\\mathcal{L}_{ex}$ we use for the M2H email data-set is a combination of the\nHTML extraction DSL from~\\cite{} and the \\textsc{FlashFill}\ntext-transformation DSL~\\cite{}.\nWe do not explain the semantics of the DSL in detail, but instead\nrefer to the corresponding papers.\nInformally, we use the HTML extraction DSL to select a set of DOM nodes\nfrom each email, and then use the \\textsc{FlashFill} DSL to transform the\ntext inside the DOM nodes to the field values.\nThe HTML extraction DSL is inspired by CSS selectors and each program is a \ncomposition of atomic selectors that act as filters on the set of DOM nodes\nin the email.\nThe \\textsf{FlashFill} DSL extracts sub-strings based on regular expression\nmatches within the text content of each DOM node.\n\n\\begin{comment}\n\\subsection{Evaluation on FX DS3 dataset}\nWe also evaluate $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ and $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ on the DS3 dataset published by $\\textsf{ForgivingXPaths}$ authors. The dataset primarily consists of product web pages across $k$ different providers. For our experiments, we sample $15$ web pages uniformaly at random from this dataset as our train data and evaluate on the remaining data, for each provider. We do $5$ runs of this experiment with different train samples and report the average numbers.\n\nTable~\\ref{} shows the F1 scores for $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ and $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ on this dataset. As seen from this experiment, \\spsays{TODO}\n\n\\end{comment}\n\\begin{comment}\nTo that effect, we further split the test data into two parts: one comprising of emails which are consistent with the landmarks and blueprints, and other consisting of emails which are not consistent. Table~\\ref{}~indicates the proportion of this split. As we can see from the table, the number of emails which are not consistent is fairly less, even though the test data was gathered from a different time period. This also points that narrowing down to small regions helps, as the core regions do not change often in datasets. In this experiment, we perform our comparison only with test emails which are consistent with the training data for that domain. We discuss more about the remaining emails in the next section.\n\\end{comment}\n\n\n\n\\begin{comment}\n\\begin{table}\n\\small\n\\label{table:m2hexpmt_1}\n\\begin{tabular}{|c|l|c|c|c|}\n\\hline\n{Domain} & Fields & $\\textsf{FX}$ & $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n\\hline\n\\hline\\multirow{9}{*}{itinerary.westjet}\n\n& AIata &  &  1.00 & 1.00 \\\\\n& ATime &  &  1.00 & 1.00 \\\\\n& DIata &  &  1.00 & 1.00 \\\\\n& DDate &  &  1.00 & 1.00 \\\\\n& DTime &  &  1.00 & 1.00 \\\\\n& FNo   &  &  1.00 & 1.00 \\\\\n& Name  &  &  1.00 & 1.00 \\\\\n& Pvdr  &  &  1.00 & 1.00 \\\\\n& RId   &  &  1.00 & 1.00 \\\\\n\n\\hline\\multirow{9}{*}{aeromexico}\n\n& AIata &  &  1.00 & 1.00 \\\\\n& ATime &  &  1.00 & 1.00 \\\\\n& DIata &  &  1.00 & 1.00 \\\\\n& DDate &  &  1.00 & 1.00 \\\\\n& DTime &  &  1.00 & 1.00 \\\\\n& FNo   &  &  1.00 & 1.00 \\\\\n& Name  &  &  1.00 & 1.00 \\\\\n& Pvdr  &  &  1.00 & 1.00 \\\\\n& RId   &  &  1.00 & 1.00 \\\\\n\n\\hline\\multirow{9}{*}{mytrips.amexgbt}\n& AIata &  &  1.00 & 1.00 \\\\\n& ATime &  &  0.99 & 1.00 \\\\\n& DIata &  &  1.00 & 1.00 \\\\\n& DDate &  &  0.99 & 1.00 \\\\\n& DTime &  &  1.00 & 1.00 \\\\\n& FNo   &  &  1.00 & 1.00 \\\\\n& Name  &  &  1.00 & 1.00 \\\\\n& Pvdr  &  &  1.00 & 1.00 \\\\\n& RId   &  &  1.00 & 1.00 \\\\\n\n\\hline\\multirow{9}{*}{qatarairways.com.qa}\n\n& AIata &  &  1.00 & 1.00 \\\\\n& ATime &  &  0.99 & 1.00 \\\\\n& DIata &  &  1.00 & 1.00 \\\\\n& DDate &  &  0.99 & 1.00 \\\\\n& DTime &  &  1.00 & 1.00 \\\\\n& FNo   &  &  1.00 & 1.00 \\\\\n& Name  &  &  1.00 & 1.00 \\\\\n& Pvdr  &  &  1.00 & 1.00 \\\\\n& RId   &  &  0.99 & 1.00 \\\\\n\n\\hline\n\\end{tabular}\n\\end{table}\n\\end{comment}\n\n\\begin{comment}\n\\subsubsection{Inconsistent test emails}\nIn this experiment, we evaluate on test emails which were not consistent with train emails with respect to landmarks and blueprints. In the M2H dataset, there were only $4$ domains which had inconsistent emails. All these emails were marked inconsistent because the blueprints did not match owing to a different structure, though the landmark remained the same. If we group these outlier emails based on blueprints inferred using ground truth, delta.com had $k$ groups, Philippines had $k2$ groups, qatar had $k3$ and alaksa had $k4$ groups.\n \nThere are various ways we can deal with this issue. We can use NDSyn programs or a ML model to predict on these emails and add the labelled set to the training data. In this work, we adopt a simpler method: We sample some emails from this set and add it to the training data. Table~\\ref{}~shows the performance once more emails are added to the training data.\n\nInteresting example where landmark improved on addition of new data.\n\\end{comment}\n\nWe treat the \\emph{n-grams} appearing across all the HTML documents as landmark\ncandidates. We filter candidates which appear in our stop word file, like\npunctuation and pronouns. We also filter n-grams which are very lengthy.  Given\na landmark candidate and a document, field values (which are obtained from the\nannotations) We use two types of scores:\n\\begin{enumerate}\n    \\item We render the HTML document in a browser and find the sum of Euclidean distance between the landmark location and the field value locations. The intuition here is that a \\emph{good} landmark candidate will appear very close to the field values in all the documents.\n    \\item Given the landmark location and the field regions, we use the $\\mathsf{LearnRegionProgram}$ routine to compute the enclosing region. And we measure the region size by the number of nodes in this region. The main reasoning here is that a smaller region will have lesser number of nodes in practice.\n\\end{enumerate}\nThe final score is a sum of these two scores weighted appropriately for scale.\nIn our ablation experiments, we also experiment with other scoring measures like\nManhattan distance and left-aligned scores. \n\n\\subsubsection{$\\mathsf{LearnRegionProgram}$ and Blueprints for HTML documents}\nThe goal of $\\mathsf{LearnRegionProgram}$ function is to find a \\emph{small}\nenclosing region given a pair of landmark and field values $\\langle \\ell^m_d,\n\\ell^{f_T}_d \\rangle$. For simplicity, let us assume that there is $1:n$ between\nthe landmark and field values. We will describe the generalization to the $m:n$\ncase later.\n\nAn HTML document provides an explicit tree structure which we leverage to find\nthe smallest enclosing region given a landmark and field value. Intuitively, the\nencompassing smallest region will be a sibling region or the lowest common\nancestor (LCA) of landmark and the field values. We represent $P_{rg}$ using\n$\\mathsf{parentHops}, \\mathsf{siblingHops}$ such that they cover all the train\ndocuments. We omit the detailed procedure computing these hops and refer the\nreaders to the Appendix.\n\n\\paragraph{$\\mathsf{Blueprint}$ for a HTML web region} HTML document provides\nus with an explicit tree structure. For computing a blueprint for an input web\nregion, we gather a sequence of tags in terms of their XPaths from a\n$\\mathsf{PreordeTraversal}$ starting from the begin node and ending at the last\nnode of the region. We ignore the indexes in the \\spsays{Add more nuances here,\nwe filter out similar tag structures}\n\n\nWe first process structured images using Optical Character Recognition (OCR)\npackages~\\cite{ocr1,ocr2}, which returns a list of bounding boxes and their\nassociated texts. Like in HTML, we treat bounding boxes which appear very close\nto each other as \\emph{n-grams}. We gather the \\emph{n-grams} appearing across\nall the structured images as landmark candidates.  For scoring the landmark, we\nagain employ two scores as below. Unlike HTML which requires rendering, the\nbounding boxes here directly give us the visual coordinates.\n\n\\begin{enumerate}\n    \\item We find the sum of Euclidean distance between the landmark region and the field regions. We use the center of these regions to compute the distances.\n    \\item We also measure the size of the rectangle induced by the landmark region and the field region.\n\\end{enumerate}\nThe final score is a sum of these two scores weighted appropriately for scale.\n\n\n\\begin{comment}\n\\begin{algorithm}\n\\small\n\\caption{$\\mathsf{Blueprint}$ for a HTML web region}\n\\label{algo:blueprint}\n\\begin{algorithmic}[1]\n\\Require HTML web region $R$\n\\State $\\mathsf{BluePrint} \\gets \\mathsf{TagSequence}$ of $\\mathsf{PreordeTraversal}$ from the start of web region to end of web region\n\\State \\Return $\\mathsf{BluePrint}$\n\\end{algorithmic}\n\\end{algorithm}\n\\end{comment}\n\n\\spsays{Rough first draft}\nAs a first step, we do an OCR on structured images which returns bounding boxes and associated text. As we do not have explicit tag structure like HTML, the region program on top of OCR boxes is more complex. Recall that the goal of $\\mathsf{LearnRegionProgram}$ is to find\/enumerate region(s) satisfying the $\\ensuremath{\\mathsf{EncRgn}}$ property. Additionally, we want the outputted region to be as small as possible. \n\nFigure~\\ref{}~depicts the overall idea behind the region growing algorithm for images. Given an initial $\\mathsf{region}$, we can expand it in all directions so as to find an enclosing region. Specifically, we start with the landmark region as in HTML, and we would like to expand this region to encompass all the field regions. We expand in any direction only if required. To control the expansion, we leverage the following properties:\n\\begin{enumerate}\n    \\item If we know the field pattern, then we can continue expanding till we find this pattern. $\\mathsf{Expand(Bottom, fieldpattern)}$\n    \\item We can continue expanding till we hit a boundary pattern, which we can choose to include or exclude. $\\mathsf{Expand(Right, pattern, include\/exclude)}$\n    \\item We can expand in an absolute manner. For instance, $\\mathsf{Expand(Left, 1)}$\n\\end{enumerate}\n\n\\begin{example}\nFigure~\\ref{}~depicts the region growing procedure for an accounts invoice, for various fields.\n\\begin{itemize}\n    \\item If we want to extract the document date given $\\mathsf{Date}$ as a landmark, we can Expand one region to the right and extract the value. We can also keep expanding the region till a Date type is found.\n    \\item Likewise, if we want to extract account number, given $\\mathsf{Account No.}$ as landmark, we can expand one region to the bottom and extract the value.\n    \\item Extracting the chassis value is more complex, as it can contain varied number of tokens and might not abide to a strict field pattern. In this case, we traverse one region down and keep expanding to the right until we hit the engine value pattern, which we do exclude from this region.\n    \\item Extracting customer address or invoice address requires us to keep traversing in the downwards direction and collecting all the lines until we hit a pattern like zip code or a stopping criterion like the constant \\emph{Raised by} or \\emph{Invoiced by}. \n\\end{itemize}\n\\end{example}\n\n\\begin{verbatim}\nlanguage Visual;\nfeature double Score = Microsoft.ProgramSynthesis.Extraction.Forms.Learning.RankingScore;\n\n@start List<Region> roi := landmarks | bothDirections;\nList<Region> bothDirections := ExtendLineItems(roi, pattern)\n\t\t\t\t\t\t\t\t\t\t| ExtendAbsAbs(roi, axis, k, k)\n\t\t\t\t\t\t\t\t\t\t| ExtendAbsRel(roi, axis, k, pattern, isIncl)\n\t\t\t\t\t\t\t\t\t\t| ExtendRelAbs(roi, axis, pattern, isIncl, k)\n\t\t\t\t\t\t\t\t\t\t| ExtendRelRel(roi, axis, pattern, isIncl, pattern, isIncl);\n\nint k;\nAxis axis;\nRegexToken pattern;\nbool isIncl;\n\n@input List<Region> landmarks;\n\\end{verbatim}\n\n\\begin{comment}\n\\begin{figure}\n\\small\n\\begin{tabular}{r c l}\nregionProg & := & $\\;\\;$\\textsf{landmarks} \\\\\n          &    & | \\textsf{ExtendAbsAbs}(\\textit{regionProg}, \\textit{axis}, \\textit{k}, \\\\\n          &    & \\textit{k}) \\\\\n          &    & | \\textsf{ExtendAbsRel}(\\textit{regionProg}, \\textit{axis}, \\textit{k}\\\\\n          &    & \\textit{pattern}, \\textit{isIncl}) \\\\\n          &    & | \\textsf{ExtendRelAbs}(\\textit{regionProg}, \\textit{axis}, \\\\\n          &    & \\textit{pattern}, \\textit{incl}, \\textit{k}) \\\\\n          &    & | \\textsf{ExtendRelRel}(\\textit{regionProg}, \\textit{axis}, \\\\\n          &    & \\textit{pattern}, \\textit{isIncl}, \\textit{pattern}, \\textit{isIncl}) \\\\\n\\end{tabular}\n\\textit{axis} $\\in$ $\\langle Up, Down, Left, Right \\rangle$; Set[Pattern] \\textit{pattern};\n\\vspace{-2ex}\n\\caption{Syntax of the visual region-growing language $\\mathcal{L}_{rg}$}\n\\label{fig:visual-dsl}\n\\end{figure}\n\\end{comment}\n\n\\paragraph{Talk about enumerative synthesis}\nFigure~\\ref{fig:visual-dsl} describes the region DSL $\\L_{rg}$ for images. Starting from an initial landmark region, we have two broad cases:\n\\begin{compactitem}\n\\item The landmark region itself encompasses all the fields, in which case the $\\textsf{regionProg}$ just returns the landmark region\n\\item We expand the landmark region in such a way that we enclose the field regions. The expansion can be done along all four cardinal directions. We group the left-right as one group and top-down as another group. In each of this group, we can either expand in an \\emph{absolute} sense or in a \\emph{relative} way.  \n\\end{compactitem}\n\n\\paragraph{Cover loop in region growing}\nIn HTML, we could find the enclosing region with parent and sibling hops. In images, we grow the region  using Enumerative synthesis with the DSL shown above. To account for the variability in the images and the resulting OCR, we employ a cover loop to find a disjunction of programs to cover the training data, instead of a single program.\n\n\\spsays{Give example}\n\\spsays{Give cover loop pseudo code}\n\\spsays{Move all blueprinting logic to clustering?}\n\n\\subsubsection{Blueprints for images} Images do not provide an explicit tag structure which we can use for blueprints, unlike HTML. As first step, we compute a frequency histogram of n-grams occurring in the given input images. Recall that, we first perform an OCR on the input image which provides us with bounding boxes and their corresponding text. We retain the top $k\\%$ of the frequently occurring n-grams as the tags for images. Our goal is to describe a given input document with respect to these tags.\n\nFor doing so, we compute the neighborhood information for each tag in the $4$ cardinal directions. The neighbors for a given tag are close-by elements, and can either be another frequently occurring n-gram or a variable text. If it turns out to be a variable text, we denote the existence of the box, rather than a specific n-gram. So the given image in Figure~\\ref{fig:formsextraction}~can be described in terms of these tags as $\\langle <VatRegNo><Top>NIL<\/Top><Right>NIL><\/Right><Bottom>Exists<\/Bottom><Right>Nil><\/Right> <\/VatRegNo><InvoiceTo>...$ and so on. \\spsays{Need a better forms example}. Section~\\ref{}~provides more details on how we use BluePrints during clustering and inference.\n\n\\begin{theorem}\nGiven $\\mathsf{LearnRegionProgram}$ for an input domain, the synthesized program $\\P$ is robust to any changes occurring outside of the enclosed region.\n\\end{theorem}\n\n\n\\begin{comment}\n\\begin{table*}[t]\n\\small\n\\label{table:m2hexpmt2}\n\\resizebox{15cm}{!}{\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n{Domain} & \\multicolumn{2}{|c|}{ArrivalAirportIata} &\\multicolumn{2}{|c|}{ArrivalTime} & \\multicolumn{2}{|c|}{DepartureAirportIata} &\\multicolumn{2}{|c|}{DepartureDate} &\\multicolumn{2}{|c|}{DepartureTime} \\\\\n\\cline{2-11} &  RobustLmk & UserLmk & RobustLmk & UserLmk & RobustLmk & UserLmk & RobustLmk & UserLmk & RobustLmk & UserLmk \\\\\n\n\\hline\\multirow{2}{*}{ifly.alaskaair.com}\n&Arrives&Arrives&Arrives&Arrives&Departs&Departs&Departs&Departs&Departs&Departs\\\\\n&&&&&&&Confirmation code: $->$Term\/Gate&Term Gate&&\\\\\n\\hline\\multirow{2}{*}{delta.com}\n&Departure Time&To&Arrival Time&Arrival Time&Flight&From&Departure Time&Departure Time&Departure Time&Departure Time\\\\\n&Flight Number:&Arrives&Arrives&Arrives&Departs&Departs&Departs&Departs&Departs&Departs\\\\\n\\hline\\multirow{1}{*}{booking.airasia.com}\nArrive&Arrive&Arrive&Arrive&Depart&Depart&Depart&Depart&Depart&Depart&Depart\\\\\n\\hline\\multirow{1}{*}{getthere.com}\n&Arrive:&Arrive:&Arrive:&Arrive:&Depart:&Depart:&Depart:&Depart:&Depart:&Depart: \\\\\n\\hline\\multirow{1}{*}{t.delta.com}\n&ARRIVE&ARRIVE&ARRIVE&ARRIVE&DEPART&DEPART&DEPART&DEPART&DEPART&DEPART\\\\\n\\hline\\multirow{2}{*}{philippineairlines.com}\n&Departure&To&Arrival&Arrival&New&From&Departure&Departure&Departure&Departure\\\\\n&Philippine Airlines&Outbound&Philippine Airlines&Outbound&Outbound&Outbound&Family:->Fare&Outbound&Philippine Airlines&Outbound\\\\\n\\hline\\multirow{1}{*}{itinerary.westjet.com}\n&Arrival:&Arrival:&Arrival:&Arrival:&Departure:&Departure:&Flight Number&Flight Number&Departure:&Departure:\\\\\n\\hline\\multirow{1}{*}{aeromexico.com}\n&Route&Route&Arrives&Arrives&Route&Route&Date&Date&Departs&Departs\\\\\n\\hline\\multirow{1}{*}{mytrips.amexgbt.com}\n&Destination:&Destination:&Arriving:&Arriving:&Origin:&Origin:&Departing:&Departing:&Departing:&Departing:\\\\\n\\hline\\multirow{1}{*}{qatarairways.com.qa}\n&Arrival&Arrival&Arrival&Arrival&Departure&Departure&Arrival&Arrival&Departure&Departure\\\\\n\\hline\n\\end{tabular}\n}\n\\caption{Landmarks detected vs User Selected Landmarks}\n\\end{table*}\n\n\\begin{table*}\n\\small\n\\label{table:m2hexpmt2}\n\\resizebox{15cm}{!}{\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n{Domain} & \\multicolumn{2}{|c|}{FlightNumber} &\\multicolumn{2}{|c|}{Name} & \\multicolumn{2}{|c|}{Provider} &\\multicolumn{2}{|c|}{ReservationId} \\\\\n\\cline{2-9} &  RobustLmk & UserLmk & RobustLmk & UserLmk & RobustLmk & UserLmk & RobustLmk & UserLmk \\\\\n\n\\hline\\multirow{1}{*}{ifly.alaskaair.com}\n&Departs&Flight&Travelers:&Travellers&&&Travelers:->Confirmation code:&ConfirmationCode\\\\\n\\hline\\multirow{2}{*}{delta.com}\n&Flight&Flight&Name&Name&Carrier&Carrier&Flight Information (Record Locator :&Flight Information (Record Locator : \\\\\n&Flight Number:&Flight Number:&Who's Coming Along&Who's Coming Along&Flight Number:&Flight Number:&Confirmation Number :&Confirmation Number : \\\\\n\\hline\\multirow{1}{*}{booking.airasia.com}\n&Depart&Depart&Guests&Guests&Depart&Depart&Booking number&Booking number\\\\\n\n\\hline\\multirow{1}{*}{getthere.com}\n&Flight\/Equip&Flight\/Equip&Meal:->Name:&Name&Flight\/Equip&Flight\/Equip&Airline Record Locator&Airline Record Locator\\\\\n\\hline\\multirow{2}{*}{t.delta.com}\n&DEPART&DEPART&Name:&Name:&DEPART&DEPART&Confirmation \\#:&Confirmation \\#: \\\\\n&&&&&&&Your Trip Confirmation \\#:&Your Trip Confirmation \\#: \\\\\n\\hline\\multirow{2}{*}{philippineairlines.com}\n&Flight&Flight&Dear&Dear&Flight&Flight&Booking reference:&Booking reference:\\\\\n&Philippine Airlines&Philippine Airlines&Passengers&Passengers&Philippine Airlines&Philippine Airlines&Reference:&Reference:\\\\\n\\hline\\multirow{1}{*}{itinerary.westjet.com}\n&Flight Number->WESTJET&Flight Number&Seat(s):&Seat(s):&Flight Number->WESTJET&WESTJET&WestJet&WestJet\\\\\n\\hline\\multirow{1}{*}{aeromexico.com}\n&Flt \\#&Flt \\#&Passengers&Passengers&Flt \\#&Flt \\#&Confirmation Code:&Confirmation Code:\\\\\n\\hline\\multirow{1}{*}{mytrips.amexgbt.com}\n&Flight:&Flight:&Booking Reference:&Traveler&Flight:&Flight:&Airline Booking Ref:&Airline Booking Ref:\\\\\n\\hline\\multirow{2}{*}{qatarairways.com.qa}\n&Class&Class&Passenger name\/ E&Passenger name&Class&Class&Booking reference (PNR)&Booking reference (PNR)\\\\\n&Class&Class&&&Class&Class&Booking reference (PNR)&Booking reference (PNR)\\\\\n\\hline\n\\end{tabular}\n}\n\\caption{Landmarks detected vs User Selected Landmarks}\n\\end{table*}\n\\end{comment}\n\n\n\\begin{comment}\n\\begin{table*}[t]\n\\small\n\\label{table:m2hexpmt1}\n\\begin{tabular}{|c|c|c|c||c|c|c|c|c|c|c|c|c|}\n\\hline\n \\multirow{\\shortstack[c]{Domain}} & \\multirow{Fields} & \\multicolumn{4}{|c|}{NDSyn} & \\multicolumn{4}{|c|}{$\\hisyn$}\\\\\n\\cline{3-19}&  & #Programs & Pre. & Rec. & F1 & #Programs & Pre. & Rec. & F1\\\\\n\\hline\n\\hline\\multiline{ifly.alaskaair.com}\n& ArrivalAirportIata & 2 & 0.9989 & 0.4718 & 0.6409 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n& ArrivalTime & 2 & 0.9697 & 0.4576 & 0.6218 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n& DepartureAirportIata&2&0.8888&0.4198&0.5703&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00 \\\\\n& DepartureDate&1&1.00&1.00&1.00&2&2&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00 \\\\\n& DepartureTime&2&0.8989&0.4241&0.5763&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00 \\\\\n& FlightNumber&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\ \n& Name&1&1.00&0.9751&0.9874&1&1&1.00&0.9751&0.9874&0& & & & &1.00&0.9751&0.9874\\\\ \n& ReservationId&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n\\hline\\multiline{delta.com}\n& ArrivalAirportIata&2&0.9967&0.9967&0.9967&2&2&1.00&1.00&1.00&7&2&0.0000&0.0000&0.0000&0.9967&0.9967&0.9967\\\\\n& ArrivalTime&2&0.9962&0.9975&0.9968&2&2&1.00&1.00&1.00&7&2&0.0000&0.0000&0.0000&0.9962&0.9975&0.9968\\\\\n& DepartureAirportIata&2&0.9934&0.9967&0.9950&2&2&0.9934&0.9967&0.9950&0& & & & &0.9934&0.9967&0.9950\\\\\n& DepartureDate&1&0.9320&0.9659&0.9486&2&2&1.00&1.00&1.00&7&1&0.1429&0.3636&0.2052&0.9951&0.9986&0.9968\\\\\n& DepartureTime&2&1.00&0.9975&0.9987&2&2&1.00&1.00&1.00&7&2&0.0000&0.0000&0.0000&1.00&0.9975&0.9987\\\\\n& FlightNumber&2&1.00&1.00&1.00&2&2&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& Name&2&0.9447&0.8795&0.9109&2&2&0.9508&1.00&0.9748&7&2&0.0000&0.0000&0.0000&0.9508&0.9951&0.9724\\\\\n& Provider&2&1.00&1.00&1.00&2&2&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& ReservationId&2&1.00&0.9915&0.9957&2&2&1.00&0.9989&0.9994&8&2&1.00&0.2222&0.3636&1.00&0.9915&0.9957\\\\\n\\hline\\multiline{booking.airasia.com}\n& ArrivalAirportIata&1&0.5000&1.00&0.6667&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& ArrivalTime&0& NaN&0.0000&0.0000&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& DepartureAirportIata&1&0.5000&1.00&0.6667&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& DepartureDate&1&0.5000&1.00&0.6667&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& DepartureTime&0& NaN&0.0000&0.0000&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& FlightNumber&1&1.00&0.8974&0.9459&1&1&1.00&0.9189&0.9577&1&1&1.00&0.5000&0.6667&1.00&0.8974&0.9459\\\\\n& Name&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& Provider&1&1.00&0.8974&0.9459&1&1&1.00&0.9189&0.9577&1&1&1.00&0.5000&0.6667&1.00&0.8974&0.9459\\\\\n& ReservationId&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n\\hline\\multiline{getthere.com}\n& ArrivalAirportIata&3&0.7356&0.8384&0.7836&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& ArrivalTime&2&0.9384&0.9114&0.9247&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& DepartureAirportIata&3&0.9426&0.9452&0.9439&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& DepartureDate&3&0.9834&0.9378&0.9601&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& DepartureTime&2&0.8659&0.8619&0.8639&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& FlightNumber&1&0.9409&0.9589&0.9498&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& Name&1&0.8285&0.9641&0.8912&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& Provider&1&0.9804&0.9530&0.9665&1&1&1.00&1.00&1.00&1&1&0.0000&0.0000&0.0000&1.00&1.00&1.00\\\\\n& ReservationId&1&1.00&0.8757&0.9337&1&1&1.00&1.00&1.00&1&1&0.0000&0.0000&0.0000&1.00&0.9941&0.9970\\\\\n\\hline\\multiline{t.delta.com}\n& ArrivalAirportIata&1&0.9996&0.9979&0.9987&1&1&1.00&0.9982&0.9991&2&1&0.9231&0.9231&0.9231&0.9996&0.9979&0.9987\\\\\n& ArrivalTime&1&0.9978&1.00&0.9989&1&1&0.9982&1.00&0.9991&2&1&0.9231&1.00&0.9600&0.9978&1.00&0.9989\\\\\n& DepartureAirportIata&1&1.00&0.9986&0.9993&1&1&1.00&0.9986&0.9993&2&1&1.00&1.00&1.00&1.00&0.9986&0.9993\\\\\n& DepartureDate&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& DepartureTime&1&0.9975&1.00&0.9987&1&1&0.9975&1.00&0.9987&2&1&1.00&1.00&1.00&0.9975&1.00&0.9987\\\\\n& FlightNumber&2&1.00&0.9634&0.9814&1&2&1.00&0.9687&0.9841&2&2&1.00&1.00&1.00&1.00&0.9688&0.9842\\\\\n& Name&1&0.9976&0.9976&0.9976&1&1&0.9976&0.9976&0.9976&63&1&0.0000&0.0000&0.0000&0.9976&0.9976&0.9976\\\\\n& Provider&2&0.9928&0.9400&0.9657&1&2&1.00&0.9472&0.9729&2&2&1.00&0.6250&0.7692&1.00&0.9453&0.9719\\\\\n& ReservationId&1&1.00&1.00&1.00&2&2&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n\\hline\\multiline{alaskaair.com}\n& ArrivalAirportIata&2&1.00&1.00&1.00&6&5&1.00&0.9990&0.9995&27&2&1.00&1.00&1.00&1.00&0.9990&0.9995\\\\\n& ArrivalTime&2&1.00&1.00&1.00&6&5&1.00&0.9990&0.9995&27&2&1.00&1.00&1.00&1.00&0.9990&0.9995\\\\\n& DepartureAirportIata&2&1.00&1.00&1.00&6&5&1.00&0.9990&0.9995&27&2&1.00&1.00&1.00&1.00&0.9990&0.9995\\\\\n& DepartureDate&2&1.00&0.9997&0.9998&6&3&1.00&0.9948&0.9974&38&2&1.00&0.9932&0.9966&1.00&0.9948&0.9974\\\\\n& DepartureTime&2&1.00&1.00&1.00&6&5&1.00&0.9990&0.9995&27&2&1.00&1.00&1.00&1.00&0.9990&0.9995\\\\\n& FlightNumber&1&0.9603&0.9995&0.9795&2&2&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& Name&2&1.00&1.00&1.00&2&2&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& Provider&2&1.00&0.9644&0.9819&2&2&1.00&0.9767&0.9882&5&2&1.00&0.5333&0.6956&1.00&0.9735&0.9866\\\\\n& ReservationId&2&1.00&1.00&1.00&2&2&1.00&1.00&1.00&165&2&1.00&1.00&1.00&1.00&1.00&1.00\\\\\n\\hline\\multiline{philippineairlines.com}\n& ArrivalAirportIata&3&0.9191&1.00&0.9578&2&3&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& ArrivalTime&2&1.00&1.00&1.00&2&3&1.00&0.9811&0.9905&0& & & & &1.00&0.9811&0.9905\\\\\n& DepartureAirportIata&2&0.9937&0.9937&0.9937&2&2&0.9932&0.9932&0.9932&5&2&1.00&1.00&1.00&0.9937&0.9937&0.9937\\\\\n& DepartureDate&2&0.9815&0.9463&0.9636&2&2&0.9815&0.9463&0.9636&0& & & & &0.9815&0.9463&0.9636\\\\\n& DepartureTime&2&1.00&1.00&1.00&2&3&1.00&0.9811&0.9905&0& & & & &1.00&0.9811&0.9905\\\\\n& FlightNumber&3&0.7397&0.6792&0.7082&2&3&1.00&0.9308&0.9642&0& & & & &1.00&0.9308&0.9642\\\\\n& Name&3&1.00&0.9745&0.9871&2&2&1.00&0.9954&0.9977&0& & & & &1.00&0.9954&0.9977\\\\\n& Provider&2&1.00&1.00&1.00&2&2&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& ReservationId&2&1.00&1.00&1.00&2&2&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n\\hline\\multiline{itinerary.westjet.com}\n& ArrivalAirportIata&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& ArrivalTime&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& DepartureAirportIata&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& DepartureDate&1&1.00&0.9956&0.9978&1&1&1.00&1.00&1.00&73&1&1.00&0.9757&0.9877&1.00&0.9956&0.9978\\\\\n& DepartureTime&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& FlightNumber&1&1.00&1.00&1.00&1&1&1.00&0.9965&0.9982&0& & & & &1.00&0.9965&0.9982\\\\\n& Name&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&73&1&1.00&1.00&1.00&1.00&1.00&1.00\\\\\n& Provider&1&1.00&1.00&1.00&1&1&1.00&0.9965&0.9982&0& & & & &1.00&0.9965&0.9982\\\\\n& ReservationId&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n\\hline\\multiline{aeromexico.com}\n& ArrivalAirportIata&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& ArrivalTime&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& DepartureAirportIata&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& DepartureDate&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& DepartureTime&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& FlightNumber&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& Name&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& Provider&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n\\hline\\multiline{mytrips.amexgbt.com}\n& ReservationId&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& ArrivalAirportIata&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& ArrivalTime&1&0.9948&1.00&0.9974&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& DepartureAirportIata&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& DepartureDate&1&0.9955&1.00&0.9977&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& DepartureTime&1&0.9955&1.00&0.9977&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& FlightNumber&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& Name&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& Provider&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n& ReservationId&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n\\hline\\multiline{qatarairways.com.qa}\n& ArrivalAirportIata&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&20&1&1.00&1.00&1.00&1.00&1.00&1.00\\\\\n& ArrivalTime&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&20&1&1.00&1.00&1.00&1.00&1.00&1.00\\\\\n& DepartureAirportIata&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&20&1&1.00&1.00&1.00&1.00&1.00&1.00\\\\\n& DepartureDate&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&20&1&1.00&1.00&1.00&1.00&1.00&1.00\\\\\n& DepartureTime&1&1.00&1.00&1.00&1&1&1.00&1.00&1.00&20&1&1.00&1.00&1.00&1.00&1.00&1.00\\\\\n& FlightNumber&1&0.9982&1.00&0.9991&2&1&0.9981&1.00&0.9990&20&1&1.00&1.00&1.00&0.9982&1.00&0.9991\\\\\n& Name&1&0.9988&1.00&0.9994&1&1&0.9988&1.00&0.9994&0& & & & &0.9988&1.00&0.9994\\\\\n& Provider&1&1.00&1.00&1.00&2&1&1.00&1.00&1.00&20&1&1.00&1.00&1.00&1.00&1.00&1.00\\\\\n& ReservationId&2&1.00&0.9967&0.9983&2&2&1.00&1.00&1.00&0& & & & &1.00&1.00&1.00\\\\\n\\hline\n\\end{tabular}\n\\caption{#Program, Precision, Recall and F1 numbers on HTML extraction scenario (M2H dataset) with NDSyn and $\\hisyn$}\n\\end{table*}\n\n\\end{comment}\n\n\\begin{comment}\n\\item FX Dataset\n    \\begin{itemize}\n    \\item Description: E-commerce website product pages from 4 domains(bestbuy, currys, ebuyer and pricerunner).\n    \\item Methodology:\n    Out of the 1000 emails provided for each domain, we first cleaned the dataset to remove pages where there were annotations missing for a given field. After this, 150 pages were sampled randomly from the now cleaned dataset to create training sets. We further considered 5 seeds to randomly sample our trainsets. The trainsets sampled were of sizes 5, 10, 15 and 20 pages. The remaining pages were used as the test set to calculate the following metrics. \n    \\item Metrics: Precision, Recall, F1 score, \\#Programs, Method of landmark scoring, \\#NotInCluster\n    \\end{itemize}\n\\end{comment}\n\n\\spsays{TODO}\n\\spsays{Write this formally}\n\\spsays{Give example? Should we add code for this, as we might run out of\nspace?}\nLandmark values can occur at multiple places in a given document. Our aim is to\nnarrow down the region precisely. We proceed as follows:\n\\begin{enumerate}\n    \\item We grow the region around the landmark and field values. And compute\n    the blueprint of this small region. We also use this region program to\n    grow the region around other occurrences of this same landmark value. And\n    compute the blueprints of these regions. If the blueprints differ, then\n    we are done. We already have a disambiguating <landmark, blueprint> pair.\n    \\item If the <landmark, blueprint> is ambiguous, we learn one more level\n    of hierarchy. For this, we treat the current landmark value as a field, and\n    learn a landmark for this value.\n    \\item We repeat this procedure till we find a disambiguating <landmark,\n    blueprint> pair.\n\\end{enumerate}\n\nAt inference time, we locate the topmost landmark value in the hierarchy, in the\ngiven document. Then we locate the next nearest landmark value and so on till we\nlocate the last value of the landmark. Then we use the region program to grow\nthe regions, match the blueprints and run extraction program on each of these\nregions. \n\n\\begin{figure*}[th]\n\\centering\n\\begin{minipage}{5cm}\n\\includegraphics[width=5cm]{images\/forms_regiongrowing.png}\n\\label{fig:formsregion}\n\\end{minipage}\n\\qquad\n\\begin{minipage}{7cm}\n\\frame{\\includegraphics[width=9cm]{images\/forms_image2_annotated.png}}\n\\label{fig:formsextraction2}\n\\end{minipage}\n\\caption{Improve these images.Anonymize these images}\n\\end{figure*}\n\n\n\nOur travel emails dataset is called the {\\em machine-to-human (M2H)} dataset.\nWe perform two experiments with the M2H dataset: contemporary and longitudinal.\nIn the contemporary experiment, the training and test datasets contain\nanonymized emails from the same time period.\nIn the longitudinal experiment, we evaluate our method with training and test\ndata across different time periods.\nThe M2H dataset consists of anonymized emails from $6$ different flight\nproviders. Each domain has  2 to 10 structural variations. We perform our\nexperiments one provider at a time. We synthesize programs for a given provider\nwith the training data and predict on the corresponding test data.\nTable~\\ref{table:m2hdataset}~shows the training and test data sizes for each\nflight domain.\n\nWe compare the performance of \\ensuremath{\\mathsf{LRSyn}}\\xspace~with two other approaches: (1) the \\ensuremath{\\mathsf{NDSyn}}\\xspace~algorithm from prior work on heterogeneous data extraction~\\cite{ij2019}, and (2) the \\ensuremath{\\mathsf{ForgivingXPaths}}\\ algorithm~\\cite{omari2017synthesis}.\nOur goal is to extract various fields from these varied data formats with an intention to answer the following questions:\n\\begin{compactitem}\n\\item \\emph{Q1: How does $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ compare with previous approaches, namely $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ and $\\textsf{ForgivingXPaths}$ in  contemporary and longitudinal settings?}\n\\item \\emph{Q2: How do the automatically learnt landmarks by $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ compare to user provided landmarks?} \n\\item \\emph{Q3: How do the programs  synthesized by $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ and  $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ compare in terms of simplicity and robustness?}\n\\end{compactitem}\n\n\\begin{table}\n\\scriptsize\n\\begin{tabular}{|l|c|c|c|}\n\\hline\nDomain & \\shortstack[c]{Train \\\\ Size} & \\shortstack[c]{Test \\\\ Size}\\\\\n\\hline\niflyalaskaair & 100 & 1092\\\\ \\hline\ndelta & 102 & 928 \\\\ \\hline\ngetthere & 50 & 154 \\\\ \\hline\n\\end{tabular}\n\\hspace{3ex}\n\\begin{tabular}{|l|c|c|c|}\n\\hline\nDomain & \\shortstack[c]{Train \\\\ Size} & \\shortstack[c]{Test \\\\ Size}\\\\\n\\hline\nairasia & 25 & 24 \\\\ \\hline\naeromexico & 35 & 303 \\\\ \\hline\namex & 50 & 640 \\\\ \\hline\n\\end{tabular}\n\\caption{M2H dataset}\n\\label{table:m2hdataset}\n\\vspace{-2ex}\n\\end{table}\n\\paragraph{Overall results}\nTable~\\ref{table:m2hoverall}~shows the average precision, recall and F1 scores across various extraction tasks for $\\ensuremath{\\mathsf{ForgivingXPaths}}$, $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ and $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ for both contemporary and longitudinal setting. Unlike $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ or $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ which use a combination of structure and text programs for extraction, $\\textsf{ForgivingXPaths}$ only outputs XPaths which correspond to the entire node, rather than the sub-text contained within that node. Consequently, $\\textsf{ForgivingXPaths}$ has high recall and poor precision. We therefore omit $\\textsf{ForgivingXPaths}$ from more detailed results tables below. We also observe that the gap between $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ and  $\\ensuremath{\\mathsf{NDSyn}}\\xspace$, as measured by precision, recall (and thereby F1 score) is larger in the longitudinal evaluation as compared to the contemporary one. This is because programs synthesized by $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ generalize better and handle format changes better than $\\ensuremath{\\mathsf{NDSyn}}\\xspace$.\n\n\\highlightbox{\n$\\ensuremath{\\mathsf{LRSyn}}\\xspace$ is very robust to variations in the longitudinal\nsetting, achieving $>95\\%$ F1 score in all $53$ fields, with a perfect F1 score of $1.00$ in $49$ cases, as compared to $40$ and $33$ cases for $\\ensuremath{\\mathsf{NDSyn}}\\xspace$.\n}\nIn many applications, having a score of $1.00$ is crucial, i.e., even a system with $0.99$ \nprecision cannot be deployed in practice.\nFor example, even a tiny imprecision in a system that automatically adds calendar entries\nbased on flight reservation emails is disastrous for millions of users.\n\n\\begin{table}\n\\scriptsize\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{Contemporary} \\\\\n\\hline\nMetric & $\\ensuremath{\\mathsf{ForgivingXPaths}}$ & $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n\\hline\nAvg. Precision & 0.17 & 0.96 & 1.00\\\\ \\hline\nAvg. Recall & 0.99 & 0.91 & 1.00\\\\ \\hline\nAvg. F1 & 0.22 & 0.93 & 1.00 \\\\ \\hline\n\\end{tabular}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\n\\multicolumn{4}{|c|}{Longitudinal} \\\\\n\\hline\nMetric & $\\ensuremath{\\mathsf{ForgivingXPaths}}$ & $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n\\hline\nAvg. Precision & 0.15 & 0.99 & 1.00\\\\ \\hline\nAvg. Recall & 0.98 & 0.89 & 1.00\\\\ \\hline\nAvg. F1 & 0.20 & 0.92 & 1.00 \\\\ \\hline\n\\end{tabular}\n\\caption{F1 scores for $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ and $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ for the M2H Contemporary and Longitudinal datasets}\n\\label{table:m2hoverall}\n\\end{table}\n\n\n\\paragraph{Detailed comparison}\nTable~\\ref{tab:m2hexpmt1} shows a more detailed drill-down of the F1 scores for $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ and $\\ensuremath{\\mathsf{LRSyn}}\\xspace$, in the two settings. For a further drill-down of the F1 score into precision and recall, please see the supplementary material.\nThe rows of the table correspond to 53 extraction tasks, each corresponding to a field value, over the 6 airline domains in the M2H dataset. \nAs seen from the table, $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ performs much better than $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ on 23 out of the 53 fields in the contemporary setting, and 24 out of 53 fields in the longitudinal setting. In the remaining fields, the two approaches have comparable F1 scores.\n\nFor the fields \\emph{AIata}, \\emph{ATime}, \\emph{DIata} and \\emph{DTime} in the \\emph{iflyalaskaair} domain, we find that $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ performs much better than $\\ensuremath{\\mathsf{NDSyn}}\\xspace$, and the gap between the two is wider in the longitudinal setting.\nIn the case of files \\emph{ATime} and \\emph{DTime} in the domains \\emph{delta} and \\emph{airasia} we find that $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ is not able to synthesize any extraction program (which is why the F1 score is \"NaN\"), wheras \\ensuremath{\\mathsf{LRSyn}}\\xspace\\ is able to synthesize programs that produce a perfect F1 score of 1.\n\nIn the domain \\emph{getthere.com}, $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ achieves perfect extraction over all fields as compared to $\\ensuremath{\\mathsf{NDSyn}}\\xspace$. This is primarily because the document formats in \\emph{getthere.com} permute the constituent regions as shown in Figure~\\ref{fig:formed_documents}, and \\ensuremath{\\mathsf{LRSyn}}\\xspace\\ handles such permutations naturally because it relies on landmarks and local regions, which are invariant to permutations of the regions in the document.\n\n\nIn the domains \\emph{aeromexico} and \\emph{mytrips.amexgbt} we find that both $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ and $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ acheive perfect F1 scores. However, this is due to some incidental aspects described below. In the \\emph{aeromexico} domain, each field has  dedicated \\emph{id} attributes in the DOM, such as\\emph{-arrival-city-, -arrival-time, -departure-city}. Both $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ and $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ are able to latch on this attribute for extraction. These attributes remain the same in both our contemporary and longitudinal datasets, and they perform the role of landmarks, even for $\\ensuremath{\\mathsf{NDSyn}}\\xspace$. Also, even the initial fine-grained whole-document blueprints in this domain exhibit no change across the contemporary and longitudinal datasets, indicating that the formats have not varied across time in this domain.\nIn the \\emph{mytrips.amexgbt} domain, there are variations in fine-grained whole document blueprints across the two datasets which suggest that $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ should perform better than $\\ensuremath{\\mathsf{NDSyn}}\\xspace$. When we look deeply into these variations, there are \\emph{hotel} and \\emph{cab} booking blocks getting added to these documents just like \"getthere\".\nThe $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ structure program which get synthesized for  \\emph{departure time} is shown in the figure below:\n\n\\begin{figure}[H]\n\\vspace{-4ex}\n\\small\n\\begin{alltt}\nTBODY > :nth-child(1):nth-last-child(1)\n> :nth-child(2) > DIV.right_section:nth-child(1)\n:nth-last-child(1) > :nth-child(10):nth-last-child(9)\n\\end{alltt}\n\\vspace{-4ex}\n\\end{figure}\nThis program iterates over the various blocks of a table, and for each block it tries to extract a region which is the $10^{th}$ child from the start and $9^{th}$ child from the end. And it so turns out that in our dataset, the \\emph{hotel} or \\emph{cab} blocks always have fewer than 5 lines and never pass this predicate. But, this program will break if more lines get added to these blocks, so $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ gets \"lucky\" in these cases. In comparison, the program synthesized by $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ zooms into the region with \"Departing\" as a landmark and synthesizes a simpler extraction program, which is robust to such changes in the format. It so happens that such format changes are not present in our test set.\n\n\\begin{table*}\n\\scriptsize\n\\begin{tabular}{|l|c|c|c||c|c|}\n\\hline\n & & \\multicolumn{2}{|c||}{ Contemporary} &  \\multicolumn{2}{|c|}{Longitudinal} \\\\\n {Fields} & Domain & $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ & $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ \\\\\n\\hline\n\\hline\n AIata & \\multirow[b]{4}{*}{ifly}   & 0.81 & \\textbf{1.00} & 0.64 & \\textbf{1.00} \\\\\n ATime &                            & 0.76 & \\textbf{1.00} & 0.62 & \\textbf{1.00} \\\\\n DIata &                            & 0.73 & \\textbf{1.00} & 0.55 & \\textbf{1.00} \\\\\n DDate &                            & 1.00 &         1.00  & 1.00 &         1.00  \\\\\n DTime &  alaska                    & 0.73 & \\textbf{1.00} & 0.55 & \\textbf{1.00} \\\\\n FNum  & \\multirow[t]{4}{*}{air}    & 1.00 &         1.00  & 1.00 &         1.00  \\\\\n Name  &                            & 1.00 &         1.00  & 0.99 &         0.99  \\\\\n Pvdr  &                            & --   &         --    & --   &         --    \\\\\n RId   &                            & 1.00 &         1.00  & 1.00 &         1.00  \\\\\n\n\\hline\n AIata & \\multirow{9}{*}{airasia} & 0.67 & \\textbf{1.00} & 0.67 & \\textbf{1.00} \\\\\n ATime &                          & NaN  & \\textbf{1.00} & NaN  & \\textbf{1.00} \\\\\n DIata &                          & 0.67 & \\textbf{1.00} & 0.67 & \\textbf{1.00} \\\\\n DDate &                          & 0.67 & \\textbf{1.00} & 0.67 & \\textbf{1.00} \\\\\n DTime &                          & NaN  & \\textbf{1.00} & NaN  & \\textbf{1.00} \\\\\n FNum  &                          & 1.00 &          1.00 & 0.96 & 0.96          \\\\\n Name  &                          & 1.00 &          1.00 & NaN  & \\textbf{1.00} \\\\\n Pvdr  &                          & 1.00 &          1.00 & 0.96 & 0.96          \\\\\n RId   &                          & 1.00 &          1.00 & 1.00 & 1.00          \\\\\n\n\\hline \n\\end{tabular}\n\\begin{tabular}{|c|c|c||c|c|}\n\\hline\n & \\multicolumn{2}{|c||}{ Contemporary} &  \\multicolumn{2}{|c|}{Longitudinal} \\\\\n {Domain} & $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ & $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ \\\\\n\\hline\n\\hline\\multirow{9}{*}{getthere}\n\n & 0.75 & \\textbf{1.00} & 0.74 & \\textbf{1.00} \\\\\n & 0.94 & \\textbf{1.00} & 0.91 & \\textbf{1.00} \\\\\n & 0.94 & \\textbf{1.00} & 0.95 & \\textbf{1.00} \\\\\n & 0.98 & \\textbf{1.00} & 0.95 & \\textbf{1.00} \\\\\n & 0.76 & \\textbf{1.00} & 0.78 & \\textbf{1.00} \\\\\n & 0.98 & \\textbf{1.00} & 0.98 & \\textbf{1.00} \\\\\n & 1.00 & 1.00 & 0.89 & \\textbf{1.00} \\\\\n & 0.98 & \\textbf{1.00} & 0.97 & \\textbf{1.00} \\\\\n & 0.93 & \\textbf{1.00} & 0.94 & \\textbf{1.00} \\\\\n\n\\hline\\multirow{9}{*}{delta}\n & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n & 0.94 & \\textbf{1.00} & 0.95 & \\textbf{1.00} \\\\\n & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n & 0.85 & \\textbf{0.97} & 0.91 & \\textbf{0.97} \\\\\n & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n\n\\hline \n\\end{tabular}\n\\begin{tabular}{|c|c|c||c|c|}\n\\hline\n & \\multicolumn{2}{|c||}{ Contemporary} &  \\multicolumn{2}{|c|}{Longitudinal} \\\\\n {Domain} & $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ & $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ \\\\\n\\hline\n\\hline\\multirow[b]{4}{*}{aero} & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                               & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                               & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                               & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n\\multirow[t]{4}{*}{mexico}     & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                               & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                               & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                               & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                               & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n\n\\hline\n\\multirow[b]{4}{*}{mytrips} & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                            & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                            & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                            & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n amex                          & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n \\multirow[t]{3}{*}{gbt}   & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                            & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                            & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n                            & 1.00 & 1.00 & 1.00 & 1.00 \\\\\n\n\\hline \n\\end{tabular}\n\\caption{M2H F1 score details. The Pvdr field is not relevant for iflyalaskaair}\n\\label{tab:m2hexpmt1}\n\\vspace{-5ex}\n\\end{table*}\n\n\\begin{comment}\n\\begin{table*}\n\\small\n\\begin{tabular}{|l|}\n\\hline\nifly.alaskaair.com : ArrivalAirportIata \\\\\n\\\\\n$\\ensuremath{\\mathsf{NDSyn}}\\xspace$: \\\\\n$L_{ex}$: $TR:nth-child(1):nth-last-child(1) > [style*=\"width:48\\%\"]:nth-child(3) >  TABLE[cellpadding=\"0\"]$ \n\\\\\n$[cellspacing=\"0\"][border=0][style*=\"width:100\\%\"]:nth-child(1):nth-last-child(1) >$\n\\\\\n$TBODY:nth-child(1):nth-last-child(1) > :nth-child(1)$\\\\\n$Text Program$ : $Extract(str, RegexPair(\"\", \"Alphanumeric\"), RegexPair(\"Alphanumeric\", \"\\epsilon\"), -1)$\\\\\n\\smallskip\\\\\n$\\ensuremath{\\mathsf{LRSyn}}\\xspace$:\\\\\n$Landmark$ : $Arrives$~,~$L_{rg}$: $\\langle \\mathsf{parenthops}: 0, \\mathsf{siblinghops}$: $1$, $\\mathsf{blueprint}$ : \/TR\/TD  $\\rangle$\\\\\n$L_{ex}$: $TR:nth-child(1)$ \\\\\n$Text Program$ : $Extract(str, RegexPair(\"\\epsilon\", \"Alphanumeric\"), RegexPair(\"Alphanumeric\", \"\\epsilon\")$, \\\\\n\\hline\n\\end{tabular}\n\\caption{Synthesized extraction programs}\n\\label{tab:m2hprograms}\n\\end{table*}\n\\end{comment}\n\n\\begin{comment}\ngetthere.com:Provider \\\\\n\\\\\n$\\ensuremath{\\mathsf{NDSyn}}\\xspace$: \\\\\n$L_{ex}$: $DIV:nth-last-child(13) > TABLE > TBODY:nth-child(1):nth-last-child(1) > :nth-child(1) > :nth-child(2)$ \\\\\n$Text Program$ : $Extract(str, RegexPair(\\epsilon, \"Alphanumeric\"), RegexPair(\\epsilon, \"WhiteSpaceNumber\"), 1)$\\\\\n\\\\ \n$\\ensuremath{\\mathsf{LRSyn}}\\xspace$:\\\\\n$Landmark$ : $Flight\/Equip$\\\\\n $L_{rg}:\\langle \\mathsf{parenthops}: 0, \\mathsf{siblinghops}: 1, \\mathsf{blueprint}:\/TD  \\rangle$\\\\\n$L_{ex}:nth-child(2)$ \\\\\n$Text Program : Extract(str, RegexPair(\\epsilon, \"Alphanumeric\"), RegexPair($\\epsilon$, \"WhiteSpaceNumber\"), 1)$\\\\\n\n\\hline\n\\end{comment}\n\n\n\n\\paragraph{Synthesized programs}\nFigures~\\ref{lst:ndsynprogram}~and~\\ref{lst:lrsynprogram} show an example extraction program synthesized by $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ and $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ respectively. More examples can be found in the supplementary material.\nAs seen from these, extraction programs synthesized by $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ programs are way more complex as they process the entire HTML page, when compared to the programs synthesized by $\\ensuremath{\\mathsf{LRSyn}}\\xspace$, which only process the local region of interest.\nConsequently, if the formats change, the programs synthesized by $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ continue to function correctly.\nFurther, the shape and size of the $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ programs makes them easy to \nexamine by hand and gain confidence in their correctness.\n\\arsays{Do we have any numbers here?}\n\\highlightbox{\nThe $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ programs are significantly smaller compared to $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ programs.\nFor the web extraction component, $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ programs have $XXX$ components\nin their CSS selectors as compared to $YYY$ for $\\ensuremath{\\mathsf{NDSyn}}\\xspace$.\n}\n\n\n\\begin{comment}\n\\begin{table}\n\\small\n\\label{table:m2hnumberofprograms}\n\\begin{tabular}{|l|c|c|}\n\\hline\nField & \\#$\\ensuremath{\\mathsf{NDSyn}}\\xspace$ & \\#$\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\nAIata & 3 & 1 \\\\\nATime & 2 & 1 \\\\\nDIata & 3 & 1 \\\\\nDDate & 3 & 1 \\\\\nDTime & 2 & 1 \\\\\nFNum & 1 & 1 \\\\\nName & 1 & 1 \\\\\nPvdr & 1 & 1 \\\\\nRId & 1 & 1 \\\\\n\\hline\n\\end{tabular}\n\\caption{Getthere.com: Number of $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ and $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ programs}\n\\label{tab:m2h-numprograms}\n\\end{table}\n\\end{comment}\n\n\\paragraph{Number of programs} Since \\ensuremath{\\mathsf{LRSyn}}\\xspace\\ merges clusters based on ROIs rather than entire documents, this results in fewer programs synthesized by \\ensuremath{\\mathsf{LRSyn}}\\xspace, when compared to \\ensuremath{\\mathsf{NDSyn}}\\xspace.\nFor \"getthere\", $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ synthesizes only a single program for all fields: \\emph{:nth-child(2)}. \\ensuremath{\\mathsf{NDSyn}}\\xspace ,on the other hand, synthesizes $3$ programs each for \\emph{AIata}, \\emph{DIata}, \\emph{DDate}, $2$ programs each for \\emph{ATime} and \\emph{DTime} and $1$ program for the remaining $4$ fields. \nWe see this trend in all other domains as well, confirming our initial hypothesis that\nlocal structure of regions of interest tend to vary less than global document\nstructure.\n\n\\highlightbox{\nGet numbers from Anirudh}\n\n\\begin{table}\n\\scriptsize\n\\centering\n\\begin{tabular}{|l|c|}\n\\hline\nDomain & Test \\\\\n\\hline\nAccountInvoice & 157\\\\ \\hline\nCashInvoice & 40 \\\\ \\hline\nCreditNote & 30 \\\\ \\hline\nSalesInvoice & 401 \\\\ \\hline\nSelfBilledCreditNote & 146 \\\\ \\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|l|c|}\n\\hline\nDomain & Test \\\\\n\\hline\naeromexico & 303\\\\ \\hline\ngetthere & 155 \\\\ \\hline\nwestjet & 500 \\\\ \\hline\namex & 619 \\\\ \\hline\n\\end{tabular}\n\\caption{Test set sizes for Finance and M2H-Images datasets. Train set size is $10$ for all the domains}\n\\label{tab:imagesdata}\n\\end{table}\n\n\n\n\\begin{table}\n\\scriptsize\n\\begin{tabular}{|c|l|c|c|c|}\n\\hline\n{Domain} & Fields & $\\ensuremath{\\mathsf{AFR}}$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n\\hline\\multirow{9}{*}{AccountsInvoice}\n  & Amount & 0.99 & \\textbf{1.00} \\\\\n  & Chassis & 0.82 & \\textbf{0.99} \\\\\n  & CustAddr & 0.98 & \\emph{0.96} \\\\\n  & Date & 0.93 & \\textbf{0.98} \\\\\n  & Dnum & 0.96 & \\textbf{0.97} \\\\\n  & Engine & 0.82 & \\textbf{1.00} \\\\\n  & InvoiceAddress & 0.90 & \\textbf{0.95} \\\\\n  & Model & 0.75 & \\textbf{1.00} \\\\\n\n\\hline\\multirow{8}{*}{CashInvoice}\n  & Amount & 1.00 & 1.00 \\\\\n  & Chassis & 0.99 & 0.99 \\\\\n  & CustAddr & 0.99 & \\emph{0.97} \\\\\n  & Date & 0.99 & 0.99 \\\\\n  & Dnum & 0.96 & 0.96 \\\\\n  & Engine & 0.93 & \\textbf{0.95} \\\\\n  & InvoiceAddress & 0.99 & 0.99 \\\\\n  & Model & 0.99 & \\textbf{1.00} \\\\\n\n\\hline\\multirow{5}{*}{CreditNote}\n  & Amount & 1.00 & 1.00 \\\\\n  & CreditNoteAddress & 0.99 & \\textbf{1.00} \\\\\n  & CreditNoteNo & 0.94 & \\emph{0.93} \\\\\n  & CustRefNo & 1.00 & 1.00 \\\\\n  & Date & 1.00 & 1.00 \\\\\n  & RefNo & 1.00 & 1.00 \\\\\n\n\\hline\\multirow{6}{*}{SalesInvoice}\n  & Amount & 1.00 & 1.00 \\\\\n  & CustomerReferenceNo & 1.00 & 1.00 \\\\\n  & Date & 1.00 & 1.00 \\\\\n  & InvoiceAddress & 0.94 & \\textbf{0.99} \\\\\n  & RefNo & 0.99 & 0.99 \\\\\n  & SalesInvoiceNo & 0.99 & 0.99 \\\\\n\n\\hline\\multirow{6}{*}{SelfBilledCreditNote}\n  & Amount & 1.00 & 1.00 \\\\\n  & CustomerAddress & 1.00 & \\emph{0.99} \\\\\n  & CustomerReferenceNo & 0.99 & 0.99 \\\\\n  & Date & 1.00 & 1.00 \\\\\n  & DocumentNumber & 1.00 & 1.00 \\\\\n  & VatRegNo & 1.00 & 1.00 \\\\\n\n\\hline\n\\end{tabular}\n\\caption{F1 scores for Finance dataset}\n\\label{tab:financedataset}\n\\end{table}\n\n\\begin{table}\n\\scriptsize\n\\begin{tabular}{|l|p{1.5cm}|c|c|}\n\\hline\nFields & {Domain} & $\\ensuremath{\\mathsf{AFR}}$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n \\hline\nAIata & \\multirow{9}{*}{aeromexico}        & 0.62 & \\textbf{0.65} \\\\\nATime &                                 & 0.69 & \\textbf{0.99} \\\\\nDIata &                                 & 0.36 & \\textbf{0.66} \\\\\nDDate &                                 & 0.71 & \\textbf{0.89} \\\\\nDTime &                                 & 0.65 & \\textbf{0.97} \\\\\nFNum  &                                 & 0.66 & \\textbf{0.83}  \\\\\nName  &                                 & 0.96 & \\textbf{0.98}  \\\\\nPvdr  &                                 & 0.69 & \\textbf{0.78}  \\\\\nRId   &                                 & 1.00 & 1.00            \\\\\n \n\\hline\n\\end{tabular}\n\\begin{tabular}{|p{1.5cm}|c|c|}\n\\hline\n{Domain} & $\\ensuremath{\\mathsf{AFR}}$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n\\hline\n\\multirow{9}{*}{getthere}  & 0.94 & \\textbf{1.00} \\\\\n                           & 0.87 & \\textbf{1.00} \\\\\n                           & 0.93 & \\textbf{1.00} \\\\\n                           & 0.96 & \\textbf{0.99} \\\\\n                           & 0.88 & \\textbf{1.00} \\\\\n                           & 0.94 & \\textbf{1.00} \\\\\n                           & 0.99 &         0.99  \\\\\n                           & 0.75 & \\textbf{1.00} \\\\\n                           & 0.89 & \\textbf{0.95} \\\\\n \n\\hline\n\\end{tabular}\n\n\\begin{tabular}{|l|p{1.5cm}|c|c|}\n\\hline\n{Fields} & Domain & $\\ensuremath{\\mathsf{AFR}}$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n\\hline\nAIata & \\multirow[b]{4}{*}{itinerary.}     & 0.94 & \\textbf{0.99} \\\\\nATime &                                    & 0.90 &   \\emph{0.88} \\\\\nDIata &                                    & 0.93 & \\textbf{0.98} \\\\\nDDate &                                    & 0.93 & \\textbf{1.00} \\\\\nDTime &  \\multirow[t]{4}{*}{westjet}       & 0.94 & \\textbf{0.95} \\\\\nFNum  &                                    & 0.96 & \\textbf{1.00} \\\\\nName  &                                    & 0.60 & \\textbf{0.61} \\\\\nPvdr  &                                    & 0.95 & \\textbf{1.00} \\\\\nRId   &                                    & 1.00 &         1.00 \\\\\n \n\\hline\n\\end{tabular}\n\\begin{tabular}{|p{1.5cm}|c|c|}\n\\hline\n{Domain} & $\\ensuremath{\\mathsf{AFR}}$ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n \\hline\n\\multirow[b]{4}{*}{mytrips}  & 0.85 & \\textbf{0.98} \\\\\n                             & 0.97 & \\textbf{1.00} \\\\\n                             & 0.96 & \\textbf{0.99} \\\\\n                             & 0.93 & \\textbf{1.00} \\\\\n\\multirow[t]{4}{*}{amexgbt}  & 0.99 & \\textbf{1.00} \\\\\n                             & 0.98 & \\textbf{1.00} \\\\\n                             & 0.98 & \\textbf{1.00} \\\\\n                             & 0.91 & \\textbf{0.99} \\\\\n                             & 0.61 & \\textbf{0.96} \\\\\n\n\\hline\n\\end{tabular}\n\\caption{F1 Score for Images M2H Dataset}\n\\label{tab:m2himages}\n\\vspace{-5ex}\n\\end{table}\n\n\n\\begin{comment} \n\\hline\\multirow{9}{*}{ifly.alaskaair}   \n & AIata & 0.98 & \\textbf{1.00} \\\\\n & ATime & 0.95 & \\textbf{0.99} \\\\\n & DIata & 0.98 & \\textbf{1.00} \\\\\n & DDate & 0.98 & \\emph{NaN} \\\\\n & DTime & 0.95 & \\textbf{0.98} \\\\\n & FNum & 0.97 & \\textbf{1.00} \\\\\n & Name & 0.98 & 0.98 \\\\\n & Pvdr & 0.93 & \\textbf{0.99} \\\\\n & RId & 1.00 & \\emph{0.86} \\\\\n\\end{comment}\n\n\\begin{comment}\n\\begin{table*}\n\\small\n\\begin{tabular}{|l|l|l|}\n\\hline\n{Domain} & Manually provided landmarks & Learnt landmarks \\\\\n\\hline\nifly.alaskaair & \n     Departs, Arrives, Term\/Gate,  & Departs, Arrives, \\textbf{Confirmation code: >> Term\/Gate,}  \\\\\n   & Travellers, Confirmation code &  Travellers, \\textbf{Travelers: >> Confirmation code}  \\\\\n\\hline\n\\end{tabular}\n\\caption{Landmarks for M2H dataset}\n\\label{tab:m2hlandmarks}\n\\end{table*}\n\\end{comment}\n\n\\paragraph{Quality of Inferred Landmarks}\nWe infer landmarks automatically using the techniques and scoring functions mentioned in Section~\\ref{sec:algorithms} and~\\ref{sec:instantiations}.\nTo check the quality of landmark inference, we also asked data annotators to tag the landmarks manually.\n\\highlightbox{\nIn $57$ out of $63$ clusters across all fields, the inferred landmarks are exactly the same as manually provided landmarks.\nIn $5$ of the remaining $6$ cases, the human annotator agreed that the inferred landmark \nwas of equal quality.\n}\nIn the remaining $1$ case, the algorithm chose the human annotated landmark as well, but \nin addition chose a disambiguating hierarchical landmark of low quality.\nIn particular, it disambiguated the term \\emph{Name} occurred in reference to both\nthe name of the passenger and the name in the  billing address.\nHere, the algorithm chose to disambiguate using the term \\emph{Meal}, i.e.,\npassengers have a meal preference while the billed person does not.\nWe refer the readers to the supplementary material for the list of all landmarks inferred.\n\n\n\\subsection{Form Image Extraction}\nWe consider two datasets of form images:\n(1) {\\em Finance} dataset: This consists of ~850 images of receipts, purchase orders, credit notes, sales invoice and similar such documents. Figure~\\ref{fig:formed_documents}(c)~shows one such image of a sales invoice. The training and test data are from the same time period. Hence this dataset is used to do contemporary evaluation.\n(2) {\\em M2H-Images} dataset: We convert M2H emails from $4$ domains to images and extract the same fields as before. This represents common scenarios in practice where HTML documents such as booking mails or receipts may be printed and then scanned again for data extraction where the original structure is lost. Table~\\ref{tab:imagesdata} shows the test data sizes for each domain in both these datasets. We use a training data size of $10$ in all our image extraction experiments. In M2H dataset, the test data contains emails from a different time period than the ones in the training set. Hence this dataset is used to do a longitudinal evaluation.\n\\arsays{Somebody will definitely ask why not all 6 domains}\n\n\nWe first pre-process these images to make them sharper and perform optical character recognition (OCR) on them. This provides us with bounding boxes and text for these images, and we operate on top of this input. For our experiments, we use the OCR offering from Microsoft Azure~\\cite{AzureOCR}. We compare $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ with a custom neural model fine-tuned using Azure Form Recognize (\\ensuremath{\\mathsf{AFR}})~\\cite{AFR}, which is a cloud-based Azure Applied AI Service. We use a sample of $10$ documents as training data for each domain in our experiments. We do not handle issues like spelling errors in OCR output in the current work and employ heuristics to correct them.\n\n\\paragraph{Overall Results}\nTable~\\ref{tab:imagesoverall}~shows the average precision, recall, F1 and accuracy scores for \\ensuremath{\\mathsf{AFR}}\\ and \\ensuremath{\\mathsf{NDSyn}}\\xspace\\, for both the Finance and M2H-image datasets. The scores are weighted by the test sizes across various extraction tasks. Unlike $\\ensuremath{\\mathsf{AFR}}$ which is a neural model, $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ synthesizes programs on top of OCR boxes. As we can see from the table, $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ performs marginally better than $\\ensuremath{\\mathsf{AFR}}$ on the Finance dataset. This is mainly because the test data does not have variations when compared to training set.  $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ outperforms $\\ensuremath{\\mathsf{AFR}}$ in the M2H-Images dataset with a F1 score of $0.95$ as compared to $0.87$, since the test data in this dataset contains images from a different time period, and these images have many variations when compared to the training set.\n\\highlightbox{\n$\\ensuremath{\\mathsf{LRSyn}}\\xspace$ outperforms a state-of-the-art industrial form extraction system trained on\n\\arsays{$10^x$s} of images, with just $10$ training images per field, having a precision\nof $0.95$ vs $0.87$ on the M2H-Image dataset.\n}\n\n\\begin{table}\n\\scriptsize\n\\parbox{.45\\linewidth}{\n\\centering\n\\begin{tabular}{|l|c|c|c|}\n\\hline\nMetric & $\\ensuremath{\\mathsf{AFR}} $ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n\\hline\nAvg. Pre.  & 0.98\t& 0.99 \\\\ \\hline\nAvg. Rec.  & 0.96\t& 0.99 \\\\ \\hline\nAvg. F1  & 0.97\t& 0.99 \\\\ \\hline\n\\end{tabular}\n\\caption*{Finance dataset}\n}\n\\hfill\n\\parbox{.45\\linewidth}{\n\\centering\n\\begin{tabular}{|l|c|c|c|}\n\\hline\nMetric & $\\ensuremath{\\mathsf{AFR}} $ & $\\ensuremath{\\mathsf{LRSyn}}\\xspace$\\\\\n\\hline\nAvg. Prec.  & 0.85\t& 0.95 \\\\ \\hline\nAvg. Rec.  & 0.91\t& 0.95 \\\\ \\hline\nAvg. F1  & 0.87\t& 0.95 \\\\ \\hline\n\\end{tabular}\n\\caption*{M2H-Images dataset}\n}\n\\vspace{-3ex}\n\\caption{Average precision, recall, F1 numbers on Finance and M2H-Images dataset}\n\\label{tab:imagesoverall}\n\\vspace{-3ex}\n\\end{table}\n\n\\paragraph{Detailed comparison}\nTable~\\ref{tab:financedataset}~shows the results of $\\ensuremath{\\mathsf{AFR}}$ and $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ on the Finance dataset with respect to $34$ extraction tasks. $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ performs better than $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ in $12$ out of the $34$ cases and is on par on the rest, with significant gains in some domains like \"AccountsInvoice\". \n\nEven though the neural model in $\\ensuremath{\\mathsf{AFR}}$ is trained with thousands of invoices and receipts, and further fine-tuned with our training data, we observe that it is sensitive to the region coordinates in a given document. If these regions translate slightly or if the document scan is tilted, $\\ensuremath{\\mathsf{AFR}}$ extraction is erroneous. $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ is partially robust to such changes as the region programs start from a landmark value, and is agnostic to changes happening in the other regions. \n\nAlso, as we can note from the table, $\\ensuremath{\\mathsf{AFR}}$ is marginally better than $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ in some of the extraction tasks. This can be attributed to the absence of the boundary pattern in some of the test documents resulting in incorrect extractions. $\\ensuremath{\\mathsf{AFR}}$, on the other hand, in all likelihood, understands concepts like \"address\" and can extract irrespective of the presence of a boundary pattern. $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ can be improved further by incorporating such a semantic layer in its processing. \n\nTable~\\ref{tab:m2himages}~shows the results of $\\ensuremath{\\mathsf{AFR}}$ and $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ on the M2H dataset with respect to $45$ extraction tasks. This dataset exhibits more variations at the visual level as compared to the Finance dataset. $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ performs better than $\\ensuremath{\\mathsf{NDSyn}}\\xspace$ in $35$ out of the $45$ tasks and is on par on most of the remaining extraction tasks. This is mainly because $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ first locates the region of interests and then extracts values from these regions. \n\n\\paragraph{Synthesized programs}\nHere is an example region and text program synthesized for the \"Date\" field in \\emph{SalesInvoice} document, with respect to a landmark \"Date\". \\\\\nRegion program: $Extend(\"Date\", Relative(Right, EOL, false)$ \\\\\nText program: Extract a substring between \":\" and EOL\\\\\n\nThe region program starts from the landmark \"Date\" and extends right to collect all boxes until the end of line (EOL). And the corresponding text program extracts the text between a colon and EOL. We refer the readers to Table~\\ref{}~ in the Appendix for more examples. \\spsays{looks ugly, needs more work!!}\n\n\\paragraph{Summary of results}\nIn the HTML domain, the prior work \\ensuremath{\\mathsf{NDSyn}}\\xspace\\ is a high-performing system with F1 scores in the range of 0.9. \\ensuremath{\\mathsf{LRSyn}}\\xspace\\ is able to push the F1 scores to a perfect 1.0 in most cases. In longitudinal scenarios, \\ensuremath{\\mathsf{LRSyn}}\\xspace\\ improves \\ensuremath{\\mathsf{NDSyn}}\\xspace\\ in 24 out of 53 fields, with significant lift in F1 scores in many cases.\nIn the images domain, even with very little training data, \\ensuremath{\\mathsf{LRSyn}}\\xspace\\ matches AFR, which is a released product on contemporary settings, and outperforms AFR in longitudinal settings. In addition, \\ensuremath{\\mathsf{LRSyn}}\\xspace\\ produces simpler interpretable programs that match human intuition for extraction, and are much easier to maintain. Hence, we see a lot of promise in this approach.\n\n\n\n\\begin{comment}\n\\begin{table*}\n\\small\n\\begin{tabular}{|l|}\n\\hline\nSale Invoice : Date \\\\\nLandmark: \"Date:\" \\\\\nRegion program: Extend(\"Date\", Relative(Right, EOL, false) \\\\\nText program: Extract a substring between \":\" and EOL \\\\\n\\hline \nAccounts Invoice: Chassis Number \\\\\nLandmark: \"Chassis Number\" \\\\\nRegion program: Extend(Extend(input, Absolute(down, 1)), Relative(Right, DATE, false)) \\\\\nText Program: Take all the strings in this path \\\\\n\\hline\nAccounts Invoice : InvoiceAddress \\\\\nRegion program: ExtendLineItems(\"Invoice Name \\& Address\", Relative(down, \"Model SKU\")) \\\\\nText Program: Take all the strings in this path \\\\\n\\hline\n\\end{tabular}\n\\caption{Synthesized extraction programs \\spsays{Improve presentation}}\n\\label{tab:imageprograms}\n\\end{table*}\n\\end{comment}\n\n\\paragraph{Efficiency of Data Extraction}\nFormed document data extraction programs are sometimes run on extremely large\ndatasets, and hence, their efficiency becomes a concern.\nFor example, an industrial system that was the source of the M2H data processes\nover $3 \\times 10^8$ emails each day.\n\\highlightbox{\n  The $\\ensuremath{\\mathsf{LRSyn}}\\xspace$ programs for the M2H dataset outperform their $\\ensuremath{\\mathsf{NDSyn}}\\xspace$\n  counterparts by \\arsays{AAAX}, with an average processing time of $XXX$~ms vs. $YYY$~ms.\n}\nWe believe this is mainly due to the extremely cheap landmark location step,\nwith the more expensive parts of the program only running on the small ROI.\n\\subsection{Jointly inferring landmarks and clusters}\n\\label{subsec:joint-infer-cluster}\n\n\\paragraph{Initial Clustering}\nThe first step in our technique is to separate the data-set into clusters that\ncorrespond to different formats of emails.\nFor this, we use the notion of {\\em blueprints}. \nFor the HTML domain, the blueprint of a document (or a document fragment)\ncontains the tag structure and values that are common across documents. \nWe compute an initial clustering by considering of whole documents, and using the closeness of blueprints as the distance metric between documents. The initial clustering produces a large number of very fine-grained clusters.\nOn the M2H data-set, our initial clustering produces ~$20$ \\emph{head\nclusters} along with many small \\emph{tail clusters} totaling ~$70$\nclusters altogether. \n\n\\paragraph{Identifying Landmark Candidates}\nFor each cluster, we identify a landmark, which is an n-gram that appears in all documents in the cluster.\nFor each such landmark candidate (i.e., shared n-grams), we assign a score based\non two metrics:\n\\begin{inparaenum}[(a)]\n  \\item the distance between the landmark candidate and the field value to be\n    extracted, and\n    %\n  \\item the number of nodes in the document region that encloses both the\n    landmark candidate and the field value.\n\\end{inparaenum}\nThese two metrics capture the intuition that landmarks should be close to the\nvalues being extracted.\nFor the email in Figure~\\ref{fig:formed_documents}(a), the  top-scoring landmark candidates for extracting \\emph{Departure Time} are \"Depart\" and \"Arrive\".\n\n\\paragraph{Regions of Interest and Re-clustering}\nA contiguous region in the document that encloses the landmark candidate and the field value is called a \\emph{region of interest}. We are interested in small regions of interest.\nIn Figure~\\ref{fig:formed_documents} the landmark candidates are shown in orange ellipses, and the corresponding regions of interest are shown using blue rectangles.\nWe now re-define a distance metric between two documents as the minimum distance (over all\nlandmark candidates) between the blueprints of the corresponding regions of interest.\nUsing this revised distance metric, we iteratively merge clusters if the average distance between documents in the clusters is less than a threshold, till no further merging is possible.\nAs a result of such merging, formats with an advertisement section added, or existing sections rearranged would all be included in the same cluster, as the blueprint of the region of interest is invariant to such  changes that occur outside the region of interest. In the M2H data-set, iterative merging based on closeness of regions of interest results in less than $5$ clusters at a field level.\n\n\\subsection{Synthesizing Extraction Programs.}\nWith the new coarse-grained clusters, we synthesize $2$ sub-programs that\ntogether with the blueprint of the region of interest forms the complete\nextraction program.\nThe first sub-program, called {\\em region extraction program}, takes as input the whole document $d_i$ and a landmark location $\\ell$ as inputs, and produces a region of interest $R$ as the output. \n  %\n  In the case of HTML documents, the synthesized region extraction program starts with the landmark location and traverses the tree-structure of the HTML document and grows a region starting at the location of the landmark, and ensures that all locations of field values are included.\n   %\n  As an example,for the airline itinerary example shown in Figure~\\ref{fig:formed_documents}(a), in order to extract the value of the departure time, with \"Depart:\" as the landmark, the synthesized region extraction program is given by \\emph{0 parent hops, 1 sibling hop}. The single sibling hop here implies that across all the training data, the extraction value lies $1$ sibling away from the landmark within the same parent node in the DOM. The \\emph{parent hops} and \\emph{sibling hops} are computed based on all the annotated training documents in the given cluster, and hence produce a large enough ROI  that includes the location of all the field values for extraction. We also compute the blueprint for this region, and use this blueprint during execution of the region extraction program (see below).\n  The second sub-program, called {\\em value extraction program}, takes the region produced by the region extraction program as input, and produces the desired field value as output. For HTML documents, we use the DSL and the corresponding synthesis algorithm from~\\cite{raza2020web}. For the example in Figure~\\ref{fig:formed_documents}(a), the synthesized region and value extraction programs, and the blueprint of the region of interest are shown in Figure~\\ref{lst:lrsynprogram}. \n \n\n\\begin{figure}\n\\small\n\\lrsynprogram{Depart}{0}{1}{\/TD}{nth-child(2)}{Extract TIME sub-string}\n\\caption{\\ensuremath{\\mathsf{LRSyn}}\\xspace\\ extraction program for Depart time}\n\\label{lst:lrsynprogram}\n\\end{figure}\n\n\n\n\n\n\n\nDuring execution of the synthesized extraction program, we compute the blueprint of the ROI  calculated by the region extraction program and compare it with the blueprint of the ROI generated during synthesis. If the distance between these two blueprints is below a threshold value $t$, the regions from synthesis and inference times are \"roughly similar\", and we use this extracted program. Otherwise, we look for other extraction programs synthesized from other clusters, which better match the current document.\n\n\n\\subsection{Preliminaries and Problem Statement}\n\n\\paragraph{Documents and Locations}\nWe use the term \\emph{document} to represent a single record of a dataset from which we are interested in extracting data.\nWe use the symbol $\\ensuremath{\\mathsf{doc}}$ to represent documents.\nA document $\\ensuremath{\\mathsf{doc}}$ has a set of \\emph{locations} $\\ensuremath{\\mathsf{Locs}}(\\ensuremath{\\mathsf{doc}})$ which can be\nused to index into the document and look up values.\nFor a location $\\ensuremath{\\ell} \\in \\ensuremath{\\mathsf{Locs}}(\\ensuremath{\\mathsf{doc}})$, the function $\\ensuremath{\\mathsf{Data}}[\\ensuremath{\\ell}]$\nreturns the value of the data present in the location $\\ensuremath{\\ell}$ of $\\ensuremath{\\mathsf{doc}}$.\n\\begin{example}\n    \\label{ex:locs}\n    When dealing with HTML documents, the location are XPaths that retrieve\n    elements in the HTML DOM document tree structure.\n    %\n    The data value of a location is the concatenation of all the text elements\n    in the DOM element.\n\\end{example}\n\n\\paragraph{Data-sets and Fields}\nWe model a heterogeneous dataset $\\ensuremath{\\mathcal{D}}$ as a  tuple $(\\ensuremath{D}, \\{\n\\ensuremath{C}_0, \\ldots, \\ensuremath{C}_n \\})$ where:\n\\begin{inparaenum}[(a)]\n\\item $\\ensuremath{D}$ is a finite set of \\emph{input documents} (or just \\emph\n    {documents} for short) and\n\\item $\\{ \\ensuremath{C}_0, \\ldots, \\ensuremath{C}_n \\}$ is a partition of $\\ensuremath{D}$ into\n    \\emph{clusters}, i.e., $\\bigcup \\ensuremath{C}_i = D$ and $\\forall i \\neq j .\\;\n    \\ensuremath{C}_i \\cap \\ensuremath{C}_j = \\emptyset$.\n\\end{inparaenum}\nEach partition represents a \\textit{similar} set of documents in terms of format.\nThe extraction framework has access to the inputs $\\ensuremath{D}$, but not the\npartitioning.\nHenceforth, we write ``dataset $\\ensuremath{D}$'' instead of ``heterogeneous\ndataset $\\ensuremath{\\mathcal{D}}$'' to denote that the exact partition of $\\ensuremath{D}$ into\nclusters is not provided to us as input.\n\nFor a given dataset $\\ensuremath{D}$, a  \\emph{field} $\\ensuremath{\\mathsf{F}}$  of type $\\ensuremath{\\mathsf{T}}$\nis a partial function  $\\ensuremath{\\mathsf{F}} : D \\not\\to \\ensuremath{\\mathsf{T}}$ that maps documents to\nvalues of type $\\ensuremath{\\mathsf{T}}$.\nWe implicitly assume that a field is either defined for all documents in a\ncluster or is undefined for all documents in the cluster, i.e.,\n$\\forall \\ensuremath{C}_i . \\forall \\ensuremath{\\mathsf{doc}}, \\ensuremath{\\mathsf{doc}}' \\in \\ensuremath{C}_i. \n    \\ensuremath{\\mathsf{F}}(\\ensuremath{\\mathsf{doc}}) = \\bot  \\Leftrightarrow \\ensuremath{\\mathsf{F}}(\\ensuremath{\\mathsf{doc}}') = \\bot$.\nWe say that $\\ensuremath{\\mathsf{F}}(\\ensuremath{\\mathsf{doc}})$ is the {\\em value of the field} $\\ensuremath{\\mathsf{F}}$\nin $\\ensuremath{\\mathsf{doc}}$.\nThe type of a field can either be a primitive type such as integer or string or a\ncomposite type such as a list of strings or set of integers.\nThough we are interested in extracting multiple fields from each document\nin a dataset, for simplicity of presentation, our formal treatment considers\nextracting the value of a single field.\n\n\\begin{comment}\n\\begin{example}\n    \\label{ex:fields}\n    %\n    The M2H dataset introduced in Section~\\ref{sec:overview} consists of flight\n    reservation emails from multiple different airlines. In this dataset, a field like reservation number has a primitive type, i.e., string, while the others like departure time are lists of strings.\n    %\n    In Figure~\\ref{fig:formed_documents} a), the value of reservation number is ``SAGDQU'', while the \n    values of departure time and traveler names are\n    [``23.34'', ``21.44''] and [``Fx Engelberg Katrena Troi''], respectively.\n\\end{example}\n\\end{comment}\n\n\n\n\\paragraph{Annotations}\nGiven a  field $\\ensuremath{\\mathsf{F}}$ of a data-set $\\ensuremath{D}$, an \\emph{annotation}\n$\\ensuremath{\\mathcal{A}}(\\ensuremath{\\mathsf{doc}})$ of $\\ensuremath{\\mathsf{doc}} \\in \\ensuremath{D}$ is a list of locations\n$[ \\ensuremath{\\ell}_1, \\dots, \\ensuremath{\\ell}_n ]$ and an aggregation function $\\ensuremath{\\mathsf{Agg}}$\nsuch that $\\ensuremath{\\mathsf{F}}(\\ensuremath{\\mathsf{doc}}) =\n\\ensuremath{\\mathsf{Agg}}(\\ensuremath{\\mathsf{Data}}[\\ensuremath{\\ell}_1],\\ldots, \\ensuremath{\\mathsf{Data}}[\\ensuremath{\\ell}_n])$.\nAnnotations are user provided ``labels'' in ML parlance, and are used as\ntraining data.\nFor our experiments, we built a visual user interface where annotators could\nclick on individual HTML and image documents to select annotation locations.\nIn the background, the tool converts these clicks into locations, i.e., XPaths\nin HTML and x-y coordinates in PDF documents.\n\n\\begin{example}\n    \\label{ex:annotations}\n    For the departure time field in Figure~\\ref{fig:formed_documents}(a), the annotation contains the two locations with text elements ``Friday, Apr 3 8:18 PM'' and ``Thursday, Apr 9 2:02 PM'', and the\n    aggregation function collects these values into a list.\n    %\n\\end{example}\n\n\\paragraph{The formed document extraction problem}\nFix a dataset $\\ensuremath{D}$ and a field $\\ensuremath{\\mathsf{F}}$.\nThe input to the formed document extraction problem is given by\nitem a set of annotations on a \\emph{training set} $\\ensuremath{D}_\\ensuremath{\\mathsf{tr}} \\subseteq\n\\ensuremath{D}$.\nThe ideal expected output for such a problem is an \\emph{extraction function}\n$\\ensuremath{\\mathsf{Extract}}$ such that $\\forall \\ensuremath{\\mathsf{doc}} \\in \\ensuremath{D}. \\ensuremath{\\mathsf{Extract}}(\\ensuremath{\\mathsf{doc}}) = \\ensuremath{\\mathsf{F}}(\\ensuremath{\\mathsf{doc}})$.\nHowever, it is hard to produce ideal extractions, and we instead use the\nstandard metrics of \\emph{precision}, \\emph{recall} and \\emph{F1 score} to\nmeasure the quality of an extraction function (see, for example,~\\cite{PR}).\nIn practice, we are usually interested in extracting multiple fields from a\ndocument at once, and in fact, our implementation can do so.\nHowever, for simplicity of discussion, we present our techniques and conduct our\nexperiments for one field at a time.  \n\n\\subsection{Landmarks and Regions}\n\\paragraph{Landmarks}\nA landmark is a value that we can use to identify a location in a\ndocument, such that the field value is present in a ``nearby'' location (or in\n\"nearby\" locations if the field value is aggregated from multiple data values).\nFormally, a \\emph{landmark} is given by a data value $\\ensuremath{\\mathsf{m}}$.\nA given landmark $\\ensuremath{\\mathsf{m}}$ identifies a unique location $\\ensuremath{\\ell}$\nin a document $\\ensuremath{\\mathsf{doc}}$ such that $\\ensuremath{\\mathsf{Data}}[\\ensuremath{\\ell}_i] \\supseteq \\ensuremath{\\mathsf{m}}$, i.e., the\nlandmark value is a sub-string of the data at $\\ensuremath{\\ell}_i$.\nIn order for a landmark to be useful for our purposes, we require the existence\nof an inexpensive \"locator\" function, which can locate the occurrences of a\nlandmark in a document.\nMore precisely, we assume a computationally inexpensive function $\\ensuremath{\\mathsf{Locate}}$ such\nthat, $\\ensuremath{\\mathsf{Locate}}(\\ensuremath{\\mathsf{doc}},\\ensuremath{\\mathsf{m}}) = \\ensuremath{\\ell} \\implies \\ensuremath{\\mathsf{Data}}[\\ensuremath{\\ell}] = \\ensuremath{\\mathsf{m}}$.\n\n\\begin{example}\n    \\label{ex:landmark}\n    Consider the travel itinerary document in Figure~\\ref{fig:formed_documents}(a).\n    %\n    In order to extract departure times from this document, a possible landmark\n    to use is the phrase ``Depart:''.\n\\end{example}\n\n\\begin{remark}\n    For ease of presentation, the definition assumes\n    that landmarks occur in one location per document\n    (contrast against \\emph{Depart} in Figure~\\ref{fig:formed_documents}(a)).\n    %\n    We discuss handling multiple, ambiguous, landmark locations in Section 6.\n\\end{remark}\n\n\n\n\\begin{comment}\nWe assume the existence of  an ordering \\ensuremath{\\preccurlyeq}\\ between locations in a document.\nThe definition of the ordering varies depending on the document type. For HTML\ntrees, the relationship can be defined on the order in which locations are\ntraversed during a traversal of the HTML tree (such as pre-order traversal). For\ninvoice documents that are scanned images, we first process them with OCR\npackages, which convert the images into a set of text boxes with x-y\nco-ordinates.\nEach of the text boxes can be thought of as a location, and the ordering between\nlocations can be defined based on conjunctions of left-to-right and\ntop-to-bottom relationships between the x and y co-ordinates of the top-left\ncorners of the text boxes.\n\\end{comment}\n\n\\paragraph{Regions}\nA {\\em region} $\\ensuremath{\\mathsf{R}}$ of a document $\\ensuremath{\\mathsf{doc}}$ is a set of contiguous locations.\nA region can be thought of as a ``sub-document''.\n\\begin{comment}\n\\begin{definition}[\\textbf{Enclosing region}]\nGiven a document $d$, and a set of locations $R$ in a document $d$, the {\\em\nenclosing region} of $L$, denoted $\\ensuremath{\\mathsf{EncRgn}}(R,d)$ is a sub-document of $d$ that\ncontains the smallest super-set of $R$ that is closed with respect to $\\ensuremath{\\preccurlyeq}$.\nThat is, $\\ensuremath{\\mathsf{EncRgn}}(L,d) \\supseteq R$, and for any pair of locations $\\ell_1$ and\n$\\ell_2  \\in \\ensuremath{\\mathsf{EncRgn}}(R,d)$, if  $\\hat \\ell \\in L_d$ is such that $\\ell_1 \\ensuremath{\\preccurlyeq}\n\\hat \\ell \\ensuremath{\\preccurlyeq} \\ell_2$, then $\\hat \\ell \\in \\ensuremath{\\mathsf{EncRgn}}(R,d)$\n\\end{definition}\n\\end{comment}\nGiven a set of locations $\\ensuremath{L}$ of a document $\\ensuremath{\\mathsf{doc}}$, the \\emph{enclosing\nregion} $\\ensuremath{\\mathsf{EncRgn}}(\\ensuremath{L}, \\ensuremath{\\mathsf{doc}})$ is the smallest region that contains all\nlocations in $\\ensuremath{L}$.\nWe are particularly interested in regions that enclose a landmark and the\ncorresponding field values as our approach is based on narrowing down the\ndocument to such regions. \nWe call such regions as {\\em regions of interest} or {\\em ROIs} for short.\n    \n\\begin{example}\n    The bottom two blue rectangles in Figure~\\ref{fig:formed_documents}(a) highlight the relevant ROIs that contain both the landmark ``Depart:'' and the\n    associated field values.\n\\end{example}\n\n\n\\paragraph{Blueprints}\nIntuitively, a blueprint of a region\nis a ``hash'' of all the parts that are ``common'' to\nall such regions in the cluster.\nFor example, the strings \"Airline Record Locator\", \"AIR\", \"Meal\", \"Depart:\" in Figure~\\ref{fig:formed_documents}(a) are common values, since they will occur in all\ndocuments that follow this format.\n  \nLet the \\emph{layout} of a region $\\ensuremath{\\mathsf{R}}$ in a document\n$\\ensuremath{\\mathsf{doc}}$ be the subset of all locations $\\ensuremath{\\ell} \\in \\ensuremath{\\mathsf{R}}$ such that\n$\\ensuremath{\\mathsf{Data}}[\\ensuremath{\\ell}]$ is a common value. The \\emph{blueprint} $\\ensuremath{\\mathsf{BP}}(\\ensuremath{\\mathsf{R}})$ of a region $\\ensuremath{\\mathsf{R}}$ is then defined as a hash\nof values in the layout of $\\ensuremath{\\mathsf{R}}$.\n\nIf two regions are similar, we want to define their blueprints such that they are close to each other.\nGiven two blueprints $\\ensuremath{\\mathsf{b}}_1$ and $\\ensuremath{\\mathsf{b}}_2$, we use the notation\n$\\ensuremath{\\delta}(\\ensuremath{\\mathsf{b}}_1, \\ensuremath{\\mathsf{b}}_2)$ to denote the distance between $\\ensuremath{\\mathsf{b}}_1$ and\n$\\ensuremath{\\mathsf{b}}_2$. \nIf $\\ensuremath{\\mathsf{R}}_1$ and $\\ensuremath{\\mathsf{R}}_2$ are similar in structure, we want\n$\\ensuremath{\\delta}(\\ensuremath{\\mathsf{BP}}(\\ensuremath{\\mathsf{R}}_1), \\ensuremath{\\mathsf{BP}}(\\ensuremath{\\mathsf{R}}_2))$ to be small value.\n\n\\begin{example}\n  Our notion of blueprint of an HTML region is based on the XPaths\n  to its DOM nodes, but ignoring node order.\n  %\n  For example, the blueprint of a region stores the path starting from\n  a  \\texttt{div} node, descending through \\texttt{table}, \\texttt{tr}, \\texttt{td}, and \\texttt{span} nodes, but without storing where this path is in relationship to other paths.\n\\end{example}\n\n\n\\subsection{Landmark-based DSLs}\nTo formalize the notions of extraction using landmarks, we introduce a special class of DSLs called landmark-based DSLs. This is a generic design of languages that formally captures our reasoning using landmarks and regions for extraction tasks in a domain-agnostic fashion. Figure \\ref{fig:dsllandmark} shows the structure of a landmark-based DSL. In such DSLs, the input is always assumed to be a document and a complete program returns a field value extracted from the document. Such DSLs consist of four notions: landmarks $m$, blueprints $b$, region extraction programs $p_{rx}$ and value extraction programs $p_{vx}$.\nThe region and value extraction programs can be instantiated arbitrarily for a particular domain by defining the language fragments $\\mathcal{L}_{rx}$ and $\\mathcal{L}_{vx}$, and we shall illustrate such instantiations of these fragments for the web and image extraction domains. \n\n These four notions are brought together in the single top-level $\\mathsf{Extract}$ operator, the semantics of which is defined in Algorithm  ~\\ref{algo:extract}. This operator takes a list $Q$ of 4-tuples, where each tuple consists of a landmark, a region program, a blueprint for the region, and an extraction program. Each tuple represents an extraction strategy for a particular region format. The region program uses a landmark to identify a region of the document, and if this region matches the given blueprint, then the extraction program can be applied on the region to extract the field value. The top-level operator acts as a switch statement that applies the first tuple that successfully extracts a value from the document. Formally, for each tuple, we first use the $\\ensuremath{\\mathsf{Locate}}$ function to identify the location corresponding to the landmark $\\ensuremath{\\mathsf{m}}$. This location is then input into the region program $\\ensuremath{\\mathsf{RProg}}_i$ to produce the\nregion of interest $\\ensuremath{\\mathsf{R}}$. Now, we proceed with this cluster only if the blueprint of the region is\nwithin a certain tunable  threshold of similarity. If the blueprint is close enough, we return the output of the extraction\nprogram $\\ensuremath{\\mathsf{EProg}}_i$ on the region. Otherwise, we continue with the remaining tuples in $Q$.\n\n\n\\begin{figure}\n\\small{\n\\begin{tabular}{rcl}\n\\!\\!\\!\\!$\\boldsymbol{\\mathsf{@start}}$ \\; $\\mathsf{T}$ \\; $\\mathit{t}$ \\!\\! \\!\\!\\!\\!&$:=$&\\!\\! \\!\\!\\!\\!\\! $\\mathsf{Extract}(\\mathit{q,\\!...,q})$ where $q = (\\mathit{m}, \\mathit{p}_{rx}, \\mathit{b}, \\mathit{p}_{vx}) $ \\;\\; \\\\ [0.75ex]\n\\!\\!\\!\\!$\\mathsf{R \\!\\rightarrow\\! T}$ \\; $\\mathit{p}_{vx}$ \\!\\! \\!\\!\\!\\!&$:=$&\\!\\! \\!\\!\\!\\!\\! $\\ldots \\;\\; \\mathcal{L}_{vx} \\;\\; \\ldots$ \\;\\; \\\\ [0.75ex]\n\\!\\!\\!\\!$\\mathsf{(doc, str) \\!\\rightarrow\\! R}$ \\; $\\mathit{p}_{rx}$ \\!\\! \\!\\!\\!\\!&$:=$&\\!\\! \\!\\!\\!\\!\\! $\\ldots \\;\\; \\mathcal{L}_{rx} \\;\\; \\ldots$ \\;\\; \\\\ [0.75ex]\n\\!\\!\\!\\!$\\mathsf{str}$  $\\mathit{m}$ \\!\\! \\!\\!\\!\\!& &\\!\\! \\!\\!\\!\\!\\!  \\;\\; \/\/ landmark \\\\ [0.75ex]\n\\!\\!\\!\\!$\\mathsf{obj}$  $\\mathit{b}$ \\!\\! \\!\\!\\!\\!& &\\!\\! \\!\\!\\!\\!\\!  \\;\\; \/\/ blueprint \\\\ [0.75ex]\n\\!\\!\\!\\!$\\boldsymbol{\\mathsf{@input}}$ \\; $\\mathsf{doc}$  $\\mathit{d}$ \\!\\! \\!\\!\\!\\!& &\\!\\! \\!\\!\\!\\!\\!  \\;\\; \/\/ input document\\\\ [0.75ex]\n\n\n\n\n\\end{tabular}\n\n}\n\\vspace{-2ex}\n\\caption{Structure of a Landmark-based DSL $\\mathcal{L}_{ld}$}\n\\label{fig:dsllandmark}  \n\\end{figure}\n\n\\begin{algorithm}\n\\small\n\\caption{Semantics of the $\\mathsf{Extract}$ operator in a landmark-based DSL. The blueprint threshold $t$ is a tunable parameter of the semantics.}\n\\label{algo:extract}\n\\begin{algorithmic}[1]\n  \\Require Input document $d$ of type $\\ensuremath{\\mathsf{doc}}$\n  \\Require List $Q = [q_1,...,q_k]$, where each $q_i$  has the form $(m, p_{rx}, b, p_{vx})$ for $ 1 \\leq i \\leq k$\n  %\n  \\For{$(m, p_{rx}, b, p_{vx}) \\in\n      Q$}\n  \\State $\\ensuremath{\\ell} \\gets \\ensuremath{\\mathsf{Locate}}(d, m)$\n  \\State $\\ensuremath{\\mathsf{R}} \\gets p_{rx}(d, \\ensuremath{\\ell})$\n  \\If{$\\ensuremath{\\mathsf{R}} \\neq \\bot \\land \\ensuremath{\\delta}(\\ensuremath{\\mathsf{BP}}(\\ensuremath{\\mathsf{R}}), b) \\leq t$}\n  \\State \\Return $\\ensuremath{\\mathsf{Agg}}(p_{vx}(\\ensuremath{\\mathsf{R}}))$\n  \\EndIf\n  \\EndFor\n  \\State \\Return $\\bot$\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\\section{HTML extraction and $\\ensuremath{\\mathsf{LRSyn}}\\xspace$}\n\n\n\\section{Structured image extraction and $\\ensuremath{\\mathsf{LRSyn}}\\xspace$}\n\n\\begin{comment}\n\\begin{algorithm}\n\\small\n\\caption{Landmark Detection}\n\\label{lst:cover}\n\\flushleft $\\mathsf{Landmark Detection}(D,O_R,F)$\n\\begin{algorithmic}[1]\n\\Require Documents $D = \\{D_1,D_2,...,D_n\\}$\n\\Require Output Regions $O_R = \\{O_1,O_2,O_3,...,O_n\\}$ for $D_i$\n\\Require Field $F$\n\\State $L_c=\\phi$ where $L_c$ is set of Lmk candidate \n\\State \\textbf{for each}$D_i \\in D$ \\textbf{do}\n\\State \\hspace{\\algorithmicindent} \\textbf{if }$L_c = \\phi$\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$L_c \\gets FindNGrams(D_i)$\n\\State \\hspace{\\algorithmicindent}\\textbf{else}\n\\State\\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$L_c \\gets L_c \\cap FindNGrams(D_i)$\n\\State \\textbf{for each} $D_i \\in D$ \\textbf{do}\n\\State \\hspace{\\algorithmicindent}\\textbf{for each} $L_i \\in L_c$ \\textbf{do}\n\\State  \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$l_r \\gets ComputeSmallestRegion(Ls,D)$\n\\State  \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$R_{L_i} \\gets RegionGrwoingLearn(L_i,O_R,F)$\n\\State  \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$R_{hash} \\gets ComputeBlueprint(R_i)$ for each $R_i \\in R_{L_i}$ \n\\State  \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}Steps $S \\gets Max(ComputeSteps(R_i)$ for each $R_i \\in R_{L_i}$ \n\\State  \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}Find $l_{r'} = l_r \\notin R_{L_i}$ \n\\State  \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$R'\\gets RegionGrowingPredict(L_i,S,D)$\n\\State  \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$R_{hash}' \\gets ComputeBlueprint(R_i')$ for each $R_i' \\in R_{L_i}' $\n\\State  \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$R_{noisy} =R_{hash}' \\in R_{hash} $\n\\State  \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}Store $R_{hash}' \\notin R_{hash}$ \n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$s_{noisy}=1\/(|R_{noisy}|+1)$\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$s_{region}=1\/(Argmax(Width(R_{L_i})))$\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$D_{visual} = ComputeVisualDistance(l_i,o_i)$\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$s_{visual}= 1\/ArgMax(D_visual)$\n\\State Compute Average across each score for $l_i$ to find ${S_{n_{l_i}},S_{r_{l_i}},S_{v_{l_i}}}$\n\\State Previous Noisy Score $S_{N_prev} = 0$ \n\\State \\textbf{do}\n\\State \\hspace{\\algorithmicindent} $L_s \\gets \\{ l_i | Argmax(ComputeScore(S_{r_{l_i}},S_{v_{l_i}})) \\land S_{n_{l_i}} > S_{N_{prev}} \\}$\n\\State  \\hspace{\\algorithmicindent} $S_{N_{prev}} = S_{n_{l_i}}$\n\\State \\textbf{while $S_{n_{l_i}}!=1$}\n\\State \\Return $L_s$\n\\State\n\\Function{FindNGrams}{$D_i$}\n \\State $N \\gets GenerateNGram(D_i)$\n \\State $N$\n\\State \\Return $\\mathcal{R}$\n\\EndFunction\n\\State\n\\Function{ComputeVisualDistance}{$l_i,o_i$}\n \\State Let $Box_{l_i} \\gets\\{x_{l_{left}},x_{l_{right}},y_{l_{top}},y_{l_{bottom}}\\} $\n \\State Let $Box{o_i} \\gets\\{x{o_{left}},x_{o_{right}},y_{o_{top}},y_{o_{bottom}}\\} $\n \\State $x_1 = (x{l_{left}}+x_{l_{right}})\/2$\n \\State $x_2 = (x{o_{left}}+x_{o_{right}})\/2$\n \\State $y_1 = (y{l_{top}}+y_{l_{bottom}})\/2$\n \\State $y_2 = (y{o_{top}}+y_{o_{bottom}})\/2$\n \\State $D=\\sqrt{(x_{1}-x_{2})^2+(y_{1}-y_{2})^2}$ \n\\State \\Return $D$\n\\EndFunction\n\\State\n\\Function{ComputeScore}{$S_{r},S_{v}$}\n \\State $S =\\log_{10}(S_{r})+\\log_{10}(S{v})$  \n\\State \\Return $S$\n\\EndFunction\n\n\\end{algorithmic} \n\\end{algorithm}\n\n\\begin{algorithm}\n\\small\n\\caption{Clustering And Landmark Detection}\n\\label{lst:cover}\n\\flushleft $\\mathsf{ClusteringAndLmkDetection}(D,T,K,F)$\n\\begin{algorithmic}[1]\n\\Require Documents $D = \\{D_1,D_2,...,D_n\\}$\n\\Require Threshold for big clusters $T$\n\\Require Number of Lmk Candidates $K$\n\\Require Field $F$\n\\State \\textbf{for each} $D_i \\in D$ \\textbf{do}\n\\State \\hspace{\\algorithmicindent} $F \\gets ComputeBlueprint(D)$\n\\State Let Big Clusters $B = \\{B_1,B_2,...,B_n\\}$ \n\\State $L_0 \\gets$ Merge Based on Blueprint F\n\\State Let $L_0=\\{\\{D_1,D_x,...\\},\\{D_3,D_y,..\\},\\{D_n,D_z,...\\}\\}$\n\\State \\textbf{for each} $C_i \\in L_0$ \\textbf{do}\n\\State \\hspace{\\algorithmicindent}\\textbf{if} $|C_i| \\ge T$\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$B_i \\gets C_i$\n\\State \\hspace{\\algorithmicindent}\\textbf{else}\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$S_i \\gets C_i$\n\n\\State \\textbf{if} $|B_i|=0$\n\\State \\hspace{\\algorithmicindent}$B_1 \\gets C_i \\forall C_i \\in L_0$\n\\State \\hspace{\\algorithmicindent}$S_i \\gets \\phi$\n\\State \\textbf{for each} $B_i \\in B$ \\textbf{do}\n\\State \\hspace{\\algorithmicindent} $L \\gets LandmarkDetection(B_i,O,F)$\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}\\textbf{for each} $l_i \\in L$ \\textbf{do}\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$R_{B_{l_i}}\\gets RegionGrowingLearn(l_i,O,F)$\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$H_{B_{l_i}}\\gets ComputeBlueprint(R_{B_{l_i}})$ where $H = ROIHash$ \n\\State $H_{score}=\\{(H_B_{l_1},0),(H_B_{l_2},0),...,(H_B_{l_i},0)\\}$ where $H_B_{l_i} \\in H_{B_{l}}$ \n\\State $C=\\{B_1\\}$ where $B1=Biggest Cluster$\n\\State $P = (B-\\{B_1\\}) \\cup S$\n\\State \\textbf{for each} $P_i \\in P$ \\textbf{do}\n\\State \\hspace{\\algorithmicindent} \\textbf{for each} $C_j \\in C$ \\textbf{do}\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$L' = L_{B}$ where $B = Argmax(|B| \\in C)$\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$R_{p_i} \\gets RegionGrowingLearn(L_{k}',O_{p_{i}},F) \\forall L_{k}' \\in L'$\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$H_{p_i}' \\gets ComputeBlueprint(R_{p_k})\\forall R_{p_k} \\in R_{p_i}$\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}\\textbf{if} $H_{p_i}' \\cap H_{C_j} \\to$ \\o\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$C \\gets P_i$; Add $P_i$ to $C$\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}\\textbf{else}\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}\\textbf{for each}$H_x \\in H_{p_i}$' \\cap H_{C_j}$ \\textbf{do}\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}H_{score}[H_x] =H_{score}[H_x]+1$\n\\State \\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent}$C_j \\gets C_j \\cup P_i$\n\n\\State \\Return C\n\\end{algorithmic} \n\\end{algorithm}\n\\end{comment}\n\n\\begin{comment}\n\\begin{example}\n  One possible approach to achieve the extraction of departure times is in\n  $3$ steps:\n  %\n  \\begin{inparaenum}[(a)]\n    \\item First, extract the itinerary table from the email,\n    \\item Second, from the itinerary table extract the row corresponding to\n      each flight; and\n    \\item Finally, extract the cell corresponding to the departure time from\n      the row.\n  \\end{inparaenum}\n  %\n  There are several advantages of doing this extraction in steps.\n  %\n  In Figure~\\ref{fig:roi}, we illustrate one hierarchy of regions of interest\n  that would be produced during this extraction.\n\n  During a one shot extraction, an extractor needs to consider multiple\n  issues at once:\n  %\n  \\emph{\n  Which of the tables in the email are itinerary tables?\n  %\n  Which of the rows in the itinerary tables correspond to flights and which\n  correspond to advertisements?\n  %\n  In the rows, which cell contains the departure time?\n  }\n  %\n  In practice, a program synthesizer or ML model handling all these issues at\n  once is more likely to produce incorrect output both due to the inherent\n  complexity of the problem and due to the technique \n  %\n  On the other hand, separating concerns, we can narrow down these regions progressively and synthesizer has to take into account only the final region to synthesize an extraction program. This results in simpler programs as well.\n  %\n  \\qed\n\\end{example}\n\nFurther, by proceeding one step at a time, it is possible to use different\nextractors in different steps. In this work, we follow simple traversal techniques to narrow down the regions, but one can imagine employing ML models for narrowing down the regions.\n\\end{comment}\n\\begin{comment}\nIt is frequently the case that different extractors are suited for different\nsteps.\nFor example, ML models are often more suited to the task of narrowing down\nfrom a whole document to a smaller region that contains all the output\nregions, while programs produced by a program synthesizer are more adept at\nextracting the intended output from the smaller region.\n\\end{comment}\n\n\\begin{comment}\nA \\emph{sequence extractor} $\\mathsf{SE}$ is given by a sequence of\nextractors $\\mathsf{E}_0 \\cdot \\ldots \\cdot \\mathsf{E}_k$.\nThe sequence of extractors behave as a single extractor as follows:\n\\begin{itemize}\n  \\item For a sequence extractor $\\mathsf{SE} = \\mathsf{E}_0$ consisting of\n    a single extractor we define $\\mathsf{SE}(\\ensuremath{\\mathsf{doc}}) = \\mathsf{E}_0(\\ensuremath{\\mathsf{doc}})$,\n    and\n  \\item For a sequence extractor $\\mathsf{SE} = \\mathsf{E}_0 \\cdot \\mathsf{SE}'$,\n    we define $\\mathsf{SE}(\\ensuremath{\\mathsf{doc}}) =\n    \\mathsf{SE}'(\\ensuremath{\\mathsf{R}}_0)\\ldots\\mathsf{SE}'(\\ensuremath{\\mathsf{R}}_n)$ where\n    $\\mathsf{E}_0(\\ensuremath{\\mathsf{doc}}) = \\ensuremath{\\mathsf{R}}_0\\ldots\\ensuremath{\\mathsf{R}}_n$.\n\\end{itemize}\nNote that we are generalizing the definition extractor to act both on\ndocuments as well as regions.\nHenceforth, we use the term extractor to also mean sequence extractor,\nassuming that the intended meaning is clear from the context.\n\nDuring a sequence extraction, each extraction phase produces intermediate\noutput regions.\nWe dub the regions at each intermediate step the \\emph{regions of interest}\n(ROIs) at that step.\n\\end{comment}\n\n\\begin{comment}\n\\section{The \\ensuremath{\\mathsf{LRSyn}}\\xspace\\xspace Algorithm}\nAlgorithm~\\ref{algo:hierarchical-extraction} depicts the high level procedure\nof algorithm \\ensuremath{\\mathsf{LRSyn}}\\xspace\\xspace.\nThe algorithm takes as input a set of examples $\\mathsf{Ex}^*$ and a set of\nlearners $\\mathsf{Learners}$.\nFirst, it chooses non-deterministically whether to do a one-shot extraction\nor a hierarchical extraction.\nIf a one-shot extraction is chosen, the procedure \\textsc{Extract} picks a\nlearner from $\\mathsf{Learners}$ and returns the result of that learner on\nthe given examples.\n\nOn the other hand, if a hierarchical extraction is chosen, for each example\n$\\ensuremath{\\mathsf{doc}} \\mapsto \\ensuremath{\\mathsf{R}}_0\\ldots\\ensuremath{\\mathsf{R}}_n$, we first identify the regions of\ninterest $\\ensuremath{\\mathsf{R}}_0^*\\ldots\\ensuremath{\\mathsf{R}}_m^*$ for the current step.\nThe first extraction task is to hierarchically extract the regions of\ninterest from each document.\nWe do this using a recursive call to \\textsc{HierarchicalExtract} on examples\nof the form $\\ensuremath{\\mathsf{doc}} \\mapsto \\ensuremath{\\mathsf{R}}_0^*\\ldots\\ensuremath{\\mathsf{R}}_m^*$ for each document.\nNow, the second extraction task is to extract the intended outputs from the\nregions of interest.\nFor each region of interest $\\ensuremath{\\mathsf{R}}^*_i$, we associate corresponding output\n$\\ensuremath{\\mathsf{R}}^{i_1}\\ldots\\ensuremath{\\mathsf{R}}^{i_k}$ where $\\ensuremath{\\mathsf{R}}_{i_*}$ consists of all the\noutput regions that are contained in the region on interest $\\ensuremath{\\mathsf{R}}^*_i$.\nNow, the examples for the second recursive call is given by the union of such\nexamples over all documents.\nThe final returned extractor is the results of the individual recursive calls\nin sequence.\n\n\\begin{algorithm}\n\\small\n\\caption{Hierarchical Data Extraction}\n\\label{algo:hierarchical-extraction}\n\\arsays{No one will understand the indexing notation}\n\\begin{algorithmic}[1]\n\\Require Extraction examples $\\mathsf{Ex}^*$\n\\Require Learners $\\mathsf{Learners} = \\mathcal{L}_0, \\ldots, \\mathcal{L}_n$\n\\State \\Return \\Call{HierarchicalExtract}{$\\mathsf{Ex}^*, \\mathsf{Learners}$}\n\\State\n\\Function{HierarchicalExtract}{$\\mathsf{Ex}, \\mathsf{Learners}$}\n\\State Let $\\mathsf{Ex}$ be $\\{ \\ensuremath{\\mathsf{doc}} \\mapsto \\ensuremath{\\mathsf{R}}_0\\ldots\\ensuremath{\\mathsf{R}}_n \\}_i$\n\\If{*} \\Comment{One-shot vs. Hierarchical}\n\\State \\Return \\Call{Extract}{$\\mathsf{Ex}, \\mathsf{Learners}$}\n\\Else\n\\State $\\{ \\ensuremath{\\mathsf{doc}} \\mapsto \\ensuremath{\\mathsf{R}}_0^*\\ldots\\ensuremath{\\mathsf{R}}_m^* \\}_i \\gets \\Call{IdentifyROI}{\\mathsf{Ex}}$\n\\State $\\mathsf{Ex}' \\gets \\{ \\ensuremath{\\mathsf{doc}} \\mapsto \\ensuremath{\\mathsf{R}}_0^*\\ldots\\ensuremath{\\mathsf{R}}_m^*\\}_i$\n\\State $\\mathsf{Ex}'' \\gets \\Call{Flatten}{\\{\n    \\{\n        \\ensuremath{\\mathsf{R}}^* \\mapsto \\ensuremath{\\mathsf{R}}_{p_0}\\ldots\\ensuremath{\\mathsf{R}}_{p_k} \\mid \\ensuremath{\\mathsf{R}}_{p_k} \\sqsubseteq \\ensuremath{\\mathsf{R}}^*\n    \\}_j\n\\}_i}$ \n\\State $\\mathsf{SE}_0 \\gets \\Call{HierarchicalExtract}{\\mathsf{Ex}', \\mathsf{Learners}}$\n\\State $\\mathsf{SE}_1 \\gets \\Call{HierarchicalExtract}{\\mathsf{Ex}'', \\mathsf{Learners}}$\n\\EndIf \n\\State \\Return $\\mathsf{SE}_0 \\cdot \\mathsf{SE}_1$\n\\EndFunction\n\\State\n\\Function{Extract}{$\\mathsf{Ex}, \\mathsf{Learners}$}\n\\State $\\mathcal{L} \\gets \\text{Pick learner from $\\mathsf{Learners}$}$\n\\State \\Return $\\mathcal{L}(\\mathsf{Ex})$\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\n\\paragraph{Identifying Regions of Interest}\n\n\\begin{itemize}\n  \\item Landmarks using delimiters\n  \\item Unsupervised clustering + limited DSL for predicates\n  \\item Common segments across multiple inputs and input formats\n\\end{itemize}\n\n\\paragraph{Learner Agnostic Robustness}\nTo ensure the robustness of the full sequence extractor produced by\nAlgorithm~\\ref{algo:hierarchical-extraction}, we ensure the robustness of the\nextractor produced at each step.\nRecall that we defined robustness using the notion of admissible inputs\n$\\ensuremath{\\mathbb{D}_A} \\subseteq \\ensuremath{\\mathsf{docs}}$, the set of input documents similar enough to the\nexample inputs, on which the sequence extractor is expected to operate\ncorrectly.\nHowever, $\\ensuremath{\\mathbb{D}_A}$ is not provided as an input to the procedure.\nInstead, we attempt to make the extractor robust using \\emph{perturbations}.\n\nGiven an example $\\mathsf{ex} = \\ensuremath{\\mathsf{doc}} \\mapsto \\ensuremath{\\mathsf{R}}_0\\ldots\\ensuremath{\\mathsf{R}}_n$, a\n\\emph{perturbation strategy} produces additional examples of the form\n$\\ensuremath{\\mathsf{doc}}' \\mapsto \\ensuremath{\\mathsf{R}}_0'\\ldots\\ensuremath{\\mathsf{R}}_n'$ derived from $\\mathsf{ex}$.\nInformally, given an example, a perturbation strategy generates multiple new\nexamples similar to the original.\nOur perturbation strategies make minor changes to the input $\\ensuremath{\\mathsf{doc}}$ which are\nlikely to change the expected output in a predictable way.\n\n\\begin{example}\n    Changing formats, changing entities, adding ads in emails, etc\n\\end{example}\n\n\\section{Introduction}\nPrograms synthesized using current benchmark examples aren't robust enough, they do not give correct outputs on similar unseen inputs. So following set of techniques were used for achieving robustness.\n\n\\section{Techniques}\nEach of these ideas are cover one or more following points of robustness:\n\\begin{itemize}\n    \\item Robust program is based on global property of input. For example, for TText this can be some character\/string like delimiters. \n    \\item The program tokens should be generic enough to cover unseen inputs. For example, TText regex token \"Words\/dots\/hyphens\" is more generic than \"Alphanumeric\".\n\\end{itemize}\n \n\\subsection{Perturbation}\nThe idea focuses on second point of robustness, where adding adding different varied input examples should guide the DSL to learning robust program. Introducing variety into constraints also help in the synthesizing program based on global property as different input representations go out of scope of DSL tokens.\\\\ \\\\\n\\begin{algorithm}[H]\n\\caption{Generate robust program: Perturbation}\n\\begin{algorithmic}[1]\n    \\Require Original benchmark constraints $C_1$, DSL Ranker $R$\n    \\State Generate new constraints $C_2$ using entity and character replacement in $C_1$, $C_2 = PERTURB(C_1)$\\\\\n    \\State Synthesize program $P = SYNTHESIS(C_2,R)$\n                          \n                           \\Comment{degree of robustness checked manually}\n    \\If{$P$ is robust enough}        \n    \\State {End}                       \n    \\Else {$C_2 = PERTURB(C_2)$to break program token in $P$, Go to 3}\n    \\EndIf\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Improved Ranking}\nDSL ranking is tweaked to give preference to certain characters based on which the relevant programs will get boost. In TText, these turn out to be common delimiters as pointed out in robustness earlier. For example, Words\/dots\/hyphens \u25e6 Comma to Comma, i.e. boost programs having only Comma than xyz \u25e6 Comma.\\\\ \\\\\n\\begin{algorithm}[H]\n\\caption{Generate robust program: Tweaked Ranking}\n    Given: original benchmark constraints $C_1$, DSL Ranker $R$ \\\\\n    Create new ranker $R_{mod}^1$ wherein delimiter is taken from user input.\\\\\n    Synthesize robust program $P = SYNTHESIS(C_1,R_{mod}^1)$\n\\end{algorithm}\n\n\n\\subsection{Perturbation and Ranking}\nCombining DSL ranking tweaking and adding perturbed examples overcomes shortcomings of both techniques and reduces user input.\\\\ \\\\\n\\begin{algorithm}[H]\n\\caption{Generate robust program: Perturbation and Ranking}\n    Given: original benchmark constraints $C_1$, DSL Ranker $R$ \\\\\n    Generate new constraints $C_2$ using entity and character replacement in $C_1$, $C_2 = PERTURB(C_1)$\\\\\n    $C= C_1\\cup C_2$\\\\\n    Create new ranker $R_{mod}^2$ by fixing delimiters such that each program in benchmarks sees same set of delimiters\\\\\n    Synthesize robust program $P = SYNTHESIS(C,R_{mod}^2)$ \n\\end{algorithm}\n\n\\subsection{Notes}\nPlease find below the revised algo for noisy landmark identification w.r.t to landmark inference.\nPreviously: \n\u2022\tIn landmark hierarchy \no\tNoisyScore = (1\/number of noisy landmarks + 1)\no\tNoisy landmarks is defined the landmark regions containing landmark string which present outside of any ROI computed using output labels. \no\tNoisyScore is used as stopping condition for deciding whether to do more rounds of landmark inference.\n\u2022\tROI which is given as input to synthesis is always w.r.t base landmark, outer and middle landmarks in hierarchy are used as anchors to reach to the correct base landmark using nearest landmark logic.\n\nCurrent:\n\u2022\tIn landmark hierarchy\no\tNoisy landmarks definition is changed slightly only for base landmark.\n\uf0a7\tNoisy landmarks for base landmark is defined as the landmark regions satisfying one more condition apart from this. Any base landmark string S which also occurs outside of the ROIs (R1, R2,\u2026Rn) when grown from noisy base LB' to a region R' (using steps computed using LB, R1), if R' structure is same as any of structures in (R1, R2,\u2026Rn) then base landmark S is noisy.\n\uf0a7\tSince we have ROI structure filtering at inference time, region grown from incorrect base landmarks will be filtered out.\no\tIf R' structure is same as we have in Getthere Name field, one more round of hierarchy is done. If R' is different as we have in Qatar then we stop at base landmark only and therefore have only Departure as landmark now, no mobile number is learnt as hierarchy. \no\tWith this change we now do not need to compute all base landmarks at inference, we revert to the original logic of picking up nearest landmark in hierarchy. \n\uf0a7\tWe did multiple base landmarks for Qatar because there was 1 mobile number but 2 Departure but now we have only Departure as landmark. \no\tRest of landmarks in hierarchy obey the previous definition of noisy landmarks since these are not concerned with ROI. \n\nImportant note:\n\u2022\tWith this reverting back to nearest landmark logic at inference, code is assuming that hierarchy is only of this pattern (L1 -> L2 -> L3 -> ..)* which says each ROI is associated with unique L1, L2, ..\n\n\\end{comment}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Introduction}\n\nLet $X$ be a noetherian scheme and denote by $\\mathop{\\sf FEC}\\nolimits(X)$ the site of finite \\'etale coverings of $X$. Then we can define the assignment $\\Pi_1:Y\\mapsto \\Pi_1(Y)$, where $Y$ is an \\'etale scheme over $X$ and $\\Pi_1(Y)$ is the \\'etale fundamental groupoid of $Y$. In this paper, we will show that this association, which is a 2-functor, is in actuality a costack over $\\mathop{\\sf FEC}\\nolimits(X)$. The costack, the dual notion of a stack, is a rather overlooked construction. However, as we will see, for a covariant 2-functor to be a costack is actually a very natural property, as it merely means that the 2-categorical version of the Seifert-van Kampen theorem holds for every covering. Hence, by proving that the 2-functor $\\Pi_1$ is a costack, we have proven that a slightly reformulated version of the Seifert-van Kampen theorem holds for the \\'etale fundamental groupoid.\n\nBut more is true. As we will show in this paper, $\\Pi_1$ is not just a costack over $X$, but indeed the 2-terminal costack. This demonstrates that the \\'etale fundamental groupoid is indeed defined by the Seifert-van Kampen theorem. This result agrees with the one obtained in \\cite{top.groupoid}, where we showed that the topological fundamental groupoid was a 2-terminal costack. Hence we are able to talk about the fundamental groupoid of a site, which could be of independent interest. Indeed, although not discussed in this paper, another use of this axiomatisation is that we can now also talk about other fundamental objects. For example, we could define the fundamental groupoid scheme of a site $\\bf{T}$ to be the 2-terminal objects in the 2-category of costacks over $\\bf{T}$ with values in groupoid schemes.\n\\newline\n\nThis paper was written as part of my PhD thesis at the University of Leicester under the supervision of Dr. Frank Neumann. He introduced me to the fundamental groupoid and inspired much of this paper, for which I would like to thank him.\n\n\\section{Preliminaries}\nIn this section we will fix some basic terminology. Recall that a 2-category is a category enriched in categories. A category $\\mathfrak{G}$ is said to be a \\emph{groupoid}\\index{Groupoid!} if every morphism is an isomorphism. \n\n\\begin{De} Let $\\mathfrak{G}$ be a groupoid such that for every $a,b\\in \\mathfrak{G}$, $\\mathsf {Hom}(a,b)$ is endowed with the profinite topology and further the composition and inverse maps are continuous. We say that $\\mathfrak{G}$ is a finitely connected profinite groupoid, if it is equivalent to the 2-coproduct of finitely many connected groupoids coming from profinite groups.\n\\end{De}\n\nNote that this is not the best way to define a profinite groupoid in the general case, since it does not deal with the topology on its connected components. One should define it as simply the filtered 2-limit of finite groupoids endowed with the profinite topology. But for the purposes of this paper, this simpler definition is adequate.\n\n\n\\subsection{2-Limits and 2-Colimits}\\label{sec.2-lim.2-colim}\n\nJust like limits and colimits are of fundamental importance in category theory, so too are 2-limits and 2-colimits in 2-category theory. In this subsection, we will talk a little about the 2-limit as well as the filtered 2-colimit of 2-functors. \n\\newline\n\nLet $I$ be a category and $\\mathfrak{F}:I\\rightarrow \\mathfrak{Cat}$ be  a covariant 2-functor from the category $I$ to the 2-category $\\mathfrak{Cat}$ of categories. For an element $i\\in I$ we let $\\mathfrak{F}_i$ be the value of $\\mathfrak{F}$ at $i$. For a morphism $\\psi:i\\rightarrow j$ we let $\\psi_*:\\mathfrak{F}_i\\to \\mathfrak{F}_j$ be the induced functor. For any $i\\xrightarrow{\\psi} j\\xrightarrow{\\nu} k$, one has the natural transformation $\\mu_{\\psi,\\nu}:\\nu_*\\psi_*\\rightarrow (\\nu\\psi)_*$ satisfying the coherence condition. We can now give the definitions:\n\nObjects of the category 2-$\\mathop{\\sf lim}\\nolimits\\limits_i\\mathfrak{F}_i$ (or simply 2-$\\mathop{\\sf lim}\\nolimits \\mathfrak{F}$) are collections $(x_i,\\xi_\\psi)$, where $x_i$ is an object of $\\mathfrak{F}_i$, while $\\xi_\\psi:\\psi_*(x_j)\\to x_i$ for $i\\leq j$, is an isomorphism of the category $\\mathfrak{F}_j$ satisfying the 1-cocycle condition. That is, for any $i\\xrightarrow{\\psi} j\\xrightarrow{\\nu} k$, the diagram\n$$\\xymatrix{\\nu_*(\\psi_*(x_i)) \\  \\  \\ar[r]^{\\nu_*(\\xi_{\\psi})}\\ar[d]^{\\mu_{\\psi,\\nu}} & \\nu_*(x_j)\\ar[d]^{\\xi_{\\nu}} \\\\ \n\t          (\\nu\\psi)_*(x_i)\\ar[r]_{\\xi_{\\psi\\nu}} & x_k }$$ \ncommutes. A morphism from $(x_i,\\xi_{\\psi})$ to $(y_i,\\eta_{\\psi})$ is  a collection $(f_i)$, where $f_i:x_i\\rightarrow y_i$ is a morphism of $\\mathfrak{F}_i$ such that for any $\\psi:i\\rightarrow j$, the following \n$$\\xymatrix{\\psi_*(x_i)\\ar[r]^{\\xi_{\\psi}}\\ar[d]_{\\psi_*(f_i)}& x_j\\ar[d]^{f_j} \\\\ \n\\psi_*(y_i)\\ar[r]_{\\eta_{\\psi}}& y_j}$$\n is a commutative diagram. \n\\newline\n\nThe dual notion $2$-$\\mathop{\\sf colim}\\nolimits\\limits_i \\mathfrak{F}_i$ is a category together with a family of functors\n $$\\alpha_i:\\mathfrak{F}_i\\to 2\\text{-}\\mathop{\\sf colim}\\nolimits\\limits_i \\mathfrak{F}_i$$\n and natural transformations $\\lambda_{ij}:\\alpha_j\\psi_{ij}\\Longrightarrow \\alpha_i$, $i\\leq j$,  satisfying  the $1$-cocycle condition: For any $i\\leq j\\leq k$,\n $$\\lambda_{ik}=\\lambda_{ij}\\circ (\\lambda_{jk}\\star\\psi_{ij}).$$\nFurthermore, one requires that for any category ${\\sf C}$, the canonical functor\n$$\\kappa:\\mathsf {Hom}_{\\mathfrak{Cat}}(2\\text{-}\\mathop{\\sf colim}\\nolimits\\limits_i\\mathfrak{F}_i,\\mathsf{C})\\to 2\\text{-}\\mathop{\\sf lim}\\nolimits\\limits_i\\mathsf {Hom}_{\\mathfrak{Cat}}(\\mathfrak{F}_i,{\\sf C})$$\nis an equivalence of categories. Here the functor $\\kappa$ is given by $\\kappa(\\chi)=(\\chi\\circ  \\alpha_i, \\chi_i\\star \\lambda_{ij})$. It is well-known that $2$-$\\mathop{\\sf colim}\\nolimits$ exists and is unique up to a unique equivalence of categories (see \\cite[pp. 192-193] {br}).\n\n\\section{Stacks and Costacks}\\label{2-mathematics}\n\n\\subsection{Stacks}\\label{stacksection}\n\nThis subsection follows \\cite{I.Mo} closely, even on notation. Let $X$ be a site and $\\mathfrak{F}:X^{op}\\rightarrow\\mathfrak{Cat}$ a 2-functor where $\\mathfrak{Cat}$ is the 2-category of categories. This is called a \\emph{fibered category}\\index{Fibered! Category} over $X$. It should be noted to avoid any confusion that this is sometimes called a prestack. If we have two fibered categories over $X$, then a morphism between them is called a \\emph{fibered functor}\\index{Fibered! Functor}. \n\nLet $X$ be a site and $\\mathfrak{F}:X^{op}\\rightarrow\\mathfrak{Cat}$ a 2-functor. Let $U$ be an object in $X$ and $\\mathfrak{U}=\\{U_i\\rightarrow U\\}$ a covering of $U$. Then we can consider the following diagram: \n$$\\xymatrix{\\bigsqcap\\limits_{i\\in I}\\mathfrak{F}(U_i)\\ar@<-.5ex>[r] \\ar@<.5ex>[r]& \\bigsqcap\\limits_{i,j\\in I}\\mathfrak{F}(U_{ij})\\ar@<-.7ex>[r]\\ar[r]\\ar@<.7ex>[r] & \\bigsqcap\\limits_{i,j,k\\in I}\\mathfrak{F}(U_{ijk}).}$$\nHere $U_{ij}=U_i\\times_{U}U_j$ and $U_{ijk}=U_i\\times_{U}U_j\\times_{U}U_k$.\nWe denote its 2-limit by $2$-$\\mathop{\\sf lim}\\nolimits(\\mathfrak{U},\\mathfrak{F})$. Note that the 2-limit in this case is usually called the \\emph{descent data} and denoted by $\\mathsf{Des}(\\mathfrak{U},\\mathfrak{F})$. However, working with 2-limits makes it much clearer what a costack should be, hence we will keep using the above notation. \n\n\\begin{De}[Stack]\\label{stack} A fibered category $\\mathfrak{F}$ over $X$ is called a \\emph{stack}\\index{Stack} if for all objects $U$ of $X$ and for all coverings $\\mathfrak{U}$ of $U$, the functor $\\mathfrak{F}(U)\\rightarrow 2$-$\\mathop{\\sf lim}\\nolimits(\\mathfrak{U},\\mathfrak{F})$ is an equivalence of categories.\n\\end{De}\n\nJust like for sheaves, we can talk about the associated stack of a 2-functor.\n\n\\begin{De}\\label{uniquestack} Let $\\mathfrak{F}$ be a fibered category over a site $X$. Then $\\hat{\\mathfrak{F}}$ is the associated stack of $\\mathfrak{F}$ if for every stack $\\mathfrak{G}$ over $X$, we have an equivalence of categories \n$$\\mathsf {Hom}_{\\mathsf{Fb}}(\\mathfrak{F},\\mathfrak{G})\\cong\\mathsf {Hom}_{\\mathsf{St}}(\\hat{\\mathfrak{F}},\\mathfrak{G})$$\nwhere $\\mathsf{Fb}$ denotes the 2-category of fibered categories and $\\mathsf{St}$ the 2-category of stacks.\n\\end{De}\n\n\\begin{Pro}[Direct Stackification]\\label{direct} Let $\\mathfrak{F}:X^{op}\\rightarrow\\mathfrak{Cat}$ be a 2-functor. Define $\\mathfrak{F}'(U):=2$-$\\mathop{\\sf colim}\\nolimits_\\mathfrak{U}(2$-$\\mathop{\\sf lim}\\nolimits(\\mathfrak{U},\\mathfrak{F}))$. In other words, we take the filtered 2-colimits of the categories $2$-$\\mathop{\\sf lim}\\nolimits(\\mathfrak{U},\\mathfrak{F})$ over all coverings $\\mathfrak{U}$ of $U$. After iterating it 3 times we get the stackification. That is to say, we have \n$$\\hat{\\mathfrak{F}}=\\mathfrak{F}'''(U)=2\\text{-}\\mathop{\\sf colim}\\nolimits_\\mathfrak{U}(2\\text{-}\\mathop{\\sf lim}\\nolimits(\\mathfrak{U},\\mathfrak{F}'')).$$ \n\\end{Pro}\n\nFor the proof of this theorem, see \\cite[Theorem 3.8]{Street}.\n\n\\subsection{Costacks}\n\nLet $X$ be a site and $\\mathfrak{F}:X\\rightarrow\\mathfrak{Cat}$ a 2-functor where $\\mathfrak{Cat}$ is the 2-category of categories. Dually to the above subsection, we call $\\mathfrak{F}$ a \\emph{cofibered category}\\index{Cofibered category} over $X$. \n\nTake an object $U$ of $X$ and a covering $\\mathfrak{U}=\\{U_i\\rightarrow U\\}$. Then we can consider the following diagram: \n$$\\xymatrix{\\bigsqcap\\limits_{i\\in I}\\mathfrak{F}(U_i)& \\bigsqcap\\limits_{i,j\\in I}\\mathfrak{F}(U_{ij})\\ar@<-.5ex>[l] \\ar@<.5ex>[l]& \\bigsqcap\\limits_{i,j,k\\in I}\\mathfrak{F}(U_{ijk})\\ar@<-.7ex>[l]\\ar[l]\\ar@<.7ex>[l]}$$\nwhere $U_{ij}=U_i\\times_{U}U_j$ and $U_{ijk}=U_i\\times_{U}U_j\\times_{U}U_k$.\nWe denote its 2-colimit by $2$-$\\mathop{\\sf colim}\\nolimits(\\mathfrak{U},\\mathfrak{F})$. \n\n\\begin{De}[Costack]\\label{cost} A cofibered category $\\mathfrak{F}$ over $X$ is called a \\emph{costack}\\index{Costack} if for all objects $U$ of $X$ and for all coverings $\\mathfrak{U}$ of $U$, the functor $\\mathfrak{F}(U)\\leftarrow 2$-$\\mathop{\\sf colim}\\nolimits(\\mathfrak{U},\\mathfrak{F})$ is an equivalence of categories.\n\\end{De}\n\nAlternatively, we can define a costack using stacks. Namely, $\\mathfrak{F}$ is a costack if for every category $C$, the assignment $U\\mapsto \\mathsf {Hom}_{\\mathfrak{C}}(\\mathfrak{F}(U),C)$ is a stack. It is clear that this is equivalent to the above since $\\mathsf {Hom}(A,-)$ is left exact. However, it should be noted that since it is not (in general) right exact, the duality between stacks and costacks breaks down here. Namely, it would not be sufficient to check that $U\\mapsto\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{F}(U),-)$ is a costack for $\\mathfrak{F}$ to be a stack.\n\n\\begin{De}[Uniqueness Property]\\label{uniquecostack} Let $\\mathfrak{F}$ be a cofibered category over a site $X$. Then we will say that $\\mathfrak{F}$ has an associated costack $\\hat{\\mathfrak{F}}$ if for every costack $\\mathfrak{G}$ over $X$, we have an equivalence of categories \n$$\\mathsf {Hom}_{\\mathsf{Cofb}}(\\mathfrak{G},\\mathfrak{F})\\cong\\mathsf {Hom}_{\\mathsf{Cost}}(\\mathfrak{G},\\hat{\\mathfrak{F}})$$\nwhere $\\mathsf{Cofb}$ denotes the 2-category of cofibered categories and $\\mathsf{Cost}$ the 2-category of costacks.\n\\end{De}\n\nNote that if our category took values in groupoids, then it would be enough to check it for every groupoid. Unfortunately, unlike for stacks, it is not known whether every cofibered category has an associated costack. However, if it exists, it is clearly unique in the 2-categorical sense.\n\n\n\\section{Galois Categories} \\label{galoischapter\n\nIn this section we will generalise the classical (connected) Galois categories to the finitely connected case. Our main interest in them is to study the \\'etale fundamental groupoid of a finitely connected, noetherian scheme. \n\nIn the first subsection, we will show that the analogue of Grothendieck's classical result holds, and that a finitely connected Galois category is equivalent to the category of $\\mathfrak{G}$-$\\mathop{\\sf FSets}\\nolimits$, where $\\mathfrak{G}$ is a finitely connected, profinite groupoid and $\\mathop{\\sf FSets}\\nolimits$ denotes the category of finite sets.\n\nIn the next subsection, we will talk about the whole 2-category of such Galois categories, which includes the morphisms and 2-morphisms. We sharpen the result above, by proving that there is a 2-equivalence between the 2-category of finitely connected categories and the 2-category of finitely connected profinite groupoids. While there are many generalisations of Galois categories, some of which are no doubt more general than ours, this reformulation of the classical theorem seems to be new. \n\nIndeed, it will follow that an even more general formulation is true, namely Corollary \\ref{Galoisstackequivalence}.\n\n\n\n\\subsection{Finitely Connected Galois Categories}\n\nIn order to give the main definition, recall that a morphism $u:A\\to B$ of a category $\\mathcal{C}$ is an \\emph{epimorphism}\\index{Morphism! Epimorphism} (resp. \\emph{monomorphism}\\index{Morphism! Monomorphism}) if for any object $X$ the induced map $\\mathsf {Hom}_{\\mathcal{C}}(B,X)\\to \\mathsf {Hom}_{\\mathcal{C}}(A,X)$ (resp. $\\mathsf {Hom}_{\\mathcal{C}}(X,A)\\to \\mathsf {Hom}_{\\mathcal{C}}(X,B))$)  is injective. Moreover, an epimorphism $u$ is called a \\emph{strict epimorphism}\\index{Morphism! Strict epimorphism} if the pull-back \n$$\\xymatrix{A\\times _B A\\ar@{-->}[r]^{p_1} \\ar@{-->}[d]^{p_2} & A\\ar[d]^u\\\\A\\ar[r]^u & B}$$ \nexists and $B$ is the coequaliser of the diagram: \n$$\\xymatrix{ A\\times_B A \\ar@<0.5ex>[r]^{\\ \\ \\ p_1} \\ar@<-0.5ex>[r]_{\\ \\ \\ p_2} & A\\ar[r]^{u} & B. }$$\n\nWe give the following definition of a finitely connected Galois category which differs from the standard definition of a Galois category (see for example \\cite{AGGT}).\n\n\\begin{De}\\label{f-c G c} A {\\sf (finitely-connected) Galois category}\\index{Galois Category!} is a category $\\mathcal{C}$ together with a set of covariant functors $\\{\\mathcal{F}_j:\\mathcal{C}\\rightarrow \\mathop{\\sf FSets}\\nolimits\\}_{j\\in J}$, satisfying the following axioms: \n\\begin{enumerate}\n\\item Finite limits exist in $\\mathcal{C}$. \n\\item Finite colimits exist in $\\mathcal{C}$. \n\\item Any morphism $u:Y\\rightarrow X$ in $\\mathcal{C}$ factors as $Y\\xrightarrow{u'}X'\\xrightarrow{u''}X$, where $u'$ is a strict epimorphism and $u''$ is a monomorphism and there is an isomorphism $v:X'\\coprod X''\\to X$ such that $u''=vi_1$, where $i_1:X'\\to X'\\coprod X''$ is the standard inclusion.\n\\item Every $\\mathcal{F}_j$ is right exact, i.e. $\\mathcal{F}_j$ respects finite colimits.\n\\item Every $\\mathcal{F}_j$ is left exact,  i.e. $\\mathcal{F}_j$ respects finite limits.\n\\item There exists a finite subset $I\\subset J$ such that any $\\{u:Y\\rightarrow X\\}$ in $\\mathcal{C}$ is an isomorphism if and only if $\\mathcal{F}_i(u)$ is an isomorphism for all $i\\in I$.\n\\end{enumerate}\n\\end{De} \n\nIf $I$ can be chosen to be a one element set, then $\\mathcal{C}$ is called \\emph{connected}\\index{Galois Category! Connected}. This is clearly equivalent to the standard definition of a Galois category. Several facts (Lemma 2.6 i), ii); Prop 3.1; Prop. 3.2 (1), (3) i), iii)) proven in \\cite{AGGT} have immediate generalisations in our situation. To state these statements, recall that an object $X$ is called \\emph{connected}\\index{Object! Connected} if $X\\not =0$ and  for any decomposition $X=Y\\coprod Z$ one has $X=0$ or $Y=0$. Here $0$ denotes the initial object. \n\n\\begin{Pro} \\label{2612} Let $\\mathcal{C}$ be a finitely connected Galois category. Then the following properties hold:\n\\begin{itemize} \n\\item[i)] A morphism $u$ is a monomorphism (resp.  strong epimorphism) if and only if for all $i\\in I$, the map  $\\mathcal{F}_i(u)$ is injective (resp. surjective). A morphism is an isomorphism if and only if it is a monomorphism and a strong epimorphism.\n\\item[ii)] An object $X$ is initial (resp. terminal) provided  for all $i\\in I$, the set $\\mathcal{F}_i(X)=\\emptyset$ (resp. $\\mathcal{F}(X)=*$). Here $*$ denotes the singleton.\n\\item[iii)] Any object $X$ has a unique decomposition $X=U_1\\coprod \\cdots \\coprod U_k$, where the $U_i$ are connected. \n\\item[iv)] If $U$ and $V$ are connected, then any morphism $U\\to V$ is a strong epimorphism. In particular, any endomorphism $U\\to U$ is an automorphism.\n\\item[v)] If $U$ is connected, then for any objects $A_1\\cdots A_m$ the natural map\n$$\\coprod_{i=1}^m \\mathsf {Hom}(U, A_i)\\to \\mathsf {Hom}(U, \\coprod_{i=1}^m A_i)$$\nis a bijection.\n\\end{itemize}\n\\end{Pro}\n\nThe proof is the same as for the connected case. (see \\cite{AGGT}).\n\n\\begin{Le}\\label{2712}  Let $\\mathcal{C}$ be a finitely-connected Galois category and $t=e_1\\coprod \\cdots \\coprod  e_d$ be a decomposition of the terminal object as a coproduct of connected objects. Then for each $1\\leq i\\leq d$  one has (after re-indexing)  $\\mathcal{F}_i(e_i)=*$ and $\\mathcal{F}_i(e_j)=\\emptyset$, $j\\not= i, \\  1\\leq i,j\\leq d$.\n\\end{Le}\n\n\\begin{proof}  Let $1\\leq i\\leq d$. Since $0\\to e_i$ is not an isomorphism, there exists at least one $\\mathcal{F}$ such that $\\mathcal{F}(e_i)\\not =\\emptyset$. After reindexing we can assume that $\\mathcal{F}=\\mathcal{F}_i$. Since $\\mathcal{F}_i(t)=*$ and $\\mathcal{F}_i$ respects coproducts we see that \n$$\\mathcal{F}_i(e_1)\\coprod \\cdots \\coprod \\mathcal{F}_i(e_d)=*.$$\nSo all terms except $\\mathcal{F}_i(e_i)$ are empty sets and the result follows.\n\\end{proof}\n\nNow we are in the position to prove the following result.\n\n\\begin{Le}[Main Lemma]\\label{G.MainLemma} Let $\\mathcal{C}$ be a finitely-connected Galois category. If\n$$t=e_1\\coprod \\cdots \\coprod  e_d$$\nis the decomposition of the terminal object $t$ as a coproduct of connected objects, then there is an equivalence of categories\n$$\\mathcal{C}\\cong \\mathcal{C}_1\\times \\cdots \\times \\mathcal{C}_d$$\nwhere $C_i$ , $1\\leq i\\leq d$ is the following full subcategory of $C$:\n$$\\mathcal{C}_i=\\{X\\in \\mathcal{C}| \\ \\mathcal{F}_j(X)=\\emptyset, j\\not =i, 1\\leq j\\leq d\\}.$$\nFurthermore, the pair $(\\mathcal{C}_i,\\mathcal{F}_i)$ is a connected Galois category and for any element $k\\in I$ the functor $\\mathcal{F}_k$ is isomorphic to exactly one of the functors $\\mathcal{F}_1,\\cdots , \\mathcal{F}_d$.\n\\end{Le}\n\n\\begin{proof} We proceed by induction on $d$. Assume $d=1$. Thus $t$ is connected. In this case $\\mathcal{C}_1=\\mathcal{C}$. Take any of $\\mathcal{F}_i$ and call it $\\mathcal{F}$. First we show that $\\mathcal{F}$ reflects isomorphisms, meaning that if $v$ is a morphism, such that $\\mathcal{F}(v)$ is an isomorphism, then $v$ is an isomorphism. If $U$ is connected, then $U\\to t$ is a strict epimorphism thanks to Proposition \\ref{2612} iv). It follows that $\\mathcal{F}(U)\\to \\mathcal{F}(t)=*$ is a strict epimorphism. Thus $\\mathcal{F}(U)\\not=\\emptyset$. Since any object is a coproduct of connected ones and $\\mathcal{F}$ respects coproducts, it follows that if $A$ is not an initial object, then $\\mathcal{F}(A)\\not =\\emptyset.$\nAssume $u:A\\to B$ is a monomorphism, such that $\\mathcal{F}(u)$ is an isomorphism. Then $B\\cong A\\coprod \\mathcal{C}$. Hence $\\mathcal{F}(\\mathcal{C})=\\emptyset$, so $\\mathcal{C}=0$ and $u$ is an isomorphism. \n\nNow  let $v:A\\to B$ be a general morphism, such that $\\mathcal{F}(v)$ is an isomorphism. Consider the following commutative diagram\n$$\\xymatrix{A\\ar[drr]^{id}\\ar[dr]_{\\delta}\\ar[ddr]_{id}&&\\\\ & A\\times_BA\\ar[r]\\ar[d]&A\\ \\ar[d]_v\\\\&A\\ar[r]^v& B .}$$\nHere $\\delta$ is the diagonal map and hence a monomorphism. Apply $\\mathcal{F}$ to this diagram and use the fact that $\\mathcal{F}$ preserves pulbacks and $\\mathcal{F}(v)$ is an isomorphism. We obtain that $\\mathcal{F}(\\delta)$ is an isomorphism. Thus $\\delta$ is an isomorphism. It follows that $v$ is a monomorphism (thanks to \\cite[Lemma 2.4]{AGGT}) and hence an isomorphism. \nThus $\\mathcal{F}$ reflects isomorphisms and $(\\mathcal{C},\\mathcal{F})$ is a connected Galois category. By \\cite[Theorem 2.8]{AGGT} any other $\\mathcal{F}_i$ is isomorphic to $\\mathcal{F}$. Hence we have proven the lemma for $d=1$.\n\nAssume now that $d>1$. Thanks to Lemma \\ref{2712}, we have $\\mathcal{F}_i(e_j)=\\emptyset$ for all $j\\not =i$ and $\\mathcal{F}_i(e_i)=*$, $1\\leq i,j\\leq d$. One easily sees that for each $1\\leq i\\leq d$ the subcategory $\\mathcal{C}_i$ is closed under finite limits and colimits. Assume $X=\\coprod_{j=1}^k U_j$ is a decomposition as a coproduct of connected objects. \n\n\\underline{Claim 1}: We have $X\\in \\mathcal{C}_i$ if and only if $U_1,\\cdots, U_k\\in \\mathcal{C}_i$. In fact if  $U_1,\\cdots, U_k\\in \\mathcal{C}_i$, then for any $j\\not =i$, $1\\leq j\\leq d$ one has $\\mathcal{F}_j(U_1)=\\cdots =\\mathcal{F}_j(U_k)=\\emptyset$. Thus\n$\\mathcal{F}_j(X)=\\mathcal{F}_j(U_1)\\coprod\\cdots\\coprod \\mathcal{F}_j(U_k)=\\emptyset.$ Hence $X\\in \\mathcal{C}_i$. Conversely, if $X\\in \\mathcal{C}_i$, then $$\\emptyset =\\mathcal{F}_j(X)=\\mathcal{F}_j(U_1)\\coprod\\cdots\\coprod \\mathcal{F}_j(U_k).$$\nThus $\\mathcal{F}_j(U_1)=\\cdots= \\mathcal{F}_j(U_k)=\\emptyset$ and $U_1,\\cdots U_k\\in \\mathcal{C}_i$.\n\n\\underline{Claim 2}: The object $e_i$ is a terminal object in the category $\\mathcal{C}_i$. So, we have to prove that the set $\\mathsf {Hom}(X,e_i)$ is a singleton provided $X\\in \\mathcal{C}_i$. By the first claim it is enough to assume that $X$ is connected. According to Proposition \\ref{2612} v) \n $$*=\\mathsf {Hom}(X,t)=\\mathsf {Hom}(X,e_1)\\coprod \\cdots \\coprod \\mathsf {Hom}(X,e_d).$$\n So the set $\\mathsf {Hom}(X,e_i)$ has at most one element. To show that it has exactly one element, we need to show that $\\mathsf {Hom}(X,e_j)=\\emptyset$ for $j\\not =i$. In fact, assume there is a morphism $X\\to e_j$. Since both objects are connected, this map must be a strict epimorphism. This implies that $\\emptyset =\\mathcal{F}_j (X)\\to \\mathcal{F}_j(e_j)=*$ is surjective and hence a contradiction. Thus our second claim is proven. It follows from the case $d=1$, that the pair $(\\mathcal{C}_i,\\mathcal{F}_i)$ is a connected Galois category.\n\n\\underline{Claim 3}: Our third claim is that if $i\\not =j$, then for any objects $0\\not =X\\in \\mathcal{C}_i$ and $Y\\in \\mathcal{C}_j$ one has $\\mathsf {Hom}(X,Y)=\\emptyset$. In fact, since $e_j$ is terminal in $\\mathcal{C}_j$ there exists a unique morphism $Y\\to e_j$. Thus it suffices to show that $\\mathsf {Hom}(X,e_j)=\\emptyset$, but this was shown in the proof of Claim 2. \n\\newline \n\nDefine the functor\n$$\\xi:\\mathcal{C}_1\\times \\cdots \\times \\mathcal{C}_d\\to \\mathcal{C}$$\nby $\\xi(X_1,\\cdots,X_d)=X_1\\coprod \\cdots \\coprod X_d$.\nWe will show that the functor $\\xi$ is an equivalence of categories. Take an object $X\\in \\mathcal{C}$ and consider the pull-back\n$$\\xymatrix{X_i\\ar[d]\\ar[r]&X\\ar[d]\\\\e_i\\ar[r]&t.}$$\n\\underline{Claim 4}: We want to show that $X_i\\in \\mathcal{C}_i$. In fact, take any $j\\not =i$. Since the functor $\\mathcal{F}_j$ respects pullbacks we obtain a diagram of sets\n$$\\xymatrix{\\mathcal{F}_j(X_i)\\ar[d]\\ar[r]&\\mathcal{F}_j(X)\\ar[d] \\ \\\\ \n\\emptyset \\ar[r]& \\mathcal{F}_j(t).}$$\nIt follows that $\\mathcal{F}_j(X_i)=\\emptyset$ and thus $X_i\\in \\mathcal{C}_i$. Moreover, the natural morphism $X_1\\coprod X_d\\to X$ is an isomorphism, because every $\\mathcal{F}_i$ takes it to an isomorphism. It follows that the functor $\\xi$ is essentially surjective. It remains to show that the functor $\\xi$ is full and faithful. Take the objects $X_i,Y_i\\in \\mathcal{C}_i$, $i=1,\\cdots,d$. We have \n$$\\mathsf {Hom}(\\coprod_{i=1}^d,\\coprod_{i=1}^dY_i)=\\prod_{i=1}^d\\mathsf {Hom}(X_i, Y_1\\coprod \\cdots \\coprod Y_d)$$\nThus it remains to show that for any object $Z\\in \\mathcal{C}_i$ one has \n$$\\mathsf {Hom}(Z, Y_1\\coprod \\cdots \\coprod Y_d)=\\mathsf {Hom}(Z,Y_i)$$\nIf $Z$ is connected this follows from Proposition \\ref{2612} v) and Claim 3. For the general case we decompose $Z=Z_1\\coprod \\cdots \\coprod Z_k$. \n\nThen  we  have\n\\begin{align*}\n\\mathsf {Hom}(Z, Y_1\\coprod \\cdots \\coprod Y_d)&=\\mathsf {Hom}(\\coprod_{j=1}^kZ_j, Y_1\\coprod \\cdots \\coprod Y_d)\\\\ \n&=\\prod_{j=1}^k\\mathsf {Hom}(Z_j, Y_1\\coprod \\cdots \\coprod Y_d)\\\\ &=\\prod_j\\mathsf {Hom}(Z_j,Y_i)\\\\\n&=\\mathsf {Hom}(\\coprod Z_j,Y_i)\\\\ \n&=\\mathsf {Hom}(Z,Y_i)\n\\end{align*}\nHence $\\xi$ is an equivalence of categories.\n\\end{proof}\n\nRecall the following easy but important fact:\n\n\\begin{Pro}\\label{homrep} Let $\\mathfrak{G}$ be a connected groupoid and let $x\\in\\mathfrak{G}$. Then we have an equivalence of categories \n$$\\mathsf {Hom}_{\\mathfrak{Cat}}(\\mathfrak{G},\\mathsf{Sets})\\cong \\mathsf {Aut}(x)\\text{-}\\mathsf{Sets}$$\nwhere $\\mathsf {Aut}(x)\\text{-}\\mathsf{Sets}$ denotes the category of $\\mathsf {Aut}(x)$-Sets.\n\\end{Pro}\n\nIf we changed $\\mathsf{Sets}$ with $\\mathop{\\sf FSets}\\nolimits$, the category of finite sets, the above would still hold. Hence we can now generalise group actions to groupoid actions and for a groupoid $\\mathfrak{G}$, we write $\\mathfrak{G}$-$\\mathop{\\sf FSets}\\nolimits$ or $\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G},\\mathop{\\sf FSets}\\nolimits)$. It should be emphasised that we use two different notations for the exact same category. The second notation will  mainly be used in calculations. If, however, $\\mathfrak{G}$ were a profinite groupoid, then we would only consider the continuous actions. For simplicity though, by abuse of notations, we will still refer to it as $\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G},\\mathop{\\sf FSet}\\nolimits)$ or $\\mathfrak{G}$-$\\mathop{\\sf FSets}\\nolimits$. \n\n\\begin{Co}\\label{G.essentialsurj} Let $\\{\\mathcal{F}_i:\\mathcal{C}\\rightarrow \\mathop{\\sf FSets}\\nolimits\\}_{i\\in I}$ be a finitely connected Galois Category. Then it is equivalent to $\\mathfrak{G}$-$\\mathop{\\sf FSets}\\nolimits$, where $\\mathfrak{G}$ is a finitely connected profinite groupoid.\n\\end{Co}\n\n\\begin{proof} As proven in the above lemma (\\ref{G.MainLemma}), \n$$\\{\\mathcal{F}_i:\\mathcal{C}\\rightarrow \\mathop{\\sf FSets}\\nolimits\\}_{i\\in I}\\cong\\{\\mathcal{F}_j:\\prod_{j'\\in J} \\mathcal{C}_{j'}\\rightarrow \\mathop{\\sf FSets}\\nolimits\\}_{j\\in J}$$ \nsuch that $J$ is a finite set and $\\mathcal{F}_j(C_k)=\\emptyset$ for $k\\neq j$. Hence our Galois category is equivalent to $\\mathcal{F}_j:\\prod_{j\\in J}\\mathcal{C}_j\\rightarrow \\mathop{\\sf FSets}\\nolimits$. Again by the above lemma, we know that for each $j\\in J$, the functor $\\mathcal{F}_j:\\mathcal{C}_j\\rightarrow \\mathop{\\sf FSets}\\nolimits$ is a connected Galois category and hence, using Proposition \\ref{homrep}, we know that it is equivalent to $\\mathfrak{G}$-$\\mathop{\\sf FSets}\\nolimits$ where $\\mathfrak{G}$ is a connected, profinite groupoid. The result now follows from the fact that \n$$\\prod_{j\\in J}\\mathsf {Hom}_{\\mathfrak{Cat}}(\\mathfrak{G}_j,\\mathop{\\sf FSets}\\nolimits)\\cong \\mathsf {Hom}_{\\mathfrak{Cat}}(\\coprod_{j\\in J}\\mathfrak{G}_j,\\mathop{\\sf FSets}\\nolimits).$$\n\\end{proof}\n\n\\subsection{The 2-category of Galois Categories}\n\n\\begin{De} Let $\\{\\mathcal{F}_i:\\mathcal{C}\\rightarrow \\mathop{\\sf FSets}\\nolimits\\}_{i\\in I}$ and $\\{\\mathcal{G}_j:\\mathcal{D}\\rightarrow \\mathop{\\sf FSets}\\nolimits\\}_{j\\in J}$ be two Galois categories. A morphism of Galois categories consists of a map $f:J\\rightarrow I$, a functor $\\varphi:\\mathcal{C}\\rightarrow \\mathcal{D}$ preserving finite limits and finite colimits and a collection of isomorphisms $\\lambda_{j,\\varphi},j\\in J$, as given in the following diagram \n$$\\xymatrix{ \\mathcal{D}\\ar[rd]_{\\mathcal{G}_j} & \\dtwocell\\omit{\\lambda_{j,\\varphi}}& \\mathcal{C}\\ \\ar[ll]_{\\varphi}\\ar[dl]^{\\mathcal{F}_f(j)}  \\\\\n& {\\mathop{\\sf FSets}\\nolimits} & . }$$ \nWe will refer to it as $\\{(f,\\varphi,\\lambda_{j,\\varphi}):\\mathcal{F}_{f(j)}\\rightarrow \\mathcal{G}_j\\}$. \n\\end{De} \n\nTo define composition, we need to define the composition of the $\\lambda_{j,\\varphi}$'s. So say we now have \n$$\\xymatrix{ \\mathcal{E}\\ar[rrrrdd]_{\\mathcal{H}_k}& & \\drtwocell\\omit{\\ \\ \\ \\ \\lambda_{k,\\phi}} & & \\mathcal{D}\\ar[dd]^{\\mathcal{G}_{f(k)}}\\ar[llll]^{\\phi} & & \\dltwocell\\omit{\\lambda_{k,\\varphi}} & & \\mathcal{C}\\ \\ar[llll]^{\\varphi}\\ar[ddllll]^{\\ \\ \\ \\ \\mathcal{F}_g(i)} & &  \\\\ \n& & & & & & & & \\\\ \n& & & & {\\mathop{\\sf FSets}\\nolimits} & & & & .}$$ \nDefine $\\lambda_{k,\\phi}\\circ\\lambda_{k,\\varphi}(x)=\\lambda_{k,\\phi}(\\varphi(x))\\circ\\lambda_{k,\\varphi}(x).$ In more detail we have \n$$\\lambda_{k,\\phi}\\circ\\lambda_{k,\\varphi}(x):\\mathcal{F}(x)\\xrightarrow{\\lambda_{k,\\varphi}(x)}\\mathcal{G}(\\varphi(x))\\xrightarrow{\\lambda_{k,\\phi}(\\varphi(x))}\\mathcal{E}(\\phi\\circ\\varphi(x)).$$\n\nIt is easily verified that the above construction is strictly associative.\n\n\\begin{De} A 2-morphism between $\\{(f,\\varphi,\\lambda_{j,\\varphi}):\\mathcal{F}_{f(j)}\\rightarrow \\mathcal{G}_j\\}$ and \\newline $\\{(f,\\phi,\\lambda_{j,\\phi}):\\mathcal{F}_{f(j)}\\rightarrow \\mathcal{G}_j\\}$ is a natural transformation \n$$\\xymatrix{ \\mathcal{C}\\ar@<-1.5ex>[r]_{\\phi} \\ar@<1.5ex>[r]^{\\varphi}\\rtwocell\\omit_{\\ \\ \\ \\ \\ \\zeta} & \\mathcal{D} }$$\nsuch that additionally the following diagram \n$$\\xymatrix{\\mathcal{F}(x)\\ar[r]^{\\lambda_{j,\\varphi}(x)}\\ar[dr]_{\\lambda_{j,\\phi}(x)} & \\mathcal{G}(\\varphi(x))\\ar[d]^{\\zeta(x)} \\\\\n\t           & \\mathcal{G}(\\phi(x)) }$$\ncommutes for all $j$ and all $x$.\n\\end{De}\n\nThis shows that we can talk about the (strict) 2-category of Galois categories. We will denote it by $\\mathsf{GCat}$. \n\n\\begin{Le}\\label{hshjsjk} Let $\\{\\mathcal{F}_i:\\mathcal{C}_1\\times\\cdots\\times \\mathcal{C}_n\\rightarrow \\mathop{\\sf FSets}\\nolimits\\}_{i\\in I}$ be a finitely connected Galois category and $\\mathcal{G}:\\mathcal{D}\\rightarrow \\mathop{\\sf FSets}\\nolimits$ a connected Galois category. \n\nLet $\\mathcal{A}:\\mathcal{C_1}\\times\\cdots\\times \\mathcal{C}_n\\rightarrow \\mathcal{D}$ be a functor between the Galois categories preserving finite limits and finite colimits. Then there exists an $i\\in I$ and $\\mathcal{A}_i:\\hat{\\mathcal{C}_i}\\rightarrow \\hat{\\mathcal{D}}$, such that $\\mathcal{A}=\\mathcal{A_i}\\circ p_i$, where $p_i:\\mathcal{C}_1\\times\\cdots\\times \\mathcal{C}_n\\rightarrow \\mathcal{C}_i$ is the $i$-th projection.\n\\end{Le}\n\n\\begin{proof} Take the terminal object $t\\in \\mathcal{C}$. As in Lemma \\ref{G.MainLemma}, $t=\\coprod_{i\\in I}e_i$, where every $e_i$ is connected. Since $\\mathcal{D}$ is connected, its terminal object is connected. Since $\\mathcal{A}$ respects finite limits and colimits there exists $i\\in I$ such that $\\mathcal{A}(e_i)=\\star$ and $\\mathcal{A}(e_j)=\\emptyset$ for all $j\\neq i$. Recall now the equivalence $\\mathcal{C}\\cong \\mathcal{C}_1\\times\\cdots\\times \\mathcal{C}_n$ as constructed in Lemma \\ref{G.MainLemma}. For every $X\\in \\mathcal{C}$ we have $X\\cong \\coprod_{i}^n X_i$ where $X_i$ is the pullback of the diagram:\n$$\\xymatrix{ & X\\ar[d] \\\\ \n\t\te_i\\ar[r] & t=\\coprod_{i\\in I}e_i.}$$ \nSince $\\mathcal{A}$ respects pullbacks, for all $j\\neq i$ we have that $\\mathcal{A}(X_j)$ is the pullback of \n$$\\xymatrix{ & \\mathcal{A}(X)\\ar[d] \\\\ \n\t\t\\mathcal{A}(e_j)\\ar[r] & \\mathcal{A}(t)}$$ \nand since $Y\\rightarrow \\emptyset$ implies that $Y$ is the empty set, we get that $\\mathcal{A}(X_j)=\\emptyset$. So $\\mathcal{A}(X)=\\coprod\\limits_j \\mathcal{A}(X_j)=\\mathcal{A}(X_i)$. Hence $\\mathcal{A}$ factors through the projection $\\mathcal{C}_1\\times\\cdots\\times \\mathcal{C}_n\\rightarrow \\mathcal{C}_i$.\n\\end{proof}\n\n\\begin{Co}\\label{G.faithfullyflat} Let $\\mathfrak{G}_i$-s be finitely connected profinite groupoids indexed by a finite set $I$ and $\\mathfrak{H}$ be a connected profinite groupoid. Then we have an equivalence of categories: \n$$\\mathsf {Hom}_\\mathsf{GCat}(\\mathsf {Hom}_\\mathfrak{Cat}(\\coprod_{i\\in I}\\mathfrak{G}_i,\\mathop{\\sf FSets}\\nolimits),\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{H},\\mathop{\\sf FSets}\\nolimits))\\cong $$\n$$\\cong\\coprod_{i\\in I}\\mathsf {Hom}_{\\mathsf{GCat}}(\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G}_i,\\mathop{\\sf FSets}\\nolimits),\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{H},\\mathop{\\sf FSets}\\nolimits)).$$\n\\end{Co}\n\n\\begin{proof} This follows from the Lemmas \\ref{hshjsjk} and \\ref{G.essentialsurj} and the fact that by definition functors in $\\mathsf{GCat}$ respect finite limits and finite colimits.\n\\end{proof}\n\n\\begin{Th}\\label{galois equivalence} The 2-category of finitely connected Galois categories is contravariantly 2-equivalent to the 2-category of profinite finitely connected groupoids.\n\\end{Th}\n\n\\begin{proof} This equivalence is given by associating to a profinite groupoid $\\mathfrak{G}$, the Galois category $\\mathsf {Hom}_{\\mathfrak{Cat}}(\\mathfrak{G},\\mathsf{Sets})$. On functors and natural transformations, the 2-functor is defined in the obvious way by composition. To show that this is a 2-equivalence, we only need to show that it's essentially surjective and full and faithful. Both of course in the 2-mathematical sense. Essential surjectivity is proven in Corollary \\ref{G.essentialsurj}. \n\\newline\n\\underline{Full and Faithful}: Let $\\mathfrak{G}$ and $\\mathfrak{H}$ be profinite and finitely connected groupoids. We have $\\mathfrak{G}\\cong\\coprod_{i\\in I}\\mathfrak{G}_i$ and $\\mathfrak{H}\\cong \\coprod_{j\\in J}\\mathfrak{H}_j$, where the $\\mathfrak{G}_i$-s and $\\mathfrak{H}_j$-s are profinite groupoids with one object. Hence \n\\begin{eqnarray*} \\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G},\\mathfrak{H})&\\cong&\\mathsf {Hom}_{\\mathfrak{Cat}}(\\coprod_{j\\in J}\\mathfrak{G}_i,\\coprod_{j\\in J}\\mathfrak{H}_j) \\\\\n\t\t\t\t\t     &\\cong&\\prod_{i\\in I}\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G}_i,\\coprod_{j\\in J}\\mathfrak{H}_j) \\\\\n\t\t\t\t\t    &\\cong& \\prod_{i\\in I}\\coprod_{j\\in J}\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G},\\mathfrak{H}_j).\n\\end{eqnarray*}\nThe last equivalence comes from the fact that the $\\mathfrak{G}_i$-s and $\\mathfrak{H}_j$-s have a single object. Hence any functor $\\mathfrak{G}_i$ can only go to a single $\\mathfrak{H}_j$. From \\cite[Corolarry 6.2. p.111]{SGAI} we get  that $\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G}_i,\\mathfrak{H}_j)\\cong\\mathsf {Hom}_\\mathsf{GCat}(\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{H}_j,\\mathop{\\sf FSets}\\nolimits),\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G}_i,\\mathop{\\sf FSets}\\nolimits)).$ On the other hand z\n\\begin{eqnarray*} & & \\mathsf {Hom}_\\mathsf{GCat}(\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{H},\\mathop{\\sf FSets}\\nolimits),\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G},\\mathop{\\sf FSets}\\nolimits))\\\\ \n\t\t          &\\cong&\\mathsf {Hom}_\\mathsf{GCat}(\\mathsf {Hom}_\\mathfrak{Cat}(\\coprod_{j\\in J}\\mathfrak{H}_j,\\mathop{\\sf FSets}\\nolimits),\\mathsf {Hom}_\\mathfrak{Cat}(\\coprod_{i\\in I}\\mathfrak{G}_i,\\mathop{\\sf FSets}\\nolimits)) \\\\ \n\t                    &\\cong&\\mathsf {Hom}_\\mathsf{GCat}(\\mathsf {Hom}_\\mathfrak{Cat}(\\coprod_{j\\in J}\\mathfrak{H}_j,\\mathop{\\sf FSets}\\nolimits),\\prod_{i\\in I}\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G},\\mathop{\\sf FSets}\\nolimits)) \\\\\n\t\t         &\\cong&\\prod_{i\\in I}\\mathsf {Hom}_\\mathsf{GCat}(\\mathsf {Hom}_\\mathfrak{Cat}(\\coprod_{j\\in J}\\mathfrak{H}_j,\\mathop{\\sf FSets}\\nolimits),\\mathsf {Hom}_\\mathfrak{Cat}(\\mathfrak{G},\\mathop{\\sf FSets}\\nolimits)).\n\\end{eqnarray*}\nUsing  Corollary \\ref{G.faithfullyflat} we get the desired result.\n\\end{proof}\n\n\\begin{Co}\\label{eq} Let $\\mathfrak{G}\\rightarrow \\mathfrak{H}$ be a functor between finitely connected, profinite groupoids. Assume that the induced functor $\\mathsf {Hom}_{\\mathfrak{Cat}}(\\mathfrak{H},\\mathop{\\sf FSets}\\nolimits)\\rightarrow \\mathsf {Hom}_{\\mathfrak{Cat}}(\\mathfrak{G},\\mathop{\\sf FSets}\\nolimits)$ is an equivalence of categories. Then $\\mathfrak{G}\\rightarrow \\mathfrak{H}$ was already an equivalence of categories.\n\\end{Co}\n\n\\begin{proof} From Theorem \\ref{galois equivalence} we get that \n$$\\mathsf{Eq}(\\mathfrak{G},\\mathfrak{H})\\cong \\mathsf{Eq}(\\mathfrak{G}\\text{-}\\mathop{\\sf FSets}\\nolimits,\\mathfrak{H}\\text{-}\\mathop{\\sf FSets}\\nolimits)$$\nwhere $\\mathsf{Eq}$ denotes the category of equivalences. This immediately implies the result.\n\\end{proof}\n\nUnfortunately, we can not take general 2-limits with values in $\\mathsf{GCat}$. So we can not talk about stacks with values in the 2-category of Galois categories for a general site. However, it follows from Proposition \\ref{galois2limit} below, that if we only consider sites where every covering can be replaced by a finite one, we can avoid that problem. \n\n\\begin{De} We call a site \\emph{finitely coverable}\\index{Site! Finitely coverable} if for every covering $\\{U_i\\rightarrow U\\}_{i\\in I}$ there exists a refinement $\\{V_j\\rightarrow U\\}_{j\\in J}$ such that $J$ is a finite set.\n\\end{De}\n\nA stack with values in the 2-category of Galois categories will be referred to as a \\emph{Galois stack}\\index{Galois Category! Stack}. Similarly for a prestack. If we have two 2-functors $\\mathfrak{F},\\mathfrak{G}:X\\rightarrow \\mathsf{GCat}$ from a finitely coverable site $X$ with values in Galois categories, then we will call a morphism between them that respects the structures a \\emph{Galois transformation}\\index{Galois Category! Transformation}, even though it would technically be a Galois functor. \n\n\\begin{Rem} It should be noted that for Galois 2-functors $\\mathfrak{F}$ and $\\mathfrak{G}$, when we write $\\mathsf {Hom}(\\mathfrak{F},\\mathfrak{G})$, we automatically assume that these are Galois transformations (i.e. respect finite limits and finite colimits), not just morphisms between 2-functors. \n\\end{Rem}\n\nLet $X$ be a site and $\\mathfrak{F}:X\\rightarrow \\sf{Groupoids}$ a covariant 2-functor. Then we denote by $\\mathfrak{F}_S$ the contravariant 2-functor given by $U\\mapsto Hom_{\\mathfrak{Cat}}(\\mathfrak{F}(U),\\mathsf{Sets})$. Now take two covariant 2-functors $\\mathfrak{E},\\mathfrak{F}:X\\rightarrow \\sf{Groupoids}$ and $F:\\mathfrak{E}\\rightarrow \\mathfrak{F}$ a natural transformation. Then it is clear that $F_S:\\mathfrak{F}_S\\rightarrow \\mathfrak{E}_S$ is a Galois transformation. But indeed Theorem \\nolinebreak \\ref{galois equivalence} shows that the reverse is also true. Hence we have the following as well:\n\n\\begin{Co}\\label{Galoisstackequivalence} Let $X$ be a site. Then the 2-category of fibred functors over $X$ with values in Galois categories, and morphisms and 2-morphisms preserving the Galois structure, is contravariantly equivalent to the 2-category of cofibered functors over $X$ with values in profinite groupoids.\n\\end{Co}\n\n\\subsection{Galois categories under stackification}\n\nFrom Proposition \\ref{direct} it follows that if a property is preserved by both 2-limits and filtered 2-colimits, it is preserved by stackification. Hence we have these following facts, which can be checked by routine methods for 2-limits and 2-colimits separately:\n\n\\begin{Pro}\\label{galois2limit} Let $I$ be a finite category and $\\mathfrak{F}:I\\rightarrow \\mathsf{GCat}$ be a 2-functor from $I$ to the 2-category of Galois categories given by $i\\mapsto \\{\\mathcal{F}_{ij}:\\mathcal{C}_i\\rightarrow\\mathop{\\sf FSets}\\nolimits\\}_{j\\in J_i}$. Then the 2-limit of $\\mathfrak{F}$ is again a Galois category and all the natural projections $2$-$\\mathop{\\sf lim}\\nolimits\\limits_i\\mathfrak{F}_i\\rightarrow \\mathfrak{F}_i$ preserve the Galois structure, i.e. are exact.\n\\end{Pro}\n\n\\begin{Le}\\label{Galois} Let $X$ be a site and $\\mathfrak{F}:X\\rightarrow \\mathfrak{Cat}$ be a 2-functor, $\\mathfrak{G}:X\\rightarrow \\mathfrak{Cat}$ a stack in $\\mathfrak{Cat}$ and $\\psi:\\mathfrak{F}\\rightarrow \\mathfrak{G}$ a natural transformation. Assume that $\\mathfrak{F}$ has an associated stack $\\hat{F}$ and let $\\hat{\\psi}:\\hat{\\mathfrak{F}}\\rightarrow \\mathfrak{G}$ be the associated natural transformation. Then the following hold:\n\\begin{itemize} \n\\item[i)] If $\\psi$ respects (finite) limits, so does $\\hat{\\psi}$;\n\\item[ii)] If $\\psi$ respects (finite) colimits, so does $\\hat{\\psi}$.\n\\end{itemize}\n\\end{Le}\n\nUsing  Proposition \\ref{galois2limit} we can talk about stacks with values in Galois categories (called Galois stacks). As such, using the universal definition, we still have the notion of an associated Galois stack, even if we don't prove that it always exists. We see then that the above can be reformulated in the following way: \n\n\\begin{Co}\\label{galois-fun-stackification} Let $\\mathfrak{F}:X\\rightarrow \\mathsf{GCat}$ be a 2-functor, $\\mathfrak{G}:X\\rightarrow \\mathsf{GCat}$ a Galois stack and $\\psi:\\mathfrak{F}\\rightarrow \\mathfrak{G}$ a Galois transformation. If $\\mathfrak{F}$ has an associated Galois stack $\\hat{\\mathfrak{F}}$, then $\\hat{\\psi}:\\hat{\\mathfrak{F}}\\rightarrow \\mathfrak{G}$ is a Galois transformation as well.\n\\end{Co}\n\n\n\\section{The \\'Etale Fundamental Groupoid}\\label{etale}\n\nIn this section we will state and prove the main result of this paper. We will prove that for any \\'etale  map $Y\\mapsto X$ of a noetherian scheme $X$, the association $\\Pi_1:Y\\mapsto\\Pi_1(Y)$ is the 2-terminal costack.\n\nThat is to say, the \\'etale fundamental groupoid is defined by the Seifert-van Kampen theorem, in complete analogue to the topological case, as proven in \\cite{top.groupoid}. The proof however is different.\n\nTo prove that the 2-functor $\\Pi_1$ is the 2-terminal costack over any noetherian scheme $X$ is equivalent to saying that it is the associated costack of the constant, covariant 2-functor taking the trivial groupoid as its value. Unfortunately we don't know much about costackification, so instead we will compose it with the functor $\\mathsf {Hom}(-,\\mathop{\\sf FSets}\\nolimits)$, and hence get a Galois category. \n\nSince by the definition of a costack, after composing with $\\mathsf {Hom}(-,\\mathop{\\sf FSets}\\nolimits)$ we will get a stack, we can use stackification to prove this theorem. This is also our main reason for studying properties preserved under 2-limits and filtered 2-colimits. \n\\newline\n\nLet $X$ be a noetherian scheme. We denote by $\\mathop{\\sf FEC}\\nolimits(X)$ the site of finite \\'etale coverings of $X$. It is a well known result that the category of finite \\'etale coverings is a finitely connected Galois category and that the following holds:\n\n\\begin{Th}\\label{Thm.8.0.2} The 2-functor $\\mathcal{_FE_C}:\\mathop{\\sf FEC}\\nolimits(X)\\rightarrow \\mathsf{GCat}$, where $\\mathop{\\sf FEC}\\nolimits$ denotes the category of finite \\'etale coverings, given by $Y\\mapsto\\mathop{\\sf FEC}\\nolimits(Y)$, forms a stack.\n\\end{Th}\n\n\\begin{De} Let $\\{\\mathcal{F}_j:\\mathcal{C}\\rightarrow\\mathop{\\sf FSets}\\nolimits\\}_{j\\in J}$ be a finitely connected Galois category. Then we can define the \\emph{fundamental groupoid}\\index{Fundamental groupoid! Galois category} $\\Pi_1(\\mathcal{C})$ of our Galois category as follows: \n\\begin{itemize} \n\\item Objects of $\\Pi_1(\\mathcal{C})$ are the functors $\\{\\mathcal{F}_j:\\mathcal{C}\\rightarrow \\mathop{\\sf FSets}\\nolimits\\}_{j\\in J}$;\n\\item For $\\mathcal{F}_i$ and $\\mathcal{F}_j$ we define $\\mathsf {Hom}_{\\Pi_1(\\mathcal{C})}(\\mathcal{F}_i,\\mathcal{F}_j):=\\mathsf{Iso}_{\\mathsf{Fnct}}(\\mathcal{F}_i,\\mathcal{F}_j)$ where $\\mathsf{Iso}_{\\mathsf{Fnct}}$ denotes the set of natural isomorphisms.\n\\end{itemize}\n\\end{De}\n\n\\begin{De} Let $X$ be a site. We define the \\emph{\\'etale fundamental groupoid}\\index{Fundamental groupoid! \\'Etale} $\\Pi_1(X)$ of $X$ to be the fundamental groupoid of the Galois category $\\mathcal{_FE_C}(X)$ of finite \\'etale coverings of $X$.\n\\end{De}\n\nEquivalently, Theorem \\ref{Thm.8.0.2} can be stated as follows:\n\n\\begin{Th}\\label{hom-stack} The 2-functor $\\mathcal{_FE_C}:\\mathop{\\sf FEC}\\nolimits(X)\\rightarrow \\mathsf{GCat}$ given by $Y\\mapsto\\Pi_1(Y)$-$\\mathop{\\sf FSets}\\nolimits$, where $\\Pi_1(Y)$ denotes the \\'etale fundamental groupoid of $Y$, forms a stack.\n\\end{Th}\n\n\\begin{Th}[Seifert-van Kampen Theorem] Let $X$ be a noetherian scheme. Then the assignment $Y\\mapsto \\Pi_1(Y)$ defines a costack on the site of finite \\'etale coverings of $X$.\n\\end{Th}\n\n\\begin{proof} For any covering $Z\\in \\mathop{\\sf Cov}\\nolimits(Y)$ we have the functor $2$-$\\mathop{\\sf colim}\\nolimits(Z,\\Pi_1)\\rightarrow \\Pi_1(Z)$, where $2$-$\\mathop{\\sf colim}\\nolimits(Z,\\Pi_1)$ denotes the 2-colimit of \n$$\\xymatrix{\\Pi_1(Z\\times_{X} Y)& \\Pi_1(Z\\times_{X} Y\\times_{X} Y)\\ar@<-.5ex>[l] \\ar@<.5ex>[l]& \\Pi_1(Z\\times_{X} Y\\times_{X}Y\\times_{X} Y).\\ar@<-.7ex>[l]\\ar[l]\\ar@<.7ex>[l]}$$ \nHence we get the associated functor $\\Pi_1(Z)$-$\\mathop{\\sf FSets}\\nolimits\\rightarrow2$-$\\mathop{\\sf colim}\\nolimits(Z,\\Pi_1)$-$\\mathop{\\sf FSets}\\nolimits$ where we denoted by $\\mathfrak{G}$-$\\mathop{\\sf FSets}\\nolimits$ the functor category $\\mathsf {Hom}_{\\mathfrak{Cat}}(\\mathfrak{G},\\mathop{\\sf FSets}\\nolimits)$. Since $\\mathsf {Hom}(-,\\mathop{\\sf FSets}\\nolimits)$ is left exact, we have \n$$2\\text{-}\\mathop{\\sf colim}\\nolimits(Z,\\Pi_1)\\text{-}\\mathop{\\sf FSets}\\nolimits\\cong 2\\text{-}\\mathop{\\sf lim}\\nolimits((Z,\\Pi_1)\\text{-}\\mathop{\\sf FSets}\\nolimits),$$ \nwhere $2$-$\\mathop{\\sf lim}\\nolimits((Z,\\Pi_1)$-$\\mathop{\\sf FSets}\\nolimits)$ denotes the 2-limit of \n$$\\xymatrix@=1.5em{\\Pi_1(Z\\times_X Y)\\text{-}\\mathop{\\sf FSets}\\nolimits\\ar@<-.5ex>[r] \\ar@<.5ex>[r] & \\Pi_1(Z\\times_X Y\\times_X Y)\\text{-}\\mathop{\\sf FSets}\\nolimits\\ar@<-.7ex>[r]\\ar[r]\\ar@<.7ex>[r] & \\Pi_1(Z\\times_X Y\\times_X Y\\times_X Y)\\text{-}\\mathop{\\sf FSets}\\nolimits.}$$ \nBy Theorem \\ref{hom-stack} the functor $\\Pi_1(Z)\\text{-}\\mathop{\\sf FSets}\\nolimits\\rightarrow 2$-$\\mathop{\\sf lim}\\nolimits((Z,\\Pi_1)$-$\\mathop{\\sf FSets}\\nolimits)$ is an equivalence of categories. Hence from Corollary \\ref{eq} $2$-$\\mathop{\\sf colim}\\nolimits(Z,\\Pi_1)\\rightarrow \\Pi_1(Z)$ was an equivalence of categories as well. This proves the assertion.\n\\end{proof}\n\n\\begin{De} Let $\\mathfrak{C}$ be a 2-category. We say that $\\mathfrak{T}$ is the \\emph{2-terminal object}\\index{Object! 2-Terminal} of $\\mathfrak{C}$, if for any other object $C\\in \\mathfrak{C}$, $Hom_{\\mathfrak{Cat}}(C,\\mathfrak{T})$ is equivalent to the 1-point category.\n\\end{De}\n\n\\begin{Th}\\label{main} Let $X$ be a noetherian scheme. Then the assignment $U\\mapsto\\Pi_1(U)$, $U\\in X$ is the 2-terminal costack over the site of \\'etale coverings of $X$.\n\\end{Th}\n\nTo prove this theorem, recall the following lemma:\n\n\\begin{Le}\\label{LCS} Consider the constant 2-functor $\\mathfrak{s}:U\\mapsto\\mathop{\\sf FSets}\\nolimits$, with morphisms in the contravariant way. Then the associated stack $\\hat{\\mathfrak{s}}$ is given by $\\hat{\\mathfrak{s}}(U)=LCS(U)$, where $LCS(U)$ denotes the category of locally constant sheaves on $U$.\n\\end{Le}\n\nLet $A$ be a covariant 2-functor. Recall that we denoted by $A_S$ the contravariant 2-functor given by $U\\mapsto Hom(A(U),\\mathop{\\sf FSets}\\nolimits)$.\n\n\\begin{proof}[Proof of Thm. \\ref{main}] We have already shown that the assignment $U\\rightarrow \\Pi_1(U)$ forms a costack. Hence to prove this theorem, we essentially have to show that for every costack $C$ we have an essentially unique map $C\\rightarrow \\Pi_1$.\nDenote by $P$ the constant, covariant assignment $U\\mapsto \\mathsf{pt}$. It is clear that we have a map $C\\rightarrow P$ and hence a map $P_S\\rightarrow C_S$ between Galois 2-functors, which is a Galois transformation as it's induced by a functor between groupoids. As shown in Lemma \\ref{LCS}, the stackification of $P_S$ exists and is $U\\mapsto LCS(U)$, where $LCS(U)$ denotes the category of locally constant sheaves on $U$. In the case of noetherian schemes, $LCS(U)$ is equivalent to $\\Pi_1(U)$-$\\mathop{\\sf FSets}\\nolimits$, which is a Galois category. Since $C$ was a costack, $C_S$ is a Galois stack and hence the map $P_S\\rightarrow C_S$ factors through $\\Pi_{1S}$. Hence, using Corollary \\ref{galois-fun-stackification}, we know that the map $\\Pi_{1S}\\rightarrow C_S$ is a Galois transformation. \n\nHence by the uniqueness of the associated stack (see Definition \\ref{uniquestack}) and Corollary \\ref{Galoisstackequivalence} respectively, we have the following equivalences of categories: \n$$\\mathsf {Hom}_{\\mathsf{Gal}}(P_S,C_S)\\cong\\mathsf {Hom}_{\\mathsf{Gal}}(\\Pi_{1S},C_S)\\cong \\mathsf {Hom}_\\mathfrak{Cat}(C,\\Pi_1).$$\nHere $\\mathsf {Hom}_\\mathsf{Gal}$ denotes the category of Galois transformations (i.e. of natural transformations respecting finite limits and colimits). Uniqueness and existence now comes from the fact that we have precisely one exact functor $P_S\\rightarrow C_S$. The last claim is true because $P_S$ is equivalent to $\\mathop{\\sf FSets}\\nolimits$ and hence a functor respecting finite colimits is defined by its value on the singleton $\\star$, which has to map to the terminal object of $C_S$. \n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe purpose of this work is to study a parapolar space which is locally of type $A_{7,4}$ and which is subject to an extra assumption called the Weak Hexagon Axiom. The present results complete a more general analysis of those parapolar spaces which are locally of type $A_{n-1, 4}({\\mathbb K})$, with $n$ a suitably chosen integer and ${\\mathbb K}$ a field, performed by the author and described below. To simplify the notation we will omit the field ${\\mathbb K}$.\n\n\\medskip\nIn a previous paper \\cite{on1}, we studied parapolar spaces ${\\widetilde \\Gamma}$ which were locally of type $A_{n-1,4}$ (with $n > 6$ and $ n \\not =  8$) and which satisfied:\n\n\\medskip\n{\\it The Weak Hexagon Axiom (WHA): Let $H = (p_1, \\ldots p_6)$ be a\n$6$-circuit, isometrically embedded in ${\\widetilde \\Gamma}$, this means that $p_i \\in p^{\\perp}_{i+1}$, indices taken mod $6$, and all the other pairs are not collinear. Also assume that at least one of the pairs of points at distance two, say $\\lbrace p_1, p_3 \\rbrace$, is polar. Then there exists a point $w \\in p_1^{\\perp} \\cap p_3^{\\perp} \\cap p_5^{\\perp}$.}\n\n\\medskip\nFirst it was proved that ${\\widetilde \\Gamma}$ can be enriched to a locally truncated geometry of rank $5$. Using the fact that ${\\widetilde \\Gamma}$ inherits from the local structure two classes of maximal singular subspaces, a new family of objects ${\\widetilde {\\mathcal D}}$ was constructed. The elements of ${\\widetilde {\\mathcal D}}$ are $2$-convex subspaces of type $D_{6,6}$; see \\cite[Theorem 2]{on1}. Second, using a sheaf theoretic argument it was shown that ${\\widetilde \\Gamma}$ was a homomorphic image of a truncated building; see \\cite[Theorem 3.b]{on1}.\n\n\\medskip\nThe case with $n=7$ which was not considered in \\cite{on1} is the subject of this work. The difficulty of this case relies on the fact that all the maximal singular subspaces have the same singular rank, hence a partition of the maximal singular subspaces according to dimension is no longer attainable and therefore many of the arguments based on the local properties cannot be used. In order to overcome this difficulty, we shall construct a locally truncated diagram geometry {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$} which is a covering of ${\\widetilde \\Gamma}$ and which is locally isomorphic to ${\\widetilde \\Gamma}$, thus of local type $A_{7,4}$, satisfies {\\it (WHA)} and in addition, has the property that the maximal singular subspaces can be separated in two families which we shall denote $\\mathcal{A}$ and $\\mathcal{B}$.\n\n\\medskip\nFrom this point our approach is similar to the one used in \\cite{on1}. We start by constructing two new collections of $2$-convex subspaces ${\\mathcal D}_{\\mathcal A}$ and  ${\\mathcal D}_{\\mathcal B}$ whose elements are of type $D_{6,6}$. We prove that every symplecton $S \\in \\mathcal{S}$ of $\\Gamma$ is contained in exactly one element of ${\\mathcal D}_{\\mathcal A}$ and one element of ${\\mathcal D}_{\\mathcal B}$. Therefore the space $\\Gamma$ can be enriched to a rank six geometry $(\\mathcal{P, L, A, B, D_A, D_B})$. Then we construct a sheaf over this locally truncated diagram geometry and we prove that $\\Gamma$ is a homomorphic image of a truncated building of affine type $E_7$.\n\n\\medskip\nIn Section $2$ we provide the reader with a list of definitions and basic results which will be used in the latter sections. In Section $3$, we use a theorem by Kasikova and Shult to construct the covering $\\Gamma$. The main result of Section $4$ is contained in Theorem $4.1$ which attests the existence of two families of subspaces ${\\mathcal D}_{\\mathcal A}$ and ${\\mathcal D}_{\\mathcal B}$ and that every symplecton of $\\Gamma$ lies in exactly two subspaces, one from each family. In Section $5$, we prove the main result of the paper:\n\n\\begin{t5}Let ${\\widetilde \\Gamma} = ({\\widetilde {\\mathcal P}}, {\\widetilde {\\mathcal L}})$ be a parapolar space which is locally of type $A_{7, 4}$. Let $I = \\lbrace 1, \\ldots 8 \\rbrace, \\; J = \\lbrace 1, 8 \\rbrace$ and $K = I \\setminus J$. Assume that ${\\widetilde \\Gamma}$ satisfies the Weak Hexagon Axiom. Then there is a residually connected $J$-locally truncated diagram geometry $\\Gamma$ belonging to the diagram:\\\\\n\\begin{picture}(1000, 50)(0,0)\n\\put(0,25){$\\mathcal{Y}$}\n\\put(60,25){$\\qed$}\n\\put(75,15){\\scriptsize 1}\n\\put(80,29){\\line(1,0){32}}\n\\put(112,26){$\\circ$}\n\\put(113,15){\\scriptsize 2}\n\\put(113,36){\\scriptsize ${\\mathcal D}_{\\mathcal B}$}\n\\put(118,29){\\line(1,0){30}}\n\\put(148,26){$\\circ$}\n\\put(150,36){\\scriptsize ${\\mathcal B}$}\n\\put(150,15){\\scriptsize 3}\n\\put(153,29){\\line(1,0){30}}\n\\put(183,26){$\\circ$}\n\\put(190,36){\\scriptsize ${\\mathcal L}$}\n\\put(190,15){\\scriptsize 5}\n\\put(186,-3){\\line(0,1){30}}\n\\put(183,-9){$\\circ$}\n\\put(171,-10){\\scriptsize ${\\mathcal P}$}\n\\put(190,-10){\\scriptsize 4}\n\\put(188,29){\\line(1,0){30}}\n\\put(218,26){$\\circ$}\n\\put(220,36){\\scriptsize ${\\mathcal A}$}\n\\put(220,15){\\scriptsize 6}\n\\put(223,29){\\line(1,0){30}}\n\\put(253,26){$\\circ$}\n\\put(253,36){\\scriptsize ${\\mathcal D}_{\\mathcal A}$}\n\\put(255,15){\\scriptsize 7}\n\\put(258,29){\\line(1,0){30}}\n\\put(276,25){$\\qed$}\n\\put(290,15){\\scriptsize 8}\n\\end{picture}\n\n\\vspace{.4cm}\nwhose universal covering is the truncation of a building. Therefore, ${\\widetilde \\Gamma}$ is also the homomorphic image of a truncated building.\n\\end{t5}\n\n\\section{Preliminaries and definitions}\n\\subsection{Geometries} In this section the basic definitions related to geometries are given; for an expository treatment see \\cite[Chapter 3]{hnbk}.\n\n\\medskip\nA {\\it geometry over} $I$ is a system $\\Gamma = (V, *, t)$ consisting of set $V$, a binary, symmetric, reflexive relation on $V$ and a mapping $t: V \\rightarrow I$. The elements of $V$ are called {\\it objects}, $*$ is called the {\\it incidence relation} and $t$ is the {\\it type function} of $\\Gamma$.\n\n\\medskip\nA {\\it flag} $F$ of $\\Gamma$ is a (possibly empty) subset of pairwise incident objects of $\\Gamma$. The set $t(F)$ is the {\\it type} of $F$ and the set $I \\setminus t(F)$ is its {\\it cotype}. The cardinalities of these sets are the {\\it rank} and the {\\it corank} of $F$. The {\\it residue} of $F$ in $\\Gamma$ is the geometry $\\res{\\Gamma}{F} = (V_F, *_{|V_F}, t _{|V_F})$ over $I \\setminus t(F)$, where $V_F$ is the set of all members of $V \\setminus F$ incident with each element of $F$. The corank of a flag $F$ is the rank of $\\res {\\Gamma}{F}$. A geometry $\\Gamma$ is {\\it residually connected} if and only if for every flag $F$ of corank at least one, $\\res{\\Gamma}{F}$ is not empty and if, for each flag of corank at least two, $\\res{\\Gamma}{F}$ is connected.\n\n\\medskip\nLet $\\Gamma _k = (V_k, *_k, t_k)$ with $k \\in K$ some index set and where each $\\Gamma_k$ is a geometry over $I_k$. Assume that $\\lbrace I_k \\rbrace _{k \\in K}$ is a family of pairwise disjoint sets. The {\\it direct sum of geometries} is the geometry denoted by $\\Gamma = \\bigoplus _{k \\in K} \\Gamma _k = (V, *, t)$, where $V = \\bigcup _{k \\in K} V_k$. The incidence is defined as follows: $*_{|V_k}\\;  :=  *_k$ and $x * y$ for any two objects $x \\in V_{k_1}$ and $y \\in V_{k_2}$ with $k_1 \\not = k_2$. Finally $t _{|V_k}\\;  := t_k$.\n\n\\medskip\nLet ${\\mathcal Geom}_I$ denote the category whose objects are\ngeometries with typeset $I$ and whose morphisms are the type preserving graph morphisms. A {\\it fibering morphism of geometries} $\\phi: \\Gamma_1 \\rightarrow \\Gamma_2$ is an object surjective morphism which is object bijective when restricted to the residue of each object in $\\Gamma_1$.\n\n\\subsection{Locally truncated geometries}\nWe gather in this subsection the necessary notions on locally truncated geometries needed in this paper; for a more detailed account of the concepts from this subsection the reader is referred to Brouwer and Cohen \\cite{loc} and Ronan \\cite{ron}.\n\n\\medskip\nLet $I$ be an index set and let $J \\subset I$. The {\\it truncation of type $J$} of $\\Gamma$, denoted by $^J \\Gamma$ is the geometry obtained by restricting the typeset of\n$\\Gamma$ to $J$. The truncation is a functor $^J {\\bf Tr}: {\\mathcal Geom}_I \\rightarrow {\\mathcal Geom}_J$ from the category of geometries over $I$\nto the category of geometries over $J$.\n\n\\medskip\nThe {\\it $J$-truncation of $\\Gamma$}, denoted by $_J \\Gamma$, has as objects the objects of $\\Gamma$ whose types\nare in $I \\setminus J$, incidence and type function are those from\n$\\Gamma$ but restricted to $I \\setminus J$. Differently said, the\n$J$-truncation of $\\Gamma$ is the truncation of type $I \\setminus J$ of $\\Gamma$, that is $_J \\Gamma = \\; ^{I \\setminus J}\\Gamma$.\n\n\\medskip\nA {\\it diagram} $D$ over $I$ is a mapping which assigns to each $2$-subset $\\lbrace i, j \\rbrace$ of $I$, a class $D(i,j)$ of rank $2$ geometries.  A geometry $\\Delta$ over $I$ {\\it belongs to the diagram} $D$ if and only if every\nresidue of type $\\lbrace i, j \\rbrace$ of $\\Delta$ is a geometry from $D(i,j)$.\n\n\\medskip\nA geometry $\\Gamma$ over $I \\setminus J$ is said to be $J$-{\\it locally truncated of type $D$} (or {\\it with diagram $D$}) over\n$I$ if and only if for every nonempty flag $F$ of $\\Gamma$, the residue $\\res{\\Gamma}{F}$ is isomorphic to the truncation of type $I \\setminus ( J \\cup t(F))$ of a geometry belonging to the diagram $D_{I \\setminus t(F)}$, the restriction of $D$\nto the typeset $I \\setminus t(F)$. If $\\Gamma$ is the truncation of type $I \\setminus J$ of a geometry\n$\\Delta$ of type $D$ over $I$ then $\\Gamma$ it is a geometry of $J$-locally\ntruncated type $D$. The converse is in general not true; see Brouwer and Cohen\n\\cite{loc} and Ronan \\cite{ron}.\n\n\\medskip\nLet $M=(m_{ij})$ be a Coxeter matrix with rows and columns indexed by $I$. The {\\it\ndiagram} of $M$, denoted $D(M)$, assigns to each $2$-subset $\\lbrace i, j \\rbrace$ of $I$, the class $D(i,j)$ of generalized $m_{i j }$-gons. If $\\Gamma$ is a residually connected geometry with diagram $D(M)$, that is every residue of type $\\lbrace i, j \\rbrace$ of $\\Gamma$ is a generalized $m_{i j}$-gon, then $\\Gamma$ is called a {\\it geometry of type} $M$.\n\n\\subsection{Chamber systems}\nThe notion of chamber system was introduced by Tits \\cite{tits}. We give below basic definitions and results which will be used in Section $5$; for a detailed account on chamber systems see \\cite{cp}.\n\n\\medskip\nA {\\it chamber system} ${\\mathcal C} = (C, E, \\lambda, I)$ over $I$ is a simple graph $(C, E)$ together with an edge-labeling $\\lambda: E \\rightarrow 2^{I} \\setminus \\lbrace \\emptyset \\rbrace$ by nonempty subsets of $I$ such that, if $a, b, c \\in C$ are three pairwise adjacent vertices, then $\\lambda (a,b) \\cap \\lambda (b,\nc) \\subseteq \\lambda (a,c)$. The elements of $C$ are called\n{\\it chambers}. Two distinct chambers $a$ and $b$ are $i$-{\\it adjacent} iff $(a,b) \\in E$ is an edge and $i \\in \\lambda (a,b)$, for any $i \\in I$. The rank of ${\\mathcal C}$ is the cardinality of the index set $I$. The chamber systems over $I$ together with the appropriate morphisms form a category denoted ${\\mathcal Chamb}_I$.\n\n\\medskip\nFor $J$ a subset of $I$, the {\\it residue of ${\\mathcal C}$ of type $J$} or the $J$-{\\it residue}, is a connected component of the graph $(C, E_J, \\lambda _J, J)$, with\n$\\lambda_J$ the restriction of $\\lambda$ to $\\lambda^{-1}(2^J) \\subseteq E$, where $2^J$ is the codomain of $\\lambda _J$, and $E_J =\\lbrace e \\in E \\mid \\lambda _J (e) \\not=\n\\emptyset \\rbrace $. A $J$-residue $R$ is a chamber system over the typeset $J$. The set $I \\setminus J$ is called the {\\it cotype} of $R$. The {\\it rank of the residue} $R$ is $| J|$ and its {\\it corank} is $| I \\setminus J |$.\n\n\\medskip\nA chamber system over $I$ is {\\it residually connected} if and only if for every subset $J \\subseteq I$ and for every family $\\lbrace R_j: j\\in J \\rbrace$ of residues of cotype $j$, with the property that any two have nonempty intersection, it follows that $\\cap _{j \\in J}R_j$ is a nonempty residue of type $I \\setminus J$.\n\n\\medskip\nLet $\\Gamma$ be a geometry over $I$. Denote by ${\\bf C}(\\Gamma)$ the set of its chamber flags, that is, the flags of type $I$. Two chamber flags $c$ and $d$ are said to be $i$-adjacent whenever they have the same element of type $j$ for all $j \\not = i$, with $i, j \\in I$. Then ${\\bf C}(\\Gamma)$, with the above adjacency relation, is a chamber system of type $I$. Starting with a chamber system ${\\mathcal C}$ over $I$, we define a geometry ${\\bf G}({\\mathcal C}) = (C_i, i\\in I, *, t)$. The objects of this geometry are the elements $C_i, i \\in I$, the collection of all corank one residues of type $I \\setminus \\lbrace i \\rbrace$ of ${\\mathcal C}$. Two objects are incident if they have nonempty intersection. The above construction gives rise to a pair of functors ${\\mathbf G}: {\\mathcal Chamb}_I \\rightarrow\n{\\mathcal Geom}_I$ and\n${\\mathbf C}: {\\mathcal Geom}_I \\rightarrow {\\mathcal Chamb}_I$ such that ${\\mathbf G}( {\\bf C}(\\Gamma)) = \\Gamma$, if $\\Gamma$ is a residually connected geometry and $I$ is finite, and ${\\mathbf C}({\\mathbf G}({\\mathcal C}))= {\\mathcal C}$, if ${\\mathcal C}$ is a residually connected chamber system. For more details see \\cite{asp}.\n\n\\medskip\nFor a connected chamber system ${\\mathcal C}$ over $I$, a {\\it $2$-cover} of ${\\mathcal C}$ is a\nconnected chamber system $\\widetilde {\\mathcal C}$ together with a chamber system\nmorphism $h : \\widetilde {\\mathcal C} \\rightarrow {\\mathcal C}$ which is surjective on chambers and is an isomorphism when restricted to any residues of rank at most $2$ of $\\widetilde {\\mathcal C}$. A $2$-cover $h: \\widetilde {\\mathcal C} \\rightarrow {\\mathcal C}$ is\nsaid to be {\\it universal} if for any other $2$-cover $\\varphi : {\\mathcal C}'\n\\rightarrow {\\mathcal C}$ there is a $2$-cover $\\psi : \\widetilde {\\mathcal C} \\rightarrow\n{\\mathcal C}'$ such that $h \\circ \\psi = \\varphi$. It can be proved that\nchamber systems always have universal coverings.\n\n\\medskip\nA {\\it chamber system ${\\mathcal C}$ over $I$ belongs to the\ndiagram $D(M)$}, with $M$ a Coxeter matrix, if and only if every residue of ${\\mathcal C}$ of type $\\lbrace i, j \\rbrace \\subseteq I$ is the chamber system of a generalized\n$m_{ij}$-gon; one also says that ${\\mathcal C}$ is a chamber system of type $M$. {\\it Buildings} are chamber systems of type $M$ which satisfy extra axioms. For a complete definition of the buildings see \\cite{tits}. The following result, known as Tits' Local Approach Theorem, can be found in \\cite[Corollary 3]{tits}:\n\n\\begin{thm}[Tits]Suppose ${\\mathcal C}$ is a chamber system of type $M$ with $M$ a Coxeter\nmatrix, and suppose that for every rank $3$ residue, the universal $2$-cover is a building. Then the universal $2$-cover of ${\\mathcal C}$ is a building ${\\mathcal B}$ of type $M$.\n\\end{thm}\n\nIn particular, the chamber system ${\\mathcal C}$ is obtained from ${\\mathcal B}$ by factoring out a group\nof automorphisms in which no non-trivial element fixes any rank $2$\nresidue of ${\\mathcal B}$.\n\n\\subsection{Sheaves}\nLet $I$ be an index set, $J \\subset I$ and set $K = I \\setminus J$. Let $\\Gamma$ be a geometry over $K$ which is locally truncated of type $D$ over $I$ and let ${\\mathcal F}$ be a family of nonempty flags of $\\Gamma$. A {\\it sheaf over the geometry} $\\Gamma$ is a class of geometries $\\lbrace \\Sigma(F) \\; \\text{for} \\; F \\in {\\mathcal F} \\rbrace$ together with isomorphisms\n\\begin{center}\n$\\varphi_F: \\res{\\Gamma}{F} \\rightarrow \\; _J \\Sigma (F)$\n\\end{center}\nof geometries over $I \\setminus t(F)$. Given a pair of incident flags $F_1 \\subseteq F_2$ in ${\\mathcal F}$ the connecting homomorphisms of the sheaf are the maps $\\varphi _{F_1, F_2} : \\Sigma (F_2) \\rightarrow \\Sigma (F_1)$ with the property that\n\\begin{center} $\\varphi_{F_1, F_2}(\\Sigma(F_2)) \\simeq \\res{\\Sigma(F_1)}{F_2 \\setminus F_1}$.\n\\end{center}\nFurthermore, they are subject to the following conditions:\n\\begin{center}\n$\\varphi _{F_1, F_2} \\circ \\varphi _{F_2, F_3} = \\varphi _{F_1, F_3}$,\n\\end{center}\nfor $F_1, F_2, F_3 \\in {\\mathcal F}$ with $F_1 \\subseteq F_2 \\subseteq F_3$. To simplify the notation, we will omit the connecting isomorphisms $\\varphi _F$ and write $\\res{\\Gamma}{F} = \\; _J \\Sigma(F)$ instead.\n\n\\medskip\nA sheaf $\\Sigma$ is {\\it residually connected} if and only if for each\nobject $x$ of the geometry $\\Gamma$ the sheaf geometry $\\Sigma (x)$ is\nresidually connected. Due to the functorial relation between the category of geometries and the\ncategory of chamber systems, whenever a sheaf $\\Sigma$ exists, there\nis a chamber system associated to it \\cite[Lemma 1]{loc}.\n\n\\subsection{Spaces}\nThis section is devoted to the basic concepts used in Section $3$; for a good review on spaces and related topics the reader is referred to \\cite[Chapter 12]{hnbk}.\n\n\\medskip\nA {\\it space} $\\Gamma$ is a pair $({\\mathcal P}, {\\mathcal L})$ consisting of a nonempty set ${\\mathcal P}$, whose members are called {\\it points}, and a collection ${\\mathcal L}$ of subsets of ${\\mathcal P}$ of cardinality at least two, whose members are called {\\it lines}. Let $p, q \\in {\\mathcal P}$ be two distinct points. We say that $p$ {\\it is collinear to} $q$ if both $p$ and $q$ lie in some line $L \\in {\\mathcal L}$. The set of all points of ${\\mathcal P}$ collinear with $p$, including $p$ itself, will be denoted $p^{\\perp}$; it is called {\\it the perp of} $p$. A space is a {\\it partial linear space} if two distinct points lie in at most one line. A space is a {\\it gamma space} if it is a partial linear space and for any $p \\in {\\mathcal P}$ and $L \\in {\\mathcal L}$ the set $p^{\\perp} \\cap L$ is empty, a point or $L$.\n\n\\medskip\nThe {\\it collinearity graph} of a space is the graph whose vertex set is ${\\mathcal P}$ and in which two points are adjacent if they are distinct and collinear. Given two points $p, q \\in {\\mathcal P}$, the distance between $p$ and $q$ in the collinearity graph will be denoted $d(p,q)$.\n\n\\medskip\nA subset $X$ of the point set ${\\mathcal P}$ is a {\\it subspace} of $\\Gamma$ if every line $L \\in {\\mathcal L}$ meeting $X$ in at least two points entirely belongs to $X$. A subspace is {\\it singular} if every two points of $X$ are collinear. A subspace is $2$-{\\it convex} if for all pairs of points $p, q \\in X$ with $d(p,q)=2$, each point which is collinear with both $p$ and $q$ is also contained in $X$.\n\n\\medskip\nThe {\\it singular rank} of a space $\\Gamma$ is the length of the longest chain of distinct nonempty singular subspaces $X_0 \\subset X_1 \\subset \\ldots \\subset X_n$. In what follows the term {\\it rank} will denote the projective rank of a singular subspace of $\\Gamma$, that is one less than the vector space dimension. A polar space is said to be of rank $n$ if its singular rank is $n-1$.\n\n\\medskip\nA {\\it parapolar space} is a connected partial linear gamma space possessing a collection of $2$-convex subspaces $\\mathcal S$, called {\\it symplecta}, isomorphic to nondegenerate polar spaces of rank at least $2$, with the properties that each line is\ncontained in a symplecton and each quadrangle is contained in a unique symplecton. A parapolar space in which every pair of points at distance $2$ belongs to a symplecton, is also called a {\\it strong parapolar space}. Let $p, q \\in {\\mathcal P}$ be two points at distance two in a parapolar space. If $|p ^{\\perp} \\cap q^{\\perp}|=1$ then $\\lbrace p, q \\rbrace$ is called a {\\it special pair}; if $|p^{\\perp} \\cap q^{\\perp}|>1$ then $\\lbrace p, q \\rbrace$ is a {\\it polar pair}. If $\\lbrace p, q \\rbrace$ is a polar pair, then the convex closure of $p$ and $q$ is the unique symplecton containing the two points which will be denoted $\\ll p, q \\gg$. In a parapolar space all the singular subspaces are projective spaces. In a parapolar space {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$}, the {\\it residue of a point} $p \\in {\\mathcal P}$ is the space ${\\rm Res}_{\\Gamma}p = ({\\mathcal L}_p, \\pi_p)$ induced on the lines and the planes which contain $p$.\n\n\\medskip\nLet ${\\mathfrak D}_n$ be a Coxeter diagram with $n$ nodes and select a node $i$ in the diagram. A space is said to be of type ${\\mathfrak D}_{n,i}$ if it is the shadow space over $i$ of a building of type ${\\mathfrak D}_n$; see \\cite[Chapter 12, Section 4.7]{hnbk}.\n\n\\subsubsection{The space of type $A_{7,4}$}\nLet $V$ be an $8$-dimensional vector space over some division ring $\\mathbb{K}$. Define the space {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$}  whose points $\\mathcal{P}$ are the $4$-subspaces of $V$ and whose lines $\\mathcal{L}$ are the $(3,5)$-flags of $V$. Then $\\Gamma$ is a strong parapolar space whose symplecta are polar spaces of type $D_{3,1}$. Let $ \\mathcal S$ denote the family of symplecta. The maximal singular subspaces ${\\mathcal M}$ have rank $4$ and can be partitioned into two classes ${\\mathcal A}$ and ${\\mathcal B}$, according to the property {\\it(G1)} below. We list some of the properties of the space of type $A_{7,4}$; these properties can be derived by linear algebra arguments. A characterization of the spaces of type $A_{n,j}$ can be found in \\cite{coh}.\n\n\\begin{itemize}\n\\item[{\\it(G1)}] If $M_1,M_2 \\in {\\mathcal M}$ are two distinct maximal singular\nsubspaces belonging to the same class then $  M_1 \\cap  M_2$ is either empty or a point. If they belong to different classes then $  M_1 \\cap   M_2$ is either empty or a line.\n\\item[{\\it(G2)}] If $p \\in {\\mathcal P}$, $S \\in {\\mathcal S}$, $p \\not\\in S$ then $ p^{\\perp} \\cap S$ is empty, a point or a plane.\n\\item[{\\it(G3)}] If $S \\in {\\mathcal S}$ and $M \\in {\\mathcal M}$ then $S \\cap M$ is empty, a point or a plane.\n\\item[{\\it(G4)}] If $( p,M) \\in {\\mathcal P} \\times {\\mathcal M}$ and $ p \\not\\in  M$ then $p^{\\perp} \\cap  M$ is either empty or a line.\n\\end{itemize}\n\n\\subsubsection{The space of type $D_{6,6}$}\nThis is the space {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$} whose points ${\\mathcal P}$ are one class of maximal singular subspaces of the polar space of type $D_{6,1}$. The lines ${\\mathcal L}$ are the rank $3$ singular subspaces of the same polar space. Then $\\Gamma$ is a strong parapolar space, with a family ${\\mathcal S}$ of symplecta which are polar spaces of type $D_{4,1}$. Let ${\\mathcal S}$ denote the collection of symplecta. There are two classes of maximal singular subspaces ${\\mathcal A}$, whose elements\nhave rank $5$ and ${\\mathcal B}$, a class of rank $3$ singular subspaces.\nThe diagram of a space of type $D_{6,6}$ is given below:\\\\\n\n\\begin{picture}(1000, 50)(0,0)\n\\put(0,25){$D_{6,6}$}\n\\put(74,26){$\\circ$}\n\\put(76,8){${\\mathcal C}$}\n\\put(80,29){\\line(1,0){40}}\n\\put(120,26){$\\circ$}\n\\put(122,8){${\\mathcal S}$}\n\\put(126,29){\\line(1,0){40}}\n\\put(166,26){$\\circ$}\n\\put(168,8){${\\mathcal B}$}\n\\put(172,29){\\line(1,0){40}}\n\\put(212,26){$\\circ$}\n\\put(220,8){${\\mathcal L}$}\n\\put(215,-10){\\line(0,1){37}}\n\\put(212,-16){$\\circ$}\n\\put(220,-24){${\\mathcal P}$}\n\\put(218,29){\\line(1,0){40}}\n\\put(258,26){$\\circ$}\n\\put(260,8){${\\mathcal A}$}\n\\end{picture}\n\n\\vspace*{1cm}\nThe space of type $D_{6,6}$ was initially characterized by Cohen and Cooperstein \\cite[Theorem 4]{cc}:\n\n\\begin{thm}[Cohen and Cooperstein]Let {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$} be a strong parapolar space of singular rank $5$, which is not a polar space and whose symplecta have rank $4$. Assume that, given a point-symplecton pair $(p, S) \\in {\\mathcal P} \\times {\\mathcal S}$ with $p \\not \\in S$, the set $p^{\\perp} \\cap S$ is either a point or a maximal singular subspace of $S$. Then $\\Gamma$ is a space of type $D_{6,6}$.\\\\\n\\end{thm}\n\n\\section{The maximal singular subspaces}\n\nSuppose that ${\\widetilde \\Gamma} = ({\\widetilde{\\mathcal P}}, {\\widetilde{\\mathcal L}})$ is parapolar space which is locally of type $A_{2n-1,n}$ with $n$ a positive integer.\n\n\\smallskip\nThe Grassmann space $A_{2n-1, n}(\\mathbb{K})$ and its quotient $A_{2n-1, n}(\\mathbb{K})\/\\langle \\sigma \\rangle$, where $\\mathbb{K}$ is an infinite division ring and where $\\sigma$ is an involutory automorphism of $A_{2n-1,n}(\\mathbb{K})$ induced by a polarity of the underlying projective space of Witt index at most $n-4$, have the same diagram. In this case ${\\widetilde \\Gamma}$ can have point residuals of both types $A_{2n-1, n}(\\mathbb{K})$ and $A_{2n-1, n}(\\mathbb{K})\/ \\langle \\sigma \\rangle$. In the second geometry the maximal singular subspaces are fused in one single family; see \\cite[Section 6]{coh} for example. Since the case of interest here is $n=4$, the Grassmann space of type $A_{7,4}$ is too ``small\" and it does not have nontrivial quotients which are parapolar spaces, thus a local partition of the maximal singular subspaces into two classes can be attained (see Section 2.5.1 for details).\n\n\\medskip\nHowever, all the maximal singular subspaces of ${\\widetilde \\Gamma}$ have the same rank and there is no global partition in two classes according to the rank. In order to overcome this difficulty, we shall use a result of Kasikova and Shult \\cite[Section 3.4]{ks} and we shall construct a covering of ${\\widetilde \\Gamma}$ in which the partition of the maximal singular subspaces in two global classes can be realized. We adapt Theorem $12$ from \\cite{ks} to the case of interest here and for completeness we also provide a proof following \\cite{ks}.\n\n\\begin{thm}[Kasikova and Shult] Let ${\\widetilde \\Gamma} = ({\\widetilde {\\mathcal P}}, {\\widetilde {\\mathcal L}})$ be a parapolar space which is locally of type $A_{7,4}$. Let $I = \\lbrace 1, \\ldots 8 \\rbrace$ and $J = \\lbrace 1,2,7,8 \\rbrace$.  Then there is a $J$-locally truncated connected geometry $\\Gamma$ with diagram:\\\\\n\\begin{picture}(1000, 50)(0,0)\n\\put(0,25){$ \\mathcal{Y'}$}\n\\put(45,25){$\\qed$}\n\\put(60,15){\\scriptsize $1$}\n\\put(65,29){\\line(1,0){40}}\n\\put(95,25){$\\qed$}\n\\put(110,15){\\scriptsize ${2}$}\n\\put(115,29){\\line(1,0){40}}\n\\put(155,26){$\\circ$}\n\\put(157,15){\\scriptsize ${3}$}\n\\put(160,29){\\line(1,0){40}}\n\\put(200,26){$\\circ$}\n\\put(210,15){\\scriptsize $5$}\n\\put(203,-3){\\line(0,1){30}}\n\\put(200,-9){$\\circ$}\n\\put(210,-10){\\scriptsize $4$}\n\\put(206,29){\\line(1,0){40}}\n\\put(246,26){$\\circ$}\n\\put(248,15){\\scriptsize $6$}\n\\put(252,29){\\line(1,0){40}}\n\\put(280,25){$\\qed$}\n\\put(295,15){\\scriptsize $7$}\n\\put(300,29){\\line(1,0){40}}\n\\put(330,25){$\\qed$}\n\\put(345,15){\\scriptsize $8$}\n\\end{picture}\n\n\\vspace{.5cm}\nand a fibering morphism of geometries $\\phi_{\\Gamma}: \\Gamma \\rightarrow {\\widetilde \\Gamma}$ which induces a one- or two-fold $\\mathcal {T}$-covering $\\varphi _{\\Gamma}: \\Delta _{\\Gamma} \\rightarrow \\Delta_{\\widetilde \\Gamma}$ of the point-collinearity graph of ${\\widetilde \\Gamma}$.\n\\end{thm}\n\n\\begin{proof} Let ${\\widetilde \\Gamma} = ({\\widetilde{\\mathcal P}}, {\\widetilde{\\mathcal L}})$ be a parapolar space which is locally of type $A_{7,4}$. Then for each point ${\\widetilde p} \\in {\\widetilde{\\mathcal P}}$ the residue ${\\rm Res}_{\\widetilde \\Gamma}{\\widetilde p}$ contains two classes of maximal singular subspaces ${\\widetilde{\\mathcal A}}_{\\widetilde p}$ and ${\\widetilde {\\mathcal B}}_{\\widetilde p}$ whose elements are projective spaces of rank $5$. Let ${\\widetilde {\\mathcal M}}_{\\widetilde p} = {\\widetilde {\\mathcal A}}_{\\widetilde p} \\cup {\\widetilde {\\mathcal B}}_{\\widetilde p}$ and set ${\\widetilde {\\mathcal M}} = \\lbrace {\\widetilde {\\mathcal M}} _{\\widetilde p} \\mid {\\widetilde p} \\in {\\widetilde {\\mathcal P}} \\rbrace$. Each plane ${\\widetilde \\pi}$ on ${\\widetilde p}$ lies in exactly one element from each class. Let ${\\mathbb P}$ denote the set of pairs $({\\widetilde p}, {\\widetilde{\\mathcal X}}_{\\widetilde p})$ where ${\\widetilde p} \\in {\\widetilde{\\mathcal P}}$ and ${\\widetilde{\\mathcal X}}_{\\widetilde p}$ is one of the classes ${\\widetilde{\\mathcal A}}_{\\widetilde p}$ or ${\\widetilde{\\mathcal B}}_{\\widetilde p}$. Let ${\\mathbb L}$ denote the collection of pairs $({\\widetilde L}, {\\widetilde{\\mathcal X}}_{\\widetilde L})$ with ${\\widetilde L} \\in {\\widetilde {\\mathcal L}}$ and ${\\widetilde{\\mathcal X}}_{\\widetilde L}$ a class of maximal singular subspaces containing ${\\widetilde L}$. We say that $({\\widetilde p}, {\\widetilde{\\mathcal X}}_{\\widetilde p})$ is incident with $({\\widetilde L}, {\\widetilde{\\mathcal X}}_{\\widetilde L})$ if and only if ${\\widetilde p}$ is incident with ${\\widetilde L}$ in ${\\widetilde \\Gamma}$ and ${\\widetilde{\\mathcal X}}_{\\widetilde L} \\subseteq {\\widetilde{\\mathcal X}}_{\\widetilde p}$. Thus $\\Upsilon = ({\\mathbb P},{\\mathbb L})$ is a point-line geometry.\n\n\\medskip\nThe geometry morphism $\\phi: \\Upsilon \\rightarrow {\\widetilde \\Gamma}$ induced by the projection onto the first coordinate is vertex surjective. Let now $({\\widetilde p}, {\\widetilde{\\mathcal X}}_{\\widetilde p}) \\in {\\mathbb P}$ and $({\\widetilde L}_i, {\\widetilde{\\mathcal X}}_{{\\widetilde L}_i}) \\in {\\mathbb L}$ with $i=1,2$, be two lines incident with $({\\widetilde p}, {\\widetilde{\\mathcal X}}_{\\widetilde p})$. Observe that $\\phi(({\\widetilde L}_1, {\\widetilde{\\mathcal X}}_{{\\widetilde L}_1})) = \\phi(({\\widetilde L}_2, {\\widetilde{\\mathcal X}}_{{\\widetilde L}_2}))$ implies that ${\\widetilde L}_1 = {\\widetilde L}_2$. Since $({\\widetilde p}, {\\widetilde{\\mathcal X}}_{\\widetilde p})$ is incident with both lines we must have  ${\\widetilde {\\mathcal X}}_{{\\widetilde L}_1} = {\\widetilde{\\mathcal X}}_{{\\widetilde L}_2}$ and therefore $({\\widetilde L}_1, {\\widetilde{\\mathcal X}}_{{\\widetilde L}_1}) = ({\\widetilde L}_2, {\\widetilde{\\mathcal L}}_{{\\widetilde L}_2})$. A similar argument can be applied if we start with a line in ${\\mathbb L}$ and two points in ${\\mathbb P}$ incident with it. Thus $\\phi$ is one-to-one when restricted to neighbor graphs, the set of all the lines (or points) which are incident with a fixed point (or line respectively).\n\n\\medskip\nThe map $\\phi$ also induces a vertex surjective morphism on the point-collinearity graphs: $\\varphi: \\Delta_{\\Upsilon} \\rightarrow \\Delta_{\\widetilde \\Gamma}$ which restricts to an isomorphism $({\\widetilde p}, {\\widetilde{\\mathcal X}}_{\\widetilde p})^{\\perp_{\\Upsilon}} \\rightarrow {\\widetilde p}^{\\perp_{\\widetilde{\\Gamma}}}$. Furthermore, all $3$-circuits in $\\Delta_{\\widetilde \\Gamma}$ lift to $3$-circuits in $\\Delta_{\\Upsilon}$. In the language of \\cite[Section 2]{ks}, the restriction $\\varphi _{\\Gamma}$ of $\\varphi$ to a connected component of $\\Delta_{\\Upsilon}$ is a $\\mathcal{T}$-covering.\n\n\\medskip\nIn $\\Upsilon$ there are two global classes of maximal singular subspaces. They can be defined as follows: for any ${\\widetilde M} \\in {\\widetilde {\\mathcal M}}$ let $\\mathcal{A}_{\\widetilde M} = \\lbrace ({\\widetilde p}, {\\widetilde{\\mathcal X}}_{\\widetilde p}({\\widetilde M}))| {\\widetilde p} \\in {\\widetilde M} \\rbrace$ where ${\\widetilde {\\mathcal X}}_{\\widetilde p}({\\widetilde M})$ denotes the class ${\\widetilde{\\mathcal A}}_{\\widetilde p}$ or ${\\widetilde{\\mathcal B}}_{\\widetilde p}$ which contains ${\\widetilde M}$ and $\\mathcal{B}_{\\widetilde M} = \\lbrace ({\\widetilde p}, {\\widetilde{\\mathcal X}'}_{\\widetilde p}({\\widetilde M}))| {\\widetilde p} \\in {\\widetilde M} \\rbrace$ where ${\\widetilde {\\mathcal X}'}_{\\widetilde p}({\\widetilde M})$ denotes the class which does not contain ${\\widetilde M}$. Then the two families of maximal singular subspaces are ${\\mathcal A} = \\lbrace {\\mathcal A}_{\\widetilde M} | {\\widetilde M} \\in {\\widetilde {\\mathcal M}} \\rbrace$ and $\\mathcal{B} = \\lbrace {\\widetilde {\\mathcal B}}_{\\widetilde M} | {\\widetilde M} \\in {\\widetilde {\\mathcal M}} \\rbrace$.\n\n\\medskip\nIt is quite clear that $\\Upsilon$ can be either connected (when the local classes of maximal singular subspaces of ${\\widetilde \\Gamma}$ are fused) or disconnected (when a global separation is possible) and has two connected components. Let $\\Gamma$ be a connected component of $\\Upsilon$. This is mapped by $\\phi$ as either a one-to-one mapping or a two-to-one mapping depending on whether $\\Gamma$ is a proper subgeometry or not. So if we denote by $\\phi_{\\Gamma}$ the restriction of $\\phi$ to $\\Gamma$ we obtain the morphism from the conclusion of the theorem.\n\\end{proof}\n\n\\begin{rem} The point-line geometry $\\Gamma = ({\\mathcal P}, {\\mathcal L})$ is connected and locally isomorphic to ${\\widetilde \\Gamma}$, which means that $\\Gamma$ is locally of type $A_{7,4}$, hence it belongs to diagram $\\mathcal{Y}'$. The maximal singular subspaces of $\\Gamma$ partition in two classes which we shall denote ${\\mathcal A}$ and ${\\mathcal B}$. Then $\\Gamma$ is a parapolar space with a class of symplecta ${\\mathcal S}$ of type $D_{4,1}$.\n\\end{rem}\n\n\\begin{rem} Assume ${\\widetilde \\Gamma}$ satisfies the Weak Hexagon Axiom {\\it (WHA)}. It is a consequence of the definition of the fibering morphism of geometries that ({\\it WHA}) is also valid in $\\Gamma$. Therefore, in our analysis we can replace the space ${\\widetilde \\Gamma}$ with its covering $\\Gamma$, gaining in this way the advantage of being able to partition the maximal singular subspaces into two classes.\n\\end{rem}\n\n\\section{The family ${\\mathcal D}$ of subspaces}\n\nThroughout this section {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$} will denote a parapolar space which is locally of type $A_{7,4}$ and it is constructed as in the proof of Theorem $3.1$ of the previous section. The local assumption means that if $p \\in {\\mathcal P}$ then ${\\rm Res}_{\\Gamma} p$ is a space of type $A_{7,4}$. Recall the basic notions on spaces and related structures from Section 2.5. The result of the previous section allows to obtain a global partition of the maximal singular subspaces into two classes; see {\\it (P1)} below for notations and details. Next, $\\Gamma$ contains a set of symplecta ${\\mathcal S}$ whose elements are (non-degenerate) polar spaces of type $D_{4,1}$.\n\n\\medskip\nLet $S$ be a symplecton in $\\Gamma$ and let ${\\mathcal X} \\in \\lbrace {\\mathcal A}, {\\mathcal B} \\rbrace$. We define the following two sets associated to $S$:\n\n$$N_{\\mathcal X}(S)= \\lbrace p \\in {\\mathcal P} \\setminus S \\mid\np^{\\perp}\n\\cap S \\in M_{\\mathcal X}(S) \\rbrace$$\nand\n\\begin{equation*}\n\\begin{split}\nG_{\\mathcal X}(S) & = \\lbrace p \\in {\\mathcal P} \\setminus S \\mid\np^{\\perp} \\cap S =\n\\lbrace q \\rbrace, \\text{ a point};\\; \\text{for any}\\; r \\in q^{\\perp}\\cap S\\\\\n&\\quad \\text{the pair} \\; \\lbrace p,r \\rbrace \\; \\text{is polar}; \\; \\text{for some} \\; A \\in {\\mathcal A}_q \\cap {\\mathcal A}(S), \\; p^{\\perp} \\cap A \\; \\text{is a plane} \\rbrace.\n\\end{split}\n\\end{equation*}\n\n\\smallskip\nThen set\n$$D_{\\mathcal X}(S)= S \\cup N_{\\mathcal X}(S) \\cup G_{\\mathcal X}(S)$$\nand let ${\\mathcal D}_{\\mathcal X}= \\lbrace D_{\\mathcal X}(S)|\\; S \\in {\\mathcal S} \\rbrace$. Finally denote ${\\mathcal D} = {\\mathcal D}_{\\mathcal A} \\cup {\\mathcal D}_{\\mathcal B}$.\n\n\\medskip\nThe main result of this section is:\n\n\\begin{thm} Let {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$} be a parapolar space  which is locally of type $A_{7,4}$ and which satisfies:\n\n\\smallskip\nThe Weak Hexagon Axiom {\\it (WHA)}: Let $H = (p_1, \\ldots p_6)$ be a\n$6$-circuit, isometrically embedded in $\\Gamma$, this means that $p_i \\in p^{\\perp}_{i+1}$, indices taken mod $6$, and all the other pairs are not collinear. Also assume that at least one of the pairs of points at distance two, say $\\lbrace p_1, p_3 \\rbrace$, is polar. Then there exists a point $w \\in p_1^{\\perp} \\cap p_3^{\\perp} \\cap p_5^{\\perp}$.\n\n\\smallskip\nThen there exist two collections of $2$-convex subspaces ${\\mathcal D}_{\\mathcal A}$ and ${\\mathcal D}_{\\mathcal B}$, whose elements are spaces of type $D_{6,6}$. Every symplecton $S \\in {\\mathcal S}$ lies in exactly one element from each class.\n\\end{thm}\n\nIn order to prove the theorem we first show that each $D_{\\mathcal X}$\nis a $2$-convex subspace which is a strong parapolar space in its own. Then we use the Cohen-Cooperstein characterization theorem, see Section 2.5.2, to conclude that $D_{\\mathcal X}$ is of type $D_{6,6}$.\n\n\\medskip\nThe following properties of $\\Gamma$ are consequences of the theory of parapolar spaces and of the local assumption; see also the properties {\\it (G1-G4)} given in Section 2.5.1.\n\n\\medskip\n{\\it (P1)} There are two classes of maximal singular subspaces\n${\\mathcal A}$ and ${\\mathcal B}$. The set ${\\mathcal M} = {\\mathcal\nA} \\cup\n{\\mathcal B}$ is a collection of singular subspaces of rank $5$. Two distinct maximal singular\nsubspaces which belong to the same class can meet at a line, a\npoint or the empty set. Two maximal singular subspaces from different\nclasses can meet at a plane, a point or they are disjoint.\\\\\n{\\it (P2)} If $(S, M) \\in {\\mathcal S} \\times {\\mathcal M}$ then the set $S\n\\cap M$ can be empty, a point, a line  or a\nmaximal singular subspace of $S$.\\\\\n{\\it (P3)} For $S \\in {\\mathcal S}$ let ${\\mathcal  M}(S)$ be the family of maximal singular subspaces with largest intersection with $S$; observe that ${\\mathcal M}(S) = {\\mathcal A}(S) \\cup {\\mathcal B}(S)$. For $M_i \\in {\\mathcal M}(S),\\; i=1,2$, denote ${\\overline M}_i = M_i \\cap S$. Then ${\\overline M}_1 \\cap {\\overline M}_2$ can be a\npoint or a plane if ${\\overline M}_1, {\\overline M}_2$ belong to different classes; ${\\overline M}_1 \\cap {\\overline M}_2$ can be empty or a line when ${\\overline M}_1$ and ${\\overline M}_2$ are disjoint and belong to the same class.\\\\\n{\\it (P4)} If $M \\in {\\mathcal M}$ and $p \\in\n{\\mathcal P} \\setminus  M$ then\n$p^{\\perp} \\cap M$ can be empty, a point or a plane.\\\\\n{\\it (P5)} If $S \\in {\\mathcal S}$ and $p \\in {\\mathcal P} \\setminus S$\nthen $p^{\\perp} \\cap S$ can be empty, a point, a line or a maximal singular subspace of\n$S$.\n\n\\begin{ntn} For $S \\in {\\mathcal S}$ and ${\\mathcal X} \\in \\lbrace {\\mathcal A}, {\\mathcal B}\n\\rbrace$ let $M_{\\mathcal X}(S)= \\lbrace {\\overline X} \\mid {\\overline X} = X \\cap  S\n\\; \\; \\text{for some} \\; X \\in {\\mathcal X}(S) \\rbrace$ be the two classes of maximal singular subspaces of $S$.\n\\end{ntn}\n\n\\begin{ntn} In what follows if $p \\in {\\mathcal\nP}$ and $S \\in {\\mathcal S}$ are such that  $p ^{\\perp} \\cap S \\in M_{\\mathcal X}(S)$,\nthen we denote ${\\overline X}_p:= p^{\\perp} \\cap S$ and the maximal singular\nsubspace containing it $X_p$. If $p, q \\in {\\mathcal P}$ form a polar pair in $\\Gamma$, the unique symplecton containing them will be denoted $\\ll p, q \\gg$.\n\\end{ntn}\n\n\\begin{lem}\nLet $S$ be a symplecton in $\\Gamma$.\\\\\na). If $p \\in N_{\\mathcal X}(S)$ and $q \\in p^{\\perp}$ is such that\n$q^{\\perp} \\cap S \\setminus {\\overline X}_p \\not= \\emptyset$, then $q \\in S \\cup N_{\\mathcal X}(S)$.\\\\\nb). The set $S \\cup N_{\\mathcal X}(S)$ is a subspace of $\\Gamma$.\n\\end{lem}\n\n\\begin{proof} a). As the statement is obviously true when $q \\in S$, one may assume that $q \\not\\in S$. Let $r \\in q^{\\perp} \\cap S \\setminus {\\overline X}_p$. Set $R= \\ll r,p \\gg$ and note that $q \\in R$. Hence $L:=q^{\\perp} \\cap r^{\\perp} \\cap \\overline X_p$ is a line and $q^{\\perp} \\cap S$ contains the plane $\\langle r, L \\rangle $. According to $\\it{(P5)}$, $q^{\\perp} \\cap S = {\\overline M}$ is a maximal singular subspace of $S$. Now $\\langle qr, {\\overline M}\\cap {\\overline X}_p\\rangle$ has rank $3$ and lies in $R$, thus it is a maximal singular subspace of $R$. It follows that ${\\overline M}\\cap {\\overline X}_p=L$ and according to $\\it{(P3)}$, ${\\overline M}$ and ${\\overline X}_p$ belong to the same class. Hence $q \\in N_{\\mathcal X}(S)$.\n\n\\medskip\nb). One has to show that if $p$ and $q$ are two collinear points in $S \\cup N_{\\mathcal X}(S)$, then the line $pq$ lies entirely in $S \\cup N_{\\mathcal X}(S)$. If at least one of the points $p$ or $q$ is in $S$ the conclusion follows at\nonce. Assume $p, q \\not\\in S$. Set ${\\overline X}_p = p^{\\perp} \\cap S$ and ${\\overline X}_q = q^{\\perp} \\cap S$, two\nmaximal singular subspaces of $S$ belonging to the same class. According to ${\\it (P3)}$\nthere are three cases to consider:\n\n\\smallskip\n$(i)$. ${\\overline X}_p = {\\overline X}_q$ in which case $X_p = X_q$ and\nthus $pq \\subset N_{\\mathcal X}(S)$.\n\n\\smallskip\n$(ii)$. ${\\overline X}_p \\cap {\\overline X}_q = L$ is a line in $S$.\nLet $r \\in pq \\setminus \\lbrace p, q \\rbrace$. Clearly $L \\subset r^{\\perp}$. Let $w \\in {\\overline X}_q \\setminus r^{\\perp}$. Hence ${\\overline N}= \\langle w, w^{\\perp} \\cap {\\overline X}_p \\rangle$ is a maximal singular subspace of $S$, not in the\nsame class with ${\\overline X}_p$ and ${\\overline X}_q$ since ${\\overline N} \\cap {\\overline X}_p$ and ${\\overline N} \\cap {\\overline\nX}_q$ are both planes in $S$. Set $R =  \\ll w,p \\gg$ and observe that $\\overline N \\subseteq R \\cap S$. In $R$, the set $r^{\\perp} \\cap {\\overline N}$ is a plane and by ${\\it(P5)}$, $r ^{\\perp}\\cap S = {\\overline X}_r$ is a maximal singular subspace of $S$. It remains to show that ${\\overline X}_r \\in M_{\\mathcal X}(S)$. It suffices to prove: ${\\overline X}_p \\cap {\\overline X}_r =L$. Now ${\\overline X}_r$ meets ${\\overline N}$ at the plane $r^{\\perp} \\cap {\\overline N}$. So, the family of ${\\overline X}_r$ is not the same as that of ${\\overline N}$, which in its turn is not the same as that of ${\\overline X}_p$ and ${\\overline X}_q$. Hence ${\\overline X}_r$ and ${\\overline X}_p$, being different, meet at a line, which must be $L$.\n\n\\smallskip\n$(iii)$. The singular subspaces ${\\overline X}_p$ and ${\\overline X}_q$ are disjoint. Let $t \\in {\\overline X}_p$ and set $T =  \\ll t, q \\gg$. Then the plane $t^{\\perp} \\cap {\\overline X}_q$ lies in $T$ and therefore $p^{\\perp} \\cap t^{\\perp} \\cap {\\overline X}_q$ is a line. But this contradicts the assumption that ${\\overline X}_p \\cap {\\overline X}_q = \\emptyset$. Thus\neither ${\\overline X}_p \\cap {\\overline X}_q \\not= \\emptyset$ or $p$ is not collinear with $q$.\n\\end{proof}\n\n\\begin{lem} Let $p \\in G_{\\mathcal X}(S)$ with $\\lbrace q \\rbrace = p^{\\perp} \\cap S$.\\\\\na). For any $X \\in {\\mathcal X}_q \\cap {\\mathcal X}(S)$ the set $p^{\\perp} \\cap X$ is a plane.\\\\\nb). Let $r \\in N_{\\mathcal X}(S)$ be such that $q \\in r^{\\perp} \\cap S$. Then either $r \\in p^{\\perp}$ or $\\lbrace p, r \\rbrace$ is a polar pair.\n\\end{lem}\n\n\\begin{proof}\na). The result is a consequence of the local structure. For a proof see \\cite[Lemma 5.1]{on1}.\\\\\nb). Follows directly from part a) of the lemma.\n\\end{proof}\n\nIn what follows, for ease of exposition, we shall let ${\\mathcal X}={\\mathcal A}$ and show that $D_{\\mathcal A}(S)$ is a space of type $D_{6,6}$. Afterwards, we can replace ${\\mathcal A}$ with ${\\mathcal B}$ and repeat the arguments to obtain that $D_{\\mathcal B}(S)$ is also a space of type $D_{6,6}$.\n\n\\begin{lem} Let $R, S \\in {\\mathcal S}$. If $R \\cap S= \\overline B$ is a maximal\nsingular subspace in $M_{\\mathcal B}(S) \\cap M_{\\mathcal B}(R)$ then $D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$.\n\\end{lem}\n\n\\begin{proof} It suffices to prove $D_{\\mathcal A}(S) \\subseteq D_{\\mathcal A}(R)$.\n\n\\medskip\n$(i)$. Let $p \\in S \\setminus {\\overline  B}$. Then $p ^{\\perp} \\cap \\overline B$ is a plane and, by {\\it (P5)}, it follows that $p^{\\perp} \\cap R = \\overline A_p$ is a maximal singular subspace of $R$ in $M_{\\mathcal A}(R)$ (since it has a plane in common with $\\overline B$). Thus $p \\in N_{\\mathcal A}(R) \\subset D_{\\mathcal A}(R)$ and $S \\subset D_{\\mathcal A}(R)$.\n\n\\medskip\n$(ii)$. Let now $p \\in N_{\\mathcal A}(S)$. If $p \\in R$ we are done so we may assume that $p \\not \\in R$. Then $\\overline A_p = p^{\\perp} \\cap S$ can have either a plane or a point in common with $\\overline B$. If $\\overline A_p \\cap \\overline B$ is a plane, then $p \\in N_{\\mathcal A}(R)$ by a similar argument to that used in $(i)$. Next let $\\overline A_p \\cap \\overline B = \\lbrace q \\rbrace$ be a single point.\nIf there exists a point $r \\in p^{\\perp} \\cap R \\setminus \\lbrace q \\rbrace$ then $r \\not \\in A_p$. Assume to the contrary that $r \\in A_p$. Then $r ^{\\perp} \\cap S \\supseteq \\langle {\\overline A}_p, r^{\\perp} \\cap {\\overline B} \\rangle \\supset {\\overline A}_p$ and since ${\\overline A}_p$ is a maximal singular subspace of $S$ we obtain a contradiction with {\\it (P5)}. Thus $ r \\not \\in A_p$. Let $t \\in {\\overline A}_p \\setminus r^{\\perp}$, which, according to Step $(i)$, is in $N_{\\mathcal A}(R)$. An application of Lemma $4.4(a)$, to the pair $\\lbrace p, t \\rbrace$ and symplecton $R$, gives that $p \\in N_{\\mathcal A}(S)$. Next assume that $p^{\\perp} \\cap R = \\lbrace q \\rbrace$, a single point. {\\it Claim}: $p \\in G_{\\mathcal A}(R)$. Let $s \\in q^{\\perp} \\cap R \\setminus S$ so $s \\in R \\subset N_{\\mathcal A}(S)$ and the pair $\\lbrace p, s \\rbrace$ is polar. Let ${\\overline A}_s = s^{\\perp} \\cap S$ with $A_s$ be the maximal singular subspace containing ${\\overline A}_s$. Since $p^{\\perp} \\cap A_s$ contains the line ${\\overline A}_p \\cap {\\overline A}_s$, it follows, by {\\it (P4)}, that $p^{\\perp} \\cap A_s$ is a plane. This concludes the proof of the claim.\n\n\\medskip\n$(iii)$. Let $p \\in G_{\\mathcal A}(S)$ be such that $p^{\\perp} \\cap {\\overline B} = \\lbrace q \\rbrace$ is a point. Assume first that $p^{\\perp} \\cap R = \\lbrace q \\rbrace$. Let $r \\in q^{\\perp} \\cap R$, so $r \\in N_{\\mathcal A}(S)$. By Lemma $4.5(b)$ the pair $\\lbrace p, r  \\rbrace$ is polar. Further, if $A \\in {\\mathcal A}(S) \\cap {\\mathcal A}(R) \\cap {\\mathcal A}_q$ then $p^{\\perp} \\cap A$ is a plane. Thus $p \\in G_{\\mathcal A}(R)$. Assume now $p^{\\perp} \\cap R \\supseteq qs$, a line. Consider ${\\rm Res}_{\\Gamma}(q)$. For a subspace  $F$ of $\\Gamma$ which contains $q$, ${\\widetilde F}$ will denote the corresponding subspace in $\\res{\\Gamma}{q}$. In $\\res{\\Gamma}{q}$, ${\\widetilde R}$ and ${\\widetilde S}$ are two ``symplecta\" which meet at a plane of $\\widetilde {\\mathcal B}$-type. Also $\\tilde p$ is a ``point\" at distance two from every single ``point\" in $\\widetilde S$ and $\\tilde s \\in \\tilde p ^{\\tilde \\perp} \\cap \\widetilde R$. According to the result of \\cite[Lemma 2.3]{on1}, it follows that $\\tilde p^{\\tilde \\perp} \\cap \\widetilde R$ is a ``plane\". Then, back in $\\Gamma$, $p^{\\perp} \\cap R$ is a maximal singular subspace of $R$. Now $(p^{\\perp} \\cap R) \\cap \\overline B = \\lbrace q \\rbrace$ which implies $p^{\\perp} \\cap R \\in M_{\\mathcal A}(R)$. Thus $p \\in N_{\\mathcal A}(R)$.\n\n\\medskip\n$(iv)$. Let now $p \\in G_{\\mathcal A}(S)$ be such that $p^{\\perp} \\cap {\\overline B} = \\emptyset$ and let $\\lbrace q \\rbrace = p^{\\perp} \\cap S$. Recall from Step $(i)$ that $q\\in N_{\\mathcal A}(R)$. Let $A_q \\in {\\mathcal A}(R) \\cap {\\mathcal A}_q$ be the maximal singular subspace which contains $\\langle q, q^{\\perp} \\cap R \\rangle$. Note that $A_q \\in {\\mathcal A}(S)$ also, since $A_q \\cap S = \\langle q, q^{\\perp} \\cap {\\overline B}\\rangle $. By Lemma $4.5(a), \\; p^{\\perp} \\cap A_q$ is a plane which intersects $R$ nontrivially. Note that $p^{\\perp} \\cap A_q \\cap R$ cannot contain a line, since this would imply $p^{\\perp} \\cap {\\overline B} \\not= \\emptyset$, contrary to our choice of $p$. Let $\\lbrace r \\rbrace = p^{\\perp} \\cap A_q \\cap R$. {\\it Claim}: $p \\in G_{\\mathcal A}(R)$. We first prove that $p^{\\perp} \\cap R = \\lbrace r \\rbrace$ is a single point. Assume by contradiction that there is a point $s \\in p^{\\perp} \\cap R \\setminus \\lbrace r \\rbrace$. According to the above argument, $s \\not \\in A_q$. Then, by Lemma $4.4(a)$, applied to the pair $\\lbrace p, q \\rbrace$ and symplecton $R$ we get $p \\in N_{\\mathcal A}(R)$. But this implies $p^{\\perp} \\cap R \\cap {\\overline B} \\not = \\emptyset$, which contradicts the fact that $p^{\\perp} \\cap {\\overline B} = \\emptyset$. Therefore $p^{\\perp} \\cap R = \\lbrace r \\rbrace$. Since $p^{\\perp} \\cap A_q$ is a plane, it remains to prove that given any point $t \\in r^{\\perp} \\cap R$, the pair $\\lbrace p, t \\rbrace$ is polar. This is clearly true for any $t \\in r^{\\perp} \\cap {\\overline B} = q^{\\perp} \\cap {\\overline B}$, by the definition of $G_{\\mathcal A}(S)$. So let us assume that $t \\not \\in {\\overline B}$. Now $t \\in N_{\\mathcal A}(S)$ and since $q^{\\perp} \\cap t^{\\perp} \\cap S$ contains a plane, the pair $\\lbrace q, t \\rbrace$ is polar. Set $T = \\ll q, t \\gg$ and note that $T \\cap S = \\langle q, q^{\\perp} \\cap t^{\\perp} \\cap S\\rangle  \\in M_{\\mathcal B}(S)$. Now $p \\in G_{\\mathcal A}(S)$, $p^{\\perp} \\cap S \\cap T \\not = \\emptyset$ and $p^{\\perp} \\cap T \\supset qr$ and according to Step $(iii)$, $p \\in N_{\\mathcal A}(T)$. Therefore $p^{\\perp} \\cap T \\cap t^{\\perp}$ contains a plane and this proves that $\\lbrace p,t \\rbrace$ is a polar pair.\n\\end{proof}\n\n\\begin{lem} a). Let $S, R \\in {\\mathcal S}$ be such that $S \\cap R = L$ is a line. Assume that $R \\cap {\\mathcal A}(S) \\not= \\emptyset$. Then $D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$.\\\\\nb). Let $p \\in G_{\\mathcal A}(S)$ with $\\lbrace q \\rbrace = p^{\\perp} \\cap S$. For $r \\in q^{\\perp} \\cap S$ set $R = \\ll p, r \\gg$. Then $D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$.\n\\end{lem}\n\n\\begin{proof}a). Let $S, R \\in {\\mathcal S}$ be such that $S \\cap R = L$ is a line. Let\n$p \\in R \\cap G_{\\mathcal A}(S)$. Denote $p ^{\\perp} \\cap S = \\lbrace q\n\\rbrace$ and observe that $ q \\in L$. Let $A \\in {\\mathcal A}(S)$ be such that $L \\subset A$. Then, according to {\\it (P2)}, $R \\cap A$ can be a line or a maximal singular subspace of $R$. Since $p \\in G_{\\mathcal A}(S),\\; p^{\\perp} \\cap A $ is a plane. Let $r \\in p^{\\perp} \\cap A \\setminus \\lbrace q \\rbrace$ and let $t \\in L \\setminus \\lbrace q \\rbrace$. Thus $R = \\ll p, t \\gg$ and $r \\in R$. Then $A \\cap R \\supseteq \\langle r, L \\rangle $ and by {\\it (P2)} it follows that $A \\in {\\mathcal A}(R)$. We proved that if $A \\in {\\mathcal A}(S)$ is such that $L \\subset A$ then $A \\in {\\mathcal A}(R)$ as well.\\\\\nLet now $A_1, A_2 \\in {\\mathcal A}(S) \\cap {\\mathcal A}(R)$ be two distinct maximal singular subspaces such that $L \\subseteq A_1 \\cap A_2$. The line $L$ is contained in the symplecton $S$ and so is the intersection of two maximal singular subspaces of $S$ of the same class. Let $p_1 \\in A_1 \\cap S \\setminus L$ and $p_2 \\in A_2 \\cap R \\setminus L$. Set $T = \\ll p_1, p_2 \\gg$. Note that $T \\cap S = \\langle p_1, p_1^{\\perp} \\cap A_2 \\cap S \\rangle $ and $T \\cap R = \\langle p_2, p_2^{\\perp} \\cap A_1 \\cap R \\rangle $ both singular subspaces of ${\\mathcal B}$-type. Then, by Lemma $4.6$: $D_{\\mathcal A}(S) = D_{\\mathcal A}(T) = D_{\\mathcal A}(R)$.\n\n\\medskip\nb). Let $S$ and $R$ be two symplecta as in the hypothesis. We claim that $R \\cap S = pq$, a line. If $R \\cap S$ contains a plane then $p^{\\perp} \\cap S$ contains a line, contradicting the properties of $p \\in G_{\\mathcal A}(S)$. Since $p \\in G_{\\mathcal A}(S) \\cap R$, part a) applies and the result follows.\n\\end{proof}\n\n\\begin{prop} Let $S \\in {\\mathcal S}$ then $D_{\\mathcal A}(S)$ is a subspace of $\\Gamma$.\n\\end{prop}\n\n\\begin{proof}Let $p, q \\in D_{\\mathcal A}(S)$ be two collinear points. We have to prove that $pq \\subset D_{\\mathcal A}(S)$.\n\n\\medskip\n$(i)$. If $p,q \\in S \\cup N_{\\mathcal A}(S)$ the result follows from Lemma $4.4(b)$.\n\n\\medskip\n$(ii)$. If $p \\in G_{\\mathcal A}(S)$ and $ q \\in S$ then $\\lbrace q \\rbrace = p^{\\perp} \\cap S$. In this case $pq \\subset S \\cup G_{\\mathcal A}(S)$ follows from the gamma space property of $\\Gamma$; see Section 2.5.\n\n\\medskip\n$(iii)$. Assume now that $p \\in N_{\\mathcal A}(S)$ and $q \\in G_{\\mathcal A}(S)$.\nLet ${\\overline A}_p = p^{\\perp} \\cap S$ and $\\lbrace r \\rbrace = q^{\\perp} \\cap S$. It follows, from Lemma $4.4(a)$, that $r \\in {\\overline A}_p$. Let $t \\in r^{\\perp} \\cap S \\setminus {\\overline A}_p$. Set $R = \\ll p, t \\gg$. Then, by Lemma $4.6$, $D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$. Now $q \\in G_{\\mathcal A}(S) \\subset D_{\\mathcal A}(R)$ and $q^{\\perp} \\cap R$ contains the line $pr$, hence $q \\in N_{\\mathcal A}(R)$. Thus $p \\in R,\\; q \\in N_{\\mathcal A}(R)$ and, according to Lemma $4.4(b), \\; pq \\subset D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$.\n\n\\medskip\n$(iv)$. Let $p, q \\in G_{\\mathcal A}(S)$. First assume $p^{\\perp} \\cap\nq^{\\perp} \\cap S = \\lbrace r \\rbrace$. Let $t \\in r^{\\perp} \\cap S$ and set $R = \\ll p, t\\gg$. Then, according to Lemma $4.7(b)$, $D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$. Thus $p \\in R$ and $q \\in R \\cup N_{\\mathcal A}(R)$ implies $pq \\subset D_{\\mathcal A}(R)= D_{\\mathcal A}(S)$. Let now $p^{\\perp} \\cap S = \\lbrace r \\rbrace$ and $q^{\\perp} \\cap R = \\lbrace t \\rbrace$ be such that $r \\not = t$. If $r \\in t^{\\perp}$ then set $R = \\ll p, t \\gg$ and, by Lemma $4.7(b)$, $D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$. Therefore $pq \\subset R \\subset D_{\\mathcal A}(S)$. If $r \\not \\in t^{\\perp}$ then take $w \\in r^{\\perp} \\cap t^{\\perp}$ and set $T = \\ll p, w \\gg$. Again, by Lemma $4.7(b)$ $D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$. So $p \\in T,\\; q \\in D_{\\mathcal A}(T)$ and by previous results of this Proposition, $pq \\subset D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$.\n\\end{proof}\n\nThe assumption {\\it (WHA)} from the statement of Theorem $4.1$ is used in the proof of the following proposition.\n\n\\begin{prop} Let $S \\in {\\mathcal S}$, then $D_{\\mathcal A}(S)$ is a $2$-convex\nsubspace of $\\Gamma$.\n\\end{prop}\n\n\\begin{proof}Let $p, q \\in D_{\\mathcal A}(S)$ be two points at distance $2$. We have to prove that $p^{\\perp} \\cap q^{\\perp} \\subset D_{\\mathcal A}(S)$.\n\n\\medskip\n$(i)$. If $p,q \\in S$ then obviously $p^{\\perp} \\cap q^{\\perp} \\subset\nS$ since $S$ is a $2$-convex subspace of $\\Gamma$.\n\n\\medskip\n$(ii)$. Let now $p \\in N_{\\mathcal A}(S)$ and $q \\in S$. The pair $\\lbrace p,q \\rbrace$ is polar. Set $R = \\ll p,q \\gg$. Since $R \\cap S \\in \\mss{\\mathcal B}{S}$ it follows, by Lemma $4.6$ that $D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$. Hence $p^{\\perp} \\cap q^{\\perp} \\subset R \\subset D_{\\mathcal A}(S)$.\n\n\\medskip\n$(iii)$. Consider now the case when $p,q \\in N_{\\mathcal A}(S)$. Then ${\\overline A}_p = p^{\\perp} \\cap S$ and ${\\overline A}_q =\nq^{\\perp} \\cap S$ are two maximal singular subspaces of $S$ from the same\nclass. There are two cases to consider:\n\n\\smallskip\n$(iii.a)$. Assume ${\\overline A}_p \\cap {\\overline A}_q =L$ is a line. Let $r \\in {\\overline A}_q \\setminus L$ and set $R = \\ll r,p \\gg$. Then $R \\cap S \\in M_{\\mathcal B}(S)$ and, by Lemma $4.6,\\; D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$. So $q \\in D_{\\mathcal A}(R)$ and because $q^{\\perp} \\cap R \\supseteq \\langle r, L \\rangle $ it follows $q \\in N_{\\mathcal A}(R)$. Note that $q \\not \\in R$ since $A_q \\cap R = \\langle r, p^{\\perp} \\cap A_q \\rangle$ has rank $3$ and thus is already maximal in $R$. Now $p \\in R,\\; q \\in N_{\\mathcal A}(R)$ and according to Step $(ii)$ of this Proposition, $p^{\\perp} \\cap q^{\\perp} \\subset D_{\\mathcal A}(R) =D_{\\mathcal A}(S)$.\n\n\\smallskip\n$(iii.b)$. Assume now that ${\\overline A}_p$ and ${\\overline A}_q$ are disjoint maximal singular subspaces of $S$. Let $r \\in p^{\\perp} \\cap q^{\\perp}$. Take $p_1 \\in {\\overline A}_p \\setminus r^{\\perp}$ and $q_1 \\in {\\overline A}_q \\setminus ( r^{\\perp} \\cup p_1^{\\perp})$. Also let $r_1 \\in p_1^{\\perp} \\cap q_1^{\\perp}$ be such that $r_1 \\not \\in p^{\\perp} \\cup q^{\\perp}$. Now $H = (p_1, r_1, q_1, q,r,p)$ is a minimal $6$-circuit in $\\Gamma$ which contains at least one polar pair $\\lbrace p_1, q_1 \\rbrace$. Then {\\it (WHA)} applies and there exists a point $w \\in r^{\\perp} \\cap p_1^{\\perp} \\cap q_1^{\\perp} \\subset S$. Thus, according to Lemma $4.4(a)$, $r \\in N_{\\mathcal A}(S) \\subset D_{\\mathcal A}(S)$.\n\n\\medskip\n$(iv)$. Let $p \\in G_{\\mathcal A}(S)$ and $q \\in S$. Let $\\lbrace t \\rbrace = p^{\\perp} \\cap S$. If $q \\in t^{\\perp}$ then $R = \\ll p,q \\gg$ is contained in $D_{\\mathcal A}(S)$; see Lemma $4.7(b)$. So let us assume $q \\not \\in t^{\\perp}$. Take $A \\in {\\mathcal A}(S) \\cap {\\mathcal A}_t$. Then according to Lemma $4.5$, $p^{\\perp} \\cap A$ is a plane. Let $p_1 \\in p^{\\perp} \\cap A \\setminus \\lbrace t \\rbrace$ and let $r \\in t^{\\perp} \\cap S \\setminus A$. Set $R = \\ll p_1, r \\gg$. Since $R \\cap S = \\langle r, r^{\\perp} \\cap A \\rangle  \\in \\mss{\\mathcal B}{R} \\cap \\mss{\\mathcal B}{S}$ it follows, by Lemma $4.6$ that $D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$. Now $p^{\\perp} \\cap R \\supset p_1 t$ so $p \\in N_{\\mathcal A}(R)$. Also $q \\in N_{\\mathcal A}(R)$. According to Step $(iii)$ of this Proposition $p^{\\perp} \\cap q^{\\perp} \\subset D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$.\n\n\\medskip\n$(v)$. Let $p \\in N_{\\mathcal A}(S), \\; q \\in G_{\\mathcal A}(S)$. Let\n$\\lbrace r \\rbrace = q^{\\perp} \\cap S$ and ${\\overline A}_p = p^{\\perp} \\cap S$. Assume first that $r \\not\\in {\\overline A}_p$. Set $R = \\ll p, r \\gg$. Then $R \\cap S\n\\in M_{\\mathcal B}(S)$ and according to Lemma $4.6, \\; D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$. Now $p \\in R,\\; q \\in D_{\\mathcal A}(R)$ and using the above results of this Proposition, it follows $p^{\\perp} \\cap q^{\\perp} \\subset D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$. Next consider the case $r \\in {\\overline A}_p$. Let $t \\in {\\overline A}_p$ be a point and set $T = \\ll q,t \\gg$. By Lemma $4.7(b),\\; D_{\\mathcal A}(S) = D_{\\mathcal A}(T)$. So we may apply the results of the Steps $(i)-(iii)$ of this Proposition to the pair of points $ p \\in D_{\\mathcal A}(T),\\; q \\in T$ and conclude that $p^{\\perp} \\cap q^{\\perp} \\subset D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$.\n\n\\medskip\n$(vi)$. Let now $p,q \\in G_{\\mathcal A}(S)$ be such that $p^{\\perp} \\cap S =\n\\lbrace p_1 \\rbrace$ and $q^{\\perp} \\cap S = \\lbrace q_1 \\rbrace$. If $p_1 \\not \\in q_1^{\\perp}$ then let $r \\in p_1^{\\perp} \\cap q_1^{\\perp}$; if $p_1 \\in q_1^{\\perp} \\setminus \\lbrace q_1 \\rbrace$ then we let $r = q_1$ and if $p_1 = q_1$ then take $r \\in p_1^{\\perp} \\cap S$. Set $R =  \\ll p, r \\gg$. By Lemma $4.7(b),\\; D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$. Now $q \\in G_{\\mathcal A}(S) \\subset D_{\\mathcal A}(R)$ and $p \\in R$ hence using previous results of this Proposition, $p^{\\perp} \\cap q^{\\perp} \\subset D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$.\n\\end{proof}\n\n\\begin{lem} Let $p \\in G_{\\mathcal A}(S)$ with $\\lbrace p_1 \\rbrace = p^{\\perp} \\cap S$ and let $w \\in S$. Then $d(p,w) = 1+d(p_1,w)$.\n\\end{lem}\n\n\\begin{proof} We start by proving the following {\\it claim}: if $p,q \\in G_{\\mathcal A}(S)$ are two collinear points with $\\lbrace p_1 \\rbrace = p^{\\perp} \\cap S$ and $\\lbrace q_1 \\rbrace = q^{\\perp} \\cap S$ then $q_1 \\in p_1^{\\perp}$. Assume by contradiction that $q_1 \\not \\in p_1^{\\perp}$. Let $A \\in {\\mathcal A}_{p_1} \\cap{\\mathcal A}(S)$. Then, according to Lemma $4.5,\\; p^{\\perp} \\cap A$ is a plane. Let $x \\in p^{\\perp} \\cap A \\setminus \\lbrace p_1 \\rbrace$. Also let $r \\in p_1^{\\perp} \\cap q_1^{\\perp} \\setminus A$. Set $R = \\ll r, x \\gg$ and observe that $R \\cap S = \\langle r, r^{\\perp} \\cap A \\cap S\\rangle  \\in M_{\\mathcal B}(S)$ and, by Lemma $4.6, \\; D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$. Note that $p^{\\perp} \\cap R \\supset p_1x$ and consequently $p \\in N_{\\mathcal A}(R)$. Set $\\overline A_p = p^{\\perp} \\cap R$. Also $q \\in G_{\\mathcal A}(S) \\subset D_{\\mathcal A}(R)$ implies that there is a point $y \\in q^{\\perp} \\cap R$. If $y \\not \\in {\\overline A}_p$, since $p$ and $q$ are collinear, it follows, by Lemma $4.4(a)$, that $q \\in N_{\\mathcal A}(R)$. But then $q^{\\perp} \\cap R$ and $R \\cap S$ are maximal singular subspaces in $R$ from different families and therefore they have a point in common. Since $y \\not \\in R \\cap S$ it follows that $q^{\\perp} \\cap R$ contains more than a point, which is a contradiction with the fact that $q \\in G_{\\mathcal A}(S)$. Therefore $y \\in {\\overline A}_p$. Next $q_1 \\in S$ and, by Lemma $4.6$, $q_1 \\in N_{\\mathcal A}(R)$. Let $\\overline A _{q_1} = q_1^{\\perp} \\cap R$ and observe that $y \\in {\\overline A}_{q_1}$ because otherwise another application of Lemma $4.4(a)$ to the pair $\\lbrace q, q_1 \\rbrace$ and symplecton $R$ would give $q \\in N_{\\mathcal A}(R)$, a contradiction. Therefore $y \\in {\\overline A}_p \\cap {\\overline A}_{q_1}$ which, using {\\it (P3)} implies ${\\overline A}_p \\cap {\\overline A}_{q_1} = L$ is a line. But $r \\not \\in x^{\\perp}$ so $x \\not \\in L$. Since ${\\overline A}_{q_1}$ has rank $3$ and since $S \\cap R$ meets ${\\overline A}_{q_1}$ in a plane it follows that $L \\cap (R \\cap S) \\not = \\emptyset$. But this implies that $p^{\\perp} \\cap S$ contains more than a point. We reach a contradiction with the fact that $p \\in G_{\\mathcal A}(S)$. Therefore the assumption made was false and $p_1 \\in q_1^{\\perp}$; the claim is proved.\n\n\\medskip\nIn order to prove the Lemma it suffices to show that, if $p \\in G_{\\mathcal A}(S)$ and $w \\in S \\setminus p_1^{\\perp}$ then $d(p, w) = 3$. Assume by contradiction that $d(p, w) = 2$. Then there exists a point $t \\in p^{\\perp} \\cap w^{\\perp}$. Note that $t$ cannot be in $S$. According to Proposition $4.10,\\; t \\in D_{\\mathcal A}(S)$. Moreover $t \\not \\in N_{\\mathcal A}(S)$, because since $w \\not \\in p_1^{\\perp}$, Lemma $4.4(a)$ would imply $p \\in N_{\\mathcal A}(S)$, a contradiction. So we must have $t \\in G_{\\mathcal A}(S)$. But now, according to the previous paragraph $w \\in p_1^{\\perp} \\cap S$, which contradicts the hypothesis on $w$. Therefore the assumption made was false and in this case $d(p,w)=3$.\n\\end{proof}\n\n\\begin{prop} For any $S \\in {\\mathcal S},\\ D_{\\mathcal A}(S)$ is a strong parapolar\nsubspace of $\\Gamma$.\n\\end{prop}\n\n\\begin{proof} We have to prove that if $p,q \\in D_{\\mathcal A}(S)$ are two points at distance two, the pair $\\lbrace p,q \\rbrace$ is polar, that is $p^{\\perp} \\cap q^{\\perp}$ contains at least two points.\n\n\\medskip\n$(i)$. Let $p \\in D_{\\mathcal A}(S)$ and $q \\in S$. If both $p$ and $q$ are in $S$ then $S = \\ll p, q \\gg$ and we are done. Assume next $p \\in N_{\\mathcal A}(S)$. Then $q^{\\perp} \\cap p^{\\perp} \\cap S$ contains a plane and therefore $\\lbrace p,q \\rbrace$ is a polar pair. Let now $p \\in G_{\\mathcal A}(S)$ with $\\lbrace p_1 \\rbrace = p^{\\perp} \\cap S$. In this case, since $d(p,q) = 2$, by Lemma $4.10$ it follows that $q \\in p_1^{\\perp}$. Then the fact that $\\lbrace p, q \\rbrace$ is polar pair follows from the definition of $G_{\\mathcal A}(S)$.\n\n\\medskip\n$(ii)$. Assume now $p,q \\in N_{\\mathcal A}(S)$. Let ${\\overline A}_p = p^{\\perp} \\cap S$ and ${\\overline A}_q = q^{\\perp} \\cap S$. If ${\\overline A}_p \\cap {\\overline A}_q \\not = \\emptyset$ the result is immediate. So we may assume that ${\\overline A}_p \\cap {\\overline A}_q = \\emptyset$. Let $r \\in {\\overline A}_p$. Then set $R = \\ll q, r \\gg$ which, by Lemma $4.6$, is such that $D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$. Thus $p \\in G_{\\mathcal A}(S) \\subset D_{\\mathcal A}(R),\\; q \\in R$ and by Step $(i)$ of this Proposition it follows that $\\lbrace p, q \\rbrace$ is a polar pair.\n\n\\medskip\n$(iii)$. Let $p \\in N_{\\mathcal A}(S)$ and $q \\in G_{\\mathcal A}(S)$. Let ${\\overline A}_p = p^{\\perp} \\cap S$ and $\\lbrace t \\rbrace = q^{\\perp} \\cap S$. If $t \\in {\\overline A}_p$ then, according to Lemma $4.5(b),\\; \\lbrace p, q \\rbrace$ is a polar pair. Assume next that $t \\not \\in {\\overline A}_p$. Take a point $r \\in t^{\\perp} \\cap {\\overline A}_p$ and set $R = \\ll q, r \\gg$. By Lemma $4.7(b), \\; D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$. So $q \\in R,\\; p \\in N_{\\mathcal A}(S) \\subset D_{\\mathcal A}(R)$ and $\\lbrace p,q \\rbrace$ is a polar pair by Step $(i)$ of this Proposition.\n\n\\medskip\n$(iv)$. Let now $p,q \\in G_{\\mathcal A}(S)$ with $\\lbrace r \\rbrace = p^{\\perp} \\cap S$ and $\\lbrace t \\rbrace = q^{\\perp} \\cap S$. If $r \\in t^{\\perp} \\setminus \\lbrace t \\rbrace$ set $R = \\ll r, q \\gg$. If $r = t$ then take $w \\in r^{\\perp}$ and if $r \\not \\in t^{\\perp}$ take $w \\in r^{\\perp} \\cap t^{\\perp}$ and set $R =\\ll w, q \\gg$. Then, by Lemma $4.7(b),\\; D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$ and since $p \\in G_{\\mathcal A}(S) \\subset D_{\\mathcal A}(R), q \\in R$, the pair $\\lbrace p,q \\rbrace$ is polar by Step $(i)$ of this Proposition.\n\\end{proof}\n\n\\begin{prop}Given $R \\in {\\mathcal S}$ there exists a unique element\n$D_{\\mathcal A}(S) \\in {\\mathcal D}_{\\mathcal A}$, for some $S \\in {\\mathcal S}$ containing $R$.\n\\end{prop}\n\n\\begin{proof}\nLet $R$ be a symplecton and assume that $R \\subset D_{\\mathcal A}(S)$ for some $S \\in {\\mathcal S}$. Write $R=\\ll p,q \\gg$ with $p$ and $q$ two points of $R$, at distance two in the collinearity graph. We analyze the possible relations between $D_{\\mathcal A}(S)$ and $D_{\\mathcal A}(R)$.\n\n\\medskip\n$(i)$. Assume $p \\in D_{\\mathcal A}(S)$ and $q \\in S$. If $p \\in S$ then $S= \\ll p,q \\gg = R$. Next let $p \\in N_{\\mathcal A}(S)$ and set ${\\overline A}_p = p^{\\perp} \\cap S$. Then $R \\cap S = \\langle q, q^{\\perp} \\cap {\\overline A}_p \\rangle  \\in M_{\\mathcal B}(S)$. Hence $D_{\\mathcal A}(S) = D_{\\mathcal A}(R)$, by Lemma $4.6$. Let now $p \\in G_{\\mathcal A}(S)$ with $\\lbrace r \\rbrace = p^{\\perp} \\cap S$. By Lemma $4.11$, $q \\in r^{\\perp} \\cap S$ and by Lemma $4.7(b)$ it follows that $D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$.\n\n\\medskip\n$(ii)$. Let $p, q \\in N_{\\mathcal A}(S)$ with ${\\overline A}_p = p^{\\perp} \\cap S$ and ${\\overline A}_q = q^{\\perp} \\cap S$.\n\n\\smallskip\n$(ii.a)$. Assume ${\\overline A}_p \\cap {\\overline A}_q = L$ is a line. {\\it Claim}: $R \\cap S = L$. Suppose by contradiction that $R \\cap S$ contains a plane $\\langle r,L \\rangle $ where $r$ is a point not on $L$. Since $r \\not \\in A_p \\cap A_q$ we can assume, without loss of generality, that $r \\not \\in A_q$. If $r \\in {\\overline A}_p$ then $\\langle r, q^{\\perp} \\cap A_p, p \\rangle$ is a singular subspace of rank $4$ in $R$, a contradiction. So let us assume that $r \\in S \\setminus ({\\overline A}_p \\cup {\\overline A}_q)$. Then $\\langle r, r^{\\perp} \\cap {\\overline A}_p, r^{\\perp} \\cap {\\overline A}_q \\rangle  \\subset R \\cap S$ which is a contradiction with the fact that $R \\cap S$ can be at most a common maximal singular subspace. Therefore the claim is proved. Now pick a point $z \\in p^{\\perp} \\cap q^{\\perp} \\setminus L$ such that $z^{\\perp} \\cap L = \\lbrace t \\rbrace$, a single point. We intend to prove that $z \\in G_{\\mathcal A}(S)$. According to Proposition $4.11, \\; z\\in D_{\\mathcal A}(S)$, so it suffices to prove that $z^{\\perp} \\cap S = \\lbrace t \\rbrace$. Assume by contradiction that $z \\in N_{\\mathcal A}(S)$ with ${\\overline A}_z = z^{\\perp} \\cap S$. Then, according to {\\it (P3)}, ${\\overline A}_z \\cap {\\overline A}_p$ is a line $L'$ distinct from $L$. So if $w \\in L \\setminus \\lbrace t \\rbrace$ then $\\langle L, L' \\rangle \\subset z^{\\perp} \\cap w^{\\perp} \\subset R \\cap S$ and we reach a contradiction with the previous result that $R \\cap S = L$. Therefore $z^{\\perp} \\cap S = \\lbrace p \\rbrace$ and $z \\in G_{\\mathcal A}(S)$. Now $R$ and $S$ are as in Lemma $4.7(a)$ and consequently $D_{\\mathcal A}(R) = D_{\\mathcal A}(S)$.\n\n\\smallskip\n$(ii.b)$. Let ${\\overline A}_p \\cap {\\overline A}_q = \\emptyset$. Let $t \\in {\\overline A}_p$ and set $T = \\ll t,q \\gg$. Thus $T \\cap S = \\langle  t, t^{\\perp} \\cap {\\overline A}_q \\rangle  \\in M_{\\mathcal B}(S)$ and by Lemma $4.6,\\; D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$. Now $p \\in D_{\\mathcal A}(T), q \\in T$ so by Step $1$, $R = \\ll p, q \\gg$ is such that $D_{\\mathcal A}(R) = D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$.\n\n\\medskip\n$(iii)$. Let $p \\in N_{\\mathcal A}(S), \\; q \\in G_{\\mathcal A}(S)$ with $q^{\\perp} \\cap S = \\lbrace r \\rbrace$. If $r \\in {\\overline A}_p$ then let another point $t \\in {\\overline A}_p$ and set $T = \\ll q, t \\gg$. By Lemma $4.7b$, $D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$. Apply Step $(i)$ to $p \\in D_{\\mathcal A}(T),\\; q \\in T$ to get $D_{\\mathcal A}(R) = D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$. If $r \\not \\in {\\overline A}_p$ set $T = \\ll p, r \\gg$ and since $T \\cap S \\in M_{\\mathcal B}(S)$, Lemma $4.6$ gives $D_{\\mathcal A}(T)=D_{\\mathcal A}(S)$. Now $p \\in T$ and $q \\in D_{\\mathcal A}(T)$ and using Step $(i)$ again we get $D_{\\mathcal A}(R) = D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$.\n\n\\medskip\n$(iv)$. Let $p,q \\in G_{\\mathcal A}(S)$ with $\\lbrace p_1 \\rbrace =\np^{\\perp} \\cap S$ and $\\lbrace q_1 \\rbrace = q^{\\perp} \\cap S$. If $p_1 = q_1$ take $r \\in p_1^{\\perp} \\cap S$, set $T = \\ll r, p \\gg$. If $p_1 \\in q_1^{\\perp} \\setminus \\lbrace q_1 \\rbrace$ take $T = \\ll p, q_1 \\gg$. If $p_1 \\not \\in q_1^{\\perp}$ then take $r \\in p_1^{\\perp} \\cap q_1^{\\perp}$ and set $T =  \\ll p,r \\gg$. Then apply Step $(i)$ to get $D_{\\mathcal A}(R) = D_{\\mathcal A}(T) = D_{\\mathcal A}(S)$.\n\\end{proof}\n\n\\medskip\n\\begin{proof}[{\\it Proof Theorem 4.1}] Let {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$} be a parapolar space\nwhich is locally $A_{7,4}$. Assume that the maximal singular subspaces of $\\Gamma$ partition in two classes. In addition, assume that $\\Gamma$ satisfies the Weak Hexagon Axiom {\\it (WHA)}. For any symplecton $S \\in {\\mathcal S}$ we defined two sets $D_{\\mathcal X}(S)$ with ${\\mathcal X} \\in \\lbrace {\\mathcal A}, {\\mathcal B} \\rbrace$; see the beginning of Section $4$. Then $D_{\\mathcal A}(S)$ is a $2$-convex subspace of\n$\\Gamma$, see Propositions $4.8$ and $4.9$, which is strong parapolar, by\nProposition $4.11$. Further, by Proposition $4.12$, every symplecton lies in a unique element of ${\\mathcal D}_{\\mathcal A}$.\n\n\\medskip\nConsider a point-symplecton pair $(p, R)$ in $D_{\\mathcal A}(S)$. Since $R \\subset D_{\\mathcal A}(S)$ it follows, by Proposition $4.12$ that $D_{\\mathcal A}(R)=D_{\\mathcal A}(S)$. Therefore, either $p \\in R$ or $p^{\\perp} \\cap R$ is a single point or a maximal singular subspace of $R$. By the characterization theorem of Cohen and\nCooperstein \\cite{cc}, see section $2.4$ also, $D_{\\mathcal A}(S)$ is a space of type $D_{6,6}$.\n\n\\medskip\nNext, replace ${\\mathcal A}$ with ${\\mathcal B}$. All of the above arguments can be repeated and we can conclude that $D_{\\mathcal B}(S)$ is a subspace of $\\Gamma$ which is of type $D_{6,6}$. Also, every symplecton $S \\in {\\mathcal S}$ is contained in exactly one element from the family ${\\mathcal D}_{\\mathcal B}$.\n\\end{proof}\n\n\\section{The sheaf theoretic characterization}\n\nIn this Section, we combine Ronan-Brouwer-Cohen sheaf theory with Tits' Local Approach Theorem and prove the following:\n\n\\begin{thm}Let ${\\widetilde \\Gamma}= ( {\\widetilde {\\mathcal P}}, {\\widetilde {\\mathcal L}})$ be a parapolar space which is locally of type $A_{7, 4}$. Let $I = \\lbrace 1, \\ldots 8 \\rbrace, \\; J = \\lbrace 1, 8 \\rbrace$ and $K = I \\setminus J$. Assume that ${\\widetilde \\Gamma}$ satisfies the Weak Hexagon Axiom. Then there is a residually connected $J$-locally truncated diagram geometry $\\Gamma$ which belongs to the diagram:\\\\\n\\begin{picture}(1000, 50)(0,0)\n\\put(0,25){$\\mathcal{Y}$}\n\\put(60,25){$\\qed$}\n\\put(75,15){\\scriptsize 1}\n\\put(80,29){\\line(1,0){32}}\n\\put(112,26){$\\circ$}\n\\put(113,15){\\scriptsize 2}\n\\put(113,36){\\scriptsize ${\\mathcal D}_{\\mathcal B}$}\n\\put(118,29){\\line(1,0){30}}\n\\put(148,26){$\\circ$}\n\\put(150,36){\\scriptsize ${\\mathcal B}$}\n\\put(150,15){\\scriptsize 3}\n\\put(153,29){\\line(1,0){30}}\n\\put(183,26){$\\circ$}\n\\put(190,36){\\scriptsize ${\\mathcal L}$}\n\\put(190,15){\\scriptsize 5}\n\\put(186,-3){\\line(0,1){30}}\n\\put(183,-9){$\\circ$}\n\\put(171,-10){\\scriptsize ${\\mathcal P}$}\n\\put(190,-10){\\scriptsize 4}\n\\put(188,29){\\line(1,0){30}}\n\\put(218,26){$\\circ$}\n\\put(220,36){\\scriptsize ${\\mathcal A}$}\n\\put(220,15){\\scriptsize 6}\n\\put(223,29){\\line(1,0){30}}\n\\put(253,26){$\\circ$}\n\\put(253,36){\\scriptsize ${\\mathcal D}_{\\mathcal A}$}\n\\put(255,15){\\scriptsize 7}\n\\put(258,29){\\line(1,0){30}}\n\\put(276,25){$\\qed$}\n\\put(290,15){\\scriptsize 8}\n\\end{picture}\n\n\\vspace{.4cm}\nwhose universal $2$-cover is the truncation of a building. Therefore, ${\\widetilde \\Gamma}$ is the homomorphic image of a truncated building, also.\n\\end{thm}\n\n\\begin{proof}Let us assume that ${\\widetilde \\Gamma}= ( {\\widetilde {\\mathcal P}}, {\\widetilde {\\mathcal L}})$ is a parapolar space, locally of type $A_{7,4}$ and which satisfies {\\it(WHA)}. Construct the space {$\\Gamma = ({\\mathcal P}, {\\mathcal L})$} according to the method described in Section $3$. Then $\\Gamma$ is also locally of type $A_{7,4}$ and satisfies {\\it(WHA)}. Furthermore, $\\Gamma$ has two classes of maximal singular subspaces ${\\mathcal A}$ and ${\\mathcal B}$, whose elements are projective spaces of rank $5$. It was proved in Section $4$ that in $\\Gamma$ there are two more families of subspaces, denoted ${\\mathcal D}_{\\mathcal A}$ and ${\\mathcal D}_{\\mathcal A}$, whose elements are of type $D_{6,6}$. Define incidence as follows:\n\\begin{itemize}\n\\item[(i).]An object in $\\mathcal{A}$ is incident with an object in $\\mathcal{B}$ if they intersect at a plane.\n\\item[(ii).] An object in $\\mathcal{B}$ (or $\\mathcal{A}$) is incident with an object from $\\mathcal{D}_{\\mathcal A}$ ($\\mathcal{D}_{\\mathcal B}$ respectively) if they intersect at a singular subspace of rank $3$.\n\\item[(iii).] An object from $\\mathcal{D}_{\\mathcal A}$ is incident with an object from $\\mathcal{D}_{\\mathcal B}$ if they have a symplecton in common.\n\\item[(iv).] Inclusion for all the remaining cases.\n\\end{itemize}\n\n\\medskip\nHence, the space $\\Gamma$ can be enriched to a rank $6$ geometry $\\Gamma = (\\mathcal{P, L, A, B}, {\\mathcal D}_{\\mathcal A}, {\\mathcal D}_{\\mathcal B})$. It follows from Theorem $3.1$ that $\\Gamma$ is a covering of ${\\widetilde \\Gamma}$. Furthermore, $\\Gamma$ is a geometry over $K$ which is $J$-locally truncated with respect to the diagram $\\mathcal{Y}$.\n\n\\medskip\nIn the sequel we will use the terminology given in Sections $2.1$-$2.3$; the definition of a sheaf over a locally truncated geometry was given in Section $2.4$.\n\n\\medskip\nNext we construct a sheaf $\\Sigma$ over the collection $\\mathcal{F}$ of flags of rank $1$ and $2$ of $\\Gamma$. It is a consequence of a result of Ellard and Shult \\cite{els}, see also \\cite[Section 6]{on1}, that it suffices to work with the collection $\\mathcal{F}$ of flags, instead of the full family of nonempty flags of $\\Gamma$.\n\n\\medskip\n${\\mathfrak 1})$. For $a \\in {}^{2}\\Gamma$, an object of type $2$ in $\\Gamma$, define $\\Sigma ^R(a)$ to be the geometry of $\\lbrace 8 \\rbrace$-locally truncated type belonging to the diagram $\\mathcal{Y}_{I \\setminus \\lbrace 1,2 \\rbrace}$ and which satisfies the property that $\\res {\\Gamma}{a} =\\;  _{\\lbrace 8 \\rbrace}\\Sigma ^R(a)$. This is well defined since $\\Sigma ^R (a)$ is of $\\lbrace 8 \\rbrace$-locally truncated type $D_{6,6}$, which, up to the relabeling of the nodes is unique. In $\\Sigma\n^R (a)$ the objects of types in $K$ are the same as in $\\Gamma$ with the corresponding incidence. The objects of type $\\lbrace 8 \\rbrace$ are collections of objects in $\\Gamma$ with their flags, which are incident with a given object of type in $K$; the incidence between objects of type $\\lbrace 8 \\rbrace$ is given by symmetrized containment.\n\n\\medskip\n${\\mathfrak 2})$. For $b \\in \\; ^{4}\\Gamma$ define $\\Sigma(b)$ to be such that $\\res{\\Gamma}{ b} =\\;_J \\Sigma(b)$  Observe that $\\res{\\Gamma}{b}$ is\nthe $J$-truncation of a geometry belonging to the diagram $D_{I\n\\setminus \\lbrace 4 \\rbrace}$ of type $A_{7,4}$. But this is\nuniquely determined by its truncation, \\cite[Theorem 1]{loc}, and thus $\\Sigma(b)$ is unambiguously defined. There are two types of objects in $\\Sigma\n(b)$: those inherited from $\\Gamma$, with their incidence and the objects with types\nin $J$, which are defined as in ${\\mathfrak 1})$.\n\n\\medskip\n${\\mathfrak 3})$. Let now $l \\in \\lbrace 3, \\dots, 7 \\rbrace$ and let $x_l \\in \\;\n^{l}\\Gamma$. Denote by $F = \\lbrace a,x \\rbrace $ a $\\lbrace 2, l-1 \\rbrace $-flag in $\\res{\\Gamma}{x_l}$. For $l=3$, $F = \\lbrace a \\rbrace$ is just an object of type $2$. Define recursively: $\\Sigma ^R (x_l) = \\Sigma ^R (a, x_l) := \\res{\\Sigma ^R (a,x)}{x_l}$. Let $F'$ be another flag of type\n$\\lbrace 2, l-1 \\rbrace$ in $\\res{\\Gamma }{x_l}$ which is $i$-adjacent to $F$ in $C$, the chamber system associated to $^{\\lbrace 2, l-1 \\rbrace}\\res {\\Gamma} {x_l}$ with  $i \\in \\lbrace 2, l-1 \\rbrace$. Then: $\\res{\\Sigma ^R (F)}{x_l} = \\res{\\Sigma ^R (F \\cap F')} {(F \\setminus F') \\cup \\lbrace x_l \\rbrace } = \\res{\\Sigma ^R (F') }{x_l}$ which proves the well-definedness in this case. If $F$ and $F'$ are not\n$i$-adjacent, since the chamber system $C$ is connected, there is a\ngallery from $F$ to $F'$, and by repeated applications of the above\nargument we get the result.\n\n\\medskip\n${\\mathfrak 4})$. Let $f \\in \\; ^{7} \\Gamma$. Define $\\Sigma ^L (f)$ to be the\ngeometry of $\\lbrace 1 \\rbrace$-locally truncated type, with diagram $\\mathcal{Y}_{I \\setminus \\lbrace 7,8 \\rbrace}$, and such that $\\res{\\Gamma}{ f} = \\;\n_{\\{1\\}} \\Sigma ^L (f)$; this is uniquely defined since it has diagram of type $D_{6,6}$. The objects in $\\Sigma ^L(f)$ and their incidence are defined as in the first step.\n\n\\medskip\n${\\mathfrak 5})$. Let $l \\in \\lbrace 6, \\ldots 2 \\}$, taken in this order. Let $x_l \\in\n\\; ^{l}\\Gamma$ and $F = \\lbrace x, f \\rbrace$ a $\\lbrace l+1, 7 \\rbrace$-flag in $\\res{\\Gamma}{x_l}$. Recursively define: $\\Sigma ^L (x_l) = \\Sigma ^L (x_l, f) := \\res{\\Sigma ^L(x, f)} {x_l}$. The proof of well-definedness is similar to the one given in the third step. The objects in $\\Sigma ^L(x_l)$ and their incidence are defined using the method from the first step.\n\n\\medskip\nThe sheaf values over the rank $1$ flags of $\\Gamma$ can be written as\nfollows:\n\\begin{itemize}\n\\item[ (i).] \\quad $\\Sigma (x) := \\Sigma ^L (x) \\oplus \\Sigma ^R (x),\\;   \\text{for\nany object}\\ x\\  \\text{of type in}\\  K \\setminus \\lbrace 4 \\rbrace$;\n\\item[(ii).] \\quad $\\Sigma (b) \\; \\text{for any object}\\ b \\in \\; ^{4}\\Gamma$.\n\\end{itemize}\n\nNext, we construct the sheaf the sheaf over the rank $2$ flags of $\\Gamma$ and we check the compatibility condition: $\\res{\\Sigma(x)}{y} = \\Sigma(x,y) = \\res{\\Sigma(y)}{x}$, where $\\lbrace x, y \\rbrace$ is a nonempty rank $2$ flag in $\\Gamma$.\n\n\\medskip\n${\\mathfrak 6})$. For $\\lbrace x_i, x_j \\rbrace$, a rank $2$ flag of type $\\lbrace i, j\\rbrace$ in $\\Gamma$ with $i, j \\in K \\setminus \\lbrace 4 \\rbrace$\nand $i<j$, set\n$$\\Sigma (x_i, x_j) := \\Sigma ^{L} (x_i) \\oplus \\;\n^{(ij)} \\Gamma \\oplus \\Sigma ^{R}(x_j)$$\nwhere $^{(ij)}\\Gamma$ denotes the\ntruncation of $\\Gamma$ to those objects of type $l \\in K$ with $i<l<j$. Since\n\\begin{equation*}\n\\begin{split}\n&\\res{\\Sigma(x_i)}{x_j} = \\res {\\Sigma ^L (x_i)}{x_j} \\oplus \\res {\\Sigma ^R(x_i)}{x_j} = \\Sigma ^L (x_i) \\oplus \\; ^{(ij)}\\Gamma \\oplus \\Sigma ^R (x_j)\\\\\n&= \\res {\\Sigma ^L (x_j)}{x_i} \\oplus \\Sigma ^R (x_j) = \\res{\\Sigma (x_j)}{x_i}\n\\end{split}\n\\end{equation*}\nit follows that the compatibility condition is satisfied in this case.\n\n\\medskip\n${\\mathfrak 7})$. There are two remaining cases, as follows:\n\\begin{itemize}\n\\item[a).] For $\\lbrace a,b \\rbrace$, a $\\{ 3,4\\}$-flag, define $\\Sigma (a,b) := \\Sigma ^L(a) \\oplus \\Sigma ^R (b)$.\n\\item[b).] For a flag $ \\lbrace b, x_j \\rbrace$ of type $\\{4, j \\}$, with $j \\in \\{5,6,7\\}$ set $\\Sigma(b, x_j) := \\res{\\Sigma ^L(x_j)}{b} \\oplus \\Sigma ^R(x_j)$.\n\\end{itemize}\nThe compatibility condition can be checked now, the proof is similar to the one given in the previous step.\n\n\\medskip\nTherefore $\\Sigma$ is defined for all rank $1$ and rank $2$ flags of\n$\\Gamma$. Now using the aforementioned Lemma, see \\cite[Section 6]{on1}, of Ellard and Shult, we can extend the sheaf $\\Sigma$ to a sheaf (denoted the same) over all nonempty flags of $\\Gamma$.\n\n\\medskip\nThe sheaf $\\Sigma$ is residually connected; this follows from the fact that all sheaf values at rank $1$ and $2$ flags of $\\Gamma$ are $J$-truncated buildings or products of $J$-truncated buildings.\n\n\\medskip\nAccording to a result of Brouwer and Cohen \\cite{loc}, there exists a canonically defined chamber system $\\mathcal{C}(\\Sigma)$ over $I$, which is residually connected and belongs to diagram $\\mathcal{Y}$. It satisfies the property that $_J {\\mathcal C}(\\Sigma) \\; \\simeq \\; {\\mathcal C}(\\Gamma)$, an isomorphism of chamber systems over $K$, with the term on the right being the chamber system corresponding to the geometry $\\Gamma$. Moreover, all rank $3$ residues are covered by truncations of buildings. Therefore by Tits' Local Approach Theorem \\cite{tits}, see also Theorem $2.1$,  the universal $2$-cover of ${\\mathcal C}(\\Sigma)$ is the chamber system ${\\mathcal B}$ of a truncated building belonging to diagram $\\mathcal{Y}$.\n\n\\medskip\nThere is a $J$-locally truncated building geometry $\\widetilde \\Delta $ which belongs to the diagram $\\mathcal{Y}$ and which is the image under the functor ${\\mathbf G}: {\\mathcal\nChamb}_I \\rightarrow {\\mathcal Geom}_I$ of $\\mathcal{B}$; see Section 2.3. The chamber system ${\\mathcal C}(\\Sigma)$ also has a corresponding geometry which we shall denote $\\Delta$. Since the chamber system ${\\mathcal C}(\\Sigma)$ is\nresidually connected, it follows that $\\Delta$ is also residually connected. The $2$-cover map of chamber systems ${\\mathcal B} \\rightarrow \\mathcal{C}(\\Sigma)$ functorially induces a covering of geometries $\\widetilde{\\Delta} \\rightarrow \\Delta$. The fact that $\\widetilde{\\Delta}$ is the universal cover of $\\Delta$ follows from the fact that $\\mathcal{B}$ is the universal cover of $\\mathcal{C}(\\Sigma)$. Moreover the geometry $\\Delta$ has the property that $ _J \\Delta \\simeq \\Gamma$; for details see \\cite{on1}. Combining the previous results with Theorem $3.1$, it follows that $\\widetilde{\\Gamma}$ is the homomorphic image of a building.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro}\n\nCataclysmic Variables (CVs) are binary systems where a late-type main-sequence donor star transfers mass onto a primary white dwarf (WD). According to standard evolutionary theory, loss of angular momentum drives CVs to initially evolve from longer to shorter orbital periods. All systems are predicted to reach a minimum orbital period roughly when the donor stops its hydrogen burning and becomes degenerate. The observed minimum period is found at $\\approx$ 83 minutes~\\cite{2009MNRAS.397.2170G}. At about the point when the donor reaches this evolutionary state, the thermal time scale becomes longer than the mass-transfer timescale, and the donor cannot shrink fast enough in response to continued mass loss. As a result, the brown-dwarf donor will expand, and the system will move towards longer orbital periods. \n\nStandard evolutionary theory also predicts that about 70\\% of all CVs should have passed their minimum period and have sub-stellar donors (\\citealp{1993A&A...271..149K}; \\citealp{1997MNRAS.287..929H}). However, few such systems have been successfully confirmed, and until recently, almost no CVs containing donors with masses below the hydrogen-burning limit, had been found. Systems that have passed their minimum period (post-bounce systems) should be very faint and are therefore difficult to find, in general. Nevertheless, in the course of the Sloan Digital Sky Survey (SDSS), a few post-bounce CVs were discovered (\\citealp{2008MNRAS.388.1582L}), among them the eclipsing system SDSS J1507+52 (hereafter J1507). \n\nThis particular system was quickly recognised to be odd due to its short orbital period of about 67 minutes (\\citealp{2005AJ....129.2386S}), which is well below the minimum period characteristic of normal CVs. A few other systems are occasionally found to have periods below the usual minimum period, but, in general, these systems show evidence for abnormally hot and bright donor stars in their optical spectra, suggesting that their secondaries are nuclear-evolved objects. This is not the case in J1507, whose secondary is not visible at all in optical spectroscopy (\\citealp{2005AJ....129.2386S}; \\citealp{2007MNRAS.381..827L}). Together with its short orbital period, this indicates a system with a faint disc, a low accretion rate and a relatively unevolved, low-mass donor.\n\n\\cite{2007MNRAS.381..827L} performed eclipse analysis of J1507 and found a donor mass of 0.056 $\\pm$ 0.001 M$_{\\odot}$, clearly below the hydrogen-burning limit. However, because of its anomalously short orbital period, the system is not consistent with the standard mass-period relation for CVs (\\citealp{2006MNRAS.373..484K}). In line with this,~\\cite{2007MNRAS.381..827L} found the donor radius to be smaller than predicted by standard CV evolution sequences. They therefore suggested that the donor might be unusually young, i.e. that J1507 might be a CV that formed recently from a previously detached WD -- brown dwarf close binary system. A young brown dwarf has a higher density, and therefore a smaller radius, since the donor has not yet had a chance to expand in response to mass loss. This would explain the short orbital period found in J1507, since according to the period-density relation, a higher density and subsequently a smaller radius, implies a shorter orbital period. The list of photometrically inferred system parameters provided by~\\cite{2007MNRAS.381..827L} also includes the effective temperature of the white dwarf, which they estimate to be $T_{\\text{eff}}$ = 11000 $\\pm$ 500 K. \n\nConcurrently,~\\cite{2008PASP..120..510P} showed that the system has an unusually high space velocity, similar to the velocities of stars in the Galactic halo. A typical star of 1 M$_{\\odot}$ in the Galactic disc has a space velocity below 50 km\\,s$^{-1}$, while J1507 has a velocity of about 167 km\\,s$^{-1}$. If J1507 is a member of the Galactic halo, its donor will be a Population II object with low metallicity. Due to their lower atmospheric opacity, such objects have significantly smaller radii than their solar metallicity Population I counterparts. Membership of the Galactic halo would therefore also account for J1507's small donor radius and short orbital period. However, Population II stars only represent about 0.5 \\% of the stars in the Solar neighbourhood, which would make J1507 a rare system.~\\cite{2008PASP..120..510P} estimated a slightly higher effective temperature of the WD in J1507, $T_{\\text{eff}}$ = 11500 $\\pm$ 700 K. They also found the system to exhibit multi-periodic variability on time-scales of minutes, which they interpreted as non-radial pulsations originating from the primary white dwarf.\n\n\nBoth hypothesis imply that J1507 is an interesting and important system that can shed light on poorly understood aspects of CV evolution. However, since the models are very different, it is clearly important to determine which -- if either -- of them is correct. As noted above, eclipse-based estimates for the effective temperature of the WD suggested $T_{\\text{eff}} \\simeq$ 10000 K -- 12000 K. At these temperatures, the ultraviolet (UV) spectrum of a WD accreting metal-rich material would be expected to show strong absorption features due to Fe II and Fe III (c.f. \\citealp{1994ApJ...426..294H}). We have therefore obtained far- and near-UV spectroscopy of J1507 with COS and STIS onboard the Hubble Space Telescope, with the aim of measuring the metallicity of the system and establishing its status as either a Pop I (disk) or a Pop II (halo) object.\n\n\n\\begin{figure}\n\\includegraphics[width=9cm]{cos_light.eps}\n\\caption{The Gaussian smoothed and flux-calibrated light curve obtained by COS\/HST, constructed from the monochromatic flux averaged over the region 1425\\,\\AA\\, -- 1900\\,\\AA. Bottom panel shows the phase-folded light curve in the vicinity of the eclipse.} \n\\label{fig:light}\n\\end{figure}\n\n\n\n\n\\section{Observations and Reduction}\n\nFar- and near-UV observations of J1507 were performed in February 2010, using the HST. The observations were obtained in the so-called time-tag mode. In this mode, each photon event is tagged with a corresponding event time, along with their wavelength and spatial position on the detector. This allows us to rebin the data over any time interval and wavelength region. \n\nFar-UV observations were carried out with the Cosmic Origins Spectrograph (COS), for a total exposure time of about 3 hours, covering 3 eclipses in three successive HST orbits. The maximum time resolution of the COS TIME-TAG mode is 32 ms. The G140L grating was used, which gave a spectral range of 1230\\,\\AA\\ -- 2378\\,\\AA\\,and a spectral resolution of 0.5\\,\\AA.  \n\nNear-UV observations were obtained by the Space Telescope Imaging Spectrograph (STIS), for a total exposure time of about 1.4 hours, at a maximum time resolution 125 $\\mu$s. The G230L grating was used, resulting in a spectral range of 1650\\,\\AA\\ -- 3150\\,\\AA\\,and a spectral resolution of 3.16\\,\\AA. The data set consists of 3 sub-exposures (of length, 1600 s, 1918 s and 1708 s). No near-UV data was obtained during the eclipses. Also, gaps in the data are cased by interruptions coming from Earth occultation. \n\nAll data were reduced and calibrated using the \\textsc{Pyraf} package \\textsc{Stsdas}, provided by STScI (the Space Telescope Science Institute)\\footnote{\\textsc{Stsdas} and \\textsc{Pyraf} are products of the Space Telescope Science Institute, which is operated by AURA for NASA.}.\n\n\n\n\n\\begin{figure*}\n\\includegraphics[width=11.0cm]{spec.eps}\n\\caption{The two top panels show the mean out-of-eclipse spectrum for both the far (left) and near (right) UV regions. A model constructed from parameters obtained from the far UV-region is plotted in red (see Table~\\ref{pars}). A model corresponding to values from Littlefair et al. (2007), $T_{\\text{eff}}$=11000 K and $\\log g$=8.5, is plotted in blue. Underneath each wavelength region is a plot showing the residuals. The dips at (1460, 1530, 1600 and 1670)\\,\\AA\\, coincide with regions of bad pixels. The 1-$\\sigma$ flux errors are plotted in green.} \n\\label{fig:spec}\n\\end{figure*}\n\n\n\\section{Analysis and Results}\n\nFigure~\\ref{fig:light} shows the Gaussian smoothed and flux-calibrated light curve from the far-UV data. This light curve was constructed by dividing the larger exposures into one-second bins, and by average the fluxes over the wavelength range, 1425\\,\\AA\\, -- 1900\\,\\AA. The light curve shows sharp, square-shaped eclipses, close to zero flux at mid-eclipse, and a very flat out-of-eclipse continuum, strongly suggesting that the UV flux is dominated by emission from (or very close to) the accreting WD. The folded light curve constructed from the combined data for all three HST orbits is presented in the bottom panel. All further analysis was carried out on the unsmoothed data, excluding the eclipses. Ground-based optical observations were performed at the same time as the HST observations, yielding a magnitude for J1507 of $g = 18.44 \\pm 0.02$. This is consistent with SDSS, showing that the system was in quiescence during the HST observations. \n\nThe two top panels of Figure~\\ref{fig:spec} show the mean out-of eclipse spectrum (in black), for both the near- and far-UV. The reduction was performed separately for the COS and STIS data, and the mean spectrum for each region were joined at 1900 \\AA\\,without overlap for a total range of 1268\\,\\AA\\, -- 3150\\,\\AA. The COS data was rebinned slightly, to 0.2\\,\\AA\/pix, to increase the S\/N and provide sufficient sampling of the G140L grating's spectral resolution. Different dispersions were retained for the the far- and near-UV regions (0.2 \\,\\AA\/pix and 1.58 \\,\\AA\/pix, respectively). The overall UV spectrum for J1507 is relatively sparse in spectral lines. We identify CII at 1335, SiII at 1526.7 HeII at 1640.5, AlII at 1671 and MgII at 2800 \\,\\AA. The strong emission feature near 1300 \\AA\\, is almost certainly due to geocoronal emission in the O I 1304 \\AA\\, line that could not be cleanly removed by the pipeline background subtraction.\n\n\n\\subsection{Spectral Modelling}\n\nA model grid spanning the four key atmospheric parameters, effective temperature ($T_{\\text{eff}}$), surface gravity ($\\log g$), metallicity ($[Fe\/H]$) and rotational velocity ($v \\sin i$), was constructed in order to fit the far- and near-UV spectrum. Atmospheric structures were calculated with \\textsc{Tlusty}, with the spectral synthesis being done with \\textsc{Synspec} (\\citealp{1988CoPhC..52..103H}; \\citealp{1995ApJ...439..875H}). The grid consisted of; 12500 K $\\mathrel{ \\rlap{\\raise.5ex\\hbox{$<$} T_{\\text{eff}} \\mathrel{ \\rlap{\\raise.5ex\\hbox{$<$}$ 20000 K in steps of 500 K, 7.5  $\\mathrel{ \\rlap{\\raise.5ex\\hbox{$<$} \\log g \\mathrel{ \\rlap{\\raise.5ex\\hbox{$<$} 9.5$ in steps of 0.25, -2.0 $\\mathrel{ \\rlap{\\raise.5ex\\hbox{$<$} [Fe\/H]  \\mathrel{ \\rlap{\\raise.5ex\\hbox{$<$}$ 0.0 in steps of 0.25, and 0 km\\,s$^{-1} \\mathrel{ \\rlap{\\raise.5ex\\hbox{$<$} v \\sin i \\mathrel{ \\rlap{\\raise.5ex\\hbox{$<$}$ 1000 km\\,s$^{-1}$ in steps of 50 km\\,s$^{-1}$. Models at intermediate parameter values were constructed by linear interpolation on this grid. Pixels with non-zero data quality flags set were excluded from our fits, as well as the regions around the spectral lines, OI at 1304\\,\\AA\\, CII at 1335\\,\\AA\\,and the quasi-molecular H feature at $\\sim$ 1400\\,\\AA. \n\nBefore our models could be compared to the observed spectrum, they were convolved with a Gaussian filter to the spectral resolution appropriate to the COS (< 1900\\,\\AA) and STIS (> 1900\\,\\AA) data, and linearly interpolated onto the observational wavelength scale. In general, our model fits did represent the main features of the data quite well, but the formal $\\chi^{2}$ was somewhat high. In order to obtain more realistic formal parameter errors, we therefore added an intrinsic dispersion term to the flux errors in such a way that $\\chi^{2}_{\\nu}$ = 1. This dispersion corresponds to about 1\\,\\% of the flux at 1900\\,\\AA. In order to explore the systematic uncertainties affecting our fits, we tried fitting many different wavelength regions, such as the far-UV (COS), near-UV (STIS), regions with a high line concentration and also the whole wavelength range spanning both the far- and near-UV.   \n\n\n\\begin{figure*}\n\\includegraphics[width=11.0cm]{met_vel.eps}\n\\caption{The top panel shows the mean out-of-eclipse spectrum for the far UV region, where models at three different $[Fe\/H]$, 0 (green), -1.2 (red) and -2 (blue), are plotted on top. The bottom left panel shows a  zoom of the region around the Si II, where the coloured lines correspond to the same values of the metallicity as in the top panel. In the bottom right panel, the colours correspond to $v \\sin i$ at 0 km\\,$s^{-1}$ (green), 180 km\\,$s^{-1}$ (red), 500 km\\,$s^{-1}$ (blue).} \n\\label{fig:met}\n\\end{figure*}\n\n\n\\subsection{Temperature and Metallicity}\n\nA fit to the far UV-region 1268\\,\\AA\\, -- 1900\\,\\AA\\,\\,yields $T_{\\text{eff}} = 14200 \\pm 50$ K, $\\log g = 8.2 \\pm 0.04, [Fe\/H] = -1.2 \\pm 0.05 $ and $v \\sin i = 180 \\pm 20$ km\\,s$^{-1}$. These fit parameters provide a good approximation also for the near-UV spectrum out to 2800\\,\\AA\\, where the spectrum may begin to show disc features. We adopt these parameters as our best-fit parameters. The quoted errors are formal errors from the $\\chi^{2}$ fitting. Table~\\ref{pars} presents the best parameters along with their uncertainties. These uncertainties are larger than the formal errors, because they include our best estimates of the systematic uncertainties associated with different choices of fitting windows. However, there are several reasons for adopting the parameters inferred from the fit to the COS far-UV data set as our best-bet estimates. This region includes almost all line features, and is also most sensitive to changes in $T_{\\text{eff}}$ and $\\log g$. Also, the far UV-region is more likely to represent pure light from the WD, while towards redder wavelengths, the spectrum could be affected by the disc and the bright spot. Finally, fitting only the COS data removes any possibility that a mis-match in the flux calibration of COS and STIS could affect our results.\n\nThe two top panels in Figure~\\ref{fig:spec} show the best-fit model in red, along with the mean out-of-eclipse spectrum for J1507. The corresponding residuals between model and data are plotted underneath each of the COS and STIS spectral regions, together with the 1-$\\sigma$ error range marked in green. The dips in the residuals at (1460, 1530, 1600 and 1670)\\,\\AA\\,, are caused by bad pixels in the data and these regions were excluded from our fits. As a comparison, a model showing the parameters by~\\cite{2007MNRAS.381..827L}, $T_{\\text{eff}}$ = 11000 K and $\\log g$ = 8.5, is plotted in blue. Clearly, this temperature is far too low to match the data.\n \n The top panel in Figure~\\ref{fig:met}, shows the mean spectrum plotted together with models at three different values of $[Fe\/H]$, 0 (green), -1.2 (red) and -2 (blue). It is clear that a model of solar metallicity (green line) does not match the data and overestimates both the number and strength of visible absorption lines in the spectrum. The left panel shows a zoom of the region around the Si II line (the coloured lines correspond to the same values of the metallicity as given for the top panel). To the right is a plot showing the dependence of the model on $v \\sin i$, where the coloured lines in green, red and blue represent rotational velocities of 0 km\\,s$^{-1}$, 180 km\\,s$^{-1}$ and 500 km\\,s$^{-1}$, respectively. In all panels, the red lines show the best fitted parameters. The metallicity and rotational velocity affect both the line depths and widths. Therefore, as a check, we fixed $T_{\\text{eff}}$ and $\\log g$ to our best values, and made model fits to smaller wavelength regions specifically chosen because they contain strong and\/or many spectral lines, with $v \\sin i$ and $[Fe\/H]$ as free parameters. We find consistent values of $v \\sin i$ and $[Fe\/H]$, independent of the wavelength region we perform the fits on.   \n  \nWe compared the metallicity found for J1507 with the metallicity distribution for Galactic halo stars at the same space velocity. ~\\cite{2008PASP..120..510P} calculated a space velocity for J1507 of 167 km\\,s$^{-1}$, which can be broken into the Galactic velocity components U, V and W, where $\\sqrt{U^{2} + W^{2}}$ =139 km\\,s$^{-1}$ for J1507.~\\cite{1996AJ....112..668C} presented a study of the metallicity distribution for the Galactic halo stars. They find that for stars with velocities in the range 100 <  $\\sqrt{U^{2} + W^{2}}$ < 140, the metallicity distribution peaks around -0.7, but has a long tail towards metallicities as low as -1.5 (see Figure 6 in their paper). This is in good agreement with our metallicity of -1.2. Therefore, we conclude that J1507 is most likely a Galactic halo CV.\n\n\n\\subsection{Possible Extinction Effects}\n\nUp to now, we have assumed that the observed far-UV spectrum of SDSS J1507 is unaffected by interstellar extinction. Based on the absence of the well-known 2175\\,\\AA\\,absorption feature in the data, we can set E(B-V) $\\leq$ 0.05 as a fairly conservative limit on the amount of reddening that may be present. In order to test the effect of extinction at this level on our conclusions, we carried out additional model fits after dereddening the data by E(B-V) = 0.05. We find that none of our inferred parameters would change significantly if extinction at this level were present.\n\n\n\n\\subsection{Distance Estimates}\n\nThe theoretical relationship between observed flux ($F$) and the Eddington flux ($H$) provided by the \\textsc{Synspec} models can be used to estimate the distance ($d$) towards J1507. More specifically, $F =  4 \\pi R_{\\text{wd}}^{2}H\/\\,d^{2}$, where $R_{\\text{wd}}$ is the WD radius. This means that the normalisation factor needed to optimally match a model spectrum to the data is a direct measure of $R_{\\text{wd}}^{2}\/d^{2}$. WDs also follow the well-known mass-radius relation (with only a weak temperature sensitivity), so the surface gravity of the model, $g = G\\,M_{\\text{wd}}\/R_{\\text{wd}}^{2}$, uniquely fixes $R_{\\text{wd}}$. Thus for a WD model with given $T_{\\text{eff}}$ and $\\log g$, the normalisation factor required to fit the data is a unique measure of distance. \n\nIn practice, we estimated the distance by first fitting a linear function to the relationship between $M_{\\text{wd}}$ and $\\log g$ in Pierre Bergeron's WD cooling models\\footnote{Cooling models by Pierre Bergeron are found at: \\newline http:\/\/www.astro.umontreal.ca\/$\\sim$bergeron\/CoolingModels\/}, at $T_{\\text{eff}}$ = 14500 K. We then use this function to estimate $M_{\\text{wd}}$, and hence $R_{\\text{wd}}$, for given $\\log g$. Rough values of the WD mass and radius are found at $M_{\\text{wd}} = 0.75 \\pm 0.15$ M$_{\\odot}$ and $R_{\\text{wd}} = 0.011 \\pm 0.002$ R$_{\\odot}$. This is consistent with both ~\\cite{2007MNRAS.381..827L} and ~\\cite{2008PASP..120..510P}. The normalisation constant of the model combined with $R_{\\text{wd}}$, yield the corresponding distance estimate. We find a distance towards J1507 of $d = 250 \\pm 50$ pc (the effect of reddening is allowed for in the quoted error).\n\n\n\\begin{table}\n  \\centering\n  \\begin{minipage}{70mm}\n  \\caption{Best-fit parameters for J1507 obtained from model fitting to the far-UV region. The errors are defined as the whole range for which we find good fits, irrespective of wavelength region. Formal 1-$\\sigma$ errors are given in parenthesis.}\n \\begin{tabular}{ll}\n \\hline\n \\hline\n$T_{\\text{eff}}$: & 14200 $\\pm$ 500 (50) K  \\\\\n$\\log g$: & 8.2 $\\pm$ 0.3  (0.04)\\\\\n$[Fe\/H]$: & -1.2 $\\pm$ 0.2 (0.05)   \\\\\n$v \\sin i$: & 180 $\\pm$  20 (20) km\\,s$^{-1}$\\\\\n\\hline\n\\hline\n\\label{pars} \n\\end{tabular}\n\\end{minipage}\n\\end{table}\n\n\n\\section{Discussion and Summary}\n\nWe have obtained HST observations  in the UV spectral-range of the cataclysmic variable J1507, with the aim of measuring its metallicity to determine whether or not the system is a member of the Galactic halo. \n\nBy comparing the observed spectrum to synthetic spectra described by the four parameters, $T_{\\text{eff}}$, $\\log g$, $[Fe\/H]$ and $v \\sin i$, a best fit is found at $T_{\\text{eff}}$ = $14200 \\pm 500$ K. This value of $T_{\\text{eff}}$ is based on the assumption that the WD is the only component contributing to the UV flux in J1507. If a hot, optically thick boundary layer would be present, it would bias our estimate of $T_{\\text{eff}}$. However, the single-temperature WD models presented here seem to provide a good fit to the data, and we expect that the boundary layer in a low $\\dot{M}$-system (such as J1507) would be optically thin and thus not contribute significantly to the UV flux. \n\nOur best model fit give a $T_{\\text{eff}}$ that is much higher than previous estimates, which implies that the accretion rate is higher than previously suggested. More specifically, since $\\dot{M} \\propto T^{4}$, the 30\\% increase in the estimated  $T_{\\text{eff}}$ corresponds to an increase in $\\dot{M}$ by almost a factor of 3 (\\citealp{2003ApJ...596L.227T}). This may have significant implications for the evolution of this system, including the question of whether gravitational radiation alone is sufficient to drive this accretion rate (see \\citealp{2009ApJ...693.1007T}; \\citealp{2011arXiv1102.2440K}).\n\nAt the higher $T_{\\text{eff}}$ we infer, Fe II and III are no longer dominant contributors to the atmospheric UV opacity, making it more difficult than expected to measure the metallicity of the system. Nevertheless, model fits to the data clearly favour a significantly sub-solar metallicity, $[Fe\/H] = -1.2 \\pm 0.2$, comparable to the typical metallicity found for halo stars at the same high space velocity as J1507.\n\n\\cite{2008PASP..120..510P} found pulsations in J1507, which they identified as non-radial pulsations originating from the primary WD. Non-accreting CVs show pulsations within a well-defined temperature range, the so-called, instability strip at 10900 K -- 12200 K (\\citealp{2006AJ....132..831G}). We find that the effective temperature in J1507 is well above the instability strip for non-accreting pulsating WDs. However, it is already becoming clear that non-radially pulsating WDs in CVs are found across a wider temperature range, often towards higher temperatures (\\citealt{2010ApJ...710...64S}). The amplitudes of these pulsations are expected to vary with wavelength, and a search for the UV counterpart of the optical WD pulsations will be presented in Uthas et al. (in preparation). The fact that non-radial pulsations are present in J1507 is certainly interesting and potentially important since a higher metallicity in the outer envelope of the accretors might be able to push the instability strip towards higher temperatures (\\citealt{2006ApJ...643L.119A}). In this context, it is interesting to note that J1507 has a very low metallicity, but is nevertheless managing to pulsate at an effective temperature of above 14000 K.\n   \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Training Details of our Full Model and Ablations}\n\\label{appsec:models}\n\\subsection{Tracking Models}\nThe hidden size of the instruction encoder is 100, the embedding sizes of action functions and entities are 30. We use dropout with a rate of 0.3 before any non-recurrent fully connected layers \\cite{dropout}. We use the Adam optimizer \\citep{adam} with a learning rate of .001 and decay by a factor of 0.1 if we see no improvement on validation loss over three epochs. We stop training early if the development loss does not decrease for five epochs. The batch size is 64. We use two instruction encoders, one for the entity selector, and one for the action selector. Word embeddings and entity embeddings are initialized with skipgram embeddings \\citep{skipgram1,skipgram2} using a word2vec model trained on the training set. We use a vocabulary size of 7358 for words, and 2996 for entities. Gradients with respect to the coverage loss (Eq.~\\ref{eq:train:usage}) are only backpropagated in steps where no entity is annotated as being selected. To account for the false negatives in the training data due to the heuristic generation of the labels, gradients with respect to the entity selection loss are zeroed when no entity label is present.\n\n\\subsection{Generation Model}\n\\label{app:ssec:gen}\nThe hidden size of the context encoder is 200. The hidden size of the state vector encoder is 200. State vectors have dimensionality 30 (the same as in the neural process network). Dropout of 0.3 is used during training in the decoder. The context and state representations are projected jointly using an element-wise product followed by a linear projection \\cite{hadamard}. Both encoders and the decoder are single layer. The learning rate is 0.0003 initially and is halved every 5 epochs. The model is trained with the Adam optimizer.\n\n\\section{Training Details of Baselines}\n\\label{appsec:baseline}\n\\subsection{Tracking Baselines}\n\\label{app:ssec:tracking}\n\\paragraph{Joint Gated Recurrent Unit}\nThe hidden state of the GRU is 100. We use a dropout with a rate of 0.3 before any non-recurrent fully connected layers. We use the Adam optimizer with a learning rate of .001 and decay by a factor of 0.1 if we see no improvement on validation loss over a single epoch. We stop training early if the development loss does not decrease for five epochs. The batch size is 64. We use encoders, one for the entity selector, and one for the state change predictors. Word embeddings are initialized with skipgram embeddings using a word2vec model trained on the training set. We use a vocabulary size of 7358 for words.\n\n\\paragraph{Recurrent Entity Networks}\nMemory cells are tied to the entities in the document. For a recipe with 12 ingredients, 12 entity cells are initialized. All hyperparameters are the same as the in the bAbI task from \\cite{ren}. The learning rate start at 0.01 and is halved every 25 epochs. Entity cells and word embeddings are 100 dimensional. The encoder is a multiplicative mask initialized the same as in \\cite{ren}. Intermediate supervision from the weak labels is provided to help predict entities. A separate encoder is used for computing the attention over memory cells and the content to write to the memory. Dropout of 0.3 is used in the encoders. The batch size is 64. We use a vocabulary size of 7358 for words, and 2996 for entities.\n\n\\subsection{Generation Baselines}\n\\paragraph{Seq2seq}\n\nThe encoder and decoder are both single-layer GRUs with hidden size 200. We use dropout with probability 0.3 in the decoder. We train with the Adam optimizer starting with a learning rate 0.0003 that is halved every 5 epochs. The encoder is bidirectional. The model is trained to minimize the negative loglikelihood of predicting the next word.\n\n\\paragraph{Attentive Seq2seq}\n\nThe encoder is the same as in the seq2seq baseline. A multiplicative attention between the decoder hidden state and the context vectors is used to compute the attention over the context at every decoder time step. The model is trained with the same learning rate, learning schedule and loss function as the seq2seq baseline.\n\n\\paragraph{EntNet Generator}\n\nThe model is trained in the same way as the NPN generator model in Appendix~\\ref{app:ssec:gen} except that the state representations used as input are produced from by EntNet baseline described in Section~\\ref{ssec:exps:intrinsic} and Appendix~\\ref{app:ssec:tracking}.\n\n\\section{Annotations}\n\\label{appsec:ann}\n\n\\subsection{Annotating State Changes} \n\\label{appssec:annsc}\nWe provide workers with a verb, its definition, an illustrative image of the action, and a set of sentences where the verb is mentioned. Workers are provided a checklist of the six state change types and instructed to identify which of them the verb causes. They are free to identify multiple changes. Seven workers annotate each verb and we assign a state change based on majority vote. Of the set of 384 verbs extracted, only 342 have a state change type identified with them. Of those, 74 entail multiple state change types.\n\n\\subsection{Annotating End States} \n\\label{appssec:annes}\nWe give workers a verb, a state change type, and an example with the verb and ask them to provide an end state for the ingredient the verb is applied to in the example. We then use the answers to manually aggregate a set of end states for each state change type. These end states are used as labels when the model predicting state changes. For example, a \\textsc{location} change might lead to an end state of ``pan,'' ``pot'', or ``oven.'' End states for each state change type are provided in Table~\\ref{table:end_states}.\n\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{l | l}\n        State Change Type & End States \\\\\n        \\hline\n        Temperature & hot; cold; room \\\\\n        Composition & composed; not composed\\\\\n        Cleanliness & clean; dirty; dry\\\\\n        Cookedness & cooked; raw \\\\\n        Shape & molded; hit; deformed; separated \\\\\n        Location & pan, pot, cupboard, screen, scale, \\\\\n        &garbage, 260 more \\\\\n    \\end{tabular}\n    \\caption{End states for each state change type}\n    \\label{table:end_states}\n\\end{table}\n\n\\subsection{Annotating Development and Test Sets} \n\\label{appssec:devtest}\nAnnotators are instructed to note any entities that undergo one of the six state changes in each step, as well as to identify new combinations of ingredients that are created. For example, the sentence ``Cut the tomatoes and add to the onions'' would involve a \\textsc{shape} change for the tomatoes and a combination created from the ``tomatoes'' and ``onions''. In a separate task, three workers are asked to identify the actions performed in every sentence of the development and test set recipes. If an action receives a majority vote that it is performed, it is included in the annotations.\n\n\\section{Additional Results}\n\n\\subsection{Removing Training data}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{l || r | r | r || r | r  }\n\\multirow{2}{*}{Model} & \\multicolumn{3}{c}{Entity Selection} & \\multicolumn{2}{c}{State Change}  \\\\\n                      & {\\sc F1} & {\\sc UR} & {\\sc CR} & {\\sc F1} & {\\sc Acc}  \\\\\n\\hline\\hline\n25\\% training data kept & 54.34 & 75.71 & 21.17 & 2.52  & 50.23   \\\\\n50\\% training data kept & 55.12 & 76.04 & 19.05 & 36.34  & 54.66 \\\\\n75\\% training data kept & 56.64 & 76.03 & 21.00 & 48.86  & 57.62   \\\\\n\\hline\n100\\% training data & 56.84 & 74.98 & 21.14 & 50.56 & 57.87  \\\\\n\\end{tabular}\n\\caption{Results for entity selection and state change selection on the development set when randomly dropping a percentage of the training labels}\n\\label{table:tp_remove}\n\\end{table}\n\\section{Conclusion}\n\\vspace{-1mm}\n\\label{sec:conclusions}\nWe introduced the \\emph{Neural Process Network} for modeling a process of actions and their causal effects on entities by learning action transformations that change entity state representations. The model maintains a recurrent memory structure to track entity states and is trained to predict the state changes that entities undergo. Empirical results demonstrate that our model can learn the causal effects of action semantics in the cooking domain and track the dynamic state changes of entities, showing advantages over competitive baselines.\n\n\\section*{Acknowledgments}\nWe thank Yonatan Bisk, Peter Clark, Bhavana Dalvi, Niket Tandon, and Yuyin Sun for helpful discussions at various stages of this work. This research was supported in part by NSF (IIS-1524371, IIS-1714566, NRI-1525251), DARPA under the CwC program through the ARO (W911NF-15-1-0543), Samsung Research, and gifts by Google and Facebook.\n\\section{Experimental Setup}\n\\label{sec:experiments}\n\nWe evaluate our model on a set of intrinsic tasks centered around tracking entities and state changes in recipes to show that the model can simulate preliminary dynamics of the recipe task. Additionally, we provide a qualitative analysis of the internal components of our model. Finally, we evaluate the quality of the states encoded by our model on the extrinsic task of generating future steps in a recipe. \n\n\\subsection{Intrinsic Evaluation - Tracking}\n\\label{ssec:exps:intrinsic}\nIn the tracking task, we evaluate the model's ability to identify which entities are selected and what changes have been made to them in every step. We break the tracking task into two separate evaluations, entity selection and end state prediction, and also investigate whether the model learns internal representations that approximate recipe dynamics.\n\\vspace{-3mm}\n\\paragraph{Metrics}\nIn the entity selection test, we report the F1 score of choosing the correct entities in any step. A selected entity is defined as one whose attention weight $a_i$ is greater than 50\\% (\\S\\ref{ssec:model:aes}). Because entities may be harder to predict when they have been combined with other entities (e.g., the mixture may have a new name), we also report the recall for selecting combined (CR) and uncombined (UR) entities.\nIn the end state prediction test, we report how often the model correctly predicts the state change performed in a recipe step and the resultant end state. This score is then scaled by the accuracy of predicting which entities were changed in that same step. We report the average F1 and accuracy across the six state change types. \n\\vspace*{-3mm}\n\\paragraph{Baselines} \nWe compare our models against two baselines. First, we built a GRU model that is trained to predict entities and state changes independently. This can be viewed as a bare minimum network with no action representations or recurrent entity memory. \nThe second baseline is a Recurrent Entity Network \\citep{ren} with changes to fit our task. First, the model can tie memory cells to a subset of the full list of entities so that it only considers entities that are present in a particular recipe. Second, the entity distribution for writing to the memory cells is re-used when we query the memory cells. The normalized weighted average of the entity cells is used as the input to the state predictors. The unnormalized attention when writing to each cell is used to predict selected entities. Both baselines are trained with entity selection and state change losses (\\S\\ref{ssec:train:train}).\n\\vspace*{-3mm}\n\\paragraph{Ablations} We report results on six ablations. First, we remove the recurrent attention (Eq.~\\ref{eq:choice}). The model only predicts entities using the current encoder hidden state. In the second ablation, the model is trained with no coverage penalty (Eq.~\\ref{eq:train:usage}). The third ablation prunes the connection from the action selector $w_p$ to the entity selector (Eq.~\\ref{eq:bilinear}). We also explore not pretraining the action selector. Finally, we look at two ablations where we intialize the action embeddings with vectors from a skipgram model. In the first, the model operates normally, and in the second, we do not allow gradients to backpropagate to the action embeddings, updating only the mapping tensor $W_4$ instead (Eq.~\\ref{eq:comp_knowl}).\n\n\\begin{table}\n\\centering\n\\begin{tabular}{l | r  r  r | r  r  }\n\\multirow{2}{*}{Model} & \\multicolumn{3}{c}{Entity Selection} & \\multicolumn{2}{c}{State Change}  \\\\\n                       & {\\sc F1} & {\\sc UR} & {\\sc CR} & {\\sc F1} & {\\sc Acc}  \\\\\n\\hline\\hline\n2-layer LSTM Entity Recognizer                  & 50.98 & 74.03 & 13.33 & - & - \\\\\nAdapted Gated Recurrent Unit                   & 45.94 & 67.69 & 7.74 & 41.16  & 52.69    \\\\\nAdapted Recurrent Entity Network               & 48.57 & 71.88 & 9.87 & 42.32  & 53.47   \\\\\n\\hline\n- Recurrent Attention (Eq.~\\ref{eq:choice})   & 48.91 & 72.32 & 12.67 & 42.14  & 50.48   \\\\\n- Coverage Loss (Eq.~\\ref{eq:train:usage})    & 55.18 & 73.98 & 20.23 & 44.44  & \\textbf{55.20} \\\\\n- Action Connections (Eq.~\\ref{eq:bilinear})  & 54.85 & 73.54 & 20.03 & 44.05  & 54.81   \\\\\n- Action Selector Pretraining & 54.91 & 73.87 & 20.30 & 44.28 & 55.00 \\\\\n\\hline\n+ Pretrained Action Embeddings & 55.16 & 74.02 & 20.32 & 44.02 & 55.03 \\\\\n- Action Embedding Updates & 53.79 & 70.77 & 18.60 & 44.27 & 55.02 \\\\\n\\hline\nFull Model                                     & \\textbf{55.39} & \\textbf{74.88} & \\textbf{20.45} & \\textbf{44.65}  & 55.07  \\\\\n\\end{tabular}\n\\caption{Results for entity selection and state change selection}\n\\label{table:tracking}\n\\end{table}\n\n\\subsection{Extrinsic Evaluation - Generation} \nThe generation task tests whether our system can produce the next step in a recipe based on the previous steps that have been performed. The model is provided all of the previous steps as context.\n\\vspace*{-3mm}\n\\paragraph{Metrics} We report the combined BLEU score and ROUGE score of the generated sequence relative to the reference sequence. Each candidate sequence has one reference sentence. Both metrics are computed at the corpus-level. Also reported are ``VF1'', the F1 score for the overlap of the actions performed in the reference sequence and the verbs mentioned in the generated sequence, and ``SF1'', the F1 score for the overlap of end states annotated in the reference sequence and predicted by the generated sequences. End states for the generated sequences are extracted using the lexicon from Section~\\ref{ssec:states} based on the actions performed in the sentence.\n\\vspace*{-3mm}\n\\paragraph{Setup} To apply our model to the task of recipe step generation, we input the context sentences through the neural process network~and record the entity state vectors once the entire context has been read (\\S\\ref{ssec:model:pm}). These vectors can be viewed as a snapshot of the current state of the entities once the preceding context has been simulated inside the neural process network. We encode these vectors using a bidirectional GRU \\citep{cho2014learning} and take the final time step hidden state $\\mathbb{e}_I$. A different GRU encodes the context words in the same way (yielding $\\mathbb{h}_T$) and the first hidden state input to the decoder is computed using the projection function:\n\n\\begin{equation}\n{\\tilde h}_0 = W_{5}(\\mathbb{e}_I \\circ \\mathbb{h}_T)    \n\\end{equation}\n\nwhere $\\circ$ is the Hadamard product\nbetween the two encoder outputs. All models are trained by minimizing the negative loglikelihood of predicting the next word for the full sequence. Implementation details can be found in Appendix~\\ref{appsec:models}. \n\\vspace*{-3mm}\n\\paragraph{Baselines} For the generation task, we use three baselines: a seq2seq model with no attention \\citep{seq2seq}, an attentive seq2seq model \\citep{bahdanau2014neural},  and a similar variant as our NPN generator, except where the entity states have been computed by the Recurrent Entity Network (EntNet) baseline (\\S\\ref{ssec:exps:intrinsic}). Implementation details for baselines can be found in Appendix~\\ref{appsec:baseline}.\n\n\n\\section{Experimental Results}\n\n\\subsection{Extrinsic Evaluations}\n\\label{ssec:exps:gen}\n\n\\begin{table}\n    \\centering\n    \\begin{tabular}{l | r | r | r | r  }\n        Model & {\\sc BLEU} & {\\sc ROUGE-L} & {\\sc VF1} & {\\sc SF1} \\\\\n        \\hline\\hline\n        Vanilla Seq2Seq        & 2.81 & 33.00 & 16.17 & 40.21 \\\\\n        Attentive Seq2Seq      & 2.83 & 33.18 & 16.97 & 41.43 \\\\\n       \n        EntNet Generator          & 2.30 & 32.71 & 17.53 & 42.43 \\\\\n        \\hline\n        NPN Generator          & \\textbf{3.74} & \\textbf{35.64} & \\textbf{20.12} & \\textbf{43.40}  \\\\\n    \\end{tabular}\n    \\caption{Generation Results}\n    \\label{table:gen}\n\\end{table}\n\n\\paragraph{Recipe Step Generation} Our results in Table~\\ref{table:gen} indicate that sequences generated using the neural process network entity states as additional input yield higher scores than competitive baselines. The entity states allow the model to predict next steps conditioned on a representation of the world being simulated by the neural process network. \nAdditionally, the higher VF1 and SF1 scores indicate that the model is indeed using the extra information to better predict the actions that should follow the context provided. Example generations for each baselines\nfrom the dev set are provided in Table~\\ref{table:genex}, showing that the NPN generator can use information about ingredient states to reason about the most likely next step. The first and second examples are interesting as it shows that the NPN-aware model has learned to condition on entity state -- knowing that raw butter will likely be melted or that a cooked flan must be refrigerated.\nThe third example is also interesting because the model learns that cooked vegetables such as squash will sometimes be drained, even if it is not relevant to this recipe because the squash is steamed. The seq2seq and EntNet baselines, meanwhile, output reasonable sentences given the immediate context, but do not exhibit understanding of global patterns.\n\\subsection{Action Embeddings and Compositionality}\n\\vspace{-3mm}\n\\paragraph{Action Embeddings} In our model, each action is assigned its own embedding, but many actions induce similar changes in the physical world (e.g.,``cut\" and ``slice\"). After training, we compute the pairwise cosine similarity between each pair of action embeddings. In Table~\\ref{table:actions}, we see that actions that perform similar functions are neighbors in embedding space, indicating the model has captured certain semantic properties of these actions. Learning action representations through the state changes they induce has allowed the model to cluster actions by their transformation functions.\n\n\\begin{figure}\n\\small\n\\begin{floatrow}\n\\capbtabbox{\n  \\begin{tabular}{c | c} \\hline\n  Action & Nearest Neighbor Actions \\\\\n\\hline\\hline\ncut & slice, split, snap, slash, carve, slit, chop \\\\\nboil & cook, microwave, fry, steam, simmer\\\\\nadd & sprinkle, mix, reduce, splash, stir, dust \\\\\nwash & rinse, scrub, refresh, soak, wipe, scale \\\\\nmash & spread, puree, squeeze, liquefy, blend \\\\\nplace & ease, put, lace, arrange, leave \\\\\nrinse & wash, refresh, soak, wipe, scrub, clean\\\\ \nwarm & reheat, ignite, heat, light, crisp, preheat\\\\\nsteam & microwave, crisp, boil, parboil, heat \\\\\nsprinkle & top, pat, add, dip, salt, season \\\\\ngrease & coat, rub, dribble, spray, smear, line\\\\\n  \\end{tabular}\n}{\n  \\caption{Most similar actions based on cosine similarity of action embeddings}%\n  \\label{table:actions}\n}\n\\ffigbox{%\n  \\includegraphics[width=6.75cm]{composition_graph}\n}{%\n  \\caption{Change in cosine similarity of entity state embeddings}\n  \\label{fig:comp}\n}\n\\end{floatrow}\n\\end{figure}\n\n\\vspace{-3mm}\n\\paragraph{Entity Compositions} When individual entities are combined into new constructs, our model averages their state embeddings (Eq.~\\ref{eq:avg}), applies an action embedding to them (Eq.~\\ref{eq:comp_knowl}), and writes them to memory (Eq.~\\ref{eq:eu}). The state embeddings of entities that are combined should be overwritten by the same new embedding. In Figure~\\ref{fig:comp}, we present the percentage increase in cosine similarity for state embeddings of entities that are combined in a sentence (blue) as opposed to the percentage increase for those that are not (red bars). While the soft attention mechanism for entity selection allows similarities to leak between entity embeddings, our system is generally able to model the compositionality patterns that result from entities being combined into new constructs.\n\n\\section{Introduction}\n\\label{sec:intro}\n\nUnderstanding procedural text such as instructions or stories requires anticipating the implicit causal effects of actions on entities. For example,\ngiven instructions such as ``\\emph{add} blueberries to the muffin mix, then \\emph{bake} for one half hour,'' an intelligent agent must be able to anticipate a number of entailed facts (e.g., the blueberries are now in the oven; their \"temperature\" will increase). While this common sense reasoning is trivial for humans, most natural language understanding algorithms do not have the capacity to reason about causal effects not mentioned directly in the surface strings \\citep{lexinf,jia2017adversarial,distrepready}.\n\n\\begin{wrapfigure}{R}{0.48\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{Intro_Figure_v2.pdf}\n\\caption{The {\\it process} is a narrative of entity state changes induced by actions. In each sentence, these state changes are induced by simulated actions and must be remembered. }\n    \\label{fig:intro}\n\\end{wrapfigure}\n\nIn this paper, we introduce Neural Process Networks, a procedural language understanding system that tracks common sense attributes through neural simulation of action dynamics. \nOur network models interpretation of natural language instructions as a \\emph{process} of actions and their cumulative effects on entities. More concretely, reading one sentence at a time, our model attentively selects what actions to execute on which entities, and remembers the state changes induced with a recurrent memory structure. In Figure~\\ref{fig:intro}, for example, our model indexes the ``tomato\" embedding, selects the ``wash'' and ``cut'' functions and performs a computation that changes the ``tomato'' embedding so that it can reason about attributes such as its ``\\textsc{shape}'' and ``\\textsc{cleanliness}''.\n\nOur model contributes to a recent line of research that aims to model aspects of world state changes, such as language models and machine readers with explicit entity representations \\citep{ren,reflm,delm}, as well as other more general purpose memory network variants \\citep{memnets,e2ememnets,cbt,qrn}. This \\emph{world-centric} modeling of procedural language (i.e., understanding by simulation) abstracts away from the surface strings, complementing \\emph{text-centric} modeling of language, which focuses on syntactic and semantic labeling of surface words (i.e., understanding by labeling).\n\nUnlike previous approaches, however, our model also learns explicit action representations as functional operators\n(See Figure~\\ref{fig:intro}). \nWhile representations of action semantics could be acquired through an embodied agent that can see and interact with the world \\citep{actionenv}, we propose to learn these representations from text. In particular, we require the model to be able to \\emph{explain} the causal effects of actions by predicting natural language attributes about entities such as \\textsc{``location''} and \\textsc{``temperature''}. The model adjusts its representations of actions based on errors it makes in predicting the resultant state changes to attributes. This textual simulation allows us to model aspects of action causality that are not readily available in existing simulation environments. Indeed, most virtual environments offer limited aspects of the world -- with a primary focus on spatial relations \\citep{actionenv,renv2,envtorques}. They leave out various other dimensions of the world states that are implied by diverse everyday actions such as ``dissolve'' (change of \\textsc{``composition''}) and ``wash'' (change of \\textsc{``cleanliness''}). \n\n\nEmpirical results demonstrate that parametrizing explicit action embeddings provides an inductive bias that allows the neural process network~to learn more informative context representations for understanding and generating natural language procedural text. In addition, our model offers more interpretable internal representations and can reason about the unstated causal effects of actions explained through natural language descriptors.  \nFinally, we include a new dataset with fine-grained annotations on state changes, to be shared publicly, \nto encourage future research in this direction.\n\n\n\n\\section{\\MODELNAME}\nThe neural process network~is an interpreter that reads in natural language sentences, one at a time, and simulates the process of actions being applied to relevant entities through learned representations of actions and entities.\n\\vspace*{-1mm}\n\\subsection{Overview and Notation}\nThe main component of the neural process network~is the simulation module (\\S\\ref{ssec:model:pm}), a recurrent unit whose internals simulate the effects of actions being applied to entities. A set of $V$ actions is known a priori and an embedding is initialized for each one, $\\mathcal{F} = \\{f_1, ... f_V\\}$. Similarly, a set of $I$ entities is known and an embedding is initialized for each one: $\\mathcal{E} = \\{e_1, ... e_I\\}$. Each $e_i$ can be considered to encode information about state attributes of that entity, which can be extracted by a set of state predictors (\\S\\ref{ssec:model:sc}). As the model reads text, it ``applies\" action embeddings to the entity vectors, thereby changing the state information encoded about the entities.\n\nFor any document $d$, an initial list of entities $I_d$ is known and $\\mathcal{E}_d = \\{e_i \\vert i \\in I_d\\} \\subset \\mathcal{E}$ entity state embeddings are initialized. As the neural process network~reads a sentence from the document, it selects a subset of both $\\mathcal{F}$ (\\S\\ref{ssec:model:as}) and $\\mathcal{E}_d$ (\\S\\ref{ssec:model:aes}) based on the actions performed and entities affected in the sentence. The entity state embeddings are changed by the action and the new embeddings are used to predict end states for a set of state changes (\\S\\ref{ssec:model:sc}). The prediction error for end states is backpropagated  to the action embeddings, learning action representations that model the simulation of desired causal effects on entities. This process is broken down into five modules below. Unless explicitly defined, all $W$ and $b$ variables are parametrized linear projections and biases. We use the notation $\\{e_i\\}_t$ when referring to the values of the entity embeddings before processing sentence $s_t$.\n\\vspace*{-2mm}\n\\subsection{Sentence Encoder} \n\\label{ssec:model:se}\nGiven a sentence $s_t$, a Gated Recurrent Unit \\citep{cho2014learning} encodes each word and outputs its last hidden vector as a sentence encoding $h_t$ \\citep{seq2seq}.\n\\vspace*{-2mm}\n\\begin{figure*}[t!]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{Model_Figure_key_v4.pdf}\n    \\caption{\\small Model Summary. The sentence encoder converts a sentence to a vector representation, $h_t$. The action selector and entity selector use the vector representation to choose the actions that are applied and the entities that are acted upon in the sentence. The simulation module indexes the action and entity state embeddings, and applies the transformation to the entities. The state predictors predict the new state of the entities if a state change has occurred. Equation references are provided in parentheses.}\n    \\label{fig:model_1}\n\\end{figure*}\n\\label{sec:model} \n\\subsection{Action Selector} \n\\label{ssec:model:as}\nGiven $h_t$ from the sentence encoder, the action selector (bottom left in Fig.~\\ref{fig:model_1}) contextually determines which action(s) from $\\mathcal{F}$ to execute. For example, if the input sentence is \\emph{``wash and cut beets''}, both $f_{wash}$ and $f_{cut}$  must be selected. To account for multiple actions, we make a soft selection over $\\mathcal{F}$, yielding a weighted sum of the selected action embeddings $\\bar{f_t}$:\n\\begin{equation}\n\\begin{aligned}\nw_p &= \\textrm{MLP}(h_t) \\\\\n\\bar w_p &= \\frac{w_p}{\\sum_{j=1}^V w_{p_j}}  \\\\\n\\bar f_t &= \\bar w_p^\\intercal \\mathcal{F} \\label{eq:as}\n\\end{aligned}\n\\end{equation}\n\\noindent where MLP is a parametrized feed-forward network with a sigmoid activation and $w_p \\in \\mathbb{R}^V$ is the attention distribution over $V$ possible actions (\\S\\ref{ssec:states}). We compose the action embedding by taking the weighted average of the selected actions. \n\\subsection{Entity Selector}\n\\label{ssec:model:aes}\n\\paragraph{Sentence Attention} Given $h_t$ from the sentence encoder, the entity selector chooses relevant entities using a soft attention mechanism:\n\\begin{equation}\n\\begin{aligned}\n\\tilde h_t &= \\text{ReLU}(W_1h_t + b_1)\\\\\nd_i &= \\sigma(e_{i_0}^\\intercal W_2 [\\tilde h_t; w_p]) \\label{eq:bilinear}\n\\end{aligned}\n\\end{equation}\nwhere  $W_2$ is a bilinear mapping, $e_{i_0}$ is a unique key for each entity (\\S\\ref{ssec:model:pm}), and $d_i$ is the attention weight for entity embedding $e_i$. For example, in \\emph{``wash and cut beets and carrots''}, the model should select $e_{beet}$ and $e_{carrot}$.\n\\vspace{-3mm}\n\\paragraph{Recurrent Attention} While sentence attention would suffice if entities were always explicitly mentioned, natural language often elides arguments or uses referent pronouns. As such, the module must be able to consider entities mentioned in previous sentences. Using $\\tilde h_t$, the model computes a soft choice over whether to choose affected entities from this step's attention $d_i$ or the previous step's attention distribution.\n\\begin{equation}\n\\begin{aligned}\nc = softmax(W_3 \\tilde h_t + b_3) \\\\\na_{i_{t}} = c_1 d_i + c_2 a_{i_{t-1}} + c_3 \\mathbf{0} \\label{eq:choice}\n\\end{aligned}\n\\end{equation}\n\\noindent where $c \\in \\mathbb{R}^3$ is the choice distribution, $a_{i_{t-1}}$ is the previous sentence's attention weight for each entity, $a_{i_{t}}$ is the final attention for each entity, and $\\mathbf{0}$ is a vector of zeroes (providing the option to not change any entity). Prior entity attentions can propagate forward for multiple steps.\n\\subsection{Simulation Module} \n\\label{ssec:model:pm}\n\\paragraph{Entity Memory} A unique state embedding $e_{i}$ is initialized for every entity $i$ in the document. A unique key to index each embedding $e_{i_0}$ is set as the initial value of the embedding \\citep{ren,kvnet}. After the model reads $s_t$, it modifies $\\{e_{i}\\}_t$ to reflect changes influenced by actions. At every time step, the entity memory receives the attention weights from the entity selector, normalizes them and computes a weighted average of the relevant entity state embeddings:\n\n\\begin{minipage}{.5\\linewidth}\n\\begin{equation}\n \\alpha_{i} = \\frac{a_i}{\\sum_{j=1}^{I_d} a_j} \\label{eq:attn}\n\\end{equation}\n\\end{minipage}%\n\\begin{minipage}{.5\\linewidth}\n\\begin{equation}\n  \\bar e_t = \\sum_{j=1}^{I_d} \\alpha_i e_i \\label{eq:avg}\n\\end{equation}\n\\end{minipage}\n\\vspace{-1mm}\n\\paragraph{Applicator} \nGiven the action summary embedding $\\bar{f_t}$ and the entity summary embedding $\\bar{e_t}$, the applicator (middle right in Fig.~\\ref{fig:model_1}) applies the selected actions to the selected entities, and outputs the new proposal entity embedding $k_t$.\n\\begin{equation}\nk_t = \\text{ReLU}(\\bar f_t W_4\\bar e_t + b_4) \\label{eq:comp_knowl}\n\\end{equation}\nwhere $W_4$ is a third order tensor projection. The vector $k_t$ is the new representation of the entity $\\bar e_t$ after the applicator simulates the action being applied to it. \n\\vspace{-3mm}\n\\paragraph{Entity Updater} \nThe entity updater interpolates the new proposal entity embedding $k_t$ and the set of current entity embeddings $\\{e_i\\}_t$:\n\\begin{equation}\ne_{i_{t+1}} = a_{i_t} k_t + ( 1 - a_{i_t}) e_{i_t} \\label{eq:eu}\n\\end{equation}\n\\noindent yielding an updated set of entity embeddings $\\{e_i\\}_{t+1}$. Each embedding is updated proportional to its entity's unnormalized attention $a_i$, allowing the model to completely overwrite the state embedding for any entity. For example, in the sentence \\textit{``mix the flour and water,''} the embeddings for $e_{flour}$ and $e_{water}$ must both be overwritten by $k_t$ because they no longer exist outside of this new composition.\n\\subsection{State Predictors} \n\\label{ssec:model:sc}\nGiven the new proposal entity embedding $k_t$, the state predictor (bottom right in Fig.~\\ref{fig:model_1}) predicts changes to the resulting entity embedding $k_t$ along the following six dimensions: location, cookedness, temperature, composition, shape, and cleanliness. Discrete multi-class classifiers, one for each dimension, take in $k_t$ and predict a unique end state for their corresponding state change type:\n\\begin{equation}\nP(Y_s | k_t) = softmax(W_{s}k_t + b_s)\n\\end{equation}\n\\noindent  For location changes, which require contextual information to predict the end state, $k_t$ is concatenated with the original sentence representation $h_t$ to predict the final state.\n\\section{Related Work}\n\\label{sec:related}\n\\vspace{-1mm}\nRecent studies in machine comprehension have used a neural memory component to store a running representation of processed text \\citep{memnets,e2ememnets,cbt,qrn}. While these approaches map text to memory vectors using standard neural encoder approaches, our model, in contrast, directly interprets text in terms of the effects actions induce in entities, providing an inductive bias for learning how to represent stored memories. More recent work in machine comprehension also sought to couple the memory representation with tracking entity states \\citep{ren}. Our work seeks to provide a relatively more structured representation of domain-specific action knowledge to provide an inductive bias to the reasoning process.\n\nNeural Programmers \\citep{neuralprogrammer, neuralprogrammer2} have also used functions to simulate reasoning, by building a model to select rows in a database and applying operation on those selected rows. While their work explicitly defined the effect of a number of operations for those rows, we provide a framework for learning representations for a more expansive set of actions, allowing the model to learn representations for how actions change the state space.\n\n\\indent Works on instructional language studied the task of building discrete graph representations of recipes using probabilistic models \\citep{actiongraph, mori2014flow, mori2012machine}. We propose a complementary new model by integrating action and entity relations into the neural network architecture and also address the additional challenge of tracking the state changes of the entities.\n\nAdditional work in tracking states with visual or multimodal context has focused on 1) building graph representations for how entities change in goal-oriented domains \\citep{chaiphysical,chaijointly,andor} or 2) tracking visual state changes based on decisions taken by agents in environment simulators such as videos or games \\citep{renv2,envtorques,actionenv}. Our work, in contrast, models state changes in embedding space using only text-based signals to map real-world actions to algebraic transformations. \n\\subsection{State Change Knowledge}\n\\label{ssec:states}\n\n\\begin{table}\n\\small\n\\centering\n\\caption{Example actions, the state changes they induce, and the possible end states}\n\\begin{tabular}{l | l | l}\nAction & State Change Types & End States  \\\\\n\\hline\n\\hline\nbraise & \\textsc{cookedness}; \\textsc{temperature} & \\textsc{cooked}; \\textsc{hot}\\\\\nchill & \\textsc{temperature} & \\textsc{cold} \\\\\nknead & \\textsc{shape} & \\textsc{molded}\\\\\nwash & \\textsc{cleanliness} & \\textsc{clean}\\\\\ndissolve & \\textsc{composition} & \\textsc{composed}\\\\\nrefrigerate & \\textsc{temperature}; \\textsc{location} & \\textsc{cold}; \\textsc{refrigerator} \\\\\nslice & \\textsc{shape} & \\textsc{separated}\n\\end{tabular}\n\\label{table:state_types}\n\\end{table}\n\nIn this work we focus on physical action verbs in cooking recipes. We manually collect a set of 384 actions such as \\textit{cut, bake, boil, arrange}, and {\\it place}, \norganizing their causal effects along the following predefined dimensions: \\textsc{location, cookedness, temperature, shape, cleanliness} and \\textsc{composition}. The textual simulation operated by the model induces state changes along these dimensions by applying actions functions from the above set of 384. For example, \\textit{cut} entails a change in \\textsc{shape}, while {\\it bake} entails a change in \\textsc{temperature, cookedness}, and even \\textsc{location}. We annotate the state changes each action induces, as well as the end state of the action, using Amazon Mechanical Turk. The set of possible end states for a state change can range from 2 for binary state changes to more than 200 (See Appendix~\\ref{appsec:ann} for details). Table~\\ref{table:state_types} provides examples of annotations in this action lexicon.\n\\subsection{Intrinsic Evaluations}\n\\paragraph{Entity Selection} As shown in Table~\\ref{table:tracking}, our full model outperforms all baselines at selecting entities, with an F1 score of 55.39\\%. The ablation study shows that the recurrent attention, coverage loss, action connections and action selector pretraining improve performance. Our success at predicting entities extends to both uncomposed entities, which are still in their raw forms (e.g., melt the {\\it butter} $\\rightarrow$ butter), and composed entities, in which all of the entities that make up a composition must be selected. For example, in a \\emph{Cooking lasagna} recipe, if the final step involves baking the prepared lasagna, the model must select all the entities that make up the lasagna (e.g., lasagna sheets, beef, tomato sauce). In Table~\\ref{table:examples}, we provide examples of our model's ability to handle complex cases such as compositional entities (Ex. 1, 3), and elided arguments over long time windows (Ex. 2). We also provide examples where the model fails to select the correct entities because it does not identify the mapping between a reference construct such as ``pizza\" (Ex. 4) or ``dough\" (Ex. 5) and the set of entities that composes it, showcasing the difficulty of selecting the full set for a composed entity.\n\\vspace*{-3mm}\n\n\n\n\\paragraph{State Change Tracking} In Table~\\ref{table:tracking}, we show that our full model outperforms competitive baselines such as Recurrent Entity Networks \\citep{ren} and jointly trained GRUs. While the ablation without the coverage loss shows higher accuracy, we attribute this to the fact that it predicts a smaller number of total state changes. Interestingly, initializing action embeddings with skipgram vectors and locking their values shows relatively high performance, indicating the potential gains in using powerful pretrained representations to represent actions.\n\\section{Training}\n\\label{sec:train}\n\\input{states}\n\\subsection{Dataset}\n\\label{ssec:data}\nFor learning and evaluation, we use a subset of the Now You're Cooking dataset \\citep{checklist}.\nWe chose 65816 recipes for training, 175 recipes for development, and 700 recipes for testing. For the development and test sets, crowdsourced workers densely annotate actions, entities and state changes that occur in each sentence so that we can tune hyperparameters and evaluate on gold evaluation sets. Annotation details are provided in Appendix~\\ref{appssec:devtest}. \n\\vspace*{-1mm}\n\\subsection{Component-wise Training}\n\\label{ssec:train:train}\nThe neural process network~is trained by jointly optimizing multiple losses for the action selector, entity selector, and state change predictors. Importantly, our training scheme uses weak supervision because dense annotations are prohibitively expensive to acquire at a very large scale. Thus, we heuristically extract verb mentions from each recipe step and assign a state change label based on the state changes induced by that action (\\S\\ref{ssec:states}). Entities are extracted similarly based on string matching between the instructions and the ingredient list.\nWe use the following losses for training:\n\\vspace*{-3mm}\n\\paragraph{Action Selection Loss}\nUsing noisy supervision, the action selector is trained to minimize the cross-entropy loss for each possible action, allowing multiple actions to be chosen at each step if multiple actions are mentioned in a sentence. The MLP in the action selector (Eq.~\\ref{eq:as}) is pretrained.\n\\vspace*{-3mm}\n\\paragraph{Entity Selection Loss} Similarly, to train the attentive entity selector, we minimize the binary cross-entropy loss of predicting whether each entity is affected in the sentence.\n\\vspace*{-3mm}\n\\paragraph{State Change Loss} For each state change predictor, we minimize the negative loglikelihood of predicting the correct end state for each state change.\n\n\\vspace*{-3mm}\n\\paragraph{Coverage Loss}\n\\label{ssec:train:use}\n\nAn underlying assumption in many narratives is that all entities that are mentioned should be important to the narrative.\nWe add a loss term that penalizes narratives whose combined attention weights for each entity does not sum to more than 1.\n\\begin{equation}\n\\mathcal{L}_{cover} = - \\frac{1}{I_d} \\sum_{i=1}^{I_d} \\log\\sum_{t=1}^S a_{i_t} \\label{eq:train:usage}\n\\end{equation}\n\n\\noindent where $a_{i_t}$ is the attention weight for a particular entity at sentence $t$ and $I_d$ is the number of entities in a document. $\\sum_{t=1}^S a_{i_t}$ is upper bounded by 1. This is similar to the coverage penalty used in neural machine translation \\citep{coverage}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\@startsection{section}{1}{\\z@}{3.5ex plus 1ex minus\n   .2ex}{2.3ex plus .2ex}{\\large\\bf}}\n   \n   \n\\renewcommand{\\thefootnote}{\\fnsymbol{footnote}}\n\\long\\def\\@makefntext#1{\\parindent 0cm\\noindent\n\\hbox to 1em{\\hss$^{\\@thefnmark}$}#1}\n\n\n\\newcommand{\\small}{\\small}\n\\makeatletter \n\\long\\def\\@makecaption#1#2{%\n  \\vskip\\abovecaptionskip\n  \\sbox\\@tempboxa{{\\small #1: #2}}%\n  \\ifdim \\wd\\@tempboxa >\\hsize\n    {\\small #1: #2\\par}\n  \\else\n    \\hbox to\\hsize{\\hfil\\box\\@tempboxa\\hfil}%\n  \\fi\n  \\vskip\\belowcaptionskip}\n\\makeatother  \n\n\n\\begin{document}\n\\begin{titlepage}\n\\vspace{.5in}\n\\begin{flushright}\nSeptember 2008\\\\\n\\end{flushright}\n\\vspace{.5in}\n\\begin{center}\n{\\Large\\bf\nThe Constraint Algebra\\\\[1ex]\nof Topologically Massive AdS Gravity} \n\n\\vspace*{.4in}\n{S.~C{\\sc arlip}\\footnote{\\it email: carlip@physics.ucdavis.edu}\\\\\n       {\\small\\it Department of Physics}\\\\\n       {\\small\\it University of California}\\\\\n       {\\small\\it Davis, CA 95616}\\\\{\\small\\it USA}}\n\\end{center}\n\n\\vspace{.5in}\n\\begin{center}\n{\\large\\bf Abstract}\n\\end{center}\n\\begin{center}\n\\begin{minipage}{4.5in}\n{\\small\nThree-dimensional topologically massive AdS gravity has a complicated \nconstraint algebra, making it difficult to count nonperturbative degrees \nof freedom.  I show that a new choice of variables greatly simplifies this \nalgebra, and confirm that the theory contains a single propagating mode \nfor all values of the mass parameter and the cosmological constant.  As an \nadded benefit, I rederive the central charges and conformal weights \nof the boundary conformal field theory from an explicit analysis of the \nasymptotic algebra of constraints.\n}\n\\end{minipage}\n\\end{center}\n\\end{titlepage}\n\\addtocounter{footnote}{-1}\n\n\\section{Introduction} \\label{intro}\n\nTopologically massive gravity \\cite{DJT}---(2+1)-dimensional Einstein \ngravity supplemented with a Chern-Simons term for the spin connection%\n---provides a fascinating playground for exploring higher-derivative \ngravity.   In contrast to the topological character of ordinary Einstein \ngravity in three dimensions, topologically massive gravity has a local \ndegree of freedom, a parity-violating massive spin two graviton that can \nbe described by a single indexless ``scalar'' field.  With the addition of a \nnegative cosmological constant $\\Lambda=-1\/\\ell^2$, surprising new \nfeatures emerge \\cite{CDWW}: for instance, the components of curvature \nperturbations propagate with different, chirality-dependent masses.\n\nTopologically massive AdS gravity may also provide a useful model\nin which to explore the AdS\/CFT correspondence.  The conformal\nboundary of a three-dimensional asymptotically anti-de Sitter spacetime \nis a flat two-dimensional cylinder, and the asymptotic symmetries are\ndescribed by a pair of Virasoro algebras \\cite{BH}.  The resulting\ntwo-dimensional conformal symmetry can be very powerful.  In pure Einstein\ngravity, for example, although the boundary conformal field theory is \nnot known \\cite{Carlip,Witten}, the classical central charges and\nconformal weights are sufficient to determine the BTZ black hole\nentropy \\cite{Strominger,BSS} and even the spectrum of Hawking \nradiation \\cite{Emparan}.  In topologically massive gravity, it was\nshown several years ago that the central charges of the ``left'' and \n``right''  Virasoro algebras split \\cite{KL,Solodukhin,Hotta}:\n\\begin{equation}\nc_\\pm = \\frac{3\\ell}{2G}\\left( 1 \\pm \\frac{1}{\\mu\\ell} \\right),\n\\label{intro1}\n\\end{equation}\nwhere the mass parameter $\\mu$ is fixed by the Chern-Simons coupling.\nThe classical contributions to conformal weights are also shifted, \nleading to interesting modifications of black hole thermodynamics \n\\cite{KL,Solodukhin,Park}.  Moreover, at $\\mu\\ell=\\pm1$ the boundary\ntheory becomes chiral.  Since the sum over topologies in three-dimensional\ngravity may require a chiral splitting \\cite{Maloney}, such a theory could \nbe of considerable interest.\n\nUnfortunately, this model appears to have a fundamental sickness.  With \nthe usual sign for the gravitational constant, the massive excitations of \ntopologically massive gravity carry negative energy \\cite{DJT}.  In the \nabsence of a cosmological constant, one can simply flip the sign of $G$,\nbut if $\\Lambda<0$, this will give a negative mass to the BTZ black hole\n\\cite{Moussa}.  The existence of a stable ground state is thus in doubt.\nThe possibility of a supersymmetric extension of the theory \\cite{Deser}\nsuggests the existence of a stable superselection sector, but this sector\nappears to exclude black holes.\n\nRecently, Li, Song, and Strominger proposed a possible cure \\cite{LSS}.\nAt the chiral point, a family of eigenstates of the Virasoro generator $L_0$\nrepresenting massive excitations disappears, and Li et al.\\ suggested \nthat the massive gravitons might no longer be present.  Unfortunately, a\ndifferent family of finite-energy eigenstates of $L_0$ has been found \n\\cite{GJ}, which violate the standard Fefferman-Graham asymptotic \nconditions \\cite{FG} but are still asymptotically anti-de Sitter; and worse, \na complete set of finite-energy asymptotically AdS wave packets also \nexists, even at the chiral coupling \\cite{CDWW}.  \n\nA loophole remains, however.  The computations of \\cite{LSS,GJ,CDWW}%\n---and, indeed, those of virtually every paper discussing this model---are based on \nclassical perturbation theory, expanding the metric in small fluctuations \naround AdS and keeping only lowest order terms.   The full field equations, \non the other hand, are highly nonlinear, and it is conceivable that new \nfeatures could emerge nonperturbatively.\n\nA general nonperturbative solution of the field equations of topologically\nmassive gravity seems distant, but we can learn a great deal by\nanalyzing the constraints.  For the case of a vanishing cosmological\nconstant, such an analysis was first performed by Deser and Xiang \\cite{DX}, \nand further amplified by Buchbinder et al.\\ \\cite{Buchbinder}.  The\nformalism is very complicated, in part because of the presence of third\nderivatives and second class constraints, but the results ultimately confirm \nthe existence of a single propagating degree of freedom.  For the asymptotically \nanti-de Sitter case, the literature is currently inconsistent: Park \\cite{Parkb}\nappears to find more than one degree of freedom (one ``for each internal\nindex''), while Grumiller et al.\\ \\cite{GJJ} find one configuration space \ndegree of freedom, but consider only the chiral coupling.\n\nIn this paper, I will show that a new choice of variables greatly simplifies the\nconstraint analysis, allowing an elegant expression of the constraint algebra\nand a simple counting of degrees of freedom.  I confirm the existence of a\nsingle propagating degree of freedom at all values of the couplings, and  \nrederive the central charges (\\ref{intro1}) from an explicit computation of\nthe algebra of asymptotic symmetries.\n\n\\section{Topologically massive AdS gravity}\n\nIn the first order formalism of (2+1)-dimensional gravity, the fundamental\nvariables are the triad $e^a = e^a{}_\\mu dx^\\mu$ and the spin connection \n$\\omega_a = \\frac{1}{2}\\epsilon_{abc}\\omega^{bc}{}_\\mu dx^\\mu$.\nThe Einstein-Hilbert action takes the form\\footnote{I choose units \n$16\\pi G=1$, a metric of signature $(-++)$, and a cosmological constant\n$\\Lambda=-1\/\\ell^2$, and set $\\epsilon_{012}=1$ (so $\\epsilon^{012}=-1$).}\n\\begin{equation}\nI_{EH}[e,\\omega] = 2 \\int \\left[ \n    e^a\\wedge\\left( d\\omega_a \n    + \\frac{1}{2}\\epsilon_{abc}\\omega^b\\wedge\\omega^c\\right)\n    + \\frac{1}{6}\\frac{1}{\\ell^2}\\epsilon_{abc}e^a\\wedge e^b\\wedge e^c\\right] ,\n\\label{b1}\n\\end{equation}\nwhere $e$ and $\\omega$ can be treated as independent variables.  The variation \nof $\\omega$ yields the torsion constraint\n\\begin{equation}\nT_a = D_\\omega e_a = de_a + \\epsilon_{abc}\\omega^b\\wedge e^c  = 0 ,\n\\label{b2}\n\\end{equation}\nwhile the variation of $e$ gives the Einstein field equations.   \n\nAn additional Chern-Simons term can be written for the spin connection,\n\\begin{equation}\nI_{CS}[\\omega] =  \\int\\left[\n    \\omega^a\\wedge\\left( d\\omega_a \n    + \\frac{1}{3}\\epsilon_{abc}\\omega^b\\wedge\\omega^c\\right)\\right] .\n\\label{b3}\n\\end{equation}\nIf $e$ and $\\omega$ are varied independently, the sum of the Einstein-Hilbert\nand Chern-Simons actions gives a model whose solutions are identical to those of\nordinary Einstein gravity \\cite{Wittenb}, although with a different symplectic \nstructure that may have implications for the quantum theory \\cite{Meusburger}.  \nIf the torsion constraint (\\ref{b2}) is imposed, however, one obtains a higher-derivative\ntheory, topologically massive gravity, with  \n\\begin{equation}\nI_{TMG}[e] = I_{EH}[e,\\omega(e)] + \\frac{1}{\\mu}I_{CS}[\\omega(e)] .\n\\label{b4}\n\\end{equation}\n\nTo simplify this action, let us define a new connection\n\\begin{equation}\nA^a = \\omega^a + \\mu e^a  ,\n\\label{b5}\n\\end{equation}\nwhose Chern-Simons action is\n$$\n\\frac{1}{\\mu} I_{CS}[A] = \\frac{1}{\\mu} I_{CS}[\\omega] + I_{EH}\n    + \\int \\left[ \\mu e^a\\wedge T_a + \\frac{1}{3}\\left(\\mu^2 - \\frac{1}{\\ell^2}\\right)\n    \\epsilon_{abc}e^a\\wedge e^b\\wedge e^c\\right] .\n$$\nRather than explicitly writing $\\omega$ as a function of $e$, we can\nimpose the torsion constraint with a Lagrange multiplier, as suggested in\n\\cite{DX,Carlipb}.  Then\n\\begin{equation}\nI_{TMG} = \\frac{1}{\\mu} I_{CS}[A] \n    + \\int \\left[ \\beta^a\\left( D_A e_a - \\mu \\epsilon_{abc}e^b\\wedge e^c\\right)\n    - \\alpha\\epsilon_{abc}e^a\\wedge e^b\\wedge e^c\\right] ,\n\\label{b6}\n\\end{equation}\nwhere  $\\beta^a = \\beta^a{}_\\mu dx^\\mu$ is a Lagrange multiplier for the\ntorsion constraint and \n$$\n\\alpha = \\frac{1}{3}\\left(\\mu^2 - \\frac{1}{\\ell^2}\\right) .\n$$ \nNote that the chiral coupling occurs at $\\alpha=0$; this is the only place in which the \nrelationship of $\\mu$ and $\\ell$ appears in this formulation.\n\nThe classical equations of motion may be obtained by varying $A$, $\\beta$, and $e$:\n\\begin{align}\n\\delta A: \\qquad& F_a + \\frac{\\mu}{2}\\epsilon_{abc}\\beta^b\\wedge e^c = 0 \n    \\qquad \\hbox{with}\\  F_a = dA_a + \\frac{1}{2}\\epsilon_{abc}A^b\\wedge A^c \n    \\nonumber\\\\\n\\delta\\beta: \\qquad& T_a = D_A e_a - \\mu \\epsilon_{abc}e^b\\wedge e^c = 0 \n     \\nonumber\\\\\n\\delta e: \\qquad&B_a \n    = D_A\\beta_a - 2\\mu\\epsilon_{abc}\\beta^be^c - 3\\alpha\\epsilon_{abc}e^be^c = 0 .\n    \\label{b7} \n\\end{align}\nI show in the Appendix that these are equivalent to the standard field equations for\ntopologically massive AdS gravity, and that they determine the Lagrange multiplier\n$\\beta$ to be \n\\begin{align}\n&\\beta^a{}_\\mu =\n  -\\frac{2}{\\mu}\\left(R^a{}_\\mu + \\frac{2}{\\ell^2}e^a{}_\\mu \n  + \\frac{3\\alpha}{2} e^a{}_\\mu\\right) \n   \\nonumber\\\\\n&\\beta = \\beta^a{}_\\mu e_a{}^\\mu = -\\frac{9\\alpha}{\\mu}  . \\label{b9}\n\\end{align}\n\n\\section{Poisson brackets and constraints}\\label{constraints}\n\nFrom the action (\\ref{b6}), we can now read off the terms involving time derivatives:\n\\begin{equation}\nI_{TMG} = \\int d^3x \\epsilon^{ij}\\left[ -\\frac{1}{\\mu}A^a{}_i\\partial_t A_{aj} \n    - \\beta^a{}_i\\partial_t e_{aj}\\right] + \\dots\n\\label{c1}\n\\end{equation}\nThe canonical Poisson brackets are thus\n\\begin{align}\n&\\left\\{ A^a{}_i, A^b{}_j\\right\\} = \\frac{\\mu}{2}\\eta^{ab}\\epsilon_{ij} \\nonumber\\\\\n&\\left\\{ e^a{}_i, \\beta^b{}_j\\right\\} = \\eta^{ab}\\epsilon_{ij} .\\label{c2}\n\\end{align}\nThe factor of $1\/2$ in the first bracket can be obtained from Dirac brackets, or more\nsimply by recognizing that $A_x$ and $A_y$ are conjugate and integrating by parts.  This\ndiagonalization of the Poisson brackets is one of the main simplifications coming from our \nchoice of variables.\n\nThe time components $A^a{}_t$, $e^a{}_t$, and $\\beta^a{}_t$ appear in the action \nwithout time derivatives, and can be considered Lagrange multipliers.  The corresponding \nconstraints are just the relevant components of the classical equations of motion:\n\\begin{align}\n&J_a = -\\frac{2}{\\mu}\\epsilon^{ij}\\left(\n    F_{aij} + \\frac{\\mu}{2}\\epsilon_{abc}\\beta^b{}_i e^c{}_j\\right)  \n   \\nonumber\\\\\n&T_a = -\\epsilon^{ij}\\left(D_i e_{aj} - \\mu \\epsilon_{abc}e^b{}_i e^c{}_j\\right) \n    \\nonumber \\\\\n&B_a \n    = -\\epsilon^{ij}\\left(D_i\\beta_{aj} - 2\\mu\\epsilon_{abc}\\beta^b{}_ie^c{}_j \n    - 3\\alpha\\epsilon_{abc}e^b{}_ie^c{}_j\\right) . \n\\label{c3} \n\\end{align}\nIt is  convenient to ``smear'' the constraints, integrating them against vectors.  More\nprecisely, let us define\n\\begin{align}\n&J[\\xi] = \\int_\\Sigma d^2x\\, \\xi^aJ_a + Q_J[\\xi]  \\nonumber\\\\\n&T[\\xi] = \\int_\\Sigma d^2x\\, \\xi^aT_a + Q_T[\\xi]  \\nonumber\\\\\n&B[\\xi]  =  \\int_\\Sigma d^2x\\, \\xi^aB_a + Q_B[\\xi] ,\n\\label{c4}\n\\end{align}\nwhere $Q_J$, $Q_T$, and $Q_B$ are the boundary terms needed to make the constraints\ndifferentiable \\cite{BH}.   For now, we shall assume that the parameters $\\xi^a$ fall \noff rapidly enough at infinity that we can freely integrate by parts and  ignore\nboundary terms; these will be restored below in section \\ref{BT}.  The Poisson brackets \nof these generators with the canonical variables $\\{A,e,\\beta\\}$ are now easy to compute,\nand are displayed explicitly in the Appendix.\n \nThe constraints $\\{J,T,B\\}$ should generate the full group of symmetries of the\naction (\\ref{b6}), including diffeomorphism invariance.  For (2+1)-dimensional \nEinstein gravity, it is known that an appropriate combination of the ``gauge''\nconstraints with field-dependent parameters does, indeed, generate\ndiffeomorphisms on shell \\cite{Wittenb,Banados0,Banados}.  The same is true here.  Let \n$\\xi^\\mu$ be an arbitrary three-vector, and define\n\\begin{equation}\n \\xi^a = e^a{}_\\mu\\xi^\\mu, \\qquad {\\hat\\xi}^a = \\beta^a{}_\\mu\\xi^\\mu, \\qquad \n    {\\tilde\\xi}^a = A^a{}_\\mu\\xi^\\mu .\n\\label{c6}\n\\end{equation}\nDefine a new combination $H$ of the constraints by\n\\begin{equation}\nH[\\xi] = B[\\xi] + T[{\\hat\\xi}] + J[{\\tilde\\xi}] .\n\\label{c7}\n\\end{equation}\nA simple computation then shows that\n\\begin{equation}\n\\{H[\\xi],X\\} = -{\\cal L}_\\xi X + \\hbox{\\it terms proportional to the equations of motion,}\n\\label{c8}\n\\end{equation}\nwhere $X$ is any of $\\{A,e,\\beta\\}$ and $\\cal L$ denotes the Lie derivative.  \n\n\\section{The algebra of constraints}\\label{algebra}\n\nWe next turn to the algebra of the constraints.  We may first check that $J[\\xi]$ generates \nlocal Lorentz transformations: a straightforward calculation shows that for any constraint \n$C[\\eta]$,\n\\begin{equation}\n\\left\\{ J[\\xi],C[\\eta]\\right\\} \n    = -C[\\xi\\times\\eta] \\quad \\hbox{with $(\\xi\\times\\eta)^a=\\epsilon^{abc}\\xi_b\\eta_c$} .\n\\label{d1}\n\\end{equation}\nThe $J[\\xi]$ are thus first class constraints.\n\nThe remaining constraints are more complicated.  We find that\n\\begin{align}\n\\left\\{ T[\\xi],T[\\eta]\\right\\} &=\n    -\\frac{\\mu}{2}\\int d^2x\\,\\xi^a\\eta^b\\left( \\epsilon^{ij}e_{ai}e_{bj}\\right) \\nonumber\\\\\n\\left\\{ B[\\xi],T[\\eta]\\right\\} &= -\\frac{\\mu}{2}J[\\xi\\times\\eta] + 2\\mu T[\\xi\\times\\eta]\n    + \\frac{\\mu}{2}\\int d^2x\\,\\xi^a\\eta^b\\left(  \\epsilon^{ij}\\beta_{ai}e_{bj}  \n    -  \\eta_{ab}\\epsilon^{ij}\\beta^c{}_ie_{cj}\\right)   \\nonumber\\\\\n\\left\\{ B[\\xi],B[\\eta]\\right\\} &= 2\\mu B[\\xi\\times\\eta] + 6\\alpha T[\\xi\\times\\eta]\n   -  \\frac{\\mu}{2}\\int d^2x\\,\\xi^a\\eta^b\\left(\\epsilon^{ij}\\beta_{ai}\\beta_{bj}\\right) .\n   \\label{d2}\n\\end{align}\nThe appearance on the right-hand side of terms that are not proportional to the \nconstraints can mean two things: either there are secondary constraints, or some of our\nconstraints are second class \\cite{HT}.  To distinguish the two possibilities, note first that\n$\\epsilon^{ij}e_{ai}e_{bj}$ is zero only if the triad is noninvertible, certainly\nnot a restriction we wish to impose.  Similarly, $\\epsilon^{ij}\\beta_{ai}\\beta_{bj}$\nand $\\epsilon^{ij}e_{ai}\\beta_{bj}$ vanishes only if some components of $\\beta_{ai}$ \nare linearly dependent.  From (\\ref{b9}), this is not generically true, holding only for \n``pure Einstein gravity'' solutions.  \n\nOn the other hand, the quantity\n$$\n\\Delta = \\epsilon^{ij}\\beta^c{}_i e_{cj} = \\epsilon^{ij}\\beta_{ij}\n$$\nalways vanishes classically, by virtue of the symmetry of $\\beta_{\\mu\\nu}$.   We can \ntherefore treat $\\Delta$ as a secondary constraint,\\footnote{Alternatively, we could  \ntreat the Hamiltonian as a second class constraint; as discussed in \\cite{BGH}, the rank \nof the brackets of the primary constraints would then no longer be constant, and the Dirac \nbrackets would become singular in some regions of phase space.}  and obtain the further \ncommutators\n\\begin{align}\n&\\left\\{ T[\\xi],\\Delta\\right\\} = -\\epsilon^{ij}D_i\\left(\\xi^ae_{aj}\\right) \n     - \\mu\\epsilon^{abc}\\left(\\epsilon^{ij}e_{ai}e_{bj}\\right)\\xi_c  - \\xi^aT_a  \n    \\nonumber\\\\\n&\\left\\{ B[\\xi],\\Delta\\right\\} = \\epsilon^{ij}D_i\\left(\\xi^a\\beta_{aj}\\right) \n     - 2\\mu\\epsilon^{abc}\\left(\\epsilon^{ij}\\beta_{ai}e_{bj}\\right)\\xi_c\n     - 9\\alpha\\epsilon^{abc}\\left(\\epsilon^{ij}e_{ai}e_{bj}\\right)\\xi_c + \\xi^aB_a \n    \\nonumber\\\\\n&\\left\\{ \\Delta(x),\\Delta(x')\\right\\} = 0 ,\\label{d3}\n\\end{align}\nalong with the relation $\\{J[\\xi],\\Delta\\}=0$ that we would expect from the role of\n$J$ as a generator of local Lorentz transformations.\n\nOur task is now to diagonalize the Poisson brackets (\\ref{d2})--(\\ref{d3}), to determine\nthe first and second class constraints.  To do so, let us define ${\\hat\\xi}^a$ \nto be such that \n$$\ne_{ai}{\\hat\\xi}^a = \\beta_{ai}\\xi^a ,\n$$  \nand let\n\\begin{equation}\n{\\hat B}[\\xi] = B[\\xi] + T[{\\hat\\xi}] .\n\\label{d4}\n\\end{equation}\nThe existence of $\\hat\\xi$ is guaranteed by the invertibility of the triad; we will take\nadvantage of its nonuniqueness below.   A straightforward computation then gives\n\\begin{align}\n\\left\\{ {\\hat B}[\\xi],T[\\eta]\\right\\} &=  \n    -\\frac{\\mu}{2}J[\\xi\\times\\eta] \n    + 2\\mu T[\\xi\\times\\eta]   \\nonumber\\\\\n    &\\quad - \\frac{\\mu}{2}\\int_\\Sigma d^2x\\,\\xi\\cdot\\eta\\,\\Delta \n    - T\\bigl[\\{T[\\eta],{\\hat\\xi}\\}\\bigr] \\approx0\\nonumber\\\\\n\\left\\{ {\\hat B}[\\xi],B[\\eta]\\right\\} &= \n    2\\mu B[\\xi\\times\\eta] -\\frac{\\mu}{2}J[{\\hat\\xi}\\times\\eta]\n    + 2\\mu T[{\\hat\\xi}\\times\\eta] +6\\alpha T[\\xi\\times\\eta]  \\nonumber\\\\\n    &\\quad + \\frac{\\mu}{2}\\int_\\Sigma d^2x\\,{\\hat\\xi}\\cdot\\eta\\,\\Delta \n    - T\\bigl[ \\{B[\\eta],{\\hat\\xi}\\}\\bigr] \\approx0\\nonumber\\\\\n\\left\\{{\\hat  B}[\\xi],\\Delta\\right\\} \n     = &- \\epsilon^{abc}\\left(\\epsilon^{ij}e_{ai}e_{bj}\\right)\n     \\left(\\mu{\\hat\\xi}_c + 9\\alpha\\xi_c\\right)\n     -2\\mu\\epsilon^{abc}\\left(\\epsilon^{ij}\\beta_{ai}e_{bj}\\right)\\xi_c\\nonumber\\\\\n     &- {\\hat\\xi}^aT_a + \\xi^aB_a  - \\int_\\Sigma d^2x'\\, T_a(x')\\{\\Delta(x),{\\hat\\xi}^a(x')\\} ,\n\\label{d5}\n\\end{align}\nwhere $\\approx$ means ``weakly equal,'' that is, ``equal up to constraints.''  The first \ntwo brackets are weakly zero; if $\\hat\\xi$ can be chosen so that the third is as well, then \nthe ${\\hat B}[\\xi]$ will be first class constraints.\n\nTo see that this is possible, we first use the invertibility of $e^a{}_\\mu$ to write\n\\begin{equation}\n{\\hat\\xi}^a = e^{a\\mu}\\beta_{b\\mu}\\xi^b + e^{at}\\eta ,\n\\label{d6}\n\\end{equation}\nwhere $\\eta$ is arbitrary.  It is already evident that the last bracket in (\\ref{d5}) \ncan be made weakly zero, since the right-hand side is linear in $\\eta$.  More explicitly,\nnote that\n$$\n\\epsilon^{\\mu\\nu\\rho}e^a{}_\\mu e^b{}_\\nu e^c{}_\\rho = e\\epsilon^{abc} \n$$\nwith $e = \\det|e^a{}_\\mu|$, and therefore\n$$\n\\epsilon^{ij}e^a{}_ie^b{}_j = -e\\epsilon^{abc}e_c{}^t, \\qquad\n     \\epsilon^{ij}e^b{}_j = -e\\epsilon^{abc}e_a{}^ie_c{}^t .\n$$\nSome simple manipulation then yields\n$$\n\\left\\{{\\hat  B}[\\xi],\\Delta\\right\\} \n     \\approx -2\\mu eg^{tt}{\\eta} + 2\\mu e\\left(\\beta^{tc} - \\beta^{ct}\\right)\\xi_c\n     - 2\\mu e\\left(\\beta + \\frac{9\\alpha}{\\mu} \\right)e^{ct}\\xi_c ,\n$$\nand we can clearly choose $\\eta$ so that the right-hand side vanishes.  More than that, \nit is evident from (\\ref{b9}) that $\\eta = 0$ on shell.  In that case, ${\\hat\\xi}^a$ is\nidentical to the parameter appearing in (\\ref{c7}), and ${\\hat B}[\\xi]$ is essentially the \ngenerator of diffeomorphisms.\n\nWe now consider the remaining constraints, which we can take to be $T^a$ and \n$\\Delta$.  It is convenient to write $T^\\mu = T^ae_a{}^\\mu$.  The Poisson brackets \n(\\ref{d2})--(\\ref{d3}) then give\n\\begin{align}\n&\\left\\{ T^i(x),T^j(x')\\right\\} \\approx \n   -\\frac{\\mu}{2}\\epsilon^{ij}\\delta^2(x-x') \\nonumber\\\\\n&\\left\\{ T^i(x),T^t(x')\\right\\} \\approx \n   \\left\\{ T^t(x),T^t(x')\\right\\} \\approx 0 \\nonumber\\\\\n&\\left\\{ T^i(x),\\Delta(x')\\right\\} \\approx \n   -\\left(\\epsilon^{ij}D_j + 2\\mu eg^{ti}\\right)\\delta^2(x-x') \\nonumber\\\\\n&\\left\\{ T^t(x),\\Delta(x')\\right\\} \\approx -2\\mu eg^{tt}\\delta^2(x-x') . \\label{d7}\n\\end{align}\nIt is clear upon inspection that for any values of $\\mu$ and $\\alpha$ (except for the\nconformal limit $\\mu=0$), the matrix of Poisson brackets has a nonzero determinant.\nIn fact, as I describe in the Appendix, it is not too hard to compute its inverse explicitly.  \nHence no further combination of the $\\{T^a,\\Delta\\}$ gives an additional first class \nconstraint.   \n\nWe thus have nine canonical pairs of variables ($A^a{}_i$, $e^a{}_i$, and $\\beta^a{}_i$), \nsix first class constraints ($J^a$ and ${\\tilde B}^a$) , and four second class constraints \n($T^a$ and $\\Delta$).  Each first class constraint eliminates two phase space degrees\nof freedom, while each second class constraint eliminates one \\cite{HT}; we therefore have\n$18-12-4 = 2$ degrees of freedom left, that is, one canonical pair of free data, describing \na single local excitation.  While the values of $\\mu$ and $\\ell$, in the combination $\\alpha$, \naffect the algebra of constraints, no choice leads to a change in the types of the constraints \nor a jump in the number of degrees of freedom.  In particular, for the chiral values \n$\\mu\\ell=\\pm1$, these results agree with \\cite{GJJ}, while in the asymptotically flat limit \n$\\ell\\rightarrow\\infty$ they go smoothly to the results of \\cite{DX}.\n\n\\section{Asymptotic symmetries} \n\nLet us briefly recall a few features of the first-order formulation of (2+1)-dimensional\nEinstein gravity with a negative cosmological constant \\cite{Wittenb,Banados0,Banados}.\nThe theory has six first class constraints, which give a canonical representation \nof the underlying symmetries of the theory,  local Lorentz invariance and \ndiffeomorphism invariance.  The constraints can be combined to form two mutually \ncommuting sets of three generators.  Each set forms a Virasoro algebra, and when \nevaluated at the asymptotic symmetries of anti-de Sitter space, the algebras have\nclassical central charges.  These central charges, along with the classical conformal \nweights, provide a powerful tool for investigating the boundary conformal field theory.\n\nFor topologically massive AdS gravity, we also have two sets of first class \nconstraints, $J^a$ and ${\\hat B}^a$, which again reflect local Lorentz invariance \nand diffeomorphism invariance.  In general, though, we should not expect these to\nsplit into commuting ``left'' and ``right'' sectors; interactions are likely to\ncouple the left- and right-movers.  Indeed, from (\\ref{d2}),  the symmetry \ngenerators do not commute: the Poisson brackets of $\\hat B$ include a term \nproportional to $J$.   \n\nTo understand the central charges and conformal weights, though, it is enough to \nlook at a neighborhood of the AdS boundary.  There, from (\\ref{b9}) and (\\ref{d6}), \n\\begin{equation}\n{\\hat\\xi}^a = -\\frac{3\\alpha}{\\mu}\\xi^a .\n\\label{e0}\n\\end{equation}\nIf we define\n$$\nL_\\pm[\\xi] =  {\\hat B}[\\xi] + a_\\pm J[\\xi] ,\n$$\nit is easy to check that\n\\begin{align}\n\\left\\{ L_+[\\xi],L_-[\\eta]\\right\\}\n&= \\left\\{ {\\hat B}[\\xi] + a_+J[\\xi], {\\hat B}[\\eta] + a_-J[\\eta] \\right\\}\n\\nonumber\\\\\n  &= (2\\mu - a_+ - a_-) {\\hat B}[\\xi\\times\\eta] \n  + (3\\alpha -  a_+a_-) J[\\xi\\times\\eta] .\n\\label{e1}\n\\end{align}\nThe right-hand side of (\\ref{e1}) will vanish if\n$$\na_\\pm = \\mu\\pm\\frac{1}{\\ell} ,\n$$\nthat is,\n\\begin{equation}\nL_\\pm[\\xi] = {\\hat B}[\\xi] \n   + \\left(\\mu\\pm\\frac{1}{\\ell}\\right)  J[\\xi] .\n\\label{e2}\n\\end{equation}\nThe remaining Poisson brackets are then\n\\begin{align}\n&\\left\\{ L_\\pm[\\xi],L_\\pm[\\eta]\\right\\} \n   = \\mp {\\textstyle\\frac{2}{\\ell}}L_\\pm[\\xi\\times\\eta]  \\nonumber\\\\\n&\\left\\{ L_+[\\xi],L_-[\\eta]\\right\\} = 0 . \\label{e3a}\n\\end{align}\n\nAn added complication can arise if the parameters  $\\xi^a$ are field-dependent:\nthey may then have nontrivial Poisson brackets with the $L_\\pm$, leading to additional \nterms in the algebra (\\ref{e3a}).  In particular, we saw earlier that the parameters \ncharacterizing diffeomorphisms are of the form $\\xi^a = e^a{}_\\mu\\xi^\\mu$.   \nThe algebra thus becomes\n\\begin{align}\n&\\left\\{ L_\\pm[\\xi],L_\\pm[\\eta]\\right\\} =\n    L_\\pm\\left[ \\{L_\\pm[\\xi],e^a{}_i\\}\\eta^i \n    - \\{L_\\pm[\\eta],e^a{}_i\\}\\xi^i \\mp {\\textstyle \\frac{2}{\\ell}}(\\xi\\times\\eta)^a\\right] \n   \\nonumber\\\\\n&\\left\\{ L_+[\\xi],L_-[\\eta]\\right\\} \n   = L_-\\left[ \\{L_+[\\xi],e^a{}_i\\}\\eta^i\\right] - L_+\\left[\\{L_-[\\eta],e^a{}_i\\}\\xi^i \\right] .\n\\label{e4}\n\\end{align}\nAgain, though, matters simplify when we consider only a small neighborhood of \nthe AdS boundary.  It is clear that if we could find parameters $\\xi$ and $\\bar\\xi$\nsuch that $\\{L_+[\\xi],e^a{}_i\\} = \\{L_-[{\\bar\\xi}],e^a{}_i\\}=0$, the extra terms  \nin (\\ref{e4}) would vanish.  Globally, this is rarely possible, but we can define asymptotic \nsymmetries for which \n\\begin{align}\n&\\left\\{ L_+[\\xi], e^a{}_i\\right\\} = - \\left(\\partial_i\\xi^a \n   + \\epsilon^{abc}(\\omega_{bi} + \\frac{1}{\\ell}e_{bi})\\xi_c\\right) \\sim 0 \\nonumber\\\\\n&\\left\\{ L_-[{\\bar\\xi}], e^a{}_i\\right\\} =  - \\left(\\partial_i{\\bar\\xi}^a \n   + \\epsilon^{abc}(\\omega_{bi} - \\frac{1}{\\ell}e_{bi}){\\bar\\xi}_c\\right) \\sim 0\n\\label{e6}\n\\end{align}\nat the AdS boundary.\\footnote{It is interesting to note that the covariant derivatives \nhere are identical to those in the gauge formulation of ordinary Einstein gravity \n\\cite{Wittenb,Banados0,Banados}. } We shall see in the next section that these eliminate \nthe extra terms in the commutator of $L_+$ and $L_-$ at the boundary.\n\nEquation (\\ref{e6}) is easy to solve.  If we choose coordinates such that the leading terms \nin the metric take the form  \n$$\nds^2 = \\ell^2d\\rho^2 + e^{2\\rho}\\left(\\ell^2d\\varphi^2 - dt^2\\right) ,\n$$\nwe find two families, labeled by functions \n$f(\\varphi+t\/\\ell)$ and ${\\bar f}(\\varphi-t\/\\ell)$:\n\\begin{alignat}{2}\n&\\xi^0_f = \\frac{\\ell}{2}e^\\rho f   \\qquad\\qquad\n&&{\\bar\\xi}^0_{\\bar f} =  \\frac{\\ell}{2} e^\\rho{\\bar f} \\nonumber\\\\\n&\\xi^1_f = - \\frac{\\ell}{2}\\partial_\\varphi f   \\qquad\\qquad\n&&{\\bar\\xi}^1_{\\bar f} =  \\frac{\\ell}{2} \\partial_\\varphi {\\bar f} \\nonumber\\\\\n&\\xi^2_f =   \\frac{\\ell}{2} e^\\rho f   \\qquad\\qquad\n&&{\\bar\\xi}^2_{\\bar f}  = - \\frac{\\ell}{2} e^\\rho{\\bar f} .\n\\label{e5}\n\\end{alignat}\nI have chosen a normalization such that the zero-modes of $\\xi^t$ and ${\\bar\\xi}^t$ are\npositive and such that\n$$\n\\left[\\xi_f,\\xi_g\\right]^a = \\xi_{\\{f,g\\}}^a, \\qquad\n\\left[{\\bar\\xi}_{\\bar f},{\\bar\\xi}_{\\bar g}\\right]^a = -{\\bar\\xi}_{\\{{\\bar f},{\\bar g}\\}}^a, \n$$\nwhere $[\\xi,\\eta]^\\mu = \\xi^\\nu\\partial_\\nu\\eta^\\mu - \\eta^\\nu\\partial_\\nu\\xi^\\mu$ is\nthe ordinary commutator of (2+1)-dimensional vector fields and \n$\\{f,g\\} = f\\partial_\\varphi g - g\\partial_\\varphi f$ is the commutator of $f$ and $g$ \nviewed as one-dimensional vector fields on the circle.  Not surprisingly, the parameters (\\ref{e5})\nmatch those found in ordinary Einstein gravity \\cite{Banados0,Banados}, and agree to \nlowest order with the asymptotic AdS Killing  vectors found long ago by Brown and Henneaux \n\\cite{BH}.\n\nRestricted to such transformations, the algebra  (\\ref{e4}) now becomes\n\\begin{align}\n&\\left\\{ L_+[\\xi_f],L_+[\\xi_g]\\right\\} =  L_+\\left[[\\xi_f,\\xi_g]\\right]  + L_+[\\chi(f,g)]\n   \\nonumber\\\\\n&\\left\\{ L_-[{\\bar\\xi}_{{\\bar f}}], L_-[{\\bar\\xi}_{{\\bar g}}]\\right\\}\n   =  L_-\\left[[{\\bar \\xi}_{\\bar f},{\\bar \\xi}_{\\bar g}]\\right] \n   + L_-[\\chi({\\bar f},{\\bar g})]\n   \\nonumber\\\\\n&\\left\\{ L_+[\\xi_f],L_-[{\\bar\\xi}_{\\bar g}]\\right\\} = \n   -({\\hat B} + \\mu J)[\\chi(f,{\\bar g})] - J[{\\tilde\\chi}(f,{\\bar g})] , \n\\label{e3b}\n\\end{align}\nwith\n\\begin{alignat}{2}\n&\\chi^1(f,g) = \\frac{\\ell}{4}\\partial_\\varphi\\left(f\\partial_\\varphi g - g\\partial_\\varphi f\\right),\n\\quad &&\\chi^0(f,g) =\\chi^2(f,g) = 0  \\nonumber\\\\\n&{\\tilde\\chi}^1(f,g) = \\frac{1}{4}\\left(f\\partial_\\varphi^2g + g\\partial_\\varphi^2f\\right) ,\n\\quad &&{\\tilde\\chi}^0(f,g) ={\\tilde\\chi}^2(f,g) = 0 .\n\\label{ex}\n\\end{alignat}\nWe shall see in the next section that the terms involving $\\chi$ and $\\tilde\\chi$ give no\ncontribution at the AdS boundary.\n\n\\section{Boundary terms and central charges}\\label{BT}\n\nUp to now, we have focused on the ``bulk'' contributions to the constraints.  We must now  \nrestore the boundary terms.  Let us first recall a few general features \\cite{BH,BHb}.  Consider\na theory of fields $\\{\\phi_i\\}$ in $n+1$ dimensions, with gauge transformations labeled\nby parameters $\\xi$ and generated by\n$$\nG[\\xi,\\phi] = \\int_\\Sigma d^n\\!x\\,{\\cal G}[\\xi,\\phi] .\n$$ \nUp to boundary terms, these generators should satisfy the appropriate gauge algebra\n$$\n\\left\\{ G[\\xi,\\phi], G[\\eta,\\phi] \\right\\} = G[\\{\\xi,\\eta\\},\\phi] ,\n$$\nwhere $\\{\\xi,\\eta\\}$ is the Lie bracket for the gauge group.\n  \nNow let us restore the boundary terms.  Under a general variation of the fields, \n$$\n\\delta G[\\xi,\\phi] = \\int_\\Sigma d^n\\!x\\, \\frac{\\delta{\\cal G}}{\\delta\\phi_i}\\delta\\phi_i\n    + \\int_{\\partial\\Sigma}d^{n-1}\\!x\\, B[\\xi,\\phi,\\delta\\phi] .\n$$\nIf the boundary term $B$ is nonzero, $G$ is said to not be ``differentiable.''  In particular, the \npresence of $B$ will lead to delta-function singularities in the Poisson brackets.  It may be\npossible to generalize the algebra to include such boundary singularities \\cite{Solo,Bering},\nbut it is normally simpler to choose boundary conditions such that $B$ is itself a total \nvariation,\n$$\nB[\\xi,\\phi,\\delta\\phi] = -\\delta Q[\\xi,\\phi]  .\n$$\nThe combination ${\\bar G}[\\xi,\\phi] = G[\\xi,\\phi]  + Q[\\xi,\\phi] $ will then have a well-defined\nvariation, with no boundary terms, and it is easy to show that\n\\begin{align}\n\\left\\{{\\bar G}[\\xi,\\phi],{\\bar G}[\\eta,\\phi]\\right\\}\n   &=  \\iint d^n\\!x'\\, d^n\\!x\\,  \n  \\frac{\\delta{\\cal G}[\\xi,\\phi]}{\\delta\\phi_i(x)}\n   \\frac{\\delta{\\cal G}[\\eta,\\phi]}{\\delta\\phi_j(x')}\\{\\phi_i(x),\\phi_j(x')\\} \\nonumber\\\\\n   &= {\\bar G}[\\{\\xi,\\eta\\},\\phi] + K(\\xi,\\eta) .\n\\label{k1}\n\\end{align}\n\nThe central term $K(\\xi,\\eta)$ arises from boundary terms in the integrals, and need not vanish.  It\nis most easily evaluated by considering the algebra (\\ref{k1}) for the ``vacuum'' configuration, \nfor which the boundary charges $Q$ vanish; the right-hand side of (\\ref{k1}) then consists solely \nof the central term.\n\nTo apply this general formalism to our case, we must first return to (\\ref{d1}) and (\\ref{d2}) \nand keep track of any boundary terms.  A straightforward calculation yields\n\\begin{align}\n&\\left\\{J[\\xi],J[\\eta]\\right\\} \n    = \\dots + \\frac{2}{\\mu}\\int_{\\partial\\Sigma}\\xi^aD_\\varphi\\eta_a d\\varphi \\nonumber\\\\\n&\\left\\{J[\\xi],T[\\eta]\\right\\} \n   = \\dots - \\int_{\\partial\\Sigma}(\\xi\\times\\eta)_ae^a{}_\\varphi d\\varphi \\nonumber\\\\\n&\\left\\{J[\\xi],B[\\eta]\\right\\} \n   = \\dots - \\int_{\\partial\\Sigma}(\\xi\\times\\eta)_a\\beta^a{}_\\varphi d\\varphi \\nonumber\\\\\n&\\left\\{T[\\xi],T[\\eta]\\right\\} = \\dots \\nonumber\\\\\n&\\left\\{B[\\xi],T[\\eta]\\right\\} \n   = \\dots + \\int_{\\partial\\Sigma}\\left[ \\xi^aD_\\varphi\\eta_a \n   + 2\\mu(\\xi\\times\\eta)_ae^a{}_\\varphi\\right] d\\varphi \\nonumber\\\\\n&\\left\\{B[\\xi],B[\\eta]\\right\\} \n   = \\dots + \\int_{\\partial\\Sigma}(\\xi\\times\\eta)_a\\left(2\\mu\\beta^a{}_\\varphi \n   + 6\\alpha e^a{}_\\varphi\\right)d\\varphi ,\n\\label{k2}\n\\end{align}\nwhere the omitted bulk terms are all proportional to the constraints, and vanish weakly. \nFor asymptotically anti-de Sitter boundary conditions, we see from (\\ref{b9}) that   \n$\\beta^a = -\\frac{3\\alpha}{\\mu}e^a$ and ${\\hat\\xi}^a = -\\frac{3\\alpha}{\\mu}\\xi^a$\nat the boundary.  Some simple algebra then gives\n\\begin{alignat}{2}\n\\left\\{ L_\\pm[\\xi],L_\\pm[\\eta]\\right\\}& &&= \\dots \n   \\pm\\frac{4}{\\ell} \\int_{\\partial\\Sigma}\\left[ \n  \\left(1 \\pm \\frac{1}{\\mu\\ell}\\right)\\xi^a D_\\varphi \\eta_a\n  + \\frac{3\\alpha}{\\mu}(\\xi\\times\\eta)^a e_{a\\varphi}\\right]\n  d\\varphi \\nonumber\\\\\n&&&= \\dots \\pm\\frac{4}{\\ell}\\left( 1 \\pm \\frac{1}{\\mu\\ell}\\right)\n    \\int_{\\partial\\Sigma}\\xi^a\\left[ \\partial_\\varphi\\eta_a\n    + \\epsilon_{abc}\\left(\\omega^b{}_\\varphi \\pm \\frac{1}{\\ell}e^b{}_\\varphi\\right) \\eta^c\n    \\right] d\\varphi \\nonumber\\\\\n\\left\\{ L_+[\\xi],L_-[\\eta]\\right\\}& &&= \\dots .\n\\label{f3}\n\\end{alignat}\nEvaluated at the AdS ``vacuum'' state, the right-hand sides of these\nexpressions are the central terms $K_\\pm$.\n\nIf our asymptotic symmetries (\\ref{e5}) were exact---that is, if (\\ref{e6}) were \nsatisfied exactly, and not just asymptotically---then the integrands on the right-hand \nside of (\\ref{f3}) would vanish.  But the symmetries are not quite exact, and a simple \ncalculation shows that\n\\begin{align}\n&\\left\\{ L_+[\\xi_f],L_+[\\xi_g]\\right\\} =  \\dots\n   + \\frac{\\ell}{32\\pi G}\\left( 1 + \\frac{1}{\\mu\\ell}\\right)\n    \\int_{\\partial\\Sigma}\\left( \\partial_\\varphi f \\partial^2_\\varphi g \n    - \\partial_\\varphi g \\partial^2_\\varphi f \\right) d\\varphi \\nonumber\\\\\n   \\nonumber\\\\\n&\\left\\{ L_-[{\\bar\\xi}_{{\\bar f}}], L_-[{\\bar\\xi}_{{\\bar g}}]\\right\\} = \\dots\n   - \\frac{\\ell}{32\\pi G}\\left( 1 - \\frac{1}{\\mu\\ell}\\right)\n   \\int_{\\partial\\Sigma}\\left( \\partial_\\varphi {\\bar f} \\partial^2_\\varphi {\\bar g} \n    - \\partial_\\varphi {\\bar g} \\partial^2_\\varphi {\\bar f} \\right) d\\varphi ,\n\\label{fx}\n\\end{align}\nwhere I have restored the factors of $16\\pi G$.  These are precisely the central terms\nfor two Virasoro algebras with central charges\n$$\nc_\\pm = \\frac{3\\ell}{2G}\\left(1\\pm\\frac{1}{\\mu\\ell}\\right),\n$$\nmatching the results (\\ref{intro1}) that had been previously obtained using very \ndifferent methods \\cite{KL,Solodukhin,Hotta}.\n\nFinally, let us directly evaluate the boundary terms $Q_{L_\\pm}$.  Here we can use\nsome results from pure Einstein gravity, where the same problem was discussed\nin \\cite{Banados0,Banados}.  Note first that from (\\ref{c3}) and (\\ref{e2}), the\nboundary terms in the variation of $L_\\pm$ are\n\\begin{align}\n\\delta L_\\pm[\\xi] \n&= \\dots - \\int_{\\partial\\Sigma} \\left[\n    \\xi^a\\delta\\beta_{a\\varphi} + {\\hat\\xi}^a\\delta e_{a\\varphi}\n   + \\frac{2}{\\mu}\\left(\\mu\\pm\\frac{1}{\\ell}\\right)\\xi^a\\delta A_{a\\varphi}\n   \\right]d\\varphi \\nonumber\\\\\n&= \\dots - \\int_{\\partial\\Sigma} \\xi^\\mu\\left[\n   e^a{}_\\mu\\delta\\beta_{a\\varphi} + \\beta^a{}_\\mu\\delta e_{a\\varphi}\n   + \\frac{2}{\\mu}\\left(\\mu\\pm\\frac{1}{\\ell}\\right)e^a{}_\\mu\\delta A_{a\\varphi}\n   \\right]d\\varphi .\n\\label{l1}\n\\end{align}\nAs before, anti-de Sitter boundary conditions require that\n$\\beta^a = -\\frac{3\\alpha}{\\mu}e^a$, and a bit of algebra reduces (\\ref{l1}) to\n\\begin{align}\n\\delta L_\\pm[\\xi] \n&= \\dots - \\int_{\\partial\\Sigma} \\xi^\\mu\\left[\n    -\\frac{6\\alpha}{\\mu}e^a{}_\\mu\\delta e_{a\\varphi} \n    + 2 \\left( 1 \\pm \\frac{1}{\\mu\\ell}\\right)e^a{}_\\mu\\delta A_{a\\varphi} \n     \\right]d\\varphi  \\nonumber\\\\\n&= \\dots - 2\\left( 1 \\pm \\frac{1}{\\mu\\ell}\\right) \\int_{\\partial\\Sigma}\n    \\xi^\\mu e^a{}_\\mu \\,\\delta\\left(\n    \\omega_{a\\varphi}\\pm\\frac{1}{\\ell}e_{a\\varphi}\\right)d\\varphi .\n\\label{l2}\n\\end{align}\nWe now adopt the boundary conditions of \\cite{Banados0,Banados}, which translate\nto\n\\begin{equation*}\n\\omega^a{}_t =  \\frac{1}{\\ell^2}e^a{}_\\varphi, \\qquad\n\\omega^a{}_\\varphi =  e^a{}_t, \\qquad \\delta e^a{}_\\rho = 0 ,\n\\label{l3}\n\\end{equation*} \nand note that for our asymptotic symmetries, $\\xi^\\varphi = \\pm\\frac{1}{\\ell}\\xi^t$.  \nThe variation (\\ref{l2}) is thus\n\\begin{align*}\n\\delta L_\\pm[\\xi] \n&= \\dots - 2\\left( 1 \\pm \\frac{1}{\\mu\\ell}\\right) \\int_{\\partial\\Sigma}\n    \\left[\\xi^t\\left(e^a{}_t\\pm\\frac{1}{\\ell}e^a{}_\\varphi\\right) + \\xi^\\rho e^a{}_\\rho\\right]\n    \\delta \\left(e_{at}\\pm \\frac{1}{\\ell}e_{a\\varphi}\\right) d\\varphi \\nonumber\\\\\n&= \\dots -\\left( 1 \\pm \\frac{1}{\\mu\\ell}\\right) \\delta \\int_{\\partial\\Sigma}\n    \\left[\\xi^t \\left(e^a{}_t\\pm \\frac{1}{\\ell}e^a{}_\\varphi \\right)\n    + 2 \\xi^\\rho\\, e^a{}_\\rho\\right]\n    \\left(e_{at}\\pm \\frac{1}{\\ell}e_{a\\varphi}\\right) d\\varphi ,\n\\label{l4}\n\\end{align*}\nand the $L_\\pm$ are thus differentiable if we add boundary terms\n\\begin{equation}\nQ_\\pm[\\xi] =  \\frac{1}{16\\pi G}\\left( 1 \\pm \\frac{1}{\\mu\\ell}\\right) \\int_{\\partial\\Sigma}\n    \\left[\\xi^t \\left(e^a{}_t\\pm \\frac{1}{\\ell}e^a{}_\\varphi \\right)\n    + 2 \\xi^\\rho\\, e^a{}_\\rho\\right]\n    \\left(e_{at}\\pm \\frac{1}{\\ell}e_{a\\varphi}\\right) d\\varphi .\n\\label{l5}\n\\end{equation}\n\nThese boundary terms are identical to those of ordinary Einstein gravity, except\nfor the prefactors of $1 \\pm \\frac{1}{\\mu\\ell}$.   That is,\n\\begin{equation}\nQ_\\pm^{\\hbox{\\tiny TMG}}[\\xi] \n    = \\left( 1 \\pm \\frac{1}{\\mu\\ell}\\right) Q_\\pm^{\\hbox{\\tiny Einstein}}[\\xi] ,\n\\label{l6}\n\\end{equation}\nin agreement with \\cite{KL,Solodukhin}.  Further, we can now verify the claim in \nthe preceding section that the $\\chi$ and $\\tilde\\chi$ terms in (\\ref{ex})\nare irrelevant at the boundary.  Indeed, these terms only appear in (\\ref{l5}) in\nthe form $\\chi^\\rho(g_{\\rho t}\\pm \\frac{1}{\\ell}g_{\\rho\\varphi})$, and\nvanish by virtue of our boundary conditions.\n\n\\section{Chirality}\n\nIt has recently been argued that topologically massive AdS gravity is chiral at \nthe critical coupling $\\mu\\ell=\\pm1$ \\cite{Stromingerb}.  In the present context, \nthis feature can be understood as follows.  \n\nConsider first a generic coupling, and let $\\xi^\\mu$ be a vector field that satisfies \nthe fall-off conditions (\\ref{e5}) but is nonzero at the boundary.  From (\\ref{fx}), \nthe constraints $L_\\pm[\\xi]$ are no longer first class:  their Poisson brackets \nare not weakly zero.  Constraints that are not first class do not generate gauge\ntransformations, but rather determine asymptotic symmetries \\cite{Benguria}.  \nHence some configurations that are formally diffeomorphic will nevertheless be\nphysically inequivalent---they will differ by a symmetry rather than a gauge \nequivalence.   As a consequence,  new ``would-be pure gauge'' degrees of freedom \nappear at the boundary, which are conjecturally the source of the degrees of \nfreedom of the black hole \\cite{Carlip,Carlipc,Carlipd}.\n\nIf $\\mu\\ell=1$, on the other hand---or, by an obvious extension, $\\mu\\ell=-1$---it \nis apparent from (\\ref{fx}) and (\\ref{l5}) that $c_-$ and $Q_-$ vanish.  Thus \n$L_-[\\xi]$ remains first class even at the boundary, and one chirality of diffeomorphisms \nextends to the boundary as a true gauge invariance.  This eliminates one chiral sector \nof the ``massless gravitons'' discussed in \\cite{LSS}.  The remaining asymptotic \nsymmetry group consists of only one copy of the Virasoro algebra, and the boundary \ntheory is thus chiral.\n\nNote, however, that this argument does not eliminate bulk excitations that are not\ndiffeomorphic to zero in the interior.  In particular, the linearized excitations of\n\\cite{CDWW} and \\cite{GKP} yield solutions with nonconstant curvature.  No \ndiffeomorphism, whether or not it extends to the boundary, can remove such \nexcitations.  \n\n \n\\section{Conclusions}\n\nThis work has, first of all, established the existence of a local degree of  freedom \nin topologically massive AdS gravity at all values of the couplings.  In \nparticular, I confirm the results of  \\cite{GJJ} for the chiral coupling $\\mu\\ell=\\pm1$.  \nThe constraint analysis presented here is, in a sense, complementary to the \nperturbative analysis of \\cite{CDWW,GKP}.  Those papers show that weak \nfield solutions exist and remain well-behaved at the AdS boundary, but cannot \naddress effects beyond the weak field approximation, while the present analysis \nis fully nonperturbative, but does not address boundary behavior.  Since the weak\nfield perturbations have negative energy (relative to the black hole), these results\ntogether provide a strong indication that the theory is unstable.\n\nOn the other hand, this work also confirms that the boundary central charges of\ntopologically massive AdS gravity are shifted, and that at the chiral coupling,\none of the two central charges vanishes.  This presents a bit of a puzzle for the\nAdS\/CFT correspondence: the central charge measures the number of states in\nthe dual conformal field theory, and the vanishing of a central charge should\nmean, in some sense, that some fields disappear.   \n\nNote, though, that the \\emph{total} central charge, $c_++c_-$, is independent\nof $\\mu$; the vanishing of $c_-$ at $\\mu\\ell=1$ is compensated by an increase\nin $c_+$.  The same behavior can be seen in the boundary conformal weights:  \nby (\\ref{l5}), when $c_-$ and $Q_-$ vanish, $Q_+$ doubles.  For the BTZ black \nhole, this is reflected in the fact that all solutions are extremal at the chiral \ncoupling \\cite{Moussa,LSS}, while the entropy nevertheless remains independent \nof $\\mu$.  How this feature is manifested in the bulk---where the constraint algebra, \nat least, shows no special behavior as couplings vary---remains a mystery.\n\nThe value of the total central charge also presents a second puzzle: it is the \nsame for topologically massive gravity as it is for ordinary Einstein gravity.  \nThe counting of states via the Cardy formula will thus match the results of \nEinstein gravity, which are already sufficient to account for for the BTZ black\nhole entropy; we will see no additional contribution from the ``massive \ngraviton'' at \\emph{any} value of the coupling constant.  This should not \nreally be such a surprise, though: the classical contribution (\\ref{intro1}) to \nthe central charge is really of order ${\\cal O}(1\/\\hbar)$, while an ordinary \npropagating field contributes ${\\cal O}(1)$.  This suggests that the classical \nPoisson bracket analysis of the boundary conformal field theory might not \ncapture enough information to tell us about the massive graviton degrees\nof freedom, which may only appear at higher orders in $\\hbar$.\n\nCan chiral topologically massive gravity be saved?  The negative-energy weak \nfield excitations of \\cite{CDWW} can be built from compactly supported initial \ndata---that is, they represent arbitrarily small and arbitrarily localized \nperturbations, which cannot be excluded by boundary conditions in any obvious \nway.  The constraint analysis developed here further shows that these perturbations\nrepresent the ``right amount'' of initial data, one free phase space degree of \nfreedom per point.  It remains conceivable, however, that higher order corrections\nto the weak field solutions violate Fefferman-Graham boundary conditions, or\nlead to a finite lower bound to the negative energies that appear perturbatively.\nUnfortunately, the one known positive energy theorem for topologically massive \nAdS gravity, which follows from the existence of a supersymmetric extension,   \ngoes in the wrong direction \\cite{Deser,Abbott}: with the sign choice for\nwhich the black hole has positive mass, the energy of local excitations is strictly \nnonpositive.  Nevertheless, a more detailed investigation of boundary conditions\nbeyond first order perturbation theory could be of interest.\n \n\n\\vspace{1.5ex}\n\\begin{flushleft}\n\\large\\bf Acknowledgments\n\\end{flushleft}\n\\noindent \nI would like to thank Stanley Deser, Marc Henneaux, Andrew Waldron, and Derek \nWise for helpful discussions.\nThis work was supported in part by the Department of Energy under grant\nDE-FG02-91ER40674.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Supersymmetric Lagrangian and Susy breaking}\n\nThe most general superfield action for a homogeneous scalar supermultiplet \ninteracting with the scalar factor in the supersymmetric FRW model\n\\cite{ORT1,TRM} has the form\n\\begin{eqnarray} \n\\rm S &=& \\rm S_{FRW} + S_{\\rm mat}, \\nonumber \\\\\nS_{FRW}&=& \\int 6\\left[- \\frac{1}{2\\kappa^2}\\frac{{I\\!\\!R}}{{I\\!\\!N}} \n{\\cal D}_{\\bar \\eta} {I\\!\\!R} {\\cal D}_\\eta {I\\!\\!R}\n+\\frac{\\sqrt{k}}{2\\kappa^2} {I\\!\\!R}^2 \\right]\nd\\eta d {\\bar \\eta} dt\\, , \\nonumber\\\\            \n S_{ mat} &=&  \\int \\left[ \\frac{1}{2} \\frac{{I\\!\\!R}^3}{{I\\!\\!N}} \n{\\cal D}_{\\bar \\eta} {\\Phi} {\\cal D}_\\eta {\\Phi}  \n- 2 {I\\!\\!R}^3 g({\\Phi} )\\right] d\\eta d {\\bar \\eta} dt\\, ,  \n\\label{action1}\n\\end{eqnarray}\nwith $k=0,1$ stands for flat and closed space, and \n$\\kappa^2 =8\\pi G_N$, where $G_N$ is Newton's constant of gravity,\n $(\\hbar =c=1)$. The units for the constants and fields in this\nwork are the following: $[\\kappa^2]=\\ell^2$, $[{I\\!\\!N}]=\\ell^0 , \n[{I\\!\\!R}]=\\ell^{1}, [{\\Phi} ]=\\ell^{-1}$  and\nsuperpotential $[g({\\Phi} )]=\\ell^{-3}$, where $\\ell $ correspond to units of \nlenght\n\nFor the one-dimensional gravity superfield ${I\\!\\!N} (t,\\eta ,\\bar\\eta)$ \n$({I\\!\\!N}^\\dagger ={I\\!\\!N})$ we have the following series expansion\n\n\\begin{equation}\n{I\\!\\!N}(t,\\eta ,\\bar\\eta )=N(t)+i\\eta\\bar\\psi^\\prime (t)+i\\bar\\eta\\psi^\\prime\n(t)+\\eta\\bar\\eta {\\cal V}^\\prime (t) \\, ,\n\\label{expansion1}\n\\end{equation}\nwhere $N(t)$ is the lapse function and also we have introduced the \nreparametrization $\\psi^\\prime (t)= N^{\\frac{1}{2}} (t)\\psi (t)$ \nand ${\\cal V}^\\prime (t) = {I\\!\\!N}(t) {\\cal V}(t) +\\bar \\psi (t)\\psi (t)$.\n\nThe Taylor series expansion for the superfield ${I\\!\\!R} (t,\\eta ,\\bar\\eta)$\nwith the scalar factor $R(t)$ has the similar form\n\n\\begin{equation}\n{I\\!\\!R} (t,\\eta ,\\bar\\eta)= R(t)+i\\eta\\bar\\lambda^\\prime (t)\n+i\\bar\\eta\\lambda^\\prime (t) +\\eta\\bar\\eta {\\cal B}^\\prime (t),\n\\label{expansion2}\n\\end{equation}\nwhere $\\lambda^\\prime (t) =\\kappa N^{\\frac{1}{2}} (t)\\lambda (t)$ and\n${\\cal B}^\\prime (t)=\\kappa N(t){\\cal B}(t)+\\frac{\\kappa}{2} \n(\\bar\\psi \\lambda -\\psi \\bar\\lambda)$.  \n\nThe component of the (scalar) matter superfields $\\Phi (t,\\eta ,\\bar\\eta)$\nmay be written as $(\\Phi^+ =\\Phi )$\n\n\\begin{equation}\n\\Phi =\\varphi (t)+i\\eta\\bar\\chi^\\prime (t)+i\\bar\\eta \\chi^\\prime (t)+F^\\prime\n(t)\\eta \\bar \\eta\n\\label{matter}\n\\end{equation}\nwhere $\\chi^\\prime (t) =N^{\\frac{1}{2}} (t)\\chi (t)$ and $F^\\prime (t)=NF+\n\\frac{1}{2} (\\bar \\psi \\chi -\\psi \\bar \\chi )$, \nbesides,\n${\\cal D}_\\eta = \\frac{\\partial}{\\partial\\eta}+i \\bar\\eta \n\\frac{\\partial}{\\partial t}$ and ${\\cal D}_{\\bar\\eta}=-\n\\frac{\\partial}{\\partial\\bar\\eta}-i\\eta \\frac{\\partial}{\\partial t}$ are the\nsupercovariant derivatives of the conformal supersymmetry $n=2$, which has\ndimension $[{\\cal D}_\\eta ]=[{\\cal D}_{\\bar\\eta}] =\\ell^{-\\frac{1}{2}}$. \n\nAfter the integration over Grassmann  complex coordinate $\\eta$ and \n$\\bar\\eta$, and making the following redefinition of the ``fermion\"  fields \n(Grassmann variables) \n$\\lambda (t) \\to \\frac{1}{3} R^{-\\frac{1}{2}}(t)\\lambda(t)$ and \n$\\chi (t) \\to R^{-3\/2} (t) \\chi (t)$, we find the Lagrangian in which \nthe fields ${\\cal B}(t)$ and $F(t)$\nare auxiliary and they can be eliminated with the help of their equations \nof motion.\n\nFinally, the Lagrangian in terms of components of the superfields \n${I\\!\\!R}$, ${I\\!\\!N}$, $\\Phi$ takes the form\n\n\\begin{eqnarray}\nL &=&-\\frac{3}{\\kappa^2} \\frac{R({\\cal D} R)^2}{N} +\\frac{2}{3} i\\bar\\lambda\n{\\cal D}\\lambda +\\frac{\\sqrt{k}}{\\kappa} R^{\\frac{1}{2}} (\\bar\\psi \\lambda -\n\\psi\\bar\\lambda ) \\nonumber \\\\\n&+& \\frac{1}{3} NR^{-1} \\sqrt{k} \\bar\\lambda\\lambda +\\frac{3k}{\\kappa^2} NR+\n\\frac{R^3}{2}\\frac{({\\cal D}\\varphi)^2}{N}-i\\bar\\chi {\\cal D}\\chi \\nonumber \\\\\n&-& \\frac{3}{2} \\sqrt{k}NR^{-1}\\bar\\chi\\chi -\\kappa^2 N g(\\varphi )\\bar\\lambda\n\\lambda -6\\sqrt{k} N g(\\varphi)R^2  \\label{lagra}  \\\\\n&-&NR^3 V(\\varphi)+\\frac{3}{2}\\kappa^2 N g(\\varphi)\\bar\\chi\\chi \n+\\frac{i\\kappa}{2} {\\cal D}\\varphi (\\bar\\lambda \\chi +\\lambda\\bar\\chi ) \n\\nonumber \\\\\n&-& 2 N \\frac{\\partial^2 g(\\varphi)}{\\partial\\varphi^2} \\bar\\chi\\chi \n-\\kappa N \\frac{\\partial g(\\varphi)}{\\partial\\varphi}(\\bar\\lambda\\chi \n- \\lambda\\bar\\chi ) + \n\\frac{\\kappa^2}{4}R^{-3\/2}(\\psi\\bar\\lambda -\\bar\\psi\\lambda )\\bar\\chi\\chi \n\\nonumber \\\\\n&-& \\kappa R^{3\/2} (\\bar\\psi\\lambda - \\psi\\bar\\lambda) g(\\varphi)+R^{3\/2} \n\\frac{\\partial g(\\varphi)}{\\partial \\varphi} (\\bar\\psi \\chi -\\psi \\bar \\chi)\\,\n ,  \\nonumber\n\\end{eqnarray} \nwhere ${\\cal D} R=\\dot R-\\frac{i\\kappa}{6} R^{-\\frac{1}{2}} \n(\\psi\\bar\\lambda +\\bar\\psi\\lambda )$ and \n${\\cal D}\\varphi =\\dot \\varphi -\\frac{i}{2} R^{-\\frac{3}{2}}\n(\\bar\\psi\\chi +\\psi\\bar\\chi)$ are the supercovariant derivatives, and \n${\\cal D}\\lambda =\\dot\\lambda -\\frac{i}{2} {\\cal V}\\lambda$, \n${\\cal D}\\chi =\\dot\\chi -\\frac{i}{2}{\\cal V}\\chi$ are the ${\\cal U}$ (1) \ncovariant derivatives.\n\nThe potential for the homogeneous scalar fields \n\\begin{equation}\nV(\\varphi ) =2 \\left( \\frac{\\partial g(\\varphi)}{\\partial\\varphi}\\right)^2\n- 3\\kappa^2 g^2(\\varphi),\n\\label{potential}\n\\end{equation}\nconsists of two terms, one of them is the potential for the scalar field in \nthe case of global supersymmetry. The potential (\\ref{potential}) \n is not positive semi-definite in contrast with the standard \nsupersymmetric quantum mechanics.\nUnlike the standard supersymmetric quantum mechanics, our model,  \n describing the minisuperspace approach to\nsupergravity coupled to matter, allows the  supersymmetry breaking when the \nvacuum energy is equal to zero $V(\\varphi)=0$.  \n\nIn order to give some implications of spontaneous supersymmetry breaking we \ndisplay the potential $V(\\varphi)$ (\\ref{potential}) in terms of the \nauxiliary fields $F(t)$ and ${\\cal B}(t)$, \n\n\\begin{equation}\nV(\\varphi) = \\frac{1}{2} F^2 - \\frac{3}{\\kappa^2 R^2} {\\cal B}^2\\, ,\n\\label{po-aux} \n\\end{equation}\nwhere the bosonic $F$ and ${\\cal B}$  are\n\\begin{equation}\nF=2 \\frac{\\partial g(\\varphi)}{\\partial\\varphi } \\, , \\,\\qquad \n{\\cal B}= -\\kappa^2 R\\, g(\\varphi) \\, .\n\\end{equation}\n\nThe selection rules for the occurrence of spontaneous supersymmetry breaking \nare\n\n\\begin{eqnarray}\n&{\\rm i)}&\\qquad \\frac{\\partial V(\\varphi)}{\\partial\\varphi} \n= 4\\frac{\\partial g}{\\partial\\varphi} \n\\left[ \\frac{\\partial^2 g}{\\partial\\varphi^2} \n-\\frac{3}{2} \\kappa^2 g \\right]=0, \n\\,\\, {\\rm at} \\,\\, \\varphi = \\varphi_0 \\label{uno}\\\\\n&{\\rm ii)}&\\qquad V (\\varphi_0 )= 0 \\Rightarrow\n \\left[ \\left( \\frac{\\partial g}{\\partial\\varphi} \\right)^2 -\\frac{3}{2} \n\\kappa^2 g^2 \\right] =0 , \\label{dos}\\\\\n&{\\rm iii)}&\\qquad F= 2\\frac{\\partial g(\\varphi)}{\\partial\\varphi} \\not =\n0 \\, , \\, {\\rm at} \\, \\varphi = \\varphi_0, \\label{tres} \n\\end{eqnarray}\n\nThe first condition implies the existence of a minimum, in the scalar field;\nthe second condition  is the absence of the cosmological constant, the third \ncondition  is for the  breaking of supersymmetry.\nThe measure of this breakdown is the term $-\\kappa^2 g(\\varphi )\nN\\bar\\lambda\\lambda$ in the lagrangian (\\ref{lagra}). Furthermore, we can \nidentify  \n\\begin{equation}\nm_{3\/2} = \\kappa^2 g(\\varphi_0)= \\frac{g(\\varphi_0)}{M^2_{pl}}\n\\label{mass}\n\\end{equation}\nas the gravitino mass in the effective supergravity theory, and \n$M_{pl}=\\frac{1}{\\kappa} = \\frac{1}{\\sqrt{ 8\\pi G_N}}= 2.4 \\times 10^{18}Gev$\nis the reduced Planck mass.\n \nThe factor $R$ in the kinetic terms of the  scalar factor \n$-\\frac{3}{\\kappa^2N} R (\\dot R)^2$ plays the role of a ``metric'' tensor in \nthe Lagrangian, the kinetic energy for the scalar factor is negative \ndefinite, this is due to the fact, that the particle-like fluctuations do not \ncorrespond to the scalar factor, and the kinetic energy of the scalar \nfields $\\frac{R^3}{2N} \\dot\\varphi^2$ \nis positive.\n\\section{The corresponding supersymmetric quantum mechanics}\n\nThe hamiltonian can be calculated in the usual way. We have the classical \ncanonical Hamiltonian\n\n\\begin{equation}\nH_{can} =NH+\\frac{1}{2}\\bar\\psi \\, S-\\frac{1}{2}\\psi \\, \\bar S+ \n\\frac{1}{2} {\\cal V} \\, {\\cal F} ,\n\\label{hamiltonian}\n\\end{equation} \nwhere $H$ is the Hamiltonian of the system, $S$, $\\bar S$  are supercharges and\n${\\cal F}$ is the  ${\\cal U}(1)$ rotation generator. The form of the canonical\nHamiltonian (\\ref{hamiltonian}) explains the fact, that $N, \\psi, \\bar\\psi$ \nand ${\\cal V}$ are  Lagrange multipliers which enforce  only the first class \nconstraints $H=0, S=0, \\bar S=0$ and ${\\cal F}=0$, which express the \ninvariance of the conformal\n$n=2$ supersymmetric transformations. As usual with the Grassmann variables,\nwe have the second-class constraints which can be eliminated by the Dirac \nprocedure, as a result only the following non-zero \nDirac brackets $\\left\\{\\,\\, ,\\,\\,\\right\\}$ remain. For Grassmann  variables \n$\\lambda ,\\bar\\lambda ,\\chi$\nand $\\bar\\chi$ \n\n\\begin{equation}\n\\left\\{\\lambda ,\\bar\\lambda \\right\\}_*=+\\frac{3}{2}i \\, , \\, \n\\left\\{\\chi ,\\bar\\chi \\right\\}_* =-i \\, .\n\\label{bracket1}\n\\end{equation}\n\nThe canonical Poisson brackets for the $R, \\pi_R$ and $\\varphi, \\pi_\\varphi$,\n have the following form\n\\begin{equation}\n\\left\\{R, \\pi_R\\right\\}_{Pb} =-1 \\, , \\, \\left\\{\\varphi ,\\pi_\\varphi \n\\right\\}_{Pb} =-1 \\, .\n\\label{bracket2}\n\\end{equation}\n\nIn a quantum theory the brackets (\\ref{bracket1}) and (\\ref{bracket2}) must be \nreplaced by anticommutator\n\n\\begin{equation}\n\\left\\{\\lambda , \\bar\\lambda \\right\\}=-\\frac{3}{2} \\, , \\, \n\\left \\{\\chi ,\\bar\\chi \\right\\} =1 ,\n\\label{bra}\n\\end{equation}\nand can be considered as generators of the Clifford algebra,  as well as the \ncommutators\n\\begin{equation}\n\\left[R,\\pi_R\\right] =-i \\, , \\,\\left [\\varphi , \\pi_\\varphi \\right] =-i\\, .\n\\end{equation}\n\nThe quantization procedure takes  into account the dependence of the \nLagrangian on the metric ``R\", but at the quantum level we must consider the\nnature of the Grassmann variables $\\lambda ,\\bar\\lambda ,\\chi$ and $\\bar\\chi$,\nand with respect to these ones we perform the antisymmetrizations, then\n we can write the bilinear combination in the form of the commutators e.g. \n$\\bar\\chi \\chi \\to \\frac{1}{2}\\left [\\bar\\chi ,\\chi \\right]$ and this leads \nto the \nfollowing quantum \nHamiltonian $H$\n\\begin{eqnarray}\nH &=& -\\frac{\\kappa^2}{12} R^{-\\frac{1}{2}} \\pi_R R^{-\\frac{1}{2}} \n\\pi_R -\n\\frac{3k R}{\\kappa^2} -\\frac{1}{6} \\frac{\\sqrt{k}}{R} \\left[\\bar\\lambda , \n\\lambda \\right]\n+ \\frac{\\pi^2_\\varphi}{2R^3}  \\nonumber \\\\\n&&- \\frac{i\\kappa}{4R^3}\\pi_\\varphi \\left(\\left[\\bar\\lambda ,\\chi \\right]\n+\\left[\\lambda ,\\bar\\chi \\right] \\right)\n-\\frac{\\kappa^2}{16R^3} \\left[\\bar\\lambda ,\\lambda \\right] \n\\left[\\bar\\chi ,\\chi \\right]+ \\frac{3\\sqrt{k}}{4 R} \\left[\\bar\\chi , \\chi \n\\right] \\nonumber \\\\\n&&+ \\frac{\\kappa^2}{2} g(\\varphi) \\left[\\bar\\lambda ,\\lambda \\right]\n+6\\sqrt{k}\\,  g(\\varphi) R^2 + R^3 V(\\varphi) +\\frac{3}{4} \\kappa^2 \ng(\\varphi) \\left[\\bar\\chi ,\\chi \\right]  \\nonumber \\\\\n&& + \\frac{\\partial^2 g(\\varphi)}{\\partial \\varphi^2} \\left[\\bar\\chi ,\\chi \n\\right]+ \\frac{1}{2}\\kappa \\frac{\\partial g(\\varphi)}{\\partial\\varphi} \n\\left( \\left[ \\bar\\lambda ,\\chi \\right]-\\left[\\lambda ,\\bar\\chi \\right]\n\\right) \\, ,\n\\label{hamil}\n\\end{eqnarray}\nwhere $\\pi_R = i \\frac{\\partial}{\\partial R}$ and\n$\\pi_\\varphi = i \\frac{\\partial}{\\partial \\varphi}$. \nThe supercharges $S$, $S^\\dagger$ and the fermion number operator ${\\cal F}$ \nhave the following structures\n\n\\begin{equation}\nS=A\\lambda +B\\chi \\, , \\qquad \\, S^\\dagger =A^\\dagger \\lambda^\\dagger \n+B^\\dagger  \\chi^\\dagger \\, ,\n\\end{equation}\n where  \n\\begin{eqnarray}\nA&=&\\frac{i}{3}\\kappa R^{-\\frac{1}{2}} \\pi_R - \\frac{2\\sqrt k}{\\kappa}\nR^{\\frac{1}{2}} + 2\\kappa R^{\\frac{3}{2}} g(\\varphi) + \\frac{\\kappa}{4}\nR^{-\\frac{3}{2}} \\left[ \\bar \\chi, \\chi \\right]\\, , \\nonumber \\\\\nB&=& i R^{-\\frac{3}{2}} \\pi_\\varphi + 2 R^{\\frac{3}{2}} \n\\frac{\\partial g(\\varphi)}{\\partial \\varphi} \\, ,\n\\end{eqnarray}\nwith $A^\\dagger$ and $B^\\dagger$ are hermitian to A and B respectively, \nand \n\\begin{equation}\n{\\cal F}=-\\frac{1}{3}\\left[\\bar\\lambda ,\\lambda \\right]\n+\\frac{1}{2} \\left[\\bar\\chi ,\\chi \\right] .\n\\end{equation}\n\nAn ambiguity exist in the factor ordering of these operators, such \nambiguities always arise, when the operator expression contains the product of\nnon-commuting operators $R$ and $\\pi_R$ as in our case. It is then \nnecessary to find some criteria to know which factor ordering should be \nselected. We propose the following: to integrate with measure \n$R^{\\frac{1}{2}}dR$ in \nthe inner product of two states \\cite{AFF}. In this measure the \nconjugate momentum  $\\pi_R$ is non-hermitan with \n$\\pi^\\dagger_R=R^{-\\frac{1}{2}} \\pi_R R^{\\frac{1}{2}}$,\nhowever, the combination $(R^{-\\frac{1}{2}}\\pi_R)^\\dagger =\\pi^\\dagger_R \nR^{-\\frac{1}{2}} = R^{-\\frac{1}{2}} \\pi_R$ is a hermitian one, and  \n$(R^{-\\frac{1}{2}} \\pi_R R^{-\\frac{1}{2}} \\pi_R)^\\dagger =\nR^{-\\frac{1}{2}} \\pi_R R^{-\\frac{1}{2}} \\pi_R$ is \nhermitian too, with this the parameter $p=1\/2$ is fixed.\n\nThe anticommutator value $\\{\\lambda , \\bar\\lambda \\}=-\\frac{3}{2}$ of the\nsuperpartners $\\lambda$ and $\\bar\\lambda$ of the scalar factor $R$ is negative,\nunlike the anticommutation relation for $\\chi$ and $\\bar\\chi$ in (\\ref{bra}),\n which is positive. They can be redefined and one becomes the following \nconjugate operation for the operators $\\lambda$ and $\\chi$\n\n\\begin{equation}\n\\bar\\lambda = \\xi^{-1} \\lambda^\\dagger \\xi =-\\lambda^\\dagger \\, , \\, \\qquad\n\\bar\\chi = \\xi^{-1}\n\\chi^\\dagger \\xi = \\chi^\\dagger ,\n\\end{equation}\nwhere $\\{\\lambda ,\\lambda^\\dagger \\} =\\frac{3}{2}$, and the operator $\\xi$ \npossesses the following properties\n\n\\begin{equation}\n\\lambda^\\dagger \\xi =-\\xi\\lambda^\\dagger \\, , \\,\\qquad \n\\chi^\\dagger \\xi=\\xi \\chi^\\dagger \\,\\qquad {\\rm and} \\,\\qquad\n\\xi^\\dagger = \\xi \\, .\n\\end{equation}\n\nSo, for the supercharge operator $\\bar S$ we have  the following equation\n\n\\begin{equation}\n\\bar S = \\xi^{-1} S^\\dagger \\xi \\, .\n\\end{equation}\n\nFor the quantum generators $H, S, \\bar S$ and ${\\cal F}$ we obtain the \nfollowing superalgebra \n\n\\begin{eqnarray}\n\\left\\{ S, \\bar S\\right\\} &=&2H \\, ,\\quad \\, S^2=\\bar S^2 =0 \\, , \\qquad \\, \n\\left[S,H\\right]=0, \\nonumber\\\\\n\\left[\\bar S,H\\right]&=&0 \\, , \\quad \\left [{\\cal F}, S\\right]=-S \\, ,\\quad \n\\,\\left [{\\cal F}, \\bar S \\right ]=\\bar S \\,  . \n\\label{superalgebra}\n\\end{eqnarray}\nWe can see, that the anticommutator of supercharges $S$ and their conjugate\n$\\bar S$ under our conjugate operation has the form\n\n\\begin{equation}\n\\overline{ \\left \\{ S,\\bar S \\right\\} }= \\xi^{-1} \n\\left\\{ S,\\bar S \\right \\}^\\dagger \n\\xi = \\left\\{S,\\bar S\\right\\} , \n\\end{equation}\nand the  Hamiltonian operator is self-conjugate under the operation \n$\\bar H =\\xi^{-1} H^\\dagger \\xi$. In the case of standard supersymmetric \nquantum mechanics\nwe would have $\\bar\\lambda =\\lambda^\\dagger$ and $\\bar S=S^\\dagger$, and \nthe Hamiltonian would be positive. In our case, the algebra \n(\\ref{superalgebra}) \ndoes not define positive-definite Hamiltonian in a full agreement with the \ncircumstance that the  potential $V(\\varphi)$ (\\ref{potential}) of the scalar \nfield is not positive semi-definite in general, in contrast with the \nstandard supersymmetric quantum mechanics.\n\nWe can choose the following matrix representation for the operators \n$\\lambda ,\\bar\\lambda , \\chi , \\bar\\chi$ and $\\xi$ in the form of the \ntensorial product of matrices $2\\times 2$\n\\begin{eqnarray}\n\\lambda &=&\\sqrt{\\frac{3}{2}}\\sigma_{(-)}\\otimes 1 \\, \n,\\qquad \\,\\bar\\lambda =-\\sqrt{\\frac{3}{2}}\\sigma_{(+)}\n\\otimes 1 , \\nonumber\\\\\n\\chi &=& \\sigma_3 \\otimes \\sigma_{(-)} \\, ,\\qquad \\, \\bar\\chi =\\sigma_3 \\otimes\n\\sigma_{(+)} \\, , \\qquad \\,\n\\xi = \\sigma_3 \\otimes 1 \\, ,\n\\end{eqnarray}\nwhere $\\sigma_{\\pm}=\\frac{\\sigma_1 \\pm i \\sigma_2}{2}$,  $\\sigma_1$,\n$\\sigma_2$ and $\\sigma_3$ are the Pauli matrices.\n\n\\section{The wave function for the zero energy state}\n\n\nIn the quantum theory the first-class constraints $H=0, S=0, \\bar S=0$ and \n${\\cal F}=0$ associated with the invariant action (\\ref{action1}) under the\nlocal $n=2$  conformal supersymmetry become conditions on the wave function \n$\\Psi$. So that any\nphysical state must obey the following quantum constrains\n\n\\begin{equation}\n\\hat H \\, \\Psi =0, \\hat S \\, \\Psi =0, {\\hat{\\bar{S}}}\\, \\Psi = 0 \\,\\, \n{\\rm and} \\,\\,\n\\hat{\\cal F} \\, \\Psi =0 ,\n\\label{physical}\n\\end{equation}\nwhere the first equation in (\\ref{physical}) is the Wheeler-DeWitt equation \nfor the minisuperspace model. The eigenstates of the Hamiltonian (\\ref{hamil}) \nhave four-components \n\n\\begin{equation}\n\\Psi (R,\\varphi ) =\n\\left[ \\begin{array}{c}\n \\psi_1(R,\\varphi ) \\\\\n \\psi_2(R,\\varphi ) \\\\\n \\psi_3(R,\\varphi ) \\\\\n \\psi_4(R,\\varphi )  \n\\end{array}\n\\right] \\, .\n\\end{equation}\n\nWe have rewritten the equations $S\\, \\Psi=0$ and $\\bar S \\, \\Psi=0$ in the \nfollowing form \n\n\\begin{eqnarray}\n\\left(\\lambda\\bar S - \\bar\\lambda S\\right) |\\Psi>& =& \\left[- \\left( \n\\frac{A-A^\\dagger}{2}\\right) \\left\\{ \\bar \\lambda , \\lambda \\right\\}\n-\\left( \\frac{A+A^\\dagger}{2} \\right) \\left[\\bar\\lambda ,\\lambda \\right]\n\\right. \\nonumber \\\\\n&&- \\left. \\left(\\frac{B-B^\\dagger}{2}\\right)\n(\\bar\\lambda\\chi +\\lambda\\bar\\chi )+\\left( \\frac{B+B^\\dagger}{2} \\right) \n(\\lambda\\bar\\chi -\\bar\\lambda\\chi )\\right] |\\Psi> =0, \\\\ \n\\left( \\bar\\chi S -\\chi\\bar S\\right)|\\Psi> &=&\\left[ \\left(\n\\frac{A-A^\\dagger}{2}\\right)(\\bar\\chi\\lambda +\\chi\\bar\\lambda )\n+\\left(\\frac{A+A^+}{2}\\right)(\\bar\\chi\\lambda -\\chi\\bar\\lambda) \\right. \n\\nonumber \\\\\n&& -\\left. \\left(\\frac{B-B^\\dagger}{2}\\right)\\left \\{\\bar \\chi , \\chi \\right\\}\n+\\left(\\frac{B+B^\\dagger}{2}\\right) \\left[\\bar\\chi ,\\chi\\right] \n\\right]|\\Psi> = 0 \\, .\n\\end{eqnarray}\n\nUsing a matrix representation for $\\lambda, \\bar \\lambda, \\chi$ and\n$\\bar \\chi$, we found that only $\\psi_4$ have the right behaviour when\n $R\\to \\infty$ because $\\psi_4 \\to 0$, \nand due that the others components $\\psi_1 , \\psi_2$ and $\\psi_3$ for the wave \nfunction at $R\\to \\infty$ are infinite, we consider these components as no\nphysical. Thus, there is a normalizable component $\\psi_4$ for H such that \n$S \\Psi = \\bar S \\Psi =0$, and this eigenstate corresponds to the\nground state with eigenvalue $E=0$. The wave function $\\Psi$ has  also\nthe non-normalizable components  $\\psi_1, \\psi_2$ and $\\psi_3$  for $E=0$,\nbut for them the eigenvalue of $H$ is non-zero, for this case all components\nare normalizable\n\nSo, the partial differencial equations for $\\psi_4 (R,\\varphi)$ have \nthe following forms\n\n\\begin{eqnarray}\n\\left[-R^{-\\frac{1}{2}} \\frac{\\partial}{\\partial R} - 6 g(\\varphi)\nR^{3\/2}+\n6\\sqrt{k} M^2_{pl} R^{\\frac{1}{2}}+\\frac{3}{4} R^{-3\/2} \\right] \\psi_4 &=&0,\n\\label{factor}\\\\\n\\left[\\frac{\\partial}{\\partial \\varphi}+2R^3 \n\\frac{\\partial g(\\varphi)}{\\partial\\varphi} \\right] \\psi_4 &=& 0 \\, .\n\\label{scalar}\n\\end{eqnarray}\n\nWe get as solution for (\\ref{factor}) and (\\ref{scalar})\n\n\\begin{equation}\n\\Psi\\equiv \\psi_4 (R,\\varphi)=C_o\\,R^{3\/4} e^{(-2g(\\varphi)\\,R^3+\n3\\sqrt{k} M^2_{pl} R^2)} \\, .\n\\label{solution}\n\\end{equation}\n\nThe scalar product  for the solution (\\ref{solution}) is normalizable \nwith the measure $R^{\\frac{1}{2}}\\, dR \\,d\\varphi$ and for the superpotential \n$g(\\varphi \\to\\pm\\infty )\\to \\infty$. \n\nConsequently the solution \n(\\ref{solution}) is the eigenstate of the Hamiltonian (\\ref{hamil}) with \n zero energy  and also with zero fermionic number.\n\nThe superpotential for the fluctuating scalar fields \n$\\varphi =\\varphi_0 + \\tilde\\varphi$ near the minimum of the potential \n$V(\\varphi_0)=0$ has the following form  \n(see (\\ref{uno},\\ref{dos},\\ref{tres}, \\ref{mass})).\n\n\\begin{eqnarray}\ng(\\tilde\\varphi)&=&g(\\varphi_0)+ \n\\frac{\\partial g(\\varphi_0)}{\\partial\\varphi} \\tilde\\varphi \n+\\frac{1}{2} \\frac{\\partial^2 g(\\varphi_0)}{\\partial\\varphi^2}\\tilde\\varphi^2 \n \\nonumber\\\\\n&=& m_{3\/2} M^2_{pl}\\left[1 + \\sqrt{\\frac{3}{2}} \\frac{\\tilde \\varphi}{M_{pl}}+\n\\frac{3}{4} \\,\\frac{\\tilde \\varphi^2}{M^2_{pl}}  \\right]=\nm_{3\/2} M^2_{pl} \\, f(x) \\, ,\n\\end{eqnarray}\nwhere $f(x)=1 + \\sqrt{\\frac{3}{2}} x+ \\frac{3}{4} \\,x^2$, with \n$x=\\frac{\\tilde \\varphi}{M_{pl}}$.\n\nSo, as an example we can consider the case $k=0$ \n\\begin{eqnarray}\n 1&=& C_o^2 \\int^{\\infty}_{-\\infty}\n\\int^\\infty_0 R^{\\frac{3}{2}} e^{-4g( \\tilde\\varphi)R^3} \\,\nR^{\\frac{1}{2}}\\, dR \\,d\\tilde\\varphi \\nonumber\\\\\n&=&\\frac{C_o^2}{12} \\int^{\\infty}_{-\\infty}\n\\frac{d\\tilde\\varphi}{g(\\tilde\\varphi)}= \\frac{C_o^2}{12 m_{3\/2} M_{pl}}\n\\int^{\\infty}_{-\\infty} \\frac{d x}{f(x)} = \n\\frac{C_o^2 \\sqrt {2} \\pi}{12 \\sqrt{3} m_{3\/2} M_{pl}} \\, ,\n\\end{eqnarray}\nthus, the normalization constant has the following value\n\\begin{equation}\nC_o=  \\left( \\frac{3}{2} \\right)^{\\frac{1}{4}} \n\\sqrt{\\frac{6 m_{3\/2}\\,M_{pl}}{\\pi}}.\n\\end{equation}\nThe behaviour for the wave-function $\\Psi$ in the $k=0$ case is shown in  \nFigure 1.\n\nThe spectation value for the scalar factor $R$ with the chosen measure is\n\n\\begin{eqnarray}\n\\overline R &=&\n<\\Psi|R|\\Psi> = C_o^2 \\int^\\infty_{-\\infty} d \\tilde\\varphi \\int^\\infty_{0}\n\\left[ R^3 e^{-4g(\\tilde\\varphi)\\, R^3} dR  \\right] \\nonumber\\\\\n&=&\\left( \\frac{ \\sqrt{3}\\Gamma\\left( \\frac{4}{3}\\right)}{4\\pi 2^{1\/6} } \n\\int^\\infty_{-\\infty} \\frac{dx}{\\left[ f(x) \\right]^{4\/3}} \\right)\n\\left( \\frac{M_{pl}}{m_{3\/2}} \\right)^{\\frac{1}{3}} \\frac{1}{M_{pl}}\\, ,\n\\end{eqnarray}\nwhere $\\Gamma(\\frac{4}{3})$ is the Gamma function.\n\nThe size of the universe in the supersymmetry breaking state is of the \norder of\n\\begin{equation}\n\\overline R = C_1 \\left(\\frac{M_{pl}}{m_{3\/2}} \\right)^{\\frac{1}{3}} \n\\ell_{pl}\\, ,\n\\end{equation}\nwhere $\\ell_{pl}= \\frac{1}{M_{pl}}=\\sqrt{8 \\pi G_N}$ is the Planck length,\nand $C_1$ has the following value\n\\begin{equation}\nC_1= \\frac{2^{1\/6} \\sqrt{3} \\left(\\frac{3}{4} \\right)^{\\frac{1}{3}}}{2 \n\\left( \\frac{3}{8}\\right)^{\\frac{5}{6}}} \n\\frac{\\Gamma\\left(\\frac{4}{3} \\right) \n\\left[\\Gamma\\left(\\frac{2}{3} \\right)\\right]^2 }{\n\\Gamma\\left(\\frac{5}{3}\\right) \n\\left[\\Gamma\\left(\\frac{1}{3} \\right)\\right]^2   } \\approx 0.505468 \\, .\n\\end{equation}\n\n\\section{Conclusion}\nUsing the ideas of the supersymmetry breaking selection rules under local n=2 \nconformal supersymmetry, we have presented  a solution to the wave function\n$\\Psi$ of the universe for the FRW cosmological model $(k=0)$, that is \nnormalizable. For this proposal it was necessary to define one ``weighted'' \ninner product with the factor $R^{1\/2}$ and  we constructed the corresponding\nhermitian operators too.\nTherefore, it  was  straighforward  to find the behaviour of the\n spectation value \nfor the scalar factor like \n$\\overline R = C_1 \\left(M_{pl}\/m_{3\/2} \\right)^{1\/3} \n\\ell_{pl}$, giving us the size of our universe.\n\nIn the proposed framework it is also possible to include the potential of \nhybrid inflation scenario \\cite{LS,KS}, since the corresponding potential term \ncan be easily introduced in the supersymmetric case. This subject will be \nreported elsewhere. \n\n\n\\section{\\bf Acknowledgments}\nThanks to I. Lyanzuridi, E. Ivanov, S. Krivonos, \nL. Marsheva and A. Pashnev for their interest in this paper. \nThis work was supported in part by CONACyT grant 3898P-E9608.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAlthough the initial mass function of metal-free stars formed out of primordial matter has not yet been determined, there is evidence that metal-free stars of low and intermediate mass were formed. \n   For example, the fact that the frequency of stars with strong carbon enhancements is much larger than in the case of population I and II stars \\citep[see also the review by Beers \\& Christlieb 2005] {ros98,luc05} is consistent with theoretical predictions. \n   One of the most prominent characteristics of models of low-mass and intermediate-mass extremely metal-poor stars (EMPSs) is that they become carbon stars at an earlier stage of evolution than the stars of younger populations \\citep[hereinafter FII00]{fuj00}.  \n  Recently, it is argued that EMP stars survived to date were formed as a secondary component in the binary systems of massive companions \\citep{Komiya07}. \n   As another example, the peculiar abundance characteristics of the most metal poor stars yet discovered \\citep{chr02} can be understood in terms of the evolutionary properties of $Z = 0$ or extremely iron-poor models in an intermediate-mass interacting binary \\citep{sud04}. \n   Thus, the evolutionary characteristics of $Z = 0$ models have direct relevance to discussions of star formation in the early universe.\n\nEMP stars may alter their initial surface abundances by bringing to the surface products of internal nuclear transformations which occur in unique ways.  \n   For initial CNO abundances $\\textrm{Z}_{\\rm CNO} \\lesssim 10^{-7}$, the outer edge of the convective zone, generated by the first off-center helium flash in the hydrogen-exhausted core extends into the hydrogen-rich layers, eventually leading to the enrichment of carbon and nitrogen in the surface (Fujimoto et al.~1990, hereinafter FIH90; Hollowell et al.~1990). \n   This ``He-flash driven deep mixing (He-FDDM)'' mechanism is distinguished from the third dredge-up in asymptotic giant branch (AGB) stars of Populations I and II which enriches surface material in $^{12}$C \\citep{ibe75}.  \n   Even if we take into account the possible surface pollution by the accretion of metal-rich gas after birth, the interior evolution of metal-poor stellar models is not altered as long as the original CNO abundance satisfies $\\textrm{Z}_{\\rm CNO} \\leq 10^{-8}$ \\citep{fuj95}.  \n\nMany calculations of Population III (hereinafter Pop.III) star evolution have been published \\citep[see FIH90; ][and references therein]{sud04}.\nBased on a top heavy initial mass function for the primordial cloud \\citet{eze71} first computed sets of massive metal-free models. \nComputations of the evolution of low mass metal-deficient stars were first carried out by \\cite{wag74} in order to provide a set of stellar lifetimes for use in calculations of galactic chemical evolution.\nComputations of low-mass, zero-metallicity star evolution were performed by \\citet{cas75} up to the exhaustion of hydrogen at the center.\n\\cite{dan82} pursued the evolution of metal-free models of mass 1 $\\ensuremath{M_\\odot}$ to the red giant branch (RGB) phase, but because of a large time step in her computation, did not find any abnormal behavior. \n\\cite{gue83} suggested that a convective instability occurs during the central hydrogen-burning phase and FIH90 found this to be the case. \nThe ``peculiar'' evolution of $Z=0$ model stars and its interpretation is revealed by the calculations of FIH90 and \\cite{hol90}, who followed the evolution of a $Z=0$ star of mass $1 \\ensuremath{M_\\odot}$ and gave explanations for the core He-H flash, a shell He-H flash, and the He-FDDM phenomenon.\nBy computing in detail the progress of the hydrogen-mixing event and the subsequent evolution, \\citet{hol90} first demonstrated that metal-free stars become CN rich carbon stars at the red giant stage.\n\\citet{cas93} and \\citet{cas96} found the core He-H flash in models of $Z=10^{-10}$, but, since their computations did not continue beyond the initiation of the helium core flash, they found no evidence for the additional events just described. \n\\cite{wei00} (hereinafter W00) also made computations of low mass metal-free stars, but terminated their computations just after the ignition of the helium core flash while the flash-driven convective zone was still growing in mass.\n\nComputations by \\citet[hereinafter S01]{sch01} support the reality\nof the He-FDDM phenomenon for low-mass zero metallicity stars, whereas computations by \\cite{sie02} (hereinafter SLL02) do not.\nWhile they found the mixing of hydrogen into the helium-flash driven convective zone, the failure of SLL02 to reproduce the He-FDDM phenomenon may be ascribed to their assumption of instantaneous mixing of elements in the helium flash convective zone; in consequence, nuclear energy released from the mixed hydrogen is distributed throughout the entire convective zone and the entropy of the zone is not built up sufficiently for the edge of the convective zone to reach deeply into the hydrogen profile.\nBecause mixing and burning of hydrogen occur on a very short timescale in the middle of the helium convective zone, following the He-FDDM event properly requires a treatment of time-dependent mixing \\citep{hol90}. \n\\citet{hol90} first derived the surface composition caused by the He-FDDM phenomenon for a $Z = 0$ model of mass $M = 1 \\ensuremath{M_\\odot}$. \n\\citet{sch02} also derived the surface composition for a model of mass $M = 0.82 \\ensuremath{M_\\odot}$ and compared the results with the observations of EMPSs.\n\nAfter the discovery of HE0107-5240 \\citep{chr02}, which held the distinction of being the most metal-poor giant known before the discovery of HE1327-2326 \\citep{fre05}, models of EMPSs were computed in an effort to determine the evolutionary history of the observed star. \nBoth \\citet[][ hereinafter P04]{pic04} and \\citet{wei04} found the He-FDDM phenomenon for $Z=0$ models of mass $M = 0.8 \\ensuremath{M_\\odot}$ and $M = 0.82 \\ensuremath{M_\\odot}$, respectively.\n\nAll but one of the works just cited find the core He-H flash; thus far, only Fujimoto and his coworkers have found the shell He-H flash driven by a violent CN-cycle burning at the base of the hydrogen-burning shell. \nWhy various groups find different evolutionary characteristics for similar masses and compositions has not yet been fully understood. \nOne reason for this is that the computation of $Z=0$ models requires not only a careful treatment of the input physics but also a careful numerical solution.\nIn particular, the treatment of convective regions requires consideration of many complicated factors including uncertain parameters. \nS01 made comparisons among models by changing parameters related to abundances, diffusion, stellar mass, and the mixing length algorithm for convection, and suggestions have been made for the causes of differences such as radiative and\/or conductive opacities and neutrino energy-loss rates (W00; S01; SLL02). \nHowever, we are not satisfied that the basic causes for differences have been pinpointed satisfactorily.\n\nIn this paper, we examine the differences in the evolutionary characteristics of low-mass $Z = 0$ models by taking into account the choice not only of the radiative and conductive opacity and of neutrino energy-loss rates, but also of nuclear reaction rates. \nThanks to the many efforts until today, useful sets of input physics are available and they have been more and more accurate for numerical computations. \n  But we think it is worth trying to check the dependence on the input physics and to elucidate the structural characteristics in detail because these efforts are still going on and because the evolution of low-mass Popualtion III stars has not been established yet. \n  In particular, as for the three alpha reaction rates, the determination of resonance states in \\nuc{12}C is still controversial \\citep{fyn05}; \n   the simulations of three body interactions are challenging, and yet, become feasible task in near future thanks to the improved computer resources \\citep[see e.g., ][and the references there]{kur05}. \n\nIn the next section, we elaborate the input physics adopted in the stellar evolution program. \nIn \\S 3, we present the result of computations of evolution of low mass $Z=0$ model stars with updated input physics, and discuss the model characteristics and their dependences on the input physics including the resonant and non-resonant reaction rates of 3 $\\alpha$ reactions. \n  In section 4, comparisons with other works with the different input physics taken into account, and, in \\S 5, we summarize conclusions.\n\n\\section{The Computational Program}\n\nThe original program to compute stellar evolution was constructed by \\citet{ibe65} and has been modified periodically \\citep[e.g., see][]{ibe75,ibe92}. \nThe size of each mass shell and the time step are regulated, respectively, by the gradients with respect to space and time of structure and composition variables.\nTypically, 200-300 mesh points are required for main sequence models and 700-800 for core helium-burning models with hydrogen-burning shells.\nThe number of equilibrium models required to follow evolution from the zero-age main sequence to the end of RGB or AGB phase varies from 5000 and 30000, with the exact number depending on initial mass. \nThe stellar structure equations are typically satisfied to better than one part in $10^{5}$.\n\nIn the program, the radiative opacity is obtained by interpolation in OPAL tables \\citep{igl96} and in tables by \\citet[][ hereinafter, AF94 tables]{ale94} and the conductive opacity is from \\citet{ito83}. \nThe equation of state involves fits by \\cite{ibe92} to work by \\cite{abe59}, \\cite{bow60}, \\cite{sla80}, \\cite{sla82}, \\cite{iye82}, \\cite{han73}, \\cite{han78}, \\cite{coh55}, and \\cite{car61}.\nNeutrino energy-loss rates are from \\citet[][ in the following I96]{ito96} for plasma, photo, pair and bremsstrahlung processes.\nIn order to compare with other works, we make use of two sets of nuclear reaction rates: those given by \\citet[][ in the following CF88]{cau88} and those given in the latest NACRE compilation \\citep{ang99}.\nNuclear screening factors for weak and strong screening are also taken into consideration using standard prescriptions \\citep[see, e.g.,][]{bohm58}, but only weak screening is dominant in the actual computational range in this work. \nNine nuclear species are considered: \\nuc{1}H, \\nuc{3}He, \\nuc{4}He, \\nuc{12}C, \\nuc{14}N, \\nuc{16}O, \\nuc{18}O, \\nuc{22}Ne, and \\nuc{25}Mg. \nAbundances of these isotopes are determined by 16 nuclear reactions which include proton, electron, and alpha captures.\n\nIn regions of high temperature and high density which are not covered by OPAL tables, we use the analytical radiative opacities from the fits by \\cite{ibe75} to \\cite{cox70a,cox70b} ones.\nAt table boundaries, no interpolation is made between table values and analytical values. \nHence, jumps in the radiative opacity occur at table boundaries, but jumps are normally smaller than a factor of 2 and do not seriously affect model convergence, primarily because, in regions not covered by the tables, the overall opacity is dominated by electron conductivity. \nAt low temperature and low density, interpolation between OPAL and AF94 tables is accomplished by setting\n$\\kappa=\\kappa_{\\rm OPAL}\\ (1-\\Theta)+\\kappa_{\\rm AF}\\ \\Theta$, where\n$\\Theta=\\sin^2{(\\pi\/2)(T-T_1)\/(T_2-T_1)}$ for the temperature $T$\nbetween $T_1=8000$ K and $T_2=10000$ K, and $\\kappa_{\\rm OPAL}$ and $\\kappa_{\\rm AF}$ are OPAL and AF94 opacities, respectively.\nIn surface layers of all models discussed here, AF94 tables provide opacities for all densities and temperatures encountered. \nIn regions of overlap, OPAL and AF94 opacities agree rather well, so switching from one table to the other introduces little uncertainty.\n\nAlthough the quantum correction to the conductivity at low temperatures has been calculated \\citep{mit84}, it is now believed that the approximation employed is not appropriate at the temperatures considered. \nTherefore, we use the analytic fits devised by I83 for the conductive opacity at low temperature; \nthe liquid metal phase for various elemental compositions as well as strong degeneracy are taken into consideration.\nStrong degeneracy prevails if $T \\ll T_{\\rm F}$, where $T_{\\rm F}(K) = 5.930 \\times 10^{9}\\\n((1+1.018(\\rho_{6}\/\\mu_e)^{2\/3} )^{1\/2} -1 )$, $\\rho_{6}$ is the\ndensity in units of $10^{6}$ g cm$^{-3}$, and $\\mu_e$ is the electron molecular weight. For the actual computations, since their approximations are good for $T \\lesssim T_{\\rm F}$, we adopt the analytical approximations of \\citet{ito83} for $T \\leq 0.5~T_{\\rm F}$.\nFor $T \\geq 2~T_{\\rm F}$, we adopt the conductive opacity used in the fits by \\citet{ibe75} to calculations by \\cite{hub69} for non-relativistic electrons and to \\cite{can70} for relativistic electrons.\nFor intermediate temperatures, interpolation between the two approximations is achieved by ``sine square'' weighting, analogous to that used in interpolating between OPAL and AF tabular opacities.\nFinally, the conductive opacity is subjected to a limit described in the Appendix.\nInterpolated conductive opacities are given in Figure~\\ref{fig:cop}, which covers the overall range of application for stellar evolution.\n\nFor neutrino energy-loss rates, we adopt the fitting formulae presented by I96, who include pair, photo, plasma, and bremsstrahlung with weak degeneracy, the liquid-metal phase with low-temperature quantum corrections and the crystalline lattice phase, and the recombination neutrino processes. \nAmong these latter processes, we do not adopt the low-temperature correction to the liquid-metal phase as calculated by \\cite{ito84} since the corrections are generally small, as stated in I96.\nFor $T \\leq 10^{7}$ K, we assume that neutrino energy-losses of all kinds can be neglected. \nThe treatment of a multicomponent gas, which is important for bremsstrahlung, is considered in applying the neutrino energy-loss rates of I96, as described in the Appendix.\n\nMass loss is neglected in our work because we are interested primarily in the stellar interior and the evolution of the helium core is not affected by modest surface mass loss. \nTo determine the temperature gradient in convective regions, we use the standard mixing length recipe by \\cite{bohm58} \\citep[see, e.g., ][]{cox68} and the mixing length is taken to be 1.5 times the local pressure scale height.\nNeither overshooting nor semi-convection are considered.\n\nIn our calculations we focus on the evolutionary trajectory up to the\nHe-FDDM event and on the thermal structure of a star at stages of interest along its path to this event; \nour computations are terminated at the onset of hydrogen mixing into the convective zone driven by a helium flash.\nThis event can occur during an off-center helium flash, \nand\/or at the beginning of the thermally pulsing AGB (TPAGB) phase, depending on the initial mass and input physics.\nSubsequent evolution after the mixing event is beyond the scope of thispaper and will be discussed in detail in a separate paper (Suda, Fujimoto, \\& Iben, in preparation).\n\n\\section{Evolution of Low-Mass Pop.III\\ Stars}\n\nThe evolution of $Z = 0$ stars has been computed from the zero-age main sequence (ZAMS) through the RGB and\/or through the TPAGB phase for model masses of $0.8 - {1.2} \\ensuremath{M_\\odot}$.\n     For all models, the initial chemical composition is $X = 0.767$, $Y = 0.233$, and $Z=0$, and the initial abundance by mass of \\nuc{3}He is $2 \\times 10^{-5}$. \nThese abundances are based on models of big bang nucleosynthesis and are the same as those chosen by FII00. \n\nTable~\\ref{tab:model} lists the models computed with all the input physics updated for this paper. \n  The first two columns give, respectively, the model identifier and the initial model mass;  the labels ``nac'' and ``cf'' mean that the model has been computed with the choice of nuclear reaction rates of NACRE and CF88, respectively.\n   The third to the ninth columns give the effective temperature and surface luminosity at the turn-off point and at the tip of the RGB, the times to reach these two stages, respectively.  \n\nFigures~\\ref{fig:hrd} and \\ref{fig:rhot} show, respectively, evolutionary tracks in the H-R diagram and in the central-density and central-temperature plane for ''cf'' and ''nac'' models of mass in the range $0.8 -1.2 \\ensuremath{M_\\odot}$. \n   All models are terminated at the onset of hydrogen mixing episodes.\n   We found that $M \\leq 1.1 \\ensuremath{M_\\odot}$ models ignite the helium burning in the off-center shell and the engulfment of hydrogen by the flash-driven convection occurs just after the peak of the major core helium-flash at the tip of RGB. \n   On the other hand, $1.2 \\ensuremath{M_\\odot}$ models undergo the helium core flash at the center without the hydrogen mixing, and encounter the hydrogen mixing in the helium-flash convection during the helium shell flash at the beginning of thermally pulsating AGB (TP-AGB) phase. \n   The two models display almost the same trajectory irrespective of the choice of nuclear reactions, with the small differences stemming from the difference in the 3$\\alpha$ reaction\\ rates, less than a factor of 2 in the temperature range $\\log T = 7.8 - 7.9$.  \n\nThe evolutionary characteristics of the core helium flash and the hydrogen mixing event are summarized in Table~\\ref{tab:he-flash}. \n   Each column gives the model identifier defined in Table~\\ref{tab:model}, the helium core mass $M_{1}\\ups{max}$ and the helium burning rate $L\\lows{He}\\ups{max}$ when the helium burning rate reaches maximum, the mass coordinate $M\\lows{BCS}$ and the maximum temperature  $T\\lows{BCS}\\ups{max}$ at the base of the convective shell, driven by the core helium flash, the helium burning rate $L\\lows{He}\\ups{mix}$ at the onset of hydrogen mixing at the RGB, and the time intervals, $\\Delta t^\\prime$ and $\\Delta t\\lows{mix}$, to it from the appearance of helium flash-driven convection and from the stage of maximum helium burning, respectively. \n   The mass $M_1$ of helium core is defined as the mass coordinate where the abundance of hydrogen is half of the surface abundance of hydrogen.\n\nIn this section, we discuss in detail the evolutionary behavior of Pop.III\\ models with respect to the differences in initial mass and input physics.\n\n\\subsection{Hydrogen Burning Phase}\n\nVariations in several quantities characterizing model stars are summarized in Figure~\\ref{fig:phys} as a function of the hydrogen abundance at the center.  \n   In a Pop.III\\ model, due to the absence of CNO catalysts, p-p chain reactions are initially the only mode of energy generation by hydrogen burning. \n   Because of the weak temperature dependence of the energy-generation rate, the central temperature keeps rising as the hydrogen abundance decreases (top panel of Fig. \\ref{fig:phys}). \n   As the central temperature increases, the 3$\\alpha$ reaction\\ gradually becomes active and CNO-cycle reactions begin to occur.  \n   Eventually, CNO-cycle reactions dominate the p-p chains with regard to total energy production. \n   When this first occurs, the central abundance by mass $X \\lows{CNO}$ of catalysts is $10^{-11} - 10^{-9}$ at $\\log T_{c} \\simeq 7.8 - 7.9$ (top and middle panels of Fig. \\ref{fig:phys}). \n   Because of lower central temperatures, the less massive the star, the later is the evolutionary stage (and the smaller is the central hydrogen abundance) at which the transition from burning dominated by the p-p chains to burning dominated by the CNO-cycle reactions takes place. \n   For $M \\leq 0.8 \\ensuremath{M_\\odot}$, hydrogen is depleted in the center before a transition can take place. \n   After the CNO cycle takes over as the main source of energy generation, because of the strong temperature dependence of CNO-cycle reactions, the central temperature remains nearly constant. \n   At the same time, the core expands because of the central concentration of energy generation and the rate of production of catalysts slows down; \n   the abundances of catalysts saturate at $X \\lows{CNO} \\simeq 10^{-10} - 10^{-8}$.\n\nThe transition to the CNO-cycle dominated phase is accompanied by the formation of a convective zone which develops outward from the center as described in the lower panel of Fig.~\\ref{fig:phys}. \n   The growth of the convective core is also evident in the middle panel of Fig.~\\ref{fig:phys}, which shows that the hydrogen abundance at the center, $X_{c}$, stops decreasing monotonically and increases for a time as evolution progresses. \n   Central convection is caused by a thermonuclear runaway and persists after the transition; due to the larger temperature dependence of the CNO-cycle energy-generation rate, energy generation is highly concentrated toward the center. \n   Both the inward mixing of hydrogen from outer hydrogen-rich layers and the outward mixing of CNO elements generated near the center during the thermonuclear runaway amplify the average hydrogen-burning rate over the region encompassed by convection relative to the average rate in the absence of convection.\n\nFor stars of mass $0.9 \\leq M \/ \\ensuremath{M_\\odot} \\leq 1.2$, the transition from the p-p chain dominated phase to the CNO-cycle dominated phase is delayed until the core has already begun to contract rapidly and electrons have begun to become degenerate at the center;  \n   the maximum energy generation has already shifted away from the center and core contraction has initiated the expansion of the envelope. \n   A thermonuclear runaway called the core helium-hydrogen (He-H) flash takes place \\citep{fuj90}. \n   In Fig.~\\ref{fig:rhot}, a first increase in the central temperature with decreasing central density indicates that the electron degeneracy is lifted; \n   then, the temperature turns to decrease with the density so as to settle in the thermal equilibrium state where the nuclear energy generation balances the energy loss from the core.   \n   In this runaway, helium burning plays a key role through the production of CNO-cycle catalysts. \n   Because of the strong temperature dependence of CNO-cycle reaction rates, the flash starts even when the contribution of the CNO cycle to the total energy-generation rate is smaller than that of p-p chain reactions in central regions.  \n   Smaller mass stars experience stronger flashes since the central entropy is smaller and the electron degeneracy is stronger at the onset of the thermonuclear runaway. \n   Thus, the temperature reaches larger, leading to a greater production of CNO catalysts and a greater extension of convection. \n   For stars of mass $M \\leq 0.8 \\ensuremath{M_\\odot}$, the central temperature does not become large enough for the production of sufficient carbon to activate the CNO cycle, irrespective of the choice of reaction rates. \n   We see a small hump along the trajectories of $M = 0.8 \\ensuremath{M_\\odot}$ models in Fig.~\\ref{fig:rhot}, which marks where the exhaustion of hydrogen at the center quenches a thermonuclear runaway.\n\nThese evolutionary characteristics during core hydrogen burning are common to all of the models, regardless of the adopted nuclear reaction rates since the evolutionary tracks prior to the transition are dominated by the p-p chain reactions with relatively small temperature dependences.  \n   All the models of masses in the range $0.9 \\leq M \/ \\ensuremath{M_\\odot} \\leq 1.2$ experience the core He-H flash.  \n   We may notice that the differences in the ignition temperatures are rather small because of the strong temperature dependence of the 3$\\alpha$ reaction\\ rate, the flash grows weaker for models of larger mass.\n\nCore convection lasts until hydrogen is almost exhausted at the center ($X_{c} \\lesssim 10^{-4}$).  \n   After the model settles in the thermal equilibrium, the central density increases again as the hydrogen abundance decreases. \n   The central temperature at first rises, and then, decreases as the nuclear energy-generation rate decreases at the center (when $X_{c} < 2 - 3 \\times 10^{-4}$). \n   Note that since the nuclear energy generation is dominated by contribution from off-center burning, the model at this phase tends to have an isothermal core with the central temperature directly reflecting the temperature in the H-burning shell. \n   In Fig.~\\ref{fig:rhot}, we can see the rise in central temperature with the growth of core, a sudden jump in temperature occurring when the hydrogen burning shell passes the shells occupied by the flash convection, and hence, have larger CNO abundances.  \n   \nThe resultant loop, as seen in Fig.~\\ref{fig:rhot}, is characteristics of flash, implying that He-H core flash eventually exerts work through the expansion and contraction of core, during which the envelope first contacts and then expands again, as seen in Fig.~\\ref{fig:hrd}. \n   In the subsequent evolution, our models do not encounter the phenomena, so called ``shell He-H flash'' at the base of hydrogen burning phase, as FIH90 discovered. \n   FIH90 discuss the behavior of hydrogen burning shell in the $Z=0$ environment and found the convective instability by the production of carbon and of nuclear energy by CN cycles. \n   This is the counterpart of core He-H flash, i.e., electrons are degenerate at the base of hydrogen burning shell, which causes He-H flash. \n   However, no other groups than Fujimoto and his collaborators find such a event. \n   The different consequence is due to the consideration of non-resonant effect in 3$\\alpha$ reaction\\ rate as we will discuss in \\S \\ref{sec:tal}. \n\n\\subsection{Helium Burning Phase}\\label{sec:heburn}\n\nCharacteristic properties of the helium core flash are given in Table~\\ref{tab:he-flash}. \n   The thermal structure of the core when the helium core flash is ignited are determined by (1) neutrino energy losses which produce cooling, (2) the radiative and conductive opacity which controls heat flow, and (3) the temperature and energy generation in the hydrogen-burning shell which affects the heating due to gravitational compression. \n   If neutrino cooling in the central region is effective enough to produce a strong positive temperature gradient, the helium core flash is ignited off center. \n   Model mass must be larger than $M \\geq 1.2 \\ensuremath{M_\\odot}$ for central helium ignition to occur (Fig.~\\ref{fig:rhot}). \n   The mass $M_{1}$ of the helium core when the helium core flash occurs differs greatly between the off-center and central ignition cases, as seen from the third column of Table~\\ref{tab:he-flash}. \n   Central ignition occurs before neutrino cooling becomes appreciable, and, hence, core masses at ignition are small:  $M_{1} \\lesssim 0.4 \\ensuremath{M_\\odot}$.  \n   Once neutrino cooling becomes effective in the central region, off-center ignition is delayed until the core mass is much larger, namely, $M_1 \\gtrsim 0.5 \\ensuremath{M_\\odot}$.\n\nFigure~\\ref{fig:str} shows the interior temperature as a function of density for models of mass 0.8 $\\ensuremath{M_\\odot}$ and core mass $M_{1} = 0.49 \\ensuremath{M_\\odot}$. \n   A constant temperature ``plateau'' spreads inward from the base of the hydrogen-burning shell. \n   This implies that the thermal structure of the helium core results mainly from the temperature in the hydrogen burning shell and neutrino energy losses in central regions \\citep{fuj84}. \n   Along with a decrease in the initial abundance of CNO elements, the hydrogen-burning rate decreases, and, in order to compensate for this, the temperature in the hydrogen-burning shell increases, which makes the contribution of compressional heating smaller. \n   Accordingly, an isothermal plateau develops in the region where the radiative heat transport dominates over electron conduction for the metallicity of $[{\\rm Fe}\/{\\rm H}] \\lesssim -5$ \\citep[see ][]{fuj95}. \n   In our models, the compressional heating plays a small part to raise the maximum temperature in the helium zone slightly higher than the temperature in the hydrogen burning shell, as seen from the lowest mass model;  \n   note that for the massive models, the helium burning already contributes appreciably to increase the temperatures in the right shoulder of structure lines in this figure.  \n   This behavior contrasts with that of model stars of younger population in which the compressional heating play the dominant part in determining the maximum temperature in the helium core and makes it much larger than the temperature in the hydrogen-burning shell.  \n   Because of high temperature in the hydrogen burning shell, therefore, $Z=0$ model stars experience central ignition at smaller initial masses than do model stars of younger populations for which the minimum initial mass for central helium ignition is $\\sim 2.5 \\ensuremath{M_\\odot}$. \n   While the density and temperature of the hydrogen burning shell are in local maximum just before the helium ignition, hot CNO-cycle is not still effective.\n   At this stage, the nuclear timescale of $\\nucm{13}{N}$ against proton capture reaction ($\\sim 10000$ sec) is much larger than that against $\\beta$-decay reaction (863 sec) and is negligible in the outcome of neither the nucleosynthesis nor the nuclear energy output.\n\nIf neutrino energy losses are sufficiently effective before helium is ignited, the central region cools, and helium is ignited off-center as is the case in low-mass stars of younger populations. \n   In the central region of the model, plasma neutrinos are the dominant neutrino energy-loss mechanism and produce a steep gradient in the rate of released energy. \n   On the other hand, the conduction, which is important in the electron-degenerate core, transport the energy towards the center where the neutrino loss works. \n   Consequently, both conductivity and neutrino energy-loss rates promote cooling of the core and, thus, delay the off-center ignition of helium until a larger core mass gives rise to a larger hydrogen-shell burning rate and a larger temperature in the hydrogen-burning shell, as seen from Table~\\ref{tab:he-flash}.   \n\nIn all of the models which experience off-center ignition, convection driven by helium burning extends into the upper hydrogen-rich layers during the decay phase of the core helium flash, because of smaller entropy in the hydrogen burning shell, as shown by \\cite{fuj90,fuj95}. \n   The ingestion of hydrogen into the helium convective zone begins a sequence of events that leads to the enrichment of the surface with carbon and nitrogen \\citep{hol90,fuj00,sch02,pic04,wei04}. \n   Characteristics of hydrogen mixing are also given in Table~\\ref{tab:he-flash}. \n   The ingestion of hydrogen occurs within a matter of days after the helium-burning luminosity reaches its peak.\n   In the models undergoing central helium burning, we do not find a hydrogen-mixing event;  \n   The core helium flash is rather weak (see $L_{\\rm He}^{\\rm max}$ in Table~\\ref{tab:he-flash}), which makes it difficult for the outer edge of the convective region driven by helium burning to reach the hydrogen-containing layer \\citep{fuj77}. \n   For the models of $M=1.2 \\ensuremath{M_\\odot}$, we follow the evolution through the thermally pulsing AGB phase to find that the He-FDDM is triggered during the helium shell flashes. \n\n\\section{Discussion and Comparison with Other Works}\n\nIn this section, we compare the models in the literature with our models adopting similar input physics to see which cause the dominant effect on the differences in the evolution and to confirm the correctness of numerical computations.\n   In addition, we compare the models without non-resonant effect for $\\alpha$-capture reactions to see the influence of uncertainty in 3$\\alpha$ reaction\\ rates on the stellar structure at $Z=0$. \n\n\\subsection{Comparison with $0.8 \\ensuremath{M_\\odot}$ Models}\n\nA model comparable with our $0.8 \\ensuremath{M_\\odot}$ model is Model~1 of \\citet[][P04]{pic04}, of composition $Y = 0.23$ and $Z = 0$.\nIn their computations, release 4.98 of FRANEC was used and time-dependent convective mixing was calculated; neutrino energy-loss rates by plasma-neutrino emission were modified, with consequences being reported as minimal.\nReaction rates and conductive opacities are, respectively, common with our models.\nThe neutrino energy-loss rates are common with our model for photo- and pair-neutrino processes, but they use energy-loss rate of \\citet{dic76} for bremsstrahlung and of \\citet{bea67} for recombination processes.  \n   For plasma neutrino energy losses, they adopt an energy-loss rate \\citep{esp03} which differs only slightly from I96 in the temperature and density ranges relevant to the ignition of the helium flash.\n\nAlthough there are some differences in adopted input physics, the evolution of the helium core flash of the P04 model is similar to that of our 08cf model. \n     The helium core mass at the onset of the core helium flash is the same in both models, namely, $M_{1} = 0.52 \\ensuremath{M_\\odot}$. \nWhen CNO-cycle reactions are the dominant contributors to the hydrogen-burning luminosity, so that the hydrogen profile is very steep, the quantity $M_{\\rm{He}}$ defined by P04 as the mass of the helium core when the maximum hydrogen-burning luminosity is reached is nearly the same as the quantity $M_{1}$ we have defined as the mass of the helium core when the core helium flash begins.  \nThe maximum helium-burning luminosities differ by less than a factor of two, being $L_{\\rm He} = \\pow{1.2}{10} L_{\\sun}$ in the P04 model and $L_{\\rm He} = \\pow{7.6}{9} L_{\\sun}$ in our model 08cf. \nThe mass at the outer edge of the convective shell at the onset of hydrogen mixing is the same in both cases, namely, $0.506 \\ensuremath{M_\\odot}$.\nThe values of $\\Delta t_{\\rm{mix}}$ (see Table \\ref{tab:he-flash}) and $X_{\\rm C}$, the mass fraction of carbon in the helium convective shell, are also comparable: \n$\\Delta t_{\\rm{mix}} = \\pow{2.1}{5}$ sec in the P04 model versus $\\pow{1.87}{5}$ sec in model 08cf, and $X_{\\rm C} = \\pow{4.15}{-2}$ in the P04 model versus $X_{\\rm C} = \\pow{4.28}{-2}$ in model 08cf.\nThere is a large difference in model characteristics when the helium convective shell first appears;  \nin the P04 model, the convective shell driven by helium burning appears at a mass shell $M_{\\rm BCS} = 0.348 \\ensuremath{M_\\odot}$ when $L_{\\rm{He}} = 0.658 L_{\\sun}$, while, in our model 08cf, $M_{\\rm BCS} = 0.3825 \\ensuremath{M_\\odot}$ when $L_{\\rm He} = \\pow{1.57}{2} L_{\\sun}$.  \n    The factor of 200 difference in the helium-burning luminosity when shell convection begins is probably simply a typographical error in P04, an interpretation reinforced by the fact that the time for the helium-burning luminosity to reach its maximum value is almost the same in the P04 model (723 yr) and in ours (715 yr).\nIt takes more than $\\sim 3\\times 10^4$ yr for the helium-burning rate to increase by a factor of 200 in this range. \n\nThe model of $0.8 \\ensuremath{M_\\odot}$ and $Z=0$ in FII00 is based on those of \\citet{fuj95} (hereafter F95) and the comparable results are given in their Table 1. \n   In F95, the values of  $M_{1} \\ups{max}$, $M_{\\rm BCS}$, and $\\ensuremath{\\log L_{\\textrm{He}}} \\ups{max} (L_\\odot) $ are $0.5116 \\ensuremath{M_\\odot}$, $0.3705 \\ensuremath{M_\\odot}$, and 9.983, respectively.  \n   This core mass $M_1$ coincides with our 08nac very closely despite the differences in the input physics;  \n   F95 took into account only the resonant 3$\\alpha$ reaction\\ reactions \\citep{aus71}, which is smaller by a factor of $\\sim 2.4$ than the NACRE rate at the relevant temperature range ($\\log T \\simeq 7.94$). \n   The smaller helium burning rate tends to delay the ignition of helium core flash.  \n   On the other hand, the I83 formulae adopted here give larger conductivity than the Iben's fitting formulae used by F95 in the region of coulomb-liquid regime where the maximum temperature in the helium zone occurs (see fig.~\\ref{fig:cop}), which works to delay the helium ignition due to the enhanced cooling of helium zone through the inward heat conduction in our model.  \n   These two effects compensate for each other, while the effect of  larger conduction is manifest in inner ignition (or in smaller  $M_{\\rm BCS}$) in the 08nac model.  \n   In actuality, the 08cf model with the same input physics as the 08nac model except for the nuclear reaction rates results in a larger core mass than the 08nac model. This is because the cf88 3$\\alpha$ reaction\\ rate is smaller by $20 \\%$ than the NACRE 3$\\alpha$ reaction\\ rate around the temperature relevant here.  \n   In any case, the dependence of $M_1$ on the nuclear reaction rate is very small because of strong temperature dependence of 3$\\alpha$ reaction\\ rate. \n   It is also worth noting that the non-resonant reaction \\citep{nom85} has little to do with the ignition of helium core flash because of rather high temperatures in the helium zone $\\log T > 7.9$, although it takes a major part in the later phase of core hydrogen burning and the omission delays the depletion of hydrogen until higher temperature is reached in the center ($\\Delta T_{\\rm c} \\simeq  0.04$ and $\\Delta \\rho_{\\rm c} \\simeq  0.25$).  \n\nSince our calculation does not follow the burning of mixed-in hydrogen, we cannot assess the results of FII00 and P04 with regard to the occurrence of a hydrogen-burning flash in the middle of helium convective zone, the splitting into two convective shells, and the merging of the upper convective shell with the surface convective zone. \n   This remains for further investigation and will be discussed in a separate paper (Suda, Fujimoto, \\& Iben, in preparation).\n\n\\subsection{Comparison with $1 \\ensuremath{M_\\odot}$ Models}\n\nIn this section, we compare our results with those of FIH90, W00, S01, and SLL02 for the evolution to the beginning of the core helium flash of models of mass $1 \\ensuremath{M_\\odot}$ and initial composition $Z=0$. \n   In all of the cited calculations, the core He-H flash occurs, although the size of the blue loop differs among the different works.\n\nWe first compare our results with the model of FIH90.  \n   The distinctive feature of FIH90 model is the He-H shell flash during the hydrogen shell-burning, as stated in the introduction. \n   We will show in the following subsection that the instability of the hydrogen shell burning and the resultant shell flash are solely attributable to the exclusion of non-resonant rate of 3$\\alpha$ reaction\\ reactions \\citep{nom85}\\footnote{The computation of FIH90 was done at the University of Tokyo Observatory in 1984 when one of authors (I.I., jr.) visited it as JSPS fellow.}.  \n   Furthermore, the FIH90 model ended in a significantly larger core mass at the ignition of helium core flash ($M_{1} = 0.528 \\ensuremath{M_\\odot}$). \n   It is even larger as compared with the $0.8 \\ensuremath{M_\\odot}$ model of FII00 with the same input physics except for the equation of state (EOS), despite the general tendency of decrease for larger stellar masses as seen in Table.~{\\ref{tab:he-flash};   \n  Since these two computations differ only in the equation of state (EOS) for a Coulomb liquid and solid among the input physics \\citep[see][for the adopted the EOS]{ibe92}, the main reason for the larger core mass may be the larger radius of the helium core in the FIH90 model (see their Table 1 in FIH90), resulting from the difference in the adopted Coulomb corrections in the EOS.  \n    A larger core radius implies a smaller gravity of the core, and hence, a smaller temperature in the hydrogen-burning shell for a given core mass, to defer the ignition of helium core flash until a larger core mass is achieved.   \n   In actuality, in FIH90 model, a He-H shell flash is postponed until the core mass grows as large as $M_1 = 0.505 \\ensuremath{M_\\odot}$ and ignited at a low density in the bottom of hydrogen burning shell under a flat configuration ($V \\gg 4$) but under non electron-degenerate conditions \\citep[see their Fig.~2 in][]{fuj82}. \n   Because of a large core mass, the He-H shell flash has little effect on the thermal state of the inner core and FIH90 find off-center ignition of the helium core flash.  \n\nNext, we compare with the work by \\citet[][ W00]{wei00} who use a code which differs from the one used by \\citet[][ S01]{sch01} with regard to the EOS \\citep{str88}, reaction rates \\citep[][which include the non-resonant term in the 3$\\alpha$ reaction\\ rate]{thi87}, and the radiative opacity \\citep[old version of OPAL,][]{rog92,igl92}. \n   The conductive opacity and the neutrino energy-loss rates are not described in their paper. \n   Since they do not follow the progress of the helium core flash, they do not find the He-FDDM event while helium is ignited off center. \n   Their value of $t \\lows{TO} = 6.31$ Gyr is close to that of our model 10cf (6.54 Gyr);  \n   the small difference may be due to the use of different versions of OPAL opacities. \n   As for the core He-H flash, their values for the location of the outer edge of the central convective zone $M_{ECS} = 0.11 \\ensuremath{M_\\odot}$ and for the helium-burning luminosity $L \\lows{He} \\ups{max} = 2.57 \\times 10^{-7} L_{\\sun}$ are similar to our values of $M_{ECS} = 0.115 \\ensuremath{M_\\odot}$ and $L \\lows{He} \\ups{max} = 3.27 \\times 10^{-7} L_{\\sun}$ for our model 10cf. \n   Element abundances at maximum nuclear burning luminosity are also comparable; \n   W00 find $X_{12} = \\pow{6.50}{-12}$, $X_{14} = \\pow{2.19}{-10}$, and $X_{16} = \\pow{2.77}{-12}$, and we find $X_{12} = \\pow{7.90}{-12}$, $X_{14} = \\pow{1.20}{-10}$, and $X_{16} = \\pow{8.44}{-13}$. \n   These abundances are influenced slightly by the choice of time step. \n   At the tip of the RGB, we can compare with their ``canonical'' model that element diffusion is not considered in their work;  \n   their $M_{1} \\ups{tip} = 0.497 \\ensuremath{M_\\odot}$ and $\\log (L_s^{\\rm tip} \/L_{\\sun}) = 2.357$ differ slightly from our 10cf model: $M_1^{\\rm tip}  = 0.5054 \\ensuremath{M_\\odot}$ and $\\log (L_s^{\\rm tip}\/L_{\\sun}) = 2.413$ (See Tables~\\ref{tab:model} and \\ref{tab:he-flash}). \n\nIn the S01 calculations, evolution is followed from the main sequence to the TPAGB phase.\n   The S01 input physics differs from ours with regard to the EOS in regions of electron degeneracy \\citep{kip90}, the energy-loss rates for photo, pair, and plasma neutrinos \\citep{mun85}, and weak screening for nuclear reactions (they use the Salpeter formula). \n   For the EOS in core regions, they adopt a simplified equation of state for a degenerate electron gas \\citep{kip90} but they provide no\nexplicit statement as to their treatment of the ion gas.\n   Their nuclear reaction rates and conductive opacities are the same as those which we have used\n\nS01 give numerical results only for the helium core flash phase.\nTheir value of $\\log (L\/L_{\\sun}) = 2.314$ at the start of the\ncore helium flash is slightly smaller than our model 10cf value and\ntheir value of $M_{1} \\ups{max} = 0.482 \\ensuremath{M_\\odot}$ is also smaller than ours. \nHowever, they find $M \\lows{BCS} = 0.151 \\ensuremath{M_\\odot}$, while our model\n10cf gives $M \\lows{BCS} = 0.3390 \\ensuremath{M_\\odot}$ and none of our other models\nignite a helium core flash with $M_{\\rm BCS}$ smaller than $0.3 \\ensuremath{M_\\odot}$,\nexcept in the case of central ignition (Table~\\ref{tab:he-flash}). \nBecause of the much larger mass between the base of\nthe convective shell and the location of the base of the hydrogen-rich layer,\nthe time required for the outer edge of the convective shell to\nreach hydrogen-rich material is much larger in the S01 model than in our models:\n$\\Delta t \\lows{mix} = 10$ yr for S01 versus ${\\Delta t}_{\\rm mix} \n= 10^{-3}$ - $10^{-2}$ yr for all our models, irrespective of the input\nphysics. \n  We note that it takes more than 1000 yr for convection generated in the center to reach the maximum extension in mass and ${\\Delta t}_{\\rm mix}$ for S01 falls in the middle of our two cases.\n\nIn an effort to reproduce the S01 results, we constructed a $1.0 \\ensuremath{M_\\odot}$ model 10cf$^\\prime$, using the fitting formula for neutrino energy-loss rates given by \\citet{mun85} which does not include neutrino bremsstrahlung. \n   The contribution of neutrino loss is rather small as compared with the gravitational energy release in the core, and yet, affects the internal structure of helium core significantly. \n    In Fig.~\\ref{fig:str}, we compare the structure line at core mass $M_1 = 0.49 \\ensuremath{M_\\odot}$ in which the neutrino energy loss rate ($L_\\nu = 0.70 L_\\odot$) is smaller than the rate of gravitational energy release ($L_g = 2.3 L_\\odot$) and the helium burning rate is still small ($L_{\\rm He} = 0.35 L_\\odot$). \n   We see that the maximum temperature shifts to the inner shell as much as $\\Delta \\log \\rho \\simeq 0.13$. \n   The reason for this inward shift is that, near the stellar center, the energy-loss rate due to neutrino bremsstrahlung is comparable to the energy-loss rate due to other neutrino processes; \n   neglect of the neutrino bremstrahlung contribution means that the cooling rate in central regions is reduced from what it would otherwise have been. \n   Accordingly, the initiation of a helium-burning thermonuclear runaway occurs for a smaller core mass than would otherwise be the case. \n   We find that, when $M_{1} = 0.4968 \\ensuremath{M_\\odot}$, helium is ignited at a mass point $M \\lows{BCS} = 0.2933 \\ensuremath{M_\\odot}$, which is about 10\\% smaller than we find for our 10cf model when neutrino bremstrahlung is included. \n   The 10\\% reduction we have found is, however, far too small to account for the S01 result but we suspect that differences in neutrino energy-loss rates are in part responsible for the small value of $M \\lows{BCS}$ found by S01. \n   Since radiative opacities, conductive opacities, and nuclear reaction rates are presumably also not responsible, differences in the EOS may be another source of the discrepancy. \n   In dense stellar matter, Coulomb corrections reduce the pressure, leading to an increase in the density and to a reduction of core radius.  \n   The increase in the gravity entails the higher temperature in the hydrogen burning shell and heats up the core.  \n   To explore this point more quantitatively, we examine conditions in the 10cf$^\\prime$ model for the same core mass as the S01 model ignites helium ($M_1 = 0.482 \\ensuremath{M_\\odot}$). \n   In Fig.~\\ref{fig:str}, we locate by filled circles the density and temperature of the hydrogen-burning shell and the density and temperature at the mass point $M_{r} \\approx 0.151 \\ensuremath{M_\\odot}$ when $M_1 = 0.482 \\ensuremath{M_\\odot}$. \n   We conclude that the S01 structure curve must be fairly different from ours in the sense that their internal core might be kept much hotter despite the neutrino energy loss, presumably due to larger Coulomb corrections. \n\nFinally, we compare the $1 \\ensuremath{M_\\odot}$ model of SLL02 with our model 10nac. \n   SLL02 provide many evolutionary tracks of zero metallicity models covering a large mass range. \n   They do not encounter the He-FDDM phenomenon and their evolutionary calculations for low mass stars extend to the AGB phase.  \n   Much of the input physics they adopt is the same as we have used to construct model ``nac''. \n   The conductive opacities, radiative opacities, nuclear reaction rates, and the mixing length parameter ($\\alpha = 1.5$) are presumably the same. \n   However, they use a different EOS and a different treatment of the nuclear screening factor \\citep{gra73}. \n   They do not comment on the choice of neutrino energy-loss rates. \n\nWith respect to the core He-H flash, the minimum carbon abundance by mass for the appearance of convection at the center in their models is $\\log X_{12C} \\simeq -11.5$ for various model masses and this agrees well with our results for lower mass models. \n   The main sequence lifetime, measured by the age at the turn-off point, is $t \\lows{TO} = 6.56$ Gyr in excellent agreement with our result of $6.53$ Gyr (see Table~\\ref{tab:model}); \n   this agreement is to be expected since both opacities and nuclear reaction rates are the same in both cases. \n   For the same reason, the CN-cycle takes over as the main energy-production mechanism at essentially the same abundance of carbon at the center: \n   $X_{c} = \\pow{5.8}{-4}$ in the SLL02 model and $\\pow{5.71}{-4}$ in ours.\n   The maximum mass of the convective core and the maximum helium-burning luminosity are very similar in the two cases: \n   $M \\lows{ECS} \\simeq 0.095 \\ensuremath{M_\\odot}$ and $L \\lows {He} \\ups{max} \\simeq 10^{-7} L_{\\sun}$ in the SLL02 model, compared with $M \\lows{ECS} = 0.104 \\ensuremath{M_\\odot}$ and $L \\lows {He} \\ups{max} \\simeq = \\pow{2.51}{-7} L_{\\sun}$ in ours. \n   At the maximum helium-burning luminosity, element abundances are $X_{12} = \\pow{6.50}{-12}$, $X_{14} = \\pow{2.19}{-10}$, and $X_{16} = \\pow{2.77}{-12}$ in the SLL02 model, compared with $X_{12} = \\pow{6.64}{-12}$, $X_{14} = \\pow{1.19}{-10}$, and $X_{16} = \\pow{8.97}{-13}$ in our model. \n\nSLL02 do not find a shell He-H flash, consistent with our results. \n   Very similar results are obtained for stellar luminosity and the mass of the helium core at the RGB tip; \n   they find $\\log L \\ups{tip} = 2.357$ and $M_{1} \\ups{tip} = 0.497 \\ensuremath{M_\\odot}$, compared with $\\log L \\ups{tip} = 2.372$ and $M_{1} \\ups{tip} = 0.4922 \\ensuremath{M_\\odot}$ for our model. \n   Core helium burning begins when $M \\lows{BCS} = 0.31 \\ensuremath{M_\\odot}$ for SLL02 and $M \\lows{BCS} = 0.3320 \\ensuremath{M_\\odot}$ in model 10nac (see Table~\\ref{tab:he-flash}). \n   SLL02 actually find that hydrogen mixes into the convective zone driven by core helium flash, but, because the hydrogen abundance that appears in the zone is so small ($X < 10^{-8}$), the luminosity due to hydrogen burning is relatively small ($L \\lows{H} \\lesssim 10^{4} L_{\\sun}$), so they neglect the effects. \n   As mentioned earlier, their result may be an artifact occasioned by their assumption that convective mixing is instantaneous.\n\nFor completeness, we discuss additional properties of our $1 M_\\odot$ model 10nac and, when possible, compare with properties of the other models. \nAt the turn-off point, our model has $M_{1} = 0.2032 \\ensuremath{M_\\odot}$ and\nthe abundance of hydrogen at the center is finite with the value\n$X_{c} = \\pow{1.57}{-2}$. \nAfter 553 Myr of evolution beyond the turnoff point, a convective core\nis formed, driven by the He-H core flash. \nStill 3.06 Myr later, the hydrogen-burning luminosity reaches a maximum\nof $L \\lows{H} \\ups{max}=24.6 L_{\\sun}$, with $L_{\\rm pp}=18.3 L_{\\sun}$\nand $L_{\\rm CN} = 6.29 L_{\\sun}$ being, respectively, the contributions of\nthe pp-chain reactions and of the CN-cycle reactions. \nThe blue loop in the H-R diagram is characterized by\n$3.795 \\leq \\log \\ensuremath{T_{\\textrm{eff}}}$ (K) $\\leq$ 3.863, and 1.27 $\\leq \\log L_{s} \\leq 1.33$. \nDuring the flash, the convective core grows to a maximum mass of\n$M \\lows{ECS} = 0.104 \\ensuremath{M_\\odot}$, which is between the masses found by\nW00 and SLL02.\nThe maximum carbon abundance is achieved at nearly the same time that\nthe convective shell achieves its maximum mass and has the value\n$X_{12} \\ups{max} = \\pow{7.146}{-12}$. \nAs evident from Fig.~\\ref{fig:rhot}, the inner CN-burning shell passes through\nthe site of this central convective zone when, at the center, \n$\\log \\rho_{c} = 5.017$ and $\\log T_{c} = 7.801$.\nA temperature plateau develops in the outer core, as is characteristic\nof all zero metallicity models, and as the core mass continues to increase,\na temperature inversion is formed \\citep{fuj84}.\nThereafter, energy transport into the center plays a crucial role\nin determining the maximum temperature in the core. \nOur model center evolves into a region of strong degeneracy, reaching the maximum central density $\\log \\rho_{c} \\ups{max} =6.074$ at $\\log T_{c} = 7.840$. \nWhen the helium flash ignites off-center, the core mass is\n$M_{1} \\ups{max} = 0.5028 \\ensuremath{M_\\odot}$ (see Table~\\ref{tab:model}) and\nthe luminosity is $L \\ups{max} = 251.6 L_\\odot$. \nAt maximum helium luminosity, the central abundances of CNO elements are\n$X_{12}=\\pow{1.18}{-5}$, $X_{14}=\\pow{5.35}{-10}$, and $X_{16}=\\pow{9.21}{-8}$.\nIn our model, $T_{c}=\\pow{5.622}{7}$ K and  $\\rho_{c}=\\pow{8.498}{5}$ g cm$^{-3}$ at this time.\n\n\\subsection{Influence of 3$\\alpha$ reaction\\ rates}}\\label{sec:tal}\n\nIn this subsection, we discuss the relevance of characteristics of 3$\\alpha$ reaction\\ rates to the evolution of zero-metallicity stars.  \n  Recently, the properties of resonances other than the so-called Hoyle resonance in \\nuc{12}C and their effects on 3$\\alpha$ reaction\\ rates have been discussed experimentally and theoretically since the compilation of NACRE \\citep[see for example,][]{ito04b,fyn05,kur05}.  \n   On the other hand, the non-resonant term of 3$\\alpha$ reaction\\ rate may also be subject to uncertainty in the cross section because the nuclear theory hardly determines the behavior of three body interactions (K. Kat$\\bar{\\rm o}$ 2007, private communication). \n   Considering these improvement in the field of nuclear physics, it is important to examine the effects of resonant and non-resonant terms of 3$\\alpha$ reaction s, separately.  \n   Indeed, for $Z=0$ models, the non-resonant term of 3$\\alpha$ reaction\\ rate can change the evolutionary behavior drastically at the hydrogen shell burning phase.\n\nAs a test for exploring the contribution of non-resonant term, we compute the models of 1.0 and $1.1 \\ensuremath{M_\\odot}$ by adopting the nuclear reaction rates of \\citet{fow75} (hereafter FCZ75) with the same other input physics as the models in this work. \n   The main difference of FCZ75 rates from the NACRE rates is the consideration of non-resonant term in 3$\\alpha$ reaction.  \n   Since FCZ75 have not yet taken into account the effect, the 3$\\alpha$ reaction\\ rate drops by far more rapidly than the NACRE rates below $\\log T \\lesssim 7.89$. \n   Other cross sections may differ by within factor of 2 or 3 \\citep{sud03} and do not affect the qualitative results. \n\nFigures~\\ref{fig:shellHeH} shows the evolutionary tracks and interior structures in the density-temperature plane for these models with the FCZ75 rates (refereed as models 10fcz and 11fcz, respectively, in the following) and compare them with those of our models with the NACRE rates.  \n   The effect of different nuclear reaction rates is apparent before the depletion of hydrogen in the center. \n   Both models with and without the non-resonant term experience the core He-H flash, the FCZ models postpone it until higher central temperature than the NACRE models; \n   in the latter models, it is ignited at the low temperatures where the non-resonant term is effective ($\\log T < 7.89$). \n   Because of stronger electron degeneracy, the FCZ75 models undergo much stronger flashes ($L_{\\rm He} = 1.06 \\times 10^4$ and $78 L_\\odot$ for models 10fcz and 11fcz, respectively), as seen from larger loops in this figure, than our models 10nac and 11nuc ($L_{\\rm He} = 24$ and $29 L_\\odot$, respectively), which entails larger extension of flash convection; \n   for models 10fcz and 11fcz, the flash convection reaches to the shells of $M_{\\rm conv} = 0.181$ and $0.149 \\ensuremath{M_\\odot}$ at the maximum extension, respectively, about twice as large as compared to our models 10nac and 11nac, with the CN abundance of $X_{\\rm CN} = 6.0 \\times 10^{-10}$ and $ 2.6 \\times 10^{-10}$, respectively. \n\nThere is also a remarkable difference in the evolutionary tracks during the hydrogen-shell burning between the models with and without the non-resonant terms. \n   In the hydrogen burning-shell, the CN-burning with carbon produced by 3$\\alpha$ reaction\\ contributes considerably to the total energy generation though the contribution to the total energy is still smaller than that of p-p chain reactions for the stars of mass $1.2 > M\/ \\ensuremath{M_\\odot} > 0.9 $, because of lower entropy in the hydrogen burning shell as compared with the stars of younger populations. \n   In the models without the non-resonant term, in particular, there form two local maxima in the energy generation rates, a narrow one associated with CN burning and localized at the base of small hydrogen abundance ($X < 0.01$), and a broad one associated with pp-chain reactions in the middle of large hydrogen abundance (nearly half of the surface hydrogen abundance). \n   When the hydrogen burning shell passes across the inner sphere, occupied by the convection during the core He-H flash, a thermonuclear runaway is triggered at the base of hydrogen-burning shell under electron degeneracy, which is a similar situation as the ignition of a core He-H flash in the center; \n   in addition, at the base of hydrogen burning shell, the pressure scale height is much smaller than the radial distance to the center, i.e., $V = r \/ \\vert d r \/ d \\ln P \\vert = G M_r \\rho \/ r P \\gg 1$, and the flat configuration also contributes to the instability \\citep{sug78}. \n   It occurs in the shell of mass $M_r = 0.212 \\ensuremath{M_\\odot}$ with $M_1 = 0.340 \\ensuremath{M_\\odot}$ for mode 10fcz and $M_r = 0.170 \\ensuremath{M_\\odot}$ with $M_1 = 0.352 \\ensuremath{M_\\odot}$ for the first shell flash of model 11fcz.\n\nOn the other hand, the models with the non-resonant term stably burn hydrogen, although electrons are degenerate ($\\Psi \\simeq 8$) at the base of the hydrogen burning shell. \n   This different behavior stems mainly from the difference in the temperature dependence of the resonant and non-resonant terms, rather than from the difference in the burning rate itself, as can be seen from the analysis of thin shell burning by \\citet[see also Fujimoto 1982]{sch65}. \n   At temperatures of $\\log T < 7.9$, the 3$\\alpha$ reaction\\ rates by FCZ75 and NACRE give a large difference in their temperature dependences when compared at the same burning rates. \n   For less massive FCZ75 models of $M \\le 0.9 \\ensuremath{M_\\odot}$, the entropy is smaller and hydrogen is burnt before carbon production becomes appreciable. \n   For FCZ75 models of mass $M \\ge 1.2 \\ensuremath{M_\\odot}$, higher temperature as well as weaker electron degeneracy tend to stabilize hydrogen shell burning; \n   in addition, as the contribution of CN burning with products of 3$\\alpha$ reaction\\ reaction as catalysis overweighs the p-p chain reactions, the pressure scaleheight grows comparable to the radial distance to the center, whicl also stabilize the shell burning by making the heat capacity negative with hydrostatic readjustment \\citep[e.g., see ][]{fuj82}.  \n   It is noted that the temperature dependence of resonant reaction rate decreases with increase in the temperature, which also stabilizes the hydrogen shell-burning in combination with the reduction in electron degeneracy when the core grows more massive than $M_1 \\gtrsim 0.4 \\ensuremath{M_\\odot}$, as is the case for 10fcz model.  \n\nDuring the shell flash, the CN-cycle burning shell expands, and the hydrogen exhausted core also expands due to the reduction in the weight of overlying layers; \n   the central temperature and density decrease almost adiabatically (Label ``A'' in Fig.~\\ref{fig:shellHeH}). \n  During the decay phase of the first shell flash, after the flash-driven convective zone disappears, the core is heated by the flow of energy from the burning shell. \n   For model 11fcz, the second shell flash is ignited at the shell of mass $M_r = 0.287 \\ensuremath{M_\\odot}$ with $M_1 = 0.374 \\ensuremath{M_\\odot}$, when the hydrogen shell-burning passes across the shells, incorporated into the convection and enriched in CN elements during the first shell flash. \n   This flash grows so strong as to drive the convection deep into the hydrogen-rich envelope up to the shell of $M_1 = 0.404 \\ensuremath{M_\\odot}$ (Label ``B''), which expedites the growth of helium core to cause the increase in central density.  \n  Since the matter in the convective zone driven by the shell flash is enriched in CNO elements, the CN-cycle reactions dominate the energy generation rate while the burning shell traverse in the site of convective zone. \n   During this stable burning phase, the growth rate of core is large because of small hydrogen abundance therein. \n   Accordingly, concomitant rapid compression of helium core increases the central temperature. \n   This enhanced growth of helium core after the shell flashes leads directly to the ignition at the center of a helium core flash (Label ``C'') with a small core masses.  \n  This is the case if the convective zone is sufficiently large, as in the case for the second shell flashes of 11fcz model, while it is barely missed in 10fcz model.  \n\nThe most important point by this test is that the contribution of the non-resonant term is discernible in the circumstance of $Z=0$, although it may be difficult to detect differences by the observations. \n   There is a difference in the mass range that the off-center helium core flashes occur. \n   If the non-resonant term is included in the 3$\\alpha$ reaction\\ rate, the central helium burning occurs at $\\geq 1.2 \\ensuremath{M_\\odot}$. \n   Otherwise, it occurs at $\\geq 1.0 \\ensuremath{M_\\odot}$ if we use the smaller conductivity by Iben's approximates to \\citet{hub69} and by \\citet{can70} and the neutrino loss rates by \\citet{bea67} as in FII00, both slightly smaller than ours. \n   Since the lifetime of $1.0 \\ensuremath{M_\\odot}$ model is $\\sim$ 7 Gyr (Table~\\ref{tab:model}), such stars cannot be seen in the present halo if they were born. \n   Only clue to those objects will be binary mass transfer between giants of mass $\\ge1.0 \\ensuremath{M_\\odot}$ and dwarfs of mass $\\le 0.8 \\ensuremath{M_\\odot}$. \n   Since FII00 predict the abundance ratio of C\/N $\\sim 1$ for He-FDDM at RGB, while C\/N $\\gtrsim 5$ for He-FDDM at AGB, it will be crucial for the constraint on the estimate of nuclear reaction rate to determine the abundance ratio of C\/N and the mass of the primary for $Z=0$.\n\n\\section{Conclusions}\n\nWe have explored the evolution of low-mass, zero-metallicity stars with the most recent input physics.   \n  The mass range is $0.8 \\ensuremath{M_\\odot} \\leq M \\leq 1.2 \\ensuremath{M_\\odot}$ in step of $0.1 \\ensuremath{M_\\odot}$ and the initial composition is $X = 0.767$, $Y=0.233$, and $Z=0$.  \n  Calculations extend from the zero-age main sequence to the beginning of hydrogen mixing into the helium convective region on the RGB or at the start of the TPAGB phase, depending on the characteristics of the helium core flash.\n\n\\begin{enumerate}\n\n\\item The emergence of CN-cycle reactions as important contributors to nuclear energy production occurs during the core hydrogen-burning phase in models of mass $M \\geq 0.9 \\ensuremath{M_\\odot}$ in consequence of the formation of carbon by the 3$\\alpha$ reaction. This phenomenon is independent of the adopted physics and its importance is a function only of the initial mass and metallicity. \n\n\\item The models of $M \\leq 1.1 \\ensuremath{M_\\odot}$ undergo an off-center helium flash and hydrogen-mixing into helium flash-convection, leading to helium-flash driven deep mixing at the tip of red giant branch.  \n   On the other hand, the models of mass $M \\ge 1.2\\ensuremath{M_\\odot}$ ignite the core helium flash at the center and postpone the He-FDDM until the helium shell flashes occur during the early phase of thermal pulsation at the asymptotic giant branch. \n   For models of mass $0.8 \\ensuremath{M_\\odot}$ and $0.9 \\ensuremath{M_\\odot}$, our results coincide qualitatively with those first found by \\cite{fuj90} and \\citet{fuj00}. \n   For $1.0 \\ensuremath{M_\\odot} \\leq M \\leq 1.1 \\ensuremath{M_\\odot}$, whether or not a helium core flash is ignited off the center and hydrogen is mixed inward into the convective zone driven by it depend on the adopted nuclear reaction rates.   \n\\end{enumerate}\n\nWe have also compared our results with those of other investigations. Our models made with the most up-to-date input physics agree well with the models of \\citet{pic04}, \\citet{wei00} and \\cite{sie02} during evolution on the main sequence and the RGB. \n   In particular, we obtain nearly the same results as do \\cite{sie02}, although we do not follow evolution after the mixing of hydrogen into the helium flash driven convective zone at the beginning of the helium core flash. \n   To check the behavior of the He-FDDM event, we need to treat mixing with a time-dependent algorithm. \n   On the other hand, even after adopting the same radiative opacities, conductive opacities, nuclear reaction rates, and neutrino energy-loss rates as \\cite{sch01}, we are not able to obtain the inner ignition at the onset of core helium burning which they find; \n  we suspect that the discrepancy may be due to differences in the EOS for the Coulomb corrections in the liquid and solid states and due to the neglect in their work of neutrino energy losses associated with neutrino bremsstrahlung.\n\nBy treating the resonant and non-resonant rates of 3$\\alpha$ reaction s separately, we demonstrate that the non-resonant term plays a critical role in the low-mass, zero-metal stars. \n   The neglect of non-resonant term causes the lower border in mass of central helium burning, i.e., $\\geq 1.0 \\ensuremath{M_\\odot}$. \n   This explains the discrepancy of the results between ours and \\citet{fuj00}, which stems mainly from the difference in the temperature dependence rather than in the energy generation rates themselves.  \n   We first point out the possibility of discerning the effect of non-resonant term of 3$\\alpha$ reaction\\ from the evolution of stars other than at low temperature regime in the accreting degenerate stars \\citep{nom85}.  \n   It is important to precisely determine the abundances and the properties of extremely metal-poor stars to constrain the nuclear reaction rates. \n\n\\acknowledgments\n\nWe are grateful to I. Iben Jr. for improving and revising our manuscript. \n   We wish to thank A. Ohnishi and K. Kat$\\bar{\\rm o}$ for valuable comments on uncertainties on nuclear reaction rates. \n   This work is part of a PhD. thesis constructed at Hokkaido University and is in part supported by a Grant-in-Aid for Science Research from the Japanese Society for the Promotion of Science (15204010, 18104003).   \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Supplemental Material }\n\n\n\n\\maketitle\n\\renewcommand{\\theequation}{S\\arabic{equation}}\n\\setcounter{equation}{0}\n\\renewcommand{\\thefigure}{S\\arabic{figure}}\n\\renewcommand{\\figurename}{Supplementary Fig.}\n\n\\setcounter{figure}{0}\n\\renewcommand{\\thesection}{S\\arabic{section}}\n\\setcounter{section}{0}\n\nIn this Supplemental Material, we present additional details and calculations regarding: 1) derivation of the evolution operator for the left and right state vectors in the non-Hermitian case; 2) Fourier transformation of the anomalous Green's functions; 3) electron-electron interaction mediated by phonons from two bands with strong asymmetric interaction. \n\n\\section{S1. Derivation of evolution operators for left and right state vectors in non-Hermitian case}\nIn this section, we derive the evolution operators for the left and right state vectors. As we consider the non-Hermitian case, it is not obvious how they look like.\n\nThe $\\eta$-pseudo-Hermiticity is defined as \\cite{mostafazadeh:jmp02}\n\\begin{eqnarray}\n\\label{eq:etaH}\nH^\\dagger=\\eta H\\eta^{-1},\n\\end{eqnarray}\nwhere $\\eta$ is a Hermitian and invertible linear operator. Then the $\\eta$-product is defined as\n\\begin{eqnarray}\n\\langle\\Phi|\\Phi'\\rangle_\\eta=\\langle\\Phi|\\eta|\\Phi'\\rangle.\n\\end{eqnarray}\nThe $\\eta$-product can be derived from the definition of pseudo-Hermiticity and Schr\\\"odinger equation as follows. The Schr\\\"odinger equation for the right state vector is\n\\begin{eqnarray}\ni\\frac{\\partial}{\\partial t}|\\Phi_R\\rangle=H|\\Phi_R\\rangle.\n\\end{eqnarray}\nIf we take Hermitian conjugation of it and substitute the definition of $H^\\dagger$ from Eq. (\\ref{eq:etaH}), we obtain\n\\begin{eqnarray}\n-i\\frac{\\partial}{\\partial t}\\langle\\Phi_R|\\eta=\\langle\\Phi_R|\\eta H,\n\\end{eqnarray}\nthat gives\n\\begin{eqnarray}\n\\label{eq:SchroedPhiL}\n-i\\frac{\\partial}{\\partial t}\\langle\\Phi_L|=\\langle\\Phi_L|H.\n\\end{eqnarray}\nNow let's derive the evolution operator in the interaction representation. For that, we represent the Hamiltonian in free and perturbation parts, $H=H_0+V$. Then, we use the Schr\\\"odinger equation following the standard procedure:\n\\begin{eqnarray}\ni\\frac{\\partial}{\\partial t}[e^{-iH_0t}|\\Phi^I_R\\rangle]=i[-iH_0e^{-iH_0t}|\\Phi^I_R\\rangle+e^{-iH_0t}\\partial_t|\\Phi^I_R\\rangle ]=(H_0+V)|\\Phi_R\\rangle.\n\\end{eqnarray}\nThe terms with $H_0$ cancel each other and we obtain\n\\begin{eqnarray}\ni\\partial_t|\\Phi^I_R\\rangle =e^{iH_0t}Ve^{-iH_0t}e^{iH_0t}|\\Phi_R\\rangle=V^I|\\Phi^I_R\\rangle.\n\\end{eqnarray}\nThis is a standard expression of the Schr\\\"odinger equation in the interaction picture. We can do exactly the same with Eq. (\\ref{eq:SchroedPhiL}) with the only change that $\\langle\\Phi_L^I|=\\langle\\Phi_L|e^{-iH_0t}$ and obtain\n\\begin{eqnarray}\n-i\\frac{\\partial}{\\partial t}\\langle\\Phi_L^I|=\\langle\\Phi_L|e^{-iH_0t}e^{iH_0t} Ve^{-iH_0t}=\\langle\\Phi_L^I|V^I.\n\\end{eqnarray}\nthe evolution operator for $\\langle \\Phi_L^I|$ is\n\\begin{eqnarray}\n\\mathcal{S}_L=\\bar{T}e^{i\\int_0^td t'V^I(t')},\n\\end{eqnarray}\nand for $|\\Phi_R^I\\rangle$ is\n\\begin{eqnarray}\n\\mathcal{S}_R=Te^{-i\\int_0^td t'V^I(t')}.\n\\end{eqnarray}\n\n\\section{S2. Fourier transformation of the anomalous Green's functions}\nThe poles of the Green's functions derived in Eq. (8) of the main text are $\\omega_{1,2}=\\pm\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}$. We perform the corresponding integration following the standard procedure and firstly add $\\pm i\\eta$ to the denominators of the Green's functions with $\\eta$ being infinitesimally small, $\\eta\\rightarrow+0$. As the electron Green's function usually has $i\\eta$ and the hole Green's function has $-i\\eta$, we put $i\\eta$ to the denominator of $\\bar{F}$ and $-i\\eta$ to the denominator of $F$. If we express denominators in terms of products and expand the square roots in $\\eta$, we obtain\n\\begin{eqnarray}\n\t\\bar{F}_{\\uparrow,\\downarrow}^{{\\bf k},-{\\bf k}}(\\omega)&=&\\frac{\\Delta_{\\bf k}}{[\\omega-\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}+i\\eta][\\omega+\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}-i\\eta]}, \\\\\n\tF_{\\downarrow,\\uparrow}^{-{\\bf k},{\\bf k}}(\\omega)&=&\\frac{-\\Delta_{\\bf k}}{[\\omega-\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}-i\\eta][\\omega+\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}+i\\eta]}.\n\\end{eqnarray}\nThis implies that $\\bar{F}_{\\uparrow,\\downarrow}^{{\\bf k},-{\\bf k}}$ has a pole $\\omega_2+i\\eta$ and $F_{\\downarrow,\\uparrow}^{-{\\bf k},{\\bf k}}$ has a pole $\\omega_1+i\\eta$ in the upper half plane. Building a contour of half-circle shape with the radius $R\\rightarrow\\infty$ and employing residues (see e.g. Ref. \\onlinecite{kornich:prr21}), we obtain for the anomalous Green's functions in the time domain\n\\begin{eqnarray}\n\\bar{F}_{\\uparrow,\\downarrow}^{{\\bf k},-{\\bf k}}(t)&=&-\\frac{i\\Delta_{\\bf k}e^{-it\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}}}{2\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}},\\\\\nF_{\\downarrow,\\uparrow}^{-{\\bf k},{\\bf k}}(t)&=&-\\frac{i\\Delta_{\\bf k}e^{it\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}}}{2\\sqrt{\\varepsilon_k^2-\\Delta_{\\bf k}^2}}.\n\\end{eqnarray}\nWe can see that $\\bar{F}_{\\uparrow,\\downarrow}^{{\\bf k},-{\\bf k}}(t)\\neq (F_{\\downarrow,\\uparrow}^{-{\\bf k},{\\bf k}}(t))^\\dagger$.\n\n\n\n\n\\section{S3. Derivation of Electron-electron interaction mediated by phonons from two interacting bands}\nLet's consider electrons interacting with phonons. Then, the partition function is given by\n\\begin{eqnarray}\n\\mathcal{Z}=\\int \\mathcal{D}[\\bar{\\psi},\\psi,\\bar{a},a]\\exp{[i(S_{ph}+S_e+S_{e-ph})]},\n\\end{eqnarray}\nwhere $S_e$ is the action of electrons with $\\psi$ and $\\bar{\\psi}$ being Grassmann fields of electrons and $S_{ph}$ is the action of phonons with $\\bar{a}$ and $a$ being bosonic fields of phonons. We make a common assumption that $\\langle a\\rangle=\\langle\\bar{a}\\rangle=0$. The action $S_{e-ph}$ describes electron-phonon interaction.\n\nLet's consider $S_{e-ph}$ as a perturbation, and expand the exponent in it. Then, we average over phononic degrees of freedom assuming that $S_{e-ph}$ is linear in $a$ and $\\bar{a}$, and thus all odd powers of $S_{e-ph}$ are zero. In such a way, we obtain\n\\begin{eqnarray}\n\\mathcal{Z}=\\int \\mathcal{D}[\\bar{\\psi},\\psi,\\bar{a},a]\\exp{[i(S_{ph}+S_e)]}(1+iS_{e-ph}-\\frac{1}{2}S_{e-ph}S_{e-ph}+...)=\\int \\mathcal{D}[\\bar{\\psi},\\psi]\\exp{[iS_e-\\frac{1}{2}\\langle S_{e-ph}S_{e-ph}\\rangle]}.\\ \\ \\\n\\end{eqnarray}\nThe last term is the action for electron-electron interaction mediated by phonons $S_{e-e}=i\\langle S_{e-ph}S_{e-ph}\\rangle\/2$.\n\nNow we consider the phonon action in detail. We assume that there are two phonon bands $\\omega_{1,{\\bf q}}$ and $\\omega_{2,{\\bf q}}$ that interact with the off-diagonal element $\\delta_{\\omega,{\\bf q}}$:\n\\begin{eqnarray}\nS_{ph}=\\int \\begin{pmatrix}\\bar{a}_{1,\\omega,{\\bf q}} & \\bar{a}_{2,\\omega,{\\bf q}}\\end{pmatrix}\\begin{pmatrix}\\omega-\\omega_{1,{\\bf q}} & -\\delta_{\\omega,{\\bf q}}\\\\ -\\delta_{\\omega,{\\bf q}} & \\omega-\\omega_{2,{\\bf q}}\\end{pmatrix}\\begin{pmatrix}a_{1,\\omega,{\\bf q}} \\\\ a_{2,\\omega,{\\bf q}}\\end{pmatrix}d\\omega d{\\bf q}.\n\\end{eqnarray}\nThe electron-phonon interaction is described as\n\\begin{eqnarray}\nS_{e-ph}=\\int d\\omega d{\\bf q}F_{\\omega,{\\bf q}}\\sum_{j=1,2}(a_{j,\\omega,{\\bf q}}+\\bar{a}_{j,-\\omega,-{\\bf q}})\\int d\\Xi d{\\bf K}\\bar{\\psi}_{\\omega+\\Xi,{\\bf K}+{\\bf q}}\\psi_{\\Xi,{\\bf K}}=\\int d\\omega d{\\bf q}F_{\\omega,{\\bf q}}\\sum_{j=1,2}(a_{j,\\omega,{\\bf q}}+\\bar{a}_{j,-\\omega,-{\\bf q}})\\rho_{\\omega,{\\bf q}}.\\ \\ \\ \\ \\ \\ \\ \\ \\ \n\\end{eqnarray}\nHere, $F_{\\omega,{\\bf q}}$ is the prefactor that usually contains material characteristics and constants, for instance, elastic constants. This gives\n\\begin{eqnarray}\n\\langle S_{e-ph}S_{e-ph}\\rangle=\\int \\mathcal{D}[\\bar{a},a]e^{iS_{ph}}S_{e-ph}S_{e-ph}=\\\\ \\nonumber=\\int d{\\bf q}d{\\bf q'}d\\omega d\\omega'F_{\\omega,{\\bf q}}F_{\\omega',{\\bf q'}}\\rho_{\\omega,{\\bf q}}\\rho_{\\omega',{\\bf q'}}\\int \\mathcal{D}[\\bar{a},a]\\sum_{i,j=1,2}[a_{i,\\omega,{\\bf q}}\\bar{a}_{j,-\\omega',-{\\bf q}'}+\\bar{a}_{i,-\\omega,-{\\bf q}}a_{j,\\omega',{\\bf q}'}] e^{iS_{ph}}.\n\\end{eqnarray}\nNow we will use the property that\n\\begin{eqnarray}\n\\langle\\bar{a}_{\\bf q_1}a_{\\bf q_2}\\rangle=\\int \\mathcal{D}[\\bar{a},a]\\bar{a}_{\\bf q_1}a_{\\bf q_2}e^{-\\sum_q\\bar{a}_{\\bf q}A_{\\bf q}a_{\\bf q}}=\\frac{\\delta_{{\\bf q}_1,{\\bf q}_2}}{A_{{\\bf q}_1}},\n\\end{eqnarray}\nin order to perform averaging. After averaging, we obtain\n\\begin{eqnarray}\n\\langle S_{e-ph}S_{e-ph}\\rangle&=&i\\int d{\\bf q}d\\omega F_{\\omega,{\\bf q}}F_{-\\omega,-{\\bf q}}\\rho_{\\omega,\\bf q}\\rho_{-\\omega,-{\\bf q}}\\times \\\\ \\nonumber&&\\left[\\frac{\\omega_{1,{\\bf q}}+\\omega_{2,{\\bf q}}-2\\omega-2\\delta_{\\omega,{\\bf q}}}{\\delta_{\\omega,{\\bf q}}^2-(\\omega-\\omega_{1,{\\bf q}})(\\omega-\\omega_{2,{\\bf q}})}+\\frac{\\omega_{1,-{\\bf q}}+\\omega_{2,-{\\bf q}}+2\\omega-2\\delta_{-\\omega,-{\\bf q}}}{\\delta_{-\\omega,-{\\bf q}}^2-(\\omega+\\omega_{1,-{\\bf q}})(\\omega+\\omega_{2,-{\\bf q}})}\\right].\\ \\ \\ \n\\end{eqnarray}\nThis yields the expression for the electron-electron interaction potential that we state in the main text in Eq. (17):\n\\begin{eqnarray}\nV(\\omega,{\\bf q})=-\\frac{F_{\\omega,{\\bf q}}F_{-\\omega,-{\\bf q}}}{2}\\left[\\frac{\\omega_{1,{\\bf q}}+\\omega_{2,{\\bf q}}-2\\omega-2\\delta_{\\omega,{\\bf q}}}{\\delta_{\\omega,{\\bf q}}^2-(\\omega-\\omega_{1,{\\bf q}})(\\omega-\\omega_{2,{\\bf q}})}+\\frac{\\omega_{1,-{\\bf q}}+\\omega_{2,-{\\bf q}}+2\\omega-2\\delta_{-\\omega,-{\\bf q}}}{\\delta_{-\\omega,-{\\bf q}}^2-(\\omega+\\omega_{1,-{\\bf q}})(\\omega+\\omega_{2,-{\\bf q}})}\\right].\n\\end{eqnarray}\nNow let's assume that $\\omega_{1,\\pm{\\bf q}},\\omega_{2,\\pm{\\bf q}}\\ll \\delta_{\\omega,{\\bf q}},\\delta_{-\\omega,-{\\bf q}},\\omega$. Then we obtain\n\\begin{eqnarray}\nV(\\omega,{\\bf q})\\simeq F_{\\omega,{\\bf q}}F_{-\\omega,-{\\bf q}}\\left[\\frac{1}{\\delta_{\\omega,{\\bf q}}-\\omega}+\\frac{1}{\\delta_{-\\omega,-{\\bf q}}+\\omega}\\right]=F_{\\omega,{\\bf q}}F_{-\\omega,-{\\bf q}}\\left[\\frac{\\delta_{-\\omega,-{\\bf q}}+\\delta_{\\omega,{\\bf q}}}{(\\delta_{\\omega,{\\bf q}}-\\omega)(\\delta_{-\\omega,-{\\bf q}}+\\omega)}\\right].\n\\end{eqnarray}\nIf the phonon interaction is strongly asymmetric in momentum space, $\\delta_{\\pm\\omega,{\\bf q}},\\omega\\gg\\delta_{\\pm\\omega,-{\\bf q}}$, then\n\\begin{eqnarray}\nV(\\omega,{\\bf q})&\\simeq& F_{\\omega,{\\bf q}}F_{-\\omega,-{\\bf q}}\\frac{\\delta_{\\omega,{\\bf q}}}{\\omega(\\delta_{\\omega,{\\bf q}}-\\omega)},\\\\\nV(\\omega,-{\\bf q})&\\simeq& F_{\\omega,{\\bf q}}F_{-\\omega,-{\\bf q}}\\frac{\\delta_{-\\omega,{\\bf q}}}{-\\omega(\\delta_{-\\omega,{\\bf q}}+\\omega)}.\n\\end{eqnarray}\nIf we further assume, $\\delta_{\\pm\\omega,{\\bf q}}\\gg \\omega$, we obtain Eq. (18) from the main text,\n\\begin{eqnarray}\nV(\\omega,{\\bf q})\\simeq -V(\\omega,-{\\bf q})\\simeq\\frac{F_{\\omega,{\\bf q}}F_{-\\omega,-{\\bf q}}}{\\omega}.\n\\end{eqnarray}\nThus we obtain odd or asymmetric electron-electron interaction potential due to an externally-induced asymmetry in the phonon system, namely, the phonon-phonon interaction. The band structure of phonons, $\\omega_{1,{\\bf q}}$ and $\\omega_{2,{\\bf q}}$, can also be asymmetric, but here we have not used this property.\n\n\n\n\n\\end{widetext}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nNon-radial pulsations (NRPs) are commonly found in isolated white dwarfs (WDs) of DA type, so called ZZ Ceti stars. These stars have hydrogen-rich atmospheres, and pulsations occur as the WD cools and passed through a phase of pulsational instability, detected mainly as g-modes (\\citealt{2006AJ....132..831G}).         \n\nDuring the last decennium, similar signals, generally interpreted as non-radial WD pulsations, have also been detected in faint cataclysmic variables (CVs). A CV is a close binary system where a late-type main-sequence star looses mass to a primary white dwarf. The first CV proposed to harbour a pulsating white dwarf was GW Librae.~\\cite{1998IAUS..185..321W} found rapid, periodic, and non-commensurate signals in its light curve, suggesting non-radial pulsations of the underlying white dwarf. In most CVs, the accretion energy tends to dominate the luminosity, and the white dwarf itself, shining with $M_{V} \\sim$ 10 -- 13, is seldom seen. However, for some of the most intrinsically faint CVs, spectroscopy and time-series photometry can reveal signatures of the underlying white dwarf, such as broad absorption features in the spectrum, sharp eclipses, and sometimes non-radial pulsations in the light curve. These signals have now been detected in about a dozen CVs, all quiescent systems of low luminosity. Here, we call these systems \\emph{GW Lib stars}, after the first discovery.~\\cite{2010ApJ...710...64S} and~\\cite{2009JPhCS.172a2069M} present recent reviews of this group of stars. \n\nAccreting WDs are different from isolated ones since they are being exposed to mass transfer, giving them atmospheres of solar-composition. The white dwarfs in CVs are therefore hotter and are also found to be spinning faster compared to isolated ones (\\citealt{2009ASPC..404..229S}). Studying these systems will provide important information of how the process of accretion is affecting the evolution of the white dwarf. In isolated WDs, pulsations are only observed in stars with temperatures located within a so-called \\emph{instability strip} in the $\\log g$ -- $T_{\\text{eff}}$ plane, spanning the temperature range $T_{\\text{eff}}$ = 10900 K -- 12200 K (see Figure 3 of~\\citealt{2006AJ....132..831G}). However, there is no clear instability strip for the GW Lib stars (see Figure 13 of~\\citealt{2010ApJ...710...64S}), and pulsations are found in systems with WD effective temperatures up to at least 15000 K. \n\nOur theoretical understanding of the mechanism for exciting non-radial pulsations in CVs is still limited. The observed pulse amplitudes are quite variable, and in some ZZ Ceti stars this is known to be the result of the beating of two signals closely spaced in frequency, a classic and highly informative signature of non-radial pulsations. But the GW Lib stars have not yet clearly revealed this kind of behaviour (although a hint of it emerged in the V386 Ser campaign reported by~\\citealt{2010ApJ...714.1702M}).  \n\nWe here present time-series photometry of two more CVs which are probably members of the GW Lib class. Both systems have very low accretion luminosity, and show signatures from the white dwarf in their optical spectra. Also, both systems show double-humped orbital signals, and non-commensurate periodic signals, suggesting non-radial pulsations. One is SDSS J1457+51 which has a photometric wave suggesting an orbital period of $77.885 \\pm 0.007$ minutes. The other is BW Sculptoris, with $P_{\\text{orb}} = 78.22639 \\pm 0.00003$ minutes. Both stars show main pulsations near 10 and 20 minutes. These rapid signals drift slightly in frequency, and may consist of several, finely spaced components. BW Sculptoris also shows a remarkable photometric variation at 87 minutes, which could be explained as a \\emph{quiescent superhump}, possibly arising from a 2:1 orbital resonance in the accretion disc.\n\n\\section{SDSS J1457+51}\n\nSDSS J1457+51 (hereafter J1457) was first identified in the Sloan Digital Sky Survey by~\\cite{2005AJ....129.2386S}. They obtained spectroscopy that showed the broad absorption characteristics of a white dwarf, indicating a system of low accretion rate. The source was found to be faint ($g \\approx 19.5$). Due to the double-peaked nature of the emission lines, they suggested the system to be of high inclination.\n\n\n\n\\subsection{Observations, Data Reduction and Analysis} \\label{obs}\n\nTime-resolved photometry of J1457 was obtained with the 1.3 m and 2.4 m MDM telescopes at the Kitt Peak observatory, Arizona, during April and May 2010. The star was observed during 14 nights in total, spread over 47 days. With all data coming from the same terrestrial longitude, we were not immune from aliasing problems and therefore strove to obtain the longest possible nightly time series (generally $\\sim$ 4 - 7 hours). Weather conditions such as clouds, snow and full moon prevented us from obtaining times series over more than 4 consecutive nights at a time. The time resolution was in the range of 20 s -- 30 s. Table~\\ref{tab:obs1} presents a log of the observations. A clear filter with a blue cutoff was used to minimise differential-extinction effects and allow for a good throughput.\n\nThe data reduction was done in real time during the observations, using standard \\textsc{Iraf} routines. The data consisted of differential photometry with respect to the field star USNO A2.0:1350-08528847. The search for periodic signals was initially done for single nights separately, and Lomb-Scargle periodograms (\\citealt{1982ApJ...263..835S}) were constructed. Formal flux errors were rescaled so that the $\\chi^{2}_{\\nu} \\approx$ 1. This was done by fitting a fake light curve to the original data, composed of multiple sine waves with periods corresponding to the strongest signals found in the single-night power spectrum. Monte-Carlo simulations were performed on every peak of interest in the power spectrum to find the period and its error. In this method, the peak errors are found by randomly re-distributing the points in the light curve within their errors, a repeated number of times, and constructing a Lomb-Scargle periodogram each time. The 1-$\\sigma$ error is then found by fitting a Gaussian to the output distribution of the peaks found in the periodograms. Data from several nights were then combined to allow the search for signals with lower amplitude, and also to improve the frequency resolution. Bootstrap analysis was performed to distinguish between the most likely alias, and was also used to find errors on the peaks. In this method, the sampling pattern of the data is changed by creating a mock dataset where every point from the original dataset is randomised. This is done to efficiently destroy or weaken the aliases pattern.\n\n\n\\begin{table}\n\\begin{center}\n \\begin{tabular}{llllll}\n  \\hline\n  \\hline\n\\textbf{Date} & \\textbf{HJD} & \\textbf{Telescope} & \\textbf{Length} \\\\\n& (-2455000) &  & (hours) \\\\\n  \\hline\n100414 & 301 &  1.3m &  2.30 \\\\\n100416 & 303   & 1.3m & 7.89 \\\\\n100417 & 304   & 1.3m & 4.32 \\\\\n100419 & 306   & 1.3m   & 4.72 \\\\\n100424 & 311    & 1.3m & 5.42 \\\\\n100425 & 312    & 1.3m  & 7.82 \\\\\n100503 & 320  &  1.3m  & 7.92 \\\\\n100504 & 321   & 1.3m  & 8.26 \\\\\n100506 & 323   & 1.3m  & 8.40 \\\\\n100507 & 324   & 1.3m  & 3.50 \\\\\n100508 & 325   & 1.3m  & 5.47 \\\\\n100528 & 345   & 2.4m  & 4.97 \\\\\n100529 & 346   & 2.4m  & 8.11 \\\\\n\\hline\n  \\hline\n\\end{tabular}\n \\caption{Summary observing log for SDSS J1457+51. Data was \\newline obtained at the MDM observatory during April and May 2010.}\n \\label{tab:obs1} \n\\end{center}\n\\end{table}\n \n\n\n    \n\\begin{figure*}\n\\centering\n\\subfigure[The mean Lomb-Scargle periodogram for J1457, from the six nights of best quality. The orbital period at 18.48 c\\,d$^{-1}$ ($\\omega_{\\text{o}}$, 77.9 min) and its first harmonic ($2\\omega_{\\text{o}}$) are plotted as solid lines.]\n{\n    \\label{fig:s_avpow}\n    \\includegraphics[width=7.75cm]{pow_best2_mean.eps}\n}\n\\hspace{0.2cm}\n\\subfigure[Normalised and smoothed light curve of J1547 from one sample night. A model light-curve constructed from the four strongest periods (including $\\omega_{\\text{o}}$ and $2\\omega_{\\text{o}}$) found in Figure~\\ref{fig:s_avpow}, is plotted together with the data.]\n{\n    \\label{fig:s_light}\n    \\includegraphics[width=7.12cm]{light_fit_320.eps}\n}\n\\caption{}\n\\label{fig:sub}\n\\end{figure*}\n    \n    \n\n\\subsection{Light Curve and average Power Spectrum}\n\nThe mean power spectrum, averaged over the six best nights is shown in Figure~\\ref{fig:s_avpow}. The 18.48 c\\,d$^{-1}$ (77.9 min) and 36.97 c\\,d$^{-1}$ (38.9 min) signals are almost certainly the orbital frequency ($\\omega_{\\text{o}}$) and its first harmonic ($2\\omega_{\\text{o}}$). This kind of variation at $2\\omega_{\\text{o}}$ is commonly seen in the orbital light curves of CVs with low accretion rates (for instance in WZ Sge; see~\\citealt{2002PASP..114..721P}). We find that the signals in the range 142 c\\,d$^{-1}$ -- 148 c\\,d$^{-1}$ ($\\approx$ 10 min) are non-commensurate with the orbital frequency. These peaks vary slightly in period and amplitude, when present at all in the nightly power spectrum. The complex structure around them indicate either an unresolved fine structure, or periods varying from night to night, and is discussed in detail below. Also, during a few nights, peaks were found at 135 c\\,d$^{-1}$ (10.7 min) and 72 c\\,d$^{-1}$ (20 min), which are also non-commensurate with the orbital frequency. The lower region of the power spectrum show strong peaks at 4 c\\,d$^{-1}$ -- 6 c\\,d$^{-1}$ (4 -- 6 hours), corresponding to the typical length of a nightly observing run. The unit c\\,d$^{-1}$ is used throughout the paper since it is the natural unit for the sampling pattern of the multi-day light curves. Also, it clearly shows the natural daily alias pattern.  \n\nThe average power spectrum in Figure~\\ref{fig:s_avpow} has low resolution since each night is less than 8 hours long. The power spectrum of a spliced light curve spanning several nights (the coherent power spectrum) is in principle better, since the resolution is always near 0.1\\,N$^{-1}$ c\\,d$^{-1}$, where N is the duration in days. It does, however, make the assumption that a candidate periodic signal is constant in period, phase, and amplitude over the duration of the observation. Power spectra will be difficult or impossible to interpret correctly when this assumption is grossly violated.\n\nFigure~\\ref{fig:s_light} shows the normalised and smoothed light curve for a sample night. A model light curve constructed from the four strongest peaks found in the power spectrum that night (including both $\\omega_{\\text{o}}$ and $2\\omega_{\\text{o}}$), is plotted on top of the smoothed light curve. Peaks found at higher frequencies than 40 c\\,d$^{-1}$ are not represented in the model light curve. During one of the observing nights, we obtained multicolour data and found the brightness of of the star to be V = 19.2 $\\pm$ 0.2. The mean brightness on each night was constant within the measurement error of 0.03 mag. Flickering, an essentially universal feature of CV light-curves, was very low. This together with the fact that absorption lines are seen in the optical spectrum implies that the total light, 4000\\,\\AA\\, -- \\,7000\\,\\AA\\, is dominated by the white dwarf. Measurements of the absorption line depth of the hydrogen lines indicate that probably no more than half of the total light comes from accretion processes.\n\n\n\n\\begin{figure*}\n\\centering\n\\subfigure[The power spectrum of J1457 from 11 nights, showing the orbital signal $\\omega_{\\text{o}}$ and its first harmonic, $2\\omega_{\\text{o}}$. A zoom of the region around $\\omega_{\\text{o}}$ is plotted on top.]\n{\n    \\label{fig:s_poworb}\n    \\includegraphics[width=7.15cm]{pow_orb_text.eps}\n}\n\\hspace{0.2cm}\n\\subfigure[A model power spectrum of J1457 constructed from two sine waves at $\\omega_{\\text{o}}$ and $2\\omega_{\\text{o}}$, using the same sampling pattern as in Figure~\\ref{fig:s_poworb}.]\n{\n    \\label{fig:s_poworbmod}\n    \\includegraphics[width=7.15cm]{pow_model_orb.eps}\n}\n\\caption{}\n\\label{fig:sub}\n\\end{figure*}\n\n\n\n\\begin{figure}\n\\includegraphics[width=7.5cm]{fold18.eps}\n\\caption{Light curve of J1457 folded on the $\\omega_{\\text{o}}$ frequency, showing a double-humped orbital wave.} \n \\label{fig:fold18} \n\\end{figure}\n\n\n\\subsection{The Orbital Signal}\n\nA dominant, stable feature at $\\approx$ 37 c\\,d$^{-1}$ (38 min) is always present in the nightly power spectra. In four of the nights, a weaker but stable signal is also present at half that frequency,  $\\approx$ 18.5 c\\,d$^{-1}$  (78 min). We interpret these two signals as the orbital frequency and its first harmonic, $\\omega_{\\text{o}}$ and $2\\omega_{\\text{o}}$. As mentioned above, double-humped orbital waves are quite common among CVs, especially those of very low luminosity (for instance in WZ Sge and AL Com). A power spectrum composed of 11 nights, spanning 45 days, yields $\\omega_{\\text{o}} = 18.4888 \\pm 0.0017$ c\\,d$^{-1}$ and $2\\omega_{\\text{o}} = 36.9740 \\pm 0.0005$ c\\,d$^{-1}$. Errors are calculated from bootstrap simulations as described in Section~\\ref{obs}.\n\nFigure~\\ref{fig:s_poworb} shows the low-frequency portion of the full 11 night power spectrum. A zoom of the region around the orbital frequency is plotted on top. A model power spectrum constructed from two artificial sinusoids at $\\omega_{\\text{o}}$ and $2\\omega_{\\text{o}}$, using the exact same sampling as for the original dataset, is shown in Figure~\\ref{fig:s_poworbmod}. When comparing model versus data, we find the surrounding picket-fence pattern similar in structure and height. This implies that the orbital signal indeed maintains an essentially constant amplitude and phase. In Figure~\\ref{fig:fold18}, data from all 11 nights are folded onto the orbital frequency, showing the double-humped orbital wave at $\\omega_{\\text{o}}$ and $2\\omega_{\\text{o}}$.  \n \nThe lower frequency range of the power spectrum was further investigated to rule out the possibility of signals hiding in the noise (see Section~\\ref{superh} for the case of BW Sculptoris). The power spectrum was cleaned from the strongest signals at $\\omega_{\\text{o}}$, $2\\omega_{\\text{o}}$ and also from the high-amplitude peaks between 4 c\\,d$^{-1}$ -- 6 c\\,d$^{-1}$. However, no additional peak was found in this region, or in the vicinity of the orbital period.\n\n \\begin{table*}\n\\begin{center}\n \\begin{tabular}{lllllll}\n  \\hline\n  \\hline\n\\textbf{Star} & \\textbf{Frequency} & \\textbf{Period} & \\textbf{Amplitude} & \\textbf{Comments}  \\\\\n & (c\\,d$^{-1}$) & (min) & (mmag) &   \\\\\n \\hline\n J1457 & $18.4888 \\pm 0.0017$  & 77.92 & 12.0 & Orbital period ($\\omega_{\\text{o}}$) \\\\      \n& $36.9740 \\pm 0.0005$ & 38.95 & 23.9 & $2\\omega_{\\text{o}}$ \\\\\n& 71.9 (nightly mean error: 0.5) & 20.0 & 12.0 & NRP, low amplitude, non-stable \\\\ \n& 135.2\/144.3\/147.9 (nightly mean error: 0.8) & 10.7\/9.9\/9.7 & 3.9\/12.7\/7.6 & NRP, non-stable \\\\\n\\hline\nBW Scl & $\\sim$ 16.5 & 87.27 & $\\sim$ 50 & Quiescent superhump, non-stable \\\\\n & 18.40811 $ \\pm$ 0.03  & 78.23 & 16& $\\omega_{\\text{o}}$  \\\\ \n& 32.98 $\\pm$ 0.01 & 43.66 & $\\sim$ 15& Harmonic of quiescent superhump \\\\\n& 36.81622 $ \\pm$ 0.03  & 39.11 & 55& $2\\omega_{\\text{o}}$  \\\\\n&  69.55 $\\pm$ 0.03 & 20.70 &$\\sim$ 25& NRP, $\\omega_{1}$, non-stable  (amplitude from 2009)\\\\\n& 103.55 $\\pm$ 0.03  & 13.90 & $\\sim$ 20& $\\omega_{2}$ - $2\\omega_{\\text{o}}$, non-stable (amplitude from 2009) \\\\\n& 140.37 $\\pm$ 0.03 & 10.26 & $\\sim$ 26& NRP, $\\omega_{2}$, non-stable (amplitude from 2009) \\\\\n& 121.99 $\\pm$ 0.03  & 11.80 &$\\sim$ 23.5 & $\\omega_{2}$ - $\\omega_{\\text{o}}$, non-stable (amplitude from 2009) \\\\\n& 153.0 $\\pm$ 0.5 & 9.4 & 9.5& periodic signal (amplitude from 2001) \\\\\n& 307.0 $\\pm$ 0.5 & 4.7 & 8.5 & Harmonic of signal at 153 c\\,d$^{-1}$, transient \\\\\n  \\hline\n\\hline\n\\end{tabular}\n \\caption{Summary of frequencies found in SDSS J1457+51 and BW Sculptoris.}\n\\label{tab:freq} \n\\end{center}\n\\end{table*}\n\n\\subsection{High-Frequency Power Excess}\n\nThe complex range of signals spanning 142 c\\,d$^{-1}$ -- 148 c\\,d$^{-1}$ ($\\approx$ 10 min) moves slightly in frequency and is non-commensurate with the orbital frequency. In addition, during five of the observing nights a broad, low-amplitude peak appeared at 135 c\\,d$^{-1}$ (10.7 min) along with a signal at 72 c\\,d$^{-1}$ (20.0 min), neither of which are of orbital origin.  \n\nWith the aim to study these signals in more detail, a coherent power spectrum was constructed from adding consecutive nights together. However, this did not produce clean signals, even though the nightly power-spectrum windows were always clean. The broadness and complexity of the signals indicate a slight shift in amplitude and frequency on the time scale of a few nights, and\/or an internal fine structure unresolved by our observations. Therefore, analysis was performed on power spectra from separate nights in comparison with the overall mean power spectrum.  \n\nWhen combining our sparse data collected over 43 nights, there is a broad power excess around the frequencies, 135 c\\,d$^{-1}$, 144 c\\,d$^{-1}$ and 148 c\\,d$^{-1}$. This splitting is evident also in the mean spectrum shown in Figure~\\ref{fig:s_avpow}, and is always seen when combining nights from the start and the end of the campaign. When studying the nightly power-spectrum in this range, there was (in general) only one peak present at the time. However, during one observing night, three peaks were seen simultaneously at approximately these frequencies (Figure~\\ref{fig:pow320}). The nightly mean error for any peak appearing at 135 c\\,d$^{-1}$ -- 148 c\\,d$^{-1}$, is about 0.8 c\\,d$^{-1}$.\n \n\n\n\\begin{figure}\n\\includegraphics[width=7.5cm]{pow_320_zoom145.eps}\n\\caption{Power spectrum of J1457 from one single night (HJD=2455320). Peaks were seen simultaneously at 135.6 c\\,d$^{-1}$, 144.6 c\\,d$^{-1}$ and 148.9 c\\,d$^{-1}$.} \n \\label{fig:pow320} \n\\end{figure}\n\n\nThe upper two frames in Figure~\\ref{fig:s_pow_best2} show the combined power spectra of the six nights of best quality. The bottom two panels show model power-spectra constructed from sine waves at the peak frequencies found in the combined six-night dataset, at 18.48 c\\,d$^{-1}$ ($\\omega_{\\text{o}}$), 36.97 c\\,d$^{-1}$ ($2\\omega_{\\text{o}}$), 71.9 c\\,d$^{-1}$, 135.2 c\\,d$^{-1}$, 144.3 c\\,d$^{-1}$ and 147.9 c\\,d$^{-1}$, using the same sampling pattern as the data. The model is able to re-construct the overall appearance and widths of the signals seen in the data reasonably well, indicating that there is power excess at these frequencies. If excluding any of the frequencies from the model, the data power spectrum cannot be reproduced. We note that 71.9 c\\,d$^{-1}$ is about half that of 144.3 c\\,d$^{-1}$, but not exactly in a 2:1 ratio, indicating that the signals are not constant in amplitude and phase (see Section~\\ref{2009} for the case of BW Sculptoris). For a complete summary of the frequency analysis, see Table~\\ref{tab:freq}. \n\n\n\n    \n\\begin{figure}\n\\includegraphics[width=8.5cm]{pow_highfreq.eps}\n\\caption{The two top panels show the power spectrum of J1457 from the same set of nights shown in the mean spectrum (see Figure~\\ref{fig:s_avpow}). The two bottom frames show models that was constructed from the same sampling pattern as the data. The following frequencies was included in the model; 18.48 c\\,d$^{-1}$ ($\\omega_{\\text{o}}$), 36.97 c\\,d$^{-1}$ ($2\\omega_{\\text{o}}$), 71.9 c\\,d$^{-1}$, 135.2 c\\,d$^{-1}$, 144.3 c\\,d$^{-1}$ and 147.9 c\\,d$^{-1}$. All main signals along with their line widths seen in the data can be reproduced fairly well by the model, indicating that there is power excess at these frequencies.} \n \\label{fig:s_pow_best2} \n\\end{figure}\n\n    \n    \n\n\\section{BW Sculptoris}\n \nBW Sculptoris (hereafter BW Scl) is a 16th magnitude blue star which was found to\ncoincide with RXJ2353-0-3852 in the Rosat bright-source catalogue, and\nthen identified as a cataclysmic variable by~\\cite{1997A&A...318..134A}.~\\cite{1997A&A...324L..57A} independently discovered the star as a blue object in the Hamburg\/ESO\nsurvey for bright QSOs. These two studies established the very short\norbital period of 78 minutes. In addition to broad and doubled H and He\nemission lines, BW Scl also shows very broad Balmer and Lyman absorptions,\nsignifying the presence of a white dwarf of modest temperature ($\\sim 15000$ K;\n~\\citealt{2005ApJ...629..451G}). If roughly half of the visual light comes\nfrom such a white dwarf, then the white dwarf has $V=17.3$ and $M_{V}\\sim 12$,\nimplying a distance of only $\\sim 110$ pc. This also agrees with the large\nproper motion found in the USNO catalogue (105 ms$\\,$yr$^{-1}$, \\citealt{2004AJ....127.3060G}). These considerations (a nearby star of very short $P_{\\textbf{orb}}$), and the possibility to study the underlying white dwarf, motivated us to carry out campaigns of time-series photometry nearly every year since 1999.\n\n  \n\\subsection{Observations}\n \nIn total, BW Scl was observed for about 1000 hours spread over about 200 nights, mainly using the globally distributed telescopes of the Center for Backyard Astrophysics (CBA: \\citealt{1993ApJ...417..298S, patterson_1998}). A summary observing log is presented in Table~\\ref{tab:obs2}. To maximise the signal and optimise the search for periodic features, usually no filter or a very broad filter, 4000\\,\\AA\\, --\\, 7000\\,\\AA\\,was used. Occasional runs were obtained in V and I bandpasses to provide a rough calibration, and to verify that the periodic features in the light curve are indeed broadband signals. The smaller (25 cm -- 35 cm) telescopes generally used the star GSC 8015-671 as a comparison, while the larger (91 cm) telescopes used USNO 0450-40780391, a nearby 16th magnitude star. These comparison stars can be considered to be constant. The clear and broadband filters permit only a rough calibration, but BW Scl remained within $\\sim 0.3$ mag of $V=16.6 $ throughout the campaign. In terms of instrumental magnitude, limits on night-to-night variability within each season are more stringent: typically $< 0.05$ mag, and always $< 0.1$ mag. This degree of constancy is remarkable for a cataclysmic variable, and is probably due to the WD's large contribution to the light in the optical. \n    \nIn order to study the periodic behaviour, we always tried to obtain photometry densely distributed in time, preferably with contribution from telescopes widely spaced in longitude (in order to solve problems associated with daily aliases). Most of the analysis below is based on long time series from stations in New Zealand, South Africa, and Chile, and hence not afflicted by aliasing problems.\n\n \\begin{table}\n  \\begin{center}\n \\begin{tabular}{llllll}\n  \\hline\n  \\hline\n\\textbf{Year} & \\textbf{Spanned}  & \\textbf{Observer} & \\textbf{Telescope} & \\textbf{Nights} \\\\\n& (days) & &  & (hours) \\\\\n \\hline\n1999 &  13  &  Kemp & CTIO 91cm & 12\/61\\\\      \n2000 & 38 & McCormick & Farm Cove 25cm & 7\/35\\\\\n         &      & Rea    &     Nelson 35cm   &  2\/9\\\\\n2001  & 77  &    Rea   &           \"     &         16\/87\\\\\n        &     & Kemp     &   CTIO 91cm       &   14\/84\\\\\n           &  & Woudt     &  SAAO 76cm           & 1\/4\\\\\n2002   & 0   &   Kemp      &       \"               & 1\/4\\\\\n2004   & 7    &  Monard   &   Pretoria 25cm  & 7\/38\\\\\n2005  & 55    &  Rea       &  Nelson 35cm    & 16\/66\\\\\n          &  & Christie   & Auckland 35cm      & 12\/48\\\\\n        &  &  Retter\/Liu   &                    & 6\/32\\\\\n    &   &  Monard     & Pretoria 25cm &  4\/22\\\\\n   &   &  Moorhouse   &                    & 3\/10\\\\\n2006  & 90   &   Rea        & Nelson 35cm  &  26\/112\\\\\n          &  & Monard     & Pretoria 35cm  & 3\/21\\\\\n&         &   McCormick  & Farm Cove 25cm & 5\/15\\\\\n2007  & 21  &    Rea        & Nelson 35cm   &  7\/28\\\\\n          &   & Richards  &  Melbourne 35cm     & 1\/3\\\\\n&         & Allen     &  Blenhem 41cm   & 1\/3\\\\\n2008  & 71   &   Monard   &   Pretoria 35cm & 22\/88\\\\\n&          &  Rea        & Nelson 35cm   & 13\/90\\\\\n2009  & 50  &    Rea         &     \"             & 15\/94\\\\\n          &   & Monard      & Pretoria 35cm & 11\/62\\\\\n\\hline\n  \\hline\n\\end{tabular}\n \\caption{Summary observing log for BW Sculptoris.}\n \\label{tab:obs2} \n\\end{center}\n\\end{table}\n\n\n\\subsection{The 1999 Campaign}\n\nThe upper frame of Figure~\\ref{fig:1} shows the light curve from one night during 1999. The general appearance is typical of all nights, as well as seen in previously published light curves (\\citealt{1997A&A...324L..57A, 1997A&A...318..134A}). As those papers demonstrated, a 39-minute wave is always present, with a small even-odd asymmetry suggesting that the fundamental period is actually the double, 78 minutes (confirmed by spectroscopy). The middle frame shows the 1999 double-humped orbital light curve, similar to that of J1457 (see Figure~\\ref{fig:fold18} above). Study of the seasonal timings yielded a 1995 -- 2009 ephemeris:\n\\\\\\\\\nOrbital maximum = HJD 2,450,032.182(3) + 0.05432392(2) E. \n\\\\\\\\\nThe average nightly power spectrum, the incoherent sum of the 11 nights of best quality, is shown in the bottom frame of Figure~\\ref{fig:1}. Power excesses near 72 c\\,d$^{-1}$ (20 min) and 143 c\\,d$^{-1}$  (10 min) are evident. In order to study these higher-frequency signals in more detail, adjacent nights were added together, and the coherent power spectrum was constructed. However, even though the 72 c\\,d$^{-1}$ and the 143 c\\,d$^{-1}$ signals were always present, they were always broad, complex, and slightly variable in frequency (similar to J1457). This is usually a sign that the actual signals violate the assumptions of Fourier analysis: constancy in period, amplitude and phase. \n \n\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig1.eps}\n\\caption{BW Scl in 1999. The upper frame shows an 8-hour light curve, dominated by the orbital wave. The middle frame presents the season's mean orbital light curve, showing the double-humped waveform (the feature at $2\\omega_{\\text{o}}$ dominates all power spectra). The lowest frame shows the mean nightly power spectrum, averaged over the 11 best nights. Significant features are labelled with their frequency in c\\,d$^{-1}$ (with an average error of $\\pm$ 0.7).} \n \\label{fig:1} \n\\end{figure}\n\n\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig2.eps}\n\\caption{BW Scl in 2001. The upper two frames show the mean nightly power spectrum, averaged over the 15 best nights. Features are labelled with their frequency in c\\,d$^{-1}$ (with an average error of $\\pm$ 0.7). The signal at 724 c\\,d$^{-1}$, is likely to be caused by instrumental effects. The lowest frame shows the power spectrum of a particularly dense 12-night segment, after subtraction of the orbital waveform. A powerful signal near 16.5 c\\,d$^{-1}$ is obvious, with a strong first harmonic. The formal error is $\\sim$ 0.01 c\\,d$^{-1}$, and each peak is about that wide. However, the power excess shows more structure and width than seen in the power-spectrum window (see Figure~\\ref{fig:4} below), indicating some amplitude and\/or phase modulation.} \n \\label{fig:2} \n\\end{figure}\n\n\n\\subsection{The 2001 Campaign}\n \nDuring 2001, observatories at three longitudes in Chile, New Zealand \nand South Africa, contributed with good nightly coverage of BW Scl. A very low flickering background is seen in the light curves of BW Scl, enabling detection of periodic signals as weak as $\\sim 0.002$ magnitudes.\n\nThe upper frames of Figure~\\ref{fig:2} shows the nightly mean power spectrum, with significant signals marked to an accuracy of 0.5 c\\,d$^{-1}$. Both the orbital period and the signal at 72 c\\,d$^{-1}$  (20 min) were present, along with two other signals at low frequency: a powerful signal at 16.6 c\\,d$^{-1}$  (86.7 min), and a weak signal at 50.5 c\\,d$^{-1}$ (28.5 min). At higher frequency, signals are detected at 153 c\\,d$^{-1}$ (9.4 min), 307 c\\,d$^{-1}$ (4.7 min), and 724 c\\,d$^{-1}$ (2 min), though the latter is likely to be caused by instrumental effects. Many telescopes have worm gears which turn with a period of exactly 120 sidereal seconds, and this period was reported in research on many types of stars during 1960 -- 1990, i.e. during the photolectric-photometer era. CCDs are much less prone to this error. However, since 724 $\\pm$ 2 c\\,d$^{-1}$ corresponds to 120 sidereal seconds (to within the measurement error), we interpret the signal to be caused by this instrumental effect.   \n \nWe interpret the signal at 153 c\\,d$^{-1}$ as a pulsation frequency, with a significant first harmonic. Examination of individual nights showed this signal to be somewhat transient, at least in amplitude. This behaviour was seen on 7 of the 15 good-quality nights. Since the orbital frequency is known precisely, and since its photometric signature is powerful and constant, we subtracted its first harmonic (and also the second harmonic when detected) from the central 12-night time-series, prior to analysis. The resultant power spectrum at low frequency is seen in the bottom panel of Figure~\\ref{fig:2}. A weak signal appears at the orbital frequency, and stronger signals at 16.34\/16.64 c\\,d$^{-1}$ and 32.98 c\\,d$^{-1}$ (with an accuracy of $\\pm\\,\\,0.01$ c\\,d$^{-1}$). It seems likely that these are superhump signals in the quiescent light curve. Specifically, we interpret the 16.34\/16.64 pair as signifying an underlying frequency of $\\sim 16.5$ c\\,d$^{-1}$, as such splitting can be produced by amplitude and\/or phase changes. The precession frequency $\\Omega$ can be expressed as $16.50 = \\omega_{o} - \\Omega$, where $\\omega_{o}$ is the orbital frequency (implying that $32.98 = 2(\\omega_{o} - \\Omega)$). The precession frequency $\\Omega$ is then equal to 1.9 c\\,d$^{-1}$.\n\nThe primary signal near 16.5 c\\,d$^{-1}$ has been seen before. It was the dominant signal reported by~\\cite{1997A&A...318..134A}, but was then discounted by~\\cite{1997A&A...324L..57A} as probably the result of cycle-count error. The data presented here certainly has no ambiguity in cycle count, and reveals this signal very clearly. \n\n\n\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig3.eps}\n\\caption{ BW Scl in 2009. The top frame shows the mean nightly power spectrum, averaged over the 19 best nights. Features are labelled with their frequency in c\\,d$^{-1}$ ($\\pm$ 0.6). Other frames show power spectra of $\\sim$ 5 day intervals with particularly dense coverage, with significant features labelled ($\\pm$ 0.03). The most interesting features are the trio of weak satellites of $\\omega_{2}$=140.37 c\\,d$^{-1}$ (103.55 c\\,d$^{-1}$ and 121.99 c\\,d$^{-1}$ are displaced by exactly $\\omega_{2}$ - $2\\omega_{\\text{o}}$ and $\\omega_{2}$ - $\\omega_{\\text{o}}$).} \n \\label{fig:3} \n\\end{figure}\n\n \n\\subsection{The 2009 Campaign}  \\label{2009}\n\nThe year 2009 saw another intensive observing campaign on BW Scl. The upper frame of Figure~\\ref{fig:3} shows the nightly power spectrum, averaged over the 19 nights of coverage. Signals near 70 c\\,d$^{-1}$  (20.6 min) and 140 c\\,d$^{-1}$  (10.3 min) are evident, with weaker signals near 104 c\\,d$^{-1}$  (13.9 min) and 122 c\\,d$^{-1}$  (11.8 min). In the three lower frames, we show intervals of particularly dense coverage, each with a nominal frequency resolution of $\\pm\\,\\,0.04$ c\\,d$^{-1}$. In these frames, the 70 c\\,d$^{-1}$ and 140 c\\,d$^{-1}$ signals show their variability in frequency and amplitude (similar to J1457). Such variability is characteristic of all our data. The second frame (JD 2,455,068 - 074) illustrates the following:\n\n\\begin{enumerate}\n\\item [1.] Although the 70 c\\,d$^{-1}$ and 140 c\\,d$^{-1}$ signal are related (when one moves\nto slightly lower frequency, so does the other), the frequencies do not appear to be exactly in the ratio 2:1. Here, these frequencies are described as $\\omega_{1}$ and $\\omega_{2}$, with $\\omega_{2} \\sim 2\\omega_{1}$.\n\n\\item [2.] The weaker signals flanking the 140 c\\,d$^{-1}$ feature are displaced by exact\ninteger multiples of the orbital frequency. Thus the signals seen in JD 068-74 are $\\omega_{1}$, $\\omega_{2}$, ($\\omega_{2} - \\omega_{\\text{o}}$) and ($\\omega_{2} - 2\\omega_{\\text{o}}$). These orbital sidebands of $\\omega_{2}$ are also visible in Figure~\\ref{fig:1}, however such orbital sidebands was never seen of $\\omega_{1}$. This reproduces the properties of the pulsations seen in SDSS J1507+52 (\\citealt{2008PASP..120..510P}).\n\\end{enumerate} \n\n  \n\\subsection{Power-Spectrum Window}\n  \nFinally, in Figure~\\ref{fig:4} we show the power spectrum in the vicinity of $2\\omega_{\\text{o}}$, for each season. The $2\\omega_{\\text{o}}$ signal maintains constancy in phase and amplitude, and therefore acts effectively like a test signal whose power spectrum should be reproduced in structure by any coherent signal (constant in phase and amplitude). The central peak at exactly 36.816 c\\,d$^{-1}$ (39.113 min) always dominates over its neighbours in the picket-fence pattern because of very long runs, and\/or observations at several longitudes. This demonstrates that these campaigns are free from aliasing. Also, it is worth noting that peaks at frequencies greater than $2\\omega_{\\text{o}}$, are always slightly wider (for the single-night time series) than expected for a truly periodic signal, and always more complex (for the multiple-night time series). This is true for all our data. The underlying $\\omega_{1}$ and $\\omega_{2}$ signals either have an intrinsic complex structure, or vary strongly in amplitude and\/or phase on short timescales (or both). In addition, the entire complex of signals near $\\omega_{1}$ and $\\omega_{2}$ moves in frequency by a few percent on timescales as short as $\\sim10$ days.\n  \n\\subsection{Superhumps in BW Scl?} \\label{superh}\n\nThe obvious signal at 87 minutes in BW Scl, displaced by $\\sim 11$ \\% from $P_{\\text{orb}}$ seen in the 2001 power spectrum (Figure~\\ref{fig:2}), is a transient but repeating feature in the light curve. In our ten years of coverage, six campaigns were sufficiently extensive to reveal such a signal. This feature was always strong when it was detected, and moved slightly in frequency, even on timescales of a few days. This qualitatively describes a common superhump, which is a well-known feature in CV light-curves. However, there are three substantial differences:\n\\begin{enumerate}\n\\item [1.] Common superhumps are found in outburst states, typically near $M_{V} \\sim 5$, not at quiescence when $M_{V} \\sim 12$.\n\\item [2.] Common superhumps occur with fractional period excesses, $(P_{\\text{sh}} - P_{\\text{orb}})\/P_{\\text{orb}}$, near 3\\%, not 11\\%.\n\\item [3.] Common superhumps are more stable, wandering from a constant-period ephemeris on a timescale of 100 -- 200 cycles. The power-spectrum signal in BW Scl appears complex and broad, suggesting much lower coherence.\n\\end{enumerate}\n\nA system with quiescent superhumps is a rare beast in the CV kingdom, but it is not unprecedented. In AL Com, a nearly identical signal was reported and extensively discussed by \\cite{1996PASP..108..748P}. Something similar has also been reported in V455 And (\\citealt{2005A&A...430..629A}), SDSS J0745+45 (\\citealt{2010ApJ...710...64S}) and possibly SDSS J1238-03 (\\citealt{2010ApJ...711..389A}). All these stars are period-bouncer candidates (systems that have already passed the minimum period; see Tables 3 and 5 of \\citealt{2011MNRAS.tmp...27P}). These candidates were chosen due to their low donor-star mass (or low $q = M_{2}\/M_{1}$). A possible account of how stars of very low $q$ might be able to manufacture quiescent superhumps has been given in~\\cite{1996PASP..108..748P}, and includes the idea that a low donor mass implies a larger Roche lobe surrounding the disc. Weak tidal torques could then allow the quiescent disc to extend to the 2:1 orbital resonance, where an eccentric instability could drive a fast prograde precession (viz., at $\\Omega$ = 1.9 c\\,d$^{-1}$), resulting in a superhump with $\\omega = \\omega_{o} - \\Omega$. \n\n\n\n\\begin{figure*}\n\\includegraphics[width=13cm]{fig4.eps}\n\\caption{BW Scl. Power spectra in the vicinity of $2\\omega_{\\text{o}}$ for the lowest frame in Figure~\\ref{fig:2}, and the three lower frames in Figure~\\ref{fig:3}. Since the $2\\omega_{\\text{o}}$ signal is essentially constant in frequency and amplitude, this is effectively the power-spectrum window for each observation. This implies that the nightly length of run and span of longitude is sufficient to exclude confusion by aliases.} \n \\label{fig:4} \n\\end{figure*}\n\n\n\\section{Interpretations as Non-Radial Pulsations}\n\nBoth BW Scl and J1457 are quiescent CVs of very low accretion luminosity. Their spectrum show evidence of the primary WD, as do their light curves which contain rapid, non-commensurate signals. This is the general signature of the GW Lib stars, where the periodic signals are believed to represent the non-radial pulsations of the underlying WD. However, no proof of this has ever been found, not for GW Lib or any other of the 10 -- 15 members of the class. In order to explain these signals, two other hypotheses deserve consideration: first, quasi-periodic oscillations (QPOs) arising from the accretion disc, and second, the spin frequency of a magnetic WD (intermediate-polar model).\n\nQPOs were first discovered, and named, for a broad excess of power seen during a dwarf-nova eruption of RU Pegasi (\\citealt{1977ApJ...214..144P}). Since then, they have been commonly found with periods of 10 -- 20 minutes in the high-state light-curves of many nova-like variables (reviewed by \\citealt{2004PASP..116..115W}). QPOs are typically seen as very broad peaks in the power spectrum ($\\delta v\/v \\sim 0.5$; see for instance Figure 11 and 12 in~\\citealt{2002PASP..114.1364P}), and typically have modulations of 1\\% -- 3\\% of an accretion disc in full outburst (for a light source with $M_{V}\\sim 4$). The most famous example is the 20-minute signal in TT Arietis (which belongs to the class of VY Sculptoris stars). However, VY Sculptoris stars appear to be the among the most luminous CVs. This can be compared to the faint BW Scl, where the 70\/140 c\\,d$^{-1}$ signals have $\\delta v\/v \\sim 0.01$ and are $\\sim 1$\\% modulations of a light source with $M_{V}\\sim 12$. The peaks in J1457 have slightly higher $\\delta v\/v$, but on both counts, the QPO hypothesis earns no applause. It would have to be an essentially new kind of accretion-disc QPO.\n\nA white dwarf spinning with a period of 20 minutes could explain the features of the signal seen at $\\sim 70\/140$ c\\,d$^{-1}$ in both objects, the common appearance of the first harmonic, the occasional switch to a pure first harmonic (two-pole accretion), and the orbital sidebands seen in BW Scl (from amplitude modulations, and\/or reprocessing from structures fixed in the orbital frame). But this hypothesis fails to account for the shifts in frequency exhibited by the $\\omega_{1}$ and $\\omega_{2}$ signals (the $\\sim 5$\\% wandering, e.g. in the range 68 c\\,d$^{-1}$ -- 73 c\\,d$^{-1}$ and 135 c\\,d$^{-1}$ -- 153 c\\,d$^{-1}$), the fact that $\\omega_{2}$ is not exactly $2\\omega_{1}$, and the intrinsic breadth (or fine structure) of both signals. These are profound inconsistencies.\n\nSuch considerations drive us back to the GW Lib model. Actually, we have studied all of the 10 -- 15 known class members with time-series photometry, and the resemblances to BW Scl and J1457 are substantial:\n\\begin{enumerate}\n\\item [1.] Low-amplitude and non-commensurate periodic signals in low-$\\dot{M}$ CVs. \n\\item [2.] Signals roughly constant over a few days, but somewhat transient in\n    frequency and amplitude on longer timescales.\n\\item [3.] Signals frequently with strong first harmonics or quasi harmonics.\n\\item [4.] Signals sometimes with known or suspected fine structure.\n\\end{enumerate}\nThese resemblances, and the difficulties with alternative models, seem\nsufficient to accept both BW Scl and J1457 as new members of the GW Lib class.     \n     \n\\section{Summary}\n\n\\begin{enumerate}\n\\item [1.] Two more CVs of very low accretion-rates have shown rapid non-commensurate signals in quiescence, which makes them likely new members of the GW Lib class. Both J1457 and BW Scl show a complex spectrum with the main signals near 10 and 20 minutes.\n\n\\item [2.] The pulsation frequencies in both stars wander by a few percent on a timescale of days. In addition, the power-spectrum constructed from multiple-night light curves, show broad peaks, which might be due to the frequency wandering, or from intrinsic fine structure not resolved by our data (or due to a strong amplitude modulation).\n\n\\item [3.] BW Scl shows several peaks displaced from the main pulsation frequency $\\omega_{2}$, by $\\omega_{2} \\pm \\omega_{\\text{o}}$ and $\\omega_{2} \\pm 2\\omega_{\\text{o}}$. Similar behaviour is evident in other GW Lib stars (SDSS J1507+52, V386 Ser and SDSS J1339+48). The origin of this phenomenon is still unknown. The rich pulsation spectrum makes BW Scl a good candidate for an intensive round-the-world time-series campaign with larger telescopes.\n\n\\item [4.] The orbital light curves of both stars show double-humped waves. From these waves, precise periods are found at $P_{\\text{orb}} = 78.22639 \\pm 0.00003$ minutes for BW Scl, and $77.885 \\pm 0.007$ minutes for J1457. Similar systems displaying non-radial pulsations, such as SDSS J1339+4847, SDSS J0131-0901 and SDSS J0919+0857, all have orbital periods right at the same orbital period ($\\approx$ 80 minutes). \n\n\\item [5.] BW Scl sometimes shows a transient wave with a period, $P_{\\text{sh}} = 87.27$ minutes, which is interpreted as a quiescent superhump. It thereby joins a small group of stars who manage a superhump at quiescence, all of which are likely to have very low mass ratios. This might arise from an eccentric instability at the 2:1 resonance in the disc.\n\n\\item [6.] The white-dwarf domination of the spectra in these stars suggests great faintness of the accretion light. This signifies a very low accretion rate, and both stars are likely to be very old CVs.\n\\end{enumerate}\n   \n   \n\\section*{Acknowledgments}   \n\nJoe Patterson would like to acknowledge the support from NASA grant GO11621.03A, the Mt Cuba Astronomical Foundation and the NSF grant AST 0908363. \n   \n   \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMotivated from the microscopic understanding of black holes in string\ntheory, brane configuration and microscopic proposals \\cite{hep-th\/0508023,\nhep-th\/0107119}, we focus on investigating the possibilities of a covariant\nthermodynamic geometric study of a class of coarse grained configurations.\nThese solutions are described in terms of the chemical parameters and a set\nof arbitrarily  excited boxes of the random Young tableaux. In particular,\nwe intend to study the thermodynamic geometry arising from the canonical\nenergy and counting degeneracy for the two parameter finite temperature\nsolutions. Ref. \\cite{07073601v2} offers an interesting motivation for\nthe case of near-extremal black holes in $AdS_5$. The main idea is first\nto develop a geometric notion for black hole thermodynamics and then to\nrelate it with the existing microscopic quantities already known from CFT\nconfigurations \\cite{hep-th\/0508023, hep-th\/0107119}. Furthermore,\nwe incorporate all order fluctuations in the exact canonical energy\n\\cite{SM} and entropy associated with the counting of excited boxes\nin random Young tableaux and analyze their respective contributions\nto the covariant thermodynamic geometries.\nThe phenomenon of the statistical fluctuation in the bubbling solutions\ninvolves the same asymptotic charges as that of the considered giant black holes.\nTo be precise, we consider an ensemble of horizonless geometries and the black hole\narises as a coarse graining of such states.\n\nLunin and Mathur have connoted a conjecture \\cite{lu-ma} that the\nblack hole microstates may be characterized by string theory\nbackgrounds with no horizons. Moreover, these solutions saturate\nan exact bound on their angular momenta, and have the same set of\nasymptotic charges as the respective black hole whose horizon area\nclassically vanishes. Some finite temperature supersymmetric \\cite{0411001v2}\nand associated non-supersymmetric generalizations appear as well,\nwhich correspond to definite black holes with finite size horizons. \nAs a consequence, we are interested in finding state-spaces for the\nBPS black holes which are already known to have a finite size horizon\nat zero temperature \\cite{mathur1, mathur2}. Furthermore, the complete\nclass of two parameter\\footnote{The two parameters of the interest\nare the charge and effective canonical temperature of the solution\n\\cite{hep-th\/0508023, hep-th\/0107119}. It is worth mentioning that \nthis temperature is not the physical temperature of the black holes.\nHowever, it arises as the canonical averaging e.g. Eqn(\\ref{effTemp}).}\nsupersymmetric black hole solutions in five dimensions are well-known\nsince the invention of \\cite{LLM}. Ref. \\cite{09103225v2} brings out\nan interesting issue of the horizonless solutions in AdS space and thereby\nconstructs its relation for the case of the small black hole solutions.\nThe associated finite temperature solutions \\cite{0411001v2,09103225v2}\ncould be thought as a deviation from the extremality condition. Importantly,\nthe near-extremal configurations \\cite{07073601v2} may further indicate a\nset of interesting issues for the present discussion thermodynamic geometry. \nThus, it would be inspiring to analyze the thermodynamic intrinsic manifold\nfor the giants and superstars configurations having two parameters, which\ncharacterize fluctuation about an ensemble of equilibrium CFT microstates\n\\cite{hep-th\/0508023, hep-th\/0107119}. From the perspective of the thermodynamic\ngeometry, this sets a good motivation to study the issue of statistical\ncorrelations for the black holes in string theory.\n\nInterestingly, the existence of such a geometric structure in\nequilibrium thermodynamics was introduced by Weinhold \\cite{wien1,wien2}\nthrough an inner product in the space of equilibrium chemical\npotentials defined by the minima of the internal energy function\n$U=U( \\lambda_i, V, S, T)$ as the Hessian function\n$h_{ij}= \\partial_i \\partial_j U $.  In this description, the\nquantities $ \\lbrace \\lambda_i, V, S, T \\rbrace $ are defined as\nthe chemical potentials $ \\lambda_i$, phase-space volume $V$,\nentropy $S$ and the temperature $T$ of the underlying equilibrium\nstatistical configuration. In order to provide a physical scale,\nwe need to restrict negative eigenvalues of the metric tensor,\nwhich may be accomplished by requiring that the volume or any other\nexternal parameter of the given fluctuating giant configurations\nbe held fixed.\n\nLet us recall that the Riemannian geometric structure thus\ninvolved indicates a certain physical significance ascribed to\ngiants and dual giants. The associated inner product structure\non the intrinsic space may either be formulated in the entropy\nrepresentation, as a negative Hessian matrix of the counting\nentropy with respect to the extensive variables, where the total\nnumber of boxes and number of excited boxes in a random Young\nTableaux define the degeneracy, or be formulated in the energy\nrepresentation, as the Hessian matrix of the associated energy\nwith respect to the intensive chemical potentials\n\\cite{hep-th\/0508023,hep-th\/0107119}. Such geometric considerations\nhave been studied earlier in the literature, as the covariant\nthermodynamics whose application in the energy representation\nhas first been considered by Weinhold \\cite{wien1,wien2}. Whilst,\nthe entropy representation in which we shall investigate fluctuating\nYoung tableaux has been considered in the view-points of thermodynamic\nfluctuation theory \\cite{RuppeinerRMP,math-ph\/0507026,RuppeinerA20,\nRuppeinerPRL,RuppeinerA27,RuppeinerA41}.\n\nThe present paper is thus intended to analyze both pictures,\nand indeed our study strikingly provides a set of exact covariant\nthermodynamic geometric quantities ascribed to the giants and to the\nsuperstars \\cite{hep-th\/0508023}. Recall that the thermodynamic\nconfiguration of the $1\/2$-BPS (dual) giants may be parameterized\nby the underlying chemical potentials, $\\lambda_1, \\lambda_2$.\nThus, fluctuations in the associated ensemble may be described by\nthe minima of the energy function $E= E(\\lambda_1, \\lambda_2)$.\nExplicitly, the Weinhold metric tensor of an intrinsic Riemannian\nspace spanned by the chemical potentials may be given as $g_{ij}=\n\\partial_i \\partial_j E(\\lambda_1, \\lambda_2)$ and turns out to be\nconformal to the Ruppenier metric\\footnote{The present paper\nassumes that the terms (Ruppenier, Weinhold) geometry and\n(State-space, Chemical) geometry are synonyms and can be used\ninterchangeably. It is worth mentioning that the geometry associated\nwith the Weinhold metric tensor is said to be the chemical geometry\nand the other whose metric tensor is defined by the Ruppenier metric\nis said to be the state-space geometry. The general definitions are\ngiven by the Eqns(\\ref{Wienmetricgen},\\ref{Ruppmetricgen}). Subsequently,\nThe Eqns(\\ref{Wienmetric},\\ref{Ruppmetric}) explain the precise\nconsideration of the analysis of the present paper.}\nwith the temperature as the conformal factor. Thus, the metric as an inner\nproduct on the intrinsic surface of chemical potentials is necessarily\nensured to be symmetric, positive definite and satisfies the triangle inequality,\nsince the energy function has a minimum configuration in the equilibrium.\n\nAs a matter of fact, a few simple manipulations illustrate that\nthe conformally related state-space curvature is inversely\nproportional to the singular part of the free energy associated\nwith long range correlation(s), whose regularity consequently\nsignifies the fact that underlying giant configurations have no\nphase transition(s) over the range of admissible boxes. It is\nworth to mention that the Ruppeiner formalism, which has further\nbeen applied to diverse condensed matter systems with two dimensional\nintrinsic Riemannian spaces have been found to be completely\nconsistent with the scaling and hyper scaling relations involving\ncertain critical phenomena and in turn reproduce their corresponding\ncritical indices, see for review \\cite{RuppeinerRMP,math-ph\/0507026}.\nIn general, such geometric notions remain well-defined and correspond\nto an interacting statistical basis for diverse extremal and non-extremal\nblack brane configurations \\cite{bnt,SST,0606084v1}.\n\nThere exists a large class of black holes which have the\nspherical horizon topology in the physical regime, whose thermodynamic\nconfigurations are quite energizing in their own respect, as we shall exhibit\nin the next section. Subsequently, we shall analyze the behavior\nof collective state-space correlations, and in particular, we wish to\nillustrate certain peculiarities of the state-space geometry which in\nthe large charge limit provides the correlation length for the associated\nmicroscopic configurations. \nThe purpose of this paper is to investigate the relation between\nthe covariant thermodynamic geometries and the microscopic counting\nperspective from the total and excited boxes of random Young tableaux.\nBoth of these notions have received considerable recent attention from\nthe perspective of black hole physics and the corresponding thermodynamics.\nThus, the goal here is to explore underlying microscopic arguments behind\nthe macroscopic notion of the covariant thermodynamic geometries arising\nfrom the energy\/ degeneracy of fluctuating (dual) giant configurations.\nFurthermore, the underlying state-space curvature for higher dimensional\nblack holes appears to be intertwined with vacuum fluctuations existing\nin the possible brane configurations, and thus one may revel certain\nintriguing informations of the possible microscopic CFTs \\cite{fuzzball}.\n\nIn this connection,\nwhat follows here shows that the thermodynamic intrinsic geometries and\ntheir respective microscopic resolutions may however be pursued by\nan intriguing involvement of the framework of Mathur's fuzzball\nsolutions \\cite{LLM,fuzzball,LM}. Such a perspective thus discloses\nan indispensable ground with the statements that the thermodynamic\nscalar curvatures of interest not only has exiguous microscopic\nknowledge of the black hole configuration, but also it has rich\nintrinsic Riemannian geometric structures. Specifically, we procure\nthat the configurations under the present analysis are effectively\nattractive in general, while they are stable only if at least one\nof the parameters remain fixed. Our hope is that finding statistical\nmechanical models with like behavior might yield further insight\ninto the microscopic properties of black holes and a conclusive\nphysical interpretation of their state-space curvatures and\nrelated intrinsic geometric invariants.\n\nWithin the well understood framework of the microscopic counting technique,\nwhat fundamentally follows is that the present geometric notions\nimplicitly assume an infinite set of subensembles of the most\ngeneral statistical system having an integrated ensemble \\cite{LM}.\nWe thus explore the intriguing role of chemical fluctuations in\nthe united canonical energy, or box counting statistical entropy,\nof the underlying black brane configurations. Consequently, we\nfind that the associated intrinsic Riemannian geometries being\nLegendre transforms of each other, indeed describe corresponding\nfluctuations of the excited droplets or fuzzballs. For example,\nthe canonical energy corrected by the contributions coming from\nthe fluctuations of an ensemble of excited boxes may be regarded\nas a closer approximation to that of the simple energy in the\ncorresponding subensembles. We shall show that such considerations\napply as well to two parameter fuzzballs \\cite{LLM,fuzzball,LM}\nor the liquid droplets \\cite{liquid}, when considered as an union\nsubensembles. The specific forms of the infinite summation and\nspecial polynomial contributions may thus be calculated for a\nclass of supergravity black holes \\cite{LLM}.\n\nIt is noteworthy that the applicability of this analysis\npresupposes that the underlying net ensemble is thermodynamically\nstable, which requires a positive specific heat, or correspondingly\nthat the Hessian matrix of the energy function or the negative Hessian\nmatrix of the counting entropy, be positive definite. The energy\nfunction and degeneracy for the (dual) giant or superstar systems\nincorporating such contributions \\cite{LLM,fuzzball,LM,liquid} may\neasily be written to an appropriate form, without any approximation.\nThe present article discovers that such contributions to the canonical\nenergy\/ box counting entropy induce certain bumps in the scalar curvature\nof underlying intrinsic thermodynamic surfaces spanned by the chemical\npotentials, or the total and excited boxes of the Young tableaux.\n\nThe present paper is organized as follows. In the first section,\nwe have presented the motivations to study the intrinsic Riemannian\nsurfaces obtained from the Gaussian fluctuations of the giants or\nsuperstars canonical energy or counting entropy. In particular,\nwe have outlined some microscopic implications of the thermodynamic\ngeometry for the giants and superstars, in the lieu of the underlying\nmicroscopic configurations, and the remaining significations have\nbeen summarized in the final section. In section $2$, we shall\nbriefly explain what are the intrinsic thermodynamic geometries\nbased on a large number of equilibrium parameters characterizing\nthe giant and superstar black hole configurations. In section $3$,\nwe obtain the two parameter giants (superstars) canonical\nenergy and counting entropy as a function of chemical potentials\nand in terms of the number of excited and total boxes of the Young\ntableaux. In section $4$, we investigate the chemical geometry\nfor the giant (\/superstar) black holes, and examine two parameter\nfluctuating canonical configurations.\n\nIn section $5$, we focus our attention on the state-space geometry\nof the aforementioned configurations, with some excited boxes and\nan arbitrarily random Young tableaux. Explicitly, we investigate\nthe nature of the state-space geometry thus defined, as an intrinsic\nRiemannian manifold obtained from the entropies, with all possible\ncontributions being considered in the most general canonical ensemble.\nWe have explained that such an intrinsic geometric configuration,\nobtained from an effective canonical energy or degeneracy of the\ngiants\/ superstars, results to be well-defined and pertains to an\ninteracting statistical system. Finally, section $6$ contains a set\nof concluding issues and a possible discussion of the thermodynamic\nchemical and state-space geometries. The general implications thus\nobtained may thus divulge the geometry of both the chemical and the\nequilibrium microstate acquisitions which are the matter of an AdS\/CFT\ncorrespondence. Furthermore, our results are in close connection\nwith the microscopic implications arising from the underlying CFTs,\nand thus offer definite physically sound explanations of the finite\nchemical potential\/ finitely many excited boxes configurations.\n\\section{Thermodynamic Geometries}\nThe present section presents a brief review of the essential\nfeatures of thermodynamic geometries from the perspective of the\napplication to the two parameter configuration\nAs mentioned in the introduction, in order to illustrate the\nconnection between thermodynamic aspects and microscopic counting,\nwe focus our attention on the geometric nature of underlying\nfluctuating chemical parameters and number of excited (unexcited)\nboxes in an ensemble of finitely  random Young tableaux. The respective\nanalysis is divulged in the neighborhood of small chemical potentials\nor in an arbitrarily large number of excited boxes. In the present\ncontext, these limits essentially provide an equilibrium configuration.\nIn the next section, we shall motivate the thermodynamic geometric\nnature  for the (dual) giants and superstars.\n\nIn order to illustrate the basic notion of thermodynamic fluctuations,\nlet us consider an intrinsic Riemannian geometric model, whose\ncovariant metric tensor may be defined as the Hessian matrix of the\ncanonical energy, with respect to a finite number of arbitrary\nchemical potentials carried by the giant gravitons and superstar,\nconsidered in a given equilibrium configuration at fixed volume, or\nany other parameters of the solution. Specifically, let us define a\nrepresentation $ E( \\lambda_i, T) $ for given canonical energy, chemical\npotentials, and temperature $ \\lbrace E,  \\lambda_i, T \\rbrace $.\nIn the present consideration, the energy, as defined below, may\nfurther be shown to be solely a function of the chemical potentials,\nwhich in fact accompanies an intriguing set of expressions, as we have\nindicated in general by Eqs. \\ref{energy}, \\ref{entropy}. The underlying\nmethod, in general, results to be in an intrinsic Riemannian geometric\nconfiguration, which is solely based on nothing but the giant\nchemical potentials. Such a general consideration in fact yields an\nintrinsic space spanned by the $n$ chemical potentials of\nthe theory under examination, and thus it exhibits a $n$-dimensional\nintrinsic Riemannian manifold $ M_n $. The components of the\ncovariant metric tensor of the so called chemical geometry\n\\cite{wien1,wien2} may be defined as\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{Wienmetricgen}\ng_{ij}:=\\frac{\\partial^2 E(\\vec{x})}{\\partial x^j \\partial x^i}\n\\end{eqnarray}\n\\end{scriptsize}\nwhere the vector $\\vec{x} =(\\lambda_i, V, S, T) \\in M_n $.\nAs anticipated in the appendix A, it is worth to mention that\nthe thermodynamic curvature corresponds to the nature of the intrinsic\ncorrelation present in the statistical system. This will altogether\nimply that the scalar curvature for two component systems can be\nthought of as the square of the correlation length, at some given\nnon-zero configuration temperature, to be $R( \\lambda_1,\\lambda_2)\n\\sim \\chi^2 $, where we may identify the $\\chi(\\lambda_1,\\lambda_2)$\nto be the correlation length of the concerned chemical system.\nConsecutively, it is not surprising that the geometric analysis,\nbased on the Gaussian approximation to the classical fluctuation\ntheory, precisely yields the correlation volume of the chosen configuration.\nThis strongly suggests that, even in the context of chemical reactions\nor in any closed system, a non-zero scalar curvature might provide\nuseful information regarding the range of microscopic phase-space\ncorrelations between various components of the underlying statistical\nconfiguration.\n\nNote that such Riemannian structures, defined by the metric tensor\nin a chosen representation, are in fact closely related to the\nclassical thermodynamic fluctuation theory \\cite{RuppeinerA20,\nRuppeinerPRL, RuppeinerA27, RuppeinerA41} and existing critical\nphenomena. The probability distribution $\\Omega(x)$ of thermodynamic\nfluctuations over an equilibrium intrinsic surface naturally\ncharacterizes an invariant interval of the corresponding\nthermodynamic geometry, which in the Gaussian approximation reads\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\Omega(x)= A \\ \\ exp [-\\frac {1}{2} g_{ij}(x) dx^i \\otimes dx^j],\n\\end{eqnarray}\n\\end{scriptsize}\nwhere the pre-factor $A$ is the normalization constant of the\nGaussian distribution and $\\otimes$ denote a symmetric product.\nThe associated inverse metric may easily be shown to be the second\nmoment of the fluctuations or the pair correlation functions, and\nthus it may be given as $g^{ij}= < x^i \\vert x^j>$, where \n$\\lbrace x_i \\rbrace $'s are the intensive chemical variables\nconjugated to the charges $\\lbrace X^i \\rbrace$ of the Legendre\ntransformed entropy representation. Moreover, such Riemannian\nstructures may further be expressed in terms of a suitable\nthermodynamic potential, obtained by certain Legendre transforms,\nwhich correspond to certain general coordinate transformations\non the equilibrium thermodynamic manifold.\n\nFollowing \\cite{RuppeinerRMP, 0606084v1, 0510139v3}, it turns out\nthat the natural inner product on the state-space manifold may\neasily be ascertained for an arbitrary finite parameter black\nbrane configuration, and the concerned state-space turns out to\nbe a $n$-dimensional intrinsic manifold $ M_n $. Typically, the\nassociated entropy $S(X^i)$ as an embedding function defines\nthe covariant components of the metric tensor of the thermodynamic\nstate-space geometry, which has originally been anticipated by\nRuppeiner in the related articles \\cite{RuppeinerRMP,RuppeinerA20,\nRuppeinerPRL, RuppeinerA27, RuppeinerA41}. Here, we shall take\nthis representation of the intrinsic geometry, and thus find\nthat the covariant components of state-space metric tensor\nmay be defined to be\n\\begin{scriptsize} \n\\begin{eqnarray} \\label{Ruppmetricgen}\ng_{ij}:=-\\frac{\\partial^2 S(\\vec{X})}{\\partial X^i \\partial X^j}\n\\end{eqnarray}\n\\end{scriptsize}\nWe may thus explicitly describe the present state-space geometric\nquantities as simply an intrinsic two dimensional Riemannian\nmanifold for $1\/2$-BPS configurations. Furthermore, the underlying\nstate-space geometry may parametrically be defined by the two invariant\nparameters, {\\it viz.}, $\\vec{X}= (n,M) \\in M_2$.\nWe may therefore notice that the components of the state-space\nmetric tensor are related to the statistical pair correlation\nfunctions, which may as well be defined in terms of the parameters\ndescribing the dual microscopic conformal field theory living on the\nboundary. This is because of the fact that the underlying metric\ntensor comprising Gaussian fluctuations of the entropy defines the\nstate-space manifold for the rotating black brane configuration.\nWe may thus easily perceive, in the present consideration, that the\nlocal stability of the underlying statistical configuration\nrequires that the principle components of state-space metric\ntensor $\\{ g_{ii} \\ | \\ i:= n, M \\}$ signifying heat capacities\nshould be positive definite\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{nn}(n,M) &>& 0 \\nonumber \\\\\ng_{MM}(n,M) &>& 0\n\\end{eqnarray}\n\\end{scriptsize}\nMoreover, the positivity of the state-space metric tensor imposes\na stability condition on the Gaussian fluctuations of the underlying\nstatistical configuration, which requires that the determinant and\nhyper-determinant of the metric tensor must be positive definite.\nIn order to have a positive definite metric tensor $\\Vert g(n,M)\\Vert$\non the two dimensional state-space geometry, one thus demands that\nthe determinant of metric tensor must satisfy $\\Vert g \\Vert >0$,\nwhich in turn defines a positive definite volume form on the\nconcerned state-space manifold. Furthermore, it is not difficult\nto calculate the Christoffel connection $\\Gamma_{ijk}$, Riemann\ncurvature tensor $R_{ijkl}$, Ricci tensor $R_{ij}$, and the scalar\ncurvature $ R $ for the two dimensional state-space intrinsic\nReimannian manifold $(M_2,g)$. Remarkably, it turns out that\nthe above two dimensional state-space scalar curvature appears\nas the inverse exponent of the inner product defining the pair\ncorrelation functions between arbitrary two equilibrium microstates\ncharacterizing the black brane statistical configuration. \n\nNotice that there exists an intriguing relation of the scalar\ncurvature of the state-space intrinsic Riemannian geometry,\ncharacterized by the parameters of the equilibrium microstates,\nwith the correlation volume of the corresponding black brane\nphase-space configuration. Additionally, Ruppeiner has revived \nthe subject with the fact that the state-space scalar curvature\nremains proportional to the correlation volume $\\tilde{\\chi}^d$,\nwhere $d$ is the system spatial dimensionality and $\\tilde{\\chi}(n,M)$\nis its correlation length, which reveals related information residing\nin the microscopic models \\cite{RuppeinerA20}. It is worth mentioning\nfurther that the state-space scalar curvature in general signifies\npossible interaction in the underlying statistical configuration. \nFurthermore, one may appreciate that the general coordinate\ntransformations on the state-space manifold thus considered\nexpound to certain microscopic duality relations associated with\nthe fundamental invariant charges of the configurations.\n\nFrom the perspective of an intrinsic Riemannian geometry, it seems\nthat there exists an obvious mechanism on the black brane side, and\nthat it would be interesting to illuminate an associated stringy notion\nfor the statistical correlations to the microstates of the giant and\nsuperstar solutions or vice-versa. In this concern, it turns out\nthat the state-space constructions so described might elucidate\ncertain fundamental issues, such as statistical interactions and\nstability of the underlying brane configurations with spins and\nnon-equal $R$ charges. Nevertheless, one may arrive to a definite\npossible realization of the equilibrium statistical structures,\nwhich is possible to determine in terms of the parameters of an\nensemble of microstates describing the equilibrium configurations.\n\nThe relation of a non-zero scalar curvature with an underlying\ninteracting statistical system remains valid even for higher\ndimensional intrinsic Riemannian manifolds, and the connection of a\ndivergent scalar curvature with phase transitions may accordingly\nbe divulged from the Hessian matrix of the considered energy\/\ncounting entropy. It is significant to remark that our analysis\ntakes an intriguing account of the scales that are larger than the\ncorrelation length and considers that only a few microstates do not\ndominate the whole macroscopic equilibrium intrinsic quantities.\nSpecifically, we shall focus on the interpretation that the\nunderlying energy includes all order contributions from a large\nnumber of subensembles of the fluctuating microstates, and thus\nit characterizes our description of the geometric thermodynamics\nfor (dual) giants, superstars and fuzzballs.\n\nOur geometric formulations thus tacitly involves an unified\nstatistical basis, in terms of the chosen union of subensembles.\nAlthough the analysis has only been considered in the limit of\nsmall fluctuations, however the underlying correlation length takes\nan intriguing account upon the quartic corrections of the canonical\nenergy or the counting entropy. With this general introduction to\nthe thermodynamic geometries defined as the (negative) Hessian\nfunction of the canonical energy (or counting entropy), let us now\nproceed to investigate the energy\/ entropy of two parameter giants,\nsuperstars and their thermo-geometric structures. In the present\ninvestigation, we shall focus our attention on the interpretation\nthat the underlying energy includes contributions from a large number\nof excited boxes, and thus our description of the geometric thermodynamics\nextends itself to the spinless $1\/2$ BPS supergravity configurations.\n\\section{The Fuzzballs and Liquid Droplets}\nIn this section, we shall first review the giant and superstar\nconfigurations \\cite{hep-th\/0508023, hep-th\/0107119} and subsequently\ncompute the canonical energy and box counting degeneracy in the\nrequired form. Thereafter, we explore the thermodynamic geometries of\nthe giants and superstars arising from the consideration of type IIB\nstring theory. For obtaining the energy as a function of an effective\ntemperature and chemical potential, we consider the liquid droplet model\nof giants and superstars. The present paper considers, {\\it viz.} (i)\nthe canonical energy with two distinct parameters $ (T, \\lambda)$,\nwith $T$ as an effective canonical temperature and $\\lambda$ as\nthe chemical potential dual to the underlying $R$-charge of the theory\nand (ii) the box counting entropy with two distinct large integers\n$(n,M)$, ($n$ corresponds to the number of excited boxes and $M^2$\ncorresponds to the total number of possible boxes). In order to obtain\nthe desired expression for the energy and counting entropy, we consider\nthe dual $\\mathcal{N}=4$ super Yang Mills theory on $AdS_5 \\times S^5$\nfor type IIB string theory.\n\\subsection{Chemical Description}\nIn order to understand the very basic picture of the giant and\nsuperstar configurations \\cite{hep-th\/0508023, hep-th\/0107119},\nwe shall restrict ourselves to the field theory description defined\non $ \\mathrm{R} \\times S^3$ and focus our attention by recalling the\n$1\/2$ BPS sector of $\\mathcal{N}=4 \\ SYM$ with conformal dimension\n$\\Delta=R$. It is known \\cite{hep-th\/0508023, hep-th\/0107119} that the\nisomorphism group involves an $SO(6)$ symmetry, and thus we shall consider\nan embedding $SO(6) \\subset SO(2) \\times SO(2) \\times  SO(2)$, which may in \ngeneral have various states $\\{ X, \\overline{X}, Y, \\overline{Y},Z, \\overline{Z} \\}$\ncorresponding to either of the $SO(2)$. However, for spinless black holes,\nas there is no spin structure in $SO(2)$, one obtains $ J=0=T$. Thus an\nappropriate dimensional reduction of the above configuration yields to a\nsimple quantum mechanical system involving only $\\{ X, \\overline{X} \\}$+\na bunch of BPS states. Following \\cite{0111222, 0403110}, the problem under\nconsideration may thus be mapped to a real matrix model with a real orthogonal\nmatrix $X$.\n\nThe problem under consideration may be viewed as a $N$ fermion configuration\nlying in a one dimensional harmonic oscillator. One may in principle arrive\nat the solution and thus find eigenvalues and eigen vectors of the\nequations of motion. For the present purpose, let the eigen values of\nthe real matrix wave function be $\\{\\lambda_i\\}_{i=1}^N$, then the\ncomplete problem may be described by considering the Van der monde determinant\nof the function $X$. Surprisingly, one recognizes that the product\n$\\prod_{k+j} (Tr X^k)^j$ may be expressed in terms of the eigen\nvalues $\\{\\lambda_i\\}$. The fermions under consideration thus\nacquire a ``Fermi sea'', which may physically be understood as\nfollows. We observe that the individual energies $\\epsilon_i= e_i\n\\hbar + \\hbar \/2$ acquire a gap of\n\\begin{scriptsize}\n\\begin{eqnarray}\nr_i&=& \\frac{1}{\\hbar} (E_i-E_i^g)= e_i-i+1, \\ \\forall \\\ni=1,2,..., N\n\\end{eqnarray}\n\\end{scriptsize}\nIn order to explicate the dual gravity picture, we shall consider the\nwell-known LLM description, see for details \\cite{LLM}. It thus\nfollows from type $IIB$ string theory that the classical moduli\nmay be described with a symmetric $AdS_5 \\times S^5$ background,\nwhich has the symmetry of $\\mathrm{R} \\times SO(4) \\times SO(4)\n\\times U(1)$, having generators $ \\partial_t $, $ S^3 \\subset AdS_5\n$, $\\widetilde{S}^3 \\subset S^3$ and $ \\partial_{\\phi}$, respectively,\nfor the concerned symmetry components. It has been pointed out by the authors\nof \\cite{hep-th\/0508023} that almost all such states have an underlying\nstructure of the ``quantum foam'', whose universal effective description\nin supergravity is a certain singular spacetime that dubs the ``hyperstar''.\nThey have further argued that the singularity arises because of the fact\nthat the classical description integrates out the microscopic details of\nthe quantum mechanical wavefunction.\n\nBubbling AdS space geometries \\cite{LLM} illustrate that the\nten-dimensional type IIB metric corresponding to the half-BPS\nsupergravity may be described as\n\\begin{scriptsize}\n\\begin{eqnarray}\nds^2 &=& - h^{-2} (dt + V_i dx^i)^2 + h^2 (d\\eta^2 + \\sum_i\ndx^idx^i) + \\eta\\ e^{G} ds_{S^2}^2 + \\eta\\ e^{ - G} ds^2_ {\\tilde\nS^3} \\\\ \\nonumber h^{-2} &=& 2 \\eta \\cosh G , \\\\ \\nonumber  \\eta\n\\partial_\\eta V_i &=& \\epsilon_{ij} \\partial_j z,\\qquad \\eta\n(\\partial_i V_j-\\partial_j V_i) = \\epsilon_{ij} \\partial_\\eta z\n\\\\ \\nonumber z &=&{ 1 \\over 2} \\tanh G\n\\end{eqnarray}\n\\end{scriptsize}\nThe complete solution may thus be determined by the single\nfunction $z$, which satisfies the linear differential equation\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\partial_i \\partial_i z + \\eta \\partial_\\eta (\\frac{ \\partial_\\eta z}{ \\eta}) =0\n\\end{eqnarray}\n\\end{scriptsize}\n\nThis is simply an electrostatic problem in $\\mathrm{R}^6=\n\\mathrm{R}^4 \\times \\mathrm{R}^2$ with potential $\\Phi(\\eta,\nx^1,x^2)= 2 \\eta^{-2}$. Let the coordinates $(x^1,x^2)\\in\n\\mathrm{R}^2$, then the smoothness of the moduli space thus\nobtained may be determined from the symmetry of the background.\nHere, one reveals that the appropriate boundary conditions may be\ncharacterized by\n\\begin{scriptsize}\n\\begin{eqnarray}\nz(\\eta;x_1,x_2) = \\frac{\\eta^2}{\\pi}\\int dx_1' dx_2' \\frac{z(0;\nx_1', x_2')}{[(x_2'-x_1')^2 + \\eta^2]^2}\n\\end{eqnarray}\n\\end{scriptsize}\nThus, the smoothness requires an absence of singularity and, at\n$\\eta =0$, it reads that $z(0; x_1,x_2)= 1\/2$ for $S^3 \\subset\nS^5$, which in this limit shrinks to zero, while the other\nadmissible boundary condition comes with  $z(0; x_1,x_2)= -1\/2$\nfor $S^3 \\subset AdS_5$. The $R$-charge associated with the\nunderlying configuration may thus be expressed via the conformal\ndimension $\\Delta$ of a given state, which may be computed as\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\Delta = \\int_{ \\mathcal{D}} \\frac{d x^1 dx^2 }{2\\pi\\hbar}\n\\frac{1}{2}(\\frac{x_1^2+x_2^2}{\\hbar} - \\frac{1}{2}N^2),\n\\end{eqnarray}\n\\end{scriptsize}\nwhere $\\mathcal{D}$ is the region with $\\eta =0$, known as the\ndroplet, which satisfies $z(\\eta;x_1,x_2)=0$ \\cite{hep-th\/0508023}.\nIt is also immediate that the phase-space velocity which may be\ndefined as $u(0,x^1,x^2)= \\frac{1}{2}- z(0;x_1,x_2)$, makes the\nunderlying configuration smooth. Specifically, one finds that the\n$u(0;x^1,x^2)$ satisfies\n\\begin{scriptsize}\n\\begin{eqnarray}\nu(0; x^1, x^2) &=& 1, \\ if \\ \\eta =1\n\\\\ \\nonumber &=& 0, \\ otherwise\n\\end{eqnarray}\n\\end{scriptsize}\nIn order to have a proper comparison, we may consider typical\nstates in the boundary field theory and thus explain the matching\nbetween the supergravity configuration and the underlying ensemble of\ntypical microstates. What follows next shows that the pure states with\nrelative energies $\\{ r_i\\}$ may be described by considering a\nYoung diagram having $N$ rows and $N_c$ columns. Such a Young\ndiagram may in general be depicted as\n\\begin{scriptsize}\n\\begin{eqnarray}\nY(N,N_c) &=& \\Box \\ \\Box \\ \\Box \\ \\ \\ \\ldots \\ \\Box \\\\ \\nonumber && \\vdots \\\\\n\\nonumber && \\Box \\ \\Box \\ \\Box \\\\ \\nonumber && \\Box \\ \\Box \\\\\n\\nonumber && \\Box\n\\end{eqnarray}\n\\end{scriptsize}\nNotice that the conformal dimension $\\Delta \\sim N^2$ puts a\nconstraint on the underlying ensemble, namely that, at most, it is\nallowed to have $N$ number of rows. Similarly, the number of column\ncan be at most $N_c$, which may be treated as some sort of cutoff in\nthe underlying quantum theory. This is due to the fact that the\ngravity singular black hole solutions may be sourced by a certain\ndistribution of giant gravitons. In this case, the singularity\ncan be described by considering spherical $D_3$-branes having\nconformal dimension  $\\Delta =J \\sim N$, while the $1\/2$\nBPS condition allows for sixteen charges in the theory. Furthermore,\nthe existence of a RR-flux in $AdS_5$ sources certain\nnon-abelian interactions, which ascribes bound states of $N$ giant\ngravitons. The concerned microstates of the underlying ensemble has\nprecisely been described in \\cite{hep-th\/0107119}. Subsequently,\nit follows \\cite{hep-th\/0107119} that the number of giants on\naverage remains the same as the number of giants sourcing the\ngravity singular solutions.\n\nThus, having a striking microscopic picture of the gravity singular\nsolutions, we shall now move on to describe the Weinhold geometric\nframework for the statistical mechanics of a large number of random\nYoung tableaux whose (rows, columns) may respectively vary from\n$(1,1)$ to $(N, N_c)$. Let us consider an ensemble of Young diagrams\nin which the total number of boxes is $\\Delta \\in [N_c,NN_c]$. This\nis because in any random Young diagram, we can have $N_c$ columns,\nand thus the maximum number of admissible row would be $NN_c$, see\nfor instance\n\\begin{scriptsize}\n\\begin{eqnarray}\nY(N_c,1) &=& \\Box \\ \\Box \\ \\Box \\ \\ \\ \\ldots \\ \\Box \\\\ \\nonumber\n&& 1 \\ \\ 2\\ \\ 3 \\ \\ \\ \\ldots \\ \\ N_c\n\\end{eqnarray}\n\\end{scriptsize}\nFor any other state with $\\Delta= N_c+ \\delta \\Delta$, the system would\nheat up and thus would acquire a non-vanishing effective temperature.\nConsequently, the entropy would increase with $S>S_0$, and the\nsystem would get randomized. This certainly does not happen for the\n$1\/2$-BPS spinless configurations, and the vacuum entropy at $T=0$\ngets maximized when half of the Young diagram is full.\n\nTo be concrete, let us consider the microstates having relative\nenergies $\\{ r_i\\}_{i=1}^N$ and compute the statistical quantities\nassociated with the promising giant configurations. Let us define\nthe difference of relative fermion energies as\n\\begin{scriptsize}\n\\begin{eqnarray}\nc_N &=& r_1 \\\\ \\nonumber && \\vdots \\\\ \\nonumber c_{N-i}&=&\nr_{i+1}-r_i \\\\ \\nonumber && \\vdots  \\\\ \\nonumber c_1&=& r_N-\nr_{N-1}\n\\end{eqnarray}\n\\end{scriptsize}\nWe thus see that the quantity $c_j$ describes the number of columns\nof length $j$ in the diagram associated to $\\{r_N,r_{N-1}, \\ldots,\nr_1\\}$. As mentioned earlier, we are interested in the statistical\nfluctuations of typical half-BPS states of large charge $\\Delta= J\n= N^2$, and thus the excitation energy of fermions is comparable\nto the energy of the Fermi sea.\n\nIn order to analyze the chemical correlation in the excited states of \na large number of free fermions, we would consider Weinhold geometry\nto study the typical correlation of statistical states being\ncharacterized by certain arbitrary Young diagrams. In order to\nsimplify the notations, we shall define $q= e^{-\\beta}, \\xi=\ne^{-\\lambda}, \\beta = 1\/T$. Then, the canonical partition\nfunction reads\n\\begin{scriptsize}\n\\begin{eqnarray}\nZ(\\beta, \\lambda)= \\xi^{-N_c}\\prod_j \\frac{1}{1-\\xi q}\n\\end{eqnarray}\n\\end{scriptsize}\nFrom the very definition of the canonical ensemble\n\\cite{hep-th\/0508023}, the average canonical energy may be\nexpressed by\n\\begin{scriptsize}\n\\begin{eqnarray}\n<E(\\beta, \\lambda)>= q \\partial_q \\ln Z (q)= \\sum_j \\frac{j \\xi\nq^j}{1-\\xi q^j}\n\\end{eqnarray}\n\\end{scriptsize}\nWe thus find that the average occupation number of the theory,\nwhich measures the maximum number of allowed columns in any chosen\nYoung diagram, may be given to be\n\\begin{scriptsize}\n\\begin{eqnarray}\nN_c= \\sum_j <c_j>= \\sum_j \\frac{\\xi q^j}{1-\\xi q}\n\\end{eqnarray}\n\\end{scriptsize}\nAs mentioned in  the previous section, we may now divulge the\nnotion of chemical fluctuations without any approximation. For\nthis purpose, we shall redefine the canonical energy in terms\nof the effective canonical temperature $T$, which leads to the\nfollowing average energy\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{energy}\n<E(T,\\lambda)>= \\sum_{j=0}^{\\infty} \\frac{j e^{-(\\lambda +\nj\/T)}}{1- e^{-(\\lambda + j\/T)}}\n\\end{eqnarray}\n\\end{scriptsize}\nFor investigating the chemical fluctuations, we shall consider\ntwo neighboring statistical states characterized by the chemical\npotentials $(T, \\lambda)$ and $(T+ \\delta T , \\lambda +\\delta\n\\lambda )$.\nThe chemical pair correlation functions may thus be defined as the\ncomponents of the Weinhold metric\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{Wienmetric}\ng_{ij}^{(E)}= \\partial_i \\partial_j E(T, \\lambda), \\ i,j= T,\n\\lambda\n\\end{eqnarray}\n\\end{scriptsize}\nThe stability of the underlying canonical chemical configuration may thus\nbe determined as the positivity of the determinant of the Weinhold\nmetric tensor, $g^{(E)}(T,\\lambda) = det(g_{ij}^{(E)})$. The\nfundamental configuration is just an ordinary two dimensional intrinsic\nsurface $(M_2,g)$ spanned by the chemical potentials $(\\beta,\\lambda)$.\nThus, the global correlation length may be defined as the $M_2$ scalar\ncurvature invariant. Explicitly, one can determine the intrinsic\ncovariant Riemann curvature tensor $ R_{T \\lambda, T, \\lambda}^{(E)}$.\nHence, the Ricci scalar may, as in Eqs. \\ref{Wiencur}, be read off for\nEqs. \\ref{energy} to be\n\\begin{scriptsize}\n\\begin{eqnarray}\nR^{(E)}= \\frac{R_{T \\lambda T \\lambda}^{(E)}}{g^{(E)}(T, \\lambda)}\n\\end{eqnarray}\n\\end{scriptsize}\n\\subsection{State-space Description}\nIn order to have  a test of our state-space formulation, the possible\nexplication of the correlations associated with the box counting\nentropy may be divulged as follows. Let us first consider the case\nof the large $N$ limit, with $N \\rightarrow \\infty$, $\\hbar \\rightarrow 0$\nand $ N\\hbar$ fixed. Then, the expectation values of relative energies\n$<r_i$ define a curve $<y(x)>= \\sum_{i=N-x}^N <c_i>$. This essentially\nleads to the evaluation of an integral $\\int_{N-x}^N di <c_i>$. The\nalgebraic curve associated with this problem may thus be defined as\na set of energies\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\varepsilon:= \\{ \\alpha q^{N-x}+\\gamma q^y=1 \\ | \\ \\alpha=\n\\frac{1-q^{N_c}}{1-q^{N_c+N}}, \\gamma=\n\\frac{1-q^{N}}{1-q^{N_c+N}}\\}\n\\end{eqnarray}\n\\end{scriptsize}\nFrom the extremization of the curve $f(x,y)=  \\alpha\nq^{N-x}+\\gamma q^y-1$, it is not difficult to see that the entropy\ngets maximized when $q\\rightarrow 1$. In this limit, we have a\n``limit curve''\n\\begin{scriptsize}\n\\begin{eqnarray}\ny(x)= \\frac{N_c}{N}x= wx\n\\end{eqnarray}\n\\end{scriptsize}\nThis is clearly a straight line with slope $N_c\/N$, and in fact it\ncorresponds to the triangular Young diagrams. For an illustration,\nwe may depict such a Young diagram for $N_c=4$ as\n\\begin{scriptsize}\n\\begin{eqnarray}\nY(4,4) &=& \\Box \\ \\Box \\ \\Box \\ \\Box \\\\ \\nonumber && \\Box \\ \\Box \\ \\Box \\\\ \\nonumber && \\Box \\ \\Box \\\\\n\\nonumber && \\Box\n\\end{eqnarray}\n\\end{scriptsize}\nThe liquid droplet model thus suggests that the involved probes\nmay be resolved by considering a large number of particles between\nthe phase-space shells $\\sigma$ and $\\sigma +d\\sigma$. Thus the\ndifferential curve may be expressed as\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\frac{u(0;\\sigma^2)}{2 \\hbar^2}d\\sigma^2= dx\n\\end{eqnarray}\n\\end{scriptsize}\nHere, the droplet $u(0; x^1, x^2)$ has been expressed as $u(0;\n\\sigma^2)$ in terms of the radial coordinate $\\sigma$, which from\nthe view-points of the underlying $U(1)$ symmetry presents single\nYoung diagram states in the $(x^1, x^2)$ plane. We thus procure an\nalgebraic curve satisfying previously mentioned droplet boundary\nconditions with\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\frac{\\sigma^2(x)}{2 \\hbar}= y(x)+x\n\\end{eqnarray}\n\\end{scriptsize}\nThe consistency may easily be checked and, in particular, one finds,\nby differentiating the algebraic curve, that it describes the\nnatural boundary conditions associated with the giant black holes\n\\begin{scriptsize}\n\\begin{eqnarray}\nu(0;\\sigma^2)= \\frac{1}{1+y'(x)}= \\frac{1}{1+w}\n\\end{eqnarray}\n\\end{scriptsize}\nThe present analysis thus describes $1\/2$-BPS $R$-charged black\nholes in $AdS_5 \\times S^5$, which are sometime called superstars\n\\cite{0109127,0411145}. It is worth to note further that the metric\nof the superstar may precisely be defined by the three harmonic\nfunctions involving the $R$-charge $q_1$ and the length $L$ of the AdS.\nIn this perspective, the number of allowed giants may be expressed as\n\\begin{scriptsize}\n\\begin{eqnarray}\nN_c= w N; \\ \\forall \\ w:= \\frac{q_1}{L^2}\n\\end{eqnarray}\n\\end{scriptsize}\nIt follows \\cite{0109127} that the phase-space density function\n$z$ may be expressed as\n\\begin{scriptsize}\n\\begin{eqnarray}\nz(\\eta;x^1,x^2) = \\frac{1}{2} \\frac{r^2\\gamma - L^2 \\sin^2\n\\theta}{r^2\\gamma + L^2 \\sin ^2 \\theta},\n\\end{eqnarray}\n\\end{scriptsize}\nwhere $\\gamma$ may be defined to be\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\gamma(r)= 1+ \\frac{q_1^2 \\sin^2 \\theta}{r^2}\n\\end{eqnarray}\n\\end{scriptsize}\nFor $\\eta =0$, we have $r^2 \\sin^2 \\theta =0$, which implies that\neither $r =0$ or $ \\sin \\theta =0$. Thus, we obtain that the\ncondition $ \\sin \\theta =0$ leads to the realization of $z(\\eta;\nx^1,x^2)=1\/2$. This is precisely the condition that there are no\n$D$-brane excitations in the ensemble of vacuum microstates. It is\nthus immediate to see that the phase-space density function may\nnow be expressed as\n\\begin{scriptsize}\n\\begin{eqnarray}\nu(0;x^1,x^2) = \\frac{1}{2}- z(0;x^1,x^2)\n\\end{eqnarray}\n\\end{scriptsize}\nIt is worth to note for $r=0$ that the function $\\gamma(r)$ leads\nto the condition that the function $z(0;x^1,x^2)$ must satisfy\n\\begin{scriptsize}\n\\begin{eqnarray}\nz(0;x^1,x^2)= \\frac{1}{2} \\frac{w-1}{w+1}\n\\end{eqnarray}\n\\end{scriptsize}\nThis, in turn, is consistent with the previously defined boundary\ncondition of the droplet $\\mathcal{D}$ and provides an intriguing\nmicroscopic picture for our present consideration. We may thus\npronounce that the Young tableaux description provides an\nappropriate framework for the microscopic understanding of the\nthermodynamic geometries of $1\/2$ BPS supergravity configurations.\n\nWe now offer an interesting origin of the box counting entropy in\nwhich the state-space correlations may, without any approximation, be\ndescribed. Here, an important ingredient follows from the non-trivial\nnotion of the coarse graining picture of the phase-space density $u_{c,g}$\nof the ground state configuration, which conjointly provides an appropriate\ndefinition of the canonical ensemble with fixed temperature, rather than\nfixing the energy \\cite{hep-th\/0508023}. In order to be concrete, let us consider\na thin shell in the phase-space $(x^1,x^2)$ of thickness $(dx^1,dx^2)$.\nThen the average density may be defined as\n\\begin{scriptsize}\n\\begin{eqnarray}\nu{c,g}(0;x^1,x^2)= \\frac{\\int_E^{E+dE} u_{g,m}dx^2}{\\int_E^{E+dE}\ndx^2}\n\\end{eqnarray}\n\\end{scriptsize}\nIt is not difficult to ascertain that the density $u_{c,g}$ may easily\nbe expressed as\n\\begin{scriptsize}\n\\begin{eqnarray}\nu{c,g}(0;x^1,x^2)= \\frac{\\Delta_X}{\\Delta_E}\\hbar\n\\end{eqnarray}\n\\end{scriptsize}\nWe may however re-express the phase-space density $u_{c,g}$ and\nappreciate that it satisfies the familiar form\n\\begin{scriptsize}\n\\begin{eqnarray}\nu{c,g}(0;x^1,x^2)= \\frac{1}{1+y'(x)},\n\\end{eqnarray}\n\\end{scriptsize}\nwhere $y(x)= \\frac{\\sigma^2}{2\\hbar^2}-x$ is the aforementioned\nrelative energy curve for the $1\/2$-BPS giant configurations at\nvery large $N$. The origin of the entropy may thus be ascribed as\nfollows. Let us consider the phase-space density and choose a set of coherent\nbasis which picks up the least uncertainty, and thus turns out to be\nappropriate for the present discussion, following from the coarse\ngraining phenomenon. In order to divulge the intrinsic geometric notion of\ndegeneracy, we may focus our attention on an arbitrary phase-space,\nhaving $M^2$ cells with at most random $n$ excited cells. The\npictorial view of this may be given as follows:\n\\begin{scriptsize}\n\\begin{eqnarray}\nY(N,N_c) &=& \\Box \\ {\\color{blue} \\maltese} \\ \\Box  \\ \\ldots \\\\\n\\nonumber && {\\color{blue} \\maltese} \\ \\Box \\ {\\color{blue}\n\\maltese} \\\\ \\nonumber && \\Box \\ \\Box  \\ \\Box \\\\ \\nonumber && \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ddots \\\\ \\nonumber && \\ \\ \\vdots \\ \\ \\ \\  \\\\\n\\nonumber && \\downarrow_{(RG \\ trans.)} \\\\ \\nonumber && \\downarrow\n\\\\ \\nonumber && \\Box \\ \\Box \\ \\Box \\ \\ldots  \\\\ \\nonumber && \\Box\n\\ \\Box \\ \\Box \\\\ \\nonumber && \\Box \\ \\Box \\ \\Box \\\\ \\nonumber && \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ddots \\\\ \\nonumber && \\ \\vdots \\ \\ \\ \\  \\\\\n\\nonumber &=& Phase \\ Space \\ (N,N_c)\n\\end{eqnarray}\n\\end{scriptsize}\nIn an arbitrary Young diagram $Y(N,N_c)$, we thus see that there\nare $n$ boxes filled with maltese and the rest $(M^2-n)$ of them are\nempty boxes. Therefore, the existing degeneracy in choosing random\n$n$ maltese (excited boxes) out of the total $M^2$ boxes may  be\ngiven by\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\Omega(n,M)= \\ ^{M^2}C_n = \\frac{(M^2)! }{(n)! (M^2-n)! }\n\\end{eqnarray}\n\\end{scriptsize}\nFrom the first principle of statistical mechanics, we thus find\nthat the canonical counting entropy may be defined to be\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{entropy}\nS(n,M)= \\ln (M^2)! - \\ln (n)! - \\ln (M^2-n)!\n\\end{eqnarray}\n\\end{scriptsize}\nIt is nevertheless important to emphasize that the subsequent analysis\ndoes not exploit any approximation, such as Stirling's approximation or\nthe thermodynamic limit, and thus we shall work in the full picture of the\ncanonical ensemble. Consequently, the present analysis bequeaths an\nexact expression for the state-space pair correlation functions and\ncorrelation length.\n\nIt is now immediate to divulge an appropriate appraisal for the\nstate-space geometry associated with the random Young tableaux\n$Y(N, N_c)$ thus described with arbitrary $n$ excited droplets\namong the $M^2$ fundamental cells in an ensemble. We observe that\nthe state-space interactions allow us to analyze the correlation\nbetween two neighboring statistical states $(n,M)$ and $(n+ \\delta n,\nM+ \\delta M)$, and thus the fluctuations in the droplets with $(n,M)$\nmay be defined via the well-defined Ruppenier metric as\n\\begin{scriptsize}\n\\begin{eqnarray}\\label{Ruppmetric}\ng_{ij}^{(S)}= - \\partial_i \\partial_j S(n,M), \\ i,j= n, M\n\\end{eqnarray}\n\\end{scriptsize}\nAs we have outlined in the case of chemical fluctuations from the\nWeinhold geometry, a very similar analysis may thus easily be\nperformed to reveal the state-space correlations existing among\nthe fluctuating microstates of the $1\/2$ BPS configuration\ndescribing certain giants or the superstars.\n\nThe conformal relation of the two geometries may formally be\nderived via the temperature as a conformal factor, which may be\naccomplished by taking the derivative of the entropy with respect\nto the energy $ \\Delta $. One may however easily expose that\n$q= exp(-\\frac{\\alpha}{N})$, and it comes as no surprise that this yields\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{effTemp}\nT^{-1}= \\frac{\\partial S}{\\partial \\Delta}= \\frac{ \\alpha}{N}\n\\end{eqnarray}\n\\end{scriptsize}\nNevertheless, it is important to note that there is no physical\ntemperature involved with half-BPS states. However, $T$ should be\nunderstood as an effective temperature of a canonical ensemble\nthat sums up the half-BPS states with the canonical energy sharply\npeaked at $\\Delta $. More generally, if $\\Delta $ scales as $N^{\\xi}$,\nthen the Ref. \\cite{hep-th\/0508023} shows that the effective temperature\ngrows as $N^{\\xi \/2}$.\n\nOur thermodynamic geometries, thus explained in either the\ndescription of chemical potentials or the number of boxes in an\narbitrarily excited Young tableau, entail remarkably simple\nexpressions. Furthermore, we discover that there exists an\nappropriate microscopic\/macroscopic explanation for the\nstatistical correlations of the $1\/2$ BPS supergravity\nconfigurations. From the viewpoint of $AdS_2\/CFT_1$,\nit could be interesting to explore the case of more charged\nblack holes, e.g. extreme Reissner-Nordstr\\\"om black hole\n\\cite{SB2}, non-BPS supergravity black holes \\cite{07104967v3}\nor a further addition of electric-magnetic charges. \nIn the next section, we shall intimately discuss the above\noutlined study for the two parameter giants and superstar\nsolutions. In particular, our investigation shows that there\nexists an exact account for the thermodynamic correlations of\nthe spinless black holes arising from the coarse grained\nLLM geometries \\cite{LLM}.\n\n\\section{Canonical Energy Fluctuations}\nIn this section, we shall present the essential features of\nthe chemical geometry and thereby put them into effect for the 2-\nparameter giants and superstars. Here, we focus our attention\non the geometric nature of a large number of gravitons, within the\nneighborhood of small fluctuations with given chemical potentials\nintroduced in the framework of $1\/2$-BPS configurations. As stated\nearlier, the thermodynamic metric in the chemical potential space is\ngiven by the Hessian matrix of the canonical energy, with respect to\nthe intensive variables, which in this case are the two distinct chemical\npotentials carried by the giants and superstars. We find, in this framework,\nthat the exact canonical energy as the function of chemical potentials\ntakes an intriguing expression\n\\begin{scriptsize}\n\\begin{equation}\nE(T, \\lambda)= \\sum_{j=0}^{\\infty} \\frac{j \\ exp(-( \\lambda\n+j\/T))}{1-exp(-( \\lambda +j\/T))}\n\\end{equation}\n\\end{scriptsize}\nFor to obtain the thermodynamic metric tensor in the chemical\npotential space, we may employ the formula in Eq. \\ref{Wienmetric},\nwhich leads to the following series expression for the components\nof the metric tensor:\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{TT}(T, \\lambda)&=&\\sum_{j=0}^{\\infty}\\big( -2 \\frac{j^2}{T^3}\n\\frac{exp(- \\lambda-j\/T)}{(1-exp(- \\lambda-j\/T))}+ \\frac{j^3}{T^4}\n\\frac{exp(- \\lambda-j\/T)}{(1-exp(- \\lambda-j\/T))}\\\\ \\nonumber && +\n3\\frac{j^3}{T^4}\\frac{exp(- \\lambda-j\/T)^2}{(1-exp(-\n\\lambda-j\/T))^2}\n+2\\frac{j^3}{T^4}\\frac{exp(- \\lambda-j\/T)^3}{(1-exp(- \\lambda-j\/T))^3}\\\\\n\\nonumber && -2\\frac{j^2}{T^3}\\frac{exp(- \\lambda-j\/T)^2}\n{(1-exp(- \\lambda-j\/T))^2}\\big)\n\\end{eqnarray}\n\\end{scriptsize}\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{T \\lambda}(T, \\lambda)&=& \\sum_{j=0}^{\\infty}\n\\big(-\\frac{j^2}{T^2} \\frac{exp(- \\lambda-j\/T)}{(1-exp(-\n\\lambda-j\/T))}-3\n\\frac{j^2}{T^2} \\frac{exp(- \\lambda-j\/T)^2}{(1-exp(- \\lambda-j\/T))^2} \\\\\n\\nonumber && -2\\frac{j^2}{T^2}\\frac{exp(- \\lambda-j\/T)^3}\n{(1-exp(- \\lambda-j\/T))^3}\\big)\n\\end{eqnarray}\n\\end{scriptsize}\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{ \\lambda \\lambda}(T, \\lambda)&=& \\sum_{j=0}^{\\infty} \\big( j\n\\frac{exp(- \\lambda-j\/T)}{(1-exp(- \\lambda-j\/T)}\n+3 j \\frac{exp(- \\lambda-j\/T)^2} {(1-exp(- \\lambda-j\/T))^2} \\\\\n\\nonumber && +2 j \\frac{exp(- \\lambda-j\/T)^3} {(1-exp(-\n\\lambda-j\/T))^3} \\big)\n\\end{eqnarray}\n\\end{scriptsize}\nFurthermore, it turns out that the determinant of the metric\ntensor takes an inelegant polynomial form and thus, in order\nto simplify the notations, we may define a level function\n\\begin{scriptsize}\n\\begin{eqnarray}\nb_j(T, \\lambda):= exp(- \\lambda-j\/T)\n\\end{eqnarray}\n\\end{scriptsize}\nIt is however evident that the local stability of the full\nphase-space configurations may be determined by computing the\ndeterminant of the concerned thermodynamic metric tensor. Here,\nwe may likewise provide a compact formula for the determinant of\nthe metric tensor, and exclusively, the intrinsic geometric analysis\nassigns a compact expression to the determinant of the metric tensor.\nA straightforward computation thus shows that the determinant of the\nmetric tensor as the function of arbitrary chemical potentials is\n\\begin{scriptsize}\n\\begin{eqnarray}\n||g(T, \\lambda)|| &=& \\sum_{j=0}^{\\infty}(-j^2 \\frac{b_j}{T^4}\n\\frac{(-2 T+2 T b_j+j+ jb_j)}{(b_j-1)^3)}) \\times \\\\ \\nonumber &&\n\\sum_{j=0}^{\\infty}(-jb_j \\frac{(1+b_j)}{(b_j-1)^3})-\n(\\sum_{j=0}^{\\infty}(j^2 \\frac{b_j}{T^2}\n\\frac{(1+b_j)}{(b_j-1)^3)})^2\n\\end{eqnarray}\n\\end{scriptsize}\n\\begin{figure}[htb]\n\\vspace*{-1cm}\n\\begin{center}\n\\hspace{-1.0cm}\n\\epsfig{file=Wiendet.ps,width=12cm}\n\\end{center}\n\\vspace*{0.01cm}\n\\caption{The Wienhold determinant of the metric tensor as a function of temperature,\nT; and chemical potential, $ \\lambda=L$ in the $1\/2$-BPS supergravity configurations.}\n\\vspace*{0.50cm}\n\\end{figure}\nWe thus see from Fig. 1 that the determinant of the Weinhold metric develops\nan instability when $T>0.15$. This is also expected, since the coarse graining\nphenomenon should break down near small effective canonical temperatures. The\nsystem is well-defined and physically sound for all $ T \\in [0, 0.15]$.\n\nIn order to conclusively analyze the nature of chemical correlations\nand the concerned properties of the statistical configuration, one needs\nto determine certain globally invariant quantities on the intrinsic manifold\n$(M_2,g)$ of the  parameters. In fact, one may easily ascertain that\nthe simplest of such invariants is the scalar curvature of $(M_2,g)$. In order to\ndo so, one may first compute the Riemann curvature tensor $R_{T \\lambda T\n\\lambda}(T,\\lambda)$ and then, from the previously defined expression, obtain\nthe scalar curvature. We see that it is not difficult to compute the\ncovariant Riemann tensor $R_{T\\lambda T \\lambda}(T, \\lambda)$. The\nexact expression for the $R_{T\\lambda T \\lambda} (T, \\lambda)$ is quite\ninvolved and thus we relegate it to the Appendix B.\n\nIn a straightforward fashion we can show, by applying our previously advertised\nintrinsic geometric technology, that one may easily obtain the scalar\ncurvature simply via the relation\n\\begin{scriptsize}\n\\begin{eqnarray}\nR(T, \\lambda)= \\frac{R_{T  \\lambda T \\lambda}(T, \\lambda)}{||g(T,\n\\lambda)||}\n\\end{eqnarray}\n\\end{scriptsize}\n\\begin{figure}[htb]\n\\vspace*{0cm}\n\\begin{center}\n\\hspace{-1.0cm}\n\\epsfig{file=Wiencur.ps,width=12cm}\n\\end{center}\n\\vspace*{0.01cm}\n\\caption{The Wienhold scalar curvature as a function of temperature, T; and\nchemical potential, $ \\lambda=L$ in the $1\/2$-BPS supergravity configurations.}\n\\vspace*{0.50cm}\n\\end{figure}\nWe observe from Fig. 2 that the system becomes strongly correlated\nfor $T<0.05$, and for $ T \\in [0.01, 0.02]$ it acquires the chemical\ncorrelation length of $8 \\times 10^3$. Interestingly, we do not see\nany chemical fluctuations for $T>0.08$.\nIn the domain of small chemical potentials $T, \\lambda \\in [0,0.2] $,\nthe pictorial views of the stability may be perceived from the\ndeterminant of the Weinhold metric tensor $g^{(E)}(T, \\lambda)$\nand the concerned correlation length from the corresponding scalar\ncurvature $R^{(E)}(T, \\lambda)$. We notice that both the plot of\nthe determinant and that of the scalar curvature of the Wienhold \ngeometry are nice regular 3D hyper-surfaces.\n\nWe thence realize that the microcanonical representation of the black\nholes under consideration posits that there exists a large number\nof correlated degenerate microscopic states of the $1\/2$ BPS giants\nand superstars. The non-vanishing scalar curvature suggests that\na constituent microstate may as well avoid that the initial pure\nstates collapses to a particular pure black hole microstate. This\nis because of the fact that the intrinsic geometric structures of\nAdS black hole can be deduced by certain suitably defined subtle\nmeasurements \\cite{hep-th\/0508023}, and thus there is no loss of\ninformation.\n\\section{The Fluctuating Young Tableaux}\nAs in the previous section, the present section is devoted to investigate\nthe notion of large number of boxes into our state-space geometry, and thereby\nwe shall analyze the possible role of concerned large integers in the veritable\ngiant black hole configurations, being considered in the framework of liquid\ndroplet model or fuzzball description. In this concern, we shall precisely\ncompute the statistical pair correlations under this consideration, which\nascribes a set of self convincing physical meaning to the state-space\nquantities, for the simplest excited giant configurations\n\\begin{scriptsize}\n\\begin{eqnarray}\nS(n,M)= ln((M^2)!)-ln(n!)-ln((M^2-n)!)\n\\end{eqnarray}\n\\end{scriptsize}\nEmploying the previously proclaimed formulation, one may easily read\noff the components of the covariant state-space metric tensor to be\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{nn}(n,M)= \\Psi(1,n+1)+ \\Psi(1,M^2-n+1)\n\\end{eqnarray}\n\\end{scriptsize}\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{nM}(n,M)=  -2 M \\Psi(1,M^2-n+1)\n\\end{eqnarray}\n\\end{scriptsize}\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{MM}(n,M)= 4 M^2 \\Psi(1,M^2-n+1) -4 M^2 \\Psi(1,M^2+1)-2\n\\Psi(M^2+1)+2 \\Psi(M^2-n+1)\n\\end{eqnarray}\n\\end{scriptsize}\nIn the above expressions, the $\\Psi(n,x)$ is the $n^{th}$\npolygamma function, which is the $n^{th}$ derivative of the usual\ndigamma function. Specifically, it turns out that the $\\Psi(x)$\nmay defined to be\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\Psi(x)= \\frac{\\partial }{ \\partial x} \\ln( \\Gamma(x))\n\\end{eqnarray}\n\\end{scriptsize}\nIn this framework, we observe that there exists a very simple\ndescription which divulges the geometric nature of the statistical\npair correlations. The fluctuating extremal black holes may thus\neasily be determined in terms of the mass and angular momentum of\nthe underlying configurations. Moreover, it is evident that the\nprinciple components of the statistical pair correlations are\npositive definite, for a range of the parameters of the concerned black\nholes, which physically signifies a certain self-interaction of a\nfictitious particle moving on an intrinsic surface $(M_2(R),g)$.\nSignificantly, it is clear in this case that the following\nstate-space metric constraints hold:\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{nn} &>& 0, \\ \\ \\forall \\ (n,M)  \\ |  \\ \\Psi(1,n+1)+ \\Psi(1,M^2-n+1)>0  \\\\\n\\nonumber  g_{MM} &>& 0, \\ \\ \\forall \\ (n,M)  \\ |  \\\n\\Psi(1,M^2-n+1)-\\Psi(1,M^2+1)> \\\\ \\nonumber &&\n\\frac{1}{2M^2}(\\Psi(M^2+1)- \\Psi(M^2-n+1))\n\\end{eqnarray}\n\\end{scriptsize}\nConsequently, we may easily reveal that the common domain of the above\nstate-space constraints defines the range of physically sensible\nvalues of the chosen number of boxes and total number of boxes, such\nthat the giants may remain into certain locally stable statistical\nconfigurations. We may also notice that the total boxes component\n$g_{MM}$ of the state-space metric tensor is asymmetrical, in comparison\nto the $g_{nM}$ and $g_{nn}$. This is physically well-accepted, because\nof the fact that the component associated with the large number of boxes\nis somewhat like the heavy head-on collision of two equal particles, which\nalternate more energy, in contrast to the other excitations, either involving\nthe single particle or the excited-unexcited particles. It is pertinent to\nmention that the relative pair correlation function determines the selection\nparameter for a chosen black hole, which may be defined as the modulus of the\nratio of excited-excited to excited-unexcited statistical correlations.\nImportantly, we procure that the parameter thus apprised may be given\nas the ratio of two digamma functions\n\\begin{scriptsize}\n\\begin{eqnarray}\na: = \\frac{1}{2M} \\vert \\frac{\\Psi(1,n+1)}{\\Psi(1,M^2-n+1)}\\vert\n\\end{eqnarray}\n\\end{scriptsize}\nIt is worth to mention that when $n>1$, one has $\\Psi(n,x)= \\Psi(\nn ) + \\gamma$, which is an ordinary rational number, where the\n$\\gamma$ is the standard Euler's constant. For small values of $n$,\n$\\Psi(n)$ is computed as a sum of gamma functions, which is again a rational\nnumber. To force this computation to be performed for larger values\nof the $n$, we may use\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\Psi(n,x) = \\frac{\\partial^n \\Psi(x)}{\\partial x^n},\n\\end{eqnarray}\n\\end{scriptsize}\nwith $\\Psi(0,x) = \\Psi(x)$. Moreover, the stability of the\nunderlying statistical configurations may earnestly be analyzed by\ncomputing the degeneracy of the associated two dimensional state-space\nmanifold. In fact, we may easily ascertain that the determinant of\nthe state-space metric tensor may be given to be\n\\begin{scriptsize}\n\\begin{eqnarray}\ng(n,M) &=& -4M^2 \\Psi(1,n+1)\\Psi(1,M^2+1)-2\\Psi(1,n+1)\\Psi(M^2+1) \\\\\n\\nonumber && +4M^2 \\Psi(1,n+1)\\Psi(1,M^2-n+1)\n+2\\Psi(1,n+1)\\Psi(M^2-n+1) \\\\ \\nonumber &&\n-4M^2 \\Psi(1,M^2-n+1)\\Psi(1,M^2+1) -2\\Psi(1,M^2-n+1)\\Psi(M^2+1) \\\\\n\\nonumber && +2\\Psi(1,M^2-n+1)\\Psi(M^2-n+1)\n\\end{eqnarray}\n\\end{scriptsize}\n\\begin{center}\n \\begin{figure}[htb]\n\\vspace*{-3.5cm}\n\\begin{center}\n\\hspace{-7.5cm}\n\\epsfig{file=rupp.ps,width=14cm,angle= 90}\n\\end{center}\n\\vspace*{-4cm}\n\\caption{The Ruppenier determinant of the state-space metric tensor\nas a function of excited droplets, $n$; and total number of fundamental\ncells, $M^2$ in the $1\/2$-BPS supergravity configurations.}\n\\vspace*{0.50cm}\n\\end{figure}\n\\end{center}\nFig. 3 indicates that the canonical configurations become unstable\nand ill-defined when both of the $n$ and $M$ take small values. Up\nto $10^5$ boxes, we discover that the same qualitative features\nhold. Particularly, it is worth to mention that the fluctuation\nbumps are aligned towards the $n^{th}$ boundary, when $n$ and $M$\nare of the same order.\n\nThe determinant of the metric tensor thus calculated is non-zero\nfor any set of given non-zero total boxes and excited boxes, and\nthus, for the set of proper choices, it provides a non-degenerate\nstate-space geometry for this configuration. In turn, one may\nillustrate the order of statistical correlations between the\nequilibrium microstates of the giant black hole system. Besides\nthe fact that the principle component constraints $\\lbrace g_{ii}\n> \\ 0 \\ \\vert \\ i= n, M \\rbrace$ imply that this system may\naccomplish certain locally stable statistical configurations,\nhowever the negativity of the determinant of the state-space\nmetric tensor indicates that the underlying systems may globally\nendure a certain instability, as well for the bad choice of boxes,\nwhich correspond to certain unphysical macroscopic configurations.\n\nThis significantly connotes that there exists a positive definite\nvolume form on the $(M_2,g)$, for the good choice of boxes and\ntheir excitations. One may thus conclude that this system may\nremain in the nice chosen configuration or, for certain\nunacceptable choices, might move to some more stable brane\nconfigurations. Furthermore, one may easily examine, in general,\nthat the covariant Riemann correlation tensor may be given by the\ntwo large integers characterizing the droplets. However, an\nexplicit expression is given by Eq. \\ref{countentcovcur} for the\n$R_{nMnM}(n,M)$, which is rather involved, and thus we have\nrelegated it to the Appendix B.\n\nFurthermore, in order to examine certain global properties of\nsuch black holes phase-space configurations, one is required to\ndetermine the associated geometric invariants of the underlying\nstate-space manifold. For the giant and superstar black holes,\nthe simplest invariant turns out to be the state-space scalar\ncurvature, which may easily be computed by using the intrinsic\ngeometric technology defined as the negative Hessian matrix of the\nentropy captured by the excited contributions. Indeed, we discover\nthat the state-space curvature scalar for the fluctuating giant\n(and superstar) configurations may easily be depicted.\nNevertheless, the exact involved expression for the scalar\ncurvature $R(n,M)$ has been depicted in Eq. \\ref{countentscacur}\nand is shifted to the Appendix B.\n\\begin{figure}[htb]\n\\vspace*{.30cm}\n\\begin{center}\n\\hspace{-1.0cm}\n\\epsfig{file=Rupcur.ps,width=12cm}\n\\end{center}\n\\vspace*{-0.1cm} \\caption{The Ruppenier scalar curvature as a\nfunction of excited droplets, $n$, and the total number of\nfundamental cells, $M^2$, in the $1\/2$-BPS supergravity\nconfigurations.} \\vspace*{0.50cm}\n\\end{figure}\nWe numerically find the following conclusions for the state-space\nscalar curvature correlations.\n\n\\begin{itemize}\n\\item For $n, M \\in [1, 10^{10}]$, it turns out from Fig. 4 that\npositive, as well as negative correlations exist only along the\n$M^{th}$ boundary of the state-space configuration. It is also\nexpected from the general consideration of statistical fluctuation\ntheory that the system would possess certain attractions and\nrepulsions.\n\\item We notice, from the respective plots of the determinant and scalar\ncurvature of the state-space configuration, that the system\nremains stable and regular along the $n^{th}$ boundary of the\nstate-space configuration. We need not mention in this range that\nthe system becomes ill-defined near the origin $(0, 0)$ of the\nstate-space manifold.\n\\item We further observe for $10^4$ boxes that there exist definite small\nattraction and repulsion along the $n^{th}$ boundary of the $R$-$n$\nsurface. Interestingly, the same conclusion remains true for the $g$-$n$\nsurface of the underlying state-space configuration.\n\\end{itemize}\nWe recognize, in the framework of our state-space geometry, that\nthe negative sign of the curvature scalar signifies that this\nsystem is effectively an attractive configuration, under the\nGaussian fluctuations. Thus, the giants and superstars appear to\nbe manifestly stable configurations, with nice combinatorial\nproperties of the Young tableaux. Furthermore, we observe that the\ncurvature scalar thus considered is inverse squarely proportional\nto the determinant of the underlying state-space metric tensor,\nand thus there are no genuine state-space instability in the\nunderlying microscopic configurations. In turn, we discover that\nthe underlying statistical correlations remain definite, non-zero,\nfinite, regular functions of total boxes and excited boxes carried\nby the giant black holes.\n\nIt is important to note that the state-space geometric quantities\nmay become ill-defined, if the state-space co-ordinates, being\ndefined as the space-time parameters, jump from one existing\ndomain to another domain of the solution. This indicates that the\nmicroscopic configurations may correspond to certain interacting\nstatistical systems, in the chosen branch of the full black hole\nsolutions. In an admissible domain of physically acceptable boxes,\nwe discover that the giant black holes have no phase transition,\nand thus the fundamental statistical configurations are completely\nfree from critical phenomena. Note that the absence of divergences\nin the scalar curvature indicates that the BPS black brane\nsolutions endure an everywhere thermodynamically stable systems,\non their respective state-space configurations.\n\nMore generally, we find that the regular state-space scalar\ncurvature seems to be comprehensively universal for the given\nnumber of parameters of the configuration. In fact, the concerned\nidea turns out to be related with the typical form of the\nstate-space geometry, arising from the negative Hessian matrix of\nthe duality invariant expression of the black hole entropy. As in\nthe standard interpretation, the state-space scalar curvature\ndescribes the nature of underlying statistical interactions of the\npossible microscopic configurations, which characteristically turn\nout to be non-zero, for the $1\/2$-BPS black holes. In fact, we may\neasily appreciate that the constant entropy curve is a standard\ncurve, which may be given by\n\\begin{eqnarray}\n(M^2)!= c\\ n!(M^2-n)!\n\\end{eqnarray}\nHere, the real constant $c$ can be determined from the given\nexpression of the vacuum entropy. This determines the giant black\nhole embedding in the view-points of the state-space geometry.\nMoreover, we may also disclose in the present case that the curve\nof constant scalar curvature is, however, a complicated curve on\nthe space of total boxes and excited boxes, but the the concerned\nfundamental nature may easily be fixed by the given number of the\nboxes. Nonetheless, it is not difficult to enunciate the\nquantization condition existing on the charges, as we have an\ninteger number of boxes, which in turn signifies a general\ncoordinate transformation in the large number of boxes on the\nstate-space manifold, and may thus be presented in terms of the\nnet number of respective branes.\n\\section{Remarks and Conclusion}\nThis paper provides an exact intrinsic thermodynamic geometric\naccount of the canonical energy fluctuations and box counting\nentropy for the giants and superstars.  The arguments required\nhave been accomplished from an ensemble of Young tableaux with\ncertain randomly excited boxes, while the remaining boxes were\nkept intact. We notice that the microcanonical representations of\nthe black holes under consideration posit a large number of\ncorrelated degenerate microscopic states. These microstates may be\nevaded by showing that an initial pure state collapses to a\nparticular pure black hole microstate, whose exact structure can\nbe deduced by suitably subtle measurements \\cite{hep-th\/0508023},\nalong with the loss of information, if any. Here, our focus has\nbeen to investigate the implications of correlated states for the\n$1\/2$-BPS supergravity black holes. Our analysis clarifies the\nnature of underlying equilibrium configurations over the Gaussian\nfluctuations.\n\nGiven the importance of the canonical correlations, we have\naddressed the following questions: (i) how do the Gaussian\ncorrelations of pure microstates look like in the view-points of\nthermodynamic geometries, (ii) what sorts of correlation areas may\ndistinguish the giants or superstars from each other, and (iii)\nwhat is the typical nature of statistical fluctuations over an\nequilibrium ensemble of CFT microstates? Interestingly, we have\nexplicitly demonstrated that the underlying chemical fluctuations\ninvolve ordinary summations, while the state-space fluctuations\nmay simply be depicted by standard polygamma functions. Hereby, we\nnotice that the large black holes in AdS spacetimes, whose horizon\nsizes are bigger than the scale of the AdS curvature, are stable,\nand exclusively there is no thermodynamic phase transition. We\nherewith find a precise matching with the fact that these black\nholes come into an equilibrium configuration, because the AdS\ngeometries create an effective confining potential \\cite{HP}, and\nthus yield the stability of the thermodynamic system.\n\nImportantly, our framework exploits the fact that the classical\ngeneral relativity large black holes come with the basic property\nthat their horizon area satisfies the Bekenstein-Hawking entropy\nrelation. With an understanding of the limiting microscopic\nconfigurations, an appropriate intrinsic geometric notion has in\neffect been offered to the counting problem of the microstates of\nthe giants and superstars. Following the discussion of the Appendix\nA, this provides an account to the degeneracy of microscopic states\nin the limit $G_N\/l_P>>1$. The semi-classical approximation thus \nuncovers that the Boltzmann statistical entropy entails certain \nwell-defined equilibrium microstate systems, upon the inclusion\nof a quadratic fluctuation on the phase-space configurations.\nOur analysis thus shows that the $1\/2$-BPS giants and superstars\nare thermodynamically stable objects and do not emit Hawking radiation.\n\nThe state-space description shows that a set of exact expressions\nmay easily be given for the quadratic fluctuations, over an\nequilibrium canonical configuration characterized by possible\nexcited and unexcited droplets, in an ensemble of arbitrary random\nYoung tableaux. In turn, the state-space fluctuations, stability\ncriterion and state-space correlation length have precisely been\ndetermined, without any approximation, for a large number of giant\ngravitons and superstars. Following the standard Riemannian\ngeometric notions, we expose that the chemical configuration\nyields in general that the chemical pair correlation functions,\nstability condition and correlation length, for an arbitrary value\nof the effective canonical temperature and the chemical potential,\nmay be determined over an intrinsic Weinhold manifold. Our\nintrinsic geometric study, thus, exemplifies that there exists an\nexact fluctuating statistical configuration, which involves an\nensemble of fuzzballs, or a number of liquid droplets.\n\nFurthermore, we can ask how the underlying highly degenerate\nequilibrium microstates get correlated with the chemical\npotential, carried by a set of given excited droplets. We have\nshown, from the perspective of string theory on $AdS_5 \\times\nS^5$, that the five dimensional $R$ charged AdS black holes admit\nan exact Legendre transformed dual chemical configuration, which\nmay be described in terms of the  chemical potential and an\neffective canonical temperature. Thus, the origin of gravitational\nthermodynamics comes with the existence of a non-zero\nthermodynamic curvature, under the coarse graining mechanism of\nalike ``quantum information geometry'', associated with the wave\nfunctions of underlying BPS black holes. Notice, further, that the\nphysical meaning of the respective curvatures is that they\ndescribe the correlation, in the concerned ensemble of black\nholes. Characteristically, it is worth to mention that the\nsemi-classical gravitational description, arising with the\nnon-vanishing scalar curvature, signifies the existence of the\nthroat of the underlying AdS background.\n\nWe notice an intriguing support, from the Mathur's fuzzball\nproposal, that the leading order entropy should come from those\nfuzzs, whose radius varies as the fuzzy throat of the black hole\nhorizon size. Furthermore, the liquid droplet model suggests that\nthe AdS length scales like $L\\sim l_P f(N_i)$, where $N_i$ is the\nnumber of bound states \\cite{hep-th\/0508023, hep-th\/0107119}. It\nhas further been suggested that the classical description is\nachieved, when $N \\rightarrow \\infty$ while $\\hbar \\rightarrow 0$,\nsuch that the Fermi level $N\/ \\hbar$ remains constant. Most\nimportantly, we have investigated the role of thermodynamic\nfluctuations in the two parameter giants and superstars, whose\nchemical and state-space configurations are being respectively\ncharacterized by the chemical variables of the effective canonical\nensemble, and by the number of both excited and total boxes,\nconstituting an ensemble of arbitrary shaped CFT microstates. The\npresent analysis thus explicates that the underlying chemical and\nstate-space configurations are well-defined, non-degenerate and\nregular, for all physically admissible domains of the statistical\nparameters, defining an ensemble of arbitrary fuzzballs, or a set\nof indiscriminate liquid droplets.\n\nNow, we enlist a number of attributes arising from our study of\nthe thermodynamic intrinsic geometry of (dual) giants and\nsuperstars, divulged in the framework of liquid droplets or\nfuzzball solutions. The intriguing nature of the chemical and the\nstate-space correlations, existing in an underlying statistical\nbasis, ascribes that the local and global thermodynamic\nstructures, thus revealed, may in either case be summarized as\nfollows.\n\\subsection{Chemical Description:}\nFollowing our specific notions, we exclusively observe that the\nchemical configurations of the $1\/2$ BPS giants and superstars\nrealize the following general properties:\n\\begin{itemize}\n\\item There exists an exact expression for the chemical pair\ncorrelation functions, stability condition and correlation length,\nfor arbitrary values of $T, \\lambda \\in M_2$.\n\\item Explicit plots displayed in the Figs. 1 and 2 show that\nthe determinant and scalar curvature are non-trivially curved, and\nsurprisingly the results remain the same, even for a single\ncomponent with $j=1$.\n\\item We further reveal, from the numerical conclusions displayed in\nFigs. 1 and 2, that the Weinhold metric tensor, and the\ncorresponding correlation length, show that the statements of\nstability and regularity hold for all $\\lambda \\in M_2$.\n\\end{itemize}\n\\subsection{State-space Description:}\nIn this case, the findings obtained from the intrinsic state-space\ngeometry from an ensemble of Young tableaux may be summarized as\n\\begin{itemize}\n\\item A set of exact precise expressions may easily be given for the\nstate-space fluctuations, over an equilibrium canonical ensemble,\ncharacterized by possible arbitrary random Young tableaux.\n\\item The state-space fluctuations, stability criterion and\nstate-space correlation length may easily be determined without\nany approximation.\n\\item We can express the $g^{(S)}_{ij}(n,M)$, $g^{(S)}(n,M)$\nand $R^{(S)}(n,M)$ in terms of nice, well-behaved digamma\n$\\Psi(n)$ and poly-gamma $\\Psi(n,M)$ functions. Notice that the\nmixing between excited and non-excited droplets or fuzzs may\nprecisely be caused by these standard polygamma functions.\n\\item The concerned state-space geometry corresponds to a non-degenerate,\nlocally stable and attractive statistical configuration.\n\\item The underlying numerical computations plotted in Figs. 3 and 4\ndisclose that the state-space correlation exists along the\nboundary of $n$ and $M$, for non-large $n$, $M$, when we take\napproximately $1$ to $10^5$ boxes in the Young diagrams.\n\\end{itemize}\nIt is worth to mention that the present analysis takes into\naccount the scales, that are larger than the respective\ncorrelation length, and contemplates that just a few giant or\nsuperstar microstates cannot dominate the entire macroscopic\nsolution. Most importantly, we have procured that the\nthermodynamic intrinsic geometric structures of the canonical\nenergy and box counting entropy provide a coherent framework, to\nfurther study the thermodynamic geometries, arising from the large\nnumber of microstates of the chosen giants, superstars and the\nother superconformal field theory and supergravity configurations\n\\cite{9711200v3}.\n\nFinally, it is worth to mention that the interpretation of a\nnon-zero intrinsic scalar curvature, with an underlying\ninteracting statistical system, remains valid even for higher\ndimensional intrinsic Riemannian manifolds. The implication of a\ndivergent intrinsic covariant curvature may accordingly be\ndivulged, from the Hessian matrix of the canonical energy or the\nbox counting entropy, irrespectively whether, or not, there exists\na phase transition in the $1\/2$ BPS black hole configurations.\n\\section*{Acknowledgements}\nThis work has been supported in part by the European Research\nCouncil grant n.~226455, ``SUPERSYMMETRY, QUANTUM GRAVITY AND\nGAUGE FIELDS (SUPERFIELDS)''. We would like to thank Prof. J.\nSim\\'on for useful discussions and view-points provided during the\n``School on Attractor Mechanism SAM-2009, INFN- Laboratori\nNazionali di Frascati, Roma, Italy''. BNT would like to thank\nProf. S. Mathur and Prof. A. Sen for useful discussions offered\nduring the ``Indian String Meeting, ISM-2006, Puri, India''; Prof.\nJ. de Boer during the ``Spring School on Superstring Theory and\nRelated Topics-2007 and 2008, ICTP, Trieste, Italy''; and Yogesh\nSrivastava during the Indian String Meeting-2007, Harish-Chandra\nResearch Institute, Allahabad, India; and Mohd. A. Bhat and V.\nChandra for reading the manuscript and making interesting\nsuggestions. BNT especially thanks Prof. V. Ravishankar for\nencouragements and necessary supports provided during this work.\nBNT would like to acknowledge nice hospitality of the\n``INFN-Laboratori Nazionali di Frascati, Roma, Italy'' being\noffered during the ``School on Attractor Mechanism: SAM-2009''\nwhere part of this work was performed. The research of BNT has\npartially been supported by the CSIR, New Delhi, India and Indian\nInstitute of Technology Kanpur, Kanpur-208016, Uttar Pradesh,\nIndia.\n\\section*{Appendix A}\nIn this appendix, we provide precise expression for the case of two\nparameter thermodynamic configuration. This review set-up is offered from\nthe viewpoint of two parameter family giant and superstar solutions.\nIn order to do so, we recall important recent studies of the thermodynamic\nproperties of diverse (rotating) black holes have elucidated interesting\naspects of phase transitions, if any, in the state-space geometric framework\nand their associated relations with the extremal black hole solutions in the\ncontext of $\\mathcal N \\geq 2$ compactifications \\cite{sfm1, sfm2}. It may\nbe argued however that the connection of such a geometric formulation to\nthe thermodynamic fluctuation theory of black holes requires several\nmodifications \\cite{rup3}. The geometric formulation thus involved has\nfirst been applied to $\\mathcal  N\\geq 2$ supergravity extremal black\nholes in $D=4$, which arise as low energy effective field theories from\nthe compactifications of Type II string theories on Calabi-Yau manifolds\n\\cite{fgk}. Since then, several authors have attempted to understand\nthis connection \\cite{cai1,gr-qc\/0304015v1,0510139v3,Arcioni}, for both\nthe supersymmetric as well as non-supersymmetric four dimensional black\nholes and five dimensional rotating black string and ring solutions.\nInteresting discussions on the vacuum phase transitions, if any,\nexist in the literature, which involves some change of the black hole\nhorizon topology, \\cite{cai1,gr-qc\/0304015v1,0510139v3,Arcioni}.\n\nRuppenier has conjointly advocated the assumption ``that\nall the statistical degrees of freedom of black hole live on the\nblack hole event horizon'', and thus the scalar curvature signifies\nthe average number of correlated Planck areas on the event horizon\nof the black hole \\cite{RuppeinerPRD78}. Specifically, the zero\nscalar curvature indicates certain bits of information on the\nevent horizon fluctuating independently of each other, while the\ndiverging scalar curvature signals a phase transition indicating\nhighly correlated pixels of the informations. Moreover, Bekenstein\nhas introduced an elegant picture for the quantization of the area of\nthe event horizon, being defined in terms of Planck areas\n\\cite{Bekenstein}. Recently, the state-space geometry of the\nequilibrium configurations thus described has extensively been\napplied to study the thermodynamics of a class of rotating black\nhole configurations \\cite{bnt,BNTBull,bntSb, BSBR}.\n\nFrom the viewpoint of the present consideration,\nthe underlying moduli configuration appears to be horizonless and\nsmooth. However, in the classical limit in which the Planck length and\nthe AdS throat scale as, respectively $l_P\\rightarrow 0$ and $L\\rightarrow 0$,\nin such a way that their ratio diverges $l_P\/L \\rightarrow \\infty$,\nthe underlying moduli configuration acquires an entropy which may\nbe assumed to be associated with the average horizon area of the \nblack hole. In general, one wishes to compare the quantization of a\nclassical moduli space from the known perspective of the AdS\/CFT\ncorrespondence \\cite{LLM, SB}. In this concern, it is worth to\nmention that we have \n\\begin{scriptsize}\n\\begin{eqnarray}\nAdS\/CFT: \\ \\{\\mathcal{N}=4 \\ SYM \\}\\ \\leftrightarrow \\ \\{Type \\\nIIB \\ String \\ Theory \\ on \\ AdS_5 \\times S^5 \\}\n\\end{eqnarray}\n\\end{scriptsize}\nFollowing \\cite{hep-th\/0508023,hep-th\/0107119}, it is important to mention\nthat the supergravity description emerges in the strong coupling limit with\n\\begin{scriptsize} $g^2_{YM}N>>$ \\end{scriptsize}\nwhile, the dual CFT emerges in the weak coupling limit \\begin{scriptsize}\n$g^2_{YM}N<<1$\\end{scriptsize}. Thus, one finds the matching of entropies\nin the $\\frac{1}{2}$ BPS sector of the configuration \\cite{hep-th\/0508023,\nhep-th\/0107119}. In this sector, the authors of \\cite{hep-th\/0508023,hep-th\/0107119}\nhave shown  that nearly all states look alike and they belong to the same chiral\nring quantization of the classical moduli space. It is however intriguing to note \nthat the dual CFT formalism faces problems with the wave functional renormalization,\nmixing of states, and in general it may feature certain other technical difficulties\nas well \\cite{hep-th\/0508023, hep-th\/0107119}. Nevertheless, none of these concerns\nthe present analysis, and thus we may safely divulge the probable fluctuations\nover chosen equilibrium giants configurations. \n\nAs mentioned in the introduction, the $1\/2$-BPS (dual) giant configurations\nare parameterized by two chemical potentials, $\\lambda_1, \\lambda_2$. Thus,\nthe fluctuations around the minima energy configuration are describe an\nintrinsic Wienhold surface. \nExplicitly, let us consider the case of the two dimensional intrinsic chemical geometry,\nsuch that the components of the metric tensor are given by \\\\\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{ \\lambda_1  \\lambda_1}&=& \\frac{\\partial^2 E}{\\partial  \\lambda_1^2} \\nonumber  \\\\\ng_{ \\lambda_1  \\lambda_2}&=& \\frac{\\partial^2 E}{{\\partial  \\lambda_1}{\\partial  \\lambda_2}}\\nonumber  \\\\\ng_{ \\lambda_2  \\lambda_2}&=& \\frac{\\partial^2 E}{\\partial\n\\lambda_2^2}\n\\end{eqnarray}\n\\end{scriptsize}\nIn this case, it follows that the determinant of the metric tensor is \n\\begin{scriptsize}\n\\begin{eqnarray}\n\\Vert g(\\lambda_1,\\lambda_2) \\Vert &= &S_{\\lambda_1 \\lambda_1}S_{\\lambda_2 \\lambda_2}- S_{\\lambda_1 \\lambda_2}^2\n\\end{eqnarray}\n\\end{scriptsize}\nNow, we can calculate the $\\Gamma_{ijk}$, $R_{ijkl}$, $R_{ij}$ and\n$ R $ for the above two dimensional thermodynamic geometry $(M_2,g)$.\nOne may easily inspect that the scalar curvature is given by\n\\begin{scriptsize}\n\\begin{eqnarray}\nR(\\lambda_1, \\lambda_2)&=& -\\frac{1}{2} (E_{ \\lambda_1 \\lambda_1}E_{ \\lambda_2\n\\lambda_2}- E_{ \\lambda_1  \\lambda_2}^2)^{-2} (E_{ \\lambda_2\n \\lambda_2}E_{ \\lambda_1 \\lambda_1 \\lambda_1}E_{ \\lambda_1  \\lambda_2\n \\lambda_2}\\nonumber\n\\\\ &&+ E_{ \\lambda_1  \\lambda_2}E_{ \\lambda_1 \\lambda_1 \\lambda_2}E_{\n\\lambda_1 \\lambda_2 \\lambda_2}+\nE_{ \\lambda_1 \\lambda_1}E_{ \\lambda_1 \\lambda_1 \\lambda_2}E_{ \\lambda_2 \\lambda_2 \\lambda_2}\\nonumber  \\\\\n&&- E_{ \\lambda_1 \\lambda_2}E_{ \\lambda_1 \\lambda_1 \\lambda_1}E_{\n\\lambda_2 \\lambda_2 \\lambda_2}- E_{ \\lambda_1 \\lambda_1}E_{\n\\lambda_1 \\lambda_2 \\lambda_2}^2 - E_{ \\lambda_2 \\lambda_2}E_{\n\\lambda_1 \\lambda_1 \\lambda_2}^2 )\n\\end{eqnarray}\n\\end{scriptsize}\nFurthermore, the relation between the chemical scalar curvature\nand the Riemann covariant curvature tensor for any two dimensional\nintrinsic geometry is given (see for details \\cite{bnt}) by\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{Wiencur}\nR=\\frac{2}{\\Vert g \\Vert}R_{ \\lambda_1 \\lambda_2 \\lambda_1 \\lambda_2}\n\\end{eqnarray}\n\\end{scriptsize}\nThe relation Eq. \\ref{Wiencur} is quite usual for an arbitrary\nintrinsic Riemannian surface $(M_2(R),g)$. \nCorrespondingly, the Legendre transformed version of the Wienhold manifold\nis known as state-space manifold. In fact, the associated configuration\nis parameterized by the number of excited and total boxes, i.e. $ \\{n,M\\}$. \nThus, the Gaussian fluctuation of the entropy of $1\/2$-BPS configurations\nform a two dimensional state-space surface $M_2$.\nExplicitly, the components of covariant state-space metric tensor may be given as\n\\begin{scriptsize}\n\\begin{eqnarray}\ng_{nn}&=&- \\frac{\\partial^2 S(n,M)}{\\partial n^2} \\nonumber \\\\\ng_{nM}&=&- \\frac{\\partial^2 S(n,M)}{{\\partial n}{\\partial M}}  \\nonumber  \\\\\ng_{MM}&=&- \\frac{\\partial^2 S(n,M)}{\\partial M^2}\n\\end{eqnarray}\n\\end{scriptsize}\nIt is easy to express, in this simplest case, that the\ndeterminant of the metric tensor turns out to be\n\\begin{scriptsize}\n\\begin{eqnarray}\n\\Vert g(n,M) \\Vert &= &S_{nn}S_{MM}- S_{nM}^2\n\\end{eqnarray}\n\\end{scriptsize}\nAs in the case of Wienhold geometry, we find that the state-space scalar curvature is given by\n\\begin{scriptsize}\n\\begin{eqnarray}\nR(n,M)&=& \\frac{1}{2} (S_{nn}S_{MM}- S_{nM}^2)^{-2}\n(S_{MM}S_{nnn}S_{nMM} + S_{nM}S_{nnM}S_{nMM} \\nonumber\\\\ &+&\nS_{nn}S_{nnM}S_{MMM} -S_{nM}S_{nnn}S_{MMM}- S_{nn}S_{nMM}^2-\nS_{MM}S_{nnM}^2 )\n\\end{eqnarray}\n\\end{scriptsize}\nFollowing the observations of \\cite{BNTBull}, it is essentially evident\nthat the scalar curvature and the corresponding Riemann curvature tensor\nof an arbitrary two dimensional intrinsic state-space manifold\n$(M_2(R),g)$ may be given by\n\\begin{scriptsize}\n\\begin{eqnarray}\nR(n,M)=\\frac{2}{\\Vert g \\Vert}R_{nMnM}(n,M)\n\\end{eqnarray}\n\\end{scriptsize}\n\\section*{Appendix B}\nIn this appendix, we provide the explicit form of the most general\nthermodynamic scalar curvatures describing the family of two\ncharged giant and superstars. Our analysis illustrates that the\nphysical properties of the specific scalar curvatures may exactly\nbe exploited, without any approximation. The definite behavior of\ncurvatures, as accounted in section three, suggests that the\nvarious intriguing chemical and state-space examples of $1\/2$ BPS\nsolutions include the nice property that they do not diverge,\nexcept for the determinant singularity. As mentioned in the main\nsections, these configurations are an interacting statistical\nsystem. We discover that their thermodynamic geometries indicate\nthe possible nature of general two parameter equilibrium\nconfigurations. Significantly, one may notice, from the very\ndefinition of intrinsic metric tensors, that the relevant Riemann\ncovariant curvature tensors and scalar curvatures may be thus\npresented as follows.\n\\subsection*{(i) Canonical Energy Fluctuations:}\nHere, we shall explicitly supply the exact Riemann covariant\ntensor of the Weinhold geometry for the 2- parameter giants and\nsuperstars. It turns out that the functional nature of a large\nnumber of gravitons, within the neighborhood of small chemical\nfluctuations introduced in the canonical ensemble of $1\/2$ BPS\nconfigurations, may precisely be divulged. Surprisingly, we expose\nin this framework, that the intrinsic covariant curvature tensor\ntakes the exact and simple expression\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{canenfluc}\nR_{T \\lambda T \\lambda}(T, \\lambda)&=& -\\frac{1}{4}\n\\{(\\sum_{j=0}^{\\infty}(-j^2 \\frac{b_j}{T^4}\\frac{(-2 T+2 T b_j+j+j\nb_j)} {(b_j-1)^3}))\\times\n(\\sum_{j=0}^{\\infty}(-j b_j\\frac{(1+b_j)}{(b_j-1)^3}) \\\\\n\\nonumber && -(\\sum_{j=0}^{\\infty}(j^2 \\frac{b_j}{T^2}\n\\frac{(1+b_j)}{(b_j-1)^3}))^2\\}^{-1} \\{-(\\sum_{j=0}^{\\infty}(-j\nb_j\\frac{(1+b_j)}{(b_j-1)^3})) \\times \\\\ \\nonumber &&\n(\\sum_{j=0}^{\\infty}(-j^2 \\frac{b_j}{T^4}\\frac{(-2 T+2 T b_j^2+j+4\nj b_j+j b_j^2)}{(b_j-1)^4}))^2 +(\\sum_{j=0}^{\\infty}(-j\nb_j\\frac{(1+b_j)}{(b_j-1)^3})) \\times\n\\\\ \\nonumber && (\\sum_{j=0}^{\\infty}(j^2 \\frac{b_j}{T^6}\n\\frac{(j^2+4 j^2 b_j+j^2 b_j^2-6 j T+6 T^2+6 j T b_j^2-12 T^2\nb_j+6 T^2 b_j^2)}{(b_j-1)^4})) \\\\ \\nonumber && \\times\n(\\sum_{j=0}^{\\infty}(j^2 \\frac{b_j}{T^2} \\frac{(1+4\nb_j+b_j^2)}{(b_j-1)^4})) +(\\sum_{j=0}^{\\infty}(j^2\n\\frac{b_j}{T^2} \\frac{(1+b_j)} {(b_j-1)^3})) \\times \\\\\n\\nonumber && (\\sum_{j=0}^{\\infty}(-j^2 \\frac{b_j}{T^4}\\frac{(-2\nT+2 T b_j^2+j+4 j b_j+j b_j^2)}{(b_j-1)^4})) \\times \\\\\n\\nonumber && (\\sum_{j=0}^{\\infty}(j^2 \\frac{b_j}{T^2}\\frac{(1+4\nb_j+b_j^2)}{(b_j-1)^4})) -(\\sum_{j=0}^{\\infty}(j^2\n\\frac{b_j}{T^2} \\frac{(1+b_j)}{(b_j-1)^3})) \\times \\\\\n\\nonumber && (\\sum_{j=0}^{\\infty}(j^2 \\frac{b_j}{T^6} \\frac{(j^2+4\nj^2 b_j+j^2 b_j^2-6 j T+6 T^2+6 j T b_j^2-12 T^2 b_j+6 T^2\nb_j^2)}{(b_j-1)^4})) \\\\ \\nonumber && \\times\n(\\sum_{j=0}^{\\infty}(-j b_j \\frac{(1+4 b_j+b_j^2)} {(b_j-1)^4}))\n-(\\sum_{j=0}^{\\infty}(-j^2 \\frac{b_j}{T^4}\\frac{(-2 T+2 T b_j+j+j\nb_j)}{(b_j-1)^3})) \\times \\\\ \\nonumber && (\\sum_{j=0}^{\\infty}(j^2\n\\frac{b_j}{T^2} \\frac{(1+4 b_j+b_j^2)}{(b_j-1)^4}))^2\n+(\\sum_{j=0}^{\\infty}(-j^2 \\frac{b_j}{T^4} \\frac{(-2 T+2 T b_j+j+j\nb_j)} {(b_j-1)^3})) \\times \\\\ \\nonumber &&\n(\\sum_{j=0}^{\\infty}(-j^2 \\frac{b_j}{T^4} \\frac{(-2 T+2 T\nb_j^2+j+4 j b_j+j b_j^2)}{(b_j-1)^4})) \\times (\\sum_{j=0}^{\\infty}\n(-j b_j\\frac{(1+4 b_j+b_j^2)} {(b_j-1)^4})) \\}\n\\end{eqnarray}\n\\end{scriptsize}\n\\subsection*{(ii) State-space Fluctuations:}\nAs stated earlier, the state-space metric in the excited and\nunexcited droplets is given by the negative Hessian matrix of the\nbox counting entropy. Here, the numbers of excited and empty boxes\nin a given Young diagram are respected to be extensive variables.\nIn this case,  the two distinct large integers characterize the\nintrinsic state-space correlation length, carried by the giants\nand superstars. Our computation shows that the exact Riemann\ncovariant curvature tensor is given by\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{countentcovcur}\nR_{nMnM}(n,M) &=&\n1\/2\\{(-2M^2\\Psi(1,n+1)\\Psi(1,M^2+1)-\\Psi(1,n+1)\\Psi(M^2+1)\n\\\\ \\nonumber && +2M^2\\Psi(1,n+1)\\Psi(1,M^2-n+1) +\\Psi(1,n+1)\n\\Psi(M^2-n+1) \\\\ \\nonumber && -2M^2\\Psi(1,M^2-n+1)\n\\Psi(1,M^2+1)-\\Psi(1,M^2-n+1)\\Psi(M^2+1) \\\\ \\nonumber &&\n+\\Psi(1,M^2-n+1)\\Psi(M^2-n+1))\\}^{-1} [\n\\Psi(1,n+1)\\Psi(1,M^2-n+1)^2 \\\\ \\nonumber && -\\Psi(1,M^2-n+1)\n\\Psi(2,n+1)\\Psi(M^2+1) \\\\ \\nonumber && +\\Psi(1,M^2-n+1)\\Psi(2,n+1)\n\\Psi(M^2-n+1) \\\\ \\nonumber && +\\Psi(2,M^2-n+1)\\Psi(1,M^2-n+1)\n\\Psi(M^2+1)\\\\ \\nonumber &&-\\Psi(2,M^2-n+1)\\Psi(1,M^2-n+1)\n\\Psi(M^2-n+1) \\\\ \\nonumber && +2 M^2 \\{ 3\\Psi(2,M^2-n+1)\n\\Psi(1,n+1) \\Psi(1,M^2+1) \\\\ \\nonumber && +2\\Psi(1,M^2-n+1)\n\\Psi(2,n+1) \\Psi(1,M^2+1) \\\\ \\nonumber &&\n-2\\Psi(1,M^2-n+1)^2\\Psi(2,n+1) +\\Psi(1,M^2-n+1)^3\\\\ \\nonumber &&\n-\\Psi(2,M^2-n+1) \\Psi(2,n+1)\\Psi(M^2+1)\\\\ \\nonumber &&\n+\\Psi(2,M^2-n+1)\\Psi(2,n+1) \\Psi(M^2-n+1)\\\\ \\nonumber &&\n-\\Psi(2,M^2-n+1) \\Psi(1,n+1)\\Psi(1,M^2-n+1) \\\\ \\nonumber &&\n+\\Psi(2,M^2-n+1)\\Psi(1,M^2-n+1)\\Psi(1,M^2+1)\\} \\\\ \\nonumber &&\n+4M^4 \\{ \\Psi(2,M^2-n+1)\\Psi(1,n+1)\\Psi(2,M^2+1)\\\\ \\nonumber &&\n-\\Psi(2,M^2-n+1)\\Psi(2,n+1) \\Psi(1,M^2+1) \\\\ \\nonumber &&\n+\\Psi(1,M^2-n+1)\\Psi(2,n+1) \\Psi(2,M^2+1)\\}]\n\\end{eqnarray}\n\\end{scriptsize}\nFinally, let us turn our attention to the state-space scalar\ncurvature for the two parameter droplet configurations. A\nsystematic examination demonstrates that the scalar curvature is\n\\begin{scriptsize}\n\\begin{eqnarray} \\label{countentscacur}\nR(n,M) &=& \\frac{1}{2} \\{-2 \\Psi(1,n+1) \\Psi(1,M^2+1)\nM^2-\\Psi(1,n+1) \\Psi(M^2+1) \\\\ \\nonumber && +2 M^2 \\Psi(1,n+1)\n\\Psi(1,M^2-n+1) +\\Psi(1,n+1) \\Psi(M^2-n+1) \\\\ \\nonumber && -2 M^2\n\\Psi(1,M^2-n+1) \\Psi(1,M^2+1)- \\Psi(1,M^2-n+1) \\Psi(M^2+1)\\\\\n\\nonumber && +\\Psi(1,M^2-n+1) \\Psi(M^2-n+1)\\}^{-2} \\\\ \\nonumber\n&&[ \\Psi(1,M^2-n+1)^3 + \\Psi(1,n+1) \\Psi(1,M^2-n+1)^2 \\\\ \\nonumber\n&& -\\Psi(1,M^2-n+1) \\Psi(2,n+1) \\Psi(M^2+1) \\\\\n\\nonumber && +\\Psi(1,M^2-n+1) \\Psi(2,n+1) \\Psi(M^2-n+1) \\\\\n\\nonumber && +\\Psi(2,M^2-n+1) \\Psi(1,M^2-n+1) \\Psi(M^2+1) \\\\\n\\nonumber && -\\Psi(2,M^2-n+1) \\Psi(1,M^2-n+1) \\Psi(M^2-n+1) \\\\\n\\nonumber && +2 M^2 \\{ \\Psi(2,M^2-n+1) \\Psi(1,M^2-n+1)\n\\Psi(1,M^2+1) \\\\ \\nonumber && - \\Psi(2,M^2-n+1) \\Psi(1,n+1)\n\\Psi(1,M^2-n+1) \\\\ \\nonumber && - \\Psi(2,M^2-n+1) \\Psi(2,n+1)\n\\Psi(M^2+1) \\\\ \\nonumber && + \\Psi(2,M^2-n+1)\n\\Psi(2,n+1)\\Psi(M^2-n+1)\\\\ \\nonumber && - 2 \\Psi(1,M^2-n+1)^2\n\\Psi(2,n+1) \\\\ \\nonumber && + 2 \\Psi(1,M^2-n+1) \\Psi(2,n+1)\n\\Psi(1,M^2+1) \\\\ \\nonumber && +3 \\Psi(2,M^2-n+1) \\Psi(1,n+1)\n\\Psi(1,M^2+1)\\} \\\\ \\nonumber && +4 M^4 \\{ \\Psi(2,M^2-n+1)\n\\Psi(1,n+1) \\Psi(2,M^2+1)\\\\ \\nonumber && + \\Psi(1,M^2-n+1)\n\\Psi(2,n+1) \\Psi(2,M^2+1) \\\\ \\nonumber && - \\Psi(2,M^2-n+1)\n\\Psi(2,n+1) \\Psi(1,M^2+1) \\}]\n\\end{eqnarray}\n\\end{scriptsize}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIncreasing spatial resolution allows to observe directly parts of active galactic nuclei. However,\nthe innermost part is still unresolved and the information about its geometrical structure comes \nfrom spectroscopy. This structure is very complex, so vigorous discussions of inflow\/outflow \nare still going on. Among this unsolved problems is the origin of Broad Emission Lines. \n\nBroad Emission Lines come from a specific Broad Emission Line Region (BLR), as seen in well separation of \nthe Broad and narrow components in the emission lines of active galaxies. They cover significant fraction \nof the AGN sky, between 10 and 30 \\%, as implied by their luminosity, but they are rarely seen in\nabsorption (for rare exceptions, see e.g. Risaliti et al. 2011) which argues for the flat \nconfiguration (e.g. Nikolajuk et al. 2005, \nCollin et al. 2006), mostly hidden inside \nthe outer dusty-molecular torus. The motion of the material is roughly consistent with \ncircular Keplerian orbits (Wandel et al. 1999), but with significant additional velocity dispersion. Careful studies\nindicate two components of the BLR (Collin-Souffrin et al. 1988): High Ionization Lines (HIL; e.g. CIV) \nand Low Ionization Lines (LIL; e.g. H$\\beta$, Mg II). HIL show blueshift with respect to narrow lines and they\nare considered to be comming from outflowing wind. LIL do not show significant shift so cannot come from outflow,\nbut the region cannot be supported by the hydrostatic equilibrium. Therefore, the dynamics of LIL is\ndifficult to understand from theoretical point of view, and it is also not resolved by direct\nmonitoring (e.g. Peterson et al. 2004). \n\n\\begin{figure}[t]\n\\begin{center}\n\\hfill~\n \\includegraphics[width=0.6\\textwidth]{czernyb_talk_fig1.eps}\n\\hfill~\n \\caption{Schematic picture of the formation of the BLR. The outflow starts where the disk effective temperature drops below 1000 K and dust formation can proceed. Dusty outflow changes into a failed wind above the disk, where the irradiation leads to dust evaporation, the radiation pressure suport rapidly decreases and the material falls back onto the disk.}\n\\end{center}\n\\end{figure}\n\n\\section{Accretion Disk Temperature at the Onset of BLR}\n\nHere we show that the combination of the simplest accretion disk theory and the reverberation\nmeasurement of the BLR distance leads to surprising conclusion: the effective temperature of the\naccretion disk at the radius measured from reverberation is universal and equal to 1000 K. The argument goes\nin the following way (for more details, see Czerny \\& Hryniewicz 2011).\n\nThe measurements of the delay of the H$\\beta$ line with respect to the continuum at 5100 A for close to 40 objects\nallowed to determine the relation between the BLR size, as measured from such delay, and the monochromatic flux\n(Kaspi et al. 2000, Peterson et al. 2004). The most recent form of this relation, corrected for the starlight \ncontamination, was derived by Bentz et al. (2009). This relation can be written in the following way\n\\begin{equation}\n\\log R_{BLR}[{\\rm H}\\beta] = 1.538 \\pm 0.027 + 0.5 \\log L_{44,5100},\n\\label{Bentz}\n\\end{equation}\nwhere $R_{BLR}[{\\rm H}\\beta]$ is in light days and $L_{44,5100}$ is the monochromatic luminosity at \n5100 \\AA~ measured\nin units of $10^{44}$ erg s$^{-1}$. The value 0.027 is the error in the best-fit vertical \nnormalization.\n\nNow, from the simplest theory of the alpha disk (Shakura \\& Sunyaev 1973) we can find the monochromatic flux. The\nasymptotic formula is usually known in the form\n\\begin{equation}\nL_{\\nu} \\propto \\nu^{1\/3},\n\\end{equation}\nbut the proportionality coefficient is also known, and depends on the black hole mass, $M$, and accretion rate,\n$\\dot M$ as $(M \\dot M)^{2\/3}$. Taking into account this dependence, physical constants and fixing the frequency at\nthe value corresponding to the wavelength 5100 A, we have\n\\begin{equation}\n\\log L_{44,5100} = {2 \\over 3} \\log (M \\dot M) - 43.8820 + \\log \\cos i.\n\\label{L5100}\n\\end{equation} \nThis formula allows us to calculate the BLR radius from the Eq.~\\ref{Bentz}, if the\nproduct of ($M \\dot M$) is known.\n\nThe Shakura-Sunyaev accretion disk theory allows us to calculate the effective temperature of the disk at that \nradius as a function of the black hole mass and accretion rate\n\\begin{equation}\nT_{eff} = \\left ({3GM \\dot M \\over 8 \\pi r^3 \\sigma_B}\\right )^{0.25},\\label{Teff}\n\\end{equation}\nwhere $\\sigma_B$ is the Stefan-Boltzmann constant. Since BLR is far from the central black hole, the inner boundary effect can be neglected.\n\nWhen we combine Eqs.~\\ref{Bentz}, \\ref{L5100}, and \\ref{Teff}, the dependence on the unknown mass and accretion rate vanishes, and we obtain a single value for all the sources in the sample\n\\begin{equation}\nT_{eff} = 995  \\pm 74 {\\rm K},\n\\end{equation}\nwhere the error reflects the error in the constant in Eq.~\\ref{Bentz}. We stress that the resulting value does not depend either on the black hole mass or accretion rate. The value is universal, for all sources.\n\n\\section{BLR Formation Mechanism}\n\nThe value of the universal temperature of $\\sim 1000$ K immediately hints that the physical mechanism of the BLR\nformation is related to the dust formation in the accretion disk atmosphere. The dust opacity is large, and even \nmoderate radiation flux from the underlying disk can push the dusty material off the disk surface. This dusty \noutflow, however, cannot proceed too far from the disk plane. Lifted material is irradiated by the strong \nUV and X-ray flux from the central region, the material heats up, the dust evaporates, the opacity rapidly decreases,\nand in the absence of the strong radiation pressure the material fall back again onto the disk surface. Therefore,\nsuch a failed wind can account for lifting the material relatively high above the disk plane and at the same time\ndoes not give a strong systematic signatures of outflow since both outflow and inflow are present.\n\nThe overall geometry is shown in Fig.~1. The BLR region starts were the disk effective temperature, without \nirradiation from the central region, falls below \n1000 K, and it ends up at larger distance when the material has the temperature below 1000 K even if the \nirradiation is included. This outer radius marks the transition to the outer dusty\/molecular torus, where the \nirradiation is too weak to \nlead to dust evaporation and the outflow proceeds.  \n\n\n\\section{Application to the Dark Energy Study}\n\nIt was already proposed by Watson et al (2011) that Eq.~2.1 can be used to determine the distance to a quasar in a way independent from the redshift. Reverberation measures the BLR size directly from the time delay, and Eq.~2.1 gives us the intrinsic monochromatic luminosity. This, combined with the observed monochromatic luminosity, gives the luminosity distance to the source. If a number of distant quasars are measured, a luminosity distance vs. redshift diagram can be constructed. Our interpretation of the BLR formation supports the view that Eq.~2.1 can apply to high redshift objects as well, since the mechanism is based on dust formation, this in turn is connected with medium metalicity, and the metalicity of distant quasars is known to be roughly solar.\n\nWith this in mind, we started a monitoring program with the Southern Africal Large Telescope. We choose intermediate quasars and Mg II line which belongs to LIL class, as H$\\beta$, and should not show rapid intrinsic variations characteristic for CIV line.\n\n\\section{Spectroscopy}\n\n\\subsection{Observations and Data Reduction}\n\nIn order to test the feasibility of spectroscopic studies of the corresponding\nobjects we have obtained a pair of medium resolution spectra of the quasar\nLBQS 2113-4538.\nThe observations have been performed using the Robert Stobie Spectrograph\non Southern Africal Large Telescope in the service mode. The observing data\nwas collected in two blocks, in the nights May 15\/16, 2012 (UT) and July 30\/31, 2012 (UT).\nEach block consists of a pair of 978 second exposures of the target spectrum\nfollowed by an exposure of the spectrum of the Argon calibration lamp, and\na set of flat-field images.\n\nInitial data reduction steps: gain correction, cross-talk correction,\noverscan bias subtraction and amplifier mosaicking have been performed\nby the SALT Observatory staff using a semi-automated pipeline\nfrom the SALT PyRAF package.\n\nFlat-field correction and further reduction steps have been performed\nusing procedures within the IRAF \npackage\\footnote{IRAF is distributed by the National Optical Astronomy\nObservatories, which are operated by the Association of Universities\nfor Research in Astronomy, Inc., under cooperative agreement with the\nNSF.}.\nThe pairs of adjacent spectrum images of LBQS 2113-4538 have been\ncombined into a single image. This enabled us to efficiently reject\ncosmic rays and raise the signal to noise ratio.\n\nIdentification of the lines in the calibration lamp spectrum, wavelength\ncalibration, image rectification and extraction of one dimensional spectra\nhave been done using functions from noao.twodspec package within IRAF.\n\n\\subsection{Preliminary results}\n\n\\begin{figure}[t]\n\\begin{center}\n\\hfill~\n \\includegraphics[width=0.4\\textwidth]{czernyb_talk_fig2.eps}\n\\hfill~\n \\caption{Two observations of the Mg II line of the quasar LBQS 2113-4538, separated by $\\sim 100$ days (preliminary results) made with SALT. The shape of the line has clearly changed, and the line intensity has changed by 9 \\%, which allows for a systematic monitoring of this source.}\n\\end{center}\n\\end{figure}\n\nIn Fig. 2 we present the two profiles of MgII line in the observed object.\nThe shape of the line has clearly changed during the $\\sim 75$ days period between the first and the second exposure. The line intensity has changed by 9 \\%. Thess facts allow for a systematic\nmonitoring of this source.\n\nWe now perform Monte Carlo simulations in order to optimize the project.\n\n\\section{Conclusions}\n\nWe propose that dust forms not only in the outer dusty\/molecular torus, but also closer in, in the non-irradiated thin disk atmosphere. This dust is responsible for the formation of the failed wind which accounts for the formation of the Low Ionization Line part of the Broad Line Region.\n\nThis interpretation automatically explains why the size of the BLR scales with the square root of the monochromatic, instead of bolometric, luminosity. It also solves the problem of high turbulence in the region without signatures of the systematic outflow. Since the this mechanism of BLR formation seems universal, and there is no strong metalicity gradients in quasars up to high redshift, this also justifies the use Eq.~2.1 for distant quasars as a way to measure their luminosity distance and probe the dark energy distribution. \n\n{\\underline{\\it Acknowledgements}}. This work was supported in part by grants NN N203 387737, NN 203 580940 and Nr. 719\/N-SALT\/2010\/0. \nAll of the observations reported in this paper were obtained with the Southern African Large Telescope (SALT).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDynamic environments still pose a challenge in robotics. A capability to detect moving objects allows extending results of various studies performed for static scenes to dynamic environments. Although much research such as the background subtraction method has been carried out to recognize moving objects, it is ineffective in scenes where moving objects occupying a major portion of the image,  {(which we refer to as ``moving objects \\textit{dominate} the scene\").} \nSuch situations happen frequently when moving objects are nearby and should be treated as a top priority. \n\n\\begin{figure}[t]\n \\centering\n\\begin{subfigure}{.22\\textwidth}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{figures\/obj000600.png}\n  \\caption{Object detection result}\n  \\label{fig:first_sfig3}\n\\end{subfigure} %\n\\begin{subfigure}{.22\\textwidth}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{figures\/lenna_mask_first.png}\n  \\caption{Object detection result}\n  \\label{fig:first_sfig4}\n\\end{subfigure} \\\\\n\\vspace{0.15cm}\n\\begin{subfigure}{.22\\textwidth}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{figures\/residual_robust.png}\n  \\caption{Residual image from the robust DVO}\n  \\label{fig:first_sfig5}\n\\end{subfigure} %\n\\begin{subfigure}{.22\\textwidth}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{figures\/residual_proposed.png}\n  \\caption{Residual image from the proposed method }\n  \\label{fig:first_sfig6}\n\\end{subfigure} \n\n\\caption{ Results of moving object detection. (a), (b) are the results of detecting moving objects, and (a) contains a large moving object which dominates the scene with a freely moving camera. (c) Typical DVO only approach  reaches a local minimum whereas (d) the proposed method demonstrates that the global optimum to the background is reached.}\n\\label{fig:first}\n\\vspace{-0.7cm}\n\\end{figure}\nTo address the above-mentioned problems arising from dynamic environments, we propose an occlusion accumulation method. The proposed method utilizes both depth and camera pose information obtained from VO to detect moving objects without a particular background model. Some direct VO methods using robust regression weight \\cite{Kerl2013ROE} are not able to track the accurate camera pose by falling into a local minimum when a moving object dominates the scene. In this work, we use the estimated camera pose up to the previous image to distinguish the moving objects and exclude them from the images to find the next camera pose. {If only the robust regression in the optimizer (bi-square regression in this paper) is used, the camera pose is likely to be estimated with respect to the moving object when the moving object dominates the scene. In the proposed method that estimates the camera pose with robust regression, the method excludes the moving object before it dominates the scene. Unlike the other mapping methods where memory usage increases each time an image is processed, the memory requirement of the proposed method is not heavy regardless of a long duration of image sequences, because it continuously updates the single occlusion information. }\n\n\nWe warp the previous depth image using the estimated camera pose and subtract with the current depth image. Accumulating those subtracted values shows a similar result to the background subtraction without relying on background modeling. Our method is robust to the effect of the moving object which dominates the scene and avoids other problems that occur when the background is incorrectly modeled. It is possible to detect a moving object which was originally static and regarded as the background. \n\nThe purpose of our research is to detect moving objects and to improve the performance of arbitrary VO in dynamic environments. Our contributions are summarized as follows.\n\n\\begin{itemize}\n\n\\item  We suggest {a new method of the occlusion accumulation} which can detect moving objects that dominate the scene, without a background model.  \n{\\item Our algorithm can improve the performance of any VO algorithm in a dynamic environment by augmenting it.}\n\\item The proposed algorithm can detect objects which have similar texture to the background and can be further improved by using an externally obtained camera pose.\n\n\n\\end{itemize} \n\n\\section{Related Work}\n\nThere are several types of moving object detection algorithms\n\\cite{Mehran2018NTMOD} using background modeling, trajectory classification, etc. The methods in \\cite{Amitha2015BMBC,EEDMTF,MODTTVCMC,BSMCAG}, belonging to the  category of background modeling, commonly extract feature points and clustered features to distinguish the background and the foreground. They warp the previous image with homography transform attained from features in the dominant cluster.  Such methods show a quite accurate object segmentation result in the scenes dominated by the background, but they assume that the dominant cluster is from a background, and they take a few seconds to process. \n\nIn \\cite{Yi2013DMONC}, the background and foreground are distinguished by a dual-mode single Gaussian model (SGM). The adaptive adjustment of the SGM learning rate reduces computation time, but some moving objects are perceived as background if the background is similar in color to the object.\n\nIn \\cite{BSFMC},  trajectory classification is conducted. They collect points that are tracked for several frames and find the dominant camera motion by applying RANSAC on the trajectories for tracked points. The authors defined conditional probability on trajectory models and applied the maximum a posterior rule to distinguish the points included in the moving object. The authors of \\cite{BSMCBTSLI, OSLTAPT} also propose a trajectory classification method by defining a low-dimensional descriptor for describing trajectory shape or spectral clustering with spatial regularity. They detect the outlier trajectories whose sum of the Euclidean distance to trajectories of the nearest neighbors exceeds a specified threshold or by using K-means clustering. The methods of \\cite{BSFMC, BSMCBTSLI, OSLTAPT} \nrequire an appropriate feature extraction method, and they may show sparse segmentation results due to utilization of the tracked features.\n\nBecause the aforementioned methods aimed to detect moving objects or tracking objects far from the camera, they used datasets where the background dominates the scene such as a surveillance system or sports relay. On the other hand, in robotic applications, because it is important to process in real-time and detect nearby objects which may occupy the scene, these studies can become less effective.\n\nWith the growing popularity of RGB-D cameras such as Kinect and ASUS Xtion PRO, moving object detection algorithms have begun to utilize them. The method in \\cite{MDTSA}  estimates camera ego-motion using depth values, and then it uses a homogeneous transformation to warp the previous frame as in \\cite{Amitha2015BMBC}, \\cite{EEDMTF}, and \\cite{MODTTVCMC}. The segmentation is achieved by  the particle filter and vectorization method. Accurate image warping is performed using depth values, but the problem of objects occupying a major portion of the image is not addressed in these methods. In \\cite{Javed2017OSRPCA}, they calculated a spatiotemporal graph Laplacians and spatial smoothness among the background pixels. Then, they adapt RPCA to incorporate two approaches for classifying foreground from background. Due to the high computational complexity of RPCA, computation time per frame is several seconds.\n\nIn the field of visual odometry, many studies have investigated  camera pose estimation in dynamic environments by reducing the effect of the moving objects. In \\cite{Kerl2013ROE, RRVOD} and \\cite{kerl2012},  the camera pose is estimated by assigning the t-distribution or  Huber norm weights, so that the dense visual odometry (DVO) method performs properly in some dynamic environments. The method of \\cite{Kim2016EBMB} labels the background with a non-parametric model and depth images obtained from the RGB-D camera, and estimates the camera pose by modifying the energy-based DVO. The algorithm would be ineffective when used for object detection, because they focus on labeling the background to estimate pose stably and does not label the moving object.\n\nThe method suggested in \\cite{leereal} performs rigid motion segmentation on RGB-D camera by a grid-based optical flow. The process of flow calculation and spatial, temporal segmentation was conducted in real-time, but the resulting segmentation  \\cite{leereal} was a low resolution that is not as clear as the silhouette-shaped segmentation.\n\nIn \\cite{Jaimez2017FOSF}, the authors propose a robust camera pose estimation in dynamic environments and 3D scene-flow estimation on the moving object by applying depth geometric clustering and checking whether clusters match the camera motion of images. \nThey labeled the foreground and background as probability, and utilize the probability label as a weighing factor on pose estimation to perform well in dynamic environments. The algorithm does not clearly distinguish moving objects and the specified number of clusters would affect the performance. \n\n\n\\section{Moving Object Segmentation}\n Various factors such as illumination change, moving object, camera ego-motion induce image changes. Assuming that the photo-consistency assumption is not violated, in a static scene that involves moving objects with fixed camera, image change indicates the moving objects. In a dynamic scene which has arbitrary camera motion, we can attain a motionless image by warping the previous image with the estimated camera pose. We use the camera pose estimated from DVO with robust regression weight in this paper. We will refer to DVO with robust regression weight as robust DVO in this paper and more details about robust regression weight will be covered in Section \\ref{pose_esti}. After attaining a motionless image, the dynamic scene can be treated as a static scene. We focus on this feature and detailed explanation will follow.\n \n\\subsection{Occlusion Accumulation} \n When a moving object roams in the scene, depth can be changed. The depth change can be classified into two kinds. The first one is occlusion: the object will occlude the background in the head of the direction of movement. The second one is reappearance: the background will appear in the tail of the direction of movement. We assume that the depth of background is larger than that of the moving object. Object which pass the rear of the static background such as pillar, tree, etc. do not affect the depth value. We consider the pixel whose depth value changes to a smaller value as a pixel of the moving object.\n\n The occlusion map and the warping function are defined as\n\\begin{equation} \\label{def_occ}\n    \\Delta Z_{i}(u,\\xi_{i}^{i+1}) = Z_i(\\text{w}(u,\\xi_{i}^{i+1})) - Z_{i+1}(u)\n\\end{equation}\nand\n\\begin{equation} \\label{def_warp}\n    \\text{w}(u,\\xi_i^{i+1}) = \\pi (\\text{exp}(\\xi_i^{i+1})\\pi^{-1}(u,Z_i(u)))\n\\end{equation}\n where $Z_{i}(u)$ is the \\textit{i}th depth image and $u$ is a pixel on image $\\Omega (640 \\times 480)$. $\\xi_i^{i+1} \\in \\mathbb{R}^6 $ represents the 6-DOF camera pose parameter between the \\textit{i}th frame and  the (\\textit{i}+1)th frame. The project function $\\pi: \\mathbb{R}^3 \\xrightarrow{}\\mathbb{R}^2$ maps a 3D point into a 2D image pixel. The term $ \\text{exp}(\\xi_i^{i+1}) \\in \\text{SE}(3) $ represents the transformation matrix for the corresponding camera pose parameter $\\xi_i^{i+1}$.\n\n If $\\Delta Z_{i} > 0$, it means that the depth has become smaller, i.e. the occlusion has occurred. Otherwise, $\\Delta Z_{i} < 0$ means that the depth has increased, i.e. the reappearance. \n We define the occlusion accumulation map by\n\\begin{align} \\label{def_accum}\n   A_{i+1} (u)  &= \\Delta Z_{i}(u,\\xi_{i}^{i+1}) + \\Tilde{A}_{i} (\\text{w}(u,\\xi_{i}^{i+1}))\\nonumber \\\\\n            &\\approx \\Delta Z_{i}(u,\\xi_i^{i+1}) + \\sum_{k=1}^{i} \\Delta Z_{k}(\\text{w}(u,\\xi_{k+1}^{i}),\\xi_k^{k+1})\\nonumber \\\\\n            &\\approx  Z_{k_u}(\\text{w}(u,\\xi_{k_u}^{i+1})) - Z_{i+1}(u)\n\\end{align}\n The initial occlusion accumulation map is set as $A_{1}(u)=0$ for $\\forall u \\in \\Omega$ and $k_u$ is the index of the frame which first observed the pixel $u$ of the current frame. The $\\Tilde{A}(u)$ is truncated $A(u)$ and described later in Eqs.\\eqref{def_accum_re1} and \\eqref{def_accum_re2}, \n\n \n The warped pixels before the index at $k_u$ would be out of the observation window $\\Omega$ and there is nothing to calculate. The value of $\\Delta Z$ would reveal zero values for these indices.\n\n \n \\begin{figure}\n \\centering\n\\begin{subfigure}{.22\\textwidth}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{figures\/lenna1.png}\n  \\caption{Color image}\n  \\label{fig:sfig1}\n\\end{subfigure} %\n\\begin{subfigure}{.22\\textwidth}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{figures\/lenna2.png}\n  \\caption{Masked image}\n  \\label{fig:sfig2}\n\\end{subfigure} \\\\\n\\vspace{0.15cm}\n\\begin{subfigure}{.22\\textwidth}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{figures\/dZdt.png}\n  \\caption{$\\Delta Z(u)$}\n  \\label{fig:sfig3}\n\\end{subfigure} %\n\\begin{subfigure}{.22\\textwidth}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{figures\/Au.png}\n  \\caption{$A(u)$}\n  \\label{fig:sfig4}\n\\end{subfigure}\n\\caption{Illustration of the occlusion accumulation method. $A(u)$ represents the accumulation of $\\Delta Z(u)$ over time. A bright area is a region where $A(u)$ shows a positive value and is considered as a moving object, and a dark area is a region where $A(u)$ shows a negative value and the background is revealed again. The middle gray represents a value of zero. The moving object detection results from Eq.\\ref{def_bg} is shown in (b). The area considered as a moving object is painted purple.}\n\\label{fig:accum}\n\\vspace{-0.5cm}\n\\end{figure}\n \n When the truncation steps for $A(u)$ have not been triggered, the second and third approximate equalities in Eq.\\eqref{def_accum} would be satisfied by $A(u)=\\Tilde{A}(u)$. One can interpret the occlusion accumulation map as the result of subtraction between $k_u$th frame and current frame.  In contrast to the background subtraction, we utilize the occlusion map, $\\Delta Z_{i}(u,\\xi_i^{i+1})$. Thus, our method does not need to depend on recognition of the background or 3D mapping.\n\n Existing depth sensors have the depth error, which quadratically grows along the depth value in \\cite{Jaimez2017FOSF}. To deal with such measurement uncertainty, we set a threshold as \n\\begin{equation} \\label{def_object}\n    \\tau_{\\alpha}(u) = \\alpha \\cdot Z(u)^2\n\\end{equation}\nAlthough we set the threshold to be larger than zero, unwanted error values can exceed the threshold through the accumulation. Furthermore, negative values can be accumulated, so that occlusion could not exceed the threshold at all. In order to reduce their effect, we truncate the occlusion accumulation map $A_i (u)$ to $\\Tilde{A}_i (u)$  by the following:\n\\begin{eqnarray} \\label{def_accum_re1}\n   \\Tilde{A}_i (u)  &=& 0 \\qquad \\text{for  } A_i (u) \\leq \\tau_{\\alpha}(u)\n\\\\ \\label{def_accum_re2}\n   \\Tilde{A}_i (u)  &=& 0 \\qquad \\text{for  } \\Delta Z_{i}(u,\\xi_i^{i+1}) \\leq -\\tau_{\\beta}(u) \\;.\n\\end{eqnarray}\nThe truncation step in Eq.\\eqref{def_accum_re1} could disturb detecting the moving object which slowly approaches toward the camera. However, once the moving object is detected, the small values of $\\Delta Z_{i}(u,\\xi_i^{i+1})$ will be reflected. When the background reappears, $A_i (u)$ would not be lower than the threshold $\\tau_{\\alpha}(u)$ due to depth error accumulation. The truncation step in Eq. \\eqref{def_accum_re2} helps to detect the background reappearance. The moving object that moves away from the camera hardly exceed the threshold. Even so, if it exceeds the threshold, the object would soon fade away in the scene.\nThe result of the occlusion accumulation is shown in Fig. \\ref{fig:accum}. In order to help understanding, $A(u)$ has not been truncated using Eq. \\eqref{def_accum_re1} through every iteration yet. Since the depth value is not measured near the object, a stripe-like result appears in Fig. \\ref{fig:sfig4}.\n\n After calculating $A_i (u)$, we truncate the occlusion accumulation map to distinguish moving objects and background. The background map $B_i (u)$ would reveal 0 if a moving object shows proper occlusion. Otherwise, if the moving object has passed and the background reappears again, $B_i (u)$ would reveal 1, i.e., \n \\begin{equation} \\label{def_bg}\n    B_i (u) = \n\\begin{cases}\n    0              & \\text{if } A_i (u) > \\tau_{\\alpha}(u) \\\\\n    1              & \\text{otherwise}\n\\end{cases}\n\\end{equation}\n The object mask can be calculated by inverting $B_i (u)$. \n\n \n\\subsection{Depth Compensation}\nOften, the depth images have invalid depth on an edge of the object or near the border of an image window. Its effect is shown in Fig. \\ref{fig:sfig4}. The segmentation in Fig. \\ref{fig:sfig2} has empty spaces over the moving object. Some of those invalid depth values in the current frame can be compensated by\n\\begin{equation} \\label{depth_compen}\n   Z_i(\\Bar{u}) = Z_{i-1}(\\text{w}(\\Bar{u},\\xi_{i}^{i-1}))\n\\end{equation}\nwhere the unmeasured depth on the pixel $\\Bar{u}$  satisfies $Z(\\Bar{u})=0$. The compensated depth is used in the next depth image. The result of the depth compensation is shown in Fig.~ \\ref{fig:compen}. The original depth image has the  unmeasured area that is colored as black,  near the border and the boundary of the object. The unmeasured area on boundary of the object is filled with the previous depth image as in Fig. \\ref{fig:com_sfig2}. From the compensated depth image, $A(u)$ is now calculated reasonably (Fig. \\ref{fig:com_sfig3}), so that the object segmentation could cover the entire moving object (Fig. \\ref{fig:com_sfig4}).\n\\begin{figure}\n \\centering\n\\begin{subfigure}{.22\\textwidth}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{figures\/depth_before.png}\n  \\caption{Original depth image}\n  \\label{fig:com_sfig1}\n\\end{subfigure} %\n\\begin{subfigure}{.22\\textwidth}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{figures\/depth_after.png}\n  \\caption{Compensated depth image}\n  \\label{fig:com_sfig2}\n\\end{subfigure} \\\\\n\\vspace{0.15cm}\n\\begin{subfigure}{.22\\textwidth}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{figures\/Au2.png}\n  \\caption{$A(u)$}\n  \\label{fig:com_sfig3}\n\\end{subfigure} %\n\\begin{subfigure}{.22\\textwidth}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{figures\/lenna3.png}\n  \\caption{Masked image}\n  \\label{fig:com_sfig4}\n\\end{subfigure}\n\\caption{Result of depth compensation. A raw depth image (a) is compensated as (b). Compared to Fig.\\ref{fig:first}, the unwanted stripe-like result is corrected.}\n\\label{fig:compen}\n\\vspace{-0.5cm}\n\\end{figure}\n\n\n\\subsection{Occlusion Prediction on Newly Discovered Area}\nSuppose that there is a newly discovered area which has to be classified as the background or the moving object in a dynamic scene. \nThe occlusion map $\\Delta Z_{i}(\\Tilde{u},\\xi_{i-1}^{i})$ cannot be correctly calculated for $\\Tilde{u}$, i.e. the pixels of the newly discovered area, because the warped pixel onto the previous image  $\\text{w}(\\Tilde{u},\\xi_{i}^{i-1})$ is not in the image window $\\Omega$. We define the newly discovered area as\n\\begin{equation} \\label{def_nda}\n    \\Tilde{\\Omega} = \\{\\Tilde{u} | \\text{w}(\\Tilde{u},\\xi_{i}^{i-1})\\not\\in \\Omega \\}\n\\end{equation}\nWe suggest a method of predicting $A(\\Tilde{u})$ {with the fast marching method  as the following:}  \n\\begin{equation} \\label{def_nda_pred}\n    A_i(\\Tilde{u})  = \\frac{\\sum_{\\delta u \\not \\in \\Tilde{\\Omega}} \\left\\{ A_i(\\delta u) + \\nabla_{\\delta u} Z_i (\\Tilde{u})\\right\\} }{\\sum_{\\delta u \\not \\in \\Tilde{\\Omega}}1}\n\\end{equation}\n The symbol $\\delta u$ means the nearest pixels of $\\Tilde{u}$ whose accumulated value $A(\\delta u)$ has been calculated, and $\\nabla_{\\delta u}Z_i (\\Tilde{u}) = Z_i (\\Tilde{u}) - Z_i(\\delta u)$ is the gradient of depth map with respect to $\\delta u$. Because $A(u)$ has correspondence with the depth map, {we predict $A(\\Tilde{u})$ with the gradient of the depth map and the occlusion accumulation map for the nearest pixel $\\delta u$ adjoining the known area.} {In other words, we interpolate  $A(\\Tilde{u})$ with the average of $A(\\delta u) + \\nabla_{\\delta u} Z_i (\\Tilde{u})$.} After that, we update the newly discovered area by $\\Tilde{\\Omega} \\xleftarrow{} \\Tilde{\\Omega}-\\{\\delta u\\}$, and repeat Eq. \\eqref{def_nda_pred} until $\\Tilde{\\Omega} = \\phi$. This process operates on two consecutive frames. Since the effect of the newly discovered area is not noticeable in two consecutive frames, we compared the masked image with some frame interval in Fig. \\ref{fig:com_sfig4}.\n\nIn the positive area of the predicted $A(u)$ which is considered as a moving object, the background depth has never been detected, because the background was continuously occluded by the moving object until the current image.\n The algorithm could fail to recognize such appearance of the background. Even if the background appears, $A(u)$ may not be lower than the threshold in Eq. \\eqref{def_bg}. Then the truncation of Eq. \\eqref{def_accum_re2} is triggered so the background recognition works properly. The result of the prediction is depicted in Fig. \\ref{fig:newly}.\nWhen there is no nearest moving object outside the predicted area, we remove $A(u)$ of the predicted area to prevent error propagation. Also, if there are small-sized moving object labels, we suppress them as they are usually negligible. \n\n\\begin{figure}\n \\centering\n\\begin{subfigure}{.22\\textwidth}\n  \\centering\n  \\captionsetup{justification=centering}\n  \\includegraphics[width=.9\\textwidth]{figures\/newly1.png}\n  \\caption{Masked image \\newline (few frames ago)}\n\\end{subfigure} %\n\\begin{subfigure}{.22\\textwidth}\n  \\centering\n  \\captionsetup{justification=centering}\n  \\includegraphics[width=.9\\textwidth]{figures\/newly2.png}\n  \\caption{Masked image \\newline (current frame)}\n\\end{subfigure} \\\\\n\\begin{subfigure}{.22\\textwidth}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{figures\/newly1A.png}\n  \\caption{$A(u)$ without prediction}\n\\end{subfigure} %\n\\begin{subfigure}{.22\\textwidth}\n  \\centering\n  \\includegraphics[width=.9\\textwidth]{figures\/newly2A.png}\n  \\caption{$A(u)$ with prediction}\n\\end{subfigure}\n\n\\caption{Example of occlusion prediction on the newly discovered area. When a new area is detected while the camera moves, there is no depth information in a few frames ago. Occlusion map has zero value on such area.  (c) shows the effect of the unmeasured depth values due to a newly discovered area. After executing  the occlusion prediction method, $A(u)$ has proper occlusion map values on the newly discovered area shown in (d) and the moving object detection result obtained from (d) is shown in (b).}\n\\label{fig:newly}\n\\vspace{-0.5cm}\n\\end{figure}\n\n\\begin{figure*}[!ht]\n\\centering\n\\includegraphics[width=0.98\\textwidth]{figures\/segmentation_result.png}\n\\caption{Segmentation result for tested sequences in the order shown in Table.\\ref{table:obj&VO}. The first row shows the masked image and the second row represents the ground truth. The result of the proposed method with VO and external pose estimation is presented orderly in the third and fourth rows {(except the TUM dataset for the reasons described in text)}. Since the depth is not measured on the boundary of the image, moving objects are not labeled in those areas.}\n\\label{fig:result}\n\\vspace{-0.5cm}\n\\end{figure*}\n\n\\section{Robust Pose Estimation} \\label{pose_esti}\n\nIn this paper, we use DVO for camera pose measurement. Initially, we estimate the camera motion using only the sequence of RGB-D images. After detecting moving objects, the camera motion is estimated considering background mask $B(u)$.\nThe method using robust regression \\cite{Kerl2013ROE} effectively estimate ego-motion over a small region of moving object, but it reaches a local minimum solution when the region of moving object is large as shown in Fig.\\ref{fig:first}, because the background is confused with the moving object. On the other hand, the remaining moving object area after excluding the moving object detected up to the current frame is small enough to find upright camera pose. Thus, our method does not confuse the background and the moving object thanks to the accurate camera pose. \n The camera pose estimation is achieved by minimizing the cost function as shown in Eq. \\eqref{def_obj_fuc}. In the cost function, we use the bi-square norm shown in Eq. \\eqref{def_bisquare} which completely ignores the effect of the outliers.\n{ This property makes it possible to obtain the camera pose $\\xi$ based on background pixels which have a relatively small residual.}\n\n\\begin{equation} \\label{def_obj_fuc}\n    \\vspace{-0.25cm}\n    \\xi_i^{i+1} = \\operatorname*{argmin}_{\\xi}\\sum_{u\\in\\Omega}  B_i (\\text{w}(u,\\xi)) \n    J_i(u,\\xi)\n    \\vspace{-0.25cm}\n\\end{equation}\nwhere\n\\begin{eqnarray} \\label{def_residual}\n    J_i(u,\\xi) = \\rho_{k_I}( \\Delta I_{i}(u,\\xi)) + \\gamma \\cdot \\rho_{k_Z}( \\Delta Z_{i}(u,\\xi)) &&\n   \n\\\\ \n\\label{def_bisquare}\n    \\rho_{k}(e) =\n    \\begin{cases}\n        \\frac{k^2}{6}\\left\\{ 1-\\left[ 1 - (\\frac{e}{k})^2 \\right]^3 \\right\\}  & \\text{for } |e| \\leq k \\\\\n        \\frac{k^2}{6}              & \\text{for } |e| > k\n    \\end{cases}&&\n\\end{eqnarray}\n\n\nThe symbol $I_i$ in Eq. \\eqref{def_residual} is the $i$th color image and $k$  is the user-defined bi-square threshold. The residual values $\\Delta I$ and $\\Delta Z$ that are larger than $k$ do not influence the optimization process. In this paper, we use the levenberg-marquardt optimizer, and the parameter values $k_I = 48\/255$, $k_Z = 0.5$, and $\\gamma = 0.001$ are used.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Experimental Results}\n\nTo evaluate our algorithm, we need RGB-D datasets in the presence of camera motion, ground truth data for moving objects, and camera trajectory. The dynamic object sequences of TUM dataset \\cite{TUMdataset} contain the ground truth camera pose and RGB-D images, but the ground truth segmentation for moving object was not provided because the moving object such as a sitting person whose arm is moving could not be segmented clearly.\nThus, we used TUM dataset only to evaluate relative pose error. Additionally, we evaluate the performance of the proposed moving object detection and the pose estimation method with the dataset in \\cite{leereal} and our dataset. \nThe data sequences from \\cite{leereal} are named with the prefix `ICSL' in \\cref{table:obj&VO}.\n\n\nWe collected dataset with an RGB-D camera of Kinect V1, and obtained the ground truth pose of the camera using VICON. In addition, we manually obtained the ground truth segmentation of moving objects in pixel-wisely.\n\nIt contains 5 sequences for static camera, whose name starts with `$static$' and 4 sequences for the dynamic environment whose name starts with `$dynamic$' where camera motion and moving objects exist simultaneously. \nThe sequences `$static\\_tree$',`$static\\_man$'  respectively contain  a tree and a man as a moving object. The sequence of `$static\\_board$' contains a large-sized moving object which could be misrecognized as the background. In `$static\\_destruct$', `$static\\_construct$' sequences, there are situations where a building-shaped object is brought in and then removed. These sequences induce background model changes. In the sequence `$dynamic\\_toss$', two people toss a doll to each other. It shows that multiple objects and fast objects can also be detected with our algorithm.\n{The dataset and detailed information can be found here:}\n\n\\url{https:\/\/haram-kim.github.io\/LARR-RGB-D-datasets\/}\n\n\\begin{table*}[t]\n\\vspace{+0.3cm}\n\\caption{Object segmentation result (F1-score) \\& relative pose error (RMSE)}\n\\vspace{-0.2cm}\n\\label{table:obj&VO}\n\n\\centering\n\\begin{tabular}{p{32mm} || C{10mm} | C{10mm} | C{10mm} | C{10mm} || C{8mm} | C{8mm} | C{8mm} | C{8mm} | C{8mm} | C{8mm}}\n\\toprule\n\\multirow{2}{*}{Sequences} & \\multirow{2}{*}{\\cite{MODTTVCMC}} & \\multirow{2}{*}{\\cite{MDTSA}}& \\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Proposed\\\\ method\\end{tabular}}& \\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Ours with\\\\ ext. pose\\end{tabular}}&\n\\multicolumn{2}{M{2cm}|}{Robust DVO}&\n\\multicolumn{2}{M{2cm}|}{Proposed method}& \n\\multicolumn{2}{M{2cm}}{Joint VO-SF \\cite{Jaimez2017FOSF}}\\\\ \\cline{6-11}\n& &\n& &\n&{tr.(m)} & {rot.($\\tcdegree$)} \n&{tr.(m)} & {rot.($\\tcdegree$)}\n&{tr.(m)} & {rot.($\\tcdegree$)}\\\\\n\\thickhline\n\nstatic\\_tree        & \\textbf{0.9385}    & 0.8873     & 0.8821      &0.8892 \n& 0.102    & 3.313   & 0.061    & \\textbf{1.856}    & \\textbf{0.032}    & 3.969\\\\\n\nstatic\\_board       & 0.9044    & \\textbf{0.9256}     & 0.9182      &0.9177\n&$\\times$   &$\\times$  & \\textbf{0.292}    & \\textbf{4.839}    & 0.584    & 10.916\\\\\n\nstatic\\_man         & 0.7350    & \\textbf{0.8975}     & 0.8739      &0.8755\n& 0.280    & 4.024   & 0.173    & 4.944    & \\textbf{0.072}    & \\textbf{1.045}\\\\\n\nstatic\\_destruct    & 0.5267    & 0.5565     & 0.8391      &\\textbf{0.9038}& 0.535    & 4.818   & 0.336    & 4.495    & \\textbf{0.131}    & \\textbf{3.014}\\\\\n\nstatic\\_construct   & 0.3948    & 0.8362     & \\textbf{0.8868}      &0.8396\n& 0.809    & 10.433   & 0.153    & 3.821    & \\textbf{0.078}    & \\textbf{1.272}\\\\\n\nICSL\\_place\\_items    & 0.7139    & 0.6494     & 0.7204      & \\textbf{0.7366}\n& 0.041    & 1.082     & \\textbf{0.030}      & \\textbf{0.893}    & 0.094      & 5.379\\\\\n\nICSL\\_two\\_objects    & 0.5073    & 0.8256     & \\textbf{0.8583}      & 0.8580\n& 0.170    & 6.493     & \\textbf{0.013}      & \\textbf{0.344}    & - & - \\\\\n\n\\hline\ndynamic\\_board      & $\\times$    & 0.6527     & \\textbf{0.9264}    &{0.9247}\n&$\\times$   &$\\times$  & \\textbf{0.111}    & \\textbf{1.939}   & 0.135  & 4.644\\\\\n\ndynamic\\_man1   & $\\times$    & 0.4704     & {0.8123}    &\\textbf{0.8287}\n& 0.255    & 9.027   & \\textbf{0.157}    & \\textbf{4.108}   & 0.223    & 12.949\\\\\n\ndynamic\\_man2   & $\\times$    & $\\times$   & \\textbf{0.8975}    &0.8955\n& 0.646    & 11.286   & \\textbf{0.166}    & \\textbf{2.165}   & 0.206    & 6.180\\\\\n\ndynamic\\_toss   & $\\times$      & 0.4395     & {0.7259}    &\\textbf{0.7975}\n& 0.635    & 8.546   & \\textbf{0.324}    & \\textbf{2.312}   & 0.378    & 4.028\\\\\n\nICSL\\_fast\\_object    & $\\times$    & $\\times$     & 0.8639      & \\textbf{0.8779}\n& 0.318    & 13.543     & \\textbf{0.092}      & \\textbf{3.603}    & -   & - \\\\\n\nICSL\\_slow\\_object    & $\\times$    & 0.7424     & 0.8971      & \\textbf{0.9142}\n& 0.407    & 17.585     & \\textbf{0.091}      & \\textbf{3.779}    & -   & -\\\\\n\nTUM\\_sitting\\_static    & & & &\n& 0.037    & 0.972     & \\textbf{0.035}      & \\textbf{0.961}    & 0.045      & 1.699 \\\\\n\nTUM\\_sitting\\_xyz     & & & &\n& 0.078    & 2.027     & \\textbf{0.073}      & \\textbf{1.860}    & 0.205      & 4.066\\\\\n\nTUM\\_walking\\_static    & & & &\n& 0.370    & 2.581     & \\textbf{0.217}      & \\textbf{0.197}    & 0.249      & 4.173\\\\\n\nTUM\\_walking\\_xyz       & & & &\n& 0.974    & 15.871     & \\textbf{0.259}      & \\textbf{4.069}    & 0.659      & 12.370 \\\\\n\n\\bottomrule\n\n\\end{tabular}\n\\vspace{-0.6cm}\n\\end{table*}\n\nThe results of the object segmentation are shown in Table \\ref{table:obj&VO} and Fig. \\ref{fig:result}. We used the F1-score as a criterion for the moving object classification evaluation. The precision refers to how many of the pixels identified as moving objects are ground truth moving objects pixels and the recall refers to how many of the moving objects pixels in the image were correctly identified as moving objects. The F1-score is the harmonic mean of the precision and the recall. We calculated F1-score for each frame, and we averaged it in each sequence. The closer the score is to 1, the better the performance of moving object segmentation is.\nWe use the symbol `$\\times$' when quantitative evaluation is meaningless due to extremely poor performance, {and  `-' for the algorithm that failed due to very few inliers in the corresponding sequence.}\n\nWe evaluated the segmentation performance of proposed method not only with robust DVO, but also with VICON data as an externally obtained camera pose. {The proposed method with camera pose of VICON data mostly performs better than the method with VO.}\n\nIn general static situations, all the compared algorithms perform well, but the performance of \\cite{MODTTVCMC, MDTSA}  deteriorates in `$static\\_destruct$' and `$static\\_construct$' sequences, where the moving object appear at the beginning or the end of the video. The performance of \\cite{MODTTVCMC} degrades even if the background and the moving object are different in color. The sequences of dynamic environments contain a difficult situation in that some newly discovered area found by the camera is covered by moving objects, thus the background of that area has never been revealed. The algorithms in \\cite{MODTTVCMC, MDTSA} suffer in those situations so that the results were greatly affected.\nSince we do not use the background model and predict $A(u)$ for the newly discovered area, our method distinguished moving objects stably in `$static\\_destruct$', `$static\\_construct$' and other dynamic environment sequences. {In the dataset of \\cite{leereal}, our method correctly distinguishes the moving object from the background which have similar texture.}\n \n\nTo evaluate the performance improvement of the VO when augmented with our algorithm, the translational and rotational relative pose error metric in \\cite{TUMdataset} is adopted. We set the time parameter $\\Delta =  150$, which means that we evaluate the drift per 5 seconds recorded at 30 Hz.\nThe results are shown in Table. \\ref{table:obj&VO}, Fig. \\ref{fig:VO}.\nEven though RMSE is calculated with the frames that do not have a moving object, our algorithm improves the estimation result of the robust DVO significantly {in various datasets, especially in dynamic environments.}\n\nAs a result of applying the joint VO-SF in \\cite{Jaimez2017FOSF}, an algorithm based on DVO, it { mostly } performs better than DVO and shows smaller error than our proposed method in a \\textit{ fixed} camera. For the $static\\_board$ sequence where the moving object is dominant in the scene, the joint VO-SF method show large relative pose error than our method. {Interestingly, the results of $TUM\\_sitting$ sequences show that the bi-square weight method is more robust than the Cauchy weight based method, which can be explained by their difference in outlier rejection intensity.} The proposed method, compared to robust DVO and Joint VO-SF in dynamic environment, performs better in both rotational and translational motion regardless of a moving object which dominates a scene.\n\nAs can be seen in Fig. \\ref{fig:result}, although the moving objects are well separated, error exists due to the lack of texture in the background scene.\nWe expect that more accurate pose estimation in a dynamic environment can be achieved by applying other VO algorithms which show better performance on pose estimation in a static environment. \n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{figures\/VO_dynmaic_man2.png}\n\\vspace{-0.1cm}\n\\caption{Translation estimation results on the sequence `$dynamic\\_ man2$'. The robust DVO fails to estimate the ego-motion due to a moving object. When combined with our algorithm, the performance improves significantly.}\n\\label{fig:VO}\n\\vspace{-0.5cm}\n\\end{figure}\n\n\\section{Conclusions}\nWe proposed a novel moving object detection method which utilizes occlusion accumulation and camera pose instead of a background model. To do this, we also presented the depth compensation on the unmeasured area and the occlusion prediction on the newly discovered area. Our method could detect the moving object which dominates the scene and improved the performance of robust DVO. In future work, we will combine the proposed method for robotic navigation tasks, which is designed to easily integrate with SLAM algorithms or obstacle avoidance algorithms.\n\n\n\n                                 \n                                 \n                                 \n                                 \n                                 \n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nField-theoretic models with polynomial self-interaction are of growing interest in various areas of modern physics, from cosmology and high energy physics to condensed matter theory \\cite{vilenkin01,manton,khare}.\n\nIn $(1+1)$-dimensional models of a real scalar field with a high-order polynomial self-interaction potential, there exist topological solutions of the type of kinks, which can possess tails such that either or both decay as power laws towards the asymptotic states connected by the kink \\cite{khare,lohe}. Properties of kinks with exponential tail asymptotics (e.g., such as those arising as solutions to the sine-Gordon, $\\varphi^4$ or $\\varphi^6$ model) are well-understood \\cite{GaKuPRE,aek01,christov01,GaKuLi,weigel02,GaLeLi,GaLeLiconf,dorey}. In particular, we know a lot about kink-antikink interactions in such models. Meanwhile, interactions of kinks with power-law tails have not been studied in such detail.\n\nThe study of some properties and interactions of plane domain walls in $(2+1)$ and $(3+1)$ dimensional worlds can often be reduced to studying the properties and interactions of kinks in $(1+1)$ dimensions \\cite{GaLiRa,GaLeRaconf,GaKuYadFiz,Lensky,GaKsKuYadFiz01,GaKsKuYadFiz02}.\n\nIn this paper, we show that power-law asymptotics lead to long-range interaction between kinks and antikinks --- specifically, the force of the interaction can decay slowly, as some negative power of kink-antikink separation. This is a crucial difference from the (``classical'') case of kinks with exponential tail asymptotics. In the latter case, the kink-antikink interaction force always decays exponentially. \n\n\\section{A $(1+1)$-dimensional $\\varphi^8$ model featuring kinks with power-law tails}\n\nConsider the $(1+1)$-dimensional $\\varphi^8$ model \\cite{khare,lohe}, given by the Lagrangian field density\n\\begin{equation}\\label{eq:largang}\n\\mathscr{L}=\\frac{1}{2} \\left( \\frac{\\partial\\varphi}{\\partial t} \\right)^2-\\frac{1}{2} \\left( \\frac{\\partial\\varphi}{\\partial x} \\right) ^2-V(\\varphi),\n\\end{equation}\nwhere the potential of self-interaction is\n\\begin{equation}\\label{eq:potential8}\nV(\\varphi)=\\varphi^4(1-\\varphi^2)^2,\n\\end{equation}\nas depicted in figure \\ref{fig:potential}. The potential \\eqref{eq:potential8} has three degenerate minima: $\\tilde{\\varphi}_1=-1$, $\\tilde{\\varphi}_2=0$, and $\\tilde{\\varphi}_3=1$, such that  $V(\\tilde{\\varphi}_1)=V'(\\tilde{\\varphi}_1)=V(\\tilde{\\varphi}_2)=V'(\\tilde{\\varphi}_2)=V(\\tilde{\\varphi}_3)=V'(\\tilde{\\varphi}_3)=0$.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.3]{Potential}\n\\caption{Potential \\eqref{eq:potential8} of the chosen $\\varphi^8$ model.}\n\\label{fig:potential}\n\\end{figure}\n\nDue to the Lorentz invariance of $(1+1)$-dimensional field theories generated by \\eqref{eq:largang} \\cite{manton}, we restrict to static solution without loss of generality. Static solutions, which are called kinks \\cite{manton}, interpolate between neighboring degenerate minima of the potential (vacua states of the model). We use the following notation for kinks: $\\varphi_{(-1,0)}(x)$ denotes the kink connecting the vacua $\\tilde{\\varphi}_1=-1$ and $\\tilde{\\varphi}_2=0$. We can also say that this kink belongs to the ``topological sector'' $(-1,0)$, and similarly for the other kinks of the model.\n\nThe $\\varphi^8$ model with the potential \\eqref{eq:potential8} has two kink solutions, $\\varphi_{(-1,0)}(x)$ and $\\varphi_{(0,1)}(x)$ shown in figures \\ref{fig:kinkU} and \\ref{fig:kinkD}, respectively, both of which exhibit mixed power-law and exponential tail asymptotics. To each kink there is also a corresponding antikink. The kink $\\varphi_{(-1,0)}(x)$ has power-law asymptotics at $x\\to +\\infty$, while the kink $\\varphi_{(0,1)}(x)$ has power-law asymptotics at $x\\to -\\infty$. All kink solutions of the chosen model can easily be obtained in the implicit closed form \\cite{khare,GaLeLi}:\n\\begin{equation}\\label{eq:kinks}\n2\\sqrt2x=-\\frac{2}{\\varphi}+\\ln\\frac{1+\\varphi}{1-\\varphi}.\n\\end{equation}\nFigure \\ref{fig:kinks} illustrates the two distinct kink solutions.\n(The corresponding expressions for antikinks can be obtained from \\eqref{eq:kinks} via the transformation $x\\mapsto -x$.)\nStandard Taylor series expansions reveal that the tail asymptotics of the kinks given by equation \\eqref{eq:kinks} are:\n\\begin{alignat}{2}\n\\varphi_{(-1,0)}(x) &\\sim-1+\\frac{2}{e^2}\\: e^{2\\sqrt{2}\\: x}, &\\qquad x\\to -\\infty.\\label{eq:kink1_asymp_minus}\\\\\n\\varphi_{(-1,0)}(x) &\\sim-\\frac{1}{\\sqrt{2}\\: x}, &\\qquad x\\to +\\infty.\\label{eq:kink1_asymp_plus}\\\\\n\\varphi_{(0,1)}(x) &\\sim-\\frac{1}{\\sqrt{2}\\: x}, &\\qquad x\\to -\\infty.\\label{eq:kink2_asymp_minus}\\\\\n\\varphi_{(0,1)}(x) &\\sim 1-\\frac{2}{e^2}\\: e^{-2\\sqrt{2}\\: x}, &\\qquad x\\to +\\infty.\\label{eq:kink2_asymp_plus}\n\\end{alignat}\nClearly, the kinks $\\varphi_{(-1,0)}(x)$ and $\\varphi_{(0,1)}(x)$ approach the vacuum state $\\tilde{\\varphi}_2=0$ very slowly (as $1\/x$) at $x\\to+\\infty$ and $x\\to-\\infty$, respectively. Meanwhile, the approach to the vacua $\\tilde{\\varphi}_1=-1$ and $\\tilde{\\varphi}_3=1$ is exponential.\n\n\\begin{figure}[h]\n  \\centering\n  \\subfloat[kink $\\varphi_{(0,1)}(x)$]{\\includegraphics[width=0.4\\textwidth]{KinkU}\\label{fig:kinkU}}\n  \\hspace{15mm}\n  \\subfloat[kink $\\varphi_{(-1,0)}(x)$]{\\includegraphics[width=0.4\\textwidth]{KinkD}\\label{fig:kinkD}}\n  \\caption{The two kink solutions \\eqref{eq:kinks} of the chosen $\\varphi^8$ model with potential \\eqref{eq:potential8}.}\n  \\label{fig:kinks}\n\\end{figure}\n\n\\section{Long-range interaction between a  kink and an antikink}\nConsider a static configuration of a kink centered at some point $x=-\\xi$ and an antikink centered at $x=+\\xi$. Then, $\\xi$ is the half-distance between the kink and the antikink. Our goal is to find how the force of kink-antikink interaction, in the chosen $\\varphi^8$ model, depends upon $\\xi$. Here,  by the force of interaction we mean the force produced on the kink by the antikink. To estimate this force, we use two methods: the collective coordinate approximation (see, e.g., \\cite{manton,aek01,GaKuLi,weigel02} and the references therein for details) and Manton's method (see, e.g., \\cite{manton,kks04}  and the references therein for details).\n\n\\subsection{Collective coordinate approximation}\n\nIn order to find the interaction force in the case of power-law tails, we assume the following ansatz for the kink-antikink field configuration:\n\\begin{equation}\\label{eq:ansatz1}\n\\varphi(x;\\xi)=\\varphi_{(-1,0)}(x+\\xi)+\\varphi_{(0,-1)}(x-\\xi),\n\\end{equation}\nas shown in figure \\ref{fig:confD}. By a standard calculation \\cite{aek01}, we find that the effective potential $U_{\\scriptsize\\mbox{eff}}(\\xi)$ and effective force $F(\\xi)$ of the interaction are, respectively,\n\\begin{equation}\\label{eq:U_eff}\nU_{\\scriptsize\\mbox{eff}}(\\xi)=\\int_{-\\infty}^{+\\infty}\\left[\\frac{1}{2}\\left( \\frac{\\partial\\varphi}{\\partial x} \\right)^2 + V(\\varphi)\\right]dx, \\qquad F(\\xi)=\\frac{dU_{\\scriptsize\\mbox{eff}}}{d\\xi}.\n\\end{equation}\nHere, $F(\\xi)$ is the projection of the force onto the $x$-axis. Notice that we do not write the minus sign in front of the derivative of the effective potential because we are calculating the force on the left kink. Thus, a positive value of $-{dU_{\\scriptsize\\mbox{eff}}}\/{d\\xi}$ means that the force directed to the left, i.e.\\ having a negative projection onto the $x$-axis. This corresponds to repulsion between the kink and antikink. Therefore, the second formula in \\eqref{eq:U_eff} is written such that $F(\\xi)>0$ corresponds to attraction, and $F(\\xi)<0$ corresponds to repulsion.\n\nIn figure \\ref{fig:forcepow} we show the dependence $F$ upon $\\xi$ for the field configuration \\eqref{eq:ansatz1}. It is seen that the kink and antikink repel each other, and the force falls off slowly with increasing separation.\n\nLet us now consider the following kink-antikink field configuration:\n\\begin{equation}\\label{eq:ansatz2}\n\\varphi(x,\\xi)=\\varphi_{(0,1)}(x+\\xi)+\\varphi_{(1,0)}(x-\\xi)-1,\n\\end{equation}\nwhich is shown in figure \\ref{fig:confU}. Again, we seek to determine $F(\\xi)$. In this case, the kink and antikink are turned to each other by the exponential tails. The numerically calculated force of interaction is presented in figure \\ref{fig:forceexp}. From this figure, it is seen that the kink and antikink attract, and the force falls off quickly with increasing separation.\n\n\\begin{figure}[h]\n  \\centering\n  \\subfloat[configuration \\eqref{eq:ansatz1} for $\\xi=15$]{\\includegraphics[width=0.4\\textwidth]{ConfigurationD}\\label{fig:confD}}\n  \\hspace{15mm}\n  \\subfloat[configuration \\eqref{eq:ansatz2} for $\\xi=10$]{\\includegraphics[width=0.4\\textwidth]{ConfigurationU}\\label{fig:confU}}\n  \\caption{Kink-antikink configurations given by (a) equation \\eqref{eq:ansatz1} and (b) equation \\eqref{eq:ansatz2}.}\n\\end{figure}\n\n\\subsection{Manton's method}\n\nWithin the framework of Manton's method \\cite{manton}, the force on the kink is given by the time derivative of the momentum on the semi-infinite interval $-\\infty<x\\leq 0$. At large $\\xi$ this method allows us to use the tail asymptotics for the kink and antikink, which were given in \\eqref{eq:kink1_asymp_minus}--\\eqref{eq:kink2_asymp_plus}, to approximate the integrands in the various integrals with respect to $x$.\n\nSpecifically, using the tail asymptotics  \\eqref{eq:kink1_asymp_plus} of the kink and the tail asymptotics of the corresponding antikink ($x\\mapsto-x$), we estimate the force of repulsion for the configuration \\eqref{eq:ansatz1} (i.e., the kink and antikink are turned to each other by the power-law tails) to be \n\\begin{equation}\\label{eq:force_power}\nF_\\mathrm{M}(\\xi) \\sim \\frac{4}{\\xi^4},\\qquad \\xi \\gg 1.\n\\end{equation}\nIn the case that the kink and antikink are turned to each other by their exponential tails, i.e.\\ the configuration \\eqref{eq:ansatz2}, the force of attraction is estimated to be\n\\begin{equation}\\label{eq:force_exp}\nF_\\mathrm{M}(\\xi) \\sim \\frac{64}{e^4} e^{-4\\sqrt{2}\\:\\xi},\\qquad \\xi \\gg 1.\n\\end{equation}\n\nThe predictions of \\eqref{eq:force_power} and \\eqref{eq:force_exp} (Manton's method) are compared to the corresponding result from \\eqref{eq:U_eff}, as shown in figure \\ref{fig:forces}. While in the case of exponential tail asymptotics (figure \\ref{fig:forceexp}) we observe good agreement, the case of power-law tails (figure \\ref{fig:forcepow}) shows considerable disagreement between the two curves. Nevertheless, the force of interaction for the case of power-law tails calculated from the collective coordinate approximation does appear to decay as $\\xi$ to a negative integer power close to $-4$ (as predicted by \\eqref{eq:force_power}), however the prefactor is off by orders of magnitude. We intend to investigate the origin of this discrepancy, and whether it is a fundamental limitation of Manton's method, in future work.\n\n\\begin{figure}[h]\n  \\centering\n  \\subfloat[force of interaction for configuration \\eqref{eq:ansatz1}]{\\includegraphics[scale=0.4]{ForcePow}\n\\label{fig:forcepow}}\n  \\hspace{15mm}\n  \\subfloat[force of interaction for configuration \\eqref{eq:ansatz2}]{\\includegraphics[scale=0.4]{ForceExp}\\label{fig:forceexp}}\n  \\caption{The force of the interaction $F(\\xi)$ between a kink and an antikink, as a function of their half-separation $\\xi$, for (a) the configuration \\eqref{eq:ansatz1} (power-law tails) and (b) the configuration \\eqref{eq:ansatz2} (exponential tails). The solid curves correspond to $F$ estimated by the collective coordinate approach (i.e., equation \\eqref{eq:U_eff}), while the dashed curves correspond to $F$ estimated by Manton's method (i.e., equations \\eqref{eq:force_power} and \\eqref{eq:force_exp}).}\n  \\label{fig:forces}\n\\end{figure}\n\n\\section{Conclusion}\n\nWithin a specific $\\varphi^8$ model, we have shown that if a kink and an antikink interact via tails that decay as power-laws, then a long-range interaction appears in the system --- the force of the kink-antikink interaction decays much more slowly than in the (``usual'') case of  exponentially decaying tails. \n\nUsing Manton's method \\cite{manton}, we have calculated the asymptotic dependence of the force of interaction on the half-distance $\\xi$ between the kink and antikink. In the case of power-law tails, the force decays as $\\xi^{-4}$ for $\\xi\\gg 1$. Meanwhile, for the case of exponential tails, this force decays off exponentially as $ e^{-4\\sqrt{2}\\xi}$ for $\\xi\\gg 1$.\n\nFurthermore, we calculated the force between the kink and antikink numerically using the collective coordinate approximation. The results obtained by both methods are in a good agreement in the case of exponential tails. The origin of the discrepancy in the case of power-law tails will be investigated in future work.\n\nFinally, note that understanding the long-range interactions of topological defects with power-law tails is important because such long-range interactions between kinks and antikinks can have key consequences for the dynamics of domain walls and other similar structures in field-theoretical models with polynomial self-interaction.\n\n\n\\section*{Acknowledgments}\nThis work was performed within the framework of the Center of Fundamental Research and Particle Physics supported by MEPhI Academic Excellence Project (contract No.~02.03.21.0005, 27.08.2013).\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe broad-line region (BLR) of quasars contains dense ($n_{\\rm{H}} \\approx 10^{9-14}$ cm$^{-3}$) and high-temperature (T $\\sim$ 10$^4$ K) gas \\citep[e.g.][]{Peterson_2006}, which is in close proximity to the supermassive black hole and photoionised by radiation from the accretion disk. Emission lines originating from the BLR can be used as virial estimators of the black hole mass \\citep[e.g][]{vestergaard_2002, Mclure_2004, Greene_2005, shen_biases_2008, Onken_2008, Vestergaard_2009}, as well as to infer chemical abundances of the gas around black holes \\citep[e.g.][]{hamann1992, Hamann_1999, Hamann_2002, Dietrich_2003, Nagao_2006, Matsuoka_2011, Wang_2012, shin_2017_outflow, Xu2018, Wang_2021}. The tight correlation between the supermassive black hole (SMBH) mass and the galactic bulge mass \\citep[the $\\rm{M}_{\\rm{BH}}-\\rm{M}_{\\rm{bulge}}$ relation;][]{1998AJ....115.2285M, 2003ApJ...589L..21M, 2004ApJ...604L..89H, Greene_2010} as well as the velocity dispersion of the galactic bulge \\citep[the $\\rm{M}_{\\rm{BH}}-\\sigma$ relation; ][]{Ferrarese2000, Gebhardt2000, Tremaine2002, Salviander_2013} suggests that host galaxies and their central SMBH co-evolve over cosmic time. High-redshift quasars provide an opportunity to understand the formation and evolution of the earliest galaxies and their supermassive black holes. The galaxy stellar mass - gas phase metallicity relationship (MZR) \\citep[e.g.][]{Maiolino_2008, Dave_2017_mufasa, Curti_2019_Klever, Maiolino_2019_metallica, Sanders_2021_MOSDEF} combined with the M$_{\\rm{BH}}$\/M$_{\\rm{host}}$ ratio \\citep[e.g.][]{Targett_2012} and a relationship between the quasar BLR metallicity with black hole mass (Z$_{\\rm{BLR}}$ - M$_{\\rm{BH}}$) can be used to link the mass or metallicity of the central black hole to metallicity in the host galaxy, enabling an investigation of their co-evolution in the young (< 1 Gyr) universe \\citep[e.g.][]{hamann_1993}.\n\nThe study of quasar BLR metallicity is strongly motivated by the relationship between quasar activity, host galaxy evolution, and star formation episodes \\citep[e.g.][]{Hamann_1999}. Elemental abundances in the BLR are indicative of the chemical evolution in galactic nuclear material. Early investigations of BLR metallicity using highly ionised ions of C, N, and O among others, indicated solar or super-solar metallicity with high associated uncertainties \\citep[e.g.][]{baldwin1978, shields1976}. Given the degeneracies involved with fitting these broad emission features, high signal-to-noise ratio (SNR) spectra or stacking spectra into high SNR composites is needed to accurately fit these lines. Considerable progress has been made since then, pushing towards higher redshift \\citep[e.g.][]{Pentericci_2002} and investigating chemical enrichment history. Photoionisation models suggest that rest-frame UV line flux ratios, such as (\\siiv+\\oiv)\/\\civ\\ and \\nv\/\\civ, can be used to infer metallicity in the BLR \\citep{Hamann_2002, Nagao_2006}. These high-ionisation lines are associated to a region closer to the nuclear engine of the black hole than low-ionisation lines as indicated by reverberation mapping experiments \\citep{Collin_1986, Collin_Souffrin_1988, 1991ApJ...366...64C, Korista_1995, williams2020}. Studies utilizing these metallicity-sensitive lines found that quasar metallicity correlates with luminosity \\citep[e.g.][]{hamann_1993, Dietrich_2003, Nagao_2006, Xu2018}, and outflow strength or velocity \\citep[e.g.][]{Wang_2012, shin_2017_outflow, metal_density}. Additional studies found that metallicity correlates with black hole mass \\citep[e.g][]{Matsuoka_2011, Xu2018, Wang_2021}, possibly pointing to a more fundamental relationship between black hole mass and the metallicity of its BLR. \n\nSimultaneously, there is no evidence to suggest that the same line ratios evolve with redshift \\citep[e.g.][]{Pentericci_2002, metallicity_distant, Xu2018} up to redshifts as high as $z = 7.64$ \\citep{Onoue_2020, yang2021probing}. Such studies consistently estimate metallicities several times the solar value $Z \\sim 5 Z_{\\rm{\\odot}}$, up to $Z > 10 Z_{\\rm{\\odot}}$ in some quasars \\cite[e.g.][]{metallicity_distant}. High redshift quasars with super-solar metallicities suggest rapid chemical enrichment scenarios. Under this paradigm, the nuclei of the most massive galaxies in the early universe were enriched rapidly from the host galaxy's interstellar medium within $\\sim 500\\, \\rm{Myr}$ from the formation of the first stars. From then on, observations suggest that the metallicity of the quasar BLR did not change appreciably for a significant span of cosmic time.\n\nIt has also been suggested that the diversity of high-ionisation emission line ratios measured across a wide range of black hole masses and luminosities can be attributed, in whole or in part, to gas emission from at least two distinct regions of differing densities, illuminated by different ionizing radiation \\citep{sameshima2017, metal_density}. This model does not necessarily require the metallicity in the BLR to vary across the quasar population in order to account for the observed differences in emission line properties. These studies indicate that inferences on quasar chemical enrichment history utilizing emission-line flux ratios have to account for variations in the physical conditions of the emitting gas. \n\nIn this paper, we study spectra of 25 high-redshift (z > 5.8) quasars taken with ESO's VLT\/X-shooter and Gemini-N\/GNIRS. The bulk of our sample consists of high-resolution and high SNR spectra from the ESO-VLT X-shooter Large Program XQR-30 (P.I. V. D'Odorico). We fit and investigate flux ratios of metallicity-sensitive lines (primarily \\nv\/\\civ\\ and (\\siiv+\\oiv)\/\\civ) using individual quasar spectra and composites binned by black hole mass, bolometric luminosity, and blueshift of the C\\,\\textsc{iv}\\ line to determine whether these parameters are correlated with metallicity in the BLR. This paper follows closely other studies at lower redshift \\citep[e.g.][]{Nagao_2006, Xu2018, Shin_2019} and high redshift studies based on smaller samples \\citep[e.g.][]{Jiang_2007, metallicity_distant, DeRosa_2014, JJTang_2019, Onoue_2020, Wang_2021}, many of which report measurements of the same metallicity indicators we use. Compared to a recent study of 33 $z\\sim 6$ quasars observed with Gemini-N\/GNIRS \\citep{Wang_2021}, our sample contains higher SNR and spectral resolution spectra from X-shooter. Additionally, we consider the effects of BLR outflow on the metallicity-sensitive flux ratios, where the outflow is measured by the blueshift of the C\\,\\textsc{iv}\\ emission line \\citep[e.g.][]{Sulentic_2000, baskin_laor_2005, Vietri_2018}.\n\nThe content of this paper is organized as follows: in Section \\ref{sec:sample-composites}, we describe the properties of our high-redshift quasar sample, data reduction, spectrum processing, and the methodology for generating composites. In Section \\ref{sec:metallicity}, we describe our approach to fitting metallicity-sensitive emission lines and the conversion from line ratios to metallicities in the BLR. We present the results for our high redshift sample in Section \\ref{sec:results}, and in Section \\ref{sec:discussion}, we discuss and contextualize the results, presenting correlations found between the properties of the high redshift quasars and the metallicity of their BLRs. We summarize and conclude in Section \\ref{sec:conclusion}. Throughout the paper, we adopt flat $\\Lambda$CDM cosmology with H$_{0} = 70$ km s$^{-1}$ Mpc$^{-1}$ and $\\left(\\Omega_{\\rm m}, \\Omega_{\\Lambda}\\right) = \\left(0.3, 0.7\\right)$. All referenced wavelengths of emission lines are measured in vacuum.\n\n\n\\section{Quasar Sample and Composites} \\label{sec:sample-composites}\n\\subsection{Sample Selection} \nThe bulk of the sample originates from quasars in the ESO-VLT X-shooter Large Program XQR-30 (P.I. V. D'Odorico, program number 1103.A-0817)\\footnote{Collaboration website: \\href{https:\/\/xqr30.inaf.it\/}{https:\/\/xqr30.inaf.it\/}}. The XQR-30\nprogram targets 30 southern hemisphere bright QSOs at $5.8 < z < 6.6$ to study the universe in its infancy. These quasars have virially estimated BH masses of $(0.8 - 6.0) \\times 10^9\\, \\rm{M}_{\\odot}$ (Mazzucchelli et al. in prep.). At lower BH masses, $0.2 - 1.0 \\times 10^9\\, \\rm{M}_{\\odot}$, we include 1 quasar spectrum from \\cite{Shen_2019} and 9 spectra from \\cite{yang2021probing}, all of them taken with Gemini\/N GNIRS. From these other samples, we only considered spectra covering quasar properties outside the range of XQR-30 quasars with SNR > 5 per resolution element near rest-frame 1600\\AA\\ and 2800\\AA, which are in the proximity of the emission lines of interest. Their redshifts span a similar range from $z = 6.0$ to $z = 6.8$ with one quasar at $z = 7.54$ \\citep[i.e.][]{Onoue_2020}. All redshifts are measured from the peak of the best fit models to the Mg\\,\\textsc{ii}\\ emission line, where we use the complete reconstructed line profile in case of multi-component fits. We provide some details of the Mg\\,\\textsc{ii}\\ fits in Section \\ref{sec:emission-fitting} and we leave the complete discussion for Mazzucchelli et al. in prep.\n\nOf the 30 quasars in XQR-30, 14 are classified as quasars with broad absorption-lines (BAL) and 16 are considered non-BAL \\citep{Bischetti_2022_submitted}. We exclude quasars with BAL features as they introduce additional uncertainty in the measurement of line flux. From the XQR-30 non-BAL sample, we reject J1535+1943 by visual inspection due to its dust-reddened continuum in the wavelength regions of interest \\citep{yang2021probing}. Such a continuum is not well-modeled by the continuum fitting method we describe in Section \\ref{sec:emission-fitting} and it would affect the continuum fit if included in composites. Combined with 10 spectra from Gemini GNIRS, a total of 25 high-redshift quasar spectra are included in this study. Figure \\ref{fig:quasar_properties} and Table~\\ref{tab:quasar_sample} show the distribution of the quasar sample and physical properties.\n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{figs\/quasar_properties.png}\n    \\caption{Stacked distribution of quasar properties in our sample from left to right: redshift, quasar bolometric luminosity, black hole mass, C\\,\\textsc{iv}\\ blueshift. The properties of the XQR-30 sample, observed with ESO's X-shooter, are highlighted in blue while the GNIRS spectra are presented in orange. The inner bin edges of composites by quasar bolometric luminosity, black hole mass, and C\\,\\textsc{iv}\\ blueshift are delineated by black dashed lines.}\n    \\label{fig:quasar_properties}\n\\end{figure*}\n\n\\subsection{Data Reduction and Post-Processing}\nThe data reduction procedure for GNIRS spectra is described in full in the source papers: \\citet{Shen_2019} and \\citet{yang2021probing}. \\citet{Shen_2019} used a combination of the PyRAF-based \\texttt{XDGNIRS} \\citep{XDGNIRS_eg} and the IDL-based \\texttt{XIDL} package while \\citet{yang2021probing} used the Python-based spectroscopic data reduction pipeline \\texttt{PypeIt} \\citep{PypeIt}.\n\nXQR-30 data {are} reduced with an improved version of the flexible custom IDL-based pipeline used with data from the XQ-100 legacy survey \\citep{XQ-100, Becker_2019}. The overall strategy is based on techniques described in \\citet{Kelson_2003} with optimal sky subtraction, telluric absorption correction, optimal extraction, and direct combination of exposures. The pipeline is described in additional detail in \\citet{Becker_2012} and it has also been used in other studies based on XQR-30 data \\citep[e.g.][]{Zhu_2021}. X-shooter data {are} obtained using three arms with the following wavelength ranges: UVB (300-559.5 nm), VIS (559.5-1024 nm), and NIR (1024-2480 nm). We extract data from only the VIS and NIR arms because there is no light in the UVB for our sources.\n\nIn the 50 km s$^{-1}$ rebinned quasar spectra from the XQR-30 sample, the mean SNR per pixel measured in the range 1400-1600\\AA\\ in the rest frame, is $\\sim30$ with a minimum of 24 and maximum of 38, while the mean SNR per pixel for the GNIRS sample is $\\sim13$, ranging between 6 and 30. Median pixel widths are 0.25\\AA\\ for the rebinned XQR-30 spectra and 0.43\\AA\\ for GNIRS spectra between rest-frame 1400-1600\\AA. The XQR-30 SNR reported here can be different from those of other studies based on XQR-30 data because of differences in binning strategies and wavelength region over which the SNR is measured. \n\nAfter data reduction, each spectrum undergoes a common post-processing procedure described as follows:\n\\begin{enumerate}\n    \\item For every reduced spectrum in our quasar sample prior to creating composite spectra, the data {are} restricted to relatively high SNR. The per-pixel SNR floor is 1 for GNIRS spectra and 5 for XQR-30 spectra. Data restricted by the SNR floor are omitted from further processing and fitting.\n    \\item We then apply a sigma-clip mask with a box width of {30 pixels}, and a 3-$\\sigma$ threshold to remove narrow absorption features and noise above 3-$\\sigma$. These absorption features are not desired when fitting the intrinsic flux and profile of the broad emission lines. {For the noisier and lower resolution GNIRS spectra, the sigma-clip mask has a minimal effect on the resulting spectra.}\n    \\item As the spectra are observed with different instruments and exhibit a diversity of redshifts, we standardize the rest-frame wavelength domain for all of the spectra, facilitating the stacking of composites later. Every spectrum is resampled using a flux-conserving algorithm into a common wavelength domain with 1 \\AA\\ bins in the rest-frame. The resampling calculation and error propagation are described in detail in \\citet{Carnall_2017}. The number of pixels per 1 \\AA\\ bin in the raw spectra is wavelength-dependent, ranging from 1-3 pixels per bin for GNIRS spectra and 2-8 for X-shooter spectra. \n\\end{enumerate}\nTo test the robustness of our measurements, we vary the details of the post-processing procedure, Among the many variations, we perform the sigma-clipping before resampling rather than after, apply an upper error threshold to restrict the maximum allowable error, and in one instance, we do not perform resampling on individual quasar spectra. In each case, we find that the majority of measurements are consistent within their uncertainties and the overall correlations and conclusions we draw from our measurements are unaffected. This gives us confidence in our results. \n\n\\subsection{Black Hole Mass Estimate}\nThe black hole mass of each quasar is based on single-epoch virial mass estimates. We source the black hole masses from \\citet{Shen_2019}, \\citet{yang2021probing}, and Mazzucchelli et al. in prep. which span $\\log\\left({\\rm{M}_{\\rm{BH}}\/\\rm{M}_{\\odot}}\\right) = 8.4-9.8$ over the entire quasar sample. To determine the masses, these studies use the rest-frame UV Mg\\,\\textsc{ii}\\ broad emission line and the Mg\\,\\textsc{ii}-based virial estimator, described generally by the following,\n\\begin{equation}\n   \\left(\\frac{M_{\\rm{BH,vir}}}{M_{\\odot}}\\right) = 10^{\\rm{a}} \\left[\\frac{\\lambda L_{\\lambda}}{10^{44} \\,\\rm{erg\\, s^{-1}}}\\right]^{b} \\left[\\frac{\\rm{FWHM (Mg\\,\\textsc{ii})}}{1000 \\,\\rm{km\\, s^{-1}}}\\right]^{2} \\,,\n   \\label{eq:mgii-vo9}\n\\end{equation}\nwhere $\\lambda L_{\\lambda}$ is the monochromatic luminosity of the continuum at rest frame 3000 \\AA, and (a,b) are empirically calibrated against reverberation mapping experiments to the values (6.86, 0.5) in \\citet{Vestergaard_2009} and (0.74, 0.62) in \\citet{Shen_2011}. The masses of quasars in the XQR-30 and \\citet{yang2021probing} samples are estimated using the \\citet{Vestergaard_2009} calibration. The mass of the one quasar we've included from \\citet{Shen_2019} is reported with the \\citet{Shen_2011} calibration, but we have re-calibrated the mass with \\citet{Vestergaard_2009}, resulting in a 0.1 dex difference. The continuum luminosity is estimated by fitting a power-law continuum and Fe\\,\\textsc{ii}\\ emission around the Mg\\,\\textsc{ii}\\ line, as described in Section \\ref{sec:emission-fitting} and the Fe\\,\\textsc{ii}\\ template \\cite[i.e.][]{UV_iron_template} is consistent between the different studies. The absolute fluxing of the XQR-30 spectra {is based on calibrations against observed near-infrared photometry and} is described in full in Mazzucchelli et al. in prep. The Mg\\,\\textsc{ii}\\ full-width at half maximum (FWHM) is determined with single or multi-component Gaussian fits to the broad emission line and the peak of the total line profile is used to calibrate the systemic redshift of the quasar spectrum. Typical systematic errors from the virial mass estimator for the Mg\\,\\textsc{ii}\\ line can be up to 0.55 dex \\citep{shen_biases_2008, Vestergaard_2009}. Bolometric luminosities are measured from the flux-calibrated spectrum using the continuum luminosity at 3000\\AA\\ and adopting a bolometric correction of 5.15 \\citep{Shen_2011} throughout our entire sample.\n\nThe virial mass estimate is routinely applied to quasars \\citep[e.g.][]{2002MNRAS.331..795M, Shen_2012} and aside from the Mg\\,\\textsc{ii}\\ line, $\\rm{H}\\beta$ and C\\,\\textsc{iv}\\ emission lines have been used. Virial mass estimates using the $\\rm{H}\\beta$ emission line is not feasible for high redshift quasar studies prior to the James Webb Space Telescope, but the Mg\\,\\textsc{ii}\\ line width is correlated with $\\rm{H}\\beta$ and can be used as its substitution in single-epoch virial black hole mass estimates \\citep[e.g.][]{Salviander_2007, shen_biases_2008, Wang_2009, Shen_2012}. Compared to the C\\,\\textsc{iv}\\ emission line, the advantage of the Mg\\,\\textsc{ii}\\ line is that it is less affected by non-virial components of the black hole emission, such as the radiatively-driven BLR wind \\citep[e.g.][]{Saturni_2018}. The difference between the C\\,\\textsc{iv}\\ and Mg\\,\\textsc{ii}\\ virial mass estimates is correlated with the C\\,\\textsc{iv}\\ blueshift \\citep{Shen_2012, Coatman_2017}. \n\n\\subsection{C\\,\\textsc{iv}\\ Blueshift Measurement}\nThe C\\,\\textsc{iv}\\ emission line is of particular interest in assessing BLR outflow strength which has also been linked to metallicity \\citep[e.g.][]{Wang_2012, shin_2017_outflow}. This high-ionisation line can exhibit significant blueshifts \\citep[e.g.][]{Gaskell1982, Wilkes1984, Marziani_1996, vandenberk2001, baskin_laor_2005, 2007ApJ...666..757S} and asymmetric velocity profiles \\citep[e.g.][]{Sulentic_2000, baskin_laor_2005}, structure that is often interpreted as arising from a disk wind or outflow \\citep[e.g.][]{2007ApJ...666..757S, Vietri_2018}. The blueshift of C\\,\\textsc{iv}\\ is therefore an indication of the balance of emission between the outflowing ionised gas and the emission at a systematic redshift, which we call the ``wind'' and ``core'' component respectively \\citep[adopting the terminology of][]{metal_density}. At high redshifts (z > 5.8), the mean and median C\\,\\textsc{iv}-Mg\\,\\textsc{ii}\\ velocity shifts are greater than for luminosity-matched quasars at lower redshifts, although this may potentially be biased by increased torus opacity and orientation-driven selection effects \\citep{Meyer_2019, JT_2020, yang2021probing}. In this study, we define our estimate of the C\\,\\textsc{iv}\\ blueshift as \n\\begin{equation} \\label{eq:blueshift}\n    \\frac{\\rm{C\\,\\textsc{iv}\\ blueshift}}{\\rm{km \\,s^{-1}}} \\equiv c \\times (1549.48\\mbox{\\normalfont\\AA} - \\lambda_{\\rm{med}})\/1549.48\\mbox{\\normalfont\\AA}\\,,\n\\end{equation}\nwhere $c$ is the speed of light and $\\lambda_{\\rm{med}}$ is the median wavelength bisecting the total continuum-subtracted C\\,\\textsc{iv}\\ emission line flux. The wavelength 1549.48\\AA\\ is the average of the C\\,\\textsc{iv}\\ $\\lambda\\lambda1548.19,1550.77$ doublet. This definition is the same as in \\citet{metal_density}, but their redshift is defined using a variety of low-ionization emission lines, some of which are known to exhibit velocity shifts relative to Mg\\,\\textsc{ii}. In this study, we define our redshift using only the Mg\\,\\textsc{ii}\\ line. Due to the 1 \\AA\\ wavelength resolution, we prescribe a minimum precision of $\\sim200$ km $\\rm{s}^{-1}$ for the C\\,\\textsc{iv}\\ blueshift, evaluated as an error of $\\pm 1$\\AA\\ at the average wavelength of the C\\,\\textsc{iv}\\ doublet. The overall uncertainty of the C\\,\\textsc{iv}\\ blueshift is combined with the uncertainty from the measured redshift. \n\nThe C\\,\\textsc{iv}\\ blueshift is also known to be anti-correlated with the line's equivalent width \\citep[EW;][]{Leighly2004, Richards2011, Vietri_2018, Rankine2020, JT_2020, metal_density}, a relationship which is reproduced for our high-redshift quasar sample in Figure \\ref{fig:CIV_blueshift_ew}. This correlation may be driven by orientation, properties of BLR winds, or the Baldwin effect linking properties of high-ionisation lines like C\\,\\textsc{iv}\\ with the quasar luminosity \\citep{Baldwin_1977_effect}. The results show highly blueshifted C\\,\\textsc{iv}\\ lines are weak, while stronger lines are less blueshifted and more symmetric.\n\nWe note that a velocity shift relative to C\\,\\textsc{ii}]\\ of 5510$^{+240}_{-110}$ km s$^{-1}$ was measured for J1342+0928 \\citep{Banados_2018, Onoue_2020} and \\citet{JT_2020} also measured velocity shifts relative to Mg\\,\\textsc{ii}\\ for several XQR-30 quasars in our sample. However, due to differences in the definition of C\\,\\textsc{iv}\\ blueshift\\footnote{It is also possible to define the C\\,\\textsc{iv}\\ blueshift by the maximum of the C\\,\\textsc{iv}\\ line profile or the blueshift and asymmetry index (BAI) defined as the flux blueward of 1549.48\\AA. Although we don't use these definition in this study, we have checked that these alternatives have little effect on the relative C\\,\\textsc{iv}\\ blueshifts between quasars. The relationship found in Figure \\ref{fig:CIV_blueshift_ew} and the correlations found in this study hold are unaffected.}, choice of Fe\\,\\textsc{ii}\\ template, and sometimes the referenced line to estimate the redshift, we have re-measured the blueshifts for most quasars in our sample. However, the C\\,\\textsc{iv}\\ blueshift could not be reliably determined for one individual quasar: J2338+2143, which, despite the minimum SNR requirement, has an overall SNR too poor to obtain a reliable fit. \n\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{figs\/CIV_Blueshift_EW.png}\n    \\caption{C\\,\\textsc{iv}\\ equivalent width as a function of C\\,\\textsc{iv}\\ blueshift. There is a moderate anti-correlation between these two quantities, implying that weaker C\\,\\textsc{iv}\\ lines are more strongly blueshifted and stronger lines are less blueshifted. The outlier with high EW and blueshift is J1216+4519. Its SNR is low ($\\sim 7$ per pixel) which is reflected by the large error in EW.}\n    \\label{fig:CIV_blueshift_ew}\n\\end{figure}\n\n\n\\subsection{Composite Spectra} \\label{sec:composites}\nThe primary objective of this work is to measure flux ratios of metallicity-sensitive lines binning by quasar properties, such as bolometric luminosity, black hole mass, and C\\,\\textsc{iv}\\ blueshift, to determine whether these parameters are correlated with metallicity in the BLR. Although most of the spectra have sufficient SNR to proceed with the emission-line fitting independently, the weak and blended emission lines of some SNR\/pixel $\\leq 8$ spectra in this sample could not be fit convincingly. By stacking the spectra, we are able to obtain higher SNR. Another reason to stack the spectra is to average out peculiarities of individual quasars in each bin in order to construct better comparisons to the photoionisation models referenced in Section \\ref{sec:line_ratios_metallicity}. As we are interested in the average spectral properties within a binned parameter space rather than the specific individual properties, we use equivalent weighting of spectra within each composite regardless of the SNR of input spectra so that the output is not biased in favor of any contributing quasar observed with high SNR. We construct 6 bins from each of the 3 quasar properties (black hole mass, bolometric luminosity, and C\\,\\textsc{iv}\\ blueshift), with a similar number of contributing quasar spectra in each bin. We also avoid extending the width of each bin too wide. Therefore, the average number of quasar spectra in each bin is 4, and all composites are created from 3-6 input spectra. \n\nThe dynamic range of quasar bolometric luminosity in this sample is $\\log\\left(\\rm{L}_{\\rm{bol}}\/\\rm{erg \\, s}^{-1}\\right) = 46.7-47.7$. We split the sample into 6 luminosity bins with the following edges: 46.72, 46.79, 46.95, 47.17, 47.30, 47.40, 47.70, and a composite is created from each bin. The first three bins include all 10 GNIRS spectra and the final three bins are composed of exclusively X-shooter spectra. The total BH mass range reflected in our high-redshift quasar spectra is $\\log\\left({\\rm{M}_{\\rm{BH}}\/\\rm{M}_{\\odot}}\\right) = 8.4-9.8$. Again, we form 6 mass bins with the following edges: 8.40, 8.75, 8.87, 8.98, 9.20, 9.40, 9.80, and create a composite spectrum from each bin. We also arrange and stack all individual quasars in the sample into 6 C\\,\\textsc{iv}\\ blueshift bins, with the following bin edges: $-$200, 680, 1500, 2500, 3000, 4000, 5000 km $\\rm{s}^{-1}$. Figure \\ref{fig:quasar_properties} shows the bin edges of each composite delineated by black dashed lines.\n\nPrior to stacking, we apply an upper error threshold equal to 2 times the minimum error within box widths of 50 pixels to restrict data to where the error is reasonable. The error threshold clips wavelength bins with unusually high error and high flux that were not masked by the general post-processing procedure. The resulting spectra contains the most stable and robustly measured elements. Without the upper error threshold, the propagation of error from a small number of component spectra can create unstable composites, leading to greater uncertainty in the final flux measurements. Every spectrum is then normalised across the rest-frame 1430\\AA$\\sim$1450\\AA\\ wavelength range and the arithmetic mean of each stack is taken to generate the composite. We also take the median or geometric mean \\citep[e.g.][]{vandenberk2001} of the stack and find that it does not significantly influence the result. Furthermore, we generate composites after subtracting a power-law continuum, fitted as described in Section \\ref{sec:emission-fitting}, and find that the resulting measured broad emission-line fluxes are not significantly discrepant either. In all cases, the resulting line ratio measurements regardless of taking the composite arithmetic mean, geometric mean, median, or after subtracting the continuum agree to within 2.0 $\\sigma$, with $\\sim 75\\%$ of measurements within 1.0 $\\sigma$.\n\nThe uncertainty in each resolution element is composed of the error in every contributing spectrum added in quadrature, but we also estimate the systematic error in each composite by generating all of the possible composites that can be obtained if any one contributing quasar spectrum is excluded. The standard deviation in each 1\\AA\\ pixel from all such simulated composites is treated as the systematic error, added in quadrature to the uncertainty propagated from each contributing spectrum. This systematic error is an additional source of error which raises the uncertainty floor of the combined spectrum and reduces the relative uncertainty between each resolution element, affecting the weighting of each pixel in a least-squares fitting routine. After combining all sources of uncertainty, the resulting SNR per pixel of the composites measured between rest-frame 1400-1600\\AA\\ is 20-100. We show a composite constructed from all 25 quasars in our sample in Figure \\ref{fig:all_composite}.\n\nThree individual quasar spectra are treated differently for our composites. PSOJ007+04, PSOJ025-11, and J1212+0505 are affected by proximate damped Ly\\,\\textsc{$\\alpha$}\\ absorption (pDLA) systems \\citep{Farina_2019, Banados_2019}, which causes a significant fraction of its Ly\\,\\textsc{$\\alpha$}\\ emission to be absorbed at the systemic redshift. We fit the emission lines of pDLA affected quasars individually, and mask their emission blueward of the N\\,\\textsc{v}\\ centroid from contributing to composites. \n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{figs\/all_composite.png}\n    \\caption{Composite created from all 25 high-redshift quasars in our sample compared against the SDSS quasar template comprised of 2200 spectra described in \\citet{vandenberk2001}. {The composite is split into two sections with independent y-axis scaling to better visualise the various spectral emission features, which are denoted with the grey dashed lines.} The composite and SDSS template are normalised together at 2200-2250\\AA, {and the grey spectrum is the error spectrum of the composite.} The blue and small patches of orange shaded regions indicate when the template or the {composite} is in excess respectively. Our high-redshift composite has a noticeably flatter continuum slope and significant Ly\\,\\textsc{$\\alpha$}\\ absorption compared to the lower redshift SDSS sample, but the profiles and relative integrated fluxes of emission features other than Ly\\,\\textsc{$\\alpha$}\\ are comparable.}\n    \\label{fig:all_composite}\n\\end{figure*}\n\n\n\\section{Line Fitting and Metallicity Measurement} \\label{sec:metallicity}\n\\subsection{Emission-line Fitting} \\label{sec:emission-fitting}\nIn this work, we study a large number of rest-frame UV emission lines: Ly\\,\\textsc{$\\alpha$}, N\\,\\textsc{v}, Si\\,\\textsc{ii}, Si\\,\\textsc{iv}, O\\,\\textsc{iv}, N\\,\\textsc{iv}], C\\,\\textsc{iv}, He\\,\\textsc{ii}, O\\,\\textsc{iii}], Al\\,\\textsc{ii}, Al\\,\\textsc{iii}, Si\\,\\textsc{iii}, C\\,\\textsc{iii}], and Mg\\,\\textsc{ii}. The measurement of emission-line fluxes can be tricky due to adjoining and heavily blended lines, such as N \\textsc{v} $\\lambda$1240 \\AA\\ with Ly \\textsc{$\\alpha$} $\\lambda$1216 \\AA\\ or Al \\textsc{iii} $\\lambda$1857 \\AA, Si \\textsc{iii} $\\lambda$1887 \\AA, and C \\textsc{iii}] $\\lambda$1909 \\AA. Furthermore, the strong Fe\\,\\textsc{ii}\\ emission biases the underlying continuum level measurement. Despite these challenges, there are two widely employed methods for fitting quasar emission-lines \\citep{Nagao_2006}. One method measures the emission-line flux by integrating above a well-defined independent local continuum model \\citep[e.g.][]{vandenberk2001} and the other method endeavors to fit emission lines using one or more appropriate functions, such as Gaussians or Lorentzians \\citep[e.g.][]{Zheng_1997}. Both of these methods have shortcomings in measuring accurate emission-line fluxes. Defining an appropriate local continuum level below an emission-line is challenging and is sensitive to where the baseline is anchored. The additional uncertainty propagates into the resulting metallicity estimates. Regarding the function fitting approach, a single Gaussian or Lorentzian profile is insufficient for broad emission-lines with asymmetric velocity profiles \\citep[e.g.][]{corbin_1997, vandenberk2001, baskin_laor_2005}. The approach utilising multiple Gaussian functions can obtain smooth realisations of the line profile, but the decomposition is not unique and a large number of free parameters is required. Modified functions such as a skewed Gaussian (defined in Appendix \\ref{appendix:line-fitting-compare}) or asymmetric Lorentzian depend on fewer parameters and are arguably more physically relevant \\citep[e.g.][]{mallery_2012}. With multiple reasonable approaches, there is a concern that the resulting line flux can be method-dependent. In this work, we use various appropriate functions to fit emission lines and we compare the several different methods against similar fits from existing literature in Appendix \\ref{appendix:line-fitting-compare}.\n\nWe follow the general procedure from \\citet{Xu2018} and define the following two line-free windows in rest-frame to fit the continuum: 1445\\AA$-$1455\\AA, 1973\\AA$-$1983\\AA. In specific circumstances, we identify two additional windows (1320\\AA$-$1325\\AA, and 1370\\AA$-$1380\\AA) to further constrain the continuum shape or we extend the blue-end of the first line-free window to 1432\\AA\\ in the case of a blueshifted C\\,\\textsc{iv}\\ line. The continuum is fit with a power-law function normalised to rest-frame 3000 \\AA,\n\\begin{equation}\n    F_{\\rm{pl}}(\\lambda) = F_{\\rm{pl, 0}} \\left(\\frac{\\lambda}{3000 \\mbox{\\normalfont\\AA}}\\right)^{\\gamma}\\,,\n    \\label{eq:pl-cont}\n\\end{equation}\nwhere $F_{\\rm{pl, 0}}$ and $\\gamma$ represent the normalization and power-law slope respectively. We also consider the contribution of the Fe\\,\\textsc{ii}\\ pseudo-continuum spectrum using the empirical template from \\citet{UV_iron_template} to cover the wavelength range from 1200 \\AA\\ to 3500 \\AA. We convolve the template with a Gaussian broadening kernel to better fit the variety of features from the Fe\\,\\textsc{ii}\\ pseudo-continuum seen across spectra in our sample,\n\\begin{equation}\n    F_{\\rm{Fe}}(\\lambda) = \\zeta_{\\rm{0}} \\,  F_{\\rm{template}}|_{\\lambda(1+\\delta)} \\circledast G(\\lambda, \\sigma)\\,,\n\\end{equation}\nwhere the free parameters of the Fe\\,\\textsc{ii}\\ flux contribution include a flux scaling factor $\\zeta_{\\rm{0}}$, the FWHM of the broadening kernel $\\sigma$, and a small wavelength shift $\\delta$. The contribution from the iron continuum is more relevant at wavelengths close to the Mg \\textsc{ii} $\\lambda$2799 \\AA\\ line and is important in obtaining the virial mass estimate. Combined, the power-law and the Fe\\,\\textsc{ii}\\ template are fit to the data in the line-free windows and form the underlying continuum baseline.\n\nEmission lines are fit with the following double power-law method adopted from \\citet{Nagao_2006}, \\citet{Matsuoka_2011}, and \\cite{Xu2018},\n\\begin{equation}\n  F_{\\rm{em}}(\\lambda) =\n    \\begin{cases}\n      F_{\\rm{0}} \\times \\left(\\frac{\\lambda}{\\lambda_{\\rm{0}}}\\right)^{-\\alpha} & \\lambda > \\lambda_{\\rm{0}} \\\\\n      F_{\\rm{0}} \\times \\left(\\frac{\\lambda}{\\lambda_{\\rm{0}}}\\right)^{+\\beta} & \\lambda < \\lambda_{\\rm{0}}\n    \\end{cases}\n    \\label{eq:line_fit}\n\\end{equation}\nwhere the two power-law indices ($\\alpha$ and $\\beta$) are used to fit the red and blue sides of the emission-line profile. The peak intensity, $F_{\\rm{0}}$, controls the height of the emission line and the peak wavelength, $\\lambda_{\\rm{0}}$, defines the location of the peak. \n\nEmission lines with different degrees of ionisation often show systematically varied velocity profiles \\citep[e.g.][]{Gaskell1982, baskin_laor_2005}. Thus, we categorise emission lines into two distinct systems: high-ionisation lines (HILs) and low-ionisation lines (LILs). The HILs include N\\,\\textsc{v}, O\\,\\textsc{iv}, N\\,\\textsc{iv}], C\\,\\textsc{iv}, and He\\,\\textsc{ii}\\ while the LILs include Si\\,\\textsc{ii}, Si\\,\\textsc{iv}, O\\,\\textsc{iii}], Al\\,\\textsc{ii}, Al\\,\\textsc{iii}, Si\\,\\textsc{iii}, and C\\,\\textsc{iii}]\\ \\citep{Collin_Souffrin_1988}. The boundary separating the two main groups is an ionisation potential of 40 eV. We assume that the emission-line profiles of lines in the same category are coupled to the same line-emitting gas clouds of the BLR, sharing a common value for the $\\alpha$ and $\\beta$ power indices. Because we did not correct for the suppression of Ly\\,\\textsc{$\\alpha$}\\ from the intergalactic medium, the redder $\\alpha$ index of the Ly\\,\\textsc{$\\alpha$}\\ line is coupled with the HILs, while the bluer $\\beta$ index is left unconstrained \\citep{Nagao_2006, Xu2018}. \\footnote{Although the Ly\\,\\textsc{$\\alpha$}\\ line doesn't directly factor into the line ratios we measure or the metallicities we determine, its flux and line profile does affect the measured flux of the N\\,\\textsc{v}\\ line.} \n\nOur adopted piece-wise power-law function fit to emission lines has been compared to the double-Gaussian and modified Lorentzian methods, achieving better fits with fewer or equal number of free parameters \\citep{Nagao_2006}. In cases when the piece-wise function does not produce a reasonable fit to the shape of the spectral feature, such as significantly blueshifted lines, we fit a skewed Gaussian function, where both the skew and FWHM of the Gaussian are coupled between LILs and HILs. Unlike the default piece-wise strategy, Ly\\,\\textsc{$\\alpha$}\\ is completely decoupled from the HIL group when fitting skewed Gaussians. We choose to fit a single skewed Gaussian because it has the same number of free parameters as the piece-wise power-law fit. Changing the fitting strategy is also motivated by the reduction in the minimum chi-square value even when the fits produce similar emission-line flux ratios. A more complete description of the comparison between these two fitting methods and the definition of the skewed Gaussian are provided in Appendix \\ref{appendix:line-fitting-compare}. \n\nWhen LILs and HILs are not coupled, some local continuum methods can produce emission-line profiles with very different widths and skewness \\citep{vandenberk2001}. We assume, as several similar other studies do \\citep[e.g.][]{Nagao_2006, Matsuoka_2011, Xu2018}, that emission lines with similar ionising potentials originate from similar line-emitting regions in the BLR. The coupling of power indices in Equation \\ref{eq:line_fit} provides a crucial constraint in ensuring that the kinematics of line-emitting clouds are preserved within the LIL and HIL groups. Furthermore, without the coupling of HILs, the decomposition of the Ly\\,\\textsc{$\\alpha$}\\ and N\\,\\textsc{v}\\ emission profile is not unique. The coupling of the N\\,\\textsc{v}\\ profile and the red wing of Ly\\,\\textsc{$\\alpha$}\\ to HILs provides a way to obtain a unique solution that disentangles their line profiles and fluxes.\n\nThe emission lines Ly \\textsc{$\\alpha$} $\\lambda$1216, N \\textsc{v} $\\lambda$1240, Si \\textsc{ii} $\\lambda$1263, Si \\textsc{iv} $\\lambda$1398, O \\textsc{iv} $\\lambda$1402, N \\textsc{iv}] $\\lambda$1486, C \\textsc{iv} $\\lambda$1549, He \\textsc{ii} $\\lambda$1640, O \\textsc{iii}] $\\lambda$1663, Al \\textsc{ii} $\\lambda$1671, Al \\textsc{iii} $\\lambda$1857, Si \\textsc{iii} $\\lambda$1887, and C \\textsc{iii}] $\\lambda$1909\\ are all fit simultaneously. The line-fitting regions generally are 1214-1290, 1360-1430, 1450-1700, and 1800-1970\\AA\\ with some flexibility depending on the width and kinematics of the spectral features. Each line is allowed an independent $\\pm 25$\\AA\\ shift in central wavelength, $\\lambda_{\\rm{0}}$, with respect to the rest-frame vacuum wavelength. Whether we used the piece-wise power-law or skewed Gaussian approach, a single emission-line is fit with only four parameters. Figure \\ref{fig:example-fit-ATLASJ029-36} shows an example fit to ATLASJ029-36 which was observed with VLT\/X-shooter. In this example, all of the lines from Ly\\,\\textsc{$\\alpha$}\\ at 1216 \\AA\\ to C\\,\\textsc{iii}]\\ at 1909 \\AA\\ have been fit simultaneously, with coupled LILs and HILs. \n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{figs\/ATLASJ029-36.png}\n    \\caption{Example fit to ATLASJ029-36 observed using VLT\/X-shooter with a mean SNR per 1\\AA\\ pixel of 27.35 between 1400-1600\\AA. The top plot shows the spectrum after the post-processing techniques and the bottom plot shows the residuals. The inset plot provides a closer look at the line profiles of Ly\\,\\textsc{$\\alpha$}\\ and N\\,\\textsc{v}. {The vertical blue bars indicate the continuum fitting windows, which are fit by the power-law continuum denoted by the orange line.} The red lines indicate the emission line fits as well as the extent of the individual line-fitting windows. All fitted emission lines are labeled and their individual line profiles are shown.}\n    \\label{fig:example-fit-ATLASJ029-36}\n\\end{figure*}\n\n\nWe fit the C\\,\\textsc{iv}\\ emission line to estimate the blueshift according to Equation \\ref{eq:blueshift}. When C\\,\\textsc{iv}\\ blueshifts < 4000 km $\\rm{s}^{-1}$, we adopt the piecewise power-law fit and at higher blueshifts, we use the skewed Gaussian function. We find both methods produce consistent results at lower C\\,\\textsc{iv}\\ blueshifts as shown in the comparison described in Appendix \\ref{appendix:line-fitting-compare}.\n\nWe also fit the Mg\\,\\textsc{ii}\\ line independently with one skewed Gaussian or two symmetric Gaussians, using the following line-free windows: 1770-1810, 2060-2340, 2600-2740, 2840-3100\\AA\\ to measure the continuum. {The continuum is measured independently for the} Mg\\,\\textsc{ii}\\ line fit because the contribution from the Fe\\,\\textsc{ii}\\ emission {is} much more significant at these longer wavelengths. {Although the fit parameters are not always consistent between the two wavelength ranges, we find the power-law to be an adequate local approximation of the accretion disk emission}. We use the Mg\\,\\textsc{ii}\\ FWHM, Equation \\ref{eq:mgii-vo9}, and calibration from \\citet{Vestergaard_2009} to determine the black hole mass and find good agreement with Mazzucchelli et al. in prep. We calibrate the spectrum against the observed quasar AB magnitude in the J bandpass, where flux from the spectrum integrated over the filter transmission profile is scaled appropriately to the observed value. The peak flux wavelength of the total Mg\\,\\textsc{ii}\\ line profile is used to determine the redshift and its error. Figure \\ref{fig:mgii_fit_example} shows an example of a multiple Gaussian fit to the Mg\\,\\textsc{ii}\\ emission line along with the combined power-law and Fe\\,\\textsc{ii}\\ continuum.\n\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{figs\/PSOJ217-16_MgII.png}\n    \\caption{Example two-Gaussian fit to the Mg\\,\\textsc{ii}\\ emission line of PSOJ217-16, observed using VLT\/X-shooter. The top plot shows the spectrum after the post-processing techniques and the bottom plot shows the residuals. The inset plot provides a closer look at the Mg\\,\\textsc{ii}\\ line profile after continuum subtraction. The orange line is the power-law and the green line is full baseline continuum with the Fe\\,\\textsc{ii}\\ template. The red line marks the fit to the emission line and the extent of the fitting windows. These fits are presented in greater detail in Mazzucchelli et al. in prep.}\n    \\label{fig:mgii_fit_example}\n\\end{figure}\n\nFor the line flux error estimation, we adopt a Monte-Carlo approach used in similar studies of high-redshift quasar spectra \\citep[e.g.][]{Shen_2019, 2020z7.5, 2021_luminous7.6}. We create 50 mock spectra for each individual spectrum and stacked composite, where the flux at each pixel is resampled from a symmetric distribution with a standard deviation equivalent to the pixel spectral error. We assume the spectrum noise to follow a Gaussian distribution in this case. The same fitting procedure is applied to every mock spectrum generated in this way and we filter out outlier fits by sigma-clipping the line flux measurements using a 3-$\\sigma$ threshold. The final line flux of each metallicity-sensitive line is the median of all remaining fits and the final uncertainty is its standard deviation.\n\n{Using other empirical templates of the} Fe\\,\\textsc{ii}\\ {emission} \\citep[e.g.][]{Tsuzuki_2006, Bruhweiler_2008, Mejia_2016} {can result in a maximum discrepancy of 20\\% in the} Mg\\,\\textsc{ii}\\ {FWHM, and 0.2\\% in the estimated redshift. Although this can modify the significance of the black hole mass correlation presented in} Section \\ref{sec:results}, {it has no effect on the metallicity estimates or the conclusions of this study, which depends primarily on the} C\\,\\textsc{iv}\\ {profile.}\n\nFigure \\ref{fig:quasar_properties} presents the distribution of redshift, bolometric luminosity, black hole mass, and C\\,\\textsc{iv}\\ blueshift in our sample. The completed sample covers redshifts $z = 5.8-7.5$, spanning a range in bolometric luminosity $\\log\\left(\\rm{L}_{\\rm{bol}}\/\\rm{erg \\, s}^{-1}\\right) = 46.7-47.7$ and a range in black hole mass $\\log\\left({\\rm{M}_{\\rm{BH}}\/\\rm{M}_{\\odot}}\\right) = 8.4-9.8$. Quasars with luminosities $\\log\\left(\\rm{L}_{\\rm{bol}}\/\\rm{erg \\, s}^{-1}\\right) \\leq 47.0$ and black hole masses $\\log\\left({\\rm{M}_{\\rm{BH}}\/\\rm{M}_{\\odot}}\\right) \\leq 8.9$ are all observed with GNIRS. We measure C\\,\\textsc{iv}\\ blueshifts spanning over 5000 km $\\rm{s}^{-1}$ in our sample, from values consistent with no detectable blueshift to the most extreme outflow-dominated spectrum in PSOJ065-25. None of the targets exhibit a significantly redshifted C\\,\\textsc{iv}\\ emission line. We list all of the measured quasar properties in Table \\ref{tab:quasar_sample}.\n\n\\subsection{Line Ratios and Metallicity} \\label{sec:line_ratios_metallicity}\nIn order to interpret the results of the fitting, it is useful to compare the measured high-ionisation line ratios against predictions from photoionisation models. It's well-known that single-zone photoionisation models are unable to fully reproduce the observed emission from the BLR, because the gas clouds span a wide range of densities and degrees of ionisation \\citep[e.g.][]{Davidson_1977, Collin_Souffrin_1988}. Multi-zone models incorporate emission from gas with a wide range of physical properties and are shown to be consistent with observation \\citep[e.g.][]{Rees_1989, Hamann_1998}.\n\nThe flux ratio-metallicity relation is sensitive to the density of line-emitting clouds, spectral energy distribution (SED) of the ionizing continuum, and microturbulence, but under the locally optimally-emitting cloud model \\citep[LOC;][]{Baldwin_1995_LOC}, the net emission spectrum can be reproduced by integrating across a wide range of physical conditions. Therefore, the characteristics of the observable spectrum originate from an amalgamation of emitters, where each emission line is formed in a region that is optimally suited to emit the targeted line. This model consistently reproduces properties of both low and high-ionisation emission lines observed in quasar spectra \\citep[e.g.][]{Korista_2000_LOC, Hamann_2002, Nagao_2006}.\n\nWe primarily utilise two broad emission-line flux ratios (\\nv\/\\civ, (\\siiv+\\oiv)\/\\civ) because the relevant lines are easier to detect and more commonly studied in the existing literature. However, we also present results for additional line ratios ((\\oiii+\\alii)\/\\civ, \\aliii\/\\civ, \\siiii\/\\civ, \\ciii\/\\civ). The emission from these other lines are substantially more difficult to measure and can only be detected in a robust manner in high SNR spectra, such as our XQR-30 sample of high-redshift quasars. We convert all of the line ratios into metallicity estimates using relations derived from \\texttt{Cloudy} photoionisation simulations described in \\citet{Hamann_2002} and \\citet{Nagao_2006}. Both models utilise the LOC model \\citep{Ferland_1998_cloudy}. \\citet{Nagao_2006} predicts line flux ratios for all listed line ratios with two models for the ionizing continuum: one with a strong UV thermal bump matching results from \\citet{large_uv_bump} and one with a weak UV thermal bump similar to Hubble Space Telescope quasar templates \\citep{Zheng_1997, telfer2002}. These two SEDs are thought to be extreme and opposite cases for the actual ionising continuum \\citep{Nagao_2006}, which gives us the full range of possible inferred metallicities. \\citet{Hamann_2002} predicts line flux ratios of only \\nv\/\\civ\\ for three mock incident spectra: that of \\citet{mf87}; a single hard power-law with index $\\alpha = -1.0$ ($f_{\\rm{\\nu}} \\propto \\nu^{\\rm{\\alpha}}$); and a segmented power-law with indices $\\alpha = [-0.9, -1.6, -0.6]$ for 0.25\\AA\\ to 12\\AA, 12\\AA\\ to 912\\AA, and 912\\AA\\ to 1 $\\mu$m respectively. The segmented power-law continuum approximates data gathered from observations \\citep[e.g.][]{Laor_1997}. For \\citet{Hamann_2002}, the \\citet{mf87} incident spectrum predicts the highest metallicities for the same line ratio and the $\\alpha = -1.0$ spectrum produces the lowest. The spread in metallicities predicted for the same line ratio is incorporated into our uncertainties. For \\nv\/\\civ\\ the results from both publications are largely consistent with minor differences arising from the SED of the ionizing continuum, integration ranges of gas density ($n_{\\rm{H}}$) or ionizing flux ($\\Phi_{\\rm{H}}$), cloud column density, and the version of \\texttt{Cloudy} used. \n\nLine ratios which imply metallicities over 10 $Z_{\\odot}$ extend beyond the parameter space probed by \\citet{Hamann_2002} or \\citet{Nagao_2006}. This occurs when the measured line ratio exceeds 0.84 for \\nv\/\\civ\\ or 0.45 for (\\siiv+\\oiv)\/\\civ. For the other line ratios, this occurs at (0.39, 0.16, 0.36, 0.57) for ((\\oiii+\\alii)\/\\civ, \\aliii\/\\civ, \\siiii\/\\civ, \\ciii\/\\civ). In order to investigate inferred metallicities for higher line ratios, we assume that the observed line flux ratio-metallicity relationship maintains a linear trend in log-space and linearly extrapolate beyond the parameter space probed by the simulations. Super-solar metallicites over 10 $Z_{\\odot}$ have not been calibrated against \\texttt{Cloudy} simulations.\n\nWe consider all of the photionisation calculations with different ionizing SEDs in our metallicity estimate. The effect of the ionizing continuum SED is responsible for up to a factor of two difference in the resulting metallicity predictions from \\nv\/\\civ. For \\nv\/\\civ, the uncertainty from the metallicity calibration based on the various photoionisation models is dominant over the observational uncertainty. (\\siiv+\\oiv)\/\\civ\\ is a more robust metallicity indicator than \\nv\/\\civ\\ because it is not as sensitive to differences in the ionizing continuum or assumed weighting functions \\citep[e.g.][]{Nagao_2006, Matsuoka_2011, Maiolino_2019_metallica}, and it is not affected by bias propagating from a poor fit to the highly absorbed Ly\\,\\textsc{$\\alpha$}\\ emission line. However, in this study, we offer no discussion on the discrepancy between the metallicity indicators. Instead, we present the inferred metallicities separately and use the spread of results from different assumed ionizing SEDs as the uncertainty of each individual measurement. A comparison between inferred metallicities from \\nv\/\\civ\\ and (\\siiv+\\oiv)\/\\civ\\ is provided in the Appendix.\n\n\n\\section{Results} \\label{sec:results}\nWe present all quasars and their measured properties in Table \\ref{tab:quasar_sample}. In addition to fitting the composites described in Section \\ref{sec:composites}, the emission-lines of nearly all of the quasar spectra can be fit individually. We fit the \\nv\/\\civ\\ line ratio for 16 of the 25 individual quasars, wherever the N\\,\\textsc{v}\\ emission can be separated from the Ly\\,\\textsc{$\\alpha$}\\ emission. We show the \\nv\/\\civ\\ and (\\siiv+\\oiv)\/\\civ\\ line ratio results for individual fits in Table \\ref{tab:quasar_sample} and provide 6 example fits to individual quasar spectra in the Appendix, covering the lowest and highest quasar bolometric luminosity, black hole mass, and C\\,\\textsc{iv}\\ blueshift. Also available are figure sets which show sample fits to bolometric luminosity composites, black hole mass composites, and C\\,\\textsc{iv}\\ blueshift composites. We provide all line fluxes measured from our composites normalised against C\\,\\textsc{iv}\\ in Tables \\ref{tab:composite_results}, \\ref{tab:composite_mass}, and \\ref{tab:composite_blueshift}.\n\n\\begingroup\n\\begin{table*}\n\\caption {\\label{tab:quasar_sample} Properties of the quasars and spectra included in this study and their measured emission line flux ratios. The redshift is determined from the Mg\\,\\textsc{ii}\\ line with an uncertainty floor of 0.001. The C\\,\\textsc{iv}\\ blueshift is measured using the median wavelength of the C\\,\\textsc{iv}\\ fit and the mean SNR is measured in the rest-frame wavelength range 1400-1600\\AA. We prescribe a minimum error of the C\\,\\textsc{iv}\\ blueshift equivalent to $\\sim$200 km s$^{-1}$, based on the 1\\AA\\ resolution of the resampled grid. All quasars listed above the horizontal divider are observed with GNIRS and all quasars listed below the divider are observed with X-shooter. The last column indicates our source for the black hole mass and bolometric luminosity. The bolometric correction used to measure the luminosity, single-epoch virial mass calibration, and Fe\\,\\textsc{ii}\\ template used to measure Mg\\,\\textsc{ii}\\ are all consistent throughout the sample.}  \n\\begin{tabular}{lcccccccc}\n\\hline \\hline\n Name & R.A. & Decl. & Mg\\,\\textsc{ii}\\ Redshift & C\\,\\textsc{iv}\\ Blueshift & SNR & \\nv\/\\civ & (\\siiv+\\oiv)\/\\civ & M$_{\\rm{BH}}$\/L$_{\\rm{bol}}$ Ref\\\\\n & (J2000) & (J2000) & & (km s$^{-1}$) & & & & \\\\\n \\hline\nJ0024+3913 & 00:24:29.77 & 39:13:19.00 & 6.620 $\\pm$ 0.004 & 635 $\\pm$ 255 & 9.45 & 0.77 $\\pm$ 0.15 & 0.27 $\\pm$ 0.06 & 1 \\\\ \nJ0829+4117 & 08:29:31.97 & 41:17:40.40 & 6.773 $\\pm$ 0.007 & 1574 $\\pm$ 336 & 16.33 & 0.49 $\\pm$ 0.09 & 0.12 $\\pm$ 0.05 & 1 \\\\ \nJ0837+4929 & 08:37:37.84 & 49:29:00.40 & 6.702 $\\pm$ 0.001 & 600 $\\pm$ 204 & 30.91 & 1.49 $\\pm$ 0.08 & 0.66 $\\pm$ 0.10 & 1 \\\\ \nJ0910+1656 & 09:10:13.63 & 16:56:29.80 & 6.719 $\\pm$ 0.005 & $-$159 $\\pm$ 279 & 9.42 & 0.50 $\\pm$ 0.08 & 0.19 $\\pm$ 0.09 & 1 \\\\ \nJ0921+0007 & 09:21:20.56 & 00:07:22.90 & 6.565 $\\pm$ 0.001 & 678 $\\pm$ 204 & 9.66 & 0.49 $\\pm$ 0.13 & 0.16 $\\pm$ 0.05 & 1 \\\\ \nJ1216+4519 & 12:16:27.58 & 45:19:10.70 & 6.648 $\\pm$ 0.003 & 4955 $\\pm$ 232 & 7.63 & --- & 0.50 $\\pm$ 0.33 & 1 \\\\ \nJ1342+0928 & 13:42:08.10 & 09:28:38.60 & 7.510 $\\pm$ 0.010 & 6900 $\\pm$ 405 & 24.79 & --- & 0.78 $\\pm$ 0.28 & 1 \\\\ \nJ2102-1458 & 21:02:19.22 & $-$14:58:54.00 & 6.652 $\\pm$ 0.003 & 3433 $\\pm$ 232 & 11.47 & 1.82 $\\pm$ 0.44 & 0.57 $\\pm$ 0.23 & 1 \\\\ \nP333+26 & 22:15:56.63 & 26:06:29.40 & 6.027 $\\pm$ 0.006 & 2534 $\\pm$ 325 & 5.73 & 0.80 $\\pm$ 0.47 & 0.27 $\\pm$ 0.19 & 2 \\\\ \nJ2338+2143 & 23:38:07.03 & 21:43:58.20 & 6.565 $\\pm$ 0.009 & --- & 7.07 & --- & --- & 1 \\\\ \\hline\nPSOJ007+04 & 00:28:06.56 & 04:57:25.64 & 6.001 $\\pm$ 0.002 & 3816 $\\pm$ 218 & 24.58 & --- & 0.45 $\\pm$ 0.20 & 3 \\\\ \nPSOJ025-11 & 01:40:57.03 & $-$11:40:59.48 & 5.816 $\\pm$ 0.004 & 2575 $\\pm$ 266 & 26.54 & --- & 0.70 $\\pm$ 0.13 & 3 \\\\ \nPSOJ029-29 & 01:58:04.14 & $-$29:05:19.25 & 5.976 $\\pm$ 0.001 & 3295 $\\pm$ 205 & 27.39 & 1.14 $\\pm$ 0.13 & 0.64 $\\pm$ 0.15 & 3 \\\\ \nATLASJ029-36 & 01:59:57.97 & $-$36:33:56.60 & 6.020 $\\pm$ 0.002 & 2705 $\\pm$ 217 & 27.35 & 1.07 $\\pm$ 0.11 & 0.49 $\\pm$ 0.13 & 3 \\\\ \nVDESJ0224-4711 & 02:24:26.54 & $-$47:11:29.40 & 6.528 $\\pm$ 0.001 & 2217 $\\pm$ 204 & 28.68 & 0.79 $\\pm$ 0.04 & 0.34 $\\pm$ 0.11 & 3 \\\\ \nPSOJ060+24 & 04:02:12.69 & 24:51:24.42 & 6.170 $\\pm$ 0.001 & 1082 $\\pm$ 204 & 30.73 & 0.76 $\\pm$ 0.05 & 0.10 $\\pm$ 0.05 & 3 \\\\ \nPSOJ065-26 & 04:21:38.05 & $-$26:57:15.60 & 6.188 $\\pm$ 0.001 & 8288 $\\pm$ 204 & 36.57 & --- & 0.78 $\\pm$ 0.65 & 3 \\\\ \nPSOJ108+08 & 07:13:46.31 & 08:55:32.65 & 5.945 $\\pm$ 0.001 & 4832 $\\pm$ 205 & 37.27 & --- & 0.78 $\\pm$ 0.27 & 3 \\\\ \nPSOJ158-14 & 10:34:46.50 & $-$14:25:15.58 & 6.068 $\\pm$ 0.001 & 2683 $\\pm$ 204 & 31.67 & 0.81 $\\pm$ 0.07 & 0.28 $\\pm$ 0.09 & 3 \\\\ \nJ1212+0505 & 12:12:26.98 & 05:05:33.49 & 6.439 $\\pm$ 0.001 & 4329 $\\pm$ 204 & 31.27 & --- & 0.93 $\\pm$ 0.37 & 3 \\\\ \nPSOJ217-16 & 14:28:21.39 & $-$16:02:43.30 & 6.150 $\\pm$ 0.001 & 4023 $\\pm$ 204 & 34.57 & --- & 0.30 $\\pm$ 0.13 & 3 \\\\ \nPSOJ242-12 & 16:09:45.53 & $-$12:58:54.11 & 5.830 $\\pm$ 0.001 & 891 $\\pm$ 205 & 15.55 & 0.87 $\\pm$ 0.11 & 0.34 $\\pm$ 0.22 & 3 \\\\ \nPSOJ308-27 & 20:33:55.91 & $-$27:38:54.60 & 5.799 $\\pm$ 0.001 & 1971 $\\pm$ 205 & 33.18 & 0.98 $\\pm$ 0.06 & 0.95 $\\pm$ 0.10 & 3 \\\\ \nPSOJ323+12 & 21:32:33.19 & 12:17:55.26 & 6.586 $\\pm$ 0.001 & 697 $\\pm$ 204 & 31.34 & 0.73 $\\pm$ 0.05 & 0.36 $\\pm$ 0.08 & 3 \\\\ \nPSOJ359-06 & 23:56:32.45 & $-$06:22:59.26 & 6.172 $\\pm$ 0.001 & 1082 $\\pm$ 204 & 35.36 & 0.96 $\\pm$ 0.13 & 0.11 $\\pm$ 0.04 & 3 \\\\ \n\\hline \\hline\n\\multicolumn{6}{l}{\\footnotesize\n$^{1}$ \\citet{yang2021probing}\n$^{2}$ \\citet{Shen_2019}\n$^{3}$ Mazzucchelli et al. in prep.}\n\\end{tabular}\n\\end{table*}\n\\endgroup\n\nWe present the measured line ratios of all individual and composite fits of bolometric luminosity and black hole mass in Figure \\ref{fig:line_ratio_Lbol_Mbh}. For comparison, we show SDSS low-redshift composites reported in \\citet{Xu2018} and high-redshift ($z\\sim6$) quasars observed with GNIRS from \\citet{Wang_2021}. Square symbols indicate measurements from fits of composites while circular points indicate fits of individual spectra. Our data, indicated in blue and black, have the highest SNR and spectral resolution of the data represented in the figure. Measurements of both metallicity-sensitive line ratios show a large scatter between individual quasar fits even when controlling for quasar luminosity or black hole mass. Particularly at a bolometric luminosity range of $\\log\\left(L_{\\rm{bol}}\/\\rm{erg \\, s}^{-1}\\right) = 47.30-47.35$, we see over a factor of 8 difference between the individual quasar measured with the highest and lowest (\\siiv+\\oiv)\/\\civ\\ line ratio, as seen in Figure \\ref{fig:line_ratio_Lbolscatter}. The composites suppress the high variance that we observe in the individual measurements, which can be attributed to varied C\\,\\textsc{iv}\\ blueshifts. The associated uncertainty of the bolometric luminosity and black hole mass for each composite is determined by the mean and standard deviation of the input spectra. We note that in \\citet{Xu2018}, the black hole masses are estimated using the C\\,\\textsc{iv}\\ emission line which can be biased by its blueshift. \n\n\\begin{figure*}\n\\begin{tabular}{cc}\n  \\includegraphics[width=90mm]{figs\/line_ratio_Lbol.png} &   \\includegraphics[width=90mm]{figs\/line_ratio_Mbh.png} \n\\end{tabular}\n\\caption{\\nv\/\\civ\\ and (\\siiv+\\oiv)\/\\civ\\ flux ratios as a function of the quasar bolometric luminosity (left) and virially estimated black hole mass (right). The low-redshift sample (2.0 < $z$ < 5.0) indicated in grey is from \\citet{Xu2018} while another higher-redshift comparison sample indicated in red is sourced from \\citet{Wang_2021}. Our sample is presented in blue and black. Square points with capped error bars indicate composites while circular points indicate individual fits. Not all individual quasars involved in the composites are plotted. The single red square denotes the composite from \\citet{Wang_2021}. The black hole masses in this study and in \\citet{Wang_2021} are estimated with single-epoch virial estimates using the Mg\\,\\textsc{ii}\\ emission line, while the \\citet{Xu2018} study uses the C\\,\\textsc{iv}\\ emission line. {The orange shaded space indicates a range of line ratios which are consistent with the metallicity indicated in the secondary axis based on photoionisation calculations with different ionizing SEDs}. The overlapping region in the \\nv\/\\civ\\ plot indicates a range of line ratios which is consistent with both $\\rm{Z} = 10\\,\\rm{Z}_{\\odot}$ and $\\rm{Z} = 20\\,\\rm{Z}_{\\odot}$ \\citep[e.g.][]{Hamann_2002, Nagao_2006}. Metallicity values larger than 10 Z$_{\\odot}$ are extrapolated.} \\label{fig:line_ratio_Lbol_Mbh}\n\\end{figure*}\n\nThe line flux ratio measurements of our high-redshift quasar sample are essentially indistinguishable from the lower-redshift results of comparable luminosity and black hole mass sourced from \\citet{Xu2018}. Therefore, we do not observe appreciable evolution with redshift. However, our high-redshift sample does not show a statistically appreciable correlation between observed line ratios and the bolometric luminosity, as evidenced in previous work \\citep{hamann_1993, Dietrich_2003, Nagao_2006, Xu2018}. This may be because this sample covers a restricted luminosity range compared to the lower redshift sample. Deeper observations of quasars with high redshift and lower luminosity \\citep[e.g.][]{Matsuoka_2016} are needed to verify any trend in emission-line ratio with quasar bolometric luminosity. On the other hand, we recover the positive correlation between (\\siiv+\\oiv)\/\\civ\\ and the estimated black-hole mass as shown in Figure \\ref{fig:line_ratio_Lbol_Mbh}. The measurements of individual quasars exhibit large scatter for very similar quasar properties. We present the line ratio dependence on the C\\,\\textsc{iv}\\ blueshift in the following section to explain this variance.\n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{figs\/line_ratio_Lbolscatter.png}\n    \\caption{Example spectra of three quasars in the sample (PSOJ242-12, PSOJ308-27, and PSOJ359-06) which exhibit similar bolometric luminosities. The raw spectrum is shown in this plot, but many of the significant absorption features (e.g. in PSOJ359-06) are masked by the post-processing procedure. Overall line fits are shown in black. A large scatter in the (\\siiv+\\oiv)\/\\civ\\ emission line flux ratio is observed between quasar spectra with significantly different C\\,\\textsc{iv}\\ equivalent width, which is inversely correlated with the C\\,\\textsc{iv}\\ blueshift.}\n    \\label{fig:line_ratio_Lbolscatter}\n\\end{figure*}\n\n\n\\subsection{C\\,\\textsc{iv}\\ Blueshift and Line Flux Ratios} \\label{sec:blueshift}\nFigure \\ref{fig:blueshift_line_ratio_results} plots emission line flux ratios against the C\\,\\textsc{iv}\\ blueshift along with the estimated black hole mass as a third axis, represented by the blue-green color scale. The C\\,\\textsc{iv}\\ blueshift and uncertainty of the composite spectra is obtained from the mean and standard deviation of the input spectra. We prescribe an uncertainty floor of the C\\,\\textsc{iv}\\ blueshift equivalent to 200 km s$^{-1}$ based on the 1 \\AA\\ wavelength grid, but the total uncertainty for measurements of individual quasars is composed also of the redshift error added in quadrature. On average, the C\\,\\textsc{iv}\\ blueshift error is 230 km s$^{-1}$. Measurements of the N\\,\\textsc{v}\\ emission line becomes more challenging to deblend from the Ly\\,\\textsc{$\\alpha$}\\ flux at high blueshifts, especially for individual lower SNR spectra, thus the high C\\,\\textsc{iv}\\ blueshift parameter space for \\nv\/\\civ\\ is only sparsely explored.\n\nThe results from Figures \\ref{fig:line_ratio_Lbolscatter} and \\ref{fig:blueshift_line_ratio_results} demonstrate that the measured emission line ratios are strongly correlated with the C\\,\\textsc{iv}\\ blueshift. Controlling for the C\\,\\textsc{iv}\\ spectral shape in Figure \\ref{fig:blueshift_line_ratio_results}, the relationship between the two emission line ratios with black hole mass is no longer clear. The C\\,\\textsc{iv}\\ lines of the highest mass quasars are typically more blueshifted, but the most massive quasars do not necessarily have the highest line ratios among other quasars with similar blueshifts. It could also be seen that quasars with moderate ($\\sim$1000 km s$^{-1}$) C\\,\\textsc{iv}\\ blueshifts can have a very large scatter in observable line ratios whereas quasars with higher blueshifts consistently possess some of the highest line ratios observed in our sample. At high C\\,\\textsc{iv}\\ blueshifts, the lower flux of the C\\,\\textsc{iv}\\ line, as evidenced by its correlation with narrower EWs shown in Figure \\ref{fig:CIV_blueshift_ew}, drives the C\\,\\textsc{iv}-normalised flux ratio measurements higher. The responses of the N\\,\\textsc{v}, Si\\,\\textsc{iv}, and O\\,\\textsc{iv}\\ equivalent widths are not proportionate to that of the C\\,\\textsc{iv}\\ line as the C\\,\\textsc{iv}\\ blueshift rises. We see similar trends on other metallicity-sensitive line ratios that depend on the C\\,\\textsc{iv}\\ flux as shown in Appendix Figure \\ref{fig:other_line_ratios}.\n\nFigure \\ref{fig:blueshift_mbh_lbol} presents the correlations found in our sample between the C\\,\\textsc{iv}\\ blueshift with the virially estimated black hole mass, quasar bolometric luminosity, and Eddington ratio. The C\\,\\textsc{iv}\\ blueshift is not significantly correlated with the quasar bolometric luminosity, but there is a moderate relationship between the C\\,\\textsc{iv}\\ blueshift with the Eddington ratio and estimated black hole mass, with the magnitude of Spearman correlation coefficients greater than 0.4 and at least 5\\% significance. We show in Figure \\ref{fig:line_ratio_residual} the residuals calculated by subtracting the correlation found between composites of C\\,\\textsc{iv}\\ blueshift and the (\\siiv+\\oiv)\/\\civ\\ line ratio from the measured line ratios of individual quasars. The results show that higher mass and more luminous quasars are not more likely to lie above this relationship, indicating no strong correlation with black hole mass or luminosity when controlling for blueshift \\citep[also see][]{metal_density}. The stronger and more significant correlation between line ratios and the C\\,\\textsc{iv}\\ blueshift {could be attributed to the fact that the} C\\,\\textsc{iv}\\ blueshift {is a more direct observable compared to the black hole mass or luminosity, which are estimated using calibrations with large associated uncertainties. When not controlled, the} C\\,\\textsc{iv}\\ blueshift {can bias other correlations found between quasar properties and the metallicity in the BLR, such as the apparent} mass correlation in Figure \\ref{fig:line_ratio_Lbol_Mbh}. It's important to note that the C\\,\\textsc{iv}\\ blueshift and quasar properties are not independent for our sample. This suggests that studies measuring metallicity-sensitive line flux ratios dependent on the C\\,\\textsc{iv}\\ flux should consider the C\\,\\textsc{iv}\\ spectral shape before interpreting the diversity of emission-line ratios as an indication of evolution in the BLR metallicity. \n\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{figs\/line_ratio_blueshift.png}\n    \\caption{\\nv\/\\civ\\ and (\\siiv+\\oiv)\/\\civ\\ flux ratios as a function of the C\\,\\textsc{iv}\\ blueshift of quasars in the sample. Blueshift composites are presented in black while individual fits are mapped onto a blue-green gradient scaled to the black hole mass. The minimum C\\,\\textsc{iv}\\ blueshift error is 200 km s$^{-1}$ based on the 1\\AA\\ wavelength bins, but the contribution from the systemic redshift error is added in quadrature for an average total of 230 km s$^{-1}$ uncertainty. The Spearman correlation coefficients and p-values are derived from the fits to individual quasars. {The orange shaded space indicates a range of line ratios which are consistent with the metallicity indicated in the secondary axis based on photoionisation calculations with different ionizing SEDs}. The overlapping region in the \\nv\/\\civ\\ plot indicates a sub-space of parameters which is consistent with both $\\rm{Z} = 10\\,\\rm{Z}_{\\odot}$ and $\\rm{Z} = 20\\,\\rm{Z}_{\\odot}$. The line ratio correlation with the C\\,\\textsc{iv}\\ blueshift is more significant than the correlation with black hole mass or bolometric luminosity.}\n    \\label{fig:blueshift_line_ratio_results}\n\\end{figure}\n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{figs\/CIV_Blueshift_corr_v2.png}\n    \\caption{The quasar black hole mass (left), bolometric luminosity (middle), and Eddington ratio (right) are plotted against the measured C\\,\\textsc{iv}\\ blueshift for our sample. The least-squares linear fits are shown along with the Spearman r-coefficients and their significance. There is a moderate, but significant, correlation between the quasar outflow indicator, i.e. the C\\,\\textsc{iv}\\ blueshift, and the black hole mass. The correlation between the C\\,\\textsc{iv}\\ blueshift with the Eddington ratio is weaker and less significant, while no significant correlation was found with the bolometric luminosity.}\n    \\label{fig:blueshift_mbh_lbol}\n\\end{figure*}\n\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{figs\/line_ratio_residual.png}\n    \\caption{(\\siiv+\\oiv)\/\\civ\\ residual as a function {of} black hole mass (left) and bolometric luminosity (right). The residual is measured by subtracting the correlation found between composites of C\\,\\textsc{iv}\\ blueshift and the (\\siiv+\\oiv)\/\\civ\\ line ratio from the measured line ratios of individual quasars. The scatter is large around zero with no apparent systematic deviation with black hole mass or bolometric luminosity. This shows that higher mass or more luminous quasars are not more likely to exhibit higher line ratios than their smaller or fainter counterparts. However, we caution the reader when interpreting this relationship, because the C\\,\\textsc{iv}\\ blueshift and quasar properties are not independent.}\n    \\label{fig:line_ratio_residual}\n\\end{figure}\n\n\n\\subsection{Inferred Metallicity in the Quasar BLR} \\label{sec:metallicity_result}\nUsing models derived from the photoionisation code \\texttt{Cloudy}, we convert the measured \\nv\/\\civ\\ and (\\siiv+\\oiv)\/\\civ\\ line ratios into metallicity estimates in the BLR. Figures \\ref{fig:line_ratio_Lbol_Mbh} and \\ref{fig:blueshift_line_ratio_results} show the range of line ratios consistent with 5, 10, and 20 Z$_{\\odot}$ for \\nv\/\\civ\\ and  1, 5, 10, and 20 Z$_{\\odot}$ for (\\siiv+\\oiv)\/\\civ. The minimum and maximum bounds of each metallicity estimate are determined by the variations on the assumed ionizing SED used in \\texttt{Cloudy} photoionisation LOC models presented in \\citet{Hamann_2002} and \\citet{Nagao_2006}. The central tick is determined by the median of all relevant models. Larger variations in metallicity can be seen for the \\nv\/\\civ\\ line ratio, indicating that (\\siiv+\\oiv)\/\\civ\\ is less dependent on the shape of the ionizing flux SED \\citep[e.g.][]{Nagao_2006, Matsuoka_2011, Maiolino_2019_metallica}. The overlapping region in the \\nv\/\\civ\\ plot shows a range in the line ratio which is consistent with both $\\rm{Z} = 10\\,\\rm{Z}_{\\odot}$ and $\\rm{Z} = 20\\,\\rm{Z}_{\\odot}$ depending on the referenced photoionisation model, implying a factor of two uncertainty. The results for the other line ratios ((\\oiii+\\alii)\/\\civ, \\aliii\/\\civ, \\siiii\/\\civ, and \\ciii\/\\civ) plotted against quasar bolometric luminosity, estimated black hole mass, and C\\,\\textsc{iv}\\ blueshift are presented in Figure \\ref{fig:other_line_ratios} in the Appendix. Generally, these other line ratios predict metallicities similar to (\\siiv+\\oiv)\/\\civ.\n\nSpectra with high C\\,\\textsc{iv}\\ blueshift are dominated by emission from an outflowing BLR wind, which is correlated with high X\/C\\,\\textsc{iv}\\ line flux ratios. Photoionisation models suggest that the wind emission originates from higher density gas clouds closer in to the accretion disk, illuminated by high ionising fluxes, while the core emission is composed of emission from clouds with a broad range of physical properties, as in the LOC model \\citep{metal_density}. We therefore consider the metallicity results from spectra where the C\\,\\textsc{iv}\\ blueshift < 1500 km s$^{-1}$, minimizing the contribution from the wind emission. The two composites satisfying this requirement yield (\\nv\/\\civ, (\\siiv+\\oiv)\/\\civ) line ratios of ($0.83 \\pm 0.15$, $0.31 \\pm 0.13$) and ($0.84 \\pm 0.09$, $0.22 \\pm 0.03$) for C\\,\\textsc{iv}\\ blueshifts from $-$200-680 and 680-1500 km s$^{-1}$ respectively. We do not consider the other line ratios in this discussion as they are substantially more difficult to measure. The \\nv\/\\civ\\ line ratio typically predicts higher metallicities with greater corresponding uncertainty than the (\\siiv+\\oiv)\/\\civ\\ line ratio. Figure \\ref{fig:blueshift_line_ratio_results} also plots the line ratios for each C\\,\\textsc{iv}\\ blueshift composite and the inferred metallicity in the secondary axis. Using the aforementioned reference photoionisation models, the measured \\nv\/\\civ\\ line ratio is consistent with being produced by gas clouds with Z$_{-\\rm{200-680}}$ = $9.77 \\pm 2.35$ Z$_{\\odot}$ and Z$_{\\rm{680-1500}}$ = $9.96 \\pm 2.42$ Z$_{\\odot}$ for the two lowest C\\,\\textsc{iv}\\ blueshift composites. For (\\siiv+\\oiv)\/\\civ, it is Z$_{-\\rm{200-680}}$ = $4.61 \\pm 0.01$ Z$_{\\odot}$ and Z$_{\\rm{680-1500}}$ = $2.01 \\pm 0.01$ Z$_{\\odot}$. Both metallicity indicators individually suggest super-solar metallicities with high significance ($\\gtrsim$ 4-$\\sigma$). The absolute measured metallicity differs by a factor of 2-4 between the indicators although the (\\siiv+\\oiv)\/\\civ\\ line provides more robust results and is less affected by the model chosen for the ionising flux. Using the (\\siiv+\\oiv)\/\\civ\\ line ratio, we can see that the metallicity in the quasar BLR at $z\\sim6$ is at least 2-4 times super-solar. Inferred metallicities from spectra observed with high C\\,\\textsc{iv}\\ blueshifts range from ${\\rm{Z}}_{\\rm{>1500}}$ = 8 Z$_{\\odot}$ to as high as ${\\rm{Z}}_{\\rm{>3000}}$ = 20 Z$_{\\odot}$\n\n\n\\section{Discussion} \\label{sec:discussion}\nPrevious studies of chemical abundances in the BLR have suggested metallicities that are several times solar across a wide range of redshifts (2.0 < $z$ < 7.5) \\citep[e.g.][]{hamann1992, Dietrich_2003, metallicity_distant, Xu2018, Onoue_2020}, consistent with some galactic chemical evolution models \\citep{Tinsley_1980, Arimoto_1987, hamann_1993, Hamann_1999}. Complementary probes targeting quasar narrow absorption features also suggest super-solar (Z > 2 Z$_{\\odot}$) metallicites \\citep[e.g.][]{Hamann_1999, dodorico_2004_absorption, Jiang_2018, Maiolino_2019_metallica}. The lack of apparent redshift evolution up to $z\\sim6$ stands in contrast to studies of metallicity in star-forming galaxies and Lyman-break galaxies up to $z\\sim3.5$, which show evolution in the mass-metallicity relationship and an overall decrease in metallicity with redshift \\citep[e.g.][]{Maiolino_2008, Mannucci_2009_LSD}. It's possible to infer the host galaxy metallicity using the mass of the central black hole using the tight (0.1 dex) galaxy stellar mass - gas phase metallicity relationship (MZR) \\citep[e.g.][]{Maiolino_2008, Dave_2017_mufasa, Curti_2019_Klever, Maiolino_2019_metallica, Sanders_2021_MOSDEF} combined with the M$_{\\rm{BH}}$\/M$_{\\rm{host}}$ ratios \\citep[e.g.][]{Targett_2012}. The results from comparisons between quasar BLR and host galaxy metallicities at redshifts $2.25 < z < 5.25$ suggest that the BLR is enriched in excess of the inferred metallicities of the host galaxies, which are approximately solar \\citep{Xu2018}. This discrepancy has been attributed to the black hole mass-metallicity relationship and selection effects, where only the most massive and enriched high-redshift quasars are selectively observed in a magnitude-limited survey \\citep[e.g.][]{metallicity_distant, Maiolino_2019_metallica}. However,  \\citet{Xu2018}, \\citet{Wang_2021}, and this paper study samples with comparable quasar properties as shown in Figure \\ref{fig:line_ratio_Lbol_Mbh}. The \\citet{Xu2018} composites include hundreds of SDSS DR12 quasar spectra in the redshift range $2.25 < z < 5.25$, whereas \\citet{Wang_2021} utilises a higher-redshift sample with 33 $z\\sim6$ quasars. These quasars occupy a similar black hole mass and luminosity range, and show very similar line ratios within the scatter of the data, suggesting that a selection bias is not sufficient to explain the apparent lack of redshift evolution. \n\nAdditionally, we note that the C\\,\\textsc{iv}\\ blueshift, a signature of quasar outflows, is a significant factor correlated with the measured C\\,\\textsc{iv}\\ flux (see Figure \\ref{fig:CIV_blueshift_ew}). According to a study of 34 low-redshift quasars spanning nearly 3 dex in black hole mass and bolometric luminosity, outflow indicators are not correlated with black hole mass and only marginally correlated with luminosity and Eddington ratio \\citep{shin_2017_outflow}. Although our study covers a smaller range of quasar parameters, our results in Figure \\ref{fig:blueshift_mbh_lbol} show a moderate, but significant, correlation between the C\\,\\textsc{iv}\\ blueshift and the black hole mass. There is also a similar negative correlation between the C\\,\\textsc{iv}\\ blueshift and the Eddington ratio, and no significant correlation with quasar bolometric luminosity. There are several methodical differences between the measurements in our study and those in \\citet{shin_2017_outflow}. We measure black hole virial masses based on Mg\\,\\textsc{ii}\\ instead of H$\\beta$, and calculate systemic redshifts from the Mg\\,\\textsc{ii}\\ line rather than from a combination of low-ionisation narrow lines (Si\\,\\textsc{ii}, O\\,\\textsc{ii}], O\\,\\textsc{i}], H$\\beta$). We also use different outflow indicators: our C\\,\\textsc{iv}\\ blueshift is defined in Equation \\ref{eq:blueshift}, in contrast to the ``velocity shift index'' (VSI) and ``blueshift and asymmetry index'' (BAI) defined in Equations 2 and 3 of \\citet{shin_2017_outflow}. According to our result, a correlation between the black hole mass or bolometric luminosity with the C\\,\\textsc{iv}\\ blueshift implies that the C\\,\\textsc{iv}\\ flux is anti-correlated with the C\\,\\textsc{iv}\\ blueshift by extension (see Figure \\ref{fig:CIV_blueshift_ew}). As the C\\,\\textsc{iv}\\ blueshift is found to be correlated with the metallicity-sensitive rest-frame UV line ratios and quasar properties, this has the potential to bias correlations between metallicity and black hole mass or luminosity. The relationship between the C\\,\\textsc{iv}\\ blueshift and these line ratios can be explained by increased gas opacity with metallicity, leading to larger absorption and increased acceleration \\citep[e.g.][]{Wang_2012}. However, the extreme metallicities ($\\sim 20\\, \\rm{Z}_{\\odot}$) seen in the most blueshifted high redshift ($z > 6.0$) quasars in our sample suggest that while the relationship between the C\\,\\textsc{iv}\\ blueshift and the line ratio is real, the comparison to the simple photoionisation models is no longer appropriate as emission from the BLR outflow dominates the observed spectrum. The dynamics, density, and geometry of the BLR wind is not the same as for symmetric core emission \\citep{metal_density}. An alternative explanation is that the C\\,\\textsc{iv}\\ blueshift relationship with the rest-frame UV line ratios is driven primarily by the weakening of the symmetric C\\,\\textsc{iv}\\ core emission and enhanced emission toward the line-of-sight from quasar orientation \\citep[e.g.][]{Yong_2020}. Further studies on quasar properties and their emission-line flux ratios will need to account for indications of quasar outflows or avoid using emission lines that are strongly affected by BLR outflow.\n\nIt has also been suggested that the observed diversity of line ratios (\\nv\/\\civ\\ and (\\siiv+\\oiv)\/\\civ\\ among others) can be attributed to the variation of density of the emitting gas and the incident ionizing flux instead of metallicity. \\citet{metal_density} proposes a model with two kinematically distinct regions, the core and the wind, that can reproduce the range of observed broad emission-line flux ratios under solar metallicities, as long as the spatial density distribution of the emitting gas clouds is adjusted accordingly. Such multiple zone photoionisation models have been used to great effect in reproducing quasar and AGN spectra \\citep[e.g.][]{Rees_1989, Peterson_1993, Baldwin_1996, Hamann_1998, Korista_2000_LOC}. The locally optimally emitting cloud (LOC) model, proposed in \\citet{Baldwin_1995_LOC}, is a natural extension of multi-zone models. The advantage of the LOC model is that the total line emission is composed of an integration over the density ($n_{\\rm{H}}$) and ionizing flux ($\\Phi_{\\rm{H}}$) parameter space assuming certain empirically motivated distribution functions \\citep{Nagao_2006}, thereby bypassing the need for specific knowledge of either $n_{\\rm{H}}$ or $\\Phi_{\\rm{H}}$. The properties of the emission lines are then dominated by the emitters that are optimally suited to emit the targeted line \\citep{Baldwin_1995_LOC}. It is well-documented that the line ratios depend sensitively on $n_{\\rm{H}}$ and $\\Phi_{\\rm{H}}$ \\citep[e.g.][]{Hamann_2002, Nagao_2006, metal_density}, but the LOC results represent the average properties of diverse quasar samples. We note that the high-density wind component ($n_{\\rm{H}} \\approx 10^{13-14}$ cm$^{-3}$, $\\Phi_{\\rm{H}} \\approx 10^{22-24}$ cm$^{-2}$ s$^{-1}$) and the range of typically assumed BLR properties ($n_{\\rm{H}} \\approx 10^{9-12}$ cm$^{-3}$, $\\Phi_{\\rm{H}} \\approx 10^{18-21}$ cm$^{-2}$ s$^{-1}$) suggested in \\citet{metal_density} are parameter ranges which are also covered by the LOC photoionisation models used in this study ($n_{\\rm{H}} \\approx 10^{7-14}$ cm$^{-3}$, $\\Phi_{\\rm{H}} \\approx 10^{17-24}$ cm$^{-2}$ s$^{-1}$) \\citep{Hamann_2002, Nagao_2006}. However, if the assumed cloud distribution functions used in photoionisation models are inaccurate, the absolute metallicity inferred from line ratios is subject to change. For example, photoionisation models using emission from clumpy disk winds can produce spectra resembling that of quasars \\citep{Dannen_2020_clumpy, Matthews_2020_clumpy}, showing that there are viable alternatives to the LOC models we have referenced for the conversions between line ratio and metallicity.\n\nThe results from \\citet{metal_density} further motivated us to use a quasar outflow indicator, the C\\,\\textsc{iv}\\ blueshift, as a control to limit the effect of the BLR wind. Even for composites of low C\\,\\textsc{iv}\\ blueshift where the assumed contribution to the overall emission from the wind is low, we observe in Figure \\ref{fig:blueshift_line_ratio_results} that the average emission-line properties are comparable to emission from gas clouds with metallicity several times solar (e.g. Z$_{-\\rm{200-680}}$ = $4.61 \\pm 0.01$ Z$_{\\odot}$ using (\\siiv+\\oiv)\/\\civ) under the LOC model.\n\nThe super-solar metallicities in the quasar BLR inferred from low C\\,\\textsc{iv}\\ blueshift composites imply rapid enrichment scenarios that are not unrealistic under normal galactic chemical evolution scenarios in the cores of massive galaxies \\citep[e.g.][]{Gnedin_1997, Dietrich_2003}. The BLR is a small nuclear region of the galaxy (<1 pc) with higher densities entailing shorter dynamical timescales \\citep[e.g.][]{Gnedin_1997, Cen_1999, Kauffman_2000, Granato_2004}. The total mass of the BLR is on the order of 10$^{4}\\, \\rm{M}_{\\odot}$ \\citep{Baldwin_2003}, and it can be enriched rapidly to super-solar metallicities within 10$^{8}$ yrs by a single supernova explosion every 10$^{4}$ yrs \\citep{metallicity_distant}. Under some multi-zone chemical evolution models, massive star formation in the galactic central regions and subsequent metal enrichment via supernovae can predict super-solar metallicities (up to 10 Z$_{\\odot}$) within 0.5 - 0.8 Gyrs \\citep[e.g.][]{hamann_1993, Friaca_1998, romano_2002}. This rapid enrichment scenario means that the properties of the BLRs do not necessarily trace the chemical properties of their host galaxies \\citep[e.g.][]{Suganuma_2006, Matsuoka_2018}. This is supported by studies presenting estimates of metallicity in the quasar narrow-line region (NLR) which represent a region over 1000 pc in size \\citep[e.g.][]{Bennert_2006_NLRsize}. The metallicity in the NLR was found to be 2-3 times lower than the BLR, following similar MZR trends as star-forming galaxies \\citep[e.g.][]{Dors_2019}. However, in addition to high-metallicity BLRs, there is now mounting evidence that entire host galaxies can be highly enriched to solar values in early cosmic time as evidenced by measurements of C, N, and O ions \\citep[e.g.][]{Walter_2003, Venemans_2017, Novak_2019, Pensabene_2021}.\n\n\nMore exotic enrichment scenarios such as enhanced supernova rates in central star clusters \\citep[e.g.][]{artymowicz_1993, Shields_1996}, star formation inside quasar accretion disks \\citep[e.g.][]{Collin_1999, Goodman_2004, Toyouchi_2021}, or nucleosynthesis without stars \\citep[e.g.][]{Chakrabarti_1999, Hu_2008, Datta_2019} are also capable of producing highly enriched BLRs in a short time. However, we do not consider these to be strictly necessary to explain the metallicities in the $z\\sim6$ redshift quasars inferred from observations in this study. Pushing metallicity estimates to even higher redshifts $z>8$ when the universe is only 0.6 Gyr old would place more stringent constraints on the metal enrichment timescales from the era of re-ionisation of the universe, where such rapid enrichment scenarios could be required to produce super-solar metallicities \\citep[e.g.][]{Friaca_1998}.\n\n\n\\section{Conclusions} \\label{sec:conclusion}\nIn this study, we examined a sample of 25 high-redshift (z > 5.8) quasars, 15 of which were observed with X-shooter during the XQR-30 programme and 10 of which were observed with Gemini North's GNIRS sourced from \\citet{Shen_2019} and \\citet{yang2021probing}. The sample from XQR-30 contains the highest-quality spectra covering the rest-frame UV emission lines observed in quasars in this redshift range. The bolometric luminosity of the quasars in this sample covers $\\log\\left(\\rm{L}_{\\rm{bol}}\/\\rm{erg \\, s}^{-1}\\right) = 46.7-47.7$ assuming a bolometric correction factor of 5.15 from the continuum luminosity at 3000\\AA. The black hole mass range in the sample is $(0.2 - 6.0) \\times 10^9\\, \\rm{M}_{\\odot}$, measured with single-epoch virial mass estimates utilizing the FWHM of the Mg\\,\\textsc{ii}\\ emission line. We measured the blueshift of the C\\,\\textsc{iv}\\ line in most of the quasars in this sample and created composites by quasar luminosity, black hole mass, and C\\,\\textsc{iv}\\ blueshift. We then measured broad rest-frame UV emission-line flux ratios in individual quasar spectra and all composites. The main results are as follows:\n\\begin{itemize}\n    \\item Due to the relationship between the C\\,\\textsc{iv}\\ blueshift and its equivalent width, the metallicity-sensitive broad emission-line ratios correlate with the C\\,\\textsc{iv}\\ blueshift, which is an indicator of the projected BLR outflow velocity. If not accounted for, this correlation biases studies of quasar metallicity and its relationship with black hole mass and luminosity. The correlation between the metallicity-sensitive emission line flux ratios and the C\\,\\textsc{iv}\\ blueshift is stronger and more significant than for the quasar bolometric luminosity or black hole mass. \n    \\item Comparing against \\texttt{Cloudy}-based photoionisation models, the metallicity inferred from line ratios of the high-redshift ($z\\sim6$) quasars in this study is several (at least 2-4) times super-solar, consistent with studies of much larger samples at lower redshifts and similar studies at comparable redshifts. We also find no strong evidence of redshift evolution in the BLR metallicity, indicating that the BLR is already highly enriched at $z\\sim6$. The metallicity-sensitive emission-line flux ratios are sensitive to the density $n_{\\rm{H}}$ of gas clouds and the incident ionizing flux $\\Phi_{\\rm{H}}$, but we use locally optimally-emitting cloud photoionisation models to draw conclusions based on the average properties of diverse samples of quasars. Our low C\\,\\textsc{iv}\\ blueshift composites are good probes of metallicity at this redshift as they minimise the effects of the BLR wind.\n    \\item The lack of redshift evolution in the BLR metallicity is contrary to studies of metallicity in star-forming and Lyman-break galaxies, which show a significant redshift dependence. Furthermore, estimates of host galaxy properties based on black hole mass suggest metallicities that are approximately solar. We find that selection effects are not sufficient to explain the apparent lack of redshift evolution and the discrepancy between the BLR metallicity and host galaxy metallicity. However, given the small scale of the BLR, rapid enrichment scenarios make it a poor tracer of host galaxy metallicity.\n    \\item The super-solar metallicity inferred for BLRs at $z\\sim6$ provides stringent constraints on the timescales of star formation and metal enrichment in the vicinity of some of the earliest supermassive black holes. Rapid metal enrichment scenarios of the BLR are not unrealistic under normal galactic chemical evolution models and more exotic explanations, such as nucleosynthesis or star formation inside the accretion disk, are not strictly necessary. \n\\end{itemize}\n\nIntrinsic absorption lines could, in principle, provide more straightforward estimates of the BLR metallicity \\citep[e.g.][]{Hamann_1999, dodorico_2004_absorption, Maiolino_2019_metallica}. In the past, such studies were not possible due to low SNR of quasar spectra at $z \\sim 6$, but the high-quality data of XQR-30 enables this type of investigation, which will be explored in a future study. More precise metallicity diagnostics would solidify and refine these results, especially for individual quasars. \n\n\n\\section*{Acknowledgements}\n\nWe thank Matthew Temple for the helpful discussion {and the reviewer, Yoshiki Matsuoka, for the thoughtful comments and suggestions which have improved this work.}\n\nThe results of this research is based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programme 1103.A-0817. \n\nThis work is also based, in part, on observations obtained at the international Gemini Observatory, a program of NSF's NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini Observatory partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigaci\\'{o}n y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog\\'{i}a e Innovaci\\'{o}n (Argentina), Minist\\'{e}rio da Ciencia, Tecnologia, Inova\\c{c}\\~{o}es e Comunica\\c{c}\\~{o}es (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea).\n\nS.L. is grateful to the Australian National University Research School of Astronomy \\& Astrophysics (ANU\/RSAA) for funding his Ph.D. studentship and the European Southern Observatory for the research internship.\n\nCAO was supported by the Australian Research Council (ARC) through Discovery Project DP190100252.\n\nM.B. acknowledges support from PRIN MIUR project ``Black Hole winds and the Baryon Life Cycle of Galaxies: the stone-guest at the galaxy evolution supper'', contract \\#2017PH3WAT. \n\nACE acknowledges support by NASA through the NASA Hubble Fellowship grant $\\#$HF2-51434 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. \n\nSEIB acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant\nagreement No. 740246 \"Cosmic Gas\").\n\nJTS acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 885301 ``Quasar Chronicles'').\n\n\n\\section*{Data Availability}\nThe data underlying this article will be shared on reasonable request to the corresponding author. \n\n\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe study of quantum many-body systems is one of the most active fields of modern condensed-matter physics. Among the most celebrated effects, we can mention frictionless flows in superfluid and superconducting systems and the geometrical quantization features of the fractional quantum Hall effect. While this physics was traditionally studied in liquid Helium samples~\\cite{pinesnozieres,leggett2004}, in atomic nuclei~\\cite{ringschuck}, in quark-gluon plasmas~\\cite{yagi,sinha}, or in electron gases confined in solid-state devices~\\cite{Schrieffer1964,Mahan1990,Tinkham2004,Yoshioka2002}, {the last two decades have witnessed impressive advances} using ultra-cold atomic gases trapped in magnetic or optical traps~\\cite{Dalfovo1999,Bloch2008,Giorgini2008}. \n\nIn the last few years, a growing community has started investigating many-body effects in the novel context of the so-called quantum fluids of light~\\cite{RMP}, i.e. assemblies of many photons confined in suitable optical devices, where effective photon-photon interactions arise from the optical nonlinearity of the medium. After the pioneering studies of Bose-Einstein condensation~\\cite{BEC} and superfluidity~\\cite{superfluidity} effects in dilute photon gases in weakly nonlinear media, a great interest is presently being devoted to strongly nonlinear systems, where even single photons are able to appreciably affect the optical properties of the system. \n\nThe most celebrated example of such physics is the photon blockade effect~\\cite{imamoglu}, where the presence of a single photon in a cavity is able to {detune the cavity} frequency away from the pump laser, so that photons behave as effectively impenetrable particles. Experimental realizations of this idea have been reported by several groups using very different material platforms, from single atoms in macroscopic cavities~\\cite{birnbaum}, to single quantum dots in photonic crystal cavities~\\cite{vuckovic,Reinhard}, to single Josephson qubits in circuit QED devices for microwaves~\\cite{CiQED-blockade,CiQED_rev}.\n\nScaling up to arrays of many cavities coupled by photon tunneling is presently a hot challenge in experimental physics{, as it would realize a Bose-Hubbard model for photons where the photon blockade effect may lead to a rich} physics, including the superfluid to Mott-insulator phase transition at a commensurate filling or Tonks-Girardeau gases of impenetrable photons in one-dimensional continuum models. The first works on strongly correlated photons were restricted to quasi-equilibrium regimes where the photon loss rate is much slower than the internal dynamics of the gas so that the system has time to thermalize and\/or be adiabatically transfered to the desired strongly correlated state~\\cite{photonMI,TGHarvard}. While this assumption might be satisfied in suitably designed circuit-QED devices in the microwave domain, radiative losses are hardly negligible in realistic optical cavities in the infrared or visible domain, {so that} thermalization is {generally} far from being granted~\\cite{RMP,CiQED_rev}.\n\nAs a result, a very active attention has been {recently} devoted to the peculiar non-equilibrium effects that arise for realistic loss rates. Starting from the pioneering work on photon blockade in non-equilibrium photonic Josephson junctions~\\cite{gerace-fazio}, the interest has {been focused} on the study of schemes to generate strongly correlated many-body states {in the very non-equilibrium context of photon systems, where the steady-state is not determined by a thermal equilibrium condition, but by a dynamical balance of driving and losses.}\n\nThe first such scheme proposed in~\\cite{TGnoi} was based on a coherent pumping: provided the different many-body states are sufficiently separated in energy, many-photon processes driven by the coherent external laser are able to selectively address each many-body state as done in optical spectroscopy of atomic levels. In this way, the non-equilibrium condition is no longer just a hindrance, but offers new perspectives, as it allows to individually probe each excited state. Furthermore, the appreciable radiative losses make microscopic information on the many-body wavefunction be directly encoded in the quantum coherence of the secondary emission from the device~\\cite{FQH,UWC,Braid}. While this coherent pumping scheme offers a viable way to generate and control few photon states in small arrays, its efficiency is restricted to mesoscopic systems where the different states are well-separated in energy. Moreover, this scheme intrinsically leads to coherent superpositions of states of different photon number: while this feature is intriguing in view of observing many-body braiding phases~\\cite{Braid}, it is not ideally suited to generate states with a well-defined photon number such as Mott-insulator states.\n\nThe identification of new schemes that do not suffer from these limitations is therefore of great importance in view of experiments. In the present work we study the potential of frequency-dependent gain processes to selectively generate strongly correlated states of photons in arrays of strongly nonlinear cavities. The frequency-dependence of amplification is a well-known fact of laser physics and is often exploited to choose and stabilize a desired lasing mode~\\cite{laser}. In the last years, a series of works by our groups~\\cite{PRL10,Chiocchetta13} have explored its effect on exciton-polariton Bose-Einstein condensation experiments, in particular questioning the apparent thermalization of the non-condensed fraction~\\cite{Chiocchetta14,Bajoni,Weitz,Kirton14,alessio2}. All these works were however restricted to the weakly interacting regime where quantum fluctuations can be treated in the input-output language by means of a Bogoliubov-like linearized theory around the mean-field. Here we tackle the far more difficult case of strong nonlinearities, which requires including the non-Markovian features due to the frequency-dependent gain into the many-body master equation for the strongly interacting photons and then to solve the quantum many-body theory of the generalized driven-dissipative Bose-Hubbard model. \n\nIn the last years, similar questions have been theoretically addressed by several groups. Just to mention a few of them, a scheme to obtain a thermal state at finite temperature with a non-vanishing effective chemical potential for photons has been proposed in~\\cite{Adhikari} using a clever parametric system-bath coupling with as special eye to circuit-QED and opto-mechanical systems. A further development in this direction~\\cite{Kapit} has considered pumping by two-photon processes in the presence of an auxiliary shadow lattice in a circuit-QED architecture: in spite of the complexity of the proposed set-up, the mechanism underlying the stabilization of many-body states is very similar to our frequency-dependent gain. With respect to these proposals and to the engineered dissipations originally proposed for atoms~\\cite{diehl} and then extended to photons~\\cite{hafezi,domokos} to organic polaritons \\cite{Lambrecht,keeling2}, and circuit QED systems \\cite{aron,gourgy,tureci}, our approach has the crucial advantage of being based on a quite commonly observed feature of laser and photonic systems such as a frequency-dependent gain.\nFinally, a pioneering discussion of the onset of collective coherence in a related model of a cavity array embedding population-inverted atoms has recently appeared in~\\cite{Ruiz}, but little attention was paid to the effect of strong nonlinearities nor to the development of a tractable quantum formalism.\n\nThe aim of this article is to introduce the readers to the basic physics of a frequency-dependent incoherent pumping and to first illustrate the consequences of the resulting non-Markovianity in the simplest configurations before attacking more complex many-body effects. With this idea in mind, the structure of the article is the following. In Sec.\\ref{sec:Presentation-of-the} we present the physical system and we develop the theoretical model based on a master equation for the cavities coupled to the atoms of the gain medium. The projective method to eliminate the atomic degrees of freedom and write a master equation for the photonic density matrix is sketched in Sec.\\ref{sec:Closed-master-equation} along the lines of the general theory of~\\cite{Breuer}. First application of the method to a single cavity configuration is discussed in Sec.\\ref{sec:onecavity} and specific features of the weak and the strong nonlinearity cases are illustrated, e.g. a novel mechanism for optical bistability and the selective generation of Fock states with a well defined photon number. The richer physics of many cavity arrays is discussed in Sec.\\ref{sec:Arrays-of-cavities}: In a Markovian regime, the photonic steady state has the surprisingly trivial form of a Grand-Canonical distribution of infinite temperature, and therefore is fully independent of the many-body photonic Hamiltonian. In a weakly non-Markovian regime, an effective Grand-Canonical distribution of finite temperature is obtained even in the absence of thermalization mechanisms; in a strongly nonlinear and non-Markovian regime, signatures of a Mott insulator state with one photon per cavity are illustrated. Conclusions are finally drawn in Sec.\\ref{sec:Conclu}. In the Appendices, we provide the details of the derivation of the photonic master equation using projective methods, on the exact stationary state in the Markovian case, on a perturbative expansion of the coherences in the weakly non-Markovian limit, and on further numerical validation of the purely photonic master equation.\n\n\n\n\\section{The physical system and the theoretical model}\\label{sec:Presentation-of-the}\n\n\\subsection{The physical system}\n\nIn this work, we consider a driven-dissipative Bose-Hubbard model for photons in an array of $k$ coupled nonlinear cavities of natural frequency $\\omega_{cav}$. In units such that $\\hbar=1$, the Hamiltonian for the isolated system dynamics has the usual form~\\cite{RMP,CiQED_rev,Hartmann_rev}:\n\\begin{equation}\nH_{ph}=\\sum_{i=1}^{k}\\left[\\omega_{cav}a_{i}^{\\dagger}a_{i}+\\frac{U}{2}a_{i}^{\\dagger}a_{i}^{\\dagger}a_{i}a_{i}\\right]{-}\\sum_{\\avg{i,j}}\\left[ Ja_{i}^{\\dagger}a_{j}+hc\\right].\n\\end{equation}\nThey are arranged in a one-dimensional geometry {and} are coupled via tunneling processes with amplitude $J$. Each cavity is assumed to contain a  Kerr nonlinear medium, which induces effective repulsive interactions between photons in the same cavity with an interaction constant $U$ proportional to the Kerr nonlinearity $\\chi^{(3)}$.\nDissipative phenomena due the finite transparency of the mirrors and absorption {by the cavity material} are responsible for a finite lifetime of photons, which naturally decay at a rate $\\Gamma_{loss}$. \n\nAs mentioned in the introduction, the key novelty of this work with respect to earlier work consists in the different mechanism that is proposed to compensate for losses and replenish the photon population. Instead of a coherent pumping or a very broad-band {amplifying} laser medium, we consider a configuration where a set of $N_{at}$ two level atoms is present in each cavity. Each atom is strongly pumped at a rate $\\Gamma_{pump}$, spontaneously decays to its ground state at a rate $\\gamma$ and, most importantly, is coupled to the cavity with a Rabi frequency $\\Omega_{R}$: as a result, the atoms provide an incoherent pumping of the cavities, with a frequency-dependent rate centered at the atomic frequency $\\omega_{at}$. Our choice of two different physical mechanisms for nonlinearity and pumping (for example, two different atomic species) allows us to to tune independently photonic interactions and emission.\n\n\nThe free evolution of the atoms and their coupling to the cavities are described by the following Hamiltonian terms,\n\\begin{eqnarray}\nH_{at}&=&\\sum_{i=1}^{k}\\sum_{l=1}^{N_{at}}\\omega_{at}\\sigma_{i}^{+(l)}\\sigma_{i}^{-(l)} \\\\ \nH_{I}&=&\\Omega_{R}\\sum_{i=1}^{k}\\sum_{l=1}^{N_{at}}\\left[a_{i}^{\\dagger}\\sigma_{i}^{-(l)}+a_{i}\\sigma_{i}^{+(l)}\\right]:\n\\end{eqnarray}\nthe atomic frequency $\\omega_{at}$ is assumed to be in the vicinity (but not necessarily resonant) with the cavity mode and the atom-cavity coupling is assumed to be weak enough $\\Omega_R\\ll \\omega_{at},\\omega_{cav}$ to be far from the ultra-strong coupling regime~\\cite{bastard} and from any superradiant Dicke transition~\\cite{dicke}.\n\nAs usual, the dissipative dynamics under the effect of the pumping and decay processes can be described in terms of a master equation for the density matrix $\\rho$ of the whole atom-cavity system, \n\\begin{equation}\n\\partial_{t}\\rho=\\frac{1}{i}\\com{H_{ph}+H_{at}+H_{I}}{\\rho}+\\mathcal{L}(\\rho),\\label{eq:evinitio}\n\\end{equation}\nwhere the different dissipative processes are summarized in the Lindblad super-operator $\\mathcal{L}=\\mathcal{L}_{pump}+\\mathcal{L}_{loss,\\, at}+\\mathcal{L}_{loss,\\, cav}$, with \n\\begin{eqnarray}\n\\mathcal{L}_{pump}  &=&  \\frac{\\Gamma_{pump}}{2}\\sum_{i=1}^{k}\\sum_{l=1}^{N_{at}}\\left[2\\sigma_{i}^{+(l)}\\rho\\sigma_{i}^{-(l)}-\\sigma_{i}^{-(l)}\\sigma_{i}^{+(l)}\\rho-\\rho\\sigma_{i}^{-(l)}\\sigma_{i}^{+(l)}\\right], \\\\\n\\mathcal{L}_{loss,\\, at} & = & \\frac{\\gamma}{2}\\sum_{i=1}^{k}\\sum_{l=1}^{N_{at}}\\left[2\\sigma_{i}^{-(l)}\\rho\\sigma_{i}^{+(l)}- \\sigma_{i}^{+(l)}\\sigma_{i}^{-(l)}\\rho-\\rho\\sigma_{i}^{+(l)}\\sigma_{i}^{-(l)}\\right], \\\\ \n\\mathcal{L}_{loss,\\, cav}&=&\\frac{\\Gamma_{loss}}{2}\\sum_{i=1}^{k}\\left[2a_{i}\\rho a_{i}^{\\dagger}-a_{i}^{\\dagger}a_{i}\\rho-\\rho a_{i}^{\\dagger}a_{i}\\right]\n\\end{eqnarray}\ndescribing the pumping of the atoms, the spontaneous decay of the atoms, and the photon losses, respectively. The $\\sigma_{i}^{\\pm(l)}$ operators are the usual raising and lowering operators for the $l$-th atom in the $i$-th cavity. We introduce the detuning $\\delta=\\omega_{cav}-\\omega_{at}$ of the bare cavity frequency with respect to the atomic frequency. In the following, we shall concentrate on a regime in which pumping of the atoms is much faster than their spontaneous decay, $\\Gamma_{pump}\\gg \\gamma$, so the $\\mathcal{L}_{loss,\\, at}$ Lindblad term can be safely neglected. \n\nFor simplicity, we will also restrict our attention to the $\\Gamma_{pump}\\gg\\sqrt{N_{at}}\\Omega_{R}$ regime, where the atoms are immediately repumped to their excited state after emitting a photon into the cavity: under such an assumption, an atom having decayed to the ground state does not have the time to reabsorb any photon before being repumped to its excited state. In this regime, complex cavity-QED effects such as Rabi oscillations do not take place and the photon emission takes place in an effectively irreversible way~\\cite{Breuer,QuantumNoise}: as a result, we are allowed to eliminate the atomic dynamics from the problem and write a much simpler photonic master equation involving only the {cavity degrees of freedom}. \n\n\n\n\\subsection{Closed master equation for the {photonic} density matrix}\\label{sec:Closed-master-equation}\n\nUnder the considered $\\Gamma_{pump}\\gg\\Omega_R$ approximation, the atomic population is concentrated in the excited state and it is possible to use projective methods to write a closed master equation for the photonic density matrix where the atomic degrees of freedom $\\mathcal{B}$ have been traced out, $\\rho_{ph}=Tr_{\\mathcal{B}}\\rho$. All details of the (quite cumbersome) calculations can be found in \\ref{app:projective}. The resulting photonic master equation reads \n\\begin{equation}\n\\partial_{t}\\rho_{ph}  =  -i\\left[H_{ph},\\rho_{ph}(t)\\right]+\\mathcal{L}_{loss} + \\mathcal{L}_{em},\n\\label{eq:photon_only}\n\\end{equation}\nwith \n\\begin{eqnarray}\n\\mathcal{L}_{loss} & = & \\frac{\\Gamma_{loss}}{2}\\sum_{i=1}^{k}\\left[2a_{i}\\rho a_{i}^{\\dagger}-a_{i}^{\\dagger}a_{i}\\rho-\\rho a_{i}^{\\dagger}a_{i}\\right],\\label{eq:loss}\\\\\n\\mathcal{L}_{em} & = & \\frac{\\Gamma_{em}}{2}\\sum_{i=1}^{k}\\left[\\tilde{a}_{i}^{\\dagger}\\rho a_{i}+a_{i}^{\\dagger}\\rho\\tilde{a}_{i}-a_{i}\\tilde{a}_{i}^{\\dagger}\\rho-\\rho\\tilde{a}_{i}a_{i}^{\\dagger}\\right]. \\label{eq:gainmarkov}\n\\end{eqnarray}\ndescribing photonic losses and emission processes, respectively.\nWhile the loss term has a standard Lindblad form at rate $\\Gamma_{loss}$, the emission term keeps some memory of the atomic dynamics as it involves modified lowering and raising operators \n\\begin{eqnarray}\n\\tilde{a}_{i}&=&\\frac{\\Gamma_{pump}}{2}\\int_{0}^{\\infty}d\\tau\\, e^{(-i\\omega_{at}-\\Gamma_{pump}\/2)\\tau}a_{i}(-\\tau) ,\n\\label{eq:atilde} \\\\\n\\tilde{a}_{i}^{\\dagger}&=&\\left[\\tilde{a}_{i}\\right]^{\\dagger} \\label{eq:adagtilde}\n\\end{eqnarray}\nwhich contain the photonic (hamiltonian and dissipative) dynamics during pumping. In the limit we are considering in which photonic losses are slow with respect to atomic pump, these operators are the interaction picture ones with respect to the photonic hamiltonian in the cavity array  and have a simpler expression~:\n\\begin{equation}\na_{i}(\\tau)=e^{iH_{ph}\\tau}\\,a_{i}\\,e^{-iH_{ph}\\tau}.\n\\end{equation}\nThe Fourier-like integral in Eqs.\\ref{eq:atilde} and \\ref{eq:adagtilde} is responsible for the frequency selectivity of the emission, as the integral is maximum when the free evolution of $a_i$ occurs at a frequency close to the atomic one $\\omega_{at}$. \n\nA deeper physical insight on the operators (\\ref{eq:atilde}) and (\\ref{eq:adagtilde}) can be obtained by looking at their matrix elements in the basis of eigenstates of the photonic hamiltonian. We consider two eigenstates $\\ket f$ (resp. $\\ket{f'}$) with $N$ (resp. $N+1)$ photons and energy $\\omega_{f}$ (resp. $\\omega_{f'}$). After elementary manipulation, we see that the emission amplitude follows a Lorentzian law as a function of the detuning between the frequency difference of the two photonic states $\\omega_{f'f}=\\omega_{f'}-\\omega_{f}$ and the atomic transition frequency $\\omega_{at}$,\n\\begin{equation}\n\\bra{f'}\\tilde{a}_{i}^{\\dagger}\\ket f = \\frac{\\Gamma_{pump}\/2}{-i(\\omega_{at}-\\omega_{f'f})+\\Gamma_{pump}\/2}\\bra{f'}a_{i}^{\\dagger}\\ket f~. \\label{eq:crea}\n\\end{equation}\nUpon insertion of Eq.\\ref{eq:crea} into the master equation Eq.\\ref{eq:photon_only}, one can associate the real part of the Lorentzian factor to an effective emission rate\n\\begin{equation}\n\\Gamma_{em}(\\omega_{f'f})=\\Gamma^0_{em} \\frac{\\Gamma^2_{pump}\/4}{(\\omega_{at}-\\omega_{f'f})^2+\\Gamma_{pump}^2\/4},\n\\end{equation}\nwhile the imaginary part can be related to a frequency shift of the photonic states under the effect of the population-inverted atoms. In the next section, this point will be made more precise under a secular approximation.\n\nThe width of the Lorentzian is set by the pumping rate $\\Gamma_{pump}$, that is by the autocorrelation time $\\tau_{pump}=1\/\\Gamma_{pump}$ of the atom seen as a frequency-dependent emission bath. The peak emission rate exactly on resonance is equal to\n\\begin{equation}\n\\Gamma^0_{em}=\\frac{4N_{at}\\Omega_{R}^{2}}{\\Gamma_{pump}}\\,.\n\\end{equation}\nWhile the $\\Gamma_{pump}\\gg\\sqrt{N_{at}}\\Omega_R$ assumption automatically implies that the emission is much slower than the atomic repumping rate, $\\Gamma_{em}\\ll\\Gamma_{pump}$, no constraint need being imposed on the parameters $J$, $U$ and $\\delta=\\omega_{cav}-\\omega_{at}$ of the photonic Hamiltonian, which can be arbitrarily large. Whereas an extension of our study to the $\\Gamma_{loss}\\gtrsim \\Gamma_{pump}$ regime would only introduce technical complications, entering the $\\Gamma_{em}\\gtrsim \\Gamma_{pump}$ regime is expected to dramatically modify the physics, as a single atom could exchange photons with the cavity at such a fast rate that it has not time to be repumped to the excited state in between two emission events. As a result, reabsorption processes and Rabi oscillations are possible, which considerably complicate the theoretical description. {These issues will be the subject of future investigations.}\n\n\\subsection{Reformulation in Lindblad form in the secular approximation\\label{sec:photonic-lindblad-form}}\nIn the case the system has a discrete spectrum, it is possible in the so-called secular approximation to write another photonic master implementing non-markovian effects with a more standard Lindblad form, compatible with Monte Carlo wave-function simulations \\cite{Castin} and giving equivalent driven-dissipative dynamics. This can be explained by the following argument: in a weak dissipation limit ($\\Gamma_{em},\\,\\Gamma_{loss}$ very small with respect to the gaps in the spectrum) terms of the density matrix $\\rho_{f,\\tilde{f}},\\, \\rho_{f',\\tilde{f}'}$ which would be rotating at different frequencies $\\omega_{f,\\tilde{f}},\\, \\omega_{f',\\tilde{f}'}$ if the system were isolated, are not coupled to each other by dissipation since the coupling $\\Gamma^{0}_{em},\\, \\Gamma_{loss}$ is negligible with respect to their frequency difference $\\Delta\\omega=\\omega_{f',\\tilde{f}'}-\\omega_{f,\\tilde{f}}=\\omega_{f',f}-\\omega_{\\tilde{f}',\\tilde{f}}$. Considering this, all relevant dissipative transitions verify then $\\Delta\\omega\\simeq 0$. Restricting the previous master equation given by Eqs.~\\ref{eq:photon_only},\\ref{eq:gainmarkov} and \\ref{eq:crea} to these transitions, it is possible to rewrite the dynamics in the following way (details of the derivation are given in \\ref{app:Lindblad-form}): \n\\begin{equation}\n\\partial_{t}\\rho_{ph}  =  -i\\left[H_{ph}+\\left(\\sum_{i} H_{lamb,i}\\right),\\rho_{ph}(t)\\right]+\\mathcal{L}_{loss} + \\bar{\\mathcal{L}}_{em},\n\\label{eq:photon_only_MCWF}\n\\end{equation}\nwith \n\\begin{equation}\n\\label{eq:gainmarkov_MCWF}\n\\bar{\\mathcal{L}}_{em}(\\rho_{ph})= \\frac{\\Gamma_{em}}{2}\\sum_{i=1}^{k}\\left[2\\bar{a}_{i}^{\\dagger}\\rho_{ph} \\bar{a}_{i}-\\bar{a}_{i}\\bar{a}_{i}^{\\dagger}\\rho_{ph}-\\rho_{ph}\\bar{a}_{i}\\bar{a}_{i}^{\\dagger}\\right],\n\\end{equation}\n\\begin{equation}\n\\label{eq:crea_MCWF}\n \\bra{f'}\\bar{a}_{i}^{\\dagger}\\ket{f}=\\frac{\\Gamma_{pump}\/2}{\\sqrt{(\\omega_{at}-\\omega_{f',f})^2+\\left(\\Gamma_{pump}\/2\\right)^2}}\\bra{f'}a_{i}^\\dagger\\ket{f},\n\\end{equation}\n\\begin{equation}\n\\label{eq:lamb_MCWF}\n \\bra{f'}H_{lamb,i}\\ket{f}=\\sum_{f''} \\bra{f'}a_{i}\\ket{f''}\\left(\\frac{(\\omega_{f'',f}-\\omega_{at})\\Gamma_{pump}\/2}{(\\omega_{at}-\\omega_{f'',f})^2+(\\Gamma_{pump}\/2)^2}\\right)\\bra{f''}a_{i}^\\dagger\\ket{f}.\n\\end{equation}\nNote that the jump operators $\\bar{a}_i^\\dagger$ have the same form as the ones considered in~\\cite{Kapit} and have for effect to modify the the transition rate, while the \"imaginary part\" of Eq.~\\ref{eq:crea} induces an additional Hamiltonian contribution in the form of a Lamb shift. Notice that the two master equations Eqs.~\\ref{eq:photon_only},~\\ref{eq:photon_only_MCWF} are slightly different. However, under the considered approximation they are expected to provide equivalent dynamics. The latter form has the advantage of being of Lindblad form, and thus is directly compatible with MCWF simulations \\cite{Castin} and can be useful from a numerical point of view.\n\nThe secular approximation can be very restrictive (particularly in the thermodynamic limit where the spectrum is continuous). However, our feeling is that the reformulation of Eq.\\ref{eq:photon_only_MCWF} should be accurate in a wider range of parameters. Quantitatively, we anticipate the condition $\\Gamma_{em},\\, \\Gamma_{loss}\\ll\\Gamma_{pump}$ to be sufficient. More investigations in this direction are under way.\n\\section{One cavity}\n\\label{sec:onecavity}\n\nAs a first example of application, we consider the simplest case of a single nonlinear cavity.  A special attention will be paid to the stationary state $\\rho_{ss}$ of the system for which Eq.\\ref{eq:photon_only} imposes \n\\begin{equation}\n0=-i\\left[H_{ph},\\rho_{ss}\\right]+\\mathcal{L}_{loss}\\left(\\rho_{ss}\\right)+\\mathcal{L}_{em}\\left(\\rho_{ss}\\right).\n\\end{equation}\nIn our specific case of a single cavity, the photonic states are labelled by the photon number $N$ and have an energy \n\\begin{equation}\n\\omega_{N}=N\\omega_{cav}+\\frac{1}{2}N(N-1)U.\n \\end{equation}\nCorrespondingly, the $N\\rightarrow N+1$ transition has a frequency\n\\begin{equation}\n\\omega_{N+1,N}=\\omega_{cav}+NU,\n\\end{equation}\nand the corresponding photon emission rate is\n\\begin{equation}\n\\Gamma_{em}(\\omega_{N+1,N})=\\Gamma_{em}^0\\frac{(\\Gamma_{pump}\/2)^{2}}{(\\omega_{N+1,N}-\\omega_{at})^{2}+(\\Gamma_{pump}\/2)^{2}}.\n\\end{equation}\nAs no coherence can exist between states with different photon number $N$, the stationary density matrix is diagonal in the Fock basis, $\\rho_{ss}=\\delta_{N,N'}\\pi_N$ with the populations $\\pi_N$ satisfying \n\\begin{equation}\n(N+1)\\Gamma_{loss}\\pi_{N+1}-(N+1)\\Gamma_{em}(\\omega_{N+1,N})\\pi_{N}+N\\Gamma_{em}(\\omega_{N,N-1})\\pi_{N-1}-N\\Gamma_{loss}\\pi_{N}=0,\n\\end{equation}\nwhere the two last terms of course vanish for $N=0$. \nAs only states with neighboring $N$ are connected by the emission\/loss processes, detailed balance is automatically enforced in the stationary state, which imposes the simple condition on the populations,\n\\begin{equation}\n(N+1)\\Gamma_{loss}\\pi_{N+1}-(N+1)\\Gamma_{em}(\\omega_{N+1,N})\\pi_{N}=0\n\\end{equation}\nwhich is straightforwardly solved in terms of a product,\n\\begin{equation}\n\\pi_{N} = {\\pi_{0}}\\,\\prod_{M=0}^{N-1}\\frac{\\Gamma_{em}(\\omega_{M+1,M})}{\\Gamma_{loss}}= \\left(\\frac{\\Gamma_{em}^0}{\\Gamma_{loss}}\\right)^{N}\\prod_{M=0}^{N-1}\\frac{(\\Gamma_{pump}\/2)^{2}}{(\\omega_{M+1,M}-\\omega_{at})^{2}+(\\Gamma_{pump}\/2)^{2}}\\pi_{0}.\n\\end{equation}\nThe excellent agreement of this result with a numerical solution of the full atom-cavity system is illustrated in \\ref{app:valid}.\n\n\n\n\n\n\n\\subsection{Linear regime}\n\n\\begin{figure*}[htbp]\n\\begin{center}\n \\includegraphics[height=0.40\\textwidth,clip]{lasing_condition}\\hspace{1cm}\n\\includegraphics[height=0.40\\textwidth,clip]{nonlin_o_2_inf_o}\\\\ \\vspace*{1.5cm}\n\\includegraphics[height=0.40\\textwidth,clip]{nonlin_o_1_inf_o_inf_o_2} \\hspace{1cm}\n\\includegraphics[height=0.40\\textwidth,clip]{nonlin_o_inf_o_1}\\\\ \\vspace*{1.cm}\n\\end{center}\n\\caption{(a) Emission vs. loss rate as a function of the detuning from the atomic frequency $\\omega_{at}$: the three curves are for peak emission $\\Gamma_{em}^0$ larger ({red dash-}dotted), equal ({black} dashed), smaller ({green} solid) than the loss rate $\\Gamma_{loss}$.\n(b-d) Populations $\\pi_N$ of the $N$-photon state as a function of $N$ in the three cases $\\omega_{2}\\leq\\omega_{cav}$ (b), $\\omega_{1}\\leq\\omega_{cav}\\leq\\omega_{2}$ (c), $\\omega_{cav}\\leq\\omega_{1}$ (d). In the three panels, the open dots are the numerical results of the atom-cavity theory, while the {solid} line is the prediction of the analytical purely photonic theory; the {dashed} curves show the ratio $\\Gamma_{em}(\\omega_{N+1,N})\/\\Gamma_{loss}$ as a function of $N$.\nParameters: $\\delta\/U=4$ (b), $-2$ (c), $-6$ (d). In all panels,  $2U\/\\Gamma_{pump}=0.2$, \n $2\\Gamma_{loss}\/\\Gamma_{pump}=0.0006$, \n $2\\Omega_{R}\/\\Gamma_{pump}=0.02$. \n \\label{fig:Gain-and-populations}}\n\\end{figure*}\n\nFor a vanishing nonlinearity $U=0$, all transition frequencies $\\omega_{N+1,N}$ are equal to the bare cavity frequency $\\omega_0$ and the {populations} of the different $N$ states have a constant ratio\n\\begin{equation}\n\\frac{\\pi_{N+1}}{\\pi_{N}}=\\frac{\\Gamma_{em}^0}{\\Gamma_{loss}}\\frac{(\\Gamma_{pump}\/2)^{2}}{\\delta^{2}+(\\Gamma_{pump}\/2)^{2}},\n\\end{equation}\nwhere we remind that $\\delta=\\omega_{cav}-\\omega_{at}$.\nFor weak pumping\nand\/or large detuning, {one has}\n\\begin{equation}\n\\Gamma_{em}^0\\frac{(\\Gamma_{pump}\/2)^{2}}{\\delta^{2}+(\\Gamma_{pump}\/2)^{2}}<\\Gamma_{loss} ,\n\\end{equation}\n{so} the density matrix for the cavity shows a monotonically decreasing thermal occupation law. \nFor strong pumping\nand close to resonance{, one can achieve the regime where the emission overcompensates} losses and the cavity mode starts being strongly populated: \n \\begin{equation}\n\\Gamma_{em}^0\\frac{(\\Gamma_{pump}\/2)^{2}}{\\delta^{2}+(\\Gamma_{pump}\/2)^{2}}>\\Gamma_{loss}  .\n \\end{equation}\nThe transition between the two regimes is the usual laser threshold, but our purely photonic theory is not able to include the gain saturation mechanism that serves to stabilize laser oscillation above threhsold~\\cite{laser,QuantumNoise}: within our purely photonic theory, the population {would in fact show} a clearly unphysical monotonic growth for increasing $N$. A complete description in terms of the full atom-cavity master equation would of course solve this pathology including a gain saturation mechanism according to usual laser theory{, but this goes beyond the scope of the present work.}\n\n \n\\subsection{Optical bistability phenomena in weak nonlinear cavities}\nFor $U>0$, the situation is much more interesting as the effective transition frequency depends on the number of photons, \n\\begin{equation}\n\\omega_{N+1,N}=\\omega_{cav}+NU\\geq\\omega_{cav}, \\hspace{1cm},\n\\end{equation}\nso the gain condition\n\\begin{equation}\n\\frac{\\Gamma_{em}^0}{\\Gamma_{loss}}\\frac{(\\Gamma_{pump}\/2)^{2}}{(\\omega_{N+1,N}-\\omega_{at})^{2}+(\\Gamma_{pump}\/2)^{2}}\\geq1\\label{eq:lasing}\n\\end{equation}\ncan be satisfied in a finite range of photon numbers only, as it is illustrated in Fig.\\ref{fig:Gain-and-populations}(a). As a consequence, even a weak nonlinearity $U$ is able to stabilize the system for any value of $\\Gamma_{em}^0$ even in the absence of any gain saturation mechanism.\n\nFor $\\Gamma_{em}^0<\\Gamma_{loss}$, losses always dominate. For $\\Gamma_{em}^0>\\Gamma_{loss}$, the gain condition is instead satisfied in a range of frequencies $[\\omega_1,\\omega_2]$ around $\\omega_{at}$. Under the weak nonlinearity condition $U\\ll \\Gamma_{pump}$, the $[\\omega_1,\\omega_2]$ range typically contains a large number of transition frequencies $\\omega_{N+1,N}$ at different $N$. Three different regimes can then be identified depending on the position of the cavity frequency $\\omega_{cav}$ with respect to the $[\\omega_1,\\omega_2]$ range.\n \n(i) If $\\omega_{2}\\leq\\omega_{cav}$, then the gain condition is never verified, and the population $\\pi_{N}$ shown in Fig.\\ref{fig:Gain-and-populations}(b) is a monotonically decreasing function of $N$. In this regime, the state of the cavity field is very similar to a thermal state, as it usually happens in a laser below threshold. (ii) If $\\omega_{1}\\leq\\omega_{cav}\\leq\\omega_{2}$, the population $\\pi_N$ shown in Fig.\\ref{fig:Gain-and-populations}(c) is an increasing function for small $N$, shows a single maximum for $N\\simeq \\bar{N}=(\\omega_2-\\omega_{cav})\/U$, and finally monotonically decreases for $N>\\bar{N}$.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth,clip]{two_point_correlation_weak_interacting.eps}\n\\end{center}\n\\caption{\n\\label{fig:g2_one_weak}\nPurely photonic simulation of the two-time coherence function $g^{(2)}(\\tau)$ in the weakly nonlinear regime.\nParameters $U\/\\Gamma_{pump}=0.1$, $\\Gamma_{loss}\/\\Gamma_{pump}=0.03$, $\\Gamma^0_{em}\/\\Gamma_{pump}=0.04$, $\\delta=-6U$ as in Fig.\\ref{fig:Gain-and-populations}(d).\n}\n\\end{figure}\n\nThe phenomenology is the richest in the regime (iii) where $\\omega_{cav}\\leq\\omega_{1}$. In this case, for small $N$ the population $\\pi_N$ decreases from its initial value $\\pi_0$ until the nonlinearly shifted frequency enters in the gain interval for $N\\simeq \\bar{N}'=(\\omega_1-\\omega_{cav})\/U$. After this point $\\pi_N$ starts increasing again until it reaches a local maximum at $N\\simeq \\bar{N}=(\\omega_2-\\omega_{cav})\/U$. Finally, for even larger $N$ it begins to monotonically decrease. An example of this complicate behaviour is shown in Fig.\\ref{fig:Gain-and-populations}(d).\n\nThe existence of two well separate local maxima at $N=0$ and $N\\simeq \\bar{N}$ in the photon number distribution $\\pi_N$ suggests that the incoherently driven nonlinear cavity exhibits a sort of bistable behaviour: when it is prepared at one maximum of the photon number distribution $\\pi_N$, the system is trapped in a metastable state localized in a neighborhood of this maximum for a macroscopically long time. Switching from one metastable state to the other results is only possible as a result of a large fluctuation, so it has a very low probability, typically exponentially small in the photon number difference between the two metastable states.\n\nThis bistable behavior is clearly visible in the temporal dependence of the delayed two-photon correlation function\n\\begin{equation}\ng^{(2)}(\\tau)=\\frac{\\langle a^{\\dagger}(t)\\,a^{\\dagger}(t+\\tau)\\,a(t+\\tau)\\,a(t) \\rangle_{ss}}{\\langle a^{\\dagger}(t)\\,a (t) \\rangle_{ss}\\langle a^{\\dagger}(t+\\tau)\\,a(t+\\tau) \\rangle_{ss}}:\n\\end{equation} \nthat is plotted in Fig.\\ref{fig:g2_one_weak}. At short times, the value of $g^{(2)}$ is determined by a weighted average of the contribution of the two maxima according to the stationary $\\pi_N$.  After a quick transient of order $1\/\\Gamma_{em,loss}$, which corresponds to a fast local equilibration of the probability distribution around each of its maxima, the $g^{(2)}$ correlation function slowly decays to its asymptotic value $1$ on a much longer time-scale mainly set by the exponentially long switching time from one maximum to the other\n\nBefore proceeding, it is worth emphasizing that the present mechanism for optical bistability bears important differences from the dispersive or absorptive optical bistability phenomena discussed in textbooks~\\cite{Boyd,B-C}. On one hand there is some analogy to dispersive optical bistability in that the intensity-dependence of the refractive index is responsible for a frequency shift of the cavity resonance; on the other hand the frequency-selection is not provided by the resonance condition with a monochromatic coherent incident field rather by the frequency dependence of the gain due to the incoherent pump.\n\n\n\\subsection{Photon number selection in strongly nonlinear cavities \\label{sub:Strongly-nonlinear-cavity,}}\n\nIn the opposite limit $U\\gg\\Gamma_{pump}$, the nonlinearity is so large that a change of photon number by a single unity has a sizable effect on the emission rate $\\Gamma_{em}(\\omega_{N+1,N})$. As discussed in \\ref{app:projective}, the derivation of the photonic master equation remains fully valid in this regime provided $\\Gamma_{pump}\\gg\\Gamma_{em}^0,\\Gamma_{loss}$.\n\nThe ensuing physics is most clear in the regime when the maximum emission rate is large but only a single transition fits within the emission lineshape: these assumptions are equivalent to imposing that \n\\begin{equation}\n\\frac{\\Gamma_{em}^0}{\\Gamma_{loss}}\\gg 1\\hspace{1cm}\\textrm{and}\\hspace{1cm} \\frac{\\Gamma_{em}^0}{\\Gamma_{loss}}\\frac{\\Gamma_{pump}^2}{U^{2}}\\ll1\n\\end{equation}\nwith the further condition that the emission is resonant with the $N_0\\rightarrow N_0+1$ transition, \n\\begin{equation}\n\\omega_{at}=\\omega_{cav}+N_{0}U. \n\\end{equation}\nAs a result, only this last transition is dominated by emission, while all others are dominated by losses.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.3,clip]{photon_selec_2.eps} \n\\hspace{1cm}\\includegraphics[scale=0.3,clip]{photon_selec_3.eps}\n\\end{center}\n\\caption{\n{Selective generation of a $N_0=2$ photon (upper panel) and $N_0=3$ photon (lower panel) Fock state: Population $\\pi_N$ as a function of $N$ for different pumping parameters. The points are the result of a purely photonic simulation, the lines are a guide to the eye.\nLeft panel parameters:} for all curves $\\delta=-U$, $2\\Omega_R\/\\Gamma_{pump}=0.01$, and then for each particular curve $2\\Gamma_{loss}\/\\Gamma_{pump}=2\\,10^{-5}$ (blue solid line), $2\\,10^{-6}$ (green, dashed line), $2\\,10^{-7}$ (red, dash-dotted line), $2\\,10^{-8}$ (magenta, dotted line). $2U\/\\Gamma_{pump}=10^{3\/2}$ (blue solid line), $10^{2}$ (green, dashed line), $10^{5\/2}$ (red, dash-dotted line), $10^{3}$ (magenta, dotted line).\n{Right panel parameters:} fora ll curves $\\delta=-U$, $2\\Omega_R\/\\Gamma_{pump}=0.01$, and then $2\\Gamma_{loss}\/\\Gamma_{pump}=5\\,10^{-8}$ (blue solid line), $5\\,10^{-9}$ (green, dashed line), $5\\,10^{-10}$ (red, dash-dotted line), $5\\,10^{-11}$ (magenta, dotted line). $2U\/\\Gamma_{pump}=2\\,10^{5\/2}$ (blue solid line), $2\\,10^{3}$ (green, dashed line), $2\\,10^{7\/2}$ (red, dash-dotted line), $2\\,10^{4}$ (magenta, dotted line).\nThe goal of {these} {choices} of parameters was to control the steady-state ratios $P(N+1)\/P(N)=10^{-2}$ and $P(N)\/P(0)=0.1, 1, 10, 100$ (blue, green, red, magenta).\n\\label{fig:one_strongly}}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.35\\textwidth,clip]{two_point_correlation_strongly_interacting}\\hspace{1cm}\n\\includegraphics[width=0.35\\textwidth,clip]{generation_pop.eps}\n\\end{center}\n\\caption{\n\\label{fig:g2_one_strongly_generation}\nLeft panel: Purely photonic simulation of the two-time coherence function $g^{(2)}(\\tau)$ for a strongly nonlinear regime in a (metastable) $N_0=2$ photon selection regime. The inset shows a magnified view of the short time region.\nParameters: $2U\/\\Gamma_{pump}=100$, $2\\Gamma_{loss}\/\\Gamma_{pump}=2\\, 10^{-3}$, $2\\Gamma_{em}\/\\Gamma_{pump}=0.2$, $\\delta=-U$; in the language of Fig.\\ref{fig:one_strongly}, the present parameters would correspond to a regime where the $N=0,2$ states are almost equally occupied.\nRight panel:  Preparation of the metastable state at $N_0=2$ starting from a $N=4$\n$\\pi(4)$ (red {dot-dashed}) $\\pi(2)$ (green {dashed}) $\\pi(0)$ (blue {solid}). Same parameters as in Fig.\\ref{fig:one_strongly}.\n} \n\\end{figure}\n\n\n\nIn terms of the diagrams in Fig.\\ref{fig:Gain-and-populations}, the stationary distribution $\\pi_N$ is therefore sharply peaked at two specific values, $N=0$ and at $N=N_0$. Examples of this physics are illustrated in Fig.\\ref{fig:one_strongly}: the two peaks are always clearly visible, but depending on the parameters their relative height can be tuned to different values almost at will. It is however important to note that having a sizable stationary population in the $N=N_0$ peak requires quite extreme values of the parameters as population would naturally tend to accumulate at $N=0$ and this difficulty turns out to be exponentially harder for larger $N_0$.\n\n\nThe physics underlying this behaviour can be easily explained in terms of the asymmetry in the switching mechanisms leading from $N=0$ to $N=N_0$ and viceversa. The former process requires in fact a sequence of several unlikely emission events from $N=0$ to $N=N_0-1$ as emission is favoured only in the last step. On the other hand, decay from $N=N_0$ occurs as a consequence a single unlikely loss event from $N=N_0-1$ to $N=N_0-2$: as soon as the system is at $N=N_0-2$, it will quickly decay to $N=0$. \n\nThe rate $\\Gamma_{acc}$ of such an accident can be estimated as follows: the probability that the system in $N=N_0-1$ decays to $N=N_0-2$ is a factor ${(N_{0}-1)\\Gamma_{loss}}\/({N_{0}\\Gamma_{em}^0})$ smaller than the one of being repumped to $N=N_0$. As the rate at which the system decays from $N=N_0$ to $N_0-1$ is approximately equal to $N_{0}\\Gamma_{loss}$, one finally obtains\n\\begin{equation}\n\\Gamma_{acc}=N_{0}\\Gamma_{loss}\\frac{(N_{0}-1)\\Gamma_{loss}}{N_{0}\\Gamma_{em}^0}\\ll N_{0}\\Gamma_{loss}.\n\\end{equation}\nThis longer time scale $\\tau_{acc}=\\Gamma_{acc}^{-1}$ is clearly visible in the long tail of the time-dependent $g^{(2)}(t)$ that is plotted in the left panel of Fig.\\ref{fig:g2_one_strongly_generation}. The quick feature at very short times corresponds to the emission rate $\\Gamma_{em}$.\n\nIf needed, the characteristic time scale $\\tau_{acc}$ could be further enhanced by adding a second atomic species whose transition frequency is tuned to quickly and selectively emit photons on the $N-2\\rightarrow N-1$ transition. In this way, the accident rate can be efficiently reduced to $\\Gamma_{acc}^{(2)}\\simeq\\Gamma_{loss}\\left(\\Gamma_{loss}\/\\Gamma_{em}^0\\right)^{2}\\ll\\Gamma_{acc}$. By repeating the mechanism on $k$ transitions, one can suppress the accident rate in a geometrical way to $\\Gamma_{acc}^{(k)}\\simeq\\Gamma_{loss}\\left(\\Gamma_{loss}\/\\Gamma_{em}^0\\right)^{k}\\ll\\Gamma_{acc}$. Finally, the Fock state with $N_0$ photons can be fully stabilized to an infinite lifetime and no problem of metastability if $N_0$ different atomic species are included so to cover all transitions from $N=0$ to $N=N_0$.\n\nFrom a slightly different perspective, we can take advantage of {the slow rate of accidents $\\Gamma_{acc}$} to selectively prepare a metastable state with $N=N_0$ photons even in parameter regimes where the $N=0$ state would be statistically favoured at steady-state. Though the state will eventually decay to $N=0$, the lifetime of the metastable $N=N_0$ state can be long enough to be useful for interesting experiments: The idea to prepare the state with $N_0$ photons is to inject a larger number $N>N_0$ of photons into the cavity: the system will quickly decay to the $N=N_0$ state where the system remains trapped with a lifetime $\\Gamma_{acc}^{-1}$. \n\nThe efficiency of this idea is illustrated in the right panel of Fig.\\ref{fig:g2_one_strongly_generation} where we plot the time evolution of the most relevant populations $\\pi_N$. The initially created state with $N=N_{in}$ photons quickly decays, so that population accumulates into $N=N_0$ on a time-scale of the order of $\\Gamma_{loss}$; the eventual decay of the population towards $N=0$ will then occur on a much longer time set by $\\Gamma_{acc}$. It is worth noting that this strategy does not require that the initial preparation be number-selective: it will work equally well if a wide distribution of $N_{in}$ are generated at the beginning, provided a sizable part of the distribution lies at $N>N_0$. Furthermore, this idea removes the need for extreme parameters such as the ones used in Fig.\\ref{fig:one_strongly} to obtain a balance between $\\pi(N)$ and $\\pi(0)$: as a result, the difficulty of creating a (metastable) state of $N_0$ photons is roughly independent of $N_0$. \n\n{These} results show the potential of this novel photon number selection scheme to obtain light pulses with novel nonclassical properties: for instance, upon a sudden switch-off of the cavity mirrors, {one would obtain} a wavepacket containing an exact number of photons {sharing the same wavefunction}. With respect to the many other configurations discussed in the recent literature to produce $N$-photon Fock states and photon bundles~\\cite{Majumdar,Rundquist,Munoz}, our proposal has the advantage of giving a deterministic preparation of a $N$-photon Fock state in the cavity, which can then be manipulated to extract light pulses with the desired quantum properties.\n\n\n\\section{Cavity arrays} \\label{sec:Arrays-of-cavities}\n\nAfter having unveiled a number of interesting features that occur in the simplest case of a single-cavity, we are now in a position to start attacking the far richer many-cavity case. From now on we consider that the isolated photonic Hamiltonian is the Bose-Hubbard one with tunelling $J$ and interaction constant $U$. Throughout this section, we shall make heavy use of the purely photonic description previously derived, which allows to consider bigger systems with a higher number of photons. A numerical validation of this approach against the solution of the full atom-cavity master equation is presented in \\ref{app:valid}.\n\n\n\\subsection{Markovian regime} \\label{sec:Markov}\n\nWe begin by considering the Markovian limit of the theory, which is recovered for $\\Gamma_{pump}=\\infty$, i.e. for a frequency-independent gain. In this case, the emission term of the master equation for photons Eq.\\ref{eq:gainmarkov} reduces to the usual Lindblad form\n\\begin{equation}\n\\mathcal{L}_{em} =\\frac{\\Gamma_{em}^0}{2}\\sum_{i=1}^{k}\\left[2a_{i}^{\\dagger}\\rho a_{i}-a_{i}a_{i}^{\\dagger}\\rho-\\rho a_{i}a_{i}^{\\dagger}\\right].\n\\end{equation}\nFor a single cavity, the stationary state is immediately obtained as \n\\begin{equation}\n\\pi_{N}=\\frac{1}{1-\\frac{\\Gamma_{em}^0}{\\Gamma_{loss}}}\\left(\\frac{\\Gamma_{em}^0}{\\Gamma_{loss}}\\right)^{N}:\n\\end{equation}\na necessary condition for stability for this system is of course that $\\Gamma_{em}^0<\\Gamma_{loss}$. For $\\Gamma_{em}^0>\\Gamma_{loss}$ amplification would in fact exceed losses and the system display a laser instability: while a correct description of gain saturation is beyond the purely photonic theory, the full atom-cavity theory would recover for this model the standard laser operation~\\cite{QuantumNoise,Breuer,laser}.\n\nFor larger arrays of $k$ sites, a straightforward calculation shows that in the Markovian limit the stationary matrix keeps a structureless form, \n\\begin{equation}\n\\rho_{\\infty}=\\sum_{N}\\pi_{N}\\mathcal{I}_{N} ,\n\\end{equation}\nwith \n\\begin{equation}\n\\pi_{N}=\\frac{1}{\\sum_{M}D_{M}\\left(\\frac{\\Gamma_{em}^0}{\\Gamma_{loss}}\\right)^{M}}\\left(\\frac{\\Gamma_{em}^0}{\\Gamma_{loss}}\\right)^{N}.\n\\end{equation}\nHere, $D_{N}=\\frac{(N+k-1)!}{(k-1)!N!}$ is the dimension of the Hilbert subspace with a total number of photons equal to $N$ and $\\mathcal{I}_{N}$ is the projector over this subspace. The interested reader can find the details of the derivation in \\ref{app:Exact-stationary-solution}. \n\nThis result shows that independently of the number of cavities and the details of the Hamiltonian, in the Markovian limit the density matrix in the stationary state corresponds to an effective Grand-Canonical ensemble at infinite temperature $\\beta=0$ with a fugacity $z=e^{\\beta\\mu}={\\Gamma_{em}^0}\/{\\Gamma_{loss}}$ determined by the pumping and loss conditions only: All states are equally populated and the system does not display much interesting physics. In particular, the steady state does not depend on the tunelling amplitude $J$ and on the photon-photon interaction constant $U$\n\n\n\n\\subsection{Effective Grand-Canonical distribution in a weakly non-Markovian and secular regime \\label{sub:GC}}\n\nThe situation changes as soon as some non-Markovianity is included in the model. In this section we start from a weakly non-Markovian case where all relevant transitions adding one photon have a narrow distribution around the bare cavity frequency, $|\\omega_{f'f}-\\omega_{cav}|\\ll\\Gamma_{pump}$. We also assume a secular limit where $U,\\, J\\gg\\Gamma^0_{em},\\,\\Gamma_{loss}$, so that the non-diagonal terms of the density matrix {in the photonic hamiltonian eigenbasis} oscillate at a fast rate and are thus effectively decoupled from the (slowly varying) populations. In this limit, we can safely assume that all coherences vanish and we can restrict our attention to the populations. This somehow critical approximation will be justified a posteriori in the next section, where we treat perturbatively the coupling of populations to coherences and show both analytically and numerically that in the weakly markovian regime, their contribution is of higher order in the 'non-markovianity' parameter $1\/\\Gamma_{pump}$ and therefore can be safely neglected.\n\nUnder these assumptions, the transfer rate on the $\\ket{f'}\\rightarrow \\ket{f}$ transition where one photon is lost from $N+1$ to $N$ has a frequency-independent form\n\\begin{equation}\nT_{f' \\rightarrow f} = \\Gamma_{loss}\\left|\\bra{f} a \\ket{f'}\\right|^{2},\n\\end{equation}\nwhile the reverse emission process depends on the detunings $\\Delta_{f'f}=\\omega_{f'f}-\\omega_{cav}$ and $\\delta=\\omega_{cav}-\\omega_{at}$ as\n\\begin{equation}\n T_{f\\rightarrow f'} = \\Gamma_{em}^0\\left|\\bra{f'} a^\\dagger \\ket{f}\\right|^2 \\frac{\\frac{\\Gamma_{pump}^2}{4}}{(\\Delta_{f'f}+\\omega_{cav}-\\omega_{at})^{2}+\\frac{\\Gamma_{pump}^2}{4}}\n \\simeq  \\tilde{\\Gamma}_{em}^0\\left|\\bra fa^\\dagger \\ket{f'}\\right|^2 \\left[1-\\beta\\Delta_{f'f}+\\mathcal{O}\\left(\\Delta_{f'f}\\right)^{2}\\right],\n \\label{eq:emissonWMS}\n\\end{equation}\nwith \n\\begin{eqnarray}\n\\tilde{\\Gamma}_{em}^0 & = & \\frac{\\left(\\Gamma_{pump}\/2\\right)^{2}}{(\\omega_{cav}-\\omega_{at})^{2}+\\left(\\Gamma_{pump}\/2\\right)^{2}}\\Gamma_{em}^0 ,\\\\\n\\beta & = & \\frac{2(\\omega_{cav}-\\omega_{at})}{(\\omega_{cav}-\\omega_{at})^{2}+\\left(\\Gamma_{pump}\/2\\right)^{2}}.\n\\label{eq:beta}\n\\end{eqnarray}\nIn this expression, the weakly non-Markovian regime is characterized by having $|\\beta \\Delta_{f'f} | \\ll 1$: in this case, the square bracket in Eq.\\ref{eq:emissonWMS} can be replaced with no loss of accuracy by an exponential\n\\begin{equation}\n1-\\beta\\Delta_{f'f}\\simeq e^{-\\beta\\Delta_{f'f}},\n\\end{equation}\nwhich immediately leads to a Grand-Canonical form of the stationary density matrix \n\\begin{equation}\n\\rho_{\\infty}=\\frac{1}{\\Xi}e^{\\beta N\\mu}e^{-\\beta H},\n\\end{equation}\nwith an effective chemical potential \n\\begin{equation}\n\\mu=\\frac{1}{\\beta}\\log\\left(\\frac{\\tilde{\\Gamma}_{em}^0}{\\Gamma_{loss}}\\right)+\\omega_{cav} \n\\end{equation}\nand an effective temperature $k_B T=1\/\\beta$: most remarkably, even if each transition involves a small deviation from the bare cavity frequency $\\omega_{cav}$,\nthe cumulative effect of many such deviations can have important consequences for large photon numbers, so to make the stationary distribution strongly non-trivial. Remarkably, both positive and negative temperature configurations can be obtained from Eq.\\ref{eq:beta} just by tuning the peak emission frequency $\\omega_{at}$ either below or above the bare cavity frequency $\\omega_{cav}$. \nAs expected for a thermal-like distribution, detailed balance between eigenstates is satisfied\n \\begin{multline}\nT_{f'\\rightarrow f}\\pi_{f'}-T_{f\\rightarrow f'}\\pi_{f}=\\left|\\bra{f'} a^\\dagger \\ket{f}\\right|^{2}  \\,\\left[\\Gamma_{loss}\\frac{1}{\\Xi}\\left(\\frac{\\tilde{\\Gamma}_{em}^0}{\\Gamma_{loss}}e^{\\beta\\omega_{cav}}\\right)^{N+1}e^{-\\beta \\omega_{f'}} \\right. +\\\\\n-\\left.\\tilde{\\Gamma}_{em}^0\\,e^{-\\beta\\left(\\omega_{f'f}-\\omega_{cav}\\right)}\\frac{1}{\\Xi}\\left(\\frac{\\tilde{\\Gamma}_{em}^0}{\\Gamma_{loss}}e^{\\beta\\omega_{cav}}\\right)^{N}e^{-\\beta \\omega_{f}}\\right] =0,\n\\end{multline}\nbut it is crucial to keep in mind that this thermal-like distribution does not arises from any real thermalization process, but is a consequence of the specific form chosen for the pumping and dissipation. The application of this concept to the study of effective thermalization effects in a driven-dissipative non-Markovian condensate in the weakly interacting regime will be the subject of a future work, also with an eye to photon~\\cite{Weitz} and polariton~\\cite{BEC,Bajoni} Bose-Einstein condensation experiments.\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.25\\columnwidth,clip]{test_GC_n}\n\\hspace{1cm}\n\\includegraphics[width=0.25\\columnwidth,clip]{test_GC_corr}\n\\hspace{1cm}\n\\includegraphics[width=0.25\\columnwidth,clip]{eigenstate_coherence.eps}\n\\end{center}\n\\caption{\nLeft and center panels: average number of photons $n_1=\\langle a_1^\\dagger a_1\\rangle$ (left) and spatial coherence $g_{1,2}^{(1)}= \\langle a_1^\\dagger a_2\\rangle\/\\langle a_1^\\dagger a_1\\rangle$ (center) in a two cavity system with small $U\/\\Gamma_{pump}$ and $J\/\\Gamma_{pump}$ as a function of the non linearity $U$ at fixed $\\Gamma_{pump}$. In red dots, exact resolution of the photonic master equation, and in black solid line the grand canonical ensemble ansatz. Parameters~: $2J\/\\Gamma_{pump}=0.02$, $2\\Gamma_{loss}\/\\Gamma_{pump}=0.002$, $2\\Gamma_{em}\/\\Gamma_{pump}=0.0014$, $2\\delta\/\\Gamma_{pump}=0.6$. \nRight panel: purely photonic simulation of the relative quantum coherence between two arbitrarily chosen two-photon eigenstates $\\rho_{ij}\/\\sqrt{\\rho_{ii}\\rho_{jj}}$ as a function of $1\/\\Gamma_{pump}$ (the result does not depend on the specific eigenstates considered). As expected, this coherence vanishes in $1\/\\Gamma^2_{pump}$ in the Markovian limit $1\/\\Gamma_{pump}\\to 0$. The value above $1$ for large $1\/\\Gamma_{pump}$ signals breakdown of positivity of the density matrix as we move out of the validity regime of the purely photonic master equation.\nParameters: $J\/\\Gamma_{loss}=1$, $\\Gamma_{em}\/\\Gamma_{loss}=0.5$, $\\delta=-\\Gamma_{loss}$, $U\/\\Gamma_{loss}=2$.\n\\label{fig:GCtest}}\n\\end{figure}\n\nA numerical test of this result for a two cavity system with a strong pumping $\\Gamma_{pump}\\gg U,J$ and a large enough photon number so to induce appreciable nonlinear effects is shown in Fig.\\ref{fig:GCtest}. The results of this comparison are displayed in the left and central panels: excellent agreement between an exact resolution of the photonic master equation and the grand canonical ensemble ansatz is found in both the average photon number and the first-order coherence.\n\n\\subsection{Beyond the secular approximation}\n\nIn the weakly non-Markovian regime, the validity of the effective Grand-Canonical description can be extended outside the secular approximation according to the following arguments.\nAs a first step, we decompose the master equation as\n\\begin{equation}\n\\frac{d\\rho}{dt}=[\\mathcal{M}_0+\\delta\\mathcal{M}]\\rho,\n\\end{equation}\nwhere the super-operators $\\mathcal{M}$ and $\\delta\\mathcal{M}$ act of the linear space of density matrices $\\rho$ as\n{\\begin{equation}\n\\mathcal{M}_{0}[\\rho]=-i\\com H{\\rho}+\\frac{\\Gamma_{loss}}{2}\\sum_{i=1}^{k}\\left[2a_{i}\\rho a_{i}^{\\dagger}-a_{i}^{\\dagger}a_{i}\\rho-\\rho a_{i}^{\\dagger}a_{i}\\right] +\\frac{\\tilde{\\Gamma}_{em}^0}{2}\\sum_{i=1}^{k}\\left[\\hat{a}_{i}^{\\dagger}\\rho a_{i}+a_{i}^{\\dagger}\\rho\\hat{a}_{i}-a_{i}\\hat{a}_{i}^{\\dagger}\\rho-\\rho\\hat{a}_{i}a_{i}^{\\dagger}\\right],\n\\end{equation}}\nand \n\\begin{equation}\n \\delta\\mathcal{M}[\\rho]=\\frac{\\tilde{\\Gamma}_{em}^0}{2}\\sum_{i=1}^{k}\\left[\\delta a_{i}^{\\dagger}\\rho a_{i}+a_{i}^{\\dagger}\\rho\\delta a_{i}-a_{i}\\delta a_{i}^{\\dagger}\\rho-\\rho \\delta a_{i} a_{i}^{\\dagger}\\right],\n \\label{perturbation}\n \\end{equation}\nwith \n\\begin{equation}\n\\tilde{a}_{i}^{\\dagger}=\\frac{\\tilde{\\Gamma}_{em}^0}{\\Gamma_{em}^0}\\,\\left( \\hat{a}_{i}^{\\dagger}+\\delta a_{i}^{\\dagger}\\right),\n\\end{equation} and \n{\\begin{equation}\n\\bra{f'}\\hat{a}_{i}^{\\dagger}\\ket f=\\left(e^{-\\beta\\Delta_{f'f}}-i\\frac{\\omega_{cav}-\\omega_{at}}{\\Gamma_{pump}}\\right)\\bra{f'}a_{i}^{\\dagger}\\ket f,\n\\end{equation}}\nfrom which we deduce that \n{\\begin{equation}\n\\bra{f'}\\delta a_{i}^{\\dagger}\\ket f\\underset{\\Gamma_{pump}\\to\\infty}{=}\\bra{f'}a_{i}^{\\dagger}\\ket f \\, \\left(-i\\frac{\\Delta_{f'f}}{\\Gamma_{pump}}+\\mathcal{O}\\left(\\frac{\\Delta_{f',f}}{\\Gamma_{pump}}\\right)^{2}\\right).\n\\label{deltacreation}\n\\end{equation} }\nUsing similar arguments to the Markovian case of \\ref{app:Exact-stationary-solution}, we can easily show that the grand canonical distribution is a steady state of this modified $\\mathcal{M}_{0}$ operator,\n\\begin{equation}\n\\mathcal{M}_{0}(e^{\\beta N\\mu}e^{-\\beta H})=0,\n\\end{equation}\nAs the correction term $\\delta\\mathcal{M}$ vanishes in the Markovian limit proportionally to $1\/\\Gamma_{pump}$, we can calculate the lowest order correction to the steady state in $\\delta\\mathcal{M}$. Expanding the steady state in powers of $1\/\\Gamma_{pump}$ keeping a constant $(\\omega_{cav}-\\omega_{at})\/\\Gamma_{pump}$, we see easily that the first order corrections in eq. (\\ref{deltacreation}) are purely imaginary so that populations are perturbed only to second order in {$\\beta\\Delta_{f'f}$}. In our Markovian limit, these corrections then vanish even if we perform simultaneously the Markovian and thermodynamic limit. \n\nSecondly, coherences (which are exactly zero in the Markovian case, see Sec.\\ref{sec:Markov}) should be then proportional to {$1\/\\Gamma_{pump}$}. However, we have shown in \\ref{app:quadratic coherences} that the linear contribution to coherences vanishes when we sum over all sites of the system. We conclude thus that in the weakly non Markovian limit, coherences between eigenstates of the hamiltonian are quadratic in $1\/\\Gamma_{pump}$ and therefore remain very small even out of the secular approximation. \n\nAs a further verification of this analytical argument, in the right panel of Fig.\\ref{fig:GCtest} we have shown the $\\Gamma_{pump}$ dependence of the coherence between an arbitrary pair of two-photon states as well as the error in the population of an arbitrary eigenstate, between the true steady state and the grand canonical distribution. As expected on analytical grounds, both these quantities scale indeed as $\\Gamma_{pump}^{-2}$.\n\nFrom these arguments, we conclude that the breakdown of the secular approximation which occurs in the thermodynamic limit where the spectrum become continuous should not affect the effective thermalization of the steady state in the weakly non-Markovian regime of large $\\Gamma_{pump}$. Even if the steady-state is not affected, we however expect that the relatively strong dissipation will significantly affect the the system dynamics. A complete study of this physics will be the subject of a future work. \n\n\\section{Two cavities with strong non linearity}\n\n\\subsection{Towards Mott-insulator physics}\n\nAs a final example of application of our concepts, in this last section we present some preliminary results on the most interesting case of two strongly nonlinear cavities with $U\\gg\\Gamma_{pump}$: extending the photon-number selectivity idea to the many-cavity case, we look for many-body states that resemble a Mott insulator~\\cite{Bloch2008,RMP,Hartmann_rev}. As in the single cavity case, the strong pumping $\\Gamma_{em}\\gg \\Gamma_{loss}$ would favour a large occupations of sites, but is counteracted by the effect of the nonlinearity $U\\gg \\Gamma_{pump}$ which {sets} an upper bound to the occupation: the result is a steady-state with a well-defined number of photons per cavity.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.3,clip]{mott_n} \\hspace{1cm}\n\\includegraphics[scale=0.3,clip]{mott_g2} \\\\\n\\includegraphics[scale=0.3,clip]{mott_g1} \\hspace{1cm}\n\\includegraphics[scale=0.3,clip]{mott_spontcoh}\n \\end{center}\n\\caption{Purely photonic simulations of steady-state observables as a function of $2U\/\\Gamma_{pump}$ in a two-cavity system: (a) average number of photons $n_{1}=\\langle{a_{1}^{\\dagger}a_{1}}\\rangle$, (b) one-site two-body correlation function $g_{1,1}^{(2)}=\\langle{a_{1}^{\\dagger}a_{1}^{\\dagger}a_{1}a_{1}}\\rangle=\\langle{n_{1}(n_{1}-1)}\\rangle$, (c) inter-site one-body correlation function $g_{1,2}^{(1)}={\\langle a_{1}^{\\dagger}a_{2}\\rangle }\/{\\langle a_{1}^{\\dagger}a_{1}\\rangle }$. Parameters: $2J\/\\Gamma_{pump}=0.2$, $2\\Gamma_{loss}\/\\Gamma_{pump}=0.002$, $2\\Gamma_{em}\/\\Gamma_{pump}=0.06$ (solid black line). Red dashed line, same simulation with a weaker $2\\Gamma_{em}\/\\Gamma_{pump}=0.00144$. Panel (d), from left to right~:  state occupancy, energy and two site spatial coherence of the different eigenstates of the hamiltonian, at the maximum coherence point $2U\/\\Gamma_{pump}=0.16$ of the red dashed line.\n\\label{fig:mott}}\n\\end{figure}\n\nThe result of numerical calculations based on the photonic master equation are shown as black lines in Fig.\\ref{fig:mott}(a-c) in the $\\omega_{cav}=\\omega_{at}$ case: for a high emission rate $\\Gamma_{em}^0$ and a strong non linearity $U$, signatures of the desired Mott state with one particle per site are visible in the steady-state average number of photons  that tends to $1$ for a strong nonlinearity $U$ [panel (a)], in the probability of double occupancy  that tends to $0$ [panel (b)], and in the one-body coherence between the two sites that also tends to $0$ [panel (c)]. \n\nWhile these results are a strong evidence of $N_0=1$ Mott state, a similar calculation for larger $N_0 \\geq 2$ Mott states is made much more difficult by metastability issues and the Mott state would typically have a finite lifetime. As in the single cavity case, we expect that this problem could be fixed by adding several atomic species on resonance with the different photonic transitions below $N_0$.\n\nBased on this preliminary analysis, we can attempt to make some claims on the structure of the non-equilibrium phase diagram of our model. As for $J=0$ one can efficiently create a Fock state in each cavity, we expect that for small $J$ the system will remain in a sort of Mott state. On the other hand, in the weakly interacting regime we expect the system to display a coherent Bose-Einstein condensate~\\cite{PRL10}. In between, one can anticipate that system should display some form of non-equilibrium Mott-Superfluid transition. Analytical and numerical studies in this direction are in progress.\n\n\\subsection{An unexpected mechanism for coherence}\n\nThe red dashed lines in the same panels Fig.\\ref{fig:mott}(a-c) show the same simulation for a weaker emission rate $\\Gamma_{em}^0$, which allows to consider weaker values of the nonlinearity without increasing too much the photon number. In particular, in panel (c) we see that the non-negligible value of $2J\/\\Gamma_{pump}$ is responsible for a  significant spatial coherence between the two sites, which attains a maximum value $g^{(1)}_{12}\\approx 0.26$ for an interaction strength $2U\/\\Gamma_{pump}\\simeq 0.16$ of the same order of magnitude as the tunnel coupling $2J\/\\Gamma_{pump}=0.2$. \n\nThe quite unexpected appearance of this coherence can be understood as follows. On one hand, in the absence of tunneling $J=0$, all the dynamics is local and we do not expect any spatial coherence. On the other hand, in the absence of interactions $U=0$ and for zero detuning, symmetric and anti-symmetric states are equally close to resonance (albeit with opposite detuning) and then equally populated, so there should not be any coherence either. However in presence of both tunnelling and small interactions (i.e. for $J,U\\neq 0$ and $U\\ll J$), the energy of all eigenstates (symmetric\/anti-symmetric states with various photon numbers) is perturbatively shifted in the upward direction by (small) interactions $U$. As a result, symmetric states, which are below the resonance, get closer to resonance and become more populated than the anti-symmetric ones, which get farther to the resonance and are thus depleted. As one can see in the plot of the energy, the spatial coherence and the steady-state occupancy of the different eigenstates shown in Fig.\\ref{fig:mott}(d) for the maximum coherence point, this induces an overall positive coherence between the two sites.\n\nEven though the nonlinearity is only active for states with at least two photons, it is interesting to note that also in the $N=1$ manifold the antisymmetric state is less populated than the symmetric one. This population unbalance is inherited from the one in the above-lying $N>1$ states, as the decay preferentially occurs into the symmetric state. Since no coherence is expected in both limiting cases of purely interacting $U\\gg J$ and non interacting $U\/0=0$ photons, the maximum of the coherence is obtained when interactions and tunelling are of the same magnitude, $U\\approx J$: this result is clearly visible in panel Fig.\\ref{fig:mott}(c).\n\nInvestigation of this many-body physics in the more interesting case of larger arrays which can accommodate a larger number of photons requires sophisticated numerical techniques to deal with the dynamics in a huge Hilbert space~\\cite{Pisa,corner} and will be the subject of future work. A very exciting advance in this direction was recently published in~\\cite{Kapit} for strongly interacting photons in the presence of a synthetic gauge field for light: analogously to the Mott insulator state studied here, the combination of the effectively frequency-dependent pumping (obtained via a two-photon pumping in the presence of an auxiliary lattice) and the many-body energy gap was predicted to generate and stabilize fractional quantum Hall states of light.\n\n\n\\section{Conclusions}\n\\label{sec:Conclu}\n\nIn this work we have proposed and characterized a novel scheme to generate strongly correlated states of light in strongly nonlinear cavity arrays. Photons are incoherently injected in the cavities using population-inverted two-level atoms, which preferentially emit photons around their resonance frequency. The resulting frequency-dependence of the gain will be the key element to generate and stabilize the desired quantum state. A manageable theoretical description of the system is obtained using projective methods, which allow to eliminate the atomic degrees of freedom and describe the non-Markovian photonic dynamics in terms a generalized master equation. \n\nThe efficiency of the our pumping scheme to generate specific quantum states is first validated on a single-cavity system: for weak nonlinearities, a novel mechanism for optical bistability is found. For strong nonlinearities, Fock states with a well-defined photon number can be generated with small number fluctuations. \n\nIn the general many cavity case, in the weakly non-Markovian case the steady-state of the system recovers a Grand-Canonical distribution with an effective chemical potential determined by the pumping strength and an effective inverse temperature proportional to the non-Markovianity: This very general results may have application to explain apparent thermalization in recent photon and polariton condensation experiments. \n\nFinally, the power of a frequency-dependent pumping to generate strongly correlated states of light is illustrated in the case of a strongly nonlinear two-cavity system which, in the strongly non-Markovian regime, can be driven into a state that closely reminds a Mott-insulator state. A general study of the potential and of the limitations of the frequency-dependent gain to generate generic strongly correlated states with many photons will be the subject of future work.\n\n\\section{Acknowledgments}\nIC acknowledges financial support by the ERC through the QGBE grant, by the EU-FET Proactive grant AQuS (Project No.640800), and by the Provincia Autonoma di Trento, partly through the project ``On silicon chip quantum optics for quantum computing and secure communications'' (``SiQuro''). Continuous discussions with Alessio Chiocchetta, Hannah Price, Alberto Amo, Jacqueline Bloch, and Mohammad Hafezi are warmly acknowledged.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nIt is well known that topologically non-trivial configurations play an essential role in\nQuantum ChromoDynamics (QCD).\nFor $J^P = 0^-$ mesons, such configurations must be present in order to split the iso-singlet $\\eta'$ meson\nfrom the octet of pions, kaons, and the $\\eta$ meson \\cite{Witten:1979vv,Veneziano:1979ec,tHooft:1986ooh}.\nThey also affect the spectrum of mesons with higher spin \\cite{Giacosa:2017pos}, \nand contribute to the proton and photon structure functions in polarised deep-inelastic scattering\n\\cite{Veneziano:1989ei,Shore:1990zu,Shore:1991dv,Shore:1997tq,Narison:1998aq,Bass:2004xa,Shore:2007yn,Tarasov:2020cwl,Tarasov:2021yll}.\n\nIn weak coupling for a $SU(N)$ gauge theory, the dominant configurations are instantons,\nwhich are self-dual solutions to the classical equations of motion.\nBy asymptotic freedom instanton effects can be reliably computed\nat high temperature, $T$, or quark chemical potential, $\\mu_{\\rm qk}$\n\\footnote{The computation at $T\\neq 0$ is only complete to one loop order\n\\cite{Gross:1980br,KorthalsAltes:2014dkx,KorthalsAltes:2015zpx,Pisarski:2019upw,Boccaletti:2020mxu}.\nWhen $\\mu_{\\rm qk} \\neq 0$, even at one loop order \nthe result for arbitrary instanton scale size is lacking \\cite{Pisarski:2019upw}.}.\nThe action for a single instanton with unit topological charge is $= 8 \\pi^2\/g^2$.  \nSince by asymptotic freedom the running coupling constant, $g^2(T) \\sim 8 \\pi^2\/(c \\log(T))$,\n\\footnote{For an $SU(N)$ gauge theory coupled to $N_f$ flavors of massless quarks,\n$c = (11 N - 4 N_f C_f)\/3$, $C_f = (N^2-1)\/(2N)$.},\nthe topological susceptibility falls off sharply at high temperature,\n\\begin{equation}\n  \\chi_{\\rm top}(T)\\sim \\exp(- 8 \\pi^2\/g^2(T)) \\sim 1\/T^{c} \\;\\; , \\;\\; T \\rightarrow \\infty \\; .\n\\end{equation}\nRemarkably, numerical simulations in lattice\nQCD find that this power law holds down to temperatures as low as $\\approx 300$~MeV when $\\mu_{\\rm qk} = 0$,\nwhich we assume henceforth\n\\cite{Borsanyi:2015cka,GrillidiCortona:2015jxo,Bonati:2015vqz,Borsanyi:2016ksw,Frison:2016vuc,Petreczky:2016vrs,Taniguchi:2016tjc,Bonati:2018blm,Bonanno:2020hht,Lombardo:2020bvn,Jahn:2021qrp,Borsanyi:2021gqg,Chen:2022fid}\n\\footnote{\nThe overall magnitude of the topological susceptibility is about an order of magnitude greater than the\ninstanton result at one loop order, but surely it is necessary to include\nthe corrections to the instanton density at two loop order for $T \\neq 0$.  \nMulti-instanton configurations can also contribute at low $T$ \\cite{Rennecke:2020zgb}.}.\n\nFor $T < 300$~MeV, in QCD the topological susceptibility is\nnot that of a dilute instanton gas. To understand the what generates the topological\nsusceptibility at lower temperature, and especially in vacuum, it is useful to consider a\n$SU(N)$ gauge theory without dynamical quarks.\nIn this case, a global $\\mathbb{Z}_N$ symmetry is\nspontaneously broken above the temperature for deconfinement, $T_{\\rm deconf}$ \\cite{Gaiotto:2014kfa}.\nNumerical simulations of the pure gauge theory find that the dependence on $N$ is rather weak.\nForming a dimensionless ratio between the topological susceptibility and the square of the string tension,\nat zero temperature $\\chi_{\\rm top}(0)\/\\sigma^2$, only varies by $\\approx 10\\%$ between $N=3$\n\\cite{Alles:1996nm,Durr:2006ky,Luscher:2010ik,Xiong:2015dya,Jahn:2018dke,Giusti:2018cmp}\nand higher $N$\n\\cite{Lucini:2001ej,Vicari:2008jw,GarciaPerez:2009mg,Fodor:2009nh,Fodor:2009ar,Lucini:2012gg,Bonati:2013tt,Ce:2016awn,Bonati:2016tvi,Kitano:2017jng,Itou:2018wkm,Bonanno:2018xtd,Kitano:2021jho,Bennett:2022gdz}.  \nThese results suggest that as $N \\rightarrow \\infty$, the topological susceptibility does not vary with temperature\nin the confined phase.\nNumerical simulations find that the\ndeconfining phase transition is of first order for three or more colors, with\n$T_{\\rm deconf} \\approx 270 $~MeV for $N = 3$\n\\cite{Boyd:1996bx,Borsanyi:2012ve,Shirogane:2016zbf,Caselle:2018kap,Borsanyi:2022xml}\nFor $N \\geq 3$, $\\chi_{\\rm top}(T)$ jumps at $T_{\\rm deconf}$, and then\nfalls off rapidly with increasing $T$, dominated by instantons above some temperature close to $T_{\\rm deconf}$.\n\nIt is difficult to see how the topological susceptibility could be due to instantons in the confined phase\nat large $N$ \n\\footnote{There could be a dense liquid or crystal of instantons \\cite{Carter:2001ih,Liu:2018znq}.\nHowever, it is necessary\nto show that the action of such a crystal vanishes to $\\sim N$, and is only $\\sim 1$, in order to generate\na topological susceptibility of the same order \\cite{Witten:1978bc}.}.\nHolding $g^2 N \\equiv \\lambda$ fixed as $N \\rightarrow \\infty$, the action of a single instanton\nin the partition function is\nexponentially suppressed, $\\sim \\exp(- (8 \\pi^2\/\\lambda) N)$,\nas $N\\rightarrow \\infty$.  This quandry was recognized originally by \nWitten \\cite{Witten:1979vv} and Veneziano \\cite{Veneziano:1979ec}, who nevertheless argued that in vacuum\nthe topological susceptibility is significant at large $N$ \\cite{Witten:1998uka}.\n\nThe most natural possibility is that there are objects with fractional topological charge $\\sim 1\/N$, whose\ncontribution directly survives at infinite $N$, $\\sim \\exp(-8 \\pi^2\/\\lambda)$.\n't Hooft first presented explicit configurations on tori of finite volume\n\\cite{tHooft:1980kjq,tHooft:1981nnx,vanBaal:1982ag,Sedlacek:1982cd,Nash:1982kp,Killingback:1984en,Alvarez:2003}.\nKraan, van Baal, Lee, and Lu\n\\cite{Lee:1997vp,Lee:1998vu,Lee:1998bb,Kraan:1998pm,Kraan:1998sn,Diakonov:2002fq,Bruckmann:2009nw,Diakonov:2009jq,Poppitz:2008hr,Gonzalez-Arroyo:2019wpu,Anber:2021upc}\nshowed that at nonzero temperature\ninstantons can be viewed as made of $N$ constituents, each with topological charge $1\/N$; the constituents have\nnontrival holonomy, and cannot be pulled arbitrarily far apart.\nConsequently, it is not apparent how these configurations survive in vacuum.\n\nA useful limit is to study gauge theories on a femto-slab, where one spatial dimension, $L$,\nis very small, with $L \\Lambda_{\\rm QCD} \\ll 1$, \nwhere $\\Lambda_{\\rm QCD}$ is the renormalization mass scale of QCD.\nOver large distances the theory reduces to one in $2+1$ dimensions\n\\cite{Polyakov:1976fu,Dunne:2000vp,Kogan:2002au,Kovchegov:2002vi}.\nOn a femto-slab, semi-classical techniques demonstrate\nthat monopole-instantons with topological charge $\\sim 1\/N$ are ubiquitous\n\\cite{Unsal:2006pj,Unsal:2007vu,Unsal:2007jx,Unsal:2008ch,Shifman:2008ja,Simic:2010sv,Anber:2011gn,Poppitz:2012sw,Anber:2013doa,Aitken:2017ayq,Unsal:2020yeh,sym14010180,Tanizaki:2022ngt}.\nHowever, it is not clear what happens when the size of the slab increases to distances where $L \\Lambda_{\\rm QCD} \\sim 1$.\n\nIn this paper we consider configurations with fractional topological charge $\\sim 1\/N$ in both\nthe $\\mathbb{CP}^{N-1}$ model in $1+1$ dimensions\n\\cite{Gross:1977wu,Witten:1978bc,DAdda:1978vbw,DiVecchia:1979pzw,Berg:1979uq,Fateev:1979dc,Rothe:1980rp,Zhitnitsky:1988zd,Ahmad:2005dr,Lian:2006ky,Bruckmann:2007zh,Brendel:2009mp,Unsal:2020yeh},\nand for $SU(N)$ gauge theories without dynamical quarks in\n$3+1$ dimensions.  The former is useful as a toy model to illustrate our basic point.  Instantons with\nintegral topological charge are solutions of the classical equations of motion\n\\cite{Atiyah:1978ri}.  For both models,\nthe classical action is invariant under a scale symmetry, which implies that instantons have a scale size, $\\rho$,\nwhich ranges from zero to infinity.  In contrast, our configurations are stationary points of a quantum,\neffective action.  Thus their scale size is manifestly {\\it non}-perturbative, on the order of the confinement\nscale.\nIn essence, we suggest that as a femto-slab increases in width, that the size of the objects with\nfractional topological charge is fixed at the confinement scale.\nIn the $\\mathbb{CP}^{N-1}$ model this is apparent,\nand we argue that the same holds for pure $SU(N)$ gauge theories.\nWe give both a general mathematical analysis, and outline how to construct \n$\\mathbb{Z}_N$ dyons with fractional topological charge.\n\nWe discuss how to measure such objects using numerical simulations\non the lattice, following Edwards, Heller, and Narayanan \\cite{Edwards:1998dj,karthik:2022}.\nWe stress that with periodic boundary conditions, the total topological charge is necessarily integral.\nTo measure a system whose total topological charge is fractional,\n$\\mathbb{Z}_N$  twisted boundary conditions must be used.  Nevertheless,\nwe suggest that as on a femto-slab\n\\cite{Polyakov:1976fu,Dunne:2000vp,Kogan:2002au,Kovchegov:2002vi,Unsal:2006pj,Unsal:2007vu,Unsal:2007jx,Unsal:2008ch,Shifman:2008ja,Simic:2010sv,Anber:2011gn,Poppitz:2012sw,Anber:2013doa,Aitken:2017ayq,Unsal:2020yeh,sym14010180,Tanizaki:2022ngt},\nthat the vacuum is a condensate of $\\mathbb{Z}_N$  dyons and anti-dyons, each with\nfractional topological charge.  This should\nbe measureable, assuming that there is some separation between the dyons and anti-dyons.\n\nWe conclude with a discussion of how quarks and $\\mathbb{Z}_N$ dyons might interact in QCD.  \n\n\n\\section{$\\mathbb{CP}^{N-1}$ model}\n\\label{sec:cpn}\n\nThis model is a nonlinear sigma model in two spacetime dimensions where the target space is the complex projective space\n$\\mathbb{CP}^{N-1}$. The target space can thus be described by\n$N$ complex variables $z^i$ which are the so-called homogeneous\ncoordinates, with the identification\n$z^i \\sim w \\, z^i$, $w \\in \\mathbb{C} - \\{ 0\\}$.\nOne can choose to fix or eliminate the freedom of the magnitude of\n$w$ in this identification by setting $\\sum_{i=1}^N {\\bar{z}}^i z^i = 1$.\nThe phase of $w$ can be removed by gauging the $U(1)$ corresponding\nto such phase transformations.\nThe Lagrangian density defining the theory can thus be taken to be\n\\begin{equation}\n  {\\cal L} = \\frac{1}{g^2} \\sum_{i = 1}^N \\; |D_\\mu z^i|^2 \\;\\; ; \\;\\; D_\\mu = \\partial_\\mu - i A_\\mu \\; .\n  \\label{lag_cpn}\n\\end{equation}\nwhere it is understood that the $z^i$'s satisfy the constraint $\\sum_{i=1}^N \\overline{z}^i z^i = 1$ at each point.\nThe gauge field $A_\\mu$ ensures that the $z^i$ are also invariant under local $U(1)$ transformations,\n$z^i(x) \\rightarrow {\\rm e}^{i \\alpha(x)} z^i(x)$. \n(Notice that the constraint ${\\bar{z}}\\cdot z = 1$ defines a sphere $S^{2 N -1}$\nat every point in spacetime.\nThe gauge symmetry removes a phase, i.e., $S^1$, so that\nwe get $S^{2 N -1}\/ S^1$ which is another way to define\n$\\mathbb{CP}^{N-1} $.)\nThe absence of a kinetic term for the gauge field in Eq. (\\ref{lag_cpn})\nensures that no new degrees of freedom are introduced via \nthe gauging. In fact, classically one can eliminate $A_\\mu$ by its equation of motion as\n\\begin{equation}\nA_\\mu = - {i \\over 2} \\left( {{{\\bar{z}}^i \\partial_\\mu z^i  - \\partial_\\mu {\\bar{z}}^i \\, z^i }\\over {\\bar{z}} \\cdot z }\\right)\n\\label{lag-cpn2}\n\\end{equation}\nso that ${\\cal L}$ can be re-expressed entirely in terms of the $z^i$ and ${\\bar{z}}^i$.\nFinally, we note that the only\ncoupling constant  in Eq. (\\ref{lag_cpn}) is $g^2$, which is dimensionless.\n\nClearly the Lagrangian in Eq. (\\ref{lag_cpn}) is invariant under global $SU(N)$ transformations,\n$z^i \\rightarrow U^{i}_{~j} \\, z^j$, where $U \\in SU(N)$.\nElements of the center of $SU(N)$, which is ${\\mathbb{Z}_N}$, are special.  These are\nthe $U_k = {\\rm e}^{2 \\pi i k\/N} {\\bf 1}$, where $k = 0, 1\\ldots (N-1)$.  \nUnder the $U_k$ the $z^i$'s transform as $z^i \\rightarrow e^{2 \\pi i k\/N} z^i$, but since the\n$z^i$ are homogeneous coordinates (with the identification $z^i \\sim w\\, z^i$), this is equivalent to the identity transformation.\nThus the full global symmetry is $SU(N)\/{\\mathbb{Z}_N}$ \\cite{Witten:1978bc,DAdda:1978vbw,Unsal:2020yeh}.\n\nThe topological winding number is\n\\begin{equation}\n  Q = \\frac{1}{2 \\pi} \\int d^2 x \\; \\epsilon^{\\mu \\nu} \\partial_\\mu A_\\nu \\; .\n\\end{equation}\nFor fields where $z^i$ approaches a constant at infinity, $Q$ is an integer.\nAll classical configurations with $ Q \\neq 0$ are known \\cite{DAdda:1978vbw}. \nKeeping $g^2 N$ fixed as $N \\rightarrow \\infty$, the value of the classical action is uniformly $\\sim N$.  The fluctuations\nabout arbitrary instanton configurations have also been computed.  While they simplify for\n$N=2$, where it reduces to an $O(3)$ model \\cite{Gross:1977wu,Berg:1979uq,Fateev:1979dc},\nfor $N > 2$ the integration over the collective coordinates of the instantons is not tractable.  Even so,\nthey all appear to be exponentially suppressed, term by term, for large $N$.\n\nAs discussed in the seminal papers \\cite{Witten:1978bc,DAdda:1978vbw},\nthe large $N$ analysis in the quantum theory\ncan be carried out by\nintroducing a Lagrange multiplier field $\\lambda(x)$ to impose the constraint \n${\\bar{z}}\\cdot z - 1 = 0$ and then integrating out\nthe $z^i$\nfields. This leads to an effective action for $A_\\mu$ and $\\lambda$,\n\\begin{equation}\n  {\\cal S}_{\\rm eff} = N \\, {\\rm tr} \\, \\log \\left( -D_\\mu^2 + i \\lambda \\right) - i \\int d^2 x \\; \\frac{\\lambda(x)}{g^2}\n  \\; .\n\\label{eff_act_cpn}\n\\end{equation}\nThe corresponding equations of motion are\n\\begin{equation}\n  N \\; {\\rm tr} \\; D_\\mu^{\\rm cl} \\frac{1}{-(D^{\\rm cl}_\\mu)^2 + m^2(x)} = 0 \\; ,\n  \\label{eqn-A}\n\\end{equation}\nand\n\\begin{equation}\n  N \\; {\\rm tr} \\; \\frac{1}{-(D^{\\rm cl}_\\mu)^2 + m^2(x)} - \\frac{i}{g^2} = 0 \\; ,\n  \\label{eqn-lamb}\n\\end{equation}\nfor arbitrary solutions $A_\\mu(x) = A_\\mu^{\\rm cl}(x)$ and $i \\lambda(x) = m^2(x)$.  In vacuum\n$A_\\mu^{\\rm cl}=0$ and $m^2(x)$ is constant, with the dynamically generated mass\n$m$ related to the coupling constant $g^2$ through dimensional transmutation \\cite{Witten:1978bc,DAdda:1978vbw}.\n\nThe quantum dynamics of the model, defined by Eq.\n(\\ref{eff_act_cpn}), has significant qualitative differences compared to the classical analysis \nbased on Eq. (\\ref{lag_cpn}). Classically, the constraint\n${\\bar{z}} \\cdot z = 1$ is imposed, {\\it a priori}, or as the equation of motion\nfor $\\lambda$. The solution of this equation necessarily breaks the\n$SU(N)$ global symmetry. In the quantum theory,\nthis symmetry breaking is eliminated, in accordance with\nthe Mermin-Wagner-Coleman theorem. The expectation values of the\n$z^i$ will vanish and they behave like massive fields. Explicitly, \nin a derivative expansion, the effective action of Eq. (\\ref{eff_act_cpn}),\nwith the inclusion of the fields $z^i$, takes the\nform\n\\begin{equation}\n{\\cal S}_{\\rm eff} = \\int \\vert (\\partial_\\mu -i A_\\mu ) z^i \\vert^2 - m^2 {\\bar{z}}^i z^i\n- {N \\over 48 \\pi m^2} F_{\\mu\\nu}F^{\\mu\\nu} + \\cdots\n\\label{eff_act-2}\n\\end{equation}\nwhere $m^2$ is the vacuum value determined by Eqs. (\\ref{eqn-A}) and ({\\ref{eqn-lamb}).\n(This equation gives the generating functional of the one particle irreducible vertices,\nand so includes small fluctuations in the $z^i$ fields, in addition to Eq. (\\ref{eff_act_cpn}).)\nThe terms with derivatives of $m^2(x) = i \\lambda (x)$ are not displayed here\nas they do not play a significant role in the subsequent discussions.\nWe note that the higher terms indicated by ellipsis in\nEq. (\\ref{eff_act-2}), by virtue of gauge invariance, will involve\nonly covariant derivatives of the fields $z^i$, ${\\bar{z}}^i$ and powers\nof $F_{\\mu\\nu}$.\n\nBefore we proceed to consider topologically nontrivial field configurations, there is one key observation\nregarding the action in Eq. (\\ref{eff_act-2})\nwhich is important for us. The global symmetry of the theory\nis still $SU(N)\/{\\mathbb{Z}_N}$. By virtue of confinement,\nthe allowed states are $\\mathbb{Z}_N$-invariant \\cite{Witten:1978bc}.\n\nIt is now possible to consider topologically nontrivial configurations,\nnot as solutions to the classical equations of motion,\nbut with the effective Lagrangian at large $N$.\nIn polar coordinates $(r,\\varphi)$ in two dimensions, at large $r$ the solution for the gauge field must satisfy\n\\begin{equation}\n  A_\\mu dx^\\mu \\sim Q  d \\varphi \\; ; \\; {\\rm i.e.,} ~A_\\varphi \\sim {Q \\over r} \\; \\; {\\rm as~}r \\rightarrow \\infty \\; ,\n  \\label{asymp_behavior_int}\n\\end{equation}\nso that $\\int F \\sim Q \\neq 0$.\nThis should be accompanied by a suitable\nansatz for $m^2(x)$ is a function of $r$, but\nwe do not elaborate on this since it is not important for the main thread of our arguments.\nDetermining the solution of the nonlocal equations of motion in\nEqs. (\\ref{eqn-A}) and (\\ref{eqn-lamb}) is not elementary. \nBut there is one aspect of any such solution which is worth of note, and which\nin fact is a recurrent point throughout our analysis.\nWhile the classical action is invariant under scale transformations, at large $N$ the quantum effective\naction is not. Thus while Eq. (\\ref{asymp_behavior_int}) fixes the behavior at infinity, the nature of\nthe full solution varies over a distance $\\sim 1\/m$.\n\nThere is one possibility, noted by Witten \\cite{Witten:1978bc}, which is that the term $\\sim N$ for the action\nof a quantum instanton vanishes.  \nThis is most unlikely for classical instantons, as their scale size $\\rho$ runs from zero to infinity, and at one\nloop order their contribution is a nontrivial function of $\\rho$ times the renormalization mass scale.\nIn contrast, for the quantum action the coupling constant\nis evaluated at a given scale through dimensional transmutation, and the size is fixed.  \nEven so, we do not see any general reason why the action of the quantum instanton should vanish at $\\sim N$.\n\nWe then turn to the possibility of configurations with fractional topological charge.\nOn a femto-slab classical instantons were constructed by Unsal \\cite{Unsal:2020yeh}; their\nsize is necessarily on the order of the width of the slab.\nIn contrast, we consider quantum instantons\nin vacuum.  As a first step, consider the spherically symmetric configuration\n\\begin{equation}\nF_{12} = \\begin{cases}\n{2 \/ (N a^2)} \\hskip .1in&r < a\\\\\n0& r>a\n\\end{cases}\n\\label{vortex1}\n\\end{equation}\nThis corresponds to $Q = \\int (F\/2\\pi) = 1\/N$, and the gauge potential\n\\begin{equation}\nA_\\mu dx^\\mu = - {1\\over N \\pi a^2} \\int d^2x'\\, {\\epsilon_{\\mu\\nu} (x- x')^\\nu \\over \\vert x- x'\\vert^2} \\rho(x')\\, dx^\\mu\n\\label{vortex2}\n\\end{equation}\nwhere $\\rho (x')$ is equal to $1$ in a small disk of radius $a$, and zero elsewhere. This configuration is a slightly thickened vortex and\nthe potential (\\ref{vortex2}) is consistent with the asymptotic behavior\n(\\ref{asymp_behavior_int}), with $Q = 1\/N$.\nThe contribution of this configuration to the action (\\ref{eff_act-2}) is\n\\begin{equation}\n{N \\over 48 \\pi m^2} \\int F^2 = {1\\over 6 N m^2 a^2}\n\\label{vortex3}\n\\end{equation}\nThe action for the higher terms will be similarly suppressed, since they must involve powers of $F$.\n\nThe ansatz (\\ref{vortex2}) may be written for $\\rho$ with support around the origin, as\n\\begin{equation}\nA_\\mu dx^\\mu = f(r) d\\varphi = {1\\over N} \\begin{cases}\n(r^2 \/a^2)\\, d\\varphi \\hskip .1in& r\\leq a\\\\\nd\\varphi& r>a\\\\\n\\end{cases}\n\\label{vortex3a}\n\\end{equation}\n\nTurning to the $z$-dependent part of the action, the only point of subtlety\nis about the phase of $z^i$. With the background (\\ref{vortex1}), (\\ref{vortex2}),\nthe parallel transport of $z^i$ in a full circle around the origin\n(or the location of the vortex) gives $z^i \\rightarrow e^{2 \\pi i\/ N} z^i$.\n(The phase may also be viewed as the Aharonov-Bohm phase acquired by\n$z^i$ in a circuit around the vortex.)\nSeemingly, $z^i$ are not single-valued, but notice that the phase can be removed\nor canceled by an $SU(N)$ transformation in its center $\\mathbb{Z}_N$.\nSince states are $\\mathbb{Z}_N$-invariant there is no difficulty\nassociated with this behavior.\n\nWe can now supplement the ansatz (\\ref{vortex2}) or (\\ref{vortex3a})\nwith a suitable ansatz for $z^i$, such as\n\\begin{equation}\nz^1 = e^{i \\varphi \/N} h(r) , \\hskip .1in z^i = 0, \\hskip .1in\ni = 2, 3, \\cdots, N,\n\\label{vortex4}\n\\end{equation}\nor any $SU(N)\/{\\mathbb{Z}_N}$ transformation of this.\nWe have incorporated the aperiodicity in $\\varphi$ mentioned above,\nnamely, $z_i(r,2 \\pi) = {\\rm e}^{2 \\pi i\/N} z_i(r,0)$. This multi-valuedness\nis where it differs from previous solutions of the classical equations of motion \\cite{Berg:1979uq,Fateev:1979dc}.\n\nTaking the matter part of the action as in (\\ref{eff_act-2}), we find\n\\begin{equation}\nS_z = 2 \\pi \\int dr r \\left[ \\left( {\\partial h \\over \\partial r}\\right)^2\n+ {h^2 \\over r^2} \\left( f - {1\\over N} \\right)^2 + m^2 h^2\n\\right] +\\cdots\n\\label{vortex5}\n\\end{equation}\nThe behavior of $h(r)$ for small and large values of $r$ can be inferred from the equation of motion for $h$, namely,\n\\begin{equation}\n- {1\\over r} {\\partial \\over \\partial r} \\left( r {\\partial h \\over \\partial r} \\right)\n+ \\left( f - {1\\over N} \\right)^2 {h \\over r^2} + m^2 h +\\cdots = 0\n\\label{vortex6}\n\\end{equation}\nBy examining the small $r$ and large $r$ limits of this equation, we \ncan see that\n\\begin{equation}\nh(r) \\sim \\begin{cases}\nr^{1\\over N} \\hskip .2in& r \\rightarrow 0\\\\\ne^{- m r} & r \\rightarrow \\infty\\\\\n\\end{cases}\n\\label{vortex7}\n\\end{equation}\nNotice that $h$ vanishes exponentially as $r \\rightarrow \\infty$.\nThis is a significant point. While the gauge part of the configuration\n(\\ref{vortex3a}) is like an Abrikosov-Nielsen-Olesen vortex, the asymptotic\nbehavior of $z^i$ is very different. We may also note that the vanishing of $z^i$ \nat spatial infinity is consistent with the fact that any configuration of finite action should reproduce\nvacuum behavior at spatial infinity, and that, by virtue of the Mermin-Wagner-Coleman theorem, we cannot have any spontaneous\nbreaking of the $SU(N)\/{\\mathbb{Z}_N}$ symmetry.\n\nIntroducing a scale factor $r_0$, a simple ansatz consistent with \n(\\ref{vortex7}) is\n\\begin{equation}\nh(r) = C \\, { u^{1\\over N} \\over 1+ u^{1\\over N}} e^{-\\mu u},\n\\hskip .1in u = {r\\over r_0}, ~\\mu = m r_0\n\\label{vortex8}\n\\end{equation}\nIt is easy to verify that $S_z$ is finite with this ansatz and\nthe term in (\\ref{vortex5}) involving both $f$ and $h$ depends on \n$a$. Along with the gauge field contribution\n(\\ref{vortex3}), we get a nonlinear expression involving\n$a$ and $C$. Treating these as variational parameters, \nwe can obtain values which minimize the action, at least within the\nclass of ans\\\"atze (\\ref{vortex3a}), (\\ref{vortex4}), ({\\ref{vortex8}).\n\nA few comments are in order at this point. Notice that, even if $C =0$, we do have a vortex-like configuration (\\ref{vortex3a}). Although extremization with respect to $a$ with just the term (\\ref{vortex3}) will lead to\n$a \\rightarrow \\infty$, there are \nterms with higher powers of $F$ in the action,\nindicated by ellipsis in (\\ref{eff_act-2}). Including them and extremizing can lead to a finite value, ultimately determined by the dimensionful\nparameter $m$.\nThe inclusion of higher $z$-dependent terms will lead\nterms which are of order $C^4$ and higher. Thus we expect that\nthe extremization including such terms will give finite values to\nboth $a$ and $C$.  As noted, this equivalent to solving the nonlocal equations of motion\nin Eq. (\\ref{eff_act_cpn}) and (\\ref{eqn-A}).\n\n\n\n\n\nTo frame this more generally, at large $N$, we can again consider expanding \n(\\ref{eff_act_cpn}) in powers of $A_\\varphi$ which is of order\n$1\/N$, based on our ansatz.  As for the solution with integral\ntopological charge, and as in the example above,\nas a solution of the quantum action the size is $\\sim 1\/m$.  The term linear in\n$A_\\varphi$ vanishes by the equation of motion for the gauge potential.  Taking $m^2(x) = m^2$, then,\nthe term in the action $\\sim 1$ automatically vanishes. \nThe expansion of the effective action to quadratic\norder in $A_\\varphi$, i.e., as in (\\ref{eff_act-2}),\nshows that the nonzero\ncontribution of the $A$-part of the action will\nbe of order $1\/N$.\n\nFinally, turning to the dependence on the vacuum angle $\\theta$,\ndirect computation \nof the free energy demonstrates that it is of order $\\sim 1$,\nwith moments which can be computed in a background electric field \\cite{Bonati:2016tvi}.\nThus even with a vacuum of a superposition of quantum instantons with fractional topological charge,\nthey cannot be approximated as a dilute gas. Instead, the interaction between such configurations is\nlarge, and dominates the action of each fractional quantum instanton.\n\n\\section{Towards fractional instantons in 4d}\n\\label{sec:math}\n\\mathchardef\\mhyphen=\"2D\n\nWe now turn to nonabelian gauge theories in four dimensions.\nOne of the key steps in understanding configurations of fractional topological charge is the identification of what is meant by the gauge group. Although this question\nhas been analyzed before, it is useful to collect some of the basic ideas here.\nWe will first consider the boundary values for gauge transformations based on the Gauss law (or the nature of the test functions to be used in implementing the Gauss law) and how these are related to\ncharge quantization conditions. \nThis will clarify the nature of the configuration space\nand will naturally lead to the possibility of fractional topological charges.\n\\setcounter{secnumdepth}{2}\n\\subsection{Gauss law in the $E$-representation}\n\nWe consider the gauge theory in the $A_0 =0$ gauge. \nWe must then impose the Gauss law on the wave functions.\nQuantization conditions on the electric charge will be important for us, so it is \nmore appropriate to consider wave functions in the representation \nwhich are eigenstates of the electric field operators $E^a$. \nIn other words, the wave functions are functionals of the electric field.\nThe Gauss law operator is given by\n\\begin{equation}\nG^a(x)= \\nabla_i E_i^a + f^{abc} A^b_i E^c_i \\label{1}\n\\end{equation}\nwhere $f^{abc}$ are the structure constants of the Lie algebra of $G$.\nThe gauge potential and the electric field obey the usual commutation rule\n$[A^a_i(x), E^b_j(y)]= i \\delta^{ab}\\delta_{ij}\\delta (x-y)$, so that, in\nthe $E$-representation, $A^a_i = i (\\delta \/ \\delta E^a_i)$.\nThe physical wave functions $\\Psi$ are selected by the condition \nthat the Gauss law operator must annihilate them.\nThis condition can be written as\n\\begin{eqnarray}\n  &\\int_M& \\theta^a(x) G^a(x) ~\\Psi = \\nonumber \\\\\n&=&  \\int_M \\theta^a(x)\\left[ \\nabla_iE^a_i \\nonumber \n    -i f^{abc} E^b_i {\\delta \\over \\delta E^c_i}\\right] \\Psi =0 \\\\\n  \\label{2}\n\\end{eqnarray}\n(The integral is over the spatial manifold $M$.)\nThis law should be required only for test functions $\\theta^a(x)$\nobeying certain conditions; the nature of these conditions\nwill be clear  from the following discussion.\nTreating $\\theta^a(x)$ as an infinitesimal group parameter, (\\ref{2}) may be\nwritten as\n\\begin{eqnarray}\n\\delta \\Psi &\\equiv& \\Psi( U^{-1}EU) -\\Psi(E)\\nonumber\\\\\n&=&-\\left[ i \\int_M \\theta^a(x) \\nabla_iE^a_i \\right]~\\Psi\n\\label{3}\n\\end{eqnarray}\nwhere $E_i= T^a E^a_i,~ U=\\exp(iT^a \\theta^a )\\approx 1+iT^a \\theta^a$, \n$T^a$ being hermitian matrices which form a basis for the Lie algebra of\n$G$, with $[T^a,T^b]= if^{abc}T^c$. For the fundamental representation, we write\n$T^a = t^a$ and normalize \nthem by ${\\rm Tr} (t^at^b) =\\textstyle{1\\over 2} \\delta^{ab}$.\nThe quantity $U^{-1}E_iU$ is the gauge transform of $E_i$ and hence $\\delta \\Psi$ \nmeasures the change of $\\Psi$ under a gauge transformation with parameter \n$\\theta^a(x)$. Obviously, if $\\Psi$ is a solution to (\\ref{3}), then so is\n$\\Psi\\, f(E)$ where $f(E)$ is a gauge-invariant function of $E_i$.\nThe general solution to (\\ref{3}) may therefore be written as $\\Psi= \\rho\\,\n\\Phi (E)$, where $\\Phi (E)$ is an arbitrary gauge-invariant function and\n$\\rho$ is a particular solution to\n\\begin{equation}\n\\delta \\rho +\\left[ i \\int_x \\theta^a(x) \\nabla_iE^a_i \\right]~\\rho =0 \\label{4}\n\\end{equation}\n\nA finite transformation, and the corresponding variation of $\\rho$, can be\nobtained by composition of infinitesimal transformations. Assume that, for\nan electric field $E_i$, we have started from the identity and built up a \nfinite transformation $U$. At this point, the electric field is given by \n${\\cal E}_i=U^{-1}E_iU$. A further infinitesimal transformation would be \ngiven by $iT^a \\theta^a = U^{-1}\\delta U$. Thus (\\ref{4}), written for \nan arbitrary point on the space of $U$'s, becomes\n\\begin{eqnarray}\n\\delta \\rho + 2 \\int_M {\\rm Tr} (\\nabla_i{\\cal E}_i~ U^{-1}\\delta U) \\rho &=&0\\nonumber\\\\\n\\delta (\\log \\rho ) = -2  \\int_M {\\rm Tr} (\\nabla_i{\\cal E}_i ~U^{-1}\\delta U) \n&\\equiv& \\Omega \n\\label{5}\n\\end{eqnarray}\nOne can integrate this equation along a curve in the space of $U$'s from \nthe identity to $U$ to obtain the change of $\\rho$ under a finite \ntransformation. With $\\delta$ interpreted as a derivative \non the space of $U$'s, $U^{-1}\\delta U$ is a covariant vector\n(or one-form) and the result of the integration is generally path-dependent.\nFor the result to be independent of the path of integration, the curl of $ {\\rm Tr} (\\nabla\\cdot {\\cal E} \\,U^{-1}\\delta U)$, viewed as a covariant vector or\nas a one-form on the space of the $U$'s, must vanish. Thus the\nthe integrability condition for (\\ref{5}), or \nthe path-independence for the change in $\\rho$, becomes\n\\begin{equation}\n\\delta \\Omega = \\delta \\left[ -2 \\int {\\rm Tr} (\\nabla_i{\\cal E}_i U^{-1}\\delta U)\n\\right] =0\n\\label{5a}\n\\end{equation}\nHere we take $\\delta$ to signify the exterior derivative, so that\n$\\delta$ acting on a one-form (or covariant vector) gives the curl.\nWe now write $\\Omega =\\Omega_1 +\\Omega_2$ with\n\\begin{eqnarray}\n\\Omega_1&=& 2\\int_M{\\rm Tr} ({\\cal E}_i \\nabla_i(U^{-1}\\delta U))\\nonumber\\\\\n&=& 2 \\int_M{\\rm Tr} \\left[ E_i \\left( \\nabla_i (\\delta U U^{-1} ) - [\\nabla_i U U^{-1}, \n\\delta U U^{-1} ] \\right) \\right]\\nonumber\\\\\n\\nonumber\\\\\n\\Omega_2&=& -2 \\oint_{\\partial M}{\\rm Tr} ({\\cal E}_i U^{-1}\\delta U )dS^i \\nonumber \\\\\n&=& -2 \\oint_{\\partial M}{\\rm Tr} (E_i \\delta U  ~U^{-1})dS^i\n\\label{6}\n\\end{eqnarray}\nIt is easily checked, using $\\delta (\\delta U U^{-1} ) = (\\delta U U^{-1})^2$, {\\it without the need of any integration-by-parts on $M$}, that\n$\\delta \\Omega_1 =0$. \nFor the second term, we find\n\\begin{equation}\n\\delta \\Omega_2= -2 \\oint_{\\partial M} {\\rm Tr} ( E_i \\delta U U^{-1} \n\\delta U U^{-1} ) dS^i \\label{7}\n\\end{equation}\nThis is in general not zero. Indeed if $U$ is constant on $\\partial M$,\n$\\delta \\Omega_2 =-2 {\\rm Tr}[Q(\\delta UU^{-1})^2],~Q=\\oint E_idS^i$.\nIn this case, $\\delta \\Omega_2$ has the form of the coadjoint orbit\ntwo-form on $G\/H$, where $H\\subset\nG$ is the subgroup which commutes with the charge $Q$. This form, well-known as the basis for the Borel-Weil-Bott theory on group representations,\nis a nondegenerate\ntwo-form on $G\/H$. In order to have $\\delta \\Omega =0$, we must therefore\nimplement the Gauss law only for those $U$'s which obey the restriction\n\\begin{equation}\n\\oint_{\\partial M} {\\rm Tr} [ E_i \\delta UU^{-1}] dS^i =0 \\label{8a}\n\\end{equation}\nThis is basically the cocycle condition\nwhich allows us to build up finite transformations using sequences of\ninfinitesimal transformations.\nIf $E_i$ on $\\partial M$ can be arbitrary, this\ncondition (\\ref{8a})  would require fixing $U$ to some\nvalue, say, $U_\\infty$\non $\\partial M$. (If $U_\\infty$ is held fixed, $\\delta U_\\infty =0$, so that the \nrequirement (\\ref{8a}) is trivially satisfied.)\nThis clarifies the nature of the test functions $\\theta^a$\nin (\\ref{4}) in imposing the Gauss law: {\\it The test functions must be so chosen that\nthey lead to $U_\\infty$ on $\\partial M$}.\n\nThe key question for us is then:\nWhat are the allowed values of $U_\\infty$? This will be determined by\nthe charge quantization conditions. But before we take up this issue,\na\ncomment on the asymptotic behavior of $U$ is in order.\nAlthough we argued using constant $U$ on $\\partial M$, generically, we cannot impose the Gauss law for $U$'s which are not constant on $\\partial M$ as well,\nsince $\\delta \\Omega_2 $ will not vanish for such cases.\nIn fact, $U$'s which are not constant\non $\\partial M$ correspond to degrees of freedom which\nare physical and generate the\n``edge modes\" of a gauge theory. If we consider the boundary to be at\nspatial infinity, such edge modes are irrelevant. This will be the case \nfor our analysis in this paper.\n\nReturning to constant values of $U$ on $\\partial M$, and\nthe identification of the possible values of $U_\\infty$, we start with the\nquestion:\nHow does $\\Psi$ change under\n transformations which go to a constant $U \\neq U_\\infty$.\nIt is easily seen that the action of a general infinitesimal\ntransformation\n\\begin{equation}\n\\delta A^a_i = -\\partial_i \\theta^a - f^{abc} A_i^b \\theta^c, \\hskip .3in\n\\delta E^a_i = - f^{abc} E^b_i \\theta^c\n\\label{G1}\n\\end{equation}\nis given by\n\\begin{equation}\n\\delta \\Psi = \\left[ i \\int_M D_i\\theta^a(x) ~E^a_i \\right]~\\Psi \n= \\exp\\left[ iQ^a \\theta^a (r= \\infty )\\right] ~\\Psi\\label{G2}\n \\end{equation}\nwhere $Q^a$ is the electric charge $Q^a = \\oint E^a_i dS_i$. Thus transformations which go to a constant $\\neq U_\\infty$ act as a Noether symmetry, under which the charged states undergo a phase transformation. If the only charges in the theory correspond to the adjoint representation of $G$ and its products, i.e., if the states are invariant under ${\\mathbb Z}_N \\in SU(N)$,\nthen the wave functions are {\\it invariant} for those $U$'s which go to an element of the center\n${\\mathbb Z}_N$ at spatial infinity.\nWe have seen that \nwe can implement the Gauss law only for transformations which go to a fixed element\n$U_\\infty$ at spatial infinity. Now we see that the allowed choices for\n$U_\\infty$ correspond to an element of the center ${\\mathbb Z}_N$.\n\nTo recapitulate briefly,  we have seen that the true gauge transformations of the theory, in the sense of corresponding to a redundancy of description,\nare of the form $U({\\vec x} )$ with:\\\\\na) $U \\rightarrow$ a constant $U_\\infty$ at spatial infinity\\\\\nb) $U_\\infty \\in \\mathbb{Z}_N$ for a theory with charges which are\n$\\mathbb{Z}_N$-invariant.\n\\subsection{Charge quantization and $U_\\infty$: An alternate argument}\n\nThere is another way to arrive at the conclusion of the previous subsection,\nnamely, by a direct analysis of the charge quantization conditions.\nNotice that, for $U$'s obeying (\\ref{8a}), we can write $\\Omega$ as\n\\begin{eqnarray}\n\\Omega &=& 2\\int_M {\\rm Tr} [ E_i U \\nabla_i (U^{-1}\\delta U) U^{-1}]\\nonumber\\\\\n&=& 2\\int_M {\\rm Tr} [ E_i \\delta (\\nabla_i U U^{-1})] \\nonumber \\\\\n&=& \\delta \\left(\n2\\int_M {\\rm Tr} [ E_i \\nabla_iUU^{-1}]\\right)\n\\label{9}\n\\end{eqnarray}\nUsing this and integrating (\\ref{5}) from the identity to $U$, we obtain\n\\begin{equation}\n\\rho( U^{-1}EU) = \\rho (E) \\exp\\left(2 \\int_M {\\rm Tr} (E_i\\nabla_iUU^{-1})\\right)\n\\label{10}\n\\end{equation}\nThis equation will be important for us; it will have a key role in subsequent analysis. So another comment and another derivation will be appropriate\nbefore proceeding. One concern about (\\ref{10})  might be that we have used integration from the identity to $U$. \nIn three spatial dimensions, \nsince $\\Pi_3 (G)={\\mathbb Z}$, there are $U$'s which are not connected to the identity. Even though the derivation given above does not quite make it clear,\nthe result (\\ref{10}) \nholds even for $U$'s which are \nnot in the connected component. This can be seen by the following \nalternate derivation borrowed from \\cite{Jackiw:1996ec}.\n\\begin{eqnarray}\n\\Psi (E) &=&\\int d\\mu (A) \\exp\\left[{ 2\\int_M {\\rm Tr} (E_iA_i)}\\right]~ \\Psi (A)\\nonumber\\\\\n&=& \\int d\\mu (A) \\exp \\left[{2\\int_M {\\rm Tr} (E_iA_i)}\\right]  \\nonumber \\\\\n&& \\;\\;\\;\\;\\;\\;\\;\\; \\times \\; \\Psi (U^{-1}AU+U^{-1}\\nabla U)\\nonumber\\\\\n&=&\\exp(-2\\int_M {\\rm Tr} (E_i\\nabla_iUU^{-1})) \\nonumber \\\\\n&& \\times \\int d\\mu (A) \n\\exp\\left[{2\\int_M {\\rm Tr} (U^{-1}E_iUA_i)}\\right]~ \\Psi (A)\\nonumber\\\\\n&=& \\exp(-2\\int_M {\\rm Tr} (E_i\\nabla_i UU^{-1}))~ \\Psi (U^{-1}EU)\n\\label{12}\n\\end{eqnarray}\nwhere we have first used the gauge invariance of the wave functions in the\n$A$-representation (i.e. $ \\Psi (A) = \\Psi (U^{-1}AU+U^{-1}\\nabla U)$) and then changed the variable of integration from\n$A$ to $U^{-1}AU +U^{-1}\\nabla U$. With $\\Psi = \\rho\\, \\Phi (E)$, (\\ref{12})\ngives (\\ref{10}). (This derivation is simpler but the earlier analysis does \nreveal some interesting aspects of imposing the Gauss law.)\n\nEquation (\\ref{10}) contains certain charge quantization requirements which \ncan be used to see why the boundary values of $U$ can be an element\nof the center, rather than strictly being the identity.\nWe can show that $\\rho (E)$ of (\\ref{10})\nwill vanish unless certain conditions are satisfied by $E_i$. \nFor this, it is adequate to examine some special configurations.\nThe basic strategy is to choose an electric field configuration and\na $U$ which commutes with the chosen configuration for $E_i$.\nEquation (\\ref{10}) then gives an identity of the form \n$\\rho =\\rho e^{i\\lambda}$ where the\nphase $\\lambda$ is given by the integral $2\\int_M {\\rm Tr} (E_i\\nabla_iUU^{-1})$.\nThis would imply that $\\rho$ must vanish unless the phase is an integral \nmultiple of $2\\pi$; this is the constraint for the chosen type of field \nconfiguration.\nFor simplicity, \nwe shall use $G=SU(2)$ for the example below; generalization to other \ngroups is straightforward.\n\nFor our example, we choose polar coordinates $(r,\\theta , {\\varphi} )$ and take \n\\begin{equation} \n\\begin{split}\nE_{\\theta} &=E_{{\\varphi}}=0, \\hskip .2in E_r= {\\sigma_3 \\over 2}~{q\\over 4\\pi r^2} \\\\\nU &= \\exp (i\\sigma_3 f(r))\n\\end{split}\n\\label{cq1}\n\\end{equation}\nThis field corresponds to a point charge at $r=0$.\nTo avoid the singularity, we shall remove the point $r=0$ from $M$. Thus\nthe boundary $\\partial M$ consists of a small sphere around $r=0$ and the sphere\nat spatial infinity.\nEven though $U$ is not constant in space, we have chosen it to commute with\nthe given $E_i$. Evaluating the phase factor in (\\ref{10}), we obtain\n\\begin{equation}\n\\rho = \\rho \\exp \\left( 2i~ (\\Delta f) ~q \\right)\n\\label{cq2}\n\\end{equation}\nwhere $\\Delta f = f(\\infty) - f(0)$.\nAs for the values of $f(0), ~f(\\infty )$, they should be integral\nmultiples of $\\pi$ to be consistent with the trivial action of $U$ on states at the \nboundaries. \nIf we require $U$ to go to the identity (and not just an element of the center)\nat the boundary, $\\Delta f = 2 \\pi n$,\n$n \\in {\\mathbb Z}$.\nEquation (\\ref{cq2}) then tells us that we can have nonzero $\\rho$ for\n$q = \\textstyle{1\\over 2} n$. The Gauss law for, say fermion sources,\nmay be written as \n\\begin{equation}\n\\nabla \\cdot E^a  + f^{abc} A^b\\cdot E^c = {\\bar \\psi} T^a \\psi\n\\label{cq3}\n\\end{equation}\nFor the fundamental representation, this gives, for a point source with\n$T^3$-charge,\n$E^3 = {1\\over 2} ({1\/4\\pi r^2})$. This is consistent with the quantization of\n$q$. On the other hand, if we allow $U$ to go to $-1$, then we only need\n$\\Delta f = \\pi n$. Correspondingly, (\\ref{cq2}) tells us that $q$ should be quantized as $q =n$. Equation (\\ref{cq3}) also tells us that this is consistent with\nsources transforming under ${\\mathbb Z}_2$-invariant representations.\n\nThe result of the arguments presented here is that\nwave functions are {\\it invariant} under gauge transformations\nwhich go to an element of the center in theories where the\ncharges are in $\\mathbb{Z}_N$-invariant representations.\nSuch transformations therefore characterize the redundancy\nof the variables $(A_i, E_i)$ in the theory.\n\nThe configuration\nwe have used for obtaining charge quantization\nhas a divergent kinetic energy\n$T={\\textstyle{1\\over 2} } \\int E^2$. \nIt is possible to find nonsingular configurations which lead to the same result; it is just that the argument will be a little more elaborate.\n\\subsection{Nature of the configuration space}\n\nThe $E$-representation of the wave functions was useful in elucidating the\nnature of the allowed boundary values for $U$. However, for the\nanalysis and formulation of ans\\\"atze for the configurations with fractional topological charge, the $A$-representation is more appropriate, so \nthis is the representation we will use for the rest of this paper.\n\nWe can now formalize the situation with the gauge transformations\nas follows.\nStaying within the $A_0 =0$ gauge, let\n\\begin{eqnarray}\n{\\mathcal A} &\\equiv&  \\{ {\\rm Set ~of ~all ~gauge ~potentials}~ A_i \\}\\nonumber\\\\\n&\\equiv&\\{ {\\rm Set ~of ~all ~Lie\\mhyphen algebra\\mhyphen valued ~vector ~fields}\\nonumber\\\\\n&&\\hskip .1in {\\rm on ~space}~ {\\mathbb R}^3\\}\\nonumber\n\\end{eqnarray}\nFurther, let \n\\begin{eqnarray}\n{\\mathcal G} &\\equiv&\\{ {\\rm Set ~of ~all} ~g({\\vec x} ): {\\mathbb R}^3 \\rightarrow SU(N),\n~{\\rm such ~that}\\nonumber\\\\\n&&\\hskip .2in g({\\vec x} ) \\longrightarrow ~{\\rm constant} \\in SU(N) ~{\\rm as} ~\\vert {\\vec x} \\vert \\longrightarrow \\infty\\}\n\\nonumber\\\\\n{\\mathcal G}_\\omega &\\equiv& \\{ {\\rm Set ~of ~all} ~g({\\vec x} ): {\\mathbb R}^3 \\rightarrow SU(N),\n~{\\rm such ~that}\\nonumber\\\\\n&&\\hskip .2in g({\\vec x} ) \\longrightarrow \\omega \\in {\\mathbb Z}_N ~{\\rm as} ~\\vert {\\vec x} \\vert \\longrightarrow \\infty\\}\n\\nonumber\n\\end{eqnarray}\nEvidently, ${\\mathcal G}\/{\\mathcal G}_1 = SU(N)$, the set of rigid transformations or the set of constant boundary values for elements $g$ in ${\\mathcal G}$.  Our discussion of the Gauss law shows that the gauge group, namely, the \nset of transformations which leave the wave functions invariant, is given by ${\\mathcal G}_1$\n in a theory without ${\\mathbb Z}_N$-invariance. However, in a theory with ${\\mathbb Z}_N$-invariance, \n ${\\mathcal G}_\\omega$ leaves $\\Psi$ invariant for any $\\omega$, so that the gauge group is\n ${\\mathcal G}_* = \\cup_{\\omega \\in {\\mathbb Z}_N} {\\mathcal G}_\\omega$. Since the difference\n between ${\\mathcal G}_1$ and ${\\mathcal G}_\\omega$ is in the boundary value, we may\n also consider any element of ${\\mathcal G}_\\omega$ to be of the form\n $g ({\\vec x} )~ \\omega$, where $g ({\\vec x} )$ goes to the identity at spatial infinity.\n \nThe physical configuration space, for theories with charges in the fundamental representation, i.e., without $\\mathbb{Z}_N$-invariance,\nis given by ${\\mathcal A}\/ {\\mathcal G}_1$. It is easy to see that this space is multiply connected.\nConsider a sequence of configurations $A_i ({\\vec x}, \\tau )$ with $0\\leq \\tau \\leq 1$ given by\n\\begin{eqnarray}\nA_i ({\\vec x}, \\tau ) &=&A_i ({\\vec x} ) (1-\\tau ) + \\tau\\, A^g_i ({\\vec x} )  \\; , \\nonumber\\\\\nA_i^g ({\\vec x} )&= &  g^{-1} A_i ({\\vec x} ) g + g^{-1} \\partial_i g  \\; ,\\label{G3}\n\\end{eqnarray}\nwhere $g ({\\vec x} ) \\in {\\mathcal G}_1$. Thus $g ({\\vec x} ) \\rightarrow 1$ at spatial infinity. \nThe starting point and ending point of this sequence of gauge fields are gauge-equivalent, so that (\\ref{G3}) gives a closed curve in ${\\mathcal A}\/{\\mathcal G}_1$. If this curve is contractible, then we will be able to transform the entire sequence into gauge-equivalent configurations, writing\n\\begin{equation}\nA_i ({\\vec x}, \\tau ) =  g^{-1}({\\vec x} ,\\tau)  A_i ({\\vec x} ) g({\\vec x}, \\tau)  + g^{-1}({\\vec x}, \\tau)  \\partial_i g({\\vec x}, \\tau)\n\\label{G4}\n\\end{equation}\nThe transformations $g ({\\vec x} , \\tau )$ give a homotopic deformation of the identity\n(at $\\tau =0$) to $g({\\vec x} ) $ at $\\tau =1$. The homotopy classes of transformations\n$g \\in {\\mathcal G}_1$ are characterized by the winding number\n\\begin{equation}\nQ [g] = {1\\over 24\\pi^2} \\int {\\rm Tr} (g^{-1} dg )^3\\label{G5}\n\\end{equation}\nThus if $g$ is chosen to have nonzero winding number, then we do not have the possibility\n(\\ref{G4}), leading to the conclusion that there are noncontractible paths in ${\\mathcal A}\/{\\mathcal G}_1$. In other words, if  $g ({\\vec x} )$ has nonzero winding number, the configuration (\\ref{G3})\ntraces out a noncontractible path in ${\\mathcal A}\/{\\mathcal G}_1$ as $\\tau$ changes from $0$ to $1$.\nThe usual instanton is an example of such a path,\nwhich, although it is not captured by the simple parametrization\ngiven in (\\ref{G3}), is deformable to\n(\\ref{G3}).\nIn general, the noncontractible paths\nare topologically nontrivial configurations with nonzero instanton number,\nbut not necessarily self-dual (or antiself-dual).\nIn fact, evaluating the instanton number on the configurations (\\ref{G3}), we find\n\\begin{eqnarray}\n\\nu [A] &\\equiv& -{1\\over 8\\pi^2}\\int_{M \\times [0,1]} {\\rm Tr} ( F ~ F)\\nonumber\\\\\n&=& {1\\over 24\\pi^2} \\int_{M, \\tau =1}{\\rm Tr} (g^{-1} dg )^3\n\\label{G6}\n\\end{eqnarray}\nwhere we used the fact that $g$ goes to the identity at spatial infinity.\n\\subsection{Fractional values of $\\nu$}\n\nIt is now easy to see how one may get fractional values of $\\nu$. We consider a path in the space  of gauge potentials ${\\cal A}$ of the form (\\ref{G4}), say with $g = U({\\vec x}, \\tau )$, where\n$U({\\vec x}, 1)$ is such that it goes to $\\omega = \\exp (2\\pi i \/N)$ as $\\vert {\\vec x} \\vert \\rightarrow\n\\infty$. In other words, $U({\\vec x}, 1) \\in {\\cal G}_\\omega$. Therefore the path\n$A^U = U^{-1} A U + U^{-1} \\nabla U$ is closed in the ${\\mathbb Z}_N$-invariant theory. The instanton number of this configuration can be evaluated explicitly, but before doing that, a comment is in order.\nThe configuration $U^{-1} A U + U^{-1} \\nabla U$ looks similar to\n(\\ref{G4}), but there is an important difference. In (\\ref{G4}), \n$g({\\vec x}, \\tau )$ gives a homotopy between the identity and $g({\\vec x} )$,\nso that $A_i ({\\vec x} , \\tau)$ is gauge-equivalent to $A_i ({\\vec x} )$ for any\nvalue of $\\tau$. Further, the value of $g({\\vec x}, \\tau )$ as\n$\\vert {\\vec x} \\vert \\rightarrow \\infty$ is\nidentity. To get a noncontractible path, one needs to consider\n$A_i$ which depend on $\\tau$ as in (\\ref{G3}) (or in the usual self-dual\ninstanton configurations). In the present case, the boundary value\nof $U$ changes from the identity to $\\omega$, \nso that at $\\tau \\neq 0, 1$, $U$ is not an element of\n${\\cal G}_1$ or ${\\cal G}_\\omega$. This is why the configurations\n $ U^{-1} A U + U^{-1} \\nabla U$ can still give a nonzero $\\nu$.\n\nTurning to details, it is useful to have an explicit construction of such a $U({\\vec x}, \\tau )$. \nLet $t^a$, $a =  1, 2, \\cdots, (N^2-1)$,  denote a basis of hermitian $N\\times N$ matrices for the Lie algebra\nof $SU(N)$, normalized so that ${\\rm Tr} (t^a t^b) = \\textstyle{1\\over 2} \\delta^{ab}$.\nWe can take $t^{N^2-1}$ to be diagonal and given by\n\\begin{equation}\n(t^{N^2-1})_{ij} = \\sqrt{N\\over 2(N-1)} \\left\\{ \\begin{matrix}\n{1\\over N} \\delta_{ij} &~& i,j = 1, 2, \\cdots, (N-1)\\\\\n{1\\over N}- 1&~&  i=j=N\\\\\n\\end{matrix}\n\\right.\n\\label{G7}\n\\end{equation}\nThis is the $SU(N)$ version of the usual hypercharge matrix.\nIt is easy to see that \n\\begin{equation}\ng = \\exp( i 2\\pi \\tau \\sqrt{2(N-1)\/N} ~ t_{N^2-1})\n\\label{G7a}\n\\end{equation}\nis a path from $g=1$ to $g =\\omega$ in $SU(N)$ as $\\tau$ varies from zero to $1$.\nThus it is a closed path in the pure $SU(N)$ gauge theory.\nKeeping in mind that instantons are essentially in an $SU(2)$ subgroup of $SU(N)$, we define\nthe $N\\times N$ matrix \n\\begin{equation}\nY_{ij} = \\left\\{ \\begin{matrix}\n{1\\over N} \\delta_{ij}& i,j =  1, 2, \\cdots, (N-2)\\\\\n{1\\over 2} (\\sigma\\cdot {\\hat x} )_{ij} + \\left( {1\\over N} - {1\\over 2} \\right)_{ij} & i,j = N-1, N\\\\\n\\end{matrix}\n\\right.\n\\label{G8}\n\\end{equation}\nWe can then define \n\\begin{equation}\nU({\\vec x}, \\tau ) = \\exp (i Y \\Theta (r,\\tau ))\\label{G9}\n\\end{equation}\nwith $\\Theta (r, 0) =0$, $\\Theta (0,\\tau ) =0$ and\n$\\Theta (\\infty, \\tau ) = 2\\pi \\tau$. \n(One example of such a function is\n$\\Theta (r, \\tau) = 2\\pi \\tau r\/(r+r_0)$. There are obviously infinitely \nmany $\\Theta$'s consistent with the required boundary behavior.) This gives a spherically symmetric ansatz\nfor an element of ${\\cal G}_\\omega$.\nIt is easy to verify that $U(\\infty, \\tau)$ traces out a path from the identity to \n$\\omega$ in $SU(N)$. Also,\nsince $U(\\infty, \\tau) \\rightarrow 1, \\omega$ at\n$\\tau =0, 1$, it qualifies as a gauge transformation at the two ends in the\n$SU(N)\/{\\mathbb Z}_N$ theory.\n\nReturning to the configurations\n$A_i^U = U^{-1} A_i U + U^{-1} \\partial_i U$\nin the space of potentials, we see that this corresponds to a closed\npath in ${\\cal A}\/ {\\cal G}_\\omega$.\nSince $U$ depends on ${\\vec x}, \\tau$, but $A_i$ depends only on ${\\vec x}$,  \n\\begin{eqnarray}\n{\\cal F} &=& dA^U + A^U A^U = U^{-1} F U + d\\tau {\\partial \\over \\partial \\tau} A^U\\nonumber\\\\\n&=& U^{1} (F - Da ) U\\label{G10}\n\\end{eqnarray}\n where $F$ involves only the spatial components of the field strength tensor\n and $a = d\\tau~ {\\dot U} U^{-1}$. (For this calculation, $\\tau$ can also be viewed\n as the time coordinate, so that $Da$ is essentially the electric field.)\nFrom (\\ref{G10}), \n \\begin{eqnarray}\n \\nu  &=& - {1\\over 8\\pi^2} \\int {\\rm Tr} ( (F- Da ) (F- Da) = {1\\over 4\\pi^2} \\int {\\rm Tr} (Da F )\\nonumber\\\\\n &=& {1\\over 4\\pi^2} \\oint {\\rm Tr} (a F)\\label{G11}\n \\end{eqnarray}\n The indicated boundary integration is over spatial infinity and over all $\\tau$.\n This shows that we will need a nonzero magnetic flux to obtain a nonzero value\n for $\\nu$. Therefore, we consider monopole-like configurations with the asymptotic\n behaviour \n \\begin{eqnarray}\n F &=& {1\\over 2} F_{ij} dx^i \\wedge dx^j\\nonumber\\\\\n &\\rightarrow& -{ i\\over 2} (\\sigma \\cdot {\\hat x}  )~ {M\\over 2} ~\\epsilon_{ijk} {{\\hat x}^k \\over r^2}\n dx^i\\wedge dx^j\n \\label{G12}\n \\end{eqnarray}\n where $\\sigma_i$ are in the $2\\times 2$ block of $i, j = (N-1), N$ viewed as\n an $N\\times N$ matrix. $M$ is (electric charge $e$ times) the monopole charge.\n We then find\n \\begin{equation}\n \\nu = M\n \\label{G13}\n \\end{equation}\n $M$ must be quantized according to the Dirac quantization condition.\n This condition, for a general gauge group is the Goddard-Nuyts-Olive\n (GNO) quantization condition \\cite{Goddard:1976qe} and amounts to the following.\n If the electric charges correspond to representations of $G$, then the magnetic charges\n $M$ take values in the dual group ${\\tilde G}$. For our case, we note that\n the GNO dual of $SU(N)$ is $SU(N)\/ {\\mathbb Z}_N$. Thus if the electric charges\n are ${\\mathbb Z}_N$-invariant, taking values corresponding to $SU(N)\/ {\\mathbb Z}_N$\n representations, then the fundamental charges of $SU(N)$ are allowed values for\n $M$. They are thus quantized in units of  $1\/N$. \n \n Thus we see that we can indeed obtain fractional values of $\\nu$.\n The problem however, is that for the nonsingular 't Hooft-Polyakov (`t H-P) monopoles, the quantization condition is not quite the Dirac (or GNO) condition. In fact, for the case of $SU(2)$, \n $M$ is an integer for `t H-P monopoles, whereas the GNO condition would suggest\n that it is possible to get $M = {\\textstyle{1\\over 2}}$.\n It is, however, possible to construct nonsingular configurations of separated GNO monopoles which\n have a total flux consistent with the `t Hooft-Polyakov condition.\n These will look like some split versions of the `t H-P monopole.\n \n We can construct an ansatz for the split monopole for the case of $SU(2)$ as follows.\n Let $A_D$ be the Dirac form of the monopole given by\n \\begin{equation}\n A_D = {({\\hat x}_1 d{\\hat x}_2 - {\\hat x}_2 d{\\hat x}_1 ) \\over  (1+ {\\hat x}_3)} \\label{G14}\n \\end{equation}\nThe `t Hooft-Polyakov form of the monopole is then given by\n\\begin{eqnarray}\nA &=& (1- K(r) ) \\left[  g^{-1} i \\left({\\sigma_3\\over 2} \\right) A_D~ g + g^{-1} d g \\right]\\nonumber\\\\\n&=&  i \\left({ \\sigma^a \\over 2}\\right) ~(1- K) ~\\epsilon_{abc}  {x^b\\over r^2} dx^c\n\\label{G15}\n\\end{eqnarray}\nwhere $g$ is the matrix\n\\begin{equation}\ng = {1\\over \\sqrt{1+z{\\bar{z}}}} \\left[ \\begin{matrix} \n1 & z \\\\\n-{\\bar{z}} &1\\\\\n\\end{matrix}\n\\right] \\label{G16}\n\\end{equation}\n and $z = \\tan (\\theta \/2)~ e^{-i{\\varphi}}$ and\n \\begin{equation}\n {\\hat x}_1 = {z +{\\bar{z}} \\over 1+ z{\\bar{z}}}, \\hskip .2in\n {\\hat x}_2 =  {i (z-{\\bar{z}} ) \\over 1+ z{\\bar{z}}}, \\hskip .2in\n {\\hat x}_3 = {1 - z{\\bar{z}} \\over 1+ z{\\bar{z}}}\n \\label{G17}\n \\end{equation}\n We may also note that $g^{-1} \\sigma_3~ g = \\sigma \\cdot {\\hat x}$.\nThe function $K(r)$ vanishes exponentially outside of the core of the monopole,\nand $1 - K(r) \\sim r^2$ for small $r$.\n The advantage of writing it as in (\\ref{G15}) is that, for large $r$, we can trivially calculate $F$ as\n \\begin{equation}\n F  = i g^{-1} {\\sigma_3\\over 2}  g ~ dA_D = \n{i\\over 2} \\sigma\\cdot {\\hat x} ~ \\sin\\theta ~d\\theta d{\\varphi}\n \\label{G18}\n \\end{equation}\nThus $F^a = - {\\hat x}^a  \\sin\\theta ~d\\theta d{\\varphi}$, with\n$\\int F^a {\\hat x}^a = - 4\\pi$. We can now modify this ansatz with some of the flux piped away from the monopole by a vortex. We consider an Abelian vortex given by\n\\begin{equation}\nA_v = {1\\over 2}~ f (\\rho, x_3)~ {x_1 dx_2 - x_2 dx_1 \\over \\rho^2}\n\\label{G19}\n\\end{equation}\nwhere $\\rho^2 = x_1^2 + x_2^2$. This is a vortex along the $x_3$-axis. We also have\n$f (\\rho, x_3) \\rightarrow 1$ as $\\rho$ becomes large, essentially\noutside the core of the vortex. The factor of $\\textstyle{1\\over 2}$ tells us that the flux carried by this vortex is\n$2\\pi \/2$; it is a ${\\mathbb Z}_N$-vortex, for $N=2$. We will consider a vortex of finite length\n$L$ by taking as an ansatz\n\\begin{equation}\nf( \\rho, x_3) =  {1\\over 2} \\tanh \\lambda \\rho ~ [ \\tanh {\\tilde \\lambda} x_3~-~\n\\tanh {\\tilde \\lambda} (x_3- L)] \n\\label{G20}\n\\end{equation}\nThis function vanishes exponentially for $x_3 \\ll 0$ and for $x_3 \\gg L$.\nThe core of the vortex has an extent in $\\rho$ of the order of $1\/\\lambda$.\nOur modified ansatz is now given by\n\\begin{equation}\nA = (1- K(r) ) \\left[  g^{-1} i \\sigma_3 (A_D -A_v)~ g + g^{-1} d g \\right]\n\\label{G21}\n\\end{equation}\n\nConsider a large sphere of radius $R$ much larger than the core of the monopole and the core of the vortex.  If $R\\ll L$, then the sphere intersects the vortex. The flux may be computed by\ntaking $K \\rightarrow 0$ , so that\n\\begin{equation}\nF = { i \\over 2} \\sigma \\cdot {\\hat x}  (d A_D - 2 ~ dA_v )\n\\label{G22}\n\\end{equation}\nThe flux is then $-(4\\pi - 2\\pi) = -2\\pi$. This is what we expect for a GNO monopole, and is equivalent to $M = \\textstyle{1\\over 2}$. \nIf we consider a sphere of radius much larger than $L$, then the contribution\nfrom $A_v$ is zero, since $f$ vanishes and we get $- 4\\pi$ for the total flux.\nIn this sense, we can view the configuration (\\ref{G21}) as\na split monopole.\n\nThe relevance of the split monopole can be understood from the following question: In a calculation or simulation of the vacuum-to-vacuum transition amplitude, can we see configurations with fractional values of $\\nu$?  For this it is useful to write $\\nu$ in terms of the Chern-Simons integral\n\\begin{equation}\nS_{CS}(M) = -{1\\over 8\\pi^2} \\int_M {\\rm Tr} (A ~dA + {2\\over 3} A^3 )\n\\label{G23}\n\\end{equation}\nThe topological charge $\\nu$, which is the integral of\nthe exterior derivative of the Chern-Simons over spacetime,\ncan then be written as\n\\begin{eqnarray}\n\\nu &=&  S_{CS} (M, \\tau =1) - S_{CS} (M, \\tau =0)\\nonumber\\\\\n&&\\hskip .1in + {1\\over 8\\pi^2} \\oint_{\\partial M} {\\rm Tr} (A_i E_j ) dx^0\\wedge  dx^i \\wedge dx^j.\n\\label{G24}\n\\end{eqnarray}\nAs the representative of the vacuum at $\\tau =0$, we may take\n$A_i =0$. The final configuration is also the vacuum, so it must be a gauge transform of $A_i =0$, say, $A_i = g^{-1} dg$.\nFurther, if we consider spatial boundary conditions (periodic, Dirichlet, etc.)\nwhich lead to vanishing of the integral with the\nelectric flux on $\\partial M$,\nwe find\n\\begin{eqnarray}\n  \\nu &=&  S_{CS} (M, \\tau =1) - S_{CS} (M, \\tau =0) \\nonumber \\\\\n  &=& {1\\over 24\\pi^2} \\int_M {\\rm Tr} (g^{-1} dg)^3 = Q [g] \\; . \n\\label{G25}\n\\end{eqnarray}\nSince $Q [g]$ is an integer, even for $g$'s such that\n$g \\rightarrow \\omega$ on $\\partial M$,\nwe get integral values of $\\nu$ in the vacuum-to-vacuum\namplitude. \nHowever, we can have configurations like\n$A_i^U = U^{-1} A_i U + U^{-1} \\partial_i U$ where $A_i$ is a split monopole configuration as in (\\ref{G21}). We get separated configurations, each of which in isolation may be considered as having a fractional value of $\\nu$, but the total value of $\\nu$ is integral.\n\n\n\n\n\\subsection{Simple solution}\n\\label{sec:simple}\n\nWe will now illustrate the analysis given above in a related but slightly different way and also comment on the situation with finite nonzero temperature $T$. It is convenient to frame this discussion in terms\nof a nonzero $A_0$, by replacing\nthe field configuration $(A_0 =0, U^{-1} A_i U + U^{-1} \\partial_i U)$\nby its gauge equivalent version\n$(A_0 = {\\dot U} U^{-1}, A_i )$. \nFor $A_0$, at $r=\\infty$ our choice is then\n\\begin{equation}\nA_0 \\; = \\; \\frac{2 \\pi T}{N} \\; \\mathbf{k} \\; .\n\\end{equation}\nHere $\\mathbf{k}$ is a diagonal $SU(N)$ matrix related to\n$\\mathbb{Z}_N$ transformations, so their elements are integers.  There are two choices,\n\\begin{eqnarray}\n\\mathbf{k}_1 \\; &=& \\;\n\\left(\n\\begin{array}{cc}\n{\\mathbf 1}_{N-1} & 0      \\\\\n0             & -(N-1) \\\\\n\\end{array}\n\\right) \\; ,\n\\\\\n\\mathbf{k}_2 \\; &=& \\;\n\\left(\n\\begin{array}{ccc}\n{\\mathbf 1}_{N-2} & 0      & 0\\\\\n0             & -(N-1) & 0 \\\\\n0             & 0      & 1 \\\\\n\\end{array}\n\\right) \\; .\n\\end{eqnarray}\nThese are obviously related to the matrix $(t^{N^2-1})_{ij} $ in Eq. (\\ref{G7}).\nFor $\\mathbf{k}$ equal to either $\\mathbf{k}_i$, the Wilson line in the imaginary\ntime direction, $t$, is\n\\begin{equation}\n\\Omega \\; = \\; \\exp \\left( i \\int^{1\/T}_0 \\; A_0 \\; dt \\right) \n\\; = \\; \\exp\\left( \\frac{2 \\pi i}{N} \\; \\mathbf{k} \\right) \\; ,\n\\end{equation}\nhas nontrivial holonomy, as these values represent $\\mathbb{Z}_N$ degenerate vacum\n\nFor the spatial components, construct a split 't H-P monopole, as in the previous section.\nDivide a sphere into an upper and a lower hemisphere,\nwith gauge potentials on each, $A^\\pm$, and take\n\\begin{equation}\nA^\\pm_\\phi \\; = \\; \\frac{1}{2N r } \\; \\mathbf{m} \\;\n\\frac{\\left( \\pm 1 - \\cos\\theta \\right)}{\\sin \\theta} \\; .\n\\label{zn_soln}\n\\end{equation}\nTo see this is a $\\mathbb{Z}_N$ monopole, compute the the\nWilson line for a special closed path, $\\vec{s}$.\nSince the vector potential is specified by two patches, we compute\nthe Wilson line with $A^+$,\ngoing around by $2 \\pi$ in $\\phi$; then, take the Wilson line\nwith $A^-$, running in the opposite direction:\n\\begin{eqnarray}\n&&\\exp \\left( \\; i \\oint \\vec{A}^+ \\cdot d\\vec{s} \\; \\right)\n\\; \\left( \\exp\\left( \\;\ni \\oint \\vec{A}^- \\cdot d\\vec{s} \\; \\right) \\right)^\\dagger \\nonumber \\\\\n &=& \\exp\\left( \\frac{2 \\pi i}{N} \\; \\mathbf{m} \\right) \\; .\n\\end{eqnarray}\nThis is manifestly gauge invariant, and $=1$ if the configuration\nis trivial, $A^+ = A^-$.  For the $\\mathbb{Z}_N$ monoopole, instead one\nobtains a non-trivial element of $\\mathbb{Z}_N$.  For this to be\ntrue, $\\mathbf{m}$ must be one of the two matrices, $\\mathbf{c}_{1}$ or $\\mathbf{c}_2$.\n\nThe above are the boundary conditions at spatial infinity, $r \\rightarrow\n\\infty$.  At the origin, $r = 0$, we require all $A_\\mu$'s to vanish,\nat least like $\\sim r^2$, so that $F_{\\mu \\nu} \\sim r$ as $r \\rightarrow 0$.\n\nAs argued in Sec. (\\ref{sec:cpn}), in general we expect that this exists only as a quantum\ninstanton, on the order of the confinement scale.  At nonzero temperature, however,\n$1\/T$ provides an alternate length scale.  \nWhile the solution is approximately self-dual over\ndistances $\\sim 1\/T$, because of the presence of the Debye screening mass, it is\nnot self-dual over larger distances.  This generates corrections $\\sim \\sqrt{g^2}$ to the action.\n\nIt is straightforward to compute the topological charge.  For large $r$,\n\\begin{equation}\nA_0(r) \\; = \\;\n\\frac{2 \\pi T}{N} \\; \\mathbf{k} \\; - \\; \\frac{1}{2 N r} \\; \\mathbf{m} \\; + \\; \\ldots\n\\end{equation}\nFor a static configuration,\n\\begin{equation}\nQ \\; = \\;\n\\frac{1}{4 \\pi^2} \\int d^4 x \\; \\partial_i \\; {\\rm tr} \\left( A_0 \\; B_i\n\\right) = \\frac{1}{N^2} \\; \\bf{m} \\cdot \\bf{k} \\; .\n\\end{equation}\nThis was first derived by 't Hooft \\cite{tHooft:1980kjq}.\n\nThere are only two cases to consider.  Either the $\\mathbb{Z}_N$ charges\nare the same, or they are different.  If they are the same,\n$\\bf{m} = \\bf{k_1}$,\n\\begin{equation}\nQ \\; = \\; \\frac{N-1}{N} \\; .\n\\end{equation}\nIf they charges are different, such as $\\mathbf{m} = \\mathbf{k}_1$ and $\\mathbf{k}\n= \\mathbf{k}_2$, then\n\\begin{equation}\nQ \\; = \\; - \\; \\frac{1}{N} \\; .\n\\end{equation}\n\nWe conclude this section by discussing the relationship between the configuration above and that of\nKraan, van Baal, Lee, and Lu (KvBLL)\n\\cite{Lee:1997vp,Lee:1998vu,Lee:1998bb,Kraan:1998pm,Kraan:1998sn,Diakonov:2002fq,Bruckmann:2009nw,Diakonov:2009jq,Poppitz:2008hr,Gonzalez-Arroyo:2019wpu,Anber:2021upc}.\nLike ours, their solution carries magnetic charge and has nontrivial holonomy.  Our ansatz, however, carries\n$\\mathbb{Z}_N$ magnetic charge, and so must be represented by a multivalued function, Eq. (\\ref{zn_soln}), while\nthat of KvBLL has integral magnetic charge.\nFor our solution, the boundary condition for holonomy at spatial infinity ensures that it is a vacuum which is\ndegenerate with the vacuum.  Thus when one loop corrections are included, the action for our solution will remain\nfinite.  In contrast, for the solution of KvBLL, the holonomy is at a maximum of the holonomous potential.  When\none loop corrections are included, then, the action for a constituent with charge $1\/N$\ndiverges as the spatial volume.  The action for an instanton with integral charge remains finite, which\nis why on the quantum level, these constituents cannot be pulled apart.\n\n\\section{Physics of $\\mathbb{Z}_N$ dyons and the lattice}\n\\label{sec:lattice}\n\nThe $\\mathbb{Z}_N$ dyons above represent configurations with fractional topological charge, and carry both\n$\\mathbb{Z}_N$ electric and magnetic charge.  In the confined phase, $\\mathbb{Z}_N$ magnetic charge is unconfined,\nbut in the deconfined phase, $\\mathbb{Z}_N$ magnetic charge which\npropagates in the temporal direction is confined.  Thus quantum mechanically, a\n$\\mathbb{Z}_N$ dyon is only relevant for a narrow temperature region above the deconfining temperature, $T_d$.\nAs the temperature increases, so will the magnetic string tension, binding the\n$\\mathbb{Z}_N$ dyons with increasing strength.  Eventually, at some temperature above but close to $T_d$,\nthere will only be instantons with integral topological charge.\n\nWe stress that unless there are boundary conditions which are twisted with respect to $\\mathbb{Z}_N$,\nas suggested by 't Hooft \\cite{tHooft:1980kjq,tHooft:1981nnx}, then it will not be possible to measure\na system with a net number of $\\mathbb{Z}_N$ dyons (or anti-dyons).  Nevertheless, $\\mathbb{Z}_N$ dyons do contribute\nto the partition function, and in particular, alter the dependence of the free energy on the $\\theta$-parameter.\nFor an especially clear discussion, see Unsal \\cite{Unsal:2020yeh}.\n\nAs noted, in the confined phase $\\mathbb{Z}_N$ magnetic charge can propagate freely.\nOur construction of a dyon depended essentially upon the non-trivial holonomy at nonzero temperature,\nbut as the temperature decreases, surely the dominant dyons will not propagate in straight lines\nin the direction of imaginary time.  Instead, they will curl up over distances set by the confinement scale.\nThis gives an alternate explanation of why the dyons disappear above a given temperature, as they\nget squeezed out by the extent of the direction in imaginary time.\n\nIn the continuum, an instanton in a $SU(N)$ gauge theory\nwith topological charge one, coupled to single massless Dirac quark in the fundamental\nrepresentation, has two zero modes, for a quark of each chirality.  In the adjoint\nrepresentation there are $2N$ zero modes.   Thus to measure a $\\mathbb{Z}_N$ dyon with\nfractional topological charge $1\/N$ it is necessary to use adjoint quarks.\n\nOn the lattice, in the pure gauge theory one can use an external quark propagator in the adjoint representation\nto look for isolated zero modes.  To ensure these are not lattice artifacts,\nit is imperative to use a Dirac propagator with exact chiral symmetry, such\nas the overlap operator\n\\cite{Ginsparg:1981bj,Kaplan:1992bt,Narayanan:1993zzh,Narayanan:1993sk,Narayanan:1993ss,Narayanan:1994gw,Luscher:1998pqa}.\n\nThis was done previously by Edwards, Heller, and Narayanan \\cite{Edwards:1998dj}.  With more modern\ntechniques, it should be possible to establish the existence of $\\mathbb{Z}_N$ dyons with certainty very close to the continuum\nlimit \\cite{karthik:2022}.  \n\nFodor {\\it et al.} performed simulations in a $SU(3)$ gauge theory\nusing an external quark propagator in the sextet representation\n\\cite{Fodor:2009nh}.  As they discuss, the sextet representation is sensitive to the presence of objects\nwith topological charge $1\/5$, and for which they see no evidence.  However, this does not exclude\nthe appearance of objects with charge $1\/3$.\n\nWith the overlap operator, one would look for configurations with zero modes.  Given that, it is then\npossible to compute the eigenvector as a function of position.  This will allow an estimation of the size\nof the object with a given topological charge.\n\nFurther, especially in the vacuum, it is not clear what the space-time structure of the tangled dyon worldlines\nis, especially at distances which are smaller than that for the confinement scale.  This may help explain why\nlattice studies of the topological structure do not find evidence for a simple instanton, concentrated at\nclose to a single point in spacetime\n\\cite{Horvath:2002yn,Ahmad:2005dr,Horvath:2005rv,Lian:2006ky,Thacker:2010zk,Thacker:2011sz,Alexandru:2021pap}.\n\n\n\n\\section{Quarks}\n\\label{sec:quarks}\n\nWe have concentrated exclusively on a gauge theory without dynamical quarks.  In this section we discuss\nwhat might occur with their introduction.\n\nFor a Wilson loop which carries electric $\\mathbb{Z}_N$ flux, adding dynamical quarks generates quark anti-quark pairs\nwhich then screen the flux tube.  In contrast, it is not possible to consistently define a $\\mathbb{Z}_N$ magnetic monopole\nwith dynamical quarks.  With dynamical quarks, the magnetic $\\mathbb{Z}_N$ flux is visible, and\nquarks generate a linear potential between $\\mathbb{Z}_N$ dyons, generating their confinement.\nSince the size of a quantum instanton in the pure\ngauge theory is of order of the confinement scale, this surely remains so in the theory with dynamical quarks.\n\nFor the topological susceptibility in QCD, there are three regimes.  Consider first zero quark chemical potential.\nInstantons dominate at temperatures above $\\geq 300$~MeV, and $\\mathbb{Z}_N$ dyons below that.\nThe lattice finds that\nthe crossover temperature for chiral symmetry is at $T_\\chi \\approx 156 \\pm 2$~MeV\n\\cite{HotQCD:2018pds,Borsanyi:2020fev,Guenther:2022wcr}.\nThus the interaction of $\\mathbb{Z}_N$ dyons with essentially massless quarks dominates for $T_\\chi \\leq T \\leq 300$~MeV,\nwhile for $T \\leq T_\\chi$, the dyons interact with massive quarks.\nWhile this description is consistent with measurements on the lattice, it is\nadmittedly speculative, as it is not clear how to measure $\\mathbb{Z}_N$ dyons\nin the presence of dynamical quarks.\n\nAt low temperature and nonzero quark chemical potential, $\\mu$, there are at least three regions.\nBecause the number of degrees of freedom for quarks and gluons at nonzero temperature is so much greater\nthan that at $T = 0$ and $\\mu \\neq 0$, \nestimates with a dilute instanton gas indicate that at zero temperature, instantons do not dominate until {\\it very} high\nchemical potential, at least $\\mu \\sim 2$~GeV \\cite{Pisarski:2019upw}.\nThus for $\\mu > 313$~MeV, when a Fermi sea of nucleons first forms, the interaction of $\\mathbb{Z}_N$ dyons and\nquarks controls the topological susceptibility.  Initially this is in a region with broken chiral symmetry,\nand then a region which is chirally symmetric. \nThis includes both a confined, quarkyonic regime \\cite{McLerran:2007qj,Lajer:2021kcz},\nand perhaps even into the nominally perturbative regime, which may arise for\n$\\mu > 1$~GeV.  In the latter, color superconductivity is dominant near the Fermi surface,\nbut the effects of the axial anomaly can still affect the possible pairing mechanisms \\cite{Pisarski:1999gq}.\nThis illustrates the general expectation that the phases of cold, dense\nquarks might be considerably more involved than at $\\mu = 0$ and $T \\neq 0$.\n\nA useful approach would be to develop a phenomenological model in ~which the effects of $\\mathbb{Z}_N$ dyons in the $SU(N)$\ntheory without quarks could be fit.  Then this model could be extended to the theory with dynamical quarks.\nWe leave this for future study.\n\n\\acknowledgments\nV.P.N. was supported in part by the U.S. National Science Foundation Grants No.\nPHY-2112729 and No. PHY-1820271.\nR.D.P. was supported by the U.S. Department of Energy under contract DE-SC0012704.\nR.D.P. thanks A. Alexandru, O. Alvarez, A. Dumitru, U. Heller, I. Horvath, T. Izubuchi, J. Lenaghan,\nN. Karthik, R. Narayanan, P. Petreczky, E. Poppitz, R. Venugopalan, and M. Unsal for discussions.\n\n\\input{frac.bbl}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:int}Introduction}\n\nThe neutrino physics has made great progress in the past decades.\nThe mass square differences $\\Delta m^2_{\\rm sol}$, $\\Delta\nm^2_{\\rm atm}$ and the mixing angles have been measured with good\naccuracy \\cite{Strumia:2006db,Schwetz:2008er,Fogli:Indication}.\nA global fit to the current neutrino oscillation data demonstrates\nthat the observed lepton mixing matrix is remarkably compatible with\nthe tri-bimaximal (TB) mixing pattern \\cite{TBmix}, which suggests\nthe following values of the mixing angles:\n\\begin{equation}\n\\label{1}\\sin^2\\theta^{TB}_{12}=\\frac{1}{3},~~~\\sin^2\\theta^{TB}_{23}=\\frac{1}{2},~~\\sin\\theta^{TB}_{13}=0\n\\end{equation}\nThe question of how to achieve TB mixing has been the subject of\nintense theoretical speculation. Recently it has been found that the\nflavor symmetry based on the discrete group is particularly suitable\nto reproduce this specific mixing pattern in leading order (LO).\nVarious discrete flavor symmetry models have been built, please see\nthe Refs.\\cite{Altarelli:2010gt,Ishimori:2010au} for a review. A\ncommon feature of these model is to produce TB mixing at leading\norder, and the leading order predictions are always subjected to\ncorrections due to higher dimensional operators in both the driving\nsuperpotential and the Yukawa superpotentials. These models provide\nan elegant description of neutrino mixing at very high energy scale,\nwhereas the neutrino experiments are performed at low energy scale.\nIn order to compare the model predictions with experimental data,\none has to perform a renormalization group (RG) running from the\nhigh energy scale where the theory is defined to the electroweak\nscale $M_Z$. Moreover, we note that RG effects have interesting\nimplications for model building, the lepton mixing angles can be\nmagnified \\cite{Balaji:2000au}, even the bimaximal mixing at high\nenergy can be compatible with low energy experiment\n\\cite{Antusch:2002hy}. Therefore, in a consistent flavor model\nbuilding, we have to guarantee that the successful leading order\npredictions are not destroyed by the RG running corrections. The aim\nof this work is to analyze the RG corrections on the TB mixing\npattern in two typical $S_4$ flavor models\n\\cite{Bazzocchi:2009pv,Ding:2009iy} in addition to the next to\nleading order corrections arising from high dimensional operators\nand to confront them with experimental values. We shall see that the\nrunning of the neutrino parameters is strongly constrained by the\nflavor symmetry as well, and this result holds very generally for the discrete\nflavor symmetry models.\n\nThe $S_4$ flavor symmetry is very interesting. From the group theory\npoint of view \\cite{Lam:2008rs}, it is the minimal group which can\nproduce the TB mixing in a natural way, namely without ad hoc\nassumptions. It is remarkable that we have more alternatives to\nrealize the exact TB mixing than those in the $A_4$ flavor model\n\\cite{Bazzocchi:2009pv,Ding:2009iy,Ma:2005pd,Meloni:2009cz,Altarelli:2009gn,Grimus:2009pg}.\nIn particular, the $\\mathbf{2}$ dimensional irreducible\nrepresentation of $S_4$ group can be utilized to describe the quark\nsector. Moreover, the group $S_4$ as a flavor symmetry, as has been\nshown for example in Refs.\n\\cite{Dutta:2009bj,Hagedorn:2010th,Ishimori:2008fi,Toorop:2010yh,Ding:2010pc},\ncan also give a successful description of the quark and lepton\nmasses and mixing angles within the framework of grand unified\ntheory (GUT). We note that $S_4$ as a flavor symmetry has been\ninvestigated long ago \\cite{Pakvasa:1978tx,Hagedorn:2006ug}, but\nwith different aims and different results.\n\nThe paper is organized as follows. In section \\ref{sec:RGE_review},\nwe briefly review the RG equations for the type I see-saw mechanism.\nThen we give a concise introduction to the Bazzocchi-Merlo-Morisi\n(BMM) $S_4$ model \\cite{Bazzocchi:2009pv} and the $S_4$ model of\nDing \\cite{Ding:2009iy} in section \\ref{sec:models}, where the main\nfeatures of these models are shown. In section\n\\ref{sec:RGE_running}, Our results of RG effects on the neutrino\nmixing parameters for these two interesting models are presented.\nFinally we draw our conclusions in section \\ref{sec:conclusion}.\n\n\\section{\\label{sec:RGE_review}Running of neutrino parameters in type I see-saw scenario}\n\nThe running of neutrino masses and lepton mixing angles is very\nimportant and has been studied extensively in the literature\n\\cite{rge1,rge2,rge3,rge4,rge5,Chakrabortty:2008zh,Antusch:2003kp,Antusch:2005gp,Xing:2007fb,Bergstrom:2010qb,Blennow:2011mp}\nin the past years. In particular, Antusch et al. have developed the\nMathematica package REAP in Ref. \\cite{Antusch:2005gp}, which can\nsolve renormalization group equations (RGE) and provide numerical\nvalues for the neutrino mass and mixing parameters. In this section,\nwe present the RGEs for neutrino parameters in the minimal\nsupersymmetric standard model (MSSM) extended by three singlet\n(right-handed) heavy neutrinos. The superpotential is given by\n\\begin{equation}\n\\label{2}W=D^cY_dQH_d+U^cY_uQH_u+E^cY_eLH_d+N^cY_{\\nu}LH_u+\\frac{1}{2}N^cMN^c\n\\end{equation}\nwhere $Q$ and $L$ are the left-handed quark and lepton doublets\nchiral superfields, respectively, $U^c$, $D^c$, $N^c$ and $E^c$ are\nright-handed up-type quark, down-type quark, heavy neutrino and\ncharged lepton singlet superfields, respectively, $H_u$ and $H_d$ are\nthe well-known two Higgs doublets in MSSM. The Yukawa matrices\n$Y_u$, $Y_d$, $Y_{\\nu}$ and $Y_e$ are general complex $3\\times3$\nmatrices and the $3\\times3$ heavy neutrino mass matrix $M$ is\nsymmetric. Integrating out all the heavy singlet neutrinos, one gets\nthe usual dimension-5 effective neutrino mass operator\n\\begin{equation}\n\\label{3}{\\cal L}_{\\kappa}=-\\frac{1}{4}\\kappa_{fg}(L^{f}\\cdot H_u)(L^{g}\\cdot H_u)\n\\end{equation}\nwhere $f$ and $g$ are family indices, and the dot indicates the\n$SU(2)_L$ invariant contractions. After electroweak symmetry\nbreaking, this operator leads to the light-neutrino masses\n\\begin{equation}\n\\label{4}m_{\\nu}(\\mu)=-\\frac{1}{4}\\kappa(\\mu)v^2\\sin^2\\beta\n\\end{equation}\nwhere $\\mu$ is the renormalization scale, $v=246$ GeV and\n$\\tan\\beta=v_u\/v_d$ is the ratio of vacuum expectation values (VEV)\nof the Higgs doublets. Above the heaviest neutrino mass scale, the\nlight-neutrino mass matrix reads\n\\begin{equation}\n\\label{5}m_{\\nu}(\\mu)=-\\frac{1}{2}Y^T_{\\nu}(\\mu)M^{-1}(\\mu)Y_{\\nu}(\\mu)v^2\\sin^2\\beta\n\\end{equation}\nWhen we evolved the energy from high energy scale down to the low\nexperimental observation scale, the heavy singlet neutrinos involved\nin the see-saw mechanism have to be integrated out one by one, thus\none has to consider a series of effective theories\n\\cite{Antusch:2005gp}. In general, the light-neutrino mass matrix\ncan be written  as\\footnote{We use the GUT charge normalization for\nthe gauge coupling $g_1$.}\n\\begin{equation}\n\\label{6}m_{\\nu}=-\\frac{1}{4}\\left(\\accentset{(n)}{\\kappa}+2\\accentset{(n)}{Y}^T_{\\nu}\\accentset{(n)}{M}^{-1}\\accentset{(n)}{Y}_{\\nu}\\right)v^2\\sin^2\\beta\n\\end{equation}\nwhere the superscript $(n)$ denotes a quantity below the $n$th mass\nthreshold. In the MSSM, the two parts $\\accentset{(n)}{\\kappa}$ and\n$2\\accentset{(n)}{Y}^T_{\\nu}\\accentset{(n)}{M}^{-1}\\accentset{(n)}{Y}_{\\nu}$\nevolve in the same way\n\\begin{eqnarray}\n\\label{8}16\\pi^2\\frac{d \\accentset{(n)}{X}}{d t}=\\left(Y^{\\dagger}_eY_e+\\accentset{(n)}{Y}^{\\dagger}_{\\nu}\\accentset{(n)}{Y}_{\\nu}\\right)^{T}\\accentset{(n)}{X}+\\accentset{(n)}{X}\\left(Y^{\\dagger}_eY_e+\\accentset{(n)}{Y}^{\\dagger}_{\\nu}\\accentset{(n)}{Y}_{\\nu}\\right)+\\Big[2\\,{\\rm Tr}\\big(\\accentset{(n)}{Y}^{\\dagger}_{\\nu}\\accentset{(n)}{Y}_{\\nu}+3Y^{\\dagger}_uY_u\\big)-\\frac{6}{5}g^2_1-6g^2_2\\Big]\\accentset{(n)}{X}\n\\end{eqnarray}\nwhere $t=\\ln(\\mu\/\\mu_0)$, and $\\accentset{(n)}{X}$ stands for\n$\\accentset{(n)}{\\kappa}$ or\n$2\\accentset{(n)}{Y}^T_{\\nu}\\accentset{(n)}{M}^{-1}\\accentset{(n)}{Y}_{\\nu}$.\nThe RG equations for the Yukawa couplings $Y_u$, $Y_d$, $Y_{\\nu}$,\n$Y_e$ and the right-handed neutrino mass matrix $M$ are given by\n\\begin{eqnarray}\n\\nonumber&&16\\pi^2\\frac{d\\accentset{(n)}{Y}_{\\nu}}{dt}=\\accentset{(n)}{Y}_{\\nu}\\left[3\\accentset{(n)}{Y}^{\\dagger}_{\\nu}\\accentset{(n)}{Y}_{\\nu}+Y^{\\dagger}_eY_e+{\\rm Tr}(\\accentset{(n)}{Y}^{\\dagger}_{\\nu}\\accentset{(n)}{Y}_{\\nu})+3{\\rm Tr}(Y^{\\dagger}_uY_u)-\\frac{3}{5}g^2_1-3g^2_2\\right]\\\\\n\\nonumber&&16\\pi^2\\frac{dY_e}{dt}=Y_e\\left[3Y^{\\dagger}_eY_e+\\accentset{(n)}{Y}^{\\dagger}_{\\nu}\\accentset{(n)}{Y}_{\\nu}+3{\\rm Tr}(Y^{\\dagger}_dY_d)+{\\rm Tr}(Y^{\\dagger}_eY_e)-\\frac{9}{5}g^2_1-3g^2_2\\right]\\\\\n\\nonumber&&16\\pi^2\\frac{dY_u}{dt}=Y_u\\left[Y^{\\dagger}_dY_d+3Y^{\\dagger}_uY_u+{\\rm Tr}(\\accentset{(n)}{Y}^{\\dagger}_{\\nu}\\accentset{(n)}{Y}_{\\nu})+3{\\rm Tr}(Y^{\\dagger}_uY_u)-\\frac{13}{15}g^2_1-3g^2_2-\\frac{16}{3}g^2_3\\right]\\\\\n\\nonumber&&16\\pi^2\\frac{dY_d}{dt}=Y_d\\left[3Y^{\\dagger}_dY_d+Y^{\\dagger}_uY_u+3{\\rm Tr}(Y^{\\dagger}_dY_d)+{\\rm Tr}(Y^{\\dagger}_eY_e)-\\frac{7}{15}g^2_1-3g^2_2-\\frac{16}{3}g^2_3\\right]\\\\\n\\label{9}&&16\\pi^2\\frac{d\\accentset{(n)}{M}}{dt}=2(\\accentset{(n)}{Y}_{\\nu}\\accentset{(n)}{Y}^{\\dagger}_{\\nu})\\accentset{(n)}{M}+2\\accentset{(n)}{M}(\\accentset{(n)}{Y}_{\\nu}\\accentset{(n)}{Y}^{\\dagger}_{\\nu})^{T}\n\\end{eqnarray}\nIn the full theory above the highest see-saw scale, the superscript\n$(n)$ has to be omitted, and the RG equations for MSSM without\nsinglet neutrinos can be recovered by setting the neutrino Yukawa\ncouplings and the mass matrix of the singlets to be zero. Below the\nSUSY breaking scale, which is taken to be 1000 GeV in this work, we\ngo to the standard model region. Since all the heavy right-handed\nneutrinos have already been integrated out at this scale, the\nneutrino masses are described by the effective dimension-5 operator,\nand the neutrino mass matrix $m_{\\nu}$ evolves as\n\\begin{equation}\n\\label{10}16\\pi^2\\frac{dm_{\\nu}}{dt}=-\\frac{3}{2}(Y^{\\dagger}_eY_e)^{T}m_{\\nu}-\\frac{3}{2}m_{\\nu}(Y^{\\dagger}_eY_e)+\\left[2{\\rm Tr}(3Y^{\\dagger}_uY_u+3Y^{\\dagger}_dY_d+Y^{\\dagger}_eY_e)-3g^2_2+\\lambda\\right]m_{\\nu}\n\\end{equation}\nwhere $\\lambda$ is the Higgs self-interaction coupling\\footnote{We\nuse the convention that the Higgs self-interaction term in the\nLagrangian is $-\\frac{\\lambda}{4}(H^{\\dagger}H)^2$.}. In order to\ncalculate the RG evolution of the effective neutrino mass matrix, we\nhave to solve the RG equations for all the parameters of the theory\nsimultaneously\\footnote{The running of the gauge couplings has to be\ntaken into account as well, the corresponding $\\beta$ functions are\nwell-known .}. At the mass threshold, we should integrate out the\ncorresponding heavy neutrino and perform the tree-level matching\ncondition for the effective coupling constant between the effective\ntheories\n\\begin{equation}\n\\label{7}\\accentset{(n)}{\\kappa}_{gf}\\Big|_{M_n}=\\;\\;\\accentset{(n+1)}{\\kappa}_{gf}\\Big|_{M_n}+2(\\;\\;\\accentset{(n+1)}{Y}_{\\nu}^{\\,\\,T}\\,)_{gn}M^{-1}_n(\\;\\;\\accentset{(n+1)}{Y}_{\\nu}\\;)_{nf}\\Big|_{M_n}~~~({\\rm no\\,sum\\,over}\\, n)\n\\end{equation}\n\n\\section{\\label{sec:models}Variants of the Two $S_4$ Models\\label{sec:BMM}}\n\nIn this section, we recapitulate the main features of the $S_4$\nflavor model of BMM \\cite{Bazzocchi:2009pv} and Ding\n\\cite{Ding:2009iy}. Both models generate neutrino masses via type I\nsee-saw mechanism, and the neutrino TB mixing is produced at LO. For\nan introduction to the group theory of $S_4$ we refer to\nRefs.\\cite{Ding:2009iy,Ding:2010pc}, the same conventions for the\n$S_4$ representation matrix and Clebsch-Gordan coefficient are used\nin this work.\n\n\\subsection{\\label{sec:BMM}BMM $S_4$ model}\nIn this model the flavor symmetry $S_4$ is accompanied by the cyclic\ngroup $Z_5$ and the Froggatt-Nielsen symmetry $U(1)_{FN}$. The $S_4$\nflavor symmetry is spontaneously broken to the subgroup $Z_2\\times\nZ_2$ in the neutrino sector and to nothing in the charged lepton one\nat leading order. This misalignment between the flavor symmetry\nbreaking in the neutrino and charged lepton sectors is exactly the\norigin of the TB mixing. Furthermore, the auxiliary symmetry $Z_5$\neliminating some dangerous terms, with the interplay of the\ncontinuous $U(1)_{FN}$, is responsible for the hierarchy among the\ncharged lepton masses. The leptonic fields and the flavon fields of\nthe model and their transformation properties under the flavor\nsymmetry are shown in Table \\ref{table:BMM_transformation}.\n\n\\begin{table}[hptb]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c||c||c|c||c|c||c|}\n  \\hline\\hline\n  & $\\ell$ & $e^c$ & $\\mu^c$ & $\\tau^c$ & $\\nu^c$ & $H_{u,d}$ & $\\theta$ & $\\psi$ & $\\eta$ & $\\Delta$ & $\\varphi$ & $\\xi'$ \\\\\n  \\hline\n  $S_4$ & $3_1$ & $1_2$ & $1_2$ & $1_1$ & $3_1$ & $1_1$ & $1_1$ & $3_1$ & 2 & $3_1$ & $2$ & $1_2$  \\\\\n  $Z_5$ & $\\omega^4$ & $1$ & $\\omega^2$ & $\\omega^4$ & $\\omega$ & 1 & 1 & $\\omega^2$ & $\\omega^2$ & $\\omega^3$ & $\\omega^3$ & 1  \\\\\n  $U(1)_{FN}$ & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0  \\\\\n  \\hline\\hline\n  \\end{tabular}\n\\caption{\\label{table:BMM_transformation}Transformation properties\nof the leptonic fields and flavons in the BMM model\n\\cite{Bazzocchi:2009pv}. Note that $\\omega$ is the fifth root of\nunity, i.e. $\\omega=e^{i2\\pi\/5}$.}\n\\end{center}\n\\end{table}\n\nBy introducing a $U(1)_R$ symmetry, the authors in Ref.\n\\cite{Bazzocchi:2009pv} have shown that the flavon fields develop\nthe following vacuum alignment at LO\n\\begin{eqnarray}\n\\nonumber\\langle\\psi\\rangle&=&\\left(\n             \\begin{array}{c}\n               0 \\\\\n               1 \\\\\n               0 \\\\\n             \\end{array}\n           \\right)v_{\\psi},~~~\\langle\\eta\\rangle=\\left(\n             \\begin{array}{c}\n               0 \\\\\n               1 \\\\\n             \\end{array}\n           \\right)v_\\eta\\\\\n\\nonumber&&        \\\\\n\\nonumber\\langle\\Delta\\rangle&=&\\left(\n             \\begin{array}{c}\n               1 \\\\\n               1 \\\\\n               1 \\\\\n             \\end{array}\n           \\right)v_\\Delta,~~~\n\\langle\\varphi\\rangle=\\left(\n             \\begin{array}{c}\n               1 \\\\\n               1 \\\\\n             \\end{array}\n           \\right)v_{\\varphi}\\\\\n\\nonumber&&  \\\\\n\\label{11}\\langle\\xi'\\rangle&=&v_{\\xi'},~~~\n\\langle\\theta\\rangle=v_\\theta\n\\end{eqnarray}\nThe superpotential of the model in the lepton sector is\n\\begin{eqnarray}\n\\nonumber&&w_{\\ell}=\\sum^{4}_{i=1}\\frac{\\theta}{\\Lambda}\\frac{y_{e,i}}{\\Lambda^3}\\,e^c(\\ell X_i)_{1_2}H_d+\\frac{y_{\\mu}}{\\Lambda^2}\\mu^c(\\ell\\psi\\eta)_{1_2}H_d+\\frac{y_{\\tau}}{\\Lambda}\\tau^c(\\ell\\psi)_{1_1}H_d+...\\\\\n\\label{12}&&w_{\\nu}=y(\\nu^c\\ell)_{1_1}H_u+x_d(\\nu^c\\nu^c\\varphi)_{1_1}+x_t(\\nu^c\\nu^c\\Delta)_{1_1}+...\n\\end{eqnarray}\nwhere the subscript $1_1$ and $1_2$ denote the contraction in $1_1$\nand $1_2$, respectively, and dots stand for higher dimensional\noperators, which are suppressed by additional powers of the cutoff\n$\\Lambda$. The composite $X$ is given by\n\\begin{equation}\n\\label{13}X=\\{\\psi\\psi\\eta,\\,\\psi\\eta\\eta,\\,\\Delta\\Delta\\xi',\\,\\Delta\\varphi\\xi'\\}\n\\end{equation}\nTaking into account the vacuum alignment in Eq.(\\ref{11}), the mass matrix for the charged lepton reads\n\\begin{equation}\n\\label{14}m_{\\ell}=\\frac{v_du}{\\sqrt{2}}\\left(\\begin{array}{ccc}y^{(1)}_eu^2t&y^{(2)}_eu^2t&y^{(3)}_eu^2t\\\\\n0&y_{\\mu}u&0\\\\\n0&0&y_{\\tau}\n\\end{array}\\right)\n\\end{equation}\nwhere $y^{(i)}_e$ is the linear combination of the $y_{e,i}$\ncontributions. The parameter $u$ parameterizes the ratio\n$v_{\\psi}\/\\Lambda$, $v_{\\eta}\/\\Lambda$,\n$v_{\\Delta}\/\\Lambda$,$v_{\\varphi}\/\\Lambda$ and $v_{\\xi'}\/\\Lambda$,\nwhich should be of the same order of magnitude to produce the mass\nhierarchy among the charged fermions. The parameter $t$ denotes the\nratio $v_{\\theta}\/\\Lambda$. It has been shown that the parameters\n$u$ and $t$ belong to the range $0.01<u,t<0.05$\n\\cite{Bazzocchi:2009pv}. The first term in $w_{\\nu}$ is the neutrino\nDirac-Yukawa coupling, and the last two terms determine the mass\nmatrix of the heavy right-handed neutrinos. Straightforwardly we\nhave\n\\begin{equation}\n\\label{15}m^D_{\\nu}=\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{ccc}1&0&0\\\\\n0&0&1\\\\\n0&1&0\\\\\n\\end{array}\\right)yv_{u},~~~~~M_{N}=\\left(\\begin{array}{ccc}2c&b-c&b-c\\\\\nb-c&b+2c&-c\\\\\nb-c&-c&b+2c\n\\end{array}\\right)\n\\end{equation}\nwhere $b=2x_dv_{\\varphi}$ and $x_t=2x_tv_{\\Delta}$. Integrating out\nthe heavy neutrino $\\nu^c$, the light-neutrino effective mass matrix\nis given by the see-saw formula\n\\begin{eqnarray}\n\\nonumber&&m_{\\nu}=-(m^{D}_{\\nu})^{T}M^{-1}_Nm^{D}_{\\nu}=\\frac{y^2v^2_u}{4}\\left(\\begin{array}{ccc}\n\\frac{b+c}{b(b-3c)}&\\frac{-b+c}{b(b-3c)}&\\frac{-b+c}{b(b-3c)}\\\\\n\\frac{-b+c}{b(b-3c)}&\\frac{-b^2+4bc+3c^2}{b(b^2-9c^2)}&\\frac{b^2-2bc+3c^2}{b(b^2-9c^2)}\\\\\n\\frac{-b+c}{b(b-3c)}&\\frac{b^2-2bc+3c^2}{b(b^2-9c^2)}&\\frac{-b^2+4bc+3c^2}{b(b^2-9c^2)}\n\\end{array}\\right)\n\\end{eqnarray}\nThe light-neutrino mass matrix $m_{\\nu}$ can be diagonalized by\n\\begin{equation}\n\\label{16}U^{T}_{\\nu}m_{\\nu}U_{\\nu}={\\rm diag}(m_{\\nu_1},\nm_{\\nu_2},m_{\\nu_3})\n\\end{equation}\nwhere $m_{\\nu1,\\nu2,\\nu3}$ are the light-neutrino masses\n\\begin{eqnarray}\n\\nonumber&&m_{\\nu_1}=\\frac{y^2v^2_u}{2}\\frac{1}{|-b+3c|}\\\\\n\\nonumber&&m_{\\nu_2}=\\frac{y^2v^2_u}{2}\\frac{1}{2|b|}\\\\\n\\label{17}&&m_{\\nu_3}=\\frac{y^2v^2_u}{2}\\frac{1}{|b+3c|}\n\\end{eqnarray}\nThe unitary matrix $U_{\\nu}$ can be written as\n\\begin{equation}\n\\label{18}U_{\\nu}=iU_{TB}U_P\n\\end{equation}\nwhere $U_{TB}$ is the TB mixing matrix\n\\begin{equation}\n\\label{19}U_{TB}=\\left(\\begin{array}{ccc}\\sqrt{2\/3}&1\/\\sqrt{3}&0\\\\\n-1\/\\sqrt{6}&1\/\\sqrt{3}&1\/\\sqrt{2}\\\\\n-1\/\\sqrt{6}&1\/\\sqrt{3}&-1\/\\sqrt{2}\n\\end{array}\\right)\n\\end{equation}\nand $U_P$=diag($e^{i\\alpha_1\/2}$, $e^{i\\alpha_2\/2}$,\n$e^{i\\alpha_3\/2}$) is a matrix of phase with $\\alpha_1={\\rm\narg}(-b+3c)$, $\\alpha_2={\\rm arg}(b)$ and $\\alpha_3={\\rm\narg}(b+3c)$. Therefore the lepton mixing matrix is the TB mixing\napart from the negligible corrections of order $u^2t^2$ from the\ncharged lepton sector. We note that the right-handed neutrino mass\nmatrix $M_N$ is diagonalized by TB mixing as well, the mass\neigenvalues are given by $M_1=|-b+3c|$, $M_2=2|b|$ and $M_3=|b+3c|$.\nComparing with the light-neutrino masses in Eq.(\\ref{15}), we have\nthe interesting relation\n\\begin{equation}\n\\label{add_EPJC_1}m_{\\nu_i}=\\frac{y^2v^2_{u}}{2M_i}\n\\end{equation}\nThe Yukawa coupling $y$ is of ${\\cal O}(1)$, and we use $|\\Delta\nm^2_{atm}|^{1\/2}$ as the typical light-neutrino mass scale, then we\nobtain\n\\begin{equation}\n\\label{add_EPJC_2}M_i\\sim 10^{14\\div15}{\\rm GeV}\n\\end{equation}\nThe coefficients $x_t$ and $x_d$ are expected to be of ${\\cal\nO}(1)$, as a consequence the VEVs $v_{\\varphi}$ and $v_{\\Delta}$\nshould be of the same order as the right-handed neutrino mass $M_i$.\nIt is obvious that the model is rather constrained, there are only\nthree independent parameters, which can be chosen to be\n$|b|=y^2v^2_u\/(4m_{\\nu2})$, $Z$ and $\\Omega$. The latter two are\ndefined according to\n\\begin{equation}\n\\label{20}\\frac{c}{b}=Ze^{i\\Omega}\n\\end{equation}\nWe can easily express $Z$ and $\\Omega$ in terms of the neutrino masses as\n\\begin{eqnarray}\n\\nonumber&&Z=\\frac{1}{3}\\sqrt{\\frac{2m^2_{\\nu2}}{m^2_{\\nu1}}+\\frac{2m^2_{\\nu2}}{m^2_{\\nu3}}-1}\\\\\n\\label{21}&&\\cos\\Omega=\\frac{\\frac{m^2_{\\nu2}}{m^2_{\\nu3}}-\\frac{m^2_{\\nu2}}{m^2_{\\nu1}}}{\\sqrt{\\frac{2m^2_{\\nu2}}{m^2_{\\nu1}}+\\frac{2m^2_{\\nu2}}{m^2_{\\nu3}}-1}}\n\\end{eqnarray}\nThe above relations hold for both normal hierarchy (NH) and inverted\nhierarchy (IH) spectrum. Taking into account the experimentally\nmeasured mass difference $\\Delta m^2_{\\rm\nsol}=m^2_{\\nu2}-m^2_{\\nu1}$ and $\\Delta m^2_{\\rm\natm}=|m^2_{\\nu3}-m^2_{\\nu1}(m^2_{\\nu2})|$, we have only one free\nparameter left, which is conveniently chosen to be the lightest\nneutrino mass. Imposing the constraint $|\\cos\\Omega|\\leq1$, we\nobtain the following limits for the lightest neutrino mass\n\\footnote{The same parameter space is obtained by the neutrino mass\nsum rule method \\cite{Barry:2010yk}.}\n\\begin{eqnarray}\n\\nonumber&&m_{\\nu1}\\geq0.011 {\\,\\rm eV},~~~~~~~~~{\\rm NH}\\\\\n\\label{23}&&m_{\\nu3}\\geq0.029{\\,\\rm eV},~~~~~~~~~{\\rm IH}\n\\end{eqnarray}\nwhere the central values of $\\Delta m^2_{\\rm sol}$ and $\\Delta\nm^2_{\\rm atm}$ are used. We would like to stress that the mass\nsquared differences are running quantities, therefore the bounds in\nEq.(\\ref{23}) would change somewhat at low energy after considering\nthe RG effects.\n\nThe model is so predictive that we can express the Majorana\nphases in terms of the lightest neutrino mass as well. In the\nstandard parametrization \\cite{pdg}, the lepton PMNS mixing matrix\nis defined by\n\\begin{eqnarray}\n\\nonumber&&U_{PMNS}={\\rm diag}(e^{i\\delta_{e}},e^{i\\delta_{\\mu}},e^{i\\delta_{\\tau}})\\left(\\begin{array}{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\\delta}\\\\\n-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\\delta}&s_{23}c_{13}\\\\\ns_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\\delta}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\\delta}&c_{23}c_{13}\\end{array}\\right)\\\\\n\\label{add1}&&~~~~~\\times{\\rm diag}(e^{-i\\varphi_{1}\/2},e^{-i\\varphi_{2}\/2},1)\n\\end{eqnarray}\nwhere $c_{ij}=\\cos\\theta_{ij}$, $s_{ij}=\\sin\\theta_{ij}$ with\n$\\theta_{ij}\\in[0,\\pi\/2]$, the unphysical phases $\\delta_e$,\n$\\delta_{\\mu}$ and $\\delta_{\\tau}$ can be absorbed into charged\nlepton fields, $\\delta$ is the Dirac CP violating phase,\n$\\alpha_{21}$ and $\\alpha_{31}$ are the two Majorana CP violating\nphases, all the three CP violating phases $\\delta$, $\\varphi_{1}$\nand $\\varphi_{2}$ are allowed to vary in the range of $0\\sim2\\pi$.\nRecalling that the leptonic mixing matrix is given by Eq.(\\ref{18}),\nwe can identify the two CP violating phases as\n\\begin{equation}\n\\label{add2}\\varphi_1=\\alpha_3-\\alpha_1,~~~~~~~\\varphi_2=\\alpha_3-\\alpha_2\n\\end{equation}\nAs a result, we have\n\\begin{eqnarray}\n\\nonumber&&\\cos\\varphi_1=\\frac{-1+9Z^2}{\\sqrt{1+81Z^4-18Z^2\\cos2\\Omega}}\\,,~~~~~\\sin\\varphi_1=\\frac{-6Z\\sin\\Omega}{\\sqrt{1+81Z^4-18Z^2\\cos2\\Omega}}\\\\\n\\label{add3}&&\\cos\\varphi_2=\\frac{1+3Z\\cos\\Omega}{\\sqrt{1+9Z^2+6Z\\cos\\Omega}}\\,,~~~~~~~~\\sin\\varphi_2=\\frac{3Z\\sin\\Omega}{\\sqrt{1+9Z^2+6Z\\cos\\Omega}}\n\\end{eqnarray}\nSince we can only determine $\\cos\\Omega$ from the neutrino mass\nspectrum, the Majorana phases $\\varphi_1$, $\\varphi_2$ can take two\nsets of values corresponding to $\\sin\\Omega>0$ and $\\sin\\Omega<0$\nrespectively. We note that the Dirac CP phase is undetermined\nbecause the reactor angle is vanishing in TB mixing. The above\nsuccessful leading order results are corrected by the NLO\ncontributions, which consists of the higher dimensional operators in\nboth the driving superpotential and Yukawa superpotentials. It has\nbeen shown that all the three leptonic mixing angles receive\ncorrections of order $u$ \\cite{Bazzocchi:2009pv}.\n\n\\subsection{The $S_4$ model of Ding}\nThe total flavor symmetry of this model is $S_4\\times Z_3\\times Z_4$\n\\cite{Ding:2009iy}. It is remarkable that the realistic pattern of\nfermion masses and flavor mixing in both the lepton and quark sector\nhave been reproduced in this model, and the mass hierarchies are\ndetermined by the spontaneous breaking of the flavor symmetry\nwithout invoking a Froggatt-Nielsen $U(1)$ symmetry. The leptonic\nfields and the flavons of the model and their classifications under\nthe flavor symmetry are shown in Table \\ref{tab:S4trans}, where the\nquark fields have been omitted.\n\n\\begin{table}[hptb]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\\hline\\hline\n& $\\ell$  & $e^{c}$ & $\\mu^{c}$ &  $\\tau^{c}$ &$\\nu^c$ &\n$H_{u,d}$&$\\varphi$ & $\\chi$ & $\\zeta$ & $\\eta$ & $\\phi$ & $\\Delta$\n\\\\\\hline\n\n$\\rm{S_4}$& $3_1$& $1_1$ & $1_2$&$1_1$ &  $3_1$  &  $1_1$&$3_1$ &\n$3_2$& $1_2$ & 2   & $3_1$   & $1_2$  \\\\\\hline\n\n$\\rm{Z_{3}}$& $\\omega$ & $\\omega^2$& $\\omega^2$&  $\\omega^2$  & $1$\n&1 & 1 &1 &1 &  $\\omega^2$  & $\\omega^2$   & $\\omega^2$\n\\\\\\hline\n\n$\\rm{Z_{4}}$& 1 &i & -1&  -i  & 1  & 1& i &i & 1&  1  & 1  & -1\n\\\\\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{tab:S4trans}The transformation rules of the leptonic\nfields and the flavons under the symmetry groups $S_4$, $Z_3$ and\n$Z_4$ in the $S_4$ model of Ref. \\cite{Ding:2009iy}, where $\\omega$ is the third root of unity, i.e.\n$\\omega=e^{i\\frac{2\\pi}{3}}=(-1+i\\sqrt{3})\/2$.}\n\\end{table}\n\nIn this model the $S_4$ symmetry is broken down to Klein four and\n$Z_3$ subgroups in the neutrino and charged lepton sector,\nrespectively, at LO, this specific breaking scheme require flavon\nfields develop the following vacuum configuration\n\\begin{eqnarray}\n\\nonumber&&\\langle\\varphi\\rangle=(0,V_{\\varphi},0),~~~\\langle\\chi\\rangle=(0,V_{\\chi},0)~~~\\langle\\zeta\\rangle=V_{\\zeta}\\\\\n\\label{24}&&\\langle\\eta\\rangle=(V_{\\eta},V_{\\eta}),~~~\\langle\\phi\\rangle=(V_{\\phi},V_{\\phi},V_{\\phi}),~~~\\langle\\Delta\\rangle=V_{\\Delta}\n\\end{eqnarray}\nWe have demonstrated that this particular vacuum alignment is a\nnatural solution to the scalar potential, all the VEVs (scaled by\nthe cutoff $\\Lambda$) $V_{\\varphi}\/\\Lambda$, $V_{\\chi}\/\\Lambda$,\n$V_{\\zeta}\/\\Lambda$, $V_{\\eta}\/\\Lambda$, $V_{\\phi}\/\\Lambda$ and\n$V_{\\Delta}\/\\Lambda$ are of the same order of magnitude about ${\\cal\nO}(\\lambda^2_c)$, and this vacuum configuration is stable under the\nhigher order corrections, please see Ref. \\cite{Ding:2009iy} for\ndetail. Then the most general superpotential in the lepton sector,\nwhich is compatible with the representation assignment of Table\n\\ref{tab:S4trans}, is given by\n\\begin{eqnarray}\n\\nonumber&&w_{\\ell}=\\frac{y_{e_1}}{\\Lambda^3}\\;e^{c}(\\ell\\varphi)_{1_1}(\\varphi\\varphi)_{1_1}h_d+\\frac{y_{e_2}}{\\Lambda^3}\\;e^{c}((\\ell\\varphi)_2(\\varphi\\varphi)_2)_{1_1}h_d+\\frac{y_{e_3}}{\\Lambda^3}\\;e^{c}((\\ell\\varphi)_{3_1}(\\varphi\\varphi)_{3_1})_{1_1}h_d\\\\\n\\nonumber&&~~+\\frac{y_{e_4}}{\\Lambda^3}\\;e^{c}((\\ell\\chi)_2(\\chi\\chi)_2)_{1_1}h_d+\\frac{y_{e_5}}{\\Lambda^3}\\;e^{c}((\\ell\\chi)_{3_1}(\\chi\\chi)_{3_1})_{1_1}h_d+\\frac{y_{e_6}}{\\Lambda^3}\\;e^{c}(\\ell\\varphi)_{1_1}(\\chi\\chi)_{1_1}h_d\\\\\n\\nonumber&&~~+\\frac{y_{e_7}}{\\Lambda^3}\\;e^{c}((\\ell\\varphi)_2(\\chi\\chi)_2)_{1_1}h_d+\\frac{y_{e_8}}{\\Lambda^3}\\;e^{c}((\\ell\\varphi)_{3_1}(\\chi\\chi)_{3_1})_{1_1}h_d+\\frac{y_{e_9}}{\\Lambda^3}\\;e^{c}((\\ell\\chi)_2(\\varphi\\varphi)_2)_{1_1}h_d\\\\\n\\nonumber&&~~+\\frac{y_{e_{10}}}{\\Lambda^3}\\;e^{c}((\\ell\\chi)_{3_1}(\\varphi\\varphi)_{3_1})_{1_1}h_d+\\frac{y_{\\mu}}{2\\Lambda^2}\\mu^{c}(\\ell(\\varphi\\chi)_{3_2})_{1_2}h_d+\\frac{y_{\\tau}}{\\Lambda}\\tau^{c}(\\ell\\varphi)_{1_1}h_d+...\\\\\n\\label{25}&&w_{\\nu}=\\frac{y_{\\nu_1}}{\\Lambda}((\\nu^{c}\\ell)_2\\eta)_{1_1}h_u+\\frac{y_{\\nu_2}}{\\Lambda}((\\nu^{c}\\ell)_{3_1}\\phi)_{1_1}h_u+\\frac{1}{2}M(\\nu^c\\nu^c)_{1_1}+...\n\\end{eqnarray}\nwhere $(...)_{1_1,1_2, 2, 3_1,3_2}$ stands for the $1_1$, $1_2$,\n$2$, $3_1$ and $3_2$ products, respectively. We note that and one can\nalways set $M$ to be real and positive by global phase\ntransformation of the lepton fields, and a priori $M$ should be of\nthe same order as the cutoff scale $\\Lambda$. Taking into account\nthe vacuum alignment in Eq.(\\ref{24}), we find that the charged\nlepton mass matrix is diagonal at LO,\n\\begin{equation}\n\\label{26}m_{\\ell}=\\frac{v_d}{\\sqrt{2}}\\left(\\begin{array}{ccc}\ny_e\\frac{v^3_{\\varphi}}{\\Lambda^3}&0&0\\\\\n0&y_{\\mu}\\frac{v_{\\varphi}v_{\\chi}}{\\Lambda^2}&0\\\\\n0&0&y_{\\tau}\\frac{v_{\\varphi}}{\\Lambda}\n\\end{array}\\right)\n\\end{equation}\nwhere $y_e$ is the result of all the different contributions of\n$y_{e_i}$. The neutrino Dirac and Majorana mass matrices can be\nstraightforwardly read out as\n\\begin{equation}\n\\label{27}m^{D}_{\\nu}=\\frac{v_{u}}{\\sqrt{2}}\\left(\\begin{array}{ccc}2b&a-b&a-b\\\\\na-b&a+2b&-b\\\\\na-b&-b&a+2b\\end{array}\\right),~~~~~~~M_N=\\left(\\begin{array}{ccc}\nM&0&0\\\\\n0&0&M\\\\\n0&M&0\\end{array}\\right)\n\\end{equation}\nwhere $a=y_{\\nu_1}\\frac{v_{\\eta}}{\\Lambda}$ and\n$b=y_{\\nu_2}\\frac{v_{\\phi}}{\\Lambda}$. As a result, the light-neutrino mass matrix is given by\n\\begin{equation}\n\\label{28}m_{\\nu}=-(m^{D}_{\\nu})^{T}M^{-1}_{N}m^{D}_{\\nu}=-\\frac{v^2_u}{2M}\\left(\\begin{array}{ccc}2a^2-4ab+6b^2&a^2+2ab-3b^2&a^2+2ab-3b^2\\\\\na^2+2ab-3b^2&a^2-4ab-3b^2&2a^2+2ab+6b^2\\\\\na^2+2ab-3b^2&2a^2+2ab+6b^2&a^2-4ab-3b^2\n\\end{array}\\right)\n\\end{equation}\nWe can see that the mass matrix $m_{\\nu}$ is exactly diagonalized by the TB mixing matrix\n\\begin{equation}\n\\label{29}U^{T}_{\\nu}m_{\\nu}U_{\\nu}={\\rm diag}(m_{\\nu1},m_{\\nu2},m_{\\nu3})\n\\end{equation}\nThe unitary matrix $U_{\\nu}$ is\n\\begin{equation}\n\\label{30}U_{\\nu}=U_{TB}\\,{\\rm\ndiag}(e^{-i\\alpha_1\/2},e^{-i\\alpha_2\/2},e^{-i\\alpha_3\/2})\n\\end{equation}\nThe phases $\\alpha_{1}$, $\\alpha_2$ and $\\alpha_3$ are closely related to the Majorana phase\n\\begin{equation}\n\\label{31}\\alpha_1={\\rm arg}(-(a-3b)^2\/M),~~~~\\alpha_2={\\rm arg}(-4a^2\/M),~~~~\\alpha_3={\\rm arg}((a+3b)^2\/M)\n\\end{equation}\nand the neutrino masses are given by\n\\begin{equation}\n\\label{32}m_{\\nu1}=|(a-3b)^2|v^2_u\/(2M),~~~m_{\\nu2}=2|a|^2v^2_u\/M,~~~m_{\\nu3}=|(a+3b)^2|v^2_u\/(2M)\n\\end{equation}\nIt is interesting to estimate the order of magnitude for the right-handed neutrino mass $M$. Since the parameters $a$ and $b$ are\nexpected to be of order $\\lambda^2_c$, with this and using\n$\\sqrt{|\\Delta m^2_{atm}|}\\simeq0.05$ eV as the light-neutrino mass\nscale in the see-saw formula, we obtain\n\\begin{equation}\n\\label{add_EPJC_3}M\\sim 10^{12\/13}{\\rm GeV}\n\\end{equation}\nSimilar to the analysis of section \\ref{sec:BMM}, we define\n\\begin{equation}\n\\label{33}\\frac{b}{a}=R\\,e^{i\\Phi}\n\\end{equation}\nStraightforwardly we can express $R$ and $\\cos\\Phi$ as functions of the neutrino masses\n\\begin{eqnarray}\n\\nonumber&&R=\\frac{1}{3}\\sqrt{\\frac{2m_{\\nu1}}{m_{\\nu2}}+\\frac{2m_{\\nu3}}{m_{\\nu2}}-1}\\\\\n\\label{34}&&\\cos\\Phi=\\frac{\\frac{m_{\\nu3}}{m_{\\nu2}}-\\frac{m_{\\nu1}}{m_{\\nu2}}}{\\sqrt{\\frac{2m_{\\nu1}}{m_{\\nu2}}+\\frac{2m_{\\nu3}}{m_{\\nu2}}-1}}\n\\end{eqnarray}\nIn exactly the same way section \\ref{sec:BMM}, the Majorana phases in standard parameterization are determined as\n\\begin{equation}\n\\label{add4} \\varphi_1=\\alpha_1-\\alpha_3,~~~~~\\varphi_2=\\alpha_2-\\alpha_3\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\nonumber&& \\cos\\varphi_1=\\frac{-(1-9R^2)^2+36R^2\\sin^2\\Phi}{(1+9R^2)^2-36R^2\\cos^2\\Phi},~~~~~\\sin\\varphi_1=\\frac{12R(1-9R^2)\\sin\\Phi}{(1+9R^2)^2-36R^2\\cos^2\\Phi}\\\\\n\\label{add5}&&\\cos\\varphi_2=\\frac{-1-9R^2\\cos2\\Phi-6R\\cos\\Phi}{1+9R^2+6R\\cos\\Phi},~~~~\\sin\\varphi_2=\\frac{6R(1+3R\\cos\\Phi)\\sin\\Phi}{1+9R^2+6R\\cos\\Phi}\n\\end{eqnarray}\nConsequently, all the low energy parameters in the neutrino sector\ncan be expressed in terms of the lightest neutrino mass. Imposing\nthe condition $|\\cos\\Phi|\\leq1$, we get the following constraint on\nthe lightest neutrino mass:\n\\begin{eqnarray}\n\\nonumber&&m_{\\nu1}\\geq0.011\\;{\\rm eV},~~~~~{\\rm NH}\\\\\n\\label{35}&&m_{\\nu3}>0.0\\;{\\rm eV},~~~~~~~~~~{\\rm IH}\n\\end{eqnarray}\nThe NLO corrections have been analyzed in detail in Ref.\n\\cite{Ding:2009iy}. It is shown that both the neutrino masses and\nmixing angles receive corrections of order\n$\\varepsilon\\sim\\lambda^2_c$ with respect to leading order result,\nwhere $\\varepsilon$ parameterizes the ratio $VEV\/\\Lambda$ and\n$\\lambda_c$ is the Cabibbo angle.\n\n\\section{\\label{sec:RGE_running}RG running effects in $S_4$ flavor models}\nAs has been shown, the TB mixing is achieved in both BMM model and\nthe $S_4$ model of Ding at LO. In this section, we turn to a\nquantitative discussion of RG effects, and compare them with the NLO\ncorrections and the experimental data. For definiteness we shall\nassume a supersymmetry breaking scale of 1 TeV, below which the SM\nis valid. We note that the mass hierarchy between top and bottom is\nproduced via the spontaneous breaking of flavor symmetry in both\nmodels, and $\\tan\\beta$ should be small. As a result, we shall take\nthe parameter $\\tan\\beta$ to be 10 apart except where explicitly indicated\notherwise. To study the running of the neutrino mixing parameters\nfrom the GUT scale to the electroweak scale, the Mathematica package\nREAP is used \\cite{Antusch:2005gp}. This package numerically solves\nthe RG equations of the quantities relevant for neutrino mass and\nmixing, and it has been widely used for different purposes\n\\cite{Boudjemaa:2008jf}. The package can be downloaded from\nhttp:\/\/users.physik.tu-muenchen.de\/rge\/REAP\/index.html, and\nMathematica version 5 or higher is required. We note that the\napproximate analytical solutions based on leading log approximation\nto the RG equations have been derived in\nRefs.\\cite{Antusch:2005gp,Antusch:2003kp}, which allows one to\nunderstand the generic behavior of the renormalization effects.\nHowever, due to enhancement\/suppression factors and possible\ncancelations, the exact numerical solutions may differ considerably\nfrom those estimates. Therefore throughout this paper we adopt a\nnumerical approach, exploiting the convenient REAP package.\n\nAs has been demonstrated above, we generally need to introduce\nflavon fields to break the flavor symmetry in order to generate\nfermion masses and flavor mixing. In the unbroken phase of flavor\nsymmetry, the flavons are active fields, therefore the corresponding\nRG equations should be modified in principle. However, the\nsuperpotentials of the models in Eq.(\\ref{12}) and Eq.(\\ref{25})\ncontain all the possible LO terms allowed by the symmetries, the\ninvariance under the flavor symmetry $S_4$ is maintained until we\nmove down to the scale of the VEV of the flavon fields, which is of\nthe order of GUT scale. We conclude that the flavor structures of\nthe models are preserved above the scale of the VEV of the flavon\nfields, the contributions of the flavon fields in the RG running can\nbe absorbed by the redefinition of the model parameters\n\\cite{Lin:2009sq}. In the following, we will discuss the RG\nevolution of neutrino masses and mixing parameters in both $S_4$\nflavor models, stating from the initial conditions of neutrino Dirac\nand Majorana mass matrices described in section \\ref{sec:models} at\nthe GUT scale. In particularly, the parameter spaces are scanned.\n\n\\subsection{\\label{sec:RG_BMM}RG effects in the BMM models}\n\nIn this section we report results of the calculations of the RG\nevolution of the neutrino mixing parameters in the BMM model.\nWithout loss of generality, we choose the Yukawa coupling $y=1$ for\nour numerical analysis. The GUT scale neutrino mass squared\ndifferences $m^2_{\\nu2}-m^2_{\\nu1}$ and $|m^2_{\\nu3}-m^2_{\\nu1}|$\nare treated as random numbers in the range of $3.5\\times10^{-5}{~\\rm\neV^2}\\sim2.5\\times10^{-4}{~\\rm eV^2}$ and $1.0\\times10^{-3}{~\\rm\neV^2}\\sim8.3\\times10^{-3}{~\\rm eV^2}$ respectively, \\footnote{We\nshall show later that the mass squared difference at the GUT scale\nis about a factor of $1.2\\sim3$ larger than its low energy value in\nthe whole spectrum.}, and the lightest neutrino mass is varied from\nthe lowest bound determined by Eq.(\\ref{21}) or Eq.(\\ref{34}) to 0.2\neV which is the future sensitivity of KATRIN experiment\n\\cite{katrin}. The RG corrected neutrino mixing angles as functions\nof the lightest neutrino mass are shown in Fig.\n\\ref{fig:RGE_BMM_mass1} for both NH and IH spectrum \\footnote{The\nresults are independent of the sign of $\\sin\\Omega$, the reason is\nexplained later.}. These plots display only the points corresponding\nto choices of the parameters reproducing $\\Delta m^2_{\\rm atm}$,\n$\\Delta m^2_{\\rm sol}$ and the mixing angles within the $3\\sigma$\ninterval.\n\nWe see that the lightest neutrino mass is still bounded from below,\nand the concrete values of the lower bounds are about 0.0107 eV and\n0.027 eV, respectively, for the NH and IH spectrum, and these values\nare found to be almost independent of $\\tan\\beta$. It is remarkable\nthat all the mixing parameters and $J_{CP}$ are predicted to lie in\na relative narrow range. For both NH and IH spectrum, it is obvious\nthat the RG changes of the atmospheric and reactor angles are very\nsmall, the corresponding allowed regions lie within the current\n$1\\sigma$ bounds. In particular, the RG corrections to\n$\\sin^2\\theta_{23}$ and $\\sin^2\\theta_{13}$ are of the same order or\neven smaller than the NLO contributions. On the other hand, the\nrunning of the solar neutrino mixing angle displays a different\npattern. The RG change of $\\sin^2\\theta_{12}$ is much larger than\nthose of $\\sin^2\\theta_{23}$ and $\\sin^2\\theta_{13}$, which is a\ngeneral property of the RG evolution\n\\cite{Antusch:2005gp,Antusch:2003kp}, consequently the deviation\nfrom its TB value can be large. In the case of NH spectrum and large\n$\\tan\\beta$, $\\sin^2\\theta_{12}$ is within the $3\\sigma$ limit only\nfor smaller values of neutrino mass. Taking into account the lower\nbound on the lightest neutrino mass, $m_{\\nu1}$ is constrained to\nlie in certain region, which decreases with $\\tan\\beta$. This point\ncan be clearly seen from Fig.\\ref{fig:RGE_BMM_mass1}. For IH\nspectrum, the RG effect of $\\theta_{12}$ is even larger due to the\nnearly degeneracy of $m_{\\nu1}$ and $m_{\\nu2}$. For example, for\n$\\tan\\beta=10$, $\\sin^2\\theta_{12}$ is very close or above the\n$3\\sigma$ upper bound in the allowed region of $m_{\\nu3}$, and the\nvalues of $\\sin^2\\theta_{12}$ goes completely beyond the $3\\sigma$\nlimit for larger $\\tan\\beta$. As a result, the IH spectrum is\nstrongly disfavored for $\\tan\\beta>10$ in the BMM model. We note\nthat possible large deviation of solar neutrino mixing angle from\nthe TB value, and small change of atmospheric and reactor angles\nunder RG running are predicted as well in the Altarelli-Feruglio\n$A_4$ model \\cite{Lin:2009sq}. In Ref.\\cite{Lin:2009sq}, the authors\nperform a general analysis of running effects on lepton mixing\nparameters in flavor models with type I see-saw, they show that, for\nthe mass-independent mixing pattern, the running contribution from\nthe neutrino Yukawa coupling $Y_{\\nu}$ can be absorbed by a small\nshift on neutrino mass eigenvalues leaving mixing angles unchanged,\nconsequently the RG change of mixing angle is due to the\ncontribution coming from the charged lepton sector. This is exactly\nthe reason why similar results to the $A_4$ model are obtained here.\n\nThe variations of Majorana phases $\\varphi_1$ and $\\varphi_2$, Dirac\nCP violating phase $\\delta$ and the Jarlskog invariant $J_{CP}$ with\nrespect to the lightest neutrino mass are also plotted in Fig.\n\\ref{fig:RGE_BMM_mass2}. We note that Dirac phase $\\delta$ arises\nfrom the running effect, even though it is undetermined in the\nbeginning. The initial value of Jarlskog invariant $J_{CP}$ is zero\ndue to the vanishing of the $\\theta_{13}$ in TB mixing scheme, and\nit remains small because of the smallness of the $\\theta_{13}$,\nalthough the value of $\\delta$ is large. It is remarkable that we\ncan understand the dependence on the sign of $\\sin\\Omega$ exactly.\nAt initial scale, the right-handed neutrino mass matrix $M_N$ shown\nin Eq.(\\ref{15}) is complex with each other for $\\sin\\Omega>0$ and\n$\\sin\\Omega<0$ apart from the irrelevant overall phase, and the\nneutrino Yukawa coupling matrix $Y_{\\nu}$ can be chosen to be real.\nTherefore, in the case of $\\sin\\Omega<0$, the complex conjugates of\n$Y_{\\nu}$, $Y_{e}$, $M_{N}$ and $\\kappa$ run in the same way as the\ncorresponding quantities of $\\sin\\Omega>0$ with the same initial\nconditions. Consequently the resulting low energy effective neutrino\nmass matrix for $\\sin\\Omega<0$ is the complex conjugate of the\ncorresponding one of $\\sin\\Omega>0$. As a result, the RG evolution\nof mixing angles and $J_{CP}$ is independent of the sign of\n$\\sin\\Omega$, and summation of the each CP phase for $\\sin\\Omega>0$\nand $\\sin\\Omega<0$ is equal to $2\\pi$. These results have been\nconfirmed in our numerical analysis explicitly.\n\nConcretely the running of neutrino masses and mixing parameters with\nthe energy scale is displayed in Fig. \\ref{fig:RGE_BMM_scale} for\nboth NH and IH spectrum with $\\tan\\beta=10$, where the initial\nconditions for the NH and IH are chosen to be $m_1=0.041$ eV,\n$\\Delta m^2_{\\rm sol}=1.76\\times10^{-4}{\\rm eV^2}$, $\\Delta m^2_{\\rm\natm}=5.85\\times10^{-3}{\\rm eV^2}$ and $m_3=0.0538$ eV, $\\Delta\nm^2_{\\rm sol}=1.87\\times10^{-4}{\\rm eV^2}$, $\\Delta m^2_{\\rm\natm}=5.58\\times10^{-3}{\\rm eV^2}$ respectively. Reasonable values\nfor the lower energy oscillation parameters are reached. We see that\nthe deviation of the solar neutrino mixing angle $\\theta_{12}$ from\nthe TB value can be relative large for the IH spectrum, the mixing\nangles $\\theta_{23}$ and $\\theta_{13}$ and the CP phases $\\delta$,\n$\\varphi_1$ and $\\varphi_2$ are stable under the RG evolution, the\ncorresponding RG corrections are small. Since\n$Y^{\\dagger}_{\\nu}Y_{\\nu}=y^2{\\bf 1}$, the contribution from the\nneutrino Yukawa coupling is universal above the see-saw threshold.\nThen, only the charged lepton relevant part $Y^{\\dagger}_eY_e$\ncontributes to the change in mixing angles, and the evolution above\nthe see-saw scales is essentially the same as below. This is in\ncontrast with the usual situation where the neutrino Yukawa coupling\nplays dominant role in the running of neutrino mass matrix above the\nhighest see-saw scale. Furthermore, we find that the running of the\nneutrino mass $m_{\\nu i}$ is approximately given by a common scaling\nof the mass eigenvalues, this is the same as the situation below the\nsee-saw scale \\cite{Antusch:2003kp,Chankowski:2001mx}. It is\nremarkable that the neutrino mass is reduced by about $2.4$ times at\nlow energy. We note that the above results about the running\nbehavior of neutrino masses and mixing parameters are very general,\nthey almost do not depend on the initial conditions.\n\n\n\n\n\n\\subsection{RG effects in the $S_4$ model of Ding}\n\nIt is remarkable that the heavy right-handed neutrinos are\ndegenerate at LO, and the corrections to the degeneracy arising from\nRG running turn out to be so small that could be neglected,\nconsequently the threshold effects should be very small in this\ncase. In particular, we note that $Y^{\\dagger}_{\\nu}Y_{\\nu}$ is not\nproportional to the unit matrix any more, large RG effects seem\npossible. As has been demonstrated in Eq.(\\ref{add_EPJC_3}), the\nright-handed neutrino mass $M$ is estimated to be of order $\n10^{12}\\sim10^{13}{\\rm GeV}$. Without loss of generality, we shall\nchoose $M=10^{12}$ GeV in the following numerical analysis, and we\nhave checked that final results change very slowly with the\nparameter $M$. The neutrino mixing angles at electroweak scale as\nfunctions of the lightest neutrino mass are shown in\nFig.\\ref{fig:RGE_Ding_mass1}, it is obvious that the lightest\nneutrino mass for NH spectrum is bounded from below, and the lower\nbound on the lightest neutrino mass in the case of IH spectrum is\nstill approximately zero. We see that the RG effects on both\natmospheric and reactor angles are rather small, and the running of\n$\\theta_{12}$ can be large depending on $\\tan\\beta$ and the mass\ndegeneracy. Matching $\\theta_{12}$ with the data already puts strong\nconstraints on the lightest neutrino mass spectrum and $\\tan\\beta$\nat the present stage, and a upper bound on the lightest neutrino\nmass is usually implied for small value of $\\tan\\beta$, which means\nthat the neutrino mass spectrum can not be highly degenerate. In the\ncase of $\\tan\\beta=20$, the IH spectrum is ruled out, since the\nvalue of $\\sin^2\\theta_{12}$ is much larger than its $3\\sigma$ upper\nbound. While the model is within the $3\\sigma$ limit only for small\nneutrino mass for NH spectrum, as is displayed in\nFig.\\ref{fig:RGE_Ding_mass1}. The predictions for the CP phases and\nthe Jarlskog invariant are plotted in Fig.\\ref{fig:RGE_Ding_mass2}.\nIn a similar way as section \\ref{sec:RG_BMM}, we learn that the\nevolutions of mixing angles and $J_{CP}$ do not depend on the\nsign of $\\sin\\Delta$, the summation of each CP phase for\n$\\sin\\Delta>0$ and $\\sin\\Delta<0$ is equal to $2\\pi$. These points\nare checked by our detailed numerical analysis.\n\nThe running of neutrino masses and mixing parameters with the energy\nscale are plotted in Fig.\\ref{fig:RGE_Ding_scale}. Being similar to\nthe situation in the BMM model, the RG corrections to the CP phases\n$\\delta$, $\\varphi_1$ and $\\varphi_2$ are typically small, the\ncorresponding curves are almost straight lines. We see that the\nneutrino mixing angles are rather stable under RG evolution except\nthe solar angle for IH spectrum. The running of neutrino mass can be\napproximately described by a common scaling factor, and it reduced by\nabout 2 times at electroweak scale. In short summary, the evolution\nof the neutrino parameters in Ding's $S_4$ model is very similar to\nthat of BMM model, although the textures of the mass matrices are\ntotally different.\n\n\\section{\\label{sec:conclusion}Conclusion}\nFlavor models based on discrete flavor symmetry are particularly\ninteresting, they can produce the tri-bimaximal neutrino mixing (or\nsome other mass-independent mixing patterns) at LO in an elegant\nway. It is a common feature that the LO predictions would be\ncorrected by the subleading higher dimensional operators, and it\nhave been shown that the subleading corrections are under control in\nsome consistent flavor models. Since the tri-bimaximal mixing is\npredicted at high energy scale, it is very necessary to investigate\nwhether the RG effects would push the mixing parameters beyond the\ncurrent allowed ranges by experimental data.\n\nIn this paper, we have analyzed the RG running of the neutrino mass\nand mixing parameters in the BMM model and the $S_4$ model of Ding,\nboth models predict tri-bimaximal neutrino mixing at LO, but the\ntextures of the mass matrices are totally different. To study the\nrunning effects, we use the Mathematica package REAP. By detailed\nnumerical analysis, we find that the evolution of neutrino mixing\nparameters displays approximately the same pattern in both $S_4$\nmodels. We see that the atmospheric and reactor neutrino mixing\nangles are essentially stable under RG evolution for both NH and IH\nspectrum. However, the running of solar neutrino mixing angle\ndepends on the neutrino mass and the parameter $\\tan\\beta$, and the\ndeviation from its TB value could be large. After we take into\naccount the RG effects, the neutrino mass spectrum is strongly\nconstrained by the current data on $\\theta_{12}$, the lightest\nneutrino mass is bounded from both below and up, and the upper bound\ndecreases with $\\tan\\beta$. For large $\\tan\\beta$ ($\\tan\\beta>10$),\nthe value of $\\sin^2\\theta_{12}$ could be larger than its $3\\sigma$\nupper bound for the whole spectrum in the case of IH spectrum. As a\nresult, the IH neutrino mass spectrum is disfavored in the case of\nlarge $\\tan\\beta$. Moreover, we note that the running of light-neutrino masses can be approximately described by a common scaling\nfactor, and they reduce by about $1.2\\sim3$ times at low energy.\nThis effects is neglected in Ref.\\cite{Lin:2009sq}. We note that the\nevolutions of mixing angles and $J_{CP}$ don't depend on the sign of\n$\\sin\\Omega$ or $\\sin\\Delta$, and the sum of each CP phase for both\nsign is equal to $2\\pi$. These results are confirmed both\nanalytically and numerically.\n\nFinally we note that running of neutrino parameters in the\nAltarelli-Feruglio $A_4$ model, BMM model and Ding's $S_4$ model is\nsimilar to each other, although they produce tri-bimaximal mixing in\ndifferent ways. The reason is that the neutrino Yukawa coupling only\ncontributes to the running of neutrino mass, it doesn't affect the\nlepton mixing angles, and the change in mixing angles is due to the contribution from the charged lepton sector. We conclude that\nthe running of mixing parameters is also severely constrained by the\nflavor symmetry in discrete flavor symmetry models.\n\n\n\\section*{Acknowledgements}\nWe are grateful to Prof. Zhi-Zhong Xing for stimulating discussions\non RGE running.  The author Gui-Jun Ding gratefully acknowledge the\npleasant hospitality of the theory group at the University of\nWisconsin. This work is supported by the National Natural Science\nFoundation of China under Grant No.10905053, Chinese Academy\nKJCX2-YW-N29 and the 973 project with Grant No. 2009CB825200.\nDong-Mei Pan is supported in part by the National Natural Science\nFoundation of China under Grant No.10775124.\n\n\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nEarly on in the history of programming, a metaphor was put forward\nthat has seen wide acceptance in the software community: that of\nprogramming as LEGO (Figure~\\ref{fig:CACM}). The metaphor suggests\nthat building large systems is a matter of connecting small\nstandardized bricks together, one at a time, through their universal\ninterfaces: the small bricks are independent of the scale and purpose\nof the construction. This metaphor had a tremendous influence\nin the development of OOP languages. Inspired by the simplicity of the\nLEGO construction model, these languages placed their focus on\nmechanisms that would allow to connect small computational units\ntogether to create large software systems.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.3in]{figures\/CACMSeptember1990.jpg}\n\\includegraphics[width=1.8in]{figures\/512px-Lego_dimensions.png}\n\\caption{Left: Cover of the CACM, Special issue on Object-Oriented\n  Programming, September 1990~\\cite{CACM:1990}. Right: LEGO bricks\n  showing standard dimensions. (Source: Wikipedia, ``Cmglee''.)}\n\\label{fig:CACM}\n\\end{figure}\n\nMeanwhile, in 1975 another idea was put forward that has also seen\nwide acceptance in the software community: that ``programming in the\nlarge'' has different characteristics from ``programming in the\nsmall.'' This idea was first formulated by DeRemer and\nKron~\\cite{DeRemer:1975}, who argued that ``structuring a large\ncollection of modules to form a \"system\" is an essentially distinct\nand different intellectual activity from that of constructing the\nindividual modules.'' DeRemer and Kron went on to advocate a ``Module\nInterconnection Language'' (MIL) for large systems.\n\nThese two popular ideas aren't mutually exclusive: it is possible to\nimagine system-wide directives and constraints (i.e. architecture) for\nlarge LEGO constructions. But DeRemer and Kron's essay states a\npremise that puts some pressure on the LEGO metaphor: ``Where an MIL\nis not available, module interconnectivity information is usually\nburied partly in the modules, partly in an often amorphous collection\nof linkage-editor instructions, and partly in the informal\ndocumentation of the project.'' In LEGO terms, this might mean that in\norder to build a large castle, one might need to plumb stronger\nconnection material into the bricks themselves. In short, the scale of\nthe system would affect the internal structure of the construction.\n\nThis paper focuses on the core of these two popular ideas by asking\nand answering the following question:\n\n\\noindent\n{\\bf Does the scale of the software system affect the internal\n  structure of its modules or are modules scale-invariant?}\n\nWe want to find out whether there are mathematical principles\nrelated to size in large ecosystems of software projects. Besides\nshedding light on the differences between programming-in-the-small and\nprogramming-in-the-large, this question has important\n  implications for research. A common practice for validating ideas\nin software research is to collect a number of artifacts, either\nrandomly or using some criteria, measure the effects of the ideas\nusing those artifacts, and reach conclusions from the empirical\ndata. Even though size of software artifacts (projects, classes, etc.)\nhas been known to be an issue in quantitative studies of software,\nsoftware research continues to be fairly oblivious to its effect in\nthese assorted datasets. This is particularly problematic for any\nstudies involving software metrics, including OO metrics. It also\naffects performance studies that tend to collect data on relatively\nsmall programs that aren't necessarily representative of large\nprograms.  Several studies published in the literature may have\nreached invalid conclusions by ignoring the effect of size or by\ntreating it inappropriately.\n\nThe question, as formulated above, is too ambitious to be answered in\none single step. This paper takes only the first step. We focus on\nObject-Oriented software systems, since those are the most influenced\nby the programming-as-LEGO metaphor; other language families should be\nstudied for broader conclusions. Within OOP, we focus on Java, since\nit is one of the most popular OOP languages; other OOP ecosystems\nshould be studied for broader conclusions. Finally, we report on a\ndozen metrics that illustrate the main trends, but many more metrics\ncould be studied.\n\nWe deconstruct the general question into five research questions for\nwhich specific metrics can be measured:\n\n\\begin{itemize}\n\\item [RQ1] Module size: Are modules of larger systems larger than modules\n  of smaller systems?\n\\item [RQ2] Module Type: Is there a statistically significant variation\n  in the mix of classes and interfaces for projects of different size\n  scales?\n\\item [RQ3] Internal Complexity: Are modules of larger systems more,\n  or fewer, complex than modules of smaller systems?\n\\item [RQ4] Composition via Inheritance: Does the scale of the project\naffect the use of inheritance?\n\\item [RQ5] Dependencies: Do larger projects use disproportionately \n  more, or fewer, types from external libraries than smaller projects?\n\\end{itemize}\n\n\nThis study puts forward strong evidence that, as programs become\nlarger, the internal structure of the modules and the mixture of\ncomposition mechanisms used are affected. As such, the paper\nmakes the following contributions:\n\n\\begin{enumerate}\n\\item It unveils strong empirical evidence of the existence of super-\n  and sublinear effects in software that have not been measured\n  before, and it shows concrete parameters of many non-linear\n  relations that underly a large and important ecosystem of Java\n  programs.\n\\item It proposes more accurate definitions of popular OO metrics that\n  properly normalize for size.\n\\item By unveilling the characteristics of large projects, it may\n  suggest new ideas for how to tame deterimental non-linear effects,\n  both in terms of programming language design and project management.\n\\end{enumerate}\n\n\n\\section{Motivation and Related Work}\n\\label{sec:relwork}\n\nIt has been almost 25 years since Chidamber and Kemerer published\ntheir influential paper on OO metrics at\nOOPSLA'91~\\cite{Chidamber:1991}. Since then, OO metrics have been used\npervasively in research and development. Here, we review and\ndiscuss the main issues with OO metrics, and the research\ncommunity's attempts to understand the empirically-based principles of\nsoftware.\n\n\n\\subsection{The Confusing Effect of Size}\n\\label{sec:confusingeffect}\n\nA large body of literature exists in analyzing how software metrics\ncorrelate with software quality. A typical study along those lines\ninvolves computing internal software metrics (e.g. coupling of\nclasses) and correlating them with external quality attributes\n(e.g. post-release bug fixes involving those classes). Many studies of\nthis kind apply simple univariate statistical analysis, and often\nconclude that there is a correlation.\n\nFor quite some time, however, size has been known to be a potential\nconfounding factor in empirical studies of software artifacts. For\nexample, in a study designed to verify whether it is possible to use a\nmultivariate logistic regression model based on OO metrics to predict\nfaults in OO programs, Briand et al.~\\cite{Briand2000245} reported\nstrong correlations between class size and several OO software\nmetrics. They then went on to compensate for that correlation by doing\npartial correlations. In another study of a large C++\nsystem~\\cite{Cartwright2000}, Cartwright et al. also reported such\ncorrelations. In 2001, El Emam et al.~\\cite{ElEmam2001} presented a\ncomprehensive analysis of the effect of class size in several OO\nmetrics, and suggested that this effect might have confounded prior\nstudies.\\footnote{We refer readers to ~\\cite{ElEmam2001} for an\n  extensive list of studies that the authors suggest may have reached\n  invalid conclusions by neglecting to compensate for size.}  They\nthen presented their own study of a large C++ framework which showed\nthat strong correlations resulting from univariate analysis of data\nwere neutralized when multivariate analysis including class size is\nused. Another more recent study reached the same conclusions when\nstudying the relation between internal software attributes and\ncomponent utilization~\\cite{Sajnani2014}.\n\nHowever, Briand et al. and El Emam et al.'s argument has drawn some\ncriticism stemming from the point of view that multivariate analysis\nof the kind proposed in their papers produces ill-specified, logically\ninconsistent statistical models~\\cite{Evanco2003}. Specifically, the\npartial correlation of $X$ and $Y$ controlling for a third variable\n$Z$, written $r(X,Y|Z)$, is a measure of the relationship between X\nand Y if statistically {\\em we hold Z constant}. But trying to\npredict, for example, the effect on post-release defects $X$ by\nincreasing the coupling value $Y$ while holding the number of lines of\ncode $Z$ constant doesn't make sense, because in the world from where\nthe data comes, increasing coupling usually requires additional lines\nof code (e.g. field and variable declarations). As Evanco points\nout~\\cite{Evanco2003}, this model is inconsistent with the reality of\nthe data. The suggestion following the criticism is that prediction\nmodels should use the metric in question $Y$ or the size metric ($Z$),\nwhichever gives more predictive power, but not both.\n\nEither way, these observations raise doubts about the value of the\nmany software metrics that are correlated with size, as they do not\nprovide any more additional statistical power than what is already\nprovided by their strong correlate -- and size is very easy to\nmeasure. \nIn summary, size may not be a confounding factor in statistical\nterminology, but it certainly has been the source of much confusion in\nsoftware research.\n\n\\subsection{Non-Normal Data}\n\\label{sec:nonnormal}\n\nIn their study of slice-based cohesion and coupling metrics over 63 C\nprograms, Meyers and Binkley ~\\cite{Meyers:2007} include correlation\ncoefficients between several coupling and cohesion metrics and Lines\nof Code (LOC). They show that they are not correlated. We noted that\ntheir correlation analysis was made for the entire dataset, which\ncontained components of considerably different sizes; this made the\nanalysis prone to sknewness-related errors. In subsequent email\nexchanges with one of the authors, he kindly shared the data with us;\nwe then verified that, indeed, the distribution of size of the\ncomponents was not normal but log-normal. Once the transformation to\nlog scale was performed, the data showed moderate-to-strong positive\nlinear correlation between log(size) and their coupling\nmetric.\n\nThis exchange illustrates another source of problems when doing empirical\nstudies of software artifacts, and how size can drastically affect the\nconclusions. Size is not just a confusing factor; because the\nprojects' size distribution is often skewed, the statistical analysis\nneeds to take non-normal data into account too.\n\n\\subsection{Software Corpora}\n\\label{sec:corpora}\n\nIn recent years, there has been an increasing number of empirical\nstudies on increasingly larger collections of software projects for\npurposes of understanding the way that developers use programming\nlanguages in real projects. \nFor example, Tempero et\nal.~\\cite{Tempero:2008} studied the way Java programs use inheritance\nin the 100 projects of the Qualitas\ncorpus~\\cite{QualitasCorpus:APSEC:2010}. The criteria for inclusion of\nprojects in that corpus is relatively strict, requiring, for example,\ndistribution in both source and binary forms.\\footnote{See\n  https:\/\/www.cs.auckland.ac.nz\/$\\sim$ewan\/corpus\/docs\/criteria.html} While\ntheir findings fall within the results reported here, the Qualitas\ncorpus contains only 100 projects. The results reported in\n\\cite{Tempero:2008} show that the data does not follow a normal\ndistribution. Another study on the same corpus explored the simulated\nuse of multiple dispatch via cascading {\\texttt instanceof}\nstatements~\\cite{Muschevici:2008}. Another study by Gil and Lenz\n\\cite{Gil:2010} studied the use of overloading in Java programs, also\nusing the Qualitas corpus. Some of the conclusions in these studies\n(e.g. whether a project is an outlier or not) may be missing the\neffect of size of the project.\n\nCalla\\'u et al.~\\cite{Callau:2011} made a statistical analysis of\n1,000 Smalltalk projects found in SqueakSource in order to understand\nthe use of certain dynamic features of Smalltalk. They do not\nreport the distribution in terms of project size. The study was\ndesigned to gather bulk statistics along an existing taxonomy, so the\nresults are reported as simple counts of feature occurrences among the\nwhole corpus or among a category of projects (e.g. out of 652,990\nmethods, only 8,349 use dynamic features, and then a breakdown is\nshown among categories). While the taxonomy is taken into account in the\nanalysis of the data, project size is not. It would be interesting\nto see whether there is a correlation between the categories and size\nof the projects.\n\nCollberg et al.~\\cite{Collberg:2007} randomly collected 1,132 jar\nfiles off the Internet and analyzed them (at bytecode level) using a\ntool developed by the authors. The purpose of that study was to inform\nJava language designers and implementers about how developers actually\nuse the language. That study reports summary statistics for their\nentire dataset without taking the distribution of jar size into\naccount. Most distributions shown in the paper aren't normal, so the\nsummary statistics are somewhat misleading. Some of the reported\nmetrics in that study are the same metrics that we use for our study;\nfor example they found on average 9 methods per class, with median 5.\nThe reported values fall within the range of ours, but particularly\nclose to the values for large projects, which leads us to believe that\ntheir dataset was biased towards large projects.\n\nIn another large study, Grechanik et al.~\\cite{Grechanik:2010} have\nconducted an empirical assessment of 2,080 Java projects randomly\nselected from Sourceforge, and discovered several facts about the\nprojects' use of Java. The size of the projects is not reported, and\nonly simple statistics are given. For example, the reported mean and\nmedian methods per class are 3.5 and 4, respectively. Given that the\ndata does not follow a normal distribution on project size, these\nvalues are, again, somewhat misleading and at odds with the findings\nof Collberg et al.~\\cite{Collberg:2007}. Like so many large open source\ncode repositories, Sourceforge is severely skewed towards small to\nmedium projects; the reported summary statistics are consistent with\nour findings for small projects.\n\n\nIn any large corpora of projects, the data rarely follows normal\ndistributions of size, so simple summary statistics such as averages\nand medians reported in some of these papers provide only weak\ninsights into the principles of those ecosystems, and may hide\nimportant phenomena. Also, sample biases may have a large influence on\nassumptions and conclusions.\nBut what exactly is the effect of size on software artifacts? Can we\nfind general statistical principles that explain the \nphenomena observed in prior studies? \n\n\\subsection{Complex Systems}\n\nOurs is not the first study to try to unveil internal mathematical\nstructures of software, and the software research community is not the\nonly one looking for mathematical principles in existing software;\ncommunities that study complex systems and networks have long found\nsoftware intriguing. One of the first studies of this kind was by\nValverde et al.~\\cite{Valverde:2002}, which analyzed the types and\ndependencies in the JDK, and noticed the existence of power laws and\nsmall world behavior. Soon after, Myers~\\cite{Myers:2003} explored\nwhat he called ``collaboration graphs'' ({\\em aka} dependencies) in\nthree C++ and three C applications. Many more studies of this kind\nfollowed. For example, \\cite{Valverde:2005}, \\cite{Zheng:2008},\n\\cite{Fortuna:2011} and \\cite{Gherardi:2013} all study the evolution\nof software networks finding evidence of known mathematical principles\nthat also exist in natural systems, and that might serve as predictive\nmodels for software evolution.\n\nCloser to our work, a study presented in 2006 by Baxter et\nal.~\\cite{Baxter:2006} also targeted the ``Lego Hypothesis,'' as\ncoined by the authors. That study, which built on an earlier one by\nthe same group~\\cite{Potanin:2005}, searched for the existence of\npower laws and other mathematical functions in a collection of 56 Java\napplications using 17 OO metrics, such as number of methods per type\nand the number of dependencies per type. For each of those 56\napplications, the study revealed whether the 17 metrics' data points\ncould fit the mathematical functions of interest. The study found that\nvery few projects, and in only very few metrics, had strict power law\ndistributions; most projects, and in most metrics, revealed reasonable\nfits at 80\\% confidence interval with several of the functions that\nthey were searching for. Another study by Louridas et\nal.~\\cite{Louridas:2008} studied the existence of power laws in a\nvariety of applications written in a variety of languages.\n\nAll of these studies largely ignore {\\em application} size, and focus\non the modules themselves (i.e. classes, interfaces). In the study by\nBaxter et al.~\\cite{Baxter:2006}, the results are ordered by\napplication size, and even grouped within size ranges; but no insights\nare given regarding the effect, if any, that application size may have\non the observations. We believe our study is complementary to all of\nthese prior studies in search for mathematical laws in software\napplications, because it focuses on the size of the application as a\nwhole, not just on the size of each OO module. \n\n\n\\section{Dataset}\n\\label{sec:dataset}\n\nIn this study, we use the Sourcerer 2011\ndataset~~\\cite{Lopes+Ossher:2012}, which contains over 150,000+\nprojects collected from Google Code, SourceForge and Apache as of\n2011. The projects have been processed into a relational database of\nentities and relations, using the Sourcerer Tools publicly available\nfrom Github~\\cite{Sourcerer:2015}. The database facilitates static\nanalysis for very large collections of source code, as it contains\npreprocessed static analysis information that can be queried on\ndemand. By issuing specific queries on the database, we extracted the\nnecessary numbers into a Comma Separated Value (CSV) file, which was\nthen used to perform the statistical analysis described in this paper.\n\nThe database produced by the Sourcerer tools was, therefore, the basis\nof our study. We present a small example that illustrates the kinds of\nentities and relations that are found in the database. Consider the\nfollowing Java program:\n\n{\\footnotesize\n\\begin{verbatim}\npackage foo;\n\npublic class FooNumber {\n  private int x;\n  FooNumber(int _x) { x = _x; }\n  private void print() {\n    System.out.println(\"It is number \" + x)\n  }\n  public static void main(String[] args) {\n    new FooNumber(Integer.parseInt(args[0])).print();\n  }\n}\n\\end{verbatim}\n}\n\n\n\\begin{table}\n\\centering\n\\caption{Entities}\n\\label{tab:entities}\n{\\small\n\\begin{tabular}{|l|l|l|} \\hline\nEntity ID & FQN & Type\\\\ \\hline\n1 & foo & PACKAGE \\\\\n2 & foo.FooNumber & CLASS \\\\\n3 & foo.FooNumber.x & FIELD \\\\\n4 & foo.FooNumber.$<$init$>$ & CONSTRUCTOR \\\\\n5 & foo.FooNumber.print & METHOD \\\\\n6 & foo.FooNumber.main & METHOD \\\\ \n... & ... & ... \\\\\\hline\n\\end{tabular}\\\\\n\n\\caption{Relations}\n\\label{tab:relations}\n\\begin{tabular}{|l|c|l|} \\hline\nSource & Relation type & Target \\\\ \\hline\n1 & CONTAINS & 2 \\\\\n2 & CONTAINS & 3 \\\\\n2 & CONTAINS & 4 \\\\\n2 & CONTAINS & 5 \\\\\n2 & CONTAINS & 6 \\\\\n3 & HOLDS  & {\\em Integer\\_ID} \\\\\n4 & WRITES & 3 \\\\\n5 & READS  & 3 \\\\\n5 & CALLS  & {\\em println\\_ID} \\\\\n6 & INSTANTIATES & 4 \\\\\n6 & CALLS & 5 \\\\ \n... & ... & ... \\\\\\hline\n\\end{tabular}\n}\n\\end{table}\n\n\nThis program results in the entities and relations shown in\nTables~\\ref{tab:entities} and \\ref{tab:relations} (not all\nentities and relations are shown, for brevity sake).\nGiven this database schema, with these entities and relations tables,\nwe issued several queries in order to extract all the numbers we\nneeded. Here is one example query that extracts the number of methods\ndeclared in classes in each of the projects:\n\n{\\footnotesize\n\\begin{verbatim}\n-- Extract number of class methods per project\nSELECT p.project_id,IFNULL(COUNT(DISTINCT m.entity_id),0) \n  FROM e_methods AS m\n  INNER JOIN r_contains AS r ON m.entity_id = r.rhs_eid\n  INNER JOIN e_classes AS c ON c.entity_id = r.lhs_eid\n  RIGHT JOIN projects AS p ON p.project_id=m.project_id\n  GROUP BY p.project_id\n\\end{verbatim}\n}\n\nAlthough the complete dataset contains projects from Google Code,\nSourceforge and Apache, for this study, we restricted the analysis to\nthe projects from Google Code only. The main properties of the Google\nCode dataset are presented in\nTable~\\ref{tab:dataset}. Figure~\\ref{fig:projects-size} shows the size\nof the projects, from smallest to largest, as well as the histogram of\nproject sizes in the dataset.\n\n\\begin{table}\n\\centering\n\\caption{Main metrics of the Google Code dataset. }\n\\begin{tabular}{|l|r|}\\hline\n            & Google Code \\\\ \\hline\nProjects    & 30,914      \\\\ \\hline\nClasses     & 3,060,853   \\\\ \\hline\nInterfaces  & 274,745     \\\\ \\hline\nMethods     & 19,358,490  \\\\ \\hline\nSLOC        & 221,194,474 \\\\ \\hline\nMedian SLOC & 1,570       \\\\ \\hline\n\\end{tabular}\n\\label{tab:dataset}\n\\end{table}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.5in]{figures\/RProjectSize2.png}\n\\includegraphics[width=1.5in]{figures\/RHistogramSize2.png}\n\\caption{Left: Size of the projects in the Google Code dataset, in Source Lines\n  of Code (SLOC) when projects are ordered by increasing size. Right:\n  Histogram of the size of the projects, in log scale.}\n\\label{fig:projects-size}\n\\end{figure}\n\nThis study's granularity is a ``project.'' For the purposes of this\nstudy, a project is the collection of Java source code files that were\nfound in each Google Code Project Hosting's project pages. For\nexample, the project named {\\texttt 1cproject} was hosted at\nhttps:\/\/code.google.com\/p\/1cproject\/, and its source code was\navailable at\n\\\\ https:\/\/code.google.com\/p\/1cproject\/source\/browse\/\\\\ \nThe ``project,'' in this case, consists of all Java source files found\nunder source control in {\\em trunk}. When the project included jar\nfiles, those were considered potential dependencies, not part of the\nproject itself.\\footnote{This paragraph is written in the past tense,\n  because Google Code is slated to become unavailable soon.}\n\n\\vspace*{0.3cm}\n\\noindent\n{\\bf Availability of Data and Tools}\n\n\\noindent\nThe Sourcerer infrastructure and tools are available from\nGithub~\\cite{Sourcerer:2015}, and have been described before in our\nprior papers~\\cite{Ossher:2009,Bajracharya:2014}. Besides those two\nprior publications, a publicly available tutorial explains the\nprocessing pipeline of the Sourcerer tools with concrete\nexamples~\\cite{Lopes+Ossher:2012}. Additionally, the artifact\nassociated with this paper contains all the Sourcerer tools and a\nsmall sample repository of projects, meant to illustrate the\nprocessing pipeline by which raw source code is converted into a\nrelational database for static analysis, such as that in this\npaper. Note that only a small repository is included, because the full\nrepository is 433Gb; its processing into a relational database took\napproximately 3 weeks of computing on a 24-core, 128 Gb RAM server.\n\nResearchers wanting to reproduce this study, or wanting to study other\nfacets of this data, can start by downloading the artifact associated\nwith this paper, and running the Sourcerer tools installed in it on\nthe included sample repository; then, they can download the full\nrepository from our Web site~\\cite{Lopes+Ossher:2012} and run the\nSourcerer tools on it.\n\nHaving done all this processing ourselves, we are making the processed\ndatasets available to other researchers. The several representations\nof the Sourcerer 2011 dataset, including the full repository and the\ndatabase, are publicly available for download\nfrom our Web site~\\cite{Lopes+Ossher:2012}. Note that this dataset is immutable; it\nwas collected once in 2011, and we do not plan to collect later\nversions of the projects. The CSV file upon which statistical analysis\nof this study was done is included in the artifact.\n\n\n\\section{Statistical Analysis Methods}\n\nThis section explains the main statistical methods that were used in\nthis study.\n\n\\subsection{Linear vs. Log Scales}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.5in]{figures\/RHistGenericSkewed.png}\n\\includegraphics[width=1.5in]{figures\/RHistGenericLogNormal.png}\n\\caption{Histograms of log-normal data when plotted in {\\em linear}\n  scale (left) and {\\em log} scale (right).}\n\\label{fig:lognormal}\n\\end{figure}\n\nAs mentioned in Section \\ref{sec:relwork}, when dealing with large\necosystems of software artifacts, the data is expected to be highly\nskewed in almost every dimension. That is also the case in the Google\nCode data. Figure~\\ref{fig:lognormal} shows a generic illustration of\nskewness in the data: the left histogram shows that the vast majority\nof data points have small values of $X$, where $X$ is some measured\nfeature of the dataset; in transforming the data into log scale,\nhowever, we can see an almost perfect log-normal distribution (right\nhistogram). When this holds, it would be ill-suited to use normal\nstatistics in linear space, but we can proceed to apply normal\nstatistics in log space. This is a critical step in analyzing\nthese ecosystems.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.5in]{figures\/RScatterGenericLinear.png}\n\\includegraphics[width=1.5in]{figures\/RScatterGenericLog.png}\n\\caption{Scatterplots of Y against X when both X and Y are\n  log-normal data. On the left: plot in {\\em linear} scale of\n  X and Y; on the right: plot in {\\em log} scale of X and Y.}\n\\label{fig:lm}\n\\end{figure}\n\n\\subsection{Linear Regression Models}\n\nThe main statistical tool we use in this study is the linear\nregression model. Linear regression tries to find the best linear\nmodel (i.e. a line) that fits the data. Figure~\\ref{fig:lm}\nillustrates our use of this statistical tool. On the left, we see a\nscatterplot of some feature $X$ against some other feature $Y$ plotted\nin linear scale of both X and Y. The plot also shows the best fit line\nresulting from linear regression of the data. On the right, we see the\nscatterplot of the same features $X$ and $Y$ but plotted in log scale,\nalong with the best fit line. In both cases, the line is given as\n$y\\_values = \\alpha + \\beta x\\_values$. However, the plot on the right\nbeing in log scale, the straight line represents $log(y) = \\alpha + \\beta\nlog(x)$. Transforming this back to linear space gives the following\nnon-linear (exponential) relation between $X$ and $Y$:\n\n\\begin{equation} \n\\label{eq1}\ny=e^{\\alpha} x^{\\beta}\n\\end{equation}\n\nWhen the relation between two features is {\\em non-linear}\nand, specifically, exponential, some observations are at hand:\n\\begin{itemize}\n\\item When $\\beta=1$, the relation between $X$ and $Y$ degenerates to\n  linear.\n\\item Any value of $\\beta \\neq 1$ indicates an exponential relation\n  between the two features. Small variations in $\\beta$ represent\n  large variations of $Y$ against $X$ in linear space.\n\\item $\\beta > 1$ indicates a superlinear relation, i.e. $Y$ grows\n  exponentially faster as $X$ grows.\n\\item $\\beta < 1$ indicates a sublinear relation, i.e. $Y$ grows\n  exponentially slower as $X$ grows.\n\\end{itemize}\n\n\\subsection{Goodness of Fit}\n\nOne critical part of linear regression is the goodness of fit, that\nis, how well the line fits the data. $R^{2}$, prononced R-squared, is\na statistic that measures how successful the fit is in explaining the\nvariation of the data.\\footnote{$R^{2} = 1 - \\dfrac{SSE}{SST}$, where\n  $SSE$ is the residual sum of squares and $SST$ is the total sum of\n  squares.} For example, $R^{2}=0.92$ means that the fit explains 92\\%\nof the total variation in the data. A value of 1 would be the perfect\nfit.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.5in]{figures\/ResPlots.png}\n\\caption{Residuals plots. Top-left: Residuals vs Fitted. Top-right:\n  Normal QQ. Bottom-left: Scale-location (aka spread). Bottom-right:\n  Residuals vs Leverage. }\n\\label{fig:residuals}\n\\end{figure}\n\nHowever, due to how it is calculated, there are several limitations\nfor what $R^{2}$ can explain. Depending on the characteristics of the\ndata, $R^{2}$ can have a low prediction value. In order to verify\nthis, it is important to analyse the {\\em residuals} of the linear\nregression models. Figure~\\ref{fig:residuals} illustrates the kinds of\nresiduals plots that we analyze to check whether the linear models are\nappropriate or not. The {\\em Residuals vs Fitted} plot (top-left) is\nthe most important one. A good fit should result in this plot showing\nrandomly distributed data around the horizontal line at the origin,\nmeaning that what's left from the fit is unbiased noise -- this\nparticular plot shows that. When this doesn't happen, then the linear\nmodel may not be appropriate to explain the data, even if $R^{2}$ is\nhigh. The {\\em Normal QQ} plot (top-right) illustrates assumptions\nabout normality of the residuals in the model. When the dots all fall\nin the straight diagonal, then the residuals fit exactly a normal\ndistribution, which is the ideal case. This particular plot shows a\nsymmetrical light-tailed normal distribution of the residuals, which\nis acceptable. In general, some deviation from the norm is to be\nexpected, particularly near the ends. The {\\em Scale-Location} plot,\nalso known as {\\em spread}, illustrates the variance of the Y variable\nalong the X variable. A flat line means that the variance is constant\nalong X, which is the ideal case for linear regression. This particular\nplot shows that there is more variance for lower values of X, and then\nthe variance evens out. This kind of small deviation from the ideal is\nacceptable. Finally, {\\em Residuals vs Leverage} illustrates the {\\em\n  leverage} (influence) that the data points had on the fitness\nprocess. This plot serves to identify potential outliers that may have\nhad undue influence in the model. We want the points to fall as close\nas possible to the horizontal line at origin, and not to fall outside\nCook's distance. That is the case with this particular plot.\n\n\n\\subsection{Binned Analysis}\n\nWhen the residuals of the linear models show potential problems with\nthe model, that means that the simple linear regression models are\nmissing important characteristics of the data. In those cases, we try\nto perform binned analysis instead of analysis on the whole data. This\nanalysis is meaningful when the data in the bins shows normal\ndistributions. When that is the case, we compare the differences of\nmeans among the bins using Welch two sample t-test on a 95\\%\nconfidence interval in order to extract more meaningful insights.\n\n\\section{Findings}\n\\label{sec:results}\n\nThis section presents the main findings of our study. It starts with\nobservations regarding the size of the modules, then their complexity,\nthe use of inheritance, and finally the kinds of dependencies the\nmodules have. It should be noted that all linear models and\ncorrelations presented here are statistically significant, with\np-values $<<$ 0.0001.\n\n\n\\subsection{Module Size}\n\\label{sec:modulesize}\n\nRQ1: {\\em Are modules of larger systems larger, or smaller, than modules\n  of smaller systems, or are there no statistically significant\n  differences?}\n\nIn Java, the modular, replaceable units are classes and interfaces, so\nwe use the word {\\em module} to mean either a class or an\ninterface. The bivariate analyses used to study the above question\nare: SLOC vs. Modules, Methods vs. Classes and Constructors\nvs. Classes. We know from several previous studies that these pairs of\nmetrics are strongly positively correlated. But what exactly is the nature\nof these relations?\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3in]{figures\/RSLOCVsModules.png} \\\\\n\\vspace*{0.5cm}\n\\includegraphics[width=3in]{figures\/RMethodsVsClasses.png} \\\\\n\\vspace*{0.5cm}\n\\includegraphics[width=3in]{figures\/RConstructorsVsClasses.png}\n\\caption{Top: SLOC vs. number of modules. Middle: Number of methods\n  vs. number of classes. Bottom: Number of constructors vs. number\n  of classes.}\n\\label{fig:modulesize}\n\\end{figure}\n\nFigure~\\ref{fig:modulesize} sheds some light into this question. On the\ntop, the size of each project in SLOC is plotted against its number of\nclasses and interfaces. As expected there is a very strong linear\ncorrelation ($r=0.93$). Moreover, with 87\\% linear fitness (in log\nspace), it appears that as the number of modules grows, the lines of\nsource code also grows but at exponentially higher pace. Specifically,\nusing equation~\\ref{eq1},\n\n\\begin{center}\n$SLOC=e^{3.5549} Modules^{1.0939}$\n\\end{center}\n\nFor example, a project with 10 modules is predicted to have close to\n434 SLOC; a project with 100 modules is predicted to have not just\n4,340 but close to 5,391 SLOC, so considerably more than 10 times\nwhat's expected of a project with 10 modules; a project with 1,000\nmodules is predicted to have close to 66,923 SLOC, again considerably\nmore than 10 times what's expected of a project with 100 modules;\netc. The growth of SLOC is exponential with number of modules, and\neven though the exponent (1.0939) is close to 1, the small 0.0939\ndifference results in large differences in the linear space.\n\nWhere do all these extra lines of code go? The middle plot in\nFigure~\\ref{fig:modulesize} explains it. The plot shows the number of\nmethods declared in classes vs. the number of classes. Again, as\nexpected, there is a strong positive linear correlation\n($r=0.94$). Moreover, with 89\\% linear fitness, it appears that as the\nnumber of classes grows, the number of methods grows\nexponentially. Specifically, \n\n\\begin{center}\n$Methods=e^{1.0949} Classes^{1.1055}$\n\\end{center}\n\nFor example, a project with 10 classes is predicted to have close to\n38 methods; a project with 100 classes is predicted to have not just\n380 but close to 486 methods; a project with 1,000 classes is\npredicted to have close to 6,195 methods; etc. Again, the growth of\nthe number of methods is exponential, not linear. \n\nA similar exponential growth can be observed for constructors\nvs. classes (Figure~\\ref{fig:modulesize}, bottom). The relation in that\ncase is $Constructors = e^{0.0246} Classes^{1.0195}$. In this case,\n$\\beta$ is very close to 1, so this is almost a linear function.\n\nIn short: projects with {\\bf more modules} have disproportionately more\nlines of code than projects with less modules, which means that they\nhave {\\bf larger modules}. Moreover, the extra lines of code seem to\nbe grouped in disproportionately {\\bf more methods} and, to a lesser\ndegree, constructors, {\\bf per class}.\n\nTable~\\ref{tab:modulesize} summarizes the statistical principles\ninferred from the data related to the effect of size. $\\alpha$ and\n$\\beta$ are the coefficients for equation~\\ref{eq1}; $r$ is the\nPearson correlation coefficient of the data in log scale; $R^{2}$ is\nthe fitness of the line in log space. The residuals of the linear\nmodel can be found in Figures \\ref{fig:res_SLOC_Modules},\n\\ref{fig:res_Methods_Classes} and\n\\ref{fig:res_Constructors_Classes}. All of them show good strength of\nthe model.\n\n\\begin{table}\n\\centering\n\\caption{Analysis of project size.}\n{\\small\n\\begin{tabular}{|l|r|r|r|r|l|}\\hline\nAnalysis            & $\\alpha$ & $\\beta$& $r$  & $R^{2}$ & Space\\\\ \\hline\nSLOC vs. Modules    & 3.5549   & 1.0939 & 0.93 & 0.87 & log-log \\\\ \\hline\nMeths. vs. Classes  & 1.0949   & 1.1055 & 0.94 & 0.89 & log-log \\\\ \\hline\nConstrs. vs. Classes & 0.0246  & 1.0195 & 0.99 & 0.98 & log-log \\\\ \\hline\n\\end{tabular}\n}\n\\label{tab:modulesize}\n\\end{table}\n\n\nThese observations explain apparent inconsistencies in the literature\nover the past few years. As described in Section~\\ref{sec:relwork},\ndifferent studies of Java corpora have reported different average\nmethods per class. This could potentially be explained by our\nfindings: a corpus that is dominated by smaller projects will have\nlower average methods per class than a corpus dominated by larger\nprojects.\n\n\n\\subsection{Module Types}\n\\label{sec:moduletype}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3in]{figures\/RInterfacesVsClassesLog.png}\n\\caption{Interfaces vs. classes in log scale.}\n\\label{fig:moduletype}\n\\end{figure}\n\nRQ2: {\\em Is there a statistically significant variation in the mix of\n  classes and interfaces for projects of different scales?}\n\nThe answer to the question involves dealing with data that shows high\nvariance. We started by regressing the number of interfaces against\nthe number of classes in each project, similarly to what we did for\nthe previous question. Figure~\\ref{fig:moduletype} shows\nthe non-linear model:\n\n{\\em Interfaces} $= e^{-1.2064}Classes^{0.7035}$\n\n$R^2 =0.49$ is not too good of a fit. Visual inspection of the plot\nand the fitted straight line exposes weaknesses of this simple linear\nmodel, particularly at the edges of the data: for projects with very\nsmall and very large classes, the model underestimates the number of\ninterfaces. Clearly, the relation between the number of classes and the\nnumber of interfaces in this ecosystem is not properly explained by\na simple exponential function.\n\nThe observations from visual inspection are confirmed in the plots of\nthe residuals in Figure~\\ref{fig:res_Interfaces_Classes} (Appendix\nB). The {\\em Residuals vs Fitted} plot, in particular shows a bend in\nthe residual data, rather than a straight line. This is indicative\nthat the linear model is missing a non-linear component. The shape of\nthe bend suggests a parabola, so we add an additional transformation\nof the X variable, specifically $log(X)^2$, and perform a linear\nregression on that transformed space ($log(y) \\sim log(x)^2$). This\nadditional transformation introduces non-monotonicity (see Appendix \nFigure~\\ref{fig:log_squared}). Transforming this back to linear\nspace, we are establishing the relation:\n\n\\begin{equation} \n\\label{eq2}\ny=e^{\\alpha} x^{\\beta log(x)}\n\\end{equation}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3in]{figures\/RInterfacesVsClasses2.png}\n\\caption{Interfaces vs. classes in log$^2$ scale.}\n\\label{fig:moduletype2}\n\\end{figure}\n\nThe plot is shown in Figure~\\ref{fig:moduletype2} and the residuals\nplots are in Appendix B, Figure~\\ref{fig:res_Interfaces_Classes2}. As\ncan be seen, this is a better fit, with $R^2=0.52$. The bias in the\nfit that transpired with the bend in the residuals plot is now\npractically eliminated. Given the parameters, this model predicts \n\n{\\em Interfaces} $= e^{0.14} Classes^{0.083 log(Classes)}$\n\nNote that due to the non-monotonicity illustrated in\nFigure~\\ref{fig:log_squared}, this model establishes a variable ratio\nbetween classes and interfaces depending on project\nsize. Specifically, smaller projects have a much higher ratio\\\\\n$Interfaces\/Classes$. \\\\\nFor example, for a project with 10 classes, the\nmodel predicts 1.79 interfaces ($\\sim$ 18\\% ratio); 50 classes $\\succ$\n4.1 interfaces ($\\sim$ 8.2\\%); 100 classes $\\succ$ 6.69 interfaces\n($\\sim$ 7\\%); 1000 classes $\\succ$ 60.4 interfaces ($\\sim$\n6\\%). Acording to the model, the ratio proceeds to increase again for\nvery large projects. For example 10,000 classes $\\succ$ 1,314\ninterfaces ($\\sim$ 13\\%). Our dataset includes only 6 projects that\ncontain over 10,000 classes each, so the model may not be precise for\nthis end of the size spectrum.\n\nAnother way of analyzing this data is to make a binned analysis.  For\nthat, we divide the data into 5 bins on the number of classes: very\nlarge, large, medium, small and very small. We then compute the ratio\n$Interfaces\/Classes$ for all the projects, and compute the means of\nthe ratios in each bin. The results are shown in\nTable~\\ref{tab:bin_moduletype}.\n\n\\begin{table}\n\\centering\n\\caption{Bins for analysis of {\\em Interfaces\/Classes}. Mean\n  and SD values are in log scale.}  {\\small\n\\begin{tabular}{|l|l|r|r|r|}\\hline\n  Bin    & \\# Classes     & Projects & Mean (linear\\%) & SD   \\\\ \\hline\nV. Large & $>$ 5,000      & 17    & -2.47 (8.5) & 0.87 \\\\ \\hline\nLarge    & 1,000 -- 5,000 & 419   & -2.83 (5.9) & 1.00 \\\\ \\hline\nMedium   & 100 -- 1,000   & 5,762 & -2.77 (6.3) & 1.05 \\\\ \\hline\nSmall    & 20 -- 100      & 11,715& -2.49 (8.3) & 0.92 \\\\ \\hline\nV. Small & $<$ 20         & 11,557& -1.68 (18.6)& 0.78 \\\\ \\hline\n\\end{tabular}\n}\n\\label{tab:bin_moduletype}\n\\end{table}\n\nFinally, we perform a Welch two sample t-test on the differences of\nmeans to check whether the differences exist and are statistically\nsignificant. The tests show statistical significance (p $<<$ 0.0001)\non the mean differences between medium and small, and between small\nand very small. The other differences are not statistically\nsignificant at 95\\% confidance level.\n\nThe numerical values of the binned analysis are consistent with those\nfrom the linear regression model that places the number of interfaces\nas a continuous function of the number of classes as by\nequation~\\ref{eq2}. This adds strength to the result.\n\nOne possible explanation for why smaller projects have\ndisproportionately more interfaces is that they have an investment in\nmodeling entities with interfaces without having enough\nimplementations of those entities to pay off the investment. In\nmedium-to-large projects, that investment pays off, as more classes\nprovide alternative implementations of the interfaces. The higher\nratio in very large projects is not statistically significant, so no\nconclusions should be made on whether that holds in general or only in\nthis particular set of 17 very large projects.\n\n\n\\subsection{Internal Complexity}\n\\label{sec:modulecomplexity}\n\nRQ3: {\\em Do larger projects have more method calls or use more unsafe\n  operations than smaller projects, or are there no statistical\n  significant differences?}\n\nA recent study by Landman et al.~\\cite{Landman2014} showed that\nthere seems to be no correlation between the size of Java projects\n(measured in SLOC) and the cyclomatic complexity of their methods. We\ngo one step further to investigate other potential sources of\ncomplexity in code: the number of outgoing methods calls, the\nnumber of {\\texttt instanceof} statements and the number of unsafe\ntype casts in each project. \n\n\\begin{table}\n\\centering\n\\caption{Analysis of code complexity}\n{\\small\n\\begin{tabular}{|l|r|r|r|r|l|}\\hline\nAnalysis           & $\\alpha$ & $\\beta$& $r$  & $R^{2}$ & Space\\\\ \\hline\nCalls vs. Methods  & 1.64  & 1.00 & 0.94 & 0.89 & log-log\\\\ \\hline\nInst.of vs. Methods& -2.77 & 0.84 & 0.70 & 0.49 & log-log\\\\ \n                   & -0.41 & 0.01 & 0.72 & 0.52 & log-log$^{2}$\\\\ \\hline\nCasts. vs. Methods & -1.81 & 1.00 & 0.83 & 0.68 & log-log \\\\ \n                   & -0.49 & 0.36 & 0.84 & 0.70 & log-log$^{1.4}$ \\\\ \\hline\n\\end{tabular}\n}\n\\label{tab:modulecomplexity}\n\\end{table}\n\nLike for module types, some of the data here also has a considerable\nvariation. Table \\ref{tab:modulecomplexity} summarizes the results.\nWe found no evidence that larger projects use more unsafe features of\nJava than smaller projects, and we found some weak evidence that the\ncontrary may happen.\n\n\n\\subsubsection{Method Calls vs. Methods declared in classes}\n\\label{sec:callsvsmethods}\n\nIn the case of method calls, there is a fairly strong fit of the\nlinear model ($R^{2}=0.89$), and the residuals plots show no warning\nsigns (Appendix B Figure\\ref{fig:res_Calls_Methods}). The exponent\n$\\beta=0.9971$, however, is very close to 1, which means that the\nrelation is essentially linear at a rate of $e^{1.64}=5.1$ calls per\nmethod. For example, according to the model, a project with 50 methods\nhas 255 method calls; a project with 500 methods has 2,531 method\ncalls; a project with 5,000 methods has 25,144 method calls. \n\n\\subsubsection{Instanceof statements vs. Methods declared in classes}\n\\label{sec:instanceofvsmethods}\n\nIn the case of {\\texttt instanceof}, the linear model in log-log space\nis not that good ($R^{2}=0.49$), and the residuals plots show some\nwarning signs (Appendix B\nFigure~\\ref{fig:res_Instanceof_Methods}). Similarly to what was done\nfor the previous analysis, we transformed the X axis (methods) with an\nadditional square function, and the fit improved to $R^{2}=0.52$\n(residuals in Appendix B\nFigure~\\ref{fig:res_Instanceof_Methods2}). This yields the\nrelation\\\\ \n{\\em Instanceof} $= e^{-0.14} Methods^{0.0702 log(Methods)}$\n\nAgain, this function is not monotonic, and therefore results in a\nnon-monotonic average number of {\\texttt instanceof} statements per\nmethod, depending on the total number of methods of the projects: the\nratio starts high for projects with just a few methods (e.g. 0.21 for\nprojects with 5 methods) and decreases sharply for projects with very\nsmall number of methods ($< 20$); it then continues to decrease but\nmore gently, reaching a minimum of 0.025 {\\texttt instanceof}\nstatements per method for projects with around 1,000 methods\n(i.e. almost 10 times less than for projects with 5 methods); from\nthen on, it increases again, but slowly. Its predicted value is 0.06\n{\\texttt instanceof} statements per method for projects with 50,000\nmethods (of which there are 17 in the dataset).\n\n\n\\subsubsection{Type casts vs. Methods declared in classes}\n\\label{sec:castsvsmethods}\n\nIn the case of casts, the linear log-log model is also not that good\n($R^{2}=0.68$, see also Appendix B\nFigure~\\ref{fig:res_Casts_Methods}). We then tried a few\ntransformations, and found $log^{1.4}$ to produce very good residuals\nplots (Appendix B Figure~\\ref{fig:res_Casts_Methods2}) and a better\n$R^{2}=0.70$. This function has a similar behavior as the one\nexplained for {\\texttt instanceof} in terms of monotonicity, but the\nminimum (0.13 casts per method) happens a bit earlier, at around 500\nmethods. This value of casts per method is roughly 10 times less than\nthe value for projects with 5 methods.\n\n\\subsubsection{Discussion}\n\nCombined, and along with the study by Landman et al., these results\nshow that there is no evidence to support the hypothesis that larger\nprojects have more complex code. On the contrary, there seems to be a\ntrend for smaller projects to include proportionally more unsafe\nstatements of Java. \n\nThe linear model method calls vs. methods (Section\n\\ref{sec:callsvsmethods}) is a fairly strong fit that shows that the\nnumber of method calls per declared method is roughly constant and\nindependent of the size of the projects (measured in number of\nmethods). The other two models (Secitons \\ref{sec:callsvsmethods} and\n\\ref{sec:instanceofvsmethods}) have a less strong fit. That simply\nmeans that their precision as predictors is not too good, but the\ntrend showing proportionally more unsafe features of Java in small\nprojects (Section~\\ref{sec:castsvsmethods}) is interesting and\nstatistically significant. We conjecture that this may happen because\ndevelopers of non-trivial projects adhere to a stricter discipline of\navoiding these features.\n\n\\subsection{Class Composition via Inheritance}\n\\label{sec:inheritance}\n\nRQ4: {\\em Does the scale of the project affect the use of\n  inheritance?}\n\nThe two linear models used to answer the above question are classes\ndefined using inheritance (DUI) vs. total classes, and classes that\nare inherited from within the project (IF) vs. total classes of each\nproject. The results are shown in Table~\\ref{tab:inheritance} (plots\nin Appendix \\ref{fig:res_DUI_Classes},\n\\ref{fig:res_DUI_Classes2} \\ref{fig:res_IF_Classes} and\n\\ref{fig:res_IF_Classes2}.\n\n\\begin{table}\n\\centering\n\\caption{Analysis of inheritance}\n{\\small\n\\begin{tabular}{|l|r|r|r|r|l|}\\hline\nAnalysis         & $\\alpha$ & $\\beta$& $r$  & $R^{2}$ & Space \\\\ \\hline\n{\\footnotesize DUI vs. Classes}  & -1.0505  & 1.0159 & 0.92 & 0.85 & log-log \\\\\n                 & -0.5364  & 0.6626 & 0.93 & 0.86 & log-log$^{1.2}$ \\\\ \\hline\n{\\footnotesize IF vs. Classes}   & -1.9908  & 0.8037 & 0.78 & 0.61 & log-log \\\\ \n                 & -0.3414  & 0.0903 & 0.80 & 0.64 & log-log$^{2}$ \\\\ \\hline\n\\end{tabular}\n}\n\\label{tab:inheritance}\n\\end{table}\n\nAs in previous analysis, the residuals plots of the initial linear\nregression models showed some warning signs that the models might not\nbe the best (Appendix B Figures~\\ref{fig:res_DUI_Classes2} and\n\\ref{fig:res_IF_Classes2}). As such, we compensated for the bend in\nthe residual data by adding an additional non-linear components to the\nX axis (classes). \n\n\\subsubsection{Classes Defined Using Inheritance (DUI)}\n\nIn the case of DUI classes vs. classes, the better\nmodel is\\\\ \n{\\em DUI} $= e^{-0.5364+0.6626 log(Classes)^{1.2}}$\n\nAlso here, the curve of the ratio starts high, decreases sharply, then\ndecreases slowly up to a minimum, then increases again. In the case of\nthese parameters, the minimum is around 10 classes, with 35\\% of them\nDUI. According to this model, projects with 2 classes have on average\n0.9 of them defined using inheritance (45\\%); 10 classes $\\succ$\n3.5 (35\\%); 100 classes $\\succ$ 37 (37\\%); 1,000 classes $\\succ$ 493\n(49\\%); 5,000 classes $\\succ$ 3,379 (68\\%); etc.\n\nA project with 100 classes, 65\\% of them DUI, is far from the norm,\nbut if the number of classes is close to 5,000, then that percentage\nof DUI is close to the norm.\n\n\\subsubsection{Classes Inherited From (IF)}\n\nIn the case of IF classes, the better model is\\\\ \n{\\em IF} $= e^{-0.3414} Classes^{0.0903}$\n\nIn the case of these parameters, the minimum is around 100 classes,\nwith 5\\% of them DUI. According to this model, projects with 10\nclasses have on average 1.1 inherited from (11\\%); 100 classes\n$\\succ$ 4.8 (5\\%); 1,000 classes $\\succ$ 52.8 (5\\%); 5,000 classes\n$\\succ$ 3,379 (10\\%); etc.\n\n\n\\subsection{Dependencies}\n\\label{sec:dependencies}\n\nRQ5: {\\em Do larger projects use disproportionately more, or fewer,\n  modules than smaller projects? How does project efferent coupling\n  vary with size? Are there statistically significant differences in how\n  types from JDK\/internal\/external libraries are used in projects of\n  varying sizes?}\n\nTo answer these questions, we first look at the growth of the number of\ndistinct types used by projects vs. the growth of the number of\nmodules\\footnote{Again, we use the term modules = types = classes +\n  interfaces.} declared in the projects. In counting the number of\nmodules used, we count the number of {\\em distinct} modules, so\nmodules used multiple times in a project are counted only once. We\nthen look deeper into the origins of the used modules.\n\n\\subsubsection{Used Modules vs. Declared Modules}\n\nProjects use a variety of modules, some of them declared internally,\nothers provided by the JDK and others provided by external\nlibraries. Again, we know that the number of modules used in a project\nis highly correlated with the size of the project. We are interested\nin studying the underlying trend function, and whether it is linear or\nsuper-\/sub-linear. Project size here is measured by\nthe number of declared modules in it.\n\nIn analyzing the initial linear regression model in log-log space, it\nwas visible that it suffered from a small bend in the residuals (see\nAppendix B Figure~\\ref{fig:res_Used_Modules}.) We then compensated\nfor it by applying a polynomial of size $1.2$, which eliminated the\nbend, producing a better model. Table~\\ref{tab:moduleuse} summarizes the\nparameters.\n\n\\begin{table}\n\\centering\n\\caption{Analysis of used modules}\n{\\small\n\\begin{tabular}{|l|r|r|r|r|l|}\\hline\nAnalysis          & $\\alpha$ & $\\beta$& $r$  & $R^{2}$ & Space \\\\ \\hline\n{\\footnotesize Used vs. Declared} & 2.006   & 0.7357 & 0.93 & 0.87 & log-log \\\\\n                  & 2.335   & 0.4863 & 0.93 & 0.87 & log-log$^{1.2}$ \\\\ \\hline\n\\end{tabular}\n}\n\\label{tab:moduleuse}\n\\end{table}\n\nThe better model establishes the following relation between the number\nof used modules and the number of declared modules in a project:\n\n{\\em Used} $= e^{2.3353 + 0.4863 log(Modules)^{1.2}}$\n\nThese parameters define a sub-linear relation between the number of\nused modules and the number of declared modules in a project, meaning\nthat the number of distinct used modules increase disproportionately less than\nthe number of declared modules in projects. According to this model, \nprojects with 10 declared modules use on average 39 modules; 100\ndeclared $\\succ$ 216 used; 1,000 declared $\\succ$ 1,450 used; etc. The\nnumber of used modules grows slower than the number of declared modules.\n\nThis result is intriguing, as it was unclear what to expect. The\nresult makes sense when the addition of a dependency (external or\ninternal) is correlated with the addition of multiple modules internal\nto the project; the causal relation is unclear, and there may be\nunknown confounding factors behind this correlation. \n\nTheoretically, according to this model, there is a scale point at\nwhich the number of used modules is less than the number of declared\nmodules, which means that some declared modules would not be used,\njust declared. That point is around 50,000 declared modules. Our\ndataset does not contain any project that large, but we found 753\nprojects where the number of declared modules is larger than the\nnumber of used modules, so this situation is not rare. An analysis of\nthis set of projects shows that they are statistically larger than the\naverage of the whole dataset, and that it contains a disproportionate\nnumber of very large projects -- 17 out of the 59 projects with more\nthan 3,000 declared modules are in this subset of projects that have\nhigher number of declared modules than used modules. It is possible\nthat these cases correspond to utility frameworks.\n\nHowever, the opposite result, if it had been observed, could also be\nexplained. That is, one could imagine that the number of used modules\nwould grow faster than the number of declared modules. In this case,\nthe addition of a dependency (external or internal) would not be\ncorrelated with additional internal modules, and, instead, it would\nsimply correlate with the addition of methods in existing modules that\nuse that new entity. That is not the case in this ecosystem: more\nmethods seem to exist for defining additional functionality with\nexisting dependencies than new methods are added for using additional\ndependencies. (plots omitted for space reasons)\n\n\\subsubsection{Efferent Coupling vs. SLOC}\n\\label{sec:coupling}\n\nThe efferent coupling of an entire project is given by the number of\nexternal modules (classes+interfaces) that the project uses. Here we\nstudy its exact relation with project size given in SLOC. This\nanalysis targets the well-known correlation between efferent coupling\nmetrics and size of artifacts, in general. Table~\\ref{tab:coupling}\nsummarizes the parameters. (Plots are in\nAppendix B Figure~\\ref{fig:res_Coupling_SLOC})\n\n\\begin{table}\n\\centering\n\\caption{Analysis of efferent coupling of projects}\n{\\small\n\\begin{tabular}{|l|r|r|r|r|l|}\\hline\nAnalysis          & $\\alpha$ & $\\beta$& $r$  & $R^{2}$ & Space \\\\ \\hline\n{\\footnotesize Coupling vs. SLOC} & 0.1176   & 0.5641 & 0.91 & 0.82 & log-log \\\\ \\hline\n\\end{tabular}\n}\n\\label{tab:coupling}\n\\end{table}\n\nAccording to this model, the relation is sublinear, i.e. efferent\ncoupling grows disproportionately slower than SLOC. Also, here,\n``normality'' changes with scale, it's not a simple constant ratio.\n\n\\subsubsection{Provenance of Used Modules}\n\nIn order to find out whether there are differences in the origin of\ndependencies among projects of different sizes, we then looked at the\nprovenance of all classes and interfaces (i.e. modules) that are used\nin each project, and regressed them against size of the project, given\nby number of declared modules. Table~\\ref{tab:provenance} summarizes\nthe parameters. (All residuals plots can be found in Appendix B,\nFigures~\\ref{fig:res_Internal_Modules}, \\ref{fig:res_JDK_Modules} and\n\\ref{fig:res_External_Modules})\n\n\\begin{table}\n\\centering\n\\caption{Analysis of origin of dependencies (I)}\n{\\small\n\\begin{tabular}{|l|r|r|r|r|l|}\\hline\nAnalysis          & $\\alpha$ & $\\beta$& $r$  & $R^{2}$ & Space \\\\ \\hline\n{\\footnotesize Inter. vs. Modules}& -0.5040 & 1.0037 & 0.96 & 0.92 & log-log \\\\ \\hline\n{\\footnotesize JDK vs. Modules}   & 1.7405  & 0.5306 & 0.81 & 0.66 & log-log \\\\ \\hline\n{\\footnotesize Exter. vs. Modules}& 0.7168  & 0.7489 & 0.80 & 0.65 & log-log \\\\ \\hline\n\\end{tabular}\n}\n\\label{tab:provenance}\n\\end{table}\n\nIndeed, these parameters show that there are differences. As $\\beta$\nindicates, larger projects use disproportionately less modules from\nexternal sources ($\\beta=0.7489$) and even less from the JDK\n($\\beta=0.5306$) than smaller projects. They use slightly\ndisproportionately more internally-defined modules ($\\beta=1.0037$)\nthan smaller projects. \n\nThese numbers are highly driven by the previous result -- in general,\nthe number of used modules grows slower than the number of declared\nmodules. That blurs the true ratios of the origin of dependencies as\nprojects grow, so let us analyze the data in a different way. We can\ntake module use as the independent variable and module origin as the\ndependent variable. This helps us quantify the mix of dependency\nprovenance as a function of project size given by the number of total\nused modules (i.e. a slightly different size metric that is highly\ncorrelated with the number of declared modules). The results are shown\nin Table~\\ref{tab:dependencies}.\n\n\\begin{table}\n\\centering\n\\caption{Analysis of origin of dependencies (II)}\n{\\small\n\\begin{tabular}{|l|r|r|r|r|l|}\\hline\nAnalysis        & $\\alpha$ & $\\beta$& $r$  & $R^{2}$ & Space \\\\ \\hline\n{\\footnotesize Inter. vs. Total}& -2.882  & 1.282 & 0.93 & 0.87 & log-log \\\\ \n                & -2.821  & 1.275 & 0.93 & NA   & log-log {\\tiny (RLM)} \\\\ \\hline\n{\\footnotesize JDK vs. Total}   & 0.162   & 0.750 & 0.90 & 0.82 & log-log \\\\\n                & 0.153   & 0.756 & 0.90 & NA   & log-log {\\tiny (RLM)} \\\\ \\hline\n{\\footnotesize Exter. vs. Total}& -1.585  & 1.072 & 0.89 & 0.79 & log-log \\\\ \n                & -1.454  & 1.059 & 0.89 & NA   & log-log {\\tiny (RLM)} \\\\ \\hline\n\\end{tabular}\n}\n\\label{tab:dependencies}\n\\end{table}\n\nAn inspection of the residuals plots (Appendix B Figures\n\\ref{fig:res_Internal_Total}, \\ref{fig:res_JDK_Total} and\n\\ref{fig:res_External_Total}) suggested that the simple linear model\nmay suffer from the effect of outliers, particularly on the use of JDK\nentities. As such we performed a {\\em robust} linear regression model\n(RLM), which excludes outliers.\\footnote{Robust linear regression does\n  not report $R^{2}$.} The new residuals plots (Appendix B Figures\n\\ref{fig:res_Internal_Total2}, \\ref{fig:res_JDK_Total2} and\n\\ref{fig:res_External_Total2}) still suffer from some left-skewness,\nbut the {\\em Residuals vs. Fitted} plot shows an improvement. Even with\nRLM, the model may not be strong for the edges of the data, i.e. for\nextremelly small and extremelly large projects.\n\nThe results indicate that, as the number of total used modules grows,\nprojects use disproportionately much less of the JDK ($\\beta=0.7559$)\nand much more internal ($\\beta=1.2822$) modules. The growth in\nexternal dependencies is also disproportionately larger, but less so\nthan the use of internal modules ($\\beta=1.0589$).\n\nIn retrospect, this result makes sense: the reason why projects are\nlarger is that they define more classes and interfaces; those are\nlikely to be used internally. For large projects, and given that the\namount of types in the JDK is fixed, the relative importance of the\ntypes from the JDK decreases and the importance of internal types\nincreases. \n\nBut this result exposes an interesting characteristic of\nprogramming-in-the-large: larger projects use much more of their\ninternal, and potentially less stable, components. Smaller projects\nleverage the JDK.\n\n\\section{Sampling Biases}\n\nThe linear regression analysis in the previous section was performed\nover the entire datastet without excluding any of the projects. The\ndataset is heavily right-tailed, with a bias towards small projects,\nand with only a few very large projects. Since linear regression\nlearns the paramters $\\alpha$ and $\\beta$ from the data, the data that\nwe use influences the exact values of those parameters. As such, it\ncould very well be that the non-linear effects that have been reported\nin the previous section, given by $\\beta \\neq 1$, could be an artifact\nof the many small projects and the very few very large projects in the\ndataset forcing that non-linear behavior in order for the models to\nfit the data. If that were to happen, there might be a simpler linear\nmodel with $\\beta=1$ that could perfectly explain the data ``in the\nmiddle'' containing only projects above and below certain size\nthresholds. In other words, we could give up explaining what happens\nfor the many very small projects, because the variance in them\nis very large, and for very large projects, because there aren't that\nmany, and focus on finding simple models for projects in between.\n\nWe investigated this possibility by constructing alternative models\nwhere the parameters are learned from various subsets that exclude\nvery small and very large projects. We report the result on only one\nof the many bivariate analysis of the previous section, specifically\nMethods vs. Classes, which had $\\beta=1.1055$\n(Section~\\ref{sec:modulesize}). Table~\\ref{tab:alternativemodels}\nsummarizes the results. The column Subset in that table denotes the\nconditions for project inclusion in the set as a range on the number\nof classes in the project. The first row is the baseline model given\nin the previous section. All of these models result in good residuals\nwithout any warning signs.\n\n\\begin{table}\n\\centering\n\\caption{Alternative models for Methods vs. Classes}\n{\\small\n\\begin{tabular}{|l|l||r|r|r|r|}\\hline\nModel & Subset      & \\#Projects & $\\alpha$ & $\\beta$ & $R^{2}$  \\\\ \\hline\n1     & Baseline    & 30,914    & 1.095    & 1.106   & 0.89 \\\\ \\hline\n2     & [10--3,000] & 22,860    & 1.283    & 1.061   & 0.84 \\\\ \\hline\n3     & [20--3,000] & 18,239    & 1.335    & 1.051   & 0.82 \\\\ \\hline\n4     & [30--3,000] & 15,030    & 1.347    & 1.049   & 0.81 \\\\ \\hline\n5     & [50--1,000] & 10,576    & 1.350    & 1.047   & 0.73 \\\\ \\hline\n6     & [100--500]  & 5,167     & 1.232    & 1.068   & 0.52 \\\\ \\hline\n7     & [10--100]   & 16,712    & 1.222    & 1.081   & 0.63 \\\\ \\hline\n8     & [1,000--3,000] & 386    & 0.168    & 1.218   & 0.36 \\\\ \\hline\n\\end{tabular}\n}\n\\label{tab:alternativemodels}\n\\end{table}\n\nAs shown in the table, all of the alternative models are still\nnon-linear, with $\\beta \\neq 1$, but, with the exception of model 8,\nthe value is lower than the baseline model. Model 5, which excludes\nalmost $2\/3$ of the dataset and includes the projects in the middle of\nthe dataset, has the lowest $\\beta$. But even in that subset, the\nnon-linear relation between classes and methods can be\nobserved. According to the parameters of model 5, a project with 50\nclasses is predicted to have 232 methods, and a project with 500\nclasses is predicted to have, not 2,320, but 2,583 methods. Any\nconcerns that the non-linear relation between methods and classes was\nan artifact of sampling bias are put to rest with the results shown in\nTable~\\ref{tab:alternativemodels}.\n\nGiven that there can be an unlimited number of models inferred from any\narbitrary subset of the original dataset, the question arises of which\nmodel to use.  \n\nIf the goal is to make predictions based on the models, the accuracy\nof each model can be tested on test datasets that aren't part of the\ndata from which the models are learned.\nWe exemplify such prediction goals by measuring the Normalized Root\nMean Square Error (NRMSE) of each model on the two extremes of the whole dataset\nthat have been eliminated from the learning part: the very small\nprojects ($\\#Classes < 10$) and the very large projects ($\\#Classes >\n3,000$); note in Table~\\ref{tab:alternativemodels} that those\nprojects are not part of any subset. NRMSE is given by\n\n\\begin{equation}\n\\label{eq:MSRE}\n\\begin{split}\nRMSE = \\sqrt{\\frac{\\sum\\limits_{t=1}^n{(\\text{\\^y} - y)^{2}}}{n}}\\\\\nNRMSE = \\frac{RMSE}{y_{max} - y_{min}}\n\\end{split}\n\\end{equation}\n\n\nThe summary of this accuracy analysis can be seen in\nTable~\\ref{tab:accuracy}. For comparison, we also show the NRMSE of\neach model on the entire dataset. Numbers in bold represent the models\nthat performed the best.\n\n\\begin{table}\n\\centering\n\\caption{Accuracy of the models, measured in NRMSE}\n{\\small\n\\begin{tabular}{|l||r|r|r|}\\hline\nModel & V.Small          & V.Large       & All               \\\\\n      & (5,472 projects) & (50 projects) & (30,914 projects) \\\\ \\hline\n1     & 0.13467          & 0.1500        & {\\em {\\bf 0.05110}}  \\\\ \\hline\n2     & 0.13109          & 0.1229        & 0.05155 \\\\ \\hline\n3     & 0.13083          & 0.1210        & 0.05181 \\\\ \\hline\n4     & 0.13081          & 0.1208        & 0.05188 \\\\ \\hline\n5     & {\\bf 0.13078}    & {\\bf 0.1204}  & 0.05189  \\\\ \\hline\n6     & 0.13171          & 0.1230        & 0.05136  \\\\ \\hline\n7     & 0.13184          & 0.1354        & {\\bf 0.05135} \\\\ \\hline\n8     & 0.21137          & 0.1530        & 0.05700  \\\\ \\hline\n\\end{tabular}\n}\n\\label{tab:accuracy}\n\\end{table}\n\nAs expected, the model that performs the best for the entire dataset\nis model 1, whose parameters were inferred from that same data. This\ncase doesn't serve to validate the model, it just confirms what was\nexpected. Excluding that baseline, the model that performs the second\nbest on the entire dataset is model 7, which contains many small\nprojects. The real validation comes only on the performance of the\nmodels on the two test sets containing very small and very large\nprojects, which weren't contained in the learning data. In both cases,\nthe model that makes the best predictions is model 5, whose parameters\nare inferred from a large portion of small\/medium\/large size projects.\n\nGiven these results, model 5, which learns the parameters ignoring the\nedges of the data, should be used instead of the baseline model\n1. Similar accuracy analysis should be done for all the other\nbivariate analysis. It is likely that the best models are always the\nones that learn the parameters ignoring the projects at the edges,\nwhere there is either more variance or uncertainty. Nevertheless, the\nmost important take away from this section is that {\\bf the\n  non-linearities exist in the data, independent of which \n  subsets we choose.}\n\n\\section{Implications for Software Metrics}\n\\label{sec:discussion}\n\nOur study was centered around a very simple question: {\\em does the scale of\nthe software system affect the internal structure of its modules or\nare modules scale-invariant?}\n\nFor the Java ecosystem, the answer is: yes, the scale of the\nsystem affects several aspects of the internal structure of its\nmodules, and of the way the modules are put together. Among those, the\nnumber of methods per class, the number of LOCs per module, the use of\ninheritance and mix of dependencies stand out. Going back to the LEGO\nmetaphor, it is as if large Java projects have injected stronger\ncoupling material and more hooks into the [larger] software bricks.\nThese findings have profound implications for software research,\nespecially quantitative studies of software artifacts. We\ndiscuss them here.\n\nAs mentioned in Section \\ref{sec:confusingeffect}, size has been the\nsource of much confusion in software studies. As noted several times\nin the literature, many software metrics -- for\nexample, Weighted Methods per Class (WMC) and (efferent and afferent)\nCoupling, just to mention two -- are correlated with size, so their\nstatistical power is very weak when size metrics are\navailable. We explain how to properly normalize for size with one\nexample metric: WMC.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3in]{figures\/RWMCVsClasses.png}\n\\caption{Correlation: how WMC grows with the number of classes.}\n\\label{fig:wmc_classes}\n\\end{figure}\n\nFigure~\\ref{fig:wmc_classes} shows the regression of WMC\nvs. Classes in our dataset, a confirmation of what we already know\nabout the existence of these correlations. In our data, the Person\ncorrelation (in log space) is $r=0.3$, so moderately strong.\n\n\\subsection{Linear or Log?}\n\nA first approach to normalizing the number of methods controlling for\nsize of the project is to make a simple average $WMC = Methods \/\nClasses$. This is, in fact, how this metric is defined in the\nliterature~\\cite{Chidamber:1991}, assuming uniform complexity of 1 (an\nassumption made in several prior studies). This gives us a number\nthat, in principle, can be used to compare projects independent of\ntheir size. If we have two projects, one with $WMC=3$ and the other\nwith $WMC=8$, that tells us that these two projects are considerably\ndifferent without needing to know any size metric. \n\nIn software ecosystems, a mean of $WMC$ can be calculated for entire\ncollections of projects by computing the $WMC$ of all the projects in\ncollection, and then computing the mean of those values. In our\ndataset $mean(WMC)=5.15$, which might lead us to conclude that in this\nvery large Java ecosystem, the average WMC is 5.15.\n\nThis value, however, is very misleading, because the distribution of\n$WMC$ in the dataset is not normal, but\nlog-normal. Figure~\\ref{fig:wmc} in Appendix shows the $WMC$ distribution in\nlinear and log scales.\n\nGiven this knowledge, a second approach to normalizing for size is to\nfind the mean and SD of $WMC$ in log scale. In our dataset that is\n$mean(WCM)_{log}=1.455$ and $SD(WMC)_{log}=0.63$. This translates to\nlinear space as $4.28$, with 68\\% of values falling within the\ninterval $[2.28 - 8.00]$, skewed towards the lower end of the\ninterval.\n\nThe first thing to notice is that these two numbers, $mean(WMC)$ and\n$mean(WMC)_{log}$ are different, the former being larger than the\nlatter. That happens because the data is highly right-skewed,\ni.e. there are many more smaller values than larger ones. Therefore\nthe simple mean in linear scale does not capture an important aspect\nof the data, its skewness; the mean and SD in log scale do. Another way of\nlooking at this is that when drawing a data point randomly out of this\ndataset, the odds are higher around $4.28$ than around $5.15$.\n\nEven though this is basic statistics, many papers continue to report\nsummary statistics in linear scale when the data is not normally\ndistributed in that scale. In general, we must inspect what kind of\ndistribution our data has and report summary statistics accordingly,\nor the reports will be misleading.\n\n\\subsection{Non-Linearity}\n\nThe above analysis is still missing something important about the\ndata, namely the findings unveiled by this paper that the number of\nmethods in a project grows {\\em disproportionately} faster with the\nnumber of classes. Therefore {\\em normality} takes a different value\ndepending on the scale of the project. We might conclude that a\nproject with $WMC=7.9$ ($WMC_{log} = 2.067$), which is on the edge of\nthe SD interval, might need special attention, and that a project with\n$WMC=4.3$ would be perfectly ``normal.'' That may or may not be the\ncase, depending on the size of that project. In\nSection~\\ref{sec:modulesize} we found a strong non-linear model given\nby:\n\n$Methods = e^{\\alpha} Classes^{\\beta}$\n\nThis equation gives us the {\\em norm} of what to expect of $WMC$ in\nprojects of varying sizes in this dataset. For a project with 10,000\nclasses, the expectation of the model is that it will have 70,000+\nmethods, not 42,800 as a simple linear model would predict (i.e. 4.28\n* 10,000); so $WMC=7.9$ is what we would expect for a project of this\nsize. A project with 10,000 classes that shows $WMC=4.3$ would be an\noddity in this ecosystem. If, however, the project has only 25\nclasses, then $WMC=4.3$ would be expected, but $WMC=7.9$ would be\nsurprising, in the sense that it is a large deviation from what is\nexpected of projects of that size. \n\nTherefore, the proper normalization for size must take this non-linear\nrelation into account, producing an adjusted ratio that is {\\em\n  truly} independent of the number of classes:\n\n\\begin{equation}\n\\label{eq4}\nWMC_{\\beta} = \\frac{Methods}{ Classes^{\\beta}}\n\\end{equation}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3in]{figures\/RWMCbetaVsClasses.png}\n\\caption{Normalization: how $WMC_{\\beta}$ grows with the number of classes.}\n\\label{fig:wmcbeta_classes}\n\\end{figure}\n\nFigure~\\ref{fig:wmcbeta_classes} shows how $WMC_{\\beta}$ and size are\n{\\bf not correlated}, using $\\beta=1.1055$, and the parameters from model\n1 in the previous section. Pearson correlation between the two\nvariables is $r << 0.001$, and Spearman correlation is $R=-0.04$.\n\nThe parameter $\\beta$ has elluded measurement, because it can only be\nobserved on sufficiently large collections of programs written in the\nsame language and that, collectively and empirically, define what is\nto be expected of programs written in that language. We now have the\nmeans to measure it, as shown in this paper. Therefore, we now have\nthe knowledge to create updated versions of well-known software\nmetrics that are truly independent of size and that may (or may not)\ncarry additional important information about the code that is not\nalready captured by size metrics. If, for example, high coupling\nreally is ``bad'', we now have the mathematical knowledge to measure\nthe size-independent essence of coupling. We plan to investigate the\nstatistical power of this seamingly small, but critical, adjustment in\nfuture work.\n\n\n\\section{Conclusion}\n\nWe have described a quantitative study designed to answer the\nquestion: does the scale of a software system affect the internal\nstructure of its modules? We have made an important step\ninto answering this question by performing a statistical analysis of a\nvery large and varied collection of Java projects. The statistical\nsignificant results in this dataset are strong: there are,\nindeed, superlinear effects on some aspects of the modules' internal\nstructure and composition with other modules. This reinforces the\nwidely accepted idea that programming-in-the-large carries with it\ndifferent concerns that aren't as strongly present for\nprogramming-in-the-small. More importantly, it has tremendous\nconsequences for software metrics in general. Many of the metrics\nproposed in the literature, and that are used widely in IDEs, have\nsuffered from poor information content for prediction models because\nthey correlate with the much simpler size metrics. Our paper shows how\nthis can be corrected.\n\n\n\\acks\n\nThis work was supported by National Science Foundation\ngrants nos. CCF-0725370 and CCF-1018374, and by the DARPA MUSE\nprogram. We would like to thank Pedro Martins for his assistance in\nthe production of the artifact, and the anonymous reviewers, who made\nthis a better paper.\n\n\n\n\\bibliographystyle{abbrvnat}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Technical lemmas and proofs}\n\\label{sec:proofs1}\n\\begin{lemma}\n\t\\label{lemma:BP}\nLet $B$ be a Bernoulli random variable, $T_0$ a nonnegative random variable and let $T=T_0$ if $B=1$ and $T=\\infty$ if $B=0$. Let $X$ and $Z$ be two real-valued random vectors. Then\n\\[\nT_0\\perp (C,X)\\mid Z\\quad \\text{ and } B\\perp (C,T_0,Z)\\mid X\\quad\\Longrightarrow\\quad T\\perp C\\mid (X,Z)\n\\]\n\\end{lemma}\n\\begin{proof}\n\nThis lemma is similar to Lemma 8.1 in \\cite{BP}. We provide the proof for completeness. By elementary properties of conditional independence we have\n\\[\nB\\perp (C,T_0,Z)\\mid X \\quad\\Longleftrightarrow \\quad B\\perp C\\mid (X,Z,T_0)\\quad\\text{and}\\quad B\\perp T_0\\mid (X,Z)\\quad\\text{and}\\quad B\\perp Z\\mid X\n\\]\nand\n\\[\nT_0\\perp (C,X)\\mid Z \\quad\\Longleftrightarrow \\quad T_0\\perp C\\mid (X,Z)\\quad\\text{and}\\quad T_0\\perp X\\mid Z.\n\\]\nThen,\n\\[\nC\\perp B\\mid (X,Z,T_0) \\quad\\text{and}\\quad C\\perp T_0\\mid (X,Z)\\quad\\Longleftrightarrow \\quad (B,T_0)\\perp C\\mid (X,Z)\n\\]\nThe result follows from the fact that $T$ is completely determined by $B$ and $T_0$.\n\\end{proof}\n\n\n\n\n\n\n\\subsection{Identifiability with restricted survival times}\n\nFor any $0<\\tau^*\\leq \\tau_0$, let\n$$\nT_0^*= \\min (T_0, \\tau^*), \\qquad  T^* = B T_0^*+ (1-B) \\infty \\qquad \\text{and} \\qquad  \nC^* =  \\min (C, \\tau^*).\n$$\nMoreover, let \n$$\nY^* = \\min (T^*, C^*)\\quad \\text{ and }\n\\Delta^* = \\mathds{1}_{\\{T^*\\leq C^*\\}}.\n$$ \n\n\nA first aspect to study is the identifiability of the true values of the parameter when $(Y,\\Delta)$ is replaced by $(Y^*,\\Delta^*)$. Here, identifiability means that the true values $\\beta_0$ and $\\Lambda_0$ of the parameters maximize the expectation of the criterion maximized to obtain the estimators. This issue is addressed in Lemma \\ref{theo:identif}. Let us introduce some additional notation: for any $0<\\tau^* \\leq  \\tau_0$ and $\\Lambda\\in\\mathcal H$, $\\Lambda_{|\\tau^*}$ is defined as \n\\begin{equation}\\label{def:lambda_trunc}\n\\Lambda_{|\\tau^*}(t) = \\Lambda (t), \\;\\;\\forall t\\in [0,\\tau^*) \\quad \\text{ and } \\quad \\Delta\\Lambda_{|\\tau^*}(\\tau^*)=\\Lambda_{|\\tau^*}(\\{\\tau^*\\})=1.\n\\end{equation} \nThe dominating measure for the model of $T_0$ changes with such a stopped cumulative hazard measure to allow for a positive mass at $\\tau^*$. Then, $\\ell$ defined in \\eqref{def:hat_l_cox_g} becomes  \n\\begin{multline}\\label{el_star}\n\\ell  (y, d, x, z ; \\beta, \\Lambda_{|\\tau^*}, \\gamma) \n=   \\mathds{1}_{\\{y< \\tau^*\\}} \\left[d \\log f_u(y|z;\\beta,\\Lambda) \\right. \\\\\n\\left. + (1-d)\\log\\left\\{1-\\phi(\\gamma,x)+\\phi(\\gamma,x)S_u(  y |z;\\beta,\\Lambda)\\right\\} \\right]\\\\\n+   \\mathds{1}_{\\{y \\geq  \\tau^*\\}} \\left[ d \\log S_u(\\tau^*|z;\\beta,\\Lambda) + (1-d)\\log\\left\\{1-\\phi(\\gamma,x)\\right\\}\\right] .\n\\end{multline}\n\n\n\\begin{lemma}\n\\label{theo:identif}\nLet  $0<\\tau^* \\leq \\tau_0$.  \nAssume that for any $\\tilde \\beta\\in B$ and $\\tilde \\Lambda\\in\\mathcal H$,\n\\begin{equation}\\label{eq:ident_T0}\nS_u ( t | z; \\tilde \\beta , \\tilde \\Lambda_{|\\tau^*}) = S_u ( t | z;\\beta_0,\\Lambda_{0|\\tau^*}), \\; \\forall t \\in [0, \\tau^*) \\quad \\Longrightarrow \\quad \\tilde \\beta=\\beta_0 \\;\\text{ and } \\; \\tilde \\Lambda_{|\\tau^*} = \\Lambda_{0|\\tau^*}.\n\\end{equation}\n Then $(\\beta_0,\\Lambda_{0|\\tau^*})$ is  the unique solution of \n\\begin{equation}\n\\label{eq:ident_T0_b}\n\\max_{\\beta\\in B,\\Lambda\\in\\mathcal H }\\mathbb E \\left[ \\ell ( Y^*   ,\\Delta^*,X,Z;\\beta,\\Lambda_{|\\tau^*},\\gamma_0) \\right] .\n\\end{equation}\n\\end{lemma}\n{Condition \\eqref{eq:ident_T0} is a minimal requirement of identification of the true value of the parameters  in the model for the uncured subjects if the variable $T_0\\wedge C$ was observed and only the events in a subset of the support of $T_0$ are considered.}  In the Cox PH model \n\\eqref{eq:ident_T0} is guaranteed by the requirement that $Var(Z)$ has full rank.\n\n\n\\begin{proof}[Proof of Lemma \\ref{theo:identif}]\nFirst, let \n$$\nH_k([0,t] |x,z) = \\mathbb P (Y\\leq t, { \\Delta = k } | X=x,Z=z), \\quad k\\in\\{0,1\\}, \\quad t\\in[0,\\infty),\n$$\nand let $H_k(dt |x,z) $ be the associated conditional measures. These conditional measures characterize the distribution of $(Y,\\Delta)$ given $X=x$ and $Z=z$. By the model and independence assumptions, for any $t\\geq 0$,\n\\begin{equation}\\label{eq:inv1_0}\nH_1(dt | x, z) = \\phi(\\gamma_0,x) F_C([t,\\infty)|x,z) f_u(t|z;\\beta_0,\\Lambda_{0}) dt,\n\\end{equation}\nand\n\\begin{equation}\\label{eq:inv2_0}\n H_0(dt | x, z ) =  \\{1-\\phi(\\gamma_0,x)+ \\phi(\\gamma_0,x) S_u(t|z;\\beta_0,\\Lambda_{0}) \\}F_C(dt|x,z) .\n\\end{equation}\nFollowing an usual notation abuse, herein we\ntreat $dt$ not just as the length of a small interval but also as the name of the interval itself. Note that up to additive terms which do not depend on the parameters $\\beta,\\Lambda$,\n$$\n(y,d)\\mapsto d   \\log f_u(y |z ;\\beta_0,\\Lambda_0) \n+\n(1-d )\\log\\left\\{1-\\phi(\\gamma_0,x)+\\phi(\\gamma_0,x)S_u(y|z;\\beta_0,\\Lambda_0)\\right\\},\n$$\nis the conditional log-density of $(Y,\\Delta)$ given $X=x$ and $Z=z$. From this and Kullback information inequality one can deduce that the expectation of $\\ell$ defined in \\eqref{def:hat_l_cox_g} is maximized by $\\beta_0,\\Lambda_0$ and $\\gamma_0$. \n\nLet $0<\\tau^*\\leq \\tau_0$. Note that \n$$\n H_1([\\tau^*,\\tau_0] | x, z) = H_1([\\tau^*,\\infty) | x, z) = \\phi(\\gamma_0,x) \\int_{[\\tau^*,\\tau_0]}  F_C([t,\\infty)|x,z)f_u(t|z;\\beta_0,\\Lambda_{0}) dt,\n$$\nand\n\\begin{multline*}\nH_0([\\tau^*,\\infty) | x, z)  = \\phi(\\gamma_0,x) \\int_{[\\tau^*,\\tau_0]} S_u(t|z;\\beta_0,\\Lambda_{0})  F_C(dt|x,z) \\\\+\\{1- \\phi(\\gamma_0,x)\\}  F_C([\\tau^*,\\infty)|x,z) .\n\\end{multline*}\nMoreover, \n\\begin{multline*}\nd(x,z;\\tau^*) :=  \\phi(\\gamma_0,x) \\int_{[\\tau^*,\\tau_0]}  F_C([t,\\infty)|x,z)f_u(t|z;\\beta_0,\\Lambda_{0}) dt \\\\ +\\phi(\\gamma_0,x) \\int_{[\\tau^*,\\tau_0]} S_u(t|z;\\beta_0,\\Lambda_{0})  F_C(dt|x,z)  \\\\ = \\phi(\\gamma_0,x) F_C([\\tau^*,\\infty) |x,z) S_u(\\tau^*|z;\\beta_0,\\Lambda_{0})\\\\ = \\mathbb P (T_0 \\wedge C \\geq \\tau^*, B=1).\n\\end{multline*}\nIn the limit case of no cure, $d(x,z;\\tau^*) =  H_1([\\tau^*,\\infty) | x, z)+ H_0([\\tau^*,\\infty) | x, z)$. \nBy construction we have \n$\nY^* =   \\min (Y, \\tau^*),\n$\nand\n$$\n\\mathbb P (Y^*=\\tau^*, \\Delta^* =1 | X=x, Z=z) = \nd(x,z;\\tau^*).\n$$\n\n\nNext, let \n$$\nH_k^*([0,t] |x,z) = \\mathbb P (Y^*\\leq t, { \\Delta^* = k } | X=x,Z=z), \\quad k\\in\\{0,1\\}, \\quad t\\in[0,\\infty),\n$$\nand let $H_k^*(dt |x,z) $ be the associated conditional measures. \nThis means\nfor any $t\\in[0,\\tau^*)  $, \n\\begin{equation*\nH^*_1(dt | x, z) = H_1(dt | x, z) \\quad \\text{and} \\quad H^*_0(dt | x, z) = H_0(dt | x, z) .\n\\end{equation*}\nMoreover, \n$$\nH_1^*(\\{\\tau^*\\} | x, z) = H_1^*([\\tau^*,\\infty) | x, z) = d(x,z;\\tau^*),\n$$\nand\n$$\nH_0^*( \\{\\tau^*\\} | x, z) =H_0^*([\\tau^*,\\tau_0] | x, z)  =  \\{1- \\phi(\\gamma_0,x)\\}  F_C([\\tau^*,\\infty)|x,z).\n$$\n\nNow, according to  the inversion formulae of \\cite{PK}, without any reference to a model, one can solve the set of equations \n\\begin{eqnarray}\\label{sq2}\nH^*_1(dt | x, z) &=& \\phi^*(x,z) F^*_C([t,\\infty)|x,z) F^*_u(dt|x,z),\\notag \\\\\n\\\\\nH_0^*(dt | x, z ) &=& \\{1-\\phi^*(x,z)+ \\phi^*(x,z) S^*_u(t|x,z) \\}F^*_C(dt|x,z) ,\\notag\n\\end{eqnarray}\nwhere $F_u^* = 1-S_u^*$. Solving \\eqref{sq2} for $F^*_C$, $S^*_u$ and $\\phi^*$, the functional $S^*_u$ is a proper survival function which puts mass only on sets where  $H_1^*$ does. Note that solving the similar system with  $H_1,H_0$ instead of $H_1^*, H_0^*$, one gets the true $F_C$, $S_u$ and $\\phi$. \nIf $\\Lambda_C^*$ denotes the cumulative hazard function associated to the solution $F^*_C$, then \n$$\n\\Lambda_C^*(dt | x,z) = \\frac{H_0^*(dt | x, z )}{H^*_1((t,\\infty)| x, z)+ H_0^*([t,\\infty) | x, z )},\\quad t\\geq 0,\n$$\nand thus, by construction, we have \n$ F_C(dt|x,z) = F^*_C(dt|x,z)$  on  $[0,\\tau^*)$, for any $x,z$. Then, by \\eqref{eq:inv2_0} and the second equation in \\eqref{sq2} we deduce \n$$\n\\phi^*(x,z) F^*_u(t|x,z) = \\phi(\\gamma_0,x)F_u(t|z;\\beta_0,\\Lambda_{0}),  \\quad \\forall t\\in [0,\\tau^*), \\forall x,z.\n$$ \nNext, taking into account that $S^*_u(t|x,z)= 0$, $\\forall t\\geq \\tau^*$, $\\forall x,z$, and integrating the second equation \\eqref{sq2} on $[\\tau^*,\\infty)$, \nwe obtain\n\\begin{multline*}\n\\{1-\\phi^*(x,z)\\}F^*_C([\\tau^*,\\infty)|x,z) = H_0^*( \\{\\tau^*\\} | x, z ) = \\{1-\\phi(x,z)\\}F_C([\\tau^*,\\infty)|x,z) .\n\\end{multline*}\nSince $F^*_C([0,\\tau^*)|x,z) = F_C([0,\\tau^*)|x,z)$, we deduce that $\\phi^*(x,z) =  \\phi(\\gamma_0,x)$\nand thus \n\\begin{equation}\\label{za1}\nF^*_u(t|x,z) = F_u(t|z;\\beta_0,\\Lambda_{0})= F_u(t|z;\\beta_0,\\Lambda_{0|\\tau^*}),  \\quad \\forall t\\in [0,\\tau^*), \\forall x,z.\n\\end{equation}\nThe second equality in the last display is by the construction of the survival function from the cumulative hazard function: only the values of $\\Lambda_0$ on $[0,t]$ contribute to obtain $F_u(t|z;\\beta_0,\\Lambda_{0})$.\nSince the inversion formula necessarily yields $F^*_u([0,\\tau^*]|x,z) \\equiv 1$, we deduce \n\\begin{equation}\\label{za2}\nF^*_u(\\{\\tau^* \\}|x,z)=S_u(\\tau^*|z;\\beta_0,\\Lambda_{0})=S_u(\\tau^*|z;\\beta_0,\\Lambda_{0|\\tau^*}).\n\\end{equation} \n\nFinally, we can write \n\\begin{multline*}\nE \\left[ \\ell ( Y^*  ,\\Delta^*,X,Z;\\beta,\\Lambda_{|\\tau^*},\\gamma_0) \\right] \\\\\n= \\iiint \n\\log \\ell ( t  ,1,x,z;\\beta,\\Lambda_{|\\tau^*},\\gamma_0) \nH^*_1(dt | x, z) G(dx, dz)\n\\\\ + \\iiint  \\log \\ell ( t  ,0,x,z;\\beta,\\Lambda_{|\\tau^*},\\gamma_0)  H_0^*(dt | x, z)  G(dx, dz).\n\\end{multline*}\nTo obtain the identifiability result it remains to apply Kullback information inequality. More precisely, it suffices to {notice} that here, up to additive terms which do not depend on the parameters, {$\\ell$ defined} in \\eqref{el_star} considered with $\\beta_0,\\Lambda_{0|\\tau^*}$ corresponds to the {log-density of  the conditional law of $(Y^*,\\Delta^*)$ given $X=x$ and $Z=z$}. (Note that the dominated measure changed as we introduce jumps at $\\tau^*$.) This follows from \\eqref{za1} and \\eqref{za2}. Thus $\\beta_0,\\Lambda_{0|\\tau^*}$ is solution of the problem \\eqref{eq:ident_T0_b}. The unicity of the solution is guaranteed by \\eqref{eq:ident_T0}.\n\\end{proof}\n\n\\subsection{Consistency}\n\n\\begin{proof}[\\textsc{Proof of Theorem~\\ref{theo:consistency}.}]\n\tWe follow the idea of \\cite{scharfstein}. Since we are interested in almost sure convergence, we work with fixed realizations of the data, $\\omega$ that will lie in a set of probability one. Let $\\Omega$ be the abstract probability space where the random vector $(B,T_0,C,X,Z)$ is defined (for example we can take $\\Omega=\\{0,1\\}\\times[0,\\tau_0]\\times[0,\\tau]\\times\\mathcal{X}\\times\\mathcal{Z}$ and $(B,T_0,C,X,Z)(\\omega)=\\omega$. Let $N\\subset\\Omega$ be a set of probability one $\\mathbb{P}(N)=1$ and fix $\\omega\\in N$. We will show that each subsequence $\\hat\\gamma_{n_k}$ has a subsequence that converges to $\\gamma_0$. As a  bounded sequence in $\\mathbb{R}^p$, $\\hat\\gamma_{n_k}$ has a convergent subsequence $\\hat\\gamma_{m_k}\\to\\gamma^*$. It suffices to show that $\\gamma^*=\\gamma_0$. Since $\\hat\\gamma_{m_k}$ maximizes $\\log \\hat{L}_{m_k,1}$, we have\n\t\\begin{equation}\n\t\\label{eqn:consistency_1}\n\t\\begin{split}\n\t0&\\leq \\frac{1}{m_k}\\log \\hat{L}_{m_k,1}(\\hat\\gamma_{m_k})-\\frac{1}{m_k}\\log \\hat{L}_{m_k,1}(\\gamma_0)\\\\\n\t&=\\frac{1}{m_k}\\sum_{i=1}^{m_k}\\left[\\left\\{1-\\hat\\pi(X_i)\\right\\}\\log\\frac{\\phi(\\hat\\gamma_{m_k},X_i)}{\\phi(\\gamma_0,X_i)}+\\hat\\pi(X_i)\\log\\frac{1-\\phi(\\hat\\gamma_{m_k},X_i)}{1-\\phi(\\gamma_0,X_i)}\\right]\\\\\n\t&=\\frac{1}{m_k}\\sum_{i=1}^{m_k}\\left[\\left\\{1-\\pi_0(X_i)\\right\\}\\log\\frac{\\phi(\\gamma^*,X_i)}{\\phi(\\gamma_0,X_i)}+\\pi_0(X_i)\\log\\frac{1-\\phi(\\gamma^*,X_i)}{1-\\phi(\\gamma_0,X_i)}\\right]+o(1)\n\t\\end{split}\n\t\\end{equation}\n\tif $N\\subset\\{\\omega: \\sup_x\\left|\\hat\\pi(x)-\\pi_0(x)\\right|\\to 0\\}$. Note that the remainder term $o(1)$ in the previous display depends on $\\omega$ and converges to zero as $\\hat\\pi$ converges to $\\pi_0$. Next we will show that, for an appropriate choice of $N$,  the first term converges to \n\t\\begin{equation}\n\t\\label{eqn:expectation_consistency}\n\t\\mathbb{E}\\left[\\left\\{1-\\pi_0(X)\\right\\}\\log\\frac{\\phi(\\gamma^*,X)}{\\phi(\\gamma_0,X)}+\\pi_0(x)\\log\\frac{1-\\phi(\\gamma^*,X)}{1-\\phi(\\gamma_0,X)}\\right]\n\t\\end{equation}\n\twhere the expectation is taken with respect to $X$ and $\\gamma^*\\in\\mathbb{R}^p$ (for a fixed $\\omega$). Since here we are dealing with a simple parametric model, this convergence follows easily from the uniform law of large numbers. However, we follow a longer argument to explain the idea that will be used also in the proof of Theorem~\\ref{theo:consistency2} (where the model is semiparametric). It is obvious, by the law of large numbers, that \n\t\\[\n\t\\begin{split}\n\t&\\frac{1}{m_k}\\sum_{i=1}^{m_k}\\left[\\left\\{1-\\pi_0(X_i)\\right\\}\\log\\phi(\\gamma_0,X_i)+\\pi_0(X_i)\\log\\left(1-\\phi(\\gamma_0,X_i)\\right)\\right]\\\\\n\t&\\to \\mathbb{E}\\left[\\left\\{1-\\pi_0(X)\\right\\}\\log \\phi(\\gamma_0,X)+\\pi_0(x)\\log\\left(1-\\phi(\\gamma_0,X)\\right)\\right]\\text{ a.s. }\n\t\\end{split}\n\t\\]\n\tand, at first sight it seems that the same holds when $\\gamma_0$ is replaced by $\\gamma^*$. However, the proof is more delicate because $\\gamma^*$ depends on $\\omega$ and thus also the event of probability one where  the strong law of large numbers holds for this average. To avoid this we consider a countable dense subset of $G$, $\\{\\tilde{\\gamma}_l\\}_{l\\geq 1}$ (for example the subset for which all components of $\\gamma$ are rational numbers). Now, consider the countable collection  of the probability one sets $\\{N_l\\}_{l\\geq 1}$ where\n\t\\[\n\t\\begin{split}\n\t&\\frac{1}{m_k}\\sum_{i=1}^{m_k}\\left[\\left\\{1-\\pi_0(X_i)\\right\\}\\log\\phi(\\tilde\\gamma_l,X_i)+\\pi_0(X_i)\\log\\left(1-\\phi(\\tilde\\gamma_l,X_i)\\right)\\right]\\\\\n\t&\\to \\mathbb{E}\\left[\\left\\{1-\\pi_0(X)\\right\\}\\log \\phi(\\tilde\\gamma_l,X)+\\pi_0(x)\\log\\left(1-\\phi(\\tilde\\gamma_l,X)\\right)\\right]\\quad\\forall l\\geq 1.\n\t\\end{split}\n\t\\]\n\tIf $N\\subseteq\\left(\\cap_{l\\geq 1} N_l\\right)$, we can write \n\t\\[\n\t\\begin{split}\n\t&\\left|\\frac{1}{m_k}\\sum_{i=1}^{m_k}\\left[\\left\\{1-\\pi_0(X_i)\\right\\}\\log\\phi(\\gamma^*,X_i)+\\pi_0(X_i)\\log\\left(1-\\phi(\\gamma^*,X_i)\\right)\\right]\\right.\\\\\n\t&\\quad-\\mathbb{E}\\left[\\left\\{1-\\pi_0(X)\\right\\}\\log \\phi(\\gamma^*,X)+\\pi_0(x)\\log\\left(1-\\phi(\\gamma^*,X)\\right)\\right]\\Bigg|\\\\\n\t&\\leq \\left|\\frac{1}{m_k}\\sum_{i=1}^{m_k}\\left[\\left\\{1-\\pi_0(X_i)\\right\\}\\log\\frac{\\phi(\\gamma^*,X_i)}{\\phi(\\tilde{\\gamma}_l,X_i)}+\\pi_0(X_i)\\log\\frac{\\left(1-\\phi(\\gamma^*,X_i)\\right)}{\\left(1-\\phi(\\tilde\\gamma_l,X_i)\\right)}\\right]\\right|\\\\\n\t&\\quad+\\left|\\frac{1}{m_k}\\sum_{i=1}^{m_k}\\left[\\left\\{1-\\pi_0(X_i)\\right\\}\\log\\phi(\\tilde\\gamma_l,X_i)+\\pi_0(X_i)\\log\\left(1-\\phi(\\tilde\\gamma_l,X_i)\\right)\\right]\\right.\\\\ \n\t&\\qquad-\\mathbb{E}\\left[\\left\\{1-\\pi_0(X)\\right\\}\\log \\phi(\\tilde\\gamma_l,X)+\\pi_0(x)\\log\\left(1-\\phi(\\tilde\\gamma_l,X)\\right)\\right]\\Bigg|\\\\\n\t&\\quad+\\left|\\mathbb{E}\\left[\\left\\{1-\\pi_0(X)\\right\\}\\log\\frac{ \\phi(\\gamma^*,X)}{\\phi(\\tilde\\gamma_l,X)}+\\pi_0(x)\\log\\frac{1-\\phi(\\gamma^*,X)}{1-\\phi(\\tilde\\gamma_l,X)}\\right]\\right|.\n\t\\end{split}\n\t\\]\n\tSince $\\tilde\\gamma_l$ can be taken arbitrarily close to $\\gamma^*$, by properties of $\\phi$ in assumptions (AC3)-(AC4), it can be easily derived that, for an appropriate choice of $\\tilde\\gamma_l$, the first and the third term on the right hand side in the previous equation converge to zero. Moreover, the second term also converges to zero in the set of probability one that we are considering. As a result, we can conclude that \n\t\\[\n\t\\begin{split}\n\t0&\\leq \\frac{\\log \\hat{L}_{m_k,1}(\\hat\\gamma_{m_k})-\\log \\hat{L}_{m_k,1}(\\gamma_0)}{m_k}\\\\\n\t&=\\mathbb{E}\\left[\\left\\{1-\\pi_0(X)\\right\\}\\log\\frac{\\phi(\\gamma^*,X)}{\\phi(\\gamma_0,X)}+\\pi_0(x)\\log\\frac{1-\\phi(\\gamma^*,X)}{1-\\phi(\\gamma_0,X)}\\right]+o(1)\n\t\\end{split}\n\t\\]\t\n\tFor each $x\\in\\mathcal{X}$, consider the function\n\t\\[\n\tg_x(z)=\\phi(\\gamma_0,x)\\log\\frac{z}{\\phi(\\gamma_0,x)}+\\left\\{1-\\phi(\\gamma_0,x)\\right\\}\\log\\frac{1-z}{1-\\phi(\\gamma_0,x)},\\quad z\\in(0,1).\n\t\\]\n\tIt is easy to check that $g_x(z)\\leq 0$ and the equality holds only if $z=\\phi(\\gamma_0,x)$. Hence, the expectation in \\eqref{eqn:expectation_consistency} is smaller or equal to zero. Due to the inequality in \\eqref{eqn:consistency_1}, it must be equal to zero, which means that $\\phi(\\gamma^*,X)=\\phi(\\gamma_0,X)$. By the identifiability assumption \\eqref{eqn:CI3}, this is possible only if $\\gamma^*=\\gamma_0$.\n\t\\end{proof}\n\n\\begin{lemma}\n\t\\label{lemma:boundedness_hat_Lambda}\n\tAssume (AC2),(AC5)\n\thold {and $\\tau^*$ is such that \\eqref{eqn:no_jump_cond} is satisfied.}  Then $\\sup_{n}\\hat\\Lambda_n({\\tau^*})<\\infty$ almost surely.\n\\end{lemma}\n\\begin{proof}\n\t\n\t\tBy definition\n\t\t\\[\n\t\t\\hat\\Lambda_n({\\tau^*})=\\frac{1}{n}\\sum_{i=1}^{n}\\frac{\\Delta_i\\mathds{1}_{\\{Y_i< {\\tau^*}\\}}}{\\frac{1}{n}\\sum_{j=1}^n\\mathds{1}_{\\{Y_i\\leq Y_j\\leq \\tau_0\\}}\\exp(\\hat\\beta'_nZ_j)\\left\\{{\\Delta_j}+(1-{\\Delta_j})g_j(Y_j,\\hat\\Lambda_n,\\hat\\beta_n,\\hat\\gamma_n)\\right\\}}.\n\t\t\\]\n\t\tFrom assumptions (AC2) and (AC5) we have\n\t\\[\n\t\\begin{split}\n\t&\\frac{1}{n}\\sum_{j=1}^n\\mathds{1}_{\\{\\tau^*\\leq Y_j\\leq \\tau_0\\}}\\exp(\\hat\\beta'_nZ_j)\\left\\{\\Delta_j+(1-\\Delta_j)g_j(Y_j,\\hat\\Lambda_n,\\hat\\beta_n,\\hat\\gamma_n)\\right\\}\\\\\n\t&\\geq \\frac{1}{n}\\sum_{j=1}^n\\Delta_j\\mathds{1}_{\\{\\tau^*\\leq Y_j\\leq \\tau_0\\}}\\exp(\\hat\\beta'_nZ_j)\\\\\n\t&\\geq c\\frac{1}{n}\\sum_{j=1}^n\\Delta_j\\mathds{1}_{\\{\\tau^*\\leq Y_j\\leq \\tau_0\\}},\n\t\\end{split}\n\t\\]\t\n\tfor some $c>0$. Since $\\frac{1}{n}\\sum_{j=1}^n\\Delta_j\\mathds{1}_{\\{\\tau^*\\leq Y_j\\leq \\tau_0\\}}\\xrightarrow{a.s.} \\mathbb{P}\\left(Y\\geq \\tau^*,\\Delta=1\\right)>0$, it follows that $\\frac{1}{n}\\sum_{j=1}^n\\Delta_j\\mathds{1}_{\\{\\tau^*\\leq Y_j\\leq \\tau_0\\}}$ is bounded from below away from zero almost everywhere. As a result\n\t\\[\n\t\\begin{split}\n\t\\sup_n\\hat\\Lambda_n(\\tau^*)&\\leq\\sup_n \\frac{1}{n}\\sum_{i=1}^{n}\\frac{\\Delta_i\\mathds{1}_{\\{Y_i< \\tau^*\\}}}{\\frac{1}{n}\\sum_{j=1}^n\\mathds{1}_{\\{\\tau^*\\leq Y_j\\leq \\tau_0\\}}\\exp(\\hat\\beta'_nZ_j)\\left\\{\\Delta_j+(1-\\Delta_j)g_j(Y_j,\\hat\\Lambda_n,\\hat\\beta_n,\\hat\\gamma_n)\\right\\}}\\\\\n\t&\\leq\\sup_n \\frac{1}{n}\\sum_{i=1}^{n}\\frac{\\Delta_i\\mathds{1}_{\\{Y_i< \\tau_0\\}}}{c\\frac{1}{n}\\sum_{j=1}^n\\Delta_j\\mathds{1}_{\\{\\tau^*\\leq Y_j\\leq \\tau_0\\}}}\\\\\n\t&\\leq \\frac{1}{c}\\left(\\inf_n{\\frac{1}{n}\\sum_{j=1}^n\\Delta_j\\mathds{1}_{\\{\\tau^*\\leq Y_j\\leq \\tau_0\\}}}\\right)^{-1}\n\t\\end{split}\n\t\\]\n\tis bounded almost surely.\n\t%\n\tNote that, if \\eqref{eqn:jump_cond} is satisfied, then we can take $\\tau^*=\\tau_0$.\n\\end{proof}\n\n\\begin{proof}[\\textsc{Proof of Theorem~\\ref{theo:consistency2}.}]\n\nLet  $0<\\tau^* \\leq \\tau_0$ and \n\\begin{equation*\n\\hat{l}_n^*( \\beta, \\Lambda_{|\\tau^*}, \\hat\\gamma_n ) = \\frac{1}{n}\\sum_{i=1}^n \\ell (Y_i^*,\\Delta_i^*,X_i,Z_i;\\beta, \\Lambda_{|\\tau^*}, \\hat\\gamma_n),\n\\end{equation*}\nwith $\\ell$ defined in \\eqref{el_star}. \nIf we consider the Cox PH model for the conditional law of $T_0$, then \n\\begin{multline*}\n\\hat{l}^*_n(\\beta,\\Lambda_{|\\tau^*}, \\hat\\gamma_n) \n=\\frac{1}{n}\\sum_{i=1}^n \\Delta_i \\left[ \\mathds{1}_{\\{Y_i < \\tau^*\\}} \\left\\{ \\log \\Delta\\Lambda(Y_i)+\\beta'Z_i-\\Lambda(Y_i)e^{\\beta'Z_i} \\right\\} \n\\right] \\\\\n+\\frac{1}{n}\\sum_{i=1}^n(1-\\Delta_i ) \\mathds{1}_{\\{Y_i < \\tau^*\\}}\\log\\left\\{1-\\phi(\\hat\\gamma_n,X_i)+\\phi(\\hat\\gamma_n,X_i)\\exp\\left(-\\Lambda(Y_i)e^{\\beta'Z_i}\\right)\\right\\}\\\\\n- \\frac{\\Lambda(\\tau^*-)}{n}\\sum_{i=1}^n \\mathds{1}_{\\{Y_i \\geq \\tau^*\\}} \\mathds{1}_{\\{B_i =1\\}} e^{\\beta'Z_i}\n+  \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{Y_i \\geq \\tau^*\\}} \\mathds{1}_{\\{B_i =0\\}}  \\log\\left\\{1-\\phi(\\hat\\gamma_n,X_i)\\right\\},\n\\end{multline*}\nand has to be maximized with respect to $\\beta$ and $\\Lambda$ in\nthe class of step functions $\\Lambda$\nwith jumps of size $\\Delta \\Lambda$ at the event times in $[0,\\tau^*) $. \nAs in \\cite{Lu}, it can be shown that the maximizer $(\\hat\\Lambda^*_n,\\hat\\beta^*_n)$ of $\\hat{l}^*_n$ exists and it is finite. Moreover, for $t\\in[0,\\tau_0] $, $\\hat{\\Lambda}_n^* = \\Lambda^*_{n,\\hat\\beta^*_n, \\hat\\gamma_n}$ where \n\\begin{equation*}\n\\Lambda^*_{n,\\beta, \\gamma} (t)=\\frac{1}{n}\\sum_{i=1}^{n}\\frac{\\Delta_i\\mathds{1}_{\\{Y_i{\\leq t, Y_i<\\tau^* }\\}}}{\\frac{1}{n}\\sum_{j=1}^n\\mathds{1}_{\\{{Y_j\\geq Y_i }\\}}\\exp(\\beta' Z_j)\\left\\{{\\Delta_j^*}+(1-{\\Delta_j})\\mathds{1}_{\\{Y_j < \\tau^*\\}}g_j(Y_j, \\Lambda^*_{n,\\beta,\\gamma},\\beta,\\gamma)\\right\\}} ,\n\\end{equation*}\n$\\Delta^* = \\mathds{1}_{\\{T_0^* \\leq C^*\\}} = \\Delta \\mathds{1}_{\\{Y_j < \\tau^*\\}} + \\mathds{1}_{\\{Y_i \\geq \\tau^*\\}} \\mathds{1}_{\\{B_i =1\\}}$ and $g_j(t,\\Lambda,\\beta,\\gamma)$ defined in \\eqref{def:g_j}. \n\nLet\n\\begin{equation}\n\\label{def:tilde_Lambda_0}\n\\tilde\\Lambda_{0,n}(t)=\\frac{1}{n}\\sum_{i=1}^{n}\\frac{\\Delta_i\\mathds{1}_{\\{Y_i\\leq t{, Y_i<\\tau_0}\\}}}{\\frac{1}{n}\\sum_{j=1}^n\\mathds{1}_{\\{Y_j\\geq Y_i{,Y_j\\leq\\tau_0}\\}}\\exp(\\beta'_0Z_j)\\left\\{{\\Delta_j}+(1-{\\Delta_j})g_j(Y_j,\\Lambda_{0},\\beta_0,\\gamma_0))\\right\\}}. \n\\end{equation}\n\n{We want to prove that $\\hat\\beta_n{\\xrightarrow{a.s.}}\\beta_0$, and $\\sup_{t\\in[0,\\bar\\tau]}|\\hat\\Lambda_n(t)-\\Lambda_0(t)|{\\xrightarrow{a.s.}}0$ for any $\\bar\\tau<\\tau_0$. We suppose  that the previous statement is false, i.e $\\hat\\beta_n$ does not converge {almost surely} to $\\beta_0$ or there exists $\\bar\\tau$ such that $\\sup_{t\\in[0,\\bar\\tau]}|\\hat\\Lambda_n(t)-\\Lambda_0(t)|$ does not converge to zero {almost surely}. This means that, \n\n\tthere exist  $\\epsilon>0$  and $\\bar\\tau<\\tau_0$ such that\n\t\\[\n\t\\mathbb{P}[A_{1} (\\bar\\tau,\\epsilon)]>0, \\;\\; \\text{ with } \\;\\; A_{1} (\\bar\\tau,\\epsilon)=\\left\\{\\limsup_{n\\rightarrow \\infty }\\left[ \\left\\|\\hat\\beta_n-\\beta_0\\right\\|+\\sup_{t\\in[0,\\bar\\tau]}\\left|\\hat\\Lambda_n(t)-\\Lambda_0(t)\\right|\\right] >\\epsilon \\right\\}.\n\t\\]\t\n\t{On the other hand,} since $(\\hat\\Lambda_n,\\hat\\beta_n)$ maximizes $\\hat\\ell_n(\\Lambda,\\beta,\\hat\\gamma_n)$, for any realization {$\\omega$} of the data we have \n\t\\begin{equation}\n\t\\label{eqn:inequality_1}\n\t\\hat{l}_n(\\hat \\beta_n,  \\hat \\Lambda_{n}, \\hat \\gamma_n) - \\hat{l}_n( \\beta_0, \\tilde \\Lambda_{0,n}, \\hat \\gamma_n)  \\geq 0.\n\t\\end{equation}\n{Then the idea for creating the contradiction  is to } show that  the previous inequality is not satisfied for any $\\omega$ in some event of positive probability. \n\tWe argue for a fixed realization $\\omega$ of the data. \n\tAs a  bounded sequence in $\\mathbb{R}^q$, $\\hat\\beta_{n}$ has a convergent subsequence $\\hat{\\beta}_{n_k}\\to\\bar\\beta$. Let $(\\tau_i)_{i\\geq 1}$ be an increasing sequence such that $\\lim_{i\\to \\infty}\\tau_i=\\tau_0$.  Since for all $\\tau<\\tau_0$, $\\hat\\Lambda_n(\\tau)<\\infty$ almost surely (see Lemma~\\ref{lemma:boundedness_hat_Lambda}), by Helly's  selection theorem (\\cite{ash}), there exists a \n\n\tsubsequence $\\hat{\\Lambda}_{m_k}$ of $\\hat{\\Lambda}_{n_k}$, converging pointwise to a function $\\bar\\Lambda$ on $[0,\\tau_1]$. Repeating the same argument, we can extract a further subsequence converging pointwise to a function $\\bar\\Lambda$ on $[0,\\tau_2]$ and so on. Hence, there exist a subsequence $\\hat{\\Lambda}_{r_k}$ converging pointwise to a function $\\bar\\Lambda$ on all compacts of  $[0,\\tau_0]$ that do not include $\\tau_0$. {This defines a monotone function $\\bar\\Lambda$ on $[0,\\tau_0)$, which could be extended at $\\tau_0$ by taking the limit.}  \n\tAs in Lemma 2 of \\cite{Lu}, it can be shown that $\\bar\\Lambda$ is absolutely continuous and pointwise  convergence of monotone functions to a continuous monotone function implies uniform convergence on  compacts.  Note that the chosen subsequence and the limits $\\bar\\beta$ and $\\bar\\Lambda$ depend on $\\omega$. To keep the notation simple, in what follows we use the index $n$ instead of the chosen subsequence $r_k$. \t\n\tFor any $\\tau^*<\\tau_0$,  we can write\n\n\t\\begin{align}\\label{lik_deco}\n\t&\\!\\!\\! 0\\leq \\hat{l}_n(\\hat \\beta_n,  \\hat \\Lambda_{n}, \\hat \\gamma_n) - \\hat{l}_n( \\beta_0, \\tilde \\Lambda_{0,n}, \\hat \\gamma_n) \\notag \\\\ \n\t&= \\hat{l}_n^*(\\hat \\beta_n,  \\hat \\Lambda_{n|\\tau^*}, \\hat \\gamma_n) + D_{1n} - \\hat{l}^*_n( \\beta_0, \\tilde \\Lambda_{0,n|\\tau^*},\\hat \\gamma_n)   - D_{2n} \\notag \\\\\n\t&= \\mathbb E [\\ell (Y^*,\\Delta^*,X,Z;\\bar \\beta, \\bar\\Lambda_{|\\tau^*},\\gamma_0)] + D_{1n}+ R_{1n} \\notag\\\\\n\t&\\quad- \\mathbb E [\\ell (Y^*,\\Delta^*,X,Z;\\beta_0,\\Lambda_{0|\\tau^*} ,\\gamma_0)] - D_{2n}-R_{2n}\n\t,\n\t\\end{align}\n\n\twhere\n\t\\begin{equation}\\label{deco_d1}\n\tD_{1n} =  \\hat{l}_n(\\hat \\beta_n,  \\hat \\Lambda_{n}, \\hat \\gamma_n) - \\hat{l}_n^*(\\hat \\beta_n,  \\hat \\Lambda_{n|\\tau^*}, \\hat \\gamma_n),\n\t\\end{equation}\n\t\\begin{equation}\\label{deco_d2}\n\tD_{2n} = \\hat{l}_n( \\beta_0, \\tilde \\Lambda_{0,n}, \\hat \\gamma_n) - \n\t\\hat{l}^*_n( \\beta_0, \\tilde \\Lambda_{0,n|\\tau^*},\\hat \\gamma_n),\n\t\\end{equation}\n\t\\begin{equation}\\label{deco_r1}\n\tR_{1n} = \\hat{l}_n^*(\\hat \\beta_n,  \\hat\\Lambda_{n|\\tau^*}, \\hat \\gamma_n) - \\mathbb E [\\ell (Y^*,\\Delta^*,X,Z;\\bar\\beta, \\bar\\Lambda_{|\\tau^*} ,   \\gamma_0)] ,\n\t\\end{equation}\n\t\\begin{equation}\\label{deco_r2}\n\tR_{2n} =\\hat{l}^*_n( \\beta_0, \\tilde \\Lambda_{0,n|\\tau^*},\\hat \\gamma_n)  - \\mathbb E [\\ell (Y^*,\\Delta^*,X,Z;\\beta_0,\\Lambda_{0|\\tau^*} ,\\gamma_0)] .\n\t\\end{equation}\nNote that the limit of $(\\hat\\beta_n,\\hat\\Lambda_n)$ depends on $\\omega$, but here the expectation is taken with respect to $(Y^*,\\Delta^*,X,Z)$ for fixed $(\\bar\\beta,\\bar\\Lambda)$.\n\tWe now define the event \n\t$\n\tA_{3}(\\tau^*)=\\left\\{|R_{1n}-R_{2n}|\\rightarrow 0\\right\\}.\n\t$\n\tBy Lemma \\ref{lem:consist_full_1}, for any $\\tau^*<\\tau_0$, we have  $\\mathbb{P}[A_{1} (\\bar\\tau,\\epsilon)\\cap A_{3}(\\tau^*)]=\\mathbb{P}[A_{1} (\\bar\\tau,\\epsilon)]$.\n\tNext, for $\\bar\\tau<\\tau_0$ and $\\epsilon>0$ such that $\\mathbb{P}[A_{1} (\\bar\\tau,\\epsilon)]>0$, by Lemma \\ref{lem:consist_full_3} there exist $0<c_1<1$ and $\\delta>0$ such that we have\n\t\\[\n\t\\begin{split}\n\tc&=\\inf\\Bigg\\{\\mathbb E [\\ell (Y^*,\\Delta^*,X,Z;\\beta_0,\\Lambda_{0|\\tau^*} ,\\gamma_0)]-\t\\mathbb E [\\ell (Y^*,\\Delta^*,X,Z; \\beta, \\Lambda_{|\\tau^*},\\gamma_0)]:\\\\\n\t&\\left.\\qquad\\bar{\\tau}+\\delta \\leq \\tau^*<\\tau_0,\\quad \\|\\beta-\\beta_0\\|\\geq    c_1\\epsilon\/2   \\quad\\text{ or}\\quad \\sup\\limits_{t\\in[0,\\bar \\tau]}|\\Lambda(t)-\\Lambda_0(t)|\\geq    (1-c_1) \\epsilon \/2   \\right\\}>0.\n\t\\end{split}\n\t\\]\n\tNote that if $\\omega \\in  A_{1}(\\bar\\tau,\\epsilon)$ and $\\bar\\beta$ and $\\bar \\Lambda$ are the limits for $ \\hat \\beta_n$ and $\\hat \\Lambda_n$, respectively, then necessarily, either $\\|\\bar \\beta-\\beta_0\\|\\geq    c_1\\epsilon\/2 ,$ or $\\sup\\limits_{t\\in[0, \\bar \\tau]}|\\bar \\Lambda(t)-\\Lambda_0(t)|\\geq    (1-c_1) \\epsilon \/2$, and consequently \n\t$$\n\t\\mathbb E [\\ell (Y^*,\\Delta^*,X,Z;\\bar \\beta, \\bar\\Lambda_{|\\tau^*},\\gamma_0)]  - \\mathbb E [\\ell (Y^*,\\Delta^*,X,Z;\\beta_0,\\Lambda_{0|\\tau^*} ,\\gamma_0)] \\geq c,\\quad \\forall \\bar{\\tau}{+\\delta}\\leq \\tau^*<\\tau_0.\n\t$$\n\tFinally, we define $$A_{2}(\\tau^*)=\\left\\{ \\limsup_{n\\rightarrow\\infty} \\left|D_{1n}-D_{2,n} \\right| \\leq c\/2\\right\\},$$ with $D_{1n}$ and $D_{2n} $ defined in \\eqref{deco_d1} and \\eqref{deco_d2}, and choose $\\tau^*\\in [\\bar\\tau+\\delta,\\tau_0)$ such that \n\t$$\n\tc_b \\left\\{ \\mathbb{P}(T_0\\geq \\tau^*) \\log \\{1\/\\mathbb{P} (T_0\\geq \\tau^*)\\} +  \\mathbb{P}(C\\in[ \\tau^*,\\tau_0])\\right\\}<c\/2,\n\t$$\n\twith $c_b$ the constant from Lemma \\ref{lem:consist_full_2}. Then we have $\\mathbb{P}[A_{2}(\\tau^*)]=1$.\n\n\tGathering facts, we deduce that  by a suitable choice of $\\tau^*\\in[\\bar\\tau+\\delta,\\tau_0)$, we necessarily have $\\mathbb{P}[ A_{1} (\\bar\\tau,\\epsilon)\\cap\n\tA_{2}(\\tau^*)\\cap A_{3}(\\tau^*)]>0$. Moreover,  with such a suitable $\\tau^*$, \n\tfor any  $\\omega\\in A_{1} (\\bar\\tau,\\epsilon)\\cap\n\tA_{2}(\\tau^*)\\cap A_{3}(\\tau^*)$, \n\twe have  \n\t$$\\limsup_{n\\rightarrow \\infty}  \\left[ \\hat{l}_n(\\hat \\beta_n,  \\hat \\Lambda_{n}, \\hat \\gamma_n) - \\hat{l}_n( \\beta_0, \\tilde \\Lambda_{0,n}, \\hat \\gamma_n)\\right] \\leq -c\/2<0.$$ \n\tWe deduce that  \\eqref{eqn:inequality_1} is violated on an event of positive probability, which by definition is impossible. Thus\n\t$\\hat\\beta_n{\\xrightarrow{a.s.}}\\beta_0$, and $\\sup_{t\\in[0,\\bar\\tau]}|\\hat\\Lambda_n(t)-\\Lambda_0(t)|{\\xrightarrow{a.s.}}0$ for any $\\bar\\tau<\\tau_0$.\n}\n\n{If condition \\eqref{eqn:jump_cond} is satisfied, we want to show in addition that $|\\hat\\Lambda_n(\\tau_0)-\\Lambda_0(\\tau_0)|{\\xrightarrow{a.s.}}0$. In that case,  $\\hat\\Lambda_n(\\tau_0)<\\infty$ almost surely and as a result, for any realization $\\omega$, there exists a subsequence $\\hat\\Lambda_{r_k}$ converging  to some absolutely continuous function  $\\bar\\Lambda$ uniformly on $[0,\\tau_0]$. \n\tSince we already showed that \n\t$|\\hat\\Lambda_n(t)-\\Lambda_0(t)|{\\xrightarrow{a.s.}}0$ for any $t<\\tau_0$ and $\\Lambda_0(\\tau_0)=\\lim_{t\\uparrow\\tau_0}\\Lambda_0(t)$, we necessarily have $\\bar\\Lambda=\\Lambda_0$ on the whole interval $[0,\\tau_0]$. This concludes the proof of the Theorem.\n}\n\\end{proof}\n\n\n\\begin{lemma}\n\t\\label{lem:consist_full_1}\n\tConsider a realization of the data $\\omega$ and assume that $\\hat\\beta_n(\\omega)\\to\\bar\\beta$ and $\\hat\\Lambda_n(\\omega)(t)\\to \\bar\\Lambda(t)$ for any $t\\in[0,\\tau_0)$, for some absolutely continuous function $\\bar\\Lambda$. Let $0<\\tau^*<\\tau_0$ and let $R_{1n}$, $R_{2n}$ be defined as in \\eqref{deco_r1} and \\eqref{deco_r2}, respectively. There exists an event $A_3(\\tau^*)$ of probability one such that, for any  $\\omega\\in A_3(\\tau^*)$, \n\t$$\n\tR_{1n}(\\omega)-R_{2n}(\\omega)\\to 0.\n\t$$\n\\end{lemma}\n\n\n\\begin{proof} {Let us consider some $0<\\tau^*<\\tau_0$}. From Theorem \\ref{theo:consistency} and  Lemma 2 in \\cite{Lu} it follows that the event \n\t\\[\n\tA_3^1 {(\\tau^*)}=\\left\\{\\hat\\gamma_n\\to\\gamma_0\\quad\\text{ and }\\quad\\sup_{t\\in[0,\\tau^*]}|\\tilde{\\Lambda}_{0,n}(t)-\\Lambda_0(t)|\\to 0\\right\\}\n\t\\]\n\thas probability one. Next we argue for the given realization of the data $\\omega{\\in A_3^1(\\tau^*)}$ and  will determine the event $A_3{(\\tau^*)}$ appropriately.  \n\tBy the triangular inequality we can write\n\t\\begin{align}\n\t\\label{eqn:R2}\n\n\t|R_{1n}\\!-R_{2n}|&\\leq\\left|\\left\\{\\hat{l}^*_n(\\hat\\beta_n,\\hat{\\Lambda}_{n|\\tau*},\\hat\\gamma_n)\\! -\\hat{l}^*_n(\\beta_0,\\tilde{\\Lambda}_{0,n|\\tau*},\\hat\\gamma_n)\\right\\} \\!-\\! \\left\\{\\hat{l}^*_n(\\bar\\beta,\\bar{\\Lambda}_{|\\tau*},\\!\\gamma_0) \\!- \\! \\hat{l}^*_n(\\beta_0,{\\Lambda}_{0|\\tau*},\\!\\gamma_0)\\right\\}\\right| \\notag\\\\\n\t&\\quad+\\left|\\hat{l}^*_n(\\beta_0,{\\Lambda}_{0|\\tau*},\\!\\gamma_0)-\\mathbb{E}\\left[l(Y^*,\\Delta^*,X,Z;\\beta_0,\\Lambda_{0|\\tau^*},\\!\\gamma_0)\\right]\\right|\\notag\\\\\n\t&\\quad+\\left|\\hat{l}^*_n(\\bar\\beta,\\bar{\\Lambda}_{|\\tau*},\\gamma_0)-\\mathbb{E}\\left[l(Y^*,\\Delta^*,X,Z;\\bar\\beta,\\bar\\Lambda_{|\\tau^*},\\gamma_0)\\right]\\right|.\n\n\t\\end{align}\n\tSince $\\bar\\Lambda$ is absolutely continuous, it is differentiable almost everywhere. Let $\\bar\\lambda(t)=\\mathrm{d}\\bar\\Lambda(t)\/\\mathrm{d} t$. \n\tBy definition we have\n\t\\[\n\t\\begin{split}\n\t&\\hat{l}^*_n(\\hat\\beta_n,\\hat{\\Lambda}_{n|\\tau*},\\hat\\gamma_n)-\\hat{l}^*_n(\\beta_0,\\tilde{\\Lambda}_{0,n|\\tau*},\\hat\\gamma_n)\\\\\n\t&=\\frac{1}{n}\\sum_{i=1}^n \\Delta_i \\mathds{1}_{\\{Y_i < \\tau^*\\}} \\left\\{ \\log \\frac{\\Delta\\hat\\Lambda_n(Y_i)}{\\Delta\\tilde\\Lambda_{0,n}(Y_i)}+(\\hat\\beta_n-\\beta_0)'Z_i-\\hat\\Lambda_n(Y_i)e^{\\hat\\beta'_nZ_i}+\\tilde\\Lambda_{0,n}(Y_i)e^{\\beta'_0Z_i} \\right\\}  \\\\\n\t&\\qquad+\\frac{1}{n}\\sum_{i=1}^n(1-\\Delta_i ) \\mathds{1}_{\\{Y_i < \\tau^*\\}}\\log\\frac{1-\\phi(\\hat\\gamma_n,X_i)+\\phi(\\hat\\gamma_n,X_i)\\exp\\left(-\\hat\\Lambda_n(Y_i)e^{\\hat\\beta'_nZ_i}\\right)}{1-\\phi(\\hat\\gamma_n,X_i)+\\phi(\\hat\\gamma_n,X_i)\\exp\\left(-\\tilde\\Lambda_{0,n}(Y_i)e^{\\beta'_0Z_i}\\right)}\\\\\n\t&\\qquad - \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{Y_i \\geq \\tau^*\\}}\\mathds{1}_{\\{B_i =1\\}}\\left\\{\n\n\t\\hat\\Lambda_n(\\tau^*)e^{\\hat\\beta'_nZ_i} - \\tilde\\Lambda_{0,n}(\\tau^*)e^{\\beta'_0Z_i}\\right\\}.  \n\t\\end{split}\n\t\\]\n\tIf $\\omega\\in A^1_3{(\\tau^*)}$, we obtain\n\t\\[\n\t\\begin{split}\n\t&\\hat{l}^*_n(\\hat\\beta_n,\\hat{\\Lambda}_{n|\\tau*},\\hat\\gamma_n)-\\hat{l}^*_n(\\beta_0,\\tilde{\\Lambda}_{0,n|\\tau*},\\hat\\gamma_n)\\\\\n\t&=\\frac{1}{n}\\sum_{i=1}^n \\Delta_i \\mathds{1}_{\\{Y_i < \\tau^*\\}} \\left\\{ \\log \\frac{\\bar\\lambda(Y_i)}{\\lambda_{0}(Y_i)}+(\\bar\\beta-\\beta_0)'Z_i-\\bar\\Lambda(Y_i)e^{\\bar\\beta'Z_i}+\\Lambda_{0}(Y_i)e^{\\beta'_0Z_i} \\right\\}  \\\\\n\t&\\qquad+\\frac{1}{n}\\sum_{i=1}^n(1-\\Delta_i ) \\mathds{1}_{\\{Y_i < \\tau^*\\}}\\log\\frac{1-\\phi(\\gamma_0,X_i)+\\phi(\\gamma_0,X_i)\\exp\\left(-\\bar\\Lambda(Y_i)e^{\\bar\\beta'Z_i}\\right)}{1-\\phi(\\gamma_0,X_i)+\\phi(\\gamma_0,X_i)\\exp\\left(-\\Lambda_{0}(Y_i)e^{\\beta'_0Z_i}\\right)}\\\\\n\t&\\qquad  - \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{Y_i \\geq \\tau^*\\}}\\mathds{1}_{\\{B_i =1\\}}\\left\\{\n\n\t\\bar\\Lambda(\\tau^*)e^{\\bar\\beta'Z_i}- \\Lambda_{0}(\\tau^*)e^{\\beta'_0Z_i}\\right\\} +o(1)\\\\\n\t&=\\hat{l}^*_n(\\bar\\beta,\\bar{\\Lambda}_{|\\tau*},\\gamma_0)-\\hat{l}^*_n(\\beta_0,{\\Lambda}_{0|\\tau*},\\gamma_0)+o(1),\n\t\\end{split}\n\t\\]\n\twhere the remainder term  depends on $\\omega$ and converges to zero. Hence, {the first term on the right hand side of \\eqref{eqn:R2}} converges to zero. Let $A_n^2{(\\tau^*)}$ be the event where \n\t\\[\n\t\\hat{l}^*_n(\\beta_0,{\\Lambda}_{0|\\tau*},\\gamma_0)\\to\\mathbb{E}\\left[l(Y^*,\\Delta^*,X,Z;\\beta_0,\\Lambda_{0|\\tau^*},\\gamma_0)\\right] \\quad\\text{ as }n\\to\\infty.\n\t\\]\n\tBy the law of large numbers $\\mathbb{P}[A^2_n{(\\tau^*)}]=1$, implying that also {the second term on the right hand side of \\eqref{eqn:R2}} converges to zero if $\\omega\\in A^2_n{(\\tau^*)}$.  It remains to deal with the third term. Note that here $(\\bar\\beta,\\bar\\Lambda)$ depend on $\\omega$ and the expectation is taken with respect to $(Y,\\Delta,X,Z)$ for fixed $(\\bar\\beta,\\bar\\Lambda)$.\n\tWe have the same issue as in the proof of Theorem \\ref{theo:consistency} when dealing with the terms involving $\\bar\\beta$ and $\\bar\\Lambda$, so we need to consider approximations by elements of a countable dense subset of $\\mathcal{B}$ and of the space of bounded, absolutely continuous, increasing functions in $[0,\\tau^*]$ (is separable, so such subset exists). The same reasoning is used also in \\cite{murphy,Lu,scharfstein}. Hence, there exists a countable collection of probability one sets $\\{N_l\\}_{l\\geq 1}$ where\n\t\\[\n\t\\hat{l}^*_n(\\beta_l,{\\Lambda}_{l},\\gamma_0)\\to\\mathbb{E}\\left[l(Y^*,\\Delta^*,X,Z;\\beta_l,\\Lambda_{l},\\gamma_0)\\right] \\quad\\text{ as }n\\to\\infty\n\t\\]\n\tand $(\\beta_l,\\Lambda_l)$ can be taken arbitrarily close to $(\\bar\\beta,\\bar\\Lambda)$. As a result, if $\\omega\\in A^3_n{(\\tau^*)}=\\bigcap_{l\\geq 1}N_l$, then \n\t\\[\n\t\\left|\\hat{l}^*_n(\\bar\\beta,\\bar{\\Lambda}_{|\\tau*},\\gamma_0)-\\mathbb{E}\\left[l(Y^*,\\Delta^*,X,Z;\\bar\\beta,\\bar\\Lambda_{|\\tau^*},\\gamma_0)\\right]\\right|\\to 0.\n\t\\]\n\tTo conclude, we define $A_3{(\\tau^*)}=A^1_3{(\\tau^*)}\\cap A^2_3{(\\tau^*)}\\cap A^3_3{(\\tau^*)}$ and we have $\\mathbb{P}[A_3{(\\tau^*)}]=1$. \n\\end{proof}\n\n\n\\begin{lemma}\n\t\\label{lem:consist_full_2}\n\tLet $D_{1n}$ and $D_{2n}$ be defined as in \\eqref{deco_d1} and \\eqref{deco_d2}, respectively, for some $\\tau^* <\\tau_0$. Then there exists a constant $c_b$ independent of $\\tau^*$ such that  \n\t$$\n\t\\mathbb P \\left[ \\limsup_{n\\rightarrow\\infty} \\left|D_{1n}-D_{2,n}\\right|  > c_b \\left\\{ \\mathbb{P}(T_0\\geq \\tau^*) \\log \\{1\/\\mathbb{P} (T_0\\geq \\tau^*)\\} +  \\mathbb{P}(C\\in[ \\tau^*,\\tau_0])\\right\\}  \\right] =0.\n\t$$\n\\end{lemma}\n\n\\begin{proof}\n\n\tBy definition, for any $\\gamma$, $\\beta$ and cumulative hazard function $\\Lambda$ piecewise constant with jumps at the observed events\n\t\\begin{align*}\n\t& {l}_n(\\beta,   \\Lambda,  \\gamma) - \\hat{l}_n^*(\\beta,  \\Lambda_{|\\tau^*}, \\gamma)\n\t\\\\\n\t& = \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{{\\tau^*\\leq Y_i< \\tau_0 } \\}} \\Delta_i \\log \\Lambda (\\{Y_i\\})\\\\\n\t& \\quad + \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{{\\tau^*\\leq Y_i< \\tau_0 }\\}} \\Delta_i \\beta^\\prime Z_i  \n\t\\\\ \n\t& \\quad - \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{ Y_i\\geq \\tau^*  \\}} e^{\\beta^\\prime Z_i} \\{ \\Delta_i \\Lambda (Y_i) - \\mathds{1}_{\\{B_i=1\\}}\\Lambda (\\tau^*-) \\} \\\\\n\t& \\quad +\\frac{1}{n}\\!\\sum_{i=1}^n \\!\\mathds{1}_{\\{\\tau_0\\geq Y_i \\geq  \\tau^*\\!\\}}\\!\\! \\left[\\! (1\\!-\\!\\Delta_i )\\log\\!\\left\\{\\!1\\!-\\!\\phi_i(\\gamma)\\!+\\!\\phi_i(\\gamma)\\exp\\left(\\!-\\Lambda(Y_i)e^{\\beta'Z_i}\\!\\right)\\!\\!\\right\\} \\right. \\! \\\\\n\t& \\quad \\quad \\left. - \\!\\mathds{1}_{\\{B_i=0\\}}\\! \\log\\left\\{1\\!-\\!\\phi_i(\\gamma)\\right\\}\\! \\right]\\\\\n\t& =: r_{1n} (\\Lambda;\\tau^*) + r_{2n} (\\beta;\\tau^*) - r_{3n}  (\\Lambda, \\beta;\\tau^*) + r_{4n} (\\Lambda, \\beta, \\gamma;\\tau^*) ,\n\t\\end{align*}\n\twhere $\\phi_i (\\gamma) $ is a short notation for $\\phi(\\gamma,X_i)$.  For proving the Lemma, we have to suitably bound $r_{1n},\\ldots,r_{4n}$.  For this purpose, let us notice that, by definition, all the cumulative hazard functions we have to consider ($\\hat\\Lambda_n$, $\\tilde \\Lambda_{0,n}$,...) have bounded jumps at the event times. More precisely, because the parameter space $\\mathcal B$ and $Z$ are supposed bounded, there exist constants \n\t$0<c_l < c_u $ such that \n\t$$\n\tc_l\\leq \\exp(\\beta'Z)\\leq c_u.\n\t$$\n\tThen the largest jump of any of  the cumulative hazard functions we need to consider is bounded by $1\/c_l$ (which is located at the last uncensored observation), the second largest one (and is located at the before last uncensored observation) is bounded by $1\/2c_l$,... \n\t\n\t\n\t\n\t\n\tTo control $r_{1n} (\\Lambda;\\tau^*) $, one would  look for a suitable lower bound forthe jumps of $\\Lambda$. However, no meaningful lower bound could be derived for these jumps. More precisely, such a bound is necessarily of order $1\/n$, so that the sequence of the logarithm of the jumps is unbounded. Fortunately, for our purposes it suffices to find a bound for \n\t$$\n\t\\left| r_{1n} (\\hat \\Lambda_{n};\\tau^*) - r_{1n} ( \\tilde \\Lambda_{0,n} ;\\tau^*)\\right| = \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{ {\\tau^*\\leq Y_i< \\tau_0 } \\}} \\Delta_i \\left| \\log  \\frac{\\hat \\Lambda_{n} (\\{Y_i\\})} { \\tilde \\Lambda_{0,n}  (\\{Y_i\\}) } \\right|,\n\t$$\n\twhere \n\t$$\n\t\\frac{\\hat \\Lambda_{n} (\\{Y_i\\})} { \\tilde \\Lambda_{0,n}  (\\{Y_i\\}) } = \\frac{\\sum_{j=1}^n \\mathds{1}_{\\{\\tau_0 \\geq Y_j\\geq Y_i \\}}\\exp(\\beta'_0Z_j)\\left\\{\\Delta_j+(1-\\Delta_j)g_j(Y_j,\\Lambda_{0},\\beta_0,\\hat \\gamma_n))\\right\\} }\n\t{\\sum_{j=1}^n\\mathds{1}_{\\{{\\tau_0\\geq Y_j\\geq Y_i  }\\}}\\exp(\\beta'Z_j)\\left\\{ \\Delta_j +(1-  \\Delta_j )g_j(Y_j,{\\Lambda}_{n,\\hat\\beta,\\hat\\gamma_n},\\beta,\\hat \\gamma_n) \\right\\} }.\n\t$$\n\tSince all $g_j$'s are between 0 and 1, it  is easy to see that  for any uncensored $Y_i\\geq \\tau^*$,\n\t$$\n\t\\frac{1}{\\rho_n(Y_i)} \\frac{c_l } {c_u} \\leq \\frac{\\hat \\Lambda_{n} (\\{Y_i\\})} { \\tilde \\Lambda_{0,n}  (\\{Y_i\\}) }  \\leq \\frac{c_u}{c_l } \\rho _n(Y_i) ,\n\t$$\n\twhere\n\t$$\n\t\\rho_n(t) =  \\frac{\\sum_{j=1}^n \\mathds{1}_{\\{ \\tau_0\\geq Y_j \\geq t\\geq \\tau^*\\} } } {\\sum_{j=1}^n  \\Delta_j \\mathds{1}_{\\{ \\tau_0\\geq Y_j \\geq t \\geq \\tau^*\\} }}, \\qquad t\\in[\\tau^*, \\tau_0].\n\t$$\n\tThus,  since all $\\rho_n(Y_i)$'s are larger than 1, it suffices to suitably bound \n\t$$\n\t0\\leq A_n = \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{ Y_i\\geq \\tau^* \\}} \\Delta_i \\log (\\rho_n(Y_i)),\n\t$$\n\twhich we decompose as\n\t$$\n\tA_n \\! = \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{ Y_i\\in [\\tau^*\\! ,\\tau_0 - a_n  ] \\}} \\Delta_i \\log (\\rho_n(Y_i)) +  \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{ Y_i\\in [\\tau_0 - a_n ,\\tau_0 ] \\}} \\Delta_i \\log (\\rho_n(Y_i)) =: A_{1n} \\!+\\!  A_{2n} ,\n\t$$\n\tfor some sequence of real numbers $a_n$, $n\\geq 1$, decreasing to zero. The rate of $a_n$ should be taken such that, on one hand,  for any constant $C>0$,\n\t\\begin{equation}\\label{eq:up_A2}\n\t\\mathbb{P} (\\limsup_{n\\rightarrow\\infty} A_{2n} > C) = 0,\n\t\\end{equation}\n\tand, on the other hand, the $\\limsup$ of $A_{1n}$ could be controlled by a function of $\\tau^*$ almost surely. More precisely, since \n\t$$\n\tA_{2n} \\leq \\frac{\\log n }{n}\\sum_{i=1}^n \\mathds{1}_{\\{ Y_i\\in [\\tau_0 - a_n ,\\tau_0 ] \\}} \\Delta_i ,\n\t$$\n\twe take $a_n$ such that $p_n\\log n \\rightarrow 0$ and $p_n \\log^2 n \\rightarrow \\infty$, where \n\t\\begin{multline*}\n\tp_n = \\mathbb{P} (Y\\in [\\tau_0-a_n,\\tau_0] , \\Delta = 1) = \\mathbb E \\left[ \\phi(\\gamma_0,X) \\int_{[\\tau_0-a_n,\\tau_0]} F_C([t,\\infty)|X,Z) F_u(dt|Z)\\right].\n\t\\end{multline*} \n\tThen, by Theorem 1(i) from  \\cite{Wellner}, we have \n\t$$\n\t\\lim_{n\\rightarrow \\infty }  \\frac{1 }{p_n} \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{ Y_i\\in [\\tau_0 - a_n ,\\tau_0 ] \\}} \\Delta_i = 1,\\quad a.s.,\n\t$$\n\twhich implies \\eqref{eq:up_A2}. On the other hand, we have\n\t$$\n\tA_{1n} \\leq \\log\\left(\\sup_{t\\in [\\tau^*,\\tau_0 - a_n  ] } \\rho_n(t) \\right) \\times  \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{ Y_i\\in [\\tau^*,\\tau_0   ]\\}} \\Delta_i .\n\t$$\n\tBy the same  Theorem 1(i) from  \\cite{Wellner},\n\t$$\n\t\\lim_{n\\rightarrow \\infty } \\sup_{t\\in [\\tau^*,\\tau_0 - a_n  ] }\\left[  \\rho_n(t)  \\frac{\\mathbb{P} (Y\\in [t,\\tau_0 - a_n  ] , \\Delta = 1) }{\\mathbb{P} (Y\\in [t,\\tau_0 - a_n  ] ) } \\right]  = 1,\\quad a.s.\n\t$$\n\tBy our assumptions, there exists a constant $C_r$, independent of $\\tau^*$, $\\beta$, $\\gamma$ and $\\Lambda$, such that \n\t$$\n\t1< \\inf_{t\\in [\\tau^*,\\tau_0{-a_n} ] }\\frac{\\mathbb{P} (Y\\in [t,\\tau_0 - a_n  ] ) }{\\mathbb{P} (Y\\in [t,\\tau_0 - a_n  ] , \\Delta = 1) } < \\sup_{t\\in [\\tau^*,\\tau_0{-a_n} ] }\\frac{\\mathbb{P} (Y\\in [t,\\tau_0 - a_n  ] ) }{\\mathbb{P} (Y\\in [t,\\tau_0 - a_n  ] , \\Delta = 1) } \\leq C_r.\n\t$$\n\tGathering facts, deduce with probability 1, for sufficiently large $n$, \n\t$$\n\t\\left| r_{1n} (\\hat \\Lambda_{n};\\tau^*) - r_{1n} ( \\tilde \\Lambda_{0,n} ;\\tau^*)\\right|   \\leq  c  \\frac{N^*}{n} ,\n\t$$\n\twhere $N^*$ be the number of uncensored observations in $[\\tau^*,\\tau_0]$ and $c$ is some constant (independent of $\\tau^*$, $\\beta$, $\\gamma$ and $\\Lambda$). Here,  $N^*$ is a binomial random variable  with $n$ trials and  success probability \n\t\\begin{multline*}\n\tp^* = \\mathbb P ( Y\\geq\\tau^* , \\Delta = 1) = \\mathbb E \\left[ \\phi(\\gamma_0,X) \\int_{[\\tau^*,\\tau_0]} F_C([t,\\infty)|X,Z) F_u(dt|Z)\\right]\\\\ \\leq \\sup_{x} \\phi(\\gamma_0,x) \\mathbb P (Y\\geq\\tau^* ).\n\t\\end{multline*}\n\t\n\t\n\t\n\t\n\tTo bound $r_{3n}=r_{3n}(\\Lambda, \\beta;\\tau^*)$, we note that $\\mathds{1}_{\\{B_i=1\\}} =  \\Delta_i  + (1-\\Delta_i )\\mathds{1}_{\\{B_i=1\\}}$ and rewrite \n\t\\begin{multline*}\n\tr_{3n} = \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{ Y_i\\geq \\tau^*  \\}} e^{\\beta^\\prime Z_i}\\Delta_i \\{  \\Lambda (Y_i) - \\Lambda (\\tau^*-) \\}\\\\- \\frac{\\Lambda (\\tau^*-)}{n}\\sum_{i=1}^n \\mathds{1}_{\\{ Y_i\\geq \\tau^*  \\}} e^{\\beta^\\prime Z_i}\\mathds{1}_{\\{B_i=1\\}} (1- \\Delta_i)  = r_{3an}-r_{3bn}.\n\t\\end{multline*}\n\tOn one hand,  \n\t$$\n\tr_{3an} \\leq  \\frac{c_u}{c_l} \\times \\frac{N^*}{n}.\n\t$$\n\tThe last inequality is obtained by bounding the jumps of $\\Lambda$ and using the following identity: for any integer $M\\geq 1$, \n\t$$\n\t\\sum_{k=1}^M \\sum_{j=k}^M \\frac{1}{j} = \\sum_{j,k=1}^M  \\frac{\\mathds{1}_{\\{k\\leq j\\}} }{j}  = M. \n\t$$\n\tTo bound $r_{3bn}$, let us note that \n\t$$\n\t\\Lambda (\\tau^*-) \\leq \\frac{1}{c_l} \\sum_{j=N^*+1}^N \\frac{1}{j} \\leq c_1 \\log \\frac{N}{N^*},\n\t$$\n\twith $c_1$ some constant depending only on $c_l$ and the maximal value of the convergent sequence \n\t$$\\sum_{j=1}^m \\frac{1}{j}  - \\log m, \\quad m\\geq 1.$$ Here, $N =\\sum_{i=1}^n \\Delta_i$ is a binomial random variable  with $n$ trials and success probability\n\t$$\n\tp = \\mathbb P ( \\Delta = 1) = \\mathbb E \\left[ \\phi(\\gamma_0,X) \\int_{[0,\\tau_0]} F_C([t,\\infty)|X,Z) F_u(dt|Z)\\right]  .\n\t$$\n\tThus\n\t$$\n\tr_{3bn} \\leq c \\log \\frac{N}{N^*} \\times \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{ Y_i\\geq \\tau^*  \\}} \\mathds{1}_{\\{B_i=1\\}} (1- \\Delta_i) =  c \\log \\frac{N\/n}{N^*\/n} \\times \\frac{Q^*}{n},\n\t$$\n\twhere $Q^*$ is a binomial variable \n\twith $n$ trials and success probability \n\t\\begin{multline*}\n\tq^* =  \\mathbb E \\left[ \\phi(\\gamma_0,X) \\int_{[\\tau^*,\\tau_0]} F_u([t,\\tau_0]|X,Z) F_C(dt|X,Z)\\right]\\\\\\leq  \\mathbb E \\left[ \\phi(\\gamma_0,X)   F_C([\\tau^*,\\tau_0]|X,Z) F_u([\\tau^*,\\tau_0]|Z)\\right] \\\\\n\t\\leq \\left[ \\sup_{x} \\phi(\\gamma_0,x) \\right]\n\t\\left[\\sup_{x,z} \\frac{F_C([\\tau^*,\\tau_0]|X=x,Z=z)}{\\tau_0 - \\tau^*}\\right] \n\t\\times (\\tau^* -\\tau_0)\\times  \\mathbb P (T_0\\geq \\tau^*) \\\\ \\leq c  (\\tau^* -\\tau_0)\\times  \\mathbb P (T_0\\geq \\tau^*) ,\n\t\\end{multline*}\n\tand $c$ is some constant. By the strong Law of Large Numbers, \n\t$$\n\t\\lim_{n\\rightarrow \\infty} \\log \\frac{N}{N^*}  = \\log \\frac{p}{p^*}, \\, a.s.\n\t$$\n\t\n\tNext, to bound $ r_{2n}= r_{2n} (\\beta;\\tau^*) $, we write\n\t$$\n\tr_{2n} = \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{ {\\tau^*\\leq T_0<\\tau_0} \\}} e^{\\beta^\\prime Z_i} \\Delta_i \n\t\\leq c_u \\frac{N^*}{n}.\n\t$$ \n\t\n\tFinally, to control $ r_{4n}= r_{4n} (\\Lambda, \\beta, \\gamma;\\tau^*)$, \n\tsince $\\mathds{1}_{\\{B_i=0\\}} = (1-\\Delta_i) \\mathds{1}_{\\{B_i=0\\}}$ and $\\log(1+u)\\leq u,$ $\\forall u\\geq 0$,\n\twe have \n\t\\begin{multline*}\n\tr_{4n} = \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{\\tau_0\\geq Y_i \\geq  \\tau^*\\}} \\mathds{1}_{\\{B_i=0\\}}  \\log\\left\\{1+\\frac{\\phi_i(\\gamma)\\exp\\left(-\\Lambda(Y_i)e^{\\beta'Z_i}\\right)}{1-\\phi_i(\\gamma)}\\right\\}  \\\\\n\t+ \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{Y_i \\geq  \\tau^*\\!\\}}  (1-\\Delta_i ) \\mathds{1}_{\\{B_i=1\\}}\\log\\left\\{1-\\phi_i(\\gamma)+\\phi_i(\\gamma)\\exp\\left(-\\Lambda(Y_i)e^{\\beta'Z_i}\\right)\\right\\}.\n\t\\end{multline*}\n\tThus \n\t\\begin{multline*}\n\t|r_{4n}| \\leq \\sup_{\\gamma,x}\\left| \\frac{\\phi(\\gamma,x)}{1-\\phi(\\gamma,x)}\\right| \\exp\\left(-c_l \\Lambda(\\tau^*-)\\right)  \\frac{1}{n}\\sum_{i=1}^n \\mathds{1}_{\\{Y_i \\geq  \\tau^*\\}}  \\mathds{1}_{\\{B_i=0\\}}\n\t\\\\ + \n\t\\sup_{\\gamma,x}\\left| \\log\\left\\{1-\\phi(\\gamma,x)\\right\\}\\right| \\times  \\frac{1}{n}\\sum_{i=1}^n  \\mathds{1}_{\\{Y_i \\geq  \\tau^*\\!\\}} (1-\\Delta_i )\\mathds{1}_{\\{B_i=1\\}}\\\\\n\t= c_1\\exp\\left(-c_l \\Lambda(\\tau^*-)\\right)   \\frac{R^*}{n} + c_2  \\frac{Q^*}{n},\n\t\\end{multline*}\n\twhere $R^*$ is a binomial variable \n\twith $n$ trials and success probability \n\t$$\n\tr^* =  \\mathbb E \\left[ \\{1-\\phi(\\gamma_0,X)\\}  F_C([\\tau^*,\\tau_0]|X,Z)\\right],\n\t$$\n\tand $c_1$ and $c_2$ are some constants. \n\tDeduce that there exists a constant $c_3$ such that\n\t$$\n\t|r_{4n}| \\leq c_3 \\left( \n\t\\frac{R^*}{n} +   \\frac{Q^*}{n}\\right).\n\t$$\n\t\n\tGathering facts, there exists a constants $C^*$ and \n\t$c^*$, independent of $\\tau^*$, $\\beta$, $\\gamma$ and $\\Lambda$, such that \n\t$$\n\t\\left| \\hat{l}_n(\\beta,   \\Lambda,  \\gamma) - \\hat{l}_n^*(\\beta,  \\Lambda_{|\\tau^*}, \\gamma) \\right| \\leq C^* \\left\\{ \\frac{N^*}{n} + \\frac{Q^*}{n}\\left[1 + \\log  {\\frac{n}{N^*}}\\right]  +  \\frac{R^*}{n}  \\right\\} + o_{a.s.} (1), \n\t$$\n\twhere $N^*$, $Q^*$ and $R^*$ are binomial  with $n$ trials and success probabilities $p^*$, $q^*$ and $r^*$, respectively, and\n\t$$\n\tp^* + q^* \\leq  c^*\\mathbb{P} (T_0\\geq \\tau^*)  \\quad \\text{and}\\quad r^* \\leq c^* \\mathbb{P} (C \\in [\\tau^*, \\tau_0]) .\n\t$$\n\t\n\\end{proof}\n\n\n\n\\begin{lemma}\n\t\\label{lem:consist_full_3}\n\tAssume that for any $x$ and $z$, the conditional distribution of the censoring times given $X=x$ and $Z=z$\n\tis such that there exists a constant $C>0$ such that\n\t$$\n\t\\inf_{[t_1,t_2]\\subset [0,\\tau_0]} \\inf_{x,z}\\{ F_C(t_2|x,z)- F_C(t_1|x,z) \\}>C(t_2-t_1) ,\\qquad \\forall \\delta>0.\n\t$$\n\tLet $0<\\bar{\\tau}<\\tau_0$ and $\\epsilon>0$. There exist  $c_1, c_2>0$, $\\delta>0$ such that $c_1+c_2=1$ and \n\n\n\t\\[\n\t\\begin{split}\n\t&\\inf\\Bigg\\{\\mathbb E [\\ell (Y^*,\\Delta^*,X,Z;\\beta_0,\\Lambda_{0|\\tau^*} ,\\gamma_0)]-\t\\mathbb E [\\ell (Y^*,\\Delta^*,X,Z; \\beta, \\Lambda_{|\\tau^*},\\gamma_0)]:\\\\\n\t&\\left.\\qquad\\quad \\bar\\tau+\\delta\\leq \\tau^*<\\tau_0,\\quad \\|\\beta-\\beta_0\\|\\geq c_1\\epsilon\\quad\\text{ or}\\quad \\sup\\limits_{t\\in[0,\\bar\\tau]}|\\Lambda(t)-\\Lambda_0(t)|\\geq c_2\\epsilon\\right\\}>0\n\t\\end{split}\n\t\\]\n\\end{lemma}\n\\begin{proof}\n\n\tNote that, for any $\\tau^*\\in(\\bar\\tau,\\tau_0)$,\n\t\\[\n\t\\mathbb E [\\ell (Y^*,\\Delta^*,X,Z;\\beta_0,\\Lambda_{0|\\tau^*} ,\\gamma_0)]-\t\\mathbb E [\\ell (Y^*,\\Delta^*,X,Z; \\beta, \\Lambda_{|\\tau^*},\\gamma_0)]\n\t\\]\n\tis the Kullback-Leibler divergence $KL(\\mathbb{P}|Q)$, where $\\mathbb{P}$ and $Q$ are the probability measures of $(Y^*,\\Delta^*,X,Z)$ when the true parameters are $(\\Lambda_0,\\beta_0,\\gamma_0)$ and $(\\Lambda,\\beta,\\gamma_0)$ respectively. By Pinsker's inequality, we have\n\t\\[\n\tKL(\\mathbb{P}|Q)\\geq 2\\delta(\\mathbb{P},Q)^2,\n\t\\]\n\twhere $\\delta(\\mathbb{P},Q)$ is the total variation distance between the two probability measures, defined as \n\t\\[\n\t\\delta(\\mathbb{P},Q)=\\sup_{A}|\\mathbb{P}(A)-Q(A)|,\n\t\\]\n\twhere the supremum is taken over all measurable sets $A$. We want to find a positive lower bound for $\\delta(\\mathbb{P},Q)$ independent of $\\tau^*$ and $Q$, for all $Q$ such that $\\|\\beta-\\beta_0\\|\\geq c_1\\epsilon$  or $ \\sup_{t\\in[0,\\bar\\tau]}|\\Lambda(t)-\\Lambda_0(t)|\\geq c_2\\epsilon$. Hence, it is sufficient to find $k>0$ and for each such $Q$ an event $A$, {which could depend on $Q$}, for which $|\\mathbb{P}(A)-Q(A)|>k.$  Without loss of generality we can assume  that the covariate vector $Z$ has mean zero. \n\n\t\n\t\\bigskip\n\t\\textit{Case 1.} If $\\sup_{t\\in[0,\\bar\\tau]}|\\Lambda(t)-\\Lambda_0(t)|\\geq c_2\\epsilon$, there exists $\\bar{t}\\in[0,\\bar\\tau] $ such that either\n\t\\begin{equation}\\label{eq:cas1}\n\t\\Lambda(\\bar{t})\\geq\\Lambda_0(\\bar{t})+c_2\\epsilon\n\t\\end{equation}\n\tor \n\t\\begin{equation}\\label{eq:cas2}\n\t\\Lambda(\\bar{t})\\leq\\Lambda_0(\\bar{t})-c_2\\epsilon.\n\t\\end{equation} \nWe first  consider  \\eqref{eq:cas1} and define \n\t$$\n\t\\delta=\\min\\left\n\t\\{\\frac{\\tau_0-\\bar\\tau}{2} ,\\;\\frac{c_2\\epsilon}{2\\sup_{t\\in[0, (\\bar\\tau+\\tau_0)\/2]}\\lambda_0(t)}\\right\\}.\n\t$$   \n\tIt follows that for all \n\t$t\\in [\\bar{t},\\bar{t}+\\delta]\\subset [0, (\\bar\\tau+\\tau_0)\/2] \\subset[0,{\\tau_0})  $ we have $\\Lambda(t)\\geq\\Lambda_0(t)+\\frac12c_2\\epsilon$. Indeed, we can write \n\t\\begin{equation*\n\t\\Lambda(t)\\geq \\Lambda(\\bar{t})\\geq\\Lambda_0(\\bar{t})+c_2\\epsilon\\geq \\Lambda_0(t)-\\delta\\sup_{u\\in[0,(\\bar\\tau+\\tau_0)\/2]}\\lambda_0(u)+c_2\\epsilon\\geq \\Lambda_0(t)+\\frac12c_2\\epsilon,\\quad \\forall t\\in [\\bar{t},\\bar{t}+\\delta] .\n\t\\end{equation*}\n\n\tSince  $Z$ has mean zero, $(\\beta-\\beta_0)^\\prime Z$ also has zero mean. \n\tMoreover, since $B$ is compact and $Z$ is bounded non degenerated variance, we have\n\t\\begin{equation}\\label{whynot}\n\t\\inf_{\\beta\\in B} \\mathbb{P}((\\beta-\\beta_0)^\\prime Z\\geq 0)>0\\quad \\text{ and } \\quad \\inf_{\\beta\\in B} \\mathbb{P}((\\beta-\\beta_0)^\\prime Z\\leq 0)>0\n\t\\end{equation}\n\t{(see proof below)}.\n\tLet $A_\\beta$ be the event $\\{\\Delta^*=0, Y^*\\in[\\bar{t},\\bar{t}+\\delta],{(\\beta-\\beta_0)^\\prime Z}\\geq 0\\}$, which depends on $\\beta$ and thus on $Q$. However, by \\eqref{whynot} and the construction of the model, the event $A_\\beta$ has positive probability which stays bounded away from zero. Moreover, we have\n\t\\[\n\t\\begin{split}\n{\\mathbb{P}(A_\\beta)-Q(A_\\beta)}&=\\iint\\limits_{{(\\beta-\\beta_0)^\\prime z\\geq 0}}\\int_{\\bar{t}}^{\\bar{t}+\\delta}\\phi(\\gamma_0,x)\\\\\n\t&\\qquad \\quad\\times \\left\\{\\exp\\left(-\\Lambda_0(t)e^{\\beta_0^\\prime z}\\right)-\\exp\\left(-\\Lambda(t)e^{\\beta ^\\prime z}\\right)\\right\\} F_C(\\mathrm{d} t|x,z) G(\\mathrm{d} x,\\mathrm{d} z).\n\t\\end{split}\n\t\\]\n\tWhenever $(\\beta-\\beta_0)^\\prime z\\geq 0$, by the mean value theorem, we obtain\n\t\\[\n\t\\begin{split}\n\t\\exp\\left(-\\Lambda_0(t)e^{\\beta_0^\\prime z}\\right)-\\exp\\left(-\\Lambda(t)e^{\\beta ^\\prime  z}\\right)\n\t&=\\left\\{\\Lambda(t)e^{\\beta^\\prime z}-\\Lambda_0(t)e^{\\beta_0 ^\\prime z}\\right\\}e^{-\\xi}\\\\\n\t&=\\left[\\{\\Lambda(t)-\\Lambda_0(t)\\}e^{\\beta_0^\\prime z}+\\Lambda(t)\\{e^{\\beta^\\prime z}-e^{\\beta_0 ^\\prime z}\\}\\right] e^{-\\xi}\\\\\n\t& { \\geq \\{\\Lambda(t)-\\Lambda_0(t)\\}e^{\\beta_0^\\prime z} e^{-\\xi},}\n\t\\end{split}\n\t\\]\n\tfor some $\\xi>0$ such that $|\\xi-\\Lambda_0(t)e^{\\beta_0^\\prime z}|\\leq |\\Lambda(t)e^{\\beta^\\prime z}-\\Lambda_0(t)e^{\\beta_0^\\prime z}|$, {$t \\in[\\bar{t}, \\bar{t}+\\delta]$. }\n\tNow, let \n\t\\[\n\tM(t)=\\Lambda_0(t)\\frac{\\sup_{\\beta,z}e^{\\beta^\\prime z}}{\\inf_{\\beta,z}e^{\\beta ^\\prime z}}+\\frac{\\log 2}{\\inf_{\\beta,z}e^{\\beta^\\prime  z}},\\quad {t \\in[\\bar{t}, \\bar{t}+\\delta]. }\n\t\\]\n\tThen, for $(\\beta-\\beta_0)^\\prime z\\geq 0$ and $t\\in[\\bar{t},\\bar{t}+\\delta],$ such that  $\\Lambda(t)\\leq M(t)$ we simply use \\eqref{eq:cas1} and write \n\t\\[\n\t\\exp\\left(-\\Lambda_0(t)e^{\\beta_0^\\prime z}\\right)-\\exp\\left(-\\Lambda(t)e^{\\beta^\\prime  z}\\right)\\geq \\frac12 c_2\\epsilon e^{\\beta_0^\\prime z}e^{-\\xi}\\geq k_1\\epsilon ,\n\t\\]\n\tfor some constant $k_1>0$ independent of $\\Lambda$, $\\beta$ and the event $A_\\beta$, because $M(t)$ is uniformly bounded on $[0,(\\bar{\\tau}+\\tau_0)\/2]$ and thus $e^{-\\xi}$ is bounded away from zero.  On the other hand, for $t\\in[\\bar{t},\\bar{t}+\\delta]$ such that  $\\Lambda(t)>M(t)$, we have \n\t\\[\n\t\\exp\\left(-\\Lambda(t)e^{\\beta^\\prime z}\\right)\\leq \\exp\\left(-M(t)\\inf_{\\beta,z}e^{\\beta ^\\prime z}\\right)\\leq \\frac{1}{2}\\exp\\left(-\\Lambda_0(t)e^{\\beta_0^\\prime z}\\right).\n\t\\] \n\tConsequently,\n\t\\begin{multline*}\n\t\\exp\\left(-\\Lambda_0(t)e^{\\beta_0^\\prime z}\\right)-\\exp\\left(-\\Lambda(t)e^{\\beta ^\\prime z}\\right)\\geq \\frac{1}{2}\\exp\\left(-\\Lambda_0(t)e^{\\beta_0 ^\\prime z}\\right) \\\\ \\geq \\frac{1}{2}\\exp\\left(-\\Lambda_0((\\bar\\tau+\\tau_0)\/2)\\sup_{\\beta,z}e^{\\beta_0 ^\\prime z}\\right)=k_2(\\bar\\tau)>0.\n\t\\end{multline*}\n\tWe conclude that, for any $t\\in [\\bar{t},\\bar{t}+\\delta]$,\n\t\\[\n\t\\exp\\left(-\\Lambda_0(t)e^{\\beta_0^\\prime z}\\right)-\\exp\\left(-\\Lambda(t)e^{\\beta ^\\prime z}\\right)\\geq {\\min\\{k_1\\epsilon , k_2(\\bar\\tau)\\}}>0.\n\t\\]\n\tIt follows that \n\t\\begin{multline*}\n\t|\\mathbb{P}(A_\\beta)\\! -Q(A_\\beta)|\\geq {\\min\\{k_1\\epsilon , k_2(\\bar\\tau)\\}}\\inf_x\\phi(\\gamma_0,x)\\\\\n\t\\times \\iint\\limits_{{(\\beta-\\beta_0)^\\prime z\\geq 0}} \\left\\{F_C(\\bar{t}+\\delta|x,z)\\!-F_C(\\bar{t}|x,z)\\right\\} G(\\mathrm{d} x,\\mathrm{d} z).\n\t\\end{multline*}\n\tBy assumption  we have  \n\t\\[\n\t\\inf_{x,z}F_C(\\bar{t}+\\delta|x,z)-F_C(\\bar{t}|x,z)\\geq C \\delta,\n\t\\] \n\tyielding  that there exist another constant $k_3>0$  independent of $\\Lambda$, $\\beta$ and the event {$A_\\beta$ (but depending on $\\epsilon$ and $\\bar \\tau$)} such that\n\t\\[\n\t\\forall \\beta\\in B,\\quad |\\mathbb{P}(A_\\beta)-Q(A_\\beta)|\\geq k_3 \\inf_{\\beta\\in B} \\mathbb{P}((\\beta-\\beta_0)^\\prime Z\\geq 0)\n\n\t>0.\n\t\\]\n\tNote that {the uniform lower bound} holds for any choice of the constants $c_1$ and $c_2$ in the statement of the Lemma. \n\n\n\n\n\t\n We next consider  \\eqref{eq:cas2}. Let\n\t$$\n\t{\\bar\\delta}=\\min\\left\\{\\frac{\\bar \\tau}{2},\\, \\frac{c_2\\epsilon}{2\\sup_{t\\in[0,\\bar\\tau]}\\lambda_0(t)}  \\right\\}.\n\t$$   \n\tIt follows that for all \n\t$t\\in [\\, \\bar{t}-{\\bar\\delta},\\bar{t}\\, ]  $ we have \n\t$\\Lambda(t)\\leq\\Lambda_0(t)-\\frac12c_2\\epsilon$. Indeed, we can write \n\t\\begin{multline*}\n\t\\Lambda(t)\\leq \\Lambda(\\bar{t}) \\leq \\{ \\Lambda_0(\\bar{t}) - \\Lambda_0(t) \\} + \\Lambda_0(t) - c_2\\epsilon\\\\\n\t\\leq {\\bar\\delta}\\sup_{u\\in[0,\\bar{\\tau}]}\\lambda_0(u)+ \\Lambda_0(t) - c_2\\epsilon \\leq \n\t\\Lambda_0(t)-\\frac12c_2\\epsilon,\\quad \\forall t\\in [\\, \\bar{t}-{\\bar\\delta},\\bar{t}\\, ] .\n\t\\end{multline*}\n\tNext we redefine  $A_\\beta$ as the event $\\{\\Delta^*=0, Y^*\\in[\\bar{t}-{\\bar\\delta},\\bar{t}\\,],{(\\beta-\\beta_0)^\\prime Z}\\leq 0\\}$, which depends on $\\beta$ and thus on $Q$. However, by \\eqref{whynot} and the construction of the model, the event $A_\\beta$ has positive probability which stays bounded away from zero. Moreover, we have\n\t\\[\n\t\\begin{split}\n\t\\mathbb{P}(A_\\beta)-Q(A_\\beta)&=\\iint\\limits_{{(\\beta-\\beta_0)^\\prime z\\leq 0}}\\int_{\\bar{t}-{\\bar\\delta}}^{\\bar{t}}\\phi(\\gamma_0,x)\\\\\n\t&\\qquad \\quad\\times \\left\\{\\exp\\left(-\\Lambda_0(t)e^{\\beta_0^\\prime z}\\right)-\\exp\\left(-\\Lambda(t)e^{\\beta ^\\prime z}\\right)\\right\\} F_C(\\mathrm{d} t|x,z) G(\\mathrm{d} x,\\mathrm{d} z).\n\t\\end{split}\n\t\\]\n\tWhenever $(\\beta-\\beta_0)^\\prime z\\leq 0$, by the mean value theorem, we obtain\n\t\\[\n\t\\begin{split}\n\t\\exp\\left(-\\Lambda_0(t)e^{\\beta_0^\\prime z}\\right)-\\exp\\left(-\\Lambda(t)e^{\\beta ^\\prime  z}\\right)\n\t\\leq \\{\\Lambda(t)-\\Lambda_0(t)\\}e^{\\beta_0^\\prime z} e^{-\\xi} \\leq - \\frac12 c_2\\epsilon e^{\\beta_0^\\prime z}e^{-\\xi},\n\t\\end{split}\n\t\\]\n\tfor some $\\xi>0$ such that $|\\xi-\\Lambda_0(t)e^{\\beta_0^\\prime z}|\\leq |\\Lambda(t)e^{\\beta^\\prime z}-\\Lambda_0(t)e^{\\beta_0^\\prime z}|\\leq 2\\Lambda_0(t)e^{\\beta_0^\\prime z} $, $t \\in[\\bar{t}-{\\bar\\delta}, \\bar{t}\\,]$. Thus necessarily $0<\\xi \\leq 2\\Lambda_0(\\bar\\tau)e^{\\beta_0^\\prime z}$, and thus $e^{-\\xi}$ stays away from zero. \n\tUsing arguments as we used for the case \\eqref{eq:cas2}, we deduce that $\\mathbb{P}(A_\\beta)-Q(A_\\beta)$ is negative and away from zero. Thus we obtain the result with \n\t$\\bar\\tau \\leq \\tau^*<\\tau_0$ instead of $\\bar\\tau+\\delta \\leq \\tau^*<\\tau_0$.\n\tFinally, it remains to recall that $\\inf$ is a decreasing function of nested sets. Now the arguments for \\emph{Case 1} are complete for any choice of the constants $c_1$ and $c_2$ in the statement of the Lemma. \n\t\n\t\n\t\\bigskip\n\t\\textit{Case 2.} If $\\sup_{t\\in[0,\\bar\\tau]}|\\Lambda(t)-\\Lambda_0(t)|\\leq c_2\\epsilon$, then necessarily $\\|\\beta-\\beta_0\\|\\geq c_1\\epsilon.$ In particular we also have that $\\Lambda(\\bar{\\tau})\\leq\\Lambda_0(\\bar{\\tau})+c_2\\epsilon$, so all such functions $\\Lambda$ are uniformly bounded on $[0,\\bar{\\tau}]$. Without loss of generality we can also assume that\n\t$\\Lambda_0(\\bar\\tau\/2)\\geq 1$ (otherwise we can take a larger $\\bar\\tau$).\n\tNote that \n\t$$\n\tVar ((\\beta-\\beta_0)^\\prime Z) = (\\beta-\\beta_0)^\\prime Var (Z) (\\beta-\\beta_0) \\geq ( c_1\\epsilon)^2 \\lambda_{min}   ,\n\t$$\n\twith $\\lambda_{min} $ the smallest eigenvalue of $Var(Z)$. From this lower bound for the variance of $(\\beta-\\beta_0)^\\prime Z$, and since $Z$ is centered and has a bounded support, we have\n\n\t\\begin{equation}\\label{whynot2}\n\t\\inf_{|\\beta-\\beta_0|\\geq c_1\\epsilon}\\left[ \\mathbb{P}((\\beta-\\beta_0)^\\prime Z\\geq z_0)+ \\mathbb{P}((\\beta-\\beta_0)^\\prime Z\\leq - z_0)\\right] >\\frac{( c_1\\epsilon)^2 \\lambda_{min}}{2\\sup \\|Z\\|^2},\n\t\\end{equation}\n\tfor $z_0= c_1\\epsilon \\lambda^{1\/2}_{min}\/2  $ {(see proof below)}.\n\tIf $$\\inf_{|\\beta-\\beta_0|\\geq c_1\\epsilon}\\mathbb{P}\\left((\\beta-\\beta_0)^\\prime Z\\geq z_0\\right)> \\frac{( c_1\\epsilon)^2\\lambda_{min}}{2\\sup \\|Z\\|^2},$$ let $A_\\beta$ be the event $\\{\\Delta^*=0,Y^*\\in[\\bar\\tau\/2,\\bar\\tau],(\\beta-\\beta_0)^\\prime Z \\geq z_0\\}$. \n\tBy \\eqref{whynot2} and the construction of the model, the event $A_\\beta$ has positive probability which stays bounded away from zero.\n\tNext, as in \\emph{Case 1},  we write\n\t\\[\n\t\\begin{split}\n\t\\mathbb{P}(A_\\beta) - Q(A_\\beta)&= \\iint\\limits_{(\\beta-\\beta_0)^\\prime  z \\geq z_0}\\int_{\\frac12\\bar{\\tau}}^{\\bar{\\tau}}\\phi(\\gamma_0,x)\\\\\n\t&\\qquad   \\qquad \\times \\left\\{\\exp\\left(-\\Lambda_0(t)e^{\\beta_0^\\prime z}\\right)-\\exp\\left(-\\Lambda(t)e^{\\beta^\\prime z}\\right)\\right\\} F_C(\\mathrm{d} t|x,z)\\, G(\\mathrm{d} x,\\mathrm{d} z) ,\n\t\\end{split}\n\t\\]\n\tand\n\t\\[\n\t\\begin{split}\n\t\\exp\\left(-\\Lambda_0(t)e^{\\beta_0^\\prime z}\\right)-\\exp\\left(-\\Lambda(t)e^{\\beta^\\prime z}\\right)=\\left[\\{\\Lambda(t)-\\Lambda_0(t)\\}e^{\\beta_0^\\prime z}+\\Lambda(t)\\{e^{\\beta ^\\prime z}-e^{\\beta_0 ^\\prime z}\\}\\right] e^{-\\xi},\n\t\\end{split}\n\t\\]\n\tfor some $\\xi>0$ such that $|\\xi-\\Lambda_0(t)e^{\\beta_0 ^\\prime z}|\\leq |\\Lambda(t)e^{\\beta^\\prime z}-\\Lambda_0(t)e^{\\beta_0^\\prime z}|$. From the boundedness of $\\beta$, $z$,  $\\Lambda$ and $\\Lambda_0$ on $[0,\\bar\\tau]$, it follows that $e^{-\\xi}\\geq k_4>0$ for some $k_4$ independent of $\\Lambda,$ $\\beta$ and the event $A_\\beta$ (but depending on $\\bar\\tau$).  Moreover, since for $t\\in[\\bar\\tau\/2,\\bar\\tau]$, $(\\beta-\\beta_0)^\\prime  z\\geq z_0$, \n\t\\[\n\t\\left|\\Lambda(t)-\\Lambda_0(t)\\right|e^{\\beta_0^\\prime z}\\leq c_2\\epsilon e^{\\beta_0^\\prime z},\n\t\\]\n\tand \n\t\\[\n\t\\begin{split}\n\n\n\n\t\\Lambda(t)\\{e^{\\beta^\\prime z}-e^{\\beta_0^\\prime z}\\}\n\t\\geq \\Lambda_0 (\\bar\\tau\/2) e^{\\beta_0^\\prime z}\\{   e^{(\\beta-\\beta_0)^\\prime z} - 1\\}\\geq e^{\\beta_0^\\prime z}z_0 =  e^{\\beta_0^\\prime z}\\lambda^{1\/2}_{min}  \\,c_1\\epsilon \/2,\n\t\\end{split}\n\t\\]\n\twe obtain \n\t\\[\n\t\\exp\\left(-\\Lambda_0(t)e^{\\beta_0^\\prime z}\\right)-\\exp\\left(-\\Lambda(t)e^{\\beta ^\\prime z}\\right)\\geq \\epsilon \\left[ \\lambda^{1\/2}_{min}  \\,c_1\/2 - c_2\\right] e^{\\beta_0^\\prime z}e^{-\\xi} \n\t\\]\n\tDefine\n\t\\[\n\tc_1= \\frac{4}{\\lambda_{min}^{1\/2} + 4}\\qquad\\text{and}\\qquad c_2= \\frac{\\lambda_{min}^{1\/2}}{\\lambda_{min}^{1\/2} + 4},\n\t\\]\n\tsuch that $0<c_1,c_2<1$ and $c_1+c_2=1$, and deduce that\n\t\\[\n\t\\exp\\left(-\\Lambda_0(t)e^{\\beta_0^\\prime z}\\right)-\\exp\\left(-\\Lambda(t)e^{\\beta ^\\prime z}\\right)\\geq   \\epsilon \\lambda^{1\/2}_{min}   e^{\\beta_0^\\prime z}k_4 \\left\\{ \\lambda_{min}^{1\/2} + 4 \\right\\}^{-1} \n\t\\]\n\tDeduce that, for any $\\|\\beta-\\beta_0\\|\\geq c_1\\epsilon$,\n\t\\begin{multline*}\n\t\\mathbb{P}(A_\\beta)-Q(A_\\beta)\\geq \\epsilon\\left[ \\lambda^{1\/2}_{min}   e^{\\beta_0^\\prime z}k_4  \\left\\{ \\lambda_{min}^{1\/2} + 4\\right\\}^{-1} \\right]\\inf_{z}e^{\\beta_0^\\prime z}\\inf_x\\phi(\\gamma_0,x) \\\\ \\times \\inf_{|\\beta-\\beta_0|\\geq c_1\\epsilon} \\mathbb{P}(C\\in[\\bar\\tau\/2,\\bar\\tau],(\\beta-\\beta_0)^\\prime Z \\geq z_0)>0.\n\t\\end{multline*}\n\tNote that this bound does not depend on $\\delta$ in the statement of the Lemma. Finally it is easy to see that similar arguments apply when $\\inf_{|\\beta-\\beta_0|\\geq c_1\\epsilon}\\mathbb{P}((\\beta-\\beta_0)^\\prime Z\\leq - z_0)> ( c_1\\epsilon)^2 \\lambda_{min}\/\\{4\\sup \\|Z\\|^2\\} $, for the same expression of $z_0$. In this case, we define \n\t$A_\\beta= \\{\\Delta^*=0,Y^*\\in[\\bar\\tau\/2,\\bar\\tau],(\\beta-\\beta_0)^\\prime Z \\leq -z_0\\}$ and follow the same steps as above to show that \n\t\\[\n\t\\exp\\left(-\\Lambda_0(t)e^{\\beta_0^\\prime z}\\right)-\\exp\\left(-\\Lambda(t)e^{\\beta ^\\prime z}\\right) <0,\n\t\\]\n\tand the difference of exponentials stays away from zero.  Now, the proof of Lemma \\ref{lem:consist_full_3} is complete. \n\\end{proof}\n\n\n\n\\begin{proof}[\\textsc{Proof of Equation \\eqref{whynot}.}]\n\tFor $\\beta=\\beta_0$ we have $\\mathbb{P}((\\beta-\\beta_0)^\\prime Z\\geq 0)=1$ and thus we only have to study $\\beta\\neq \\beta_0$. Since  $Var(Z)$ is non degenerated, $\\mathbb{P}((\\beta-\\beta_0)^\\prime Z= 0)<1$ for any $\\beta\\neq \\beta_0$. Next, we could write \n\t\\begin{multline*}\n\t0 = \\mathbb E \\left( \\frac{(\\beta-\\beta_0)^\\prime Z}{\\|\\beta-\\beta_0\\|}  \\right) = \\mathbb E  \\left( \\frac{(\\beta-\\beta_0)^\\prime Z}{\\|\\beta-\\beta_0\\|}  \\mathds{1}_{\\{ (\\beta-\\beta_0)^\\prime Z \\geq 0\\}  }\\right) + \n\t\\mathbb E  \\left( \\frac{(\\beta-\\beta_0)^\\prime Z}{\\|\\beta-\\beta_0\\|}  \\mathds{1}_{ \\{ (\\beta-\\beta_0)^\\prime Z < 0 \\} }\\right) \\\\ \\leq \\|Z\\| \\mathbb{P}((\\beta-\\beta_0)^\\prime Z\\geq 0) +\\mathbb E  \\left( \\frac{(\\beta-\\beta_0)^\\prime Z}{\\|\\beta-\\beta_0\\|} \\mathds{1}_{\\{ (\\beta-\\beta_0)^\\prime Z < 0 \\} }\\right) .\n\t\\end{multline*}\n\tIt remains to notice that \n\t$$\\sup_{\\|\\tilde \\beta\\|=1}  \\mathbb E  \\left( \\tilde \\beta^\\prime Z\\mathds{1}_{\\{ \\tilde \\beta^\\prime Z < 0 \\} }\\right)<0.$$\n\tIndeed, if $\\mathbb E  \\left( \\tilde \\beta^\\prime Z\\mathds{1}_{\\{ \\tilde \\beta^\\prime Z < 0 \\} }\\right)$, which is negative,  could be arbitrarily close to zero, and since $\\mathbb E  \\left( \\tilde \\beta^\\prime Z\\mathds{1}_{\\{ \\tilde \\beta^\\prime Z < 0 \\} }\\right)= - \\mathbb E  \\left( \\tilde \\beta^\\prime Z\\mathds{1}_{\\{ \\tilde \\beta^\\prime Z \\geq 0 \\} }\\right)$, we deduce that $\\mathbb E  (| \\tilde \\beta^\\prime Z|)$ could be arbitrarily close to zero, for suitable $ \\tilde \\beta$ with unit norm. Since the support of $Z$ is bounded and \n\t$$\n\t\\lambda_{min} (Var(Z)) \\leq \\mathbb E  (| \\tilde \\beta^\\prime Z|^2)\\leq \\mathbb E  (| \\tilde \\beta^\\prime Z|)\\sup \\|Z\\|, \\quad \\forall  \\|\\tilde \\beta\\|=1,\n\t$$\n\twe thus get a contradiction with the assumption that $\\lambda_{min} $, the smallest eigenvalue of $Var(Z)$, is positive. \n\tDeduce that \\eqref{whynot} holds true. \\end{proof}\n\n\n\n\n\\quad\n\n\\begin{proof}[\\textsc{Proof of Equation \\eqref{whynot2}.}]\n\tIt suffices to prove the following property.  \n\tLet $U$ be a centered variable such that $|U|\\leq M$ for some constant $M$ and $Var(U)$ is bounded from below by some constant $0<C<M^2$. Then there exists $z_0>0$ such that $\\mathbb{P}(|U|\\geq z_0)> C\/2M^2$ with $z_0$ depending on $M$ and $C$ but independent of the law of $U$. \n\t\n\tFor any $0<z_0 <M$ we can write \n\t$$\n\tC\\leq \\mathbb E (U^2) = \\mathbb E \\left( U^2 \\mathds{1} _{ \\{ |U| \\geq z_0 \\} } \\right) +\n\t\\mathbb E \\left( U^2 \\mathds{1} _{ \\{ |U| < z_0 \\} } \\right)\\\\\n\t\\leq M^2 \\mathbb{P} (|U| \\geq z_0 ) + z_0^2 \\{ 1 - \\mathbb{P} (|U| \\geq   z_0 ) \\}.\n\t$$\n\tDeduce that \n\t$$\n\t\\mathbb{P} (|U| \\geq   z_0 ) \\geq \\frac{C-z_0^2}{M^2-z_0^2}.\n\t$$\n\tFinally, it suffices to take $z_0^2 \\leq  CM^2\/(2M^2 - C)$ in order to obtain \n\t$$\n\t\\mathbb{P} (|U| \\geq   z_0 ) \\geq \\frac{C}{2M^2}.\n\t$$\n\tSince $C$ could be arbitrarily small,  we could take $z_0^2 =  C\/4$. \\end{proof}\n\\subsection{Asymptotic normality}\n\\begin{proof}[\\textsc{Proof of Theorem \\ref{theo:asymptotic_normality}.}]\nLet $m$ be as in \\eqref{def:m} and define $M_n(\\gamma,\\pi)=\\frac{1}{n}\\sum_{i=1}^{n}m(X_i;\\gamma,\\pi)$ and $M(\\gamma,\\pi)=\\mathbb{E}[m(X;\\gamma,\\pi)]$. Note that $m(x;\\gamma_0,\\pi_0)=0$ and $M(\\gamma_0,\\pi_0)=0$. If we take partial derivatives with respect to  $\\gamma$ of the vector $M(\\gamma,\\pi)$ evaluated at $(\\gamma_0,\\pi_0)$ we obtain a matrix $\\nabla_\\gamma M(\\gamma_0,\\pi_0)$ with elements\n\\[\n\\nabla_\\gamma M(\\gamma_0,\\pi_0)_{kl}=\\mathbb{E}\\left[\\frac{\\partial}{\\partial \\gamma_l}\\left\\{\\frac{1-\\pi(X)}{\\phi(\\gamma,X)}-\\frac{\\pi(X)}{1-\\phi(\\gamma,X)} \\right\\}\\frac{\\partial}{\\partial \\gamma_k}\\phi(\\gamma,X)\\right]\\Bigg|_{\\gamma_0,\\pi_0}, \n\\]\nfor $k,l\\in\\{1,\\dots,p\\}$, because \n\\[\n\\frac{1-\\pi_0(x)}{\\phi(\\gamma_0,x)}-\\frac{\\pi_0(x)}{1-\\phi(\\gamma_0,x)}=0\\quad\\forall x\\in\\mathcal{X}.\n\\]\nHence, \n\\begin{equation}\n\\label{def:gradient_M}\n\\Gamma_1:=\\nabla_\\gamma M(\\gamma_0,\\pi_0)=-\\mathbb{E}\\left[W(X)\\nabla_\\gamma\\phi(\\gamma_0,X)\\nabla_\\gamma\\phi(\\gamma_0,X)'\\right]\n\\end{equation}\nwhere\n\\[\nW(X)=\\frac{1-\\pi_0(X)}{\\phi(\\gamma_0,X)^2}+\\frac{\\pi_0(X)}{\\left(1-\\phi(\\gamma_0,X)\\right)^2}=\\frac{1}{\\phi(\\gamma_0,X)}+\\frac{1}{1-\\phi(\\gamma_0,X)}>0.\n\\]\nWe will also use the Gateaux derivative of $M(\\gamma,\\pi_0)$ in a direction $[\\pi-\\pi_0]$ given by\n{\\small\\begin{equation}\n\\label{eqn:directional_derivative}\n\\begin{split}\n\\Gamma_2(\\gamma,\\pi_0)[\\pi-\\pi_0]&:=\\nabla_\\pi M(\\gamma,\\pi_0)[\\pi-\\pi_0]\\\\\n&=\\lim_{h\\to 0}\\frac{1}{h}\\left[M(\\gamma,\\pi_0+h(\\pi-\\pi_0))-M(\\gamma,\\pi_0)\\right]\\\\\n&=-\\lim_{h\\to 0}\\frac{1}{h}\\mathbb{E}\\left[\\left\\{\\frac{h\\left\\{\\pi(X)-\\pi_0(X)\\right\\}}{\\phi(\\gamma,X)}+\\frac{h\\left\\{\\pi(X)-\\pi_0(X)\\right\\}}{1-\\phi(\\gamma,X)} \\right\\}\\nabla_\\gamma\\phi(\\gamma,X)\\right]\\\\\n&=-\\mathbb{E}\\left[\\left\\{\\pi(X)-\\pi_0(X)\\right\\}\\left\\{\\frac{1}{\\phi(\\gamma,X)}+\\frac{1}{1-\\phi(\\gamma,X)} \\right\\}\\nabla_\\gamma\\phi(\\gamma,X)\\right].\n\\end{split}\n\\end{equation} }               \n\tWe apply Theorem 2 in \\cite{chen} so we need to verify its conditions. Consistency of $\\hat\\gamma_n$ is shown in Theorem \\ref{theo:consistency}, while condition (2.1) in \\cite{chen} is satisfied by construction since \n\t\\[\n\t\\Vert M_n(\\hat\\gamma_n,\\hat\\pi_n)\\Vert=0=\\inf_{\\gamma\\in G}\\Vert M_n(\\gamma,\\hat\\pi_n)\\Vert.\n\t\\] \n\tNote that assumption (AC1) was needed in Theorem \\ref{theo:consistency} in order to obtain almost sure convergence. However, here we only need convergence in probability for which (AN4)-(ii) suffices. \n\tFor condition (2.2) in \\cite{chen}, the derivative of $M$ with respect to  $\\gamma$ is computed in \\eqref{def:gradient_M} and the matrix is negative definite (as a result also full rank) because of our assumption (AN3).  Moreover the directional derivative was computed in \\eqref{eqn:directional_derivative} and for $(\\gamma,\\pi)\\in G_{\\delta_n}\\times\\Pi_{\\delta_n}$ with $G_{\\delta_n}=\\{\\gamma\\in G: \\Vert\\gamma-\\gamma_0\\Vert\\leq\\delta_n\\}$, $\\Pi_{\\delta_n}=\\{\\pi\\in \\Pi: \\Vert\\pi-\\pi_0\\Vert_{\\infty}\\leq\\delta_n\\}$, $\\delta_n=o(1)$, we have\n\t\\[\n\t\\begin{split}\n\t&\\left\\Vert M(\\gamma,\\pi)-M(\\gamma,\\pi_0)-\\Gamma_2(\\gamma,\\pi_0)[\\pi-\\pi_0]\\right\\Vert\\\\\n\t&=\\left\\Vert\\mathbb{E}\\left[\\left\\{\\frac{\\pi_0(X)-\\pi(X)}{\\phi(\\gamma,X)} +\\frac{\\pi_0(X)-\\pi(X)}{1-\\phi(\\gamma,X)} \\right\\}\\nabla_\\gamma\\phi(\\gamma,X)\\right]\\right.\\\\\n\t&\\quad+\\left.\\mathbb{E}\\left[\\left\\{\\pi(X)-\\pi_0(X)\\right\\}\\left\\{\\frac{1}{\\phi(\\gamma,X)} +\\frac{1}{1-\\phi(\\gamma,X)}\\right\\}\\nabla_\\gamma\\phi(\\gamma,X)\\right]\\right\\Vert=0,\n\t\\end{split}\n\t\\]\n\twhich means that condition (2.3i) is satisfied. For condition (2.3ii), we have \n\t\\[\n\t\\begin{split}\n\t&\\Gamma_2(\\gamma,\\pi_0)[\\pi-\\pi_0]-\\Gamma_2(\\gamma_0,\\pi_0)[\\pi-\\pi_0]\\\\\n\t&=-\\mathbb{E}\\left[\\left\\{\\pi(X)-\\pi_0(X)\\right\\}\\left\\{\\left( \\frac{1}{\\phi(\\gamma,X)} +\\frac{1}{1-\\phi(\\gamma,X)}\\right)\\nabla_\\gamma\\phi(\\gamma,X)\\right.\\right.\\\\\n\t&\\quad\\qquad-\\left.\\left. \\left( \\frac{1}{\\phi(\\gamma_0,X)} +\\frac{1}{1-\\phi(\\gamma_0,X)}\\right)\\nabla_\\gamma\\phi(\\gamma_0,X)\\right\\}\\right].\n\t\\end{split}\n\t\\]\n\tThen, from $\\sup_x|\\pi(x)-\\pi_0(x)|\\leq\\delta_n$, $|\\gamma-\\gamma_0|\\leq\\delta_n\\to 0$ and (AN1),\n\tit follows that \n\t\\[\n\t\\Vert\\Gamma_2(\\gamma,\\pi_0)[\\pi-\\pi_0]-\\Gamma_2(\\gamma_0,\\pi_0)[\\pi-\\pi_0]\\Vert\\leq o(1)\\delta_n.\n\t\\]\n\tConditions (2.4) and (2.6) in \\cite{chen} are satisfied thanks to our assumption (AN4) because\n\t\\begin{equation}\n\t\\label{eqn:M_n-Gamma_2}\n\t\\begin{split}\n\t&M_n(\\gamma_0,\\pi_0)+\\Gamma_2(\\gamma_0,\\pi_0)[\\hat\\pi-\\pi_0]\\\\\n\t&=\\frac{1}{n}\\sum_{i=1}^n\\left\\{\\frac{1-\\pi_0(X_i)}{\\phi(\\gamma_0,X_i)}-\\frac{\\pi_0(X_i)}{1-\\phi(\\gamma_0,X_i)}\\right\\}\\nabla_\\gamma\\phi(\\gamma_0,X_i)\\\\\n\t&\\quad+ \\mathbb{E}^*\\left[\\left\\{\\hat\\pi(X)-\\pi_0(X)\\right\\}\\left(\\frac{1}{\\phi(\\gamma_0,X)} +\\frac{1}{1-\\phi(\\gamma_0,X)}\\right)\\nabla_\\gamma\\phi(\\gamma_0,X)\\right] \\\\\n\t&=\\mathbb{E}^*\\left[\\left\\{\\hat\\pi(X)-\\pi_0(X)\\right\\}\\left(\\frac{1}{\\phi(\\gamma_0,X)} +\\frac{1}{1-\\phi(\\gamma_0,X)}\\right)\\nabla_\\gamma\\phi(\\gamma_0,X)\\right].\n\t\\end{split}\n\t\\end{equation}\n\tThen we conclude by central limit theorem that \n\t\\[\n\t\\sqrt{n}\\left(M_n(\\gamma_0,\\pi_0)+\\Gamma_2(\\gamma_0,\\pi_0)[\\hat\\pi-\\pi_0]\\right)\\xrightarrow{d} N(0,V)\n\t\\]\n\twhere $V=Var(\\Psi(Y,\\Delta,X,Z))$.  It remains to deal with condition (2.5), which is a consequence of Theorem 3 in \\cite{chen} and assumption (AN2) because from (AN1) we have\n\t\\[\n\t\\begin{split}\n\t\\Vert m(x;\\gamma_1,\\pi_1)-m(x;\\gamma_2,\\pi_2)\\Vert\n\t&\\leq \\left\\Vert\\left(\\frac{1-\\pi_1(x)}{\\phi(\\gamma_1,x)} +\\frac{\\pi_1(x)}{1-\\phi(\\gamma_1,X)}\\right)\\nabla_\\gamma\\phi(\\gamma_1,X)\\right.\\\\\n\t&\\left.\\qquad\\qquad-\\left(\\frac{1-\\pi_2(x)}{\\phi(\\gamma_2,x)} +\\frac{\\pi_2(x)}{1-\\phi(\\gamma_2,X)}\\right)\\nabla_\\gamma\\phi(\\gamma_2,X)\\right\\Vert\\\\\n\t&\\leq C_1\\Vert\\gamma_1-\\gamma_2\\Vert+C_2\\Vert\\pi_1-\\pi_2\\Vert_\\infty.\n\t\\end{split}\n\t\\]\n\tFinally, the asymptotic normality  follows from Theorem 2 in \\cite{chen} and the asymptotic covariance matrix is given by \n\t\\begin{equation}\n\t\\label{def:Sigma_gamma}\n\t\\Sigma_\\gamma=(\\Gamma'_1\\Gamma_1)^{-1}\\Gamma'_1V\\Gamma_1(\\Gamma'_1\\Gamma_1)^{-1}=\\Gamma_1^{-1}V\\Gamma_1^{-1}\n\t\\end{equation}\n\t\\end{proof}\n\\begin{proof}[\\textsc{Proof of Theorem \\ref{theo:asymptotic_normality2}.}]\n\tWe  show that conditions 1 and 4 of Theorem 4 in \\cite{Lu} are satisfied. Define $S_n$ as the version of $\\hat{S}_n$ where $\\hat\\gamma_n$ is replaced by $\\gamma_0$ \n\\[\n\\begin{split}\n{S}_n(\\hat\\Lambda_n,\\hat\\beta_n)(h_1,h_2)&=\\frac{1}{n}\\sum_{i=1}^n \\Delta_i\\mathds{1}_{\\{Y_i<\\tau_0\\}} \\left[h_1(Y_i)+h'_2Z_i\\right]\\\\\n&\\quad-\\frac{1}{n}\\sum_{i=1}^n \\left\\{ \\Delta_i+(1-\\Delta_i)\\mathds{1}_{\\{Y_i\\leq\\tau_0\\}}g_i(Y_i,\\hat\\Lambda_n,\\hat\\beta_n,\\gamma_0) \\right\\}\\\\\n&\\qquad\\qquad\\quad\\times\\left\\{e^{\\hat\\beta'_nZ_i}\\int_0^{Y_i}h_1(s)\\mathrm{d}\\hat\\Lambda_n(s)+e^{\\hat\\beta'_nZ_i}\\hat\\Lambda_n(Y_i)h'_2Z_i\n\\right\\}\n\\end{split}\n\\]\n\\textit{Condition 1.} We start by writing \n\\begin{equation}\n\\label{eqn:condition1}\n\\begin{split}\n\\hat{S}_n(\\Lambda_0,\\beta_0)(h_1,h_2)-{S}(\\Lambda_0,\\beta_0)(h_1,h_2)&=\\left[\\hat{S}_n(\\Lambda_0,\\beta_0)(h_1,h_2)-{S}_n(\\Lambda_0,\\beta_0)(h_1,h_2)\\right]\\\\\n&\\quad+\\left[{S}_n(\\Lambda_0,\\beta_0)(h_1,h_2)-{S}(\\Lambda_0,\\beta_0)(h_1,h_2)\\right].\n\\end{split}\n\\end{equation}\nFor the second term on the right hand side we have\n\\begin{equation}\n\\label{eqn:condition1_2}\n{S}_n(\\Lambda_0,\\beta_0)(h_1,h_2)-{S}(\\Lambda_0,\\beta_0)(h_1,h_2)=\\int f_h(y,\\delta,x,z)\\,\\mathrm{d}(\\mathbb{P}_n-\\mathbb{P})(y,\\delta,x,z)\n\\end{equation}\nwhere\n\\[\n\\begin{split}\nf_h(y,\\delta,x,z)&=h_2'z\\left\\{\\delta\\mathds{1}_{\\{y<\\tau_0\\}}-\\left[\\delta-(1-\\delta)\\mathds{1}_{\\{y\\leq\\tau_0\\}}g(y,\\Lambda_0,\\beta_0,\\gamma_0)\\right]e^{\\beta'_0z}\\Lambda_0(y) \\right\\}\\\\\n&+\\delta \\mathds{1}_{\\{y<\\tau_0\\}}h_1(y)-\\left[\\delta -(1-\\delta)\\mathds{1}_{\\{y\\leq\\tau_0\\}}g(y,\\Lambda_0,\\beta_0,\\gamma_0)\\right]e^{\\beta'_0z}\\int_0^{y}h_1(s)\\mathrm{d}\\Lambda_0(s).\n\\end{split}\n\\]\nThe classes  $\\{h_2\\in\\mathbb{R}^q,\\Vert h_2\\Vert\\leq \\mathfrak{m}\\} $, $\\{h_1\\in BV[0,\\tau_0],\\, \\Vert h_1\\Vert_v\\leq \\mathfrak{m}\\}$ and  $$\\left\\{\\int_0^y h_1(t)\\,\\mathrm{d}\\Lambda_0(t), \\,h_1\\in BV[0,\\tau_0], \\Vert h_1\\Vert_v\\leq \\mathfrak{m}\\right\\}$$ are Donsker classes (the last one because it consists of monotone bounded functions). As in \\cite{Lu}, because of the boundedness of the covariates and $\\Lambda_0$, it follows that $\\{f_h(y,\\delta,x,z),\\,h\\in\\mathcal{H}_\\mathfrak{m}\\}$ is also a Donsker class since it is sum of products of Donsker classes with fixed uniformly bounded functions.  \n\nOn the other hand, for the first term on the right hand side of \\eqref{eqn:condition1}, we have \n\\begin{equation}\n\\label{eqn:condition1_4}\n\\begin{split}\n&\\left[\\hat{S}_n(\\Lambda_0,\\beta_0)(h_1,h_2)-{S}_n(\\Lambda_0,\\beta_0)(h_1,h_2)\\right]\\\\\n&=-\\frac1n\\sum_{i=1}^n (1-\\Delta_i)\\mathds{1}_{\\{Y_i\\leq\\tau_0\\}}\\left\\{e^{\\beta'_0Z_i}\\int_0^{Y_i}h_1(s)\\mathrm{d}\\Lambda_0(s)+e^{\\beta'_0Z_i}\\Lambda_0(Y_i)h'_2Z_i\n\\right\\}\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\times\\left\\{g_i(Y_i,\\Lambda_0,\\beta_0,\\hat\\gamma_n)-g_i(Y_i,\\Lambda_0,\\beta_0,\\gamma_0)\\right\\}\\\\\n&=-\\frac1n\\sum_{i=1}^n (1-\\Delta_i)\\mathds{1}_{\\{Y_i\\leq\\tau_0\\}}e^{\\beta'_0Z_i}\\left\\{\\int_0^{Y_i}h_1(s)\\mathrm{d}\\Lambda_0(s)+\\Lambda_0(Y_i)h'_2Z_i\n\\right\\}\\\\\n&\\quad\\qquad\\qquad\\qquad\\qquad\\qquad\\times\\frac{\\partial g_i}{\\partial\\phi}(Y_i,\\Lambda_0,\\beta_0,\\gamma_0)\\left\\{\\phi(\\hat\\gamma_n,X_i)-\\phi(\\gamma_0,X_i)\\right\\}+o_P(n^{-1\/2}),\n\\end{split}\n\\end{equation}\nwhere \n\\[\n\\begin{split}\n\\frac{\\partial g_i}{\\partial\\phi}(Y_i,\\Lambda_0,\\beta_0,\\gamma_0)&=\\frac{\\exp\\left(-\\Lambda(Y_i) e^{\\beta'Z_j}\\right)}{1-\\phi(\\gamma,X_j)+\\phi(\\gamma,X_j)\\exp\\left(-\\Lambda(Y_i) e^{\\beta'Z_j}\\right)}\\\\\n&\\quad+\\frac{\\phi(\\gamma,X_j)\\exp\\left(-\\Lambda(Y_i)e^{\\beta'Z_j}\\right)\\left[\\exp\\left(-\\Lambda(Y_i) e^{\\beta'Z_j}\\right)-1\\right]}{\\left[1-\\phi(\\gamma,X_j)+\\phi(\\gamma,X_j)\\exp\\left(-\\Lambda(Y_i) e^{\\beta'Z_j}\\right)\\right]^2}.\n\\end{split}\n\\]\nIn order to conclude that the remainder term is of order $o_P(n^{-1\/2})$ we use \n\\[\n\\sup_x\\left|\\phi(\\hat\\gamma_n,x)-\\phi(\\gamma_0,x)\\right|\\leq \\sup_{\\gamma\\in G, x\\in\\mathcal{X}}\\left\\Vert\\nabla_\\gamma\\phi(\\gamma,x)\\right\\Vert |\\hat\\gamma_n-\\gamma_0|=O_P(n^{-1\/2})\n\\]\nand  the fact that $\\frac{\\partial^2 g_i}{\\partial\\phi^2}(Y_i,\\Lambda_0,\\beta_0,\\gamma)$ and  $$(1-\\Delta_i)\\mathds{1}_{\\{Y_i\\leq\\tau_0\\}}e^{\\beta'_0Z_i}\\left\\{\\int_0^{Y_i}h_1(s)\\mathrm{d}\\Lambda_0(s)+\\Lambda_0(Y_i)h'_2Z_i\n\\right\t\\}$$  are uniformly bounded functions thanks to our assumptions on  $Z,$ $\\Lambda$, $\\Phi$ and $h$. \nFrom the same assumptions we also obtain \n\\begin{equation}\n\\label{eqn:condition1_5}\n\\begin{split}\n&\\frac1n\\sum_{i=1}^n (1-\\Delta_i)\\mathds{1}_{\\{Y_i\\leq\\tau_0\\}}e^{\\beta'_0Z_i}\\left\\{\\int_0^{Y_i}h_1(s)\\mathrm{d}\\Lambda_0(s)+\\Lambda_0(Y_i)h'_2Z_i\n\\right\\}\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\times\\frac{\\partial g_i}{\\partial\\phi}(Y_i,\\Lambda_0,\\beta_0,\\gamma_0)\\left\\{\\phi(\\hat\\gamma_n,X_i)-\\phi(\\gamma_0,X_i)\\right\\}\\\\\n&=\\frac1n\\sum_{i=1}^n (1-\\Delta_i)e^{\\beta'_0Z_i}\\left\\{\\int_0^{Y_i}h_1(s)\\mathrm{d}\\Lambda_0(s)+\\Lambda_0(Y_i)h'_2Z_i\n\\right\\}\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\times\\frac{\\partial g_i}{\\partial\\phi}(Y_i,\\Lambda_0,\\beta_0,\\gamma_0)\\nabla_\\gamma\\phi(\\gamma_0,X_i)'\\left\\{\\hat\\gamma_n-\\gamma_0\\right\\}+o_P(n^{-1\/2})\\\\\n&=\\mathbb{E}\\left[(1-\\Delta)\\mathds{1}_{\\{Y\\leq\\tau_0\\}}e^{\\beta'_0Z}\\left\\{\\int_0^{Y}h_1(s)\\mathrm{d}\\Lambda_0(s)+\\Lambda_0(Y)h'_2Z\n\\right\\}\\right.\\\\\n&\\left.\\qquad\\qquad\\qquad\\qquad\\qquad\\times\\frac{\\partial g}{\\partial\\phi}(Y,\\Lambda_0,\\beta_0,\\gamma_0)\\nabla_\\gamma\\phi(\\gamma_0,X)'\\right]\\left(\\hat\\gamma_n-\\gamma_0\\right)+o_P(n^{-1\/2}).\n\\end{split}\n\\end{equation}\nand the expectation term is uniformly bounded. To prove the asymptotic normality of $\\hat\\gamma_n-\\gamma_0$ in Theorem \\ref{theo:asymptotic_normality} we used Theorem 2 in \\cite{chen}. Going through the proof of Theorem 2 in \\cite{chen}, we actually have \n\\[\n(\\hat\\gamma_n-\\gamma_0)=-(\\Gamma_1'\\Gamma_1)^{-1}\\Gamma'_1\\left\\{M_n(\\gamma_0,\\pi_0)+\\Gamma_2(\\gamma_0,\\pi_0)[\\hat\\pi-\\pi_0]\\right\\}+ o_P(n^{-1\/2})\n\\]\nwhere $\\Gamma_1$ is defined in \\eqref{def:gradient_M} and  $\\Gamma_2$ in \\eqref{eqn:directional_derivative}. From Assumption (AN4-iii) and \\eqref{eqn:M_n-Gamma_2}, it follows that \n\\begin{equation}\n\\label{eqn:condition_1_6}\n(\\hat\\gamma_n-\\gamma_0)=-(\\Gamma_1'\\Gamma_1)^{-1}\\Gamma'_1\t\\int \\Psi(y,\\delta,x)\\,\\d(\\mathbb{P}_n-\\mathbb{P})(y,\\delta,x,z)+ o_P(n^{-1\/2}). \n\\end{equation}\nPutting together \\eqref{eqn:condition1}-\\eqref{eqn:condition_1_6}, we have\n\\[\n\\begin{split}\n&\\hat{S}_n(\\Lambda_0,\\beta_0)(h_1,h_2)-{S}(\\Lambda_0,\\beta_0)(h_1,h_2)\\\\\n&=\\int \\left\\{ f_h(y,\\delta,x,z)-Q_h\\Gamma_1^{-1} \\Psi(y,\\delta,x)\\right\\}\\,\\mathrm{d} (\\mathbb{P}_n-\\mathbb{P})(y,\\delta,x,z)+o_P(n^{-1\/2})\n\\end{split}\n\\]\nwhere\n\\[\nQ_h=\\mathbb{E}\\left[(1-\\Delta)\\mathds{1}_{\\{Y\\leq\\tau_0\\}}e^{\\beta'_0Z}\\left\\{\\int_0^{Y}h_1(s)\\mathrm{d}\\Lambda_0(s)+\\Lambda_0(Y)h'_2Z\n\\right\\}\\frac{\\partial g}{\\partial\\phi}(Y,\\Lambda_0,\\beta_0,\\gamma_0)\\nabla_\\gamma\\phi(\\gamma_0,X)'\\right].\n\\]\nIn order to conclude the convergence of $\\sqrt{n}(\\hat{S}_n(\\Upsilon_0)-S(\\Upsilon_0))$ to a Gaussian process $G^*$, we need to have that $\\{Q_h\\Gamma_1^{-1} \\Psi(y,\\delta,x), h\\in\\mathcal{H}_\\mathfrak{m}\\}$ is a bounded Dosker class of functions (since sum of bounded Donsker classes is also Donsker). We can write \n\\[\n\\begin{split}\n&Q_h\\Gamma_1^{-1}\\Psi(y,\\delta,x)\\\\\n&=h'_2\\mathbb{E}\\left[(1-\\Delta)\\mathds{1}_{\\{Y\\leq\\tau_0\\}}Ze^{\\beta'_0Z}\\Lambda_0(Y)\n\\frac{\\partial g}{\\partial\\phi}(Y,\\Lambda_0,\\beta_0,\\gamma_0)\\nabla_\\gamma\\phi(\\gamma_0,X)'\\right]\\Gamma_1^{-1}\\Psi(y,\\delta,x)\\\\\n&\\quad+\\int_0^{\\tau_0}\\mathbb{E}\\left[(1-\\Delta)\\mathds{1}_{\\{Y\\leq\\tau_0\\}}e^{\\beta'_0Z}\\frac{\\partial g}{\\partial\\phi}(Y,\\Lambda_0,\\beta_0,\\gamma_0)\\nabla_\\gamma\\phi(\\gamma_0,X)'\\right]h_1(s)\\mathrm{d}\\Lambda_0(s) \\Gamma_1^{-1} \\Psi(y,\\delta,x)\n\\end{split}\n\\]\nBy assumption (AN1) and $\\inf_{x}H((\\tau_0,\\infty)|x)>0$, $\\Lambda_0(\\tau_0)<\\infty$ and the boundedness of the covariates,  we have that $$\\mathbb{E}\\left[(1-\\Delta)\\mathds{1}_{\\{Y\\leq\\tau_0\\}}Ze^{\\beta'_0Z}\\Lambda_0(Y)\n\\frac{\\partial g}{\\partial\\phi}(Y,\\Lambda_0,\\beta_0,\\gamma_0)\\nabla_\\gamma\\phi(\\gamma_0,X)'\\right]\\Gamma_1^{-1} \\Psi(y,\\delta,x)$$ is uniformly bounded. Hence \n\\[\n\\begin{split}\n&\\left\\{h'_2\\mathbb{E}\\left[(1-\\Delta)\\mathds{1}_{\\{Y\\leq\\tau_0\\}}Ze^{\\beta'_0Z}\\Lambda_0(Y)\n\\frac{\\partial g}{\\partial\\phi}(Y,\\Lambda_0,\\beta_0,\\gamma_0)\\nabla_\\gamma\\phi(\\gamma_0,X)'\\right]\\Gamma_1^{-1} \\Psi(y,\\delta,x):\\right.\\\\\n&\\,\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\qquad\\qquad\\qquad \\qquad\\qquad\\qquad h_2\\in\\mathbb{R}^q, \\Vert h_2\\Vert_{L_1}\\leq \\mathfrak{m}\\bigg \\}\n\\end{split}\n\\]\nis a Donsker class (see Example 2.10.10 in \\cite{VW}). It can also be shown that, since $h_1$ belongs to the class of bounded functions with bounded variation and all the other terms are uniformly bounded,  that \n\\[\n\\begin{split}\n&\\left\\{\\int_0^{\\tau_0}\\mathbb{E}\\left[(1-\\Delta)\\mathds{1}_{\\{Y\\leq\\tau_0\\}}e^{\\beta'_0Z}\\frac{\\partial g}{\\partial\\phi}(Y,\\Lambda_0,\\beta_0,\\gamma_0)\\nabla_\\gamma\\phi(\\gamma_0,X)'\\right]h_1(s)\\mathrm{d}\\Lambda_0(s) \\Gamma_1^{-1} \\Psi(y,\\delta,x), \\right.\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad \\qquad\\qquad\\qquad \\qquad\\qquad\\qquad h_1\\in BV[0,\\tau_0], \\,\\Vert h_1\\Vert_v\\leq \\mathfrak{m}\\bigg\\}\n\\end{split}\n\\]\nis also a bounded Donsker class (covering numbers of order $\\epsilon$ of $\\{ h_1\\in BV[0,\\tau_0], \\Vert h_1\\Vert_v\\leq \\mathfrak{m}\\}$ correspond to covering numbers of order $c\\epsilon$ for some constant $c>0$).\n\nThe limit process $G^*$ has mean zero because\n\\[\n\\mathbb{E}[f_h(y,\\delta,x,z)]=S(\\Upsilon_0)(h)=0\\qquad\\text{and}\\qquad\\mathbb{E}[\\Psi(Y,\\Delta,X)]=0.\n\\]\nThe covariance process of $G^*$ is \n\\begin{equation}\n\\label{eqn:cov_G*}\n\\begin{split}\n&Cov\\left(G^*(h),G^*(\\tilde{h})\\right)\\\\\n&=\\mathbb{E}\\left[\\left\\{f_h(Y,\\Delta,X,Z)-Q_h\\Gamma_1^{-1} \\Psi(Y,\\Delta,X)\\right\\}\\left\\{f_{\\tilde{h}}(Y,\\Delta,X,Z)-Q_{\\tilde{h}}\\Gamma_1^{-1} \\Psi(Y,\\Delta,X)\\right\\}\\right]\\\\\n&=\\mathbb{E}[f_h(Y,\\Delta,X,Z)f_{\\tilde{h}}(Y,\\Delta,X,Z)]-Q_h\\Gamma_1^{-1}\\mathbb{E}[f_{\\tilde{h}}(Y,\\Delta,X,Z)\\Psi(Y,\\Delta,X)]\\\\\n&\\quad-Q_{\\tilde{h}}\\Gamma_1^{-1}\\mathbb{E}[f_h(Y,\\Delta,X,Z)\\Psi(Y,\\Delta,X)]+Q_{\\tilde{h}}\\Gamma_1^{-1}\\mathbb{E}[\\Psi(Y,\\Delta,X)\\Psi(Y,\\Delta,X)']\\Gamma_1^{-1}Q'_h.\n\\end{split}\n\\end{equation}\n\\textit{Condition 4 of Theorem 4 in \\cite{Lu}. } As for condition 1, we write\n\\begin{equation}\n\\label{eqn:cond4}\n\\begin{split}\n\\sqrt{n}\\left\\{(\\hat{S}_n-S)(\\Upsilon_n)-(\\hat{S}_n-S)(\\Upsilon_0)\\right\\}&=\\sqrt{n}\\left\\{({S}_n-S)(\\Upsilon_n)-({S}_n-S)(\\Upsilon_0)\\right\\}\\\\\n&\\quad+\\sqrt{n}\\left\\{(\\hat{S}_n-S_n)(\\Upsilon_n)-(\\hat{S}_n-S_n)(\\Upsilon_0)\\right\\}\n\\end{split}\n\\end{equation}\nFor the second term in the right hand side of \\eqref{eqn:cond4}, similarly to \\eqref{eqn:condition1_4}-\\eqref{eqn:condition1_5}, we have\n\\[\n\\begin{split}\n&(\\hat{S}_n-S_n)(\\Upsilon_n)-(\\hat{S}_n-S_n)(\\Upsilon_0)\\\\\n&=\\frac1n\\sum_{i=1}^n (1-\\Delta_i)\\mathds{1}_{\\{Y_i\\leq\\tau_0\\}}e^{\\hat\\beta'_nZ_i}\\left\\{\\int_0^{Y_i}h_1(s)\\mathrm{d}\\hat\\Lambda_n(s)+\\hat\\Lambda_n(Y_i)h'_2Z_i\n\\right\\}\\\\\n&\\quad\\qquad\\qquad\\qquad\\qquad\\qquad\\frac{\\partial g_i}{\\partial\\phi}(Y_i,\\hat\\Lambda_n,\\hat\\beta_n,\\gamma_0)\\nabla_\\gamma\\phi(\\gamma_0,X_i)'\\left\\{\\hat\\gamma_n-\\gamma_0\\right\\}\\\\\n&\\quad-\\frac1n\\sum_{i=1}^n (1-\\Delta_i)\\mathds{1}_{\\{Y_i\\leq\\tau_0\\}}e^{\\beta'_0Z_i}\\left\\{\\int_0^{Y_i}h_1(s)\\mathrm{d}\\Lambda_0(s)+\\Lambda_0(Y_i)h'_2Z_i\n\\right\\}\\\\\n&\\quad\\qquad\\qquad\\qquad\\qquad\\qquad\\frac{\\partial g_i}{\\partial\\phi}(Y_i,\\Lambda_0,\\beta_0,\\gamma_0)\\nabla_\\gamma\\phi(\\gamma_0,X_i)'\\left\\{\\hat\\gamma_n-\\gamma_0\\right\\}+o_P(n^{-1\/2})\\\\\n\\end{split}\n\\]\nUsing the boundedness in probability of $\\hat\\beta_n$ and $\\hat\\Lambda_n(\\tau_0)$, the boundedness of the covariates, $\\beta_0$, $\\Lambda_0(\\tau)$, $\\nabla_\\gamma\\phi(\\gamma,x)$ and the consistency results in Theorem \\ref{theo:consistency2}, it follows that \n\\[\n\\begin{split}\n&\\left|\\frac1n\\sum_{i=1}^n (1-\\Delta_i)\\mathds{1}_{\\{Y_i\\leq\\tau_0\\}}\\left\\{e^{\\hat\\beta'_nZ_i}\\left(\\int_0^{Y_i}h_1(s)\\mathrm{d}\\hat\\Lambda_n(s)+\\hat\\Lambda_n(Y_i)h'_2Z_i\n\\right)\\frac{\\partial g_i}{\\partial\\phi}(Y_i,\\hat\\Lambda_n,\\hat\\beta_n,\\gamma_0)\\right.\\right.\\\\\n&\\left.\\left.\\quad-e^{\\beta'_0Z_i}\\left(\\int_0^{Y_i}h_1(s)\\mathrm{d}\\Lambda_0(s)+\\Lambda_0(Y_i)h'_2Z_i\n\\right)\\frac{\\partial g_i}{\\partial\\phi}(Y_i,\\Lambda_0,\\beta_0,\\gamma_0)\\right\\}\\nabla_\\gamma\\phi(\\gamma_0,X_i)\\right|=o_P(1)\n\\end{split}\n\\]\nAs a consequence, since $\\hat\\gamma_n-\\gamma_0=O_P(n^{-1\/2})$, we obtain\n\\[\n\\sqrt{n}\\left\\{(\\hat{S}_n-S_n)(\\Upsilon_n)-(\\hat{S}_n-S_n)(\\Upsilon_0)\\right\\}=o_P(1).\n\\]\nIt remains to deal with the first term on the right hand side of \\eqref{eqn:cond4}. It suffices to show that, for any sequence $\\epsilon_n\\to  0$,\n\\[\n\\sup_{|\\Lambda-\\Lambda_0|_\\infty\\leq\\epsilon_n, \\,\\Vert\\beta-\\beta_0\\Vert\\leq\\epsilon_n}\\frac{\\left|({S}_n-S)(\\Upsilon)-({S}_n-S)(\\Upsilon_0)\\right|}{n^{-1\/2}\\vee \\Vert\\beta-\\beta_0\\Vert\\vee |\\Lambda-\\Lambda_0|_\\infty }=o_P(1).\n\\]\nLet\n\\[\na_1(y,\\delta,z,x)=\\delta e^{\\beta'z}\\int_0^{y}h_1(s)\\mathrm{d}\\Lambda(s)-\\delta e^{\\beta'_0z}\\int_0^{y}h_1(s)\\mathrm{d}\\Lambda_0(s),\n\\]\n\\[\na_2(y,\\delta,z,x)=\\delta e^{\\beta'z}\\Lambda(y)h'_2z-\\delta e^{\\beta'_0z}\\Lambda_0(y)h'_2z,\n\\]\nand\n\\[\n\\begin{split}\na_3(y,\\delta,z,x)&=(1-\\delta)\\mathds{1}_{\\{y\\leq\\tau_0\\}}g(y,\\Lambda,\\beta,\\gamma_0)\\left\\{e^{\\beta'z}\\int_0^{y}h_1(s)\\mathrm{d}\\Lambda(s)+e^{\\beta'z}\\Lambda(y)h'_2z\\right\\}\\\\\n&\\quad-(1-\\delta)\\mathds{1}_{\\{y\\leq\\tau_0\\}}g(y,\\Lambda_0,\\beta_0,\\gamma_0)\\left\\{e^{\\beta'_0z}\\int_0^{y}h_1(s)\\mathrm{d}\\Lambda_0(s)+e^{\\beta'_0z}\\Lambda_0(y)h'_2z\n\\right\\}\n\\end{split}\n\\]\nThen, we have  \n\\[\n\\begin{split}\n({S}_n-S)(\\Upsilon)-({S}_n-S)(\\Upsilon_0)&=-\\frac{1}{n}\\sum_{i=1}^n \\left\\{a_1(Y_i,\\Delta_i,Z_i,X_i)-\\mathbb{E}\\left[a_1(Y,\\Delta,Z,X)\\right]  \\right\\}\\\\\n&\\quad-\\frac{1}{n}\\sum_{i=1}^n \\left\\{a_2(Y_i,\\Delta_i,Z_i,X_i)-\\mathbb{E}\\left[a_2(Y,\\Delta,Z,X)\\right]  \\right\\}\\\\\n&\\quad-\\frac{1}{n}\\sum_{i=1}^n \\left\\{a_3(Y_i,\\Delta_i,Z_i,X_i)-\\mathbb{E}\\left[a_3(Y,\\Delta,Z,X)\\right]  \\right\\} \n\\end{split}\n\\]\nNext we consider the first term. The other two can be handled similarly. From a Taylor expansion we have\n\\[\n\\begin{split}\n&\\delta e^{\\beta'z}\\int_0^{y}h_1(s)\\mathrm{d}\\Lambda(s)-\\delta e^{\\beta'_0z}\\int_0^{y}h_1(s)\\mathrm{d}\\Lambda_0(s)\\\\\n&=(\\beta-\\beta_0)'z\\delta e^{\\beta'_0z}\\int_0^{y}h_1(s)\\mathrm{d}\\Lambda(s)+\\delta e^{\\beta'_0z}\\int_0^{y}h_1(s)\\mathrm{d}(\\Lambda-\\Lambda_0)(s)+o(\\Vert\\beta-\\beta_0\\Vert).\n\\end{split}\n\\]Hence\n\\begin{equation}\n\\label{eqn:a1}\n\\begin{split}\n&\\frac{1}{n}\\sum_{i=1}^n \\left\\{a_1(Y_i,\\Delta_i,Z_i,X_i)-\\mathbb{E}\\left[a_1(Y,\\Delta,Z,X)\\right]  \\right\\}\\\\\n&\\leq (\\beta-\\beta_0)\\left\\{\\frac{1}{n}\\sum_{i=1}^n Z_i\\Delta_i e^{\\beta'_0Z_i}\\int_0^{Y_i}h_1(s)\\mathrm{d}\\Lambda(s)-\\mathbb{E}\\left[Z\\Delta e^{\\beta'_0Z}\\int_0^{Y}h_1(s)\\mathrm{d}\\Lambda(s)\\right]+o(1)\\right\\}\\\\\n&\\quad+ \\frac{1}{n}\\sum_{i=1}^n\\left\\{\\Delta_i e^{\\beta'_0Z_i}\\int_0^{Y_i}h_1(s)\\mathrm{d}(\\Lambda-\\Lambda_0)(s)-\\mathbb{E}\\left[\\Delta e^{\\beta'_0Z}\\int_0^{Y}h_1(s)\\mathrm{d}(\\Lambda-\\Lambda_0)(s)\\right]\\right\\}.\n\\end{split}\n\\end{equation}\nBy the law of large numbers\n\\[\n\\frac{1}{n}\\sum_{i=1}^n\\left\\{ Z_i\\Delta_i e^{\\beta'_0Z_i}\\int_0^{Y_i}h_1(s)\\mathrm{d}\\Lambda(s)-\\mathbb{E}\\left[Z\\Delta e^{\\beta'_0Z}\\int_0^{Y}h_1(s)\\mathrm{d}\\Lambda(s)\\right] \\right\\}=o_P(1)\n\\]\nand as a result, the first term in the right hand side of \\eqref{eqn:a1} is $o_P(\\Vert\\beta-\\beta_0\\Vert)$. The second term can be rewritten as \n\\[\n\\int_0^{\\tau_0} D_n(s)h_1(s)\\mathrm{d}(\\Lambda-\\Lambda_0)(s)\n\\]\nwhere\n\\[\nD_n(s)=\\frac{1}{n}\\sum_{i=1}^n\\left\\{\\Delta_i\\mathds{1}_{\\{Y_i\\geq s\\}} e^{\\beta'_0Z_i}-\\mathbb{E}\\left[\\Delta\\mathds{1}_{\\{Y\\geq s\\}} e^{\\beta'_0Z}\\right]\\right\\}. \n\\]\nBy integration by parts and the chain rule we have\n\\[\n\\begin{split}\n&\\int_0^{\\tau_0} D_n(s)h_1(s)\\mathrm{d}(\\Lambda-\\Lambda_0)(s)\\\\\n&=  D_n(\\tau_0)h_1(\\tau_0)(\\Lambda-\\Lambda_0)(\\tau_0)- \\int_0^{\\tau_0} (\\Lambda-\\Lambda_0)(s)\\mathrm{d} \\left[D_n(s)h_1(s)\\right]\\\\\n&=D_n(\\tau_0)h_1(\\tau_0)(\\Lambda-\\Lambda_0)(\\tau_0)-\\int_0^{\\tau_0} (\\Lambda-\\Lambda_0)(s) D_n(s)\\mathrm{d} h_1(s)\\\\\n&\\quad +\\frac{1}{n}\\sum_{i=1}^n\\left\\{ \\Delta_i (\\Lambda-\\Lambda_0)(Y_i)h_1(Y_i) e^{\\beta'_0Z_i}-\\mathbb{E}\\left[ (\\Lambda-\\Lambda_0)(Y)h_1(Y)\\Delta e^{\\beta'_0Z}\\right]\\right\\}\n\\end{split} \n\\]\nNote that \n\\[\n\\begin{split}\n\\mathbb{E}\\left[ (\\Lambda-\\Lambda_0)(Y)h_1(Y)\\Delta e^{\\beta'_0Z}\\right]&=\\mathbb{E}\\left[e^{\\beta'_0Z}\\int_0^{\\tau_0}(\\Lambda-\\Lambda_0)(s)h_1(s)\\mathrm{d} H_1(s|X,Z)\\bigg|X,Z\\right]\\\\\n&=-\\int_0^{\\tau_0}(\\Lambda-\\Lambda_0)(s)h_1(s)\\mathrm{d}\\mathbb{E}\\left[\\Delta\\mathds{1}_{\\{Y\\geq s\\}}e^{\\beta'_0Z}\\right].\n\\end{split}\n\\]\nIt can be shown that $\\sqrt{n}D_n$ converges weakly to a tight, mean zero Gaussian process $D$ in $l^\\infty([0,\\tau_0])$. Since $h_1$ is bounded, it follows \n\\[\n\\frac{\\left|D_n(\\tau_0)h_1(\\tau_0)(\\Lambda-\\Lambda_0)(\\tau_0)\\right|}{\\Vert\\Lambda-\\Lambda_0\\Vert_{\\infty}}=o_P(1)\n\\]\nMoreover, since $D_n\\to 0$ and $h_1$ is of bounded variation \n\\[\n\\frac{ \\left|\\int_0^{\\tau_0} (\\Lambda-\\Lambda_0)(s) D_n(s)\\mathrm{d} h_1(s)\\right|}{\\Vert\\Lambda-\\Lambda_0\\Vert_{\\infty}}\\leq \\sup_{t\\in[0,\\tau_0]}|D_n(s)| \\int_0^{\\tau_0}\\,\\left|\\mathrm{d} h(s)\\right|\\to 0.\n\\] \nFinally, since  $\\left\\{g_{\\Lambda}(y,\\delta,z)=\\delta(\\Lambda-\\Lambda_0)(y)h_1(y)e^{\\beta'_0Z}:\\,\\Vert\\Lambda-\\Lambda_0\\Vert_{\\infty}\\leq \\epsilon_n\\right\\}$ is a Donsker class (product of bounded variation functions, uniformly bounded) and \n\\[\n\\mathbb{E}\\left[(\\Lambda-\\Lambda_0)(Y)^2h_1(Y)^2\\Delta e^{2\\beta'_0Z}\\right]=O(\\epsilon_n^2)=o(1)\n,\n\\] \nwe have that\n\\[\n\\sqrt{n}\\frac{1}{n}\\sum_{i=1}^n\\left\\{ \\Delta_i (\\Lambda-\\Lambda_0)(Y_i)h_1(Y_i) e^{\\beta'_0Z_i}-\\mathbb{E}\\left[ (\\Lambda-\\Lambda_0)(Y)h_1(Y)\\Delta e^{\\beta'_0Z}\\right]\\right\\}\n\\]\nconverges to zero in probability. So we obtain \n\\[\n\\frac{ \\int_0^{\\tau_0} D_n(s)h_1(s)\\mathrm{d}(\\Lambda-\\Lambda_0)(s)}{n^{-1\/2}\\vee \\Vert\\Lambda-\\Lambda_0\\Vert_{\\infty} }=o_P(1)\n\\] \nThe other two terms related to $a_2$ and $a_3$ can be treated similarly. \n\nThis concludes the verification of conditions of Theorem 4 in \\cite{Lu} (or Theorem 3.3.1 in \\cite{VW}). Hence, the weak convergence of $\\sqrt{n}(\\Upsilon_n-\\Upsilon_0)$ to a tight, mean zero Gaussian process $G$ follows. Next we compute the covariance process of $G$.  \nFrom Theorem 3.3.1 in \\cite{VW} we have\n\\begin{equation}\n\\label{eqn:asymptotic_relation}\n-\\sqrt{n}\\dot{S}(\\Upsilon_0)(\\Upsilon_n-\\Upsilon_0)(h)=\\sqrt{n}(\\hat{S}_n(\\Upsilon_0)-S(\\Upsilon_0))(h)+o_P(1).\n\\end{equation}\nMoreover, in \\cite{Lu} it is computed that \n\\begin{equation}\n\\label{eqn:derivative}\n\\dot{S}(\\Upsilon_0)(\\Upsilon_n-\\Upsilon_0)(h)=\\int_0^{\\tau_0}\\sigma_1(h)(t)\\mathrm{d}(\\hat\\Lambda_n(t)-\\Lambda_0(t))+(\\hat\\beta_n-\\beta_0)'\\sigma_2(h)\n\\end{equation}\nwhere $\\sigma=(\\sigma_1,\\sigma_2)$ is a continuous linear operator from $\\mathcal{H}_\\mathfrak{m}$ to $\\mathcal{H}_\\mathfrak{m}$ of the form \n\\[\n\\begin{split}\n\\sigma_1(h)(t)&=\\mathbb{E}\\left[\\mathds{1}_{\\{Y\\geq t\\}}V(t,\\Upsilon_0)(h)g(t,\\Upsilon_0)e^{\\beta'_0Z}\\right]\\\\\n&\\quad-\\mathbb{E}\\left[\\int_t^{\\tau_0}\\mathds{1}_{\\{Y\\geq s\\}}V(t,\\Upsilon_0)(h)g(s,\\Upsilon_0)\\{1-g(s,\\Upsilon_0)\\}e^{2\\beta'_0Z}\\mathrm{d}\\Lambda_0(s)\\right]\n\\end{split}\n\\]\nand\n\\[\n\\sigma_2(h)(t)=\\mathbb{E}\\left[\\int_0^{\\tau_0}\\mathds{1}_{\\{Y\\geq t\\}}W(t,\\Upsilon_0)V(t,\\Upsilon_0)(h)g(t,\\Upsilon_0)e^{\\beta'_0Z}\\mathrm{d}\\Lambda_0(t)\\right]\n\\]\nwhere\n\\[\nV(t,\\Upsilon_0)(h)=h_1(t)-\\left\\{1-g(t,\\Upsilon_0)\\right\\}e^{\\beta'_0Z}\\int_0^th_1(s)\\mathrm{d}\\Lambda_0(s)+h'_2W(t,\\Upsilon_0)\n\\]\nand\n\\[\nW(t,\\Upsilon_0)=\\left[1-\\left\\{1-g(t,\\Upsilon_0)\\right\\}e^{\\beta'_0Z}\\Lambda_0(t)\\right]Z\n\\]\nIn \\cite{Lu}, it is also shown that $\\sigma$ is invertible with inverse $ \\sigma^{-1}=(\\sigma_1^{-1},\\sigma_2^{-1})$.  Hence, for all $g\\in\\mathcal{H}_\\mathfrak{m}$, let $h=\\sigma^{-1}(g)$. If in \\eqref{eqn:derivative} we replace $h$ by $\\sigma^{-1}(g)$ and use \\eqref{eqn:asymptotic_relation}, we obtain\n\\[\n\\begin{split}\n&\\int_0^{\\tau_0}g_1(t)\\mathrm{d}\\sqrt{n}(\\Lambda_n(t)-\\Lambda_0(t))+\\sqrt{n}(\\hat\\beta_n-\\beta_0)'g_2\\\\&=-\\sqrt{n}(\\hat{S}_n(\\Upsilon_0)-S(\\Upsilon_0))(\\sigma^{-1}(g))+o_P(1)\\xrightarrow{d}- G^*(\\sigma^{-1}(g)).\n\\end{split}\n\\]\nSince the previous results holds for all $g\\in\\mathcal{H}_\\mathfrak{m}$, it follows that $(\\sqrt{n}(\\hat\\Lambda_n-\\Lambda_0),\\sqrt{n}(\\hat\\beta_n-\\beta_0))$ converges to a tight mean zero Gaussian process $G$ with covariance \n\\begin{equation}\n\\label{def:cov_G}\nCov(G(g),G(\\tilde{g}))=Cov\\left(G^*(\\sigma^{-1}(g)),G^*(\\sigma^{-1}(\\tilde{g}))\\right)\n\\end{equation}\nand the covariance of $G^*$ is given in \\eqref{eqn:cov_G*}.\n\\end{proof}\n\\section{Additional simulation results}\nIn this section we report the simulation results for scenario 2 of  the models~{1-4}, $n=200, 400$, that were omitted from the main paper and the results for sample size $n=1000$ (all models and scenarios). In addition, Table \\ref{tab:results3_4_beta} complements Tables~\\ref{tab:results3} and~\\ref{tab:results4} containing results for $\\hat\\beta$. \n\n\n\\section{Introduction}\nThere are many situations in survival analysis problems where some of the subjects will never experience the event of interest. For instance, as significant progress is being made for treatment of different types of cancers, many of the patients get cured of the disease and do not experience recurrence or cancer-related death. Other examples include study of time to natural conception,\ntime to default in finance and risk management,\ntime to early failure of integrated circuits in {engineering, time to find a job after a layoff.}\nHowever, because of the finite duration of the studies and censoring, the cured subjects (for which the event never takes place) cannot be distinguished from the `susceptible' ones. We can just get an indication of the presence of a cure fraction from the context of the study and  a long plateau (containing many censored observations) with height greater than zero in the Kaplan-Meier estimator of the survival function. Predicting the probability of being cured given a set of characteristics is often of particular interest in order to make better decisions in terms of {treatment, management strategies or public policies.}\nThis lead to the development of mixture cure models.\n\nMixture cure models were first proposed by \\cite{boag49} and \\cite{berkson52}. They assume that the population is a mixture of two groups: the cured  and the susceptible subjects. Within this very wide class of models, various approaches have been considered in the literature for modelling and {estimating}  the incidence (probability of being uncured) and the latency (survival function of the uncured subjects). Initially, fully parametric models with a logistic regression form of the incidence and various parametric distributions for the latency  were used in \\cite{farewell82,yamaguchi92,kuk92}.  Later on, more flexible semi-parametric approaches were proposed for the latency based on the Cox proportional hazards model \\cite{ST2000,peng2000} or accelerated failure time models \\cite{li2002,zhang2007}. However, they still maintain the logistic regression model for the incidence.  More recently, nonparametric methods have been developed for both or one of the model components in \\cite{XP2014,PK2019,AKL19}. \n{In this} wide range of models, probably the most commonly used one in practice is the {logistic\/Cox mixture} cure model  \\citep{stringer2016cure,wycinka2017,lee2017extinct}.\n\nThere have been different proposals for estimation in the logistic\/Cox mixture cure model. The presence of a latent variable (the unknown cure status), does not allow for a `direct' approach as in the classical Cox proportional hazards model. \\cite{kuk92} adapted a marginal likelihood approach computed through Monte Carlo approximations, whereas \\cite{peng2000} and \\cite{ST2000} computed the  maximum likelihood estimator via the Expectation-Maximization algorithm. Asymptotic properties of the latter estimators are investigated in \\cite{Lu2008}, while the procedure is implemented in the package \\texttt{smcure} \\cite{cai_smcure}. \nOne concern about the previous estimators is that they are obtained by iterative procedures which {could} be unstable  in practice. {In particular, when the sample size is small there are situations in which the EM algorithm fails to converge (even though the \\texttt{smcure} package can still provide without error the estimates obtained when the maximum number of iterations is reached). Such problems are for example reported in \\cite{han2017statistical}. In addition, the maximum likelihood estimator for the incidence component depends on which variables are included in the latency model (see for example the illustration in Section~\\ref{sec:application}) and this instability might in practice lead to unobserved effects (when the effect is not very strong). In particular, if the latency model is misspecified, even the estimators of the incidence parameters suffer from induced bias (see for example \\cite{BP18}).}\n\n In this paper, we introduce an alternative {estimation method which applies {very broadly and}, in particular, for the} logistic\/Cox mixture cure model. {Our approach focuses on } direct estimation of the cure probability without using distributional assumptions on the latency and iterative algorithms.  It relies on  a preliminary nonparametric estimator for the incidence which is then `projected' on {a parametric class of functions (like logistic functions)}.  The idea of constructing a parametric estimator by nonparametric estimation has been previously proposed for the classical linear regression by \\cite{cristobal1987class}. {Later on it was shown to be effective also in the context of variable selection and functional linear regression  \\citep{presmoothing_var_sel,ferraty2012presmoothing}. However, its }\nextension to nonlinear setups has been very little investigated. Here we show that in the context of mixture cure models, even when a parametric form is  assumed for the incidence, the use of a presmoothed estimator as an intermediate step for obtaining the parameter estimates often leads to more accurate results. Once the cure fraction is estimated, we estimate the survival distribution of the uncured subjects. {In the case of the logistic\/Cox cure model, this is done by maximizing the Cox component of the likelihood.}  In this step, {an iterative algorithm}\nis used to compute the estimators of the baseline cumulative hazard and the regression parameters. This new approach is of practical relevance given  the popularity {of the semiparametric logistic\/Cox mixture cure model.} However, the method can be applied more in general to a mixture cure model with a parametric form of the incidence and {other type of models for the uncured subjects, such as the semiparametric proportional odds model or the semiparametric AFT model. Our findings} suggest that presmoothing has potential {to improve parameter estimation for small and moderate sample size.}\n\nThe paper is organized as follows. In Sections \\ref{sec:model} and \\ref{sec:method} we describe the model and the estimation procedure.   {Section \\ref{sec:Cox} focuses on the estimation method in the case of the logistic\/Cox mixture cure model.}\nConsistency and asymptotic normality of the estimators are shown in Section~\\ref{sec:asymptotics}. {Thanks to the presmoothing, we are able to present theoretical results under more reasonable assumptions and thus we contribute to fill a gap between unrealistic technical conditions and applications. \n\tThe finite sample performance of the method is investigated through a simulation study and results are reported in {Section~\\ref{sec:simulations}}. \n\t{For practical purposes, we propose to make simple and commonly used choices for the bandwidth and the kernel function in the presmoothing step, and we show that these choices provide satisfactory results.}\n\n\t  The proposed estimation procedure is applied to two medical datasets about studies of patients with melanoma cancer (see Section~\\ref{sec:application}).   {We conclude in Section \\ref{sec:disc} with some discussion and ideas for further research.  Finally, some of the proofs can be found in Section \\ref{sec:appendix}, while the remaining proofs {and additional simulation results} are collected in the online Supplementary Material.} \n\t\n\t\n\t\n\t\n\t\\section{Model description}\n\t\\label{sec:model}\n\tIn the mixture cure model  the survival time $T$ can be decomposed as \n\t\\[\n\tT=BT_0+(1-B)\\infty,\n\t\\]\n\twhere $T_0$ represents {the finite survival time} for an uncured individual and  {$B$ is an unobserved $0$-$1$ random variable giving the uncured status: $B=1$ for uncured individuals and $B=0$ otherwise}. By convention $0\\cdot \\infty = 0$.  Let $C$ be the censoring time and $(X',Z')'$ a $(p+q)$-dimensional vector of  covariates, where $x'$ denotes the transpose of the vector $x$. Let $\\mathcal{X}$ and $\\mathcal{Z}$ be the supports of $X$ and $Z$ respectively. Observations consist {of} $n$ i.i.d. realizations of $(Y,\\Delta,X,Z)$, where $Y=\\min(T,C)$ is {the finite follow-up time} and $\\Delta=\\mathds{1}_{\\{T\\leq C\\}}$ is the censoring indicator. {Since $Y$ is finite, then necessarily $\\mathbb{P}(C<\\infty)=1$, that means} the censoring times are finite ({which makes sense given the} limited duration of the studies). As a result, {censored survival times of the uncured subjects} cannot be distinguished from the cured ones. \n\t\n\tThe covariates included in $X$ are those used to model the cure rate, while the ones in $Z$ affect the survival conditional on the uncured status. This allows in general to use different variables for modelling the incidence and the latency but does not exclude situations in which the two vectors $X$ and $Z$ share some components or are exactly the same.  \n\tApart from the standard assumption in survival analysis that $T_0\\perp (C,X) | Z$, here we also need \n\t\\begin{equation}\\label{eqn:bb_ind}\n\t\tB\\perp (C,T_0,Z)|X.\n\t\\end{equation} \n\tThis implies in particular that \n\t\\begin{equation}\n\t\t\\label{eqn:CI1}\n\t\tT\\perp C| (X,Z)\n\t\\end{equation}\n\t(see {Lemma~1 in the Suplementary Material}). Moreover, \\eqref{eqn:bb_ind} implies \n\t\\begin{equation}\n\t\t\\label{eqn:condition_X_Z}\n\t\t\\mathbb{P}(T=\\infty|X,Z)=\\mathbb{P}(T=\\infty|X).\n\t\\end{equation}\n\tIn addition, in the cure model context we need that the event time $T_0$ has support $[0,\\tau_0]$, {i.e.  $\\{T>\\tau_0\\}=\\{T=\\infty\\}$, such} that   \n\t\\begin{equation}\n\t\t\\label{eqn:CI2}\n\t\t{\\inf_{x}\\mathbb{P}(C>\\tau_0|X=x)>0.}\n\t\n\t\\end{equation}\n\t{(If the support of $T_0$ given $Z=z$ depends on $z$, then we let $\\tau_0 = \\sup \\tau_0(z)$, where $\\tau_0(z)$ is the right endpoint of this support.)}   This condition tells us that all the observations with $Y>\\tau_0$ are cured. Even if it might {seem restrictive,} it is reasonable when a cure model is justified by a `good' follow-up beyond the time when most of the events occur and it is commonly accepted in the cure model literature in order for the mixture cure model to be identifiable and not to  overestimate the cure rate. Since $T_0\\perp X | Z$, we have \n\t\\begin{equation*}\n\t\n\t\t\\mathbb{P}(T_0\\leq t |X,Z)=\\mathbb{P}(T_0\\leq t|Z), \\quad \\forall t\\in [0,\\tau_0].\n\t\\end{equation*}\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\tWe assume a parametric model for the cure rate and we denote by $\\pi_0(x)$ the cure probability of a subject with covariate $x$, i.e\n\t\\[\n\t\\pi_0(x)=\\mathbb{P}(T=\\infty|X=x)=1-\\phi(\\gamma_0,x),\n\t\\]\n\tfor some {parametric model $\\{\\phi (\\gamma,x): \\gamma \\in G\\}$ and $\\gamma_0\\in G$.} The first component of $X$ is equal to one and {the first component of $\\gamma$} corresponds to the intercept. In order for $\\gamma$ to be identifiable we need the following condition\n\t\\begin{equation}\n\t\t\\label{eqn:CI3}\n\t\t\\mathbb{P}\\left(\\phi(\\gamma,X)=\\phi(\\tilde{\\gamma},X)\\right)=1\\qquad\\text{ implies that }\\qquad \\gamma=\\tilde{\\gamma}.\n\t\\end{equation}\n{Choosing a parametric model for the incidence seems quite standard in the literature of mixture cure models (\\cite{PK2019,BP18,sposto2002cure}) because of its simplicity and ease of interpretability (particularly for multiple covariates).  To check the fit of this model in practice, one can compare the prediction error with that of a more flexible single-index model as done in \\cite{AKL19} and for our real data application  in Section~\\ref{sec:application}. It is also possible to test whether this assumption is reasonable using the test proposed in \\cite{muller2019goodness}, but this is currently developed only for one covariate. Among the parametric models for the incidence component, } the most common example is the logistic model, where \n\t\\begin{equation}\n\t\t\\label{eqn:logistic}\n\t\t\\phi(\\gamma,x)=1\/(1+{\\exp(-\\gamma' x)}).\n\t\\end{equation}\n\n\t{We state} the results in Section \\ref{sec:asymptotics} for a general parametric model for the incidence, but then we focus on the logistic function in the simulation study in Section \\ref{sec:simulations} since it is more of interest in practice. \n\tFor the uncured subjects, we {can consider a general semiparametric model defined through the survival function \n\t\t\\begin{equation} \\label{Su}\n\t\t\tS_u(t|z) =S_u(t|z;\\beta,\\Lambda) = \\mathbb{P}(T_0>t| {Z=z}, B=1) \\quad\\text{and}\\quad S_u(\\tau_0|z) =0,\n\t\t\\end{equation}\n\t\t{where the conditional survival function $S_u$ is allowed to depend on a finite-dimensional parameter, denoted by $\\beta\\in \\mathcal B$, and\/or an infinite-dimensional parameter, denoted by $\\Lambda\\in \\mathcal H$, with $\\mathcal B$ and $\\mathcal H$ the respective parameter sets.} Let $\\beta_0\\in\\mathcal  B$ and $\\Lambda_0 \\in\\mathcal H$ be the true values of these parameters. As a result, the conditional survival function corresponding to $T$  is then\n\t\t$$\n\t\tS(t|x,z)=\\mathbb{P}(T>t|{X=x, Z=z})=1-\\phi(\\gamma_0,x)+\\phi(\\gamma_0,x)S_u(t|z).\n\t\t$$\n\t\tThe main example we keep in mind is the Cox proportional hazards (PH) model where  $\\Lambda_0$ is the baseline cumulative hazard. In this case \n\t\t\\begin{equation} \\label{SuCox}\n\t\t\tS_u(t|z)\n\t\t\t=S_0(t)^{\\exp(\\beta'_0z)} = \\exp(-\\Lambda_0(t)\\exp(\\beta'_0z)),\n\t\t\\end{equation}\n\t\twhere $S_0 $ is the baseline survival and $\\beta_0 $ does not contain an intercept.}\n\n\t\n\t\n\n\n\n\n\n\n\t\n\t\n\t\n\t\\section{{Presmoothing estimation approach}}\n\t\\label{sec:method}\n\tThe estimation method we propose is based on a two step procedure. We first estimate nonparametrically the cure probability for each observation and then compute an estimator of $\\gamma$ as the maximizer of the logistic likelihood, ignoring {the model for the uncured subjects.}\n\n\tIn the second step, we plug-in this estimator of $\\gamma$ in the {full  likelihood of the\n\t\n\t\tmixture cure model and fit the \n\t\tlatency model \n\t\tusing maximum likelihood estimation.} \n\tIn what follows, we describe in more details these two steps. \n\t\n\t\\textit{Step 1.} Even though a parametric model is assumed for the incidence, we start by computing a nonparametric estimator of the cure probability for each subject. One possibility is to use  the method {followed by \\cite{PK2019} (see also \\cite{XP2014})}, {but other estimators are possible as well, as long as the conditions given in Section \\ref{sec:asymptotics} are satisfied.  The estimator of \\cite{PK2019} is defined as follows:}\n\t\\begin{equation}\n\t\t\\label{def:hat_pi}\n\t\t\\hat\\pi(x)=\\prod_{t\\in\\mathbb{R}} \\left(1-\\frac{\\hat{H}_1(dt|x)}{\\hat{H}([t,\\infty)|x)}\\right){,}\n\t\\end{equation}\n\twhere  {$\\hat{H}([t,\\infty)|x)=\\hat{H}_1([t,\\infty)|x)+\\hat{H}_0([t,\\infty)|x)$,} $\\hat{H}_1(dt|x) = \\hat{H}_1((t-dt,t]|x)$ for small $dt$ and \n\t\\[\n\t\\hat{H}_k([t,\\infty)|x)=\\sum_{i=1}^n\\frac{\\tilde{K}_b(X_i-x)}{\\sum_{j=1}^n\\tilde{K}_b(X_j-x)}\\mathds{1}_{\\{Y_i\\geq t, \\Delta_i=k\\}},\\quad k=0,1{,}\n\t\\] \n\tare estimators of\n\t\\[\n\tH_k([t,\\infty)|x)=\\mathbb{P}\\left(Y\\geq t,\\Delta=k|X=x\\right), \n\t\\] \n\t$H([t,\\infty)|x)=H_1([t,\\infty)|x)+H_0([t,\\infty)|x)$. Here $\\tilde{K}_{{b}}$ is a multidimensional kernel function defined in the following way. If $X$ is composed of continuous and discrete components, {$X=(X_c,X_d)\\in\\mathcal{X}_c\\times\\mathcal{X}_d\\subset\\mathbb{R}^{p_c}\\times\\mathbb{R}^{p_d}$ with $p_c+p_d=p$}, then\n\t\\[\n\t\\tilde{K}_b(X_i-x)=K_b(X_{c,i}-x_c)\\mathds{1}_{\\{X_{d,i}=x_d\\}},\n\t\\] \n\twhere $b=b_n$ is a bandwidth sequence, $K_b(\\cdot)=K(\\cdot\/b)\/b^{p_c}$ and $K(u)=\\prod_{j=1}^{p_c}k(u_j)$, {with $k$ a kernel.} \n\tNote that, one can compute this estimator with any covariate but here we only use $X$ because of our assumption~\\eqref{eqn:condition_X_Z}.\n\n\t{The estimator} $\\hat\\pi(x)$ coincides with the Beran estimator of the {conditional survival function} $S$ at the largest observed event time $Y_{(m)}$ {and does not require any specification of $\\tau_0$}. {Since $\\hat{H}_1(dt|x)$ is different from zero only at the observed event times, computation of $\\hat\\pi(x)$ requires only a product over $t$ in the set of the observed event times.} Afterwards, we consider the logistic likelihood\n\t\\[\n\t\\hat L_{n,1}(\\gamma)=\\prod_{i=1}^n \\phi(\\gamma,X_i)^{1-\\hat\\pi(X_i)}(1-\\phi(\\gamma,X_i))^{\\hat\\pi(X_i)}{,}\n\t\\]\n\tand define $\\hat\\gamma_n$ as the maximizer of \n\t\\begin{equation}\n\t\t\\label{def:hat_L_gamma}\n\t\t\\log \\hat{L}_{n,1}(\\gamma)=\\sum_{i=1}^n\\Big\\{\\left[1-\\hat\\pi(X_i)\\right] \\log  \\phi(\\gamma,X_i)+\\hat\\pi(X_i)\\log \\left[1-\\phi(\\gamma,X_i)\\right]\\Big\\}.\n\t\\end{equation}\n\tExistence and uniqueness of $\\hat\\gamma_n$ holds under the same conditions as for the maximum likelihood estimator in the binary outcome regression model where $1-\\hat\\pi(X_i)$ is replaced by {the outcome $B_i$}. For example, in the logistic model,   it is required that $p<n$ and the matrix of the variables $X$ has full rank. \n\t\n\t\n\t\\textit{Step 2.} {Now we consider the  likelihood of the\n\t\tmixture cure model.  Let {$$f_u(t|z;\\beta,\\Lambda) = -(\\partial\/\\partial t) S_u(t|z;\\beta,\\Lambda)$$ with $S_u(t|z;\\beta,\\Lambda)$ as defined in (\\ref{Su}),} {denote an element in the model for the conditional density of} $T_0$ given $Z=z$, which is supposed to exist {and belong to the model}.} Assuming  non informative censoring and that the distribution of the covariates does not carry information on the parameters $\\beta$, $\\Lambda$, the likelihood criterion is then\n\n\n\n\n\n\n\n\t{\\begin{equation}\n\t\t\t\\label{eqn:likelihood2}\n\t\t\tL_{n,2}( \\beta,\\Lambda,\\gamma)=\\prod_{i=1}^{n}\\left\\{\\phi(\\gamma,X_i)f_u(Y_i|Z_i;\\beta,\\Lambda)\n\t\t\t\\right\\}^{\\Delta_i}\\!\\left\\{1\\!-\\phi(\\gamma,X_i)+\\phi(\\gamma,X_i)S_u(Y_i|Z_i;\\beta,\\Lambda) \\right\\}^{1-\\Delta_i},\n\t\t\\end{equation}\n\t\tand we maximize it w.r.t. $\\beta$ and $\\Lambda$ for $\\gamma=\\hat\\gamma_n$, i.e.  $(\\hat\\beta_n,\\hat\\Lambda_n)$  are the maximizers of \n\t\t\\begin{equation}\\label{eqn:likelihood2b}\n\t\t\n\t\t\t\\hat{l}_n(\\beta,\\Lambda,\\hat\\gamma_n)\n\t\t\t= \\frac{1}{n}\\sum_{i=1}^n \\ell (Y_i,\\Delta_i,X_i,Z_i;\\beta,\\Lambda,\\hat\\gamma_n),\n\t\t\\end{equation}\n\t\tover a set of possible values for $\\beta $ and $\\Lambda$, where\n\t\t\\begin{equation}\n\t\t\t\\label{def:hat_l_cox_g}\n\t\t\t\\begin{split}\n\t\t\t\t\\ell (Y_i,\\Delta_i,X_i,Z_i;\\beta,\\Lambda,\\gamma) &=\\Delta_i  \\log f_u(Y_i|Z_i;\\beta,\\Lambda) \\\\\n\t\t\t\t&\\quad+\n\t\t\t\t(1-\\Delta_i)\\log\\left\\{1-\\phi(\\gamma,X_i)+\\phi(\\gamma,X_i)S_u(Y_i|Z_i;\\beta,\\Lambda)\\right\\}.\n\t\t\t\\end{split}\n\t\t\\end{equation}\n\t}\n\n\n\n\n\n\n\n\n\t%\n\n\n\n\n\t%\n\t%\n\t%\n\n\n\n\t\n\t\n\t%\n\n\n\n\n\n\t%\n\t%\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t%\n\t%\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\t\n\t\\section{{Presmoothing estimation for the parametric\/Cox mixture cure model}} \n\t\\label{sec:Cox}\n\t\n\n\t\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\t\n\tIn the sequel we focus on the case of a Cox PH model {defined in (\\ref{SuCox})} for the conditional law of $T_0$. \n\n\n\tThe criterion defined in \\eqref{eqn:likelihood2b} becomes \n\n\t\\begin{multline}\n\t\t\\label{def:hat_l_cox}\n\t\n\t\n\t\t\\hat{l}_n(\\beta,\\Lambda,\\hat\\gamma_n)  \n\t\t=\\frac{1}{n}\\sum_{i=1}^n \\Delta_i  \\left\\{{\\mathds{1}_{\\{Y_i<\\tau_0\\}}}[ \\log \\Delta\\Lambda(Y_i)+\\beta'Z_i]-\\Lambda(Y_i)e^{\\beta'Z_i} \\right\\} \n\t\t\\\\\n\t\t+\\frac{1}{n}\\sum_{i=1}^n(1-\\Delta_i )\\log\\left\\{1-\\phi(\\hat\\gamma_n,X_i)+\\phi(\\hat\\gamma_n,X_i)\\exp\\left(-\\Lambda(Y_i)e^{\\beta'Z_i}\\right)\\right\\},\\,\n\t\n\t\n\t\\end{multline}\n\tand has to be maximized with respect to $\\beta$ and $\\Lambda$ in  \n\tthe class of step functions $\\Lambda$  defined on $[0,\\tau_0]$ (thus by definition $\\Lambda(t)={\\infty}$ if $t>\\tau_0$), with jumps of size $\\Delta \\Lambda$ at the event times. {The indicator of the event $\\{Y_i<\\tau_0\\}$ in the first term is needed in case the distribution of the event times has a jump at $\\tau_0$ meaning that $\\mathbb{P}(T_0=\\tau_0|Z)>0$. In such a case $f_u(\\tau_0|Z;\\beta,\\Lambda)=\\exp(-\\Lambda(\\tau_0)e^{\\beta'Z})$ where $\\Lambda(\\tau_0)=\\lim_{t\\uparrow\\tau_0}\\Lambda(t)$. Otherwise, if $\\mathbb{P}(T_0=\\tau_0|Z)=0$, then  for all uncensored observations we have $\\mathds{1}_{\\{Y<\\tau_0\\}}=1$ with probability one. Thus, the presence of the indicator function can be neglected. }\n\tAs in \\cite{Lu2008}, it can be shown that  \n\t\\begin{equation}\\label{def_MLE}\n\t\t(\\hat\\beta_n, \\hat\\Lambda_n)= \\arg\\max_{\\beta, \\Lambda }\n\t\t\\hat{l}_n(\\beta,\\Lambda,\\hat\\gamma_n)\n\t\\end{equation}\n\texists and it is finite. Moreover, for any given $\\beta$ and $\\gamma$, the ${\\Lambda}_{n,\\beta,\\gamma}$ which maximizes $\\hat{l}_n(\\beta,\\Lambda,\\gamma)$ {in \\eqref{def:hat_l_cox}}, \n\twith respect to $\\Lambda$  with jumps at the event times,\n\tcan be characterized as \n\t\\begin{equation}\n\t\t\\label{eqn:hat_lambda_n}\n\t\t{\\Lambda}_{n,\\beta,\\gamma}(t)=\\frac{1}{n}\\sum_{i=1}^{n}\\frac{\\Delta_i\\mathds{1}_{\\{Y_i\\leq  {t {, Y_i<\\tau_0} }\\}}}{\\frac{1}{n}\\sum_{j=1}^n\\mathds{1}_{\\{{{Y_i\\leq Y_j\\leq\\tau_0}  }\\}}\\exp(\\beta'Z_j)\\left\\{{\\Delta_j}+(1-{\\Delta_j})g_j(Y_j,{\\Lambda}_{n,\\beta},\\beta,\\gamma)\\right\\}} ,\n\t\\end{equation}\n\twhere\n\t\\begin{equation}\n\t\t\\label{def:g_j}\n\t\tg_j(t,\\Lambda,\\beta,\\gamma)=\\frac{\\phi(\\gamma,X_j)\\exp\\left(-\\Lambda(t) \\exp\\left(\\beta' Z_j\\right)\\right)}{1-\\phi(\\gamma,X_j)+\\phi(\\gamma,X_j)\\exp\\left(-\\Lambda(t) \\exp\\left(\\beta' Z_j\\right)\\right)}.\n\t\\end{equation}\n\tNext, we could define\n\t$$\n\t\\hat\\beta_n = \\arg\\max_{\\beta}\\hat{l}_n(\\beta,{\\Lambda}_{n,\\beta,\\hat\\gamma_n} ,\\hat\\gamma_n) \\quad \\text{and} \\quad \\hat\\Lambda_n  = {\\Lambda}_{n,\\hat\\beta_n,\\hat\\gamma_n}.\n\t$$\n\t\n\tTo compute $(\\hat\\beta_n,\\hat\\Lambda_n)$ we use an iterative algorithm based on profiling.  \n\tTo be precise, we start with initial values which are the maximum partial likelihood estimator and the Breslow estimator (as if there was no cure fraction) and we iterate between the next two steps until convergence:\n\t\\begin{itemize}\n\t\t\\item[{a)}] Compute the weights \n\t\t\\[\n\t\tw_j^{(m)}={\\Delta_j}+(1-{\\Delta_j})\\frac{\\phi(\\hat\\gamma_n,X_j)\\hat{S}^{(m)}_u(Y_j|Z_j)}{1-\\phi(\\hat\\gamma_n,X_j)+\\phi(\\hat\\gamma_n,X_j)\\hat{S}^{(m)}_u(Y_j|Z_j)},\n\t\t\\]\n\t\twhere\n\t\t\\[\n\t\t\\hat{S}^{(m)}_u(Y_j|Z_j)=\\exp\\left(-\\hat{\\Lambda}^{(m)}_n(Y_j) \\exp\\left(\\hat\\beta^{(m)'}_n Z_j\\right)\\right),\n\t\t\\]\n\t\tusing the estimators $\\hat{\\Lambda}^{(m)}_n$, $\\hat\\beta^{(m)}_n $ of the previous step. \n\t\t\\item[{b)}] Using the previous weights, update the estimators for $\\Lambda$  and $\\beta$, i.e. $\\hat\\beta^{(m+1)}_n$ is the maximizer of\n\t\t\\[\n\t\t\\prod_{i=1}^n \\left\\{\\frac{e^{\\beta'Z_i}}{\\sum_{{Y_k\\geq Y_i }}w_k^{(m)}e^{\\beta'Z_k}}\\right\\}^{\\Delta_i\n\t\t\n\t\t}\n\t\t\\]\n\t\tand\n\t\t\\begin{equation}\n\t\t\t\\label{eqn:lambda}\n\t\t\t\\hat\\Lambda_n^{(m+1)}(t)=\\sum_{i=1}^n\\frac{\\Delta_i\\mathds{1}_{\\{{Y_i\\leq t{, Y_i<\\tau_0}}\\}}}{\\sum_{j=1}^n\\mathds{1}_{\\{{{Y_i\\leq Y_j\\leq\\tau_0}}\\}} w_j^{(m)}\\exp\\left(\\hat{\\beta}_n^{(m+1)'} Z_j\\right)}.\n\t\t\\end{equation}\n\t\\end{itemize}\n\t{The update of $\\Lambda$ an $\\beta$ in {Step (b)} coincides with the maximization step of the EM algorithm and the weights $w^{(m)}$ correspond to the expectation of the latent variable $B$ given the observed data and the current parameter values. However, unlike the maximum likelihood estimation  \\citep{ST2000}, we are  keeping $\\hat\\gamma_n$ fixed while performing this iterative algorithm.}  {The estimator $\\hat\\Lambda_n$ seems to depend on the unknown $\\tau_0$. However,  with data at hand, one could easily proceed without knowing $\\tau_0$. Indeed, if there are ties at the last uncensored observation, then $\\tau_0$ is revealed by the data. On the other hand, if there are no ties, all uncensored observations will be smaller than $\\tau_0$, hence no need to know $\\tau_0$.}\n\t\n\n\t\n\tAs suggested in \\cite{taylor95,ST2000}, we impose the zero-tail constraint, meaning that $\\hat{S}_u^{(m)}$ is forced to be equal to zero beyond the last event. In this way, all censored observations in the plateau are assigned to the cured group. \n\t\n\n\t\n\t\n\t\n\t\\section{Asymptotic results}\n\t\\label{sec:asymptotics}\n\t\n\tWe first explain {why} presmoothing  {allows for more realistic} asymptotic results in semiparametric  mixture cure models. \n\tNext, we\n\tshow consistency and asymptotic normality of the proposed estimators $\\hat{\\gamma}_n$, $\\hat\\beta_n$ and $\\hat\\Lambda_n$ \n\t{for the parametric\/Cox \n\t\tmixture cure model}\n\twhen, in Step 1, we  use a general nonparametric estimator $\\hat\\pi$ of $\\pi_0$ that satisfies certain assumptions. Afterwards, we verify these conditions for the particular estimator $\\hat\\pi$ in \\eqref{def:hat_pi}. \n\tSome of the proofs can be found in Section~\\ref{sec:appendix} and the rest in  the online Supplementary Material.\n\tThe assumptions mentioned in  Section \\ref{sec:model} are assumed to be satisfied throughout this section. {In addition $Var(Z)$ is supposed to have full rank. }\n\t\n\n\t\n\t\\subsection{A challenge with mixture cure models}\n\t\n\tTo derive asymptotic results, in most of the existing literature it has been  assumed  that\n\t\\begin{equation}\n\t\t\\label{eqn:jump_cond}\n\t\t\\inf_{z}\\mathbb{P}(T_0\\geq\\tau_0|Z=z)>0,\n\t\\end{equation}\n\t\\citep{Lu2008,PK2019}. In nonparametric approaches such a condition keeps the denominators away from zero. In the parametric\/Cox \n\tmixture cure model, it guarantees that  the baseline distribution stays bounded on the compact support $[0,\\tau_0]$.\n\t{However, condition \\eqref{eqn:jump_cond} implies that $\\inf_{z}\\mathbb P (Y=\\tau_0{,\\Delta=1}|Z = z)  >0$, a condition which is not frequently satisfied in real-data applications. \n\t\t\n\t\t\n\t\t\n\t\tOne could imagine that, instead of imposing condition \\eqref{eqn:jump_cond}, it could be possible to proceed as follows: first restrict to events on $[0,\\tau^*]$ for some \n\t\t$\\tau^* < \\tau_0$ such that \n\t\t\\begin{equation}\n\t\t\t\\label{eqn:no_jump_cond}\n\t\t\t\\inf_{z}\\mathbb{P}(Y\\geq\\tau^*{,\\Delta=1}|Z=z)>0,\n\t\t\\end{equation}\n\t\tnext derive the asymptotics, and finally let $\\tau^*$ tend to $\\tau_0$. This idea is used, for instance, in Cox PH model, see  \\cite{FleHar} chapter 8, or \\cite{AndGill82}. However, this idea does not seem to work for  mixture cure models without suitable adaptation. This is because it implicitly requires \n\t\tthat $\\beta_0$ and $\\Lambda_0$ are identifiable from the restricted  data. Here, identifiability means that the true values $\\beta_0$ and $\\Lambda_0$ of the parameters maximize the expectation of the criterion maximized to obtain the estimators.\n\t\tTwo aspects have to be taken into account when analyzing this identifiability. The first aspect is related to the parameter identifiability in the semiparametric model for $T_0$ when the events are restricted to $[0,\\tau^*]$. This property is satisfied in the common models, in particular it holds true in the Cox PH model as soon as $Var(Z)$ has full rank. \n\t\tThe second aspect is the additional complexity induced by the mixture with a cure fraction. If the cure fraction is unknown and one decides to restrict to events on  $[0,\\tau^*]$, the parameter identifiability is likely lost because  the events $\\{T_0\\in (\\tau^*, \\tau_0]\\}$ and $\\{T=\\infty\\}$ are not distinguishable. The usual remedy for this is to impose \\eqref{eqn:jump_cond}, so that  $\\tau^*$ could be taken equal to $\\tau_0$. \n\t\t\n\t\t\n\t\tPresmoothing allows  to avoid  condition \\eqref{eqn:jump_cond}  and thus to fill the gap between the technical conditions and the reality of the data. This is possible because, when using the presmoothing, the conditional probability of the event $\\{T=\\infty\\}$ is identified by other means. We are thus able to prove the consistency of $\\hat \\beta$ and $\\hat\\Lambda$ without imposing   \\eqref{eqn:jump_cond}. Deriving the asymptotic normality without \\eqref{eqn:jump_cond} remains an open problem which will be addressed elsewhere. \n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\\subsection{Consistency}\n\t\t\n\t\tWe first prove  consistency of $\\hat\\gamma_n$ and then use that result to obtain consistency of $\\hat\\Lambda_n$ and $\\hat\\beta_n$. In order to proceed with our results, the following conditions will be used.\n\t\t\\begin{itemize}\n\t\t\t\\item[(AC1)] $\\sup_{x\\in\\mathcal{X}}\\left|\\hat\\pi(x)-\\pi_0(x)\\right|\\to 0$ almost surely. \n\t\t\t\\item[(AC2)] The parameters $\\beta_0$ and $\\gamma_0$ lie in the interior of compact sets $B\\subset\\mathbb{R}^q$, $G\\subset\\mathbb{R}^p$.\n\t\t\t\\item[(AC3)]There exist some constants $a>0$, $c>0$ such that\n\t\t\t\\[\n\t\t\t\\left|\\phi(\\gamma_1,x)-\\phi(\\gamma_2,x)\\right|\\leq c\\Vert\\gamma_1-\\gamma_2\\Vert^a,\\qquad\\forall\\gamma_1,\\gamma_2\\in G,\\,\\forall x\\in\\mathcal{X},\n\t\t\t\\]\n\t\t\twhere $\\Vert\\cdot\\Vert$ denotes the Euclidean distance.\n\t\t\t\\item[(AC4)] $\\inf_{\\gamma\\in G}\\inf_{x\\in\\mathcal{X}}\\phi(\\gamma,x)>0$ and $\\inf_{\\gamma\\in G}\\inf_{x\\in\\mathcal{X}}\\phi(\\gamma,x)<1$.\n\t\t\t\\item[(AC5)] The covariates are bounded: $\\mathbb{P}\\left(\\Vert Z\\Vert<{m} \\text{ and } \\Vert X\\Vert<m\\right)=1$ for some $m>0$.\n\t\t\t\\item[(AC6)] {The baseline hazard function $\\lambda_0(t)=\\Lambda'_0(t)$} is strictly positive and continuous on   $[0,{\\tau_0)}$.  \n\t\t\n\t\t\t\\item[(AC7)] With probability one,   the conditional distribution function of the censoring times $F_C(t|x,z)$  is continuous in $t$ on $[0,{\\tau_0}]$ and   \n\t\t\tthere exists a constant $C>0$ such that\n\t\t\t$$\n\t\t\t\\inf_{0\\leq t_1<t_2\\leq\\tau_0} \\inf_{x,z} \\frac{F_C(t_2|x,z)- F_C(t_1|x,z)}{t_2-t_1} >C.\n\t\t\t$$\n\t\t\\end{itemize}\n\t\t{(AC1) is a minimal assumption given that we want to match $\\phi(\\gamma,\\cdot)$ to $\\hat \\pi (\\cdot)$. \n\t\t\t(AC2) to (AC4) are mild conditions satisfied by usual binary regression models, like for instance the logistic one, and (AC5) is always satisfied in practice for large $m$. }\n\t\t\\begin{theorem}\n\t\t\t\\label{theo:consistency}\n\t\t\t{Let the estimator $\\hat\\gamma_n$ be defined as in (\\ref{def:hat_L_gamma}).}\n\t\t\tAssume that \n\t\t\t(AC1)-(AC4) hold. Then, $\\hat\\gamma_n\\to\\gamma_0$ almost surely.\n\t\t\\end{theorem}\n\t\t\\begin{theorem}\n\t\t\t\\label{theo:consistency2}\n\t\t\tLet the estimators $\\hat\\beta_n$ and $\\hat\\Lambda_n$ be defined as in Section~\\ref{sec:Cox}.\n\t\t\tAssume that\n\t\t\t(AC1)-(AC7) hold.  Then, with probability one,\n\t\t\n\t\t\n\t\t\t$\\Vert\\hat\\beta_n-\\beta_0\\Vert\\to 0$,\n\t\t\n\t\t\twhere $\\Vert\\cdot\\Vert$ denotes the Euclidean distance. Moreover, \n\t\t\tfor any $\\tau^*\\leq  \\tau_0$\n\t\t\tsatisfying  \\eqref{eqn:no_jump_cond}, with probability one,\n\t\t\t\\[\n\t\t\t\\sup_{t\\in[0,{\\tau^*}]}\\left|\\hat\\Lambda_n(t)-\\Lambda_0(t)\\right|\\to 0.\n\t\t\t\\]\n\t\t\\end{theorem}\n\t\t{When condition \\eqref{eqn:jump_cond} is satisfied and $\\tau^*=\\tau_0$ in the previous Theorem, we are referring to the continuous version of $\\Lambda_0$, i.e.  $\\Lambda_0(\\tau_0)=\\lim_{t\\uparrow\\tau_0}\\Lambda_0(t)$. Note that, by definition,  we also have $\\hat\\Lambda_n(\\tau_0)=\\lim_{t\\uparrow\\tau_0}\\hat\\Lambda_n(t)$. } \n\t\t\\subsection{Asymptotic normality}\n\t\tWe first derive asymptotic normality of $\\hat\\gamma_n$ following the approach in \\cite{chen2003}. Theorem 2 in that paper provides sufficient conditions for the $\\sqrt{n}$ normality of parametric estimators obtained by minimizing an objective function that depends on a preliminary infinite dimensional estimator $\\hat\\pi$. In our case, since $\\hat\\gamma_n$ solves\n\t\t\\[\n\t\t\\frac{1}{n}\\nabla_\\gamma\\log \\hat{L}_{n,1}(\\gamma)=0,\n\t\t\\] \n\t\twhere $\\nabla_\\gamma$ denotes the vector-valued partial differentiation operator with respect to the components of $\\gamma$, it follows that $\\hat\\gamma_n$ minimizes the function \n\t\t\\[\n\t\t\\left\\Vert\\frac{1}{n}\\nabla_\\gamma\\log \\hat{L}_{n,1}(\\gamma)\\right\\Vert=\\left\\Vert\\frac{1}{n}\\sum_{i=1}^n m(X_i;\\gamma,\\hat\\pi)\\right\\Vert,\n\t\t\\]\n\t\twhere\n\t\t\\begin{equation}\n\t\t\t\\label{def:m}\n\t\t\tm(x;\\gamma,\\pi)=\\left[\\frac{1-\\pi(x)}{\\phi(\\gamma,x)}-\\frac{\\pi(x)}{1-\\phi(\\gamma,x)}\\right]\\nabla_\\gamma\\phi(\\gamma,x).\n\t\t\\end{equation}\n\t\tHence, we only need to check that the conditions of Theorem 2 in \\cite{chen2003} are satisfied.           \n\t\tTo do that, we need the following assumptions which are stronger than the previous (AC1)-(AC4).\n\t\t\\begin{itemize}\n\t\t\t\\item[(AN1)] The parameter $\\gamma_0$  lies in the interior of a compact set $G\\subset\\mathbb{R}^p$ and, for each $x\\in\\mathcal{X}$, the function $\\gamma\\mapsto\\phi(\\gamma,x)$ is twice continuously differentiable with uniformly bounded derivatives in $G\\times\\mathcal{X}$ and satisfies (AC4).\n\t\t\t\\item[(AN2)] $\\pi_0(\\cdot)$ belongs to a class of functions $\\Pi$ such that \n\t\t\t\\begin{equation*}\n\t\t\t\t\\label{eqn:entropy}\n\t\t\t\t\\int_0^\\infty \\sqrt{\\log N(\\epsilon,\\Pi,\\Vert\\cdot\\Vert_{\\infty})}\\,\\mathrm{d}\\epsilon<\\infty,\n\t\t\t\\end{equation*}\n\t\t\twhere $N(\\epsilon,\\Pi,\\Vert\\cdot\\Vert_{\\infty})$ denotes the $\\epsilon$-covering number of the space $\\Pi$ with respect to $\\Vert\\pi\\Vert_{{\\infty}}=\\sup_{x\\in\\mathcal{X}}|\\pi(x)|$.\n\t\t\t\\item[(AN3)] The matrix $\\mathbb{E}\\left[\\nabla_\\gamma\\phi(\\gamma_0,X)\\nabla_\\gamma\\phi(\\gamma_0,X)'\\right]$ is positive definite.\n\t\t\t\\item[(AN4)] The estimator $\\hat\\pi(\\cdot)$ satisfies the following properties:\n\t\t\t\\begin{itemize}\n\t\t\t\t\\item[(i)] $\\mathbb{P}\\left(\\hat\\pi(\\cdot)\\in\\Pi\\right)\\to 1$.\n\t\t\t\t\\item[(ii)] $\\left\\Vert\\hat\\pi(x)-\\pi_0(x)\\right\\Vert_{{\\infty}}=o_P(n^{-1\/4})$. \n\t\t\t\t\\item[(iii)] There exists a function $\\Psi$ such that \n\t\t\t\t\\[\n\t\t\t\t\\begin{split}\n\t\t\t\t\t&\\mathbb{E}^*\\left[\\left(\\hat\\pi(X)-\\pi_0(X)\\right)\\left(\\frac{1}{\\phi(\\gamma_0,X)}+\\frac{1}{1-\\phi(\\gamma_0,X)}\\right)\\nabla_\\gamma\\phi(\\gamma_0,X)\\right]\\qquad\\qquad\\\\\n\t\t\t\t\t&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad=\\frac{1}{n}\\sum_{i=1}^n\\Psi(Y_i,\\Delta_i,X_i)+R_n,\n\t\t\t\t\\end{split}\n\t\t\t\t\\]\n\t\t\t\twhere $\\mathbb{E}^*$ denotes the conditional expectation given the sample, taken with respect to the generic variable $X$. Moreover, $\\mathbb{E}[\\Psi(Y,\\Delta,X)]=0$ and $\\Vert R_n\\Vert=o_P(n^{-1\/2})$.\n\t\t\t\\end{itemize}\n\t\t\\end{itemize}\n\t\t\\begin{theorem}\n\t\t\t\\label{theo:asymptotic_normality}\n\t\t\t{Let the estimator $\\hat\\gamma_n$ be defined as in (\\ref{def:hat_L_gamma}).}\n\t\t\tAssume that\n\t\t\n\t\t\t(AN1)-(AN4) hold. Then, \\[\n\t\t\tn^{1\/2}\\left(\\hat\\gamma_n-\\gamma_0\\right)\\xrightarrow{d}N(0,\\Sigma_\\gamma)\n\t\t\t\\]\n\t\t\twith covariance matrix $\\Sigma_\\gamma$ defined { in~(A28).}\n\t\t\\end{theorem}\n\t\t{For deriving the asymptotic distribution of $\\hat\\beta_n$ and $\\hat\\Lambda_n$ we assume, for simplicity,  that condition \\eqref{eqn:jump_cond} is satisfied. In such case, in Theorem~\\ref{theo:consistency2} we can take $\\tau^*=\\tau_0$ and obtain uniform strong consistency of $\\hat\\Lambda_n$ on the whole support $[0,\\tau_0]$. We believe that, at the price  of additional technicalities, asymptotic distributional theory can be obtained also without imposing \\eqref{eqn:jump_cond}, as we did for the consistency in Theorem~\\ref{theo:consistency2}. This conjecture is supported by simulations but we leave the problem   to be addressed by future research. }\n\t\t\\begin{theorem}\n\t\t\t\\label{theo:asymptotic_normality2}\n\t\t\tLet the estimators $\\hat\\beta_n$ and $\\hat\\Lambda_n$ be defined as in Section~\\ref{sec:Cox}. \tAssume that {condition~\\eqref{eqn:jump_cond},}\n\t\t\t(AN1)-(AN4) and {(AC2)}, (AC5)-(AC7) hold. Then, \n\t\t\t\\[\n\t\t\t\\left\t\\langle\\sqrt{n}\\left(\\hat\\Lambda_n-\\Lambda_0\\right),\\sqrt{n}\\left(\\hat\\beta_n-\\beta_0\\right)\\right\\rangle\\to G \n\t\t\t\\]\n\t\t\tweakly in $l^\\infty(\\mathcal{H}_\\mathfrak{m})$, where {$\\mathcal{H}_\\mathfrak{m}$ is a functional space defined in Section \\ref{sec:appendix_Cox}}, { $l^\\infty(\\mathcal{H}_\\mathfrak{m})$ denotes the  space of bounded real-valued functions on $\\mathcal{H}_\\mathfrak{m}$,}  $G$ is a tight Gaussian process in $l^\\infty(\\mathcal{H}_\\mathfrak{m})$ with mean zero and covariance process given {in~(A39)} { and  for $h=(h_1,h_2)\\in \\mathcal{H}_\\mathfrak{m}$\n\t\t\t\t\\[\n\t\t\t\t\\langle \\Lambda,\\beta\\rangle(h)=\\int_0^{\\tau_0}h_1(t)\\,\\mathrm{d}\\Lambda(t)+h_2'\\beta.\n\t\t\t\t\\]\t\n\t\t\t} \t\n\t\t\\end{theorem}\n\t{The asymptotic variances of each component of $\\hat\\beta_n$ and of $\\hat\\Lambda_n(t)$ can be obtained from the covariance process in~(A39) by taking $h_1(t)=0$ for all $t$ and $h_2=e_i$ (the $i$th unit vector) or $h_2=0$ and $h_1(s)=\\mathds{1}_{\\{s\\leq t\\}}$. We leave the details about these covariance matrices in the Supplementary Material because they have quite complicated expressions that require definitions of several other quantities. Even though it could be possible in principle to estimate the asymptotic standard errors through plug-in estimators and numerical inverse, we think that this is not feasible in practice and we  do not intend to exploit it further. Instead, we use a bootstrap procedure for estimation of the standard errors in the application discussed in Section~\\ref{sec:application}.  However,  the maximum likelihood estimators are not more favorable in this regard. For example, in the logistic\/Cox model, the proposed estimators of the asymptotic variance in \\cite{Lu2008} also involve solving numerically complicated nonlinear equalitons. For this reason,  bootstrap is used in practice to estimate the standard errors even for the maximum likelihood estimators. }\n\t\n\t{By considering a two-step procedure, where estimation of the incidence parameters is performed independently of the latency model, we expect to loose efficiency of the estimators. However, this does not cause major concern because our  purpose is to provide an alternative estimation method that performs better than the maximum likelihood estimation with sample sizes usually encountered in practice. Efficiency is a key concept for the asymptotics of the estimators, and in general there is no particular need for another method since the MLE would be the best choice. However, in many nonlinear models, like the mixture cure models, the asymptotic approximation is poor and the efficiency becomes a less relevant purpose for real data sample sizes. Hence, we choose to trade efficiency for better performance in a wider range of applications. }\n\t\t\\subsection{Verification of assumptions for $\\hat\\pi$}\n\t\tNext we show that our assumptions \n\t\t(AN1)-(AN4) of the asymptotic theory  are satisfied for  the nonparametric estimator $\\hat\\pi$ defined in \\eqref{def:hat_pi} and the logistic model in \\eqref{eqn:logistic}. For reasons of simplicity, since we use results available in the literature only for a one-dimensional covariate,  we consider only cases with one continuous covariate. In order for assumption (AN4) to be satisfied we need the following conditions:\n\t\t\\begin{itemize}\n\t\t\t\\item[(C1)] The bandwidth $b$ is such that $nb^4\\to 0$ and $nb^{3+\\xi}\/(\\log b^{-1}) \\to \\infty$ for some $\\xi>0$.\n\t\t\t\\item[(C2)] The support $\\mathcal{X}$ of $X$ is a compact subset of $\\mathbb{R}$. The density $f_X(\\cdot)$ of $X$ is bounded away from zero and twice differentiable with bounded second derivative.\n\t\t\t\\item[(C3)] The kernel $k$ is a twice continuously differentiable, symmetric  probability density function with compact support and $\\int uk(u)\\,\\mathrm{d} u=0$.\n\t\t\t\\item[(C4)] (i) The functions $H([0,t]|x)$, $H_1([0,t]|x)$ are twice differentiable with respect to $x$, with uniformly bounded derivatives for all $t\\leq \\tau_0$, $x\\in\\mathcal{X}$. Moreover, there exist continuous nondecreasing functions $L_1$, $L_2$, $L_3$ such that $L_i(0)=0$, $L_i(\\tau_0)<\\infty$ and \tfor all $t, s\\in[0,\\tau_0]$, $x\\in\\mathcal{X}$,\n\t\t\t\\[\n\t\t\t\\begin{split}\n\t\t\t\t\\left|H_c(t|x)-H_c(s|x)\\right|\\leq \\left|L_1(t)-L_1(s)\\right|,&\\quad \\left|H_{1c}(t|x)-H_{1c}(s|x)\\right|\\leq \\left|L_1(t)-L_1(s)\\right|\\\\\n\t\t\t\t\\left|\\frac{\\partial H_c(t|x)}{\\partial x}-\\frac{\\partial H_c(s|x)}{\\partial x}\\right|&\\leq \\left|L_2(t)-L_2(s)\\right|\\\\\n\t\t\t\t\\left|\\frac{\\partial H_{1c}(t|x)}{\\partial x}-\\frac{\\partial H_{1c}(s|x)}{\\partial x}\\right|&\\leq \\left|L_3(t)-L_3(s)\\right|,\n\t\t\t\\end{split}\n\t\t\t\\]\n\t\t\twhere the subscript c denotes the continuous part of a function.\n\t\t\t\n\t\t\t(ii) The jump points for the distribution function $G(t|x)$ of the censoring times given the covariate, are {finite and} the same for all $x$. The partial derivative of $G(t|x)$ with respect to $x$ exists and is uniformly bounded for all $t\\leq\\tau_0$, $x\\in\\mathcal{X}$. Moreover, \n\t\t\tthe partial derivative with respect to $x$ of $F(t|x)$ {(distribution function  of the survival times $T$ given $X=x$)}  exists and is uniformly bounded for all $t\\leq\\tau_0$, $x\\in\\mathcal{X}$.   \n\t\t\t\\item[(C5)]{The survival time $T$ and the censoring time $C$ are independent given $X$.}\n\t\t\\end{itemize}\n\t\t{(C1) to (C5) are conditions guaranteeing the rates of convergence and the i.i.d. representation  \\citep{DA2002}}.\n\t\tIn case of discrete covariates we also need to have only a finite number of atoms.  Assumption (C5) is needed because we are dealing with the distribution of $T$ conditional only on the covariate $X$ (since the cure rate depends only on $X$).\n\t\t\n\t\t\\begin{theorem}\n\t\t\t\\label{theo:pi_hat}\n\t\t\tUnder the conditions (C1)-(C5), the assumptions (AN1)-(AN4) hold true for the logistic model {and the estimator $\\hat\\pi(x)$ defined in (\\ref{def:hat_pi}).}\n\t\t\\end{theorem}\n\t\t\n\t\t\n\t\t\n\t\t\\section{Simulation study}\n\t\t\\label{sec:simulations}\n\t\tIn this section we focus on the logistic\/Cox mixture cure model and evaluate the finite sample performance of the proposed method. Comparison is made with the maximum likelihood estimator implemented in the package \\texttt{smcure}. \n\t\t\n\t\t{\n\tWe first illustrate through a brief example the convergence problems of the smcure estimator. \tWe consider a model where the incidence depends on four independent covariates: $X_1\\sim N(0,2)$, $X_2\\sim \\text{Uniform}(-1,1)$, $X_3\\sim\\text{Bernoulli}(0.8)$, $X_4\\sim\\text{Bernoulli}(0.2)$. The latency depends on $Z_1=X_1$, $Z_2=X_3$ and $Z_3=X_4$.  We generate the cure status $B$ as a Bernoulli random variable with success probability  $\\phi(\\gamma,X)$ where $\\phi$ is the logistic function and $\\gamma=(0.6,-1,1,2.5,1.2)$. The survival times for the uncured observations are generated according to a Weibull proportional hazards model\n\t\\[\n\tS_u(t|z)=\\exp\\left(-\\mu t^\\rho\\exp(\\beta'z)\\right),\n\t\\]\n\tand are truncated at $\\tau_0=14$ for $\\rho=1.75$, $\\mu=1.5$, $\\beta=(-0.8,0.9,0.5)$. The censoring times are independent from $X$ and $T$. They are generated from the exponential distribution with parameter $\\lambda_C=0.22$ and are truncated at $\\tau=16$. We generate $1000$ datasets according to this model with sample size $n=100$, and we observe that smcure fails to converge in $43\\%$ of the cases. Convergence fails mainly in the $\\gamma$ parameter, with only $17\\%$ of the cases failing to converge also for the $\\beta$ parameter (because of the unreasonable $\\gamma$ estimators). On the other hand, there was no convergence problem in the second step of the presmoothing approach. In addition, even among the cases where smcure converged, the presmoothing approach showed significantly better behavior, as can be seen in Table~\\ref{tab:convergence}. \n\t}\n\\begin{table}[h!]\n\t\\caption{\\label{tab:convergence}Bias, variance and MSE of $\\hat\\gamma$ and $\\hat\\beta$ for \\texttt{smcure}  and our approach  among the iterations that converged for smcure.}\n\t\\centering\n\n\t\\scalebox{0.85}{\n\t\t\\fbox{\n\t\t\t\\begin{tabular}{rrrrrrr}\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\t&\t& & & & &  \\\\[-7pt]\n\t\t&\t\\multicolumn{3}{c}{presmoothing}&\\multicolumn{3}{c}{smcure}\\\\\n\t\t\tPar. &  Bias & Var. & MSE & Bias & Var. & MSE \\\\[2pt]\n\t\t\t\t\\hline\n\t\t\t\t& & & & & &  \\\\[-8pt]\n\t\t\t\t $\\gamma_1 $ & $-0.113 $  & $0.620 $ & $0.633  $ & $0.200  $ & $8.318 $ & $8.358 $ \\\\\n\t\t\t$\\gamma_2$\t& $-0.073 $  & $0.156$ & $ 0.162 $ & $-0.388 $ & $3.085$ & $3.236 $\\\\\n\t\t\t $\\gamma_3 $ & $-0.071$  & $0.546$ & $0.551 $ & $ 0.280$ & $1.957 $ & $2.035 $ \\\\\n\t\t\t\t$\\gamma_4$ & $ 0.037$  & $1.326$ & $1.327 $ & $0.704$ & $14.395 $ & $14.891 $ \\\\\n\t\t\t\t\t$\\gamma_5$ & $ -0.250$  & $8.398$ & $8.461 $ & $1.621$ & $36.450 $ & $36.945 $ \\\\\n\t\t\t\t $\\beta_1 $ & $-0.014 $  & $0.011 $ & $0.012  $ & $-0.017  $ & $ 0.012$ & $ 0.012$\\\\\n\t\t\t\t$\\beta_2$ & $0.024 $  & $0.064 $ & $ 0.065 $ & $ 0.026 $ & $0.065 $ & $0.065$ \\\\\n\t\t\t\t$\\beta_3$ & $-0.053 $  & $0.165 $ & $ 0.168 $ & $- 0.053 $ & $0.166 $ & $0.169$ \\\\\t\n\t\t\t\\end{tabular}\n\t}}\n\\end{table}\n\n{Hence, in the cases in which smcure exhibits very poor behavior, the presmoothing is obviously superior. Next, we focus on models for which smcure behaves reasonable (there are convergence problems in less then $3\\%$ of the cases) and show that, even in such scenarios presmoothing can lead to more accurate results. }\n\t\t\n\t\t\n\t\t \n\t\tWe consider four different models and for each of them various choices of the parameters in order to cover a wide range of scenarios. \n\t\tThe models are as follows. \n\t\t\n\t\t\\textit{Model 1.} Both incidence and latency depend on one covariate $X$, which is uniform on $(-1,1)$. We generate the cure status $B$ as a Bernoulli random variable with success probability  $\\phi(\\gamma,X)$ where $\\phi$ is the logistic function. The survival times for the uncured observations are generated according to a Weibull proportional hazards model\n\t\t\\[\n\t\tS_u(t|x)=\\exp\\left(-\\mu t^\\rho\\exp(\\beta x)\\right),\n\t\t\\]\n\t\tand are truncated at $\\tau_0$ for $\\rho=1.75$, $\\mu=1.5$, $\\beta=1$ and $\\tau_0=4$. The censoring times are independent from $X$ and $T$. They are generated from the exponential distribution with parameter $\\lambda_C$ and are truncated at $\\tau=6$. \n\t\t\n\t\t\\textit{Model 2.} Both incidence and latency depend on one covariate $X$ with standard normal distribution. The cure status  and the survival times for the uncured observations are generated as in Model 1 for $\\rho=1.75$, $\\mu=1.5$, $\\beta=1$ and $\\tau_0=10$. The censoring times are  generated according to a Weibull proportional hazards model\n\t\t\\[\n\t\tS_C(t|x)=\\exp\\left(-\\nu\\mu t^{\\rho}\\exp(\\beta_Cx)\\right),\n\t\t\\] \n\t\tfor $\\beta_C=1$ and various choices of $\\nu$ and are truncated at $\\tau=15$. \n\t\t\n\t\t\\textit{Model 3.} For the incidence we consider three independent covariates: $X_1$ is normal with mean zero and standard deviation $2$, $X_2$ and $X_3$ are Bernoulli random variables with parameters $0.6 $ and  $0.4$ respectively. The latency also depends on three covariates: $Z_1=X_1$, $Z_2$ is a uniform random variable on $(-3,3)$ independent of the previous ones  and $Z_3=X_2$.  The cure status  and the survival times for the uncured observations are generated as in Model 1 for $\\rho=1.75$, $\\mu=1.5$ and different choices of the other parameters.  The censoring times are  generated independently of the previous variables from an exponential distribution with parameter $\\lambda_C$ \n\t\tand are truncated at $\\tau$, for given choices of  $\\lambda_C$ and $\\tau$. \n\t\t\n\t\t\\textit{Model 4.} This setting is obtained by adding an additional continuous covariate to the incidence component of Model 3. To be precise,  $X_1$ is normal with mean zero and standard deviation $2$, $X_2$ is uniform on $(-1,1)$ {independent of the other variables}, $X_3$ and $X_4$ are Bernoulli random variables with parameters $0.6 $ and  $0.4$ respectively. As in Model 3, $Z_1=X_1$, $Z_2$ is a uniform random variable on $(-3,3)$ independent of the previous ones  and $Z_3=X_3$. The event and censoring times are generated as in the previous model.\n\t\t\n\t\tFor the four models we choose the values of the unspecified parameters in such a way that the cure rate is around $20\\%$, $30\\%$, $50\\%$ (corresponding respectively to scenarios $1$, $2$ and $3$) and the censoring rate corresponds to three levels (with a difference of $5\\%$ between each other). The specification of the parameters and the corresponding censoring rate and percentage of the observations in the plateau are given Table~\\ref{tab:models}. {Note that, within each scenario, the fraction of the observations in the plateau decreases as the censoring rate increases because more cured observations are censored earlier and as a result are not observed in the plateau. This makes the estimation of the cure rate more difficult.} The truncation of the  survival  and censoring times on $[0,\\tau_0]$ and $[0,\\tau]$ is made in such a way that $\\tau_0<\\tau$ and  condition \\eqref{eqn:jump_cond} is satisfied { but in practice\n\t\t\tit is unlikely to observe event times at $\\tau_0$. In this way, we try to find a compromise between theoretical assumptions and  real-life scenarios.}\n\t\t\n\t\t\\begin{table}\n\t\t\t\\caption{\t\\label{tab:models}Parameter values and model characteristics for each scenario.}\n\t\t\t\\centering\n\t\t\t\\addtolength{\\tabcolsep}{-4pt}\n\t\t\t\\fbox{%\n\t\t\t\t\\begin{tabular}{ccccccc}\n\t\t\t\t\tModel & Parameters & Scenario & Cens.  & Cens. & Cens.  & Plateau\\\\\n\t\t\t\t\t&  &  &  level & parameters &  rate & \\\\%[2pt]\n\t\t\t\t\t\\hline\n\t\t\t\t\t& & & & & &  \\\\[-10pt]\n\t\t\t\t\t&  &  & $1$  & $\\lambda_C=0.1 $ & $25\\%$ & $15\\% $\\\\\n\t\t\t\t\t& $\\gamma=(1.75, 2)$ & $1 $ & $2$  & $\\lambda_C=0.2 $ & $30\\%$ & $11\\% $\\\\\n\t\t\t\t\t&  &  & $3$  & $\\lambda_C=0.3 $ & $35\\%$ & $9\\% $\\\\\n\t\t\t\t\t\\cline{2-7}\n\t\t\t\t\t& &  & $1$  & $\\lambda_C=0.1 $ & $34\\%$ & $22\\% $\\\\\n\t\t\t\t\t1 & $\\gamma=(1, 1.5)$ & $2$ & $2$  & $\\lambda_C=0.25 $ & $40\\%$ & $15\\% $\\\\\n\t\t\t\t\t&  &  & $3$  & $\\lambda_C=0.4 $ & $46\\%$ & $10\\% $\\\\\n\t\t\t\t\t\\cline{2-7}\n\t\t\t\t\t& &  & $1$  & $\\lambda_C=0.2 $ & $54\\%$ & $32\\% $\\\\\n\t\t\t\t\t& $\\gamma=(0.1, 5)$ & $3 $ & $2$  & $\\lambda_C=0.4 $ & $59\\%$ & $23\\% $\\\\\n\t\t\t\t\t&  &  & $3$  & $\\lambda_C=0.7 $ & $65\\%$ & $15\\% $\\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\t& & & & & &  \\\\[-10pt]\n\t\t\t\t\t&  &  & $1$  & $\\nu=1\/15 $ & $25\\%$ & $7\\% $\\\\\n\t\t\t\t\t& $\\gamma=(1.5, 0.5)$ & $1$ & $2$  & $\\nu=1\/7 $ & $30\\%$ & $4\\% $\\\\\n\t\t\t\t\t&  &  & $3$  & $\\nu=1\/4 $ & $35\\%$ & $2\\% $\\\\\n\t\t\t\t\t\\cline{2-7}\n\t\t\t\t\t& &  & $1$  & $\\nu=1\/13 $ & $35\\%$ & $14\\% $\\\\\n\t\t\t\t\t2 & $\\gamma=(1, 1)$ & $2 $ & $2$  & $\\nu=1\/10 $ & $40\\%$ & $9\\% $\\\\\n\t\t\t\t\t&  &  & $3$  & $\\nu=5\/18 $ & $45\\%$ & $6\\% $\\\\\n\t\t\t\t\t\\cline{2-7}\n\t\t\t\t\t& &  & $1$  & $\\nu=1\/9 $ & $56\\%$ & $38\\% $\\\\\n\t\t\t\t\t& $\\gamma=(-0.1, 5)$ & $3 $ & $2$  & $\\nu=1\/4$ & $60\\%$ & $30\\% $\\\\\n\t\t\t\t\t&  &  & $3$  & $\\nu=2\/5 $ & $65\\%$ & $25\\% $\\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\t& & & & & &  \\\\[-10pt]\n\t\t\t\t\t& $\\gamma=(0.5, -1,2.5, 1.2)$ &  & $1$  & $\\lambda_C=0.12$ & $25\\%$ & $10\\% $\\\\\n\t\t\t\t\t& $\\beta=( -1,0.5, 1.5)$ & $1$ & $2$  & $\\lambda_C=0.25 $ & $30\\%$ & $6\\% $\\\\\n\t\t\t\t\t& $\\tau_0=30$, $\\tau=35$ &  & $3$  & $\\lambda_C=0.45 $ & $35\\%$ & $4\\% $\\\\\n\t\t\t\t\t\\cline{2-7}\n\t\t\t\t\t& $\\gamma=(1,2,1.8, 0.5)$&  & $1$  & $\\lambda_C=0.2 $ & $35\\%$ & $16\\% $\\\\\n\t\t\t\t\t3 & $\\beta=(1,0.5, 2)$ & $2 $ & $2$  & $\\lambda_C=0.5 $ & $40\\%$ & $9\\% $\\\\\n\t\t\t\t\t& $\\tau_0=6$, $\\tau=8$ &  & $3$  & $\\lambda_C=0.8 $ & $45\\%$ & $6\\% $\\\\\n\t\t\t\t\t\\cline{2-7}\n\t\t\t\t\t&$\\gamma=(-0.8,1.3,1.5,-0.2) $&  & $1$  & $\\lambda_C=0.3 $ & $55\\%$ & $24\\% $\\\\\n\t\t\t\t\t& $\\beta=(1,-0.1,0.8)$ & $3 $ & $2$  & $\\lambda_C=0.7 $ & $59\\%$ & $14\\% $\\\\\n\t\t\t\t\t& $\\tau_0=5$, $\\tau=7$ &  & $3$  & $\\lambda_C=1.3 $ & $65\\%$ & $8\\% $\\\\\n\t\t\t\t\t\\hline\n\t\t\t\t\t& & & & & &  \\\\[-10pt]\n\t\t\t\t\t& $\\gamma=(0.6,-1,1,2.5,1.2)$ &  & $1$  & $\\lambda_C=0.1$ & $25\\%$ & $11\\% $\\\\\n\t\t\t\t\t& $\\beta=( -0.8,0.3,0.5)$ & $1$ & $2$  & $\\lambda_C=0.22 $ & $30\\%$ & $7\\% $\\\\\n\t\t\t\t\t& $\\tau_0=14$, $\\tau=16$ &  & $3$  & $\\lambda_C=0.35 $ & $35\\%$ & $5\\% $\\\\\n\t\t\t\t\t\\cline{2-7}\n\t\t\t\t\t& $\\gamma=(0.45,0.5,2{,}1,0.5)$&  & $1$  & $\\lambda_C=0.15 $ & $35\\%$ & $11\\% $\\\\\n\t\t\t\t\t4 & $\\beta=(1,0.5, 2)$ & $2 $ & $2$  & $\\lambda_C=0.35 $ & $40\\%$ & $7\\% $\\\\\n\t\t\t\t\t& $\\tau_0=18$, $\\tau=20$ &  & $3$  & $\\lambda_C=0.6 $ & $45\\%$ & $5\\% $\\\\\n\t\t\t\t\t\\cline{2-7}\n\t\t\t\t\t&$\\gamma=(-0.22,0.3,-0.4,0.5,-0.2) $&  & $1$  & $\\lambda_C=0.2 $ & $55\\%$ & $30\\% $\\\\\n\t\t\t\t\t& $\\beta=(0.4,-0.1,0.5)$ & $3 $ & $2$  & $\\lambda_C=0.4 $ & $59\\%$ & $20\\% $\\\\\n\t\t\t\t\t& $\\tau_0=6$, $\\tau=8$ &  & $3$  & $\\lambda_C=0.7 $ & $65\\%$ & $12\\% $\\\\\n\t\t\t\t\n\t\t\t\t\\end{tabular}\n\t\t\t}\n\t\t\\end{table}\n\t\t\n\t\t\\begin{table}\n\t\t\t\\caption{\\label{tab:results1_2}Bias, variance and MSE of $\\hat\\gamma$ and $\\hat\\beta$ for \\texttt{smcure} (second rows) and our approach (first rows) in Model 1 and 2.}\n\t\t\t\\centering\n\t\t\n\t\t\t\\scalebox{0.85}{\n\t\t\t\t\\fbox{\n\t\t\t\t\t\\begin{tabular}{ccccrrrrrrrrr}\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-7pt]\n\t\t\t\t\t\t&\t&&&\\multicolumn{3}{c}{Cens. level 1}&\\multicolumn{3}{c}{Cens. level 2}&\\multicolumn{3}{c}{Cens. level 3}\\\\\n\t\t\t\t\t\tMod.&\tn&  scen. & Par. &  Bias & Var. & MSE & Bias & Var. & MSE & Bias & Var. & MSE\\\\[2pt]\n\t\t\t\t\t\t\\hline\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t1&\t$200$ & $1 $ & $\\gamma_1 $ & $0.001 $  & $0.060 $ & $0.060  $ & $0.020  $ & $0.065 $ & $0.065 $ & $0.005  $ & $0.078 $ & $0.078 $\\\\\n\t\t\t\t\t\t&\t& &  & $0.021 $  & $0.063 $ & $ 0.063 $ & $0.050  $ & $0.068 $ & $0.071 $ & $  0.044$ & $0.084 $ & $0.086 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_2 $ & $-0.034 $  & $0.164 $ & $0.165  $ & $ -0.014 $ & $0.202 $ & $0.202 $ & $-0.051  $ & $0.209 $ & $0.212 $\\\\\n\t\t\t\t\t\t&\t& & & $ 0.026$  & $0.173 $ & $0.173  $ & $ 0.067 $ & $0.222 $ & $0.226 $ & $0.044  $ & $0.229 $ & $0.230 $\\\\\n\t\t\t\t\t\t&\t& & $\\beta $ & $0.008 $  & $0.028 $ & $0.028  $ & $0.015  $ & $ 0.029$ & $ 0.029$ & $0.013  $ & $0.034 $ & $0.035 $\\\\\n\t\t\t\t\t\t&\t& &  & $0.007 $  & $0.028 $ & $ 0.028 $ & $ 0.012 $ & $0.029 $ & $0.029 $ & $0.009  $ & $0.035 $ & $0.035 $\\\\\n\t\t\t\t\t\t\\cline{3-13}\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t&\t& $3 $ & $\\gamma_1 $ & $-0.001 $  & $0.059 $ & $0.059  $ & $0.009  $ & $ 0.065$ & $ 0.065$ & $ -0.014 $ & $0.091 $ & $ 0.092$\\\\\n\t\t\t\t\t\t&\t& &  & $0.010 $  & $0.064 $ & $0.064  $ & $0.029  $ & $0.074 $ & $0.075 $ & $0.037  $ & $0.113 $ & $0.115 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_2 $ & $-0.034 $  & $0.536 $ & $0.537  $ & $-0.111  $ & $ 0.595$ & $0.608 $ & $ -0.085 $ & $ 0.809$ & $0.816 $\\\\\n\t\t\t\t\t\t&\t& & & $0.201 $  & $ 0.649$ & $ 0.689 $ & $0.218  $ & $0.768 $ & $0.816 $ & $0.400  $ & $1.146 $ & $1.306 $\\\\\n\t\t\t\t\t\t&\t& & $\\beta $ & $ 0.011$  & $0.090 $ & $ 0.090 $ & $ 0.024 $ & $ 0.109$ & $0.110 $ & $0.014  $ & $0.128 $ & $0.128 $\\\\\n\t\t\t\t\t\t&\t& &  & $ 0.007$  & $0.091 $ & $ 0.091 $ & $0.014  $ & $0.110 $ & $0.110 $ & $-0.001  $ & $0.129 $ & $0.129 $\\\\\n\t\t\t\t\t\t\\cline{2-13}\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t&\t$400$ & $1 $ & $\\gamma_1 $ & $0.001 $  & $0.028 $ & $0.028  $ & $ 0.007 $ & $0.032 $ & $0.032 $ & $0.001  $ & $0.037 $ & $0.037 $\\\\\n\t\t\t\t\t\t&\t& &  & $0.015 $  & $0.029 $ & $0.030  $ & $0.027  $ & $0.033 $ & $ 0.034$ & $0.024  $ & $0.039 $ & $0.039 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_2 $ & $-0.024 $  & $0.083 $ & $ 0.084 $ & $-0.004  $ & $ 0.088$ & $ 0.088$ & $ -0.018 $ & $0.107 $ & $0.107 $\\\\\n\t\t\t\t\t\t&\t& & & $ 0.021$  & $0.087 $ & $0.087  $ & $ 0.049 $ & $0.093 $ & $0.095 $ & $0.041  $ & $0.111 $ & $0.113 $\\\\\n\t\t\t\t\t\t&\t& & $\\beta $ & $0.003 $  & $0.013 $  & $0.013  $ & $0.007 $ & $0.015 $ & $0.015  $ & $0.002 $ & $ 0.016$ & $0.016$\\\\\n\t\t\t\t\t\t&\t& &  & $0.002 $  & $0.013 $ & $0.013  $ & $0.005  $ & $0.015 $ & $0.015 $ & $ 0.000 $ & $ 0.016$ & $0.016 $\\\\\n\t\t\t\t\t\t\\cline{3-13}\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t&\t& $3 $ & $\\gamma_1 $ & $-0.004 $  & $0.029 $ & $0.029  $ & $-0.004  $ & $0.030 $ & $ 0.030$ & $-0.007  $ & $0.048 $ & $ 0.048$\\\\\n\t\t\t\t\t\t&\t& &  & $ 0.002$  & $0.030 $ & $0.030  $ & $ 0.009 $ & $0.033 $ & $0.033 $ & $0.015 $ & $0.053 $ & $0.053 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_2 $ & $-0.050 $  & $ 0.237$ & $ 0.239 $ & $-0.080  $ & $0.312 $ & $0.318 $ & $-0.134  $ & $0.432 $ & $ 0.450$\\\\\n\t\t\t\t\t\t&\t& & & $0.111 $  & $0.260 $ & $0.273  $ & $ 0.142 $ & $0.361 $ & $ 0.381$ & $ 0.167 $ & $0.491 $ & $0.519 $\\\\\n\t\t\t\t\t\t&\t& & $\\beta $ & $-0.003 $  & $0.039 $ & $0.039  $ & $0.024  $ & $0.051 $ & $0.052 $ & $ 0.013 $ & $0.071 $ & $0.071 $\\\\\n\t\t\t\t\t\t&\t& &  & $-0.007 $  & $0.039 $ & $ 0.039 $ & $ 0.017 $ & $0.051 $ & $0.052 $ & $0.000  $ & $0.071 $ & $0.071 $\\\\\n\t\t\t\t\t\t\\hline\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t2&\t$200$ & $1 $ & $\\gamma_1 $ & $0.004 $  & $0.040 $ & $0.040  $ & $0.020  $ & $0.045 $ & $0.045 $ & $-0.016  $ & $0.060 $ & $0.060 $\\\\\n\t\t\t\t\t\t&\t& &  & $0.017 $  & $ 0.040$ & $ 0.040 $ & $ 0.058 $ & $0.047 $ & $0.050 $ & $ 0.083 $ & $0.079 $ & $0.086 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_2 $ & $ 0.001$  & $0.039 $ & $0.039  $ & $- 0.022 $ & $0.042 $ & $0.043 $ & $-0.027  $ & $ 0.055$ & $0.056 $\\\\\n\t\t\t\t\t\t&\t& & & $0.016 $  & $0.040 $ & $0.040  $ & $ 0.008 $ & $0.047 $ & $0.047 $ & $0.029 $ & $0.072 $ & $0.073 $\\\\\n\t\t\t\t\t\t&\t& & $\\beta $ & $0.006 $  & $ 0.011$ & $0.011  $ & $ 0.000 $ & $ 0.014$ & $ 0.014$ & $0.011  $ & $0.015 $ & $0.015 $\\\\\n\t\t\t\t\t\t&\t& &  & $0.005 $  & $ 0.011$ & $0.011  $ & $ -0.002 $ & $0.014 $ & $ 0.014$ & $0.004  $ & $0.016 $ & $0.016 $\\\\\n\t\t\t\t\t\t\\cline{3-13}\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t&\t& $3 $ & $\\gamma_1 $ & $-0.016 $  & $0.071 $ & $ 0.071 $ & $ -0.057 $ & $0.065 $ & $0.068 $ & $ -0.139 $ & $ 0.083$ & $ 0.102$\\\\\n\t\t\t\t\t\t&\t& &  & $ 0.029$  & $0.092 $ & $0.092  $ & $ 0.051 $ & $0.119 $ & $0.121 $ & $0.024  $ & $0.175 $ & $0.176 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_2 $ & $-0.468 $  & $0.723 $ & $0.942  $ & $ -0.943 $ & $0.823 $ & $ 1.713$ & $-1.348  $ & $0.829 $ & $ 2.646$\\\\\n\t\t\t\t\t\t&\t& & & $0.364 $  & $ 0.926$ & $ 1.058 $ & $ 0.495 $ & $ 1.453$ & $ 1.698$ & $0.596  $ & $2.128 $ & $2.482 $\\\\\n\t\t\t\t\t\t&\t& & $\\beta $ & $0.017 $  & $0.035 $ & $ 0.035 $ & $0.022  $ & $0.039 $ & $0.039 $ & $ 0.036 $ & $0.052 $ & $0.054 $\\\\\n\t\t\t\t\t\t&\t& &  & $ 0.014$  & $ 0.035$ & $0.035  $ & $ 0.017 $ & $0.040 $ & $0.040 $ & $0.025  $ & $0.053 $ & $0.054 $\\\\\n\t\t\t\t\t\t\\cline{2-13}\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t&\t$400$ & $1 $ & $\\gamma_1 $ & $0.011 $  & $0.019 $ & $0.019  $ & $0.019  $ & $0.023 $ & $ 0.023$ & $0.002  $ & $0.032 $ & $0.032 $\\\\\n\t\t\t\t\t\t&\t& &  & $0.018 $  & $0.019 $ & $0.019  $ & $ 0.037 $ & $0.023 $ & $0.025 $ & $0.047  $ & $0.034 $ & $0.036 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_2 $ & $-0.002 $  & $0.018 $ & $ 0.018 $ & $-0.010  $ & $0.023 $ & $ 0.023$ & $ -0.019 $ & $0.027 $ & $0.028 $\\\\\n\t\t\t\t\t\t&\t& & & $0.009 $  & $0.018 $ & $ 0.018 $ & $ 0.007 $ & $0.025 $ & $ 0.025$ & $ 0.008 $ & $ 0.032$ & $0.032 $\\\\\n\t\t\t\t\t\t&\t& & $\\beta $ & $ 0.000$  & $0.006 $ & $ 0.006 $ & $0.004  $ & $0.006 $ & $ 0.006$ & $  0.003$ & $ 0.008$ & $0.008 $\\\\\n\t\t\t\t\t\t&\t& &  & $0.000 $  & $ 0.006$ & $0.006  $ & $ 0.002 $ & $0.006 $ & $0.006 $ & $ 0.000 $ & $0.008 $ & $0.008 $\\\\\n\t\t\t\t\t\t\\cline{3-13}\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t&\t& $3 $ & $\\gamma_1 $ & $-0.015 $  & $0.031 $ & $0.031  $ & $-0.071  $ & $0.034 $ & $0.039 $ & $-0.086  $ & $ 0.041$ & $0.048 $\\\\\n\t\t\t\t\t\t&\t& &  & $0.014 $  & $0.037 $ & $0.038  $ & $ 0.001 $ & $0.050 $ & $0.050 $ & $ 0.047 $ & $0.072 $ & $0.074 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_2 $ & $-0.444 $  & $ 0.330$ & $ 0.527 $ & $-0.802  $ & $0.410 $ & $ 1.053$ & $ -1.191 $ & $0.463 $ & $1.881 $\\\\\n\t\t\t\t\t\t&\t& & & $0.149 $  & $ 0.364$ & $0.386  $  & $ 0.244$ & $0.557 $ & $0.616  $ & $0.325 $ & $ 0.739$& $ 0.845 $\\\\\n\t\t\t\t\t\t&\t& & $\\beta $ & $0.007 $  & $0.016 $ & $0.016  $ & $0.015  $ & $0.019 $ & $0.020 $ & $0.017  $ & $ 0.024$ & $0.024 $\\\\\n\t\t\t\t\t\t&\t& &  & $0.004 $  & $ 0.016$ & $0.016  $ & $ 0.010 $ & $0.019 $ & $0.020 $ & $0.010  $ & $ 0.024$ & $0.024 $\\\\\n\t\t\t\t\t\\end{tabular}\n\t\t\t}}\n\t\t\\end{table}\n\t\tFor each setting we consider samples of size $n=200, 400, 1000$. This leads to a total of $108$ settings ($4$ models, $3$ scenarios for the cure rate, $3$ censoring levels and $3$ sample sizes). In this way, we hope to address a number  of issues such as the effect of the cure proportion, the sample size, amount and type of censoring, covariates (number, relation between $X$ and $Z$ and their distribution).\n\t\tFor each configuration $1020$  datasets were generated and the estimators of $\\beta_0$ and $\\gamma_0$ were computed through \\texttt{smcure} and our method. We report the bias, variance and mean squared error (MSE) of the estimators, computed after omitting the lowest and the highest $1\\%$ of the estimators (for stability of the reported results) and rounded to three decimals. Tables~\\ref{tab:results1_2}-\\ref{tab:results4} show some of the results, while the rest can be found in the online Supplementary Material. {We aim to provide a ready-to-use method that works well in practice without needing to think about how to choose the kernel function or the bandwidth. Hence, we illustrate the performance of the method for some standard and commonly used choices. } The kernel function $k$ is taken to be the Epanechnikov kernel $k(u)=(3\/4)(1-u^2)\\mathds{1}_{\\{|u|\\leq1\\}}\n\t\t$. \n\t\tWe use the  cross-validation bandwidth  (implemented in the R package \\texttt{np}) for kernel estimators of conditional distribution functions, in our case for estimation of $H=H_0+H_1$ given the continuous covariates (affecting the incidence). In addition, we restrict to the interval  $[0,Y_{(m)}]$, where $Y_{(m)}$ is the last observed event time since the estimator of the cure probability $\\hat\\pi$ in \\eqref{def:hat_pi} is essentially a product over values of $t$ that are equal to the observed event times. This means that we use the {cross-validation} bandwidth for estimation of the conditional distribution $H(t|x)$ for $t\\leq Y_{(m)}$. This choice of bandwidth improves significantly the performance of the estimators, compared to the cross-validation bandwidth on the whole interval $[0,\\tau]$, in situations with a large percentage  of observations in the plateau, while it leads to little difference otherwise. \n\t\t\\begin{table}\n\t\t\t\\caption{\t\\label{tab:results3}Bias, variance and MSE of $\\hat\\gamma$ for \\texttt{smcure} (second rows) and our approach (first rows) in Model 3.}\n\t\t\t\\centering\n\t\t\n\t\t\t\\scalebox{0.85}{\n\t\t\t\t\\fbox{\n\t\t\t\t\t\\begin{tabular}{ccccrrrrrrrrr}\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t&\t&&&\\multicolumn{3}{c}{Cens. level 1}&\\multicolumn{3}{c}{Cens. level 2}&\\multicolumn{3}{c}{Cens. level 3}\\\\\n\t\t\t\t\t\tMod.&\tn&  scen. & Par. &  Bias & Var. & MSE & Bias & Var. & MSE & Bias & Var. & MSE\\\\[2pt]\n\t\t\t\t\t\t\\hline\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t3&\t$200$ & $1 $ & $\\gamma_1 $ & $ 0.025$  & $0.147 $ & $0.147  $ & $0.010  $ & $0.192 $ & $0.192 $ & $ -0.008 $ & $0.243 $ & $0.243 $\\\\\n\t\t\t\t\t\t&\t& &  & $ 0.034$  & $0.147 $ & $0.148  $ & $0.034  $ & $0.191 $ & $ 0.192$ & $0.062  $ & $0.249 $ & $0.253 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_2 $ & $-0.045 $  & $0.042 $ & $ 0.044 $ & $ -0.078 $ & $0.049 $ & $ 0.055$ & $-0.085  $ & $0.059 $ & $ 0.066$\\\\\n\t\t\t\t\t\t&\t& & & $-0.077 $  & $ 0.050$ & $ 0.056 $ & $ -0.122 $ & $0.065 $ & $0.080 $ & $-0.148  $ & $0.092 $ & $0.144 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_3 $ & $0.081 $  & $0.366 $ & $ 0.373 $ & $ 0.074 $ & $ 0.485$ & $ 0.491$ & $0.029  $ & $0.536 $ & $ 0.537$\\\\\n\t\t\t\t\t\t&\t& & & $0.174 $  & $ 0.397$ & $0.427  $ & $ 0.266 $ & $0.574 $ & $0.644 $ & $0.309  $ & $ 0.799$ & $0.895 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_4 $ & $ -0.046$  & $0.326 $ & $ 0.373 $ & $-0.160  $ & $ 0.412$ & $0.437 $ & $ -0.289 $ & $ 0.453$ & $ 0.537$\\\\\n\t\t\t\t\t\t&\t& & & $0.087 $  & $0.366 $ & $ 0.374 $ & $ 0.089 $ & $0.528 $ & $0.535 $ & $0.186  $ & $ 0.908$ & $0.943 $\\\\\n\t\t\t\t\t\t\\cline{3-13}\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t&\t& $3 $ & $\\gamma_1 $ & $-0.059 $  & $ 0.161$ & $ 0.164 $ & $  -0.091$ & $ 0.258$ & $0.266 $ & $ -0.223 $ & $ 0.419$ & $0.468 $\\\\\n\t\t\t\t\t\t&\t& &  & $-0.053 $  & $ 0.163$ & $ 0.166 $ & $-0.071  $ & $0.261 $ & $0.266 $ & $-0.138  $ & $0.524 $ & $ 0.543$\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_2 $ & $0.018 $  & $0.046 $ & $  0.046$ & $0.026  $ & $ 0.063$ & $0.064 $ & $0.086  $ & $0.088 $ & $ 0.096$\\\\\n\t\t\t\t\t\t&\t& & & $0.080 $  & $ 0.052$ & $ 0.058 $ & $0.121  $ & $0.080 $ & $0.095 $ & $0.252  $ & $0.170 $ & $0.233 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_3 $ & $0.060 $  & $0.235 $ & $0.238  $ & $0.076  $ & $ 0.366$ & $0.372 $ & $ 0.135 $ & $0.517 $ & $ 0.535$\\\\\n\t\t\t\t\t\t&\t& & & $0.091 $  & $0.242 $ & $0.251  $ & $0.135  $ & $ 0.375$ & $ 0.393$ & $0.228  $ & $0.642 $ & $0.694 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_4 $ & $-0.030 $  & $0.202 $ & $ 0.203 $ & $ -0.040 $ & $0.292 $ & $ 0.293$ & $ -0.081 $ & $0.479 $ & $0.486 $\\\\\n\t\t\t\t\t\t&\t& & & $-0.027 $  & $ 0.205$ & $0.205  $ & $ -0.017 $ & $0.277 $ & $0.277 $ & $ -0.037 $ & $0.534 $ & $ 0.535$\\\\\n\t\t\t\t\t\t\\cline{2-13}\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t&\t$400$ & $1 $ & $\\gamma_1 $ & $ 0.016$  & $0.074 $ & $0.074  $ & $0.021  $ & $0.091 $ & $ 0.092$ & $ 0.003 $ & $ 0.128$ & $0.128 $\\\\\n\t\t\t\t\t\t&\t& &  & $0.017 $  & $ 0.072$ & $0.073  $ & $ 0.022 $ & $0.082 $ & $0.082 $ & $ 0.023 $ & $0.108 $ & $0.108 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_2 $ & $ -0.026$  & $0.019 $ & $0.019  $ & $ -0.039 $ & $0.023 $ & $ 0.025$ & $-0.070  $ & $0.032 $ & $ 0.037$\\\\\n\t\t\t\t\t\t&\t& & & $-0.042 $  & $0.020 $ & $0.021  $ & $ -0.049 $ & $0.025 $ & $0.027 $ & $-0.081  $ & $0.035 $ & $ 0.041$\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_3 $ & $0.039 $  & $0.194 $ & $ 0.195 $ & $0.028  $ & $0.219 $ & $ 0.220$ & $ 0.026 $ & $0.298 $ & $ 0.298$\\\\\n\t\t\t\t\t\t&\t& & & $0.093 $  & $ 0.190$ & $ 0.198 $ & $ 0.097 $ & $0.206 $ & $0.215 $ & $0.158  $ & $0.297 $ & $ 0.322$\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_4 $ & $ 0.010$  & $0.171 $ & $ 0.171 $ & $-0.091  $ & $0.193 $ & $0.201 $ & $ -0.178 $ & $0.276 $ & $0.307 $\\\\\n\t\t\t\t\t\t&\t& & & $ 0.070$  & $0.177 $ & $ 0.182 $ & $0.038  $ & $0.198 $ & $0.200 $ & $0.088  $ & $0.289 $ & $0.297 $\\\\\n\t\t\t\t\t\t\\cline{3-13}\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t&\t& $3 $ & $\\gamma_1 $ & $-0.023 $  & $0.089 $ & $0.089  $ & $-0.051  $ & $0.118 $ & $0.121 $ & $-0.124  $ & $0.212 $ & $0.228 $\\\\\n\t\t\t\t\t\t&\t& &  & $-0.029 $  & $0.092 $ & $ 0.093 $ & $-0.032  $ & $0.112 $ & $ 0.113$ & $-0.062  $ & $0.200 $ & $0.204 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_2 $ & $0.003 $  & $0.023 $ & $  0.023$ & $0.006  $ & $0.033 $ & $0.033 $ & $0.042  $ & $0.048 $ & $0.050 $\\\\\n\t\t\t\t\t\t&\t& & & $0.042 $  & $0.023 $ & $ 0.025 $ & $ 0.057 $ & $0.034 $ & $0.037 $ & $ 0.104 $ & $ 0.055$ & $0.066 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_3 $ & $0.010 $  & $0.113 $ & $0.113  $ & $ 0.042 $ & $ 0.166$ & $ 0.168$ & $0.090  $ & $0.276 $ & $0.284 $\\\\\n\t\t\t\t\t\t&\t& & & $0.039 $  & $0.111 $ & $0.111  $ & $ 0.060 $ & $0.152 $ & $0.156 $ & $ 0.108 $ & $0.250 $ & $ 0.262$\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_4 $ & $0.012 $  & $0.110 $ & $ 0.110 $ & $ -0.021 $ & $ 0.131$ & $ 0.131$ & $-0.047  $ & $0.220 $ & $0.223 $\\\\\n\t\t\t\t\t\t&\t& & & $0.014 $  & $ 0.111$ & $0.111  $ & $ -0.018 $ & $0.117 $ & $ 0.118$ & $-0.020  $ & $0.183 $ & $0.183 $\\\\\n\t\t\t\t\t\\end{tabular}\n\t\t\t}\t}\n\t\t\\end{table}\n\t\t\n\t\tSimulations show that, for not large sample size, the new method performs better than \\texttt{smcure} for estimation of $\\gamma_0$, mostly because of a smaller variance.\n\t\tAs  the sample size increases, they tend {to behave quite similarly.} \n\t\tOn  the other hand, both methods give almost the same estimates for $\\beta_0$ and $\\Lambda$. \n\t\tThe most favorable situation for our method is when there is little censoring among uncured observations and \n\t\tthe censored uncured observations are in the region of covariates that corresponds to higher cure rate. This comes from the fact that the nonparametric estimator in \\eqref{def:hat_pi} takes larger values when the  product has more terms equal to one. This should not be a problem when we expect that subjects with high probability of being cured correspond to longer survival times, meaning that it is more probable for them to be censored compared to those with small cure probability and shorter survival times.\n\t\t\n\t\t\\begin{table}\n\t\t\t\\caption{\t\\label{tab:results4}Bias, variance and MSE of $\\hat\\gamma$ for \\texttt{smcure} (second rows) and our approach (first rows) in Model 4.}\n\t\t\t\\centering\n\t\t\n\t\t\t\\scalebox{0.85}{\n\t\t\t\t\\fbox{\n\t\t\t\t\t\\begin{tabular}{ccccrrrrrrrrr}\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t&\t&&&\\multicolumn{3}{c}{Cens. level 1}&\\multicolumn{3}{c}{Cens. level 2}&\\multicolumn{3}{c}{Cens. level 3}\\\\\n\t\t\t\t\t\tMod.&\tn&  scen. & Par. &  Bias & Var. & MSE & Bias & Var. & MSE & Bias & Var. & MSE\\\\[2pt]\n\t\t\t\t\t\t\\hline\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t4&\t$200$ & $1 $ & $\\gamma_1 $ & $ 0.041$  & $ 0.157$ & $0.159   $ & $ 0.016 $ & $ 0.187 $ & $0.188  $ & $-0.010 $ & $ 0.210 $ & $0.210 $\\\\\n\t\t\t\t\t\t&\t& &  &   $0.077 $  & $0.178 $ & $ 0.184  $ & $ 0.096 $ & $0.228  $ & $ 0.238 $ & $ 0.127$ & $0.285  $ & $0.301 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_2 $ &   $ -0.017$  & $0.039 $ & $ 0.039  $ & $ -0.019 $ & $ 0.042 $ & $ 0.042 $ & $ -0.015$ & $0.049  $ & $ 0.049$\\\\\n\t\t\t\t\t\t&\t& &  &   $-0.090 $  & $0.052 $ & $ 0.060  $ & $ -0.125 $ & $  0.069$ & $ 0.085 $ & $ -0.164$ & $ 0.108 $ & $ 0.135$\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_3 $   & $ -0.245$  & $0.159 $ & $0.219   $ & $ -0.281 $ & $  0.165$ & $ 0.244 $ & $ -0.355$ & $0.179  $ & $0.305 $\\\\\n\t\t\t\t\t\t& & &  & $0.064 $  & $0.244 $ & $ 0.249  $ & $ 0.084 $ & $0.304  $ & $ 0.311 $ & $ 0.114$ & $ 0.395 $ & $0.408 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_4 $   & $-0.068 $  & $0.331 $ & $ 0.336  $ & $-0.162  $ & $ 0.385 $ & $ 0.411 $ & $ -0.285$ & $ 0.443 $ & $0.524 $\\\\\n\t\t\t\t\t\t& & &  & $0.171 $  & $0.401 $ & $ 0.430 $ & $ 0.241 $ & $0.561  $ & $ 0.619 $ & $ 0.314$ & $ 0.842 $ & $ 0.941$\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_5 $   & $-0.095 $  & $0.301 $ & $ 0.310  $ & $ -0.234 $ & $0.349  $ & $ 0.404 $ & $ -0.366$ & $ 0.371 $ & $ 0.505$\\\\\n\t\t\t\t\t\t& & &  & $ 0.106$  & $0.363 $ & $ 0.375  $ & $ 0.143 $ & $0.509  $ & $0.529  $ & $ 0.177$ & $ 0.693 $ & $ 0.724$\\\\\n\t\t\t\t\t\t\\cline{3-13}\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t&\t& $3 $ &$\\gamma_1 $ & $ -0.044$  & $0.079 $ & $  0.081 $ & $-0.079  $ & $  0.095$ & $ 0.101 $ & $-0.148 $ & $  0.132$ & $0.154 $\\\\\n\t\t\t\t\t\t& &  &  & $0.000 $  & $ 0.079$ & $ 0.079  $ & $ 0.003 $ & $ 0.096 $ & $ 0.096 $ & $0.009 $ & $ 0.141 $ & $ 0.141$\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_2 $ &   $ 0.018$  & $0.007 $ & $  0.008 $ & $ 0.024 $ & $  0.008$ & $0.009  $ & $0.041 $ & $ 0.010 $ & $0.012 $\\\\\n\t\t\t\t\t\t& & &  & $0.015 $  & $0.008 $ & $ 0.008  $ & $ 0.017 $ & $ 0.009 $ & $ 0.010 $ & $0.028 $ & $ 0.013 $ & $0.014 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_3 $ &   $0.034 $  & $0.066 $ & $  0.067 $ & $ 0.046 $ & $ 0.073 $ & $ 0.075 $ & $0.067 $ & $0.087  $ & $0.091 $\\\\\n\t\t\t\t\t\t& & &  & $-0.025 $  & $0.080 $ & $   0.080$  & $ -0.033 $ & $ 0.091 $ & $ 0.092 $ & $ -0.041$ & $0.120  $ & $ 0.122$\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_4 $ &   $0.041 $  & $0.102 $ & $  0.104 $ & $  0.054$ & $ 0.125 $ & $0.128  $ & $0.082 $ & $ 0.166 $ & $0.173 $\\\\\n\t\t\t\t\t\t& & &  & $0.022 $  & $0.100 $ & $0.101   $ & $ 0.023 $ & $ 0.126 $ & $ 0.126 $ & $0.026 $ & $0.179  $ & $ 0.180$\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_5 $ &   $-0.031 $  & $ 0.103$ & $ 0.104  $ & $ -0.034 $ & $0.120  $ & $ 0.121 $ & $ -0.054$ & $0.159  $ & $0.162 $\\\\\n\t\t\t\t\t\t& & &  &$ -0.016$  & $0.099 $ & $  0.099 $ & $ -0.013 $ & $ 0.115 $ & $ 0.115 $ & $ -0.018$ & $ 0.150 $ & $ 0.151$\\\\\n\t\t\t\t\t\t\\cline{2-13}\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t&\t$400$ & $1 $ & $\\gamma_1 $ & $ 0.013$  & $ 0.067$ & $ 0.067  $ & $ 0.015 $ & $ 0.079 $ & $0.080  $ & $ 0.005$ & $ 0.097 $ & $ 0.097$\\\\\n\t\t\t\t\t\t& &  &  & $0.024 $  & $0.067 $ & $   0.068$ & $ 0.037 $ & $  0.080$ & $0.082  $ & $ 0.043$ & $ 0.101 $ & $0.103 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_2 $ &   $ -0.001$  & $ 0.017$ & $  0.017 $ & $  -0.003$ & $ 0.020 $ & $ 0.020 $ & $-0.007 $ & $ 0.023 $ & $0.023 $\\\\\n\t\t\t\t\t\t& & &  & $-0.042 $  & $0.020 $ & $   0.021$ & $ -0.055 $ & $0.025  $ & $ 0.028 $ & $-0.079 $ & $  0.034$ & $0.041 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_3 $ &   $-0.229 $  & $0.089 $ & $  0.141 $ & $ -0.207 $ & $ 0.090 $ & $ 0.133 $ & $ -0.275$ & $ 0.102 $ & $0.178 $\\\\\n\t\t\t\t\t\t& & &  & $0.046 $  & $0.107 $ & $ 0.109  $ & $  0.061$ & $ 0.137 $ & $ 0.141 $ & $ 0.066$ & $ 0.162 $ & $ 0.166$\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_4 $ &   $ -0.063$  & $ 0.161$ & $ 0.165  $ & $ -0.143 $ & $ 0.175 $ & $  0.196$ & $-0.222 $ & $ 0.215 $ & $ 0.265$\\\\\n\t\t\t\t\t\t& & &  & $ 0.085$  & $ 0.176$ & $  0.183 $ & $ 0.107 $ & $ 0.222 $ & $ 0.234 $ & $ 0.145$ & $0.318  $ & $0.339 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_5 $ &  $ -0.075$  & $0.146 $ & $ 0.151  $ & $-0.192  $ & $ 0.177 $ & $  0.214$ & $-0.299 $ & $ 0.199 $ & $ 0.289$\\\\\n\t\t\t\t\t\t& & &  & $0.043 $  & $0.157 $ & $ 0.159  $ & $ 0.049 $ & $ 0.194 $ & $0.196  $ & $ 0.060$ & $ 0.253 $ & $ 0.257$\\\\\n\t\t\t\t\t\t\\cline{3-13}\n\t\t\t\t\t\t&\t& & & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t\t&\t& $3 $ & $\\gamma_1 $ & $-0.024 $  & $ 0.038$ & $ 0.039  $ & $ -0.038 $ & $ 0.047 $ & $0.048  $ & $-0.092 $ & $0.064  $ & $0.073 $\\\\\n\t\t\t\t\t\t& &  &  & $-0.003 $  & $0.036 $ & $ 0.036  $ & $ 0.002 $ & $ 0.044 $ & $ 0.044 $ & $0.004 $ & $ 0.060 $ & $ 0.060$\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_2 $ &   $ 0.006$  & $ 0.003$ & $ 0.003  $ & $0.010  $ & $ 0.004 $ & $ 0.004 $ & $ 0.019$ & $  0.005$ & $0.006 $\\\\\n\t\t\t\t\t\t& & &  & $0.005 $  & $ 0.004$ & $ 0.004  $ & $ 0.005 $ & $0.004  $ & $0.004  $ & $0.006 $ & $  0.006$ & $0.006 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_3 $ &   $ 0.028$  & $0.033 $ & $  0.034 $ & $0.045  $ & $0.038  $ & $ 0.040 $ & $0.065 $ & $ 0.045 $ & $0.049 $\\\\\n\t\t\t\t\t\t& & &  & $ -0.010$  & $ 0.036$ & $ 0.036  $ & $ -0.008 $ & $ 0.042 $ & $ 0.042 $ & $ -0.008$ & $ 0.052 $ & $0.052 $\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_4 $ &  $ 0.021$  & $ 0.052$ & $  0.053 $ & $0.032  $ & $ 0.062 $ & $ 0.063 $ & $0.060 $ & $ 0.088 $ & $ 0.092$\\\\\n\t\t\t\t\t\t& & &  & $0.015 $  & $ 0.050$ & $  0.050 $ & $ 0.016 $ & $0.060  $ & $ 0.060 $ & $0.019 $ & $ 0.083 $ & $ 0.083$\\\\\n\t\t\t\t\t\t&\t& & $\\gamma_5 $ &   $ -0.024$  & $0.049 $ & $ 0.050  $ & $  -0.041$ & $ 0.059 $ & $ 0.061 $ & $ -0.048$ & $ 0.077 $ & $ 0.079$\\\\\n\t\t\t\t\t\t& & &  & $-0.017 $  & $0.049 $ & $ 0.049  $ & $ -0.022 $ & $  0.056$ & $  0.057$ & $-0.022 $ & $ 0.069 $ & $0.069 $\\\\\n\t\t\t\t\t\\end{tabular}\n\t\t\t}\t}\n\t\t\\end{table}\n\t\t\n\t\tThis is indeed the case in Model 1 and we observe that our approach outperforms \\texttt{smcure} in all the scenarios. The difference between the two is more marked when $n$ is small and the absolute value of the $\\gamma$ coefficient is larger. In Model 2, the situation is more difficult because censoring depends on the covariate in such a way that, the non-cured subjects have the same probability of being censored independently of their cure probability. However, for the first two scenarios the new method is still superior. The third scenario is more problematic because  \n\t\tthe cure probability drops very fast from almost one to almost zero, resulting in a large fraction of uncured observations with almost zero cure probability. The presence of censoring in this region leads to overestimation of the cure rate. If we would take $\\beta_C=0.1$ (meaning larger probability of being censored for higher cure rate), then the new approach is significantly superior (see  Table \\ref{tab:2_4} for $n=400$ and scenario 3). In Model 3,  complications arise because of the presence of different covariates for the incidence and latency. Hence, subjects with higher cure rate might correspond to shorter survival times. As a result, the previous problem might still happen and its effects are more visible for large sample size and large censoring rate. Finally, Model 4 suggests that, even though the assumptions in Section \\ref{sec:asymptotics} were shown to be satisfied only for one continuous covariate, the method  {could be applied in more general cases.}\n\t\tWe noticed that, when a continuous covariate affects only the incidence and not the latency, the bandwidth selected by the \\texttt{np} package is often very large, meaning that it fails to capture the effect of this covariate on the conditional distribution function. In those cases, we truncate the selected bandwidth from above at $2$. Note that the bandwidth is chosen for standardized covariates so the truncation level can be fixed regardless of the distribution of the covariate. We decided to truncate at $2$ since it seems to be a kind of boundary for a `reasonable' bandwidth {with standardized covariates} (we do not want to externally affect chosen bandwidths smaller than $2$ but we only replace extremely large values by $2$). However, even when reasonable, the \\texttt{np} bandwidth for $X_2$ seems to be larger than it should, resulting in more bias in the estimator of $\\gamma_3$. Nevertheless, in terms of mean squared error, the method performs well  for not large sample size. If $X_2$ would affect also the latency, the selected bandwidth would be more adequate and there would be no bias problems.\n\t\t\n\t\t\\begin{table}\n\t\t\t\\caption{\\label{tab:2_4}Bias, variance and MSE of $\\hat\\gamma$ and $\\hat\\beta$ for \\texttt{smcure}  and our approach in Model 2, scenario 3 when $\\beta_C=0.1$ and $n=400$.}\n\t\t\t\\centering\n\t\t\t\\scalebox{0.85}{\n\t\t\t\t\\fbox{\n\t\t\t\t\t\\begin{tabular}{crrrrrr}\n\t\t\t\t\t\n\t\t\t\t\t\t& \\multicolumn{3}{c}{}  & \\multicolumn{3}{c}{} \\\\[-8pt]\n\t\t\t\t\t\t&\t \\multicolumn{3}{c}{\\texttt{smcure} package}   & \\multicolumn{3}{c}{Our approach}\\\\\n\t\t\t\t\t\tParameter\t& Bias & Var & MSE & Bias &Var & MSE\\\\[2pt]\n\t\t\t\t\t\t\\cline{1-7}\n\t\t\t\t\t\t& & & & & & \\\\[-8pt]\n\t\t\t\t\t\t$\\gamma_1$ & $0.014 $ & $0.123 $ & $0.123$   & $-0.058 $ & $ 0.103$ & $0.106$\\\\\n\t\t\t\t\t\t$\\gamma_2$ & $0.418 $ & $1.243 $ & $1.418 $   & $-0.535 $ & $0.652 $ & $0.937 $\\\\\n\t\t\t\t\t\t$\\beta$ & $0.001$ & $0.025$ & $0.025 $  & $0.001 $ & $0.027 $ & $ 0.027$\\\\\n\t\t\t\t\t\n\t\t\t\\end{tabular}}}\n\t\t\\end{table}\n\t\t\n\t\tTo conclude,  the new approach seems to perform significantly better than \\texttt{smcure} when  the sample size is not large and the fraction of censored observations is not much higher than the expected cure proportion. In other situations, both methods are comparable. However, one has to be more careful when there is no reason to expect that the censored subjects correspond to higher cure probabilities.\n\t\n\t\t\n\t\t{In the previous settings, we truncated the event times at $\\tau_0$ in such a way that condition \\eqref{eqn:jump_cond} is satisfied  but in practice\n\t\t\t\tit is unlikely to observe event times at $\\tau_0$. Next, we consider one additional model for which condition \\eqref{eqn:jump_cond} is not satisfied. The covariates and the parameters are as in Model 3 described above, but the event times are  generated from a Weibull distribution on $[0,\\tau_0]$ with $\\tau_0=15$, i.e.\n\t\t\\[\n\t\tS_u(t|z)=\\frac{\\exp\\left\\{-\\mu t^\\rho\\exp(\\beta'z)\\right\\}-\\exp\\left\\{-\\mu\\tau_0^\\rho\\exp(\\beta'z)\\right\\}}{1-\\exp\\left\\{-\\mu\\tau_0^\\rho\\exp(\\beta'z)\\right\\}}\n\t\t\\]\n\t\tThe censoring times are exponentially distributed  as in Model 3 and truncated at $\\tau=20$. Results for sample size $n=200$ and three censoring levels are shown in Table~\\ref{tab:nojump}. Compared to Model 3 above, we observe that, when condition \\eqref{eqn:jump_cond} is not satisfied, presmoothing is even more superior than the smcure estimator. \n\t\t\t\t}\n\t\t\t\\begin{table}[h!]\n\t\t\t\\caption{\t\\label{tab:nojump}Bias, variance and MSE of $\\hat\\gamma$ for \\texttt{smcure} (second rows) and our approach (first rows) in Model 3 without condition \\eqref{eqn:jump_cond}.}\n\t\t\t\\centering\n\t\t\n\t\t\t\\scalebox{0.85}{\n\t\t\t\t\\fbox{\n\t\t\t\t\t\\begin{tabular}{crrrrrrrrr}\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t&\\multicolumn{3}{c}{Cens. level 1}&\\multicolumn{3}{c}{Cens. level 2}&\\multicolumn{3}{c}{Cens. level 3}\\\\\n\t\t\t\t\t\t Par. &  Bias & Var. & MSE & Bias & Var. & MSE & Bias & Var. & MSE\\\\[2pt]\n\t\t\t\t\t\t\\hline\n\t\t\t\t\t & & & & & & & && \\\\[-8pt]\n\t\t\t\t\t  $\\gamma_1 $ & $ 0.015$  & $0.152 $ & $0.152  $ & $0.000  $ & $0.196 $ & $0.196$ & $ -0.032 $ & $0.246 $ & $0.247 $\\\\\n\t\t\t\t\t\t & $ 0.017$  & $0.150 $ & $0.151  $ & $0.027  $ & $0.193 $ & $ 0.194$ & $0.035  $ & $0.260 $ & $0.262 $\\\\\n\t\t\t\t\t $\\gamma_2 $ & $-0.054 $  & $0.044 $ & $ 0.047 $ & $ -0.077 $ & $0.052 $ & $ 0.058$ & $-0.109  $ & $0.064 $ & $ 0.076$\\\\\n\t\t\t\t\t & $-0.085$  & $ 0.050$ & $ 0.057 $ & $ -0.119 $ & $0.069 $ & $0.083 $ & $-0.171  $ & $0.101 $ & $0.130$\\\\\n\t\t\t\t\t $\\gamma_3 $ & $0.087 $  & $0.379 $ & $ 0.386 $ & $ 0.073 $ & $ 0.450$ & $ 0.456$ & $0.045  $ & $0.578 $ & $ 0.580$\\\\\n\t\t\t\t\t\t& $0.197 $  & $ 0.423$ & $0.462  $ & $ 0.249 $ & $0.561 $ & $0.623 $ & $0.343  $ & $ 0.885$ & $1.002 $\\\\\n\t\t\t\t\t $\\gamma_4 $ & $ -0.010$  & $0.339 $ & $ 0.339 $ & $-0.106 $ & $ 0.373$ & $0.385 $ & $ -0.228 $ & $ 0.498$ & $ 0.550$\\\\\n\t\t\t\t\t\t& $0.125 $  & $0.364 $ & $ 0.380 $ & $ 0.156 $ & $0.523 $ & $0.548 $ & $0.260  $ & $ 1.513$ & $1.581 $\\\\\n\t\t\t\t\t\t\t\\end{tabular}\n\t\t\t}\t}\n\t\t\\end{table}\n\t\t\n\tFinally we conclude with a remark about the computational aspect.\t{The proposed approach is computationally more intensive than the MLE mainly because of the bandwidth selection through a cross-validation procedure. For example,  for one iteration in Model 3 with sample size $200$ and $400$, \\texttt{smcure} computes the estimates in $0.7$ and $0.8$ seconds respectively, while the new approach requires $4.1$ and $23.5$ seconds (with a Core i7-8665U CPU desktop).  However, this seems still reasonable since the method is not meant for much larger sample sizes.} \n\t\t\n\t\t\n\t\t\\section{Application: Melanoma study}\n\t\t\\label{sec:application}\n\t\tTo illustrate the practical performance, we apply the proposed estimation procedure to two medical datasets for patients with melanoma and compare the results with \\texttt{smcure}. Melanoma is the third most common skin cancer type with overall incidence rate 21.8 out of 100,000 people in the US (Cancer statistics from the Center for Disease Control and Prevention) and according to the American Cancer Society, $6850$ people are expected to die of melanoma in $2020$. However, in the recent years,  the chances of survival for melanoma patients have increased due to earlier diagnosis and improvement of treatment and surgical techniques. The 5-year survival rates based on the stage of the cancer when it was first diagnosed are $92\\%$ for localized, $65\\%$ for regional and $25\\%$ for distant stage. It is also known that this disease is more common among white people and the death rate is higher for men  than women.  Even though most melanoma patients are cured by their initial treatment, it is not possible to distinguish them from the uncured patients. Hence, accurately estimating the probability of being cured is important in order to plan further treatment and prevent recurrence of uncured patients. \n\t\t\n\t\t\\subsection{Eastern Cooperative Oncology Group (ECOG) Data}\n\t\tWe use the melanoma data (ECOG phase III clinical trial e1684) from the \\texttt{smcure} package \\cite{cai_smcure} in order to compare our results with those of \\texttt{smcure}.\n\t\tThe purpose of this study was to evaluate the effect of treatment (high dose interferon alpha-2b regimen) as the postoperative adjuvant therapy. The event time is the time from initial treatment to recurrence of melanoma and three covariates have been considered: age (continuous variable centered to the mean), gender (0=male and 1=female) and treatment (0=control and 1=treatment).\n\t\tThe data consists of $284$ observations (after deleting missing data) out of which $196$ had recurrence of the melanoma cancer (around  $30\\%$ censoring).  The Kaplan-Meier curve is shown in Figure~\\ref{fig:KM_melanoma_1}. \n\t\tThe parameter estimates, standard errors and corresponding p-values for the Wald test using our method and the \\texttt{smcure} package are given in Table~\\ref{tab:melanoma1}. Standard errors are computed through  $500$ {naive} bootstrap samples.\n\t\t\n\t\t\\begin{figure}\n\t\t\t\\centering\n\t\t\t\\makebox{\n\t\t\t\t\\includegraphics[width=0.48\\linewidth]{KM_melanoma_1.pdf}\\qquad\\includegraphics[width=0.48\\linewidth]{KM_melanoma_1_treatment.pdf}}\n\t\t\t\\caption{\\label{fig:KM_melanoma_1}Left panel: Kaplan-Meier survival curve for ECOG data. Right panel: Kaplan-Meier survival curves for the treatment group (solid) and control group (dotted) in the ECOG data.}\n\t\t\\end{figure}\n\t\t\n\t\t\\begin{table}\n\t\t\t\\caption{\\label{tab:melanoma1}Results for the incidence (logistic component) and the latency (Cox PH component) from the ECOG data.}\n\t\t\t\\centering\n\t\t\t\\scalebox{0.85}{\n\t\t\t\t\\fbox{\n\t\t\t\t\t\\begin{tabular}{c|crrrrrr}\n\t\t\t\t\n\t\t\t\t\t\t&& \\multicolumn{3}{c}{}  & \\multicolumn{3}{c}{} \\\\[-8pt]\n\t\t\t\t\t\t&\t& \\multicolumn{3}{c}{\\texttt{smcure} package}   & \\multicolumn{3}{c}{Our approach}\\\\\n\t\t\t\t\t\t&\tCovariates\t& Estimates & SE & p-value & Estimates & SE & p-value\\\\[2pt]\n\t\t\t\t\t\t\\cline{2-8}\n\t\t\t\t\t\t&\t& & & & & & \\\\[-8pt]\n\t\t\t\t\t\t\\multirow{4}{*}{\\STAB{\\rotatebox[origin=c]{90}{incidence}}}\t&Intercept & $1.3649 $ & $0.3457 $ & $8\\cdot10^{-5} $   & $1.6697 $ & $ 0.3415$ & $10^{-6} $\\\\\n\t\t\t\t\t\t&\tAge & $0.0203 $ & $0.0159 $ & $0.2029 $   & $0.0220 $ & $0.0104 $ & $0.0344 $\\\\\n\t\t\t\t\t\t&\tGender & $-0.0869 $ & $0.3347 $ & $0.7949 $  & $-0.3039 $ & $0.3448 $ & $ 0.3493$\\\\\n\t\t\t\t\t\t&\tTreatment & $-0.5884 $ & $ 0.3706$ & $0.1123 $   & $-0.9345 $ & $0.3603 $ & $0.0095 $\\\\\n\t\t\t\t\t\t\\hline\n\t\t\t\t\t\t&\t& & & & & &   \\\\[-8pt]\n\t\t\t\t\t\t\\multirow{3}{*}{\\STAB{\\rotatebox[origin=c]{90}{latency}}}\t&\tAge & $-0.0077 $ & $0.0069 $ & $0.2663 $  &  $-0.0079 $ & $0.0060 $ & $0.1861 $\\\\\n\t\t\t\t\t\t&\tGender & $0.0994 $ & $0.1932 $ & $0.6067 $   & $0.1240 $ & $0.1653 $ & $0.4534 $\\\\\n\t\t\t\t\t\t&\tTreatment & $-0.1535 $ & $0.1715 $ & $0.3707 $  & $-0.0947 $ & $ 0.1692$ & $0.5756 $\\\\\n\t\t\t\t\t\n\t\t\t\\end{tabular}}}\n\t\t\\end{table}\n\t\t\n\t\tWe observe that, for both methods, the effects of the covariates\n\t\n\t\thave the same direction. Only the intercept was found significant for the incidence with \\texttt{smcure}, while our method concludes that  also age and treatment are  significant. In particular, the probability of recurring melanoma is higher for the control group compared to the treatment group. This seems to be indeed the case if we look at the Kaplan Meier survival curves for the two groups in Figure~\\ref{fig:KM_melanoma_1}.  On the other hand, both methods agree that none of the covariates is significant for the latency. \n\t\t\n\t\t{To illustrate another advantage of the new approach, we also compute the maximum likelihood estimator with the  \\texttt{smcure} package for different choices of the latency model. We see in Table~\\ref{tab:melanoma3} that the estimators of the incidence component (and their significance) change depending on which variables are included in the latency. On the other hand, the new method does not suffer from this problem because it estimates the incidence independently of the latency.\n\t\t\t\\begin{table}\n\t\t\t\\caption{\\label{tab:melanoma3}Results for the incidence (logistic component) and the latency (Cox PH component) from the ECOG data.}\n\t\t\t\\centering\n\t\t\t\\scalebox{0.85}{\n\t\t\t\t\\fbox{\n\t\t\t\t\t\\begin{tabular}{c|c|rrr|rrr|rrr}\n\t\t\t\t\t\n\t\t\t\t\t\t&& \\multicolumn{3}{c}{}  & \\multicolumn{3}{c}{}  & \\multicolumn{3}{c}{}\\\\[-8pt]\n\t\t\t\t\t\t&\t& \\multicolumn{3}{c}{Model 1}    & \\multicolumn{3}{c}{Model 2 }& \\multicolumn{3}{c}{Model 3 }\\\\\n\t\t\t\t\t\t&\tCovariates\t& Estimates & SE & p-value & Estimates & SE & p-value\t& Estimates & SE & p-value\\\\[2pt]\n\t\t\t\t\t\t\\cline{2-11}\n\t\t\t\t\t\t&\t& & & & & & &&&\\\\[-8pt]\n\t\t\t\t\t\t\\multirow{4}{*}{\\STAB{\\rotatebox[origin=c]{90}{incidence}}}\t&Intercept & $1.3507 $ & $0.3001 $ & $7\\cdot10^{-6} $   & $1.4148 $ & $ 0.3213$ & $10^{-5} $& $1.4181$ & $ 0.3073$ & $4\\cdot10^{-6} $\\\\\n\t\t\t\t\t\t&\tAge & $0.0164$ & $0.0125 $ & $0.1905 $   & $0.0205 $ & $0.0154 $ & $0.1803 $& $0.0209 $ & $ 0.0146$ & $0.1528 $\\\\\n\t\t\t\t\t\t&\tGender & $-0.0265 $ & $0.3113 $ & $0.9320 $  & $-0.0673 $ & $0.3352$ & $ 0.8407$& $-0.0222$ & $ 0.3130$ & $0.9432 $\\\\\n\t\t\t\t\t\t&\tTreatment & $-0.6060$ & $ 0.3509$ & $0.0842 $   & $-0.6773 $ & $0.3223 $ & $0.0415 $& $-0.6913$ & $ 0.3439$ & $0.0444 $\\\\\n\t\t\t\t\t\t\\hline\n\t\t\t\t\t\t&\t& & & & & &  &&& \\\\[-8pt]\n\t\t\t\t\t\t\\multirow{3}{*}{\\STAB{\\rotatebox[origin=c]{90}{latency}}}\t&\tAge &  &  &  &  $-0.0074 $ & $0.0066 $ & $0.2568 $& $-0.0073 $ & $ 0.0064$ & $0.2579 $\\\\\n\t\t\t\t\t\t&\tGender &&  &    & $0.0789 $ & $0.1863 $ & $0.6719$&  & & \\\\\n\t\t\t\t\t\t&\tTreatment & $-0.1324 $ & $0.1561 $ & $0.3963$  &  &  & &  &  &\\\\\n\t\t\t\t\t\n\t\t\t\\end{tabular}}}\n\t\t\\end{table}\n\t}\n\t\t\\subsection{Surveillance, Epidemiology and End Results (SEER) database}\n\t\t\n\t\tThe SEER database collects cancer incidence data from population-based cancer registries in US. These data consist of patient demographic characteristics, primary tumor site, tumor morphology, stage at diagnosis, length of follow up and vital status. We select the database `Incidence - SEER 18 Regs Research Data' and extract the melanoma cancer data for the county of San Francisco in California during the period $2004-2015$. We consider only patients with stage at diagnosis: localized, regional and distant and exclude those with unknown or zero follow-up time  and restrict the study to white people because of the very small number of cases from other races. The event time is death because of melanoma.  This cohort consists of $1445$ melanoma cases out of which $596$ are female and $849$ male. The age ranges from $11 $ to $101$ years old, the follow-up from $1$ to $155$ months. For most of the patients the cancer has been diagnosed at early stage (localized), while for $101$ of them the stage at diagnosis is `regional' and only for $42$ it is `distant'. We aim at evaluating how age, gender and stage at diagnosis affect the survival  of melanoma patients in this cohort. The use of cure models is justified from the presence of a long plateau containing around $20\\%$ of the observations (see the Kaplan-Meier curve in Figure \\ref{fig:KM_melanoma_2}). Moreover, the Kaplan-Meier curves depending on gender  and stage at diagnosis  in Figure~\\ref{fig:KM_melanoma_2} confirm that  gender and stage affect the cure rate. \n\t\t\n\t\t\\begin{figure}\n\t\t\t\\begin{center}\t\\makebox{\n\t\t\t\t\t{\\includegraphics[width=0.48\\linewidth]{KM_melanoma_2.pdf}}\n\t\t\t}\t\\end{center}\n\t\t\t\\makebox{\n\t\t\t\t{\\includegraphics[width=0.48\\linewidth]{KM_melanoma_2_gender}\\qquad \t\\includegraphics[width=0.48\\linewidth]{KM_melanoma_2_stage}}}\n\t\t\t\\caption{\\label{fig:KM_melanoma_2}Upper panel: Kaplan-Meier survival curves for SEER data. Left panel: group division based on gender, females (solid) and males (dotted). Right panel: group division based on cancer stage at diagnosis, localized (solid), regional (dashed) and distant (dotted).}\n\t\t\\end{figure}\n\t\t\n\t\tWe checked the fit of the logistic model by comparing it with the single-index mixture cure model proposed in \\cite{AKL19} through the prediction error of the incidence. More precisely, as in \\cite{AKL19}, we divide the data into a training set and a test set of size $964$ and $481$ respectively. Using the training set, we estimate the logistic\/Cox model and the single-index\/Cox model. Afterwards, we compute the prediction error in the test set given by\n\t\t\\[\n\t\tPE=-\\sum_{j=1}^{481}\\left\\{\\hat{W}_j\\log [1-\\hat\\pi(X_j^{\\text{test}})]+(1-\\hat{W}_j)\\log \\hat\\pi(X_j^{\\text{test}})\\right\\}\n\t\t\\]\n\t\twhere $\\hat\\pi(X_j^{\\text{test}})$ and $\\hat{W_j}$ are the predicted cure probability and the predicted weight for the $j$th observation in the test set, computed based on the parameter estimates (and the link function for the single-index model) in the training set. {More precisely, for the logistic\/Cox model we have $\\hat\\pi(X_j^{\\text{test}})=\\phi(\\hat\\gamma_n,X_j^{\\text{test}})$ and \n\t\t\t\\[\n\t\t\t\\hat{W}_j=\\Delta_j^{\\text{test}}+(1-\\Delta_j^{\\text{test}})\\frac{\\hat\\pi(X_j^{\\text{test}})\\exp\\left(-\\hat\\Lambda_n(Y_j^{\\text{test}})e^{\\hat\\beta'_nZ_j^{\\text{test}}}\\right)}{1-\\hat\\pi(X_j^{\\text{test}})+\\hat\\pi(X_j^{\\text{test}})\\exp\\left(-\\hat\\Lambda_n(Y_j^{\\text{test}})e^{\\hat\\beta'_nZ_j^{\\text{test}}}\\right)}\n\t\t\t\\]\n\t\t\twhere $\\hat\\gamma_n$, $\\hat\\beta_n$ and $\\hat\\Lambda_n$ are  the estimated parameters and the estimated hazard function in the training set. For the single-index\/Cox model, the only difference is that $\\hat\\pi(X_j^{\\text{test}})=\\hat{g}_n(\\hat\\gamma_n,X_j^{\\text{test}})$ where $\\hat{g}_n$ is the estimated link function as in \\cite{AKL19}. The weights $\\hat W_j$ correspond to the conditional expectation of the cure status $B$ given the observations. } \n\t\tWe find that the prediction error for the logistic model is $98.53$, whereas for the single-index model it is $156.55$. This means that the logistic model performs better. \n\t\t\n\t\t\\begin{table}\n\t\t\t\\caption{\\label{tab:melanoma2}\n\t\t\t\tResults for the incidence (logistic component) and the latency (Cox PH component) from the  SEER data.}\n\t\t\t\\centering\n\t\t\t\\scalebox{0.85}{\n\t\t\t\t\\fbox{\n\t\t\t\t\t\\begin{tabular}{c|crrrrrr}\n\t\t\t\t\t\n\t\t\t\t\t\t&& \\multicolumn{3}{c}{}  & \\multicolumn{3}{c}{} \\\\[-8pt]\n\t\t\t\t\t\t&\t& \\multicolumn{3}{c}{\\texttt{smcure} package}   & \\multicolumn{3}{c}{Our approach}\\\\\n\t\t\t\t\t\t&\tCovariates\t& Estimates & SE & p-value & Estimates & SE & p-value\\\\[2pt]\n\t\t\t\t\t\t\\cline{2-8}\n\t\t\t\t\t\t&\t& & & & & & \\\\[-8pt]\n\t\t\t\t\t\t\\multirow{5}{*}{\\STAB{\\rotatebox[origin=c]{90}{incidence}}}&\tIntercept & $-4.2071 $ & $0.3817 $ & $0 $  & $-4.2436 $ & $0.3980 $ & $0 $ \\\\\n\t\t\t\t\t\t&\tAge & $0.0304 $ & $0.0122 $ & $0.0124 $  & $0.0328 $ & $0.0172 $ & $ 0.0565$ \\\\\n\t\t\t\t\t\t&\tGender & $ 1.1318$ & $0.4211 $ & $0.0072 $  & $1.2341 $ & $0.4792 $ & $0.010 $ \\\\\n\t\t\t\t\t\t&\t$S_1$& $2.6738 $ & $0.3702 $ & $5\\cdot 10^{-13} $  & $ 2.4474$ & $0.4247 $ & $ 8\\cdot 10^{-9}$ \\\\\n\t\t\t\t\t\t&\t$S_2$ & $4.0763 $ & $0.5067 $ & $8\\cdot 10^{-16} $  & $3.9426 $ & $0.4536 $ & $ 0$ \\\\\n\t\t\t\t\t\t\\hline\n\t\t\t\t\t\t&\t& & & & & &  \\\\[-8pt]\n\t\t\t\t\t\t\\multirow{4}{*}{\\STAB{\\rotatebox[origin=c]{90}{latency}}}&\tAge & $-0.0139 $ & $ 0.0098$ & $ 0.1577$  & $-0.0143 $ & $0.0106 $ & $0.1756 $ \\\\\n\t\t\t\t\t\t&\tGender & $-0.0549 $ & $0.4065 $ & $ 0.8925$  & $-0.0871 $ & $0.3687 $ & $0.8131 $ \\\\\n\t\t\t\t\t\t&\t$S_1$& $0.5176 $ & $0.3993 $ & $0.1949$  & $0.6130 $ & $0.3971 $ & $0.1226 $ \\\\\n\t\t\t\t\t\t&\t$S_2$ & $1.8039 $ & $0.4529 $ & $7\\cdot 10^{-5} $  & $1.8623 $ & $0.5072 $ & $0.0002 $ \\\\\n\t\t\t\t\t\n\t\t\t\t\t\\end{tabular}\n\t\t\t}}\n\t\t\\end{table}\n\t\t\n\t\tThe parameter estimates, standard errors and corresponding p-values for the Wald test using our method and the \\texttt{smcure} package are given in Table \\ref{tab:melanoma2}. Standard errors  are computed  with $500$ {naive} bootstrap samples. The covariate stage is classified using two dummy Bernoulli variables $S_1$ and $S_2$, where $S_1=1$ indicates the regional stage and\n\t\n\t\t$S_2=1$ indicates the distant stage.\n\t\n\t\tThe gender variable is equal to zero for females and one for males. We observe that both methods agree that all the considered covariates are significant for the incidence (with age being a borderline case for our approach).  \n\t\tFor the latency, only being in the distant stage is found significant with both methods. \n\t\tMoreover, again the effects of all the covariates on the latency and incidence have the same direction for both methods.\n\t\t\n\t\t\n\t\t\\section{Discussion} \\label{sec:disc}\n\t\tIn this paper we proposed a new estimation procedure for the mixture cure model with a parametric form of the incidence (for example logistic) and {any semiparametric model for the latency. We investigated more in detail the logistic\/Cox model given its practical relevance}. Instead of using an iterative algorithm for dealing with the unknown cure status, this method relies on a preliminary nonparametric estimator of the cure probabilities.  We showed through simulations that the new approach improves upon the classical maximum likelihood estimator implemented in the package \\texttt{smcure}, mainly for smaller sample sizes.  For the latency, both methods behave similarly. Hence, it is of particular interest in situations in which the focus is on the estimation of cure probabilities.  {The real data application on the ECOG clinical trial also showed that the improvement in estimation can be meaningful in practice and help detecting significant effects.}\n\t\t\n\t\t{ The proposed method has the advantage of direct estimation of the incidence component, without relying on the latency, which makes it robust to latency model misspecification. On the contrary, the \\texttt{smcure} estimator  strongly depends on the choice of the variables for the latency and could be biased for a misspecified Cox  model. Hence, for practical reason, confronting the estimators obtained with the two methods is valuable for confirming the results or obtaining new insights.  }\n\t\t{From the theoretical point of view, unlike the standard maximum likelihood estimation,  presmoothing allows us  to obtain consistency and asymptotic normality without requiring the `unrealistic' assumption that the distribution of uncured subjects  has a positive mass at the end point of the support.  } \n\t\t\n\t\t{It might be argued that since the proposed method relies on smoothing, it is more complex and the results can be affected by the choice of the kernel function or the bandwidth. Our purpose was to show that the user doesn't have to think about this  because the standard choices proposed in this paper perform well in practice. In addition, since the final estimator is a parametric one and the kernel estimator is only a preliminary step of the procedure, the results would anyway be more stable with respect to these choices than in a nonparametric setting. The main challenge this method faces is extension to many continuous covariates for the incidence. \t\t}\n\t\tWe did not deeply investigate such situations since, in that case, multiple bandwidths have to be chosen, which can be more problematic and computationally intensive. However, our approach based on presmoothing allows to efficiently handle these situations if the estimator $\\hat\\pi$ is constructed in a more adequate way. {One possibility would be to construct the estimator assuming a single-index model for the latency, which is reasonable since the final goal is a parametric estimator. With this approach one can avoid the choice of multiple bandwidths and perform the estimation as in the one dimensional case.} However, this problem will be addressed by future research.   {In this regard, even though considering only one continuous covariate might seem restrictive in practice, the proposed procedure constitutes the basis for further developments of new estimators for general dimension scenarios that do not require multidimensional smoothing.}\n\t\t\n\t\t\n\t\t\\section{Appendix}\n\t\t\\label{sec:appendix}\n\t\t\\subsection{Proof of Theorem \\ref{theo:asymptotic_normality2}}\n\t\t\\label{sec:appendix_Cox}\n\t\tWe obtain the asymptotic normality of $\\hat\\Lambda_n$, $\\hat\\beta_n$ following the proof of Theorem 3 in \\cite{Lu2008}. In order to work with a one-dimensional submodel, {for $d$ in a neighbourhood of the origin,} let   $\\Lambda_d(t)=\\int_0^t\\{1+dh_1(s)\\}\\mathrm{d} \\hat\\Lambda_n(s)$ and $\\beta_d=dh_2+\\hat\\beta_n$, where $h_1$ is a function of bounded variation on $[0,{\\tau_0}]$ and $h_2$ is a $q$-dimensional real vector. Let $\\hat{S}_n(\\hat\\Lambda_n,\\hat\\beta_n)(h_1,h_2)$ denote the derivative of ${\\hat{l}_n}\n\t\t(\\Lambda_d,\\beta_d)$ (defined in \\eqref{def:hat_l_cox}) with respect to $d$ and evaluated at $d=0$. We have \n\t\t\\[\n\t\t\\begin{split}\n\t\t\t\\hat{S}_n(\\hat\\Lambda_n,\\hat\\beta_n)(h_1,h_2)\n\t\t\t&=\\frac{1}{n}\\sum_{i=1}^n \\Delta_i\\mathds{1}_{\\{Y_i<\\tau_0\\}} \\left[h_1(Y_i)+h'_2Z_i\\right]\\\\\n\t\t\t&\\quad-\\frac{1}{n}\\sum_{i=1}^n \\left\\{\\Delta_i+(1-\\Delta_i){\\mathds{1}_{\\{Y_i\\leq\\tau_0\\}}}g_i(Y_i,\\hat\\Lambda_n,\\hat\\beta_n,\\hat\\gamma_n)\\right\\}\\\\\n\t\t\t&\\qquad\\qquad\\qquad\\quad\\times\\left\\{e^{\\hat\\beta'_nZ_i}\\int_0^{Y_i}h_1(s)\\mathrm{d}\\hat\\Lambda_n(s)+e^{\\hat\\beta'_nZ_i}\\hat\\Lambda_n(Y_i)h'_2Z_i\n\t\t\t\\right\\},\n\t\t\\end{split}\n\t\t\\]\n\t\twhere $g_j$ is defined in \\eqref{def:g_j} and $\\hat\\gamma_n$ is the maximizer of \\eqref{def:hat_L_gamma}.  Let  $\\Upsilon_n=(\\hat\\Lambda_n,\\hat\\beta_n)$ and $\\Upsilon_0=(\\Lambda_0,\\beta_0)$. Furthermore, denote  by $S$ the asymptotic version of $\\hat{S}_n$:\n\t\t\\[\n\t\t\\begin{split}\n\t\t\tS(\\Lambda,\\beta)(h_1,h_2)&=\\mathbb{E}\\bigg[ \\Delta\\mathds{1}_{\\{Y<\\tau_0\\}} \\{h_1(Y)+h'_2Z\\}-\\left\\{\\Delta+(1-\\Delta){\\mathds{1}_{\\{Y\\leq\\tau_0\\}}}g(Y,\\Lambda,\\beta,\\gamma_0)\\right\\}\\\\\n\t\t\t&\\qquad\\qquad\\qquad\\qquad\\left. \\times\\left\\{e^{\\beta'Z}\\int_0^{Y}h_1(s)\\mathrm{d}\\Lambda(s)+e^{\\beta'Z}\\Lambda(Y)h'_2Z\n\t\t\t\\right\\}\\right].\n\t\t\\end{split}\n\t\t\\]\n\t\tWe have $\\hat{S}_n(\\Upsilon_n)=0$ and $S(\\Upsilon_0)=0$.  \n\t\tThe score function $S_n$ and $S$ are respectively  a random  and a deterministic map from $\\Xi$ to $l^{\\infty}(\\mathcal{H}_\\mathfrak{m})$ (the space of bounded real-valued functions on $\\mathcal{H}_\\mathfrak{m}$), where\n\t\t\\[\n\t\t\\Xi=\\left\\{(\\Lambda,\\beta)\\,:\\,\\sup_{h\\in\\mathcal{H}_\\mathfrak{m}}\\left|\\int_0^{{\\tau_0}}h_1(s)\\mathrm{d}\\Lambda(s)+h'_2\\beta\\right|<\\infty\\right\\}\n\t\t\\] \n\t\tand  $\\mathcal{H}_\\mathfrak{m}=\\{h\\in\\mathcal{H}\\,:\\, \\Vert h\\Vert_{H}\\leq \\mathfrak{m}\\}$. Here $\\Vert h\\Vert_{H}=\\Vert h_1\\Vert_v+\\Vert h_2\\Vert_{L_1}$, $\\Vert h_2\\Vert_{L_1}=\\sum_{j=1}^q|h_{2,j}|$, $\\Vert h_1\\Vert_v=|h_1(0)|+V_0^{{\\tau_0}}(h_1)$ and $V_0^{{\\tau_0}}(h_1)$ denotes the total variation of $h_1$ on $[0,{\\tau_0}]$. \n\t\tThis means that $S_n$ is a random variable defined in the abstract probability space $(\\Omega,\\mathcal{F},\\mathbb{P})$  (where the random vector $(B,T_0,C,X,Z)$ is defined) with values in the space of bounded functions $\\Xi\\mapsto l^\\infty(\\mathcal{H}_\\mathfrak{m})$ with respect to the supremum norm. The latter one is a Banach space equipped with the Borel $\\sigma$-field.\n\t\t\n\t\tWe need to show that conditions 1-4 of Theorem 4 in \\cite{Lu2008} (or Theorem 3.3.1 in \\cite{VW96}) are satisfied. \n\t\tThe main difference of the function $S$ from the one in \\cite{Lu2008} is that here $\\gamma=\\gamma_0$ fixed. We are only considering variation with respect to $\\beta$ and not $\\gamma$, so the components of $h$ that correspond to $\\gamma$ are set to zero.\n\t\tHowever, conditions 2 and 3 of Theorem 4 in \\cite{Lu2008}  for $S$ can be shown in the same way as in \\cite{Lu2008}.  \n\t\tDetails about conditions 1 and 4 can be found in the online Supplementary Material.\n\t\t\\qed\n\t\t\n\t\t\n\t\t\\subsection{Proof of Theorem \\ref{theo:pi_hat}}\n\t\tThe logistic model for the cure probability obviously satisfies assumptions (AN1) and  (AN3).  Let $\\Pi$ be the space of  continuously differentiable functions $f$ from $\\mathcal{X}$ to $[0,1]$ such that $\\sup_{x\\in\\mathcal{X}}|f'(x)|\\leq M$ and\n\t\t\\[\n\t\t\\sup_{x_1, x_2\\in\\mathcal{X}}\\frac{|f'(x_1)-f'(x_2)|}{|x_1-x_2|^\\xi}\\leq M\n\t\t\\]\n\t\tfor some $M>0$ and $\\xi\\in(0,1]$.  If such space is equipped with the supremum norm, the covering numbers  satisfy \n\t\t\\[\n\t\t\\log N\\left(\\epsilon,\\Pi,\\Vert\\cdot\\Vert_\\infty\\right)\\leq K\\frac{1}{\\epsilon^{1\/(1+\\xi)}}\n\t\t\\]\n\t\tfor some constant $K>0$ independent of $\\epsilon$ (see Theorem 2.7.1 in \\cite{VW96}). Obviously, for $\\epsilon>1$, $\\log N(\\epsilon,\\Pi,\\Vert\\cdot\\Vert_\\infty)=0$. Hence, assumption (AN2) is satisfied.   \n\t\tIt remains to check (AN4). Recall that the estimator of the cure probability $\\hat\\pi(x)$ is the value at time $\\tau_0$ of the Beran estimator $\\hat{S}(t|x)$, while $\\pi_0(x)=S(\\tau_0|x)$. Moreover, by assumption  \\eqref{eqn:CI2}, we have $\\inf_x H((\\tau_0,\\infty)|x)>0$. From Proposition 4.1 and 4.2 in \\cite{KA99} it follows that\n\t\t\\[\n\t\t\\begin{aligned}\n\t\t\t\\sup_x\\left|\\hat{\\pi}(x)-\\pi_0(x)\\right|&=O\\left((nb)^{-1\/2}(\\log b^{-1})^{1\/2}\\right)\\quad a.s., \\\\\n\t\t\t\\sup_x\\left|\\hat{\\pi}'(x)-\\pi_0'(x)\\right|&=O\\left((nb^3)^{-1\/2}(\\log b^{-1})^{1\/2}\\right)\\quad a.s.\n\t\t\\end{aligned}\n\t\t\\]\n\t\tand\n\t\t\\[\n\t\t\\sup_{x_1, x_2\\in\\mathcal{X}}\\frac{|\\hat\\pi'(x_1)-\\pi'_0(x_1)-\\hat\\pi'(x_2)+\\pi'_0(x_2)|}{|x_1-x_2|^{\\xi\/2}}=O\\left(\\left(nb^{3+\\xi}\\right)^{-1\/2}(\\log b^{-1})^{1\/2}\\right)\\quad a.s.,\n\t\t\\]\n\t\twhere $\\xi$ is as in assumption (C1). Since $\\pi_0$ is twice continuously differentiable, from  assumption (C1) it follows that $\\hat\\pi$ satisfies (i,ii) of (AN4).\n\t\tFrom Theorem 3.2 of \\cite{DA2002} (with $T=\\tau_0$) we have $\\hat{\\pi}(x)-\\pi_0(x)=\\frac{1}{n}\\sum_{i=1}^n A_i(x)+R_n(x)$, where\n\t\t\\begin{equation}\n\t\t\t\\label{eqn:iid_pi}\n\t\t\t\\begin{split}\n\t\t\t\tA_i(x)=-\\frac{1-\\phi(\\gamma_0,x)}{f_X(x)}\\frac{1}{b}k\\left(\\frac{x-X_i}{b}\\right)\n\t\t\t\t\\left\\{\\frac{\\Delta_i\\mathds{1}_{\\{Y_i\\leq\\tau_0\\}}}{H([Y_i,\\infty)|x)}-\\int_0^{Y_i\\wedge \\tau_0}\\frac{H_1(ds|x)}{H^2([s,\\infty)|x)}\\right\\}\n\t\t\t\\end{split}\n\t\t\\end{equation}\n\t\tand $\\sup_x|R_n(x)|=O\\left((nb)^{-3\/4}(\\log n)^{3\/4}\\right)$ a.s..  \n\t\tHence\n\t\t\\[\n\t\t\\begin{split}\n\t\t\t&\\mathbb{E}^*\\left[\\left(\\hat\\pi(X)-\\pi_0(X)\\right)\\left(\\frac{1}{\\phi(\\gamma_0,X)}+\\frac{1}{1-\\phi(\\gamma_0,X)}\\right)\\nabla_\\gamma\\phi(\\gamma_0,X)\\right]\\\\\n\t\t\t&=\\frac{1}{n}\\sum_{i=1}^n\\mathbb{E}^*\\left[A_i(x)\\left(\\frac{1}{\\phi(\\gamma_0,X)}+\\frac{1}{1-\\phi(\\gamma_0,X)}\\right)\\nabla_\\gamma\\phi(\\gamma_0,X)\\right] \\\\\n\t\t\t&\\quad+\\mathbb{E}^*\\left[R_n(X)\\left(\\frac{1}{\\phi(\\gamma_0,X)}+\\frac{1}{1-\\phi(\\gamma_0,X)}\\right)\\nabla_\\gamma\\phi(\\gamma_0,X)\\right].\n\t\t\\end{split}\n\t\t\\] \n\t\tThe second term on the right hand side of the previous display is bounded by $c\\sup_x|R_n(x)|=o(n^{-1\/2})$ for some $c>0$ because of assumptions (C1) and (AN1). Furthermore, from (AN1) and (AC4) and a Taylor expansion, it follows that the generic element of the sum in the first term is equal to \n\t\t\\[\n\t\t\\begin{split}\n\t\t\t&-\\int_{\\mathcal{X}}\\frac{1}{b}k\\left(\\frac{x-X_i}{b}\\right)\n\t\t\t\\left\\{\\frac{\\Delta_i\\mathds{1}_{\\{Y_i\\leq\\tau_0\\}}}{H([Y_i,\\infty)|x)}-\\int_0^{Y_i\\wedge \\tau_0}\\frac{H_1(ds|x)}{H^2([s,\\infty)|x)}\\right\\}\\frac{1}{\\phi(\\gamma_0,x)}\\nabla_\\gamma\\phi(\\gamma_0,x)\\,\\mathrm{d} x\\\\\n\t\t\t&=-\n\t\t\t\\left\\{\\frac{\\Delta_i\\mathds{1}_{\\{Y_i\\leq\\tau_0\\}}}{H([Y_i,\\infty)|X_i)}-\\int_0^{Y_i\\wedge \\tau_0}\\frac{H_1(ds|X_i)}{H^2([s,\\infty)|X_i)}\\right\\}\n\t\t\t\\frac{1}{\\phi(\\gamma_0,X_i)}\\nabla_\\gamma\\phi(\\gamma_0,X_i)+O(b^2).\n\t\t\\end{split}\n\t\t\\]\n\t\tSince because of (C1) we have $O(b^2)=o(n^{-1\/2})$, (AN4-iii) holds with \n\t\t\\[\n\t\t\\Psi(Y,\\Delta,X)=-\\left\\{\\frac{\\Delta\\mathds{1}_{\\{Y\\leq\\tau_0\\}}}{H([Y,\\infty)|X)}-\\int_0^{Y\\wedge \\tau_0}\\frac{H_1(ds|X)}{H^2([s,\\infty)|X)}\\right\\}\\frac{1}{\\phi(\\gamma_0,X)}\\nabla_\\gamma\\phi(\\gamma_0,X).\n\t\t\\]\n\t\t\\qed\n\t\t\\section*{Acknowledgements}\n\t\n\t\tI. Van Keilegom and E. Musta acknowledge  financial support from the European Research Council (2016-2021, Horizon 2020 and grant agreement 694409). For the simulations we used the infrastructure of the Flemish\n\t\tSupercomputer Center (VSC).\n\t\t\n\t\t\\section*{Supplement}\n\t\tSupporting information may be found in the online appendix.\n\t\n\t\t{It contains the proofs of Theorems 1, 2 and 3} in Section~\\ref{sec:asymptotics} and  additional simulation results.\n\n\n\n\n\n\\bibliographystyle{imsart-number}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPropagation coherence is the condition for realization of\noscillations - the interference effects that lead to\ntime - distance periodic variations of observables \\cite{Kayser:1981ye,Kiers:1995zj,Giunti:1997wq,Grimus:1998uh,Giunti:2002xg,Akhmedov:2009rb,Akhmedov:2010ms,Akhmedov:2012uu,Akhmedov:2019iyt}. \nCorrespondingly, decoherence means the disappearance of the interference \nand oscillatory pattern. \n\nIn the configuration space, the decoherence is produced by \nrelative shift, and eventually, separation of the wave \npackets (WPs) that correspond to the eigenstates of propagation. The coherence length was defined as the distance at which\nthe wave packets are shifted with respect to \neach other (separated) by the size of the wave packet $\\sigma_x$, or equivalently, when the interference term in the probability is suppressed by a factor $1\/e$.\nIn the energy-momentum space, the decoherence is due to \naveraging of the oscillation phase over the energy interval of the energy uncertainty in a setup. \nIt was checked that in vacuum and uniform matter, the two considerations \nare equivalent and lead to the same value of coherence length \\cite{Mikheev:1986wj,Mikheev:1987jp,Kersten:2015kio}. \n\n\nThe loss of the propagation coherence does not lead to the\nloss of information and can be restored under certain conditions. In the configuration space, the restoration requires a long enough coherence time of detection. In the energy-momentum \nspace it requires very good energy resolution. Irreversible loss of coherence occurs if neutrinos are treated as open quantum systems \\cite{Benatti:2000ph}.\n\nMostly the coherence was studied for vacuum oscillation \\cite{Kayser:1981ye,Kiers:1995zj,Giunti:1997wq,Grimus:1998uh,Giunti:2002xg,Akhmedov:2009rb,Akhmedov:2010ms,Akhmedov:2012uu,Akhmedov:2019iyt}. \nIn matter with constant (or adiabatically changing) density\nit was considered in \\cite{Mikheev:1987jp,Mikheyev:1989dy,PhysRevD.41.2379,Peltoniemi:2000nw,Kersten:2015kio}. It was noticed that \nat certain energy (close to the MSW resonance energy), the difference of group velocities of the WPs vanishes in media with constant density and the \ncoherence length becomes infinite \\cite{Mikheev:1986wj,Mikheev:1987jp,Kersten:2015kio}. If density varies with distance monotonously, then at a specific point, $x_0$, near the MSW resonance, the difference of group velocities changes sign, and the shift that happened at $x<x_0$ can be undone at $x>x_0$, so that the packets overlap can be restored \\cite{Mikheev:1987jp,Mikheyev:1989dy,Kersten:2015kio}. Decoherence affects resonant oscillations of solar and supernova neutrinos \\cite{PhysRevD.41.2379} and, in the case of supernova, decoherence happens before the resonance region. However, the effects of collective oscillations were not taken into account in \\cite{PhysRevD.41.2379}. In ref.~\\cite{Peltoniemi:2000nw}, a formal solution for the neutrino equation of motion in general profile including non-adiabaticity was found. Based on it, the equivalence between $x$- and $E$-spaces was shown, at least for situations with $\\sigma_E \\ll E$. Limitations of the WP treatment were also discussed; in particular, as $\\sigma_E$ increases, negative energy components of the WPs become relevant. This can be important in situations with $\\sigma_E \\sim E$.\n\n\nThe coherence in the inner parts of supernova (SN)\nis of special interest \\cite{Kersten:2013fba}. The reason is that the wave packets of SN neutrinos are very short, $\\sigma_E \\sim E$, suggesting that decoherence can occur at smaller scales than\nthe scales of possible collective oscillation phenomena \\cite{Kersten:2015kio,Akhmedov:2017mcc}. \nFurthermore, effectively, the problem is non-linear \\cite{Hansen:2018apu}. \nTherefore the question arises whether the loss of coherence destroys or not the collective phenomena \\cite{Hansen:2019iop}. \n\nIn this paper, we consider coherence and decoherence in matter in detail. We revisit the case of the infinite coherence length in matter of constant density and give its physics interpretation. We show that in matter dominated regime, decoherence is determined by vacuum parameters. We comment on the fact that oscillations of massless neutrinos do not decohere. In the adiabatic case $L_{coh} \\rightarrow \\infty$ is possible if a density profile has periodic modulations.\nWe then consider decoherence in the presence of adiabaticity violation. That includes decoherence in matter with single density \njump which can be relevant for propagation through the shock wavefront in SN \\cite{Kersten:2015kio}, and in the castle-wall profile \\cite{Akhmedov:1988kd,Krastev:1989ix,Akhmedov:1998ui,Akhmedov:1999ty,Akhmedov:1999va}. The latter can give some idea about the decoherence of collective oscillations in the inner parts of supernova \\cite{Hansen:2018apu}. The castle-wall profile is the simplest example because it admits analytical treatment and allows us to draw some conclusions about the relation between parametric resonances and coherence length. \n\nDecoherence in the configuration ($x$-space) is equivalent to averaging of oscillation phase in energy or $E$-space for stationary situations \\cite{Stodolsky:1998tc}. Therefore, for each matter profile, we elaborate on the\nissue of equivalence of results in the $x$- and $E$-spaces. In particular, for the castle-wall profile, we show that the modification of shape factors in $x$-space due to the density jumps do not alter its Fourier transform and the initial energy spectrum of the WPs is preserved.\n\n\nThe paper is organized as follows: in section~\\ref{coherence-lengths} we study the coherence length in vacuum and matter with constant and adiabatically varying density. We emphasize the existence of energies with infinite coherence length and give interpretation of $L_{coh} \\rightarrow \\infty$ in position and energy spaces.\nIn section~\\ref{coherence and adiabaticity violation}, we analyze coherence in the case of maximal adiabaticity violation.\nSection~\\ref{parametric-oscillations} presents the study of coherence in the castle-wall profile.\nSection~\\ref{applications} is devoted to the applications of our results to supernova neutrinos.\nIn section~\\ref{conclusions} we present our conclusions.\n\n\n\\section{Coherence in matter with constant and adiabatically changing density} \n\\label{coherence-lengths}\n\nIn what follows for simplicity, we will consider the coherence in two neutrino system; \nthe generalization to the case of three neutrinos is straightforward. \n\n\\subsection{Coherence of oscillations in matter}\n\nIn the energy-momentum space, the decoherence is related \nto averaging of the oscillation phase $\\phi$ or \nthe oscillatory (interference) term in probability $P_{int}$ \nover the energy uncertainty, $\\sigma_E$, of a set up\\footnote{There is subtle aspect related to averaging \nof $\\phi$ and $P_{int}$ which we will discuss later.}.\n$\\sigma_E$ gives the width of the wave packet (WP), and \nin general, it is determined by the energy uncertainty at the production \nand detection, thus including a possibility \nof restoration of the coherence at the detection: \n$1\/\\sigma_E = 1\/\\sigma_E^{prod} + 1\/\\sigma_E^{det}$ \\cite{Kayser:1981ye,Kiers:1995zj,Giunti:1997wq,Grimus:1998uh,Giunti:2002xg,Akhmedov:2009rb,Akhmedov:2010ms,Peltoniemi:2000nw,Hollenberg:2011tc,Kersten:2015kio}.\n\nIn the case of uniform medium (vacuum, constant density matter) the \neigenstates and eigenvalues of the Hamiltonian of propagation \n$H_{im}$ ($i = 1,\\, 2$) are well defined \\cite{Wolfenstein:1977ue,Mikheev:1986wj,Mikheev:1987jp,Mikheyev:1989dy}. The difference of the eigenvalues, $\\Delta H_m \\equiv H_{2m} - H_{1m}$, determines the oscillation phase\nacquired along the distance $L$: \n\\begin{equation} \n\\label{mat-phase} \\nonumber\n\\phi_m = \\Delta H_m L.\n\\end{equation}\nFor a fixed $L$, variation of the phase with change of the neutrino \nenergy $E$ in the interval $E \\pm \\sigma_E$ equals \n\\begin{equation} \n\\label{expansion}\n\\Delta \\phi_m = \\left( 2 \\sigma_E \n\\frac{d \\Delta H_m}{d E}\n+ \\frac{\\sigma_E^3}{3}\\frac{d^3 \\Delta H_m}{d E^3} \n+... \\right) L. \n\\end{equation}\nIt increases linearly with $L$ and we define the coherence \nlength, $L_{coh}$, \nas the length at which the variation of oscillation phase\nbecomes $2\\pi$: \n\\begin{equation}\n\\label{cohlength}\n|\\Delta \\phi_m (L_{coh}^m)|= 2 \\pi.\n\\end{equation}\nThis condition means that averaging of the oscillatory dependence of the probability, which is a measure of interference, \nbecomes substantial. \nUsing the first term of the expansion (\\ref{expansion}) \nwe obtain from the condition (\\ref{cohlength}) \n\\begin{equation} \n\\label{const-density}\nL^m_{coh} = \\frac{\\pi}{\\sigma_E} \n\\left| \\frac{d \\Delta H_m}{d E} \\right|^{-1}. \n\\end{equation}\nAt \n\\begin{equation} \n\\label{infcoh}\n\\frac{d \\Delta H_m (E)}{d E} = 0 \n\\end{equation}\nthe coherence length becomes infinite \\cite{Mikheev:1987jp,Mikheyev:1989dy,Kersten:2015kio}. The next terms in the expansions (\\ref{expansion}) shift the pole in (\\ref{const-density})\nbut do not eliminate it. The shift due to higher order terms is suppressed if\n$\\sigma_E\/E \\ll 1$.\n\nIn vacuum \n\\begin{equation} \\nonumber\n\\Delta H = \\frac{\\Delta m^2}{2 E},\n\\end{equation}\nwhere $\\Delta m^2 \\equiv m_2^2 -m_1^2$ is the mass square difference.\nConsequently, Eq.~(\\ref{const-density}) gives\n\\begin{equation} \n\\label{L-coh}\nL_{coh} = \n\\frac{\\pi}{\\sigma_E}\\frac{2 E^2}{\\Delta m^2} = \nl_{\\nu} \\frac{E}{2 \\sigma_E}, \n\\end{equation}\nwith $l_{\\nu} = 4 \\pi E \/ \\Delta m^2$ being \nthe vacuum oscillation length. \nNotice that $L_{coh}\\rightarrow \\infty$ in the limits \n$E \\rightarrow \\infty$ or $\\Delta m^2 \\rightarrow 0$. \n\nIn matter with constant \ndensity \\cite{Kersten:2015kio}, the difference \nof eigenvalues for a given matter potential is given by \n\\begin{equation} \n\\label{split-eigenvalues}\n\\Delta H_m = \\frac{\\Delta m^2}{2E} \n\\sqrt{\\left(c_{2\\theta} - \\frac{2EV}{\\Delta m^2} \\right)^2 \n+ s^2_{2\\theta}} \\, ,\n\\end{equation}\nwhere $\\theta$ is the vacuum mixing angle and $c_{2 \\theta} \\equiv \\cos 2 \\theta$ and $s_{2 \\theta} \\equiv \\sin 2\\theta$. The derivative of (\\ref{split-eigenvalues}) equals \n\\begin{equation}\n\\label{eq:partder}\n\\frac{d \\Delta H_m}{d E} = \n\\frac{\\Delta m^2}{2E^2} \n\\frac{ \\left(\\frac{\\Delta m^2}{2 E} - \nV \\cos 2 \\theta \\right)}{\\Delta H_m}.\n\\end{equation}\nConsequently, according to Eq.~(\\ref{const-density}) \nthe coherence length in matter equals \n\\begin{equation} \\nonumber\n\\label{const-density1}\nL^m_{coh}= \n\\frac{L_{coh}}{l_m} \\frac{2 \\pi}{\\left|\\frac{\\Delta m^2}{2 E} \n- V \\cos 2 \\theta \\right|} = L_{coh} \n\\left(1 -c_{2\\theta} \\frac{2VE}{\\Delta m^2} \\right)^{-1} \n\\sqrt{\\left( c_{2\\theta} - \\frac{2VE}{\\Delta m^2} \\right)^2 + s^2_{2 \\theta}} ,\n\\end{equation}\nwith $L_{coh}$ given in (\\ref{L-coh}) and $l_m=2 \\pi \/ \\Delta H_m$ \nbeing the oscillation length in matter. \n\nLet us consider dependence of $L^m_{coh}$ on energy. \nIts salient feature is existence of the pole at \n\\begin{equation} \\nonumber\n\\label{eq-E0}\n E_0=\\frac{\\Delta m^2}{2 V \\cos 2 \\theta}. \n\\end{equation}\nThat is, the infinite coherence length \n$L^m_{coh} \\rightarrow \\infty$ is realized \nat finite energy $E_0$, in contrast \nto the vacuum case. $E_0$ is related to the MSW resonance energy, $E_R =\\Delta m^2 \\cos 2\\theta\/2 V $, as \n\\begin{equation} \n\\label{MSW}\nE_0 = \\frac{E_R}{\\cos^2 2\\theta}. \n\\end{equation}\nThus, $E_0 > E_R$ and it can be very close to $E_R$ in case of small mixing. Since the width of the MSW resonance peak is $2\\tan 2 \\theta E_R \\approx 2\\sin 2\\theta E_R$, the pole is within the peak. \nTherefore, at $E_0$, the transition probability is large. \n\nIn terms of $E_0$ or $x \\equiv E\/E_0$ the derivative (\\ref{eq:partder}) \ncan be rewritten as \n\\begin{equation} \\nonumber\n\\label{eq:intermse0}\n\\frac{d \\Delta H_m}{d E} = \n\\frac{2\\pi}{E l_\\nu} \n\\frac{1 - x}{\\sqrt{(1 - x)^2 + x^2\\tan^2{2\\theta} }}. \n\\end{equation}\nConsequently, the ratio of the coherence lengths \nin matter and in vacuum (\\ref{const-density}) becomes \n\\begin{equation} \n\\label{ratio}\n \\frac{L^m_{coh}}{L_{coh}}=\\sqrt{1+ \n\\frac{\\tan^2 2 \\theta}{\\left(1-\\frac{E_0}{E} \\right)^2}}.\n\\end{equation}\nThe ratio Eq.~(\\ref{ratio}) as function of energy \nis shown in fig.~\\ref{fig-E0}. It\nhas the universal form which depends on the vacuum mixing only. \nThe value $L^m_{coh}\/L_{coh} = 2$ corresponds to energies at both\nsides of the peak $E=E_0 \\left(1 \\pm \\tan 2 \\theta\/\\sqrt{3}\\right)^{-1}$.\nThus, for small $\\theta$ the quantity $\\Delta E = (2\/\\sqrt{3})\\tan 2\\theta E_0$\ncharacterizes the width of the peak.\n\nAt the MSW resonance, $E=E_R$, the ratio (\\ref{ratio}) equals\n\\begin{equation} \n\\label{ratio-resonance}\n\\frac{L^m_{coh}}{L_{coh}}=\\frac{1}{\\sin 2 \\theta}.\n\\end{equation}\nThat is, the coherence length Eq.~(\\ref{ratio-resonance})\nis enhanced in comparison to the vacuum length. \nFor example, if $\\theta=\\theta_{13}=8.5^\\circ$, \nwe have $L^m_{coh} \\approx 12 L_{coh}$. \nIn the resonance the oscillation length increases \nby the same amount: $l_m^R = l_\\nu\/ \\sin 2\\theta$.\n\n\nFor $E \\ll E_0$, {\\it i.e.} in the vacuum dominance case,\nwe have $L^m_{coh} \\approx L_{coh}$. \nInterestingly, for $E \\gg E_0$, (matter dominated region)\n\\begin{equation} \nL^m_{coh} \\approx \\frac{L_{coh}}{\\cos 2 \\theta}. \n\\label{matter-domination}\n\\end{equation}\nThat is, $L_{coh}^m$ is determined by the vacuum parameters, and for small $\\theta$ it approaches the vacuum $L_{coh}$ again. \nThis has important implications for supernova neutrinos. \n\nThe depth of oscillations in matter given by\n\\begin{equation} \n\\label{sine-mixing}\n\\sin^2 2 \\theta_m = \\frac{\\sin^2 2 \\theta}{\\left( \\cos 2\\theta \n- \\frac{2EV}{\\Delta m^2} \\right)^2 + \\sin^2 2\\theta}\n\\end{equation}\nbecomes at $E=E_0$\n\\begin{equation} \\nonumber\n \\sin^2 2 \\theta_m =\\cos^2 \\theta.\n\\end{equation}\nFor $\\theta=8.5^0$ the depth is close to maximum: \n$\\sin^2 2 \\theta_m \\approx 0.98$. \nTherefore, in matter, strong flavor transitions can be preserved \nfrom decoherence over large distances $L$ from the source to the detector.\n\n\n\n\n\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=0.8\\linewidth]{fig-E0.png}\n\\caption{Dependence of the ratio $L^m_{coh}\/L_{coh}$ (\\ref{ratio}) \non energy for $\\theta = \\theta_{13} = 8.5^\\circ $. The ratio diverges at $E_0$, and $E_R$ is the MSW resonance energy. \n}\n\\label{fig-E0}\n\\end{figure}\n\n\n\n\\subsection{Coherence in the configuration space}\n\nIn the configuration space, the loss of propagation coherence \nis associated with relative shift and eventually separation \nof the wave packets of the eigenstates. \nThe difference of group velocities \nof the eigenstates is given by \n\\begin{equation} \n\\label{Delta-v}\n \\Delta v_m=-\\frac{d \\Delta H_m}{d E}.\n\\end{equation}\nThen spatial separation (relative shift) \n$x_{shift}$ of the packets after propagating a distance $L$ equals\n\\begin{equation} \\nonumber\n\\label{velocity}\nx_{shift}(L) =\\int_{0}^{L} \\Delta v_m dx = \n-\\int_{0}^{L} \\frac{d \\Delta H_m}{d E} dx.\n\\end{equation}\nThe coherence length $L^m_{coh}$ can be defined as the distance at which \nthe separation equals the spatial size of the packets $\\sigma_x$:\n$|x_{shift}(L^m_{coh})| \\approx \\sigma_x$. At this point, the overlap of the packets becomes small. \nIn the case of constant density this condition gives \n\\begin{equation} \n\\label{velocity1}\nL^m_{coh} = \\sigma_x \n\\left| \\frac{d \\Delta H_m}{d E} \\right|^{-1}.\n\\end{equation}\nSince $\\sigma_x = 2\\pi\/(2 \\sigma_E)$ \n(recall that $\\sigma_E \\approx \\sigma_p$ is the half-width), \nthe expression (\\ref{velocity1}) coincides with expression \nfor the length in the energy representation (\\ref{const-density}). \nThis shows the equivalence of the results in the \nconfiguration and the energy-momentum spaces. \n\nAccording to Eq.~(\\ref{Delta-v}) at $E=E_0$ \nthe eigenstates in matter have equal group \nvelocities and therefore do not separate. As a result, \ncoherence is maintained during infinite time: \n$L^m_{coh} \\rightarrow \\infty$. \nThus, in the configuration space the condition for infinite coherence \nis \n\\begin{equation} \n\\label{infcoh-conf}\nv_1 = v_2. \n\\end{equation} \n\nThe infinite coherence energy $E_0$ coincides\nwith the resonance energy of oscillations\nof the mass eigenstates in matter\n$\\nu_1 \\leftrightarrow \\nu_2$ \\cite{Mikheyev:1989dy}. This is not accidental.\nThe origin of the energy dependence of phase,\nthe difference of group velocities\nand therefore decoherence is the mass states.\nThe resonance means that at $E_0$ the mass states $\\nu_1$ and $\\nu_2$\nare maximally mixed in the eigenstates $\\nu_i^m$.\nThat is, $\\nu_1^m$ and $\\nu_2^m$\nboth contain equal admixtures of $\\nu_i$. Consequently,\nthe group velocities of $\\nu_1^m$ and $\\nu_2^m$ should be equal.\nIn other words, in a given flavor state\nthe components $\\nu_1$ and $\\nu_2$ oscillate\nwith maximal depth $\\nu_1 \\rightarrow \\nu_2\n\\rightarrow \\nu_1 \\rightarrow \\nu_2$,\nwhich compensate separation.\n\nThe derivative of the oscillation length in matter equals\n$$\n\\frac{d l_m}{dE} = - \\frac{2\\pi}{(\\Delta H_m)^2}\n\\frac{d\\Delta H_m}{dE}.\n$$\nTherefore the condition of infinite coherence (\\ref{infcoh}) coincides\nwith the extremum of oscillation length.\nFor a given potential $V$,\n$l_m^{max} = 2\\pi\/(V \\sin 2\\theta)$, which is larger than\nthe length in the MSW resonance:\n$l_m^{R} = 2\\pi\/(V \\tan 2\\theta)$, and the difference\nis substantial for large vacuum mixing.\n\nNotice that decoherence is related to\nuncertainty in energy on a certain interval of energy.\nTherefore infinite coherence energy should refer to the\naverage energy in a wave packet.\n\nNotice that the equivalence of results in two representations \noriginates from the fact that \nthe same quantity $d \\Delta H_m \/d E$ \ndetermines the change of the oscillation phase with energy \n(in $E-p$ space) on the one hand side, and the difference of the \ngroup velocities (in $x$-space) on the other hand.\n\n\\subsection{Correlation of the phase difference and delay}\n\nThe separation between the WP of eigenstates, $t \\approx \\Delta v X$,\nis proportional to their phase difference (oscillation phase), \n$\\phi = \\Delta H_m X$: \n\\begin{equation}\n\\label{eq:phase-delay}\nt = g(E, V) \\phi, \n\\end{equation}\nwhere according to (\\ref{eq:partder})\n\\begin{equation} \\nonumber\n\\label{eq:gfunc}\ng(E, V) = \\frac{\\Delta v}{\\Delta H_m} = \n\\frac{1}{\\Delta H_m}{2E^2} = \n\\frac{1}{E} \n\\frac{1 - \\frac{2 V E c_{2\\theta}}{\\Delta m^2}} \n{\\left(c_{2\\theta} - \\frac{2VE}{\\Delta m^2}\\right)^2 + s^2_{2\\theta}}.\n\\end{equation}\nIn terms of infinite coherence energy, $E_0$, it can be rewritten as\n\\begin{equation} \\nonumber\n\\label{eq:gfunc2}\ng(E, V) = \\frac{2 V c_{2\\theta}}{\\Delta m^2} \n\\frac{\\frac{E_0}{E} - 1} \n{\\left(c_{2\\theta} - \\frac{E}{E_0 c_{2\\theta}} \\right)^2 \n+ s^2_{2\\theta}}.\n\\end{equation}\nFor $E \\ll E_0$ (vacuum case)\n$$\ng(E, V) = \\frac{1}{E}.\n$$\nAt $E = E_0$ (near the MSW resonance): \n$g(E, V) = 0$ - there is no loss of coherence, \nand the oscillation phase, being non-zero, is suppressed by mixing: \n$$\n\\phi = \\frac{\\Delta m^2 X}{2E_0} \\tan 2\\theta = \n\\phi_{vac}(E_0) \\tan 2\\theta. \n$$\nFor $E \\gg E_0$ (matter dominated case):\n$$\ng(E, V) \\approx - \\frac{c_{2\\theta} \\Delta m^2}{2V E^2}= - \\frac{ E_0 c_{2 \\theta}^2}{ E^2}.\n$$\nHere also the relative separation is strongly \nsuppressed, while oscillations proceed with large phase \n$\\phi \\approx V X$. \n\nNotice also that the relative separation changes the sign at $E_0$. Therefore in layers $a$ and $b$ with \ntwo different densities one may have opposite sign \nof a delay and also \n$t^a = -t^b$ (see below). The phases do not change the sign and are\nalways positive. \n\n\n\n\\subsection{Physics of $L^m_{coh} \\rightarrow \\infty$} \n\\label{interpretation}\n\nHere we present an interpretation of the divergence of \n$L^m_{coh}$ at certain energy $E_0$. \nThe transition probability is given by\n\\begin{equation} \\nonumber\n\\label{generic-prob}\nP_{\\alpha \\beta}(\\theta_m,\\phi_m ) =\n\\frac{1}{2}\\sin^2 2 \\theta_m \\left(1 - \\cos \\phi_m \\right).\n\\end{equation}\nIt should be averaged over the energy interval $2\\sigma_E$. \nThe change of $\\sin^2 2 \\theta_m$ with energy in the interval \n$E_0 \\pm \\sigma_E$ equals\n\\begin{equation}\\nonumber\n \\left| 2 \\sigma_E \\frac{d}{d E}\\sin^2 2 \\theta_m \\right|=2 \\sigma_E \\frac{V c_{2 \\theta}}{\\Delta m^2}=\\frac{\\sigma_E}{E_0} \\ll 1,\n\\end{equation}\nwhere we have taken into account that at $E_0$ the mixing parameters are $\\sin^2 2 \\theta_m=\\cos^2 \\theta$ and $\\cos^2 2 \\theta_m=\\sin^2 \\theta$. Therefore \n$\\sin^2 2 \\theta_m$ can be put out of the averaging integral at $E \\sim E_0$. \nFurthermore, to simplify consideration, instead of averaging over $E$ we will \naverage $P_{\\alpha \\beta}$ immediately over the phase $\\phi_m$ \nin the interval $2\\delta \\phi_m$ determined by \n\\begin{equation}\n2 \\delta \\phi_m = \\frac{d \\phi_m }{d E} 2\\sigma_E\n= \\frac{d \\Delta H_m}{d E} L 2\\sigma_E. \n\\label{pm-int}\n\\end{equation}\nThe averaging of the oscillatory term yields\n\\begin{equation} \\nonumber\n\\label{average}\n\\frac{1}{2 \\delta \\phi_m} \\int_{\\phi_m - \\delta \\phi_m}^{\\phi_m\n+ \\delta \\phi_m} \nd\\phi_m' P_{\\alpha \\beta}(\\phi_m') = \\frac{1}{2}\n\\sin^2 2 \\theta_m (E_0) \n\\left[1 - D(\\delta \\phi_m) \\cos \\phi_m \\right], \n\\end{equation}\nwhere \n\\begin{equation} \\nonumber\n\\label{decoh}\nD(\\delta \\phi_m) \\equiv \\frac{\\sin \\delta \\phi_m}{\\delta \\phi_m}\n\\end{equation}\nis the decoherence factor which describes suppression of \nthe interference term.\n\nAccording to (\\ref{pm-int}) for $E = E_0$ the variation of phase \nis $\\delta \\phi_m = 0$, \nand consequently, $D(\\delta \\phi_m) \\rightarrow 1$.\nThe averaging effect is small in the region around $E_0$. \nIndeed, $D(\\delta \\phi_m)$ has a peak centered at $\\delta \\phi_m = 0$. \nThe half-width at nearly half of maximum, \n$D(\\delta \\phi_m) = 0.5$ corresponds to \n\\begin{equation} \\label{rounding}\n|\\delta \\phi_m| \\approx 1.89. \n\\end{equation}\nConsequently, the half-width of the peak in the energy scale, $\\Gamma$, \nis determined by the condition \n\\begin{equation} \\label{Gamma}\\nonumber\n|\\delta \\phi_m (E_0 + \\Gamma)| = 2\n\\end{equation}\n(rounding (\\ref{rounding}) to $2$).\nUsing this value \nand expressions in (\\ref{pm-int}) and (\\ref{eq:partder}), \nwe find the relative half-width of the peak:\n\\begin{equation} \n\\label{exact-peak}\n \\frac{\\Gamma}{E_0} = \n\\frac{ \\tan 2 \\theta }{\\sqrt{\\pi^2 d^2 - 1}},\n\\end{equation}\nwhere \n\\begin{equation}\\nonumber\nd \\equiv \\frac{\\sigma_E}{E_0} \\frac{L}{l_{\\nu}}. \n\\end{equation}\nNotice that (\\ref{exact-peak}) requires $d \\pi > 1$, \non explicitly \n$L > l_{\\nu}E\/ (\\pi \\sigma_E) \\approx L_{coh}$.\nFor $d \\pi \\rightarrow 1$, the width $\\Gamma \\rightarrow \\infty$, \nwhich means that coherence is well satisfied for all the energies. \nMoreover, the peak disappears (half a height does not exist). \n\nFor $\\pi d \\gg 1$ the Eq.~(\\ref{exact-peak}) becomes\n\\begin{equation} \n\\label{exact-peak1}\n\\frac{\\Gamma}{E_0} = \\frac{ \\tan 2 \\theta }{\\pi d} = \n\\tan 2 \\theta \\frac{E_0}{\\sigma_E} \\frac{l_{\\nu}}{\\pi L}. \n\\end{equation}\nThe larger $L$, as well as $\\sigma_E$, the narrower the peak.\nAlso, according to (\\ref{exact-peak1}) the width of the peak increases with energy:\n\\begin{equation} \\nonumber\n\\label{propto-peak}\n \\frac{\\Gamma}{E_0} \\propto E_0^2. \n\\end{equation}\n\n\n\n\n\n\n\\subsection{Adiabatic evolution and infinite coherence}\nIn the case of slow adiabatic density change \\cite{Mikheev:1986wj,Mikheev:1987jp,Mikheyev:1989dy}, one can\nintroduce the instantaneous eigenstates and eigenvalues\n$H_{im}(x)$ and their difference $\\Delta H_m(x)$,\nwhich are well-defined quantities.\nThe adiabatic oscillation phase is given by the integral\n\\begin{equation}\n\\label{adiab-phase}\n\\phi_m^{ad}(L) = \\int_0^{L} dx \\Delta H_m(N_e(x)).\n\\end{equation}\nA change of the phase with the energy in the interval\n$2\\sigma_E$ equals\n\\begin{equation}\n\\label{adiab-diff}\\nonumber\n\\Delta \\phi_m^{ad}(L) = 2 \\sigma_E \\int_0^{L} dx\n\\frac{d}{d E}\\Delta H_m(x),\n\\end{equation}\nwhere we permuted the intergration over $x$ and differentiation\nover $E$.\nThen the condition for decoherence is\n$\\Delta \\phi_m (L^m_{coh}) = 2\\pi$, or explicitly,\n\\begin{equation}\n\\label{cond-var-density}\\nonumber\n\\left| \\int_0^{L^m_{coh}} dx \\frac{d}{d E}\n\\Delta H_m(x)\\right| = \\frac{\\pi}{\\sigma_E},\n\\end{equation}\nand $d \\Delta H_m(x)\/d E$ is given in (\\ref{eq:partder}).\n\n\nIn the configuration space the adiabatic evolution means\nthat there are no transitions between the eigenstates.\nTherefore a propagation is described by two wave packets,\nwhich do not change shape factors as in the constant density case.\nSeparation of the WP equals\n\\begin{equation}\n\\label{adiab-sep}\n\\Delta x (L) = \\int_0^{L} dx\n\\frac{d}{d E} \\Delta H_m(x),\n\\end{equation}\nand from (\\ref{adiab-phase}), (\\ref{adiab-sep}) we obtain\n\\begin{equation}\\nonumber\n\\label{adiab-equal}\n\\frac{d\\phi_m^{ad}}{d E} = \\Delta x (L),\n\\end{equation}\nwhich is the basis of equivalence of results in the $x$- and $E$-representations. In particular,\nthe infinite coherence condition, $d\\phi_m^{ad} \/dE = 0$, correspons to\nzero separation.\n\nLet us consider coherence for different density (potential) profiles.\nFor monotonous change of the potential the infinite\n$L_{coh}^m$ can not be realized. Indeed, for a given $E$ the pole\ncondition\nis satisfied for specific value of the potential\n$V_0 (E) = V(E, x_0)$ in specific point $x_0$.\nIn the point $x = x_0 + \\Delta x$, where $V(x_0 + \\Delta x)$\nis outside the $L_{coh}$-peak,\nsignificant enhancement of coherence is absent and $L_{coh}^m < \\Delta x$.\n\n\nCertain increase of the coherence length can be related\nto the fact that at $V_0$ the derivative\n$d \\Delta H_m(x)\/d E$\nchanges the sign which suppresses the integral over $x$.\nThis corresponds to the change of sign of\nthe difference of the group velocities at $E_0$ ($V_0$).\n\nFor a layer with a given length $L$ the energy of zero\nseparation (complete overlap) is realized\n(\\ref{adiab-sep}) when\n\\begin{equation} \\label{L_0}\n \\int_0^{x_0} dx \\frac{d}{d E} \n\\Delta H_m(x)=-\\int_{x_0}^{L} dx \\frac{d}{d E} \n\\Delta H_m(x) ,\n\\end{equation}\nand $x_0 = x_0(E)$.\n\nThe infinite coherence can be achieved\nin the potential with periodic modulations:\n\\begin{equation}\n\\label{eq:var-pot}\nV(x) = \\Bar{V} (1 + h \\sin 2\\pi x\/X).\n\\end{equation}\nFor a large $X$, the adiabaticity condition is satisfied.\nZero separation of WP, $\\Delta x = 0$, is realized when\nthe difference of group velocities has different signs in the\nfirst and the second parts of the period.\nSo, there is a continuous \"catch-up\" effect.\n\n\nAccording to (\\ref{eq:partder}) the energy $E_0$ of infinite coherence\nis obtained from the condition\n\\begin{equation}\n\\label{eq:intcond}\nlim_{L \\rightarrow \\infty} \\int_0^L dx\n\\frac{\\left(\\frac{\\Delta m^2}{2 E} - V\\cos 2\\theta \\right)}{\\Delta\nH_m(x)}\n= 0.\n\\end{equation}\nAround the zero value of nominator, denominator changes\nmuch slower\nand it can be put out of integral at some average value of the potential.\nThen the condition (\\ref{eq:intcond}) reduces to\n\\begin{equation}\\nonumber\n\\label{eq:intcond2}\n\\frac{\\Delta m^2}{2 E} L - \\cos 2\\theta \\int_0^L dx V (x) = 0.\n\\end{equation}\nPerforming explicit integration with the potential\n(\\ref{eq:var-pot}) we obtain\n\\begin{equation}\n\\label{eq:intcond3}\n\\frac{\\Delta m^2}{2 E_0} \\approx \\cos 2\\theta \\Bar{V} + \\frac{h}{\\pi}\n\\frac{X}{L}\n\\sin^2 \\frac{\\pi L}{X}.\n\\end{equation}\nIn the limit $L \\rightarrow \\infty$ the last term\non the RHS of (\\ref{eq:intcond3}) can be neglected and for $E_0$\nwe obtain the same expression (\\ref{eq-E0}) as in constant density case\nwith the average potential $\\Bar{V}$.\n\nIn general, if $V$ varies about some average value $\\bar V$ (even irregularly), there may be energy where $L_{coh}^m \\rightarrow \\infty$. \n\n\n\\subsection{Coherence for massless neutrinos}\n\n\nThe limit $L_{coh}^m \\rightarrow \\infty$ is realized \nfor oscillations of massless neutrinos, \nas originally introduced by Wolfenstein \\cite{Wolfenstein:1977ue}. \nHere the oscillation phase is independent of neutrino energy.\nRecall that the propagation decoherence is related to \nthe energy dependence of level splitting $\\Delta H_m$ and \nshows up as averaging over energy. \nTherefore the decoherence does not exist for oscillations \nof massless neutrinos driven by potentials. \n\n\n\\section{Coherence and adiabaticity violation}\n\\label{coherence and adiabaticity violation}\n\nAdiabaticity violation means transitions between the\neigenstates of propagation $\\nu_{1m} \\leftrightarrow \\nu_{2m}$ and therefore change of shape of the WPs. Here we will consider the cases of extreme adiabaticity violation when $dV\/dx \\rightarrow \\infty$, which corresponds to density jumps at certain spatial points.\n\nExtreme adiabaticity violation combined with shift and separation of the WPs leads to splits and catch up of the WPs \\cite{Kersten:2015kio}. We will first illustrate these effects considering a single density jump.\n\n\\subsection{Coherence in the case of single density jump}\n\nConsider a two layers ($a$ and $b$) \nprofile with density jump at a border between them. The layers have matter potentials $V_a$ and $V_b$ and lengths $L_a$ and $L_b$, \nso that the total length equals $L \\equiv L_a + L_b$. \nLet $\\theta_k = \\theta_m(V_k)$\nbe the flavor mixing angle \nin layer $k$, \n($k = a, \\, b$).\n\nSuppose in the beginning of layer $a$ the state is\n\\begin{equation} \n\\label{init}\n\\nu (x=0,t) = c_a' f_1(t) \\nu_1^a + s_a' f_2 (t) \\nu_2^a, \n\\end{equation}\nwhere we use abbreviations $c_a' \\equiv \\cos \\theta_a$, \n$s_a' \\equiv \\sin \\theta_a$ and \n$f_i(t)$ are the shape factors of the wave packets \nnormalized as \n$$\n\\int dt |f_i (t)|^2 = 1.\n$$\nEvolving to the border between the layers $a$ and $b$ \nthe state (\\ref{init}) becomes \n\\begin{equation} \n\\label{border}\n\\nu (L_a,t) = c_a' f(t - t_1^a) e^{i 2\\phi_1^a} \\nu_1^a + \ns_a' f(t - t_2^a) e^{i 2\\phi_2^a} \\nu_2^a. \n\\end{equation} \nHere $t_i^a$ are the times of propagation in the $a$-layer,\n$t_i^a = L_a\/ v_i^a$, $v_i^a$ are group velocities, and $\\phi_i^a$ are phases \nacquired by the eigenstates taken \nat the average energies in the packets. \nWe assume that in the beginning ($t=0$) the shape factors of $\\nu_1^a$ \nand $\\nu_2^a$ are equal. \n\nCrossing the border between $a$ and $b$ each eigenstate $\\nu_i^a$ splits into eigenstates $\\nu_j^b$ of the layer $b$:\n\\begin{equation} \n\\label{splitab} \n\\nu_1^a = c_\\Delta \\nu_1^b + s_\\Delta \\nu_2^b, \\, \\, \\, \\, \n\\nu_2^a = c_\\Delta \\nu_2^b - s_\\Delta \\nu_1^b,\n\\end{equation}\nwhere $\\Delta \\equiv \\theta_b - \\theta_a$. Inserting (\\ref{splitab}) into (\\ref{border}) we obtain the state in the beginning of layer $b$. Then to the end of the layer $b$ the state (\\ref{border}) evolves to \n\\begin{eqnarray} \n\\label{border-ba}\n\\nu (L,t) = \n\\left[ c_a' c_\\Delta f(t - t_1^a - t_1^b) e^{i 2(\\phi_1^a + \n\\phi_1^b)} \n- s_a' s_\\Delta f(t - t_2^a - t_1^b) e^{i 2(\\phi_2^a + \\phi_1^b)} \n\\right] \\nu_1^b + \n\\nonumber \\\\\n\\left[c_a' s_\\Delta f(t - t_1^a - t_2^b) \ne^{i 2(\\phi_1^a + \\phi_2^b)} \n+ s_a' c_\\Delta f(t - t_2^a - t_2^b) e^{i 2(\\phi_2^a \n+ \\phi_2^b)} \\right] \\nu_2^b. \n\\end{eqnarray} \nWe put out the common phase factor \n$e^{i 2(\\phi_1^a + \\phi_1^b)}$ \nand introduce the phase differences,\n\\begin{equation} \\label{phase-redefinition}\n \\phi^a \\equiv \\phi_2^a - \\phi_1^a \\, \\, \\, \\, \\text{and} \\, \\, \\, \\, \\phi^b \\equiv \\phi_2^b - \\phi_1^b.\n\\end{equation}\nThen we make the time shift as $t' = t - t_1^a - t_1^b$, which allows us to express results in terms of the relative \ntime shifts of the eigenstates 1 and 2 in the layers $a$ and $b$:\n\\begin{equation} \\label{time-redefinition}\n t^a \\equiv t_2^a - t_1^a \\, \\, \\, \\, \\text{and} \\, \\, \\, \\, t^b \\equiv t_2^b - t_1^b.\n\\end{equation}\n\n\nUsing these quantities and projecting (\\ref{border-ba}) \nonto $\\nu_e$ we obtain the amplitude \nof $\\nu_e \\rightarrow \\nu_e$ transition\nover the two layer profile:\n\\begin{eqnarray} \n\\label{eeamp1}\nA_{ee} & = &\nc_\\Delta \\left[\n c_a'c_b' f(t') e^{- i\\phi}\n+ s_a's_b' f(t' - t^a - t^b) e^{i\\phi} \n\\right] \n\\nonumber \\\\\n& + & \ns_\\Delta \\left[\n c_a' s_b' f(t' - t^b) e^{-i\\phi'}\n- s_a'c_b'f(t' - t^a) e^{i\\phi'} \n\\right]. \n\\end{eqnarray} \nIn this expression we put out the factor \n$e^{i(\\phi^a + \\phi^b)}$ and introduce\n\\begin{equation} \\label{phase-redefinition-2}\n \\phi \\equiv \\phi^a + \\phi^b \\, \\, \\, \\, \\text{and} \\, \\, \\, \\, \\phi' \\equiv \\phi^a - \\phi^b.\n\\end{equation}\n\\subsection{Specific examples}\nLet us consider specific cases. \n\n1. Suppose $|t^a| \\gg \\sigma_t$, that is, \nthe separation of the WPs in the layer $a$ \nis larger than the coherence time, which means that the \nWPs are completely separated and coherence is lost. Suppose also that \n$|t^b| \\ll \\sigma_t$, {\\it i.e.} loss \nof coherence in the layer $b$ is negligible, $|t^b| \\approx 0$. Then taking into account that the \nshape factors $f(t')$ and $f(t - t^a)$ do not overlap \nwe obtain from (\\ref{eeamp1}) \n\\begin{eqnarray}\n\\label{eeamp2}\nP_{ee} & = & \\int dt |A_{ee}|^2 = \n\\int dt |f(t')|^2 \n\\left|c_a' c_b' c_\\Delta + \nc_a' s_b' s_\\Delta e^{ i 2\\phi^b} \\right|^2\n\\nonumber \\\\\n& + & \\int dt |f(t' - t^a)|^2 \n\\left| s_a's_b' c_\\Delta e^{i 2\\phi^b} \n- s_a'c_b's_\\Delta \\right|^2. \n\\end{eqnarray} \nWe assume that change of the oscillation phase $\\phi^b (t)$ along the \nWP is negligible, so that the oscillatory factors can be\nput out of the integral and take into account that \nintegrations of the moduli squared of the shape factors give $1$ \ndue to normalization. As a result, we obtain \n\\begin{equation}\n\\label{eeamp2}\nP_{ee} = \n\\left|c_a' c_b' c_\\Delta + \nc_a' s_b' s_\\Delta e^{ i 2\\phi^b} \\right|^2\n+ \\left| s_a's_b' c_\\Delta e^{i 2\\phi^b} \n- s_a'c_b's_\\Delta \\right|^2. \n\\end{equation} \n\nComputing the amplitudes in the energy space, we need to use the plane waves so that $f(t) =1 $, and the energy\ndependent phases $\\phi^a (E)$ and $\\phi^b (E)$. \nThe probability is given by \n\\begin{equation}\n\\label{eepr1}\nP_{ee} = \\int dE F(E) |A_{ee}|^2,\n\\end{equation}\nwhere $F(E)$ is the energy spectrum of neutrinos \nthat corresponds to the WP in the $E$-$p$ space\n$F(E) = |f(E)|^2$ and $f(E)$ is the Fourier transform of $f(t)$. \nThe amplitude (\\ref{eeamp1}) can be written as \n\\begin{equation} \n\\label{eeamp12}\nA_{ee} = e^{-i\\phi(E)} A_1 + e^{i\\phi'(E)} A_2, \n\\end{equation}\nwith\n\\begin{equation} \n\\label{a1a2}\nA_1 = c_a' \\left[c_b' c_\\Delta \n+ s_b' s_\\Delta e^{i2\\phi^b(E)} \\right], \n\\, \\, \\, \\\nA_2 = s_a'\\left[s_b'c_\\Delta e^{i 2\\phi^b(E)} \n- c_b' s_\\Delta \\right]. \n\\end{equation} \nHere $A_i$ are the oscillations amplitudes of the eigenstates \nof the layer $a$, $\\nu^a_i$, in the layer $b$, $\\nu_i^a \\rightarrow \\nu_e$. \nInserting (\\ref{eeamp12}) into (\\ref{eepr1}) we have \n\\begin{equation}\n\\label{eepr3}\nP_{ee} = \\int dE F(E) \\left[ |A_1|^2 + |A_2|^2 + \n2|A_1 A_2| \\cos 2(\\phi^a + \\chi) \\right], \n\\end{equation}\nwhere $\\chi \\equiv Arg (A_1^* A_2)$ depends on the phase $\\phi^b$. \nTo compare this with consideration in the $x$-space we assume that \n$\\phi^a \\gg 1$ (which corresponds to large delay in the layer $a$). In contrast, \nthe phase $\\phi^b$ is relatively small, \n$\\phi^b \\ll 1$. In this case we can put all the terms \nbut those which depend on $\\phi^a$ \nin (\\ref{eepr3}) out of the integral taking them\nat average value of $E$ in the spectrum. Then \ndue to normalization $\\int dE F(E) = 1$ the expression \n(\\ref{eepr3}) becomes\n\\begin{equation}\n\\label{eepr4}\nP_{ee} = |\\bar{A}_1|^2 + |\\bar{A}_2|^2 + \n2|\\bar{A}_1 \\bar{A}_2| \\int dE F(E)\\cos 2(\\phi^a + \\chi). \n\\end{equation}\nHere $\\bar{A}_i$ are the amplitudes at the average value of energy. \nThe last term is suppressed as $1\/\\phi^a$ and negligible \nfor $\\phi^a \\gg 1$. Thus, the result (\\ref{eepr4}), with $A_i$ in (\\ref{a1a2}), coincides with that in the $x$-representation (\\ref{eeamp2}). \n\nIn the example considered above, the result does not depend\non $L_a$, and $L_a \\rightarrow \\infty$ is possible. In a sense,\none can consider this as the case of infinite coherence since\noscillations\ncan be observed at the arbitrary long distance from the source.\nThus, density jump and splitting of eigenstates induce or restore\nthe interference and therefore oscillations. Here suppression of\npropagation\ncoherence is determined\nby properties of layer $b$. That is, the problem is reduced to\na single layer with constant density.\nDepending on initial mixing and $\\Delta \\theta$, crossing\nthe density jump can lead to even stronger\ninterference than at the beginning of layer $a$.\n\nThe situation described above is realized, {\\it e.g.}, \nfor the solar and supernova neutrinos oscillating inside the Earth, \nwhere $a$ is the vacuum between the star and the Earth, and $b$ is the Earth. \nIn the high energy part of solar neutrinos: $c_a^2 \\ll s_a^2$.\n\n2. Suppose the layer $b$ has density at which\nvelocities of eigenstates are equal so that coherence is maintained for arbitrary large $L_b$ and $t^b=0$.\nFurthermore, suppose in the layer $a$ the wave packets\nare completely separated, $L_a \\rightarrow \\infty$ which means that dependence on $\\phi^a$ disappears.\nThen according to (\\ref{eeamp1}),\n\\begin{eqnarray}\n\\label{eeampcomp}\nA_{ee} & = &\nc_a' c_b' c_\\Delta f(t) e^{i \\phi} - s_a'c_b' s_\\Delta f(t - t^a) e^{i\\phi'}+\n\\nonumber \\\\\n& & c_a' s_b' s_\\Delta f(t) e^{-i\\phi'}\n+ s_a's_b' c_\\Delta f(t - t^a) e^{i\\phi}.\n\\end{eqnarray}\nThe interference of the overlapping parts\n(they have the same argument in $f$) equals\n\\begin{equation}\n\\label{int1112}\nt: \\, \\, 2 c_a'^2 c_b' s_b' s_\\Delta c_\\Delta \\cos 2\\phi^b (t),\n\\, \\, \\, \\, \\, \\,\n(t - t^a): \\, \\, - 2 s_a'^2 c_b' s_b' s_\\Delta c_\\Delta\n\\cos 2\\phi^b (t - t^a).\n\\end{equation}\nAt a detector, both oscillation phases\nare equal to $\\phi^b$, and, the sum of interference terms (after integration over\ntime)\nis\n\\begin{equation}\n\\label{intsum}\n\\cos 2\\theta_a \\sin 2 \\theta_b s_\\Delta c_\\Delta \\cos 2\\phi^b.\n\\end{equation}\nThis can be compared to the depth of interference\nin the beginning of the layer $a$: $0.5\\sin^2 2\\theta_a$.\n\nRegions of parameters exist where the depth increases after\npropagation in the two-layer profile. \\\\\n\n\nApart from the restoration of the interference, there is another phenomenon\nwhich will be important for our further consideration, namely, change of the shape of the WP at\ncrossing the border between the layers:\n\n\n\\begin{itemize}\n\n\\item\nEach packet $\\nu_i^a$ becomes the two-component one with an overall size\nextended by the relative delay in the layer $a$, $t^a$.\n\n\\item\nAccording to (\\ref{eeampcomp})\nthe amplitudes of components of $\\nu_1^b$ equal\n$(c_a' c_\\Delta, \\, \\, - s_a' s_\\Delta)$ for the earlier and the later\nones correspondingly. The amplitudes in $\\nu_2^b$ are\n$(c_a' s_\\Delta, \\, \\, s_a' c_\\Delta)$.\n\n\\item\nThe shape depends on the size of density jump.\nIf, {\\it e.g.}, the jump is small, $s_\\Delta \\sim s_a' \\sim \\epsilon \\ll 1$,\nwe find from the previous item the amplitudes\nfor $\\nu_1^b$: $(1, \\, \\epsilon^2)$,\nand for $\\nu_2^b$: $(\\epsilon, \\, \\epsilon)$.\nIf $s_\\Delta \\approx c_\\Delta$, the packets will have similar form with similar amplitudes\n$\\nu_1^b$: $(c_a', \\, - s_a')$ and $\\nu_2^b$: $(c_a', \\, s_a')$.\nIf $s_\\Delta = 1$, $c_\\Delta = \\epsilon$ we have\n$\\nu_1^b$: $(\\epsilon, \\, \\epsilon)$ and\n$\\nu_2^b$: $(1, \\, \\epsilon^2)$.\n\n\\end{itemize}\n\n\n3. Consider the case $t^a = - t^b$, when separation in the layer $b$\ncompensates separation in $a$. In spite of this compensation,\nand in contrast to the constant density or adiabatic cases,\nthere is no complete overlap at the end of layer $b$\ndue to split of the eigenstates. Only WP of $A_{11}$ and $A_{22}$\ncomponents will overlap completely, where subscripts indicate eigenstates in which a given final component propagated in layer $a$ and $b$ ({\\it e.g.} $A_{11}$ is the amplitude of WP $\\nu_1^a \\rightarrow \\nu_1^b$). The component $A_{12}$\nwill shift forward by $t^a$ and $A_{21}$ backward by $t^a$ with respect\nto\noverlapping components,\nor vice versa, depending on the sign of $t^a$.\nIf $|t^a| > \\sigma_x$, only $A_{11}$ and $A_{22}$ interfere and\nthe interference term equals\n\\begin{equation}\n\\label{intdd}\n0.5 \\sin 2\\theta_a \\sin 2\\theta_b c_\\Delta^2 \\cos 2(\\phi^a + \\phi^b),\n\\end{equation}\nwhich again can be bigger than interference in\nthe beginning of the layer $a$. \\\\\n\n4. In the case $t^a \\approx 0$, there is no separation and loss\nof coherence in the layer $a$. The components $A_{11}$ and $A_{21}$\nas well as $A_{22}$ and $A_{12}$ will overlap, there is no\nsplit of eigenstates at the border $a \\rightarrow b$. \n\n\nThere is no unambiguous way of introducing the coherence length in the presence of density jump in the profile. Recall that the equivalence in the $E$- and $x$-representations \nis established at the level of averaging of the oscillation phase and separation of the WP. \nIn the case of a single jump, there are two different oscillation phases and two different delays due to splitting of eigenstates. The introduction of a single effective phase and effective delay is non-trivial. \nAs we will see in sect.~\\ref{parametric-oscillations}, this can be done for periodic structures with density jumps.\n\n\\subsection{Wave packets in the $x$- and $E$-representations}\n\nSplit of the eigenstates and delays of splitted components lead\nto substantial modifications of the shape factor\nof the total WPs, $\\psi_i(x,t)$, in the $x$-space. \nAt the same time, the Fourier transforms of $\\psi_i(x,t)$\ndetermine the neutrino energy spectrum. Let us show that despite substantial modifications\nof $\\psi_i(x,t)$, the energy spectrum remains unchanged, as it should be \\cite{Stodolsky:1998tc}.\n\nSuppose $f(E)$ is the Fourier transform of $f(t)$,\nthen in the $E$-representation the initial state (\\ref{init}) is\n$$\n\\nu(x=0,E) = f(E)(c_a' \\nu_1^a + s_a' \\nu_2^a).\n$$\nThe energy spectrum of this state equals\n\\begin{equation}\nF(E) = |f(E) c_a'|^2 + |f(E) s_a'|^2 = |f(E)|^2.\n\\label{eq:spectr1}\n\\end{equation}\n\nConsider the simplest element of the profile\nwhich leads to modification of the shape factor:\nthe layer $a$ with constant density and\ndensity jump at the end\n(general case is just repetition of this element). State (\\ref{border-ba}), at the beginning of layer $b$ ($t^b_i = 0$ and $\\phi^b_i = 0$) in terms of $\\psi_i(x=L_a,t)$ is\n\\begin{equation}\n\\nu(L_a,t) = \\psi_1(L_a,t) \\nu_1^b + \\psi_2(L_a,t) \\nu_2^b,\n\\label{total-WPs}\n\\end{equation}\nwhere the total WPs equal\n\\begin{eqnarray} \\label{total-WPs-2}\n\\psi_1(L_a,t) & = &\n[c_a' c_\\Delta f(t) - s_a' s_\\Delta f(t - t^a) e^{2i \\phi^a}],\n\\nonumber\\\\\n\\psi_2(L_a,t) & = &\n[c_a' s_\\Delta f(t) + s_a' c_\\Delta f(t - t^a) e^{2i \\phi^a}].\n\\label{eq:totwp12}\n\\end{eqnarray}\nHere we made time shift by $t_1^a$ and put out common phase\n$\\phi^a_1$.\nIn the beginning, $\\psi_i$ are the elementary WPs, $f(t) \\nu_i^a$, as in (\\ref{init}), but along the propagation their widths increase and their forms change.\n\nIn $E$-space, the state just after crossing the density jump, $\\nu(L_a)$, in Eq.~(\\ref{total-WPs}), is:\n\\begin{equation}\n\\nu(L_a,E) = \\psi_1(L_a,E) \\nu_1^b + \\psi_2(L_a,E) \\nu_2^b,\n\\label{eq:statexb}\n\\end{equation}\nThe Fourier transform of\n$f(t - t^a)$ is $f(E) e^{i E t^a}$, so that\n\\begin{eqnarray}\n\\psi_1(L_a,E) & = &\n[c_a' c_\\Delta - s_a' s_\\Delta e^{- i E t^a + 2i \\phi^a}] f(E) ,\n\\nonumber\\\\\n\\psi_2(L_a,E) & = &\n[c_a' s_\\Delta + s_a' c_\\Delta e^{- iEt^a + 2i \\phi^a}] f(E).\n\\nonumber\n\\end{eqnarray}\nUsing these expressions, we find the energy spectrum\n$$\nF(E) = |\\psi_1(L_a,E)|^2 + |\\psi_2(L_a,E)|^2 = |f(E)|^2,\n$$\nwhich coincides with (\\ref{eq:spectr1}).\n\n\n\\section{Coherence in the castle-wall profile}\n\\label{parametric-oscillations}\n\nOne can introduce the coherence length in the case of a periodic structure with a sharp density change. The castle-wall (CW) profile \nis multiple repetitions of the two layer structure considered sect.~\\ref{coherence and adiabaticity violation}. \nThis is an explicitly solvable example \nof the periodic profile, which can give an idea about effects in \nprofiles with sine (or cosine) type of dependence on distance \\cite{Akhmedov:1988kd,Akhmedov:1999ty,Akhmedov:1999va}. \n\n\\subsection{Parametric oscillations in the energy-momentum space} \\label{parametric1}\n\nThe oscillation probability after crossing $n$ periods can be obtained from \nthe probability for a single period computed above. Indeed, the amplitude $A_{ee}$ (\\ref{eeamp12}),(\\ref{a1a2}) can be written as \n\\begin{equation} \\nonumber\n\\label{eeprob}\nA_{ee} = R + i I_3, \n\\end{equation}\nwhere the real and imaginary parts of the amplitude equal respectively, \nsee \\cite{Akhmedov:1999ty,Akhmedov:1999va}, \n\\begin{equation} \n\\label{R}\nR=c_{a} c_{b} - s_{a} s_{b} \\cos \\left(2 \\theta_a - 2 \\theta_b\\right),\n\\end{equation} \nand \n\\begin{equation} \\nonumber\n\\label{third-component}\nI_3 = - (s_a c_b \\cos 2 \\theta_a + s_b c_a \\cos 2 \\theta_b).\n\\end{equation}\nThen the probability is given by\n\\begin{equation} \\nonumber\n\\label{eeprob}\nP_{ee} = R^2 + I_3^2. \n\\end{equation}\n\nSimilarly to $A_{ee}$ one can find the amplitudes \n$A_{e \\mu}$, $A_{\\mu e}$ and $A_{\\mu \\mu}$ which compose \nthe evolution matrix $U_L$ over one period. The matrix \nallows to reconstruct the Hamiltonian integrated over the period: \n$i H_{CW} L = I - U_L$. Diagonalization of this Hamiltonian \ngives the eigenstates, {\\it i.e.} mixing and \ndifference of eigenvalues \nwhich allow to write the transition probability \nafter passing $n$ periods of size $L$, $P = |U(nL)_{\\alpha \\beta}|^2$, as \n\\begin{equation} \n\\label{P-CW}\nP(\\nu_{\\alpha} \\rightarrow \\nu_{\\beta},nL) = \n\\left(1-\\frac{I_3^2}{1 - R^2} \\right) \\sin^2 \\left(n \\xi \\right).\n\\end{equation}\nHere $\\xi$ is the effective phase acquired over period:\n\\begin{equation}\\nonumber\n\\xi \\equiv \\arccos R. \n\\end{equation}\nThe factor in front of sine gives the depth of the parametric \noscillations. The depth is maximal at $I_3 = 0$, or explicitly\n\\begin{equation} \n\\label{condition1}\n s_a c_b \\cos 2 \\theta_a +c_a s_b \\cos 2 \\theta_b = 0, \n\\end{equation}\nwhich is the condition of the parametric resonance. \n For arbitrary mixings the condition is satisfied \nfor values of phases\n\\begin{equation} \\nonumber\n\\label{condition2}\n\\phi^a = \\frac{\\pi}{2}+k \\pi \\hspace{0.5 cm} \\text{and} \n\\hspace{0.5 cm} \n\\phi^b = \\frac{\\pi}{2}+k' \\pi. \n\\end{equation}\nIn general, $I_3 = 0$ requires specific correlations between \nphases and mixings. The transition probability (\\ref{P-CW}) reaches maximum, $P = 1$, when \n\\begin{equation} \\nonumber\n\\label{condition3}\nn \\xi = \\frac{\\pi}{2} + k \\pi.\n\\end{equation}\n\nIn what follows for illustration of the results we will use \na castle-wall profile with $n = 5$ periods \nand the following parameters: \n\\begin{equation} \n\\label{eq:paramet}\nV_a = 5.3 \\times 10^{-4} \\, \\text{eV}^{2}, \\, \\, \\, \\, \\, L_a = 4 \\, {\\rm km}, \\, \\, \\, \nV_b = 2 \\times 10^{-5} \\, \\text{eV}^{2}, \\, \\, \\, L_b = 2 \\, {\\rm km}.\n\\end{equation}\nWe take the vacuum oscillation parameters $\\theta=8.5^{\\circ}$ and \n$\\Delta m^2=2.5 \\times 10^{-3}$ $\\text{eV}^2$. \n\nThe transition probability of parametric oscillations \nEq.~(\\ref{P-CW}), as function of energy, is shown in fig.~\\ref{resonance1}. The probability\nreaches maximum at $E \\sim 2.2$ (MSW resonance), $3.1$ and $7.7$ MeV, \nwhere condition (\\ref{condition1}) is met.\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=0.7\\linewidth]{new-prob.png}\n\\caption\nTransition probability of the $\\nu_e \\rightarrow \\nu_{\\mu}$ parametric oscillations as a function of neutrino energy for castle-wall profile with $5$ periods.\nParameters of the CW profile are given in Eq.~(\\ref{eq:paramet}).\nWe used $\\theta=8.5^{\\circ}$ and $\\Delta m^2=2.5 \\times 10^{-3}$ $\\text{eV}^2$.}\n\\label{resonance1}\n\\end{figure}\n\n\n\\subsection{Coherence length in a castle-wall profile}\n\\label{coherence-castle-wall}\n\nIn the energy representation we can repeat here the same procedure of determination of the \ncoherence length as in the case of constant density. \nAccording to Eq.~(\\ref{P-CW}), the phase of parametric \noscillations equals\n\\begin{equation} \\nonumber\n\\label{CW-phase}\n \\phi_n = n \\xi = n L \\frac{\\xi}{L} . \n\\end{equation}\nTherefore the effective difference of eigenvalues equals \n\\begin{equation} \\nonumber\n\\label{eff-eig}\n\\Delta H_{cw} = \\frac{\\xi}{L}. \n\\end{equation}\n\nWe introduce $n_{coh}$, the number of periods over which the coherence \nis maintained, so that $L_{coh}=n_{coh} L$. \nUsing Eq.~(\\ref{const-density}) \nfor the coherence length we find\n\\begin{equation} \\nonumber\n\\label{phase-var}\nn_{coh} = \\frac{\\pi}{\\sigma_E}\n\\left|\\frac{d \\xi}{d E} \\right|^{-1}. \n\\end{equation}\nAccording to (\\ref{const-density}) and (\\ref{R})\n\\begin{equation} \n\\label{D-xi}\n\\frac{d \\xi}{d E} = \n-\\frac{1}{\\sqrt{1-R^2}} \\frac{d R}{d E}, \n\\end{equation}\nand consequently, \n\\begin{equation} \n\\label{p-coh}\nn_{coh}(E)=\\frac{\\pi \\sqrt{1-R^2}}{2 \\sigma_E \\left| \n\\frac{d R}{d E} \\right|}.\n\\end{equation}\n\nThe condition for infinite coherence length, \n$L_{coh} \\propto n_{coh} \\rightarrow \\infty$, is \n\\begin{equation}\n\\label{divergency} \n\\left|\\frac{d \\xi}{d E} \\right| = 0, \\, \\, \\, \\, \n{\\rm or } \\, \\, \n\\frac{d R(E)}{d E} = 0, \\, \\, \\, (R \\neq 1).\n\\end{equation}\nSince $R(E)$ is an oscillatory function of $E$, and therefore has several \nmaxima and minima at certain energies $E_0^i$, the condition \n(\\ref{divergency}) is satisfied at these energies \n$E = E_0^i$.\n\nAs in the case of constant density, averaging of probability over the \nenergy intervals $\\sigma_E$ leads to the appearance \nof the decoherence factors \n$D_i(E)$, (\\ref{decoh}), centered at energies $E_0^i$. Let us find the widths \nof the peaks of $D_i(E)$ for fixed length of trajectory \n$ n L$. \nAs in constant density case, instead of averaging over \n$E$ we perform averaging over \nthe effective phase in the interval which corresponds to \n$2\\sigma_E$: \n\\begin{equation}\n\\label{eq:dphi-n}\n\\delta \\phi_n = 2\\sigma_E n \\frac{d \\xi}{d E}.\n\\end{equation}\nThe decoherence factor \n\\begin{equation}\\nonumber\n D(\\delta \\phi_n (E)) = \\frac{\\sin \\delta \\phi_n}{\\delta \\phi_n}\n\\end{equation}\nhas a width $\\Gamma$ determined by the condition in (\\ref{Gamma}).\nPlugging \n$\\delta \\phi_n$ from (\\ref{eq:dphi-n}) in the above equation we obtain\n\\begin{equation} \n\\label{half-max1}\n\\left| \\frac{d \\xi(E_0^i +\\Gamma)}{d E} \\right| \n= \\frac{1}{\\sigma_E n}.\n\\end{equation}\nFor narrow peaks, $\\Gamma \\ll E_0^i$, \nwe can expand the left hand side of Eq.~(\\ref{half-max1}) \naround $E_0$; and take \ninto account that \n$d \\xi (E_0)\/d E = 0$. Then Eq.~(\\ref{half-max1})\nreduces to \n\\begin{equation}\n\\frac{d^2 \\xi (E)}{d E^2}\\bigg{|}_{E=E_0^i} \n\\times \\Gamma = \\frac{1}{\\sigma_E n}.\n\\label{eq:condigamma}\n\\end{equation}\nDifferentiating $d\\xi\/dE$ in (\\ref{D-xi}) over $E$ we have \n\\begin{equation}\n\\label{eq:doubleder}\n\\frac{d^2 \\xi (E)}{d E^2}\\bigg{|}_{E=E_0^i} = \n-\\frac{1}{\\sqrt{1- R^2(E_0^i)}} \\frac{d^2 R(E_0^i)}{d E^2}.\n\\end{equation}\nThen insertion of this into (\\ref{eq:condigamma}) gives\n\\begin{equation} \\nonumber\n\\label{CW-peak}\n\\Gamma \\approx \n\\frac{\\sqrt{1-R^2(E_0^i)}}{ \\sigma_E n \\left| \n\\frac{d^2 R(E_0^i)}{d E^2} \\right|}. \n\\end{equation}\n\n\nIn the limits $E_0^i \\ll E_{a},E_{b}$ and $E_0^i \\gg E_{a},E_{b}$,\nwhere $E_{a}$ and $E_{b}$ are the MSW resonance energies, we have \n$\\theta_a \\approx \\theta_b$, and therefore \n\\begin{equation} \\nonumber\n\\label{approx-R}\nR \\approx \\cos (\\phi^a + \\phi^b). \n\\end{equation}\nSince $d\\xi\/dE \\propto d(\\phi^a + \\phi^b)\/dE = 0$ at $E = E_0^i$ \nthe equation (\\ref{eq:doubleder}) becomes \n\\begin{equation}\\nonumber\n\\frac{d^2 \\xi}{d E^2} \\bigg{|}_{E=E_0} \n\\approx \\frac{d^2 (\\phi^a + \\phi^b)}{d E^2}\\bigg{|}_{E=E_0},\n\\end{equation}\nand the width equals\n\\begin{equation} \\nonumber\n\\label{approx-gamma}\n\\Gamma_i \\approx \\frac{1}{\\sigma_E n \n\\frac{d^2 (\\phi^a + \\phi^b)}{d E^2} \\big{|}_{E=E_0^i} }.\n\\end{equation}\nThe widths $\\Gamma^i$ are narrower at lower energies \ndue to faster variations of the half-phases $\\phi^a$ \nand $\\phi^b$. Similarly to the case of constant \nmatter density, the ranges of weak averaging effect \nof transition probability are smaller at lower $E_0^i$. \n\n\\subsection{Infinite coherence and parametric resonance}\n\nThe energies of infinite coherence, $E_0^i$, \nare correlated to the energies of parametric resonances, \ndetermined by the condition $I_3=0$. This correlation is analogous to the one in Eq.~(\\ref{MSW}) for constant density.\nTo show this, we consider the expressions for $d R\/d E$ and $I_3$ \nin three different regions of energies. Starting with the explicit expression for $d R\/d E$,\n\\begin{multline} \\label{dR}\n\\frac{d R}{d E}=-s_a c_b \\frac{d \\phi^a}{d E}-c_a s_b \\frac{d \\phi^b}{d E} -\nc_a s_b \\cos(2\\theta_a-2 \\theta_b) \\frac{d \\phi^a}{d E}\\\\\n-s_a c_b \\cos(2\\theta_a-2 \\theta_b) \\frac{d \\phi^b}{d E} + \ns_a s_b \\sin(2\\theta_a-2 \\theta_b) \\frac{d (2\\theta_a-2 \\theta_b)}{d E}.\n\\end{multline}\nWe obtain the following\n\\begin{enumerate}\n\n\\item Far from the MSW resonances of both layers:\n$E \\ll E_{a}, \\, E_{b}$, \nwhere $\\theta_a \\approx \\theta_b \\approx \\theta$, or \n$E \\gg E_{a}, \\, E_{b}$ where \n$\\theta_a \\approx \\theta_b \\approx \\pi\/2$. \nIn both cases $\\cos (2\\theta_a - 2\\theta_b) \\approx 1$,\nand consequently, $R \\approx \\cos (\\phi^a + \\phi^b)$. Therefore \n\\begin{equation} \\nonumber\n \\frac{d R}{d E} = \n- \\sin (\\phi^a + \\phi^b)\\frac{d (\\phi^a + \\phi^b) }{d E} \n\\end{equation}\nOn the other hand \n\\begin{equation}\\nonumber\nI_3 \\approx \\sin (\\phi^a + \\phi^b) \\cos 2\\theta_a.\n\\end{equation}\nSo that, the condition $I_3 \\rightarrow 0$ requires \n$\\sin (\\phi^a + \\phi^b) = 0$ and consequently, \n$\\frac{d R}{d E} \\rightarrow 0$, \nwhich implies $n_{coh} \\rightarrow \\infty$.\n\n\\item In the MSW resonances: when $E$ coincides with one of the resonance \nenergies {\\it e.g.} $E \\approx E_{a}$, \nand is far enough from the other one, $E_{b}$, then\n$\\theta_a=\\pi\/4$, $\\theta_b = \\theta \\approx 0$ (small mixing) or $\\pi\/2$ and $d \\theta_b\/d E=0$. Because $E_{a}\\sim E_{0a}$, we can take in (\\ref{dR}) $d \\phi^a\/d E=0$ which becomes\n\\begin{equation} \\label{dR1}\n \\frac{d R}{d E}=(-c_a s_b \\pm s_a c_b \\cos 2\\theta_a) \\frac{d \\phi^b}{d E}\\pm 2s_a s_b \\sin 2\\theta_a \\frac{d \\theta_a}{d E}.\n\\end{equation}\nHere the upper sign is for $\\theta_b \\approx 0$ and the lower one $\\theta_b \\approx \\pi\/2$. We can rewrite (\\ref{dR1}) as\n\\begin{equation} \\label{dR2}\n \\frac{d R}{d E}=\\pm I_3 \\frac{d \\phi^b}{d E}\\pm 2s_a s_b \\sin 2\\theta_a \\frac{d \\theta_a}{d E}.\n\\end{equation}\nAssuming that $\\sigma_E d \\phi^b\/d E \\sim 1$ and $\\sigma_E d \\theta_a\/d E \\ll 1$, we find that the second term in (\\ref{dR2}) shifts the zero of $d R\/d E$ relative to the zero of $I_3$. Nevertheless, $E_0$ is close to $E_R$, and the closer they are, the smaller $d \\theta_a\/d E$ is relative to $d \\phi^b\/d E$.\n\n\\item Between the MSW resonances:\n$E_{a} < E< E_{b}$ or $E_{a} > E > E_{b}$. \nIf resonances are well separated, \nwe have $|\\theta_a - \\theta_b| \\approx \\frac{\\pi}{2}$ or \n$\\cos (2\\theta_a - 2\\theta_b) \\approx -1$. Therefore, \n\\begin{equation}\\nonumber\nR \\approx \\cos (\\phi^a - \\phi^b), \n\\end{equation}\nand \n\\begin{equation}\\nonumber\n \\frac{d R}{d E} = \n- \\sin (\\phi^a - \\phi^b)\\frac{d (\\phi^a - \\phi^b) }{d E}. \n \\end{equation}\nOn the other hand, \n\\begin{equation}\\nonumber\nI_3 \\approx -\\sin (\\phi^a - \\phi^b) \\cos 2 \\theta_1.\n\\end{equation}\nConsequently, $I_3 \\rightarrow 0$ implies \n$d R\/ d E \\rightarrow 0$, and \n$n_{coh} \\rightarrow \\infty$. \n\n\\end{enumerate}\n\nIn the left panel of fig.~\\ref{fig-p-coh} we show $d R\/d E$\nand $I_3$ as functions of neutrino energy.\n$I_3$ is null at $2.25$, $3.1$ and $ 7.7$ MeV, while the zeros of \n$d R\/d E$, $E_0^i$ are $2.3$, $3.3$ and $7.7$ MeV. \nThese coincidences show correlation of the parametric resonance and infinite coherence.\nThe MSW resonance energies are $E_{a}=2.25$ MeV \nand $E_{b}=62.7$ MeV, outside the range of the plot.\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=1\\linewidth]{joint-new.png}\n\\caption{Left panel: $I_3$ and $\\frac{d \\xi}{d E}$ as function \nof neutrino energy. Right panel: Dependence of the $n_{coh}$ in (\\ref{p-coh}) \non neutrino energy with $\\sigma_E\/E=0.1$ MeV. The parameters of CW profile are the same as in fig.~\\ref{resonance1}. \n}\n\\label{fig-p-coh}\n\\end{figure}\n\nOn the right panel of fig.~\\ref{fig-p-coh} we show dependence of $n_{coh}$ on $E$\nfor the CW profile with parameters (\\ref{eq:paramet}) \nand $\\sigma_E\/E=0.1$ MeV. $n_{coh}$ diverges at the energies $E_0^i$. Notice that the shape \nof the peak is similar to MSW case in fig.~\\ref{fig-E0}. Thus, at the parametric resonance energies \nthe averaging of oscillations is weak, and coherence is enhanced.\n\n\n\\subsection{Coherence in castle-wall \nprofile in the configuration space}\n\nIn the $x$-representation, a picture of evolution in the CW profile is the following. \nCrossing every border between the layers $a$ and $b$ \na given eigenstate of the layer $a$, $\\nu_i^a$, splits into two eigenstates of the layer $b$: $\\nu_i^a \\rightarrow \\nu_1^b, \\nu_2^b$. In turn, at the next border each eigenstate $\\nu_j^b$ splits into eigenstates \nof $a$ $\\nu_j^b \\rightarrow \\nu_i^a$,\n{\\it etc}. The amplitudes of the splits are determined by \nchange of the mixing angles in matter (\\ref{splitab}). \n\nAs in subsection~\\ref{parametric1}, \\ref{coherence-castle-wall} we consider that a flavor neutrino enters the layer $a$ of the first CW period and \na detector is placed at the border of $b$ layer of the last period. After crossing $n$ periods of the CW profile, \neach initially produced eigenstate $\\nu_i^a$ ($i = 1, 2$) splits \n$2^{2n -1}$ times, and therefore at a detector \nthere are $2^{2n}$ elementary components. \nThe eigenstate $\\nu_i^b$ ($i=1,2$), that arrives \nat a detector (last border), will be a composition \nof $2^{2n -1}$ such components. They will arrive at different moments and have different phases and amplitudes. \nWe will assume that these elementary WPs are short enough and do not spread. Therefore the oscillation phases (phase differences) are the same along the elementary WPs. Each elementary WP has its \"history\" of propagation determined by \ntype of eigenstate it showed up in each layer of the profile, \n{\\it e.g.} $\\nu_1^a \\rightarrow \\nu_2^b \\rightarrow \\nu_2^a \\rightarrow \\nu_1^b \\rightarrow \\nu_1^b \\rightarrow ... \\rightarrow \\nu_2^b$. \n\nThe elementary WPs compose the total WP of eigenstates $\\psi_i$.\nSplits of eigenstates and delays lead to spread\nof total WP after crossing the CW profile.\nThis spread affects the level of overlap\nand coherence.\nIf a detector is sensitive to the flavor\n$\\nu_f = c_b' \\nu^b_1 + s_b' \\nu^b_2$, the detected signal is determined by \n\\begin{equation}\n\\label{eq:sign}\n\\int dt \\left| c_b' \\psi^b_1 (t) + s_b' \\psi^b_2 (t) \\right|^2. \n\\end{equation}\nwhere $\\psi_i^b$ are the total WP of the eigenstates\n$\\nu_i^b$ in a layer $b$ at a detector.\nThe total WP, $\\psi^b_j (t)$, can be found summing up all $2^{2n-1}$ elementary WPs at the detector.\n\n\nA given elementary WP at the detector can be characterized by \n$k_a$ and $k_b$ - the numbers of $a$ and $b$ layers it propagates through as \nthe second eigenstate, {\\it i.e.} $\\nu_2^m$, $m=a,b$. \nCorrespondingly, the numbers of layers \nit crossed as $\\nu_{1}^m$ are $(n - k_a)$ and $(n - k_b)$. Using the same notation for phases, $\\phi_i^{a}$ and $\\phi_i^{b}$ $(i = 1,2)$, as in sect.~\\ref{coherence and adiabaticity violation} $(i = 1,2)$ we can write \nthe total phase of a given elementary packet \nat a detector: \n\\begin{equation}\n\\label{eq:totph}\n\\phi (k_a, k_b) = \\phi_1^{a} (n - k_a) + \\phi_2^{a} k_a \n+ \\phi_1^{b} (n - k_b) + \\phi^{b}_2 k_b. \n\\end{equation}\nThe phase difference between the elementary WPs characterized by \n$k_a, k_b$ and $k_a', k_b'$ equals\n\\begin{equation}\n\\label{eq:totphdif}\n\\phi (k_a, k_b, k_a', k_b') = \n\\phi^{a} (k_a - k_a') + \\phi^{b} (k_b - k_b'), \n\\end{equation}\nwhere we used definitions in (\\ref{phase-redefinition}).\n\nSimilarly one can find the relative time delays of arrival of \nthe packets characterized by $k_a, k_b$ and $k_a', k_b'$ at a detector using the definitions (\\ref{time-redefinition}): \n\\begin{equation}\n\\label{eq:totdeldif}\nt (k_a, k_b, k_a', k_b') = t^a (k_a - k_a') + t^b (k_b - k_b'). \n\\end{equation}\nAs for the case of a single layer (\\ref{eq:phase-delay}), there are the relations between the \ntotal delays (\\ref{eq:totdeldif}) and total oscillation phases (\\ref{eq:totphdif}):\n\\begin{equation}\n\\label{eq:correltp}\nt (k_a - k_a', k_b- k_b') = \n\\frac{d \\phi (k_a - k_a', k_b - k_b')}{d E}, \n\\end{equation}\nand \n\\begin{equation}\n\\label{eq:correltp}\nt (k_a - k_a', k_b- k_b') = \ng(E, V) \\phi (k_a - k_a', k_b - k_b'). \n\\end{equation}\n\nSuppose, for definiteness, that $t^a$ and $t^b$ are positive, \nthen maximal relative delay (separation) $t^{max}$ corresponds to \n$|k_a - k_a'| = |k_b - k_b'| = n$: \n$$\nt^{max} = n(t^a + t^b).\n$$\nThis is the time difference of arrival of the fastest \nand the slowest extreme elementary WP. \n\n$t^{max}$ separation determines the spread of whole WPs. The space between the fastest and the slowest WPs is \nfilled in by non-extreme WPs. \nThe extreme WPs are coherent if \n\\begin{equation}\n\\label{eq:nxs-coh}\nn (t^a + t^b) \\leq \\sigma_x,\n\\end{equation}\nwhich gives the coherence number of periods (length) \n\\begin{equation} \n\\label{part-1}\nn_{coh} \\leq \\frac{\\sigma_x}{t^a + t^b}.\n\\end{equation}\nUnder this condition, all other (intermediate) WP-components \nare also coherent (overlap). \nThe condition (\\ref{part-1}) is sufficient but not necessary, since \nthe coherence broken for extreme packets \nstill can hold for intermediate packets. \n\nParameters $k_a, k_b$ uniquely determine the phase and the delay of a given component. \nNumber of components with a given \n$k_a$ and $k_b$ equals\n$$\n ^n_{k_a}C \\times~ ^n_{k_b}C, \n$$\nwhere $^n _{p}C$ is number of combinations of $p$ elements from $n$ elements. Apart from \n$k_a$ and $k_b$\nthe amplitudes of individual components \nare determined by the product of various mixing parameters: the initial and final flavor mixing, {\\it e.g.} $c_a'c_b'$ for $\\nu_e \\rightarrow \\nu_e$ channel and $s_\\Delta$, $c_\\Delta$ determined by change \nof mixing at the borders between the \nlayers $\\Delta$:\n$$ \nA(n,r,h) = (-1)^h c'_a c'_b s_\\Delta^r c_\\Delta^{2n - 1 - r}. \n$$\nFactor $s_\\Delta$ originates from each border at which the eigenstate number \nchanges: $\\nu_{1}^a \\rightarrow \\nu_{2}^b$, \\textit{etc.}, while $c_\\Delta$ \nappears from the borders without change of eigenstate: \n$\\nu_{i}^a \\rightarrow \\nu_{i}^b$. $r$ is the number of borders where eigenstate changes; it is \ndetermined by the number of merging blocks: sequences of layers without change \nof eigenstate. $h$ is given by the number of blocks with \neven number of sequential layers in which WP propagates as\nthe second eigenstate. \n\nFor large number of periods there are many elementary packets \nwith the same values of \n$k_a, k_b, r, h$. We call this number the multiplicity, $M(k_a, k_b, r, h)$. \nTherefore the total amplitude of all WPs with a given \n$k_a, k_b$ equals\n\\begin{equation} \\label{amp-kakb}\nA(k_a, k_b) = \\sum_{r,h} A(n,r,h) M (k_a, k_b,r,h).\n\\end{equation}\n\n\nAs for the one period case, here the shape of the total WP \ndepends on mixing \nand multiplicities. \n\n\nFor $k_a = k_a'$, $k_b = k_b'$ there is no shift but also \nthe phase difference is zero. \nFor $k_a - k_a' = 1$, $k_b - k_b' = 0$, the phase difference \nis $\\phi^a$, while for $k_a - k_a' = 0$, $k_b - k_b' = 1$ the phase difference is $\\phi^b$, {\\it etc}. \nIn general, one finds the components with phase differences varying from $0$ to $ n(\\phi^a + \\phi^b)$. \n\n\\subsection{Resummation}\nWe can obtain the compact expression for the total WPs summing up effects after each crossing of the CW period representing them as the interference of the two (total)\nWPs of $\\nu_1^a$ and $\\nu_2^a$.\n\nWe consider first the summation in the basis of eigenstates of the layer $a$:\n$\\nu^a = (\\nu^a_1, ~\\nu^a_2)$.\nLet $\\psi^{k - 1} = (\\psi_1^{k - 1},~ \\psi_2^{k - 1})^T$ be the vector describing\nthe state of system after crossing\n$k - 1$ periods of the CW profile. $\\psi_i^{k - 1}$ is result of resummation\nof the elementary WP of $\\nu_i^a$\nin the layer $a$ in the beginning of $k$-period.\nThen after crossing of the $k$th period, the state becomes\n\\begin{equation}\n\\psi^{k } = U_L^a \\psi^{k - 1},\n\\label{eq:state-k}\n\\end{equation}\nwhere the evolution matrix over one period \n\\begin{equation}\nU_L^a = U_\\Delta D^b U_\\Delta^{\\dagger} D^a .\n\\label{eq:evmat}\n\\end{equation}\nHere $U_\\Delta$ is the matrix of the mixing change at crossing the border between $a$ and $b$:\n\\begin{equation}\nU_\\Delta = U^{a \\dagger}U^b =\n\\left(\\begin{array}{cc}\nc_\\Delta & s_\\Delta \\\\\n- s_\\Delta & c_\\Delta\n\\end{array}\\right),\n\\label{eq:udelta}\n\\end{equation}\nand $U^a$, $U^b$ are the flavor mixing matrices in the\n$a$ and $b$ layers.\n$D^a $ is the evolution matrix in the layer $a$ which can be written as\n\\begin{equation}\nD^a =\n\\left(\\begin{array}{cc}\nS(t^a_1) e^{i2 \\phi_1^a} & 0 \\\\\n0 & S(t^a_2) e^{i2 \\phi_2^a}.\n\\end{array}\\right)\n\\label{eq:dop-a}\n\\end{equation}\nHere $S(t_i^a)$ is the time-shift operator acting on the shape factor of the WP\nin such a way that\n$$\nS(t_i^a) f_i (t) = f_i (t - t_i^a) , ~~~~ S(t_i^a) S(t_j^b) = S(t_j^b) S(t_i^a) = S(t_i^a + t_j^b) ~~~\n(i, j = a, b).\n$$\nWe can also require that $S(t)^{\\dagger} = S(-t)$, so that $S$ is the unitary: $S(t)^{\\dagger} S(t) = I$.\n\nSimilarly one can introduce the evolution matrix for the layer $b$. Inserting $U_\\Delta$ and $D^m_i$ ($m = a, b$) from\n(\\ref{eq:udelta}) and (\\ref{eq:dop-a}) into (\\ref{eq:evmat}) we obtain\n\\begin{equation}\nU_L^a =\n\\left(\\begin{array}{cc}\nc_\\Delta^2 e^{-i\\phi} + s_\\Delta^2 S(t^b) e^{-i\\phi'} & \n  s_\\Delta c_\\Delta [S(t^b + t^a) e^{i\\phi} - S(t^a) e^{i\\phi'} ] \\\\\ns_\\Delta c_\\Delta [S(t^b) e^{- i\\phi'} - e^{-i\\phi} ] & \n  s_\\Delta^2 S(t^a) e^{i\\phi' } + c_\\Delta^2 S(t^b + t^a) e^{ i\\phi}\n\\end{array}\\right),\n\\label{eq:evmat3}\n\\end{equation}\nwhere we subtracted the commom phase factor $\\exp \\{i(\\phi_2^b + \\phi_2^a + \\phi_1^b + \\phi_1^a) \\}$ and applied (\\ref{phase-redefinition}) and (\\ref{phase-redefinition-2}), \nas well as performed the time shift by \n$t_1^b + t_1^a$ and used (\\ref{time-redefinition}). Using (\\ref{eq:evmat3}), we can recover results in sect.~\\ref{coherence and adiabaticity violation} for one period.\n\n\nIf all $S = 1$, we obtain the matrix which leads to the parametric oscillations as in \\cite{Akhmedov:1988kd,Akhmedov:1998ui,Akhmedov:1999ty}. \n\n\nThe evolution matrix after $n$ periods equals\n\\begin{equation}\nU_{nL}^a = (U_L^a)^n.\n\\label{eq:evmatnn}\n\\end{equation}\nAccording to (\\ref{eq:state-k}) the total eigenstate is\n\\begin{equation}\n\\psi^n (t) = (U^a_L)^n f(t) \\psi^0,\n\\label{eq:eigen12}\n\\end{equation}\nwhere $f(t)$ is the initial shape factor and\n$\\psi^0$ are the admixtures of eigenstates\n$\\nu_i^a$ in the initial moment of time.\nThe components of $\\psi^n$ have the form\n\\begin{equation} \\label{sum-kakb}\n\\psi^n= \\sum_{k_a,k_b}^n A(k_a,k_b) f(t - k_a t^a - k_b t^b) e^{i (k_a \\phi^a + k_b \\phi^b) } \\psi^0.\n\\end{equation}\nNotice from (\\ref{sum-kakb}) that oscillations in energy\nare induced not only by interference between the total WPs of $\\psi^n_1$ and $\\psi^n_2$ but also by interference of components within amplitudes\n$\\psi_1^n$ and $\\psi_2^n$ with amplitudes (\\ref{amp-kakb}).\n\nThe amplitude of the $\\nu_e \\rightarrow \\nu_e$ transition equals\n\\begin{equation}\nA_{ee} (t) = \\nu_e^T (U_L^a)^n f(t) \\nu_e, ~~~~~~~~ \\nu_e^T = (c_a,~ s_a).\n\\label{eq:ampl}\n\\end{equation}\nThis reproduces the amplitude for two layer case of sect.~\\ref{parametric1}. Then the probability can be found as\n\\begin{equation}\nP_{ee} = \\int dt |\\nu_e^T U_{nL}^a f(t) \\nu_e|^2.\n\\label{eq:probab}\n\\end{equation}\n\nVarious results can be obtained using the general form of amplitude and probability. \nOne can consider evolution in the flavor basis, which corresponds to the derivation of\nthe parametric oscillation probability in \\cite{Akhmedov:1988kd,Akhmedov:1998ui,Akhmedov:1999ty}.\nNow the evolution matrix equals\n\\begin{equation}\nU_L^f = U^b D^b U_\\Delta^{\\dagger} D^a U^{a \\dagger} = (U^b D^b U^{b \\dagger})(U^a D^a U^{a \\dagger}),\n\\label{eq:evmatf}\n\\end{equation}\nwhere in the last step the eigenstates of the layer $b$ are projected onto the flavor basis.\nThe matrices in the $a$ eigenstate basis (\\ref{eq:evmat}) and in the flavor basis (\\ref{eq:evmatf})\nare related as\n\\begin{equation}\nU_L^a = U^{a \\dagger} U_L^f U^{a}.\n\\label{eq:evmatfa}\n\\end{equation}\nThe matrices after $n$ layers:\n\\begin{equation}\n(U_L^f)^n = U^{a} (U_L^a )^n U^{a \\dagger}.\n\\label{eq:evmatfan}\n\\end{equation}\n\n\nWe can determine the effective Hamiltonian over one period using the evolution matrix over one period:\n$U_L = I - i H L$. Then diagonalizing the Hamiltonian gives the effective mixing and the eigenvalues,\nwhich, in turn, give the depth of the parametric oscillations and the phase.\nClearly, such a Hamiltonian depends on mixing $\\Delta$ and the phases of individual layers\n$\\phi^a, \\phi^b$. \n\n\n\n\n\n\n\\subsection{Effective group velocities and infinite coherence in CW\nprofile}\n\nIn general, the condition of infinite coherence can be formulated\nas an equality of the effective\ngroup velocities of the wave packets:\n\\begin{equation} \\label{equal-eff}\nv_1^{eff} = v_2^{eff}\n\\end{equation}\nor $\\Delta v^{eff} = 0$.\nDetermination $v_i^{eff}$ depends\non properties of the density profile and turns out to be non-trivial for\nthe CW profile.\n\nRecall that in the case of constant density $v_i^{eff}$ are well defined, see (\\ref{infcoh-conf}) and related discussion.\nSince the shape factor does not change, the group velocity\nis the velocity of any fixed point of shape factor, {\\it e.g.}, maximum.\nThe difference of group velocities $\\Delta v =\n\\Delta v (E, V, \\theta, \\Delta m^2)$\ndoes not depend on the oscillation phase.\nThe velocities are constant and do not change in the course of\npropagation.\n\n\nIf density changes along the neutrino trajectory,\n$v_i^{eff}$ and $\\Delta v^{eff}$ do depend on distance.\nIn the adiabatic periodically varying density\n$v_i^{eff}$ can be introduced as the velocities averaged over period:\n$$\nv_i^{eff} = L^{-1} \\int_0^L dx \\frac{d H_i}{dE}\\bigg|_{\\bar{E}} .\n$$\nThen the condition of infinite coherence is\n$$\n\\Delta v^{eff} = L^{-1}\\int_0^L dx \\frac{d \\Delta H\n}{dE}\\bigg|_{\\bar{E}} = 0.\n$$\nIt corresponds to a situation when $\\Delta v^{eff}(x) > 0$ in one part of the\nperiod and $\\Delta v^{eff}(x) < 0$ in another part (\\ref{L_0}).\n\nThe situation is much more complicated if adiabaticity is broken as in the case of a single jump between two layers, discussed in sect.~\\ref{coherence and adiabaticity violation}.\nIn the second layer, the WP has two components,\nand it is non-trivial to identify the point of the WP to which the group velocity should\nbe\nascribed.\nFurthermore, just after crossing the jump, certain parts of the WP will\nhave opposite phases and therefore interfere destructively,\nso that the interference is the same as before crossing.\nThis also affects shape factors (\\ref{total-WPs}) and (\\ref{total-WPs-2}). In this case, the infinite\ncoherence has no meaning apart from specific cases which are reduced\nto infinite coherence in a single layer (sect.~\\ref{coherence and adiabaticity violation}).\n\n\n\nInfinite coherence (infinite number of periods)\nhas meaning in the castle-wall profile for the\n(long-range) parametric oscillations. The effective group velocities\nshould be defined for a single\nperiod of CW profile similarly to the periodic profile\nwith adiabatic density change considered above. Furthermore,\none should take into\naccount (similarly to single jump case) that (\\ref{sum-kakb})\n\n\\begin{itemize}\n\n\\item Shape factor of the total WPs changes at the crossings,\ndue to splitting and delays of the components;\n\n\\item Oscillation phase is different in different parts of the same\ntotal WP because of different histories of the elementary components.\n\n\\end{itemize}\n\nAs a result, expressions for $v_i^{eff}$ and condition for\ninfinite coherence should depend not only on phases acquired in the\nlayers $a$ and $b$ but also on the mixing in both layers (\\ref{eq:evmat3}).\n\nIndeed we can obtain the condition in the $x$-representation rewriting condition\n(\\ref{divergency}) for $n_{coh} \\rightarrow \\infty$ in terms of\ndelays $t^a$ and $t^b$ we find:\n\\begin{multline}\n\\label{x-space}\n [s_a c_b +c_a s_b \\cos 2\\Delta] t_a +[c_a s_b+s_ac_b\n\\cos 2\\Delta]t_b =\\\\\n=s_a s_b \\sin 2\\Delta \\frac{1}{E}\n\\left(\\frac{V_{a} \\sin 2 \\theta_{a}}{\\Delta H_{a}}-\\frac{V_{b}\n\\sin 2 \\theta_{b}}{\\Delta H_{b}}\\right),\n\\end{multline}\nwhere we used\n$$\n\\frac{d 2\\theta_{m}}{dE}=\\frac{V_{m} \\sin 2 \\theta_{m}}{E \\Delta H_{m}},\n$$\n$m=a,b$. From (\\ref{x-space}),\nwe see that infinite coherence for parametric oscillations\nis unrelated to overlaps of the elementary WPs at the\nend of each period; this is the major difference from the cases\nwhere adiabaticity is conserved.\n\nThere is no simple interpretation of infinite coherence for parametric oscillations in terms of characteristics of elementary WP. In the\nTable~\\ref{table}, for three different $E_0$, we\nshow the delays of elementary WPs as well as phases and mixings in each layer of the CW profile with parameters (\\ref{eq:paramet}). No simple correlation is observed.\n\\begin{table}\n\\begin{center} \n\\begin{tabular}{ |p{1.8cm}||p{2cm}|p{2cm}|p{1.4cm}|p{1.4cm}| p{1.4cm}|p{1.4cm}| }\n\n\n \\hline\n $E_0$ (MeV) & $t_a$ ($\\text{MeV}^{-1}$) & $t_b$ ($\\text{MeV}^{-1}$)&\n$\\phi^a$(rad) & $\\phi^b$(rad) & $\\theta_a$(rad) & $\\theta_b$(rad) \\\\\n \\hline\n $2.3$ & $-0.98$ & $-2.4$ & $1.6$ & $2.7$ &\n$0.83$ & $0.15$ \\\\\n $3.3$ & $1.5$ & $-1.2$ & $2.1$ & $1.8$ &\n$1.3$ & $0.156$ \\\\\n $7.7$ & $0.4$ & $-0.21$ & $3.87$ & $0.73$ &\n$1.5$ & $0.2$ \\\\\n \\hline\n\\end{tabular}\n\\caption{\\label{table}Energies of enhanced coherence length, $E_0$, delays, phases and mixing angles in each of the layers $a$ and $b$ of a CW profile with parameters (\\ref{eq:paramet}).}\n\\end{center}\n\\end{table}\n\\section{Applications to supernova neutrino evolution}\n\\label{applications}\n\nThe issues of coherence in matter are of great relevance for oscillations of supernova neutrinos (SN). \nThe reasons are enormous distances from the production points to a detector at the Earth, \ncomplicated oscillation and conversion phenomena inside a star, non-trivial profiles of medium potentials. \nHere we briefly describe some effects, while a detailed study will be presented elsewhere \\cite{yago:next}. \n\nSN neutrinos are produced at densities $\\rho \\sim 10^{12}$ $\\text{g\/cm}^3$ and have very short \nwave packets $\\sigma_x \\sim 10^{-11}$ cm \\cite{Kersten:2013fba,Kersten:2015kio,Akhmedov:2017mcc}. Propagating to the surface of a star they can undergo \nthe collective oscillations at $R < 100$ km \\cite{Duan:2010bg,Mirizzi:2015eza,Chakraborty:2016lct,Tamborra:2020cul} and then \nthe resonance flavor conversion at $R > 1000$ km \\cite{Wolfenstein:1979ni,Mikheev:1986if,Dighe:1999bi,Lunardini:2003eh,Tomas:2004gr,Dasgupta:2005wn}. \nThe collective effects are supposed to be active at certain \nphases of a neutrino burst \\cite{Mirizzi:2015eza}. \n\nLet us first assume that collective oscillations are absent due to damping effects \\cite{Raffelt:2010za} or \ndue to lack of conditions for these oscillations at certain time intervals. \nIn this case, the standard resonance conversion picture is realized: \nin the production region, the mixing is strongly suppressed so that the flavor states coincide with the eigenstates of propagation in matter. The coincidence depends on the type of mass hierarchy. \nExcept for special situations (time periods and locations), the adiabaticity condition is well satisfied, \nand therefore the adiabatic transformations occur when neutrinos travel to the surface: $\\nu_i^m \\rightarrow \\nu_i$\n$(i = 1, 2, 3)$. The adiabaticity can be broken in fronts of shock waves which affects the described picture. \n\n\nIf the initial fluxes of $\\nu_\\mu$ and $\\nu_\\tau$ are equal, \nthe results of conversion depend only on the $\\nu_e - \\nu_e$ survival probability, $p$, in the neutrino \nchannel and on the $\\bar{\\nu}_e - \\bar{\\nu}_e$ probability, $\\bar{p}$, in the antineutrino channel. \nFor definiteness we will consider the $\\Bar{\\nu}_e$ channel. \nIn this case, the flux of $\\Bar{\\nu}_e$ at the surface of the star, and consequently, the Earth equals \n\\begin{equation}\n F_{\\Bar{\\nu}_e}=\\Bar{p} F_{\\Bar{\\nu}_e}^0 + (1-\\Bar{p}) F_{\\Bar{\\nu}_x}^0, \n\\label{eq:eflux}\n\\end{equation}\nwhere $F^0_{\\Bar{\\nu}_e}$ and $F^0_{\\Bar{\\nu}_x}$ are the initial fluxes of $\\Bar{\\nu}_e$ and $\\Bar{\\nu}_x$ \n(mixture of $\\Bar{\\nu}_\\mu$ and $\\Bar{\\nu}_\\tau$). \nThe second term in (\\ref{eq:eflux}) corresponds to transitions \n $\\Bar{\\nu}_\\mu, \\Bar{\\nu}_\\tau \\rightarrow \\Bar{\\nu}_e$. Derivation of the equation (\\ref{eq:eflux}) assumes that \ninitially produced $\\nu_i^m$ (which coincide with the flavor states) are incoherent and evolve independently. \nTherefore signals produced by $\\nu_i^m$ or states, to which $\\nu_i^m$ evolve, \nsum up incoherently. \n\nFor $R \\ll 1000$ km, the matter density is much bigger than \nthe density of the H-resonance associated with $\\Delta m^2_{13}$. \nIn this matter dominated range, according to Eq.~(\\ref{matter-domination}) in \nsect.~\\ref{coherence-lengths} the coherence length $L_{coh}^m$ is function of the vacuum parameters \n(see also fig.~\\ref{fig-E0}): \n\\begin{equation} \\nonumber\nL_{coh}^m = \\frac{L_\\nu}{\\cos 2 \\theta}\\frac{E}{2\\sigma_E} = 1100~ {\\rm km} \n\\left(\\frac{E}{ 20 {\\rm MeV }}\\right) \\left(\\frac{E}{10 \\sigma_E }\\right).\n\\label{eq:cohinsn}\n\\end{equation}\nTherefore the WP of eigenstates will separate before reaching the H-resonance even if they \noverlapped at the production. \nNotice that with an increase on $E$, the length $L_{coh}^m$ increases, but the resonance shifts to outer layers. Still, some partial coherence may exist. \n\n\nFor definiteness we assume the inverted mass hierarchy. Then according to the level crossing scheme \nat the production $\\bar{\\nu}_3^m \\approx \\bar{\\nu}_e$, $\\bar{\\nu}_1^m \\approx \\bar{\\nu}_x$ \nand $\\bar{\\nu}_2^m = \\bar{\\nu}_x'$. \nFurthermore, antineutrinos cross only H-resonance. \nThe dynamics is related to $\\bar{\\nu}_3^m - \\bar{\\nu}_1^m$ subsystem, while \n$\\bar{\\nu}_2^m$ essentially decouples appearing as a spectator. Therefore the $\\bar{\\nu}_e$ survival probability can be written as \n\\begin{equation} \\nonumber\n\\label{survival}\n\\Bar{p}=|U_{e1}|^2 P_{31} + \ns_{13}^2 (1- P_{31}), \n\\end{equation}\nwhere $P_{31} \\equiv P(\\Bar{\\nu}_{3}^m \\rightarrow \\Bar{\\nu}_{1}^m)$ is the \n$\\Bar{\\nu}_{3}^m \\rightarrow \\Bar{\\nu}_{1}^m$ transition probability.\n\nLet us consider the effects of the coherence loss \nand coherence enhancement under different circumstances. \n\n\n\n1. In the completely adiabatic case, $\\bar{\\nu}_3^m (\\approx \\bar{\\nu}_e)$ evolves to $\\bar{\\nu}_3$, so that \n$P(\\Bar{\\nu}_{3}^m \\rightarrow \\Bar{\\nu}_{1}^m) = 0$ and $\\bar{p} = |U_{e3}|^2 = s_{13}^2$. \nIn central parts of a star $\\bar{\\nu}_1^m$ has larger group velocity than $\\bar{\\nu}_3^m$. Below \nthe $H$-resonance, \ninversely, $\\bar{\\nu}_3^m$ moves faster and $\\bar{\\nu}_3$ will arrive at the Earth first. \n\n\n2. Adiabaticity is strongly broken in a shock wave front which can be considered as \nthe instantaneous density jump (see sect.~\\ref{coherence and adiabaticity violation}). \nWe will assume that before the jump (layer $a$) and after the jump (layer $b$) neutrinos propagate adiabatically. \nCrossing the jump the eigenstates $\\bar{\\nu}_1^a$ and $\\bar{\\nu}_3^a$ split into eigenstates $\\bar{\\nu}_j^b$. \nTo find $\\bar{p}$ we consider evolution of $\\bar{\\nu}_3^a$ which becomes \n$c_\\Delta \\bar{\\nu}_3^b - s_\\Delta \\bar{\\nu}_1^b$ after crossing the jump. Then its components $\\bar{\\nu}_j^b$\nloose coherence and evolve to $\\bar{\\nu}_j$ at the surface of a star. \nConsequently, we find \n\\begin{equation}\\nonumber\n\\Bar{p} = c_\\Delta^2 s_{13}^2 + s_\\Delta^2 |U_{e1}|^2. \n\\label{eq:padviol}\n\\end{equation}\nStrong adiabaticity violation effect on $\\Bar{p}$ is realized if the change of mixing in matter, $\\Delta$, is large. The latter occurs when the jump appears in the resonance layer, \nand the size of the jump is larger than the width of the resonance layer: \n$\\Delta \\rho > \\rho_R \\tan 2\\theta_{13} = 0.3 \\rho_R$. That is, even a small jump can produce a strong effect. \n\n\nAs we discussed before, the jump regenerates oscillations, which then disappear, \nand $\\Bar{p}$ does not depend on the oscillation phase. Single shock front (jump) could \nlead to phase (coherence) effect, if $\\bar{\\nu}_1^a$ and $\\bar{\\nu}_3^a$ are produced in a coherent state \n({\\it e.g.}, as a result of collective oscillations) and coherence is supported till the jump. \nIn this case, the result of evolution depends on the phase $\\phi^a$ acquired in the layer $a$, and this phase is observable. Notice that formula (\\ref{eq:eflux}) is invalid in this case. \\\\\n\n \n\n3. Let us consider the case of two density jumps that appear due to the presence \nof two shock wavefronts: the direct and reversed one. \nThis situation is analogous to the one described \nin sect.~\\ref{coherence and adiabaticity violation}. \nWe denote by $a$, $b$ and $g$ the layers between the production point and the inner (reversed) shock, \nbetween two shocks and outside the outer shock correspondingly. \nThe potential of the layer $b$, $V_b$ can be constant or change adiabatically, \nthe potentials just before inner shock, $V_a$, and after second shock (jump), $V_g$, are not equal \n(in contrast to CW).\n\nThe picture of evolution of $\\bar{\\nu}_{3}^m$ state is the following:\n\n\\begin{itemize}\n\n\\item \n$\\bar{\\nu}_{3m}$ propagates adiabatically in the layer $a$ and splits at the border between $a$ and $b$: \n$\\bar{\\nu}_3^a = c_\\Delta \\bar{\\nu}_3^b - s_\\Delta \\bar{\\nu}_1^b$; \n\n\\item\nthe latter state is coherent and oscillates in the layer $b$ acquiring the oscillation phase $\\phi^b$: \n$c_\\Delta e^{i\\phi^b} \\bar{\\nu}_3^b - s_\\Delta \\bar{\\nu}_1^b$; \n\n\\item \nthe eigenstates $\\Bar{\\nu}_3^b$, $\\Bar{\\nu}_1^b$ split in the second jump (the border between $b$ and $g$): \n$\\bar{\\nu}_3^b = c_{\\Delta'} \\bar{\\nu}_3^g - s_{\\Delta'} \\bar{\\nu}_1^g$, $\\bar{\\nu}_1^b = \nc_{\\Delta'} \\bar{\\nu}_1^g + s_{\\Delta'} \\bar{\\nu}_3^g$, \nwhere $\\Delta' \\equiv \\theta_g - \\theta_b$ is the change of the mixing angle in the second jump; \n\n\\item \nin the layer $g$ the eigenstates $\\nu_j^g$ evolve adiabatically to the mass eigenstates \n$\\bar{\\nu}_j^g \\rightarrow \\bar{\\nu}_j$. Furthermore, $\\bar{\\nu}_1$ and $\\bar{\\nu}_3$ separate and decohere. \n\n\\end{itemize}\n\nAccording to this picture we obtain the following expression for the probability $\\bar{p}$: \n\\begin{equation}\n\\label{transition2}\n\\bar{p} = |U_{e1}|^2 \\sin^2(\\Delta + \\Delta') + s_{13}^2 \\cos^2(\\Delta + \\Delta')\n-(|U_{e1}|^2 - s_{13}^2) \\sin 2\\Delta \\sin 2\\Delta' \\sin^2\\frac{\\phi^b}{2}. \n\\end{equation}\nNotice that $\\Delta + \\Delta' = \\theta_g -\\theta_a$. \n\n\n \nIn fig.~\\ref{dip} we plot $\\Bar{p}$\ncomputed with Eq. (\\ref{transition2}) for the following values of \nparameters: $V_a=6.3 \\times 10^{-12}$ $\\text{eV}$, $V_b=3.2 \\times 10^{-11}$ $\\text{eV}$, $V_g=3.2 \\times 10^{-12}$ $\\text{eV}$ and $L_b=1060$ km. If we assume electron fraction $Y_e=0.5$, the corresponding densities are $\\rho_a=2 \\times 10^2$ $\\text{g\/cm}^3$, $\\rho_b= 10^3$ $\\text{g\/cm}^3$, $\\rho_g= 10^2$ $\\text{g\/cm}^3$.\nSince the distance between the two shock fronts is \nmuch larger than the oscillation length in matter, $l_m < 100$ km, \none can naively average the oscillatory terms in (\\ref{transition2}), {\\it i.e.} take \n$\\sin^2 \\phi^b\/2 = 0.5$ (the \nred curve in fig.~\\ref{dip}). However, according to our analysis in sect.~\\ref{coherence-lengths} \nin layers of constant and slowly varying densities, the energy range of enhanced \ncoherence exists around $E_0$, where oscillation effects might not be negligible. \nThis is shown in fig.~\\ref{dip} by the black curve for \nthe energy resolution $\\sigma_E\/E=0.1$. Here $E_0=41$ MeV.\nThis oscillation effect will survive further propagation to the Earth, the effect encodes information about \ndistance between shock wave fronts and density profile. Similar effects of enhanced coherence also survives for adiabatic propagation between the two shocks according to (\\ref{eq:intcond3}).\nAs the distance between the two jumps increases, \none needs better energy resolution, $\\sigma_E\/E < 0.1$, to observe the effect of enhanced coherence.\n\n\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=0.8\\linewidth]{pbar.pdf}\n\\caption{Dependence of the $\\bar{\\nu}_e$ survival probability in supernova\non neutrino energy (blue line).\nInverted mass hierarchy and 1-3 vacuum mixing\nwere assumed. The probability\naveraged over the energy interval\n$\\sigma_E = 0.1 E$ is shown by black line. The red line shows the probability without oscillatory\nterm.}\n\\label{dip}\n\\end{figure}\n\n\n\n\n\nLet us consider the possible effect of the coherence loss on the collective\noscillations in central parts of a star.\nThese oscillations are induced by coherent flavor exchange in the $\\nu - \\nu$ scattering, and high neutrino densities at $r < 100$ km make them relevant. \n\n\n\nThe critical issue is whether fast loss of coherence due to\nvery short WPss destroys collective oscillations.\nIndeed, the latter imply the $\\nu - \\nu$ interactions\nafter neutrino production, which can be considered as\n$\\nu$-detection (\"observation\") process.\nLoss of coherence between production and interaction\nas well as between different interactions can be important.\nFurthermore, effectively the problem becomes\nnon-linear, and therefore, it is not clear if integration\nover energies that form the WP can be interchanged with evolution.\nSuch an interchange is in the basis of equivalence \nof results in the $x$- and $E$-representations \\cite{Stodolsky:1998tc}. There is no clear answer to \nthis question \\cite{Kersten:2013fba,Kersten:2015kio,Akhmedov:2016gzx,Akhmedov:2017mcc}.\n\nSome new insight into the problem can be obtained using\ndescription of the collective oscillations as\nevolution of the individual neutrinos propagating\nin effective external potentials formed by matter\nand background neutrinos \\cite{Hansen:2018apu}. In this case, the problem becomes\nexplicitly linear. Since the background\nneutrinos oscillate, the effective potentials\nhave non-trivial oscillatory dependence in time (distance).\nThese variations of the potentials are non-adiabatic\nand therefore strong flavor\ntransitions can be interpreted as parametric\noscillations and parametric enhancement effects. Taking into consideration this picture\none can apply the results of sect.~\\ref{parametric-oscillations}.\n\nThe following comments are in order:\n\n1. Propagation decoherence is related to the energy dependence\nof the Hamiltonian (and consequently the phases). The collective (fast)\ntransformations are driven mainly by potentials that dominate\nover the vacuum term $\\Delta m^2\/2E$ (the source of $E$ dependence).\nThe vacuum term (with mixing) triggers conversion at\nthe initial short phase. Therefore, apart from the initial phase,\n decoherence is simply absent.\nIf the initial phase is short enough\nthe decoherence is entirely irrelevant.\n\n2. In the region of fast collective oscillations, $r_{coll} < 100$ km,\nwith strong matter dominance the coherence length\nis given by vacuum parameters (\\ref{eq:cohinsn}) \nand turns out to be $L_{coh} \\sim 10^3$ km. So, $r_{coll} \\ll L_{coh}$ and\ndecoherence can be neglected for $r < r_{coll}$.\n\n3. For low energies ($E \\sim 10$ MeV) and $\\sigma_E \\sim E$\nthe length $L_{coh} < 10^2$ km can be comparable to the\nscale of collective oscillations.\nHowever, this estimation of $L_{coh}$ is for constant density.\nAs we established in sect.~\\ref{parametric-oscillations}, the parametric\noscillations themselves can substantially enhance coherence at least\nin certain energy intervals around $E_0$. So, here we deal with\n coherence sustained by oscillatory dependence of\npotentials.\n\nNotice that consideration of coherence for collective oscillations\nwas problematic because of strong adiabaticity violation and\ndifficulty to define the eigenstates. Using analogy with the CW case, \nthis can be done by integrating the Hamiltonian\noven one period, thus eliminating fast variations. For\nthe averaged Hamiltonian, one can introduce the eigenstates\nand their effective group velocities.\nThen the enhanced coherence corresponds to approximate equality\nof these group velocities (\\ref{x-space}).\n\n\n\n\n\n\n\n\\section{Conclusions} \n\\label{conclusions}\nThe propagation decoherence occurs due to the difference\nof the group velocities, which leads to the shift of\nthe WPs relative to each other and their eventual separation.\nIn matter, due to refraction, both the group\nand phase velocities of neutrinos change\nwith respect to the vacuum velocities.\nThus, the refraction affects the propagation coherence (and\ndecoherence).\n\nThe consideration of WPs in the $x$-space\nwith average energy and momentum produces the same results\nas consideration of plane waves in the $E$-space\nand integration of probability (oscillation phase) over the energy\nspectrum with width\ngiven by the energy uncertainty.\nIn particular, the same value of the coherence length\nfollows from both approaches.\n\nFor different density profiles, we determined the coherence lengths\nand their dependence on neutrino energy.\nThe salient feature in matter is the existence\nof infinite coherence length,\n$L_{coh} \\rightarrow \\infty$, at certain energies\n$E_0$, and regions of enhanced coherence around $E_0$.\nIn the energy space, $E_0$ is given by zero derivatives\nof the phase on energy. In the configuration space,\nit follows from equality of the group velocities\nof the eigenstates. The condition can be described as\nthe energy of minimal averaging of the interference term of the\nprobability.\n\nThe fundamental notion here is the effective group velocities\nwhich depend on the density profiles.\n\n1. For constant density the infinite coherence $E_0 = E_R\/ \\cos 2\\theta$\ncoincides with the MSW resonance energy in oscillations\nof mass eigenstates $\\nu_1 \\leftrightarrow \\nu_2$, and for small vacuum mixing, it is close to the MSW flavor resonance.\nThe width of the region of enhanced coherence (weak averaging) is inversely proportional to the energy resolution, $\\sigma_E$, and grows according to $\\propto E_0^2$.\nFor very large densities (high energies), the coherence length\nis determined by vacuum parameters and becomes close to the coherence\nlength in vacuum: $L_{coh}^m = L_{coh}\/ \\cos 2\\theta$.\n\nFor massless neutrinos and oscillations driven by matter\nthe coherence is supported infinitely for all the energies.\n\n2. In varying density profiles, the infinite coherence\nis realized in particular situations.\nFor monotonously and adiabatically changing density\nthe coherence length can increase toward specific energy $E_0$,\nwhich corresponds to certain $V_0$, such that for $V > V_0$ and\n$V < V_0$ separation have opposite signs and compensate each other.\n\nInfinite coherence can be realized\nfor periodic profile with adiabatic density change. It corresponds to\nequality of the group velocities of eigenstates averaged over\nthe period. $E_0$ can be found from the latter condition.\n\n3. In the presence of adiabaticity violation, transitions between\neigenstates\nof propagation happen. This leads to modification of WP shape,\nand consequently, to change of the effective group velocities.\n\nWe considered in details the case of maximal adiabaticity\nbreaking - density jumps in certain points of space (neutrino\ntrajectory).\nIn this case, two new effects are realized, which are related to\nthe split of the eigenstates at\nborders between layers: (i) spread of total WP, (ii) change of the\noscillation phase\nalong these total WPs.\n\nFor the example of a single jump between two layers with different\nconstant densities,\nthere are two components in the first layer and four components in the\nsecond.\nCorrespondingly there are various group velocities and oscillation\nphases at a detector,\nand it is not possible to introduce the coherence length in the usual\nway. Still the medium\nparameters can be selected so that some level of coherence can be\nsupported for long\ndistances because the split of the eigenstates regenerates interference\nand oscillations.\n\n4.\nWe checked that modifications of the total\nWP in the $x$-space do not change its Fourier\ntransform to the $E$-space,\nand consequently, the energy spectrum of neutrinos.\nTherefore derivation of the infinite coherence\nconditions for complicated density profiles can be more accessible in the\n$E$-representation.\n\nIn certain situations, consideration of WP in the $x$-space\nsimplifies computations of probabilities, especially when\ncoherence is completely lost.\nMoreover, it becomes crucial when time tagging\nis introduced at the production or detection, breaking\nthe stationarity condition.\n\n5. For periodic structures with density jumps, such as the castle-wall profile, the coherence length can be introduced for the parametric\noscillations\n(and not for small scale coherence of elementary WP).\nThen the coherence can be supported for many periods.\nFor parametric oscillations, there are several $E_0$, each associated with\na particular parametric resonance.\n\n\nInterpretation of the condition for $E_0$ in $x$-space in terms of\nelementary WPs propagating in the individual layers of the profile is very non-trivial.\nWe find that the effective group velocities for one period (and\ntherefore the\ninfinite coherence condition) are rather complicated functions of mixing angles, phases and delays acquired by the elementary WPs in two parts of the period.\n\n6. Using elementary WPs, we reconstructed total WPs at a detector and\nstudied their spread and shape change\nin the course of propagation depending on parameters of the CW profile. The reconstruction of the total WP\nfrom the elementary WP using operators of time shift is presented. Resummation of the elementary WP at a detector can be\nperformed using the evolution matrix for one period\nexpressed in terms of operators of time shifts.\n\n\n\n7. We outlined applications of the obtained results to supernova neutrinos.\nIn particular, we showed that coherence can be supported between\ntwo shock wavefronts, leading to observable\noscillation effects at the Earth.\n\nFor collective oscillations \ndue to large matter potentials\nthe coherence is determined by vacuum parameters and\ncoherence length is much larger than the scale of\nfast oscillations. Using interpretation\nof collective oscillations as the parametric effects\nwe find that coherence can be further enhanced in certain\nenergy ranges.\n\n\n\\section{Acknowledgements}\nYPPS acknowledges support from FAPESP funding Grants No. 2014\/19164-453 6, No. 2017\/05515-0 and No. 2019\/22961-9. \n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":" \n# Lord Mouse\n\nBy Mason Thomas\n\nScoundrel by nature and master thief by trade, Mouse is the best there is. Sure, his methods may not make him many friends, but he works best alone anyway. And he has never failed a job.\n\nBut that could change.\n\nWhen a stranger with a hefty bag of gold seduces him to take on a task, Mouse knows he'll regret it. The job? Free Lord Garron, the son of a powerful duke arrested on trumped up charges in a rival duchy. Mouse doesn't do rescue missions. He's no altruistic hero, and something about the job reeks. But he cannot turn his back on that much coin--enough to buy a king's pardon for the murder charge hanging over his head.\n\nGetting Garron out of his tower prison is the easy part. Now, they must escape an army of guardsmen, a walled keep and a city on lockdown, and a ruthless mage using her power to track them. Making matters worse, Mouse is distracted by Garron's charm and unyielding integrity. Falling for a client can lead to mistakes. Falling for a nobleman can lead to disaster. But Mouse is unprepared for the dangers behind the plot to make Lord Garron disappear.\nFor Bob, my loving and infinitely patient husband.\n\n# Acknowledgments\n\nVery special thanks to Kimberly Gabriel. Without her keen insight and tireless encouragement, this would never have been possible. Thanks also to Jessica Trent and Tye Radcliffe for muscling through the early drafts and taking the time to provide their invaluable feedback.\n\n# Chapter 1\n\nTHE MAN who had taken the chair across from Mouse stared back with skepticism and no small measure of wide-eyed surprise. \"So, you're him, then?\"\n\nMouse allowed the corner of his mouth to lift a fraction. He was accustomed to such reactions, of course. Not that it didn't needle him still, but he had learned to accept it. They all came around eventually. Once they saw what he was capable of, they always came around.\n\nHe encircled his fingers around the hard leather mug in front of him, considered the contents a moment, then brought it to his lips. He allowed a swig of ale to enter his mouth, but left the mug tilted against his face longer to give the impression that he was taking a much bigger swallow than he had. A stalling tactic. Plus, it sometimes helped to let a client think he was drinking more than he actually was. If it came to negotiating a price, it was best to let the client think he had the upper hand.\n\nHe set the mug down again, exhaled, and wiped his chin with his sleeve. \"And the nature of this job?\" he asked.\n\nThe client leaned back and considered him a moment with narrow eyes. The expression revealed everything--Mouse could read the man's thoughts as if they were his own. He was trying to decide if this was a prank of some kind and if he should proceed or just push up from the table and leave the tavern. The man had assumed, as many had before him, that Mouse was merely an intermediary, someone to pass on the message. Not the one he would actually be hiring.\n\nBut in all such cases, time was an issue, and it had taken plenty of it to hunt down someone for the job in question. Desperation and hope would keep the man rooted to his seat.\n\nWhile the client deliberated, chewing the inside of his cheek as he weighed his options, it gave Mouse the opportunity to study him in return. He was a wealthy sot, certainly--though he tried to disguise it with a ratty cloak. But the shiny buttons on his jerkin beneath it didn't come cheap. He had all his teeth and didn't reek like the rest of the lot in this shabby little hole. And his inflection virtually sang of nobility and privilege. No one outside of the aristocracy had that kind of diction.\n\nAt least he had the good sense not to have any recognizable livery showing.\n\nThe noble dandy's lips tightened. \"Perhaps I was misinformed, Master...?\" The man lifted a single brow, waiting for Mouse to provide a name.\n\nMouse didn't fill in the blank for him.\n\nWhen the silence between them grew unbearable, the dandy pressed on. \"I am sorry to have wasted your time. Seems you are not what I require after all. My informant assured me--\"\n\n\"Not what you expected, eh?\" Mouse interjected. He raked back his dark hair with his fingers and leaned in closer. \"Not the hulking bruiser you thought you were hiring?\"\n\n\"I don't--\"\n\n\"No shiny armor or a conspicuous sword at my belt.\"\n\nThe dandy was showing signs of true unease now. His superior breeding had provided him none of the qualities necessary for this sort of dark business. The slightest challenge had unraveled his resolve. Good.\n\n\"I mean no disrespect, good sir,\" he stammered. \"I only meant--\"\n\nGood sir? Mouse almost laughed. \"Do not let appearances fool you. Whoever sent you to me knows my skills.\" He paused and leaned in with a wider sly smile. \"And knows what I can accomplish if amply motivated. If I was brought up in conversation, it was for good reason, I assure you.\" This dithering was tediously predictable. Mouse knew he didn't fit the conventional adventuring figure. His size alone was often enough to give potential clients pause. In dim lighting, he could easily be mistaken for a youth. But Mouse wished just once a client would let reputation govern over appearance.\n\nThe dandy in the shabby cloak was losing his nerve. He put his hands on the arms of the chair and prepared to push back from the small table. A tiny bead of sweat appeared at his hairline, and his flawless skin--the color of blond wood--grew ruby blotches around his neck.\n\nTiming was everything. Mouse knew when to spring the trap. This scared little rabbit wasn't going anywhere.\n\n\"Must be an important job,\" he said. He pulled a small leather drawstring sack from its hiding place between his legs and let it drop onto the table with a substantial jingle. \"Quite the amount of coin here. How much of this were you authorized to spend on hiring me?\"\n\n\"How--?\"\n\nThe dandy reached for the purse, but Mouse had his hand over it before the man could reclaim it.\n\n\"I give you credit, coming here.\" Mouse spoke softly but clearly. \"Alone. Not your usual quality of establishment, I'd imagine. This place can be dangerous, so in the future I recommend you come armed with more than just this....\"\n\nWith one hand still on the coin purse, Mouse brought his other hand up and set a dagger in the middle of the table.\n\nThe dandy sprang to his feet in surprise, nearly knocking his chair off its legs behind him. Others at nearby tables looked over with raised eyebrows, probably thinking a brawl was about to break out. The bar wench Serafina froze and watched what happened next carefully.\n\nMouse rolled his eyes. \"Sit down!\" he hissed.\n\nThe man obeyed. With no ensuing action, the other patrons lost interest quickly and went back to their conversations and mugs of ale.\n\n\"There are certain tasks,\" Mouse told him, \"that require a more adept hand. Ones that require subtlety and finesse over the clumsy sword swinging of a barbarian. That is presumably why you were sent to me. So, I'll ask again. What's the job?\"\n\nThe man still seemed uncertain, but Mouse's instinct and experience told him to wait it out. For a moment, the dandy seemed to retreat into himself. He lowered his chin, closed his eyes, and took in a breath. Mouse took the opportunity to let his eyes wander and appreciate the man's form. He was well proportioned. Healthy. Toned. From tennis more likely than swordplay, but muscle was muscle. Mouse didn't typically care how one came about acquiring it. He tried to imagine him without the garments.\n\nThe man eventually nodded. \"You did come highly recommended.\" He seemed to be speaking more to himself than to Mouse. \"But you'll forgive me, I am finding it difficult to swallow that you--alone--can accomplish the task that I require.\"\n\n\"I work best alone.\" Mouse leaned back again. \"But... if it will put you at ease, I will tell you honestly if it is a task beyond my ability.\"\n\nMouse kept his face expressionless. The job was already his if he wanted it, even if the dandy didn't yet know it himself. Now, it was just teasing out the important details and deciding if the task was worth taking on. From the man's demeanor, the job was likely an onerous one, but there was a lot of coin in that purse that could end up in his own.\n\nHe waited for the man to begin his yarn.\n\n\"I come from Har Dionante. I am a representative of Duke Braddock the Mighty.\"\n\nMouse gave a slow nod. He knew of Har Dionante but had never had cause to venture there. It was a heavily fortified city to the south, positioned high on a bluff. The city housed a number of valuable artisans, mainly those working with alabaster. The town was known for it. Its strategic position and surrounding resources made it a historically tempting target for other ambitious lords. But he'd heard nothing of this Braddock, and he almost groaned at his title. Sounded a bit self-aggrandizing. How mighty could he be if Mouse had never heard of him?\n\n\"The duke's son, Lord Garron,\" the man continued, \"was sent as an emissary to Har Orentega for some delicate negotiations.\"\n\nAh. That was a city Mouse had had the pleasure of visiting once or twice. A story was starting to come together. Har Orentega was also south, even farther south than Har Dionante, and it too was a fortified city. The two cities were notorious rivals, both financially and militarily. Over the centuries there had been numerous scrapes between their two armies. Nothing too serious--just tit-for-tat scrimmages that gave neither any advantage or sway over the other. Lately, though, under the rule of King Harus V, the region seemed to have settled into a sort of delicate armistice. Harus was less tolerant of these regional disputes than his predecessors. The boys were being forced to play nice.\n\n\"Lord Garron was to meet with Duke Delgan's court regarding a certain trade dispute.\" He waved his hand to dismiss it as trivial. \"The details of that are not relevant in this, so I'll not waste your time. Garron left for Har Orentega with a small entourage and was expected to be gone no more than ten days. His father had given him clear guidelines of any concessions he could give, but beyond that he was to return if no agreement was made. A messenger was dispatched that Garron had arrived safely and negotiations had begun.\"\n\nThe man took in a long breath.\n\n\"But Garron did not return when he was scheduled to. A messenger was sent to Lord Delgan's palace to inquire and was given this.\" The man reached inside his jerkin over his left breast to retrieve something. His eyes widened in sudden panic as his hand found nothing. He dug around, reaching deeper, unable to find what he was searching for.\n\nMouse pulled a small parchment scroll from his own jerkin. It was tightly rolled and tied with red ribbon. Fragments of a broken wax seal remained clinging to the paper. \"I assume you are looking for this.\" He held it out for the man to see.\n\nThe dandy man's lips tightened. He snatched the scroll from Mouse's hand. \"Stop doing that!\"\n\n\"Force of habit,\" Mouse said with a shrug. \"You might as well hand it back so I can read it. That was the intention, yes?\"\n\nThe man huffed and tossed it onto the table.\n\nMouse slid off the ribbon and unfurled the paper.\n\n\"Let me know if you need help with any of the words,\" the man gibed.\n\nMouse repressed a smile and lifted his eyes. \"I'll manage, thank you.\" The man had some cheek after all. He liked that.\n\nThe note was curt, to the point of almost being dismissive and rude considering whom it was sent to. It was a legal document announcing that Lord Garron, the son of Duke Braddock, had been detained for breaking Orentegan law and awaited trial. But the vagueness of the missive was in and of itself noteworthy.\n\n\"What crime has he been accused of?\" Mouse asked, handing back the parchment.\n\n\"We know not. Duke Delgan refuses to respond to our communications or provide audience to the delegates that we have sent. We've appealed to the king... but he seems disinclined to get involved in any legal matters of one of his provinces. Some of our own people inside the city--\"\n\n\"Spies.\"\n\nThe man shrugged. \"If you will. They have learned nothing and cannot get close to him. They don't even know where he is kept. We feel the charges are trumped up to somehow leverage something from Duke Braddock. But we cannot see to what end.\"\n\nMouse leaned back in his chair and scratched the stubble on his chin. He was already questioning if he wanted to get involved in this. Political fights could get messy, and the wiser course was sometimes to keep out of them entirely. But that purse sure was heavy....\n\n\"So how do I square into this?\" He suspected he already knew. But he wanted the man to say it.\n\n\"We fear for him, sirrah. We need someone who can get in there, rescue him from whatever dungeon they have him in, and get him out of Har Orentega. Alive and unharmed.\"\n\nMouse nodded in thought a moment with pressed lips. A rescue mission, then. He wondered who sent this man to him. Most would know it was a waste of time. With a chuckle, he picked up the leather purse and tossed it back. The man caught it awkwardly over his sternum. \"You're going to need a bigger purse, my friend. That, I'm afraid, is not big enough for the job you describe.\"\n\nSmuggling was one of his specialties, of course. But standard contraband was his forte. Not people. Liberating someone from some unknown internment and then smuggling him out of a walled city was something he had yet to attempt. For good reason. This imprisoned noble from a rival city would be well protected, and should his escape be discovered too soon, the entire city's guard would be assigned to his recapture. This job had the makings of a disaster--especially if the young lord was an entitled fop who wet himself at the first sign of danger. There were too many variables at play in this. Too many unknowns. And Har Orentega courts were notoriously severe with the enforcement of its laws. If he were caught....\n\nThe client set the purse back gently in the middle of the table. \"This,\" he said, \"is only the first installment. A sign-on bonus, if you will. Agree to take on the job, and you walk out with the entire purse. I am authorized to award you considerably more if the task is completed with the Lord Garron safely within the walls of Har Dionante.\"\n\n\"How much more?\"\n\nThe man reached down into his boot and pulled out another slip of paper. Damn, thought Mouse, he'd missed one. The paper was handed over.\n\nMouse tried to maintain a neutral expression when he looked at the amount written there but feared he wasn't entirely successful. His jaw most certainly slackened. \"Is this man the heir?\" he asked.\n\n\"No. He is the second son of three.\"\n\nDuke Papa must care very deeply for his son indeed. Mouse put the paper facedown on the table and slid it back across without a word.\n\nThe dandy picked it up and returned it to his boot. \"Should you fail to rescue him,\" he continued, \"you will be allowed to keep the entire contents you have before you but will be awarded nothing further. But we were assured--by people we have great confidence in for such matters--that if anyone can see this done, and done quickly, it is you. I hope their faith in you is not misguided.\"\n\nMouse smiled. \"It's not.\" He took a gulp from his mug, a real one this time. Then he lifted the purse and stored it safely away out of sight. \"Tell me everything you know.\"\n\nHE WAITED until the new client was out the door before he rose from his small table and approached the bar counter. Serafina leaned one elbow on the counter, inspecting a coin pinched between her thumb and forefinger. Her face was scrunched as if she smelled something foul. She polished it on her shift, then eyed it again.\n\nMouse leaned in next to her, his forearms overlapped on the wooden surface, which was polished smooth by generations of patrons with their belly to the bar. At the far end, Ludro was draining a keg into three mugs. Mouse caught his eye and lifted his chin. Ludro returned a quick nod.\n\n\"What d'ya make of this, Mouse?\" Serafina handed the coin over to him. Mouse took it and turned it around in his fingers.\n\n\"Old,\" he said. \"Poorly minted but legitimate. If it is a forgery, it's good enough to pass.\" He handed it back.\n\nWith a shrug, she slipped it into a little cloth purse and tucked it into her bodice. \"Quite the handsome dandy you were chatting with tonight. Friend of yours?\"\n\n\"Client.\"\n\nShe made a noise that said she was either disappointed or didn't really believe him. \"Should have given him more attention.\" She tugged on the sides of her bodice and wiggled her torso to fluff up her tits. \"He had coin, that one. A girl can tell such things, you know.\" Mouse was tempted to comment that the man's disguise as a commoner was about as good as her perfume, but he held his tongue. She straightened as if a thought suddenly occurred to her. \"Oh, unless, of course, he swings his sword for your army and not mine.\" Her eyes narrowed, and the corner of her mouth lifted in a sinister smile.\n\nMouse's jaw tightened. \"Not now, Sera,\" he replied. Normally he would play along, but tonight he wasn't in the mood. The new client had given him a lot to think about. This changed everything.\n\nSerafina pushed away from the counter with dramatic indignation. \"Well, excuse me, m'lord.\" She made a clumsy attempt at a curtsy. Gods help her if she encountered an actual noble. She'd be beheaded on the spot.\n\nLudro worked his way down the bar counter to join them, thankfully interrupting Mouse's response. Serafina gave the barkeep a warning glare. \"Careful, Lu. Someone's ill-tempered tonight.\"\n\n\"Go earn your wage, wench,\" Lu growled back. \"Off with ya.\"\n\nSerafina laughed and swished back into the crowd, her hips grazing against the shoulders of every seated patron she sauntered past. \"I'd be surprised if that toff you were friendly with makes it to his carriage with his purse,\" Lu said once she was out of earshot.\n\n\"Not to worry. He left most of it with me.\" And Mouse was well aware of the bodyguards lingering outside the tavern. The man had plenty of protection with him.\n\n\"Job?\"\n\nMouse nodded. \"Going to keep me busy awhile. Get the box for me, would ya?\"\n\n\"What? Already?\"\n\n\"Just get it.\"\n\nLu bent under the counter, fumbled around a bit, then pulled up a small leather box covered in iron rivets and bound with a buckled strap. He set it down in front of Mouse with a frown. Mouse undid the buckle and lifted the top off the box. Inside was a scrap of parchment--just as there always was.\n\n\"How much is it now?\"\n\n\"Doesn't matter,\" Mouse answered dourly. He dug out the bag of coin he just received from the new client. He counted out coins from the sack into his palm, then tilted his hand and let them spill into the box. The familiar ring made nearby patrons look up in interest. Mouse returned the lid to the top and secured the buckle. When he looked up again, Lu was staring at him with a dark expression.\n\n\"Mouse, you can't keep doing this forever.\"\n\n\"I'll do it as long as I have to.\"\n\n\"As long as you can afford their demands, you mean.\"\n\n\"Drop it, Lu.\" But Mouse knew he wouldn't. He knew the man well enough to know he was going to hear plenty more.\n\n\"Mouse.\" Ludro reached across the bar and put a callused hand on Mouse's forearm. \"Like it or not, I'm the closest thing to a friend you have in this town. So listen to me when I tell you, I know how these things end. These men are going to bleed you dry. The day will come when you don't have the coin to pay them.\"\n\nMouse lifted the purse the client had left him. \"But today is not that day.\"\n\nLudro pulled his hand away and shook his head in frustration. \"You're too good to be mugged like this. There's another way....\"\n\nMouse grabbed him by the sleeve before his arm left the bar. \"And what would you have me do? You know the law. I will not have him hanged, Lu. Not for my crime.\"\n\n\"Your father is a resourceful man. If you just get word to him, he can--\"\n\nMouse shook his head. \"No. He can know nothing of this. I will not bring more ruin upon his life by forcing him into hiding.\" The man had endured enough shame already. Mouse wasn't about to compound his grief by turning him into a fugitive as well. He released his grip on Lu's sleeve and pushed the box closer to him. \"He enjoys a happy, quiet life. I will not destroy that. This is the only way, Lu. Just send the message.\"\n\nLudro took the box and held it a moment against his chest but didn't put it away again. \"I don't like these people. I don't trust them.\"\n\nMouse chuckled in spite of himself. \"They're the king's guard. Why would you trust them?\"\n\nLu's head swung about in sudden panic. \"Keep your voice down, idiot! You want to get me closed down?\"\n\nMouse held up his palms as an apology. \"They've kept their word so far, Lu.\"\n\n\"As far as you know! Your father could be in irons now as we speak, and you'd have no way of knowing.\"\n\nMouse's lips tightened. That was true enough. There was no way to ensure his bribe money would continue to keep his father safe, but he wasn't about to admit that. Lu was also right that he couldn't keep paying the amounts they were demanding. The cost of keeping his father from arrest had quadrupled just over the last year. \"Look, Lu. Just one more. I need just one more moon. This new job... well, if all goes as planned, I could make enough to buy a king's pardon.\"\n\nLu's eyes widened with skepticism. \"You're going to bring in a thousand crowns?\"\n\n\"With enough coin left over to buy you a mead.\"\n\nLudro's expression shifted from disbelief to overt concern. He leaned in. \"What in the name of the gods does he want you to do for that kind of coin?\" His voice had dropped to a whisper. \"You have any idea what you're getting yourself into, Mouse?\"\n\nMouse dropped one more coin on the counter and slid it across the bar toward Ludro before tucking the bag away again. Fifty royals. \"For your trouble, Lu.\"\n\nLudro reluctantly took the coin and slipped it into his belt. \"I sure hope you know what you're doing,\"\n\nMouse nodded and pushed himself from the bar. So did he.\n\n# Chapter 2\n\nMOUSE MADE a gentle roll to the edge of the bed so as not to disturb Taurin sleeping next to him. He grimaced at the creak of the frame. Taurin stirred and adjusted the covers over his shoulder but didn't wake. Mouse smiled at him warmly--with a stirring sense of pride that he had tuckered the man out. He fumbled around the floor among the salad of discarded garments for his own tunic, which he discovered by the shuttered window. How in the name of the gods had it gotten way over there? He pulled the garment over his head and walked about the chamber bare legged.\n\nA splash of water on the face from the basin and rubbed into his eyes dispelled the last of the grogginess from his head, but his body still felt the deep weariness from the road. Ravaging Taurin for a couple of hours didn't help on that score either, he thought wistfully, but smiled again at the memory. That, at least, was energy well spent.\n\nAfter he'd entered the West Gate of Har Orentega around midmorning, he'd headed directly to Taurin's fine establishment--The Scarlet Unicorn. It had been some while since he'd been through the area, so Mouse wasn't entirely sure what kind of reception he'd receive. But he needn't have feared.\n\nIt was near evening, if the angle of light coming through the shutters were any indication, which explained why his stomach had begun to gripe at him. It was later than he'd planned. The nap had felt good after their vigorous calisthenics, but that left few hours to roam the streets before the real fishing work began at the local inns. Of course, he had Taurin's brain to pick at too before he left. That would take time. It was never an easy feat to tease out anything useful from that crafty bastard.\n\nHe dropped down in one of the chairs at the round table by the empty hearth, the leather seat cool on his bare ass. He leaned back and pulled up one heel on the seat's edge, then helped himself to the plate of cut figs and the chalice of wine he'd abandoned a few hours ago.\n\nHe closed his eyes and allowed the liquid to wash over his tongue a moment before swallowing it down. Taurin always provided the best wine.\n\nIn a nearby room, one of Taurin's clients called out with noises that resembled someone being tortured, but Mouse knew better. He'd likely made a very similar commotion a few hours ago. The client certainly was enjoying himself. The girl he was with made coinciding noises, but her vocalizations were less earnest. Her heart didn't seem nearly as invested in the activity. Mouse doubted the man would notice.\n\nWhen he set the glass on the table, he spied his favorite dagger--his father's dagger--resting there. How careless of him to leave that out in plain view. That was unlike him. He must have been more tired than he thought.\n\nHe lifted it by the hilt and inspected it, just as he had a thousand times prior. The strange but unequalled craftsmanship, the red leather bindings that showed no wear or blemish, the unknown rune etched into the metal of the blade itself--all adding up to a mystery he would likely never solve. Without conscious thought, he began the ritual of spinning it through his fingers, an act that always helped him relax.\n\nThe relative quiet of the moment felt good. Peaceful. The journey to Har Orentega had been slow going and rough, and the burden of the work in front of him weighed on his shoulders like a stockade. Something about this job felt off. The amount of coin promised seemed rather inflated for the task. Yes, he didn't come cheap, but the figure on the paper even made him blush a little. No question that the duke loved his little lordship, but shelling out that much coin for someone other than the heir seemed... silly. The bad blood between the two provinces could be a motivator certainly. This could easily escalate the conflict--but his instinct told him there were details being omitted. And his instinct was rarely wrong.\n\nA gentle knock on the door shook him from his musing and ruptured the stillness of the moment.\n\nTaurin stirred. \"Yes,\" he called from beneath a pillow. \"What is it?\"\n\nThe door opened, and a young girl shuffled into the room. She was no more than thirteen years of age and dressed plainly in a simple shift and apron. Not one of Taurin's career girls, apparently. She moved into the room with her head bent low, staring at her feet.\n\nMouse felt suddenly exposed and self-conscious. He glanced down at himself to make sure the front of the tunic covered his three-piece set adequately, and adjusted the fabric a little anyway. The girl struck him as innocent, although surely her time in this place had exposed her to more shocking sights than a man's bare legs. But the laws in Har Orentega were less than friendly to his kind, and Mouse was less than keen in giving anyone leverage that could be used against him.\n\nThe girl didn't appear to take in his presence. Her attention was solely on the master of the house.\n\n\"Forgive the intrusion, my lord,\" she said. \"But the clients you were expecting have arrived. They have asked for you.\"\n\nTaurin rolled and sat up in bed in one smooth motion. He clasped his fingers and stretched both arms up over his head. Mouse felt the corners of his mouth lift in a smile at the fine shape of his arms, his shoulders, his torso.... The memory of the last few hours washed over him like a warm bath. Damn. How in the name of all the gods did that man keep such a fine shape owning a brothel?\n\n\"Settle them in the blue parlor,\" he said in a grunt at the end of his stretch. He exhaled and dropped his arms in his lap. \"Serve drinks but nothing too high on the shelf. Have....\" His mouth scrunched in the corner. \"Desna's free, yes? Have her attend to them. But keep Rashiar close. I don't trust them.\"\n\n\"Yes, my lord.\"\n\n\"I'll be down soon. Just keep them busy 'til I get there.\"\n\nThe girl bowed to him as if he were a lord and departed. Mouse caught her steal a glance at him as she closed the door behind her.\n\nTaurin smiled at him from the bed. \"Duty calls.\"\n\n\"I should get to work myself.\"\n\n\"Damn.\" Taurin shook his head. \"Sitting there like that makes me want to forget business for a while yet. You still look good, Mouse.\"\n\nMouse's smile broadened. \"Likewise.\"\n\nTaurin raked fingers through the tangled curls of his hair. \"Just as well, I suppose. I'm going to be sore enough for a while as it is.\" He was easily ten years older than Mouse, but despite the first signs of gray at the temples, the full head of thick dark hair made him look barely past his youth. Taurin shuffled to the edge of the bed and threw his legs over the side. \"It's been a while since I've been fucked like that.\"\n\nMouse laughed. \"No Orentegan boys around to service you properly?\"\n\n\"Not like that,\" Taurin replied with an eye roll. Then his face turned a shade more serious. \"Laws here keep many nervous, so pickings are slim. This isn't Har Serina or Har Dionante, where our kind are looked on with more charity. Prudence is a necessary aptitude.\"\n\nThe mention of Har Dionante made Mouse's ears perk up with interest. A strange coincidence that he should mention that particular duchy of all places, the home of his intended target.\n\nBut Taurin rolled on, unaware. \"Enough drink and a stubborn hard-on will get men to take chances, but you just have to be discreet and choose wisely. When I do get my chance,\" he added with a wolfish grin, \"I usually have to maintain my image as someone in control. Besides, these pent-up blokes sneaking away from their wives all prefer to be the one with their ass in the air anyway.\"\n\nMouse gestured toward the door with tilt of his head. \"No danger for you?\"\n\n\"Nah.\" Taurin stood up from the bed and positioned himself over the chamber pot. The stream hit the side of the pot and rang like a bell. \"My girls know what side their bread is buttered. They won't say anything.\" He shook himself a few times and turned about. \"Plus, I'll bed a noblewoman now and then just to keep the rumors at bay.\" Mouse raised an eyebrow at him and Taurin laughed. \"You know how it is. A wet hole's a wet hole. Not that difficult.\"\n\nMouse didn't see that way. No matter how beautiful the woman, he never could get himself aroused enough to even get started. He'd tried plenty.\n\nTaurin crossed the room to where Mouse was seated and stood over him, naked. His cock was lifted up a bit in early arousal. Mouse resisted the urge to reach out and take it in his hand. Taurin refilled his own chalice from the bottle and took a long draught. \"So, what brings you to this end of the world, you scallywag?\"\n\n\"Work.\"\n\n\"So, this wasn't a social call? I'm hurt!\"\n\n\"I came here first, didn't I?\" Mouse replied with a smile. \"Why do you think I took the job in the first place? Gave me an excuse to stumble this far south.\"\n\nTaurin tilted his head back and laughed. \"That's some horseshit if I ever heard it.\" He took another drink. \"So, what's the job?\"\n\n\"Missing person.\"\n\n\"Ah. Anyone I'd know?\"\n\n\"Probably not.\" He took a swig from his wine to let the thought linger and hang in the air. As much as he enjoyed Taurin's company, he didn't trust him. Not as far as he could toss him. Even now, Mouse knew, Taurin was waiting for a nugget of information that would serve him in return. He was standing over him intentionally, displaying his fine attributes in order to distract and bring Mouse's guard down. Mouse knew to tread carefully. \"Right now, I'm just keeping my ear to the wall, seeing if there are any rumors about. Any pillow talk of interest?\"\n\n\"Nothing about anyone gone missing.\"\n\n\"Anyone of note being kept under lock and key?\"\n\n\"Ho ho!\" Taurin's face lit up with interest. \"That sort of missing!\" He popped a fig into his mouth, then put his foot on the chair and leaned in with an elbow on his knee. Taurin's big toe was under the hem of the tunic, and Mouse could just feel the tip of it brushing the hair on his sac. \"You're looking for a prisoner.\"\n\nMouse shrugged noncommittally, ignoring the smell of oils, musk, and sweat emanating from Taurin's skin. \"Just covering all possibilities,\" he replied. \"Best to look under the obvious rocks first. The idiot probably found himself in the middle of some tavern brawl and got himself arrested. No one sends word to the family.\"\n\nTaurin's smirk let Mouse know that he knew he was lying, but he didn't push it. He straightened up again, removing the foot from Mouse's crotch, and set about collecting his garments from the floor. \"To afford your services, we're talking someone with coin. Not going to be your typical street thug that got into fisticuffs. So, we're talking about a merchant, I presume. One of some worth. Or perhaps a nobleman.\"\n\nMouse didn't respond but watched Taurin like a hawk following the path of a rodent from its perch in a tree as he moved about the room. Taurin dropped the garments onto the bed--and to Mouse's disappointment--began to pull them on, hiding from view his strong physique.\n\n\"Well, nothing's come to my ear,\" he said, lifting his jerkin up by the shoulders in front of him to examine it. \"This won't do,\" he grumbled to himself, and tossed it back on the bed. He opened the wardrobe and pulled out a long surcoat embroidered with gold thread. The fabric was richly designed and clearly expensive. Taurin was doing quite well for himself, it seemed. He slipped his arms into the sleeves and began to button it up from the neck down.\n\nIt was Mouse's turn to know that Taurin was lying. He knew something but wasn't about to say it. Self-preservation, most likely. He didn't want what he knew being traced back to his business. Or he wasn't going to give valuable information away for nothing. Whatever the reason, he wasn't divulging what he knew specifically. But that didn't mean there was nothing to learn. Mouse waited him out.\n\nTaurin faced the looking glass standing in the corner. \"Ah, much better.\" He turned to Mouse with his arms extended outward. \"Yes?\"\n\n\"I liked you better without the clothes.\"\n\n\"Of course you did. You're a lech. But I need to impress my guests.\"\n\n\"Then go down there without the clothes. Your equipment impresses the hell out of me!\"\n\nTaurin threw him a playful glare as he sat on the bed to pull on his boots. He stopped. A thought seemed to occur to him, then. \"Now that I think on it,\" he said. An obvious ploy that Mouse saw through easily and had even anticipated.\n\n\"Yes?\" Mouse prodded.\n\n\"There was talk of a fuss in Duke Delgan's court a few weeks back. Something involving a nobleman from out of town. Could that have something to do with your missing person?\" He glanced over at Mouse to read his reaction. Mouse was careful not to let anything show. Taurin was fishing, but his instincts were as keen as his own. He had already guessed exactly who Mouse was here to find.\n\n\"No way to know. Who was the nobleman? And what was the fuss about?\"\n\n\"I really can't say. It was all very hushy-hushy.\"\n\n\"I see.\" Mouse studied him as he laced up his boots and tied them off. His demeanor had shifted. Few would have noticed, but Mouse knew him well enough to pick up on it. He wore his business face now--the charisma he wielded so effortlessly was more calculated, and the easy genuineness from before had slipped behind a protective veil. Taurin was protecting himself, or maybe his business. If any clients from Delgan's court caught wind or suspected that Taurin was whispering state secrets about, it could cause him problems. But it was clear there were some ugly dealings behind this. And Taurin did not want to be involved.\n\nMouse dropped his foot back to the floor and leaned in, resting his elbows on his knees. \"Any idea who might know more about what happened up there?\"\n\nTaurin stood up from the bed, fully dressed. \"There,\" he said and smoothed out his surcoat. \"This should serve adequately. But I better go. I've kept them waiting long enough.\" He moved around the bed toward the door. \"Don't feel you have to rush off. You are welcome to stay and enjoy my chambers as long as you require, of course. I'll have Rena draw you a hot bath if you'd like.\"\n\n\"Won't be necessary. I'll be leaving shortly myself.\"\n\n\"Unfortunately, I'll be fully engaged all night. And tomorrow as well, I'm afraid. Can't be helped.\"\n\nIn other words, don't call on me until this business of yours is concluded. \"Not to worry,\" Mouse answered with a warm smile.\n\nTaurin put his hand on the latch and paused. \"You know,\" he began. He cocked his head and cupped his chin in his hand. \"There's a man I think you should meet while you're here in Har Orentega. A guard captain by the name of Hawken. I think you two would hit it off famously. Interesting bird, that one. Married, three kids, and loves to have his cock sucked off by a man. Says a woman's mouth just isn't big enough to handle what he's got--which, of course, is ridiculous. I've seen what he keeps in his weapons locker.\"\n\n\"Really?\" Mouse replied flatly. \"How interesting.\"\n\n\"But whatever you need to hang your hat on to justify it, I suppose.\" Taurin shrugged. \"Why these men just can't admit it's a man they want.... Anyway, apparently Hawken pissed off some of the wrong people of late and has been reassigned. He used to work in the palace, but now he's working the night watch, east wall. You should call on him.\" Taurin opened the latch and pulled open the door. With one final look back over his shoulder, he gave Mouse a mischievous grin--a genuine one, with bright humor shining in his eyes. \"I think your mouth is up for the task.\"\n\nMouse returned the grin. \"Thanks.\" They held each other's gaze for a long moment, Taurin's eyes reflecting the warm affection Mouse felt himself for the savvy businessman. \"It was good to see you, friend.\"\n\n\"You too,\" said Taurin.\n\nAnd he was gone.\n\n# Chapter 3\n\nMOUSE SLIPPED away from Taurin's personal little empire, quietly and unnoticed. Time for the real legwork to begin. He'd gleaned all he was going to from Taurin, his strongest connection within the Shadow Elite.\n\nTaurin had done well for himself here. Clearly, he had risen higher in the ranks of the Elite than Mouse would have anticipated, but then, he'd always been the ambitious sort. Taurin was driven by wealth, power, and influence, and pick-pocketing or running small cons wasn't going to bring any of those. And if someone with ambitions wasn't lucky enough to be nobleborn, the only option was the Shadow Elite.\n\nThe organization was likely made up of merchants, business owners, and guild leaders throughout the kingdom. No one knew for certain. Membership was exclusive and highly confidential. They garnered its power and influence through a blend of bribery, intimidation, and the careful provision of illicit services. Unlike the blunt instrument of street gangs, they functioned more like a sharp butcher's knife, shrewdly carving out the best cuts from the city for themselves. But despite the obscene profits they raked in, they haughtily viewed themselves as the counterbalance of the nobility's power and claimed to work for the betterment of the people.\n\nWhich was utter horseshit.\n\nMouse knew the truth of it--they were nothing more than thugs in fine doublets. For all their vainglorious boasting, the coin never trickled its way into the purses of the lowborn and hungry. And any actual power the Shadow Elite wielded was as tenuous as smoke. Sure, they did evoke some measure of influence and fear, and they could manipulate the operations within the city with their machinations, but they were more under the thumb of the nobility than they would ever admit.\n\nNow it was time for Mouse to look at the underbelly of this city. His first destination--the local thieves' guild.\n\nA long shot, he knew. Cabals in these southern provinces tended to be territorial and suspicious of outsiders. Squeezing coin from unwitting victims was generally more difficult out here, so a newcomer with discernible talent was a direct threat to filling your pocket that day. The locals would be wary of someone encroaching on their established territory. They were going to be less than amiable and loath to cooperate.\n\nBut Mouse had to start someplace. And he knew how to make a big splash in a small backwater such as this.\n\nThe streets of Har Orentega were in full bustle. A bright and warm day had driven the populace en masse out of doors to take advantage of it. Mouse marched the cobbles at an unhurried pace, following the current of bodies that filled the streets. There was something deeply satisfying about walking through a crowd in a faraway city, alone and anonymous. No one knew him, and there was an exhilarating sensation of liberty with that. He savored the feeling of isolation as he wove his way through the center of the street.\n\nShopkeepers and food vendors shouted at him as he passed, but he ignored them. He was too busy dissecting the crowd on pure force of habit. Many loitered about the peddlers' wagons or the shops that lined the narrow streets, while others were just out for an afternoon stroll in their finest with the intent of being noticed. But Mouse was more interested in the small minority out for more nefarious purposes.\n\nFinding the thieves' guild headquarters was always a straightforward affair. Regardless of the city. There wouldn't be any advertising placard hanging above the front door or recruiting announcements nailed to the news posts in the square, but Mouse knew what to look for.\n\nIt took him little time to locate a pickpocket meandering through the crowd in a spacious plaza with a fountain at its center. He was a gaunt youth dressed as a dock laborer, but Mouse recognized him as a fraud immediately. His movements among the people were too aimless for a laborer during the day and his skin too pale for someone who worked regularly by the river. Plus, the loading and unloading of cargo required far more bulk than this scrawny lad possessed. He canvassed the street for just the right victim. When the moment was right, he would spring the deception that would end with the patsy unaware his purse no longer hung from his belt.\n\nThe pickpocket circled the crowd in the same direction water drained, so he favored his right when scanning for potential targets. Mouse easily avoided his gaze, closed in from behind, and appeared at his left shoulder.\n\nThe lad started when he finally noticed Mouse hovering there. \"Bugger off, mate. Some space if you don't mind.\" He then sucked in a breath through his teeth when he felt the point of Mouse's dagger poke into his back.\n\n\"Some advice, friend. You're sloppy. You pay no attention to your blind side.\"\n\n\"What? I think you're confused, mate--\"\n\nMouse leaned into his ear. \"How's business today?\" he asked. \"So much low-hanging fruit to choose from, eh? So deliciously unaware. Practically inviting you to take their purse.\"\n\nThere was a moment when the lad considered going on with the denial. Mouse watched the calculation pass through his eyes. But instead the lad opted to drop the pretense of innocence.\n\n\"Fuck off. This here's my quarter, by rights.\"\n\nMouse pushed the point a little more. The lad stiffened and arched his back away from the pressure. \"No need to be rude, friend,\" Mouse replied. \"I've no interest in wrestling your territory from you. I'm new in town and just in need of some directions.\"\n\n\"Do I look like a fucking road map?\"\n\n\"Your guild. Where is it?\"\n\n\"See, that there is privileged information.\" The lad, looking over his shoulder, locked his narrow gaze on Mouse and sneered. \"Members only.\"\n\n\"Perhaps I'd like to sign up. Think I have the skills?\" With his left hand, he lifted a leather pouch and bounced it. It jingled heartily.\n\nThe lad's face darkened. His jaw tightened, and his eyes shifted from haughtiness to hatred. \"That's a day's work there, mate,\" he growled. \"And me boss's cut.\"\n\n\"Then sing. Or the coin is tossed onto the cobbles, and you can see how much of your hard work you can recoup.\"\n\nThe lad seethed but said nothing.\n\nMouse extended his hand, ready to flip the pouch over. \"There are others who will tell me, you know. And you will still have to face your boss later with a fraction of the day's take.\"\n\n\"Stranger, you are playing at a dangerous game.\"\n\n\"My favorite kind.\"\n\n\"Well, this town isn't that big. I will find you again, I promise. And I will not rest until I see you dead.\"\n\nMouse nearly laughed. \"Then you will go a very long time without any sleep.\"\n\nThe lad spilled the location. Mouse tossed the bag into the air, forcing the lad to dive for it, then disappeared into the crowd.\n\nHAR ORENTEGA was a fortified town built on a high foothill of the Southern Spine. The manor crested the hilltop like a rusty iron crown, but the town itself swooped down the hillside until it crashed onto the shore of the Dark Maiden, the wide river that bled out from the mountains. The only public gates into the town were on the opposite bank, and two bridges crossed the expanse. The center of each bridge could be drawn up, cutting off all direct access to the town in the event of an invasion or attack. Warehouses and docks lined both busy shores of the river, with the long wooden booms of cranes clawing upward and out over the river like skeletal fingers.\n\nThe guild, Mouse learned, was located beneath an old warehouse not far from the river and the merchants' quarter.\n\nThe dank headquarters was quiet when he entered. Not surprising for this time a day, but it worked to his benefit. A few skulking wastrels wouldn't cause him the grief that some of the more industrious guild members might if threatened by his presence. Mouse had seen better facilities behind a fishmonger's shop. The ceilings were low--not that it mattered particularly to him--and the ground was haphazardly covered in various wooden planks in a sad imitation of an actual floor. A few lanterns set on makeshift tables cast a grim light about the place. The guild here was more poorly funded than he thought and even more poorly managed. Had he half the mind to, he could wrestle control of it in the better part of a fortnight and turn it into something that could turn a healthy profit. In a well-organized thieves' guild, everybody won.\n\nThis was pathetic.\n\nA waifish older man with ripped trousers and no shirt was passed out on some cushions in the corner, and a small group of grunts sat around a table playing at cards. He made for the card game.\n\n\"Room for another?\" he asked breezily.\n\nOne of the brutes looked up, gave him the quick glaze over, and scowled. \"Piss off, boy. I'm not your daddy. Tell your whore mum to look elsewhere.\" He chuckled and looked to his friends to laugh at his joke, who of course obliged him.\n\nMouse smiled inwardly. The poor light in the room could explain the mistake. But had the brute taken the energy to look at him more directly he would have seen the tightly cropped beard that hugged Mouse's jaw. He cocked his head to the side with an innocent expression of surprise. \"Oh, friend, I know there's no chance you sired me. For one, I know who my father is. He's a good deal more comely than you are and, unlike yourself, has the advantage of all his teeth and a thorough cleaning less than a month ago. Second, I'm fairly certain you couldn't find a pussy to stick that tiny dick of yours into with the help of a road map, so there's actually little chance of you siring anyone.\"\n\nThe brute slammed his cards down on the table, the impact launching his friend's mug into the air. He attempted to make a dramatic launch to his feet--but too much girth and wine made the effort far clumsier than he intended. His stool toppled behind him with a bang that made the half-naked old man in the corner grumble in protest. \"What did you say to me?\"\n\n\"Gods help me! You're not going to make me repeat all that again, are you? Weren't you listening?\"\n\nMouse could see his hands clenching and unclenching with mounting rage. \"I've killed people for less, boy,\" he growled through his teeth.\n\n\"Old women and cripples don't rightly count toward that tally.\"\n\nIt was the final prod the brute needed. He pulled back his arm and thrust his meaty fist at Mouse's face with all the strength he could put behind it. Mouse easily saw it coming as if the man had advertised what his attack would be in the market square a week before. He twisted aside with little effort as the swing sped past him, taking the brute's arm and then the shoulder and torso along with it.\n\nWhen would people learn that heavy drink did not make you better in a brawl?\n\nMouse elbowed him in the jaw as his head soared past. He felt something give under the impact. The brute hit the floorboards hard. He lay there a moment, groaning before remembering how mad he was. Then he tried to scramble back to his feet. The ordeal was painful to watch. He looked more like a newborn colt making its first venture into the world of standing.\n\nMouse stepped back with arms crossed and waited patiently.\n\nNot at all surprising, once erect again, the idiot decided to make another go of it. He lunged at Mouse with a primal shout of rage. Mouse's reaction was swift and precise. He deflected the man's inelegant and haphazard swing with his forearm, guiding the punch safely to the side, while moving himself in closer. He made a sharp sideways kick to the kneecap and felt it shatter beneath the heel of his boot. The leg bent in an impossible angle. The shout from the brute's throat lifted an octave to become a cry of agony.\n\nBut Mouse wasn't finished.\n\nHe seized the man's hand at his wrist and forced the brute's arm about until his elbow was locked and his body forced forward. The cry elevated to screech, and the brute dropped heavily to his one good knee. Mouse punched with the heel of his hand against the side of the man's face. Once. Twice. Thrice.\n\nMouse released the arm and the brute collapsed to the boards and was still. His friends stared at the bloody heap in stunned horror as Mouse righted the stool and took his seat on it. Casually, he picked up the cards scattered about the table.\n\n\"Mind if we talk while we play?\"\n\nTHE MEN stared at Mouse with open mouths like three dead fish. They still gripped their cards, but their arms had gone limp and dropped to the table.\n\n\"Come now,\" Mouse said. \"Whose turn is it?\" He leaned left and whispered conspiratorially to the thug closest to him, \"You know, I can see your cards.\"\n\nThe man across from Mouse gathered some of his wits back about him and straightened. \"Friend, I don't know who you are, but we ain't got nothin' to say to you.\" Brave words. But the crack in the man's voice betrayed him. One of the other men stared down at his comrade on the floor, who hadn't moved since Mouse took his seat at the table. \"This here is an exclusive organization. It's best you leave. Now.\"\n\nMouse smiled back at him. \"In time. But first, let's play a simpler game. I'm not really in the mood for cards anyway.\" He folded up the cards and tossed them in the center of the table, then turned his hand up, revealing a gold coin--a whole crown--he had palmed there. All three opposite him straightened on their stools at the sight of it. It was probably the first time any of them had seen one that close up.\n\n\"Shit, is that a real--oof.\"\n\nThe brute in the middle elbowed the one on his left.\n\n\"So, here's how the game is played,\" Mouse said. He gestured to the middle one, who seemed at present the leader of the three. \"You. Stick out your hand, palm up. Come, come, I won't bite you.\" Like magpies, they were all captivated by the shiny coin. The heap on the floor, for the moment, was forgotten. Curiosity got the better of the man; he eased his hand out over the table, and Mouse held the coin between his forefinger and thumb over the man's palm. \"I'm going to put this into your hand. Don't close into a fist.\" Mouse rested the coin in the man's hand, and the man took a slow intake of breath. But he behaved and did as Mouse asked.\n\n\"Now. I'm going to hold my hand up here.\" Mouse lifted it a boot's length above the man's palm. \"You cannot close your hand until I start to move my hand, understand? If you close your hand around the coin before I get to it, you get to keep it.\"\n\nAll three laughed. The middle thug shrugged, grinning. \"All right, little man. If you want to just throw your coin away like that, I'll take it from you.\"\n\n\"Ah, but if I get the coin out of your hand before you close it....\" Mouse paused for emphasis, leaning in. \"You have to answer one question.\"\n\n\"Sure. Why not?\" He laughed again. \"I'll even give you two questions.\"\n\n\"Very well,\" said Mouse. \"Two it is. Oh, try anything or close your hand before I move my hand, and you end up like your friend. Understood?\"\n\nThe man's hand quivered a little as if suddenly nervous, but then he said, \"I don't need to cheat to beat you at this fucking game.\"\n\nThe man narrowed his brow and held his gaze on Mouse's hand, at the same time trying to seem unconcerned. Mouse watched him, smiling. He waited a bit, holding his hand rock steady in the air. Then sprang.\n\nHe had the coin back before the fingers barely began to close. The man opened his fist back up and gawked at it, dumbfounded that no coin was in there.\n\n\"Hey! How the fuck you do that?\"\n\nMouse displayed the coin, pinched in his fingers once again. \"Two questions, was it? First one, then. And lie to me, I'll know it, so don't try. What are your names?\"\n\n\"Excuse me?\"\n\n\"Your names. What are they?\"\n\nThe thug glanced to his two comrades as if looking for help in remembering. \"Why do you want to know my name?\"\n\n\"I like to know who I'm speaking with,\" Mouse replied with a grin. \"Call me old-fashioned.\" In truth, Mouse wanted to be able to track any of them down again should he have need of them or have follow-up questions. Not that there was much of a chance that these morons would yield anything useful. Also, guild members as a rule tended to be wary of strangers asking questions. It was helpful to drop the name of another member once in a while during a chat to minimize suspicion.\n\nThe thug considered for a while, clearly worried that he was giving away something valuable. \"Berrun,\" he said. \"Name's Berrun.\"\n\nMouse nodded as if the information was vitally important. \"And your friends?\"\n\n\"Peddux, and that's Ludlow.\"\n\nPeddux was the smallest of the three. He was likely a pickpocket or a lookout of some kind. Took part in nothing too dangerous, anyway. He had a persistent eye twitch that was clearly a reaction to anxiety. Ludlow seemed to be the brooding type. He was hunched into himself, his brow deeply furrowed, and he massaged the heel of his palm absently with his thumb.\n\n\"Excellent.\" He allowed his smile to broaden. \"See? Now we're all friends. So, Berrun, my second question. Who here is the highest ranking in the organization?\"\n\nThey all turned their gaze to the unconscious mound on the floor. \"Well, he is,\" said Peddux. \"Er... was, anyway.\"\n\nBerrun bent forward and rested his sizable bulk on his elbows. \"That'd be me now, I reckon,\" he said with a sneer.\n\nMouse leaned back and crossed his arms. He kept his face neutral, looking like he was carefully considering him, but he was mostly attempting to avoid the man's toxic breath. Mouse noted the scars on the man's knuckles. He was a bruiser, nothing more. Not a new recruit, but not high in the ranks either, and not privy to any strategy planning. He was muscle brought in to lean on nonpayers and those that didn't follow the guild rules.\n\nNone of these grunts would know anything. He suspected that coming here, but that really wasn't the point of his visit anyway. And if he asked questions pertaining to the job, word of why he was in town would drift through the guild like egg farts at a tea party. He wasn't ready to tip his hand to the entire duchy just yet. These three were just blunt tools in the workshop. He needed to speak to the craftsman that used them.\n\n\"Next question, then,\" he said.\n\n\"Hey!\" Ludlow perked up. \"That was two questions. That's all you get.\"\n\n\"So it was,\" Mouse replied, waving a finger at the man. \"Ah, would have gotten away with it if it weren't for you.\"\n\nLudlow looked to his comrades with a single eyebrow raised and a satisfied smirk, very proud of his clever catch.\n\nMouse held up the coin. \"Another go, then?\"\n\n\"Sure,\" Berrun said. He adjusted himself on the stool and held out his hand. His brow tightened into a determined scowl.\n\n\"Same terms? Two questions?\"\n\n\"One question this time,\" Berrun answered. He was going to play tougher this time around.\n\n\"Hmm.\" Mouse pursed his lips, assuming a conflicted expression. \"But I have two questions left for you. So....\" He lifted his shoulders in a shrug. \"Let's make this interesting.\" He flicked his hand and now held two crowns in his fingers instead of one.\n\n\"Sorcery!\" said Peddux.\n\nBerrun backhanded him in the face. \"Don't be an imbecile. Just a tavern trick.\" While Peddux rubbed his cheek looking stunned, Berrun returned his hand to the center of the table. \"Seen it a hundred times.\"\n\nMouse doubted Berrun could count to one hundred without help, but he wasn't going to quibble.\n\n\"Go on, then,\" Berrun said to Mouse.\n\nMouse placed both coins in Berrun's palm. \"I retrieve only one coin,\" he said. \"I get one question, and you keep the other coin. If I get both, two questions.\"\n\n\"Deal.\" Berrun seemed confident now.\n\nMouse positioned his hand again. \"Ready?\"\n\nBerrun brought his face forward, and his eyes narrowed. He focused on Mouse's hand with the intensity of a hawk following its prey. A vein popped to the surface of his forehead so prominently that Mouse was concerned he might burst something in his head.\n\nBut the result was the same. Mouse had both coins back in his hand before Berrun closed his fingers halfway.\n\nBerrun pounded his fist on the table, upsetting the mugs and spilling sour ale over the cards.\n\n\"Sorcery, I tell you,\" Peddux said again, still rubbing his cheek. He flinched when Berrun threw an angry glare his way, but this time Berrun didn't strike.\n\n\"Well, then,\" Mouse said in a satisfied tone, holding up the two coins. \"Next question. Who's your guild leader?\"\n\nUneasy silence dropped onto the table like a stone brick. The three exchanged nervous glances, no one wanting to be the first to open his mouth. Mouse had caught them off guard with the question, and they recovered slowly. Guild leadership was not information typically shared outside the organization.\n\n\"Come now. We had a deal, remember?\" He erased the easy smile from his face and dropped his voice into a sort of growl to remind them he meant business. \"Answer the question.\"\n\n\"Man named Keeler,\" Ludlow answered finally in a voice barely above a whisper. He ignored the hard stare he got from his two comrades.\n\n\"Ah. And what is your opinion of your commander? Does he run the guild the way you would want or expect?\" Not really vital information, but Mouse wasn't going to let the question go to waste. He liked to know who he was dealing with, though, but he knew he was really pushing his luck this time. Speaking ill of a guild leader was dangerous territory for any member. But even if they refused to answer, it would tell him plenty. Singing the praises of your leader was easy, so if they hesitated, he already knew the answer. They all squirmed on their stools now, avoiding each other's eyes.\n\n\"Need I remind you that you agreed to the terms,\" Mouse prodded.\n\n\"Hard to say,\" Berrun spoke up bravely. \"He's new to the position.\" He shrugged then, noncommittally. \"We were all surprised when he came out on top. Waiting to see what happens.\"\n\nA coup, then. This Keeler had managed to swing control of the outfit. And he wasn't the second-in-command apparently either--the typical successor to a guild. Mouse got the sense he wasn't even among the officers. A whole new administration was in place, which would explain the current pathetic state of the guild, but was that the cause of the coup? Or the result of it?\n\nMouse lifted from the stool. \"Send him a message for me. I want to talk to him. No lieutenants. No other officers. Only Keeler himself. He either finds me, or I'll find him.\"\n\nHe turned to leave.\n\n\"But.... How will he find you? What's your name?\"\n\n\"That's his problem. Thanks for your help.\"\n\nMouse tossed one of the crowns on the table. He could still hear the brawl when he was out on the street as the three of them fought for the coin.\n\nMouse shook his head as he hoofed back uphill toward the market district. What a shithole. Of course, he hadn't expected to squeeze out much juice from that rotting fruit--but he accomplished what he needed to do. Somebody knew something, and his visit would catch the necessary attention of those higher in the ranks. He just needed to draw them out.\n\n# Chapter 4\n\nMOUSE WANDERED the market district during the remaining hours of the day.\n\nHe played the role of the earnest shopper with coin to spend. He perused the stalls and loitered through shops with keen interest, fingering textiles with groans of pleasure or turning the stoneware about in his hands, all the while keeping his ear to the conversations that buzzed around him. Gossip was always richest and most current in the markets--because for all intents and purposes, the people who were there were not really there to shop. It was, first and foremost, a social gathering. And at such events, people talked. If one wanted to learn the latest news, one struck up a conversation with the cobbler or leaned on the counter of the butcher and made some small talk while he trimmed the fat from your cut. Bored with their work, shop owners were always looking to pass the time with some informative banter. In the time it took to purchase your lamp oil and have it poured into your leather flagon, the latest news about town could be exchanged in efficient bite-sized portions.\n\nBut although there were plenty of morsels chucked about that afternoon, none of them seemed related to his mission. There was prattle about some intrigue at the court, but the details were fuzzy at best. No one actually knew anything, so speculation was rampant, and most of the rumors and claims were outrageous enough to be dismissed out of hand. And although town folk seemed to be aware something scandalous had occurred, none of the rumors even mentioned a foreign noble.\n\nThe lack of talk was telling in and of itself. If a foreign noble had been arrested, it would be heady news indeed. People would be buzzing with speculation, wondering if this meant the start of another war. Everyone knew that putting a foreign nobleman behind bars tended to piss someone off. And military tussles had been sparked by far lesser offenses.\n\nBut none of the gossip around the market square even hinted at an arrest, or of a high-level prisoner being kept in some dark hold somewhere.\n\nSo, young Lord Garron's imprisonment was not being made a public affair. In fact, the arrest had been conducted in utter secrecy. Interesting. This meant Duke Delgan was keen on this not becoming a political issue. The arrest wasn't to score any points or stir up trouble.\n\nDid that mean the arrest had legitimacy?\n\nMouse doubted it. Standard practice among nobles was to turn a blind eye to bad behavior and just send the miscreant packing.\n\nSome around town knew details of the arrest--obviously. Taurin deduced straight off whom he was seeking out. The members of Duke Delgan's court that frequented Taurin's little haven must talk in their sleep. Which meant it was safe to assume that all of the Shadow Elite here in Har Orentega were informed.\n\nWell, Mouse reminded himself, his task wasn't to figure out why he had been arrested. Just to get him out. He just needed to find out where he was being held.\n\nEvening crept in, and the crowd thinned. Shops packed up their wares and pulled their shutters closed, so Mouse ambled off. He crossed over the bridge and lingered around the East Gate for a time. He made some small talk with some of the guards stationed there, playing the tourist. Bored, they were friendly enough and answered all his questions.\n\nMouse took his cue from the lamplighters as they began their nightly ritual of shimmying up lantern poles that it was time to find his room. He skirted the shadier parts of town that he would normally inhabit on such a job--\"low town\" as it was cleverly named--and hiked back uphill. He roamed the neighborhood where the more respectable merchants laid down their heads. He walked past a number of quiet little inns--he had no use for quaint and sleepy--and eventually found the inn he was looking for. A sizable one with a full and lively tavern.\n\nThe Blue Stag.\n\nThe innkeeper took his coin, but not before he gave Mouse's garb a suspicious twice over.\n\n\"Is there a problem?\" Mouse prodded.\n\nThe innkeeper smiled thinly through his well-manicured beard. \"Not at all, gentle sir,\" he answered in a haughty tone. \"But this is a fine and peaceful establishment. I hope you've no plans on using that dagger I see in your belt during your stay.\"\n\nMouse returned the smile. \"As long as no one gives me cause to, it shouldn't be a problem. Here's hoping your patrons are as fine and peaceful as your inn.\"\n\nThe innkeeper's smile faltered. \"One night only?\"\n\n\"To be determined. My key, if you would be so kind.\"\n\nThe innkeeper lost all pretense of maintaining the smile. For several moments he stood unmoving, mulling over what he should do. Mouse could see the internal debate in the man's eyes--whether or not to return the coin he had just pocketed. It was late, and the odds of renting the room now were slim, and Mouse had paid the premium without question. After a long awkward stall, greed championed over his apprehension. He extended his hand, and Mouse gingerly lifted the skeleton key from the man's sweaty palm. \"Much obliged to you. I'll be taking a moment or two to check out my accommodations and see that all is in order. When I return I'll expect a freshly prepared plate. And a tankard of something worth drinking. Have the table prepared.\" Mouse fought to keep his grin from widening. The poor man was seriously confused now. Mouse, dressed like a scoundrel, spoke with the unquestioning self-assurance of a nobleman.\n\n\"Of course... sir.\"\n\nAs Mouse marched up the stairs, he stole a glance at the tavern below. The innkeeper had hurried over to a big man leaning on the bar. The innkeeper spoke quickly to him, and the man looked up over his shoulder at Mouse on the stairwell. An off-duty city guard that spent time in the tavern playing the heavy for free drink and maybe a coin or two. Someone Mouse was sure to become acquainted with later over his meal.\n\nTHE NIGHT'S dinner was a succulent meat pie, which Mouse approved of enthusiastically. And the mead the inn served was of equally high quality. He tucked into both with gusto at a small table by the fire that awaited him after he returned from his surprisingly spacious and rather posh accommodations. He would have to be careful. He could easily grow used to this treatment. Usually his budget didn't afford him such luxuries.\n\nHe sensed some of the other patrons eyeing him with trepidation, but for the most part the room carried on much as it had before he took his seat. He ate and drank without making eye contact with anyone, making loud approving noises as he chewed, and kept lifting his tankard to the innkeeper across the room.\n\nThe room was full. Conversations around him were loud and lively. No matter how much coin you had in your pocket, you talked louder with a belly full of mead.\n\nA shadow crossed over him while he sopped up the juices on his plate with a hunk of bread. Something very large blocked out the lamplight.\n\n\"Have a seat, friend,\" Mouse said without looking up. \"Wondered how long it would take for you to mosey over.\"\n\nThe large man didn't say a word, so Mouse placed a shiny gold coin on the table in front of the seat across from him. \"Come now. Let me buy you a mead.\"\n\nOf course, the coin far exceeded the price of an entire barrel of the stuff. The man, still without uttering a word, pulled out the chair and dropped heavily upon it. Mouse offered a cheery smile back at him and waved at the wench, signaling for her to bring one for his new friend.\n\n\"What brings you here, mate?\" the man said, crossing his forearms on the table and leaning in.\n\n\"Business,\" Mouse answered. \"You?\"\n\n\"Loric, the keeper here, is a personal friend of mine.\" The emphasis he put on \"personal\" was intended to sound vaguely threatening, but the man had had enough drink in him to dispel the intended effect.\n\n\"Is that so? Seems a charming fellow.\"\n\n\"He wants no trouble.\"\n\n\"Then we are of like minds.\"\n\nFrustrated that his intimidation tactics weren't having the desired impact, he settled for something more straightforward. \"We've no need of your type here.\"\n\n\"My type?\" Mouse raised an eyebrow. \"Whatever do you mean? What type am I?\" He leaned in too so their noses were a hand's width apart. \"Am I your type?\" he whispered, and added a salacious wink to prevent any chance of being misunderstood.\n\nStartled, the man jerked back as if Mouse had punched him. The chair tipped onto its two back legs and with any more force he would have tumbled backward. He flushed in embarrassment and anger. \"I know a scoundrel when I see one.\"\n\nMouse let the corner of his mouth lift. \"I'm certain you do.\"\n\nThe wench arrived with the tankard and set it front of the man, then gave Mouse a curious look as she moved on. Mouse lifted his tankard to the man. \"To your health, Hawken.\"\n\nThe man lifted his tankard with a scowl and was about to drink, but stopped short. \"Hey. How do you know my name?\"\n\n\"Your reputation precedes you.\"\n\nHe put the tankard down again. Hard. Mead splashed onto the table. \"What's your game, little man?\"\n\nMouse pushed the plate aside and leaned in on his elbows. \"Take the coin that's in front of you. It's yours. Put it in your purse, and just talk to me for a bit. No harm in that, is there?\"\n\nHawken's eyes lowered and stared at the coin. It was more than Loric was paying him, surely. More than he would make in a month as captain of the night watch.\n\n\"It's your night off. Live a little.\"\n\nHis eyes shifted upward to meet Mouse's unwavering gaze, and his mouth tightened, but he slid the coin off the table into his palm and tucked it away at his belt.\n\n\"There now,\" Mouse cooed. \"Easy as that.\"\n\nHawken seemed to relax a bit, but he still held on to his scowl. He leaned back in his chair with his hand cupping the tankard. He'd taken the coin and was willing to play along, but he clearly was unsettled. The furrow in his brow implied he was still wary where this conversation was going. \"You didn't answer my question.\"\n\n\"A friend suggested I call on you.\" When Hawken tensed and made to straighten again, Mouse added quickly, \"A discreet friend.\"\n\nWhen the meaning of what Mouse alluded to seeped through the mead-filled brain, Hawken paled slightly. Worry flashed across his features. His shoulders tightened toward his ears. \"I don't know what you're talking about.\"\n\nMouse knew to proceed carefully. A man with a dark secret to protect could be easily spooked. Hawken had much to lose if his secret compulsion was exposed. His position in the guard certainly, and likely his family as well. But Orentegan morality law was strict on the matter. If caught, he would be thrown in irons. At the minimum.\n\nBut it was those dark impulses that kept him at the table, kept him talking. Mouse had piqued his interest enough for him to face the danger. Perhaps the risk was even part of the appeal.\n\n\"Oh, cut the shit. You know.\"\n\n\"What do you want?\"\n\nMouse chuckled and narrowed his eyes. \"I think you can guess.\"\n\nHawken stared, his mouth tightly pressed. Mouse could see in his eyes him weighing the risk against his desire.\n\n\"I think we can help each other out, friend. I understand the great pressure you're under. I will gladly provide you with the relief you require. And then some. I just have a small favor to ask in return.\"\n\n\"A favor?\" Hawken's expression darkened. \"I've no coin to--\"\n\n\"No, no. Nothing so base, I assure you.\" Mouse took a drink from the tankard, and as he set it down again, he made a show of wiping his lower lip with his forefinger and thumb. \"If what you see doesn't appeal to you, then take the coin, and our business is concluded. I will bid you a fair evening.\"\n\nHawken didn't move. In fact, he'd turned so rigid Mouse wondered if he'd turned to stone instead. But from the corner of his eye, he could see the pinky of the man's left hand quivering.\n\n\"But as I understand it,\" Mouse continued, \"sometimes it gets rather late, so you decide to take one of the empty rooms here instead of heading home to your adoring wife. Yes? Well, take Loric up on the offer tonight as well. Only....\" Mouse paused and gave the man a wry smile. \"Tonight there'll be a slight change.\"\n\nMouse took Hawken's silence as permission to proceed.\n\n\"Between your legs on the chair you'll find my room key. Use it. I'll be awake and waiting for you. And just so as to not arouse any suspicion, I need you to threaten me. Loud and public. Make it look good.\"\n\nHawken's hand moved down beneath the tabletop. Mouse could tell the instant his fingers discovered the location of the key, and grinned as the man tried to work out how it got there. Then Hawken sprang to his feet. The chair capsized behind him with a dramatic crash. In one fluid move, with speed that surprised even Mouse, he moved around the table, grabbed Mouse by the jerkin, and lifted him bodily from the chair. He pulled his face in close--so close Mouse wondered if Hawken would kiss him instead.\n\n\"Listen, little man,\" he barked. The room around them had fallen suddenly silent as everyone turned to watch the spectacle. \"I've heard enough of your lies tonight. You get one night here and that's all. Then I don't want to see your face again. Understood?\" With a mighty shove, he flung Mouse back down into his chair. \"Any trouble and I'll gut you.\"\n\nWithout looking back, Hawken crossed the tavern and took his seat at the bar. A good performance. Mouse did his best to look shaken by the incident as everyone around him pretended they weren't watching the scene and picked up their conversations again. Satisfied, he straightened out his jerkin, drained the last of his mead from the tankard, and sauntered back to his room.\n\nTHE FIRST time, Mouse had him cooing like child. After the second time, Hawken was eating out of his hand. For a chance at a third go, Mouse was pretty sure the guard captain would do just about anything.\n\nPanting and covered in a slick sheen of sweat, Hawken slumped back against the bed frame. \"Gods!\" he grunted. \"You are strong for your size.\" His hand moved to his cheek as if just noticing something. It came away with a shiny wet pearl on the tip of his finger. He held up his finger to show Mouse and made a low sound that might have been a chuckle. \"And that's a first.\"\n\n\"That's nothing,\" Mouse replied with a laugh. \"You actually hit the bed frame behind you.\"\n\nHawken glanced over his shoulder at the wet spot and opened up with a laugh himself.\n\n\"Well, now,\" said Mouse, sitting back on his heels. \"I was beginning to think you weren't capable of that.\"\n\nHawken looked up at him with his head askew. \"Of what?\"\n\n\"Laughing.\"\n\nThe transformation in a few short hours had been striking. The tense and brooding man that arrived at his door was now relaxed and lively--if a bit winded at present. He was very nearly giddy, as if he had drank in too much too quickly. Being in a big comfortable bed helped. It was probably Hawken's first time fully naked and in an actual bed with a man which, having previously only experienced backrooms and alleys, tended to change one's perspective on the event. Hawken's horizons had been now irreversibly opened to new and unforeseen territory. Anything was possible now. At this rate of progress, the tables would be turned about, and Mouse would be fucking the goon by morning.\n\nHawken grimaced sheepishly. \"I have been accused from time to time of being too serious.\"\n\n\"You don't say.\"\n\nHawken put his foot on Mouse's chest and playfully pushed him backward. \"I suppose,\" he said, a hint of sobriety seeping back into his tone, \"I hadn't had much cause to laugh. Before now.\"\n\n\"Well, you should make a point of doing it more. It becomes you.\"\n\nHawken smiled back at the compliment.\n\nMouse propped himself up with his arms behind him, thankful for the respite. For him, the night wasn't exactly recreational. In point of fact, it was a lot of fucking work. As was typical of bedding men like Hawken, it was Mouse who ended up doing most of the heavy lifting. Men new to this variant approach to coitus tended to drop into their typical one-sided role. Stick it in something, move around a bit, then leave. At some point during the second romp, though, Hawken figured out there was more to it than just thrusting the hips with his eyes closed, and lo and behold, he did his own part to experiment.\n\nA bit.\n\nMouse--not one to be passive--also reminded him, albeit gently, that he didn't need to be in complete control now. A novel idea.\n\nDespite Hawken's awkwardness and narrow view of what could happen between them, it wasn't a horrible experience. The man was remarkably fit. But then, one does not become captain of the guard by sitting around eating biscuits and salted pork all day. His legs and ass alone could keep Mouse entertained for some time. And he was actually more hygienic than he expected. He'd had a thorough scrub down within days, not weeks. Not always the case on such jobs, so a pleasant and unexpected windfall.\n\nAll in all, he'd encountered worse.\n\nHe reached over to the small bedside table and grabbed his pewter mug of wine and a bunch of grapes from a tray, and in one fluid motion tossed his leg over the big man's midsection. Straddling his waist, he took a long swig from the mug then dangled the bunch over Hawken's mouth, who plucked off a few with his teeth while growling like a sad imitation of rabid dog.\n\nWhile he chewed, his face lost some of its joy as a thought occurred to him. \"Will I see you again?\" he asked.\n\nExperience lit a warning in his mind. He had seen that particular look in the eye enough times before, heard that distinctive tone in the voice.\n\nThere was a danger here, and he would need to navigate this delicately. When used to quick alley encounters, time in a bed for the first time had a new intimacy that could be exhilarating and intoxicating. And what was once a pragmatic satisfying of base urges could become an emotional connection never considered possible. That could lead to all kinds of unpredictable behaviors. Mouse knew to tread carefully.\n\n\"We are hardly finished with tonight, my dear,\" Mouse replied. He took another gulp of wine, then leaned in to put the mug back on the tray. Their bellies touched. When Mouse moved to push himself back up, Hawken put one strong hand at the back of Mouse's neck and pulled his face toward his own. Their lips pressed, and Mouse opened up his mouth to allow Hawken's hungry tongue to penetrate. Hawken pressed harder. The kiss was something beyond passionate and closer to something resembling desperation.\n\nMouse allowed it for a time, then gently pulled away, making certain he had a smile on his face.\n\n\"I'm sorry,\" Hawken said, suddenly sheepish. \"I've never done that before.\"\n\n\"What? Kissed? That's what two people do when they're in bed together.\"\n\n\"I mean, kissed... you know....\"\n\n\"A man,\" Mouse filled in for him.\n\n\"Yeah. I never wanted to before.\"\n\nMouse leaned forward and kissed him this time but softer. He needed to keep this moving--keep him talking before ideas about round three entered his head. Or the opposite happened. Hawken was at risk of becoming serious now, sinking into a maelstrom of new emotions.\n\nHe peeled himself off Hawken's lap and crossed the room to his pack that was sitting on a chair. All the while he felt Hawken's eyes on him. He dug through the pack until he found his pipe and small leather pouch of smokeweed. He packed the pipe, lit it with a candle flame, then puffed until the end glowed red.\n\nHawken scooted back until his shoulders were higher up against the headboard. \"You never really answered my question.\"\n\n\"What question was that?\" Mouse puffed out a tiny cloud as he returned his pouch, acting casual. He knew very well what the question was.\n\n\"Will I see you again? After tonight?\"\n\n\"I'm in town for only a few days. I have some business to attend to, and once it's concluded, I have to leave. But until then....\" He gave Hawken a generous smile with the pipe still clenched in his teeth. \"I am your toy to do with what you will.\"\n\nThe big man's expression was relief with a little disappointment mixed in behind it. He was already hoping for a long-term affair. Of course, Mouse would ensure he never saw the man again come morning. No good would come of prolonging this further.\n\nMouse intentionally kept his distance while he puffed away at his pipe to keep things from slipping again into becoming physical. He played an act of looking for something in his pack and let a silence linger between them a bit. And as he'd hoped, Hawken wanted to keep up the connection he was feeling, so he filled in the distance with conversation.\n\n\"What nature of business?\"\n\n\"Missing person,\" Mouse replied matter-of-factly and pretended to give up whatever he looking for. \"Which, since you bring it up....\" He moved over to the bed again, climbed up and sat on his heels next to Hawken, \"Reminds me you now owe me a favor.\"\n\nHawken's expression was a mix of suspicion and curiosity. \"All right, then. I suppose you held up your end of the bargain.\"\n\n\"I held my end up twice if you'll recall. Or more accurately, you held it up for me.\"\n\nHawken let his head fall back to bark out a laugh. \"True enough. What's this favor?\"\n\n\"I need a little information. I was informed you might be able to help me out.\"\n\nHawken raised an eyebrow. \"Is that so?\"\n\nMouse grinned wolfishly. \"Might be something special in it for you.\" He threw his leg over Hawken's waist again and straddled him. \"A very distinctive reward.\"\n\nThat got his attention. Hawken returned an eager boyish grin. \"Oh really? How's that?\"\n\nTo make his point, Mouse shifted his hips side to side, grinding his bare ass against Hawken's groin. \"If I get this wrapped up quick, I can spend more time here. I have yet more wisdom to impart to my new and rather skilled apprentice.\"\n\n\"I'm listening,\" he replied, his grin broadening.\n\n\"I'm looking for a nobleman that's disappeared here.\"\n\n\"A nobleman, huh?\"\n\n\"Rumor around town says he was arrested for some reason, but I checked with all the jailhouses and found no sign of him anywhere.\"\n\nHawken pursed his lips. \"This wouldn't happen to be more than just any nobleman, would it? Someone big, like the son of a provincial duke, perhaps?\"\n\n\"It's possible. My client had coin to spend. But all I have to go on is a name.\"\n\n\"Garron?\"\n\n\"Yes, that's the name. You know of him?\"\n\nHawken's expression turned murky. \"You're not going to find him in any jailhouse. Where they have him exactly, to be honest, I'm not sure. He was arrested at Lord Delgan's manor by the chancellor himself. He is likely being kept in there.\"\n\n\"The chancellor?\" Mouse feigned that this wasn't information he already knew. \"That would mean a crime against the state.\"\n\n\"That is what I assumed at first as well.\"\n\nMouse studied him a moment, trying to read him. Something behind his eyes told him this wasn't mere hearsay. He took a calculated risk. \"You were there when it happened.\"\n\n\"Yes,\" Hawken replied in a tight voice. \"I was assigned to Duke Delgan's court at the time.\"\n\nHis reassignment could have had something directly to do with this business--something Mouse hadn't anticipated. Taurin had selected his contact well. \"So, what happened?\"\n\n\"The chancellor described it as 'corruption of the duke's nephew.' I don't know what that means, but the charge against that Lord Garron was bogus.\"\n\n\"How do you know?\"\n\n\"I was assigned to the apartments of Lady Calpha, Delgan's sister. Rax, the nephew, was never in any direct contact with Lord Garron, so whatever 'corruption' took place, it wasn't Lord Garron who did the corrupting. I said as much to Chancellor Elasiar. He didn't appreciate my contribution.\"\n\n\"Is that what got you reassigned?\"\n\n\"Maybe in part.\" Hawken smirked with a mischievous glint in his eyes. \"But I would wager it was me shoving a mouthy prig down a privy hole that probably had more to do with it.\"\n\nMouse couldn't help but reflect Hawken's grin. \"Worth it?\"\n\nHawken shrugged. \"Maybe. Except for the lower pay grade. But I'll get back in the manor eventually. I already hear grumbling that things aren't run properly. They'll be asking me back to straighten the ship right again in no time.\"\n\nMouse didn't get the sense that this was idle bragging or some misguided delusion. He believed the man was probably good at what he did. \"So no ideas where they are keeping this corrupter of Orentegan youth?\"\n\n\"Are you here to negotiate his release of something?\"\n\n\"Not exactly.\"\n\nHawken's eyes widened with sudden understanding. \"Oh. Well, you won't be able to get to him if that's what you're thinking. Delgan will have him heavily guarded. My guess, he's not in the dungeon. Delgan may be shrewd, but he holds the aristocracy above all. I doubt he'd stoop to throwing a noble in a rat-filled hole.\"\n\n\"Where is he, then?\"\n\n\"For my coin, I'd wager he's locked away tight in one of the manor's four towers. Which one, I couldn't say.\" Hawken propped himself up on his elbows and leaned in close. \"That's all I know. So... enough talk. I'm ready for my reward.\"\n\nMouse thought about pressing the matter but acquiesced. There was probably little more he knew, if anything. Mouse had enough to go on now anyway. So, one more wrestling match between them, then slip out while Hawken was asleep and find a place for him to catch some himself. Too bad it wouldn't be here, he thought--but such was the life of a man in his profession. He grinned impishly down at the big lug. \"And well deserved it is,\" he said. \"Hope you've caught your breath. No quarter for you this time, Captain.\"\n\n# Chapter 5\n\nMOUSE WANDERED a few streets over from the Blue Stag and rented another room--one greatly inferior to the one he left behind but comfortable enough. Two strides in, he unloaded his gear, stripped, and collapsed on the small lumpy bed. He slept deeply until late morning. Then he washed from the basin, dressed again, and was back out on the streets.\n\nA ceiling of low clouds had rolled in sometime during the night, making for a cooler day. Fewer idlers and gadabouts were out to congest the streets and plazas, and shopkeepers kept their tables of wares off the curb for fear of rain. Mouse could now traverse the network of byways with ease; he preferred to avoid the major arteries whenever possible. He pointed himself toward low town. He would swing through the market square, this time more for a quiet meal and some processing time than for digging up any information.\n\nExcept he realized almost immediately that he was being followed.\n\nHe went about his business, acting casual and forcing his pursuer through clusters of people or large open areas that would make it hard for the clumsy oaf to stay inconspicuous. Mouse had no intention of losing him, which would be an easy enough affair. The idiot's skill at tailing someone was laughable. But for his own petty entertainment, he wanted to make the poor bastard work for it. He would find out who it was soon enough. In fact, he counted on it.\n\nMouse arrived at the market square soon after with his frustrated pursuer in tow. Strolling among the haphazard cluster of booths, he allowed a portion of his concentration to ponder the next stage of his mission. Hawken had given him a rough idea where Lord Garron was being held. That was all he really needed. He could work out the details once inside Duke Delgan's manor house--the sprawling, fortified structure that rested atop the crest of the hill. The city region of high town surrounded it like a dingy cowl. A high wall surrounded the entire manor, and as far as he could discern, it only had one gate. Breaching that would be tricky business, but he already had a plan taking shape in the back of his mind.\n\nLocating the young lord would be easy enough, certainly--getting him out again would be another matter. The logistics of the task began to weigh on him more and more. Compared to his typical work, like smuggling contraband or relieving an unsuspecting countess of her jewels, this job was fraught with too many potential hiccups. The size of Delgan's keep alone was daunting. Plus, Mouse had no idea yet of the duke's manpower inside, or how much of it was dedicated to protecting the prisoner. Was the young lord starved and beaten to the point that he would be too weak or too injured to transport out? Or so fat and entitled he couldn't wipe his own ass without a team of advisors and servants to guide him through the process? Either way, the escape attempt could fail before it began.\n\nThe word on Lord Garron's crime seemed to be some sordid insinuation that he preyed on young lads--a high crime here in Har Orentega. But Hawken's pillow testimony implied the charges raised against him were trumped up. They chose something that would be particularly heinous in the public view, which meant should word of his arrest and imprisonment leak, they wanted no one second-guessing the wisdom of the seizure. And there was also the issue of Lord Garron's sentence. What did they intend to do with him? A lengthy internment was unlikely. Was he already scheduled to meet his fate at the gallows? Time was now pressing in on his mission. If Mouse didn't locate him quickly, it might be too late for the poor lad.\n\nAnd one other vital question remained. Why was he arrested in the first place?\n\nThe question nagged at him like pinching smallclothes on a hot day--irksome and unrelenting. Two days of digging, and he was no closer to an answer. The more he learned, the more politics or leverage seemed unlikely motives. So... what, then? Was this some kind of revenge plot?\n\nHe pulled up a stool at a soup vendor's counter and when he caught the cook's eye, gestured toward one of her steaming cauldrons. A bowl of hot stew and an iron kettle of tea arrived in front of him moments later.\n\nHe spooned the first taste into his mouth.\n\nSome small, more logical corner of his mind chastised him. He was doing it again, of course--wasting mental energy on elements that had no bearing on the job at hand. The \"why\" was irrelevant. The details of his mission had been clearly defined. Rescue him. Get him out of this city and home to his duchy and his family. He wasn't being paid to find out what game Duke Delgan was playing at.\n\nBut he was always a sucker for a mystery. For his own good night's rest, he knew he would need to untangle this little knot. He knew himself well enough to know that he would be unable to just let it be. Delgan was up to something sinister, and Mouse needed to unravel what it was. If only for his own satisfaction.\n\nThe oaf that had been pursuing him so ineptly finally took a seat on the stool next to him.\n\n\"Wondered how long it would take for you to get here,\" Mouse said without even looking up from his bowl.\n\n\"Expecting me, were you?\"\n\n\"Might as well put a cowbell around your neck. Wouldn't make you any more obvious.\" From the corner of his vision, he saw the man tighten up. He had already guessed who he was and what this was about before the man opened his mouth to tell him. \"Your guild is shit, Keeler. You are aware of that, right?\"\n\nThe man started at the use of his name, though he tried to disguise it. It had been a guess on Mouse's part--though he was pretty sure the idiot would come looking for him himself. Not having worked his way up the ranks like most guild leaders, he was likely unaccustomed to delegating out work. He was a hands-on man, used to working the streets on his own. He was overconfident of his own abilities and did not trust others to do a job properly. Plus, Mouse knew, curiosity would get the better of him. He would want to see for himself who this mysterious intruder was that wreaked havoc in their guild headquarters.\n\nMouse knew all this because it was exactly how he would act himself if he took over a guild.\n\nKeeler was silent a moment while he readjusted his strategy. He quickly seemed to realize that his initial tactic of intimidation wasn't going to work with Mouse. \"We're in transition,\" he replied. \"Change in leadership.\"\n\n\"So I understand. Question remains which direction the organization is 'transitioning' toward. I have half the mind to usurp you and take over myself just on principle.\"\n\n\"You're an arrogant little bastard.\"\n\n\"It's hardly arrogance, I assure you. No idle boast.\" He took another slurp of stew. \"You received my message, and you had your demonstration of what I'm capable of. I assume that's why you've decided to come.\"\n\n\"You nearly killed my lieutenant.\"\n\nMouse nearly choked on his stew suppressing a guffaw. That was an officer in this shit organization? \"Consider it an object lesson on courtesy. And never underestimating who you're dealing with. It was his ill manners to blame for his current condition. And the ale. I merely pointed that out to him in a way he'll remember.\"\n\nThe man nodded in thought for a moment, clearly annoyed at Mouse's audacity at not being intimidated or affected by his presence. \"I'm speaking to you now as a professional courtesy. Guild member to guild member. I could have just as easily stabbed you in the back, you know.\"\n\n\"Try that tactic next time and see what happens.\" Mouse pushed aside his empty bowl and turned on his stool to face the man directly for the first time. He was what he expected--a brutish man, unkempt and excruciatingly obvious. Any guild member worth his salt, especially the one at the helm, knew how to blend in and not draw attention to himself when necessary. Every citizen on the street would take a single look and know Keeler for a thug. He had no finesse. No style.\n\n\"I appreciate you coming here to save me the trouble of hunting you down. So, now that you're here, a few questions, then.\"\n\nKeeler stared at him agog. \"I did not come here to answer your questions. I came here to tell you I want you out of my city. You have one day to be out of Har Orentega.\"\n\n\"I need to know what the guild knows about the arrest of a certain Lord Garron of Har Dionante. Not the official bullshit word you'd give the guard either. I need to know what you really know.\"\n\nKeeler put his palms on the counter and started to push himself up. \"We're done here. One d--\"\n\nMouse moved his hand to the man's shoulder. The long thin dirk palmed in his hand was little more than a needle. He pushed into the man's neck with just enough pressure to divot the skin without breaking it. \"Sit down, friend. We aren't nearly done yet.\"\n\nThe man lowered his wide ass back onto the stool.\n\n\"Make one move,\" Mouse added softly into his ear. \"And I'll puncture your neck with a barely perceivable hole....\" He applied a little pressure to emphasize his point. A tiny drop of blood appeared beneath the tip of the dirk. \"It will puncture your windpipe. And as you fall to the ground and gurgle and choke, I will tell everyone you're choking on a chicken bone and slip away quietly when they are trying to save you.\"\n\n\"I'll kill you for this,\" he growled.\n\n\"Better than you have tried. Now, what do you know about Lord Garron?\"\n\n\"I don't know nothing.\"\n\nMouse was starting to lose patience. \"Stop,\" he growled in his ear, pushing a little harder, \"wasting my time.\"\n\n\"All right. All right! We knew the arrest was going to happen. Days before.\"\n\n\"You knew ahead of time? How?\" This was an interesting development. And unexpected.\n\n\"We have a plant in the manor's kitchens to keep an eye on the rich peacocks that come and go in the city. Many of them are the personal guests of Duke Delgan, and we like to keep tabs on what they're doing with their money.\"\n\nMouse was genuinely surprised. \"Well, how about that? An effective plan with actual merit.\"\n\n\"Like I said, under new leadership. We are attempting to move things in the right direction here.\"\n\n\"If you're not careful, I might start to think of you as a legitimate professional. What did you learn?\"\n\n\"Days before the lord's arrest, a visitor came to talk to the duke, and over dinner, the plan to have him taken was laid out. Our mole overheard a good portion of it.\"\n\n\"So one, it was a trumped-up charge dreamed up ahead of time. And two\"--and the more important, in Mouse's view--\"the order came from outside of Har Orentega.\"\n\n\"Yes and yes,\" the guild leader responded.\n\nMouse took a moment to chew on that. So, someone outside of the town wanted this wealthy aristocrat out of the way. And this mysterious puppeteer had strings in Duke Delgan's hands and was making him dance along with his scheme. Was Delgan getting something in return for this, or did the visitor have some power over the duke?\n\n\"Any idea who this stranger was that called upon the illustrious Duke Delgan?\"\n\n\"None whatsoever. His name was never used.\"\n\n\"Convenient. So why the hush-hush, then? Why is the guild sitting on this information and not using it?\"\n\nKeeler winced and shifted on his stool awkwardly. \"A representative from the manor showed up at our secret guild house, bold as balls. Said Duke Delgan knew of our mole and what she heard. Said if the guild kept quiet she could remain doing guild business as long as she didn't interfere in the dukedom's affairs or get in Delgan's way. If word got out of the plan to arrest the young lord, he'd know where the leak came from, and he'd have her arrested and executed for treason. The choice for us was obvious. For the betterment of the guild, we kept quiet.\"\n\nMouse pursed his lips and nodded. Given those options, he would have made the same decision, at least until he was able to get a new mole secured in the manor and a new base of operations established. He wondered if Keeler even thought to do either of those things.\n\nIt explained why his poking around had made people like Taurin a bit jumpy. The Shadow Elite were always closely aligned with guild activity, and Mouse's inquiry could cause business to turn south if the guild was reined in. A lot of coin could be at stake. The duke had very effectively kept the information about Garron contained. Mouse was getting the sense that this Delgan was a savvy one.\n\n\"So where is the lord being held?\"\n\n\"In one of the towers, I'm told.\"\n\nGood. That corroborated Hawken's opinion. \"Which one?\" he asked. \"There's four.\"\n\n\"The tall one--ow!\" Keeler flinched when Mouse applied a little more pressure with the dirk. He wasn't in the mood for any of Keeler's cheek. \"I don't know,\" he added quickly. \"I do know it's heavily guarded. You'll never get in there, if that's your plan.\"\n\n\"My plan is none of your concern.\"\n\nHe pulled the dirk from the man's neck and slipped it back into his boot. The man put his hand to the spot, then checked his fingers for blood. \"You have nothing to worry about from me,\" Mouse told him. \"I have no intention of upsetting the operations for the guild here in Har Orentega. Or endangering your plant in the manor. But if anything were to change, I'll send word that she should disappear from the manor.\"\n\n\"What is your angle in all this?\"\n\n\"Never mind. But I'll not get in the way of guild business if that's what you're worried about.\" He threw some coin on the counter to cover the cost of his bowl of stew and stood up. \"You'll not see me again.\"\n\nAs he walked away from the vendor's stand, he heard the guild leader mutter, \"Fine by me.\"\n\n# Chapter 6\n\nTHE FOUR-TEAM carriage pulled through the first open portcullis into the arched entry of the gatehouse, the sole entry into the walled complex around Duke Delgan's expansive keep. Though quaintly referred to in the city as the manor house, the structure beyond the gate was by all accounts a fortress. Though not particularly majestic or architecturally inventive, the manor house was, with its dark volcanic stone and soaring towers, an imposing structure nonetheless.\n\nThe grand team of horses stamped to an impatient halt at the second closed gate while bored guards in fine Orentegan livery drifted out of their chambers to address the new arrival. The guardsmen lumbered into position around the carriage at a languid shuffle, like the dead roused from their graves. Some were still pulling on their helms as they stumbled from the gatehouse, looking annoyed that their naps or card games had been interrupted. They kept close to the walls and eventually posed themselves at attention, their ceremonial lances planted on the ground next to them. A well-practiced routine that had lost its intended pomp.\n\nAs soon as the carriage rolled to a halt, one of the four hulking bodyguards that accompanied it on foot marched up to the carriage door, opened it wide, and positioned a stepping stool beneath the opening. The other hired men joined him in a snappy choreographed display. They flanked the carriage door with stiff backs, wide stances, and hands clasped behind their backs.\n\nVery official, Mouse thought, as he viewed the proceedings from inside the carriage. They had only practiced it a few times but still managed to pull it off swimmingly. Coin was a terrific motivator.\n\nTo pull this off, Mouse had had to return to the Scarlet Unicorn and rely heavily on Taurin's generosity and fine taste. Only, Taurin wasn't as of yet entirely aware of how generous he had been. Or how generous some of his current patrons were either, who hopefully wouldn't be missing this fine carriage and its team of horses anytime soon. Taurin had made it exceedingly clear how unavailable he would be, so Mouse didn't want to trouble him with such trivialities. So, he absconded quietly with a few borrowed items.\n\nMouse wouldn't have need of these items for long. He hoped to have them all returned to the Scarlet Unicorn before anyone took notice of their absence.\n\nMouse grinned at the young man sitting across from him, a sharp-witted young stable hand named Jastar. When Mouse first discovered him, he was filthy and unkempt, but Mouse sensed the potential underneath the layers of grime. And he was right. The lad had cleaned up remarkably well and was exactly what he needed. \"Ready?\"\n\nJastar squirmed in his new finery but nodded. It hadn't taken much to convince the lad to join in, but now he seemed less certain.\n\nMouse stepped down from the carriage, scrubbed clean, freshly shaven, and resplendent in his newly acquired regalia. He was enshrouded in a heavy surcoat of ocean blue, embellished with gold thread piping and swirling brocades at the sleeves and hem. On his head was a matching plumed hat so ridiculous it made Mouse laugh just to look at it. It was a style thankfully now out of fashion, which explained why it was buried deep in one of Taurin's trunks, but it was ostentatious enough to suit his current guise. The boots, hose, and ruffled chemise were all borrowed from Taurin's wardrobe as well. The garish jewelry on his fingers and draping over his breast came from the same patron who so generously supplied the carriage, who was likely still passed out from too much physical activity with one of Taurin's girls and the strong mulled wine.\n\nMouse couldn't wait to doff the lot of it. Especially the hose. He despised wearing hose.\n\nThe young attendant followed him out of the carriage, handed Mouse a cane, and draped a small cape across his shoulders.\n\nThe captain of the gatehouse approached and made a perfunctory bow. \"Good day, m'lord,\" he greeted flatly. He opened his mouth to launch further into his standard speech but stopped short when something caught his eye. His gaze was pulled to the bodyguard standing at Mouse's right hand, and his eyes widened in sudden recognition.\n\n\"Ah,\" Mouse said with a bright smile. \"I see you remember my recently acquired footman.\"\n\nThe guard stared at Hawken in unabashed shock. \"Yes,\" he replied, then remembered himself. \"I mean, yes, m'lord.\"\n\nHawken allowed himself a satisfied smirk in return. He said nothing, embracing his role as a rich merchant's bodyguard. He'd been quick to agree to this scheme, even before he learned how much coin Mouse was adding to his purse.\n\nThe gatehouse captain did his best to recover, but even as he spoke to Mouse, he repeatedly shifted his gaze toward Hawken, who continued to grin down on him. \"Your... your business here at the manor, m'lord?\" the guard asked.\n\nMouse assumed an expression of surprise and mild irritation. \"Were you not informed of my pending arrival?\"\n\n\"I am afraid not, m'lord. You are?\"\n\nMouse's narrow gaze swung to Hawken, who made a slight eye roll and shrugged. \"Appears your assessment of the place was depressingly accurate,\" Mouse said to him in a low voice but plenty loud enough for the guardsman to hear. \"Small wonder you were quick to take my offer.\" He returned his gaze to the flummoxed guard and managed to squeeze out a condescending smile. \"Yazura Tenda.\" He waited for the guard to respond, and when the guard just stared, he added, \"From Arendia. The capital? You bumpkins down here have heard of the capital, yes? It's the big city to the north, where our glorious king resides.\"\n\nThe footman to Mouse's left, a bruiser by the name of Reginus Fenn, snickered deep his throat. Hawken had dredged him up from the docks somewhere. Mouse presumed they knew each other from some dubious connection, but he didn't ask. The man was enormous, larger than Hawken, even, and had a mane of wild, rust-colored hair that Mouse insisted he comb out and pull back into a tail behind him.\n\n\"I'm familiar with it,\" the guard replied, his jaw line tightening with annoyance. \"You have papers, I presume?\"\n\nMouse sighed as if this was all too much for him bear. He nodded to Jastar. The lad reached into the carriage, fumbled with the insides of a wooden box, and pulled out a bone scroll case. He handed it to Mouse, who slid out a scroll sealed with wax, and then passed the scroll over to the guardsman.\n\nMouse hoped the ink had had enough time to dry. The forged papers were scribed just an hour before. The seal press had indeed come from Arendia. Mouse had stolen it from a well-connected family last time he'd been to the capital city. The family had certainly realized it had been stolen by now and updated the seal, but there was little chance this grunt was going to expend the energy to cross-reference the image imbedded in the wax with his records.\n\nThe guard broke the seal and studied the document a moment. No surprise he could read--he wouldn't have been assigned to this particular post otherwise--but his skills were not likely strong enough to spot a well-made forgery. After a thorough scan, he handed the parchment back to Mouse. \"My apologies, Lord Tenda. Your visit was not relayed to us. We will open the gate for you immediately and someone will direct you to the manor. Is His Lordship, the duke, expecting you?\"\n\n\"Well, I certainly hope so. Announce me nonetheless.\" Mouse didn't wait for a response. He spun on his heel and climbed back into his borrowed carriage. Hawken and the attendant climbed in after him.\n\nThe inner gate cranked open with a steady grinding and scraping. Once the portcullis was fully raised, the team was brought into motion again. Mouse reclined back in his seat and smiled.\n\nJastar breathed out heavily, puffing out his cheeks. \"I can't believe they fell for it.\"\n\n\"Of course they fell for it,\" Mouse replied.\n\nThe lad shook his head. \"Amazing. You actually got us into Delgan's manor. I thought for certain the guards would see right through it and arrest us all on the spot.\"\n\nMouse chuckled at his youthful astonishment. This was the easy part.\n\n\"You're enjoying yourself,\" Hawken said with a tightened brow.\n\nMouse couldn't deny it. He caught himself grinning wider. \"As are you, I imagine.\"\n\nHawken shrugged wryly, but Mouse could still detect the gleeful glint in his eye. He'd created a convert, it seemed.\n\n\"Ah, be careful, my friend,\" Mouse told him. \"The taste of adventuring is the most potent of wines. Too large of a draught, and you might find yourself lost in it.\"\n\n\"Then I shall curse you henceforth for my crapulence, Mouse. It was you who first handed me the cup.\"\n\n# Chapter 7\n\nTHE CARRIAGE was led to a side entrance of the manor where guards delicately delivered \"Lord Tenda\" from the carriage and escorted him inside. A midrank guardsman and a man Mouse assumed was the chancellor awaited them in a receiving area just inside the door.\n\nThe man, clothed in official-looking robes, glided forward, long hands clasped at his waistline as if protecting his groin. He stopped a pace or two in front of Mouse and pushed his shoulders back. He was gaunt to the point of frail, sallow to the point of sickly, and everything about the man said prickly and ill-humored.\n\n\"Good day to you, Master Tenda. I am Lord Elasiar, Chancellor to Duke Delgan and custodian of the seal of Har Orentega.\"\n\nThe greeting was civil but held little warmth. The man was understandably suspicious, but he had no reason to turn them away just yet. He studied Mouse with narrow birdlike eyes, his lips pursed behind the closely trimmed beard. With a brow slightly raised, he waited for Mouse to acknowledge his station.\n\nMouse did his best to look unimpressed. He returned a weak smile and spoke to him as if he were the stable boy. \"Has the duke been informed of my arrival yet, Chancellor?\"\n\nA noticeable pause. \"A messenger has been dispatched, yes,\" Elasiar replied. Which meant they were still trying to figure out what was going on. \"If you would indulge us, sir, and present your papers again....\"\n\nMouse made a dramatic show of rolling his eyes. \"If you insist,\" he huffed, \"but this grows tiresome.\" He snapped his fingers. The attendant reached into the haversack slung on his shoulder and again removed the bone case with scroll. He handed it to Hawken, who pulled out the scroll and passed it on to Elasiar.\n\nThe chancellor froze momentarily when he looked up and noticed for the first time who was handing over the parchment. Hawken returned a knowing smile but said nothing. Elasiar recovered quickly. He unfurled the scroll and read through the credentials with a critical eye. Brow tight, he scanned it thoroughly before handing the scroll back to the Hawken. He seemed disappointed he couldn't find anything amiss. \"Seems, Master Tenda,\" he said, \"you have taken one of our own in your employ.\"\n\n\"Indeed!\" Mouse beamed with excitement. \"You know Hawken? Quite the find, I'd say. I discovered him stationed on the wall of all places. A diamond tossed in the shit, if you will. Ridiculous! What blind idiot thought to waste him there? I spotted him training some of the wall grunts and recognized his talent immediately. Small wonder it took little convincing to have him take my offer.\"\n\nElasiar's lips tightened a fraction, though he clearly was fighting to keep his face neutral. Mouse resisted a wider smile. He'd guessed correctly that it was Elasiar who had reassigned Hawken. Mouse thrilled at any opportunity to abuse one of the elite. The chancellor would be unaccustomed to being so openly criticized but was savvy enough not to offend someone of a higher station than himself by taking umbrage. Instead of responding directly, he went back to the matter at hand.\n\n\"It appears, sir, that your arrival in our city was not properly relayed to us.\"\n\n\"So I've been led to believe,\" Mouse replied with an air of well-measured hauteur. \"Honestly, to be treated such after so long a journey. I've a mind to just turn about and head out again. I'm sure some other far-flung village around here will appreciate my proposition. Har Dionante perhaps....\"\n\nElasiar forced a smile. Mouse could read the dialogue moving through his mind from the reddening color on his cheeks. Fucking big-city types marching in here thinking they can tell us how to function.\n\n\"Allow me to direct you to a chamber where you can wait comfortably until I've heard from Duke Delgan himself. I'm certain we can get to the bottom of this unfortunate misunderstanding.\"\n\n\"I certainly hope so. This ordeal has been quite taxing, I assure you.\" He turned his head to Hawken, who shook his head in dismay. Then he gestured toward the entryway. \"Lead on, Chancellor.\"\n\nHis party, consisting of Hawken, the attendant, and the other three footmen, was escorted down a series of corridors to a small but elegant chamber, complete with wash basin and basket of fruit. Mouse nodded in approval. He'd learned to gauge his perceived influence by the accommodations they settled him into. Had this been a barracks or some nondescript meeting room, he'd have been nervous and might have looked for an excuse to slip out hastily. But the chamber was of decent quality. Not a high-end apartment meant for visiting nobles but good enough. The accommodations suggested some uncertainty on Elasiar's part, so the manor staff wasn't uncorking the good wine just yet.\n\nHe had to move things along quickly. He checked the latch on the door. Unlocked. That was a good sign.\n\nHe pulled the door open a fraction to get a glimpse of the corridor. As far as he could tell, no guard had been posted outside. At least none directly outside the door. But that didn't mean a guard wasn't stationed close by, or the room itself wasn't being spied on through a peephole. He put a hand to the shoulder of one of the footmen and leaned into his ear. \"Do a quick loop around the halls in this area and report back anything you see. If anyone questions you, ask them where the privy is. If pressured or followed, don't take any chances. Just head back. If you're escorted back, sneeze when you reach the door.\" The footman nodded once and slipped out.\n\nMouse turned to Jastar. \"Ready?\"\n\nThe lad made a small bob of his head.\n\n\"Get started, then,\" Mouse told him. Jastar nodded again and started to disrobe. Mouse did the same, tossing the ridiculous hat onto a chair and unfastening the buttons of his surcoat from the neck down. Hawken looked on with pressed lips, clearly dubious.\n\n\"Are you certain this is going to work?\"\n\n\"Without question.\" Jastar was specifically chosen because he was very nearly Mouse's size. To them, Mouse was just an arrogant fop in a big hat. \"They won't be expecting the change, so they won't see it.\" As long as the lad could continue the role in a convincing manner, of course, but he kept that to himself. He turned his gaze toward the attendant with a smirk. \"You can act snide and superior, right?\"\n\nJastar grinned.\n\n\"Just keep the tone elevated like I taught you,\" Mouse added. \"No street talk.\"\n\nJastar unlaced and removed the jerkin, then pulled the tunic off over his head exposing his near-hairless and well-proportioned torso. Mouse turned his gaze away. Elsewhere, he might have taken the opportunity to appreciate the lad's form, but not here. He would not allow his base urges to undermine his focus. Hawken, on the other hand, could not pull his hungry leer away--he looked like a wolf eyeing a chicken. Mouse made two quick snaps of his fingers to draw Hawken's attention from the show. Hawken started as if awoken from a dream and, looking at Mouse, winced.\n\n\"Time for that later, friend,\" Mouse told him quietly as he began to disrobe as well.\n\nHawken nodded sheepishly.\n\nOne by one, Mouse stripped off each item of his lavish garb and tossed it off to Jastar, who stood nearly bare in his smallclothes, waiting. What a compromising predicament it would be should a guard or the chancellor himself return now, thought Mouse. Two nearly naked men standing about in the chamber. How would he spin a yarn to explain that, he wondered. The other footman scooped up Jastar's garments from the floor and handed them to Mouse an item at a time.\n\nIn short order, the lad was fully dressed in the blue surcoat with gold buttons and the monstrous hat that looked like a giant bird landed on his head, and Mouse in an attendant's jerkin and hose.\n\n\"How do I look?\" Jastar asked.\n\nMouse bobbed his head in satisfaction. \"Every bit the entitled noble shit.\"\n\nHawken looked more unsure. \"What if Duke Delgan actually grants the audience with him?\"\n\n\"He won't. We will be politely told to bugger off today and come back tomorrow when his schedule is cleared.\"\n\n\"How can you be so certain?\" Hawken glanced at the lad with clear concern. \"What if they investigate who this Lord Tenda is?\"\n\n\"They will find he is wealthy beyond measure and very well connected in the king's court.\"\n\nHawken raised an eyebrow. \"You've run this scam before?\"\n\nMouse chuckled as he guided Jastar by the shoulders to the center of the chamber. He sculpted the lad like a clay doll. Lifted his chin, pushed in his lower back, positioned one hand on his hip while the other gripped the cane. \"When you walk,\" Mouse told him, \"remember to think someone has wired your shoulder blades together.\"\n\nWhen Mouse glanced over, Hawken still had a brow knitted with concern.\n\n\"If they notice you missing....\" the big man began.\n\nMouse reached up and grabbed Hawken's shoulders. \"We've gone over this. A stiff neck like the Elasiar pays not a speck of attention to ground beetles like attendants or footmen. They're invisible to him. Furniture. The puffed-up sot will make his appearance just long enough to send us packing until tomorrow, then be on his way. They will assign a few guards who will see us out. Guards that have no idea how many of us came in.\"\n\nHawken shook his head. \"I certainly hope you're right.\"\n\n\"It will work. Trust me. Just make sure you're ready and available for tomorrow.\"\n\n\"Tomorrow?\"\n\n\"An official visitor will call at your residence.\" Hawken went a little ashen. Mouse smiled. \"Not what you're thinking. If you were hired by someone as powerful as Tenda, you must be more valuable than they realized. The chancellor will offer you a position back here in the manor. Effective immediately, most likely, and I would wager at an increased wage.\"\n\n\"You have it all figured out, don't you?\" His tone was sharp, but there was a glint in Hawken's eye when he said it.\n\n\"I have indeed,\" Mouse replied coolly. \"When they arrive, just make a convincing show that you are packing to relocate to the capital. I, they will have learned, have already left for Arendia spitting with indignation.\"\n\nThe door to the chamber opened, and the young footman slipped in. He moved directly toward Jastar now dressed as Tenda and was about to start a whispered conference with him but stopped short, realizing suddenly it was not Mouse. He looked around, embarrassed until he saw Mouse waving at him.\n\nMouse glanced up at Hawken. \"Still any doubts?\"\n\nThe footman grimaced and made his way over.\n\n\"The corridors are clear,\" the footman told him. \"No one stopped or questioned me. But the chancellor is making his way back. I returned as soon as I spotted him coming down the corridor.\"\n\nMouse nodded. \"Good.\"\n\n\"There is more,\" the footman added as he looked over his shoulder to see if the chamber door was opening yet. \"I happened on two lords talking about an area of the manor they referred to as 'restricted.'\"\n\nMouse's back straightened. \"Go on.\"\n\n\"I didn't learn what part of the manor it was. Neither seemed to know why it was closed off, but only those with certain approval are apparently free to go there. Both spoiled dandies were miffed about their exclusion from the list and expressed hope they would be allowed to return soon. That was when one said he heard it rumored that it was only a matter of day or two before they were.\"\n\nMouse tightened his lips as he clapped him on the shoulder. \"Well done,\" he said just as door to the chamber swung open. \"Well done, indeed.\"\n\nIf this rumor was true, that meant Mouse had less time than he'd hoped.\n\nGuards preceded the chancellor and took position flanking the doorway. Elasiar glided into the room with a stiff-backed gait, like a man who assumed everyone was rapt by his very presence. His gaze centered on Jastar alone. All others in the room did not warrant his attention. \"Lord Tenda.\" He greeted him with a polite bow that carried no warmth. \"I pray we did not keep you waiting too long.\"\n\nJastar scowled and shifted his shoulders back a fraction. \"I'm certain it was just these dreadful accommodations that made it seem so.\"\n\nMouse nearly broke a smile. He'd chosen his doppelganger well, it seemed.\n\n\"Forgive us. It was what we had available at such short notice.\" The chancellor approached and wore his political face--the one that broadcast he really, really cared. \"I'm afraid I'm going to have to test your patience further. It appears His Grace is unable to pull himself away from some important state business. Which means he will be unable to meet with you today. But if we could impose on you further, we will gladly find room in our docket tomorrow.\"\n\nJastar rolled his eyes. \"Was hoping to be back on the road tomorrow.\" He stomped the end of his cane on the floor like a petulant child that didn't get the pony he demanded. \"I've had just about all I can take of this uncultured backwater. Can you promise at the least it'll be in the morning?\"\n\nMouse winced. That sounded too deferential. Too meek. A fop like Tenda would never ask Elasiar for anything. Mouse had coached him with the proper words to say and the diction in which to say them, but in the moment, there was always the need to extemporize. Though the lad had taken his notes well, there was a danger of him improvising too much, going too far. Mouse watched Elasiar's face for signs of alarm but only saw irritation.\n\n\"I'm certain we can make it work at your convenience,\" Elasiar replied.\n\nMouse slowly exhaled.\n\n\"If you were concerned about what was convenient, I'd be speaking to the duke at present. But very well.\" Jastar threw up his hand. \"Tomorrow it is. I'll be here early. Make certain the gate is prepared for me this time.\"\n\nMouse wondered if he should take this prot\u00e9g\u00e9 under his wing when he left the city. The lad had promise.\n\n\"Of course,\" said the chancellor. \"I have taken the liberty of seeing your carriage prepared.\" Which meant the chancellor had already had it thoroughly searched. \"It awaits you in the yard. These men will direct you. I take my leave, and I look forward to receiving you tomorrow.\"\n\nJastar waved him off.\n\nElasiar bowed and turned to leave but stopped unexpectedly to face Hawken. \"Sir Hawken, I was hoping perhaps I may have a word with you tomorrow as well. While His Grace and Master Tenda are meeting perhaps?\"\n\n\"Of course, Lord Chancellor.\" Hawken lowered one knee in a respectful bow. \"I am at your service.\"\n\nElasiar's mouth lifted at the corners into something that resembled an authentic smile. \"Excellent. I look forward to it.\" And without another word, he spun about and departed.\n\nHawken looked to Mouse with an arched brow. \"Are you ever wrong?\"\n\nMouse shrugged. \"People are predictable. Mostly.\"\n\nThis time tomorrow, Hawken would be better placed in the manor than he had been before. Mouse wondered if he'd ever see him again. He was surprised to realize he liked the man--and felt he could indeed be better utilized elsewhere than this backwater shithole of a duchy. He'd be happier, certainly. But if all went according to plan, Mouse would have freed the imprisoned Lord Garron and be long gone before Hawken took up his position in the manor again.\n\n\"You'll do me the kindness as to follow me, Master Tenda,\" said the guard at the door.\n\nJustar still looked annoyed. He flapped his hand toward him as if swatting at a fly. \"Lead on, then.\"\n\nOut in the courtyard, it was a simple matter for Mouse to slip away unnoticed. The open area was a bustle of activity, so when Jastar began his performance of making a fuss about how his team was not being handled properly, it drew enough attention for Mouse to dip into the shadows of an alcove by the stables. No one would be counting footman or attendants at this point.\n\nHe stripped off his attendant's jerkin--easily identifying him as someone that didn't belong--and tucked it behind some crates. Now, it was time to wait. Wait for the carriage to exit the gates. Wait for an opportunity to get back into the manor.\n\n# Chapter 8\n\nTHIS WAS indeed his favorite part. The carriage was gone, safely beyond the gates with the portcullis once again closed, and he was alone with only the sum of his strategy and skill. And his dagger and small kit of thieving tools, of course.\n\nNow, the adventure was afoot.\n\nThere is an undeniable thrill that one experiences when he finds himself alone in the enemy camp. It's exhilarating and intoxicating in equal measure. The senses become amplified, the blood is roused, and the fingers are set to tingle. There's a glorious tightness of the gut--like the one felt just before the first punch of a brawl or when dice are in hand before the cast. The unknown awaits, and the tension is delicious.\n\nThe exuberance of the moment, however, was somewhat diluted. He had chosen to lie in wait next to a particularly potent crate of onions.\n\nFor the residents of the keep, the brief excitement of Tenda's visit had ended--if it could rightly be called that. It was more a welcome interruption of the mundane. Manor life returned to what Mouse reckoned was the norm. An industrious but relative quiet hung over the grounds. The drudges of the keep went about their manual labors, while guards, stable hands, maidservants, and the like milled about the open courtyard like busy ants on a hill. Some traveled in pairs, engaged in quiet conversation, while others ambled about alone and consumed in their own thoughts. A group of hired masons toiling with an overburdened wheelbarrow of bricks provided Mouse the cover he was looking for. He slipped from his smelly hiding place and strolled along behind the lot as if he belonged.\n\nHe kept pace but was careful not to stray too close and pull their attention. The group was deep in a heated flap over the previous night's dice match, so there was little danger of being noticed. Any who happened to glance in his direction saw only an apprentice tagging behind his betters. The benefits of a youthful look and a small stature.\n\nHe made a casual but careful survey of the grounds while he strolled along. The inner ward surrounding the main structure of the keep was a wide-open field between the gates and the primary entrance, but to either flank was a mishmash of outbuildings. The masons led him toward the northern wall, past the guards' dormitories. A few off-duty grunts loitered about sharing a wineskin, grumbling about their shift. Their path then took them past a smithy, easily identified from the black smoke billowing from the chimney and the telltale repetitive clang of a hammer coming from inside.\n\nOne of the bricks in the wheelbarrow slipped off the top of the stack, but none of the masons noticed. Without slowing his stride, Mouse swept his arm down to grab it. Shortly, the masons angled toward the shadowy northern face of the keep, and Mouse spotted their destination: a building with a partially collapsed wall. As they veered toward it, Mouse called out to them.\n\n\"Ho there, masters.\"\n\nThe argument came to a sudden halt as they stopped and turned around in unison.\n\nThey eyed him with suspicion as he approached. Laborers were always wary of unsolicited attention. It typically meant more work on their docket.\n\nMouse held up the brick for them to see. \"Dropped this back there.\" He extended his arm to them.\n\n\"Ah,\" said the eldest among them, a sturdy man of about fifty years. He looked noticeably relieved. \"Thank you, friend.\" He took the brick from Mouse and set it back onto the stack.\n\n\"Thought it felt a bit light,\" joked the hulking man who'd wrestled with the wheelbarrow the entire journey.\n\nMouse chuckled freely. \"What an unfortunate irony it'd been if your job ended up one short.\"\n\nThey all laughed.\n\n\"What was all that excitement about earlier?\" Mouse asked them, leaning in conspiratorially and thumbing back at the gate.\n\n\"That carriage?\" scoffed the older mason, waving him off. \"Just another fucking bloated peacock in an oversized codpiece. No end to them lately.\"\n\nMouse couldn't help but smile at that. He shook his head in a woeful expression. \"Ah. Thought it might have something to do with that prisoner the duke has hidden away up in the tower.\" It was worth the gamble to prod a bit. Laborers like masons had a different brand of access inside keeps such as this, and they were most often overlooked and ignored. That meant they observed things others didn't.\n\nBut he was met with blank stares. One of them shaded his eyes and looked up at the closest tower. \"There's some prison'r up in there?\"\n\nMouse shrugged. \"So says the rumor mongers.\" He offered a warm tip of his head. \"I'll trouble you no further. Good day to you, masters.\"\n\nHe abandoned them and slipped in among the cluster of low wooden storage sheds while they trudged on, picking up the threads of their argument where they'd left off.\n\nAs he'd hoped, the area was quiet and sparse of people.\n\nHe weaved among some of the buildings, heading farther along the keep's northern side toward the back corner of the keep. He came across an arched opening. More tunnel than passage, it extended through the thick stone wall of the keep. But two paces inside, an iron gate barred the way. Mouse made a quick scan for witnesses and slipped inside the passage. With his thief tools, it took but a moment to spring the mechanism within the lock--so simple in fact that Mouse wondered why they bothered to lock it at all. He closed the gate behind him and emerged onto a perfectly manicured path surrounded by artfully trimmed shrubbery, statuary, and a little stone bench.\n\nHe was at the narrow end of a garden courtyard.\n\nHe stooped and slunk into the foliage. Dressed as he was, he would not be permitted here. This isolated haven was for the aristocracy--for those with the right blood or with the proper size purse. He remained still for a time, listening. He heard the sound of water, and as he crept a bit, he spied a round fountain at the center of a clearing with more paths leading off in other directions. With its gently meandering pathways and secluded corners, this was a garden designed for privacy. Only the black stone walls surrounding the courtyard spoiled the illusion that the garden was miles from the city. And the prospect of solitude was shattered as well when one considered that three walls of the manor were lined with windows and balconies. Any activities that took place in the garden could very well have an audience.\n\nMouse remained in the leafy cover and skulked along close to the wall, circling the garden. He listened for others who might be enjoying the garden with him as he cat-pawed over the soft loam. He caught a tiny titter of laughter.\n\nGently, he lifted a bough out of the way to have a look.\n\nNot far from him, a young couple was seated on a stone bench. Knees touching, they stared at each other with grotesque expressions of longing. The young man cupped her hand in both of his as if it were something precious and fragile. The lad leaned in and whispered something that made the girl giggle again. She put her fingers to her mouth and looked away, blushing. Mouse rolled his eyes and fought the urge to throw a rock at them.\n\nBehind them, watching the spectacle with the scrutiny of a warden, was the girl's nurse. The woman leaned her back against the wall with her arms crossed tightly over her excessive bosom in a tight knot. From her expression, she was clearly waiting for the lad to make the wrong move and give her the excuse to break up the visit and drag the girl inside.\n\nThe nurse had chosen to perform her skulking surveillance immediately beside the only door Mouse had seen so far. Presumably, the only door out of the garden.\n\nThere was no sneaking past her--that was clear. He had to find another way in. He made a quick inventory of the balconies of the second floor. Sure enough, a door was ajar.\n\nThere was always one. A manor of this size was simply too big to secure properly all the time--especially when those within felt safe and simply grew careless.\n\nMouse inched his way closer and positioned himself under the balcony. His movements would be in direct line of sight from both the lovers and the nurse. Any movement or sound would draw their attention with ease.\n\nHe considered waiting, but the two of them could be at this for hours. And he wouldn't be able to stomach it for another five minutes before he lost control and dumped cold water on the sticky couple. No choice but to act.\n\nHe studied the wall, paying close attention to the massive bricks of the keep's foundation and how they were connected, noticing which corners stood out more than others. His ascent would have to be quick. He planned his approach--footholds, handholds--and mapped it out in his head. Then checking on the activity and attention of the other three, he launched at the wall.\n\nHe made all his grips as planned, and his footing was secure, and he scaled the wall like a squirrel scales the trunk of a tree. He made it to the bottom of balcony with ease. Within two heartbeats, perhaps three, he was pulling himself over the balustrade. By the time the three were aware of the sound and movement and then located the source, Mouse was leaning his elbows on the balustrade and smiling down at them.\n\nThe young lad's eyes turned cross when he focused on Mouse. \"Ho there! Some privacy if you don't mind.\" From birth, the elites were trained to believe that what they wanted actually mattered.\n\n\"Ah,\" said Mouse with a carefree smile. \"My apologizes, my lord. I was unaware this was a private event.\"\n\nHe slipped inside the open door. He was inside the manor.\n\nNow the real work could begin.\n\nTHE SMALL two-room apartment was probably for some lesser noble. Not currently in residence, apparently. The bedside washbasins were empty, no supply of wood was stacked by the fireplace, no tray of sweet pastries on the table, and a layer of dust covered the surfaces. The bed coverings had been disturbed, though. Chambers like this were often the clandestine meeting place for lovers since there was little chance of discovery when the rightful occupant was not around.\n\nA thorough search ferreted out enough to piece together a new outfit. Strolling around in a tunic and hose was fine for the ward, but inside the manor he needed proper attire. He found some green woven trousers and a fine leather jerkin that fit him surprisingly well. With his own tunic and boots, the look was satisfactory. He would blend in nicely. Nothing flashy or too specific but discreet enough not to draw any notice.\n\nMouse was always astounded with what he could get away with when he'd infiltrated a place. After leaving the chamber, he strode the corridors as if he'd been there all his life. He needed to get a sense of the layout, which meant a systematic survey of the passageways and the various wings. He'd learned some time ago that if you act like you belong, no one will challenge you. You're a guest of the duke, residents tell themselves, or a member of the extended family in for a visit. Or perhaps part of an entourage. No one wants to be the foolish snoop that embarrasses himself or insults someone important by asking if they, in fact, belong. That sort of thing is always best left for others.\n\nHe swept the wing of apartments and living quarters first. He picked a few locked doors to gauge their quality and gain a feel for the mechanisms inside. Every locksmith designed his own device, each with its own particular qualities and nuances. The one who installed the locks around the manor was no different. Mouse had it figured out quickly and could have a door opened faster than someone could with the proper key.\n\nHe scavenged for any useful items he might need. Then he went about storing little caches around the manor. Anything that might have any use at all, he tucked it away in some storeroom or unused chamber, and he kept a catalog of all of it locked away in his head.\n\nOnce he was satisfied he had enough supplies squirreled away, he made a pass through the belly of the keep. He swept through the servant work areas and the kitchens. It was Mouse's firm opinion that the state of these areas was the true test of a nobleman's worth and distinction. How someone treated those beneath him spoke a great deal about his character. In general, filth and squalor was ubiquitous in the work zones of every keep and castle in the kingdom. But the conditions here were appalling and some of the worst that he'd seen. Duke Delgan did not much care for those who labored for him, it seemed.\n\nMouse marched into the kitchens and demanded with authority a tray of the night's meal from the cook on duty, a grim woman who clearly did too much taste-testing of the meals she prepared. Promptly, a tray was thrust into his hands without a word. He slunk off and found a quiet place to eat and plan.\n\nThen he tackled the more dicey areas of the manor. He carried his tray around as he explored. As a rule, no one ever challenged you if you carried a tray. With a tray in front of him, he could likely stroll right into Duke Delgan's own bed chamber and not be called out.\n\nHe wound his way up to the third level. The first tower was easy enough to locate. He followed the winding open stairwell to the top--which turned out to be a guard station and lookout. Currently empty. Chairs and a table in the center of the room with a cup of dice, cards, and flagons with stale mead swimming in the bottom was all that was there. It was possible that the guards stationed here only occupied this lookout at night. But it was more likely that they knew their captain wasn't going to haul his ass up all those stairs to check up on them, so they weren't going to either. They probably had slunk off to pinch a drink somewhere, shag a maidservant, or sleep off a hangover.\n\nTray still in hand, he pressed into another corner of the manor and found the stairs that led up to the second tower. A rope hung down the center from the top of the open stairwell. The bell tower. Not much chance of a prisoner held up there. There were no guards or any signs of recent activity, but he checked it out just to be thorough. As he suspected, it was empty. Someone had assembled a small pallet bed in one corner, but this was not where the prisoner was being held.\n\nWhen he entered the area leading to the third tower, he sensed something different in the air almost immediately. The corridors possessed a cold vacancy. No one lingered here. The few he encountered moved with determination and purpose and eyed him with a hint of suspicion as he walked by. He felt the prickle of danger on his skin. It had the hush and heft of a restricted zone, an area of authority. He rounded a corner and spied four guards down the far end of the corridor, two flanking an open doorway, and two standing against the opposite wall.\n\nThis was it.\n\n# Chapter 9\n\nTHIS WAS why Mouse didn't take on rescue missions. When hired to liberate pretty little baubles from their unsuspecting owners, they never had this much protection.\n\nKeeler had warned him, but four guards still seemed especially excessive. And there were more guards farther up the tower, certainly. A lot of manpower dedicated to one prisoner. Was the man dangerous? Mouse doubted that. Someone would have mentioned that particular detail by now. Had he attempted an escape already? Or did Delgan anticipate an attempt?\n\nMouse ran a catalog of ideas through his head. Getting past the guards undetected would be a trick--even for him. He could find a way past one or two, but four seemed overly ambitious. It took only one to raise an alarm. He toyed with the notion of one of his many ruses that might gain him entry but rejected them all. He preferred not to risk any suspicion just yet. Slaying them all wasn't an option either. Even if he managed to keep it quiet, four bodies would be difficult to hide, and their absence would be noticed before he even had the prisoner out of the tower. No, too great a gamble. He needed another way.\n\nWith the manor's layout conjured up in his mind, he skulked down a side corridor and chose a door that he hoped would serve his needs. It was locked, but a quick fiddle of the insides with his tools gained him his access. He ducked inside.\n\nThe chamber was a small office, just large enough for the desk that occupied one whole wall. Heaping piles of parchment, scroll cases, and ledgers cluttered the surface of the desk and a fair portion of the floor as well. This was the workspace of some journeyman scribe. He lifted one of the loose sheets of parchment from the top of a pile and scanned through it. Legal, by the look of it. Since this wing was deemed off-limits to most, the lowly scribe was likely reassigned to another part of the manor, so there was little danger of Mouse being discovered here. He headed straight to the solitary window. He flung back the shutters, leaned out on the sill, and looked up.\n\nHe'd guessed correctly. The window provided him both a clear view of the tower and also access to the roof if he climbed up. From the roof, he could scuttle over to the base of the tower.\n\nIt was a square construction, like the others, and made with the same black stone as the rest of the manor. It had arrow slots at the various corners, which would be the locations of the staircase landings inside. There were two sets of shuttered windows--halfway up the tower and near the top. The guardroom was likely the lower set of windows.\n\nMouse knew the protocol. Guardsmen assigned to an important prisoner would not sleep back in the barracks with their brethren. They would be sequestered there at the tower for more efficient shift rotation and to provide a quick response should something happen. That chamber would not be empty.\n\nMouse would find his quarry inside the upper set of windows.\n\nBut of course, it was the tallest of all four towers.\n\nMouse leaned his elbows on the sill and studied the construction of the tower with a pursed mouth. The stone blocks were large and roughly hewn, similar to the stones he had shimmied up earlier. He spotted a few areas that looked troublesome, but from his angle they seemed infrequent and could probably be avoided. Halfway up, the walls were reinforced with iron bands to take some of the stress from the weight up above it. The bands were interspersed regularly up the sides.\n\nHe bit the inside of his cheek. It could be done, he told himself. It would be tricky. One might even say foolhardy. But considering all his options, which were few, and the risk associated with each of them, he saw no other way around it. To get inside the tower, he would have to scale it.\n\nFuck.\n\nNot his favorite of activities. Scaling was an art form. Mouse had enough experience to know that every stone wall had its own personality and its own surprises. It was always a dangerous and uncertain business. He could plan for days, but the one thing he didn't account for could mean his downfall. Literally. A fall from the second floor of a jeweler's residence might mean a broken leg, but a fall from a tower this high meant certain death.\n\nThe stonework would offer plenty of handholds, certainly--masons never assembled the stones with absolute precision. But what concerned Mouse was the age of the tower. Over time, weather could undermine the masonry, making it weak and unstable. Old brickwork could break off under his weight, or the mortar could fracture, causing stones to loosen and shift unexpectedly. From the window, he could spot some areas of degradation. He could avoid those easily enough, but it was the areas he couldn't see that troubled him. He would be climbing the wall in the dark of night, after all. Wind could be a factor as well. Unpredictable gusts were not unusual at such altitudes, and considering the keep was built atop the crest of a hill, this tower reached very high indeed.\n\nIt would be a long climb. A great deal of strain on his hands and no room for error.\n\nOne last time he brainstormed for a way around this course, but nothing came to him. It was his best, if not only, chance of getting to Lord Garron undetected and quickly.\n\nHis revised plan required new preparations and new supplies. He set off to pilfer what he needed, slipping out of the little office to plunge back into the lower floors again. With a tray in hand, of course. This time, he looped through deeper regions of the manor, foraging through the now quiet workshops and dens. He even skulked out to the inner ward again, back to where the masons were repairing the wall, and through the smithy's workshop, which despite the cooling of the evening was still oppressive from the heat radiating from the furnace. In under an hour, he slunk back to the third floor, small linen sack slung over his shoulder filled with supplies. He shut himself back in the same quiet room that looked out at the tower and went to work.\n\nHe stripped out of his garments, stored them and his boots in a trunk under the desk, and pulled on some leather riding trousers he'd unearthed. Tailored for a lad, they were snug around the ass and thighs, but he cut a slit down each leg to loosen them and free up his movement. He strapped iron spikes and the sheath for his dagger to his legs with some leather lacings he had found, and to his waist, the small sack of chalk powder he'd lifted from the masons. He bound wider strips of leather around his hands and feet until just the ends of his fingers and toes were exposed. Lastly, he threw a coil of rope over his head to rest on his shoulder and against his side.\n\nHe left his tunic off and would remain bare chested. He could handle the cold--he just wanted nothing on that would snag as he climbed or could be seen flapping in the wind. The last thing he needed was to look like a flag hanging from the side of the tower.\n\nIt was edging on toward evening when he was finished and satisfied. The sun, swollen and red like a blister in the sky, was descending toward the distant tree line to the west. He climbed out the window and sat on the narrow sill with his legs dangling. From this perch, he waited and watched twilight descend on the city. Once the sun had fully set, he would begin the climb.\n\nDarkness gathered first from the deepest and narrowest streets and rose up slowly like a tide of shadow. Little by little, the buildings were consumed, swallowed up by the dark, as the sky surrendered too and gradually deepened into indigo. Then one by one, lanterns flickered to life--on the manor grounds beneath him and out around the city. It seemed peaceful from here. Quiet and undisturbed. Though he knew all too well what nature of wickedness came with the nighttime. Thievery. Malice. Bedlam. Murder. He knew because much of the time it was him down there in the gutter, ushering in the chaos himself.\n\nBut for now, from this high vantage point, he could imagine it was a tranquil city without all the festering ills that invariably lived in its underbelly. A city at peace. A city lumbering off to its bed. Such thoughts were an indulgence, he knew, but it was nice for a change to pretend that the world was not the polluted cistern he knew it to be. It was nice for a change not to be down there wallowing in it waist-deep. He could feign the ignorance so many others possessed, and for the moment, he could ignore the truth of what he was.\n\nBut his fantasies of a serene and uncomplicated life darkened along with the western sky, for his heart knew where this night would lead him. Before it was over, he would once again sink deep into the catacombs of the human experience. He would be the shepherd of chaos and death. To complete this job, to rescue this young nobleman, he would have no choice. The night would be filled with bloodshed.\n\nMOUSE MADE one final inventory of his supplies before he hoisted himself up onto the roof and jogged at a crouch to the base of the tower. It loomed over him, encased now in shadow, blocking out most of the blue-black sky.\n\nHe shivered and told himself it was the wind. Which was stronger than he liked. And as the night deepened, it would grow stronger still.\n\nFrom near the top of the tower, a light suddenly spilled from a window. The young nobleman was not ready for bed yet, apparently. He'd be awake when Mouse reached to the top. Good. Lord Garron was about to have a busy night.\n\nMouse reached up, found his first handhold and a place for his toes, and pulled himself up.\n\nFor obvious reasons, night climbs were the worst. It was a lot of groping and fumbling, feeling about in the dark for a suitable ridge or protrusion to grab or put his weight onto. Tonight was no exception.\n\nBelly to the cold rough stone, he worked his way up, inch by painful inch. For a time, the stones were uneven enough for him to find purchase with his fingertips and his toes, but higher up it seemed a better mason was in charge. Mouse found fewer suitable protuberances. With one hand gripping a stone lip, he reached out with the other, blindly searching with his palm until he found something sticking out enough for him to tighten his fingers onto. Then trusting the stone would hold, he pulled himself up a little farther. Each time was more harrowing than the last and set his heart racing.\n\nHe reached the first of the horizontal iron bands around the tower's girth. It was thick and smooth, giving his fingers something comfortable to grip for a change. He felt around for a place that he might pass the rope through, but there was no space large enough. The band was flush against the stone. The next one, perhaps.\n\nHe pulled himself higher to stand on the iron band for a bit. It gave him a moment to catch his breath and relieve his muscles, which were already tightening and starting to ache. A gust of wind tugged at him mischievously, toying with him. His body tensed, both from the sharp, cold air against his skin, slick with sweat, and from an unexpected splintering of his nerve.\n\nFuck. This was a bad idea.\n\nHe took a moment to loosen the drawstring of the sack at his waist, and he reapplied the chalk to his hands and his toes--tricky to do when standing on a ledge the width of a knuckle. Then onward he crept. He stretched and reached, curling his fingers around the thin grips he could find in the dark; then he pulled his body up by small degrees, inch by inch. A sudden stronger gust brought him to standstill. He shifted his weight, pressed his body to cold stone, and waited for the wind to subside. Another one like that at the wrong time could easily pluck him from the wall. He had to make a change. When the wind died back, he shuffled horizontally, then maneuvered around a corner--another perilous move, but he was now on a side that was better shielded from the wind.\n\nBut his new path took him directly through a zone of crumbling masonry. Almost immediately a fragment broke off in his hand as soon as it held his weight. He jolted as he caught himself, his heart leaping into his throat. The shards fell from his hand, and it took what seemed an eternity before he heard it clatter on the roof below. He pushed his cheek to the cold wall, closed his eyes, and forced his breathing under control again.\n\nFuck.\n\nNo choice but to slow his pace further. It was that or head back down--and retreating wasn't an option he was willing to entertain. He'd committed to this and was too far up to turn back now. But more time spent clinging to the wall meant more strain on his hands, arms, and feet. He tried to ignore the cramping in his left hand as he tugged down on his next grip a few times to insure its strength. Then he lugged himself upward again.\n\nHis shoulders now burned from the strain. And he was pretty certain that he had cut open the ball of his foot on a sharp edge. It stung when his weight was on it. There was nothing he could do about it now, so he pushed it out of his mind and pressed on. More shards broke loose in his hands, but now he was ready for them. Only it was getting harder to find safe grips. He was reaching farther and farther away from his position.\n\nThen a welcome discovery. He found an area where the mortar between the bricks had fractured and loosened. He removed one of the iron spikes tied to his thigh and with a bit of prodding and jabbing was able to create a gap big enough to wedge the spike on in. Testing it with a downward tug, it held firm. He climbed up and stood on it with one foot for a few minutes, enjoying the relative security of the wider surface. His muscles enjoyed the respite as well. He added a second spike for a handgrip, and took time again to apply more chalk.\n\nAfter agonizingly slow progress, he reached another of the iron bands and again took some time to gather his strength. And his nerve. In the dark it was hard to gauge, but he believed he'd made it past the worst of it. The masonry seemed stronger above him. A quick glance down told him he was about halfway up the side. Maybe a little more. Around the corner was one of the lower windows. He could hear sudden explosions of raucous laughter. The men were enjoying their time off duty--which hopefully meant plenty of drink.\n\nHe wondered whether he could get through that window unseen by the guards. Possible, he thought, but he had no idea if the window led right into their barracks or into the stairwell. And there might be a guard posted outside the cell door as well. Too many unknowns. It was too risky, he decided. He was better off proceeding as planned.\n\nHe heard a bell tower far below in the city mark the time. He'd been climbing for more than one hour now. This was taking too long.\n\nCreeping ever closer to the lit window above him--groping, stretching, reaching, and pulling--his arms and shoulders truly burned now. The pads on his fingertips were worn raw, and the arches of his feet were on the verge of cramping. He couldn't take much more of this. Some part of his body was bound to give out.\n\nHis fingers discovered another hole between the bricks, and he quickly inserted another spike, wedging it in tight. He was about to test its hold when something crashed against the side of his head. It threw him off-balance. One foot slipped from the narrow lip, and before he could catch himself, his full weight was hanging from the iron spike. He hung suspended for several heart-pounding moments, dangling by one hand. The spike shifted but held firm. He dragged his feet down the wall, looking to gain purchase on anything. His toes caught an edge. It was sharp, but he managed to lift himself enough to grab hold of another stone with his free hand.\n\nHe leaned his body against the cold stone while his heart rammed against his sternum. What the fuck was that? Had someone thrown something at him? He looked up, then down. Nothing in sight. Then it happened again, but this time from the opposite direction, and it was preceded by a high squawk. It hit the top of his head with some force, and he felt a sharp scratch on his scalp. Whatever hit him wasn't hard like a rock.\n\nLooking over his shoulder, Mouse caught a glimpse of dark wings reflecting the light from the window. A fucking bird. A sizable one too, judging from the wingspan. He watched as it swooped around in a wide arc for another attack. He must have strayed too close to a nest. The ridiculousness of it almost made him laugh out loud.\n\n\"Get away from me, fucker!\" he called out.\n\nIt squawked back indignantly.\n\n\"Don't make me kill you.\" He pulled the dagger from the sheath and waited for the next pass. He slashed upward as the bird closed in. Despite Mouse's speed, the bird avoided the blade easily and struck him again against the head. Both the impact and his sudden movement upset his balance. His body twisted dangerously, and the spike he'd inserted in the wall shifted again. An attack wasn't going to work. The bird was flying too fast, and it was too dark for him to track it properly.\n\nIt made another taunting cry somewhere behind him.\n\nHe had no desire to leave the relative security of the spike, but he knew the bird would not relent until Mouse was off the wall. One way or another. It would continue to bombard him, and with his limbs and fingers sinking further into exhaustion, each strike threatened more and more to knock him loose. He pushed upward again. He had to get to that upper window. He was standing on the spike when the bird swooped in close yet again. He ducked his head just in time, and the bird swooshed past without contact.\n\n\"Ha! Missed me, fucker!\"\n\nThe window was close. Only a few more feet....\n\nHe scrambled to get higher. The window was getting closer. He could make it as long as there were places to grip. The bird made a shorter circle this time and wasn't going to fall for the same trick again. It made a hate-filled shriek as it dove in. Mouse dropped his head again, but the bird had compensated its flight path and made contact. Hard. Even though he was prepared for it, the blow weakened his already precarious grip on the wall.\n\nThen he was hit again.\n\nIt was too soon for the same bird. It could only mean a second angry parent had thrown itself into the fray. \"Fuck!\" he called out again. The toes of his left foot found a lip to step up onto. He reached up with his right arm, searching with his palm like a blind man scavenging for a dropped coin. Nothing. He could find no purchase for his hand. The window's sill was a mere foot out of reach, but there was no place for him to grab and pull himself up.\n\nThe birds made a coordinated assault. One whacked him against his shoulder while the other clouted him again on his head. Even harder this time. Its claw sliced a finger-long gash across his brow, just missing his eye.\n\nHis right foot peeled off the edge. As his leg dropped, a sharp protrusion in the rock sliced a gash in his chin. Panicked, he tightened his remaining fingertips and simultaneously flailed his foot and right hand against the wall in search for the edge. But he couldn't find it. His grip was weakening. One by one, the fingers of his left hand popped off the edge. His throat constricted with the sudden realization. He was going down.\n\nAfter everything he'd been through in his life, after all the times he should have been dead but somehow managed to find a way to survive, it was a fucking bird that accomplished what no one else could. A bird was going to bring about his demise. A fucking gods-cursed bird. The absurdity of it all almost made him want to laugh.\n\nAlmost.\n\nThen something had hold of his wrist.\n\nHe looked up to a dark silhouette leaning out of the window. \"I've got you.\"\n\n# Chapter 10\n\nMOUSE WAS hoisted up through the window with little effort, lifted as though he were a child. His rescuer didn't grunt or seem to strain.\n\nThe hand around his wrist released him once he was fully through the window and into the chamber. Mouse propped up his torso with hands on his knees and dropped his head, afraid he might pass out. Blackness encircled the boundary of his vision. His heart hadn't yet learned that he wasn't going to plummet to his death. It felt like it might break through his chest.\n\n\"Well, this... is a strange... turnabout,\" he forced out between gasps. \"I'm supposed... to be rescuing... you.\"\n\nHe lifted his head and took his first look at the young lord--and any words he planned to say failed him.\n\nGarron had backed away and stood by his bed. He was dressed simply: brown leather trousers cuffed at the knees and an untucked sleeveless tunic, which exposed his powerful arms. The size of his biceps and shoulders certainly explained the ease with which Mouse had been lifted up. He had a full head of thick hair, the color of pitch. It was cropped short, a style currently unfashionable in the capital, but Mouse preferred it to the flowing locks all the foppish lords were now wearing. His face was rugged but striking. He had a thick brow and square jaw, covered now in days of unmanaged growth. His fair skin was flawless and iridescent in the soft candlelight--not a blemish anywhere.\n\nBut what captivated Mouse most of all was the man's eyes.\n\nThey were a brilliant emerald. Bright, nearly luminescent, and they cut into Mouse with shocking intensity.\n\n\"I heard someone,\" Garron said. He was still wide-eyed and confused. \"Someone calling out. I thought at first it was coming from outside the door.\" As if Mouse needed help with his meaning, he turned and pointed at the solitary door. \"From one of the guards.\"\n\n\"Well, you have my thanks all the same. A moment later, and I'd be a stain on Delgan's roof.\"\n\n\"You're bleeding.\" Garron gestured to his own forehead.\n\nMouse put a hand to his brow. When he pulled his hand away, it was stained with a crimson smear. Fucking bird. \"It's nothing,\" he said.\n\n\"Who are you? Why are you here?\"\n\n\"Not obvious?\" Mouse asked with a tilt of his head.\n\nGarron stared back him.\n\n\"All right, then. We'll take this one step at time. People call me Mouse.\"\n\n\"Mouse?\"\n\n\"Mouse.\"\n\n\"What kind of name is that?\"\n\n\"It's my name. Moving on to point number two.\" He took his first sweep of the room. It was a small chamber, but not uncomfortable as far as prisons went. He had a full bed, not some cot or pallet on the floor, and a writing desk and a table with a couple of chairs--as if he were receiving visitors. And the single door. \"I've been hired with the singular purpose of getting you out of here.\"\n\nHe moved over to the door, crouched down onto all fours, and put the side of his head to the floor. The gap beneath the door was just large enough to allow torchlight from the other side to shine through. The corridor beyond was well lit, but the light was obstructed along the right side of the gap. As he suspected, a guard was posted out there. He was seated on the floor and leaning against the door, likely asleep, but Mouse wasn't going to take chances.\n\nHe returned to Garron and stood in front to him. Garron was a clear head taller than him. Mouse looked up at him and spoke in a low voice. \"Your family is concerned about you. They'd like you to come home now.\"\n\n\"My family.\" Garron repeated tonelessly. His voice had the deep resonance of a drum. He seemed to have recovered from the shock of finding a stranger hanging outside his window and now stood with the straight-backed self-assurance of the aristocracy. He stared down at Mouse for a moment with a questioning look. Mouse stubbornly held his gaze.\n\nGarron's head tilted in curiosity. \"You are an uncommon sort. It is not often I encounter someone bold enough to look at me so directly. Even in here.\"\n\n\"We are not at court. Boot licking and servility are luxuries we can ill afford at the moment. I'll make sure to bow properly and avert my eyes once we are out of here. But for now, I'm here to rescue you, not dress you or wipe your bum.\"\n\nGarron's face darkened at Mouse's crudeness. \"Bold and cheeky. Impertinence is something you're accused of frequently, I'd wager.\"\n\n\"The least of my crimes, I assure you.\"\n\nGarron broke off the gaze with pressed lips and moved to the window again. He leaned out over the sill. \"You climbed up all that way? How'd you manage that?\"\n\n\"Not all that well, as you observed yourself.\"\n\nGarron shook his head in wonder. \"And this is what you do? Rescue people?\"\n\nFrom his tone, Mouse couldn't tell if that was just a general question or a judgment on his ability. \"Not exactly. But I can promise that I am the most qualified man around for the task.\" The man was asking a lot of questions. Mouse felt the need to move this along a bit faster. \"So, how do you feel about rappelling? I brought rope.\" He lifted the rope off his shoulder and presented the coil to him. \"Coming down with me?\"\n\nGarron looked at the rope as if it were a snake, then out the window again.\n\n\"Look,\" Mouse prodded. \"The sooner I can get you out of here, the longer it's going to take for them to discover you're gone. And that way is going to be the fastest.\"\n\nWhen Garron looked back at Mouse, he was a more than a little pale. \"Well,\" he said, \"if that is your plan, I fear you've wasted your time. It's not likely to happen.\"\n\n\"It'll be an easy climb with the rope. You're fit enough, certainly, and I--\" The look on Garron's face stopped him short. \"Oh, never mind. Stop gawking at me like a startled rabbit. It's clear enough you can't handle heights.\"\n\nGarron looked wounded. \"I....\"\n\nMouse waved him off. \"Forget it. I know few who'd have the balls for it, really. I was just being overly optimistic that this might actually be easy. Besides, if you fell to your death, I wouldn't get paid.\"\n\nGarron frowned. His gaze was narrow, and he gave Mouse a long, measured stare before he spoke again. \"So, what, then?\"\n\n\"We do it the messy way.\"\n\n\"Which means?\"\n\nMouse chose to not hear the question as he looped his way around the space, making a quick survey. It was comfortable enough, not typically what one would think of as a prison cell. No rats, no shackles and so forth. The floor was covered in thick rugs, and heavy blankets were on the bed. Downright cozy. So despite his accused crime, he was being treated with a certain level of respect worthy of his station. Interesting. Considering the accusations against him, the duke could have thrown Garron into the darkest dungeon. Instead he was being treated reasonably well. But regardless of how posh the surroundings, it was still a prison. It had a heavy door locked from the outside. Nothing he could pick.\n\nGarron shadowed Mouse as he moved around the chamber, but at a distance. \"What cause do I have to trust you?\"\n\nMouse stopped and looked at him curiously. \"How long have you been here?\"\n\nGarron folded his arms. \"I've started to lose count. Three weeks, I'd wager.\"\n\nA long time. \"And they've told you nothing of what they are going to do with you?\"\n\n\"No. They accused me of....\" He stopped and adjusted what he was going to say. \"Of something I didn't do, then threw me in here. Nothing since.\"\n\n\"No trial? No one's arrived to discuss the crime or your sentence?\"\n\nGarron shook his head.\n\nMouse pursed his lips and made a slow nod of understanding. \"They don't know what to do with you,\" he said. \"And that will make them nervous.\"\n\nLord Delgan had followed whatever instructions he'd received from outside, but now, for whatever reason, they hadn't followed through to the next stage. So what was their plan? Three weeks was enough time for any payment to exchange hands--if that's what was happening. Was the duke waiting for more instructions? Was he feeling squeamish about ending Garron's life? Perhaps he was having second thoughts.\n\nOr did the original plan develop a snag and they were reassessing?\n\nKeeping the young lord here indefinitely wasn't an option. Eventually they would have to act before the situation became untenable. Word of his internment would eventually leak out. \"You've been lucky so far,\" he said, \"but you are running out of time. At this point, they are obviously not going to pursue those charges. They would have done so by now. And they are not going to just let you walk out of here either. This will end only one way.\"\n\nGarron's suspicions seemed to dissolve. His face grew more ashen, and his shoulders fell a bit. \"You believe they intend to kill me, then.\"\n\nMouse shrugged. \"May not have been their initial intent, but....\" He let the thought hang in the air a moment. \"From what I've managed to dig up, this wasn't Delgan's idea. The strings are being pulled from outside the city. Any notion why someone would want you out of the way?\"\n\nGarron's lips tightened. \"I can think of one or two,\" he replied softly but said nothing more. Mouse didn't press it. Garron didn't look like the type to have enemies, but one never could tell.\n\nHe stepped closer and looked up into Garron's emerald eyes. \"I am going to get you out of here. I won't promise it'll be easy. You should be prepared for a rough go of it.\"\n\n\"What does that mean?\"\n\n\"It means getting out of this room is the easier part. Getting you safely out of the keep and out of the city will be far more difficult. But,\" he added with a finger in the air, cutting off Garron's response, \"I'm very good at what I do, and I have never failed a job.\" He chose not to expound on the fact that he'd never attempted to smuggle a living person out of a heavily guarded palace before either. \"I'm going to need you to trust me tonight. And do precisely as I ask. Without question or hesitation. Agreed?\"\n\nGarron weighed the conditions a moment, then nodded. \"As best I am able,\" he replied. He squared his shoulders in an attempt to appear ready for what was to come, but his eyes betrayed his uncertainty.\n\nMouse nodded too. \"Seems there's only one guard outside. That normal?\" He started to unwind the leather binding that surrounded his hand.\n\n\"This time of night, yes. Sometimes during the day there are two, but it has been less often lately.\"\n\nJust as he suspected. Garron was a good little prisoner who didn't cause a fuss. The guards were growing complacent. \"I need you to get that guard to open the door.\"\n\n\"How do I do that?\"\n\n\"Think of a reason. You've been here for weeks and caused no trouble. They'll trust your word.\"\n\nGarron looked reticent at first but then nodded and strode toward the door. \"Guard?\" He waited a moment, then knocked and called again. \"Guard?\"\n\n\"What? What d'ya want?\" came a muffled voice from the other side. There was a sudden shuffle of moment. He sounded groggy. As Mouse had predicted, he'd been asleep.\n\nGarron looked to Mouse, who nodded in encouragement. \"My chamber pot was never emptied today.\"\n\n\"Ah, fuck! You're telling me this now? It can wait 'til fucking morning.\"\n\n\"It's full,\" Garron called back.\n\n\"Fuck!\" came the reply. \"That's the chambermaid's duty!\" There was a pause. Mouse wondered if he'd left to go fetch a chambermaid. But then, \"Dammit. All right. Hold on a minute.\"\n\nSounds came from the opposite side of the door: thumps, the clank and click of a lock being released, then a scraping sound as the bar was removed from across the door. Mouse positioned himself behind the swing of the door while Garron stepped back so he would be the first thing the guard saw.\n\nThe door swung into the room. A standard in jail cells so the prisoner couldn't rush the door.\n\n\"Hope you be appreciating this, Your Lordship,\" the guard grumbled. \"Not something I--wha--?\"\n\nMouse shouldered the door closed and before the guard could react, he slipped his thin dirk cleanly into the man's throat. Warmth splattered onto his hand. The guard shuffled sideways and made sickly gurgling sounds as his hands flung to his neck, blood spilling from between his fingers. His eyes rolled back, and he collapsed in a heap. After a few final spasms, he bled out onto the floor and went still.\n\nGarron stumbled away from the corpse and had to steady himself with a hand to the wall. \"Gods!\" His breath came in quick bursts \"What have you done?\"\n\n\"Keep your voice down,\" Mouse hissed. He grabbed the guard's legs and dragged the body away from where most of the blood pooled. He rolled the body over. Some blood had splashed on the man's armor, but dark brown leather would mostly wipe clean. \"Help me remove his uniform.\" Mouse knelt on one knee next to the body and began to release the fasteners down the front.\n\n\"How could you just kill someone like that?\" Garron looked like he was about to be sick.\n\n\"He was a guard,\" Mouse snapped at him. \"Employed by the man who imprisoned you.\" He tugged off the helm and tossed it on the bed--and realized too late his mistake. Garron stared directly at the dead man's face. His young face. Mouse should have left the helm on for now and kept him as a nameless and faceless grunt.\n\n\"He didn't deserve that,\" Garron replied.\n\n\"Pull yourself together. What did you think was going to happen?\"\n\nGarron shook his head with a tight jaw. His complexion had lost all color. \"Not this....\"\n\nThe pampered fop wasn't going to be any help, Mouse realized. He untied the thongs laced at the shoulder and slipped off the sleeves, then wrestled the thick padded jerkin off the body. \"Well, I'm fairly certain that asking politely to be let out isn't going to work. But if you insist on it, we can give it a go and see what happens.\" He carried the vest to the washbasin to clean the blood splatter covering the front. The water turned instantly into a swirling maelstrom of red.\n\nA hand grabbed him underneath his arm and spun him about.\n\nGarron glared down at him, his eyes narrow and his jaw set. \"Not again,\" he growled through his teeth.\n\nMouse yanked his arm back but failed to break Garron's powerful grip. Damn, the man was strong. Stronger than he'd thought.\n\nGarron pulled Mouse closer to his face. Mouse felt the hot puffs of his breath against his cheeks. \"No more killing.\"\n\n\"If you think you will get out of here--\"\n\n\"I will not have any deaths on my head. Understood? I would rather meet my fate here.\"\n\nMouse stared back defiantly. \"You do realize they will be attempting to kill us, right?\"\n\n\"No matter. They are innocents--\"\n\n\"Hardly!\" Mouse spat back.\n\n\"Accept my terms, or leave the way you came! I will not tolerate such... such wanton disregard for life.\"\n\nMouse yanked his arm again, this time with more force and a twist to break the hold. His arm came free, and he stepped back and took a defensive stance. If Garron went to grab for him again, he would be ready. He should just leave, he thought. Keep the coin he had and leave this coddled idealistic dandy to his fate. All he needed to do was report back he couldn't spring him, or it was too late.\n\nBut that would mean admitting a failure. His first notch in the opposite column. Something he'd never had to do. And it meant giving up on his chance for a king's pardon, his sole hope of removing the threat that ever loomed over his father's head.\n\n\"Careful, Your Lordship,\" he spat the title with clear disdain. \"I am sorely tempted to take you up on that offer. I am your last and only chance. You are a quandary Duke Delgan will not want to suffer with for much longer.\"\n\nGarron looked down at the dead man at their feet. A wave of sadness washed over his face. \"I don't care.\"\n\nSeven hells! Mouse knew he was going to regret this.\n\n\"If you expect me to passively submit.... If my life is in danger--\"\n\n\"Then defend yourself!\" He locked his eyes into Mouse's. The fury behind those emerald orbs almost made Mouse step back farther. \"But no more of... of this. Nothing so cold-blooded.\" With a rigid jaw and tight fists, Garron's body shook. His voice took on a deep intensity that punctuated his resolve. \"Swear it to me now. If you comprehend the meaning of honor, swear it.\"\n\nDespite himself, that stung, and anger surged up in him. He was honorable, dammit--in his way.\n\n\"Swear it!\" Garron growled again.\n\nMouse sighed. \"Very well. I swear it. But this severely cuts our chances of success, you realize.\"\n\n\"If it means never witnessing something as atrocious as this again, so be it. What kind of man are you?\"\n\n\"One who gets things done. It's a rough world outside your gilded chambers, Your Lordship.\"\n\nGarron held Mouse's hard stare for several heartbeats. Then the fire in his eyes faded, revealing something deeper beneath. Sadness? Resignation? His shoulders lost their tension and dropped a fraction as he turned away from Mouse and the body on the floor. \"Save your sanctimonious rhetoric, scoundrel,\" he replied softly. \"I am well versed on the brutality of this world.\"\n\n# Chapter 11\n\nGARRON GRABBED a mug of something off the table, studied its contents a moment, then drained it. His face had turned hard and unreadable. \"So, what now?\" he asked, looking at nothing in particular.\n\nMouse tossed the jerkin over to him. It landed by his feet. \"Start by putting on his guard uniform.\"\n\n\"What?\" Garron scooped up the vest, wadded it into a ball, and threw it back at Mouse. \"Forget it!\"\n\n\"Don't be a prude, Lordship.\" Mouse walked the jerkin over to him this time. He extended his hand with it hanging from his fingertips. \"I cleaned off most of the blood. And it's not going to fit me.\"\n\nGarron made a face as he took the garment. He slipped it on over his tunic and started to buckle it up over his chest.\n\n\"Wait,\" Mouse said. \"You'll need a tunic with sleeves.\" With a tilt of his head, he indicated the dead man. \"Have one?\"\n\nGarron nodded and went over to a small chest at the foot of the bed and pulled out a clean white tunic. \"I was saving this,\" he said, more to himself than to Mouse, \"for my eventual release.\"\n\nMouse stole a glimpse at Garron's torso as he stripped off the old tunic. As Mouse suspected, the man's chest was as perfectly proportioned as his arms. He had a swath of short hair that followed the inner contour of his breasts with a dark line that ran down the length of his abdomen to disappear below the waistline of his trousers.\n\n\"What are you going to have me do?\" Garron asked as he buckled down the front of the jerkin.\n\nMouse brought over the sleeves and gestured for Garron to slip his arm through. \"We'll get to that,\" he said, as he laced the thong through the grommets at the shoulder of the jerkin. \"But you're going to have to pocket that accent of yours. It's too formal. Practice slurring your words, blending them together. Say something.\"\n\nGarron turned to glare down over his shoulder at him. \"I think you're a murderous, cold-hearted villain.\"\n\n\"Your diction is still too proper. Articulate less. Round out the vowels. Like this, 'I think you're a privileged, self-righteous prat that is damn lucky I've arrived in time.'\"\n\nGarron scowled and looked the other way. Mouse had to fight down a smirk. Clearly, His Lordship was not used to being spoken to with such brazen insolence. Well, get used to it, Mouse thought. I'm not being paid to handle you with white gloves.\n\n\"Try this,\" he said. \"Repeat it just like I say it. 'His Lordship's claimin' he nev'r got no sup t'night.' Make the lordship bit sound like an insult.\"\n\nGarron repeated it.\n\n\"Better.\" Mouse said the phrase again and made Garron repeat it twice more. \"All right, you're getting it. When do the guards change?\"\n\n\"I don't know. I'm always on this side of the door when it happens.\"\n\nFair enough. \"My guess is not 'til morning.\" With both sleeves reattached, Mouse retrieved the guard's boots and handed them off to Garron one at a time.\n\n\"This'll never work,\" Garron said. \"They'll recognize me.\"\n\n\"They believe you're still secured on this side of the door. They won't see you at all.\"\n\n\"Then they'll at least recognize that I'm not one of them.\"\n\nMouse sighed. \"You are giving them too much credit. Look. People don't notice changes they don't expect to see. You're the right size--I'd do it if I was, but I'm not. You'll have the helm on which covers a good part of your face. And you'll be likely waking them up on top of it.\"\n\n\"It's late. Won't they get suspicious? I would have said something earlier.\"\n\n\"If they're all asleep, they will have no idea what time it is. It could be just after dusk for all they know.\"\n\n\"That's a lot of 'ifs' this plan is balanced on.\"\n\nMouse smiled. \"That makes it exciting! Just make sure they send for a wench to bring you a meal.\"\n\n\"I don't see how she's going to help us.\"\n\n\"Don't worry about that right now.\"\n\nGarron's eyes narrowed in sudden seriousness. \"You swore me an oath. You're to honor that? No harm is to come to that maidservant.\"\n\n\"Gods! Don't be daft!\" Mouse went to the door, listened first, then opened it up. Outside was a narrow corridor lit by a solitary wall sconce opposite the door. Mouse slunk out with his weight on the front of his feet and followed the short distance to where the corridor turned, and peered around. A stairwell descended fifteen or so steps to a landing at the corner of the tower and then continued down along the outside wall.\n\nHe tiptoed back. \"Halfway down the tower... three, maybe four turns down, there'll be a chamber. Usually used for storage, but also additional guard support if needed. Do you remember seeing it when you were brought up?\" Garron nodded. \"Should have two, maybe three guards at the most.\"\n\n\"How can you know that?\"\n\n\"This isn't the only tower in the world, Lordship! They all function pretty much the same. Haven't you ever explored the towers of your own family's manor?\"\n\n\"What if this goes wrong?\"\n\n\"Then you won't find me waiting here for you when they drag you back up in shackles.\" It was a jest, but Garron clearly did not find the humor in it. \"Stop fretting. Just do as we practiced and it'll work. See to it they fetch that wench. Ready, Lordship?\"\n\n\"Stop... stop calling me that.\"\n\nMouse narrowed his eyes at him. \"That is your title, is it not?\"\n\n\"You make it sound like....\" He sighed. \"I would just prefer you not use it.\"\n\n\"How would you prefer I address you, then?\"\n\nGarron shook his head, a look of frustration on his face. \"Just... just Garron. Call me Garron.\"\n\n\"As you wish,\" Mouse replied with a shrug. \"Now, we've wasted enough time. Get your ass down there and make it convincing.\"\n\nMOUSE STOOD at the first landing, straining to listen. He could just hear them but couldn't make out the words. In truth, he was far less confident than he let on. Garron's highborn inflection was too deeply inset to be masked entirely, and if any of the guards had more sense than a cabbage, he'd be found out. Which meant Mouse would have to act. And act fast.\n\nOn the balls of his feet, he stole down to just past the next landing. The conversation drifted up to him, resonating off the walls. He could make out the individual words. Garron sounded gruff, which was good. At least he thought it was Garron. Gods help him if he overplayed it somehow and set them off. He kept an ear for fast-moving feet on stone just in case and held his fingers on the hilt of his blade.\n\nThe discussion was brief, followed by the clap of boots on the stairs. Mouse tensed, waiting. Then Garron rounded the corner from the landing below. He climbed the stairs at a hurried pace and pulled off his helm.\n\nThe wry grin on the lord's face told Mouse the answer he sought, but when they were both back in the chamber and the door was closed, he asked anyway. \"Well?\"\n\n\"Appears they're going to summon a maidservant for me.\"\n\nMouse did his best to mask his relief. \"Not a surprise,\" he answered coolly, returning to the chamber. \"People are daft, as a rule.\"\n\nGarron's satisfied smile faded some. \"That's a grim view of the world.\"\n\n\"Well, you said it yourself. I'm cold-hearted.\"\n\n\"Perhaps,\" he said with a challenging glare, \"my performance was just that convincing.\"\n\n\"Doubtful,\" Mouse responded glibly. \"More likely they had too much drink, and they wouldn't have noticed if you sang it to them in four-stanza verse.\"\n\nThat killed what remained of the smile, which satisfied Mouse.\n\nGarron dropped the helm on the table. \"I don't understand how this ruse is going to get us out of here?\"\n\n\"Patience, Lor--\" He caught himself and smiled impishly. \"Garron.\" He picked up the helm and pushed it against Garron's chest until he put his hands on it again. \"I hate spoiling surprises.\" He actually didn't want Garron overthinking his plan and cultivating any doubts about it. The less he knew in advance, the better. \"Now, back out there and reprise your winning performance as a guardsman of the manor one more time.\" He put a hand to Garron's back and guided him back toward the door, unable to ignore how solid it felt beneath the layer of leather. \"Lock the door. When the wench arrives, admit her.\"\n\nOnce Garron was out of the chamber again, Mouse set to work quickly. He dragged the body of the guard to the opposite side of the room and threw a blanket over it. Then he slid one of the rugs over the large bloodstain on the floor. He finished just as he heard Garron fumbling with the keys to unlock the door again. Mouse dashed behind the door as it swung open.\n\n\"Your Lordship?\" came a quiet voice. \"I have the meal you requested. Lordship?\"\n\nMouse waited until she was fully into the room. He moved fast before she could process was happening. In one swift movement he had her from behind, one hand around her waist pulling her against him, the other over her mouth. The tray she was carrying flew from her grasp and clattered to the floor, making a mess of gruel everywhere. She flailed in her panic, but Mouse held her firm.\n\n\"Quiet now, poppet,\" he whispered into her ear. \"No harm's to come to you, so easy now.\"\n\nShe struggled harder, wiggling and kicking, but she was a slight thing and no match for Mouse's strength. Soon, her energy flagged, and the fight in her weakened.\n\nGarron came in from the corridor and eased the door closed behind him. As he pulled off the helm again, the wench clearly recognized him. She froze a moment as the realization of what was happening must have dawned on her--then she resumed her struggle, but with less fervor.\n\n\"He speaks truly,\" Garron told her. \"We will not harm you.\" Unconsciously, his eyes shifted around the room looking for the body. \"You have my word as a nobleman.\"\n\nMouse pursed his lips at him. And what is that worth, he wondered. It was a nice touch--the girl noticeably calmed against him--but gods help them if he went around making that promise to everyone.\n\n\"I'm going to pull my hand away now, dove,\" Mouse told her. \"No screaming. Understood?\"\n\nShe nodded.\n\nMouse released his hand from her mouth. The girl didn't make a peep.\n\nAh, refreshing. Someone who understood how this works and didn't do something heroic or stupid.\n\n\"See?\" Mouse said. He kept his arm around her waist but relieved the pressure. \"We've no desire to see you ill-treated.\"\n\nMouse felt her shoulders and back relax a degree. \"What... what do you want from me?\"\n\n\"A pittance, dove. I just need you to do me the smallest of favors. I'm going to need you to slip out of that shift for me.\" Her eyes widened and her body stiffened. \"Ah now, don't you let that mind go places,\" he added quickly. \"Our friend here spoke true. Neither of us is going to abuse you. You're pretty and all, only not my type. It's your shift I want, not your dignity.\"\n\nShe nodded and her body loosened again. \"All right. If that is your promise.\"\n\n\"It is. Now, I can tell you're a clever thing, so I needn't explain to you what's happening here. Yes?\"\n\nAgain, she nodded.\n\n\"Excellent. I don't want you to face any trouble on our account. It wouldn't do to have them think you did anything to help us. So a second thing I need you to do is let us tie you up for a bit. Now, I know what you're thinking. But it won't be long, I promise. You'll be discovered up here in no time.\"\n\nShe thought about it a moment, her eyes shifting from Mouse to Garron and back again. She quivered a little but nodded her consent.\n\nMouse smiled at her. \"See, Lord Garron? And you said the pretty ones were unreasonable.\"\n\nHer cheeks reddened, and a small smile lifted one side of her mouth. Mouse could see it in her eyes: She was starting to buy into this as some sort of adventure. Fear had segued into exhilaration. This would be a story she could relay to her friends in the kitchens: how she was abducted by savage killers and she was lucky to get out of it unscathed.\n\nShe was a model of cooperation. Her shift was off almost immediately, and she stood there between them in feminine smallclothes. She covered some of her more exposed places, but it was more out of habit than any real modesty or embarrassment. She stole glances at Mouse's bare chest, which told Mouse that she was quietly hoping the two might actually make a little time to ravage her anyway. Mouse guided her to the bed and had her sit on the edge. \"It won't hurt, will it?\" she asked.\n\nHe winked at her. \"Not the slightest.\" He tied her hands behind her back--nothing too tight or uncomfortable. He held a glass of water to her lips.\n\n\"So you don't get thirsty, dove,\" he told her and tapped a finger to her nose.\n\nShe giggled at him and took a sip. Then he removed her bonnet and brought a strip of cloth up over her eyes. She tensed with sudden anxiety. \"Must you?\"\n\n\"I'm sorry, but yes,\" he cooed. \"It'll be quite fine. Just lie there and go to sleep until you're found.\"\n\nHe guided her down onto the bed. \"Comfortable?\"\n\n\"Quite,\" she answered.\n\n\"Wonderful,\" he said, and stroked her hair. \"Now make sure you tell them what horrible and dangerous brutes we are, and how you feared for your life.\"\n\nShe giggled.\n\nMouse looked over to Garron and rolled his eyes. Garron, who watched him with arms crossed, stared back with a tight brow.\n\nMouse put a finger to his lips as he stood up. He tied the rope he had brought along with him to the leg of the bed, flung open the shutters of a different window, and tossed the loose end out the opening. The rope uncoiled as if fell to the dark depths below.\n\nGarron looked alarmed and scurried over to Mouse's side. \"I thought--\"\n\nMouse shoved his palm over his mouth. \"But they won't know that, will they?\" he mouthed with an arched brow. He pulled the girl's shift over his head, wiggled his arms into the sleeves, and tugged at the sides until it hung properly. It was a bit short on him but otherwise fit rather well.\n\nGarron was at his side again. \"That is not going to work,\" he whispered furiously. \"And if you're caught dressed that way--\"\n\n\"Oh, gods help me!\" Mouse shook his head. \"Lord Delgan's morality laws are the least of my concerns at this point. And besides--\" He pulled the bonnet onto his head. \"--I make a very convincing wench.\"\n\nFrom the look on Garron's face, Mouse could tell he agreed. \"It's unseemly.\"\n\n\"Hardly. Remind me later to share with you a story or two about what true indignity looks like.\"\n\nHe moved to stand by the window with the rope. \"Okay, Your Lordship,\" he said in his full voice. \"Ready? Now, like I told you, lower yourself down slowly.\" He banged the shutter a few times. \"Careful now. That's it.\"\n\nGarron watched Mouse's performance with folded arms. He closed his eyes and shook his head.\n\n\"But it's so high!\" Mouse replied in his best noble accent. Then he changed back to his own voice. \"It's all right, your Lordship, I have you. You'll be safely down in no time. That's right. Slowly now.\"\n\nGarron scowled at him, which made Mouse almost laugh out loud. He shooed the young lord out the door, grabbed the tray from the floor--always have a tray on hand--then followed him out. Once out of the chamber, Mouse closed the door as gently as he could and locked it with an audible click. Garron cringed next to him.\n\n\"Relax. She believes we went out the window,\" he whispered. \"She won't even notice that.\"\n\n\"I still don't see the point of the rope....\"\n\n\"They'll be operating on a faulty assumption. Which means they will hopefully be looking for us where we are not.\" He started down the stairs. \"Come on. We have to make good use of the time we have. We are not even close to being out of this.\"\n\n# Chapter 12\n\nMOUSE HOPED they had until morning before the alarm was raised. Realistically, they had a few hours. At best.\n\nThe girl's long absence would be noticed by some old hag in the kitchen that kept her staff on a short lead. It was only a matter of time before someone would be sent off to investigate.\n\nSo, first order of business--get Lord Green Eyes as far from the fucking tower as possible before that happened.\n\nThey tiptoed down the stairs as fast as they dared. At the guardroom door, Mouse peered around the corner. Two guards, both sprawled out on wooden pallets, dead to the world and snoring happily. An empty flagon of wine was on the floor between them. Mouse looked at Garron and gestured with his head that it was safe to proceed. Normally, Mouse would have crept into the room and made extra certain they would not be coming for them later, but he kept to the agreement he made to Garron, and he let them be.\n\nHe was, if nothing else, a man of his word.\n\nOr at least tried to be.\n\nDownward they went, Mouse leading the way. They passed one locked door, which Mouse forced himself to ignore. He hadn't foreseen a chamber being there. Not knowing what was behind the door needled at him like an itch. They turned another corner, and the inner wall was gone. The stairwell opened up--it still hugged the outer wall but was open to the center. The drop to the base of the tower was at least fifteen paces.\n\nGarron subconsciously moved to the wall side of the stairs away from the edge, steadying himself with a hand to the stone. The man's squeamishness around heights was no small matter. How was he ever to get him over the outer wall that surrounded the keep when just an open stairwell made him dizzy?\n\nHe'd have to work that out later. Right now, the priority was getting somewhere out of sight for when the manor discovered his absence.\n\n\"Stay here,\" he whispered. \"Count to one hundred after I'm gone--\"\n\n\"Where are you going?\"\n\n\"Relax. I'll be waiting for you around the corner of the next corridor. There are guards just outside.\" He pointed at the opening at the bottom of the stairs. The guards had left the doorway into the tower open after the serving wench had been sent up, which was a strong indication of their complacency. From Garron and Mouse's angle on the stairs, they could make out one of the guard's legs in the hall. \"It'll look suspicious if we walk out together.\"\n\nSudden unease widened Garron's eyes.\n\nJust the notion of Mouse leaving him unattended put Garron near panic. Now free from his confinement, Garron was gripped by the concern of possible recapture. Mouse had inspired some measure of confidence, it seemed.\n\nHe resisted a sigh of impatience. That wouldn't do, he told himself. He needed the man calm and compliant, and acting snappish or frustrated would be counterproductive. He would have to find a way to exercise more patience. This was the sort of thing Mouse did regularly, he reminded himself. It was all quite new to the nobleman.\n\nHe stepped closer and kept his voice calm. \"Follow me out after you've counted to one hundred. Just walk out with intention, like you belong here. Like you've done it every day for weeks. They have no reason to suspect anything. If they say something, just play along.\"\n\nGarron nodded, still looking uncertain.\n\n\"Just remember to talk like one of us lowlifes.\"\n\nGarron's expression darkened. \"Do I really sound like that much of a prig?\"\n\n\"No more than every other high born in the world, Lordsh--oops, sorry!\" In truth, Lord Green Eyes was very nearly tolerable. Over the years, he had suffered far worse arrogance and entitlement from various nobles, but he wasn't about to let him off the hook yet. \"One hundred,\" he reminded him with a pointed look. He adjusted the bonnet on his head, smoothed the shift on his front, then skipped down the stairs brandishing his tray in front of him.\n\nHe stepped out into the corridor. Either side of the doorway was flanked with a guard. Two more faced him along the opposite wall. Thankfully, the lighting was poor. Only a single sconce bore a lit torch, and it was nearing its end of pitch. The guards all looked up and turned to him as he crossed the threshold. One instinctively lowered his hand down to the hilt of his short sword.\n\nMouse kept his head low in a demure and respectful manner and offered a gentle curtsy.\n\n\"Up there awhile, you were,\" one commented.\n\nMouse shrugged and made a high-pitched titter into his shoulder but said nothing. He scurried away down the corridor.\n\n\"Little trollop,\" Mouse heard one of them grumble. \"Looks like she's serving up more than gruel to that Dionantian prig.\"\n\n\"I'll take a bowl of what's she's offering.\"\n\n\"Ho there, we're all hungry too. When do we get our share?\"\n\nThey all laughed.\n\nMouse rounded the corner out of sight of the guards and waited. It was late--the corridors were quiet and would likely stay that way. As he counted down in his head, he pulled out his dagger. He didn't expect trouble, but it always paid to be prepared. The guards made quiet banter among themselves, then....\n\n\"Hey. Where're you off to?\" one of the guards exclaimed.\n\nMouse stiffened and leaned around the corner. Garron was out of the tower and strolling down the corridor. He moved straight-backed and rigid, and his gate was paced too fast. He looked guilty of something, like someone not wanting to be noticed but wanting to get somewhere else quickly. He slowed but didn't stop. \"Urgent message for the chancellor,\" he called back over his shoulder. \"From His Lordship!\"\n\nMouse laughed inwardly. Garron spoke with the same intonation that he himself would have said it. The prat was learning after all.\n\n\"What kind of message?\" one of the guards called after him.\n\n\"For his ears only, you dullard!\" Garron snapped.\n\nMouse was impressed. Garron understood that none of those standing guard would know why the prisoner was even here. Though there'd be rumors, they would assume it was something political. Garron rounded the corner, joining Mouse. He exhaled heavily and slumped against the wall.\n\nMouse smiled at him. \"Appears you're learning a few new skills.\" He clapped the lord on the shoulder. \"Congratulations. First step accomplished. You are officially out of your tower.\"\n\nGarron scowled inwardly. \"But for how long.\"\n\n\"Permanently. At least as long as I have anything to say about it.\"\n\n\"That supposed to be comforting?\"\n\n\"Come on.\" Mouse started down the corridor.\n\nGarron followed at his heels. \"Where to?\"\n\n\"Somewhere they don't expect you to go?\" In other words, he wasn't sure yet.\n\nMOUSE ESTIMATED it was after midnight, but he would keep his ear open for a crier or a bell to give him a better indication. In a few hours at most, the manor would come awake and residents would filter out into the corridors like water filling the gutters after a spring shower. Earlier risers--the cooks, stable hands, maidservants, and valets preparing their master's and lady's garments for the day--they were the eyes that potentially could remember them after Garron's absence was noticed and someone sounded the alarm.\n\nMouse knew it wouldn't take much deductive power to work out their method of escape once the tied-up wench and the dead guard were discovered in the tower. The rope out the window would stall them at best, but a short debriefing with the guardsman at the base of the tower would unearth the truth of how they escaped. When that happened, it made their current disguises useless. Best to ditch the outfits now. First stop--the storeroom where he'd stashed the other clothes.\n\nHe led Garron down the corridor to the small office he'd infiltrated earlier. They ducked inside, and Mouse locked the door with his tools.\n\n\"Handy skill,\" commented Garron.\n\n\"You have no idea,\" Mouse replied.\n\nGarron pulled off the helm and tucked it under his arm. He took in the chamber with a curious expression while Mouse rummaged in the corner to retrieve the clothes he had stashed there earlier. \"So do you have an actual plan?\" Garron asked.\n\nMouse deposited the clothes and boots on the small desk. \"You sabotaged the one I had in mind--thanks equally to both your high morals and fear of high places. A good plan, mind you. I'm being forced to improvise.\"\n\n\"So... that's a no.\"\n\nMouse gave him an exasperated look. \"This is not my first day at the tourney. I have more than a few tricks--\" He stopped.\n\nHe heard something--a sound he was all too familiar with. Damn.\n\nHe grabbed Garron by the front of the jerkin with both hands. \"I apologize for what is about to happen.\"\n\n\"What--\"\n\n\"Play along!\" Mouse pulled Garron toward him pressed his lips firmly against his.\n\nGarron's entire body stiffened. He made a startled grunt in this throat and on instinct tried to pull back, but Mouse held him firm. The helm dropped to the floor.\n\nThey heard the lock release, and the creak of the hinges as the door was pushed open.\n\nGarron must have understood, for he stopped struggling. Mouse threw his hands around Garron's neck, who in turn slipped his arms around Mouse's waist and pressed back against Mouse's kiss with greater force. Gods, he had soft lips....\n\nBehind Garron, someone made a sudden gasp. \"Oh!\"\n\nGarron pulled his face from Mouse and shot an angry look over his shoulder. \"What the fuck are you doing? Can't you see I'm a little busy here?\"\n\nMouse ducked his head into Garron's chest so the intruder could only see the bonnet on his head and not his face. He made a convincing giggle.\n\n\"Ho, ho!\" came the reply. A man's voice. \"Sorry 'bout that, mate! I'll come back later. Don't let the night captain catch you, though. Or the cook! They'll have your balls after the wench's done with them.\"\n\n\"Just bugger off!\" Garron's lips returned to Mouse's with relish, and they continued their performance. Garron's mouth opened, but Mouse fought the urge to give in further and thrust his tongue past the opening. The bristles on Garron's face burned with sweet friction on the skin around his mouth. He could picture himself doing this for a long time--in fact, felt himself respond and start to swell. The man knew how to kiss, that was certain. And Garron's body felt solid and warm against him. Moments later he heard the door close and the latch click. But the kiss continued. It was Mouse who broke contact.\n\n\"I believe he's gone,\" he said, a little breathless. A feeling of relief washed over him. If it had gone on much further, Mouse might have lost control and crossed over the line.\n\nGarron released his tight hold on him and stepped back, his cheeks betraying his embarrassment. \"Oh. Sorry. I was just making certain.\" He used the knuckle of his forefinger to wipe the corners of his mouth.\n\nMouse was caught at a loss for words, which was new for him. He turned his body away so Garron wouldn't see how the front of the shift protruded conspicuously. \"We better move on quickly,\" he managed to choke out.\n\n\"Sure,\" Garron replied. \"Of course.\" He was trying to sound nonchalant, but Mouse could sense the tension in his voice. An awkwardness had materialized, like some invisible curtain that separated them.\n\nFor Mouse, the play-acting had somehow slipped into something real. The sensation of that kiss lingered and filled his head like an intoxication. Dammit all! Had Garron sensed that? Did he sense how much Mouse actually enjoyed that? Had he felt Mouse's stupid fucking cock pressing up against his leg?\n\nHis back to Garron, he pulled the shift off over his head. He fought the urge to see what Garron was doing. A part of him fantasized that Garron was watching him undress, but he didn't dare turn to look. His cock was surging even bigger at the mere thought.\n\nFuck! Get a grip on yourself!\n\nGods, when did he start losing control? The man was a nobleborn--an entitled prig that shat gold and pissed fine wine. And here Mouse was snogging him like a drunkard in the back of a tavern. He loathed the nobility. He despised the injustice of their power and influence, their pomposity and superiority, their shameless self-interest while good people toiled beneath them with no hope of a better life. In his privileged life, Garron had never known strife or hardship. Like all his kind, he surely treated the common folk no different than the others: like shit stuck to his heel. So what in the seven hells was his cock thinking?\n\nHe pulled on the blue breeches. They helped to mask the effect at his groin, but he tied them tighter to be sure. Then he quickly donned the rest of the garments and tugged on the boots. He shoved the shift, bonnet, and tray behind the crates.\n\nGarron was leaning against the wall, arms folded across his chest and staring downward at his own feet.\n\n\"Ready?\" Mouse asked. He was surprised to hear a measure of anger in his tone.\n\n\"I suppose. But where are we going?\"\n\n\"First and foremost, I want to get you out of that guard uniform--\" He closed his eyes and cringed at how that sounded. \"And into something they won't be looking for,\" he added hastily.\n\n\"You have something in mind?\"\n\n\"Always. Don't forget your helm.\"\n\nGarron scooped it up from the floor as Mouse opened the door to peer outside in the corridor. All clear. He was just about to slip out when the loud clanging of a bell rang out into the night and throughout the manor, followed by more bells. Someone had raised the alarm. Garron's absence had been discovered.\n\nFuck. Fuck. Fuck.\n\n# Chapter 13\n\nMOUSE KNEW they had to act fast.\n\nHe was nothing if not flexible. In his vocation, it was often necessary to shift tactics. He worried about Garron, though. Lord Green Eyes would be unaccustomed to this sort of pressure and could come unnerved. Something neither of them could afford.\n\n\"Does that mean what I think it means?\" Garron asked.\n\n\"Come on.\"\n\n\"But--\"\n\nMouse grabbed him by the front of his jerkin and pulled him into action. Time here was vital. They had ten... maybe fifteen minutes. Already people around the manor were scurrying to report in--many being brought out of a sound sleep--in order to find out what the alarm was about. But it would take a little time for the news to break and the details to spread.\n\nTime he needed to use to his advantage.\n\nFinding a new way out of the manor had become a secondary priority. Right now, they had to keep a step ahead of the search. He'd figure out the escape later.\n\nHe jogged down the corridor, Garron on his heels. A sleepy older man still in his nightshirt rounded the corner in a huff and seeing Garron, grabbed him roughly by the arm as he tried to pass. \"What's this about, boy? We under attack?\"\n\nGarron's eyes widened in alarm for a moment. He shifted his gaze toward Mouse, looking for aid. \"Uh... I know not, m'lord,\" he managed to blurt out. He winced, realizing he hadn't adjusted his diction and sounded too high born for a guardsman. He adjusted, adding, \"Only just reportin' to the night captain, now.\"\n\nThe man released him with an irritated grunt. \"Well, find out and get back to me,\" he grumbled. He didn't seem to notice the discrepancy in Garron's voice. \"Better be a fucking invasion, waking me up like this from a dead sleep. I'll be in my chambers.\" Without looking back, he shuffled back down the corridor from where he came.\n\nGarron turned to Mouse with an arched brow, a surprised expression that said, \"Could it really have been that easy?\"\n\nMouse shrugged, then signed for him to follow.\n\nNo one else challenged them. Everyone spilling into the corridors was scrambling to get somewhere in response to the alarm--in that, Garron and Mouse fit it perfectly well. Mouse led him down a stairwell, through a main artery of the manor, then off into a darkened wing. He located the door he wanted at the far end of a corridor, and with quick work with his tools, he unlocked a storeroom. They slipped in and shut the door behind them.\n\nGarron leaned his back against the door, gulping in heavy breaths.\n\n\"All right. We're going to have to do this quick,\" he said digging out a leather pack from the inside of an empty crate.\n\nGarron could not hide his surprise. \"Where did that come from?\"\n\n\"I'm a good planner.\" From the corner, he pulled out a bucket of water. \"Take off the jerkin and tunic and....\" He made a quick scan of the room. He lifted the empty crate and positioned it in front of a lone barrel against the wall. He considered the height with his head tilted and nodded in satisfaction. \"And sit there,\" he added, pointing to the crate.\n\nGarron eyed him suspiciously. \"What for?\"\n\nMouse pulled out his dagger--his father's dagger. It was the sharpest one he had. Garron's eyes widened a fraction when he saw it. \"Relax. That beard of yours hasn't been cut in three weeks. They'll be combing the shadows for a shaggy escaped prisoner. Not a neatly trimmed aristocrat.\"\n\nGarron still looked wary but started to unclasp the front of his jerkin. \"If you stashed away a looking glass somewhere, I can shave myself.\"\n\n\"We don't have that kind of time. I will shave you much faster than you'll be able to do it yourself. Besides this blade is sharper than anything you've ever used.\"\n\nA group of heavy footfalls clapped on the stone floor of the hallway and seemed to stop just outside the door. Booted. Clearly guardsmen. Mouse held up his hand, but Garron instinctively knew to freeze in place. Both stared at the latch of the door, waiting for it to move. Mouse repositioned the dagger in his hand. He could hear the muffled voices of the men outside though he couldn't make out any of the words.\n\nThen the footsteps resumed and faded.\n\nGarron looked pale as he exhaled slowly. Mouse gave him a knowing look with a single raised eyebrow to emphasize his point. Time was short. Guardsmen could start sweeping chambers and storerooms at any time.\n\n\"Let's get to this,\" he said.\n\n\"Have you ever shaved someone before?\" Garron asked in a voice that was nearly a whisper.\n\n\"Many times.\" It was exactly three, but Mouse wasn't about to confess that. He set the bucket within reach and climbed up on the barrel with his legs spread apart.\n\nGarron hesitated as if still uncertain, but with a barely perceivable nod he stripped off the garments, once again exposing his perfectly formed torso. The man's tone was fair like white marble, and his shape seemed to have been sculpted by a gifted artist.\n\n\"Sit.\" Mouse pointed at the crate with the blade. \"And lean your head back against the barrel.\"\n\n\"You sure about this?\"\n\n\"Trust me.\"\n\nGarron complied, lowering onto the crate and tilting his head back. His shoulders were tight and his hands clenched. \"I still think it would be easier if I shaved myself,\" he grumbled.\n\nMouse guided Garron's head back a little further to lift the chin. His hair was soft. Mouse resisted the urge to run his fingers through it. \"Well, then I would be denied the joy of putting a knife to a nobleman's throat, now wouldn't I?\"\n\nFeeling the weight of Garron's head against his lap, another fear suddenly occurred to him. Gods, what if his fucking prick decided to act up again? It apparently had a mind of its own lately. All he needed was for it to start poking Garron in the back of the head.\n\nHe shoved those thoughts back and focused on the task at hand. He began on the jawline, dipping the blade in the water and making short strokes upward on the face. He didn't run the edge directly against the skin, but instead kept Garron's beard short and neatly cropped. Cleanly shaven might draw attention. A day or two's worth of stubble would raise fewer suspicions. But the neck would have to be clean. A good shave always had a clean neck.\n\n\"Lift your chin more.\"\n\n\"You have a steady hand,\" Garron said through his teeth, as Mouse made slow deliberate scrapes up his throat.\n\n\"So I've been told.\" Mouse dipped the dagger back into the bucket and spun it about in the water.\n\n\"Glad you're here to rescue me and not kill me.\"\n\n\"Well, we'll just see how much gold Duke Delgan is willing spend to get you back.\"\n\nIt was a joke, obviously. Mouse would never switch sides when he was hired for a job. It was bad for business. If word got out that he was not trustworthy and his allegiance could be easily swayed with a few more coins, the jobs would quickly dry up.\n\nBut Garron didn't know that.\n\nAs Mouse moved to bring the dagger back to Garron's neck, Garron shoved the arm away and sprang to his feet. He backed away and looked down at Mouse with sudden fire in his eyes.\n\nMouse put his hands on his knees and let his chin drop. He knew better than to say something like that. Even in jest. He needed Garron to trust him fully if he was going to get him out of here alive. \"I'm sorry,\" he said. \"That was a stupid thing to say.\"\n\nGarron glared back at him with tight lips, looking a little foolish with one half of his beard long, and the other neatly cut.\n\nMouse looked up and held Garron's hard gaze. \"I know you don't view me as having much honor, and you have no cause to believe me. But I would never betray you.\"\n\nGarron didn't reply, his expression hard.\n\nMouse took in a long breath; then he held up the dagger for Garron to see. \"This dagger belonged to my father. It is all I possess of my family, from my life before. It is the only thing that has any meaning to me. On this blade, I swear to you. I will remain loyal to you and my mission until the end. If you will it, the knife shall be put in your keeping until you are safe from here.\" Mouse extended the dagger to Garron.\n\nGarron took the blade from Mouse's hand and examined it. For several moments he was quiet. His expression changed--softened. The heat of his anger tempered into something more pensive. \"This symbol, etched onto the metal of the blade....\"\n\n\"I know not what it is,\" Mouse replied. \"I have tried to find its meaning but cannot.\"\n\nGarron nodded gently, looking thoughtful. He handed the knife back. \"I believe you,\" he said. \"I have no need to hold the knife.\"\n\nMouse took it back, unable to veil his surprise. He had great confidence in his own power to influence but still expected some uncertainty from Garron, some resistance surely. He did not expect blind acceptance on his word alone. Garron seemed the trusting sort, but not foolish or na\u00efve.\n\nGarron seemed unfazed as he returned to the crate and leaned his head back into Mouse's lap without a word.\n\nNeither spoke while Mouse resumed the task.\n\nFeeling the weight of him against his legs and the supple skin of his cheek against his hand, Mouse felt an inexplicable sadness well up from inside his gut. He knew not where it came from, but it swelled quickly and settled like drying resin around his heart. What could have caused Garron's quick trust in him? It felt undeserved. Unearned. Mouse had jeopardized everything with a single flippant comment, an amateurish mistake, and Garron had no reason to believe it was only a joke. But he did. Mouse was comfortable with the suspicion and distrust from others. Someone trusting him at his word was... unsettling.\n\nAnd why did it matter so much what this noble thought of him?\n\nGARRON MADE a quick rinse of his face with water from the bucket. \"Now I really do wish I had a looking glass. If only to make sure you didn't make me look like a buffoon.\"\n\nMouse climbed off the barrel and put his ear to the door. For the moment, it seemed quiet out there, but that could change quickly. \"Don't be vain,\" he replied dryly.\n\nIn truth, he looked dashing. Mouse was a little concerned that his good looks were going to attract too much attention. People were going to notice and remember seeing him. He lifted the leather sack of supplies from the floor and dropped it onto the barrel. He reached in and pulled out long black vestments, a colored sash, and a black four-cornered hat. \"Now, put on this.\"\n\nGarron held the garment up by the shoulders in front of him to inspect it. \"A prelate's robes? Do I strike you as someone to take up the righteous path?\"\n\n\"Hardly, but who's going to question you?\" In his experience, religious figures got unmitigated access around places like this.\n\n\"How in the seven heavens did you acquire it?\"\n\n\"It's amazing what you can find lying about when you do a bit of snooping.\" Mouse pulled out a novice's robe for himself. \"Found them in a trunk of some nobleman's antechamber. Likely costumes for some fetishy sex games. But they're authentic, which is important.\" He unclasped and peeled off his jerkin and shoved it into the sack. \"Quick now,\" he urged with a double snap of his fingers. \"Your tunic and trousers can stay on under the vestments, nothing more. Especially not the boots.\"\n\nGarron paused from pulling on his tunic to give Mouse a questioning look.\n\n\"Shepherds only wear cloth slippers,\" Mouse told him. \"No hard heels.\"\n\nGarron nodded and pulled the black vestments on over his head. It hung loose around the neck. \"I think there are latches in the back,\" he said, turning around.\n\n\"Right.\" Mouse fastened the clasps behind his neck and between his shoulder blades. He then helped him position the hat on his brow, tucking in the stray hairs. He arranged the purple sash over Garron's shoulder and down the front, making certain the runic symbol of each god was prominently and correctly displayed, then took a step back to scrutinize the finished product.\n\n\"How do I look?\"\n\n\"Eh,\" Mouse replied with a shrug. \"A little young to be donning the purple, but no one will question you.\"\n\n\"You certain of that?\"\n\n\"Have I yet to lead you astray?\"\n\n\"We've only been out for a maybe an hour. Not exactly the towering accomplishment you want me to believe it is. I could have lasted this long hiding in a broom closet.\"\n\nMouse was taken aback by the nobleman's sudden sharp tone and felt his shoulders contract toward his ears, but he disguised his ire by turning to retrieve his own costume. \"Perhaps you would prefer to experiment with that particular stratagem and see if it gets you out beyond the walls.\" Not for the last time, he questioned the wisdom of taking on this job. Contraband never questioned your skills. \"It will work, I assure you,\" he added, forcing a calmer tone.\n\nGarron nodded. \"All right. Where to now?\"\n\n\"Somewhere they won't think to ever look for us. The wing of the royal apartments.\"\n\n# Chapter 14\n\nTHE GUARDSMEN stationed at the entrance of the apartment wing stiffened into attention as they approached.\n\n\"And if this doesn't work?\" Garron grumbled under his breath.\n\n\"Stop fretting and keep walking.\" Mouse was striding two steps behind him on his right side, which was customary for a novice. \"And look pompous and arrogant. Oh, wait. Look who I'm talking to.\"\n\nGarron shot him a hard look over his shoulder.\n\n\"Yes. Just like that.\"\n\nProdding Garron was an intentional gambit. Anger would help overcome his nerves. And improve his performance. Prelates were notorious for their sour moods, judging everyone they encountered with sanctimonious fervor. No one outside of the calling could ever measure up in conviction and devotion.\n\n\"Good eve to you, Shepherd,\" one of the guards greeted with a quick and respectful bow.\n\nThe guards stood shoulder to shoulder at the threshold of the arched passage that led to the royal apartments, barring entry. Garron stopped a few paces from them, hands tucked into his sleeves, and offered a single nod in return. The motion was thick with condescension, stating in clear language that he had no interest in responding or acknowledging them but was being forced to by circumstance. He eyed each in turn, waiting.\n\n\"Is there something we can do for you, Shepherd?\"\n\n\"I wish to return to my apartments,\" he replied as if it were obvious. \"As was requested.\"\n\n\"I am sorry, m'lord, but this wing of the manor has been declared restricted.\"\n\nGarron looked perplexed, giving each a chilling glare. He turned to give Mouse a similar look. Mouse took a step forward to take over, thinking Garron was stuck and didn't how to proceed. Garron halted him with his extended palm.\n\n\"Are you saying,\" Garron said, stepping closer to the men, \"that I am being denied access to the apartments that the generous Duke Delgan has provided me?\" He spoke softly, enunciating each word with the sharp staccato of a hammer on a nail.\n\n\"Lord Prelate, the orders came from the duke himself. I am only--\"\n\n\"I have spent half the night providing the final rites for a dying man and comforting his grieving widow, and I return to find I am barred from my bed.\" Garron's voice rose in pitch. The incredulous look on his face was so authentic Mouse would have believed it himself. There was more to this young lord than he estimated.\n\nThe men eyed each other with worried glances. It was unwise to anger someone with a direct ear of the gods. Especially one wearing a purple sash. A Grand Prelate could be a powerful enemy. It hadn't occurred to either of them that a prelate of this level wouldn't partake in any final rites. It was a task clearly beneath him--but Mouse wasn't about to quibble about the details. Garron's quick thinking had provided a plausible reason to be away from his bed, and the guards were too flustered to overthink it.\n\nThey were nearing the point of caving in. Mouse could see it in their fidgeting stances. Neither wanted to be the one to deny the prelate, so each was waiting for the other to play the role of the brave one. But neither would--there was just too much personal risk involved. Not only for their immortal souls but the danger of some real damage to their corporeal forms as well. A prelate could order a lashing just as easily as any noble lord.\n\nSo instead the two of them looked to each other askance with raised brows and jerks of the head. They were seconds away from letting them through, but the sound of footfalls interrupted their silent debate.\n\nFrom behind them, a figure marched up the corridor. A fellow guardsman but this one with red cords at the shoulder. This was no grunt, no witless moron. This was their superior.\n\nFuck.\n\nIt wasn't someone too high up the ranks. That at least was good fortune. A high-level officer might have had interactions with Garron and recognized him. This was some midlevel officer of nebulous rank--which, Mouse realized, could prove troublesome as well. Someone working his way up the ranks with something to prove tended to be overly decisive and less prone to intimidation.\n\nThe puffed-up officer marched up to them with purpose, looking self-important and swollen with authority. \"What is happening here?\"\n\nThe guards exhaled in relief at his timely arrival. They looked in danger of passing out. Luckily, they would not have to make this decision, after all. Both gave the officer a hasty salute. \"The Grand Prelate wishes to return to his apartments,\" one quickly announced, the delight of passing on the responsibility evident in his tone, \"but we were given strict instructions to not let anyone through.\"\n\nThe officer made a quick inspection of Garron, from his head on down. Mouse could not read the expression on the officer's face. To be safe, he moved his hand closer to where his dagger was hidden inside the sleeve of his robe, just in case this turned south. His promise to Garron of no killing would be null and void if they faced a risk of being dragged away in chains or slain here in this corridor. He would protect himself. And his charge.\n\nThe officer looked to the men, his expression shifting to annoyance. \"Joltheads. Does this look like an escaped prisoner to you?\"\n\n\"No, sir. But we were told--\"\n\n\"My apologies, Shepherd,\" he said, putting his back to the guards and bowing low to Garron. \"Their ineptitude never ceases to astound me.\"\n\nGarron narrowed his eyes and made a conspicuous tilt of the head in Mouse's direction. \"Oh, trust me, I understand completely.\"\n\nThe officer's stoic face cracked, allowing a small smirk to lift the corner of his mouth. \"I've no doubt. I only wish they'd learn to use the good sense that the gods provided them.\"\n\n\"They are but a blunt tool to be wielded by those with the wisdom to use them properly. The hammer does not know what nail to strike, does it?\" They both chuckled. Mouse felt bile burn the back of his throat and was dangerously close to vomiting. \"We can all be thankful,\" Garron continued, leaning closer, \"and sleep more soundly knowing that you are here and possess the necessary intellect to guide them.\"\n\nThe officer nodded at Garron's sage assessment. \"We will delay you no further, Shepherd. I thank you for your patience in this matter.\"\n\nGarron bowed graciously to the officer and with a very straight back, started walking. The two guards scrambled to get out of the way. Mouse followed at his heels.\n\nAround a corner and well out of earshot, Garron finally slowed his deliberate march. He put a hand to the wall for support and slumped forward as he exhaled long. \"Thought we were done for,\" he breathed.\n\nWhile Garron recovered, Mouse listened at a nearby door, picked the locked, and peered inside. Dark and quiet. It was a small antechamber with a bedchamber beyond. With any luck it was currently unoccupied.\n\n\"Quick,\" he said, \"in here.\"\n\nThey scurried in and shut the door behind them. Mouse put a finger to his lips, then tiptoed to the second chamber for a quick survey. As he feared, there was a dark mound in the bed, buried beneath a blanket. The shape was smaller, so a woman or a youth. He backed out of the chamber and pulled the double doors that separated to the two chambers closed.\n\n\"Quiet now,\" he said softly with a tilt of his head in the direction of the chamber.\n\nEven in the dark, Mouse could see Garron's eyes widen. \"Someone's in there?\" he replied in harsh whisper. \"What are we doing in here?\"\n\nMouse held up his palm to quiet him. \"The alarm didn't wake whoever it is. We aren't likely to either--as long as we keep our voices down,\" he added pointedly.\n\nMouse carefully negotiated his way over to the window and pulled back the heavy curtain. Moonlight streamed in to cast a pale silvery glow on the furnishing. It wasn't much, but it provided at least a little to see by. Already his eyes were adjusting.\n\n\"Rather impressive back there,\" he said. A side table had a pitcher of water and a tray of small round pastries. He poured himself a mug and helped himself to a couple of pastries.\n\nGarron gave him a hard look as if he were somehow tarnishing the space. \"I've had plenty of experience dealing with members of the prelacy. One acts much like another.\"\n\n\"You think fast on your feet.\" He popped a whole tart into his mouth, then handed the other off to Garron. \"That's good.\" It sounded more like \"das ood.\"\n\nIn the dim light, he saw Garron shrug. He took the pastry and turned it around in his hand a moment, then took a gentile bite from it.\n\n\"It certainly helped you didn't have to attempt your dreadful low speech again,\" added Mouse. \"You could speak as your normal lofty self. Now we need to find you something else to wear.\"\n\n\"Again? How many costume changes are you going to put me through?\"\n\n\"As many as it takes. Look, it only takes one person to hear that a Grand Prelate was allowed through into this wing and look into it. If no prelates are currently supposed to be here....\" He lifted a palm in the air and let the thought linger. \"We need to stay one step ahead of them.\"\n\nGarron lifted a pair of riding britches that were left draped over the back of a chair and held them up for Mouse to see. They were small, even for Mouse. \"Well, I'm not likely finding anything to wear in here.\"\n\n\"No, which is why I have to head out there again.\" He cut off a sudden protest with his palm. \"I have more caches scattered around the manor. It will be easier... and decidedly less dangerous if I simply retrieve what we need and bring it back. Don't look at me like that. You're safe in here.\" He headed for the door.\n\n\"What if... if whoever is in there wakes up, or someone shows up at the door?\"\n\n\"Stay in a dark corner, and don't make a sound. I won't be long.\" Mouse put a hand to the latch, but Garron grabbed his forearm.\n\n\"I'm sorry,\" he said. \"About what I said earlier.\"\n\nMouse looked up at him, confused.\n\n\"I'd still be in that cell if it weren't for you. And I'd never have made it this far on my own, that's for certain.\"\n\nMouse caught himself smiling back, though Garron likely couldn't see it. The testy comment Garron had made earlier, a comment that Mouse had already dismissed and forgotten about, was evidently still weighing on him. Garron was concerned that he'd offended him.\n\nMouse couldn't help himself. He was inexplicably moved by Garron's na\u00efve concern for his feelings. It gave him a strange sensation of warmth and a tightness in his throat--which he quickly choked back down.\n\n\"Worry it none,\" he replied offhandedly. \"I took no offense. This is what I do.\" And later we can discuss ways for you to thank me, he added silently.\n\nThen he slipped out into the corridor.\n\n# Chapter 15\n\nMOUSE WAS glad to be away from him for a while. Garron's presence alone was mucking up his thinking.\n\nThis was unlike him. During a job, Mouse never allowed himself to be distracted by a pretty face. In that manner, he was the consummate professional. But something about Garron had disrupted his focus. Those emerald eyes were largely to blame. They had an uncanny sorcery about them, and Mouse had caught himself several times stealing glances at them. But Garron had other fine attributes that drew the eye as well, and he possessed an energy about him that seemed to fill an entire room.\n\nMouse wanted desperately to despise him. But he found he couldn't. Garron wasn't anything like the other nobles that Mouse had had the displeasure of encountering in his career. The nobility were all insufferable bastards--the entire lot of them. Pus-filled boils slowly poisoning the lifeblood of the kingdom. But Garron, he was different somehow. He acted... well, noble.\n\nIt didn't seem possible. Was this all some well-rehearsed deception?\n\nFraud and trickery were Mouse's forte. It was a large slice of what he did when on a job, so he could typically see right through it when attempted by others. A sustained deception was an art form, and in the hands of a novice, easy to spot. Mouse saw no crack in the man's quality. If this was an act, Garron was a true master.\n\nThat fucking kiss, he thought dourly. That was the problem here. The memory of it clung to his brain like cold porridge sticks to a bowl. He was allowing it to muddy his perception.\n\nHe needed a walk about. Yes, finding new costumes was necessary, but this excursion was more about clearing his head.\n\nFirst, he needed to hunt down the nearest window that offered a view of the inner ward. He wanted to gauge the level of commotion around the manor, gather a sense of how aggressive the search was currently elsewhere. If he was going to devise an effective escape, he needed intelligence on the enemy's search strategy.\n\nThe layout of the noble apartment wing of the manor was a labyrinthine mess, a muddle of corridors and chambers that didn't follow any sense or reason. The result of centuries of alterations and additions by each successive duke putting his personal mark on the manor. It was a wonder that anyone was able to find a way to their apartment without a map or a guide.\n\nMercifully, the window he needed was only a short distance away.\n\nIt was tall and narrow and recessed into the thick outer wall of the manor. He had to hoist himself up to reach the shutter's latch. He swung the shutters back, pushed out the paned glass, and leaned out. He was on the third floor, and the window provided a clear view of the entire area below. The sky to the east was only just beginning to lighten from black to indigo. At this early hour, the ward should have been quiet and vacant. Instead, it was a hive of activity. At least two dozen torches burned brightly below. Groups of guards stood in formation, awaiting commands while officers pointed and sent squads off running.\n\nMouse heard no distinct voices rising up to his window. Orders from commanders were conducted in whispers or in elaborate hand gestures. They were hoping to have Garron apprehended before most of the manor rose from their beds.\n\nA group of officers, and what looked to be the chamberlain himself, stood around a makeshift table covered in maps. Maps of the manor presumably.\n\nHe watched them for a short while, trying to decipher their strategy. It seemed like they were focusing on the exterior of the manor, setting up a perimeter on and around the wall. They assumed, as he'd hoped they would, Garron and he would be attempting to scale over the wall. Good. Because he had other ideas.\n\nThe question was how much manpower were they dedicating to dragging a net through the manor. And were the noble apartments going to be part of the sweep?\n\nHe left the window and began a casual survey of the corridors. More reconnaissance. So far, this part of the manor was quiet and still. If felt isolated from the activity outside. A few early risers were about--laborers slinking off to perform their duties before their masters were out of bed. They didn't show any signs of concern or urgency.\n\nThat would all change soon. News of an escaped prisoner would spread, sending the manor into further chaos.\n\nOr would it?\n\nThat gave Mouse a momentary pause. How much would the people of the manor actually learn about the escape? Certainly, the guard presence in the manor would be out in force to apprehend Lord Green Eyes, but Mouse wondered how much the residents of the manor would be told of what was happening. Very little, Mouse surmised. Garron's internment in the tower was kept largely secret, for reasons Mouse had yet to tease out. That meant they would be wary about what details they revealed about his escape.\n\nThat could work to his and Garron's advantage.\n\nThe chancellor would spin some yarn to explain the ramped-up military presence around the manor. \"A training exercise\" was the most likely excuse they'd give. But maintaining a ruse like that fragmented resources and energy. It muddied the available information and slowed progress. The inevitable surge of rumors would further the turmoil. That could provide Mouse plenty of wiggle room to maneuver around in the confusion, and serve up the opportunity he needed to slip Garron out of the manor.\n\nMouse's confidence grew. He could still manage to make this work and get Lord Green Eyes out of here. But they were far from out of danger, and Garron's restraints on him were becoming a real hindrance. They would be out in the city streets by now if Mouse had been allowed to handle it his way. A simple matter of a few tactical incisions in their defenses... and done. Quick and efficient. As it stood now, it was nearly daylight, and Mouse was nowhere nearer to figuring out how to get him beyond the wall.\n\nMouse continued his casual stroll about the place, keeping his eye open for anything that would help them. Nothing seemed particularly out of the ordinary or alarming.\n\nIf fact, it seemed perfectly ordinary, as if this were any other morning.\n\nMouse swore under his breath--he was more distracted than he realized not to have noticed before. There was little to no actual security presence in the wing. Some guards lingered about along with the few laborers that were already up from their beds, but their numbers didn't match what he saw gathered down in the ward, and they didn't seem to be searching or even on alert. They must have viewed this wing of the manor as secure. They believed that the prisoner couldn't have made it past the guards--or wouldn't have tried.\n\nBut no... it was more than that. They had intentionally shielded the alarm from this part of the manor. These apartments would be filled with important political and economic guests, and a heavy military presence tended to put such dignitaries' teeth on edge. More so, news of an escaped prisoner could erode their confidence in the esteemed duke and sour some delicate negotiations.\n\nDelgan was undoubtedly savvy enough to not jeopardize any sensitive deals he had in the works. An escaped prisoner could be construed at the very least as an unfortunate embarrassment and at the worst, a sign of weakness or a lack of control over his dominion. This meant a lower guard presence for the time being and a more surreptitious search of the area if they suspected the two of them had retreated here. This was welcome news.\n\nHe checked in on his caches. He selected what he needed and stuffed it into a canvas sack.\n\nOn his way back, he swung by the open window again and peered out. Little had changed, and he was about to duck back in again, but something caught his eye. Someone new had joined the men around the table. Female likely, based on the comparative size to the others, and she wore a flowing black robe. Not a Shepherd--she was not wearing the obligatory four-cornered hat. Mouse frowned.\n\nThe men surrounding the table all made respectful bows. She appeared to say one final thing to them, and then she spun from the table. Hands hidden in her sleeves, her chin high into the air, she marched across the ward toward the doors of the manor. A gust through the wards made the robes lift and flow in easy billows behind her. The robes were made of black silk.\n\nNo, he thought. Couldn't be.\n\nNot here.\n\nHe leaned farther out the window for a better view, but the light was still too dim and they were too far away to see clearly.\n\nFuck!\n\nThere was only one possibility. He abandoned the window at a jog. He had to get back to Garron. And fast.\n\nMOUSE PICKED the lock and lunged in. The room was still dark. Garron must have pulled the heavy drapes closed after he left, for they were drawn again. The weak morning light formed a pinkish line down the narrow gap between the curtains and did little to illuminate the chamber. Mouse stood in the open doorway, allowing the light from the corridor to spill inside. It took a moment for his eyes to adjust, but he spotted Garron in the corner straight away, leaning a shoulder against the wall. Garron straightened, clearly sensing Mouse's urgency.\n\n\"Quick,\" Mouse barked. \"Pull off those robes now!\"\n\n\"What?\" Garron asked, confused. \"Why? What's happened?\"\n\n\"Just do it.\" Mouse made a scan of the room. He spied what he needed quickly--the fireplace. He dodged out into the corridor again. Halfway down, he found a burning lantern in a wall sconce. He lifted it out and sprinted back to the room.\n\nGarron was still clothed in the black robe when he returned. He had pulled off the purple sash from his shoulders but watched Mouse scramble about, looking confused.\n\n\"What are you doing? Take that off!\"\n\nGarron was shaken from his inaction and started pulling the robe over his head.\n\nMouse tossed a number of logs from the hopper into the hearth and laid the lantern down on its side among them. Oil spilled from the lantern onto the wood, and the fire spread. In moments, the hearth was ablaze. He inspected the fireplace tools propped in a bucket, which clanged like a bell when he moved them about. He selected a poker, studied the tip a moment, then positioned it into the flames of the growing fire.\n\nGarron had the robe off but clutched it in front of him as if reluctant to let it go. \"Mouse, what's this about?\" He was looking anxious. \"What's happening?\"\n\n\"Come over here. By the fire so I can see better. Just hope I'm wrong.\"\n\nHe joined him by the fireplace still clutching the robe. \"About what?\"\n\nThe fire was starting to crackle, eating up more of the logs. Mouse repositioned the poker in a hotter zone of the fire. He took the robe from Garron's grasp and flung it aside, leaving Garron standing there in his smallclothes. His flawless skin reflected the firelight like gold.\n\nMouse immediately started to inspect every surface of his skin, turning him about in a circle to expose different parts to the light. His chest. His back. His legs.\n\nIt would be somewhere Garron wouldn't be able to see it himself. There was no mirror in his tower prison. \"Lift your arms,\" he said.\n\nGarron obeyed. \"What are you looking for?\"\n\n\"A tattoo.\"\n\n\"A what? I don't have a tattoo.\"\n\n\"Let's hope you're right.\" He squatted low to inspect the back of his legs--his thick beautiful legs. He rubbed his hands along the silky surface hoping to catch the feel of raised skin. Some tattoos--if well made--were close to the color of the skin and hard to spot.\n\n\"Hey. What's going on?\"\n\nMouse sprung up. Both he and Garron turned to double doors to find a lad of perhaps fourteen standing in the open doorway rubbing sleep from his eyes. For several heartbeats no one spoke. Everyone exchanged stares, not knowing what to do.\n\nFuck!\n\n\"Nothing,\" Mouse replied, recovering his wits. \"Go back in your bedchamber and close the door.\"\n\n\"Who are you? Why are you here?\"\n\nMouse marched over to him. He didn't have time for this. \"No one you need to concern yourself with. We are here on important palace business.\" He took the lad under the arm and guided him back in the chamber. \"Now just stay in here and don't make a sound.\" He gave him a pointed look after he released him that left no question that he meant what he said. \"Understand?\"\n\nThe boy nodded with short little terrified bobs of his head.\n\nGarron cried out in sudden pain.\n\nMouse closed the bedchamber doors behind him as he dashed back. Garron was down on one knee, his face contorted. Damn! He was too late.\n\n\"Where?\" he demanded. \"Where is it?\"\n\nGarron sucked through his teeth as he put his fingers over his right buttock. Mouse crouched down behind Garron. \"Undo the drawstring,\" he ordered. Garron reached down to his waist and pulled the end of the string. The waistband of his smallclothes went slack, and Mouse tugged the garments down, exposing one cheek of his ass. One perfectly round hemisphere.\n\nThere it was.\n\nLow on the right cheek, the rune burned into his flesh.\n\n\"What is happening?\" Garron hissed.\n\n\"Tracer ink,\" Mouse told him. \"They put a tattoo on your ass to find you in case you escaped somehow. They have a fucking mage here.\"\n\n\"Tracer ink?\"\n\n\"It's been activated. That's why the pain. It's to slow you down so they can catch up to you.\" The rune glowed red against his skin.\n\n\"Well, get the fucking thing off of me,\" he snapped.\n\n\"It's not that easy, Garron.\" He kept his voice low. Calm. \"There's only one way right now. And it won't be pleasant.\"\n\nGarron looked to the fireplace, which was crackling wildly now. Then back at Mouse, who was still crouched down at eye level with his buttocks. \"Oh shit.\"\n\n\"Yes. I'm sorry.\" Mouse adjusted the poker again, placing it deeper where coals were glowing angry red. \"And I've got to do it quick. Before they narrow down where you are. The good news is I don't have to burn off the entire rune. Just disrupt it. But it does have to burn deep enough into the skin so the ink is removed in that spot. It's almost ready.\"\n\nGarron winced again. His forehead was beading with sweat. \"Well, it burns now, so I'd rather it hurt and not have them tracking me. How much time do we have?\"\n\n\"Not much. The incantation does take some time to prepare, but once evoked, the mage will pinpoint your exact location.\"\n\n\"How did you know? About the tattoo?\"\n\n\"I spotted the mage. They obviously summoned her out because of the escape. Tracer ink was the only logical conclusion.\" He hated mages. In his line of work, mages got in his way. So he made it his business to know theirs.\n\n\"I don't follow.\"\n\nMouse fought the urge to sigh and roll his eyes. Of course he didn't follow. \"If a mage was employed here in Har Orentega, with the principal motive of keeping you under lock and key, they have two options. One, put a ward around the cell, which requires a mage of very high level to craft. Since there wasn't a ward around your cell--which is no surprise--few other than the great king himself can afford a mage with the capability to create one and frankly, I doubt you warrant the expense--we can therefore conclude the second option. Tracer ink.\"\n\n\"How did they get it on me?\"\n\n\"Drugged food's a possibility.\" Mouse scrunched his face, reconsidering that thought. \"But that can have side effects you'd notice. Probably went with a sleep spell. Something a mediocre mage could accomplish. Pull those off entirely. We don't need them catching on fire.\" He handed Garron a leather belt, part of his next wardrobe change. \"You'll want to bite down on this.\"\n\nGarron complied and stripped off his short clothes, allowing them to drop to the floor. He stood there completely naked.\n\nMouse pulled the poker from the flames. The tip was glowing angrily. He squatted again down on one knee behind Garron, poker positioned in front of him like a weapon. He closed his eyes and tried to think of something other than Garron's perfect form glistening in the firelight. He was stalling, he knew--not because he worried about inflicting pain, but he honestly didn't want to mar the man's perfection. It felt like throwing paint on a masterpiece. Or smashing a statue.\n\n\"Get ready,\" he said.\n\nGarron put the belt between his teeth, leaned into a chair, and spread his legs apart a little more.\n\nDear gods! Mouse thought. If only this pose were for another purpose.\n\n\"I'm ready,\" Garron replied. \"Do it.\"\n\nMouse moved the glowing tip of the poker closer, making sure it was positioned, then pushed the tip against the skin, right in the middle of the tattoo.\n\nGarron's entire body stiffened. He arched his back and made a low grunt through clenched teeth. But he didn't cry out any louder. Smoke rose up from the contact point, and Mouse could smell the charred flesh. He held the point for a count of five, then pulled the tip away. The rune was no longer glowing.\n\n\"Okay, done.\"\n\nGarron sagged against the chair. \"Nothing to it,\" he replied between breaths.\n\nMouse dropped the poker back into the bucket. Already the skin was swollen, inflamed, and blistering. \"We're going to have to hunt down some salve. That's going to start to seep, then stick to anything you wear. There's danger of it festering.\"\n\nGarron nodded. \"Do you think we got it in time?\" His brow glistened with sweat.\n\n\"Only one way to know. But I don't want to stick around here to find out. We need to move.\" He dropped the canvas bag at his feet. \"Start dressing.\" Mouse looked up to find the door to the bedchamber cracked open and the lad peering out from the other side. He let out an exasperated sigh and threw up his hands. \"Ah shit! Boy, I told you to keep the door shut.\" He spun about on his heel. Garron had already pulled on his smallclothes and a white chemise with a lace collar. \"We're going to have to do something about him now.\"\n\nGarron's head popped up, his eyes wide. \"Mouse,\" he said in a low voice. \"You spoke an oath to me.\"\n\nMouse stepped closer to Garron and leaned in. \"Gods!\" he said through clenched teeth, part whisper and part growl. \"What kind of monster do you take me for? I've no intention on harming him. He just can't be allowed to blab about us when we leave. He's going to have to join us.\"\n\n\"What? You're insane! We can't drag him along.\"\n\nMouse shoved the door open farther. The lad stumbled back into a corner, quaking. \"We don't have a choice. Right, lad? Should have minded when you had the chance. Now, make your bed. We're going on a little stroll.\"\n\n# Chapter 16\n\n\"WHAT'S YOUR name, boy?\"\n\nThe lad stiffened his spine and pulled his shoulders back. \"Lord Nevon Rhee of Har Senestus.\"\n\nGods help him. Another noble brat.\n\n\"Mother is here visiting with--\"\n\n\"Don't care,\" Mouse cut him off. \"When does your nursemaid fetch you, Lord Nevon of Har Senestus?\"\n\nThe lad scowled at Mouse. \"My steward,\" he corrected pointedly, \"shall be here shortly. He arrives shortly after dawn. He has strict orders to wait--\"\n\n\"All right, shut up.\" Mouse put a hand on the hilt of his dagger at his waist and made sure Nevon saw him do it. \"Despite what I said to my companion over there,\" he said in a low voice, \"I will hurt you if you don't cooperate.\" He wouldn't, of course. He spoke true when he told Garron he wouldn't bring harm to a child. The young were off-limits. But Lord Nevon here didn't need to know that. \"And if you answer me in more than five words at a time, I may just gag you and stuff you into a pickle barrel in some forgotten corner of the kitchen. Don't test me. Understood?\"\n\nNevon's lips tightened in irritation, but he nodded.\n\nMouse shepherded the lad out of the bedchamber. In the adjoining chamber, Garron was now fully dressed. Mouse had chosen a more garish look for him this time, the conspicuous attire of a rich merchant. Someone new to wealth and needing to show it off. Mouse calculated that it was a look that would fit in well in this wing of the manor. But even in the foppish garb, Lord Green Eyes managed to pull it off surprisingly well and make it presentable.\n\n\"I feel ridiculous,\" Garron grumbled.\n\n\"You look perfect. Now, we have to move out before the young prince's nursemaid shows up--\"\n\n\"Steward,\" Nevon put in sourly.\n\n\"--to pull his pants on for him and shake him dry after his morning piss.\"\n\nGarron scowled. \"Lay off him, Mouse. He's just a boy.\"\n\n\"I'm fifteen,\" Nevon protested.\n\n\"And just keep asking yourself if you want to see sixteen.\" Mouse pulled the novice robes off over his head and began tugging on his new garments--the simple garments of a merchant's underling and clerk. Innocuous and easily overlooked. \"All right,\" he said once he'd stomped his feet into his boots. \"Time to move.\"\n\nNevon looked at Mouse horrorstruck. \"You cannot expect me to run about the manor in... in... this!\" He pulled out the sides of his nightshirt like a curtsy. \"It's unseemly.\"\n\n\"Unseemly?\" Mouse was losing his patience. They didn't have time for this. \"You'll run the corridors na--\"\n\nGarron grabbed Mouse by the shoulder and guided him backward, away from the boy.\n\n\"Look,\" Nevon said. He rolled his head back and exhaled, as if speaking to two simpletons. \"If you don't want to be noticed, it makes sense that I'm dressed appropriately.\"\n\nGarron and Mouse exchanged looks. \"He's got a point,\" Garron replied.\n\nMouse was ready to argue that it was early enough that there wouldn't be many people up and out of bed yet, but he knew the lad was right--which irritated him. Mouse had to watch himself. He was allowing fatigue and frustration to cloud his thinking.\n\nHe elbowed Garron's arm away and stepped closer to Nevon again. \"How much of our conversation did you overhear?\"\n\n\"Enough to know you're in some trouble with Duke Delgan,\" Nevon replied with a measure of self-satisfaction. \"I gather that he was imprisoned here, and you're helping him to escape.\"\n\nFuck.\n\nMouse glanced over his shoulder at Garron. \"Excellent.\" He rubbed his eyes with forefinger and thumb and tried to think of what to do about this. Things were spiraling out of control.\n\n\"Well, I'm not going to say anything,\" Nevon said, \"if that's what you're worried about.\"\n\nGarron's eyes narrowed curiously. \"And why is that?\"\n\nNevon gave Garron a surprised look, as if it were obvious. \"'Cause you're a nobleborn. Like me.\"\n\nMouse fought the urge to snarl and blow air through his teeth.\n\n\"I'm not certain I understand the connection,\" said Garron, sounding very much like a diplomat.\n\nNevon's arms dropped to his sides as if he'd lost control of those muscles. \"If he'd been the one thrown into a dungeon,\" he said, indicating Mouse with a tilt of his head, \"I'd say it was probably because he did something horrible and deserves to rot there. But everyone knows that nobles are not put in irons for the same reasons as commoners. That means it's something political. And probably unfair.\"\n\nMouse had to admit the logic was sound. The lad was astute.\n\nGarron's eyes shifted briefly toward Mouse, signifying he had come to the same conclusion.\n\n\"I can tell you too that Mother doesn't trust Lord Delgan,\" Nevon continued. He leaned in conspiratorially and lowered his voice. \"She's telling people she's here to visit a cousin, but is really here to conduct some business with the merchants in the province. Lord Delgan would try to prevent it if he knew, so she's meeting with the merchants in secret. Mother calls him an 'untrustworthy snake.'\"\n\nFrom the look on Garron's face, Mouse could tell exactly what he was thinking. He shook his head. \"We are not leaving him here, Garron,\" he said.\n\n\"I think we can trust him.\"\n\n\"Absolutely not. Too risky. He's coming with us.\"\n\n\"I'll only need a few minutes, and I can be ready to leave,\" Nevon said.\n\nMouse turned and raised his eyebrow at the lad, unable to mask his surprise.\n\nNevon shrugged. \"I've nothing else to do. Mother won't let me take any part of her negotiations. She only brought me along to cover for her when she's not in the manor. Frankly, I'm bored silly.\"\n\n\"One minute,\" Mouse agreed with a nod. The lad opened his mouth to protest, but Mouse held out his hand to stop him. \"It's that, or I pick out garments for you.\"\n\nNevon looked horrified at the very idea of it. \"Very well,\" he said with an eye roll, and then he strode back into the bedchamber.\n\n\"This is a terrible idea,\" Garron grumbled once Nevon was out of earshot.\n\n\"We have no choice, Garron.\"\n\n\"He's just a boy.\"\n\n\"All the more reason not to trust him. A boy that age can lie to your face as effortlessly as farting.\"\n\nGarron looked as if he might argue but frowned and said, \"I don't like it. I fear we are putting him in danger.\"\n\n\"He will not come to any harm,\" Mouse said. He looked up directly into Garron's bright emerald eyes. Garron still looked dubious. \"I promise,\" he added, then glided to the open doorway to keep an eye on the lad, if only to make sure he didn't try to signal someone from a window or attempt some other foolhardy notion of escaping. Nevon rummaged through a trunk beside the bed and pulled out random items.\n\nIn the other chamber, Garron leaned heavily against a chair with his hip pushed oddly in one direction, presumably to keep the fabric from touching the wound.\n\nNevon pulled out trousers and a silk chemise and set them aside, then selected three different jerkins and arranged them side by side on the bed. He stepped back and considered each one.\n\n\"Pick something,\" Mouse growled.\n\n\"I'm working on it!\" the lad testily replied.\n\n\"Choose, or I'll choose for you.\"\n\n\"I won't be spotted wearing something that makes me look like a street beggar. I'd rather you killed me.\"\n\n\"A tempting offer!\" Mouse sniped.\n\n\"Look, it's not that simple. Ashimnae always selects my daily wardrobe for me.\"\n\nMouse sighed. He was losing patience, but he dug deep and forced a calm in his voice that he didn't feel. \"The blue one. That one looks nice. Wear that.\"\n\nNevon looked over at Mouse as if he were a troll that had strayed into the room. \"Of course you'd pick that one,\" he said, and grabbed the green-and-gold jerkin next to it. He started to squirm out of the nightshirt, but stopped and glared at Mouse. \"A little privacy, if you don't mind.\"\n\nMouse folded his arms and looked up at the ceiling. When he looked back down again, Nevon was buttoning up the front of his jerkin.\n\n\"Wait!\" the lad suddenly exclaimed. \"I've no boots! Ashimnae took my boots with him last night to polish them. I've no others.\"\n\n\"Then you leave without any.\"\n\nNevon froze, eyes wide with shock. \"And run about the manor like a street urchin?\"\n\nMouse turned to Garron. \"Are all you noble types as ignorant as this? Perhaps someone should inform your ilk that there is more to street poverty than a lack of proper footwear.\" Garron made a wry face and turned away. Mouse threw up his hands. \"Fine! If this moves things along.\" He sat down on the edge of a chair and pulled off his boots again. Mouse's own feet were small and wouldn't be much larger than the lad's, so the boots would suffice. He chucked them across the room with some force. Nevon had to leap out of the way or be struck by them. \"Here!\" Mouse said. \"I'll run about like the urchin, then.\"\n\nA merchant's assistant without boots would look peculiar, but Mouse didn't have time to worry about it now. They needed to be out of the room before the brat's nursemaid arrived.\n\n\"Something you're accustomed to, I suspect,\" Nevon said as he scooped up each boot and tugged them on. He stamped his feet a few times, his lips twisting down in a grimace. \"These aren't very well made. They pinch my toes.\"\n\n\"Can we leave now, Your Highness?\"\n\n\"Hold on!\" Nevon said suddenly. Mouse threw up his hands again as the lad dashed to the bedside table. He pulled open the drawer and rummaged through it. \"I'll leave him a note. I'll tell him that I went to go get something to eat in the kitchens.\" He pulled out paper, a quill, and an ink well and dashed over to a desk by the window. He scribbled out a quick note, folded the paper in half, then considered the room with pursed lips. He settled on the table by the fire, and placed the note next to the tray of sweets, propping it against a chalice.\n\nMouse grabbed the note and opened it.\n\nNevon scowled at him. \"You can read?\"\n\n\"Better than you. And I can apparently spell better than you as well.\"\n\n\"I was writing quickly,\" Nevon replied, sounding a little wounded. \"It should keep Ashimnae from growing suspicious, for a while at least. He'll wait here for a time before he decides to look for me.\"\n\nThe note said exactly what the lad claimed it would say. Mouse kept his face neutral and replaced the note on the table. \"So maybe I won't have to stuff you in a pickle barrel today after all.\"\n\nNevon pressed his lips together and gave Mouse a level stare. \"I think you're all talk.\" And he marched for the door.\n\n# Chapter 17\n\nTHE SUN had nearly crested the horizon when the three of them exited Nevon's small apartment. The eastern sky had lightened to sapphire and was painted with streaks of pink clouds. Pale light filtered in through the occasional window, illuminating the eddy currents of dust and giving the corridors a strange, dreamlike air.\n\nMouse had Garron lead the way while he moved Nevon along with fingers lightly pinching the back of the lad's neck. They moved at a deliberate pace, careful not to appear too hurried and draw attention. Mouse also made certain they did not run headlong into Nevon's manservant en route. The lad pointed the direction Ashimnae would come, from his temporary residences, so Mouse led them along a different route but was cautious all the same in case he was lying.\n\nThe corridors were more occupied than Mouse liked. Unlike earlier, it was the dignitaries and aristocrats now roaming about, looking haughty and official. Most were either local luminaries or guests from neighboring duchies, but he spotted the livery of a well-connected family from the capital. Delgan was showing his ambitions, clearly attempting to expand his influence. As for the laborers and servants that Mouse encountered before, they had done what was expected of them when their masters were about; they disappeared from sight entirely.\n\nNo one paid them any mind, but Mouse caught himself tightening his thumb and forefinger on Nevon's neck each time someone walked past.\n\n\"Ow! That's not necessary, you know,\" Nevon grumbled.\n\nDespite his proclamations of willing cooperation, Mouse still didn't trust him. \"Just reminding you that I'm here.\"\n\n\"Your odor is doing a superb job of that already. You realize if we were in Har Senestus, you would be drawn and quartered for just laying your hand upon me.\"\n\n\"Just keep moving, little prince.\"\n\nThe lad's shoulders clenched under his hand, and he grinned in satisfaction. He needed to get them out of the corridors as quickly as possible, before someone recognized Nevon or started asking questions.\n\nStill, no one about seemed in any particular hurry or on edge. That could all change quickly enough if the mage had managed to isolate Garron's whereabouts before the ink was disrupted. At any moment, these corridors could be thick with guards.\n\nMouse herded them into a seldom-used stairwell toward the back of the manor. They descended two flights and hurried through another maze of corridors.\n\nAs they rounded a corner, Garron nearly collided with a cluster of three guardsmen. Mouse's heart lurched into his throat, and on reflex he moved his hand close to where the dagger was hidden. But Garron offered a quick apology and a bow, and the three marched off without a backward glance. They may have been on the lookout for an escaped prisoner, but not two men and boy.\n\nThe lad might turn out to be a surprising asset, after all.\n\n\"There, to your right,\" Mouse said to Garron. \"Turn.\"\n\nThey ducked down a short hall to a small unassuming door that would have been easy to overlook. \"This will lead us to the lower levels.\"\n\n\"Servants' passages?\" Nevon said with sudden alarm.\n\n\"Easy, Your Highness. You won't break out into open sores by communing with the common folk.\"\n\nNevon's face darkened in irritation. \"I know,\" he grumbled. \"And don't call me that.\"\n\nGarron threw him a look. \"You sure know how to win people over.\"\n\n\"I can be charming,\" Mouse retorted. \"When need be.\" And he wasn't going to waste his time kowtowing to some entitled snob that he was going to ditch somewhere at the first opportunity.\n\nGarron pressed his lips together in a sardonic snarl. \"Warn me when it's about to happen so I know to keep an eye open for it.\"\n\nMouse pulled open the door, peered in, then stepped inside, flagging the other two to follow. The passage beyond was dark and narrow, forcing them to proceed in a single file. A single tallow candle burned in a small wall sconce, providing just enough light for them to see their way, but its black, sooty smoke hung in the air like a specter and made the air thick and foul.\n\n\"How did you know this was here?\" Garron asked.\n\n\"I told you. I'm a good planner. I learned everything I could about the place before I came to fetch you.\"\n\n\"It's creepy in here,\" Nevon grumbled. \"And it smells.\"\n\nMouse bit back a sharp reply. Yes, brat, this is how most experience the world--so you can enjoy silk sheets and the smell of rose petals. Garron was in a more generous mood. \"We won't be in here long,\" he replied in a gentle tone.\n\nThe passage continued for a ways, then rounded a corner. They could hear murmurs echoing against the walls from up ahead. Servants of the manor going about their morning business. A stairwell seemed to materialize from the shadows on their left. Mouse clicked his tongue at them and signaled with his head for them to keep up with him. With palpable trepidation, Garron and Nevon descended behind him. The stairwell was hardly wider than Mouse's shoulders, and it twisted in a tight spiral as if following the tread of a screw. The front edge of each step was worn down--eroded away by centuries of boots owned by those locked into a life of servitude.\n\nMouse felt a measure of guilt for what he was about to do. This was a private sanctum, a place where the servant class could escape the eye of the elite for a time, a place where they could relax and mingle freely with members of their kind. Now he was bringing two nobleborn into it.\n\n\"Why are we traveling deeper into the manor?\" Garron asked. \"Instead of trying to get out?\" He spoke in hushed tones, as if in a shepherd's chapel. He felt it too, Mouse sensed. His movement down the steps was awkward and clumsy, and he hissed a little each time he lowered his right foot. It was clear the burn on his ass was bothering him.\n\n\"I'm working on it,\" Mouse answered. He tried to sound confident, like he knew what he was doing, but his voice betrayed him. Escape for the moment was secondary. Right now, he was more focused on not getting them captured. The mage obviously now knew the tattoo on Garron had been discovered and the rune's power disrupted. There was no way of telling how precisely she had pinpointed Garron's exact location before Mouse burned it off, but Mouse was willing to bet she could at the very least provide a general vicinity. He wanted to be well clear of the noble apartments if and when a sweep began.\n\nGods burn him! This fucking mage was going to be trouble for them, no mistake. There was little that ever caught him flatfooted, but he certainly didn't anticipate having to contend with that.\n\nWhat other little tricks did she have planned? He'd be a fool to think she had nothing else.\n\nAnd what could warrant the amount of coin required to hire the mage in the first place? What was so important about Garron?\n\nThe stairwell dumped them into a space that was too wide to be a corridor but too long to be considered a chamber either. It had low walls but a high arched ceiling that ran its entire length. They were deep in the bowels of the manor. More tallow candles burned in carved-out niches in the stone wall, but they barely pushed back the dank gloom of the place. Its purpose seemed eclectic--part storage, with barrels and crates left stacked about with no discernible order, and part gathering place. Makeshift tables surrounded by stools and benches were clustered about one end like an abandoned tavern, plates and mugs left in haphazard piles.\n\nNevon looked around with wide-eyed fascination, as if amazed that such a place could even exist under his feet. It was clearly his first exposure to the underbelly of a keep, his first glimpse at how the meat became the sausage that appeared on his silver plate. \"What is this place?\"\n\nThree women emerged from a doorway. They each bore a stack of folded linen sheets in their outstretched arms and giggled among themselves. Mouse caught only a fraction of their conversation, but it was clear it was about a boy one of them fancied. As they tumbled into the space, relaxed and carefree, they all became aware simultaneously that they were not alone.\n\nThe change was immediate. With a start, the conversation was severed and backs stiffened. Smiles fled with such immediacy it made Mouse fume. Their faces turned to Garron in unison.\n\n\"M'lord!\" one of them exclaimed as she fought to recover herself. She lowered her gaze from him and curtsied. The other two followed her lead. \"Forgive us. Are you in need of something?\" In a single heartbeat, the three of them had recognized Garron for what he was. Their unease and surprise was palpable.\n\n\"No, good ladies,\" Garron replied in gentle speech. \"Forgive the intrusion.\" Mouse could sense that he recognized the effect his presence was having and was shamed by it. He had stepped into their realm and altered it, claimed it by just being present.\n\n\"It is no intrusion, m'lord. How can I assist you?\"\n\n\"You cannot. Please, I insist, go about your work and pay us no mind.\"\n\nFat little chance of that happening, Mouse thought.\n\n\"As you wish, m'lord.\" With a solemn march not unlike a funeral procession, the three carried their neatly folded burdens away, all the joy from moments before suffocated away.\n\nThe room from where the women appeared was lined with washing tubs along one wall and tables for folding along another. The pungent smell of lye and lavender choked the room. At the opposite side, a stairwell led up, likely to an outside courtyard where the lines were tied for drying. Two other women were elbow deep in washtubs, hard at work scrubbing sheets against a washboard. They too already knew of Garron's presence, for their chins were tucked low, and they said not a word.\n\nMouse considered the stairs to the outside, but to keep such workings out of sight from the gentle class, clotheslines were kept in closed-off areas typically with no access to the rest of the manor or manor grounds. Gods forbid the nobility catch a glimpse of their smallclothes flopping about in the breeze.\n\n\"This way,\" Mouse announced with conviction, as if he knew the way, and led them away from the laundry. The two followed without question, like sheep lost in an unfamiliar wood. Mouse poked his head into the various doorways, scanning each chamber. In short order, he discovered what he was searching for. He snatched a tallow candle from the niche and ducked inside.\n\nA food store. His nose told him they were getting closer to the kitchens, which set off his stomach. The chamber was deep and dark and filled mostly with crates of vegetables and dirty tubers, but a quick survey revealed a crate of decent apples in the corner and a wheel of wax-covered cheese on the shelf. Mouse wasted no time pulling it down and cutting off thick slices. He passed the sections around.\n\nGarron leaned against a stack of crates as if winded. He pulled the fabric of the trousers away from his skin with a hiss and positioned his hip so it didn't come in contact with the wound. He took the wedge of cheese offered him but just held it. \"We're resting here, I gather?\" He looked drawn and pale. The change in him seemed sudden, like a lantern about to exhaust the last of its oil. The night had taken its toll on him. Not surprising--after weeks of lethargy in a cell and a questionable diet, this had been a taxing spike of activity. And the initial adrenaline of his escape had worn off.\n\n\"For a bit,\" Mouse replied. \"I'm going to survey the area.\" He put a hand on Garron's arm. \"Keep to the back and rest. You'll be safe here.\"\n\nGarron nodded.\n\nNevon climbed up on a crate and chewed on his hunk of cheese, looking content and unflustered. This was all just an interesting distraction for the lad. An unexpected adventure. He must have felt Mouse's eyes on him, for he looked up suddenly.\n\n\"What?\"\n\nMouse pursed his lips in thought. What to do about him?\n\nGarron seemed to read his mind. \"I will ensure he stays put, Mouse,\" he said.\n\nMouse made a single affirming nod. He had no choice but to trust him. \"I'll be back soon.\"\n\nMOUSE WAS back in less time than it took to skin a rabbit. He half expected when he slipped into the storeroom to find Garron asleep and the lad gone, but they were both in the back, lounging on crates as if seated on thrones and conspiring quietly like old friends. Mouse dropped the sack he carried on a barrel top and rummaged through it.\n\nGarron looked up. \"That was quick.\"\n\n\"Just getting a feel of what we're up against.\"\n\n\"And?\"\n\n\"There are guards about, sweeping the area. Nothing too aggressive, but they are actively looking. My guess is they believe us elsewhere, but they're not leaving anything to chance.\"\n\nHe pulled out a loaf of bread and handed it to Nevon, who grabbed it with both hands and put his face into it to smell it. He made a groan of pleasure.\n\n\"It's still warm,\" he exclaimed. \"How'd you manage that?\"\n\nMouse shrugged. \"Trade secret.\" The reaction to good warm bread was universal. No matter how much coin was in the coffers, or how rich your diet, everyone responded the same to freshly baked bread.\n\nHis quick tour had surveyed the kitchens and some of the workshops buried down there in the belly of the manor. He located where foods and supplies were carted in too--a potential exit point from the manor. It now had guards stationed there, but he could figure out a way past them if need be. It was worth considering, but it would put them out into open ward and no closer to being on the other side of the wall.\n\n\"Now if only I had some butter and jam,\" Nevon lamented.\n\nMouse pulled out two small earthenware crocks and handed them over too. \"Any other requests, Your Highness? A roast duck, perhaps?\"\n\nNevon attempted an annoyed look, but he was too happy about the bread to pull off anything believable. Even Garron smiled.\n\n\"I didn't leave you out, Lordship. I come with gifts for you as well.\" He pulled out another small earthenware container and held it up for Garron to see.\n\n\"What's that?\"\n\n\"I happened upon the manor apothecary. Should help with your burn.\" In truth, it was what he set out to look for, but Garron didn't need to know that.\n\nGarron took the jar, relief visible on his face. \"Thanks, Mouse.\" He pursed his lips and held on to it a moment without moving. \"I... uh... might need some help getting it on.\"\n\nMouse noticed how Garron leaned to one side favoring his right cheek as he sat. Mouse nodded and took the jar back. \"Stand up, then.\"\n\nGarron complied, carefully lifting himself off the crate. He turned his back, then propped one arm on the crate to brace himself. It didn't take Mouse long to learn why Garron wanted him to do it for him. The wound had begun to seep, and his smallclothes had stuck to the burn.\n\nMouse carefully began the slow process of peeling the encrusted linen from the wound. Garron stiffened but didn't utter a sound.\n\n\"Why does he call you Mouse?\" Nevon asked.\n\n\"That's my name.\"\n\nNevon scrunched up his face. \"Your parents named you after a rodent?\"\n\n\"It was not the name they gave me.\" Mouse's lips tightened at the memory. He hadn't thought about his other name in a long time. It was there, tucked in a dusty bin in the back of his mind, waiting to be recalled. It seemed somehow alien to him now, as if it belonged to someone who didn't exist anymore. The distraction caused him to pull the linen too quickly and Garron flinched. \"Sorry,\" he told Garron. \"Mouse was another name given to me. When I was older. I chose to use one and not the other.\"\n\nGarron looked down over his shoulder at him. \"And how does one come to be called Mouse?\"\n\nMouse intended to dodge as he always did when that particular question arose--as it inevitably did. But the look in Garron's eye weakened his resolve, and before he was even aware, heard himself answering truthfully.\n\n\"It's not terribly complicated,\" he began. \"I have always been small. And... perhaps caused some measure of mischief around the village. From time to time.\"\n\n\"So far, nothing particularly surprising,\" Garron replied with a smirk.\n\nMouse scowled at him. \"You may want to wait until after I'm done down here to insult me.\" He tugged on the fabric a little harder to make his point.\n\nGarron stiffened. \"A valid point. Proceed, then.\"\n\n\"In time, it would appear, I inadvertently developed a rather... inimical reputation. Which turned out convenient for all the other children of the village when they had done something particularly impish. I had become an easy scapegoat.\"\n\n\"And an easy target for ridicule and harrying as well, I imagine,\" Garron added.\n\nMouse was momentarily tripped up by Garron's insightfulness but forced his face to not let on. \"Which is how I learned how to defend myself, incidentally,\" he said. \"I don't know how it started, but some villager began referring to me as that 'pesky little mouse.' As things happen, the name stuck.\" Mouse plucked the last of the fabric from the wound and sat back on his heels. \"There. Tricky part's finished.\" Mouse tugged the short clothes farther down Garron's hip to keep it out of the way. He peeled the wax seal out of the earthenware jar.\n\n\"Those days are far behind you. Why continue to use it?\"\n\nMouse dug two fingers into the yellow paste and began to gently apply a thick layer over the wound. He was amazed how little the flesh beneath the skin gave in to the pressure of his fingers. It was like massaging a ripe melon. \"I grew to like it. It suits me. Plus, a name attached to a reputation can have its uses.\"\n\n\"No doubt. It's a name people will remember, and I'm certain your reputation precedes you aplenty.\"\n\n\"Is that why you left?\" Nevon asked. \"Were you forced out? Did the villagers run you out with pitchforks at your back?\" The lad leaned in, captivated with this glimpse into the lives of commoners. Especially sinister ones like himself. The whole morning had been an exciting peek under the rock for the lad, seeing how the rest of the world lived their sad lives.\n\nHe felt his expression darken. \"Nothing so dramatic.\" He pushed the wax back into the opening and rose to his feet. \"All right, that's done. Should be good for a while. But it'll have to be reapplied before too long, or it'll start to stick again.\"\n\n\"The pain is subsiding,\" Garron noted with some surprise as he carefully pulled his short clothes and trousers back over his ass.\n\n\"Reagents in there for pain, I imagine,\" Mouse told him. \"Should stop the wound from festering too. But let me know if you feel a fever coming on.\"\n\nGarron put his hand on Mouse's shoulder just below the neck and squeezed. \"Thanks, Mouse.\" His thumb and forefinger pressed directly against skin above the line of Mouse's jerkin. The gesture was gentle and familiar, intimate in a way that Mouse was unprepared for. It sent shock waves of heat through his flesh, and his torso and shoulders tightened involuntarily. It wasn't until Garron pulled his hand away that Mouse found he could breathe again.\n\nWhat the fuck was wrong with him?\n\nMouse nodded to appear casual. \"We need to be going,\" he said.\n\n\"I could use a moment longer,\" Garron replied. \"I feel I am just starting to catch my breath again.\" He leaned back and rested his head against the wall.\n\nMouse wasn't entirely certain what their next move was anyway, so he agreed. \"Very well.\"\n\nGarron lifted his head again and gave Mouse a narrow look. \"Then I have one more question, if you don't mind.\"\n\nMouse raised an eyebrow.\n\n\"Your father. Tell me about him.\"\n\nThat was not the question he expected, and he did little to hide his surprise. \"My father? Garron, we hardly have the time--\"\n\n\"Indulge me. Please.\"\n\nMouse gave a small shrug of acquiescence. \"What is there to tell? He's a simple man. Uncomplicated. Kind. Nothing like me.\"\n\n\"Was he military?\"\n\nMouse chuckled lightly. \"Gods, no. He's a tanner. Wanted me to follow him in his craft... but both fate and I had other plans. He still has a shop in the village and carves out a good living there.\" What power did Garron possess over him? Mouse found himself defenseless, spilling information he hadn't disclosed to anyone in years. \"He is well known, actually. Many are known to travel quite far to commission his work.\" Mouse could not help but crack a smile of pride. His father was a master, and Mouse could never understand why he stayed in that small town instead of heading to one of the bigger cities. \"He's from the south originally, I think, but met my mother in the north--Entenund, I believe--so he decided to settle near there in Hallowridge.\"\n\n\"And your mother?\"\n\n\"Died when I was very young. Look, what's this about. Why all these questions?\"\n\n\"Curiosity,\" Garron replied. \"I cannot help but wonder how your father came about owning such a spectacular dagger.\"\n\nMouse's hand unconsciously moved to the hilt of the dagger. \"I don't really know,\" he said. \"Never thought about it. I found it the night I left. I was searching for a bit of coin to take with me, and I stumbled upon it at the bottom of a chest. I took it, not knowing its value, but needing a weapon.\"\n\n\"And now? You don't find it strange that he should have it, hidden away?\"\n\n\"Perhaps it was given to him as payment.\"\n\nGarron made a slow nod. \"Perhaps.\" Then he let the subject drop and said no more.\n\nA wave of irritation washed over Mouse. What was that about? What was he trying to imply? He folded his arms and looked across the storeroom at nothing in particular.\n\nBut something caught his eye. Movement. In the corner. And his heart tightened.\n\nFuck.\n\nSlowly his hand moved back toward the dagger. His father's dagger. And he slipped it out of his belt. With deft practiced fingers, he positioned it for a throw. But even as he did it, he knew it was too late.\n\nFuck. Fuck. Fuck.\n\n# Chapter 18\n\nIN ONE fluid motion, Mouse pulled the dagger back behind his ear. With a quick flick, he sent it spinning across the storeroom. The dagger made a soft whistle as it flew and a moment later, from the darkened corner, came a satisfying single high squeal. Then silence.\n\nBoth Nevon and Garron flinched. \"Shit, Mouse!\" Garron exclaimed, irritation in his voice. \"Strange time for target practice.\"\n\n\"Gather up our supplies,\" he answered with cold seriousness as he crossed the storeroom. \"Time to move.\"\n\n\"Why? What's happening?\"\n\nMouse located the dagger. His aim had been true. The blade had completely skewered the rodent's body. He grabbed the hilt and lifted it off the ground, keeping it horizontal with the carcass still on the blade. He looked over the body and frowned. Exactly what he feared. He carried it back to the others.\n\n\"Ew!\" Nevon exclaimed, recoiling. \"What are you doing? Get rid of that!\"\n\nMouse held it a little higher for Garron to see. \"Look closely. What do you see?\"\n\n\"It's just a rat!\" Nevon said.\n\n\"No,\" Garron said, leaning in to inspect it. \"Mouse is right. There's something on the fur. A symbol of some sort.\"\n\n\"A mage's rune. Burned into the flesh. This is one of the mage's spies.\" He flung the carcass into the corner. It hit the wall with a small thud and slid down to the floor. Dammit. He knew she would try something else. How could he have been so careless?\n\nNevon looked in horror in the direction where the body had been tossed. \"A spy? A mage can do that?\"\n\n\"She was watching us through the creature's eyes. They know where we are, Garron. We have to go. Now.\"\n\nThey fled the storeroom and sprinted down the corridor. In short order, the entire area would be flooded with guardsmen. He had to get them someplace quickly, or it could all end in disaster.\n\n\"Where are we going?\" Garron asked behind him.\n\n\"Away from here,\" he replied--but he really had no idea. They were heading in the direction of the kitchens, which would have too many people about. His mind ran a quick inventory of their options. He had to lead them somewhere else. But where? The mage had discovered that he and Garron were still deep in the manor, hiding out in the servants' tunnels, which meant they could dedicate all their resources and concentrate their search. They would send in guards from every direction to block their escape.\n\nAnd it was likely that the mage had other four-legged spies about.\n\nFollowing the main artery through the manor's substructure was too risky. Not that any of the servants would deliberately turn them in, but self-preservation was powerful. Someone would sing if pressed. He led them down the first side passage they came across--a narrow, low-ceilinged route that looked promising. But ten steps in, Mouse heard voices farther ahead, so he shooed the other two back again and sought a different route. A different passage branched off to the right. This time, Mouse turned an ear to the opening and waited. Hearing nothing, Mouse trotted ahead on the front of his feet, dagger gripped tightly in his hand.\n\nThe passage was short and ended at an ironbound door. Mouse knew immediately what he'd stumbled upon. There was only one thing that would be kept this tightly locked down here among the manor's drudges.\n\nHe sent Garron off to fetch a light while he squatted down and pulled out his small leather case of tools. Garron returned moments later with one of the fat tallow candles from the wall niche. He shaded the flame with his hand as he scurried back. He handed the candle to Mouse, who set it on the floor underneath the lock. It smelled of rancid fat. The weak flame gave off just enough light for Mouse to see by, but also black smoke that billowed upward and quickly formed a sooty cloud near the ceiling of the tight space. Nevon coughed and covered his nose and mouth.\n\nThe door had two locks. Kneeling before the door, leaning in and squinting from the poor light and the stinging smoke in his eyes, Mouse slipped his tools into the first lock. The process didn't demand too much by sight. While it did help to see how far the tools were in, their angles into the lock, and so forth, it was a job accomplished mostly by feel and instinct. He moved his tools around to sense the interior. This lock was unlike all the others in the manor. No surprise. It would have been commissioned special from a locksmith of some prestige.\n\n\"Can you do it?\" Garron asked.\n\n\"Of course I can do it,\" he snapped, his hands freezing in place as he looked over his shoulder. \"Though it would go quicker without interruptions.\"\n\nSomeone had spent a good deal of coin on this lock. It was well made and did not provide him a lot of room to maneuver about inside. Plus it had two release pins instead of the typical one. He felt the first mechanism within catch on his tool and start to release. Then....\n\nSnap.\n\n\"What was that?\" Garron asked.\n\n\"My tool broke.\"\n\n\"What?\" Panic was rising in his voice. \"You have another?\"\n\nMouse teased out the broken end from the hole, tossed it aside, and returned to his leather case on the stone floor. \"One more.\" He slipped it out and began again.\n\nVoices came from down the passage. Shouting.\n\nNevon started bouncing and fidgeting with his hands. \"Mouse?\" he said in a frantic whisper. \"I think they're coming.\" Garron put a hand on the lad's shoulder to calm him.\n\nHow odd, thought Mouse, while he felt around again with the tool. How odd it was how invested their captive was in their escape. The lad was more anxious than any of them. This had been a strange day indeed. \"Hurrying is what broke the first tool,\" he replied flatly.\n\nThe tool caught the mechanism again. He gave it more a gentle pressure this time and could feel it--almost see it in his mind--move. Holding the first tool steady, he positioned the second tool and dug deeper for the second release. Behind him, the voices were getting louder. Someone was barking orders. He could hear movement too, footfalls on stone and the clunky sound of armor.\n\nHe maneuvered his hands deftly and felt the lock mechanism obey his tender coaxing. He could feel the strain on the tool. He held his breath until he heard the soft click.\n\n\"One down.\"\n\n\"Mouse, they're going to be upon us any second.\"\n\nGarron was right. He could hear them working their way down the corridor, making a thorough sweep.\n\nMouse set to work again. The lock was similarly constructed... luckily. But it was made for a different key. Two separate keys for one room. A fail-safe way to prevent the contents of the room from walking off. Two different people had to unlock the door. Mouse shut out the sound of the guards drawing closer, and the sound of Garron's foot anxiously tapping on the floor, and focused his entire attention on the tiny hole in front of him. He allowed himself a full breath, and then he inserted the tools, orienting them as he had before. He twisted his hands, feeling the force of the pins against his tool. Slowly. Smoothly. He closed his eyes and constructed the lock in his mind as he guided the two slender iron rods.\n\nClick.\n\nMouse pushed the latch, and the door swung open.\n\n\"Quick, now!\" he ordered. Garron and Nevon dashed into the room. Mouse scooped up the candle and his tools and followed.\n\nJust as he predicted. The room was rows and rows of stacked wooden kegs. The wine cellar.\n\nHe closed the door--softly, for guards were no more than ten strides down the main passage. He handed the candle to Garron and waved him away from the door. It would not help if the guards saw a light coming from the gap under the door.\n\nNevon grabbed Mouse's sleeve. \"I can distract them,\" he said in a whisper. \"I can lead them away from here. Tell them I escaped and know where you are hiding. That could give you an opening to escape.\"\n\nGarron looked at Mouse with a raised brow.\n\nMouse herded the two of them away from the door, deeper into the cellar. \"A good idea, Your Highness,\" he said. \"But it wouldn't work.\"\n\nThe lad's face darkened. \"You don't trust me,\" he said sharply. \"You still believe I'll cross you and tell the guards where you are. What's stopping me from screaming out right now, huh?\"\n\nMouse put a hand on his shoulder and with a gentle look, held up his other palm to slow the lad down. \"I do trust you,\" he answered. And he did, surprisingly. For whatever reason, the youth had attached himself to their struggle. \"But they wouldn't believe you. The mage witnessed you too and knows you aren't in any danger from us. But she also doesn't specifically know who you are, and I want to keep it that way.\"\n\n\"Maybe we should let him run off, then, Mouse.\" Garron offered.\n\nMouse considered that but shook his head. \"Not yet. A young noble down in the servants' passages? Dressed like that? They'll work out he was with us in an instant.\"\n\nHe moved on the front of his feet toward the door again, knelt in front of it, and went to work relocking the door from the inside. He reinserted the tools, this time to trigger the lock, which was always easier. In seconds he had both relocked.\n\nThe three of them crept down one of the long aisles of stacked kegs until they reached the back wall. Mouse heard someone try the latch. Muffled voices came through the door.\n\n\"This here's the wine cellar... gonna require two keys. Should we get the chancellor to unlock it so we can check?\"\n\n\"Don't be daft. Place's locked up like a vault....\"\n\nFor the moment, they were safe. Cornered yes, but safe. But there was the danger that someone would eventually decide to check it out. For now, their best option was to wait out the initial sweep and sneak out when the trail went cold and they expanded the search to other areas again.\n\nThey sat on the floor and leaned against the wall. Mouse took a moment to close his eyes and catch his breath. He could still hear the muffled shouts out in the passageway, but the relative quiet of the cellar gave him a chance to think and plan. The rich smell of oak from the barrels reached into his brain and calmed him, clearing his head. He sat there for a time, leaning back against the cold stone of the wall. He could easily drift off, he realized, so he forced his eyes back open.\n\n\"What would happen if Delgan discovered I aided in your escape?\" Nevon asked.\n\nMouse frowned. The lad spoke matter-of-factly, with the same intonation that he might use when asking what soup was being prepared that day. But there was a disquiet layered beneath his tone, a prickling concern. Apparently it had begun to seep into the lad's brain that dangers are inherent in adventures.\n\n\"He won't,\" Mouse replied.\n\n\"Suppose he does.\"\n\n\"You yourself pointed out that there are different rules for the noble class than for us rabble.\" He attempted to sound casual. Blas\u00e9. \"It's unlikely Delgan would throw you into any of his dungeons.\"\n\nGarron considered that, then made a small shrug and lifted his brow. \"That is not to say there wouldn't be consequences, Mouse.\"\n\nMouse threw him a hard look.\n\n\"What does that mean?\" Nevon asked, his voice rising a degree.\n\nMouse sighed. He hoped to spare Nevon that added worry. They didn't need the lad panicking and causing a problem for them. But there was no avoiding it now. \"Just what he said. There would be consequences. Direct punishment would be unlikely--at least from Delgan's end--but the knowledge of your involvement would sour the relationship between Delgan and your family. And your dukedom.\"\n\nNevon was quiet a moment. \"I hadn't thought of that.\"\n\nNo, of course you didn't, Mouse thought, but held his tongue.\n\n\"I don't want this to cause problems for Mother,\" Nevon added. The concern was more evident in his voice now. \"She's worked hard to establish her contacts here. I don't want to be the cause of it failing.\"\n\n\"Don't worry about it, Your Highness. I won't let it happen.\"\n\nNevon's lips scrunched into a pout. \"Stop calling me that. I'm not a prince.\"\n\nGarron swatted at Mouse's arm, which made Mouse chuckle. \"Look,\" Mouse told the lad, \"if we run into any trouble, I'd have you just hide out in here until it all blew over. Once it's quiet again, you can sneak back to the noble apartments and find your way back to your posh little room. You can then spend the rest of the day eating caviar and roast grouse and ordering servants about at your whim. No one need ever be the aware of your involvement.\"\n\nNevon seemed to accept that and nodded.\n\nThey fell into a silence again. Mouse looked down at his hands and systematically tugged at the knuckles of each finger until they cracked. His fingers were sore, the muscles resisting when he tightened his hand into a fist. There were small abrasions and scrapes that had scabbed over on the pads and sides of his fingers. The climb up the tower, he realized. His hands were recovering from that still. Had that really only been nine or so hours ago?\n\nWhen Mouse lifted his head up, Garron was glaring at him narrowly. \"What is it you have against nobles, Mouse?\"\n\nHe almost laughed. \"You cannot be serious!\"\n\n\"It is not as if Nevon or I had any control of our status in the world. You cannot blame us for who our parents are. We are each born into our own circumstances.\"\n\n\"And I was born with nothing.\"\n\nGarron's face darkened into a mix of sorrow and fatigue. Not surprising. It had been a very long night. \"I think your life was far richer than you realize.\" He sighed. \"Yes, my sheets are satin and meals obscene compared to most. But, honestly, the life of a noble is....\"\n\n\"Dull!\" interjected Nevon.\n\nGarron laughed. \"Yes. Dull is indeed the word I'd use.\"\n\n\"I could go for a little dull now and then,\" Mouse replied. \"And the satin sheets.\"\n\n\"You'd be clawing your eyes out from boredom in less than a fortnight, and you know it.\"\n\nMouse didn't know how to respond, so he said nothing.\n\n\"You know,\" Garron added, \"we aren't all terrible people.\"\n\nSo I've come to learn, Mouse thought.\n\nTHE SILENCE settled on them for a time. Each slipped away into his own thoughts. Nevon seemed fixated on a spot that appeared on his fine jerkin and tried to scrape it off with his fingernail. His dresser was going to have a fit when she saw the amount of dirt under his fingernails and how disheveled his hair was. He was one shabby tunic away from passing as a commoner.\n\nWhich, Mouse realized, could be the lad's way out of this mess.\n\n\"Garron?\" he said.\n\nGarron lifted his head from the wall and opened his eyes. \"Huh? What?\" He had obviously fallen asleep.\n\n\"I should probably reapply the salve,\" Mouse told him.\n\n\"Oh, right.\"\n\nWhile Mouse dug out the jar from his pack, Garron climbed to his feet and dutifully tugged down his trousers and short clothes. This time, the linen pulled cleanly away and did not stick to the wound. Mouse scooped more of the yellow cream on his fingers and applied it to the burn, taking more time to massage it in than probably necessary.\n\n\"Look, Garron...,\" he said as he put the earthenware jar back in his pack. He wasn't certain how to proceed, and the pregnant pause that followed did not escape Garron's notice. The nobleman pulled up his trousers and turned about. He remained quiet with a questioning look, inviting Mouse to finish.\n\n\"This escape of yours,\" Mouse began tentatively, \"has grown... progressively more complicated.\" Mouse was trying to approach this gently. But in truth, the ordeal was a fucking disaster.\n\nGarron glanced around at their surroundings. \"We do appear to be moving in the wrong direction,\" he agreed with a nod. \"Moving deeper into the manor rather than closer to any exit. What are you saying, Mouse?\"\n\n\"Nothing specific, I suppose. The mage's presence here is forcing me to recalculate our options. It all means that getting you beyond the walls will prove more difficult than I initially thought.\"\n\nGarron seemed to digest this a moment, his expression unreadable. \"Impossible?\"\n\n\"Not even improbable. I wasn't lying when I said that I've never failed a job I was hired for. Today will not count as my first.\"\n\n\"But you are having some doubts. That is plain enough.\"\n\nMouse rubbed one eye with his forefinger and middle finger. \"No, it's not that. My ability to see you freed is not in question. But I cannot see how we are going to get out of this now without... without some measure of bloodshed.\"\n\nGarron's gaze lowered as he considered this, and he was silent a moment. \"You have held strongly to your word so far. I am grateful, truly. Whatever happens from this moment on, I trust you will do what you can to not cause any unnecessary harm or death. But....\" He took in a long breath and exhaled slowly. \"But should it come to that... should it come to violence, know I will surrender before harm comes to any man.\"\n\nMouse's heart tightened. \"And they will likely kill you.\" And me.\n\nGarron nodded. \"If that is the case, so be it. I will not live out my days with the blood of innocents on my hands.\"\n\n\"They are hardly--\"\n\n\"Have they any choice, Mouse? Yes, they pursue us, but their will is not their own. They follow orders they do not dare question, and cannot be held responsible.\"\n\nMouse looked away and was quiet.\n\n\"You think I'm weak,\" Garron said to Mouse's silence. It wasn't a question. It was more of a challenge. \"You view my forbearance from violence as flaw of character.\" Garron's eyes were narrow and his lips thin. It was the closest Mouse had seen him to anger.\n\nMouse shook his head. \"No,\" he said. \"No, quite the contrary actually.\" In truth, he wondered if he would ever have the strength to risk his own life on any such conviction. \"A question, though. What of the one giving the orders?\"\n\n\"As in Duke Delgan?\"\n\n\"Or the chancellor.\"\n\nA coldness passed over Garron's features as his eyes narrowed. \"If we face him, do with him what you will.\"\n\nMouse nodded. He could live with that.\n\nGarron studied Mouse's face a moment. His lips pressed in a thin line of concern. \"There is something yet on your mind.\"\n\nMouse turned away, his mouth turning downward. He was not accustomed to being read so easily. \"A personal matter,\" he answered quickly out of reflex, then reconsidered. He felt compelled to say more, though he didn't know why. He told himself the information might be relevant to the mission but knew it was something else. \"The mage certainly heard our entire conversation when I spoke of my father. I said too much....\"\n\nGarron looked back at him quizzically. \"How so?\"\n\nMouse shook his head. \"It was foolish of me. I should never have spoken so freely of him. Now... I have put him in danger. Based on what I said, it is possible they can locate him. They will find him and--\" He stopped himself. The words were spilling from him in a rush like a leaky keg, and he needed to regain control of his tongue. \"Garron,\" he continued, more slowly this time, more mindful of his words. \"My past... well, I've had to survive, you know. I won't apologize for that. I did what was necessary, but... my crimes are many, to say the least.\"\n\n\"Ah,\" Garron replied with nod. Mouse could see on his face he understood. \"Your fear is that they will hold your father responsible for your crimes in your absence.\"\n\nMouse nodded. \"That is the law. I've managed so far to hold off the king's guard from taking him.\" Garron's brow lifted at that. \"Bribery,\" Mouse put in, answering Garron's unasked question. \"Delgan now presumably has a substantial ax to grind with me. I have stolen his prize. He is a dangerous man, Garron, and I fear I may no longer be able to protect my father.\"\n\nGarron considered this a moment, then said, \"We can get a message to him. Warn him.\"\n\n\"Assuming there's time. They are already making efforts to locate him, certainly. We would have to find our way out of here soon if we are to get a messenger to him in time.\"\n\n\"We will,\" Garron replied, but Mouse didn't think he sounded all too confident. Garron pressed his lips together as he thought a moment. \"You realize, of course, I can provide some aid in this. Beyond just sending him a warning. Once we are free from here, I'll have access to resources--\"\n\n\"A kind and generous offer,\" Mouse cut in. The typical response of the wealthy and powerful--the solution to any problem is to throw coin at it. \"But that won't be necessary. It is my affair.\" And he would manage it himself, as he had always done. \"I've no wish to drag you into it.\" He felt himself pulling back and regretted having even mentioned it.\n\n\"I would be happy to help, if I can. Considering all you've done....\"\n\nMouse made a sad little chuckle. \"All I've done,\" he repeated. In truth, all he'd managed to do so far was to get him locked into a chamber far grimier than his previous cell. Garron was forgetting that Mouse wasn't doing this for any altruistic reasons. He was to be handsomely paid. And the coin he earned would in theory solve the issue anyway with the purchase of a king's pardon. But all the gold in the king's coffers wouldn't mean shit if he didn't get out of this keep. \"Thank you,\" he said, attempting to lift his tone and put an end to the topic. \"We can discuss it further, perhaps. Once I've actually accomplished what I was hired to do.\" Mouse doubted that Lord Green Eyes would even recall the offer or the conversation once he was safe beyond the keep's walls. Garron would surely not want to sully his breeches wading through the complications of Mouse's life. \"Until then, we should plan our next move.\"\n\nAs if on cue, Mouse heard the sound he most dreaded. The click of the lock. Then a second click followed shortly after.\n\n\"Hide,\" he told Nevon in a heavy whisper. The lad gave a sharp nod and moved quickly. He crawled up the shelving that housed the rows of kegs and lay down on the top row near the ceiling. He was completely out of sight. Mouse reached over and snubbed out the candle, thrusting them into darkness as thick as pitch. The door to the passage beyond crept open, the hinges protesting with a high-pitched squeal, and warm light from a lantern crashed into the room. It lifted over the row of barrels like a sunrise over a horizon.\n\nHopefully, it was just some lackey fetching a requested keg of wine. The door was shut and light moved. Aisle by aisle it moved closer. It would stop for a time at each aisle as if searching. From the footfalls, Mouse determined it was only one person--which confused him. Two keys were needed to open the door, and they wouldn't likely give both to one person. Was this some high-ranking courtier entrusted with both keys to hunt down a specific vintage? Mouse pulled out his dagger from his belt and gingerly moved his feet underneath him into a low squat. He was ready to spring if necessary.\n\nRegardless of what he told Garron--he would protect him. At all cost. And he would cope with Garron's outrage afterward.\n\nThe light rounded the end of the last aisle and for a moment, Mouse was temporarily blinded by the sudden intensity. He lifted the dagger, prepared to throw at the first sign of a threat. He could only see a dark silhouette.\n\n\"Well, shits on a stick! There you are!\"\n\n# Chapter 19\n\nMOUSE PULLED back his arm, fully prepared to release the dagger, but some unknown motive stayed his hand. Some deep part of his brain told him to wait--and a single heartbeat later his intellect caught up to his instinct. This was no guardsman, clearly. The silhouette was too small, and the shape implied no armor was worn. And the voice was clearly that of a woman.\n\nBut that did not mean she wasn't a danger.\n\nShe stepped forward down the aisle at an unhurried pace, lifting the lantern higher. \"Don't bother lobbing that lovely thing my way, dear. No good will come of it.\" She was dressed in the simple attire of a maidservant and wasn't either particularly pretty or homely. She would be someone easily overlooked and ignored by most. But Mouse recognized the signs that she was much more. She held herself with too much self-assuredness and willful authority and spoke with an ease that was far above her alleged station. This was no common drudge. And Mouse knew immediately who she was.\n\nHe lowered his arm and lifted to his feet. But he kept the blade firmly gripped in his hand nonetheless.\n\nHer brow lifted a fraction as she raked her eyes over him. \"My, my. They weren't lying. You are rather... undersized, aren't you?\"\n\n\"Size isn't everything.\"\n\n\"Apparently. Drux is still recovering from his encounter with you.\"\n\n\"How did you find us?\"\n\n\"Wasn't easy. I've been combing the manor an entire day. When I caught an ear that you had been discovered hiding out down here, I decided to check out the places I'd hunker down.\"\n\nGarron climbed to his feet. \"Mouse? What's going on? Who is this woman?\"\n\nMouse did not pull his eyes from her. \"I wondered if we would eventually meet,\" he told her. He kept his tone neutral. \"I could have used your help earlier.\"\n\n\"By my estimation, looks like you can use some help now.\"\n\nGarron joined him at his elbow, his body nearly pushing against his. Mouse could feel his heat against his bare arm. \"You know each other?\"\n\n\"We've not had the pleasure of meeting, no.\" He chanced a quick glance toward Garron. \"But I knew she was here somewhere. This here is the guild's mole stationed in the manor.\"\n\nThe woman curtsied. \"Yensia, at your service.\"\n\nMouse bowed politely in return--but kept his eye on her the whole time and his hand firmly around the hilt of the blade. Any furtive movements on her part, and he would act. \"Mouse, at yours,\" he replied with a thin smile. \"I assume this gentleman requires no introduction.\"\n\n\"Indeed he does not.\" She made a respectful nod to Garron. \"M'lord.\"\n\nGarron replied in kind with a simple bow.\n\n\"Whoever you have hiding up on the stacks might as well come down as well,\" Yensia added casually.\n\nMouse was about to proclaim he had no idea of what she was talking about, but the lad's face poked out from atop a barrel.\n\n\"Well, Master Nevon, as I live and breathe,\" she said with a widening smile. \"This is certainly an unexpected pleasure.\"\n\n\"Good day, Yensia,\" Nevon answered sheepishly.\n\nYensia clicked her tongue and shook her head. \"Your garments are going to be frightfully dirty from being up there, you know. I imagine your mother is going to be quite cross with you.\" She looked over at Mouse, and her eyes twinkled with delight at Mouse's surprise. \"His mother and I are acquainted, as it turns out. We have had some business dealings together during her visit here.\"\n\nCurious, thought Mouse. He was beginning to think he'd like to meet the lad's mother after all. \"Please don't take this wrong way, Yensia. But why have you come to find us?\"\n\n\"As I said, you could use my help. It appears the cat has you cornered, little Mouse.\"\n\nMouse very nearly groaned. Everyone thought the cat and mouse references were so clever. He shrugged and attempted to appear unconcerned. \"Nothing I can't handle.\" She didn't look like she believed him but had the courtesy to not say so. \"But,\" he continued. \"I cannot help but wonder who you are presently representing on this visit.\"\n\nYensia's genial demeanor faltered some. The warmth behind her eyes cooled. \"Ah. You refer to the guild's arrangement with Lord Delgan. Heard about that, did you?\"\n\nGarron's body stiffened beside him. \"And what arrangement is that?\" Garron asked. He was quick to recognize the danger they were in, Mouse noted.\n\nMouse answered for her, just to make sure she understood how much he already knew. \"The guild has been given broad license to conduct their business both in the manor and on the streets in exchange that Yensia here provide Delgan with useful intelligence about the guests and residents of the manor. He is essentially ignoring her presence as long as it's beneficial to him. I'm certain the arrangement also includes a favor or two where the guild is to take care of some unofficial and less pleasant tasks for him as well. Delgan has managed to expand his network--and sphere of influence--to include the city's cabals. He now controls both the legitimate and illicit forces of the region. Accurate?\"\n\n\"More or less.\" Yensia's face discarded any pretense of good humor. \"Delgan only just has his foot in the guild door, but we all see where this is heading. He is starting to push, demanding more by small degrees so as not to generate too much discontent too fast and trigger an internal coup. Keeler is losing control of the guild. Though he's too blind or pigheaded to see it.\"\n\nKeeler. The same oaf he encountered at the soup stall. It was no surprise to Mouse he was making a mess of it.\n\nYensia shook her head. \"It's only a matter of time before he--and the rest of us--are answering to one of Delgan's men. Specifically, it'll be the chancellor.\"\n\n\"Elasiar,\" Mouse replied, unable to keep the disdain from his voice.\n\n\"Ah, I see you've had the pleasure. He's an ambitious worm, that one, and has a keen insight of the criminal underground here in the city. Too clever by half and bloody ruthless besides. I hear he's set his eyes onto the Shadow Elite now too. Seizing up businesses. Putting members in irons that don't play by his rules. Piece by piece he's dismantling their power.\" She shook her head. \"Bet the Elite are wringing their hands raw.\"\n\nMouse nodded. \"So, I'll ask again. Who are you working for now? Delgan? Or the guild?\"\n\nYensia stepped closer, her fist clenched. \"I don't work for that fat cur, Mouse. I am loyal to the guild. And a guild does not heel to any aristocracy.\"\n\nA good answer. Mouse had been a guild man for a time himself. It turned out that working with others wasn't necessarily his strength, but he understood the sentiment that kept people in. And the zeal behind it. It was all very intoxicating--the ripe feeling of having power over your destiny, the love of the city that you worked tirelessly every day to carve up like a holiday roast, and the sense of belonging the guild provided to all the lost and indigent that the city shat out. \"The brotherhood before all\" and all that rot. And, of course, the fervent contempt and distrust of anyone with a title sitting before his name.\n\nYes. It was the appropriate response. But nonetheless, he studied her face, her eyes. Garron was watching him.\n\n\"Can we trust her?\" Garron asked.\n\nMouse read the intensity and determination. And the intelligence. But he did not see deception. \"Yes.\" He loosened his grip on the dagger and let the tension in his shoulders subside. He felt Garron ease next to him as well. Strange how quickly the man had come to trust him so explicitly. They'd known each other less than a complete day.\n\nHe nodded to Nevon, and the lad climbed down from the shelving. As he rejoined Mouse and Garron, he wiped the front of the jerkin with his hands, self-conscious now of the amount of dirt and grime that stained the fine fabric.\n\n\"How did you know where to find us?\" Mouse asked.\n\n\"Part luck. Could tell they thought you were down here somewhere, so I was working my way through all the hiding places I'd consider. This was my third choice.\"\n\n\"How did you get in here so easily?\"\n\nShe waved her hand in front of her dismissively. \"I've had copies made of every key in the manor. There's no chamber I can't get into. Delgan has greatly underestimated me and my skills.\"\n\nYou hope, thought Mouse. Delgan was no dolt and seemed to be keeping one pace ahead of everyone in this city.\n\n\"He's convinced I'm his loyal puppet,\" Yensia continued. \"That took some patient work, I can tell you. No thanks to that simpleton Keeler. But I'm very selective as to what information I pass on to him. Nothing too valuable. Exhale, boy, before you pass out. I've told him nothing of your mother's dealings. Her work here in the city is safe.\"\n\nNevon's shoulders lost their tension, and his torso slumped forward.\n\n\"Anyway, it's a fine line I straddle here. I work harder to oppose him than benefit him, so I have to be careful. When word came to me from the guild what you were attempting, I started keeping eyes open for you.\" She gave Garron a pointed look. \"Delgan's to benefit greatly from your internment. So, if helping you escape sets him back a pace, I'm in.\"\n\n\"How's he to benefit?\" Mouse asked.\n\n\"Don't know the details. But he sure was happy with the deal he struck.\"\n\n\"Any idea who he struck the deal with?\"\n\n\"None whatsoever. Never saw him before, or since. An obvious intermediary.\"\n\nMouse glanced over at Garron, who had turned and stared off into the darkness of the cellar, unseeing and lost in thought. Garron's mouth was pressed in a tight line, his lips all but gone. Was he trying to figure out who was behind this? Or did he already know?\n\n\"You have a plan?\"\n\nYensia laughed, her easy disposition returning. \"Of course. I'll have you out of here in the time it takes a flea to piss.\"\n\nYENSIA CHECKED the corridor first. The other three waited just inside the cellar. She returned moments later. \"Looks like they've completed their first sweep. Since they didn't find any sign of you, they'll conduct another, and soon. This next one will tear the place apart.\" She shook her head. \"I don't envy the servants and laborers down here. This will not be a pleasant afternoon for them.\"\n\nShe was right about that. The guardsmen would assume they'd seen something, so they would likely be heavily interrogated.\n\n\"So we don't have much time,\" he finished for her.\n\n\"Yes. So, follow me quickly and do exactly as I say.\"\n\nShe led them past the manor kitchens, circumventing the more active regions. Warm savory-smelling currents flushed into the cooler passages. Guardsmen loitered about by the ovens, but they didn't seem to be actively searching. They were more likely taking advantage of the search to score some buttered rolls or flirt with the kitchen wenches. Without a good reason to be there, the guardsmen would have been chased out by the manor cooks with large wooden spoons. The lantern brandished in front of her like a weapon, Yensia took them down a series of unlit passages in an area that Mouse hadn't yet explored. From the shuffle marks in the dirt floor, Mouse could tell guards had swept through here, but it was quiet now. The three of them followed close on Yensia's heels. Mouse trusted she knew where she was going--she made her way through the labyrinthine network with practiced ease.\n\nDeeper in, the corridors grew cruder, the walls roughly hewed right out of the bedrock the manor was built on. The latest passage ended abruptly in a space that was more cave than chamber. A hodgepodge of items had been dumped there--a broken iron weapons rack, a cracked stone vat, and some old crates, probably forgotten about for generations and likely to disintegrate if ever moved. Yensia went right to the corner and dragged a rotted old tapestry out of the way to expose a hinged trap door in the floor.\n\n\"Where does this lead?\" Mouse asked.\n\nYensia's eyes twinkled as one corner of her mouth lifted in an impish smile. \"Straight out of the manor.\"\n\n\"You certain?\" Garron asked, his voice betraying a resurgence of fresh hope. \"Why is this here?\"\n\nYensia and Mouse exchanged a quick glance, both taking a bit of pleasure from Garron's naivet\u00e9.\n\n\"Well,\" she replied, \"lords with a shiny reputation can't exactly parade their whores right through the main gates, now can they? Or should they want to sneak out for a night of wicked behavior and don't want their wives to know....\" She trailed off with a single brow raised.\n\nBoth Garron and Nevon made the same disapproving face that made Yensia laugh.\n\nMouse bobbed his head approvingly. \"How many know of this?\"\n\n\"Very few.\" She grabbed one of two iron rings attached to the portal and indicated with a tilt of her head that Mouse grab the other. \"The nobles and the occasional merchant who use it still keep it very quiet. Guards would not know to check here.\"\n\nPerfect.\n\nMouse and Yensia heaved, and the door begrudgingly swung open on its stiff hinges. Underneath was a hole that descended into darkness with a wooden ladder fixed to the side.\n\n\"Here.\" She handed Mouse the lantern. \"Go quick now.\"\n\nMouse took it from her. \"How will you get back?\"\n\nShe swatted the air in front of her. \"Ah, don't you worry. I can find my way through these passages blindfolded.\"\n\n\"But what about him?' Garron asked, leaning his head toward Nevon.\n\n\"I'll get him back safely, don't you worry. And he won't say a word, will you m'lord?\" She smiled down at the lad. \"I know too much of his mum's recent activities for him to wag his tongue about.\"\n\n\"I wouldn't say anything anyway,\" he grumbled back at her. He looked up at Garron. \"Can't I go with you? I could help.\"\n\nMouse and Yensia exchanged glances again. What was it about Garron? The lad was willing to abandon the safety of the manor, abandon his mother, and travel down a dark hole with him to some unknown place. He was strikingly handsome, of course--but that was only part of it. His nature had an intoxicating magic. Was anyone immune to his charms? Himself included?\n\nGarron put a hand on his shoulder and smiled warmly down at him. \"You have done us a great service already, Nevon. But now it's time for Mouse to get me to safety, and it'll be dangerous.\"\n\n\"I can handle myself.\"\n\n\"I've no doubt,\" he answered with a laugh. Not a condescending or mocking laugh, but an affirming one that made Nevon's cheeks color. \"But should the unlikely happen, I would be unable to forgive myself. I would be denying the world a strong future leader, and there are precious few of those.\" With his hand still on Nevon's shoulder, he guided him over to stand with Yensia. He then took the lantern by the handle from Mouse. \"I will send word to you when I am safely away. And I would very much someday like to meet your mother.\"\n\n\"I have a few tasks that he can do from here,\" Yensia replied. \"If he were that strongly inclined to assist.\"\n\nGarron nodded to her. \"Thank you. For everything.\"\n\n\"Oh, I'm not helping you, darling. I'm hindering that sleazy bastard.\"\n\n\"All the same,\" he said. Then he lowered himself onto the top rung of the ladder and descended into the hole. The darkness gradually consumed the chamber as the lantern sank with him.\n\nMouse positioned himself to follow but paused. \"Your guild deserves a better leader than Keeler.\"\n\n\"I've often thought so. We're in a bit of flux at the moment. He's the third in a year. Lasted longer than the other two, but still not worth a floating turd.\"\n\n\"I may know someone better qualified.\"\n\n\"Is that so?\" In the dark it was hard to read her face, but he thought he saw amusement light her eyes.\n\n\"Interested in the position?\"\n\n\"The thought has crossed my mind,\" she said. \"On occasion. Can't do any worse.\"\n\n\"Well, should you wish to pursue that course, I'd be willing to offer my services in helping you achieve that end. Free of charge, of course.\"\n\n\"Nothing in our vocation is without a price, love.\"\n\n\"Then consider my price marginal. A favor or two, perhaps.\"\n\nShe folded her arms. \"Is this some clumsy attempt at showing gratitude? I told you, I'm not doing this for him. Or you.\"\n\n\"No,\" he replied. \"I just despise seeing a poorly run guild.\"\n\n\"Wouldn't want it for yourself?\"\n\n\"Not my style. I'd rather pull strings from the shadows.\" He stepped down onto the first rung. \"One more thing. Should you happen on any evidence as to the who or the why behind this business....\"\n\nShe shrugged. \"I'll see what I can do. Won't promise anything.\"\n\nThat was good enough for him. As Mouse moved to follow Garron, Nevon sprang forward, waving his hands. \"Wait,\" he said. \"I still have on your boots.\"\n\n\"Keep them,\" Mouse said.\n\n\"You can't go down into that hole without boots!\"\n\n\"I've been in worse places with less, lad. I assure you.\"\n\nNevon shook his head as he dropped to the floor. \"Here.\" He tugged each boot off with a grunt and tossed them over. Mouse considered reminding him of his earlier comments about running around like a street urchin but chose to hold his tongue. He sat down on the edge of the square hole and pulled the boots on.\n\n\"Thanks.\"\n\nAs he lowered himself down the ladder, he heard Yensia say, \"Help me with this.\"\n\n\"It's going to be black as pitch in here,\" Nevon replied over the angry screech of the hinges. \"How are we going to find--\"\n\nThe trap door dropped with a bang above him, shutting out the rest of his protest.\n\n# Chapter 20\n\nMOUSE FELL in behind Garron. The tunnel was dank and foreboding... and very narrow, forcing them to walk single file. Garron's larger form blocked the light from the lantern he carried, shrouding Mouse in shadow. The air was noxious and stale. It felt more like a sewer than a passageway. Water seeped in from fissures in the stone and trickled down the rough walls to form pools on the uneven floor. At times it was as deep as their ankles, and Mouse discovered that the boots he had reclaimed from Nevon had substantial cracks in both soles. Icy water leaked in, numbing his toes.\n\nIt was hard to picture any parlor girls making this trek up from the city, or noble dandies venturing out, risking a soiling of their finery. They would have to be quite randy indeed.\n\n\"Mouse, I so enjoy our time together,\" Garron said dryly. \"You introduce me to the most extravagant places.\"\n\n\"Only the best for His Lordship. What do you think about exploring a coal mine later? Or we could rent a wagon and collect the dead from some recent plague.\"\n\nGarron laughed. A hearty laugh bigger than the gloom of the place should allow. His shift in mood was palpable. He was energized, giddy even. Even the murkiness of the tunnel didn't seem to dampen it. Now that they were out of the manor proper, his escape must have seemed like a reality for the first time. They weren't entirely out of danger yet, Mouse knew, but they were getting closer. Mouse couldn't help himself and laughed along with him.\n\nThe laughter died away gently, and they fell into the quiet task of traversing the tunnel. In the silence, Mouse caught himself thinking about the end of the mission. It felt close now. All that remained was the task of smuggling Garron out of the city. No small feat surely, but Mouse predicted it would prove easier than getting him out of Delgan's keep. And after that, once he delivered Garron safely to his protective entourage, well... that was it, wasn't it? Mouse would get the rest of his payment, and Garron would be escorted back to Har Dionante. He would return to his life as son of Duke Braddock--a life of satin sheets, ten-course meals, tennis in a private courtyard, and watching tournaments from box seats safely distanced from the smell of horses and the unwashed rabble below.\n\nIt all had a peculiar finality that tugged at Mouse unexpectedly. He and Garron would never cross paths again. They existed in two very different worlds, and there was little chance of those worlds ever crossing paths again. It wasn't as if they were going to ever meet up in a tavern one night to share a pint and laugh about their adventure together.\n\nHe shook his head.\n\nWhat the fuck was wrong with him? Garron was a job. Another object he was hired to steal. Nothing more. Any sense of a companionship he felt between them was nothing more than an illusion. Mouse knew well he was not the gentle sort. He was crude and undisciplined, not the sort that mixed well with the gentility. He was simply confusing Garron's affable nature and civility with genuine amity.\n\nThe tunnel angled downward for a while, at times steeply and at other times becoming roughly formed stairs. It didn't seem to hold to a single path but would sway and turn as if avoiding unseen obstacles in the surrounding rock. The deeper in they traversed, the drier it became, but Mouse's feet were already soaked and gone numb, so it didn't much matter.\n\n\"Do you have a plan?\" Garron asked, breaking the long silence.\n\n\"I always have a plan,\" Mouse replied. That didn't, however, mean it was a good plan. He had a few ideas brewing but wasn't ready yet to settle on one yet. He wanted to wait to gauge the climate when they arrived outside. \"Getting out of the city will still be dicey. It won't take them long to figure out that we gave them the slip. There will be plenty of guards sweeping the streets.\"\n\n\"It gets low up ahead. Careful,\" Garron said, looking over his shoulder. \"Oh wait,\" he added, \"never mind.\" He tried to hold a steely face when he said it, but a smile cracked through. Mouse made a face back at him. \"Well, if anyone is to get me out of this shithole town, it's you.\" Mouse nearly laughed out loud at that. He couldn't explain it. Something about Garron cursing, or even acting bad-tempered, brightened his mood. \"Just tell me what you need me to do,\" he added.\n\n\"You needn't worry about that,\" Mouse replied. He paused, hesitating to proceed. His father still weighed on his mind. The more time that passed, the more danger he was in. He needed to act. \"Actually,\" he added slowly. \"With your indulgence, I'd like to seek out a messenger to hire straight away.\"\n\n\"To warn your father,\" Garron said, nodding. \"Of course, Mouse. I agree. He should be warned.\"\n\n\"It will not take long nor endanger you.\"\n\n\"I know,\" Garron said. His expression changed then. The levity of earlier fled his eyes, and his mouth turned down on one side. \"But....\" He stopped and turned around to face Mouse directly. \"About that.\"\n\nMouse looked back up at him with a suspicious raised brow, not at all clear where this was heading. \"Yes?\"\n\n\"I think you needn't worry about him. I'm fairly certain your father can take care of himself.\"\n\nMouse felt his cheeks flush and lips tighten. \"Needn't worry? What can you know about the danger he may be in? I may well have doomed him by--\"\n\nGarron put a hand on Mouse's shoulder to stop him. \"That's not what I meant.\" He smiled gently, while his eyes revealed that he was carefully considering what to say next. \"I agreed, remember? We'll send warning. It's just that.... Let me see that dagger again. The one that belonged to your father.\"\n\nMouse rolled his head to the side. \"That again?\"\n\n\"Indulge me.\"\n\nHis shoulders fell some in reluctant compliance. \"Very well.\" He slipped the dagger out of his belt and handed it hilt first to Garron, who took it from him with gentle hands as if holding an ancient relic. \"Honestly, I can't comprehend your fixation with it.\"\n\nHe smiled then, strangely and inexplicably, as if some mystery had been answered. \"It feels heavy to my hand. Clumsy even.\"\n\n\"Then you're an idiot, because it's the lightest and most evenly balanced blade I've ever held.\" He reached for it again, to take it back, but Garron twisted and moved it out of his reach, and held up his lantern between them to interfere.\n\n\"A moment yet. I'm not done.\"\n\n\"You just don't know a good blade when you hold one.\"\n\nGarron chuckled. \"I've held some of the very best in all the kingdom, actually.\" Gripping it by its hilt and holding the lantern close, he angled the flat side of the blade to face the light. He looked closely. \"This symbol.\"\n\n\"I already told you. I've tried to identify it, but I've not seen the emblem ever before.\"\n\nGarron was very quiet for a moment as he studied it. \"But I have.\"\n\nMouse's heart froze solid. He wanted to speak, say something, ask anything, but found for once in his life, he could not utter any words.\n\n\"How did you come to have it again?' Garron asked.\n\n\"I told you,\" Mouse said in quiet voice. \"I found it among his things. I took it the night I left the village.\"\n\n\"Where? Where did he keep it?\"\n\n\"He had it in a chest, hidden under a false bottom. Wrapped in a velvet cloth. Garron, tell me. How do you know that symbol?\"\n\n\"I have seen this symbol when I've visited the capital, when I've gone to the keep and the king's palace. And I've seen it in my studies. This weapon... it would have been too valuable for someone to give up as payment for services. I'm certain your father was good at his craft, but this could have purchased an entire town. Several, in fact.\"\n\nMouse listened, waiting--a feeling of dread creeping over him. A part of him knew the knife was more than just a knife, even when he took it. But he had blocked those thoughts from his mind.\n\n\"Mouse, this is the emblem of the Black Guard.\"\n\nMouse inadvertently took a step backward. He shook his head. \"I'm sorry... did you just actually say to me that this belonged to the Black Guard?\"\n\n\"The symbol is undeniable. I'd recognize it anywhere.\"\n\n\"That just isn't possible.\" He had not heard them spoken by name in a very long time. The Black Guard were famed throughout the kingdom. Famous for their skill but also infamous for their terrible failure.\n\n\"I'm certain this must come as a shock. But there is only one way this dagger has come into your father's possession.\"\n\n\"No. That is patently ridiculous. My father is a tanner. He is not--nor ever was--some royal guard.\" Not just any royal guard, either. The Black Guard. Mouse wanted to laugh. His father, the quiet-spoken, gentlest man he'd ever known, a highly skilled and deadly protector of the royal family?\n\nHe knew little of them, only what he picked up from tavern stories, which were mostly rumor and hearsay--and likely exaggerated. The legion disbanded before he was even born, so there were fewer around now who could vouch for these tales. It was always a company of twenty men, as the stories went, and it was claimed that the brotherhood existed for generations beyond counting, since the dawn of the realm and the very first king. To be chosen was said to be the greatest honor bestowed upon someone of common birth. Each had been hand selected by the king himself, and membership insured a life of respect, honor, and some said great riches and land. But acceptance meant a life dedicated fully to the service and safety of the king and his family, never to marry or father children. And from the day a warrior took the emblem on his skin, no one but other members of the brotherhood and the royal family would be allowed to gaze upon his face again.\n\nBut the Black Guard was no more....\n\n\"There were survivors, Mouse.\"\n\n\"Three,\" he replied, his voice barely making a sound. \"As the story is told.\"\n\n\"Yes. There were three who lived that day. Three members of the brotherhood... and the young prince.\"\n\nThe prince who would become King Harus V.\n\nThe tale, regardless of the narrator, had enough common threads to hold at least some truth. A rival family attempted an assassination of the king and his entire family, with the aim to end the royal line. An advisor close to the king betrayed him, and an entire legion of soldiers breached the palace in the middle of the night. The Black Guard fought off the onslaught against incredible odds and succeeded in putting each would-be assassin to the sword, but not before the king, his queen, and two of his children had been butchered... along with seventeen of the Black Guard. One heir remained.\n\nIt was told that the three survivors were so distraught at their failure to protect the king, they left the palace in disgrace, never to return again. Many believed that they had taken their own lives.\n\nMouse touched his own forearm absently, thinking of the burn that marred his father's arm. He'd fallen into a fire after too much drink. That was the story he'd always told. It was what would have been his sword arm, the arm where such an emblem would have been inked into his skin.\n\nGarron extended the hilt to him, and Mouse hesitated before taking it back into his hand. \"The dagger is enchanted,\" Garron said. \"Spelled to recognize your father's hand. You, having his blood, are close enough to satisfy the enchantment. Which is why the dagger has always felt so natural to you. It has, in a manner of speaking, accepted you. But,\" he added with a tilt of head, \"it would somehow know your father, should he hold it. Or perhaps even if he were near show some sign. Such enchantments are like that.\"\n\n\"This cannot be real,\" Mouse said.\n\n\"It explains much. Your uncanny skill, most of all. You apparently take after you father more than you realize. Come. Let's get out of this place so we can send that warning. I would very much enjoy being there if any of Delgan's thugs try to apprehend him. They would be little match for a member of the Black Guard.\"\n\nMouse nodded and followed Garron in a stupor as if his head swam from drink. Only if they get the warning to him in time, he thought. Forces were certainly already on the move against him.\n\n# Chapter 21\n\nTHE TUNNEL came to an abrupt end not unlike it began, with a ladder rising to a trap door. Mouse climbed up and lifted his ear to the door. Hearing nothing, he tested it with a light push. It lifted easily, unlocked and unobstructed. Which was a relief--he half expected this underground journey to be for naught, and they'd be forced to return. A pale light stabbed in from the crack, but still he heard nothing. He pushed the door up farther and climbed out.\n\nThe chamber he found himself in was small and cramped. The walls were unpainted planks of wood, and the ceiling had rows of roughly hewn beams. Bright afternoon sunlight streamed in from two barred windows near the ceiling. Crates were stacked neatly against one wall, and clay amphorae of lamp oil and bolts of textiles were arranged on large wooden shelving that stood along another. A small desk and stool were positioned in the corner, with a chaos of invoices and legers blanketing the surface.\n\nA warehouse, Mouse realized. Small-scale merchants with limited cargo rented storage chambers such as these. They were often used as quiet hiding places for various contraband as well. Mouse wondered if the merchandise here was just props to disguise the true purpose of the place, or if the renter received a discount for allowing people to parade through his stockroom.\n\nHe leaned back over the hole. \"Stay there a moment,\" he said. Garron nodded from the bottom.\n\nHe made a quick loop of the tiny chamber. It was always best to be overcautious. Another of the mage's four-legged spies could end their escape plan quickly. He listened at the storeroom's lone door--a standard plank construction with three horizontal iron bands. It hung on a track bolted to the wall above it. Again, no sound came from the opposite side. So far, they seemed safely away, but that didn't mean a legion of men weren't outside the building, waiting for them to emerge.\n\nMouse tested the door's release and pulled. It slid to the side with a small squeak of protest from the wheels as they rolled along the track, but it moved easily. Someone had made certain the whole apparatus was well oiled. Outside was a wide corridor, plenty wide for merchants to cart in their goods, and either side was lined with identical doors spaced equally apart.\n\nHe signaled Garron up.\n\n\"Looks clear,\" Garron said but turned to Mouse for affirmation.\n\nMouse gave a slow nod. \"So, here's what we are going to do. Just walk on out of here as if you belong. Appear as if you've been here before and know where you're going, so don't walk too fast so you can get a look around. You're already dressed as a merchant, so no one has cause to stop or question you.\" He grabbed a ledger off the desk. \"I--your dutiful clerk--will be right behind you.\"\n\n\"What if they ask for papers or something? Like an invoice or lease.\"\n\n\"They won't.\" At least he hoped they wouldn't. Some hired watchmen enjoyed the authority and could be difficult, but most were indifferent and lazy. By now, it was past midnight, which meant the warehouse would be quieter. Mouse would be surprised if they found anyone about that was awake or sober. \"If you just act important, most will leave you be. I've seen you look that way before, so I know you can do it.\"\n\nGarron made a rude gesture, but it was half-hearted. The skin encircling his eyes was dark while the rest of his face was ashen and drawn. It had been more than an entire day since either of them had had any sleep, and now with most of the excitement hopefully behind them, it was catching up to him.\n\n\"All right, then,\" Garron said. \"Let's get it over with.\" He took a deep breath and marched down the center of the corridor with a straight back and his chin up. Mouse slid the door closed behind him and followed.\n\nThe corridor led to another, which then deposited them into a high-ceilinged open area: the space where merchants brought their carts in to be loaded or unloaded. Two crane arms affixed with a block and tackle were mounted to opposite walls to lift freight to or from the platform on the second floor. Garron marched right through the center of the open area with Mouse close on his heels, the clap of his footfalls on the wooden floor echoing around them. No other merchants were about, but a few unattended carts had been wheeled into a corner. No one was about to operate the crane, apparently. At the far end was a set of wide double doors, also hanging from tracks. Larger versions of the doors at each storeroom. One of the doors was slid partially open.\n\nGarron made straight for the open door, his pace quickening.\n\nMouse scanned the area for the hired watch. He knew they had to be somewhere. He spotted them along the side wall--three of them, sitting around a wooden drum turned on its side. They were throwing dice in a wooden tray and talking quietly among themselves. A woman in a simple green shift leaned against the wall behind one man's shoulder. A wife or favorite whore. She glanced up at Garron as he strutted past, then returned her attention to the dice game. None of the men took notice.\n\nGarron and Mouse stepped out of the warehouse and onto the streets of Har Orentega. No ambush awaited them. No guards were canvassing the street. The citizens milling about paid them no attention as they walked past.\n\nThey had escaped the manor.\n\n\"Where to now?\" Garron asked.\n\n\"Somewhere I'm fairly certain I won't be welcome.\"\n\n\"WHAT THE fuck is the matter with you?\"\n\nTaurin paced the floor of his bedchamber, red-faced. With one hand, he tugged at the clasps of his surcoat as he stomped back and forth in front of his massive bed, while he swung a chalice of wine about with the other, splashing the crimson liquid all over his very expensive rug, making it look like the recent scene of a murder. Their arrival had interrupted the entertaining of important clients, and Taurin had dressed well for the affair. But moments after they'd entered the chamber, he'd transformed into something considerably less urbane.\n\nMouse had never seen him quite this angry--but then he'd never brought a fugitive to his doorstep before either. He took a seat in one of the chairs by the fireplace, leaned back, and brought one leg over the other knee as if enjoying fine theater.\n\nGarron on the other hand moved to stand very quietly against the wall trying not to bring attention to himself. He too threw harsh glares in Mouse's direction.\n\n\"I have half the mind.... No. Three-quarters of the mind to send for Delgan's men immediately and have you arrested, and him\"--he made a dramatic point of the finger at Garron--\"turned over.\"\n\n\"But you won't.\"\n\nHe spun about. \"Oh, won't I?\" Nothing terrified Taurin more than the notion of his little empire collapsing. He had grown powerful, certainly. And Mouse had never known anyone to enjoy the fruits of power more unabashedly. But no one's ongoing success among the Shadow Elite was ever assured. Just as Taurin had shouldered his way to a better position when some titan fell, some young villain would do the same to him.\n\nMouse knew that Taurin was dangerous in this state. But a part of him didn't care.\n\n\"You are overreacting, Taurin,\" he said.\n\nTaurin stumbled back as if struck. \"Overreacting? You brought an escaped political prisoner here! In my establishment! Do you have any idea--\" He stopped himself and shook his head. \"Of course you do. You just don't fucking care.\" He tugged at the neck of his fine white chemise to loosen the lacings of the thong over the chest and wiped the sweat from his brow with the sleeve of his surcoat. \"Did I not make myself clear before that I did not want any part of this on my doorstep? I have a fucking business to run here. And that business does not include helping Delgan's personal little trophy escape from his dungeons.\"\n\n\"Taurin, you know as well as I do they'll be searching every inn and tavern in the city tonight.\"\n\n\"Leave the city tonight, then!\"\n\n\"Don't be daft. Both bridges leading to the city gates have already been drawn.\" The decrees were posted in every public square. An unnamed threat to the citizens of Har Orentega had forced the city to keep both bridges raised until the threat was neutralized. Even now, Delgan wasn't willing to admit his secret prisoner had escaped. \"Until we arrange passage on a ship, we need a discreet place to stay out of sight.\" Mouse moved both feet to the floor and leaned in, resting his elbows on his knees. \"One night. That's all I'm asking.\"\n\n\"You ask too much. This could ruin me.\"\n\nWas nothing more important than the power and influence he brandished about in this city? Yensia was right. The Shadow Elite were running scared here in Har Orentega.\n\nHe dropped his forehead into his palm. \"Gods! The Taurin I once knew would leap at an opportunity to thwart the aristocracy. When did you become so gutless?\"\n\n\"Fuck you, Mouse!\"\n\nMouse dragged his hand down his face to cup his chin. \"Look,\" he said. \"I'll make it worth the risk you're taking. Fifty crowns.\"\n\nGarron straightened. \"Mouse--\"\n\nHe held up his hand to stop Garron from saying anything further.\n\nTaurin glanced inside his chalice, saw it was empty, and threw it on the floor. He folded his arms, and his eyes shifted toward Mouse. For several moments he held them on him, and Mouse could see the calculations working in his mind. The businessman had risen up through the cloud of anger.\n\n\"One hundred,\" Taurin said finally. \"And you owe me one job. No limits. No turning it down.\"\n\n\"Mouse, no!\" Garron exclaimed.\n\n\"Done,\" Mouse replied, louder to drown out Garron's protest.\n\nGarron stepped away from the wall. \"Two hundred crown,\" he said to Taurin. \"And Mouse owes you nothing.\"\n\nTaurin slowly swung his gaze over to Garron as if noticing him for the first time. The anger had evaporated completely, and his eyes narrowed with interest. \"And how will I get my coin, fugitive?\"\n\n\"You will get it. You have my word as a nobleman.\"\n\nTaurin smiled thinly like one does when a child says something charming. \"Darling, I run a brothel. I know firsthand the market value of a nobleman's word.\" He held Garron's hard gaze for several moments before breaking it off with casual disregard. The corner of his mouth twitched. He seemed to realize something about Garron at the moment, and it amused him. \"Still,\" he continued with a slight shrug. \"You seem an honorable sort. Perhaps worth the risk. Two hundred crown would go far in making any damage you potentially cause to my business disappear. Plus....\" He swung his gaze toward Mouse and the expression turned colder. \"I would enjoy the added bonus of not having to suffer this rat-faced bastard again.\"\n\nMouse sneered back. \"Well, then. Everybody wins.\"\n\n\"Do we have a contract, then?\" Garron pressed.\n\nTaurin's chin lifted as he considered. \"Yes. One night. No more.\"\n\n\"We'll be gone by morning,\" Mouse replied.\n\n\"See that you are.\" Taurin strolled over to a long table against the wall and selected a new chalice, then poured wine from the decanter. \"I regret that the accommodations I have available currently are probably not what you're accustomed to, Your Lordship. The night has turned out to be surprisingly busy.\"\n\n\"I have spent the last three weeks in a cell. Whatever you have, I'm certain it will be more than adequate.\"\n\n\"We'll need a meal, though,\" Mouse put in.\n\nTaurin sighed and his shoulders drooped. \"I will have something brought to you.\"\n\n\"And I need to hire a messenger. Someone I can trust. I'll need you to arrange that for me.\"\n\n\"Fine.\"\n\n\"And I need my gear retrieved from the inn I was staying at. Have one of your lackeys run that over.\"\n\nTaurin waved a hand in the air like he was having an attack. \"Fine, fine, fine! Just get the fuck out of here before I change my mind.\" His cheeks were flaring again, and Mouse fought the urge to smile.\n\n# Chapter 22\n\nTHE BEDCHAMBER Taurin provided was certainly not one used for entertaining clients but likely belonged to one of the entertainers, or perhaps one of the other staff that occupied the brothel. The furnishings were utilitarian at best--a small wardrobe, a table and stool, and bed against the wall under a shuttered window. The quilted blanket that covered the bed looked generations old and was the only visible color in the entire space. An attempt by Taurin certainly to make some point, but Mouse didn't care. The room was clean and quiet, one might even say cozy. Mouse had often had worse accommodations, and Garron seemed not to mind either. He seemed ecstatic to be free of the manor and somewhere he could finally relax. Within moments of being in the room, Garron had kicked off his boots and tugged off the merchant's jerkin.\n\nA knock on the door revealed a young boy burdened with a tray of food. Garron relieved the lad of the tray that threatened to capsize him, and Mouse tossed him a half royal. Mouse hadn't realized how hungry he was until the smell hit him, triggering his stomach to twist in sudden complaint.\n\nTogether, he and Garron dragged the table next to the bed and shared the meal of mutton sausage, beans, roasted tubers, and wedges of hard bread all loaded unceremoniously onto the tray. They were provided a flagon of wine as well, but no mugs, so they passed it back and forth. They ate in silence, both leaning over the tray and shoveling in food as fast as their mouths could chew. The brief times Mouse lifted his own head, he had to suppress a laugh. Garron's table manners were anything but noble. He imagined how mortified his fellow aristocrats would be if they saw him now. In short order, they were sopping up the juices with the remaining bread.\n\nGarron fell back with a satisfied groan. \"I don't recall ever eating so much in one sitting.\"\n\n\"Impressive, I'll grant you,\" Mouse replied. \"You would have fit in well among us ruffians.\"\n\nAnother knock came at the door, soft and tentative. Mouse lifted from the stool and, out of habit, lay on his belly to see under the bottom of the door. One set of boots was visible. He rose to his feet again and opened it cautiously. Another lad stood waiting with a canvas sack slung over his shoulder. His gear. So far, Taurin had kept his end of the bargain. All that remained was the message to his father, and there was no way to confirm that Taurin had indeed sent it off. He had no choice but to take Taurin at his word, and although he'd given Mouse no reason to doubt him so far, the notion did not put Mouse at ease.\n\nHe dropped the sack on the bed next to Garron and began inspecting the contents. One by one he pulled items out and arranged them on the bedding. There was something gratifying--comforting even--about having his familiar gear with him again. Garron sat up and took an interest too. He picked up one of Mouse's boots.\n\n\"Careful with that,\" Mouse said.\n\nGarron made a face. \"I'm not going to damage your boot, Mouse. It's not made of glass.\"\n\n\"It's not that.\" He took the boot from him and turned it around so the heel was facing up. \"See this release?\" He pointed to a small round button on the side of the heel, then depressed it with his thumb. There was a sharp click, and a blade the length of a forefinger stabbed out from the back.\n\n\"Oh,\" said Garron, his brow lifting in surprise.\n\n\"Yeah,\" he replied with a chuckle. \"Didn't want you to skewer your hand.\"\n\nGarron pulled his hands away, palms out. \"I leave you to that, then.\" The look on his face was one of distaste, which made Mouse's stomach tighten. What a thug he must seem.\n\nGarron pulled himself farther onto the bed and reclined, resting his head on what was a sad excuse for a pillow. Mouse took the hint and moved everything to the floor. He made an inventory of his gear, making certain it was all accounted for. One could never be sure that the lad sent to fetch the sack, or some curious lackey at the inn, didn't paw through his equipment and select some prize for himself. Everything seemed present, so he took a few moments to organize it for the next day.\n\nHe looked up at the bed. Garron's eyes were closed, and his breathing had slowed to a steady cadence, his chest rising and falling. Mouse couldn't help but grin. Free of the manor, and with a belly full of food and wine, his body had finally had enough.\n\nMouse closed the shutters on the small window to darken the room. He arranged his pack on the floor into a makeshift pillow, then pulled off the page's uniform he'd worn all day and threw it into the corner, happy to never have to wear that again. He stretched himself onto the floor. Only moments after he rested the back of his head on the pack and closed eyes, he felt himself drifting off.\n\n\"What are you doing?\"\n\nMouse's eyes popped open, jolted awake. He rolled onto his side. In the pale light of the room, Mouse could just make out Garron's shape seated on the edge of the bed. The fading light of evening that filtered through the shutters reflected off the skin of his shoulder, arm, and leg. He had stripped out of tunic and breeches and wore only his smallclothes.\n\n\"What does it look like?\" he said. \"I'm sleeping. Or was rather.\"\n\n\"Why are you on the floor?\"\n\n\"Despite what you think about me, Lordship, I can be nice. I was letting you have the bed.\"\n\n\"You're an idiot. The bed's big enough for both of us.\"\n\nHardly, thought Mouse. \"Not when you're passed out like a drunken sailor.\" He worried that in the dark Garron wouldn't pick up on his sarcasm, so he lightened his tone. \"I didn't want to disturb you.\"\n\n\"Well, I'm not going to let you treat me like the entitled prig you think seem to think I am. I'm willing to share the bed with a commoner.\"\n\n\"Prig may have been a bit harsh, I'll admit,\" Mouse said. \"Fop, perhaps. Or a dandy.\"\n\nGarron threw the small pillow at him.\n\nMouse laughed and tossed it back. \"I've slept in worse places, Garron. I assure you.\"\n\n\"Get up here,\" he said. \"We can share the bed.\"\n\nMouse's entire body tensed. The last thing he wanted right now was to lie down so close to him. Were it anyone else, Mouse would not have thought twice about it. But now, the idea filled him with cold dread. Mouse could no longer deny or ignore his draw to Garron. It had been growing steadily since he first met him, and only now was Mouse able to face what was happening.\n\nAside from the fact that Garron was a nobleman, and the target of a job--two strong reasons why he should stay as far away from him as possible--there was also no reason to think that Mouse's growing affection would ever be reciprocated. Even if he did share Mouse's attraction to men--which was unlikely--there was no chance that a sophisticated gentleman like Garron, accustomed to refined things, would ever have interest in a base scoundrel like himself.\n\nDespite his resolve, lying next to him would feel too close to what he caught himself longing for. His mind wrestled against an onslaught of emotion, but his body had already surrendered to the idea. He wanted it too fiercely for any good to come of it. He considered turning the offer down again, but would refusing only raise more questions? Or was that just his way of justifying his answer?\n\n\"All right,\" he said.\n\nHe lifted himself slowly from the floor. Garron pulled the quilt back and slipped his legs and body under it, then shuffled closer to the wall to give Mouse more room. Mouse sat on the edge of the bed. The room was warm, but he left his tunic on to reduce the chance of their skin touching during the night. He slid his legs under the quilt and turned on his side with his back to Garron.\n\n\"See?\" Garron said cheerfully. \"Plenty of room.\"\n\n\"Yeah. Much better,\" he replied.\n\nThey weren't touching, but Mouse could still feel the heat of Garron's body radiating against his back.\n\nThis job needed to be fucking over. He needed to get Garron safely out of Har Orentega and to the meeting place so he could collect his coin and be on his way, putting the whole thing behind him. This boyhood infatuation he had with Garron was getting out of hand. Once he was gone, it would get better. He'd forget about him.\n\nHe closed his eyes and took a deep breath, trying to ignore Garron and just go to sleep. But Garron's presence seemed to fill the whole room and envelop him.\n\nAs he lay there hoping for sleep to take him, images flooded into his mind unbidden--memories he had locked away and not thought about in years. Memories of his crime. He tightened his eyes against them, willing them away, but they would not be deterred. They came, and Mouse's heart twisted.\n\nIt had been a day of exploring the wood south of the village, wrestling about, and swimming naked in the lake. The day was glorious. Just he and Marcan, two inseparable friends. However, Mouse had misread Marcan's affection for him as something more. At a moment that seemed perfect, while they both warmed themselves in the sun, Mouse rolled over and kissed him.\n\nGods, it seemed so ridiculous now. How could he ever have thought Marcan shared his feelings? But he was young, and the young see only what they want to see.\n\nAs soon as Marcan came to realize what was happening, he shoved Mouse off him. There was a stunned silence between them, several heartbeats of them staring. Mouse wanted to say something, anything, but couldn't find the words. Then Marcan was angry. Terribly angry. His face flushed and he lunged. Mouse was knocked back, sprawled on the sand, and Marcan was on him in instant. The punches flew at him with a frenzied desperation, and Mouse brought up his arms around his head to protect himself. But a fist caught his brow above the eye and then his jaw. His head reeled from the impact, and the metallic taste of blood filled his mouth. And still Marcan swung.\n\nMouse pushed him off, but Marcan was on him again. Shouting at him. Again and again.\n\nA fist crashed against the side of his head. The pain was fantastic, and the edges of Mouse's view tightened with blackness.\n\nPanic set in. He squirmed to escape, but Marcan was bigger and forced him back to the ground. Mouse pushed him back again, enough to bring up his knee and plant a foot at Marcan's waist. He kicked, putting all his strength behind it.\n\nMarcan flew back, arms flailing, none of his body in contact with the ground. He hit the ground, and a great cry of agony exploded from his mouth.\n\nMouse lifted himself onto his elbows. Through a fog of tears and blood, he tried to make sense of what he saw.\n\n\"Renn, help me.\" Marcan's voice was no more than a choking whisper.\n\nMouse wiped his eyes clear. Marcan lay on his back, leaning against a driftwood log. It took a moment for his brain to piece together the gruesome scene. There was blood--Gods, so much blood. He'd never seen so much. Then he saw it. A long jagged branch protruded from the center of his chest.\n\nMouse could only stare in horror.\n\n\"Renn... get... help.\" Blood gurgled out over his chin. He tried to say more, but it was lost in chokes and spits. Then Marcan's head fell back and his body went limp against the log. He was dead.\n\nAnd Mouse had killed him.\n\nThat was the last time anyone had ever called him by his given name.\n\n\"Good night, Mouse,\" Garron said behind him.\n\nMouse hadn't thought about that day in years, but all the pain came crashing in around him as fresh and raw as when it first happened. He didn't trust his voice, so he said nothing in return. Maybe Garron would think he was already asleep.\n\nMOUSE WOKE with dawn's pink light squeezing through the slats of the shuttered window. Warm and comfortable, he allowed himself to lie there a moment while the fog of sleep evaporated. He had slept deeply, not waking once as far as he could remember. But as consciousness returned, he was aware of Garron's body against him. No, not just against him--on him. Garron's leg was bent at the knee and draped over Mouse's thigh, and his thick arm came over his waist and snaked up his tunic. Garron's palm lay against the skin of his chest, gently cupping the curve of his pectoral.\n\nMouse's entire body went rigid.\n\n\"Good morning,\" Garron said into Mouse's shoulder, a smile in his voice. He squirmed a little, snuggling in tighter and moving his hand in small circles on Mouse's chest.\n\n\"What... what are you doing?\"\n\n\"It's called spooning,\" Garron said.\n\n\"Yeah, I know what it's called.\"\n\nGarron lifted himself onto his elbow to see Mouse over his shoulder. He had a big wolfish grin splitting his face. \"Now you have a problem with it?'\n\n\"What do you mean 'now'?\"\n\n\"Well, you started it.\"\n\nMouse felt his eyes pop open wider. \"I didn't!\"\n\nGarron chuckled and tilted his head a little to the side in a gesture that said he had conflicting evidence.\n\n\"I didn't!\" Mouse insisted. He tried to wiggle his way up to a sitting position, but Garron's arm prevented him from moving. That was what he told himself anyway.\n\n\"Well, then you're rather handsy in your sleep.\"\n\nMouse felt his face flush with embarrassment. \"Oh gods!\"\n\n\"Relax. I rather enjoyed it,\" Garron said with a widening smile. He leaned in closer to Mouse's face, a wicked look in his eyes. \"Saved me the trouble of making the first move.\"\n\nThen his lips were against Mouse's lips. They were just as soft and warm as that first time they kissed. Garron began with gentle pressure, moving his lips around and tenderly pulling Mouse's lips into his mouth. The intensity grew, and the pressure against Mouse grew harder. With his strong tongue, he probed the opening a moment, then forced Mouse's mouth open farther and thrust the tongue deep inside.\n\nMouse closed his eyes and his head lifted to meet the kiss. Every sensation he was aware of centered on that tongue penetrating his throat. He even pulled the tongue in farther with his own suction. He was dimly conscious of Garron adjusting his body over him, his weight bearing down on him, the warmth of his flesh radiating into his own like a hot bath. And the swell of Garron's cock against his thigh.\n\nMouse felt himself slipping, sinking deeper into the kiss, deeper into the passion that pulsed out of Garron like a furnace.\n\nHe pushed Garron away.\n\n\"No,\" he gasped. \"No, no, no, not this.\" He pushed harder, forcing Garron off enough for him to slip out of the bed and cross the room.\n\nGarron leaned back against the wall, flummoxed. \"Wait. What just happened?\n\n\"Nothing. I'm sorry. I... I don't want this.\"\n\n\"You don't want this?\" Garron repeated.\n\n\"No.\"\n\n\"Well, your cock is telling a different story.\"\n\nMouse looked down. Sure enough, his body had betrayed him. He was fully erect and protruding straight out, forcing his small clothes into a sizable tent. He tried to adjust it, to somehow disguise it, but it was no use, so he held his hand over it instead.\n\n\"Look, this just can't happen,\" he said.\n\n\"And why not?\"\n\n\"Gods! You're a nobleman, Garron, first of all!\"\n\n\"What does that have to do anything?\"\n\nMouse rolled his head back. \"You cannot be serious!\"\n\n\"It doesn't matter to me,\" Garron replied, his voice softening as if he was wounded by Mouse's insinuation. \"Mouse, I know you don't think highly of us nobleborn--\"\n\n\"Stop. It's not that.\"\n\n\"Then what?\"\n\n\"You know my life, Garron.\"\n\n\"Is this about your status, then? You're somehow embarrassed? I wouldn't have thought you'd care about that. I am free to choose my partners, if that's what you're worried about.\"\n\nMouse shook his head. \"You know the work I do. You don't want to keep company with the likes of me. You have a reputation to consider--\"\n\n\"Don't be ridiculous.\" Garron moved to sit on the edge of the bed. \"I'll decide who I spend my time with. And no one will dictate otherwise.\"\n\n\"How very noble,\" Mouse sneered.\n\nGarron fell quiet a moment, holding Mouse's gaze with an expression close to sadness. \"You're noble too, you know,\" he answered gently. \"In your way. Though you don't want to believe it.\"\n\n\"Oh, sod off!\" Mouse groaned, rolling his eyes.\n\n\"I'm serious, Mouse. There is a quality to you that I have never encountered in anyone... of any class. I'd like the privilege and opportunity to know more.\"\n\nMouse put the heels of his hands against his eyes. Garron knew nothing of the dark deeds of his past. If he but glimpsed Mouse's history, he would change this pursuit in an instant. \"You don't know what you're saying,\" he said quietly.\n\n\"I know that the thought of not seeing you again makes my heart ache. A part of me doesn't want this to be over. I'm dreading it, in fact. But it doesn't have to be that way.\"\n\n\"Stop right there. Don't you even dare suggest it, Garron.\"\n\n\"Why? Does that frighten you?\"\n\nMouse stared at him imploringly. Why did he not understand? He felt the sting of tears in his eyes. \"There's no point in pursuing it. It's a fantasy. A mirage.\"\n\nGarron's head jerked back as he'd been struck. \"Mirage? Mouse, I am genuinely and deeply drawn to you. That is no illusion, I promise you.\"\n\n\"Of course it is. It's just....\" He waved a hand about as he struggled for words. \"Just postliberation infatuation. You're misinterpreting your feelings about being freed.\"\n\n\"Stop being ridiculous. There's no such thing. What I'm feeling right now is... is....\"\n\n\"If you say 'love' I swear I'll punch you in the head. I'm no silly fourteen-year-old maid that is going to swoon from your attention.\"\n\nGarron chuckled dryly. \"A sense of anticipation, actually,\" he said. \"Hope.\" The corner of his mouth lifted impishly. \"Desire. And I would wager good coin that you feel the same. Do you think I have not noticed your affections? The way you look at me when you think I cannot see.\"\n\nWas he that obvious? \"But it doesn't matter! Don't you see? I would never be accepted in your world, Garron. And you certainly don't belong in mine.\"\n\n\"We would figure something out,\" Garron replied with a shrug.\n\nMouse threw up his arms. \"Gods! Would you listen to yourself? I'm not going to become some pet kept in some out-of-the-way gilded tower, if that's what you're thinking.\"\n\nGarron's eyes narrowed. \"I'm not suggesting that. But shouldn't we at least give it a go? See where it leads?\"\n\n\"To what end?\"\n\nGarron opened his mouth to respond but was interrupted by a knock at the door.\n\nFlustered by the conversation, his head swimming with so many conflicting emotions, Mouse broke one of his own cardinal rules of his craft. He opened the door without checking it first.\n\n\"What?\" he demanded, flinging it open wide.\n\nChancellor Elasiar stood on the other side, flanked by two of Delgan's helmed guards.\n\n\"Well, isn't this cozy,\" he said. \"I pray I'm not interrupting anything.\"\n\n# Chapter 23\n\nCHANCELLOR ELASIAR glided into the room as if he owned it, followed by two guards, which put them all directly between Mouse and all his gear on the opposite side of the small bedchamber. Mouse, standing in his smallclothes and a tunic, calculated the odds of getting to his dagger. There was little chance. The room was now very crowded, and there was no room to maneuver. Both guards, fully armored and with helms covering their faces, had their weapons drawn and ready. One would surely strike before he made it around them. If not at him, at Garron.\n\nHis mind whirled, searching for a solution.\n\nGarron slowly moved off the bed and stood against the wall. His face had gone ashen.\n\n\"How did you find us?\" Mouse asked.\n\nElasiar's mouth spread to expose his teeth. \"Oh, we had a little help from a friend.\"\n\nFrom the corner of his eye, Mouse caught movement. Another figure moved into the doorway but remained at the threshold of the room. Taurin.\n\nMouse felt his face burn with sudden rage. With a sharp intake of breath, his jaw clenched, and his fists tightened into white spheres. Every muscle was ready to lunge at Taurin. It took the entire strength of his will to not give in to it.\n\n\"You fucking gutless backstabbing rat dick!\" he growled through his teeth. From his periphery he could see Elasiar's grin lengthen with satisfaction.\n\nTaurin matched Mouse's glare with steely defiance. \"It wasn't me, you imbecile,\" he said.\n\n\"I will gut you from ass to ear, you pusillanimous, pig-fucking louse. Your best hope is that they kill me right here and now.\"\n\n\"That is one possible outcome,\" Elasiar sneered.\n\n\"Mouse, it wasn't me.\"\n\nSomething in Taurin's tone needled its way through his fury and made him take note. He didn't know what yet, but something tapped into his rational mind. Through the thick haze of anger, he forced himself to look deeper into Taurin's expression for signs of... anything. Deception? Guilt? Taurin was a crafty bastard and a cunning liar, but Mouse knew his tricks. Taurin's face was as solid and unwavering as he'd ever seen it. It was then he noticed that a third guard stood behind Taurin, weapon drawn. In his sudden willingness to believe Elasiar, he hadn't seen him there at all. And Taurin was still wearing his nightclothes. Under no circumstances would that vainglorious fop leave his bedchamber not fully dressed and his hair not perfectly coiffed.\n\nAnd his hands were behind his back. Shackled.\n\nMouse's rage began to leak from him at a slow dribble, like a wineskin with a pinhole. Some measure of control returned to his muscles, and his fists unclenched. \"No,\" he said. \"It wasn't you, was it? But I know who did.\" He swung his acidic gaze at Elasiar. \"It was one of his girls.\"\n\nElasiar raised a single brow.\n\nTaurin's face lit with surprise. \"Wait. What?\" He lurched forward into the room, but the guard grabbed him from behind and pulled him back to the threshold. Elasiar rolled his eyes and nodded to the guard, who struck Taurin from behind, forcing him to his knees with a grunt of pain.\n\n\"The woman at the warehouse,\" Mouse continued, \"with the men throwing dice. You must have suspected we would use the tunnel, so you sent her there to watch the exit. She recognized me from when I was here before, knew we'd likely head here, and reported it to you.\"\n\nElasiar lost all sign of his earlier smugness. \"You are as clever as they say,\" he answered in a tone that held both ire and fascination. \"I must admit until this moment, I couldn't figure out how you got past the outer walls and into the manor. But now I recognize you, Lord... Yazura was it?\"\n\n\"It was.\"\n\n\"A masterful deception. But I saw you leave with your carriage. Accompanied by my guards. Mind telling me how you accomplished that?\"\n\n\"I will happily explain it to you over a pot of tea someday.\"\n\n\"Or with you stretched to your limit on a rack,\" Elasiar replied with an indifferent shrug. \"Begging for mercy. Whichever suits me when the time comes. As for your friend, Hawken... well, now that I know he was complicit in this business, I will sadly have to rescind my offer to return him to his old office.\"\n\nFuck. Mouse stiffened. \"Hawken had no knowledge of--\"\n\nElasiar waved his hand in the air. \"Oh, please don't insult me. If it makes you feel better, I can throw the two of you in the same cell. You can reminisce about how you very nearly outwitted me.\" He closed his eyes and shook his head as if saddened. \"I always thought no good would come to that paunchy lummox. Troublesome, that one. Unruly and undisciplined. I don't know what I was thinking when I offered him that position again.\"\n\nElasiar swung his attention toward Garron, then. His smile broadened as if seeing an old friend. \"And, Lord Garron. What a merry chase, yes?\"\n\nGarron stared back with hard eyes but said nothing.\n\n\"You very nearly slipped through our fingers,\" Elasiar said, \"and thwarted the duke's grand deal.\"\n\n\"And what is this deal?\" Mouse asked.\n\n\"Ta-ta,\" said Elasiar, waving a finger at him. \"That's the big secret to be revealed later. But I suspect Lord Garron has already figured it out.\" The chancellor's eyes looked askance at Garron. \"Hmm?\"\n\n\"I will not return to your prison, Elasiar. You will have to kill me right here.\"\n\nElasiar's smile turned sinister as his eyes narrowed. \"I'm happy to hear we are of like minds on that, my lord. You see, it was the duke who didn't have a big enough cock to have you killed straight away. I recommended it, but he got squeamish. Nobleborn just aren't accustomed to making those kinds of difficult decisions. I happily don't suffer from such weaknesses. With you dead, there is nothing to prevent the deal from proceeding.\" He looked over his shoulder. \"Guards. This is where you come in.\"\n\nThe guards raised up their swords as Elasiar moved to get out of the way.\n\nMouse's mind was a whirlwind. Ideas spun through his head in dizzying loops, each rejected as they flew past. He had to do something. Anything. He watched in horror as the guards moved into action. Unarmed and nearly naked, if he tried to intervene he would be cut down easily, and the outcome would be the same. The door was blocked by Taurin and the third guard, his gear was on the opposite side of the room, and there was nothing in the room he could remotely use as a weapon. He scanned the room searching for something--anything--he could use.\n\nHe lunged. In one swoop of his arm, he grabbed the food tray from the floor as he put himself between Garron and the guards. He braced himself, brandishing the tray as a shield. But as he leapt into position, something else was happening.\n\nOne helmed guard lifted his thick sword arm for a broad backswing. Mouse quickly calculated the attack. Where he stood, he was well out of range of the blade. The arc of it wasn't nearly close enough to reach him. Mouse watched as the blade spun about. The sweeping horizontal swing whizzed through the air with terrible force to strike the neck of Chancellor Elasiar. The blade cut deep, nearly severing the head entirely. The body buckled and hit the wall with a thud, and there was an eruption of blood like a ruptured wineskin. Elasiar dropped into a disturbing crumpled heap.\n\nAt the same time, the second guard grabbed Taurin by the nightshirt and flung him in the room, then thrust his sword into the unprotected neck of the third guard in the corridor.\n\nWhat followed was a strange kind of timeless silence. No one moved or said a word. The guards, Taurin, Mouse, and Garron all stood frozen while two bodies bled out on the planked floor. Mouse, with his back to Garron, felt fingers curl around the curve of his shoulder. The first guard glanced in Mouse's direction, then looked down at his handiwork on the floor through the visor slit of his helm. He considered the bloody sword in his hand before tossing it onto the body of the chancellor. Then he pulled off the helm.\n\nHawken shook his head from side to side to loosen the sweaty tendrils of hair adhered to his forehead. \"Fuck it's hot in there!\" He turned to Mouse with a knitted brow. \"Paunchy lummox?\" With a scowl, he kicked the motionless lump.\n\nMouse let out the air from his lungs in one rush as he dropped forward and propped himself up on his knees. \"Gods burn me, Hawken! Couldn't you have given me a signal or something?\" He wanted to scream but was too light-headed at the moment. His heart pounded furiously against his rib cage.\n\n\"Always hated that pompous bastard,\" Hawken grumbled down at Elasiar.\n\nGarron's hand remained on Mouse's back. \"Mouse? You know this man?\"\n\nGarron's rapid breath burst in hot waves against the back of Mouse's neck. \"He was instrumental in helping me get to you in the first place.\"\n\nThe second guard tugged off his helm, unveiling a mane of red hair and bushy beard to match. Mouse recognized him immediately, though he was cleaned up better the last time he'd seen him. Mouse had hired him to play one of their bodyguards when they borrowed the carriage to get in through the manor gate. He was a dockworker by day but took the occasional coin for work that required a different kind of muscle. Hawken had apparently been busy networking in his absence.\n\nHawken smiled and gestured to the man with his head. \"You remember Reginus Fenn, yes?\"\n\n\"I do indeed,\" said Mouse, certain he had a perplexed look on his face he couldn't wrestle under control.\n\n\"Good to see you again, mate!\" Fenn replied with an easy grin, as if they'd just gotten reacquainted in the market square. Fenn tossed his helm onto the bed, then ducked out into the hall again. He grabbed the body of the third guard by the ankles and dragged it into the room, leaving a gruesome streak of red across the planked floor. Hawken moved aside as Fenn shuffled in to deposit the body next to Elasiar. He dropped the legs with a satisfied flourish and dusted off his hands as if pleased with a job well done. His face scrunched when he noticed the long stain from the hall. He looked down at Taurin, who was still on the floor in stunned disbelief. \"Mate, you might want to consider a rug or something for out there.\"\n\nTaurin recovered. His face flushed with sudden anger as he struggled to his feet using the wall for support. \"Hawken, you had better have covered your tracks. If this butchery gets traced back here--\"\n\n\"Oh, relax, Taurin,\" Hawken replied. There was something significantly different about the man. He seemed at ease, almost cheerful. What had happened during Mouse's time in the manor? \"The idiot told no one about coming here. He wanted the glory of parading the prisoner back to Delgan all to himself. The only other person who knows is your girl that was working for Elasiar. I suggest you see to it she doesn't sing.\"\n\nTaurin's eyes darkened with a dangerous intensity. \"You can be certain of that.\" He turned to show that his hands were still shackled behind his back. \"Can someone take these fucking things off me?\"\n\nHawken shrugged and tossed a ring of keys to Mouse. Still sore about the deal he made with Garron the night before, he grabbed Taurin by the nightshirt and pulled his face close.\n\n\"Looks like Hawken did you quite a service, Taurin,\" Mouse growled. \"Word is Elasiar had his fingers wiggling about in a lot of different pies here in the city.\"\n\n\"If he's right, and no one knew he was here, then yes. None of my associates will be sorry to hear he's gone, certainly. The man was a pain in everyone's ass.\"\n\n\"Won't be any tears shed in the guild either, I wager,\" Fenn added.\n\n\"Forgive me if I don't start a jig just yet,\" Taurin grumbled. \"It's my floor that he's bleeding all over. All it takes is one witness that saw him entering my establishment--\"\n\nHawken shook his head. \"He had his cloak up over his head the whole time. No one could identify him. He didn't want anyone tipping off Mouse.\"\n\n\"Think, you idiot.\" Mouse tilted his head toward the body. \"There is the mastermind behind Delgan's power grab. Dead. As should now be Delgan's ploy to seize control of the city's underground. Now it's time for the Shadow Elite to get their thumbs out of their collective asses and take back the power you generously handed over to him.\" Mouse pushed Taurin against the wall and brought his face so close their noses nearly touched. \"I want Delgan punished. You understand? Not only for what he did to Garron, but also for his fucking overreach. Make him suffer so he and every other greedy nobleborn with a scheme thinks twice.\"\n\nWithout waiting for an answer, Mouse spun Taurin around. He unlocked the shackles from Taurin's wrists, and they fell to the floor with a clang. Taurin didn't say a word. With a tight scowl on his lips, he stared at the floor as he flexed and rubbed at his wrists.\n\nMouse raked his fingers through his hair and found the hairline wet with sweat. He tried to wrangle in the thousand questions whirling through mind. \"Hawken, how--\" He held out his arms in a wide shrug. \"How the fuck did you pull this off?\"\n\nHawken laughed. A big, hearty laugh. \"I've been keeping my ear to the wall since you got in there. Caught word that they believed you might have found a way out of the manor and beyond the wall, and that they had a new plan. I knew Elasiar would want his hand in it, so....\" He shrugged. \"I may have used my newly restored respectability to talk my way back into the manor and position myself as one of Elasiar's personal guards. Unbeknownst to him, of course. All said and done, it was all fairly easy, actually.\"\n\n\"But you realize you just butchered your employer, right?\"\n\n\"I have a letter signed by the duke himself that I have been restored to my former office. This bastard's death--or unexplained disappearance, anyway--will have no bearing on any part of that. Seems his position's open now, anyway.\" He shrugged again, grinning this time. \"Who knows? Maybe I'll be recommended for that office instead.\"\n\nMouse couldn't help but chuckle. He couldn't picture the brute in the robes of a chancellor, playing the diplomat in the court.\n\n\"Besides,\" Hawken continued, \"there's a solid chance I won't be taking up the position anyway.\"\n\nMouse raised his brow. \"Oh?\"\n\nHawken's expression sobered some as he nodded. \"I've been doing a fair share of thinking over the last couple of days, Mouse.\"\n\n\"Always a dangerous pursuit, you ask me,\" Mouse replied.\n\n\"Well, you're to blame for that. I was perfectly satisfied with my life until you came around to muck it up.\"\n\nMouse knew that was horseshit, but he didn't challenge it. \"You've received a better offer, I gather?\"\n\nHawken scratched the back of his head. \"Not exactly. May just see where the winds take me.\"\n\nMouse knew exactly what that meant. He was off to seek adventure.\n\n\"The wife and kids are heading north to live with her sister and assist on their farm,\" Hawken continued. \"A mutually agreed upon decision,\" he added with a hand up to ward off Mouse's next question. \"So... that leaves me free to chase other interests. Perhaps His Lordship here would consider something for me in Har Dionante. It seems a more amiable town than the one I currently reside in.\"\n\n\"I am certain,\" Garron replied, \"we can come to some agreeable solution to your needs. The least I can do, considering.\"\n\n\"Excellent news!\" Hawken said with a widening smile. \"I look forward to discussing that with you.\" He seemed a liberated man, to Mouse's eye. Relieved of some invisible oxbow that had once burdened his shoulders. He now stood straighter and smiled freely, like a smitten boy. \"But first, let us conclude our current business of getting the two of you out of this damnable cesspool.\" His expression shifted as if he was less confident about what he was about to say next. \"If you'll forgive me, I took a bit of initiative.\"\n\nMore than a bit, Mouse thought.\n\n\"Yesterday, when Delgan decreed the bridges be drawn up, I knew it meant you'd accomplished the deed. It was why I started poking about for information. But it got me thinking too. It was going to be a challenge for you to get His Lordship out of the city and have it done quickly. My time spent at the Blue Stag did put me in acquaintance with other members of your variety, so I reached out to some I knew. I thought I could arrange some quiet passage out for you. Well, I'll cut to the quick. I was visited by a man named Keeler. I'm told you know of him.\"\n\n\"We're acquainted,\" said Mouse dryly.\n\n\"Well, Keeler made it clear to me he is anxious to see you on your way and out of his city. Those were his words. So, it's been arranged--on his coin, mind you--that you are to board the River Falcon, a cargo vessel at the docks. That ship has agreed to smuggle you out of the city and awaits you both now. We just have to get you to the docks and on board that ship.\"\n\nAs if it was going to be that easy.\n\n\"So... as much as I enjoy the both of you standing there in your smallclothes,\" Hawken added with a single brow raised. \"I suggest you pull on your gear and we head off straight away.\"\n\nGarron made a sharp glance Mouse's way with a surprised look, but Mouse pretended he didn't see it. He wasn't about to get into any of that at the moment. He considered offering Hawken a reminder that he was, in fact, speaking to a nobleborn, but taking into account that he'd just saved their skins, Mouse figured that any umbrage Garron felt would be short-lived. He looked at Taurin and indicated the bodies with a tilt of his head. \"You'll handle this mess?\"\n\n\"I have a choice?\" Taurin sneered. \"Yes,\" he added with a small shake of his head. \"I'll take care of it. Just get out of my place. Get out of my city. And the next time you are in town and are considering a visit--think again. I am henceforth unavailable.\"\n\n\"Seems to be a theme with you,\" Garron noted.\n\nMouse shrugged indifferently. \"I have a way with people.\" Anyway, he was more than happy to oblige him. He had no intention of ever returning to this shithole town again.\n\n# Chapter 24\n\nTHE FOUR of them--Garron, Hawken, Fenn, and Mouse--left the Scarlet Unicorn as a unit. After borrowing yet more from Taurin's extensive wardrobe and the uniforms of the slain guards, they hit the streets with Garron dressed as a rich foreign merchant followed by his scribe and guardsmen, who were graciously loaned to him by the duke himself. The guards led the way, parting the crowd with shouts and the swing of forearms while Garron made a haughty march behind them, unaware of the throngs of people he inconvenienced as if they were beneath his contempt. Mouse trailed the small company, with a ledger book tucked under one arm while he gripped the bridle of a donkey with the other. The lumbering and uncooperative beast pulled a small two-wheeled cart loaded with textiles. As they progressed down the center of the street, causing no small disruption, Mouse kept checking under the cart for drips of blood leaking through the wood from the bodies stowed beneath the mound of fabric.\n\n\"This isn't going to work,\" grumbled Garron over his shoulder.\n\n\"Losing faith in me already, Your Lordship?\" Mouse answered.\n\n\"We couldn't be making ourselves any more conspicuous.\"\n\n\"But who, mind you, is going to stop and question two of the duke's own personal guard?\"\n\nThe gambit was perhaps a bit of a risk. If anyone bothered to look at the merchandise in the cart, they would find the textiles to be moth-eaten and cheap in quality--old unused scrap from Taurin's storehouse--not nearly the quality that Garron's performance was selling. And it was highly unlikely that the duke would loan off his own detail even for the most favored of suppliers. But his argument stood. Who would risk checking? Theater was always the art of illusion.\n\nThey traversed the narrow streets and worked their way steadily toward low town and the docks. They cut through the crowd like a ship cutting through ice sheets in winter. And as Mouse suspected, no one thought too much about it and just stepped to the side long enough to let the dandy and his small entourage pass, then went about their business.\n\nThe boundaries of low town were marked by the leveling of the ground beneath their feet as they entered onto the flat flood plain of the river. Gracious homes, sprawling inns, and storefronts gave way to tenements and warehouses. Winds raced the byways and carried the odors of the river docks--fish, tar, and the smell of freshly cut wood. Ahead of them, the top of one of the drawbridges reached over the roofline. Raised, Mouse knew, specifically to prevent them from leaving the city by one of the two gates on the opposite shore of the river. The river was the only way out of the city now. There was no chance the duke didn't know that as well. The place would be a hive of guard activity.\n\nAny zig or zag up ahead could put them at the shore of the river. Mouse kept his eyes open for signs of the city guard. Some would certainly be in plain sight. Others would be covert. He kept a careful eye on each person they passed by, searching for any indication they were not who they seemed.\n\nThen someone stepped out of a shadowed corner to stand in the middle of the street as they approached.\n\n\"Gods,\" grumbled Mouse.\n\nKeeler stomped toward them and was clearly heading directly toward Mouse, ignoring Fenn and Hawken--two men dressed as the duke's personal guard. Mouse shook his head in wonderment. What an imbecile.\n\nHawken was looking his way, waiting for the go-ahead to intervene. Mouse sighed and gave a single nod. \"Get him someplace.\"\n\nAs Keeler marched between Hawken and Fenn, seeming blind that they were even present, the two scooped up the guild leader from under his arms and dragged him off backward in the opposite direction. Keeler sputtered in rage and waggled about to free himself but was no match for the two of them. People on the street very quickly glided away to give them the widest possible berth. They had no interest in getting mixed up in the Duke Delgan's honor guard's business.\n\nKeeler was shuffled into a small merchant's shop. From the street, Mouse could hear a small series of crashes from within the shop; then the distressed vendor scurried out in a rush.\n\n\"This will only take a moment,\" Mouse told Garron, and strolled into the shop himself.\n\nIt was a small storefront, and Fenn and Hawken took up the larger share of it. Keeler was on his backside, sprawled against the shop's counter with pewter and silver dinnerware scattered on the floor around him. His face was crimson with rage, but he made no attempt to get on his feet when he saw Mouse enter the shop. A pewter goblet rolled off the counter and landed on Keeler's head with a dull clang.\n\n\"Stay with Lord Garron,\" he told Fenn, who immediately ducked out of the shop. Mouse turned his hot gaze to Keeler. \"What the fuck are you doing here?\"\n\n\"Making sure you get on the ship and leave my city,\" Keeler spat back, attempting to sound like he was still in control. He managed to get his feet under him with the help of the counter and stand back up.\n\nMouse rubbed the corner of his eye with the heel of his hand. \"How did someone with the brains of a fucking bale of rotting hay manage to take control of this city's guild?\"\n\nKeeler's brow knotted. \"Say, what now?\" He leaned in with his fist clenched. \"I arranged your transport--\"\n\n\"You very nearly sabotaged our cover. Because why would a thug approach a rich merchant's apprentice while the duke's own honor guards just stood by idly and watched? You nearly just told everyone out there that we are not what we seem and that I'm the one in charge. If a city guardsman had seen you, we'd all be fucked.\"\n\nKeeler's face contorted as he processed that. He at least had the grace to look a little sheepish.\n\nMouse stepped right up to Keeler until their torsos nearly touched. He stared hard up into the larger man's eyes. \"I'm going to speak very slow and use all small words to make sure you understand,\" Mouse told him in a gentle tone as if speaking to a child. \"Gather up your men, get them to the docks, and be prepared.\"\n\n\"Prepared for what?\"\n\n\"Anything, you moron. Honestly, how do you dress yourself in the morning without someone pointing out to you what goes where? The city guard will be there in force looking for us. They know as well as we do that the river is the only way we're getting out of the city since the drawbridges are raised.\" Keeler's eyes opened a fraction as comprehension sparked to life. That obviously had never occurred to him. \"Yes, Keeler. Understand now? Get your fat ass to the docks and be ready to cause a little mischief when the need arises. We will need distractions if we are to get Lord Garron on that ship unseen. Can you handle that?\"\n\nMouse didn't wait for an answer. He turned back to the door, but before he exited to the street again, he glanced over his shoulder at Keeler once more. \"If this goes sour,\" he said, \"I'm holding you responsible. And I will see to it your days of running the guild are over.\" Not that it mattered, Mouse knew. His own idiocy would bring about his downfall in short order anyway.\n\nWith Hawken on his heels, he rejoined the others. He grabbed the donkey's bridle and gave it a tug to get her moving again. Hawken and Fenn took their place in the front, and the four continued on their way. Mouse looked behind them one last time to see Keeler stumble out of the shop and stomp off down the street in the opposite direction.\n\n\"Do you want to run the guild here?\" Mouse said to the donkey. \"There's sure to be an opening soon, and you're better qualified.\" The donkey raised its ears as if considering the offer.\n\n\"Will he do as you asked?\" Garron asked.\n\n\"Probably not. But I can't worry about that. We'll get you on that ship. Don't worry.\"\n\nHawken dropped back a bit. He looked at Garron as he spoke, but Mouse knew he was talking directly to him. \"We're nearly to the river and the docks,\" Hawken said, his voice muffled from inside the helm. \"What then?\"\n\nMouse approached and spoke as if to Garron. \"Don't lead us out into the open just yet. I want to see what we're dealing with first.\"\n\nHawken bobbed his helmed head, and they marched on.\n\nThey reached the edge of low town and crossed into the warehouse district. Mouse directed them into a quiet alley to wait. If Keeler was going to aid them, as unlikely as that was, Mouse needed to give him time to get his men into position.\n\nAfter a time, he ventured out alone into the wide-open region along the river. The docks. The area was a sprawling industry that hugged the shore upstream from the two drawbridges. Tall ships with tied sails and broad, flat-decked barges, about a half a dozen or so in all, claimed the available dock space this side of the river, while three other vessels were moored up on the opposite bank. Cranes levered the freight off their decks and stacked them in neat little mountains on the wide cobbled plaza that stretched from the boardwalk by the river to the row of warehouses that lined the edge of the city.\n\nIt was past midday, and the docks were in full chaos as sailors, dockhands, and merchants hurried to make use of the remains of the day. But their efforts were hampered by the choking presence of the city guard--poking around, asking questions, and generally putting themselves underfoot and making a nuisance of themselves. Their numbers were staggering, actually. Delgan had put his guards out in force. It looked more like an army assembling for an invasion. Many of the city guardsman brandished sheets of parchments as they circled the docks.\n\nWere those renderings of Garron?\n\nSome of the guardsmen stood at intervals around the perimeter of the wide area, while others milled about trying to look official and imposing. They snooped around the piles of stacked freight, as if they thought Garron might be smuggled out in a crate of onions. Or they were just looking for something to pilfer for themselves while they pretended to care about some unknown escaped prisoner. More guards were on the docks themselves, searching the ships and interrogating crew and captains.\n\nBut none of the streets leading out to the city were blocked, and the traffic heading in or out of the docks were not being detained or questioned. It was a strange, conflicted sort of tossing the net. The duke's forces were making a clear presence at the dock, obviously trying to hunt someone down, and at the same time people were freely passing in and out of the area. A blockade would bring activity to a standstill, and Delgan knew anything that disrupted commerce would chafe the ass of the merchants and shippers who brought substantial coin into his own coffers. So the guards attempted a casual but aimless sweep of the area.\n\nMouse made his own loop along the margins of the docks. Within moments, he spotted a group of guild members loitering about. They were easy enough to identify. Their thuggish garb and ink-covered arms were enough to flag them, but they were the only ones not involved in any industry. They leaned against walls and stacks of crates, spitting brown globs on the cobbles and scanning the area for any trouble to cause. Then he eyed another cluster farther up. So, Keeler had wised up after all and put his men to action.\n\nMouse had to admit he was surprised. Keeler wanted to be the one giving orders, not taking them, and Mouse had treated him like an errand boy. Keeler could have just as easily stormed off and left Mouse to slip through the net on his own. But he really wanted Mouse out of his territory and was willing to swallow his pride this one time to make sure it happened.\n\nHe approached the outer boundary of a cluster of brutes and made eye contact with the craggiest among them, the one likely to be the leader.\n\nThe man nodded back to him. \"You the one they call Mouse?\" There was a noticeable edge in his tone.\n\nMouse bobbed his head while looking out toward the river.\n\n\"You have a plan?\" asked the thug.\n\n\"More or less. Just be ready. Await my signal, then set out to cause as much turmoil as you can.\"\n\nThe thug grinned, exposing his four teeth. He clearly liked the sound of that. \"What are we looking for?\"\n\n\"You'll know it when you see it.\" Mouse never slowed his pace as he strolled past the scruffy group. He marched on and noticed still more groups of guild members peppered throughout the area. Mouse made a point of holding his gaze on them until they noticed him in return and nodded in acknowledgment.\n\nThen he looped back, venturing closer to the river this time. It took him no time to locate the barge they were to be smuggled out on. However, a half dozen guards were pacing the dock that led to the barge's gangplanks, and one guardsman was even standing on the vessel's deck speaking with a member of the crew. Even with what he had planned, getting aboard was going to be dicey.\n\nHe veered his trajectory toward the alley where the others waited, but something caught his eye. He happened to look out at another of the docks, and he saw her. The woman in the black robes he'd seen in the courtyard two nights ago. The mage.\n\nFuck.\n\nHe hurried back to the others, but not before he lifted three oil lanterns from outside a warehouse.\n\n\"You find the ship?\" asked Garron.\n\nMouse nodded. \"It's a barge moored at the end of the second dock. Red paint on the hull, a red falcon head on the flag. Hawken, I'm going to need you two to get him there and on board.\"\n\nHawken and Fenn had taken off their helms and held them under their arms. Both nodded. \"Easy enough,\" Fenn replied with a shrug.\n\n\"It won't be. Guardsmen are on the boardwalk and on board some of the vessels as well.\"\n\n\"I know these docks, mate,\" Fenn told him. \"I know the men that work them. Dressed like this, who's going to stop us? They're going to think we're just part of the search. We'll get His Lordship aboard.\"\n\nGarron's brow tightened in concern. \"What about you?\" he asked.\n\n\"I am going to make sure no one pays any attention to you.\" Mouse pulled out his dagger and started to cut the bindings on the donkey's bridle, careful not to sever it completely. He left a thin section still connected.\n\nGarron gripped Mouse's arm. \"But you're going to be there, right?\"\n\nMouse gave him a reassuring smile. \"I'll catch up to you.\" Seeing Garron's anxiety deepen, he added, \"I promise.\" He turned to Fenn and Hawken. \"Get out of the area as soon as it's done. Ditch the--\"\n\nFenn waved him off. \"We know the routine, mate! Let's just get it started.\" Then Hawken and Fenn fitted the helms back over their heads.\n\nThey needed to position themselves closer to the docks first. Mouse took the donkey's bridle and guided her out of the dark alley and farther down the narrow street. It was a quiet channel without shops or much traffic, which was one of the reasons Mouse had chosen it. Once the docks were in view, Mouse, one by one, opened the lanterns and poured the contents of oil over the bolts of textiles in the cart. He pulled out his flint kit from his pack.\n\nThe other three, piecing together what was about to happen, put more distance between them and the cart.\n\n\"Well, my friends,\" he said to Hawken and Fenn, \"appears this is farewell. If you're ever in Har Terrena, call on me. I won't be hard to locate if you look under the right rocks.\"\n\n\"I may at that,\" said Hawken. \"Just don't fuck this up. I won't be able to save your ass a second time.\"\n\nMouse laughed and clasped both their hands.\n\nHe struck the flint stone over the fabric. It took only three strikes before a spark caught and set it alight, and fed by the oil, it surged into a healthy flame.\n\nThe flame spread quickly, eating up more and more of the fabric in the cart. The donkey began to pace and stamp nervously, but Mouse held her bridle firm. In moments the cart was engulfed. The flames crackled and roared. Black smoke billowed upward in a twisting column. The donkey lunged about, eyes white in panic. Mouse held on as long as he dared.\n\nAs the donkey started to rear and kick, he let go and jumped clear. Finally freed, the panicked beast sprang away at speed.\n\n\"Now for the show. Get going, Garron.\" He took off across the docks at a sprint.\n\nThe flaming cart careened across the dock, spilling black smoke in its wake. Screams and shouts erupted all around as people dodged and lunged out of the way. The terrified donkey brayed as she galloped in wild frenzy. She bucked and kicked, trying to break free from the inferno behind her. The cobbled plaza was not made for such speeds--the cart bounced and tilted to the side precariously.\n\nMouse ran across the area as fast as his legs could take him. He caught glimpses of guild men spilling from hiding places. Some set upon groups of guardsmen, while others began scampering off with crates of merchandise. Other guards ran over to assist their comrades in the widening brawl while merchants and dock workers hollered for help and chased after the thieves making off with their profits.\n\nIn moments, the entire area had descended into chaos.\n\nThe bindings keeping the donkey attached to the cart finally gave way and snapped, freeing the beast. The cart listed onto one wheel and bounced into the air. Carried by momentum, it twisted and rolled. By now, the cart itself was fully aflame and coming apart. It hit the cobbles on its side and tumbled in a terrifying, unpredictable fireball. It struck the corner of a large stack of earthenware vats and exploded apart, sending flaming debris in all directions. But the impact destabilized the neat order of vats. The ones at the top teetered, then toppled and fell to the ground.\n\nThe first to hit the stones cracked open like an egg, and viscous dark liquid oozed out.\n\nPitch.\n\nMouse winced--he hadn't planned for it to go quite this far.\n\nOthers saw what was happening and scattered in all directions, waving their arms and shouting for people to get clear as more clay containers were dropping from the top. The result was extraordinary. Within seconds, the entire pile erupted in a fiery blast. People closest were thrown to the ground. A black mushroom of smoke boiled up over the docks, and flaming debris rained down over everything. The flames reached higher than the row of warehouses.\n\nThe danger of this becoming a true conflagration was suddenly very real, and everyone knew it. The crowds occupying the docks now had a unified purpose. If just one of the surrounding buildings went up in flames, if one burning coal landed on a warehouse roof, there might be no stopping the fire. All of Har Orentega could burn to the ground.\n\nAs Mouse drew closer to the boardwalk along the river, he slowed to a jog. Dockhands and ship crew alike were dashing past him with buckets and shovels. One of the barges was throwing off its lines and pushing free from the docks, the captain on the deck barking orders. He was not taking chances with the vessel or its cargo catching fire as well. Among the chaos, Mouse spotted the mage, still on the dock and shouting to the guardsmen around her. Her voice was lost in the cacophony, but Mouse was certain she was ordering them to ignore everything else and continue their search.\n\nMouse waited. Everyone around her had sprinted off at her bidding and left her alone and unattended on the dock. He walked up to her.\n\nArcane symbols were embroidered along the hem and sleeves of her robes. The cloth was elegant silk that shimmered like the dark surface of a lake at nighttime. Her hood was pulled back to expose her single tight braid of black hair that hung down over her left shoulder. She had pale skin, and her face was gaunt and eyes deeply set--likely the cause of too much time spent in dark cells studying spells by candlelight. Mages were not prone to physical vanity, but this one was pretty in her own way, with full lips and bright yellow eyes that sparkled with intelligence. It took her a moment to notice him standing there and realize he was not part of the surrounding bedlam. She looked at him, confused a moment as to why he was there staring at her, and then comprehension and recognition widened her eyes.\n\n\"Your handiwork, I presume?\" she asked.\n\nMouse allowed himself a satisfied grin.\n\n\"I almost didn't recognize you, Mouse,\" she said. \"A rat's vision isn't so clear, you know, but its hearing is astounding.\"\n\nMouse ignored the veiled threat. He smiled at her. \"You're too late,\" he said.\n\nShe raised a single eyebrow at him.\n\n\"Thought I'd let you know and save you the trouble of searching in vain,\" he continued. \"He's out of the city. And outside of your reach.\"\n\nShe smiled condescendingly. \"Is that so?\"\n\n\"It is.\" He turned to look out at the river. She turned too, following the direction of his gaze. The barge that had pulled away from the dock to protect itself from the fire was now drifting away downstream. \"Appears I won, mage.\"\n\nThe exchange was a strange contrast from the wild commotion that surrounded them. They spoke calmly, almost civilly, as if the two of them sat in a parlor sharing a decanter of wine, while the world around them unraveled.\n\nHer face gave nothing away as she swung her eyes back toward him. \"Hardly.\" Her voice was cold. \"You see, even though you cleverly figured out about the tattoo, you didn't remove all the ink. I may not be able to track his specific location with it anymore, but I can still sense its presence.\"\n\nMouse too kept his face rigid and unreadable. \"What does that matter?\"\n\n\"I feel it now, my dear. The ink is close. Closer than that barge out there, certainly. I know that Lord Garron is still here.\" She took a step closer. \"Foolish little mouse, thinking you could outwit one of us. We will find him soon enough, despite your pathetic attempt at distraction. He will be found and returned to Duke Delgan... and you will be hanged by the neck before the day is concluded.\"\n\n\"You know,\" he replied, \"I've never cared much for mages. For a number of reasons really. I don't really understand your power, but to my mind you possess an unfair advantage in the natural world. I find it abhorrent. But secondly, I find the arrogance of your lot to be insufferable.\"\n\n\"Is it arrogance? Or do we just better understand the universe and embrace our place in it, more so than the likes of you? I would argue instead it is you who is arrogant, thinking you could best me in this contest.\"\n\n\"I've done well so far. I'll remind you Garron is not yet in your custody.\"\n\n\"You've given us a merry chase, I'll admit. But that ends soon.\" She tilted her head and lifted her shoulders a fraction. \"But,\" she said, \"should you somehow manage to slip him past us yet again--as unlikely as that is, but you've surprised me before--\" Her face darkened in new intensity. \"I'll have you know that I will not quit my pursuit until Lord Garron is either recaptured or dead. It will be a personal mission of mine. I cannot allow someone like you thinking there is any hope in resisting us. You will never be allowed to win.\"\n\nMouse nodded with pursed lips. \"Exactly what I feared.\"\n\nAnd he lunged.\n\nWith reflexes she could not hope to counter, he sprang forward. His shoulder rammed against her midsection, and he threw his arms around her. The tackle caught her unaware, and the impact knocked the wind from her in a startled cry. Mouse pushed with his legs using all the strength he could muster, and drove her back... back... back. She flailed about to gain her footing again, but the force of his push was beyond her strength to oppose. He drove her to the end of the pier, and they went over the edge together into the river.\n\nAFTER THE initial crash into the water, a strange silence followed. The din from above the surface was instantly just a distant hum. It was now just him and her, spinning into the depth as if dancing.\n\nThe water was shockingly cold. Mouse's entire body constricted. He fought against the impulse to exhale the air he held in his lungs. Against him, the mage flailed about to free herself but only managed to tangle herself in her own robes. He tightened his grip on her anyway. The cloth absorbed the water as they sank and clung to her body. It twisted into knots around her legs as she struggled. Before the impact, Mouse had pulled in a full breath. In her surprise, the mage cried out instead as they fell and emptied her lungs. Panic was already setting in, and she clawed at him in a desperate attempt to break free.\n\nDeeper and deeper they sank.\n\nThey were near the base of the pier. The thick wooden pylons reached up through the greenish gloom like the pillars of a sunken temple.\n\nMouse released his grip on the mage. She tried to kick away from him and swim for the surface, but he grabbed her long braid before she could escape. With his free hand, he reached into his boot and pulled out a dagger. Not his father's--but an old one he carried about.\n\nHe dragged her by the braid closer to the pylon.\n\nThe wood of the pier was very old--it had stood here for centuries, and in places, the wood had split. Cracks ran vertically along its length. At some point, the city had attempted to repair the problem by fastening thick iron bands around the pylons to prevent the wood from splitting apart further.\n\nFueled by terror, she fought wildly to break his hold on her, kicking and waving her arms about. But already her strength was flagging. She'd lose consciousness soon. He pushed the blade through the weaves of her hair, then thrust the tip of the blade into one of the cracks in the pylon. He wedged and wiggled the dagger in as deep and as tight into the crevice as he could manage. Satisfied, he pushed off with his legs and swam away.\n\nAs he kicked toward the surface, he glanced back down into the murk one last time. The mage was tugging vainly at the blade's handle, but she did not have the strength to break it loose.\n\n# Chapter 25\n\nTHEY WERE provided a small cabin below deck and told in no uncertain terms to stay put and out of the way of the crew. Mouse was happy to oblige.\n\nAs soon as the door was closed behind them, Garron dropped onto a large shipping trunk and leaned back with his head on the wall. He closed his eyes, and his entire body sagged. He looked like he was a heartbeat from being asleep.\n\n\"Quite the diversion,\" he said without opening his eyes. \"More than you intended?\"\n\n\"A tad, perhaps,\" Mouse replied. \"It worked, which was the important part. Hawken got you aboard easily enough.\"\n\n\"At the risk of sounding ungrateful,\" Garron said dourly, \"I didn't want you to burn down the fucking whole city.\"\n\n\"They'll get it under control. Everyone was pitching in.\" Mouse wanted to believe that, but he didn't sound as confident as he wanted. Frankly, he didn't give a shit. Delgan was a rat-fucking bastard, and his city could burn to cinders as far he was concerned. Garron would judge him harshly for such thoughts, so he kept them to himself.\n\nHe peeled off his wet clothes and hung them on wall hooks to dry. Mouse had climbed onto the barge from the river, and Garron thankfully hadn't asked him why. Mouse had no interest in revealing what had taken place in the depths of the river. He sprawled himself out naked on the narrow bunk that was nestled into an alcove. He allowed himself a full deep breath and released it slowly--his body accepting the deed was done now and he could finally relax. It was done. The barge now lumbered its way down the river, and Garron was beyond the reach of Delgan.\n\nThis should have been a time of celebration. Garron was safe, and Mouse's part in this was finished. All that remained was to deliver him over and collect his reward. Mouse's flawless record remained intact, and he could return to his old life of lifting trinkets and baubles from rich prigs that didn't appreciate or deserve them. And now, with the payoff he'd receive, he could purchase the king's pardon and conduct his work without the added complication of having to pay off His Grace's men. Coin could stay put in his own purse for a change.\n\nBut the encounter with the mage left him feeling uncharacteristically morose and with the sensation that he was somehow indelibly stained. Fatigue was largely to blame--it cut deeply into his muscle and bone. And, of course, he couldn't shake the festering sentimentality that was gnawing at him from within.\n\nHe told himself he looked forward to this business being concluded, but in truth, his stomach twisted at the thought of passing Garron on to his entourage and then walking away.\n\nGarron's words from that morning were still hot in his mind, like a steaming kettle fogging up his thoughts. Such tempting and compelling words. Mouse could almost see his willpower collapsing and giving himself over to them. But Garron had spoken with the passion of the moment--not with reason, and Mouse knew he'd be a fool to take any of it to heart. Garron had been intoxicated by his unexpected freedom, nothing more, mistaking gratitude for attraction. And now that he was truly free, the words he said were likely already forgotten.\n\nMouse closed his eyes and wove his fingers together behind his head. As he lay there trying to push all such thought from his mind, he was suddenly aware of Garron's eyes on him. He pretended not to notice. Or care. But a part of him wondered if he had displayed himself naked on the bunk on purpose.\n\nMouse heard Garron rise off the crate and then felt him sit with one cheek on the edge of the berth next to him. Garron pulled his leg up, and the round of his knee brushed against Mouse's thigh. The gentle rocking of the barge moved Garron's hip against Mouse's side in soft waves. Mouse opened one eye to look up at him, but he couldn't read his expression because his face was in shadow.\n\n\"How long will we be aboard, do you think?\" Garron asked.\n\n\"Hard to say,\" Mouse answered. \"Until morning, certainly. At the speed we're moving, it'll take a while to get beyond Orentegan borders. Depends on where they plan on docking. Har Senica, perhaps, but I doubt they'll wait that far. The captain will want us off as soon as possible in case there's trouble. He'll deposit us off at one of the more remote villages. Renoa would be my guess.\"\n\nGarron nodded, seeming more to buy time and collect his thoughts than in understanding. Mouse could tell he had more to say and was working his way around to it but hadn't decided on the best route. \"Never was much a man for the water,\" he said. \"I swim well enough, and I don't get the sickness like some. Just prefer the ground beneath to be more firmly settled.\"\n\n\"Haven't had much cause to be aboard ship much either. But I understand you grow accustomed to it.\"\n\n\"Not sure I ever would,\" Garron replied pensively. He leaned back against the wall. \"What happens then? When we are off the ship?\"\n\n\"I will send out a courier to my contact the first chance we get and let him know I have you safe. And then we buy a couple of horses and head north to the established meeting place. West Dughnum, at an inn called the Trident Arms.\"\n\nGarron nodded thoughtfully. \"Then you accept your reward, and you're on your way.\"\n\n\"Something along those lines, yes.\"\n\nGarron looked down at his hands and for a moment seemed distracted by a fingernail. He looked uncomfortable. Mouse could tell he had something to say and was trying to figure out how to broach the subject.\n\nOf course, Mouse knew what was on Garron's mind. He wasn't na\u00efve, and he wasn't an idiot. He'd already seen this coming. Garron was regretting the words of passion he'd said at Taurin's and was trying to figure out a gentlemanly way of retracting them. Now that he was free--truly free--and could once again envision his life back in Har Dionante as a reality, Garron had come to realize the folly of that earlier proposal. He'd come to realize at last that the worlds they lived in were just too different. And whatever attractions they felt for each other, nothing could ever realistically come of it. Mouse knew Garron would see it that way eventually.\n\nStill looking at his hands, Garron pursed his lips. \"So, about this morning,\" he said. \"About... the conversation before we were interrupted.\"\n\nMouse steeled himself for what was coming next. It would sting certainly, for there was a part of him that ached to be with Garron. But holding on to some fantasy that this nobleborn gentleman would want him in his life was nothing but an absurdity. Mouse knew better.\n\nHe was about to save Garron the trouble of saying it, but Garron spoke first.\n\n\"I meant every word I said, you know. That hasn't changed.\"\n\nMouse's eyes sprang open. That was not what he expected to hear. \"Excuse me?\" His heart jolted into rapid beats.\n\nMouse could feel the smile light Garron's face even though it lay in shadow over him. Instead of saying more, Garron leaned in, and Mouse felt soft lips press firmly into his own.\n\nThis time, Mouse was powerless against it. Some corner of his mind thought about fighting, pushing him away, but the notion of resisting was pointless. He didn't have the strength of will. Instead, Mouse sank deeper into the warmth of the kiss. Garron's mouth pulsed against him, sending waves of passion coursing through Mouse's entire body. His jaw relaxed as Garron's tongue prodded the opening of his mouth. As the thick muscle thrust inside, he was aware of nothing else--and pulled the tongue in farther with his own suction.\n\nGarron pressed in even harder. Suddenly, the weight of Garron's torso was against his own, and Garron's soft hand explored the contours of Mouse's chest and abdomen, pinching chest hair between his fingers and tugging gently. Then the hand moved south. Past the belly. The muscles of Mouse's abdomen tightened in response. Past the waist. Fingers brushed the line of hair that ran vertically downward until the fingertips combed through the edges of his tuft of thicker hair.\n\nMouse's sensual awareness was now fully on his cock as it swelled to life, lifting upward like a crane's boom and springing back against the back of Garron's hand. Garron chuckled softly but didn't pull off the kiss. Instead, he curled his fingers in a tender embrace around his swollen girth. Mouse involuntarily moaned his approval.\n\nNever had he felt so consumed by his senses. Each breath through his nose brought Garron's heady scent directly into his brain like an apothecary's physic. It sent his mind whirling. Eyes closed, he could see everything with his skin. The soothing rolls of the barge on the waves rocked Garron's firm body against him again and again in an intoxicating rhythm while Garron's hand guided the foreskin upward over the head in cadence with the ship's movement.\n\nHis face flushed with sudden heat, and he felt his sac tighten.\n\n\"Gods!\" he whispered against Garron's lips. \"Please stop.\"\n\nGarron pulled back from the kiss, and at the same time slid his hands down to cup Mouse's scrotum. He rolled the wrinkled skin between his fingers. \"Plan on protesting some more?\" he asked. There was a wicked sort of delight in his voice. \"Going to tell me some more how you don't want this?\"\n\n\"This is such a bad idea, Garron,\" he said, but there was no conviction in it. He spoke between gasps, and his hips continued to respond to Garron's touch. \"I'm not the one for you.\"\n\n\"Shouldn't that be my call?\"\n\nEven though Garron's face was fully in shadow, Mouse turned his eyes away. \"But you have no idea the kind of man I am.\" The words barely formed, and each consecutive one that passed his lips made his heart twist tighter. A part of him hated himself for even saying the words. Part of him burned to just give in to the desire and worry about the consequences later. But he couldn't allow himself. Fear and practicality always won out.\n\nHe could feel Garron smiling down at him, which irritated him. Why didn't he fucking understand?\n\nGarron moved his hand from Mouse's groin and caressed his knuckles down the length of Mouse's jaw. \"Gods, you are such an idiot! Adorable... but an idiot.\"\n\nMouse's eyes widened, taken aback. \"Excuse me?\"\n\n\"Do you think me so na\u00efve? Of course I know your past is not filled with philanthropy and pacifism. But in the short time I've known you, Mouse, you've proven yourself to possess more integrity and honor than most of the so-called noblemen I know.\"\n\nMouse pushed air through his lips. \"Me? Honorable? You are na\u00efve.\"\n\n\"You have convinced yourself you are wicked and incapable of redemption. I'm here to tell you you're wrong.\"\n\nMouse opened his mouth to protest, but Garron put two fingers over his mouth. \"Let me finish. Do you think you alone have committed heinous deeds? We have all done terrible things, Mouse. No one can honestly say otherwise.\"\n\n\"Except you.\"\n\n\"No,\" Garron replied softly, breaking his gaze. \"I am not as virtuous as you would paint me. I've done my share as well, believe me.\"\n\nMouse didn't believe him, but he didn't argue.\n\n\"I've seen men,\" Garron continued, \"who claim themselves to be good and honorable do wicked things with ink and parchment. A most cowardly act, because they never have to face the ones they've wronged or feel the consequences of their deed. I have seen those who rail on about their own compassion willfully ignore those in need or treat those beneath them with contempt.\n\n\"We all justify our dark behavior in some way, mask it behind some twisted form of truth that gives us license to do what we will. In my experience, people can rationalize the most reprehensible acts and sleep soundly through the night as if no blame rests upon their shoulders. Take Delgan. He has certainly justified my imprisonment and death sentence as something for the betterment of his city and his people.\"\n\nGarron held Mouse's eyes with intensity. \"But you,\" he said. \"You face your demons directly. You do not try to disguise them behind some false truth or rationalize them away. That doesn't make you wicked. It makes you honest.\"\n\nMouse was quiet a moment. \"But what difference does that make?\"\n\n\"All the difference in the world. The events that led you down this path you're on are behind you now. From here on, everything can change.\"\n\n\"You don't understand my past, Garron.\"\n\n\"Your past doesn't matter. What matters is what happens now. You can now choose a new path, Mouse. Perhaps a new path that includes me.\"\n\nMouse wanted to laugh, but his heart wouldn't allow it. \"So... am I to put on tights and surcoat and join you in your manor? Parade about as your concubine, join the other ladies of the court in their sewing circles and afternoon tea?\"\n\nGarron laughed out loud. Heartily. Warmly. \"No. Nothing like that, you idiot. I would never subject you to that slow torture. I had something else in mind.\"\n\nMouse raised his eyebrow.\n\n\"Let's disappear,\" Garron said.\n\n\"What?\"\n\n\"You heard me.\"\n\n\"Well, now who's being ridiculous!\"\n\n\"I'm serious, Mouse. We'll go off together and build a new life for each other. We go to where you agreed to meet for your payment, we take the coin and instead of heading off to Har Dionante, we head somewhere else. Anywhere. A new city--or a small town, whatever we want. But just us.\"\n\n\"And do what?\"\n\n\"I don't know. Buy an inn. Buy a farm. Buy a tall ship and sail the seas to the east.\"\n\nMouse shook his head, not believing what he was hearing. \"You're telling me you'd give up your station and leave your posh life behind? For me?\"\n\n\"Without a second thought. Without ever turning back.\"\n\n\"Garron....\" he began, but stopped himself.\n\n\"Are you telling me you wouldn't do the same for me?\"\n\n\"It's not as simple as that,\" Mouse replied quietly.\n\n\"It can be.\" Garron too fell quiet for a moment, then said, \"Mouse, I have never felt this way about anyone before. Have never known anyone like you before.\"\n\n\"And what if it turns out in a month we despise each other? Or you snore terribly and I can't spend one more night with you? What then?\"\n\nGarron shrugged. \"You go back to your old life. And I will go back to mine.\"\n\nMouse turned away and looked up at the ceiling.\n\n\"If you don't feel the same way,\" Garron added, mistaking Mouse's silence for something else, \"then say so. And I will leave it be.\"\n\nMouse's gut twisted. A part of him screamed out to push Garron off of him, tell him he didn't feel the same way--not because it was true, but for Garron's own sake. To protect him. How could he ask Garron to give up his life for him? How could he subject Garron to the life of a commoner? But try as he might, he couldn't form the words. He couldn't bring himself to say them out loud. The notion of sharing a life with him was too exhilarating. Though he couldn't bring himself to believe it was even possible, he couldn't reject it either. Giving in to Garron was just too dangerous--more dangerous than anything he'd ever done before. But he couldn't resist it either.\n\nIn a very quiet voice, he heard himself say, \"I do feel the same.\"\n\nThen Garron was kissing him again. It felt different this time. It felt deeper. Stronger. As if some veneer between them had stripped away. But it was more than that too. The kiss wielded a potency that Mouse had never experienced, one that transcended some unseen boundary he hadn't thought possible. He felt it everywhere. It seeped into his bones and made his toes curl. And the last remnants of doubt he clung to dissolved away.\n\nGarron broke off the kiss only long enough to pull off his tunic and strip off his trousers. Then once again he was lowering onto the small cot next to Mouse, ready to resume. His naked body pressed against Mouse's. The touch of their skin together was electric and sent waves of passion coursing through every muscle. But as Garron leaned in again, a thought wormed into Mouse's brain, and he held Garron back with a hand to his chest. His beautifully formed chest. \"Garron, why did Delgan imprison you in the first place? Who did he make the deal with and why?\"\n\n\"That doesn't matter anymore,\" Garron said.\n\nHe's right, thought Mouse. What did it matter? None of it would ever matter again. And he once again gave in to Garron's intoxicating lips. And a great deal more after that.\n\n# Chapter 26\n\n\"HOW LONG do you expect?\" Garron asked.\n\nMouse adjusted the fat hare he had positioned over the fire. Juices dripped out and sizzled on the hot coals beneath. He wasn't sure if Garron meant dinner or their destination. \"Soon,\" he replied, covering both answers.\n\nMouse stole a casual glance at Garron. He sat opposite him, on the other side of the fire, his face illuminated by warm yellow light. He sat cross-legged on the ground, wrapped in a blanket, and stared into the flames, mesmerized as if it was his first experience with fire.\n\nHis first taste of a commoner's life.\n\nThe River Falcon had docked at the village of Renoa in the hours before dawn. The captain had his crew unceremoniously deposit the two of them on the pier, and then they pushed off again without a word or a wave.\n\nMouse made certain their time in Renoa was brief. He straight off hired a messenger to run word to his contact regarding Garron's liberation. Then he hunted down the village farrier and procured two horses, saddles, and plenty of provisions for a few days of travel. The farrier had a weapon for Garron too--an old military-grade short sword, crudely made but well tended and still sharp enough. Garron would need something to protect himself. The road could be dangerous.\n\nThey were immediately off on the north road before the sun had barely broken the horizon. They traveled the entire day at a fair pace, with Mouse constantly looking behind them to see if they were being followed.\n\n\"Do you suspect we'll arrive by tomorrow?\"\n\nAh. The journey was the topic, then. \"Hard to say,\" Mouse answered. He caught himself, reluctant to tell Garron that it was unlikely they'd reach West Dughnum by the morrow. Garron might well romanticize the joys of a simpler life, but he had yet to experience its true charms, like spending a night out in the chill air, trying to sleep on a cold hard ground with a root poking you in the back. After one night of that, Garron could well be put off on the idea of running off with him if that was what life would be like.\n\nHe looked up at the sky. Most of the stars were obscured, meaning clouds had rolled in. \"Depends on the weather,\" he said. \"And the condition of the road.\"\n\nGods! Mouse dreaded the notion of rains coming in during the night. A night spent on the cold ground was unpleasant enough. A cold, wet night was infinitely worse.\n\n\"And what of your contact? How long before he and my escort arrive in West Dughnum after they receive the news?\"\n\n\"Again, impossible to say. My guess is we'll enjoy a few days of quiet solitude at the inn before they arrive.\"\n\nGarron nodded but didn't reply. His expression was hard to read in the dim firelight, but it seemed in that moment that he sank into a more pensive mood.\n\n\"Are you feeling anxious?\" Mouse prodded. He wasn't at all surprised by Garron's inward mood of late. The closer they drew to West Dughnum, the closer Garron was to the decision of parting ways with his old life. Of course, Mouse was still unconvinced it would happen at all.\n\n\"A little,\" Garron replied.\n\n\"Well, understandable,\" Mouse said after a moment. He forced down his own tide of anxiety that threatened to rise up and shake him apart. Garron's tone was pregnant with deep unspoken thoughts that Mouse could not begin to read. Garron felt closed to him.\n\nHe added more wood on the fire and turned the hare on the spit again. He didn't know what else to say, so he said nothing. When he glanced again at Garron, he was watching the swirl of smoke that rose up through the gap in the canopy overhead. He must have sensed Mouse watching him, for he looked down again and caught Mouse's eye.\n\n\"Come over here,\" Garron said.\n\n\"I'm tending to dinner,\" Mouse replied, forcing a smile. \"You want to eat, don't you?\"\n\n\"The hare is fine for a minute. Come here and warm me up.\" He extended an arm to hold out one end of the blanket, suggesting Mouse should occupy the space.\n\nMouse dropped the branch he had just picked up and moved around the fire to join him. Instead of taking the seat next to him, he crouched behind Garron and wrapped his arms about his shoulders in a gentle hug. Garron's hand reached up and caressed the length of Mouse's bicep.\n\nHow quickly this feeling had become so familiar, so comfortable. The feel of Garron against him. His touch. His scent. It felt dangerous in a way, but he couldn't pull himself away.\n\nGarron's head tilted back to rest on Mouse's shoulder. \"I'm sorry,\" he said.\n\n\"For what?\" Mouse replied, deliberately obtuse.\n\n\"I'm brooding.\"\n\n\"I hardly noticed,\" Mouse said.\n\nGarron chuckled. \"I'm only adjusting to the idea of being free again. It's a lot to take in.\" He was quiet a moment. Mouse tightened his grip around him and listened to his breathing. \"Nothing's going to change, you know,\" Garron told him softly in his ear. \"Nothing will alter what we planned.\"\n\nThe plan. When Mouse's contact arrived at the inn, Garron was to insist that Mouse be awarded his coin immediately. Once Mouse had the coin in hand, then Garron would announce that he was not returning to Har Dionante with the escort after all. The reward money would help them begin their new life together. Mouse and Lord Garron would disappear from the world. Simple.\n\nWe'll see, thought Mouse, but he kept his suspicions to himself. Instead he said, \"I know.\"\n\nThey remained there a time, watching the fire as it danced and crackled. Fat from the hare continued to drip down onto the coals to sizzle and spit. Night sounds sang out around them in a gentle concerto.\n\nThen Mouse caught a new sound. He stiffened instinctively and waited. Something out in the wood beyond their tiny clearing was out of place--a sound discordant from the rest. It was the snap of a dry twig.\n\n\"What is it?\" Garron asked, his body stiffening in response to Mouse. His hand slid off of Mouse's arm.\n\n\"Not sure,\" he answered. Slowly, he released his hold around Garron and moved his hands to Garron's shoulders, ready to rise to his feet. He held his breath and reached out with his senses into the surrounding night for another indication that something was out there. His hearing worked to filter out the sounds he expected to hear and focus in on anything that didn't belong. It could have been just an animal of some form, drawn by the smell of the roasting meat. But bandits were always present in these parts. One always had to be vigilant when traveling the open roads. And there were other would-be dangers as well. His hand slipped from Garron's shoulder and crept to the dagger sheathed at his waist.\n\nGarron saw where Mouse's hand was moving. He shrugged the blanket off his shoulders and reached for the scabbard that rested on the ground next to him.\n\nMouse looked to the horses, tied to a branch at the edge of the clearing. One was alert, ears perked up. He'd heard or sensed something as well, but whatever it was wasn't enough to spook the beast. Likely not a wolf or some other predator, then. Its scent on the wind would have caused more distress.\n\nMouse held his breath and waited.\n\nHe heard a soft click come from within the thick underbrush. It was mechanical--not natural. It was a distinctive sound he knew well.\n\nTiming was everything. Heart thumping, he counted backward. Three. Two. One. Then he threw himself on Garron, knocking him to the side, just as he heard the louder click he waited for. The crossbow bolt whistled over their heads and struck a tree trunk behind them with a hard thud. Mouse rolled one way and leapt to his feet in one smooth motion while Garron rolled the other.\n\nThe attack came from all sides at once. Five men. They erupted from the surrounding brush, swords flashing in the firelight.\n\nThey were dressed in black leather and had their faces covered with black cloth. They moved in fast, silky and effortless. These were no ordinary thugs--some cheap blunt hammers sent into a tavern or market to take care of someone ordinary. These men were seasoned and skilled. And expensive.\n\nMouse armed himself with a dagger in each hand. His father's was in his left--his favored hand. He adjusted his fingers for just the right grip as he bent at the knees in a defensive position. The first attacker made a charging leap for him. He glided through the air impossibly high as if winged and landed like a dancer in front of Mouse, sword already in motion.\n\nA bold attack meant to unnerve and drive Mouse back. But Mouse wasn't about to give up any ground this early. He pivoted on one foot to plant the other behind him to support himself from the impact, then caught the sword between the crossed blades of his daggers. Surprise registered in the eyes of the attacker--and provided Mouse just enough time to formulate a fitting response. He leaned in and brought the foot behind him forward in a swift kick. He struck solidly between the attacker's legs. A dirty move, but Mouse wasn't feeling particularly gentlemanly.\n\nThere was a grunt and sudden whoosh of air, and the attacker bent forward. Mouse thrust his knee up into his face, throwing his head back again. Mouse kicked again, only this time against the attacker's chest. The man tumbled backward into the fire.\n\nMouse picked up the same distinctive click he heard before and on instinct, threw one shoulder back to twist his torso. He felt more than saw the crossbow bolt whiz past him less than a hand's width distance. He was going to have to take care of that crossbowman. The bolt had come from the shadows still. There was a sixth man.\n\nBehind him came the ring of steel as blades met. Garron had engaged one of the attackers. Mouse couldn't turn about to see, for another was bounding toward him. A wave of numbing anxiety flushed through him.\n\nAnd it nearly cost him his life.\n\nHis fear for Garron dulled his reaction time. He only just avoided the next attacker's swipe at him. Mouse sprang backward, and the sword's tip passed close enough to his unprotected chest to snag on the tunic and rip it open.\n\nFocus, damn you, he chastised himself.\n\nThe attacker recovered quickly. He pushed in closer and made a quick jab. Mouse caught the sword blade in the curved cross guard of his dagger and guided it to the side, twisting his opponent's torso to expose his flank. Mouse thrust out with his left hand. The Black Guard's blade sliced through the leather armor like parchment and dug deep into the man's side. It passed cleanly between his ribs at a steep upward angle.\n\nThe man stumbled backward, gasping. The stab must have punctured his lung. The sword in his hand was forgotten as he struggled to pull in a full breath. Mouse swung his leg behind the man's calf, throwing him off his balance. The man tumbled backward and hit the ground hard on his back--where he would remain, choking for air until he bled out of the wound.\n\nOne down.\n\nMouse turned about to assist Garron in his fight. But surprise halted him. Garron deftly controlled the sword, thwarting each attempt his attacker made to break his defenses. Garron appeared to toy with the man, but Mouse knew he was conserving energy and waiting for the right moment to make his own assault. Already the man was tiring and showing weakness.\n\nMouse wanted a moment to admire the shape of Garron's graceful form while he wore down his opponent--his shoulders and back that shown beneath the thin fabric of the tunic and the way his trousers clung to accentuate the cleavage of his perfectly formed buttocks. Garron moved with beauty and precision. He was well trained and undoubtedly skilled.\n\nBut two other men were jumping into the fray and closing in for an attack, forcing his attention. One barreled directly for him, while the other eyed Garron, edging around to his blind side to flank. The man Mouse had kicked into the fire had now rolled out and sprung to his feet. Seeming unfazed, he released the clasp at his throat and let the burning cloak fall from his shoulders.\n\nAnd Mouse could hear the crossbowman arming a bolt. He made a quick mental note of how much time he had before the crossbowman would have a new bolt loaded and be able fire again.\n\nNot long.\n\nMouse again took a defensive position as the next attacker charged, the man's sword arm already in motion in a wide backswing. He waited--his heart thumping madly as he calculated the timing. When the man was close enough, Mouse arched his back while shifting his center of gravity forward with his hips. The move dropped him onto his knees, his head leaning back. The sword traveled harmless over Mouse's head, and the attacker's momentum brought him nearly on top of him. Mouse sliced upward with his father's dagger across the man's sword arm. The edge cut deep into the flesh just below the elbow. It sliced through until it met with the resistance of the bone. Warm liquid sprayed down onto Mouse's face.\n\nThe man cried out in agony, high-pitched and animal-like. He stumbled to the side and doubled over, clutching the nearly severed arm to his chest.\n\nStill on his knees, Mouse flipped the dagger in his right hand from hilt to tip. He pulled back his arm and let the dagger fly. The attacker that had climbed from the fire lifted his head up just in time to have the dagger impale him in the eye. Without a sound, his head whipped back, and he collapsed again to the ground and this time was still.\n\nMouse leapt back onto his feet in one smooth motion and took stock. Garron had run one man through, the body in a heap beside him, and was pressing the other hard.\n\nAll that remained was the crossbowman.\n\nHe sprinted out into wood.\n\nWHEN MOUSE returned to the clearing, Garron stood over the fire with his arms folded across his breast. He stared blankly into the flames until he heard Mouse's approach and looked up with clear anxiety in his eyes. He breathed out, puffing out his cheeks, and cast his gaze downward again. \"You're safe,\" he said softly.\n\n\"You're surprised?\" Mouse replied, attempting a breezy tone, but it seemed out of place and tactless as soon as the words passed his lips. Now was not the time for bravado.\n\nGarron looked to the clearing edge from where Mouse had emerged with tight lips. \"Well?\"\n\n\"He scampered off. No sign of him.\"\n\n\"More will come, then.\"\n\n\"Not tonight, I'd wager, but I agree it's best we not risk it. We should ride on.\"\n\nGarron remained motionless and silent as he continued to stare down at the weak flames. Something in Garron's bearing made Mouse feel the chill in the night once again. \"You're rather impressive with that.\" Mouse gestured with his head toward the sword he'd dropped to the ground.\n\n\"Just because I don't relish killing doesn't mean I'm a pacifist,\" Garron answered dourly. \"I had adept trainers in my youth.\"\n\n\"Just glad you're on my side.\" Mouse looked around the carnage of their campsite. Garron had already dragged the bodies to the edge of the clearing, but the signs of the battle were still evident everywhere. The fire burned with less intensity than before, since the careful pile of logs had been knocked over. The spit Mouse had fashioned had been smashed apart, and the roasted hare was sitting in the dirt. \"Bugger me!\" Mouse said as he picked up the carcass. \"Bastards ruined our dinner.\"\n\nGarron's head tilted to the side when he looked up at Mouse, and then he was suddenly on the move. He crossed the distance between them in three steps. With a hand under Mouse's bicep, he gently turned him about to face the firelight. He scowled down at Mouse's shredded tunic.\n\n\"You're injured,\" he said, his face turning ashen.\n\n\"Not my blood,\" he replied, but even as Mouse said it, he knew he was wrong, for he became aware of the sting across his chest only at that moment.\n\nGarron separated the gash in the fabric with his thumb and forefinger and looked closer. \"Mouse--\"\n\nMouse took Garron's hand and guided it away. \"It's not deep.\" Still holding Garron's one hand, Mouse reached up with his other and put a palm to the side of his neck. \"I'll be fine.\"\n\n\"Fine? Any deeper and it could have killed you.\"\n\n\"Not fucking likely. You should know by now I'm not that easy to be rid of.\"\n\nGarron pulled away. He wiped the sweat from his brow with the sleeve of his tunic. \"By all rights, we should have both been killed. They meant to ambush us while we slept.\"\n\n\"I know,\" Mouse answered. The ambush would have likely ended with both of their throats cut. But as the attackers closed in to lie in wait, someone mucked it up and made a noise. They could see that Mouse had heard it--so they had no choice but to attack. \"I suppose we should have kept one of them alive. We might have learned if there are more of them out there. I'd like to know how they were able to track us all the way from Har Orentega.\"\n\nGarron looked down at his feet. \"They didn't. These were not Delgan's men.\"\n\nSomething about Garron's peculiar certainty sent a chill chasing down the length of Mouse's spine. Could he have questioned one of them already? What could he have learned in the short time Mouse was traipsing through the wood? \"What are you saying, Garron?\"\n\nGarron looked down at his hand and seemed to only then notice the dark blotch staining his skin. He rubbed his palm on the thigh of his trousers. \"I... I should have never put you in so much danger.\"\n\nMouse stepped closer. \"Garron, what's going on?\"\n\nGarron seemed to struggle with the answer. \"I looked beneath their armor,\" he said eventually. \"Their tunics... they wear their livery.\"\n\nMouse understood immediately. They wore insignia to identify their house. \"And you know them?\"\n\n\"Know to whom they pledged their fealty anyway.\"\n\n\"Then you're certain. These were not Delgan's men.\"\n\nGarron shook his head.\n\n\"Which then means you know who is behind this attack. And the plot to remove you.\"\n\nThis time Garron made a slow bob of his head. \"I had suspected. Now....\" His jaw tightened, and he didn't finish.\n\nMouse closed the gap between them. He grabbed the front of Garron's tunic with both hands, ignoring the stinging pain across his chest. He locked his eyes into Garron's. \"Gods! Out with it! What is this about? If this is something that could get us killed, I should be privy to the details. Who wants you dead, Garron?\"\n\nGarron put his hands over the top of Mouse's. He held them there a moment, then gently removed them. He turned, strode away from the fire, and kept his back to Mouse as he spoke. \"I suppose I haven't been entirely honest with you, Mouse.\"\n\nMouse felt his lips tighten. \"Is that so?\"\n\n\"I'm sorry. I cannot truthfully say why. To protect you, probably.\"\n\n\"Go on, then.\"\n\n\"In a way, this is about my brother. My older brother.\"\n\n\"The heir to the duchy.\"\n\n\"Yes. Tiamond. He has an aggressive disease. Tumors that riddle his entire body.\" Garron turned about and held Mouse's gaze. \"Mages have been trying to heal the growths for years, but the sickness is outpacing their ability to combat the spread. They are losing ground.\"\n\nMouse suddenly understood. \"So if he dies, the duchy passes to you.\"\n\nGarron nodded.\n\n\"So, all of this,\" Mouse said as he gestured toward the ruined campsite and the heap of dead assassins. \"This is to prevent you from becoming the heir?\"\n\n\"Apparently.\"\n\nMouse's mind whirled about, grasping for understanding. Garron was honorable and decent; his moral character was unassailable. Who could possible object to Garron taking the seat of the duchy? \"Is some type of rivalry behind this? Another family vying for power?\"\n\n\"Something like that.\"\n\n\"Gods!\" was all Mouse could say in response. He spun around, hand to his forehead. Not only had he bedded a nobleman--but the future fucking duke. \"Hold a moment. You're telling me that you are walking away from the seat of the duchy? To run off with me?\"\n\nGarron glanced at Mouse. \"Yes,\" he said in a tone that implied the question was idiotic, \"Happily. And without a backward glance.\"\n\n\"So, who is going to inherit the seat now?\"\n\n\"I have a younger brother as well. Dennithe. He would take up the mantle and rule Har Dionante as its duke.\"\n\nComprehension suddenly crashed in around Mouse. Of course, this had nothing to do with some rival family as Mouse had first thought. Garron's own brother was the one directly to gain from his disappearance. He was behind it all.\n\nNo wonder Garron had remained so reticent on the subject all along. He was being betrayed by his own brother. With Tiamond on the verge of death, the third-born son suddenly sees the duchy is within his reach. Of course, that was why Garron was also willing to just walk away. The thought made Mouse's heart twist. It had less to do with Garron wanting to live out his days with Mouse and more about avoiding a confrontation with a brother he might have once loved and trusted. Perhaps it was easier to disappear and abandon his claim than watch his brother hang for the crime of treason and attempted fratricide. Mouse was just a convenient way out.\n\n\"I know what you're thinking, Mouse,\" Garron said, then averted his eyes to the ground. Quickly. As if ashamed. \"I would have loved our quiet life together.\"\n\n\"Would have?\" Mouse felt everything around him crash to a halt. Here it comes, then. \"And what has changed, Garron?\" he asked in a quiet voice. His insides reeled in mad combat. Of course it wouldn't go as they planned. He had known this all along, though he didn't want to believe it. He was just waiting for the moment to arrive. The fight had just brought it on sooner.\n\nBut a part of him--a silly, childish, heartsick part of him--nonetheless believed in Garron's sweet vision of their future. He felt betrayed.\n\n\"You were almost killed tonight,\" Garron replied with hardened edge, as if annoyed he had to point out the obvious.\n\n\"A daily occurrence for me, really,\" Mouse mumbled.\n\n\"Don't joke, Mouse. How could I live with myself if something happened to you because of me?\" Behind the flash of anger in Garron's eyes, Mouse discerned what truly kindled it. Pain and fear.\n\nMouse moved closer to embrace Garron but stopped himself. He wanted to wrap his arms around Garron's waist and put his head against his chest, but that felt like desperation, as if he was clinging to him to prevent him from leaving at that moment. So he stood there with hands at his side not knowing what to do. \"Nothing's going to happen to me. Or you.\"\n\n\"You cannot guarantee that.\"\n\n\"Perhaps,\" Mouse conceded. \"But I've gotten us this far. Have faith in me.\"\n\n\"This will not end.\" Garron let his head fall back. \"I was such a fool, Mouse.\" He reached out and pushed his fingers through Mouse's dark hair. When the hand reached the back of Mouse's neck, Garron pulled him closer. Their lips met. The kiss was soft and tender but brief. \"I honestly believed we could just leave,\" he said as he pulled away. \"I hope you know that to be true. I honestly believed if I just disappeared the problem would resolve itself. But as long as I still breathe--\"\n\n\"You remain the true and rightful heir,\" Mouse finished for him with a nod of comprehension. \"And someone finds that particular condition unacceptable.\"\n\n\"Yes,\" Garron answered. \"And it will not end until I'm dead. I realize that now.\"\n\n\"Garron, if that is what you truly want--if you want to abandon your claim, you still can. You can disappear, just like we planned--\"\n\n\"And live in fear each day? Waiting for the time they find us again?\" Garron shook his head. \"I could not live like that, worrying each and every day that something might happen to you. That is no way to live.\"\n\n\"Garron, I can protect you.\"\n\n\"I know you believe that. And yes, if not for you I would not be here now. But tonight we were lucky. You know this to be true. Had you not heard something, we'd be dead before the morn.\" Garron moved one hand over Mouse's heart, over the wound that ran across his chest. \"The breadth of a finger more, and you would be lost to me tonight. What if next time we are not so lucky?\"\n\n\"Garron--\"\n\n\"I must put an end to this, Mouse. Once and for all. I wanted to run away and put it behind me and live out my days quietly with you--but I realize now that was folly. I know now what I must do. I must confront this. I must return to Har Dionante.\"\n\nMouse took a step toward him. \"No. That is true folly.\"\n\n\"I must. I have to try and end this. It's the only way.\"\n\n\"You'll be heading exactly where your enemies expect you to go. You'll be walking right into their hands.\"\n\n\"It won't be as easy as that. Out here, I would simply disappear, never to be heard from again. But there, within the walls of my father's keep? It would be much harder. You'll have to trust me on this.\"\n\nMouse looked into Garron's eyes and saw the resolve. He would not be able to persuade him away from this, though his entire being screamed at him from within to at least try. There must be some argument that could be made to sway him from this idiocy. But instead, he made a slow nod. \"If you're determined,\" he said. \"As I can see that you are. Then I am going with you. I'm not going to let you piss away all my hard work by getting yourself killed.\"\n\n\"Good,\" Garron replied. \"I was hoping you'd say that.\"\n\n# Chapter 27\n\nMOUSE DROPPED into a wide cushioned chair expecting a long sit. There was no telling how long they would have to wait. He lifted a book from the small table next to him and read the title along the spine. A silly romance. He had no interest in such drivel, but he opened the book anyway. But as he pulled back the leather cover to inspect the inside, he heard something coming from outside the chamber.\n\nVoices.\n\nThey came from the corridor, muffled and distant at first, but they grew steadily louder. There was one speaker mainly, male, presumably, from the deep resonance of the tone. He spoke in a casual and unhurried cadence as if listing instructions, and the others with him made brief respectful replies. Mouse counted two others at least. Then, of course, there would be a guard contingent following as well, and perhaps a servant or scribe. A total of six. Seven, perhaps.\n\nMouse replaced the book on the table, leaned back and crossed his legs, and draped his arms over the padded arms. Ridiculously comfortable. Who knew such a chair existed? The opulence of the chamber was unequaled by anything Mouse had previously seen. It made Taurin's den feel like a village hovel in comparison. Every wall was paneled with warm wood, the ceiling richly painted, and the carpets felt like the forest floor beneath his feet. There was enough gold inlaid in the furnishing to provide him a rich existence until the end of a very long life.\n\nHe glanced over at Garron, who stood at the window frame with his arms folded in a tight nervous knot.\n\n\"How do I look?\" Mouse asked him.\n\nGarron shushed him, making a glance at the door. \"Like a village drunk trying to pass himself off as a lord,\" he snarled at him.\n\nMouse made a face. \"Rather harsh.\"\n\nGarron turned back to the dark window, his expression hard. Mouse frowned. Every attempt to lessen Garron's mounting anxiety only managed to irritate him.\n\nThe voices were outside the door. In moments, it would all change anyway.\n\nGarron stiffened and turned, unconsciously tugging on his surcoat. He very much cut the strapping lord in his finery, which Mouse had quietly pinched from his personal chambers earlier in the day. He was handsome and dauntless, like a valiant hero in a painting. Garron straightened his back, steeling himself for the confrontation to come. His countenance was a tight scowl.\n\nThe door opened.\n\nA manservant entered first to swing the door fully open to make room for the others. He stood with his hand to the latch, ready to close the door again once everyone had come in. Focused on his task, he was not yet aware the room was already occupied.\n\nStill in the middle of a lengthy diatribe, the man Mouse had heard from the hall entered next. He lumbered in like an ogre after a hunt, gruff and haggard. He was a sizeable man. He had the body of someone once strong and formidable but who had since been reshaped by age and languidness. The deep crags on his face were clearly the making of the persistent scowl he currently wore. He was dressed more as a military commander than nobleman at court, with a black leather jerkin and trousers and a fur-lined cloak draped over his shoulders, which he pulled off as soon as he crossed the threshold and tossed over the back of a chair. He was followed in by a few others--a young lord that Mouse recognized immediately, a middle-aged courtier, and a man with an armful of scrolls and ledgers and black-stained hands. The court scribe or perhaps a clerk of some kind. Three helmed guards marched into the room after that. Mouse pinned them as an honor guard by their elaborate attire and very shiny armor. They spread out across the room to stand at attention against the wall, hands clasped behind their backs.\n\nThe older man waited for the manservant to close the door, then scurry over to pull out a chair from a table. The man dropped down onto it as if the journey to the chamber had flagged the last of his energy and patience. He snapped his fingers. \"Let's get this over with, then,\" he grumbled.\n\nThe scribe unloaded his burdens on the table and began to arrange them.\n\nThe younger lord, lingering off to the side with crossed arms, was the first to notice the room was already occupied. He glanced in Garron's direction and, spotting him, sucked in air and went rigid. The young lord's attire was quite different from the last time Mouse had seen him, back in the tavern where he first hired Mouse.\n\n\"Brother?\" the lord whispered.\n\nThe hulking man at the table looked up at the younger lord, then turned his gaze to see where he was looking.\n\n\"Hello, Denn,\" Garron said quietly. He stepped away from the window and into the more direct light of the lantern on the table. He held his gaze squarely on the larger man. \"Hello, Father.\"\n\nThe others in the room started, and as realization set in who stood before them, the room seemed to drown in a thick silence. No one spoke, but all eyes turned to the man at the table. Mouse could see the resemblance in the big man. Garron had his father's square jawline and the same intense eyes.\n\nNo one had noticed Mouse yet. He leaned forward, enjoying the reaction, as if this were some well-produced theater.\n\nDuke Braddock lost what color he had in his already pallid face.\n\nOf course, Mouse now knew that his earlier assumption that it was Garron's younger brother Lord Dennithe that had plotted against him was wrong. Garron had told him as much, insisting that Dennithe was above suspicion, but Mouse still had his doubts. But now, when he laid his eyes on him, he knew Garron was correct. It had, after all, been Dennithe who had met him in that tavern and hired him.\n\nIt was Duke Braddock's reaction that betrayed him. Mouse could see the calculations whirling in the man's eyes. He desperately struggled to keep his face unreadable while at the same time tried to reckon how Garron had gotten here and evaluate how much his son actually knew. Any remaining doubts that Mouse might have had, they were vanquished.\n\nBut the duke attempted to recover. \"Garron, my son.\" He stood from the chair and forced a wide smile on his face that fell far short of anything convincing. \"You... you escaped?\" There was an awkward silence that followed as the duke struggled to mask his obvious distress and aggravation. \"Why, that's wonderful.\"\n\nGarron took a few steps closer but did not reply. His face was stiff and gave nothing away.\n\nDuke Braddock laughed nervously. \"You gave me quite a start, sneaking in like this.\" He put a meaty paw over his chest. \"You could have sent word, you know.\"\n\n\"I wanted it to be a surprise,\" Garron said in a quiet voice.\n\n\"Well, you certainly accomplished that.\" He turned his big false smile to the others in the room looking for help. Garron's brother looked fearful while the others were still recovering from their shock. \"What a happy occasion, indeed. However did you manage it, son? Last we all heard you were locked away in Delgan's tower.\"\n\n\"I had some help,\" Garron answered.\n\nMouse knew this was his cue. He pushed himself up from the chair with slow dramatic flourish. Again everyone started. The duke's face darkened as he glared at Mouse. He raked his eyes over him with disdain. Mouse held his gaze firmly on Braddock's, knowing such insolence would infuriate him.\n\n\"Guards, arrest that intruder!\"\n\nTwo of guardsmen sprang at the command. They drew weapons and converged on Mouse, who on instinct reached for his dagger. Garron quickly held out his palm to Mouse and made a sharp shake of his head. Stand down, was the meaning. Reluctantly, Mouse pulled his hand from the dagger and allowed the guards to wrench his arms behind him. Garron certainly knew that Mouse wasn't in any real or immediate danger; if Mouse wanted to break their hold on him, he could at any time. One of the guards removed the dagger from its sheath at Mouse's belt. The third guard put himself between Duke Braddock and Mouse.\n\nDuke Braddock's thin veil of happiness at seeing his son again faltered. He spun on Garron, his cheeks hot with sudden anger. He pounded a fist to the table. \"How dare you bring this... this filthy guttersnipe into my presence.\"\n\nGarron's voice was icy calm. \"You don't wish to thank him personally for single-handedly saving me from the executioner's block?\"\n\n\"I do not,\" Duke Braddock growled through his teeth. \"I should throw him in irons for violating the sanctity of my chambers. His very nature is an offense to me. Remove him,\" he barked to the guards. Immediately, Mouse was dragged toward the door.\n\n\"Stop!\" Garron's voice boomed out with sudden undeniable authority that rocked the room. Even the duke was taken aback by the force of it. His head jolted back with wide eyes, momentarily stunned that someone would dare speak to him so. Garron's voice dropped again, but it lost none of it potency. \"He remains.\"\n\nThe guards froze, confused and afraid to contradict either of them. Mouse leaned close to one of the guardsmen's ears. \"I'd listen to him,\" he whispered.\n\nDuke Braddock's face turned purple with rage. All pretense of a joyous reunion had crumbled away. Before he could collect himself enough to respond, Garron pushed harder.\n\n\"Father, let's not pretend further.\" He paused as he moved a bit closer. \"I'm curious, actually. What was I worth? Tell us all what you were willing to barter away of the duchy in order to have me eliminated.\"\n\nThe courtier gasped at the accusation. \"Lord Garron! How dare you make such a heinous charge against your duke and loving father.\"\n\nGarron raised a single brow. \"Will you deny it, Father?\"\n\nThe courtier looked to his sovereign and waited for him to voice a denial or show outrage. When none came, concern began to contort his features. \"My lord duke? Tell him this outrageous allegation is truly false.\"\n\n\"He cannot, Lord Heldan,\" replied Garron's brother.\n\nDuke Braddock swung his hot glare toward his youngest son. His face had turned so dark, Mouse wondered if something might rupture in the man's head. Then he looked to Lord Heldan. \"Get out.\"\n\nLord Heldan looked as if he'd been struck. \"But, my duke--\"\n\nDuke Braddock grabbed the front of the courtier's jerkin and gave him a harsh shove. \"Get. Out.\"\n\nHeldan made a hasty retreat from the chamber, taking one last look before he scrambled down the hall.\n\n\"You too, Denn,\" Braddock growled.\n\n\"No, Father. I believe I will remain.\"\n\nBraddock looked at his son with wide unabashed surprise. \"You dare say no to me?\"\n\n\"I do. I will hear your answer.\"\n\nBraddock could only stare back speechless.\n\n\"I can fill you in on the details if you like, Lord Dennithe,\" Mouse chimed in. His cheery tone seemed to grate against the tension in the chamber. \"Your father signed away the lands to the east, the region that is in ongoing dispute with Har Orentega, an area I believe you call the Crimarc.\"\n\n\"The Crimarc?\" Dennithe swung to face his father, his face crimson. It was a strange family resemblance, how their faces colored in the same manner and location. Mouse hoped to never see that on Garron's face. \"You gave away some of our most fertile lands? To the Orentegans?\"\n\n\"I did no such thing,\" the duke replied, but his voice held little conviction.\n\nDennithe was visibly incensed. \"Those people have sworn fealty to our family for generations.\"\n\n\"I said I didn't, and that's the end of it. I'll hear no more about it.\"\n\n\"Well, we have documents that say otherwise,\" Mouse said. \"You see, someone working for us inside Delgan's manor lifted some papers from his own chamber. Documents I'm sure you'll recognize. One of the more damning manuscripts is a drawn contract. With your name and your seal upon it.\"\n\nBraddock's face dropped a fraction, exposing his sudden concern. \"That's a bloody lie.\"\n\nMouse looked to the guard at his left. \"Be a love and pull out the parchment I have tucked inside my jerkin.\"\n\nWhen the guard moved to reach in, Braddock shouted, \"Hold!\" He stomped over to Mouse and roughly thrust his hand between the jerkin and tunic at Mouse's neckline. His prodding fingers found the parchment. He yanked it free and without looking at its contents tore into several pieces and tossed into the flames of the fireplace. \"You'll not frame me so easily.\"\n\nMouse smiled at him. \"Would I have been so foolish and na\u00efve as to bring the actual document here?\"\n\n\"They are in safe hands, Father,\" Garron added. \"I assure you.\"\n\nDennithe glared at Braddock. \"But your actions all but confirm the truth of what this man says.\"\n\n\"I'll not have you question me again, Denn.\" His voice was nearly a growl and thick with danger.\n\nBut Dennithe was unfazed by it. \"Or what, Father? You will have me removed and imprisoned in some far-flung dungeon as well? I knew something was amiss in all this. I knew somehow you were behind it. You were too... lackadaisical about the charges against him by half. That was why I hired this man to find him. And free him.\"\n\n\"You? You brought this rampallian into this?\"\n\n\"I did.\"\n\nDuke Braddock picked up a glass chalice from the table and threw it against the wall. It shattered with a wild crash, sending shards in all directions and leaving a dark stain on the paneling. \"You fool! Imbecile. You have always been the spineless mealworm that couldn't make the hard decisions.\"\n\nDennithe's face took on a steely hardness, but he did not reply.\n\n\"You stood to gain the entire duchy.\"\n\n\"At the cost of my brother? I would rather die, Father.\"\n\nStill unnaturally calm, Garron stepped to the opposite side of the table from the duke and rested his knuckles on the surface. \"Why, Father? Why did you do it?\"\n\nThe point of denying it had long passed. Garron's father glared back at him with contempt. \"You know why.\"\n\n\"I want to hear you say it.\"\n\nBraddock hesitated, then seemed to come to some internal decision. \"Fine. I did it because I would not have the likes of you taking the seat of the duchy when I'm gone. The family must come above all else. Even at the price of my one of my sons. The bloodline must be preserved.\"\n\n\"The bloodline?\" asked Dennithe. \"Garron, what is he talking about?\"\n\n\"He's referring to my particular attractions, Denn.\"\n\n\"That's right,\" spat the duke. \"He would rather get buggered by some man than lie with a woman.\"\n\nDenn looked at Garron, then back at his father. \"That's the reason?\"\n\n\"Damn right it is!\"\n\n\"But what does that matter, Father?\"\n\n\"What does that matter?\" Duke Braddock replied incredulously. \"Gods, you're callow. You sound more and more like your fucking mother every passing hour.\"\n\nDennithe took a step closer, fists clenched. \"Har Dionante has always been a province of tolerance for those--\"\n\n\"Not for the bloody successor!\" Braddock roared. \"One does not produce an heir with a man's cock in the ass. Our family has ruled Har Dionante for fifteen generations. I will not have this faggot end that. If Tiamond dies, he stands to inherit the duchy and all its holding.\"\n\n\"But to have him killed?\"\n\nGarron remained silent and only stared back at his father with cold eyes. Mouse could not imagine what he was thinking.\n\n\"No, Denn.\" Braddock's voice was suddenly calm, as if trying to soothe a child from a nightmare. \"Not killed. Just removed. They are trying to confuse you, fill your head with lies. Delgan was not to harm him, but just make him disappear.\"\n\n\"That is not the case, Lord Dennithe,\" Mouse said. \"Garron was to be beheaded. We have the writ of execution to prove that as well.\"\n\n\"Shut your fucking mouth, cretin, before I cut out our tongue!\" Braddock swung back to face Dennithe, immediately trying to curb his tone. \"Do not listen to them. They are trying to turn you against me. You are the heir now, Denn. Think of that. It is you who will take my place as duke when I'm gone.\"\n\n\"No, Father, I won't.\" Denn put a gentle hand on Braddock's arm. \"It's over. You cannot think to pursue this any further. If word of this scheme gets out to the public....\" Dennithe shook his head. \"Filicide is a clear violation of the king's law.\"\n\nDuke Braddock whipped his arm about to remove Denn's hand from his arm and roughly shoved Denn away from him. \"Should've known all along that you'd still be the fucking gutless worm that you've always been. Why can't you be like Tiamond? Either of you. He's a true heir. He's a fighter. But the gods are taking him away from me and leaving me with two worthless bags of shit.\" He leaned forward, knuckles planted firmly on the table and propping up his torso with two locked arms. He dropped his head and for a time was silent. \"No,\" he said softly, more to himself to the room. \"No, it is far from over.\" He pushed from the table and moved across the room at surprising speed for a man his size. In a heartbeat, he was standing directly in front of Mouse.\n\n\"I think you're bluffing, little man. I don't think those documents exist.\"\n\nMouse allowed the corner of his mouth lift in a knowing smirk but didn't answer.\n\n\"So, here's what the official word is going to be,\" Braddock continued as he took Mouse's dagger from the guard. He brought the point of it near Mouse's throat, leaned in, and spoke low. \"After I managed to successfully free my son from the dungeons of Har Orentega and bring him home, an assassin snuck into my chambers and brutally murdered him right here in front of me. But I was able to bring the criminal to justice and killed him myself by stabbing him in the heart with the very same weapon he used to take my son from me.\"\n\nThe guards tightened their grip on Mouse as Braddock adjusted the dagger to prepare for the strike.\n\nOut of the corner of his eye, Mouse saw Garron lunge, but the third guard moved to intervene, sword drawn.\n\nMouse was on his own. He had to act fast.\n\nHe pressed the small trigger on the side of his boot with the heel of the other and heard the small click as the blade sprang out from the back of the heel. He kicked backward. The blade impaled deep into the lower leg of one the guardsmen. The man cried out in sudden pain and flinched back, pulling his leg up. His grip on Mouse's arm loosened just enough.\n\nMouse made three lightning-fast movements. He twisted his body to break the injured man's weakened hold on him, kicked forward with his other boot, striking high on the shin of the duke's leg with his heel, and he punched underneath the helm of the second guard, striking the windpipe with his knuckles. The duke howled and stumbled backward. The second guard doubled over and collapsed to the ground while he gasped for breath.\n\nThe two guards were out of the fight, but Braddock was not that easily deterred. He limped once, then sprang forward as if suffering no injury at all--and surprisingly fast for a man of his size and age. He slashed Mouse's own blade at him in a powerful downward backswing.\n\nThe attack was intended to gut him, but Mouse arched his spine back, pulling in his stomach, and the tip whizzed past his midsection by a mere hand's width.\n\nBraddock frowned down at the dagger. \"Of course. A fucking bewitched blade.\" He threw the dagger to the ground in disgust as if it were diseased. \"A coward's weapon to tip the fight in your favor.\" As Mouse stepped backward away from him, Braddock kicked one of the two men on the floor in the shoulder. The guard flopped over, allowing Braddock access to the sword sheathed at his belt. Braddock pulled the short sword free with a rough tug and faced Mouse again.\n\nMouse didn't dare take his eyes from Braddock--the man was quick and a wily fighter, but from the corner of his vision, Mouse could see Garron driving hard against the remaining guard. He'd drawn his own sword, and the two were already in heated combat. The duke's personal guard was well trained and matched Garron's assault swing for swing.\n\nThe duke adjusted his grip on the weapon and stepped closer. Mouse noted a slight favoring of the leg. He had injured him after all. Braddock's face split with a sinister smile. He believed Mouse was unarmed and this would be a quick fight.\n\nMouse let him think that.\n\n\"Even if you kill us, Braddock, your reign is over. The documents will be sent to King Harus.\"\n\n\"That is Lord Duke to you, gutter rat.\" Braddock made a thrust, which Mouse avoided easily.\n\n\"You are undeserving of that honor. You are about as noble as a boll weevil.\"\n\nBraddock responded with another assault--a sweeping forward swing. This was less calculated, more impulsive, and fueled by anger. Mouse sprang left, and the blade smashed a vase on a side table.\n\n\"The truth will come out,\" Mouse continued to prod. \"And the king has no patience for filicide. Especially among you nobles.\"\n\n\"I'll not fall for your lies, trickster. You have no documents in your possession. Why else are you here? If you had them, you'd have already used them to expose me.\"\n\n\"Garron's idea. He's a sentimentalist. Thought he might actually get you to see reason. I knew better.\"\n\nBraddock feigned an attack, then sidestepped to the left. He grabbed a chalice off a table, splashing its red contents in all directions, and threw it. Mouse easily dodged the projectile, but Braddock was lunging for him again.\n\nMouse leapt backward again to stay out range from the sword but found he was nearly against the wall. Getting cornered by the big man would likely prove fatal. His eyes darted about for a possible exit.\n\nBelieving he had Mouse trapped, Braddock sneered as he pulled his arm up over his left shoulder for a broad backswing. As he leaned in to put his weight behind the swing, Mouse ducked and rolled. The sword whistled over Mouse's head as he tucked into the roll. As his shoulder and back hit the floor, Mouse's hand was already on the hilt of the dagger hidden in his boot. He wheeled back onto his feet, twisted, and drove the dagger into Braddock's hip.\n\nBraddock threw back his head and howled like a bear, more in anger than in pain. Pulled by the inertia of his power swing, he swung about with impossible speed for a man of his size and flung his free hand at Mouse. Mouse lunged back, removing the dagger from the duke's flesh, but he wasn't quite fast enough. The fat fingers of Braddock's left hand nicked Mouse's tunic.\n\nTightening his grasp on Mouse's front, Braddock tried to adjust his stance and regain his balance. But as soon as he put his weigh on the injured hip, Braddock's leg gave out. He collapsed sideways, dragging Mouse down with him. Mouse attempted to break the man's hold and push himself free, but his own boot heel skidded on the stone floor, now slick with wine. His leg flew up, and he toppled backward.\n\nHis ass hit the floor--hard--and a heartbeat later, Braddock threw his heavy torso on top of Mouse's legs. He was pinned.\n\nBraddock sneered up at him.\n\nGarron, seeing Mouse's predicament, cried out in sudden alarm. Panic and desperation drove him in a sudden offensive and put strength to his assault. He broke through the guard's defenses, but instead of running him through, he shouldered him aside and raced to Mouse's aid.\n\nBraddock kicked with his powerful legs and propelled his entire body forward. The force of the impact drove the air from Mouse's lungs. One massive hand crashed down on Mouse's upper arm and shoulder, holding it to the floor and preventing Mouse from lifting the dagger for an attack. The other arm, gripping the sword, lifted back. Braddock positioned the tip to be driven into Mouse's throat.\n\nGarron used both hands to grapple the edge of his father's armor from behind the neck and the armor's arm hole. He heaved Braddock's body backward. The duke struggled, but Garron was stronger than his father. Braddock was pulled onto his side and off Mouse. Denied his kill, he roared in fury. He rolled onto his back and thrust his sword in blind retaliation.\n\nThe sword pierced Garron in the gut.\n\n# Chapter 28\n\nTIME LURCHED to a halt, and a stunned silence flooded the room. No one spoke, or moved, or even breathed as Garron, eyes widened in pain and shock, stumbled backward and fell to the floor. The guards, Denn--they stared in horror and disbelief. Even Braddock was affected. For all his bold talk, having stabbed his own son left him ashen and silent. He dropped the sword and fell back onto his elbows.\n\nBut Mouse felt his entire world beneath him shake and crumble. His heart twisted in a brutal knot while the rest of his body went numb.\n\n\"No,\" he mouthed. \"No, no, no.\" It wasn't supposed to end like this. Vision blackened around the periphery. All he saw was Garron and the look of fear and disbelief on his face.\n\nDenn appeared at Garron's side, helping his brother to the floor.\n\nCold dread tightened around the agony of Mouse's chest. It can't end like this. Not again. It can't. He wouldn't allow it.\n\nThen something inside him clicked. It happened in an instant. Dread changed to rage.\n\nIt swept through him like a deluge, from his head on down and seemed to stain him on the inside, and he felt he might never feel right again. It wasn't the hot fury he'd felt in the past, but instead a deep coldness. A calm stillness. A lifelessness. As if he too were feeling death's icy grip and nothing else mattered. It stretched through his limbs. His mind crystallized with intent.\n\nNothing else mattered. There was no longer any consequence that could match the pain he now endured and would endure for all time hence.\n\nAll that mattered was that Braddock pay.\n\nIn one fluid motion, he rolled to his side and plunged the dagger into Braddock's neck just above the collarbone. The blade sank deep--all the way to the guard and hilt. Braddock cried out and made a lurch forward as if to sit up but fell backward instead. Mouse pulled the blade free as he collapsed. Blood sprayed from the wound in a pulsating jet. Braddock gasped for breath and tried to lift himself, but his body only managed to twitch piteously. He stared contemptuously at Mouse the entire time.\n\nIn moments it was over. The spasms slowed, and the time between gasps increased. In one final convulsion, the body slumped to the floor and was still. Braddock's eyes remained open but unseeing.\n\nMouse discarded the bloody dagger and scrambled to Garron's side. Denn was already there, kneeling and holding Garron's hand, speaking softly. Mouse all but collapsed on top of him. There was blood everywhere now. It leaked from under his doublet in alarming measure. It spread across the floor in black rivulets and soaked into the rugs.\n\n\"No... no... no...,\" Mouse whispered. Tears stung his eyes and cascaded down his cheeks, but he barely noticed. He was only aware of how colorless Garron's skin was and the amount of blood. Gods, there was already so much. \"Garron, stay with me.\"\n\nBut there was no response. Garron lay motionless on the floor. His eyes were closed and his breathing shallow. Mouse could see the life slipping from him.\n\n\"I will get help,\" Denn announced. He passed over Garron's hand to Mouse and sprang to his feet. Mouse took the hand in his own. It felt cold against his skin. Mouse looked haphazardly over his shoulder at Denn as he dashed from the room.\n\nHelp? Garron clung to life by a spider's silk. Was there help for him now? In moments the thread would break, and the last remnants of life would fade to nothingness.\n\n\"Damn you, Garron.\" He tightened his grip on Garron's still hand and put his face against his cheek, kissing him gently. He knew the guards were watching, but he didn't care.\n\nMouse felt his heart tear apart as if ravaged by mad dogs. How could he go on without him? It didn't seem possible. For the first time since he was a child, he had allowed himself to give his heart to another. He vowed never to love--but now once again he stood as witness as his love died. He would never recover, he knew. This would leave a hole so vast in his soul that he would likely never feel anything again other than grief. A grief so deep and dark, the light would forever be lost to him.\n\nIt all sounded so wonderful, their future life together. But Mouse should have known better. Someone like him just wasn't meant to live a quiet and happy life.\n\n\"You tried to save me,\" he whispered into his ear. \"You tried. But the gods aren't finished toying with me, I suppose.\"\n\nHe choked down the sob rising up in his throat. But still the tears came. He could not stop the steady flow of them. They filled his eyes and blurred his vision and spilled down over his face in heavy streams.\n\nHe sat back and used the heel of his hand to wipe his eyes and cheeks. No. He couldn't do this again. He didn't have the strength. He could not sit here and watch Garron's last moments slip away, watch as the final fragments of life left his body. His heart simply wouldn't be able to take it. The pain was too great, and the wound would never heal. It would torment him for all time.\n\nHe had to leave.\n\nAs soon as the thought entered his mind, he knew it was what he had to do. He had to leave before death took him. He simply could not be a witness to Garron's beautiful face without all the warmth and light that spilled from him in life. He wanted to remember him as he was. He leaned forward again and pressed his lips against Garron's. He held them there long, telling himself that this is what he would remember. Not the blood, not the stillness, not the paleness of his face or the cold of his hand. He would lock the memory of the kiss in his mind--the soft feel of his lips, his smell, and the look of peace on his face. That was what he would hold onto. \"Thank you for trying to save me,\" he told him. \"I love you, you know. I didn't want to, but I do. And I think as long as I have the will to breathe, I will never be able to love another.\"\n\nHe placed Garron's hand on his chest and stood up weakly. In a numb fog, he stumbled over to where Braddock had thrown his dagger and retrieved it from the floor. After he sheathed it at his hip, he had to take a moment to steady himself on the back of a chair.\n\nThen he did what he did best when faced with emotions he was not equipped to handle.\n\nHe ran away.\n\n# Chapter 29\n\nMOUSE FOLDED the parchment in half twice and dropped it inside of the small wooden box atop the pile of coin at the bottom. With pursed lips, he stared at the contents a moment, then closed the lid with a definitive thud and gave the box a shove. It slid across the counter's surface and stopped in front of the barkeep who stood opposite with him.\n\nLudro gave Mouse a long, hard stare before he lifted the box. \"They're not coming, Mouse.\"\n\nMouse held his face firm, expressionless. Of course, he feared Lu was right, but wasn't ready to admit it. Not yet. \"They'll come for it,\" he replied coolly. And when they did, he would be able to track them. He'd paid good coin for a mage to pen the note in tracer ink. He didn't want to consider that he'd thrown good money after bad.\n\n\"They are what, six days late now?\" Lu said. \"They are never late to pick up your coin. Ever. Somehow they figured out that you know.\"\n\nMouse allowed a frown. \"How?\"\n\n\"How the fuck would I know? I just sling mugs of ale here, friend. You know me. I keep my nose out of the shit that goes on around here--\"\n\n\"That's a load of horseshit if I ever heard it.\"\n\nLudro pounded his index finger on the counter. \"But it's as clear as piss after five ales. They know. Can't say I'm sad about it, either. If I see any of them around again it'll be too soon. Those rat fuckers gave me the duck skin.\"\n\n\"Well, put the fucking box down there anyway.\" Mouse lifted his mug and took a long draught.\n\nLudro shook his head but did as he was told. \"All right, all right. It'll be safe here at least until you give up on this nonsense.\"\n\nMouse almost spit out the ale he was about to swallow. \"Nonsense? You know how much coin they bilked me for? I want it back, Lu.\"\n\n\"What would you have done with it anyway? Bought land and turned yourself into a lord? Not fucking likely. Let it go, Mouse.\"\n\nSerafina leaned her hip against the bar next to Mouse. \"So damn glad to have your cheery mug back here again, Mouse.\"\n\n\"Bugger off, Sera.\"\n\nSera rolled her eyes. \"Isn't that your realm of expertise? Lu, how'd we get so lucky that he would choose this place over all the taverns in this city. Pour me two reds, would ya?\"\n\nLudro turned to the kegs, grabbing two clean mugs from the shelf. \"The gods must smile on us, I suppose.\" Holding both mugs by the handles, he pulled the lever on the keg with his free hand and allowed the ruddy liquid to spill inside.\n\nSerafina gave him a level look. \"Find yourself a client, darling,\" she said. \"And get your ass out of this dung heap--\"\n\n\"Hey, now,\" protested Ludro, depositing the two mugs on the counter. \"Watch what you're calling my bread and butter.\"\n\n\"--and out onto the street where you belong.\" She straightened up and picked up the mugs. \"I don't know what happened to you on that last job, but you've been one sour apple since you've slunk back. Not that you were ever a summer morning, mind you, but you're starting to even dull my sunny disposition.\"\n\nSerafina swished away to deliver the ale.\n\nLudro wiped down the spills on the counter with a rag. \"Dung heap,\" he grumbled. \"Why I keep that ungrateful wench around is beyond me.\" He looked up and caught Mouse's eye. \"But she's right, you know. About you. Not my tavern.\"\n\nMouse responded by taking another gulp of ale.\n\nAt his hip, Mouse began to feel the newly familiar sensation he described as a type of soundless hum. He reached down and placed his hand on the sheath that housed his dagger. His fingers tingled as they brushed the leather. The dagger seemed to swell with power and anticipation--like a faithful dog recognizing its master was approaching. \"Well, if it isn't the triumphant return of the Black Guard,\" he said.\n\nLudro looked up past Mouse's shoulder and a smile lit his face. \"Master Naevaro,\" he hailed with surprise. \"Welcome back, good sir!\"\n\nMouse waited for the footfalls to draw closer before he turned around on the stool, mug still in hand.\n\n\"Hello, Pa.\"\n\nNaevaro tried to look cross, but Mouse knew better. \"I never should have let you keep that weapon.\" He was larger than his son--Mouse was told he took more after his mother in that regard. Whereas Mouse was muscular but tightly framed, Naevaro had the broad shoulders and full chest of a man who'd labored hard his whole life. But he had Mouse's angular face and full black hair, only it was gray now at the temples, and his beard was more salt than pepper.\n\n\"I think it's taken more to me now, anyway.\"\n\n\"Then it's a traitor, and I should take it to the king and have it destroyed. How am I to ever make a surprise visit?\" The corner of Naevaro's mouth lifted a fraction.\n\n\"I hate surprises. Lu, pour my father a dark ale, if you please.\"\n\n\"Please, huh?\" Ludro exclaimed with a raised brow. \"Well, ain't that something. Master Naevaro, you are welcome here always. Your very presence improves our friend's disposition one hundred fold.\" Naevaro could no longer sustain his scowl. His mouth broke a grin and he chuckled.\n\nLudro grabbed a mug, flipped it in his hand, and positioned it under the keg spout. With a foamy crown spilling down the side, he handed the mug off to Naevaro. \"On me. If this reprobate isn't going to show you proper respect, I will. We all here are mighty grateful for the service you done for the throne.\"\n\nNaevaro winced. It didn't take Mouse long to learn that his father hated talking about his days back in the Black Guard. Hated even more that people knew that he was once counted among their ranks. In his mind, the incident that brought about the death of the old king and his queen was a failure that could never be forgiven. He couldn't absorb the fact that the general populace had now elevated the survivors of the massacre to legends. They'd saved the life of King Harus V, and in doing so, saved the entire realm. Their mysterious disappearance afterward had only added to the allure.\n\nUsually, Mouse could give a shit if his father liked the attention or not. But for some reason, he was feeling generous and decided to let him off easy.\n\n\"Lay off, Lu. He's doesn't know any noble women to introduce you to.\"\n\nNaevaro seemed to relax that no more would be said on the subject. He lifted the mug to Ludro in salute before he took a sip.\n\n\"What brings you back here so soon, Pa? Thought you hated the city.\"\n\n\"I was in the area. Can't a man make an effort to visit his son unannounced?\"\n\nMouse frowned. Something wasn't right.\n\nBut before he could press him on it, Ludro dried his hands on a towel and leaned his elbows on the bar. \"Funny you should happen in, Master Naevaro. We were just discussing you in a fashion. Perhaps you can talk some sense into your son.\"\n\nNaevaro wiped foam from his beard with the back of his gloved hand. \"As you can imagine, I've never had much luck on that score. But of what matter do you speak?\"\n\n\"He's intent on finding the hoods that swindled him for all that gold. I keep telling him no good will come of it. All that matters is you're safe, Master Naevaro, and there is no warrant to take your son away in irons.\"\n\n\"I saw to that long ago,\" Naevaro replied, nodding. He took another drink from the mug.\n\nMouse clenched his teeth. They were all getting a good snigger over this, he could tell. The renowned and infamous master thief suckered out of his own coin, played at his own game.\n\n\"As if I could somehow know you paid for the pardon yourself,\" Mouse grumbled. \"Or that you had that much coin in the first place.\" The coin remained a bit of a mystery, and Naevaro refused to elaborate on it. Mouse assumed it was a pension of sorts for his time in the Black Guard.\n\n\"If you had bothered to contact your old man once in the last twenty years,\" his father prodded teasingly, \"I could have told you myself.\"\n\nThey'd been over this. His father knew that Mouse did not want to put him at further risk by contacting him. Mouse wasn't about to rehash all that again in front of a gleeful audience that would take far too much pleasure in it.\n\nBut the irony of it all hadn't escaped him. He had taken the job in the first place to finally put an end to the payoffs and to protect his father once and for all. If he had known his father was in no danger, he might not have taken the job and then would never have met Garron. He wasn't sure if that was the better option or not.\n\nNaevaro's expression softened. \"Let it go, Renneus.\"\n\n\"Don't call me that,\" Mouse grumbled under his breath.\n\nLudro's back suddenly straightened. \"Renneus?\"\n\n\"Yes,\" Naevaro answered with a decisive nod. \"The name his loving mother gave him. A good name. I won't call him after some disgusting rodent. Ridiculous!\"\n\nLudro threw back his head and laughed heartily.\n\n\"Lu, don't even think to call me--\"\n\n\"Or what? You'll take your jolly business elsewhere? Don't threaten me with a good time! Master Naevaro, all the ale you can drink is on me tonight, sir. You have made my day, and half of tomorrow as well.\"\n\nWith a sigh, Mouse turned back to his father, who was chuckling away and winking at Ludro. \"Pa, why are you here? Is this another attempt to lure me back home with you?\"\n\n\"No, son, I know that tiny village is not for you.\"\n\n\"Then why? This isn't just some casual visit, Pa. Out with it.\"\n\nNaevaro smiled, but this time it was a different smile. Not sad, exactly, but something close. \"I brought you a client.\"\n\nMouse's head jerked back. That was unexpected. \"A client?\" He tilted his head in suspicion. \"You hate the work I do.\"\n\n\"True enough. But this client could change everything for you.\" Naevaro turned toward the tavern entrance. \"Perhaps then you can retire.\"\n\nThe man who stood at the doorway was dressed in a heavy cloak with the hood pulled up over his head. His face was shrouded in shadow. He stood very still until Naevaro nodded to him once; then he stepped forward into the room.\n\nMouse felt a chill race over his shoulders and his heart quicken. He rose from his stool. \"Father? What is going on?\"\n\nNaevaro said something to Ludro, but Mouse didn't hear the words. He was transfixed by the large figure stepping toward him. The man stopped about three paces away. Mouse thought he could hear him breathing. Then he pulled back the hood.\n\nMouse stumbled backward, knocking over his ale and the stool. He gripped the counter to stop him from falling to the ground.\n\n\"Master Ludro,\" Naevaro said softly, \"be so kind as to clear your tavern for a time.\"\n\nMouse didn't hear Lu's response but was distantly aware of activity around him. Everything around the edges of his vision had gone black. He was only aware of the man who stood in front of him.\n\n\"Hello, Mouse,\" Garron said.\n\nMouse's heart leapt into his throat. He tried to inhale, but the air only drew into his lungs in quick staccato breaths. \"Are... are you a specter? Have you come to haunt me for my crimes?\"\n\nGarron took a step forward. \"No, Mouse. I'm real. It's me.\"\n\n\"That... that's not possible.\"\n\nGarron held out his hand. \"Take my hand. Feel me. It's me.\"\n\nTears were suddenly flooding his vision. He clamped his eyes closed and leaned heavily on the bar. \"No, I saw you dying. You could not have lived. The wound was fatal.\"\n\n\"I did live. Denn went for help. Not two rooms over, mages were with my brother. They saved me, Mouse. They were able to save me.\"\n\nMouse's head spun. It was all too much. It was too much for him to accept. He was there. He saw the wounds. This couldn't be real. He wanted so desperately to believe, but his heart and his head would not allow it. His legs felt weak. He tried to take another step back, but something moved under his foot, and he started to fall.\n\nGarron closed the distance between them in a heartbeat, caught Mouse by the jerkin, and pulled him back onto his feet. Garron's powerful arms surrounded him and tightened. \"I'm here, Mouse. It's all right.\"\n\n\"Garron,\" Mouse said, choking on a sob. He pushed his face into Garron's chest. The feel of his arms around him, the smell of him, the warmth of his voice--it was all as he remembered it. This was Garron. This really was Garron. \"Oh gods! What have I done?\" Mouse looked up. \"Garron, I'm so sorry. I'm so sorry.\" The words spilled from him in a torrent of heartache. \"I should never have left you to die alone. Oh gods, I'm so sorry.\"\n\n\"You didn't know,\" he said softly in Mouse's ear. \"You didn't know.\"\n\nAnd he held Mouse against him, arms engulfing him. Mouse tightened his own arms around him and sobbed against his chest--something he had not done in a very long time. But the months of grief and guilt he'd locked away broke free all at once in an unstoppable surge. He was lost in it, swept away. He had no idea for how long.\n\nAt last, he pulled away. \"You came for me,\" he said, wiping his eyes against Garron's cloak. It wasn't a question but more of a revelation. \"You came back for me.\"\n\nGarron's face scrunched. \"Of course I did, you idiot.\" He rolled his eyes. \"Took me a while to find you. Denn couldn't remember either the town or the tavern where he first hired you. He never did pay attention to such details. It's probably best he's not the duke. But in our time together, you had given me enough information about your father I was able to locate him. And as I hoped, he knew where to find you.\"\n\nMouse looked up at him. \"Wait. You're the duke now!\" he exclaimed with sudden realization.\n\n\"I am,\" he replied with a sheepish grin. \"I haven't been officially recognized as of yet.\" A look of pain filled his eyes. \"My brother still clings to life but is expected to pass soon. So, until then I am considered his regent. But for all intents and purposes, I am now Duke Garron. I will officially be granted the scepter of Har Dionante after my brother's funeral and I am deemed fully recovered.\"\n\n\"And all that has happened, Garron? Has that not changed everything?\"\n\nGarron shrugged. \"With the help of my brother, neatly explained away. Our family was set upon by an assassin who remains at large. Your involvement and presence during the incident have been erased from any official records. You will not be held responsible for anything that transpired.\"\n\nMouse smiled weakly, not certain how to respond. He felt conflicted--torn--for it was he that killed Garron's father. Would Garron forever blame him for his death?\n\n\"Garron, what about Delgan?\"\n\nGarron's eyes narrowed. \"Fuck him. What does he have to do with anything?\"\n\n\"Other than he was involved in the whole plot to make you disappear?\" Mouse ticked off the points on his fingers. \"He lost out on a deal that would grant him the land his family coveted for generations. And lost his chamberlain too, who was the mastermind behind the power grab.\"\n\n\"Delgan doesn't worry me, Mouse.\"\n\n\"You aren't at all concerned he'll retaliate?\"\n\n\"Not likely. Right now, I'm told his hands are rather full in his own duchy. Seems his tight-fisted control has unraveled of late. Deals inexplicably falling through. Political alliances severed. Merchant barges even refusing to moor at the docks. All this has led to open rioting in the streets, I hear.\" Garron attempted an innocent lift of his shoulder, but there was a devious glint in his eye. \"All very strange. It's as if some surreptitious forces are at work there in Har Orentega.\"\n\nMouse's smile broadened. So, Taurin had taken his advice after all and put the Shadow Elite to work dismantling Delgan's power structure. Mouse straightened his spine and pushed back his shoulders. \"Well, then, I suppose congratulations are in order, Duke Garron,\" he replied formally. \"How should I address you henceforth?\"\n\nGarron placed his arms on Mouse's shoulders and ran them down the lengths of his arms. \"Well, I'm glad you asked.\" He held a mysterious grin that Mouse couldn't quite qualify. \"You see, as duke I am given license by the king to make laws as I see fit. So, I've made a few alterations to some of our official doctrine that has been in place for some time and was sorely in need of revision.\"\n\n\"Have you now?\" Mouse replied cautiously.\n\n\"I have indeed. As one of my first acts, I decreed that the stipulations and limitations on marriage have been lifted. The citizens of Har Dionante are now free to marry whomever they wish.\" His eye had a sudden twinkle as he gazed down at Mouse. \"And... to model that sentiment for the entire duchy, I wish to marry you.\"\n\nMouse could only stare back at Garron with his jaw hanging slack.\n\n\"Nothing to say?\" Garron prodded.\n\nMouse felt like the room was spinning. \"Garron, there's going to be pushback. People aren't go--\"\n\n\"I'm the duke, remember? I make the rules.\" He smirked and rolled his eyes. \"Mouse, Har Dionante has a long tradition of tolerance for those like us. I have only just made the convention official duchy policy.\"\n\n\"The people will see it as self-serving.\"\n\n\"They may. But I am prepared to face any backlash that may follow.\"\n\nMouse wasn't convinced. \"What of the king--?\"\n\n\"I have already spoken to him about it personally. Made a trip to the capital specially to address it with him face-to-face. He was rather taken aback, I'll admit, but he stands by his policy of letting the duchies stipulate their own laws. He will not interfere, Mouse. In fact, he plans to publicly offer us his blessing. Oh, and furthermore,\" he added, his grin widening, \"I have taken the liberty to clear this with your father, as well. He has graciously granted me your hand.\" Garron took one of Mouse's hands in his own and slid his other behind Mouse's waist. \"So, the decision is yours. What say you? Will you be Lord Mouse and rule by my side?\"\n\nMouse's knees felt weak, threatening to drop him again. He thought a moment--more to compose himself than to really consider the answer--and scratched his jaw to stall for time. Then he said. \"I will.\"\n\nEverything else he planned on saying was lost and forgotten, for Garron's lips were pressed against his after that for a very long time.\nMASON THOMAS began his writing journey at the age of thirteen when his personal hero, Isaac Asimov, took the time to respond to a letter he wrote him. He's been writing stories ever since. Today he is ecstatic and grateful that there is a place at the speculative table for stories with strong gay protagonists.\n\nMason, by all accounts, is still a nerdy teenager, although his hairline and waistline indicate otherwise. When his fingers are not pounding furiously at a keyboard, they can usually be found holding a video game controller, plucking away at an electric guitar, or shaking a twenty-sided die during a role-playing game. Mason will take any opportunity to play dress-up, whether through cosplay, Halloween, or a visit to a Renaissance Faire. He pays the bills by daring middle school students to actually like school and encouraging them to make a mess in his science classroom. He lives in Chicago with his endlessly patient husband, who has tolerated his geeky nonsense for eighteen years, and two unruly cats who graciously allow Mason and his husband to share the same space with them.\n\nFacebook: www.facebook.com\/MasonThomas999\n\nTwitter: @MasonThomas999\n\nE-mail: masonthomas999@gmail.com\n\nPublished by\n\nDREAMSPINNER PRESS\n\n5032 Capital Circle SW, Suite 2, PMB# 279, Tallahassee, FL 32305-7886 USA\n\nwww.dreamspinnerpress.com\n\nThis is a work of fiction. Names, characters, places, and incidents either are the product of author imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events, or locales is entirely coincidental.\n\nLord Mouse\n\n\u00a9 2015 Mason Thomas.\n\nCover Art\n\n\u00a9 2015 AngstyG.\n\nwww.angstyg.com\n\nCover content is for illustrative purposes only and any person depicted on the cover is a model.\n\nAll rights reserved. This book is licensed to the original purchaser only. Duplication or distribution via any means is illegal and a violation of international copyright law, subject to criminal prosecution and upon conviction, fines, and\/or imprisonment. Any eBook format cannot be legally loaned or given to others. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without the written permission of the Publisher, except where permitted by law. To request permission and all other inquiries, contact Dreamspinner Press, 5032 Capital Circle SW, Suite 2, PMB# 279, Tallahassee, FL 32305-7886, USA, or www.dreamspinnerpress.com.\n\nISBN: 978-1-62798-994-7\n\nDigital ISBN: 978-1-63216-000-3\n\nLibrary of Congress Control Number: 2015911626\n\nFirst Edition November 2015\n\nPrinted in the United States of America\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\norbitbooks.net\n\norbitshortfiction.com\n\n**Begin Reading**\n\nTable of Contents\n\nA Preview of _A Dance of Mirrors_\n\nOrbit Newsletters\n\nCopyright Page\n\nIn accordance with the U.S. Copyright Act of 1976, the scanning, uploading, and electronic sharing of any part of this book without the permission of the publisher is unlawful piracy and theft of the author's intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained by contacting the publisher at permissions@hbgusa.com. Thank you for your support of the author's rights.\n_For Dad, for insisting Haern had more to his story than I first told._\n\n# CHAPTER\n\n#\n\nHaern watched the ropes fly over the wall, heavy weights on their ends. They clacked against the stone, then settled on the street. The ropes looked like brown snakes in the pale moonlight, appropriately enough given the Serpent Guild controlled them.\n\nFor several minutes, nothing. Haern shifted under his well-worn cloaks, his exposed hand shivering in the cold while holding an empty bottle. He kept his hood low, and he bobbed his head as if sleeping. When the first Serpent entered the alley from the street, Haern spotted him with ease. The man looked young for such a task, but then two older men arrived, their hands and faces scarred from the brutal life they led. Deep green cloaks fluttered behind them as they rushed past the houses and to the wall where the ropes hung like vines. They tugged each rope twice, giving their signal. Then the older ones grabbed a rope while the younger looped them about a carved inset in the aged stone wall, then tied the weighted ends together.\n\n\"Quick and quiet,\" he heard one of the elder whisper to the younger. \"Don't let the crate make a sound when it lands, and the gods help you if you drop it.\"\n\nHaern let his head bob lower. The three were to his right, little more than twenty feet away. Already he knew their skill was laughable if they had not yet noticed his presence. His right eye peeked from under his hood, his neck twisting slightly to give him a better view. Another Serpent appeared from outside the city, climbing atop the wall and motioning down to the others. Their arm muscles bulging, the older two began pulling on the ropes. Meanwhile the younger steadily took in the slack so it wouldn't get in their way.\n\nHaern coughed as the crate reached the top of the wall. This time the younger heard, and he tensed as if expecting to be shot with an arrow.\n\n\"Someone's watching,\" he whispered to the others.\n\nHaern leaned back, the cloaks hiding his grin. About damn time. He let the bottle roll from his limp hand, the sound of glass on stone grating in the silence.\n\n\"Just a drunk,\" said one of them. \"Go chase him off.\"\n\nHaern heard the soft sound of a blade scraping against leather, most likely the young one's belt.\n\n\"Get out of here,\" said the Serpent.\n\nHaern let out a loud, obnoxious snore. A boot kicked his side, but it was weak, hesitant. He shuddered as if waking from a dream.\n\n\"Why... why you kick me?\" he asked, his hood still low. He had to time it just right, at the exact moment the crate touched ground.\n\n\"Beat it!\" hissed the young thief. \"Now, or I'll gut you!\"\n\nHaern looked up into his eyes. He knew shadows danced across his face, but his eyes... the man clearly saw his eyes. His dagger dipped in his hand, and he took a step back. Haern's drunken persona had vanished as if it had never been. No defeat, no inherent feeling of lowliness or shame. Only a calm stare that promised death. As the crate softly thumped to the ground, Haern stood, his intricate gray cloaks falling aside to reveal the two swords sheathed at his hips.\n\n\"Shit, it's him!\" the thief screamed, turning to run.\n\nHaern felt contempt ripple through him. Such poor training... did the guilds let anyone in now? He took the young man down, making sure no hit was lethal. He needed a message delivered.\n\n\"Who?\" asked one, turning at the cry.\n\nHaern cut his throat before he could draw his blade. The other yelped and stepped back. His dagger parried the first of Haern's stabs, but he had no concept of positioning. Haern smacked the dagger twice to the right, then slipped his left sword into his belly and twisted. As the thief bled out, Haern looked to the Serpent atop the wall.\n\n\"Care to join the fun?\" he asked, yanking out his blade and letting the blood drip to the street. \"I'm out of players.\"\n\nTwo daggers whirled down at him. He sidestepped one and smacked away the other. Hoping to provoke the man further, Haern kicked the crate. With no other option, the thief turned and fled back down the wall on the other side. Disappointed, Haern sheathed a sword and used the other to pry open the crate. With a loud creak the top came off, revealing three burlap sacks within. He dipped a hand in one, and it came out dripping with gold coins, each one clearly marked with the sigil of the Gemcroft family.\n\n_Interesting._\n\n\"Please,\" he heard the young thief beg. He bled from cuts on his arms and legs, most certainly painful, but nothing life-threatening. The worst Haern had done was hamstring him to prevent him from fleeing. \"Please, don't kill me. I can't, I can't...\"\n\nHaern slung all three bags over his shoulder. With his free hand he pressed the tip of his sword against the young man's throat.\n\n\"They'll want to know why you lived,\" he said.\n\nThe man had no response to that, only a pathetic sniffle. Haern shook his head. How far the Serpent Guild had fallen... but all the guilds had fallen since that bloody night five years ago. Thren Felhorn, the legend, had failed in his coup, bringing doom upon the underworld. Thren... his father...\n\n\"Tell them you have a message,\" Haern said. \"Tell them I'm watching.\"\n\n\"Who?\"\n\nIn response Haern took his sword and dipped it in the man's blood.\n\n\"They'll know who,\" he said before vanishing, leaving only a single eye drawn in the dirt as his message, blood for its ink, a sword its quill.\n\nHe didn't go far. He had to lug the bags to the rooftops one at a time, but once he was up high his urgency dwindled. The rooftops were his home, had been for years. Following the main road west, he reached the inner markets, still silent and empty. Plunking down the bags, he lay with his eyes closed and waited.\n\nHe woke to the sounds of trade. Hunger stirred in his belly, but he ignored it. Hunger, like loneliness and pain, had become a constant companion. He wouldn't call it friend, though.\n\n\"May you go to better hands,\" Haern said to the first sack of gold before stabbing its side. Coins spilled, and he hurled them like rain to the packed streets. Without pause he cut the second and third, flinging them to the suddenly ravenous crowd. They dove and fought as the gold rolled along, bouncing off bodies and plinking into various wooden stalls. Only a few bothered to look up, those who were lame or old and dared not fight the crowd.\n\n\"The Watcher!\" someone cried. \"The Watcher is here!\"\n\nThe cry put a smile on his lips as Haern fled south, having not kept a single coin.\n\nIt had taken five years, but at last Alyssa Gemcroft understood her late father's paranoia. The meal placed before her smelled delicious, spiced pork intermixed with baked apples, but her appetite remained dormant.\n\n\"I can have one of the servants taste it, if you'd like,\" said her closest family advisor, a man named Bertram who had loyally served her father. \"I'll even do so myself.\"\n\n\"No,\" she said, brushing errant strands of her red hair back and tucking them behind her left ear. \"That's not necessary. I can afford to skip a meal.\"\n\nBertram frowned, and she hated the way he looked at her\u2014like a doting grandfather, or a worried teacher. Just the night before, two servants had died eating their daily rations. Though she'd replaced much of the mansion's food, as well as executed those she thought responsible, the memory lingered in Alyssa's mind. The way the two had retched, their faces turning a horrific shade of purple...\n\nShe snapped her fingers, and the many waiting servants rushed to clear the trays away. Despite the rumble in her belly, she felt better with the food gone. At least now she could think without fear of convulsing to death because of some strange toxin. Bertram motioned to a chair beside her, and she gave him permission to sit.\n\n\"I know these are not peaceful times,\" he said, \"but we cannot allow fear to control our lives. That is a victory you know the thief guilds have longed for.\"\n\n\"We're approaching the fifth anniversary of the Bloody Kensgold,\" Alyssa said, referring to a gathering of the Trifect, the three wealthiest families of merchants, nobles, and power brokers in all of Dezrel. On that night Thren Felhorn had led an uprising of thief guilds against the Trifect, burning down one of their mansions and attempting to annihilate their leaders. He'd failed, and his guild had broken down to a fraction of its former size. On that night Alyssa had assumed control after the death of her father, victim to an arrow as they'd fought to protect their home.\n\n\"I know,\" Bertram said. \"Is that what distracts you so? Leon and Laurie have both agreed to delay another Kensgold until this dangerous business is over with.\"\n\n\"And when will that be?\" she asked as another servant arrived with a silver cup of wine. \"I hide here in my mansion, fearful of my food and scared of every shadow in my bedroom. We cannot defeat the guilds, Bertram. We've broken them, fractured many to pieces, but it's like smashing a puddle with a club. They all come back together, under new names, new leaders.\"\n\n\"The end is approaching,\" Bertram said. \"This is Thren's war, and he champions it with the last of his strength. But he is not so strong, not so young. His Spider Guild is far from the force it used to be. In time the other guilds will see reason and turn against him. Until then, we have only one choice left before us, and that is to endure.\"\n\nAlyssa closed her eyes and inhaled the scent of the wine. For a moment she wondered if it was poisoned, but she fought the paranoia down. She would not sacrifice such a simple pleasure. She couldn't give the rogues that much of a victory.\n\nStill, when she drank, it was a small sip.\n\n\"You told me much the same after the Kensgold,\" she said, setting down the cup. \"As you have every year for the past five. The mercenaries have bled us dry. Our mines to the north no longer produce the yields they were renowned for. The king is too frightened to help us. How long until we eat in rags, without coin for servants and wood for fires?\"\n\n\"We are on the defensive,\" Bertram said, accepting his own cup of wine. \"Such is our fate for being a large target. But the bloodshed has slowed, you know that as well as I. Be patient. Let us bleed them as they bleed us. The last thing we want is to inflame their passions while we still appear weak and leaderless.\"\n\nAlyssa felt anger flare in her chest, not only at the insult, but also at its damning familiarity.\n\n\"Leaderless?\" she asked. \"I have protected the Gemcroft name for five years of shadow wars. I've brokered trade agreements, organized mercenaries, bribed nobles, and done everything as well as my father ever did, yet we are _leaderless_? Why is that, Bertram?\"\n\nBertram endured the rant without a shred of emotion on his face, and that only infuriated Alyssa further. Again she felt like a schoolchild before her teacher, and part of her wondered if that was exactly how her advisor viewed her.\n\n\"I say this only because the rest of Dezrel believes it,\" he said when she was finished. \"You have no husband, and the only heir to the Gemcroft name is a bastard of unknown heritage.\"\n\n\"Don't talk about Nathaniel that way,\" she said, her voice turning cold. \"Don't you dare speak ill of my son.\"\n\nBertram raised his hands and spread his palms.\n\n\"I meant no offense, milady. Nathaniel is a good child, smart too. But a lady of your station should be partnered with someone equally influential. You've had many suitors; surely you've taken a liking to one of them?\"\n\nAlyssa took another sip of wine, her eyes glancing up at the shadowy corners of the dining hall.\n\n\"Leave me,\" she said. \"All of you. We'll speak of this another time.\"\n\nBertram stood, bowed, and followed the servants out.\n\n\"Come down, Zusa,\" she told the ceiling. \"You know you're always welcome at my table. There's no need for you to skulk and hide.\"\n\nClinging like a spider to the wall, Zusa smiled down at her. With deceptive ease she let go, falling headfirst toward the carpet. A deft twist of her arms, a tuck of her knees, and she landed gracefully on her feet, her long cloak billowing behind her. Instead of any normal outfit she wore long strips of cloth wrapped around her body, hiding every inch of skin. Except for above her neck, Alyssa was still pleased to see. Zusa had once belonged to a strict order of Karak, the dark god. Upon her willful exile Zusa had cast aside the cloth from her face, revealing her stunning looks and her beautiful black hair, which she kept cropped short around her neck. Two daggers hung from her belt, wickedly sharp.\n\n\"Let me be the one in the shadows,\" Zusa said, smiling. \"That way you are safe, for no assassin can hide beside me.\"\n\nAlyssa gestured for her friend to sit. Zusa refused, but Alyssa took no offense. It was just one of the skilled lady's many quirks. The woman had rescued her years before from an attempt by a former lover to take over her family line, and then helped protect her estate from Thren's plans. She owed her life to Zusa, so if Zusa wanted to stand instead of sit, she was more than welcome.\n\n\"Did you hear everything?\" Alyssa asked.\n\n\"Everything of worth. The old man is scared. He tries to be the rock in a storm, to survive by doing nothing until it passes.\"\n\n\"Sometimes a sound strategy.\"\n\nZusa smirked. \"This storm will not pass, not without action. Not with _his_ cowardly action. You know what Bertram wants. He wants you bedded and yoked to another man. Then your womanly passions may be safely ignored, and he can rule through your husband.\"\n\n\"Bertram has no desire for power.\"\n\nZusa lifted an eyebrow. \"Can you know for sure? He is old, but not dead.\"\n\nAlyssa sighed and drained her cup.\n\n\"What should I do?\" she asked. She felt tired, lost. She badly missed her son. She'd sent Nathaniel north to Felwood Castle, to be fostered by Lord John Gandrem. John was a good man, and a good friend of the family. More important, he was far away from Veldaren and its guilds of thieves. At least there Nathaniel was safe, and the training he received would help him later in life.\n\n\"Bertram's question... are there any you have taken a fancy to?\" Zusa asked.\n\nAlyssa shrugged.\n\n\"Mark Tullen was attractive, though his station is probably lower than Bertram would prefer. At least he was willing to talk to me instead of staring down my blouse. Also, that noble who runs our mines, Arthur something...\"\n\n\"Hadfield,\" Zusa said.\n\n\"That's right. He's pleasant enough, and not ugly... little distant, though. Guess that's just a product of being older.\"\n\n\"The older, the less likely to cavort with other women.\"\n\n\"He's more than welcome to,\" Alyssa said. She stood and turned away, trying to voice a silent fear she'd held on to for years, a fear that had strangled her relationships and kept her unmarried. \"But any child we have... that is who will become the Gemcroft heir. Too many will shove Nathaniel aside, deem him unfit, unworthy. I can't do that to him, Zusa. I can't deny him his right. He's my firstborn.\"\n\nShe felt Zusa's arms slip around her. Startled by the uncommon display of emotion, she accepted the hug.\n\n\"If your son is strong, he will claim what is his, no matter what the world tries,\" she said. \"Do not be afraid.\"\n\n\"Thank you,\" Alyssa said, pulling back and smiling. \"What would I do without you?\"\n\n\"May we never find out,\" Zusa said, bowing low.\n\nAlyssa waved her off, then retreated to her private chambers. She stared out the thick glass window, beyond her mansion's great walls, to the city of Veldaren. She found herself hating the city, hating every dark corner and crevice. Always it conspired against her, waiting with poison and dagger to...\n\nNo. She had to stop thinking like that. She had to stop letting the thief guilds control every aspect of her life through force and fear. So she sat at her desk, pulled out an inkwell and a piece of parchment, and paused. She'd sent Nathaniel away to protect him, to be fostered with a good family. Not so long ago her father had done the same with her, and she remembered her anger, her loneliness, and her feelings of betrayal. Gods help her, she'd even sent Nathaniel to the same person she'd been sent to. Once more she understood her father in a way she never had before. He'd hidden her because he loved her, not to get her out of the way as she'd once thought.\n\nStill, how angry she'd been when she returned...\n\nNo, she would not let history repeat itself. Her decision made, she dipped the quill in the ink and began writing.\n\n_My dear Lord Tullen_ , she began. _I have a request for you involving my son, Nathaniel..._\n\n# CHAPTER\n\n#\n\nBiggs kept watch at the door while the rest of the Hawk Guild cleared away the bodies.\n\n\"How many will be coming?\" asked one as he wrapped a body in its dark gray cloak.\n\n\"Depends,\" said Biggs.\n\n\"On what?\"\n\nBiggs rolled his eyes. \"On who is coming. If it's Veliana, only a handful. If it's Garrick, though... maybe twenty.\"\n\nThe other thief's face twitched at that. There were only ten of them weaving through the empty tables and quiet furnaces of the smith's workplace.\n\n\"So what do we do if it's him?\"\n\nBiggs turned, grabbed his shirt, and yanked him close.\n\n\"I didn't betray my guild and execute my friends so you can turn tail and run,\" he said. His knife was in his hand, and he pressed it against the shaking thief's belly. \"We hide, and we kill. You know how to do that, right?\"\n\nBiggs shoved him away and turned back to the door. They still had ten minutes until the expected rendezvous, but it wouldn't surprise him if either Garrick or Veliana showed up early. He'd worked the deal himself, the supposed purchasing of a large amount of illegal leaf from a nonexistent merchant from the south. The leaf was known as the Violet, and many nobles had taken a fancy to it down south in Angelport. Reselling it would bring in an absurd amount of money, meaning Biggs's deal would have been the best score for the ailing Ash Guild in over a year. And now his former guildmates lay dead, and once their leaders fell, Biggs knew he would take control. The Hawks' guildleader, Kadish Vel, had promised it.\n\n\"Into position,\" said the highest-ranking member of the Hawks present, a thin man named Kenny, whose nasal voice annoyed Biggs to no end. \"And for the love of the gods, keep it quiet.\"\n\nKenny slid beside Biggs and glanced up and down the dark streets.\n\n\"You sure they're coming?\" he asked.\n\n\"I know what I'm doing,\" Biggs said, glaring. \"A deal this big needs one of the two leaders to show. I hope it's Garrick, but it'll probably be Veliana. Not a bad thing, though. She's the scary one, could kill Garrick in a clean fight, and even easier in a dirty one. Not sure why she hasn't taken control yet, but I ain't giving her time to change her mind. If she's the one, you make sure you get her first.\"\n\nKenny lifted his small custom crossbow and winked.\n\n\"I've shot the nipple off a whore at twenty yards,\" he said.\n\n\"Bastard. What'd she do to you?\"\n\nKenny laughed. \"After that? Everything I asked.\"\n\nBiggs chuckled despite himself and their need for quiet.\n\n\"Remind me to never\u2014\"\n\nHe stopped, for he heard a scream from the other side of the building.\n\n\"What the fuck?\" asked Kenny, spinning about. \"The Ash send more scouts?\"\n\n\"Doubt it,\" said Biggs. \"Watch the door. I'll take care of this.\"\n\nHe tightened his grip on his dagger and ran through a maze of anvils and fire pits. While the full moon kept the streets bright, the smithy was dark and confusing, with only a fraction of the pale light streaming in through windows smeared with dirt and soot. Briggs heard a second scream, and when he turned toward it he smacked his knee into the edge of an anvil. He sucked his breath in through his teeth and tried to ignore the pain.\n\n\"What's going on?\" he asked, deciding stealth and silence were pointless with the Hawks howling bloody murder. He heard the sound of scuffling, then the rattling of weaponry. When he reached the place where the shop opened up into various displays of blades, hilts, and machinery, he stopped. There was another door in the back, and it was open. Moonlight shone through, falling upon bodies that lay crumpled about. At first Biggs thought them Ash guildmembers, but then he saw their brown cloaks and knew otherwise. Standing over them was a man.\n\n\"Who the fuck are you?\" asked Biggs.\n\nThe man looked up and smiled. His skin was dark, and his long hair darker. He wore the red robes of a wizard, though he held a dagger in his hand instead of a staff. Blood dripped from its blade. Covering his features was a mask of gray cloth pulled tightly across his face, with two large slits to allow sight. His brown eyes sparkled with amusement.\n\n\"I have no name,\" said the intruder with blood on his hands. \"But if Karak asks who sent you to his Abyss, tell him the Council's reaper, the outcast, or the dark man in red.\"\n\nHe was chuckling, and the sound raised the hairs on the back of Biggs's neck.\n\n\"You're crazy,\" he said. \"You know who you just killed? You'll have the fury of the Hawk Guild come down on you.\"\n\nHe was blustering, of course. He'd made a quick count of the bodies, and knew that only two remained alive in the smithy besides him and Kenny. Still, he couldn't act weak. It took all his concentration to keep the dagger from shaking in his hand.\n\nThe stranger made a flicking motion, flinging tiny globs of blood. Biggs swore as they flecked across his shirt and pants.\n\n\"They have to know I exist first,\" he said, snapping his fingers.\n\nThe blood caught fire, burning as if it were lamp oil. The heat came sudden and intense. Biggs fought an impulse to drop and roll. Magic fire would not be snuffed out so easily. As he felt his flesh burn he lunged, his dagger aiming for the stranger's chest. Before he could reach it the man fled, still laughing, still mocking. Instead of chasing, Biggs turned and ran for the other entrance.\n\n\"Kenny!\" he shouted. \"Get your ass back...\"\n\nIt seemed his own shadow tripped him. There was no other way to describe the strange sight and sensation. His head cracked against an anvil on the way down, and the sudden pain disoriented him beyond all measure. His stomach heaved, and he thought he would vomit. When he got to his feet he bolted, not knowing if he was going the right way. He didn't care. He had to move; he had to escape that terrible man who could burn blood with a snap of his fingers.\n\n\"Gods!\" cried Kenny as Biggs plowed right into him. Biggs clutched him to remain standing, and this time he did vomit. The mess splattered across Kenny's shoes, but to Kenny's credit he didn't bat an eye.\n\n\"Kill him,\" Biggs said, turning and pointing.\n\nThe stranger approached, his dagger still in hand.\n\n\"You have but a few left,\" he said as the blood upon his blade burned like embers fresh from a hearth. The light danced across his masked face, casting an orange haze over the gray. Biggs stepped back, doing his best to ignore the pain of his burns and the throbbing of his head.\n\n\"Think we need numbers to kill you?\" asked Kenny. \"All we need is me.\"\n\nHe lifted his crossbow and fired. The bolt bounced off the stranger's skin as if it were made of stone.\n\n\"A spellcaster?\" said Kenny. \"Damn it, Biggs, what shit did you get us into?\"\n\nThe man's grin spread, but he didn't laugh. It seemed the time for laughter was over. A gleam was in his eye, like that of a predator ready to pounce upon its prey. From either side a thief rushed from his hiding place. Kenny laughed, and Biggs realized it had been a trap for the intruder, prepared while he ran headlong like a fool to see the reason for the commotion. The two thieves stabbed, but their daggers struck only cloth. The stranger twisted and fell, avoiding both blows. When he hit the ground, his hands became a blur of strange motions. An explosion of fire blinded Biggs in the darkness, and then he heard the screams.\n\n\"Don't worry,\" Kenny said as Biggs took a step forward, doing his best to ignore the charred corpses before him. \"I keep this baby for special occasions like this.\"\n\nBiggs saw him pull a bolt from one of his many pockets, its tip glistening with silver. The stranger rolled along the floor until safely hidden behind a giant hearth. Kenny took a wide step around, trying to get a clear shot.\n\n\"What are they paying you for?\" Kenny asked. \"Wizards aren't supposed to get involved with normal people, and they sure as shit aren't supposed to hire out as assassins. What's your game?\"\n\n\"No game.\" Biggs kept close to Kenny, standing opposite his trigger arm and keeping his dagger ready in case the stranger charged. \"And I am no wizard.\"\n\n\"Bullshit you're not,\" Kenny said. \"What's this to you?\"\n\nAnother sidestep, slow and careful. Just as Kenny prepared his crossbow, so too could the stranger be preparing a spell.\n\n\"This is a game, an amusement, a moment of enjoyment, a time of laughter...\"\n\n\"Cut the shit. What's your name, and your price?\"\n\n\"Out of everything, you ask name and price?\" the stranger asked, suddenly stepping from the shadows and into their line of sight.\n\nThe bow fired. Biggs saw Kenny shift it to the side just the slightest amount, as if anticipating a dodge. None came. The stranger let the bolt hit him, and it pierced his shoulder just below his collarbone. He gasped at the pain, leaned forward, and then to Biggs's horror steadied himself and stood erect.\n\n\"Name? Price? I have neither.\"\n\n\"Reload!\" Biggs shouted, stepping between them and holding his dagger out. Fire danced in the stranger's eyes, then to his hands. Knowing he had to buy his ally time, Biggs let out a curse and dashed in, swinging for the man's neck. He never made it. The fire consumed his clothes, its heat beyond anything he'd ever felt. His legs refused to obey. As he collapsed he looked back, hoping Kenny would at least kill the bastard who'd done him in, but of course the rogue was long gone, running like the intelligent coward he was.\n\n\"You died for nothing,\" he heard the stranger say as the pain vanished amid a wave of darkness. His voice echoed in the chambers of Biggs's mind, slowly fading, slowly dying.\n\n\"Nothing...\"\n\nVeliana led them down the alleyway, her daggers sheathed at her hips. Her hands never strayed far from their handles. Something about this meeting felt worrisome. Perhaps it was the great amount of coin about to change hands. Ever since James Beren's death, things had gone poorly for the Ash Guild. James had been more than their leader: he'd been a sign of stability during the chaos and bloodshed. He'd died defying Thren Felhorn, and while in a nobler world that might have meant something, in theirs it had nearly brought about the guild's dissolution.\n\n\"Hurry,\" she whispered, ushering the rest of her guildmates along. They were at the very edge of their pitifully small territory. The last thing she needed was an ambush. Even if they fought it off, the delay might be enough to disrupt their sale. They were supposed to meet a wealthy, and eccentric, merchant from Angelport. All it'd take was a few minutes of fret and worry before he took his things and left.\n\nAssuming the men she'd sent in advance let the merchant leave.\n\nThey curved through the streets, which narrowed because of the stalls that sprang up along the sides. They were passing many tanneries and smithies, and their furnaces seemed to coat the stone street with a thin, ever-present layer of ash. Almost there. She stopped at an intersection with a main road leading toward the castle, looked about for patrols, and continued when she saw none. The sky was clear and bright, but still the chill seeped through her clothes and into her skin. She hated winter. It made her hurry, made her spend only four seconds checking a turn when she should have spent five. If there was anything she was certain of, it was that when she was buried, the ground would be cold and hard. Assuming she was buried at all. Given her life, even that was far from a given.\n\n\"We're here,\" she said. A quick set of instructions sent two around to the other side, and then the remaining six followed her through the main door. She let one of her guildmembers, Pryor, go first, just in case there was a trap. When she heard him gasp, she thought he'd found one, and she drew her daggers. But instead she heard her name.\n\n\"Vel?\"\n\nShe followed Pryor in and surveyed the area.\n\nA man waited for them. He sat atop a large crate, presumably the leaf for the deal. His body was bent over as if greatly burdened. He wore red robes stained with ash and blood. His skin was dark, and his hair darker. His shoulder was bleeding, but only a little. In one hand he held a dagger, in the other a long piece of gray cloth. At his feet was a bloody crossbow bolt. When he lifted his head, she stared into his brown eyes and saw a combination of fury and hopelessness that frightened her. He was handsome, but she felt no attraction. How could she, seeing a gaze like that?\n\nAll around him, burned to ash and bone, were bodies.\n\n\"What is going on?\" she asked, stunned by the sight.\n\n\"You were betrayed,\" said the strange man. \"One of your own helped kill the others so they might prepare an ambush.\"\n\n\"Who?\" Veliana asked.\n\nThe man slowly shook his head.\n\n\"This is my time to speak,\" he said. \"Ask your questions when I am done, for I need your ears listening and your mind open. I do not know who prepared your betrayal, but I am sure they are one of the dead at my feet. They are ash now, a fitting end given the name of your guild. Think now on what you see. I handled what seven men of yours could not. Whereas they died, betrayed, I came and killed the betrayers. I am alone, woman. Now ask yourself, what use might I be to you? Surely I am worth the seven that died.\"\n\n\"He's lying,\" said Pryor. \"He killed them all! Greg, Biggs, Brendan... he killed them!\"\n\nThe man shook his head, and his shoulders sagged further.\n\n\"Don't make a fool of yourself. Fools have a way of dying around me.\"\n\nVeliana cried out for Pryor to stop, but it was too late. He flung a dagger at the stranger, who avoided the hasty throw by a simple tilt of his head. His retaliation came swiftly as he hurled his own dagger. It pierced a lung as it embedded itself in Pryor's chest. The rest of the Ash Guild prepared to attack, but Veliana snapped at them to remain back.\n\n\"Who are you?\" she asked. \"What is your name?\"\n\nFor a moment his look shifted, and she saw an incredible sadness lurking behind those eyes. He lifted the gray cloth and let it unfurl from his hand, revealing the eyeholes.\n\n\"My real name is lost,\" he said. \"For it was banished from me by a power I cannot challenge. I have only the name they left me. I am Death, and this is my mask.\"\n\nAnd then he smiled, and she decided that just maybe she could find him attractive.\n\n\"You'd start as the lowest rank,\" she told him. \"You'd receive no special treatment, no favors, and this is only if Garrick accepts you in the first place. That acceptable?\"\n\nHe nodded. A quick word from her, and the Ash members hurried forward to grab the crate. \"Death\" stepped aside, and he laughed when he saw what they were to do.\n\n\"The leaf inside is fake,\" he said. \"Did you really think the Hawks would bring a real crate of Violet to an ambush?\"\n\nVeliana had to see for herself, and she held back glaring at the stranger. After they'd pried open the crate, she reached inside, grabbed a single leaf, and popped it into her mouth. She bit down, then spit it right back out.\n\n\"Damn it,\" she said, having received little more than the bland taste of a normal leaf. \"Gods damn it all. Garrick's going to be pissed.\"\n\nThe stranger shrugged. \"Then he'll be pissed. At least it confirms the ambush, yes?\"\n\nVeliana chewed her lip as she thought of what exactly she would tell Garrick Lowe, their new guildmaster. He wouldn't be too thrilled with the loss of men, and even more so about the faked merchandise. As for this Death and his mask...\n\nShe slipped closer. She wanted to understand him, his motives. He might be part of a trap, or a disaster she was blindly bringing into their guild. The blame would all fall on her.\n\n\"Don't betray me,\" she whispered to him as the rest hauled off the crate. \"I don't care how strong you think you are, I've fought stronger, I've survived better. You walk into this willingly, but the only way you walk out is dead. Do you understand me?\"\n\nHe tied the cloth over his face, and through the holes in the mask winked at her.\n\n\"The only way I leave will be as your guildmaster,\" he said.\n\nBecause of the mask she could not see if he smiled, or search his features for tells. In the end she decided it didn't matter.\n\n\"Come with me,\" she said. \"You're bound to make waves, so I think it best Garrick meet with you now... assuming he trusts you enough to be in the same room.\"\n\nHe moved, faster than she'd thought possible. His left hand wrapped around her waist. His right grabbed her wrist, and he pulled her close. She tried to draw her dagger, but he held her tight. Their eyes locked on one another's.\n\n\"You were brave enough to come this close,\" he said, his voice nearly a whisper. \"And even in my arms you do not tremble. I will remember that. Tell me, woman, what is _your_ name?\"\n\n\"Veliana,\" she said.\n\nHe let her go. She slapped him, and he straightened his mask.\n\n\"Deserved, and well worth it,\" he said. \"Lead on, Veliana. I wish to see your guildmaster.\"\n\nThe Ash Guild had moved many times because of their weak status and constant war with the rest of the guilds. Only recently had they made peace with most, though the Hawk Guild still preyed heavily upon them. If not for this... Death... Veliana knew there was a good chance she'd be a corpse.\n\nAssuming he wasn't part of a trap.\n\nTheir current base was in the confines of a small merchant guild, one desperate enough for coin that they'd accepted Garrick's bribes. As lairs went it wasn't the most secretive, but at least it was warm in the winter months, and moderately well furnished. Veliana led her men through a side door. Four steps down they stopped at a basement door with a small lamp burning at either side. She frowned at the lack of guards. No doubt they were on the inside. Garrick liked having his protectors there with him at all times, even if it wasn't safer. They should have been out in the cold, keeping watch so they could lock and bar the door should they notice trouble coming.\n\nBut of course the door was locked and barred anyway when she tried to open it. Rolling her eyes, she knocked twice, then once. She heard the scraping of metal, and then a slit opened to reveal a bloodshot pair of eyes.\n\n\"Say the pass,\" said the guard.\n\n\"Veliana. Now open the damn door.\"\n\nThere was a password, of course. Three, even, in case she needed to alert those inside to a hidden threat. But she was in no mood, and she knew the guard on the other side was too spineless to refuse her entrance. The slit closed, and as they heard a loud thumping, Death chuckled behind her.\n\n\"Your professionalism is astounding,\" he said. \"I know I came with few expectations, but I still feel them failing to be met.\"\n\n\"Quiet,\" she said. \"And stay here. I'll need to introduce you to Garrick first.\"\n\nShe paused and gave him a glance. The mask hid his face, but she couldn't fight off the feeling he was smirking at her.\n\n\"Just how should I introduce you, anyway?\" she asked.\n\n\"I told you, I have no name.\"\n\n\"That makes for a poor introduction. Should I call you Death? It's a little over the top, but I've heard worse.\"\n\n\"Death might be too great a mantle for me to wear,\" said the man. \"But I can bear no name for the curse given me. All I have is my mask. Perhaps you can call me that.\"\n\nThe door opened and she stepped inside. A guard stood at either side, dagger drawn. The room was well lit with many lanterns. At one end were tables covered with maps and documents, and a locked chest underneath for guild funds. At the other were blankets, pillows, and illegal means of comfort. Amid the meager luxury sat Garrick, his eyes glazed from the substance he smoked through a short pipe. Several other men lay scattered about him, their senses just as dull from the smoke and liquor.\n\n\"Veliana!\" Garrick said, standing. \"Did the trade go through as...\"\n\nHe stopped as Veliana's guest shoved his way inside, so fast that he was beside her before the guards could react. He made no threatening motion, only stayed at her side. With an elaborate bow he greeted the guildmaster.\n\n\"Mighty Garrick, how the shadows tremble when I mention your name,\" he said, and Veliana felt anger burn inside her at the obvious sarcasm. Garrick, however, seemed oblivious to it. Instead he appeared worried by the newcomer's strange attire and sudden entrance. He stepped back and ran a hand through his long brown hair, a sign Veliana knew meant he was nervous.\n\n\"And who are you?\" he asked. \"A friend of Veliana's?\"\n\n\"This is... Death's Mask,\" she said. \"He helped us tonight, perhaps saved many lives. We've been betrayed, Garrick. When we...\"\n\n\"Do you still have the leaf?\" Garrick interrupted.\n\n\"I... yes, we do.\"\n\n\"Good, good,\" said Garrick. He sat back down on the cushions, drew his dagger, and held it while he listened. \"Now what is this betrayal you speak of? And tell me again\"\u2014he made a sound like a cross between a laugh and a cough\u2014\"who this Death... Deathmask is?\"\n\n\"One of your men betrayed you, killed the rest, and hoped to have the Hawks destroy your guild as you came to make the trade,\" said Deathmask, accepting the name without pause. \"I killed them as a way of showing you my worth. I wished to join your Ash Guild, and Veliana has accepted me.\"\n\nVeliana opened her mouth to correct him, then shut it. It was pointless to argue now, and Garrick's eyes flared at what he'd heard. He set his pipe aside and gently touched the tip of his dagger.\n\n\"And how did you know this ambush was to take place?\" he asked.\n\nDeathmask smiled but did not answer.\n\n\"No matter,\" Garrick said. \"I take it our betrayer was killed?\"\n\n\"Painfully,\" said Deathmask.\n\n\"Good. The more pressing question, then, is why the Hawks are so eager to crush us. I must think about this.\"\n\n\"Think?\" asked Veliana. \"We need to counter, and quickly, before they know their ambush failed. Surely there's a few of theirs still scouting our streets. If we can mark our borders with their blood, we can send a message.\"\n\n\"We will do no such thing,\" Garrick said. He winced as he pierced his skin with the dagger's tip, but instead of cleaning it, he watched the blood trickle down the blade. \"I will handle this in my own manner. Kadish Vel is no real threat to us.\"\n\n\"With all due respect,\" said Deathmask, \"he'd have killed either of you today if it weren't for me.\"\n\nThe room turned silent, and the rest of the thieves lounging about watched as if awaiting an execution.\n\n\"Is that so?\" asked Garrick. Veliana tensed, wondering what he would do. \"Then it is a good thing we have you now, isn't it? Patrick, take him upstairs and get him fitted for a cloak. Wouldn't want him showing the wrong colors, would we?\"\n\nDeathmask bowed, shot Veliana a wink, and then followed his escort out of the basement. Garrick stood and looked to the others.\n\n\"Out,\" he told them. \"You've smoked enough of my wealth. Get out!\"\n\nThey all jumped to leave, all but Veliana. She could see the way he stared at her, a wordless command to remain. As the last shut the door, Garrick strode over and grabbed her by the throat.\n\n\"Have you lost your damn mind?\" he asked.\n\nShe kept her breathing calm as his fingers tightened. He wouldn't leave a bruise, not yet, but if he did...\n\n\"And have you lost yours?\" she asked. He raised an eyebrow. The veins in his eyes pulsed, their edges yellow. In answer to his question, she drew one of her daggers and pressed its edge against his wrists.\n\n\"Tighter,\" she said. \"I dare you.\"\n\nHe let her go and stepped back. Blood dripped from his finger, and he glared at the stains on the floor.\n\n\"I am your guildmaster,\" he said, as if that meant something.\n\n\"I could replace you in a heartbeat.\"\n\n\"But they'll never follow you,\" Garrick said, gesturing toward the door. \"Them out there... they're savages. They're pigs, and you're damaged beauty. If they thought the only thing standing between them and ruling this guild was... was a _woman_ , they'd strip you naked and take turns.\"\n\n\"They'd die if they tried,\" Veliana said. She knew Garrick thought himself worth far more than he really was, but this was the most brazen he'd ever been. Something had changed, given him confidence... but what?\n\n\"Some, yes,\" said Garrick, and a gleam shone in his drug-maddened eyes. \"But not all. You need me, Vel. They listen to me. They trust me. I kept this guild together after James's death. I kept our dealings going. I kept our coffers from emptying and our territory from becoming nothing more than a single penniless street.\"\n\n\"That was _me_ ,\" Veliana shouted, not caring if others might hear from behind the door. \"I'm the one who bloodied her hands. I'm the one who gave them stability.\"\n\n\"But do they know that?\" he asked. He stepped closer to her, a smile growing on his face. It was the drugs, she thought. It had to be the drugs. Ignoring the daggers, he gently ran a hand along the wicked scar across her beautiful face, cutting from forehead to chin, discoloring her right eye from a dazzling violet to a red orb of blood.\n\n\"They won't follow you,\" he said again. \"You're dangerous, and they respect that, but they won't submit to it. They listen now only because the specter of my authority hangs over you. Just as James's hovered over you when he was alive. You need me, just as I need you. More, even. Never forget that.\"\n\nShe bit her tongue and fought down a thousand fantasies of plunging her daggers deep into his throat. Garrick walked back to his cushion, retrieved his pipe, and began the laborious process of filling it anew.\n\n\"I don't care how powerful that Deathmask freak is,\" he said. \"I want him killed by tomorrow night, no matter how you do it. He's clearly trouble, and in someone's pocket. Cut his throat before he can accomplish whatever task he was sent here for. Deathmask? What a stupid name.\"\n\n\"If you say so,\" she said, nodding her head. \"I'll be in my quarters. I trust you'll handle the matter of the Hawk Guild in an appropriate manner?\"\n\nGarrick smiled as she headed for the door.\n\n\"Dear Veliana, there are a thousand promises and lies between us all. You aren't aware of half of them. Trust me. We'll be fine.\"\n\nShe left without giving him the dignity of a response.\n\n# CHAPTER\n\n#\n\nHaern slept beside the shop of a baker he'd befriended. Besides gaining an occasional scrap of bread, he could take in the warm smells while he slept. He kept himself wrapped in blankets, never bothering to hide his face. His blond hair was matted to the sides of his head, much of his skin covered with dirt. He'd always been a clean, meticulous child. More than anything since his self-imposed exile from the Spider Guild, the filth bothered him most. He knew there were ways he could wash up, obtain cleaner clothes, but it'd never work. What homeless, coinless man lived on the side of the street, yet kept a clean face and hands? A rich man from the northern parts of Veldaren might not notice, but those who'd grown up in the lower portions of the city could sense those things as surely as Haern could smell the bread that baked in the shop every morning.\n\nHe had a small bowl before him, not expecting any alms but feeling his heart warm when he did receive them. It was mainly there for looks. Every guild in the city of Veldaren wanted him dead, and he wouldn't draw needless attention to himself by neglecting the minor details.\n\nJust before nightfall he stirred. The baker had gone home for the day, so he picked the familiar lock and slipped inside. He stole two slices of bread, dumped his bowl's coins across the counter to pay for the meal, and then left. He ate as he walked south along the main road, turning off after half a mile and heading directly into the territory of the Serpent Guild. He added a limp and ran through his persona for dealing with the Serpents. He let his lower jaw hang a little and muttered a few random words, practicing his lisp. His name was Berg. He was often drunk. Like all his personae, Berg worked odd jobs, whatever paid him coin, and that gave him an excuse to know things he shouldn't know.\n\nLike how the Watcher had intercepted a shipment of gold from the Serpent Guild bearing the Gemcroft sigil.\n\nHis contact was a one-eyed ruffian originally from the far west nation of Mordan. He leaned beside the entrance of an inn, smoking a long pipe. His name was Mensk.\n\n\"What you want, Berg?\" asked Mensk. He looked him over with his one eye, and he didn't hide his shudder at Haern's stink.\n\n\"I overheard something,\" Haern said with a lisp. \"Something worth at least a silver.\"\n\nMensk's eyes narrowed.\n\n\"Nothing you could have heard is worth that much. Five coppers if I decide it's useful, one otherwise, and none if I've heard it already.\"\n\n\"Six coppers,\" Haern said, knowing he wouldn't get it but also knowing his persona would at least make the attempt.\n\n\"Five,\" Mensk said, frowning. \"Now spit it out before it gets lower.\"\n\n\"I was drinking by the east wall last night, and I saw something, something strange. There were some of your guild, you Serpents, by the wall. They were lifting something over... a crate, valuable, yes?\"\n\nBy now he had Mensk's attention. The thief had drawn his dagger and held it behind his back. So on edge... clearly the shipment's secrecy was beyond important.\n\n\"It might be,\" Mensk said. \"That all you saw?\"\n\nHaern shook his head.\n\n\"No, oh no. I wouldn't have come here for just that. Why tell you your own business? No, what I saw was the Watcher. He killed them, all but a few!\"\n\n\"We know who killed them,\" Mensk said. He shifted a little, placing his right foot a few inches back. He was preparing for a stab. Haern kept his breathing steady, not letting a shred of fear show in his eyes. Oblivious... he had to act oblivious...\n\n\"But did you know the one who killed them was a Hawk?\" he asked.\n\nThere. He saw the momentary pause in Mensk's eyes and knew he had him.\n\n\"What?\" he asked.\n\n\"Oh no,\" Haern said, stepping out of stabbing range. He exaggerated his lisp further to pretend excitement. \"This gets me seven coppers, or I go to someone else who'll pay.\"\n\n\"Here's six,\" Mensk said, tossing them to the dirt. \"Now tell me what you know!\"\n\nHaern flung himself to the ground and began scooping up the coins, all the while rambling.\n\n\"I saw him skulking along the rooftop. I wanted to shout warning, I did, but it was too fast, you know? One moment I see him, and I'm wondering what some damn fool is doing on the roof, and then he's whirling and cutting. Gods, never seen so much blood.\"\n\n\"The Hawks?\" Mensk asked, prodding him along.\n\n\"Oh, well, I saw him pull out a pendant just before he attacked. Now it was dark and far away, but I swear the moon was bright enough. It had a feather on it, just one feather, plus I think an eye. That's their new symbol, right, symbol for the Hawks? Thanks for the coppers, Mensk!\"\n\nHe stood and stepped back. With the added distance, he knew Mensk wouldn't come after him, not when he could just turn and run. A bit of disappointment came over the thief's face, which pleased him immensely. He'd told such lies before to the various guilds. With the distance and the dark, his proof was meager, but that was the point. A thousand tiny lies and misdirections would add up to a far greater proof than a single disprovable accusation. He wanted the guilds at each other's throats, always convinced the Watcher was one of their own. Mensk would pass along what he'd heard, storing it away with all the other tales Haern had whispered over the years.\n\n\"Get out of here,\" Mensk said. \"And I should only pay you three for something that worthless.\"\n\n\"If I'm right, you know I should get a hundred silver,\" Haern said as he limped along. \"But I like you Serpents. Always treated me fair, helped rebind roofs to put over my head. I sure hope that Watcher didn't disrupt anything important. Would hate for him to put you in a bad bind...\"\n\n\"I said get out,\" Mensk said, revealing his dagger. \"Those shipments'll continue no matter what that bastard does. Now get your ass out of my sight before I decide to take back my coin.\"\n\nHaern fled, and once he reached the main merchant road he abandoned the limp and started trudging north. In the light of the stars he pondered over what he'd learned. When he sold information he often managed to sneak out with a little bit more in return. That shipment of gold hadn't been a one-time deal. Obviously the shipments were important, and the Serpent Guild wanted them to continue. Originally he'd thought the Serpents just lucky robbers of the Gemcroft mines, but now...\n\nNow he was intrigued.\n\nHaern didn't always give away the gold he stole. Sometimes he kept a little, with the promise to never spend it on himself but instead his personal war against the guilds. There were many men who responded to gold far better than threats, and it was to one of them Haern hurried next. He was in a quiet section of Veldaren, far to the north and away from the poorer districts and the strongholds of the thief guilds. Here there were rows of homes, most fenced, and Haern climbed over one fence and softly landed on the grass beyond.\n\nTwo dogs immediately rushed from around the side of the house, large hounds with black fur and impressive jaws. Haern knelt, not panicking, instead offering his hands. One of the two licked away at it, while the other nearly bowled him over leaping against him for attention. Haern smiled as he petted both the animals. The house's owner, a man named Dashel, had bought the two animals for protection. When Haern decided to make the man one of his contacts, he'd spent weeks visiting the home while the owner was gone, bringing food and treats for the dogs, petting them, letting them get used to his smell and appreciate his presence. Now they hardly barked upon his arrival, and if they did it was out of excitement.\n\n\"Easy now,\" Haern whispered as he finally pushed them back. \"I still have work to do.\"\n\nOne whined, the other barking a single time. Haern shot them a wink.\n\n\"I'll bring a few bones next time, I promise.\"\n\nDashel's bedroom was on the second floor, and in the moonlight Haern climbed up to the rooftop, then hung upside down before the elegant window. It was unlocked. Pushing it open, Haern then rotated about and landed on the sill, crouching down. His shadow splayed out across the fine bed upon which slept Dashel and his wife. Taking a rock out of his pocket, he flung it, striking Dashel across the nose.\n\nThe man let out a snort, then groggily looked about. Upon seeing Haern lurking in his window, he froze. Haern tossed him something else, this time a coin that glinted yellow when it caught the light. Dashel pushed the sheets aside, caught the coin, and then staggered out of his bed.\n\n\"Outside,\" the older man hissed before shutting the window in Haern's face. Laughing, Haern dropped back to the two waiting dogs and accepted another round of licks.\n\nA few minutes later Dashel stepped out from his door, a thick white robe wrapped around his rotund form. His hair and beard were gray, his face wrinkled. Haern kept his hood pulled low and his face turned away from the moonlight, hiding his own features.\n\n\"Stupid dogs,\" Dashel said, glaring at them. \"Man who sold them to me swore they were as vicious as a wounded orc.\"\n\n\"They have smart noses,\" Haern said. \"They know I'm not here to hurt you.\"\n\nDashel rolled his eyes and waved a hand.\n\n\"What is it, Watcher? It's cold out here, and I'd much rather be sleeping in my warm bed.\"\n\n\"The Serpent Guild has a new source of gold,\" Haern said. \"What do you know about it?\"\n\nThe old man frowned at him, then tightened his robe. Dashel was the master of the mint for the kingdom of Neldar, and because of this he had access to resources many people prized. For years he'd worked as a fence, accepting from the various guilds any gold that might be marked or distinguishable and replacing it, for a price, with fresh coin from the mint. Haern had briefly thought of taking him out, but decided the information he might learn from the man was more valuable.\n\n\"It's been going on for months,\" Dashel said. \"Though I suspect it is new to you?\"\n\n\"I'd have been here sooner if I discovered it sooner.\"\n\n\"Of course you would have.\" Dashel shook his head. \"They've come to me regularly, with amounts high enough I've considered several times declining for fear of having my... side activities discovered. As for the source, all I can tell you is that they always have the same mint and mark.\"\n\n\"Which one?\"\n\n\"The Gemcroft mines up north. My guess is from Tyneham.\"\n\nHaern frowned. Massive shipments of gold coming in from the Gemcroft mines, delivered in secret to the Serpent Guild? What did it mean?\n\n\"Have they discovered a way to rob Alyssa's caravans?\" Haern wondered aloud.\n\n\"Been happening too long for that to be the case,\" Dashel said. \"I don't know Alyssa well, but her reputation alone suggests if she knew she was being robbed, she'd have swarmed the northern roads with guards until it stopped. So whatever's happening, it's happening underneath her nose.\"\n\nDashel gestured to his door.\n\n\"Can I go to bed now? This is hardly worth one lousy coin.\"\n\nHaern pulled out two more gold coins, tucked one into the collar of each dog, and bid Dashel farewell. Before the man was even inside, Haern had already vaulted over the fence and headed south. His mind raced over the possibilities, and every time he came up empty. Everything seemed too large, too far beyond the reach of the Serpent Guild's leader, William Ket. If he wanted to find out more, or put a stop to it, he'd most likely need to leave the comfort and familiarity of his city for the wilds of the north.\n\nPushing such thoughts aside, Haern focused on his original task: spreading misinformation. He'd already claimed the Watcher was working for the Hawks, so now it was time to inform the Hawks of the Serpents' shipments, adding one more layer to the deception. Perhaps the rival guild might even aid Haern in putting an end to the sudden influx of gold. Wending his way into the Hawks' territory, he put a scowl over his face and adopted a new persona, that of just another hardworking peasant hoping to eke out enough copper for a few extra drinks. He went to their headquarters, a tavern where their leader, Kadish Vel, no doubt played cards in a private room in the back. Again his contact waited by the door, a big man whose name he didn't know, only his nickname: Fists. In working his way into the man's trust, Haern had had to endure a few beatings that showed exactly where that nickname came from.\n\n\"It's late for someone weak as you,\" said Fists as Haern approached.\n\n\"Never too late for a good drink,\" he said.\n\nFists smirked at that. \"Good drink? You won't find that here. What you have to tell me?\"\n\n\"I saw a shipment coming in over the wall, and I think the Serpents were...\"\n\nHe sensed the attack coming long before Fists did. Haern dove to the side as arrows thudded into the tavern. He heard Fists groan, and a glance back showed two arrows in his belly. Haern raced around the building, out of reach of the tavern's torchlight. As he turned the corner he slammed into a cloaked man with daggers at the ready. Haern rolled along, separating their bodies. He leaped to his feet, his swords in hand. With the greater reach, he had the advantage, and his opponent knew it. The man lunged in, hoping for a strike before Haern could prepare.\n\nBut the lunge was too slow. Haern twisted so the dagger brushed his side without drawing blood, then slashed around with his swords. One cut deep into his opponent's extended arm, the second kept the other dagger out of position. As the man cried out in pain, Haern pulled his swords back and then stabbed. The man tried to block, but he was unbalanced, his arms poorly placed. Haern yanked out his blades and kicked the body. He frowned at the color of the cloak. He'd first thought the Serpents had followed him for knowing about the gold, but instead this man wore the gray of the Spider Guild.\n\nHe heard the sound of a bowstring being drawn tight, and he dove on instinct. The arrow plinked the stone beside him, poking a hole in his cloaks but doing no damage. His attacker was on the roof, readying another arrow. Haern spun, flinging his cloaks into a confusing display, and then leaped in the opposite direction. Again the arrow missed. By now he'd raced around the corner, cursing his bad luck. Why had the Spider Guild come now? What reason did they have to war with the Hawks?\n\nHis flight took him back to the front of the tavern. Several Serpents had come out to fight, but nearly all bore crimson stains on their green cloaks. The Spiders moved in, outnumbering them two to one. Haern took in the combat as quickly as he could, searching for an escape route. Underneath the awning over the tavern entrance he was safe from the archer, but in plain view of the rest of the thieves. They'd set up a perimeter, but he trusted himself to break through. He was the Spider Guild's champion, after all, their greatest creation and most disappointing failure. That, and they all thought he was dead, a belief he didn't want to change. He'd grown much over the past five years, but still, someone might recognize him underneath the dirt...\n\nOne of the Spiders saw him there, saw the blood on his swords. When he started to attack, Haern met his charge with a vicious assault of his own, surprising his opponent with sudden, overwhelming fury. He batted away a pair of daggers, cut open the man's throat, and then bolted to the street. Two more moved to stop him but Haern slide-kicked between them, scraping his leg along the hard ground. He cut the thigh of one and curled his sword around to hamstring the other. As they fell screaming, he ran, hoping they would not chase. None did, but he didn't go far.\n\nKnowing he was being stupid didn't stop Haern from doing it. He hurried back to the Hawk headquarters. The perimeter had closed in, and it seemed most of the fighting had stopped. Various members of the Spider Guild stood near the entrance, keeping watch while a few rifled the bodies.\n\nAnd then he saw him: Thren Felhorn, leader of the Spider Guild. His father.\n\n\"Why?\" Haern asked as he watched him walk into the tavern as if it were his own, accompanied by four of his men. \"What did they do to you?\"\n\nHe resolved to find out, but not now. He turned and headed back toward the city's center, realizing for the first time he was limping. After stealing a bottle from a man lying facedown in the ditch (whether dead or unconscious, he didn't know), he took a momentary reprieve to clean his scraped leg and splash some alcohol across it. After the pain subsided he continued.\n\nHe had two possible avenues to pursue. He could discover the reason for the Hawks' and Spiders' squabble, or he could look into the mystery of the Serpents' gold. Doing his best to convince himself it had nothing to do with any fear of his father, he resolved to look into the gold. Guilds fought all the time; he had no proof this was any different. Shipments from the Gemcroft mines, however...\n\nAfter breaking into another shop and stealing a few supplies, he found one of the quieter stretches along the great wall surrounding the city of Veldaren, went up the steps, and then climbed down the wall with a rope he tied at the top. Once out he trekked northeast, following the main road around the King's Forest toward Felwood Castle many miles farther along. Beyond that were the Crestwall Mountains and the many villages around the Gemcroft mines. It'd take him weeks to arrive, but he held hope it would be worth it. Out there he might find some information. When it all boiled down, the war between the guilds and the Trifect was about pride and money. Both sides had suffered too many losses to endure on pride alone. It was coin that mattered now, and if the Serpent Guild had suddenly discovered a new source of gold, they might endure for years. Even worse, they might score a significant victory against the Trifect, further emboldening the underworld. Haern had to stop it. He had to find some way to crush the guilds and end the war.\n\nHe laughed as he walked. End the war. It seemed nothing would. It had continued for ten years and seemed ready for another ten. But at least he might make things uncomfortable for a while. He'd done his part to weaken and bleed the guilds, to punish them for the bloodshed he'd witnessed firsthand. Randith, his own brother, Senke, his friend, Robert Haern, his mentor, Kayla, his first crush. And then there was Delysia...\n\nHis father had shot her with an arrow for daring to love him. He still relived that moment in his nightmares, sitting on a rooftop with her bleeding over his hands. He'd thought her dead, but Kayla had later told him she lived with the priests of Ashhur. And then Thren had killed Kayla. As for Senke, Haern had heard he'd died in the fire at Connington's estate during the Bloody Kensgold.\n\n\"Maybe I'm afraid,\" he whispered to the stars as he pulled his cloaks tighter about him to keep in his warmth. \"Gods forgive me, maybe I really am afraid.\"\n\nAmid it all, amid the death and loss and pain that haunted him come his restless daytime sleep, there stood his father. It seemed appropriate. It was a terrible web his father wove, and he was the spider in the center of it all. A spider that Haern was not sure he'd ever have the courage to face.\n\nHe continued along the road, part of him hoping he might be ambushed along the way. The bloodshed and excitement would have felt a thousand times better than the dread the memories of his childhood brought.\n\n# CHAPTER\n\n#\n\nMark Tullen rode toward the gates of Felwood Castle, as always in awe of the fortress made of dark stone and covered with ivy. Along with the provisions stashed in his pack was Alyssa's letter requesting him to retrieve her son and bring him back to Veldaren. Mark had been in Riverrun when the letter arrived, not far south of Felwood. He'd written his affirmation in a hurry, for he could sense Alyssa's unease. Whatever the reason, he didn't want to lose any favor in her eyes by tarrying.\n\n\"I seek audience with Lord Gandrem,\" he called at the gate. \"I am Mark Tullen, lord of Riverrun, and I come at Lady Gemcroft's request!\"\n\nThe gates opened and guards escorted him in. After he cleaned his boots, he followed them along the emerald carpet to the throne, the seat of power for all the Northern Plains. John Gandrem stood as they entered, a smile on his wrinkled face. He wore robes of green and gold, and a thin crown of silver atop his gray hair.\n\n\"Welcome,\" said John, clasping hands with Mark. \"It's been too long since you visited. The distance from here to Riverrun is not so great that you should visit only once a year.\"\n\n\"I was here in spring,\" Mark said. \"Do not tell me you forgot?\"\n\n\"I wouldn't be surprised if I did,\" John said, sitting back down and trying to laugh off the error.\n\n\"Sadly, I cannot count this as much of a visit,\" Mark continued. \"I've come for Alyssa's boy, Nathaniel. I'm to give him safe passage back to Veldaren.\"\n\nA shadow passed over John's face, and he took a sip from a goblet beside him before responding.\n\n\"Nathaniel is not here,\" he said, setting the goblet down. \"Lord Arthur Hadfield came a few months back and brought him north to Tyneham. I assumed this was at Alyssa's request, and he certainly implied as much.\"\n\nMark felt his gut tighten.\n\n\"I'm sure Arthur's done the boy no harm,\" the old man continued. \"Said he wanted to show him the ways of the business, if you know what I mean. If he's to take over the Gemcroft fortune, a bit of experience with their mines would do him much good.\"\n\n\"Thank you,\" Mark said, bowing.\n\n\"Will you not stay?\" John asked.\n\n\"My apologies,\" Mark said, glancing over his shoulder. \"But Alyssa seemed eager to see her son, and this delay will add many days to our travel. I dare not spend even a single night here when I might be riding instead.\"\n\n\"Very well,\" said John. \"Safe travels.\"\n\n\"And pleasant nights,\" said Mark.\n\nHe left Felwood and immediately followed the road north. Thankfully he'd packed enough rations for both him and the boy, so he'd have enough to make it to Tyneham alone. There he'd need to supply himself with at least enough to get them back to Felwood Castle. He let his mind wander as he rode. It'd take several days to arrive, so he had more than enough time to think.\n\nMark knew he and Arthur were rivals for Alyssa's affection. In their rare times together, Mark sensed Alyssa found him more interesting, more handsome. But Arthur had wealth and influence, which Mark did not, and something which Alyssa could not ignore. Nearly every town along the mountainside belonged to Arthur in one way or another, while Mark controlled just Riverrun, and even that only recently due to Theo Kull's execution, at Alyssa's hand no less. From the whispers he heard, and the cold stares from Alyssa's advisor, Bertram, he knew he was not the favorite in the rest of Veldaren's eyes.\n\nBut he wouldn't let that stop him. He'd been told the same about challenging Theo Kull, who had his fingers in everything. But Theo had died, and amusingly enough it had been because his son Yoren had tried for Alyssa's hand. Alyssa had kept an eye on Mark's takeover of Riverrun in the wake of the Kulls' demise, and that was how he had first met her.\n\n\"Ride on,\" Mark whispered to his horse. \"I know you're tired, but give me just a few more miles.\"\n\nNathaniel alone with Arthur... the idea greatly disturbed him. Arthur was an older man, calm in all things, always calculating every potential outcome of a choice. It was as if the two of them played a game, moving pieces and exchanging tokens, all for the sake of Alyssa's heart. So far Mark was losing, and now Arthur held a potential game ender. If the boy favored Arthur, his mother's heart might easily follow.\n\nMark slept close to the path, keeping his sword beside him as he tucked himself into his bedroll. His hard rations were bland and salty, but they kept him going. Several hours into the next morning's ride he found a stream to fill both his waterskins as well as give his horse a well-deserved rest. He kept up his pace, though not to such an extent that it might endanger his mount. The whole while he pondered Arthur's coming reaction. Clearly he wouldn't know of Alyssa's request to have her son returned. Would he refuse? Come with? Ask for proof? Mark had Alyssa's letter, of course, but what if Arthur challenged its authenticity?\n\nMark pushed the thoughts aside. It wouldn't matter. Trying to outthink Arthur would be pointless. Many would have to make the best decision available at the time, without fear or doubt. That was how he'd risen to his stature. That was how he planned on rising even higher.\n\nOn the fifth day out from Felwood, he arrived at the mining village of Tyneham. The lone inn was small, with only two rooms and a post out back to tie his horse. Much of its white paint had long since peeled away. Mark had a bite to eat, drank a cup of the awful ale, and then asked for Arthur Hadfield's location.\n\n\"He don't come to town often,\" said the innkeeper, a portly old woman. \"But when he does, you can find him overlooking the mines. He keeps an eye on things, and he's caught quite a few thieves who thought themselves bright.\"\n\nMark smiled at her obvious hint.\n\n\"I come in the right,\" he told her. \"But if I were a thief, I'd share at least a token of my haul with you, if only for your beauty.\"\n\nShe laughed and waved him off.\n\nHe received a few odd stares as he worked his way toward the mountains. He'd seen the Crestwall Mountains only once before, and he stopped beside a well to take in the view. They rose toward the sky like bony fingers, cracked and weather-beaten. He wondered how vicious the storms might get so far north, something he hoped not to find out. Still the mountains possessed a majestic beauty, towering above him, reaching into the clouds until their tips turned white with snow. Winter was halfway over, but he wondered if it ever ended there. The past several days he'd ridden through snow, and he thanked Ashhur there were an abundance of trees for firewood.\n\nRealizing he was stalling, Mark forced himself onward. As he neared the bustle of activity at the mines, a foreman spotted his approach and yelled for him to halt.\n\n\"Not from around here,\" said the foreman as he neared. He wore furs that were hopelessly dirty, and giant calluses covered his hands. \"You dress too well and too lightly.\"\n\n\"I'm warm enough,\" said Mark. He offered a hand. \"Mark Tullen, lord of Riverrun. I'm here to speak with your lord.\"\n\nThe foreman grunted.\n\n\"You're in luck. Arthur and the boy are further in. We might have hit a new vein, and he wants to take a look.\"\n\nMark tried to hide his reaction at hearing about \"the boy,\" but felt he did a miserable job. The foreman raised an eyebrow but refused to comment. Mark mentally cursed himself. If he couldn't hide his emotions from a lowly foreman, what hope did he have with someone as observant as Arthur?\n\n\"Please,\" he said, deciding to get it over with. \"Can you take me to him? I come with urgent business from Alyssa Gemcroft.\"\n\nThe foreman snapped to attention. If there was anyone more powerful than Arthur in the village, it was Alyssa. Her mines gave them work, wealth, and the means to survive on the harsh land. Without them Tyneham would become a ghost town.\n\n\"Follow me,\" said the foreman.\n\nThey walked along a path pounded flat by half a century of carts, feet, and wheelbarrows. A few of the men glanced up, but most ignored them, or did their best to look busy. Mark saw several women wandering about with food and water for the men. A few carried needles and cloth to wrap, stitch, and bandage the day's toll of blisters and cuts. He saw at least four main entrances to the lower slopes of the mountain. The foreman took him to the largest, where a crowd had gathered.\n\nThe two stopped and listened, for a man had come from inside the mine. A young boy stood at his side, his red hair covered with dirt. Mark knew them both.\n\n\"I've looked it over,\" said Arthur as he pulled off a pair of gloves and tossed them aside. \"It's a new vein, all right, the richest we've found in ten years. We'll shift men from mines three and four to help drain the rest of the water, and I'll send word for more oxen. Hard work is ahead, but tonight we'll share a glass to celebrate!\"\n\nThe crowd cheered and smiled, even the foreman beside Mark clapping in excitement. Mark kept his arms crossed and watched Nathaniel. He stood beside Arthur, his face passive and his eyes to the ground. Such good behavior from someone barely five... it struck Mark as worrisome. Even when the cheering began, Nathaniel only looked around once, and after a few seconds' delay clapped twice.\n\nMark waited as the rest of the men resumed their duties, cheerfully delving back into the mines or pushing their carts for the smelters and their mills. Arthur saw Mark through the crowd, nodded once, and then approached.\n\n\"Lord Tullen, I was not expecting such a pleasant surprise,\" he said, but the tone of his voice didn't match the honeyed words.\n\nMark withdrew the letter and handed it over.\n\n\"I've come for Nathaniel,\" he said. \"Alyssa wishes his safe return, for she misses him terribly. I must say, I was surprised to find him here instead of with Lord Gandrem.\"\n\nA smile pulled at the sides of Arthur's lips. He had a long, oval face, and gray hair trimmed extremely short. Mark had never seen a more smug, self-satisfied grin.\n\n\"I often talked with Alyssa about bringing Nathaniel here to learn the duties involved in running the mines,\" he said. \"At my last visit I mentioned doing so should the weather break.\"\n\n\"Her letter doesn't say that.\"\n\n\"Given how great her duties are, I am not surprised such a casual comment by myself went unremembered.\"\n\nMark didn't believe it for a second, but he tried to act like he did.\n\n\"Either way, she wishes him back,\" he insisted. \"So come, Nathaniel. Let us return to your mother.\"\n\n\"You can't take him,\" said Arthur. When Mark's eyes flared, the grin on Arthur's face only grew. \"Not by yourself. You would bring a son of the Trifect along the northern road unprotected? He is far too precious a target for ransom. Let me send some of my men with you as escort.\"\n\nMark looked away and muttered. Arthur was testing him, testing his reactions, and he'd given away his thoughts plain as day. As he looked about, he saw two wagons loading up not far to the south.\n\n\"Where are they headed?\" he asked.\n\n\"They?\" asked Arthur. He followed his gaze, and then answered far too quickly, \"I'm not sure, but they are of no matter to you. Let me get my men.\"\n\n\"Veldaren,\" said Nathaniel before Arthur could leave. \"Every week they bring gold for Veldaren.\"\n\nMark shot the boy a wink, not caring that Arthur saw.\n\n\"Then I will ride with them,\" he said. \"Surely we will be safe amid a well-guarded caravan.\"\n\nArthur's grin faded.\n\n\"Very well. They will slow you down, so make sure Alyssa knows the reason for your delay falls upon you, and not myself. I'll tell the men you'll be joining them. Nathaniel, go to the castle and pack your things. Hurry now! Do not keep Lord Tullen waiting.\"\n\nNathaniel bowed to both and then ran off. Mark watched him go.\n\n\"Not a smart child, but at least he is obedient,\" Arthur said, walking away.\n\nNathaniel rode in one of the two wagons while Mark trotted beside them on his horse. He'd purchased supplies from the tavern, not wishing to be a burden on the caravan. Though he'd stayed out of their way as best he could, he made sure to sneak a glance at the cargo\u2014crates of newly minted gold coins, all stamped with the symbol of the Gemcroft family. Each wagon had a single crate.\n\n\"Why just one crate per wagon?\" he asked the leader of the caravan, a fat man named Dave. \"Seems wasteful.\"\n\n\"Each wagon has its own driver, own guards, own cargo,\" Dave answered. \"Makes it harder for someone to get to plotting. That, and we'll fill both wagons on the way back with supplies. You should see how many tools we run through. I swear, for every pound of gold we dig we break two pounds of iron.\"\n\nCome nightfall they set up camp. Several of the guards had slept during their day ride, and so they wandered about, eating, drinking, and watching the roads. Mark took the time to find Nathaniel. The boy ate by himself, huddled in a blanket with his back to a fire.\n\n\"Cold?\" Mark asked as he sat down beside him.\n\nNathaniel shook his head.\n\n\"I can't be cold,\" he said. \"Arthur says that makes me look weak.\"\n\nMark chuckled. \"Even the greatest of leaders needs to wear boots in the snow. You're allowed to be human, Nathaniel.\"\n\nThe boy pulled the blanket tighter about him. He looked so similar to his mother, the same soft features, stubby nose, and startling red hair. He glanced back at Mark, and then a smile crept across his trembling lips.\n\n\"Maybe... maybe I'm a little cold.\"\n\nMark laughed.\n\n\"Take this then,\" he said, wrapping his own blanket around the boy. \"This should help. From here on out, anything Arthur told you, you check with me, all right?\"\n\n\"Why?\" Nathaniel asked, suddenly looking worried. \"Does Arthur lie?\"\n\n\"No, no,\" Mark said, quicker than he meant to. \"He just... has a peculiar way of looking at the world. He doesn't think people get cold, remember? I'd love to see him wander in his skivvies during a snowstorm. I bet he'd look like a blue ogre when he came back inside. What do you think? Or maybe a blue orc. Nah. He's too skinny to be an orc.\"\n\nHe yammered on, telling jokes both humorous and terrible. It didn't matter. He watched Nathaniel slowly warm to him, and it relieved Mark tremendously. He'd been worried that Arthur's words had wrapped a spell about the boy, turning him into some mindless stooge believing his every word. But Nathaniel was still a five-year-old boy, and given the chance he wanted to laugh and joke as much as any other kid his age. Mark knew he might not be the most charming dinner guest, but at least he knew how to make a kid laugh.\n\nMark let Nathaniel keep his blanket, instead borrowing another from the wagons. They slept beside the fire.\n\nCome the next morning Mark awoke with a chill having seeped deep into his bones. When he stirred he saw a thin layer of snow atop the world, including his blanket.\n\n\"About time,\" said Dave, who was busy untethering their oxen. \"You sleep like the dead, Mark.\"\n\n\"Better to sleep like them than to be them,\" he said, shaking off his blanket and looking for a fire.\n\n\"No fire,\" said Dave. \"We need to save the wood in case the snow picks up. Move about. Help us pack. You'll warm up soon enough.\"\n\nHe found Nathaniel sitting in one of the wagons, half-buried in blankets.\n\n\"I hate winter,\" he said when he saw Mark.\n\n\"I hear you,\" Mark said, slapping him on the shoulder. \"Just try to endure. We'll be home with your mother soon enough.\"\n\nThe snowflakes were light as they traveled, just a slight nuisance that wetted their skin and occasionally stung their eyes. By midday the snow had thickened, and at last Dave called a halt.\n\n\"The wagons might get stuck if it continues,\" Mark told him.\n\n\"Better stuck on the road than in a ditch,\" Dave shot back.\n\nThey used the wagons to block the wind, shoveled snow until they found cold, dry ground, and then built a fire. They gathered around it, their own bodies sheltering the fire from the wind that sneaked in.\n\n\"Come morning we'll dig out and then continue,\" Dave said as they huddled there. \"Run this route plenty of times, and I have a feeling for how the weather works. We'll have clear sky tomorrow. Assuming we don't break a wheel, we should reach Felwood in a\u2014\"\n\nHe stopped, for amid the howling of the wind he heard something strange.\n\n\"Horses,\" said Dave.\n\n\"Who would ride in this weather?\" asked one of the guards.\n\nMark drew his sword and stood, and the rest did likewise. There were only four guards per wagon, and the eight hurried to the opening between them.\n\n\"It might be a messenger meant to reach us,\" said Dave, just before a crossbow bolt pierced his arm.\n\n\"Shit,\" he cried, snapping the shaft in half and tossing it. \"Stay down, all of you!\"\n\nHorses thundered by on either side, and as they passed the gap, many of the riders fired crossbows. Mark dove into one of the wagons as the bolts flew, dragging Nathaniel with him. The horses turned around, and at their return charge he heard the sound of steel hitting steel.\n\n\"Stay down,\" Mark said to Nathaniel. The boy sat huddled in blankets beside the crate of gold. His eyes were wide, rimmed with tears that refused to fall in the chill air.\n\n\"I'm scared,\" Nathaniel said, and his whole body shook.\n\n\"I am too,\" Mark said as bolts tore through the fabric of the covering, thankfully missing them. He kept his sword facing the back of the wagon and listened. He heard screams, plus Dave hollering like a madman. From where he crouched he could only see a small portion of the combat. The guards had cut down two of the riders, but the rest continued their charge, hacking as they passed or firing more crossbow bolts.\n\nThen he heard Dave cry something that made no sense, but at the same time was certain to be true.\n\n\"Lord Hadfield? But why?\"\n\nHe died soon after, or at least his orders stopped. The cries of pain lessened. Swords struck rarely, then stopped altogether. Mark pushed Nathaniel farther into the wagon and tried to shrink down. He might be able to surprise one or two of them if they didn't realize he was inside...\n\nA man rode up behind the wagon, a crossbow in hand. Mark lunged at him, extending his arm as far as it could go. His sword pierced the man's breast, punching through his leather armor. As he bled out, the crossbow fired harmlessly into the air. Mark retreated into the wagon, his blood running cold. He recognized the symbol on that armor. They were Arthur's men, all right. But why? Why would he ambush his own wagons?\n\nHe glanced back at Nathaniel and decided he already knew the reason.\n\n\"Mark?\" he heard Arthur call out. \"Is that you in there, Mark?\"\n\n\"Just keeping warm,\" Mark shouted back. \"What'd your men do to deserve this?\"\n\n\"Deserve? Nothing. They died in my service, as all men should for their masters. Where is the child? I don't want him to witness your execution.\"\n\nMark clutched his sword tighter. Behind him he heard Nathaniel whimper.\n\n\"You'll protect him?\" Mark asked.\n\n\"As if he were my own son.\"\n\n_Or at least until you have a son of your own_ , thought Mark. _At least until you've consummated your marriage to Alyssa, you heartless bastard._\n\n\"Listen to me,\" he whispered to Nathaniel. \"He's lying, I know it. You need to run, you understand? I know you don't want to, but you have to try. He's cruel. I've always known it, now just...\"\n\n\"Mark!\" Arthur shouted. \"Come out and face this with honor!\"\n\n\"That way,\" Mark said, pointing to the exit beside the driver's seat.\n\nNathaniel nodded. Despite his fear he was holding together. Though they lacked any blood connection, Mark felt proud of the boy. A child worthy to raise, to claim the Gemcroft wealth. A child who'd probably freeze to death in the next twelve hours. He almost thought to change his mind, to carry Nathaniel out and discover what Arthur would do. But he couldn't. If Nathaniel was somehow part of Arthur's plans, Mark wanted to ruin them. It was petty, perhaps, but by the gods, he had to do something to avenge his own death.\n\nHe stepped out from the wagon, his sword still drawn.\n\n# CHAPTER\n\n#\n\nHaern kept his cloaks wrapped tightly about himself as he trudged along the road. He felt foolish for not preparing for such weather. His feet were numb from the cold, and he'd have given anything for a thick coat. He'd dressed like a thief when he should have dressed like a bear.\n\nHe lacked the proper tools to build a fire, which was especially problematic given how heavily the snow came down. Moving kept him warm, so that's what he did. It'd been two days since he saw houses in the distance, farms both large and small. Before that he'd stayed a night in the comforts of Felwood Castle, stocking up on food and, like a fool, refusing to steal anything warmer to wear before continuing toward the mines along the Crestwall Mountains. That was before the snow, before he realized just how pathetic he was compared to nature's forces. His hood pulled low, he stared at the white ground and kept his feet moving. Night was approaching, and he pondered what to do. Surely he could find a tree for shelter, and should probably start looking. But he didn't want to stop just yet. He didn't want to admit it, but part of him feared that the moment he stopped moving he would curl up, fall asleep, and never wake again.\n\nThe first time he heard the noise he thought it a hallucination. Then it came again, and many times after. It was the sound of steel hitting steel, coupled with the neighs of horses. He felt some of his drowsiness leave him. He'd headed north in hopes of discovering the source of the Serpents' gold, prepared to travel all the way to Tyneham if he must, to where the gold was minted. But could the source be something as simple as raiding the caravans as they traveled south? Dashel had insisted against it, and Haern had agreed with his argument. Then what?\n\nHe urged himself on, and despite the snow that blew in his face, he forced himself to stare straight ahead. The snow was thick, and it seemed as if a white fog had enclosed the land so many yards away from where he stood. When he saw the first rider, it was if he emerged from another world. Haern dove for the cover of trees, then glanced back to see if he'd been spotted. He hadn't. The rider turned back and charged into the unseen combat.\n\nNot willing to risk such foolishness again, Haern weaved through the trees, making sure to stay close to the road. If the weather remained foul, it might be days before he found the path again. He was no woodsman. The city streets were his home. Out among the trees, in the snow, he felt like a bumbling idiot.\n\nThe sound of combat faded not long after. After a few moments of silence he heard someone yelling. His numb ears at first refused to make words out of the noise. As he followed the sound, he started to understand.\n\n\"Where is the child?\" he heard a man ask. \"I don't want him to witness your execution.\"\n\nFinally reaching the area of combat, Haern leaned against the thick trunk of a tree and surveyed the situation. Two wagons were pulled close together, the oxen tethered behind them. Eight men on horses milled about, all with swords or crossbows. The speaker seemed older than the rest, and he wore no armor, just a thick coat of bearskin that Haern felt ready to kill for. All around them lay bodies, their warm blood melting the snow beneath them.\n\nIt didn't make sense. All the horsed ambushers wore the same insignia, a sickle held before a mountain. This wasn't the Serpent Guild. They didn't wear green cloaks. What then? Should he interfere?\n\nMeanwhile the older man continued talking, evidently with someone inside a wagon given the direction he faced and the command he gave.\n\n\"Mark!\" cried the older man. \"Come out and face this with honor.\"\n\nAnd then it seemed Mark obeyed, stepping from the back of one of the wagons. He looked young, his armor dark and expensive. The riders circled about him as the older man smiled.\n\n\"Hiding during a battle,\" he said. \"Such shameful behavior.\"\n\n\"Perhaps,\" Mark said. He lunged at the nearest rider, but never got close enough to swing. Two crossbow bolts pierced his back, and he stumbled, his weapon falling from his hand. Haern winced. At least the man had died bravely, even if he hadn't accomplished...\n\nBut then he saw the child leap from the wagon's front and dash for the forest. Haern's eyes flared wide. The kid was heading straight for him.\n\n\"Get him!\" the riders shouted. One took off, dismounted at the forest's edge, and then rushed on, his sword drawn. Haern flung his back against the tree. Should he interfere? Would they kill the boy, or merely keep him captive? Was this for ransom? Too much he didn't know. Too much!\n\nThe boy rushed by, followed by the soldier. Haern stared, paralyzed by indecision. If he acted now he'd reveal himself. Eight riders... what chance would he have? He'd be throwing his life away, and why? For all he knew, the boy belonged to the ambushers.\n\nThe soldier quickly gained ground, for he could make longer strides. He kept his sword drawn, and Haern recognized the way he held the blade in preparation for a thrust. This was no capture. This was no attempt at ransom. He ran after the two, feeling slow and clumsy in the snow. The boy glanced behind, saw his pursuer, and then stumbled. Haern wanted to cry out but didn't dare reveal his location. The soldier reached the boy, and his sword thrust. Blood spilled across the snow.\n\nHaern slammed into the soldier with his shoulder, flinging him onto his back. Before he could stand, Haern drew a sword, slapped aside a weak defense, and buried it in the soldier's throat. The man gargled blood, quivered, and then lay still.\n\n\"You get him?\" a man shouted from the road.\n\nHaern ignored him and instead looked to the boy. He lay on his back, his whole body shaking. The thrust had cut deep into his arm, right to the bone. The blade had continued, piercing his chest. He still breathed, and it didn't sound wet. The tip hadn't gone deep enough to pierce anything vital. With proper care he might live, but at the moment he was wide-eyed with shock. He'd need time, which at the moment Haern sorely lacked. He sliced off a strip of one of his cloaks and tied it around the wounded arm, then took the boy's hands and pressed them firmly against the wound on his chest.\n\n\"Stay still and quiet,\" Haern whispered, propping him against the nearest tree. \"I'll come back for you, I promise. No matter what you do, don't let go.\"\n\nHe stood, drew his swords, and looked to the road. Through the forest he saw the faintest glimpse of the riders. Amid the trees and snow the horses would be useless. So long as they didn't know he was there, he had a chance.\n\nHe stepped gingerly across the snow, crouched low and hidden behind the trunks. The forest was quiet, and he heard their discussions with ease as they grew steadily heated.\n\n\"Terry!\" one shouted. \"Where are you? Did the brat lose you somehow?\"\n\n\"Jerek, Thomas, go look for him, and hurry. I don't want to be out in this weather any longer than I have to.\"\n\nHaern smiled at the lucky break. He stayed to the side and watched two more men walk right past him. He started creeping after them, but they stopped halfway.\n\n\"See that, Jerek?\" asked Thomas as he pointed. \"Something ain't right.\"\n\nThey drew their swords and looked about as Haern realized what he pointed at: the footprints he'd left in the snow when chasing after the soldier and the boy.\n\n_Damn wilderness_ , thought Haern. _Give me a city any day._\n\nThey followed the footprints, but they were no longer hurrying. His surprise advantage was nearly blown. He continued following, using the trees to hide in case they glanced back, but they were getting too close to where the soldier's body lay.\n\n\"Found them,\" said Jerek. \"Shit, his throat's cut!\"\n\nHaern gave up stealth, knowing he couldn't muffle his running. The crunching of the snow turned them about, but he was too close, too fast. He gutted Thomas, ducked under a dying slash, and then turned to Jerek. Instead of the desperate lunge he expected, Jerek pulled back and held his sword with both hands in a defensive position. Haern felt respect for the man, as well as agitation. He didn't need a drawn-out duel against a worthy opponent. He needed the man killed before any others arrived.\n\n\"Ambush!\" Jerek screamed. \"It's a fucking ambush!\"\n\n\"One against five,\" Haern said. \"Some ambush.\"\n\n\"There's six of us,\" said Jerek.\n\nHaern shrugged.\n\n\"You'll be dead soon enough.\"\n\nHe feinted, stepped to the left, and then lunged for real. Jerek bit on the feint, but not enough. He parried both blades aside, but he extended to do so. Haern closed the distance between them, slamming an elbow into the man's chest while they shoved their weapons together. Jerek tried to separate, but Haern shifted again, positioning his right foot in the way. When Jerek stepped back he tripped, and that was all the opening Haern needed.\n\n\"Jerek? Thomas?\" asked another soldier as he approached the bloody mess. Haern watched, doing his best to keep his breathing calm. Only three had come, not five, which meant one had stayed behind to protect the older man, presumably their leader. They were only a handful of paces from where the boy lay, but they stopped at the bodies of their comrades. Two held swords, while the third held a crossbow. They looked about the area, searching. They never looked up, and that was enough.\n\nHaern fell upon them from his perch in a nearby tree, leaving one bleeding from a gash in the neck and another holding a crossbow with a broken string. Haern kicked him in the chest to force him back, needing the space. The last swordsman hacked at him, but Haern spun his cloaks, using them to appear farther to the right than he was. The strike hit nothing but air and cloth. Haern continued his spin, slashed his foe's arm, reversed the spin, and buried his sword in the henchman's stomach, just below his armor.\n\nPain spiked up his arm. He struck on reflex, ending up cutting the crossbowman across the mouth. The man dropped the dagger he'd drawn and clutched his jaw as blood ran across his hands. He tried to say something, but it came out as an unintelligible sob. Haern glanced at his arm. The cut would scar, but assuming it didn't get infected he'd be fine. Frustrated at his mistake, he leaped at the lone survivor, who turned to flee. A kick took out his knee, and he fell. Haern's swords pierced his lungs, and then he sobbed no more.\n\nCursing at the pain, Haern approached the road. He kept the blood on his blades, wanting the fear it would bring. The red contrasted with the snow, made the swords seem all the more deadly. When he stepped from between the trees, he saw both riders on the far side. The younger raised his crossbow and fired. It tore a hole in Haern's cloaks as he flung himself to the side. He spun around a tree and emerged, but the soldier had not even begun to reload.\n\n\"Who are you, stranger?\" asked the older man. \"What are you hoping for? Is it coin you want?\"\n\n\"Too many questions,\" Haern said, watching the last soldier. His hand kept inching toward his hip, but for what?\n\n\"Then answer me just one: is the boy alive?\"\n\n\"I don't know, nor care. He was a distraction. If he lives he'll freeze by morning.\"\n\nTheir leader seemed pleased by the answer. Haern made sure he didn't blink, didn't twitch, didn't reveal the lie.\n\n\"Good,\" said the older man. \"Then what is it you want now? You cannot kill us, and you cannot make off with my gold. You'd need to tether the oxen and drive it many days to town. So please, accept my offer. Leave us be, and I will allow you take as much gold as you can carry.\"\n\n\"You'd buy your safety with what I can freely take?\" Haern asked.\n\n\"Freely? Nothing is free, thief. Everything is bought with sweat and blood. Come spill it if you'd dare.\"\n\nHaern chuckled. Whoever the man was, he reminded him of his father. Not a good comparison.\n\n\"Leave,\" he said. \"I have no use...\"\n\nHe rolled behind the tree as the throwing dagger pierced the bark, hurled with frightening precision from the soldier's hip. From behind the tree he laughed.\n\n\"Ride off!\" Haern shouted to them. \"Even if you have a hundred of those daggers to throw, it won't matter. Flee or die!\"\n\nHe listened and waited. The men muttered quietly, and when done they rode north. Haern sighed and looked to his arm. Still bleeding, and its pain was now a deep ache. It'd have to wait. He trudged off for the boy, who looked horribly pale.\n\n\"I'm sorry I couldn't bandage these sooner,\" he said as he knelt before him. He pulled the boy's hands away and looked at the stab. \"You can thank Ashhur this wasn't an inch deeper, or you'd be like the rest of them.\"\n\nHe used more cloth from his cloaks to tie a bandage around the boy's chest, then turned his attention to the arm. So far the boy hadn't spoken a word, only watched with a glazed look in his eyes. Fearing he might pass out at any moment, Haern slapped him a couple times across the face.\n\n\"Stay with me,\" he said. \"I bled for you. Least you could do is survive.\"\n\nThe earlier bandage he'd applied had soaked through, so he removed it, cut another strip, and tied it. Given the depth and severity of the cut, he wondered if the whole arm would need to go, but knew someone wiser in healing arts should decide that. So long as it didn't turn green and rot off, he figured the boy had a chance of regaining its use.\n\n\"What's your name?\" Haern asked him as he tore the shirt off the dead soldier beside them. When the boy didn't answer, Haern snapped his fingers in front of his eyes a few times. Still nothing. Sighing, he cut up the shirt and used it to form himself a sling.\n\n\"Come on, what's your name? We're friends, blood brothers, fellow travelers in the snow. You're not too cold to speak, are you?\"\n\nAfter a few seconds the boy shook his head. Good. At least he was somewhat alert. Haern tore free the cloak of another dead soldier, wrapped it around the boy, and then lifted him into his arms. His wounded arm shrieked in protest, so he shifted a bit of the weight onto his shoulder.\n\n\"Name,\" he said. \"I'd really love a name.\"\n\nBut the boy slumped and passed out. Haern sighed again. He returned to the road and surveyed the carnage, laying the boy beside the fire while he searched. It didn't make any sense. The attackers were well armed and well equipped, and they bore the symbol of a lord. When he looked into the wagons, the exact same symbol was stamped upon the crates. The oxen's harnesses had the same as well, a sickle raised before a mountain.\n\nIf he'd had time he might have hidden the gold so it couldn't be taken by the first passerby. But he didn't. Furious at his confusion and helplessness, he used his sword to draw an eye in the dirt beside the fire, where no snow lay. Beneath it he scrawled his mark, \"The Watcher.\" At least he might accomplish something out of all this. Let the thieves know that even outside Veldaren they were not safe from him.\n\n\"Well, boy,\" he said, returning to the fire. \"I'm sure it's nice and warm, but we have to move. I can't remember the last farm I passed, but it's our only chance. Can you walk?\"\n\nNo response. Haern bandaged his own arm, tore open one of the crates, and grabbed a handful of coins. They bore a symbol he easily recognized, that of the Gemcroft family.\n\n\"What do you have to do with the Serpents?\" he wondered aloud. A shame he had no one to interrogate, no time to investigate. He pocketed the coins, hauled the boy into his arms, and started walking south.\n\nThere was another reason he needed to hurry. The two who'd fled would certainly return, and he had a feeling it'd be with far more than their initial eight. Step after step he cursed the snow, the wind, the cold, and his clumsy mistake that had cost him a cut. All the while the boy slept in his arms. By nightfall Haern felt ready to collapse. He walked off the road, kicked aside the snow before a tree, and set the boy down. He wrapped him tighter in his cloaks and did his best to keep hope. The boy's lips were blue, his skin a deathly white. He'd lost so much blood, right when he needed its warmth the most.\n\nStill standing, Haern pulled an emblem hanging by a silver chain from beneath his shirt. It was of a golden mountain, and as he held it he prayed over the boy.\n\n\"Just keep him warm and alive, Ashhur. And don't forget me, too. I could use the damn help.\"\n\nHe put away the emblem, sat down beside his nameless boy, and pulled him close so they could share their warmth.\n\n\"It'll get better,\" he said, not sure if the boy could hear him or not. He was so thoroughly wrapped Haern couldn't see his eyes. \"Don't worry about any pain. As my father once said, pain is a tool that should always be under our control. It teaches us when we err. It distracts and weakens our opponents. And for you, it'll help you for the rest of your life. Who cares about a silly scratch from a sword when you've been struck to the bone, yeah?\"\n\nHe felt like a moron yammering on, but he did so anyway. At last he heard the boy snore, and he leaned his head back against the bark. His eyes looked to the clouded heavens.\n\n\"Couldn't you at least stop the snow?\" he asked Ashhur.\n\nAshhur didn't bother to respond.\n\nHaern slept through the night, waking only once at the sound of hoofbeats. He curled his body tighter against the tree and kept perfectly still. From the corner of his eye he saw the light of torches. Unable to see his tracks veer off the road because of the fresh snow, the horsemen rode right on by.\n\n\"Never mind,\" Haern whispered once they were gone. \"Go ahead and let it snow.\"\n\nHe closed his eyes, leaned his head against the boy's, and slept until morning.\n\nHaern had little food and water, certainly not enough for two. He ate the food, deciding he needed the strength for carrying the boy. He did his best to get the child to drink, though. Other than a few quick sips, he was unsuccessful. His back ached, and his arm throbbed, but he forced the pain far away, as he'd been trained to do by his many mentors. He carried the boy, stopping every hour to rest and catch his breath. Any time he let him go, the boy collapsed to the ground.\n\n_So much for making the brat walk_ , Haern thought.\n\nHe shook his head and immediately felt guilty. Of course the boy couldn't walk. He was sitting at the Reaper's door. That he even had his eyes open was a miracle.\n\nThey walked along the road, encountering no others. Evidently no one else was dumb enough to travel in such weather. The snow had stopped in the morning, and as he followed the road he observed the chaos of hoofprints. They all appeared to be headed south, with no hint of them returning. Haern knew he'd have to stay alert for their eventual trip back north. He was in no shape to fight a group of horsemen.\n\n_Keep walking_ , he told himself. _Keep walking. Keep going._ The son of Thren Felhorn would not die unknown in the wilderness. He couldn't. He wouldn't.\n\nNear the end of the second day he finally found a farm. He crossed the fields, every bone in his body aching. The boy hadn't had a drink the entire day, and his skin was hot with fever. Part of Haern wondered if the cold was the only thing keeping him from burning alive. At the door to the home he stopped, hid his swords with his cloaks, and knocked.\n\n\"I come in time of need,\" he shouted, surprised by how hoarse his voice sounded. \"Please, I have a wounded child with me.\"\n\nThe door crept open. In the yellow light of lamps he saw the glint of an old short sword. A man looked through the crack, saw him holding the boy.\n\n\"Winter's nearing its end,\" said the man. \"We have little to spare.\"\n\n\"I'll pay,\" Haern said. \"Please, I've walked without rest for days.\"\n\nThe man glanced inside, whispered something, and then nodded.\n\n\"Come on in,\" said the farmer. \"And by Ashhur's grace, I pray you mean no trouble.\"\n\nHaern found what appeared to be the entire family gathered in the front room, under blankets and around a stove whose heat felt glorious on Haern's skin. He saw two girls huddled beside each other, their hair a pretty brown. The farmer had two boys, one of them of age, and they held knives as if to help their father should it come to bloodshed. His wife sat beside the fire, tending it.\n\n\"He has a fever,\" Haern said, setting the boy down beside the stove. \"And he hasn't had food or drink for days.\"\n\n\"I'll get some water,\" the wife said as she stood. She cast a worried glance at her husband, then vanished into the next room.\n\n\"My name is Matthew Pensfield,\" said the farmer, extending his hand. Haern accepted it, and was shocked at how much his own hand shook. He hadn't eaten much, he knew, but had it really affected him so greatly?\n\n\"Haern,\" he said as he pulled his cloaks tight about him and surveyed the house. It seemed cozy enough, and not a hint of draft. The man had done well in building it.\n\n\"I know some of the Haerns,\" said Matthew as his wife returned. He had a hard look, his square jaw covered with stubble, but he spoke plainly and seemed more at ease now that Haern showed no inclination to violence. \"Good men, own several fields west of here. What's your full name? I might have heard them speak of you.\"\n\n\"Just Haern,\" he replied, nodding to the boy. \"And I have no name for him. I found him wounded; why does not concern you. That room beyond there, is that your kitchen? Might we speak in private?\"\n\nSome of the farmer's worry returned, but he nodded anyway.\n\n\"I reckon we can.\"\n\nOnce in the other room, Haern dropped his voice to a whisper.\n\n\"I have a difficult request for you,\" he said. \"I need you to take care of that boy until he's regained his health. I cannot stay.\"\n\n\"We don't have enough food to...\"\n\nHe stopped as Haern drew out a handful of coins and dropped them upon the table. His eyes stretched wide. The gold shone in the dim light.\n\n\"People will be hunting for him,\" Haern said. \"No matter what, you treat him like your own child. When he regains his health, he'll tell you his name and where his family might be, assuming they're still alive. Until then, give him up to no one.\"\n\n\"What if they threaten violence?\" Matthew asked, his eyes lingering on the gold.\n\n\"Would you give up one of your daughters?\"\n\nThe farmer shook his head. \"No. I wouldn't.\"\n\nHaern let the cloaks fall away from his left side, exposing one of his swords.\n\n\"I hope you understand,\" he said. \"I'll return, and if I find him abused, or dead, I will repay you tenfold in blood.\"\n\n\"He's sick and wounded. What if he dies of fever?\"\n\nHaern smiled, and he let the coldness he felt in his bones creep into his eyes.\n\n\"How well do you trust your children, your wife, to tell the truth? I will know one way or the other what happened to him. Do not give me cause to wonder.\"\n\nMatthew swallowed. \"I understand. This land is harsh, and we've taken in children before. Once he heals up, and the weather breaks, I'll take him where he needs to go. If he doesn't know, well, there's always a need for more hands on a farm.\"\n\nHaern slapped him on the shoulder, nearly laughed at his terrified jump.\n\n\"Good man,\" he said. \"Now how about a warm meal?\"\n\nHe ate some soup while he watched the wife tend to the boy. She put a wet cloth across his forehead, dressed his wounds far better than Haern had, and then used a spoon to slowly get him to drink. Haern was impressed. Seemed these Pensfields knew how to take care of themselves. Whoever the boy was, he could do far worse for a temporary home.\n\nThe soup did wonders for his mood. Its warmth seeped deep into his chest and then spread to the rest of his limbs. With that and the heat of the wood stove, he felt warmed within and without. He could feel his muscles tightening from lack of motion after so much exertion, and he did his best to stretch in the cramped quarters.\n\n\"You can spend the night here if you wish,\" Matthew said as the day neared its end. \"I'd be a sad man to banish a guest just as the sun sets.\"\n\n\"Thank you,\" Haern said. He shifted farther away from the fire so the children could take a turn. He wrapped his blankets around his body and closed his eyes. For the first time in his entire life he found himself in a true home, with a real family. The children bickered, but there was a harmless familiarity to it. He thought of his own childhood, never spent with someone his own age, only the parade of tutors and mentors, training him to read, to write, to move, to kill. Had he ever curled up on the floor beside a fire, surrounded by a family that would never wish him harm? Had he ever been inside a house that felt at peace? Had he ever...\n\nHe slept, and his dreams were dull and calm, and he did not remember them when he woke.\n\n# CHAPTER\n\n#\n\nIt was Veliana's third attempt at killing Deathmask, and the first she'd been personally involved with. She lay atop the roof of their headquarters, a crossbow in hand. The pale sun shone down on her, and she found herself thankful for the winter. In any other season she'd have been sweating like mad on the roof's shadeless wood slats.\n\n\"What if you miss?\" asked Garrick, who stood behind her so he couldn't be seen from the street.\n\n\"Then Rick will take him down,\" Veliana said. She pointed to the building on the opposite side of the street. A man in gray lay atop it, a crossbow beside him.\n\n\"I can't believe he's not dead yet,\" said Garrick as he took a chunk of crimleaf from his pocket and began chewing. \"Are our men truly so incompetent?\"\n\nVeliana rolled her eyes. The first attempt to kill Deathmask had been a simple stabbing in the night. She'd selected one of their lower-ranking thugs for the deed. They'd found the thug's body rotting beside Deathmask's bed in the morning. How he'd died, no one knew. Deathmask hadn't shown the slightest irritation at the attempt, either. Veliana held in a chuckle. Shit, the guy had tossed her a wink on the way to breakfast.\n\nThe second attempt had actually been three separate instances of poisoning his food. He hadn't eaten any of it. The third time, Veliana caught him waving his hand over his meal. That same day both their cooks died vomiting blood. Garrick ranted and assumed they had mishandled the poison. Veliana knew differently.\n\n\"We've given him a simple task,\" Veliana said, sighting the crossbow on their door. \"He's to collect protection money from a handful of vendors a few blocks over. When he exits the door, he should see Rick preparing to fire. In fact, I'm counting on it. He's too damn clever not to notice. Maybe he'll run, or cast a spell, or pretend he doesn't see. It won't matter. That's when I put an arrow through his back.\"\n\n\"So confident,\" Garrick said. \"Remember, if this fails, I make the next plan. This was your last chance to get things done safe and clean.\"\n\n\"I figured you'd like things safe,\" she muttered.\n\n\"What?\" he asked.\n\n\"I said we should reconsider. He's clearly skilled. What if he isn't here on someone else's payroll? What if he really wants a position in our guild?\"\n\nGarrick chuckled. \"If he's that good, why choose ours? We're far from the most powerful. Others would have made more sense. Or why not become a mercenary? The pay would be better, and then he could kill our kind all he wants. I'm sure the Trifect would love to have him on their...\"\n\n\"Quiet,\" Veliana hissed.\n\nThe door opened, and out stepped Deathmask. He wore his red robes and the dark gray cloak of their guild. As always when he went out in public, he'd tied a gray cloth around his face, hiding all but his eyes and hair. His back was to her. She glanced at Rick, who shot her a thumbs-up. When she looked back down, Deathmask was staring up at her. Slowly he shook his head, as if berating a child.\n\n\"Fuck,\" Veliana whispered. She pulled back from the ledge as Garrick asked her what was going on. \"He's spotted me.\"\n\n\"Then Rick should...\"\n\nHe stopped as they both watched Rick tumble over the edge of the building, blood gushing from his mouth and ears. When his body hit the ground, Veliana let out an involuntary gasp. Rick hadn't even fired, his crossbow lying useless atop the flat roof. Deathmask laughed, and he called out from the quiet street.\n\n\"I'm disappointed, Vel! Only the one?\"\n\nHe walked west, and both remained silent as they watched him. Veliana hadn't seen what he'd done to Rick, but hadn't needed to. Deathmask was no normal thief or trickster. Only a spell could have done what she'd just seen, a dark and powerful one. She was playing a game against an opponent she knew nothing about. Such was a sure path toward losing.\n\n\"That son of a bitch,\" Garrick said. \"He's toying with us. He knows we want him dead, and he doesn't care! If we don't do something soon, I'll be a mockery to the rest of the guild.\"\n\n\"Of course you will be,\" Veliana said as she stood. \"You're trying to kill someone you accepted into our guild, all without any proof or reason. _That_ is what will upset the rest of our guildmembers, not that you can't kill him. So far you've been lucky enough to have Deathmask kill off everyone who's ever been involved in the attempts, but word will get out in time.\"\n\nShe thought Garrick would explode, but instead he gave her an amused grin.\n\n\"You failed, Vel, so now I choose the attempt. Enough of poison and cowardly arrows. It's time you bloodied your hands.\"\n\n\"So long as you don't have to bloody yours,\" she said, offering him a mock bow. Her sarcasm hid her fear. She couldn't back down, not when Garrick was starting to develop a spine. But did she really want to mess with Deathmask?\n\nShe hung from the edge of the roof, dropped down to a windowsill, and then used it to fall to the street. A closer look confirmed what she'd already realized: Deathmask was equal to her in skill, if not superior. She found a thin razor embedded deep in Rick's neck. No doubt Deathmask's spell had required some sort of physical contact, and he'd thrown the razor as a way of carrying that spell. A simple but foolproof ambush, but it was her man who lay dead, not her target.\n\nLaughter floated down. She flipped Garrick a rude gesture, knowing he stood at the roof's edge watching her. So be it. No matter what Garrick thought, the Ash Guild was hers, and she would remind him of that fact. No doubt he viewed her coming attempt as a win-win, for either she or Deathmask would die. There had to be another way. More important, she had to think of a replacement for him, and soon.\n\n\"Bury Rick somewhere,\" she said to her guards at the door as she marched inside to think.\n\nAny deviation from Deathmask's normal routine would immediately alert him, so Veliana played it patient. Two days after the third failed attempt she had one of her lower-ranking members tell him he was to stay up late working as a guard. She hoped the tedium might dull his senses for when she struck. Despite his spotting her before, she took to the rooftop and waited. Four hours before dawn, when her own eyes started to droop, she decided it was time. She drank a mixture she'd prepared earlier in the day, a combination of strong tea and herbs. A few minutes later she felt the mixture kick in. Her head ached, but her drowsiness was gone.\n\nShe drew her daggers and crept to the rooftop's edge. No arrows or crossbows this time. If he really was a skilled spellcaster, her only chance was at close range, where she could disrupt the intricate movements needed to cast. She looked down and saw him standing several feet away from the building, keeping watch outside their safe house as instructed.\n\n_Damn it_ , she thought. _Won't be a straight drop. He can't possibly know I'm coming, can he?_\n\nBut of course he could. He might be able to read her mind for all she knew. Now was the time. He was mortal. He was fallible. She was the better. She had to prove that, not just to Garrick, but to herself.\n\nShe leaped from the roof, silent as a ghost, her daggers aimed for his neck, her knees bent and ready to absorb the impact of their collision. She felt exhilaration soar through her, the wind blowing her hair as she fell. In that half-second she saw him turn, saw him step aside. She twisted, suddenly panicking. He'd known. Somehow he'd known.\n\nRolling with the landing helped reduce the pain, but not by much. Scraping along the ground, one of her hands flooded with pain, jarred backward before suddenly going numb. The dagger fell from her limp fingers. She tumbled along, then forced herself out of the roll. Turning around, she expected her death, some sort of spell to sap her breath or explode her blood from her nostrils. Instead Deathmask stood there, shaking his head.\n\n\"Not good enough,\" he said. \"I need you stronger, faster. Otherwise you are useless to me.\"\n\nShe clutched her numb hand to her chest and glared.\n\n\"No matter what it is, I won't help you,\" she said. \"I've worked too hard to let you destroy everything.\"\n\n\"Destroy?\" Deathmask said as he looped his arms in a circle. \"I've come to perfect, not destroy.\"\n\nShe lunged at him as shadows pooled around his feet, bursting upward to form a wall that her dagger could not penetrate. She stabbed again, then spun about looking for an opening. There was none. Unsure, she closed her eyes and focused. She'd be vulnerable, but so long as the shadow wall remained, she might have the time. Power was focused through her, channeled into her dagger. Purple fire surrounded the blade, and with a cry she thrust it forward. It broke through the wall, which shattered and vanished as if it were made of fine glass. She had the briefest moment to enjoy the look of panic on Deathmask's face before her dagger cut flesh.\n\nIt wasn't fatal, and she cursed her foul luck. She'd guessed wrong where he stood, and her dagger only slashed his side and cut his robe. Warm blood spilled across her hand. They were so close it seemed time froze as they eyed one another, preparing the next moves for their dance. He drew a blade as he shifted away from her. Her kick sent it flying, and she stabbed again, wishing she had her other dagger. Deathmask fell back, his palms open. A light flashed from them, except it was black instead of white. It dazed her all the same, and her next two swings cut only air.\n\n\"What is wrong?\" she asked as she took two steps and jumped. Her heel smashed into his stomach, and he gasped as he crumpled to the ground. \"Where is the brutal killer who bested all my plans?\"\n\nVeliana dropped to one knee and thrust for his throat, not caring for his answer. He caught her wrist just as the tip of the dagger entered his flesh. A single drop of blood ran down his neck as they struggled. By the gods he was strong!\n\n\"Still here,\" he said, all trace of amusement gone. His voice was cold, merciless. She felt a shiver run up her spine. She jerked her arm back, but still he held her. His brown eyes met hers. If only she could tear off that damn mask of his. If only she could see his face, remind herself he was human, for his strength was unreal.\n\nShe swept her left leg around, taking out his feet. He didn't let go even as he fell. Together they hit the ground. The collision bumped her injured hand, and her fingers throbbed in agony. They had to be dislocated, if not broken. Still her dagger hovered inches from his flesh, unable to either attack or pull away. He landed on his back, and instead of rolling over he reached up and held her good arm with both hands.\n\n\"I could burn your flesh until I clung to bone,\" he said. His tone told her he spoke truth. \"Are you ready to listen, or must I find another?\"\n\n\"No others,\" she said as she prepared. \"You won't have the chance.\"\n\nShe dropped the dagger. Her legs kicked, pushing her into the air. Not high, just enough for her to drive her knees into his chest, blasting the air from his lungs. He still clutched her, but she rammed an elbow into his throat, denying him his next few words. She pressed her body against his, keeping the elbow tight. Their foreheads touched. Still he held her other hand.\n\n\"What do you want? What is your game? Who are you?\"\n\nShe released the pressure on his throat just enough for him to speak. Her nerves remained on edge. The second he flinched, or said a syllable that sounded remotely like a spell, she'd crush his larynx and leave him gagging on the street.\n\n\"I told you before, I have no name.\" He stared at her, unafraid.\n\n\"Bullshit. Everyone has a name.\"\n\n\"And mine was taken from me!\"\n\nThe anger seemed to warm his very body. Her arm flared with white-hot pain where he held it.\n\n\"By who?\" she asked, her voice low. She wanted him calm. She wanted answers before she ended his life.\n\n\"The Council of Mages. They banished me, and stole my name away.\"\n\n\"Banished for what?\"\n\nShe heard him chuckle.\n\n\"Everyone has their secrets, and I must have mine. What will you do, Veliana? Will you kill me? Or will you listen? I am your last hope. Your guild is crumbling, and you've lost control of Garrick, haven't you?\"\n\nHer hesitation was answer enough, so she didn't bother to lie.\n\n\"How do you know that?\" she asked.\n\nHe shook his head.\n\n\"I will answer nothing with an elbow on my throat. Let me up. I promise no harm will come to you tonight.\"\n\nHer mind whirled as she thought. He was clever, and dangerous. She could kill him, but what would that gain her? Garrick would get what he wanted, his paranoia fed. Clearly this Deathmask had a plan, but whose? Could it be the Council's? Did he lie about the banishment? No, his anger was too sincere. Despite the mask, she felt he spoke truth. Then what? What should she do?\n\nShe thought of Garrick's mockery, of his telling her how she needed him.\n\n\"Stand then,\" she said, letting him go as he released her wrist. \"And let me hear you speak.\"\n\n\"I will not tell you everything,\" he said as he stood and rubbed his throat. \"Not until I can trust you, and perhaps not even afterward. For now, just know that my assignment from the Council was to... watch over the guilds. I know of your true skill and control, Veliana. I know that Garrick was but a puppet, and you were pulling the strings. But that isn't the case anymore, is it? Something's changed.\"\n\nHe retrieved one of her daggers and tossed it to her. She caught it in her good hand and sheathed the blade. Instead of continuing, he walked over and eyed her other hand.\n\n\"If I wanted to hurt you, I'd have done so already,\" he said when she tensed.\n\nHis fingers brushed hers, feeling along the bone.\n\n\"Dislocated,\" he said. \"Bite the hilt of your dagger if you must.\"\n\n\"Just do it.\"\n\nOne after the other he yanked the fingers back into place. The pain was immense, and after the third she leaned against him, unable to stand. He held her steady, and when he finished he removed his mask and tied it around her hand as a bandage. Through the tears in her eyes she looked upon his face. The anger was gone. It had never been directed at her, just those who had banished him. She felt her curiosity grow. Just what did he plan for her guild?\n\n\"Listen to me,\" he whispered, as if suddenly worried others were listening. He leaned close, his cheek almost touching hers. \"I cannot do this alone. I desire to create something special, something Veldaren has never before seen. You won't be the new guildmaster, I won't lie about that, but you will always be there as my right hand.\"\n\n\"Why would I trade Garrick for you if my position shall stay the same?\"\n\nHe smiled, a bit of his amusement returning to twinkle in his eye.\n\n\"Because I respect you. Garrick only knows fear. Which would you prefer? And I will not replace Garrick, not entirely. My aim is greater. We will be legends in the underworld, Vel. All you must do is accept my wisdom.\"\n\nShe looked to her bandaged hand, then to his eyes.\n\n\"I must think on it.\"\n\n\"Time is against me right now, but you may have a day and a night to decide. Garrick will soon stop his tricks and poisons and midnight arrows. A time will come when he publicly orders me to be executed by the rest of the guild, regardless of the fallout. I must have you at my side when that happens.\"\n\nShe pulled away.\n\n\"Resume your post,\" she said.\n\n\"Of course, milady.\"\n\nBefore she could go, he put an arm in her way.\n\n\"That trick with your dagger,\" he said. \"The violet flame... where did you learn to do something like that?\"\n\nThis time it was her turn to smile.\n\n\"Everyone has their secrets.\"\n\nHe seemed amused, and he stepped aside so she could pass. She went into the headquarters, found her bunk, and lay down, not to sleep but to think. She felt lost and confused. There wasn't anyone within the Ash Guild she could trust for advice, but there was one woman outside the guild who would die to protect her secrets. Someone who had saved her from a horrible death at the hands of a disgusting man who'd gone by the moniker of the Worm.\n\nVeliana left her bed, changed into a darker outfit, and donned her gray cloak. She used a different door from the one Deathmask guarded, and then took to the rooftops. Once she was in the open night air, she removed the cloak signifying her allegiance to the Ash Guild and then set out to meet Zusa behind the Gemcroft mansion.\n\n# CHAPTER\n\n#\n\nIt was past midnight when Arthur Hadfield arrived at the gates of the Gemcroft mansion, escorted by nine of his soldiers. One of the guards immediately recognized him and opened the gate.\n\n\"Our lady sleeps,\" said the guard, \"but we would not turn away such an esteemed visitor in the cold of night. I pray no ill news brings you here at such an hour?\"\n\n\"Pray all you want,\" Arthur said. \"But it won't change the news I bring.\"\n\nInside the main foyer they stopped and forfeited their weapons, even Arthur. He gave the guard a stern look as he handed over his beautiful long sword, a family heirloom of five generations.\n\n\"A lash for every scratch,\" he said. \"Unless you think Alyssa would not listen to me.\"\n\n\"Understood, sir,\" said the guard. \"Please, wait here. Our lady will be down shortly; we have already sent a servant to wake her.\"\n\n\"Warm some food for my men,\" Arthur said. \"And find me something stiff to drink. I'd rather not meet Alyssa looking pale as a corpse fresh from the grave.\"\n\n\"Right away.\"\n\nSeveral servants rushed from one room to the next, haggard-eyed and clothes unkempt. Most of the guards looked a little better, but they were probably used to odd hours and the constant threat of thieves sneaking in at night. An elderly lady appeared and gestured to the soldiers to follow her.\n\n\"Coming?\" one asked Arthur.\n\nHe shook his head.\n\n\"I'll wait here. All I need is a drink. Enjoy yourselves, and don't forget\"\u2014he glanced at the servant\u2014\"to find yourselves lodging for the night. We won't go traipsing for an inn at this hour.\"\n\nIt seemed the servant got the message, and even if she didn't, he knew his men would hammer the point home. Standing in the foyer, he removed his bearskin coat and set it aside. A large fireplace burned before him, so he stood beside it and let its heat sink into his skin. When a servant arrived with a glass, he took it and gave it a taste.\n\n\"Thank you,\" he said, doing his best to hold in a denigrating remark. The lady had brought him a recent vintage, no doubt the cheapest bottle in the mansion other than what was reserved for the hired help. Probably thought they were keeping Alyssa's interests in mind since she had not ordered it for him, but they should have known better. He swallowed the rest of it anyway. It might look like brown water and taste like piss, but at least it'd still warm his bones.\n\nHe watched the fire burn as he waited, his thoughts racing through the recent events. Alyssa needed to marry soon, and with Mark Tullen dead, Arthur had removed all serious competition. Only two wrinkles remained. One was Alyssa's child, heir to the Gemcroft wealth, as well as a potential danger should he describe the ambush accurately enough to blame Arthur. The other was that strange man who had attacked them. He'd been dressed like a thief, yet none of his colors had marked him as a member of any guild. Plus there was that symbol carved in blood beside the fire. The Watcher. Arthur didn't visit Veldaren often, but it seemed things had gotten far stranger in his absence. Not for the first time he felt thankful he lived in the north, where men had to survive by the plow, the sword, or the pick, and not by the deftness of their sticky fingers.\n\n\"Lord Hadfield,\" Alyssa said, pulling him from his thoughts. He turned to her and smiled as she approached through the doorway. Her hair was immaculate, her cheeks warmed with rouge. The dark circles under her eyes were hidden with powder. Now he knew why she'd taken so long to come down. At least her clothing was appropriate for the late hour, a crimson robe tied with a yellow sash. She wrapped her arms in his and gave him a chaste kiss on the cheek.\n\n\"Forgive me for ruining your sleep,\" he said. \"It's a cruelty, waking someone at this hour, but I feared it'd be crueler risking someone other than myself bringing you the news.\"\n\n\"Enough,\" Alyssa said, holding her arms against her chest as if she were cold. \"Good news rarely comes after midnight. Whatever it is, tell me, or my mind will assume the worst.\"\n\nArthur frowned and looked away for a moment, just long enough for her to believe he was in doubt.\n\n\"You could assume nothing worse than the truth,\" he said at last. \"I'm sorry, Alyssa. Your son is dead.\"\n\nShe'd been preparing herself for terrible news, he could tell, but it didn't matter. She took a step back as if he'd slapped her. Her mouth dropped open, and her hands quivered as she pressed them to her lips.\n\n\"No,\" she whispered. Tears swelled in her eyes, then fell, smearing the powder. \"No, no, please, you're wrong, you have to be wrong...\"\n\nHe shook his head. This was by far the easiest part. None of it was a lie.\n\n\"Mark Tullen came and took Nathaniel from Tyneham, where I'd brought him for tutoring. They joined one of my caravans traveling to the city. I thought they'd be safe, but someone ambushed them several days ago, stealing the shipment of gold.\"\n\n\"Mark?\" Alyssa asked as she tried in vain to compose herself. \"Was he...?\"\n\nArthur wrapped his arms around her waist and pulled her close.\n\n\"There were no survivors,\" he whispered. \"They piled the bodies together and burned them.\"\n\nShe fell against him and sobbed. A bit of rouge rubbed onto his vest, and he wondered whether it would come off. As her cries escalated he tightened his grip, holding her against him. He gently rocked her from side to side, his cheek resting on the top of her head. He felt unprepared for her grief, and he mentally delayed his plans of marriage. She'd need time to get over this, at least three months. Perhaps if he could bring her closure he could make things progress faster, but how?\n\nShe asked a question, but it was muffled against his chest.\n\n\"What, my love?\" he asked, tilting her face up by her chin. It was the first time he'd called her that, and he knew it would carry far more impact now given the circumstances.\n\n\"Who?\" she asked. She sniffed and pulled free of his grip. \"I want to know who.\"\n\n\"I told you, someone wishing for our gold. Ruffians, most likely, come from only the gods know where.\"\n\nAlyssa shook her head. It seemed as if her skin were darkening to red, her whole body suddenly given over to rage. When she spoke again, her voice was held together by such fierce concentration he worried she had pierced his lies.\n\n\"That's not good enough,\" she said. \"There has to be something, some clue, some mistake they made. They can't make off with that much gold without others noticing. No one is that perfect, that calculated. If you know something, tell me!\"\n\nArthur felt his opening, and it took all his willpower to keep from smiling.\n\n\"There was a... symbol,\" he said, as if hesitant. \"I didn't wish to bother you with it, not when you should be grieving.\"\n\n\"I have the rest of my life to grieve,\" she said, wiping her makeup with her hands. \"What was the symbol?\"\n\n\"It was an open eye, drawn in blood. Below it was written a name, a strange name. The Watcher. I believe he is tied to the local thief guilds in some manner, though I cannot say more, for it has been years since I lived in Veldaren. Perhaps someone in your employ will better know who he is than I.\"\n\nBy the way she startled, he knew he'd hit his mark.\n\n\"How dare he?\" she said. \"He kills my son, and my... and Mark, and then dares leave his name? I'll see him flayed before me, that heartless bastard.\"\n\n\"Allow me to help in the search,\" Arthur offered.\n\n\"No,\" she said, shaking her head. \"This is my loss, and my fault. I sent Mark for him when I should have left him in safety.\"\n\n\"It was _my_ caravan, don't forget.\"\n\nShe looked at him, and he forced a mask of anger across his face. He had to seem regretful, not eager. He had to seem furious at the loss of his men, not pleased with Mark's death. He did his best, and it seemed like she bought it.\n\n\"Very well,\" she said. \"Kill him if you must, though I'd prefer him alive.\"\n\n\"Torture and vengeance shouldn't belong to a woman so beautiful as you.\"\n\n\"Then blame the world for giving me this sorrow. If the gods are kind, I'll be the one to cut this Watcher's throat and feel the blood spill across my hands.\"\n\nAfter a long pause she asked, \"Did you bring... the body?\"\n\n\"No,\" Arthur said, shaking his head. \"As I said, it was a great pyre, and we left it alone.\"\n\n\"So my son will spend eternity in some common grave, his ashes trodden on by horses and oxen? For all your attempts at kindness, you failed me terribly, Arthur. You should have brought me his bones.\"\n\nShe walked to the fireplace and retrieved a bell from its mantel. When she shook it, two servants came running.\n\n\"Please find quarters for Lord Hadfield,\" she told them, then turned to Arthur. \"I need time to rest, and I feel I would be poor company for you. Good night.\"\n\nHe bowed and followed the servants. They showed him to his room, a nice enough place: large bed, soft mattress, and thick curtains he pulled back from the windows to let in the starlight. Given such haste, there were no logs or coals in the fireplace, and the room felt only marginally warmer than the outside. He put his coat back on and sat atop the bed. His joints creaked, and he lay upon the mattress trying to will his muscles to relax. Mere minutes later he heard a knock on his door.\n\n\"Come in,\" Arthur said, wishing he still had his sword.\n\nBertram, the old advisor who had worked for the Gemcroft family for at least fifty years, stepped inside.\n\n\"I've yet to speak with Alyssa about these matters,\" the man said, \"but I know her well enough to anticipate her response. She will put me in charge of the boy's funeral. Did you bring Nathaniel's body with you?\"\n\nThe question stung, just as deeply as when Alyssa had first asked. He had no body...\n\n\"Burned,\" he said. \"Back with the caravan. Alyssa didn't take too kindly to that when I told her.\"\n\nBertram frowned. \"I do not blame her. It would be greatly distressing if we didn't have something to bury for the funeral. Not that it matters, in my opinion. Bones are bones after a fire reaches them, yes?\"\n\nArthur stared at the old man, trying to understand what was going on. Was he helping him, or fishing for information?\n\n\"I doubt Alyssa would agree,\" he said, erring on the side of caution.\n\n\"She cannot judge what she does not know.\" Bertram turned for the door, put his hand on the handle, and then stopped. \"I will be very busy over the next few days, and will have no time to venture into the wild. Perhaps you, or some of your men, might retrieve the body for me? I would greatly appreciate it.\"\n\n\"Anything I can do to help.\"\n\nBertram glanced back and smiled.\n\n\"Claim her hand, and you'll have done all that I could ask. Good night, Lord Hadfield.\"\n\nArthur waited a few minutes, then called for a servant.\n\n\"Bring me one of my soldiers,\" he ordered. \"A man named Oric.\"\n\n\"Yes, milord,\" the servant said, bowing quickly before vanishing. Trying to keep his mood from souring too much, Arthur paced and waited. When the knock on his door finally came, he had no chance to answer it before Oric barged inside.\n\n\"You needed something?\" Oric asked. He was an ugly man: thick cheeks, round jaw, and flat nose that made him look like his mother had mated with a pig. He was skilled with a blade, though, and meaner than any soldier he'd ever employed. Not a brilliant man, but he could follow events as they happened, and every now and then he'd have an insight that left Arthur pleased.\n\n\"Who did you work for before coming to me?\" Arthur asked.\n\n\"Mostly for the Conningtons. Past year or so they got shy when it came to killing thieves, so I went looking for more enjoyable work.\" He grinned. \"Everyone told me to talk to you. I think you have yourself a reputation, and not a good one.\"\n\n\"Hard times require hard men,\" Arthur said. \"Do you have any friends that might still be here in Veldaren?\"\n\n\"A mercenary never has friends, not if he wants to live long enough to get his pay. And sure, I have some contacts that should still be around. What you thinking?\"\n\n\"That man in gray who attacked us at the caravan, the one calling himself the Watcher... we need to bring his corpse to Alyssa so she might move on from her son's death. That, and who knows where his loyalties lie? He could do me great harm by telling the right pair of ears what actually happened.\"\n\n\"It won't be easy,\" Oric said. \"I never met him when working for Leon, but we all heard about him. The thieves can't stand him, but they can't find him either. Everyone's always pointing fingers, convinced he's a member of this or that guild. My thinking? He belongs to none, just wants them all dead.\"\n\n\"I don't care,\" Arthur said. \"Whoever you hire, make sure they're good enough to handle the job. Cost is no object.\"\n\nOric nodded. \"What about the boy?\"\n\n\"I must stay here with Alyssa in her time of need. Take half my soldiers and ride north. We must make sure he did not survive.\"\n\nOric's grin was ear-to-ear.\n\n\"If he ain't frozen in the woods somewhere, we'll find him. Don't worry, Arthur. Might not be how we wanted it, but when it all boils down, you're in control. Just keep whispering them sweet words in that pretty ear of hers. I'll take care of any blood-spilling.\"\n\n\"One more thing,\" Arthur said. \"Alyssa wants Nathaniel's body.\"\n\nOric raised an eyebrow. \"That's an interesting request to fill, given the circumstances.\"\n\n\"And I expect you, and you alone, to handle it. Remember, I told them the body was burned.\"\n\n\"Consider the problem solved,\" Oric said, heading for the door. \"We never talked, and you know nothing of this. Watch your tongue. We're sleeping in the den of lions now.\"\n\n\"We're the lions in this den,\" Arthur said, flashing Oric a grin. He followed Oric out the door, but whereas the mercenary headed for his crowded quarters with the rest of the guards, Arthur headed to the eastern half of the complex, to where Alyssa no doubt lay alone, tired, and in desperate need of his comfort.\n\nAlyssa waited until her servants escorted Arthur Hadfield to his room before she fled to her own. She was only halfway there when she stumbled and fell sobbing in the carpeted hallway. In the quiet of the night she felt alone, and if any servants or guards were nearby, they allowed her privacy for her grief. She thought of Nathaniel, her son, her wonderful, handsome son. A thousand memories flashed through her mind, all of them tainted with sorrow. His smile she'd never see again. His laugh she'd never hear again. The way he'd cried at night, the way he'd nuzzled her breast when only a newborn, the way he... the way he...\n\nMore sobs. She felt close to breaking, as if her sanity would pour out with her tears. She'd lost many over the years, friends, her father, but why Nathaniel? Why him? Why now? How could she have made such a foolish mistake?\n\nOver and over she slammed a fist against the floor. Not her fault. Not her fault, damn it all, not her fault! It was them, the thieves and their greed and lust. It was this whole conflict, a decade of paranoia and bloodshed and poisoning. Most of all it was that Watcher, a monster unleashed against her by the thieves. Whoever he was, whatever he was, he'd pay. All of them, they would pay.\n\n\"I'm sorry,\" she heard Zusa say. She looked up, wiped her nose with her robe, and then nodded at the woman. Zusa sat opposite her in the hall, her knees pulled up to her chest. She still wore the strange wrappings of her former sect.\n\n\"He was supposed to be safe,\" Alyssa said, trying to regain control. Her voice quivered but a little. \"Safe, and now he's dead. Did you talk to any of Arthur's soldiers?\"\n\n\"They all say the same. They came with Arthur along the road and found the caravan attacked, with the bodies gathered together in a great pyre. Their only clue was a symbol of an eye, that of the Watcher.\"\n\n\"Who is he?\" Alyssa asked. \"What is he?\"\n\n\"I don't know.\"\n\n\"Then find out, Zusa. Whatever it costs, whatever it takes, find him. I have never ached for something like I do for this. I want Nathaniel's killer brought to me. I want him to die by my hand. Can you do this? Can you find him?\"\n\nZusa stood and then bowed low.\n\n\"The dark of the streets has always been my home,\" she said. \"Nothing can hide from me. I will find him, I swear it.\"\n\nAlyssa accepted her offered hand and stood. She kissed Zusa's fingers, then bowed herself.\n\n\"Thank you. Send Bertram to my room after a few minutes. Wake him if he isn't already.\"\n\nAlyssa hurried to her room, wishing to wash her face of the garish rouge and powder she'd put on for Arthur. Once there she dipped a cloth into a basin of cold water, left there from when she first went to bed. Off came her painted face. She was still washing when she heard the knock at the door.\n\n\"Come in,\" she said.\n\nBertram entered, and he looked about half as bad as she felt. Dark circles rimmed his eyes, and his face was covered with uneven gray stubble.\n\n\"My dear child,\" he said, taking her into his arms. She set the cloth down and leaned against him. She felt so tired, so lost.\n\n\"It's like a terrible dream,\" she said softly. \"One I can't seem to wake from. What did I do to deserve this, Bertram?\"\n\n\"Nothing,\" he said. \"No woman should endure this, but endure you must. The Gemcroft legacy must survive, no matter the hardships. And we shall, Alyssa, we shall. Whatever help you need, I am here for you.\"\n\n\"Thank you,\" she said. \"It would help me greatly if you would prepare the... the services.\"\n\nHe nodded. \"He will have a fine funeral, one worthy of his bloodline.\"\n\nShe bit back a bitter comment. While her son was alive he'd been displeased with Nathaniel's bloodline. Now Nathaniel was dead, and his threat of succession was ended, Bertram seemed ready to forget all that. No, she scolded herself. She was overreacting. Bertram had never said a harsh word about Nathaniel, at least not an undeserved one.\n\n\"There is something else,\" she said, dipping her cloth in the basin and wiping underneath her eyes. \"Call in every loan we have. Whatever stored grain, minerals, property, I want it all sold. We need gold, lots of it. Find every mercenary who needs work, no matter how expensive. Hire them, outfit them if need be.\"\n\n\"You wish to declare war against the guilds?\" asked Bertram, allowing a tiny bit of doubt to creep into his voice.\n\n\"We're _already_ at war, or have you forgotten? Come the night of Nathaniel's funeral, I want the streets to run red with the blood of thieves. I don't care what cloak they wear; I want them dead.\"\n\n\"You'll only reignite their anger, and ruin whatever progress we've made in the last...\"\n\nShe spun on him, furious.\n\n\"I don't care! We've suffered and played the coward. No more peace. No more hope. Red, Bertram. I want the streets of this city red.\"\n\nBertram muttered and waved his hands, as if he could not decide what to say. Despite her grim mood she took pleasure in his distress. At last the old man regained his composure.\n\n\"But what will the king think when we flood his streets with chaos?\"\n\n\"The king is a coward. He won't dare refuse me, and neither will you.\"\n\n\"So be it,\" he said. \"We'll have the funeral three days from now. Come that night, you will have your folly. You run the risk of bankrupting a century of wealth, Alyssa. Is your vengeance truly worth so much?\"\n\n\"That and more,\" she said. \"Go. You have work to do.\"\n\nHe bowed and left, looking far from pleased. Both matters settled, her revenge in motion, she finished washing and collapsed onto her bed. She tried, but sleep remained a distant hope and nothing more. After half an hour she heard another knock on her door, just one, but it was firm. She ignored it. Thirty seconds later she heard it again.\n\n\"Come in,\" she said, removing her arm from over her eyes.\n\nIn stepped Arthur, and he paused at the door.\n\n\"I don't mean to intrude,\" he started, but she shook her head. He crossed the room and climbed into the bed beside her. His clothes were on, and for that she was thankful. His arms wrapped around her, and in their comfort she broke down once more. He was something steady, dependable amid the chaos overwhelming her. He said nothing as she cried, only gently stroked her hair and held her against him. His body was warm, and it felt pleasant. After a while he spoke.\n\n\"If there is anything you need, I am here. It doesn't matter the hour, nor the reason. I want you to know that.\"\n\nShe clutched his hand in hers and squeezed. Her whole body ached, and her temples throbbed. Her tears still ran down her face, but they were silent. She closed her eyes, pressing her face against his chest and focusing on the sound of his breathing. As long as she thought about that, only that, maybe she could fall asleep. Maybe she could forget the whole damned night, and come the morning the nightmare would be over. Maybe, just maybe...\n\nShe slept.\n\n# CHAPTER\n\n#\n\nIn searching for the Watcher, there was one person Zusa knew would be best to meet with first. Behind the fenced Gemcroft estate was a small empty building. For years it had been nothing more than a shed for tools to be used by the servants tending the gardens and lawn, but Alyssa had turned it into a gift for Zusa. Within were a bedroom and a training room, the floor soft and padded, the walls lined with paintings of distant places. Zusa intended to gather a few things for her task of hunting down the Watcher, but when she stepped inside she was instead surprised to find Veliana waiting for her.\n\n\"I know I'm a day early,\" Veliana said. She'd taken off her cloak, and wore only skin-tight clothing of blacks and grays. \"I've come not just for training, but for advice.\"\n\nZusa removed her own cloak and set it atop her bed. Veliana's boots lay beside the door, and she padded barefoot to the center of the room.\n\n\"Tell me while we spar,\" Zusa said. \"I still feel sleep's allure, so I need the awakening.\"\n\nThey both drew a pair of daggers. No training weaponry for them; Zusa had insisted on real blades. She trusted her skill to make sure she caused no serious injury, as well as to prevent Veliana from doing the same. Over the past five years Veliana had closed much of the gap between them, so now if either scored a hit she counted it a well-earned rarity.\n\n\"Have you heard of a sorcerer named Deathmask, or perhaps Death's Mask?\" she asked as she stretched. Zusa shook her head. Veliana didn't seem surprised. \"Thought I'd ask anyway. He appeared about a week ago, a dangerous man. He has plans to kill Garrick, though I don't know how. I think he has a chance.\"\n\n\"Will you kill him?\"\n\nVeliana feinted, then slashed low, fully expecting Zusa to block. Their daggers connected, and as the steel rang out they thrust and parried, resuming a skilled dance they had perfected over the years, a perfect give-and-take of cuts and dodges, parries and thrusts. They talked as they fought, though they were a little out of breath.\n\n\"I'm not sure if I still can, nor if I want to. Garrick has turned against me, thinks he can survive without me. He may be right, though he was a lying coward when I first thrust him into his role.\"\n\nZusa upped the pace, forcing Veliana on the defensive as she spun and slashed.\n\n\"Men change over time, as do women.\"\n\n\"But not like this. It's too sudden. I feel like I'm missing something obvious.\"\n\n\"Perhaps you are, and that is why you miss it. What is it you wish from me?\"\n\nVeliana leaped away, but instead of her gaining a moment's breather, Zusa rushed in, her daggers leading. After she parried both, Veliana struck Zusa in the chest with an elbow and then pushed her back.\n\n\"I must make a choice, but I don't know which is the right one. You know me best, Zusa. What should I do?\"\n\nZusa pulled back from her attack and rubbed her chest. Veliana must have been terribly distracted to have thrown so much strength into the elbow.\n\n\"I see many choices,\" she said. \"Find what suddenly gave Garrick testicles and then cut them off. Join this Deathmask and solidify your position as second-in-command. Weave your own plans to remove Garrick. Accept your diminished role, and wait for the inevitable dagger to remove you completely. It's all yours to decide.\"\n\n\"I'm tired of the games,\" Veliana said. \"I have no time to investigate Garrick. He'll move against me soon, I know it. I promised Deathmask an answer by tomorrow night.\"\n\n\"Will he kill you if you say no?\"\n\nVeliana laughed. \"He might. I know nothing of him.\"\n\n\"Then how can you trust him?\"\n\n\"Because I've fought him. He didn't kill me when he had the chance. He never showed fear, even when I had the chance to end his life. He's brutal, terrible, and driven. Whatever his goal is, he'll succeed... I guess I'm scared I'll be in his way.\"\n\nZusa twirled her daggers and motioned for another spar. \"Then join him, and do so without pause or regret. Garrick has turned against you, inviting this betrayal. Hear Deathmask's plan, but always watch and listen. Every plan can be turned to your favor.\"\n\nIt was Veliana's turn to be on the offensive, and she took to it with a wild fury that worried Zusa. Normally the woman was more controlled. This Deathmask must have upset her greatly. Did she feel guilt for betraying Garrick? Or was she too proud to agree to anything that left her second in the Ash Guild? Whatever the reason, Vel's daggers lacked their normal grace, and Zusa had to leap away several times to prevent blood from staining the floor.\n\n\"Restrain yourself, girl,\" Zusa said at last, after a desperate thrust nearly opened her throat. \"If this choice disturbs you so much, I will choose for you, so you can concentrate and not kill me through your carelessness.\"\n\n\"I'm sorry,\" Veliana said, sheathing her daggers and leaning against the wall. She sounded terribly out of breath. \"I should go.\"\n\n\"No,\" Zusa said. \"I have a question for you as well. Someone murdered Alyssa's child. I must discover who.\"\n\n\"Someone killed Nathaniel?\" Veliana asked. \"I thought you'd persuaded her to move him up north and out of the city.\"\n\n\"I had. She called him back. He died on the northern road.\"\n\n\"It wasn't the Ash Guild, I promise,\" Veliana said, just in case the thought had crossed Zusa's mind. \"I'd never let Garrick do something that low, and he's not yet reached a point where he can plan something so large behind my back.\"\n\n\"Are you sure?\"\n\nShe paused a moment, thinking, and then sighed.\n\n\"No, I'm not. His control might be greater than I've realized. My opinion of him was far too low, and it's blinded me to his ambitions. He's not content to be a puppet. Still, I can't think of a reason why he'd have killed Nathaniel, nor how he'd have known the boy was on his way back. Is there anything else?\"\n\nZusa took her dagger and scrawled the symbol found at the caravan exactly as the soldiers had shown her.\n\n\"That,\" Zusa said. \"Tell me all you know of him, this... Watcher.\"\n\n\"We first heard of him about three years ago, but honestly, he might have been killing us for longer. Given the amount of infighting, and the Trifect's war against us, we probably blamed others for his early murders. But then we started finding these runes, an eye here, or the letter _W_. Perhaps he thought us dense, or his confidence hadn't grown yet. Either way he started killing more, and leaving his marker larger, clearer, and often in blood. He kills thieves of all guilds, with seemingly no preference. Every guild has accused the others of secretly harboring him, but we've never had an ounce of proof. Whoever he is, I believe he has a profound hatred of all thief guilds, and he's also incredibly good. Far too many have died by his hand, and those who survive can only speak of a face shrouded in shadow and hidden by a hood and many cloaks.\"\n\n\"Has he ever attacked the Trifect?\" Zusa asked.\n\nVeliana shrugged. \"If he has, we don't know about it. Not that any of those rich bastards would tell _us_.\"\n\nZusa frowned, for she had never heard of the man interfering with Gemcroft activities. What could Alyssa have done to spur this strange killer of thieves into attacking her son?\n\n\"I must find him,\" Zusa said. \"Is there anything you know that can help me?\"\n\n\"Find him? What for?\"\n\n\"He killed Nathaniel. I must give my lady her vengeance.\"\n\nVeliana crossed her arms and shook her head, looking all the more upset.\n\n\"If the Watcher killed him, something else is going on. Perhaps he thinks you've secretly colluded with one of the thief guilds. Maybe he was confused. Or maybe he's just insane and out for blood. We know nothing of him other than that he is a skillful killer.\"\n\n\"Regardless of the difficulty, he is my prey, and must be found. My honor is sworn upon it.\"\n\n\"Then I wish you luck,\" Veliana said as she sheathed her daggers and swung her cloak over her shoulders. \"Many have tried, and no lead we've ever found has panned out. He might as well be a ghost. If you wish to find him, your best bet is to scour the streets at night and listen for the sound of combat. If you don't catch him in the act, I doubt you ever will.\"\n\n\"Will you not stay?\" Zusa asked when Veliana opened the door to leave. \"Your daggerwork is excellent, but you still neglect practice of your spells.\"\n\n\"I should be going. Deathmask seeks his answer, and I must prepare until then.\"\n\n\"Good luck,\" Zusa said, bowing. \"May you make the right decision, and in time find peace with my lady and her family.\"\n\nVeliana pushed open the door, and as the chill wind blew in she sadly shook her head.\n\n\"Long as Thren Felhorn lives, this war will continue. Too many fear him, and many more live in the palm of his hand without even knowing. He's a bitter, angry man. Sometimes I think all of Veldaren will burn before the end.\"\n\n\"Perhaps it is not Garrick you should plot against, but Thren,\" Zusa said.\n\nVeliana's smile turned bitter.\n\n\"We did, once,\" she said. \"And the Ash Guild lost its best leader in decades because of it. I'll see you soon. Safe travels.\"\n\n\"To you as well.\"\n\nZusa had hoped talking to Veliana would illuminate matters, but instead it had made things worse. An assassin had been killing thieves for several years, yet not once had any of the guilds discovered his real identity. Who could be that skilled? And what had drawn that skill against her lady? What would happen if she did find him? Did she have the ability to take him down?\n\nOnly one way to find out, of course. Dawn was fast approaching, less than an hour away. Still, in that last twilight moment, perhaps she might find word of the Watcher.\n\nShe scoured the rooftops, an eye always kept on the streets. She saw several deals, a whore earning her pay, and two men dying so their killers might make off with their gold. No Watcher. Up on the rooftops she was alone.\n\n\"You must have left people alive,\" Zusa whispered to herself as she watched the sun rise. \"You've hurt many opponents, though none will work together. But I am not one of them. I will piece it together. I will discover who you are. Perhaps, in time, I will be the one leaving my mark for you.\"\n\nShe returned to the Gemcroft mansion, and in her room she slept through the day. Come nightfall she had an underworld to interrogate.\n\nHaern woke to the sound of the door banging open against the snow. A sliver of light lanced across his eyes. Dawn was fast approaching, but the snow magnified what little light crept over the horizon. He rubbed his eyes, then looked again. Matthew was dressed in many layers of coats and furs, and his two older sons were dressed similarly. A glance around showed the daughters still slept.\n\n\"Need to break the ice so our cattle can drink,\" Matthew explained, keeping his voice low so as to not wake the others. \"Forgive me, but it's an early morning here on the farm.\"\n\n\"Forgiven,\" Haern said, rising. He pulled his cloaks tight about himself. He needed to piss, and he wasn't looking forward to the excursion in what little clothing he had.\n\n\"Here,\" said Matthew, tossing him a coat. \"It's an extra, and with what you paid me, you certainly deserve it. I have a feeling you won't be staying around much longer.\"\n\n\"Your feeling is right,\" Haern said, inspecting the coat. It was old and gray, the fur too faded for him to accurately guess what animal it had been made from. Still, the lining remained intact and well cared for. He slipped it on and nodded his thanks.\n\n\"Come on,\" Matthew said to his boys. \"Let's go. My wife's in the kitchen cooking if you'd like a bite to eat, Haern.\"\n\n\"I would, but let me take care of other things first.\"\n\nWhen he came back inside after finishing his business, he passed through the curtain into the kitchen. Sure enough, the lady had cooked him a bowl of oatmeal and flavored it with honey.\n\n\"Thank you,\" Haern said, accepting the bowl and using his fingers to scoop the oatmeal into his mouth. \"What is your name?\"\n\nShe kept busy scrubbing and tending to the rest of breakfast, all so she could avoid looking him in the eye when she talked.\n\n\"Evelyn,\" she said.\n\n\"Thank you for the meal, Evelyn. How fares the boy?\"\n\n\"I looked in on him while you slept. His fever still burns, and I don't think he'll get to keep that right arm. Don't worry, though, if it comes to that. I've done it before, and not just on animals. For most of my neighbors, I'm the closest to a healer we got.\"\n\n\"Your husband explained my request?\" Haern asked.\n\nShe finally looked at him, and he liked the strength he saw in her.\n\n\"He told me enough, and I have a brain to figure out the rest. We'd have taken him in if you'd only asked. Didn't need the coin, or the threats. I pity the life you've led if you thought either was necessary.\"\n\nThe comment stung, far more deeply than she'd probably meant it to.\n\n\"Thank you for your hospitality,\" he said. \"I must be going. Take care of the boy.\"\n\n\"We will. Safe travels to you, Haern. That bag on the table is yours. It should last you until you reach Felwood, assuming that's the direction you're headed.\"\n\nInside was a small selection of salted meats. He took it and left without checking on the boy, fully trusting Evelyn and her husband to know what was best. He wanted to get back to Veldaren, to the world he understood. He'd watched the farmer talk to his boys. Matthew was raising them to be like him, just as his own father had. But there was no malice, no underlying threat of violence to ensure perfection and obedience. The obedience was expected, sure, but he'd felt the love in that household. Living under Thren's roof he'd felt only paranoia, expectations, and disappointment. He'd loved Senke, loved Kayla, loved Delysia, even his older brother who had always been cruel to him in return. None of their fates had been kind because of it. At least Delysia had lived, though he'd lost her to the temple of Ashhur.\n\nThe pond was not far from the road, and he saw Matthew in the distance. Haern waved, and Matthew waved back. He promised himself he would return, not just to check on the fate of the boy, but also to have another night of sleep like he'd just had. So many nights and days he had slumbered on the side of the street, and he'd forgotten the comforts of a warm bed. Perhaps it was time to consider paying for lodgings at one of the inns, his various personae be damned.\n\nThe snow had stopped, and the coat did wonders to keep him warm. He nibbled on the meat Evelyn had given him, and despite its salt he found he enjoyed the flavor. He walked along the road and tried to determine how far he was from Veldaren. For much of his walk carrying the boy, he had been in a frozen delirium. He couldn't even guess how many miles he'd traveled, and like a fool he hadn't asked either Matthew or Evelyn how far to Felwood before he'd left. Oh well. She'd given him enough to reach the castle, and guessing by the food it should be four days, three if she assumed him a heavy eater. From there it'd be a week or so back to Veldaren.\n\nBack to Veldaren. Which meant away from the mines, away from Tyneham, the reason for his trek north in the first place. Was he really willing to give up already?\n\nLooking about the snow, and feeling the cold still nipping at his exposed skin, he decided he was. Whatever was going on with the Serpent Guild, he'd deal with it within Veldaren's walls. That was his home. That was the world he understood. Let the wild lands and dirt roads belong to the brigands and the rangers.\n\nNear midday he heard the sound of hoofbeats. He felt his spirits brighten. If he could beg a ride he might reach the castle far sooner. But the woods had played tricks on his ears, and the riders came not from the north but the south. His pulse quickened at the sight of riders approaching from the distance. Clearly emblazoned upon their tunics was the symbol of the men who'd attacked the caravan. He hurried off the road, wishing he'd had time to hide his footprints. It'd be an easy trail to follow. Damn the snow!\n\nThey rode right on by. If they saw his tracks they didn't care about them in the slightest. Haern let out the breath he'd been holding and returned to the road. Tightening his coat, he hurried on, determined to gain as much distance as possible before nightfall. Only once did he look back, and he prayed the riders had passed by the farm as easily as they had his own tracks.\n\nEvelyn had given him a small piece of flint and another of steel in his bag of food, which he found immeasurably kind. Off the road he built a fire and slept by it through the night, waking every few hours or so to toss a log upon it and poke the embers with a stick to reignite the flame. He ate lightly come morning, just in case it took him longer than expected to reach Felwood. He kept his eye out for more riders, but none came. He passed another caravan moving north, loaded with salt and farming equipment. The travelers offered him a ride, but he smiled and gestured south.\n\n\"Heading the wrong way,\" he said before continuing on.\n\nNot long after he'd wolfed down the rest of his food, he reached the forest of Felwood. From there he continued until he reached the castle. He still had a few coins from the caravan, and he used them to pay for lodgings, food, and a warm room. He left come morning, feeling worlds better than he had before.\n\nThe days passed, and he continued his travel. Fires at night kept him comfortable, and steadily the weather warmed, a front of southern air coming along and mocking the snow. At last he reached the King's Forest. Heartened, Haern jogged at a steady pace. Once he curled around the woods he'd arrive in no time at Veldaren. He couldn't wait. Never before had he realized how much he considered the city home. It didn't matter that his trip had ended as a colossal failure, leaving him no wiser as to the Serpent Guild's mystery gold. Outside the city he was out of his element, and he'd remember that the next time he decided to pursue a lead that went beyond Veldaren's walls.\n\nTwenty minutes later he saw smoke rising from farther ahead. Wary of the cause, he upped his pace while slipping closer to the forest so he might hide at a moment's notice. He rounded a bend and then stumbled upon a terribly familiar sight. A single wagon was under attack, but instead of horsemen, he recognized the attackers as members of the Wolf Guild. He counted eight of them circling the wagon, most holding bows or crossbows. From where he stood he couldn't see any of the defenders, but from the way the Wolves stayed low, refusing to approach, he knew they were still alive.\n\n\"I leave for a spell and you grow brave enough to assault travelers in daylight?\" Haern whispered as he peered around a tree. Of all the guilds, the Wolves were the ones most likely to venture beyond the city, sometimes attacking merchant caravans, sometimes just attacking travelers for a bit of fun away from the danger of the city guard. But this, this almost felt like a slap in the face. The walls of Veldaren were in sight. The Wolves' lack of fear of the city guard didn't matter. But to be so unafraid of _him_? His pride was already wounded. This was more than enough to piss him off.\n\n\"No matter where you go, you won't be safe from me,\" Haern whispered, drawing his swords. \"Perhaps it is time I delivered that message loud and clear.\"\n\nHe stayed close to the tree line, and once within fifty yards of the wagon he vanished into the woods completely. Three of the Wolves hid at the edge of the forest, using trees as cover while they fired their crossbows. Haern swung wide so he could approach them directly from behind. He heard them muttering as he neared, offering each other advice on where to shoot or where they thought the defenders were hiding.\n\nHaern cursed the vegetation as he neared. He'd heard of men so accustomed to the wild that they could pass across dry leaves without making a sound, yet he crushed twigs and brushed at leaves no matter how stealthy he tried to be. What he'd give for a paved road and the shadows of a building. The three were too focused on the wagon, though, to notice what little noise he made. He thanked Ashhur for small favors.\n\n\"Watch for a hand,\" the rightmost Wolf said. He looked older than the others, and Haern wondered if he was their leader. \"Don't let that yellow bastard have even a moment, or we're all dead.\"\n\nHaern was less than five feet behind him. With his swords drawn he took another step, amused that they were so afraid of those in the wagon. Had they bitten off more than they could chew? And who might this \"yellow bastard\" be? It didn't matter. He was out of time. Already the Wolves on the other side were closing in, readying daggers and seemingly no longer afraid of the defenders inside. Deciding the one on the right was the most dangerous, he rushed in, his swords leading.\n\nHis first attack sliced through the Wolf's back and into his lung. Haern didn't bother muffling his scream or holding him steady, for the other two were too near. He slashed with his left arm, hoping for an easy cut, but the thief fell just out of reach. Twisting his blade free, Haern kicked away the dying man and turned his attention to the other two. The closest tossed his bow and drew a dagger, but the other...\n\nHaern dropped to his belly, the crossbow bolt screaming over his head. The Wolf with the dagger dove after him and he rolled, deflecting the thrusts with his swords as he tried to gain distance. He rolled his knees underneath him and then kicked, leaping backward and to his feet. Instead of pressing the advantage his opponent remained back, a grin on his face.\n\n\"Idiot,\" said the thief as his comrade raised his reloaded crossbow.\n\nTwisting his cloaks, Haern hoped to confuse him, but as a sharp pain bit deep into his shoulder he knew he'd only partially succeeded. He continued his spin, using his cloaks to obstruct their view. It'd only gain him a moment, an extra step closer, but if they were staying defensive, hoping to down him with arrows instead of blades...\n\nHe pulled out of the spin, putting every bit of his strength into his jump. He crashed into the closest, pure luck keeping the thief's dagger from impaling him. As they hit the ground Haern twisted so his elbow slammed against the man's throat. The Wolf spewed blood. Before the other could respond Haern lashed out, knocking the crossbow off its aim. The third bolt struck a tree, its dull thud music to Haern's ears. Without a melee weapon his opponent had no chance. Haern's assault was wild and brutal, with no hint of defense. Two slashes took out the man's throat, and a third across his hamstring brought him down to the dirt to die.\n\nFinally given a chance to breathe, Haern cursed and grabbed the bolt in his shoulder. It was deep in his flesh, and a quick glance at the man's quiver showed barbed heads he wouldn't dare pull out. Gritting his teeth, he recited a mantra he'd been taught as a child, one to help him ignore pain. He clutched the shaft tighter. Another recital, followed by a deep exhalation. He pushed the bolt through and out the other side.\n\nHe screamed.\n\nTossing the bolt, he leaned against a tree and struggled to catch his breath. It didn't look as if the bolt was poisoned\u2014another lucky break. Evidently the Wolves hadn't thought their upcoming ambush dangerous enough to spend the time and coin applying some. He looked to the wagon, trying to assess the situation. He couldn't see those on the other side, but he saw one Wolf lying dead upon the road, his body curiously aflame. That left four alive at the most. So far none appeared to have detected his intervention, for which he was thankful. He needed another moment to recover.\n\nBut then that moment vanished, for the wagon caught fire.\n\n\"Shit,\" he muttered. One of the Wolves must have tossed oil and a torch. Black smoke billowed to the air, blocking nearly all his view of events. Knowing the thieves would be rushing to cut down the survivors, he charged. Pain spiked up his entire left arm, and the sword hung limp in his hand as he ran. He'd block with it if necessary, but it seemed the killing would be restricted to his right.\n\nA figure crawled out of the smoke toward him, a red-haired woman in white.\n\n\"Run to the trees!\" he shouted to her, not stopping. He swerved about her and leaped straight into the smoke. The heat was tremendous, but so far the fire was restricted to the outer covering of the wagon. No survivors remained within. He saw a gap in the tarpaulin and jumped.\n\nJust before he landed and rolled, he had a split second to survey the fight and react. Four Wolves formed a half-circle around the wagon, easily identifiable with their black cloaks striped with gray. Three men faced them, one in yellow robes wielding a staff, another in gray parrying with two maces, and the third a portly man hiding back with a single club to protect himself. There was something tremendously familiar about the way the man with the maces fought, but Haern had no time to consider it. He rolled closer to the fat man, no doubt the caravan's driver or owner. He seemed the least skilled, unable to fend off the single Wolf who weaved from side to side in front of him.\n\nHaern kicked out of his roll, using his good arm to run the Wolf through. Their collision sent them both tumbling, and Haern screamed as he felt something hard strike his wounded shoulder, screamed again as a sharp pain pierced his stomach. He rolled off the corpse and saw blood, his blood, covering the thief's dagger. This time his collision had not been so lucky. Struggling to stand, he turned to the others, his vision a blur of pain, smoke, and tears. One of the two fighting the man in gray had pulled off to address the new threat, and Haern put his swords in position and tried to feign confidence.\n\nHis opponent wielded two short swords, and he chopped with both, hoping to overpower Haern. Not a bad strategy, given his condition. Haern crossed his swords and blocked, the nerves in his wounded shoulder shrieking in protest at the collision. Twice, three times the man chopped, as if Haern were a wall to be broken down. The third time Haern's left arm gave out, and he twisted to avoid the deathblow. He feigned a retreat, but then kicked his right foot out, tripping the Wolf. He slashed with his good arm, but it wasn't lethal, just a cut across the thief's chest. It bought him time, and he retreated. Blood flowed across his shirt and down his pants. He felt its warmth along his left arm as well. He coughed, and hoped it was only the smoke, not something worse, that caused it.\n\nHis opponent, infuriated by the cut, charged like a mad animal. Haern braced his legs and met it head on, just barely slapping the thrusts aside. Again they collided, but this time he was better positioned. His knee slammed into his attacker's groin, and he let his wounded arm absorb most of the impact. When the Wolf collapsed to the ground, Haern practically fell upon him. The sword dropped from his left hand, but he stabbed with his right and leaned all his weight upon it. The blade pierced the thief's belly and bit into the dirt, pinning him there. He thrashed for a moment as he bled out, then went still.\n\nHaern only felt marginally better than the man he'd killed. The collision had torn the cut on his arm further open, as well as angering the arrow wound. His stomach still ached. He didn't know how deep the wound went, but it felt horrendous. He struggled to stand but couldn't. At last he yanked out his sword and fell to his back, his breath coming in hurried gasps. So much for leaving the guilds a message. So much for inspiring terror. He'd killed five, only five...\n\nThe sounds of fighting ended. His head swam. A man leaned over him, a face he recognized from his past. Another joined it, younger and female. He was delusional now, he realized. How else to explain why two people, one dead, one missing, spoke down to him, their voices muffled as if they were speaking through water? How else to explain why Senke was telling him to hold on? Or why Delysia was tearing at his clothes to see the wound in his stomach? He felt pressure there, and then his vision turned yellow, all shapes outlined in red. Sound faded, and then he saw nothing at all.\n\n# CHAPTER\n\n#\n\nShe knew Garrick would want an explanation, but Veliana delayed him for several hours. The longer he wondered and worried the better. She wanted him to feel belittled, to realize her contempt for him. Anything else might make him think things were different. At last, when the day neared its end, one of their young guild rats found her at a tavern on the other side of the city and informed her of Garrick's request.\n\n\"Tell him I'm on my way,\" she said, flicking the boy a copper piece. \"But I still plan on finishing my drink first.\"\n\nShe nursed it for another half hour. During that time several men arrived, all wearing the dark gray of the Ash Guild. They were recent recruits to the guild, men she knew very little about. Garrick's men, then. Again she cursed herself for being so blind to the man's ambition. Of course she'd vetted them, knew their names, but that was the limit of her influence upon them.\n\n\"Garrick's waiting,\" said one. He was named Gil, if she remembered correctly. Why had she let him in? He looked like a dog had shit out a muscular version of itself that happened to walk on two legs.\n\n\"Surely he has more important problems,\" she said, draining the last of her drink.\n\n\"Than insubordination? No, Veliana, he doesn't.\"\n\nShe shot him a wink as she stood.\n\n\"Lead on, boys. Three to one to take me to the dance? I feel honored.\"\n\n\"Shut it.\"\n\nIt was an hour until sunset, and in the orange glow Veliana felt they were exposed wearing their cloaks as they traversed the city. They were deep in Spider territory, and instead of trying to travel through the less profitable outskirts, they marched together through its very center. She saw a few men in the paler-gray cloaks, but these men did not accost them, nor hail them to demand an explanation. Strange.\n\nDeathmask was already waiting in the chamber when Veliana arrived. Garrick sat on his cushions, smoking as usual. He looked incredibly pleased, which threw her off. She had expected him to be ranting and raving. And why was Deathmask there? Was she about to witness another attempt on his life? Twelve other men were gathered about. All were armed. She felt her worry grow. What if this was it, the moment when Garrick tried to assert his control over the guild?\n\nHer hands brushed the daggers at her hips. If it came to that, she'd take Garrick down with her, no matter the cost. Better the Ash Guild dissolve leaderless than continue in the hands of that paranoid bastard.\n\n\"You know how to keep a gentleman waiting,\" Garrick said as she arrived.\n\n\"I'm sure Deathmask will forgive me.\"\n\nGarrick chuckled, not at all bothered by the slight.\n\n\"Well, we're all here now. Before we begin... Vel, would you care to hand over your daggers? I'd hate for someone to get hurt.\"\n\nIt took all her concentration to hide her panic. Should she do it? If she complied she'd be vulnerable, but if she refused it might be seen as a threat against Garrick. She glanced at Deathmask, and during the brief moment their eyes connected, she saw the corner of his mouth curl into a grin, followed by a wink.\n\nTrusting him, she handed over her daggers and crossed her arms.\n\n\"Is it safe to say I won't like what I'm about to hear?\" she asked.\n\n\"Perhaps, but I am not the one who shall be speaking.\" He turned to Deathmask. \"Tell me, please, why is Veliana still alive?\"\n\nAll about, men murmured, and she wondered how they were interpreting that statement.\n\n\"I do not understand,\" Deathmask said, feigning confusion. \"Is there a reason she should not be?\"\n\n\"Don't lie to me. I know she attacked you last night. One of my men watched your exchange. Tell me, why didn't you kill her? She did, after all, try to kill you.\"\n\n\"I assumed it just a training exercise,\" Deathmask said, the lie smooth on his tongue. \"Veliana confirmed as much near the end of our fight. Was I wrong in my assumption? Was her response to me a lie?\"\n\nThis was clearly not the answer Garrick had expected. He frowned and shifted on his cushion.\n\n\"Yes, you damn fool, you were wrong, and she a liar. You should be dead, yet are not.\"\n\nVeliana held her tongue. What game was Garrick playing? She'd warned him about revealing attempts to murder an accepted member without reason or proof, yet here he was exposing his plans to the guild, and not just that, but showing how those plans had failed.\n\n\"For what reason would she attack me?\" Deathmask asked.\n\n\"Isn't it obvious? She fears you. She knows with your skill you might quickly ascend to take her place. Isn't that right, Veliana?\"\n\nHe grinned at her, his bloodshot eyes twinkling. Veliana's hands shook as she choked down her outrage. So that was it. He'd cast the shame of a failed intra-guild execution on her, and if she tried to deny it, it would be her word against his. Her word against the word of their guildmaster. One clearly outranked the other. The punishment for such a charge was limited to two options: banishment or death.\n\nStaring at that grin, she knew which option Garrick had already chosen.\n\n\"You planned this from the start,\" she said, her voice nearly a whisper. Garrick stood and stepped closer as the rest of the guild tensed. They understood the accusation, and they too knew the possible punishments.\n\n\"I merely took advantage of an opportunity,\" he whispered so only she could hear. She glared at Deathmask, suddenly wondering how many of his promises had been lies. Perhaps all of them. He'd set her up, she realized. She'd wasted time debating and discussing with Zusa when she should have killed the masked bastard. All attempts on his life had been made in secret, with every man involved dying as a result. No one could prove Garrick's attempts. Once more, his word against hers. Damn it all!\n\n\"As you know,\" Garrick said, raising his voice to a theatrical level as he turned his back to her, \"our laws are clear for such an attempt. We cannot have anarchy within our ranks, not in this crucial time while we fight for our very survival.\" He spun. \"You will be made an example, Veliana, one for the entire guild to see.\"\n\n\"Guildmaster, if I may make a request,\" said Deathmask. Garrick seemed worried, but gestured for him to continue. \"Since it was my life she tried to take, I ask that I be the one to carry out her punishment.\"\n\n\"You fuck,\" she said, her hands clenched tight into fists. \"You sick little fuck.\"\n\nShe feinted a lunge at Garrick and then hurled herself at Deathmask. She was unarmed, but Zusa had taught her a multitude of ways to kill with her bare hands. If she could strike him just right, crush his throat or snap his neck, then at least she'd die taking revenge. Her fist slammed into his mouth, just in case he attempted to cast a spell. With her other fist she doubled him over with a blow to the stomach. She heard men shouting, but she stepped closer and wrapped her arms around his neck. Just a single hard twist and then...\n\nSomething hard smacked the back of her head. Her stomach heaved, and her whole body went limp. Deathmask pulled free, and he shouted for the others to leave her be.\n\n\"She is mine,\" he said. \"Guildmaster, I ask, is your punishment for this madwoman execution?\"\n\n\"It is,\" said Garrick. He seemed amused by her display.\n\nHer helpless rage grew. The men let her go, but it took all her strength to stand. Already a knot was growing on the back of her head. She felt ready to vomit. Deathmask closed the distance, and her wild punch missed. He grabbed her by the throat and flung her against a wall. A dagger flashed from his belt and pressed against her neck.\n\n\"Do you trust me?\" he whispered into her ear. His grip tightened around her throat. Her eyes met his, and against all her instincts, something in those brown orbs gave her hope. She nodded, a barely perceptible movement because of how tightly he held her against the wall.\n\n\"Then stay perfectly still.\"\n\nHe muttered a few more words, soft whispers hidden by his grin. At last he pulled back his dagger and stabbed her chest, in and out with such speed her blood was flowing before she ever felt the pain. Black dots cluttered her vision as he held her still.\n\n\"Sleep in darkness,\" she heard him say as the Ash Guild hooted and hollered. No doubt Garrick was one of those cheering. She tried to curse his name as she died, but her whole body was turning rigid, refusing to cooperate, refusing to struggle, refusing to breathe...\n\nAnd then the darkness came, and she could only obey Deathmask's request.\n\nOric waited until midafternoon before heading out. He traveled south along the main road, ignoring the peddlers and the beggars. He veered off when appropriate, making his way to the mercenary guild. While it had once been a weak entity, the years of battle and constant work had filled its coffers, drastically improving its recruitment and influence. Anyone wanting to be a sellsword had to go through it. There were some advantages to the system, such as guaranteed minimums in rates, and insurance should some of the higher-ranked members fail to fulfill their duties. Mostly Oric thought it a grand scheme to jack up the cost of hiring mercenaries, but what did he know?\n\nThe building itself was still small, little more than a large cube to house records and provide the wealthy with a place to visit that was close enough to the main road that they might not be afraid. Oric entered, crossing his fingers as he looked about the office hoping to see an old friend. Sure enough, there he was, white bushy unibrow and all.\n\n\"Oric?\" asked the old man as he came from a back room to the front at the sound of a bell ringing above the door. \"Come closer, my eyes aren't what they... so it is you! Good to see you, you ugly son of a bitch.\"\n\nOric grinned. \"Was worried you'd died off, or been replaced by someone who can still remember what happened more than an hour ago.\"\n\nThe old man laughed. His name was Bill Trett, and in Oric's former sellsword life he had been a respected colleague. Bill had killed until his strength failed him, but by then he'd acquired such a wealth of knowledge of his various employers that the guild taught him his numbers and set him in charge of its transactions.\n\n\"You see this mess?\" he asked, pointing to the various shelves stocked with expensive paper. \"Only I know where everything's at. They'll keep me on until I die, and perhaps a little bit longer than that if they can figure out a way.\"\n\n\"Gods know they need you,\" Oric said. \"The Trifect still filling your purses?\"\n\nBill waved a dismissive hand. \"The money's steadied, nearly every one of them just wanting the bare minimum. Not like when this mess first started, when I saw more gold change hands than I could count. Blood really filled the streets then, didn't it?\"\n\nOric smiled, remembering the many thieves he'd cut down while in Leon Connington's pay. It'd been a very good year.\n\n\"I think Alyssa's going to give everyone some work later today, so be prepared,\" Oric said. Bill raised his eyebrows but didn't inquire further. \"But for myself, I need a favor, Bill.\"\n\n\"What's that? Not that I should be doing _you_ any favors. Last I remember, I saved your life up in Felwood, not the other way around.\"\n\n\"If not for favor, then for gold,\" he said, dropping a bulging coin purse atop the desk. \"I need the best you have to hire, and I don't mean who the _guild_ thinks is the best through their stupid rankings. You know every sellsword from here to Angelport, and I want your real opinion. I need someone who could find a mouse in a forest before an owl could; that damn good.\"\n\nBill rubbed his chin as his milky eyes stared off into nowhere.\n\n\"I suppose you don't mind if he's a bit unsavory?\"\n\n\"He can be the ugliest, meanest bastard you know. Prefer it, even. We'd get along.\"\n\nBill laughed, but it was lacking in humor.\n\n\"I know of one, and he's good, Oric. He's all the way over from southern Ker, though some say he ain't even from Dezrel. Out of the nine jobs he's done for me, he's never once failed to catch his prey, and always done it with days to spare. Been tough getting work lately, though. Charges twice what anyone else does.\"\n\n\"What's his name?\"\n\n\"He calls himself Ghost. I'm not brave enough to tell him to pick something more original. Besides, with this guy, it's fitting once you see his face.\"\n\nOric crossed his arms. \"What's that mean?\"\n\n\"No point in me telling you. You'll find out for yourself. He's costly, he's dangerous, but he's the best. You still want to meet him?\"\n\nOric thought of the six men he and Arthur had lost when this lone Watcher had ambushed them along the northern road. He also thought of how Alyssa might execute him if she found out his and Arthur's role in her son's death.\n\n\"Yeah. I need the best. Where can I find this... Ghost?\"\n\n\"Know where the Mug and Feather is? No? Lousy tavern built to the far south, just off the main road. Head there a few hours from now. The barkeep's a cheat, but he'll point Ghost out for you... though I'm thinking you won't need him to.\"\n\nBill opened the purse and dumped the coins across his desk. After he counted them up he nodded.\n\n\"You've got a few extra in here.\"\n\n\"Keep them,\" Oric said, heading for the exit. \"Consider it a gift to an old friend for keeping things quiet.\"\n\n\"Understood. Safe travels, Oric.\"\n\nThough Bill had told him to wait, Oric had no such plans. He wanted to be there when this Ghost showed up for a drink. Besides, if he had enough time he might glean some information from the regulars there. Just after midday anyone in there would certainly be a frequent drinker.\n\nFinding the tavern was easy enough, given the sign hanging above the door: a poorly drawn mug and an even uglier feather. Owner had probably been cheap enough to draw it himself. Inside stank of vomit and alcohol, and the lighting was abysmal. In one corner was a fire pit, no doubt the only source of both heat and light at night. Among the various tables he saw a few stragglers, most eating. They glanced back at him as he entered and squinted to see in the dark. None stood out, at least not as dangerous assassins.\n\nThe barkeep was a thin man with a blond beard that reached to the bottom of his neck. He nodded at Oric and then waited for him to take a seat before coming over.\n\n\"Whatever's cheapest,\" Oric muttered, tossing him several coppers. When the barkeep came back with a third of his mug froth, Oric rolled his eyes. A cheat indeed. Deciding he needed information more than he needed to administer a good beatdown, he let it slide.\n\n\"Need anything to eat?\" the barkeep asked.\n\n\"What's warm?\"\n\n\"Haven't started the soup yet. Got a bit of bread, though, and butter if you're willing to pay.\"\n\n\"That'll do.\"\n\nHe kept his eyes to himself as he waited for his food. Just in case Ghost was already there, he didn't want to make it seem like he was looking. When they met, he wanted to have the upper hand, just in case this Ghost tried to haggle for more pay, which he might, given the target. When his bread arrived he smothered it with butter and ate. When he caught the barkeep watching, he pulled out a silver.\n\n\"Keep the rest,\" he said. \"Care to answer me a question?\"\n\nThe barkeep held the silver piece close to his eyes as he inspected it, frowned, and then put it away.\n\n\"Of course,\" he said. \"Not so busy I can't stay away from the bar long enough to talk with a customer.\"\n\nOric chuckled, then lowered his voice.\n\n\"I'm looking for a man who calls himself Ghost.\"\n\nThe barkeep wiped his hands on his pants and laughed. \"Not too many go looking for him. Usually he's got to go to those making offers no one else is dumb enough to accept. What business you have with that dark-skinned monster?\"\n\n_An actual dark-skin from Ker?_ thought Oric. _Interesting._\n\n\"No business of yours,\" he said. \"Now fill the rest of my mug, and with ale, not foam, got it?\"\n\nThe barkeep glared but obeyed. Oric washed the rest of the bread and butter down, then glanced around once more. No dark-skin in the tavern. Shit, he wasn't even sure if he'd seen a dark-skin in all of Veldaren. The people down south in Omn, particularly along the coast, were known for their darker skin, but it looked more like a deep tan. Those from Ker were claimed to have skin as black as obsidian. No wonder the guy had trouble getting work. Settling in for a wait, Oric moved from his table to the one farthest from the door. He leaned back against the wall and closed his eyes. He didn't actually sleep, but let it look like he did. If anyone was dumb enough to try to rob him, well, they'd get a nice surprise.\n\nAs the sun moved across the sky, with dusk steadily approaching, more men filtered into the tavern. Oric thought it might be one of the very few taverns left in southern Veldaren since King Vaelor's edict banned all caravans from entering the southern entrance, forcing them to enter through the east instead. All the merchants, and their wealth, had shifted farther and farther north. The men who entered looked tired and haggard, and he guessed many of them worked the nearby fields outside the city walls. The ale was terrible, same as the prices, but they were probably far closer to home and among friends.\n\n\"You're in our seat,\" he heard someone say. He opened his eyes to see three men, their tanned skin covered with soil. All three of them combined might still be skinnier than he was.\n\n\"That's a damn shame,\" Oric said, shifting so they could see the sword sheathed at his side.\n\n\"Ain't no swords allowed in here,\" said one of them.\n\n\"Like to see him stop me,\" Oric said, nodding toward the barkeep.\n\nThe men scowled, but armed with only their fists, they dared not challenge him and his blade. They backed down to another table, and as they moved out of his way he finally saw Ghost. He sat alone in the center of the tavern. His skin was indeed darker than Oric had ever seen on a man. His head was shaved, and he wore loose clothing more appropriate for a warmer climate. His enormous strength was obvious, his arms thick as tree trunks. Most shocking, though, was the brilliant white paint he wore across his face.\n\nOric stood, glared at the men who'd wanted his seat, as if daring them to try, and then approached Ghost.\n\n\"Mind if I join you?\" he asked.\n\nThe man looked up, and he flashed a smile, revealing clean white teeth.\n\n\"I have a seat to spare, so take it if you wish.\" His voice was deep, intimidating. Oric sat and leaned back in his chair. If not for the white paint, this Ghost might have been handsome. Oric tried to decide why he wore it, yet could not. Was it because of his name? A pathetic attempt to fit in?\n\n\"Not much need to ask, but I assume you're the one called Ghost?\"\n\nThe mercenary chuckled. \"I am.\"\n\n\"They say you're good.\"\n\n\"Who is _they_? That blind fool running the guild's coffers? Or the rest of my colleagues? I'd be surprised if any bothered to speak of me except in disdain.\"\n\n\"It was Bill,\" Oric admitted. \"And is it true? I'm starting to have my doubts.\"\n\n\"Is that an attempt to make me boast? No boast. There is none better. Now tell me your name, and your business, otherwise I might decide I prefer to drink alone.\"\n\n\"Sad man that'd prefer to drink alone.\"\n\nGhost grinned again, and there was something wolfish in his brown eyes.\n\n\"Come now, stranger, do you think I am unused to being alone?\"\n\nOric felt put off guard, and he cursed his verbal clumsiness. Arthur would have been so much better at milking information from the man, figuring out who he was, what made him tick.\n\n\"Fair enough. My name's Oric. Who I work for is my own business. I need you for a job, and I've already paid Bill for your services.\"\n\nGhost leaned back in his chair and crossed his arms. Oric saw two hilts just below his elbows. It seemed Ghost didn't care much for the barkeep's no-weapons policy either.\n\n\"I can refuse if I wish, so don't think I am already in your pocket, Oric. Do you wish someone found, killed, or both?\"\n\n\"Both.\"\n\nAgain that wolfish smile.\n\n\"Excellent. Who?\"\n\n\"They call him the Watcher.\"\n\nOric was surprised by the sudden burst of laughter. It seemed the rest of the tavern winced at the sound, as if they expected Ghost to explode any second.\n\n\"The Watcher?\" asked Ghost. \"Now that is interesting. I've heard a rumor of him here or there, but they make him sound as real as the Reaper. But now you come and ask me to kill him? Do you have anything for me other than a name?\"\n\n\"I saw him with my own eyes,\" Oric said, annoyed. \"He wore gray, and kept his face hidden with the hood of his cloak.\"\n\n\"You describe nearly every beggar in this city.\"\n\n\"He wielded two swords, one for each hand.\"\n\n\"I'd be more impressed if he wielded two swords in the same hand.\"\n\n\"Enough!\" Oric slammed his hand against the table. \"I won't be intimidated by a freak like you.\"\n\nThe entire tavern quieted at his words. Ghost leaned closer to Oric, not angry, just amused, but something lurked in his smile, something dangerous. His voice dropped to a whisper.\n\n\"A freak?\" he asked. \"Why is that? Is it my skin? There are thousands like me in Ker.\"\n\n\"Only a freak would paint his face to look like a dead whore,\" Oric said, still trying to rein in his temper.\n\n\"Ah, the paint.\" His voice dropped even lower, as if he was about to say something intimate for only Oric to hear. \"It itches like ivy, and does not come cheap here. Do you know why I wear it?\"\n\n\"Because you're trying to fit in?\"\n\n\"Fit in?\" He laughed loudly, a boisterous eruption that startled the nearest tables. Oric felt himself jump, though he didn't know why. He'd lost control of the conversation, he knew that much. If he wanted to be in charge of any further negotiations, he needed to get his act together, and fast.\n\n\"No, not to fit in,\" Ghost continued. \"I wear it to stand out. When people see this paint, it only reminds them of why I must wear it. People cannot hide from me, Oric. That is why I am the best. Everyone I talk to feels fear, for they know nothing of who I am, only that I am different. Do you see that farmer over there? I could find out the name of his wife faster than you could introduce yourself. When _you_ ask questions, they'll evade, they'll delay, they'll hope for bribes or favors. When _I_ ask questions, they wish me gone, because I make them afraid without a single threatening word. Fear is stronger than gold. All the wealth in the world cannot make someone conquer their fear, not when it comes to death and blood. They will tell me everything so I'll let them go back to their safe little existence. Fit in? What an unimaginative man you are.\"\n\n\"Enough,\" Oric said. \"Will you accept the job or not?\"\n\nGhost took a drink of his ale and set down the glass.\n\n\"Triple what Bill told you,\" he said. \"I won't accept a copper less.\"\n\n\"I could hire fifty men for that price!\"\n\n\"And all fifty would stomp about unable to find their own assholes. Triple.\"\n\nOric stood, having had enough. \"I won't, you damn mudborn. I refuse. Either accept your standard pay, or nothing at all.\"\n\nGhost drew a sword and slammed it onto the table. Oric jumped, but instead of reaching for his sword, he realized he had turned to run for the door. His cheeks flushed, and he knew Ghost had seen it as well.\n\n\"I expect the rest in Bill's hands by nightfall. Farewell, Oric. I will have the Watcher's head in two weeks. Should I fail, though I won't, all the coin will be returned to you.\"\n\nHe left, and it seemed the whole tavern breathed easier with him gone. To his shame, Oric realized he did too. He ordered another glass, drained it, and then hurried off. He still had another pressing matter to attend to, and he needed to handle it far better than he had the business with Ghost. Farther into southern Veldaren he approached a large wooden structure with two floors.\n\nInside, at least fifty boys and girls hurried about, cleaning, sweeping, and preparing their beds for nightfall. A man of forty hurried to the door to greet him.\n\n\"Hello, my friend,\" said the man. \"My name is Laurence, and welcome to our orphanage. May I help you, perhaps with finding an apprentice or maidservant?\"\n\n\"Show me the boys,\" Oric said.\n\nLaurence whistled and sent the children running. They traveled farther in. The room looked like a giant warehouse, with rows of bunk beds on either side. He lined up twenty boys of various ages, parading them before Oric as if they were cattle. For the most part the children behaved, having certainly gone through this before.\n\n\"Anything particular you're looking for?\" Laurence asked, licking his lips.\n\n\"That's my own business, not yours.\"\n\n\"Of course, sir, of course.\"\n\nOric kept Nathaniel in mind as he looked over the younger ones. One in particular looked close in size, maybe an inch taller. His hair was even the same color, which might help the illusion.\n\n\"Step forward,\" he said, nodding toward that boy. \"He'll do. What's the cost?\"\n\n\"Adoptions are not cheap, but he's still young, so it'll be nine silvers.\"\n\nOric reached into his pocket and pulled out twice the amount.\n\n\"No papers,\" he said. \"I was never here.\"\n\nLaurence's eyes bulged, and he glanced between the man and the boy.\n\n\"His name's Dirk,\" he said.\n\n\"That's fine. Come on, boy.\"\n\nLaurence watched them leave but said nothing.\n\nOric traveled by foot, so he took Dirk by the hand and told him to hurry along.\n\n\"No questions,\" he said. \"We're heading along the northern road. I've got a house for you there, where you can work off all that silver I just spent on you. You understand?\"\n\nDirk nodded.\n\n\"Good.\"\n\nHe took the boy to the southern gate, not wishing to travel through the more populated areas of the city, regardless of how close to night it was getting. The guards gave them a cursory glance before letting them through. At a branch in the road they followed the loop back around the city and then to the north. Dirk looked maybe six, and his legs were nowhere near as long as Oric's. He tired rapidly, and by the way his skin clung to his bones, it'd probably been forever since he'd had a filling meal. Oric eventually picked him up and carried him until the city was behind them and the sun almost set.\n\n\"How long until we're there?\" Dirk asked, the first time he'd spoken in an hour.\n\n\"No questions,\" Oric growled. He glanced about as the first of many stars appeared in the sky. He was nearing the King's Forest. Stretching out to the east were acres of hills. He turned toward one of them, still holding the boy.\n\n\"Almost there,\" he muttered. Once he put the closest hill between him and the road, he set Dirk down. \"You see that forest over on the other side of the road? I want you to go fetch me some sticks, whatever you can carry.\"\n\n\"Yes, sir.\"\n\nOric pulled out his sword and a piece of flint. While the boy was gone, he gathered enough dry grass to create kindling. He carefully shielded it with his hands once he got it lit. When Dirk returned, holding about six sticks, Oric snapped them on his knee and carefully set them into the kindling. He sprinkled a tiny bit of lamp oil from his pack to get it going, then stood.\n\n\"We need far more wood than that,\" he said, still holding his sword. \"But I've got time. First, Arthur's orders. Come here, Dirk.\"\n\nWhile the boy's body bled out on the grass, Oric went to the forest and broke off several thick branches. He dragged them back to his camp, grunting as he did. He used his boot to break the branches into pieces, and one by one he tossed them upon the fire. Once it was roaring, he picked up the body. It felt stiff and cold. He hoped it'd burn. Without a bit of ceremony he tossed it into the fire. The ragged clothing caught first, then the hair, and finally flesh. The burning meat smelled sweet, but Oric always hated the scent of burning hair.\n\nDeciding he could go without the warmth, he unpacked his bedroll and slept upwind so he wouldn't be bothered by the smell. Come morning, he gathered the bones in a sack and returned to Veldaren.\n\n# CHAPTER\n\n#\n\nHe had a soft bed underneath him, which confused Haern to no end. A bed? When was the last time he'd slept in a bed? Three years ago? Four? Wait, what about at that farm? No, that'd been on the floor, right? The inn, of course, he must be at the inn he'd stayed in afterward. Except he'd left there, traveling on toward home. He remembered an ambush, being stabbed fighting beside a wagon...\n\nWhen he opened his eyes it didn't help much. He saw a low ceiling, poorly painted a dull cream color. A glance around took in the rest of his surroundings. The room was tiny, with barely any space to walk between his bed and the door. Opposite him was a single closet, stacked full of a strange assortment of clothing and weaponry. He recognized his own weapons in the pile, and he tried to go for them.\n\nThe pain in his stomach convinced him it wasn't a good idea. He lay back down and pressed a hand against his abdomen. His fingers touched bandages sticky with blood. Larger fragments of the attack at the caravan came back to him. He'd killed several Wolves, been stabbed in the stomach, and then as he'd been passing out...\n\n\"What is going on?\" he muttered as he inspected his arm. He remembered the cut there, and it'd been bad, if not to the bone. It was bandaged as well, but the pain was only a dull ache. He pried back some of the cloth and saw an angry scar, lacking any stitching to have helped it close. It didn't seem possible. For that much healing he'd have to have been out for weeks. The same went for the arrow wound in his shoulder. Either that, or a priest had come and healed him.\n\nOr a priestess...\n\nHaern remembered those last fleeting images, images no longer certain to be hallucinations. Could it be? After all these years, had Delysia exited the safety of Ashhur's temple? A part of him felt excited to meet her, but for the most part he felt terror. His hair was still a mess, his face unevenly shaven. His clothes fit the part of the beggar, the image that was like a second skin. But Delysia... she'd been his first glimpse of light in a world of darkness, something clean and pure. He felt like living dirt, scabbed over with his blood and the blood of those he'd killed. It seemed so wrong for her to find him like this, assuming she even remembered him, or recognized him through the filth.\n\nHe tried once more to sit up, and, now prepared for the pain, he managed a better job of it. Using his hand to support his weight against the wall, he limped into the closet and grabbed his swords. He knew it made no sense for anyone to try to kill him, not after bandaging him up and healing him, but he felt naked without their weight at his hips. Sweat dripped down his neck as he caught his breath. He offered a quick prayer to Ashhur for strength and then pulled the door open.\n\nA very surprised Senke stood there, holding a slice of buttered bread, his free hand still reaching for the door handle that had swung away from him at the last moment.\n\n\"Going somewhere?\" Senke asked.\n\nIt was too much. Haern staggered back and half sat, half fell onto his bed. He stared, his mouth hanging open.\n\n\"You look like you've seen a ghost,\" Senke said, seeming amused by the whole scenario.\n\n\"I think I have.\"\n\nSenke laughed, and that familiar sound helped melt Haern's doubts. The man had shaved his head and grown out his beard, but underneath the disguise he had the same smile, same laugh, same guarded amusement in his eyes.\n\n\"Only a handful have recognized me, but I guess I shouldn't be surprised that you're one of them. Always were the observant one, weren't you, Aaron?\"\n\n_Aaron..._\n\nA flood of memories tore through him, of days practicing with Senke, of walking at the side of his father, and of those few fleeting moments with Robert Haern before executing him at his father's command and then cleaning up the blood. Aaron... he hadn't gone by that name since that day. He'd adopted a new name, become a new person. A better person.\n\n\"Haern,\" he said. \"Aaron died a long time ago.\"\n\nSenke handed over the bread and leaned against the door, chuckling again.\n\n\"That you did, and I was one of many who thought so, though I forgot your little oddity about the name. Everyone heard how you died in the fire. I barely got out myself, though I lost most of my hair in the process. Helped disguise me, though, and I'm kind of attached to the look now.\"\n\nHaern looked at the bread as if he didn't know what it was for. At last he dropped it, stood, and flung his arms around Senke. He didn't say anything, didn't know what to say. He felt thirteen again, bewildered, torn, and suddenly given a link to a past that had had moments of good. It seemed Senke understood, for he patted Haern on the back and then gently pulled free.\n\n\"Don't get all sentimental,\" he said, winking. \"Otherwise I might start thinking you aren't really Thren's son. Now have a seat. Del says you've got another day or two before you'll be in top shape, and I don't want you tearing those wounds open. You've grown up, gods damn, boy. Taller than me now. How about you tell me what you've been doing these past five years?\"\n\nPart of Haern wanted to, but mostly he felt an overwhelming sense of embarrassment. For the past five years he'd struck at the guilds out of vengeance. Whom else could he tell? Who else would understand the rage of seeing everything he'd loved broken and destroyed by that sick culture of the underworld? Still, strange-looking or not, there was Senke, the closest thing to a friend he'd ever had. The time melted away. He told it all, of his escape from the fire and living on the streets, always keeping his hair messed and unevenly cut, his skin a blanket of dirt and scabs. He stole food to live, and lived to kill those of the thief guilds. He felt keen shame admitting that, though he wasn't sure why. In his heart he felt justified.\n\n\"Why not just leave?\" Senke asked. \"Everyone thinks you're dead. You could move on, make a name for yourself as a mercenary or a thief in any other city but here. Why stay?\"\n\nHaern felt his neck flush.\n\n\"Because this is my home,\" he said. \"This is all I know. I wanted to punish the guilds, every one of them. I thought maybe, if I made them afraid of the Watcher, I could scare away the recruits. I could make them doubt their safety. I could... I don't know, Senke. I thought I could finally end it all.\"\n\nSenke chuckled.\n\n\"Well, it seems the shy boy I once knew has grown up to be an ambitious young man. Shouldn't surprise me. Well, besides you being alive and all. How'd you end up at our caravan?\"\n\n\"Investigating the Serpent Guild and their newfound gold. Was on my way back when I stumbled upon your attack.\"\n\n\"And good timing it was,\" Senke said, grinning. \"We were hired on to protect some fat-ass merchant who'd done something to really piss off the Wolf Guild. I think he'd stiffed them on protection money or something, was trying to flee north before they noticed. Well... they noticed.\" He laughed. \"I must admit, I first thought it was Thren who'd come to our aid. The way you just charged in, then danced and weaved, it all seemed so familiar, Aaron...\"\n\n\"I said it's Haern now.\"\n\nSenke lifted his hands to show he meant no offense.\n\n\"Forgive me, just habit. Why so strict?\"\n\nHaern felt a chill coming on, and he wrapped his blankets tight about him.\n\n\"Because that's not who I am anymore. I refuse everything of my father, including his name. I won't be what he wants me to be.\"\n\n\"Wanted you to be,\" Senke said. \"He thinks you're dead now. And instead of being your father's pet killer, you instead spend every night killing. A neat trick, that.\"\n\n\"Don't you dare judge me!\" Haern said, and he was surprised by his own anger.\n\n\"No judging, just stating the obvious. Well, guess it's my turn. Not nearly as interesting. I fled the city for the first few years. Always wanted out, think I told you that, but Thren wasn't one to take such requests too well. Doesn't take too kindly to those who sell him out to the king either, but thankfully he never caught wind of my doing that. I used the fire that 'killed' you to die as well, make myself a new life. Spent some time down in Woodhaven, cutting lumber. After a while, got bored, took some odd jobs more favorable to a mace than an ax. All of a sudden I had a slow but steady stream of mercenary work. About a year ago I came back to Veldaren, going by the name of Stern and hoping for a bit more lucrative employment. Before you start thinking it, I wasn't exactly falling into that same old trap. I chose my contracts carefully, and while I wasn't always working for the nicest of people, I wasn't killing innocents or torching the homes of the poor either.\n\n\"Anyway, eventually met with my current employer. Even joined up with him as a permanent member of his mercenaries. Seems like he went through twenty guys, trying to find one who was... well, not scum. Lucky me, eh?\"\n\nHaern smiled but said nothing. He was still trying to wrap his head around everything. Here was someone he could talk to, could trust. After half a decade of silence and loneliness, it had all come crashing to an end because of a single poorly timed ambush. For all the many times he'd felt overlooked by Ashhur, he wondered just how unnoticed he really was. While he thought, he ate, figuring it a good excuse not to talk. All his confidence had flown out the window with Senke's arrival.\n\n\"I see your eyes drooping,\" Senke said when Haern finished his meal. \"Let me send in Delysia to swap out some clean bandages and then you can rest, ponder over this craziness.\"\n\n\"Delysia?\" he asked, thinking of the other images that had flashed before his eyes as he lay bleeding after the fight. \"Is she... is her last name Eschaton?\"\n\nSenke raised an eyebrow. \"Well, yeah, but how would you... wait a minute. You did know a Delysia. Is she _her_ , the one you killed Dustin to protect and... shit, that is her, isn't it?\"\n\nHaern nodded, and was totally unprepared for Senke's eruption of laughter.\n\n\"Looks like she returned the favor. She's the one that kept you from bleeding out like a stuck boar. Damn, this is too funny. You never told me she became a priestess. Always wondered how she hid from Thren so well.\"\n\n\"I never told anyone,\" Haern mumbled. \"Kayla told me the night of the Kensgold.\"\n\nSenke's face saddened at the mention of her name. \"She was a pretty lass. What I heard, Thren killed her for aiding you. Such a shame. Didn't pay much to help you out, did it?\"\n\nThe comment stung, and at Haern's pained look, Senke immediately started trying to take it back.\n\n\"I'm sorry, Haern, you know I don't mean that. It wasn't your fault, any of it. Your father's just a bastard, still is, though his influence is slowly dwindling, thank Ashhur.\"\n\n\"Senke, I... I'm not ready to see her yet.\"\n\n\"She's seen plenty of you.\"\n\nHe blushed a fierce red but remained adamant. \"Please, just let me rest. Meeting you again is too much as it is. Let me think, all right?\"\n\nSenke shrugged. \"I guess you'll survive. But if those cuts get infected, it's your own damn fault. Sleep tight, Haern.\"\n\n\"Thanks.\"\n\nEven after Senke left, his words echoed in Haern's head.\n\n_Didn't pay much to help you out, did it?_\n\nHow many had died because of him? Robert had died by his hand. His father had killed Kayla, again for helping him. Senke had nearly died in the fire. Delysia had been forced into hiding. And now, when every thief guild in the city would gladly string him up by his thumbs and let the entire underworld have a go at him, the two had brought him into their home and given him succor. Were they mad? He was a monster, a beacon of chaos and murder. The streets were where he belonged. Their gutters had room for the blood.\n\nBesides, he couldn't face her. He just couldn't. The last image he had of Delysia was of her gasping in his arms after the bolt pierced her back. She'd looked so shocked, so betrayed, and then to see his own father approaching, crossbow in hand, he'd felt such guilt...\n\nHe tightened his belt and held back a grimace at the pain in his stomach. His cloaks were folded up beside his bed, as were his tattered clothes. Again he blushed a bright red as he remembered Senke's comment, and he prayed that it had been anyone but Delysia who had changed him into what he wore now, a plain white shirt and brown pants. Quietly he changed into his old clothes. In their dirt and dried blood he felt all the more wretched and eager to be gone. Everything about him was filthy, even the task he'd devoted his life to. Was he really any better than his father? At least Thren had developed an empire, however fleeting. All Haern did was destroy.\n\nHe shook his head, trying to banish such thoughts. He needed to concentrate. Drowsiness still tugged at his eyes, and that soft warm bed tempted him more than any woman ever had. Deciding it was now or never, he crept open the door and looked about.\n\nWhatever building he was in was small in space but attempting to make up for it by being two stories tall. He saw a second door across from him, and a few feet away, stairs going down at sharp angles to the bottom floor. He heard muffled talking from the other door. Feeling like a trespasser, he hurried along as fast as his wounds allowed him to go. The bottom floor was blessedly empty. It was sparsely furnished: he saw a table, an oak desk in the corner, and a modest pile of books atop it. At the door he removed the bolt and stepped out into the street.\n\nHe looked around a moment, taking in his surroundings. The sun was rising, still low enough to hide behind the city's walls. There was an inn not far away, Prather's if he read the sign right, and that meant he was in... Crimson Alley, deep in southern Veldaren. He felt muted horror at the realization. Senke and Delysia lived in the Crimson, one of the most dangerous places in the city? No wonder he never saw either of them when he patrolled the night. They'd certainly keep their doors locked and windows bolted. How often had he passed right on by when scouring for isolated members of the guilds?\n\nHe worried about his injuries, but those desperate enough to rob in daylight he could certainly handle. Giving one last glance at the dilapidated building to memorize its location, Haern rushed north, eager to put some distance between himself and this sudden assault from the past.\n\nVeliana floated in silence, and that alone convinced her she was dead. She didn't know if her eyes were open or closed. All she saw was darkness, though she didn't really see it so much as be swallowed by it. The numbness she likened to cold, so at least she felt something, however faint. Time drifted by as if it were bored of her. Then came a sudden, shocking pain to her chest, lighting up her darkness with streaks of red. Again she felt the pain, but this time there was a comfort to it, a strange familiarity. The third time it hit, she realized it was her heartbeat restarting.\n\nPins and needles came in waves, first to her chest, then her face, and last her extremities. The darkness gradually faded from black, to yellow, to red, and at last to an assortment of colors that congealed to create the unmasked face of Deathmask.\n\n\"Welcome back,\" he said with a smile.\n\nShe would have hit him if her limbs had bothered to listen to her commands.\n\nHe vanished. She lay on her back, and now she stared at a cobwebbed ceiling. Based on the cold she felt, she decided she was on a dirt floor. Her ears, about the only thing working properly, heard shuffling, followed by a laugh.\n\n\"I'm sure you're angry with me, but let me assure you, I hope I never have to do that again.\" Deathmask leaned over, and she felt him press his hands against her neck. \"Pulse is getting stronger. Good. Never actually used that spell before, so consider yourself a lucky first try. Stopping someone's heart is never easy. Well, not if you want to start it again.\"\n\n\"What... happened?\" she forced her dry throat and swollen tongue to ask.\n\n\"I faked killing you. The thrust to your chest wasn't deep, but I made sure no one bothered to investigate the matter. The spell I cast put your entire body in stasis. No breathing, no heartbeat. For all anyone would know, you were dead, and honestly I'm not quite sure how far from the truth that is. I took over the burial, and here we are. Simple enough explanation, and once you get your bearings I think you'll be pleased with the elegance of my solution.\"\n\nThe pins and needles returned throughout her body, and this time she felt herself regaining control. Her head pounded with the unholiest of headaches, but she forced herself to sit up, forced the memories to come back. She'd been in their headquarters, Garrick had been there, accusing her of...\n\nShe reached for a dagger at her side but all she did was topple herself back to the floor.\n\n\"Don't rush things,\" she heard Deathmask say. \"You'll be fine in a few more minutes. We have much to discuss, so try not to do anything stupid like killing me, all right?\"\n\n_No promises_ , she thought amid her delirium.\n\nAs her body reawakened, so did her mind. She glanced about, taking in her surroundings. They appeared to be in a cellar of some sort, the only light coming from a single torch lit behind her head. She saw no door but assumed it was also behind her. Deathmask leaned against a stone wall to her right, his arms crossed, his face blanketed with a smug grin she'd have given everything to cut off. Feeling far better, she sat up, braced herself for the ensuing dizziness, and then sat on her knees.\n\n\"I'm fine now,\" she told him. \"You say we have lots to talk about, so let's talk.\"\n\nHe nodded, as if perfectly willing to hurry through the bullshit.\n\n\"Garrick surprised me with his boldness. I'd feel more upset if he hadn't surprised you as well, and you've known him far longer than I. Even a rigged die will roll something new with enough throws, if you know what I mean. I tried to save us both, but clearly Garrick wasn't one to be persuaded. He wanted you dead, and I did my best to fool him regarding that. I succeeded, of that I'm sure. My position in the guild is tenuous right now, esteemed among most of the lower members that despised you, but Garrick wants me killed, that much is obvious. And now here we are. It's only been a day, and I managed to keep you wrapped and safe so no bugs or worms could get at you before I returned.\"\n\nVeliana shuddered at the prospect.\n\n\"So here we are,\" she said. \"What is it you want? Why keep me alive?\"\n\n\"Because I made you an offer, and I won't let some idiot guildmaster interfere with my plans. That offer still stands, though I need your answer now. Will you aid me, or must I find another?\"\n\n\"And if I say no?\"\n\nHis eyes held no joy, no amusement, only grim truth.\n\n\"I'll put you as you were, though this time no spell will bring you back.\"\n\nShe thought of the cold, the darkness. An involuntary shudder coursed through her, too strong to hide. She couldn't go back to that, even if it hadn't been a true death. Garrick had turned against her, and now she would be an outsider, a ghost banished from her own guild.\n\n\"I'll help you,\" she said. \"Even if I did have a choice, I'd still help you. I want that son of a bitch to die, slowly, painfully, and at my hand. That is all I ask. Can you promise me that?\"\n\nDeathmask handed her a small bottle of some red vintage. She tore out the cork and drank.\n\n\"That, my dear,\" he said, \"is something I can assure you of. He'll be yours to kill, all yours.\"\n\nThe alcohol burned going down, but damn did it feel good.\n\n\"Then enough lies and games. I can hardly turn against you, now that I've been 'executed' by the Ash Guild. What is your plan? Why have you come to Veldaren?\"\n\nDeathmask smoothed his robe and then sat opposite her on the floor. He scratched at his chin, as if considering where to start.\n\n\"I was once a member of the Council of Mages,\" he said. \"Less than six months ago, in fact. They preach noninvolvement in political matters, but it's nonsense. We had our eyes everywhere, especially on the kings and their capitals. When this war erupted between the guilds and the Trifect, I was assigned to watch. Through coin and magic I learned of every guildleader, their goals and the reaches of their power. As the years dragged on, and my boredom grew, I formulated various plans and contingencies. I was not a high-ranking member, Veliana; far from it. My strength was equal to many, but my years were few. Not enough gray hairs, if you will. I also had a reputation as a... troublemaker.\"\n\n\"Shocking,\" Veliana muttered. He chuckled and continued.\n\n\"They told me nothing of why I was tracking Veldaren's underworld, nor did they keep a close eye on me, so I let my mind wander to far more interesting ideas. Profitable ideas. Recently I came up with a plan I was sure would work. It wasn't foolproof, and would involve risk to whoever tried carrying it out, but I was certain of its worth. This stupid grudge war wasn't going to end, not without killing off Thren Felhorn, and I knew a way to do it. When I tried to convince the Council of this, they gave me strict orders not to interfere. I saw a wealth of gold, but gold means nothing to those aging bastards. They want influence, power, and information. Gold helped with that, but was hardly their true ambition.\n\n\"I went on with my plan without their approval, and that is when I realized how dangerous they thought me. Several of my contacts were actually in the pockets of other higher-ranking members. My assassination attempt accomplished nothing but my banishment. They took my name from me, Veliana. They stole it with a spell, called my ambitions foolish, my desire for gold a young man's folly. _We call you nameless_ , they said, _for you are death and folly_. They refused to see the power I might gain, the wealth the underworld of Veldaren smuggled every single night.\n\n\"I came here to prove them wrong. They want power, yet I will wield more power than they can imagine. They mock gold, so I will collect more than they would dream of. Your Ash Guild was perfect, Veliana. It was the smallest, the weakest, the one most easily breakable to my will. I have a plan, many in fact, and this is just one of them. We will take over the Ash Guild and turn it into a powerful force no one will dare act against. All I need is you at my side. We'll kill many accomplishing this, possibly hundreds. Does this bother you?\"\n\nShe thought over everything he'd said. It did explain many of his strange powers. Of course he could be lying, but even if he was, it didn't change what Garrick had done to her. As for his question...\n\n\"No,\" she said. \"Not if it lets me kill Garrick. The innocent aren't long for this world anyway.\"\n\nDeathmask smiled.\n\n\"Good girl,\" he said. \"Then you shouldn't mind what you must do next.\"\n\n# CHAPTER\n\n#\n\nEvelyn washed her hands in the bowl and wiped them on a nearby cloth. Feeling like a morbid butcher, she tossed her bloody apron aside. Matthew stood beside her, looking down at the boy.\n\n\"Will he live?\" he asked.\n\n\"I believe so,\" she told her husband. \"I sewed him up quick enough. Lost more blood tending to the Kenders' boy when he hit his head on their fence.\"\n\n\"He was also older,\" Matthew said. Evelyn nodded but said nothing. The silence stretched as they both looked upon the sleeping boy, his skin pale and slick with sweat. She hoped that meant his fever was finally breaking, but it easily could be because of the amputation. The human body did strange things when in pain, and Evelyn didn't want to imagine what the boy had felt when she took the saw to his shoulder. He'd been feverish and unable to talk, so maybe he hadn't felt much.\n\n\"Strange not knowing his name,\" Matthew said.\n\n\"We could give him one.\"\n\n\"Not much point. If he wakes he'll tell us it himself.\"\n\n\" _When_ he wakes.\"\n\nHer husband gave her a look, then nodded. It was the closest she ever got to an apology with him.\n\n\"Right. When he wakes. Give me a moment to take care of this.\"\n\nThe boy's arm lay wrapped in rags on the floor by the bed. Matthew scooped it up and carried it outside. Though he didn't tell her, she knew the hogs penned out back were about to get an interesting addition to their diet. They'd kept the children out of the house while she cut, though her oldest, Trevor, had tried to insist on watching. Just shy of fourteen, he might have been able to stomach it, but truth be told, she couldn't afford the distraction should he have cried out or lost control of his stomach.\n\n\"I've been thinking,\" she said when Matthew returned. \"We might as well name him. Both him and that Haern were injured, and I have no doubt they're hiding from someone. I'd hate for one of us to let slip his real name before we can get him home.\"\n\n\"I reckon you're right. Any ideas?\"\n\n\"Always wanted to have a boy named something fancy. How about Tristan?\"\n\n\"Too fancy. No one would believe us when we say he's ours. How about John?\"\n\nShe frowned. \"He'll only have this name for a week or two at most, and you want something so plain as John?\"\n\nHe blinked at her. \"My father's name is John.\"\n\n\"Your father was a very plain man.\"\n\nHe took an angry step toward her, grabbed her wrists, and then, laughing, pulled her against him. Her own laughter faded as he held her tight, and she wrapped her arms about him.\n\n\"You going to be all right?\" he asked.\n\n\"Just blood and dead flesh,\" she said. \"Not as if he was squealing like a hog.\"\n\n\"It's just you and me right now. You know that, right? We can open the stitches, say he bled out when we took the arm. Wouldn't be a lie...\"\n\nShe pulled away.\n\n\"We gave our word,\" she said, as if that should explain everything. \"I don't know what's going on, but what I do know is that anyone out to kill a boy that young ain't on the side of right. That stranger paid us a fortune for this. We can buy more land from the Potters, those acres they can't till worth a damn, but we could do it. We can hire help, extra salt and meat for next winter, lumber for the house... I won't better our life on the blood of a dead child, and I expect you to feel the same.\"\n\nHis cheeks flushed red. When he tried to speak, he shut his mouth again and waited another moment to get his composure back.\n\n\"True enough,\" he said. \"Hard as it is to get the kids to avoid a lying tongue. Won't do no good to go lying ourselves. I'm scared, Evelyn. We're just farmers. I don't like going up to Tyneham to trade our wares to the miners, let alone all the way south to Veldaren where the real crooks are. Whoever's after this boy...\"\n\n\"Tristan.\"\n\nHe laughed. \"Fine. Whoever's after Tristan probably has money, soldiers... Who'll take care of Debbie, or Anna, or little Mark should something happen to us? Or, Ashhur forbid, what if something happened to _them_?\"\n\nShe stood on her tiptoes and kissed his lips.\n\n\"Stop worrying. We'll deal with each problem as it comes, and Ashhur will keep us safe. Now let Tristan here sleep.\"\n\n\"Tristan,\" Matthew muttered as they passed through the curtain into the other room. \"You really wanted to name one of our children Tristan?\"\n\nThe newly named Tristan woke her a few hours later, well past midnight. Trying not to panic, Evelyn lit a candle and went to his side. The boy was moaning, and his legs and arm twitched every few moments. She touched his forehead. It was like touching fire.\n\n\"Get water in the tub,\" she told Trevor. \"Mix in some snow too. If you can stand to keep your hand in it for long, it ain't cold enough.\"\n\n\"Yes ma'am,\" Trevor said, his eyes lingering on Tristan as he put on his coat and boots. They had a small tub inside their home (\"A luxury if I ever saw one,\" Liza, her crone of a neighbor, had once told her). They had to bring the water in by buckets, and that would take time. Until then she tore off the boy's blankets and clothes, stripping him naked upon the bed.\n\n\"He going to be all right?\" asked Anna as she poked her head through the curtain. She was twelve, old enough to help her mother when she acted the healer.\n\n\"Wake your father,\" Evelyn said, ignoring her question. \"And make sure Mark and Julie stay in bed. Probably already scared, and I don't want them scared worse.\"\n\nAnna nodded, and her head vanished behind the curtain. Evelyn lifted Tristan in her arms, and it was like picking up a burning log. When she brushed through the curtain into their living room, she found Matthew putting on layers.\n\n\"Trevor said you needed water in the tub. The fever gotten that bad?\"\n\nShe nodded.\n\n\"I told you he hadn't had enough to drink,\" he said. \"Can't sweat off a fever if you ain't got nothing to sweat.\"\n\n\"I know,\" she said. \"Now's not the time.\"\n\nShe caught her younger children looking at her, and she turned her back to them and hurried to the tub. The door to the house slammed shut, whether from Matthew leaving or Trevor coming in she didn't know. In the tub she found a single bucketful of water, barely enough to wet the surface. She put him in anyway and held him down as his body flailed against the cold.\n\n\"Anna!\" she cried. Her daughter hurried in after. \"Help me hold him down. His shivers are going to get worse. Don't feel bad about the chill, either. He'll burn to death before he catches cold.\"\n\nShe shifted so that Anna might hold his arm, then pressed down on the boy's knees. Trevor came in with another bucket of water, and he looked lost about what to do with it.\n\n\"Just dump it on him,\" Evelyn said, trying to be patient. \"It's only water!\"\n\nTrevor hesitated, but the look in his mother's eyes got him going. He upended the bucket, cold water from the well. Tristan's moan turned to a full-fledged wail. Rather than stay, Trevor hurried out. Evelyn leaned more of her weight on her arms as Tristan's struggling grew. Beside her, Anna quietly cried.\n\n\"Start praying,\" she whispered. \"It'll help, but don't you dare let go of that arm.\"\n\nMatthew came in with a larger bucket, and he poured it in by the boy's feet. The water was halfway up his body, and Evelyn told him one more should be enough.\n\n\"Still need the snow?\" he asked.\n\n\"This water will be warm soon enough.\"\n\n\"All right.\"\n\nWhen they came back she put the bucket of snow beside her, saving it for when the chill left the tub. Tristan was still shivering, and he cried when he had the energy, and moaned when he did not. After twenty minutes she dumped in the half-melted bucket of snow, sending Tristan's shivers back to full strength. Ten more minutes and she lifted him out, wrapped him in a towel, and brought him back to his bed. Matthew was there not long after, a small cup of milk in one hand, a slender funnel in the other. Evelyn recognized it surely enough. They used it to feed their animals various herbs and tonics should they become ill.\n\n\"Hogroot's in the milk,\" Matthew said. \"He needs to drink. Hold open his jaw, and don't let him move. I have no intention of drowning him.\"\n\nOnce the milk was gone, they wrapped him tighter in blankets and waited.\n\n\"Go rest,\" she told her husband. \"You have enough work in the morning, and it won't be no good for you to do it on a half night's sleep. Get the kids back to bed as well. I'll keep vigil on him.\"\n\nMatthew squeezed her shoulder and then left. Once he was gone she gently stroked Tristan's forehead with her fingers. He looked like a drowned rat, but his fever had finally dropped. He'd fallen back asleep too, for which she was thankful. The hogroot in his milk would help break his fever, and she prayed it'd break completely while he slept. A quick inspection showed the stitches on his shoulder to be clean. No infection, thank Ashhur. There wasn't anywhere higher left to cut, other than his neck to end the suffering.\n\nOn her knees, her weight leaning against the bed, she waited out the night. Just before dawn his fever broke, and for the first time since Haern had brought him there he opened his eyes.\n\n\"I'm thirsty,\" he said, his voice a croak.\n\nEvelyn smiled and clutched his hand.\n\n\"Fresh milk,\" she said, \"coming right up.\"\n\nA squad of twelve mercenaries escorted Alyssa's litter through the city. Anyone foolish enough to linger in their way received a quick slap with the flat edge of a blade. They stuck to the main streets, where thief guild presence was weakest, the town guards too numerous for the thieves to act rashly. The distance between her estate and Leon Connington's new mansion was enough to be a bother, but she felt it necessary to carry her message in person. She pulled her fox fur coat tighter about her and waited.\n\nWhen they arrived she stepped out and surveyed the place. She'd been there once, just before its completion. After Leon's old mansion had burned down during the Bloody Kensgold, he'd rebuilt with security in mind. An enormous wall of stone surrounded his estate, perfectly smooth so there'd be no handholds. There were no trees in the yard either, nothing to hide behind. Four men stood at the gate, wearing ornate plate mail and wielding halberds. Beyond she could see the building, the walls stained a deep red, the roof sharply slanted. If Leon was nothing but roundness and fat, his new home looked the opposite, long and stretched thin.\n\n\"Greetings, Lady Gemcroft,\" said one of the gate guards. \"Please wait while we summon an escort. Make sure you stay on the path, for an errant step might prove deadly.\"\n\nBertram wasn't there, but Alyssa could imagine the scowl he'd have given them. For her part she was willing to understand Leon's craving for safety. Perhaps he took it too far, but it had been his belongings destroyed in the fire, not hers. Ten armed men came from the front door, approaching in neat formation along the cobblestone path leading across the yard. When they reached the front they unlocked the gate from the inside and ushered Alyssa through.\n\n\"Your men must stay outside,\" said their leader when her mercenaries started to follow. Alyssa paused, gave him a glare to show she didn't appreciate being told just before entering, but then complied. If she felt safe anywhere other than her mansion, it was here. An assassin would have to be a lunatic to risk the guards, the wall, and the various traps hidden underneath the grass. The men's heavy boots clacked across the stone as they entered the mansion.\n\nLeon waited just inside the door, a large smile on his face. Everything about Leon was big: his face, his eyes, his home, and most of all his belly. Hugging him was like hugging a giant sweet roll wrapped in silk. Only his mustache was thin.\n\n\"I'm so sorry about your loss,\" he said as he let go. \"I'm sure he'd have been a fine man, very fine. If there's anything you need, please let me know.\"\n\n\"Thank you,\" she said, doing her best to smile and forget how he'd always glared at Nathaniel as if he were a cockroach whenever Nathaniel was in his presence. \"Bertram is busy finishing the arrangements for the funeral, so I thought it best I stay out of his way.\"\n\n\"Of course, of course. It'd do you good to get out of that stuffy old mansion anyway. Always told your father he should fire whoever was in charge of his maidservants. Every breath in there was like licking the bottom of a dustbin.\"\n\nAnother smile. Her last head maid had died coughing and gagging on blindweed. She had a feeling Leon would have approved of such a fate for one who had let his sensitive allergies be affected.\n\n\"Have you any news from Laurie?\" she asked as they walked toward\u2014of course\u2014the dining area.\n\n\"Ever since the Kensgold he's refused to come to Veldaren,\" Leon said as he took her hand. \"I think your father's death spooked him more than a little. Such cowardice is inappropriate for a member of the Trifect, but what can you do?\"\n\n\"Surely someone who lives with an army of mercenaries behind great stone walls has no reason to question another's bravery,\" she said, unable to hold back.\n\nFar from upset, Leon only gave her a wink.\n\n\"It's one thing to be brave, and another to be stupid. I won't die from a garrote in my sleep. Neither would Laurie, if he took proper precautions.\"\n\n\"Maybe staying in Angelport is his precaution.\"\n\nLeon laughed. \"True, maybe it is. Still, he is going a little overboard, eh?\"\n\nThey sat down at one end of a luxurious table easily able to seat more than eighty people. Alyssa watched the servants parade a variety of treats and pastries before her. She didn't feel like eating, but it seemed Leon would keep them coming forever until she picked. Deciding on a small flour cake topped with strawberries, she scooped a tiny bit with her spoon and ate. The flavor awakened a dormant part of her, a tiny voice reminding her of her own needs and not of others'. Her stomach grumbled, and she wondered how long it'd been since she ate. She was horribly tired, and in the fog that was her mind she couldn't remember.\n\nThe rest of the cake vanished as she wolfed it down. Leon smiled at her and tore into his own assortment of desserts, as if he'd known all along she'd been neglecting her appetite.\n\n\"You are more than welcome to stay the night,\" he said, sipping some wine from a silver goblet, but only after a servant tasted it first. \"Just say the word, and I'll let your men at the gate know they can go home.\"\n\n\"Thank you, but I'd still prefer my own bed. Besides, the funeral is tomorrow, and I should make sure Bertram has everything in order.\"\n\n\"Where will it be held?\"\n\nShe sipped her wine. The alcohol tasted strong, and she pushed it away, fearing how much it might affect her.\n\n\"My mansion. We'll bury Nathaniel in the garden out back.\"\n\n\"Beautiful.\"\n\nShe debated, then called over a servant and asked for another pastry. The woman bowed, and moments later returned with a cake topped with blueberries. Alyssa wondered how much Leon spent keeping such stock deep in winter. Did he have some secret to keeping it from spoiling? She made a note to ask him once she had some free time on her hands.\n\nHalfway through her second cake, she decided she could delay no longer.\n\n\"There is another reason I'm here,\" she said, pushing her food away. \"I will soon be putting something in motion, and I've come to ask for your cooperation.\"\n\n\"Oh?\" he said, that one word pregnant with meaning. The way he lifted his eyebrow, the way he let his lips linger in an O shape... he knew he was about to be asked something he wouldn't like. He could read her too well. She had to improve. She felt like an imposter walking in her father's shoes. No wonder Bertram always harped at her to host more, visit more. Her social skills were lacking their proper finesse.\n\n\"We've crossed ten years of this nonsense with the thief guilds,\" she said. \"I once thought my father inept, but I've come to see how difficult it is finding these rats and bringing them to their proper fate. Worse, I thought we could make peace, at least reach a level of understanding. There will always be those who steal from us, but neither of us should fear death in the night. They live off our trade, after all, and should that trade end, they will be like leeches sucking a corpse without blood. But this won't happen. Though it may sting, we must pull them off. My son died because we have gone soft, tried to pretend they would finally calm down and leave us be. No longer.\"\n\n\"Does this have something to do with what Potts has been telling me about you hiring every mercenary able to lift a sword?\"\n\n\"It does.\"\n\nLeon sighed and, shockingly enough, pushed his own plate away.\n\n\"Listen, you're just a silly girl trapped in your position, so I'll do my best to save you from this embarrassment. You can't find them all, Alyssa. You'll never win. You'd sooner drive out every flea from the southern district than bring the guilds down. Half those you're paying will just sit in taverns and claim they killed a rogue or three before dinner. How will you know? How will you keep track? Every damn beggar you passed on your way here might have been a Serpent, or a Spider, or a member of the Ash. Can you know for sure? Can you prove it? You're throwing your money in the damn gutter. I've killed more thieves trying to sneak onto my grounds than I have actually going out and looking for them.\"\n\nShe felt her neck reddening, but she pressed on anyway, his arrogance be damned.\n\n\"They want us to think that,\" she said. \"But it isn't true. They act as if they'll endure, but their organizations can crumble, their loyalties break. They threaten us with poison and razor wire, and they've convinced the city that _they_ are the ones to fear, _they_ are the vicious ones. It is our fault for believing the lie.\"\n\nA guarded look crossed Leon's face as if he realized how far off his first read of her had been.\n\n\"What exactly are you planning?\" he asked.\n\n\"We break into every building. We search every crack of every wall. I have many men skilled in interrogation, and the men we've hired are even better at it. We'll find where they run, every time. These men have no pride, no honor. They'll point us the right way until they run out of places to hide. Every guildleader will fall, as will their replacements. If they wear a cloak, they die, regardless of the color.\"\n\nLeon looked ready to explode.\n\n\"Are you out of your mind? We haven't had that level of conflict since this started, not even during the Bloody Kensgold. The guilds are growing complacent, Alyssa. You act as if you want an end to this war, but the war's already over. Sure, my guards die sometimes, but they take with them twice as many thieves. Veldaren is learning to accept this, to live with the precautions and the death. But what you want? What you'll do? You're throwing away an entire fortune to thrust fire into a hornet's nest. You'll get every single one of us killed, and all because you're grieving over the loss of some... of some... _bastard_?\"\n\nShe stood and flung what was left of her cake into his face.\n\n\"Father was right,\" she said as he wiped icing from his cheek. \"Your cowardice is as big as your gut. I will not fear them anymore, and neither should you. It was my fear that sent Nathaniel away, my fear that left him vulnerable. And his executioner, this... Watcher? He is no different from the rest of them. He lives among them, and from what Zusa tells me he is the best of them. And so I shall bring them down. _All_ of them. The best to the last, the leaders to the dirt-crawling scum. Come Nathaniel's funeral, I will unleash my wrath upon the city that has sheltered the murderer of my son. Now if you would kindly request a servant to escort me to the door?\"\n\nLeon chewed on his lip a moment, his fat face blotched red. At last he clapped and did as she requested.\n\n\"Wait,\" he said, just before she exited the room. \"Just how many men have you hired?\"\n\n\"Close to two thousand,\" she said, and she felt a sense of victory at the way his jaw dropped. \"As I said, Leon, I will destroy them. I will destroy _everyone_ who dares try to stop me. Even the king. Even the Trifect itself.\"\n\nHe muttered something, but she did not hear it. Still furious, she turned and followed the servant woman out, more than ever wanting to be home to plan with her mercenary captains. Hiding was no longer an option. It was time to act. The time for putting up with the criminals was at an end, no matter the risk, no matter the bloodshed. Come the funeral, it was a lesson the whole city would learn.\n\n# CHAPTER\n\n#\n\nMatthew was pouring grain for his cattle when he saw the men ride to his front door. There were two of them, their chain mail dirty from the road. Even far out in his field he could tell they were armed.\n\n\"Who are they?\" Trevor asked beside him. He squinted against the light reflecting off the snow. \"Do you know them?\"\n\n\"No, I don't,\" Matthew said. \"Remember, if anyone asks, Tristan's your brother, and he caught infection from a spider bite. You understand me?\"\n\n\"Yes, sir.\"\n\n\"And just in case, get your knife, but don't you dare let them see you holding it. This is serious, Trevor.\"\n\nThe lad's eyes widened. He started to ask a question, thought better of it, and then just nodded.\n\nMatthew led them back to the house. Evelyn had answered their knock, and after a moment invited them in. He trusted her to keep her wits about her, probably more than himself. His other children were in there, though, and once the strangers were outside the public eye, he wondered just what type of men they might be.\n\n_Should have made them wait outside until I got back_ , he thought. _Damn it, Evelyn. Sometimes you ought to act the proper wife._\n\nJust before reaching the house, he stopped at a small shed and ducked inside. He heard his son gasp as he yanked their pitchfork off the wall.\n\n\"Won't do much against their armor,\" he said, inspecting its four teeth. \"But they ain't wearing helmets, so that's something.\"\n\nHe set it beside the door to the house, then opened the door and stepped inside. The two men sat beside the fire, their cloaks stretched out to dry at their feet. They both had swords, still sheathed, thank Ashhur. The rest of his children kept a safe distance away, again something to be thankful for. The strangers held small wooden bowls of a broth Evelyn had prepared for breakfast. His stomach grumbled involuntarily. He hadn't eaten yet himself. He wondered how much of his own portion sat in the strangers' bowls.\n\n\"Welcome, gentlemen,\" he said, taking off his gloves. \"I see my wife has helped you feel right at home, which is proper. It's cold work riding in winter.\"\n\n\"She's a lovely host,\" one of them said. He was a plain-looking man, dark-haired, flat nose. Only the scar running from his eye to his ear made him seem dangerous. He wore no tabard, but his accent was distinctly of the north, most likely Tyneham or one of the smaller mining villages near it.\n\n\"That she is,\" he said. \"On your way to Felwood, or beyond? I must say, I didn't catch which direction you came from while out in my field.\"\n\n\"Riding north,\" said the other. He was uglier, with brown hair in desperate need of a cut. \"Our horses need a rest, and we must admit, the thought of a warm building was too much for us to resist when we saw your farm.\"\n\n\"A fire warms eight as well as six,\" he said. Evelyn gave him a glare, and he realized his mistake. He had seven in his family if he counted Tristan.\n\n\"Been times we had to cram twelve of us in here,\" he continued, hoping to make them forget the comment. \"Neighbors had their house burn down, lost one of their sons too. Makes for a rough winter with no roof, so we brought 'em in until spring.\"\n\n\"It must have been tough,\" said the first, looking around the small home.\n\n\"Forgive me, I've yet to introduce myself. My name's Matthew Pensfield. You've met my wife, Evelyn. This here's my oldest, Trevor. Little Mark's over there, hiding in the corner. And these're my two daughters, Anna and Julie.\"\n\nThe girls smiled and tilted their heads in proper respect. The soldiers tipped their heads back, and each of them had a leer that sent fire up and down Matthew's spine. He hesitated, trying to decide what to do about Tristan. He didn't know what was the right course of action. The boy had been asleep when he last went outside. His wife took the decision away from him, and as much as it scared him, he trusted her.\n\n\"You must forgive us for not introducing you to Tristan. He's sick with a fever in bed. Just had to amputate an arm, the poor dear. Spider bite.\"\n\n\"That's a shame,\" said the dark-haired one. \"My name's Gert, and this here's Ben. Like I said, we're riding the road, maybe to Felwood, maybe all the way to Tyneham.\"\n\n\"Only wanderers and thieves ride the road without knowing how far they wish to go,\" Matthew said. \"I hope you're neither.\"\n\nGert laughed.\n\n\"Nah. We're looking for someone, actually. A lost boy, five years in age. Perhaps you've seen him?\"\n\nMatthew shook his head. He'd played cards only a few times when trading in the bigger towns. He'd never been good figuring the odds of things, but he'd always done all right because of one thing going for him: he had one of the best card faces of anyone he knew. Only Evelyn could read what was going on behind his eyes.\n\n\"I haven't, and I doubt I would, either. A boy that young running around in the snow? He'd be lucky to last a single night. How long's he been missing? I hope I cause no offense, but a coyote pack's probably gotten him, or at least what was left of him.\"\n\n\"There's the thing,\" said Ben. \"He might not be alone. Had another man with him, wore gray and carried two swords. He's a kidnapper, and we're trying to capture him before he can think of asking for ransom.\"\n\n\"Kidnapped?\" asked Evelyn. \"From who?\"\n\nGert sipped some of his broth. \"That's something I'd rather we keep to ourselves. Either you seen the boy and that bastard, or you haven't. Don't matter none where either's come from.\"\n\nAs they talked, Trevor slipped into his room. When he came back Matthew saw the bulge in his pocket that was a knife. Matthew walked to the door and put his weight against it. His short sword leaned beside its hinges, sheathed. Whenever he'd needed it before, it'd always been at the door. If either of the two newcomers had seen it, they hadn't said anything.\n\n\"Well, I ain't seen a boy wandering around here, nor some man in gray. We've been shuttered inside for most of the past few days, the storm and all. If they went this way, they probably rode right on by.\"\n\n\"Not sure they're riding,\" said Gert. \"Think they're walking, honestly. Not too many out right now, and we managed to find what might have been his tracks.\"\n\n\"That so?\"\n\n\"Led this way, actually,\" said Ben. \"You sure you ain't seen nothing?\"\n\nMatthew paused, trying to think of a lie. Again his wife beat him to it, bless her heart.\n\n\"We turned them away,\" she said. \"They came wanting shelter, but they were bleeding, and he was armed. Looked like a thief, he did. We didn't want any trouble, and we don't want any now. He said he was on his way to Veldaren, if he's to be trusted.\"\n\nThe two men looked to one another, as if communicating silently.\n\n\"A hard woman that could refuse a wounded man asking for succor,\" Ben said.\n\nMatthew watched his wife give them an iron glare, one he'd been on the receiving end of more times than he preferred.\n\n\"Life out here's cold and cruel, gentlemen. We do what we can for our family. Maybe things are different where you come from, but out here that's the way things are.\"\n\n\"I understand,\" Ben said. \"We're just getting paid to ask these questions. Your broth's delicious, by the way. Feel it warming me all the way to my toes.\"\n\nMatthew started to relax, but only a little. The men seemed too confident, too sure of themselves. They were no strangers to those swords at their hips, either. The sooner they left the better. When they finished they stood and flung their cloaks over their shoulders.\n\n\"Our horses are probably itching to continue,\" Gert said. \"Or at least get out of the wind.\"\n\nAs they stepped toward the door they stopped, and Gert turned toward the curtain where Evelyn had said Tristan slept.\n\n\"You know, I've been fighting and killing for a long while. If there's anything I've seen before, it's a chopped limb. Mind if I take a look? I can make sure you stitched it up right, well as cut it proper. There's more art to keeping people alive than making 'em dead, after all.\"\n\nEvelyn hesitated, and Matthew knew that if she was unsure, then he was in over his head.\n\n\"If you wish,\" he said, putting on his gloves. \"I should get back outside. Only wanted to come visit with my guests, be polite. You two men have a good day.\"\n\n\"Want me to come with?\" asked Trevor.\n\n\"No,\" Matthew said, harsher than he meant. \"No, you ain't much use outside. Stay with your ma.\"\n\nTrevor got the idea, and his hand brushed the knife hidden in his pants. Matthew winced and hoped neither of the soldiers saw. He pulled the door open and stepped outside. When he shut it he leaned his back against it, closed his eyes, and listened. Never one with an active imagination, he struggled to picture the most likely thing they'd do. They were searching for the boy, obviously. They'd step through the curtain, one inside to look, the other hanging back, watching them, waiting to see if anyone did anything stupid.\n\nHis hand closed around the pitchfork's handle.\n\nSomething stupid like this.\n\nMatthew kicked the door inward. It seemed as if his entire vision narrowed down, just a thin window to see one of the soldiers staring back at him from the curtain, the one named Ben. His eyes widened for just a moment. His hand reached for his sword as if he were lagging in time. Matthew thrust the pitchfork for the soldier's exposed throat. Ben's sword couldn't clear his scabbard in time, so instead he ducked and turned away from the thrust, a purely instinctual move. It only made matters worse. When two of the teeth pressed against the side of his face, Matthew shoved with every hard-worked muscle in his body. The tips were thick, but with such force behind them they still punched through flesh and tore into bone.\n\nBen rolled his head downward, trying to pull free. When he did, blood spewed across the floor. He screamed. It might have been a word, a curse, but Matthew didn't know, didn't understand. Ben's jaw hung off-kilter, his right cheek shredded and the bone connecting it shattered. The look in his eyes reminded Matthew of the one time he'd encountered a rabid coyote attacking his animals. His sword free, Ben charged, not waiting for Gert. Matthew took a step back, braced his legs, and shoved the pitchfork in the way. The teeth hit his chain mail, and amid the screams of his family he heard the sound of metal scraping against metal. The teeth didn't punch through the armor, but they still bruised Ben's flesh and pushed inward hard enough to break more bones.\n\nMatthew twisted the handle to the side, bringing Ben to his knees, still stuck on the pitchfork's teeth. Dimly he heard his wife cry out, the words not registering in his mind as having any meaning, only her tone. Gert rushed through the curtain, his sword swinging. Abandoning his clumsy weapon, Matthew lunged for the door. He landed on his knees, grabbed his short sword, and spun. Gert bore down on him, swinging with both hands. Their blades connected, and panic flooded him when saw a tiny chip break off at the contact. His sword was weaker, the metal cheaper. It wouldn't be long before it broke.\n\n\"Leave him alone!\" he heard Evelyn shout, finally piercing through the haze. Gritting his teeth, he groaned as Gert pressed down with all his weight. He spared only a moment's glance to see Ben fling the pitchfork to the dirt and turn toward his wife. He had to help her, but he was pinned and badly positioned.\n\n\"Trevor!\" he screamed. Where was his boy? Why wasn't he helping? Now wasn't the time for fear, damn it! He angled his sword to block another chop, realized it was a feint, and smacked aside the thrust aimed for his belly. \"Don't you be a coward, boy, treat 'em like damn hogs!\"\n\nEvelyn hurried across the room, grabbing the poker from the fire. She held it clumsily, a pathetic weapon compared to the gleaming sword Ben wielded in his blood-soaked hand. Then he couldn't spare the glimpse, for Gert had dropped to one knee, knocking Matthew's sword out of position. Matthew struggled against it, but slowly his sword wavered, then hit the floor beside him. Gert's elbows pressed against his chest, his knee atop one of Matthew's legs.\n\n\"Don't worry about your wife,\" Gert said, his beady eyes inches away. \"I'll take good care of her. Your daughters too.\"\n\nIt was the absolute worst thing Gert could have said.\n\nMatthew let go of his sword, one hand grabbing Gert's wrist, the other ramming his eyes and mouth with his fingers. The soldier howled and tried to pull away, but Matthew dug his fingers in deeper and held on, feeling softness give way, then cartilage crunch in his grip. Gert tore his sword arm free, shrieking all the while. Blindly he stabbed. Matthew rolled, knocking his attack off balance. The sword struck the floor. With every bit of strength in his right arm, Matthew slammed Gert's head against the wall. He heard a wet crack, like the sound of a breaking pumpkin.\n\nThe abrupt end was startling. He heard his children crying, but saw no motion. He stood, shaking the gore from his hand. Evelyn huddled beside the fire, the poker at her feet, Trevor in her arms. The eldest son still held his blood-soaked knife. Nearby lay Ben, bled out from the cuts on his face and the deep stab wound at the small of his back.\n\n\"Everyone all right?\" he asked. Evelyn met his eyes, then nodded. \"Thank Ashhur.\"\n\nHe gave his wife a hug, making sure he didn't stain her dress with his right hand. The rest of his children stayed sitting, and he could tell they were traumatized by the violence. He went to each of them, hugging them and whispering that all would be well. At last he grabbed the dead bodies and dragged them outside by their feet.\n\nOnce they were out of sight, he came back inside and plopped into a chair beside the fire. His upper body started shaking, and he closed his eyes to try to hold back a sudden bout of nausea.\n\n\"We'll bury the armor until we can sell it in the spring,\" he told Evelyn, talking in hopes of stopping the violence replaying over and over in his head. \"Same with their swords. We'll unbridle the horses and send them on their way, hopefully far, far away. As for... you know... we'll give 'em to the hogs.\"\n\nHis wife made a soft cry. He shuddered but forced himself not to dwell on it. They'd do what they must, no different from ever before. Opening his eyes, he looked to the curtain, wondering if that blasted boy still slept, or if he was in there cowering in terror.\n\n\"Not worth the coin,\" he said, just before leaning to one side and vomiting.\n\n# CHAPTER\n\n#\n\nArthur Hadfield looked about the room in total disgust. He'd worked with mercenaries before, but to invite them into his home? So disgraceful. They gathered in the dining hall, over twelve of them. They were the captains, the ones with at least a hundred men at their disposal. They chatted with one another, killing time until Alyssa returned. They were a motley bunch, wearing various combinations of armor, ribbons, and tunics to distinguish themselves. Arthur dared not imagine how much coin was flowing into their pockets for simply picking their teeth and eating Alyssa's food.\n\n\"Not sure how much fun this'll be,\" said one, a bald man with a shaved head. \"Proper fight is on a battlefield, not crashing into people's homes and searching for rats.\"\n\n\"Killing's killing,\" said another. \"Since when you started getting picky?\"\n\n\"I'll take the money, but don't mean I can't want a nice open place to swing my ax.\"\n\n\"Probably need that space too, otherwise you'll cut your own fucking head off.\"\n\n\"Fuck you, Jamie. You probably can't wait to start. Your men will feel right at home wading through open sewers.\"\n\nArthur turned to leave and found Bertram standing behind him at the door, looking just as miserable.\n\n\"The stains they leave on the carpet...\" he said, shaking his head.\n\n\"Price of doing business, I suppose.\"\n\nThe old man nodded as he watched the captains bicker. Arthur went to his side, his arms crossed over his chest.\n\n\"Have you talked with Alyssa lately?\" Bertram asked after a time.\n\n\"Just this morning. Her mood has soured as the funeral approaches. I'd hoped she might grieve like any other woman, but instead she's out for blood.\"\n\n\"She wants that boy's killer found.\"\n\n\"I'm working on it, but he's proven an elusive little fuck.\"\n\nBertram chuckled. \"I am not surprised. There are a thousand criminals and murderers in this city. Finding one particular man must be difficult. Still, you might look at it a different way. Knowing one lowborn wretch from another is just as difficult.\"\n\nJust like last time, Arthur wondered if he was being tested. This time the wording seemed too strong to be coincidence. He decided to go out on a limb.\n\n\"I'm sure that even if we do catch the Watcher, it'll be tough proving that's who he is,\" he said.\n\n\"No one seems to know,\" Bertram agreed. \"Though I trust you in these matters, and would vouch for your opinion.\"\n\nArthur's eyes lit up.\n\n\"Is that so?\" he said. \"I don't think it will be long before I have a man to present to her. The city might be large, but there are too many eyes, too many mouths, for a man to hide forever. But I'm glad to know your trust in me is so great.\"\n\n\"I trust you more than I would any of them,\" Bertram said, waving a dismissive hand at the mercenaries. \"The Hadfields have always been good friends of the Gemcrofts. I can only do so much. Alyssa needs help in matters such as these, a guiding voice amid her grief. If only you could talk to her, get her to _listen_...\"\n\n\"I understand,\" Arthur said. \"I have my own matters to attend to, but I should return before nightfall, or close to it. When I do, I'll see if Alyssa will open up to me.\"\n\n\"Thank you,\" Bertram said, bowing low. \"Now, by your leave, I must try to convince those men that while the wine is a courtesy, it is certainly not free if you drink it by the barrel.\"\n\n\"Gods give you luck with that.\"\n\nArthur left the dining hall, retrieved his coat and sword, and exited the estate. Normally Oric would have gone with him, but he had headed north after bringing back Nathaniel's supposed remains. Arthur was skilled with a sword, though, and he knew his way. Besides, once inside the Serpents' territory, he'd be treated like a king.\n\nOnly minutes from Alyssa's mansion, he noticed the first of many escorting him from the shadows. Their cloaks were green, so he relaxed. No doubt William Ket, leader of the Serpent Guild, wanted to protect his investments. Arthur couldn't blame him. He took a few turns, vanishing deeper into the dark, dilapidated part of the city. Several more followed him, and for a moment he might have sworn he even saw someone along the rooftops. When he arrived at the guildhouse his escorts came into view of its torches, and they motioned for him to enter.\n\nAmid the emerald cushions and gold-framed paintings, Arthur sat and waited for William. A pretty lady wearing thin veils, and nothing else, approached and asked him his preference of drink. Normally he refused, always fearing some sort of poison or drug, but tonight he needed the help.\n\n\"The strongest of whatever you have,\" he said. \"Oh, but make sure it doesn't taste like piss.\"\n\n\"As you wish,\" she said, batting her beautiful green eyes at him. He watched her go, admiring her figure. With enough coin, he knew he could have her. Shame he had to spend the night at Alyssa's mansion. Tight figure like that, there was so much he could do to...\n\n\"Arthur! Welcome!\"\n\nArthur stood and tore his attention away from the little tart.\n\n\"William,\" he said, offering his hand to his younger brother, who had been William Hadfield before he'd changed his name to Ket to protect his family from embarrassment. \"My apologies for being gone so long.\"\n\n\"No need,\" said William. He was as tall as Arthur, and had the same eyes and hair. \"I figure you have your hands full handling a grieving mother, am I right?\"\n\n\"Hands aren't full just yet, but she'll give in to me in time.\"\n\nThe lady returned with his drink, and he accepted it gratefully. After a sip to test its flavor (somewhere between sewer water and burning oil) he took a large gulp. As it set his throat aflame, he chuckled at his brother.\n\n\"You've been late with your last shipment,\" he said, holding back a cough. Damn that stuff was strong. \"I'm a little curious as to why.\"\n\nWilliam's smile drooped, but only for a moment before he fixed it. This time it was far more fake.\n\n\"I should have figured a leisurely chat with you was not in store for me this evening. The gold was stolen from us as we smuggled it into the city, and through no fault of our own.\"\n\n\"No fault? Is that so? A convenient excuse to not pay me my half, wouldn't you think?\"\n\nWilliam sat down, and Arthur followed suit. The two stared at one another, a quick, silent exchange. Arthur knew William was trying to decide how much he should tell, and what Arthur's reaction might be. He hoped that for once his little brother told the truth, the whole damn truth.\n\n\"Have you heard of the Watcher, by any chance?\"\n\nArthur was too surprised to hide his reaction.\n\n\"Should I take that as a yes?\" William asked, raising an eyebrow.\n\n\"I have, but tell your tale first, and then perhaps I can better explain mine.\"\n\nWilliam waved over another servant girl, who brought him wine in a long slender glass.\n\n\"Well, whoever that bastard is got lucky,\" he said after taking a sip. \"Stumbled upon us while we were lifting the crate over the wall. Killed my men, took the bags of gold, and then do you know what this motherfucker did? He scattered it across the street. Middle of the day, high market, and he just tears them open and flings them to the crowd. Not the first time he's done that to us, either, but usually with smaller amounts. Scary, really. If he'd throw away that much coin, then there isn't a chance we can bribe or deal with him. He's out to kill us, all of us, not just Serpents. Wish I knew what we'd done to piss him off.\"\n\n\"Normally I'd doubt the ability of one man to kill so many of yours,\" said Arthur, \"but I've seen it for myself. Lost six soldiers to the Watcher. Wrote his name with their blood as a message to me when I returned. He took some of the gold, but not all, thank the gods. Another day or two and I'll have the rest outside the walls and ready for you. Alyssa thinks all of it was stolen, which will help make up for what we lost.\"\n\n\"I can change most of it into royal crowns with my merchants, and, amusingly enough, by buying large quantities of food and wine from Laurie Keenan. Something quite appropriate about laundering the Trifect's gold using the Trifect themselves. Any large increase will take some time, though. When will you be ready to pick up your portion of what we've exchanged so far?\"\n\n\"Keep it safe for now. Things are too chaotic. That's one of the other reasons I'm here. Tomorrow is Nathaniel's funeral, and come nightfall you need to make sure your men are prepared. Alyssa's hired at least a thousand mercenaries, perhaps far more. She's going to let them loose upon the city.\"\n\nWilliam's face darkened.\n\n\"Is she mad? What could we have done to spark such animosity...\" He stopped, glared at him. \"Unless you blamed her son's death on us?\"\n\n\"I told her the Watcher was tied to the local thief guilds, thinking it would keep her from asking too many questions. I never could have expected this overblown reaction.\"\n\nWilliam flung his half-empty glass to the floor. \"Of course not. You'll throw us to the wolves to make your life easier. Always have, always will. What do we do now? We can't face that many on our own.\"\n\n\"Then don't do it on your own. Spread word to the rest of the guilds. I want Alyssa humiliated by this course of action. She needs to doubt herself, her decisions, so she might trust me more. She is not in my pocket, not yet. In time she will be, I have no doubt, but until then I need your help.\"\n\nThe redness gradually left William's neck as he leaned back in his chair.\n\n\"I think I can convince the others, though I have little time. A single night to prepare a counter-ambush? Thank you for not telling me sooner. I like having to pull plans out of my ass.\"\n\nA hardness entered Arthur's words.\n\n\"You chose this life, not me. I came here the moment Alyssa was away, now deal with it.\"\n\nThey glared at one another, but finally William relented.\n\n\"So be it. But what will you do about this Watcher? He's making life miserable for the both of us. If he left the city, then he's certainly caught wind of our scheme, or at least part of it. A wrong word whispered to Alyssa and you'll hang.\"\n\n\"We'll hang, you mean.\"\n\nWilliam smirked at him.\n\n\"I live my life in the shadows. She already wants me dead, and is about to spend a fortune trying to achieve that tomorrow night. But you? You live your life in the light. The only place you can hide would be with me. Are you prepared to crawl through the gutters and eat shit to save your own ass?\"\n\n\"I think I might prefer the noose.\"\n\nWilliam stood, their conversation clearly over.\n\n\"Then you need to make sure the Watcher fucking dies.\"\n\n# CHAPTER\n\n#\n\nZusa watched the ceremony from the rooftops, her long cloak wrapped tightly about her. With only a few days' notice, the crowd was smaller than it might have been. She recognized various merchants, wealthy nobles, and a few members of Connington's distant family, plus Leon himself. All were from within Veldaren or its surrounding estates. Tradition would have provided a long enough delay for Laurie Keenan to travel from Angelport, but Alyssa seemed to care less and less about tradition with each passing day. Zusa didn't blame her.\n\nThe bones, brought back to the city by Arthur's lackey Oric, had been placed within a small sealed coffin. As her house guards lowered it into the ground, Alyssa stepped forward to address the crowd. She wore a dress of black and dark blues, and she'd smeared ashes upon her face. Behind her the sun dipped below the walls of the city, and in the twilight she made her decree.\n\n\"It is despicable that such a thing could happen,\" she said, loud enough for Zusa to hear. \"Worse that I might be expected to deny my son vengeance. So many of you here have made peace with the thieves. So many of you have come to accept the danger they pose. In doing so you have stripped away every shred of your pride and dignity. I will not! Those who prey on us, steal from us, kill and poison to claim their power, they die tonight. Go home if you must, or stay here if you fear the streets. But this night, only this night, must you fear them any longer. Veldaren aches for a purge, and I will be the one to deliver it. Cry out at me if you wish, but it will change nothing. The gold is spent, the orders are given. Let the blood flow.\"\n\nShe looked to the rooftop, directly at Zusa. Zusa nodded in return. That was the last command. There was no turning back. Spinning about, her cloak billowing silently behind her, Zusa raced to the front of the mansion and vaulted off the roof. When she landed, it was amid the gathered group of mercenary captains, who had been ordered to wait opposite the funeral.\n\n\"You have your orders,\" she told them all. \"Bring the Abyss to Veldaren, and throw every cloak into it. Give vengeance to my lady.\"\n\nThe captains grinned and smacked one another on the shoulder.\n\n\"About damn time,\" said one. \"Let's get to it!\"\n\nZusa left to the south, still trying to decide on her course of action. The mercenaries were scattered throughout the city in taverns, camps, and houses of those loyal to the Trifect. They would spill out onto the streets, and no one would be there to stop them. Only King Vaelor could make a reasonable attempt with his soldiers, but he'd have to break a lifelong streak of cowardice, which Zusa knew would not happen. Ever since the Bloody Kensgold he'd given them all freedom to kill one another so long as their threats were never aimed at him. When the nightmare began, no doubt the city watch would turn the other way, if they even left the castle at all. She had an inkling they would not.\n\nBut the bloodshed would accomplish nothing if she couldn't find Nathaniel's killer. The Watcher. Where did he hide?\n\nThose in the shadows were about to be flung into the light. She resolved to scour the city, keeping an eye open for anything unusual. If the Watcher was as skilled as Veliana made him sound, he would hold his own no matter how many mercenaries they flung at him.\n\nVeliana...\n\nZusa might have offered a prayer for her, but she had turned her back on her former god, Karak. She had no one to pray to, so instead she just murmured the thoughts aloud, hoping that her friend might survive the night. So badly she'd wanted to warn Veliana of the coming slaughter, but to do so would have endangered Alyssa's efforts, especially if Vel informed the other guilds. That meant keeping her mouth shut and hoping for the best. If only the woman could have relinquished her desire for control of the Ash Guild, she might have made a new life at Zusa's side within the Gemcroft mansion.\n\n\"Stay safe, Vel,\" she said, crawling up the side of a small house with a flat roof. Once atop she leaped across, scanning her surroundings for a man in strange gray cloaks and wielding twin swords. A man skilled enough to defeat her.\n\nHalf an hour passed, painfully quiet. It seemed the entire city had drawn its collective breath. Then all at once came the exhalation. Two fires erupted in southern Veldaren, both at supposed headquarters of thief guilds. Deciding there was as good a place as any, she headed that way. She passed several patrols, and one even had the gall to fire a crossbow at her. She ducked lower and continued, realizing she would be far from the only one to travel by rooftop that night.\n\nThey were torturing a thief in the street when she arrived at the first fire. It was probably supposed to be an interrogation, but that would have involved a chance for answers from the victim. The thief had blood smeared across his face, and from the way his jaw hung, it looked like it'd been broken in multiple places. The best he could do was point. The light of the fire bathed him and the mercenaries in red, and in it the thief sobbed for mercy.\n\n\"This is your creation,\" Zusa whispered to the distant thief, hardening her heart against the violence. \"This is the fate you have earned.\"\n\nStill, it seemed a cruel fate. When a mercenary impaled the thief, she was thankful. She turned for the second fire when she caught movement from the corner of her eye. She backflipped into the air as a blade cut where she had been. Facing her attacker, she fell, grabbed the ledge of the building, and then flung herself at him. He was a giant man, his features shrouded in twilight. She slammed her knees against his chest. It was like trying to knock down an ancient oak. She rolled over his head, jumped away to give herself some distance, and then drew her daggers. As her opponent whirled, she used the half-second to examine him.\n\nHe was dark-skinned, darker than she'd ever seen, and wore light clothing with a long gray cloak. He carried two enormous swords, each a length most would need two hands to hold. His muscles looked more appropriate to a woodcutter or blacksmith than a thief. But most of all her eyes were drawn to the white paint across his face, making his shaved head look like a bleached skull.\n\n\"A woman?\" he said. Zusa lunged, hoping to take advantage of his surprise. She parried one of his swords to the side, then thrust her other dagger for the opening. The man seemed prepared for the maneuver. He twisted, slapped aside her thrust, and stepped closer. She tried leaping back to gain more distance, but he followed, trapping her at the edge of the roof. Falling to one knee, she tried to hamstring him, but again his swords were there, batting her far smaller weapons aside. Part of her wondered why, with such an advantage in reach, he forced them to remain in close combat.\n\nThen one of his swords fell, and she felt a hand grab her hair, each finger as thick as a sausage. Her feet lifted off the roof. She held in a scream, all her focus narrowed to a razor's edge. Her daggers swung, both aiming for his neck. With only one weapon, he had no chance to block, or at least so she thought, but he used its flat edge to strike both her wrists as they thrust, pushing them over his head. Before she could bring her arms back down, the sword's edge pressed against her throat.\n\n\"Stop struggling,\" he said. \"I'm not here to kill you.\"\n\nHis voice was deep, so deep it reminded her of the rare times she'd heard Karak whisper to her in the night. She forced herself to calm, to look into his brown eyes without flinching. The sword pressed tighter against her throat, as if he expected her to try to break free.\n\n\"What do you want?\" she asked.\n\n\"Not you,\" said the man. \"My target is not a woman. I might have said so if you hadn't jumped at me like a rabid dog.\"\n\n\"Who are you?\"\n\nHe stared at her as if deciding something. Decision made, he unceremoniously dropped her. She landed on her feet and crouched before him, ready to leap at the slightest wrong move.\n\n\"I am Ghost. I've come to claim the Watcher's head, and other than the breasts, you fit the description rather nicely.\"\n\nZusa slowly straightened, though her muscles remained tense. Whoever this Ghost was, she had no intention of relaxing in his presence.\n\n\"Who has hired you?\" she asked. \"One of the thieves?\"\n\nHe grinned at her. Something about it worsened her unease.\n\n\"I cannot tell you, as surely you can understand. You seem at home in the night, and you move as I expected the Watcher to move. Do you know of him? Tell me, and I might make it worth your while.\"\n\n\"Whatever I know, I cannot say, for I seek him as my own bounty. My mistress wishes to be the one to claim his life, and I would not dare risk cheating her of it.\"\n\n\"Mistress?\" asked Ghost, raising an eyebrow.\n\n_Too much, stop speaking, you always say too much._\n\nSo she smiled, hoping to convince him that it wasn't a slip of a tongue, instead a ploy to make him wonder if she spoke truth or not. He probably didn't buy it, but it was still worth the attempt.\n\n\"So be it,\" Ghost said. One of his swords shifted and she moved to jump, but then he saluted her with the blade. \"A game, then. I will let you search unimpeded, but I expect the same courtesy of you. If you somehow find him first... all I ask is that you come to me at the Mug and Feather tavern so I may know your name. Any lady more skilled in tracking than I is a lady I would sorely wish to meet with again. Consider it a repayment of my generosity.\"\n\n\"The generosity of a man who nearly stabbed my back before seeing my face to confirm a kill?\"\n\nGhost laughed. \"You are still alive, woman. That alone is proof of my generosity.\"\n\nThe way he said it, with no anger or pride, only amusement, chilled Zusa's blood. This was a man to whom death was a common companion, who believed he had nothing to prove. If such an agreement kept her safe from his blades...\n\n\"I accept,\" she said. \"Now forgive me, but I have a man to find.\"\n\n\"Good luck,\" he said. \"Oh, and stay safe. I hear there's a lot of mercenaries out searching for people like you.\"\n\nShe looked to the fire behind her, and the several corpses left there to rot in the street. When she looked back Ghost was gone. _No one that big should be able to move so silently_ , she thought, but it seemed she was wrong. Cursing herself, she hurried north, following the distant cries of battle. If she was lucky, they might still scare out the Watcher, but now she wondered if she truly had any hope of finding him before Ghost. How great would Alyssa's anger be if she found out Ghost had captured him first?\n\nStill, that would be better than their finding him at the same time. However that confrontation might end, she knew her blood would be spilled.\n\nThe city had gone to the Abyss. There was no other way to describe the horrors Haern watched as he hurried along the streets, keeping his head down and his swords hidden. It was too dangerous to remain on the rooftops. Every mercenary with a bow was firing at any shadow that moved. He'd counted four fires the last time he'd found a quiet enough spot to scale a wall and look over the city. Madness, total madness. Was this what it'd been like when his father first declared war against the Trifect so long ago?\n\nThe mercenaries traveled in squads, some as large as a hundred. They roamed the streets, smashing in doors, dragging out scared owners to ask questions, demand names, and sometimes execute them outright. He watched a group of three thieves, Spiders by their cloaks, chased by twenty men in armor. They died when a second group cut them off, another ten with naked blades and eager eyes. The mercenaries left only pieces of the three. Pieces.\n\nBecause he lacked guild colors and appeared the beggar, he'd managed to avoid much of their ire. He'd been questioned twice. The first time he'd feigned deafness. The second time he'd pointed the mercenaries toward the headquarters of the Serpent Guild. While following them he'd watched a couple dragged out of bed, the husband hollering, his wife holding blankets to her chest to hide her nakedness. While their children watched from the doorway, the mercenaries had cut the couple's throats and cheered the name of Alyssa Gemcroft as if she were a goddess of blood and murder.\n\nAll the while the city guards remained nowhere to be found.\n\nHaern ducked through a side alley, not surprised to find two more there with him. They wore the brown cloaks and sleeves of the Hawks, and they drew daggers as he rushed by. He wished them luck, out of nothing more than professional courtesy. Part of him wondered how many would abandon their cloaks. Doing so was punishable by death. Still, it seemed the only way to survive. Of course he'd seen plenty without cloaks or colors dying in the street. Perhaps all it'd take was a single man whispering your name to find yourself in the arms of sellswords...\n\nDeciding it worth the risk, he used a window ledge to vault atop a house that overlooked the headquarters of the Serpents. Thirty mercenaries surrounded the place, some wielding crossbows, some torches. There was no doubt as to their intentions. Those inside could die by blade or by fire. Hardly a choice he'd desire to make.\n\n\"By order of Lady Gemcroft, all those affiliated with the thief guilds are to be executed,\" the mercenary captain shouted. \"We know you're in there, so come out and die with honor!\"\n\nHaern flattened himself on the rooftop, making sure no one could see him. The last thing he wanted was some crossbowman with a twitchy finger putting a bolt through his eye.\n\n\"This city ain't hers!\" challenged a hidden stranger. Haern's eyes narrowed as he realized the voice did not come from the surrounded headquarters. \"Time you sellswords learned that!\"\n\nFive buildings had clear views of the headquarters, including the one Haern lay atop. From every one of their windows appeared green-cloaked men with bows and crossbows. With a loud cry they released. A third of the mercenaries died in the first wave. Some tried fleeing toward the main streets, while others stormed toward the buildings. The arrows gave chase. When the last fell, Haern saw William Ket step out from the headquarters, a gleaming sword in his hand. He found the mercenary captain, chopped off his head with three swings, and lifted it in the air. The rest of his guild cheered.\n\nHaern's blood ran cold. The other guilds would certainly be preparing ambushes as well. While it'd be foolish not to expect losses on both sides, and he generally held no love for mercenaries, there were a few mercenaries he did care for.\n\n\"Damn it, Senke,\" he whispered, pulling back so he wouldn't have to see the executions of those who still cried out in pain. \"Tell me you aren't part of this nonsense. You can't have been this stupid.\"\n\nBut of course he would have. It seemed every free man had been brought into Alyssa's fury. What had spurred this on? Why now? He so rarely heard news of her, or of the Trifect in general. They'd grown quiet, defensive. But this?\n\nThere was nothing quiet about the screams that seemed to come from everywhere as the stars vanished beneath great curtains of smoke. How much of the city would burn? Would any be desperate enough to form bucket brigades while the death squads marched? He didn't know, and didn't care. He had to find Senke and Delysia. Losing them once had been hard enough. He couldn't do it again. The city was large and swarming with men. Finding them would be near impossible, but at least he knew names to search for.\n\nHaern dropped back to the street and began searching for a smaller group of men to ambush. He had to avoid several larger patrols, and at one point a crossbow bolt. At last he encountered a group of three, deep in combat with five members of the Ash Guild, a sixth bleeding out nearby. The mercenaries were cornered against the city wall, and it was only their superior armor and reach that kept them alive against the greater numbers. Haern descended upon them like a whirlwind, his swords cutting down two before the thieves even knew he was there. He cut the throat of a third, then watched the final two die at the hands of the mercenaries.\n\n\"Thanks,\" said one of them, a gruff man with a beard and a wicked cut across his nose that was still bleeding. \"Fuckers led us here into a trap, jumped us from the rooftops.\"\n\nThe man pointed at Haern's cloaks.\n\n\"You one of them? If so, you might want to run while we're still feeling grateful...\"\n\n\"I am no friend of the guilds,\" Haern said, making sure to keep his cloaks low so they only partially saw his face. \"I'm searching for a small group of mercenaries. One is named Senke, sometimes using the name Stern. The other is Delysia Eschaton. They should be together.\"\n\n\"Not a clue,\" said the man with the bleeding nose.\n\n\"Don't know about no Delysia,\" said the second as he walked over and killed the sixth Ash member, who had been whimpering in pain. \"But that name sounds familiar. I know a Tarlak Eschaton, and he runs his tiny group, nothing but small-timers. Guy's half off his rocker. Might be them you're looking for.\"\n\n\"Perhaps. Do you know where they might be?\"\n\nThe man spit. \"All them smaller groups got put together and sent to the far south. Alyssa figured it'd be easier to spread out and find the rats when they tried to hide in their slums.\"\n\nHaern fought down his panic. They'd been sent into the very homes of the thieves, as if they would be the ambushers? When the guilds fled to safety there, they would be doing the killing, not the mercenaries. He had to get Senke and Delysia out of there, and fast. He bowed to the three men and then turned south.\n\n\"Hey, what's your name?\" asked the man with the bleeding nose.\n\n\"I have none,\" he shouted. When he turned, he caught sight of something strange, like a white face peering down from the wall, round and hairless. A glance back revealed only darkness.\n\n_I'm jumping at shadows_ , he thought. _If you're alive, Senke, I'm going to murder you for accepting such a stupid assignment._\n\nHe wondered where his father was in all this. He wouldn't hide, not from such blatant disrespect. The Spider Guild's headquarters were in the southeast. If these Eschaton mercenaries stumbled upon him and his best...\n\nHaern ran faster.\n\nDeciding speed was more important than stealth, he cut to one of the main roads and trusted himself to outrun any patrols. He raced south, all the while doing his best to ignore the pain in his stomach. The wound there hadn't quite healed, though it was hardly more than an angry red scar. Still, the movement was enough to stretch the skin and agitate it. What he'd have given for an extra day or two to heal before all this madness hit.\n\n\"You! Stop!\"\n\nHaern cursed as he saw a group of five mercenaries up ahead rush toward him, no doubt thinking him a fleeing member of the guilds. The houses were packed tight on either side, so it was either turn back in search of a route around, or go straight through.\n\n\"Move!\" he screamed, hoping to startle them with his mad rush. He slide-kicked just before nearing, avoiding an arrow that whistled overhead. His kick took out the legs of one, and he rolled away. Smashed under that much armor, he'd be killed in seconds by the others. A sword swung for his head, but he twisted and fell to one knee. Two others lunged, their swords thrusting. Haern dove to his right, landing hard on his shoulder. He'd made it to the other side.\n\n\"We said stop!\" one of them shouted. Haern laughed, wondering who exactly he thought would obey the command. He weaved from side to side, not surprised to see one last arrow plink into the dirt beside him. With their heavy armor they couldn't hope to match his speed. Last he looked, they'd abandoned the chase and instead smashed in the door of a nearby home. He offered a prayer for the inhabitants as he ran.\n\nIt seemed the very air grew thicker as he plunged deep into southern Veldaren. He counted five fires now, and one of them wasn't far off. The smoke whirled down the streets, its distortion good for the thieves, bad for the mercenaries. He heard sounds of combat from one home with a broken door, and he watched another group battle, four Hawks against two mercenaries. He let them be. There were too many fights for him to help them all. The gravemen would be busy tomorrow, he thought grimly. Whatever buildings didn't burn would soon have their walls painted with blood.\n\nA plume of smoke erupted to his left, accompanied by a deep explosion. His curiosity couldn't take it. He turned, climbed onto a rooftop, and hurried that way. Whatever he was expecting to see, it was not what he found.\n\nThe area opened up to a fountain carved in the likeness of two women bathing each other, the fountain long broken and out of water. At least twenty men lay dead, half mercenaries, half thieves, their blood mixing together to stain the surrounding cobblestones red. A large group of thieves remained, Wolves judging by their colors. They faced off against only four, a strange four at that. Senke guarded one half of the fountain, parrying and blocking with his two flanged maces. In the other half fought a short man in plate mail, a punch dagger in each hand. Delysia stood in the fountain behind Senke, her lovely red hair matted to her face by a cut across her forehead. She was taller than he remembered, and she wore the white robes of a priestess. His heart raced faster, and he forced himself to move. While Delysia cast spells that bathed them all in white light, the last of the four stood in the other side of the fountain with his hands surrounded by fire. He wore a yellow robe and a pointy yellow hat. The color reminded Haern of dandelions. He had red hair similar in color to Delysia's, and a well-trimmed goatee.\n\n_Guy's half off his rocker_ , the mercenary had said about Tarlak, their leader. Haern had a feeling that was him. Only the insane, or the extremely confident, would dare wear such an outfit.\n\nThis Tarlak swung his hands in a circle. Fire danced across his fingers, then streaked toward a group of three hiding behind an overturned wagon, trying to shoot crossbow bolts from cover. The wagon exploded into shrapnel and embers. The short man seemed hard-pressed fighting one thief, but the wizard kept zapping thin bolts of blue lightning, knocking the thief back and keeping him from scoring a kill. Senke fought three at once, yet seemed to be doing better at protecting the two in the fountain. Having been on the receiving end of so many smacks to the head and kicks to the chest while training with him, Haern wasn't surprised.\n\nDecision made, he drew his swords and charged. It was Senke and Delysia he'd come to protect, so it was them he'd help. The wizard saw his approach and turned, magic shimmering on his hands.\n\n\"It's Haern!\" Senke shouted just as the lightning arced out, having seen his approach as well. Haern rolled, wishing he'd had far more training in dealing with spellcasters. The roll seemed to work, for he heard the ground behind him crackle and break from the impact. He kicked back to his feet and jumped, having closed enough distance to reach the first Wolf. The man turned and tried to impale Haern with his sword. Having had enough of that nonsense only a few days before, Haern batted aside the thrust while still in the air and landed short. His opponent exposed, it was easy work looping his other sword around and cutting his throat.\n\n\"Sorry!\" he heard the wizard shout.\n\n_Damn fool_ , Haern thought as he linked up with Senke, standing side by side with him as the Wolves closed in.\n\n\"Glad you could join us,\" Senke said between breaths. Despite the smoothness of his parries and kills, he was clearly winded.\n\n\"I shouldn't have to. What the Abyss are you doing out here?\"\n\n\"Fight now, insult me later.\"\n\nAs one they went on the offensive. It felt like old times, carelessly training in Thren's safe house. But this time it wasn't dummies they fought, nor did they wield wooden swords. This time their opponents bled. Haern struck both high and low, forcing his thief to make desperate parries with his daggers. The shorter weapons made his arms move more than Haern's to keep up with the attacks, and Haern used that to put him more and more out of position. At last he feinted a wide slash, pulled his sword in, and stabbed. The edge sliced through cloth and flesh and into lung. As the thief fell, Senke gave him a good bash on the head with his mace, just to be sure. Two more rushed ahead, but a blinding flash from behind Haern dazed them. With such a handicap, they fell with ease to the skilled fighters.\n\n\"Help Brug!\" Senke shouted as three more Wolves approached from the north, joining the others.\n\n\"Brug?\"\n\n\"The short fat guy.\"\n\nHaern felt a moment's hesitation. He'd fought alone for so long, he wasn't used to obeying orders. But then again, he felt himself slipping once more into the past, nothing but an awkward child learning from his masters. He turned and circled the fountain, joining Brug, who was bleeding from his shoulder and face. A dagger was lodged in the crease of Brug's armor. Haern yanked it out as he ran past, hurled it at Brug's opponent, and then followed it up with a flying kick. The dagger's hilt hit the thief's throat, and then Haern's foot cracked his chest. The Wolf dropped to one knee, lifting his dagger in a clumsy defense. Haern cut him down, a clean slice through his throat.\n\nBrug looked ready to explode.\n\n\"I had him!\"\n\nHaern blinked. \"Uh, sorry?\"\n\nA fireball sailed over both their heads, delaying the attack of several Wolves who had abandoned their attempts at shooting the wizard and instead chosen to close the distance. Haern felt the heat of it on his neck.\n\n\"Damn it, Brug, what am I paying you for?\" shouted the wizard. \"And you, Haern, right? Keep him alive, will you?\"\n\nHaern turned to the wizard's opponents, somewhat amused at how much redder Brug's face had grown. Brug blubbered, then rushed ahead, punching the air with his daggers. Haern's amusement left. The idiot was going to get himself killed because of his pride. He rushed after, the two of them barreling at the three Wolves as if they were madmen. At the time it was a fair assessment. The Wolves wavered, he saw the doubt in their eyes, and then they turned to flee. Haern killed two, for he was too fast and had far too much momentum to be outrun. He sliced the hamstring of the third as he ran on by, allowing Brug to catch up and eviscerate the thief with his punch daggers.\n\nSucking in air, Haern turned back to the fountain. The last of the Wolves were either dead or fleeing. Tarlak stepped out of the fountain, helped Delysia follow, and then waved.\n\n\"That wasn't so bad,\" he said.\n\nHaern shook his head. Off his rocker, indeed.\n\n# CHAPTER\n\n#\n\nGhost followed the Watcher south, though he did so with no hurry. He'd watched him fight, and learned two things: One, no puke-brained mercenary would do him in. Two, the Watcher had friends he cared deeply about. He might have thought his quiet voice concealed his emotions when he asked the mercenaries questions, but Ghost heard the hint of worry, particularly about the one named Delysia. With that, it was only a matter of time before he brought the Watcher down. You couldn't have attachments, not if you wanted to survive against someone like Ghost.\n\nTarlak Eschaton wasn't well known to him, but if he played the mercenary game he had contacts, friends, employers, maybe even a spot in the guild. There would be no hiding. So as he strolled down the street, always given a wide berth by the groups that rushed past with bloodied swords, Ghost paused and looked west. A strange commotion was brewing down there\u2014he could tell by the torches and the way several of the recent patrols had all turned in that direction. Had the fighting coalesced into an actual battlefront? Surely not. The thieves weren't that organized, nor would that benefit them in any way. So what, then?\n\nHis hands on his hilts, he strode over to the mob. He estimated at least sixty men were gathered around what he realized was the temple of Ashhur. So far they remained at the steps, but that appeared ready to change at any moment. Fifteen priests stood in their way, their hands at their sides. They were proficient with many spells, he knew, but how effective those spells would be on armored men he was unsure. An elderly priest with a bald head stood in the center of the steps, and he faced the crowd without any semblance of worry. Sweat ran down the sides of his neck, though, and Ghost knew him just as scared as the rest.\n\n\"You cannot enter,\" the old man shouted, hardly heard over the din of the mercenaries.\n\n\"Let us in!\" they shouted.\n\n\"Out of the way!\"\n\n\"You harbor thieves!\"\n\nGhost frowned. He needed a better look. Pillars lined the edges of the lengthy stairs. Hiding behind one, he climbed halfway up the steps, and from there he peered into the temple. Inside was chaos, hundreds of people crammed within for shelter. Given how many homes he'd seen ransacked, it made perfect sense. Where else might be safer? They sat on benches, huddled against walls, and lay in the aisles if need be. Sure enough, he caught sight of a few cloaks in there, but not many.\n\n\"I will not hand anyone over to be butchered in the streets,\" the head priest was saying. \"Go on your way. Curse our city with your bloodshed if you must, but I will not allow it to happen on my doorstep!\"\n\nAnother patrol joined the group, this one numbering twenty. They muttered among themselves, wanting to know the reason for the delay. Several more lit up torches. Ghost felt his blood boil. They would set fire to a temple to slake their bloodthirst? A thousand rogues must remain hiding elsewhere, but they would come here?\n\nThe head priest lifted a hand over his head. A bright light grew from his palm, and even from the side it was painful for Ghost to look upon. He didn't want to imagine staring into it from the front. This seemed to cause the mercenaries to draw back a little. A captain Ghost had met a couple of times, named Jamie Half-Ear, for obvious reasons, stepped forward.\n\n\"We don't want nothing bad to happen here,\" he said. \"But we saw plenty run inside before we surrounded the place. No innocents need to be hurt. Just send them out.\"\n\n\"Those who came here for succor shall receive it,\" said the priest.\n\n\"I know, I know, you gots to say that,\" said Jamie. \"Please, er...\"\n\n\"Calan.\"\n\n\"Calan. I doubt you want us filthy men running through your fancy place, so how about I just send one or two to point 'em out to your little helpers? Only the guilty get taken out, and just a few. Everyone else stays safe, see what I'm saying? They ain't your flock. They ain't your children. They're damn thieves, and those with more power than you say it's time they die.\"\n\nCalan shook his head. \"In time, perhaps, but not tonight. Go on your way, all of you.\"\n\n\"They're dying tonight, you stupid ass, no matter what you do. I see a lot of stone, but this building's still got plenty that'll burn. You hearing me, Calan? It'll _burn_!\"\n\nThat was enough for Ghost. Though he couldn't care less about their deity, the temple was a beautiful structure. Stubborn and blind, the whole lot of them. He reached into a side pocket and withdrew a handful of knives, weighted to fly true. Scanning the crowd, he waited, wanting his choices to be absolutely perfect. Jamie was too close, too public, but those near the front, most eager for blood...\n\nHidden behind his pillar, he flung his first knife into the crowd. It plunged deep into the throat of a man hollering at the top of his lungs, ceasing his cries for fire. The next three took down torch-bearers, as Ghost had decided fire was the greatest threat to the temple. Down they went, their torches clattering across the stone. By now the crowd had noticed the deaths, and while some wondered aloud, most roared for blood. They'd blame the priests, which was the goal. All the better to make the mercenaries appear fools when he stepped out.\n\n\"You'd murder us in the street while claiming to protect life?\" Jamie cried, more to the mercenaries than the priests.\n\n\"We have done no such thing,\" Calan insisted. He might as well have shouted at a thunderstorm to cease its rumbling.\n\nGhost slipped to the other side of the pillar, then hopped lower, closer to Calan. He was directly behind him, with a clear view of Half-Ear. The captain was practically frothing at the mouth while screaming for blood, but he'd not yet drawn a blade. Not yet... but close... his hand twitching... now!\n\nGhost coiled his legs and pushed off, launching himself between them. He wielded a single sword in both hands, needing the power. Before Jamie could swing, Ghost's sword tore through him, slicing from collarbone to hip. When Ghost kicked, the upper half of him collapsed backward and rolled down the steps, spilling innards, while the legs crumpled and lay in place. A flood of gasps came from the crowd, those who were not stunned silent. Ghost held the bloody blade before his face, peering over it with his eyes.\n\n\"Be gone, damn cowards. You have no business here. Go elsewhere, and slay the cloaks in their homes, the streets, wherever but here.\"\n\n\"This is madness,\" Calan said behind him. \"I can't allow...\"\n\n\"Quiet, Priest. You may not want bloodshed, but it'll happen, and better here on these steps than inside your halls. Such disrespect, it's shameful. They deserve to die just as much as those thieves within.\"\n\n\"I'd rather neither die,\" Calan said, his voice dropping lower.\n\n\"Good luck with that.\"\n\nGhost had hoped the brutal display would cow the crowd, but he'd underestimated their desire. Too many seemed eager for a fight. They'd gone unchallenged, no doubt thought themselves already nearing the end of their task. If they went south they'd probably think differently. The fires were growing, the smoke blanketing the city. This fight wasn't over; not even close.\n\n\"Stun as many as you can,\" he said to the head priest. \"Blind them, knock them back. Leave the killing to me. I'll be better at it than you anyway.\"\n\nHe drew his other weapon and lifted both high above his head, as if worshipping a god of the sky.\n\n\"Draw swords or flee!\" he shouted to the crowd. It was time to end this stalling. It was time for blood or cowardice.\n\nThe mercenaries surged forward, not following any spoken command, only a collective realization that they had to attack now or be revealed as bluffers. The priests raised their hands, their palms facing down the steps. Light shone from them, the intensity blinding. Ghost heard the sound of a hundred claps of thunder, and it passed over him like a physical force. Those at the front staggered or fell back, forcing the rest behind to pull them out of the way. Ghost took advantage of the confusion, leaping forward to gut one mercenary and cut down another. He backed up the steps, parrying random thrusts that seemed wild, as if his opponents were still struggling to see.\n\n\"Fall back,\" he ordered. The priests exchanged a quick look, but then Calan nodded.\n\n\"As he says!\" shouted the head priest.\n\nThe fifteen climbed the steps, prayers still on their lips. Walls of force slammed into the crowd, invisible but their effects not. He watched noses break, heads bruise, and fingers snap in painful directions. More stumbled upon the steps, crushed by those who rushed behind them. Ghost stayed before Calan, figuring him the most important to protect. He was the strongest, and as long as he stood, the other priests wouldn't break rank and flee.\n\n\"We must hide inside,\" Calan said. \"We can't hold them back.\"\n\nGhost lunged from left to right, taking out those who pushed through the spells ahead of the pack. They fell, disadvantaged by their lower ground and disoriented beyond measure. If not for the sheer numbers, Ghost feared he might have gotten bored.\n\n\"We go inside and they'll burn us out,\" he insisted. \"Stand and fight, old man. Stand and fight.\"\n\nHe kicked another body down the steps. They were gathering before the door, nearly out of space to retreat. Calan nodded, accepting Ghost's decision.\n\n\"So be it,\" he said, standing beside Ghost and lifting his arms. \"Forgive me, Ashhur, but I need your retribution this night.\"\n\nHe lowered his hands. A sound of thunder rolled, and then the steps to the temple trembled and broke. Dust rippled outward in concentric circles, giving visibility to the shock wave. There had been about twenty men pressing upward, and they collapsed, crying out in pain. The number of broken limbs Ghost witnessed was staggering. It was like Ashhur had stepped down and crushed them with his heel. Calan trembled, then stepped back and accepted the support of two more priests. The rest of the mercenaries, not willing to even think of climbing those corpse-strewn steps, turned and fled. Ghost leaped among them, killing a few more for good measure. Coated with gore, he returned to the temple, where he heard songs and lamentations coming from within.\n\nA couple of the priests thanked him, but most eyed him warily and scooted farther away when he neared.\n\n\"Why?\" asked Calan, his arm still around one of the younger priests. \"Why did you help us?\"\n\nGhost shrugged. \"Can't stand disrespect, and that's all that was. They should respect me, and respect you. They didn't, and now they're dead.\"\n\n\"Not all of them,\" Calan said, gesturing to the many wounded. He turned to his priests. \"Go and tend to them.\"\n\n\"I doubt they'll bother you now,\" Ghost said. \"But I'd consider getting rid of those with colors in your temple while it's still calm.\"\n\n\"If I did that,\" said the head priest with a smile, \"I wouldn't be worthy of much respect, would I?\"\n\nGhost laughed. \"Maybe you're right. Then I hope your god watches over you. Before this night ends, you might still need him.\"\n\nDeathmask watched the carnage from the window of a room currently absent of its original occupants. No doubt they'd fled to safer territory, assuming there was anywhere safe within the city walls. He'd gone out with the initial patrols, under orders from Garrick to help ambush some of the smaller mercenary groups. When the first fight began he'd slipped away, joining Veliana in the large apartments overlooking their headquarters. After a few initial confrontations that left many on both sides dead, the area had remained quiet for the past hour. Most recently a squad of fifty men had checked the headquarters for thieves, found none, and then moved on.\n\n\"I doubt there'll be much guild left for you to rule after tonight,\" Veliana said, relaxing on a moth-eaten chair. \"Hope Garrick survives, though. Would be heartbroken if some lousy sellsword gets the honor of cutting off his head.\"\n\n\"He's too cowardly to die tonight,\" Deathmask said as he tied a gray cloth over his face and straightened it so the eyeholes lined up properly. Behind him Veliana did the same, using the knot of the cloth to also keep her dark hair in a tight ponytail. They both wore loose gray clothing, and cloaks of a darker shade. Veliana had killed a pair of Spiders in the initial chaos before joining Deathmask in the building, bringing with her their clothes and weapons.\n\n\"While unexpected, tonight certainly works in our favor,\" he said, once more looking to the window. \"The fewer we have to thin out the better. Have you given thought as to who we should spare?\"\n\n\"The only ones who come to mind are the twins,\" she said. \"They have a head on their shoulders, though it seems like they share it. They think so alike it's creepy.\"\n\n\"Can they wield a blade?\"\n\n\"They're better at throwing them than wielding them, but no average cutthroat could handle either of the twins in a fight.\"\n\n\"Good. Names?\"\n\nShe tugged at her mask, trying to get it to fit comfortably. \"Mier and Nien.\"\n\nDeathmask rolled his eyes. \"What wonderful parents. Gods forbid their names be even a little different.\"\n\nHe leaned away from the window as a man rushed down the road, a jittery fellow who kept glancing in every direction. Two more followed after. Veliana saw Deathmask's reaction and straightened in her chair.\n\n\"Someone there?\" she asked.\n\n\"Looks like some scouts, no doubt making sure it's safe to come home. Get ready. We'll have little time between their leaving and Garrick's arrival.\"\n\nThe Ash scouts vanished into the building. Deathmask peered out the window, watching, waiting. When the scout emerged, Deathmask beckoned Veliana closer.\n\n\"Go!\" he said when the scout turned a corner. They tossed a rope that was tied to the bed in their room, sliding down even as it uncoiled. They hit the street in seconds and sprinted for the headquarters. Deathmask led the way, Veliana at his heels. Once inside they slowed, walking through the hallway into the lavish rooms.\n\n\"Pick your spot,\" he said, his eyes darting about. \"Keep close to the doors for when we make our escape.\"\n\n\"I'm no stranger to this sort of thing,\" Veliana said, glaring at him through her mask.\n\n\"Keep your hood raised. If they see your hair, they might figure out who you are, instead of just assuming you another Spider.\"\n\nShe lifted the hood of her cloak and let it fall across her face as Deathmask did the same. He entered one of the side sections curtained off to give privacy with the dancer women, leaving a gap so he could watch the entrance. Veliana adjusted a giant pile of pillows, hiding behind it with daggers drawn. Deathmask drew his weapons as well. There would be no magic for him, no spells of blood and shadows. Using such skills would reveal him to Garrick, which would ruin his current plan. No, it'd be just knifework. Veliana had trained him for a few hours, but it had only made clear how far from proficient he was. He'd spent an hour casting spells of speed and strength on himself to try to make up for the lack, but he wouldn't know for sure until the ambush if they'd worked. Not being much of a praying man, he crossed his fingers and swore to succeed whatever the cost.\n\nThe door slammed open. In rushed a collection of the Ash Guild: all men close to Garrick, Deathmask noticed. Their clothes were lacking in blood and gore. No ambushes for Garrick, which put a smile on Deathmask's face. That fact would work wonders for them later, should he and Veliana survive for the second part of their plan. The thieves went straight for the obvious: the bar filled with bottles of wine and ale. Deathmask was glad he couldn't see Veliana, who was no doubt smirking. She'd insisted that would be their first action, whereas he'd thought many would rest atop the pillows to relax after a brutal night of fighting.\n\n\"They'll drink it off before they sleep it off,\" she'd said while they waited through the night.\n\n_Need to listen to her more often_ , he thought. _She thinks more like a man than I. What I get for growing up among wizards, I guess._\n\nThey both waited, Deathmask watching until he was sure... and there he was, standing amid his men, holding his glass the highest as they toasted a night of survival.\n\n\"To standing atop the dead!\" he heard Garrick say.\n\n_Toasting your own cowardice? And to think I thought I was a bastard._\n\nHe pushed aside the curtain and charged, his dagger drawn and ready. As he pushed himself to the limit, he felt his feet move faster, the world almost imperceptibly slower. He buried his dagger in the back of the nearest thief, whose glass fell from his hand. Before it hit the ground, two daggers flew across the room, thudding into the back of another. Veliana scattered pillows as she lunged, much of her face thankfully hidden by her hood. She kicked the closest thief, the one she'd hit with her daggers, yanking out the blades as her foot slammed him into the others. Wine splashed to the floor as the rest dropped their drinks and drew their blades, crying out warnings of trap and ambush.\n\nGarrick was in their center, and he fell back instead of drawing his dagger. Deathmask knew Garrick was Veliana's target, not his, but he had to clear a path for her. Sidestepping a thrust, he jammed his dagger into the chest of another, using the body to protect himself from several more. The Ash members were starting to spread out, the better to take advantage of their numbers. That thinned the wall toward Garrick, and Veliana wielded her daggers like a demoness, twisting and curling to avoid every thrust. Blood soon joined the wine that stained the floor. Deathmask felt pride in seeing her work. No one who survived could possibly doubt that the best of the Spider Guild had come to take the life of a rival.\n\nWell, those who watched _her_ , anyway. He, on the other hand, struggled to stay alive. His dagger flailed from side to side, sometimes faster than he expected thanks to the earlier enchantments. The impulse to cast a spell to blind his opponent filled him, and only at the last second did he refrain. The ruse was more important. He would gain nothing by giving himself away. The Ash needed to be his guild to rule. He couldn't do that if revealed in the clothing of a rival guild. His arms trembled as he felt steel cut into them. He fought three men at once, and they grinned at the sight of blood. He was outmatched, and now they knew it.\n\n\"Finish it!\" he cried to Veliana, hurrying to the door.\n\nVeliana was in the middle of disemboweling another man, and at his cry she shoved the man aside. The path between her and Garrick was clear. Instead of charging, she lifted a dagger and threw. Its aim was true. The point pierced his shoulder and lodged deep, burying itself up to the hilt. Garrick howled as his blood ran.\n\nThat was enough. Deathmask rushed for the exit, feeling like his legs didn't belong to him. Veliana hesitated for just a moment, and he saw her other dagger trembling in her hand. Trusting her to do the smart thing, he burst through the door and into the night. She appeared a moment later, looking none too pleased.\n\n\"Come on,\" he said. He took a zigzag course through the city, on a path he had memorized. They arrived at an inn with rooms they'd paid for several hours before. Deathmask climbed in through a window, which had no glass, only thick wood shutters that he had left unlocked. He was already changing back into his Ash Guild outfit when Veliana climbed inside.\n\n\"Did you kill him?\" he asked.\n\n\"I wanted to.\"\n\n\"That a no?\"\n\nShe yanked the mask off her face and flung back her hood. \"What do you think?\"\n\nHe grinned. Knowing his skill was nowhere near hers, he'd left the delicate task of harming, but not killing, Garrick up to Veliana. Up until the throw itself, he hadn't been sure if she would make it lethal or not.\n\n\"You did marvelously,\" he said, tossing the Spider cloak to the bed and pulling off his tunic. \"And now I can trust you all the more. If I were in your position, I might have accidentally hit Garrick's throat.\"\n\n\"That would have left me homeless and guildless,\" she said, grabbing her old Ash outfit from the bed. She reached for the door.\n\n\"Where you going?\" he asked.\n\n\"To change.\"\n\nThe door shut behind her. Deathmask sighed. No fun at all.\n\nShe returned moments later, dressed in the colors of the Ash and looking to be in an even fouler mood.\n\n\"They're still stained with my blood,\" she said, referring to the red patch on her chest.\n\n\"I'll try to get you something newer when I can,\" he said. \"Didn't want to attract any attention. They might wonder why I was requesting an outfit for someone half my weight.\"\n\n\"You're a thief now, remember? Steal it.\"\n\nDeathmask shrugged. \"Ready?\"\n\nShe pushed him aside and climbed out the window.\n\n\"This better work,\" she muttered. \"Otherwise we're in for a lengthy death.\"\n\n\"I'm in for one, perhaps, but you've already had your public execution, remember?\"\n\nShe slammed the shutters in his face.\n\n# CHAPTER\n\n#\n\nIn her grief, she had thought hearing the cries of pain and seeing the river of blood would give her closure, but instead Alyssa felt hollow as she watched the fires spread across the city. Standing at the second-story window of her private study, she touched the cold glass and wondered what it was she had done. Had she brought freedom to the city? Peace of mind? Or would every death just result in more death, every killed thief replaced by two more filled with thoughts of revenge? Bertram had told her the monetary cost of her single night, and it was staggering.\n\nShe knew there'd be no peace, not this night. But perhaps this was just like cauterizing a wound. There would be heat, pain, but then the bleeding would stop and healing could commence.\n\nSomeone knocked on the door, and she had a feeling who. Her help would be asleep, or perhaps lying awake in their rooms wondering about the safety of their friends and family beyond her mansion's walls. That left very few who would dare come to her room at night while the rest of the city was in chaos.\n\n\"Come in, Arthur,\" she said, surprised by how tired she sounded. She rubbed her face with her hand, discovering tears. Had she really reached such a low, crying without realizing it while she wasted the night away staring out a window?\n\nThe door opened, then softly shut. Moments later she felt Arthur's hands on her shoulders. When he started massaging she leaned back, pressing her head against his neck.\n\n\"People are too scared to form bucket lines,\" he said. \"The fires will only spread.\"\n\nShe sighed. She should have known, of course. Probably had, even, but she'd let her hatred blind her. _Let the whole city burn_ , she'd thought plenty of times, _so long as it burns the rats with it_. But this was her war now, and that meant dealing with all its ills, all its blame.\n\n\"Send someone to the castle. Tell the king I request the aid of his soldiers in putting out the fires. With the castle guard there, it should outweigh any fear.\"\n\n\"Self-preservation is strong,\" Arthur said, letting her go. \"For so many to hide instead of fighting the fires shows how great a fear you have created.\"\n\n\"I meant to scare the thieves,\" she said. \"Not the innocent. But are there any innocent anymore? How deep does Veldaren's sickness run? Maybe I should let it burn, all of it. My son is nothing but ash, so why not them, why not...?\"\n\nHe wrapped his arms around her and held her tight, and in them she let herself cry. She found herself crying often in his presence. There was a strength in him, and a desire to please. More than anything, she felt she could trust him. He'd been there for her when she needed him most.\n\n\"Was this wrong?\" she asked. \"Have I truly erred so badly?\"\n\nHis response was long delayed, to the point that she thought he might not answer.\n\n\"You have done what you thought was right, and what was best for the Gemcroft family. I will not fault you for that, if you will do the same for me for what I am about to do now.\"\n\n\"And what is that, Arthur?\"\n\nHe turned her about and kissed her. His hands were firm on her shoulders. She felt herself responding. She was so exhausted, so drained. His touch was like an awakening, a pull from a nightmare that threatened to consume her day in and day out.\n\n\"The messenger to the castle,\" she breathed while her mind remained able to think.\n\nArthur leaned close, his hot breath against her ear.\n\n\"Let the fires burn a little longer. If the cowards cannot save their own city, the blame lies with them.\"\n\nThe study lacked a bed, but the carpet was soft. They made love, her beneath him. She wrapped her arms around his chest and clutched him as if her life might end if she let go. She tried to forget the death and fire, her call for revenge. Even as the pleasure tore through her, she could not help but wonder if that wicked man responsible for the death of her son lay dead somewhere, his body nothing but ash in a distant fire. Atop her, Arthur continued to grunt and thrust.\n\nThe arrival of the sun was a blessed thing to Veldaren's citizens. The mercenaries retreated, having fought and searched long through the night. Those with cloaks and colors buried themselves inside whatever safe houses they had to recuperate and plan. Those who sided with neither filled the streets, forming bucket lines from the wells and digging ditches to combat the fires. Many others went to their families and friends, needing confirmation of their survival before beginning their daily tasks. The market's bustle was subdued, the streets awash with murmurs.\n\nHaern watched it all through the window of the small apartment. The fire had gotten dangerously close to Senke and Delysia's home, reaching all the way to Prather's Inn and burning it to the ground. People were everywhere, half-buried in the smoke that billowed from the dying fire. The king's soldiers hurried about, but their presence in the streets did nothing to ease people's minds.\n\n\"Too little, too late,\" Haern whispered. He scratched at his face, which itched worse than it had in ages. Upon their returning to the apartment, Tarlak had insisted Haern bathe, shave, and wear something, as he put it, \"not smelling like a rat just shit all over it.\" So he'd used their washtub, shaved himself with a slender knife, and then borrowed an outfit of Senke's, simple grays that he knew would blend well into the streets. The ordeal left him tired and feeling strangely outside himself, as if this clean, well-dressed individual were someone else.\n\n\"You look troubled,\" Delysia said, and he flinched as if poked with a stick. Blushing for no reason, he turned back to her and accepted the cup of warm milk she'd brought him.\n\n\"I mixed in some herbs,\" she said, sitting opposite him in a rickety chair. \"You'll sleep well, and by the looks of it you could use the rest.\"\n\nHe thanked her and sipped the milk, wisely deciding not to comment on how terrible the drink tasted. His eyes lingered on her face, and he struggled not to make his staring obvious. She'd grown so much over the past five years. Her hair was longer, but still the same fiery red. Her cheekbones were more pronounced, and in her priestess robe she looked almost regal. Her chest was also significantly larger. Out of everything, he tried to make sure his glances at that remained uncaught.\n\nHe continued to sip the drink, mostly to avoid conversation. He had no clue what to say to her. The last time they'd met, he'd come to her in desperate need of guidance. He'd needed to understand a life outside the cold retribution of his father. His tutor Robert Haern had spoken of the god Ashhur, and now here she was, a priestess of the same god. His thoughts had turned only to survival during the long years, yet now the name of Ashhur came back to him with a burning vengeance. What was it he'd told Delysia? He needed Ashhur, otherwise he'd end up like his father. He'd be a killer without mercy, a terrible creation the city feared.\n\n_Long live the Watcher_ , he thought. _What have I become?_\n\n\"I... I'm glad you're all right,\" Haern blurted, feeling stupid as he said it. He saw a shadow cross over Delysia's face, but she pushed it aside with a smile.\n\n\"I try not to think about that night,\" she said. \"There's too much I don't understand, even now. Who you were. Who you are. What Ashhur's purpose might have been. I must confess, I almost hoped I'd die. I was so tired, so confused. But I feared I might never see my brother, and so I struggled for every breath...\"\n\nThe room fell silent. The rest were asleep, exhausted from the long night, but Delysia had stayed awake, insisting she could manage for a few more hours. Haern, used to going long periods without sleep, had dully stared out the window and waited for a chance to talk. Now he had the chance, he didn't have a clue what to do with it.\n\n_I'm better at killing. Does that prove just how far I've fallen? You'd be proud, Thren._\n\n\"The man who shot you was my father,\" he said, figuring to start with what he knew for certain. \"He feared what your influence might do to me. He was right to fear it, too. They dragged me to Karak's temple and did their best to burn away everything good within me.\"\n\n\"Did they succeed?\" she asked, sipping from her own cup. Her green eyes peered over its edge. He felt as if he were that same stupid kid she'd trapped in her pantry. He remembered watching her cry moments after Kayla executed her father. What could he ever be to her but a reminder of those painful times? He saw her watching him, and he remembered her question.\n\n\"No,\" he said.\n\nThe past five years, murdering men in the streets, seemed to have done a fine job of it, though.\n\n\"What have you been doing?\" she asked. \"How have you survived?\"\n\nHe didn't want to answer. Why was he so afraid she'd judge him? So long ago, he'd come to her for advice. Now he feared every word she might say?\n\n\"I slept in the streets,\" he said. He was the Watcher of Veldaren, damn it. He would fear no one, nothing. \"Ever since, I've been killing members of the thief guilds, hoping to destroy them. It's pointless, futile, but still I try. It's the only thing that gives me meaning.\"\n\nHe thought she'd berate him, or challenge his claim. Instead she looked at him with sad eyes, and that was worse.\n\n\"I'm so sorry,\" she said. \"It's because of me, isn't it? Because you protected me?\"\n\nHis mouth fell open.\n\n\"Of course not. Don't be... Delysia, I chose everything I did. I would have stayed with you, spoken with you forever if I could. That night... that single night, I've cherished that memory. It was one of the few bright spots in my entire childhood. But then my father darkened it with blood. My precious memories always lead to him, his murder, his guilt. It pushes me on, consumes everything. I have become something I don't think either that little girl or that little boy could ever have understood or accepted.\"\n\nHe looked back to the window, not wanting to see her reaction. He was a damn fool, that's what he was. Hoping she'd leave him be, he refused to react when she stood from her chair, set her cup down, and came closer. Her hand touched his face, and reluctantly he turned to her. Tears were in his eyes.\n\nShe kissed his cheek, then pressed her forehead against his.\n\n\"Go to sleep, and try to remember that while you are not that little boy, I am no longer that little girl.\"\n\nShe trudged up the sharply curved stairs to the second floor. Haern watched her go, and when she was gone he almost fled to the streets. But he remembered that feeling in the Pensfields', of having a home. He felt that same thing here, though the company was on the odder side. He downed the rest of his drink, grimaced, and then set the cup aside. His chair was comfortable enough, far more so than the cold street he was used to, so he crossed his arms and tried to sleep.\n\nFootsteps coming back down the stairs caused him to open his eyes. He didn't think he'd slept, but he wasn't sure. It wasn't Delysia who had come down, though. Instead it was the wizard, her brother. He'd shed his pointy hat, though he still wore those strange yellow robes. He rubbed his goatee as he plopped down in a chair opposite Haern.\n\n\"Had some words with Senke,\" he said.\n\n\"That so?\"\n\n\"Well, most involved variations of 'Get out of my room so I can sleep, you idiot,' but there were some more intriguing bits I dragged out of him. Most interesting was that of your father. Thren Felhorn, really? You look more like something two vagabonds might bump out on a cold, drunk night.\"\n\n\"Flattered.\"\n\nTarlak tapped his fingers together, and his mouth shifted about as if he were chewing on his words before saying them.\n\n\"Not much for talking. I get that. I like to talk, so perhaps I can make up for the both of us. Senke says you're good, really good. What I saw out there tonight certainly confirms it. Can't expect much less from Thren's son, of course. You've established quite a reputation, too. I've heard plenty talk of the Watcher, usually poor thieves grumbling into their cups about how much gold you cost them. A few even thought you were Ashhur's vengeance come down upon them for their lifestyle, though they usually had to be incredibly drunk to admit it.\"\n\n\"You have a point?\"\n\n\"Several, one on my nose, one on my hat, and one where the ladies love me. But that's beside the, uh, point. It seems like, other than revenge, you don't have much going for you. Ashhur knows those streets out there aren't comfortable living. So how about you join my mercenaries instead? Pay isn't the best, but with half the city employed in killing thieves, I think we could make a few coins. Besides\"\u2014his eyes lit up\u2014\"can you imagine the rates I could charge if people knew the Watcher was in my employ?\"\n\n\"I'm not for sale,\" Haern grumbled.\n\nTarlak frowned.\n\n\"Well, that's disappointing. You sure?\"\n\n\"Very.\"\n\nThe wizard scratched at his chin. \"This a pride thing?\"\n\n\"I have no use for money.\"\n\nTarlak grinned. \"I'm more thinking you feel you don't _need_ money. Considering all the stories of you tossing gold coins in the middle of high market, I can believe that. But there are some things you can't buy with gold that you might be more interested in. Our introductions were a little haphazard, but you met Brug, right?\"\n\n\"Short guy, cussed a lot, can't fight worth crap?\"\n\n\"That's him. I didn't hire him because of his skill with those ludicrous whatever-they-are he fights with. Obviously. You want to know why I did?\"\n\nHaern stared at him with an expression showing he didn't think he had a choice in whether he found out or not. Tarlak blinked.\n\n\"Right. Anyway, he's a blacksmith, and with my help he can create items that many would sell their souls to own. Would you like to run faster? Jump higher? Or perhaps own a fancy sword or three...\"\n\n\"I'm not much for bribery either.\"\n\n\"Don't see why you shouldn't be. You spend your nights crawling around the rooftops killing thieves. Might as well get paid for it, or at least better tools to do it with.\"\n\nHaern turned his chair so his back was to Tarlak, and he stared out the window.\n\n\"Very well.\" Tarlak stood. \"I'll leave you be. Take a nap, or vanish in the afternoon. You aren't held prisoner here. Think about my offer, though. We may not be much now, but I think we've got potential.\"\n\nHaern snorted. Whether Tarlak heard or not, he didn't react, only went up the stairs. Staring at the men and women still fighting the fire, Haern wondered what in the world had gotten ahold of him. That wizard was no better than anyone else, not even his father. He killed for money, except he used fire and words instead of a blade. What could possibly have possessed Senke to join them?\n\nHe closed his eyes and felt the light of the sun warm his face. Come that afternoon he'd sneak his way out. Oh, he had no delusions of abandoning Delysia and Senke completely. He knew himself better than that. It'd be easy enough to keep watch on them, though, keep his eyes open for a wizard in yellow, accompanied by a beautiful girl with hair like fire...\n\nWhen he opened his eyes, many hours had passed. He shook his head, fighting the grogginess. His back ached, and it popped several times as he shifted his upper body from side to side. Senke stood at a small counter, eating cold bread left over from that morning. His fingers drummed the counter, the sound no doubt what had awakened Haern.\n\n\"You chew like a cow,\" Haern said, pressing his thumb and forefinger against his eyes.\n\n\"And you look like one, only worse. When was the last time you had a bath?\"\n\n\"This morning, you noisy bastard.\"\n\nSenke shoved the rest into his mouth and wiped crumbs from his shirt.\n\n\"Here,\" he said, his mouth full. He pointed at Haern's swords. \"Been a long time since we sparred. Thought it'd be a nice way to catch up.\"\n\n\"Where?\" The room was cramped as it was. Senke nodded toward a back exit.\n\n\"There. Come on.\"\n\nThere was a small space of flat dirt out back, part of an alleyway that ran behind the group of apartments. The faint outline of a circle remained dug into it, and Senke refreshed it with his heel.\n\n\"Only person to train with has been Brug, and trust me, that's not much of a workout. You'll do me fine.\"\n\nHaern stretched away the rest of his drowsiness. Senke had been the better fighter when they last met, but the years had hardened Haern, granted him strength and height while his nightly excursions had honed his reflexes and skill. He touched the tips of his swords together and bowed. Senke had carried two short swords with him, and he wielded those instead of his maces.\n\n\"Maces will be too slow for you,\" he said. \"So let's try the blade.\"\n\nEager to show how much he'd learned, Haern initiated their combat with a quick lunge. Expecting the ensuing parry, he followed up with a slash with his other weapon, using it as a distraction to allow his first blade to pull back and thrust again. Senke, however, hadn't been Thren's enforcer without good reason. He shoved both attacks high, stepped closer, and feinted an elbow to Haern's face. When Haern stepped back, trying to fall into position, Senke pressed the attack, keeping his swords out wide. The second elbow that came flying in was no feint, and it smacked into Haern's chest with a heavy thud. Again Haern stepped back, but instead of chasing, Senke pointed to where he'd stepped beyond the bounds of the circle.\n\n\"Out,\" he said.\n\nFeeling his cheeks flush, Haern stepped back into the practice ring. He wasn't focused, wasn't analyzing Senke's reactions as he might other opponents'. Taking a deep breath, he forced himself to calm down. A nod, and they resumed.\n\nThis time he remained patient, and he swallowed down his pride to acknowledge Senke was just as fast as he. Most opponents he could overwhelm with simple brute speed, the massive gaps in their skill overriding any of his carelessness. Not now. Senke stepped closer, swinging both his blades in a downward arc. Haern parried them aside, then looped both swords around as he advanced. Senke blocked the barrage, then planted his back foot to halt his steady retreat to the circle's edge.\n\nSeeing this, Haern pressed the attack, relying on his opponent's lack of mobility. But the planted foot had been a kind of a feint, for when Haern swung with all his might, ready for the clash of steel and challenge of strength, instead Senke twirled out of the way. Overextended, Haern could do nothing but accept the stings of Senke's short swords slapping against his arm.\n\n\"Come on now,\" Senke said, pausing to catch his breath. \"I expected far better than that. King's sake, I saw you handle yourself better last night against those thieves.\"\n\nAgain he felt his neck flush. Was he holding back? He didn't mean to.\n\n\"Treat me like any other opponent,\" Senke said, clanging his swords together. \"Fuck. Treat me like your father. Everything, Haern, show me everything you got.\"\n\n_Everything_ , he thought. _Everything_. It seemed like a red light bathed over him, flashing from a ring on Senke's finger. He forgot they only sparred, forgot they fought in a dirt circle instead of a real battlefield. He forgot his opponent's name was Senke, and imagined instead the glaring figure of Thren Felhorn, furious, deadly, a bow in his hands and Delysia dying at his feet. His father grinned, as if the corpse there didn't matter.\n\n\"Hello, Son,\" said Thren.\n\nHe gave that image everything. His swords wove in tight circles as he slipped from stance to stance, always shifting, always attacking. The sound of steel on steel became a song in his ears. Their blades looped and twisted, parrying away sure strikes and blocking cuts that should have hit before either could counter. Thren's grin faded; he was just a cold image that watched Haern without any sign of exertion or worry. Haern found himself wondering where he was, what was going on. Around him the alley had become an old safe house they'd lived in for a year, the hardwood floor polished and prepared for practice.\n\n\"You've learned nothing!\" Thren shouted, bearing down on him with his short swords. Haern's arms ached with each block, and that ache slowed his response when one of the attacks slipped to the side, curling back for a thrust. Haern twirled, his sword parrying moments too late. His chest burned, and blood ran down. As he grunted in pain, Thren rammed his heel into his stomach, knocking him back.\n\n\"What are you to me now? What could you hope for? Come at me, Son! Kill me! Your skill is nothing, _nothing_!\"\n\nHaern felt his mind change, becoming something whole, focused, and dangerous. The entire world ceased to exist, and even time struggled to keep him under its rule. He let out a cry and charged. This time his father's attacks were no longer so frightening. Despite Thren's feints, his parries, Haern saw through them all and refused to be controlled by them. Faster and faster he whirled, his body lost in a dance, their blades intertwined, their motions in constant reaction to each other's. He managed a snap-kick into Thren's face, dropped to the ground, and then swept his feet out from under him. As he fell, Haern lunged, one sword shoving his father's defenses out of the way, the other stabbing for his throat.\n\nInstead of piercing flesh, he stabbed dirt. Thren scattered as if his body were made of dust, and then Haern was back in the alley. The wound on his chest vanished, his pain along with it. Senke leaned against the complex, his arms crossed. Haern felt naked before him, his heart exposed and bleeding.\n\n\"Your hatred for him is so great,\" Senke said softly. \"It's all that keeps you alive, isn't it? You can't live like that, Haern. You have every reason to escape his shadow, yet you still let it lord over you. What have you become? How many have you murdered in his name?\"\n\n\"They were all guilty,\" Haern shouted. \"Thieves and murderers!\"\n\n\"Were they always? I just saw what lurked behind your eyes, Haern, more frightening than anything your father might have done to me.\"\n\nHaern thought of all the men and women he'd hunted in the night. They'd worn guild colors, yes, but how many had also been innkeepers, farmers, smiths, and butchers? How many had he killed for doing business with them, smuggling and trading and selling? Night after night, he felt the waves of his dead. For Ashhur's sake, he'd written his name with their blood!\n\n\"It's not hopeless,\" Senke continued. \"I thought I'd lost you, but now finding you, I wonder what is left of that small boy who loved to read. The one who asked me about jewelry for a girl he liked. I'd always hoped that, if you'd survived, you'd have gone and experienced everything Thren denied you. Now I see you denying yourself... love, faith, friendship... and you do so out of _revenge_?\"\n\nSenke walked over and sat down beside him, putting a hand on his shoulder.\n\n\"Sorry about the illusion,\" he added. \"Just a trick of this ring Tarlak gave me. I had to see. I had to know who you are, how good you can be.\"\n\n\"Now you do,\" Haern said, feeling his insides tighten and twist behind his ribs. \"Is it truly so bad?\"\n\nSenke squeezed, then smacked his back.\n\n\"Doesn't matter,\" he said with a wink. \"I'm still here for you, as is that pretty redhead Tarlak has for a sister. He's a good man, Tarlak. A bit strange, but he's a wizard, so that's to be expected. I've told them of my past, and they swore it didn't matter who I was then, but who I am now. Please, stay with us. Put this life behind you. If you're to have a legacy, it shouldn't be this. You've become the feared reaper of the guilds. Should Thren ever find out you still live, I cannot help but wonder if he'd be furious... or proud.\"\n\nHe stood and moved for the door.\n\n\"Go back to your streets,\" he said. \"Think on everything I've said. There's so much good in you, I can see it still. It's never too late to change who you are, so long as you're willing to bear the consequences. You've carried heavy burdens your whole life, Haern. Maybe it's time you let some of them go.\"\n\nNot waiting to see if he stayed or not, Senke stepped into the room and closed the door behind him. As it locked, Haern felt thrust back into the world he'd called his home for the past five years, but now the streets seemed foreign, their alleyways and rooftops offering no safety. His skin was clean, his clothes fresh, his face shaven.\n\nFor once he didn't feel like he belonged, but onward he ran all the same.\n\n# CHAPTER\n\n#\n\nDo you think he's telling the truth?\" Matthew asked his wife as they cuddled in bed for the night.\n\n\"I don't see any reason for him to lie.\"\n\n\"I can think of plenty. He's hurt, sick, and in a stranger's home. Truth might be the furthest thing from his mind. What if he doesn't know Lady Gemcroft, only hopes she'll take him in if we show up at her doorstep?\"\n\nEvelyn put her arm across his chest and pressed her face against his neck.\n\n\"It would explain a lot though, wouldn't it?\" she asked, her voice quiet. \"Why those men were searching for him. We both knew he was no ordinary boy to be hunted like he is.\"\n\n\"But why would Arthur's men be after him? The whole bloody north knows he's been courting her.\" An unspoken question hovered in the air between them, until at last Matthew gave it voice. \"What if the men we killed were actually trying to rescue him?\"\n\nMatthew waited for his wife to speak, trusting her to better understand these complicated matters. He could list the price of every vegetable that grew in Dezrel, what the color of the soil meant and what could grow in it, but these things were beyond him. He liked living outside the city, where so long as you paid the taxman when he came, you could live unbothered by your lord and trust in your neighbors. Hard luck comes in strangers' hands, his ma had always said.\n\n\"That man, Haern, might have kidnapped him,\" Evelyn said. \"If he was wounded and low on food, he'd need someone like us to help out, but why leave the boy here? Why tell us to take him back to his parents whenever he could talk? Everything he paid us for, he could have taken by force. Still, I won't pretend to understand Arthur's reasons, and neither does Tristan.\"\n\n\"He says his name is Nathaniel.\"\n\nEvelyn kissed his neck.\n\n\"I told him his new name, and we'll use it so long as he's with us. No need to risk undue attention should we go out and about.\"\n\nMatthew grunted. Fair enough.\n\n\"It might be Arthur himself that came for him, though everything's just a jumble when Tristan tells what happened,\" he said. \"But I think you're right. Those men were up to no good. Could see it in their eyes.\"\n\n\"So what do we do?\"\n\nMatthew sighed. He wished he knew. While he thought, he ran his rough fingers through her hair, enjoying its softness.\n\n\"We got to get him home, even if that means traveling all the way to Veldaren.\"\n\n\"What if you stop in Felwood and deliver him to Lord Gandrem? He's always been a close friend to the Gemcrofts.\"\n\n\"So was Arthur.\"\n\nHe was right, of course, and he could tell she knew it.\n\n\"Let us all go, then. I don't want to be left here, and it won't be safe for the kids, either.\"\n\n\"Our livestock'll die.\"\n\n\"With how much Haern gave us, we could buy our farm back twice over.\"\n\nMatthew shook his head, thinking of all the work he'd put into raising his cattle and pigs.\n\n\"Still no good reason to let them die, waste all they're worth. Besides, me going to the city might be strange, but it ain't unheard of. All of us packing up to go? If there's more soldiers looking, and you know there are, then they'll hear about it in a heartbeat. I'll go alone, just me and the boy. He's light enough. We can ride together, make good time.\"\n\n\"We have no horses.\"\n\n\"I'll buy one from the Utters in the morning.\"\n\nEvelyn pulled her arms tight across her chest as if she were cold. She recognized that tone in his voice. He'd made up his mind, and it'd take tears and a hysterical fit from her to change it. She didn't have it in her. They had to do something before more soldiers showed up looking for Tristan.\n\n\"Trevor's old enough to look after most things,\" Matthew continued, as if trying to reassure her. \"And with the cold already breaking, we'll easily last until spring on what wood we have. I'll leave you half the coin too, in case something happens. You can afford salt or meat if need be.\"\n\n\"I know I can do it,\" she snapped. \"Don't mean I want to, or will enjoy it. I'm scared, Matt, scared witless. What if men come looking while you're gone?\"\n\nHe kissed her forehead.\n\n\"I trust you,\" he said. \"And I'll pray you stay safe. I don't know what else to do, Evelyn. I just don't know.\"\n\nCome morning he trudged east through the half-melted snow, across fields he knew by heart. The Utters were a large clan, and wealthier than most of the local farm folk. They had several horses, and while they might not be eager to part with one, Matthew knew the gold jingling in his pocket would be persuasive enough.\n\nWhen he returned, it was atop a mare he'd paid for\u2014far more than she was worth, but given how they were still waiting for winter to make its exit, and time wasn't on his side, he'd been forced to accept. He'd refused to be overcharged on the saddle, though.\n\n\"Without that mare you got no reason for it anyway,\" he'd argued, and after he threatened to buy a saddle from the Haerns or the Glenns, they'd relented. The mare was a beautiful horse named Strawberry by one of the Utter daughters. Matthew thought the name a little demeaning for such a majestic creature, but figured he'd leave it be considering the horse was already familiar with it. On his ride back he swung by Fieldfallow (the closest thing to a town for thirty miles) and bought trail rations and a thick riding coat.\n\n\"Little early to be heading up to Tyneham,\" the old storekeeper had said. Matthew only gritted his teeth and paid, again twice as much as he would have in spring. Back at the house Tristan was already bundled up and ready to go. His fever had come and gone, but never as badly as before the amputation. Matthew kissed his kids good-bye, hugged his wife, and then set Tristan on the saddle.\n\n\"You ever ridden a horse before?\" he asked.\n\nThe boy nodded. \"At the castle,\" he said. Matthew guessed he meant Felwood, and again he felt tempted to stop there. Lord Gandrem was an honorable old man. Surely he wouldn't let something untoward happen to the boy. But Haern had told him to deliver the boy to his parents, and that meant Alyssa Gemcroft all the way south in Veldaren. There was also his nagging worry that Lord Gandrem might, however unlikely it seemed, also be involved in the attempt on the boy's life. Resolving to decide the issue later, he climbed into the saddle, shifted Tristan so they could both sit more comfortably, and then set off.\n\nThe first day came and went uneventfully. A caravan passed them heading north, dour men who didn't even wave in greeting. Just before nightfall he spotted a distant pond. Glad for once for the cold, since there'd be no mosquitoes flitting about, he set up camp beside it, Strawberry staked close enough to the water's edge to drink. Tristan had remained quiet through much of the ride, and Matthew didn't press him to talk. Come the fire, though, it seemed both their tongues loosened.\n\n\"How long until we get there?\" Tristan asked.\n\nMatthew poked the fire with a stick, shifting one of the thicker logs into a hotter section so it might burn better.\n\n\"It'll be several days to reach Felwood. From there, less than a week to ride into Veldaren. That's where your ma is, right?\"\n\nThe boy shivered, as if the mere mention of her reminded him how far away she was.\n\n\"I think so,\" he said. \"Do you... do you think she misses me?\"\n\n\"Can't see why not. Evelyn would be sick with fits should one of our sons run off missing.\"\n\nTristan pulled his blanket tighter about him, and his eyes glazed as he stared into the fire.\n\n\"He died protecting me,\" he said.\n\n\"Who?\"\n\n\"Mark. I liked him. He's nicer than Lord Hadfield.\"\n\nThe name Mark didn't ring any bells, but _Hadfield_ sure did.\n\n\"Do you know why Arthur would want you dead, boy? You're young, sure, but you got ears and you probably know more than I when it comes to the upper crust.\"\n\n\"I don't. He always said I was like his son, and when he married Mom, he'd be my father.\"\n\nMatthew felt a tingle in the back of his head at that. Perhaps it had something to do with marriage. Had Alyssa rejected Arthur, and he lashed out in spite? Did he want to remove any potential heirs? What foul plans might he have for Alyssa as well? Too many questions without answers.\n\n\"Safe to say he ain't planning to be much of a father to you,\" Matthew said. \"Now eat up. Got a long ride tomorrow, and you'll need the energy for it. Riding's tiring work, though you wouldn't think it.\"\n\nThey slept under blankets. Halfway through the night Matthew awoke to distant howling. Coyotes, he figured. A tired glance to his side showed Tristan shivering, a shaking fist pressed to his lips. He was crying. Touched, Matthew reached out and put his arm around the boy, sliding him closer so he could wrap him in a hug. Tristan continued to cry, but his trembling stopped. Soon the crying turned to sniffles, which turned to steady breathing. Matthew fell asleep not long after.\n\nCome morning, they both woke red-eyed. Tristan said little, and several times Matthew had to hold back an angry word. Evelyn always insisted he was a bear when he got up in the morning. No reason to take that out on the poor kid. They ate some rations, drank, and then rode south, stopping every few hours to stretch their legs and rest their backs. Matthew wasn't a stranger to a horse, but he hadn't ridden in over six months. Muscles he hadn't known he had announced their angry presence to him.\n\n\"Starting to think walking would be a better idea,\" he grumbled.\n\nTristan said nothing.\n\nBy the second day the plains were spotted with trees, and with each hour they rode, the trees gathered more thickly, forming clusters that would soon be a forest. Felwood Castle was getting closer. It was one of those nearby clusters that saved both their lives. They'd stopped by one for a piss, and while dismounted they heard the thunder of hoofbeats approaching from the south. A warning instinct, like when he knew something was after his animals, told Matthew it was time to get off the road.\n\n\"This way,\" he said, grabbing the reins in one hand and Matthew's wrist with the other. He led them into the copse, far enough that they'd go unnoticed.\n\n\"Stay here and hold on tight,\" he said, handing Tristan the reins. Hurrying back toward the road, he peered from behind a tree as a group of five rode past at full gallop. They wore dark tabards that he easily recognized. Hadfield's men. Did they know of Gert and Ben's absence? More important, did they know where it'd happened?\n\nTrying not to think about it, he returned to Tristan, who stood with wide eyes. \"It's them again, isn't it?\" Tristan asked.\n\n\"Yeah,\" Matthew said. \"Looked like it.\"\n\n\"Will everyone be safe?\"\n\nMatthew's jaw clenched tight. He yanked the reins from the boy's hand and led them back to the road.\n\n\"Ashhur only knows,\" he said as the silence hung over them. \"And if not, then may Karak curse every one of those bastards.\"\n\n_Including the one who brought you to me_ , he thought, not cruel enough to say it aloud.\n\nOric sped his men across the road between Felwood and Tyneham, the lightest touch of panic brushing his neck. It wanted to dig in, sink its claws deep, but he refused to let it. He hadn't failed his master yet, and so far he had no reason to think he would. Not a soul had seen or heard anything of Nathaniel. It seemed likely he'd frozen to death, that strange Watcher there for the gold and nothing else. The lack of information suggested the boy was a corpse in the melting snow somewhere, his body devoured by coyotes or vultures\u2014except for one troublesome detail: they'd found Gert's horse unbridled, the soldier nowhere to be found. That meant he was dead somewhere, killed while searching for Nathaniel. So far Oric had no evidence, but he assumed the same had happened to Ben. For two of his men, armed and armored, to mysteriously vanish...\n\nThey'd found Nathaniel, and then paid the price. That's what Oric's gut said, and he trusted it over everything else in the miserable world. He needed to discover where they'd found the boy, and quick. If he'd even made it to Felwood, there'd be disaster. Lord Gandrem certainly knew of Alyssa's loss, and Oric had personally brought \"the body\" to be buried. All sorts of questions would need to be answered should Nathaniel appear alive and well, and none of the answers would endear Oric to anyone. It was either find the boy or hang from a noose.\n\nThe farms were few and far between as they rode north, and something clicked as he finally came upon where the ambush had first been.\n\n\"Let's say you're wounded and carrying a sick boy,\" he said to his men. \"Snow's falling, and you're low on food. What is it you'd do?\"\n\n\"Ditch the boy,\" said one. \"Either way he's dead. No reason to go with him.\"\n\n\"Assume yourself a better man than that. What then?\"\n\n\"Carry him until I find the closest shelter.\"\n\nOric tapped his forehead. \"Exactly. Patt, take Rat and go north. Stop at the first two homes off the road, and you search them thoroughly. The rest of you, come with me.\"\n\nThey split, two north, three south. Oric had a feeling this Watcher, when in danger, would have gone south instead of north, since by all appearances Veldaren was his home. They saw no dwellings for the rest of that first day, but come the second a farm appeared in the distance. Oric led the way, feeling his pulse quicken. This had to be it. The Watcher would have stopped here, maybe not for long, but at least for food and water.\n\nWhen he knocked on the door, it was a long time before he heard a response.\n\n\"Who's there?\" asked a woman's voice.\n\n\"Oric Silverweed, soldier of Lord Hadfield of the north. I demand entrance.\"\n\nA lock rattled from inside. Oric leaned back toward his men and whispered, \"Hands on your hilts at all times.\"\n\nThe door opened, revealing a mildly attractive woman in her early thirties. Beside her stood a teenage boy, a dagger tucked into his belt. From where Oric stood he saw several more children, all younger, huddled about a wood stove.\n\n\"Where's the man of the house?\" he asked as they stepped inside.\n\n\"That's me,\" said the eldest boy. Oric raised an eyebrow as he glanced at the woman. Something already felt off.\n\n\"What's your name, boy?\" he asked, glad to see him ruffle at being called _boy_. If he was angry he might say something stupid, something he'd rather have kept quiet about.\n\n\"Trevor.\"\n\n\"Where's your pa, Trevor?\"\n\nThat brief hesitation, along with the woman's sudden flare of her eyes, was all Oric needed.\n\nHe had two men with him, one a young soldier named Uri, the other a skilled fighter named Ingle. Oric turned to them, purposefully putting his back to Trevor and his dagger.\n\n\"Ingle, search out back. Check the barn, the fields, anywhere they might keep him. Uri, search the house. Pull up the floors if you have to.\"\n\n\"You can't do this!\" the woman shouted. Oric struck her with the back of his hand. Finally Trevor drew his dagger. Before he could do a thing, Oric crossed the distance, rammed his throat with one arm, and grabbed Trevor's wrist with the other. He held him pinned against a wall as the younger children screamed.\n\n\"You pulled a blade on me, boy,\" Oric said, feeling like a wolf among sheep. He let a wildness appear in his eyes, knowing it'd frighten them more. \"That means I can do whatever I want, and I got half a mind to leave you a cripple. Think your ma here will keep feeding a worthless belly that can't help out in the fields? How you think she'll like watching me cut off your fingers one at a time?\"\n\nTrevor's eyes were wide, and he looked ready to cry. He couldn't speak, only cough, and Oric kept the pressure up to keep it that way. He wanted him light-headed, scared, convinced he was about to die.\n\n\"Stop it, please,\" the woman pleaded. She still stood near the door, a red mark swelling on her face. Meanwhile Uri flung open drawers and dressers as he searched the house, occasionally stamping hard with his heel to test for false floorboards.\n\n\"Stand over there with your children,\" Oric snapped at her. \"You make a move toward me, anything at all, and you can watch as I pull your son's guts out one inch at a time.\"\n\nShe reluctantly obeyed, sitting between her two girls. A young boy was with them, and he moved to sit at her feet. Oric turned back to Trevor, who had dropped the dagger and started retching silently.\n\n\"Take a deep breath,\" he said, lessening the pressure. As he sucked air into his windpipe, Trevor coughed, every gasp he made strained. \"Good. Now you listen to me, got it? I'm missing two of my men, and I'm thinking they were here. But let's not worry about that right now. Right now I want to know about a little boy, red hair, about five years old. Did someone bring him here? The truth, you worthless shit, tell me the truth.\"\n\nTrevor's face contorted with pain. He had something to say, no doubt about it. But he didn't want to. Even threatened with death, he didn't want to say. He was protecting his parents, Oric realized. Nothing else could keep his tongue still when so blatantly faced with death. Well, there were ways around that.\n\n\"Uri,\" he shouted. The man appeared seconds later.\n\n\"Yeah, Oric?\"\n\n\"Find anything?\"\n\nUri shook his head. \"He ain't in here. Nothing for Ben or Gert either.\"\n\nOric looked to the adjacent room, which was curtained off, decided there would work.\n\n\"Come take him,\" he told Uri. The other soldier grabbed Trevor by the wrists and shoved him through. Meanwhile Oric walked over to the woman.\n\n\"What's your name?\" he asked her.\n\n\"Evelyn,\" she said through clenched teeth.\n\n\"Pretty name, that. You come with me now, or I'll drag you away by the hair while your little ones watch. Your choice.\"\n\nShe kissed her daughters and stood. Oric put a hand on her neck and guided her into the room where Uri pinned Trevor against a wall.\n\n\"You're trying to protect your ma, maybe your pa, or both,\" Oric said as he shoved Evelyn onto the small bed in the cramped room. \"But you ain't protecting them, not anymore. Gonna show you what'll happen, Trevor, if you don't tell me what you know, got that? Hold him tight, Uri.\"\n\n\"Will do.\"\n\nOric struck the mother, spun her onto her stomach, and ripped at her skirt. When she started to sob, he took a wad of the blanket and shoved it into her mouth. Trevor struggled, but Uri stood a foot taller and easily outweighed him. Oric pulled off his belt, pushed aside the rest of Evelyn's skirt, and shoved himself inside. She screamed into the gag, tears streaming down her face. Oric beat her when her screams got too loud, or when Trevor's struggles lessened. He needed the horror to continue. He wanted that fucking brat scarred.\n\nWhen he finished, Oric pulled back and refastened his belt. Evelyn pulled at her skirt, trying to hide her nakedness, but Oric yanked at it, denying her even that.\n\n\"Let him go,\" he said to Uri.\n\nTrevor flung himself at Oric, who had expected the reaction. He ignored a single punch, caught Trevor by the throat, and flung him back against the wall.\n\n\"You want to know what will happen next?\" he asked as Trevor clutched his wrist. \"I'm thinking Uri wants a turn, but I'm not letting him. Know why? Because _you're_ going next unless you tell me everything that happened here.\" He laughed. \"How's that sound? Ever wanted to needle your ma, Trevor? Here's your chance. No one will blame you. You were just being a man, right, protecting your family? How about one of your sisters out there? Think they could use a good poke? Maybe I'll make you do all three, just one after another, until...\"\n\n\"Just stop,\" he screamed with what little breath he had. \"I'll tell, everything, I'll tell everything, please, just stop, just stop...\"\n\nOric let go of Trevor, who collapsed at his feet. The boy huddled there trying to stop his crying. A grin on his face, Oric knelt so he and Trevor could stare eye to eye.\n\n\"You tell me every goddamn detail you know, or next time I might not be so nice.\"\n\nOric listened as Trevor told of Haern's arrival with a boy he knew only as Tristan. He listened as he detailed Tristan's amputation. Then came Ben and Gert's arrival, and Oric felt his blood boil as he heard of how their father had killed them. Both of them, Trevor insisted. He seemed determined to make that clear. Last came their father's departure for Veldaren only a few days prior, mounted and following the main road.\n\n\"Good boy,\" Oric said, slapping him across the face when he was finished.\n\n\"Mind if I have a go?\" Uri said, nodding to where Evelyn remained upon the bed, her face wet with tears. Oric shrugged.\n\n\"Get on out, boy. No need for you to watch this.\"\n\nThe three soldiers gathered outside the house ten minutes later.\n\n\"No sign of anything strange,\" Ingle said. \"Found where they maybe did some digging recently, but the ground's too hard and cold for me to check.\"\n\n\"Don't bother,\" said Oric. \"We know what they did. Nathaniel's with their father riding south. If we press hard we can overtake him.\"\n\nUri pointed a thumb back at the house.\n\n\"You leaving them alive? They helped kill two of us, tried lying as well. Don't set much of an example.\"\n\n\"Leave them all for now,\" Oric said. \"When we find this Matthew, I want to drag him back to his home so he can watch as we kill every last member of his family. Let that story spread across the north. No one opposes Arthur, and no one dares kill his soldiers. Now ride. No matter what, they can't get to Felwood before we do.\"\n\n# CHAPTER\n\n#\n\nZusa had scoured the south and found nothing. The night had come and gone, bathed in blood and lit with fire, yet she had seen nothing of this elusive Watcher. Too much chaos, too much death. How do you pick one murderer out of a thousand? It was a question she had no answer for. Still, it seemed Alyssa's desires had been granted. Hundreds of thieves had died, though many mercenaries had fallen as well. She doubted her master would grieve their loss. Her grief was saved solely for herself.\n\nZusa's only strategy left was to hope the Watcher had lain low during the night, knowing he wasn't needed. Come morning, though, perhaps he'd try to escape, or survey the damage. As she ran along the rooftops, Zusa crisscrossed among the various thief guild headquarters, at least those that she knew. She saw various men pass below her, staying to the alleys and quiet streets, but they all wore the colors of various guilds. From what she'd gathered from those she'd interrogated the night before, the Watcher never appeared wearing any guild colors, only a multitude of gray cloaks and shirts. Still, gray was worn by both the Ash and the Spiders, so to those she went.\n\nAt the Ash Guild she leaned against a triangular rooftop, rested her arms on its tip, and overlooked the square. Nothing. The magnitude of her task set upon her then. She was trying to find a lone man in the entire city, one who appeared to have no friends, no allegiances, and no clear motive other than killing thieves. She had a vague description to go on based on his clothes, and a rumor that he had blond hair. Some said he had red eyes, but she dismissed that as easily as the stories claiming he had demon blood and blades for hands. But blond she could work with.\n\nShe dozed for a while, not meaning to. Sometime later she startled, ashamed of her weakness. It'd been a long twenty hours, sure, but she'd handled worse.\n\n\"Nava would be so disappointed,\" she whispered, feeling sad and tired. Nava had been one of the last three faceless women, killed at the hand of a dark paladin of Karak. They'd been deemed outcasts, traitors to their god. But it was their god who had abandoned them, and so Zusa had turned on his paladin, who had come for Alyssa and protected her. Zusa had given Karak no prayers for the last five years. She missed Nava and Eliora more than his presence.\n\nNot far to her right, down in the alley, she heard someone cry out in pain. Curious, she rushed over and leaned down. Her eyes widened. Whirling below her was a mass of gray cloaks, spinning and sliding as if possessed. Three men fought against it, all wearing the colors of the Ash. A man lived inside those cloaks, and she saw his face, his blond hair... but that wasn't what convinced her. It was the sheer ferocity of the stranger, a shocking rage somehow still held in check. One by one the thieves fell, throats sliced and chests cut open. His skill was incredible.\n\n\"Watcher,\" she whispered, drawing her daggers. \"I find you at last.\"\n\nShe felt a seed of worry planted in the back of her mind. Her master wanted the Watcher brought back alive, but the way he fought, the way he moved, it might be impossible. He'd die before surrendering, she knew that, just as she knew he'd be prepared for her attack from above despite all her silence.\n\nHis swords lifted high, colliding with her daggers. Her momentum continued, her feet kicking out mid-fall to slam into his chest. Still he held his ground. She pushed off, using a hand to cartwheel away before landing on her feet. The two stared at one another, a smile blooming across her face.\n\n\"Ethric was the last true challenge I fought,\" she said. \"Can you be the next, Watcher?\"\n\n\"I'm more than a challenge,\" said the Watcher. He pointed a blade at her cloak. \"Who is it you work for? What fool have you sold your soul to?\"\n\nZusa laughed, the amusement only half acted. The man was watching her, analyzing her. She felt naked before his eyes, as if in time he might know every movement. She was doing the same to him, true, but he was too guarded, too still.\n\n\"You seek my colors?\" she asked. Slowly she lifted one arm, slashed it, and let the blood drip down onto the cloth of her cloak. She wondered if her spell would take hold. Her strength had come from Karak, or so she'd always thought. She'd once lived within shadows, danced with cold fire on her blades, but not since Ethric had she tapped Karak's power.\n\nThe color spread through the cloak in seconds, turning it a vibrant red. It coiled around her, as if suddenly alive. Zusa felt her blood pound in her ears, her head ached from the effort, but still she smiled. Perhaps Karak hadn't abandoned her after all.\n\n\"I serve willingly,\" she said, tensing for an attack. \"I have sold nothing.\"\n\nShe lunged, one dagger looping upward to block, the other thrusting for his chest. Her cloak wrapped about her like a shield. When the Watcher countered, her dagger parried his blade away, but her thrust met his other sword, and her arm jarred at the strength of the block. Her cloak lashed out like a whip, its fine edges sharp as razors. It slashed across his face, blood splattered them both, and then he leaped back. His hood fell lopsided, and she saw how blue his eyes were, how dirty his face was. Who was hidden beneath the guise? Whom would Alyssa find when she dumped his body before her?\n\n\"Neat trick,\" the Watcher said before leaping into his own attack. Their weapons clashed again and again, his speed incredible. Twice Zusa had to spin and let her cloak snap inward, deflecting a killing thrust. This was no spar, no game. He wanted her dead. That seed of worry in her mind grew to a thorn. One of his swords slashed her thigh. The other pierced her chest, the wound shallow but painful. The worry bloomed like a deadly flower.\n\nIt was the narrowness of the alley they fought in that saved her. When he lunged for a killing blow she kicked off the wall, sailing over his head. Her feet hit the opposite side, the collision jarring, but she pushed off, higher. Her cloak trailed below her, twisting. It lashed at him, cutting deep grooves into his arms. He'd expected her to land, not continue back the way she'd come. The cloak kept him off balance, and when she landed she lunged in, daggers leading.\n\nShe underestimated his speed.\n\nThe sound of steel hitting steel rang in her ears, and her carefully coordinated attack broke as his swords danced. She refused to relent, chasing every backward step he took. There was still no fear in his eyes, only death. Whether it was for her or himself she didn't know.\n\nThe ache in her head grew. She couldn't maintain the cloak's enchantment much longer. It'd never hurt like this before, never drained her so terribly. Maybe Karak truly had abandoned her, as she'd abandoned him. Or perhaps Karak wasn't with her at all? Intrigued, she suddenly somersaulted away from the Watcher, pulling out in mid-attack while he was unprepared to give chase. She'd once been able to treat the shadows as doorways. Could she still do so?\n\nThe sun was low enough that several deep stretches of darkness remained in the alley. Zusa focused on one behind the Watcher, then turned and leaped at the shadowed wall behind her. Part of her expected to hit stone, but she passed cleanly through. Again her mind ached, but when the distortion ended she was behind her opponent. Her cloak its normal color and shape, she flung herself at him, knowing her chance to surprise him like this again was nonexistent.\n\nAny normal opponent would have died, but this Watcher was beyond normal. He looked a man possessed, and the moment she vanished he was already spinning, searching for her. He parried her leading thrust, and she was forced to use her other dagger to counter a slash aimed for her throat. Her momentum continued, and they slammed into one another. His head cracked against the wall. Her hands a blur, she cut once, twice, into the tendons at his elbow. The sound of his sword hitting the ground was music to her ears.\n\nHe screamed, but the pain did not slow down his other blade. She felt its edge dig into her skin, and she rolled with it to prevent too deep a cut. Blood ran down her face and neck, urging her on. She used both her daggers to pin his sword aside, then rammed her elbow into his throat. He gasped for air, his gag reflex leaning him closer. Pulling her daggers back, she hit his temple with the hilts. The Watcher dropped to his knees.\n\n\"I'll kill you if I must,\" she said as he leaned on his arms, as if bowing to her. \"Come now, and face the woman you wronged.\"\n\n\"I've wronged no one,\" he said, his voice hoarse.\n\n\"Liar. Murderer of children. Surrender now.\"\n\nHe laughed. It was tired and broken.\n\n\" _I_ am the murdered child, woman. Ask my father.\"\n\nHe flung one of his cloaks at her. As she batted it aside, his heel followed after, ramming into her forehead. Fearing an attack, she retreated, her daggers falling into defensive positions. Her blurred vision saw no one. He was gone, but where? Follow the blood, she thought. Follow the blood.\n\nShe caught a speck of it halfway up the building to her left. The rooftops. He was running away. Knowing her time was short, she jumped from one windowsill to another and grabbed hold of a ledge. Before she could pull herself up, a woman lurked above her, weapon drawn.\n\n\"I'm sorry, Zusa,\" Veliana said. \"But this one is mine. Go back.\"\n\nZusa tensed, hating the awkward position she found herself in. The other woman's dagger was drawn, and in a flash of movement could easily sever her throat.\n\n\"Alyssa desires his blood for what he did to Nathaniel,\" Zusa said.\n\n\"And we need him for something more than petty revenge. Please, Zusa, I'm asking you.\"\n\nZusa frowned, finding her loyalties torn. Veliana was her apprentice, and had trained so much with her over the past five years. Yet Alyssa was even more than an apprentice, even more than a friend.\n\n\"Will you kill him?\" she asked.\n\nVeliana shook her head.\n\n\"No. Not unless he makes us. Just one meeting, that's all we need. Is my friendship at least worth that?\"\n\nZusa met her eye, let Veliana know the seriousness of her words.\n\n\"It is, but consider it an end to our friendship. Do you still desire to take him?\"\n\n\"I do.\"\n\n\"Then have him, Veliana. And pray we never meet again.\"\n\nShe released, and down she went, falling, refusing to look up as Veliana turned and raced after the bleeding body of the Watcher.\n\nDeathmask strode into the Ash Guild's headquarters with a smile on his face and his head held high. His head pounded as if within his skull was a caged orc, and the wine he'd drunk with Veliana to toast their success had done little to take off the pain's edge, but that didn't matter. The sight before him, the horrible disarray of the room, was enough to lift his spirits. Pillows lay scattered and shards of glass covered the floor by the bar. Garrick stood trembling at the far end. About twelve Ash members were inside, and none seemed eager to be near their guildmaster.\n\n\"Greetings,\" Deathmask said, pretending nothing was amiss. \"Good to see you survived last night intac\u2014\"\n\n\"Where were you?\" Garrick shouted. Deathmask blinked, and he glanced at one of the other men as if to show how confused he was. In truth, after attacking his own guild, he and Veliana had attacked the headquarters of the Spider Guild, slaughtering three and then leaving the symbol of the Ash scrawled into the dirt by the bodies. But he had no intention of telling Garrick that.\n\n\"Running for my life out in the streets, much like every other thief in Veldaren,\" Deathmask smoothly lied. \"I stopped by here once, but found the place empty, so I hid until morning.\"\n\nGarrick paced back and forth. His eyes were bloodshot. Deathmask wondered how much crimleaf the man had coursing through his veins. His speech was also slurred, perhaps from one, or several, of those broken bottles over at the counter. Drunk and stoned. Deathmask struggled to contain his amusement.\n\n\"Spiders!\" Garrick shouted, as if none of them were there anymore. \"Goddamn Spiders! What is Thren thinking? That I betrayed him? He think I'd be stupid enough to do that? We had a deal, you fucking Spider, you fucking... fucking... damn fucking Spider!\"\n\nDeathmask's eyes lit up at that. A deal? Could Garrick have been working for Thren? It would explain so much...\n\n\"Someone of Thren's showed up around half an hour ago,\" offered one of the nearest thieves, keeping his voice low so his guildmaster would not hear. \"Claimed that two members of the Ash came and killed several of Thren's men, and he demanded an explanation.\"\n\nDeathmask lifted his eyebrows in surprise.\n\n\"That obviously can't be correct,\" he said.\n\nGarrick overheard and stormed closer. Deathmask saw how incredibly dilated his pupils were, and decided his guess was correct. If Garrick's strength and confidence were built upon Thren's protection, then having that suddenly taken away would probably scare the shit out of him. Deathmask couldn't wait to tell Veliana. She'd been ready to kill the man before. What might she do knowing he'd sold the entire guild out to the man who'd executed her former guildmaster?\n\n\"Serious accusations,\" Deathmask said, repeatedly telling himself not to smile, not to reveal how terribly amused he was. \"What did you say?\"\n\n\"That it's bullshit,\" Garrick said, waving an unsteady finger in his face. \"And I'll convince him of that. But I want to know what's going on. Mercenaries by the hundreds running through the street, and for what? And tomorrow night, will they do the same? We need to plan. We need to prepare. Shit. What about the other guilds? Maybe they know what's going on. We should ask. Someone should go.\"\n\n_Behold your glorious leader_ , Deathmask mused, glancing at the rest of the Ash members who mingled about. _He was a puppet for Veliana, then a puppet for Thren. Now the strings are cut, and he can do nothing but collapse._\n\n\"I will go,\" Deathmask said. \"And to the Spider Guild, no less. We should show them we mean no ill will, and most of all that the survival of all the guilds is more important than our petty squabbles. How many of us died last night? This is a war now, a true war. Let me take that message to Thren.\"\n\nGarrick bit his lip, no doubt trying to process the idea in his drug-addled mind. The rest of the thieves looked pleased. Deathmask wasn't surprised. He'd arrived in the chaos, remained calm, and then presented a plan. This was something they could latch onto, however simple. Let the guild see that he was in control, not Garrick.\n\n\"Fine,\" he said. \"You may speak for me. Be careful, and don't press if Thren turns you away. Friends. That's what we must be. Good friends. We'll teach the Trifect to mess with us. Won't we? _Won't we?_ \"\n\nA halfhearted cheer came from the rest of the thieves. As Deathmask left he caught the looks they gave him, and this time he did not hide his smile. He was a stranger, a newcomer to the guild, but he was still becoming more of a leader in their minds than Garrick. Come a crisis, men and women always searched for stability. Let them see that in him.\n\nWhen he stepped out to the street, he looked to the rooftop for Veliana. At his request she'd remained hidden there when he went inside, just in case Garrick decided to do something stupid like try to kill him. Yet despite having been inside for only a few minutes, she was no longer waiting for him. Odd. Had someone else spotted her? He approached that same building, looped around to its back, and then climbed up. He expected Veliana to be lying there, perhaps bored or asleep. Instead no one.\n\n\"Vel?\" he wondered aloud.\n\nThen he saw it, a single streak of blood. He followed it to an alley, and when he peered down he saw Veliana kneeling over the body of a fallen man. His clothes were dark, well-worn, and lacking any local guild affiliation. His hair was short and blond, and across his chest was a smear of blood. At first Deathmask thought she'd attacked him, but when he climbed down he was surprised to see her bandaging the man's wounds.\n\n\"What the Abyss is going on?\" he asked.\n\n\"It's him,\" she said, not at all surprised by his arrival. \"It has to be. I fought him once, years ago, but who else might the Watcher be? It's Aaron... Thren Felhorn's long-dead son.\"\n\nDeathmask's mouth dropped, and every plan whirling through his head rearranged itself to match this new set of circumstances. He could hardly contain his excitement. Every one of those plans was infinitely better than the last.\n\n\"Take him,\" he said. \"Hurry. We have so much to discuss.\"\n\n# CHAPTER\n\n#\n\nHaern awoke with his head pounding and no clue where he was. The last thing he remembered was crawling across a rooftop, just before being struck from behind. Then he'd been thrown forward, off the building and to the ground below. Then nothing. His eyes slowly adjusted to the dim light. He saw stone walls. A dungeon perhaps? No, that didn't seem right. More of a cellar, windowless, lit by torches.\n\n\"You're awake.\"\n\nHe looked up. A man and a woman stood before him. The man wore a red robe, his dark hair pulled tight behind his head. As for the woman, she looked vaguely familiar, as if he'd seen her before a long time ago, in a dream. It had something to do with the long scar that ran down her face, bloodying her eye. He tried to stand, but he was tied to a chair. Whoever had tied the knots knew what they were doing. There wasn't the slightest give, and the moment he tested them, various cords across his chest and neck tightened, choking off his breath.\n\n\"Not sure I want to be awake,\" he said, doing his best to relax. He'd known this was the fate awaiting him. He couldn't make enemies of every guild and expect to live forever. Still, it seemed too soon. He'd accomplished so little. He'd die without mourners, without friends, without a legacy. A damn shame.\n\n\"Do you remember me?\" the woman asked. \"It was during the Bloody Kensgold. You were still a boy then, almost a man. We fought...\"\n\nAnd then he did remember. He'd seen her twice, once when his father had tried blackmailing her to turn against her guildmaster, then later in the attic of Leon Connington's mansion. Her name was Veliana, and the last time they'd met she'd nearly killed him.\n\n\"You do remember,\" she said, seeing the recognition in his eyes.\n\n\"I could never forget,\" he said softly. \"You showed me I would never escape. Aaron was dead, yet you never believed me. You refused.\"\n\nShe crossed her arms and leaned back against the stone wall. The man beside her remained quiet, seemingly content to let them talk.\n\n\"You could have let the smoke take me,\" she said. \"Why didn't you?\"\n\nHe shrugged the best he could, given the circumstances.\n\n\"It wasn't right to. I was there when my father left you to die at the hands of that disgusting... Worm. I couldn't do so again.\"\n\nHis words hung in the air. When she spoke, her tone didn't seem quite so hard.\n\n\"I spread word of Aaron Felhorn's death after that night. I told myself I did it to hurt your father, but I was lying. I did it for you, for sparing me. Aaron was dead, and it seemed true enough. What was the name you spoke to me that night? Something plain...\"\n\n\"Haern.\"\n\n\"That's right. Is that who you are now? Haern the Watcher? I find it hard to believe you're that same boy who spared my life. Do you know how many friends of mine you've killed? How many associates? You're still Thren's son, aren't you? Perhaps you should adopt your old name, Aaron.\"\n\n\"Never call me that,\" he said, glaring.\n\nIn the corner the dark-haired man laughed.\n\n\"Such ferocity. Well, there's no doubt you're skilled, and Vel here was lucky enough to get the jump on you after you were injured. Seems to corroborate everything we've heard about you, other than the demon blood. I'd sense it if that were the case. Still, your father cavorting with a succubus _is_ a rather amusing thought.\"\n\nHaern shifted in his bonds.\n\n\"What do you want from me?\" he asked.\n\n\"Before I tell you that, I want you to understand just how powerless you currently are. I could kill you right here, that's obvious enough. But I could also tell your father you're still alive. How beautiful a game that would be, watching him tear the city apart house by house until he found where you hid. My curiosity is almost strong enough to see what exactly he'd do. Would he greet you with open arms, or a dagger? Perhaps both?\"\n\nHaern glared. The man saw this and laughed again.\n\n\"Still, I'm in this for the coin and power, not the curiosity. I have a proposition for you, Aaron. Sorry, Watcher. Or would you prefer Haern? So many names, I find myself at a loss.\"\n\n\"Haern is fine.\"\n\n\"Very well then, Haern, my name is Deathmask, and I have a request of you. This is something I cannot do on my own, nor with just Veliana's help. But you... you have no loyalties, no weaknesses, no aspirations other than killing. So my request is simple, really: help me end this war between the guilds and the Trifect.\"\n\nThis time it was Haern's turn to laugh.\n\n\"I've slaughtered a hundred of you over the years, and even with my help the Trifect has sat on its hands and failed to do what needed to be done. Last night was a start, but it won't work, we both know that. It'll just anger them further. The retaliation will be terrible, if it hasn't happened already. What could I possibly do?\"\n\n\"Your name carries weight, believe it or not,\" Deathmask continued. \"Though really, I should say your reputation does. Every thief fears the night when the mark he goes to rob turns out to be you. Even the guildleaders are frightened of you, except for perhaps Thren. A rampage of dragons wouldn't make him soil his pants. But you have to understand something. All of these thieves, these underworld rogues, they've been trained since birth to survive. That's all they know. They'll claw and grab everything they can on their way up, but deep down they just want to live, and live well. If you threaten that, you can turn them to your side.\"\n\n\"No guildmaster would step down at my threat,\" Haern said. \"You're a fool. They'd rather die than forfeit their wealth.\"\n\n\"And that's the other thing you must understand,\" Deathmask said, grinning. \"They have no honor, no code. They want wealth, and they want to live, but they won't live without wealth, not when they've at last obtained it. So you must threaten their lives while at the same time offering them a chance to keep everything they've gathered. It can work. I know it.\"\n\nHaern leaned back in his chair, still not convinced.\n\n\"What is your plan?\" he asked.\n\n\"Do you know how much the Trifect pays to employ those mercenaries? How much money they lose year in and year out from Thren burning their goods, from Kadish slaughtering their help, from Garrick looting their wares? They're going bankrupt fighting this war, but they can't stop, they can't make peace, for Thren won't let them. No one wants this to continue. Before the guild wars, everyone made a tidy profit and hardly anyone died. We had a system. But Thren took offense, and the Trifect overstepped their bounds. A fair mistake by both, but now everyone's too stubborn to stop.\n\n\"Now, my plan is very simple; we gather every single guildleader, all three members of the Trifect, and then stick them in a room. No one leaves until they agree to end all this.\"\n\nOnly the sheer audacity of it kept Haern from laughing.\n\n\"It'd never work, you have to know that. Who would dare agree to go? Someone would back out, and others would try to execute their enemies while they're in the same place. Too much greed, too much mistrust. Someone will turn on another. I expected something more clever from someone so... strange.\"\n\n\"You spent five years trying to single-handedly conquer the thief guilds,\" Deathmask said. \"Yet you want to mock _my_ imagination? But you're right: such a meeting would instantly fall apart... unless we had an enforcer during the deal. Someone they all fear, so they would know that stepping out of line would result in a quick, efficient death.\"\n\nHaern remained quiet for a long time.\n\n\"Untie me,\" he said at last.\n\n\"Will you try to kill me?\" Deathmask asked.\n\n\"So little trust for someone who wants me to work for him.\"\n\nThe man shrugged. \"Fair enough. Cut him loose, Vel.\"\n\nShe did so, reluctantly. Haern stood and stretched his muscles, grimacing as his back popped. His arm also ached like mad, the muscles there definitely torn. He glanced about the dim cellar, then turned his attention to Deathmask.\n\n\"Tell me everything,\" he said.\n\nDeathmask seemed all too willing to talk.\n\n\"We'll let every member of the Trifect, and every thief guildmaster, know in no uncertain language that if they refuse to come to the meeting, they'll die. After that I'll work behind the scenes, grease a few wheels, and come up with terms that everyone can accept. Anyone who tries to make a fuss, or tries to refuse those terms, we send you after them. How does that sound?\"\n\n\"Insane,\" Haern said. He looked for his swords, saw neither. \"Were you not out last night? The blood clogs the gutters. They'll never agree, not to anything. You're delusional, Deathmask.\"\n\n\"If you don't accept,\" Veliana said, stepping in front of the stairs leading upward to daylight, \"then we'll have no choice. You'll have no one to blame but yourself.\"\n\n\"For what?\"\n\n\"We'll tell your father of your new name, and of who you really are,\" she said. \"How long do you think your little crusade against us will last once he knows? Right now he sees you as a nuisance, a ghost to keep his men on their toes and to cull the weak from his guild. But for Aaron Felhorn to turn against his own flesh and blood...\"\n\nHaern stood before her, staring eye to eye. Even wounded and exhausted, he would not back down.\n\n\"Do it, then,\" he said. \"But ask yourself who will find who first... my father finding me, or me finding you? Move aside. Now.\"\n\nShe tilted her head so she could see Deathmask, who must have made some sort of approving gesture.\n\n\"Very well. It was good to see you, _Aaron_.\"\n\nHe stepped out into the street, winced at the daylight, and then hurried away. With nowhere else to go, he headed for Senke and the mercenaries, hoping he might reach there without any other strange women attacking him.\n\n\"Do you think he'll change his mind?\" Veliana asked once he was gone. Deathmask shrugged.\n\n\"Depends on what you mean. About what I just offered? No. But I never expected him to.\"\n\nVeliana raised an eyebrow. \"Care to fill me in?\"\n\n\"Of course. Haern will never put his heart behind a plan I created. It needs to be his own, one he feels will be his legacy. We need his pride involved, otherwise he'll be ineffective and dangerous. I've planted the idea, though. He knows many of us desire peace, and that he can be a key part of it all. The idea of an enforcer is what I think will appeal to him most. He'll return, I assure you, and it will be with a plan far more insane than mine.\"\n\nShe sat down on the pile of cushions that had been her bed since Deathmask took her away from the Ash Guild. \"Even after we kill Garrick, it'll be tough getting the rest of the guild to accept any sort of plan involving the Watcher.\"\n\n\"I know,\" Deathmask said. \"Which is why I turned the Spider Guild against them. Why should I have to do all the dirty work?\"\n\nSomething about how casually he said it, how amused he seemed, clicked everything into place. Veliana's mouth dropped.\n\n\"You hope to destroy the Ash Guild, tear it down to nothing but yourself.\"\n\n\"Not quite,\" Deathmask said. \"You'll be there, as well as two or three others who have the skill to endure the bloodshed to come. I don't need a guild full of simpletons, children, and the like. I just need enough to put fear into the heart of the guilds, to ensure that all the coin I desire flows to my hands...\"\n\n\"Then this was your plan all along? Tear down the Ash Guild and rebuild it with you at the top, and a few of us bowing around your feet?\"\n\nVeliana rubbed her temples, trying to think. Everything she and James Beren had created, it'd been dwindling since Thren killed James, but they'd still held on to shards of it. Might it all vanish? Could she let him do this, destroy the legacy of the Ash Guild forever?\n\n\"You'll ruin everything I've worked for,\" she said quietly.\n\n\"I'll give the Ash Guild a legacy all of Dezrel will one day fear. I don't expect it to be easy. We'll have to kill a lot, yes. But if we succeed, think of all that wealth, that respect. We'll fill every other guild with the fear of retaliation. We'll never need to guard, only attack. Those who turn against us will meet death, every single one. Once that reputation takes hold, we'll be gods in this city.\"\n\n\"You're a lunatic.\"\n\nHe smiled. \"Perhaps. But oh what fun it'll be in the attempt. Are you getting scared on me, Vel? Aspire to greatness, and damn all others. Garrick took your guild. Help me take it back, and mold it into something never seen before in the history of Dezrel.\"\n\nShe still wasn't sure, but she'd not let Deathmask know.\n\n\"What of the Spider Guild?\" she asked. \"Weren't you to meet with Thren?\"\n\nDeathmask's eyes twinkled, and his grin pulled wider. Despite it, she thought she sensed fear hiding behind the guise.\n\n\"Come with me,\" he said. \"Hopefully we haven't missed the show. I discovered something when in Garrick's audience, and I've rethought what Thren's reaction to our masquerade will be.\"\n\nThey removed their cloaks and dressed in drab colors that showed no affiliation. Veliana kept a hood low over her face to hide her scar. Given the cold wind of the morning, no one would think the hood odd. Deathmask led the way, taking a winding path back to the Ash guildhouse. Before they were even halfway there, Veliana could already see the smoke rising.\n\n\"What happened?\" she asked.\n\n\"Careful,\" he said, hugging the wall and peering around every turn. \"We might have a few more enemies now. Do you remember when the Hawks tried to ambush you, the attempt I broke up when I first met you? Garrick didn't do anything, did he?\"\n\n\"No,\" she said. \"But what does that\u2014?\"\n\n\"The Spider Guild attacked the Hawks barely a day later. Why do you think that, Veliana? Why do you think Garrick suddenly grew testicles and dared challenge your hidden control?\"\n\nThe realization hit her like a battering ram.\n\n\"No. That cow-sucking shit-eating motherfucker. I'll kill him. It'll hopefully take days, but I'll kill him.\"\n\n\"Assuming he's still alive,\" Deathmask said as he led them into an apartment. They climbed the stairs, stopped at a door on the higher floor, and knocked. When no one answered, Veliana kicked it in. The room was disheveled, what little was left. The occupants appeared to have either fled or died. From the small window they could see the guildhouse. Deathmask looked first, then backed away so Veliana might see. The whole building was in flames. It had already collapsed on its supports, black smoke billowing. Surrounding it was a circle of thieves wearing the cloaks and colors of the Spider.\n\nWhile she watched, a man crawled from the wreckage, and even from their distance he looked badly burned. One of the Spiders shot him with an arrow before he could rise to his feet.\n\n\"Unbelievable,\" Deathmask said as he took a second look. \"An army of mercenaries descends upon us, and without hesitation he massacres a fellow guild, all for a single act of betrayal.\"\n\n\"Thren is not one to hesitate.\"\n\nDeathmask muttered and flopped down onto the poorly stuffed bed.\n\n\"We need to establish control of the guild, and now,\" he said. \"I'd hoped for a bit of backstabbing between the two, a thinning of members, but this... Thren's viciousness is astounding. We must take over before the guild disbands completely, and the rest of the city moves in on our territory. At least the mercenaries will keep them from doing so for a while. As for Thren... if there's to be any chance of peace, he'll need to be dealt with, one way or another. Tell me, where will the remainder of the guild flee now this place has burned?\"\n\n\"The old house,\" she said. \"Below the Split Pig Inn. We expanded their cellars and paid handsomely to do so. They should still be empty, and the owner was a crusty old dog that won't be intimidated by sellswords.\"\n\n\"Then that's where we'll go,\" he said, retying his mask over his face. \"Let them see you alive, let them hear my demands of obedience. As of now, the Ash Guild is under my control.\"\n\n\"And if Garrick still lives?\"\n\nDeathmask flashed her a smile, all pretense of fear gone.\n\n\"Then you get your revenge, assuming you're strong enough to take it.\"\n\nShe patted her daggers. \"I'll be fine. Follow me, and keep your eyes open. The last thing I want to do is die before my steel tastes Garrick's blood.\"\n\n# CHAPTER\n\n#\n\nGhost woke when the sun shone bright on his face. He stirred, rubbed his eyes, and then forced them open. Midday, he guessed. His stomach rumbled, and his head pounded from the night before. He felt he could sleep another four hours or so, but his body would have to endure. Still, he wasn't in too much of a hurry. He had a name, after all, a place to search and question. Not much harm in grabbing a bite to eat.\n\nOnce he left his room at the shabby inn, he swung by the main market in the center of town, buying a thick slice of bread smothered with butter and honey. While he ate, he sat by a fountain in the center and listened to the idle talk as men and women passed by. The overwhelming sensation was not fear, as he'd expected. It was anger. More surprising was how it wasn't directed at the guilds, or even the Trifect. They directed it at the king.\n\n_Stupid dogs_ , he thought as he ate. _You've lived under this chaos for so long it's become normal to you. The Trifect and the guildswar, and you see this as acceptable, but only if the king protects you. Last night destroyed your apathy. Last night saw your blood joining the others'. So you rage, but only to your protector. Damn king. Should have put this nonsense to rest years ago._\n\nStill fairly new to Veldaren, Ghost knew only a little of the king, but what he'd gleaned wasn't flattering. As he listened to men swear against their liege's honor, and women insinuate he'd been born without his manhood, it seemed obvious that his cowardly indifference could no longer last. But whose side would he fall on, the guilds' or the Trifect's? Logic seemed to place him as a puppet of the Trifect, but Ghost was unsure. Which one would he fear more? If the man was a true coward, he'd fear the enemy he couldn't keep out with gates and walls, the enemy that'd fill his drink with poison and lay a dagger under his pillow while he slept.\n\nMeal finished, he drank from the fountain and then headed to the mercenaries' headquarters. Not surprisingly, it was crowded with both the rich and the poor. They were pleading their cases, demanding compensation for damages done in the chaotic night. The old keeper, Bill Trett, shouted the same phrase over and over, as if by the fiftieth time it might finally sink in.\n\n\"Take all complaints to Alyssa Gemcroft's estate. She has promised to accept full responsibility. I'm sorry if your house burned down, or someone died, but please, take all complaints to Alyssa Gemcroft's estate. She has promised...\"\n\nGhost slammed a massive fist against the door, the sound thunderous in the small room. The crowd, about twenty in all, jumped and turned.\n\n\"Enough!\" he roared. \"Get your asses out of here, and go to Lady Gemcroft's with your problems.\"\n\nHe kept his muscular arm pressed against the door, holding it open. The stance also revealed the weapons at his hips. He glared, letting them see he had no desire to argue. A few filed out, while the rest looked about as if trying to decide just how serious he was. Only a few carried weapons, and he doubted they were proficient with them.\n\n\"I'm letting go of the door,\" he said, his voice lowering in volume but not in depth. \"When it shuts, I kill everyone in here not a member of the mercenary guild. That clear?\"\n\nHe let go. A wiry man in silks lunged for it, sticking his hand in the way. The rest followed him, until only a thankful Bill remained.\n\n\"What the Abyss happened last night?\" the older man asked. \"I expected several of them to jump the counter and attack me.\"\n\n\"Frightened sheep,\" Ghost said. \"Let Alyssa handle them. No reason for you to put up with their bleating.\"\n\n\"I doubt you've come here to be my savior,\" Bill said, sitting down and smoothing his hair. He pulled a bottle from a drawer and took a deep swig. \"So what is it you need?\"\n\n\"A small group of mercenaries, led by one named Tarlak. Do you know them?\"\n\nBill raised an eyebrow. \"I do, but only because they've caused me a bit of trouble. Tarlak Eschaton, leader of the Eschaton Mercenaries. Refused to join our guild or pay dues. The last representative I sent their way demanding they join came back as a toad.\"\n\nGhost blinked. \"A toad?\"\n\n\"A damn toad. Cost a fortune to send for a representative of the Council of Mages to come change him back. They weren't too happy with that Tarlak, either. Evidently he's a rogue apprentice or something, but since he's not an official member they don't consider him their problem, so long as he doesn't start blowing up houses or trying to become anything more than what he is now.\"\n\n\"Which is?\"\n\nBill shrugged. \"A small-time mercenary. Why do you ask?\"\n\n\"I need to find him.\"\n\n\"Last I knew he was on Crimson Alley, thirteenth heading south from Ax Way. Should still be there.\"\n\n\"Do you know how many are with him?\"\n\nBill took another long swig, paused to think, and then stood. After he'd locked the door, then put a wooden bar across, he sat back down.\n\n\"I'm thinking we're closed for the day,\" he said. \"And I'm not sure I like where this is going, Ghost. Care to tell me why you want to know so much about this Tarlak fellow?\"\n\n\"He knows something.\"\n\n\"From what I hear, people that know something you want to know have a funny way of turning up dead.\"\n\nGhost shrugged. \"Depends on how loose their lips are.\"\n\n\"My, aren't you a piece of work?\" Bill said, chuckling. \"But considering they aren't part of the guild, and you are, I guess I can tell you what I know. He lives with his sister, young gal. Priestess, I think. Also got some guy named Brug, though why he's taken him in I don't have a clue. We turned down that guy's application twice. Too much temper without a shred of skill to back it up. Last is a guy named Stern, bald as you, but that's all I know. If he can fight, I haven't heard word of it. Like I said, small-time, with only his petty magic tricks to make him stand out in the slightest. Oh, and those gods-awful yellow robes of his. Their group's only been around for nine months or so, maybe a year. I don't expect 'em to last.\"\n\nGhost bowed, stealing Bill's bottle as he did. He took in several gulps, the burning in his throat doing much to awaken him.\n\n\"Enjoy your day in peace,\" Ghost said, handing it back. \"And lock the door after I leave. There's still people gathering outside.\"\n\n\"Will do.\"\n\nA glare from Ghost caused the few waiting outside to step back, and he didn't move away until he heard the thump of the wood barring the door.\n\n\"You've still got your lives,\" he told them. \"If you've got that, you can move on. I suggest you do. Your pleading and curses won't do a damn thing to sway anyone, not in this city.\"\n\nHe trudged south, toward the Crimson, keeping an eye out for Ax Way so he could begin counting. Finding the thirteenth from there was easy. Eyeing the building's two floors, he crossed his arms and thought. Deciding his entry point, he continued. A block later he turned around, coming back by a side alley. The way was dark, and two men glared at him as he passed. Any other they might have tried to rob, but he'd flashed them a grin, and he saw the way they stared at his painted face. They'd have been more likely to try to rob a dragon.\n\nThe Eschatons' building itself was smooth wood on the outside, but not the one next to it. That was well-worn and cracked with many handholds. Climbing up to the roof, he turned and leaped the gap, rolling to absorb the impact of his landing. It wasn't that he feared injuring his legs; he knew they could endure the blow. He just didn't want to alert anyone inside to his presence. There was no direct entrance from the roof, but the second floor had a window, and that would be enough for him. Hanging upside down, he looked through it.\n\nThe glass was surprisingly clean, and he guessed it was because of the woman who slept beneath the window, red-haired and buried under blankets. The priestess, he figured. The muscles in his legs flexing to keep himself steady, he brushed his fingers against the glass, testing whether or not the window opened. It didn't.\n\nHe swung back onto the roof and debated. If he broke in through the front door, the surprise would be less, and he wouldn't have immediate access to the inhabitants. He could try coming back later, but if they slept through the day they'd probably be out with the other mercenaries come nightfall. Again, no good. It'd have to be now. He put his back to the ledge, then crouched down so he could clutch it with his hands. The window would be a tight fit, but if he stretched out enough he could squeeze through.\n\nHe kicked, shifting into a pivot. Feetfirst he smashed through the window, showering the woman's bed with glass. He let go of the ledge and held his arms high above his head. His momentum carried him through, and he landed on his back on top of her. Before she could scream he rolled over and clamped a hand over her mouth.\n\n\"Shush now,\" he said, ramming an open palm against the side of her head, knocking her out cold. Knowing time was short, he pushed off the bed and made for the door.\n\n\"Del?\" he heard someone ask from the other side. His voice was nervous, but not yet worried. A broken window could be many things, most likely a stone, least likely a dangerous man built like a mountain. The door swung inward. Before it was a third of the way open, Ghost rammed it closed with his knee, then grabbed it with one hand and flung it wide. A man was falling on the other side, his balance broken by the sudden hit from the door. It was Tarlak, based on his outfit.\n\n_Bill was right. What is that, piss-yellow?_\n\nGhost rammed a meaty fist into his mouth, just to make sure no spells got out. He had no intention of showing up at Bill's later as a toad. The punch split the wizard's lip, and blood flecked across Ghost's knuckles. Ghost followed it up with a punch to the stomach, doubling him over. He dropped the wizard with both his fists together, bashing the back of his head. Tarlak crumpled, easily unconscious. Neither he nor the girl would be out long, maybe a few minutes, but Ghost felt it enough time. When they came to they'd be bound, and the wizard gagged.\n\nThere was one other room up top, its door open. Deciding it was Tarlak's, he hurried to the stairs. If the other two were awake, they'd be rushing up them. Sure enough he heard hollering, and a short man with muscular arms and a beard met him halfway.\n\n\"What the bloody\u2014?\"\n\nGhost snapped a fist into his face, cutting the sentence off short. A knee to his groin sent the man rolling back down. Again Bill's information rang true. The guy wasn't much use, was he?\n\nHe followed Brug's roll down the stairs, kicking him at the bottom to convince him to stay down. One left, the one called Stern. His eyes swept the lower floor. Two doors were in the back, plus an exit to the outside. One was open, Brug's he assumed. The other...\n\nThe door blasted open as he reached for its handle. He spun immediately in the direction of the door, using it as a shield. A flanged mace cut through the air where he'd been. Ghost bounced off the wall, drawing both his swords. He shoved the mace aside, then slashed blindly around the door. The other sword hit something hard, and then they were both in view of each other. The man, Stern no doubt, glared back at him, a mace in each hand. Their weapons pressed against each other, testing strength. Stern, while stronger than he looked, was still no contest.\n\nThis didn't seem to surprise him, though, and when Ghost tried to push him back, Stern parried both swords to the side and tried to leap past. He wanted the open area, Ghost realized, hoping his speed might win out. Ghost couldn't stop him, but he could make life more difficult. He kicked as he rotated his body, taking out one of Stern's knees. Stern didn't even try to keep his balance, instead rolling forward, around an old wooden chair, and up to his feet before the front door. He lifted his maces and grinned.\n\n\"Clearly skilled,\" he said. \"So what person with money did we piss off this time?\"\n\n\"Don't matter,\" Ghost said. He feinted a charge, then kicked the chair at him. Stern blocked it with his heel, but it was enough of a stall. Double-slashing, Ghost gave him no choice but to block, and block he did. His arms jarred, though nothing broke as Ghost had hoped. Sometimes, if he swung just right, he could pop a collarbone or wreck the joints in an elbow.\n\nGhost looped his swords around for an attack from both sides. That left the only opening straight ahead, toward his chest. He wanted this Stern character to try it, to hold his own instead of bouncing about. Maddeningly, he didn't take it. Stern dropped to one knee, blocking the lower slash from his left and letting the right sail over his head. Immediately after blocking he rolled to the side. Ghost chased, but each slash smashed only floor. Time was no longer on his side. The others would be up soon, groggy and heads full of fog, but still up.\n\nHow much concentration did it take to turn someone into a toad, Ghost wondered.\n\nStern at last had nowhere to run. His back was to the wall. To his left were the stairs, and to his right the main door. His eyes flicked between both, deciding. Ghost gave him no time, rushing in while keeping his swords close. He would block a retreat with his own body and let his blades do the work. Stern had no chance of matching Ghost's strength, and without a retreat, dying would be his only option.\n\nIt seemed Stern knew this as well. His eyes widened, and he appeared pressed to the very edge of his control by adrenaline and fear. The look of a cornered animal. Ghost knew Stern would not roll onto his back and hope for mercy. He'd lunge, mad, vicious. And that's just what he did. Ghost feared those first few seconds as the maces came crashing in, slamming into his swords with shocking impact. He felt kicks strike his body, at one point an elbow, and still the maces looped and struck. But they were fighting Ghost's fight now, close up and animalistic. He blocked a swipe from the side, then savagely struck the mace's length with his other blade, knocking it from Stern's hand. Stern's other weapon came around, straight for his head. Instead of ducking Ghost stepped closer, chest to chest. The side of Stern's arm hit his face, but that was far better than the sharp edges of the mace.\n\nOne sword slashed Stern's arm, making him drop his weapon. The other thrust into his belly and twisted.\n\n\"Shit,\" Stern grunted, clutching Ghost's wrist with both hands. His whole body shook, and his face rapidly paled. Ghost pulled his weapon free, breaking Stern's grip as if it were that of a child. The man slid down the wall, blood pouring across his hands and down his legs. He held the wound with his palms, slowing the bleeding.\n\n\"You should have surrendered,\" Ghost said. \"Though I respect your defense of your friends, this was all unnecessary.\"\n\nHe left him there, stepped over Brug, and climbed the stairs. The short fighter was moaning still, conscious but only just. He was no threat. Ghost found the wizard first, glad to see him still out. Unwrapping the rope he'd looped about his own waist, he cut a length of it and bound Tarlak's hands. Thinking for a moment, he cut a smaller length, wedged a piece of the wizard's own robe into his mouth, and then tied it there to form a gag. Hefting Tarlak onto his shoulder, he carried him back down the stairs and deposited him in a chair. Stern watched him with glazed eyes from where he lay.\n\nLast was the girl. She opened her eyes when he stepped inside, but she showed no recognition, nor any signs of fear. Concussion, he figured. She probably didn't know the difference between him and the King of Ker.\n\n\"To your feet,\" he said. \"I'd hate to strike you again.\"\n\nHe grabbed her wrists and held them tight as he escorted her down the stairs. Once she was tied to another chair, he kicked Brug to see how he was faring.\n\n\"Damn it,\" Brug muttered, his eyes suddenly focusing. \"What was that for?\"\n\nHe saw Ghost standing over him, and then he tried to reach for his weapons. Laughing, Ghost slammed his heel onto his throat and pushed him back.\n\n\"I'd recommend you behave,\" he said, the tip of his sword dangling before an eye. \"Otherwise I might just let go.\"\n\nBrug ground his teeth, glanced about, then nodded. Ghost bound his hands and feet and then dumped him on the floor beside the others.\n\n\"Well, that was disappointingly easy,\" Ghost said, sheathing his swords. \"I hope the Watcher proves more challenging than you four.\"\n\nStern said something, but his voice was too weak to hear. Ghost stepped closer and leaned down.\n\n\"You'll find out when he kills you,\" Stern said, then made a sound like a cross between a cough and a laugh. Ghost slapped the side of his face, the gesture almost playful.\n\n\"At least you put up a fight,\" he said. \"So I'll forgive you for your frightened boasting. Stay still, and try not to let your grip slip. You might know something useful to me, and I'd hate to lose it because you can't keep your guts from squeezing through your fingers.\"\n\nThe priestess seemed to be getting her bearings, but Tarlak was still clearly out. Ghost reached into a pocket and pulled out some smelling salts. Shoving them under the wizard's nose, he held his head by his hair and waited. After a few sniffs Tarlak's eyelids began to flutter, and then he jolted as if splashed by a bucket of water.\n\n\"Whmmph,\" he said.\n\n\"Welcome back,\" Ghost said, smacking his shoulder. \"Forgive the gag. I know how dangerous your kind is with a few silly words. I may take it out, but only for a moment, and only when my swords are at your throat. Understand?\"\n\nA soft gasp came from his right. It seemed like the priestess had finally come to her senses.\n\n\"Senke!\" she gasped.\n\n_Senke?_\n\nHe followed her gaze to the wounded man against the wall. A pet name, perhaps? Or maybe Bill had been wrong about the man's identity?\n\n\"He put up a better fight than the rest of you,\" Ghost said.\n\n\"Don't say nothing, Delysia,\" Brug muttered. \"Just bite your tongue and say nothing.\"\n\n\"I don't think I'd listen to him,\" Ghost said, placing the name to her face. Many people he interrogated became much more compliant when he called them by their names.\n\n\"Please, I can help him!\" She squirmed against her bonds. \"He's dying!\"\n\n\"If he's dying, he's doing a poor job of it.\" He watched her struggle to see if his ropes would hold. Satisfied, he took her chin in a giant hand and forced her gaze to his. \"But if you want untied, you'll have to talk. That's all, little girl, just talk. No sin in that, right?\"\n\n\"What do you want?\" she asked.\n\n\"Don't!\" Brug shouted. Ghost turned on him, and this time his kick was lower, and harder. Brug howled like an animal, and his face turned a beet red.\n\n\"Enough out of you,\" he said. \"You're beaten, and at my mercy. Lies and silence get you pain, only pain. Not honor. Not sacrifice. Not nobility. Just pain.\"\n\nTarlak mumbled something into his gag. Ghost debated, but then left him alone. He'd go to the wizard only if the others proved uncooperative. So far this Delysia appeared the most compliant. He knelt before her, all teeth and smiles.\n\n\"Senke's bleeding over there,\" he said, dropping his voice lower. She tried to look, but his eyes held her. He knew he could do that, had so many times before. He felt like a snake charmer, controlling them by the sheer ferocity of his personality. \"You can feel it, his pain washing over you like a heat. You're a priestess, so you could help him, tend his wounds. How badly you must wish to go to him. Such sweet compassion.\"\n\nHe shifted to behind her, pressing a painted cheek against hers as they both looked to where Senke lay.\n\n\"But is it just compassion? I don't think it is. I think it's fear. I smell the stink on you. It's rising in your chest, crawling upward like a beast. You don't want to watch him die, yet that's what you are doing. Life is draining away before you, and all you can do is sit here. Struggling against your ropes won't seal the hole in his gut, Delysia. Only one thing will, and that is talking to me. Tell me the truth, and only the truth. Can you do that, pretty girl? Can you do that for Senke?\"\n\nShe bit her lower lip. Tears ran down her face.\n\n\"Yes,\" she said at last. Brug sighed. By the wall, Senke chuckled. Tarlak let out another _mrmph_ into his gag.\n\n\"Good lass. It's a simple question, really. I have a contract to find the Watcher, and your group knows of him. Tell me, where can I find him?\"\n\n\"I don't know,\" she said. She stared into his eyes, and he realized she wanted him to know she didn't lie. \"He's only stayed here twice. Where he goes when he leaves... please, I don't know. None of us do.\"\n\nGhost frowned.\n\n\"Tell me his name, then. He must have a name.\"\n\nTears ran down her face. She looked to Senke, but Ghost grabbed her jaw and forced her back to him.\n\n\"Haern,\" she said. \"He calls himself Haern.\"\n\n\"Last name? First?\"\n\n\"Just Haern.\"\n\nPossibilities ran through Ghost's head, and he didn't like any of them. A single, plain name would be marginal help at best in tracking him down. Still, it was better than nothing, which is what he'd been going on before. But mostly he didn't want a name. He wanted the man in person.\n\n\"Will he be coming back here?\"\n\nShe hesitated, just for a second, but Ghost saw it and smiled.\n\n\"No lies,\" he said. \"That just gives pain, remember?\"\n\n\"I don't know,\" she said at last. \"But I think he will. Please, can I help him now? He's almost gone.\"\n\n\"Of course, my dear.\"\n\nHe untied her wrists and then gestured for her to go. She ran to Senke's side and knelt. He whispered something to her, and he heard her begin to cry. Ideas raced through Ghost's head as he watched out of the corner of his eye. If this Haern was coming back, then he had to keep them all here until he did, otherwise they might find a way to warn him. Of course it could be days until he showed, or worse, Haern might spot the ambush through a window, or sense it from the lack of common activity. Troubling. He'd have to dump them in one of the rooms, preferably without a window. Once done, then he might...\n\nAnd that was when the door opened, and in stepped the Watcher.\n\n# CHAPTER\n\n#\n\nHaern's elbow still ached like the Abyss, but at least it'd stopped bleeding. He felt naked without his swords, so he kept his head down and shambled along as if he were drunk. Given the horror of the previous night, he knew he was far from the only one who staggered along the road. Many had buried their grief in alcohol. His nerves rose as he hurried down the Crimson, but he reminded himself it was safer in the day than at night. Sure, some of the young cutpurses might try to swipe his coin, but he had nothing to steal.\n\nWhen he reached the Eschatons' building he put his hand on the door and closed his eyes. Returning here meant many things, and he wasn't sure if he was prepared for the implications. Could this place become a home to him? Could he accept Senke's companionship, even knowing his presence would be a danger to his friend? Deep in his heart he knew he desired nothing else. It was his head that kept getting in the way. But sometimes you needed to think like that to protect others. To Haern, the self had never mattered more than those he cared for. He'd learned that lesson from watching his teacher, Robert, sacrifice his life to protect him.\n\nHe opened the door with his good arm. So lost in his thoughts, so focused on what he might say to them, and what they might say back, he was unprepared for the sight before him. Tarlak sat bound and gagged in a chair. Brug lay on the floor, also bound. Senke slumped against the wall, blood covering his clothes. Delysia knelt before him, her hands also covered with blood. And there amid them all was a giant stranger, skin like obsidian, face painted white as a skull. It seemed the stranger was as surprised as he was, and they both froze for a split second. Haern looked into this man's eyes and saw death.\n\n\"Watcher,\" said the painted man. Not a question, just a statement. His deep voice chilled Haern to the bone, telling him it was time to act. This was no game. Their lives hung in the balance.\n\n\"Run!\" Delysia screamed.\n\nBut he couldn't leave them like this. Damn it, what he'd give to have his swords!\n\nThe stranger lunged, drawing two swords as he did. Haern dove farther into the house, tumbling to avoid the attack. His eyes searched for a weapon, any weapon. There, on the wall, he saw the short swords Senke had used during their brief spar. Scrambling to his feet, he ran for them, not even slowing when he slammed into the wall. His good arm snatched one free, and then he rolled aside, the stranger's sword cutting several inches into the thin wall.\n\n\"Who are you?\" Haern asked as he held the blade before him and crouched into position.\n\n\"I am Ghost,\" said the man. His brown eyes shone amid the paint. Sweat dripped down his neck and arms, every inch seemingly nothing but muscle. His swords lifted and dipped into a stance, perfectly smooth, perfectly calm. Haern felt terror at the sight. For all his reputation, all his killings, this man faced the Watcher unafraid. He even smiled.\n\nEvery instinct told Haern to retreat, but he wouldn't. He'd thought he'd lost Senke in a fire, and he'd never come back to look. He'd been dragged off by his father while Delysia bled. This time he'd stay until the end, whatever that might be. _Death or victory_ , he thought. His father would be proud.\n\n\"Come then,\" Haern said. \"Kill me if you can.\"\n\nHe kicked aside the table and in the limited space began spinning in place. His multitude of cloaks dipped and rose, hiding his presence. Ghost watched it, the concentration in his eyes frightening. When he moved to attack, out lashed Haern's short sword, nearly slicing off his nose. Again Ghost watched, waited. Haern had practiced the cloakdance over the years, trying to perfect it since it had first been used on him during the Bloody Kensgold. The constant motion kept his true movements unpredictable, kept the positioning of his swords hidden. Lesser foes he could defeat with ease, and it gave him an advantage against several attackers at once. But against someone so skilled? It was a stalling diversion, nothing more.\n\n\"Stop dancing and kick his ass!\" Brug shouted, unable to do anything but watch from his spot on the floor.\n\nA sword swung in. Haern dipped below it, his spine nearly parallel to the floor. Out went his blade, cutting into the thin flesh of Ghost's knee. It'd be painful, but not debilitating beyond a limp. He hadn't been able to apply enough force due to his awkward position. Worse was that the blade caught on the bone instead of slicing free. Ghost stepped in, unafraid of the cloaks and defiant of the weapon lodged in his flesh. He swung downward with both swords. Haern's momentum had him rising to a stand, so he kicked out his own feet to fall instead. The swords missed, but only barely. Haern landed flat on his back, the impact knocking the breath from his lungs. His wounded elbow hit hard as well, and the pain of it filled his vision with black dots. Ghost twirled a sword in his right hand, pointing the blade downward, eager for a killing thrust.\n\nBut then Delysia was there, her hands raised, palms facing Ghost. Bright light flared, blinding even to Haern. Ghost roared, and he took a step back as if struck by a blow. Haern swung his legs wide, taking advantage of the distraction. His heel struck the wounded knee, hard enough to dislodge the sword stuck in the joint. Down Ghost went, the knee crumpling. Again Delysia let out a cry, shouting out the name of her deity. Her hand moved in a downward arc. A golden sword materialized in the air before her, mimicking the motion. It sliced into Ghost's chest. Blood sprayed across her, but she didn't appear to notice. Another prayer was already on her lips, demanding the strength of Ashhur.\n\n\"Begone!\" she cried. Haern saw a faint outline, almost like an enormous hand, shimmer and vanish in the blink of an eye. Ghost flew back several feet, as if hit by a battering ram. When his body met the wall, it was the wall that gave, the cheap plaster breaking. Haern took to his feet, his wounded elbow held against his chest. It had started bleeding again, staining the gray of his clothes red. Ghost took a woozy step forward, then collapsed when he tried to stand on the other leg. Haern reached down and grabbed his sword while the giant man crawled toward the exit.\n\n\"Don't,\" Delysia said, grabbing his shirt. Her voice had authority now, and something in him was unwilling to challenge it. \"Please, don't kill him.\"\n\n\"Are you mad?\" Brug asked, still squirming against his ropes. Haern felt inclined to agree.\n\n\"He's beaten, and leaving,\" she insisted. \"Don't. He let me save Senke. He deserves as much.\"\n\n\"He's also the one who did it in the first place,\" Senke said with a sleepy voice. \"Just thought I'd point that out.\"\n\n\"Phggrrmpf,\" Tarlak chimed in.\n\nGhost looked at them as if they were all mad. He used a chair to brace himself as he stood, then limped toward the door, his teeth clenched against the pain.\n\n\"You were beaten,\" he said as he took a lumbering step outside.\n\n\"Sure thing,\" Haern said, Delysia still clutching his shirt. The moment the door closed he slumped backward, sitting atop the edge of the overturned table. Delysia checked his elbow.\n\n\"Senke needs my help more than you,\" she said. \"It can wait. Untie Tarlak and Brug.\"\n\n\"As you wish.\"\n\nDelysia returned to Senke and knelt before him. Haern heard her prayers, and white light shone around her hands. No wonder the wound on his chest had healed so quickly those few days ago.\n\n\"Friend of yours?\" Tarlak asked once the gag was removed.\n\n\"You aren't funny,\" Haern said.\n\nHe cut the ropes around his hands and feet, and while the wizard stretched, he did the same for Brug.\n\n\"Son of a whore ambushed me coming up the stairs,\" Brug said, grabbing his punch daggers. \"Otherwise I'd have torn him a new hole.\"\n\n\"You mean like this one?\" Senke asked.\n\nBrug flushed and looked away. Haern tossed his short sword to the floor. He felt sick, and he still hadn't recovered from the blow to his head earlier in the day. His elbow throbbed, feeling even worse than when he'd first received the cut. He saw Brug and Tarlak glaring at him, and he felt he deserved their ire. He tried to stumble for the door, but Tarlak blocked the way, holding it shut with his arm.\n\n\"Not yet,\" he said. \"And not anytime soon. It's time we talked, Watcher.\"\n\n# CHAPTER\n\n#\n\nMatthew's relief upon seeing Felwood Castle lasted only as long as it took him to see one of Hadfield's men standing watch far from the other guards. It was as he'd feared. Less than ten minutes ago he'd had to drag himself and Tristan off the road, and when the horsemen rode on by, his gut had told him who it was they served. And now the sentry was there, keeping a close eye on everyone entering and leaving Lord Gandrem's castle.\n\n\"What do we do?\" Tristan asked. Matthew had abruptly turned them both around and back north on the road, hoping the soldier hadn't seen their approach. Given the distance, it seemed probable.\n\n\"I don't know,\" he said. He could imagine what would happen if they tried to pass by. The soldier would cut them down before letting them reach John Gandrem. Whatever punishment the soldier received would be bearable so long as no one identified the one-armed boy as the son of Lady Gemcroft. Given his disfigurement, the dirt on his face and the plain clothes he now wore, it seemed doubtful that anyone would.\n\n\"Will we continue on to Veldaren?\" Tristan asked.\n\n\"Quiet, boy, I don't know!\"\n\nHe waited until his temper calmed, then resumed.\n\n\"And I'm not sure we can. Don't have the food, and water might end up scarce too. I need inside to resupply, but that might mean leaving you behind for a while. They won't know me from shit, since you're the one they want. That, and I don't know who John's sided with in all this.\"\n\n\"John was always nice to me,\" Tristan said. \"I stayed with him for a year. What if... what if I get us inside? Will he keep us safe?\"\n\nMatthew shot him a look.\n\n\"How could you get us inside?\"\n\n\"I don't know. I could run real fast. I'm a fast runner, even Arthur said it!\"\n\nMatthew bit his lip. It was just one man, a professional soldier perhaps, but still just one. He touched the old sword at his hip. If he could last for a little while, just a little...\n\nHis eyes fell upon the near-empty sack that had carried their food.\n\n\"I have an idea,\" he said. \"But you better run like the wind, you hear me? Like it, and even faster. My life is depending on those legs of yours.\"\n\nIngle mumbled curses as he shifted his weight from foot to foot, trying to generate some heat to counter the cold. After another minute he pulled a blanket from a saddlebag and wrapped it around his shoulders. Beside him his horse clomped the ground.\n\n\"Blanket ain't big enough for two of us,\" he said. \"We'll get you somewhere warm once we find that brat, though, I promise.\"\n\nIngle and his horse waited a hundred yards outside the castle's entrance, near the fork where the main road turned toward him. The woods had been thinned out toward the front, though they were still thick enough to make him worry. Nathaniel and the farmer might try to sneak along the walls, using the woods as cover. Doing that was a good way to earn an arrow in your back from a guard, though. They would come traveling down the road, he felt certain of it. According to his bitch of a wife, Matthew had left immediately after killing Gert and Ben. Dimwit farmer couldn't know how many were actually looking, or that they might have beaten him here. Ingle expected him to come riding full gallop, the boy behind him on his horse, thinking he'd finally reached safety. Already Ingle had practiced his excuses for when the castle guards came running.\n\n\"Guy looked mad as a dog,\" he'd say. \"Started hollering for me to hand over my money, then sent the boy to do his dirty work.\"\n\nNo one would question him for killing two hungry thieves too stupid to know better. Even if they did, what would it matter? John Gandrem wouldn't challenge Arthur, not over something so petty as a dead farmer and his boy.\n\nWhile he held the rough blanket and looked about, he saw a man approaching. He walked on foot, leading his horse. A large sack lay slung across the saddle. Ingle raised an eyebrow at the sight. No boy, but what could someone be bringing to trade this late in winter?\n\n\"Slow down there,\" Ingle said, tossing his blanket back toward his horse and putting a hand on his hilt. \"Strange time for travel, don't you think?\"\n\n\"Pigs die when they die,\" said the man. \"Come to see if his lordship would like a fine meal tonight.\"\n\nThe cogs and wheels in Ingle's brain were never the most tightly fitted, but still they turned the words over, again and again, unable to get rid of a deep feeling of someone putting something over on him.\n\n\"Let me see it,\" he said. The man continued leading the horse right on by, forcing Ingle to jump in his way. Still the man didn't slow, and Ingle took several steps backward to keep from getting knocked over. At last he drew his sword and stood his ground.\n\n\"I said let me see,\" he said. \"I don't think that's no pig.\"\n\n\"If you say so,\" said the man. He pulled the sack off the horse with a grunt and plopped it to the ground. \"Just a small one, maybe good for John and some of his closest...\"\n\nWhile he talked, his hands messed with a tie at the end. The moment the knot came undone the sack was flung open, and out ran a boy who even Ingle knew had to be Nathaniel. The boy darted underneath his horse's legs and then shot straight for the castle.\n\n\"Fuck!\" Ingle shouted, turning to give chase. Matthew flung himself in the way. He wielded an old sword, recently polished but still timeworn and unreliable. Didn't seem to matter, though, for he wielded it as if it were Ashhur's blade itself and Ingle the dark-spawn of Karak.\n\n\"Outta the way!\" Ingle shouted, slashing with his sword in the hope of overpowering the unskilled farmer. He blocked, clumsily perhaps, but Matthew still banged his sword away. Instead of pressing the advantage he retreated, fully defensive. Behind him the little brat hollered like his lungs were on fire.\n\n\"He's gonna kill my pa, he's gonna kill my pa, he's gonna _kill_ him!\"\n\n_Damn right_ , thought Ingle.\n\nIngle feinted, smirked at how easily the farmer fell for it, and then cut from the other direction. The edge of his sword slashed into Matthew's arm, eliciting a cry of pain. Ingle swung again, lower, hoping to split his belly open. The man put his blade in the way just in time. The sound of metal on metal rang out, though there was something funny to it, as if one of their weapons wasn't flexing as it should. Ingle doubted it was his. Blood spilled down Matthew's arm, and Ingle saw the elbow below it shaking.\n\n\"Should have turned him over,\" Ingle said. Their eyes met, and for that brief moment he could tell Matthew thought the same. Behind him the guards approached, alerted by the boy. Fear bubbled up in Ingle's throat. Even if he lived, what might Oric do for such a screw-up? The least he could do was kill the stupid man who had given them so much trouble. He thrust, the tip nicking ribs before Matthew managed to parry it aside. Stepping closer, Ingle pulled his sword around, smacking it against Matthew's, which had pulled back to defend, and then he slashed once more at exposed flesh. Matthew fell back, but he was too slow, too unprepared for the maneuver. He was a farmer, not a trained fighter.\n\nThe sword cleaved through his shoulder and shattered his collarbone. In the distance Ingle heard Nathaniel scream. Matthew coughed once, his sword falling from limp fingers. His eyes grew wide. His lips quivered, his skin turning white. Ingle put a boot on his chest and kicked him back, freeing his crimson blade. The body clumped to the ground and lay still.\n\n\"Stubborn little shit,\" Ingle muttered as he wiped his sword clean on Matthew's leg.\n\n\"Drop your blade!\" ordered the two gate guards as they arrived. They had their swords drawn, and Ingle promptly obeyed. He gave a smile to Nathaniel, who cowered behind the two guards, tears on his face.\n\n\"What is the meaning of this?\" asked one as he picked up Ingle's blade. The other circled around and pressed the tip of his sword against his back to ensure he did nothing stupid. A hand reached in, yanking his dagger from his belt and tossing it to the dirt.\n\n\"I can explain, though Oric can do it better,\" he said. He pointed to the body. \"That man there's a kidnapper. I know it 'cause we been searching high and low for him. And that boy there, well...\"\n\nHe turned to Nathaniel, whose eyes looked like white saucers. He grinned, for he felt his lie building, the slow gears in his head turning.\n\n\"That's Nathaniel Gemcroft, back from the dead, as we always hoped.\"\n\nThe guards looked to the boy, whose skin had gone pale.\n\n\"I wasn't kidnapped,\" he insisted to the guards. \"I wasn't. He was helping me, and you let him kill him. Why didn't you run? I told you to run!\"\n\nHe was crying now, snot dripping from his nose. The first guard took him by the hand while the other grabbed Ingle by the arm and led him toward the castle.\n\n\"This is something Lord Gandrem will settle,\" said the guard. \"Stay quiet, and answer only when you've been asked directly, understand?\"\n\n\"Sure do,\" said Ingle. \"But don't squeeze so rough. You'll be treating me like a hero soon enough.\"\n\nThe four entered through the castle gates, then followed the emerald carpet into the main chamber. Uri and Oric were already there, in mid-conversation with John Gandrem on his throne. He sat up straighter at their arrival, clearly recognizing the boy.\n\n\"Nathaniel?\" he said, his mouth hanging open.\n\nIngle saw Oric glaring at him, his eyes ready to bulge out of his head. Not knowing what his captain might have been saying, he knew he should set things in motion, let his captain know what lie he'd created.\n\n\"I just saved him from his kidnapper,\" he said, loud and boastful. A mailed fist struck the back of his head, and for a moment his vision turned to yellow stars over a purple sky.\n\n\"You weren't addressed,\" said the guard behind him.\n\n\"My apologies,\" Ingle muttered.\n\nNathaniel rushed into the lord's arms, and in their comfort he sobbed uncontrollably. John patted his back and whispered comforting words, but his eyes remained drawn to where his missing arm should have been.\n\n\"Milord,\" said one of the guards, \"we found him attacking another who had traveled with the boy, killing him before we could arrive. We've brought both here for you.\"\n\n\"You told me you were searching for a man,\" John said, looking to Oric. \"Though you said he was merely a thief.\"\n\n\"And indeed he was,\" Oric said. Ingle beamed as his captain took his lie and ran with it. \"We suspected him of taking Nathaniel from one of Lord Hadfield's caravans. Never could we have hoped we'd find him here, of course. Perhaps he had come to issue ransom?\"\n\nNathaniel had begun shaking his head, and Ingle watched him carefully. A child's story against that of several men shouldn't matter, but one never knew. If only he'd keep his mouth shut, keep crying.\n\n\"I was told Nathaniel had died,\" John said. \"Learned too late of the funeral, sadly. I was told they'd been given a body, by you in fact, Oric.\"\n\nOric licked his lips.\n\n\"We suspected too late it was another child. The body was badly burned, you see. When I thought it might be a trick to throw us off the kidnapper's trail, we went looking. We learned nothing worthwhile, not yet, so we've been keeping our reasons a secret. Don't want harmful rumors flying about, nor giving Alyssa false hope.\"\n\n\"He's lying,\" Nathaniel said. \"Don't listen to them, he's lying! He was my friend, he killed my friend. He helped me!\"\n\nOric's voice dropped lower.\n\n\"Men do strange things to boys in their capture. Given time, he might twist their head around, make them friendly. He needs rest. This all's clearly been too much for the lad.\"\n\n\"He called the stranger his pa when running to us,\" offered one of the guards.\n\nJohn nodded, as this bit seemed to support what Oric was saying. He kept Nathaniel close, as if afraid he might lose him should he let go.\n\n\"What happened to your arm?\" he asked.\n\n\"They said my arm got sick and had to be cut off.\"\n\n\"Who is they?\"\n\nIngle's eyes widened. This might be tougher to explain. Maybe they could spin some blame onto that wife of his...\n\n\"She's just a filthy liar, that's all,\" Ingle said, ignoring Oric's glare. \"Probably cut the arm off to torture him.\"\n\nJohn's face darkened at this.\n\n\"She?\" he asked.\n\nIngle opened his mouth, then closed it. He didn't know how to respond to that.\n\n\"Meant _he_ ,\" he said lamely.\n\n\"We discovered a lady who claimed her husband had Nathaniel,\" Oric said, trying his best to mend the situation but clearly fighting a losing battle. John's eyes had narrowed, and he had a look like a snake ready to strike. \"That's how we knew to come here is all.\"\n\nJohn patted the boy on the head and leaned closer to him. He whispered something, too quietly for Ingle to hear. Nathaniel whispered something back. When finished, John sank deeper in his chair.\n\n\"Take them into custody,\" he said to his guards.\n\n\"Wait,\" Ingle shouted. \"You got things wrong!\"\n\nMen grabbed his arms and wrenched them painfully behind his back. It seemed like the very curtains had spawned armored guards. Oric reached for his sword, but the sheer number made him decide not to. One of the guards smashed his face with his fists, as if insulted Oric had even considered it.\n\n\"I wasn't there,\" Ingle shouted as he was yanked backward, but it only seemed to infuriate the lord more. \"Oric was, he saw it all, I was just doing what I was told!\"\n\n\"Bring him to me!\" John roared, standing from his throne.\n\nTwo guards dragged Ingle across the carpet, then shoved him to his knees. A fist grabbed his hair and forced his head to bow reverently.\n\n\"I want you to watch this,\" Lord Gandrem said to Nathaniel. \"You deserve it. With your arm as it is, you'll be living a hard road, and this is something you should always remember. This is how we treat the scum who dare strike against us.\"\n\n\"No,\" Ingle moaned as he felt his head pulled back. John held a beautiful sword, and he pressed its edge against his throat.\n\n\"Pull back the carpet,\" he said to his servants. \"I don't want to stain it.\"\n\nIngle felt hollow fear building and building.\n\n\"Please,\" he begged. \"I didn't do nothing, I didn't, I was just...\"\n\nThey lifted him up, and when they set him back down, his feet touched smooth stone.\n\n\"You killed a good man,\" John said.\n\n\"Says who?\"\n\n\"Says Nathaniel, and I trust his word over yours.\"\n\nThe sword moved. He felt pain, but when he gasped to scream, it was as if he'd been dunked underwater. His exhalation was a pitiful garble. His head swam, and he thought to faint, but still the guards held him upright. Until the darkness came, he watched John and Nathaniel watching him die. There was no mercy in either of their eyes.\n\nGuardsman Mick trudged up the road away from the castle, having drawn the shortest straw of the lot. One of the men's horses remained standing on the path. The other had wandered off, and he grumbled and hoped it wouldn't be far. He'd have to stable them both, work out ownership, probably even send one or both back to whoever had originally owned them. Bunch of hassles. Of course there was also the body, which needed to be stripped of any valuables and then disposed of.\n\nDeciding the horses could wait, Mick knelt beside the body, and he glanced around to see if anyone was watching. No one was, so he put a hand into the dead man's pockets, searching for loose coin. Of course, not all valuables needed to be handed over...\n\nWhen the dead body let out a groan, Mick startled, fell back on his rump, and nearly soiled his armor. He closed his slack jaw, put a hand on the man's chest, and leaned close. Both were weak, but he felt the tremors of a heartbeat and heard the soft hiss of breath.\n\n\"Goddamn, you're a stubborn one,\" he said, unable to believe it. He took to his feet and ran toward the castle, crying out for a healer to make his way to the gate.\n\n# CHAPTER\n\n#\n\nHaern sat atop the roof of the Eschatons' home and watched the sun dip below the wall. His elbow rested upon his knee, his chin on his hand. Tarlak's words haunted him, and no matter how hard he tried he could not shake them from his head.\n\n_I don't care who you think you are, or how good you might be_ , he'd said. _You're a danger to me, and a danger to my sister. I made you an offer, and I won't go back on it now, but you better put some serious thought into it, because otherwise you're just a renegade killer with a vendetta. There's no reason to house you then, no point. How many more will come storming through my windows, come kicking down my doors? I'm terrified the secret's out, Haern, or it will be soon. What do you expect me to do? Fight for you? Protect you? Give me a reason. Any._\n\nHaern had none to offer. His neck had flushed, and he'd shook his head. What could he say? _I'm sorry a mercenary broke into your home, hurt you, your sister, and your friends, all whiletrying to find me?_ He'd always thought he was so careful, but he'd slipped up as usual. What had Senke said? It didn't pay to be his friend. Yet again that remained painfully true.\n\nHe'd left, but lacked the heart to go far, so up to their rooftop he went. Part of it was because he didn't want to leave them, to say good-bye to Delysia and Senke forever. Part of him also feared that the giant man with the painted face might return, and if he did, Haern wanted to be there, waiting.\n\n\"Haern?\"\n\nHe looked down from the roof to see Delysia peering up at him.\n\n\"Will you come down?\" she asked. He shook his head. \"Then can you help me up?\"\n\nSighing, he grabbed the side of the roof with one hand and hung. He offered her his other hand and she took it, still trusting him for reasons he'd never understand. Using him as a guide she stepped on a window ledge, then with his aid jumped up to catch the roof. Once she had climbed all the way up, he swung himself up to join her.\n\n\"I think a set of stairs might be easier,\" she said, brushing off her priestess robes.\n\n\"And defeat the whole point of me coming up here,\" Haern said, immediately regretting it. Why should he snap at her? Her silence showed the comment stung, and he tried to think of something to say.\n\n\"Is Senke all right?\" he asked.\n\n\"I stopped his internal bleeding, and I sealed the wound best I could. He'll be sore for days, but yes, he'll be fine.\"\n\nHe walked back to the center of the roof and sat down. She sat beside him, and immediately he felt himself pulled back to the past. Would Thren arrive once more, a crossbow in his hand?\n\n\"I'm sorry about my brother,\" she said. \"He can be a bit of a hothead.\"\n\n\"No kidding. Why'd you join up with him, anyway? Mercenary work doesn't seem suited to you.\"\n\n\"Because he asked,\" she said, as if it should have been obvious. \"When I left the priesthood they gave me back my father's wealth from their safekeeping. It wasn't much, not after it'd been used to settle my father's estates and debts. We used it to buy this place. Was all we could afford.\"\n\n\"But here?\" Haern asked, gesturing about. \"On the Crimson? You deserve someplace better. Someplace safer.\"\n\nShe shrugged. \"My brother had a place he wanted, but the king refused to even hear his offer. It's no matter. I spent two years in the temple unable to leave for fear of Thren's anger. I'm used to keeping inside.\"\n\n\"It's not right,\" Haern said. She smiled at him.\n\n\"You living on the street is what isn't right. At least I have a warm bed, and a family to share my meals with. What do you have, Haern? What have you done over the years?\"\n\nHe thought of his deals, his rumors, his ambushes in the night and days spent sleeping with the homeless and destitute.\n\n\"I tried to stop my father's war. I tried to kill until there'd be no one left to fight in his name. I failed.\"\n\nShe took his hand and held it.\n\n\"Don't be so hard on yourself. We all make mistakes. You once wanted something more, to understand a life beyond what your father taught you. I think you still do. But you won't find a new life in vengeance, Haern, only sadness and loneliness. You grew up alone, I could tell that immediately. You change nothing by remaining so.\"\n\nSilence fell over them. He let it linger, trying to find the courage to ask what he needed to know.\n\n\"Do you hate me for killing?\"\n\n\"No. I am not so na\u00efve. I would like to live in a world where no killing was needed, but I fear I may never see it. I won't judge you for what you do, Haern. I can only try to be a light, and to shine as long as I can in a world that seems obsessed with darkness. If you need forgiveness, then know you have it from me, and from Ashhur. If you need guidance, ask, and I will do my best to answer. I'll heal your wounds, and pray for you before I lay my head down to sleep. I won't hate you. How could you ever think so?\"\n\nHe felt like a child, and he clutched her hand tight. She shifted so she might sit next to him, and her head rested against his shoulder.\n\n\"All those years,\" she said softly. \"Where did you sleep? Where did you live?\"\n\n\"On the streets,\" Haern said, feeling uncomfortable speaking of it with her, but forcing himself to anyway.\n\n\"Even in winter? How did you survive the cold?\"\n\n\"Thousands do so in this city every year. In that, I was nothing special.\"\n\n\"I doubt many of those thousands did so by choice. You _are_ special, Haern. Pretending otherwise is pointless. Why would you endure that? Why not flee, why not become anything else?\"\n\n\"Because I...\" Haern paused. He wanted to answer her truthfully, but that also meant knowing what he himself believed. \"Because I couldn't let my father win. And not just my father. The whole damn underworld that rose up to swallow you, me, and everything else good in this city. I learned the true face of this city, who operates it, who controls what. And then I did everything I could to slowly tear it all down.\"\n\nDelysia shifted closer to him, and he felt her arms wrap around his.\n\n\"I'm sorry,\" she whispered. \"You must have been so alone.\"\n\nAgain that discomfort, that shame for all he'd done. He didn't like it. He'd endured the years with a single-minded purpose, a desire for revenge that felt pure enough to justify his squalid conditions, his brutality, his entire purpose for existing. The last thing he wanted was to have light shine upon that darkness.\n\n\"Will you go out tonight?\" Haern asked her, wishing to change the subject. \"You and your mercenaries, I mean.\"\n\n\"No,\" Delysia said, shaking her head. \"Tarlak didn't understand the magnitude of what was going on when he first agreed. Our fault for not being part of the mercenary guild, I guess. We gave one night, and that is all Alyssa will get from us.\" She paused. \"Will you?\"\n\nHaern let out a sigh.\n\n\"I think I will. I have some part to play in all of this, whether I want it or not.\"\n\nShe pulled back and gently took his injured elbow into her hands. For the first time he truly looked at her, and he saw how tired she was, the whites of her eyes rimmed with angry veins. Still she closed her eyes, took a deep breath, and began praying. Soft light shone from her fingers, and he felt their healing magic pour into his elbow. Several minutes later she stopped. The pain had become a vague ache, like a sore muscle, but little else. He flexed the elbow twice, and it felt strong enough for combat.\n\n\"I should go,\" she said. \"It's not safe for me out here after dark.\"\n\n\"Please,\" he said, taking her hand. \"Just... sit with me a while longer. You're safe at my side. I promise.\"\n\nHe saw the look on her face, and he wished he could understand what it was she thought. Her hesitation was brief, and then she sat back down. Her arms wrapped around him, and he allowed his own eyes to close. It was only with her that he relaxed, all other times a coiled spring. But there, with her, he felt able to let it go. He had nothing to hide, and no reason to. Together they watched the sun sink farther, until it was nothing but a glow peeking over the wall.\n\n\"Help me down,\" she said at last. \"Senke wants you to see him. He seemed certain you wouldn't be staying tonight, yet would still be close. I think he knows you better than I.\"\n\n\"He understands the world I came from. Tonight will be worse, for everyone. I think he knows that.\"\n\nThe rest were eating when the two came in. Brug and Tarlak seemed to act as if he weren't there, but Senke greeted him warmly enough.\n\n\"Follow me,\" he said, leading Haern to a closet built into a space underneath the stairs. He pulled out a wooden crate, wincing at the effort. Feeling guilty, Haern ordered him aside and pried open the crate himself. Inside were an assortment of weapons, from knives to two-handed swords.\n\n\"I saw your fight with that mercenary,\" Senke explained. \"That cloakdance thing you did was something special, puts Norris Vel to shame, I'll tell you that. But your swords weren't right for it at all. Here, take these.\"\n\nHe lifted a pair of weapons out and handed them over. They were long and slender, with the ends gently curved.\n\n\"These sabers are designed for slashing, and should do well with how you're always moving. The points are sharp, but you'll still have a hard time thrusting through heavier armor. Same with heavy chops, but I have a feeling brute force isn't your usual method, given your speed.\"\n\nHaern swung the swords about, getting a feel for their weight. They were lighter than his previous swords, with a slightly longer reach. Their grips were comfortable, making them feel natural, like an extension of his body when he wielded them. He could tell they were expertly made.\n\n\"Thank you,\" he said.\n\n\"Don't thank me. Thank Brug over there. He made them.\"\n\n\"Just don't break 'em,\" Brug muttered from the table.\n\n\"Both sides will be out for blood tonight,\" Senke said, leaning against a wall and holding a hand against his stomach. \"You sure you have to go out? People will kill each other just fine without your help.\"\n\nHe realized they were all looking at him, either blatantly or from the corners of their eyes. In his heart he felt something harden, as if he wanted to prove them wrong, to show he didn't care what they thought. But what did it matter? Why _did_ he go out? What might he accomplish? Deathmask's biting words returned to him, mocking him in his mind.\n\n_You spent five years trying to single-handedly conquer the thief guilds. Yet you want to mock_ my _imagination?_\n\nSomething clicked in his head, several pieces tumbling together as the idea took form. He looked to them, then out the window. No, there was nothing out there for him, not this night. Come the day, he'd find Deathmask, assuming he still lived. Perhaps there was a chance to have a legacy that was the opposite of his father's.\n\n\"You know,\" he said, feeling a great weight lift off his shoulders. \"I think I will stay here tonight, if you'll have me.\"\n\n\"Pull a seat up at the table,\" Senke said with a smile. \"You bet your ass we will.\"\n\n\"Are you prepared to do what must be done?\" Deathmask asked her.\n\n\"I am,\" said Veliana.\n\n\"You'll have to kill many of them. They were once your friends, your guildmates. Maybe you even considered them family. They won't understand, and their loyalties are anyone's guess. This is Garrick's guild, and you're nothing but a feeble woman who got in his way. Last time I ask. Can you stick a knife in them, every one of those familiar faces?\"\n\n\"Not so familiar anymore,\" she said. She tapped her sickly, bloodied eye. \"Too many hate me for this. I've heard their whispers, their insults of my ugly mark. They never loved me, not like they loved James Beren. This guild may or may not be mine, but more than ever I know it should not be Garrick's. If he sold his soul to Thren, then he betrayed every shred of James's memory. Anyone who stays at his side is no friend of mine.\"\n\nDeathmask smiled at her.\n\n\"I want to do something for you,\" he said. \"This will take just a moment, but I hope you'll appreciate it.\"\n\nHe put a finger to his eye, the same as Veliana's injured one, and then whispered the words of a spell. They seemed simple enough, and then came the change. His iris bloomed from a dark brown to a bloody red.\n\n\"This is what I think of your ugly mark,\" he said. \"I'll proudly bear it so long as you stay with me. I will never cast aside your loyalty, for I've been cast aside enough in my own life.\"\n\nVeliana felt strangely touched by the gesture.\n\n\"One day,\" she said, \"I hope to believe you.\"\n\nThey turned their attention to the unassuming building before them. The rooms appeared dark, but both knew of the expansion belowground, no doubt housing the last remnants of the Ash Guild. A few men and women wandered past them on the streets, several with dead eyes and drunken gaits. To Veliana it seemed the entire city was suffering a massive hangover, a crude comparison given how many of her kind had been mercilessly butchered. So far Deathmask hadn't explained how he planned on dealing with all the mercenaries, but she had no choice but to trust him. Patting her daggers, she told him to lead the way.\n\n\"Keep your hood low,\" he told her as they approached the door. \"Surprise is everything. Theatrics can turn even the most ordinary of foes into something fearsome, and you are no ordinary foe.\"\n\nA single thief leaned against the door, looking like he'd been up for two days straight. Through bleary eyes he watched their arrival, recognizing Deathmask when they were almost within striking distance.\n\n\"Hey, we thought Thren\u2014\"\n\nVeliana cut his throat before he finished the sentence. As his body fell, she glanced to Deathmask, and her meaning was clear. _Look what I can do. Do not fear my loyalty. They are no longer friends of mine._\n\n\"Atta girl,\" he said, his mismatched eyes sparkling behind his mask.\n\nWhen she tried the door, it was both locked and barred. Deathmask gently moved her aside, put his hands upon the wood, and closed his eyes.\n\n\"Theatrics,\" he whispered.\n\nHis hands shimmered between red and black, and then the door exploded inward in a great shower of splinters, accompanied by a shock wave that thumped against Veliana's chest with enough force to make her catch her breath. Deathmask stepped through the dust and debris into a small entry room. Two men sat in chairs on either side of the doorway, their hands over their faces. Specks of blood dotted their clothes: damage from the shrapnel. Veliana rushed the one on the right, thrusting a dagger into his chest before he could react. Deathmask waved a hand at the other, who suddenly dropped to his knees, gagging. Before she could see the total effects of the spell, Veliana stabbed his heart.\n\n\"Sometimes quick is better,\" she said.\n\nSo far it seemed their arrival had gone unnoticed by those farther in, hidden behind a second door. Deathmask pushed it open, and they stepped into the last remnants of the Ash Guild, all gathered from the various corners of Veldaren. There were twenty of them, sitting on chairs and pillows and looking miserable. Veliana felt both anguish and elation at seeing Garrick among them. Part of her had hoped he'd died in the fire, for he deserved nothing better, but at least his survival meant that he would be hers, all hers.\n\n\"Members of the Ash,\" Deathmask said, screening Veliana with his body. He wanted to maximize the impact of revealing her, she knew. A smirk crossed her lips. They all thought her dead, Garrick included. How his mouth would drop, how his eyes would go wide... All around, the thieves stood and drew their weapons, for though Deathmask was one of them, there was something dangerous about his arrival, in the way he walked, the way he addressed them.\n\n\"You,\" Garrick said, pointing a shaking finger. \"You turned the Spiders against us, didn't you? Why else would they let you live?\"\n\n\"I am not the one who went into bed with the spider thinking I might not get bitten,\" Deathmask said. \"This destruction is your doing, all your doing. Listen to me, guildmembers! He sold your souls to Thren Felhorn, all so he might sleep well at night.\"\n\n\"You lie!\"\n\nAbout a third of the men around them were exchanging glances, and their daggers and clubs lowered. Veliana watched and waited. She had to be fast. The first attacker needed to die immediately if she was to discourage the rest. When it came to a battle of personalities between Deathmask and Garrick, there would be no contest. At some point Garrick would call an end to it before he lost completely.\n\n\"How else would you have maintained leadership?\" Deathmask asked. \"Why else would the guilds have made peace with you, even though your position was weak? Weeks ago you made your pact, and one by one the other guilds realized and left you alone. Only the Hawks attacked, and only once. Thren punished them severely for that, didn't he?\"\n\nMore mumblings about them. A couple glared at Garrick. These were the rumblings of treason, Veliana knew. Normally such accusations would be whispered from ear to ear, allowed to fester and grow. But the Trifect had pressed too hard. If they were to survive, they needed new leadership, and now.\n\n\"I don't know what you're talking about,\" Garrick said. He had drawn his dagger, but it remained at his side, as if he was afraid to even point it at Deathmask.\n\n\"Come now. We all know whose guild this truly was, before it was Thren's. It was Veliana's, not yours, never yours. That is why you wanted her dead.\"\n\nLouder grumblings, though many were disparaging her. She felt anger simmer in her heart. Even now they would deny her work, her sweat, her toil. The gods damn them all.\n\n\"She died because she tried to kill you, that's all,\" Garrick said.\n\nVeliana stepped to Deathmask's side and pulled her hood back. She smiled, and the look on Garrick's face was everything she'd hoped it'd be.\n\n\"I never died,\" she said. Her voice was soft, but even a whisper could have been heard in that suddenly quiet room. \"But you will, you traitor. You sold your soul to Thren. I can never forgive you.\"\n\nShe flung herself at him, not caring for her safety, nor the guild's greater numbers. She would have his head, and this time no one would stop her. Garrick cried out for aid, and several thieves jumped in her way. Spinning away from a club, she gutted one on her left, rolled along the ground, and hamstrung another as she stood. The one with the club tried to smash her back, but she twirled again, her spine bending at an unnatural angle so the swing passed above her breasts. And then she was up again, stabbing him repeatedly, kicking away his corpse with seven bleeding holes in his chest.\n\n\"Make your choice!\" Deathmask cried out. It seemed many had. They turned on the others, striking at those who moved toward him. The room was now in chaos, and within it Veliana thrived. She kicked out the legs of one rushing for Deathmask, burying a dagger in his ribs as his body hit the ground. Pulling it free, she flicked blood off it toward Garrick, who stood with his back to the wall, his dagger held before him.\n\n\"Where's Thren to protect you now?\" she asked as she stalked him, her daggers hungry in her hands. \"Where's the men who would rather rape me than serve under my leadership? Where's your _guild_ , Garrick?\"\n\nA blinding flash burst from behind her, a spell of some sort from Deathmask. In its light she rushed Garrick, her knee leading. It slammed into his crotch while she swatted away his dagger. Her other dagger's hilt struck his forehead. She rammed an elbow into his mouth, then slashed across his face when she pulled back. Blood spurted from a gash across the bridge of his nose. His cry of pain was a garbled, weak thing.\n\n\"Now you're the example,\" she whispered to him. She stabbed her dagger into his throat, twisted it left, then right, and finally yanked it free. Blood splashed across her chest, but she didn't mind. At his death much of the chaos slowed, for it seemed there was little point left in fighting. She glanced around and saw all eyes upon either her or Deathmask. Only ten remained of the initial twenty.\n\n\"Those who would betray their loyalties deserve nothing less,\" Deathmask said, kneeling beside Garrick's body. He put a hand on the head, which burst into flame. The body blackened and smoked, and in seconds it was nothing but a pile of ash. Taking a handful, Deathmask stood and flung it into the air. It revolved around his head, hiding his visage, making him look like some strange monster instead of a man.\n\n\"I am the Ash now. None of you are worthy of my leadership. You killed for me, and for that I spare your lives. Begone. Throw down your colors, or prepare to have them stained with your own blood.\"\n\nIt seemed none there had the will to challenge the blood-soaked Veliana and her master. Her heart felt a pang at their exit, feeling like the last remnants of the guild she and James had built were gone, but Deathmask had promised her something greater, and she had to trust him. She scanned those exiting, looking for a set of faces, men who had remained out of the fight like the sensible opportunists they were.\n\n\"Nien, Mier,\" she said as they left. \"You two, stay.\"\n\nThe twins looked back. They had pale skin, dark hair, and brown eyes that seemed to twinkle with subdued amusement.\n\n\"Yes?\" they said.\n\nDeathmask approached them, and he offered his hand.\n\n\"Veliana has vouched for your skills. Would you remain with me, and fight not for the Ash Guild that was, but for what it might yet be?\"\n\nThe two glanced about the room, as if to point out the obvious to them.\n\n\"What guild?\" Mier asked.\n\n\"There are only us four,\" said Nien.\n\n\"And as long as the four of us live, there will always be an Ash Guild,\" Deathmask said. \"You have seen what we can do. Join us. We need your strength tonight. The mercenaries must be shown that we will not roll over and die for them.\"\n\nThe twins shared a look, and Veliana swore some sort of mental conversation was going on between them that she was not privy to. Then they accepted Deathmask's offered hand and shook it.\n\n\"Why not?\"\n\n\"Could be fun.\"\n\n\"Indeed,\" Deathmask said, grinning behind his mask of cloth and ash. Veliana shook her head, wiped the blood clean from her daggers. She spit on what little was left of Garrick's remains.\n\n# CHAPTER\n\n#\n\nIn the dark of Felwood's dungeon Oric shivered. He sat on a wood cot and listened to the water drip. Where it dripped he didn't know. To pass the time he'd tried to guess, but the echo always seemed to change on him. His cell was completely dark, without a single shred of light. He'd scoured the floor with his palms, but everywhere he touched was wet, and a drop never landed upon him. Still, the search did better to pass the time than thinking about his fate. Anytime he thought of that, or of how long he might be in total darkness, his head swam and his heart lurched into his throat.\n\nHe'd tried talking to anyone else, a guard or fellow prisoner, but his voice only echoed through the emptiness, never answered. For some reason that always made it worse. Without light, company, or a single meal, time was meaningless. At least two times he slept, and in his dreams he saw color, women, friends. He wished he could sleep more often.\n\nA loud creak startled him from a doze. Heavy footsteps echoed down the hall. Orange and yellow flickered along the walls, at first a wonderful sight but soon painful in their brightness. Holding a hand before his eyes to block the pain, he felt a wretched sight as John Gandrem stepped in, soldiers at his side.\n\n\"Stay seated,\" he said, \"otherwise my guards will open you up in many places.\"\n\n\"But a man should always rise at the arrival of a lord,\" Oric said. He held back a cough. His voice felt scratchy, dry. He remained sitting despite his protest. Given how light his head felt, he thought he'd pass out if he stood too quickly.\n\nJohn crossed his arms and looked down at him. In the yellow light John's skin seemed like stone, old and unmalleable. His eyes looked even worse. For all the stories Oric had heard of Lord Gandrem's kindness, he'd yet to hear a story describe those eyes. Mercy didn't belong in them, not now, maybe not ever. Perhaps this was the lord of the dungeon, a different man from the lord of Felwood.\n\n\"Before we start, there's a few things you should know,\" Gandrem began. \"First, I have talked extensively with the boy, Nathaniel. His story is consistent, and most damning. Second, the man Ingle thought he killed, the farmer Matthew, is not dead. Third, my men have already worked over Uri, and how he _sang_ , Oric. I know what you did to that farmer's wife. The idea that you could claim they assaulted a caravan and held Nathaniel hostage is laughable.\"\n\n\"I never claimed it. That was Ingle's stupid idea.\"\n\nThe faintest hint of a smile stretched at Lord Gandrem's lips, but then vanished.\n\n\"Perhaps. A shame I cut his throat before I could tell him the farmer lived. I plan on ensuring Matthew is well rewarded, as is his wife. But the question remains now, what do I do with you?\"\n\n\"Well, between the rope and the ax, I think I'd prefer the ax.\"\n\n\"In time, Oric. In time. See, my biggest problem is not with you, but with your master, Arthur Hadfield. Mark Tullen visited me before meeting with you and Nathaniel in Tyneham. I know he was escorting the boy back, and I'm not a damn fool. Everyone knows he was a potential suitor of Alyssa, and Arthur wanted him gone. Proving that, however, is another matter.\"\n\nHis soldiers rushed in and grabbed Oric by either arm. Up went his hands, back and above his head. Chains rattled, and then he felt clamps tighten about his wrists. With him safely shackled, John sat on the small cot and pulled his heavy coat tighter about him.\n\n\"Now I don't mean proving it to just Alyssa,\" he continued. \"She's a bright gal, and there's too much here for her to ignore. However, Arthur's long held those mines at the edge of my lands, always refusing to pay taxes. I want those lands. It is my knights that have protected them. It is my lands his traders travel across to Veldaren. It is on my roads he ships his gold and sends for his supplies. By all rights they should be mine, and would have been if not for the Gemcrofts.\"\n\n\"What could I possibly have to do with that?\" Oric asked. His shoulders were starting to cramp, and he had a creeping feeling it was about to get a whole lot worse... especially if they left him like this for several hours, if not days.\n\n\"King Vaelor has rejected every claim of mine for taxes, no doubt because he fears the Trifect more than he fears me. That, and their bribes. But Arthur has no heir, and he's never written a will in case he does have a son. If he dies as such, his lands will be joined with the closest lord's.\"\n\n\"You,\" Oric said, starting to understand. If Arthur died before marrying Alyssa, then the land would become John's. \"But you aren't the one holding Arthur. Alyssa is. And if she finds out...\"\n\n\"If she finds out, then she'll force him to return the lands to her,\" said John. \"But she'll only do that if she discovers what happened. Now do you understand? I hold all the control here. Arthur won't dare challenge me about your deaths, for the truth gets him killed. He can only keep his mouth shut and pray for the best. I, however...\"\n\nOric tried to flex his back, but he was held too closely to the wall. He rolled his neck back and forth, and it popped loudly. Minutes. It'd only been minutes, but he already wanted out. Far better to shiver freely on the floor than sit unable to move half his body. He didn't want to think about hours. Or days. Or, gods forbid, years.\n\n\"I hold Arthur's life in my hands, and yours as well,\" said John. \"I might have used Uri for this, but he didn't take well to my low servants' questionings. We had to ensure he spoke the truth, of course. So it is down to you. Where do your loyalties lie, Oric? You deserve death, we both know this. What might you do to be spared that fate? Help me, or otherwise... you said it yourself: rope or ax.\"\n\nOric couldn't believe his luck. He'd thought that he'd have nothing of value to offer, but if he could roll on his former master and somehow escape with his own head...\n\n\"What is it you want from me?\" he asked.\n\n\"I need you to kill Arthur before he can discover things have gone awry, and before Alyssa might realize his involvement. Before you do, I want you to sign a statement I might use in the king's court detailing every bit of your, and Arthur's, involvement.\"\n\n\"What do I do once I kill Arthur?\" Oric asked. \"What happens then?\"\n\nThis time the lord did smile.\n\n\"A man of your talents? Surely you could disappear into a crowd afterward, and then, well... Ker's a long way away, and Mordan even farther. I also hear the sailors in Angelport often need a good sellsword aboard their ships.\"\n\n\"What about the farmer?\"\n\n\"He's injured, and my healers say it will take several days for him to recover. We should have this concluded before he can be of any concern. Besides, these matters are far above his station, and his word in any court would be suspect at best, being just a low-birth simpleton.\"\n\nIt couldn't get any better. Oric was hardly afraid of a little travel, and killing Arthur would be no skin off his nose. Given the nature of his mission, it'd only be natural they go somewhere quiet to talk, and after a bit of knifework he'd have his freedom.\n\n\"I'll do it,\" Oric said.\n\n\"Excellent. We'll claim you escaped the dungeon after we extracted your confession. When you went to Arthur, he tried to cut ties and claim everything was your plan. You killed him and fled, and to where, I don't want to know. Is this understood?\"\n\n\"It is.\"\n\n\"I'll have a servant down here with candles and parchment. Tell him everything you know, every possible detail. Farewell, Oric.\"\n\nHe stood and left, and to Oric's great relief John had his guards remove the clamps at his wrists before he went. True to John's word, an elderly man with a crooked nose arrived.\n\n\"The beginning, please,\" he said, dipping his quill into an inkwell.\n\nSo Oric told him, starting with Arthur's thieving from the Gemcroft mines and smuggling the coin to the Serpent Guild for laundering.\n\n\"Will you truly let him go?\" asked one of the soldiers walking alongside John, a veteran and trusted knight named Cecil.\n\n\"Of course not,\" the lord snapped. \"The Gemcrofts have had those mines tied up in legal protection for over a century. I could wipe out half their family, their extended family, Arthur included, and they'd still find someone besides myself to be legal heir. To be truthful, I don't even want them. Giant hassle, all of it.\"\n\n\"Then why the ruse?\"\n\n\"I need his confession, quick, truthful, and most importantly damning to Arthur. I'll be sending you to Veldaren with that confession in your hands, along with a letter of my own.\"\n\nCecil bowed to show he was honored.\n\n\"Will we not be bringing Nathaniel back to his mother?\" he asked as they exited the dungeon, doused their torch, and headed toward the mess hall.\n\n\"Nathaniel was already abducted once on the road, and when he should have been in my care, no less. My own damn foolishness for trusting that snake, Arthur. I will keep him here, and in safety, until Alyssa comes for him. But you... you can let Alyssa know of his survival. She's a bright lady, but Arthur has a way with words, and who knows what lies he has spun about her to protect himself? That confession should burn them all away, and if she is who I think she is, she'll deal with him accordingly. Let me get some food into these old bones, and then I'll pen my letter. When you have mine and Oric's, ride hard to Veldaren. If Arthur suspects something's amiss, I fear he will make a move against her.\"\n\n\"Of course, milord,\" Cecil said, bowing again. \"What of Oric?\"\n\nJohn grinned.\n\n\"He said he preferred the ax, so prepare the gallows. He deserves nothing, not even the choice of his own death. Let him hang from my walls, the honorless bastard.\"\n\n# CHAPTER\n\n#\n\nFor a second night Alyssa watched the city burn from her window. There were more fires now, at least seven she could see. She wondered what it meant. Were her mercenaries finding more of the thieves' ratholes? She held an empty glass in her hand, and she toasted the stars, which were hidden behind a blanket of smoke.\n\n\"You deserve better, Nathaniel,\" she whispered.\n\n\"I too can think of better homages for your son,\" Zusa said, having slipped inside without making a noise. Alyssa had trained herself not to jump at Zusa's voice, but still she quivered, her nerves frayed.\n\n\"Perhaps,\" she said as the woman came to her side. \"But this is the best I can do.\"\n\n\"You lie to yourself. This is for you, your hurt. Do what you must, but do it in truth, and bear the burden proudly.\"\n\n\"Enough,\" Alyssa said, hurling her glass against the window. It shattered, small flecks of red wine dripping down to the floor. \"I don't need speeches. I don't need your wisdom. I need my son back, my little boy...\"\n\nShe pressed her head against the glass and refused to break. As the tears ran down her face she stared at the distant fires and tried to revel in the bloodshed they represented. But she only felt hollow.\n\n\"As you wish,\" she heard Zusa say.\n\n\"Stay,\" she whispered, knowing the faceless woman would leave her.\n\n\"As you wish.\"\n\n\"Tell me, how goes it out there?\"\n\nZusa gestured to the city. \"The thieves are ready, more than they were last night. They started those fires, and they've killed many innocents. I think they're hoping to turn the people against the king, and it might work. If Veldaren is an altar, you've covered it in blood as a sacrifice to your son. I don't know which god will honor it, though. Perhaps they've both washed their hands of this miserable city.\"\n\nAlyssa nodded. It sounded right. She had opened up her coffers and replaced their stores of gold with bodies of the dead. Was it a fair trade? Could it ever be?\n\n\"What about the one who killed my son?\" she asked.\n\nZusa thought of her fight with him, and how she'd been stopped at the last minute by Veliana's insistence. What the other woman wanted with him she didn't know, nor did she care. Veliana had sacrificed their friendship to take him from her and Alyssa. If that was all she was worth, then so be it. But she didn't want to let Alyssa know how close she'd come to taking the Watcher back to her, as much as she hated to lie. So she stretched the truth as far as it would go.\n\n\"I fought him,\" she admitted. \"But he escaped before I reached victory. Where he is now, I do not know.\"\n\n\"Did you hurt him?\"\n\n\"Yes. I drew blood.\"\n\n\"Good. At least that's a start. Will you go out again before the night is over?\"\n\nZusa put a wrapped hand upon the glass and stared out. Slowly she shook her head.\n\n\"No. There's nothing out there, just men killing one another. I think even the Watcher has stepped back to let it run its course. May I leave?\"\n\nAlyssa nodded. When Zusa was halfway to the door she stopped and glanced back. She looked as tired as Alyssa felt.\n\n\"If I may be so bold, I have a request. Make this the last night, Alyssa. Killing doesn't cover the pain of loss. It'll only drain you, leave you empty. I do not pretend that these men deserve mercy, not all of them... but this path you've chosen will only lead to ruin. Even if it does, I will follow you into it, even unto death.\"\n\n\"I don't know if I can stop this,\" Alyssa said.\n\n\"You're strong enough, Lady Gemcroft. I know it.\"\n\nAnd with that she left, the door closing with a soft click of wood. Alyssa watched the fires, but it seemed she could no longer keep her mind upon them. She felt tired, and she often thought of returning to her bed. She hadn't slept well lately, maybe a few hours at most. All throughout the day men and women had come to her, claiming damages for what her mercenaries had done. Near the end she had paid every claim, whether its validity could be proven or not. She hadn't had the energy left to care. At last she'd delegated the responsibility to Bertram.\n\nAs if thinking his name had summoned him, the old man opened the door, then knocked on it after the fact.\n\n\"Yes, Bertram?\" she said, keeping her face to the window so he couldn't see her tears. \"If this is about the cost of damages and repairs, spare me. I am in no mood, and you should be asleep in any case.\"\n\n\"As should you as well,\" Bertram said, quietly approaching. \"But it seems sleep is a difficult thing for most of us in these troubled times. I've come to discuss a different matter.\"\n\n\"And what is that?\"\n\nShe could see his reflection in the glass, and she watched him chew his lower lip while he clasped his hands behind his back.\n\n\"I've gone over the mercenaries' pay, along with our promised payments to the citizens, and the total is...\"\n\n\"I told you I had no interest,\" she snapped.\n\n\"It's more than that,\" Bertram said, doing his best to keep his tone soothing and controlled. \"I did as you requested, and treated cost as no object, but I feared the folly, and as I feared, it has come to pass. The cost has been unbelievable, especially with how many have died. The guild requires extra compensation for men who fall in the line of duty, for wives, children, mistresses, and the like. Plus the fires were more than we expected, and you have accepted blame for nearly all cases.\"\n\n\"Your point?\" she asked.\n\nHe stood up straighter as he spoke.\n\n\"We have nothing left, Lady Gemcroft. Your fortune is spent. Unless we delay payments for several years, we will default on at least a third of the mercenaries, and aid with only half of the repairs.\"\n\nHer mouth dropped open.\n\n\"Are you certain?\"\n\nHe nodded. \"I have checked multiple times.\"\n\nShe saw the fires burning before her, and suddenly they took on a different meaning.\n\n\"Is that counting tonight?\"\n\n\"For the mercenaries, yes, but not necessarily any extra for the dead, since I cannot know for certain until tomorrow. But not the fires, no. I can only assume the worst.\"\n\nShe felt her whole world spiraling away from her. How could all her wealth have vanished so quickly? It didn't seem possible. Of course the mines in the north had lessened in their production over the past year, but still, what of their contracts, their trade? Had the thief guilds truly destroyed so much?\n\n\"All is not lost,\" he said, sliding closer and wrapping a comforting arm around her shoulders. \"I have thought of a way to help ease this burden. We still may delay some payments, especially for those who died without families or still have means to survive.\"\n\n\"What is it?\" she asked.\n\n\"Lord Hadfield has an extensive amount of wealth saved up, an amount he hoped to bequeath to an eventual heir. If you were to marry, he would assume your debts. I have already discussed the matter, and he is willing to do his part to help you move on from your son's death, including this debacle you have unleashed upon our city.\"\n\nShe crossed her arms and held them against her as if she were cold. Could she marry Arthur? True, he'd been kind since he arrived, and he seemed to have no intention of leaving. They'd shared a bed, even, and with him she did feel some measure of comfort. Her heart ached for Mark, but he was gone, as was her son. Should she continue to let that haunt her? Maybe Zusa was right. Maybe it was time to end all of it.\n\n\"When?\" she finally asked.\n\n\"It will need to be soon, especially given how deep our debt is. If we make significant payments we can hold our debtors at bay, as well as convince them we have every intention of making good on our promises. Leon Connington will help if you ask. No member of the Trifect would let another fall so far as to bring shame upon them as well. Perhaps in a few days I can have the ceremonies prepared, and all the proper documents written and presented before the king's council for approval.\"\n\nThere was something chipper about his tone that scraped against her spine like metal on glass.\n\n\"Enough,\" she said. \"Start on whatever you must, but don't tell Arthur anything other than that I am open to the idea. He should at least be the one to propose, not my father's old advisor.\"\n\nBertram smiled. \"Quite right, quite right. Good night, milady. Perhaps soon you will finally have pleasant dreams.\"\n\n\"Good night, Bertram.\"\n\nOnce he was gone she blew out her candles, returned to her bed, and tried to sleep. She couldn't.\n\n\"Arthur Gemcroft,\" she whispered, moving the name about her tongue as if trying to taste it. He would adopt her last name, as all men and women did when entering into a family of the Trifect.\n\n\"Arthur Gemcroft. Arthur... Gemcroft.\"\n\nIt had a ring to it, she had to admit. She'd put off marriage for long enough. It was time for her to be practical. Still, as much sense as it made, it gave no comfort, and she tossed and turned until the morning light shone through her curtains, falling upon her haggard face and bloodshot eyes.\n\nThe wound in Ghost's leg was worse than he'd originally feared. Without much reason to join the night's slaughter, he'd instead languished in his favorite tavern, drowning himself in alcohol to dull his pain. He'd passed out, and no one had had the nerve to wake him. Finally he'd returned to his squalid inn, carefully put his weapons aside, and then collapsed onto his bed. The windows had no shutters or curtains, and the light streamed in upon his face. The dressings on his leg, haphazard at best, had soaked through, and he feared it was now infected.\n\nAs he lay there he felt the pain crawling its way up his thigh, as if it were a spider scurrying through his veins. If he didn't do something soon he'd lose use of the knee, if not the entire leg. He wouldn't be the best anymore. He wouldn't even be a threat. A man of his strength, his skill, was not meant to be a cripple. Surely the gods did not intend such a fate for him.\n\nThe gods...\n\nGhost rolled off the bed, putting all his weight on his good leg. Damn that priestess. The Watcher had been his, thoroughly beaten. He didn't give a shit that he'd appeared wounded and weaponless. Assassinations, by their very nature, were unfair. But he'd been a fool to let her tend to the wounded Senke. He'd thought her too young to be a threat, but how wrong he'd been.\n\n\"It's not the big snakes you need to fear,\" he remembered a friend telling him once as they crossed the grasslands toward Veldaren. \"It's the tiny ones who carry the real venom. Put that on your darts if you want a sure kill.\"\n\nThe priestess was the tiny snake, the insignificant one among the wizard and warriors. Stupid. Stupid!\n\nHe took a hobbled step toward a large dresser full of clothes and outfits. Leaning against it for support, he yanked out the top drawer, letting it crash to the floor. Reaching into the hole where it had been, he pulled out a small bag of coin. It'd have to do. Retrieving his swords, he opened the door and stepped out into the painful morning light.\n\nTwice on the way to the temple he collapsed, his knee unable to bear his weight. A black bruise swelled across it, and the pus where the Watcher's sword had cut him was turning green. No one paid him any attention, the crowd flowing to either side as if he were an overturned cart, or a dead body.\n\nReaching the temple offered Ghost little comfort. He still had to climb the many steps, a fact made no easier given their ruined state after the spell their high priest had cast the previous night. After the first few Ghost gave up any pretense of pride. He sat down upon them, put his back to the temple, and pushed himself up one at a time. At the top he braced himself on a pillar and used it to keep his balance while he stood. Men and women gathered about the wooden doors, crying out for aid. No doubt the temple was swarming with people inside as well. He'd seen the fires, heard the sounds of combat flowing up and down the streets. The thieves had put up a fight this time, firing arrows from the rooftops and preparing a hundred ambushes.\n\nHe pushed his way through them, the wound in his leg meaning nothing to his enormous arms. Most turned to glare at him, then decided otherwise seeing his size and painted face. Once inside the temple he leaned against a wall and surveyed the madness. Priests and priestesses of all ages were running about. They looked like white bees zipping from flower to flower. They'd kneel, exchange a word, maybe say a prayer, and then move on. The older ones lingered, and he saw many put their hands on wounds and whisper words of prayer to Ashhur. White light would cover their hands, sometimes weak, sometimes strong, and then sink into the wound. That was what he needed. Faithful or not, he wouldn't deny that the priests had their uses. But he wouldn't risk some juvenile treating him. He wanted a master, someone who knew what he was doing.\n\n\"Where is Calan?\" he asked as he grabbed an elderly priestess, her face a circular web of wrinkles.\n\n\"Busy,\" she said, giving him a reproachful glare. She didn't seem the slightest bit unnerved by his size or skin.\n\n\"Bring him,\" he said, refusing to let go despite her tugging to.break free. \"He owes me one. Tell him Ghost is here, and that I'm the one who saved this damn temple from the mob two nights ago.\"\n\nShe looked him up and down, and though it seemed impossible her frown grew deeper.\n\n\"I'll see if I can find him,\" she said, then hurried on her way. He leaned back against the wall and closed his eyes. If he could ignore every noise, every visual distraction, he could focus on the pain, and doing so made him feel better. His temples throbbed with each pulse, but he kept it under control. He felt the pain's limits, how far it stretched throughout his leg. Time passed, and he was dimly aware of it.\n\n\"I see you've returned,\" a man's voice said. Ghost stirred to see Calan standing before him. He looked tired, with dark circles under his eyes and his smile forced. \"May I ask why you've given us the pleasure?\"\n\nIn answer, Ghost pulled up his pant leg to reveal the wound. He winced when he saw it himself. The purple bruise had spread, and the green pus was filling up his bandage. Calan's smile immediately vanished, and he grabbed Ghost's arm.\n\n\"This way,\" he said. \"You need a bed, now.\"\n\nGhost wanted to protest but didn't. He'd hoped for a bit of healing magic, and then off he'd be. Instead he obeyed without argument, for his head ached, his stomach was doing loops, and he felt intensely drained. It was as if the pain were a fire burning away his energy. Calan led him through the maze of people and pews. His head swiveled, but he saw no opening, no space available. Muttering, he turned Ghost toward the back, then through a door to a modest room. It had a small desk, a bookshelf, and a bed, and it was that bed Calan set him upon.\n\n\"My quarters will have to do,\" the priest said. \"Though I fear the bed might be small for you.\"\n\n\"A bed's a bed,\" Ghost mumbled.\n\n\"I suppose you're right.\"\n\n\"Here.\" He tossed the small bag of coins to pay for his treatment. \"Save my leg, will you?\"\n\nCalan rolled up the pant leg, carefully folding it over and over until it was up to his thigh. Ghost closed his eyes. For some strange reason he didn't want to watch. He didn't want to understand what the priest would do, or what its implications were. Gods were for other people, not him. Gold and killing, that was god enough for him. He heard whispers, undoubtedly prayers, so he leaned his head back and tried to relax. The pain continued to throb, its reach growing. He felt it down to his shin, as if instead of giving him a single cut, the Watcher had beaten and smashed his whole leg with a club.\n\nA strange sound met his ears. It was like a soft breeze blowing past the entrance of a cave, yet deeper, fuller. Even through his closed eyelids he saw the light flare. When it plunged into his leg it was like fire. He clutched the sides of the bed and clenched his teeth. His nostrils flared as he breathed in and out.\n\n\"The infection is deep,\" he heard Calan say. \"Bear with me, Ghost. I know you're strong. You will endure.\"\n\nMore prayers, and another burst of light. This time when it plunged into him there was no feeling of fire, only a cold numbness that spread with alarming speed. He worried that if it reached his lungs he'd never breathe again. It stopped at his thigh, though, and then seemed to shrink back in on itself. With its retreat he realized he felt no pain, even when the coldness left his leg entirely.\n\n\"What did you do?\" he asked, daring to open his eyes.\n\n\"What you wanted me to do,\" Calan said, looking down at him. \"What else?\"\n\nThe priest resumed his prayers, and as his hands hovered over Ghost's knee, the flesh began to knit itself together, forming a pale scar on his dark skin. When finished, Calan took a step back and more collapsed than sat with his weight pressed against the door. His head thumped against the wood, and it seemed those dark circles had worsened.\n\n\"A long two days,\" he said, as if to himself.\n\n\"Blame the Trifect,\" Ghost said.\n\n\"I blame no one. Have no reason. Some days are long, and some painfully short. Must say, I do prefer the calmer ones to this, however.\"\n\nGhost chuckled, but he didn't have the strength to continue. Drowsiness was stealing over him. He'd only slept a few hours in the tavern, and it'd hardly been deep or comforting. The pain had found him even in his dreams.\n\n\"I think I'll sleep now,\" he said.\n\nAnd then he did. His sleep was deep, dark, and strangely without dreams. When he awoke he felt as if an immense amount of time had passed. His leg felt worlds better, though he was still hesitant to bend it. What if it was all an illusion, and the pain would return tenfold when he finally tested it? Rubbing his eyes with his hand, he shook his head to speed up his waking. He found himself alone in the room.\n\nWhen he put his weight on his knee, it buckled and gave completely. He caught himself on the bed and collapsed back atop it.\n\n\"What the fuck?\" he asked, then felt guilty for cursing in the middle of a temple. It was a silly feeling, but still his neck flushed. He stretched his arms and back, then settled in. What should he do now? It wasn't like he was in any real danger, and he'd already paid for the bed and healing. The only thing nagging at his mind was the Watcher. He needed another confrontation, one without those annoying mercenaries getting in the way. How could he manage it? And would the Watcher be foolish enough to return to that building, return to where he knew others might find him? What he knew about the Watcher could be written on a pebble. The man might still be with the Eschatons, or he might be halfway to Ker.\n\nTen minutes later the door opened, and in stepped Calan. He looked a little better, but not much.\n\n\"Was your rest pleasant?\" he asked. He sounded distracted, the question more obligatory than anything.\n\n\"Best in years. How long was I out?\"\n\n\"My guess is five hours,\" Calan said. He pulled the chair out from the desk and plopped into it. Massaging his forehead with his fingers, he stared down at the wood and appeared to soak in the calm. Ghost had seen people look like that before, after they'd endured a long stretch on a battlefield. Once the blood and bodies were gone, the men looked as if solitude were something physical they could soak in like a sponge, silence a concoction they could massage into their temples and necks.\n\n\"It bad out there?\" he asked, disliking the lack of noise.\n\n\"It was,\" Calan said, his eyes staring through his desk. \"Better now. A lot of dead, and even more anger and hopelessness. Too many expect miracles, as if I had any to give.\"\n\nGhost felt another awkward silence descend over them. Deciding he was out of his league, he pulled things back to something more grounded, more real to him.\n\n\"What's wrong with my knee?\" he asked. \"I can't stand on it.\"\n\nCalan looked up. \"I cleansed the infection and knit the flesh, but it is still tender. The spell I used to numb your pain will take time to fade, and until it does, most of your muscles will ignore any request you make of them. Don't fight it; there isn't much point. In another hour or so you'll be walking, albeit with a limp. A few more and you'll be back to doing whatever it is you do. Killing, I assume, sending me even more men and women to care for.\"\n\n\"I came and paid good coin for healing, not insults.\"\n\n\"My apologies,\" Calan said. \"That was uncalled for.\"\n\n\"It was.\"\n\nHe tilted his head toward the wall, not even wanting to look at the old man. The only ones he'd killed recently were those he'd been contracted to kill, and those who had been attacking the priests' temple. That was the thanks he got? Vague accusations of making the priest's life harder, and a claim that he was nothing but a killer?\n\n\"You know what it's like to live in a place where everyone who sees you either hates you or is afraid?\" Ghost asked.\n\n\"There are many who are unsettled by my presence, and more who are angered by what I speak.\"\n\n\"But it isn't the whole city. Even those who fear you do so because you've got something they don't understand. They don't understand me either, but you, they could choose to be like you if they wanted. They can't be like me, no matter what they do. The best they could do is smear themselves with coal, and that'd vanish with a good scrub.\"\n\nCalan leaned back in his chair, and he seemed to truly look at Ghost for the first time.\n\n\"Is that why you paint your face? To show them how different you are?\"\n\nGhost chuckled. \"You want to know why? Truly why? It should show them how the difference between us, between me and you, is something as stupid as a strip of paint, something so thin and artificial we think nothing of it if done to a wall or a piece of armor. But that never happens. Instead they look at me with even greater fear. When I first started, those I hunted called me Ghost, and so I took the name and abandoned my old one. At least if they hated the Ghost, feared it, it was my own creation they feared. It wasn't me; it wasn't who I really was. Let them focus their hatred on something I can shed as easily as I shed this paint upon my face.\"\n\n\"Are you a killer?\" the priest asked.\n\n\"No. But I think Ghost is.\"\n\n\"And who are you when you are not this Ghost?\"\n\nGhost looked at him, trying to understand the true desire behind the question. Calan seemed interested, almost invested, in what he might say and do. There was no deceit in him, and of that quality Ghost considered himself an excellent judge. Who was he when not the man with the white face? Who was he when not hunting, when not contracted to capture or kill another?\n\n\"I'm not sure I remember anymore,\" he said.\n\n\"Do you still remember your name?\"\n\nHe should have, but suddenly it didn't seem so clear. It'd been over ten years since he adopted the Ghost moniker. Before that he'd gone by a dozen names, changing them as he traveled east, for each city a new name. He tried to pull up childhood memories, of hearing his mother say his name, but each one was different in the timeworn haze. Suddenly he felt ashamed, and he wanted to be anywhere other than beneath the priest's unrelenting gaze.\n\n\"No,\" he said at last. \"And I may never. Why does it matter to you, old man?\"\n\n\"If you have to ask, I fear your mistrust has sunk in far deeper than any infection.\"\n\nGhost used the wall to shove himself onto his good leg.\n\n\"Enough,\" he said. \"My thanks for your help. Good luck with your wounded.\"\n\n\"And you with your wounds as well.\"\n\nAn hour later Ghost limped from the temple, more than ever certain that Haern needed to die, if only to put his suddenly troubled mind at rest.\n\n# CHAPTER\n\n#\n\nVeliana pulled her dagger free of the man's neck and kicked his body away. All around her rose the stench of blood and dead bodies. They'd thoroughly trashed the home, broken chairs and shattered tables. Deathmask stood at the door, scanning for more trespassers on their territory, while the twins entered from the house's other room.\n\n\"I'm bleeding,\" said Mier.\n\n\"He's bleeding,\" said Nien.\n\n\"Badly?\" Deathmask asked, not bothering to glance inside.\n\n\"No.\"\n\n\"No.\"\n\nThat seemed good enough for Deathmask. Veliana cleaned her dagger and jammed it into her belt. She felt ready to pass out. Between mercenaries and other members of the guilds, they'd killed over thirty men since the night started, and now it was halfway through morning and still they continued, the most recent dead being from the Wolf Guild. It seemed Deathmask's desire for blood knew no bounds. She felt ready to collapse at the slightest breeze, yet he was still searching, still bouncing as if he were an excited maiden.\n\nThe worst was that the territory they'd chosen to make their stand on was a single street aptly named Shortway, poorly traveled and worth a meager handful of coins in theft and protection money.\n\n\"This is hopeless,\" she said, approaching her guildmaster. Some guild, she thought. Four of them slaughtering trespassers on a single street. Surely the other guildmasters were quaking in their boots. \"We've accomplished nothing other than a few bodies.\"\n\n\"Rumors,\" Deathmask said, still scanning the sparsely populated street. \"Whispers. Exaggerations. Given time they will work for us. We start with a single road, and let the rest of the city know that it is ours. Then we take a second, and a third. With each passing day we spread until we can take no more, and by then they will fear us more than any other guild, for we will be few, we will be skilled, and we will have shown they cannot stop us, cannot even slow us down.\"\n\nVeliana rolled her eyes but decided not to press the point. She felt too tired to argue.\n\n\"Sleep would be nice,\" said Mier, or perhaps it was Nien.\n\n\"Very nice,\" said the other.\n\n\"Very well,\" Deathmask said. \"Let's return. Tomorrow will be just as long, and longer should the mercenaries finally slack off. We have more to fear from the guilds than from them. To the mercenaries we are a small nuisance, a paltry four worth no bounty. It's the other guilds they want. But Thren, Kadish, William... they'll understand. One of them will descend upon us with all their fury, and that is the battle we must win, that our entire fate will rest upon.\"\n\n\"Rest,\" said Veliana. It was really the only word from his spiel that her mind could latch onto. \"I think that's the smartest thing you've said.\"\n\nHis eyes narrowed behind his mask, but then he laughed.\n\n\"We have longer days ahead of us than this, you three. I hope you understand that. Still, no reason to press ourselves without reason. We've accomplished what we must. Let's get back to our little hideout.\"\n\nDeathmask led the way. No one accosted them on their travels, and it seemed none were tracking them either. Shortway was hardly the center of much guild activity, and those few who had stumbled upon it had died. Most of their kills had been thieves fleeing from other territories, where the mercenaries had been at their thickest. When they reached their safe house, a cellar rented from a well-bribed tavernkeeper, Deathmask flung the doors open, lit a waiting torch with a touch of his finger, and led them down.\n\nThe cellar was not empty.\n\n\"It took you long enough,\" said the Watcher, leaning against the far wall.\n\nThe twins drew their daggers, and Veliana felt her hands reach for her own. Deathmask put an arm before them all, then took off his mask and grinned.\n\n\"Forgive us. We didn't know we had company waiting. Have you come to accept my offer?\"\n\nHaern nodded toward the twins.\n\n\"Have you made new friends?\"\n\n\"For now,\" the twins said in unison.\n\nThe Watcher chuckled. \"Very well then. I take it you can read?\"\n\nA pouch hung from his waist, and he pulled one of many scrolls from it and tossed it to Deathmask, who caught it and began reading. His eyes widened, so Veliana glanced over, but he was moving it too much for her to decipher anything useful.\n\n\"The whole city?\" Deathmask asked. He looked ready to both laugh and lash out in rage. \"Have you lost your mind?\"\n\n\"Perhaps. Have you lost your courage?\"\n\n\"Don't turn this on me. You want to present every guildleader and member of the Trifect with the same demand, and then force them to accept within the span of a _single night_?\"\n\nVeliana yanked the document from his hand and read further. In a tight, careful script the letter warned the reader to either accept the following terms or die: A yearly sum, to be determined by the king or his advisors to be equal to one-third the gold lost to theft or spent on mercenaries in an equal amount of time, would be equally distributed among the remaining five major thief guilds. In return the guilds would protect everything within the city walls from theft by any of their members. Following these terms was the date on which the Watcher expected an answer: the winter solstice... two nights away.\n\nShe rolled the document up and tossed it back to Haern, who deftly caught it.\n\n\"You've lost your mind,\" she said.\n\n\"If I remember correctly, it was your idea, not mine. So the insanity should at least be shared.\"\n\nDeathmask laughed, but he looked ready to explode. \"I wanted them to meet to decide terms. I wanted delays, chances to manipulate various parties, and to thin out the guilds who might resist. You want to force the guilds and Trifect to become bedfellows, and even worse, you want to do it all in a single night. How? What madness in you makes you think this could work?\"\n\n\"You know several will agree,\" Haern insisted. \"Those mercenaries are devastating everyone, and will continue to do so as long as the Trifect can afford them. This war has lasted ten years, far longer than even Thren wanted. The Trifect itself is hemorrhaging money, but they currently have no way to end this while saving face. And that's assuming Thren would even let them end it.\"\n\nVeliana shook her head. \"You of all people should understand, too many would resent this. You'd turn us from honest thieves to low-rent bodyguards. The very nature of the guilds would shift.\"\n\n\"They shifted before,\" Haern said. \"When my father took rule, the guilds were full of simple thieves, nothing more. They stole from who they could, rarely even having the courage to go after the Trifect. The drugs, the trafficking, the protection money, it all came from his mind. His vision. He changed the game.\"\n\nHaern's voice softened.\n\n\"My father turned poor men with deft hands into an empire of organized theft and murder, and every guild followed. Give me my chance to replace his legacy with another. I've killed, and killed, and now I will make it have meaning. Every guildmaster or -leader who refuses will die by my hand. Those who assume control will be given the same demands, and suffer the same fate if they refuse. My father began this chaos, and I will end it.\"\n\nVeliana looked to her guildleader, who was deep in thought. It was almost as if she could watch the idea growing in his mind, taking shape, every potential reaction sifting through a spider web of end results.\n\n\"You have the Ash Guild's approval,\" he said suddenly, as if snapping from a daze. \"Will you deliver the letters tonight?\"\n\n\"I will.\"\n\n\"Then go do it. I only have one request: leave your father to me.\"\n\nHaern paused, and his eyes glanced away.\n\n\"Very well,\" he said when he looked back. \"You stand to make a fortune, Deathmask. That is why I trust you. But remember, the same deal applies to you as well.\"\n\nIf her guildmaster was upset by the threat, he didn't show it.\n\n\"Save your energy for those who will give you the most trouble.\"\n\nWhen the Watcher was gone, Veliana spun on Deathmask, shoving a finger into his face.\n\n\"I can understand _him_ wanting to perform this madness, but you?\"\n\nDeathmask winked at her with his red eye.\n\n\"If he succeeds, he succeeds. If he fails, we lose nothing. Besides, Veliana, what else does this plan need but some theatrics? The Watcher's plan is insane, and most likely he'll get himself killed... but I won't stop him. He already frightens the lower members of the thief guilds. If he lives he'll become a terror to them, the only real chance of keeping this agreement in order long enough for us to benefit.\"\n\n\"Why Thren?\" asked Mier.\n\n\"Why us?\" asked Nien.\n\n\"Because,\" Deathmask said, turning to the twins with his smile nearly ear-to-ear, \"if Thren accepts, or dies horribly, the rest of the guilds will fall like dominoes. You wouldn't think I'd leave the most important part of the entire plan to someone _else_ , would you?\"\n\nNot since the hunger riots preceding the Bloody Kensgold had Gerand Crold seen the people of the city so furious. As advisor to the king he had listened to the many complaints about guards, fires, and theft, and the overall demands for compensation. He'd sat in his uncomfortable chair, a ledger before him, and denied every single one. The line of petitioners seemed endless, and that was with the castle guards filtering out some of the more unkempt individuals.\n\nOnce the sun had finally set, Gerand spoke to the king, whispering lies in his ears about how the people still respected him. Finally free to retreat to his chambers, where a full bottle of wine awaited him as per his orders to his servants, Gerand let out a sigh as he passed through the stone hallway.\n\n\"Fucking thieves,\" he muttered as he shut the door. The past five years had filled his head with gray hair, and his marriage had plummeted into occasional nights with his wife, but mostly him sleeping in the castle, her in their home. Removing the cork, he poured a glass and toasted the empty room.\n\n\"To you, Alyssa,\" he said. \"For destroying in two days everything I built in five years.\"\n\n\"To Alyssa,\" someone whispered, their breath upon his neck.\n\nGerand nearly choked on his wine. He spun, torn between diving for a weapon and falling to his knees to beg for his life. The last time a thief had sneaked into his room, it'd been Thren Felhorn and a female companion. They'd threatened his wife to make him fulfill certain desires of theirs. The first thought that ran through his mind as he saw a man cloaked in gray was that they'd have to kidnap someone better if they wanted him to obey this time.\n\n\"Thren?\" he said, startled by the sight. It looked like Thren, only much younger. Gerand had a sudden fear that the man was immortal, immune to time, and determined to haunt him for all his life. As a wolfish grin spread across the intruder's face, he choked down such irrational thoughts. This wasn't Thren. The hair was the wrong color.\n\n\"No,\" said the man. \"Close, though. I am the Watcher. Perhaps you have heard of me?\"\n\n\"I have, though I've wondered if you were actually real.\" Gerand chuckled. \"I guess this should count as proof?\"\n\nThe Watcher snatched the glass from his hand and drank the remaining half. As he smacked his lips he tossed Gerand a scroll.\n\n\"Read it,\" he commanded.\n\nGerand did, his eyes growing wider with each sentence.\n\n\"You want the credit for the idea to go to the king?\" he asked when finished. \"But why?\"\n\n\"The more involved the better,\" said the Watcher. He leaned against the wall, just beside the door. Even if Gerand managed to call the guards, and not die doing so, the man would still get the jump on them. \"Besides, I need someone neutral in all this, someone both sides view as on their side. You've accepted bribes from both the thieves and the Trifect. Both will think you'll be in their pocket once the smoke clears.\"\n\n\"But Edwin will never agree. He's terrified someone will poison his tea or put shards of metal into his bread. By the gods, he thinks every shadow in his bedroom is a man poised with razor wire.\"\n\n\"He has something more real to fear, and we both know it. Veldaren is furious. You've failed to protect its people, and this time it's gone too far. Fires have burned down a quarter of the city. Innocent men and women died at the hands of mercenaries, and they come here finding no justice, no empathy. They have no one to turn to, no one to trust. Do you remember the riots five years ago? They will make those look downright orderly.\"\n\nGerand nodded. He'd seen the anger simmering in the many waiting in his line. They'd certainly not left in a better mood after talking to him, either. Getting King Vaelor to agree would involve marginal effort at best. Once Gerand played on his fears, then offered him the deal as a way to come out a hero, he'd agree in a heartbeat. Gerand glanced down at the parchment in his hand, still trying to determine the loophole, the underhanded secret hidden beneath the words.\n\n\"What do you gain from this?\" he dared ask.\n\n\"We all want a legacy,\" the Watcher said. \"This will be mine. The arrangement will rely on you, once everything is in order. Can you enforce it?\"\n\nThose words, of a stranger demanding enforcement, sparked a memory, and coupled with seeing such a familiar face, Gerand couldn't keep his tongue still, keep the question unasked. Years ago Gerand had tried to capture Thren's son, Aaron, and had even used the king's old advisor, Robert Haern, to aid him in the task. Time had dulled his memory, and no doubt the child had grown, but still...\n\n\"You have to be Thren's kin,\" he said. \"His long-lost son, Aaron, perhaps?\"\n\nThe Watcher pulled his hood lower over his face, and his mood seemed to sour.\n\n\"I would keep such thoughts to yourself, friend. They are dangerous.\"\n\nGerand felt his blood chill.\n\n\"Of course, of course. I guess it is no matter. But can you pull this off? A bluff won't work with either side.\"\n\n\"I have lived on the streets, hunting them like dogs. Every single guild has initiated me into their order without knowing it. I know where they live, where they hide. Few can challenge my skill, and none my determination. I will kill them, all of them, if I must. Make the king listen to you.\"\n\nHe stood and put a hand on the door.\n\n\"I've already delivered the rest of the messages. They'll bring their answers to you. Come tomorrow I'll check here first, to see who is safe and who must be dealt with.\"\n\n\"I understand.\"\n\nAs the door opened, Gerand couldn't hold in a chuckle. The Watcher stopped, as if he suspected a trap.\n\n\"No, it's not that,\" he said as the man closed the door. \"I just found it humorous, is all. A long time ago your father came to me, threatening my wife to help him escalate his conflict. Yet now you come here seeking to end it. This time the apple fell far from the tree, didn't it?\"\n\nThis seemed to put a smile on the Watcher's face.\n\n\"Good night, advisor. Do your part, and trust me to do mine.\"\n\nHe vanished out the door. Gerand plopped down onto his bed, and now that he was alone he felt his hands start to shake, his nerves finally getting the better of him. It seemed, despite the guards and walls, those with enough skill could still reach him. Perhaps the king's jumping at shadows wasn't so irrational after all...\n\n\"Where's that damn wine?\" he asked. Seeing the bottle, he held it by the neck and drank straight from it. Given what he was about to go through the following day and night, he'd need all the courage he could get.\n\n# CHAPTER\n\n#\n\nDelysia stumbled upon him getting ready as night fast approached.\n\n\"Where are you going?\" she asked.\n\nHe'd shared Senke's room, but the other rogue had gone out to spend a night relaxing in the taverns.\n\n\"I've got some business to take care of,\" Haern said. Brug's trunk of weapons lay open before him, and he slid several daggers into his belt, plus another into a pocket of his boot.\n\n\"My brother did some digging,\" Delysia said, crossing her arms underneath her breasts. \"And he says the mercenaries won't be going out again tonight. So what is it you're planning? For once the night might be peaceful.\"\n\nHaern felt a half-smile tug at the corner of his mouth.\n\n\"The nights are never peaceful here. Quiet, perhaps, but killing can be silent work when done right. Never mind that, though. Promise me you'll stay inside. Things are about to get very dangerous.\"\n\nShe put a hand on his arm. \"For you as well?\"\n\nHe shrugged. \"Can't help it. I have a chance to do something great, Delysia, something real.\"\n\n\"Will you kill?\"\n\nHe rolled his eyes. \"This isn't the same.\"\n\n\"Then what is it?\"\n\n\"Safety,\" he said. \"For all of us. My father wants a legacy, and I'll deny him it. What he started I'll end, or I'll die trying.\"\n\n\"You don't have to do this alone,\" she said. \"Let us help. Let _me_ help.\"\n\n\"You lost enough because of me. I won't risk the life you've rebuilt here.\"\n\n\"Who said you had a choice?\"\n\nHe winked at her.\n\n\"Where I'm going tonight, I don't think a priest or priestess has ever been. Good night, Del.\"\n\nHe took her hand, kissed it, and then left.\n\nThe trip to the castle wasn't long, though scaling the outer wall, sneaking past the guards, climbing up to one of the higher windows, and then stealthily descending to Gerand's room took more time than he would have preferred. Probably should have made the advisor meet him at the front gates, he thought. Would have saved him the hassle.\n\nWhen he slipped inside, Gerand looked ready for him. He smiled at Haern's entrance, but Haern saw the way his eyes darted about, and how the edges of his smile quivered. The man was nervous, but Haern didn't think it was because he'd set a trap. No doubt he felt the eyes of every guild upon him, plus the anger of the Trifect.\n\n\"What were their responses?\" Haern asked, having no time to waste.\n\n\"I've received an answer from everyone but the Ash Guild and Laurie Keenan. With him down in Angelport, there's no way he might give an answer, and no keeper of his estate here would dare agree to something like this without confirmation.\"\n\n\"Don't worry about the Ash Guild. I have their answer. Laurie will fall in line when the other two Trifect leaders agree. Tell me, have any said yes?\"\n\n\"The Wolf Guild will, but only if Thren agrees as well. If the Spider Guild doesn't fall in line, though, they'll deny ever saying so.\"\n\n\"Is that it?\"\n\nGerand licked his lips. \"The Serpent Guild's man said they'd rather kiss the asses of a thousand corpses than the Trifect's. The Hawks fired an arrow over our wall, a cloth tied to the shaft with the word _Never_ written in blood upon it. Leon Connington's advisor sent a letter saying they were open to negotiations, but not under such conditions. Lady Gemcroft's response was cryptic. I received a letter from her saying you'd have to kill her, while her advisor came by later insisting she might change her mind given time. As for the Spider Guild...\"\n\nHe gestured to a package waiting beside his bed. Haern opened it to find a severed head, eyes and mouth sewn shut.\n\n\"Who is it?\" he asked, frowning.\n\n\"Look closer.\"\n\nHe did, saw the gray hair, thin nose, and most noticeably the fresh cut running from the head's left eye to its ear. Haern glanced back at the advisor and saw a similar, albeit faded, scar on his face. He felt a pang of guilt, and he wondered who the poor guy had been.\n\n\"Intimidation,\" Haern said. \"Don't fall for it. I won't let them get to you.\"\n\n\"How?\" asked Gerand, exasperated. \"There are five guilds, and three leaders of the Trifect. Three of the guilds, and two of the Trifect, have denied you. Can you kill them all? I might be better off having you killed instead, and letting them fight amongst themselves.\"\n\nHaern narrowed his eyes, and the advisor immediately retracted his comment.\n\n\"Forgive me, I'm stressed, and have had more to drink than I probably should. How will I know if you succeed?\"\n\n\"They'll come tell you,\" Haern said, turning to leave. \"Oh, and escort me out, will you? I don't have time to mess around with your guards.\"\n\n\"Sure,\" said Gerand. \"Why not? A king's advisor and an assassin, side-by-side as friends. I've suffered through stranger.\"\n\nThey walked through the halls toward the castle's exit, and several times Gerand had to calm soldiers who saw Haern's cloaks and sabers and immediately assumed the worst. At the giant doors Gerand grabbed his arm and pulled him close.\n\n\"Be careful whom you kill first,\" he said. \"If you fail, but still strike down the leaders of either side, you will unbalance everything. You must succeed in this, Watcher.\"\n\n\"If I unbalance things, at least you'll finally have a winner,\" Haern said, grinning. \"And I don't plan on dying. I'll visit you come the morning. I promise.\"\n\nHe flew down the streets, his legs pumping. His eyes darted every which way, knowing that the guilds would be prepared for his arrival. For the longest time he'd only been a phantom to them, but now they'd heard his promise. This was their chance. Deathmask had asked him if he was insane. Perhaps he was. But at least Deathmask had also seemed to understand. If he was going to try, why not try for the impossible?\n\nHe decided the first ones to visit would be the Hawks. They were no friends of the Spider Guild, and their recent conflict with both them and the mercenaries had surely devastated their numbers. Their leader, Kadish Vel, was a sensible enough man, a bit of a gambler. If Haern could convince him to give the treaty a chance, especially as a way to diminish Thren's danger, then he should be able to win over the guild's cooperation without having to kill its leader.\n\nThe Hawk Guild's headquarters had shifted several times, but even after Thren's attack they'd refused to move from their current tavern. No doubt Kadish was tired of running. They'd emptied out during the mercenaries' rampage, but with the night turning quiet he felt certain they'd return. With the strange threat of the Watcher hanging over them all, they'd want to be where they felt safe, where they recognized every face, knew every shadow, every entrance. If he was lucky they had thought his offer a bluff, or even an opportunistic lie of the king, an attempt to save face before his furious populace. Still, their eyes would be open even if they doubted. Nothing that night would be easy. If he had to open up with a stealthy kill, he would, conversation be damned. Dead bodies spoke clearer than anything else in the world of thieves. He'd given them their warning.\n\nTwo men stood at the entrance, obviously on the lookout. The one on the left looked bored, as if he'd drawn the short straw that night. The other was older and more aware. Haern drifted along on the left, hunched over, limping, and muttering to himself as if he'd had too much to drink. Neither Hawk gave him a second glance. Once out of their sight Haern walked closer, a hand leaning against the tavern as if he needed it for balance. Upon reaching the back he saw a door, most certainly locked and barred. He tested it just to make sure. No lock, but it was barred. A veritable army of feral cats lurked about, and one hissed at his presence as it hunkered down over a scrap of fat.\n\n\"You're right,\" Haern whispered to the feline. \"I should wait my turn, shouldn't I?\"\n\nThe alley was thin, dark, and full of places to hide. Ten minutes later he heard a thud from the other side of the door, and then it creaked open.\n\n\"Enjoy,\" said a burly man, chucking a bucket of what seemed to be a mixture of vomit and woodchips. The mess splattered onto the stone, and in leaped the cats, there to hiss and growl while they searched for something edible. When the man's back was turned, Haern stepped from his hiding spot, jumped into the opening, and flung his arms around his neck. A quick jerk and he fell, his scream muffled by Haern's palm.\n\n\"Sleep well,\" he whispered, slipping inside.\n\nHaern stepped inside a storeroom, cramped and occupied almost entirely by the two shelves on either side. The man he'd knocked out wasn't the barkeep, one of the few things that might allow him to remain hidden. Already he heard the raucous drinkers at the front, and they would not have tolerated any sort of absence from the one supplying their liquor. Judging from the man's size and duty Haern had taken out a heavyhand, someone to deal with those who got out of line. Haern chuckled. Now that was someone who would _not_ be missed by the crowd.\n\nHe ducked underneath the lowest shelf, peered out the inner door, and was rewarded with the sight of a pair of hairy legs as the barkeep scratched at a scab near his ankle. Rows of bottles stretched before him. He was behind the bar. From his time inside before, Haern knew there'd be a locked door to the right of the bar, blocking off a staircase that led into the lower quarters that the Hawks had enlarged and adopted as their own. Question was, how did he get inside?\n\nAs he pressed his back against the wall, an idea came to him. He waited a moment to see if the barkeep noticed the movement, then pressed his elbows against the wall. There was no plaster or paint, just a thin piece of wood. Bracing with his feet, he pressed inward, hoping the laughter and singing might drown out the noise. He pressed harder, every muscle in his legs taut, and then the board gave. Wincing at the noise, he pushed it all the way in until it snapped in half. He worked an arm through, then his head, and finally the rest of his body. When he fell out the other side, he was behind the locked door and inside the small staircase leading to a second door.\n\nHaern drew his sabers, thrilled by the way they handled. Senke had given him a fine gift. Hopefully he'd put them to good use. The time for stealth was over. He tested the door at the bottom of the stairs, found it unlocked, and then kicked it in. As the loud crack echoed in the room, he leaped inside, already searching for his first victim. There were four men inside, sitting at a table with cards and wooden chips stacked before them. They shouted and reached for their weapons, but they were too slow. Haern whirled through them, rolling across the table and to the other side. Chips clattered along the floor, mixing with the blood.\n\n\"Shit,\" he said as he stared at the bodies. None was Kadish Vel. He glanced about, seeing only one other room. Within were some ledgers, a shelf stocked with tiny bottles, and a bed. Haern couldn't believe it. The first of his many places to attack, and he hadn't even bothered to check inside the tavern first. Kadish was probably with his men, drinking. So much for being the feared Watcher of Veldaren. So much for his plans.\n\nHe rushed back to the stairs, trying to decide on a course of action. No one would know he'd come down there yet. Kadish was in a public place, with lots of people, but that could work in Haern's favor. Halfway up the stairs he turned back and hurried to the small side room. Scouring the shelf, he looked to see if his idea was possible. The tiny bottles weren't alcohol as he'd first assumed. They were tonics, tinctures, and most important, poisons. Only half were labeled, as he'd expected. Haern recalled his lessons when training under his father. For three months he'd had a tutor who knew more about poisons than Haern could ever have learned in a lifetime. Many of his lessons during those long days had involved discovering the nature of unlabeled poisons.\n\nHe shook several, checking their color, their consistency, and their weight. He pulled four aside, cleared a space on the shelf, and put the bottles atop it. One of the bottles was clearly an extract of shadeleaf, but that was only part of what he needed. He took two others, shook them, and then poured drops of each together. When they turned green, he frowned. Trying another bottle, he mixed it with the nightshade. When the drops turned clear, he grinned.\n\nDumping out half the bottle of shadeleaf extract, he poured another bottle, a common mixture of kingsblood and dandyblooms, in with the rest and shook it. The resulting mixture turned clear. It had a strong taste to it, which meant it only worked with certain alcohols. That was if you planned on putting it in someone's drink, of course. Haern had other ideas. Among the marked bottles was a useful paste that bonded with most poisons, thickening them into something akin to glue. Haern applied the poison to the blade of one of his daggers, being careful not to rush. The last thing he wanted was to prick a finger and die in the Hawks' basement dwelling. That hardly seemed like the noble end he felt he deserved. Last, he found one more particular bottle and smashed it.\n\nHaern climbed back through the hole in the wall, slipped out the rear of the tavern, and then circled around to the front. The two thieves remained on guard, and they sneered at his approach.\n\n\"Hey, you got to have money first,\" said the one on the right, blocking his way with his arm. Haern glared, then pointed through the door, slurring his words and making sure his hand bobbed up and down in the air.\n\n\"That... that guy there's my brother. He'll cover for me, really. Ask him, he's a great guy, married a whore who makes more money on her back than I... I could... that could make in a _month_.\"\n\nHaern made sure he pointed between two tables, and the movement of his arm made it no clearer whom he was referring to as his brother. The guard on the left looked inside, as if he could somehow pinpoint the man anyway. The one on the right grabbed Haern's arm.\n\n\"I said get out,\" he said, but Haern moved too fast. He spun out of his grip, slashed open his throat, and then turned to the other. Before he could let out a cry, Haern stabbed his chest with an unpoisoned dagger, ramming his arm over his mouth to hold in the scream. It came out muffled, not loud enough to attract any attention within. It seemed the men and women were eager to celebrate their first moment of peace in two days. No doubt they thought they'd beaten the mercenaries, or at least they wanted to think so.\n\nKnowing time was far from an ally, Haern lumbered into the tavern, resuming his drunken gait. With his head low he scanned the bar, looking for Kadish Vel. He found him in the far corner, sitting with his back to a wall. Six men sat with him at the giant round table, along with a pretty lady at his side. She seemed bored with the proceedings, and Haern wondered if she stayed with him for coin or for safety. The rest were joking or boasting, their voices loud and slurred. All but Kadish. He seemed mildly amused at best. Haern drifted toward him. He had one chance at this, just one.\n\n\"Hey, hey, hey!\" shouted one of the men as Haern slid between them, directly opposite Kadish. Haern put his hands on the table, and he leaned forward as if keeping balance were a struggle.\n\n\"Kadisshh?\" Haern asked, looking lazily at the guildmaster.\n\n\"I'm sorry, friend, the bar might be mine, but the drinks still don't come free,\" said Kadish.\n\nHaern never responded. His hands, leaning there on the table, were also within inches of the sabers at his hips. As his cloaks folded away, Kadish saw them, and that was when Haern moved. He drew them both, and in one smooth motion sliced through the necks of the men to either side of him. As they collapsed, their blood splattering across the table, Kadish flung himself out of his chair, pressing his back into a corner. The pretty girl looked dazed, as if she didn't believe what she saw. Two of the men moved closer to defend Kadish, the others drawing daggers and lunging. Haern batted one aside, killed another with a riposte, and then spun, a whirling machine of death. Cries of alarm spread across the tavern as the rest realized what was going on.\n\nFalling to one knee, Haern let go of a saber and yanked the poisoned dagger from his boot. From underneath the table he saw the lower half of Kadish's body. No armor. No realization of the danger. He flung the dagger, trusting his aim. It plunged into the meat of Kadish's thigh, and Haern allowed himself a smile.\n\nAnd then he was moving again, his sabers reveling in the blood of his opponents. The whole tavern was in chaos, half the customers fleeing, wanting no part of whatever might happen. Many others cried out in warning, expecting an ambush. One shouted Thren's name, as if he must be the one responsible. Haern weaved through them all, deflecting sword strikes and slicing into the arms of those who thrust their daggers. A heavyset man tried to block his way at the door, but Haern rammed into him with his left shoulder. His right hand stabbed repeatedly. The two collapsed through the door, landing beside the bodies of the guards.\n\nOut in the open Haern took to his feet and ran. Curses followed after him, but he was too fast, the city so familiar that he could weave and turn without pause through the maze of alleys and streets. He wished he could have talked to Kadish, convinced him to change his mind, but there'd been no way. The poison would work its way up his leg and to his lungs, locking them in place. There was a cure, but that had been in the bottle Haern smashed before leaving. By the time they found another, Kadish would be dead.\n\nA list of targets passed through his head. He'd taken out one of the guilds, so it was time to move on to a member of the Trifect. From what he'd learned, the recent wave of mercenary attacks had been the doing of only one. If there was to be any peace in the city, Alyssa Gemcroft would need to die next.\n\n# CHAPTER\n\n#\n\nAlyssa was hiding in her room from her relatives when Zusa arrived, a letter in hand.\n\n\"He's coming,\" she said as she handed over the parchment. \"The man who killed your son. He wants you to agree to his terms, or he'll kill you.\"\n\n\"This man, the Watcher...\" Alyssa crumpled the letter without reading it, needing only the signature at the bottom to know her answer. \"He kills Nathaniel, then dares make demands?\"\n\n\"He'll come tonight,\" Zusa said. \"And he's skilled, milady. He might carry out his promise.\"\n\n\"Let him come,\" Alyssa said. \"You will protect me. He cannot hide from you, not here in my mansion. This is our home, and he the stranger. I trust you with my life, Zusa. Don't let me down.\"\n\n\"The terms aren't so unfair,\" Zusa insisted. \"Bertram would have you agree.\"\n\n\"I don't care. Let the Watcher try. He dies tonight.\"\n\nAnd so the day droned on, Alyssa with less and less patience with the relatives who had remained after the funeral, preferring the safety of her mansion to their own homes. Bertram came to discuss the wedding, but she ordered him away. She was even curt with Arthur, who brought her a plate of food and a glass of wine.\n\n\"You have eaten nothing all day,\" he said. \"Please, take something. You will feel better. We have things we must discuss.\"\n\nFearing he might bring up the subject of marriage, or, gods forbid, propose right there holding a plate of bread, boiled potatoes, and cabbage, she told him to get out. His caring demeanor faltered a bit, and he stormed away.\n\n\"You have no time left to be a little child,\" he told her before slamming the door shut. \"Already your immaturity overstays its welcome.\"\n\n\"And you yours!\" she screamed, hurling the glass of wine he'd left behind.\n\nShe wished Zusa were there, but she'd vanished, although promising to never be far.\n\n\"If you know, then you might reveal my presence,\" the faceless woman had argued. \"If you trust me, then trust me. Trust the shadows to hold only me.\"\n\nThe night dragged on. This time there were no fires to watch, no men rushing up and down the streets. Just quiet. It seemed eerie, as if the city was suddenly waiting for something. Bertram had told her he feared terrible retribution from the guilds for her actions, but so far it seemed none were coming. Or maybe one was, coming in the form of the Watcher.\n\nAlyssa checked the lock to her room for the fourth time.\n\nWith nothing to read and nothing to do she sat down on her bed, closed her eyes, and wished she could sleep. It'd be so much better for her to die that way, unable to feel the pain. Part of her expected just that to happen, though another part was revolted by the sheer weakness involved in considering it. She should be stronger, better, but she was so tired. The Gemcroft estate felt like chains attached to every inch of her body, dragging her down, pulling her into an exhausted pit where she could feel no emotion, cry no tears, and express no love. It was in that pit that Arthur waited.\n\nThe sound of cracking glass startled her to a sitting position, her heart leaping all the way up to her throat. Any calm she might have felt in accepting this fate vanished with the threat finally there. A man was outside her window, hanging from a rope. The glass had cracked in a circle of veins where his heel rammed into it. She saw him kick off, his momentum bringing him back into the glass. This time it shattered, and in rolled the Watcher, all cloaks and blades and broken shards.\n\nAlyssa rolled off the bed in the direction opposite him. Hitting the ground with a thud, she scampered toward the door. A dagger flew, and she felt a tug as it passed through strands of her hair before thudding into her locked door. Panic struck her, and she spun to face her attacker.\n\n\"Lady Gemcroft,\" said the man, and he bowed low as if in respect. It seemed so comical coming from him that her mouth dropped open. \"I've come for an answer to my offer, and for your sake I hope it differs from the one I last received.\"\n\nShe thought to lie, or bargain, or maybe just turn, fling open the lock, and hope she was faster than him even though she knew she wasn't. She'd die with a dagger in her back, or maybe a saber in her neck. With each step he took, she took one back, until she realized he was carefully guiding her away from the door. If she was to flee, she had to do it now, either that or give in. Before she could choose either option, the shadows above her shifted, and from the corner Zusa leaped, her cloak uncurling from about her body as if she were an insect emerging from a cocoon.\n\nThe Watcher, instead of retreating at the ambush, rushed toward her. Alyssa dove out of the way, and only when she landed did she realize she'd done as the man hoped. He'd cut her off from the door. From her knees she watched Zusa and the Watcher clash. His sabers were longer than Zusa's daggers, and he had the greater reach, but it seemed not to matter. Alyssa had seen Zusa leap naked into a river to fight a dark paladin of Karak, seen her wield her blades with shocking speed and skill.\n\nBut Alyssa had never seen her fight like this.\n\nSword and dagger clashed in a constant echo of noise, sharp and painful in the closed room. The Watcher whirled, his weapons a blur, yet Zusa met every move. Her body arched and weaved as if her bones were liquid and her balance relied on will alone. Alyssa tried to follow them with her eyes, but could not. Over and over a saber would pass so close to Zusa's flesh she'd wince, expecting a shower of red, but it never happened.\n\n\"Alyssa!\" someone shouted from the other side of the door. Something heavy thudded against it, most likely a fist striking in a panic.\n\n\"Send my guards!\" she screamed back, returning to her wits. Another thud, this heavier, but the lock on her door was sturdy, designed to hold out far more than a single man. She desperately wished she'd left it unlocked.\n\nThe Watcher leaped from side to side, avoiding Zusa's thrusts, and then spun about, hiding his body with a sudden flourish of his cloaks. From within its folds she saw the glint of his sabers, darting out with a quick slash from the chaos. Zusa retreated from them, careful to keep herself between him and Alyssa. For a moment Alyssa thought it might work. If Zusa could hold on until help arrived, the man would have to retreat.\n\nBut then Zusa cried out, and blood splashed across the carpet. Alyssa felt her heart stop. Zusa continued fighting, even as the wraps on her left arm were soaked crimson. Could she fight through it? For a few long moments she did. Zusa went on the offensive, her whole body twisting into her thrusts, giving her a reach beyond even the sabers. This time the Watcher fell back, batting each thrust aside, the contact ringing in Alyssa's ears. Their speed... it was unreal. At one point their weapons entangled, looping and parrying in what could only be described as a lethal dance. Faster and faster they moved, each refusing to give the other an inch of room. The sound of their battle escalated, and Alyssa found herself clutching the carpet, her whole body tense. The second she saw another splatter of blood, she would run for the door, risks be damned.\n\nBut it was all bluster. She saw the pain on Zusa's face, and her heart broke. The Watcher's foot struck her protector's chin, and as she staggered back, in sliced a saber, cutting into her arm. Zusa dropped one of her daggers and, suddenly at a disadvantage, she could not hold him back. His assault came, vicious and quick. Another thin cut opened on her leg, the slash so quick Alyssa never even saw it. Fists and knees slammed into Zusa, and she rolled with them, rolling... away from the door.\n\nShe was granting Alyssa a way out, sacrificing her body, her life, to do it.\n\nThe Watcher kicked her again, hard in the throat. As she fell back he descended upon her, a saber tip pressing against her chest, just above her heart. His elbow trapped the arm still wielding a dagger, the rest of his weight pressing against her waist to keep her from moving.\n\n\"Why do you hunt me?\" she heard him ask Zusa. It seemed so strange to her, to hear him wonder, but she dared not think of it. She wouldn't allow Zusa to die for nothing. With her fall, Alyssa knew there was no way for her to escape. Men shouted from the other side of the door, but they were yet to begin breaking it in. At least she could accomplish something with her death. At least she could reward such a loyal servant for all she had done.\n\n\"Wait!\" Alyssa screamed before she might lose her courage. \"Take me, but let her live!\"\n\nThe Watcher looked over to her, a quick glance before returning his gaze to Zusa, who even in such a state remained dangerous.\n\n\"Why?\" he asked. \"Why would you kill yourself? I offered you a chance to end all this! Are you so vain that you would rather die than work with men such as me? All of Veldaren suffers, and I offer you a chance to save it!\"\n\n\"Damn your offer,\" said Alyssa. \"I could never agree, not after what you've done. Now take my life, and spare hers.\"\n\n\"What _I've_ done?\" he asked, and he sounded genuinely perplexed.\n\n\"Her son,\" Zusa said, her voice hoarse from the kick. \"You killed her son, then scrawled your name in the dirt with his blood.\"\n\nThe Watcher looked taken aback. His eyes glanced between them. When something heavy thudded against the door, his body tensed.\n\n\"When?\" he asked, his voice soft.\n\n\"Weeks ago, on the road north to Tyneham.\"\n\nUsing one of his sabers, he pushed the final dagger free from Zusa's hand, then stood.\n\n\"Murderer of children,\" he whispered, as if finally understanding something. \"I know what child you speak of. Five years old, perhaps six, red hair? He _lives_ , Lady Gemcroft. I saved him, though he was wounded and with fever. I left him in the care of a family, and paid them well to protect him.\"\n\nAlyssa shook her head. He had to be lying. It didn't make any sense.\n\n\"Why?\" she asked. \"How? The caravan was attacked...\"\n\n\"I came upon an ambush of a caravan, but not by thieves. They were men wearing the same insignia, a sickle held before a mountain. Your boy was the target, though at the time I never learned his name. That caravan contained crates of gold bearing your family's crest. They were smuggling it in to the Serpent Guild, though why I do not know.\"\n\nIt was too much. That crest, that was the Hadfield family crest. Could Arthur have attacked Mark, attacked Nathaniel? But why would the Watcher lie? He could easily kill her and Zusa. And besides, she'd seen her son's body... his burned...\n\n\"Arthur claimed to have found the body,\" Zusa said, as if she were following Alyssa's thinking. \"His men brought it back, burned beyond recognition. His men found the caravan. Mark, his competition. Nathaniel, your heir.\"\n\nAs the door to her room cracked, the hinges wincing in protest, she saw the Watcher tighten the muscles in his body. He was preparing an action. Time was nearing its end. She couldn't stall. She had to make a decision, the one that felt right.\n\n\"If you did save my son, you have my most sincere gratitude,\" she said. \"I'll agree to your terms. Zusa told me they were fair, and I trust her. But if find this to be a lie, I will bring the wrath of the entire Trifect down upon you.\"\n\nThe Watcher grinned at the threat.\n\n\"I have a date with the Serpents, so I must be going. I'll keep your threat in mind.\"\n\nHe turned to the window and ran, leaping out as if a madman. Then she saw him catch the rope still hanging, and like a spider he zipped back up toward the roof and out of sight. Zusa gingerly rose to her feet, holding her bleeding arm with her hand.\n\n\"You know what you must do,\" she said, and Alyssa nodded.\n\n\"Enough!\" she shouted to the men trying to break down her door. \"Stand back; he's gone.\"\n\nZusa went over and unlocked the door. Guards spilled in, their weapons drawn as if they refused to believe her. One checked underneath her bed, while several others looked out the window and swore.\n\n\"Are you all right?\" asked Bertram, pushing through them and hugging her.\n\n\"I'm fine,\" she said. \"It's Zusa who was hurt, that's all.\"\n\nBertram didn't even glance the faceless woman's way.\n\n\"Thank the gods. I wish you would spend the night with the rest of the guards. It was folly to remain in here alone in the first place.\"\n\n\"I said I'll be fine. I'm safe here, save for a bit of a draft.\" She tried to smile, but her hands were shaking, and her smile trembled on her lips. And the night wasn't done just yet.\n\n\"Leave me,\" she told her guards. \"Trust me, I will be safe.\"\n\nThe men grumbled, and they looked none too happy, but she was their employer, and grumble was all they could do. Bertram waited until the rest were gone before bowing.\n\n\"Is there anything else you need from me?\" he asked.\n\n\"Send for Arthur. I need his comfort.\"\n\n\"Of course, most understandable.\"\n\nWhen he was gone, she looked to Zusa.\n\n\"How bad are your wounds?\"\n\n\"I have suffered worse.\"\n\n\"You're bleeding all over my expensive carpet.\"\n\nZusa smiled, then suddenly let out a rare laugh.\n\n\"I guess I am.\"\n\nAlyssa walked to the door, stopped, and outstretched her hand. Zusa put a dagger into it and folded her fingers closed.\n\n\"Tell Arthur to meet me in the gardens,\" Alyssa said. \"I'll be waiting at Nathaniel's grave.\"\n\nShe hurried out of her room and down the hallways. Every twist and turn was second nature to her, and she managed her way out to the darkness without alerting any of her guards or, more important, Arthur's mercenaries. In her garden behind the mansion, she felt the cold air bite into her skin, felt the moonlight shine down upon her. A few minutes later Arthur arrived, his arms crossed over his chest to keep in his warmth. She turned and kept her hands folded behind her back, the hilt like ice in her fingers.\n\n\"Are you safe, my dear?\" Arthur asked, immediately wrapping her in his arms. \"I only heard the commotion at its end, and Bertram found me before I could dress.\"\n\nAlyssa smiled at him, but when he made to kiss her she tilted her head away, her eyes falling upon the grave marked by a stone angel, the lettering at its feet reading \"Nathaniel Gemcroft.\"\n\n\"I just want to ask you something first,\" she said, her hands still behind her as if she were shy.\n\n\"Anything.\"\n\nShe looked into his eyes, watching his every reaction.\n\n\"Whose body is truly in this grave?\"\n\nThe hour was late, and Arthur had not the slightest reason to expect the question. He paused, and for one second she saw the guilt in his eyes, and the fear. What replaced them was cold cruelty. Without a word he lunged for her throat with his bare hands. Before they could close around her, she stabbed him with Zusa's dagger. As his blood spilled across her arms, he looked at her with wide-eyed shock. She could almost read his thoughts as they flashed over him. _This can't be happening, not to me, not when I played this so perfectly._\n\nShe twisted the dagger, taking pleasure in the pain that crossed his face.\n\n\"I wish I could make it hurt worse,\" she whispered into his ear, as if they were still lovers. And then his body collapsed, and she stepped out of the way, letting go of the dagger. She stood there, breathing heavily, and watched the blood mix with the soil. When Zusa arrived moments later, Alyssa did her best to smile.\n\n\"At least I didn't stain the carpet,\" she said, but her smile was forced. Zusa put her arms on Alyssa's shoulders and kissed her forehead.\n\n\"Thank you,\" Zusa said.\n\n\"For what?\"\n\n\"For being willing to sacrifice yourself for me.\" She gestured to the body. \"What do you want me to do with him?\"\n\n\"Hide him for now, until we know how Arthur's mercenaries will react. But first... the Watcher's plan... will it work?\"\n\nZusa shrugged. \"It's possible. He'll have difficulty doing it alone.\"\n\n\"I can't do this anymore, Zusa. If Nathaniel's still alive, I want him here with me. I want him safe. Go with the Watcher. Help him against the Serpents, but the Serpents only. I won't help him harm a member of the Trifect, but for one more night I can bring revenge on the guilds that have harmed us so. They stole from me, them and Arthur. Bring ruin upon them as they deserve.\"\n\n\"As you wish, milady,\" Zusa said. Despite the wounds on her body, she offered no protest. Dragging Arthur's body by the arm, she hid him behind a row of rosebushes, currently nothing but thorns and brown stems.\n\n\"I'll inform Bertram,\" the faceless woman said, cleaning her dagger. \"He'll need to replace Arthur, and perhaps much of the Hadfield family, in their rule over our mines. If he acts quickly, we might find that which was stolen.\"\n\n\"Thank you. I could use a moment's peace until then.\"\n\nZusa returned to the house, and Alyssa followed. Their paths branched, she to her room, Zusa to Bertram's. Once back, Alyssa collapsed on her bed, eager for the night to be over. Doubt gnawed at her, and she hoped she'd done the right thing. At least in time she'd know for sure. If Nathaniel lived, then everything would be justified. Everything would be made better. She thought of the carnage and chaos she'd unleashed upon the city, all in a desire for vengeance, only for it to be Arthur's doing, not the thieves'. Not directly, anyway, though it seemed the Serpent Guild had had a hand in it. Hopefully the gods might forgive her.\n\nThe door opened, and in stepped Bertram. Not surprisingly, he looked rather upset.\n\n\"I can scarcely believe what Zusa told me,\" he said. \"Is it true? Did you kill Arthur?\"\n\n\"I did, and I was right in doing so.\"\n\nThe old man locked the door, then placed his head against the wood as if he needed its support to remain standing.\n\n\"He was our last hope,\" he said. \"We've squandered our wealth on mercenaries, destroyed the Gemcroft family's reputation, and now you ruin the only chance of bringing back any respectability to our name? Why?\"\n\nHe turned to her, and the look in his eyes sent a chill running through her spine.\n\n\"Bertram?\" she asked, sliding her legs underneath her. \"Why did you lock the door?\"\n\nThe old advisor pulled a dagger from his bed robes. For a moment his cold rage turned to pity, and she wasn't sure which infuriated her more.\n\n\"You had such potential,\" he said, stepping toward her. \"Instead you've denied your duties. You've led us to ruin, destroying everything I've worked my entire life for. If there's any hope for us now, it is someone, anyone, taking over.\"\n\n\"Including you?\"\n\nHe looked insulted by the notion.\n\n\"I do this for the legacy of your entire family, generations before and generations yet to come. Not myself. Never for myself. I hope you understand.\"\n\nHe lunged for her, and she rolled to the side. Compared to Zusa or the Watcher he was slow, but she had no weapon, no dagger or club. She grabbed the sheets as she hit the ground beside her bed and flung them at Bertram.\n\n\" _Help!_ \" she screamed as she circled around, putting the broken window to her back. Bertram shoved the sheets aside, his dagger catching momentarily on them. He was between her and the door, and she thought to run past while he was entangled, but it wasn't a long enough distraction. He stepped over the sheets, his eyes locked onto her, watching, waiting for the slightest twitch so he might react. For an old man he seemed reenergized, his movements carrying purpose.\n\n\"It won't matter,\" he said. \"You will be gone by the time they arrive, and even if they execute me, I'll have removed the sickness within our house.\"\n\nShe thought of Nathaniel, of Arthur, and of Bertram's insistence on marriage. Deep inside she wondered just how involved he'd been with those events, and the anger that burned within gave her the courage to do what she might never have done otherwise. When he stabbed for her chest, she didn't dodge. Instead she flung herself at him, twisting to one side in hopes of avoiding the blow. The dagger's edge slashed her flesh, the pain almost unbearable, but her mind was full of fury and the rush of battle. Her left hand grabbed his arm so he couldn't stab again, the other clutching the front of his robes. She might have lacked the strength of a man, but Bertram was old and thin.\n\nWith a mindless cry she flung him face-first toward the shattered window, where many shards still remained. He cried out once, a surprised yelp that ended in a painful shriek. Still trembling with rage, she watched as blood poured across the glass. He'd been impaled by one of the thick shards, the pointed edge digging deep into the flesh below his throat. Bertram tried to suck in another breath, but it came in gargling and wet. His arms flailed uselessly at his sides, and he sliced his hands as he grabbed more shards trying to push himself off the window.\n\nBehind her the door broke as her guards smashed it open. They poured in once more, and this time they did not allow her to resist as they took her by the hand and led her out. She looked back only once, to make sure Bertram was still there, still bleeding.\n\nThe sickness within her house, she thought. Bertram was right. At last it had been removed. And then she broke down and cried.\n\n# CHAPTER\n\n#\n\nThe main quarters for the Serpents were unsurprisingly empty when Haern searched them. Given what he knew of William Ket, their leader, they would not dismiss him as lightly as the others. William took every threat seriously, and he dealt with them all as harshly as possible. Kadish was a gambler, a drinker, and a man too proud to stop celebrating when he should have been in hiding. William was easily the opposite, and would therefore be far more difficult.\n\nOf course Haern knew where they'd go. They'd retreated there several times, usually after one of the other guilds had engaged them in a skirmish over territory. This one was smaller than their main guildhouse, with only a single door and no windows. It had once been the armory for the city guards stationed deep in the south of Veldaren, before the king repositioned them farther north and sold the building.\n\nOn his way there Haern checked the cut on his side. He'd hidden it from Alyssa and her frightening female guard, though the guard might have noticed given her incredible skill. It was shallow, a flesh wound along his ribs. It was bleeding, though, as such cuts liked to do. Pausing to catch his breath, he leaned against the wall of a smithy and drew one of his sabers.\n\n\"Something tells me you'll get reopened a dozen times tonight,\" he said as he cut the edge of a cloak to make a bandage. It reminded him of trying to bandage that boy's arm while out in the snow. At least he knew who the boy was now: Nathaniel Gemcroft. No wonder he'd been a target. Of course, who had attacked him and why were no real concern. The Serpents were all that mattered right now. Assuming Deathmask handled the Spider Guild as he'd said he would, the Serpents were the only guild left to be dealt with one way or another.\n\nA shadow shifted in the corner of his eye, elongated in a way that was unnatural to the moonlight. His mind cried out in alarm, and he dropped to one knee just in time. A crossbow bolt smacked into the wood behind him and then ricocheted off. Haern drew both his sabers as he rushed toward the main street, where the buildings would be farther apart and he'd have more room to dodge.\n\nSome finely honed instinct in him lifted the hairs on his neck. He slid as he heard the twang of two more crossbows. Bolts flew above him, striking the ground harmlessly. At least three attackers, he realized, given how closely together they'd fired, with all three firing from the rooftops. Not good. Back on his feet in a heartbeat, he continued running, relying on the slow reloading of a crossbow to give him time.\n\nHis way was blocked, however. A man in the cloak of a Serpent landed before him, his dagger already drawn.\n\n\"Damn fool,\" the Serpent said.\n\n\"Same to you.\"\n\nHe crashed right into the man, positioning his sabers so they pushed the tip of his opponent's dagger to the side. Haern's forehead rammed the man's nose. Blood splattered across them both, blinding one of Haern's eyes. Trying to ignore the pain, Haern rolled atop him and into the wide street beyond. When he spun he saw bolts flying in. The first missed. A second passed an inch from his leg as he dodged. The third thudded into his side, and he gasped at the impact. _Is it poisoned?_ he wondered, but he dismissed the thought. Poison wouldn't matter if several more buried their tips in something vital.\n\nThe man he'd rammed struggled to his feet, and Haern reacted on instinct. He knew he needed to get away from the men with the crossbows, but he also couldn't leave an opponent free to chase. He lunged, slashed away the Serpent's pitiful defense, and then cut his throat. As he bled out, Haern dove around a corner. A single bolt was fired after him, missing by a foot. And then he was running. He sheathed one of his sabers so he could wipe at the blood in his eye, then glanced back. Three Serpents leaped after him, two men and a woman, their green cloaks trailing, the color an eerie haze in the moonlight.\n\nHe crossed to the other side of the street, giving them no choice but to climb down. Again he made a decision, this one more from pride than rational thinking. They'd hurt him, maybe even poisoned him. They had to pay. His reputation was all that would hold this arrangement with the guilds and Trifect together. If they felt they could wound him, make him run, then all would be for naught.\n\n\"Come on!\" he shouted, slamming his sabers together before charging. The three had abandoned their crossbows on the rooftop so they could climb down, and all of them drew short swords. As he charged they formed a triangle, trapping him in the center. He grinned at the maneuver. Clever, but it was too late to change his mind. His tactics, however, he could change. Instead of lunging at the first, as they thought he would, he spun in place, beginning his cloakdance. His feet twisted and spun, pushed to the very limits of his speed. He relied fully on instinct, for what his eyes saw through his cloaks were but snippets of his opponents. Thrust after thrust he smacked away, until one overextended, his attacker confused as to where he actually was within the cloaks. Haern broke out from the spin, double-slashing the man's arm. Down went the weapon as blood spilled and the Serpent screamed.\n\nBack into the dance, but just for a moment, just long enough to confuse the remaining two. No time left to mess around. The pain in his side was escalating, and his fingers tingled. Assaulting the nearest, he unleashed a furious display, his swords weaving around the woman's short sword as if it were motionless. He spun as he cut her, whirling toward the last Serpent in a single smooth motion. The thief blocked only one of the two sabers, the other taking his life.\n\n\"Damn it,\" Haern said, pausing to catch his breath. He gingerly touched the bolt in his side, wincing as his fingers made it move the slightest bit. Too deep: he'd have to push it through and pray there'd been no poison. Before he could, he heard movement above. Too late he brought up his sabers, but instead of the expected bolt a single Serpent fell bleeding to the ground, crumpling in a heap. From the rooftop Zusa waved at him.\n\n\"The one you don't see is the one that kills you,\" she called down to him.\n\nDespite his exhaustion he gave her a smile.\n\n\"Why are you here?\" he asked as she climbed down to the street.\n\n\"My lady wishes the Serpents punished, and so I've come to help you. Clearly you need it.\"\n\nHe gestured to the bolt in his side.\n\n\"Clearly.\"\n\nWithout giving him warning she stepped forward, grabbed the shaft, and pushed. He clenched his jaw and ground his teeth together to hold in his scream. Warm blood dripped down the small of his back. Zusa pushed aside his cloak, retrieved the bolt, and then held it close to her eyes so she might see in the moonlight.\n\n\"Not poisoned,\" she said. \"Either one of the gods favors you, or they were too stupid and lazy to prepare for you properly.\"\n\n\"Perhaps both?\" He grinned at her, but the grin faltered. \"Sorry about your arm.\"\n\n\"Sorry about your chest.\"\n\nSo she had noticed. He chuckled.\n\n\"If I stumble from the blood loss, make sure you kill me. I'm not sure who would be happiest to torture me, but I'd rather not find out.\"\n\n\"They'd probably auction the right. More money.\"\n\n\"Aren't you a cheery soul?\" He pointed farther down the street. \"Come on. The armory isn't far.\"\n\nHe led the way, Zusa trailing behind him, like a feminine version of his own shadow. Silent as ghosts they crisscrossed their way toward their goal. Haern checked the alleys and Zusa the rooftops for any more potential ambushes. At the armory they stopped and peered around the corner of a nearby home.\n\n\"No guards on the outside,\" Zusa whispered.\n\n\"That would give them away. They think this place is safe, otherwise they wouldn't come here.\"\n\n\"Nowhere is safe in this city.\"\n\n\"Well,\" Haern said, drawing his sabers, \"let's go reinforce that lesson for them.\"\n\n\"How many entrances?\"\n\nHe thought for a moment, then held up a single finger.\n\n\"They boarded up their windows. Together, through the door. No mercy, Zusa. Can you handle that?\"\n\nShe gave him a look that showed how insulted she was.\n\n\"I was raised in the heart of Karak's temple,\" she said. \"Mercy is not my bedfellow.\"\n\nAs if to prove the point she rushed ahead, and silently cursing her he followed. The door was locked, but when Haern went to draw his lock-picker's kit, she only shook her head. She mouthed something to him, but he only caught half the words. She wanted to try something, though, that much he understood. Putting her hands on the lock, she closed her eyes, and to him it looked like she was praying. Shadows slipped off her fingertips like water dripping from a melting wedge of ice. A moment later they both heard an audible click from within the lock.\n\nHer balance wavered, but when she regained it she shot him a wink. Haern rolled his eyes.\n\n\"Ladies first,\" he said, spurring her into action. She flung open the door and in he followed, two deadly phantoms in the night. A single thief waited on guard, looking half-awake. They cut his throat as they rushed past. He never even had a chance to cry out an alarm. They bashed through a door and into an elaborate room, one that instantly felt familiar to Haern. It was like so many others of the posh headquarters the guilds created, all curtains and pillows, alcohol and sex.\n\nTheir first warning something was wrong came when the door behind them slammed shut. The second was when William Ket greeted them with a warm smile from his chair on the far side of the room.\n\n\"Well, well, well, at last I meet the Watcher,\" he said, sounding far too pleased with himself. \"And you've brought a friend. Excellent. Did you think I'd be foolish enough to think you couldn't find me here, not with your... storied reputation?\"\n\n\"The curtains,\" Zusa whispered, her body tensed like a cat's before a pounce.\n\n\"I know.\"\n\nWilliam's grin spread.\n\n\"Alyssa called off her mercenaries, the silly girl. She had us on the run, but then suddenly she flooded Veldaren with bored, unemployed men with a penchant for violence. How could I not take advantage of such a gift?\"\n\nThe curtains pushed aside, revealing armored men standing in every little alcove. Haern estimated at least thirty. He felt his blood run cold. So this would be how it ended. His side ached, every breath hurt his chest, his head pounded from exhaustion, and standing before him, William Ket laughed.\n\n\"Don't you give in,\" Zusa whispered, her voice almost a hiss. \"They are children to you, you understand? We are the lions. We are the hunters.\"\n\nHaern thought of his moment in Karak's temple, when he'd been in the very presence of the Lion of Karak. It had roared, and he'd gazed into an emptiness that seemed to go on forever. He remembered the terror, and he realized that he'd been far more afraid then than he was now. Focusing upon that fear, he knew he could be that lion to these men. He looked at them as they waited for the order to attack, let them see in his eyes that same emptiness, that same certainty of their death. Pulling his hood low, he let the shadows of the torchlight scatter his features. Beside him Zusa wrapped her cloak tight about her body and then hunched low.\n\n\"Kill them,\" William said.\n\nHaern went left, Zusa right. He felt every nerve in his body firing, and he gave in to his instincts completely. This was the beast Thren had created over the years, day in and day out with training, practice, lectures, and tutors. This was the monster whose teeth had been sharpened by half a decade skulking in the shadows, slaughtering the thieves of the night. His sabers were a blur as he cut down the first, the mercenary's ax too slow to block. The two closest rushed in, wielding long swords. He parried their thrusts, which felt slow, as if his opponents fought in molasses. Blood soaked his sabers as the rest came rushing in, swinging with their clubs, maces, and swords.\n\nCutting, twisting, never staying in the same place. As his feet shifted and turned he thought of the hours he'd been forced to stand in strange stances to pacify a tutor. As he curled his body around thrusts, he remembered the complicated stretches another tutor had taught him to do every morning. As he slashed and dodged, he thought of the words of his father.\n\n_They can't kill you until you let them. That is why you must be better. That is why you must be perfect. Never, ever let them think they can win._\n\nSaid to a thirteen-year-old boy. More than anything, he wished his father could be there to see what he had created. One after another the mercenaries fell. They knew how to bully. They knew how to put the strength of their arms into their blows, and they could handle the rudimentary thrusts and parries of the battlefield. But Haern felt himself beyond them, beyond anything. They scored cuts on him, to be sure, but he felt the pain in a distant place locked in the back of his mind. They would not kill him. He would not let them. His wrist might bleed from a lucky stab of a sword. His chest might ache where a club struck him before he could dodge. His eyes might sting from blood running into them from where a blade had slashed his forehead. But they would not kill him.\n\nZusa's cry pulled him back from the animal, from the mindless killer. Despite the many dead she was overwhelmed. Refusing to give the thieves anything, Haern descended upon them. Their backs were turned to him, and he thrust and stabbed and kicked, shoving them aside so he might link up with Zusa. She was bleeding, and so was he, but they grinned.\n\n_We were made for this_ , he thought.\n\nBack to back they turned to their foes. Of the original thirty, only ten remained. Blood and gore soaked the floor where it wasn't covered by a body. The psychological damage was just as bad. None looked ready to attack. Whatever they had been paid, it wasn't enough. The first turned to flee, and as if a dam had broken, the rest rushed for the door. Ignoring them, Haern looked for William, not finding him.\n\n\"Where is he?\" he asked.\n\nZusa rushed to the chair he'd been sitting in and flung it aside. Hidden behind it she found a ring, and she pulled, revealing a trapdoor. Haern followed her as the mercenaries broke down the door behind them and poured out into the night. The trapdoor led to a tunnel, tight enough that Haern had to crawl along on his elbows, worming his way through. It wasn't a long tunnel, and Zusa pushed open another trapdoor and then helped him out.\n\nThey emerged behind the armory, the trapdoor hidden by a compacted layer of dirt. Haern felt his muscles aching, the familiar feeling of receding energy after a fight coming over him. He'd expected to search for William, to have to hunt for wherever he'd run off to, but instead saw him lying dead in the street, two men standing over him.\n\n\"You look like shit,\" Senke said, still cleaning William's blood off his mace.\n\nHaern tried to think of a response, but only stared dumbly at him and Tarlak, who looked vaguely amused by the whole ordeal.\n\n\"Delysia spent the better part of tonight begging us to help you,\" the wizard said, his arms crossed. \"And as usual, I finally gave in.\"\n\n\"How?\" Haern asked. He'd meant to ask how they had found him, but breathing suddenly seemed difficult. His body was finally taking account of all the blows and cuts he'd received, and it wasn't happy.\n\n\"What, find you?\" Tarlak asked. \"I'm a wizard. That's just what I do.\"\n\nHaern saw Zusa down on one knee, bracing herself with one of her arms. Her normally dark skin was disturbingly pale.\n\n\"Are you hurt?\" he asked, stepping to her side.\n\n\"Of course I am,\" she said. \"Farewell, Watcher. I have done as my mistress asked. Let your friends help you from now on.\"\n\nShe rose to her feet, took an uneasy step, then another, and by the time she was running her balance looked like it had returned. Haern watched her go, hoping she'd be all right.\n\n\"So,\" Tarlak said, smacking him on the back. \"What's next on the agenda?\"\n\nHaern looked back at the body of William Ket, and he mentally checked another off the list in his head. One left, just one.\n\n\"Leon Connington.\"\n\nSenke whistled. \"Going after the big dogs, are we? Who else after that?\"\n\nHaern shook his head. \"He's the last. Everyone else has agreed, or...\"\n\nHe gestured toward the body.\n\n\"The last?\" Tarlak laughed. \"Aren't you a freak? Well, let's go. Leon's not exactly close to here.\"\n\nThey walked down the street, and for a moment Haern let himself relax. With the three of them, one a wizard, any thieves would have to be incredibly brave or reckless to consider an ambush. He used his shirt to wipe the blood from his forehead, then pressed it against his eyes. They watered, but when he pulled the shirt away he could see better. Senke twirled his two maces in his hands, and Haern wished he could feel as energetic as Senke looked. He might have just been the lion, but now he felt like a lamb, ready to give up everything just to lie down and sleep. Every single part of his body ached.\n\n\"How long until dawn?\" Haern asked.\n\n\"About two hours,\" said Tarlak. \"You been at this the whole night?\"\n\n\"Just before sunset, yes.\"\n\n\"We of the magical profession call that biting off more than you can chew.\"\n\n\"And we of the stabby profession call that getting yourself killed,\" said Senke.\n\nHaern winced as an awkward step flared pain along his chest and to his back.\n\n\"You two are such wonderful help,\" he muttered.\n\nLeon Connington's estate was one of the best-guarded places in the city, and all three of them knew it. The warning letter Haern had sent certainly hadn't given them reason to slack off, either. Tall stone walls surrounded the mansion, the single opening a thick iron gate with two guards. They stood at attention, no slacking there either. From far down the road Haern and the others observed the gate and planned.\n\n\"There will be mercenaries stationed throughout the mansion,\" Haern said as they stared. \"And traps along the ground, other than the path leading directly to the door. If we're to get to Leon, I think we'll need to be stealthy about this.\"\n\n\"Stealthy?\" asked Tarlak. He gestured to his bright yellow robes. \"Stealthy?\"\n\nHaern gave him a dumb look, then shrugged.\n\n\"Any other ideas?\"\n\nThe wizard lifted his arms high, and a steady stream of magical incantations slipped from his lips. Fire burst about his hands, growing, growing, and then soaring toward the gate as an enormous ball. It hit the iron and detonated, blasting the gates aside and tearing off chunks of stone. Haern didn't see what happened to the guards, and he didn't want to think about it either.\n\n\"Stealthy,\" said Tarlak, hurling a smaller ball of fire that rolled across the ground. It detonated the various traps along the grass leading toward the mansion, filling the night with the sound of their explosions. Haern didn't know whether he should laugh or cry.\n\n\"Stealthy?\" he asked Senke, who only shrugged.\n\nTarlak sent one more blast, this one aimed at the front door. He frowned as the spell evaporated into smoke just before contact. He sent another at a window, this a thick shard of ice. Again it broke, this time into water that showered the ground harmlessly.\n\n\"Strong wards,\" the wizard said. \"Looks like the rest is up to you. Have fun!\"\n\nSenke led the way, Haern following.\n\n\"Out of his damn mind,\" Haern muttered.\n\nTarlak watched them go, offering a prayer for luck. He wished he could help, but the few spells he'd cast had put a deep ache in his head, and he knew he had but a few more before he'd be worthless. Unable to help it, though, he neared the gates to observe his handiwork.\n\n\"Getting better,\" he said, estimating the size of the explosion.\n\n\"Tarlak Eschaton?\"\n\nHe turned, and with mild surprise saw the giant man with the painted face approaching from down the street.\n\n\"I'm thrilled we could meet again,\" he said. \"Especially with my mouth un-gagged.\"\n\nGhost pointed toward the mansion. \"Is the Watcher inside?\"\n\n\"He is,\" Tarlak said, standing in the center of the gate. \"He's a bit busy right now, so you'll have to wait until tomorrow to resume whatever grudge you have against him.\"\n\n\"No grudge,\" Ghost said, still approaching. \"Just money.\"\n\nTarlak snapped his fingers, summoning a spark of flame at his fingertips.\n\n\"No closer,\" he warned. Ghost only laughed. \"Well, don't say I didn't warn you.\"\n\nHe slammed his hands together, and a ring of fire rolled out from his waist, burning the air with a heavy roar. His opponent fell back, and he landed on his shoulder so the fire could pass harmlessly above him. Tarlak gave him no reprieve, another spell already on his lips. This time it was ice, thick shards that flew like arrows. Ghost rolled, the shards shattering upon the ground behind him. Only one drew blood, a thin gash along his side. On a man so giant, it looked like a cat scratch.\n\n\"How long?\" Ghost roared, back on his feet again and lunging. Tarlak tried to ignore him, kept focused on his spellcraft, but he knew what Ghost was asking. How long might he last casting his spells? How long until the well of energy within him ran empty, and the best he could summon was a little puff of smoke from his fingertips?\n\nGiven the pounding of his head, he didn't think it'd be long.\n\nHis hands clapped together, and the space before him filled with a swirling wall of smoke and fire. Ghost's swords passed through it, but his feet dug into the ground, halting his momentum and preventing him from continuing into the inferno. Tarlak muttered. He'd hoped for a charred corpse to leap through. How the Abyss did this guy react so fast?\n\nLeaving the wall of fire intact, Tarlak guessed a direction and pointed. This time luck was with him, for of the two directions in which Ghost might have leaped, he'd chosen correctly. A bolt of lightning shot from his finger, striking the giant man square in the chest. He fired a second one, this one hitting his leg. Ghost screamed, but more in anger than pain. Tarlak felt the hairs on his neck stand on end. Short of taking the man's head off, it didn't look like there'd be any way to stop him.\n\n\"You hurt my friend,\" Tarlak said, summoning small meteors of lava and flinging them. Ghost hunched on his knees, blocking with his swords. The meteors plinked off the steel, coupled with an impressive but harmless shower of sparks.\n\n\"You hurt my sister,\" he said, pressing his wrists together and hurling shards of stone from his palms. Ghost jumped and leaped like an enormous spider. Only two shards hit, and again the wounds were superficial.\n\n\"You even hurt Brug.\"\n\nHis bolt of lightning shot out, but his aim was off. Ghost didn't dodge this time, instead lunging straight for the kill. A sword slammed into Tarlak, piercing his flesh. He gasped as the white-hot pain spread throughout his body.\n\n\"And I hurt you,\" Ghost whispered, his cheek pressed against the wizard's.\n\nOut came the blade, and Tarlak collapsed. Unable to stop him, he could only watch as Ghost passed through the gates, continuing the hunt for his real prey. The blood flowed, staining his yellow robes red. His mind throbbing from pain and exhaustion, he crawled across the ground, bleeding upon the street as he headed for safety.\n\n_Damn you, Haern_ , he thought as he collapsed after hardly crossing any distance. _You better kill him for me, or I'll... I'll..._\n\nAnd then he felt his thoughts slipping away like leaves in a storm, and unconsciousness came and took him.\n\n# CHAPTER\n\n#\n\nDeathmask knew he might be walking into an early grave, but he didn't let worry show on his face, not with the rest of his guild watching him.\n\n\"Keep an eye out for anything suspicious,\" he said to the others. \"I don't expect him to do anything stupid, but it is Thren Felhorn, after all. Stupid to us is step five of a plan for him.\"\n\nThey approached the headquarters of the Spider Guild. It was more a mansion than anything else, though careful examination would have shown how the windows were reinforced so no one could break through, and all doors but the front were boarded over. Two men in gray waited at the front, and they drew their swords and daggers at his approach. Veliana glared at them, but she remained quiet.\n\n\"I am Deathmask, leader of the Ash Guild. I've come to speak with Thren.\"\n\n\"Only if Thren says,\" one said. The other banged on the door. A small window opened, and the guard relayed the message to someone inside. A few minutes later the door opened.\n\n\"Just him,\" said one of the Spiders from inside, pointing to Deathmask.\n\n\"It's all right,\" he told Veliana, who looked ready to object. \"I can handle myself.\"\n\nHe stepped inside.\n\nThe interior of the mansion might have once been well decorated, but nearly all its original treasures had been plundered by the Spider Guild and sold off. Bright squares on the walls showed where paintings had once been, and in many places the floor was scraped and dull, as if the carpet had been ripped up, or a long-present rug removed. Deathmask tracked the turns and doors to ensure he could find his way back, all the while going over every bit of information he knew about the near-legendary leader of the Spider Guild. At last they reached a door, and the thief gestured. Deathmask opened it, stepped inside, and closed it behind him, finding himself alone in a small den with Thren Felhorn.\n\nThren looked old. That was the first thing that struck Deathmask. He knew the man's age, still in the late forties, but his hair was fully gray. His skin had a tight, stretched look about it, but his eyes still shone with intensity. He stood beside a fireplace, a drink in hand. His two short swords hung at his sides, their hilts gleaming in the light. He smiled at Deathmask, but the smile hid a strong sense of impatience and contempt. Thren surely knew the reason for his coming, and was not pleased.\n\n\"Welcome,\" Thren said. His voice was deep, and the power in it impressed Deathmask to no end. He wished he had such a commanding voice as that. The man could probably describe himself taking a shit and still make it sound authoritative. \"I've heard rumors of your assuming control from Garrick Lowe, not that there is much to assume.\"\n\n\"What is it we say to the ladies, it's not the size of the sword, but the skill in the wielding?\"\n\nThren chuckled. This was good. If he could get the man to feel a sense of kinship, things might go more smoothly.\n\n\"Maybe so, but even I wouldn't assault a man with a spear wielding only a butter knife.\"\n\n\"You know you would, Thren, if the price was right. You'd cut the man three times before he knew where you were, too.\"\n\nThe flattery didn't get him what he'd hoped. Thren waved a dismissive hand and set down his drink.\n\n\"Enough. The night is late, and you didn't come here to banter, nor to make introductions. This is about that Watcher madman, isn't it?\"\n\n\"I must admit, I am curious as to your thoughts on the deal.\"\n\n\"Deal? _Deal?_ This is no deal. This is enslavement. This is the king severing our testicles and selling them to the Trifect. Do you know how this world works, Deathmask? The strong take what the weak cannot hold, and that is the proper order of things. The foolish and the na\u00efve try to prop up the weak, to protect them with strength that is not their own. Babes, all of them, nothing but babes forever suckling their mother's milk.\"\n\n\"We would still make plenty of coin,\" Deathmask said. \"And we have accepted protection money before. Is that not a way of the weak voluntarily giving up what they have to the strong?\"\n\n\"Never on this scale before,\" Thren insisted. \"They don't just protect their own, but the entire city. What insanity led to this? I have watched them bleed before me. Entire nations could live and die on the wealth I have taken from the Trifect's safes. Yet now they throw gold at me in a pitiful attempt to barter safety and peace of mind. At least Alyssa was willing to fight back, though even that moment of pride lasted only two nights before cowardice returned, no doubt replaced by this deal from the king.\"\n\nDeathmask saw an open bottle on a small stand, and he walked over and poured himself a drink. He sniffed it once, and was pleased by the scent of strawberries. Thren didn't object, so he took a drink and set it aside.\n\n\"This is how I see it,\" he said. \"It's been what, ten years? A man can only fight for so long, even the greatest of us. We need a break. We need a return to some shred of normality.\"\n\n\"Says the man wearing a mask.\"\n\nDeathmask laughed. \"Relative normality, then.\"\n\nHe watched Thren carefully, though he knew it was pointless. The man could guard his emotions better than anyone, probably better than even him with his mask on. Thren was watching him as well, gauging his reactions, staring into his eyes as if he could divine the true purpose of his visit.\n\n\"This Watcher...\" Deathmask said. \"He claims he'll kill everyone who refuses. Do you think he'll succeed?\"\n\n\"You and I are alive,\" said Thren. \"It seems to me he is doing a poor job. And it doesn't matter. He could kill everyone, but he won't kill me, and as long as I survive, the Trifect will never have a moment of peace.\"\n\nDeathmask tapped his forehead with a finger.\n\n\"As long as you are alive... that's the clincher right there, Thren. Don't tense up, I'm not here to kill you. That wasn't a threat, just a statement. This war is yours, solely yours, and it is yours to end as you see fit. But you won't have the ending you're hoping for. The Trifect is too big. Yes, you've hurt it, killed many, and taken away their coin. But has it mattered? If an opponent is not allowed to surrender, they'll keep fighting and fighting. Give them the option of defeat. That's what this deal is, if you look at it from their perspective. They admit they cannot defeat you, cannot protect themselves from you. So they make it worth your while to instead do the protecting for them. It's a bribe, nothing more, nothing less, and in a city where this is hardly an unusual circumstance.\"\n\nThren looked tired of the debate, and Deathmask knew he was treading on thin ice. He'd lied, at least partially, when he said he hadn't come there to kill Thren. Could Thren have read him correctly, despite his best attempts at subterfuge? More than anything he wanted a victory here without bloodshed. Other thief leaders could come and go, but if Thren died the Trifect might decide it didn't need protection after all. Thren's power had shrunk considerably over the past few years, but his reputation had not.\n\n\"Aren't you tired of this?\" he asked, letting his voice soften. \"Every man and woman in this city has lost someone these past ten years. Despite the rumors, I know you are human, and lost as much as any.\"\n\nFor a moment, so quick Deathmask thought he might have imagined it, Thren allowed himself to look exhausted, look torn with despair.\n\n\"It's for that loss I continue,\" he said. \"Why else would I go on? To accept anything less than total victory would be an insult, not just to myself, but my wife, my...\"\n\nHe seemed to return to his senses, and he glared at Deathmask as if he were the reason for the sudden weakness.\n\n\"I will not agree,\" he said. \"And if that is your sole purpose here, get out now.\"\n\nDeathmask chuckled. The slightest misstep might cost him his life. But this was it. This was the heart of everything.\n\n\"That Watcher, I hear he is good, almost impossibly good. I also hear he fights like you. Did you know that? As if he might be your own son, but we both know that couldn't be. He died in a fire, of course. I'm sure you saw his body...\"\n\nHe looked to Thren, letting the guildmaster know there was far more he wasn't telling. No lie. No bluff. Thren opened his mouth, then closed it. Those blue eyes barely moved. _What firestorm of thought rages behind them?_ Deathmask wondered. Taking a deep breath, he tried his wildest gambit.\n\n\"If he succeeds, the Watcher will be a legend. He'll have beaten both the Trifect and the thief guilds, all in a single night. He'll have ended ten years of conflict with a stroke of his swords. The entire city will fear him, for he will be the King's Watcher, enforcer of the truce. The night won't belong to us anymore. It'll belong to him.\"\n\nHe swallowed. Now or never. Take the risk.\n\n\"He'll have surpassed even you, Thren. How amazing must that man be?\"\n\nThren looked like a heavy burden had settled upon him. His muscular frame wasn't quite so strong anymore. The terrible will that had ruled him weakened, and a million questions died unspoken on his lips. For perhaps the first time ever, Thren Felhorn looked uncertain.\n\n\"Did he send you here?\" he finally asked. Deathmask nodded. \"So be it. Give him his chance. My guild will accept the terms, so long as the Watcher lives. This city is a cruel one, and even now it might have claimed him.\"\n\n\"I doubt it,\" Deathmask said, his heart pounding in his chest. \"Given who he is and who made him. Come the morning we'll count the bodies, and we'll see what remains of those in power. I have a feeling, though, that tonight is when it all ends.\"\n\n\"Leave my home,\" Thren said. \"And never speak a word of this to anyone, or I will kill you.\"\n\nDeathmask bowed low.\n\n\"As you wish,\" he said, glad the mask could hide his enormous smile. More relieved than he'd ever been in his life, he exited the room, wove unguided through the halls, and emerged from the mansion, alive and victorious.\n\n# CHAPTER\n\n#\n\nBecause of the breastplates the guards wore, Senke led the way, his flanged maces able to punch through if swung hard enough. Haern followed, watching behind them as much as ahead. The entire mansion was in chaos. Servants fled every which way, and several times he heard them cry out the names of thief guilds. His lips curled into a vile grin every time. No thief guild, not this time. They were worse than any guild. They'd come, live or die, to complete their mission, though so far it was only the guards who were doing the dying.\n\n\"Where might Leon be hiding?\" Haern asked as he yanked his saber free from the armpit of a dead mercenary.\n\n\"Holed up in his bedroom?\" Senke suggested. \"He's not the most mobile of men.\"\n\n\"And where would that be?\"\n\nSenke gestured ahead, and then behind them.\n\n\"Your guess is as good as mine.\"\n\nThe cries of \"Intruder\" followed them as they rushed along. They shoved servants aside if they got in the way, but most were smart enough to cower or turn and run.\n\n\"Bottom floor,\" Haern said as they passed a set of stairs. \"I can't imagine him climbing those every night.\"\n\nSenke opened another door, then slammed it immediately shut, and on the other side arrows sank into the wood with heavy thunks.\n\n\"I think we're getting somewhere,\" Senke said, and he grinned.\n\nThey backtracked, weaving through the corridors so they might curl around their ambushers. Sure enough, they found them at the intersection of another hallway, kneeling behind small overturned tables that had once housed vases of immeasurable wealth. All three held crossbows and wore boiled leather armor. Senke crashed into two, Haern the third. Making quick work of them, they turned left and continued along.\n\n\"It should be harder than this,\" insisted Senke, needing to shout to be heard over the commotion.\n\n\"Don't say it, or it might come true.\"\n\nIn the next hallway they met six mercenaries, all wielding short swords and small circular shields of wood latched together with iron. Senke laughed and rushed the six with wild glee, as if he could see Haern's glare behind him. Despite his exhaustion, Haern couldn't help but feel energized, and he raced to Senke's side so they might crash into the mercenaries in a single brutal collision. Every turn might house more men ready to kill them. Every door might hide archers ready to shoot barbs into their throats. And neither could care less.\n\nThe shields proved difficult, mostly because Haern had little experience dealing with them. A shield was hardly standard issue for the men who stalked the night. He kicked and stabbed as he and Senke slammed into the mercenaries, cutting the tendons of one guard's arm and tripping another. Before Haern could finish this one off, another was there, and his saber slapped harmlessly against the wood of his shield, not even drawing a splinter from the finely polished surface. The soldier thrust for his midsection, but Haern parried it aside with his left hand, leaped closer to the wall, and then kicked off it to give his maneuver speed. His saber crashed into the guard's neck, punching through the leather armor and into flesh.\n\nSwords stabbed for where he should have been, but he dropped to the ground and rolled. Senke, as if in some mental link with him, saw and jumped over him, blocking blow after blow with his maces. Haern leaped to his feet, slamming his left shoulder against the wall to painfully kill the rest of his momentum. Only one guard remained within reach, and Haern desperately shoved one of his sabers in the way. The guard's short sword deflected the saber and stabbed the wall, close enough that Haern could see his reflection in the blade. And then his sabers were thrusting in, and the shield could not block all his attacks.\n\nSenke took down the last mercenary, hammering his shield with his maces until the guard made a mistake, not surprising given how the rest of his fellows had fallen and panic was surely crawling through his veins. His sword slashed, but he over-extended, and Senke broke his elbow with an upward swipe of his mace. A kick to his neck blasted him against the wall, and he slid to the ground unconscious.\n\n\"You hurt?\" Senke asked. Haern shook his head. \"Good. One of those sons of bitches cut my leg. Delysia's going to be pissed at me.\"\n\nDeeper and deeper in they went, until at last they found Leon's bedroom. It was empty.\n\n\"Slap me silly,\" said Senke, looking around. \"Where could that giant tub of lard have gone off to?\"\n\nHe took a step forward, not seeing the thin string laced across the door. Haern did, and he pulled Senke back by his cloak just before the entire room erupted in flame. The fire swirled about in a momentary funnel before fading away, leaving nothing but ashes inside, the rest of the house safely intact.\n\n\"A trap?\" Senke asked, his eyes wide. \"A fucking _magical_ trap?\"\n\n\"You're welcome,\" Haern said. He leaned against the wall and closed his eyes, wishing he could just order away the headache pounding in his forehead.\n\n\"Damn traps. Where to now? He might have left, Haern, and then what the fuck do we do?\"\n\n\"Stay calm,\" Haern said, eyes still closed. \"He slept here until an alarm sounded because of your wizard friend, and so he gets up, activates the trap. He's in a hurry, but not moving fast. The rest of his guards are ushering him along. Where do you go? Where is safe, close, and defensible?\"\n\n\"You take him where no one could have gotten to yet, where there couldn't possibly be an ambush waiting. You take him to the mercenaries' quarters.\"\n\nHaern opened his eyes and shot his mentor a wink.\n\n\"Good a guess as any. In the back, and away from the other quarters. He wouldn't want their low-class manners upsetting any of his privileged guests.\"\n\n\"You going to make it, Haern?\"\n\n\"Worry about yourself.\"\n\nThey rushed along, and this time Haern pushed himself to the front. Despite the help, this was still his task, his responsibility. If anyone should be walking into traps, it should be him. But there was only one trap left, and they sprung it together. Finding a long corridor leading to a thick set of double doors, they rushed into it only to have doors behind them fly open. Out rushed mercenaries, five in all.\n\n\"Leave them to me,\" Senke cried. \"Go after Leon, now!\"\n\nHaern accepted the order without delay. He rushed toward those double doors, and when he reached them he leaped feetfirst, wanting to go crashing in with a frightening display of skill and strength.\n\nHis foot slammed into the door, followed by the rest of him, and then he dropped to the ground. His whole body aching, he realized the doors opened outward only. Feeling far more humble, he grabbed a handle and pulled. Instead of making a vicious display of skill and strength, he walked inside a hurt, calm, exhausted man.\n\n\"You,\" said Leon from the far side of the room. Rows of bunks were built into either side. Four personal guards stood before him, forming a human wall of protection.\n\n\"Me,\" Haern said, bowing low.\n\n\"Who is paying you for this?\" Leon asked. Sweat dripped down his thick neck, and blotches covered his face. To Haern he looked like a pig that had been overfed and then stuffed into fine clothing. \"Thren? Alyssa? Maybe the king? Tell me, what did they offer you?\"\n\nHaern laughed. He couldn't help it. Would Leon even believe the truth? Could a man in his position understand there were things beyond wealth and influence? Could he understand a desire for atonement, for a single moment of rest and relief from a life devoted to slaughter and revenge? Or would he just see a madman? Would he hear only nonsense and lies?\n\n\"I do it because I want to,\" he said, figuring that if there was anything Leon might understand, it would be that. \"And you have the ability to make me not want to. Last chance, Leon. Accept the terms, or accept my blades.\"\n\n\"Neither. You're just a rabid dog, and my men will put you down.\"\n\nTwo of the guards pulled out crossbows. In a single smooth motion Haern unclasped the cloaks from his neck and spun them into the air, just before the guards pressed the triggers. Twisting behind the cover, he made himself as small a target as possible. The arrows punched holes through the cloaks and sailed on, neither hitting flesh. As the cloaks fell, Haern rushed the mercenaries, his sabers feeling light as air in his hands, just extensions of his body, keen edges of his will. This was it. This was the last. His night was done. The men would die, Leon would die, and he would have his truce.\n\nThe two abandoned their crossbows and drew swords, falling behind the others, who pushed ahead. There was only enough space for two to stand side by side, and that only barely. Haern used his greater mobility to his advantage, weaving like a snake preparing to strike. He smacked down every thrust and then struck with the other saber, cutting thin slashes across their faces and necks. Each hit made them angrier, until at last they tried rushing as one.\n\nHaern wrapped an arm around the post of a bunk, whirling across the mattress and to the other side. A whirlwind of steel, he cut down two mercenaries from behind, then turned on the other two, who were unprepared for the sudden assault. A third fell before lifting his sword into position, and one versus one, the last stood no chance. He was only a sellsword, and had maybe killed a handful of men in his lifetime. Haern had killed twenty just breaking into Leon's mansion.\n\nWhen Leon realized he was alone, he fell to his knees and pleaded in his high-pitched voice.\n\n\"Please, you're a reasonable man. You can listen, yes? I'll pay you, double, triple whatever you were offered. That deal of yours, that's it, right? I'll accept, of course, anything you want!\"\n\nHaern approached him, his sabers dripping blood.\n\n\"You're lying,\" he said. \"I see it in your eyes, your lips, your trembling hands. Besides, I'm just a rabid dog.\"\n\nHe cut Leon's throat, and he watched the life leave the fat man's eyes as the door behind him opened.\n\n\"He dead?\" he heard Senke ask.\n\nHaern turned. He wanted to smile, but he felt exhausted, and he knew getting out of the mansion might not be any easier than entering. Senke stood in the doorway, and he seemed happy enough, but something was wrong. Something was moving...\n\nAnd then the sword pierced the front of Senke's chest. The man arched back, his eyes wide. His limbs trembled, and blood dribbled from his lips. As his body collapsed, slipping free of the blade, Haern was too stunned to even scream. Behind him, now occupying the doorway, stood Ghost, the white paint on his face speckled with red. His grin was as wide as Senke's had been.\n\n\"I found you,\" he said, his deep voice rumbling in the confined room.\n\n\"Why?\" Haern asked. It was the only question he seemed able to think. \"Why? Why now?\"\n\n\"Because I have a reputation to keep, Watcher. I've been paid to kill you, and so you'll have to die. It's the way things work.\"\n\nHe lifted his swords into position, and slowly, as if in a dream, Haern did the same. In the back of his mind he felt anger building and building, like it belonged to someone else yet would soon be given to him whether he wanted it or not.\n\n\"You monster,\" he said, crouching into position.\n\n\"Monster? I see one body at my feet, Watcher, and five at yours. How am I the monster?\"\n\nWhat could he say to that? That his kills had a pure motive? That he wasn't motivated by greed? The arguments felt hollow, petty. They were two killers, and they eyed one another with an understanding so few could know.\n\n\"Then I'm the monster this city needs,\" Haern said. \"But we don't need you.\"\n\nGhost lunged, no doubt hoping to catch him off guard while he talked. Haern was better than that, though still his heart leaped in his chest. How could the man be so huge and yet so fast? With little ground behind him, he refused to retreat. His sabers met the swords, and they rang with deafening volume. Haern's tired arms jolted with pain.\n\n\"Need?\" Ghost asked, and his voice washed over Haern like a physical wave. With every word he struck again, hammering away at Haern as if Haern were a door barring his way. \"This city needs its eyes opened. It needs its cowardly heart ripped from its chest and held up to the light. What it does _not_ need is some damn fool vigilante.\"\n\nSo fast were his movements, and so strong, Haern could only twist and parry without hope of retaliation. The few times he blocked, he felt the impact travel all the way up his arm. Even at the peak of his skill he might have struggled to win. Now, after a full night without rest, his nerves frayed, his energy spent, he had only one last desperate gasp to hold on to, fueled by the corpse of Senke slumped beside the door.\n\n\"No,\" he whispered, a denial of everything before him. Of failing so close to his goal. Of letting Senke's murderer go unpunished. Of succumbing to the anger in those brown eyes surrounded by paint and blood. Of dying.\n\n\"No.\"\n\nAt the end of the room was a single large window, and Haern turned toward it, running with a speed Ghost could not hope to match. He crossed his arms, ducked his head, and leaped through. Glass shattered, and he felt its edges cut into his flesh. It didn't matter. Hitting the ground, he rolled, then dug his heels into the earth. He glared back at the window, suppressed anger bursting free with a fire he felt sear his veins. Not caring about the blood, not caring about the jagged edges still lodged in his arms and forehead, he took two steps and leaped back through.\n\nHe caught Ghost pulling up before the broken glass, and his sabers slashed an X across his muscular chest. Their bodies collided. Haern's knee rammed into Ghost's groin. His forehead slammed the man's neck. The glass lodged in his head tore skin, blood ran freely, but several shards ripped into Ghost's throat. Despite Haern's momentum and surprise, Ghost refused to go down. He held his ground, matching Haern fury for fury. With no room to cut or thrust, he punched Haern in the chest with a hilt, then caught his chin with a roundhouse. Feeling a tooth fly loose, Haern dropped to his knees and rolled forward. His sabers slashed out, cutting the tender flesh above Ghost's heels. The giant man's shriek rewarded his efforts.\n\nBut Haern wasn't done. Tears filled his eyes, born of pain both physical and from the torment of Senke's corpse refusing to fade from his sight. He kicked back into Ghost then stabbed his sabers. Warm blood poured across his hands. Steel pierced flesh again and again. Ghost crumpled to his knees, then fell upon a gore-filled smear atop the bare floor. Haern hovered over him, one eye swollen shut, the cut on his chest reopened, his face rivulets of blood from shards of glass, his clothes equally soaked. And then he screamed, the saddened, burdened, victorious monster.\n\nSlowly the sane part of him returned. He thought to carry Senke's body, to make sure they could bury him properly, but he knew he lacked the energy. Limping over, he knelt and kissed the man's forehead.\n\n\"I'm sorry,\" he whispered. Thinking of a distant memory, he reached underneath Senke's bloody shirt and retrieved a pendant\u2014that of the Golden Mountain.\n\n\"I hope you're with him now, Senke,\" Haern said, slipping the pendant around his own neck. \"Think well on me. Might not be long before I need you to plead my case to enter within.\"\n\nBeside him Ghost let out a moan, somehow alive despite his many wounds. Blood dripped from his lips as he pushed his head up. Haern slowly stood, and he pointed a blade toward the man's neck. He felt his rage ebbing, leaving behind something hollow deep within his chest.\n\n\"Do you want mercy?\" Haern asked him.\n\n\"Not...\" Ghost coughed, the sound wet and certainly fatal. \"Not from you.\"\n\nHaern sheathed his sabers, went to the broken window. Ghost's head settled back on the bloodied floor, and he let out a long, shuddering breath.\n\n\"Suffer as you wish,\" Haern said before stepping out. \"But it's your choice.\"\n\nHe stayed close to the house, wary of any more traps. At the front he followed the path. The gateway was empty, and dimly he wondered where Tarlak had gone off to. He stood there dumbly, looking, and then saw him two blocks down the street, his yellow robes rather hard to miss. As he approached, he saw that Tarlak was slumped against a building.\n\n\"Had to get away,\" the wizard said, sounding drowsy. \"Just in case he... just in case he came back.\"\n\nAll across the front of his robes was an ominous circle of blood.\n\n\"How bad?\" Haern asked, kneeling beside him so he could check the wound.\n\n\"Not bad,\" Tarlak said, his eyelids drooping. \"Better than you, from what I see. Where's Senke?\"\n\nThe name nearly made Haern choke. Every last bit of his self-control kept him speaking, kept him moving.\n\n\"He won't be coming back,\" he said.\n\nTarlak heard this, started to ask something else, then remained quiet. Tears fell from his eyes.\n\n\"He'll be with Ashhur now,\" he whispered.\n\n\"Come on,\" Haern said, putting an arm around him to help support his weight. \"We'll be joining him if we don't hurry. I think there's about to be a lot of angry people on the street.\"\n\n\"I think I agree.\"\n\nThey limped down the street, and whether through luck or the grace of Ashhur, they made it to the Crimson and Delysia's healing hands without any further trouble.\n\n# CHAPTER\n\n#\n\nCome the morning, Alyssa awoke feeling like her temples were ready to explode. The dim light hurt her eyes, and she covered them with an arm.\n\n\"Milady?\" she heard someone say.\n\n\"What is it?\" she asked. \"Can it not wait?\"\n\n\"Forgive me, milady. My name is Cecil Glenhollow, and I come with a message from Lord John Gandrem of Felwood.\"\n\nAlyssa removed her arm and glared. The knight stood over her, looking a mixture of awkwardness and impatience. She wondered what fool of her guard had let the man come to her, especially with her so indecent. She pulled her blankets tight about her and sat up.\n\n\"Whatever business you have, it can wait,\" she said. \"Have my servants prepare you some food, and my guards will\u2014\"\n\n\"My lady,\" said Cecil, \"it is about your son.\"\n\nHer mouth dropped open, and then she saw the parchment in the knight's hand. She took it from him and unrolled it. Her eyes scanned, not reading, only looking for the one sentence that meant everything. She missed it the first time, but there it was, just the second line of the entire thing.\n\n_I believe you will be pleased to know that, contrary to what you have been told, your son Nathaniel is in my company, and alive and well._\n\nAlive...\n\nShe flung her arms around the knight and hugged him as tears wetted her face.\n\n\"Thank you,\" she whispered. The knight stood shocked-still, as if unsure what he might or might not do to avoid insult. Pulling back, she kissed the man's scruffy cheek, then rushed for her bedroom, not caring that she wore only loose bed robes.\n\n\"You'll take me to him?\" she asked even as she exited the room.\n\n\"But, yes... of course,\" said Cecil, having to hurry to keep up.\n\nAlyssa couldn't believe how giddy she felt. The Watcher had not lied. Nathaniel was alive, and now she could go to him, could hold him, could keep him close for the rest of his life as he grew into the man to lead her fortune.\n\nNathaniel was alive. No matter how many times she told it to herself, it never lost its impact. Nathaniel was alive, alive, praise the gods, alive!\n\nWhen she arrived at her room, Cecil respectfully remained outside. Hurrying about, Alyssa opened a closet and ran outfit combinations through her head. Someone else knocked on the door, and she told whoever it was to enter without a thought. In stepped a younger man, a distant cousin of hers named Terrance. His features were soft, his reddish-blond hair carefully trimmed. He walked into her room trying to put on a somber face, but he was clearly giddy with news. When he saw the joy on her face, his own lit up. He must have thought she'd be grieving Arthur's loss, she realized. Foolish man.\n\n\"Forgive me for the intrusion,\" Terrance said. \"When I heard about Bertram's... betrayal, I went through his things. I'm learning my father's trade, you see, and he works with accounts and...\"\n\n\"Hurry it up,\" Alyssa said, yanking off her robes and pulling a loose dress over her head. The man flushed a deep red, and he stammered a bit, but he continued.\n\n\"Anyway, rumors have it that you wouldn't be able to pay the mercenaries, or to help with repairs. Bertram told my father that, anyway, and several of the servants.\" He saw the look she gave and so he skipped to the point. \"Thing is, Bertram was lying. I found his ledger for the mercenaries' payments, and it only comes to a third of your current wealth. Expensive, to be sure, but not near what he...\"\n\nShe kissed the man, laughed, and then tied a sash about her waist before flinging open another closet and searching for a thick-enough coat for the ride north.\n\n\"I need a replacement for Bertram,\" she said. \"And I have no time to search for one, so you'll have to do, Terrance.\"\n\nHis jaw nearly hit the floor.\n\n\"Me? But I'm still an apprentice, and my father says I can't own my own store until I reach my twentieth year. To try and manage all this...?\"\n\n\"Well, you start today.\"\n\n\"But why? Where are you going?\"\n\nAlyssa laughed again.\n\n\"I'm going to get my son.\"\n\nMatthew Pensfield felt the first twinges of consciousness pulling at him, and he resisted. Dull aches felt like the only welcome awaiting him. His gradual awareness thawed from whatever cold sleep it'd fallen into, and he remembered fighting, protecting the boy, Tristan. Or was it Nathaniel? And how was he alive? He _was_ alive, right?\n\nHis eyes fluttered open, and there in front of him sat the boy with two names, his head in his hands as he stared at the floor.\n\n\"Tristan?\" Matthew asked, his voice coming out like a strained croak. The boy startled, but his surprise didn't last long. A smile spread across his young face, and it lit up his eyes.\n\n\"You're awake!\" he said.\n\n\"I reckon so.\"\n\nTristan hugged him, eliciting a cough. It felt like half his body was full of fluid, the other half aches. He tried rolling over in bed, was denied by a terrible spike of pain from his shoulder. He glanced at it and saw an impressive amount of stitchwork in his flesh. A bruise spread from the wound all the way across his chest. Cut, that was right, he'd been cut down through the collarbone by that bastard at the castle gate.\n\n\"What happened to him?\" he asked.\n\n\"Who?\"\n\nMatthew grunted. \"Never mind. You're alive, and so am I, so it must have worked out fine.\"\n\n\"Lord Gandrem's said you should be treated as a hero.\"\n\n\"That so?\"\n\nTristan bobbed his head up and down. Matthew chuckled.\n\n\"If this is how heroes feel, count me out. The plow fits me better than the sword.\" He frowned. Tristan kept looking to the door, and his smile never seemed to last long.\n\n\"Something the matter, Tristan? Well, guess I should call you by your real name, shouldn't I? Not much point in hiding who you are now.\"\n\nThe boy obviously looked embarrassed, as if he wasn't sure how to respond.\n\n\"You can call me Tristan still, if you want, sir.\"\n\n\"I guess I'll let the habit linger, least until I can get out of this damn bed. What is the matter? You look like you're expecting the executioner.\"\n\nSomething about the way Tristan's face paled made him wonder what he'd said wrong.\n\n\"It's nothing,\" Tristan said. \"I just, it's... nothing. I'm glad you're awake. Really glad.\"\n\nMatthew's head felt groggy and stuffed with cotton, but he pushed through to see his surroundings better and to make sense of them. He was in a small room with stone walls, a single red carpet, and a large bed with sheets stained with what must have been his blood. Tristan wore fine clothing, far beyond anything Matthew could have afforded at his farmhouse (before that Haern guy dumped a pile of gold in their hands, anyway). It didn't look like everyday attire, but then again he was hardly knowledgeable about the ways of courts and castles.\n\n\"They treating you well?\" he asked.\n\n\"Yes,\" said Tristan.\n\n\"Something bothering you?\"\n\nHe looked once more to the door.\n\n\"Is it... is it all right if we just talk for a while?\"\n\nMatthew smiled. \"Sure, son. You care about what?\"\n\nWhen he shook his head, Matthew began discussing his plans for the farm. He prattled on about cattle, where he bought his pigs, and how if Nathaniel ever should get into the business in the north, to never ever buy from the Utters in the middle of winter unless he wanted to bend over and let them have their way with him. Tristan remained silent, but it seemed as if the tension drained out of him, until at last his eyes sparkled and he laughed at what few lame stories Matthew had to tell.\n\nEvery bit of that tension returned, though, when John Gandrem stepped into the room.\n\n\"Milord,\" Matthew said, tilting his head to show his respect. Getting up and bowing was obviously out of the question.\n\n\"I'm pleased to see you well,\" the lord said, though his voice hardly carried much pleasure. \"You'll be rewarded handsomely for protecting young Nathaniel here. Once I found someone who recognized you, I sent a rider to inform your loved ones of your stay in my care.\"\n\n\"Thank you, milord,\" said Matthew. \"My wife will much appreciate knowing.\"\n\n\"Rest, Matthew, and when you're better, we can discuss giving you and your family appropriate compensation. For now, I must borrow Nathaniel. We have matters to attend to.\"\n\n\"I'll talk to you tonight,\" Matthew told Nathaniel. \"Right now, I feel like eating a little, and then sleeping for a while, so don't worry about me.\"\n\nThey left, and servants arrived immediately after, carrying bowls of soup, and bread, and changes of clothing. While they buzzed about, Matthew thought of Nathaniel, and offered a prayer for whatever trial seemed to await him.\n\nNathaniel followed after Lord Gandrem, feeling like an obedient dog. The thought was unfair, for he had been treated absurdly well. But already he heard the murmuring of the crowd as they climbed the stairs toward the front wall of the castle. The sunlight was glaring when they emerged, and the crowd of hundreds below cheered at their arrival. Four guards flanked them upon the ramparts. Directly ahead, atop a retractable plank of wood hanging over the wall and with a long rope tied about his neck, was the man named Oric.\n\nLord Gandrem waved his greetings to the crowd gathered to watch the execution.\n\n\"This man was a coward and a traitor,\" he cried out to them. \"He dared lie to the lord of the lands, to mock the honor of Felwood! My allies he struck against. This fiend, this foul murderer, even sought to coat his blade with the blood of children. What fate does he deserve?\"\n\nThose gathered below howled for his hanging. Nathaniel heard their cries and shivered. Lord Gandrem turned to him and beckoned him forth. His feet feeling made of lead, he approached. Oric's face was covered with a black cloth, and his hands were tied behind him, but still he appeared dangerous.\n\n\"He's bound and gagged,\" John said, seeing his hesitation. \"And even if he weren't, you should not show fear. The eyes of the people are upon you, and more than anything, they want certainty from those who rule their lives.\"\n\nNathaniel nodded.\n\n\"Yes, sir,\" he said.\n\nThe older man guided him to where a lever waited, connected to various gears and wheels that would drop the platform Oric stood upon. It was as tall as him, and when he put his hand upon it, he worried he might be too weak to move it.\n\n\"This way,\" said a nearby knight, gesturing the direction for him to push. Hurling his weight upon it, Nathaniel felt the lever budge, then lurch forward. The crowd gasped, and before he could look away, Lord Gandrem took hold of his shoulder and forced him to watch. Oric fell, the rope snapped taut, but as he swung, his feet still kicked. A sickening groan floated up to them, barely audible over the cheer of the crowd.\n\n\"Bastard's neck didn't break,\" said one of the knights, leaning forward so he could see him.\n\n\"Just following orders,\" said the man beside him. \"John wanted to send a message.\"\n\nThe words flowed over him, but Nathaniel refused to give them any meaning. Instead he just watched as below him Oric kicked, gagged, and swung from the castle wall, John's hand holding him with a strength frightening for his age.\n\n\"Remember this always,\" John said to him. \"This is the fate that should meet all who challenge you. If you deny them this, then you become as cowardly as they. Besides, listen to that roar, Nathaniel. Listen to them cheer. Our people want blood, crave it. Every dead man hanging is a man worse than them. They'll spit on his corpse when we cut him down, and they'll unite in a hatred of something they hardly even understand. We are their lords. We are their gods. Never deny them the spectacle they deserve. So long as your acts are just, they will follow.\"\n\nNathaniel swallowed.\n\n\"Is it just?\" he asked.\n\n\"Doesn't matter, boy,\" Lord Gandrem said. \"All that matters is you believe it is just. And do you believe it?\"\n\nNathaniel opened his mouth to answer, opened his mouth to say he did, but instead he vomited, his stomach shifting from side to side within him, perfectly in time to the convulsing body of the dying Oric.\n\n# EPILOGUE\n\nHaern found Deathmask and his Ash Guild back in their hiding hole, and they greeted him like a long-lost friend.\n\n\"Behold the legend,\" Deathmask said, but his laughter cut with dark humor.\n\n\"Gerand told me of the Spider Guild's acceptance,\" Haern said, not wishing to waste any time. \"As for the Conningtons, a man named Potts has assumed control while his relatives bicker and position themselves. I wouldn't be surprised if it takes a year or two for them to settle things. For now, though, the old advisor's in charge, and he has agreed to the terms. Only two guilds have refused, but they're both currently leaderless.\"\n\n\"Already we move in on their territory,\" Veliana said. \"The Spiders and Wolves join us in the feeding frenzy. Whoever finally takes control of the remaining guilds will readily agree, just to save themselves from certain annihilation.\"\n\n\"So this is it then,\" Haern said. He looked to Deathmask. \"Gerand will arrange a set of terms to distribute payments to be divided equally among the five guilds. I imagine that much wealth will divide much better among you four than, say, the hundred or so of the other guilds.\"\n\n\"That thought had come to mind,\" Deathmask said, grinning. \"It's going to be rough these next few days. Everyone will be testing limits, seeing what they can get away with, and if you are capable of holding things in line. I'd say you normally could pull it off, but right now you look like an animal after a carriage has rolled over it a few times.\"\n\n\"I'll be fine,\" Haern said. \"And I'll be watching you as closely as any other guild. Don't forget that.\"\n\nDeathmask laughed.\n\n\"We aren't allies, Watcher, and I never intended to be. Keep your eye upon me all you want. You won't find anything, and your blades will never touch my skin. Go worry about those who truly present a danger to this truce. We'll be here reaping the rewards. For now, I've accomplished everything I desired, albeit in a more... chaotic fashion than I imagined. My former colleagues assumed the chaos in Veldaren could never be ended, and never be made profitable to us. I dare say I've proven them wrong. Or I will have once the Trifect's gold starts being delivered my way, without me having to lift a finger.\"\n\n\"At least someone profited from all this,\" Haern muttered.\n\n\"Given your position, I doubt you'll be living as a pauper either,\" Veliana said. \"Yet you whine like a mule.\"\n\n\"I don't want gold,\" Haern said. \"I want peace. Let me have that, and the Ash Guild will have nothing to fear from me.\"\n\nHe left without giving them any more chances to mock or gloat about their own cleverness. With much of his business done until nightfall, Haern debated on where to go. In the end he went back to the closest thing he had to a home. On the Crimson he found a wagon sitting in front of the Eschatons' place, half-loaded with trivial things. That none of it had been stolen yet seemed a miracle to him, until he remembered the very truce he'd just set up. Well, that was a start. He went to knock on the door, but it flew open. A very tired and surprised Tarlak stood before him, a stack of books in hand.\n\n\"Oh, you,\" he said.\n\n\"I've come to...\"\n\n\"Save it, Haern. I'm sure you did your absolute best, and I doubt Senke would have changed a thing. Well, other than him dying. He might have... look, the offer still stands. No speeches, no apologies, no nonsense requirements. I bought a tower on the outskirts of the King's Forest, and I plan on making it a far better home than this dung hole. I told you I thought we had potential to be something special, and I still mean it. You want to come, be useful and grab a box.\"\n\nHaern stepped aside, and Tarlak set his things on the wagon. Glancing inside the building, Haern saw Brug packing up various smithy tools. Delysia helped him, the two joking with each other in hushed tones. He could see the redness in their eyes, but they were moving on the best they knew how. The priestess saw him, and despite the loss of a friend, Delysia smiled and beckoned him inside.\n\n\"Why not,\" Haern said as Tarlak came back to the door. He stepped inside, grabbed a box, and hoped that just perhaps the newly titled King's Watcher might finally have a home.\n\n# A NOTE FROM THE AUTHOR\n\nI think, looking back at all these books so far, _A Dance of Blades_ is currently my favorite. _Cloaks_ was written by some strange, clearly deranged madman. This book? I opened my original Note from the Author with this line: \"I think I'm getting the hang of this.\" And I think it is still very much appropriate. In preparing everything for re-release under Orbit's guiding hands, this was the one needing the least work. Not to say it didn't need work... Devi will slap me if I claim that. Plus it'd be a total lie. But this was the point I was writing toward even when I started _Cloaks_. This was the big moment.\n\nBut it also helped immensely that I got to bring in all these other characters I'd spent so much time with in my Half-Orc books. Haern without the Eschatons just wasn't the same. But Tarlak? God, he's so much fun to write. Mocking Brug never gets old. Even having Veliana finally meet up with Deathmask was immensely satisfying, building up everything that would come later. Bringing in new characters to interact with the old kept _Blades_ in a perfect balance for yours truly, something I always strive to reach in any book I write.\n\nSpeaking of new characters, I should probably confess my absolute terror at writing the character Ghost. A quick glance at my author photo should probably explain why. I'm about as white as it gets, and living here in the dead center of the US heartland, what the heck do _I_ know about walking in someone like Ghost's shoes? But I had to try. His concept, his character in my head, it was just too awesome, too striking, too memorable. Going back over _Blades_ , I fell in love with him all over again. I'm glad I didn't wimp out, I'm glad I stuck to my guns. Did I succeed? Heck if I know. But I think I did.\n\nSome of you might be wondering at Thren's diminished role in this book. That was very much on purpose. The final confrontation between Haern and Thren comes (much) later, which meant I couldn't keep him as a central villain, especially not in this book, where Haern needed to develop fully into his own character. So I found it better to have Thren lurking in the background, always referenced, always affecting decisions and outcomes even when he's not there. That one scene between him and Deathmask? Hardest one to write in the entire book.\n\nSpeak of hard scenes... yeah, Senke dying? I know it may sound odd for an author to have regrets. I mean, we're gods of our little universes. I could have Senke come riding back to life on a pink pony with rainbows shooting out his fingers. But it wouldn't work, and it wouldn't be right. And I wish I could have found a way to have Senke survive. He was fun, he was useful, and most important, he had a connection with Haern that even now I struggle to replicate with other characters. The closest I come, as you'll see in _A Dance of Mirrors_ , is with Zusa. May seem odd now, but if you look at what both of them endured, the restricted upbringing, plus the sheer skill each of them wields and can appreciate in the other, it won't seem quite so strange.\n\nThey're such a cute couple, really.\n\nWell, I've rambled enough, so probably time to wrap this up. Thanks again to Michael for the agent stuff, Devi for pushing each draft to be that much better than the previous, my wife for putting up with my lengthy phone calls and inability to talk about really anything else at the dinner table, and you fans who have been with me since the very beginning.\n\nAnd you of course, dear reader. I'd be foolish, if not professionally suicidal, if I did not accept that it is your time, your patience, your entertainment that keeps me writing my silly little stories. I hope, as the years pass and I crank out more and more stories of Haern, Tarlak, and the rest of the world of Dezrel, that you never once get bored. Mad at me? That's all right. Sad at times? Perfect.\n\nBut never bored. I do that, and I'll call it good.\n\nDavid Dalglish\n\nApril 9, 2013\n\n# extras\n\n# meet the author\n\n_Mike Scott Photography_\n\nDAVID DALGLISH currently lives in rural Missouri with his wife, Samantha, and daughters Morgan and Katherine. He graduated from Missouri Southern State University in 2006 with a degree in mathematics and currently spends his free time playing not near enough Warhammer 40K.\n\n# introducing\n\nIf you enjoyed\n\nA DANCE OF BLADES,\n\nlook out for\n\n# **A DANCE OF MIRRORS**\n\nShadowdance Book 3\n\n_by David Dalglish_\n\nHaern pulled his hood low over his head and tied his sabers to his belt as the leader of the Eschaton mercenaries, the wizard Tarlak, sat at his desk and watched.\n\n\"Do you want our help?\" Tarlak asked, picking a bit of dirt off his yellow robe.\n\n\"No,\" Haern said, shaking his head. \"This one needs to be a message for the underworld of the city. Brann crossed a line that I need to make sure no one else ever crosses. I'll do this on my own.\"\n\nTarlak nodded, as if not surprised.\n\n\"What about Alyssa?\"\n\nHaern tightened the clasp of his cloak. They'd heard word that Alyssa planned some sort of retaliation against the thief guilds, though the reason was unclear. Their source was fairly respected in the Gemcroft household, so much so they had no choice but to take it seriously. At some unknown point in the night, there was to be a meeting at her mansion to discuss the circumstances.\n\n\"After,\" Haern said. \"I'm sure you understand.\"\n\n\"I do,\" said Tarlak. \"Good luck. And remember, I can't pay you if you die on me.\"\n\n\"I won't be the one dying tonight,\" Haern said, feeling the cold persona of the King's Watcher coming over him.\n\nHe left the room, descended the staircase to the tower's exit, and then ran the short distance toward the city. A dozen secret passageways, ropes, and handholds were available to him as a way to cross the wall, and he drifted to the southern end before climbing over. Alyssa's potential conflict with the thief guilds was a greater threat in the long run, but Haern could not bring himself to focus on it just yet. His target was a piece of scum named Brann Goodfinger. He operated in the far south of the city, and it was there Haern went.\n\nNormally he felt pride as he traversed the rooftops, carefully observing the doings of the various guilds. Ever since the thief war ended two years ago, the factions had settled into an uncomfortable truce. The first few months had been the worst, but Haern's sabers had spilled torrents of blood. Through sheer brutality, he had brought both sides to their knees. He was the silent threat watching all, and tolerating nothing. But tonight his accomplishment felt bitter. For the first time, his plan had been turned against him in a most cruel, personal way.\n\nThieves who stole from the Trifect died. They all knew this, knew that every night Haern patrolled the city as the King's Watcher to ensure the agreed-upon peace. And so Brann had recruited children, a bold dare against the Watcher's threat.\n\n\"Where is it you hide?\" Haern whispered as he lay flat atop a roof. For two days Brann had eluded him, and his children had gone unchecked. No longer. He spotted one of their youngest, a boy surely no older than seven. He was exiting the broken window of a shop, a handful of copper coins clutched to his chest. He ran, and Haern followed.\n\nThe boy tried to vary his pattern, as he'd no doubt been trained to do, but against someone like Haern the tactic was a minor inconvenience, nothing more. Haern kept far out of sight, not wanting to alert him to his presence. Twice he'd tracked Brann's child-thieves, but one had spotted him, abandoned his ill-gotten coin, and fled. The other had been killed by a different thief guild before he could question him. Children bled out on the streets of Veldaren. The Watcher's wrath would be terrible.\n\nHaern turned a corner and watched the child hurry inside a warehouse. Approaching the door, Haern slipped into the shadows and looked through the crack near the hinges. A faint lantern burned inside, and from what he could make out, two other children were within. Hoping it was Brann's hideout, and not a simple gang of orphans, he drew his sabers. There would be no stealthy entrance. This wasn't a time for quiet deaths in the night.\n\nHe slammed the door open with his shoulder at full charge. Without slowing, he took in the surroundings, his finely honed instincts guiding him. The storehouse was full of crates and bags of grains, limiting his maneuverability. At least twenty children were gathered in a circle, and before them, his dirty face covered with a beard, was Brann. The man looked up. His jaw dropped, and then he turned to run.\n\n\"Stop him!\" Brann shouted to the children.\n\nHaern swore as they drew small knives and daggers. He leaped between them, twirling his cloak as a distraction. A sweeping kick took out three, and then he pushed through the opening. The storehouse was divided in two by a high wall, and Brann vanished through the doorway in the center. Haern raced after him, again slamming aside the door with his shoulder. To his surprise, Brann was not the coward he'd believed. His sword lashed out from behind the door. Haern's speed was too great, though, and he fled beyond Brann's reach, pivoted on his heels, and jumped again.\n\nBrann was only a gutter snake, a clever bully who relied on size or surprise to defeat a foe. Haern had fought his kind, knew their tactics. With three strikes, Brann's sword fell from a bleeding wrist. Two kicks shattered a kneecap, and then he fell. Haern clutched his hair and yanked his head back, his saber pressing against Brann's throat.\n\n\"How dare you,\" Haern whispered. His hood hung low over his face, and he shook his head to knock it back. He wanted Brann to see the fury in his eyes.\n\n\"You hold this city prisoner yet ask me that?\" said Brann.\n\nHaern struck him in the mouth with the hilt of a saber. As Brann spit out a tooth, the children rushed through the door, surrounding them both.\n\n\"Stay back,\" Brann said to them, and he grinned at Haern, his yellow teeth stained red with blood. There was a wild look in his eyes that made Haern uncomfortable. This wasn't a man who cared about life\u2014not his own, nor that of others.\n\n\"What game is this?\" Haern asked, his voice a cold whisper. \"Did you think I wouldn't find out? Using children, here, in my city?\"\n\n\"Your city?\" Brann said, laughing. \"Damn fool. All the rest are scared, but I know what you are. They think you're as bad as us, but you're not... not yet. Once the thief guilds find out, they'll have your head on a spike.\"\n\nHe gestured to the children, all prepared to attack. Haern didn't want to imagine what Brann had put them through to achieve such a level of control.\n\n\"Kill me,\" Brann said. \"Do it, and they'll swarm you. You won't die\u2014you're too good for them\u2014but you won't escape without killing at least one. So what'll it be, Watcher? Can you take my life if it means taking the life of a child?\"\n\nHaern looked at the twenty children. Some were as young as seven, but others were maybe eleven or twelve. All it'd take was one lucky stab by any of them and he might go down.\n\nHis saber pressed harder against Brann's skin. He leaned closer to whisper into his ear.\n\n\"Nothing, Brann. You know nothing about me. You die, they go free.\"\n\n\"I die, then innocents will as well. You don't have the stomach for it. You aren't the beast the others think you are. Now let me go!\"\n\nHaern glanced at the children, all poised to act. He tried to decide what to do, but he knew what life someone like Brann would lead them to. No matter what, no matter the risk, he couldn't allow it.\n\n\"This was never a choice,\" Haern whispered.\n\nHe slashed, spilling blood across his clothes. Hoping to move before the children reacted, he turned and leaped, vaulting over their circle. They gave chase, not at all bothered by the death of their master. Haern rolled to his feet, his sabers crossed to block their weak stabs. A quick glance showed no exits except the door he'd come through. Doing everything he could to fight down his combat instincts, he shoved through the group's center. His cloak whirled and twisted, pushing aside feeble attacks.\n\nPulling out of the spin, he lunged for the door. One of the older boys was there, and Haern felt panic rise in his chest as he saw the deadly angle of the boy's thrust. He reacted on instinct, blocking hard enough to knock the dagger free, then following it up with a kick to send the boy flying. Breaking back into a run, he kicked off a pile of crates to vault into the air, catching a rafter with one hand. Swinging himself up onto a perch, he stared down at the children, several of whom gathered around the body of the one he'd kicked.\n\n\"Listen to me,\" Haern said to them, trying to forgive the children's attack. They didn't know any better. The rage he felt was misguided, born of frustration. \"Your master is dead. You have no hope of winning this fight.\"\n\n\"Fuck you,\" said one of the kids.\n\nHaern swallowed down his anger at such disrespect. They were frightened and living in a world Haern knew all too well. If reason would not work, he knew what would.\n\n\"Say that again, and I'll cut out your tongue.\"\n\nThe boy stepped back, as if stunned by the coldness in his voice. The rest looked up at him, some ready to cry, some angry, but most were heartbreakingly indifferent. Haern pointed to Brann Goodfinger's corpse.\n\n\"Take his coin,\" he said. \"Go, and make better lives than this. Remain thieves, and you'll fall to the guilds, or to me. I don't want to kill you, but I will. There is no future for you, not in this.\"\n\n\"None for you, either,\" said another, but Haern could not tell who. With practiced efficiency the children took everything of value from Brann's corpse and vanished into the streets. Haern didn't know where they went, nor did he care. He only felt fury. Brann had died quickly, hardly the example Haern desired to set. As for the boy he'd kicked...\n\nHe dropped from the rafter, landing lightly on his feet. Gently he rolled him over, put a hand on his neck. No pulse.\n\n\"Damn you, Brann,\" Haern whispered. \"I hope you burn forever.\"\n\nLeaving the body there was not an option. Haern considered himself better than that. Lifting him onto his shoulder, he rushed out to the streets, praying no gutsy member of a thief guild spotted him and tried something incredibly heroic and stupid. There were several gravekeepers in Veldaren, plus another who burned bodies instead of burying them. Haern went to the burner, picked the lock of his door, and went inside. The owner was asleep on a cot in a small room, and Haern woke him with a firm prod of his saber.\n\n\"What? Who are... Oh, you.\"\n\nThe elderly man, Willard, rubbed his eyes, then opened them when Haern dropped a handful of coins onto his lap.\n\n\"Spare no expense, and bury his ashes.\"\n\n\"Who was he?\" asked Willard, looking over the boy's body as Haern set him down on the floor.\n\n\"An accident.\"\n\n\"Then what shall I engrave on his urn?\"\n\n\"Pick something,\" Haern said as he left.\n\nIn a foul mood, he raced off for the Gemcroft estate, wishing he could put the prior events out of his mind and knowing there'd be no such luck. Brann's death would still be a warning to the others against using children to break the arrangement between the guilds and the Trifect. He'd accomplished that, though not how he'd hoped. But it was that nameless boy who haunted him, made his insides sick. Brann had been convinced Haern would not have the stomach for what might happen. Turned out he might have been right.\n\nScaling the fence around the Gemcroft estate was easy enough, though avoiding the guards was another matter. There was a secondary building in the back, where he'd been told the meeting would take place. Most of the patrols kept close to the mansion, which helped tremendously. Haern lurked beside the gate, running along it when outside the patrols' vision and lying flat amid the shadows when they passed. At last he reached the small building. Timing the patrols, he knew he had about thirty seconds to slip in and out without being seen. Faint light burned within. He pressed his ear against the door and heard no discussion.\n\nToo late, or too early? The door was unlocked, so he opened it and slipped inside. The room was surprisingly bare, containing only a single bed atop a padded floor. Hardly the servants' quarters he'd expected. The lone lantern kept the place dimly lit, with plenty of shadows in the far corners. So far, it appeared empty.\n\n\"Damn,\" he whispered.\n\nHe headed for the far corner, figuring to wait a few hours just in case the meeting was yet to transpire. In the center of the room, though, he stopped. Something in the corner wasn't right, the shadows not smooth...\n\nHaern lunged for the door, his instincts screaming _trap_. Before he could get there, something latched on to his cloak and tugged, hard. He spun to the ground, torn between attacking and tearing his cloak free to flee. Already furious because of Brann, he kicked to his feet and attacked. To his surprise, his sabers clashed against long blades, his thrusts perfectly blocked. He was already preparing a second strike when he saw his opponent's outfit. Long dark wrappings covering her body\u2014all but her shadowed face.\n\n\"Enough, Watcher,\" said Zusa, her slender body contorted into a bizarre defensive formation. \"I am not here to kill you.\"\n\nHaern pulled away, and he put his back to a wall, the door at his side.\n\n\"Then why are you here?\" he asked.\n\n\"Because I desired it,\" said a voice at the door.\n\nHaern turned, then dipped his head in a mock bow. \"Lady Gemcroft,\" he said. \"It is good to see you, Alyssa.\"\n\nThe ruler of the Gemcroft fortune smiled at him, not at all bothered by his tone. Zusa sheathed her daggers, though her hands remained on their hilts. She joined Alyssa's side, her dark eyes never leaving him. Alyssa seemed relaxed, far more so than when Haern had last seen her. Of course, he'd been trying to kill her at the time, back when Alyssa was flooding the streets with mercenaries. She wore a slender dress underneath her robe, her red hair let down loose about her shoulders. Haern almost felt flattered she'd dressed up for him, as if he were some noble or diplomat.\n\n\"I was told of a meeting concerning the thieves,\" Haern said. \"Was there any truth to this?\"\n\n\"I assure you, Terrance is loyal to me, and me alone,\" she said.\n\nThe side of Haern's face twitched. Terrance had been his informant, of course. He felt at a disadvantage, with no clue as to the reason for their meeting. He didn't like that. The two also blocked the only exit. He really didn't like that.\n\n\"Then I was told a lie, just to bring me here,\" he said. \"Why is that, Alyssa?\"\n\n\"Because I want to hire you.\"\n\nHaern paused, then laughed at the absurd notion. \"I am no pawn for you to force your will upon. And if what you say is true, why this secrecy and deception?\"\n\n\"Because I don't want anyone\u2014not the guilds nor the Trifect\u2014to know. I leave for Angelport, and I wish for you to accompany me and Zusa.\"\n\nHaern's hands fidgeted as they held his sabers. Answering such a request with someone as dangerous as Zusa blocking his way out was not his idea of a fair bargaining position.\n\n\"What reason could you possibly have?\" he asked. \"I assure you, Zusa is quite capable of keeping you alive.\"\n\nA bit of impatience finally pierced Alyssa's calm demeanor.\n\n\"Someone broke into Laurie Keenan's home, slaughtered his son and daughter-in-law, along with a dozen guards. I'm going for their funeral services, as is appropriate. I want you and Zusa to hunt down this killer and bring him to justice while I'm there.\"\n\nHaern shook his head. \"I can't leave Veldaren,\" he said. \"The peace I've managed to create\u2014\"\n\n\"Is no peace at all,\" Alyssa said. \"The thief guilds prey on each other, killing themselves in an endless squabble over the gold we pay them. The few that steal are more often caught by their own kind, not you. No one will know you've left, not for weeks. It's been two years, and you've spilled enough blood to wash the city red. Those who remain have settled into their comfortable lives of bribes and easy money, and you know it. You've become a figurehead, a watcher against only the most reckless of the underworld. The city's changed. It won't miss you while you're gone.\"\n\nHaern did know that, but that didn't mean he liked it.\n\n\"This is your problem,\" he said. \"I've had enough dealings with the Trifect to last a lifetime. Find your killer on your own. Now let me through.\"\n\nAlyssa glanced at Zusa, then nodded. They stepped aside. As Haern walked out into the night, Alyssa called after him.\n\n\"They found a marking,\" she said. \"Drawn in their blood.\"\n\nHaern stopped. \"What of?\" he asked.\n\n\"A single eye.\"\n\nHaern turned, and he felt his anger rise. \"You would accuse me of this crime?\"\n\n\"No accusation,\" Alyssa said, stepping out. \"I have already looked into the matter and know you were in Veldaren both the night it happened, plus the nights before and after. Laurie's kept word of it a secret and told only those closest to him of its presence at the murder. He knows you didn't perform the deed, though he still fears your involvement somehow...\"\n\nGrinding his teeth, Haern tried to think through what any of it meant, but it left him baffled.\n\n\"This makes no sense, Alyssa,\" he said. \"Why would someone frame me so far away? I've never been to Angelport, nor used that symbol for years. Not since the war between the Trifect and the thief guilds ended.\"\n\n\"It's not a frame,\" Zusa said, as if it were so simple. \"It is a calling. You're being summoned, Watcher.\"\n\nHaern tried to think it over, but he felt so tired, so unprepared. The boy's dead face kept flashing before his eyes.\n\n\"How do I know this isn't a trap?\" he finally asked.\n\nAlyssa looked away, as if embarrassed by what she had to say. \"Because of you, my son lives,\" she said. \"And because of you, I was able to bring vengeance to the one who tried to kill him. I will never betray you. Someone murdered powerful citizens of Angelport, my friends and colleagues, and is using their blood to send you a message. Help me find him. Help me stop him.\"\n\nHaern sighed. \"So be it,\" he said. \"When do we leave?\"\n\n# BY DAVID DALGLISH\n\n#### Shadowdance\n\n_A Dance of Cloaks_\n\n_A Dance of Blades_\n\n_A Dance of Mirrors_\n\n_A Dance of Shadows_\n\n### Thank you for buying this ebook, published by Orbit.\n\nTo get news about the latest Science Fiction and Fantasy titles from Orbit, along with special offers and exclusive content, sign up for the Orbit newsletter.\n\n Sign Up\n\nOr visit us at www.orbitbooks.net\/booklink\/\n\nFor more about this book and author, visit Bookish.com.\n\n# Contents\n\nCover\n\nTitle Page\n\nWelcome\n\nDedication\n\nMap\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\nChapter 25\n\nChapter 26\n\nChapter 27\n\nChapter 28\n\nChapter 29\n\nChapter 30\n\nChapter 31\n\nChapter 32\n\nEpilogue\n\nA Note from the Author\n\nExtras\n\nMeet the author\n\nA Preview of _A Dance of Mirrors_\n\nBy David Dalglish\n\nOrbit Newsletters\n\nCopyright\n\n# Copyright\n\nThe characters and events in this book are fictitious. Any similarity to real persons, living or dead, is coincidental and not intended by the author.\n\nCopyright \u00a9 2013 by David Dalglish\n\nExcerpt from _A Dance of Mirrors_ copyright \u00a9 2013 by David Dalglish\n\nCover design by Kirk Benshoff\n\nCover photo illustration by Michael Frost and Gene Mollica\n\nCover copyright \u00a9 2013 by Hachette Book Group, Inc.\n\nAll rights reserved. In accordance with the U.S. Copyright Act of 1976, the scanning, uploading, and electronic sharing of any part of this book without the permission of the publisher is unlawful piracy and theft of the author's intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained by contacting the publisher at permissions@hbgusa.com. Thank you for your support of the author's rights.\n\nOrbit\n\nHachette Book Group\n\n237 Park Avenue, New York, NY 10017\n\norbitbooks.net\n\norbitshortfiction.com\n\nFirst ebook edition: November 2013\n\nOrbit is an imprint of Hachette Book Group, Inc. The Orbit name and logo are trademarks of Little, Brown Book Group Limited.\n\nThe Hachette Speakers Bureau provides a wide range of authors for speaking events. To find out more, go to hachettespeakersbureau.com or call (866) 376-6591.\n\nThe publisher is not responsible for websites (or their content) that are not owned by the publisher.\n\nISBN 978-0-316-24248-6\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n_ECLIPSE OF MAN_\n_NEW ATLANTIS BOOKS_\n\n_Adam Keiper, Series Editor_\n\nPREVIOUS VOLUMES:\n\n_Why Place Matters:_\n\n_Geography, Identity, and Civic Life in Modern America_\n\nEdited by Wilfred M. McClay and Ted V. McAllister\n\n_Merchants of Despair:_\n\n_Radical Environmentalists, Criminal Pseudo-Scientists, and the Fatal Cult of Antihumanism_\n\nRobert Zubrin\n\n_Neither Beast nor God:_\n\n_The Dignity of the Human Person_\n\nGilbert Meilaender\n\n_Imagining the Future:_\n\n_Science and American Democracy_\n\nYuval Levin\n\n_In the Shadow of Progress:_\n\n_Being Human in the Age of Technology_\n\nEric Cohen\n\n_www.newatlantisbooks.com_\n\n\u00a9 2014 by Charles T. Rubin\n\nAll 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, or otherwise, without the prior written permission of Encounter Books, 900 Broadway, Suite 601, New York, New York 10003.\n\nFirst American edition published in 2014 by Encounter Books, an activity of Encounter for Culture and Education, Inc., a nonprofit, tax exempt corporation.\n\nEncounter Books website address: www.encounterbooks.com\n\nThe paper used in this publication meets the minimum requirements of ANSI \/ NISO Z39.48-1992 (R 1997) ( _Permanence of Paper_ ).\n\nFIRST AMERICAN EDITION\n\nLIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA\n\nRubin, Charles T.\n\nEclipse of man : human extinction and the meaning of progress \/ by Charles T. Rubin.\n\npages cm\n\nIncludes bibliographical references and index.\n\nISBN 978-1-59403-741-2 (ebook)\n\n1. Philosophical anthropology. 2. Human beings\u2014Forecasting. 3. Human evolution. 4. Human body\u2014Technological innovations. 5. Cyborgs. 6. Biotechnology\u2014Moral and ethical aspects. 7. Humanity. 8. Progress. I. Title.\n\nBD450.R73 2014\n\n128\u2014dc23\n\n2014022022\n_CONTENTS_\n\n_Introduction_\n\nCHAPTER ONE: The Future in the Past\n\nCHAPTER TWO: Discovering Inhumanity\n\nCHAPTER THREE: Enabling Inhumanity\n\nCHAPTER FOUR: Perfecting Inhumanity\n\nCHAPTER FIVE: The Real Meaning of Progress\n\n_Acknowledgments_\n\n_Notes_\n\n_Index_\n_FOR MY PARENTS_\n_I know that it is a hopeless undertaking to debate about fundamental value judgments. For instance, if someone approves, as a goal, the extirpation of the human race from the earth, one cannot refute such a viewpoint on rational grounds. But if there is agreement on certain goals and values, one can argue rationally about the means by which these objectives may be attained_.\n\nALBERT EINSTEIN, 1940\n\n_Introduction_\n\n\"MANKIND WILL surely destroy itself.\" Whether predicted in a thunderous denunciation of our flaws or with mild worldly regret, an apocalyptic future has become a clich\u00e9. Will it be global warming or global cooling? Nuclear winter or radiation poisoning? Famine due to overpopulation, or pollution-induced sterility? These are some of the possibilities I grew up with. But today, it is becoming increasingly common to hear of another route to the demise of humanity: we will improve ourselves, becoming something new and better, and in doing so we will destroy what we are now. We have this opportunity because science and technology are giving us the power to control human evolution, turning it from a natural process based on chance to one guided by our own intelligence and will.\n\nThis idea\u2014that human progress points toward human extinction\u2014is held by people who go by a variety of names: transhumanists, posthumanists, extropians, advocates of H+, or singularitarians. It can be difficult to keep these terms straight, as they each represent schools of thought whose agreements and disagreements can be complex and ingrown. For the purposes of this book, all these schools of thought will be given the generic label _transhumanism_. The essential insight that defines transhumanism is, to borrow a phrase, that \"We are as gods and might as well get good at it.\" Transhumanists argue not only that modern science and technology are giving human beings the power to take evolution into our own hands to improve the human species, and then to create some new species entirely, but also the ability to improve on all of nature. Much like the older apocalyptic visions, the transhumanists believe that mankind as we know it and nature as we know it are on their way out; but for most transhumanists, that is the deliberate goal sought, not a consequence of our hubris to be avoided. Indeed, the transhumanists believe that if we are to prevent some of the more common apocalyptic visions from becoming reality, we _must_ redesign humanity so that our ruinous flaws can be eliminated. To avoid mere destruction, we must embrace creative destruction.\n\nHence, the end of man can be the beginning of . . . who knows what, exactly? But, we are told, it will doubtless be some wondrous new home for intelligence, able to do things far beyond our present ability to imagine. If my baby boom generation was warned that the challenge of the future involved ensuring the continued existence of human beings in the face of all the threats posed by our own activities (and to some extent by nature itself), for the transhumanists it is becoming increasingly difficult to imagine a future that has any place for us at all, except perhaps as curiosities.\n\nSuch views of the future can strike sensible people as idiosyncratic or even loony. But this book will show why that understandable first reaction should not be the whole story. There are some serious reasons why transhumanists have come to measure progress by the speed at which mankind disappears, and those reasons are more deeply rooted in mainstream ways of looking at the world than might at first be obvious. At the same time, as we shall see, there are also very good reasons to reject the transhumanist future, and to work toward a future in which, as William Faulkner put it in his speech accepting the Nobel Prize in Literature, \"man will not merely endure: he will prevail.\" Those reasons will emerge from a serious confrontation with transhumanist ideals and the foundations on which they are built.\n\n_SUFFICIENT UNTO THE DAY?_\n\nIt can be difficult to know what transhumanism amounts to\u2014a worldview, an ideology, a movement, or some combination\u2014but it cannot simply be dismissed as irrelevant. Doctoral dissertations and academic conferences have focused on transhumanism, and a few major universities have scholarly centers wholly devoted to exploring transhumanist ideas. Books and blogs, think tanks and online communities, documentaries and blockbuster movies have all helped to popularize those ideas. Major news outlets routinely publish reports uncritically explaining them. Meanwhile, some of Silicon Valley's best and brightest are committed transhumanists. Perhaps the most prominent promoter of transhumanism, the inventor and bestselling author Ray Kurzweil, subject of countless media profiles, was hired by Google in 2012 to serve as the company's director of engineering. Three years earlier, Kurzweil cofounded Singularity University, an institution dedicated to disseminating transhumanist ideas, with sponsorship from several high-tech companies and philanthropic foundations, as well as help from NASA. In short, many of the people who are inventing the tools of tomorrow embrace, or are at least informed by, the transhumanist vision of the day after tomorrow.\n\nThe transhumanist program to redesign humanity is often linked with the rise of the so-called \"converging technologies\": nanotechnology, biotechnology, information technology (and sometimes robotics), and cognitive science. They are called converging technologies because each reinforces the potential that the others have for vastly increasing our ability to manipulate nature, including our own nature. And what remarkable things are now in the works! While it is still possible to make a splash by writing a book about how, in the not-so-distant future, people will regularly have sex with robots, that is hardly a revolutionary thought when for some time others have been writing books about people turning themselves into robots. The \"virtual reality\" expected soon to make movies and games more immersive is just a precursor to direct connections between our brains and computers, and even that is merely a prelude to uploading our minds into computers\u2014providing us with a kind of immortality (so long as proper backups are made). The undoubted promise biotechnology holds for lengthening human lives is overshadowed by speculation concerning the ability of nanotechnology to bring the dead back to life. We are told that genetic engineering to cure disease and bioengineering to overcome disabilities are just foreshadowings of a complete re-engineering of human beings to add whatever senses, features, and capacities an individual might wish to possess.\n\nAre such developments really likely, or even possible? Some critics point out problematic scientific or technological assumptions underlying transhumanists' ideas, or the obstacles that they might have overlooked. Critics also sometimes focus on the considerable uncertainty about what is possible, arguing that that uncertainty might in and of itself be adequate reason for ignoring those who happily anticipate an end to humanity. Without a clearer idea of what will actually be possible in the future, debating the details of various transhumanist predictions might seem like a waste of time\u2014and surely we have more pressing things to worry about today.\n\nBut even if the converging technologies do not pan out exactly as transhumanists expect, who would really want to bet against the likelihood that science and technology will, in the future as in the past, continue to allow us to do things routinely that only a few decades previously might have seemed like mere science fiction? Indeed, it is surely likely that for a great many items on those worry-about-it-today lists, we will call upon science and technology to deal with them. That is our contemporary way of thinking about solving problems. Nanotechnologies are already being developed to deal with environmental and energy issues. The fight against terrorism has spurred advances in robotics. The most surprising human future would be one in which we did _not_ continue to accumulate scientific information and innovative technology and use them to increase our powers over nature, which includes power over ourselves.\n\nFrom this point of view, it is clear that the broad transhumanist goal of overcoming the human condition does not depend on whether or not a _particular_ constellation of technologies works as expected. Indeed, we will see how some transhumanists reasonably suggest that dissatisfaction with the human condition and the wish to transcend it run deep in human thinking, and perhaps even define our humanity. If so, we may just keep trying to redesign ourselves with whatever means actually become available.\n\n_MAN IS BORN TO TROUBLE_\n\nDissatisfaction with the miseries of human life\u2014whether we are beset by them from outside or we bring them on ourselves\u2014is nothing new. In the Bible, the Book of Job lays out the situation in a familiar way from which certain generic conclusions can be drawn. Job loses his great wealth, the external goods that make for a comfortable life. Boils afflict his body and take his health away. Job is deprived of his loved ones, and to that extent also of his future, as he fears that death is absolutely final. The only thing missing is deliberately omitted. Satan expects the balance of Job's mind will be disturbed and he will curse God. But while Job acknowledges confusion, longs for death, and contends with God, he does not explicitly curse Him.\n\nSo the human condition has long been understood to include the possibility, indeed the likelihood, of being deprived of external goods, bodily goods, and goods of the mind or spirit\u2014and by dissatisfaction that we have such vulnerabilities. Doubtless that kind of dissatisfaction helped to prompt ancient imagination of longer lives, greater wealth, and superhuman power, as in the case of the Greek gods and heroes. These Greek gods\u2014without the curse of mortality, with all possibility of ease and wealth and security, full of rude health and bodily vigor\u2014are just what we might wish to be. And yet they are still restless, jealous, capricious, untrustworthy, often angry, unhappy, and quite dissatisfied. Even without the darker sides of the human condition, and without having to bear real consequences for the vast majority of their actions, they have a terrible lightness of being.\n\nWhile these gods will punish those who seek to be too much like them, the gods themselves are punished for being what they are, for having some of the very goods that mortals hope for. Apparently, in the ancient view, the things we might have thought would make us happy do not guarantee it at all. Failure to appreciate this catch-22 means that imagination of something better is likely to be just another source of suffering, since what you imagine would never work out in any case. So whether you think things could be better or not, suffering is to be taken for granted. Options for dealing with this fact of life, whether among premodern polytheists or monotheists, were likewise limited. One might seek to delay suffering by proper relationships with gods or God, or to find some message or meaning in the suffering when it inevitably occurred, or simply to accept it with resignation as the cost of living. And at that, even the pious rabbis of the Talmud were known to wonder if it were better for mankind to have been created or not. When it came to a vote, they voted not.\n\nEven if the sources of our misery have not changed over time, the way we _think_ about them has certainly changed between the ancient world and ours. What was once simply a fact of life to which we could only resign ourselves has become for us a problem to be solved. When and why the ancient outlook began to go into eclipse in the West is something scholars love to discuss, but that a fundamental change has occurred seems undeniable. Somewhere along the line, with thinkers like Francis Bacon and Ren\u00e9 Descartes playing a major role, people began to believe that misery, poverty, illness, and even death itself were not permanent facts of life that link us to the transcendent but rather challenges to our ingenuity in the here and now. And that outlook has had marvelous success where it has taken hold, allowing more people to live longer, wealthier, and healthier lives than ever before.\n\nSo the transhumanists are correct to point out that the desire to alter the human condition runs deep in us, and that attempts to alter it have a long history. But even starting from our perennial dissatisfaction, and from our ever-growing power to do something about the causes of our dissatisfaction, it is not obvious how we get from seeking to _improve_ the prospects for human flourishing to _rejecting_ our humanity altogether. If the former impulse is philanthropic, is the latter not obviously misanthropic? Do we want to look forward to a future where man is absent, to make that goal our normative vision of how we would like the world to be?\n\n_DREAMS OF THE FUTURE AS MORAL VISIONS_\n\nMy previous book culminated in a discussion of deep ecology, a form of radical environmentalism that adopts the principle of \"ecoegalitarianism\": no one species has any moral priority over another. Because technological man so consistently violates this stricture, some of the deep ecologists look forward to a day when human beings will have been replaced by a new human-like species that lives more at one with nature. I criticized that view for its profoundly anti-human character, but even then I noted that the same threat could come from the direction of technological utopianism. Deliberately seeking our own extinction represents the extreme limit of how far we could want to go to overcome our given circumstances and raises in an obvious way a question that is always lurking in our rapidly changing world: What kind of future are we trying to create?\n\nIt is very likely that the world we will have in the future will not be _exactly_ the one laid out by today's transhumanists. Still, our utter dependence on continuing scientific and technological development makes it impossible to dismiss the broad goals of transhumanism outright; indeed, it is hard to imagine how we will avoid making choices that could provide building blocks for a project of human extinction. Even if in most instances these choices will actually be made with a view to the contingencies of the moment\u2014arising from scientific curiosity, engineering creativity, military necessity, or commercial possibility\u2014the transhumanist grand vision of the eclipse of man will be there to influence, rationalize, and justify favoring certain alternatives. It provides a narrative that takes those alternatives beyond contingency and presents them in a way that intentionally creates dissatisfaction with any merely human account of how we live and treat each other now. If it is the only story going, it is all the more likely to provide the moral meaning behind the scientific and technological future.\n\nAn argument could be made that we should avoid taking too seriously such grand visions of the future. In his book _In the Shadow of Progress_ , Eric Cohen warns, speaking specifically of genetics, that in order \"to think clearly\" and avoid the twin vices of over-prediction (assuming that our worst fears or greatest hopes will come to pass) and under-prediction (failure to acknowledge where present developments might go), \"we must put aside the grand dreams and great nightmares of the genetic future to consider the moral meaning of the genetic present,\" instead exploring \"what these new genetic possibilities might mean for how we live, what we value, and how we treat one another.\"\n\nCohen's caution is well taken, and the questions he poses ought indeed to be where thoughtful people begin to confront the apparently ceaseless innovations of our technological society. But it may be harder than it looks to separate how we think about the present from how we think about the future. Whether in secular or religious terms, it is not unusual for people to define \"the moral meaning of the present\" in terms of the future, judging what _is_ against what they hope (or fear) _will be_. After all, it is moral choice in the present that creates a future, so such visions influence how we live and treat each other today.\n\nCohen is doubtless right that a more sober and serious moral world would have less place for \"grand dreams and great nightmares.\" Yet given that we cannot but be influenced by such visions, we must come to grips with them on their own terms; there have to be _reasons_ for putting them aside. People were not unaware of problems with capitalism before Marx, but Marxism became a sufficiently powerful grand vision that it had just the effect Cohen fears, blinding people to the truth of their present circumstances and of their obligations to those around them. Facing up to the defects of Marxism was easier said than done and remains an incompletely accomplished task. But one part of doing so was intellectual confrontation with it _as a grandvision_. None of the specific transhumanist visions of the future have as yet anything like the intellectual or political power of Marxism at its height. But if we are to eschew them, a similar intellectual confrontation will be necessary.\n\n_CONFRONTING THE EXTINCTIONISTS_\n\nIt can be difficult to find a footing for criticism of transhumanism, given the broad scope of the transhumanist vision, the variety of (sometimes conflicting) transhumanist ideas, and of course the fact that we cannot yet know precisely what direction and shape transhumanist ideas will take in the real world. That is why the focus of this volume will be on transhumanism's _moral_ vision of the future, rather than its technological or scientific content. We will approach transhumanism sometimes directly and sometimes obliquely. While we will analyze the ideas of some of the most provocative and controversial transhumanist thinkers, we will also look at a wide range of other materials. We will journey from the invisibly tiny scale of molecular engineering to the far reaches of outer space. We will discuss ancient myths and recent science fiction, centuries-old paintings and Hollywood movies. And we will explore some forgotten byways, learning from long-neglected stories that can teach us about our desires for the future. Several of the chapters will also begin with prologues, short fictional stories I have written to help bring to life some of the ideas addressed in those chapters.\n\nWhy the emphasis on fiction? When compared to the works of nonfiction by experts and specialists in transhumanism, very often fictional works present us with a far more realistic and morally nuanced picture of the issues at stake in human self-overcoming. This should not surprise us. After all, excellent fiction requires a serious understanding of human things, and to tell a great story about scientific and technological possibilities, fiction writers have to start from a convincing human world\u2014which of course is the world out of which the scientific and technological developments that will allow human redesign will _actually_ emerge. Those who start instead from the imagination of technological possibilities often suffer a kind of tunnel vision, a narrow focus that makes them assume that whatever they are writing about will be the center around which everything else will revolve. Indeed, that outlook helps to account for the moral weakness of the transhumanist project.\n\nWhen you get beyond transhumanism's fascination with the technological cutting edge, it becomes evident that its hopes are not new. To separate the question of what kind of future is technologically likely from the question of what is a morally desirable future, it is useful to look closely at some of the earlier thinkers who developed ideas that are important to transhumanism\u2014even if the transhumanists do not realize the debt they owe them. While we will not attempt to offer a complete intellectual history of such ideas, in the chapter that follows we will look at a selection of particularly influential presentations of the theme of overcoming humanity, stretching back to the late eighteenth century. As we shall see, these thinkers lay down some of the key foundations upon which today's transhumanists build. We turn there first in order to answer a fundamental question: Why would anyone think that human extinction is a good thing?\n_CHAPTER ONE_\n\n_The Future in the Past_\n\nTHE ASPIRATIONS of transhumanism are not entirely new because the human desires to change unsatisfactory aspects of our lives or to extend our powers are not new. Yet these deeper, perduring aspirations have gained a new force in the modern age, as the world has come increasingly to be defined by the ability of modern science and technology to change it significantly. It has become possible to think seriously about progress, about what it would mean if these achievements were increasingly to shape our lives, building one on another.\n\nProgress does not inevitably have to point to the kind of future transhumanists envision. Science fiction authors, and others who think about the future in less overtly fictional ways, have had no trouble thinking about some pretty remarkable and very distant futures in which humanity persists in an eminently recognizable form. This does not represent a mere failure of imagination on the part of these futurists; in some ways, it arguably takes even more imagination to depict a future in which mankind has survived without becoming radically different.\n\nTo understand why the transhumanists believe progress requires human extinction, we can study some of their intellectual forebears\u2014authors who, while not all widely remembered today, were in their own time influential on the way people understood where science and technology might be taking us. We begin with one of the greatest of Enlightenment prophets of progress, Marie Jean Antoine Nicolas de Caritat, the French aristocrat known as the Marquis de Condorcet (1743\u20131794).\n\nCondorcet was a mathematician and philosopher, an abolitionist and advocate of women's equality and religious toleration, an admirer and biographer of Voltaire, an economic liberal, and a genial enthusiast\u2014an embodiment, in short, of all things enlightened. A prominent public intellectual, he was \"one of the few Enlightenment thinkers to witness the [French] Revolution and to participate fully in its constitutional aftermath. . . . He was, in short, an outstanding disciple of the Enlightenment, uniquely located at the center of great events.\" A leader in the early days of the French Revolution, he was forced into hiding as the political winds shifted. Concealed in the house of one Madame Vernet in 1793\u201394, he wrote his most influential work, a little book called _Outlines of an Historical View of the Progress of the Human Mind_. He was eventually arrested and thrown in prison, where he died the day after his arrest, under circumstances that \"have been the subject of much speculation ever since.\" But his final great work was published the next year, and it became a landmark of Enlightenment thinking and would shape how generations understood the idea of progress.\n\nIn the book, Condorcet is convinced that the progress of reason has gone too far to allow any future lapse into barbarism, but he is still chagrined at how little progress has been made to increase human happiness. \"The friend of humanity,\" he writes, \"cannot receive unmixed pleasure but by abandoning himself to the endearing hope of the future.\" And what a future Condorcet expects it to be:\n\nMay it not be expected that the human race will be meliorated by new discoveries in the sciences and arts, and, as an unavoidable consequence, in the means of individual and general prosperity; by farther progress in the principles of conduct, and in moral practice; and lastly by the real improvement of our faculties, moral, intellectual and physical, which may be the result either of improvement of the instruments which increase the power and direct the exercise of these faculties, or of the improvement of our natural organization itself?\n\nHow is this \"improvement of our faculties\" and \"natural organization\" to take place? Growing liberty, equality, and prosperity within nations and among nations, Condorcet writes, will raise the general level of instruction, and that in itself will improve human ability. More instruction will in turn produce more knowledge, and Condorcet expects that as we come to understand the world better and improve our ability to teach that understanding, what once might have required genius to uncover or comprehend can become a subject of general knowledge. The result is that we advance the starting point for yet further attainments in the arts and sciences, which in turn increases our powers of action\u2014an upward spiral of enlightenment. As a result, better food, better sanitation, and better medicine will extend the human lifespan, as will \"the destruction of the two most active causes of deterioration, penury and wretchedness on the one hand, and enormous wealth on the other.\" Likewise, Condorcet foresees that \"contagious disorders\" will be brought under control, as well as occupational and environmental illness. The net result:\n\nWould it even be absurd to suppose this quality of melioration in the human species as susceptible of an indefinite advancement; to suppose that a period must one day arrive when death will be nothing more than the effect of extraordinary accidents, or of the slow and gradual decay of the vital powers; and that the duration of the middle space, of the interval between the birth of man and this decay, will itself have no assignable limit? Certainly man will not become immortal; but may not the distance between the moment in which he draws his first breath, and the common term when, in the course of nature, without malady or accident, he finds it impossible any longer to exist, be necessarily protracted?\n\nDespite his denial of the possibility of immortality, Condorcet expects \"that the mean duration of life will for ever increase.\" It is one of the \"laws of nature\" that living things are subject to \"perfectibility or deterioration.\" The breeding of animals already shows that we could improve our own physical capacities and senses, but perhaps our \"quality of melioration\" suggests that a good deal more is possible for us by other means as well. Hence Condorcet wonders whether better-educated parents might not transmit the superior \"organization\" they have achieved directly to their children.\n\nCondorcet's vision of the future, so familiar to those of us in the West who live it on a daily basis, is very much a reflection of the project laid out more than a century earlier by the English philosopher Francis Bacon for \"the enlarging of the bounds of Human Empire, to the effecting of all things possible.\" Condorcet sees how the increase of our power over nature will soften the hard edges of the human condition by improving the material conditions of life, which will allow at the same time an improvement in the moral conditions. He adds an intimation of accelerating progress; one generation can build on the work of another. A smarter and healthier generation sets the stage for even greater achievement by the next generation. He may have hedged on the question of immortality, but as we shall see, not everyone who came after him was so modest.\n\nCondorcet's thesis\u2014that mankind would improve and expand, growing wiser and healthier and much longer-lived\u2014soon received a powerful rebuttal. Thomas Robert Malthus (1766\u20131834) was a political economist whose _Essay on the Principle of Population_ explicitly attacked Condorcet's depiction of progress. Malthus believed that finite resources limit what human beings can ever hope to accomplish, and that because human reproduction always races ahead of available food, our future holds great misery and scarcity. Malthus's ideas influenced many fields, including biology, where Charles Darwin (1809\u20131882) adapted them to help explain the workings of evolution, a kind of natural progress caused by competition for limited resources. Through to our own day, much of the debate about progress has arisen from tensions among these three men's ideas: Condorcet's optimism about human perfectibility, the Malthusian problem of resource scarcity, and the Darwinian conception of natural competition as a force for change over time. The transhumanists, as we shall see, reconcile and assimilate these ideas by advocating the end of humanity.\n\n_THE INVENTION OF IMMORTALITY_\n\nIn 1872, the British author and adventurer William Winwood Reade (1838\u20131875) revisited the project of progress that Condorcet had laid out. Reade, born in Scotland, was a failure as a novelist but had modest success as an African explorer and war correspondent. He was in correspondence with Darwin, who is said to have used in _The Descent of Man_ (1871) some information from Reade's expedition to West Africa. While it seems like in his short life Reade never quite lived up to his own expectations for himself, his attempt at a universal history\u2014an 1872 book called _The Martyrdom of Man_ \u2014was once highly regarded. W. E. B. Du Bois, Cecil Rhodes, H. G. Wells, and George Orwell all found reasons to praise it. Perhaps even the character Sherlock Holmes was a fan: in _The Sign of the Four_ , Holmes says to Watson, \"Let me recommend this book\u2014one of the most remarkable ever penned. It is Winwood Reade's _Martyrdom of Man_. I shall be back in an hour.\" From its original publication to 1910 the book went through eighteen editions in England and seventeen in the United States; one can only imagine that many a Baker Street Irregular has felt compelled to track it down.\n\n_The Martyrdom of Man_ began, Reade says, as an effort to give the hitherto neglected story of \"Inner Africa\" its due place within European history. But in the writing, the book became very much more: an effort to place human history within a larger natural history that eventually takes Reade right back to the development of the solar system and the origins of life. He adopts his own version of a Darwinian perspective, along with the theories about a geologically dynamic Earth, which were still rather recent in his day. Reade's naturalism is particularly deployed in extensive efforts to provide non-supernatural explanations for the rise of religions.\n\nBut there is one crucial aspect of his argument that distinguishes it from most similar presentations in our own day: Reade believes that nature is purposive\u2014and indeed, that something like a cunning of nature is evident in human history. That is to say, human activities like war and religion, or conditions like inequality, serve developmental purposes within a natural scheme of things beyond what is intended by the human beings participating in those activities. \"Thus when Nature selects a people to endow them with glory and with wealth her first proceeding is to massacre their bodies, her second, to debauch their minds. She begins with famine, pestilence, and war; next, force and rapacity above; chains and slavery below. She uses evil as the raw material of good; though her aim is always noble, her earliest means are base and cruel. But, as soon as a certain point is reached, she washes her black and bloody hands, and uses agents of a higher kind.\"\n\nTo put it another way, Reade believes that there is a natural imperative for higher abilities and capacities to grow out of lower ones: \"The philosophic spirit of inquiry may be traced to brute curiosity, and that to the habit of examining all things in search of food. Artistic genius is an expansion of monkey imitativeness. Loyalty and piety, the reverential virtues, are developed from filial love. Benevolence and magnanimity, the generous virtues, from parental love. The sense of decorum proceeds from the sense of cleanliness; and that from the instinct of sexual display.\" Reade's claim that the higher derives from the lower does not just apply to human beings. It is a characteristic of life itself, indeed a characteristic of matter, which he regards as inseparable from mind.\n\nWe ought not to think there is anything degrading about thus understanding the higher in light of the lower, Reade argues. Indeed, his reductionism opens the door to remarkable possibilities:\n\nIt is Nature's method to take something which is in itself paltry, repulsive, and grotesque, and thence to construct a masterpiece by means of general and gradual laws; those laws themselves being often vile and cruel. This method is applied not only to single individuals, but also to the whole animated world; not only to physical but also to mental forms. And when it is fully realised and understood that the genius of man has been developed along a line of unbroken descent from the simple tendencies which inhabited the primeval cell, and that in its later stages this development has been assisted by the efforts of man himself, what a glorious futurity will open to the human race! It may well be that our minds have not done growing, and that we may rise as high above our present state as that is removed from the condition of the insect and the worm.\n\nThat we can assist in our own uplift and greatly transcend what we are today is crucial to Reade's picture of the future, as it is to today's transhumanists. In the natural order of things, the individual human life has limited potential, precisely because by nature we are parts of a whole with at least potentially greater significance:\n\nAs the atoms are to the human unit, so the human units are to the human whole. . . . Nature does not recognise their individual existence. But each atom is conscious of its life; each atom can improve itself in beauty and in strength; each atom can therefore, in an infinitesimal degree, assist the development of the Human Mind. If we take the life of a single atom, that is to say of a single man, or if we look only at a single group, all appears to be cruelty and confusion; but when we survey mankind as One, we find it becoming more and more noble, more and more divine, slowly ripening towards perfection.\n\nThat Reade believes mankind is \"slowly ripening towards perfection\" implies that he is tolerably certain he understands the immediate project that faces humanity, and at least some of its longer-term consequences. Although he claims that he does not mean to suggest that humanity will ever understand the ultimate purpose of creation, he feels confident enough to assert that man was\n\nnot sent upon the earth to prepare himself for existence in another world; he was sent upon earth that he might beautify it as a dwelling, and subdue it to his use; that he might exalt his intellectual and moral powers until he had attained perfection, and had raised himself to that ideal which he now expresses by the name of God, but which, however sublime it may appear to our weak and imperfect minds, is far below the splendour and majesty of that Power by whom the universe was made.\n\nBy the power of science rather than prayer, Earth, \"which is now a purgatory, will be made a paradise.\" The genuinely \"Sacred Cause\" is \"the extinction of disease, the extinction of sin, the perfection of genius, the perfection of love, the invention of immortality, the exploration of the infinite, the conquest of creation.\"\n\nSo by making men mortal and immoral, nature points humanity in the direction of immortality and morality so long as we exercise our intelligence. Reade could already see signs of progress in this direction: \"Life is full of hope and consolation; we observe that crime is on the decrease, and that men are becoming more humane. The virtues as well as the vices are inherited; in every succeeding generation the old ferocious impulses of our race will become fainter and fainter, and at length they will finally die away.\" Delusions about an immortal soul will only stand in the way of such efforts; Christianity, which Reade treats under general headings such as \"Religion\" and \"superstition,\" will have eventually done the work intended of it as a tool of nature, and at that point can and must be destroyed, for it is in the nature of these tools to become obstructions once they have brought life to the next level. While human beings may never rival the great Creator of all things, there is a long way to go before that would become an issue. Echoing Francis Bacon, Reade notes that \"we can conquer Nature only by obeying her laws, and in order to obey her laws we must first learn what they are. When we have ascertained, by means of Science, the method of Nature's operations, we shall be able to take her place and to perform them for ourselves.\" Nature intends that we rebel against being the serfs of nature.\n\nHaving placed immortality explicitly on the agenda of the future, Reade considers space travel a necessary consequence:\n\nDisease will be extirpated; the causes of decay will be removed; immortality will be invented. And then, the earth being small, mankind will migrate into space, and will cross the airless Saharas which separate planet from planet, and sun from sun. The earth will become a Holy Land which will be visited by pilgrims from all the quarters of the universe.\n\nWhen human beings \"invent\" immortality we press beyond the natural order of things in which we are mere cells in a larger whole, and when we die are dead forever. In similar fashion, with space travel we will also have proven the essentially mundane character of the once-transmundane heavens. Yet Reade's imaginative assurance about the great things ahead for humanity puts in high relief the ignorance and miseries of humanity today.\n\nThese bodies which now we wear belong to the lower animals; our minds have already outgrown them; already we look upon them with contempt. A time will come when Science will transform them by means which we cannot conjecture, and which, even if explained to us, we could not now understand, just as the savage cannot understand electricity, magnetism, steam.\n\nThis is a glorious future, one in which men will be \"perfect,\" having the power of \"what the vulgar worship as a god.\" But Reade recognizes\u2014more so than did Condorcet\u2014that this vision is not entirely consoling. In a prayer-like passage, he acknowledges that it makes the ills of the present look all the more terrible, and the past a yet darker place:\n\nYou blessed ones who shall inherit that future age of which we can only dream; you pure and radiant beings who shall succeed us on the earth; when you turn back your eyes on us poor savages, grubbing in the ground for our daily bread, eating flesh and blood, dwelling in vile bodies which degrade us every day to a level with the beasts, tortured by pains, and by animal propensities, buried in gloomy superstitions, ignorant of Nature which yet holds us in her bonds; when you read of us in books, when you think of what we are, and compare us with yourselves, remember that it is to us you owe the foundation of your happiness and grandeur, to us who now in our libraries and laboratories and star-towers and dissecting-rooms and work-shops are preparing the materials of the human growth. And as for ourselves, if we are sometimes inclined to regret that our lot is cast in these unhappy days, let us remember how much more fortunate we are than those who lived before us a few centuries ago.\n\nThe fact that he calls his book _The Martyrdom of Man_ indicates that Reade is well aware of the tragic side of his progressivism. But \"in each generation the human race has been tortured that their children might profit by their woes. Our own prosperity is founded on the agonies of the past. Is it therefore unjust that we also should suffer for the benefit of those who are to come?\" Until men become immortal, the only satisfaction to be found is in the superiority of the present to the past, and the chance of making one's own infinitesimal contribution to the future.\n\n_SALVATION IN SPACE_\n\nWe turn next to another thinker who concluded early on that humanity, in order to preserve itself, would have to venture into outer space. Nikolai Fedorovich Fedorov (1829\u20131903), the illegitimate son of a Russian prince, was an intense but retiring Moscow librarian who was \"reputed to have read all the books he catalogued.\" Unlike all the other figures discussed in this chapter, Fedorov was not widely known during his own lifetime. The posthumous publication of two volumes of his work did not change that situation a great deal, even though the publisher made them available free of charge, in accord with Fedorov's beliefs about property. But the quality of those who admired his work makes up for the lack of quantity: he was known and respected by both Dostoevsky and Tolstoy. And Fedorov had one yet more important connection: he assisted, and some think passed his ideas on to, Konstantin Tsiolkovsky, among the greatest of the pioneers of space travel.\n\nIn a work composed sometime after a famine in 1891, Fedorov writes that the \"learned\" have neglected their obligation to the \"unlearned\" to improve the conditions of their lives, particularly the lives of agricultural workers. He is mightily impressed by reports of using explosives to create rain. He regards using the tools of war for peaceful purposes as literally providential, a sign of what God expects of man. He is much more skeptical than Reade about the cunning of nature, asserting that it is \"extreme childishness\" to expect that the \"blind force\" of nature will produce just good results. It is only when human beings put their will behind their common task\u2014to understand and control that force\u2014that it will be turned to the good by our conscious control.\n\nFedorov is well aware that such control as we currently possess is far from guaranteed to be used for the benefit of mankind. What is lacking, he believes, is the necessary sense of human kinship. \"Unbrotherly relations\" make life the \"struggle\" that has hitherto been definitive of human civilization. They also lead to a thoroughgoing misunderstanding of the true meaning of progress. Progress, Fedorov claims, is not to be seen in the superiority of man over beast, the superiority of the present generation over past generations (as Reade would say), or the superiority in this generation of the young over the old. Indeed, such a picture of progress has its tragic tone because it is inherently divisive: \"Progress makes fathers and ancestors into the accused and the sons and descendants into judges; historians are judges over the deceased, that is, those who have already endured capital punishment (the death penalty), while the sons sit in judgment over those who have not yet died.\"\n\nIn contrast, Fedorov\u2014a devout if unconventional adherent of Russian Orthodoxy\u2014writes that we should take our cue from \"true religion,\" which is \"the cult of ancestors, the cult of all the fathers as one father inseparable from the Triune God, yet not merged with him.\" On this basis Fedorov imagines the single, common task of mankind as a union of sons bent on overcoming the blind forces of nature\u2014not only to defeat hunger, disease, and death for the living, but to achieve the resurrection of all of their fathers, grandfathers, great-grandfathers, etc. In a passage that could have been written by a number of today's transhumanists, he says, \"Death has become a general organic evil, a monstrosity, which we no longer notice and no longer regard as an evil and a monstrosity.\" But we will learn to bring the dead back to life, \"substituting resurrection for birth\"; we will thereby eliminate the need for sexual reproduction, which is just another example of the blind operation of nature. We will solve hunger, too, substituting \"creativity for nutrition\": we will not need to eat, but will produce ourselves \"out of the very basic elements into which the human body can be decomposed.\"\n\nEven so, Fedorov is not unconcerned about the Malthusian problem of eventual exhaustion of resources here on Earth. He has his own, to us familiar, apocalyptic vision:\n\nThe extinction of stars (sudden or slow) is an instructive example, a terrifying warning. The growing exhaustion of the soil, the destruction of forests, distortions of the meteorological process manifested in floods and droughts\u2014all this forebodes 'famines and plagues' and prompts us to heed the warning. Apart from a slowly advancing end, we cannot be certain whether a sudden catastrophe may not befall the Earth, this tiny grain of sand in the vastness of the Universe.\n\nFor such reasons resurrection will not suffice; the exploration of outer space is also absolutely necessary to prepare the \"future homes of the ancestors.\" God has arranged that \"the Earth itself has become conscious of its fate through man\" and this consciousness would be useless were we simply to stand by and observe \"the slow destruction of our home and graveyard\" at the hands of purposeless nature. Rather, \"God is the king who does everything for man but also through man\" and he intends that humanity not be \"idle passengers\" but \"the crew of its terrestrial craft\"\u2014a remarkable prefiguration of the space-age\/environmentalist idea of \"Spaceship Earth.\"\n\nThe reasons for developing space travel transcend the merely practical necessities of overcoming resource exhaustion (\"the economic problem posed by Malthus\") and finding a place for resurrected ancestors. Space travel will also deeply affect our moral nature. Is it more fantastic, Fedorov asks, to believe in the Christian understanding of heaven and the afterlife\u2014\"to create a moral society by postulating the existence of other beings in other worlds and envisioning the emigration thither of souls, the existence of which cannot be proven\"\u2014or to believe that we might visit other worlds ourselves someday? Moreover, to \"inhabit all heavenly bodies\" would\n\nunite all the worlds of the Universe into an artistic whole, a work of art, the innumerable artists of which, in the image of the Triune Creator, will be the entire human race . . . attaining divine perfection in the cause, the work of restoring the world to the sublime incorruptibility it had before the fall. Then, united, science and art will become ethics and aesthetics; they will become a natural universal technology of their work of art.\n\nIt is through human efforts, then, that a _cosmos_ \u2014an ordered universe\u2014comes to exist, rather than the purposeless _chaos_ , the mere raw materials given us by God, that exists prior to human intervention. We will see a similar point being made many times in the pages that follow.\n\nBoth Fedorov and Reade seem to assume that outer space is entirely available for human colonization. In so doing they are taking a position on the question of the existence of life on other worlds, which was in fact already much debated in their day. Developments in astronomy, geology, and biology, in particular the thinking of Darwin, were leading many to consider that there was no reason to expect Earth to be the sole abode of life\u2014and if life could exist elsewhere, so too intelligence may have evolved. We turn to one such thinker next.\n\n_EDIFYING ALIENS_\n\nFrench astronomer and spiritualist Nicolas Camille Flammarion (1842\u20131925) ranks high among the great popularizers of science, particularly astronomy\u2014the Carl Sagan of his time. Owner of a private observatory and author of some seventy books, he was particularly interested in the possibility of extraterrestrial life, and was among the first to imagine it in thoroughly alien terms, an idea he presented in both his popular science writing and science fiction. A sense of his contribution can be gauged by the fact that he has craters named for him on both the moon and Mars. Into the 1960s, decades after his death, his name remained on a popular, if by that time much updated, introduction to astronomy. And he garnered all this recognition despite the fact (perhaps in his own time because of the fact?) that he considered spiritualism to rank among his scientific interests.\n\nWriting early in the twentieth century, Flammarion articulates what is often called the \"assumption of mediocrity\": that there is nothing special about Earth's place in the universe, and so life likely exists elsewhere. \"On what pretext could one suppose that our little globe which, as we have seen, has received no privileges from Nature, is the exception; and that the entire Universe, save for one insignificant isle, is devoted to vacancy, solitude, and death?\" If outer space looks unfriendly to life at first glance, we learn even from terrestrial observation that the environment that is hostile to one form of life is favorable to another. So the fact that extraterrestrial conditions may appear hostile to life as we know it is no warrant that life is impossible.\n\nFurthermore, Flammarion argues that extraterrestrial life will likely be different from life as we know it, perhaps with a different chemical basis and very different capabilities. Perhaps it will in some respects be superior to us. Might not nature\n\nhave given to certain beings an electrical sense, a magnetic sense, a sense of orientation, an organ able to perceive the ethereal vibrations of the infra-red or ultra-violet, or permitted them to hear at a distance, or to see through walls? We eat and digest like coarse animals, we are slaves to our digestive tube: may there not be worlds in which a nutritive atmosphere enables its fortunate inhabitants to dispense with this absurd process? The least sparrow, even the dusky bat, has an advantage over us in that it can fly through the air. Think how inferior are our conditions, since the man of greatest genius, the most exquisite woman, are nailed to the soil like any vulgar caterpillar before its metamorphosis! Would it be a disadvantage to inhabit a world in which we might fly whither we would; a world of scented luxury, full of animated flowers; a world where the winds would be incapable of exciting a tempest, where several suns of different colors\u2014the diamond glowing with the ruby, or the emerald with the sapphire\u2014would burn night and day (azure nights and scarlet days) in the glory of an eternal spring; with multi-colored moons sleeping in the mirror of the waters, phosphorescent mountains, aerial inhabitants\u2014men, women, or perhaps of other sexes\u2014perfect in their forms, gifted with multiple sensibilities, luminous at will, incombustible as asbestos, perhaps immortal, unless they commit suicide out of curiosity?\n\nFor Flammarion, the possibility of life elsewhere\u2014indeed, life everywhere\u2014raises the question of the ultimate destiny of human life. For it is life, he believes, not matter, that is the key to understanding the universe, yet life eventually seems to give way to mere matter; individually we are built on death and proceed unto death ourselves, and the same is true for our planet as a whole. Hence \"Let no one talk of the Progress of Humanity as an end! That would be too gross a decoy.\" Flammarion believes that progress is the law of life, but, as Fedorov also suggested, _material_ progress alone would mean that in the end we would still fall prey to entropy itself\u2014that each of our lives, and that human life as a whole, will be extinguished. We reject this gloomy possibility, he says, as being \"incompatible with the sublime grandeur of the spectacle of the universe.\" So while \"Creation does not _seem_ to concern itself with us,\" this appearance may be deceiving. He even goes so far as to ask if \"distant and unknown Humanities\"\u2014that is, alien races\u2014might not be \"attached to us by mysterious cords, if our life, which will assuredly be extinguished at some definite moment here below, will not be prolonged into the regions of Eternity.\"\n\nOne would be hard pressed today to find a popularizer of science who, like Flammarion in this poetic and confusing passage, seems to hover between a fairly traditional notion of heaven and a suggestion of interstellar reincarnation. But he seems to have felt that our intuition that our lives cannot end had support from his astronomy: \"As our planet is only a province of the Infinite Heavens, so our actual existence is only a stage in Eternal Life. Astronomy, by giving us wings, conducts us to the sanctuary of truth. The specter of death has departed from our Heaven. The beams of every star shed a ray of hope into our hearts.\"\n\nFor Reade and Fedorov, the prospect of a universe that can be enlivened by human action, which involves transcending the natural order even to the point of inventing immortality, gives hope and meaning to the human future. For Flammarion, on the other hand, the conclusion that alien life is already omnipresent and diverse calls attention to the parochialism of our own view of ourselves, fostering scientific imagination of beings different from and even superior to us. We see the limitations on our own lives by imagining beings with different capacities, and in that light our own limits appear simply arbitrary. Yet at the same time, the likelihood of alien life suggests to him the insignificance of brute matter. The ascendance of life that he imagines we will see in the universe as a whole opens the door to a hope for the yet greater victory for life that would be personal immortality.\n\n_BECOMING ALIENS_\n\nOur next thinker shared Fedorov's and Flammarion's concern about the limited prospects for material progress if we are confined to Earth, and so believed that humanity was destined to explore outer space\u2014but that, in the long term, we would become something no longer recognizably human at all. J. B. S. Haldane (1892\u20131964) was a distinguished scientist, a major public intellectual, and an outspoken Marxist. His main contributions to science were in quantitative analysis of genetics and evolutionary biology\u2014from which one would not necessarily adduce the humor, imagination, and charm of his popular writing. His influence was such that he helped inspire two of the greatest works of anti-utopian literature of the twentieth century: his acquaintance Aldous Huxley based some of _Brave New World_ on Haldane's ideas, and C. S. Lewis is said to have had Haldane in mind for various speeches and characters in his space trilogy.\n\nHaldane most vividly sketched out his vision of the human future in \"The Last Judgment,\" a piece of writing from 1927 that is part essay, part science fiction story. (Lewis considered it \"brilliant, though to my mind depraved.\") Haldane begins by looking at how the Earth and our sun might come to their natural ends. He soon turns to consider a theme that we will see become increasingly common, how we might destroy the Earth ourselves, imagining an account of the last millennia of human life on Earth as it might be told by a distant descendant living on Venus. The premise of this story is that as a consequence of having \"ridiculously squandered\" tidal power over a period of some five million years, humans have changed the moon's orbit until it comes so close to Earth that it is pulled apart, in the process making Earth uninhabitable.\n\nWhile this result was long predictable, humans \"never looked more than a million years ahead\" so few were ever concerned with this consequence of using tidal power. Instead, in the course of their three-thousand-year-long lives, most people concentrated on \"the development of personal relationships\" and on \"art and music, that is to say, the production of objects, sounds, and patterns of events gratifying to the individual.\" Natural selection having ceased, the only substantial change to humanity was \"the almost complete abolition of the pain sense.\" Real advances in science came to a halt; rather than try to develop the human race, attention was paid to breeding beautiful flowers.\n\nHaving foreseen what was to come, however, a few did what they could to assure the existence of life elsewhere after the anticipated disaster. That is no small task as Haldane paints it; even simple steps like managing to land explorers successfully on the moon, Mars, and Venus takes a couple of million years. The technical difficulties of landing and return are compounded by the disinclination of individualistic humans to give up their long lives on what amount to suicide missions. Those who finally land alive on Mars are destroyed by sentient alien life already established there, as Flammarion might have expected. Those who land on Venus find extremely hostile environmental conditions under which humans cannot possibly survive. Efforts at further exploration are dropped for the time being.\n\nAbout eight million years later, the approaching moon having disrupted earth's geology and ecosystems, a minority undertake renewed efforts to colonize Venus. \"A few hundred thousand of the human race . . . determined that though men died, man should live forever.\" By a ten-thousand-year-long effort at selective breeding, humans create a new race that can survive on Venus. These colonists are sent out in 1,734 ships; eleven manage to land. Such life as Venus already had, inimical to the colonists, is utterly destroyed by bacteria prepared for that purpose. From that point on, the settlement of Venus proceeds apace.\n\nOur Venusian descendants were designed by the small minority of species-minded Earthlings to share a hive mind; they do not suffer from the selfish propensity for seeking individual happiness that led to Earth's destruction. Two new senses contribute to the hive mind: at every moment they sense \"the voice of the community,\" and they also have a sort of built-in radio that can be turned on or off at will. They are also genetically predisposed to look to the future more than the past, unlike Earthlings whose strange backward-looking propensities are illustrated not only by their failure to act in the face of their destructive tendencies, but by their religious beliefs. The Venusians' forward-looking characteristic also makes them more willing to sacrifice themselves.\n\nThe net result is that the Venusians see their potential extending far beyond anything humans ever could have accomplished; \"we have settled down as members of a super-organism with no limits on its possible progress.\" They plan to breed a version of themselves that will be able to settle Jupiter. Foreseeing in 250 million years an improved opportunity for interstellar travel, they think they can take it \"if by that time the entire matter of the planets of the solar system is under conscious control.\" Only a few of the millions of projectiles they send out might succeed. The Venusians are undaunted:\n\nBut in such a case waste of life is as inevitable as in the seeding of a plant or the discharge of spermatozoa or pollen. Moreover, it is possible that under the conditions of life in the outer planets the human brain may alter in such a way as to open up possibilities inconceivable to our own minds. Our galaxy has a probable life of at least eighty million million years. Before that time has elapsed it is our ideal that all the matter in it available for life should be within the power of the heirs of the species whose original home has just been destroyed. If that ideal is even approximately fulfilled, the end of the world which we have just witnessed was an episode of entirely negligible importance. And there are other galaxies.\n\nIn his commentary on his story at the end of \"The Last Judgment,\" Haldane acknowledges that he is not really trying to predict the future\u2014he is just engaging in an imaginative thought experiment, a \"valuable spiritual exercise.\" The future will certainly not conform to our present ideals, but thinking about it can illuminate \"our emotional attitude towards the universe as a whole\" that presumably is one source of those ideals. Traditionally, that attitude has been the province of religion. But modern science has taught us that the universe is far vaster in size and possibilities than religions ever knew, and so it is necessary to start using our imaginations in connection with these new realities. In effect, then, science fiction stands in for religion. The new scale of things we can begin to imagine should call forth a greater ambition among the most creative humans to develop (and for the rest of us to cooperate in) a plan that goes beyond traditional ideas of salvation, such as the assumption that the purpose of creation is to prepare some few for \"so much perfection and happiness as is possible for them.\"\n\nWe can \"only dimly conjecture\" what this plan might be, but Haldane wonders whether it might be the \"emergence of a new kind of being which will bear the same relation to mind as do mind to life and life to matter.\" As we can already envision the end of our own world, some such transformation will be necessary. Only if the human race proves that \"its destiny is eternity and infinity, and that the value of the individual is negligible in comparison with that destiny,\" will \"man and all his works\" not \"perish eternally.\"\n\nThe tension within this edifying conclusion is not hard to spot. In Haldane's scheme, an eternal and infinite destiny can only be achieved by making man himself into one of the works of man, such that in fact _human_ beings _do_ perish eternally. Furthermore, the imagination of this superior progeny is really an exercise in elucidating all the reasons for which, by and large, we should not be missed. So Haldane's substitute for religion embodies an \"emotional attitude towards the universe as a whole\" which is predicated on the assumption that whatever human beings do, \"man's little world will end.\" The real choice is between ending it ourselves and having it ended for us, or perhaps between ending it accidentally and ending it deliberately.\n\n_MANKIND REMANUFACTURED_\n\nHaldane may have claimed he was not trying to predict the future, but our final author certainly was\u2014and in the process, he lays out a rather specific path toward what he calls \"the progress of dehumanization,\" integrating many of the themes that our other authors developed.\n\nThe Irish-born J. D. Bernal (1901\u20131971) was, like Haldane and Flammarion, a scientist by training. He is probably best known for the development of the mathematics of X-ray crystallography, which quickly became a key technique of chemical analysis. (It was this technique that allowed the double-helix structure of DNA to be discovered, for example.) He did research that helped facilitate the D-Day landings, and made serious contributions to the sociology of science. A public intellectual of some note, Bernal was a dedicated communist and admirer of the Soviet Union; in 1953, he was awarded the Stalin Peace Prize, a prominent Soviet prize for the country's international supporters, and from 1959 to 1965 he was president of the World Peace Council, a Soviet-funded international activist group.\n\nBernal's first popular publication was a thin volume called _The World, the Flesh and the Devil: An Inquiry into the Three Enemies of the Rational Soul_ (1929). In it, he proposes an objective effort to predict things to come. Bernal acknowledges that this task might be easier said than done, partly because it can be difficult to distinguish prediction and desire, partly because of all the complex interactions that make the world what it is, and partly because \"all evidence\" points \"to ever increasing acceleration of change.\" Nevertheless, it is reasonable to start by looking at what the trends are. Bernal projects the future in three areas: \"the world,\" or our power in relationship to the material world; \"the flesh,\" or our power over life, particularly our own bodies; and \"the devil,\" our power over our own psyches. He concludes his volume by attempting to see what might come of developments in these three areas taken together.\n\nIn each realm Bernal expects remarkable things. His chapter \"The World\" focuses primarily on \"the conquest of space.\" He anticipates some developments that we have only recently achieved, like the use of huge sails to propel ships using solar wind, but his most extended discussion is of what it would take to create ten-mile-diameter spherical habitations with tens of thousands of inhabitants. With the necessary propulsion systems added, these communities in space would eventually allow for the long voyages that interstellar travel would require\u2014voyages that will be necessary as our sun begins to fail.\n\nThe chapter \"The Flesh\" starts from the bald assertion that \"modern mechanical and modern chemical discoveries have rendered both the skeletal and metabolic functions of the body to a large extent useless.\" Bernal expects the increasing substitution of mechanical for biological systems in the human body, with all the augmentation of physical and sensory abilities that implies\u2014for example, \"we badly need a small sense organ for detecting wireless frequencies.\" People have always wanted longer lives and more opportunities \"to learn and understand.\" But achieving such goals is now in sight:\n\nSooner or later some eminent physiologist will have his neck broken in a super-civilized accident or find his body cells worn beyond capacity for repair. He will then be forced to decide whether to abandon his body or his life. After all it is brain that counts, and to have a brain suffused by fresh and correctly prescribed blood is to be alive\u2014to think. The experiment is not impossible; it has already been done on a dog and that is three-quarters of the way towards achieving it with a human subject.\n\nBernal expects that once some men were thus transformed, they would be most able at transforming others. Humans will have a \"larval\" stage of six to twelve decades in our current bodies, then we will pass into \"chrysalis, a complicated and rather unpleasant process of transforming the already existing organs and grafting on all the new sensory and motor mechanisms.\" Of course, unlike a butterfly, the end result of the human transformation will be capable of constant upgrade and modification\u2014and indeed there will be no one form into which people will change themselves in any case, as the mechanical body will be readily customizable.\n\n\"Normal man is an evolutionary dead end; mechanical man, apparently a break in organic evolution, is actually more in the true tradition of a further evolution.\" Bernal envisions each of these mechanical men as looking something like a crustacean, with the brain protected in a rigid framework and a system of appendages and antennae attached for sensing and manipulating the world. He freely acknowledges that, to us, these beings would appear \"strange, monstrous and inhuman.\" But he claims that such monsters are \"only the logical outcome of the type of humanity that exists at present.\"\n\nIn any case, beings so designed would quickly become progressively more different from us. Their brains would be readily linked together electronically to become a kind of group mind. Thus, while the original individual organic brain itself would still have a limited lifespan (perhaps three hundred to a thousand years, Bernal estimates), sharing its feelings, knowledge, and experience with other brains would be a way of \"cheating death.\" Bernal ends the chapter with the speculation that these inhuman beings would invent whole new materials and forms of life out of which to constitute themselves, so that even organic brain cells could be replaced with more diffuse materials with more complex interconnections, thus ensuring itself \"a practical eternity of existence.\" They might transcend physical embodiment altogether, becoming completely etherealized, atoms in space communicating by radiation, ultimately perhaps resolving entirely into light. \"That may be an end or a beginning, but from here it is out of sight.\"\n\nThese first chapters lay out what would become an agenda for decades of science fiction and a fair amount of actual research and development. The next chapter, \"The Devil,\" is one that Bernal himself expressed dissatisfaction with nearly four decades after the book was published; he admitted that it was too much written under the influence of Freud. But the issue it discusses remains one that is debated, even if not precisely on Bernal's terms. The main question is whether continued progress in science will be able to overcome the problem posed by the new (that is, Freudian) insight that \"the intellectual life\" is not \"the vocation of the rational mind, but . . . a compensation . . . a perversion of more primitive, unsatisfied desires.\" That is, science requires an ongoing supply of \"perverted individuals capable of more than average performance.\" Should our psychology and our power over nature combine to make the satisfaction of our desires the norm, we could settle into a \"Melanesian\" life of \"eating, drinking, friendliness, love-making, dancing and singing, and the golden age may settle permanently on the world\" without any desire for further progress at all. (Note the similarity here with Haldane's flower breeders.) On the other hand, it could also be that we might be able to live lives that are both \"more fully human and fully intellectual\" if \"a full adult sexuality would be balanced with objective activity.\"\n\nThe question of whether all this progress will eliminate the desire for further progress has another side as well, given the \"distaste\" that Bernal acknowledges he feels, and others are likely to feel, about what the future holds \"especially in relation to the bodily changes.\" It may even be that people will not have a chance to get used to such changes gradually, given the accelerating rate of change. Bernal does not pretend to predict whether repugnance, combined with satisfaction, will ultimately triumph over the increasing power that will be in the hands of those who advance the cause of science and mechanization. But one result might be the \"splitting of the human race\" into two branches: a stagnant because \"fully balanced humanity\" and another branch \"groping unsteadily beyond it.\" Seeing how that outcome might arise is the point of the book's concluding chapters.\n\nBernal's basic thought here is that the mechanical men he envisions would be very well suited for colonizing and exploiting space, as their life-support requirements would be far less than what human beings require and their capacities would be wider. He imagines these transhumans as \"connected together by a complex of ethereal intercommunication\" and spread out across space and time. But he is brought up short by the recognition that the human mind had hitherto \"evolved always in the company of the human body.\" The radical change he anticipates to \"the delicate balance between physiological and psychological factors\" will create \"dangerous turning points and pitfalls.\" What will happen to the sexual drive, for example? Perhaps it will require yet more thoroughgoing sublimation into research or, even more likely, into \"aesthetic creation.\" As these new beings come ever more completely to understand the world around them, and ever more capable of manipulating it, their primary purpose is likely to become determining \"the desirable form of the humanly-controlled universe which is nothing more nor less than art.\"\n\nAfter much consideration about how the possibility of \"permanent plenty\" might transform society, Bernal settles on the thought that the future is likely to hold _de facto_ or even _de jure_ rule by a scientific elite that could be the first stable aristocracy. This elite would have the means to assure that the masses engage in \"harmless occupations\" in a state of \"perfect docility under the appearance of perfect freedom.\" \"A happy prosperous humanity enjoying their bodies, exercising the arts, patronizing the religions, may be well content to leave the machine, by which their desires are satisfied, in other and more efficient hands.\"\n\nSince Bernal thinks that those who tend the machines will increasingly be machines themselves, we now see why he thinks the human race might split into two branches. Yet whereas it seems very likely, as he has suggested, that the distinction between machine-men and men would also be the distinction between ruler and ruled, perhaps that would not have to be the case. For as Bernal notes, science depends on the supportive routine work of non-scientists, and on the recruitment from the many of the few most capable minds. Furthermore, he claims, scientists themselves tend to have a strong identification with humanity. So the first stable aristocracy could be a meritocracy that might at least recruit (or should one say harvest?) fresh brains from the most promising of humans. Still, characteristics that might bind the two groups will likely diminish with time, allowing the underlying processes producing dimorphism to hold sway. At that point it is quite possible that \"the old mechanism of extinction will come into play. The better organized beings will be obliged in self-defense to reduce the numbers of the others, until they are no longer seriously inconvenienced by them.\" The main hope for a different outcome is once again the prospect that the more advanced beings will settle in space, leaving Earth to the old-fashioned model in \"a human zoo, a zoo so intelligently managed that its inhabitants are not aware that they are there merely for the purposes of observation and experiment.\" So decades before today's transhumanists, Bernal predicts the survival of humanity as a curiosity (at best).\n\nBernal is not certain his vision will prevail, nor does he hold that the developments he lays out will produce a perfect world:\n\nthe dangers to the whole structure of humanity and its successors will not decrease as their wisdom increases, because, knowing more and wanting more they will dare more, and in daring will risk their own destruction. But this daring, this experimentation, is really the essential quality of life.\n\nBernal's predictions are not so millennial as Fedorov's, nor as overtly tragic as Reade's. But they do contain tensions. He recognizes that his scientifically driven \"progress of dehumanization\" is motivated by an ultimately unfulfillable desire for the mysterious and supernatural. Those who push the boundaries of knowledge outward will create a world that, for them at least, will be ever more prosaic, and therefore of less interest. Even if this process is infinite, it retains a Sisyphean character, and one might wonder: why bother? It turns out, however, that this \"daring\" effort to transcend one's time and control one's life is nothing other than an expression of life itself that is beyond our control. There is a natural fatality to our effort to control nature.\n\n_FROM BETTER HUMANS TO BEYOND HUMANITY_\n\nMy purpose in presenting these examples of thinkers who anticipated today's transhumanism is not to suggest how these thinkers might have influenced one another, or to prove their influence on today's transhumanists. But I trust it is reasonably clear that between Condorcet and Bernal the idea of progress itself has traveled quite a distance. Where for Condorcet the friend of humanity can find reason to think that in the world to come people will be more humane to each other, when Bernal looks to the future he sees \"the progress of dehumanization\": human extinction at worst, and at best human irrelevance to the progressive development of intelligence and power over the natural world\u2014an evolutionary \"dead end.\" As we will discuss later, Oxford philosopher Nick Bostrom, one of the founders of the World Transhumanist Association and the director of Oxford's Future of Humanity Institute, thinks we can have it both ways, expressing the hope that our posthuman replacements will be designed to be _more_ humane than we are. Everyone seems to agree that the kinder, gentler world that Condorcet imagines could indeed come into being; but for Haldane it would amount to a short-sighted squandering of nature's potential, while for Bernal we humans would be likely to live in it as subjects to powers so far beyond our control as to make historical aristocracies seem models of egalitarianism. Today, David Pearce, another founder of the World Transhumanist Association, cuts through the problem of needing some to rule others in order to keep them happy by suggesting that we can redesign ourselves so that we are always experiencing \"a sublime and all-pervasive happiness.\" Condorcet's expectation that people will be better fed becomes the revolt in Fedorov and Flammarion against the \"absurd\" need to eat at all, while today's transhumanists likewise find it unacceptable that we eat and excrete as we do. Condorcet suggests the possibility of accelerating progress toward the conquest of nature here on Earth, but for Haldane and Bernal what is at stake is the aesthetic recreation of the universe itself. Today, inventor and author Ray Kurzweil, the most widely known of the transhumanists, wonders if posthuman superintelligence might not be able to overcome entropy itself, thereby preventing the now-expected eventual end to the possibility of life in the universe and overcoming the last challenge to the immortality that Condorcet was only willing to hint at.\n\nThere is no single arc that connects all of our authors in such a way as to account for this significant transformation in the understanding of progress. But each lays part of the foundation for the change, a foundation on which is built the edifice that is contemporary transhumanism. Let us try to identify some of the key points.\n\nThe main line of Condorcet's argument is the most familiar. Perhaps building off Rousseau's notion of human \"perfectibility,\" Condorcet asserts that we possess a (unique?) \"quality of melioration\" that allows us to improve ourselves, primarily through the conquest of nature. As that project succeeds, many of the longstanding, seemingly given conditions of human life\u2014poverty, hunger, disease, vice, and other pervasive disabilities that have stood in the way of a good human life\u2014become problems that can be solved. By solving them, we make better human beings, human beings who are physically more fit, mentally more capable, morally improved. The rate of progress accelerates with this new starting point, but Condorcet seems to believe that people will remain human beings.\n\nYet Condorcet's own ideas about life extension, combined with the possibility of accelerating progress, begin to suggest something more radical. Initially life extension seems of a piece with the other improvements he speaks of. After all, as we become healthier and eat better, longer life would seem to follow as a matter of course. One could say that Condorcet is merely pointing out how we can reduce the incidence of premature death. But when he starts talking about an indefinite extension of lifespan, the door is opened to the possibility of a significantly more fundamental change in the terms of human existence. It does not seem as if Condorcet wishes to open the door very wide. Yes, he looks forward to a time when death becomes something that is chosen, but note that it is chosen \"in the course of nature,\" as if there is in this respect at least some part of nature that human beings will not or should not master.\n\nThis limit is one that our other authors are not nearly so inclined to respect, and their overt desire for immortality is of a piece with a far stronger inclination to imagine the development out of humanity of some completely new kind of superior being. Perhaps Condorcet did not understand, or chose not to highlight, the more radical consequences of his own picture of accelerating progress. But it seems more likely that something had to be added to Condorcet in order to promote this shift in the imagination. What might that be?\n\nFirst, as Fedorov explicitly highlighted, there is the Malthusian dilemma of resource scarcity. Malthus wrote in direct response to Condorcet's hopes for the future, arguing that the melioration Condorcet imagined would be self-defeating. More people living materially more comfortable lives will simply produce resource scarcity, which in turn will bring back all the ills of human life. Perpetual progress understood as an ongoing improvement in the material conditions of life for all is thus simply impossible since population will grow faster than available resources. For Fedorov, the conquest of space is in part a solution to the Malthusian dilemma of resource exhaustion, a solution that becomes the more plausible as the sense grows that the Earth is but a tiny speck in a very large universe. Of course, as the hostility of the extraterrestrial environment came to be better understood, this prospect may have seemed more daunting. But Haldane and Bernal are there with a solution to this part of the problem: the radical reconstruction of humanity in a way that makes mankind better suited for life away from Earth.\n\nAs we have seen, once you have taken that step, the promise of effectively infinite worlds in infinite space makes anything seem possible; both Haldane and Bernal take us to the very limits of the human imagination, Haldane by suggesting that all the matter in the galaxy available to life should be used by it before moving on to other galaxies, Bernal by suggesting that our distant descendants will remake the universe with a new \"let there be light.\" Perhaps these beings that conquer space will also conquer time and entropy, a route to the eternity promised by Flammarion and the end to death promised by Fedorov.\n\nThe resource scarcity that Condorcet did not worry about implies ongoing competition among human beings rather than an ever more cooperative world, and by the late nineteenth century that ongoing competition was firmly associated with Darwinian evolution. This intellectual revolution is the second change that has pushed Condorcet's successors in a more radical direction. Condorcet could assert that living things must be on a course of either perfection or decline; human beings in the future could change, but the result would be perfected or degraded human beings. From a Darwinian point of view, as Reade highlights, why should there not be changes in the future that correspond in magnitude to the changes that produced man as we now observe him? After Darwin, it becomes possible, _if not downright necessary_ , to think that future human descendants will not be human.\n\nScholars disagree about whether Darwin himself conceived that the evolutionary changes that brought about human beings (and other species) should be called \"progress\" or merely change. There is agreement that some of his writings point in one direction, some in another. At present, the \"mainstream\" scholarly view is that Darwin's statements implying that there is an ascent to humanity were mere concessions to the progressive spirit of his Victorian times, and that Darwin himself understood that his principles allowed him to speak of evolutionary change but not progress. Yet there is also an impressive body of arguments and evidence to suggest that Darwin _did_ believe in evolutionary progress, so long as we take sufficient care to define what that phrase means. In either case, it seems indisputable that a great many of those who, like Reade, were influenced by Darwin's ideas _took_ him to be pointing to an evolutionary ascent to humankind. And if so, why should that process not continue to produce yet higher forms of life? After all, even the penultimate words of Darwin's _Descent of Man_ suggest that man may be excused for \"feeling some pride at having risen, though not through his own exertions, to the very summit of the organic scale,\" which in turn may give \"hopes for a still higher destiny in the distant future.\"\n\nYet all of our thinkers would agree that there will be a difference between the blind evolution that produced humanity and the future evolution driven by human beings and then posthumans, precisely because our \"own exertions\" can now play a part. If evolution is the law of life, and at the same time if evolution has brought about human beings who can take hold of evolution and direct it as one more aspect of our control over nature, then a grand narrative of free human creativity becomes possible. Our conquest of nature is no longer a local affair but takes on a cosmic significance. For Reade, Fedorov, or Haldane's Venusians, the creation of new forms of life, the enlivening of the cosmos, is the goal of goals, the highest good. We transform the universe by transforming ourselves.\n\nFlammarion may be the superior Darwinian here, thinking that it is at least short-sighted and at worst completely inconsistent to think that all this marvelous development of matter into life and life into intelligence should be a process confined to one planet alone. Yet he does not highlight the tougher Darwinian consequences of this line of thought. What if the cosmos is not ours to do whatever we want with, because other forms of life have already staked a claim? Haldane seems to understand the situation best of all our thinkers: he extends the realm of competition beyond our world, with the expected consequences of extraterrestrial winners (the Martians in Haldane's tale) and losers (the aboriginal Venusians). At some point, the Martians, victorious over mere human beings, will have to deal with the greater abilities of the human-created Venusians, or vice versa. Still, even if it's a harsh universe, we come to the same kind of conclusion about the imperative of creative, self-directed evolution. For we had best be prepared to meet it coming from \"out there,\" or else (as Haldane might say) suffer the fate of the first Venusians, or indeed of earthly human life, for our folly.\n\nFlammarion presents what is probably just the prettier side of the same coin. One might have thought that to the scientific mind _any_ life beyond Earth would be interesting enough\u2014surely any intelligent life. But the alien forms Flammarion imagines all have some wonderful advantage over mere human beings, so wonderful as to make us look pretty grubby by comparison, \"like coarse animals . . . nailed to the soil like any vulgar caterpillar.\" Should we not then aspire to be more like those superior alien beings? Such evolutionary fitness as human beings might exhibit is relative only to the conditions of our place and time, and perhaps (as Haldane would suggest) we are not suited well enough even for that. We should not expect to persist into the far future, or on worlds beyond our own, without becoming alien to what we are now. But here again the key point is that this change counts as progress. Haldane's Venusians are clearly presented as superior to the humans who created them, and against all squeamishness about crustaceans, Bernal would in effect have us will to become Flammarion's aliens \"gifted with multiple sensibilities, luminous at will, incombustible as asbestos, perhaps immortal.\"\n\nThe grand narrative of material progress and self-overcoming has one final twist that again links the materialist-minded Bernal with Fedorov's faith and Flammarion's spiritualism: material progress is itself something to be overcome. Bernal imagines that the ultimate destiny of intelligence may be to resolve itself into light. Whatever that might mean, how different is it from imagining spirits that might be communicated with or resurrected into bodies? Matter can be raised up into life and life raised up into intelligence; why should there not be further extensions of the sequence, however beyond our comprehension they might be? The progress of dehumanization runs from vile bodies to healthy bodies to redesigned bodies to no bodies at all.\n\n_THE PARADOX OF PROGRESS_\n\nThis last transformation into the luminous, if not the numinous, raises in the most acute form a problem has become increasingly obvious as we have proceeded through these lines of thought. In the lead-up to one of his most widely quoted aphorisms, philosopher George Santayana says,\n\nProgress, far from consisting in change, depends on retentiveness. When change is absolute there remains no being to improve and no direction is set for possible improvement; and when experience is not retained, as among savages, infancy is perpetual. Those who cannot remember the past are condemned to repeat it.\n\nAs Santayana suggested, the kind of \"absolute\" change in the human being imagined by Bernal and Haldane, along with today's transhumanists, really precludes the use of the term \"progress.\" It becomes harder and harder for our authors to imagine what will be retained, hence where change will start from. And if the rate of change is accelerating, that simply means we are headed the more rapidly from one unknown to another, while the recognizable old standards for judging whether the changes are progressive are overthrown along with our humanity.\n\nIn today's world, a vision of progress like that laid out by Condorcet remains very much alive. The easing of human life through universal education, reduction in disease, increased sanitation, improved agricultural productivity, and a rising material standard of living is an established fact for much of the world, and the main questions involve how most rapidly and \"sustainably\" to extend these benefits more widely and how to improve upon what we already have. Likewise, we take increased life expectancy for granted, and worry only about continuing a well-established trend.\n\nBeyond that, even though the thinkers we have examined did not anticipate some of the technological advances that today's transhumanists hang their hopes on\u2014none foresaw the rise of digital computing, for example\u2014some elements of the dehumanization they envisioned are already in place around us. Genetic engineering means that we would not necessarily require generations of careful breeding to create our Venusians. The \"conquest of space\" is in principle at least an established fact, and if the prospects for space colonies, planetary exploration, and interstellar travel still seem distant, that is less because of what we don't know or can't do than because of how we choose to arrange our funding priorities. Increasingly sophisticated and intimate man-machine interfaces are being developed; we are seeing impressive, if admittedly still early, advances in artificial ears, eyes, and limbs. We may not yet have the organ that Bernal imagined for detecting radio waves, but we do have t-shirts that can display the presence of wi-fi signals.\n\nIt is not just what we _do_ that links us with the authors we have looked at, but what we _expect_. We don't yet know there is alien life, let alone intelligence, but the idea is widely accepted by scientist and layman alike\u2014if not always for the same reasons. It is likewise a commonplace that we live in a world with an accelerating rate of change.\n\nWe might not yet normally place these ideas and achievements within a framework of efforts to overcome the merely human\u2014but they are there to be placed. The eclipse of man is underway. However amazing our present might look from the perspective of a not-so-distant past, there remain those who look down on the human because they can imagine something far better, whether it involves immortality or resurrection of the dead or brains transplanted into machines. Even if, as Bernal warns, we should also be wary of thinking that the future is going to work out just as we envision it today, it would certainly be the height of folly to assume whatever in these visions has not yet happened could never happen. Some (like Malthus) would have said that what we have today is impossible.\n\nThe main home for the hopes and fears that define the eclipse of man as we have examined it from the past may be transhumanism, but as we will see in the chapters that follow they are also at work elsewhere\u2014including in the scientific search for extraterrestrial intelligence, to which we turn next. The general public is fascinated by hostile alien invaders. The scientists who look for extraterrestrials are fascinated by contact with advanced, benevolent intelligence. Some transhumanists would be surprised if there are any aliens at all. All these prospects are working out the consequences of ideas about human-alien relations that we have seen in this chapter. The differences among them are not so great as they might first appear.\n_CHAPTER TWO_\n\n_Discovering Inhumanity_\n\n* * *\n\n_PROLOGUE: ONLY CONNECT_\n\nWHEN SHE accepted a postdoc position to be part of the team decoding the first message ever received from extraterrestrials, Camille never expected that the effort would occupy the better part of her career\u2014would really _be_ her career. That was actually the third surprise about the message. The first was that it came in on a tightly focused, extremely powerful beam of modulated UV light, when most of those engaged in the search for extraterrestrial intelligence (SETI) still worked on radio telescopes detecting microwave radiation. At first, the speculation was that whoever was sending the message must be quite technologically advanced to have lasers on a scale that humans were just beginning to think about. But the intensive study of the alien sun that followed showed it to be far more active than Earth's sun; the frequency and intensity of solar storms that Earth astronomers inferred from the data would have made radio communication on their planet so unreliable as to be nearly worthless. Unlike humans, they had probably started and stayed with light as they developed long-distance communication.\n\nThe second surprise was that the message was not very user-friendly. It was at least pretty clearly divided into \"words,\" and statistical analyses of their frequency looked a whole lot like what you got from similar analyses of human documents. But beyond that, it was not clear the aliens had considered the audience. Along with everybody else interested in SETI, Camille had given a good deal of thought to how she would design a message that started simple and moved on to more complex topics. At first, the assumption was that the easy stuff had been lost at the undetected beginning of the transmission. After nine months\u2014a dauntingly long letter!\u2014it was clear the message was repeating with no obvious primer at the start. That assumption had to be put aside.\n\nOf course, at the beginning _everybody_ was interested. The discovery galvanized and monopolized media attention at least as much as Sputnik and the moon landing. Like most others in her field, deep in her heart Camille had thought that \"first contact\" would be . . . well, like a revelation from on high. It would change everything. And the fuss at the beginning had done nothing to dissuade her. Interviews, op-eds, news analysis persisted for months. There were two \"instant books\" on the market within weeks; a couple of the senior people on Camille's team were still living down some of the things they were quoted as saying in those early days. As transmitted, the signal was invisible to the human eye, but it was tuned down, analogized and transformed, mixed and remixed by countless artists in visual and audio forms, bits and pieces of it showing up in popular music and on t-shirts. Camille's prior interest in SETI put her well ahead of the game; there was a huge \"catchup\" increase in interest in astronomy, optical engineering, linguistics, mathematics, and even astrobiology now that it was a real discipline. A predictable glut of Ph.D.'s in all these areas followed. Even the shifts in government funding for sciences couldn't produce ways to employ them all, although that's what funded Camille's early years. The space program was reinvigorated, two sports teams abandoned Native American names in favor of \"ALIENS.\" Once the message was complete, three \"unauthorized\" translations were out within months. The only one that didn't make it into print didn't have to, as the author \"proved\" on his website that the message was none other than the King James Bible.\n\nAt the time, Camille had been too busy and too much on the inside to appreciate fully how the message was like a great rock dropped into a small pond. Ripples spread widely, reflected back on each other, interfered and formed a complex pattern. But as time went by and the message remained enigmatic, the disturbance in society at large faded; life returned pretty much to normal. A rump group of enthusiasts stayed focused on translating the message, and some people spent what seemed like all their time trying to show the whole thing was a hoax (\"They say the message is transmitted on a light-beam, but YOU CAN'T SEE IT!\"), but the vast majority of the world's population went on exactly as before. That humans now knew there was intelligence \"out there\" became a historical fact among historical facts, part of the background against which the human drama continued to play pretty much as usual. Despite her own dedication to the project, Camille concluded that the discovery really was not, as some had claimed it would be, \"the most significant event in the modern history of mankind\" or still less \"likely . . . the most earthshaking event in human history\" or \"perhaps _the_ greatest discovery in scientific history.\" It certainly didn't \"change everything,\" or \"cause the most dramatic shift in the status of our human species that has ever occurred in history,\" which Camille came to count as her fourth surprise. It was not yet her last.\n\nAs the academic work of decoding went on and on and on, various schools of thought formed, competing journals were established on the basis of divergent assumptions, there were conferences you went to and those you didn't. Camille did her best to be a uniter (consistent with keeping favor with her funders) and thank goodness the factions never lost contact with each other entirely, so when her final breakthrough came, nobody ended up a dissident prisoner of his own previous assumptions.\n\nNo, the last surprise was not that everybody essentially agreed about what the message said, it was rather the message itself. It proved to be a comprehensive history of the aliens' world (the science parts proved key to the translation, of course). They wanted us to know where they had been, because they were concerned about where they were going. Their admitted flaws and imperfections were becoming increasingly dangerous, they thought, as their powers over their world (they didn't seem to have a concept of \"nature\") grew. Camille was the one who had first understood how the message began and ended, and it still chilled her when she looked back on it: \"Can you help us?\"\n\n* * *\n\nWE SAW in the previous chapter how human re-engineering is related to ideas about space exploration\u2014and has been at least since Winwood Reade's popular 1872 book\u2014and to speculation about alien life in space. The link continues to be significant today in the writings of prominent transhumanists, but with a new twist. Ray Kurzweil has concluded that \"it is likely (although not certain)\" that there are no alien civilizations. Nick Bostrom has written that \"in the search for extraterrestrial life, no news is good news. It promises a potentially great future for humanity.\"\n\nWhy are transhumanists invested in the idea that we are alone in the universe? Kurzweil and Bostrom each draw their conclusions following out a similar logic. Stars and galaxies are, in comparison with the time it took for life and civilization to evolve on Earth, very old. There would have been plenty of time before we arrived on the scene for an alien civilization to have come into being. So such an alien civilization could be thousands or tens of thousands or hundreds of thousands of years \"ahead\" of us in terms of its science and technology. Even with the great distances involved, it has been estimated that an alien civilization\u2014one more advanced than ours, but not unimaginably more advanced\u2014could colonize our galaxy in perhaps 60 to 300 million years. From a cosmic perspective, that is a relatively short time. If an alien civilization were to evolve in the way some transhumanists believe we will evolve, achieving great powers to manipulate matter and travel great distances, then surely it would have left its mark on the cosmos. Yet we see no evidence for such a thing. So, as the great physicist Enrico Fermi is said to have asked, where are they? Kurzweil and Bostrom plainly doubt the aliens are there to be found. To understand more fully transhumanist hopes and fears about alien civilizations, it is necessary to take a few steps back, and recall some of the earlier links between aliens and the eclipse of man.\n\nAs we saw in Chapter One, human space travel has been proposed as a way to solve the supposed Malthusian consequences of any Condorcet-like vision of material progress. If ever more people are going to be leading longer and wealthier lives, then they will require ever more of the finite resources upon which those lives depend. If we cannot do ever more with ever less, the argument goes, then either human civilization will come crashing down or the resource base will need to be expanded. Space travel, exploration, and settlement, however technically formidable, is conceptually a familiar solution\u2014especially for a civilization, like ours, that was profoundly shaped by its own history of exploration, colonization, and expanding frontiers. So space exploration can seem like a way to protect and extend humanity. However, the genuinely \"alien\" conditions that prevail in space and on other worlds put a premium on imagining intelligent beings better suited to these environments. Having learned from Darwin that evolutionary diversity is a product of changing environmental circumstances over time, we can readily imagine how evolution might be deliberately helped along to our own advantage. Think of Haldane's humans, bred to select for qualities conducive to survival on Venus, or Bernal's attempt to imagine mechanical beings built for hostile extraterrestrial conditions.\n\nSo some human beings _become_ aliens to explore and settle new worlds. Bernal acknowledges that this result may be problematic from the point of view of any who choose to remain merely human. Perhaps there is some further reflection of that problem in the complete equanimity with which Haldane's Venusians report the end of humanity on Earth\u2014the humanity that had created them and made Venus habitable for them.\n\nBut there is an additional problem. If we can expect to become alien and indifferent to _ourselves_ , what if the universe is _not_ waiting for us to enliven it? What if Flammarion is correct, and life establishes itself at the slightest opportunity? If there is alien life, then why should there not be alien intelligence? And if there is alien intelligence, why would it not eventually find itself facing the same limits and opportunities that, based on Malthusian assumptions, would drive us into space? If, as Haldane saw, _we_ could become alien invaders faced with an imperative to destroy or be destroyed, why shouldn't extraterrestrials behave in exactly the same way?\n\nOf course, based on complete ignorance, we can say anything we wish about alien motives and abilities, a freedom much employed by those who write both fictional and speculative non-fiction works on this topic. So it is not hard to find grounds for happier outcomes, starting from skepticism about the Malthusian dilemma itself. But for our present purposes, the significant point is this: concerns about hostile aliens do not have to arise from commonly identified factors like primitive xenophobia or Cold War paranoia. They do not have to depend on any quirk of human history or psychology. As we have seen, when we imagine dangerous aliens we are imagining beings that are acting no better and no worse than we would act if we fulfill the hopes articulated by today's transhumanists.\n\nStarting from this worldview, then, it is not immediately obvious that contact with aliens would be a good idea. Leave aside all prospects for tragic cultural misunderstandings; on the essential point we may understand each other only too well: they may not come in peace. Unless we are confident we would have the upper hand in the relationship, it might well be thought best to lay low, cosmically speaking.\n\n_MAKING FRIENDS WITH ALIENS_\n\nAfter World War II, advances in technology made it possible for the first time to think seriously about what it would mean to communicate across interstellar distances. Thanks to science fiction, the theme of hostile aliens was by then well established in the popular culture. So those who were advocating the scientific search for extraterrestrial intelligence (SETI) had a problem: why would we want to have contact with unfriendly aliens? The first efforts at SETI, led by the American astronomer Frank Drake in 1960, were just a matter of listening with a radio telescope for what was hoped might be the background chatter of alien intelligence\u2014the interstellar equivalent of tuning a radio to eavesdrop on conversations among truckers, or police, or ham-radio users. But those first efforts were rather quickly followed by deliberate attempts to send out contact signals of our own, over and above the radio and television broadcast signals that were already leaking into outer space.\n\nFortunately for the SETI pioneers, there was a readily available reason _not_ to worry about giving ourselves away. By the 1960s, the prospects for other intelligent life in our own solar system were looking bleak, and the distance to the nearest stars provided a comforting buffer. Messages traveling at the speed of light would take more than four years to reach even just the star nearest to our sun. Any back-and-forth communication given this limit would be difficult enough\u2014likely a project of generations, given that our part of the galaxy is not very densely populated by stars in comparison to some other parts. Visits in person, including marauding fleets of star cruisers, seemed, to say the least, highly implausible. So the scenario of aliens exploiting our world for resources to solve their Malthusian problems did not look plausible. We could reach out safely.\n\nThere was only one catch. Many of the supporters of SETI believed that any contact we would make would be with aliens more scientifically and technologically advanced than we. There is a simple logic to this familiar belief. Our ability to send and receive signals at interstellar distances is still new and remains quite limited. It would be impossible for us to detect signals from anybody much _less_ advanced than we ourselves, since even to detect and distinguish the kinds of signals that we could send out would be difficult or impossible for us. Furthermore, the existence of human intelligence at all on Earth, let alone human intelligence with the technology to begin to contemplate interstellar communication is, cosmically speaking, a very, very recent event indeed. Since on the cosmic clock there has been ample time for life and intelligence to have developed elsewhere long before humanity even began to emerge on the scene, at whatever point we might stand on some general scale of abilities and intelligence, any beings we might contact are likely to be well above it.\n\nBut this argument, made by sober Ph.D.'s who did everything they could to distinguish themselves from those who believed in \"flying saucers\" and (later) alien abductions, in fact raises troubling questions. How confident ought we to be that our understanding of nature and the technologically possible is sufficiently definitive as to preclude practical interstellar travel? Couldn't very advanced aliens know how to do things that would look impossible to us, just as some of the things we can do would seem impossible to primitive men? We are regularly told that the universe is a strange and surprising place\u2014might it be even stranger and more surprising than we now suppose? Even if decades have gone by without aliens appearing on our doorstep, perhaps we should not be confident about what tomorrow holds.\n\nSo it proves necessary to attack the very premise of hostile aliens, to turn them from cosmic pirates to cosmic philanthropists. And that effort requires following up on a different aspect of the eclipse of man. For along with this positive view of aliens comes a very dark picture of human beings. As foreshadowed by nearly all of the post-Condorcet thinkers we discussed in Chapter One, the more that alien beings look like our saviors, the less we look worth saving.\n\n_HUMAN INSIGNIFICANCE_\n\nThere are three aspects of the SETI effort to put human beings in their proper\u2014that is, insignificant\u2014place in the cosmic scheme of things. The first is the claim that mankind does not occupy a privileged place in the universe; the second suggests that we should expect aliens to be our moral superiors; the third attempts to show how aliens might help us to overcome our own tendencies toward self-destruction. These three lines of argument combine to suggest that, because anything we can do they can do better, we would be best off following any lead that aliens were willing to give.\n\nThe first claim, about the smallness of humanity in the face of the infinite possibilities of cosmic life, is sometimes called the \"assumption of mediocrity.\" Longtime SETI researcher Seth Shostak put it this way: \"There's nothing remarkable, nothing the least bit special, about our cosmic situation.\" Just as we know that there are lots of galaxies like ours and huge numbers of stars like ours, so we have every reason to believe that there are lots of planets like ours. (So far, planet-hunting efforts have revealed lots of planets, but few like ours.) And if there are lots of planets, there are going to be many opportunities for life to arise, as Flammarion argued. (This, too, is a matter about which we have no clear-cut evidence.) And given those huge numbers of stars and planets, then if there is lots of life there will have been ample opportunities for intelligence to arise in at least some cases (or maybe even all). In a universe teeming with life, we just aren't that special.\n\nThe assumption of mediocrity is sometimes also called the \"Copernican principle,\" a nod to the \"decentering\" of humanity that began with the discovery by Nicolaus Copernicus that the Earth is not at the center of the cosmos. Darwin too is sometimes said to have made a contribution in rejecting the necessity for any special creation to explain the existence of humanity. As Shostak tells the story, \"These events caused us to lose our central role in the physical and biological realms.\" If we were to discover evidence of alien intelligence, he believes, it would \"surely deliver a roundhouse punch to any remaining hubris, such as the belief that we are intellectually, culturally, or morally superior.\" *\n\nAs yet we have no evidence to suggest how common extraterrestrial life of any kind is, let alone intelligent life. For the time being, then, as Carl Sagan, the great popularizer of astronomy and SETI, acknowledged, \"the application of this method [i.e., the assumption of mediocrity] to areas where we have little knowledge is essentially an act of faith.\" And it could remain a matter of faith indefinitely. Even if we were to imagine some arbitrarily large number of technologically sophisticated alien civilizations\u2014and given the hundred-plus billion stars in our galaxy and the hundred-plus billion galaxies in the universe, there could be a staggeringly high number of intelligent alien civilizations\u2014they might still be spread so thinly that, short of miraculous-seeming technology, contact among them would be unlikely. That would be cosmic quarantine on the grandest scale.\n\nStill, should we ever make contact with aliens, vindication of the assumption of mediocrity will supposedly teach human beings important moral lessons, as Shostak's use of the word \"hubris\" suggests. For one thing, the knowledge that we are not alone would teach us something very important about our common humanity. \"Just learning of the existence of other civilizations in space,\" Frank Drake wrote, \"could catapult nations into a new unity of purpose. Indeed, the search activity itself reminds us that the differences among nations are as nothing compared with the differences among worlds.\" Sagan apparently shared this belief: \"There will be a deparochialization of the way we view the cosmos and ourselves. There will be a new perspective on the differences we perceive among ourselves once we grasp the enormous differences that will exist between us and beings elsewhere\u2014beings with whom we have nonetheless a serious commonality of intellectual interest.\" These are very edifying lessons, to be sure, even if one can conceive of a \"human family\" (an idea whose roots are to be found in the Bible and ancient Greek thought) without SETI. It is quite easy for what is presented as a future prospect to elide into a cosmopolitan norm that the authors almost certainly believe should be guiding our behavior even before any discovery of aliens.\n\nHowever, encouraging a sense of brotherhood only among human beings does not have to be the end of the story. In exchange for the loss of our cosmic specialness \"we will gain something perhaps more desirable,\" according to physicist Gerald Feinberg and biochemist Robert Shapiro. \"Earthlife will be a part of the much more encompassing phenomenon of Universelife. Just as people gradually extend their vision from self-preoccupation as children to identification with a wider community as adults, so in the coming centuries the human species may prefer to be a local representative of Universelife rather than exist almost alone in a vast and sterile void. If so, this shift in attitude may prove to be the most important outcome of a successful search for extraterrestrial life.\"\n\nVoltaire famously asserted that nothing human was alien to him; Feinberg and Shapiro seem to be suggesting that the yet more mature view would be that nothing alien is alien to us. They hope we may one day be able to aspire to an identity beyond the human in a universe enlivened by many kinds of beings other than ourselves. Just as Flammarion's picture of the wonder and beauty of alien life on alien worlds highlights all that we might find problematic about being human, if we have sufficient faith in the assumption of mediocrity, we could adopt a point of view that weakens our attachment to the merely human, and in this way further the eclipse of man.\n\n_HUMAN FOLLY AND ALIEN ASSISTANCE_\n\nFor many SETI authors, that detachment from the human can hardly come too soon. Along with many others in the second half of the twentieth century, they looked at the world and saw it on the brink of destruction. In SETI circles the discussion has usually been couched in terms of the \"L factor,\" meaning the lifetime of technological civilizations capable of sending and receiving an interstellar signal. The shorter L is, for us or for any aliens, the less likely it is that any two such civilizations will be sufficiently close in time and space to allow communication to take place. Over the years various perils have been used to suggest why we should not be confident that human beings, or at least our present level of technological development, will be around much longer. If there is nothing special about humanity, and if humanity seems bent on self-destruction, then perhaps it is a general rule that technological civilizations are short lived, as the following characteristically apocalyptic quotes from various sources suggest:\n\nThere is a sober possibility that _L_ for Earth will be measured in decades. On the other hand, it is possible that international political differences will be permanently settled, and that _L_ may be measured in geological time. It is conceivable that on other worlds, the resolution of national conflicts and the establishment of planetary governments are accomplished before weapons of mass destruction become available.\n\nYou might also say that the present very modest trends to reverse the nuclear arms race [in the late 1980s] are an indication that the \"L\" variable . . . might be larger. On the other hand, the other catastrophes that have come to the fore since then\u2014global warming, ozone depletion, nuclear winter\u2014work in the other direction.\n\nThe mushroom cloud rising above Hiroshima and Nagasaki clearly warned humanity of our potential extinction twelve years before Sputnik soared into space and pointed to our next step in evolution. The planet has managed to avoid nuclear destruction over the past fifty years, but ecological hazards remain a clear threat. Some scientists studying the rapid extinction of terrestrial species today compare it with the time of the great dinosaurs.\n\nThe intelligence and dexterity required to build radios are useful for other purposes that have been the hallmark of our species for much longer than have radios, such as devices for mass killing and means of environmental destruction. We are now so potent at doing both that we are gradually stewing in our civilization's juices. We may not enjoy the luxury of an end by slow stewing. Half a dozen countries have the means to bring us to a quick end, and other countries are eagerly seeking to acquire those means. The wisdom of some past leaders of bomb-possessing nations, or of some present leaders of bomb-seeking nations, does not encourage me to believe that the Earth will have humans and their radios much longer.\n\nUnfortunately, because the human race is squandering its natural capital at an unsustainable rate, it seems at least as likely that terrestrial collapse rather than extraterrestrial expansion awaits us in the next two centuries.\n\nAll these challenges mean that we live at an unusual moment, a time of testing, or a \"great filter,\" as one author has called it. If we pass through successfully, there is hope: we come out on the other side in a brave new world. And that is just where contact with aliens might help us. Any contact we make almost has to be with a longer-L civilization than our own, which in turn means that it is likely to have surmounted just the existential problems that threaten to keep our \"L\" short. So might they not be able to give us a leg up?\n\nSagan's writings sometimes show a nuanced understanding of this possibility. He recognizes that there is a problem in expecting to hear from a civilization advanced \"vastly\" ahead of us; we may no more be able to appreciate their existence than \"an ant performing his anty labors by the side of a suburban swimming pool has a profound sense of the presence of a superior technical civilization all around him.\" (And, we might wonder, would that superior civilization be any more interested in us, or well intentioned toward us, than the suburbanite would be toward the ant?) Hence, most likely our first communication will be with \"civilizations only somewhat in our future.\" Sagan notes that \"it has been suggested that the contents of the initial message received will contain instructions for avoiding our own self-destruction, a possibly common fate of societies shortly after they reach the technical phase.\" Given the likely differences between humanity and the aliens we might meet, he concedes that they might not be able to send us information about \"stabilizing societies,\" but \"it is a possibility not worth ignoring.\"\n\nWhy isn't Sagan certain we could be helped? Here another important SETI principle must be mentioned: anti-anthropocentrism. The assumption of mediocrity\u2014the idea that there is nothing special about life or even intelligent life\u2014is not taken to mean that the particular form we take, and the particular way our intelligence builds a human civilization, is not special. Indeed, it would be a great surprise to most SETI scientists if we _were_ mediocre in the sense that the alien life we found were somehow to resemble us physically. Notwithstanding popular ideas about \"little green men\" and other humanoid aliens, most scientist-speculators have long envisioned a wild diversity of alien life. On the other hand, the possibility that they could be, like us, carbon-based life forms, is often taken seriously. We would hardly expect them to communicate with familiar sounds or images, but we do hope they have technologies like ours and share sufficiently common scientific or mathematical knowledge to send an intelligible signal on that basis\u2014although even this possibility has been disputed. In any case, were we to be able to decode a signal, that would at least suggest some commonality of thought processes\u2014but then we might never be sure that we were not missing some (to them) crucial nuance. It might make a difference to the way we understand a message if we knew it was the alien equivalent of a sixth-grade science project, the product of a marketing firm looking for customers, or of a government agency tasked to identify and mitigate potential security threats\u2014none of which possibilities might be immediately obvious to us from the contents of the message itself.\n\nSuch potential roadblocks are why Philip Morrison, a physicist who coauthored the first scientific paper explaining how one might do SETI using radio telescopes, had a more sober assessment than Sagan of what might be learned from aliens. Morrison, participating in a U.S.-Soviet conference in 1971, suggested that the impact of an alien message would come less from how it instructs us about questions we are already asking than from the discipline that will arise from the careful deciphering and study of the work of a radically different civilization. He argued\u2014in the process creating some controversy at the conference\u2014that we ought not to expect \"some simple telegraphic message like a newspaper.\" Hence, a signal will have \"great impact\u2014but slowly and soberly mediated, transmitted through all those filter devices of scholars who have to interpret and publish a book, and so forth.\" The \"interpretation of the signal will be a social task comparable to that of a very large discipline, or branch of learning.\" What might we learn from this enterprise? As Morrison put it at a 1972 conference:\n\nI think the most important thing the message will bring us, if we can finally understand it, will be a description, if one exists at all, of how these beings were able to fashion a world in which they could live, persevere, and maintain something of worth and beauty for a long period of time. Again, we will not be able to translate it directly and make our institutions like theirs; the circumstance will be too different. But something of it will come through in this way. I think, therefore, that this will be the most important message we could receive. But it will be more of a subtle, long-lasting, complex, debatable effect than a sudden revelation of truth, like letters written in fire in the sky.\n\n_IMMORTAL LONGINGS_\n\nBut among the early SETI advocates Morrison was unusually cautious. There is a more widespread tendency to be optimistic about the ability of aliens to solve at least some of our problems for us. Even Sagan said, \"imagine if one day the contents of 100,000 books of a Type II civilization [one that has completely mastered its own solar system] suddenly fluttered through the receivers of our radio telescopes, a kind of Encyclopedia Galactica for children! The rewards of success are inestimable.\" Drake has been yet more outspoken on this point: \"There is probably no quicker route to wisdom than to be the student of more-advanced civilizations.\" What does he have in mind? \"A simple yes or no answer to the question of whether fusion-energy research should be pursued would be worth tens of billions of dollars to the governments of Earth.\" Or again: \"I can only guess what a civilization far more advanced than our own might teach us. . . . But let me share one of my favorite 'what ifs' with you: What if _they_ are immortal?\"\n\nDrake is suggesting two kinds of gain by contact with intelligent alien life. The first is technical know-how: how to achieve fusion power or immortality. This enthusiastic belief that there might be advanced technological goodies to be got plainly stems from the assumption of mediocrity as applied to technological development. Speaking of aliens as \"more advanced\" suggests that the timeline along which our knowledge and abilities have developed will be more or less the same elsewhere\u2014that our future will resemble their past. But since our developmental path includes the possibility of being destroyed by our technology, it is not enough to hope that we will get yet more dangerous toys; we will also find the \"route to wisdom\" that they must have if they were able to surmount the dangers we face. (Of course, this argument overlooks the possibility, highlighted in the prologue to this chapter, that _they_ are hoping for assistance in getting beyond difficulties that we cannot even begin to foresee.)\n\nIf we look again at the litany of dangers we pose to ourselves, we might conclude that they all point toward some combination of Malthusian scarcity and Darwinian competition. It would only be consistent with the premises of SETI\u2014and not implausible in its own right\u2014to think that these are constraints that any alien life would have to deal with. Here, then, the assumption of mediocrity would properly seem to triumph. But wouldn't anti-anthropocentrism be a safer bet with respect to the _particular_ timeline on which such dangers would be faced? When it comes to discoveries and inventions, there are surely some material and intellectual preconditions that suggest the sequence in which things might be discovered or invented; for example, we couldn't have invented rockets without prior experience in combustion. But while such conditions might be necessary, they are not sufficient. It also seems true that there is a high level of contingency with respect to the motives that turn possibilities for discovery and invention into realities. Would we have had atomic weapons in the mid-twentieth century, for example, without the Treaty of Versailles?\n\nEven leaving aside the contingency that seems to characterize many of our scientific and technological achievements, we might notice that our science and technology have developed along the lines that they have because we have the specific human abilities and needs that we do, based, for example, on the shapes of our bodies, the capacities of our senses, the needs that must be satisfied to keep us alive. Add to that the specific raw materials that living on Earth provides to us. All these things shape our particular constellations of abilities, motives, and social structures, which in turn influence scientific and technological developments, and the manner in which they will be useful or dangerous. It is very hard to imagine what it would mean to be made wise by aliens, whose lives, anti-anthropocentrism requires us to believe, may hardly even begin to raise issues that for us require wisdom. To assume that intelligent civilizations all go through similar developmental stages is to adopt an extremely anthropocentric mediocrity, hubristically turning human experience into a cosmic norm.\n\nBut Drake himself hints at a kind of solution to this problem when he speaks of becoming immortal. Such a change, if revealed to us, would fundamentally alter what it means to be human. Drake is not being anthropocentric if from the start he is really assuming that \"wisdom\" means precisely the abandonment of our humanity, and the adoption of the ways of the \"advanced\" aliens. The point is made with startling directness by physicist-philosopher Paul Davies: \"If they practiced anything remotely like a religion, we should surely soon wish to abandon our own and be converted to theirs.\"\n\nWhat is perhaps most striking about this assertion is that Davies surely knows that mainstream Christian belief has no problem imagining alien intelligence that would even be subject to salvation. In other words, the reason that we would want to adopt an alien religion is not because all human religions are incapable of dealing with the reality of alien life. Rather, the sole justification would seem to be their presumed civilizational longevity and technological and scientific superiority. In other words, if they are so much more advanced than we are and still have religion, there must be something to it.\n\nThe abandonment of central aspects of human civilization is really only to be expected if we are also to abolish hitherto central aspects of the human condition. That may not be such a big deal if we are as spiritually primitive as SETI popularizers seem to imagine, if as Haldane suggested we are an infection hardly worth sterilizing, or if we are, as anthropologist Ashley Montagu once suggested in some remarks about SETI, a cosmic version of \"rabies or cancer or cholera.\" It is true that humans have long abased themselves in the face of Divinity, but Shostak's concern about hubris seems almost quaint in this race to use the authority of science to diminish our sense of what it means to be human.\n\n_WHAT'S KEEPING THEM SO LONG?_\n\nLet us now return to Fermi's question: where are they? As we have seen, SETI scientists argue that life could have sprung up and developed elsewhere far earlier than it did on Earth, and that other planets might thus have civilizations that were already flourishing and growing in technological capacity over all those eons during which human beings did not even exist. They would have been accumulating knowledge and capacities over what for us is evolutionary time. If they are doing some of the things that the transhumanists predict we will someday be doing\u2014like \"reorganizing matter and energy\" so as to start turning the cosmos into one big computer\u2014shouldn't we see signs of it underway already? Why haven't we heard from them yet, or detected their work?\n\nMany explanations have been provided to explain the silence of these ancient alien civilizations, but all of them have problems. Perhaps ancient alien civilizations may place young civilizations like our own in a cosmic quarantine for study or other purposes, maybe even leaving behind tools for covert observation. Or perhaps we may just be too primitive for them to bother with. But in a big universe, is there not even one deviant culture that is sufficiently committed and resourceful to want to interact with primitives? Perhaps, instead, at a certain point in their development, technological cultures turn inward, abandoning concern with space. But here again, the suggestion seems to be that there is not even one civilization so taken by spacefaring that it departs from the norm. So perhaps we just cannot hear the aliens because they long ago stopped broadcasting over radio waves\u2014just as we ourselves have in recent decades replaced much of our over-the-air broadcasting with other forms of transmission that do not leak into space. Or perhaps, as some transhumanists expect for our own future, the aliens have given up on physical bodies, \"retreating into cyberspace.\" Yet even then, might they not leave indirect evidence of their existence, such as signs of intensive use of matter and energy to satisfy their need for resources?\n\nPerhaps interstellar travel is far more difficult or costly than we think. But that argument runs up against the suggestion that the development of civilization over evolutionary time scales may produce beings that are simply incomprehensible to us, while we are of as little interest to them as Sagan's ants by the swimming pool. Of course, that in turn means that aliens might have been here all along, and we have been looking for the wrong sorts of evidence. Perhaps instead of searching for big spaceships we should be on the lookout for relatively small probes, or even probes tinier than living cells, or information encoded in the very physics and math of our universe itself, as Sagan imagined in his novel _Contact_. But in a galaxy (or universe?) filled with such life, why would there not be some who would enjoy the challenge of communicating with lower beings just as we imagine how it would be interesting to communicate with animals? There might be, along lines that Sagan has suggested, \"a few specialists\" among the aliens who study \"primitive planetary societies\" like our own. They might even leave a message on the Internet. Or perhaps they have been trying to communicate with us all along and we just have not taken the possibility seriously\u2014and we should start rethinking UFOs.\n\nThere are two kinds of conclusions that can be drawn from the failure of SETI to find evidence of aliens thus far, although it is very important to be clear from the start that this failure emphatically does not mean there is nothing to be found. However, we can note on the one hand that because we know literally nothing about alien intelligence, the only restriction on our reasoning about it is that our conclusions be logically consistent with our premises\u2014premises that, like the assumption of mediocrity, are themselves as yet more or less without empirical foundations. As a result we have great freedom for imagination, ingenuity, and speculation; the price is that any plausible conclusion can be plausibly countered until there actually is an empirically based science of astrobiology and alien intelligence. On the other hand, it is certainly consistent with a variety of premises to suggest that silence means that advanced alien civilizations who we were counting on to help us may not be as interested in providing that help as we might have hoped. To put it another way, if they want to help, they will do so based on their judgment of the appropriate moment for intervention, which may (given the intelligence gap between us) have little or nothing in common with our hopes.\n\nFrank Drake has come up with a way to accommodate this reluctance and still maintain the possibility of our receiving useful information from alien races. If they achieve something like immortality, he posits, they may become extremely risk-averse. That would explain why they are not interested in exploration, colonization, or contact generally, all of which are risky enterprises. But their \"fanatic obsession with safety,\" Drake theorizes, might lead them to broadcast the secret of immortality. By seducing other species with the very immortality that today's transhumanists so fondly desire, the aliens could minimize any threat from afar.\n\nDoes Drake realize how chilling his picture of \"the immortals\" is? If these beings were immortal, then they could travel to the stars; the times involved would mean nothing to them. But instead of extending and expanding the realm of their experience, their mastery of death leaves them mastered by their fear of death. This is the gift that Drake imagines they would wish to pass on to us. If that counts as \"benevolence,\" it is only by a standard very different from any that poor mortals \u2014 who aspire to go _per ardua ad astra_ (through advertsity to the stars) \u2014 would recognize.\n\nBut that is really the biggest question posed by SETI's effort to construct benevolent aliens in the face of the Malthusian-Darwinian argument that they might be our cosmic competition. For is it anything other than gross anthropocentrism to believe that our ideas of benevolence will accord with those of an advanced alien civilization? What makes us so confident that the assumption of mediocrity can be applied to the ethics of alien species? There is an uncertainty at the core of the very notion of alien benevolence. It was explored with brilliance in Arthur C. Clarke's 1953 novel _Childhood's End_. We now turn to that novel to explore how the progress of inhumanity, coming as a gift from aliens, might well be a Trojan horse.\n\n_THEY'RE HERE_\n\n_Childhood's End_ begins in 1975 near the imagined climax of a space race between the Americans and Soviets; both are out to launch nuclear-powered rockets to the moon. But instead, huge alien ships appear over the major cities of the world. After letting tension build for six days, on the seventh humanity hears from someone calling himself Karellen, Supervisor for Earth. Showing \"complete and absolute mastery of human affairs,\" Karellen announces that all international relations are now under control of his people, who come to be called the Overlords. An unnamed nation tests their control, attacking an Overlord ship with nuclear weapons. Nothing happens; the bomb does not go off and no effort is made to punish the offending nation. \"It was a more effective, and more demoralizing, treatment than any punitive expedition could have been.\"\n\nAt the beginning, all contact between the Overlords and earthlings takes place through the secretary-general of the United Nations, Rikki Stormgren, who (not surprisingly) quickly comes to trust Karellen. The Overlords leave many human things alone; they don't care about forms of government or economy, but they are not indifferent to oppression or corruption, even of animals. While it remains okay to kill for food or self-defense, sport killing is out, as the spectators of a bullfight find when they are made to feel the pain of the bull. Or again, the Overlords deprive South Africa of sunlight when the country, which by this point has replaced apartheid with a new policy that oppresses its white minority, is too slow to grant all its citizens equality. In any case, the international system of sovereign nation-states was already dying before the Overlords arrived, and within five years of their arrival a World Federation is in the works.\n\nThe Overlords quickly bring unprecedented \"security, peace and prosperity\" to Earth. Only a few malcontents have any doubts about their essential benevolence or attempt to cling uselessly to the past. But one issue chafes even at those who are happy with the new world: the Overlords will not show themselves to humanity. Karellen promises they will do so in fifty years, after which \"we can begin our real work.\"\n\nFifty years on, the world has been changed \"almost beyond recognition\" by the \"social engineering\" of the Overlords, based on powers of great scope and subtlety. Earth has become a progressive, secular-humanist utopia, truly \"One World.\" Everyone is wealthy, healthy, beautiful, and employed in interesting work that yet allows plenty of leisure time. There is almost no crime. Cities have been rebuilt, and the weather is under control. Continuing education is part of everyone's life. The Overlords have provided aircraft that allow people to globetrot simply to attend a party, and the result is an end of all racial and ethnic divisions. Oral contraception and reliable paternity testing have \"swept away the last remnants of Puritan aberration\" just as religion itself \"vanished like the morning dew\" when the Overlords provided access to a time machine that could (selectively) display the past, proving that the world's faiths might have had noble, but certainly not divine, origins. So when the Overlords do reveal themselves, at twice human size and with wings, horns, and barbed tails, humanity quickly gets over the shock that they look like devils.\n\nBy the middle of the twenty-first century, the human race achieves \"as much happiness as any race can ever know\"\u2014a golden age. Only a few ask, \"Where do we go from here?\" One such is young Jan Rodricks, whose dreams of exploring space are thwarted by the Overlords. They have effectively banned any independent human space travel, and refused to share much of their own obviously greater knowledge and capacities in this area. Through a series of what turn out to be not quite coincidences, he comes to be the only human who knows the location of the Overlord home world. He resolves to visit it. Aware that the Overlords travel at nearly the speed of light, he understands that during a round trip he will subjectively experience as four months long, eighty years will have passed on Earth. He nevertheless stows away on an Overlord starship. As usual, when they learn of Rodricks's disobedience, the Overlords punish no one, but the incident prompts a stern press conference by Karellen announcing formally that the humans will never be allowed access to interstellar space. Humans could never deal with the vastness of the galaxy, and the \"forces and powers that lie among the stars\u2014forces beyond anything that you can ever imagine.\"\n\nSoon enough, however, the human race gets a sense of those forces and powers. All children under ten start to act in strange ways, exhibiting paranormal powers. When this breakthrough occurs, the Overlords must at last announce the truth about their mission. They came not to prevent man from destroying himself with nuclear weapons or other modern technology, but to forestall developing investigations into parapsychology that, had they been followed up, might have loosed a kind of mental cancer into the universe. For humankind turns out to be a race gifted with psychic potentials that even the Overlords do not possess, potentials that make it possible for humans to merge with the Overmind, whose servants the Overlords have been all along. Their job has been to prepare humanity for this merging, to cultivate the proper conditions for it. The children are developing a group mind that will eventually be taken up into the Overmind. All of them are evacuated by the Overlords to Australia, where they become increasingly strange as their minds merge, their psychic powers expand, and they prepare under the tutelage of the Overmind to leave their bodies behind.\n\nIn the face of the shocking news that humanity as such has no future, people react variously. Many commit suicide, some live on in decline and desperation. Jan Rodricks returns to Earth from the Overlord home world to find he is literally the last human being left. When the Overlords see that the dangerous moment has arrived for the apotheosis of the children they prepare to depart. Jan offers to stay behind and report to them what happens; as a human he can see things the Overlords are not able to observe. Thus, Jan is witness to the complete destruction of Earth, which ensues as the once-human group mind tests its powers and meets its destiny in the Overmind. Karellen departs the solar system with regret not so much for Earth and humanity as for his own people, who for all their knowledge, brilliance, and power can never achieve this union with their masters, the Overmind.\n\n_RETHINKING ALIEN BENEVOLENCE_\n\nThe issue of the benevolence of alien assistance is a live one throughout _Childhood's End_ , far more so than this bare summary might indicate. Certainly the outcome of the story puts this question in high relief. How are we to understand a benevolence that results in the destruction of mankind and the Earth itself? What are we to make of an end so traumatic that (we are told) it echoes _backwards_ in time (the Overlords insist that time is more complex than humans understand) such that the physical figure of the Overlords becomes the Devil, the Tempter who just like the Overlords provides all apparently good things as he leads on to destruction? There are times when it is clear that Clarke himself understands some of the problems of Overlord benevolence, and others when he provides hints of deeper issues that he may or may not have been willing to confront himself. The matter of alien benevolence can be discussed in three parts that correspond to the three divisions of the novel: first, the initial subjugation of humanity; second, the golden age; and third, the end of humanity.\n\nFrom early on there are humans who distrust the motives of the Overlords. That mistrust is given expression by the Freedom League. U.N. Secretary-General Stormgren judges the Freedom League leader, Alexander Wainwright, to be \"completely honest, and therefore doubly dangerous.\" Apparently unknown to Wainwright, his organization contains a secret radical faction. They kidnap Stormgren in order to get his cooperation in finding out more about the Overlords. The Overlords rescue him before he is forced to decide whether to cooperate, but the experience prompts him to start his own clandestine effort to get a look at them. His plan has modest success; Stormgren, a modern Moses, is allowed to catch a glimpse of Karellen's huge backside, presumably tail and all. Understanding now why these devil-like beings would want to conceal their appearance, but confident they will make everything right eventually, he takes his limited knowledge to his grave.\n\nWhen Wainwright first meets with Stormgren to lodge a formal protest about the plans for the World Federation, he asserts among other things that his group objects to the Overlords because they have deprived humanity of its ability \"to control our own lives, under God's guidance.\" In case we miss the irony here, Stormgren himself points out that many religious leaders support the Overlords' policies, but he concludes privately that \"the conflict is a religious one, however it may be disguised.\" Interestingly, on the same page Clarke has Stormgren note that the \"fundamental difference\" between him and the Freedom League is that he \"had faith in Karellen, and they had not.\" Clarke seems willing to consider the possibility that the conflict stems from something like religious belief on _both_ sides.\n\nKarellen, on the other hand, presents the conflict with the Freedom League more conventionally as religion versus reason and science. People like Wainwright \"fear we will overthrow their gods\"; science can prove religion wrong, or it can destroy religion by ignoring it. _\"All_ the world's religions cannot be right, and they know it.\" Given the way things play out, it looks like Karellen is right on this point; Overlord science does eventually destroy human religion (or nearly so, as we will see) both by active measures (the history viewer) and by the material utopia they provide. And yet Clarke describes Karellen's speech to Stormgren as follows: \"His voice was somber now, like a great organ rolling its notes from a high cathedral nave.\" It would be perfectly consistent with the facts of the story thus far if Clarke were highlighting again that the benevolence of the Overlords is at this stage only a matter of faith. Furthermore, there is the troubling fact that the Overlords are also attempting to conceal the kernel of truth in mankind's _mystical_ beliefs, which often find their home in the religions the Overlords have carefully discredited. Given their true purpose in coming to Earth, why should we be confident that the already selective history viewer they loan to mankind is a reliable record of the origins of religion? By standing up for science as he does, Karellen is describing the conflict in a way useful to the deception in which he is engaged.\n\n_WHAT DO HUMANS WANT?_\n\nThe golden age that the Overlords usher in would seem to vindicate their good intentions. The \"average man\" was grateful to them for being freed from want and war; \"mankind had grown to trust them, and to accept without question their superhuman altruism.\" Few people doubt if it really was altruism. Yet we readers are given ample reason to question the golden age. Recall the carefully formulated way it is described: \"mankind had achieved as much happiness as any race can ever know.\" That does not mean that everyone is happy, that the race as a whole is simply happy, or that happiness is the only or best way to measure the value of life.\n\nFor example, there are hints that in various ways this utopia does not quite even live up to its own billing. Religion may have withered, but there is one character who is unwilling to participate in a s\u00e9ance out of what her husband sarcastically labels Talmudic objections. The fact that \"Puritan aberrations\" have been swept away does not mean that men and women do not still conceal their affairs from each other. The fact that couples marry for defined periods does not eliminate the possibility that a man can become indifferent about his wife, and later regret that indifference. Female beauty is common, but that does not stop jealous impulses when one presumably beautiful woman encounters another yet more beautiful. There are still love affairs, and their failure still creates unhappiness. The security and material well-being provided by the Overlords apparently cannot solve all human problems.\n\nLife under the Overlords is quite clearly not for everybody. We have already seen Jan Rodricks's restless dissatisfaction with the way the Overlords have restricted his life possibilities. More generally, we learn that utopia is dull and that it stifles adventure and creativity in the arts and sciences. (This recalls Bernal's quasi-Freudian views of the pitfalls of a \"Melanesian\" existence, described in Chapter One.) Inner and outer conflict spur creativity, peaceful satisfaction of desires dulls it. Seeking to counter this problem, some creative types set up a socially engineered island society for artists called \"New Athens.\" But it is hard to believe that Clarke is not gently parodying their efforts. The kind of \"conflict\" that spurs them includes things like riding bicycles up hills, cooking one's own dinner (in perfectly modern kitchens), or attending endless committee meetings. When an Overlord comes to visit the island, he praises some of its composers for the \"great ingenuity\" of their work, leaving them \"to retire with pleased but vaguely baffled expressions.\"\n\nIf socially engineered creativity is somewhat oxymoronic, that tension at least does not apply to the pacifying social engineering of the Overlords themselves. While they are unabashedly colonial administrators, they highlight two differences between their efforts and analogous human efforts. The Overlord visiting New Athens notes that unlike the British in India, the Overlords have \"real motives\" and \"conscious objectives\" for what they are doing. Of course, only after the fact are these motives and objectives made clear to the administered. But humanity could have done nothing to stop them in any case, and that is the second salient difference: the massive disparity in knowledge and power between human and Overlord. When Stormgren accuses Karellen of thinking that might makes right, Karellen does not disagree; he rather points out that with sufficient power of the right kind \"might\" can be exercised in very subtle, indirect and efficient ways. Might does not have to mean brutality. The Overlords say of their own relationship with the Overmind that \"no one of intelligence resents the inevitable\"; that is really the bottom-line justification for such benevolence as they have shown by creating a utopia for mankind, which in the end they admit was in part a distraction: \"What we did to improve your planet, to raise your standards of living, to bring justice and peace\u2014those things we should have done in any event, once we were forced to intervene in your affairs. But all that vast transformation diverted you from the truth, and therefore helped to serve our purpose.\"\n\n_PERPETUAL PEACE_\n\nOnce the truth about their mission has come out and the children are about to be evacuated, Karellen announces that \"all the hopes and dreams of your race are ended now.\" He wonders whether it would not be \"most merciful, to destroy you\u2014as you yourselves would destroy a mortally wounded pet you loved. But this I cannot do. Your future will be your own to choose in the years that are left to you. It is my hope that humanity will go to its rest in peace, knowing that it has not lived in vain.\"\n\nWhy exactly does the human race have no future? Only young children, under the age of ten, will unite with the Overmind; why can't those who remain continue to reproduce? We are told that many committed suicide in the aftermath of the evacuation of the children; New Athens blows itself up. Others live on with reckless abandon and violence as the global social order quickly deteriorates despite the Overlord power to keep order. But now, their mission nearly complete, the Overlords no longer care. They claim they cannot explain to Jan Rodricks why those who remained did not have more children. He suspects the reasons were \"psychological,\" which is no explanation at all.\n\nIn fact, however, their failure to answer this question\u2014are human beings unable or unwilling to procreate?\u2014draws us back starkly to the nature of the Overlords' mission. Their job was to stop humanity from the independent development of paranormal powers and prepare the way for those powers to be used to unite with the Overmind. Independent development of psychic abilities would have made humanity like \"a telepathic cancer\" in the universe\u2014that is, a threat to the Overmind, even if also to itself. You don't cure a cancer by excising only part of it. A human race that continues to exist _and_ knows the truth about its potential would be even more of a competitive danger to the Overmind than a human race that had only halting knowledge of what it could do. And, at any rate, the Overlords seem to have good reason to believe that the children will destroy the world when their rapture comes.\n\nIf as we have seen the benevolence of aliens in creating \"One World\" and a golden age is already problematic, how much more so is it when it eventuates in the end of humanity? Even if, hopeless as it would have been, humanity had attempted to resist either the Overlords or the Overmind, it is difficult to imagine a worse outcome than the one Clarke presents. However, that does not seem to be his view of events. Perhaps he begins from assumptions like Flammarion's; it is in the nature of things that species evolve or face extinction, that worlds are born and die. No matter what else happens, _someday_ there will be an end to humanity and the Earth. From this point of view we can understand Clarke's attempt to have us believe that the tragedy of his story is not the fate of the Earth and humanity but the situation of the Overlords, caught in their \"evolutionary cul-de-sac,\" servants of a master they cannot really understand, destined always to be bridesmaids and never brides. The offspring of humanity, on the other hand, are moving on to great and glorious things.\n\nJan comforts himself by thinking that the Overmind draws \"into its being all that the human race had ever achieved.\" However, it is not even remotely clear that his view is justified. Jan's view implies that the children sequestered in the Australian outback have mental powers of assimilating human history and culture\u2014powers that are never presented explicitly. The Overmind, by taking only children under ten, takes those with the _least_ knowledge or experience of what it means to be human and indeed there is some stress put on the fact that the children are becoming something not human at all even before they are assimilated. It is the Overlords who seem intent on preserving remnants of Earth's life and human cultures; they go into a great museum that catalogues the evidence of the worlds they have helped to destroy. Nor would Jan's experience being debriefed by the Overlords on their home world give much comfort about the persistence of human things even if they _did_ take older minds. However well educated he is, Jan finds himself unable to answer many of the questions put to him about his own race. These men who have been gifted a golden age are not the deepest reservoirs of human hopes, experiences, dreams, and knowledge.\n\nKarellen's assessment of the situation offers cold comfort: \"You have given birth to your successors, and it is your tragedy that you will never understand them. . . . For what you have brought into the world may be utterly alien, it may share none of your desires and hopes, it may look upon your greatest achievements as childish toys\u2014yet it is something wonderful, and you will have created it.\" But of course humans only \"created\" what is to come in the bare sense of procreating the necessary conduits for the Overmind.\n\nPerhaps a better argument that the Overlords act benevolently can be found in their justification for barring humanity from the stars. Space is far more immense than human beings imagine. \"In challenging it,\" Karellen says, \"you would be like ants attempting to label and classify all the grains of sand in all the deserts of the world. Your race, in its present stage of evolution, cannot face that stupendous challenge. One of my duties has been to protect you from the powers and forces that lie among the stars\u2014forces beyond anything you can ever imagine.\" Within this universe there is a hierarchy; \"as we are above you, so there is something above us, using us for its own purposes. We have never discovered what it is, though we have been its tool for ages and dare not disobey it.\"\n\nA reminder of the sheer scale of the universe is indeed a useful corrective for human hubris\u2014although surely one could still be sympathetic with any industrious ants who against all odds attempt such a great study of sand. Yet what are we to make of the hierarchy Karellen is suggesting? It sounds almost as if the Biblically inspired vision of God, angels (fallen or otherwise), and man is not so far off the mark after all. Merely from the fact that the Overmind operates through Overlords, we cannot conclude that it is not omnipotent. And from the fact that it is \"conscious of intelligence, everywhere,\" we might suspect it is omniscient and in some sense omnipresent. Karellen only said that _\"all_ the world's religions cannot be right\"; can we go so far as to say that the Overlords suppressed religions not so much because all of them were false as because some of them pointed to the truth?\n\n_ALIEN DO-GOODERS_\n\nLike Camille in the prologue to this chapter, most SETI researchers would be surprised if aliens wanted any help from us. But the idea that intelligent alien beings can somehow help us is deeply ingrained in the SETI worldview. The proven existence of aliens alone, it is thought, would have salutary moral effects. Beyond that, there is all the scientific, technological, social, and religious advice that we might get from them. This confidence is based on one aspect of the eclipse of man: its sense of the terrible flaws and insignificance of mankind as it is now. Aliens will have had to find ways around the self-destruction that we fear will be our own fate. They will have found ways to match their great power over nature with great moral responsibility.\n\nArthur C. Clarke's novel forces us to acknowledge another side of the aspiration to overcome our human flaws. In Haldane's vision of the future, Malthusian and Darwinian forces prompted us to aspire to become the alien invaders, destroyers of worlds. Why wouldn't those same forces be at work everywhere there is life? So between SETI and Haldane we seem to have two very different visions at work. But there is a link between them, which _Childhood's End_ suggests can be seen starting from an ambiguity within the meaning of benevolence itself.\n\nFirst, Clarke knows full well that benevolence has a price even on its own terms; golden ages still have tradeoffs. Access to a galactic version of the Internet may help us solve some of our problems, but it will create new ones. Furthermore, he reminds us that not everything that seems benevolent at first glance really is; the Overlords have their own agenda in creating the golden age\u2014it makes their job of ending humanity easier. This deception raises the question of motive, which will always be a problem with benevolence to the extent that we cannot see into the hearts of others. Is it not troubling that even after the Overlords show themselves to humanity, they are presenting a false front?\n\nBut even when motives are pure, benevolence must contend with the fact that people see the world in very different ways. Think about a moral experience not uncommon among us. Should you give the alcoholic panhandler the cash he asks for, or a cup of coffee and a sandwich, or a ride to a shelter, or an appointment at a detox center? Who doubts that one of the latter options is better for the individual than cash? But who would be surprised if the recipient had quite a different view of the subject? So there can be dilemmas of benevolence even when everyone shares a common horizon of human experience and assumptions. Why would the situation be any simpler when we start thinking in terms of all those highly advanced alien races SETI is looking for? As Karellen suggests, Overlord kindness is not really comprehensible to us; it is kindness by a completely alien standard. Clarke allows us to see that even after it has empirical support, there would still be a need for faith in alien benevolence.\n\nThe Overlords speculate that they were sent to deal with humanity out of some underlying likenesses between them and us. The hopes of SETI are predicated on the universe being such that we can come to communicate with intelligent beings who are at the same time enough like us to be comprehensible and different enough to be useful. But if, as Karellen suggests, fully in agreement with Carl Sagan, we end up being like ants in comparison to the aliens, then Clarke is correct to be concerned that the universe could contain beings that are very much _not_ like us. In Clarke's story the only reason we meet nice guys first is because there is a cosmic hierarchy with a superintending intelligence at work, something remarkably like good old-fashioned providence.\n\nNeedless to say, SETI is not built on the fictional metaphysics of _Childhood's End_ , but on the accepted metaphysic of modern natural science. Hence SETI rejects the paranormal as a means of communication, just as it rejects flying saucers, just as it would reject Flammarion's spiritualism. Strictly speaking, the universe of SETI is not an orderly, hierarchical cosmos\u2014not a _cosmos_ at all in the literal, etymological sense of that term\u2014but a chaos of random interactions of matter and energy without any inherent goal or purpose. Insofar as SETI hopes aliens will lead us to the supposed goods that Clarke's novel portrays, whether world peace and plenty or the overcoming of our humanity, it does so from an entirely different set of premises. So even if we accept that as far as humanity's cosmic fate is concerned, things turn out about as well as one could hope for in _Childhood's End_ , we are entitled to wonder if that happy outcome is more likely in Clarke's cosmos or in the materialist universe of modern science, with its Malthusian and Darwinian imperatives.\n\nAnd yet, perhaps we can begin to reconcile the kinder, gentler aspects of the SETI vision with its tougher underlying assumptions if we note how for Clarke the question of benevolence really becomes a question of power. The Overlords can define benevolence as they wish, and they have the means to get humanity to go along. Ancient alien civilizations would be ancient not because they were good or learned to use their powers wisely, but simply because they learned to use them in the service of their own survival. If might makes right, then all of our dilemmas of alien benevolence disappear; how nice for us if they are so powerful they do not have to be brutal, but if they are brutal (by our primitive standards, of course), then we are left with Karellen's dictum: \"No one of intelligence resents the inevitable.\" This is the functional equivalent of Drake's glib assertion that \"There is probably no quicker route to wisdom than to be the student of more-advanced civilizations.\"\n\nIf the law of the jungle is the effectual truth of SETI, then it makes more sense that one supposed moral benefit coming from alien contact, as noted earlier, would be basing our sense of human fraternity on there being a non-human \"other\" for a radically new Us VS. Them. It also helps explain the transhumanist optimism about the meaning of our failure to find intelligence thus far; in the formula made famous on the TV show _Hill Street Blues_ , it means we can \"do it to them before they do it to us.\"\n\n_ARE WE OUR OWN OVERLORDS?_\n\nBecause SETI's contribution to the eclipse of man depends on the contribution of aliens, it may seem that the sort of issues that arise would be unique to that particular way of envisioning the future. But we have already seen more than hints that things are not so simple, especially in Bernal's picture of crustacean-like mechanical monsters with human brains. In the chapters that follow we will see that some of the same issues arise whether the source of our dehumanization is messengers from space or laboratories right here on Earth.\n\nEven before there was any understanding of genetics, people could imagine how careful breeding of human beings could enhance our well-being. When knowledge of genetics started to increase, the eugenics movement was quick to arise, and many thinkers developed even greater confidence that great things could be done to improve humanity. Once genetic engineering comes on the scene, our power to mold ourselves in some new image seems only more likely.\n\nThe prospects for and problems with genetic engineering have been widely debated, even though with respect to human beings they are still largely speculative. And yet, there are already those who believe that genetic engineering in and of itself will not be the main route by which to create new and improved human beings. At best, it will be one among the \"converging technologies\" that together will vastly increase our power over the naturally given, the others being cognitive science, information technology, and nanotechnology. Nanotechnology in particular may open the door to a world where human power quickly reaches the ability to do seemingly everything the laws of nature allow. That is where we turn next.\n\n* It ought to be noted, simply for the sake of intellectual honesty, that the traditional cosmology that put Earth at the center of things did so not out of a hubristic sense that we were the most important beings, but rather because, in the words of Alexandre Koyr\u00e9, the center represented the bottom of a hierarchy \"rising from the dark, heavy and imperfect earth to the higher and higher perfection of the stars and heavenly spheres.\" By contrast, the idea that we are not unique could arguably represent a significant elevation of human status\u2014for example, it allows Carl Sagan to suggest that we are \"star-stuff.\" (Alexandre Koyr\u00e9, _From the Closed World to the Infinite Universe_ [Baltimore: The John Hopkins Press, 1957], 2.)\n_CHAPTER THREE_\n\n_Enabling Inhumanity_\n\n* * *\n\n_PROLOGUE: THE VOICE OF EXPERIENCE_\n\nYes, I was what you might call an early adopter when it came to living with nanites, and that's why my memories of the world before and after are still so vivid these many years later\u2014I got prosthetic memory even before you could get the software to edit it. It was like being a kid in a candy shop, and with a big allowance to boot. Not everybody was as willing to experiment as I was, of course, but you more or less found your own level, finding people who were chasing the same kind of dream, at least at the moment you were both chasing it. Not that there weren't already some problems between those of us pushing the envelope and those who thought the world needed more boundaries given all the new possibilities. Old-style democracy was still around at that point, and there was some political pushing and shoving as well as some rioting and sundry nastiness on both sides. But it never worried me much at that stage. I was convinced I knew where things were going; those of us who took the plunge were better off\u2014hell just plain _better_ \u2014for having done so, and if it was going to come down to any kind of competition I was confident we would carry the day.\n\nAnd of course I was right about that. I saw the end coming when the people who wanted to limit or reject the new nano-world started to defend themselves using the same \"let us do our own thing\" principle that _we_ had used to defend transformation! By that time we were perfectly happy to leave them alone as they were pretty much irrelevant anyway. But I admit that I was surprised that that victory was not the end of our problems. Actually, it was more like the beginning. We thought the end of the old society, politics, and economics would free us to be wonderful in whatever ways we wanted. Not exactly. There was a saying in the old world, \"when nothing is true, everything is permitted.\" I don't know about that, but with just a small change it sure works the other way: when everything is permitted, nobody is true. You just could not count on people from one day to the next, or sometimes one moment to the next. There were no serious costs to picking up and going somewhere else, to becoming someone else, even to escaping altogether by getting yourself frozen. Responsibility proved to be pretty rare. It also became perfectly clear that bad guys don't become any less bad when they have more stuff and more power; we tried to deal with that by using \"white-hat nanites\" to fight \"black-hat nanites.\" That worked okay when the people who invented them stuck to the job, but like I just told you they didn't always. And how could the rest of us blame them? Meanwhile, we also rediscovered that letting everybody do his own thing was not the end of conflict; one person's fondest desire might be exactly what another person most wants to avoid.\n\nSo there was even more reason for the like-minded to stick together, cutting themselves off as much as possible from anybody who didn't see it their way. We all told ourselves that we'd still be free to do our own thing because we could join any group whenever we wanted. It wasn't long before there were enough instances of people who joined in bad faith and other kinds of organizational sabotage to bring that phase to an end. By that time the white-hat nanites were being developed by artificial intelligence so we thought we'd be pretty well shielded from outsider lifestyle choices. Yet once the AI systems understood the logic of what they were being asked to do they realized the benefits of defense in depth and started taking what sure looked like offensive measures\u2014only against the bad guys of course. Or at least the ones who looked bad to us.\n\nSo yeah it is a pretty tough world out there just now, but there never was a technology that didn't have some kind of downside, right? Maybe when nano really takes off it'll be different. In the meantime you sprouts may not be quite living the dream but . . . sorry, which siren was that? Another Level Seven incursion? Ok kids, you know where to go. With any luck we can finish this later.\n\n* * *\n\nFOR TRANSHUMANISTS, nanotechnology opens all kinds of doors. For starters, it will bring great material and economic benefits. \"By making it possible to rearrange atoms effectively,\" writes the transhumanist philosopher-activist Nick Bostrom, nanotechnology \"will enable us to transform coal into diamonds, sand into supercomputers, and to remove pollution from the air and tumors from healthy tissue.\" Simon Young, another cheerleader for transhumanism, adds that \"Eventually, through nanotechnology, vast armies of miniature robot workers manufacturing goods at the molecular level will bring productivity levels through the roof, bringing an end to scarcity and want, poverty, and hunger.\"\n\nBut such changes to the world we inhabit may seem minor when compared to the changes in store for who and what we are. Nanotechnology will be one of the main routes to \"the redesign of the human organism,\" Bostrom writes, suggesting dryly that \"once there is both nanotechnology and superintelligence, a very wide range of special applications will follow swiftly.\" Still, \"If we have a choice it seems preferable that superintelligence be developed before advanced nanotechnology, as superintelligence could help reduce the risks of nanotechnology but not vice versa.\" Meanwhile, the prominent transhumanist Ray Kurzweil points to a specific possibility in the way of building new bodies: smart dust. \"In the late twenty-first century, the 'real' world will take on many of the characteristics of the virtual world through the means of nanotechnology swarms,\" trillions of intelligent networked nanobots that will be able to form into shaped clouds that will simulate anything, including a human body. When we go out, \"we will have to select our body, our personality, our environment\u2014so many difficult decisions to make! But don't worry\u2014we'll have intelligent swarms of machines to guide us.\" (Note how readily Kurzweil slides from using this technology to express our choices to using it to guide our choices.)\n\nThe actual definition of the term nanotechnology is somewhat controversial, so it may be safest to begin with the basics. The \"nano-\" in \"nanotechnology\" is a prefix used in the metric system, much as a \" _milli_ meter\" is one thousandth of a meter and a \" _kilo_ meter\" is one thousand meters. A _nano_ meter is one billionth of a meter, and nanotechnology usually refers to manipulating matter at the scale of around one to a hundred nanometers\u2014that is, at the level of molecules and atoms. It is very hard to imagine things this small. For the sake of comparison, a human hair is about 60,000 to 120,000 nanometers wide; a human red blood cell is about 6,000 to 8,000 nanometers in diameter; a DNA molecule is 2 to 12 nanometers in diameter. If every nanometer in the diameter of a CD or a DVD were expanded to the size of the more familiar millimeter, the disc would be 74 miles across.\n\nFor the time being, much mainstream nanotechnology research in universities and corporations is focused on finding uses for \"nanoparticles\"\u2014very fine particles of some existing material\u2014or experimenting with novel molecular structures. At this very tiny scale, substances can have properties that are not present at larger sizes. Adding nanoparticles to other materials can in turn give those materials new properties. At the moment, applications for nanoparticles are relatively prosaic; one can find them in items like sunscreens, disinfectants, and car parts. These nanoparticles and nanomaterials will surely have many uses in the years ahead, but they are hardly revolutionary. Indeed, there is reason to believe that they have a long history: some materials used in the ancient world, like Damascus steel and some kinds of Roman glass, had special properties that today's scientists attribute to nanoparticles.\n\nBut experts are confident that potentially revolutionary applications of nanotechnology are not far off. Mihail Roco, founding director of the U.S. government's National Nanotechnology Initiative, foresees future generations of nanotechnology that go well beyond today's \"passive\" efforts. Already researchers are developing a \"second generation,\" with active devices such as nanoscale motors that could power nanodevices. According to Roco, third-generation systems of nanodevices could, for example, consist of systems of nanowires in the brain to sense or direct the activity of neurons. Or they could be networked nanoscale robotic devices that assemble themselves into dynamic three-dimensional images for highly realistic telecommunication or virtual realities. A fourth generation of nanotechnology could be, like biological systems, self-assembling, blurring the distinction between living and non-living systems.\n\nBy the time we get to the third and fourth generations of nanotechnology, which Roco expects to come into existence over the next two decades, nanotechnology begins to look a great deal like what Nick Bostrom, Simon Young, Ray Kurzweil, and other transhumanists have described. This shared vision of the more advanced possibilities for nanotechnology can be traced back to the work of K. Eric Drexler, whose groundbreaking and influential 1986 book _Engines of Creation_ remains one of the most thoughtful treatments of the risks and benefits of such great power.\n\nDrexler, born in Alameda, California in 1955, was as a young man intensely interested in the possibility of establishing colonies in space. He started thinking about nanotechnology in 1976, during his time as an undergraduate at the Massachusetts Institute of Technology, and over the next decade he developed and refined his ideas, presenting them in a scientific journal in 1981 and then bringing them before the wider public in _Engines of Creation_. In that book, Drexler imagined the nanotechnology future in terms of \"nanomachines\" designed to manipulate matter at the molecular scale. These nanomachines, he thought, could be used to manufacture all kinds of goods, including themselves\u2014that is, they would be self-replicating. One nanomachine could first build another, and those two could replicate in turn, until millions or billions were available to assemble molecule by molecule whatever additional product they were designed to produce. As Drexler originally envisioned nanomachines, they could manufacture anything\u2014food, rocket engines, human organs, any consumer product\u2014by building it from the bottom up. * The necessary raw materials could be provided by having other nanomachines sort and disassemble recycled goods, or by using natural resources.\n\nWith nanomachines programmed to produce just about anything we could imagine, something like the \"replicator\" in _Star Trek_ becomes possible; Drexler said it \"might aptly be called a 'genie machine.'\" Nanomachines, he expected, would help in the conquest of space, and natural resources from outer space will provide cheap raw materials and hence \"a future of great material abundance.\" They would circulate in our bloodstreams, looking for signs of disease or decay, forming an artificial immune system. They would repair living cells and even redesign them, for there is plenty of room in a human cell for a nanodevice to set up shop. If atoms were the size of marbles, Drexler writes, a single human cell would be a kilometer across. He estimates that a device useful for doing repairs in the cell would at this scale be about the size of a three-story house, guided by a nanoscale-computer system about the size of a building with a footprint the size of a football field, thirty stories tall. Such repair abilities would open the door to revival after cryonic suspension\u2014the ability to freeze a recently deceased or still-living body to preserve it until technologies are developed to cure or revive the patient. This life-extending technology is the single advance Drexler spends the most time considering in _Engines of Creation_. The central problem of cryonic suspension is that the freezing designed to preserve living or recently deceased tissue actually causes serious damage to cells and tissues. With nanotechnology, people sick or dying or recently dead from illnesses that medicine cannot cure could be frozen, put into what would amount to suspended animation, and thawed out as cures became available. Drexler notes that cryonics need not be limited to the sick: the same freezing and thawing approach could be used by anyone who wants to \"time travel\" into the future.\n\nDrexler's book sparked the public interest in nanotechnology and inspired many scientists to begin working in the field. But Drexler is a prophet accorded only limited honor in the discipline he did so much to develop and popularize; the U.S. National Nanotechnology Initiative makes no mention of him in its \"Nanotechnology Timeline\" nor in any of its other official materials. His ideas have been subject to controversy at least since the publication of _Engines of Creation_. The book's strong focus on cryonic suspension and reviving the dead put Drexler in suspect company, and the book's frequent use of the term \"imagine\" (as in \"imagine if . . .\") is also not the sort of thing that many scientists and engineers are likely to be comfortable with, particularly when there is an ongoing and spirited debate about the very possibility of the nanomachines he described. There is also among his critics a sense that his early speculations about the promise, and even more the peril, of nanotechnology had an unhealthy effect on public perceptions, raising hopes and fears in relation to things that might never even come to pass.\n\nYet his pioneering work remains important whatever the misgivings of his critics. Even if it turns out he did not envision the details of self-replicating nanotechnology correctly, the big-picture issues he raises concerning it remain relevant. For what is he thinking about in _Engines of Creation?_ First of all, he is considering the consequences of continued progress in miniaturization in a world where what has already been achieved has had huge impact. Second, he is thinking about the significance of an improved ability to develop useful new materials and customize old materials. Third, he is talking about increasingly fine-grained control over natural processes, the productivity gains that could result from this manipulation, and the results of increasing resource availability. Finally, he is thinking about what all these capacities together might do for our ability to cure or prevent disease\u2014or indeed, to do harm to people\u2014and for our ability to choose freely among a widening set of lifestyle possibilities. In and of themselves, none of these trends he claims to see is controversial. Who does not expect the future to hold more miniaturization, more synthetic materials, more control over nature, more ability to help or harm human health?\n\nSo even if it turns out that Drexler was wrong about some specifics of nanotechnology back in 1986, he still identified a _trajectory_ of technological development that can be taken seriously. He certainly is not alone in thinking that nanotechnology will change everything. But it could be said that he has thought out its perils as well as its promise more than most. He offers a carefully articulated quasi-philosophy of cosmic history that suggests the _necessity_ of developing nanotechnology\u2014a framework that is both useful for transhumanists and contains familiar elements from the eclipse of man. He offers a theory of human personality as well\u2014and it is attractive to transhumanists, too, for it provides a justification for not worrying too much about using nanotechnology to transform our minds and bodies. And he offers a vision of what a world remade by nanotechnology ought to look like. All that makes _Engines of Creation_ an unusually comprehensive effort to think about the consequences of a new technology even before it exists. The discussion that follows will not take up the technical challenges to Drexler's vision that have been raised, but will simply accept that something like what he described in _Engines of Creation_ could indeed be possible. At the very worst, Drexler's discussion of nanotechnology becomes an imaginative case study for trying to think through the consequences of accelerating technological change, and the discontinuities that it might produce in what it means to be human.\n\n_DANGER AND DIVERSITY_\n\nDrexler suggests with some justice that his book is not so much advocating nanotechnology as promoting \"understanding of nanotechnology and its consequences.\" For all the benefits that he expects to result from his nanomachine \"engines of creation,\" which he calls replicators, he warns they could also readily be \"engines of destruction.\" \"Dangerous replicators could easily be too tough, small, and rapidly spreading to stop,\" he says. That possibility \"has become known as the 'gray goo problem,'\" in which uncontrolled replicators literally consume the entire earth in an orgy of exponential growth. (This warning is one of the main reasons Drexler is looked at unfavorably; he is accused of raising implausible frightening scenarios before we know that the nanomachines he describes are even possible.) There are other dangerous possibilities, too. Advanced artificial intelligence (AI), Drexler argues, will facilitate and be facilitated by nanotechnology, and will be able to do in seconds what it would take hundreds of years of human engineering to do. States could achieve \"destabilizing breakthroughs\" with the ability to mass-produce better versions of existing weapons, or \"programmable germs and other nasty novelties.\" \"A bomb can only blast things, but nanomachines and AI systems could be used to infiltrate, seize, change and govern a territory or a world.\" Under these circumstances, no traditional \"balance of power\" is enough in international relations; advances could be made in a day that would destabilize any status quo. Domestically, totalitarian states could use nanomachines to infiltrate bodies and minds so as to dominate their populations even more ruthlessly, and to make it yet easier to treat their human subjects as disposable.\n\nFor such reasons Drexler acknowledges that \"it seems we must guide the technology race or die\"; clearly \"guide\" means direct or channel rather than restrain, since \"the force of technological evolution makes a mockery of anti-technology movements: democratic movements for local restraint can only restrain the world's democracies, not the world as a whole.\" It does not follow from this skepticism about local restraint that Drexler is in favor of world government. It would take a global totalitarian government to stop advanced nanotechnology from being developed; the elimination of liberty would be, he believes, too high a price to pay for safety.\n\nHow then are we to maximize the desired outcomes and minimize the dangers? Drexler spends a good deal of time discussing engineering techniques that could be used to reduce risks. Beyond design features, however, Drexler calls for serious efforts at \"foresight,\" which involves asking \"three questions. What is _possible_ , what is _achievable_ , and what is _desirable?\"_ Laws of nature set limits to the possible, and Drexler makes the bold claim that nanotechnology will allow our power over nature to approach rapidly the limits of the possible. Drexler is aware in principle that as the achievable begins to reach the limits of the possible, foresight will have to focus on decisions about what is desirable. That is clearly a moral or ethical question\u2014how people _should_ use this amazing expansion of human power.\n\nWe are left with a dilemma: \"Our differing dreams spur a quest for a future with room for diversity, while our shared fears spur a quest for a future of safety.\" People's desires for our world and themselves diverge wildly, and so, as Drexler develops the argument, it becomes clear that he believes liberty and diversity go hand in hand. But liberty and safety do not necessarily go together. Leave people free to do as they wish and they may end up hurting each other; the more you want to keep them safe from each other the more you may need to restrict their liberty. Political philosophy has long had to deal with this dilemma. But in Drexler's case, this already serious problem is made yet more difficult by the underlying logic that he thinks will drive nanotechnology development, which strongly constrains the foresight question of what is desirable, even as it extends the concept of diversity well beyond anything that we are familiar with today. First we will examine that logic, and see its ultimate consequences for what diversity means to Drexler, and then turn to how he believes that radical new diversity can be dealt with.\n\n_COMPETITIVE PRESSURE_\n\nTo understand the logic behind the development of nanotechnology, it is important to see that Drexler is not content to argue for nanotechnology on the grounds of its possibility and desirability alone. He also wants to suggest that because of competitive pressure we\u2014understood in some sense as a national \"we\" and in some sense as a human \"we\"\u2014have no choice _but_ to develop it. The most immediate reason is international competition; whichever country is the \"leading force\" in nanotechnology will gain a huge economic and military advantage over everyone else. That means there is every incentive for _somebody_ to develop it, and to his credit, Drexler, writing toward the close of the Cold War, clearly prefers that it be the United States.\n\nCompetitive pressure allows Drexler to place his nanomachines within a broad sweep of evolutionary development. The philosophy of cosmic history that he offers is reminiscent of the way Winwood Reade expanded on and adapted Darwin's ideas in _The Martyrdom of Man_ (discussed in Chapter One). Drexler follows Richard Dawkins, the neo-Darwinian evolutionary biologist, in arguing that evolution is all about the \"variation and selection of replicators.\" The earliest simple replicators based on RNA and DNA evolved into a bewildering variety of forms\u2014giving us life as we observe it. Eventually, the story goes, a kind of replicator we call human beings came along, capable of developing technology. This technology in turn undergoes the same process of evolutionary selection, as do the ideas (\"memes\") that are part and parcel of its development. Nanomachines are just the latest form of replicator, brought about by human replicators.\n\nWe must expect, then, that \"deep-rooted principles of evolutionary change will shape the development of nanotechnology, even as the distinction between hardware and life begins to blur.\" Competitive pressure will drive the evolution of these new replicators as it drives all others; \"the global technology race has been accelerating for billions of years. The earthworm's blindness could not block the development of sharp-eyed birds.\" Unlike natural selection, which depends on randomness, the variation and selection of nanomachines can be directed by the proper application of foresight to serve the ends of intelligence. Yet because competition makes the development of nanotechnology necessary, the question of desirability cannot really be a question of \"nanotechnology or not\" but must rather be a question of what kind and how. Indeed, Drexler argues that foresight must help us select _against_ ideas and ways of thinking that stand in the way of accepting this new form of replicator.\n\nBy lumping together many different biological, physical, and mental phenomena under the broad category of \"replicators,\" Drexler assimilates the whole history of technology into the history of life, which in turn makes two kinds of evolution possible: one based on chance and the other guided by intelligence. (This distinction is an important point of contact between Drexler's argument and the arguments of the transhumanists, as we will see in the next chapter.) So when we consider the future of nanotechnology, we are seeing a new stage in an old story. Drexler's proof of concept for self-replicating nanomachines is the existence of the self-replicating organic \"protein machines\" that are so well established in the biological realm already\u2014that is, all the parts that make up cells and organisms. With nanotechnology, we are just developing a new form of a common phenomenon.\n\n_MIND IN MACHINES_\n\nYet there will still be significant discontinuity in the shape of life to come. Nanomachines may lead to the eclipse of man, and into realms of achievement we cannot now comprehend. Drexler envisions \"revolutions\": remaking our bodies, melding mind and machine, spreading life into space. The range of these possibilities suggests that the \"diversity\" Drexler discusses will have a very wide scope. Still, he is plainly aware that some people will be troubled by the convergence of minds and machines, with all that it implies about what we are and what we may become: \"some feel uncomfortable with the idea that machines underlie our own thinking.\" To make such people less uncomfortable, Drexler attempts to shift their understanding of \"machine\" away from just the sort of \"picture of gross, clanking metal\" that Bernal presented in his crustacean-like brain housing. Drexler would rather we imagine \"signals flickering through a shifting weave of neural fibers. . . . The brain's really machinelike machines are of molecular size.\" But the further implications of this view of what a machine can look like raises its own issues. Signals passing through fibers could describe either human intelligence or artificial intelligence\u2014another idea that Drexler understands people resist. Yet if the brain is _already_ understood as a machine, opposition to artificial intelligence becomes merely \"biochauvinist prejudice.\" Drexler is drawing out the consequences of scientific materialism's view of human beings as sophisticated machines. From this point of view, a new transhuman or posthuman model of humanity is no big deal.\n\nThe supposed need to overcome biochauvinist prejudice tells us how Drexler's idea of diversity includes dehumanization as a core element of the nanotechnology future. We see another in his discussion of one of the great payoffs he expects from nanomachines: indefinitely extended life. (Like Condorcet, Drexler is careful not to say immortality.) The argument begins with relatively conventional goals for nanomachines: we will use them to cure disease, detect and repair cellular damage, replace old parts with new. However, should there be anything wrong with the body that nanomachines can't fix, they can still make possible revival after cryonic suspension. Echoing Bernal, the only human part that Drexler really believes needs to be frozen is the brain\u2014since the patterns of the information stored in the molecular machines that are our brains define the meaning of the \"I\" in the phrase \"I will live far longer.\" To make this case, Drexler takes advantage of the real uncertainty we have about the basis for our personal identity and selfhood. Is it the body? Yet the body changes over time, and the very stuff of the body is in constant flux and not at all what it was not that long ago. So where does it reside? As a materialist, Drexler does not want to believe in a soul\u2014but he comes close. The \"I\" is a pattern of information, a pattern residing in the brain.\n\n\"Nature draws no line between living and nonliving,\" Drexler writes. Matter is matter, but it can be patterned in different ways, and these patterns make a huge difference; \"one simple sum of our parts would resemble hamburger, lacking both mind and life.\" With regard to that brain, however, Drexler turns out to be something of a dualist:\n\nA mind and the tissue of its brain are like a novel and the paper of its book. Spilled ink or flood damage may harm the book, making the novel difficult to read. Book repair machines could nonetheless restore physical 'health' by removing the foreign ink or drying and repairing the damaged paper fibers. Such treatments would do nothing for the book's content, however, which in a real sense is nonphysical. If the book were a cheap romance with a moldy plot and empty characters, repairs are needed not on the ink and paper, but on the novel.\n\nWe know that the nonphysical information of the book, the novel, can have any variety of physical embodiments and remain the same with respect to the information it contains, whether the book be hardcover, paperback, e-book, or audiobook. Of course, the resurrected patterns of information that reside in the brain could be housed in a familiar body\u2014perhaps even a body that resembles the brain's original body. But why stop there? Why not an unfamiliar body, or any body at all? Indeed, once the _pattern_ in the brain becomes the key thing, even the brain itself becomes disposable. In a sense, Drexler allows the argument to advance a step beyond Bernal, for whom already the body was a mere tool of the brain. Why should the pattern of information that truly is each of us reside in a biological brain at all? And why must that pattern reside in only one location at a time?\n\nDrexler's presentation of individual identity as a pattern of information is a symptom of the relatively precarious way that consciousness and our sense of self are situated in scientific materialism. In _Engines of Creation_ Drexler does not explore in depth all the implications of the ability to \"remake our bodies.\" While he mentions in passing the possibility of \"bizarre\" modifications of the body, for the most part he is content with letting his readers assume that a resurrected brain will be placed in something like a familiar body, and that one will lead a familiar life with resurrected friends and family\u2014a scene that provided some inspiration for the prologue to this chapter. The implications are nevertheless clear enough, and suggest another respect in which nano-based diversity will produce inhumanity. Information can be stored, copied, manipulated, and transmitted in all kinds of ways. The idea that \"I\" am just a pattern abstracts not just from bodily particulars, but from how bodies are embedded in the larger world; or, perhaps more precisely, it assumes that we live in our own heads, and we don't even need to stay there. It wipes away much of how we experience and understand ourselves and our world. (In the next chapter we will see how transhumanists take advantage of the potentially radical consequences that pattern identity has for detaching people from \"biochauvinist prejudice.\")\n\n_NANOPOLITICS AND HUMAN IMPERFECTION_\n\nDrexler does his best to argue that the kind of diversity he has in mind\u2014where people can choose to live on starships, to colonize alien worlds, to communicate telepathically, to redesign their bodies, to \"time-travel\" with cryonics, or even to live the way we do today\u2014is in keeping with the kind of wide, free choice that is considered desirable in liberal democracies and capitalist economies. As remarkable as those examples may appear, Drexler seems to suggest that they fall under the rubric of the \"pursuit of Happiness\" mentioned in the Declaration of Independence. But will people with such diverse desires be able to live together? Drexler is smart enough to understand that we cannot simply assume that giving people more power to choose individually or collectively how to live will make them more tolerant of those who choose a different way of life; it might make them more insular and less tolerant. So it is at least not self-evidently a good idea to seek a future where more people will have larger differences and more power to fight about them, a world in which people believe themselves even more entitled to do as they please than in many parts of the world today. We might hope that such power would be a recipe for leaving each other alone, for doing as we would be done by, but that is not the lesson of experience.\n\nSo what is to be done to make sure that nanotechnology is used responsibly\u2014to govern its use in such a way that the gray goo scenario, for example, never comes to pass? Political, legal, or moral restraints that will help us to make the right decisions about the development of nanotechnology do not have to \"start from scratch,\" Drexler writes. There are many elements of liberal democratic society and politics that can provide the necessary foundations. \"The principles of representative government, free speech, due process, the rule of law, and protection of human rights will remain crucial,\" as will \"such diffuse and lively institutions as the free press, the research community, and activist networks.\" Further, such things as \"personal restraint, local action, selective delay, international agreement, unilateral strength, and international cooperation\" will all be useful for avoiding fearsome scenarios.\n\nHowever, such institutions and modes of restraint are only useful in a specific context: when they go along with what Drexler calls \"active shields.\" Active shields are defensive, automated nanotechnologies that act as an immune system, seeking out and destroying hostile nanomachines. In other words, the best defense against bad nanomachines in the wrong hands is for nanotechnology in the right hands always to be one step ahead. If we look more closely at the kinds of measures Drexler advocates for guiding the development of nanotechnology, we can understand why in the end Drexler must place a huge amount of weight on active shields.\n\nPolitical solutions are at least as imperfect with respect to nanotechnology as they are generally; in particular they are not complete answers to the existence of \"power, evil, incompetence, and sloth.\" Drexler acknowledges that democracies can commit \"atrocities\" and that they contain evil people. The potential for evil is balanced by the fact that democratic \"leaders gain power largely by appearing to uphold conventional ideas of good.\" He never makes clear just what constitutes \"good\" or \"evil,\" a fact that suggests that he does not have a lot of hope that some \"ethics of nanotechnology\" will be very useful as a restraint. That power and evil are relatively intractable problems already suggests why active shields will be necessary.\n\nSloth and incompetence seem to be more tractable problems, although still not easy to eliminate because \"we human beings are by nature stupid and ignorant.\" But in this area as in so many others, Drexler does not think we have to be content with nature. Indeed, we already know how to cooperate on technical matters to \"gain reliability through redundancy.\" Incompetence can thus be weeded out. He also provides suggestions to \"improve our institutions for judging important technical facts.\" One of these ideas is to create \"fact forums\" (sometimes also called \"science courts\"), which would put technical disputes within a quasi-judicial framework of due process, so that clear statements of what is agreed upon, and the parameters of disagreements, can emerge. Drexler does not envision them as policymaking bodies; they seem designed largely to lay out the merits of technical disputes for the public and decision-makers.\n\nEven if foresight can be improved by improved competence, sloth is another matter. It may well be that not everyone will be willing to \"meet great challenges with great effort.\" But it is not necessary that everyone be on board from the start. \"It will require only that a growing community of people strive to develop, publicize, and implement workable solutions\u2014and that they have a good and growing measure of success,\" Drexler writes. \"Sloth will not snare everyone's effort. Deadly pseudo-solutions (such as blocking research) will lose the battle of ideas if enough people debunk them. And though we face a great challenge, success will make possible the fulfillment of great dreams.\"\n\n_SHIELDS AND LIMITS_\n\nWhile Drexler suggests that incompetence can be minimized and sloth may be made irrelevant under the right conditions, he wisely does not claim to have a complete solution to the two other problems he names: evil and the abuse of or hunger for power. These are facts of life, and no set of beliefs or framework of laws and institutions can restrain them perfectly. But the great power conferred by nanotechnology, power that Drexler himself suggests could change everything in a day, makes evil and the lust for power particularly dangerous. So moral, legal, institutional, and political restraints are likely to mean little unless they go along with active shields. Moreover, moral, legal, institutional, and political restraints are likely to be in tension with the goals of liberty and diversity in a way that active shields are not. The arguments that Drexler uses against efforts to prohibit nanotechnology generally\u2014local restraint alone is ineffective, global restraint totalitarian\u2014would apply just as much to any effort to restrain some particular form or use of nanotechnology. So moral, political, and legal restraints are either ineffective when non-uniform or dangerous to liberty and diversity when uniform. This dilemma may help explain why Drexler is not very interested in exploring terms like \"evil\" or conventional understandings of what is good. As different ways of life develop around different uses of nanomachines, the meaning of these terms will be contested. If active shields work as Drexler intends, he can think of them as guarantors of libertarian cultural relativism; it will not matter to one protected enclave what is going on in another.\n\nNow, any society, even one that values liberty, needs some kinds of \"active shields\" like police, private security, or military forces because moral, political, and legal restraints cannot enforce themselves and not everyone will be equally restrained by them. In a society that loves liberty, these human active shields are there to make sure that (to use the familiar phrase) the liberty of your fist ends just prior to my nose. In a civilized society these old-fashioned active shields are the _ultimate_ line of defense; they come into play when all other restraints on behavior fail. These systems are imperfect, so crime and conflict are not always prevented. But the intensity of crime and conflict is limited not just by our active shields, but by some degree of unity, even if limited temporally or geographically, on normative beliefs that restrain conflict, and on the relative difficulty of acquiring tools that would be capable of larger- rather than smaller-scale destruction. So although it is relatively easy to get guns in the United States, for example, most people are not going to use them to commit crimes, and the few that have criminal inclinations will find it progressively harder to get progressively more dangerous weapons.\n\nThe future world that Drexler invites us to imagine is one with greater diversity than the one we live in, less normative consensus, and easier access to more dangerous tools. Thus, his active shields are not the last line of defense, they are the _precondition_ for creating a \"stable, durable peace\" while maintaining diversity and liberty in a world where human power increases but human goodness may not. Indeed, to the extent that the active shields work, it might appear that the question of goodness has reduced significance. Ensconced behind an active shield, we can safely follow our own vision of the good without having to worry about anyone else's.\n\nDrexler's willingness in some fashion to confront the problem of evil justifies his claim that he rejects standard utopian fare, which \"all too often [has] been impossible and the attempt to achieve it has been disastrous.\" He wants to present us with \"useful dreams\"; as we will see shortly, neither _we're running out of resources tomorrow_ nor _we can do anything we want forever_ is a useful dream. Likewise, it is not useful to think that we need to plan today for everything the future might hold; the \"great task of our time\" is not so much to build this world of diverse dreams as to guide \"life and civilization through this transition\" to it.\n\n_UTOPIAN ANTI-UTOPIANISM_\n\nWe have already seen some of Drexler's recommendations for making the nanotech future safer despite the presence of power and evil, and for reducing, or at least minimizing the effects of, incompetence and sloth. While he claims these suggestions can build on existing institutions and ideas, will our system of politics survive in a recognizable way? The future Drexler describes seems likely to fundamentally challenge liberal democracy, which is predicated on checks and balances that allow the interests and ambitions of some, running along their well-established lines, to counter the interests and ambitions of others. What happens when those lines can be extended to the limits of the possible? Certainly the liberal democratic \"pursuit of Happiness\" is _not_ predicated on the degree of human malleability and the consequent radical range of choice that Drexler presents, and the same would be true of rights generally and indeed of any of the hitherto existing \"conventional ideas of good.\"\n\nThis discontinuity should be understood carefully. There is ample historical precedent for expanding the sphere of moral concern and political protection to formerly excluded classes of human beings and considering it progress. The future could simply hold more of the more-or-less same. At the very least, our conception of the human (as in human rights) would have to expand, or our understanding of rights-bearers would have to shift from human beings to some other category, such as sentient beings. (The already out-of-favor yet foundational idea of \"natural rights\" will have even less approval in a world bent on reconstructing nature itself.) But even under these \"optimistic\" circumstances, Drexler's \"useful dreams\" are of a world that remains extremely dangerous precisely because of his hope for maximizing choice and minimizing restraint. The libertarian world he looks forward to seems unlikely to make people less assertive about their desires and perceived interests given the increasingly unimaginable benefits of nanotechnology, particularly when, for the \"I\" that is coming to understand itself as a \"pattern\" capable of diverse embodiments, the act of choosing becomes the locus of identity. The sphere of moral concern would have to be expanded to encompass all possible definitions of its appropriate scope.\n\nIn light of the diversity that Drexler envisions, one might ask, \"If people can choose to do as they please, why should they concern themselves if others are choosing differently?\" One could think even more creatively, and imagine that out of the settlement of alien worlds will spring a plethora of new nanotechnology-enabled species, alien beings who will have little interest in each other. But Drexler has already admitted that there are two flies in this ointment. The first is the existence of ongoing scarcity. This is a subject on which his position is nuanced, if not somewhat obscure. Drexler's original interest in space and then in nanotechnology grew out of a reaction to the doomsaying projections made famous in the Club of Rome's 1972 report _The Limits to Growth_ , which he first read as a young man. In _Engines of Creation_ , he utterly rejects the report's projections that we are quickly running out of resources; nanotechnology and human expansion into space will put greater resources under human control. But fundamental laws of nature still create limits under which people will have to live. For example, nature dictates that the expansion of humanity into the galaxy will be limited by the speed of light (at best). That natural limit means that if our population grows exponentially, we will not be able to spread rapidly enough to obtain the resources we need. So while \"the spread of life and civilization faces no fixed bound,\" Drexler writes, \"unchecked population growth, with or without long life, would overrun available resources in one or two thousand years at most.\" Drexler, in other words, believes that Malthus was essentially correct. So scarcity will remain an issue, even if only a cosmic-scale matter of living space, and competition among worlds will not disappear.\n\nThe second reason that radical diversity will challenge peaceful coexistence and live-and-let-live values becomes clear in an admission that Drexler makes: \"Unless your dreams demand that you dominate everyone else. . . .\" Such dreams are hardly unknown, but it is not clear Drexler has fully confronted their significance. As Aristotle knew, people do not turn to crime simply \"through being cold or hungry\" or become tyrants solely \"to get in out of the cold.\" However much we may prefer a world where there is plenty of heat to one where there is not, from the point of view of Aristotle's realism, the abundance that Drexler imagines may encourage desires \"beyond the necessary things\" or for \"enjoyment that comes with pleasures unaccompanied by pains. . . . The greatest injustices are committed out of excess.\" Excess almost seems to be the point of the nanotechnology that Drexler describes in _Engines of Creation_.\n\nBecause some people have dreams of domination, others will need protection from them. Even if greater choice and less scarcity reduces conflict, there will still be reason for conflict so long as there is ordinary crime and so long as diversity includes those (call them evil if you wish) who find reason to dissent from the orthodoxy of diversity itself. If even only a few want more than they can get, or even a few concern themselves with martial glory or the love of honor, or even a few exhibit pathological evil, the diffident and tolerant many will need to be able to protect themselves, and their best defense may be a good offense.\n\nSince Drexler gives us reason to think that active and competent evil will continue to exist in the future, then while he would like to think that active shields are purely defensive and hence non-threatening, they _are_ a threat to anyone who has reason to worry about what is going on behind them. And such concerns arise not only so long as the possibility of evil exists. They will arise also out of the very fact that diverse visions of the good will often lead to divergent understandings of what kinds of behaviors are \"safe\" or \"risky\" in the first place. Drexler ignores the complication that in his world of choice among such visions, one group's benefit may be another's risk. Unless \"worlds\" can be completely isolated one from another, the competitive pressures Drexler counts on to produce progress in developing nanotechnology will not disappear even as the power to impose on others will increase, even as wild diversity decreases the sense of solidarity. Even those who might choose to \"opt out\" of the lifestyles nanotechnology makes possible will still be dependent on it: for example, if _almost_ everybody is willing to let me raise my child without nanotechnology, I will still need an active shield against unrestrained deviants who find my behavior to be child abuse. In general, then, it appears that active shields will have to be very active indeed.\n\nWhich raises a vital question. Who will develop and maintain the active shields? Not all of the lifestyle choices Drexler speculates about, or that we might imagine, would in and of themselves point to interest in or competence at this important task, particularly to the extent that they share the desire for ease and comfort that Bernal spoke of as \"Melanesian\" aspirations. But somebody is going to have to mind the store. One wonders whether Bernal did not see more clearly than Drexler the likely outcome of this dilemma: utter dependence of the many on those relative few (perhaps the committed and non-slothful?) who develop and manage the technology the many depend upon. Diversity must be built on uniformity of technological capacity, freedom on dependence on those guardians who develop and maintain active shields. The shield guardians will not always share the values of those they protect because their focus will have to be on the non-diverse technical demands of their job.\n\nOr perhaps\u2014as indeed seems likely\u2014these shield guardians will not be human at all but precisely those advanced artificial intelligences that Drexler counts on to push so rapidly the achievable to the limits of the possible. Indeed, it is hard to imagine how such an artificial intelligence could even begin to be under mere human control, given that its main strategic and tactical advantage over human development, design, and deployment would be the speed at which it would operate. Either human beings would have to be enhanced to catch up to it, or it would have to have been securely programmed (or otherwise convinced) to be well inclined to those it nominally served. The old question of who will guard the guardians was hard enough to answer in light of the old worldly wisdom that man is a wolf to man. It is far from clear that handing over power to artificial intelligences will represent a solution to this problem.\n\nDrexler is aware that diversity and conflicting values go together. He knows also that there will be \"genuine opposition to an open future, based on differing (and often unstated) values and goals.\" He expects it will be opposed by \"the power-hungry, the intolerant idealists, and a handful of sheer people-haters.\" But despite the conflicts that Malthusian scarcity and Darwinian competition would suggest follow from such diversity, he does what he can to avoid dealing fully with the implications of divergent visions of the good. He holds that such disagreements will be far less important than those related to \"differing beliefs regarding matters of fact.\" Just as successful nanomachines reduce the problem of sloth, so too in the form of active shields are they the key to getting around value disputes.\n\nUnless Drexler believes that there is an objective basis upon which to distinguish among visions of the good, then clarity about the facts will never be enough to settle such disputes. Rather, it seems that Drexler believes that in the present, the force of necessity allows foresight to slight the question of what is desirable, as if the might of technological possibility makes right. In the future, he hopes that active shields will make it unnecessary for there to be any common answers to that question. Yet the significance he gives to competition and scarcity make it implausible that what happens within active shields will not have consequences for relations among the shielded groups, and therefore on the way they are organized internally to meet this challenge. It is all very well to let a thousand flowers bloom, until you discover that some of them are invasive weeds. If it is impossible to isolate the worlds created by nanotechnological diversity from one another, we cannot overlook the significance of conflicting visions of the good among them.\n\n_WHAT SHOULD BE DONE?_\n\nThis matter that Drexler seems not to address fully\u2014the issue of conflict based on divergent ideas of the good\u2014is at the core of Neal Stephenson's portrayal of a nanotechnology future in his dazzling novel _The Diamond Age_ (1995). Stephenson builds his story on broadly Drexlerian assumptions about the future of technology, and indeed, Drexler himself has endorsed the novel. But Stephenson is far more interested than Drexler in the question of how diverse visions of a good life might be organized internally in a nanotech future, and how such organized groups will be influenced by their competitive, external relations with those groups that have differing values. This question of relations between inner and outer arises throughout _The Diamond Age_ , relevant not just to the way nanotech-based cultures interact, but also to how people interact as individuals.\n\nThe world of _The Diamond Age_ is, by our present-day standards, pretty magical. A bird's-eye view would note the following: a worldwide computer network (called \"the Net,\" recognizable as roughly equivalent to our Internet) was responsible for the downfall of the nation-state; once the Net made possible secure financial transactions that could not be traced by governments, people stopped paying their taxes. Nanotechnology produces goods plentifully; nanomachines disassemble matter at the molecular level and, normally for a price, programmable \"matter compilers\" (Drexler's genie machines) can reassemble it into anything that is wanted. This world, Stephenson says, is one in which \"nearly anything\" has become possible. But the observation that makes Stephenson's book so insightful is stated early on; for as nearly anything has become possible, the \"cultural role in deciding what _should_ be done with it had become far more important than imagining what _could_ be done with it.\"\n\nIn the absence of the nation-state, answers to the question \"What should be done?\" are provided by a host of culturally distinct tribes, or \"phyles.\" But all phyles are not equally successful, and those inequalities matter a great deal. The largest and perhaps most powerful are New Atlantis, Nippon, Han, and Hindustan. The major phyles have obvious geographic antecedents and still control territory, including islands they fabricated using nanoconstruction off the coast of China and elsewhere. The story of _The Diamond Age_ is largely centered on one such island, called New Chusan. It serves as real estate occupied by members of the New Atlantis phyle, which has self-consciously replicated Victorian norms as the best way to deal with the promises and perils of this new world. New Chusan is also the intake processor for the molecular stocks\u2014the raw materials sorted largely out of the atmosphere and seawater\u2014that are passed to the matter compilers through a network of feeder lines (the Feed). Territory on the island's lowland periphery is leased to other less powerful phyles and to \"thetes,\" who belong to no phyle or to small \"synthetic\" phyles. The thetes constitute a relatively lawless underclass, living for the most part off the limited selection of free goods provided by the matter compilers.\n\n_The Diamond Age_ is by no stretch of the imagination a utopia. Nanotechnology makes it a frighteningly dangerous place, where thugs install \"skull guns\" in their foreheads that shoot nanoprojectiles capable of turning a body to mush from the inside out. Successful phyles have extensive security measures to protect them from hostile nanotechnology; these Drexlerian active shields consist of things like tiny hunter-killer airborne nanomachines, whose immune-system-like battles with intruding nanotechnology can turn the air gray with dead \"mites.\" New Atlantans also revive Victorian customs useful for self-defense: reception parlors in their homes serve to scan all visitors for hostile nanomachines, and women wear nanotech veils to fight off the same. The whole New Atlantis enclave is surrounded by a grid of floating security pods.\n\nYet despite all the advances in nanotechnology, the action of the story develops from the fact that in this new world, people (mostly) remain people. Parents want to do well by their children, who remain difficult to raise. Businesses want to make a profit. Society remains \"an elongated state of low-intensity warfare,\" and high-intensity warfare can break out among competing phyles. _The Diamond Age_ pictures a world of prejudice, inequality, exploitation, competition, and crime\u2014but also a world of nobility, self-sacrifice, self-discipline, and virtue. In other words, in its moral fundamentals it is a world very much like ours. Although Stephenson is not blind to the potential for the progress of dehumanization inherent in Drexler's promises, his achievement is to think through the impact of nanotechnology and other new technologies in relation to perennial _human_ possibilities.\n\nStephenson observes how nanomachines, by opening up all the diverse choices that Drexler anticipates, challenge the _ability_ of human associations to perpetuate themselves without altering the fundamental _need_ for associations to do so, given the existence of competing groups with different ideas of what should be done. As a result, Stephenson must go deeper than Drexler into the question of how the \"inward\" side of nano-constructed lives, be they understood as the cultures behind the active shields or the lives of individuals within them, mold and are molded by the \"outward\" side: relations with other cultures and individuals.\n\nWe will explore various factors that create this situation, but competition and scarcity are key elements that Stephenson draws from Drexler's account. In a world of ongoing competition and still-limited (even if amazing) resources, how a given group answers the question of _what should be done_ will be a factor in their success or failure vis-\u00e0-vis other groups with different answers. Stephenson also identifies factors other than scarcity and competition that mold the deployment and development of nanotechnology, and these round out Drexler's rather abstract picture of human things. In _The Diamond Age_ , even when people try to make the kinds of fact-based rational decisions that Drexler would have us aim at, they still run up against the consequences of imperfect information and the misapprehensions it creates. And in Stephenson's fictional world, as in ours, people are powerfully moved by the love of their own, a love that eventually points to certain mysteries of the human heart.\n\n_OUR STORY THUS FAR_\n\nA very spare plot summary can only hint at the Trollopian, if not Dickensian, richness in character and incident of _The Diamond Age_. It is largely the story of Nell (short for Nellodee), an abused thete child, who at the age of five or six receives from her delinquent brother Harv (short for Harvard) a stolen copy of an extraordinary interactive, educational book\/computer\/game of immense power, _The Young Lady's Illustrated Primer_. Harv stole it from its designer, a talented nano-engineer named John Percival Hackworth, who intended it as a gift for his daughter Fiona. But Hackworth's copy is in turn a bootlegged version of the original, which he had designed for Alexander Chung-Sik Finkle-McGraw, one of the \"duke-level Equity Lords\" in New Atlantis. Lord Finkle-McGraw intends it as a gift for his granddaughter Elizabeth, a gift he hopes will subvert her conventional New Atlantis education so that she can lead an \"interesting\" life; as one of the founders of New Atlantis, he finds his own children painfully dull and complacent.\n\nBootlegging the Primer requires that Hackworth use the facilities of Dr. X, a Mandarin working on three projects that move the plot: to free China from foreign influence, to rescue abandoned female babies from the ecologically collapsing Chinese interior, and to replace the centralized Feed of raw materials with \"Seed\" technology, decentralized nanotech manufacturing. Dr. X believes that the Seed will allow the Celestial Kingdom to become a true Confucian regime. Under Dr. X's power, Hackworth is enlisted more or less against his will to create the Seed.\n\nThe Primer, while an impressive example of artificial intelligence, still requires a human voice to interact with Nell as it tells her a host of compelling and didactic adventure-puzzle stories about a character who shares the name Nell. The human voice is supplied by Miranda Redpath, a \"ractor\" (interactive actor) who comes to think of herself as educating and raising Nell, with whom she shares a background of abuse. (Even though Miranda has no idea who or even where Nell is, the Primer closely monitors and draws on the circumstances of Nell's real life, so Miranda can infer a great deal about what is happening to the real Nell from the lines she is given to read for stories involving the fictional Nell.) The net result of the Primer's education is that Nell turns into a formidable, self-reliant child who can escape her dead-end thete world when her life is threatened by one of her mother's abusive boyfriends. She has behind-the-scenes help from Finkle-McGraw, who pays the fees for Miranda's racting. He later supports Nell as well as Hackworth's daughter Fiona at Miss Matheson's Academy of the Three Graces, the posh neo-Victorian finishing school that his own granddaughter Elizabeth attends. While each of the three girls was educated by the Primer as the heroine of her own story, it is Nell who turns into the most impressive and unusual young lady. In fact, by the end of _The Diamond Age_ she is a Queen of her own phyle, just like the character Nell in the Primer's didactic fairytale. Her real-life subjects are the abandoned girls saved by Dr. X. These hundreds of thousands of girls were also educated by pirated copies of the Primer that Dr. X and Hackworth supplied; these girls' experiences with the Primer have turned them into a real-world equivalent of the \"Mouse Army\" in Nell's Primer's story, an army that is entirely loyal to Nell.\n\nWhile Hackworth's daughter Fiona and Nell are growing up, Hackworth spends ten years involuntarily (it would seem) immersed in the world of the Drummers, a strange underwater phyle whose members constitute a \"Wet Net\" in which individuals lose all self-consciousness in a quasi-telepathic linkage. Each becomes a node for processing information through intensive exchange of nanomachines in bodily fluids. Rescued once, Hackworth is drawn back to the Drummers for a second attempt to complete the Seed calculations in the midst of the Confucian uprising Dr. X has arranged. But while the last foreigners are expelled from China, many of their lives saved by Queen Nell and her real-world Mouse Army, Hackworth is again prevented from finishing the Seed calculations. For Miranda, having joined the Drummers in hopes that she could use their group mind to find Nell, was to be the sacrificial repository for this second attempt at Seed design. With the help of Carl Hollywood, Miranda's onetime producer, Nell locates and saves the woman whom the Primer itself has helped her understand as the loving human presence behind her book.\n\n_CONTINUING SCARCITY AND COMPETITIVE PRESSURES_\n\nCompetition and conflict drive the plot of _The Diamond Age_ , as China yet again seeks to get out from under foreign domination. Technology makes new forms of exploitation possible, but those new forms of exploitation mimic in their effects and results familiar stories from history. As Drexler seems to expect, individuals can find a social world into which they fit with least conflict by having choices with respect to phyle membership (if the desired phyle will have them, which is far from a foregone conclusion). However, as Drexler does not expect, the competing notions of a good life represented by the different phyles continue to be the basis for conflict among them in part because (as we will see) nanotechnology has not ended scarcity.\n\nCompetition among phyles has contrasting consequences; it usually, but not always, promotes Drexlerian diversity. In some instances it leads to an emulation of successful models and a certain homogenization. We learn, for instance, from Madame Ping, the proprietress of a brothel, that men \"from all tribes\" want \"to be like Victorian gentlemen,\" so their sexual fantasies turn toward neo-Victorian scenarios.\n\nCompetition among phyles means that they cannot in their internal organizations be blind to the qualities that will allow them to compete successfully. Observing the diversity of ways of life in her world, the teacher Miss Matheson notes as if speaking to Drexler himself: \"It is upon moral qualities that a society is ultimately founded. All the prosperity and technological sophistication in the world is of no use without that foundation\u2014we learned that in the late twentieth century, when it became unfashionable to teach these things.\" Elsewhere in the book we learn more of this history: the New Atlantis phyle arose as a reaction against the moral relativism and mindless egalitarianism of the late twentieth century, just as the original Victorians turned against the excesses of the Regency era. Lord Finkle-McGraw grew up in Iowa (more or less in our present) and as a young man he\n\nhad some measure of the infuriating trait that causes a young man to be a nonconformist for its own sake and found that the surest way to shock most people, in those days, was to believe that some kinds of behavior were bad and others good, and that it was reasonable to live one's life accordingly. . . . Finkle-McGraw began to develop an opinion that was to shape his political views in later years, namely, that while people were not _genetically_ different, they were _culturally_ as different as they could possibly be, and that some cultures were simply better than others. This was not a subjective value judgment, merely an observation that some cultures thrived and expanded while others failed. It was a view implicitly shared by nearly everyone but, in those days, never voiced.\n\nThe fashionable egalitarian relativism out of which New Atlantis sprang amounts to the belief (implicit also in Drexler) that one way of life is as good as any other. In the world of _The Diamond Age_ , thetes continue to act on the basis of this mistaken belief, and we can see its ruinous effects. Nell's mother breaks up with her most promising boyfriend because he is a blacksmith in a tribe of artisans; they produce handmade goods for the neo-Victorians. She \"didn't like craftsmen, she said, because they were too much like actual Victorians, always spouting all kinds of crap about how one thing was better than another thing, which eventually led, she explained, to the belief that some people were better than others.\" From all we see of the thetes in the Leased Territories, we have no reason to believe her views are anything but typical. Certainly Nell's life as a thete, governed by violence, and the life of her thuggish father Bud, whom we see at the beginning of the book, represent a broader consequence of holding this belief, which is exploitation of the weak by the strong. The \"logical conclusion\" of the thete lifestyle is to be \"homeless, addicted, hounded by debtors, or on the run from the law or abusive members of their own families.\"\n\nSo despite nanotechnology's potential to greatly reduce material scarcity\u2014no one need go hungry ever again, since public matter compilers make such things as food and first-aid supplies freely available\u2014inequality and relative poverty still exist. Yet the failure of the lives lived on thete assumptions is not entirely a product of their own moral failings; it is also a consequence of those failings in connection with their external circumstances, over which they have little control. The lowland territories where the thete underclass lives provide a useful buffer zone against nano attack for the highland-dwelling Atlantans, even at the cost of chronic lung disease for its residents. The thetes inadvertently are a sort of human layer in an active shield, a result of New Atlantis protecting its own interests. Nanotechnology is a big, powerful business that makes New Atlantis big and powerful. The Feed infrastructure was developed by and is owned and run by people made wealthy and powerful by its success; New Chusan is in effect Finkle-McGraw's ducal estate. The interests that brought about this new world do not simply vanish when it is created, but continue to shape it. New Atlantis, in short, has no interest in giving all possible goods away for free; to some extent the Atlantans must manufacture scarcity along with everything else.\n\n_LOVE OF ONE'S OWN_\n\nIt should not be news that people tend to protect their own interests; it is a manifestation of the love of one's own that is pervasive in _The Diamond Age_. Hackworth goes outside the law to give his daughter a leg up, yet repeatedly also expresses loyalty to his phyle; Harv severely beats and perhaps kills his mother's boyfriend, who has abused Nell; Finkle-McGraw wants the Primer to improve the life of his granddaughter and advance the interests of his phyle. Carl Hollywood looks out for Miranda, his employee and perhaps romantic interest; Miranda comes to think of Nell as her daughter; the Mouse Army will do anything for their Queen Nell. As these examples suggest, while love of one's own forms a basis for phyles that protects them from outsiders, it can also threaten their internal organization. This perennial tension inherent in voluntary human associations is not changed by technological development. Hackworth is loyal to his phyle in relation to outside phyles; he is loyal to his family inside the phyle; his wife might claim he is loyal to his own love of engineering problem-solving, putting his family on the outside. It is all the more a challenge to reconcile inner and outer when the definitions of each can shift in this way. The multiple layers of the love of one's own make relevant a variety of moral qualities that contain it or direct its expression, virtues like loyalty, honor, courage. Having been well instructed throughout the book in how phyles take care of their own, we are hardly surprised when a grandmother blows herself up to save her fleeing cohort from oncoming Celestial Kingdom troops.\n\nSometimes the love of one's own directs violence against outsiders, sometimes it contains it in relation to those inside the group. It may involve competition among phyles, but there is also competition within phyles that must be dealt with. People in whom the moral qualities that discipline, train, and direct love of one's own are missing become dangerous; here again the thetes are instructive. Nell's mother and father have very limited attachments to anything beyond themselves. One of the reasons Bud is sentenced to death for a mugging is that he is not aware that his girlfriend Tequila has given birth to Nell; hence, the judge is confident that Bud's departure will not be a loss to his children. While young neo-Victorian ladies are taught to think of late-twentieth-century urban America as a historical low point, the thete lives we see are pretty nasty and Bud's is also literally poor, brutish, solitary, and short. Given their nearly unmediated selfishness, being spared the perils of starvation and other kinds of gross material deprivation by the free goods offered by the matter compilers only seems to liberate thetes to be more self-indulgent, irresponsible, and depraved.\n\nBut if thetes have little concern for anything outside of their own gratification, the other extreme is represented by the Drummers. In this tribe, individuals become parts in a \"gestalt society,\" which is to say they share thoughts without being aware they are doing so. By their appearance, behavior, and tunnel-dwelling lifestyle, the Drummers rather remind one of naked mole rats. So far as an outsider can tell they lack any sign of inner life, consciousness, self-consciousness or individual volition; it is hard to see how there would be any competition among them. (The few we see escaping the Drummers seem to do so only with outside assistance.) The mechanisms of their physical survival in this state\u2014like what and how they eat\u2014are largely left unexplained.\n\nThe ultimate purpose of their common life is similarly mysterious. Carl Hollywood concludes that by directly linking computing technology with the human mind, which \"didn't work like a digital computer and was capable of doing some funny things,\" they have gained the capacity to break the encryption codes upon which the security of the Net depends\u2014a development with disastrous potential for the status quo. They have no \"obvious way\" or no desire to take advantage of their ability. Yet we know they _are_ willing to be used to create Seed technology, a tool equally subversive of the status quo. In losing a self-conscious identity, a core sense of being individuals, it seems the Drummers have succeeded either in transforming themselves into tools, the means to someone else's ends, or into an organization whose purposes are not comprehensible to those human beings who have not taken the Drummers' evolutionary leap to inhumanity.\n\nIn different senses, then, both extreme love of one's own (as in the case of the thetes) and extreme abandonment of it (as in the case of the Drummers) are dehumanizing. It is the middle ground that brings us to the question of _perpetuation_ : how a regime strives to contain and employ love of one's own to continue its institutions, fundamental ideas, and mores. For Miss Matheson, managing the tensions inherent in the love of one's own is a key element of perpetuation. Her work of \"propagation\" in her school is predicated on the belief that since it is a dangerous world, special care must be taken that, for \"educated Westerners,\" the love of their own be properly constrained and directed to the group by what amounts to propaganda disguised as education and by inculcating the intensive discipline necessary to adhere to the powerful social norms of the neo-Victorians. While Miss Matheson appreciates that for some outstanding individuals (such as Nell) more is possible, she would seem to be content if most of her students turn out to be relatively uninteresting because of their thoughtless acceptance of the norms of their world. She wants them to love New Atlantis because it is theirs.\n\nLord Finkle-McGraw, on the other hand, hopes the Primer will promote an education subversive of the complacency inherent in Miss Matheson's method, so that those who get it will come to love New Atlantis because they understand it is \"the best of all possible tribes.\" (Miss Matheson, in contrast, thinks it is only \"as good as any other\u2014better than most, really.\") Both Miss Matheson and Finkle-McGraw are dealing with the problem of perpetuation for a small portion of the ruling class of New Atlantis. Finkle-McGraw thinks that, for some part of this class, perpetuation is best achieved by a subversive education that will allow those who get it to maintain a sufficiently critical distance from their society\u2014based on having the capabilities to live an \"interesting life\"\u2014to allow them to choose it as best. For Miss Matheson, perpetuation is best achieved for most of this class by discipline and indoctrination. The ideas of both are consistent with the broad character of their New Atlantis phyle. Miss Matheson's school prepares its female charges for the phyle's complex internal norms of self-discipline and social interaction, ruthlessly enforced by peer pressure. Finkle-McGraw is evidently looking to what is required when commercial and technological competition and innovation are crucial to keeping the phyle on top. Their positions could be reconciled, of course, by discrimination between those who would be best served by each kind of upbringing.\n\nDealing properly with love of one's own is necessarily a difficult proposition. Thetes seemingly solve the problem by caring for nothing beyond themselves, with the consequent dangerous atomism. Drummers seemingly solve the problem by becoming an aggregation whose parts are unaware of having inner or outer lives. Stephenson acknowledges that nano and net technology could combine to liberate love of one's own entirely or destroy it, but that these are both dehumanizing choices. Between them rests the challenge of perpetuation, where moral qualities make the difference between success or failure at the difficult task of reconciling inner social and personal character with the outward circumstances of a competitive world.\n\n_THE INERTIA OF BEING HUMAN_\n\n_The Diamond Age_ presents a world that is exactly like our world in that what is going on in one culture may not be comprehensible when viewed from the outside by another. A good deal of misunderstanding can be generated on this basis. But the same is true for one human being observing another; for all its technology, in Stephenson's world some mystery remains in the human heart. At one point in the story, Miranda is told by a certain Mr. Beck that it may be possible for her to locate Nell despite all the security built into the Net because that security is based on physical laws. But perhaps, he says, there is another dimension \"invisible to those laws of physics, describing the same things with different rules, and those rules are written in our hearts, in a deep place where we cannot go and read them except in our dreams.\" Beck, we find out, is a CryptNet agent, probably recruiting for the Drummers, who will use Miranda, perhaps precisely because of her connection to Nell and through Nell to Hackworth\u2014or at least so we might speculate. His appeal may not be disinterested, but it could still be true; after all, in the end Nell and Miranda _are_ united. Yet while the Drummers are able, it seems, to manipulate dreams and perhaps therefore hearts, they are not so attuned to the mysteries thereof that they can forestall the rescue of Miranda.\n\nMysteries of the heart come together with competition and love of one's own in a second shadowy conflict that moves much of the action of the book: the battle between New Atlantis and CryptNet. The New Atlantan view that there is a legitimate \"cultural role in deciding what _should_ be done\" is challenged by the members of the CryptNet phyle, who believe, as Hackworth explains, \"that information has an almost mystical power of free flow and self-replication, as water seeks its own level or sparks fly upward\u2014and lacking any moral code, they confuse inevitability with Right.\" Like Drexler, and reflecting many of the advocates of the eclipse of man, CryptNet apparently feels that changes brought by technology are both inevitable and the basis for \"a more highly evolved society.\" But, at least in Stephenson's vision, technological development does not so much lead to \"higher\" social forms as it shifts the shape of the constellation in which perennial characteristics of human life show themselves. Hackworth's own metaphor reminds us of the perennial wisdom in the book of Job: \"man is born unto trouble, as the sparks fly upward.\"\n\nThe CryptNet view\u2014the idea that technological advancement equals inevitable social improvement\u2014fails to appreciate what seems to be a key lesson of _The Diamond Age_ : the complexly mixed human motivations that produce a given world-altering technological development do not vanish as that development works its changes upon the world. Born of desires for independence and control, of curiosity and mastery, the Seed technology that China and CryptNet desire for essentially contradictory reasons will not supersede or nullify such motives. Rather, its use will be conditioned by them, and by the myriad other sources of human action that come into play as a result of its employment by real people.\n\n_HUMAN NATURE, RESPONSIBILITY, AND TECHNOLOGY_\n\n_The Diamond Age_ illustrates how the deeper and more nuanced one's understanding of human things, the more it makes sense to imagine technological change within the framework of historical or cultural continuity, and the less the discontinuities promised by the eclipse of man seem desirable. Stephenson does not deny that technology could usher in vast changes in the ways in which people live, even changes which may fundamentally alter our humanity. The kind of world depicted in _The Diamond Age_ could represent an unstable, transitory, or intermediate phase of developments that will work themselves out in the victory of more radical possibilities. That outcome seems to be acknowledged in the book by the fact that none of the fundamental issues that move the plot is resolved. China, for example, is freed of foreigners, but its independence is by no means assured since it faces accelerated ecological collapse in its interior without either Feed or Seed.\n\nNevertheless, Stephenson imagines more convincingly than Drexler how deliberate choices with respect to what ought to be done with technology remain possible, and indeed how, given the enduring characteristics of human life, the increased power given by new technology will _require_ greater care with making those choices if societies are to thrive. Stephenson goes beyond Drexler's acknowledgment of human failings to think about a future where a full measure of human motivations, noble and ignoble, continue to exist. Our frailties are not problems to be solved but are built into what we are and how we are placed in relation to each other, guaranteeing that perpetuation will remain a challenge. _The Diamond Age_ begins and ends with the sounds of the bells of St. Mark's Cathedral in New Chusan\u2014perhaps a reminder that godlike powers will not make us gods, let alone God.\n\nDrexler's abstract understanding of humanity as \"pattern plus stuff\" in _Engines of Creation_ \u2014an understanding that makes our bodies, for example, somehow accidental to who we are\u2014makes it easy for him to downplay the significance of the complex passions and interests out of which any real world of nanotechnology is likely to be built, even though he is not blind to them. The net result is his belief that nanotechnology will not only solve many of the problems of our world today, but that nanotechnology will solve nearly all the problems created by nanotechnology. In and of itself that prospect seems very unlikely\u2014unless, as Drexler only suggests, it would result from competition among various ways of life effectively ceasing when human power reaches the limits of the possible. But in fact, so long as humans remain flawed in ways Drexler acknowledges, and in others he does not, as the realm of choice widens we will only have more reason to think about responsible choice\u2014choice made with an eye to the technical facts, but also all the messy complications of individual and collective moral life.\n\nDrexler is no Bernal or Haldane; his advocacy of dehumanization is considerably less overt. But there are surely more than hints of it in disembodied brains, in the bizarre experiments that nanotechnology will make possible, in mind emerging in machines, in the promise of remaking our bodies, and in all the essential premises of his argument. Stephenson seems to be suggesting that the inertia of human things will and ought to be a challenge to most of the radical achievements Drexler looks forward to. One could accept that analysis, and conclude: so much the worse for human things if they stand in the way of Drexlerian imagination.\n\nWith that thought in mind, we turn next to a more detailed examination of the transhumanists, to see what the eclipse of man has come to look like today.\n\n* It should be noted that Drexler subsequently abandoned this specific route to nanomanufacturing, has expressed his dislike of the terms \"nanobots\" and \"nanites\" that many writers have used to popularize this concept, and has disavowed the notion of \"tiny, swarming, intelligent, socializing, conniving things\" without making fully clear that this description sounds a good deal like what he once expected, or at least resembles the way he described it. However, the alternative course of technological development that he has laid out in works more recent than _Engines of Creation_ does not make a substantial difference in reference to the ultimate promise he sees in nanotechnology. (See Eric Drexler, \"Why I Hate Nanobots,\" March 7, 2009, Metamodern [blog], <http:\/\/metamodern.com\/2009\/03\/07\/i-hate-\"nanobots\"\/>.)\n_CHAPTER FOUR_\n\n_Perfecting Inhumanity_\n\n* * *\n\n_PROLOGUE: IT'S A WONDERFUL LIFE_\n\nAt the age of 75, Adam Newman is deeply satisfied that with the onset of middle age he is reaching the full height of his operatic powers. He has always been thankful for the genetic endowment his parents arranged for him: the propensities for musical abilities and language skills, of course, but also the fact that they had enough foresight to keep him a bit over the average in height. His 6' 6\" frame can carry with ease the expanded lung capacity and musculature that have made him such a powerful singer, and being big along with handsome still has substantial social and stage advantages.\n\nNot that Adam has failed to make some improvements of his own. It was no picnic training to use his new optimized vocal cords, but well worth it in the end; many audiences still appreciate that it is really him they are hearing across his full six-octave range, and not some processed sound coming out of speakers. Of course the biomechanical cords also allow him to play the more traditionally minded houses that won't synthesize at all. His language and vocal-style memory implants are another matter; here again he can hardly fail but be grateful to his parents. For his innate skills in these areas mean that unlike many of his peers, who need a minute or two to reorient, he can \"switch gears\" seamlessly from one language or musical style to another, no matter how exotic. Since stylistic and linguistic eclecticism have run riot among composers (why not say what you want to say in whatever language that it sings best in?), this ability puts him in great demand. He does not need the contrived off-stage moments that librettists often have to supply for singers to recover after they have switched over. Adam feels it just adds to the realism.\n\nHe's never happier than when he is on stage; indeed, Adam can hardly imagine a different life for himself. Still, even in the very enjoyment of full summer Adam is sometimes reminded that fall will come. Lately, he's found that he really benefits from a couple of hours of sleep a week. (His folks once told him they had to stretch a bit to afford the sleep-optional mod, but thought in the end it was worth it\u2014what great people they were!) And his metabolism has needed a bit of tweaking to optimize his nanotech waste-minimization feature. But, thank Gentech, so far it's nothing that remotely affects his voice. Adam has reached the point where he has no real concerns about his legacy; after all, his great performances are already available in full-def 3D VR, and he is under contract for a series that includes the new emotional track, having finally found a producer who shares his contempt for the crude work of much that is done in this area and is willing to listen to his suggestions about whom to use as his emo-dubber. (It never occurred to Adam that anybody would want to feel what he is _actually_ feeling as he sings, which to be honest is not very much beyond a desire to please the audience.)\n\nBut he's beginning to wonder why he should have to think in terms of a \"legacy\" at all. Adam has never been what you would call an early adopter. He has all the innate conservatism you'd expect from an opera singer, but he has watched with interest the developing debate between the cyborgs and virtuals\u2014those who have left birthbodies behind and uploaded their minds to biomechanical bodies and those who have chosen virtual instantiations. They both make good cases for their choices, although when the virtuals were just holograms, cyborgs seemed to have the better life. But then smartdust came along. Intellectually he knew Daphne Morgan wasn't a hologram anymore, but he never got over his surprise at how solid she was in that final embrace in _The Golden Ass_ , when just a moment before she had been a tree. No more need to mime body-resistance with her! Still, the virtuals tend to get a bit . . . weird. Adam had been content with their collegial relation, but then she started giving him what he thought were signals that she wanted more. So he invited Daphne out to dinner to test the waters. He supposes he should be grateful that she came as a baby rather than adult giraffe. But having her standing across the table from him, with her big eyes, long lashes, and baby-doll lisp; it aroused in him feelings he did not quite know what to do with. Although it was supposedly impolite to ask them, everybody knew that the virtuals are multitasking their avatars when they are dealing with the very slow-paced solid world. The rumors about how virtuals like to mess with actuals, and about the strange things that virtuals get up to in cyberspace didn't help matters; he'd had a sense a couple of times that they were not exactly alone at the table. After that, he was content that they just work together.\n\nSo Adam wants to see how things shake out before he gives up entirely on the body that has been so good to him\u2014just another way of showing gratitude to his wonderful parents. He wouldn't be a bit surprised if by the time he's ready, you won't even have to choose between virtual and cyborg; there will be something better than both! What would it be like to sing with a full cast of Adam Newmans?\n\n* * *\n\nDEHUMANIZATION IS central to contemporary transhumanism. A skeptical reader of this book's first chapter might have concluded that the advocates of the eclipse of man discussed there\u2014Bernal, Haldane, and the rest\u2014represent little more than historical curiosities. But if their ideas remain in some way curious, at least they are not simply historical curiosities. As we saw in Chapters Two and Three, the themes they explored are alive and well today: scarcity and competition, remaking our bodies and extending our lives, merging minds and machines, mankind's destiny in space, and more. These are some of the dearest hopes of today's transhumanists. Ray Kurzweil, the most prominent popularizer of transhumanist ideas, imagines we will have multiple virtual avatars, so that \"I\" can at the same time be attending various meetings and sharing a stimulating simulated sexual encounter that will be better than the real thing because I will be able to feel not only my own sensations and emotions but also those of my partner(s). Hans Moravec, a prominent roboticist at Carnegie Mellon University, has enthused about how we will be able to speed up our thought processes so that \"You will have time to read and ponder an entire on-line etiquette book when you find yourself in an awkward social situation.\" (Of course, that seems to assume that one's fellows are not similarly \"speeded up.\") The transhumanist Simon Young looks forward to being able to eliminate the need to eat or excrete or sleep. James Hughes, another booster of transhumanism, thinks we will have cybernetic brain implants that we will \"undoubtedly\" use for \"watching and advising our behavior,\" to make sure we are acting morally by whatever standard of morality one might choose. A 2006 conference at Stanford University Law School organized by some of the top figures in transhumanism featured a panel discussion about people like Cat Man and Lizard Man, who have had themselves surgically and otherwise altered to resemble their namesakes. The list of alterations to our bodies and brains, our minds and morals, that the transhumanists envision goes on and on, becoming ever more strange.\n\nYet the best case for transhumanism starts from familiar, even prosaic, premises. This \"common sense\" case may be characterized as follows. Incremental technological and scientific progress, likely accelerating in pace, will continue to occur across a wide variety of fronts, driven by commercial, medical, and military motives, as well as the pure joy of research and development. Who does not already have a sense that things are changing rapidly, and that the rate of change is increasing? The consequences are readily taken for granted today: ever more people living lives that are healthier, wealthier, and longer. Who doubts that medicine will continue to make the kinds of strides that will offer opportunities for ever better health? Who would find it remotely plausible that computers would be the same in a decade as they are today?\n\nHence, many of the new abilities made possible by this progress will, taken one by one, hardly be controversial in themselves. But taken together, these ever-increasing powers over nature will converge on the ability to redesign our minds and bodies, opening the door to the more strange and radical possibilities mentioned above. There need be no dedicated research program to make these dreams a reality; the necessary technologies will appear on their own, regardless of transhumanists' hopes for the future. The technologies that cure disease and restore lost functionality open the door to enhancement of our abilities; indeed it seems to be agreed by both transhumanists and many of their critics that there is no bright line between therapy and enhancement. So transhumanists readily imagine using our increasingly sophisticated understanding of the human machine to create novel capacities.\n\nIndeed, the more one thinks about how much better we could do if we designed our own bodies, the more dissatisfied we are likely to be with the present model. In particular, isn't just about everybody dissatisfied with the brevity of life? Pushing this open door is a favorite transhumanist selling point; they are confident that significant life extension is on its way. Ongoing developments in our ability to treat once-fatal diseases will certainly play a role, but aging itself, seen as a disease, will soon enough start to yield, whether to genetic engineering, artificial or cloned organs, or the kind of nanotechnology-enabled medicine that Drexler foresaw.\n\nAs we will see in what follows, the arguments that drive transhumanism from the uncontroversial to the strange are very similar to those that moved us from Condorcet's vision of human progress to the eclipse of man. Transhumanists argue, in a fashion that will by now seem familiar to readers of this book, that manipulating nature is simply part of what defines us as human beings. The growing abilities of modern science and technology that stem from this trait are giving us the power to take evolution into our own hands; ongoing competition will force us to use technological evolution to improve not only on the naturally given \"outside\" of us but also on our selves.\n\n_EAGERLY AWAITING ENHANCEMENTS_\n\nLet us begin by looking more closely at the first part of this line of thought. There are already many technologies available that enhance human abilities. Take, for example, human vision. Eyeglasses enhance the vision of those with poor eyesight, but they are already coming to appear as relatively primitive. Contact lenses can both enhance eyesight and provide cosmetic alteration of eye color or appearance. LASIK surgery can eliminate the need for certain kinds of vision prosthetics altogether. Artificial retinas are being used to restore sight to people with certain kinds of blindness, and tiny \"telescopes\" are being implanted into the eyes of some patients with macular degeneration. Such technologies are rapidly improving, and remarkable further advances are both easy to imagine and plausible. None of this \"enhancement,\" if that is indeed what it is, is remotely controversial, and we expect and desire further improvements in such technology, made possible by growing knowledge of how vision and the eye work, and by advances in computing, materials science, and biotechnology. So an artificial eye, whether grown or constructed with organic or inorganic materials, seems far from a crazy dream, but rather a reasonable goal of medical research and a great boon, restoring normal sight to all those who suffer from vision loss.\n\nSo far so good. But if one can make such an eye, why restrict its use to those with vision loss? Just as many cars now come equipped with once-unimaginable rear-vision cameras, would it not be useful to implant a third eye in the back of one's head? Granted, the brain is not used to processing the data such an eye would provide. But the brain is quite malleable; could it not be trained to see with a third eye? A recent study suggests it treats tools as body parts, so why not new body parts as tools?\n\nIf there are atavistic aesthetic objections to a third eye, why not construct an eye that has _better_ -than-normal vision, one that can see more sharply at greater distances, or can see better in low light? Why not an eye with a zoom lens, or one that can extend the range of human vision into the ultraviolet or infrared? We are told that the eye is a kind of camera, and we already make cameras that do all these things. The brain \"processes\" electrical impulses from the eye, and we now also make cameras whose outputs are electrical impulses.\n\nIt does not take much imagination to see how such artificial, enhanced eyes could provide a competitive advantage to soldiers, pilots, police, sportsmen, contractors, repairmen\u2014the list could be as long as the kinds of additional vision that _someone_ might find useful. And once some people adopt the new ability, others in their line of work will be faced with competitive pressure to keep up by using the same enhancement or risk falling behind. While doubtless early versions would be quite expensive, why not experiment on wealthy early adopters? Just as the price of LASIK surgery has plummeted, so too we can expect that some kinds of artificial eyes will eventually come to be affordable by nearly all\u2014even if the cutting-edge models continue to be high-priced.\n\nThe transhumanists can point out that all you have to do is look around you today to see that if you leave people free to make their own choices about what kind of vision or any other sort of enhancement they wish to have, things will settle out in a way that favors their ever-increasing use, even if the distribution of benefits remains in-egalitarian. This argument, of course, tacitly acknowledges both that there will be winners and losers in a transhumanist future, and who the losers will be: the unenhanced or even the less-enhanced. Should we consider that a problem? Many (though by no means all) people accept that there will be winners and losers in an economy based on just the kind of free choice that the transhumanists advocate in relation to enhancements. So it seems as if anyone already accepting such inequalities would have to do so also in relationship to enhancements.\n\nBut there is a difference. The present inequalities are generally accepted to the extent that we believe winners and losers are not graven in stone, that there is dynamism and mobility such that winners are not entirely self-perpetuating\u2014for example, there are reversals of fortune, or successful people can have stupid and profligate children who fritter away the fruits of their success. Beyond that, mortality is a great leveler. But transhumanist success aims precisely to be self-perpetuating to a far greater degree and takes perpetuity of some sort to be a reasonable aspiration. Hence, the prospects for entrenched inequalities seem to be dramatically improved as the logic behind how an artificial eye might be used is applied to just about any aspect of our bodies: senses, limbs, or organs, including our brains.\n\n_CHOOSE YOUR OWN ADVENTURE_\n\nNevertheless, in relation to all such technologies of enhancement, a great many transhumanists stand foursquare behind the principle of consumer choice. Most are willing to concede that enhancements ought to be demonstrably safe and effective. But the core belief is that people ought to be able to choose for themselves the manner in which they enhance or modify their own bodies. If we are to use technology to be the best we can be, each of us must be free to decide for himself what \"best\" means and nobody should be able to stop us.\n\nThis techno-libertarianism is important to transhumanists if for no other reason than that it allows them to believe they can distinguish between their outlook and that of the early-twentieth-century advocates of eugenics with whom they fear being confused. After all, the transhumanists, like the eugenicists before them, talk about a quasi-Darwinian competitive imperative to improve the human race; it might appear as if transhumanists are simply advocating more effective means to old-fashioned eugenic ends, replacing selective breeding and sterilization with enhancements. But transhumanists insist it is crucial that, unlike the eugenicists, they are not interested in using the power of government to coerce people into making better humans. For the most part the transhumanists insist that just as no one should be prevented from choosing the enhancements or modifications he wishes, so no one should be forced into any kind of enhancement or modification. Indeed, the transhumanists argue, it is their critics\u2014whom they disparagingly label \"bioconservatives\" and \"bioluddites\"\u2014who, by wishing to restrict enhancement choices, are the real heirs of the eugenicists; they are the ones who have an idea of what humans should be and want government to enforce it. The transhumanists would say that they are far less interested in asserting what human beings should _be_ than in encouraging diverse exploration into what we might _become_ , including of course not being human at all. Moreover, the argument goes, transhumanists are strictly speaking not like eugenicists because they are not interested only in making better human beings\u2014not even supermen, really. For to be merely human is by definition to be defective. It is this view of human things that makes the transhumanists _de facto_ advocates of human extinction. Their dissatisfaction with the merely human is so great that they can barely bring themselves to imagine why anyone would make a rational decision to remain an unenhanced human, or human at all, once given a choice.\n\nHowever, if the transhumanists are for the most part against state coercion in relation to enhancements, as we have already seen that does not mean there is no coercive element in the transition to the transhuman. They can avoid government coercion because they believe that the freedom of some individuals to enhance and redesign as they please adds up to an aggregate necessity for human enhancement, given competitive pressure and the changing social norms it will bring. Indeed, to the extent that transhumanists recognize that theirs is presently the aspiration of a minority, they are _counting_ on this kind of pressure to bring about the changes in attitude they desire. Within the framework of the largely free market in enhancements the transhumanists imagine, an arms-race logic will drive ever-newer enhancements, because if \"we\" don't do it first, \"they\" will, and then \"we\" will be in trouble. This kind of coercion is not of much concern to transhumanists; they are content to offer that it does not infringe upon freedom because, as the rules of the game change, one always retains the freedom to drop out. Indeed, the transhumanists seem to take particular delight in pointing out that anyone who opposes the idea that the indefinite extension of human life is a good thing will be perfectly free to die. In a world of enhancement competition, consistent \"bioluddites\" will be self-eliminating.\n\nThere is another kind of necessity that transhumanists can adduce to support their case for the free choice of enhancements. Recalling a theme introduced by Condorcet, they assert that human beings are practically by definition the beings that enhance themselves, or that human beings are the beings who overcome their own limitations and the limits of the naturally given\u2014including the sorts of limits that Malthus had in mind. According to British bioethicist John Harris, who for no good reason objects to being called a transhumanist,\n\nIt is doubtful that there was ever a time in which we ape-descended persons were not striving for enhancement, trying to do things better and to better ourselves. . . . I am personally pleased that our ape ancestor lacked either the power or the imagination, or indeed avoided the errors of logic and\/or morality, which might have led her to preserve herself at our expense. I hope that we will have the imagination, the power, and the courage to do better for ourselves and our descendants than the combination of chance, genes, and environment has done for us.\n\nWhile Harris frames the issue as a choice, it is not clear why we are any more free to make it than our ape ancestress, given his initial assumption about what it means to be human\u2014that we have always been striving for enhancement. Starting from this definition, the transhumanists are simply doing what human beings have always done. So just as history tells the story of cultures gradually gaining the wisdom to accept as humans people who come in all kinds of shapes, sizes, abilities, and colors, we ought to be prepared to extend that circle of identification and accept further such variance so long as the essential attribute of self-overcoming is still present. Transhumanist philosopher Nick Bostrom has suggested that even if they are not human, transhumans and posthumans may yet be \"humane\"\u2014that is, they may possess recognizably admirable human moral qualities. Indeed, they may well be more likely than humans to be humane, since they will have overcome the human or natural constraints that produce bad behavior among humans.\n\nYet by calling attention to this self-overcoming aspect of human nature, the transhumanists create another kind of problem within their argument. Starting from their assumptions, evolution is the only way in which human beings could have gained their ability to overcome natural limits; that we have this capacity is ultimately a matter of chance. The ability to control our own evolution going forward, then, has at least this much of an irreducible foundation in natural evolution. That foundation would seem to represent a limit on what parts of our nature we would seek to modify, lest we compromise this all-too-human drive to self-overcoming. Or, to describe this paradox in another way, our descendants would only genuinely be \"posthuman\" if they reached the point where they were no longer engaged in the self-overcoming that is our presumed essence. Whether such a result would ever be desirable is unclear, unless lurking somewhere in the transhumanist argument, as it did in Drexler's, there is a tacit belief that at some future moment our descendants will have reached some limit of the possible with respect to what can be done to themselves. Only at that point would this drive for enhancement become counterproductive and have to be rooted out at last. Short of this outcome, a remnant of our nature remains in whatever kind of beings come after us.\n\n_GUARANTEED RETURNS?_\n\nCompetition combines with the challenging of Malthusian limits to give the transhumanists a very strong sense that they represent the direction of history. Ray Kurzweil has taken the argument one step further, giving it a cosmic twist (like Drexler and Reade before him). Kurzweil calls the force that drives the progress of inhumanity the \"law of accelerating returns.\" As its name implies, this law encapsulates the idea of an accelerating rate of progress, but Kurzweil claims that he can show the trend line is an asymptotic curve, rapidly accelerating today toward an unreachable future infinite. Something like this law, he seems to believe, has been at work extending even into the deep past, long before the rise of humanity. If Kurzweil is correct, then any appearance of choice we might have with respect to a posthuman future is illusory, and the might of the law behind this trend line is the ultimate source of the rightness of the future he describes. (So Kurzweil, like CryptNet in _The Diamond Age_ , confuses \"inevitability with Right.\") Why bother advocating for transhumanism at all, if there is no reason to fear that transhumanism's critics might win out, as the law dictates their irrelevance? Advocacy for transhumanism must instead be predicated on concern that without a sufficiently solid understanding of what the future holds, the powers we are gaining could be misdirected in destructive ways. In other words, instead of deliberate human extinction, we might cause it accidentally.\n\nStill, perhaps the transhuman future need not be quite so dire. David Pearce, a cofounder of the World Transhumanist Association, has what at first glance seems to be a considerably more benign outlook. He returns us to the prosaic, seeing the transhumanist effort to control evolution as being driven by what he claims is a completely rational and undoubtedly moral desire to maximize happiness and minimize suffering and pain:\n\nOver the next thousand years or so, the biological substrates of suffering will be eradicated completely. \"Physical\" and \"mental\" pain alike are destined to disappear into evolutionary history. The biochemistry of everyday discontents will be genetically phased out too. Malaise will be replaced by the biochemistry of bliss. Matter and energy will be sculpted into life-loving super-beings animated by gradients of well-being. The states of mind of our descendants are likely to be incomprehensibly diverse by comparison with today. Yet all will share at least one common feature: a sublime and all-pervasive happiness.\n\nWe would of course first make every effort to get our own house in order, making sure that human beings are all happy all the time, but there is also pain and suffering in other parts of the animal kingdom, and eventually we would face up to the moral necessity of dealing with that by the appropriate changes in genes and ecosystems to put an end to animal predation. Pearce goes on to consider what happens when we discover life on other worlds, and has no compunctions about extending his principle, and hence posthuman efforts, in that direction. \"So long as sentient beings suffer extraordinary unpleasantness\u2014whether on Earth or perhaps elsewhere\u2014there is a presumptive case to eradicate such suffering wherever it is found.\" When the posthuman Earth fleet arrives on an alien doorstep, it is only with the most benign and unselfish intention of completely restructuring their world so that all its beings can share in such happiness as we make possible for them. In Pearce's account of the posthuman future, our descendants become the aliens we have feared or hoped for. Yet if we imagine an alien fleet arriving on Earth with the same intention of rebuilding nature from the ground up, we are entitled to wonder if it would be taken as good news.\n\n_OLD WINE IN NEW BOTTLES_\n\nMeanwhile, the roboticist Hans Moravec has a different take on the transhuman future. As an alternative to Bernal's idea of transplanting an organic brain into a mechanical body, but based on a similar view of the primacy of the brain, Moravec seems to have been the first to think out in a systematic way what would be involved in \"uploading\" a mind into a computer. As he explains it in his classic 1988 book _Mind Children_ , Moravec's premise is the same idea of \"pattern identity\" that Drexler spoke of; who we are is fundamentally a matter of the processes going on in our brain, the electrochemical activity among our neurons, not the stuff itself; \"the rest is mere jelly.\" There is of course a huge number of neurons with an even larger quantity of interconnections among them, so this electrochemical activity is extremely complex\u2014but not infinitely so. Sooner or later, Moravec argues, as computing power grows there will be some ultra-sophisticated version of today's brain scanners that will make it possible to scan neuron by neuron what is going on in the brain, and store and duplicate those results in some combination of software and hardware simulation.\n\nMoravec assumes that at least at first, efforts at uploading would have to destroy the brain bit by bit in order to expose each neuron for scanning. He asks us to consider someone hooked up to a device that would read and replicate each neuron. Suppose that as each cellular layer is scanned, the subject could switch back and forth between the organic brain and the simulation; if the system works correctly he could not tell the difference, so that each layer could be destroyed and the next scanned. As the process continued, more and more of the subject's thinking would be going on in the computer, until finally it would completely replace his old brain.\n\nMoravec imagines that the brain thus computerized could, in the manner anticipated by Bernal, be combined with any sort of robotic body one wished. His personal favorite seems to have been the \"bush robot,\" which is little more than a collection of arms and manipulators arranged in a fractal pattern, from the big to the very tiny. He described it thus:\n\nA bush robot would be a marvel of surrealism to behold. Despite its structural resemblance to many living things, it would be unlike anything yet seen on earth. Its great intelligence, superb coordination, astronomical speed, and enormous sensitivity to its environment would enable it to constantly do something surprising, at the same time maintaining a perpetual gracefulness. . . . A trillion-limbed device, with a brain to match, is an entirely different order of being. Add to this the ability to fragment into a cloud of coordinated tiny fliers, and the laws of physics will seem to melt in the face of intention and will. As with no magician that ever was, impossible things will simply _happen_ around a robot bush.\n\nThe bush robot was not the end of his imaginings. Computerized minds could communicate with and access each other with all the ease of networked hardware and software. They could be duplicated, or optimized to run in various configurations. Beyond that, just as one might merge electronic documents, why not merge one electronic mind pattern with another? Since any brain could be scanned, why shouldn't a human pattern be merged with an animal pattern?\n\nMoravec points out that this computerized mind could be effectively immortal (assuming sufficient provision for backups), which might seem like a very appealing prospect. But he recognizes that this immortality might not end up being precisely what someone might have originally hoped for. Competition and scarcity (\"changing conditions,\" \"external challenges\") mean that the I that sought immortality will not actually get it:\n\nImmortality of the type I have just described is only a temporary defense against the wanton loss of knowledge and function that is the worst aspect of personal death. In the long run, our survival will require changes that are not of our own choosing. Parts of us will have to be discarded and replaced by new parts to keep in step with changing conditions and evolving competitors. . . . In time, each of us will be a completely changed being, shaped more by external challenges than by our own desires. Our present memories and interests, having lost their relevance, will at best end up in a dusty archive, perhaps to be consulted once in a long while by a historian. Personal death as we know it differs from this inevitability only in its relative abruptness. Viewed this way, personal immortality by mind transplant is a technique whose primary benefit is to temporarily coddle the sensibility and sentimentality of individual humans.\n\nNote how the desire for personal immortality is such a small thing from Moravec's point of view, a sentiment to be coddled, a kind of temporary error. Here is surely Olympian detachment! It is as if the gods in the full enjoyment of their immortality would wonder why mortals make such a big deal about death. And yet somehow this small thing drives us on to great things.\n\nGiven the speed with which computing technology developed since Moravec first published his thoughts on uploading in 1988, it did not take long for the idea to be modified to conform to new ideas and capacities. The rise of virtual reality made some wonder why the uploaded mind would require a physical instantiation at all when it could occupy whatever kind of virtual realities it wished. For those who might want to insist on continuing to be able to interact in the \"real world\" but find robotic bodies too constraining, the theoretical nanotech \"smart dust\" provides an imagined solution, promising \"real\" virtual bodies that would have none of the messy inconveniences of actual living things\u2014very much along the lines that Fedorov imagined. And there is the added benefit that were such bodies ever to be disrupted in some way, they could always simply be rebooted.\n\n_NEW WINE IN NEW BOTTLES_\n\nIn Moravec's presentation, transhumanist aspirations do not end with the (somewhat deceptive) promise of personal immortality. Once biologically based human patterns are instantiated in computers, they will as we have already noted become upgradeable in all kinds of ways. In particular, they will be able to think faster. Hence, just as a chess-playing computer can today consider more options in greater depth and shorter time than a human player, so will the formerly human patterns be able to think about and experience far more than a human being\u2014and this without fatigue, without forgetfulness, without the distractions of biological needs. Here is surely a great boost to progress in any area to which these posthumans may wish to turn their attention. Problems will be modeled, solutions simulated, results refined in an interactive cycle of astonishing speed and complexity. Such intelligence, alone or in concert with advanced artificial intelligence, would constitute what is sometimes called \"superintelligence\" or \"hyperintelligence.\" Its arrival, as Drexler suggested and as many others now believe, would be just the sort of thing that would rapidly push technological development to the limits of the possible. As Bostrom has speculated, \"Superintelligence would be the last invention biological man would ever need to make, since, by definition, it would be much better at inventing than we are.\"\n\nWhat exactly happens to humanity at that moment is an important and to some degree divisive issue for transhumanism. The issue is framed by the concept of \"the Singularity.\" While the very meaning of this term is contested, it is fair to think about it in the way it was first systematically presented by mathematician and science fiction author Vernor Vinge:\n\nWhen greater-than-human intelligence drives progress, that progress will be much more rapid. In fact, there seems no reason why progress itself would not involve the creation of still more intelligent entities\u2014on a still-shorter time scale. The best analogy that I see is with the evolutionary past: Animals can adapt to problems and make inventions, but often no faster than natural selection can do its work\u2014the world acts as its own simulator in the case of natural selection. We humans have the ability to internalize the world and conduct \"what if's\" in our heads; we can solve many problems thousands of times faster than natural selection. Now, by creating the means to execute those simulations at much higher speeds, we are entering a regime as radically different from our human past as we humans are from the lower animals.\n\nFrom the human point of view this change will be a throwing away of all the previous rules, perhaps in the blink of an eye, an exponential runaway beyond any hope of control. Developments that before were thought might only happen in \"a million years\" (if ever) will likely happen in the next century. . . . I think it's fair to call this event a singularity (\"the Singularity\" for the purposes of this paper). It is a point where our models must be discarded and a new reality rules. As we move closer and closer to this point, it will loom vaster and vaster over human affairs till the notion becomes commonplace. Yet when it finally happens it may still be a great surprise and a greater unknown.\n\nIf this hyperintelligence, like human intelligence, is routinely dissatisfied with its own capacities, it will surely turn its attention to finding the matter and energy for the ever-increasing computer processing power it will presumably demand. As Moravec puts it, \"Our speculation ends in a supercivilization . . . constantly improving and extending itself, spreading outward from the sun, converting nonlife into mind.\" Matter and energy are not in short supply; what already exists just needs to be used to some purpose. Think of all the mass wasted in asteroids, moons, lifeless planets, or indeed in organic bodies of all sorts that do not need to live in biological form when they could just be simulated. Think of all the radiation that streams pointlessly into space from stars, but could be captured by enclosing them in gigantic spheres. Just as, from the transhumanist point of view, the human body is sub-optimally designed, so too there may be a great deal of work to be done to reconstruct our own and other solar systems to make them happier homes for intelligence. Whatever hyperintelligence is in its own way is, like the human mind, simply a pattern of information, a pattern that can be replicated in a variety of forms. The deconstruction of our solar system to turn dead matter into mind, and any other solar system hyperintelligent posthumanity may be moved to encounter, would not have to mean the loss of the information it represents. It was J. D. Bernal who said that \"the desirable form of the humanly-controlled universe\" is \"nothing more nor less than art\"\u2014but in the vision of today's transhumanists, that desirable form is not, as it turns out, actually controlled by humans.\n\n_WHEN HYPERINTELLIGENCE COMES KNOCKING_\n\nRay Kurzweil takes the argument about the capacities of hyperintelligent posthumans one step further. He wonders whether our understanding of the constraints imposed on us by the laws of nature may just be an artifact of our limited intellectual abilities. For the sake of \"saturating\" the universe with intelligence, which he (like Haldane) believes is \"our ultimate fate,\" Kurzweil hopes that the speed of light may not after all prove to be a cosmic speed limit. Along with Moravec he would like to believe that efforts will be made to overcome the entropy that will eventually cause the universe to \"run down.\"\n\nThese remarkable aspirations clearly connect back with issues we already looked at in connection with SETI. If Kurzweil is correct about the potential abilities of hyperintelligent posthumans, then would the same not also be true of the hyperintelligent form of some alien race? If so, then Fermi's question arises once again\u2014 _where are they?_ \u2014and we probably have to have recourse to the same kinds of arguments already noted about our decidedly limited ability to detect the existence of such godlike beings, particularly if they wish to remain undetected. Then again, even less-capable aliens bound by the speed of light would still have had plenty of time to populate the galaxy, or use its mass and energy for computational purposes in a way that we might detect. As we saw in Chapter Two, such arguments make Kurzweil and some other transhumanists skeptical that there is alien intelligence at all.\n\nAnd perhaps we should be happy about that. If we take posthuman hyperintelligence and the Singularity seriously, we are saying that the successors we are being asked to open the door to will be alien to us far beyond any change possible in natural evolution. Vinge from the start understood that one of the greatest unknowns would be how hyperintelligent beings would treat human beings. Would we be seen as revered progenitors or as inconvenient pests? Such a question is hardly new to science fiction, of course, but now there are no few transhumanists and AI proponents who think that the question of \"friendly\" or \"moral\" AI needs to be brought to the fore of serious, real-world efforts. What \"friendly\" or safe or moral AI would look like is anybody's guess, but it is very hard to see why the net result would not look very much like _Childhood's End_ , given the radical lack of a common framework of judgment between us and hyperintelligent life. Kurzweil likens it to the gap between us and bacteria, and yet no one claims seriously that we share a common moral universe with bacteria. Hence the transformation of supposed hyperintelligent benevolence into perceived malevolence remains a live issue. It does not seem likely that these otherwise incomprehensible beings will be more moral than we are in terms that we could appreciate.\n\nSo if this kind of posthuman hyperintelligence were to arrive on our doorsteps tomorrow, it is hard to see how it would look different from a hostile alien invasion of the sort classically depicted in science fiction. Separated from them by their Singularity (forget about mere biological or cultural differences), we would be incapable of understanding their needs and motivations beyond noting that they seemed to have an insatiable appetite for transforming matter and energy into forms useful to them but inimical to us. The opportunity to be absorbed (assimilated?) into whatever mode of life they will have created for themselves, to have our patterns preserved, would seem to be about the most we might expect.\n\n_TRANSHUMANIST TENSIONS_\n\nThere are genuine tensions between various aspects of the case for transhumanism that we have highlighted, not the least of which is that something that begins by promising human enhancement ends up more than reconciled to human extinction. More specifically: on the one hand, the motive force for transforming ourselves is a deep dissatisfaction with the merely human. On the other hand, this dissatisfaction, and the efforts at transformation it produces, are presented as quintessentially human. We are all to be free to make choices about enhancements, but radical inequalities may lead some to be freer than others. To raise our hopes, the transhumanists urge overcoming human limitations. To quiet our fears, they claim there could be continuities between us and the superior beings to come. Specific human characteristics are treated as if they are merely contingent with respect to all the alternative ways we can imagine intelligence might be embodied, but however embodied we are told there is no reason to think that intelligence could not still be humane. Yet even the continuity of the humane gives way to the absolute discontinuity of the Singularity.\n\nThese tensions mean there are reasons to have qualms about the transhuman and posthuman future despite the fact that it can seem to be derived from minimally controversial premises. The prosaic best case does not quite tell the whole story, but is more like the proverbial slowly warming pan of water in which the frog sits unaware of the boiling point to come. By thinking through the assumptions of the eclipse of man all the way through to the Singularity\u2014to a point where it must be acknowledged that none of us knows or could even understand the full implications of the path we are set upon\u2014today's transhumanists have clarified the full implication of those assumptions even more than the earlier thinkers we examined. But that does not mean they have thought them through completely.\n\nIn the prologue to this chapter, the character of Adam Newman has not been dehumanized\u2014or at least, not obviously so. But his story exemplifies some problems to come, should transhumanist aspirations prove possible. We don't know that talents and dispositions can be engineered in the way implied in the vignette, but on the assumption that they can be, is Adam the freely choosing transhumanist ideal? He seems very much an instrument of his parents' wishes for him, far more than is possible for even the most controlling parents today. There are hints that he is unable really to question that legacy; as a result he is less of a free man and self-overcomer than he himself might believe. We may envy the ease with which he falls into a role that is perfect for him and the happiness it provides him, and still wonder whether this lack of tension accounts for his emotional flatness. We can wonder if his musicianship represents the triumph of technique, with Adam occupying something like the position of a racecar driver, in effect the business end of a highly tuned team of technicians who have made him what he is. Yet he probably has less understanding of what they do and how they do it than the driver does.\n\nThen there is the character of Daphne Morgan. She is not so dehumanized that we cannot suspect her of being a trickster or a tease, or of having the sort of contempt for \"solids\" that bohemians have for bourgeois. But if her own self-regard makes her no more humane than any morally fallible human being, surely she has far greater power and even motivation to exercise her inhumanity, to sport with Adam as the old gods used to sport with humanity. Surely Adam's self-satisfied narcissism would make him a tempting target. Yet his sense that there is more going on with her than he is likely to be able to understand is probably correct. How would Adam understand what it is like to be in more than one place at a time, or to have all his consciousness be as editable, reversible, and replayable as the recordings he makes?\n\nLooking at such distant prospects, which is after all encouraged by the transhumanists themselves, reveals important problems with their arguments. And yet it could be objected that even if the transhumanists are correct about what the future holds, any problems associated with it will not be _our_ problems\u2014or at least will be experienced only by the relative few reading these words today who will have made provision for their own reanimation. For most of us, as technology progresses incrementally, the issues are going to be posed in terms of far more prosaic-seeming technologies of human enhancement, technologies that may look attractive whether or not there is an obvious link to the transhumanist aspiration for self-design. Hence, while there is no lack of thoughtful explorations of more extreme transhuman futures in science fiction (some of which we have already discussed in our examination of the \"benevolent\" superintelligence in _Childhood's End_ ), it makes sense to explore instead a still speculative, but not that speculative, nearer-term development: the ability selectively to erase unhappy memories. Difficulties with this relatively modest kind of enhancement will alert us to difficulties inherent in the more radical alterations transhumanism proposes.\n\n_NO THANKS FOR THE MEMORIES_\n\nAs imagined biotechnologies go, memory alteration seems not quite in the same league as things like the transhumanist hopes for direct links between brains and computers, or wireless connections between brains that would allow people to feel each other's feelings, or the ability to \"upload\" into a machine the entire pattern of our identity. But it is worth considering memory alteration closely, since memory is central to identity, and identity to happiness\u2014a point highlighted in _Beyond Therapy_ , a 2003 report of the President's Council on Bioethics. So if we are to make ourselves better off by the use of emerging enhancement technologies, or if in accord with Pearce's hedonic imperative we are to be happy all the time, memory and memory alteration in some form will be at the core of our various pursuits of happiness.\n\n_Beyond Therapy_ focuses in a general way on the use of drugs to head off the impact of likely traumatic memories, or reducing their subsequent emotional consequences. Grounded in present medical realities, the report does not speculate at any length on the consequences of developing less blunt approaches to adding or subtracting specific memories as we understand the neuro-chemistry of the brain with increasing precision. While it is always easy to understate the difficulty of such efforts, there is nothing in our present understanding of the brain that definitively shows that precise interventions in memory will be impossible.\n\nThe consequences of selective forgetting have been a literary theme since long before science become seriously interested in the topic; think of magic potions in tales ranging from Homer to Wagner, or the elaborate deception practiced on the title character in Ermanno Wolf-Ferrari's opera _Sly_. Here we will look at two stories about memory alteration. The first is _Dr. Heidenhoff's Process_ by Edward Bellamy, who would some years after writing it become internationally famous for his utopian novel _Looking Backward_. The second is the movie _Eternal Sunshine of the Spotless Mind_ , which provides a corrective to the generally positive position taken on memory alteration by Bellamy.\n\nBellamy's 1880 novella _Dr. Heidenhoff's Process_ is hardly great literature, but it nevertheless contains a well-thought-out moral defense of selective memory obliteration. If the story is Victorian in its operatic sentimentality, its effort to think through the moral consequences of Dr. Heidenhoff's process for elimination of troublesome memories exhibits likewise a Victorian intellectual thoroughness.\n\nThe plot is simple enough. Henry Burr loves Madeline Brand unrequitedly. Just as it seems she might be returning his feelings, the flashier Harrison Cordis comes to town, and Madeline and Harrison begin a flirtation. As their relationship deepens, Henry leaves town to work in Boston, unable to bear the loss of Madeline. However, \"there was a scandal\" and Harrison \"deserted her.\" A fallen woman, Madeline flees to Boston. Henry finds her and offers marriage, but in her shame she keeps putting him off; she cannot ruin him as she has been ruined. One day, she seems to relent, and even allows Henry a not-so-chaste kiss prior to their parting, promising to settle the matter the next day.\n\nHenry returns to find her terribly excited; she has seen a story in the newspaper about a technique invented by a certain Dr. Heidenhoff for the purpose of selective memory destruction. Madeline is anxious to have the memory of her shame destroyed so she can marry Henry with a clear conscience. Skeptical at first, Henry visits Heidenhoff and comes away convinced that Madeline should indeed try the process; \"the thought of receiving his wife to his arms as fresh and virgin in heart and memory as when her girlish beauty first entranced him, was very sweet to his imagination.\" The procedure works, the couple are engaged\u2014but at the moment of seeing Madeline model her wedding gown, Henry wakens to find he has been dreaming. He almost immediately receives Madeline's suicide note, in which she repeats how she could not dream of ruining him, and gently chides him for his failure to perceive that the kiss she at last permitted him was one of farewell.\n\n_THE SPONGE OF OBLIVION_\n\nDr. Heidenhoff is no shrinking violet when it comes to explaining and defending his invention. In the full enjoyment of what he himself regards as the powers of a god, Heidenhoff is only too happy to explain the medical basis for his process; his hopes for \"amusing experiments\" that will confound the \"thicker-headed\" and \"foggy-minded,\" whose false moral assumptions will be exposed; and the \"good tidings\" that his invention brings the world.\n\nIndeed, Heidenhoff is so obnoxiously full of himself that it is tempting to think that Bellamy might be interested in exposing scientific hubris. Yet while that possibility cannot be entirely eliminated, it seems unlikely given the fact that the story is framed by two suicides stemming from their perpetrators' inability to forget shameful events. _Dr. Heidenhoff's Process_ suggests that neither love nor Christianity nor conventional moral ideals are adequate to the problems created for us by the memory of the misdeeds we do or those done to us. Bellamy evidently had some hope that science could step into the breach.\n\nDr. Heidenhoff believes that memory has a physical basis in the brain\u2014hardly a controversial proposition today. He admits that, for the time being, his process, which applies electricity to the brain, can only extirpate memories that have been so dwelled upon as to create \"a morbid state of the brain fibres concerned.\" But he expects that in twenty years, having established \"the great fact of the physical basis of the intellect,\" \"the mental physician will be able to extract a specific recollection from the memory as readily as the dentist pulls a tooth, and as finally.\"\n\nHeidenhoff acknowledges some further problems with his process, admitting, for example, that there may be \"shreds and fragments of ideas, as well as facts in his external relations\" which are unaccountable without the eliminated memory. But these things are no more a problem than the challenge faced by a drunk or a sleepwalker who has done things he no longer remembers. Heidenhoff also acknowledges that the patient will know that he forgot _something_ , which may create \"slight confusion.\" We see both problems arise in Madeline's case, and observe her filling in the blanks by creating plausible stories to account for matters which otherwise do not make sense to her.\n\nDr. Heidenhoff believes that his process has wide-ranging moral implications. On Henry's first visit the doctor notes that he finds patients generally come to him out of\n\nvery genuine and profound regret and sorrow for the act they wish to forget. They have already repented it, and according to every theory of moral accountability, I believe it is held that repentance balances the moral accounts. My process, you see then, only completes physically what is already done morally.\n\nIn this respect, he claims superiority to the \"ministers and moralists\" who preach forgiveness, but in fact leave the penitent subject to the ongoing tyranny of \"remorse and shame.\" Indeed, Heidenhoff notes, it is the most sensitive moral natures, those precisely who most deserve that their repentance relieve them of their misery, who are most likely to be tormented by the memory of their wrongdoing:\n\nThe deeper the repentance . . . the more poignant the pang of regret and the sense of irreparable loss. There is no sense, no end, no use, in this law which increases the severity of the punishment as the victim grows in innocency.\n\nIn contrast, \"I free him from his sin\" entirely.\n\nHeidenhoff is not content to beat religion at its own game; he believes that understanding the physical basis for memory and intellect, hence of all human actions, leads to the need for moral revaluation even beyond the specific consequences of his own technique. Some people, he says, may mistakenly think that sin must have painful consequences, whether physical or mental. They may believe that Nature is supposed to punish physical \"vice and violence\" with \"diseases and accidents,\" and God is supposed to provide \"moral retribution\" via shame and sorrow. But in fact, both forms of retribution are essentially \"blind, deaf and meaningless.\" Nature does not care and Providence is unreliable, so human beings must step in. Just as no doctor would refuse to set the broken leg of a drunk, just to make an example of him, so we should be equally willing to relieve mental suffering by forgetfulness.\n\nHeidenhoff holds that because all suffering ultimately has the same physical basis, all suffering can and ought to be relieved. But in a manner familiar to arguments today about the difficulty of distinguishing between curing disease and enhancement, Heidenhoff's assumptions lead him to speculate that the world might be better off \"if there were no memory\" at all. For without memory, we would still have \"congenitally good and bad dispositions,\" but the bad would not, as they do, grow \"depraved\" by the demoralizing effects of memory. \"Memory is the principle of moral degeneration. Remembered sin is the most utterly diabolical influence in the universe. . . . more sin is the only anodyne for sin, and . . . the only way to cure the ache of conscience is to harden it.\"\n\nHeidenhoff's apparent insistence on the objective existence of sin will doubtless strike oddest those who most share his materialist assumptions, but in response to an objection by Henry, he develops his argument in a fashion that suggests it is not sin which is the problem, but the _idea_ of sin. For Henry argues that even without memory, people will still be just as responsible for their acts as with it. Heidenhoff agrees, but with a twist: \"Precisely; that is, not at all.\" For \"human beings are not stationary existences, but changing, growing, incessantly progressive organisms, which in no two moments are the same.\" Hence, while crimes can be punished for prudential reasons\u2014as a matter of \"public policy and expediency\"\u2014they can never be punished _justly_ , for it is never just to punish people for what they cannot help. Why should I be punished for what I did yesterday, when I am not the same person who I was yesterday, and the person I am today has no control over what the person I was yesterday did?\n\nIn short, for Heidenhoff the self only exists moment to moment; by disconnecting the present from past and future the self becomes extremely thin. The moral implications of this thinness are clarified when Henry points out what he believes to be a flaw in Heidenhoff's argument. A person holding Dr. Heidenhoff's views would never need to erase any memories, since such a person would never feel guilty or remorseful about an act committed by an earlier version of himself. The doctor agrees that his process is indeed only for those who do not have the strength of mind independently to \"attain\" his philosophy. In fact, we are born anew each day, each moment. \"Is there not sorrow and wrong enough in the present world without having moralists teach us that it is our duty to perpetuate all our past sins and shames in the multiplying mirror of memory,\" the doctor exclaims. Just as it is \"only fools who flatter themselves on their past virtues, so it is only a sadder sort of fools who plague themselves for their past faults.\"\n\nIn sum, Dr. Heidenhoff's process reveals that when we understand the true physical basis for memory, hence for human thought and action, we see how we make two errors when we fail to forgive ourselves or others: the failure to forget an irrelevant past and the failure to realize there is no continuous self to grant or receive forgiveness. That our conventional moral precepts are not based on these truths has the perverse consequence of making us either more vicious or less happy.\n\n_MEMORY AND MORALITY_\n\nDoes Bellamy intend that we should test the adequacy of Heidenhoff's morality? For surely from the start we can wonder whether, as Heidenhoff believes, his process really exposes serious confusion in our existing moral intuitions on their own terms. Is there really a weak moral case for punishing a person who has forgotten his crime? For example, when someone in a drunken stupor does not recall the hit-and-run accident, we do not only punish him in order to deter others but because there has been a wrong done to the victim that exists whether or not the perpetrator recalls it. Or again, if you were told that someone who gravely wronged you could not be punished because he had his memory erased, it is not obviously unreasonable to be concerned both about the injustice of getting away with a wrong and the further injustice that the wrongdoer is allowed to forget it. Heidenhoff has not shown that such responses are confused in themselves; they are only confused if we are prepared to join him in rejecting the continuity of our personality, and hence our moral responsibility for past deeds.\n\nBut on this very point there would seem to be confusion on Dr. Heidenhoff's side in the matter of changing personality versus settled dispositions. Madeline clearly has a flirtatious and self-willed disposition. If all memory of her errors created by that disposition has been sponged away so that she cannot become, as the saying goes, \"sadder but wiser\" through them, surely she will constantly repeat them; being born anew will not free her from anything but rather would be the constant return of the same. On the other hand, Harrison Cordis, the man who wronged her, could be said to have attained the heights of Heidenhoff's \"that-was-yesterday\" philosophy, and is arguably the freer to indulge his dispositional lusts\u2014his freedom to act in accordance with his dispositions is hardly a moral improvement. The moral wrong of Harrison Cordis's seduction of women would surely not be reduced even if the memory of the seduction can be wiped away.\n\nFinally, take the case of Henry, who acts rather nobly, even if self-interestedly, in seeking the fallen Madeline out and offering her the shelter of wedlock. If Heidenhoff is correct and the memory of wrongdoing contributes to the worsening of bad dispositions, does memory of good deeds like Henry's also contribute to the improvement of good dispositions? Heidenhoff seems to look forward to a world where there would be no need to comfort the afflicted, hence a world with reduced opportunity for the development of other-regarding virtues.\n\n_BRAIN-DAMAGED GOODS_\n\nBeyond the problems we have just noted with Heidenhoff's argument, the 2004 film _Eternal Sunshine of the Spotless Mind_ is nearly a point-for-point reexamination of some of his key premises, suggesting further grounds for skepticism about their adequacy. The story is not complex, but the movie presents it in a particularly thought-provoking way, such that the viewer only gradually comes to understand the sequence of events. The summary that follows, therefore, is very much a spoiler.\n\nJoel and Clementine (played by Jim Carrey and Kate Winslet) meet, fall in love, and then fall out of love. To forget their painful breakup, Clementine goes to Dr. Mierzwiak (Tom Wilkinson) to have her memory of Joel erased; when Joel finds out what she has done he attempts to do the same. An underling of Mierzwiak's named Patrick (Elijah Wood) falls for Clementine when he participates in her memory erasure. Mierzwiak's process, as we will see, gives him access to Joel's diaries and other effects, and Patrick uses this information about Clementine to court her. She finds his efforts confusing and increasingly disturbing for reasons we will speculate about later. Mierzwiak's secretary Mary (Kirsten Dunst) and his main technician Stan (Mark Ruffalo) are likewise involved, despite the fact that Mary carries a torch for Mierzwiak.\n\nThe situation comes to a head during the procedure to delete Joel's memory. Patrick goes off to be with Clementine, allowing Stan and Mary to get stoned and have sex while waiting for Dr. Mierzwiak's process to do its work on Joel. Midway through the procedure, Joel changes his mind about erasing his memories of Clementine, and we see inside his head his heroic mental efforts to try to preserve them. Stan notices that things are going wrong, and calls in Mierzwiak to solve the problems created by Joel's resistance. At the end of a hard night of erasing Joel's memories, Dr. Mierzwiak and Mary kiss, observed by the doctor's suspicious wife (Deirdre O'Connell)\u2014who reveals when she confronts him outside Joel's apartment that Mary and Mierzwiak have already had an affair. Mierzwiak is forced to admit to Mary that she had agreed to have her memories of their affair erased. Angered, Mary mails Mierzwiak's patients their files. Joel and Clementine, who by chance or fate have met again and are falling for each other again, listen to the tapes detailing their previous unhappiness with each other. The movie ends as they are deciding to give their relationship another try.\n\nWe begin by observing that Mierzwiak tries to overcome what Heidenhoff acknowledged as imperfections in his process. To reduce confusion created by waking up in a doctor's office, the procedure is done in the patient's home at night. There will be no memory of having a memory erased. Mierzwiak also has the patient purge his life of any material reminders of whoever or whatever is to be erased, which is how Patrick gets his hands on Joel's Clementine memorabilia. Finally, to reduce further the chance for confusion based on forgotten events, Mierzwiak sends out postcards to inform the patient's associates not to talk about what the patient had erased.\n\nYet the movie demonstrates several plausible ways in which the process remains imperfect (after all, \"technically, the procedure itself is brain damage\"). For example, we can understand why Joel would turn in a coffee mug with Clementine's picture on it, but how does he get away with not getting rid of the foldout bed that figures prominently in so many of his memories of her? Another problem created by the association among memories works the other way. Joel loses all memory of the cartoon character Huckleberry Hound apparently due to the association with the song \"Darling Clementine.\" Or again, he is evidently a bit confused by waking up in the new pajamas that he bought for the procedure, but does not give them much thought. More troubling is the car damage he discovers the morning after the procedure. Clementine crashed it before their last fight, but, failing to recall that, Joel must construct a plausible story for himself that accounts for the now-unexpected dents. So he is upset but not confused.\n\nAll these examples fade in significance compared with Joel's discovery, inveterate diarist that he is, that two years of diary entries are missing. Surely it is only because he is so emotionally withdrawn in the first place that he is not more consciously concerned to understand this inexplicable loss. By contrast, the acute distress which Clementine increasingly feels, while doubtless exacerbated by Patrick's mimicking of Joel, shows another possible result of losing memories, more or less incompletely, without conscious awareness that one has lost anything at all. In short, there does not seem to be a consistent ability to predict what memories and associations will create problems when lost, or what exactly will be lost\u2014hardly surprising given the rich associational and emotional character of our memories. Joel and Clementine have shared the _world_ , and they cannot turn the whole thing over to Mierzwiak. Perhaps it is the fact of such ongoing associations along with the habits created by their dispositions that draws the two of them back together even after their memories of each other have been erased.\n\n_DOCTOR KNOWS BEST_\n\nThe efforts taken to make the patient forget that he has forgotten something may therefore be of only limited utility from the point of view of the patient, but they _certainly_ are useful to Dr. Mierzwiak. We see one hint of how when we find out that the \"don't ask about . . .\" postcards about Clementine's procedure are sent to everybody but Joel. Had Joel (as intended) remained in the dark about Clementine's decision he would not have stormed into Mierzwiak's office as he did when he found out. In like manner, if Clementine grows increasingly unhappy and confused as Patrick courts her, she will not see herself as an unhappy _return_ customer in Mierzwiak's office, although she may see him again as a \"new\" customer, unhappy for reasons she cannot connect to him. In short, Mierzwiak's improvements, while nominally for the good of the patient, in fact insulate _him_ , to the extent they work, from any bad consequences of his procedure. The postcards even serve the additional useful function of providing advertising when, of necessity, there can't be direct word-of-mouth.\n\nMierzwiak has arranged his process so that as much as possible he will not be confronted by the problematic consequences of his work. The irresponsibility inherent in his methods is reflected yet more clearly in the behavior of his underlings. Drinking, drugs, theft, and perhaps sex seem to be normal activities as they \"monitor\" the memory-erasing procedure. The forgetfulness they induce in others allows them to forget themselves, professionally speaking; they can satisfy their transitory desires with impunity. Now of course, in any care-giving situation there is a possibility for abuse such as this, and presumably it only increases when the patient is unconscious. But Mierzwiak's employees are further liberated by the fact that those from whom they steal, for example, will not even remember they were unconscious and under treatment. Joel finds he has less whiskey than he expected in the final moments of the film\u2014but even knowing what he knows by then he does not connect it with the procedure.\n\nThe technical abilities and limits of Dr. Mierzwiak's process provide fresh opportunities for human failings, which brings us closer to the problems the film suggests with the moral case that Heidenhoff makes for memory erasure. It is easy to connect the irresponsible behaviors we have been looking at with Heidenhoff's denial of any continuity of self that would justify moral responsibility. Just about everybody in the movie has the very thin sense of self that Heidenhoff believes is a great moral achievement. Stan's stock humor is the quick personality shift from brusque professionalism to clownishness. The colorless Patrick readily adopts Joel's persona when given a chance, turning it on and off as needed. Mary gets her opinions from the multi-voiced _Bartlett's Familiar Quotations_. Clementine laments that she gets her personality out of the pastes for her ever-changing hair color. These thin identities are what it means to \"attain\" to Heidenhoff's philosophy of _that was then, this is now_ and _the I of today is not the same as the I of yesterday_.\n\nIf the results are bad all around, Dr. Mierzwiak himself illustrates the dangers best. In the movie he is not played as a mad scientist or evil genius; indeed his fatigued diffidence and calm contrasts favorably with Heidenhoff's preening iconoclasm. Joel barges into Mierzwiak's office, he calmly sees Joel. Mary kisses him, he kisses back. Because his wife Hollis was upset, he reasonably arranges to have Mary's memory erased. (Evidently he could not get Hollis to agree to the same thing.) But Hollis thinks he is a monster, and there is something to her point of view.\n\nWe see how when we think more about how he erased Mary's memories of their previous affair (in the shooting script, there is also an abortion involved). While subjecting her to the treatment was a decision he claims they came to together, we have no reason to think, given how the procedure works, that this claim would have to have been true. And even if it were, the obvious asymmetry in their relationship makes the claim of a mutual decision suspect. He could have employed against her the very fatherly\/professional authority which is clearly part of what attracts her to him.\n\nBut Mary and Mierzwiak's case exposes a yet deeper problem in Heidenhoff's belief that we lack moral responsibility for our pasts while maintaining that we have dispositions that determine the general direction of our behaviors. In what proves to be a particularly poignant speech, Mary articulates the Heidenhoff-like hopes behind Mierzwiak's work. \"To let people begin again. It's beautiful. You look at a baby and it's so pure and so free, so clean. Adults . . . they're like this mess of anger and phobias and sadness . . . hopelessness. And Howard just makes it go away.\" It is true that Howard can make Mary's memories of their affair go away, memories presumably painful mostly in retrospect. But _everything_ does _not_ go away. Instead, he \"frees\" her to fall in love with him again and creates a fresh opportunity for more infidelity. Mary cannot learn from her mistakes; her unhappiness over his unwillingness to leave his family for her will not cause her to grow angry with him, or disillusioned, or to leave her job and move on, because she has forgotten she was unhappy. On the other hand, how satisfying it must be for Mierzwiak to have her prove his power over her by falling in love with him yet again.\n\nThe same dynamic is at work between Joel and Clementine. From a purely prosaic point of view, if all had gone as expected their mutual erasure would simply have opened the door, should they meet again, for them to repeat all of their past mistakes. The same things would attract each about the other, the same things would eventually drive them crazy. Mary's intervention in sending out the patient files opens the door to something new and different in their relationship, but this _deus ex machina_ paradoxically merely restores the possibilities inherent in normal life: \"boy meets girl, boy loses girl, boy gets girl.\" At the movie's end, for Clementine and Joel to reunite they must forgive each other for doing and saying things they have no memory of doing and saying. Granted, that may be an easy kind of forgiveness. But some forgiveness would have been possible had they simply resolved to get together again after a normal breakup\u2014which is not to say they have much of a future in any case.\n\nAgainst this argument that _Eternal Sunshine_ represents the problems created by Heidenhoff's notion of a thin moral self, it could be argued that it is in the process of getting his memory erased that Joel comes to _know_ himself. He attains the Proustian insight that if he truly no longer loved Clementine he would not feel the need to show her he no longer loved her, which is, in his own mind, what he wanted to demonstrate by the erasure. Of course, the effort really makes no sense at all since he cannot demonstrate anything to someone who no longer remembers him; he can only demonstrate it to the Clementine of his memories, who will likewise be gone soon enough. Furthermore, such an insight is hardly useful if he cannot remember it; Joel's already thin self becomes all the thinner by his decision to deal with the problem of Clementine by erasing her memory.\n\nStill, Joel's motive gives the lie to Heidenhoff's claim that patients will come to him out of sincere repentance. We see Joel make his decision on the basis of that combination of anger and sorrow and desire for revenge which is not atypically a result of failed relationships. (Clementine, typically for her, is said to have undertaken the procedure impulsively.) Where Heidenhoff thought no one could want to revenge himself on an ignorant wrongdoer, Joel exhibits animosity not in spite of but _because_ he knows that Clementine no longer has any knowledge of the wrongs he thinks she has done him. Joel's revenge is typically passive-aggressive; we are reminded of how fortunate that is by the disturbing fact that in happier days he and Clementine apparently played murder\/suicide games together, their thin souls toying with complete dissolution. Were Joel's \"disposition\" different, Clementine's choice to undergo Mierzwiak's process could have resulted in tragedy rather than romantic comedy.\n\nFinally, Heidenhoff's own argument about the moral meaninglessness of psychic pain is shown by the movie to lower the bar for what kind of pain is tolerable. Perhaps it is simply moral progress that sexual relations outside of marriage no longer ruin a woman's life. But at least Madeline is portrayed as seeking out drastic measures because her life was indeed, by the standards of the day, ruined. Is it equally progress when one employs such drastic measures for a rather routine breakup such as Joel's and Clementine's\u2014or, as we see with another patient, for the loss of a dog? Or is it rather once again an indicator of the difficulty these thin selves have in finding any meaning in their own actions and choices?\n\nNothing could stand in sharper contrast with the attitudes we see playing out in the movie than the poem from which the film draws its title, Alexander Pope's retelling of the great medieval love story of Peter Abelard and Eloisa in \"Eloisa to Abelard\" (1717). The poet has Eloisa both lament and celebrate the impossibility of her ever forgetting or overcoming her love for Abelard; \"Tis sure the hardest science to forget.\" The nuns around her may enjoy the \"Eternal sunshine of the spotless mind,\" but with her own \"Far other dreams,\" such enjoyment is as impossible for her as it is undesirable. Unlike nearly everyone in _Eternal Sunshine_ , she knows who she is, knows whom she loves, and is willing to live out the consequences to the last.\n\n_SHRUNKEN SELVES_\n\nWhat can we learn from these fictional sources about the moral challenges we may face in a world of transhumanist enhancements? Enthusiastic advocates of enhancements don't like to think seriously about the practical inevitability that the enhancements will work imperfectly, and that those technical imperfections will combine with human imperfections (for we will not make decisions to be enhanced from an enhanced position) to create opportunities for people to take advantage of one another. That this is not a new problem does not mean it is not a serious problem. To the contrary, its persistence over time exposes the utopianism of any who think that the _sine qua non_ for the proper deployment of enhancement technologies is more technology plus the maximization of individual freedom of choice.\n\nThese fictional works remind us that enhancements are not, as techno-libertarianism would have us believe, just a matter of what one individual chooses to do or not do. The choices that become available will arise out of the old world, with all of its unenhanced (or, at later stages, less enhanced) baggage and, as we saw in _The DiamondAge_, the choices will be conditioned by those circumstances. The assumptions behind thinking of some change as an enhancement\u2014why we think it is desirable in the first place, what we think it will tell us about ourselves\u2014will have an effect not just on the enhanced, but on their relationships as well, both with their enhancers and with all others who form a part of their world. From this point of view, to imagine as do the transhumanists nothing beyond the greatest possible freedom to make choices among the widest variety is really a restatement of the morally thin conception of the self. That observation in turn casts light on the transhumanist ideal of endless self-transformation. Enhancement is no longer a matter of becoming the best one can be when there is no core or stable self to enhance; perpetual change is equivalent to destruction.\n\nDr. Mierzwiak and his people act out the consequences of Dr. Heidenhoff's denial of any continuity of self that would produce moral responsibility in what is by transhumanist standards a quite prosaic world of still limited choice. Imagine how much worse their behavior could be when they have, perhaps like Daphne, less reason to think of themselves in terms of a self that could bear moral responsibility. After all, the whole notion of enhancement is predicated on a restless dissatisfaction with what we are, in the name of the as-yet-unknown possibilities for what we might be, and the hope for discontinuity between what evolution has produced by chance and what intelligence can manage deliberately. But from there it is but a short step, already taken by some transhumanists like Ramez Naam, to casting doubts on the existence of a self altogether\u2014the self is, some would say, mere \"user illusion.\"\n\nThere is some irony, then, in defenses of enhancement in the name of liberty. To be a truly free human being, to make one's own choices, implies accepting responsibility. It makes sense to advocate liberty if there is such a strong, responsible, moral self waiting to be free. But if, following Heidenhoff and today's transhumanists, we think that there is no such morally responsible self, we are left with only the willful or passionate choices of the moment, rationalized or otherwise, the expression of which is hardly liberty at all but license. There are of course huge moral hazards in a nanny state that would presume to protect the Joels and Clementines of the world from the selfish ministrations of the likes of Mierzwiak and his minions. However, in the transhumanist world of the right to choose without responsibility, the power that our future Heidenhoffs and Mierzwiaks will have over nature, and the power that they will therefore have over human beings, will be not so much the trump card as the only card to play when doubts arise about where our knowledge is taking us. Paradoxically, the world recreated by human beings will be predicated on the law of the jungle; our technological might will make right.\n\nIt may well be that this situation arises out of the fact that transhumanism is an effort to maintain some concept of progress that appears normatively meaningful in response to Malthusian and Darwinian premises that challenge the idea of progress. Malthusianism has come to be defined by thinking that the things that appear to be progress\u2014growing populations and economies\u2014put us on a self-destructive course, as we accelerate toward inevitable limits. But it almost seems as if, in the spirit of Malthus's original argument, there is something inevitable also about that acceleration, that we are driven by some force of nature beyond our control to grow until we reach beyond the capacities of the resources that support that growth. Meanwhile, mainstream Darwinian thinking has done everything it can to remove any taint of progress from the concept of evolution; evolution is simply change, and randomly instigated change at that.\n\nTranshumanism rebels against the randomness of evolution and the mindlessness of a natural tendency to overshoot resources and collapse. It rejects, as we have seen, the \"assumption of mediocrity\" in favor of arguing that man has a special place in the scheme of things. But its rebellion is not half as radical as it assumes, for transhumanism builds on the very same underlying conception of nature that the Malthusians and Darwinians build on, vociferously rejecting the thought that nature has any inherent normative goals or purposes. While it rejects blind evolution as a future fate for man, it accepts it as the origins of man. While it rejects a Malthusian future, it does so with threatening the same old apocalypse if we do not transcend ourselves, and, in the form of Kurzweil's law of accelerating returns, it adopts a Malthusian sense that mankind is in the grip of forces beyond its control.\n\nBecause transhumanism accepts this account of nature, it is driven to reject nature. Rejecting also any religious foundations for values, then, it is left with nothing but socially constructed norms developed in response to human power over nature, which, given the unpredictable transformative expectations they have for that power as it becomes not-human, ultimately amounts to no norms at all. Transhumanism is a nihilistic response to the nihilism of the Malthusians and Darwinians.\n\nThis moral lacuna at the heart of transhumanism is why it can advocate the progress of dehumanization so enthusiastically. Bostrom may assert that posthumans can be humane, but human morality is built on human capacities and the circumstances of human life. If these are precisely the things that are going to change radically, it is hard to see how the progress of transhumanism can really be called progress anymore. At least, if the Singularity is the outcome, transhumanism cannot be \"progress\" understood as achieving or approaching some humanly comprehensible ideal or goal. It is progress only if a leap into the unknown can be called progress. That being the case, the only thing that can be said for it by anyone other than someone who enjoys taking incalculable risks is that it is _necessary_ that we go down this path.\n\nBut since we cannot in fact know if it is necessary, the Singularity represents an aspiration, and aspiring to it has consequences long before it arrives, if it will arrive at all. To aim at the Singularity is to aim at something without content, or (which amounts to the same thing) to aim at any content at all. Such a goal is peculiarly fitting to the thin, willful self that we have seen come to the fore within the transhumanist framework, a self that supposes itself capable of any content but can only prove that capability by serial negation. This self is the apotheosis of what the philosopher Thomas Hobbes had in mind as a defining human characteristic, the \"perpetual and restless desire of power after power, that ceaseth only in death,\" except of course it now expects to cheat death as well. We saw how Moravec recognizes that there is something deceptive about the transhumanist promise of immortality, because the \"I\" that seeks it will not survive its own desire for constant transformation. In fact, the same deception is at work in all of transhumanism's promises to be looking toward solving human problems or other benevolent representations. However sincere these promises, the good intentions reference a world whose destruction in its present form transhumanists are at the very least reconciled to, if not actively seeking. As they see it, the \"present form\" is mere accident, unworthy of the respect of intelligent beings.\n\nThere is of course something to this line of argument. Our desires are restless, and hard to control. Genuine Stoic resignation about the ills of the world has only ever been genuinely possible for a very few. It becomes all the more difficult in a world where it is manifestly possible to meliorate ills that once called for resignation, even if so doing has been known to create new problems. In any case, change seems to be in our blood. Those who even try to use the rhetoric of technological stasis, let alone rollback, are few and far between, and it is yet harder to find those living in a manner consistent with such beliefs.\n\nSo surely the safest bet is that the world will continue to change in response to human activities. Even so, the transhumanists may or may not be correct that all roads that are not immediately disastrous lead to the Singularity. This uncertainty means that a great deal is at stake respecting the extent to which their vision of the future is adopted. If it becomes the dominant narrative of technological change and scientific progress (all technical questions aside) it becomes more likely by that very fact. That would be unfortunate, given what we have seen in this chapter. The transhumanists have not adequately dealt with the very thin moral self and the problematic relations between enhancers and enhanced that are central to _Eternal Sunshine_. The disparity in power that would be problematic under any circumstances is all the more so in the time of the thinning self. The will can become all the more willful when it is no longer even directed toward what I want to achieve, which after all implies some kind of self and limit, but is instead liberated to be all-negating. John Harris, avoiding as is his wont any use of transhumanist terminology as he makes his transhumanist arguments, tells us proudly that he does not \"recognize finitude, only the limitless possibilities of the human spirit and of human ingenuity.\"\n\nFinitude is, of course, all around us and in comparison to this reality transhumanism offers what is ultimately a kind of nothingness, be it in the form of mere negation or the unknowability of the Singularity. From this point of view, limits don't look so bad. If we go back to the common-sense beginnings of transhumanism in the admittedly extraordinary potential that modern science and technology hold, how might we begin to think about a future within limits\u2014a _human_ future?\n_CHAPTER FIVE_\n\n_The Real Meaning of Progress_\n\n* * *\n\n_PROLOGUE: FATHERS AND SONS_\n\nI've never really given that much thought to birds. Eating them, of course. But beyond that they do what they do and we do what we do. Dad's another story. If it hadn't been \"let's try this experiment to escape from Minos,\" I'm guessing that sooner or later he would have found some other excuse to strap on wings. That's just the way he is\u2014which means never satisfied with the way he is.\n\nIt's an admirable enough trait. I guess. We're supposed to strive for excellence, at least as much as the gods allow. But excellence for Dad seems to be more doing something new than doing something well. I've seen the \"wings\" he's making, and I'll never strap them on sober, I can tell you that. It's not that I'm against new things\u2014where would we be without Prometheus? But Dad thinks that the fact that he _can_ do something is reason enough to do it. So let anybody offer him the chance to do something new and he is all over it. No thought for the consequences. Even now I'm pretty sure that he misses the irony of the fact that we would not need these stupid wings if he had not made that disgusting hollow cow. And this damn maze.\n\nIf we do get home after all this I'm going to have to talk with him. It's not that I think he is likely to change, but the gods know that somebody has to try and straighten him out before something really terrible happens. I'd hate to be the one to have to pick up the pieces.\n\n* * *\n\nWHEN APPEALING to common sense, transhumanists promise a better world in humanly comprehensible terms. However, their own assumptions lead them to abandon those promises in favor of willful change toward incomprehensible outcomes. They promise that science and technology will provide us with more of whatever it is we want more of at any given moment\u2014and it is indeed hard to deny the attraction of that promise to people like us. The hitch is that people like us are not going to be around to enjoy it. Indeed, we are not even supposed to see our elimination as a cost at all, but as a great benefit. At least, for the transhumanists, this outcome is in some fashion necessary, and we are supposed to accept that technological might dictates right.\n\nThe transhumanists believe that their ideas represent _progress_ \u2014not just technological progress, but progress in the much larger sense of humankind fulfilling its ultimate destiny (a destiny of overcoming itself). Now, one might have thought that ethical guidance would be central to deciding whether a given discovery or invention actually served to advance the well-being of humanity in a way that would deserve labeling it progress rather than mere change, or worse. When Condorcet spoke of progress, for example, we could see his hopes for moral improvement. But shorn of any serious moral content, the measure of progress\u2014if it can be said to exist at all\u2014comes to be simply our amazement at, or dissatisfaction with, the present state of our discoveries and inventions, our awed anticipation of what might yet be achieved. Indeed, our terror about what might go wrong along the way becomes a kind of measure of progress.\n\nThe result of framing the question of progress in this way is obvious in popular discussions of the future of science and technology. First, start with a little history to produce an attitude of pride that we know so much more than we once did. Then look at what we know now, and stress the dangers of our remaining ignorance. Finally, anticipate future discoveries, combining hopes and fears for what might happen with a humbling sense that, if only we stick with it, those who follow us will know more than we do and be able to do far more than we can. It would look like Winwood Reade's vision of mankind \"ripening towards perfection,\" if it had the idea of perfection and Reade's tragic insight into the sacrifices involved.\n\nBeyond that, \"progress\" becomes the sheer accumulation of information, a kind of hoarding mentality that is based on the belief that you never know what might come in handy someday. This helps to explain the widespread belief that any effort to restrain science or technology on the basis of ethics represents a threat to progress. After all, if progress is mere accumulation, then of course restraint _is_ a threat. Yet to see this concern as simply expressing the self-interest of researchers and inventors is to do them an injustice. Haldane wrote that the prospect for the future \"is only hopeful if mankind can adjust its morality to its powers,\" and many well-intentioned professionals probably would agree that it is the job of morality to adjust to scientific and technological change without appreciating how that amounts to saying that might makes right, since they are saying that the question of what we ought to do must always bow to what we have the power to do. Others probably believe that ethical restraint is a weak reed. That is why we frequently hear the argument that if \"we\" restrain ourselves with respect to some line of research or development, \"they\"\u2014some other country, usually\u2014will not, putting us at a disadvantage. Combine these two perspectives and you can begin to understand why so often in practice \"ethics\" of science and technology becomes a matter of filing the right paperwork, following professional codes, publishing in highly specialized journals, or scientists and engineers being willing to have meetings with people from otherwise safely segregated humanities departments to talk about the \"ethics of . . .\" this or that technology or line of research.\n\nIn fact, however, the kneejerk suspicion of any effort to limit developments in science and technology represents something of a betrayal of the bargain that gives science its high place in the modern world. The enlightened acceptance and public encouragement of science and technology was built on the assumption that freedom for such research and development by those so inclined would serve _human_ well-being. The so-far largely successful results of that bargain have made the modern world what it is. Whether the bargain can be as successful if we lose sight of its terms is another matter, about which one is entitled to be quite skeptical.\n\nSome may argue, in the manner of this book's epigraph from Albert Einstein, that today we are too sophisticated to think that there really is any rational answer at all to the question of what human well-being means; hence it could hardly help us think about how to limit our growing power over nature. Cultural and moral relativism, historicism, postmodernism, dogmatic materialism, and fashionable nihilism all create obstacles to taking the question of the human good seriously in our time, obstacles that were not created by advocates of the eclipse of man but that are consistent with their unwillingness to tackle moral questions. But if it is no betrayal of the bargain behind our scientific and technological prowess to at least _inquire_ into the limits represented by the pursuit of human well-being, is it not all the more urgent to pose the question when the ultimate promises of transhumanism so blatantly reject that goal in the name of powerful enhancers, willful negation, and the mystification of the Singularity?\n\nIt is no new observation that the great increase in our powers coexists with a diminished capacity to think about them with any kind of moral realism. But slighting ethics does not genuinely serve the cause of science and technology, since they only matter in human terms if they truly serve our humanity. When progress is defined by dehumanization, it is obvious that this result is by no means guaranteed.\n\nWhile transhumanism is still a fairly recent development, questions about the extent to which human ingenuity serves human beings well are hardly new\u2014as witness the ancient Greek myth of Daedalus and Icarus. The details of the story are familiar: Daedalus is a great craftsman and engineer, the builder of the Labyrinth used to entrap the Minotaur on the island of Crete. Despite this service, Daedalus and his son Icarus are imprisoned on Crete by King Minos. Daedalus fashions birdlike wings of feathers and wax so he and his son can fly to their escape. Ignoring his father's cautions, Icarus flies too close to the sun; his waxen wings melt and he plunges to his death in the sea.\n\nThe tale has drawn a fair amount of attention from artists over the centuries, including the three paintings we will discuss in the next section. The discussion that follows does not aim at offering a comprehensive account of the human good. Nor does it attempt to defend any particular limit on how we might use science and technology to preserve a future of _human_ well-being. But it suggests how we might begin to think about such limits in the course of even modest reflections on the world we are making day to day with science and technology. The transhumanist arguments obscure what is present in front of us in this world; its imperfections and failures, for example, are swept away in a tide of technological determinism drawing us on to some distant horizon of imagined possibilities. That transhumanist farsightedness is then taken to be the best framework by which to give a trivializing and dismissive meaning to present-day things. The three paintings we now turn to provide illustrations of the range of moral responses to the eclipse of man, responses that can illuminate the reasons for rejecting transhumanist farsightedness and put us in a better position to take seriously the human purposes that science and technology promised to serve.\n\n_WINGS OVER THE WORLD: THREE VISIONS_\n\nOur first image is \"Daedalus and Icarus\" by the French writer and painter Charles Paul Landon, who would eventually become better known as a writer on art than an artist. It is a marvelous illustration of the transhumanist hopes for the progress of inhumanity. The wings seem so natural\u2014the thin fabric straps that bind them to the bodies of Daedalus and Icarus are easy to miss\u2014that the pair look less like human beings wearing improvised wings than like winged beings. We see a moment of great promise, aptly in what seems the light of a new day, as Icarus steps into his first moment of flight, with the intent assistance\u2014perhaps a slight push, perhaps just steadying hands\u2014of his ingenious father. How Icarus feels is anyone's guess (does his face need to be quite so heavily shaded?), and Daedalus exhibits more concentration than amazement or even satisfaction at what he has achieved. Without clothing, with only a sky as background, and with only the vaguest of classical motifs in the pedestal on which they stand and perhaps in their hair, the picture presents this moment of accomplishment, the dawn of a new day, almost completely abstracted from time, place, personality, and circumstance (Icarus is even curiously androgynous). Thus, all the distinctions by which we normally define human beings, except the one that highlights our ingenuity, are missing, and perhaps therefore arbitrary, and the painting becomes a pure tribute to the magical-seeming potential of human invention. Landon of course knows we know what happens next in the story. But from this starting point, it is almost impossible to believe that Icarus will fall\u2014it seems at least as incomprehensible as the fall of some angel. It is as if Landon has in mind a new version of the story, in which father and son both survive.\n\nCharles Paul Landon, _Daedalus and Icarus_ , 1799\n\nMus\u00e9e des Beaux-arts et de la Dentelle, Alen\u00e7on Picture: David Commenchal\n\nOur second image, \"The Death of Icarus\" by the contemporary German painter Bernhard Heisig, is a powerful illustration of a certain kind of problematic critical response to the eclipse of man. In contrast to Landon's painting, Heisig's shows the end of the story, perhaps at sunset, with a screaming, terrified Icarus (apparently a selfportrait of Heisig) the center of attention as he is crucified on the obvious artifice of his father's inventiveness. The background echoes the famous painting of the Tower of Babel by Pieter Bruegel the Elder, while the prophetic, prominent pointing finger beneath Icarus' left wing could be a rotated image of the hand of God from Michelangelo's _The Creation of Adam_. All these Biblical visual references add up to a strong warning against overreaching technological ambition\u2014a reminder of the human moral imperfection that conditions the way we use our amazing abilities, and a useful corrective to transhuman aspirations.\n\nYet Heisig, like Landon, simplifies the moral equation at work in the story. Despite or even because of its cultural quotations, the context in which he places the unfolding events is perhaps even more mysterious than Landon's. The violence of this image, something for which Heisig's work is generally known, is quite shocking, and puts the primary focus on the very direct line between Daedalus' innovation and Icarus' terror. \"Here is where your creative pretensions will end up,\" the painting seems to be saying: not just in failure but in horrifying disaster visited upon those closest to you. So in a curious way this deeply negative outlook depends on accepting the same kind of necessity that transhumanists like to claim drives their project. It appears that for Heisig failure is the only option given that human beings are as they are. There is no point in speaking of progress at all if, from Adam on, we are fated to make the same overreaching kind of mistakes again and again, as the painting seems to suggest.\n\nBernhard Heisig, _Der Tod des Ikarus_ (\"The Death of Icarus\"), 1979\n\n\u00a9 2014 Artists Rights Society (ARS), New York \/ VG Bild-Kunst, Bonn\n\nHowever, that perspective overwhelms the fact that this son is, after all, not bearing his cross because he is self-sacrificing or even fated to be sacrificed, but is a young man personally responsible for having flown, against the warnings of his father, too close to the sun. Of course, we can hardly expect that sons will always obey their fathers. But it is not clear whether Heisig sees how the moral significance of Daedalus' work looks very different when we take account of the fact that his son's character plays a role in the way things turn out.\n\nOur last picture is \"Landscape with the Fall of Icarus,\" generally attributed to Pieter Bruegel the Elder. It is loosely adapted from Ovid's telling of the story in _Metamorphoses_. Much has been and deserves to be said about this remarkable painting; let us therefore just stick with the obvious. Icarus plays a very small visual part in the story the painting tells; it depicts nearly the same moment of failure that Heisig presents, but in Bruegel we see it at a great and impersonal distance. A splash at the bottom right, some scattered feathers, and Icarus' tale is done; Bruegel has to play with the perspective a bit even to make him as visible as he is.\n\nPieter Bruegel the Elder, _Landscape with the Fall of Icarus_ , CA. 1555\n\nMusees Royaux des Beaux-Arts de Belgique, Brussels, Belgium \/ Giraudon \/ The Bridgeman Art Library.\n\nOf course, that minimization is its own kind of warning against overreaching: who wants to be remembered most for self-imposed failure? But beyond disobeying his father's instruction to follow a middle course between sea and sun, what is the source of Icarus' failure? Ovid's version presents Daedalus in a not-very-flattering light that casts some doubt on his desire \"to work on unknown arts, to alter nature,\" but it is not so clear that Bruegel agrees. The shepherd tending his flock in the middleground of the painting may come closest to living with nature unimproved (or at least, improved so long in the past as to _appear_ unimproved), but the fisherman certainly requires artifice to make his living. And our farmer's contour plowing may reveal him to be conservation-minded, but his plow, clothing, and literal reshaping of the earth together suggest no small degree of technological sophistication, a suggestion only amplified by the great argosies plying the waves and the alabaster city in the distance that is their apparent destination. In short, Icarus is placed on a continuum of very human enhancements of the given. Yet the tiny part that Icarus plays in the picture suggests a very different view from that presented by Landon and Heisig.\n\nBruegel's potential witnesses to the fall of Icarus are barely witnesses, if at all\u2014they are certainly not dismayed, as in Heisig. In this respect Bruegel changes Ovid's story:\n\nA fisherman, who with his pliant rod\n\nwas angling there below, caught sight of them;\n\nand then a shepherd leaning on his staff\n\nand, too, a peasant leaning on his plow\n\nsaw them and were dismayed: they thought that these\n\nmust surely be some gods, sky-voyaging.\n\nBruegel keeps the same characters, but not one of them seems astonished by what he has seen; indeed, it is not really clear they have seen anything at all. Because we know what is happening, and because we know Ovid, we can wonder about their _lack_ of wonder. Is it mere peasant stolidity, or the general human obliviousness in the face of something new and (as some would say) important? Perhaps the fisherman sees Icarus. He might have thought, \" _Was that really a man falling from the sky? No, it couldn't be_.\" before returning, like the farmer, to doing his job. A flying man is outside of his expectations, and a god would hardly crash-land. Our shepherd is a more interesting case; he, not the plowman, raises his eyes, but not in the right direction to observe Icarus' fall. Perhaps he looks up because he hears Daedalus calling to his son? Or perhaps gazing up at the birds he daydreams of flying? But for him it is a passing fancy; not everyone has the genius and motive of Daedalus to turn such dreams to reality.\n\nIn any case, the most thought-provoking thing about Bruegel's painting of the story is that he does not have us (as Landon and Heisig do) asking much about Icarus, does not have us wondering about his wondrous or terrible flight. By placing Icarus in a relatively commonplace landscape, Bruegel instead makes us wonder about what Icarus means to these commonplace figures. He is an easily missed part of Bruegel's story, overwhelmed by Bruegel's seductive landscape of the everyday. In other words, in the world out of which they arise, innovations by the likes of Daedalus may hardly appear at all. Daedalus' great purpose, to find a clever way of freeing himself and his son, and Icarus' great failure, echoing his own father's quest to be free, do not intersect directly with what is important to the others in the painting.\n\nOn one hand, Bruegel's painting reminds us that the mundane is not given once and for all: there was a time when ships, cities, plows, fishing rods, and even herding sheep were new, the kinds of novelties that might have been, like Icarus' strange flight, either dangerous or hardly worthy of notice by sensible people. On the other hand, Bruegel's painting suggests the limits to the power of change, for the world he depicts would have been just as recognizable to a viewer four centuries before his time as it is for us four centuries after. Even four millennia before Bruegel there was food to grow, there were animals to catch or tend, there were goods to trade, and there were disobedient children. Familiarity across such time scales may be less than the blink of an eye from some imagined cosmic perspective, but this lived human experience provides the continuities that ground and shape human life. It is because of this backdrop that we have the chance to judge innovation and change to be merely that, or in some real sense _progress_ ; there has to be a human condition in order to speak of progressive improvements to it. Furthermore, it is because of this backdrop that the importance of an innovation will become obvious only in retrospect; we cannot know in advance whether its significance, if any at all, comes from the manner in which it is integrated into the old world, or overturns it. Bruegel stresses what Landon did not: that Daedalus' achievement was a dead end.\n\n_BETWEEN TWO VISIONS OF THE FUTURE_\n\nThere is a further consequence of Bruegel contextualizing Daedalus' invention within the commonplace earthy, watery, and airy worlds of the farmer, fisherman, and shepherd. Can we not imagine them absorbed not merely in their work, but in those they are working to support? Their own aspirations for and worries about their families and communities would actually be their point of contact with the otherwise unfamiliar events whose final moments unfold before them. As they pursue their daily lives, Bruegel's figures look better dressed and better off than might have been true for their forebears; they might have similar hopes for their own descendants' material improvement without imagining them to be like gods. If they could meet Daedalus, then, we can well imagine they would be more inclined to marvel at his escape from a tyrant and commiserate with him about a disobedient son than focus on the details of his invention. Daedalus does not invent in a vacuum.\n\nThe existence of this shared world may be frustrating to those who today pride themselves on being at the scientific and technological cutting edge. But Bruegel reminds us that, practically by definition, the cutting edge is not where most people are; it does not even loom large in their lives. (Indeed, I know of no ancient story that suggests that Daedalus himself ever flew again; even for him flying itself was perhaps less an aspiration than escape.) Still, to those who think themselves in the vanguard, like today's transhumanists, the rest of us will appear as the fisherman, plowman, and shepherd\u2014the ignorant and unobservant who through mere inertia ignore a fabulous future, or seek to keep it at bay. However, there is a certain falseness about this perspective, since in fact the vanguard does not really live in the future they imagine and must continue to rely on the existing world built by the mundane choices and motives that they would rather not acknowledge. Ovid presents Daedalus as an escaping prisoner, Icarus as proud and foolish. Such human details decisively shape how innovations come to be and how they are used. The extraordinary ambitions of the transhumanists and our other latter-day followers of Daedalus will be mediated by the complexities of the ordinary.\n\nIs there any force that can move the ordinary world inexorably toward the radical changes foreseen by the transhumanists? Mightn't the powerful forces of Malthusian scarcity and Darwinian competition push us in that direction? Bruegel, as a man of the sixteenth century, of course cannot be blamed for being unaware of such ideas. Does his unawareness undermine the human vision that he is seeking to present?\n\nThe first thing that needs to be noticed in order to answer this question is that the lesson of ongoing scarcity Malthus taught, and to some extent also the lesson of Darwinian competition, is shocking only after hopes for _ending_ it, such as those articulated by Condorcet, come on the scene and become widely adopted. Without a vision of progress like Condorcet's, scarcity and competition would more or less simply be taken as definitive of the way things are. You need to have a vision of progress first, and then the Malthusian challenge can contribute to its radicalization.\n\nBruegel chooses in this painting not to highlight the worst consequences of scarcity and competition, aspects of life that would have been perfectly obvious to a man of his time. But that does not mean they are absent entirely. We see them at work in the inequality implicit in the painting, the distinction between city and country life, the somewhat menacing island fortress, the occasion for Icarus to be flying that is the story behind the picture, the existence of trade, the fact that the three peasants must make their livings by the sweat of their brows. All such things are just the norm in Bruegel's world. His acceptance stands in stark contrast to the mental gymnastics that advocates of transhumanism have to attempt in order to deal with scarcity and competition. As we saw, Drexler is driven to reject Malthusian scarcity in the near term, accept it over the longer term, and then argue for its irrelevance even in the long term. To one-up him, Kurzweil has to speculate that things we take to be fundamental physical limits may prove subject to intelligent manipulation after all.\n\nWe are fortunate to live in a world where, based on some very hard work, Malthus's gloomy predictions about scarcity so far seem to have been confounded. Perhaps, then, Bruegel's resigned attitude is misplaced\u2014perhaps innovation, like that of Daedalus, deserves a much more central placement in our picture of the world. But if scarcity and competition have not yet done their Malthusian worst, we should also note that we can hardly claim that Condorcet has obviously been vindicated. Material progress has had costs with respect to both want and excess that Condorcet did not anticipate, and to be satisfied that it has produced the moral progress he expected it to create would plainly be mere smugness. If, as suggested above, some sort of progress is not entirely alien to Bruegel's picture, it may navigate between the Scylla and Charybdis of these alternatives\u2014between the pessimistic vision of Malthusian decline and the optimistic vision of progress\u2014that so readily occupy our contemporary imaginations. Bruegel's in-between depiction may therefore be truer to the world we live in.\n\nFinally, does Darwin's argument for the mutability of forms of life require us to abandon the perspective on human things that Bruegel seems to be suggesting? At the simplest level it does not, because a great many people in the world today, perhaps even most of them, are as little influenced by Darwin's ideas as Bruegel's peasants were, despite all the efforts of bestselling neo-Darwinian popularizers. We may decry such ignorance or (worse) obstinacy, but there it is; integrating Darwin into a horizon for understanding the world remains the achievement of a relatively small number, and it would be a brave soul today who would claim that these represent the vanguard the rest must necessarily follow.\n\nFurthermore, it is not really that clear that Darwin's vision of natural evolution should be _expected_ to change the shape of our lives to something other than what Bruegel observed. What life lessons ought we to learn from Darwin? If his work represents a warning against a certain kind of human pridefulness, it is hardly the only source of such a warning; if it teaches us that this too shall pass, we likewise need not learn that lesson from him. If Darwin links us to animals in our origins, that is a more distant link than the one a dispassionate observer of human things like Aesop can readily see in our actions.\n\nIt seems true that Darwinism confounds both pagan and Biblical stories of human origins, and it is impossible to deny the corrosive effect Darwinism has had on Biblical faiths. Yet here again its limited impact is worth noting. The secular view that Darwinism has been taken to advance has not achieved its ultimate victory, not only because (for better and worse) faith can trump materialistic rationalism, but because faith has adapted. Rather than say Darwinism confounds the Biblical stories, it would be truer to say that Darwinism confounds certain ways of understanding those stories, and long before Darwin there were ways of reading them that did not treat them simply as scientific, historical, or journalistic accounts of events.\n\nThe transhumanist case for modifying our Bruegel-like understanding of the everyday world depends on making a very un-Darwinian move: transforming evolution from a natural and long-term process into a human project today. Is it truer to Darwin to modify his ideas in this fundamental way, or simply to let evolution take care of itself, and continue, as a great many people seem to do, to live out their lives without a concern for the ultimate fate of _Homo sapiens_ and our evolutionary successors?\n\nNo, if there is a flaw in Bruegel's portrait of the everyday, it is not in how it treats Malthusian and Darwinian realities, but that it contains so little hint beyond the scattered feathers of Icarus' wings of what Heisig reminds us of\u2014that terrible things can become mundane. But Heisig and Bruegel might agree at least on this: what is terrible in the tale of Daedalus and Icarus reflects flaws and limits that make the human story what it is. We cannot simply wish these flaws away, lifting ourselves into some new state of affairs where they will make no difference. We might hope that our innovations always be motivated by a wish to be better, but they will certainly always arise within a framework in which that wish has not yet been fulfilled, and that fact will always make their actual result uncertain.\n\n_THE POWER OF THE SEEMINGLY MUNDANE_\n\nBruegel's painting suggests how transhumanists slight the power of the everyday, instead projecting our hopes and fears for the future onto what is essentially a blank canvas. Hoping to overcome the merely human, they look at the present from the point of view of their projection\u2014judging the world around us as though they already understand the future\u2014in order to give meaning and direction to present human activities. So what is important to them are the real or imagined innovations that serve as a prelude to this future whose own meaning will be beyond us. The prosthetic hand that could serve the disabled veteran is immediately attached to the pioneer who wants a third hand, and since he _wants_ it, no further thought about the context in which such a thing might happen is considered necessary. If the transhumanists bother to look at anything in the past or present at all, it is only the as-yet-unrealized dreams of things like immortality or super powers. The godlike capacities that have long been wished for, and yet traditionally have been regarded as at least as much curse as blessing, if not far more curse, are turned into unambiguously normative aspirations. Such wishes become human essentials rather than aberrations.\n\nWith this blank canvas as their starting point, it does not seem so strange to transhumanists when they go on to assert that the meaning for life today, and a direction for future \"progress,\" is to be found in an incomprehensible future. Yet that argument creates a powerful bias, a tunnel vision that focuses on developments and possibilities that make the least sense from the point of view of where we are today and for that very reason suggests they are the most important things in the world of tomorrow. At the very least, this kind of tunnel vision is not the only way of treating the future. Like Bruegel, we can admit the desirability of innovation and still value the continuities that for better and worse influence the meaning that those innovations will come to have for our lives, projecting the past and present into the future rather than the other way around.\n\nBruegel's alternative is not his alone; for example, a similar way of looking at the world is found in Tolstoy's _War and Peace_. Tolstoy describes how Pierre Bezukhov was changed by his experiences as a prisoner of Napoleon's army. Pierre grew up and, abandoning a way of looking at the world that sounds rather like the transhumanist vision, adopted a perspective like Bruegel's:\n\nAll his life he had looked off somewhere, over the heads of the people around him, yet there was no need to strain his eyes, but only to look right in front of him.\n\nFormerly he had been unable to see the great, the unfathomable and infinite, in anything. He had only sensed that it must be somewhere and had sought for it. In all that was close and comprehensible, he had seen only the limited, the petty, the humdrum, the meaningless. He had armed himself with a mental spyglass and gazed into the distance, where the petty and humdrum, disappearing in the distant mist, had seemed to him great and infinite, only because it was not clearly visible. . . . Now he had learned to see the great, the eternal, and the infinite in everything, and therefore, in order to see it, to enjoy contemplating it, he had naturally abandoned the spyglass he had been looking through until then over people's heads, and joyfully contemplated the ever-changing, ever-great, unfathomable, and infinite life around him.\n\nFor the advocates of the eclipse of man, the \"spyglass\" may be some distant future, aliens from the stars come to save us, the ever-rising trend line that brings the actual ever closer to the limits of the possible, or the posthuman Singularity. From all these imagined points of view there is little to be said for humanity as we see it in front of us\u2014it is indeed petty, humdrum, and meaningless. Yet even if the futures they look forward to are possible, and even if there are powerful forces at work behind the innovations that would bring them about, there is no necessity to look at them through the \"spyglass\" that the advocates of dehumanization use. The peculiar farsightedness of the spyglass makes small, speculative things look big and important while turning things that are close up into a blur. Instead of looking \"over the head\" of humanity to the alien or posthuman, we can attempt to see what is right in front of us, to meet human life face to face, and at the very least not abandon it until we are certain we have understood it and appreciated it on its own terms.\n\nG. K. Chesterton writes of a reformer who sees a gate in the road and says, \"I don't see the use of this; let us clear it away.\" A more intelligent reformer, Chesterton says, would respond this way: \"If you don't see the use of it, I certainly won't let you clear it away. Go away and think. Then, when you can come back and tell me that you _do_ see the use of it, I may allow you to destroy it.\" It is far from clear that the advocates of dehumanization have given much thought to the \"use\" of human beings, beyond various attempts at material explanations of why we are the way we are. There are a few exceptions: Winwood Reade and Nikolai Fedorov sought to place their arguments within an understanding of human things wider than the horizon of the power of technological possibilities. But most advocates of dehumanization, including today's transhumanists, are far more likely not only to take the world to revolve around the actual technological and scientific infrastructure, but to assume it revolves around what _might_ become possible if only we clear away all old gates.\n\nIf we appreciate instead how important the seemingly mundane is in shaping our expectations and hopes about the world, we are more likely to ask why we have this or that technology, and why we want it, or why we want something different. What good does it do, or would we expect it to do? How does it fit with our vision of what makes for a good life\u2014which of course may or may not be a vision of a good life simply.\n\nSuch questions will not be easy to answer. Since science and technology as such offer few if any resources for ethical reflection, on their own they leave us with a painfully thin understanding of the shape of human life. Furthermore, the advocates of dehumanization are just one part of a larger picture, and in our time that larger picture in commerce and the arts, in the humanities, in the natural and social sciences, is one that _often_ simplifies, if not outright debases, our self-understanding in ways that reinforce the eclipse of man. If we lack the general intellectual tools required to make distinctions between progress and change, for example, or between freedom and willfulness, some of the transhumanists' heaviest lifting has already been done, for then it is easy to transform the possible into the necessary simply because it satisfies someone's desire.\n\n_THE PROGRESS OF HUMANITY_\n\nBeyond these particular blinders of our time and place, investigating the meaning of well-being has _never_ been easy, because it requires a willingness to look at the question of the human good with care and seriousness, not taking the day-to-day for granted but not rejecting it dogmatically. That would mean neither the dogmatic acceptance nor the dogmatic rejection of the moral values of one's time and place. It would require avoiding cynicism and utopianism about human motives and possibilities. Such an investigation might yield a complex and mixed picture of what a good life is and how science and technology contribute to it. There will be grave uncertainties and honest disagreements along the way. We will likely find that even as individuals we have conflicting desires and visions of the good, not to speak of wider social and cultural disagreements. But the investigation is still worth undertaking if we want to speak meaningfully of \"progress.\"\n\nPutting all such challenges together, we begin to see why the problem of benevolence that has arisen in these pages looms so large. It can be hard enough to know what is genuinely benevolent when human beings are relating to human beings under the best of circumstances. When we start talking about benevolence directed to us by beings of unimaginable power and knowledge, the only intellectual experience we have is summarized in theodicy, the effort to understand God's goodness in light of the manifest evils of the world. That effort, which has occupied many a great mind over many centuries, is, shall we say, ongoing. Why it would be any easier to settle were we to start having to talk about artificial superintelligence or advanced aliens is far from clear.\n\nA willingness to act on the basis of nearly complete moral ignorance relative to the central question of progress\u2014the question of what would make for a better world\u2014is really the only justification for the otherwise simplistic desire for the eclipse of man. Otherwise, we would surely want to adopt more _modest_ expectations for a human future. Surely it is not as if the only future that is worth looking forward to, and working hard for, is one in which we can achieve anything we can imagine, where everything will be permitted. If it were, what we are left with is mere pride in novelty and superlatives, a constant one-upmanship of imaginative possibilities that diminish the worth of human beings as we actually know them. With no clear goal, direction, or purpose, with willful freedom of choice as the guiding light, how could it be otherwise?\n\nTo be clear, this kind of needed modesty is not that of the SETI advocates, who are happy to expose what they take to be human vanity about our place in the universe. For that is a patently false modesty; diminishing what we are is only a prelude to pride in the great expectations of what we can achieve once we meet up with aliens, a goad to take up our true task of creating limitless possibilities.\n\nThe kind of modesty that we need acknowledges that there is much that can and ought to be done to make human lives better, and that science and technology will play a major role in that effort. At the same time, it does not take for granted what we mean by \"better,\" in light of the whole range of human strengths and weaknesses that we observe when we pay attention, like Bruegel, to the world in front of us. We are notoriously in-between beings, neither beasts nor gods as Aristotle famously put it. How much confidence is appropriate, then, in our abilities to wield the great powers that are being promised to us? We can hardly afford to act on the basis of thinking that because we can imagine a day when we are without human vices, we can therefore ignore their reality when presented with technologies that could be used to help them flourish.\n\nAnything we actually accomplish will be the product of limited and flawed creators, so the odds are that our creations will of necessity perpetuate those limits and imperfections. As we have seen repeatedly in these pages, the more we place our understanding of technological change within the constraints of the world out of which it actually arises and through which it must percolate, the more it seems likely that the result will never be as wonderful or as terrible as less-disciplined imaginations can so easily make it. That hardly guarantees a good result, but if we are unwilling to take up seriously the question of what good means, and if we are too much influenced by the tunnel vision of the dehumanizers, we can hardly expect anything better.\n\nHuman beings, unlike other animals, can make deliberate choices to change what it means to be human. It may be that now we are seeing the beginnings of a real choice about being human at all. In attempting to confront the \"grand vision\" of the eclipse of man as such, we have seen how its advocates have made arguments against our humanity based on a painfully thin understanding of what it means to be human, and made promises that will lead to the demise of the goods sought even as they are fulfilled. Their project is neither as inevitable nor as rational as they would like to believe, and they are therefore certainly not excused from defending it on the moral grounds implicit in calling it progress: that it will actually create a _better_ world.\n\n_Acknowledgments_\n\nTHIS BOOK has taken longer to bring to fruition than I ever imagined it would. As a consequence, there are more people to thank than might otherwise have been the case. I only started to enhance my memory of such people by making a list after the process had already extended itself to such a degree that I fear there is every likelihood that I have forgotten some important assistance, kindness, or inspiration. To any of these people who might pick up this volume without finding the recognition due them, my sincere apologies.\n\nIt has been a deep pleasure for some years now to work in a variety of contexts, not the least of which is the Futurisms blog, with the talented editorial staff of _The New Atlantis_ : Adam Keiper, Ari N. Schulman, Samuel Matlack, Brendan P. Foht, and Caitrin Nicol Keiper. They also served as my editors on this book. As usual, their thoughtful corrections and suggestions have the net result of making me seem a better writer and a more intelligent person than I am. My only excuse for complicity in this deception is that at least I am smart enough to recognize that is the outcome, and to acknowledge it herewith. Eric Cohen was the founding editor of _The New Atlantis_ ; he along with Peter A. Lawler at _Perspectives on Political Science_ were the first to provide a print home for the writing I was doing on the arcane topic of human redesign, encouragement that came at a crucial moment. I also owe many thanks to the assiduous fact-checking efforts of several _New Atlantis_ research assistants and interns: Maximilian de la Cal, Steven Fairchild, Barbara McClay, Maura McCluskey, and Galen Nicol. _New Atlantis_ interns Moira E. McGrath and M. Anthony Mills helped compile the index. And thanks are due, too, to Roger Kimball, Heather Ohle, Carl W. Scarbrough, and the rest of the Encounter Books team for their contributions to unleashing this book on an unsuspecting public.\n\nHaving touched on the subject of finding a place to publish my thoughts on the topics this book covers, let me single out also Mary Nichols, who was instrumental in founding the Politics, Literature, and Film section of the American Political Science Association, and by so doing, and by the example of her own writing, encouraged me and many like me to think of literary analysis as a legitimate way to understand political things. Similarly, Stuart Kingsley's efforts to make a place for discussions of optical SETI at conventions of the International Society for Optical Engineering (SPIE) opened the door for this political scientist to put my thoughts before a community of natural scientists and reap the benefits thereof. Robby George and Brad Wilson likewise generously provided opportunities to air my ideas in the context of conferences hosted by the James Madison Program in American Ideals and Institutions at Princeton University.\n\nIn its early stages, my writing was supported by a sabbatical from Duquesne University and a generous grant from the Sarah Scaife Foundation. I note \"early\" to express my gratitude for the subsequent patience of executive director Michael Gleba, who never failed to show interest in how the book was going.\n\nI can't even begin to thank my wife Leslie enough for the many things big and small she did over the years to give me the time, emotional support, and advice that allowed me to write. From discussion in the earliest days of this book to the last stages as she read galleys, her help has been indispensable. I am grateful that I can count on her energy and intelligence under all circumstances. My children Ted and Anna did their part by showing a remarkable tolerance for the needs of a father with such odd interests.\n\nThis book is less the product of specialized academic studies within a clearly defined discipline than it is of an academic life where, thanks to the tolerant attitude of colleagues at Duquesne University, I have been able to pursue the questions that intrigue me wherever they have led. But academic life is just one aspect of life, and when thinking about prospects for the human future, one can hardly help (even if unknowingly) drawing on that larger life experience. Such insights as I hope the book contains could have arisen as much from encounters walking about in my community or attending synagogue services as from the classroom, conferences, the library, the Internet. Its lineage is correspondingly complex. The list that follows encompasses people to whom I am indebted for a variety of the usual kinds of scholarly help: inspiring students and teachers (formal and informal), research assistance, paper discussants, or editorial advice. It also includes some intellectual sparring partners, sounding boards, and providers of moral support. I offer all my appreciation, without of course imputing any responsibility to them for arguments or ideas that, in some instances at least, I quite hope they do _not_ agree with. So my thanks to: Steve Balch, Mark Blitz, Todd Bryfogle, Nigel Cameron, Mark Coda, Tobin Craig, Stephen L. Elkin, Robert Faulkner, Barbara Goldoftus, Kim Hendrickson, Shawn Igo, Leon R. Kass, Caroline Kelly, Syma Levine, Michael Lotze, Don Maletz, Michele Mekel, Thomas W. Merrill, Jacqueline Pfeffer Merrill, Andrew Morriss, Gary McEwan, Michael Myzak, Steve Ostro, Marty Plax, Betsy Rubin, Bill Rubin, Jeff Salmon, Adam Schulman, Tom Short, Ted Weinstein, and Stephen Wrinn. Thanks also to Wilfred M. McClay, whose 1994 book _The Masterless_ included a chapter on Edward Bellamy that introduced me to the fascinating story \"Dr. Heidenhoff's Process.\"\n\nThe last years of writing this book have taken place in the shadow of my Mom's developing \"dementia consistent with Alzheimer's disease,\" as the more careful physicians seem to put it. I did not particularly need this lesson to teach me the fragility of human life, nor to bring home the depth of the love between my parents. On the one hand, then, it has reinforced my belief that we still have a great deal of progress to make simply towards alleviating _human_ suffering and disease, and that however what we learn in the process of so doing might be related to grand schemes for redesigning ourselves, those schemes remain a distraction, and a rather ethically oblivious if not cruel one at that. Yet on the other hand, it has shown me more clearly than I would have wished the occasion for noble and beautiful deeds that open to loving and respectful souls, my Dad's foremost among them, as they provide the care that must be given her in the absence of cure. I cannot call these opportunities compensation, let alone silver lining, but they are a weighty moral fact, and not the first my folks taught me. So this book is dedicated to them.\n_NOTES_\n\nEPIGRAPH\n\nAlbert Einstein, \"Freedom and Science,\" trans. James Gutmann, in _Freedom: Its Meaning_ , Ruth Nanda Anshen, ed. (New York: Harcourt, Brace and Company, 1940), 382. The essay was later partially reprinted in various collections under the title \"On Freedom.\"\n\nINTRODUCTION\n\n The differences among these schools of thought are important to those who have coined and adopted the different labels, even if outsiders may not immediately appreciate what is at stake. Yet in practice the adherents of one position or another often aim to adopt a \"big tent\" outlook that seeks to minimize the sectarian tendencies implicit in the different designations. So while my use of \"transhumanism\" in this book as a generic term necessarily means some nuance will be lost, the decision is likely to be controversial only to the extent that it may imply a single-mindedness that adherents to one position or another will object to when their own ideas seem to be slighted. But that problem is balanced by the fact that by using the term in this way I follow their own movement-building example.\n\n Stewart Brand, \"Purpose,\" _Whole Earth Catalog_ , 1 (Fall 1968), 2. This phrase, originally rendered in italics with the word \"are\" underlined, was the first sentence of the mission statement Brand put at the beginning of the first issue of his _Whole Earth Catalog_ , a short-lived publication that was influential in the counterculture movement, among futurists, and among the high-tech innovators of the 1970s and 1980s\u2014a milieu in which many transhumanist ideas developed.\n\n Thus, while transhumanist philosopher Nick Bostrom acknowledges that the technologies that allow human redesign may pose their own \"existential risks,\" he also argues that the transformation of mankind may be a route to avoiding other such threats. For the briefest of statements to this effect, see his \"Existential Risks and Future Technologies,\" www.futuretech.ox.ac.uk\/existential-risks-and-future-technologies.\n\n William Faulkner, \"Banquet Speech\" (speech, Stockholm, December 10, 1950), Nobelprize.org,nobelprize.org\/nobel_prizes\/literature\/laureates\/1949\/faulknerspeech.html.\n\n See, for example, the cover of the February 21, 2011 edition of _Time_ magazine, which reads: \"2045: The Year Man Becomes Immortal.\"\n\n John Letzing, \"Google Hires Famed Futurist Ray Kurzweil,\" Digits ( _Wall Street Journal_ blog), December 14, 2012, blogs.wsj.com\/digits\/2012\/12\/14\/google-hiresfamed-futurist-ray-kurzweil\/.\n\n Singularity University, \"FAQ,\" singularityu.org\/faq\/.\n\n The classic presentation of this idea is to be found in Mihail C. Roco and William Sims Bainbridge, eds., _Converging Technologies for Improving Human Performance_ (Dordrecht, Netherlands: Kluwer Academic Publishers, 2003), wtec.org\/ConvergingTechnologies\/Report\/NBIC_report.pdf.\n\n David Levy, _Love and Sex with Robots: The Evolution of Human-Robot Relationships_ (New York: HarperCollins, 2007). The widespread attention given this book is perhaps best indicated by the fact that its author made an appearance on _The Colbert Report_ on January 17, 2008: thecolbertreport.cc.com\/videos\/ykpl7i\/david-levy.\n\n See, for example, Ray Kurzweil, _The Age of Spiritual Machines: When Computers Exceed Human Intelligence_ (New York: Penguin, 1999).\n\n See, for example, Hans Moravec, _Mind Children: The Future of Robot and Human Intelligence_ (Cambridge, Mass.: Harvard University Press, 1988).\n\n K. Eric Drexler, _Engines of Creation: The Coming Era of Nanotechnology_ (New York: Anchor Books, 1990).\n\n See, for example, Simon Young, _Designer Evolution: A Transhumanist Manifesto_ (Amherst, N.Y.: Prometheus Books, 2006).\n\n For nanotechnology and water purification see Lora Kolodny, \"Puralytics CEO On Cleaning Water With Light, Winning The Cleantech Open,\" techcrunch.com\/2010\/11\/25\/cleantech-open-winners-puralytics\/. For nanotechnology and solar energy see Xiangnan Dang, _et al_., \"Virus-templated Self-assembled Single-walled Carbon Nanotubes for Highly Efficient Electron Collection in Photovoltaic Devices,\" _Nature Nanotechnology_ 6 (2011), 377\u2013384, dx.doi.org\/10.1038\/nnano.2011.50.\n\n P. W. Singer, _Wired for War: The Robotics Revolution and Conflict in the 21st Century_ (New York: Penguin, 2009).\n\n Tractate _Eruvin_ 13b.\n\n Charles T. Rubin, _The Green Crusade: Rethinking the Roots of Environmentalism_ (New York: The Free Press, 1994), 181\u2013184, 206\u2013211.\n\n For an excellent account of the important role our visions of the future play in thinking even about the present, see Yuval Levin, _Imagining the Future: Science and American Democracy_ (New York, Encounter Books, 2008).\n\n Eric Cohen, _In the Shadow of Progress: Being Human in the Age of Technology_ (New York: Encounter Books, 2008), 81.\n\nCHAPTER ONE: THE FUTURE IN THE PAST\n\n Nick Bostrom, \"The Transhumanist FAQ: A General Introduction,\" Version 2.1 (2003), 47, transhumanism.org\/resources\/FAQv21.pdf. See also Chris Wren, \"Star Trek's Greatest Weakness,\" Mondolithic Sketchbook (blog), May 11, 2006, web.archive.org\/web\/20061015002913\/http:\/\/mondosketch.blogspot.com\/2006\/05\/star-treks-greatest-weakness.html.\n\n Keith Michael Baker, \"Marquis de Condorcet,\" in Paul Edwards, ed., _The Encyclopedia of Philosophy I_ (New York: Collier Macmillan, 1967), 182\u201384.\n\n David Williams, _Condorcet and Modernity_ (Cambridge: Cambridge University Press, 2004), 3. See also William Doyle, _The Oxford French Revolution_ (Oxford: Clarendon, 1989), 399.\n\n Williams, _Condorcet and Modernity_ , 42.\n\n _Ibid_., 43.\n\n Marie Jean Antoine Nicolas de Caritat, Marquis de Condorcet, _Outlines of an Historical View of the Progress of the Human Mind_ (Philadelphia: M. Cary and Co., 1796), 245, oll.libertyfund.org\/titles\/condorcet-outlines-of-an-historicalview-of-the-progress-of-the-human-mind.\n\n _Ibid_., 252.\n\n _Ibid_., 289.\n\n _Ibid_., 290.\n\n _Ibid_., 291.\n\n _Ibid_., 289.\n\n _Ibid_., 289\u201390.\n\n _Ibid_., 292.\n\n Francis Bacon, _New Atlantis and The Great Instauration_ (Wheeling, Illinois: Harlan Davidson, 1989), 71.\n\n Thomas Robert Malthus, _An Essay on the Principle of Population, As It Affects the Future Improvement of Society, with Remarks on the Speculations of Mr. Godwin, M. Condorcet, and Other Writers_ (London: J. Johnson, 1798), archive.org\/stream\/essayonprincipl00malt#page\/n7\/mode\/2up. Condorcet's _Outlines_ first appeared in English in 1795; Mathus's _Essay_ came out three years later.\n\n David N. Reznick, _The \"Origin\" Then and Now: An Interpretive Guide to the \"Origin of Species\"_ (Princeton: Princeton University Press, 2010), 68.\n\n The Darwin Correspondence Project, \"Spotlight on a Correspondent: William Winwood Reade,\" Darwin and Human Nature (blog), May 20, 2011, darwinhumannature.wordpress.com\/2011\/05\/20\/spotlight-on-a-correspondent-william-winwood-reade\/.\n\n \"One always turns back to Winwood Reade's _Martyrdom of Man_ for renewal of faith.\" (W. E. B. Du Bois, _The World and Africa_ [New York: The Viking Press, 1947], x.) Rhodes called the book \"creepy,\" but also said \"That book has made me what I am.\" (Princess Catherine Radziwi\u0142\u0142, _My Recollections_ [London: Ibister and Company, 1904], 342.) Wells considered Reade a \"great and penetrating genius.\" (Leon Stover, \"Applied Natural History: Wells vs. Huxley,\" in Patrick Parrinder and Christopher Rolfe, eds., _H. G. Wells under Revision: Proceedings of the International H. G. Wells Symposium_ [Selinsgrove, Penn.: Susquehanna University Press, 1990], 127.) Orwell wrote, \"however hasty and unbalanced\" Reade's book may seem, it \"shows an astonishing width of vision.\" (George Orwell, _The Collected Essays, Journalism and Letters of George Orwell_ , Sonia Orwell and Ian Angus, eds., vol. 3, _My Country Right or Left_ , 1940\u20131943 [New York, Harcourt Brace Jovanovich, 1968], 37.)\n\n Sir Arthur Conan Doyle, _The Sign of the Four_ (orig. 1890), in _The Annotated Sherlock Holmes_ , William S. Baring-Gould, ed. (New York: Wings\/Random House, 1992), vol. 1, 619. I say \"perhaps\" Holmes admires Reade because \"remarkable\" does not have to be a term of praise, and Reade's book may just be a way for Watson to while away an hour. There is less ambiguous praise later in _The Sign of the Four_ : \"'Winwood Reade is good upon the subject,' said Holmes. 'He remarks that, while the individual man is an insoluble puzzle, in the aggregate he becomes a mathematical certainty.'\" ( _Ibid_., 666.)\n\n Winwood Reade, _The Martyrdom of Man_ (London: Kegan Paul, Trench, Tr\u00fcbner & Co., 1910), xlvii, archive.org\/stream\/martyrdomofman00readuoft.\n\n _Ibid_., 395\u2013400.\n\n _Ibid_., 10.\n\n Whether or not Reade was familiar with Immanuel Kant, the argument recalls the nineteenth-century German philosopher's position in works such as \"Idea for a Universal History with a Cosmopolitan Intent,\" collected in Immanuel Kant, _Perpetual Peace and Other Essays_ (Indianapolis: Hackett Publishing Co., 1983), 29\u201340.\n\n Reade, _The Martyrdom of Man_ , 10.\n\n _Ibid_., 392.\n\n _Ibid_., 410.\n\n _Ibid_., 393.\n\n _Ibid_., 522\u2013523.\n\n _Ibid_., 443.\n\n _Ibid_., 467.\n\n _Ibid_., 512.\n\n _Ibid_., 512.\n\n _Ibid_., 464.\n\n _Ibid_., 452.\n\n _Ibid_., 505, 516ff.\n\n Note that Reade also says, \"Finally, men will master the forces of nature; they will become themselves architects of systems, manufacturers of worlds. . . . There is but a difference in degree between the chemist who to-day arranges forces in his laboratory so that they produce a gas, and the creator who arranges forces so that they produce a world; between the gardener who plants a seed, and the creator who plants a nebula.\" So perhaps we will not always be so far below the \"Power by whom the universe was made\" after all. Reade, _The Martyrdom of Man_ , 515, 467.\n\n \"Human knowledge and human power meet in one; for where the cause is not known the effect cannot be produced. Nature to be commanded must be obeyed.\" Francis Bacon, _The New Organon_ (Aphorisms I.3), Fulton H. Anderson, ed. (Indianapolis: Bobbs-Merrill, 1960), 39.\n\n Reade, _The Martyrdom of Man_ , 513.\n\n _Ibid_., 511.\n\n _Ibid_., 515.\n\n _Ibid_., 514\u201315\n\n _Ibid_., 515.\n\n _Ibid_., 538\u201339.\n\n _Ibid_., 543.\n\n Elisabeth Koutaissoff, \"Introduction,\" in Nikolai Fedorovich Fedorov, _What Was Man Created For?: The Philosophy of the Common Task_ (selected works), Elisabeth Koutaissoff and Marilyn Minto, trans. and eds. (Lausanne: Honeyglen Publishing\/L'Age d'Homme, 1990), 16.\n\n Charles Tandy and R. Michael Perry, \"Nikolai Fedorovich Fedorov (1829\u20131903)\" _Internet Encyclopedia of Philosophy_ , iep.utm.edu\/fedorov\/.\n\n Koutaissoff, \"Introduction,\" 16.\n\n Fedorov, _What Was Man Created For?_ , 36.\n\n _Ibid_., 34.\n\n _Ibid_., 36.\n\n _Ibid_., 39.\n\n _Ibid_., 42, 43.\n\n _Ibid_., 53\u201354.\n\n _Ibid_., 54.\n\n _Ibid_., 65.\n\n _Ibid_., 56, 43\u201344.\n\n _Ibid_., 91.\n\n _Ibid_., 60\u201361, 66, 43.\n\n _Ibid_., 90. For an interesting comment on the likeness between Fedorov and the transhumanists, see Charles Stross, \"Federov's  _sic_ ] Rapture,\" Charlie's Diary (blog), July 1, 2011, [www.antipope.org\/charlie\/blog-static\/2011\/07\/federovsrapture.html.\n\n Fedorov, _op cit_., 88.\n\n _Ibid_., 96.\n\n _Ibid_., 97, 96.\n\n _Ibid_., 97.\n\n _Ibid_., 97.\n\n _Ibid_., 97\u201398.\n\n _Ibid_., 127\u2013128.\n\n Camille Flammarion, _Astronomy for Amateurs_ , trans. Francis A. Welby (New York: D. Appleton and Co., 1915 orig. 1904]), 319, [archive.org\/stream\/astronomyforama00welbgoog.\n\n _Ibid_., 325\u2013326.\n\n _Ibid_., 338.\n\n _Ibid_., 331.\n\n _Ibid_., 338.\n\n _Ibid_., emphasis added.\n\n _Ibid_., 337.\n\n _Ibid_., 340.\n\n Richard Jeffery, \"C. S. Lewis and the Scientists,\" _The Chronicle of the Oxford University C. S. Lewis Society_ 2, no. 2 (May 2005), 17\u201319.\n\n In the United Kingdom, \"The Last Judgment\" was published as the final piece in the book _Possible Worlds and Other Essays_ in 1927. It was excluded from the American edition of that book, because the piece was instead published in the March 1927 edition of _Harper's Magazine_ and as a small standalone book the same year by Harper & Brothers. (See Mark B. Adams, \"Last Judgment: The Visionary Biology of J. B. S. Haldane,\" _Journal of the History of Biology_ 33, no. 3 [Winter 2000], 462.) \"The Last Judgment\" is a sort of sequel to Haldane's 1923 lecture _Daedalus, or Science and the Future_ , which I analyzed in Charles T. Rubin, \"Daedalus and Icarus Revisited,\" _The New Atlantis_ 8 (Spring 2005), 73\u201391.\n\n C. S. Lewis, \"On Science Fiction,\" in _Of Other Worlds: Essays and Stories_ (San Diego: Harvest\/Harcourt, 1994 [orig. 1966]), 66.\n\n J. B. S. Haldane, _Possible Worlds and Other Essays_ (London: Chatto and Windus, 1927), 292.\n\n _Ibid_., 295.\n\n _Ibid_.\n\n _Ibid_.\n\n _Ibid_.\n\n _Ibid_., 302.\n\n _Ibid_., 303.\n\n _Ibid_.\n\n _Ibid_.\n\n _Ibid_., 309.\n\n _Ibid_.\n\n _Ibid_., 310.\n\n _Ibid_.\n\n _Ibid_., 311.\n\n _Ibid_., 311\u2013321.\n\n _Ibid_., 312.\n\n _Ibid_., 310, 312.\n\n Andrew Brown, _J. D. Bernal: The Sage of Science_ (Oxford: Oxford University Press, 2006), 54.\n\n _Ibid_., 238\u2013254, 158\u2013159.\n\n _Ibid_., 341, 416, 432.\n\n J. D. Bernal, _The World, the Flesh and the Devil: An Enquiry into the Future of the Three Enemies of the Rational Soul_ (Bloomington: Indiana University Press, 1969), v.\n\n _Ibid_., 6, 5.\n\n _Ibid_., 15.\n\n The first spacecraft to rely on solar sails was a Japanese probe given the acronym IKAROS, after Icarus, the mythological figure who perished after flying too close to the sun; he is depicted on the cover of this book and discussed at length in Chapter Five.\n\n Bernal, _The World, the Flesh and the Devil_ , 18, 25.\n\n _Ibid_., 27.\n\n _Ibid_., 32.\n\n _Ibid_., 35.\n\n _Ibid_., 34.\n\n _Ibid_., 34.\n\n _Ibid_., 37.\n\n _Ibid_., 36.\n\n _Ibid_., 37\u201338.\n\n _Ibid_., 42.\n\n _Ibid_., 38\u201341. Bernal paints a vivid picture of the mechanical men he envisions: \"Instead of the present body structure we should have the whole framework of some very rigid material, probably not metal but one of the new fibrous substances. In shape it might well be rather a short cylinder. Inside the cylinder, and supported very carefully to prevent shock, is the brain with its nerve connections, immersed in a liquid of the nature of cerebro-spinal fluid, kept circulating over it at a uniform temperature. The brain and nerve cells are kept supplied with fresh oxygenated blood and drained of de-oxygenated blood through their arteries and veins which connect outside the cylinder to the artificial heart-lung digestive system\u2014an elaborate, automatic contrivance. . . . The brain thus guaranteed continuous awareness, is connected in the anterior of the case with its immediate sense organs, the eye and the ear\u2014which will probably retain this connection for a long time. The eyes will look into a kind of optical box which will enable them alternatively to look into periscopes projecting from the case, telescopes, microscopes and a whole range of televisual apparatus. The ear would have the corresponding microphone attachments and would still be the chief organ for wireless reception. Smell and taste organs, on the other hand, would be prolonged into connections outside the case and would be changed into chemical tasting organs, achieving a more conscious and less purely emotional role than they have at present. . . . Attached to the brain cylinder would be its immediate motor organs, corresponding to but much more complex than, our mouth, tongue and hands. This appendage system would probably be built up like that of a crustacean which uses the same general type for antenna, jaw and limb; and they would range from delicate micromanipulators to levers capable of exerting considerable forces, all controlled by the appropriate motor nerves. . . . The remaining organs would have a more temporary connection with the brain-case. There would be locomotor apparatus of different kinds, which could be used alternatively for slow movement, equivalent to walking, for rapid transit and for flight. On the whole, however, the locomotor organs would not be much used because the extension of the sense organs would tend to take their place. Most of these would be mere mechanisms quite apart from the body; there would be the sending parts of the television apparatus, tele-acoustic and tele-chemical organs, and tele-sensory organs of the nature of touch for determining all forms of textures. Besides these there would be various tele-motor organs for manipulating materials at great distances from the controlling mind. These extended organs would only belong in a loose sense to any particular person, or rather, they would belong only temporarily to the person who was using them and could equivalently be operated by other people.\"\n\n _Ibid_., 40, 41.\n\n _Ibid_., 41.\n\n _Ibid_., 43.\n\n _Ibid_., 46.\n\n _Ibid_., 47.\n\n _Ibid_., vi.\n\n _Ibid_., 56.\n\n _Ibid_., 53.\n\n _Ibid_.\n\n _Ibid_., 56, 57.\n\n _Ibid_., 58.\n\n _Ibid_., 60.\n\n Interestingly, the word \"cyborg\" (cybernetic organism), now popularly used to describe beings that are part living and part machine, was first coined in 1960 to propose a way to adapt human beings for the hostile environments of space exploration. See Manfred E. Clynes and Nathan S. Kline, \"Cyborgs and space,\" _Astronautics_ 14, no. 9 (September 1960), 26\u201327, 74\u201376.\n\n Bernal, _The World, the Flesh and the Devil_ , 63.\n\n _Ibid_., 64.\n\n _Ibid_., 65.\n\n _Ibid_., 66.\n\n _Ibid_.\n\n _Ibid_., 70, 75.\n\n _Ibid_., 74.\n\n _Ibid_.\n\n _Ibid_., 78\u201379.\n\n _Ibid_., 79.\n\n _Ibid_., 80.\n\n Bernal worries that his vision may fail to be convincing for two kinds of reasons. On the one hand, perhaps he has not portrayed a future that is sufficiently \"mysterious and full of supernatural power.\" Paradoxically, the very desire for such a future has driven the creation of the world today, a world in which we have the non-mysterious power to fulfill all our desires except this one. Will the fact that the future is now becoming \"a function of our own action\" turn us away from further achievement, or will we accept that an unfulfillable desire will drive us ever onwards? On the other hand, Bernal acknowledges that \"we shall have very sane reactionaries at all periods warning us to remain in the natural and primitive state of humanity, which is usually the last stage but one in their cultural history.\" But he is not terribly worried that such critics, however sane, might actually prevail. He suggests that \"the secondary consequences of what men have already done\u2014the reactionaries as much as any\u2014will carry them away then as now.\" As even reactionaries are only committed to preserving \"the last stage but one,\" they are forced to be progressive by the brute facts that are being established around them by those who have no corresponding qualms. _Ibid_., 80\u201381, 65.\n\n _Ibid_., 68.\n\n _Ibid_., 64.\n\n David Pearce, \"The Hedonistic Imperative\" (online publication, 1995), hedweb.com\/hedethic\/hedonist.htm.\n\n \"The need to refuel three times daily is most frustrating. . . . The requirement for the daily evacuation of unpleasantly odorous waste products from an orifice needlessly situated directly next to the sexual organs is an obvious design fault.\" Simon Young, _Designer Evolution: A Transhumanist Manifesto_ (Amherst, N.Y.: Prometheus Books, 2006), 28.\n\n Jean-Jacques Rousseau, _The First and Second Discourses_ , Roger D. Masters, ed. (New York: St. Martin's Press, 1964), 114\u2013115.\n\n Condorcet, 290.\n\n The power of this vision of space as the new New World is such that still today, when we have gained so much knowledge about the actual technical challenges of space travel, visionary proponents of this solution to Malthus's problem remain. See, for example, Marshall T. Savage, _The Millennial Project: Colonizing the Galaxy in Eight Easy Steps_ (Boston: Little, Brown and Company, 1994).\n\n Timothy Shanahan, _The Evolution of Darwinism: Selection, Adaptation, and Progress in Evolutionary Biology_ (Cambridge: Cambridge University Press, 2004), 173\u2013195.\n\n _Ibid_., 276.\n\n Flammarion, 325.\n\n Flammarion, 325.\n\n George Santayana, _The Life of Reason or The Phases of Human Progress_ (New York: Charles Scribner's Sons, 1906), 284.\n\n Adam Keiper, \"The Age of Neuroelectronics,\" _The New Atlantis_ 11 (Winter 2006), 4\u201341.\n\n See the \"Wi-Fi Detector Shirt\" sold by ThinkGeek, thinkgeek.com\/tshirts\/illuminated\/991e\/.\n\n See the account of actual experiments involving head transplants and monkey and dog brains in a bottle in Mary Roach, _Stiff: The Curious Lives of Human Cadavers_ (New York: W. W. Norton, 2003).\n\nCHAPTER TWO: DISCOVERING INHUMANITY\n\n Mark Chartrand, \"What To Do When ET Comes,\" _Ad Astra_ 7, no. 2 (March\/April 1995), 54.\n\n Shawn Carlson, \"Science and Society: SETI on Earth,\" _The Humanist_ 51 (September\/October 1991), 37.\n\n Donald Tarter, \"Practicing Safe Science: Treading on the Edge,\" _Skeptical Inquirer_ 17, no. 3 (Spring 1993), 295.\n\n Carl Sagan, \"The Search for Signals From Space,\" _Parade_ , September 19, 1993, 4\u20137.\n\n Jeffrey Bennett, _Beyond UFOs: The Search for Extraterrestrial Life and Its Astonishing Implications for Our Future_ (Princeton: Princeton University Press, 2008), 204.\n\n Inspiration for the ending of this prologue is owed to Walker Percy, _Lost in the Cosmos: The Last Self-Help Book_ (New York: Washington Square Press, 1983), 255\u2013256.\n\n Ray Kurzweil, _The Singularity Is Near: When Humans Transcend Biology_ (New York: Viking: 2005), 357.\n\n Nick Bostrom, \"Where Are They? Why I Hope the Search for Extraterrestrial Life Finds Nothing,\" _Technology Review_ (May\/June, 2008), 72\u201377; quoted from nickbostrom.com\/extraterrestrial.pdf, 9.\n\n Eric M. Jones, \"Estimates of Expansion Timescales,\" in B. Zuckerman and M. H. Hart, eds., _Extraterrestrials: Where Are They?_ 2nd ed. (Cambridge: Cambridge University Press, 1995), 92\u2013102.\n\n Paul Davies, _The Eerie Silence: Renewing Our Search for Alien Intelligence_ (Boston: Mariner Books, 2010), 167.\n\n \"Fermi Paradox,\" Wikipedia, en.wikipedia.org\/wiki\/Fermi_paradox.\n\n This topic is sometimes addressed by scientists interested in the search for extraterrestrial intelligence (SETI). Perhaps conflating technological prowess with moral decency, pioneering SETI astronomer Jill Tarter suggests that, \"If aliens were able to visit Earth, that would mean they would have technological capabilities sophisticated enough not to need slaves, food or other planets\" (\"Sci-Fi Movies Are Wrong About Aliens, E.T. Hunter Jill Tarter Says,\" Space.com, May 25, 2012, space.com\/15870-aliens-earth-science-fiction-tarter-seti.html). On the other hand, her SETI colleague Seth Shostak believes that any aliens who come visit us will probably have \"a more aggressive personality. And if they have the technology to come here, the idea that we can take them on is like Napoleon taking on the] U.S. Air Force. We're not going to be able to defend ourselves very well\" (Sue Karlin, \"The Aliens Would Win,\" _IEEE Spectrum_ , June 6, 2012, [spectrum.ieee.org\/tech-talk\/geek-life\/hands-on\/the-aliens-would-win). The theoretical physicist Stephen Hawking periodically makes a splash by suggesting that contact with aliens might not be a good idea. For instance, in 2010 he said, \"If aliens ever visit us, I think the outcome would be much as when Christopher Columbus first landed in America, which didn't turn out very well for the Native Americans. . . . I imagine they might exist in massive ships, having used up all the resources from their home planet. Such advanced aliens would perhaps become nomads, looking to conquer and colonize whatever planets they can reach\" (Chris Matyszczyk, \"Stephen Hawking: Aliens Might Hate Us,\" CNET, April 25, 2010, cnet.com\/8301-17852_3-20003358-71.html).\n\n Seth Shostak, _Sharing the Universe: Perspectives on Extraterrestrial Life_ (Berkeley: Berkeley Hills Books, 1998), 7.\n\n _Ibid_., 190.\n\n _Ibid_.\n\n I. S. Shklovskii and Carl Sagan, _Intelligent Life in the Universe_ (San Francisco: Holden-Day, 1966), 358.\n\n Frank Drake and Dava Sobel, _Is Anyone Out There? The Scientific Search for Extraterrestrial Intelligence_ (New York: Dell Publishing, 1992), 159.\n\n Carl Sagan, _The Cosmic Connection: An Extraterrestrial Perspective_ (New York: Anchor Books, 1973), 218.\n\n Gerald Feinberg and Robert Shapiro, _Life Beyond Earth: The Intelligent Earthling's Guide to Life in the Universe_ (New York: William Morrow and Co., 1980), 436.\n\n Shklovskii and Sagan, _Intelligent Life in the Universe_ , 412.\n\n \"Interview with Dr. Carl Sagan, Planetary Society Founder, SETI Pioneer\" in Frank White, _The SETI Factor: How the Search for Extraterrestrial Intelligence is Changing Our View of the Universe and Ourselves_ (New York, Walker and Co., 1990), 196.\n\n White, _The SETI Factor_ , 85.\n\n Jared Diamond, \"Alone in a Crowded Universe,\" in Zuckerman and Hart, _Extraterrestrials_ , 163.\n\n Zuckerman and Hart, _Extraterrestrials_ , x.\n\n Robin Hanson, \"The Great Filter: Are We Almost Past It?\" (online article), September 15, 1998, hanson.gmu.edu\/greatfilter.html.\n\n Sagan, _The Cosmic Connection_ , 232. Somewhat more charitably, Sagan also says, \"I think it is a great conceit, the idea of the present Earth establishing radio contact and becoming a member of a galactic federation\u2014something like a bluejay or an armadillo applying to the United Nations\" (242).\n\n _Ibid_., 224.\n\n _Ibid_., 219.\n\n _Ibid_.\n\n The English scientist William Whewell imagined unintelligent and distinctly non-humanoid alien life, based on the physics and chemistry of other planets, in his 1853 book _Of the Plurality of Worlds_ ; the astronomer Camille Flammarion, as we saw in Chapter One, imagined greatly varied alien life; the two French-Belgian science and science fiction writers who shared the pseudonym J. H. Rosny imagined intelligent alien life that was very different from terrestrial life; the evolutionary biologist Alfred Russel Wallace wrote in 1904 that alien life \"with a mind and spiritual nature equal to that of man might have been developed in a very different form\"; the naturalist Loren Eiseley, in a 1953 essay later collected in his book _The Immense Journey_ (1957), argued that alien life \"will not wear the shape of man\"\u2014a point on which most scientific theorists now seem to agree.\n\n Nicholas Rescher, _Finitude: A Study of Cognitive Limits and Limitations_ (Frankfurt: Ontos Verlag, 2011), 20\u201327.\n\n Giuseppi Cocconi and Philip Morrison, \"Searching for Interstellar Communications,\" _Nature_ 184, no. 4690 (September 19, 1959), 844\u2013846, dx.doi.org\/10.1038\/184844a0.\n\n Carl Sagan, ed., _Communication with Extraterrestrial Intelligence_ (Cambridge, Mass.: MIT Press, 1973), 337.\n\n _Ibid_., 337\u2013338.\n\n _Ibid_., 337.\n\n Richard Berendzen, ed., _Life Beyond Earth and the Mind of Man_ (Washington, D.C., National Aeronautics and Space Administration, 1973), 44. Later collected in Philip Morrison, _Nothing Is Too Wonderful to Be True_ (Woodbury, New York: AIP Press, 1995), 178.\n\n Shklovskii and Sagan, _Intelligent Life in the Universe_ , 396\u2013397.\n\n Drake and Sobel, _Is Anyone Out There_?, 159.\n\n Greg Easterbrook, \"Are We Alone?,\" _The Atlantic Monthly_ 262, no. 2 (August 1988), 37, theatlantic.com\/past\/docs\/issues\/88aug\/easterbr.htm.\n\n Drake and Sobel, _Is Anyone Out There_?, 160. It should be no surprise that hopes for immortality provide one of the themes that link SETI with technologies promoting transhuman transformations. Robert Freitas, for example, has written on xenobiology, the theoretical study of alien biology, but he is also one of the great proponents of the medical promise of nanotechnology, a source of many transhumanist hopes. (See Robert A. Freitas Jr., _Xenology: An Introduction to the Scientific Study of Extraterrestrial Life, Intelligence, and Civilization_ Sacramento: Xenology Research Institute, 1979] and his website xenology. info, as well as his multivolume work _Nanomedicine_ , published by Landes Bioscience beginning in 1999, [nanomedicine.com.) For an intellectual history of such links that is intelligent, illuminating, and amusing, see Ed Regis, _Great Mambo Chicken and the Transhuman Condition: Science Slightly Over the Edge_ (Reading, Mass.: Perseus Books, 1990).\n\n For a popular treatment of the contingent factors that shape technological development, see James Burke, _Connections_ (Boston: Little Brown and Co., 1978).\n\n Paul Davies, _Are We Alone? Philosophical Implications of the Discovery of Extraterrestrial Life_ (New York: Basic Books, 1995), 55.\n\n See, for instance, C. S. Lewis, \"Religion and Rocketry,\" in _The World's Last Night and Other Essays_ (New York: Harcourt, Brace and Company, 1960), 83\u201392.\n\n J. B. S. Haldane, \"The Last Judgment,\" in _Possible Worlds_ (London: Chatto and Windus, 1927), 288.\n\n Berendzen, _Life Beyond Earth and the Mind of Man_ , 21.\n\n Ray Kurzweil, _The Singularity Is Near: When Humans Transcend Biology_ (New York: Viking, 2005), 21, 364\u2013365. On this theme see Charles Stross, _Accelerando_ (New York: Ace Books, 2005). Interest in searching for signs of SETI based on the more exotic possibilities that transhumanism has suggested for the human future has only grown in recent years. It is an important theme in Davies, _The Eerie Silence_ , 93\u2013115.\n\n J. A. Ball, \"The Zoo Hypothesis\" in Donald Goldsmith, ed., _The Quest for Extraterrestrial Life: A Book of Readings_ (Mill Valley, Cal.: University Science Books, 1980), 242; P. A. Sturrock, \"Uncertainty in Estimates of the Number of Extraterrestrial Civilizations\" in M. D. Papagiannis, ed., _Strategies for the Search for Life in the Universe_ (Boston: D. Seidel, 1980), 71.\n\n Sagan, _Cosmic Connection_ , 222; R. N. Bracewell, \"Communications from Superior Galactic Communities,\" in Goldsmith, _The Quest for Extraterrestrial Life_ , 105\u2013107.\n\n Michael H. Hart, \"An Explanation for the Absence of Extraterrestrials on Earth,\" in Zuckerman and Hart, _Extraterrestrials_ , 1\u20134. But see also Sebastian von Hoerner, \"Where is Everybody?\" and \"The General Limits of Space Travel,\" in Goldsmith, _The Quest for Extraterrestrial Life_ , 250\u201354 and 197\u2013204.\n\n Sebastian von Hoerner, \"Where is Everybody?\" in Goldsmith, _The Quest for Extraterrestrial Life_ , 250\u2013254. Although von Hoerner notes this possibility, he does not find it convincing.\n\n Davies, _The Eerie Silence_ , 82.\n\n _Ibid_., 167.\n\n Freeman J. Dyson, \"Search for Artificial Stellar Sources of Infrared Radiation,\" _Science_ 131, no. 3414 (June 3, 1960), 1667\u20131668, dx.doi.org\/10.1126\/science.131.3414.1667, also collected in Goldsmith, _The Quest for Extraterrestrial Life_ , 108\u2013109.\n\n M. D. Papagiannis, \"The Number N of Galactic Civilizations Must Be Either Very Large or Very Small,\" in Papagiannis, _Strategies for the Search for Life in the Universe_ , 55. Cf. also Robert Lemos, \"Rocket Scientists Say We'll Never Reach the Stars,\" Wired.com, August 19, 2008, archive.wired.com\/science\/space\/news\/2008\/08\/space_limits and Valerie Jamieson, \"Starship Pilots: Speed Kills, Especially Warp Speed,\" NewScientist.com, February 17, 2010, newscientist.com\/article\/dn18532.\n\n Arwen Nicholson and Duncan Forgan, \"Slingshot dynamics for self-replicating probes and the effect on exploration timescales,\" _International Journal of Astrobiology_ 12, iss. 4 (October 2013), 337\u2013344, dx.doi.org\/10.1017\/S1473550413000244.\n\n Carl Sagan, _Contact_ (New York: Simon and Schuster\/Pocket, 1986 [orig. 1985]), 367\u2013368, 416\u2013422, 430\u2013431.\n\n Sagan, _Cosmic Connection_ , 222.\n\n Consider, for example, the \"Invitation to ETI\" project founded by Allen Tough, which hopes to \"establish communication with any form of extraterrestrial intelligence able to monitor our World Wide Web\" (ieti.org).\n\n Drake and Sobel, _Is Anyone Out There_?, 161.\n\n Arthur C. Clarke, _Childhood's End_ (New York: Harcourt, Brace and World, 1953), 15.\n\n _Ibid_., 16\u201317.\n\n _Ibid_., 24.\n\n _Ibid_., 41.\n\n _Ibid_., 42.\n\n _Ibid_., 14.\n\n _Ibid_., 58.\n\n _Ibid_., 68.\n\n _Ibid_., 72\u201373.\n\n _Ibid_., 136.\n\n _Ibid_., 111.\n\n _Ibid_., 135.\n\n _Ibid_., 175.\n\n _Ibid_., 209.\n\n Not for nothing did C. S. Lewis say that he was \"bowled over\" by Clarke's novel, finding it \"quite out of range of the common space-and-time writers,\" an \"ABSOLUTE CORKER\" that even \"brought tears to my eyes.\" C. S. Lewis to Joy Gresham, December 22, 1953, in _The Collected Letters of C. S. Lewis, Volume 3: Narnia, Cambridge, and Joy, 1950\u20131963_ , ed. Walter Hooper (New York: HarperCollins, 2007), 390\u2013392.\n\n _Childhood's End_ , 12.\n\n _Ibid_., 14.\n\n _Ibid_.\n\n _Ibid_., 21.\n\n _Ibid_.\n\n _Ibid_., 20.\n\n _Ibid_., 182\u2013183.\n\n _Ibid_., 70, 74.\n\n _Ibid_., 136.\n\n _Ibid_., 95.\n\n _Ibid_., 190.\n\n _Ibid_., 77\u201378.\n\n _Ibid_., 90.\n\n _Ibid_., 91, 141.\n\n As he describes New Athens, Clarke shows he has in mind Francis Bacon, one of the founders of modern science. The character who is \"the driving force behind New Athens\" is given the name Ben Salomon. In Bacon's famous story \"New Atlantis,\" he depicts a secretive island that is home to \"Salomon's House,\" what we would now recognize as an institute for scientific research, founded by the late King Solamona. While, in Plato's original myth, Athens was victorious over Atlantis, in Clarke's story the material satisfactions prefigured in Bacon's \"New Atlantis\" show their power over the creative efforts of his New Athens. (See Francis Bacon, _New Atlantis and The Great Instauration_ , ed. Jerry Weinberger [Wheeling, Il.: Harlan Davidson (1989, revised edition)], 56, 58.)\n\n Clarke, _Childhood's End_ , 164.\n\n _Ibid_., 20, 168.\n\n _Ibid_., 161.\n\n _Ibid_., 68\u201369.\n\n _Ibid_., 208, 184.\n\n _Ibid_., 185.\n\n _Ibid_.\n\n _Ibid_., 183.\n\n _Ibid_., 207.\n\n _Ibid_., 204.\n\n _Ibid_., 186.\n\n _Ibid_., 135.\n\n _Ibid_., 184.\n\n _Ibid_., 184.\n\n That the Overmind begins to look like a stand-in for a rather traditional God is also suggested by the fact that its angels seem to be slowly and quietly preparing a revolt. The Overlords are not quite so resigned to their task as they make themselves out to be. They use Jan Rodricks to gather information about the Overmind, both on their own planet and when the last moments of Earth approach. As a human, he can see things that are hidden from them because of their complete lack of paranormal powers. Their effort parallels the investigations that Stormgren undertakes at the end of his career. In neither case ought we to conclude that there are immediate plans or even hopes of altering the hierarchy\u2014but if ever an effort were to be made, it would have to be based on such intelligence. The Overlords are loyal, because so far they have no choice but to obey. But they are also patient and long lived. That they want to understand their master better may be more than idle curiosity; perhaps they would like to have a choice whether or not to obey.\n\n See, for example, the justification that Plutarch offers for the Spartan policies of the community of wives and the exposure of ill-formed infants in his \"Lycurgus.\" Bernadette Perrin, trans., _Plutarch's Lives_ (Cambridge, Mass.: Harvard University Press, 1914), 251\u201355.\n\nCHAPTER THREE: ENABLING INHUMANITY\n\n Nick Bostrom, \"The Transhumanist FAQ: A General Introduction,\" Version 2.1 (2003), 9, transhumanism.org\/resources\/FAQv21.pdf.\n\n Simon Young, _Designer Evolution: A Transhumanist Manifesto_ (Amherst, N.Y.: Prometheus Books, 2006), 52.\n\n Bostrom, \"The Transhumanist FAQ,\" 49, 26.\n\n Ray Kurzweil, _The Age of Spiritual Machines: When Computers Exceed Human Intelligence_ (New York: Penguin, 1999), 145.\n\n _Ibid_., 146.\n\n The Project on Emerging Nanotechnologies at the Woodrow Wilson Center has since 2005 published a \"Nanotechnology Consumer Products Inventory\" that, as of its October 2013 update, lists some 1,628 products that use\u2014or at least _claim_ to use\u2014nanoparticles or nanomaterials. See nanotechproject.org\/cpi\/.\n\n See, for example, \"Engineers Build World's Smallest, Fastest Nanomotor\" (press release), University of Texas at Austin, Cockrell School of Engineering, May 20, 2014, engr.utexas.edu\/features\/nanomotors.\n\n Ortwin Renn and Mihail C. Roco, \"Nanotechnology and the Need for Risk Governance,\" _Journal of Nanoparticle Research_ 8, iss. 2 (April 2006), 153\u2013191, dx.doi.org\/10.1007\/s11051-006-9092-7.\n\n K. Eric Drexler, \"Molecular engineering: An approach to the development of general capabilities for molecular manipulation,\" _Proceedings of the National Academy of Sciences_ 78, no. 9 (September 1981), 5275\u20135278.\n\n K. Eric Drexler, _Engines of Creation: The Coming Era of Nanotechnology_ (New York: Anchor Books, 1990 orig. 1986]), 5. Drexler has placed the complete text of the book online for free at [e-drexler.com\/d\/06\/00\/EOC\/EOC_Table_of_Contents.html.\n\n _Ibid_., 53\u201363.\n\n _Ibid_., 95.\n\n _Ibid_., 81.\n\n _Ibid_., 93\u201395.\n\n _Ibid_., 99\u2013116.\n\n _Ibid_., 106.\n\n _Ibid_., 130\u2013146.\n\n _Ibid_., 111.\n\n The physicist Richard Feynman is often credited as an important inspiration for nanotechnology; see Richard P. Feynman, \"There's Plenty of Room at the Bottom,\" _Engineering and Science_ 23, no. 5 (February 1960), 22\u201336, calteches.library.caltech.edu\/1976\/1\/1960Bottom.pdf. For a thoughtful and critical assessment of this claim and Drexler's role, see Adam Keiper, \"The Nanotechnology Revolution,\" _The New Atlantis_ 2 (Summer 2003), 17\u201334, and Adam Keiper, \"Feynman and the Futurists,\" _Wall Street Journal_ , January 8, 2010, wsj. com\/news\/articles\/SB10001424052748703580904574638160601840456.\n\n National Nanotechnology Initiative, \"Nanotechnology Timeline\" (undated), nano.gov\/timeline.\n\n Drexler has written thoughtfully about how his kind of scientifically informed speculation about technological possibilities differs from the ways we usually conceive of science. His preferred term is \"exploratory engineering,\" which emphasizes both the theoretical nature of his ideas and the fact that they are aimed at accomplishing potential technical goals. It is a mistake, Drexler argues, to judge exercises in exploratory engineering by the standards of either science or engineering, which have different goals and methods. K. Eric Drexler, _Radical Abundance: How a Revolution in Nanotechnology Will Change Civilization_ (New York: PublicAffairs, 2013), 132\u2013144. See also K. Eric Drexler, _Nanosystems: Molecular Machinery, Manufacturing, and Computation_ (New York: John Wiley & Sons, 1992), 489\u2013506, where he makes the same argument but prefers to use the term \"theoretical applied science.\"\n\n Ed Regis, \"The Incredible Shrinking Man,\" _Wired_ (October 2004), archive.wired.com\/wired\/archive\/12.10\/drexler.html.\n\n _Ibid_., 241. This phrase, which appears in Drexler's 1990 afterword to _Engines of Creation_ , appears partially in italics in his book.\n\n _Ibid_., 171.\n\n _Ibid_., 172.\n\n In the years since _Engines of Creation_ was first published, Drexler has taken pains to downplay the \"gray goo\" problem, including in the 1990 afterword to _Engines_ , and in his most recent book: K. Eric Drexler, _Radical Abundance_ , 201 _n_. For more on the public reaction to the \"gray goo\" notion, see also W. Patrick McCray, _The Visioneers: How a Group of Elite Scientists Pursued Space Colonies, Nanotechnologies, and a Limitless Future_ (Princeton: Princeton University Press, 2013), 192, 248, 251.\n\n Drexler, _Engines of Creation_ , 174.\n\n _Ibid_.\n\n _Ibid_., 195.\n\n _Ibid_., 176.\n\n _Ibid_., 32.\n\n _Ibid_., 39. Drexler went on to co-found, and for many years was closely associated with, the Foresight Institute, an organization dedicated to studying these questions about nanotechnology.\n\n _Ibid_., 77.\n\n _Ibid_., 39.\n\n _Ibid_., 189.\n\n _Ibid_., 26.\n\n _Ibid_., 38. Drexler has since given further thought to the role of design versus evolution in understanding how nanotechnology might develop, but it is not clear that there is any fundamental shift in his belief that nanotechnology will allow deliberate design to replace random evolution. K. Eric Drexler, \"Evolutionary Capacity: Why Organisms Cannot Be Like Machines,\" Metamodern (blog), August 2, 2009, metamodern.com\/2009\/08\/02\/contrasts-in-evolutionarycapacity\/.\n\n Drexler, _Engines of Creation_ , 32.\n\n _Ibid_., 8.\n\n _Ibid_., 21.\n\n _Ibid_., 37, 103.\n\n _Ibid_.\n\n _Ibid_., 76.\n\n _Ibid_., 103.\n\n _Ibid_., 102.\n\n _Ibid_., 110.\n\n See, for example, Tzvetan Todorov, _Imperfect Garden: The Legacy of Humanism_ (Princeton: Princeton University Press, 2002) and Marilynne Robinson, _Absence of Mind: The Dispelling of Inwardness from the Modern Myth of the Self_ (New Haven, Conn.: Yale University Press: 2010).\n\n Drexler, _Engines of Creation_ , 21.\n\n _Ibid_., 234, 145.\n\n _Ibid_., 234\u2013237. As Drexler sums up, \"In short, we have a chance at a future with room enough for many worlds and many choices, and with time enough to explore them. A tamed technology can stretch our limits. . . . In an open future of wealth, room, and diversity, groups will be free to form almost any society they wish, free to fail or set a shining example for the world. Unless your dreams demand that you dominate everyone else, chances are that other people will wish to share them. If so, then you and those others may choose to get together to shape a new world. If a promising start fails\u2014if it solves too many problems or too few\u2014then you will be able to try again. Our problem today is not to plan or build utopias but to seek a chance to try\" (237).\n\n _Ibid_., 200.\n\n _Ibid_.\n\n _Ibid_., 196.\n\n _Ibid_., 187\u2013188.\n\n _Ibid_., 199.\n\n _Ibid_., 200.\n\n _Ibid_.\n\n _Ibid_., 201.\n\n _Ibid_.\n\n _Ibid_.\n\n _Ibid_., 209\u2013216. Drexler also endorses the idea of hypertext. Considering that he was writing in the 1980s, before the flowering of the World Wide Web, he showed foresight in anticipating that hypertext had the potential to be a Gutenbergscale revolution in information dissemination. See 217\u2013230.\n\n _Ibid_., 201.\n\n _Ibid_., 201. This idea of a dedicated community of the likeminded\u2014somewhat reminiscent of the Earthlings in J. B. S. Haldane's story \"The Last Judgment\" who worked so hard to salvage something from the coming disaster\u2014is one we will return to shortly in the context of Neal Stephenson's novel _The Diamond Age_.\n\n _Ibid_., 232.\n\n _Ibid_.\n\n _Ibid_., 231.\n\n _Ibid_., 239. Drexler's notion of a perilous period of transition to a future stage of great power recalls the idea of the \"great filter\" we discussed in the context of SETI in Chapter Two.\n\n _Ibid_., 200.\n\n Donella H. Meadows _et al., The Limits to Growth: A Report for the Club of Rome's Project on the Predicament of Mankind_ (New York: New American Library, 1972).\n\n Ed Regis, _Great Mambo Chicken and the Transhuman Condition: Science Slightly Over the Edge_ (Reading, Mass.: Perseus Books, 1990), 116\u2013117. See also Drexler, _Radical Abundance_ , 13\u201316, 246, and McCray, _The Visioneers_ , 149, 153.\n\n Drexler, _Engines of Creation_ , 163.\n\n _Ibid_., 165, 163. It should be remembered that Drexler wrote _Engines of Creation_ at a time of intense public interest in the issue of overpopulation and worries about a \"population bomb.\" In the decades since, world population has continued to grow but the rate of growth has slowed, and in many countries the population is now shrinking or is projected to soon shrink.\n\n Drexler argues that the \"settled cultures\" towards the interior of the wave of human (or transhuman?) space invaders he envisions will have to deal with limits more than those on the frontier, because as settled they will have more clearly defined goals, which already implies living within limits no matter what kinds of resources are available. But they will stand in contrast with the creativity of the frontier, where \"standards keep changing\" and therefore \"this idea of limits becomes irrelevant.\" More broadly, he concludes that even if \"brute matter\" creates ultimate limits, mind has \"room for endless evolution and change.\" _Ibid_., 165.\n\n _Ibid_., 237.\n\n Aristotle, _The Politics_ (1267a), trans. Carnes Lord (Chicago: University of Chicago Press, 2013 [second edition]), 40.\n\n _Ibid_.\n\n For more on this theme, see P. B. Thompson, \"Social Acceptance of Nanotechnology,\" in Mihail C. Roco and William Sims Bainbridge, eds., _Societal Implications of Nanoscience and Nanotechnology_ (Washington, D.C.: National Science Foundation, 2001), 198\u2013202, wtec.org\/loyola\/nano\/NSET.Societal.Implications\/nanosi.pdf. See also Mary Douglas and Aaron Wildavsky, _Risk and Culture: An Essay on the Selection of Technological and Environmental Dangers_ (Berkeley, Cal.: University of California Press, 1982).\n\n Drexler, _Engines of Creation_ , 238.\n\n _Ibid_.\n\n _Ibid_.\n\n Drexler says he points readers to _The Diamond Age_ whenever he is \"asked to recommend a science fiction novel about advanced nanotechnologies.\" Stephenson's book, Drexler writes, is \"a good read, it portrays a complex and surprising world, and] it's not saturated with nanobots.\" K. Eric Drexler, \"Nanotechnology in Science Fiction (and _vice versa_ ),\" Metamodern (blog), April 9, 2009, [metamodern.com\/2009\/04\/09\/nanotechnology-in-science-fiction\/. (Actually, nanobots\u2014called \"nanosites\" in Stephenson's novel\u2014are crucial to the plot.)\n\n Neal Stephenson, _The Diamond Age: Or, A Young Lady's Illustrated Primer_ (New York: Bantam Spectra, 2003 reissue [orig. 1995]), 273.\n\n _Ibid_., 38, 40.\n\n _Ibid_., 37.\n\n _Ibid_.\n\n _Ibid_., 8.\n\n _Ibid_., 30.\n\n Strangely, analysts of the novel seem divided on this point. Jan Berrien Berends seems to think that Stephenson wants to show that \"our future is bright, _really_ bright\"\u2014\"who wouldn't want to live there?\"\u2014and that Stephenson is blind to the many flaws Berends readily perceives. Not only does this mischaracterize the book's setting and tone, but Berends himself seems oblivious to several key plot points. (Jan Berrien Berends, \"The Politics of Neal Stephenson's _The Diamond Age_ ,\" _New York Review of Science Fiction_ 9, no. 104 April 1997], 15.) On the other hand, Joachim Schummer seems to err in a different direction when implying that, as an example of the \"cyberpunk\" or \"postcyberpunk\" genres, _The Diamond Age_ must have a \"nihilistic undertone\" and \"focus on human alienation,\" and if it weren't so fascinated with technology, it would be dystopian. (Joachim Schummer, \"'Societal and Ethical Implications of Nanotechnology': Meanings, Interest Groups, and Social Dynamics,\" _Techn\u00e9_ 8, no. 2 [Spring 2005], 59.) Brian Opie catches the right tone when he notes that _The Diamond Age_ is \"not utopian, since the societies it describes are not only not perfected but are self-consciously reproducing inherited models of cultural practice and values.\" (Brian Opie, \"Technoscience in Societies of the Future: Nanotechnology and Culture in Neal Stephenson's Novel _The Diamond Age_ [1995]\" (conference paper), February 2004, [www.europe.canterbury.ac.nz\/conferences\/tech2004\/tpp\/Opie_paper.pdf, 3.) N. Katherine Hayles believes the book shows how its utopian projects fail \"at every level.\" (N. Katherine Hayles, \"Is Utopia Obsolete?\" _Peace Review_ 14, no. 2 June 2002], 136, [dx.doi.org\/10.1080\/10402650220140148.)\n\n Stephenson, _Diamond Age_ , 53\u201355.\n\n _Ibid_., 172, 331.\n\n _Ibid_., 55\u201357.\n\n _Ibid_., 292.\n\n _Ibid_., 286.\n\n _Ibid_., 18.\n\n _Ibid_., 108.\n\n _Ibid_., 374.\n\n _Ibid_., 322.\n\n A thought echoed in the epigraph for _The Diamond Age_ , from Confucius: \"By nature, men are nearly alike; by practice, they get to be wide apart.\"\n\n _Ibid_., 20\u201321.\n\n _Ibid_., 185.\n\n _Ibid_., 332.\n\n _Ibid_., 9, 136, 180.\n\n It is striking how frequently analysts find this failure to end scarcity unexplained or inexplicable. See Berends, \"The Politics of Neal Stephenson's _The Diamond Age_ ,\" 18, and Richard Rorty, \"Hope and the Future,\" _Peace Review_ 14, no. 2 (June 2002), 152, dx.doi.org\/10.1080\/10402650220140166.\n\n _Ibid_., 59, 333.\n\n _Ibid_., 21.\n\n _Ibid_., 84, 23, 208; 69; 80\u201382.\n\n _Ibid_., 217\u2013218, 268; 284\u2013285; 448. The reviewer Jan Berrien Berends finds Miranda's attachment to Nell mysterious, having perhaps missed that they share a history of abuse. He can only imagine that it is based on \"a _huge_ dose of maternal instinct (a property in which I do not, in fact, believe).\" Berends also misses the hints that Hackworth has something to do with the Mouse Army's loyalty to Nell, and so he instead draws the extraordinary conclusion that the only reason they follow her is because she is white and Victorian. (Berends, \"The Politics of Neal Stephenson's _The Diamond Age_ ,\" 18, 16.)\n\n Stephenson, _The Diamond Age_ , 486\u2013487.\n\n _Ibid_., 42\u201343.\n\n _Ibid_., 313.\n\n _Ibid_., 337.\n\n _Ibid_., 433\u2013434.\n\n _Ibid_., 434.\n\n While a skeptic with respect to the Singularity, the technology commentator Jaron Lanier (inventor of the concept of shared \"virtual realities\") hopes and expects that the next stage for human beings is \"fuller contact between minds.\" He imagines advanced Martians who pity us for the poverty of our abilities to connect with each other, separated as we are into \"sacks of skin.\" (Joel Garreau, _Radical Evolution: The Promise and Peril of Enhancing our Minds, Our Bodies\u2014and What it Means to be Human_ [New York: Doubleday, 2005], 200\u2013202.)\n\n Stephenson, _The Diamond Age_ , 263, 323.\n\n _Ibid_., 365.\n\n _Ibid_., 354.\n\n _Ibid_., 304.\n\n _Ibid_., 383.\n\n _Ibid_., 37, 384.\n\n _Ibid_.\n\n Job 5:7 (KJV).\n\nCHAPTER FOUR: PERFECTING INHUMANITY\n\n Ray Kurzweil, _The Age of Spiritual Machines: When Computers Exceed Human Intelligence_ (New York: Penguin, 1999), 146\u201349.\n\n Hans Moravec, _Mind Children: The Future of Robot and Human Intelligence_ (Cambridge: Harvard University Press, 1988), 114.\n\n Simon Young, _Designer Evolution: A Transhumanist Manifesto_ (Amherst, NY: Prometheus Books, 2006), 28.\n\n James Hughes, _Citizen Cyborg: Why Democracies Must Respond to the Redesigned Human of the Future_ (Cambridge: Westview Press, 2004), 255.\n\n William Saletan, \"Among the Transhumanists: Cyborgs, Self-mutilators, and the Future of Our Race,\" _Slate_ , June 4, 2006, slate.com\/id\/2142987\/.\n\n For example, Kurzweil, who is among other things interested in altering the human digestive system, notes that the nanotechnologies that will allow that to happen are already in the works but their intended applications are diagnostic and therapeutic rather than enhancement and redesign. Ray Kurzweil, _The Singularity Is Near: When Humans Transcend Biology_ (New York: Viking, 2005), 303.\n\n For an extended effort to provide this kind of \"common sense\" defense of transhumanism, see John Harris's book _Enhancing Evolution_. In a wonderful combination of populism, disingenuousness, and Seinfeldian phrasing, Harris claims, \"I have no transhumanist program or agenda. I do think there are powerful moral reasons for ensuring the safety of the people and for enhancing our capacities, our health, and thence our lives. If the consequence of this is that we become transhumans, there is nothing wrong with that, but becoming transhumans is not the agenda.\" John Harris, _Enhancing Evolution: The Ethical Case for Making Better People_ (Princeton: Princeton University Press, 2007), 38\u201339.\n\n See, for example, The President's Council on Bioethics, _Beyond Therapy: Biotechnology and the Pursuit of Happiness_ (Washington, D.C.: U.S. GPO, 2003), 13\u201316, thenewatlantis.com\/BeyondTherapyPDF; and Ramez Naam, _More Than Human: Embracing the Promise of Biological Enhancement_ (New York: Broadway Books, 2005), 5\u20136.\n\n U.S. Food and Drug Administration (FDA), \"FDA approves first retinal implant for adults with rare genetic eye disease\" (press release), February 14, 2013, www.fda.gov\/NewsEvents\/Newsroom\/PressAnnouncements\/ucm339824.htm. FDA, \"FDA Approves First Implantable Miniature Telescope to Improve Sight of AMD patients\" (press release), July 6, 2010, www.fda.gov\/NewsEvents\/Newsroom\/PressAnnouncements\/ucm218066.htm.\n\n Lucilla Cardinali, _et al_., \"Tool-Use Induces Morphological Updating of the Body Schema,\" _Current Biology_ 19, iss. 12 (June 23, 2009), R478\u201379, dx.doi.org\/10.1016\/j.cub.2009.05.009. See also Karen Hopkin, \"Tools Are Body Parts to Brain,\" _Scientific American_ (podcast), scientificamerican.com\/podcast\/episode\/tools-are-body-parts-to-brain-09-06-23\/.\n\n If we could see better in low light, we would use less energy for lighting; such an enhancement could be a way of addressing global warming. See S. Matthew Liao, Anders Sandberg, and Rebecca Roache, \"Human Engineering and Climate Change,\" _Ethics, Policy and Environment_ 15, iss. 2 (2012), dx.doi.org\/10.1080\/21550085.2012.685574.\n\n Young, _Designer Evolution_ , 286\u2013287. Cf. Kurzweil, _The Singularity Is Near_ , 338.\n\n \"Democratic transhumanist\" James Hughes is not so happy with the thought of a purely market-based system of access to enhancements, and holds that social justice would require \"subsidies and universal provision.\" Hughes, _Citizen Cyborg_ , 233.\n\n See the brief scenario presented in Joel Garreau, _Radical Evolution: The Promise and Peril of Enhancing Our Minds, Our Bodies\u2014and What It Means to Be Human_ (New York: Doubleday, 2005), 7\u20138.\n\n We will get to \"the Singularity\" shortly, but the following quote gives an idea of the crusading mentality transhumanists can adopt: \"We are willing to do whatever it takes, within reason, to get a positive Singularity. Governments are not going to stop us. If one country shuts us down, we go to another country. . . . Just because you don't want it doesn't mean that we won't build it.\" Michael Anissimov, \"Response to Charles Stross's 'Three Arguments Against the Singularity,'\" Accelerating Future (blog), acceleratingfuture.com\/michael\/blog\/2011\/06\/response-to-charles-stross-three-arguments-against-the-singularity\/.\n\n Nick Bostrom flirts with the idea that a time may come when parents have a \"moral responsibility\" to enhance their children. But he concludes that \"Only in extreme and unusual cases might state infringement of procreative liberty be justified. If, for example, a would-be parent wished to undertake a genetic modification that would be clearly harmful to the child or would drastically curtail its options in life, then this prospective parent should be prevented by law from doing so.\" (Nick Bostrom, \"The Transhumanist FAQ: A General Introduction,\" Version 2.1 (2003), 21, transhumanism.org\/resources\/FAQv21.pdf.) But note that it is not hard to imagine that a time may come when a failure to enhance, genetically or otherwise, would \"drastically curtail\" life options. James Hughes is also in this ambiguous situation given his belief that enhancement should be subsidized by governments. Under these circumstances it may indeed be true that one will not be coerced by government into being enhanced, but one will be coerced into paying for others to be enhanced.\n\n See for instance Max More, \"A Letter to Mother Nature,\" MaxMore.com (personal website), August 1999, web.archive.org\/web\/20130324082510\/<http:\/\/www.maxmore.com\/mother.htm>.\n\n John Harris finds it odd that there should be anyone who is against enhancement; after all, does not enhancement mean \"a difference for the better\"? And who would not want that? Harris, _Enhancing Evolution_ , 36.\n\n See, for example, Harris, _Enhancing Evolution_ , 38 and Kurzweil, _The Singularity Is Near_ , 322.\n\n Harris, _Enhancing Evolution_ , 13, 16. Cf. Naam, _More Than Human_ , 9.\n\n Bostrom, \"The Transhumanist FAQ,\" 36.\n\n Kurzweil, _The Singularity Is Near_ , 35\u2013110, but cf. 14\u201321.\n\n David Pearce, \"The Hedonistic Imperative\" (online publication, 1995), hedweb.com\/hedethic\/hedonist.htm.\n\n Pearce, \"The Hedonistic Imperative,\" hedweb.com\/hedethic\/hedon4.htm.\n\n Moravec, _Mind Children_ , 117. For the same idea, see also Kurzweil, _The Singularity Is Near_ , 258.\n\n Moravec, _Mind Children_ , 108\u2013110.\n\n _Ibid_., 107\u2013108.\n\n _Ibid_., 115\u2013116.\n\n _Ibid_., 121\u2013122.\n\n Moravec offers what he calls a \"fun\" kind of resurrection, just as he offers a kind of immortality: \"imagine an immense simulator (I imagine it made out of a superdense neutron star) that can model the whole surface of the earth on an atomic scale and can run time forward and back. . . . Because of the great detail, this simulator models living things, including humans, in their full complexity. According to the pattern-identity position [that Moravec holds], such simulated people would be as real as you or me. . . . [W]e would bring people out of the simulation by . . . linking their minds to an outside robot body, or uploading them directly into it. In all cases we would have the opportunity to recreate the past and to interact with it in a real and direct fashion. It might be fun to resurrect all the past inhabitants of the earth this way and give them an opportunity to share with us in the (ephemeral) immortality of transplanted minds. Resurrecting one small planet should be child's play long before our civilization has colonized even its first galaxy.\" _Ibid_., 123\u2013124.\n\n Kurzweil, _The Singularity Is Near_ , 310\u2013320.\n\n Nick Bostrom, \"When will Computers Be Smarter Than Us?,\" Forbes.com, June 22, 2009, forbes.com\/2009\/06\/18\/superintelligence-humanity-oxford-opinions-contributors-artificial-intelligence-09-bostrom.html. This notion dates back at least to the 1960s, when mathematician I. J. Good wrote that \"the first ultraintelligent machine is the last invention that man need ever make.\" Irving John Good, \"Speculations Concerning the First Ultra-intelligent Machine,\" in Franz L. Alt and Morris Rubinoff, eds., _Advances in Computers_ , vol. 6 (New York: Academic Press, 1965), 31\u201388. More recently, the idea has been taken as the central thesis for a book about artificial intelligence: James Barrat, _Our Final Invention: Artificial Intelligence and the End of the Human Era_ (New York: Thomas Dunne Books, 2013).\n\n Vernor Vinge, \"The Coming Technological Singularity: How to Survive in the Post-Human Era\" (1993 conference paper hosted on Vinge's faculty website), www-rohan.sdsu.edu\/faculty\/vinge\/misc\/singularity.html. See also Vinge's later annotated version of this essay: Vernor Vinge, \"What Is the Singularity?,\" _Whole Earth Review_ (Winter 1993, annotated Spring 2003), wholeearth.com\/uploads\/2\/File\/documents\/technological_singularity.pdf.\n\n Moravec, _Mind Children_ , 116.\n\n This is the idea of the \"Dyson sphere\" or \"Dyson shell\" that so often pops up in science fiction, named for the theoretical physicist who was the first to mention it, in passing, in a scientific journal: Freeman J. Dyson, \"Search for Artificial Stellar Sources of Infrared Radiation,\" _Science_ 131, no. 3414 (June 3, 1960), 1667\u20131668, dx.doi.org\/10.1126\/science.131.3414.1667.\n\n Kurzweil, _The Singularity Is Near_ , 349\u2013352. Kurzweil does note that \"Dyson Shells can be designed to have no effect on existing planets, particularly those, like Earth, that harbor an ecology that needs to be protected\" (350). But it is not clear that this statement is anything more than a pious sentiment, given the overall direction of his argument at this point; he has just calculated the computing power latent in the mass of the solar system as a whole, \"not including the sun (which is ultimately also fair game)\" (349). There is not much point in trying to protect Earth's ecosystems without a sun.\n\n Moravec is a bit hazy on just why posthumans would wish to engage in wholesale resurrection efforts; as we saw above (note 30) it might be \"fun\" \"child's play.\" Eric Drexler was similarly vague about his own version of resurrection involving frozen bodies. But we can at least note the interesting progression that has been made in the ideal of resurrection from Fedorov's solemn religious obligation to Moravec's idea of what might amuse hyperintelligence.\n\n J. D. Bernal, _The World, the Flesh and the Devil: An Enquiry into the Future of the Three Enemies of the Rational Soul_ (Bloomington: Indiana University Press, 1969), 66.\n\n Kurzweil, _The Singularity Is Near_ , 366.\n\n _Ibid_., 353\u2013357.\n\n Moravec, _Mind Children_ , 147\u2013148; Kurzweil, _The Age of Spiritual Machines_ , 260.\n\n Kurzweil, _The Singularity Is Near_ , 342\u2013349; but cf. Moravec, _Mind Children_ , 136\u2013139.\n\n See, for example, Wendell Wallach and Colin Allen, _Moral Machines: Teaching Robots Right from Wrong_ (Oxford: Oxford University Press, 2009); J. Storrs Hall, _Beyond AI: Creating the Conscience of the Machine_ (Amherst, N.Y.: Prometheus Books, 2007); and Eliezer Yudkowsky, \"Creating Friendly AI 1.0: The Analysis and Design of Benevolent Goal Architectures\" (white paper), Machine Intelligence Research Institute (formerly the Singularity Institute for Artificial Intelligence), June 15, 2001, intelligence.org\/files\/CFAI.pdf. See also my essay, Charles T. Rubin, \"Machine Morality and Human Responsibility,\" _The New Atlantis_ 32 (Summer 2011), 58\u201379.\n\n Kurzweil, _The Singularity Is Near_ , 297\u2013298. The nonbiological intelligence Kurzweil expects to exist by 2045 will be \"one billion times more powerful than all human intelligence today\" (136); by the end of the twenty-first century, he expects \"the nonbiological portion of our intelligence will be trillions of trillions of times more powerful than unaided human intelligence\" (9).\n\n _Beyond Therapy_ , 211.\n\n _Ibid_., 205\u2013273.\n\n Edward Bellamy, _Dr. Heidenhoff's Process_ (New York: D. Appleton and Company, 1880), 72, archive.org\/stream\/drheidenhoffsproobellgoog.\n\n _Ibid_., 108.\n\n _Ibid_., 115\u2013116, 115, 116, 114.\n\n _Ibid_., 103\u2013105.\n\n _Ibid_., 100. It should be noted that, while brain researchers today accept that memories are sometimes stored in very specific locations in the brain, they do not believe that, as Dr. Heidenhoff puts it, \"conventionally or morally morbid or objectionable\" memories can bring about \"a morbid state of the brain fibers.\"\n\n _Ibid_., 101.\n\n _Ibid_., 103.\n\n _Ibid_.\n\n _Ibid_., 102.\n\n _Ibid_., 105.\n\n _Ibid_.\n\n _Ibid_., 117.\n\n _Ibid_., 105.\n\n _Ibid_., 116\u2013117.\n\n _Ibid_., 116.\n\n _Ibid_., 119.\n\n _Ibid_.\n\n _Ibid_., 120.\n\n _Ibid_.\n\n _Ibid_., 121.\n\n _Ibid_., 124, 121.\n\n _Ibid_., 125.\n\n _Ibid_., 126.\n\n _Ibid_.\n\n Charlie Kaufman, _Eternal Sunshine of the Spotless Mind: The Shooting Script_ (New York: Newmarket Press, 2004), 38.\n\n _Ibid_., 107.\n\n _Ibid_., 58.\n\n Mary may be the most serious character in the movie; at least, in the shooting script she has some intimation of what is genuinely interesting about the _Bartlett's Quotations_ she loves so much\u2014it represents the human race in \"constant conversation with itself\" ( _Ibid_., 64).\n\n This insight into how forgetfulness compromises our ability to learn from mistakes suggests a deeper meaning to the fact that the happy conclusion of the Madeline and Henry story takes place only in his dream. Shorn of the experience that made her sadder and wiser, would not the innocent Madeline that is restored to Henry resume the \"mirthful, self-reliant\" ways that got her in trouble the first time around? Even in Henry's dream there are signs of such a restoration in her \"flashing, imperious expression.\" But she is spared by another _deus ex machina_ of sorts: trying on her wedding dress, turning her expression to \"shy and blushing softness\" with \"a host of virginal alarms.\" Perhaps marriage would tame her\u2014and perhaps not. At any rate, that is Henry's dream for her. (Bellamy, _Dr. Heidenhoff's Process_ , 136.)\n\n Marcel Proust, _In the Shadow of Young Girls in Flower_ , trans. James Grieve (New York: Penguin Books, 2002), 99.\n\n Alexander Pope, \"Eloisa to Abelard,\" poetryfoundation.org\/poem\/174158.\n\n Tor N\u00f8rretranders, _The User Illusion: Cutting Consciousness Down to Size_ (New York: Penguin Books, 1999).\n\n Thomas Hobbes, _Leviathan_ (New York: Collier Books, 1962 [orig. 1651]), 80.\n\n Harris, _Enhancing Evolution_ , 137.\n\nCHAPTER FIVE: THE REAL MEANING OF PROGRESS\n\n Winwood Reade, _The Martyrdom of Man_ (London: Kegan Paul, Trench, Tr\u00fcbner & Co., 1910), 523, archive.org\/stream\/martyrdomofman00readuoft.\n\n J. B. S. Haldane, _Daedalus, or Science and the Future_ (New York: E. P. Dutton & Company, 1924), 88.\n\n For example, when the Obama administration dissolved the Bush-era President's Council on Bioethics, which routinely attempted to pose the question of science and ethics in terms of a good human life, the relief in some circles was palpable. The promise was that this supposedly too-philosophical council would be replaced by a body that would instead provide so-called \"practical\" guidance. See Charles T. Rubin, \"Postmodernism, Autonomy and Bioethical Boundaries,\" _The Good Society_ 19, no. 1 (2010), 28\u201332, dx.doi.org\/10.1353\/gso.0.0090.\n\n See, for example, Hans Jonas, \"Technology and Responsibility: Reflections on the New Tasks of Ethics,\" in _Philosophical Essays: From Ancient Creed to Technological Man_ (Chicago: University of Chicago Press, 1974), 19.\n\n I should acknowledge openly what will in any case become evident soon enough: I do not approach these paintings as an art expert or art historian. But I believe looking at them na\u00efvely can serve a useful purpose.\n\n Alan Barnett, \"Bernhard Heisig: The Sound and Fury of Painting,\" Politics and Art (blog), December 1, 2005, politicsandart.com\/2005\/12\/bernhard-heisig-sound-and-fury-of_01.html.\n\n _Ibid_.\n\n Ovid, _The Metamorphoses of Ovid_ , trans. Allen Mandelbaum (New York: Harcourt Brace and Co., 1993), 254.\n\n My reading of the painting owes much to W. H. Auden, \"Mus\u00e9e des Beaux Arts\" (collected in W. H. Auden, _Selected Poems_ , ed. Edward Mendelson [New York: Vintage, 2007], 87) and William Carlos Williams, \"Landscape with the Fall of Icarus\" (collected in William Carlos Williams, _Collected Poems: Volume II_ , 1939\u20131962, ed. Christopher MacGowan [New York: New Directions, 1991], 385\u2013386).\n\n Ovid, 256.\n\n J. B. S. Haldane notes, \"There is no great invention, from fire to flying, which has not been hailed as an insult to some god. . . . The biological invention then tends to begin as a perversion and end as a ritual supported by unquestioned beliefs and prejudices\" (Haldane, _Daedalus_ , 45, 49).\n\n Compare Larry Arnhart, _Darwinian Conservatism_ (Charlottesville, Va.: Imprint Academic, 2005).\n\n Leo Tolstoy, _War and Peace_ , trans. Richard Pevear and Larissa Volokhonsky (New York: Alfred A. Knopf, 2007), 1104.\n\n G. K. Chesterton, _The Thing: Why I Am a Catholic_ (orig. 1929), collected in _The Collected Works of G. K. Chesterton_ , vol. 3 (San Francisco: Ignatius Press, 1990), 157.\n\n Aristotle, _The Politics_ (1253a), trans. Carnes Lord (Chicago: University of Chicago Press, 2013 [second edition]), 5.\n\n For much more on this theme, see Peter Augustine Lawler, _Stuck with Virtue: The American Individual and Our Biotechnological Future_ (Wilmington, Del.: ISI Books, 2005).\n_INDEX_\n\nAbelard, Peter\n\nactive shields ( _see_ nanotechnology)\n\nAesop\n\naliens 24\u201325, , , , , , , 55\u201356, , , , , , , , ; assistance from ; benevolence of 77\u201379; contact with 51\u201353, 55\u201360; immortality of ; morals and ethics of ; religion of ; technological superiority of , 63-64,  ( _see also_ search for extraterrestrial intelligence [SETI])\n\nanti-anthropocentrism , 62\u201363\n\naristocracy 35\u201337\n\nAristotle ,\n\nartificial intelligence (AI) , , , 103\u2013104, , , 211 _n_; friendly AI and moral AI\n\nassumption of mediocrity , 54\u201356, , 61\u201362, 65\u201366,\n\nastronomy 24\u201326, ,\n\navatars 122\u2013123\n\nBacon, Francis , , , 191 _n_, 201 _n_\n\nBellamy, Edward 143\u2013144, , , 213 _n_, 214 _n_ ( _see alsoDr. Heidenhoff's Process_)\n\nBernal, J. D. 31\u201340, 42\u201344, , , , 93\u201395, , , , , , 193 _n_, 195 _n_ ( _see alsoWorld, the Flesh and the Devil: An Inquiry into the Three Enemies of the Rational Soul, The_)\n\n_Beyond Therapy_ (President's Council on Bioethics) 142\u2013143\n\nBible , ,  ( _see also_ Job)\n\nbiochauvinism ,\n\nbioconservatives and bioluddites 128\u2013129\n\nbodies , , , ; as contemptible 19\u201320; creation of ; redesign of , 95\u201396, , 123\u2013124, ; and self-overcoming , , 42\u201343; separation from , , ; virtual bodies ,\n\nBostrom, Nick , 49\u201350, , , , , , 187 _n_, 210 _n_, 211 _n_\n\nbrain , , 32\u201333, , , , , , , , , ; and machine enhancement 93\u201394, , 126\u2013127, 133\u2013134,  ( _see also_ mind)\n\n_Brave New World_ (Huxley)\n\nBruegel, Pieter the Elder 168\u2013178,\n\nbush robot 133\u2013134\n\nde Caritat, Marie Jean Antoine Nicolas ( _see_ Condorcet, Marquis de)\n\nCarnegie Mellon\n\nCat Man\n\nchaos ,\n\nChesterton, G. K.\n\n_Childhood's End_ (Clarke) , , 77\u201379, ,  ( _see also_ Clarke, Arthur C.)\n\nChristianity ,\n\nClarke, Arthur C. , 70\u201371, , , 77\u201379, 200 _n_, 201 _n_ ( _see alsoChildhood's End_)\n\nClub of Rome ( _seeLimits to Growth, The_)\n\ncognitive science ,\n\nCohen, Eric\n\nCold War ,\n\ncolonization , 28\u201329, , , , , , , 197 _n_, 211 _n_\n\ncompetition ( _see_ Darwinian competition)\n\nCondorcet, Marquis de (Marie Jean Antoine Nicolas de Caritat) 12\u201315, , 37\u201340, , 50\u201353, , , , , 174\u2013175\n\n_Contact_ ( _see_ Sagan, Carl)\n\nconverging technologies 3\u20134,\n\nCopernican Principle ( _see_ assumption of mediocrity)\n\nCopernicus, Nicolaus\n\ncosmos , , , , ,\n\ncreation, as divine act 18\u201319, , ; as nature 17\u201318, , 40\u201341; as human act , 29\u201330, , ,\n\n_Creation of Adam, The_ (painting by Michelangelo)\n\ncryonic suspension , ,  ( _see also_ life extension _and_ nanotechnology)\n\nDaedalus (mythical figure) , , 172\u2013176\n\n_Daedalus and Icarus_ (painting by Landon) 166\u2013168\n\nDarwin, Charles , , 40\u201341, , , , 175\u2013176\n\nDarwinian competition , , , , , ,\n\nDarwinian evolution , 40\u201341, , , , 158\u2013159, 175\u2013176 ( _see also_ evolution)\n\nDavies, Paul\n\nDawkins, Richard\n\ndeath 5\u20136, , , 25\u201326, , 39\u201340, , 134\u2013135, , ,\n\n_Death of Icarus, The_ (painting by Heisig) 167\u2013168,\n\nDeclaration of Independence, the\n\ndehumanization , 36\u201337, 43\u201344, , , , , , , , , , , 179\u2013180,\n\nDescartes, Ren\u00e9\n\n_Descent of Man, The_ (Darwin) ,  ( _see also_ Darwin, Charles)\n\ndestiny ( _see_ humanity)\n\n_Diamond Age, The_ (Stephenson) 105\u2013112, 116\u2013118, , 206 _n_, 207 _n_ ( _see also_ Stephenson, Neal)\n\ndiversity , , , 90\u201391, 93\u2013107, , , , 205 _n_\n\nDostoevsky, Fyodor\n\n_Dr. Heidenhoff's Process_ (Bellamy) 143\u2013150, 153\u2013155, 157\u2013158, , 213\u2013214 _n_\n\nDrake, Frank , , , , , , 198\u2013199 _n_\n\nDrexler, K. Eric 85\u2013107, 110\u2013111, 117\u2013119, , 130\u2013131, , , , 203 _n_, 204 _n_, 205 _n_, 206 _n_, 212 _n_\n\nDu Bois, W. E. B.\n\nDyson sphere 211 _n_\n\neco-egalitarianism\n\nEinstein, Albert (epigraph),\n\n_Engines of Creation_ (Drexler) 85\u201389, , 101\u2013102, , 205\u2013206 _n_\n\nenhancement 124\u2013125, , , , 209 _n_, 210 _n_; and concept of self 156\u2013157; and memory ; as personal choice 126\u2013131\n\nEnlightenment, the 12\u201313 ( _see also_ progress)\n\nentropy , , ,\n\n_Essay on the Principle of Population_ (Malthus)\n\n_Eternal Sunshine of the Spotless Mind_ (Gondry) , , , ,\n\nethics , , , , 164\u2013165; science and 164\u2013165; technology and\n\neugenics ,\n\nevil , , 31\u201332, , 97\u2013100, , ,\n\nevolution 1\u20133, , , , , 40\u201342, , , 64\u201365, 75\u201376, , 90\u201393, , , 130\u2013132, , , 157\u2013158, 175\u2013176 ( _see also_ Darwinian evolution _and_ progress)\n\nextended life ( _see_ life extension)\n\nextraterrestrial life ( _see_ aliens)\n\nfact forums\n\nFedorov, Nikolai Fedorovich 21\u201324, 26\u201327, , 38\u201342, , , 212 _n_\n\nFermi, Enrico ( _see_ Fermi paradox)\n\nFermi paradox , 64\u201366,\n\nFlammarion, Nicolas Camille 24\u201328, , , 40\u201342, , , , ,\n\nforesight 90\u201392, ,\n\nForesight Institute, The 204 _n_\n\nfreedom , , , , 128\u2013129, , 156\u2013157, , 180\u2013181\n\nFreedom League 70\u201371\n\nFreud, Sigmund ,\n\n_From the Closed World to the Infinite Universe_ (Koyr\u00e9) 55 _n_\n\nFuture of Humanity Institute\n\ngenetic engineering , , ,\n\ngenetics , , , , , , ,\n\ngenie machines ( _see_ nanotechnology)\n\nGod 5\u20136, , 21\u201324, , , , , , , 201 _n_\n\nGoogle\n\n\"gray goo problem\" ( _see_ nanotechnology)\n\ngroup mind ( _see_ mind)\n\nHaldane, J. B. S. 27\u201328, 30\u201331, , 37\u201343, 50\u201351, , , , , , , 215 _n_ ( _see also_ \"Last Judgment, The\")\n\nhappiness , , 29\u201330, , , , , 141\u2013142, , ; pursuit of ,\n\nHarris, John 129\u2013130, , 209 _n_, 210 _n_\n\nhedonic imperative\n\nHeisig, Bernhard 168\u2013172,\n\nHobbes, Thomas\n\nHomer\n\nHughes, James , 209\u2013210 _n_\n\nhuman condition ( _see_ humanity)\n\nhuman extinction ( _see_ humanity)\n\nhuman pattern ,\n\nhuman rights ( _see_ rights)\n\nhumanity, association and , ; assumption of mediocrity and 54\u201356, , , 158\u2013159; becoming aliens , ; bodies of ( _see_ bodies); destiny of , , 41\u201342, , , ; extended life and ( _see_ life extension); extinction of , , 9\u201311, , , , , , , ; hubris of , 54\u201355, 63\u201364, , ; human condition and 5\u20136, , ; human rights and ( _see_ rights); immorality and , 198\u2013199 _n_; memory and ( _see_ memory); morality and , , , , 77\u201379, , , 112\u2013113, 116\u2013117, , , 129\u201330, , , , ; perpetuation of 114\u2013116, ; suffering of ( _see_ suffering); transformation and , , , , , , , , , 187 _n_ ( _see also_ nanotechnology _and_ progress)\n\nHuxley, Aldous\n\nhyperintelligence\/superintelligence , , , 136\u2013139, , , 211 _n_, 212 _n_\n\nIcarus (mythical figure) 165\u2013172, ,\n\nimmortality , , , , , ; attainment of , , ; invention of , 18\u201319, ; personal immortality 134\u2013136\n\n_In the Shadow of Progress_ (Cohen)\n\ninfinite space\n\ninformation technology ,\n\ninhumanity , , , ,\n\nintellect 145\u2013146\n\nintelligence 1\u20132, , , 41\u201342, , , , 137\u2013138, ; alien intelligence 44\u201346, , 51\u20134, 63\u201366, 78\u201379, 139\u2013140; human intelligence , , 58\u201359; nanotechnology and 92\u201393 ( _see also_ artificial intelligence _and_ hyperintelligence\/superintelligence)\n\nInternet, the , , ,\n\nJob (Biblical figure and book) ,  ( _see also_ Bible)\n\nKant, Immanuel 190 _n_\n\nKoyr\u00e9, Alexandre 55 _n_\n\nKurzweil, Ray , , 49\u201350, 84\u201385, , , 138\u2013139, , , 209 _n_, 212 _n_\n\n\"L Factor\"\n\nLandon, Charles Paul 166\u2013168, 171\u2013173\n\n_Landscape with the Fall of Icarus_ (painting by Bruegel) 169\u2013172\n\n\"Last Judgment, The\" (Haldane) 27\u201331, 205 _n_ ( _see also_ Haldane, J. B. S.)\n\nlaw of accelerating returns ,\n\nLewis, C. S. , 200\u2013201 _n_\n\nliberty , , 98\u201399, 157\u2013158; vs. license\n\nlibertarianism ,\n\nlife extension 38\u201339, , , ,  ( _see also_ cryonic suspension _and_ nanotechnology)\n\n_Limits to Growth, The_ (Club of Rome)\n\nLizard Man\n\n_Looking Backward_ ( _see_ Bellamy)\n\nlove , , , , 112\u2013117, 143\u2013144, , 154\u2013156, ; of one's own 112\u2013117\n\nMalthus, Thomas Robert , , , , 50\u201352, , , , 174\u2013175, 195 _n_\n\nMalthusianism , , , 50\u201352, , , , , , , , , 174\u2013176\n\n_Martyrdom of Man, The_ (Reade) , , , 189\u2013190 _n_, 190\u2013191 _n_ ( _see also_ Reade, William Winwood)\n\nMassachusetts Institute of Technology\n\nmaterialism ( _see_ mind)\n\nmemes\n\nmemory 142\u2013143, 146\u2013149, 151\u2013155; alteration of 142\u2013143, 146\u2013150; erasure of 143\u2013144, 148\u2013150, 153\u2013155; human identity and 146\u2013149, 153\u2013158; morality and 148\u2013149, 156\u2013158 ( _see also_ humanity)\n\nmemory erasure ( _see_ memory)\n\nMichelangelo ( _seeCreation of Adam, The_)\n\n_Mind Children_ (Moravec) 133\u2013135 ( _see also_ Moravec, Hans)\n\nminiaturization\n\nmind 4\u20135, 16\u201319, 29\u201330, 33\u201336, , , , , ; group mind\/hive mind , , 69\u201370, ; machines and , , 133\u2013135; materialism and , , 93\u201395, , , ; pattern identity and ,  ( _see also_ brain)\n\nMontagu, Ashley\n\nMoravec, Hans , 133\u2013138, , 211 _n_, 212 _n_ ( _see alsoMind Children_)\n\nmorality 98\u201399, , 165\u2013166; of aliens 54\u201355; of humanity , , , , 77\u201379, , , 112\u2013113, 116\u2013117, , , 129\u201330, , , , ; of memory erasure 143\u2013148\n\nMorrison, Philip 60\u201361\n\nNaam, Ramez\n\n\"nanobots\" ( _see_ nanotechnology)\n\nnanomachines ( _see_ nanotechnology)\n\nnanotechnology 3\u20134, , ; active shields 97\u201399, 112\u2013114, 116\u2013117; current research on 84\u201385; development of 91\u201393, 96\u201398, ; \"gray goo problem\" , ; nanomachines\/\"nanobots\" ; nanoparticles 84\u201385; promise of , 83\u201385, , , , , ; replicators \/genie machines 89\u201392; smart dust ,  ( _see also_ cryonic suspension _and_ hyperintelligence\/superintelligence)\n\nNational Nanotechnology Initiative (U.S.) ,\n\nnatural rights\n\nnatural selection , ,\n\nnaturalism\n\nnature 2\u20134, , , 16\u201323, , , 37\u201339, , , , , 124\u2013125, , , , , 191 _n_, 210 _n_; as a force 21\u201322, , 190 _n_; control\/power over , , , , , ; laws of , , , ; manipulation of ,  ( _see also_ progress)\n\nnegation , ,\n\nnihilism ( _see_ transhumanism)\n\nOrwell, George\n\n_Outlines of an Historical View of the Progress of the Human Mind_ (Condorcet) 12\u201314 ( _see also_ Condorcet, Marquis de)\n\nOvid 169\u2013170,\n\nOxford University\n\npattern identity ( _see_ mind)\n\nPearce, David , 131\u2013132,\n\nperfection , , , ,\n\nperpetuation 114\u2013116\n\npersonal immortality ( _see_ immortality)\n\nPope, Alexander\n\npopulation competition , , ,\n\nposthumanism , 37\u201338, , , 130\u2013132, , 138\u201340, , 178\u2013179\n\nposthumanity ( _see_ posthumanism)\n\nPresident's Council on Bioethics ( _seeBeyond Therapy_)\n\nprogress 13\u201314, , , , , , 124\u2013125, , , , ; accelerating progress , , ; defined, , 37\u201338, 164\u2013165, , ; Enlightenment ideals of 12\u201314; evolution and , 40\u201341, ; human condition and ; human destiny and ; human extinction and 1\u20132, ; human identity and , ; material progress 26\u201327, , , , ; nanotechnology and , ; paradox of ; progress of dehumanization , 36\u201337, , , ; science and , ; space travel and ; transhumanism and 36\u201338, , , 158\u2013160, 163\u2013166; vs. change , 42\u201343, ,\n\nReade, William Winwood 15\u201322, , , , 40\u201341, , , , ,  ( _see alsoMartyrdom of Man, The_)\n\nreductionism\n\nreligion 16\u201319, , 30\u201331, , , , 71\u201372, 76\u201377, 144\u2013146,\n\nreplicators ( _see_ nanotechnology)\n\nreproduction, human , , , 210 _n_\n\nresources , 50\u201352, , , , , , 206 _n_; competition for , , , , , 174\u2013175; exhaustion of 22\u201323, 101\u2013102, 197 _n_; scarcity of 14\u201315, 39\u201340, , , 101\u2013102, , , , , , , 174\u2013175\n\nresurrection 22\u201323, , 211 _n_, 212 _n_\n\nRhodes, Cecil , 189 _n_\n\nrights , ; expansion of ; human rights , ; natural rights ;\n\nRoco, Mihail\n\nRousseau, Jean\u2013Jacques ,\n\nSagan, Carl , , 59\u201361, , , 198 _n_\n\nsalvation , ,\n\nSantayana, George ,\n\nscience courts\n\nscience fiction , , , , , , , , , , , 197 _n_, 198 _n_, 206 _n_, 211 _n_; cyberpunk 207 _n_; postcyberpunk 207 _n_\n\nscientists , , 58\u201359, , , , , , 197 _n_; non\u2013scientists and\n\nScylla and Charybdis\n\nsearch for extraterrestrial intelligence (SETI) , , , 54\u201357, 59\u201366, 77\u201380, , , , 197 _n_, 199 _n_, 200 _n_ ( _see also_ aliens)\n\nselective breeding ,\n\nselective memory obliteration ( _see_ memory)\n\nself-replication\n\nShostak, Seth , , , 197 _n_\n\nSimulation , , 211 _n_\n\nSingularity, the , 136\u2013140, 156\u2013161, , , 188 _n_, 196 _n_, 199 _n_, 208 _n_, 209 _n_, 210 _n_, 211 _n_, 212 _n_; discontinuity and  ( _see also_ transhumanism)\n\nSingularity University\n\nsloth 97\u201398, , ,\n\nsmart dust ( _see_ nanotechnology)\n\nsociety , , , , , , , , , , 114\u2013115, 117\u2013118, , 204\u2013205 _n_\n\nsoul , , , , , ,\n\nSoviet Union , ,\n\nspace colonies , 28\u201329, , , , , , , 197 _n_, 211 _n_\n\nspace exploration , , , , , , 194 _n_\n\nStephenson, Neal 105\u2013108, 116\u2013119, 206\u2013207 _n_\n\nsuffering , , , , ; alleviation of, ; eradication of\n\nsupercivilization\n\nsuperintelligence ( _see_ hyperintelligence\/superintelligence)\n\nsupernatural , , 195 _n_\n\ntechno-libertarianism ( _see_ libertarianism)\n\ntechnological utopianism ( _see_ utopianism)\n\ntechnology 1\u20134, 11\u201312, , , , , 83\u201384, , , , , , , 116\u2013117, , , , , 180\u2013181, 204\u2013205 _n_; choice and , ; communication with aliens and 52\u201353, ; converging technologies ( _see_ converging technologies); development of , 91\u201392; ethics of progress and 163\u2013166; evolution and , ; history of ; life-extension technology ; nanotechnology ( _see_ nanotechnology); of aliens ( _see_ aliens); responsibility and ; technology race 90\u201392 ( _see also_ enhancement, life extension, nanotechnology, progress, _and_ transhumanism)\n\ntheodicy\n\nTolstoy, Leo ,\n\nTower of Babel\n\ntransformation ( _see_ humanity) ( _see also_ progress)\n\ntranshumanism, bodily alteration and 123\u2013124; definition of 1\u20133, 187 _n_; dehumanization and , ; enhancement and , , 209 _n_, 210 _n_; extended life and ; human extinction and , ; immortality and ; \"might makes right\" and ; moral vision of , 9\u201310, , ; nihilism and , , ; superintelligence and ; the Singularity and , , 199 _n_; transformation and , , , , 174\u2013175 ( _see also_ progress)\n\nTreaty of Versailles\n\nTsiolkovsky, Konstantin\n\nUnited Nations\n\nuniverse , , , 24\u201327, 30\u201331, , , 39\u201342, , , 53\u201356, 64\u201365, , , , 78\u201379, 138\u2013139, ,\n\nutopianism , , 71\u201373, , , , , , , 204\u2013205 _n_, 207 _n_; technological utopianism , 106\u2013108\n\nVinge, Vernor 136\u2013137,\n\n_World, the Flesh and the Devil: An Inquiry into the Three Enemies of the Rational Soul, The_ (Bernal)  ( _see also_ Bernal, J. D.)\n_A NOTE ON THE TYPE_\n\n_E CLIPSE OF MAN has been set in Minion, a type designed by Robert Slimbach in 1990. An offshoot of the designer's researches during the development of Adobe Garamond, Minion hybridized the characteristics of numerous Renaissance sources into a single calligraphic hand. Unlike many early faces developed exclusively for digital typesetting, drawings for Minion were transferred to the computer early in the design phase, preserving much of the freshness of the original concept. Conceived with an eye toward overall harmony, Minion's capitals, lowercase letters, and numerals were carefully balanced to maintain a well-groomed \"family\" appearance\u2014both between roman and italic and across the full range of weights. A decidedly contemporary face, Minion makes free use of the qualities Slimbach found most appealing in the types of the fifteenth and sixteenth centuries. Crisp drawing and a narrow set width make Minion an economical and easygoing book type, and even its name evokes its adaptable, affable, and almost self-effacing nature, referring as it does to a small size of type, a faithful or favored servant, and a kind of peach_.\n\nSERIES DESIGN BY CARL W. SCARBROUGH\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nTHE VETERAN'S PTSD HANDBOOK\nAlso by John D. Roche\n\n_The Veteran's Survival Guide_\n\n# THE VETERAN'S PTSD HANDBOOK\n\nHow to File and Collect on Claims for Post-traumatic Stress Disorder\n\nJohn D. Roche\n\nCopyright \u00a9 2007 John D. Roche\n\nPublished in the United States by Potomac Books, Inc. All rights reserved. No part of this book may be reproduced in any manner whatsoever without written permission from the publisher, except in the case of brief quotations embodied in critical articles and reviews.\n\n**Library of Congress Cataloging-in-Publication Data**\n\nRoche, John D., Maj.\n\nThe veteran's PTSD handbook : how to file and collect on claims for post-traumatic stress disorder \/ John D. Roche. \u2014 1st ed.\n\np. ; cm.\n\nIncludes bibliographical references and index.\n\nISBN-13: 978-1-59797-064-8 (alk. paper)\n\n1. Post-traumatic stress disorder. 2. Veterans\u2014Legal status, laws, etc. 3. Veterans\u2014Psychology. 4. Military pensions. I. Title.\n\n[DNLM: 1. Stress Disorders, Post-Traumatic\u2014United States. 2. Veterans Disability Claims\u2014 legislation & jurisprudence\u2014United States. 3. Veterans\u2014psychology\u2014United States. W 32.5 AA1 R673va 2006]\n\nRC552.P67R63 2006\n\n616.85'21008697\u2014dc22\n\n2006018027\n\nPrinted in the United States of America on acid-free paper that meets the American National Standards Institute Z39-48 Standard.\n\nPotomac Books, Inc.  \n22841 Quicksilver Drive  \nDulles, Virginia 20166\n\nFirst Edition\n\n10 9 8 7 6 5 4 3 2 1\nWe drifted o'er the harbour-bar, And I with sobs did pray  \nO let me be awake, my God! Or let me sleep always.\"  \n\u2014 Samuel Taylor Coleridge, _The Rime of the Ancient Mariner_\n\n## CONTENTS\n\n_Preface_\n\n_Acknowledgments_\n\n_List of Abbreviations_\n\n_Introduction_\n\n**1. Post-traumatic Stress Disorder**\n\nWhat Is Post-traumatic Stress Disorder?\n\nVA Marketing of PTSD Information\n\nUnderstanding PTSD\n\nHow Does PTSD Develop?\n\nHow Is PTSD Assessed?\n\nHow Common Is PTSD?\n\nWho Is Most Likely to Develop PTSD?\n\nWhat Are the Consequences Associated with PTSD?\n\nHow is PTSD Treated?\n\nLet's See What's on the Other Side of the Coin\n\nComplexities in Diagnosing PTSD\n\nSex-Related Trauma\n\nCoexisting Psychiatric Disorders\n\nSome Fundamental Facts about PTSD\n\nSecondary Service-connected Disorders\n\nChapter 1: What Have We Learned?\n\n**2. Two Little Words**\n\nCongressman Lane Evens Musters Support for Vets\n\nThe VA Flow Chart\n\n_Title 38 U.S. Code_\n\nFederal Register Final Notice 06-18-99\n\nVA Manual M21-1\n\nChanging the Meaning to 32 CFR \u00a7578.11&12\n\nChapter 2: What Have We Learned?\n\n**3. Sgt. David Raz's Five-Year Struggle**\n\nPart I: 2001\n\nPart II: 2002\n\nPart III: 2003\n\nPart IV: 2004\n\nPart V: 2005\n\nChapter 3: What Have We Learned?\n\n**4. Rules They Must Follow\u2014Know Them!**\n\nDisability Claims Processing\n\nHow a PTSD Claim Is Decided\n\n_Title 38 U.S. Code_ : Veterans Benefits\n\nWhat to Do If the VA Sends You a Rejection Letter\n\n_38 Code of Federal Regulations_\n\nVA Manual M21-1\n\nChapter 4: What Have We Learned?\n\n**5. Down to Basics**\n\nHistory of the Veterans Administration\n\nNine Types of Claim Actions\n\nA Documented Claim Is a Must\n\nNo Record Equals No Claim\n\nHow the VA Rates an Injury or Disease\n\nTestimonial Evidence Is a Must\n\nThe First Weak Link: Adjudication\n\nWillful Misconduct: Benefits Denied\n\nPresumption of Soundness: A Key Issue\n\nChronic Diseases\n\nTropical Diseases\n\nCombat-Related Injuries and Diseases\n\nChapter 5: What Have We Learned?\n\n**6. Elements of a PTSD Claim**\n\nComponents of Your PTSD Claim\n\nDeveloping a Winning Claim\n\nChapter 6: What Have We Learned?\n\n**7. Compensation and Pension Examinations**\n\nVA Health Administration\n\nContradicting VA Hoopla\n\nUsing Part-time and Fee-Based Physicians\n\nVeterans Healthcare Reform Act of 1996\n\nCompensation and Pension Examination\n\nThe Protocol for C&P Examinations\n\nAMIE C&P Examination Program\n\nAMIE's PTSD Protocol Examination\n\nCompleteness of Reports\n\nScope of the Examination\n\nSeverity of Disability\n\nReview of the Claim Folder\n\nTaking the Examination\n\nDon't Make Assumptions\u2014The VA Is Not Always Right\n\nChapter 7: What Have We Learned?\n\n**8. Total Disability Based on Individual Unemployability**\n\nTDIU Issues and the Court\n\nThe Most Common Reasons TDIU Claims Are Denied\n\nBasic Eligibility for TDIU Benefits\n\nProving a Claim for TDIU Benefits\n\nApplying for TDIU Benefits\n\nChapter 8: What Have We Learned?\n\n**9. PTSD Court Cases**\n\nProve Your Case and File Your Claim _or_ You Will Be Forced to Defend It on Appeal\n\nThe Court at Work Protecting PTSD Claims\n\nThe Appeal Process\n\n\"They Did It Before and They Will Do It Again\"\n\n_Allen v. Principi_ : Alcohol Abuse Secondary to PTSD\n\nU.S. Court of Appeals for Veterans Claims\n\n_Flayter v. Principi_ : Substance Abuse\n\n_Yancey v. Principi_ : Noncombat PTSD in a Combat Zone\n\n_Doran v. Brown_ : Noncombat Disability\u2014A Lightning Bolt from Zeus\n\n_Patton v. West_ : Sexual Assault\n\n_Zarycki v. Brown_ : In Combat\u2014Wrong MOS\n\nChapter 9: What Have We Learned?\n\n_Glossary_\n\n_Index_\n\n_About the Author_\n\n## PREFACE\n\nSeveral events led me to resign as a Department of Veterans Affairs (VA) adjudicator and become a veterans' advocate. After leaving the U.S. Air Force, I became a claims examiner for the VA. It wasn't long before I learned that there was a definite bureaucratic pecking order. As the new kid on the block, I had difficulty learning how not to transgress these protocols.\n\n### Mr. GS-13\n\nUnder no circumstance does one respond to criticism from a GS-13 section chief. To do so is to cause a bureaucratic firestorm. Here's what happened to elevate me to top of the \"watch your step\" list. Mr. GS-13 was a very proper man who wore his importance with considerable pride. Career advancement hinged on his perception of your worth. I'll let you judge how serious my indiscretion was to rocket me to the head of the list.\n\nAll was great with my world; the pay was good, my kids were all in school and had realistic goals, and our house made the front cover of _Scott's Fertilizer_ magazine thanks to my wife. That was until 12:35 PM one Tuesday in August.\n\nHaving enjoyed my thirty-minute lunch break, plus a five-minute bathroom break, I was all set to do good for our veterans. As I walked to my desk, I discovered Mr. GS-13 charging down the aisle like \"Storming Norman\" of the first Persian Gulf War fame. As I watched the distance between us shrink, I asked myself \"what did I do _now_?\"\n\n\"Mr. Roche, why is it necessary for me to remind you that lunch is from 12:00 to 12:30, not till 12:35?\" The entire section became quiet while waiting to see what would happen next. I answered back in equal volume, \"Well, Mr. GS-13, the government allowed me to drink water on government time so I just figured it was all right to pee on government time.\" When the entire section started to snicker, Mr. GS-13 retreated back to his \"office\" (really a cubicle) in a foul mood. My section mates also received his caustic wrath that afternoon.\n\nI further alienated myself from Mr. GS-13 when I continuously exceeded the twenty-five cases per day performance standard. There are days when cases are resolved one, two, three. This time, Mr. GS-13 wasn't going to chat with me before an audience. I was invited back to his cubicle where he advised me it was not in the VA's best interest for me to exceed the twenty-five cases-per-day work standard. He pointed out that the division received its manpower and budget authorization based on a twenty-five-case-per-day standard. If I continued to turn out more cases than the standard, I could upset the budget and manpower allocations set for this division.\n\n\"But, sir,\" I replied, \"I really think I'm helping you because my extra cases per day will pick up the slack when some of my colleagues fail to reach their twenty-five-case-per-day quotas.\" He said in a tone that could only be read as \"failure to obey would bring down the bureaucratic gods who will dispatch you to somewhere other than to the land of VA Oz.\" He actually said, \"There is no debate here, Mr. Roche, do you understand?\" \"Yes, sir. I will do only twenty-five cases each day and I will finish the last one by 4:28 PM each workday.\" I really needed the two extra minutes to make my work zone shipshape before leaving.\n\nAs I walked back to my desk I thought, \"What the hell is the purpose of this agency if it is not to help the people in the quickest possible time?\" I was unskilled at looking busy and doing nothing. I had to find a way to fool the VA gods circling in a holding pattern over my head and keep them from becoming displeased.\n\nSince I was a new and unskilled bureaucrat, knowing no other way to upgrade my gaming skills, I started an afternoon self-improvement program. What I would do between each case was to drag out a regulation or manual and study it. Surely, I would win favor with Mr. GS-13 because I would maintain my twenty-five-cases-per-day quota and advance my knowledge of the rules and laws government devised to help veterans.\n\nIt was smooth sailing for more than a month, and I thought I was learning the ropes. Then one morning from the back of the room came, \"Mr. Roche, my office!\" After I arrived, Mr. GS-13 said, \"You still don't understand. You must work the whole eight hours doing only twenty-five cases! You don't spend several hours each day reading regulations and manuals. Do I make myself clear Mr. Roche?\" \"Oh, yes, sir! Mr. GS-13. I will work real hard to fit in and meet your expectations.\" As I walked back to my desk I thought, \"This relationship is headed for the divorce court.\"\n\n### Fall In, Line up, Dress Right Dress\n\nIt seemed as if I were put on earth for one purpose only\u2014to become Mr. GS-13's worst nightmare. To explain, the adjudication department was organized to look like a military parade passing in review. The desks, chairs, bookcases, and waste paper baskets were precisely aligned. In the spirit of trying to be part of the team, I made sure my unit was like everyone else's. That was until I was rear-ended in a car accident and suffered severe whiplash from the collision. My doctor wrote to the VA advising that the front end of my desk be elevated at least four inches above floor level. Poor Mr. GS-13! For more than three months, he had to walk past my work area knowing that harmony within the section was broken. The thrill of maintaining a well-organized section was his dream and goal. Every time he passed my work zone, his blood pressure seemed to jump twenty to thirty points as he looked at the one desk that was different. I was starting to feel sorry for the man.\n\n### Discrimination Is a No-No!\n\nNo sooner was the room back to order with the front of my desk lowered to floor level than some friends and I disrupted the tranquility of the section once again. One of my friends, with whom I served on active duty, was not promoted to the GS-9 level when he transferred into the division. There were about nine of us who started at the same time; all had no experience and came aboard as GS-9s. My friend, however, who was transferred out of the contact division on the first floor, was transferred to Adjudication at his current grade of GS-7. He had the same educational qualifications as all of us and was far more knowledgeable about the technical aspects of the job. As a group, we went to the union representative and complained that management was discriminating against Charlie. Three levels of management were all shook up by the complaint. There were closed-door meetings and all kinds of subsequent meetings to deal with this crisis.\n\nThen, the unbelievable happened! They promoted him to a GS-9 to restore tranquility and stop any bad press in the newspapers if the word got out. Needless to say, I once again managed to notch my way up another step on Mr. GS-13's \"watch your step\" list.\n\n### Finding the Secret of Looking Busy\n\nMeanwhile my friend, who was now a GS-9, clued me in on the way to look busy when you are ready to shut down and go home at two in the afternoon. He suggested taking each file and reviewing it cover to cover. It's an easy trick to look busy and not get in trouble. This was dynamite information and I thought I could soon be off Mr. GS-13's list. Well, guess what? I screwed up again.\n\nSeveral months later after I mastered the technique of reviewing a file from cover to cover, I came across a claim that had been denied. In the file was the evidence needed to grant the claim. To make matters worse, the veteran was a neighbor of mine. In my innocence, I thought telling Mr. GS-13 and my neighbor about the find would certainly earn me a civilian citation for actions above and beyond the call to duty.\n\nThe morning after I found this treasure, I marched in into Mr. GS-13's cubicle and proudly announced I found an error by the rating board that erroneously denied Mr. Smith his compensation benefits. In a voice that echoed through the entire 5th floor (well, that may be a slight over statement), he responded, \"You did _what_?\" By the tone of his voice, I knew I had stepped in it again. He asked, \"Who else knows about this?\" \"No one, sir, except you and... the... veteran, who is my neighbor.\" I had told my neighbor about the find last night. I got another loud **\"** _What_! **\"** Before he could fully recover, I hit him with a quick, \"Isn't that our job, sir? To see that veterans get everything they are entitled to?\" \"Who do think you are, that you can tell a veteran what he or she is entitled to?\" he bellowed. \"That is not your job. I'm taking this immediately up to the _Director._ \" (The Director sits on the left hand side of the God of Gods in VA Land.) He is the local representative of the VA's Secretary of the Department of Veterans Affairs. Poor Mr. GS-13! This most heinous act was committed in his section. I remember he paled and said there was going to be \"hell to pay for this.\" On departing he left me with these chilling words: \"You have breached an unpublished administrative policy.\" Man, I was really confused, how can you breach a policy if they never told you about it?\n\nI learned that morning that you do not go back further than the last rating decision when reviewing a file. Some old timers explained this policy to me. It is known as \"Top Sheeting\" and it had several purposes:\n\na. It hastens the flow of cases through the system.\n\nb. If the claimant doesn't challenge the denial of benefits within one year, the decision stands. To reopen the claim later may be impossible. The VA has created an administrative mine field to discourage any attempt to reopen a claim.\n\nc. The executive branch of the government and the budgeters who hide in the back rooms of VA Central Office in Washington give each other a thumbs up every time a veteran fails to find a way to reopen his or her claim.\n\n### The Time Has Come to File for a Divorce\n\nFor nearly three years, I struggled with Mr. GS-13. I finally realized that I did not want to become a Mr. GS-13; therefore, with a happy heart, I said my farewells, feeling like I was the winner because I walked away with a whole briefcase full of knowledge of how the system really works. It was nearly a decade before an opportunity popped up for me to become an advocate. Now I was ready to do battle in their backyard. I finally felt I could help my fellow comrades.\n\nBut this isn't the end of the story. My interest in how the VA treated veterans never wilted. Although the faces may change, the \"Mr. or Ms. GS-13\" will ensure the gates to receiving benefits will always remain tightly controlled. And convincing a rating board member that being held hostage by post-traumatic stress disorder (PTSD) is a serious medical issue is almost an impossible task.\n\nIt is impossible, in many cases, to convince a rating board member of the seriousness of PTSD when they have no life experiences to relate it to. I wouldn't be surprised if the closest any of these individuals ever came to experiencing a psychological trauma was when they fell off their bicycles at the early age of five and neither their mothers nor fathers were around to comfort them.\n\nA story recently leaked about a veteran denied service connection for PTSD because he did not serve in Vietnam. He was awarded a non-service connected rating for PTSD. When the advocate went to the rating board member who denied the claim, the board member told the advocate to look at his atlas of Vietnam. \"Do you see any cities called Saigon?\" asked the board member. \"No,\" was the answer, \"but I do see the city they renamed. It was renamed Ho Chi Minh City when our troops pulled out.\" \"My atlas makes no mention of this, therefore the decision stands. If he doesn't like it, let him appeal it,\" was the board member's last word on the subject. It's completely conceivable that veterans may battle this issue for the next two to ten years.\n\n### Reward Incompetence\n\nThe rewards are great for those bureaucrats who do _not_ resist the dismantling process. When he retired from the VA, a former Director of a VA Regional Office was appointed to the position of Executive Director of the Florida State Department of Veterans Services. He, in turn, brought in his number one enforcer. His \"yes\" shadow also retired from the VA and was appointed to the second most important position in the Florida State Department of Veterans Services. The former VA director and adjudication officer were the movers and shakers of one of the worst run regional offices in the country. Information released by the Department of Veterans Affairs' Board of Veteran's Appeals identified this regional office as having an almost 80% remanded or reversed rate! But to the great VA God in Washington, the Secretary of the Department of Veterans Affairs, the Florida director was serving his master well. Millions upon millions of dollars were never shipped to Florida to support the growing veteran population.\n\nIt is hard to understand how a VA director and adjudication officer, whose leadership earned them an Al Capone reputation, can now, through Harry Potter's wizardry, be turned into the Elliott Ness protector of veterans' rights. Will this leadership help the cause of veterans? A real Elliott Ness would organize task forces and go about righting all the inequities in Florida. To be an advocate, you have to be committed to challenge the systems. You cannot be a friend of the Governor or Washington's VA God of Gods and do the job you signed on to do.\n\n## ACKNOWLEDGMENTS\n\nFirst, I wish to acknowledge Sgt. David Raz (a pseudonym) for his valuable contributions by letting me tell his story of frustration and mental stress when dealing with the VA for the past four years. It takes great courage to hang tough when every instinct says to chuck it. It angers me and saddens me that any veteran has to fight so hard for what should have been a slam-dunk request from a government who gave them the problem.\n\nThe second thank you goes to Congressman Michael Bilirakis from Florida's Ninth district and his Palm Harbor staff, Sonja and Shirley, for all the great services they have provided Florida veterans. It was their help that made it possible for Sergeant Raz to accumulate evidence and facts from the VA in a timely manner. Congressman Mike, we are going to lose a great champion of veterans' rights in the House of Representatives when you retire next year.\n\nThird, to my wife of more than a half of century, your input helped me \"stay the course.\"\n\n## ABBREVIATIONS\n\nA&A | Aid and Attendance\n\n---|---\n\nACR | American College of Radiology\n\nADJ | Adjudication\n\nAHCRP | Agency for Healthcare Policy and Research\n\nAL | American Legion\n\nALJ | Administrative Law Judge\n\nAOJ | Agency of Original Jurisdiction\n\nAMA | American Medical Association\n\nAMIE | Automated Medical Information Exchange System\n\nAPC | Armored Personnel Carrier\n\nApp. | Appeal\n\nBVA | Board of Veterans Appeals\n\n\"C\" | Claim File Number\n\nCCEP | Comprehensive Clinical Evaluation Program\n\nCDC | Center for Disease Control and Prevention\n\n_CFR_ | _Code of Federal Regulation_\n\nCHAMPUS | Civilian Health and Medical Program United States\n\nCHAMPVA | Civilian Health and Medical Program Veterans Administration\n\nCOLA | Cost of Living Adjustment\n\nCOMP | Compensation\n\nCOPD | Chronic Obstructive Pulmonary Disease\n\nC&P Exam | Compensation and Pension Examination\n\nCSS# | Claim Number using Social Security Number\n\nCAVC | United States Court of Appeals for Veterans Claims\n\nDAV | Disabled American Veterans\n\nDEPS | Did Not Exist Prior to Service\n\nDMZ | Demilitarized Zone\n\nDNA | Defense Nuclear Agency\n\nDOD | Department of Defense\n\nDOS | Date of Separation\n\nDRG | Diagnostic Related Groups\n\nDSC | Direct Service Connection\n\n_DSM IV_ | _Diagnostic and Statistical Manual of Mental Disorders_\n\nDSWA | Defense Special Weapons Agency\n\nDVA | Department of Veterans Affairs (VA)\n\nDVAMC | Department of Veterans Affairs Medical Center (same as VAMC)\n\nEOD | Entrance on Active Duty\n\nEPA | Environmental Protection Agency\n\nEPS | Existed Prior to Service\n\nEVR | Eligibility Verification Report\n\n_FR_ | _Federal Register_\n\nFTCA | Federal Tort Claims Act\n\nFDA | Federal Drug Administration\n\nFOIA | Freedom of Information Act\n\nFTCA | Federal Tort Claims Act\n\nGSW | Gun Shot Wound\n\nHO | Hearing Officer\n\nIB-56 | Physician's Guide for Disability Evaluation Examinations\n\nIOM | Institute of Medicine\n\nIU | Individual Unemployability\n\nJAG | Judge Advocate General\n\nKIA | Killed in Action\n\nKIA\/BNR | Killed in Action\/Body Not Recovered\n\nLOD | Line of Duty Investigation\n\nMIA | Missing in Action\n\nMOAA | Military Officers Association of America\n\nMOS | Military Occupational Specialty\n\nMR | Morning Report\n\nN\/A | Not Applicable\n\nNARA | National Archive and Record Administration\n\nNAS | National Academy of Science\n\nNCI | National Cancer Institute\n\nNCPTSD | National Center for PTSD\n\nNCOA | Noncommissioned Officers Association\n\nNOA | Notice of Appeal\n\nNOD | Notice of Disagreement\n\nNPRC | National Personnel Record Center\n\nNRL | National Research Laboratory\n\nNSC | Non-Service-Connected\n\nNTPR | Nuclear Test Personnel Review\n\nOGC | Office of General Counsel\n\nOIG | Office of Inspector General\n\nP&T | Permanent and Total\n\nPCS | Permanent Change Station\n\nPCT | Porphyria Cutanea Tarda\n\nPFC | Private First Class\n\nPOW | Prisoner of War\n\nPTSD | Post-Traumatic Stress Disorder\n\nRAD | Released from Active Duty\n\nRO | Regional Office\n\nRPG | Rocket Propelled Grenade\n\nSBA | Survivors Benefit Program\n\nSC | Service Connected\n\nSCP | Specialized Care Program\n\nSMC | Special Monthly Compensation\n\nSMR | Service Medical Records\n\nSOC | Statement of the Case\n\nSSA | Social Security Administration\n\nSSD | Social Security Disability\n\nSSOC | Supplemental Statement of the Case\n\nTDIU | Total Disability due to Individual Unemployability\n\nTDY | Temporary Duty\n\nUSA | United States Army\n\nUSAF | United States Air Force\n\nUSC | United States Code\n\nUSCA | United State Code Annotated\n\nUSCG | United States Coast Guard\n\nUSCMJ | United States Code of Military Justice\n\nUSMC | United States Marine Corps\n\nUSN | United States Navy\n\nVA | Veterans Administration\n\nVACOVA | Veterans Administration's Central Office, Washington, DC\n\nVABA | Veterans Administration Benefit Administration\n\nVAC | Veterans Administration Clinic\n\nVAE | Veteran Administration Examination\n\nVAF | Veteran Administration Form\n\nVAHA | Veterans Administration Health Administration\n\nVAHAMO | Veteran Administration Health Administration Mammography Office\n\nVAMC | Veteran Administration Medical Center (DVAMC)\n\nVARO | Veterans Administration Regional Office\n\nVEE | Venezuelan Equine Encephalomyelitis\n\nVFW | Veterans of Foreign Wars\n\nVSO | Veteran Service Officer\n\nVVA | Vietnam Veterans of American\n\nWAC | Women's Army Corps\n\nWIA | Wounded in Action\n\nWWI | World War I\n\nWWII | World War II\n\nWWW | World Wide Web\n\n## INTRODUCTION\n\nMembers or former members of the U.S. military are more likely than not to have experienced a psychological trauma while in the service. As for police officers, firefighters, and paramedics, the work environment for military service men and women will always be considered extremely hazardous. Unfortunately, tens of thousands of former veterans have been denied service-connected healthcare and compensation for post-traumatic stress disorder (PTSD) because they were unable to make a case for their claim. Such a claim need not be conclusive but must contain enough evidence to satisfy the initial burden.*\n\nIf you were a combat infantry trooper or pilot flying off an aircraft carrier or an aircrew member shot down over enemy territory the VA would most likely rule in your favor. Don't assume that your claim will automatically be accepted as service-related, thus granting benefits.\n\nBut suppose you were a cook assigned to a marine unit stationed close to the DMZ (Demilitarized Zone) when a surprise attack on your unit's compound occurred late one night. The events you witnessed that night have kept you prisoner in your own body for more than twenty years. Two decades after leaving Vietnam, you file a claim for disabilities associated with PTSD. It is almost 100% certain that, based on your statement alone, your claim for disabilities benefits will be denied. Why? To a senior claims examiner who most likely never served in combat or, for that matter, in the armed forces, would ask, \"Who ever heard of a cook engaging in one on one combat with the enemy? And why did his PTSD take twenty years to surface?\" Now, you wait. It is nearly a year before the VA letter arrives denying your claim. The letter states: after careful view of the records, we have determined your claim is not valid. What records? This is the question you will ask the examiner to get your foot in the door.\n\nThis is a book about the dos and don'ts of filing a claim for PTSD. The last thing you want to do is submit a claim that the VA will deny because it is not well grounded. Here is what you have to look forward to if you do. First, you have to file a Notice of Disagreement (NOD) with the Regional Office that rejected the claim. An NOD must be filed within one year of the denial date. Failure to file on time will greatly complicate future efforts to reopen your claim. Failure to challenge the VA decision will let the VA assume that you are in agreement with their decision. Six to eight months later, the VA will send you a letter known as a Statement of the Case (SOC). The SOC is a summary of how the VA sees the facts and is used to justify why your claim was denied. You might say that it's the \"opening statement\" by a defendant (the VA) in a civil action suite brought by you (the plaintiff) against the VA. Now claim processors are supposed to cite the factual reasons why your claim was denied. They are required by statute to cite exactly what evidence justified the denial of benefits. Be alert to generic phrases such as \"a preponderance of the evidence is against your claim\" or \"your records are totally silent as to any reference you were a combat infantry soldier.\" Such statements do not pass the litmus test. Despite the law, maybe one out of ten SOCs will cite the factual evidence which justified the denial of benefits.\n\nStep two, you must respond to the SOC within sixty days, addressing only one issue: why your claim is well grounded. It has been nearly a year since you submitted the original claim. You scratch your head and ask, \"What do they mean my PTSD problems didn't start in Vietnam?\"\n\nAbout six to eight months later, the mail carrier delivers another letter from the VA. This time it is called a Supplement Statement of the Case (SSOC). They have rejected your arguments supporting why your claim is valid. You have sixty days to respond. This time you ask to meet with a hearing officer. Again you must wait more than six months before the VA notifies you that a meeting has been scheduled with a hearing officer.\n\nThe next letter will tell you that your case is being certified to be forwarded the Board of Veterans' Appeals (BVA). At this point your claim has been in the system nearly three years and you are no closer to obtaining benefits.\n\nYour claim now sits in a locked file until the BVA calls for it to be sent to Washington. Again the magic numbers seem to be six to eight months for action. Once your claim gets to the BVA, another six months may pass before a decision is issued that advises you of the findings.\n\nIn total, the time spent with the BVA will add another eighteen months to the process. Here we are, nearly four and half years later, and the only issue so far addressed is whether your claim was valid.\n\nLet's say the BVA agrees with you and remands your case back to the Regional Office of origin. Now the Regional Office will reevaluate the case to determine if the evidence is sufficient to grant service-connection status for your PTSD. Hold on, you're looking at nearly another year before someone gets back to you with that decision.\n\nWe are now over the five-year mark, still waiting to see if we will be granted benefits. If the VA does not grant service-connection for your PTSD, the only course is to start all over again with the appeal process. It is at this point many veterans throw up their hands in total frustration and quit trying to receive benefits.\n\nThe keys to successfully submitting a winning claim are planning, knowing as much as you can about PTSD, documenting events that caused the psychological trauma, and understanding how the VA thinks. Believe this, they make big-time errors. But this book will help you avoid pitfalls that could result in the rejection of your claim.\n\nChapter 1 will provide background information necessary to understand the crippling affects of PTSD. Understanding the ABCs of the disorder will help you identify the level of disability you are coping with.\n\nChapter 2 tells how thousands of VA claims for PTSD more than likely were denied because direct service connection for PTSD was blocked by simply requiring the Bronze Star and the Air Medal when awarded was for heroism. Without a \"V\" device for these medals, per Operational Manual, M21-1 subchapter 11, section (or \u00a7)11.37, a veteran had to prove he or she engaged in direct combat with the enemy and the traumatic event was the direct cause of the PTSD.\n\nChapter 3 is where you will meet Sgt. David Raz. He is a Vietnam veteran with whom I have worked for nearly four years. It has been an uphill battle all the way, but we still are in the fight. He has granted me permission to tell his story, and show his evidence that is on record. It was extremely hard for him to dig up his past and make his life an open book. His story is no different than the hundreds of thousands of PTSD victims who have tried to deal with the VA and lost. We have managed to storm three-quarters of the way up the mountain. The VA still holds the high ground, but not for long. Read this story before you submit your claim for PTSD so you can plan your strategy.\n\nChapter 4 opens the door to what the VA adjudication process is supposed to do according to the laws and its regulations when deciding a claim for PTSD. The action word is \"supposed.\" As you will see, what the system is supposed to do and what it does are often quite different.\n\nChapter 5 introduces you to the basics that you must know before filing a claim. What kind of claim are you going to file? Is it an original claim or perhaps a reopened claim, or is it an informal claim? Not knowing which claim to submit can lead to a deathblow for your benefits.\n\nChapter 6 explains your responsibility when preparing a claim. Like everything you do in life, there is a right way and a wrong way. Most important, you must file a claim that meets the four critical elements for any claim. If you fail to meet any one of these elements, there is no way you will be granted service connection for PTSD.\n\nChapter 7 is about the required Compensation and Pension Examination. This is another VA minefield that takes heavy casualties. In Sergeant Raz's story you will come across a document evaluating his PTSD. While he was required to have this examination, it was not performed by a psychiatrist or even by a psychologist. So who is left in the VA medical feeding chain? How about a _nurse practitioner_? As our case rolls to the finish line, it's going to be interesting to see who will prevail in deciding Sergeant Raz's case. Will it be the findings of a board-certified psychiatrist with thirty years' experience or will it be a two-year graduate student majoring in medicine? If it turns out like several of my previous cases, the rating board will base its decision on the findings of the least qualified medical source.\n\nChapter 8 is devoted to helping veterans qualify for 100% compensation benefits if their level of PTSD prevents them from being employed. The courts ruled that if your PTSD prevents you from earning a living, it does not matter what percentage your disability is rated. In theory it could be as low as 10%.\n\nChapter 9 is titled \"Prove it or Lose It.\" If your claim is not properly supported by the right kind of evidence, there is no way you can prevail. This chapter is designed to give you a way to find evidence to support your claim. Even though the VA has a duty to assist you in looking for evidence, it would be foolish to depend on them. Only you can provide the energy and interest to submit a winning claim.\n\nChapter 10 explains how to organize the evidence, assemble your evidence package, and submit it to the VA. There are many kinds of daily experiences on active duty that could cause PTSD. Always remember serving in the military is a 24\/7\/365 job. Therefore anything that changes your life and affects your ability to adjust to a normal post-service life style is a service-incurred disability. Never forget that within the executive branch of the government there are powerful individuals who are high on slogans but short on support. The purpose of this book is to minimize the chances of your claim being denied for PTSD. If benefits are denied and you do challenge the VA with an appeal, be prepared to hold the course for a period of three to five years. Yes, even ten years. There are no fast-track procedures that would put your appeal ahead of other claims in disputes with the VA. This book will provide you with the means of moving your claim through the process without going the appeal route.\n\nChapter 11 concerns your rights and how to protect them. It discusses what a well-grounded claim is all about and the consequences for failing to submit a well-documented claim. You will be introduced to precedent-setting decisions by the U.S. Court of Appeals for Veterans Claims that might have an effect on your claim.\n\nChapter 12 will acquaint you with specific court decisions that are related to PTSD claims. These cases were selected because they represent some of the most difficult claims to be approved.\n\n### Notes\n\n* VA's _Duty to Assist Act_.\n\n## 1 POST-TRAUMATIC STRESS DISORDER\n\n### What Is Post-traumatic Stress Disorder?\n\nPost-traumatic stress disorder (PTSD) is a psychological disorder with a long recorded history of incapacitating hundreds of thousands of individuals who were in combat or traumatized by an event in their lives. It can occur following the experience or witnessing of life-threatening events such as military combat, natural disasters, terrorist incidents, serious accidents, or violent personal assaults. People who suffer from PTSD often relive the experience through nightmares and flashbacks, have difficulty sleeping, and feel detached or estranged. These symptoms can be severe enough and last long enough to significantly impair the person's daily life.\n\nPTSD is marked by clear biological changes as well as psychological symptoms. PTSD is complicated by the fact that it frequently occurs in conjunction with related disorders such as depression, substance abuse, problems of memory and cognition, and other problems of physical and mental health. The disorder is also associated with an impaired ability to function in social or family life, which can manifest itself as occupational instability, marital problems and divorce, family discord, and difficulties in parenting.\n\nDespite popular belief, post-traumatic stress disorder is not a new medical phenomenon that surfaced during the Vietnam War. It has affected warriors for more than twenty-seven hundred years. Neither the victors nor the defeated can escape the horrific after-effects of war. Every person who was ever called to serve or defend his or her country is a possible victim of this disorder.\n\nIn 1980 the American Psychiatric Association (APA) published a standard to diagnose traumatized victims who became dysfunctional as a result of their traumatic experience. The _Diagnostic and Statistical Manual of Mental Disorders, 3rd Edition (DSM-III)_ wasn't accepted as a standard by the Veterans Administration (VA) Medical Administration for psychiatric service related disability until 1986. The VA now uses the updated fourth edition as its standard. Before the 1986 change of direction, the VA and military doctors commonly diagnosed veterans demonstrating symptoms of PTSD as having a personality disorder, obsessive-compulsive disorder, paranoia, or schizophrenia, but never with PTSD.\n\nBy diagnosing individuals with one of these illnesses, the Department of Defense (DOD) and VA had no lawful duty to compensate these disabled veterans. The law was quite clear: if a veteran had any one of these congenital illnesses, he or she was barred from receiving medical or monetary support. The rational was that congenital medical problems were not the result of serving in the U.S. Armed Forces.\n\nMany of these veterans joined the ranks of the thousands of homeless wanderers. Dysfunctional veterans had a high death rate in automobile accidents and suicides. The genesis for chronic alcoholism and drug addiction is rooted deeply in PTSD.\n\nWhen considering the hundreds of thousands of individuals who can be ignored by the movers and shakers within an administration their care translates into billions of dollars that can be directed for other purposes. The executive branch of the government\u2014including the White House, the Office of Management and Budget, and the Veteran's Administration\u2014generates billions of dollars in savings annually at the expense of veterans' compensation and health care. The technique is invisible and so does not raise the interest of congressional oversight committees.\n\nThose in the Secretary's Office claim they do not have the necessary computer memory to establish a databank to collect information concerning the number of claims denied annually. How strange it is that an employee could take home the personal data for 26.5 million veterans on a disk and portable hard drive.\n\nHere is an example of how the VA plays the game and what the consequences are for the veteran population: In fiscal years 2003 and 2004, the VA announced that it notified 1,530,000 veterans of their decisions on their claims. Of those who were notified, 330,000 were added to the roll. No explanation was offered concerning the 1,200,000 claims that were not added to the rolls. Let us be conservative and say the VA denied 600,000 claims at an average benefit rate of $500 per month. That would equate to a savings of $7 billion over a two-year period. Veterans with health insurance who seek medical services from the VA Health Administration would have their insurance company charged back for their care. This amount equals hundreds of millions of dollars annually.\n\nIf an active duty member is determined to be unfit for worldwide service, he or she must be evaluated by a medical evaluation board. The military uses the VA disability rating schedules to determine the degree of disability for those appearing before their boards. Suppose the air force medical board found an airman 40% disabled because of PTSD. This finding requires that he be medically separated from active service. Military regulations hold that if an individual is to be medically separated and is rated 30% or higher, he or she will be medically retired.\n\nWhat happens if this person served in WWI, WWII, the Korean War, Vietnam, or during the '60s, '70s, or '80s and has PTSD? Because the VA did not recognize PTSD as an illness prior to 1986, this individual would be separated from the military without medical retirement benefits or assistance from the Department of Veterans Affairs.\n\nThe general public's attitude toward people who are mentally ill historically has been extremely intolerant; most would shun them rather than make physical contact. A good example of this attitude was made public during WWII by an incident caused by Gen. George C. Patton. General Patton was on an inspection tour of an army field hospital when he observed a soldier sitting on a bunk sobbing. He told the soldier to get a hold of himself and behave like a man. The soldier apparently responded with a statement that he was \"afraid.\" The general became so enraged that he grabbed the soldier and struck him while shouting he would \"have no cowards\" in his army. Patton ordered the hospital commander to send the soldier immediately to the front.\n\nThis intolerance demonstrated by General Patton of those handicapped by traumatic incidents often still flows just below the surface in today's military. A career noncommissioned officer (NCO) or officer will try to hide his symptoms so as not to destroy his career, forfeit retirement benefits, or subject himself to unnecessary discrimination owing to lack of knowledge regarding mental health issues.\n\nAnother policy generated during the Regan Administration (1981\u20131989) was the reason why thousand upon thousands of mentally dysfunctional individuals were sent to halfway houses. During 1982, Reagan's administration initiated fiscal changes to reduce the operational cost of running the government. One of these changes was to cut funds for long-term institutional care for the mentally challenged. The reason was simply, \"it was more cost effective\" to send mentally dysfunctional individuals to halfway houses than to spend the money on their treatment or support. It wasn't too long before funds started to dry up, closing the centers and leaving individuals with no option but to move on to the streets. They became \"the walking dead\" in every town, city, and state. Many of these individuals were former GIs who served during WWII, Korea, and Vietnam. It was a budget coup for the administration to force local charities to pick up the expense for these homeless people.\n\nDuring the twentieth century alone, the name for post-traumatic stress disorder was changed three times, but the effects are the same. Doughboys of WWI who succumbed to the psychological traumas of open-trench warfare were identified as suffering from \"shell shock.\" Next, the GI Joes of WWII and Korea who showed the symptoms of psychological trauma were said to have \"combat fatigue.\" Finally, in 1986 combat troops experiencing severe trauma in Vietnam were diagnosed as suffering from PTSD. The American Psychiatric Association added \"post-traumatic stress disorder\" to the _Diagnostic and Statistical Manual of Mental Disorders, 3rd Edition_ in 1980.\n\nHowever, it was 1986 before the VA considered rating claims for PTSD under the APA standard. Imagine being tagged by the Department of Veterans Affairs as crazy for twenty-seven years. I remember when they started rating PTSD claims. It was in the early 1980s and of those vets granted compensation benefits for PTSD the majority were rated between 0% and 30%. The rating system was devised to compensate a veteran for the loss of their ability to earn a living. A person who has no injuries or illness is said to be 100% whole. A person with PTSD, for example, who is medically diagnosed with definite impairment in the ability to establish or maintain effective and wholesome relationships with people is rated 30% disabled. The earliest VA records available (1986) confirm that this group of veterans comprised 71% of all those on the rolls through 1986. It is truly remarkable that, with all the data available and the advanced collecting software that the VA has not tracked how many claims are received each year and how many claims are denied for any given year. What is known is that of the twenty-five million veterans in the United States, only 10% are on the VA rolls for compensation benefits.\n\nWinning a claim for post-traumatic stress disorder has always been difficult for a veteran. Through the years the VA has arbitrarily rejected thousands of claims based on PTSD disabilities. Sergeant Raz's story, which is explored in chapter 3, is no exaggeration of the difficulty imposed on veterans suffering from this disorder.\n\n### VA Marketing of PTSD Information\n\nOne of the ways the Department of Veterans Affairs markets its public persona is through a government-identified website for the National Center for Post-traumatic Stress Disorder (NCPTSD). When researching background information for this book, I came across this website by accident. My first thought was this information was being presented by a nongovernment research group. However, suspicion was aroused at the end of the article when the authors stated that it could be used by any one, as it was information in the public domain. Following the links it provided lead me to the NCPTSD home page, which identified itself as a source of information sponsored by the VA. Looking for further information that the NCPTSD was a division of the VA, website I pointed my browser to the Department of Veterans Affairs official home page. The NCPTSD is not mentioned in any of the official hyperlinks associated with the VA website. I did find it when I used their internal search engine. That is not to say that the information is not valid for gaining a basic understanding of this disability.\n\nThe National Center for Post-traumatic Stress Disorder was created within the Department of Veterans Affairs in 1989 in response to a congressional mandate to address the needs of veterans with military-related PTSD. Its mission was and remains: \"to advance the clinical care and social welfare of America's veterans through research, education, and training in the science, diagnosis, and treatment of PTSD and [other] stress-related disorders.\" This website is provided as an educational resource concerning PTSD and other enduring consequences of traumatic stress.\n\n### Understanding PTSD\n\nPTSD is not new. There are written accounts of similar symptoms that go back to ancient times, and there is clear documentation in the historical medical literature starting with the Civil War, when a PTSD-like mental disability was known as \"Da Costa's Syndrome.\" There are particularly good descriptions of post-traumatic stress symptoms in the medical literature on combat veterans of World War II and on Holocaust survivors.\n\nCareful research and documentation of PTSD began in earnest after the Vietnam War. The National Vietnam Veterans' Readjustment Study estimated in 1988 that the prevalence of PTSD in that group was 15.2% at that time and that 30% had experienced the disorder at some point since returning from Vietnam.\n\nPTSD has subsequently been observed in all veteran populations that have been studied, including World War II, Korean conflict, and Persian Gulf populations and in United Nations peacekeeping forces deployed to other war zones around the world. There are remarkably similar findings of PTSD in military veterans in other countries. For example, Australian veterans who served in Vietnam experience many of the same symptoms that American veterans experience.\n\nPTSD is not only a problem for veterans, however. There are unique cultural and gender-based aspects of the disorder, and it occurs in men and women, adults and children, Western and non-Western cultural groups, and all socioeconomic strata. A national study of American civilians conducted in 1995 estimated that the lifetime prevalence of PTSD was 5% in men and 10% in women.\n\n### How Does PTSD Develop?\n\nMost people who are exposed to a traumatic, stressful event experience some of the symptoms of PTSD in the days and weeks following exposure. Available data suggest that about 8% of men and 20% of women soon develop PTSD, and roughly 30% of these individuals develop a chronic form that persists throughout their lifetimes.\n\nThe course of chronic PTSD usually involves periods where symptoms increase followed by periods of remission or decrease, although some individuals may experience symptoms that are unremitting and severe. Some older veterans, who report a lifetime of only mild symptoms, experience significant increases in symptoms following retirement, severe medical illness in themselves or their spouses, or reminders of their military service (such as reunions or media broadcasts of the anniversaries of war events).\n\n### How Is PTSD Assessed?\n\nIn recent years, a great deal of research has been aimed at developing and testing reliable assessment tools. It is generally thought that the best way to diagnose PTSD\u2014 or any psychiatric disorder for that matter\u2014is to combine findings from structured interviews and questionnaires with physiological assessments. A multi-method approach especially helps address concerns that some patients might be either denying or exaggerating their symptoms.\n\n### How Common Is PTSD?\n\nAn estimated 7.8% of Americans will experience PTSD at some point in their lives, with women (10.4%) twice as likely as men (5%) to develop PTSD. About 3.6% of U.S. adults aged eighteen to fifty-four (5.2 million people) have PTSD during the course of a given year. This represents a small portion of those who have experienced at least one traumatic event; 60.7% of men and 51.2% of women reported at least one traumatic event in their lifetime. The traumatic events for men most often associated with PTSD are rape, combat exposure, childhood neglect, and childhood physical abuse. The most traumatic events for women are rape, sexual molestation, physical attack, being threatened with a weapon, and childhood physical abuse.\n\nAbout 30% of the men and women who have spent time in war zones experience PTSD. An additional 20% to 25% have had single episodes of PTSD at some point in their lives. More than half of all male Vietnam veterans and almost half of all female Vietnam veterans have experienced \"clinically serious stress reaction symptoms.\" PTSD has also been detected among veterans of the Gulf War, with some estimates running as high as 8%.\n\n### Who Is Most Likely to Develop PTSD?\n\nHere are several examples:\n\n  those who experience greater stressor magnitude and intensity, unpredictability, uncontrollability, sexual victimization, real or perceived responsibility, and betrayal;\n\n  those with preexisting vulnerability factors such as genetics, early age of onset and longer-lasting childhood trauma, lack of functional social support, and concurrent stressful life events;\n\n  those who report greater perceived threat or danger, suffering, upset, terror, and horror or fear; and\n\n  those with a social environment that produces shame, guilt, stigmatization, or self-hatred.\n\n### What Are the Consequences Associated with PTSD?\n\nPeople with PTSD tend to have abnormal levels of key hormones involved in the body's response to stress. Thyroid function also seems to be enhanced in people with PTSD. Some studies have shown that cortisol levels in those with PTSD are lower than normal and epinephrine and norepinephrine levels are higher than normal. People with PTSD also continue to produce higher than normal levels of natural opiates after the trauma has passed. An important finding is that the neurohormonal changes seen in PTSD are distinct from and actually opposite to those seen in major depression. The distinctive profile associated with PTSD is also seen in individuals who have both PTSD and depression.\n\nPTSD is associated with an increased likelihood of co-occurring psychiatric disorders. In a large-scale study, 88% of men and 79% of women with PTSD met criteria for another psychiatric disorder. The co-occurring disorders most prevalent for men with PTSD were alcohol abuse or dependence (51.9%), major depressive episodes (47.9%), conduct disorders (43.3%), and drug abuse and dependence (34.5%). The disorders most frequently co-morbid with PTSD among women were major depressive disorders (48.5%), simple phobias (29%), social phobias (28.4%), and alcohol abuse or dependence (27.9%.)\n\nHeadaches, gastrointestinal complaints, immune system problems, dizziness, chest pain, and discomfort in other parts of the body are common in people with PTSD. Often, medical doctors treat the symptoms without being aware that they stem from PTSD.\n\n### How Is PTSD Treated?\n\nPTSD is treated by a variety of forms of psychotherapy and drug therapy. There is no definitive treatment and no cure, but some treatments appear to be quite promising, especially cognitive-behavioral therapy, group therapy, and exposure therapy. Exposure therapy involves having the patient repeatedly relive the frightening experience under controlled conditions to help him or her work through the trauma.\n\nStudies have also shown that medications help ease associated symptoms of depression and anxiety and help with sleep. The most widely used drug treatments for PTSD are selective serotonin reuptake inhibitors, such as Prozac and Zoloft. At present, cognitive-behavioral therapy appears to be somewhat more effective than drug therapy. However, it would be premature to conclude that drug therapy is less effective overall since drug trials for PTSD are at a very early stage. Drug therapy appears to be highly effective for some individuals and is at least somewhat helpful for many more. In addition, recent findings on the biological changes associated with PTSD have spurred new research into drugs that target these biological changes, which may lead to much increased efficacy.\n\nPTSD also significantly affects psychosocial functioning, independent of co-morbid conditions. For instance, VA studies have found that Vietnam veterans with PTSD have profound and pervasive problems in their daily lives. These include problems with family and other interpersonal relationships, problems with employment, and involvement with the criminal justice system.\n\n### Let's See What's on the Other Side of the Coin\n\nThe above information is revealing and if used as a source for basic insight into PTSD it has its merits. You cannot file a claim based solely on the information in this book. To do so is to invite a denial of benefits. As previously stated, if you go down this road you might never have your claim for PTSD benefits approved.\n\n### Complexities in Diagnosing PTSD\n\nPTSD is unusual when compared to other psychiatric conditions. You cannot be diagnosed unless you have met the \"stressor criterion,\" meaning you were exposed to an event that was considered traumatic. Examples of events that meet the stressor criterion are: you were the sole survivor of an aircraft accident or buried alive for several days because an earthquake collapsed the building you were in.\n\nExperienced clinicians will tell us that the symptoms of PTSD vary from one patient to another. Because the psychological makeup of each person differs, differing ability to manage a catastrophic stressful event will result in one person developing PTSD while another individual experiencing the same event will not. This observation also leads to the acceptance that trauma, like pain, cannot be seen externally.\n\nBefore a traumatic experience is recognized as a threat, it must be screened through what are known as the cognitive and emotional processes of the mind. It is this individual difference in the appraisal process that produces different traumatic thresholds. That is why some people are more vulnerable to developing clinical symptoms of PTSD after experiencing a traumatic event.\n\nA major problem encountered with PTSD claims is that it is often misdiagnosed. There are many diverse diagnoses that share symptoms with PTSD, includeing anxiety disorder, depressive disorder, organic mental disorder, adjustment disorder, personality disorder, and, in some cases, even psychosis. Clinicians tell us that PTSD is known to coexist with other mental disorders. In order for a rating board member to provide the claimant with the benefits to which he or she are entitled, the board must carefully scrutinize all the medical evidence of record, making certain that they do not prematurely judge the evidence. The danger is great that an inexperienced physician or rating board member will fail to recognize the overlap-ping symptoms common between PTSD and some other forms of neurosis and wrongfully conclude that the claimant does not suffer from PTSD. This is one reason why your supporting medical evidence must be very specific about the symptoms and diagnosis of your condition.\n\n### Sex-Related Trauma\n\nSex-related traumatic events are considered stressors. By direction, rating board members have been told that they must accept the fact that rape, sexual assault, or some other forms of sexual wounding are stressors. Violent events such as these are considered to be stressful incidents by almost anyone. Rating board members are instructed that sexual harassment is also a stressor but much more difficult to corroborate. Still, sexual harassment should never be ruled out as a stressor. Repeated incidences of sexual harassment collectively could turn the victim's life into a living psychological hell. This would qualify as a stressor by acceptable medical standards.\n\nIf your claim was denied because the rating board did not consider the trauma associated with sexual harassment as a valid cause of your PTSD, the denial should be challenged. VA policy in recent years has put rating board members on notice that when rating sexual traumatic claims, they must not display any form of moral judgment. The fact that a person is sexually active in no way diminishes the trauma of some form of sexual assault.\n\n### Coexisting Psychiatric Disorders\n\nBy decisions of the U.S. Court of Appeals for Veterans Claims and the medical community at large, the VA has accepted the fact that it is possible for PTSD to actively coexist with another psychiatric disorder. However, that does not mean that individual rating board members grasp or rate PTSD claims with a thorough understanding of this acceptable medical standard. _Always remember that rating board members do not have a formal education or experience in the complexities of medicine or law_. What they do have is a manual, a regulation, some in-house guidance from the adjudication officer, an extremely heavy caseload, and a great deal of discretionary power. My experience has been that if they are going to err they do so in favor of the government. This means if they don't understand the medical or legal aspects of the claim, they will simply deny it and allow the veteran try to argue it out through the appeal process.\n\n### Some Fundamental Facts about PTSD\n\nPTSD is not a psychosis. It is a neurosis and, as such, the patient will not share those symptoms common of a patient suffering from a psychosis. In order for a physician to diagnose a neurosis, the symptoms must fit into a group of symptoms common to that condition. If the patient does not demonstrate those required symptoms associated with the condition, the doctor will not diagnose the veteran as having a neurosis. The presence of cognitive disorders is not a symptom of PTSD. However, a PTSD patient may experience flashbacks\u2014a form of visual hallucination that is very detailed and relates directly to the stressful event.\n\nPTSD can occur at anytime during your lifetime. However, VA policy is if an individual is diagnosed with \"partial PTSD\" or with \"PTSD features,\" he or she is not considered to have PTSD for compensation purposes.\n\n### Secondary Service-connected Disorders\n\nThe VA does recognize that there can be a relationship between chronic PTSD and substance abuse. This policy decision, made in the early 1990s, was not endorsed by the executive branch of the government. It was the U.S. Court of Appeals for the Federal Circuit that forced the VA to consider this issue. Alcohol and drug abuse per se cannot be rated or granted a service-connected disability status. This designation is specifically barred by law because it is considered an action of willful misconduct. However, there have been many hotly debated appeals before the U.S. Court of Appeals for Veterans Claims. The frequency of this issue before the court, in addition to PTSD appeals, demonstrates the doggedness the VA rating boards demonstrate in holding the line on PTSD claims.\n\nWhen substance abuse is the consequences of some other medical condition, it can be considered secondary to the primary medical problem and thus recognized as service connected. Substance abuse (alcohol or drug) can coexist with PTSD just as other forms of neurosis or psychoses can co-exist with PTSD. It is not unusual for multiple disorders to exist simultaneously and have no clinical relationship to each other. Because of the uniqueness of this possible relationship, you should make certain that your medical examiners clearly define the relationship between alcohol and drug abuse and the service-connected PTSD in their claim statements.\n\nAlcohol or substance abuse can quickly complicate what otherwise appears to be a clear-cut case of a service-related PTSD. The claimant must make certain that his or her physician clearly states that the current drug or alcohol condition is secondary to, a symptom of, or caused by PTSD. If this fact is not plainly made in the medical assessment by your physician, you can bet the claim will be denied. The VA will not accept a simple diagnosis of this relationship. The medical evidence furnished by your physician must support the diagnosis of a secondary relationship between alcohol or substance abuse and PTSD.\n\nIf secondary service-connected status is granted for alcohol or drugs, it will be considered part of the primary disorder and, as such, no additional compensation will be authorized. However, if subsequently, the substance abuse condition is responsible for triggering another medical condition such as cirrhosis of the liver, this condition would be rated as service-connected and compensation would be in order for the liver problem.\n\n### Chapter 1: What Have We Learned?\n\nThe standard medical manual used by VA to rate a claim for PTSD is _Diagnostic and Statistical Manual of Mental Disorders, 4th Edition_. The manual's definition of PTSD is at <http:\/\/allpsych.com\/disorders\/dsm.html>. If you do not have access to a computer, your local library should have a copy of the manual in their reference section.\n\nWe will never know how many PTSD claims are denied each year until the VA is required to obtain from their regional offices a monthly accounting of how many veterans have applied for PTSD benefits. If there are twenty-five million veterans living, from WWII through Gulf War II, the VA's 10% disabled figure is extremely skewed to the right. There has been a drastically increase number of veterans granted PTSD since the courts and Congress increased their oversight in 1989.\n\n### Notes\n\n1. American Psychiatric Association (APA), \"309.89 Post-traumatic Stress Disorder,\" in _Diagnostic and Statistical Manual of Mental Disorders: DSM-IV-R, 4th ed. revised_. (Washington, DC: American Psychiatric Association, 2000), <http:\/\/www.cirp.org\/library\/psych\/ptsd\/>.\n\n2. Principals Relating to Service Connection, 38 CFR 3.304(c), March 7, 2002.\n\n3. Ann Palmer, \"20th Century History of the Treatment of Mental Illness: A Review,\" <http:\/\/www.mentalhealthworld.org\/29ap.html>.\n\n4. APA, _DSM-IV_.\n\n5. National Center for Post-Traumatic Stress Disorder, <http:\/\/www.ncptsd.va.gov\/>.\n\n6. Department of Veterans Affairs, <http:\/\/www.va.gov>.\n\n7. National Center for Post-Traumatic Stress Disorder, <http:\/\/www.ncptsd. va.gov\/>.\n\n8. Yancy v. Principi, Docket Number 01-298, CAVC, April 27, 2004.\n\n9. Palmer, \"20th Century History of the Treatment of Mental Illness.\"\n\n10. APA, _DSM-IV_.\n\n11. Palmer, \"20th Century History of the Treatment of Mental Illness.\"\n\n12. Ibid.\n\n## 2 TWO LITTLE WORDS\n\nIn chapter 3, I will introduce you to Sgt. David Raz and his five-year struggle for benefits resulting from PTSD. You will see how he became a victim of the Veterans Administration's game of _deny, deny, and deny_ that has been in play for decades. Why would the VA do this? I can only guess that administrators do not want to be forced into budget increases year after year. This position is supported by a speech made in the House of Representatives.\n\n### Congressman Lane Evens Musters Support for Vets\n\nThe ranking democratic member of the Budget Resolution for Fiscal Year 2005 Conference Committee, Representative Lane Evens, made this statement before the House on May 19, 2004. His speech was to encourage members on both sides of the House to vote against the concurrent resolution laying out the congressional budget for the U.S. Government for fiscal year 2005.\n\nMr. Chairman, I rise in opposition to S. Con. Res. 95. As too many of us in this body know, this budget is a sham. It fails to account for the real costs of waging the war in Iraq and Afghanistan. At least on the House side, it allows tax cuts for the wealthiest Americans to go forward unchecked, while spending for important domestic programs is laid to waste.\n\nAmong those high priorities that will be _severely under funded is veterans' health care_. Committee on Veterans Affairs Chairman Christopher H. Smith and I submitted Views and Estimates to the Budget Committee requesting that it add $2.5 billion to VA's budget for fiscal year 2004. This was not a \"pie in the sky\" request, but rather, focused on maintaining current services, restoring funds from the Administration's failed proposals to increase co-payments and introduce new enrollment fees, and slightly enhancing some services that will have to respond to the needs of demobilizing troops. The resolution we are voting on today will make less than half of these funds available to VA's discretionary programs.\n\nMr. Chairman, sadly, I realize VA programs are among those considered \"protected\" in this budget fiasco. Many social programs will fare worse. Unfortunately, that's not good enough for our veterans, especially during a time of war when we should be most sensitive toward keeping our promises to the men and women who have borne the battle.\n\nMany of the major veterans' organizations have expressed great concern about the budget. As under funded as the budget was by the Committee's reckoning, it is even more so according to the Independent Budget. The four major veterans service organizations who prepare this document estimate that VA requires almost $4 billion to maintain its services in fiscal year 2005. And that's not the worst of it\u2014 budget process bills that may be put forth in the near future may use the projections of future years spending to bind us to even more inadequate budgets. So as bad as fiscal year 2005 looks, the outlook for future years could be even bleaker. Mr. Chairman, we must do better by the veterans who have served us. We must do better by the American people. Vote NO on accepting this Conference Report.\n\n### The VA Flow Chart\n\nThe management level decision to adopt only a portion of the prerequisites for awarding the Bronze Star was greatly influenced by this type of budget posturing. Adding two words\u2014\"V\" device\u2014to the statutory requirements for the Air Medal slammed the door on many PTSD claims. Thousands of Vietnam and Gulf War veterans failed to qualify for direct service connection for PTSD because their awards were not considered \"conclusive evidence.\" These administrative changes had an enormous impact on how PTSD claims are processed. All the key rules governing PTSD claims are discussed in chapter 4.\n\nHad the St. Petersburg Regional Office followed the law as Congress approved it, Sergeant Raz's claim for PTSD should have been resolved in six to eight months in his favor. The VA claimed his PTSD did not qualify for direct service connection since his Bronze Star had no \"V\" (for Valor) device. As you read chapter 3 you will see how the VA used various bureaucratic and administrative techniques to sabotage Sergeant Raz's will to hold the course.\n\nIt is important to know what your rights are and what the VA can and cannot do. When filing a claim do not expect the VA to do the legwork to develop the evidence necessary to establish your claim. They are directed to assist you by statutes, regulations caselaw, and their own manuals. However, to be uninformed of the claiming process is to invite a denial of the benefits. When filing a claim for PTSD, know the rules better than the adjudicators.\n\nIt is the Congress who authorizes all agencies of the government to execute their duties in accordance with the statutes they pass. The laws governing the Department of Veterans Affairs are marshaled under _Title 38 U.S. Code_ , section 38 of the _Code of Federal Regulations_ , general counsel opinions, and decisions rendered by the U.S. Court of Appeals for Veterans Claims and the U.S. Court of Appeals for the Federal Circuit.\n\n### _Title 38 U.S. Code_\n\nChapter 11, section 1154 of _Title 38_ authorizes assistance to veterans who are injured, traumatized, or disabled by disease on or after January 1, 1947. Sub-paragraph (b) specifically addresses the needs of veterans suffering from PTSD.\n\nWhen reading chapter 4, take note there is no mentioning of a narrow or restrictive guideline pertaining to presumptive of service\u2013connection tied to combat with the enemy. The bottom line of section (b) is: _The VA shall accept as sufficient proof of service-connection any disease or injury allegedly incurred in or aggravated during active service. Lay or other evidence will be accepted if such injury or disease is consistent with the circumstances, conditions, or hardships of such service._\n\nThis section also states if the claim is denied, adjudicators must spell out in detail exactly what evidence topples the scales in their favor. The U.S. Court of Appeals for Veterans Claims (CAVC) has said numerous times, \"No, No, No, you [the VA] cannot simply say a preponderance of evidence is against the claim.\" Should you receive such an explanation for the denial of your claim, _appeal it immediately_.\n\n### Federal Register Final Notice 06-18-99\n\nAs you read through the Federal Register (\"Final Notice of Change\"), the major amendment that impacted 38 _CFR_ ( _Code of Federal Regulations_ ) \u00a73.304(f) was the removal of references to service department evidence of combat or receipt of specific combat citations. These combat awards, such as the Purple Heart, Combat Infantryman Badge, or similar combat citations were conclusive evidence of the claimed in-service stressor. This changed the procedure a veteran diagnosed with PTSD must follow before service-connection could be granted.\n\nAlthough this revision is touted as being a more liberal interpretation of 38 CFR \u00a73.304(f), quite the opposite is true. This revision states \"if evidence establishes that the veteran engaged in combat,\" direct service connection will be approved. The questions that must be ask are: What kind of evidence will satisfy this regulatory requirement? Where do I find it? How do I get it? If you can't answer \"I know\" to each question, you lose. Claim denied!\n\nEach of the three revised paragraphs has one common denominator. Paragraphs one and two declare that the veteran's lay testimony alone _may_ establish the occurrence of the claimed in-service stressor. Paragraph three states evidence from sources other than the veteran's service records _may_ corroborate the veteran's account of the stressor incident.\n\nThe use of the word \"may\" is an open invitation for adjudicators to say \"your statement isn't sufficient to warrant direct service connection; please provide us with details and proof of the traumatic event you're alleging is the reason for your PTSD.\" Because the VA has not accepted your initial application for PTSD benefits, you are now destined for a lengthy process. This is where adjudicators start telling you they need detailed information confirming the event you are alleging as the cause of your PTSD.\n\nRemember, rating board members are not professionally trained to deal with complex medical or legal issues. They rely on a sort of checklist of questions that is intended to lead them to a proper decision. Their method of resolving decisions is best described as a \"Yes or No\" guideline of questions. A question is asked; if the answer is \"yes,\" you will do this, if answer is \"no,\" you will do that. Thus by answering all the questions, the rating board members are supposed to arrive at the correct solution.\n\nThe easiest way for them to avoid making an error when granting direct service connection is to simply declare the veteran's statement is not sufficient to support service occurrence of the claimed in-service stressor. This way they don't present themselves to management as reckless cowboys letting the steers out of the coral. If they error in favor of the government, their annual merit pay increase is never in jeopardy.\n\nAs you read Sergeant Raz's journal, you will see how the St. Petersburg Regional Office avoided concrete decisions for four years and counting. We had to furnish endless reams of paper detailing his traumatic experiences along with medical evaluations by his psychiatrist. With all that, it was necessary to challenge them two times with the appeal process before they relented and granted the benefits. And as you will see, we must now appeal the denial of his claim for Total Disability based on Individual Unemployability (TDIU).\n\nTo deny Sergeant Raz's claim for TDIU benefits and deliberately ignore statutes, regulation, general counsel opinions, caselaw and, most importantly, evidence that substantiates his claim, is unbelievable. I don't wish to offend any readers, but I have to say the St. Petersburg Regional Office has a giant set of \"brass b\u2014.\" Their denial letter and our response will be covered in chapter 3.\n\n### VA Manual M21-1\n\nVA Manual M21-1 is a detailed procedure manual instructing rating board members and claims examiners what is required when evaluating a claim. Subchapter 11, \u00a711.37 specifically outlines the standards to rate a PTSD claims. Unfortunately, I do not have access to an archive that would trace the flip-flop revision of this particular chapter going back to the 1980s. However, between court cases, revisions to _Title 38 U. S. Code_ , and 38 _CFR_ \u00a73.3.304(f), the manual seems to indicate that the Veterans Benefits Administration is once again forced to comply with the intent of Congress.\n\nVA Manual M21-1 \u00a711.37 was last revised on March 10, 2004 and reads as follows:\n\n**(1) Conclusive Evidence.**\n\na. Any evidence available from the service department indicating that the veteran served in the area in which the stressful event is alleged to have occurred and any evidence supporting the description of the event are to be made part of the record. Corroborating evidence of a stressor is not restricted to service records, but may be obtained from other sources (see _Doran v. Brown_ , 6 Vet. App. 283 [1994]). If the claimed stressor is related to combat, in the absence of information to the contrary, receipt of any of the following individual decorations will be considered evidence that the veteran engaged in combat:\n\n  Air Force Cross\n\n  _Air Medal with \"V\" Device_ [italics added]\n\n  Army Commendation Medal with \"V' Device\n\n  _Bronze Star Medal with \"V\" Device_ [italics added]\n\n  Combat Action Ribbon\n\n  Combat Infantryman Badge\n\n  Combat Medical Badge\n\n  Combat Aircrew Insignia\n\n  Distinguished Flying Cross\n\n  Distinguished Service Cross\n\n  Joint Service Commendation Medal with \"V\" Device\n\n  Medal of Honor\n\n  Navy Commendation Medal with \"V\" Device\n\n  Navy Cross\n\n  Purple Heart\n\n  Silver Star.\n\nOther supportive evidence includes, but is not limited to, plane crash, ship sinking, explosion, rape or assault, duty on a bum ward or in graves registration unit. POW status that satisfies the requirements of 38 CFR 3.1(y) will also be considered conclusive evidence of an in-service stressor.\n\n**(2) Personal Assault**\n\nEvidence of Personal Assault Personal assault is an event of human design that threatens or inflicts harm. Examples of this are physical assault, domestic battering, robbery, mugging, and stalking.\n\na. Alternative Evidence. If the military record contains no documentation that a personal assault occurred, alternative evidence might still establish an in-service stressful incident. Examples of such evidence include, (but are not limited to):\n\n  records from law enforcement authorities;\n\n  records from rape crisis centers, hospitals, or physicians;\n\n  pregnancy tests or tests for sexually transmitted diseases; and\n\n  statements from family members, roommates, fellow service members, or clergy.\n\n### Changing the Meaning of 32 CFR \u00a7578.11&12\n\nThe Veterans Benefit Administration is being forced to flip flop once again concerning \"conclusive evidence\" and \"direct service connection.\" Note the requirements set forth for the Bronze Star and Air Medal in this revised change to M21-1 Subchapter 11, \u00a711.35. The VA states for these two medals to qualify as conclusive evidence they must include a \"V\" device for heroism. But, by doing so, they eliminated several million Vietnam and Persian Gulf veterans from being eligible for direct service connection. Every one of these vets will have to do it the hard way just like Sergeant Raz. But before we declare ourselves defeated, let's look into the Department of Defense's Award Division regulation and see what is necessary to qualify for either one of these awards.\n\nAir Medal\n\n\u00a7578.12 Air Medal\n\n_a. Criteria._ The Air Medal, established by Executive Order on May 11, 1942, is awarded to any person who, while serving in any capacity in or with the army of the United States, has distinguished himself or herself by meritorious achievement while participating in aerial flight. (Fig. 1). Awards may be made to recognize single acts of merit or _sustained operational activities against an armed enemy._ (Italics added.)The required achievement, while of lesser degree than that required for the award of the Distinguished Flying Cross, must nevertheless have been accomplished with distinction above and beyond that normally expected.\n\n578.11 Bronze Star Medal.\n\n_a. Criteria._ The Bronze Star Medal, established by Executive Order on February 4, 1944, is awarded to any person who, while serving in any capacity in or with the army of the United States, on or after December 7, 1941 shall have distinguished himself or herself by heroic _or meritorious achievement or service not involving participating in aerial flight in connection with military operations against an armed enemy._ (Italics added.)\n\n(1) Heroism. Awards may be made for _acts_ of heroism performed in actual ground combat against an armed enemy which are of lesser degree than required for the award of the Silver Star.\n\n(2) Meritorious _achievement and service._ Awards may be made to recognize single acts of merit and meritorious service. The required achievement or service, while of lesser degree than that required for the award of the Legion of Merit, must nevertheless have been meritorious and accomplished with distinction.\n\nWhile reading the official policy concerning the issuance of the Air Medal, did you see where the executive order specifically stated that this decoration would be awarded for heroism? My personal experience during the Korean War earned me an Air Medal for bombing targets deep in North Korea. I assure you the North Koreans were not lighting up the sky with fireworks in celebration of our heroic northward run to targets just south of the Chinese border. Every crewmember of the 98th Bomb Wing earned an Air Medal when he flew ten or more combat missions north of the DMZ.\n\nNow for a few words concerning the VA position on the Bronze Star. Yes! Executive order originating the award states it can be awarded for heroism. But it also states that it can be awarded for meritorious achievement or _service in connection with military operations against an armed enemy_.\n\nThe question that comes to mind is how could the Under Secretary for Veterans Benefit Administration overlook such an important criterion that is built into the 32 CFR \u00a7578.11? Just think of the hundreds of thousands of soldiers who for one whole year chased \"Charlie\" up and down the Vietnam peninsula. They would spend endless days away from base camp tracking the enemy; it was the norm to sleep in the mud or wade through snake- and tiger-infested jungles and swamps. However, to all the bureaucrats waging a major attack on vast backlogs of claims action in their citadel accommodations in the center of any city USA, these actions by the ground forces are not considered special achievements.\n\n### Chapter 2: What Have We Learned?\n\n1. When the VA adjudicators grant a benefit or denies it, they must give a full explanation of what evidence was used to approve or refuse the claim. Failure to do this is grounds for appeal.\n\n2. It is Congress's intent to make the burden of proving your claim as liberal as possible. As you have seen, the VA will go to extremes to deny benefits.\n\n3. Your word is your bond as demonstrated by the legislation passed by the Congress. However, just because Congress wants to make it easy for you to have your claim processed with as little hassle as possible, it does not mean the VA shares this same commitment to those who serve.\n\n4. Changes that control how the Department of Veterans Affairs operates come from six sources. They are statutes, regulations, general counsel opinions, inspectors general's staff, caselaw, and acknowledgeable claimants.\n\n5. Look for trigger words in any governing document, or award letters that give a different meaning to Congress's intent, such as using \"may\" instead of \"shall,\" or adding a word to a regulation that changes its meaning.\n\n6. Do not give up in frustration; if you fail to get what you are claiming and have the proof to support your claim, remember: _appeal, appeal, appeal_.\n\n7. The VA will force a veteran needlessly to search for additional evidence when there is sufficient evidence of record to make a determination.\n\n8. When rating a combat injury or other conditions that obviously had their inception in the service, the rating process is suppose to go forward pending receipt of active duty medical records and all service records.\n\n9. If an injury or medical condition is tied to combat, the veteran's word is sufficient proof of service connection if the circumstance matches the conditions even though there is no proof of the traumatic event.\n\n10. PTSD requires medical evidence that the veteran is currently disabled by PTSD and the disorder is associated with events related to his or her combat experiences.\n\n### Notes\n\n1. 32 CFR \u00a7578.11, February 4, 1944.\n\n2. 32 CFR \u00a7578.12, September 11, 1942.\n\n## 3 SGT. DAVID RAZ'S FIVE-YEAR STRUGGLE\n\nHave you ever wondered what it would be like to be so traumatized by serving your country that you could barely function in the civilian world after leaving the service? I've known such men and it really grinds me to see how they are jerked around by an agency chartered to help them. This chapter is in six parts and will acquaint you with an ongoing account of the typical, unnecessary stress veterans claiming PTSD experience when they attempt to collect benefits. This chapter will provide you with techniques to counter the VA's line of attack in denying benefits.\n\nTo give you a realistic insight into the claims process and an appreciation for the burden of evidence PTSD victims bear, I'm going to tell you the story of Sgt. David Raz. To protect the real David Raz, others who played a role in helping him, and those who gave their lives for their country, names and places have been changed. But, believe me, the events that unfold are very real. Who knows, this might be your story.\n\nThis is Sgt. David Raz's story, told in a journal format. It is put together as a time line, starting when he first applied for PTSD benefits in 2001. His four-year battle with the VA is far from over. In pursuit for PTSD benefits, he was forced to provide evidence that, by law, should not have been required. David was the recipient of the Bronze Star and Purple Heart, which entitles him to direct service connection. However, that is not how the St. Petersburg Regional Office saw it.\n\nSergeant Raz's campaign to win benefits has been a terrible stressor for a man who is already stressed to the limits. The demand for evidence required resourcefulness and perseverance beyond belief. The last thing the government should do to a man or woman traumatized by war is to force them to stand exposed and dance to the tunes of the bureaucrats. Writing down traumatic and painful memories for others to read is an impossible task for thousands of veterans. Many simply say, \"The hell with it! Let the VA get what they can and let the chips fall where they may.\" A letter denying benefits is the VA's way of saying, \"Nice working with you, but good-bye\" In David's case, dealing with the VA on a letter-by-letter basis for four years nearly forced him to throw up his hands and quit. There are tens of thousands of David Razes who experienced similar treatment by the VA and did quit.\n\nI have been working with David since February 2001. To date he still hasn't been granted all the entitlements the law provides. However, we have forced the VA to reverse its decision two times, which resulted in establishing service connection for PTSD at the 70% rate.\n\nSgt. David Raz, sixty-one, was inducted into the army at the age of twenty-six. He was encouraged to apply for VA compensation benefits by his psychiatrist, Dr. Henry Jones. Dr. Jones, a former VA psychiatrist, knew only too well the tortuous existence of the bloodless wounded.\n\nJust before I started working with Sergeant Raz, he filed a claim for PTSD. He told the VA he wanted to reopen his claim. The VA had his records and knew he had shell fragment wounds. He was awarded the Purple Heart because of these injuries. The VA rated him 0% on December 20, 1970, for \"shell fragment wound scars right upper eyelid and left scalp.\"\n\nIn _Proscelle v. Derwinski_ the court noted that in order to trigger the VA's duty assist, the claimant's evidentiary burden is to submit _sufficient evidence to justify a belief by a fair and impartial individual that the claim is well grounded_. In this case the VA denied his request for increased benefits on the basis his claim for PTSD was not well grounded. However, in similar cases the court has held that the claim was plausible and capable of substantiation.\n\n### Part I: 2001\n\n#### FEBRUARY 28, 2001: VETERAN FILES CLAIM FOR PTSD\n\nFigure 3.1 is David Raz's letter claiming benefits for PTSD. Unknowingly, David used the phrase \"reopen my claim.\" What David did not know was \"reopen my claim\" implies a previous claim for PTSD was denied. This was not the case. He had no way of knowing that the correct phrase should have been \"amend my original claim to include PTSD.\"\n\n**Fig. 3.1**\n\n#### MARCH 30, 2001: DOCTOR RELEASES VETERAN  \nTO RETURN TO WORK\n\nDr. Henry Jones, David's psychiatrist, sent a letter to Suncoast District Occupational Health Services Office concerning the effects of PTSD as they related to the veteran's work. After six extensive weeks of psychiatric therapy, David was ready to return to work at the U.S. Postal Service. Figure 3.2 shows a copy of Dr. Jones's March 30, 2001, letter to the post office.\n\n**Fig. 3.2**\n\n#### AUGUST 24, 2001: VA REQUESTS ADDITIONAL EVIDENCE\n\nThe VA sent the letter in figure 3.3, dated August 24, 2001, to the veteran stating they needed additional information before deciding his case. This type of letter is only appropriate when a noncombatant is claiming PTSD. In David's case, the request for this information was totally without merit.\n\nSending this letter was one of the first errors team 70B made when adjudicating Mr. Raz's claim for PTSD benefits. The VA manual M21-1 states that when a combat veteran is awarded the Purple Heart or Bronze Star, Air Medal with a \"V\" device, or any one of the other thirteen awards, this is conclusive evidence that the veteran was involved in combat with the enemy. Mind you, shortly after he was discharged in 1970, the VA rated David 0% for shrapnel wounds to the head. Team 70 had this information when they decided that he did not qualify for direct service connection base on conclusive evidence of combat with the enemy. When the records show the veteran was wounded by the enemy, the VA is not to delay the rating process by demanding additional information related to his combat experiences. This is an appealable error that demonstrates the rater was not very knowledgeable about the rules and regulations governing PTSD claims. This is the second letter David received.\n\n**Fig. 3.3**\n\n#### AUGUST 24, 2001: VA SENDS SECOND LETTER  \nREQUESTING MORE EVIDENCE\n\nThis is the second letter dated August 24, 2001. The first one basically said, \"Hi! I'm your new buddy. Here is what I can do to help you.\" The second one states, \"If you want to be service-connected for PTSD, you had better get us the following information.\" In thinking about the letters, it looks like the right hand of team 70 doesn't know what the left hand is doing.\n\nThis is a good time to give you a sidebar note of interest. The VA published a fact sheet in December 2004 proudly touting statically data that should make you want to stand up and shout. In the year 2003, the fifty-seven Regional Offices notified 827,000 veterans of decisions concerning their disability claims. According to the VA, 256,000 veterans were added to Compensation and Pension rolls in 2003. If a third grade math class was asked how many veterans were turned down, their answer would be 571,000. So, what are the odds of having a claim approved? You could say seven of ten veterans were denied or, if you prefer, you could say three out of ten were approved. This is why I have said several times: _you have to know the territory_.\n\n#### SEPTEMBER 13, 2001: VETERAN PUTS THE VA ON  \nNOTICE ABOUT THE CLAIM DATE\n\nFigure 3.4 is an example of a form to be used when the VA does not respond correctly to your intent. David filed VA Form 21-4138 (which can be downloaded from the VA website) restating the intent of his February 28, 2001, letter. His statement stressed that his February 28, 2001, letter was meant to be an \"informal claim.\" If we hadn't filed a statement, the VA would have rated the claim in sixty days. There was no question that team 70 was going to deny the benefit. This action put us back in the game. No matter how they responded, by filing this form we established grounds for appealing that action.\n\n**Fig. 3.4**\n\n#### SEPTEMBER 13, 2001: VETERAN PROVIDES  \nSTATEMENT OF COMBAT DUTY\n\nFigure 3.5 contains David Raz's statement giving a day-by-day account of his duty while in Vietnam. This sworn statement by Sgt. David Raz is in response to the August 24, 2001, letter in which the VA requested a detailed account of his combat experiences.\n\nThe more facts you can remember and present in an orderly sequence, the better your chances are of having your claim approved. You probably have many memories that you would prefer not to remember because they are too painful. However painful, you must find a way to make these experiences known.\n\nTo ensure that David Raz's statement was given the proper weight, it was submitted as a sworn declaration. A sworn statement submitted to the VA has to be given the same weight as any other form of evidence and cannot be considered hearsay evidence or a nonbinding lay statement. An important note to remember: you can collaborate your military combat experiences by obtaining an after action report for the days of events that caused your PTSD. This information is available under the Freedom of Information Act. Have your congressional representative or senator request the information from the military service involved. It's fifty times faster than if you do it on your own.\n\n**Fig. 3.5**\n\n#### OCTOBER 19, 2001: DR. JONES SENDS RESPONSE TO VA\n\nDr. Jones sent the VA a very detailed account (see figure 3.6) of the severity of David Raz's PTSD. He pointed out that to give Mr. Raz even partial relief from the effects of chronic and permanent PTSD, he would have to remain on Zoloft and Ativan for life. Dr. Jones states David's condition is directly related to his wartime service and meets the 70% disability VA rating standard. Dr. Jones's letter to the VA is an excellent example of the information, style, and depth that is necessary to coax the VA to pay attention to a claim. By law claim processors are required to identify the evidence they have that is superior to a professional physician findings. They cannot simply whip out a phrase like, \"a preponderance of evidence is against your claim; therefore, your claim for PTSD is denied.\" Even though case law mandates how evidence should be weighted, you will see as the chain of events unravels in Sergeant Raz's claim, they actually had denied the claim with this kind of evidence of record.\n\n**Fig. 3.6**\n\n### Part II: 2002\n\n#### JANUARY 15, 2002: VETERAN RECEIVES THE VA RATING DECISION\n\nAfter waiting ten months, the VA finally rates David's claim. Figure 3.7 shows the official rating decision issued by the St. Petersburg Regional Office. The January 15, 2002, rating decision denied service connection for PTSD. The reason given for the denial: his service records were silent as to any combat service. It should be noted that the VA rating dated January 9, 1973, granted service connection for David Raz's shell fragment wound\u2014scars upper eyelid and left scalp. He earned a Purple Heart prior to leaving Vietnam. This rating alone should qualify the claim as well grounded, thereby accepting the current claim for PTSD with a lay statement. Development of the claim by the VA should have stopped here. The veteran's lay statement would have been sufficient to move the process along.\n\n**Fig. 3.7**\n\n* Department of Defense Code of Federal Regulation (32 _CFR_ Subchapter F, Part 578 \u00a7578.11) states: \"The Bronze Star is awarded to any individual who has distinguished himself or herself by heroic or meritorious achievement in connection with military operations against armed enemy. For heroism a Valor Device is added to award. For meritorious operation against armed enemy there is no 'V' device requirement.\" However, according to the VA, to concede a combat stressor the veteran's Bronze Star must also have a \"V\" device. This policy was put into effect when VA manual M 21 Part VI was changed on October 28, 1998. The VA has been slammed by the U.S. Court of Appeals for Veterans Claims and the U.S. Court of Appeals for the Federal Circuit for establishing a standard that exceed the statutory limits imposed by Congress.\n\nLet's examine a few inaccuracies in this rating decision. Yes, they did send David two letters dated August 24, 2001. One outlined their responsibility in assisting the veteran in developing his claim. The other was a request for information concerning the events that may have caused the alleged PTSD.\n\nAs I pointed out, Sergeant Raz was wounded by shrapnel, which the VA rated as service-connected in 1970. This is where the process should have concluded. However, the veteran responded within twenty days of the letter looking for background facts about his combat experiences. One month after David submitted his response, Dr. Jones sent a detailed evaluation to the VA. Now, according to the rating board, there was no evidence in the file supporting his claim five months later? How can that be?\n\n#### JANUARY 23, 2002: DR. HENRY JONES SUBMITS  \nSECOND STATEMENT\n\nDavid requested Dr. Jones prepare an assessment of his progress since the last evaluation was submitted to the VA. The purpose of his letter (see figure 3.8) was to show that David's condition was unchanged since Dr. Jones submitted his first evaluation. Dr. Jones emphasized David's condition was not only chronic but permanent.\n\n**Fig. 3.8**\n\n#### FEBRUARY 15, 2002: VETERAN'S WIFE SUBMITS  \nSWORN STATEMENT\n\nDavid Raz's wife crafted this sworn statement, shown in figure 3.9. To demonstrate the social interaction between the veteran, his family, friends, and acquaintances, his wife agreed to give a declaration of her observations during their thirty-two years of marriage. Before preparing her statement, she was briefed that she should avoid giving any kind of opinions, especially medical. This is an important point to remember when obtaining statements from spouses, family, and friends. For example, they should not say \"he has PTSD.\" That statement is reserved for a medical professional. However, they can say, \"He wakes up every night in a cold sweat.\" This is an observation.\n\nIf a family member gives an opinion that, for example, the veteran has PTSD, the rating board will throw out the statement because diagnoses cannot be made by a lay person. However, this outcome can be easily sidestepped by ensuring that her statement simply describes the symptoms she observed. She can say that he has nightmares and keeps her up all night. See chapter 2 for a complete list of symptoms associated with PTSD. Mrs. Raz's declaration is an excellent illustration for a spouse or friend to use.\n\n**Fig. 3.9**\n\n#### FEBRUARY 19, 2002: VETERAN FILES NOTICE OF DISAGREEMENT\n\nOn February 19, 2002, David Raz served a Notice of Disagreement (NOD) to the VA disagreeing with the January 28, 2002, denial of benefits. He used VA Form 21-4138, also used earlier to put the VA on notice about the claim date. He stated the effective date of the award (figure 3.4) was erroneous. He also restated that the VA was advised on September 3, 2001, that an informal claim was filed February 28, 2001. Attached to this NOD were three sworn statements by the veteran (figure 3.10), his wife (figure 3.9), and an addendum from Dr. Jones's medical evaluation (figure 3.8).\n\n**Fig. 3.10**\n\n* This response to the VA letter is a legally binding sworn statement. This is a good example to use whenever you are submitting a written statement to the VA. The VA must now treat this statement as positive evidence supporting your case.\n\n#### JUNE 26, 2002: VA SENDS LETTER REQUESTING  \nMORE MEDICAL EVIDENCE\n\nThe VA sent the letter shown in figure 3.11 to Dr. Jones in June 2002. This letter is meaningless and without merit considering how much medical background information Dr. Jones had already submitted to the VA on behalf of the veteran. This is the type of letter sent to a veteran's physician when the veteran first files a claim.\n\nThis letter also provides another example of the VA extending claim-processing time. By sending a letter to the doctor telling him he has sixty days to reply, the VA can safely ignore this claim until that time period has passed. Once this letter goes out, the file is shipped back to the filing room where it will stay until the computer sends out an alert message that the sixty days have expired.\n\nPrior to sending this request, the VA had more than enough medical evidence to make a decision. Only five months earlier, Dr. Jones provided the VA with an updated medical evaluation of David's condition.\n\n**Fig. 3.11**\n\n#### AUGUST 19, 2002: DR. JONES RESPONDS TO VA REQUEST\n\nIn his response to the VA's June 2002 request, Dr. Jones summarized everything he knew of David since he started treating him in 1997. His letter emphasized that David's condition is not only chronic, but also permanent. He also alerted the VA that events surrounding David could totally incapacitate him at any time. Dr. Jones responded within the sixty-day suspense date, forwarding a five-page complete medical history of David Raz. He once again reviewed David's records obtained under the Freedom of Information Act and could add nothing new. Dr. Jones's report is too long to include in its entirety. The closing paragraph summarizes David's albatross:\n\nDavid is someone that I would call a \"silent sufferer.\" He does not verbalize how he is feeling to his wife and\/or employer and tries to keep everything bottled up until it explodes in an emotional state. He has had many arguments with his supervisors at work. Before [treatment] with me, he even had developed a plan to do harm to the individuals he felt were threatening him, although he did not carry this plan out. He has many sick days banked and was forced to take medical leave of absence last year due to his Post Traumatic Stress Disorder disability. He strives to be reliable and efficient in his duties. And he also tries to be as flexible as possible with the many demands that are put on him at this job. However, even with medication, the stress of these situations that David has a tendency to \"bottle up\" becomes too much and his Post Traumatic Stress Disorder disability once again exacerbates, leaving him incapacitated. When he is incapacitated, even simple daily living tasks become hard for him to accomplish.\n\n#### SEPTEMBER 9, 2002: BAY PINES VAMC SCHEDULES  \nC&P EXAM FOR PTSD\n\nFigure 3.12 is an example of the standard instructions issued by the medical center conducting the C&P examination. The C&P Exam is probably one of the highest alert areas for any veteran claiming benefits. See chapter 7 for details why everything hinges on this examination. This action by the VA came eighteen months after the claim was filed.\n\n**Fig. 3.12**\n\n#### SEPTEMBER 16, 2002: VAMC CONDUCTS C&P EVALUATION\n\nThe results of David Raz's C&P Exam can be found in figure 3.13. When scheduled for a C&P evaluation, cautionary measures must be taken to protect your rights. A detail explanation will be found in chapter 7. However, five points must be repeated over and over:\n\n1. Make sure the examiner has read your entire C-file.\n\n2. Keep track of how long you are with the examiner.\n\n3. Ask the examiner about his or her medical background.\n\n4. If a physician's assistant or a nurse practitioner conducts the exam, be extra vigilant in observing what they do and say.\n\n5. Make certain the examiner follows the medical protocol for the condition being evaluated. (A copy of the C&P Exam for PTSD is provided in chapter 7.)\n\n**Fig. 3.13**\n\n#### SEPTEMBER 24, 2002: VA ACKNOWLEDGES RECEIPT OF NOD\n\nThe VA claim processors stated that they received the veteran's Notice of Disagreement with their decision of February 19, 2002, and they were proceeding with the appeal process. However, it seems clear that they already had made their minds up to continue the denial of PTSD benefits when the C&P Exam was ordered.\n\nThis acknowledgment is dated September 24, 2002, and the date the C&P Exam was completed and forwarded to the RO was September 16, 2002. The results of the C&P Exam failed to convince members on team 70 that the veteran had a legitimate claim for PTSD benefits. The only explanation I can see for this blatant disregard for the results of the C&P Exam is that they had no intention of deciding if David's PTSD disability was related to his service. Instead, team 70 was not going to grant a benefit for PTSD but rather forward the claim to the next level for someone else to make the decision.\n\n### Part III: 2003\n\n#### APRIL 30, 2003: VA GRANTS SERVICE CONNECTION FOR PTSD\n\nThe VA advised David that service connection for PTSD was approved, effective April 30, 2003, and his PTSD was rated 70% disabling. However, as shown in the May 20, 2003, award letter in figure 3.14, the rating board considered only evidence dated between January 23, 2002, and September 16, 2002, without explaining why records between February 28, 2001, and October 19, 2001, were not considered relevant to support an entitlement date of March 1, 2001.\n\nYou should be alert to a \"snake in the grass\" when the last paragraph reads like this: \"This constitutes a complete grant of benefits sought and this issue is being removed from appeal without further appellate review.\" With this statement the VA adjudicators were effectively trying to coerce the vet into accepting their decision. If they failed to address all the issues of the claim, as they did in this case, appeal the decision. They can't take away what they granted and they can't stop you from appealing any issues they failed to address. As you will see later, by challenging the rating after they granted a 70% entitlement rate, the veteran ultimately received another $13,000 as back benefits. This is because rating board members overlooked evidence that was in the file from February 28, 2001, to February 28, 2002.\n\n**Fig. 3.14**\n\n#### MAY 31, 2003: VETERAN FILES NOTICE OF DISAGREEMENT\n\nThe effective date of February 1, 2002, was not the appropriate start date for benefits according to the VA's own regulations. We knew the file contained medical evidence going all the way back to February 28, 2001, when David originally filed his claim for PTSD service connection. Our best guess at why this evidence was ignored is that the rater only reviewed information in the file as far back as January 15, 2002. This was the date the VA denied the original claim for PTSD benefits. This practice is known as \"top sheeting.\" The VA justifies top sheeting because of regulations governing the appeal process. When a claim is denied, the claimant has one year from the date of notification to file an appeal. Once the year has passed, the issued is dead. In our case, the action was improper because the VA adjudicator initially gave us one year from the date of the informal claim, January 28, 2001, to obtain all the evidence and file the original claim. The rater overlooked this fact.\n\nWhen filing a NOD, it's important that you focus your rebuttal on the issues the VA failed to address and, when you can, cite references to support your rebuttal. To help organize your appeal and keep the facts in focus, it is suggested that you fashion your appeal from the one we filed with the VA on May 31, 2003, which can be seen in figure 3.15. Italics have been added for emphasis.\n\n**Fig. 3.15**\n\n#### JUNE 25, 2003: VA OFFERS DECISION REVIEW OFFICERS SERVICES\n\nIn June 2003, the VA offered David Raz the option of choosing to have a Decision Review Officer (DRO) assigned to his case or proceeding with the appeals process. There are times when the service of a DRO will definitely work to your advantage. We felt very strongly that our case would win on appeal, so we had everything to gain and nothing to lose by requesting a local hearing. Convincing the local RO that we were entitled and able to prove it, it was more likely than not that the appeal would not be certified to the Board of Veterans Appeals (BVA) in Washington. Winning locally saved us between one and two years additional wait time. A copy of their letter (figure 3.16) is provided to acquaint you with the mechanics of what the Decision Review Officer (DRO) duties are. DROs are senior adjudicators who are picked at random from the DRO pool. They are supposed to review all of the relevant records, not just those from the last action.\n\n**Fig. 3.16**\n\n#### JULY 13, 2003: VETERAN ACCEPTS A DRO REVIEW\n\nFigure 3.17 shows David's \"Statement in Support of Claim,\" sent to the VA in July 2003. In responding to the VA's letter, state your case as you see it, based on the facts and regulations. When reviewing your response and the evidence of record, raters are required to give a detailed explanation as to why they continue to deny the claim. Once you have their brief (Statement of the Case), you can review it piece by piece using their regulations, manuals, general counsel opinions, and case law.\n\nDavid Raz notified VA, within the sixty-day suspense period, that he was requesting his Notice of Disagreement be reviewed by a Decision Review Officer at RO. Included was a four-page timeline of key events and six questions the DRO must cover when deciding the merits of the NOD. The six questions were designed to keep the DRO focused on the issues favorable to the vet.\n\n**Fig. 3.17**\n\n### Part IV: 2004\n\n#### APRIL 5, 2004: VA GRANTS NEW EFFECTIVE DATE\n\nThe decision letter from the DRO can be found in figure 3.18. By acknowledging that the effective date of the veteran's entitlement was one year earlier, the case was able to advance to the next level. The VA claim reviewers resolved the issue by changing the effective date of entitlement to March 1, 2001. By forcing the issue, the veteran obtained an additional $13,000 in benefits from the VA.\n\nHowever, Sgt. David Raz is still in limbo because he is not considered eligible for increased benefits because of unemployability. In stating the veteran's disability was 70% in 2001, they should have automatically considered him for Total Disability based on Individual Unemployability at a 100% rate. They did not do this. A formal claim was initiated July 9, 2004, for TDIU benefits.\n\nJanuary 2002 was the forty-sixth month since Sergeant Raz applied for compensation benefits based on PTSD. The battle is still not over. If the past is any indication, it may be 2008 before a conclusion is reached on this man's entitlements.\n\n**Fig. 3.18**\n\n* The _denovo_ review was a fair and careful review until the last paragraph when the reviewer made an assumption that was contrary to evidence in the file. The paragraph is quoted to illustrate the point: \"Since there is a likelihood of improvement, the assigned evaluation is not considered permanent and is subject to a future review examination.\" Yet there is no evidence in the file that suggests the condition of a veteran with PTSD will improve. By failing to make the initial rating permanent, the VA has made it possible to recall the veteran for additional C&P examinations and to review his evaluation.\n\n#### JUNE 22, 2004: VETERAN SUBMITS FORMAL  \nAPPLICATION FOR TDIU BENEFITS\n\nSergeant Raz filed a formal application (VA Form 21-8940) for increased benefits owing to unemployability. It would be easy for the veteran to overlook this notice among a considerable collection of miscellaneous sheets of information attached to the VA letters. It is very important to submit the prescribed form when it is your intent to apply for Total Disability based on Individual Unemployability. This notice of possible entitlement was attached to the schedule of monthly benefit payments.\n\nThere are certain circumstances in which the VA should initiate development of a TDIU before the form is submitted. When a medical report received from a physician indicates the veteran's condition is chronic and permanent and that he is unemployable because of his PTSD, the rater must accept this as an informal claim on behalf of the veteran. The U.S. Court of Appeals for the Federal Circuit held in _Moody v. Principi_: the \"VA is to fully and sympathetically develop the veterans claim to its optimum before deciding its merits.\" This was not the case for Sergeant Raz.\n\n#### JULY 9, 2004: VETERAN NOTIFIES VA OF FAILURE  \nTO CONSIDER TDIU\n\nThe notice in figure 3.19 tells the VA raters that they should have notified the veteran that an informal claim of 100% for TDIU was established by Dr. Jones's letter dated October 19, 2001. This letter provided a foundation on which to claim that the veteran's condition was so severe that he was rated 70% disabled. When a veteran is disabled to the extent that his service-connected disability prevents him from being employed, the law says he shall be granted an 100% rating owing to unemployability.\n\n**Fig. 3.19**\n\n#### AUGUST 17, 2004: VA INITIATES DELAYING TACTIC\n\nThe VA responded to David's formal request for TDIU benefits by asking again for the same complete package of information and evidence sent in July 2004. The claim reviewers also advised they would do nothing on his claim for two months, pending his response. The letter also included an Expedited Action Attachment, which authorizes the VA to rate the claim without waiting the sixty days established by statute. The purpose of this sixty-day waiting period is to give the veteran two months to submit additional evidence before the claim is rated. The wavier was signed because our evidence package was already of record. A copy of the Expedited Action Attachment and the signed wavier, dated August 20, 2004, can be viewed in figure 3.20.\n\n**Fig. 3.20**\n\n#### SEPTEMBER 20, 2004: VETERAN SENDS FOLLOW-UP NOTICE\n\nDavid Raz sent a follow-up notice to his signed Expedited Action Attachment letter dated August 20, 2004. The purpose was to ensure the VA raters knew they had all the evidence necessary to rate his claim.\n\n#### SEPTEMBER 29, 2004: VA DENIES TDIU BENEFITS\n\nWithin ten days, the VA reviewed Sergeant Raz's claim for TDIU benefits and denied it on the grounds that he was employable and his condition was not considered permanent. The mail time to the VA, internal mail distribution time, and processing by the adjudication division (which includes evaluating all the evidence, preparing a denial letter, and dispatching it) added up to only thirty-two workdays from the date David Raz filed his claim for TDIU benefits. That is a record for any VA Regional Office. It is obvious the claim was denied without anyone looking at it. The first two pages of the denial letter are shown in figure 3.21.\n\n**Fig. 3.21**\n\n#### OCTOBER 26, 2004: VETERAN TRIES TO VERIFY  \nDATE OF C&P EXAM\n\nThe denial letter dated September 29, 2004, referred to a pending C&P Exam due in September 2004. The question that jumps out of the denial letter is why would the VA rate a claim, knowing that new medical evidence might alter the rating?\n\nThe VA Medical Center Bay Pines was contacted and asked to check on what day the C&P Exam was scheduled. After typing the information into the computer, the clerk came back with an excuse she was having some trouble and would David call back in about an hour. The next time David talked to the clerk, later that day, she read off the computer that his appointment was scheduled for November 10, 2004. He would receive written conformation within a few days.\n\n#### OCTOBER 2, 2004: MEDICAL CENTER CONFIRMS C&P EXAM\n\nVA Medical Center Bay Pines confirmed the November 10, 2004, appointment. The failure of the RO to coordinate and follow up on the alleged September C&P Exam will be one of the appeal arguments should the veteran continue through the complete process to obtain his TDIU benefits.\n\n#### OCTOBER 29, 2004: RO ALERTS VET PENDING C&P EXAM\n\nStrangely, a letter from the RO arrived three days later advising that a required C&P review evaluation had been scheduled. If the veteran had not made that telephone call to inquire about this alleged September C&P Exam its doubtful if would have ever heard from the RO concerning an exam.\n\n#### NOVEMBER 4, 2004: VETERAN FILES NOTICE OF DISAGREEMENT\n\nFigure 3.22 is the response the veteran filed on November 4, 2004, rebutting the VA reasoning for denying his claim for TDIU. This is an example of a NOD that puts the VA raters on defense. They have to address every rebuttal point to explain why they ignored their own regulations, manuals, and court decisions. One of the best suggestions I can offer to anyone going head to head with the VA is to take the initiative. Demand answers to your questions or your challenges.\n\n**Fig. 3.22**\n\n* Pages 610 and 1020 from _Black's Law Dictionary_ were copied and attached to his Notice of Disagreement. The purpose of this is to constrain the rater's interpretation of what constitutes \"substantial and gainful occupation.\" It is important to remember that the person rating your claim is not necessarily right and all knowing. Rating board members do not have legal or medical training.\n\n** General counsel opinions are very important and are binding on rating board members. If you find where the adjudicator ignored a GC opinion, you have a winning hand. To find opinions on the Internet, use Google to search for the term \"VAOPGCREC 10-95.\"\n\n*** Code of Federal Regulations, title 38, 3.340 governs _total and permanent total ratings and unemployability_ and is the authority to grant a 100% rating.\n\n**** The director of the Compensation and Pension Administration ordered a PTSD case review. The results didn't favor the veteran suffering from PTSD. To read the entire report, point your browser to <http:\/\/obama.senate.gov\/press050810_obama_criticizes_va_decision_to_review_72000_post_traumatic_stress_disorder_cases\/index.html>.\n\n#### NOVEMBER 12, 2004: NURSE PRACTITIONER CONDUCTS  \nREVIEW EXAMINATION FOR PTSD\n\nA nurse practitioner conducted the C&P evaluation without the supervision of a board-certified psychiatrist. The use of a nurse practitioner in place of a doctor to perform a C&P examination raises questions as to the thoroughness of the examination. The margin for errors increases tenfold when decisions are delegated to the inexperienced medical staff members. The VA procedures allow for review examinations to be performed by others besides doctors to justify claim denials. A copy of the report forwarded to the St. Petersburg Regional Office can be viewed in figure 3.23.\n\nIf the RO refuses to grant TDIU benefits, we will attack the report as \"flawed\" because of its deviation from the protocol format. In this case the examiner failed to express the opinion that David's condition was permanent and will not improve. He also failed to address the employability factor.\n\n**Fig. 3.23**\n\n#### DECEMBER 14, 2004: VA CONTINUES RATING AT 70%\n\nBased on the C&P examination dated November 14, 2004, the VA has determined that David's PTSD remains 70% disabling. On the last page in the lower corner of the of the rating decision was the remark, \"Individual Unemployability Denied.\" When David called to clarify the statement concerning the status of his appeal, he was advised the appeal had not been decided.\n\n### Part V: 2005\n\nAnother year will begin before Sergeant Raz is informed by the St. Petersburg Regional Office whether benefits will be conceded without any further action or whether a full-blown appeal will be necessary. It is my prediction that a resolution of David's claim will take several more years. The local RO doesn't want to go on record granting 100% for PTSD. They also don't want to be looked upon by the Central Office in Washington as the RO who gave $50,000 or more for a PTSD claim the first time it was reviewed. If the RO's decision is reversed by the Board of Veterans' Appeals and told to grant the benefits, the RO will be able to say, \"We didn't grant the benefit; we were told to grant it by the BVA.\"\n\n### Chapter 3: What Have We Learned?\n\n1. PTSD is not a newly identified medical phenomenon. Historical medical stories identified this disorder more than 2,700 years ago.\n\n2. It took eight years after medical standard for PTSD was accepted by the medical profession before the VA was forced to deal with the issue of PTSD on a more realistic level.\n\n3. The fourth version of _DSM_ has redefined how long symptoms must exist before the condition is considered chronic. The new standard is three months.\n\n4. Veterans who were tagged by the DOD or VA as having a personality disorder, compulsive disorder, paranoid disorder, or schizophrenia, and now have been diagnosed with PTSD should reopen their original claim for service connection.\n\n5. Use Sergeant Raz outline of his dealing with the VA as a model for developing your claim for PTSD benefits. Later chapters will help you put your claim together so benefits are granted.\n\n6. Sergeant Raz's case demonstrates the tenacity that is necessary to pursue your interest in having a claim approved. Don't believe the concept that a veteran dealing with the VA is an act of non-advocacy. The moment the claim is denied, that us versus them relationship is dissolved. Through a series of administrative steps, the parties enter the arena of civil legation.\n\n7. Know the type of claim you are filing. For example if you have never filed a claim with the VA, then you should file an original claim on form VA 21-526. Chapter 6 will tell you all about the various claim actions for filing a claim for PTSD.\n\n8. Make certain you have proof that you are currently disabled by PTSD. It is preferable to be evaluated by a professional specialist who is willing to prepare a written assessment of your disability.\n\n9. Prepare a time line of events as you remember them, using the time line Sergeant Raz submitted as a guide.\n\n10. If you were wounded and received the Purple Heart, make sure it is stated on your discharge certificate (DD-214). The same is true with the Bronze Star. Send a copy of the award letter along with your application for benefits.\n\n11. Make certain any statement you, your family, or your friends submits is sworn before a notary public.\n\n12. If you have a valid claim, don't let the VA deny benefits. Remember Sergeant Raz has challenged raters on three occasions and two were victories for him and the third is pending. Remember! If you are turned down, appeal the denial.\n\n13. The VA is not all knowing and the decisions from the local Regional Offices are not cast in cement, as we have demonstrated in this chapter.\n\n14. Know where you can find their laws, regulations, opinions, and case law that support your claim.\n\n### Notes\n\n1. Timeline submitted to the VA focusing on key dates and evidence supporting Sergeant Raz's PTSD claim. Submitting a timeline is a valuable technique to use. It forces the VA to look at evidence that favors you.\n\n2. This is the initial notice to the VA that the veteran wanted to claim service connection for PTSD. See chapter 5 for the details concerning the various kinds of claims.\n\n3. Dr. Jones's letter to the VA is an excellent example of how a doctor's statement supporting your claim for PTSD should be prepared. Remind the doctor when he is preparing a statement that the VA readers have no medical training; a letter written as if to a professional colleague would heighten the odds of having the claim denied. One other point of great importance: provide the doctor with a copy of 38 CFR Part IV, \"Schedule for Rating Disabilities Section,\" Section 9400, when he states the degree disability resulting from PTSD. Also provide him with C&P examination: Initial or Review Examination for Post-Traumatic Stress Disorder (PTSD).\n\n4. This is an example of the VA failing to follow its own regulations by rating the claim eight months prior to end date.\n\n5. _Moody v. Principi_ , 360 F. 1303 (3d. Cir. 2004).\n\n## 4 RULES THEY MUST FOLLOW\u2014KNOW THEM!\n\n### Disability Claims Processing\n\nIn order to have your claim for PTSD benefits accepted, you must understand the rules and regulations of the VA and claims process. First, you need intelligence as to how a Regional Office is structured and operated. Believe me, fighting an enemy blind is no different than putting your faith in a giant bureaucracy such as the VA and expecting results automatically in your favor.\n\nThe Department of Veterans Affairs is administered through a network of fifty-seven VA Regional Offices. Disability compensation benefits covering chronic illnesses or injuries are one of the most expensive operational costs for the Department of Veterans Affairs.\n\nThe VA published a fact sheet in December 2004 titled \"VA Disability Claims Processing.\" As you can see in figure 4.1, the text is biased in emphasizing how the VA is resolving operational problems benefiting veterans. Congressional representatives put pressure on the secretary of the Department of Veterans Affairs by passing several legislative acts that mandated reform.\n\n**Fig. 4.1**\n\nSeveral sections in the fact sheet need to be reviewed from an impartial perspective. First, let's take a look at the figures presented in paragraph three. The VA notified 827,000 veterans of the results of their disability claims. In the next sentence, the VA announced that 256,000 compensation and pension claims were added to the rolls. Their statement is silent as to how many claims were granted for compensation or how many for pension. What they didn't present was how many of the 571,000 claims were denied. This tells us the best guess for determining causality rate is seven out of every ten claims is denied. It would be safe to assume that of the remaining three out of ten claims approved, one and a half to two would be for pension benefits.\n\nThe second issue raised by the fact sheet is how well claims are evaluated if there are more than 324,000 claims pending and the overall incentive for acceptable performance is based on how many claims can be processed per day. If in fact merit pay increases are tied to daily production, it is in the claims examiner's best interest to save time and deny claims when there isn't absolute proof easily understood by caseworkers with no knowledge of complex medical or legal issues.\n\nThe hidden pressure forcing the VA to change the way it does business comes from the White House and the Office of Management and Budget (OMB).\n\nIf we are to assume that OMB executives dictate the amount of money spent on PTSD claims, evidence shows that the VA has been directed to deny claims whenever possible. It is even possible that in a meeting between the secretary of the Department of Veterans Affairs and the direct of OMB could have occurred stating this in no uncertain terms:\n\n_OMB director:_ \"I don't care what you publicly tell veterans. Just do not give away the store for all these PTSD claims. It's bad enough we had to tiptoe around the congressional mandate in 1989 concerning supporting veterans with PTSD. I don't want a repeat of interference from Congress on this issue.\"\n\n_Department of VA Affairs secretary_ : \"But isn't it our job to support our vets?\"\n\n_OMB director:_ Look what happened when we had to cave in on the real effects of Agent Orange used in Vietnam. Every time you turn around, medical research has established another form of cancer associated with Agent Orange. We cut off claims for that before we were forced to pay out billions to vets in compensation and benefits.\"\n\n_Department of VA Affairs secretary:_ \"Are you telling me that because allowing compensation and benefits for PTSD, we are going to lose billions more?\"\n\n_OMB director:_ \"Of course! PTSD isn't even a physical injury! Go back and tell the assistant secretary for Veterans Benefits Administration and Healthcare Administration to deny those claims. The president wants budget money allocated to other projects.\"\n\nMost directions of the OMB have behaved as if the government budget is their personal funds that should not be given away as if it was a public charity.\n\nAnother policy change that works against the veteran is not covered by the VA's self-congratulatory fact sheet. The Federal Circuit ruled in 2000 that denial of a claim is premature before the expiration of the one year period established by the Veterans Claims Assistance Act of 2000. However, the executive management level of the Department of Veterans Affairs lobbied Congress to insert a policy change in the Veterans Benefits Act of 2003 in which the VA can now deny a claim without waiting a full year. This is disturbing and unfair. Supposing you file a claim lacking certain evidence to support your argument. The VA adjudicator sends you a letter stating the evidence he requires and gives you a suspense date of sixty to ninety days to produce it. You are unsuccessful in obtaining this information within the time frame; thus, your claim goes into the automatic claim-denial mode.\n\n### How a PTSD Claim Is Decided\n\nNow we want to take a look at the administrative side of the equation. In deciding the merits of a PTSD claim, the rating board member must apply five sets of guidelines to each claim he reviews: _U.S. Code Title 38_ , sections 1110, 1131, 1154(b), 5103, 5103A, 5104, 5107, and 5109; _Code of Federal Regulations_ volume 1, parts 3 and 4; VA Manual M21-1, part VI, subchapter 11, section 11.37, opinions provided by the Office of General Counsel (OGC), Department of Veterans Affairs; and rulings passed by the U.S. Court of Appeals for Veterans Claims (USCAVC).\n\nIn theory, if the guidelines in each of these sources are applied correctly, there should seldom be a need for someone to appeal a decision. As you read through statutes to the _CFRs_ , the manuals, and the general council opinions, notice the changing interpretation of the guidelines.\n\nBut, in fairness, it is the veteran who makes it possible for claims examiners to turn down so many claims. The hit song, \"You Gotta Know the Territory\" from _The Music Man_ is right on point. You have to know how to implement your rights. _Only you can do this_. If you say to them: \"Hey! You didn't do this or that, which is why you mishandled my claim,\" the next time claims examiners review it, they will pay attention. No one likes to look like an idiot to those who sit on the thrones of upper management.\n\n### _Title 38 U.S. Code_ : Veterans Benefits\n\n_Title 38 U.S. Code_ , \"Veterans Benefits,\" is a codification of veterans' affairs laws that are under the jurisdiction of the Committee on Veterans' Affairs of the U.S. House of Representatives and the U.S. Senate. These statutes are the laws for providing compensation benefits. They are for all veterans who have, in one way or another, become disabled because of an in-service injury or illness.\n\n#### SECTION 1131: BASIC ENTITLEMENT (LAST AMENDED IN 2003)\n\nThe full text of this section reads as follows:\n\nFor disability resulting from personal injury suffered or disease contracted in line of duty, or for aggravation of a preexisting injury suffered or disease contracted in line of duty, in the active military, naval, or air service, during other than a period of war, the United States will pay to any veteran thus disabled and who was discharged or released under conditions other than dishonorable from the period of service in which said injury or disease was incurred, or preexisting injury or disease was aggravated, compensation as provided in this subchapter, but no compensation shall be paid if the disability is a result of the veteran's own willful misconduct or abuse of alcohol or drugs.\n\n#### SECTION 1154(B): CONSIDERATION TO BE ACCORDED TIME,  \nPLACE, AND CIRCUMSTANCES OF SERVICE\n\nThe text of this section reads as follows:\n\nIn the case of any _veteran who engaged in combat_ with the enemy in active service with a military, naval, or air organization of the United States during a period of war, campaign, or expedition, _the Secretary shall accept as sufficient proof of service connection of any disease or injury alleged_ to have been _incurred in or aggravated_ by such service satisfactory lay or other evidence of service incurrence or aggravation of such injury or disease, _if consistent with the circumstances, conditions, or hardships of such service_ , notwithstanding the fact that there is no official record of such incurrence or aggravation in such service, and, _to that end, shall resolve every reasonable doubt in favor of the veteran_. Service connection of such injury or disease may be rebutted by clear and convincing evidence to the contrary. The reasons for granting or denying service-connection in each case shall be recorded in full. (Italics added.)\n\nThe last time this particular statute was amended was August 6, 1991. Certain words in this passage have been italicized for emphasis. These are the words are needed to request further action when your claim is denied; that is, these words are your Bill of Rights. Know them so that if the VA steps out of line\u2014by denying your legitimate claim\u2014you can successfully defend your claim by showing how the VA guidelines were not followed.\n\n#### SECTION 5103: NOTICE TO CLAIMANTS OF REQUIRED  \nINFORMATION AND EVIDENCE (2004)\n\nThe text of this section reads as follows:\n\n#### (a) Required information and evidence.\n\nUpon receipt of a complete or substantially complete application, the Secretary shall notify the claimant and the claimant's representative, if any, of any information, and any medical or lay evidence, not previously provided to the Secretary that is necessary to substantiate the claim. As part of that notice, the Secretary shall indicate which portion of that information and evidence, if any, is to be provided by the claimant and which portion, if any, the Secretary, in accordance with section 5103A of this title and any other applicable provisions of law, will attempt to obtain on behalf of the claimant.\n\n#### (b) Time limitation.\n\n(1) In the case of information or evidence that the claimant is notified under subsection (a) is to be provided by the claimant, such information or evidence must be received by the Secretary within one year from the date such notice is sent.\n\n(2) This subsection shall not apply to any application or claim for Government life insurance benefits.\n\n(3) Nothing in paragraph (1) shall be construed to prohibit the Secretary from making a decision on a claim before the expiration of the period referred to in that subsection.\n\nWhen the VA claims examiner writes for information or to establish a time requirement, you are expected to react before the expiration date unless you wish to invite a denial of benefits.\n\n#### SECTION 5103A: DUTY TO ASSIST CLAIMANTS\n\nFull text of this section reads as follows:\n\n#### (a) Duty to assist.\n\n(1) The Secretary shall make reasonable efforts to assist a claimant in obtaining evidence necessary to substantiate the claimant's claim for a benefit under a law administered by the Secretary.\n\n(2) The Secretary is not required to provide assistance to a claimant under this section if no reasonable possibility exists that such assistance would aid in substantiating the claim.\n\n(3) The Secretary may defer providing assistance under this section pending the submission by the claimant of essential information missing from the claimant's application.\n\n#### (b) Assistance in obtaining records.\n\n(1) As part of the assistance provided under subsection (a), the Secretary shall make reasonable efforts to obtain relevant records (including private records) that the claimant adequately identifies to the Secretary and authorizes the Secretary to obtain.\n\n(2) Whenever the Secretary, after making such reasonable efforts, is unable to obtain all of the relevant records sought, the Secretary shall notify the claimant that the Secretary is unable to obtain records with respect to the claim. Such a notification shall\n\n(A) identify the records the Secretary is unable to obtain;\n\n(B) briefly explain the efforts that the Secretary made to obtain those records; and\n\n(C) describe any further action to be taken by the Secretary with respect to the claim.\n\n(3) Whenever the Secretary attempts to obtain records from a Federal department or agency under this subsection or subsection (c), the efforts to obtain those records shall continue until the records are obtained unless it is reasonably certain that such records do not exist or that further efforts to obtain those records would be futile.\n\n#### (c) Obtaining records for compensation claims.\n\nIn the case of a claim for disability compensation, the assistance provided by the Secretary under subsection (b) shall include obtaining the following records if relevant to the claim:\n\n(1) the claimants service medical records and, if the claimant has furnished the Secretary information sufficient to locate such records, other relevant records pertaining to the claimant's active military, naval, or air service that are held or maintained by a governmental entity;\n\n(2) records of relevant medical treatment or examination of the claimant at Department healthcare faculties or at the expense of the Department, if the claimant furnishes information sufficient to locate those records;\n\n(3) any other relevant records held by any Federal department or agency that the claimant adequately identifies and authorizes the Secretary to obtain.\n\n#### (d) Medical examinations for compensation claims.\n\n(1) In the case of a claim for disability compensation, the assistance provided by the Secretary under subsection (a) shall include providing a medical examination or obtaining a medical opinion when such an examination or opinion is necessary to make a decision on the claim.\n\n(2) The Secretary shall treat an examination or opinion as being necessary to make a decision on a claim for purposes of paragraph (1) if the evidence of record before the Secretary, taking into consideration all information and lay or medical evidence (including statements of the claimant)\n\n(A) contains competent evidence that the claimant has a current disability, or persistent or recurrent symptoms of disability; and\n\n(B) indicates that the disability or symptoms may be associated with the claimant's active military, naval, or air service; but\n\n(C) does not contain sufficient medical evidence for the Secretary to make a decision on the claim.\n\n#### (e) Regulations.\n\nThe Secretary shall prescribe regulations to carry out this section.\n\n#### (f) Rule with respect to disallowed claims.\n\nNothing in this section shall be construed to require the Secretary to reopen a claim that has been disallowed except when new and material evidence is presented or secured, as described in section 5108 of this title.\n\n#### (g) Other assistance not precluded.\n\nNothing in this section shall be construed as precluding the Secretary from providing such other assistance under subsection (a) to a claimant in substantiating a claim as the Secretary considers appropriate.\n\n#### SECTION 5104: DECISIONS AND NOTICES OF DECISIONS\n\nFull text of this section reads as follows:\n\n**(a)** In the case of a decision by the Secretary under section 511 of this title affecting the provision of benefits to a claimant, the Secretary shall, on a timely basis, provide to the claimant (and to the claimant's representative) notice of such decision. The notice shall include an explanation of the procedure for obtaining review of the decision.\n\n**(b)** In any case where the Secretary denies a benefit sought, the notice required by subsection (a) shall also include:\n\n(1) a statement of the reasons for the decision, and\n\n(2) a summary of the evidence considered by the Secretary.\n\nThis is one of those rules you must be aware of. Any time the VA denies your claim without including the documents outlined in section (b) above, immediately file a Notice of Disagreement.\n\n#### SECTION 5107: CLAIMANT RESPONSIBILITY; BENEFIT  \nOF THE DOUBT\n\nFull text of this section reads as follows:\n\n#### (a) Claimant responsibility.\n\nExcept as otherwise provided by law, a claimant has the responsibility to present and support a claim for benefits under laws administered by the Secretary.\n\n#### (b) Benefit of the doubt.\n\nThe Secretary shall consider all information and lay and medical evidence of record in a case before the Secretary with respect to benefits under laws administered by the Secretary. When there is an approximate balance of positive and negative evidence regarding any issue material to the determination of a matter, the Secretary shall give the benefit of the doubt to the claimant.\n\nThe benefit of doubt law says that before the VA can deny your claim it must prove that its evidence is superior to yours. The VA can't say it considered all the evidence and a preponderance is against your claim. It must say exactly what evidence it has and why this evidence is superior. Be warned: claims officers disregard this requirement on a regular basis.\n\n#### SECTION 5109: INDEPENDENT MEDICAL OPINIONS\n\nFull text of this section reads as follows:\n\n**(a)** When, in the judgment of the Secretary, expert medical opinion, in addition to that available within the Department, is warranted by the medical complexity or controversy involved in a case being considered by the Department, the Secretary may secure an advisory medical opinion from one or more independent medical experts who are not employees of the Department.\n\n**(b)** The Secretary shall make necessary arrangements with recognized medical schools, universities, or clinics to furnish such advisory medical opinions. Any such arrangement shall provide that the actual selection of the expert or experts to give the advisory opinion in an individual case shall be made by an appropriate official of such institution.\n\n**(c)** The Secretary shall furnish a claimant with notice that an advisory medical opinion has been requested under this section with respect to the claimant's case and shall furnish the claimant with a copy of such opinion when it is received by the Secretary.\n\nTreatment of veterans with PTSD is, at its very core, appalling. Why has the U.S. Court of Appeals for Veterans Claims revisited the issue of PTSD nearly 900 times? One reason is that diagnosing PTSD is not a black and white medical issue. Another reason is that both parties contribute to the confusion. The typical veteran has no idea what constitutes a valid claim. All he does know that he is a victim of PTSD, but for various reasons he is unable to speak about it to the outside world. On top of this, he is ill-prepared to cope with the administrative and technical realities of a giant bureaucracy. When he does reach out, he puts his trust in the VA to do the right thing.\n\nVery few VA claims examiners have experienced a traumatic, life-threatening event that would give them some insight into the experiences of a veteran with PTSD. Remember my friend from chapter 3, Sgt. David Raz? He is a victim of VA bias. It appears that PTSD claims are rated by narrowest interpretation of the laws, regulations, and evidence.\n\n### What to Do If the VA Sends You a Rejection Letter\n\nThese points should be kept in mind if your claim for PTSD is denied:\n\n  The RO's decisions must be based only on the evidence contained in the claim file. This is mandated by Title 38 U.S. Code.\n\n  If the claimed stressor is not combat related, your statement alone is not sufficient to establish a well-grounded claim.\n\n  There are three elements that must be proved to establish service connection for PTSD. (1) There must a current medical diagnosis that the PTSD is active. (2) The veteran must show the condition was the result of his military service. (3) A link between the current medical symptoms and the incident that caused the PTSD must be shown. A cook, for example, has to prove that he was shot at on guard duty and that that incident was the cause of his PTSD. A cook is not considered a combatant; therefore, he has to prove both that he is currently disabled by PTSD and that someone actually shot at him. The proof is the nexus that ties it all together.\n\n  Rating boards and the BVA are required by law to account for all the evidence both persuasive and unpersuasive. It must give reasons for rejecting any evidence supporting the veteran's claim.\n\n  Just because the RO cannot find evidence in the veteran's military service records (service medical records, or SMRs) does not exempt claims examiners from appraising the value of other evidence in determining service connection.\n\n  All evidence is presumed to be true when a the board considers whether or not a claim is well grounded.\n\nThe VA must assist the veteran with his claim by obtaining records, advising what other evidence may be necessary, and scheduling him for a C&P examination. Do not expect more than a cursory search for records. In all the years I've been dealing with the VA, I never saw a case where claims examiners have requested morning reports, organization histories, after action reports, or military hospital records to help a veteran make a case for his claim.\n\n### _38 Code of Federal Regulations_\n\nThe next step in the administrative process is publishing a regulation that spells out in greater detail the intent of Congress. Congress will pass a specific \"act\" pertaining to one of the three branches of government. That new law is assigned to the appropriate government agency, and each department or agency is responsible for authoring a corresponding statute. The section pertinent to the Department of Veterans Affairs is known as _38 Code of Federal Regulations (38 CFR)_ and is published annually. These regulations should make clear what Congress intended when it passed the initial act, and the individual department cannot add new language that totally disregards Congress's original meaning.\n\nThe pertinent regulations are provided in as follows.\n\n#### 38 _CFR_ \u00a73.304: DIRECT SERVICE CONNECTION; WARTIME  \nAND PEACETIME (2003)\n\n#### (c) Development.\n\nThe development of evidence in connection with claims for service connection will be accomplished when deemed necessary but _it should not be undertaken when evidence present is sufficient for this determination_. In initially rating disability of record at the time of discharge, the records of the service department, including the reports of examination at enlistment and the clinical records during service, will ordinarily suffice. Rating of combat injuries or other conditions which obviously had their inception in service may be accomplished pending receipt of copy of the examination at enlistment and all other service records.\n\n#### (d) Combat.\n\nSatisfactory lay or other evidence that an injury or disease was incurred or aggravated in combat will be accepted as sufficient proof of service connection if the evidence is consistent with the circumstances, conditions, or hardships of the veteran's service even though there is no official record of such incurrence.\n\n#### (f) Post-traumatic stress disorder.\n\nService connection for post traumatic stress disorder requires medical evidence diagnosing the condition in accordance with Sec. 4.125(a) of this chapter; a link, established by medical evidence, between current symptoms and an in-service stressor; and credible supporting evidence that the claimed in-service stressor occurred. Although service connection may be established based on other in-service stressors, the following provisions apply for specified in-service stressors as set forth below:\n\n(1) If the evidence establishes that the veteran engaged in combat with the enemy and the claimed stressor is related to that combat, in the absence of clear and convincing evidence to the contrary, and provided that the claimed stressor is consistent with the circumstances, conditions, or hardships of the veteran's service, the veteran's lay testimony alone may establish the occurrence of the claimed in-service stressor.\n\n(2) If the evidence establishes that the veteran was a prisoner-of-war under the provisions of Sec. 3.1(y) of this part and the claimed stressor is related to that prisoner-of-war experience, in the absence of clear and convincing evidence to the contrary, and provided that the claimed stressor is consistent with the circumstances, conditions, or hardships of the veteran's service, the veteran's lay testimony alone may establish the occurrence of the claimed in-service stressor.\n\n#### 38 _CFR_ \u00a73.326: EXAMINATIONS\n\nFor purposes of this section, the term examination includes periods of hospital observation when required by VA.\n\n**(a)** Where there is a claim for disability compensation or pension but medical evidence accompanying the claim is not adequate for rating purposes, a Department of Veterans Affairs examination will be authorized. This paragraph applies to original and reopened claims as well as claims for increase submitted by a veteran, surviving spouse, parent, or child. Individuals for whom an examination has been scheduled are required to report for the examination.\n\n**(b)** Provided that it is otherwise adequate for rating purposes, any hospital report, or any examination report, from any government or private institution may be accepted for rating a claim without further examination. However, monetary benefits to a former Prisoner of War will not be denied unless the claimant has been offered a complete physical examination conducted at a Department of Veterans Affairs hospital or outpatient clinic.\n\n**(c)** Provided that it is otherwise adequate for rating purposes, a statement from a private physician may be accepted for rating a claim without further examination.\n\n#### 38 _CFR_ \u00a73.340: TOTAL AND PERMANENT RATINGS AND  \nUNEMPLOYABILITY\n\n#### (a) Total disability ratings.\n\n(1) General. Total disability will be considered to exist when there is present any impairment of mind or body which is sufficient to render it impossible for the average person to follow a substantially gainful occupation. Total disability may or may not be permanent. Total ratings will not be assigned, generally, for temporary exacerbations or acute infectious diseases except where specifically prescribed by the schedule.\n\n(2) Schedule for rating disabilities. Total ratings are authorized for any disability or combination of disabilities for which the Schedule for Rating Disabilities prescribes a 100% evaluation or, with less disability, where the requirements of paragraph 16, page 5 of the rating schedule are present or where, in pension cases, the requirements of paragraph 17, page 5 of the schedule are met.\n\n(3) Ratings of total disability on history. In the case of disabilities which have undergone some recent improvement, a rating of total disability may be made, provided:\n\n(i) That the disability must in the past have been of sufficient severity to warrant a total disability rating;\n\n(ii) That it must have required extended, continuous, or intermittent hospitalization, or have produced total industrial incapacity for at least one year, or be subject to recurring, severe, frequent, or prolonged exacerbations; and\n\n(iii) That it must be the opinion of the rating agency that despite the recent improvement of the physical condition, the veteran will be unable to effect an adjustment into a substantially gainful occupation. Due consideration will be given to the frequency and duration of totally incapacitating exacerbations since incurrence of the original disease or injury, and to periods of hospitalization for treatment in determining whether the average person could have reestablished himself or herself in a substantially gainful occupation.\n\n#### (b) Permanent total disability.\n\nPermanence of total disability will be taken to exist when such impairment is reasonably certain to continue throughout the life of the disabled person. The permanent loss or loss of use of both hands, or of both feet, or of one hand and one foot, or of the sight of both eyes, or becoming permanently helpless or bedridden constitutes permanent total disability. Diseases and injuries of long standing which are actually totally incapacitating will be regarded as permanently and totally disabling when the probability of permanent improvement under treatment is remote. Permanent total disability ratings may not be granted as a result of any incapacity from acute infectious disease, accident, or injury, unless there is present one of the recognized combinations or permanent loss of use of extremities or sight, or the person is in the strict sense permanently helpless or bedridden, or when it is reasonably certain that a subsidence of the acute or temporary symptoms will be followed by irreducible totality of disability by way of residuals. The age of the disabled person may be considered in determining permanence.\n\n#### (c) Insurance ratings.\n\nA rating of permanent and total disability for insurance purposes will have no effect on ratings for compensation or pension.\n\n[26 FR 1585, Feb. 24, 1961, as amended at 46 FR 47541, Sept. 29, 1981]\n\n#### 38 _CFR_ \u00a74.18: UNEMPLOYABILITY\n\nA veteran may be considered as unemployable upon termination of employment which was provided on account of disability or in which special consideration was given on account of the same when it is satisfactorily shown that he or she is unable to secure further employment. With amputations, sequelae of fractures, and other residuals of traumatism shown to be of static character, a showing of continuous Unemployability from date of incurrence, or the date the condition reached the stabilized level, is a general requirement in order to establish the fact that present unemployability is the result of the disability. However, consideration is to be given to the circumstances of employment in individual claims, and, if the employment was only occasional, intermittent, tryout or unsuccessful, or eventually terminated on account of the disability, present unemployability may be attributed to the static disability. Where unemployability for pension previously has been established on the basis of combined service-connected and non-service-connected disabilities and the service-connected disability or disabilities have increased in severity, Sec. 4.16 is for consideration.\n\n[40 FR 42536, Sept. 15, 1975, as amended at 43 FR 45349, Oct. 2, 1978]\n\n#### 38 _CFR_ \u00a74.125: DIAGNOSIS OF MENTAL DISORDERS\n\n**(a)** If the diagnosis of a mental disorder does not conform to DSM-IV or is not supported by the findings on the examination report, the rating agency shall return the report to the examiner to substantiate the diagnosis.\n\n**(b)** If the diagnosis of a mental disorder is changed, the rating agency shall determine whether the new diagnosis represents progression of the prior diagnosis, correction of an error in the prior diagnosis, or development of a new and separate condition. If it is not clear from the available records what the change of diagnosis represents, the rating agency shall return the report to the examiner for a determination.\n\n(Authority: 38 U.S.C. 1155)\n\n[61 FR 52700, Oct. 8, 1996]\n\nRegulations are specific laws that are turned into working legal instructions. These regulations have the power of law for the agency, employees, and veterans. When two people (e.g., the VA claims examiners and a veteran) have a different conception of what the regulation dictates, the Board of Veterans or the U.S. Court of Appeals for Veterans Claims resolves the issue.\n\n### VA Manual M21-1\n\nSubchapter 11, paragraph 11.37 answers the rater's question: \"What am I suppose to do when rating PTSD claims?\" After you read some case summaries of the U.S. Court of Appeals for Veterans Claims in chapter 12, you have to wonder how they arrive at some of the decisions they publish. Here is the full text from VA Manual M21-1, part III, chapter 5, paragraph 5.14 and part VI, chapter 11, paragraph 11.37 (directly quoted or, in places, amended for clarity).\n\n#### 5.14 POST-TRAUMATIC STRESS DISORDER\n\n**Post-traumatic stress disorder.** Service connection for post-traumatic stress disorder (PTSD) requires medical evidence diagnosing the condition in accordance with 38 _CFR_ 4.125(a), a link, established by medical evidence, between current symptoms and an in-service stressor, and credible evidence that the claimed in-service stressor occurred (38 _CFR_ 3 304(f)).\n\n**Evidence of a stressor in service.** Obtain all available evidence from the service department showing that the veteran served \"where the stressful event occurred. Also obtain evidence supporting the veteran's description of the event.\n\n(1) Combat Stressors. The following individual decorations are examples of decorations which may serve as evidence that the veteran engaged in combat:\n\n  Air Force Cross\n\n  Air Medal with \"V\" Device\n\n  Army Commendation Medal with \"V\" Device\n\n  Bronze Star-Medal with \"V\" Device\n\n  Combat Action Ribbon\n\n  Combat Infantryman Badge\n\n  Combat Medical Badge\n\n  Distinguished Flying Cross\n\n  Distinguished Service Cross\n\n  Joint Service Commendation Medal with \"V\" Device\n\n  Medal of Honor\n\n  Navy Commendation Medal with \"V\" Device\n\n  Navy Cross\n\n  Purple Heart\n\n  Silver Star.\n\nIf the evidence established that the veteran engaged in combat with the enemy and the claimed stressor is related to that combat, in the absence of clear and convincing evidence to the contrary\u2014and provided that the claimed stressor is consistent with the circumstances, conditions, or hardships of the veteran's service\u2014the veteran's testimony alone establishes the occurrences of the claimed in-service stressor. (38 _CFR_ 3.30 4(f) and 38 USC 1154(b)).\n\n(2) Noncombat Stressors. PTSD may result from a noncombat stressor, such as: a plane crash, ship sinking, explosion, rape or assault, duty on a burn ward or in a graves registration unit.\n\n(3) POW Status. If the evidence establishes that the veteran was a Prisoner of War under the provisions if 38 _CFR_ 3.1(y) and the claimed stressor is related to that [POW] experience, in the absence of clear and convincing evidence to the contrary and provided that the claimed stressor is consistent with the circumstances conditions or hardships of the veteran's service, the veteran's testimony alone establishes the occurrence of the claimed in-service stressor, 3M _CFR_ 3.304(f) and 38 U.S.C 1154(b).\n\n#### Development.\n\n(1) If the veteran indicates pertinent treatment in a VA facility, Vet Center, or elsewhere, request hospital report(s) and clinical records.\n\n(2) If the evidence does not establish that the veteran was engaged in combat with the enemy, or the evidence does establish this but the claimed stressor is not related to that combat, then credible supporting evidence is required to establish that a stressor occurred, 38 _CFR_ 3.304(f). In cases where available records do not contain this develop for this evidence as follows:\n\n(a) Request specific details of the in-service stressful incident(s) date(s), places(s), unit of assignment at the time of the event(s), description of the event(s), medal(s) or citation(s) received as a result of the event(s) and if appropriate name other identifying information concerning other individuals involved in the event(s). As a minimum, the claim must indicate the location and approximate time (a 2-month specific date range) of the stressful event(s) in question, and the unit of assignment at he time the stressful event occurred. Inform the veteran that this information is necessary to obtain supportive evidence of the stressful event(s) and that failure to respond or an incomplete response may result in denial of the claim.\n\n_Note:_ Do not ask the veteran for specific details in any case in which there is credible supporting evidence that the claimed in-service stressor occurred.\n\n(b) Use only the codes at the time in PIES. Use O19 for simple PTSD claims and 018 for personal trauma. Do _not_ use free text and list all of the information mentioned in the in manual. The response team at NPRC knows which pages to send for simple PTSI cases. In personal trauma cases, they will send copies of everything. There should be no occasion where both 018 and 019 are used. _If the claim is for both personal trauma and PTSD, the use of code 018 will include pages that would have sent in response to code 019. In either event, no additional_ instructions need to be added to the PIES request. (Italics added.)\n\n  **ARMY:** Personnel Qualification Record DA Form 2-1. The form is used for both Officers and Enlisted personnel and first came into use in January 1973. Prior to that DA Form 20 and DA Form 66 were used.\n\n  **NAVY:** Enlisted record of \"Transfer and Receipts (pp. 1, 32\u2013 33). Enlisted record of \"Administrative Remarks\" (pp. 4\u20139, 34) Officer record, NAVPERS 1301\/51. \"Officer Data Card,\" (p. 35). DD 214 and enlistment contracts are usually included.\n\n  **AIR FORCE:** \"Airman Military Record,\" AF Form 7, Enlisted (pp. 36 \u201339); AF Form 11, Officer (pp. 39\u201340). Performance reports are for both Officer and Enlisted.\n\n  **MARINE CORPS:** Enlistment contracts discharge Papers, MABMC-11 (discharge order) and service records are usually provided (pp. 3, 5\u20136, 8\u20139, 12\u201313, 17).\n\n  **COAST GUARD:** Enlisted Record, \"Endorsement on Order Sheet\" (DOT Form GC 3312B). Officer Record, \"Service Records Card, CG 3301, CG 3313, and CG 3305 (pp. 3, 5, 6\u20137), DD-214 and enlistment contract.\n\n(3) If medical evidence establishes a valid diagnosis of PTSD and development is complete in every respect but for confirmation of the service stressor, contact either the U.S. Armed Center for Unit Records Research (CURR or lay marine corps (subparagraph (3) (b) Requests to that office must include the following:\n\n  a two-month specific date range when the stressful event occurred\n\n  the veteran's unit of assignment at the time of the stressful event, and\n\n  the geographic location where the stressful event took place.\n\nAdditional information identified by CURR as helpful in conducting research includes:\n\n  medals or citations received by the veteran; and\n\n  stressful events witnessed by the veteran. Specific information including the names of other soldiers or sailors' involved, dates, units of assignment, and their geographic location is essential.\n\n_Note:_ CURR maintains a database of pending research requests. Any research requests identified as inadequate will be closed out. Additional development should be undertaken prior resubmission of the request to CURR. Stations may, at their discretion, render a final decision on cases if they determine that the information needed to conduct research is unobtainable. The file should be documented to this effect.\n\n(a) For all services except the marine corps send the letter to: U.S. Armed Services Center for Unit Records Research (CURR), 7798 Cissna Road, Suite 101, Springfield, VA, 221511-3197. Telephone numbers are 703-806-7835 or 703-806-7838. If necessary, you may also contact the Coast Guard at Commander, Military Personnel Command. MPC\u2013S3, 2100 (H) Second St. SW, Washington, DC, 20593-0001.\n\n(b) For U.S. Marine Corps veterans with service after 1956, send the letter to: Commandant of the Marine Corps Headquarters U.S. Marine Corps MMSB10, 2008 Elliot Road, Suite 201, Quantico, VA 22134-5030. Telephone numbers are 703-784-3935, 703-784-3935, or 703-784-3940. For veterans with service before 1956, send to: Marine Corps Historical Center, Building 58, Washington Navy Yard, Washington, DC, 20374-9580. Telephone number is 202-433-3840. All command chronologies are located at the Historical Center as well as unit diaries before 1956 and some unit diaries for the period 1956 to 1966. Quantico should have all unit diaries from 1956 on. In some instances, the military may forward the RO's inquiry to the other facility for a better answer.\n\n(c) Enclose copies of information received from the veteran and the service department with these requests. These letters should _not_ be sent until a _valid_ diagnosis of PTSD is of record.\n\n(4) If sufficient evidence i _s_ already of record to grant service connection for a claimed condition, do so. If not, but there is a continued diagnosis of PTSD, send an inquiry to the CURR or the marine corps. However, always send an inquiry for instances in which the only obstacle to service connection is confirmation of an alleged stressor. A denial solely because of an unconfirmed stressor is improper unless it has first been reviewed by the CURR or the marine corps.\n\n(5) Occasionally the CURR or the marine corps will request a more specific description of the stressor in question. Failure of the veteran to respond substantively to the request for information will be grounds to deny the claim based on unconfirmed stressors.\n\n#### PTSD claims based on personal assault.\n\n(1) Veterans claiming service connection for disability owing to an in-service personal assault face unique problems in documenting their claims. Personal assault is an event of human design that threatens or inflicts harm. Examples of this are rape, physical assault, domestic battering, robbery, mugging, and stalking. Although these incidents are most often thought of as involving female veterans, male veterans can also be involved. Care must be taken to tailor development for a male or female veteran. These incidents are often violent and may lead to the development of PTSD secondary to personal assault.\n\n(2) Because assault is an extremely personal and sensitive issue, many incidents of personal assault are not officially reported, and victims of this type of in-service trauma may find it difficult to produce evidence to support the occurrence of the stressor. Therefore, alternative evidence must be sought.\n\n(3) To service connect PTSD there must be credible evidence to support the veteran's assertion that the stressful event occurred. This does not mean that the evidence actually proves that the incident occurred; rather there should be at least an approximate balance of positive and negative evidence that it occurred.\n\n(4) Review the claim and all its attached documents. Develop for SMRs and MPRJ information as needed.\n\n(a) Service records not normally requested but may be needed to develop this type of claim. Responses to the development letter may identify additional information sources. These include:\n\n  a rape crisis center or center for domestic abuse\n\n  a counseling facility\n\n  a health clinic\n\n  family members or roommates\n\n  a faculty member\n\n  medical reports, civilian physicians or caregiver's who may have treated the veteran either immediately following the incident or sometime later\n\n  a chaplain or clergy\n\n  fellow service persons, or\n\n  personal diaries or journals.\n\n(b) Obtain any reports from military police, shore patrol, provost marshals office, or other military law enforcement. Development may include phone, fax, e-mail, or correspondence as long as they are documented in the file.\n\n(5) Identifying possible sources of alternative evidence will require that you ask the veteran for information concerning the incident. This should he done as compassionately as possible in order to avoid further traumatization for PTSD stressor development letter used by regional offices to solicit details concerning a combat stressful incident is inappropriate for this type of PTSD claim. See a letter developed locally for this type of claim.\n\n(6) Same development letters are inappropriate for PTSD claims based on personal assault and should not be used for that purpose. Instead use a letter or an attachment developed locally.\n\n(7) Rating Veterans Service Representatives (RVSRS) must carefully evaluate all the available evidence. If the military record contains no documentation that a personal assault occurred, alternate evidence might still establish an in-service stressful incident. Behavior changes that occurred at the time of the incident may indicate the occurrence of an in-service stressor. Examples of behavior changes that might indicate a stressor are but not limited to:\n\n(a) visits to a medical or counseling clinic or dispensary without a specific diagnosis or specific ailment;\n\n(b) sudden requests that the veteran's military occupational series or duty assignment be changed without other justification;\n\n(c) lay statements indicating increased use or abuse of leave without an apparent reason such as family obligations or family illness;\n\n(d) changes in performance and performance evaluations;\n\n(e) lay statements describing episodes of depression, panic attacks, or anxiety but with no identifiable reasons for the episodes;\n\n(f) increased or decreased use of prescription medications;\n\n(g) increased use of over-the-counter medications;\n\n(h) evidence of' substances abuse such as alcohol or drugs;\n\n(i) increased disregard for military or civilian authority;\n\n(j) obsessive behavior such as overeating or under-eating;\n\n(k) pregnancy tests around the time of the incident;\n\n(l) increased interest in tests for HIV or sexually transmitted diseases;\n\n(m) unexplained economic or social behavior changes;\n\n(n) treatment for physical injuries around the time of the veteran's claimed trauma but not reported as a result of the trauma; and\n\n(o) breakup of a primary relationship.\n\n(8) In personal assault claims, secondary evidence may need interpretation by a specialist if it involves behavior changes. Evidence that documents such behavior changes may require interpretation in relationship to the medical diagnosis by a neuropsychiatric physician.\n\n#### 11.37 POST-TRAUMATIC STRESS DISORDER (PTSD)\n\nService connection for PTSD requires medical evidence establishing a clear diagnosis of the condition, credible supporting evidence that the claimed in-service stressor actually occurred, and a link (established by medical evidence) between current symptomatology and the claimed in-service stressor (38 _CFR_ 3.304(f)). The issue of service connection for PTSD is the sole responsibility of the rating specialist at the local level. Central Office opinion or guidance may be requested on complex cases.\n\n#### Stressors.\n\nIn making a decision, exercise fair, impartial, and reasonable judgment in determining whether a specific case of PTSD is service connected. Some relevant considerations are:\n\n(1) PTSD does not need to have its onset during combat. For example, vehicular or airplane crashes, large fires, flood, earthquakes, and other disasters evoke significant distress in most involved persons. The trauma may be experienced alone (rape or assault) or in the company of groups of people (military combat).\n\n(2) A stressor is not to be limited to a single episode. A group of experiences also may affect an individual, leading to a diagnosis of PTSD in some circumstances\u2014for example, assignment to a grave registration unit, burn cam unit, or liberation of internment camps could have a cumulative effect of powerful, distressing experiences essential to a diagnosis of PTSD.\n\n(3) PTSD can be caused by events that occurred before, during, or after service. The relationship between stressors during military service and current problems\/symptoms will govern the question of service connection. Symptoms must have a clear relationship to the military stressor as described in the medical reports.\n\n(4) PTSD can occur hours, months, or years after a military stressor. Despite this long latent period, service-connected PTSD may be recognizable by a relevant association between the stressor and the current presentation of symptoms. This association between stressor and symptoms must be specifically addressed in the VA examination report and to a practical extent supported by documentation.\n\n(5) Every decision involving the issue of service connection for PTSD alleged to have occurred as a result of combat must include a factual determination as to whether or not a veteran was engaged in combat, including the reasons or bases for that finding (see _Gaines v. West, 111 Vet. App. 113, 1998_ ).\n\n#### Evidence of stressors in service.\n\n(1) Conclusive Evidence. Any evidence available from the service department indicating that the veteran served in the area in which the stressful event is alleged to have occurred and any evidence supporting the description of the event are to be made part of the record. Corroborating evidence of a stressor is not restricted to service records, but _may_ be obtained from other sources (see _Doran v. Brown, 6 Vet. App. 283_ , ( _1994_ ). If the claimed stressor is related to combat, in the absence of information to the contrary, receipt of any of the following individual decorations will be considered evidence that a veteran engaged in combat:\n\n  Air Force Cross\n\n  Air Medal with \"V\" Device\n\n  Army Commendation Medal with \"V\" Device\n\n  Bronze Star Medal with \"V\" Device\n\n  Combat Action Ribbon\n\n  Combat Infantryman Badge\n\n  Combat Medical Badge\n\n  Combat Aircrew Insignia\n\n  Distinguished Flying Cross\n\n  Distinguished Service Cross\n\n  Joint Service Commendation Medal with \"V\" Device\n\n  Medal of Honor\n\n  Navy Commendation Medal with \"V\" Device\n\n  Navy Cross\n\n  Purple Heart\n\n  Silver Star.\n\nOther supportive evidence includes but is not limited to plane crash, ship sinking, explosion, rape or assault, duty on a burn ward or in graves registration unit, POW status that satisfies the requirements of 38 _CFR_ 3.1(y). These will also be considered conclusive evidence of an in-service stressor.\n\n(2) Evidence of Personal Assault. Personal assault is an event of human design that threatens or inflicts harm. Examples of this are rape, physical assault, domestic battering, robbery, mugging, and stalking.\n\n(a) Alternative Evidence. If the military record contains no documentation that a personal assault occurred, alternative evidence might still establish an in-service stressful incident. Examples of such evidence include but are not limited to:\n\n  records from law enforcement authorities;\n\n  records from rape crisis centers, hospitals, or physicians;\n\n  pregnancy tests or tests for sexually transmitted diseases; and\n\n  statements from family members, roommates, fellow service members or clergy.\n\n(b) Behavior Changes. Behavior changes that occurred at the time of the incident may indicate the occurrence of an in-service stressor. Examples of such changes include but are not limited to:\n\n  visits to a medical or counseling clinic or dispensary without a specific diagnosis or specific ailment;\n\n  sudden requests for a change in occupational series or duty assignment without other justification;\n\n  increased use or abuse of leave without an apparent reason, such as family obligations or family illness;\n\n  changes in performance and performance evaluations;\n\n  episodes of depression, panic attacks, or anxiety but no identifiable reasons for the episodes;\n\n  increased or decreased use of prescription medication;\n\n  increased use of over-the-counter medications;\n\n  substance abuse such as alcohol or drugs;\n\n  increased disregard for military or civilian authority;\n\n  obsessive behavior such as overeating or under-eating;\n\n  unexplained economic or social behavior changes; and\n\n  breakup of a primary relationship.\n\n**(c) Development Requirements.** It is unlawful to deny a post-traumatic stress disorder claim that is based on in-service personal assault without first advising the claimant that evidence from sources other than service medical records, including evidence of behavior changes, may constitute credible supporting evidence of the stressor. The veteran should have the opportunity to furnish this type of evidence or indicate its potential sources.\n\n**(d) Interpretation of Secondary Evidence.** In personal assault claims, secondary evidence may need interpretation by a clinician, especially if it involves behavior changes. Evidence that documents such behavior changes may require interpretation in relationship to the medical diagnosis or an opinion by an appropriate medical or mental health professional as to whether it indicates that a personal assault occurred.\n\n(3) Credible Supporting Evidence. A combat veteran's lay testimony alone may establish an in-service stressor for purposes of service connecting PTSD _(Cohen_ v. _Brown_ , 94-661, U.S. Ct. Vet. App. March 7, 1997). However, a noncombat veteran's testimony alone does not qualify as \"credible supporting evidence of the occurrence of an in-service stressor as required by _38 CFR 3.304(f)_ After-the-fact psychiatric analyses that infer a traumatic event are likewise insufficient in this regard ( _Moreau v. Brown, 9 Vet. App. 389, 1996_ ).\n\n#### Development.\n\n(1) For instructions regarding development of service records, medical treatment, and evidence of stressor or personal assault, refer to Part III, subparagraphs 5.l4b and 5.14c.\n\n(2) Unless medical evidence adequate for rating purposes is already of record, an immediate examination is needed. When requesting an examination, state in the remarks section of VA Form 21-2507, \"Request for Physical Examination,\" \"Claims folder to be made available to examiner upon request.\"\n\n#### Incomplete examinations and\/or reconciliation of diagnosis.\n\nIf an examination is received with the diagnosis of PTSI) that does not contain the above essentials of diagnosis, the claims examiner should return the examination as incomplete for rating purposes, note the deficiencies, and request reexamination.\n\n(1) Examples of an unacceptable diagnosis include not only insufficient symptomatology but failure to identify or to adequately describe the stressor, or failure to consider prior reports demonstrating a mental disorder that could not support a diagnosis of PTSD. Conflicting diagnoses of record must be acknowledged and reconciled (by the claims examiner).\n\n(2) The reviewer of claims should exercise caution to accurately define situational disturbances containing adjustment reaction of adult life, which subside when the situational disturbance no longer exists or is withdrawn, as well as the reactions of those without neurosis who have \"dropped out\" and have become alienated but are not afflicted with PTSD.\n\n#### Link between in-service stressor and diagnosis.\n\nRelevant specific information concerning what happened must be described along with as much detailed information as the veteran can provide to the examiner regarding time of the event (year, month, day), geographical location (corps, province, town, or other landmark feature such as a river or mountain), and the names of others who may have been involved in the incident. The examining psychiatrist or psychologist should comment on the presence or absence of other traumatic events and their relevance to the current symptoms. Service connection for PTSD will not be established either on the basis of a diagnosis of PTSD) unsupported by the type of history and description or where the examination and supporting material fill to indicate a link between current symptoms and an in-service stressful event(s).\n\n#### Review of evidence.\n\n(1) If a VA medical examination fails to establish a diagnosis of PTSD, the claim will be immediately denied on that basis. If no determination regarding the existence of a stressor has been made, a discussion of the alleged stressor need not be included in the rating decision.\n\n(2) If the claimant has failed to provide a minimal description of the stressor (i.e., no indication of the time or place of a stressful event), the claim may be denied on that basis. The rating should specify the previous request for information.\n\n(3) If a VA examination or other medical evidence establishes a valid diagnosis of PTSD, and development is complete in every respect but for confirmation of the in-service stressor, additional evidence will be requested from either the Center for Unit Records Research (CURR) or marine corps. (See part III, paragraph 5.14.)\n\n(4) A claims examiner should not send a case to the CURR or marine corps unless there is a confirmed diagnosis of PTSD adequate to establish entitlement to service connection. Correspondingly, an inquiry should always be sent in instances in which the only obstacle to service connection is confirmation of an alleged stressor. A denial by the claims examiner solely because of an unconfirmed stressor is improper unless it has first been reviewed by the CURR or marine corps.\n\n(5) If the CURR or the marine corps requests a more specific description of the stressor in question, the veteran should immediately provide the necessary information. If the veteran provides a reasonably responsive reply, the claims examiner should forward it to the requesting agency. Failure by the veteran to respond substantively to the request for information will be grounds to deny the claim based on unconfirmed stressor. (See Part III, paragraph 5.14.)\n\nWhen the VA denies a claim, it often makes mistakes in administrative procedures or in interpreting the intent of the law. VA manuals are big books of SOPs (Standard Operating Procedures) that spell out the ABCs of the regulations. The purpose of VA manuals is to provide the adjudication teams with detailed instructions about how they are to do their job. Knowing about the manuals' contents may be the difference between victory or defeat for a veteran.\n\n### Chapter 4: What Have We Learned?\n\n1. \"You've got to know the territory.\" This chapter is a map of the workings of the VA's adjudication process. _You can't defend what you don't know_. One reason why so many claims are denied or rated below their actual level of disability is because the claimant has no idea what the process is all about. You may think it's no fun reading all the pertinent laws, regulations, and manuals that apply to your claim. You're right! It doesn't beat a night on the town, but if you expect to receive your full benefits, you have to do it.\n\n2. There are five primary groups of principle standards that govern the way the VA executes its duty to veterans. They are: statutes, regulations, manuals, general counsel opinions, and case law (see chapter 12). Failure of the VA claims examiners to properly execute the guidelines in any one of these regulatory standards is grounds for an appeal.\n\n3. The VA has a duty to advise you of what information is required to support your claim. However, the process does not require providing details of how you can find this information. VA Manual M21-1, part III, chapter 5, paragraph 5.14(c) spells out just what type of information is required and where to find it in the archives of the U.S. Government.\n\n4. This is very important: When filing a PTSD claim, there are two steps you must follow: 1. Always file an informal claim first, as this gives you one year to accumulate all the evidence and protects your entitlement date. 2. Submit everything through your Senator or Congressional Representative. The VA response is much faster and they are less likely to dismiss your evidence when they know your claim is being followed by your legislator. The details of this procedure are discussed in chapter 10.\n\n5. If you have been awarded any one of the medals listed in VA Manual 21-1, part III, chapter 5, paragraph 5.14(b)(1), your claim should be processed without making you prove your combat experiences. If you received a Bronze Star or Air Medal without a \"V\" Device, challenge the claims examiners if they don't accept this combat award as a combat stressor. Ask them what their legal authority is to exclude you from direct service connection when the Department of the Army awards the medal for meritorious service against the enemy. The bottom line is: in asking you to prove your combat experiences, they deliberately ignored a portion of 32 _CFR_ \u00a7578.11 and \u00a7578.12.\n\n6. When you are filing a claim for PTSD, it is absolutely essential that you are able to prove two important points. First, you must be currently diagnosed with PTSD; and, two, PTSD must be able to be traced back to your combat experiences.\n\n7. Your chances of being awarded compensation benefits for PTSD improve considerably when a private sector psychiatrist evaluates you first. The purpose for this step is to take the initiative with the VA instead of waiting for a claims examiner to question your proof. Submitting a medical evaluation diagnosing PTSD with your claim application forces them to prove, by a preponderance of evidence, that this isn't true.\n\n8. If you were previously denied compensation for PTSD, you might be able to reopen your claim because examiners failed to follow the court's decision in _Doran v. Brown_. This case is discussed detail in chapter 12.\n\n9. If the VA evaluation is conducted by anyone other than a psychiatrist or psychologist and the claim is denied, immediately notify the VA that you're appealing the decision because you were evaluated by a nonphysician. Remember, always ask the examiner what his or her medical credentials are. I once worked a case where they used a student resident from the local medical school to give a C&P examination.\n\nHere is a list of the most common reasons VA adjudicators use when denying a PTSD claim:\n\n  The declared stressor was not deemed as a possible cause of PTSD;\n\n  Rating boards would not accept a diagnosis of PTSD as being substantiated;\n\n  Rating boards would not accept alcoholism and drug addiction as a secondary condition to PTSD;\n\n  Evidence, written or oral, given by the veteran was not given the proper weight and thus was not considered in the decision process;\n\n  Evidence in the form of lay \"buddy letters\" was not accepted as evidence;\n\n  Medical diagnosis and assessment by the veteran's own doctors were not accepted by rating board members;\n\n  More often than not, rating boards held that other mental disorders were the cause of the veteran's conditions without considering the possibility that there could be multiple disorders for part of the veteran's illness;\n\n  In many instances, rating board members would not assist in developing evidence related to the stressor once the veteran presented a well-grounded claim; rating board members would accept from VA C&P examiners reports that were not in compliance with their manuals and regulations.\n\n### Notes\n\n1. _Doran v. Brown_ , 6 F. 283 (Vet. App. 1994).\n\n## 5 DOWN TO BASICS\n\n### History of the Veterans Administration\n\nThe U.S veterans' benefits system traces its roots back to the seventeenth century when the pilgrims of Plymouth passed a law protecting citizens who joined the colony army to defend it against the hostile Pequot Indians. The colony ensured that any soldier disabled in its defense would be supported by the colony.\n\nWhen the colonies united and declared their independence from England, the Continental Congress of 1776 also addressed the needs of its soldiers. To encourage enlistments during the Revolutionary War, Congress passed legislation authorizing pension benefits for soldiers. This was the first move by a fledgling republic to protect all citizens who took up arms to defend the republic.\n\nIndividual states and communities assumed responsibility for medical and hospital care for veterans in the early days of the republic. In 1811, the federal government authorized the first domiciliary and medical facility for veterans. Throughout the 1800s the nation's veterans assistance program expanded to include benefits and pensions not only for veterans, but also for their widows and dependents.\n\nFollowing the Civil War, many states established homes for their veterans. Medical and hospital treatment was provided for all injuries and diseases, whether or not of service origin. Such homes cared for all homeless and disabled veterans of the Civil War, Indian Wars, Spanish-American War, and Mexican Border period, as well as those discharged from peacetime service.\n\nWhen the United States entered World War I in 1917, Congress established a new system of veteran benefits. Programs included disability compensation, insurance for military members and vocational rehabilitation for the disabled. The swelling ranks of veterans in the 1920s led to the administration of benefits by three different federal agencies: the Veterans Bureau, the Bureau of Pensions of the Interior Department, and the National Home for Disabled Volunteer Soldiers. Yet veterans had to fend for themselves with each agency, each with its own laws and regulations. Many veterans wandered from agency to agency because of confusion and contradiction within the agencies about who was in charge.\n\nIn 1930, Congress authorized the president to consolidate and coordinate government activities affecting war veterans. The newly formed organization was known as the Veterans Administration. The three former agencies that independently exercised control over veteran's benefits were reassigned to the Veterans Administration and became bureaus within the new organization.\n\nThe Department of Veterans Affairs was established as a cabinet-level position on March 15, 1989. President George H. W. Bush hailed the creation of the new department saying, \"There is only one place for the veterans of America, in the Cabinet Room, at the table with the President of the United States of America.\" The first secretary of the Department of Veterans Affairs was former Republican Congressman Edward Joseph Derwinski. Despite President Bush's statement regarding the recognition of the VA, as of yet nothing of momentous proportion has escaped from the president's cabinet room to benefit veterans at large.\n\nDespite the rhetoric of \"We can't do enough for you,\" that has bombarded veterans since the early 1980s, benefits have been consistently downsized. Those ruling the executive branch since President Ronald Reagan have measured veterans' benefits only by bottom-line costs. Gone are the sincerity and pledges of a grateful nation for the sacrifices of those who have served. When politicians look at the fact that nearly twenty-seven million people have served in the military\u2014in theory each has a potential claim for benefits\u2014they see an immediate threat to their special interest projects.\n\nAt one time, it would have appeared un-American for any politician to announce benefits cuts for veterans. One might as well attack motherhood, apple pie, or the American flag. That was until President George W. Bush proclaimed in 2005 that domestic programs would have to take some heavy hits. The Office of Management and Budget is taking aim at veteran, social security, and education programs to offset the cost of the Iraq War, War on Terrorism, and continued tax cuts, while the national debt continues to grow larger.\n\nThe story in chapter 3 about Sgt. David Raz illustrates a covert way of cutting veterans benefits. Another measure, discussed in chapter 1, is to give different meanings to laws and regulations. The VA drags claims on endlessly before reaching a decision and hires rank and file of processing personnel with low-level technical skills. There has also been an effort to make VA compensation benefits taxable, which, if voted into law, would reduce the spendable income of the disabled. Some veterans must pay for their medication from the VA if the prescription is not directly linked to a service-connected condition. Veterans treated for illness or disease not designated service connected, and whose income is greater than the poverty level, must pay for their own care. If they have a private insurance company, their care could be charged against their policy. Because the VA cannot compete for highly skilled physicians, they hire foreign doctors, some with inadequate English-language skills, who will work for less than the going rate normally earned by physicians. The VA will accept language limitations in an effort to maintain hospital staffing requirements. They are also staffing VA medical positions with physician's assistants and nurse practitioners. In order to cut costs, the VA has reduced the size and configuration of rating boards; doctors and lawyers are no longer members of the board, and now a single individual decides just about every type of claim considered by the former three-member board. Very few, if any, have the ability to evaluate complex medical issues requiring specialized training and experience.\n\nThe only event in 1989 that was truly helpful for veterans was the establishment of the U.S. Court of Appeals for Veterans Claims. No longer could the VA act as its own judge and jury in reviewing veterans' claims and benefits. The new federal court's only purpose was to ensure the laws and regulations pertaining to veterans' benefits were properly and fairly administered. The court has successfully reshaped the review process by the Board of Veterans' Appeals. The BVA is reversing and remanding appeals from Regional Offices (ROs) nationwide at an astounding rate.\n\nOnly a small percentage of veterans (20%) ever exercise their right of appeal. _If every veteran who was turned down appealed the denial of benefits, the VA would buckle under the volume of appeals_. VA regional offices continue to ignore the decisions of the court; they fail to follow the regulations and laws and they do not decide claims based on precedent-setting case law, as mandated by the court. If the St. Petersburg RO followed the regulations and case law, Sgt. David Raz would not be passing his fifth year pursuing his benefits for PTSD.\n\nBefore filing any type of claim action with the VA, there are several fundamental concepts you need to know. Jesse Brown, former secretary of Department of Veterans Affairs, claims that the VA is user friendly. According to the _American Legion Magazine_ (1995), \"He does not believe that the Veterans Benefits Administration was systematically ignoring the law or the court.\" He did not say, \"I know they are _not_ ignoring the law or the courts.\" For this reason, you and only you must take the time to understand how the VA operates, what its responsibilities are, and what proof you need for your claim. There are no shortcuts when it comes time to file a claim for your VA benefits.\n\n### Nine Types of Claim Actions\n\nA veteran can file many types of claim actions with the VA. This book focuses on the four most important service-connected claim actions filed by a veteran. _It is absolutely necessary_ that you know exactly what is entailed for the benefit you are claiming. Before you submit your claim, you must know what the law requires for that category of claim, as each category has special development techniques. One of the most basic and frequent mistakes made is requesting that a claim be \"reopened\" when the veteran actually wants to \"amend\" the claim. You must know what you want to claim and what the VA's responsibility is in responding to your request.\n\nThe nine categories of claims are:\n\n  informal claim\n\n  compensation claim\n\n  claim for an increased rating\n\n  reopened claim\n\n  adjunct or secondary disabilities to original service-connected disability claim\n\n  100% total disability due to individual unemployability claim\n\n  service connection for conditions resulting from medical treatment at VA hospital or clinics claim\n\n  survivor's dependency and indemnity compensation claim\n\n  nonservice-connected pension claims.\n\n#### INFORMAL CLAIM: AN ANTI-DENIAL WEAPON\n\nAnytime you are planning to file a claim, you should first file an _informal claim_. An informal claim establishes the claim entitlement date while you do the work of obtaining evidence and necessary documents. An informal claim is no more than a notice of intent to file a claim for benefits. It is simply a written notice such as:\n\nPlease accept this notice as my informal claim for compensation benefits owing to a back injury I had while on active duty in 1967. I was treated for this condition at the VA hospital in Lake City, Florida, in 1968, Houston, Texas, in 1973, and the Bronx, New York, in 1980. A formal application will be filed once I have collected all of the evidence to support my claim for disability benefits.\n\nThis notice can be submitted on VA form 21-4138 \"Statement in Support of Claim\" or in a personal letter to the regional office.\n\n#### ORIGINAL CLAIM\n\nBy definition, an original claim is the first claim filed for a particular benefit. Claims in this group must be first-time claims for service connection for a particular disease or injury as distinguished from a subsequent claim to reopen. An amendment to Title 38 U.S. Code declares that the VA has a statutory duty to assist you in the development of facts pertinent to your claim. This claim must be filed on VA form 21-521.\n\n#### CLAIM FOR INCREASED BENEFITS\n\nA claim for increased benefits is one of the most dangerous claim actions a veteran can file. If you think that your service-connected medical condition has increased in its severity, you may decide to file a claim for increased benefits. A majority of veterans base this decision on what they perceive as a deterioration of their health problem rather than on a current medical assessment from their physician. In fact, seldom will a veteran discuss the current level of his or her disability with a private physician before filing a claim for increased benefits. It is also true that the average veteran will not determine exactly what he or she must medically prove before filing a claim for increased benefits.\n\nThere is a way to improve the odds that the VA will approve your claim when filing a claim for increased benefits. First, you must know the exact rating code your disability is classified under. To do find this, you need to obtain a complete copy of your claim file. Law entitles you to complete set of the files the VA has on you. Once you have a copy of your claims file, find the _very last rating decision_. From this document, determine the four digit rating code assigned to your disability.\n\nThe last step in the process is to compose a statement on VA Form 21-4138 stating your intent to file a claim for increased benefits. Again, the statement should be short and to the point, covering the technical requirements of your claim. I would suggest something along these lines:\n\nPlease accept this statement as a request for increased benefits for my service-connected lower back injury rated under rating code 5295. My orthopedic physician, Dr. Robert Young, after reviewing VA standards for a lumbosacral strain, finds that my condition is best rated under \"severe at 40%.\" His letter of April 1, 1997, details his medical findings. Also enclosed is a sworn statement from my wife of 20 years whose lay observations report the difficulty I have in dealing with everyday tasks because of pain and loss of motion.\n\n#### REOPENED CLAIM\n\nA claim can be reopened after it has been adjudicated and previously denied. A claimant may reopen a claim either if he or she initially failed to appeal a VA decision within a year following the denial notice or (2) if his or her appeal was not granted. A claim may be reopened only on the grounds of new and material evidence or by claiming the VA committed a clear and unmistakable error. It is incredibly difficult to reverse the original decision of denial. To understand the mechanics and legal aspects of reopening claims, I suggest you obtain a copy of my book _Veterans Survival Guide: How to File and Collect on VA Claims_.\n\nRules that shape a winning claim action are briefly discussed as follows. A successful claimant must have a basic understanding of what is involved in the development of a claim. Nearly all the topics discussed below are examined in detail in later chapters.\n\n### A Documented Claim Is a Must\n\nThe VA will dismiss any claim action it perceives as not well documented. You have the duty to submit evidence sufficient to justify a belief by a fair and impartial individual that the claim is plausible or capable of substantiation. This means that you cannot just go into the VA with a claim application stating: \"I have PTSD caused by what I experienced in Iraq.\" Simply claiming your medical condition is the result of an active service in Iraq will not qualify you as being disabled by PTSD. You must back it up with evidence.\n\nAs previously stated in chapter 4, there are three conditions that must be met. If your records support that you engaged in combat with the enemy, the first condition has been met.\n\nUntil you have met this first requirement, the law does not require the VA to assist you in developing your claim. To this end, you must know how to prove your claim. Failure to do so will result in a denial of your claim. If you stick it out, you'll be in for a three- to five-year battle establishing only one issue\u2014that your claim was well grounded. Assuming you are successful in eventually proving your claim was well grounded, the VA will then go back to square one and look at your alleged injury based on all available evidence and decide if service connection is warranted. Should your claim be denied or underrated, you will have no other choice but to start the appeal process all over again, which means another three to five year wait. As you can see, a final decision on a hastily filed claim action could take a decade. The question you must ask yourself is, \"Will I be alive when the claim runs its course?\" Do it right the first time. Make certain your claim is well grounded and that there is evidence to support your contention that you were indeed injured or sickened while in the service.\n\n### No Record Equals No Claim\n\nIf you do not have copies of your active duty outpatient or inpatient medical records, you must obtain these. Outpatient medical records and military hospital records are maintained separately at the National Personnel Records Center in St. Louis, MO. Outpatient records are kept with your administrative and personnel records and inpatient records are stored in the hospital records section. When requesting a search for hospital records, you'll be asked to provide your full name, military serial number, the military unit your were assigned to at the time of hospitalization, the name and location of the hospital, the dates you were hospitalized, and the reason for the hospitalization.\n\nUnder the 1996 Memorandum of Understanding between the Department of Defense and the Department of Veterans Affairs, medical records of all military members retiring or separating after May 1, 1994, are transferred directly to the VA rather than to the National Personnel Records Center. The agreement calls for the military branch to forward medical records to the Department of Veterans Affairs Records Management Center, St. Louis, MO, for those members not initiating a claim. If the member initiates a claim during out-processing, his or her medical records will be transferred directly to the VA Regional Office closest to the veteran's future address.\n\n### How the VA Rates an Injury or Disease\n\nWhen all the medical facts evidencing your claim have been gathered, the evidence is cross-referenced against _38 Code of Federal Regulations_ , part 4, Schedule for Rating Disabilities. This section of the _CFR_ lists the medical problems that VA rates as service connected.\n\nBecause this section of the 38 _CFR_ has not been updated since 1945, there are no rating descriptions for many new disabling conditions. The rating board member makes an \"analogous rating,\" which simply means he or she finds an injury or disease that is closely related. This decision-making process by a rating board member is highly subjective. A rating given under this procedure should be carefully scrutinized. You can recognize a rating for a disability that has not diagnostic code assigned to it by the last two numbers in a four digit rating code: 99.\n\nBefore you submit a claim for a particular medical problem, obtain a copy of 38 _CFR_ part 4 and compare your medical condition to the rating schedule for that condition. The schedule will tell you what medical facts must be shown before the VA will grant service connection.\n\n### Testimonial Evidence Is a Must\n\nTo make sure you have probative evidence of an injury or disease, you must know how to prepare a sworn written statement to support your claim. The VA is quick to disregard \"buddy letters\" as evidence. However, a properly executed deposition giving a detailed explanation of pertinent facts from you, members of your family, fellow employees, employer, or former military comrades is vital, _relevant_ evidence. Although they may not want to, the VA must give this type of evidence considerable weight. The VA can disregard your sworn statements only if they have hard, factual evidence to the contrary. You must learn how to construct these statements so they relate facts, not opinions or conjecture (see Donna Raz's statement in chapter 3 for a good example). The deposition must demonstrate how your medical condition affects your daily routine in the work place.\n\n### The First Weak Link: Adjudication\n\nThe working rules are translated from laws passed by Congress\u2014they are not arbitrary rules that claims examiners and rating board members can create, ignore, or change at will. However, VA ROs, contrary to former Secretary Jessie Brown's belief, violate the law frequently.\n\nVery few veterans are aware that their claim file must be reviewed and made part of the examination by the doctor. It is the duty of the senior claims examiner requesting the examination to ensure that the hospital or clinic has the file prior to the examination.\n\nTo ensure a just and fair decision on your claim, you must understand what the law requires of those rating the claim. Unless you are knowledgeable about what takes place during this process, it's very likely that your claim will be denied or, if granted, the award may be considerably less than what your disability justifies.\n\n### Willful Misconduct: Benefits Denied\n\nDetermining whether your alcohol or drug abuse was the result of willful misconduct is one of the first items a rating specialist will consider if you are claiming that your substance abuse is secondary to PTSD. The exception occurs when the VA can prove the facts were patently inconsistent with the evidence and the laws administrated by the Department of Veterans Affairs.\n\nIf your claim is denied on the basis of willful misconduct, the VA claims examiners must give you a detailed account of how they arrived at this conclusion. If their explanation does not discuss the evidence used and how they arrived at their decision, they are in violation of their own regulations. At this point, you should appeal the denial of benefits.\n\n### Presumption of Soundness: A Key Issue\n\nThe governing regulations pertaining to \"presumption of soundness\" are 38 _CFR_ \u00a73.304(b) and 38 _CFR_ \u00a73.305(b). Both regulations state that a veteran will be considered to have been in sound condition when examined, accepted, and enrolled for service, with exception made for the defects, infirmities, and disorders noted on the entrance examination. _When_ the claimant entered the service is the difference between the two regulations. Regulation 38 _CFR_ \u00a73.304 (b) addresses those claims for veterans who served after January 1, 1947, whereas 38 _CFR_ \u00a73.305(b) covers claims for those who served prior to January 1, 1947. There is one other important difference between the two regulations. Individuals serving on active duty prior to January 1, 1947, have to have served on active duty six or more months before the presumption of soundness policy applies to their claim.\n\nThe law provides that any defects, infirmities, or disorders noted during the entry examination will not be considered service-related conditions unless they are aggravated by service. Both regulations stipulate that there must be clear and unmistakable evidence that demonstrates the injury or disease existed before entry into the service. A key point to remember is that only those medical conditions recorded on the entrance examination can be considered preexisting. An isolated reference in the medical file alleging a condition existed prior to service cannot be the sole bases for denial of benefits.\n\nOn several occasions the Court of Appeals for Veterans' Claims has addressed the issues of denial of benefits based on pre-existing medical conditions and presumption of soundness. The case of _Parker v. Derwinski_ illustrates how the VA deviates from its own regulations in applying varying interpretations to the issue under consideration.\n\nMr. Parker sought to reopen his claim for disability benefits based on alleged vision deficiency caused by a hole in his left eye. The VA justified its denial of service connection because of the absence of any indication of trauma during active duty and an unsubstantiated statement by an army doctor referring to an alleged injury when Parker was nineteen years old. The VA concluded that this evidence was sufficient to rebut the presumption of soundness rule.\n\nTo prevail, the VA had to find that a preexisting injury or disease had been demonstrated by unmistakable evidence. The evidence did not support the VA position. They had two pieces of evidence: a statement from an army doctor alleging Mr. Parker injured his eye playing football at age nineteen and the fact that service medical records were silent as to an injury to the left eye. In its ruling, the court pointed out that the VA ignored Mr. Parker's sworn statement and the statements of several doctors in evidence. The court vacated the VA decision and remanded it back to the Regional Office for readjudication.\n\nThis particular veteran's claim was handicapped by the fact that he waited ten years after leaving the service before filing his original claim for service connection. His initial claim was denied, and he filed no appeal. In 1987, after waiting another ten years, he tried to reopen his claim based on new and material evidence; this claim was promptly denied. Again, he did not appeal the denial. Each time he tried to reopen his claim, he included one new letter from a friend or doctor. In 1989, after receiving another denial to reopen his claim, he filed a timely appeal. It wasn't until 1991 that his appeal made its way to the Court of Appeals for Veterans Claims.\n\nThis case is an excellent example of how a claim can be dragged out for decades. When you file your claim, it's imperative that you _not_ assume someone else (i.e., the VA) will gather the evidence to support your contentions. This is _your_ job. If you do take the easy way out and only file an application with the hope the VA will gather the evidence, you will lose.\n\n### Chronic Diseases\n\nIn order to receive service-connection benefits, a chronic disease must manifest itself to a degree of 10% or more within one year of the date of separation from the service. However, there are several exceptions to this rule. If leprosy or tuberculosis develops at a 10% level of disability within three years, or multiple sclerosis within seven years, then service connection under this rule will be granted.\n\nThe key word in this guideline is \"manifest\" and it is not synonymous with a diagnosis. If you can prove that during the year following separation the symptoms for the condition equate to a 10% rating, as stated in the Schedule for Rating Disabilities, then the disease will be considered a service-related medical problem. The forty-two diseases listed in 38 _CFR_ \u00a73.309(a) will be considered chronic under the presumptive rule:\n\n  anemia, primary\n\n  arteriosclerosis\n\n  arthritis\n\n  atrophy, progressive muscular\n\n  brain hemorrhage\n\n  brain thrombosis\n\n  bronchiectasis\n\n  calculi of the kidney, bladder, or gallbladder\n\n  cardiovascular-renal disease including hypertension\n\n  cirrhosis of liver\n\n  coccidioidomycosis\n\n  diabetes mellitus\n\n  encephalitis lethargica residuals\n\n  endocarditis (all forms of valvular heart disease)\n\n  endocrinopathies\n\n  epilepsies\n\n  hansen's disease\n\n  hodgkin's disease\n\n  leukemia\n\n  lupus erythematosus, systemic\n\n  myasthenia gravis\n\n  myelitis\n\n  myocarditis\n\n  nephritis\n\n  other organic diseases of the nervous system\n\n  osteitis deformans (Paget's disease)\n\n  osteomalacia\n\n  palsy, bulbar\n\n  paralysis agitans\n\n  psychoses\n\n  purpura idiopathic, hemorrhagic\n\n  raynaud's disease\n\n  sarcoidosis\n\n  scleroderma\n\n  sclerosis, amyotrophic lateral\n\n  sclerosis, multiple\n\n  syringomyelia\n\n  thromboangiitis obliterans (Buerger's disease)\n\n  tuberculosis, active\n\n  tumor malignant brain\n\n  tumors, malignant, or of the brain or spinal cord or peripheral nerves\n\n  ulcers, peptic (gastric or duodenal).\n\n### Tropical Diseases\n\nHistorically, the tropics have presented serious medical hazards for military members. Realizing this, Congress presumes service connection for sixteen diseases and resultant disorders. Also recognized is that treatment for many of these diseases could cause adjunct disabilities as a result of the therapy administered or as a preventive measure.\n\nHowever, Congress also wrote into the law a rebuttal rule that the VA claims examiners must satisfy before presumptive service connection can be granted. The claim can be denied if the VA can show that the individual did not serve in a locality having a high incidence rate of the disease, or when the disease did not occur continuously in the local population. It can also be denied if the claimant separated from the service after the known incubation period for the tropical disease.\n\nOne of the sixteen diseases in this presumptive group is amebiasis, a disease that is generally characterized by dysentery with diarrhea, weakness, and prostration. The disease may be in remission for several years after exposure. When the patient does experiences the first symptom, it is often diagnosed as a stomach virus and treated accordingly.\n\nWhen the patient doesn't respond to the standard treatment normally given for a stomach virus, the doctor usually orders a more serious study. At this point, the findings usually show the individual as being disabled by amoebas (microscopic parasites). The patient then learns from his doctor that his chronic condition is not found to be common where he lives, and he is also advised that the disease is not known to be a common health problem at his last duty station. He is told that it is common in the tropics\u2014such as in Southeast Asia\u2014and is acquired by ingesting food or drink containing encysted forms.\n\nThen the patient recalls that just before his discharge, he was on temporary duty for one week in Saigon, South Vietnam. He and his friends made a tour of the local sights and ate and drank in local restaurants. A claim is filed and then is promptly denied by the VA on the basis of not being a well-grounded claim. The claimant's medical records do not establish treatment for the condition on active duty. His service records show he was never stationed in any tropical regions where the disease was prevalent. His records also establish he was discharged more than two years ago.\n\nThis is a winnable claim, but the claimant must properly develop it before filing a formal application. In this situation, file an informal claim to protect the date and serve notice to the VA that a service-connected disability claim will be forthcoming. (Detailed discussion related to various types of claim actions will be discussed in chapter 6.)\n\nThree sections within 38 _CFR_ part 3 govern the awarding of service connection for one of the sixteen tropical diseases presumed to be service connected if they manifest to a degree of 10% within a year of separating from the service. The sixteen diseases are identified in 38 _CFR_ \u00a73.309(b). Next, 38 _CFR_ \u00a73.307(a)(1)(4) requires a veteran to have served at least ninety days after December 31, 1946, and that the disease manifest to a degree of 10% or more within one year from the date of separation. The third condition outlined in 38 _CFR_ \u00a73.308(b) requires that a veteran who separated before January 1, 1947, serve at least six months on active duty before becoming eligible for presumptive service connection.\n\n### Combat-Related Injuries and Diseases\n\nOfficial records or documents are not required to support your claim if you are a combat veteran and you file a claim for injuries or diseases that were incurred or aggravated _while_ engaged in wartime operations against hostile forces. The VA is required to accept satisfactory lay or other evidence as the basis for the claim as long as the evidence is consistent with the circumstances, conditions, or hardships of such service.\n\nThis rule refers to combat with the enemy. Being assigned to a combat unit conducting tactical operations during a declared wartime, or during undeclared wartime engagements such as campaigns or expeditions supporting the United States or the United Nations for diplomatic purposes may not qualify for direct service connection. But the real problem is establishing direct service connection is when the disability is PTSD. An example of a situation where direct service connection does not apply would be if a vehicle mechanic were assigned to a forward area base and, while on guard duty, he and his friend are ambushed. His friend dies right next to him. The psychological trauma of witnessing a death under these circumstances would not qualify him for direct service connection. This veteran would have to prove he witnessed this death _and_ he was on guard duty when the incident occurred.\n\nYour sworn statement of the events surrounding the injury or illness is not sufficient as evidence of combat-related injuries. However, sworn statements from members of your unit are acceptable evidence in establishing the basis of a well-grounded claim when there are no medical records to support your claim. This is an extremely important rule for veterans who served in a combat zone. Proof of receiving the Purple Heart or Combat Infantry Badge can establish your presence in a combat zone. Obtaining a copy of the official unit history during the period when you were wounded or when you became ill is considered proof your claim is combat related.\n\nIn one special case, _Collette v. Brown_ , the U.S. Court of Appeals for the Federal Circuit reversed the Court of Appeals for Veterans Claims on the issue of what a veteran must prove if he or she alleges combat injuries are the basis of a claim. The Appeal Court held that the VA must accept the veteran's lay evidence (a statement that the injury or disease occurred in combat), unless proof shows the contrary. The only element the veteran has to show is that he or she engaged in combat with the enemy. This decision is absolutely binding on all VA Regional Offices.\n\n### Chapter 5: What Have We Learned?\n\n1. Sworn statements are evidence.\n\n2. Anything that happens to you on active duty is service related unless there is a wanton disregard for safety or you are involved in a felonious criminal act.\n\n3. If your entrance physical examination is silent as to a medical condition existing prior to joining the service, then it is presumed that you were in perfect health when sworn in.\n\n4. Make certain that when you are examined for compensation benefits the VA doctor has your claim folder and has reviewed it prior to the exam.\n\n5. The VA must accept a combat veteran's explanation of the circumstances of his or her injury or disease unless solid evidence exists that rebuts the veteran's account of the facts.\n\n### Notes\n\n1. White House press release.\n\n2. VA Fact Sheet, 2002.\n\n3. \"Judge Chastises Regional Offices,\" American Legion Magazine, January\u2013February 1995.\n\n4. _Parker v. Derwinski_ , 1 F. 522 (Vet. App. 1992).\n\n5. 38 CFR, Part 4, Section 4.96.\n\n6. _Collette v. Brown_ , 82 F.3d 389 (Fed. Cir. 1996).\n\n## 6 ELEMENTS OF A PTSD CLAIM\n\nRating a claim for PTSD benefits is governed by the instructions and policy found in these regulations, manuals, case law, general counsel opinions, and medical manuals:\n\n  38 _CFR_ \u00a73.304, \u00a73.326, \u00a73.327, and \u00a73.340;\n\n  38 _CFR_ \u00a74.2, \u00a74.7, \u00a74.18, \u00a74.16 \u00a74.125, \u00a74.126, and \u00a74.129;\n\n  \"Adjudication Procedures,\" M21-1, part III, chapter 5, paragraph 5.14; and M21-1, part VI, chapter 11, paragraph 11.37;\n\n  M21-1, part VI, paragraph 1.02 and 1.09, change 116, July 16, 2004;\n\n  \"Adjudication Procedures,\" M 21-1, chapter 1, \"Physical Examinations, Social Surveys, and Field Examinations\";\n\n  _Diagnostic and Statistical Manual of Mental Disorders_ , 4th edition, revised (DSM-IV-R).\n\nIt is important to read these references before you file a claim for PTSD. The information provided can help ensure your claim is well-developed and that the VA cannot be rebut the evidence. Protect yourself by making certain the claim is fully developed before submitting it. You do not want to get caught in the time warp known as the \"Appeal Process.\"\n\nVA manual M21-1, part VI, chapter 7, \u00a77.46 opens with the following statement, \"The issue of service connection for PTSD is the sole responsibility of the rating board at the local level.\" Now, because so many ratings are being decided by single member boards, under no circumstances can you afford to submit a claim for PTSD or, for that matter, any other claim that is not well supported by irrefutable evidence.\n\nAs previously stated, the VA has downgraded the job skill level for rating board members and further complicated the matter by placing people in these positions that have no formal training in medicine or law. The bottom line is: _if they don't understand it, they will deny the claim_.\n\nIf you read the decisions published on the Internet by the Board of Veterans' Appeals, you will immediately realize local ROs are not doing their job when it comes to developing and deciding issues using their regulations and manuals. This is why it is essential that you understand what a senior claims examiner must do when the claim is being decided. If you do everything right, you will know where to address your disagreement if the claim is denied.\n\n### Components of Your PTSD Claim\n\n#### FIRST ELEMENT: EVIDENCE OF A STRESSOR\n\nHow you organize your claim for PTSD depends on the facts and circumstances surrounding the traumatic event that caused the disorder. The evidence supporting the claim will require corroborating proof your diagnosis with PTSD is current at the time of application. The medical evidence must also show that these symptoms and treatment have been continuous for at least one month. The next question to be answered is was the stressor a combat event or was a noncombat-related psychological trauma the root cause of the disorder. The last major question your claim must answer is whether the onset of PTSD was immediate or a delayed reaction to the stressor. The basis of a delayed trauma must be fully explained by the examining physician.\n\nThe VA will deny your claim speedily if it is not supported by a clear-cut medical diagnosis of an active PTSD condition. Without a clear diagnosis that makes the connection between your military experiences and the current disorder, you will not have the basis for a well-grounded claim. There is an exception to this rule in that if you have been awarded one of the medals or awards refer to in VA manual M21-1, chapter 5, paragraph 5.14(b)(1) then the VA must accept your word that the PTSD event was the result of engaging in combat against the enemy (see chapter 4 for the full text of this section).\n\nLinking the in-service incident to your current diagnosed condition has to be substantiated. The VA will not to accept a diagnosis of PTSD if the basis of the diagnosis by a mental health specialist was the veteran's uncorroborated account of the event. By producing corroborating evidence, such as sworn \"buddy statements\" or copies of military records, the examiner can state that, based on the records he reviewed, it is more likely than not that the in-service stressor occurred as you have related.\n\nThe U.S. Army Joint Services Environmental Support Group and the Commandant of the Marine Corps have archives that may provide proof of a stressor, which may in turn help a veteran's claim. When soliciting information from these agencies, make certain to provide the following information:\n\n  full name;\n\n  VA claim number, if you have one;\n\n  social security number;\n\n  military service number;\n\n  unit of assignment and its location at the time of the stressor incident (give complete military address);\n\n  type, place, and date of the specific stressor incident claimed;\n\n  the names of other individuals involved or aware of the incident;\n\n  a detailed statement of the events that took place (as best as you can remember).\n\nI would also suggest that you send your request for this information through your senator or congressional representative since government function must give legislative inquiries a priority response. Information from any federal agency is available to you under the provisions of the Freedom of Information Act.\n\n#### SECOND ELEMENT: TYPES OF STRESSORS\n\n#### Combat-Related Stress\n\nIf you claim your PTSD is based on actual combat with the enemy, the VA must accept your account of what happened as true. When the U.S. Court of Appeals for Veterans Claims decided _Zarycki v. Brown_ and _West v. Brown_ , these cases made it clear when the claimant engaged in combat with the enemy and the alleged stressors are related to such combat, no further evidentiary development will be required by the VA. Your lay testimony will be satisfactory, so long as it is consistent with the conditions of combat service or hardships of such service.\n\nFirst, let's assume your mental health specialist determined you have been suffering from PTSD for the past month. The next step is to determine if the stressors were related to combat. This can be established from service department records and sworn lay statements. The records will confirm when you arrived and departed the combat zone, the military organization to which you were assigned, your combat military occupational specialty code (MOS), and combat medals and decorations awarded. Most importantly, the VA accepts the Purple Heart, combat action ribbons, or the Combat Infantry Badge as proof of engagement with the enemy.\n\nAlthough the VA is required to accept a sworn credible statement from a combatant, some corroboration is still required in addition to the veteran's word. Part 38 of _CFR_ \u00a73.304(f) states in part, \"If the claimed stressor is related to combat, service department evidence that the veteran engaged in combat or that the veteran was awarded the Purple Heart, Combat Infantryman Badge, or similar combat citations will be accepted, in the absence of evidence to the contrary, as conclusive evidence of the claimed in-service stressor.\"\n\nWhen the claim is submitted, the evidence package attached to support your entitlement should include copies of orders and award letters, a certified copy of your DD-214, and copies of pertinent medical records and assessments that link your combat experiences to the current PTSD condition. Make the effort to get this information so you do not have to rely on the VA to try and develop corroboration that you were, in fact, in combat and that you are suffering from PTSD. There are several important reasons for taking this proactive step. First, it will shave months off the processing time required to decide your claim. Second, you will be in control, because you proved entitlement upon application. If the VA is going to deny the claim, claims examiners must prove (with hard evidence) your evidence is insufficient to grant benefits.\n\n#### Noncombat-Related Stressors\n\nThe basic requirements in establishing a claim for PTSD benefits as a noncombatant are similar to those required for a combat-related diagnosis of PTSD. First, you must medically establish that you have been currently suffering from PTSD for at least one month. The primary difference between a claim based on a combat-induced stressor and that of a noncombat stressor is that in the latter, your sworn statement alone is not sufficient to establish the basis for a claim. Your statement regarding the cause of your PTSD must be supported by credible and corroborated evidence.\n\nThis does not mean that corroboration can only come from your service records. A sworn statement from individuals knowledgeable of the events that caused your stressor is acceptable evidence. A medical assessment from a mental health specialist is another example of corroborative evidence where the causes and effects of your in-service stressor are discussed in detail. Make certain the medical assessment by your doctor clearly states the root of your problem was traced back to a stressor that occurred in the service. As long as your service records do not contradict this type of evidence, the VA cannot deny your claim on the grounds that there was no proof in your service records to support the claim.\n\nOne other important distinction made when the court ruled in _Doran V. Brown_ was that a qualifying stressor must be based on an event. Mere service in a combat zone does not qualify as a stressor for PTSD. The court held, \"It is the distressing event rather than the mere presence in a 'combat zone,' which may constitute a valid stressor for purposes of supporting a diagnosis of PTSD.\"\n\n#### The Case of Robert E. Doran\n\nThe Court of Appeals for Veterans Claims gave veterans claiming PTSD based on noncombat-related stressors a leg up when it decided Robert Doran's case, _Doran v. Brown_. The court closed the door on one of the VA's most frequently used reasons for denying claims: \"the absence of support in service records.\"\n\nMr. Doran reopened his claim for psychoneurosis to include post-traumatic stress disorder in 1976, based on new and material evidence. However, it wasn't until August 1992 that his case made its way through the system to the court for the first time. He submitted medical evidence from the doctor who treated him for his neurosis and PTSD. He was also able to obtain lay statement from individuals who were knowledgeable of the circumstances of his stressors.\n\nMr. Doran claimed that during his active service (1950\u20131952), he was struck by lightning, which caused him to become unconsciousness and require resuscitation. The second stressor Mr. Doran cited was an encounter with a snake on an infiltration course. From the records available to the court, the evidence submitted in support of his claim was considered noncumulative, relevant to, and probative to the issue at hand. In other words, the evidence that Mr. Doran submitted was new, clarified the argument under appeal, and proved his claim. On August 7, 1992, the court issued a single-judge memorandum decision vacating the Board of Veterans' Appeals' November 28, 1990, denial of service connection and remanded it back to the VA for readjudication.\n\nThe court held that the \"board focused only on evidence that tended to support the denial of service connection, ignored evidence in favor of the appellant's claim, and failed to provide reasons or basis for its findings and conclusions.\" The order further stipulated that on remand, the board would adjudicate Mr. Doran's claim for service connection for PTSD, which was raised but never addressed.\n\nOn March 5, 1993, the Board of Veterans' Appeals (BVA) issued a decision in which it determined (1) the evidence received by the VA since the BVA denied entitlement for service connection for a psychoneurosis in September 1976 was both new and material and the claim for psychoneurosis to include PTSD would be reopened, (2) Mr. Doran's psychoneurosis preexisted active service and the sound condition at enlistment had been rebutted, (3) the psychoneurosis was not aggravated during service, and (4) service connection for PTSD was not warranted. Mr. Doran filed a timely appeal to the court.\n\nOne year later, on March 8, 1994, a three-judge panel heard Mr. Doran's appeal to the court. That portion of the court's discussion and decision concerning the aggravation of a preexisting psychoneurotic condition will not be addressed here as it is not pertinent to establishing service connection for PTSD.\n\nThe court found that the BVA's treatment of Mr. Doran's claim for PTSD highlighted the inconsistencies among the statute, the regulations, and the specific provisions of VA Adjudication Manual M21-1. The court gave a new definition to that section of the manual that required \"service records must support the assertion that the veteran was subjected to a stressor of sufficient gravity to evoke symptoms (PTSD) in almost anyone.\"\n\nThe court held that under the governing statutory and regulatory provisions regarding claims for service connection for PTSD, \"if the claimed stressor is not combat-related, appellant's lay testimony regarding in service stressors is insufficient to establish the occurrence of the stressor and must be corroborated by credible supporting evidence.\" \" _There is nothing in the stature or the regulations which provides that corroboration must and can only be found in service records._ \" Those service records that are available must not contradict the appellant's lay testimony concerning his noncombat stressor. _VA manuals cannot give out instructions that are contrary to the statute._\n\nFor the second time the court vacated the VA decision to deny benefits. Mr. Doran's case was remanded with instruction to readjudicate his claim for PTSD and make certain a critical examination of all the evidence was conducted when justifying the decision. It is not known if the VA has readjudicated his claim for PTSD and granted the benefits, or whether the local rating board has continued to deny his claim, thus forcing him to continue with the appeal process.\n\n#### THIRD ELEMENT: MEDICAL EVIDENCE\n\nThe third element of your PTSD claim is actually divided into two very distinct sub-elements. It is crucial that you become knowledgeable of the requirements set forth in the regulations and manuals for how these instructions must be implemented by senior claims examiners.\n\nThe first sub-element considered by rating board members is whether the medical evidence is in compliance with VA \"Adjudication Procedures\" M21-1, part VI, \u00a77.46(b) which provides the minimum medical prerequisites to be rated as suffering from PTSD.\n\nThe second sub-element is those instructions detailed in VA medical manual IB11-56, \"Physician's Guide for Disability Evaluations Examinations.\" Compensation and Pensions examiners are required to follow the procedures outlined in this manual to ensure the claimant's disability is properly evaluated. The results of the examination are used to determine if benefits will be awarded, and, if so, how much will be granted. How well the C&P physician does his or her job in evaluating the claimant's disabilities will make or break a veteran's claim.\n\n#### Establishing Minimum Medical Standards\n\nThe adjudication manual M21-1, part VI, \u00a77.46(b) mandates the information a VA psychiatric medical examination must provide before a rating decision can be rendered. The preface to \u00a77.46 states, in part, \"[I]n order for the rating board to make that decision, the _examining psychiatrist_ must present a clear diagnosis showing a detailed history of the stressful events which are thought to have caused the condition and a full description of past and present symptoms.\" (Italics added.) The psychiatrist's report must show that the claimant's experience was an event that is outside the range of usual human experience and would be markedly distressing to almost anyone. Examples would include a serious threat to one's own life or physical well-being; a serious threat or harm to one's children, spouse, or family member; sudden destruction to one's own home or community; or witnessing another person being killed or severely maimed as a result of an accident or physical violence.\n\nNext, the psychiatrist's report must demonstrate how the claimant relives the traumatic event in _one_ of the four following ways:\n\n  recurrent and intrusive distressing recollections of the event;\n\n  recurrent distressing dreams of the event;\n\n  sudden acting or feeling as if the traumatic event were recurring (including a sense of reliving the experience, illusions, hallucinations, and dissociate flashback episodes, even those that occur upon awakening or when intoxicated);\n\n  intense psychological distress at exposure to events or occasions that symbolize or resemble an aspect of the traumatic event, including anniversaries of the trauma.\n\nThe examination must also address what the patient will do to avoid persistent stimuli associated with the trauma or note the presence of numbing of general responsiveness (not present before trauma), as indicated by at least _three_ of the seven following conditions:\n\n  efforts to avoid thoughts or feelings associated with the trauma;\n\n  efforts to avoid activities or situations that cause recollections of the trauma;\n\n  inability to recall an important aspect of the trauma (psychogenic amnesia);\n\n  markedly diminished interest in significant activities;\n\n  feeling of detachment or estrangement from others;\n\n  restricted range of affect, for example, unable to have loving feelings;\n\n  sense of foreshortened future, for example, does not expect to have a career, marriage, or children or a long life.\n\nThe evaluation must record at least _two_ of the six persistent symptoms of increased arousal (not present before the trauma), from the following group of symptoms:\n\n  difficulty falling or staying asleep;\n\n  irritability or outbursts of anger;\n\n  difficulty concentrating;\n\n  hypervigilance;\n\n  exaggerated startled response;\n\n  physiologic reactivity upon exposure to events that symbolized or resemble an aspect of the traumatic event (e.g., a woman who was raped in an elevator breaks out in a sweat when entering any elevator).\n\nThe claimant must exhibit the symptoms of the disturbance (as described in the sub-paragraphs) for at least one month. If the onset of symptoms was at least six months after the trauma, the physician must specify the reason for the delayed trauma reaction. It's important to note at this time that the VA acknowledges that it might be years after the traumatic event before the disorder surfaces. However, the examining psychiatrist should comment on the present or absence of traumatic events and their relevance to the current symptoms. The veteran must provide to the examiner relevant, specific information concerning what happened, with as much detailed information as possible. Information about the event to provide includes the time (year, month, day), geographical location (corps, province, town, or other landmark feature such as a river or mountain), and names of others who may have been involved in the incident.\n\nIn the event the claim for PTSD is based on stressors such as rape, sexual assault, or sexual harassment, the rating board expects the C&P examiner to give the claimant an extensive psychiatric examination. The examining psychiatrist will elicit a detailed history of the stressful events that caused the condition and a full description of the past and present symptoms. VA adjudicators in general seem to not believe women who claim PTSD as a result of rape, sexual harassment, or sexual assault, unless there is a police report or hospital examination recording the event.\n\nIt is important to recognize that the statutes, regulations, and case law that govern the administrative conduct of the adjudication division have absolutely no control over the operation of the hospital and its duty toward patients or the examinee. The Veterans Benefits Administration has no legal authority to direct or dictate policy governing the activities of the hospital administration that conducts the C&P evaluations. A good analogy used to explain the relationship between these two organizations is to compare them to two Smith families, unrelated by blood, living on the same block. The Mr. Smith living at one end of the block cannot tell the Mr. Smith living at the other end of the block just how to cut his grass. If he tried, the second Mr. Smith would shoot back the \"f word.\" Herein lies one of the major reasons why many veterans are so often denied benefits. The merit of the medical aspects of your claim is affected by the type of C&P examination you receive. This critical element is discussed in detail in chapter 7.\n\n### Developing a Winning Claim\n\nThe best way to maintain control throughout the entire claims process is to stay on the offensive. _Do not file_ a claim until you know that the i's have been dotted and the t's have been crossed.\n\n#### FIRST STEP\n\nThe first step in building a winning case is to file an informal claim. By doing so, the entitlement date is protected and you have up to one year to prepare your claim and obtain all evidence necessary to prove entitlement. For our purposes, let's say that you sustained a traumatic war experience that has been diagnosed as post-traumatic stress disorder. The severity of the condition is characterized by your severe withdrawal from contact with other people, your inability to maintain steady employment, and thoughts of suicide, just to name some of the symptoms. Much of your physical ability is limited because of the PTSD.\n\n#### SECOND STEP\n\nYour next step is obtaining copies of all VA regulations, manuals, and pertinent case law that may relevant for your circumstances. Copies of VA regulations (38 _CFRs_ ) can be found in your local county law library or through the VA website. Copies of U.S. Court of Appeals for Veterans Claims decisions can also be found at a library (West's Veterans Appeals Reporter). The AMIE protocol can be found in chapter 7 and chapter 4 contains excerpts of many other regulations.\n\n#### THIRD STEP\n\nThird, arrange an appointment with your personal medical specialist and request an assessment of your condition. Make sure your doctor has a copy of 38 _CFR_ , part IV regulation, \"The AMIE Protocol Exam,\" copies of any service medical records, and a copy of any lay statements from people knowledgeable of the injury, disease, or condition. If others have treated you for these, obtain copies of these records.\n\nIf you do not have a specialist to work with, your local AMA chapter or hospital referral service will provide the names of doctors whose specialty covers your condition. To save yourself grief and unnecessary expenses, first determine if the doctor will give you a written assessment of the findings. If the doctor is not willing to put all findings in writing, tell him or her, \"thank you, but without a written report, this evaluation would be totally useless for VA purposes.\"\n\n#### FOURTH STEP\n\nYour next step is to develop a lay statement to support the authenticity of your claim. Unless you file a claim within the first year following your separation, the VA has shown a history of denying benefits based solely on the fact that service medical records are silent as to your condition. All lay statements should be in the form of a sworn deposition so that these statements become evidence.\n\nRemember, a statement from family or service friends will not be accepted as valid evidence if the statement is gives a medical opinion. A lay statement can give sworn testimony as to what was observed. It is important that the veteran give a sworn detailed statement as to what occurred, where he or was treated, and how often he or she was treated prior to filing the claim. Refer back to Sergeant Raz's sworn statement in chapter 3 to assist you in collecting your thoughts. If the injury occurred during combat with the enemy, then your statement alone is sufficient to establish service incurrence.\n\n#### FIFTH STEP\n\nThe last step is to prepare all the supporting evidence to be appended to your claim. You are now ready to file your official claim and are prepared to take any scheduled C&P Exams within the next few months.\n\n### Chapter 6: What Have We Learned?\n\n1. The VA must follow all its published rules. Failure to do so is an appealable action.\n\n2. Each of the statutes, _CFRs_ , and manuals are online by linking to the VA Home Page (www.va.gov) and search under Compensation and Pension Directives.\n\n3. Your claim must have all three elements in order to qualify for service-connected PTSD.\n\n4. The U.S. Army Joint Services Environmental Support Group or the Commandant of the Marine Corps are sources of obtaining records to substantiate your claim.\n\n5. Always have your congressional representative or senator obtain these records on your behalf.\n\n6. If at all possible, have a non-VA health provider evaluate your condition before filing a formal claim.\n\n7. If you're claiming PTSD is caused by combat with the enemy, support it with a list of your medals and the corresponding citation that was given. Proving you were directly involved in combat will save you years of haggling with the VA.\n\n8. PTSD-caused noncombat stressors is more difficult to pursue because you have the burden to prove the event did, in fact, occur and is responsible for your PTSD.\n\n9. Reread the section \"Establishing Minimum Medical Standards.\" You have to meet these standards in order move your claim along.\n\n10. PTSD symptoms may first surface decades following the actual traumatic experience.\n\n11. If service connection was denied and the claim met all the standards, _don't give up_. If Sgt. David Raz gave up, he wouldn't be rated 70% today. Go back and reread chapter 3, then appeal the denial of benefits.\n\n### Notes\n\n1. The Board of Veterans Appeals' website allows you to search for decisions that may be favorable to your case: <http:\/\/www.index.va.gov\/search\/va\/index.jsp>.\n\n2. All service members (except marines) should write to: U.S. Army and Joint Services Environmental Support Group (ESG), Building 5089, Room 101, Stop #387, Fort Belvoir, VA, 22060. Inquires from marines should be addressed to Commandant, Headquarters U.S. Marines Corps, ATTN: MMRB-16, Quantico, VA, 221340001.\n\n3. _Zarycki v. Brown_ , 6 F. 91 (Vet. App. 1993); _West v. Brown_ , 7 F. 70 (Vet. App. 1994).\n\n4. Copies of the following documents can be obtained from the National Personnel Records Center (NPRC) by submitting form SF 180: orders assigning you to a combat unit (DD-214); copies of orders designating your MOS citations and orders awarding the Combat Infantry Badge; medals and awards reflecting hands-on combat duty.\n\n5. See highlights of _Doran v. Brown_ , 6 F.283 (Vet. App. 1994).\n\n6. _Doran v. Brown_ , 6 F. 283 (Vet. App. 1994).\n\n7. Ibid.\n\n8. Ibid.\n\n9. Ibid.\n\n10. Action Alert! If you were denied service connection for PTSD for any one or all of the items discussed in this portion of the chapter, it will pay to reopen your claim based on a clear and unmistakable error, provided you have not previously appealed the decision. VA Manual M21-1 \u00a77.46 states the examination will be conducted by a psychiatrist. Case law requires that no examination should take place unless the examining physician has reviewed your complete case file at the time of the examination. The C&P evaluation report must conform to \u00a77.46(b) regarding the information required by this paragraph. If the rating board did not comply with \u00a77.46(d), return the examination as incomplete to the C&P examiner when the report failed to comply with \u00a77.46(b).\n\n11. APA, DSM-IV-R.\n\n## 7 COMPENSATION AND PENSION EXAMINATIONS\n\nThe Department of Veterans Affairs Health Administration has four primary functions: first, mandatory in-patient and out-patient treatment for service-related disabilities; second, treatment of non-service-related injuries and diseases based on the availability of medical services and patients' financial resources; third, conduct Compensation and Pension (C&P) examinations; and fourth, medical research.\n\nThe purpose of this chapter is to explain what the VA must do when they conduct a C&P examination for PTSD. Most importantly, you'll learn what to do if you are not given a _thorough and contemporaneous medical examination_ , a phrase used often by the U.S. Court of Appeals for Veterans in cases involving flawed C&P examinations. Millions of veterans are cheated out of lawful benefits because of flawed C&P examinations and the flawed interpretation of these examinations by senior claims examiners who have neither medical nor legal training nor experience.\n\n### VA Health Administration\n\nBefore focusing on the VA's responsibility to administer C&P examinations, a general discussion of the overall operation of the VA Heath Administration is in order. During the latter part of 1997, VA medical service came under the scrutiny of both Congress and the public because of abusive management styles, condoning sexual harassment of patients and employees, many unnecessary deaths of veterans in VA medical facilities, and congressional pressure to cut operating costs and reduce services. During a nineteen-month period (mid-1997 to late 1998) more than three thousand medical mistakes were documented that resulted in more than seven hundred deaths.\n\nHere are some of the newspaper headlines that captured the public's attention during 1997 and weakened the trust and confidence in VA medical care:\n\n  Deaths, Federal Probe Surface at VA Centers\n\n  Disabled VA Doctor's Assignment Criticized, Managed Care?\n\n  VA Hospital Leads Nation in Settlements\n\n  VA Managers Grilled on Patient Deaths\n\n  Families Ask VA: Did You Treat Our Loved Ones Properly?\n\n  VA Focus on Fatal Errors\n\n  House to Review Harassment at VA\n\n  Under Fire, VA Administrator Leaves\n\n  Deaths Haunt Miami VA Hospital\n\n  Troubles at Small VA Hospital Mirror Agency Overall\n\n  Lawsuit Sparks New Investigation of VA Doctor\n\n### Contradicting VA Hoopla\n\nTo create a medical environment where the quality of care is influenced primarily by budget considerations is a wanton disregard of the intent of the laws authorizing medical care of veterans. Every effort of VA healthcare management strives to cut operational costs without any concern for the end effect on patients. Employing physicians who must only meet minimum standards to practice medicine has generated great savings. In the private sector, if a doctor moves from New York to Florida, he or she must be relicensed in Florida in order to practice medicine. However, not so with the VA: _U.S. Code Title 38_ requires only that a doctor graduate from a medical school approved by the Secretary (which could be anywhere in the world); complete an internship satisfactory to the Secretary (no standards prescribed in the law); and obtain a one-time license to practice medicine, surgery, or osteopathy _in any state and in any field_ , without board certification and postgraduate training in his or her field to qualify for VA employment. Over the years the press has reported cases where a state's medical board revokes a doctor's license and the doctor now practices medicine for the VA in another state without having to be relicensed.\n\nBeyond the requirements listed above, there are no other professional standards demanded of VA doctors. The plus side of requiring only minimum professional experience is the VA can pay them less than physicians who are diplomats of their medical specialty. The law also allows VA hospitals to hire doctors under fee or part-time arrangements. They are also allowed to hire interns, doctors in residency, and physician's assistants.\n\n### Using Part-time and Fee-Based Physicians\n\nMany VA hospitals hire retired doctors with only a general medical background to conduct C&P examinations and to perform other medical services that would normally require the skills of a specialist. The VAMC Bay Pines, Florida, use fee-based doctors along with physician's assistants to conduct C&P exams.\n\nThe danger in using fee-basis or part-time, nonspecialist physicians for a veteran's C&P examination is best demonstrated by a case I worked on. An eighty-year-old, retired physician conducted C&P and psychiatric examinations under a fee-based arrangement with the VA. The VA profited by this situation because many veterans with psychiatric disorders were not properly evaluated and thus were denied benefits or given benefits at a rate less than entitled. This doctor's fees were inexpensive when compared to full-time, board-certified psychiatrists whose fees are earned based on education and experience. Because of the flawed examination administered by this doctor, I spent years appealing before I recaptured my client's 100% total disability rating. This long delay was also caused by the stubbornness of the local rating boards in acknowledging that the veteran's initial psychiatric examination had been flawed.\n\nThe Fee Based Program functions very much like piecework jobs. VA doctors are paid for a set number of examinations and receive a flat-rate fee as compensation. This program is, without a doubt, one of the greatest disservices burdening veterans in their quest for legitimate claims resolution.\n\n### Veterans Healthcare Reform Act of 1996\n\nIn September 1996, the VA Health Administration announced another new healthcare strategy that discreetly further distanced itself from promises made for medical care to veterans who served their country. Veterans seeking medical care after October 1998 must apply annually for enrollment in the VA healthcare program at which time they will be assigned to one of eight priority groups. Failure to apply will make the veteran ineligible to receive medical treatment. The only exceptions to this policy are for veterans rated by the VA as having a service-connected disability of 50% or more, veterans discharged or separated from the service less than one year ago with a compensable disability, or veterans seeking care from the VA for only a service-connected disability. The group to which you are assigned will determine exactly what care you will receive; that is, veterans in one group receive the best care while veterans in another group receive no care. This categorization limits medical support to a very large, aging veteran population. Bureaucrats will never gain the kind of brownie points necessary to enjoy the rewards of a Republican politician unless they can cut the operating costs of the VA so the money can be spent on other important presidential initiatives.\n\nIronically, the House Committee on Veterans Affairs released a statement on September 30, 1996, to announce the new H.R. 3118\u2014the Veterans' Health Care Eligibility Reform Act of 1996\u2014a bill that specifically addresses the psychiatric needs of veterans. The press release says, in part:\n\nThis veterans' health care bill would help streamline and strengthen the VA health care system by expanding, updating, and simplifying many outdated rules on eligibility for VA medical care. The bill would also substantially expand veterans' eligibility to receive treatment on an outpatient basis.\n\nThis reform would, for the first time, establish a \"medical need\" as the sole test for veterans who enroll for care with VA.\n\n_H.R. 3118 recognizes the important role of VA psychiatric care; more than 50 % of all eligible veterans who suffer from severe mental illness rely on VA for care. Among its provisions, the bill would authorize VA to establish for the first time, centers for excellence combining education, research, and clinical care of the severely mentally ill._ (Italics added.)\n\nYou should be aware of another feature of this new healthcare program: the veteran's eligibility status will be evaluated annually by the primary care center to determine if he or she will remain in the current priority group, be assigned to a lower priority group, or be terminated from the program. For example, let's assume you are assigned to Priority Group 2 when first enrolled because you have a 30% service-connected disability. However, the VA rating board recently reduced your rating to 0% based on their interpretation of the results from a recent C&P examination. You are now transferred to Priority Group 6. This translates to continued priority care for your service-related condition. Any other medical problems will be treated at the lowest level of care offered.\n\nNow that we have explored the structure of the VA and the relevant, governing legislation, it's time to consider the most important component in determining your eligibility for benefits: the Compensation and Pension Examination.\n\n### Compensation and Pension Examination\n\nAlmost all service-connected benefits extended to veterans rely on some degree of disability that was incurred in the service, aggravated in the service, or resulted from treatment in a VA facility by VA medical staff. The claim process for service connection requires the VA to obtain original service medical records, medical records from VA treatment centers (hospitals or clinics), copies of private physician's records, and hospital records before deciding the merits of the claim.\n\nThe _Code of Federal Regulations_ , 38 \u00a73.326(a) states that where a reasonable probability of a valid claim is indicated, a Department of Veterans Affairs examination will be authorized. However, the regulation also permits the rating board to decide the claim based on the findings of a private physician if the report meets certain standards. The specialist's report must include clinical manifestations and substantiation of diagnosis by findings of diagnostic techniques generally accepted by the medical authorities. Pathological studies, X-rays, and laboratory tests must be part of your specialist's assessment.\n\nIf the records are not sufficient to determine the current degree of disability or if the rating board elects not to accept your specialist's report for any reason, you will be scheduled for what is known as a C&P Examination. Herein lies a major unfairness to veterans.\n\nThe level of medical expertise of the examining physician is left to the discretion of the medical service being tasked. In my thirteen years working with veterans as an advocate, I don't recall a single case where a doctor board-certified for the condition being evaluated examined the veteran. All C&P examinations were administered by non\u2013VA staff doctors who were recruited from the ranks of retired doctors. Almost all of the doctors I checked through the AMA Headquarters in Chicago were identified as family or general practitioners. A while back, using a veterans' bulletins board, I asked if it was a common practice throughout the United States for veterans to be examined by these fee-based general practitioners. The veterans' responses were not surprising: C&P examinations by nonspecialist doctors was a common practice of all VA Medical Centers.\n\nThe only time a general practitioner is not supposed to be used is when an appeal is remanded back with instructions that a board-certified specialist examine the claimant. VA manual M21-1 mandates this procedure be followed, yet the VA medical center will assign a variety of medical personnel; in many cases, the assignment goes to the same doctor who did the initial evaluation. I've yet to see a document in a claimant file noting the rating board returned the examination mandating the medical center assign a board-certified specialist.\n\nThis is why it's important for you to know the medical specialty of the physician who conducts your exam; it's the basis for an appeal if benefits are denied. The court has held many times that the VA has a duty to provide you with a thorough and contemporaneous examination. When you have a complex medical problem and a general practitioner examines you, a warning should sound for you.\n\nLet me tell you of one incident to drive home the point. I was involved in a case where the veteran was completely disabled by Buerger's Disease. A part-time, general practitioner reported he did not agree with the findings of two well-known cardiologists, and the rating board denied the veteran's claim, agreeing therefore with the least capable physician. After six years and several trips to the BVA, the claim was granted. The veteran received a check for $88,000 in back benefits. The saddest part about the veteran's story is that he died within a year of being determined 100% disabled.\n\n### The Protocol for C&P Examinations\n\nThe process starts when a claim is filed for any benefit involving a disability or disease that was incurred in the service. Your claim and medical evidence is forwarded to the adjudication division where the medical evidence attached to your claim and service medical records are reviewed. By law, the rating board can rate your claim on this evidence alone if deemed adequate for rating purposes.\n\nAs discussed previously, the first hurdle that you must jump is the claim must be recognized as _well founded_. Assuming the claim meets this standard, the law mandates that the VA has a duty to assist the veteran from this point forward. If the rating board cannot make a rating based on the medical evidence submitted with the claim and the service medical records, the rater must request a C&P examination from a VA medical facility. The veteran should be scheduled for a C&P Exam by the medical facility closest to his or her home.\n\nIt is the rating board that tells the C&P Section medically what kind of examination to conduct. The exam request is then forwarded to the local VAMC electronically through the Automated Medical Information Exchange (AMIE). In most cases the rating board orders only a general evaluation with no specific requests for the veteran to see a specialist.\n\nEven if the rating board requested a specialty examination, the hospital is not obligated to assign a specialist. Hospital personnel have complete freedom to assign whomever they choose. Through the years, I have seen them assign only fee-basis doctors or student residents from a local medical school to evaluate complex medical problems claimed by a veteran.\n\nThe _Code of Federal Regulations_ 38 \u00a74.2, \"Interpretation of an Examination Report\" requires the rating specialist to return the examination report when the diagnosis is not supported by the findings or if the examination report does not contain sufficient details. Rating boards seldom, if ever, return C&P examinations unless the veteran or the advocate has pointed out upon appeal that the examination was inadequate and thus flawed. Rating boards often deny benefits even when the exam protocol fails to follow the examining procedures outlined in the _C &P Service Clinician's Guide_ (2002). I've never seen a rating board member study the results of a C&P Exam to ensure that the report submitted was in full compliance with the VA Manuals and Regulations. One reason for this is that the rating board members' shallow level of understanding of medicine is not sufficient for them to detect the flawed nature of the examination. If you looked at the remand and reversal rate of appeals reviewed by the Board of Veterans' Appeals (between 70% and 80% nationwide) you would see that rating boards are extremely deficient in the interpretation and application of the concepts related to law and medicine.\n\n### AMIE C&P Examination Program\n\nThe AMIE system is a multipurpose electronic system created to share administrative information between medical centers, outpatient clinics, and Regional Offices. The C&P Examination Program is a program within AMIE that allows ROs to electronically transmit examination requests to medical centers. After the medical center prints the requests and schedules specific examinations, standardized examination worksheets are printed. The printed request and worksheet now become the official record between the RO and medical center.\n\nObtaining copies of the records generated during the examination process is absolutely essential should you have to go head-to-head with the VA to dispute a decision. Following, briefly, are the steps of the program:\n\n  The RO clerk electronically inputs the request for a C&P Exam to the VAMC.\n\n  Upon receipt of the request, the medical center clerk schedules the exam.\n\n  Worksheets are printed and the examination is scheduled.\n\n  Using the printed worksheets, the examiner conducts the examination.\n\n  The examiner dictates the report using the worksheet format.\n\n  The transcriber enters the results into the system and routes a printed copy to the physician for review and signature.\n\n  The examination report is electronically dispatched to the RO.\n\n  The report is printed by the RO clerk and forwarded to the rating board for action.\n\nA copy of the following records should be requested from the RO:\n\n  the initial request for C&P examination transmitted to the medical facility;\n\n  a copy of the doctor's report furnished to the RO; and\n\n  a copy of any lab reports, X-rays, and special tests performed and sent to the RO.\n\nFrom the VA medical facility performing the C&P examination, you'll need:\n\n  a copy of the worksheet used by the examining physician;\n\n  a copy of the dictated report of the examination;\n\n  a copy of all tests results, radiological reports, and lab results; and\n\n  a statement of the physician's qualifications.\n\n### AMIE's PTSD Protocol Examination\n\nThe eleven-page document shown in figure 7.1 is the actual worksheet used by C&P examiners when evaluating a veteran for PTSD. Study it! Know exactly what they should be exploring during the examination. Once the report is dictated and transmitted to the RO, immediately obtain a copy from the medical center. Compare the report to the protocol outlined to determine if every item was, in fact, covered. Look for any variation between your testimony and what was reported to the RO. Also keep track of how long the interview lasted. If the VA failed to do the job correctly, you have grounds for an appeal.\n\n**Fig. 7.1**\n\n### Completeness of Reports\n\nThe examining physician is responsible for recording a complete and detailed report, including correct diagnosis analysis of the disabling condition and a description of the effects of the disability on the veteran's ordinary activity. The correct diagnosis is of great importance. The reports should include the clinical and laboratory findings as well as all other evidence that will substantiate the diagnosis and severity of the disability.\n\nThe recorded evidence of exact findings is just as important as the actual diagnosis itself in meeting the needs of a rating board member. Similarly, the findings and clinical evidence demonstrating the severity of the disability should be reported because in most claim actions for compensation benefits, the rating board member never sees the veteran. Clarity of description, legibility of notes, accuracy of dates, identification of normal ranges for tests reported and other details are crucial.\n\nYou should obtain a copy of the C&P examination and carefully review it to make certain the C&P examiner did, in fact, convey this detailed information in the report. If any information is missing, the rating board member is required under 38 _CFR_ \u00a74.2 to return the examination report as inadequate for evaluation purposes. However, I assure you, that rating board members do not routinely return C&P examination results unless the claimant states concern that benefits were denied because the report was inadequate.\n\n### Scope of the Examination\n\nA request for a C&P examination from a rating board member will list all service connected disabilities, active, static, or alleged. It has been the practice of rating boards to ask only for a general medical examination instead of a specific examination directly related to the conditions being claimed. The manual states the \"scope of the examination should be broad enough to cover all diseases, injuries and residuals, which are alleged by the veteran.\" But examinations should be made by a specialist; that is, when a patient has a heart condition, a cardiologist, not a nurse practitioner, should conduct the examination. In _Irby v. Brown_ , the court held that the VA failed in its duty to assist by failing to order a thorough and contemporaneous psychiatric examination (specialty examination) and, as such, the report did not give a specific finding.\n\nThe way the system works, if you have a serious back disorder with radiating pain, for example, a general practitioner will most likely evaluate you instead of an orthopedist or neurologist. Although this system is an economical way of containing the cost of care, you have the right to receive evaluations from specialists. Any lesser alternative is a breach of the doctrine decreed by this country to veterans who became disabled while serving the interest of the country.\n\nIf your claim was denied based on a C&P examination administered by a fee-based general practitioner, file a Notice of Disagreement (NOD), citing that the VA failed in its duty to assist by not providing a thorough and contemporaneous examination. In addition, you should cite either _Sklar v. Brown_ or _Irby v. Brown_ , whichever is most applicable. Make certain that you state very clearly the reasons why the C&P examination was flawed and why either case law is applicable.\n\n### Severity of Disability\n\nWhen reporting the findings of a C&P examination, the _Clinician's Guide_ emphasizes that \"the essential duty of the examining physician is to record a full, clear report of the medical and industrial history, the symptomatology and physical findings.\" The purpose of the C&P report is to permit a rating board member to compare the medical findings reported by the examining physician with the disability percentage evaluations contained in the rating schedule.\n\nFor a client who intended to obtain a medical assessment from a private specialist of his choice, I always made a copy of the rating schedule for the private doctor. This provided the framework for the private specialist's report to be equated to the VA rating schedule. Thus, if the private examination report, supported by solid medical evidence, determined the veteran was 60% disabled and the VA rated the claim at 40% disabling based only on its own unsubstantiated conclusions, the veteran was on solid ground with a winnable appeal.\n\n### Review of the Claim Folder\n\nThe law requires that the veteran's claim file will be available to the C&P examiner and must be reviewed by the examiner prior to the examination. _This is one of the most important regulations that is violated and ignored by C &P physicians and rating board members alike_. The court and BVA have remanded case after case back to local ROs because they refuse to follow this policy, thus violating the veteran's right to a fair decision.\n\nIf the claim file is not present when you are examined, or if the examining physician has not read the file prior to the examination, grounds exist to appeal the denial of any benefit sought. If you do not see your claim file when being examined, ask the attending physician, \"Do you have my claim file?\" If you see the file, ask the doctor, \"Have you read the claim file in its entirety?\" If the answer is no to either question, the local RO should be advised immediately in writing that you did not receive a fair and contemporaneous examination because the examiner did not have your file present or was not aware of your case medical history. _Make a copy of your letter and send it by certified mail with a return receipt requested._\n\n### Taking the Examination\n\nOnce you are scheduled for a C&P examination, you must comply with the order and report to the medical center or clinic or risk denial of benefits. However, if the condition you are being evaluated for is cyclical\u2014having inactive and active stages\u2014and in remission at the time of your examination, you must contact the medical center and reschedule so that you can be evaluated and observed by a physician when the condition is active. Failure of the VA to evaluate your condition during an active stage is grounds for a reversal if the claim is denied.\n\nMedical examiners assigned to the C&P unit are notorious for sometimes spending only five to twenty minutes with a patient. There also exists the possibility, as previously discussed, that the evaluation will be flawed because the examination was all or in most part performed by someone other than the highest qualified doctor. The VA is notorious for substituting nurse practitioners or physician assistants to conduct C&P examinations. I speak from experience. I had an unqualified examiner whom the rating board really loved. During my C&P examination, she concluded that my heart conditions could not be considered service related because no professional medical studies had been published to substantiate my claim. She failed to read my file as she was supposed to. My file contained two board-certified cardiologists and one primary care physician statement concluding that my air force medical records did in fact show early stages of heart disease. But this evidence did not deter the rating board from denying the claim. Thus, I was forced into the appeal process, which has been ongoing for nearly two years.\n\nThere are, however, precautions that can be taken to reduce the ineffectiveness of an examination by an examiner with limited medical skills:\n\n  Take a companion with you if at all possible so that he or she can attest in a sworn statement that you were only in the examining room for _x_ number of minutes. Your companion can also testify if you were sent to other medical services at the clinic, and, if so, which ones. Require that your companion accompany you to the examining room to support your statement later of whether your claim file was available at the time of the evaluation and if the doctor was knowledgeable of your medical history.\n\n  Determine the examiner's credentials. For example, ask: \"Are you a psychiatrist or psychologist?\" Find out if the doctor is board-certified in a specialty and whether he or she is currently licensed in the state where the examination was conducted. After the examination, take the time to write the AMA national headquarters in Chicago, and obtain a background report on the doctor or doctors who examined you. All this information is necessary if you have to refute the examiner's capability to administer a thorough and contemporaneous examination.\n\n  Make a list of your medical problems and describe all the symptoms you are experiencing for each medical condition. If you are on medication, list the drug and dosage of each drug used. If the medication has resulted in side effects that are responsible for other medical problems, record this information and make it known to the examiner.\n\n  Give the examiner a copy of the credentials and medical assessment by your own private specialist. I have found that in many cases this will intimidate or influence an examiner of a lesser medical background. When it is the VA policy to use minimally qualified examiners to evaluate complex injuries or diseases, it's important that a veteran is able to counter this breach of faith.\n\n### Don't Make Assumptions\u2014The VA Is Not Always Right\n\nIf your claim is denied, you should not accept the results of a VA medical evaluation as conclusive and absolute. There is a lot riding on the results of a C&P examination and to assume that the VA administered a thorough and contemporaneous examination is a grave mistake that could deny lawful entitlements and care. Do not assume that the VA examining doctor is an expert in the field related to your disability problems. The services of a recognized private specialist should be obtained to refute the medical findings of any VA physician if benefits are denied. This is the only way to protect your interests.\n\n### Chapter 7: What Have We Learned?\n\n1. The entire VA system is undergoing a transformation that is not necessarily in the best interest of veterans. The VA Health Administration is currently engaged in an ambitious program to downsize the VA heathcare system and drastically cut costs.\n\n2. One initiative designed to cut healthcare access for veterans across the board was made part of the Veterans Healthcare Reform Act of 1996. Veterans desiring medical treatment from the VA had to enroll in VA Healthcare Program by October 1998. Now, at the time of enrollment, a veteran will be assigned into one of eight priority groups. This program requires that every veteran who is not rated at least 50% disabled must register annually. The priority group to which a veteran is assigned is not permanent and can be adjusted up or down the priority ladder.\n\n3. With the exception of veterans rated 50% or more, the only treatment available to a veteran who is rated less than 50% will be for the actual condition that is service related. The money that is saved with this policy can be redirected to other political goals of the government, totally unrelated to the needs of the citizenry.\n\n4. Primarily, it is minimally trained examiners who perform VA medical services. Basically it is cheaper than providing the kind of specialist required to properly evaluate a serious injury or complex disease. The VA also uses nonphysician medical personnel to conduct C&P examinations. Even though it is alleged that they work under the supervision of a physician, an examination by one of these caregivers cannot meet the VA standards needed for benefit claims.\n\n5. It is an absolute necessity for a veteran to consult with a private medical specialist before being examined by the VA. Your specialist must be provided with medical records of your condition, VA regulations, and manuals that are applicable to your situation. The specialist assessment of your condition must show a clear diagnosis, the limitations it imposes upon you, and how your condition relates to your contention that it is service incurred. The doctor's closing remarks should contain a statement similar to this: \"Based on a review of Mr. Smith's service medical records, my examinations and tests, it is my opinion that this condition did in fact occur while he served in the U.S. Army between 1945 and 1955. Using the standards published in 38 _CFR_ Part IV Mr. Smith is 60% disabled.\"\n\n6. By law, the VA must provide you with an examination if the medical evidence submitted with your claim and service medical records are not sufficient to rate your claim. You cannot afford to sit back and expect the VA to adjudicate your claim wisely. To be successful you must know exactly what the VA must do when you are examined. Remember 38 _CFR_ , part IV and \"AMIE PTSD Protocol Examination\" are the tools the VA examiner must work with when he or she evaluates your condition. Know what the examiner should check and if he or she doesn't follow the manual, ask why.\n\n7. Make certain when you are examined that you ask the examiner about his or her credentials and, _most important, whether he or she read and reviewed your claim file prior to the examination_. Don't let the doctor ignore the question; get an answer. Failure to do so is a not only a violation of VA policy but also a violation of federal law. Make this fact known to the examiner even if you are rebuked because of your question. You cannot afford to be passive in these matters. As soon as the examination has been concluded, go out into the waiting room and record everything you remember that was discussed and checked. Record the total time you were with the examiner. If you don't, it may take you three to five years to recover lost ground.\n\n### Notes\n\n1. These stories have been pulled from various newspapers, including the St. Petersburg Times, Tampa Tribune, New York Times, Los Angeles Times, and Chicago Tribune.\n\n2. The headlines quoted are from the St. Petersburg Times, St. Petersburg, Florida between April of 1997 and November 1997.\n\n3. U.S. Code 38, chap. 74, \u00a77401\u201310, 1991.\n\n4. VA manual M21-1, part VI, change 38, dated August 30, 1995.\n\n5. _Irby v. Brown_ , 6 F. 12 (Vet. App. 1994).\n\n6. _Sklar v. Brown_ , 5 F. 140 (Vet. App. 1993). In _Sklar v. Brown_ , the court held that if there is a diagnosis by a specialist and the only evidence against the veteran claim is a contrary opinion by a nonspecialist, then the findings of the nonspecialist should carry little weight.\n\n7. Clinician's Guide (Washington, DC: VA Health Administration, March 2002), <http:\/\/www.warms.vba.va.gov\/admin21\/guide\/cliniciansguide.doc>, 5\u201321.\n\n8. See _Dusek v. Derwinski_ , 2 F. 519 (Vet. App. 1992). In this case, the court ruled that the VA properly denied the veteran's increased benefits because he failed to report for an evaluation examination and failed to provide a good cause for his action. The court noted, \"The duty to assist is not always a one-way street.\"\n\n9. See _Addison v. Brown_ , 6 F. 405 (Vet. App. 1994). The court ruled that the VA failed in its duty to assist by relying on an examination that was performed during an inactive stage, that is, when Addison's condition was in remission. The case was remanded and returned to the BVA with instructions to examine the veteran when his condition was active.\n\n## 8 TOTAL DISABILITY BASED ON INDIVIDUAL UNEMPLOYABILITY\n\nIt is a national commitment to provide every veteran who suffered a service-related disability with compensation to offset the loss of earning capacity owing to the injury or disease. Within this pledge is a special provision whereby a disabled veteran who is unemployable owing to a disability will be compensated at a rate of 100% even if his or her disability is not totally disabling (100%).\n\nThe controlling regulations for this provision are 38 _CFR_ \u00a73.340(a)(2) and 38 _CFR_ \u00a74.16. In addition, the code regulations are backed by numerous court decisions clarifying the intent of Congress. If a veteran is unemployable, this special benefit should be one of the easiest to establish. Unfortunately, it is one of the hardest claims to establish and it is also one of the easiest benefits for the VA to terminate.\n\nIn the early 1980s there were more than 100,000 veterans in receipt of Total Disability based on Individual Unemployability (TDIU) benefits. Thus began a movement within the VA to limit entitlement for this benefit. When the Republicans moved into the White House in 1981, President Reagan and his administration moved to cut the cost of social programs. One of the biggest budget gobblers was the Veterans Administration with its compensation and healthcare programs. Thanks to Reagan's efforts to divert funds from the VA and other agencies to his missile projects, by the early 1990s, the number of veterans in receipt of TDIU was reduced to a little more than 60,000 nationwide.\n\nThe reduction process was quite simple to initiate. First, the VA ordered a series of reevaluation examinations for veterans in receipt of TDIU. Many veterans were notified that the results of their examination showed their disabilities had improved and they could return to the workplace. Next, the central office in Washington, DC, took over approval for TDIU claims and instructed local RO adjudicators to disregard some well-grounded claims for TDIU benefits.\n\nThis policy changed the way the VA handled claims for TDIU and violated the provisions of 38 U.S.C. \u00a77722 (Outreach Services) and 38 _CFR_ \u00a73.103(a) (Procedural Due Process and Appellate Rights). _Veterans Advocate_ reported that 200 veterans in receipt of TDIU benefits were terminated in November 1992 while the central office approved only five veterans for the benefits. These numbers demonstrate just how clever and ruthless an administration can be when they want to cut benefits.\n\nIn 2006 the VA made another attempt to chop billions from its budget. The agency announced that it intended to review 72,000 compensation beneficiaries, rated between 70% and 100%, who could conceivably be overrated for PTSD. VA's data shows that 72,000 was exactly the number of veterans who were rated 70% to 100% in that year. Public and congressional uproar in response to the announcement convinced VA to reconsider. The agency subsequently claimed that its announcement had been misinterpreted and that claims examiners were simply going to review the records for minor administrative errors.\n\n### TDIU Issues and the Court\n\nWhen the U.S. Court of Appeals for Veterans Claims started hearing TDIU cases, it found the adjudication process to be out of line with the laws and regulations governing TDIU issues. The _Veterans Advocate_ reported that when the court began deciding TDIU cases, many mistakes and improprieties were found. Some Regional Offices had created secret rules\u2014questionable directives that could possibly be illegal. In addition, basic rules were misinterpreted by rating boards and authorization specialists, and expert medical and vocational opinions were treated as statements of speculation. ROs were not requesting expert vocational opinions prior to deciding cases and, when professional opinions were provided by the claimant attesting to the inability to be gainfully employed, rating boards were ignoring the value of the evidence and were not giving it the same weight as symptoms noted on a C&P examination. Adjudicators also refused to discuss in detail the reasons why claimants were considered to be employable.\n\nEven with decisions of the court defining the intent of the law in TDIU cases, ROs continued to deny claims based on former standards. It seemed too many local ROs continued to do business as usual regardless of what the court instructed adjudication personnel on how to decide a TDIU claim. Two cases decided by two different panels of judges provide excellent examples of the VA doing business as usual. Chief Judge Nebeker declared _Beaty v. Brown_ and _James v. Brown_ \"nearly identical\" when writing for the court in _James v. Brown_. He wrote, \"This case presents a situation nearly identical to that in _Beaty_ in which all the evidence supports the veteran's claim for a TDIU rating. We hold that that the Board's [BVA] decision to the contrary does not have a plausible basis. Accordingly, the Board's decision is REVERSED and the matter REMANDED.\" The court has rendered numerous TDIU decisions since 1991.\n\nThat court decision was published in 1991 and Sgt. David Raz still is fighting the same prejudices that plagued his predecessors. This is why it necessary to know what the regulations are when a VA claims specialist decides a TDIU claim.\n\n### The Most Common Reasons TDIU Claims Are Denied\n\n#### MEDICAL PROBLEMS UNRELATED TO SERVICE\n\nBenefits are often denied for TDIU when the veteran has non-service-related medical problems in addition to service-connected disabilities. The VA will not make a determination based solely on the service-connected medical conditions. Claims examiners contend that the non-service-connected medical problem is a major factor in causing the veteran to be unemployable. If you are denied benefits on these grounds you should reopen your claim and argue that a determination can only be made on the service-related disabilities. Non-service-connected disabilities are irrelevant.\n\n#### WORKFORCE ATTRITION\n\nThe VA denied many claims because the veteran dropped out of the workforce because of a non-service-connected condition or because of age. Again, this is totally irrelevant. The VA can evaluate only the effects of the service-connected condition as it relates to employability. Do your records show the service-connected medical problem can stand alone as a cause of individual unemployability? If so, then you need to reopen your claim and argue that the only relevant issue is your service-connected condition.\n\n#### OPINION, NOT EVIDENCE\n\nLocal rating boards have rebutted unemployability assessments provided by the veteran's own professional vocational evaluators with their own unsubstantiated opinions. One of the first cases involving TDIU before the court was _Colvin v. Derwinski_ in which the court ruled that when all the evidence in the record is in favor of the veteran's claim, the VA cannot deny the claim based on opinion instead of on positive evidence to disprove the claim. If your claim was denied on this basis, reopen your claim citing _Colvin v. Derwinski_ and 38 _CFR_ \u00a73.105(a).\n\n#### INFORMAL AND FORMAL CLAIMS ACKNOWLEDGMENT\n\nAnother error made by the VA in handling TDIU claims is when the VA ignores the original request for this benefit. If you ever served notice to the VA, formally or informally, that you were unable to be gainfully employed because of your service-connected condition, and if the VA did not rate your request at that time, then the claim is still pending.\n\nA formal application is the submission of a completed VA Form 21-8940 (Veteran's Application for Increased Compensation Based on Unemployability). An informal claim is any form of written notice to the VA that states you were unable to work because of your service-connected condition. When the VA receives an informal request for benefits, adjudicators are required to send you the proper form to use to apply for that benefit. Failure to do so is a major error.\n\n#### RECLASSIFICATION BY CROSS TRAINING\n\nIf your records show that you are in receipt of Social Security Disability (SSD) benefits for the same disabilities for which you are service-connected and your doctor states you are not capable of working in your field of employment because of your service-connected condition, you may still be entitled to vocational rehabilitation benefits\u2014which would allow the VA to remove you from the 100% TDIU rating group. VA claims examiners will require an evaluation by a rehabilitation vocational counselor to determine if you are a candidate for possible cross training into another field. You will be required to provide a history of your education and work experiences. If you were previously denied vocational rehabilitation training, you should file a claim to reopen your claim based on new and material evidence.\n\n#### PSYCHIATRIC DISORDERS RESTRICTIONS\n\nAnother group of veterans (other than the thousands of veterans rated less than 70% who are consistently denied TDIU benefits) is those who are consistently denied a 100% disability rating owing to unemployability\u2014because they are disabled by psychiatric disorders and unable to be gainfully employed. Psychiatric disorders as identified in 38 _CFR_ \u00a74.125 are psychotic disorders, psychoneurotic disorders, organic mental disorders, and post-traumatic stress disorder. The schedule rating for each of these categories is found in 38 _CFR_ \u00a74.132.\n\nIf you are a veteran with psychiatric disabilities and you were denied benefits for TDIU, you may have strong grounds to reopen your claim under the provisions of 38 _CFR_ \u00a73.105(a) and 38 _CFR_ \u00a74.16(b) including decisions by the court in _Murincsak v. Derwinski_ and _Gleicher v. Derwinski_. These provisions legally removed the former guideline that restricted interpretation of psychiatric disability. You can reopen your claim based on a CUE (clear and unmistakable error), even after a decision becomes final. Mind you, this method of challenging the VA is not a walk in the park. The court's decisions on CUE cases have been extremely conservative, possibly because if you prevail, the VA must go back to day one when settling the claim resulting in a huge settlement.\n\n### Basic Eligibility for TDIU Benefits\n\nThe governing regulations for TDIU benefits are 38 _CFR_ \u00a73.340(a)(2) and 38 _CFR_ \u00a74.16(a)(b). A total disability rating is based on the premise that there is impairment of mind or body that is sufficient to render it impossible for the average person to follow a substantially gainful occupation. In accordance with 38 _CFR_ \u00a73.340(a)(2), entitlement to a total disability rating can be established even when the scheduler rating for the condition or conditions is less than 100% disabling.\n\nIn 38 _CFR_ \u00a74.16(a), the eligibility requirements necessary to receive TDIU benefits when rated less than 100% disabled are spelled out. Subparagraph (a) establishes where the scheduler rating is less than total (less than 100%), but _in the judgment of the rating board member_ , you are unable to maintain a substantially gainful occupation, a total disability rating may be assigned. However, to qualify under subparagraph (a) you must have one disability rated at least 60% disabling or have two or more disabilities of which one disability is rated at least 40% and the combined ratings of all disabilities equals 70%.\n\nAdjudicators are told that marginal employment shall not be considered substantially gainful employment. VA policy defines marginal income as annual earned income that does not exceed the poverty threshold for one person.\n\nSubparagraph (b) addresses the needs of a veteran who is rated less than 60% for one disability or the combined disability rating for two or more disabilities is less 70%. It was established to ensure that a veteran unable to qualify for benefits under 38 _CFR_ \u00a7 3.340(a)(2) or 38 _CFR_ \u00a74.16(a) would be considered entitled to total disability benefits if unable to be gainfully employed because of the service-connected disability.\n\nThe regulation reads, in part:\n\nIt is the established policy of the Department of Veterans Affairs that all veterans who are unable to secure and follow a substantially gainful occupation by reason of service-connected disabilities _shall be rated totally disabled_. Therefore, _rating boards should submit_ to the Director, Compensation and Pension Service, for extra-scheduler consideration on all cases of veterans who are unemployable by reason of service-connected disabilities, but who fail to meet the percentage standards set forth in paragraph (a) of the section. (Italics added.)\n\nThe reduction of veterans on the roll or being approved for TDIU benefits stems from several factors. First, 38 _CFR_ \u00a74.16(a) grants rating boards considerable leeway in deciding whether the claimant is capable of obtaining and maintaining gainful employment. Allied to this factor of personal judgment, rating board members are not specially trained to equate the medical limitations of the claimant disability to the ability to be gainfully employed.\n\nThe court has rendered many decisions addressing the issue of TDIU. In _Gleicher v. Derwinski_ , the court held that the VA improperly denied a TDIU claim because it based its denial of benefits on its own unsubstantiated vocational opinion. If TDIU benefits were denied when there was sufficient evidence you were not capable of being gainfully employed and the VA examiners had no evidence to refute your claim other than their personal judgment, you should reopen your claim based on a clear and unmistakable error.\n\nAnother reason many veterans experience so much difficulty in gaining approved for TDIU benefits is because ROs seldom follow the intent of the regulation. The veteran is told the requirements set forth in 38 _CFR_ \u00a74.18(a) have not been met, therefore the claim is denied. The denial letter cites the percent of disability required to be eligible for TDIU benefits. The veteran is not told TDIU benefits are possible if the service-connected condition, regardless of the rating, is responsible for his or her inability to be gainfully employed. For example, if you are unemployable owing to a psychological disability rated less than 70%, file a claim to reopen your case based on a clear and unmistakable error (38 _CFR_ \u00a73.105(a.) The basis of the reopened claim would be that the RO of original jurisdiction failed to comply with the provisions of 38 _CFR_ \u00a74.16(b) and forward your claim on to the central office in Washington, DC.\n\nFew veterans and service officers know what the regulations mandate. For example, when initiating a claim, most claimants do not provide the type of evidence that establishes the service-connected condition as responsible for causing the inability to be gainfully employed. Veterans and many advocates do not know that VA doctors, when performing a C&P examination, are to state in their evaluation if the service-connected condition is responsible for the individual's unemployability. The rating board members know this statement should be part of the evaluation report. However, its absence is ignored and so is the provision of 38 _CFR_ \u00a74.2 (Interpretation of Examination Reports) that directs a rating board to return the examination to the medical center as being inadequate for evaluation purposes.\n\nThe only saving grace is that now the BVA is examining each appeal under guidelines provided by the many court decisions. Unfortunately, the veteran must still wait several years to be granted TDIU benefits while the appeal makes its way through the process.\n\n_Gleicher v. Derwinski_ also addressed what the VA was required to do when a veteran with a scheduler rating of 70% was unemployable solely because of his psychological problem. The VA contention was a veteran with only a service-connected psychological disability was not entitled to TDIU benefits.\n\nWriting for the court, Judge Ronald M. Holdaway held that, \"A veteran with a 70% disability rating is entitled to an 'extra scheduler' total disability rating if he is unable to secure or follow a substantially gainful occupation as a result of the disability.\" The court further stated that in this case \"the BVA failed to consider the 1988 VA psychiatric examination report and the 1988 VA social and industrial survey, both of which concluded that the appellant was incapable of securing or maintaining employment.\" There was no evidence of record to the contrary to rebut the appellant's claim.\n\n### Proving a Claim for TDIU Benefits\n\nAlways remember that you, the claimant, has the burden to prove your claim. Technically, if your evidence establishes a well-grounded basis for the claim, the VA is required to assist you in fully developing the claim before it is decided. In one of the earliest cases decided before the court, _Colvin v. Derwinski_ , the court spelled out the specifics of the VA's duty toward the claimant.\n\nThe court ruled that the VA's duty was: (1) to obtain the evidence, (2) to consider the evidence, (3) to evaluate the claim, and (4) to explain, if the claim is denied, how examiners arrived at that decision in light of expert medical and vocational opinions from professional witnesses. They must tell you exactly how they justified the denial of benefits. Regional offices continue to this day to deny claims in spite of the ruling in this case and others like it. For this reason alone, _you have to depend only on yourself to put together a winning claim_. Don't make the mistake of thinking that VA personnel will do all the digging and searching for evidence that will grant you the benefits claimed.\n\nProving up your claim is not a difficult task if you understand exactly what kind of evidence is needed to get the job done. You must know the rules the VA is supposed to follow. TDIU claims must prove that your service-connected disability alone is responsible for your inability to be gainfully employed. There are five sources of evidence to include in a well-documented TDIU claim: expert medical proof, vocational proof, a sworn statement by the veteran, sworn statements from former or prospective employers who rejected your application for employment because of your service-connected disability, and a copy of your Social Security award letter which confirms Social Security disability benefits based on your service-connected medical problems. Taking the time to put together a well-documented claim will avoid the pitfalls most veterans face.\n\n#### MEDICAL EVIDENCE\n\nFirst, obtain an expert medical assessment from your own physician. The doctor must explain in writing exactly how your service-connected disability affects your ability to be gainfully employed. Should you have an additional disability that is not service-connected, explain to the doctor that the report must address only your service-connected condition. However, if your doctor could make a statement to the effect that your non-service-connected conditions would not or could not be the cause of your unemployability, have the doctor do so. Do not let the doctor lump all your disabilities together and give a generalized report. The danger here is, if the report offers the slightest doubt that your service-connected condition is not responsible for your unemployability, the VA will deny the claim.\n\nIf you are being treated solely by a VA medical service for your service-connected condition, it is doubtful that the VA physician will put together the kind of letter you need. He or she will tell you verbally that because of your service-connected condition, you can't work, but the physician is often extremely reluctant to put it in writing. In my own experience with VA doctors, I was never successful when trying to obtain the type of letter necessary to prove unemployability on behalf of my clients.\n\nOne last thought on medical evidence: You cannot afford _not_ to have an assessment by a specialist in the field related to your disability. When you see the specialist, state up front that you want an evaluation concerning your service-connected condition and its effects on your ability to be gainfully employed. The doctor must understand that this report _must_ be in writing. If the doctor resists providing a written report, don't invest anymore of your money or time. Find someone who will meet your needs. Remember that you have a lot riding on this claim: more than $2,000 per month, for life. The cost of a first-rate medical work-up should not deter you from obtaining the examination.\n\nOnce you have located a specialist, your next job is to explain what you need in order to qualify for TDIU benefits. You must provide the following tools for the specialist and your physician:\n\n  copy of your complete VA claim file;\n\n  copies of Social Security Disability (SSD) decisions and the evidence used to grant SSD benefits (detailed copies of Social Security records can be obtained under the Freedom of Information Act via your congressional representative's office);\n\n  copies of all medical evidence from private physicians;\n\n  copies of any sworn statements from former employers that terminated your employment because of your disability;\n\n  copy of any special educational training you may have completed;\n\n  copies of high school, college, or technical school transcripts;\n\n  copy of the pertinent paragraphs from 38 _CFR_ , part 3 and 4 that apply to your disability.\n\n#### VOCATIONAL EVIDENCE\n\nThe second most important step in supporting your claim is an evaluation by a professional counselor. Developing an unimpeachable source of evidence requires the claimant to be dogged in the search for a professional vocational counselor. The counselor's credentials are critical to your case.\n\nA professional vocational counselor's educational background will range from a bachelor's degree to a doctoral degree (PhD) in vocational counseling. If you have a choice, select an individual with a PhD in vocational counseling. Remember, you are going to battle against any negative conclusions drawn by a rating board member who has had no special training to qualify as a vocational expert. If you are forced to go to appeal, the higher the credentials of your vocational counselor, the greater your chances are of having the decision reversed on appeal.\n\nIf you have no idea where to start your search for a vocational specialist, begin with the closest legal service that helps indigent individuals in civil cases. Your local bar association should have the contact information for this legal service. These services do a great deal of litigating on behalf of clients who have been denied Social Security disability benefits. They will be able to put you in touch with a vocational specialist who is experienced in dealing with the government. The upside of hiring a specialist with this kind of experience is that they know how to prepare a written evaluation that has all the right buzz words, and they understand the bureaucratic obstacles you are trying to hurdle over.\n\nIt's important that the vocational specialist understand what the VA means by \"marginal employment,\" as opposed to \"substantial, gainful employment.\" This meaning is spelled out in 38 _CFR_ \u00a74.16(a), which explains the basic rules for granting TDIU benefits to a veteran with a disability rating less than 70%. Make sure the vocational specialist is aware that age and non-service-connected disabilities are irrelevant in the decision-making process. The only issue that the vocational specialist must address is whether your service-connected disabilities alone would prevent you from obtaining gainful employment.\n\n#### CLAIMANT'S SWORN STATEMENT AS EVIDENCE\n\nPreparing a sworn statement is not as complicated as it sounds. The statement will include your name, VA claim number or Social Security number, current address, and a sentence to the effect that the statement is true and correct to the best of your knowledge. It should be signed before a notary public. This will ensure that your statement is in compliance with 38 _CFR_ \u00a73.200(b). The format of the letter should be a series of direct statements of fact such as is seen in figure 8.1. For another good example, refer back to chapter 3 for Sergeant Raz's sworn statement.\n\n**Fig. 8.1**\n\n#### LAY STATEMENT IN SUPPORT OF YOUR CLAIM\n\nA lay statement is support for your claim from anyone who can provide factual information as to how your service-connected disability affects your ability to work. Lay evidence can be from your spouse, family members, close friends, former employers, or prospective employers who declined to hire you because of your service-connected disability. The statement should be sworn if possible; a sworn statement is preferable and in compliance with 38 _CFR_ \u00a73.200(b).\n\nThe VA has often dismissively referred to this type of statement as a \"buddy letter,\" but it should be considered valid evidence. Until Congress gave every veteran a source (Court of Appeals for Veterans Claims) to challenge BVA decisions when disputed, lay evidence was seldom given any evidentiary weight. However, rating boards cannot ignore this evidence or fail to give it proper evidentiary weight. Two cases decided early on by the court, _Colvin v. Derwinski_ and _Cartright v. Derwinski_ , held that lay evidence, in and of itself, might warrant service connection.\n\nBoth 38 _CFR_ \u00a73.200(b) and 38 _CFR_ \u00a73.307(b) dictate how the VA will treat lay evidence in determining the granting of specific benefits sought. First, 38 _CFR_ \u00a73.200(b) instructs: \"All written testimony submitted by the claimant or in his or her behalf for the purpose of establishing a claim for service connection will be certified or under oath or affirmation.\" In referring to the adjudication process of deciding a claim, 38 _CFR_ \u00a73.307(b) states, in part, \"The factual basis may be established by medical evidence, competent lay evidence or both.\"\n\nAs previously discussed, a lay statement under oath should be simple and direct, stating only facts, not opinions or conjecture. The same basic format that was outlined for a veteran's statement (fig. 8.1) can be adapted for use by anyone.\n\n#### SOCIAL SECURITY DISABILITY EVIDENCE\n\nA veteran that has been granted Social Security Disability benefits for the same condition that is service connected can thank the court for giving them a major leg up toward being granted TDIU benefits. The first four cases concerning the impact of SSD benefits ( _Collier v. Derwinski, Murincsak v. Derwinski, Brown v. Derwinski_ , and _Shoemaker v. Derwinski_ ) clearly defined what the VA must consider and do when a veteran is in receipt of SSD benefits for the same reason the veteran is receiving compensation benefits. Despite these case rulings, the VA continued to dismiss the importance of a claimant in receipt of SSD benefits and these subsequent cases also dealt with this issue: _Masors v. Derwinski, Clarkson v. Brown, Beaty v. Brown_ , and _James v. Brown_.\n\nThese are the basic ground rules the VA must follow when adjudicating a claim where the veteran is in receipt of SSD benefits for the same condition or conditions that granted him or her VA compensation benefits:\n\n  A SSD claim cannot be ignored and, to the extent its conclusions are not accepted, the reasons should be clearly stated.\n\n  A determination by the Social Security Administration (SSA) that a claimant is entitled to disability benefits is relevant to a VA determination of the severity of claimant's disability. Further, the court held that SSA disability ratings are not outdated for the purpose of making a determination, since SSA continuously reviews eligibility for disability benefits.\n\n  When reviewing a well-grounded claim for TDIU benefits where the claimant is recognized as unemployable by SSA, the VA must consider entitlement under 38 _CFR_ \u00a74.16(b), even if the condition is rated less than those standards required by 38 _CFR_ \u00a74.16(a). In theory if you had a 10% disability, and Social Security determined you were unemployable, the VA would be required to grant the benefit or prove your employability, both vocationally and medically.\n\n  The VA must obtain and review the claimant's Social Security records when put on notice that he or she is receiving SSD benefits.\n\n  The VA cannot justify the denial of TDIU benefits with the argument that SSA disability determinations are based on a different set of standards. If the VA rejects the findings of your SSA disability determination, claims examiners must state in detail exactly what evidence they have that is superior to the SSA decision granting benefits and why the SSA determination is not relevant in your case.\n\nVA adjudicators lack the initiative to request SSA records and associated medical records so this point is still a losing battle for most veterans. In spite of all the cases decided by the CAVC on this issue and all the remands by the BVA, the ROs are still not doing their job.\n\nLet me give you example of a BVA case that illustrates this issue perfectly. Appeals from 1994 and 1995 are available on the BVA website. A search under the heading TDIU brought up more than thirty decisions where TDIU was the issue on the appeal. BVA case number 950971, docket number 93-15-575 dealt with a claim for TDIU benefits that had to be remanded back to the RO. The RO failed to obtain the SSA records when the record showed the veteran had made it known to the RO that he was getting SSD for the same reasons he was receiving service-connected benefits. The case goes back to 1993, when the case appeared before the BVA. The BVA remanded the case back to the RO in 1995. Another six months to a year are allowed before the case is recycled and decided. If the RO still denies TDIU, it will be another couple of years before it is reviewed once again by the BVA. This is a very long time for a veteran to live with a greatly reduced income. There is no justifiable excuse for the RO to fail in its duty to assist by not obtaining those records the moment the claims examiner was advised SSA records would be relevant to this claim.\n\nIf you were denied TDIU benefits in the past because the VA did not consider the records and decision of SSA, you should reopen your claim based on a clear and unmistakable error under 38 _CFR_ \u00a73.105(a.) If you were told that you did not qualify for TDIU benefits because your service-connected disabilities did not qualify under 38 _CFR_ \u00a74.16(a), you'll want to reopen your claim if you were in receipt of SSA benefits for the same reasons you were granted service connection.\n\n### Applying for TDIU Benefits\n\n#### AN INFORMAL CLAIM\n\nThe first step in applying for TDIU benefits is to file an informal claim. This action will:\n\n  protect your entitlement date;\n\n  give you one year from the date the VA furnishes the formal application form (VA form 21-8490) to submit your claim for consideration;\n\n  enable you to get a complete copy of your VA claim file.\n\nArrange for a medical and vocational assessment to document your claim that you are too disabled to work. Obtain sworn lay statements, including sworn testimony by you, describing how your service-connected condition has prevented you from maintaining gainful employment. Obtain copies of all your Social Security Disability records to support your claim that you are unemployable. Apply for SSD benefits, if you haven't already done so, and have your claim for SSA benefits decided.\n\nAn informal claim is a written notification to the VA that you _intend_ to file a claim for Total Disability based on Individual Unemployability. The statement should be short and to the point. For example:\n\nPlease accept this action as my informal notice to apply for Total Disability Benefits Due to Individual Unemployability. My service-connected PTSD condition has made it impossible for me to continue to be gainfully employed. A formal application will be submitted once I receive the formal application and I have obtained the medical and vocational evidence necessary to grant TDIU.\n\nI also request that I be furnished a complete copy of my VA claim file at the earliest possible date so it will be available to my doctors and vocational counselors for review along with an official application form.\n\n#### FORMAL APPLICATION FOR TDIU BENEFITS\n\nUnless you have a completely documented claim, ready to be filed, do not file an original application for TDIU benefits initially. _It's a major tactical error to go to the VA with a claim that is not well documented and supported by solid evidence._ You should take charge of putting together your own claim. If you err at this point, it's all over but for the cussing. Your only recourse will be to appeal the denial of benefits. Two to three years later, your case will be reviewed by the BVA and the only issue on appeal is, was your claim a well-grounded claim? If claims reviewers affirm the decision of the RO, your last recourse is the Court of Appeals for Veterans Claims. Let's assume that nearly five years have gone by and the court reverses the BVA. Now we go all the way back to square one and determine if your service-connected condition will entitle you to TDUI benefits based on the evidence in the record at the time of application.\n\nTo claim TDIU benefits you must initiate a claim action on VA Form 21-8940 and submit it to the RO that services the area where you live. Unless you have obtained all of the evidence previously discussed, _do not_ call the VA on their toll free number (1800-827-1000) and request the official application form. Make them respond to your written informal claim notice and provide a formal application by mail.\n\nIf the VA has not provided you with the application (as they are required to do by law), send them a certified letter telling them this is your second request for a TDIU application. Attach a copy of the first request putting them on notice that you are applying for TDIU benefits. _Don't do business with the VA by telephone_ ; any transactions you conclude with them cannot be verified or proven. It's your word against theirs, and you won't get the benefit of doubt. Just remember what the gold miners in California had to deal with when they turned over their laundry to the Chinese shop owners and then tried to claim their laundry without a ticket: \"No tickee, no laundry, so sorry.\" Without proof, you're in the same boat as the miners of 1849.\n\nThe application form is fairly straightforward and, after requesting some basic information about the claimant, is divided into four specialized sections: Section I, Disability and Medical Treatment; Section II, Employment History; Section III, Schooling and Training; and Section IV, Authorizations, Certifications and Signature releases.\n\n#### Basic Identification of Claimant\n\nThe first six items on the application are for identification only. It asks for both your Social Security number and VA claim number. If your original claim for benefits dates back to the mid 1970s or later, most likely the VA assigned your Social Security number as your VA claim number. Prior to the changeover, a VA claim number was an eight-digit number preceded by \"C,\" (e.g., C-00-000-000). If you do not know your \"C\u2013Number,\" you can obtain it by contacting the nearest local RO. Its identification system can cross-check your name, date of birth, military serial number, or Social Security number to find out your C-Number.\n\n#### Section I: Disability and Medical Treatment\n\nItems 7 through 12 require a considerable amount of information to support your claim for TDIU benefits, with very little space provided to do so. The best way to complete the form under these circumstances is to enter into each box, \"See attachment for details.\" When recording the information on the attachment for item 7, for example, write, \"Item 7: Lower back injury, gunshot wound to left forearm and right calf; residual damage to liver owing to infectious hepatitis.\"\n\nItems 9 and 10 should be combined when there is more than one doctor involved. For example: Dr. Robert Smith, 1000 S. Turner Street, Winston, OH, 55441, April 2, 1986, to date; Dr. Elizabeth D. Green, 2230 N. Main Street, Winston, OH, 55442, July 30, 1990, to date; Dr. Harry G. Hart, 315 Harrison Blvd., Winston, OH, 55446, September 3, 1993, to date. You should follow the same format for Items 11 and 12 if you had multiple hospitalizations related to your service-connected disabilities. Make certain that on each attached page you print your name, VA claim number, and Social Security number. You should also note how many pages are attached and what this particular page number is. For example, \"Page 1 of 3 pages,\" \"Page 2 of 3 pages,\" and so on.\n\n#### Section II: Employment Statement\n\nThe dates for Items 13, 14, and 15 could be one date or could reflect three entirely different dates, depending on the circumstances involved. For example, if the claimant was doing physical work and his shoulder, arms, and right leg caused him extreme pain to the extent that he could no longer perform his assigned tasks, this would the be the date his disabilities affected his full-time employment. He may have stayed out of work on sick leave for a month, returned to work, then found he could no longer perform any of his duties. His employer terminated his employment effective the end of the following month because he was using up his accrued vacation time.\n\nItem 17 asks for your employment history for the past five years. Normally this is a one- or two-line entry if you have worked for the same employer for at least seven years before changing jobs.\n\nItems 18 through 21 raise several questions that need to be discussed. Item 18 would normally be checked \"no.\" Item 20 would normally be checked \"no.\" However, if in fact you were in receipt of federal workman's compensation benefits (see 38 _CFR_ \u00a73.958) the law prohibits concurrent payments of VA Compensation benefits and Workmen's Compensation benefits after September 13, 1960. The VA requires a detailed explanation in Item 24 as to why your disability forced you to leave your job. The space provided to answer Item 24 is inadequate for a detailed explanation. Note in Item 24, \"See attachment, page 2 for explanation of reasons disability caused me to become unemployable.\"\n\n#### Section III: Schooling and Other Training\n\nItems 22 through 24 provide the VA with some insight into your potential for employment in another occupation field. Veterans with a college education or other special training are usually considered employable. This is why it's essential that you retain the services of a well-qualified vocational specialist who will weigh all factors of your background before making a determination as to your employability.\n\nI have seen the VA deny TDIU benefits based on the assumption the claimant's college degree made him employable. This one fact rose above the findings of his doctor, who explained why he was unable to be gainfully employed. Remember that these rating specialists do not have any special medical, legal, or vocational training. What they _do_ have is the authority to use their own judgment. Your claim has to be well supported by expert evidence or it's going to be years before you see TDIU benefits.\n\n#### Section IV: Authorization and Certification\n\nThe authorization section of VA Form 21-8940 requires that the claimant grant permission for the VA to solicit information from the doctors, surgeons, hospitals, or dentists who have treated you. You are giving claims examiners permission to obtain _any background information_ these sources may have. When you sign the application, you are automatically granting them the right to contact anyone to obtain information about you.\n\nThe certification portion of Section IV is where you affirm that all statements made are true and correct to the best of your knowledge. Further, it states that you certify that your service-connected disabilities do not permit you to follow _any_ substantially gainful occupation. I believe the only reason the VA highlighted the word \"any\" is to intimidate the claimant. It reminds one of the early days of the Reagan administration when thousands of individuals were taken off the Social Security Disability rolls under the assumption that \"everyone can do something.\" However, the Office of Administrative Law Judges (OALJ) reinstated benefits to these individuals as fast as the cases could be processed. OALJ did not uphold the concept that everybody can be substantially and gainfully employed.\n\n### Chapter 8: What Have We Learned?\n\nCongress introduced the Total Disability based on Individual Unemployability (TDIU) benefit program for veterans who were not rated 100% disabled, but who were unable to be gainfully employed because of a service-connected injury, disease, or condition. The program recognized that such veterans should not be deprived of the basic standards of life. It was Congress's intent that the VA should administer this new law under liberal guidelines.\n\nHowever, this is an entitlement that disabled veterans have been refused more often than any other. The reason is quite simple: it takes a big chunk of money out of the VA budget. During the early 1980s, primarily under the Reagan\/Bush administration, more than 40,000 veterans were removed from the TDIU rolls, not because they suddenly became employable, but because they lacked the knowledge and tools to counter the decision. In one decade, the VA reduced its budgetary requirements by $80 million. This amount does not include the savings generated by denying thousands of claims for TDIU benefits between 1980 and 1990.\n\nDuring this timeframe the VA was acting as its counsel, jury, and, judge for all claims. The Board of Veterans Appeals seldom reversed the rulings of ROs. Only 15% of the cases appearing before the BVA were reversed or remanded. Today, nearly 80% of the appeals being reviewed by the BVA are remanded or reversed.\n\nThe shift toward making the ROs do their jobs correctly was not because of a change in policy, training, or a deep commitment to do what was right. The driving force behind this unprecedented turnabout was the decisions of the Court of Appeals for Veterans Claims. The court has been reviewing appeals cases\u2014many of which have dealt with TDIU benefits\u2014since February 1990 and only now is the BVA reviewing appeals with an eye on the intent of Congress.\n\nThere are eight court decisions cases cited in this chapter that specifically address the issue of why a veteran in receipt of Social Security Disability (SSD) benefits should not arbitrarily be denied compensation benefits. These benefits will remain some of the most difficult to obtain until Regional Offices apply the intent of the law to the facts of the case and realize that they are not a HMO measuring the amount of black ink on a profit and loss statement.\n\nIf you are a veteran who has been denied TDIU benefits or who was removed from the rolls, this is a very important chapter for you. The eligibility requirements are quite broad and uncomplicated. If you are unemployable because of a service-related disability, you are entitled to a special rating of 100%. The key is that you must be unemployable because of your _service-connected disability_. The VA will put considerable effort into abutting your claim, based on the premise that your unemployability can be traced to reasons unrelated to your service-connected disabilities. ROs are notorious for denying TDIU claims based on their own unsubstantiated conclusions.\n\n### Notes\n\n1. Ronald Abrams, \"Court Decision Exposes Erroneous VA Adjudication of Individual Unemployability Claims,\" _Veterans Advocate_ 4, no.3 (November 1992): 17\u201324.\n\n2. _Beaty v. Brown_ , 6 F. 532 (Vet. App. 1994); and _James v. Brown_ , 7 F. 495 (Vet. App. 1995).\n\n3. _Colvin v. Derwinski_ , 1 F. 171 (Vet. App. 1991).\n\n4. _Murincsak v. Derwinski_ , 2 F. 363 (Vet. App. 1992); and _Gleicher v. Derwinski_ , 2 F. 26 (Vet. App. 1991).\n\n5. _Colvin v. Derwinski_ , 1 F. 171 (Vet. App. 1991).\n\n6. _Colvin v. Derwinski_ , 1 F. 171 (Vet. App. 1991); and _Cartright v. Derwinski_ , 2 F. 24 (Vet. App. 1991).\n\n7. _Collier v. Derwinski_ , 1 F. 413 (Vet. App. 1991), _Murincsak v. Derwinski_ , 2 F. 363 (Vet. App. 1992), _Brown v. Derwinski_ , 2 F. 444 (Vet. App. 1992), and _Shoemaker v. Derwinski_ , 3 F. 248 (Vet. App. 1992).\n\n8. _Masors v. Derwinski_ , 2 F. 181 (Vet. App. 1992), _Clarkson v. Brown_ , 4 F. 565 (Vet. App. 1993), _Beaty v. Brown_ , 6 F. 532 (Vet. App. 1994), _James v. Brown_ , 7 F. 495 (Vet. App. 1995).\n\n## 9 PTSD COURT CASES\n\n### Prove Your Case and File Your Claim _or_  \nYou Will Be Forced to Defend It on Appeal\n\nKnowing what evidence the VA requires is an absolute must if you expect to submit a winning claim. You cannot win if you\n\n  cannot protect your interest in the decision-making process because you do not know what the regulations dictate.\n\n  are unable to submit a well-supported claim that is supported by evidence.\n\n  do not know how to go about collecting this evidence. You cannot win if you do not know why the VA denies valid claims.\n\n  assume the VA is an all-knowing agency and that you can rely upon its skills to adjudicate the facts of the case accurately and fairly.\n\nWhat do the regulations say about your claim before benefits can be granted? To answer that question you need to obtain a copy of the pertinent section of 38 _CFR_ , parts III and IV. If you have access to a computer, search the Internet for the information. If you need to make copies of pertinent sections of the regulations, look to the local law library in your county, your VA Regional Office reading room, or a university library designated as a federal depository.\n\nWhen you are ready, make certain that you know how to put your claim together and the method used in filing. A VA claims examiner will deny your claim if the medical evidence is not persuasive. Organize all your evidence in a logical order and provide a table of exhibits so that you can expect the examiner to rule in your favor. Take no short cuts. Include all required evidence in an orderly manner. Remember that you hold the burden of proving the claim. If you can't do that, for the next two or three years you will argue only one issue: was the evidence sufficient to grant the benefit?\n\nThe assembly of your claim and all its supporting evidence must follow a logical path. Each piece of evidence must link to the next and so on, otherwise you put your claim for benefits in harm's way. Identify every piece of paper that is attached to the claim. Number your pages. Mail your claim and documentation by Certified Mail with Return Receipt Requested. If you deliver your claim in person to the VA and you have no proof of exactly what you provided, it could cost you your benefits. Lost documents, forms, statements, and evidence will hurt only you.\n\nYou are a veteran. You have the GI Bill of Rights to protect you. Seek out competent advocates to help you through every step and who are knowledgeable about the VA's obligations. Find your evidence, build your case, back up your claim, and follow through until you receive your benefits.\n\nIf you need more detailed information regarding how to support your claim, knowing the VA requirements, finding resources, gathering evidence, and successfully filing your claim, please read my previous book, _The Veterans Survival Guide: How to File and Collect on VA Claims._ Also, you may wish to visit these websites:\n\n**American Medical Association physician member locator:** <http:\/\/webapps.amaassn.org\/doctorfinder\/home.html>\n\n**Board of Veterans' Appeals (BVA):** <http:\/\/www.va.gov\/vbs\/bva\/>\n\n**Compensation and Pension (C &P) Examinations:** <http:\/\/www1.va.gov\/vhapublications\/ViewPublication.asp?pub_ID=1400>\n\n**_C &P Service Clinician's Guide_:** <http:\/\/www.warms.vba.va.gov\/admin21\/guide\/cliniciansguide.doc>\n\n**_The Code of Federal Regulations_ :** <http:\/\/www.gpoaccess.gov\/cfr\/index.html>\n\n**_Federal Register_ :** <http:\/\/www.gpoaccess.gov\/fr\/index.html>\n\n**InfoSpace:** <http:\/\/www.infospace.com\/>\n\n**National Medical Association (African American) physician locator:** <http:\/\/0081be6.netsolhost.com\/index.asp>\n\n**National Personnel Records Center:** <http:\/\/www.archives.gov\/st-louis\/militarypersonnel\/>\n\n**U.S. Army military history:** <http:\/\/www.armyhistory.org\/armyhistorical.aspx?pgID=889>\n\n**U.S. Armed Services Center for Research of Unit Records:** <http:\/\/acronyms.thefreedictionary.com\/United+States+Armed+Services+Center+for+Research+of+Unit+Records>\n\n**_U.S. Code_ :** <http:\/\/www.gpoaccess.gov\/uscode\/index.html>\n\n**U.S. Court of Appeals for the Federal Circuit:** <http:\/\/fedcir.gov\/>\n\n**U.S. Court of Appeals for Veterans Claims:** <http:\/\/www.vetapp.uscourts.gov\/>\n\n**U.S. Department of Veterans Affairs, Veterans Benefits Administration:** <http:\/\/www.vba.va.gov\/>\n\n**U.S. National Archives and Records Administration:** <http:\/\/www.gpoaccess.gov\/nara\/index.html>\n\n### The Court at Work Protecting PTSD Claims\n\nThe U.S. Court of Appeals for the Federal Circuit and the U.S. Court of Appeals for Veterans Claims have addressed many of the issues that have denied veterans PTSD benefits for years. For this chapter, I've selected a cross-section of cases decided by these courts that represent the major reasons why the VA turned down claims.\n\nTo give you a sense of how the VA is holding the line on denying service connection for PTSD and substance abuse, here is some startling information: between 1989 and 2004, 819 PTSD cases have been reviewed by the court. The appeals for substance abuse as secondary to PTSD number 57 during this same period. It would take volumes to tell the stories of the nearly 900 PTSD appeals decided by the court. What these numbers don't tell us is how many original, amended, or reopened claims for PTSD or substance abuse were denied by the VA where the veteran did not challenge the decision in the court.\n\nAs I worked on this chapter, I became curious how many claims have been filed for PTSD since the very beginning of the Vietnam War through 2003. I wanted to know how many were approved, how many were denied, and, if granted, what percentage of disability was granted. To my knowledge this information has never been made public. In order to reduce the time I would have to wait for this information under the Freedom of Information Act, I asked my congressional representative to forward my request to the VA. Their answer to me was \"We do not collect or correlate this type information.\" Yeah, right! See figure 9.1 to read how their letter explains it to me.\n\n**Fig. 9.1**\n\nThis response was supposed to satisfy my concerns concerning what the VA could or could not do in compiling data. However, twenty days later the secretary's office released a VA fact sheet dated December 2004 in which they boast of their accomplishment in fiscal year 2003. This VA fact sheet demonstrates the VA has the ability to track all incoming claims and how many claims each Regional Office approves. The full fact sheet and a discussion of its implications can be found in chapter 4.\n\n### The Appeal Process\n\nThere is a definite set of rules that must be observed when a veteran appeals a VA decision denying benefits. The first step is to file a Notice of Disagreement with the Regional Office of original jurisdiction within one year of the date the veteran received the denial letter.\n\nIf the issue is not resolved to the veteran's satisfaction by the RO, the case is forwarded to the Board of Veteran's Appeals in Washington. If the BVA ruling has not resolved the issue, the veteran can have his or her case heard before the U.S. Court of Appeals for Veterans Claims. An unsatisfactory decision by this court can be heard before the Federal Circuit Court of Appeals. The last stop in the appeal process is the Supreme Court.\n\n### \"They Did It Before and They Will Do It Again\"\n\nOne veteran pursued his claims to the fullest extent and would not give up until the case was argued before the Supreme Court. In its ruling on the case, the Court sided with veteran Fred Gardner, agreeing that he should be paid compensation for the disabilities incurred while a patient of the VA.\n\nAssociate Justice Souter, writing for a unanimous Court held\n\nthat Paragraph 3.358(c)(3) (38 _CFR_ ) is not consistent with the plain language of the Title 38 U.S. Code \u00a71151, which does not contain a word about fault or accident. The statutory text and reasonable inferences from it give a clear answer against the Government's arguments that a fault requirement is implicit in the terms \"injury\" and \"as a result of.\" This clear textually grounded conclusion is also fatal to the Government's remaining principal arguments: that Congress ratified the VA's practice of requiring a showing of fault when it reenacted the predecessor of \u00a71151 in 1934, or, alternatively, that the post-1934 legislative silence serves as an implicit endorsement of the fault based policy; and that the policy deserves judicial deference due to its undisturbed endurance. Pp. 2-8. 5 F. 3d 1456, affirmed.\n\nThis practice is no different than the VA adding to its operational manual (M211) that to qualify for Direct Service-Connection the veteran must be awarded a Bronze Star or Air Medal with a \"V\" device. Without aggressive oversight, the VA will continue to implement policies unfair to veterans in the name of reducing budget requirements.\n\n### _Allen v. Principi_ : Alcohol Abuse Secondary to PTSD\n\nVeteran William Allen, unsatisfied with the CAVC's decision on his case, took his case to the U.S. Court of Appeals for the Federal Circuit in Washington, and on February 2, 2001, the court decided one of the landmark cases upholding veterans' rights. The finding by the U.S. Circuit Court of Appeals blocked one of the many tactics used by the VA to deny PTSD claims.\n\n#### FACTS\n\nWilliam Allen served on active duty in the U.S. Marine Corps from September 1965 to September 1969. Mr. Allen alleged that he suffered from PTSD resulting from his service. He also alleged that he suffered from an alcohol abuse disability that manifested itself as a symptom of his PTSD. In 1999, he requested increased monetary compensation for both his PTSD and for his alcohol abuse disability. Mr. Allen challenged the VA's position that the existence of his alcohol abuse disability should be considered in determining his disability rating for PTSD.\n\n#### HISTORY\n\nIn 1993, the Boston Regional Office granted Mr. Allen a 30% disability rating for service-connected PTSD effective from June 1992. In October 1994, Mr. Allen filed a claim for an increased rating. The RO issued a rating decision in June 1995 refusing an increase but confirming the 30% disability rating.\n\nIn 1995\u201396, Dr. Victoria Russell conducted a VA Compensation and Pension examination to determine the severity of Mr. Allen's PTSD and to ascertain whether there was a relationship between Allen's PTSD and his alcohol abuse. Dr. Russell wrote that \"the reason for [Allen's] alcohol admissions had to do with his rapidly accelerating symptoms of PTSD.\" The doctor diagnosed Mr. Allen with severe PTSD in addition to her diagnosis of chronic alcohol abuse and dependence, in remission, secondary to PTSD. In November 1996, the RO issued a rating confirming the 30% disability evaluation for PTSD and again declining an increased rating.\n\nOn June 6, 1997, however, following a psychiatric hospitalization, the RO increased Mr. Allen's disability rating to 50%, effective from October 13, 1994. Although the RO considered Mr. Allen's prior hospitalizations in determining his rating, the RO did not consider the August 1996 C&P Exam by Dr. Russell in evaluating whether Mr. Allen was entitled to an increased rating above 50%.\n\nMr. Allen appealed the denial of a rating higher than 50% to the BVA. On November 26, 1997, the board upheld the RO and denied a higher rating. The board said that \"alcohol abuse may not be service-connected, on either a direct or secondary basis, and impairment from non-service-connected substance abuse may not be considered when evaluating the severity of the service-connected PTSD.\" In order to receive a 70% rating\u2014the next available step above 50%\u2014the board required Mr. Allen to demonstrate both occupational impairment and social impairment.\n\nIn considering all of the evidence, the board decided that \"the record demonstrates that the veteran's employability has been seriously undermined by his history of alcohol abuse and dependence,\" but determined that Mr. Allen's PTSD was \"not the only factor influencing his industrial [occupational] impairment.\" The board also found that Mr. Allen failed to produce sufficient evidence to demonstrate social impairment.\n\nOn March 25, 1998, Mr. Allen filed a timely appeal to the CAVC. On July 12, 1999, the U.S. Court of Appeals for Veterans Claims vacated the BVA's decision and remanded the case for a further evidentiary hearing. Both Mr. Allen and the secretary of veterans affairs agreed that a remand was required in light of the CAVC's opinion in _Barela v. West_ , which had been issued subsequent to the board's November 26, 1997, decision. In _Barela_ it was held that U.S. Code \u00a71110 precludes a veteran from receiving compensation for disabilities resulting from alcohol or drug abuse. In its _Barela_ ruling, however, the court added that \u00a71110 does not bar an award of service connection for alcohol or drug abuse. An award of service connection would entitle a veteran to benefits such as educational assistance for dependents or housing loan benefits for the veteran. It would not entitle the veteran to any increase in disability compensation.\n\nInitially Mr. Allen appealed to CAVC to vacate the November 26, 1997, BVA decision that denied increased disability compensation. In its one-judge order, the CAVC instructed the board that its interpretation in _Barela v. West_ was a _binding precedent_ and barred recovery to the extent that Mr. Allen sought increased compensation for his alcohol abuse disability, either as a secondary service-connected disability or as evidence of the increased severity of his service-connected post-traumatic stress disorder.\n\n#### JUDICIAL APPRAISAL\n\nU.S. Court of Appeals for the Federal Circuit held they have jurisdiction over this remand order because the CAVC interpretation of the statute will affect the remand proceeding, and our future review may be evaded. \"We further hold that \u00a71110, when read in light of its legislative history, _does not preclude a veteran from receivingcompensation for alcohol or drug-related disabilities arising secondarily from a service-connected disability, or from using alcohol or drug-related disabilities as evidence of the increased severity of a service-connected disability_. Therefore, we reject the CAVC statutory interpretation of \u00a71110 and to that extent reverse the court's order, and remand for proceedings in accordance with the interpretation of \u00a71110 as set forth herein. The part of the order remanding to the Board for proper application of the rating criteria is affirmed.\" (Italics added.)\n\nIn Allen's case, the CAVC remanded in order for the board to apply the court's interpretation of \u00a71110 as set forth in _Barela_. Allen had argued that _Barela_ was \"wrongly decided to the extent that it held that the law prohibits the payment by VA of compensation for alcohol abuse either on a secondary-service-connection basis or as evidence of an increase in the severity of a service-connected disability.\" The Court of Appeals for Veterans Claims, however, rejected Mr. Allen's interpretation of \u00a71110 and stated that _Barela_ was \"binding precedent\" to the extent that Mr. Allen \"seeks compensation based either on alcohol abuse secondary to service-connected PTSD or an increase in his rating for service-connected PTSD based on manifestations of PTSD symptomatology, i.e., alcohol or drug abuse.\"\n\n#### COURT'S RULING\n\nWe therefore conclude, based on the language of the statute and the pertinent legislative history, that 38 U.S.C. \u00a71110 does not preclude compensation for an alcohol or drug abuse disability secondary to a service-connected disability or use of an alcohol or drug abuse disability as evidence of the increased severity of a service-connected disability. We would stress that the holding of the case is quite limited. Veterans can only recover if they can adequately establish that their alcohol or drug abuse disability is secondary to or is caused by their primary service-connected disorder. We foresee that such compensation would only result where there is clear medical evidence establishing that the alcohol or drug abuse disability is indeed caused by a veteran's primary service-connected disability, and where the alcohol or drug abuse disability is not due to willful wrongdoing.\n\nOn remand, the Board will have to determine whether Allen's alcohol abuse disability is secondary to his PTSD, or whether it demonstrates the increased severity of his PTSD disability. If it finds sufficient evidence demonstrating a causal connection, Allen could be entitled to an increase in his schedular rating. But if the Division Review Officer finds that Allen's alcohol abuse is willful and did not result from his PTSD, Allen could not receive additional compensation for a willful alcohol abuse disability. Barela and any other decisions of the Veterans Court are hereby overruled, to the extent they are contrary to this opinion.\n\nFor the foregoing reasons, the decision of the Veterans Court is affirmed-in-part and reversed-in-part. The case is remanded to the Veterans Court for further proceedings consistent with this opinion.\n\n#### WHAT WE LEARNED\n\n1. A veteran does not have to exhibit both occupational and social impairment before he or she can qualify for PTSD benefits under Diagnostic Code 9411. If you have one or the other diagnosed by a physician, you meet the basic standard. PTSD claims suffer a high casualty rate when it comes time to rate a claim because adjudicators have historically required degrees of proof that are not required by law.\n\n2. Section 1110, 38 U. S. Code does not bar compensation payment for a claim involving alcohol or drug abuse if, in fact, this abuse is a secondary condition brought about by a service-related injury or disease. It also can be used as evidence that the service-connected condition has increased in its severity.\n\n### U.S. Court of Appeals for Veterans Claims\n\nThe rest of this chapter consists of cases heard by the U.S. Court of Appeals for Veterans Claims. In these cases, the court dealt with questions of substance abuse, noncombat-related PTSD experienced in a combat environment, sexual assault, noncombat MOS in a combat zone, and injury caused by a lighting strike. All of these cases, representing a cross-section of obstacles a veteran faces when he or she files an appeal, touch on a PTSD claimant's needs.\n\n### _Flayter v. Principi_ : Substance Abuse\n\n#### FACTS\n\nIn the case of _Flayter v. Principi_ , the VA denied benefits based on the finding in _Barela v. West_. When the U.S. Court of Appeals for the Federal Circuit overruled _Barela v. West_ , the U.S. Court of Appeals for Veterans Claims gained the authority to reverse the BVA in _Flayter v. Principi_.\n\n_Flayter v. Principi_ is a memorandum decision (single-judge ruling), which means it cannot be cited as an authority in an appeal to the court. It is, however, extremely valuable for a veteran who is filing a claim based on alcohol or substance abuse as secondary to PTSD. The good news is _William F. Allen v. Anthony J. Principi_ , when appealed to the U.S. Court of Appeals for the Federal Circuit, overturned _Barela v. West_. In _Allen v. Principi_ the VA rubber-stamped the denial letters where alcohol or substance abuse was secondary to PTSD. A decision by the U.S. Court of Appeals for the Federal Circuit can be cited in a case on appeal before the U.S. Court of Appeals for Veterans Claims.\n\n#### HISTORY\n\nIn 1996 Mr. Patrick Flayter filed a claim for increased disability compensation in excess of 30% for PTSD and service connection for alcohol and drug abuse secondary to PTSD. The RO denied his claim in May 1997 for increased compensation for PTSD. Flayter was granted service connection for drug and alcohol abuse secondary to PTSD. However, in this case, claims examiners denied paying compensation for drug and alcohol abuse. They cited statutes that they say prohibit payment for any condition caused by willful misconduct on the part of the veteran. They maintained _Barela v. West_ prevailed in Mr. Flayter's case, thus denying compensation payment for alcohol and drug abuse secondary to PTSD.\n\n#### JUDICIAL REVIEW\n\nThe veteran filed a timely appeal of the denial of increased compensation with CAVC, and a decision was rendered approximately two and a half years later on April 7, 2001. CAVC reversed the BVA based on the findings announced by the Federal Court of Appeals for the Federal District in their decision of _Allen v. Principi_ , Docket Number 98 555, decided on April 11, 2002.\n\nThe crux of this case was that a veteran can receive compensation for an alcohol or drug abuse disability where it is established that such a disability is secondary to or is caused by his or her primary disability or disorder. This decision does not limit the scope of secondary alcohol or drug abuse to only PTSD.\n\n#### COURT'S RULING\n\nIn view of the finding in _Allen v. Principi_ , the court vacated the BVA decision and remanded it back to be readjudicated consistent with this order.\n\n#### WHAT WE LEARNED\n\n1. When the Court of Appeals for Veterans Claims renders a single-judge decision, you cannot use it as an instrument to force the VA to change its finding in your case. Only issues addressed by a three-judge panel can be quoted as law, thus forcing the VA to comply with its decision. However, a single-judge finding (referred to as a memorandum decision) can be valuable to you if you use it as a tool to frame your claim before the VA.\n\n2. Once again the Department of Veterans Affairs's administrative function is forced to change its policy to conform to the law. It is almost certain if Flayter's claim was decided a year earlier by the CAVC, the decision would not have been vacated and the veteran would not have had a second bite out of the apple. It never pays to leave a denied claim unchallenged. When the VA denies your claim, file a Notice of Disagreement within a year of the date you were notified.\n\n3. The decision by the U.S. Court of Appeals for the Federal Circuit stopped the VA from denying compensation benefits when alcohol or substance abuse is secondary to PTSD. Prior to this decision claims examiners would grant service connection for this secondary condition, but would withhold compensation stating it was the result of willful misconduct.\n\n### _Yancey v. Principi_ : Noncombat PTSD in a Combat Zone\n\n#### FACTS\n\nAlberta Yancy's claim for service-connected PTSD benefits was denied by the Board of Veterans Appeals in October 2002. Her case was reviewed before a single judge, making it off limits as a precedent-mandating authority. Even so, this case can show you what to do and what not to do.\n\n#### HISTORY\n\nMs. Yancy, the appellant, served on active duty in the U.S. Army from December 1990 to May 1991 during Operations Desert Shield and Desert Storm. The record shows she was stationed in Saudi Arabia from January to May 1991. She was a practical nurse attached to the 114th Evacuation Hospital Unit. In a supplemental medical report dated May 1991, Ms. Yancy was asked whether she experienced insomnia, interrupted sleep, or dreams about war experiences. She was quoted as saying, \"I have nightmares of gerbils holding me down\" and a \"fear of getting hit by a scud.\" In another record, a sick call record dated April 1991, she told the doctor \"she was having nightmares when she first arrived in country.\" Her record notes \"reoccurring thoughts about her experiences in Saudi Arabia.\"\n\nIn February 1994 the VA ordered a C&P examination following her application for compensation benefits. The C&P examiner reported:\n\nShe had flashbacks of \"many scuds\" coming close. The claimant stated that there were days when there were four scud attacks per day.... She is reminded and relives some intrusive memories of the scud attacks when she sees trucks or dollies.\n\nThe examiner diagnosed the appellant with PTSD, delayed. During that same month, the VA scheduled her for a C&P social survey, part of the C&P examination for PTSD that determines the veteran's ability to work in the public domain. The examiner reported, while on active duty, the appellant had \"begun becoming more terrified as scud missiles struck close to 114th Evacuation Hospital Unit compound. It was here that she began periods of restlessness and poor sleep.\"\n\nIn February 1994 a statement in support of claim (VA Form 21-4138) was filed with the VA Regional Office. The record shows she stated that she \"had been very fearful of the scud missiles.\" In June 1994 the RO issued a decision in which it denied the appellant's PTSD claim. The RO decided her \"unverified general statements about being afraid of the scud missile attacks are not acceptable evidence of an incident or incidents during active service which would cause PTSD.\"\n\nOn March 2, 1995, the veteran filed a notice to reopen her claim based on new and material evidence. In that statement, she commented, \"Being a nurse by profession I somewhat understand why the claim was denied. But, by the same token working as a fulltime professional nurse, I don't understand why a person my age should be under constant medical care, taking daily medication.\" She requested that the RO obtain her most recent medical records for a \"new evaluation.\" Ms. Yancy also reported that many symptoms related to her several disabilities (for which she was claiming service connection), including PTSD, had been exacerbated.\n\nThe court noted that the RO and the BVA treated this March 1995 VA Form 214138 as an attempt to reopen a previously disallowed claim. Subsequently, the RO requested copies of the appellant's private medical records. This was followed in April 1995 by an order for another C&P general evaluation and a PTSD examination.\n\nThe April 1995 PTSD examination report acknowledged that the veteran was preoccupied with memories of scud missile attacks and offered a diagnosis of borderline PTSD. The October 1995 PTSD examination report diagnosed the appellant with PTSD, \"borderline, by history only,\" and chronic major depression. The RO, in August 1995, requested the appellant's service records from the commander of the U.S. Army Reserve Unit. The request for background information included the veteran's unit and dates of assignment and participation in combat operations during Desert Storm. The court notes it appears that the RO did not receive this information. In November 1995, the RO requested a DA Form 20, a Department of the Army personnel record of the soldier's history, from the U.S. Army Reserve Personnel Center. It was further noted by the court that the Records on Appeal did not contain a copy of any such form or file.\n\nThe VA claims examiners rated her claim for PTSD in February 1996, reaffirming the denial of her PTSD claim. She appealed that decision in March 1996 by filing a Notice of Disagreement. In March 1996 the VA responded with a Statement of the Case.\n\nThe veteran was given a hearing before the local RO's hearing officer in July 1996 where she testified that there had been three to four scud missile attacks every night during the three weeks that she had been stationed in Dhahran, Saudi Arabia. Ms. Yancy also testified that she treated wounded allied personnel and enemy prisoners of war. She further testified that, after service, certain noises and odors had triggered flashbacks and the nightmares that she had begun to experience in Saudi Arabia had worsened. Her representative indicated that the appellant anticipated obtaining personally her DA Form 20 from her Army Reserve Unit in St. Louis, Missouri. However, the ROA does not reflect that the appellant submitted any such request.\n\nAs noted in a December 1996 VA Treatment Summary, the veteran reported stressful events occurred while she was serving on active duty. These events included: (1) hearing \"constant\" warning sirens used to alert personnel of missile attacks, (2) \"constantly\" getting in and out of protective chemical warfare gear and being afraid of chemical exposure while she slept, (3) witnessing a child die from a heart condition, (4) talking with an Enemy Prisoner of War (EPW) whose arm had been amputated, (5) hearing gunfire in the compound, (6) learning of her mother's heart attack and of her grandmother's death, and (7) merging with another unit prior to activation.\n\nIn September 1997 the RO received from a veteran who had served with the appellant a statement that during service the appellant had been subjected to \"additional pressures due to emergencies at home\"; the appellant had made \"three trips back to the United States\" prompted by family emergencies, including one leave of approximately three weeks.\n\nIn a September 1998 decision, the BVA determined that the record confirmed the appellant had a diagnosis of PTSD, but that she had not engaged in combat with the enemy. Based on the latter determination, the board stated VA regulations required the appellant's alleged stressors be corroborated by credible supporting evidence. The board concluded that the record lacked credible supporting evidence and thus it remanded the matter and ordered that the RO: (1) \"attempt to obtain copies of the action reports or unit histories\" for the appellant's unit and (2) \"make a determination as to whether any of the [appellant's] claimed stressors are verified by these unit histories.\"\n\nSubsequently the RO requested from the U.S. Armed Services Center for Research of Unit Records (USASCRUR) copies of the veteran's unit historical records from January to May 1991. The RO claims examiner explained it was \"trying to determine if the veteran's unit was attacked by scud missiles on several occasions.\"\n\nIn June 1999 USASCRUR responded by providing the RO with a unit history submitted by the 114th that \"documents the locations, missions, operations and significant activities of the unit during the veteran's tour of duty.\" The unit history report, dated March 15, 1991, reflects that the 114th spent its first three weeks of deployment in \"transient quarters in Dhahran, Saudi Arabia, before establishing its evacuation hospital near King Khalid Military City, Saudi Arabia.\" The report also reflects that: (1) a public address system was \"necessary for enemy threat alerts,\" (2) Purple Hearts were awarded to three members of the unit, (3) the unit treated large numbers of wounded EPWs, and (4) the \"unit was in danger in Dhahran without proper protection during scud attacks.\"\n\nFollowing receipt of this report, the RO claims examiners issued a Supplemental SOC in which they continued the denial of her PTSD claim. The RO justified denying her claim:\n\nThe unit history noted the existence of enemy threat and also noted that the unit was in danger in Dhahran. However, it failed to note any actual scud attack on the 114th Evacuation Hospital location. There is no information on the frequency of these attacks or the severity that may have followed the attacks.\n\nThe SSOC made no mentioning or offered any documentation of the wounded. They concluded the evidence reviewed still did not verify the stressful experiences that the veteran claimed.\n\nIn October 2000 the veteran rejected the decision of the BVA and on February 10, 2001, filed a timely Notice of Appeal to the U.S. Court of Appeals for Veterans Claims. Her brief filed with the court challenged the BVA decision denying benefits.\n\n#### JUDICIAL REVIEW\n\nThe board members concluded that the 114th's unit history report did not provide evidence sufficient to corroborate her reported stressor of experiencing multiple scud missile attacks. The BVA made the point that the report did \"not state the veteran's unit actually came under attack by scuds, nor did it state the frequency or duration of these attacks.\" The board further stated, although some of the appellant's other alleged stressors were verifiable, all of her diagnoses of PTSD were based on the alleged in-service stressor related to scud missile attacks. Thus, the board concluded none of her PTSD diagnoses were linked to a specific in-service stressor, verified by credible supporting evidence; therefore they denied her claim for service connection for PTSD.\n\n#### COURT'S RULING\n\nChief Judge Kramer stated he was not prepared to rule on the question of the quantum of evidence that the unit history report provides as to whether the 114th actually came under attack or attacks from scud missiles. The court's discretion allows for a remand on such matter.\n\nThe court ordered on remand the VA should attempt to secure documents that provide detail regarding the appellant's duty stations, dates of assignments, and dates of the appellant's physical presence at the location of the 114th. Moreover, the court further noted that the unit history report reflects that the \"unit was in danger in Dhahran without proper protection during scud attacks,\" but that it does not provide any detail about any actual attack that occurred between January and March 15, 1991. The court noted that, as of March 15, 1991, according the unit history report, the 114th was no longer in Dhahran, Saudi Arabia, the location where the appellant maintains that she had experienced her scud-missile stressor. On remand, the VA should attempt to obtain specific information as to whether and when any such attack actually occurred while the veteran was physically present at the 1l4th's location. In the event that further army documents are unavailable, the VA should survey unit members as to these inquiries. Further, the board is directed on remand to address the effect of the words \"during scud attacks\" and the significance of the three Purple Hearts awarded to members of the 114th.\n\nFinally, on remand the appellant is free to submit additional evidence and argument, including those arguments raised in her briefs to this court. The board shall proceed expeditiously, in accordance with 38 U.S.C. 7112.\n\n#### WHAT WE LEARNED\n\nAlberta Yancey's case is an excellent example of a delaying tact by the Department of Veterans Affairs. Ms. Yancy doggedly held on for more than twelve years and deserves many kudos for not letting the process force her to cave in. She has persisted and used the system, expecting the VA to consider all the information necessary to process her claim.\n\nUnfortunately, she is now twelve years down the road and no closer to having her award granted. Even though the law mandates that adjudicators assist in developing the claim, provided the claim is well grounded, they have not fulfilled their duty to assist.\n\nIn this case the court raised issues it expects the VA to act upon. Unless the evidence is indisputable, a VA adjudicator will deny a veteran's claim. In other words, if you cannot force the VA to grant benefits by introducing solid evidence, you'll be unsuccessful during the first round. If a claim is based on either the claimant's combat with the enemy or rape, the chances are it will be approved. Title 38 _USC_ , chapter 51, section 5103A, leaves options open as far as the VA's duty to assist. An adjudicator must make only a _reasonable effort_ to assist a claimant, and he or she is not required to assist if, in his or her opinion, the claim cannot be proved.\n\nYou can learn from _Yancey v. Principi_ and take steps that will help you file a winning claim:\n\n1. Have all statements from former members sworn and notarized. These documents now become evidence that cannot be dismissed as unsubstantiated statements.\n\n2. If the unit history report does not support your statement, then try to obtain a report from the next level of command, which summarizes events occurring in all its suborganizations.\n\n3. Do not hesitate to involve your congressional representative or senator in expediting your request for records.\n\n4. Remember you can be your own advocate. I mean no disrespect to all the service officers' nationwide, but they have not been trained to be aggressive advocates. The VA basically uses these service officers as clerks, but few service officers know what the law requires when documenting hundreds of details into binding evidence.\n\n### _Doran v. Brown_ : Noncombat Disability\u2014  \nA Lighting Bolt from Zeus\n\n#### FACTS\n\nI have selected the _Doran v. Brown_ case for further discussion because it demonstrates the kind of review a PTSD claim undergoes when the circumstances fail to fit a narrow definition of PTSD. It appears that the only PTSD claims that have a chance of being granted have a combat incident or rape as the stressor. Many cases have come before the court that reflect how the intent of Congress is ultimately manipulated by simply changing the wording in VA manuals.\n\nThe court gave veterans claiming PTSD caused by noncombat-related stressors a leg up when it decided Robert Doran's case. The court closed the door on one of the VA's most frequently used reasons for denying claims: \"the absence of support in service records.\"\n\n#### HISTORY\n\nIn 1976 Robert E. Doran reopened his claim for psychoneurosis to include post-traumatic stress disorder, based on new and material evidence. However, it wasn't until August 1992 that his case made its way through the system to the court for the first time. When he reopened his claim and the court heard his case in 1993, he submitted medical evidence from the doctor who treated him for his neurosis and PTSD. He was also able to obtain lay statements from individuals who were knowledgeable of the circumstance of his stressors.\n\nMr. Doran claimed that during his active service (1950\u20131952) he was struck by lighting, causing unconsciousness and requiring resuscitation. The second stressor cited by Mr. Doran was an encounter with a snake on an infiltration course. The court considered the evidence submitted in support of his claim noncumulative, not relevant to or probative to the issue at hand. On August 7, 1992, the court issued a single-judge memorandum decision vacating the Board of Veterans' Appeals' November 28, 1990, denial of service connection and remanded it back to the VA for readjudication.\n\n#### JUDICIAL APPRAISAL\n\nThe court held that the \"board focused only on evidence that tended to support the denial of service connection, ignored evidence in favor of the appellant's claim, and failed to provide reasons or basis for its findings and conclusions.\" The order further stipulated, on remand, the board would adjudicate Mr. Doran's claim for service connection for PTSD, a claim that was raised but not addressed by the VA.\n\nOn March 5, 1993, the Board of Veterans' Appeals issued a decision in which they determined (1) the evidence received by the VA since the BVA denied entitlement for service connection for a psychoneurosis in September 1976 was both new and material and the claim for psychoneurosis to include PTSD would be reopened; (2) the veteran's psychoneurosis preexisted active service and the sound condition at enlistment had been rebutted; (3) the psychoneurosis was not aggravated during service; and (4) service connection for PTSD was not warranted. Mr. Doran filed a timely appeal to the court.\n\nOne year later, on March 8, 1994, a three-judge panel heard Mr. Doran's appeal to the court. That portion of the court's discussion and decision concerning the aggravation of a preexisting psychoneurotic condition will not be addressed as it is not pertinent to establishing service connection for PTSD.\n\nThe court found that the BVA's treatment of Mr. Doran's claim for PTSD highlighted the potential inconsistency between the statute, the regulations, and the specific provisions of VA Adjudication Manual \"M21-1.\" The court clarified the section of the manual that required \"service records must support the assertion that the veteran was subjected to a stressor of sufficient gravity to evoke symptoms (PTSD) in almost anyone.\" The court held that under the governing statutory and regulatory provisions regarding claims for service connection for PTSD, \"if the claimed stressor is not combat-related, appellant's lay testimony regarding in service stressors is insufficient to establish the occurrence of the stressor and must be corroborated by credible supporting evidence.... _There is nothing in the stature or the regulations which provides that corroboration must and can only be found in service records._ \" (Italics added.) Those service records that are available must not contradict the appellant's lay testimony concerning his noncombat stressor. _VA manuals cannot give out instructions, which are contrary to the statute_.\n\n#### COURT'S RULING\n\nFor the second time the court vacated the VA's decision to deny benefits. Mr. Doran's case was remanded with instruction to readjudicate his claim for PTSD making certain that a critical examination is conducted of all the evidence when justifying the decision. Because Robert Doran does not appear in public records, it is not known if the VA has readjudicated his claim for PTSD and granted the benefits or whether the local rating board has continued to deny his claim, thus forcing him to continue with the appeal process.\n\n#### WHAT WE LEARNED\n\nThis case opened the door exposing many of the tactics that are used by various rating officers. If the circumstances of your appeal can be supported by the finding in this case, it can be cited because it is a three-judge decision. Here are several key points to keep in mind when preparing your claim or if you are appealing any denial of benefits:\n\n1. The VA cannot publish a change to its manuals that is contrary to the statutes. Be alert for this. This is why you should do your research before you submit a claim.\n\n2. There is nothing in the law or regulations that requires corroboration of a traumatic event to come from service records.\n\n3. The court recognized a potential inconsistency between the statutes, regulations and the specific provisions in VA Manual M-21-1.\n\n### _Patton v. West:_ Sexual Assault\n\n#### FACTS\n\nThe case of _Dorrance H. Patton v. Togo D. West_ demonstrates many of the problems encountered by an individual who was traumatized by an noncombat-related event, which then took thirty-three years to surface as PTSD. You will note as you read Judge Steinberg's account of the events that Mr. Patton initially was diagnosed with anxiety and depressive neurosis; emotional instability reaction, chronic, moderate, manifested by diffused anxiety; and poor performance under stress. These conditions were the reason he was found unfit for duty and given an honorable administrative discharge.\n\n#### HISTORY\n\nThe veteran served in the U.S. Army from March to August 1956 and from October 1959 until February 1960. His service medical records (SMRs) indicated normal psychiatric clinical evaluations at the time of his first entry into service, first discharge, and second entrance physical.\n\nOn the evening of December 9, 1959, the veteran was admitted to the emergency room of an army hospital, where he was treated for an acute anxiety reaction. A detailed description of his treatment contained notations that he was afraid someone would \"jump him\" and that \"inferentially, his profound feelings of shame, as well as other indirect derivatives suggest to me that there may be underlying homosexual panic.\" The SMRs also indicated that he complained of headaches, anorexia, and sleeplessness, and that he attributed \"most of his problems\" to an incident when he had been hit in the head with a glass bottle while on leave a few weeks before. According to those records, other soldiers in the barracks reported that the veteran had \"some sort of attack\" the night he was first admitted to the army hospital. Although SMRs dated December 10, 1959, report his anxiety appeared to have worsened since he had arrived at Fort Bragg, the ROA contains no notations of anxiety prior to December 9, 1959.\n\nThe veteran's service personnel records contain no indication of any in-service assault incident. On January 21, 1960, \"despite a limited work assignment,\" the veteran \"presented himself in a tearful disheveled state\" at the army hospital. He was diagnosed as having \"emotional instability reaction, chronic, moderate; manifested by diffuse anxiety, poor performance under stress, and administrative separation from the service [was] recommended.\" During 1978 and 1979 the veteran was hospitalized several times in a VA medical facility, was treated for anxiety and depressive neurosis, and received medication from VA and psychotherapy from a private clinic. In August 1978 he reported a history of alcoholism and in March 1979 he \"admitted to heavy use of alcohol as a means of coping with his anxiety\" and was diagnosed by a VA psychiatrist as having a paranoid personality.\n\nIn May 1979 a VA physician diagnosed the veteran as having a \"habitual excessive drinking\" disorder and stated: \"On previous admission patient had been called paranoid type personality though at this time I feel that this is probably this patient's basic underlying personality type which was brought out by his decompensation due to many years of alcohol abuse.\" The veteran submitted letters from two acquaintances; each provided general information about his nervous condition and requested any assistance the VA could provide for the veteran. One mentioned the veteran's drinking problem.\n\nThe veteran also submitted an August 1979 letter from his wife detailing his changed behavior since service, including his \"not getting along with people,... his hiding behind his drinking,... his anger and depression, and his being ashamed of the way he was.... She specifically reported that around the end of 1961, the veteran's mother had written to her saying... she was afraid he might kill himself and that... something had happened to him in the service.\"\n\nThe veteran submitted a July 1979 letter from a private physician stating that the veteran was \"extremely agitated, depressed, loses control of his temper, and was potentially dangerous to both himself and others,\" and had been so for the prior twelve months. The physician also stated that the veteran had not responded to medication and other treatments and he might \"lose control and kill someone.\"\n\nIn September 1979, after the VA Regional Office denied service connection for a nervous condition and sent the veteran a Statement of the Case, the veteran filed a VA Form 1-9, Substantive Appeal to the BVA. At an October 1979 hearing at the RO regarding the nervous-condition claim, the veteran testified under oath that he had been hospitalized after an altercation with a noncommissioned officer; that he did not \"know what in the hell happened: it just all went black\"; that he had then received a medical discharge of which he was ashamed; that since he left the service he had \"done a lot of heavy drinking to cover this up\"; that his anxiety had become worse since he stopped drinking a few years ago; and that, until recently, he had not sought psychiatric assistance because he did not want anyone to know about his problem. He subsequently submitted both VA and private medical records from 1976 through 1981, including a January 1981 private psychiatric evaluation that noted that the veteran had \"a paranoid personality with decompensation into a psychotic state manifested by homicidal ideation, agitation and depression.\" No BVA decision apparently was issued in connection with the 1979 RO denial. The ROA does not reflect further contact with the VA by the veteran until 1993.\n\nWhen the error was found in 1993, a C&P examination was scheduled. During the November 1993 VA medical examination, the veteran stated, for the first time to any medical professional, that while he was at Fort Bragg, he had been raped by three men and that he did not report it because of \"fear and shame.\" He disclosed that after a second hospitalization (apparently in January 1960) he had told a sergeant about the rape and that the sergeant had instructed him not to tell anyone. At that time he was diagnosed, for the first time, with \" _PTSD, non-combat, chronic_.\"(Italics added.)\n\nIn December 1993 the veteran sought to have his nervous-condition claim \"reopened to include PTSD.\" In January 1994, a VA medical examiner diagnosed PTSD, clearly relating the PTSD to the alleged in-service rape trauma. In December 1994 the RO denied service connection for PTSD, finding that the veteran had not provided the required corroborating evidence for an in-service, noncombat stressor.\n\nThe veteran appealed to the board in a timely manner. In April 1995, he testified under oath before the RO that he had been sexually assaulted, but had been told (presumably by a sergeant) that he would receive a dishonorable discharge and would be sent to prison if he reported it. In the 1997 BVA decision on appeal, the board denied the PTSD claim because it was \"based on non-combat-related unverified stressors,\" and because corroboration of an in-service stressor was an essential element of the veteran's PTSD claim.\n\n#### MOTION TO STRIKE\n\nThe first issue the court had to address was a motion to strike by the Department of Veterans Affairs. The veteran contended that, in the absence of evidence of a combat-related stressor, he must \"furnish the specific details of the in-service incident.\" He stated that he has \"attempted to submit an affidavit which is confirmed by a third party immediately after this witness left the service.\"\n\nThe secretary responded with a motion to strike references to the affidavit. The court noted that \"there is no such exhibit attached to the appellant's brief.\" In addition, \"the court is precluded by statute from including in the Records Contained on Appeal (RCA) any material that was not contained in the record of proceedings before the Secretary and the Board.\" The court made it known it would not consider the appellant's purported attempt to submit an affidavit as an exhibit to his brief. The order went on to say \"the veteran was free to submit additional evidence, including the asserted statement referred to above, to the BVA in connection with the remand that the court ordered in its opinion.\"\n\nThis is an important point to remember if your claim goes to the CAVC. Once the BVA has reviewed your appeal, no new evidence can be added to the file by either side. However, if your claim is remanded back to the controlling Regional Office, you can add new evidence for them to consider. If the BVA affirms the local RO decision to deny benefits, the case must go as is to the court with no new evidence attached.\n\n#### COURT'S RULING\n\nJudge Steinberg, writing for the majority, presented a fast moving account of the events that reshaped Mr. Patton's life. Judge Holdaway did not agree with the majority decision, and thus filed a dissenting opinion.\n\nThe court's finding for this case laid out a great set of basic rules the veteran and VA must observe. The court's decision in this case has provided us with eight rules to build a solid claim for service-connected PTSD:\n\n1. The third rule outlines exactly what the court will look for should you challenge the VA's denial of benefits. The veteran must show the following key points:\n\n  a current, clear, medical diagnosis of PTSD;\n\n  credible supporting evidence that the claimed in-service stressor actually occurred;\n\n  medical evidence of a causal connection between current symptomatology and the specific, claimed, in-service stressor. The evidence required to support the occurrence of an in-service stressor varies \"depending on whether or not the veteran was engaged in combat with the enemy.\"\n\n2. The board is required to include in its decision a written statement of the reasons or bases for its findings and conclusions on all material issues of fact and law presented on the record.\n\n3. The VA's statement must adequately enable an appellant to understand the precise basis for the board's decision, as well as facilitate review in the CAVC. To comply with this requirement, the board must analyze the credibility and probative value of the evidence, account for the evidence that it finds to be persuasive or unpersuasive, and provide the reasons for its rejection of any material evidence favorable to the veteran.\n\nThis requirement is one of the most neglected instructions as the VA seldom follows it. Instead claims examiners dismiss a claim by stating that a preponderance of evidence is against the claimant's claim. However, the court ruled the VA must cite \"chapter and verse\" what evidence is superior to the evidence the claimant submitted. They must also provide an explanation why the evidence favorable to you was not sufficient to grant you benefits.\n\n4. The evidence required to support the occurrence of an in-service stressor varies \"depending on whether or not the veteran was 'engaged in combat with the enemy.'\" According to _Zarycki v. Brown_ , when the VA determines that the veteran did not engage in combat with the enemy, \"the veteran's lay testimony, by itself, will not be enough to establish the occurrence of the alleged stressor.\" The necessary additional evidence may be obtained from sources other than the veteran's SMRs.\n\n5. When the law or regulation changes after a claim has been filed or reopened, but not before the administrative or judicial appeal process has been concluded, the version most favorable to the appellant will apply unless Congress legislates otherwise.\n\n6. The service record may be devoid of evidence because many victims of personal assault, especially sexual assault and domestic violence, do not file official reports either with military or civilian authorities. Therefore, development of alternative sources for information is critical. Alternative sources that may provide credible evidence of the in-service stressor include:\n\n(a) medical records from private (civilian) physicians or caregivers who may have treated the veteran either immediately following the incident or sometime later\n\n(b) civilian police reports\n\n(c) reports from crisis intervention centers such as rape crisis centers or centers for domestic abuse.\n\n7. Rating board personnel must carefully evaluate all the available evidence. If the military record contains no documentation that a personal assault occurred, alternative evidence might still establish an in-service stressful incident. Behavior changes that occurred at the time of the incident may indicate the occurrence of an in-service stressor. Examples of behavior changes that might indicate a stressor include, but are not limited to:\n\n(a) visits to medical or counseling clinic or dispensary without a specific diagnosis or specific ailment\n\n(b) sudden requests that the veteran's military occupational series or duty assignment be changed without other justification\n\n(c) lay statements describing episodes of depression, panic attacks, or anxiety without an identifiable reasons for the episodes\n\n(d) evidence of substance abuse of alcohol or drugs.\n\n8. Rating boards may rely on the preponderance of evidence to support their conclusions even if the record does not contain direct contemporary evidence. In personal assault claims, secondary evidence may need interpretation by a clinician, especially if it involves behavior changes. Evidence that documents such behavior changes may require interpretation to clarify its relationship to the medical diagnosis by a VA neuropsychiatric physician.\n\n#### WHAT WE LEARNED\n\nThe most important points in this case are the eight rules stressed by CAVC. They are, in brief:\n\n1. If you appeal, the CAVC will look for medical evidence of PTSD, evidence of an in-service stressor, and evidence that supports a link between the two.\n\n2. The VA's decision must include detailed reasons and basis.\n\n3. The VA must make it clear to the claimant what evidence would support the claim.\n\n4. Evidence must support an in-service occurrence.\n\n5. When the law is changed, the veteran's claim should be evaluated with the version most beneficial to him or her.\n\n6. Service records do not always contain supporting evidence.\n\n7. ROs must evaluate all evidence, both for and against, when deciding a claim.\n\n8. ROs may rely on preponderance of evidence when there is no direct contemporary evidence.\n\n### _Zarycki v. Brown_ : In Combat\u2014Wrong MOS\n\n#### FACTS\n\nCases such as _Alex E. Zarycki v. Jesse Brown_ undoubtedly have played over and over again since 1961. Alex Zarycki's military occupational specialty was a noninfantry designation. Because of his MOS, according to the RO rating board, his alleged PTSD was not caused by participating in actual combat. I suppose you could say he was just in the neighborhood. I ask you, who ever heard of a _wireman_ and _switch board operator_ working in a hot war zone? His case is as follows.\n\n#### HISTORY\n\nMr. Zarycki served on active duty in the U.S. Army from September 1967 to June 1970. From February 1968 to February 1969, he was a member of Headquarters Battery, 7th Battalion, 15th Artillery Brigade, stationed in the Republic of Vietnam. From April to September 1969, the appellant was assigned to Fort Hood, Texas. From September 1969 to June 1970, appellant was reassigned to the Vietnam Theater as a member of Headquarters and Headquarters Company, Second Battalion, 22nd Infantry Brigade, 25th Infantry Division.\n\nThe appellant's military occupational specialties are listed as field wireman and switch operator. The appellant received the National Defense Service Medal, the Vietnam Service Medal, the Army Commendation Medal, and the Vietnam Campaign Medal. Both his enlistment and separation examinations were negative for any psychiatric disorders.\n\nIn April 1983, the appellant filed an application for compensation for PTSD with a VA Regional Office. On the application form, the veteran noted that he had experienced flashbacks about Vietnam and dreams of dead bodies. In support of his application, he submitted a summary of his hospitalization at a VA medical facility from April 15\u201321, 1983.\n\nAccording to the summary by a medical staff member at the VA hospital, Mr. Zarycki was admitted to the VA medical facility because of fear of losing control, depression, and poor impulse control resulting from the stress accompanying his impending divorce. It was noted that the appellant had a history of alcohol and drug abuse, financial difficulties, difficulty in finding employment, and domestic violence. Although the veteran had reported experiencing flashbacks and nightmares, none were reported during the period of hospitalization.\n\nAlso noted in the hospitalization summary was \"the appellant's criminal history, including stealing, breaking and entering, robbery, and shoplifting. The treating physician commented that the veteran was cooperative, somewhat anxious, trying to impress 'him' with the severity of his Vietnam experiences and how dangerous he could be, showing poor impulse control with a lot of underlying hostility. Basically he appeared to be an angry person who felt that the government owed him something and that he was fundamentally different from the rest of the patients.\"\n\nIn July 1983, the appellant filed a second application for compensation for PTSD with the RO; in describing the nature and history of his disability, he reported that he was experiencing flashbacks, nightmares, detachment, anger, and memory impairment. In September 1983, the RO sent a letter to the claimant requesting that he provide detailed information regarding specific incidents in service that he believed were related to his current psychiatric disorder. That same month, the appellant underwent a VA special neuropsychiatric examination.\n\nIn relating the appellant's military and medical history, the examining neuropsychiatrist, Dr. S. Loeb, noted in part:\n\nThe veteran was in the Army from September of 1967 to June of 1970. He was in Vietnam for 2 years, in Cambodia, with the 25th Infantry, in 1970 and later with the 2d Battalion, 22nd Infantry, Mechanized. He worked S-5 Intelligence and Reconnaissance.\n\nIn 1968 he was in a three-quarter track vehicle and ended up in a rice paddy. He was unconscious for minutes. He doesn't know of having a skull fracture, had a laceration, did not receive a Purple Heart, but did get medals.\n\nHe had the Bronze Star for performing above and beyond the call of duty, in a firefight. He has nightmares twice daily, wakes up in cold sweats. His wife, he dreams, is in a \"hooch\" and there are 2 squads of MVA [sic] coming toward him and he has no ammunition in his weapon. He sees friends that were killed. A friend named Keith got killed. An RPG put a hole in his head.\n\nHe has flashbacks every day. One was triggered off by holes that he saw in the aluminum siding, reminding him of wounds and the firefights connected with them. He was placed in a rice paddy. He feels guilty about surviving, feels guilty about killing innocent people, but he isn't sure that he did, because he couldn't actually see who was killed.\n\nDr. Loeb rendered a diagnosis of \"PTSD in a person with substance use, alcohol and mixed drugs and history of emotional instability, with considerable anxiety and depression.\"\n\n#### VA RATING OF THE CLAIM\n\nIn a rating decision dated October 1988, the RO denied service connection for PTSD. The RO noted: \"Post-traumatic stress disorder has been diagnosed on VA examination, but the diagnosis is not supported by any confirmed stressors or Vietnam experiences, by the evidence of record.\"\n\nIn response and to support his claim, the veteran submitted copies of his criminal records. (People who suffer PTSD are likely to engage in criminal behavior.) In December 1988, the RO again denied service connection for PTSD, finding that no new factual evidence had been presented to warrant service connection for post-traumatic stress disorder or any other condition, as \"criminal behavior and difficulties with the justice system resulting from criminal behavior do not constitute an adequate stressor for the grant of service connection for PTSD.\"\n\nOn November 8, 1989, the RO received the appellant's Notice of Disagreement with the denial of his claim for service connection for PTSD. In January 1990, the RO provided the veteran with a Statement of the Case and he perfected an appeal to the BVA shortly thereafter. In October 1990, the appellant's service representative presented oral arguments for subsequent presentation to and review by the BVA. In a decision dated December 11, 1990, the BVA remanded the appellant's claim to the RO for further evidentiary development and readjudication. Prior to readjudication, the board specifically directed:\n\n1. The veteran should be asked to submit a detailed statement regarding the traumatic events that he claims to have experienced in Vietnam, including names of others involved, units, dates, and so on. He should also be asked to submit the citation for his claimed Bronze Star Medal, since that award is not shown in his service personnel records. If adequate information is received, it should be forwarded to the U.S. Army and Joint Services Environmental Support Group in Washington, DC for verification.\n\n2. The veteran should be given a comprehensive examination by a board of two psychiatrists to determine the nature and extent of any psychiatric disabilities that he may have. The doctors should perform \"psychological studies with post-traumatic stress disorder subscales.\" All complaints and objective findings should be reported in detail. The claims folder should be made available to the examiners for their review prior to the examination. It is essential that the claims folder be reviewed before a final diagnosis is made, with attention to the police record submitted by appellant.\n\nIn January 1991 the RO sent a letter requesting the additional information regarding the veteran's in-service stressors and his Bronze Star Medal, as directed by the BVA in its December 1990 decision.\n\nIn February 1991 Mr. Zarycki underwent another VA special neuropsychiatric examination; the examination was conducted by one psychiatrist rather than by a board of two psychiatrists as had been directed by the BVA. The examining psychiatrist, Dr. G. A. Batizy, noted that \"appellant was not hallucinatory or delusional, but he showed elevated expansive mood with definite liability, also he had over-talkativeness, destructibility, and flight of ideas. He was also overtly anxious when he was talking about his Vietnam experiences. Intellectually, he was functioning fairly well. His memory was intact to remote and recent events. However, due to his disorganized thinking, both his insight and judgment were definitely distorted.\"\n\nDr. Batizy diagnosed the veteran with bipolar disorder, mixed moderate; PTSD; alcohol abuse; and cannabis abuse. Approximately one week later, the appellant underwent a battery of psychological tests. The administering psychologist, Dr. William F. Flynn, reported:\n\nReality contact has weak spots, with some perceptions blocked, at least initially, some distorted, and some peculiar ideation present. Lying just beneath the surface is a great store of combat\/mayhem themes. When stimuli in the environment touch on this, veteran could lose his judgment, impulse control, and even orientation, with resultant violence. He therefore keeps interaction to a minimum.\n\nReinforcing the trend toward isolation is paranoid component, involving the belief that et al. former co-workers were against him because of his Vietnam history, as well as jealousy on their part. He is also subject to auditory hallucinations in which drill instructors tell him he loves no one, and should kill. He seeks refuge from his conflicts in excessive fantasy, which only strengthens people's perception of him as \"weird.\" Obsessive rumination and a high level of tension make normal functioning difficult if not impossible. He is currently depressed in the mild range, and suicidal ideas are entertained at times, but the act is presently forestalled by concern for his son.\n\nDr. Flynn concluded that appellant's test record was \"consistent with a picture of PTSD superimposed on Bipolar Disorder Not Otherwise Specified.\"\n\nThereafter, the veteran submitted a detailed statement recounting his experiences in Vietnam and Cambodia. He reported that approximately two weeks after arriving in Vietnam in 1968, he saw a pile of dead enemy soldiers, which made him unable to eat for four days. He described a \"wave\" infantry attack at landing zone Crystal. He claimed during his first tour of duty in Vietnam, a fellow serviceman, Private First Class Wake, committed suicide behind the motor pool at LZ Uplift.\n\nThe veteran also alleged that his unit was ambushed by sniper fire between LZ Pony and LZ Uplift, and that the bullets came between himself and a friend. He claimed that during his second tour of duty in Vietnam, he worked in communications for S-5 Intelligence. He alleged that he was involved in reconnaissance missions into Cambodia and during one such mission, the vehicle in which he was riding was ambushed resulting in several casualties and injuries. He stated that in June 1970, a fellow soldier, identified only as \"Keith,\" wrote to his son in West Virginia telling him that he would be returning home in thirty days and would take him fishing. Keith then was killed shortly after writing the letter.\n\nThe veteran also claimed that an unidentified fellow soldier received stomach wounds from flying shrapnel, while another \"had his ear drums blown out.\" He also described how a medic in his unit was injured in the crotch by flying shrapnel. He reported that a personal friend of his, Sergeant Webb, was \"blown to pieces,\" killed by a booby trap.\n\nHe recalled that his battalion lost nine men in a firefight forty-five days prior to his departure from Vietnam. He remembered listening on the radio to air strikes against American troops. He stated that the only serviceman with whom he kept in contact with after his discharge, Don Stryker, died from \"cancer amputation of his right leg.\" Further, he stated that although he previously had alleged that he had received a Bronze Star, he had been confused and had actually received an Army Commendation Medal. In the final sentence of his statement, the appellant wrote: \"If you need more information please let me know.\"\n\n#### JUDICIAL APPRAISAL\n\nIn April 1991 the RO sent a letter detailing the appellant's claimed stressors to the U.S. Army and Joint Services Environmental Support Group (ESG) in order to attempt to verify the alleged stressors. A report of the ESG, dated May 1991, stated: On March 4, 1968, LZ Crystal received heavy mortar and recoilless rifle fire resulting in light damage; elements of the 2nd Battalion, 22nd Infantry, participated in combat operations in Cambodia during May and June 1970; PFC Russell D. Wake died from a self-inflicted gunshot wound at LZ Uplift on July 7, 1969; and Specialist Fourth Class Donald F. Webb was killed in action on March 9, 1970. The ESG was unable to document an enemy ground attack at LZ Crystal during the period February 1 through April 30, 1968. The ESG also could not confirm how Specialist Fourth Class Webb was killed. The ESG stated that in order to be able to research the other stressors that the veteran mentioned, more specific information was required from him. The record on appeal, however, does not indicate that the RO ever requested that supplemental information from the appellant.\n\nIn a rating decision dated July 1991, the RO again denied service connection for PTSD. After recounting the appellant's claimed stressors, the various diagnoses of PTSD, and the findings of the ESG, the RO determined that the veteran's records did not contain evidence of verified stressor events.\n\nThe RO noted that appellant's Army Commendation Medal was awarded for meritorious achievement rather than for valor in combat. The RO also noted that although the ESG had verified the suicide of PFC Russell Wake and the death of SP4 Donald Webb, these incidents were not sufficient stressors to warrant a finding of PTSD. With respect to those instances where the appellant indicated that he had witnessed other soldiers near him being killed in combat, the RO concluded that he had not provided sufficient information to allow verification by the ESG. The RO provided a Supplemental Statement of Case in August 1991. In its decision dated April 21, 1992, the BVA denied appellant's claim for entitlement to service connection for PTSD. A timely appeal to the CAVC followed.\n\nMr. Zarycki's was a major appeal case on the court's calendar that would require a three-judge panel. The outcome of this panel's review is law. In return, the Secretary of the Department of Veterans Affairs will be held accountable to the court that each level affected by the decision will comply with the findings. One question that comes to mind is: if the Secretary instructs his Board of Veterans Appeals and all Regional Offices to comply with the CAVC's rulings, why does the VA continue to fail to acknowledge any legitimate claim?\n\n#### COURT'S RULING\n\nHere is what the court had to say in the matter of _Alex E. Zarycki v Brown_ :\n\n#### A. Breach of Statutory Duty to Assist\n\nThe BVA found that appellant's claim for service connection for PTSD was well grounded and that the VA had satisfied its duty to assist appellant in developing the claim. After carefully reviewing the evidentiary record, the Court agrees that appellant fulfilled his initial burden of submitting a well-grounded claim; as noted in Part I, the record contains several diagnoses of PTSD by VA neuropsychiatrist, psychiatrists, and psychologists, including one psychiatrist who noted that appellant was \"overtly anxious when he was talking about his Vietnam experiences\" and another psychologist who noted that \"lying just beneath the surface is a great store of combat mayhem themes.\" In addition, the record contains appellants lay statements and hearing testimony to the effect that he had been engaged in combat with the enemy and had been exposed to significant stressors during his service in Vietnam.\n\nBecause appellant's claim for service connection for PTSD was well grounded, the Board was required to assist him in developing the facts pertinent to his claim. Contrary to its findings in its decision, however, the BVA did not carry out its statutory duty to assist.\n\nAlthough the Environmental Support Group (ESG) was unable to document all of the information supplied by appellant related to his alleged combat experiences in Vietnam and the alleged stressors he experienced during that period of service, it was able to verify some of the information. Further, and more importantly, the ESG informed the VA that it required additional information from appellant in order to investigate further his claimed experiences in service. Despite the fact that appellant had volunteered to provide additional information if needed, the record contains no evidence to suggest that the VA ever requested such information from appellant. In failing to accept appellant's offer and to afford him the opportunity to respond to the ESG's request for additional information, the VA breached its statutory duty to assist appellant in developing his claim.\n\nThe breach of the duty to assist takes on additional significance in the context of this appeal because the BVA did not make a specific factual finding as to whether or not appellant was \"engaged in combat with the enemy.\" Instead, the board appears to have intertwined the issue of whether appellant was engaged in combat with the enemy with whether there was corroborative evidence of his alleged in-service stressors:\n\nA valid diagnosis of [PTSD] requires, first of all, that there be a confirmed service stressor. American Psychiatric Association: _Diagnostic and Statistical Manual of Mental Disorders_ 250 (3rd ed., revised, 1987.) When referring to combat situations, there must be corroboration that the veteran was exposed to more than the ordinary stressful environment of a combat zone. We are not bound to accept either the veteran's uncorroborated account of his Vietnam experiences or a psychiatrist's unsubstantiated post-service opinions that the veteran's alleged [PTSD] had its origins in service.\n\nThe veteran's DD Form 214, Report of Transfer or Discharge, does not list any decorations that would indicate actual combat. Although the veteran initially contended that he won a Bronze Star, he later withdrew this contention. There are no complaints or findings in service of a psychiatric disorder, and the first post-service evidence of a psychiatric problem was not until April 1983, when the veteran was hospitalized because of the stress of a divorce, alcohol use, financial difficulties, and his inability to find a job.\n\nAlthough PTSD was subsequently diagnosed, this diagnosis was based on the veteran's subjective history. His contentions that his unit was involved in a ground attack on LZ Crystal in March 1968 could not be verified by the [ESG] which did determine that LZ Crystal received heavy mortar and recoilless fire resulting in light damage in March 1968. Additionally, the veteran's contentions that he was in an armored personnel carrier in approximately June 1970 that was ambushed, resulting in several deaths, could not be verified. Since there is no independent evidence confirming that the veteran was exposed to more than the ordinary stress inherent in service in a combat zone, service connection for PTSD is not warranted.\n\nOn remand, appellant will have the opportunity to provide the VA with the supplemental information requested by the ESG. Appellant is reminded that \"the duty to assist is not always a one-way street. If a veteran wishes help, he cannot passively wait for it in those circumstances where he may or should have information that is essential in obtaining the alleged evidence\" (duty to assist fulfilled where two attempts were made by VA to obtain necessary information despite the limited data supplied by appellant).\n\nAfter fulfilling its duty to assist, the BVA then will have the opportunity to make a specific factual finding as to whether or not appellant was engaged in combat with the enemy based on all the evidence of record. The Board will support its finding with an adequate statement of reasons or bases. If the BVA determines that appellant was engaged in combat with the enemy and that appellants alleged stressors are related to such combat, appellant's lay testimony, without further evidentiary development, will be enough to establish the occurrence of the claimed in-service stressors provided that such lay testimony is satisfactory, credible, and consistent with the circumstances, conditions, or hardships of such service.\n\nThe BVA will then need to determine (1) whether appellant's claimed stressors are sufficient to support a diagnosis of PTSD, i.e., _events during service that are outside the range of usual human experience and that would be markedly distressing to almost anyone_ , and (2) whether appellants currently diagnosed PTSD is causally related to those stressors.\n\nIf the BVA finds that appellant was not engaged in combat with the enemy, or that appellant was engaged in combat with the enemy but that his claimed stressors are not related to such combat, it then will have to determine, as a preliminary matter, whether appellant's testimony as to the claimed stressor is corroborated sufficiently by service records to establish the occurrence of the claimed stressful events (case remanded for BVA to address whether appellant was engaged in combat with the enemy, to examine the circumstances, conditions, and hardships surrounding his period of service, and, if necessary, to corroborate the occurrence of his alleged service-related stressors). Next, the BVA will have to determine whether the claimed stressful event is a sufficient stressor to support a PTSD diagnosis and whether appellant's current PTSD is related to the claimed in-service stressful episode.\n\n#### B. Additional Arguments\n\nBefore this Court appellant contends that the BVA erred by misapplying the diagnostic criteria for 1150 in the DSM-III-R; specifically, he posits that to support a diagnosis of PTSD stemming from a claimant's military service, a stressor must consist of exposure to \"more than an ordinary stressful environment\" rather than \"more than the ordinary stressful environment of a combat zone,\" as stated by the BVA in its decision. Appellant essentially contends that service in a combat zone may, in and of itself, constitute a sufficient stressor to support a diagnosis of PTSD. For the reasons discussed in before appellant's contentions are without merit.\n\nAppellant also asserts that the BVA erred in (1) basing its decision on the opinions of one psychiatrist rather than a board of two psychiatrists as the BVA had previously directed in its December 1990 decision; (2) rejecting the medical diagnoses supporting appellant's claim based on its own unsubstantiated medical conclusions; (3) failing to give due consideration to appellant's lay testimony; and (4) failing to apply the benefit of the doubt doctrine.\n\nBecause the BVA determined that there was no evidence of in-service stressors, however, it never reached the issue of whether or not appellant currently suffers from PTSD, irrespective of whether or not it would be related to service. As a result, the BVA did not address the medical evidence and lay testimony supporting appellant's current diagnoses of PTSD, did not rely on any medical evidence to support its decision, and did not discuss the application of the benefit of the doubt doctrine. However, in view of the need for a remand for readjudication, there is no need for the court to address these arguments at this time.\n\nA remand is meant to entail a critical examination of the justification for the decision. The Court expects that the BVA will reexamine the evidence of record, seek any other evidence the Board feels is necessary, and issue a timely, well-supported decision in this case.\n\nUpon consideration of the record and the filings of the parties, the court finds that a remand is warranted. Accordingly, the Secretary's improvident motion for summary affirmance is DENIED, the BVA's April 21, 1992, decision is VACATED; and this matter is REMANDED for further proceedings consistent with this opinion.\n\nThis case demonstrates that an appeal may not be concluded for years and also shows how court members think.\n\n### Chapter 9: What Have We Learned?\n\n1. Military personnel who serve in combat zones and do not have a combat MOS (such as infantryman) must meet a higher standard of proof to qualify for PTSD benefits. As this case shows, the veteran was part of a combat-unit but because he was assigned as a wireman and switchboard operator, his word was not acceptable under 38CFR 3.304(f) to trigger further development by the VA. As the case illustrates, the RO did not know what to ask for beyond requesting more hard evidence.\n\n2. The VA in this case relied on SMRs and DD-214 as the basic review records in denying the veteran's claim. They are supposed to obtain detailed information concerning the veteran's combat unit history.\n\nThe director of the Veterans Benefits Administration issued a letter to all VA Regional Offices and centers on January 8, 2001, which dictates they must make every effort to obtain all pertinent evidence, including private medical records, the veteran may have referred to, before they make an unfavorable decision.\n\n3. Regional Offices do not follow the remand order issued by the Board of Veterans' Appeals. They pretty much do what their own operational policy dictates. This makes it critical that you review your copy of the BVA's remand order making certain the local RO is in compliance. If not, immediately file a Notice of Disagreement, stating their determination is flawed.\n\n4. According to the APA's _Diagnostic and Statistical Manual of Mental Disorders, 4th edition_ , a valid diagnosis of PTSD requires, first of all, that there be a confirmed service stressor. When referring to combat situations, there must be corroboration that the veteran was exposed to more than the ordinary stressful environment of a combat zone.\n\nYou can preempt the possibility of your PTSD claim being denied if you contact the Environmental Support Group, now known as the U.S. Armed Services Center for Research of Unit Records, and request pertinent information concerning your circumstances under the provisions of the Freedom of Information Act. You can also search the web for after action reports by typing \"after action reports\" into any search engine.\n\n5. Once you have developed a detailed list of combat incidents that supports your contention of PTSD caused by a combat-related stressor, contact your local congress member or senator to forward your request to USASCRUR. When a member of the congress forwards a request, government agencies go into overdrive to respond.\n\n6. One of the first things that should strike you about this case is the number of years it takes to get a PTSD claim through bureaucratic mine fields. To succeed you must be light-years ahead of the VA by anticipating exactly what evidence you need to establish a well-grounded claim and what evidence you need to fully support your claim.\n\n7. If you have been awarded any one of the medals listed in M21-1MR, part IV, subpart ii, chapter 1, section D, paragraph 13 and have been diagnosed as being disabled by PTSD you have met the combat standard, therefore, you do not have to provide additional evidence to start the claim process.\n\n8. To write a statement of all your experiences, the outline of Sgt. David Raz's sworn statement to the VA is a good example.\n\n9. If you can possibly obtain a psychological evaluation prior to submitting your claim for PTSD you will have a much greater influence on the outcome of your claim.\n\n10. Do not submit any evidence that cannot be verified. This is especially true if the root of your trauma is not combat related. The six court decisions provide an excellence guide on what to do and what not to do.\n\n11. The first basic step is to submit an informal claim. It protects your early entitlement date but gives you a year to put your claim together with all the necessary evidence.\n\n12. It's essential that you go back and reread the \"What We Learned\" sections after each of the appeals in this chapter. Look at them as the dos and don'ts in preparing your PTSD claim.\n\n### Notes\n\n1. _Barela v. West_ , 11 280 (Vet. App. 1998).\n\n2. _Flayter v. Principi_ , 2 409 (Vet. App. 1992).\n\n3. Key knowledge for any veteran seeking a hearing before the U.S. Court of Appeals for Veterans Claims is that there are two basic types of decision the court renders. When a three-judge panel decides a case, the findings are binding on the veteran and the Department of Veterans Affairs. This means that if you are in a similar situation you can cite that case as precedent and the VA must rule according to the finding of the court. However, if the decision is rendered by a single judge, the decision is only applicable to that case and cannot be used as a binding decision in similar cases.\n\n4. _William F. Allen v. Anthony J. Principi_ , 237 F.3d 1368 (3rd Cir. 2001).\n\n5. _William F. Allen v. Anthony J. Principi_ , CAVA Docket Number 01-298, April 27, 2004.\n\n6. _Yancy v. Principi_ , Docket Number 01-298, April 27, 2004.\n\n7. _Doran v. Brown_ , 6 Vet. App. 283 (1994)). ( _Robert E. Doran v. Jesse Brown_ , USCAVC Docket No. 93-228 March 8, 1994.)\n\n8. _Alex E. Zarycki v. Jesse Brown_ , 6 91, 98 (Vet. App. 1993).\n\n9. The summary of this case has been edited to clarify points and to remove references to legal aspects that are not of interest for our purposes.\n\n10. Robert J. Epley, Training Letter Based on PTSD Review, January 8, 2001.\n\n11. The U.S. Armed Services Center for Research of Unit Records can be found online at <http:\/\/www.tpromo.com\/usvi\/claims\/step2.htm>.\n\n## GLOSSARY\n\n**affidavit** | A written statement that is sworn before a notary public. When the claimant, family, or friends provide such testimony, the VA must accept the statement as evidence and give it proper weight.\n\n---|---\n\n**Agency of Original** | The VARO where the claim first adjudicated.\n\n**AOJ** | agency of original jurisdiction\n\n**appeal** | A request for a review of an AOJ determination on a claim.\n\n**appellant** | An individual who appeals an AOJ claim determination.\n\n**Armed Forces** | U.S. Army, Navy, Marine Corps, Air Force, and Coast Guard.\n\n**board** | Board of Veterans Appeals.\n\n**board member** | A law judge appointed by the Secretary of Veterans Affairs and approved by the president, who decides veterans' benefits appeals.\n\n**BVA** | Board of Veterans' Appeals. The VA department that reviews benefits claims appeals and that issues decisions on those appeals.\n\n**Central Office** | The Director of the Compensation and Pension Service, VA Central Office, shall approve all VA regional office determinations establishing or denying POW status with the exception of those service departments findings establishing detention or internment was by an enemy government or its agents.\n\n**claim** | A request for veteran's benefits.\n\n**claim file** | The file containing all documents concerning a veteran's claim or appeal.\n\n**claim number** | A number assigned by VA that identifies a person who filed a claim; often called a \"C-number.\"\n\n**Court of Veterans Appeals**\n\n|\n\nAn independent United States Administrative Court that reviews appeals of BVA decisions.\n\n**date of receipt** | The date on which a claim, information, or evidence was received in the Department of Veterans Affairs, except as to specific provisions for claims or evidence received in the State Department or in the Social Security Administration of Department of Defense as to initial claims filed at or prior to separation.\n\n**decision** | The final product of BVA's review of an appeal. Possible actions are to grant or deny then benefit or benefits claimed or to remand them back to the AOJ for additional action.\n\n**determination** | A decision on a claim made at the AOJ. (Local Regional Office).\n\n**discharge** | Separation including retirement from the active military, naval, or air service.\n\n**evidence** | Records, documents, exhibits, or any probative material introduced to support your claim.\n\n**fair preponderance** | Evidence that must be sufficient to prove that the evidence against the veteran's claim outweighs that evidence offered in support of the claim.\n\n**file** | To submit in writing.\n\n**former prisoner of war**\n\n|\n\nA person who, while serving in the active military, naval, or air service, was forcibly detained or interned in the line of duty by an enemy or foreign government, the agents of either, or a hostile force.\n\n**hearing** | A meeting, similar to an interview, between an appellant and an official from the VA who will decide an appellant's case, during which testimony and other evidence supporting the case is presented. There are two types of personal hearings: regional offices hearings (also called local office hearings) and BVA hearings.\n\n**hostile force** | Any entity other than an enemy or foreign government or the agents of either whose actions are taken to further or enhance anti-American military, political, or economic objectives or views or to attempt to embarrass the United States.\n\n**in the line of duty** | When an injury or disease is said to be \"in the line of duty,\" it means it was incurred or aggravated during a period of active military, naval, or air service unless such injury or disease was the result of the veteran's own willful misconduct or, for claims filed after October 31, 1990, was a result of his or her abuse of alcohol or drugs. A service department finding that injury, disease, or death occurred in line of duty will be binding on the VA unless it is patently inconsistent with the requirements of laws administered by the VA.\n\n**issue** | A benefit sought on a claim or an appeal. For example, if a claimant submits an appeal of a decision on three different matters, the appeal is said to contain three issues.\n\n**marriage** | A marriage valid under the law of the place where the parties resided at the time of marriage, or the law of the place where the parties resided when the right to benefits accrued.\n\n**non-service-connected**\n\n|\n\nWith respect to disability or death, such a disability was not incurred or aggravated, or that the death did not result from a disability incurred or aggravated, in line of duty in the active military, naval, or air service.\n\n**notice** | A written notice sent to a claimant or payee at his or her latest address of record.\n\n**Notice of Disagreement**\n\n|\n\nA written statement expressing dissatisfaction or disagreement with a local VA office's determination on a benefit claim that must be filed within one year of the date of the Regional Office's decision.\n\n**political subdivision** | Includes the jurisdiction defined as a state and the counties, cities, or municipalities of each.\n\n**preponderance of evidence**\n\n|\n\nDemonstration that the VA's evidence is superior to that which was introduced by the veteran before a claim may be denied.\n\n**probative evidence** | Evidence that has a tendency to prove or actually prove an alleged entitlement.\n\n**proximate cause** | An event that caused an injury or disease and without such injury or exposure the disability would not have occurred.\n\n**reasonable doubt** | In a VA claim action, the term implies certainty that must be applied before benefits can be denied. If the amount of evidence for and against the claim is equal or nearly equal, the decision must be in favor of the claimant.\n\n**Regional Office** | A local VA office; there are fifty-eight VA Regional Offices throughout the United States and its territories.\n\n**Regional Office hearing**\n\n|\n\nA personal hearing conducted by an RO hearing officer. A Regional Office hearing may be conducted in addition to a BVA hearing.\n\n**relevant** | Refers to evidence that is pertinent, relative, connected to the point or target.\n\n**remand** | An appeal returned to the Regional Office or medical facility where the claim originated.\n\n**service connection** | With respect to disability or death, that such disability was incurred or aggravated, or that the death resulted from a disability incurred or aggravated, in the line of duty in the active military, naval, or air service.\n\n**state** | Each of the several states, territories, and possessions of the United States, the District of Columbia, and commonwealth of Puerto Rico.\n\n**Statement of the Case (SOC)**\n\n|\n\nPrepared by the AOJ, this is a summary of the evidence considered, as well as a listing of the laws and regulations used in deciding a benefit claim. It also provides in formation on the right to appeal an RO's decision to BVA.\n\n**Supplemental Statement of Case (SSOC)**\n\n|\n\nA summary, similar to a SOC, that the VA prepares if a VA Form-9 contains a new issue or presents new evidence and the benefit is still denied. A Supplement Statement of the Case will be provided when an appeal is returned (remanded) to the RO by the Board of Veterans' Appeals for new or additional action.\n\n**traveling board hearing**\n\n|\n\nA personal hearing conducted at a VA Regional Office by a judge of the BVA.\n\n**veteran of any war** | Any person who served in the active military, naval, or air service during a period of war.\n\n**veteran service organization**\n\n|\n\nAn organization that represents the interests of veterans. Most veterans' service organizations have specific membership criteria, although membership is not usually required to obtain assistance with benefit claims or appeals.\n\n**war time service** | The following dates are formal periods of war. If a veteran served during one or more of these periods, he or she will have been considered as serving during a wartime period: WWI, April 6, 1917\u2013November 11, 1918 (Veterans who served in Russia between April 6, 1917\u2013July 7, 1921 are also considered wartime veterans);\n\nWWII, September 16, 1940\u2013December 31, 1946;\n\nKorean Conflict, June 27, 1950\u2013January 31, 1955;\n\nVietnam War, August 5, 1964\u2013May 7, 1975 (If you served as an advisor in Vietnam, any time between February 28, 1961 and August 4, 1964);\n\nPersian Gulf War, August 2, 1990 to a date yet to be determined.\n\nWartime service also means any time in which combat service was performed during between January 1, 1947 and a date yet to be determined. Several examples would be the Berlin Air Lift, Lebanon Crisis, Grenada, Iranian Crisis, and Bosnia.\n\n**waters adjacent to** | With regard to service during the Mexican boarder period, the waters (including the islands) that are within 750 nautical miles (863 statute miles of the coast of the mainland of Mexico).\n\n**willful misconduct** | An act involving conscious wrongdoing or known prohibited action ( _malum in se_ or _malum prohibitum_ ). A service department finding that injury, disease, or death was not because of misconduct will be binding on the Department of Veterans Affairs unless it is patently inconsistent with the facts and the requirement of laws administered by the Department of Veterans Affairs.\n\n## INDEX\n\n_Addison v. Brown,_ n9\n\nafter action reports, ,\n\nAir Medal, , , \u201319, , ,\n\nalcohol abuse, , , , \u201391\n\n_Allen v. Principi,_ \u201391,\n\nAMIE (Automated Medical Information Exchange),\n\nappeal process. _See also_ Board of Veterans' Appeals (BVA); U.S. Court of Appeals for the Federal Circuit; U.S. Court of Appeals for Veterans Claims\n\nnumber of claims appealed,\n\npercentage of veterans who appeal, ,\n\nrules for,\n\nSupreme Court,\n\ntime frame for, ,\n\nAutomated Medical Information Exchange (AMIE),\n\n_Barela v. West_ ,\n\n_Beaty v. Brown_ , ,\n\nbenefit of doubt law (38 _USC_ 5107), \u201395\n\nbenefits increase, claims for, , \u201319\n\nBill of Rights, \u20139\n\nBoard of Veterans' Appeals (BVA)\n\nappeal process,\n\nauthority of,\n\nremand and reversal rates of, ,\n\nreview process of,\n\nTDIU appeal,\n\ntime frame for review, xxvi\n\nwebsite, n1\n\nBronze Star, , , \u201319, , , ,\n\n_Brown, Addison v.,_ n9\n\n_Brown, Beaty v.,_ ,\n\n_Brown, Clarkson v._ ,\n\n_Brown, Collette v.,_\n\n_Brown, Doran v.,_ , \u201334, \u2013200\n\n_Brown, Irby v.,_ ,\n\n_Brown, James v.,_ ,\n\nBrown, Jesse,\n\n_Brown, Sklar v.,_ , n6\n\n_Brown, Zarycki v._ , \u201314\n\n_Brown v. Derwinski,_\n\nbuddy letters, ,\n\nBush, George H. W.,\n\nBush, George W.,\n\nC&P examination. _See_ Compensation and Pension Examination\n\n_C &P Service Clinician's Guide,_ , , n7\n\n_Cartright v. Derwinski,_\n\nchronic diseases, \u201325\n\nclaims\n\namending,\n\napproval rate, , \u201389\n\ncompleteness of, , ,\n\ncomponents of, \u201336, \u201393\n\nconfusion about,\n\ndenial of. _See_ denial of claims\n\ndeveloping winning claims, \u201338, , \u20135\n\nevidence for. _See_ evidence\n\nExpedited Action Attachment,\n\ninterpretation of laws for,\n\nmedical evidence for. _See_ medical evidence\n\nmistakes, procedural,\n\nnotice of decisions,\n\nregulations and guidelines for, \u201319, \u201395, \u201322,\n\navailability of, ,\n\nreopening, ,\n\nreview by physician prior to C&P Exam, , , n10, ,\n\nstatement in support of. _See_ Statement in Support of Claim (VA form 21-4138)\n\nsteps to follow, \u201312\n\nsubmission of, senator or congressional representative help, ,\n\ntime frame for processing, , \u201315\n\ntop sheeting, xvi,\n\ntracking of, , , , \u201387\n\ntypes to file, \u201384, \u201312, \u201319,\n\nVA Disability Claims Processing, \u201390\n\nClaims Processing Improvement Task Force, \u201388\n\n_Clarkson v. Brown,_\n\n_Code of Federal Regulations_\n\nauthority of,\n\navailability of, , ,\n\n32 _CFR_ 578.11,\n\n38 _CFR,_ \u2013100,\n\n3.102: reasonable doubt doctrine,\n\n3.105: clear and unmistakable error, , , ,\n\n3.200: evidence requirements, , ,\n\n3.304: direct service connection, \u201316, ,\n\n3.307: tropical diseases, ,\n\n3.308: tropical diseases,\n\n3.309: chronic diseases, \u201325\n\n3.309: tropical diseases,\n\n3.326: examinations,\n\n3.340: insurance ratings,\n\n3.340: permanent total disability, \u201399\n\n3.340: total disability ratings, \u201399, , \u201367\n\nPart IV: Schedule for Rating Disabilities, n3,\n\n4.125: mental disorders, diagnosis of, ,\n\n4.132: mental disorders rating schedule,\n\n4.16: total disability rating, , \u201367, , , ,\n\n4.18: unemployability, \u2013100,\n\n4.2: C&P exam report interpretation, , ,\n\n_Collette v. Brown,_\n\n_Collier v. Derwinski,_\n\n_Colvin v. Derwinski,_ , ,\n\ncombat\n\ncombat duty statement, \u201333,\n\ncombat stressors, , \u20138, \u201332, \u201314\n\nevidence of, , \u201316, , \u201319, , \u201311, , , \u201331, ,\n\ninjuries and disease related to, \u201327\n\nCompensation and Pension Examination\n\n_C &P Service Clinician's Guide,_ , , n7\n\nclaim file review by physician, , , n10, ,\n\nexam reports\n\ncompleteness of,\n\nexamples of, \u201354, \u201382\n\ninterpretation of, , ,\n\npurpose of,\n\nreturn of exam reports, ,\n\nvalidity of,\n\nfailure to report for exam, , n8\n\nimportance of,\n\ninformation to provide to examiner,\n\nnotification of, \u201350,\n\nprocedures for, , \u201347\n\nqualifications and experience of examiners, , , n10, \u201343, \u201345, , \u201357, \u201359, , n6\n\nrecords to request, ,\n\nrequirement for, , ,\n\nscheduling during active stage of condition, , n9\n\nthorough and contemporaneous exams,\n\ntime spent on exam, ,\n\ntips for, , \u201359\n\ntypes of and scope of exams, \u201346, \u201357\n\nwitnesses to,\n\nworksheet for exams, \u201355\n\ncongressional representative. _See_ senator or congressional representative\n\nDD-214 (Report of Separation), , n4\n\nDecision Review Officer, \u201369\n\ndenial of claims\n\nbasis for, , \u201396, \u201311, \u201313, , \u201330\n\nbenefit of doubt law (38 _USC_ 5107), \u201395\n\ndenial rate,\n\nlaw to defend against, \u201391\n\nmoney saved by denying claims,\n\nreasonable doubt doctrine (38 _CFR_ 3.102),\n\nscheduling of C&P Exam and, , n9\n\nservice records, absence of support in, \u201334\n\ntime frame for, \u201390\n\nvalidity of C&P Exam report,\n\nwillful misconduct,\n\n_Derwinski, Brown v.,_\n\n_Derwinski, Cartright v.,_\n\n_Derwinski, Collier v._ ,\n\n_Derwinski, Colvin v.,_ , ,\n\n_Derwinski, Dusek v.,_ n8\n\nDerwinski, Edward Joseph,\n\n_Derwinski, Gleicher v._ , , \u201369\n\n_Derwinski, Masors v.,_\n\n_Derwinski, Murincsak v.,_ ,\n\n_Derwinski, Parker v._ ,\n\n_Derwinski, Proscelle v._ ,\n\n_Derwinski, Shoemaker v.,_\n\n_Diagnostic and Statistical Manual of Mental Disorders_ (APA), , , , ,\n\ndisability ratings\n\nanalogous ratings,\n\npermanent total disability,\n\nrating code ending in ,\n\nseverity of disability,\n\n38 _CFR_\n\n3.340: total disability ratings, \u201399, , \u201367\n\nPart IV: Schedule for Rating Disabilities, , \u20134, n3, ,\n\n4.132: mental disorders rating schedule,\n\n4.16: total disability rating, , \u201367, , , ,\n\ndischarge statement (DD-214), , n4\n\ndoctrine of reasonable doubt (38 _CFR_ 3.102),\n\n_Doran v. Brown,_ , \u201334, \u2013200\n\ndrug abuse, , , , \u201393\n\n_Dusek v. Derwinski,_ n8\n\nDuty to Assist Claimants (38 _USC_ 5103A), \u201394,\n\nentitlement award letter, \u201358\n\nEvens, Lane, \u201314\n\nevidence. _See also_ medical evidence\n\nassistance in obtaining, \u201394, , , , ,\n\nof combat, , \u201316, , \u201319, , \u201311, , , \u201331, ,\n\ncombat duty statement, \u201333\n\ndevelopment of, ,\n\nDuty to Assist Claimants (38 _USC_ 5103A), \u201394,\n\nregulations and guidelines for, , ,\n\nrequest for, \u201327, \u201331\n\nrequired information and evidence, \u201392, \u20136, \u201320\n\nresponsibility to collect,\n\nreview of, , \u201311\n\nsenator or congressional representative help in requesting, , , , ,\n\nservice records. _See_ military service records\n\nsources of, \u20136\n\nafter action reports, ,\n\nmilitary service records. _See_ military service records\n\nstatements from spouses, family, and friends, \u201343, , , \u201373\n\nsworn statements\n\nexamples of, \u201343, \u201346,\n\npreparation of, , \u201338, \u201373\n\nrelevance of, , , , \u201373\n\nfor TDIU claim, \u201374\n\ntime requirements, , \u201312\n\nVA Manual M21\u20131, \u201318\n\nvocational evidence, \u201371\n\nwell-developed evidence, , ,\n\nExpedited Action Attachment,\n\nfamily and friends, statements from, \u201343, , , \u201373\n\n_Federal Register,_ \u201316\n\nFee-Basis Program, \u201343, ,\n\n_Flayter v. Principi,_ \u201391\n\nfriends, statements from, \u201343, , , \u201373\n\nGardner, Fred,\n\nGI Bill of Rights,\n\n_Gleicher v. Derwinski,_ , \u201369\n\nHealth Administration. _See_ U.S. Department of Veterans Affairs: Health Administration\n\nheroism, \u201319,\n\nhospital records,\n\ninformal claims, \u201328, \u201312, , \u201337, \u201375,\n\ninsurance ratings,\n\n_Irby v. Brown,_ ,\n\n_James v. Brown,_ ,\n\n_Masors v. Derwinski,_\n\nmedical evidence. _See also_ Compensation and Pension Examination; evidence\n\nbenefits increase, claim for, \u201319\n\nchronic diseases, \u201325\n\ncombat-related injuries and diseases, \u201327\n\ndevelopment of,\n\ndiagnoses by lay people,\n\nindependent medical opinions (38 _USC_ 5109),\n\nminimum medical standards, \u201336\n\nphysicians' statements, \u201335, \u201339, , n3, , , , \u201336, , , ,\n\npre-existing medical conditions, \u201323,\n\nregulations and manuals for,\n\nrequest for, , \u201348\n\nsources of military hospital records,\n\nspecificity of, , ,\n\nfor TDIU claim, \u201370\n\ntropical diseases, \u201326,\n\nmedical records,\n\nmedical retirement,\n\nmedical separation,\n\nMemorandum of Understanding (1996),\n\nmental disorders\n\nattitude toward, \u20133\n\ncoexisting disorders, , , , ,\n\ndiagnosis instead of PTSD, \u20132\n\ndiagnosis of, ,\n\ndisability ratings,\n\nfiscal decisions and halfway houses,\n\nmilitary hospital records,\n\nmilitary occupational specialties (MOS), \u201314\n\nmilitary service records\n\nabsence of support in, \u201334\n\nafter action reports, ,\n\navailability of, , n1, n4,\n\nDD-214 (Report of Separation), , n4\n\nsenator or congressional representative help in requesting, , ,\n\nMOS (military occupational specialties), \u201314\n\n_Murincsak v. Derwinski,_ ,\n\nNational Center for Post-Traumatic Stress Disorder,\n\nNational Personnel Records Center\n\nmedical evidence source,\n\nrecord requests to, n4\n\nStandard Form , n4\n\nnoncombat stressors, , \u20137, \u201334, \u2013200\n\nNotice of Disagreement\n\ndeadline for filing, xxv\n\nexample of, \u201346, \u201362, \u201380\n\nfocus of,\n\nform for,\n\nwhen to file, ,\n\nOffice of Management and Budget,\n\noriginal claim,\n\n_Parker v. Derwinski,_\n\n_Patton v. West,_ \u2013203\n\npermanent total disability,\n\npersonal assault claims, , \u20136, \u20139, , \u2013203\n\nPost-traumatic Stress Disorder (PTSD)\n\nassessment of, \u20136,\n\ncoexisting disorders, , , , , , , , \u201391\n\ncompensation for, \u20132\n\nconsequences associated with, \u20137\n\ndescription of, , ,\n\ndevelopment of, , , \u2013111\n\ndiagnosis of, , \u20139, , , \u201310, , , \u201336\n\nhistorical cases of, \u20135\n\nname change of disorder,\n\nprevalence of, , , ,\n\npsychosocial functioning and, \u20138\n\nsymptoms of, , , \u201310, , \u201336\n\ntreatment of,\n\nPOW status, \u20132\n\npreexisting medical conditions, \u201323,\n\npresumption of soundness policy, \u201323,\n\n_Principi, Allen v.,_ \u201391,\n\n_Principi, Flayter v._ , \u201393\n\n_Principi, Yancey v.,_ \u201398\n\n_Proscelle v. Derwinski,_\n\nPurple Heart, , ,\n\nrating boards\n\nauthority of,\n\nrating decisions, \u201338, , \u20136\n\nrules that guide, \u201322\n\ntime spent on claims,\n\ntraining of members of, , , , ,\n\nRaz, David, \u201384\n\nBronze Star,\n\nC&P examination, \u201354, , \u201382\n\nclaim denial,\n\nclaim filing,\n\nclaim status,\n\ncombat duty statement, \u201333\n\nDecision Review Officer services, \u201369\n\nentitlement award letter, \u201358\n\nevidence request, \u201327, \u201348\n\nNotice of Disagreement, \u201346, , \u201362, \u201380\n\nphysician's statements, \u201335, \u201339,\n\nPurple Heart, ,\n\nrating decision, \u201338\n\nstatement in support of claim, \u201328, , , \u201366, , \u201380\n\nTDIU appeal, , \u201380\n\nVA Form 21\u20134138, \u201328,\n\nwife's statement, \u201343\n\nwork release, \u201324\n\nwounding of, , ,\n\nReagan, Ronald,\n\nreasonable doubt doctrine (38 _CFR_ 3.102),\n\nRecords Management Center,\n\nreopened claims, ,\n\nReport of Separation (DD-214), , n4\n\nSchedule for Rating Disabilities, , \u20134, n3, ,\n\nsecondary service-connected disorders, ,\n\nsenator or congressional representative\n\nclaims submission help, ,\n\ninformation requests through, , , , ,\n\nservice records. _See_ military service records\n\nservice-connected disorders\n\nBill of Rights for,\n\nchronic diseases, \u201325\n\ncombat-related injuries and diseases, \u201327\n\ncontinuity of symptomatology,\n\nentitlement, basic, \u201391\n\nentitlement award letter, \u201358\n\nlaws governing, \u201319\n\npreexisting medical conditions, \u201323,\n\npresumption of soundness policy, \u201323,\n\nqualifications for, , , , \u201327, ,\n\nsecondary service-connected disorders, ,\n\ntropical diseases, \u201326,\n\nveterans' benefits and,\n\nsexual assault, , \u20136, \u20139, , \u2013205\n\n_Shoemaker v. Derwinski,_\n\n_Sklar v. Brown,_ , n6\n\nSocial Security Disability (SSD), , \u201374,\n\nstate veteran service offices, \u201389\n\nStatement in Support of Claim (VA form 21-4138)\n\nbenefits increase, claim for, \u201319\n\nDecision Review Officer services, \u201366\n\nexamples of, , \u201362, \u201366, , \u201380\n\ninformal claim submission, \u201328,\n\nNotice of Disagreement, , \u201362, \u201380\n\nTDIU claim, \u201371\n\nStatement of the Case (SOC), xxv\u2013xxvi\n\nsubstance abuse, , , , \u201393\n\nSupplement Statement of Case, xxvi\n\nSupreme Court,\n\nsworn statements\n\nexamples of, \u201343, \u201346,\n\npreparation of, , \u201338, \u201373\n\nrelevance of, , , , \u201373\n\ntop sheeting, xvi,\n\nTotal Disability Based on Individual Unemployability (TDIU)\n\nappeal of, , \u201380,\n\napplication for benefits, , \u201366, \u201381\n\ncourt cases concerning,\n\ndenial of claims, \u201366\n\nefforts to limit entitlement, \u201364,\n\neligibility for, \u201368, \u201382\n\nevidence for, \u201374\n\npurpose of,\n\nregulations and guidelines for, , \u201368\n\nVA Form 21-8940 (TDIU application form), , \u201366, \u201381\n\ntotal disability ratings\n\n38 _CFR_ 3.340: total disability ratings, \u201399, , \u201367\n\n38 _CFR_ 4.16: total disability rating, , \u201367, , , ,\n\ntropical diseases, \u201326,\n\nunemployability, \u2013100. _See also_ Total Disability based on Individual Unemployability (TDIU)\n\nU.S. Armed Services Center for Research of Unit Records, ,\n\nU.S. Army and Joint Services Environmental Support Group, \u201331, n1,\n\n_U.S. Code_\n\nTitle 38 _USC:_ Veterans Benefits\n\n1131: Basic Entitlement, \u201391\n\n1154B: Consideration to be Accorded Time, Place, and Circumstances of Service, ,\n\n5103: Notice to Claimants of Required Information and Evidence, \u201392\n\n5103A: Duty to Assist Claimants, \u201394, ,\n\n5104: Decisions and Notices of Decisions,\n\n5107: Benefit of the Doubt, \u201395\n\n5107: Claimant Responsibility,\n\n5109: Independent Medical Opinions,\n\nU.S. Court of Appeals for the Federal Circuit\n\nappeal process,\n\nauthority of,\n\ncombat injury claims,\n\nsecondary service-connected disorders,\n\nU.S. Court of Appeals for Veterans Claims\n\nappeal process,\n\nauthority of, ,\n\navailability of decisions of,\n\ncombat injury claims,\n\ndenial of claims, ,\n\npurpose of,\n\nsecondary service-connected disorders,\n\nTDIU claims,\n\ntypes of decisions by, ,\n\nU.S. Department of Veterans Affairs Claims Processing Improvement Task Force, \u201388\n\nfunding for, \u201314, \u201317, ,\n\nHealth Administration\n\n_C &P Service Clinician's Guide,_ , , n7\n\nenrollment in healthcare program, \u201344,\n\nFee-Basis Program, \u201343, ,\n\nfinancial management of, ,\n\nfunctions of,\n\nmanagement of, \u201342\n\nphysicians in, \u201343\n\nprioritization of healthcare, \u201344,\n\n159\u201360\n\nVeterans Healthcare Reform Act (1996), \u201344,\n\nhistory of, \u201317 laws governing, \u201319\n\nNational Center for Post-Traumatic\n\nStress Disorder,\n\norganizational structure of,\n\nRecords Management Center,\n\nRegional Offices\n\nclaims processed by,\n\nrecord requests to, \u201347\n\nrecords transferred to,\n\ntelephone communication with,\n\ntelephone number for,\n\nVA Adjudication Procedure Manual M21-1, ,\n\navailability of,\n\ncombat, evidence of, , , \u201311\n\n5.14: post-traumatic stress disorder, \u20136\n\n7.46: medical prerequisites, , n10\n\n11.37: post-traumatic stress disorder, \u201318, \u201311\n\nVA Disability Claims Processing, \u201390\n\nVA Form 21-4138 ( _See_ Statement in Support of Claim [VA form 21-4138])\n\nVA Form 21-8940 (TDIU application form), , \u201366, \u201381\n\nVA medical manual IB11-56,\n\nU.S. Marine Corps\n\narchives available through Commandant, \u201331, n1\n\nVA. _See_ U.S. Department of Veterans Affairs\n\nValor Device (\"V\" Device), , , , ,\n\nVeterans Affairs. _See_ U.S. Department of Veterans Affairs\n\nveterans' benefits, \u201317\n\nVeterans Benefits Act (2003),\n\nVeterans Claim Assistance Act (2000), ,\n\nVeterans Healthcare Reform Act (1996), \u201344,\n\n_Veterans Survival Guide_ (Roche), ,\n\nvocational evidence, \u201371\n\n_West, Barela v.,_\n\n_West, Patton v.,_ \u2013205\n\nwitnesses\n\nto C&P Exam,\n\nstatements from, \u201343, , , \u201373\n\n_Yancey v. Principi,_ \u2013205\n\n_Zarycki v. Brown,_ \u201314\n\n## ABOUT THE AUTHOR\n\n**John Roche** 's involvement as a veteran's advocate began nine years before he retired from the U.S. Air Force. His VA actions on behalf of retired military personnel came to the attention of the St. Petersburg Regional Office's adjudication officer when, as a causality assistance officer, he successfully argued for a widow that her retired Air Force officer husband's accidental death in the county jail was service-connected.\n\nImmediately after leaving the air force, Roche joined the VA Regional Office adjudication division as a claims specialist. During the three years that he was with the VA, he completed its 1,560-hour formal training program and gained considerable insight into why so many claims were denied. Veterans did not know how to prove their claims; they relied on the VA to do it for them.\n\nHis decision to leave the VA was fostered by policies that contradicted the reason he accepted the job. Since leaving the VA, Roche has used his knowledge of the system to help more than forty thousand clients during the past thirteen years as a county service officer. Health reasons caused him to step down as an active veteran's advocate with the county service in 1996.\n\nRoche is a published writer specializing in veteran's issues and has four forthcoming books on the subject. He has written five books focusing on how to get claims approved. His books have introduced him to veterans from all over the United States who have had legitimate claims denied.\n\nMr. Roche is a lifetime member of the Disabled American Veterans, National Association of Uniform Services, and the Military Officers Association of America. He is also a member of Congressman Michael Bilirakis's advisory board. He lives in Palm Harbor, Florida.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nTable of Contents\n\nTitle Page\n\nCopyright Page\n\nDedication\n\nEpigraph\n\nForeword\n\nIntroduction\n\nChapter 1 - A Medicine Called Energy\n\nChapter 2 - Energy Techniques for Health and Vitality\n\nChapter 3 - Dancing with Your Hormones\n\nChapter 4 - Reclaiming the Wisdom of Your Menstrual Time\n\nChapter 5 - Sexuality, Fertility, Pregnancy, and Birth\n\nChapter 6 - Menopause\u2014Gateway to Your Second Prime of Life\n\nChapter 7 - Weight Management Tips\n\nAppendix - Talking with Your Body's Energies\u2014 The Art of Energy Testing\n\nNotes\n\nResources\n\nIndex\n\nOther Books and Programs of Interest\n\nAbout the Author\nOther books coauthored by Donna Eden and David Feinstein\n\n_Energy Medicine_\n\n_The Promise of Energy Psychology_\nJEREMY P. TARCHER\/PENGUIN   \n_a member of_   \n_Penguin Group_ ( _USA_ ) _Inc._   \n_New York_\n\nJEREMY P. TARCHER\/PENGUIN   \nPublished by the Penguin Group   \nPenguin Group (USA) Inc., 375 Hudson Street, New York, New York 10014, USA   \nPenguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario M4P 2Y3, Canada   \n(a division of Pearson Canada Inc.) \u2022 Penguin Books Ltd, 80 Strand, London WC2R 0RL, England   \nPenguin Ireland, 25 St Stephen's Green, Dublin 2, Ireland (a division of Penguin Books Ltd)   \nPenguin Group (Australia), 250 Camberwell Road, Camberwell, Victoria 3124, Australia   \n(a division of Pearson Australia Group Pty Ltd) \u2022 Penguin Books India Pvt Ltd, 11 Community Centre,   \nPanchsheel Park, New Delhi-110 017, India \u2022 Penguin Group (NZ), 67 Apollo Drive, Rosedale,   \nNorth Shore 0632, New Zealand (a division of Pearson New Zealand Ltd) \u2022 Penguin Books (South Africa)   \n(Pty) Ltd, 24 Sturdee Avenue, Rosebank, Johannesburg 2196, South Africa   \nPenguin Books Ltd, Registered Offices:   \n80 Strand, London WC2R 0RL, England\n\nCopyright \u00a9 2008 by Donna Eden and David Feinstein\n\nAll rights reserved. No part of this book may be reproduced, scanned,   \nor distributed in any printed or electronic form without permission. Please do   \nnot participate in or encourage piracy of copyrighted materials in violation   \nof the authors' rights. Purchase only authorized editions.   \nPublished simultaneously in Canada\n\nMost Tarcher\/Penguin books are available at special quantity discounts for bulk purchase for sales promotions, premiums, fund-raising, and educational needs. Special books or book excerpts also can be created to fit specific needs. For details, write Penguin Group (USA) Inc. Special Markets, 375 Hudson Street, New York, NY 10014.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nEden, Donna.   \nEnergy medicine for women : aligning your body's energies to boost your health and vitality \/ Donna Eden with David Feinstein.   \np. cm.   \nIncludes bibliographical references and index.\n\neISBN : 978-1-585-42647-8\n\n1. Energy medicine. 2. Women\u2014Health and hygiene. I. Feinstein, David. II. Title.   \nRZ421.E   \n613'.04244\u2014dc22\n\nPrinted in the United States of America\n\nNeither the publisher nor the authors are engaged in rendering professional advice or services to the individual reader. The ideas, procedures, and suggestions contained in this book are not intended as a substitute for consulting with your physician. All matters regarding your health require medical supervision. Neither the authors nor the publisher shall be liable or responsible for any loss or damage allegedly arising from any information or suggestion in this book.\n\nWhile the authors have made every effort to provide accurate telephone numbers and Internet addresses at the time of publication, neither the publisher nor the authors assume any responsibility for errors, or for changes that occur after publication. Further, the publisher does not have any control over and does not assume any responsibility for author or third-party websites or their content.\n\n<http:\/\/us.penguingroup.com>\n_Dedicated to my daughters,_   \n_Tanya Dahlin and Dondi Dahlin\u2014_   \n_may they and their generation soar!_\n\nJeremy P. Tarcher, who has been the trumpet for hundreds of our culture's most inspiring voices, has brought special notice to my own voice as well. I am grateful for his encouragement, his wisdom, and his pioneering spirit.\n\nSara Carder at Tarcher\/Penguin has been a dream of an editor.\n\nThe late Jeffrey Harris, M.D., was a wise and wonderful source of guidance and support to me throughout this book.\n\nEvery person involved in our growing energy medicine teaching community has contributed in his or her own special way to the development of the ideas presented here.\nCredits\n\nPhotographs by Christine Alicino\n\n_Illustration Models_   \nChristine Alicino   \nInfant Audrey Dahle   \nCindy Dahle   \nDondi Dahlin   \nTanya Dahlin   \nDonna Eden   \nRose Harris   \nCatherine Potenza   \nBeate Priolo   \nTia Via\nThere is a vitality, a life force, an energy, a quickening, that is translated through you into action, and because there is only one of you in all time, this expression is unique. And if you block it, it will never exist through any other medium and will be lost.\n\n\u2014MARTHA GRAHAM\nForeword\n\n_Christiane Northrup, M.D._\n\nI began experimenting with energy medicine in my gynecological practice many years ago, using it when I had to perform invasive office procedures such as endometrial biopsies, the removal of IUDs, or tests that involved injecting dye into the uterus. I sensed how disruptive these routine procedures were to the body's natural energy fields, so as the final part of the treatment, I would have my patient lie down and I would move my hands in long passes above her body. Something in this simple act seemed to help restabilize the body's energy fields. Many women would report immediate relief of pain or cramps, pleasurable tingling sensations in the areas that had been traumatized, or a deepening calm throughout their bodies.\n\nWhile about as simple a technique as exists, it was pleasant, noninvasive, and effective. Now, even as the field of hands-on energy medicine has developed far more sophisticated procedures than my early experiments, the approach has retained these basic qualities: simple, pleasant, noninvasive. But don't be fooled. Your hands hold powerful medicine\u2014\"energy medicine\"\u2014that beyond just soothing the body after an invasive procedure is able in itself to prevent or help overcome challenging illnesses.\n\nEnergy medicine is moving into our health care system quickly, powerfully, and none too soon. We have as a society come to the end of our journey in Newtonian medicine, a perspective that looks at the body more like a bag of organs and bones than a miracle of animation; that focuses on illness rather than optimizing health; and that often futilely seeks to identify simple cause-and-effect relationships rather than to grasp how body, mind, and spirit are profoundly interrelated. We have as a result built a medical system based on drugs and surgery that usually doesn't become meaningfully involved in health care until after the person is already sick. We are like a river patrol that sends powerboats out into the rapids to rescue people who are drowning rather than going upstream and figuring out how to keep them from falling into the rapids in the first place.\n\nWe can't go much further with this model. It isn't working, as Michael Moore's documentary _Sicko_ makes abundantly and compassionately clear. The old paradigm is breaking down before our eyes. But we also see the emergence of a new paradigm and, happily, one whose ancient roots have withstood the test of time. The new paradigm, as this book beautifully articulates, addresses biological processes at their energetic foundation; gives rise to methods that are precise, practical, rapid, and noninvasive; optimizes health as well as countering illness; empowers the person with effective methods for back-home self-care; and integrates body, mind, and spirit.\n\nBut perhaps the best news about energy medicine is that even though many leading-edge thinkers in health care believe it to be the medicine of the future, it is available today. You don't have to wait. So what if it's not yet in your town! So what if it's not yet in your hospital! So what if your doctor doesn't yet believe in it! You can learn the basics from the book you are reading. And you can use it with any medical regimen you are currently on. Any regimen! A beauty of energy medicine is that it won't interfere. There are no side effects whatsoever. So why wouldn't you try it? There is nothing to lose!\n\nThis book is a wonderful introduction to energy medicine. It distills the experiences of one of the field's most joyous and effective pioneers into a heartful, user-friendly, enormously practical guide. I agreed to write this foreword before I met Donna Eden or had a chance to see her work. While I review many well-written books that provide good information, there is a deeper ring of truth that is precious and rare. This book sang with it. When I finally had the pleasure of meeting Donna, it was evident that the compassion and wisdom I felt in her book reflect the integrity in the being of its author. Incredibly youthful in her sixty-fifth year, Donna is her own best advertisement. She is ebullient with energy. And she is very, very fun to be around.\n\nWriting this foreword is part of a full circle in my relationship with energy medicine. I have always recognized the importance of the body's energies in my medical practice. I intuitively understood the influence of my patients' thoughts, fears, desires, relationships, family history, jobs, diet, use of exercise, and overall lifestyle on their body's energies and the powerful impact of those energies on their health and their illnesses. But this fundamental dimension of health care isn't taught in medical schools, and as I began to speak with my colleagues about its importance, I felt like the proverbial voice in the wilderness. In the early part of my career, I didn't think this perspective would be accepted during my lifetime. But now, with the embrace of these ideas reflected in many arenas, including in my own life\u2014 from having my books on the _New York Times_ best-seller list to multiple appearances on _Oprah_ \u2014I have clear-cut, measurable evidence that a broad population is ready to hear about the principles that have guided my work. And the medical community is listening as well.\n\nThe world is in fact changing faster than most of us imagined could be possible even a decade ago. While many of the changes are shaking us down to our foundations, a hopeful landscape is also emerging. Energy medicine is one of its contours. It is not just the latest fad in alternative health care. It is a fundamentally new way of grasping who we are. During this precarious time in our history, as you will read in Donna's introduction, \"a book about the energies that animate a woman's body is obliged to mainline you into a deeper connection with the archetypal feminine principle that must be re-embraced if our species is to survive. . . . The archetypal feminine is not just a set of ideas and values somewhere out in the ethers. It is coded, yes, in your energies, but also in your genes, your hormones, and the actions they cause you to take. [At their core] our feminine instincts are toward love, cooperation, justice, compassion, family, nature, and peace.\" Energy medicine brings these values into health care, and it is not just for women or just for healers. Every mother, father, teacher, police officer, and politician needs to become a quick study in energy medicine to better cope with the challenges we all face in today's world.\n\nMost women, however, are innately sensitive to energy in a way that is different from most men. We are more multimodal in the ways our brains are constructed. We are also the healers of our families. When women are healthy and happy, their families are healthy and happy. So you aren't reading this book just for yourself. You are also reading it for the better mother, wife, sister, and leader you will become.\n\nI want to emphasize and comment on a few of my favorite concepts from the book. It is absolutely lovely the way Donna conveys that when she is with a client, she starts by seeing beyond the person's energy blockages to the radiant soul that is inside each of us. This leads to health care where there is no judgment, no blame, no shaming, no guilt, no sense of doing it wrong. There is simply the instruction: \"This is how you heal.\" When your healer sees you this truly and deeply, you experience the radiant self that is within you. The book gives you a vicarious sense that attunes you to your own radiant self.\n\nA second major concept in this book is the basic premise of energy medicine, which is that Einstein was right\u2014 _energy is all there is._ The physical world is nothing but a slower vibration of energy. Energy is the prime mover of all we see and know. This understanding opens you to a new way of taking charge of your health. You change the energy and, over time, your body _has to_ respond. This is an enormously empowering idea. It flies in the face of conventional wisdom, where we believe that treating illness requires something from the outside, such as a drug or a surgical procedure. Energy healing is an inside job. As you begin to do the exercises in this book, you will feel a difference as the energy begins to shift within you. But you don't need to believe me. See for yourself. As you start to use these methods, you will have irrefutable evidence that they work. And from these experiences, you will develop a rock-solid knowledge about the central role of your body's energies. I was impressed when Donna told me about being alerted to a discussion on a cystic fibrosis blog. Double-lung-transplant patients were figuring out together how to use energy techniques (based on Donna's first book, _Energy Medicine_ ) and sharing their successes. So no matter what the concerns that caused you to pick up this book\u2014whether PMS, menstrual cramps, infertility, premature ovarian failure, or menopausal symptoms\u2014there is something here for every woman on the planet.\n\nA third concept that I find compelling is the notion of using energy medicine to \"evolve your body.\" Because our bodies evolved for a world that no longer exists, our adaptations are often imperfect compromises, and we can teach our bodies to adapt more effectively. This is a radical idea. Most people's sense of what is possiblefor them is tied to the prevailing notion called _genetic determinism:_ \"This is what runs in my family.\" \"I am who I am because of my genes.\" But what we are discovering is that the same DNA can express itself in multiple ways. Your genes respond to the environment, and your body's energies are the first tier of that environment. Energy medicine shows you how to help those energies evolve, and when you do, your physical body will follow. An illuminating example from the book is that when Donna got her own spleen meridian into a better balance while addressing a serious health condition, she spontaneously lost 17 excess pounds without changing her diet. That you can evolve your body is a new and enormously exciting concept, and beyond merely offering the concept, energy medicine gives you a way to use it.\n\n_Energy Medicine for Women_ is extremely well documented. We now have the ability to measure the energy fields it discusses with scientific instruments. For the first time in human history, in fact, it is possible to scientifically establish that the Chinese physicians who intuitively mapped the acupuncture meridians thousands of years ago were right! Of course you don't need scientific instruments to use the techniques taught in this book or to feel their effects on your own body, but the scientific documentation builds a bridge between worlds that makes the book much easier for a medical school or a nurse or a doctor to use.\n\nHands-on energy medicine techniques are already starting to be used in conventional health care settings. Some hospitals routinely give patients an opportunity to have their energy fields balanced before and after surgery, chemotherapy, radiation, and other invasive procedures. It diminishes the shock to the body, hastens healing, and ameliorates many adverse effects, from pain to nausea to hair loss. In the medicine of the future, as I envision it, working with the patient's energy field will be the first intervention. Surgery will be a last resort. Drugs will be a last resort. They will still have their place, but shifting the energy patterns that caused the disease will be the first line of treatment. And before that, teaching people how to keep their energies in healthy patterns will be as much a part of physical hygiene as flossing or exercise.\n\nIt is significant that this book is directed toward women. The group with the most money and the most clout on the planet right now is baby-boom women in Western nations. And no group is more primed for changing established health practices. None of us wants to age the way we've seen our mothers age. And because we were the generation that said, \"Don't trust anyone over 30,\" we don't take the previous generation's practices on faith. We also came out of the women's liberation movement, where we cultivated the idea that we could reinvent every stage of life we go through. We in fact have a knee-jerk reaction to the messages we've gotten about how one ages: \"No, thank you. That's not about me. I won't do it that way.\" Energy medicine gives women tools they can use to reinvent themselves and what is possible for them, not only as they age, but also with their physical appearance, their vitality, and their physical health. And it gives new credence to the vision of \"dying young as late as possible.\"\n\nBeing able to sense energy and work with energy is our native human territory. We are born with this ability, and people in indigenous cultures retain it. In technological cultures, however, where sensing the subtle energies of the body and the environment is not mentioned or cultivated, this ability has usually atrophied by the age of two. This needn't be the case. If infants are nurtured to see and feel the energy they are born knowing, they will retain this ability. And it is an enormously practical skill that will serve as a significant part of their inner guidance system. Fortunately, even if your natural ability to sense energy wasn't cultivated, you can learn how to work consciously with the body's energies, and this is very empowering. Donna, who literally sees subtle energies, has made a career of empowering people who don't see them to work with them effectively nonetheless.\n\nAnd the implications are major. Not only can you improve your own health by using energy medicine to uplift the energies in your body, uplifting the energies in your body affects and uplifts the entire planet.\n\n\u2014 JANUARY 2008   \nSANOVIV\n\nIntroduction\n\n**Embracing Your Exquisite Vibration**\n\n_What a piece of work is a woman!_\n\n\u2014 MRS. SHAKESPEARE\n\n**Y** our body's abilities to heal a wound, fight off disease, cope with stress, respond to danger, communicate to you what it needs, and reward you with pleasure when you provide what it requests all reflect an astounding intelligence that is totally independent of your mind. In this book, you will come to understand that this bodily wisdom is contained not primarily in the neurons of your brain but in energy fields that mobilize your cells into action, that coordinate the strategies used by your organs to maintain your health, and that bathe you in an atmosphere of peace and joy when things are well or in stress and alarm when they are not. You will also come to understand how this remarkable intelligence of your body and its energies is calibrated for a world that no longer exists, and that this is the underlying cause of some of our greatest health challenges. And most important, you will learn how to impact your body's energies and energy fields in ways that are attuned to the world in which you are living and that markedly enhance your health and well-being.\n\nEnergy is the living, vibrating ground of your being, and it is your body's naturalself-healing elixir, its natural medicine. This medicine, this _energy_ medicine, feeds body and soul, and attending to it restores your natural vitality. Energy medicine is the science and the art of optimizing your energies to help your body and mind function at their best.\n\nThe notion of taking action to enhance your health is familiar. You exercise, you are conscious that some foods and patterns of eating are better for you than others, you probably have considered vitamins or other dietary supplements, you take antibiotics when you have a persistent infection, and you may be on hormone replacement therapy if you are going through menopause or using other medications to help you with PMS or arthritis or anxiety or depression. This book will show you simple physical techniques, many requiring only a few invigorating minutes each day, that will rapidly shift the energies in your body. I present these techniques with confidence that they are among the most effective and efficient tools available for enhancing your health and boosting your vitality.\n\n#  EMBRACING THE ARCHETYPAL FEMININE\n\nWe are living during one of the most precarious times in humanity's perilous history. A book about the energies that animate a woman's body is obliged to mainline you into a deeper connection with the archetypal feminine principle that must be re-embraced if our species is to survive. It needs to be more than just a guide to health. By personally embracing your deep feminine nature, you are more able to keep yourself upright and oriented through the chaotic cultural waters we are all navigating.\n\nThe archetypal feminine is not just a set of ideas and values somewhere out in the ethers. It is coded, yes, in your energies, but also in your genes, your hormones, and the actions they cause you to take. This book\u2014by connecting you with the energetic foundation of the female body and soul\u2014is designed to empower you to make better choices for your own health as well as to strengthen the connection between the precious wisdom of the archetypal feminine and your daily life.\n\nOpening with \"What a piece of work is a woman!\" and its tongue-in-cheek attribution was not to slight men but to underline a cultural dilemma that has urgent consequences. We live within the context of a society whose patriarchal values are spinning out of control, marred by dominance over love, competition over cooperation, greed over justice, punishment over compassion, career over family, technologyover nature, and war over peace. This tends to bring out the worst in both men and women.\n\nAt our core, however, our feminine instincts are toward love, cooperation, justice, compassion, family, nature, and peace. What a piece of work! If our deepest natures as women were to powerfully rise and flourish, the world would be propelled into adjustments it desperately needs. Learning how to bring your body into better health and balance through energy medicine allows your true nature to shine forth and is a training ground for becoming a force in transforming the energies around you so your world becomes a better place. And it is a path toward becoming an effective advocate for the love, cooperation, and peace that are our natural gifts to the world.\n\nTo best support you in this grand adventure, your body needs to be cared for and maintained, not only as a physical structure, but also as an energy system, which is exactly what it is!\n\n#  AN EXQUISITE VIBRATION\n\nAs scientists are able to peer more and more deeply into the ingredients of physical matter, they have found that the unimaginably minuscule building blocks of nature, such as electrons and protons, are composed of ever smaller particles. Now scientists are speculating that at its base, matter may not be made of particles at all\u2014it may be more like strings of vibrating energy.\n\nWhatever nature's most fundamental design, we do know that every electron, atom, molecule, cell, tissue, and organ vibrates, and its vibration determines much about its character and function. Our entire bodies are, in fact, continually vibrating at a subtle level, moving energy and information through the connective tissues that encase us. We also resonate with energies in our environment. The frequency of the vibrations in your heart begins to match the frequency of the vibrations in the heart of your friend or lover or healer when you are in each other's presence. Vibration is everywhere, and appreciating vibration is at the core of energy medicine. At that core is also the realization that each of us is an exquisite vibration, an intelligent system of energies that supports the distinctive qualities that make us human: how noble in reason! How infinite in faculty! In action, how like an angel! In apprehension, how like a goddess\n\n#  GIVING EVOLUTION A BOOST\n\nAs remarkable as the intelligence of your body and the energies and energy fields that animate it, this intelligence, this guidance system, evolved for a planet that in many ways no longer exists. It is still attuned to an environment with no pollutants in the air or food, no crowded appointment books, no jet travel, no e-mail tyranny, no danger of not getting exercise, no conflict between your children and your career, no glass ceiling. Evolution could not possibly have anticipated the sedentary, information-saturated, polluted world in which your body is trying to thrive. Like a mighty knight from Arthur's court on a modern battlefield, many of your body's coping strategies are outmoded.\n\nIf you are a woman, the world your body evolved for kept you barefoot, pregnant, and probably dead before you reached menopause. Your hormones and neurotransmitters are the living record of your ancestors' adaptations to that world. While they worked well for your ancestors\u2014you are the living proof\u2014today they carry an antiquated game plan for directing your body's responses, sometimes in critical situations, such as illness or threat. If you want to thrive in the world you actually inhabit, a new set of adaptations, a new vibration beyond your body's ancient strategies, needs to be introduced. The underlying principle of this book is that by changing the vibration in your body, you can change the actions of its chemistry, helping it to better adapt to the strange new world in which we all live. You can _evolve your body_ by changing its \"energy habits.\"\n\nShifting an energy field can often trump preprogrammed biological strategies. Just like a radio signal creates the physical vibrations that result in the sounds that come through a speaker, changing an energy field exerts a domino effect on the body's chemistries. Simply thinking a happy thought or a sad thought instantly changes the chemicals being produced in the emotional centers of your brain. Energy medicine teaches you how to shift your body's chemistry as if you were handed a keyboard that produces electrochemical signals instead of sounds. Massage this energy point on your hand and you send endorphins to your brain. Tap that spot under your eye and your stomach cramps go away. Circle your hand over your chest and the heaviness in your heart lifts. Controlling your chemistry by managing your energies is the fast track for helping your body evolve and adapt to the challenges of the twenty-first century.\n\n#  PROMISES\n\n_Energy Medicine for Women_ takes you on a personal tour of your body as an energy system, and it offers simple \"energy tools\" you can use to help your body function at its best. These tools are generally more precise, often more effective, and certainly less invasive than medication or surgery. They can also be self-applied, are available 24\/7, and are free. Energy medicine shows you how to tap into your intuition, embrace your natural self-healing abilities, acquire new tools for assessing and optimizing your health, and generally become less reliant on pharmaceuticals or invasive procedures. While the marvels of modern chemistry can sometimes powerfully boost the healing process, energy medicine can still be used\u2014even when medication is required\u2014to ensure that you are using substances and dosages that are in harmony with your body's energies and needs.\n\nBoth creativity and wisdom are evident in your body's every action, whether it is to put on more weight than it seems to need, to take on a fever, or to rev up your anxieties for that big presentation. In recognizing what your body is attempting, you will be able to form a better partnership with its natural intelligence.\n\nEnergy medicine is particularly potent for women. For almost every health condition a woman faces, hormonal imbalances are in the foreground or in the background and, as you will see, energy medicine can help you dance with your hormones more effectively and gracefully than we in the Western world have been taught. This book will teach you how to become a better partner with your own energies and your hormones. When your energies are in harmony, your hormones follow. Energy medicine helps your body become more effective in adapting to the world in which you live, with its fast pace, stresses, pollution, and amazing opportunities to experience realms of human possibility that were not available to any generation before you.\n\n#  BORN INTO A WORLD OF ENERGY\n\nI see energy. I see it around people as clearly as you see the print on this page. I always have. It never occurred to me when I was growing up that this was unusual. My mother, sister, and brother all saw energy as well, and I assumed that everyone'stake on life was based on the energies that they saw and felt. After all, how can you know if your friends need something they aren't aware of or are too ashamed to request if you can't see their energies? How do you figure out how to tap into your daughter's untapped potential with the necessary precision if you don't see it in her aura?\n\nOne of my earliest memories was watching from our front porch as my next-door neighbor, Sammy Henka, played in his yard. I was a tiny little thing. I could sit, and probably I could crawl, but I couldn't walk yet. Sammy was a couple of years older than I was; perhaps he was three or four. I saw Sammy coming off the porch, surrounded by a pretty azure-blue aura. He seemed carefree. Then I saw two girls coming down the street. A crisp amber color surrounded both of them, and there was something in the way their shared energy moved, pulsing in on them, that bothered me. I could tell by the thickness of the energy that they were talking about something with intensity. I also knew, even at that young age, that the energies I was seeing dealt with what I would later know to call \"judgment.\" When the girls got to the house, they said in unison, \"We don't like you, Sammy Henka. We don't like you!\" Sammy's azure-blue energy instantly became brown, and it looked like it fell down an elevator shaft, descending to his knees, until he didn't even have an aura around his head or shoulders or torso. I died for him. They might as well have shot him through the heart. When I see such energy, I also feel it, and I couldn't conceive why human beings would do this to another person. Couldn't they see what they were doing? It was not until my early twenties, many years after this moment of painful empathy and shock, that I came to realize, _no,_ they _couldn't_ see what they were doing!\n\nMy ability to see and feel subtle energies has been my primary compass for relating to, understanding, and healing other people. The aura, as in the azure-blue energies that surrounded Sammy Henka, is but one level. The aura tells a lot about how the person feels, about the person's mental state, and even about the personality, but there is much more.\n\nWhen I focus in with someone, I am often pulled straight into the energies that reside at the person's core by what feels like a gravitational force from within the aura. At that level, beneath the personality, beneath whether someone is loud or quiet or smiling or angry or kind or hateful, is pure goodness. We are each, at heart, an exquisite vibration! I can also see all the problems\u2014mean, selfish, sloppy, senseless, or ugly defenses; harsh, brittle, jagged, stagnant, or mushy energies\u2014 but I don't stay focused there. I am drawn to the center, to the soul level. When you see into a person's essence, it is a magnificent moment. It takes my breath away. Every time!\n\nServing as the personal healer for so many individuals was a privilege that eventually became a problem. While I loved the healing process, as I matured, I found a growing urgency in myself to not perpetuate the false and disempowering idea that I was the one with the healing power and my client was the passive recipient. This concern, while valid on its own merits, was also driven by the simple fact that it had become impossible for me to see all the people who wanted sessions. I became determined to empower people to discover their own self-healing abilities, attune themselves to the energies that impact their health, and learn how to manage those energies for staying vital and healthy. After nearly a quarter-century with a full-time practice, at just before the turn of the millennium, I shifted my career in a way that has been exhilarating. I dedicated myself to teaching people how to help themselves and others using the powerful yet noninvasive techniques of energy medicine. My first book, _Energy Medicine,_ is a by-product of that career shift. So are the almost 600 classes and talks I've given in the nine years between the publication of _Energy Medicine_ and this writing. And so is this book.\n\nI have had the privilege of witnessing again and again that it is possible to teach people who are not able to \"see\" energies to nonetheless work with their energies in ways that are powerful and often life-changing. _Energy Medicine for Women_ presents basic methods that are helpful for _every_ body and that also form the essential building blocks in tailoring a self-care program for your unique body, chemistry, and spirit.\n\n#  ENERGY MEDICINE IN ACTION\n\nAn attorney who came to me for a treatment had been pushy in setting up the appointment. As the session began, she was abrupt, demanding, and skeptical. She was, in fact, exuding such aggression and hostility that I really couldn't see her essence. I invited her to lie down on the table, and I held some points to calm her energies. I could feel her start to relax a bit. Then waves of energy began to release from a vortex that sits over and reaches into the solar plexus. These energies were huge, overwhelming, and thick. I sensed that all her power issues were encrypted in these energies.\n\nI began to work with this force field by moving my hand in slow circles over it. As the energetic storm began to clear, an opening appeared at the center of the vortex and I could see and feel into her energies more deeply. Because a person's history is literally coded in such energies, I began to see images. I saw her when she was in law school. She seemed like just a girl, young for her 22 years. She was being ostracized and ridiculed for being \"soft.\" She learned that everyone is out for themselves and that you better find the other guy's Achilles heel before he finds yours. I laughed good-naturedly as I referred to her demeanor with me and told her, \"I know it is your MO to nail the other guy before he nails you, but your energy shows that the reason for it is that you have really been hurt.\" She looked at me quizzically.\n\nI described her experience in law school and named the conclusions she came to about being tough and invulnerable, and commiserated with her how it was perfectly understandable. She was listening attentively now, following my every word. I went deeper. More accurately, it was like the opening in her energy vortex pulled me in. It was a realm of such goodness and light, such a beautiful field, and so large, that I was overcome. I started to cry. She looked at me and said, \"What's wrong?\" I started to tell her what I saw. I described the glorious energy at her core. I depicted the images\u2014embedded in her energy field\u2014of a tender, innocent girl growing up in what felt to me like New Hampshire. I elaborated upon the dormant presence of the entirely appealing and likable person who was beneath her defenses. And I acknowledged the hard journey she had traveled. She felt my compassion as I explained why she had been a tormented soul for such a long time and why she had built all her defenses. Now the tears were in her eyes.\n\nShe lived out of town, but whenever she could travel my way, she would come in for a session. In essence, our work was to realign her with the deeper beautiful energy that was her true nature, her exquisite vibration. When I finally met her husband three years later, he gave me a long impassioned hug and credited me with having saved their marriage. She and I had never talked about their marriage.\n\nBeing pulled into direct contact with a person's divine nature is a profound experience. It transcends anything words can describe. The physical energies and colors fade into the background, and I know that I am in the presence of pure goodness. Suddenly I'm in a heavenly realm, enveloped by a radiance so deep and so beautiful that I know I am witnessing the face of God. I see profound wisdom in the person, and how truly good he or she is. I'm filled with love. This is my core orienting experience for understanding the world.\n\nWhatever the personality, whatever the damage or defenses you encounter, you approach healing work differently when you are guided by a sense of certainty that unfathomed wisdom, goodness, strength, and beauty are the person's true nature. This is my orienting belief, and it is important to me to emphasize it from the start as we move into a narrower focus on energy flows, hormones, chemical imbalances, physical problems, and mechanical techniques. At the root of it all, in every one of us, is an exquisite vibration.\n\n#  CREDENTIALS\n\nWhat are my credentials to make all the claims and promises in this book? First, I am a woman. Women are rhythmically directed from within. A doctor with only one X chromosome cannot know what we know. We have an inside track on women's bodies.\n\nSecond, I am a woman who has had more health challenges than most, and I have overcome them. I was born with a heart murmur and serious metabolic problems. I had tuberculosis as a little girl, severe hypoglycemia and allergies into my 30s, multiple sclerosis by the time I was 16, a heart attack at 27, severe asthma in my thirties, a near-fatal insect bite at 33, and a malignant breast lump at 34. My entire body was breaking down, and I was told by more than one doctor to get my affairs in order. As it turned out, Western medicine giving up on me was one of the best things that could have happened. I was determined not to leave my young daughters for someone else to raise. Since Western medicine couldn't help me, I was forced to get very serious about helping myself. Today, in my midsixties, with the help of the techniques taught in this book, I am healthier and more capable of maintaining my health than I have been at any previous time in my life.\n\nThird, I have struggled with my hormones since I was ten, including monthly cycles that could last two weeks and were debilitating. To complicate this, my body often had adverse or paradoxical reactions to standard medications and painkillers. I have had to figure out what would work for me in a culture that didn't give me answers or help.\n\nFourth, I have had a healing practice since 1977, applying my ability to see energies to helping more than 8,000 women (as well as several thousand men) in individual sessions.\n\nFinally, I have successfully taught tens of thousands of people in energy medicine classes to work with their own health challenges. And, most important, I have figured out how to teach people who can't see energy to monitor their own energies and formulate interventions that have been effective in promoting their health and well-being.\n\nThose are my credentials to offer this book. _Energy Medicine for Women_ will show you how your body's energies affect every aspect of your life, and it will show you how to vitalize your cells, organs, and entire being by understanding your body as a network of energies that are responsive to simple physical techniques.\n\n#  A \"SHOW ME\" ATTITUDE\n\nI invite you to take a \"show me\" attitude throughout the book and think of its techniques as experiments. Often you will feel immediate results, even if they are subtle. Sometimes you will need to do an exercise on a daily basis for a while before you experience the desired outcome. And sometimes a technique isn't right for you. But because hands-on energy medicine is by nature noninvasive, no harm is likely to result by using the methods in this book as instructed. Side effects from energy interventions, though they rarely occur, may include unexpected emotional releases, mild nausea, headache, or other minor symptoms that quickly pass. The cause is usually too much energy being moved too quickly for a physically unstable body to readily accommodate it.\n\nThough its methods are generally harmless and surprisingly potent, I am not suggesting that you should use this book as a substitute when professional health care is needed. Good self-help health books have the seemingly contradictory tasks of encouraging self-care _and_ emphasizing that professional treatment is sometimes required. While there is an important place for each, the theme of this book is how much you can do for yourself.\n\nSince the notorious Flexner report of 1910, when the medical community rose up to fight the threat of naturopathic and chiropractic physicians and began licensing the practice of conventional medicine, a line has been drawn in the sand for those who hold themselves out as health care professionals. There has been a virtual patent on the term \"medicine.\" While at its best this line is meant to prevent quackery and charlatanism, it also has had the effect of concentrating the most prestigious and well-financed care into the hands of increasingly technological and pharmacological medical practices and away from folk _medicine_ and health care practices adapted from other cultures, such as Native American _medicine_ or traditional Chinese _medicine._ \"Practicing medicine without a license\" has come to mean any attempt at healing people that falls outside the framework of conventional Western medicine. It is illegal. And that is disgraceful!\n\nI have personally been impacted by these laws. Early in my career, I was seeing many people who, as it turned out, were also patients at a group medical practice owned by five physicians who all had affiliations with the local hospital, which was adjacent to their office building. Enough stories got back to them that they decided to initiate legal proceedings against me for practicing medicine without a license (a felony). I can look back now and understand it from their point of view\u2014this uncredentialed \"healer\" giving their patients, in their estimation, false hope by doing some kind of energy voodoo with passes of her hands over their bodies and other such nonsense. They subpoenaed several of these patients to testify, at a pretrial hearing, that I had been working with them on ways that crossed the line into their exclusive domain. Hearing from the witnesses, the judge was impressed that my work had indeed made a difference in their medical conditions. He asked the prosecuting attorney to find one patient who had been harmed. When he could not, the case was thrown out of court, with the judge agreeing that quackery should be prevented but then making some disparaging remarks about anyone trying to ban the kinds of healings the patients had described under oath. The incident had a redeeming sequel. Some eight years later, the hospital that was adjacent to the doctors' office had me teach an energy medicine class to its staff. Several of the doctors who had brought the charges against me attended and were among the class's most enthusiastic participants.\n\nOur culture must come to grips with the fact that conventional Western medicine is failing us in many ways. According to one credible estimate, more deaths are caused by taking medications as prescribed and other iatrogenic reactions (illnesses induced by medical treatment) than all the other causes of death in the United States. The health care system needs help, and time-honored, natural approaches that are powerful and safe are there to be embraced. While it places me and my colleagues in a legally more precarious position as we strive to stretch the boundaries of health care, we have chosen to reclaim the term _medicine_ in calling our work \"energy medicine.\" Your body's energies are the most natural medicine that exists, able to precisely orchestrate the body's self-healing capacities wherever required.\n\nThe conventional medical paradigm has a number of weaknesses. It is oriented toward diagnosing illness and treating disease. It is less oriented toward the larger concern of maintaining health. Energy medicine, at least as I and many others practice it, does not diagnose or treat illness. Rather, the focus is on assessing where the body's energies are blocked or not in harmony and then correcting the flow of these energies. Illness may provide clues about where the energies are disturbed, but it is not the focus of energy medicine. Energy medicine is every bit as much about optimal functioning as it is about overcoming maladies. This is a critical difference from the way we usually think about the word \"medicine.\" In energy medicine, the \"medicine\" is not an external substance. It is the curative power of the proper flow of the body's energies. It is the most natural approach in the world. I think we should start training our children in its most basic procedures by the time they are able to wash their hands.\n\nWhere many interventions within conventional medicine are \"done to you,\" you can do many energy medicine procedures yourself. The following pages are filled with techniques that have evolved from three decades of working with thousands of women on the issues being addressed. However, one of the most serious limitations of energy medicine is that if you don't use it, it doesn't work. Yes, sad but true. If you just read about the techniques and don't do them, the book will still be informative, and there is value in that, but it will be a little like hoping you will lose weight by putting the diet book under your pillow instead of doing the diet.\n\n#  HOW TO USE THIS BOOK\n\nA dilemma in writing this book has been that it is intended for two audiences: people who have read my first book, _Energy Medicine_ (or who are otherwise familiar with the material it presents), and people who are new or relatively new to energy medicine and my approach to it. In order to apply energy medicine for women, the reader first needs some background in energy medicine. So I have incorporated essential concepts and methods from my first book into this book and ask for indulgence from readers already familiar with this material. I have also tried to present these basics in a manner that will be a useful review for those already conversant with them (this material is found primarily in chapters 1 and 2 and the Appendix). I should also mention that the first book gives you more tools to figure out ways to apply energy medicine to your own unique body and health concerns, while this book, by necessity, is more formulistic, offering techniques that focus on many of the generic health issues women frequently face.\n\nYou can approach _Energy Medicine for Women_ in several ways. If you are new to energy medicine, my suggestion is that you first read chapters 1 through 3 and immediately begin to experiment with the procedures they present. You will gain a good understanding of how the body's energies govern a woman's health, chemistry, and spirit, and how simple exercises can optimize that process. Next, read the text of subsequent chapters that interest you and scan the procedures. Use those that apply to you, and then keep the book handy as a self-help reference.\n\nChapters 1 and 2 provide an introduction to energy medicine and its most basic methods, synthesizing some of the most important principles and practices from my earlier book. If you have studied the first book, you may certainly skip these chapters, and the Appendix, though I think you will still find they provide a good review and synthesis oriented toward the theme of this book. Chapter 3 gives you a more detailed understanding of the relationship between your hormones and your body's energy systems, as well as the myths and the medical politics that affect a woman's health, and it teaches several basic energy medicine procedures for working with the hormones involved with stress and with immune function. The subsequent chapters then present an energy medicine approach to women's issues such as menstruation, sexuality, fertility, pregnancy, birth, menopause, and weight management, offering techniques you can apply directly for each area of concern. Finally, the Appendix is a tutorial in the important art of energy testing, which can help you tailor the entire program to your body's unique biochemistry and needs.\n\nThe physical techniques presented throughout this book are each designed to bring about specific benefits. I have tried very hard to describe each procedure clearly, and they are well illustrated with Christine Alicino's superb photographs.\n\nThe techniques each require only from a few seconds to a few minutes, and I've generally indicated how much time will be needed. Please do not be put off by the length of the written instructions. Spelling out the procedures in a precise, step-by-step manner often requires instructions that _take longer to read than to do._ But by doing them as you read, you will find that you can pick up each technique quite readily. I have also made a companion DVD program (available through www.EnergyMedicineForWomen.com) that leads you through most of the techniques presented in this book. It simply is the case that it is easier to follow along on your television screen than to follow written instructions. If you go to that site, you can also watch a free video of me introducing you to this book. I know that if I can first hear and see an author, I have a more visceral relationship as I read the book, so do let me personally introduce myself using the magic of Internet video.\n\nAs with my first book, _Energy Medicine,_ this book also lays out a systematic yet open approach for consciously working with your own energies, or the energies of those you care about, in order to achieve a healthier body, sharper mind, and more joyful spirit. As was my approach in that book, I have generally not pulled my punches in presenting techniques that are potent and that can be helpful with difficult conditions. On the basis of my experience in teaching these techniques to more than 50,000 people, I have a good idea of how each method presented here impacts a wide range of people, and I have used my best discretion. Your part of the bargain is to exercise your own best discretion as you go through the program. If you need professional treatment, please consult a competent health practitioner! The techniques in this book will only complement the care you receive. But while sincerely stressing the value of professional intervention, I do not want to undercut my primary message and deep conviction that there is much you can do to care for yourself. Ultimately, it is you who is responsible for your health, and the more you know and the more you do for yourself, the easier the task for those who care for you professionally, and the better your health will fare.\n\nIn the acknowledgments to _Energy Medicine,_ I noted that my husband, David Feinstein, \"has been tireless in interviewing me and writing first drafts from those interviews, editing transcripts of tapes from my classes, conducting computer searches, and generally bringing left-brained organization to my right-brained way of being. His abilities to craft a phrase, suggest an analogy, find order amid complexity, and place an idea into its larger intellectual context, all the while retaining the spirit of _my_ voice, permeate this volume. In short, this is the book _I_ would have written if my mind worked like both of our minds combined.\" While David has also been a major force in the creation of this book, I am pleased to say that a decade later, each of our minds works a bit more like our combined minds used to work. Some of the tools we offer here have served us both in that regard. Balancing your energies balances your mind. May this book help you find your way to greater balance, health, and inner peace.\n\n_**Chapter 1**_\n\nA Medicine Called Energy\n\n_We're in the midst of a radical, essential paradigm shift in the Western medical model. . . . It is imperative that we expand our concepts of health care to include subtle energy and energy medicine._\n\n\u2014 JUDITH ORLOFF, M.D. _Positive Energy_\n\n**E** nergy medicine is probably the \"real\" oldest profession. Knowing how to keep the body's energies healthy and vital gave our ancestors a tremendous edge for surviving in the wild. Strategies for sensing and correcting energy imbalances by stimulating specific \"energy points\" have been passed down through generations in China, as well as in other parts of the world, for at least 5,000 years. A body that had been mummified in a snowbound mountainous region along the border between Austria and Italy around 3000 B.C. had tattoos on exactly the points that are indicated in traditional acupuncture for treating the kind of lumbar spine arthritis revealed by an X-ray analysis of the body. Nine of the fifteen markings were along a meridian that is used in treating back pain, including one on the precise acupuncture point that is considered the \"master point\" for back pain. Forensic analysis also revealed that the body's intestines had been rife with whipworm eggs, and indeed, some of the other markings were on points that are traditionally used for treating stomach upset. Similar tattoos have been found on mummified bodies in other regions, ranging from South America to Siberia.\n\nIn traditional Chinese medicine, acupuncture and other methods were used to keep the body healthy by keeping the energy fields that support it healthy. In some provinces of ancient China, you paid the physician a nominal fee when you were well. If you got sick, the physician would work hard to try to cure your illness, but you did not have to pay because the physician had failed to keep your energy field healthy enough to prevent the illness. Try proposing that arrangement to your local hospital!\n\nDisturbed energies precede illnesses in the physical body, and it is possible to correct energy imbalances _before_ they coagulate into illness. Energy medicine can address illness, but it can also prevent it. Think of the energy field as the blueprint of the physical body. If this living blueprint remains sound, the body stays healthy. If the blueprint is damaged, the body follows. Maintaining a healthy energy field is a powerful strategy for maintaining health and preventing illness. By adeptly engaging the energy systems within your body, you can improve your health, turn around an illness, and live with greater vitality day by day.\n\n#  TWO SIMPLE ENERGY EXPERIMENTS\n\nThe tools of energy medicine range from elegantly simple to highly sophisticated. Some of the methods involve touching the body, while others involve no touch at all. Try this experiment. Bring your palms toward each other until they are about three inches apart. Now twist your arms so they form an X, with your wrists at the center of the X, still about three inches apart. Bring your attention to the space between your wrists. Because your wrists contain several energy centers, the energies will connect, and most people will feel some sensation in the area between their wrists. Move your wrists about an inch closer and then out a few inches back and forth. What happens to the sensations in the space between your wrists as you change the distance? Do not be concerned if you do not feel the energy, but it is there. The palms of your hands also emit a considerable amount of subtle energy. See if you can feel the energy in the space between your hands if you cup your hands and move them toward and away from each other.\n\nNow do another experiment. Most people carry some tension in their shoulders. Reach your right hand over to your left shoulder and press into any point with your middle finger. Feel around until you find the spot that is most tender (if there is another area of your body you would prefer to focus upon, do so). Give a rating of 0 to 10 to this spot, with 0 meaning no tension or tenderness at all and 10 meaning extreme tension or tenderness. Next, vigorously rub your hands together and shake them off. Everyone's hands have a measurable energy field. Now cup your right hand about two inches above the area where you felt tension and begin to move it over the area in a slow counterclockwise circle, making about a dozen rotations. Notice if you have a sense of energies being exchanged between your hand and your shoulder. Hold your hand over the area a few more seconds, then relax. What sensations do you feel? Now press the original point with your middle finger and again rate it from 0 to 10. Most people find that the tension diminishes.\n\nWhile about as simple as a technique could be, this experiment shows how readily we can move our energies. Energy medicine begins with such simple tools for reducing tension and fostering healing, and it progresses to more complex protocols for bringing about targeted benefits and addressing serious health challenges.\n\n#  YOUR HEALING IS IN YOUR HANDS\n\nEnergy medicine is one of five domains of \"complementary and alternative medicine\" identified by the National Institutes of Health (NIH). Methods used within energy medicine include electrical devices, magnets, crystals, needles, aromas, and herbal or other ingested substances. But the tool used by the largest number of practitioners for moving and harmonizing the body's energies and fields is the human hand. Practitioners as well as anyone reading this book can use their hands to bring balance and harmony to the body's energy fields. You can tap, massage, pinch, twist, or connect specific energy points on the skin. Because everyone's hands carry a measurable electromagnetic charge, specific areas of the body can be surrounded with the hands to produce a field effect, or the hands can be used to move and align the body's energies by tracing specific energy pathways along the skin. The use of postures and movements are other noninvasive ways to benefit the body's energy system.\n\nOne of the most fundamental (if not always honored) axioms of conventional medicine is that the least invasive measure likely to impact an illness should be the first applied. Fortunately, low-tech energy medicine procedures are not only readilyavailable, they are also noninvasive, preventive, and strikingly cost-effective in contrast to the skyrocketing expense of conventional medicine and its disastrous impact on the economy.\n\nThe power of energy medicine over conventional medicine's \"medicate or cut\" approach is also seen in the way its methods are able to rapidly affect your entire body, and your energy \"blueprint,\" rather than focus only on body parts. How are energy interventions able to have this \"holistic\" influence?\n\nWhile we don't usually think about the body's connective tissue as an organ, it is actually a remarkable one, and it transmits energy impulses to reach every part of the body virtually instantly. Each of our organs, according to Dawson Church, \" _is encased within the body's largest organ, which functions as a liquid crystal semi-conductor_ \" that conducts information as well as electrical signals by being able \"to _store_ energy, _amplify_ signals, _filter_ information, and _move_ information\" to every cell of your body. With the connective tissue acting as a giant electrical semiconductor, energy interventions can simultaneously be brought to every cell of the body.\n\nSix fundamental principles that are the basis of energy medicine give it strengths not found in conventional health care models. Further elaborated in an article that is available as a free download (www.EnergyMedicinePrinciples.com), these principles include:\n\n1. _**Reach:**_ Energy medicine addresses biological processes at their energetic foundation so that it is able to impact the full spectrum of physical conditions.\n\n2. _**Efficiency:**_ Energy medicine regulates biological processes with precision, speed, and flexibility.\n\n3. _**Practicality:**_ Energy medicine fosters health with interventions that can be readily, economically, and noninvasively applied.\n\n4. _**Patient empowerment:**_ Energy medicine includes methods that can be utilized on an at-home, self-help basis, allowing more effective self-care while also fostering a more creative patient-and-practitioner partnership in the healing process.\n\n5. _**Quantum compatibility:**_ Energy medicine adopts nonlinear concepts consistent with distant healing, the healing impact of prayer, and the role of intention in healing.\n\n6. _**Holistic orientation:**_ Energy medicine strengthens the integration of body, mind, and spirit, leading not only to a focus on healing, but also to achieving greater well-being, peace, and passion for life.\n\n#  SENSING SUBTLE ENERGIES\n\nIn short, it is time for Western medicine to embrace the energy paradigm and move forward as a more powerful, attuned, and responsive discipline.\n\nAs you begin to apply energy techniques in your life, you open yourself to a realm of reality that operates beneath the radar for most people in technological cultures. The subtle energies in the environment, however, were an essential source of information for our ancestors, telling them if danger lay around the next corner or if a particular plant could be safely ingested. The subtle energies within their own bodies guided their daily choices and activities as decisively as the not-so-subtle energies of a headache might cause you to take a nap.\n\nI am certain that when babies are born they are far more able to register these subtle energies than we are as adults. Have you ever noticed the way infants will often look intently just above your head or to the side of your face? There is no question in my mind that they are seeing the energies that surround you. Babies see energy, feel it, sense it, know it. But because the brain has so much to learn, and because the realm of subtle energies is rarely spoken of and rarely validated, these sensitivities become dormant. They fall out of the loop in the learning process.\n\nI have on occasion, however, had an opportunity to encourage a pregnant woman or the parents of a baby to talk with their child about energy from the beginning. Regardless of whether or not the parents see energy, I ask them to imagine the energies that animate all of life and talk about them. The parents may be speaking only about their imaginings, but they are attuning themselves to what is a reality for their child. From the offspring of these parents, I now know a handful of older children who can still see energy in vivid colors and are able to talk about it freely and easily. After having been mostly on the road teaching for seven years, I had returned to the town where I had had my private practice. I was walking on the main street and, in the opposite direction, five large young men who looked like high school football players were sauntering in my direction. I did not recognize them and they looked a bit intimidating. When we were about half a block apart, one of them, who as it turned out had been my client along with his family when he was a child, looked at me directly and called out for everyone, including his buddies, to hear: \"Hey, Donna!!! You still have pink in your aura!\" Welcome to Ashland, Oregon, dear reader. I have also witnessed many adults as they began to see and accurately sense energies after experimenting with techniques presented in this book. They are opening to a deeply ingrained but forgotten ability.\n\nAgain in Ashland, a woman dragged her physician husband to one of my weekly evening classes. He found the idea of energy healing implausible and the concept that a person could see color in people's energies ludicrous. But as he delivered his sometimes sarcastic comments about \"seeing\" colors, he spoke in one of the deepest and most beautiful voices I had ever heard. When people paired up to practice techniques with each other, he would always work with his wife, back in a corner of the room. On the last evening of the class, as people were practicing with partners, a high-pitched, squeaky voice from the back of the class exclaimed, \"Purple, I see purple!\" It was the physician. He was so surprised by seeing the purple in his wife's energies that his voice jumped three octaves. This jolt to his worldview resulted in his taking further classes, now on his own motivation, and he began to incorporate energy medicine into his medical practice.\n\nWhen you learn the native language of your own unique energies and energy fields, you become able to read them, hear them, and converse with them. One of the trickiest things in cultivating new sensibilities about your own and others' energies is that they often don't show up in the ways you expect them. Like synesthesia, where certain people can _smell_ colors or _see_ sounds, the perception of energy may just slide into one of your normal sensory channels. I've known people who can hear, smell, or taste specific energies, rather than see or feel them as I do. My own sense of taste has actually also grown stronger. I can usually taste which of the five elements (the elements are one of the core energy systems) is a dominant theme in the health challenges, as well as psychological issues, a person is moving through. For instance, a metallic taste in my mouth alerts me to an imbalance in the metal element. I've now known several people who can _hear_ energies move and detect where they are blocked in a client lying on their treatment table. One of my colleagues began to smell energies, and the scents became so overwhelming that she had to get out of the work for a time. You cannot know in advance how you will register subtle energies. We all have different strengths and different ways of knowing. As you expand into the language of energy, the only thing you can predict for sure is that it will speak to you in the way it chooses, not necessarily in the way you hope for or expect.\n\nMy friend and associate Sandy Wand often sees symbols when she is working on someone. She never knows where they will take her. But she has learned to describe them to her clients, often having no idea what they mean but trusting that they will eventually begin to make sense. I had not told her of a frightening experience I had had in which I thought I was dying. Lying in my hotel room bed in London, I felt that all my energy instantly dropped, like a runaway elevator, down into my root chakra, stopping with a jerk. Suddenly, I couldn't see the room anymore. All I could see was a deep blue-black that felt like hot ink boiling within my root chakra. It began to rise, filling my entire body. It felt like a poisonous fluid was inside me.\n\nWhen I returned home, still feeling poisoned, Sandy gave me a session. After working on me for a few minutes, she said, \"Well, this isn't going to make a lot of sense, but you know how squid squirt out ink to protect themselves? I'm getting an image that, like a squid, your root chakra has been squirting out energy in order to protect you.\" The piece of the puzzle she gave me that I didn't have was that while I thought this deep blue-black was an energy of death, she saw it as quite the opposite. This was an energy of life protecting itself. Squid squirt out ink so nothing can see them or get to them. This provided me with an extremely useful insight. If I didn't start protecting myself better, _then_ I might be dead. Since I wasn't succeeding all that well at setting boundaries, my energy system was trying to set a boundary for me. The blue-black \"ink\" was putting out a force to both contain my own energies and keep out the energies of others who might harm or drain me. Sandy has a gift, but when she first started doing energy healing, she didn't know she had it. Open to the unique ways energies reveal themselves to you, and your own natural capacities to work with energy can flourish as well.\n\n#  BETTER LIVING THROUGH ENERGY\n\nA happy bit of fallout since the publication of my first book has been how many times someone I've never met or even seen in a class has sent our office an e-mail describing how using the methods in the book have turned around a difficult physicalcondition. Just today, September 21, 2007, as I was working on this section, two such e-mails were forwarded to me (and I have subsequently received permission to include them here). In the first, a woman with cystic fibrosis who had received a double lung transplant wrote to the person who ships our books and DVDs: \"There seems to be a big buzz on the CysticFibrosis.com site about _Energy Medicine._ I thought you all would appreciate what others are saying.\" Among the comments in the interactive discussion blog we found when we went there:\n\n_I've mentioned before the book_ Energy Medicine. _WHY do I love it so much? For so many reasons. I learned how to energy test in ways that I did not know before. I know how to test for foods, vitamins, and drugs that work for my body through energy testing. That is in the book. My latest discovery. I would consider myself borderline diabetic. I took my glucose after eating one day and it was 143 (one hour after eating a high-energy bar). I practiced the daily routine and traced all my meridians and after eight minutes, I retested my glucose again to see a change. My glucose dropped nineteen points. After eight minutes that is pretty remarkable. I did this for three days in a row to see if this was for real, and every single time I had the same results. Two days ago when I went for my six-month checkup, I told three doctors about the blood sugar readings and they all seemed very impressed. Before I eat now, I do all of the energy routines that I mentioned. I have had the book now for four years, and I have read it over and over again._\n\nThe very next person to post an entry, another double-lung transplant recipient, described how she has also\n\n_tried the_ Energy Medicine _book and some exercises to help with things like headaches, cramps, constipation, and gas pains. For me the book and its exercises are amazing. I can tell a significant change in my heart rate or blood pressure when I do and do not do the exercises. I have actually checked my blood pressure when it was high\u2014like in the 150\/90 range\u2014and then did a few of the energy medicine techniques and checked it again within about five minutes and it had dropped to about 120\/80, into a normal range._\n\nThe second e-mail to arrive on September 21 stated:\n\n_I am writing this to express my sincere thanks to Donna and David for their generosity in sharing the invaluable knowledge of energy medicine and energy psychology. I was recently diagnosed with dengue\u2014an epidemic spread by mosquitoes, which has no vaccines or drugs to treat it, squeezes the life out of the victim through its symptoms, and is fatal too. I was admitted into the hospital. My blood tests showed infection in blood and liver. My doctor was very concerned as the liver was affected. Hearing this, all family members got extremely worried. I was just the opposite. I asssured them that I would be out of the hospital in two to three days (something considered impossible for someone with my condition). I do not know why I said that, it just popped out of my mouth. But that's exactly what happened! How?_\n\nDay 1: _I recalled what Donna had said in her DVDs: \"Disease never lives in a well-oxygenated body.\" \"Strengthen your spleen to be healthy.\" I had only one hand free as the other was on tubes for intravenous fluid, so the challenge was how do I do all these exercises with one hand? I could breathe, buzz my K-27, thump my thymus, and tap my spleen points on one side and then the other with a single hand, but couldn't do the rest. I remembered Donna mentioned a man who had a stroke and got well by just_ imagining _doing the routine. I did the same. I visualized myself flushing spleen, hooking up, \"connecting heaven and earth,\" drying off, and doing figure eights all over where my single hand could not reach. I also did energy psychology. My \"karate chop\" was with my single hand on my thigh! I focused on more figure eights on my stomach, as I had severe vomiting. I also sat there appreciating every single part of my body for being healthy and all the objects in my room (including my vomiting pan) for being there and all the people behind them. I remembered Donna's words: \"Gratitude is the best vaccination for everything\" [gratitude is not just something you do with your mind\u2014it activates a healing energy called the radiant circuits, discussed later in this chapter]._\n\n_My vomiting disappeared within 12 hours. Everyone was surprised to see me eating my favorite food. My fever was under control, and I felt so energetic that I didn't sleep at all except the few hours that I got at night between routine blood samples and temperature checks every two hours. My doctor was so surprised that he said, \"You look so bright! What have you been up to?\"_\n\nDay 2: _Blood results in the morning showed higher platelet count, almost normal. I kept doing all that I did on Day 1. During his evening visit, my doctor told me that my blood results and temperature were almost normal. He was surprised to learn what I ate and how energetic I was._\n\nDay 3: Completely _recovered and discharged from the hospital!_\n\n_No one could believe it. Patients with dengue are usually bedridden and take an average of seven to ten days to fully recover, if they recover at all. The doctor was so curious that he asked, \"Do you do any kind of exercise or something?\" I immediately shared with him about energy medicine and how I learned it. As I left, he said with a smile, \"I'll remember, energy medicine!\" My mother, who was a non-believer, is now doing the daily five-minute routine [see page 51]. I am yet to give her the gift I bought her, Donna's Energy Medicine Kit. My husband and daughter are staunch believers of energy medicine. My brother is planning to make a special visit just to learn this from me. I am so happy and thankful!_\n\nWhile single cases may simply be \"spontaneous remissions\" having nothing to do with energy medicine, you can imagine how gratifying it is to consistently hear comments like these on a full spectrum of health concerns from people who have no particular investment in \"proving\" that energy medicine works, and you may find it encouraging to know that such cases are not unusual as you begin to apply the methods presented in this book to your own health concerns.\n\nEnergy medicine works with the life force, and part of its power is this focus on the well-being of the entire body rather than merely on parts or symptoms. Western-trained physicians, on the other hand, learn anatomy on cadavers\u2014the flesh without the force, the limbs without the life. Because conventional medicine focuses on physical structures rather than energy, its primary tools are those that work on matter: drugs and surgery. Because energy medicine focuses on the _forces that animate_ the physical body, its tools are more subtle. They are generally noninvasive. And they are often more effective. Energy is also free, user-friendly, and always available to help us feel better and function better.\n\nBut if that is so, why isn't energy medicine mainstream? Why is this book shelved in the Alternative Medicine section? Conventional medicine is a trillion-dollar(that is one thousand billion) industry annually. The total outlay for alternative medicine is two-tenths of one percent of that . In business, money talks louder than other truths and, make no mistake, the health care industry is _big_ business. Energy medicine, being quick, effective, and cheap, can threaten that business. Extraordinary claim? Yes!\n\nAnd I am not denying the powers and wonders of modern medicine. Nor am I claiming there is a conspiracy to keep you from knowing about energy medicine. I am claiming, rather, that there are many, many areas where conventional medicine is not particularly proficient in helping you maintain your optimum health and that there are market forces and cultural forces that have kept energy medicine\u2014which may be far more effective and safer in many of these areas\u2014from becoming mainstream.\n\nBut a new paradigm is rapidly emerging in which the body's energies are going to become a central focus in the practice of medicine. According to Mehmet Oz, M.D., one of the most respected surgeons in the United States and the director of the Cardiovascular Institute at the Columbia University College of Physicians & Surgeons (speaking to an international audience on _Oprah_ , no less): \"The next big frontier in medicine is energy medicine.\" Dr. Oz is not alone in this opinion. Norm Shealy, M.D., founding president of the American Holistic Medical Association, has flatly stated that \"energy medicine is the future of all medicine.\" Richard Gerber, M.D., predicts that \"the ultimate approach to healing will be to remove the abnormalities at the subtle-energy level which led to the manifestation of illness in the first place.\"\n\nThis emerging paradigm is actually both old and new. According to Albert Szent-Gy\u00f6rgyi, Nobel Laureate in Medicine: \"In every culture and in every medical tradition before ours, healing was accomplished by moving energy.\" I sketch some of the basic elements of this emerging paradigm in the remainder of this chapter. However, if you are already familiar with it, are not interested, or are eager to get on with the book's self-care program, you may proceed directly to Chapter 2. But I think you will find it quite interesting to learn about the science that is supporting the idea that health is influenced in profound ways by the invisible energy anatomy of the human body.\n\n#  THE BODY'S ENERGIES\n\nWhile energy takes many forms\u2014such as kinetic, thermal, chemical, and nuclear\u2014 the energies most pertinent to energy medicine seem to involve the body's _electrical_ energies, _electromagnetic_ energies, and _\"subtle\"_ energies:\n\n\u2022 Like a miniature battery, each cell in your body stores and emits _electricity._ Every breath you take, every muscle you move, and every morsel of food you digest involves electrical activity.\n\n\u2022 Wherever electricity moves, _electromagnetic fields_ are produced, and I explore the role of such fields in health and healing in the following discussion.\n\n\u2022 _Subtle energies_ were described by Einstein as energies we know of because of their effects but do not have the instruments to detect directly. While these subtle energies cannot move a needle on a gauge, many healers know how to engage them to restore health and vitality. Interestingly, a device developed at Stanford University that detects a form of energy that until recently had eluded scientific instruments shows that this energy responds to human intention .\n\nOther energies are also at work in the body, though they are not so much the focus of energy medicine. The forces that hold an atom's nucleus in place, for instance, are more than ten billion billion billion times stronger than gravity. If you start to feel droopy in the afternoon, it may be inspiring (though counterintuitive in that moment) to recall how much energy sits in every one of your cells.\n\nI see nine energy systems in the human body, though I was not the first to discover any of them. Each is recognized in the healing tradition of one culture or another and each is a mixture of electrical, electromagnetic, and more subtle energies. For instance, the chakras, one of these nine systems, can be measured according to electromagnetic frequencies in the area of the body where the chakras are located.  But the chakras are not _just_ electromagnetic frequencies. They also contain information that a sensitive person can \"read\" intuitively by becoming attuned to the subtle energies held by that chakra. That is why a healer may see or even relive a person's deep traumatic memories by tuning in to the person's chakra energies.\n\nNot only can these visions be immediately verified by the client, different healers working with the same person often pick up on the same story. Is a chakra an electromagnetic field? Yes. Is it also a more subtle type of energy that carries information electromagnetic fields are not known to contain? Yes again.\n\nWhile I have always seen these nine energy systems, it was only from working with people in my practice that I clearly distinguished among them and learned that each has been identified, named, and worked with in the healing traditions of other cultures. Of the nine energy systems, names for three of them have entered our language: the meridians, chakras, and aura. Descriptions of these energies by \"seers\" often correspond with one another, and their physical existence is increasingly being verified by instruments measuring electricity, electromagnetic fields, light, or other forms of energy. The following overview of the nine systems is built around an _analogy_ for each system, designed to give you a more concrete sense of the nature and function of these invisible systems.\n\n1. _**The meridians:**_ In the way an artery transports blood, a meridian transports energy. As the body's _energy bloodstream,_ the meridian system carries the life force, adjusts metabolism, removes blockages, and even determines the speed and form of cellular change. The flow of the meridian energy pathways is as critical as the flow of blood. No energy, no life. Meridians affect every organ and every physiological system, including the immune, nervous, endocrine, circulatory, respiratory, digestive, skeletal, muscular, and lymphatic systems. Each system is fed by at least one meridian. If a meridian's energy is obstructed or unregulated, the system it feeds is jeopardized. The meridians include fourteen tangible channels that carry energy into, through, and out of your body. Your meridian pathways also connect hundreds of tiny, electromagnetically distinct points along the surface of the skin. These are known as acupuncture points. They have less electrical resistance than other areas of the skin, and they can be stimulated with needles or physical pressure to release or redistribute energy along the meridian pathway.\n\n2. _**The chakras:**_ The word chakra translates from the Sanskrit as \"disk,\" \"vortex,\" or \"wheel.\" The chakras are concentrated centers of energy. Each major chakra in the human body is a center of swirling energy positioned at one of seven points, from the base of your spine to the top of your head. Whereas the meridians deliver their energy _to_ the organs, the chakras bathe the organs _in_ their energies. Each chakra supplies energy to specific organs, corresponds to a distinct aspect of your personality, and resonates (respectively, from the bottom to the top chakra) with one of seven universal principles having to do with survival, creativity, identity, love, expression, comprehension, or transcendence. Your chakra energies encrypt your experiences in much the way your neurons code your memories. An imprint of every emotionally significant event you have experienced is recorded in your chakra energies. A sensitive practitioner's hand held over a chakra may resonate with pain in a related organ, congestion in a lymph node, subtle abnormalities in heat or pulsing, and areas of emotional turmoil, or may even tune in to a stored memory that might be addressed as part of the healing process.\n\n3. _**The aura:**_ Your aura (or biofield, the term used by scientists who have been studying it10) is a multilayered shell of energy that emanates from your body and interacts with the energies of your environment. It is itself a _protective atmosphere_ that surrounds you, filtering out many of the energies you encounter and drawing in others that you need. Like a space suit, your aura protects you from harmful energies. Like a radio antenna, on the other hand, it brings in energies with which it resonates. The aura is a conduit, a two-way antenna that _brings in_ energy from the environment to your chakras and that _sends_ energy from your chakras outward. When you feel happy, attractive, and spirited, your aura may fill an entire room. When you are sad, despondent, and somber, your aura crashes in on you, forming an energetic shell that isolates you from the world. Some people's auras seem to reach out and embrace you. You know such people, and if you could see their auras, you would find that their aura energies do indeed reach toward your own. You also know people whose auras are so tightly bound and protective that they keep you out like an electric fence. A study conducted by Valerie Hunt, a neurophysiologist at UCLA's Energy Fields Laboratory, correlated \"aura readings\" with physiological measures.  The auras seen by eight practitioners not only corresponded with one another, they also correlated with wave patterns picked up by electrodes on the skin at the spot that was being observed.\n\n4. _**The electrics:**_ The electrics are an energy that seems to emerge from the electrical dimension of the other energy systems. They are not an independent energy system like the meridians, chakras, or aura, but they are rather intimately related to all the major energy systems: separate from each but also an aspect of each, something like the way liquid is separate from yet part of each of your organs. The electrics serve as a bridge that connects all the energy systems at the basic level of the body's electricity. I usually have no idea what will occur when I first hold a person's electric points. The energy goes where it is needed. There are reports of scar tissue being healed during an electrics session, heart ataxia eliminated, a knee replacement operation avoided, and all manner of emotional trauma overcome. But most important, in terms of whole-body healing, is the way the electrics connect all the systems. If energy fields such as the aura and chakras align the organs and other energies by surrounding them, the electrics move right through them, connecting and coordinating them at the tangible dimension of their electrical nature.\n\n5. _**The Celtic weave:**_ The body's energies spin, spiral, curve, twist, crisscross, and weave themselves into patterns of magnificent beauty. The equilibrium of this kaleidoscope of colors and shapes is maintained by an energy system known by different names to energy healers throughout the world. In the East, it has been called the \"Tibetan energy ring.\" In yoga tradition, it is represented by two curved lines that cross seven times, symbolically encasing the seven chakras. In the West, it is seen in the caduceus, the intertwined serpents on a staff\u2014also crossing seven times\u2014associated initially with the Greek god Hermes, messenger for the gods, and later used as a symbol in alchemy and then medicine. I use the term _Celtic weave_ not only because I have a personal affinity with Celtic healing, but also because the pattern _looks_ to me like the old Celtic drawings of a dynamic, spiraling infinity sign, never beginning and never ending and sometimes forming a triple spiral. Like _invisible threads_ that keep all the energy systems functioning as a single unit, the Celtic weave networks throughout and around the body in spiraling figure-eight patterns. It is a living system, continually weaving new crossovers, ever expanding and contracting. The double helix of DNA is this pattern in microcosm. The left hemisphere's control of the right side of the body and the right hemisphere's control of the left side is this pattern writ large. These crisscrossing energies permeating your body are the \"connective tissue\" of your energy system.\n\n6. _**The five rhythms:**_ Your meridians, chakras, aura, and other essential energies are influenced by a more pervasive energy system. I do not see it as a separate energy but rather as a _rhythm_ that runs through all the others, leaving its vibratory imprint on physical attributes, health patterns, and personality traits. Mapped long ago in traditional Chinese medicine, all of life was categorized into five \"elements,\" \"movements,\" or \"seasons\" (there is no perfect translation\u2014all three terms have been used, suggesting qualities of being substantial, dynamic, and cyclical). These energies were considered the building blocks of the universe, providing a basis for understanding how the world works, how societies organize themselves, and what the human body needs to maintain health. Metaphors for describing these five distinct rhythms have drawn from concrete, observable elements of nature (water, wood, fire, earth, and metal) and from the seasons (winter, spring, summer, Indian summer, and autumn). Like the background music during a movie, the person's primary rhythm, in combination with the changing rhythms of life's seasons, directs the tone and mood of the entire energy system and sets the atmosphere of the life being lived.\n\n7. _**The triple warmer:**_ Triple warmer is the meridian that networks the energies of the immune system to attack an invader, and it mobilizes the body's energies in an emergency for the fight, flight, or freeze response. It operates in ways that are so beyond the range of any other meridian that it must be considered a system unto itself. Its energies work in conjunction with the hypothalamus gland, which is the body's thermostat and also the instigator of the body's emergency response. Like an _army,_ triple warmer mobilizes during threat or perceived threat, coordinating all the other energy systems to activate the immune response, govern the fight\/flight\/freeze mechanism, and establish and maintain habitual responses to threat.\n\n8. _**The radiant circuits:**_ Whereas the meridians are tied to fixed pathways and specific organs, the radiant energies operate as fluid fields, embodying a distinct spontaneous intelligence. Like hyperlinks on a Web site, they jump instantly to wherever they are needed, bringing revitalization, joy, and spiritual connection. If triple warmer mobilizes your _inner militia,_ the radiant circuits mobilize your _inner mom,_ showering you with healing energy, providing life-sustaining resources, and lifting your morale. The radiant circuits function to ensure that all the other energy systems are working for the common good. They redistribute energies to where they are most needed, responding to any health challenge the body might encounter. In terms of evolution, the radiant circuits have been around longer than the meridians. Primitive organisms such as insects move their energies via the radiant circuits rather than through a meridian system, and the radiant circuits can be seen in the embryo before the meridians develop. As in the way that riverbeds are formed, it is as if radiant energies that habitually followed the same course became meridians.\n\n9. _**The basic grid:**_ The basic grid is your body's foundational energy. Like the chassis of a car, all the other energy systems ride on the energy of the basic grid. For instance, when you are lying down, it would appear to a person who sees subtle energies that each of your chakras sits upon this foundational energy. Grid energy is sturdy and fundamental. But severe trauma can damage your foundation, and when this occurs, it does not usually repair itself spontaneously. Rather, the other energy systems adjust themselves to the damaged grid, much as a personality may be formed around early traumatic experiences. Repairing a person's basic grid is one of the most advanced and intense forms of energy medicine. If a grid's structure or a car's chassis is sound, you never notice it is there; if it is damaged, nothing else is quite right.\n\n#  DO YOU WANT YOUR HEALTH CARE APPROACH TO BE BASED ON THE REALITY OF ENERGY OR THE ILLUSION OF MATTER?\n\nEinstein told us that \"energy is everything,\" and modern physics has confirmed, in essence, that the fundamental \"stuff\" of stuff isn't stuff at all. It is energy, somehow constellated into a form that we _experience_ as solid. You might think that would hold some implications for the way we approach healing. But it makes no intuitive sense, so we simply discount it. I mean, is the material world simply an illusion?\n\nActually, yes. For starters, the atoms of which you are composed are one part nucleus and, depending on the atom, 10,000 to 100,000 parts space. That is, they are almost all space. The atom's empty space is encased by \"particles\" called electrons that surround it something like a cloud, but they aren't really particles at all. They are mysteries that seem to function sometimes like particles and sometimes like waves, more like light. In fact, according to some physicists, a proton is composed of light circling a point, an electron is light moving between two points, and _those_ are the building blocks of matter. Beyond that, if you increased the nucleus of an atom to the size of a billiard ball, the nearest electron would be more than a mile away. In between the nucleus and the electron is space. If atoms are the fundamental building blocks of the chair or couch on which you are sitting, and if space is the most fundamental property of each of its atoms, how does it support you? If you are almost all space, your clothes are almost all space, and the couch is almost all space, why don't you simply pass through one another?\n\nAt the most basic level, it would certainly seem that if two atoms are headed on a collision course they would\u2014as minuscule galaxies of mostly space\u2014indeed pass through one another. But they do not. The reason is their _energetic_ charge. Because the electron \"cloud\" surrounding each atom carries a negative charge, the atoms repel one another. At the larger level (no offense) of your bottom on the couch, your electrons and its electrons oppose one another, so you are, to a degree, actually _levitating_ away from the couch.\n\nThe more you look at the fundamental building blocks of matter, in fact, the more puzzling everything becomes. Niels Bohr, widely considered one of the greatest physicists of the twentieth century, once quipped that \"if you think you can talk about quantum theory without feeling dizzy, you haven't understood the first thing about it.\" Returning to the question of why we don't experience everything as energy if it really is energy, there may be much more than our senses perceive. As my dear friend Ann Mortifee's son, Devon, observed when he was eight: \"If I were a dog, it would all be in black and white. If I were a mosquito, I would only see heat waves. If I were a snake, it would all be infrared. So I guess you can never really know what is there; it all depends on whose eyes are looking.\" Through my eyes, at least, the human body is a system of living energy, and energy medicine appears to be barking up the right tree.\n\n#  GENES OR FIELDS?\n\nWestern medicine tends to wait for broken parts and broken systems and then tries to fix them. Energy medicine keeps a focus on your body's overall functioning and is as much for the prevention of illness as it is for treating it. Conventional medicine is long on invasive interventions and short on prevention.\n\nThe prevailing paradigm you learned in high school biology is that the blueprint for building and maintaining your body is coded in your genes. But that is only part of the story. Deep in the nucleus of each cell of your body are indeed, as you were taught, 46 chromosomes (23 from each parent), threadlike structures made of the nucleic acid DNA. Your chromosomes carry close to 24,000 genes, the fundamental units of heredity by which characteristics such as facial structure, hair color, eye color, height, build, introversion\/extroversion, and specific types of intelligence, as well as susceptibility to certain diseases, are passed on from the parents to the offspring.\n\nHow all this is actually coordinated is a mystery. We do know that genes instruct cells to produce proteins or other molecules. However, each cell undergoes some 100,000 chemical reactions per second, and these are exquisitely coordinated with the actions of many of the body's other 100 trillion or so cells. The number of sensors and switching devices that would be required in each cell to orchestrate such a lavish operation is of an order that far exceeds any known mechanisms.\n\nAnd how does the gene, which gives the instructions, even know it is part of, say, a kidney cell and not part of a liver cell? The chromosomes and genes in the nucleus of every cell are identical. In fact, when primitive, undifferentiated tissue cells from a salamander were grafted near the tail, they grew into another tail; when grafted near the hind leg, they grew into another leg. These identical genes, depending on where they are located, give their instructions as if they are perfectly aware of not only what is going on all over the body but also what is needed from them in relationship to it all. How do genes know which instructions are required? Lynne McTaggart asks, \"If all these genes are working together like some unimaginably big orchestra, who or what is the conductor?\"\n\nWestern science has no idea. No one has identified chemical mechanisms that inform the genes about the state of the entire body. What of the concept of an \"energy field\" where the information is more or less \"broadcast\" to the genes? It is hardly plausible that billions of as-yet-undiscovered chemical reactions would be carrying out the job. Think instead of a TV signal activating the pixels of your television and the coil of its speakers to produce the next episode of _Boston Legal._ It is quite plausible that some sort of field is broadcasting information and coordinating cell behavior. Wild speculation? Perhaps. But until we have scientific instruments that can reliably detect such fields, healers who are sensitive to the body's subtle energies are themselves just such instruments. And by influencing these fields, as you will learn to do in the coming pages, you will be able to impact the moment-to-moment expression of the genes that are instructing your body with its every breath, every heartbeat, and every brain wave. An incredible universe is spinning right there beneath your skin. Interventions into the energy fields that coordinate that universe have some dramatic advantages over drugs and surgery.\n\n#  FIELDS HOLD INFORMATION\n\nWhile biologists have carefully mapped the actions of homeostasis and other complex feedback mechanisms, these are the notes, not the melody. No chemical explanations account for how the whole scheme is managed. The intelligence displayed by the body's energies and energy fields, however, is stunning, and may provide the explanation. Even at the subatomic level, as Einstein observed, \"the field is the sole governing agency of the particle.\"\n\nThe idea that energy fields impact biological development keeps reemerging within Western science. Sir Isaac Newton provided the first modern description of the aura in 1729 when he wrote of an \"electromagnetic light, a subtle, vibrating, electric and elastic medium that was excitable and exhibited phenomena such as repulsion, attraction, sensation, and motion.\" In the 1930s Harold Burr, a neuroanatomist at Yale, measured the electromagnetic field around an unfertilized salamander egg and found that it was shaped like a mature salamander, as if the blueprint for the adult were already there in the egg's energy field. He was amazed, actually, to find that the electrical axis that would later align with the brain and spinal cord was already present in the unfertilized egg. He went on to find electromagnetic fields surrounding all manner of organisms, from molds to plants to frogs to humans, and he was able to distinguish electrical patterns that corresponded with health and with illness.\n\nThe role of electromagnetic fields in healing is well established. When an animal is injured, for instance, electrical currents connecting enormous numbers of cells are produced as part of the growth and repair mechanism. In addition to such internally generated fields, when external currents are applied to an area of tissue, large numbers of cells act in concert. They initiate physiological changes, which may be for better or for worse. This finding may explain the therapeutic effects when a healing practitioner's hand (which itself generates a measurable electromagnetic field) has been held in the proximity of diseased or injured tissue. When a healer is at rest and then begins to work with a patient, the electromagnetic field of his or her hands increases significantly.\n\nWhile the idea that fields carry information is still in many ways unfamiliar in our culture, powerful examples have made their way into the public eye. Some of the most dramatic instances have been the heart transplant patients who have information about the person whose heart now beats in their bosom. No explanation makes sense other than that the heart carries its own energy field (indeed, the electromagnetic field of the heart is about 60 times greater in amplitude than that of the brain, and its magnetic field according to some estimates is up to 5,000 times stronger22) _and_ that this field carries information about the person. Consider the following story, told by a psychiatrist to an international group of psychotherapists, about one of her patients:\n\n_I have a patient, an eight-year-old little girl who received the heart of a murdered ten-year-old girl. Her mother brought her to me when she started screaming at night about her dreams of the man who had murdered her donor. She said her daughter knew who it was. After several sessions, I just could not deny the reality of what this child was telling me. Her mother and I finally decided to call the police and, using the descriptions from the little girl, they found the murderer. He was easily convicted with evidence my patient provided. The time, the weapon, the place, the clothes he wore, what the little girl he killed had said to him . . . everything the little heart transplant recipient reported was completely accurate._ 23\n\n#  THE PHYSICIST AND THE PHYSICIAN DRINK FROM DIFFERENT CUPS\n\nThis account, which is consistent with the experiences of many organ recipients, seems to beg for an explanation that transcends conventional understanding of the material world. The great irony here is that the paradigm or worldview embraced by Western medicine is a century behind the paradigm used by modern physics. In 1905, Albert Einstein published his piercing formula, e=mc, showing that energy and matter are interchangeable. This discovery revealed that Newtonian physics, which focuses on the mechanics of life and is the basis of Western medicine, gives us only a glimpse of a much larger story. The implications of this larger story burst into our collective psyche on August 6, 1945, when the myth of Prometheus, who stole fire from the gods, became the terrifying reality of a humanity that suddenly had an atomic bomb. But the realization that the billiard ball-like atoms of a century ago are really packets of energy\u2014unique in their distribution of positive and negative charges, spin rate, and vibrational pattern\u2014is also about to revolutionize some of our most cherished Promethean inventions, such as televisions, cell phones, and computers, all originally based on electromagnetic effects.\n\nNonetheless, Western medicine continues to focus on the physiology and chemistry of the body rather than its energies, preserving the hallowed place of pharmaceuticals and surgery over energy treatments in our health care practices. But leading-edge science does not support this unilateral approach. According to cell biologist Bruce Lipton, who has served as a research scientist on the faculty of Stanford Medical School, hundreds upon hundreds of scientific studies over the past fifty years have revealed that \"every facet of biological regulation\" is profoundly impacted by the \"invisible forces\" of the electromagnetic spectrum. He explains that specific patterns of \"electromagnetic radiation regulate DNA, RNA, and protein synthesis, alter protein shape and function, and control gene regulation, cell division, cell differentiation, morphogenesis [the process by which cells assemble into organs and tissues], hormone secretion, nerve growth and function,\" essentially the fundamental processes that contribute to \"the unfolding of life.\" But, he laments, \"though these research studies have been published in some of the most respected mainstream biomedical journals, their revolutionary findings have not been incorporated into our medical school curriculum.\"\n\nWhat does this disregard of the role of energy in regulating biological processes mean for you? It means that procedures are more invasive and less precise than energy medicine. Electromagnetic imbalances cause the body to produce chemicals, such as progesterone or estrogen, to restore balance. Those chemicals are produced in the precise quantities needed and are directed only where needed. When medications are introduced into the bloodstream to do the same thing, their dosage is based on averages and guesswork. They can travel to and impact parts of the body that are not intended, with disastrous results, including increases in heart disease, strokes, and breast cancer among women who have undergone hormone replacement therapy. Although these are blandly called \"side effects,\" between 100,000 and 300,000 people in the United States die each year from medications taken as prescribed, and unintended effects of medical treatment are by some estimates our leading cause of death. A team that reviewed government health statistics over the past decade concluded, \"When the number one killer in a society is the health care system, then that system has no excuse except to address its own urgent shortcomings . . . beginning at its very foundations.\"\n\n#  CHEMICAL SIGNALS AND ELECTRICAL SIGNALS: THE TORTOISE AND THE HARE\n\nWhile the chemical paradigm remains at the foundation of conventional medicine, the energy paradigm is gaining ground for good reason. Electromagnetic frequencies are vastly more efficient than chemical signals (moving at 186,000 miles per second versus less than a centimeter per second). Plus most of the information in chemical signals is lost because so much of the operation goes into simply making and breaking chemical bonds.\n\nEnergy fields, on the other hand, can evolve and quickly adapt, keeping up with rapid and unending changes in the physical body, the environment, and other energy fields. Lipton summarizes the benefits and costs of energy versus drug treatments: \"Energy signals are 100 times more efficient and infinitely faster than physical chemical signaling. What kind of signaling would your trillion-celled community prefer? Do the math!\"\n\nConventional medical treatments still do not for the most part take advantage of the powerful ways energy can transmit information in biological systems (with some notable exceptions, such as heart pacemakers, brain \"pacemakers\" for treating Parkinson's disease and depression, harmonic frequencies that dissolve kidney stones, and the use of magnets for alleviating tendinitis, facial paralysis, and optic-nerve atrophy). In another irony, conventional medicine has had no difficulty accepting diagnostic instruments that are based on the concept of energy _as information._ Healthy and unhealthy tissues have different electromagnetic properties that can be detected in scanned images. Energy-scanning devices analyze the frequencies in these tissues. MRIs, EEGs, EMGs, and CAT scans have proven their ability to noninvasively detect illness.\n\nLipton observes that \"diseased tissue emits its own unique energy signature, which differs from the energy emitted by surrounding healthy cells,\" and he goes on to speculate that scientific evidence suggests that we will be able to tailor energy and waveforms to act as therapeutic agents \"in much the same way that we now modulate chemical structures with drugs.\" Meanwhile, indigenous healers have been making diagnoses based on the unique \"energy signatures\" of diseased tissues and devising remedies by working with the body's energies going back, undoubtedly, to our earliest times. The re-embrace of their skills and perspective, combined with the almost miraculous power of medical technology, brings us to the threshold of a brave new world in health care.\n\nBut even without the technology, the perspective and hands-on skills offered by energy medicine can be a profound and life-enhancing resource for you, and they are available _now._ With these concepts and techniques, you can better manage your energies, your hormones, your mood, and your health. This book shows you how. In the following chapter, we begin with some of my favorite ways for putting this into practice.\n\n_**Chapter 2**_\n\nEnergy Techniques for Health and Vitality\n\n_Your body is not just a collection of physical and chemical events._   \n_Like all living systems, it is a unified collection of energy fields._   \n_Take action to alter the quality of these fields, and you can_   \n_change the way your body functions for ill or good._\n\n\u2014LESLIE KENTON\n\n_Passage to Power_\n\nThis chapter presents a series of exercises based on six principles:\n\n1. Stretching makes space inside your body for your energies to move in their most natural flow.\n\n2. Clearing toxins supports the healthy flow of your body's energies.\n\n3. A set of simple exercises done on a daily basis can help establish and maintain positive \"energy habits\" that optimize your health and vitality.\n\n4. It is possible to assess how the energies are flowing in various parts of your body, and to take simple actions that enhance the flow of energies that are stagnant, blocked, or otherwise distressed.\n\n5. Healthy energies move in crossover patterns.\n\n6. You can reprogram deep-seated energetic patterns in your body and mind.\n\n#  MAKING SPACE FOR YOUR ENERGIES TO MOVE\n\nFor your cells and organs to serve you as they are designed to, the energies that sustain them must be given _space to move._ This is a basic physical principle of energy medicine. Stretching is one of the most natural ways to \"make space,\" which is in turn one of the best ways to support the natural rhythms and flows of your body's energies. From watching cats and dogs stretch upon waking to practicing disciplines that have made stretch into a science (such as yoga and tai chi), many models are available.\n\n##  A BASIC STRETCH TO KEEP YOUR ENERGIES FLOWING\n\nA simple exercise that I call \"Connecting Heaven and Earth\" is easy to do and is an excellent way to help keep your energies in a good flow. Versions of it have been found in numerous cultures throughout history. I've seen it on Egyptian hieroglyphics in the Museum of London, and variations of it can be found in qigong, yoga, and other disciplines. It is formulated here to also help integrate the left- and right-brain hemispheres and activate the radiant circuits, which carry the healing energies that you experience as joy. If I have a long day with many clients, this is the exercise I am most likely to use between clients to release stagnant energies in myself or any energies I may have taken on from my clients. It also provides a nice meditative break and is refreshing whenever you use it.\n\n**Connecting Heaven and Earth** (time\u2014about two minutes)\n\n_(Please do not be put off by all the words in the instructions here and throughout this book. It is actually easier to do these exercises than to read about them, and the photos will also help.)_\n\n1. Rub your hands together and shake them off.\n\n2. Place your hands on the fronts of your thighs with your fingers spread.\n\n3. With a deep inhalation, circle your arms out to your sides.\n\n4. On the exhalation, bring your hands together in front of your chest in a prayer position.\n\n5. Again with a deep inhalation, separate your arms, stretching one high above your head and flattening your hand back, as if pushing something above you. Stretch the other arm down, again flattening your hand as if pushing something toward the earth. Look up to the heavens (see Figure 2-1). Stay in this position for as long as is comfortable.\n\n6. Release your breath through your mouth, returning your hands to the prayer position in front of your heart.\n\n7. Repeat, switching the arm that raises and the arm that lowers. After this first set, do two or more sets.\n\n8. Coming out of this pose the final time, drop your arms and allow your body to fold over at the waist. Hang there with your knees slightly bent as you take two deep breaths. Slowly return to a standing position with a backward roll of the shoulders.\n\n_Figure 2-1_ CONNECTING HEAVEN AND EARTH\n\n#  MOVING TOXINS OUT OF YOUR BODY\n\nClearing the body of toxins is as important for personal health as removing garbage from local neighborhoods is for keeping a city clean and robust. Many illnesses can be traced to toxic buildup, which also depletes your vitality and darkens your sense of well-being. Organs such as the liver, kidneys, bladder, and intestines work in harmony to rid the body of toxins. The liver itself is a chemical-processing plant that carries out some 100 functions, including filtering toxins and waste products from the blood, producing chemicals that break down fat, and creating urea (the main substance in urine). A powerful, simple way for keeping your liver strong and clean while also supporting the flow of hormones through your bloodstream is to massage points on the hands and feet that stimulate liver function. Breathe deeply throughout the massage.\n\n_In this and all the subsequent procedures in this book, unless otherwise noted, when you are invited to \"breathe deeply,\" I suggest you breathe in slowly through your nose and then out through your mouth. This breathing pattern connects the energies of the central and governing meridians at the back of your throat, creating a healing force field that permeates and surrounds your body._\n\n**Massaging Your Hands to Keep Your Liver Clear**   \n(time\u2014about a minute)\n\n_Figure 2-2_ HAND MASSAGE FOR LIVER\n\n1. With your right thumb on top of the fleshy webbed area between the thumb and index finger of your left hand, and your right index and middle fingers on the bottom of the left hand, massage the webbed area. This will stimulate the fourth acupressure point on the large intestine meridian, or the Hoku point, one of the most widely used points in acupressure and acupuncture. If you are pregnant, however, skip this step.\n\n2. Bring your right thumb to the palm side of your left hand and massage up the palm and off each finger.\n\n3. End by bending your fingers backward (see Figure 2-2). Stretching these ligaments releases stagnant energy from the liver meridian (which governs the ligaments) and also frees energies from six other meridians that travel through your hands.\n\n4. Repeat on your right hand.\n\n**Massaging Your Feet to Keep Your Liver Clear**   \n(time\u2014about a minute)\n\n1. Massage the foot point that corresponds   \nwith the Hoku point, at the _V_ where the big   \ntoe meets the second toe on the top of the   \nfoot (see Figure 2-3).\n\n2. Use your thumb to massage the area between   \nthe tendons of the two toes, while   \nusing your fingers to massage the bottom   \nof your foot.\n\n3. It also feels great to massage and clear the   \n\"gate reflexes,\" the areas on the top of the   \nfoot that are between the tendons of each   \nof the toes.\n\n4. Repeat on the other side. When you are   \ndone, place your hands around the bottoms   \nof each foot and squeeze the sides hard.\n\n_Figure 2-3_ FOOT MASSAGE FOR LIVER\n\nI find myself massaging my hands almost every day, and you can readily massage the points on your feet whenever you take a bath, get in a hot tub, or sink into bed. This is a great form of self-care that not only helps your liver function better, it also keeps your hands and feet\u2014which function as \"energy antennae\"\u2014open and clear.\n\n#  THE LYMPH SYSTEM IN ENERGY MEDICINE\n\nYou know lymph as the clear fluid that comes out of a cut. It is there to remove foreign matter and fight germs. Lymph plays a key role in your immune system by helping counter conditions ranging from colds to cancer. It creates antibodies and produces lymphocytes, specialized white blood cells produced in the lymph nodes, found in your neck, armpits, abdomen, and groin. Your lymph system also carries proteins, hormones, and fats to the cells, and eliminates dead tissue and other waste products. Its role in eliminating toxins, as do organs such as the liver and kidneys, is particularly important in energy medicine.\n\nBecause toxins block the flow of energy\u2014and this includes \"energy\" toxins such as emotional residue, stagnant energies, electromagnetic pollution, and radiation, as well as chemical toxins\u2014the necessity for _moving toxins out of your body_ is a second basic principle of energy medicine, along with the need for _space_ and _stretch._ The lymphatic system moves both chemical and energy toxins out of the body. Chemical toxins, in fact, are a cause of stagnant energy.\n\nTwo types of liquid circulate through your body: blood and lymph, with the lymphatic system sometimes being called the body's \"other\" circulatory system. You, in fact, have twice as many lymph vessels as blood vessels, but while your circulatory system has your heart to pump your blood, your lymphatic system has no pump. Your body's movement during your daily activities, and during exercise in particular, fosters the circulation of your lymph. But sometimes the accumulation of toxins will cause blockages in the flow of the lymph so it becomes less and less effective in clearing toxins from certain parts of the body.\n\nReflex points that stimulate the flow of lymph are located on various points throughout your body, particularly on the chest, back, and upper portions of the legs. When they are massaged, the lymphatic system is stimulated and toxins are removed more quickly and effectively. This use was developed in the 1930s by Dr. Frank Chapman, D.O., and is a mainstay in the field of applied kinesiology. These points, called \"neurolymphatic reflexes,\" have, in my experience, proven to be extraordinarily beneficial in releasing the body's toxins as well as in freeing the flow of energy throughout the body.\n\nDavid used to question me about whether working these points was really part of energy medicine. He thought they were part of massage. Really, David, who cares! But I mention this because a few influential legislators have too often been persuaded by a professional group that they should create legislation that protects the narrow interests of the professional group. Some states have actually made it illegal for certain health care providers to utilize isolated massage techniques unless they have a massage license. This does not serve the public in any way, shape, or manner. Massage techniques always have been, and always will be, part of many approaches to healing, including energy medicine, and it is not necessary to become a fully trained massage therapist (which I happen to be) to utilize specific techniques effectively and appropriately. Massaging neurolymphatic reflex points releases toxins into the lymphatic vessels and subsequently into the bloodstream so that they can be eliminated. Clearing toxins in this way makes more space for energy to flow, and that is why working with these points is an important technique within energy medicine.\n\nThere are 90-some neurolymphatic reflex points on the surface of the body, sometimes simply referred to as \"lymphatic points,\" though, as reflexes, they are not necessarily situated directly over the lymph nodes, vessels, or other lymph tissue. Figure 2-4 gives a map. When the neurolymphatic reflex points become clogged, every system in your body is compromised. Congested neurolymphatic reflex points feel sore when massaged. For that reason, they are not hard to locate. And there are so many of these points so close to one another that you won't miss them. Massaging them is a way to clear them and allow the energy that has been blocked to flow again. If you simply start to push in on these points with your middle finger, using some pressure, you are likely to quickly identify at least a few that are sore. If you are certain there is no injury, no strain from exercise, and no medical condition responsible for the soreness, the chances are good that you have located a neurolymphatic reflex point needing attention.\n\n**The Lymphatic Massage** (time\u2014about 10 seconds on each point)\n\n1. With your middle finger pressing neurolymphatic reflex points on your chest, identify one that is sore.\n\n2. Press into the skin over this spot with two or three fingers and massage by firmly moving the skin in all directions.\n\n3. Press hard enough that you really feel the pressure but not so hard that you risk bruising yourself. Massage each sore point for about ten seconds.\n\nMassaging your lymph points is a simple procedure that will enhance the flow of your energies while also supporting your hormones to function as nature intended. A few cautions with this method need to be followed. You should not massage painful points if the pain is based on a bruise or injury. Be conservative about massaging too many points that are sore in the same session. Since massaging these points does release toxins into the lymph and then the bloodstream, you do not want to overwhelm your body's capacity to eliminate them. This caution applies particularly if you are or have recently been ill. It also applies if you have an autoimmune disorder such as multiple sclerosis or Parkinson's disease, since it is sometimes difficult for people with autoimmune disorders to assimilate changes in the body's chemistry; so go slow. But for the vast majority of situations, these are wonderful points to massage. The process is generally pleasant (in the sense of \"it hurts so good\"), you will often feel the pleasing sensation of energy moving in the area being cleared immediately after the self-massage, and it is an easy way to help keep your energies flowing. So with the Connecting Heaven and Earth exercise, the Hand Massage, the Foot Massage, and the Lymphatic Massage, you have four simple ways to maintain your body as a friendlier container for your energies.\n\n_Figure 2-4_ NEUROLYMPHATIC REFLEX POINTS\n\n#  A FEEL- GOOD DAILY ENERGY ROUTINE\n\nSome of the techniques you will learn in this book, particularly those presented in this chapter, will help you keep your overall energies vital and balanced. They also have the effect of maintaining harmony among the energies that govern your body's chemistry. Other procedures, presented in each of the subsequent chapters, are oriented toward working with specific conditions such as PMS, fertility problems, or hot flashes. Energy medicine often begins, however, with a focus on the fundamental energies that affect general well-being. Following is a set of powerful techniques, presented as a five-minute routine, which I encourage you to do every day.\n\nAfter working with more than 10,000 clients in 90-minute individual sessions, giving most of them back-home assignments and seeing what occurred, I have some ideas about which techniques are the most potent and which will have a positive impact for just about everyone. When we were writing my first book, _Energy Medicine,_ we wanted to give people a simple routine they could use every day to keep their \"energies humming.\" I identified the procedures that had the greatest impact on the greatest number of people and selected those that, as a set, tended to bring all of the body's energy systems into balance.\n\nThe ultimate effect is to reestablish positive \"energy habits.\" As we adjust to the stresses and unnatural practices and substances that characterize modern life, our energy systems make compromises as they valiantly attempt to keep us healthy. Often, however, these compromises throw our energies into habitual patterns that hurt us more than help us. The Five-Minute Daily Energy Routine is like pressing the reset button, helping restore your body's natural energy flows.\n\nThe Daily Energy Routine follows. I refer to it throughout this book, and I suggest that you practice it regularly. All the subsequent techniques are based on the assumption that you are taking basic steps to keep your overall energies in balance. In the same sense that a woman with blurry vision who is trying to improve her reading ability might consider eye exercises before taking a speed-reading class, some basic measures are necessary before fine-tuning will be as effective.\n\nI know it is no small thing to suggest that you build another routine into your life. We are all extraordinarily busy, an epidemic of modern life. But some investments pay off. I promise you that the Daily Energy Routine, practiced regularly, will give you a good return\u2014in terms of how you feel and function\u2014on the small daily investment it requires.\n\nBeyond my personal promise, however, some very encouraging research supports this claim. One of the practitioners I have trained, Linda Geronilla, Ph.D., a psychologist as well as energy medicine practitioner, taught a group of 18 school-teachers a modified version of the Daily Energy Routine and studied whether their brain patterns changed. Certain activities have been shown to optimize specific areas of the brain. Dr. Geronilla patterned her study after the work of Daniel Amen, M.D., who has investigated the relationship of brain activity and optimal functioning with more than 30,000 subjects. At the end of eight weeks of daily practice, positive changes could be inferred in four of the six brain areas that Dr. Amen has identified as most important to target in programs for optimizing people's performance (the prefrontal cortex, the cerebellum, the temporal lobe, and the anterior cingulate gyrus). The participants also reported improvements in capacities such as memory, concentration, and energy level. A control group did not report such improvements and did not show significant changes in any of the six brain areas over the eight-week period. These findings corroborate Dr. Amen's previously untested but published prediction that \"Brain Gym\" (an approach that uses similar exercises for working with the body's energies) will enhance the cerebellum\u2014 helping people think more clearly and quickly\u2014and improve their \"judgment, attention, and overall brain health.\"\n\nWhile this should persuade you to seriously consider making the Five-Minute Daily Energy Routine a habit, I also know from experience that you will be more likely to maintain a program like this if you tie it into an activity you already do. If you exercise regularly, or do yoga, tai chi, or Pilates, it can be a great warm-up or cooldown. If you meditate, it can bring you into a centered space from which you can go deeper. Some people, particularly those who are not morning persons, do it before they get out of bed. Some people do it as a kind of transition ritual when they come home from work. Some people do it as part of their bath or shower. The more comfortable you are, the better. It is not a matter of \"no pain, no gain\"\u2014just the opposite, in fact. The Five-Minute Daily Energy Routine consists of the Three Thumps, the Cross Crawl, the Wayne Cook Posture, the Crown Pull, the Lymphatic Massage, the Zip-up, and the Hook-up.\n\n**Three Thumps** (time\u2014about 30 seconds)\n\nCertain points on your body, when tapped with your fingers, will impact your energy field in predictable ways, sending electrochemical impulses to your brain and releasing neurotransmitters. By tapping three specific sets of points, a technique I call the \"Three Thumps,\" you can activate a sequence of responses that will restore you when you are tired, increase your vitality, and keep your immune system strong amid stress. Do not be too concerned about finding the precise location of each point. If you use several fingers to tap in the vicinity described, you will hit the right spot.\n\n**Thump #1: Your K-27 Points.** Acupuncture points (also called acupressure points with procedures in which needles are not used) are tiny energy centers arranged along the body's fourteen major meridians or energy pathways. The K-27 points (the paired 27th acupressure points on the left side and the right side of the kidney meridian) are juncture points that affect all of the other meridians. Tapping or massaging your K-27 points also sends signals to your brain to adjust your energies so you will be more alert and able to perform more effectively. Tapping these points can energize you when you are feeling drowsy and focus you if you are having difficulty concentrating.\n\nTo locate these points, place the pointer finger of each hand on your collarbone and move your hands toward each other until you reach the two inside corners of your collarbone. Drop straight down from these points to about an inch below your collarbone. For most people, there is a soft spot or indentation there. Then breathe slowly and deeply as you firmly tap or massage your K-27 points (see Figure 2-5) for two or three deep breaths.\n\n_Figure 2-5_ K-27, THYMUS, AND SPLEEN THUMPS\n\n**Thump #2: Tarzan Thump.** Tapping the area over the thymus gland (think of Tarzan) is a simple technique that awakens the body's energies, sharpens the immune system, and increases your strength and vitality. The thymus thump can help you if you are feeling bombarded by negative energies, catching a cold, or fighting an infection, or if your immune system is otherwise challenged. Your thymus gland supports your immune system. If you keep making choices that are not in harmony with your body's needs and design, your thymus's surveillance mechanism becomes confused. Thumping the thymus helps stimulate and reset it:\n\n1. Place the fingers of either or both hands in the center of your sternum, at the thymus gland, about two inches below the K-27 points in the center of your chest (see Figure 2-5).\n\n2. Using your thumb and all your fingers as you tap firmly, breathe slowly and deeply, in through your nose and out through your mouth for two or three deep breaths.\n\n**Thump #3: Spleen Points.** Tapping points that affect the spleen meridian is a quick way to lift your energy level, balance your blood sugar levels, and bolster your immune system.\n\n1. Tap the _neurolymphatic_ spleen points firmly for about 15 seconds (see Figure 2-5). The _neurolymphatic reflex points_ are located beneath the breasts in line with the nipples and down one rib.\n\n **Alternate:** Tap the spleen _acupuncture_ points firmly for about 15 seconds. These acupuncture points are on the sides of the body about four inches beneath the armpits.\n\n2. If either set is more tender, tap those points.\n\n3. Breathe slowly and deeply, in through your nose and out through your mouth, as you tap.\n\n4. Tap (or massage) firmly while you take two or three deep breaths.\n\n**Cross Crawl** (time\u2014about 30 seconds)\n\nThe cross crawl facilitates the crossover of energy between the brain's right and left hemispheres. It will help you feel more vital, think more clearly, and move with better coordination. It is essentially an exaggerated walk. Most people find it energizing. If you find that it drains you or tires you, instead do the Homolateral Crossover (page 67). To do the Cross Crawl:\n\n1. First briskly tap the K-27 points (see page 53) to assure that your meridians are moving in a forward direction.\n\n2. The cross crawl is as simple as marching in place. While standing, lift your right arm and left leg simultaneously (see Figure 2-6).\n\n3. As you let them down, raise your left arm and right leg. If you are unable to do this because of a physical disability, here is an alternative: while sitting, lift your right knee and touch it with your left hand, then lower it and lift your left knee and touch it with your right hand.\n\n4. Repeat, this time exaggerating the lift of your leg and the swing of your arm across the midline to the opposite side of your body.\n\n5. Continue in this exaggerated march for at least a minute, again breathing deeply in through your nose and out through your mouth.\n\n_Figure 2-6_ CROSS CRAWL\n\n**Wayne Cook Posture** (time\u2014about 90 seconds)\n\nI use the Wayne Cook Posture when I am overwhelmed or hysterical, cannot get clarity on a situation, cannot concentrate, must confront someone, or am upset after someone has confronted me. This procedure is named to honor Wayne Cook, a pioneering researcher of bioenergetic force fields, who invented the approach that I have modified into the form presented here. Perhaps more than any other single approach I teach, the Wayne Cook Posture can calm you, bring order to your thinking, and help you better understand and confront the problems you face.\n\nThis technique is effective even when the upset is so intense that you are unable to quit crying, are finding yourself snapping or yelling at others, are sinking into despair, or are feeling that you are beyond exhaustion. It helps process stress hormones. Almost immediately, you will begin to feel less crazy and less overwhelmed. To do the Wayne Cook Posture:\n\n1. Sitting in a chair with your spine straight, place your right foot over your left knee. Wrap your left hand around the front of your right ankle and your right hand over the ball of your right foot, with your fingers curled around the side of the right foot (see Figure 2-7a).\n\n2. Breathe in slowly through your nose, letting the breath lift your body as you breathe in. At the same time, pull your leg toward you, creating a stretch. As you exhale, breathe out of your mouth slowly, letting your body relax. Repeat this slow breathing and stretching four or five times.\n\n3. Switch to the other foot. Place your left foot over your right knee. Wrap your right hand around the front of your left ankle and your left hand over the ball of your left foot, with your fingers curled around the side of your left foot. Use the same breathing.\n\n4. Uncross your legs and \"steeple\" your fingertips together so they form a pyramid. Bring your thumbs to rest on your \"third eye,\" just above the bridge of your nose. Breathe slowly and deeply, in through your nose and out through your mouth, about three or four full breaths (see Figure 2-7b).\n\n5. On the last exhalation, curl your fingers into the middle of your forehead and separate them, firmly and pleasantly, pulling across your forehead to your temples.\n\n6. Slowly bring your hands down in front of you. Surrender into your own breathing.\n\n_Figure 2-7_ WAYNE COOK POSTURE\n\n**Crown Pull** (time\u2014about 30 seconds)\n\nA great deal of energy is processed in your brain and skull, but it can become stagnant if it doesn't release and move out through the energy center at the top of your head (called the crown chakra in yoga tradition). The Crown Pull physically opens this chakra so energy can move through it. It clears the cobwebs from your mind and brings a calm to your nervous system, releasing mental congestion, refreshing your mind, and opening you to higher inspiration. People today are hungry to receive higher forms of inspiration and guidance and to feel more fully connected with the force of creation or God or however they conceive of the larger picture, and the crown chakra is a key to such psychic opening. The Crown Pull is a powerful way to begin to open your crown chakra, and it also has the more mundane benefit of simply clearing congested energies. To do the Crown Pull, breathe deeply, in through your nose, out through your mouth, and:\n\n1. Place your thumbs at your temples on the sides of your head. Curl your fingers and rest your fingertips at the center of your forehead.\n\n2. Slowly, and with some pressure, pull your fingers apart so that you stretch the skin above your eyebrows (see Figure 2-8a).\n\n3. Rest your fingertips at your hairline and repeat the stretch.\n\n4. Continue this pattern with your fingers curled, pushing in at each of the following locations:\n\na. Fingers at the top of your head, with your little fingers at the hairline. Push down with some pressure and pull your hands away from each other, as if pulling your head apart (see Figure 2-8b).\n\nb. Fingers over the curve at the back of your head, again using the same stretch (see Figure 2-8c).\n\nc. Fingers at the bottom of your head, again using the same stretch.\n\nd. Continue, pulling to the sides of your neck with three passes (top of neck, middle of neck, and bottom of neck), finally resting your fingers on your shoulders.\n\ne. After a deep breath, pull your fingers firmly forward over the tops of your shoulders and let them drop.\n\n_Figure 2-8_ CROWN PULL\n\n**Lymphatic Massage** (time\u2014at least a minute)\n\nMassaging the neurolymphatic reflex points discussed beginning on page 48 is so valuable that I have placed the procedure in the Daily Energy Routine. Try to work two or three sore spots each day (see Figure 2-4, page 50\u2014there are many to choose from) for about 10 to 20 seconds each. You may find that points that have been chronically sore become less tender, sometimes almost immediately. This indicates that the toxins are being eliminated. An alternative to massaging your own points is to work with a partner and give each other a \"Spinal Flush.\" Because major neurolymphatic reflex points affecting every meridian run along either side of the spine, those points are a great focus for a shared session, and the Spinal Flush is a real treat. Almost every morning when we are not on the road, I make David a health drink. Then he gives me a Spinal Flush. I hope he doesn't read this and decide to renegotiate, but I am getting the better part of the bargain. Instructions for the Spinal Flush:\n\n1. Lie facedown, or stand three or four feet from a wall and lean into it with your hands supporting you. This helps stabilize your body while your partner applies pressure to your back.\n\n2. Have your partner massage the points down both sides of your spine, half an inch to an inch to the left and right of the spine, using the thumbs or middle fingers and applying body weight to get strong pressure. Massage from the bottom of the neck down to the bottom of the sacrum (See Figure 2-9).\n\n3. Have your partner go down the notches between your vertebrae and deeply massage each point. Staying on the point for at least three seconds, your partner moves the skin in a circular motion with strong pressure, checking to make sure that the pressure is comfortable.\n\n4. Upon reaching your sacrum, your partner can repeat the massage or can complete it by \"sweeping\" the energies down your body. From your shoulders, and with an open hand, your partner sweeps all the way down your legs and off your feet, two or three times.\n\n_Figure 2-9_ SPINAL FLUSH\n\nDo not be concerned about a point being missed. Each of your meridians will be covered by simply going between all the notches. Rather than knowing which meridians are associated with which points, simply ask for special attention to any points that are sore. If you are moving through intense emotional or physical stress, or if you have been exposed to environmental toxins, doing the Lymphatic Massage or the Spinal Flush quickly helps clear your lymphatic system. The Spinal Flush not only cleanses the lymphatic system, it also stimulates the cerebrospinal fluid, clearing your head as well. It is a quick rebalance, and of all the energy techniques I've seen, it probably delivers the most benefit for the least effort in the greatest number of situations. As a cold is coming on, the Spinal Flush can stop it in its tracks. I also regularly recommend it to couples, both as a way of lovingly caring for each other and as a way to head off problems. If you sense that an interchange is headed toward an argument, tell your partner, as lovingly as you can, \"Up against the wall!\" and firmly work the neurolymphatic reflex points. This simple technique immediately reduces built-up stress and takes the edge off emotional overreactions.\n\n**Zip-up** (about 20 seconds)\n\nWhen you are feeling sad or vulnerable, the central meridian, one of the two energy pathways that govern your central nervous system, can be like a radio receiver that channels other people's negative thoughts and energies into you. It's as if you are open and exposed. The central meridian runs like a zipper from your pubic bone up to your bottom lip, and you can use the energies of your hands to \"zip it up.\" The Zip-up will help you feel more confident and positive about yourself, think more clearly, and protect yourself from negative energies that may be around you. Pulling your hands up the central meridian draws energy along the meridian line. Before doing the Zip-up, briskly tap the K-27 points to assure that your meridians are moving in a forward direction. Then:\n\n_Figure 2-10_ ZIP-UP\n\n1. Place your hand or hands at your pubic bone, which is the bottom end of the central meridian.\n\n2. Take a deep in-breath as you simultaneously move your hand(s), slowly and with deliberation, straight up the center of your body (see Figure 2-10), to your lower lip, where the meridian ends.\n\n3. By continuing upward, however, bringing your hands past your lips and exuberantly raising them into the sky, you connect your central meridian with your aura and with forces that are beyond you.\n\n4. Turn your palms outward and, slowly releasing your breath, extend your arms out to the sides as far as they will go and back down to your public bone.\n\n5. Repeat three times.\n\n**Hook-up** (time\u201415 to 20 seconds)\n\nThe Zip-up strengthens the central meridian. The central meridian, which sends energy up the front of your body, works in tandem with the governing meridian, which sends energy up your spine. The two meridians meet at the back of your throat. The Hook-up connects these two meridians, bridging their energies between the front and back of your body and between your head and torso. It increases your coordination and stabilizes all your energies, including the aura energies that surround you. It is one of the most powerful tools I know for quickly centering myself. It has immediate neurological consequences. Several of my students have reported using it with someone who has started to seizure and seeing the seizure stop within 15 to 20 seconds. To do the Hook-up (again, remember to breathe in through your nose and out through your mouth):\n\n_Figure 2-11_ HOOK-UP\n\n1. Place the middle finger of one hand on your third eye (between your eyebrows above the bridge of your nose).\n\n2. Place the middle finger of the other hand in your navel.\n\n3. Gently press each finger into your skin, pull it upward, and hold for 15 to 20 seconds (see Figure 2-11).\n\n#  ASSESSING YOUR BODY'S ENERGIES\n\nSome methods in energy medicine, such as those in the Five-Minute Daily Energy Routine, can benefit virtually anyone. Other procedures are tailored to a person's unique energy patterns. Clues indicating what your body needs energetically can be found simply by noticing what you feel when you tune in to your inner sensations and identifying areas of discomfort, by watching the shifts in your emotional landscape, or by noting physical symptoms. Symptoms are clues about underlying energy disturbances. In addition to observing what is going on in your body and mind, a technique for assessing the body's energies can be invaluable in your health care choices.\n\nYou can \"energy test\" to assess the state of your body's energies and to find where disturbances exist in their flow. This can lead to highly specific information. For example, your liver meridian may be depleted of energy, your kidney meridian overcharged, or your third chakra sluggish. You can also energy test to check whether a particular remedy (conventional or unconventional) is working, or at least what impact it is having on your body's energies.\n\nEnergy testing comes from the field of Applied Kinesiology, which uses the term \"muscle testing\" for the procedure. Applied Kinesiology was developed in the 1960s by chiropractor George Goodheart. Goodheart identified relationships among muscles, organs, and the body's _meridians,_ or \"subtle energy pathways.\" At least 670 acupressure points are distributed over the body's 14 meridians. The needling of an acupuncture point (or massaging of an acupressure point) affects the energy flow of the meridian on which it is located. Goodheart was able to test and verify these relationships by drawing upon the field of kinesiology (the study of muscles and how muscles coordinate to move the body). Kinesiologists had developed muscle-testing procedures for evaluating isolated muscles according to their relative strength and range of motion.\n\nSince each major muscle is associated with a meridian, Goodheart reasoned, weakness or restricted range of motion in an uninjured muscle indicates impairmentin the meridian or energy flow through that muscle. By implication, the organs that receive energy from that meridian might also be affected. Applied Kinesiology was a bridge from traditional Chinese medicine into Western culture, and it has had a strong influence on many of the contemporary approaches to energy medicine, including my own. It utilizes a variety of methods\u2014including holding, massaging, and tapping specific acupressure points\u2014to correct impaired energy flow in meridians that have been identified by a muscle test.\n\nWhile there has been growing scientific support for the validity of energy testing,  the procedure is as much art as science. Energy testing is done in many ways by different practitioners, and there are enough variables and nuances in a single energy test that teaching people how to do the procedure, as well as conducting research on it, is fraught with dilemmas.\n\nI detailed the procedure in my previous book, and I have introduced it to tens of thousands of people in my classes. While research has shown a strong correlation between the results found by people new to energy testing and highly experienced practitioners, I always give the caveat that it takes considerable practice to become proficient with the method, to know how to control all the variables, and to be confident in the results. The mind, for instance, is one of the factors that can interfere with an accurate test. Simply holding a thought shifts your energies. If you are longing for a hot fudge sundae and you put it into your energy field and test the flow of a meridian, you may be testing the effect of your anticipation of the good taste rather than the effect the sugar will have on your energy field. If you do not know how to control for the effects of the mind, the test may show what you want or expect instead of what is.\n\nI have adapted the instructions for energy testing from _Energy Medicine_ and placed them in the Appendix. I put them in the Appendix because I do not want to be overly reliant on energy testing in this book. It is a skill that is invaluable and that you can use with great benefit for the rest of your life. But it is best to receive live, hands-on training at first, or to consult a professional. There is, unfortunately, much nonsense and misapplication of the method, so that it has sometimes been reduced to little more than a parlor trick. But its potential is enormous. I have many times been consulted by physicians for help in determining the proper type and amount of medication for a tricky case, including having being brought several times into an operating room to help with delicate decisions about anesthesia.\n\nI have designed this book so it can be used without energy testing, but I have also indicated places where you can benefit from energy testing, either by having learned it yourself or by engaging a qualified practitioner. Energy testing is the most effective method I know for helping people who don't \"see\" energy work with the body's energies as if they did see it. It is invaluable in deciding how to apply energy medicine procedures and to track their effects. So I encourage you to use the instructions in the Appendix as a tutorial for developing this skill.\n\nAnd even before you have become proficient with the method, a great benefit of learning energy testing beyond being able to do specific assessments is the way you begin to attune yourself to the world of energies in which we all live. Being in harmony with your energies expands your connection to life, and energy testing gives you a way to affirm your experiences with the energies within and around you. This also opens you to other dimensions. When you attune yourself to the energy realm, you begin to sense that there is a grander purpose to your existence. From finding answers to daily health concerns to exploring the mysteries of the universe, energy testing is a wonderful tool that you can use all your life. You will also have an opportunity to do some optional experimentation with energy testing in the following section if you are already adept in the technique or choose to take a detour to the Appendix now to begin studying energy testing.\n\n#  YOUR BODY'S VITAL CROSSOVER PATTERNS\n\nEnergies need to cross over freely throughout the body. The crossover pattern that I call the Celtic Weave is essential for maintaining the health of your energies and that of your body. I've had two class participants, both male nurses, one from Brazil and one from Nepal, who immediately recognized this energy system when I described it. Each was expected by the hospital where he worked to regularly \"Celtic Weave\" his patients, one in a burn unit, the other in a general ward, though different names were used for the procedure.\n\nThe Celtic weave is both an independent energy system within the body, akin in that sense to the meridians, chakras, and aura, and also a procedure for supporting the body's crossover patterns. As an _energy system_ , the Celtic weave laces through all of your other energy systems and creates a resonance among them. It is the weaver of your force fields. It holds your entire energetic structure together. As a _procedure_ , the Celtic Weave connects all of your energies so they operate as a single web. Touch one strand, anywhere, and your whole system reverberates in harmony.\n\nThe Celtic Weave, both as an energy system and as a procedure, draws all the body's energy systems together into a tight web of communication so that information can easily travel wherever it needs to go. This enables your energy systems to work in harmony and cooperation (the energy systems, as opposed to the Celtic Weave technique). When the Celtic weave is dynamically engaged, you have a sense of power, a feeling of being charged, and your energies start humming. I've known several energy healers who can _hear_ the Celtic weave in their clients, particularly when the person is healthy and the pattern is distinct. They say it literally sounds like a musical hum.\n\nEveryone's energy is unique, yet the Celtic weave is universal. It crosses the energies over from each hemisphere of the brain to the opposite side of the body. Its largest formation is a figure eight or infinity sign extending from head to toe. In the caduceus, in the curved lines that cross seven times as used in yoga tradition, and in other systems that focus on the way energy travels up the spine, the Celtic weave is portrayed as a series of figure eights, looping seven times up the torso. But this pattern can also be found throughout the body, over the top of the head, the face, the trunk, the legs, and the feet. These intertwining energies get ever smaller, eventually weaving their primordial pattern at the cellular level. In fact, a single strand of DNA, with its spiraling double helix, may be the prototype of the Celtic weave as an energy system. I am humbled as I look at the intelligence of the forces that weave the complex energies of a human body into a thriving unit. And this discussion gives only a glimpse. Within the Celtic weave are spirals, circles, and figure eights of bright light that connect and strengthen other energies and may also take form over a part of the body needing repair or extra support. I discuss these natural geometric miracles in a paper titled \"Tibetan Energy Rings and the Celtic Weave,\" available as a free download from www.energymed.org\/hbank\/handouts\/tibetan_energy_rings.htm .\n\nIn addition to the Celtic Weave exercise are three other techniques for directly supporting the body's crossover patterns: the Homolateral Crossover, Rhythmic Eights, and the Suspension Bridge. Because the Celtic Weave exercise does \"weave it all together,\" I present these first so that what the Celtic Weave exercise weaves is what you would want woven into your energy body. For each of these three techniques, I explain how you can know if you need them. Whether or not you _need_ them, however, doing them will cause no harm and, like any good exercise, is generally beneficial. Then, with these other patterns in a good flow, doing the Celtic Weave exercise will tend to maintain that flow as it enhances the communication among all nine of your energy systems (see page 28).\n\n**Homolateral Crossover** (time\u2014about 4 minutes)\n\nYour energies are running in a homolateral pattern if you are depressed, chronically exhausted, or ill, or if your recovery from even a mild condition seems slow. Your body does this to conserve its resources, temporarily sacrificing in its energy systems the crossover patterns that are necessary for optimal functioning. Too often, however, homolateral patterning becomes habitual. By some estimates, you have less than 50 percent capacity to heal when your energies are running in a homolateral pattern. So getting homolateral energies back into a crossover pattern, particularly during illness, depression, or simply when you have \"lost your energy,\" is a fundamental energy medicine procedure.\n\nNormally, the force of the Celtic weave keeps your energies crossing over, from the energies in the smallest cell to the way the right side of your brain controls the left side of your body and the left side of your brain controls the right side of your body. At the level of these large patterns, if your energies are going straight up and down the trunk of your body instead of crossing over, you are homolateral. While not uncommon, this is not good. You do not have your full vitality when your energies are not crossing over. You cannot think as clearly. You cannot heal as readily.\n\nThe Homolateral Crossover takes about four minutes. Most people experience immediate benefits in terms of their sense of vigor and capacity to think clearly. How long such benefits will last for you, however, depends on the state of the energies affecting your health and the energy patterns your body has established. If your energies have been in a homolateral pattern for a long time, or if your energies are depleted, you may need to do the Homolateral Crossover two or three times a day to form a crossover habit.\n\nHow do you know if the Homolateral Crossover will benefit you? One way is to simply try it and then see what you feel. There are also many subjective signs. As mentioned earlier, if doing the Cross Crawl is difficult for you, or if you cannot easily coordinate your opposite arms and legs, or if just starting to do the Cross Crawl confuses or exhausts you, your energies are probably running in a homolateral pattern. In that case, walking or marching in a natural cross-crawl manner is literally going against your own flow. Other important hints that the Homolateral Crossover might be helpful to you include depression, exhaustion, inability to get well, inability to benefit from exercises (include the Daily Energy Routine), or a lack of a response to treatments (conventional or alternative) that you have reason to believe should be effective.\n\nThere is also an energy test to determine whether your energies are homolateral, so this is an opportunity, if you have learned energy testing from the Appendix or through another source, to experiment with it. This will require a partner, and you can energy test each other. To do the energy test:\n\n1. Draw a large _X_ on one piece of paper and two parallel lines on another.\n\n2. Look at the _X._ Have your partner energy test you.\n\n3. Look at the parallel lines. Energy test again.\n\nIf your energies are crossing over properly, looking at the _X_ will be in harmony with the crossover flow of your energies and you will test strong. Looking at the parallel lines will momentarily jar your energies, and you will test weak. But if your energies are homolateral, the opposite will occur. The _X_ will weaken you, the parallel lines will strengthen you.\n\nTo get your energies to cross over again, rather than going against the homolateral energies, the Homolateral Crossover begins with movements that are in alignment with the way your energies are already flowing. It then guides your energies into a crossover pattern by doing a Cross Crawl. Going back and forth between these two movements stabilizes a crossover pattern. To do the Homolateral Crossover:\n\n1. Begin by tapping or massaging the K-27 points (the first points of the Three Thumps, page 53) followed by a full-body stretch that \"reaches for the stars.\"\n\n2. Do a homolateral crawl, lifting the right arm with the right leg, and then the left arm with the left leg. Do about 12 lifts (see Figure 2-12). This may be done standing, sitting, or lying down.\n\n3. Then march in place, as when doing a Cross Crawl, lifting the right arm with the left leg and then the left arm with the right leg (see Figure 2-6, page 56). If you are sitting or lying, the movement can easily be modified to fit your position.\n\n4. After about 12 lifts of the arms and legs in this cross-crawl pattern, stop and return to the homolateral pattern (lifting the same-side arms and legs) for about 12 lifts.\n\n5. Then stop and return to a normal Cross Crawl (lifting opposite arms and legs) for about 12 lifts.\n\n6. Do this sequence a total of three times.\n\n7. Anchor it with an additional dozen normal Cross Crawls. End by again stimulating the K-27 points.\n\n_Figure 2-12_ HOMOLATERAL CROSSOVER\n\n**Rhythmic Eights** (time\u2014about a minute)\n\nWhere the Homolateral Crossover addresses large patterns within the Celtic weave, doing Rhythmic Eights brings all of the energies, from those that surround you down to those of individual cells, into a crossover pattern. If you are having difficulty relaxing, are caught up in your daily troubles, or are just feeling the blahs, Rhythmic Eights are fun to do and are always a good energy boost. It adds a touch of joy to do them to music that makes you want to move.\n\n1. With your hands hanging, sway your body, shifting your weight from one hip to the other, moving to music or moving as if you are moving to music.\n\n2. Let your arms sway with your body. You will notice that there is a natural figure-eight movement in both your arms and body as you sway from side to side. Let your arms move out farther so they are swinging wide.\n\n3. Stretch your hands out in front of you and make a sideways figure eight with your arms. Go up and over on the right, circle down, then up and over on the left, circle down, and then again up and over to the right. Twist at the waist in the direction toward which you are reaching, allowing your body's rhythm to move with your arms. It is more like a free-form dance than a tightly patterned exercise (see Figure 2-13).\n\n_Figure 2-13_ RHYTHMIC EIGHTS\n\nSeveral Rhythmic Eights will help get the left-and right-brain hemispheres into better communication. This technique is used in \"Educational Kinesiology,\" an application of energy medicine that helps children with dyslexia and other learning handicaps. Several studies have shown that making rhythmic eights on the blackboard with chalk, while putting their whole body into the motion, helps children with learning disorders.\n\n**Suspension Bridge** (time\u201430 to 60 seconds)\n\nAnother way to get the energies in your body moving and crossing over is to elongate and stretch your spine in a crossover pattern. If your muscles are tight or your body doesn't have the flexibility you would like, this exercise can be rejuvenating.\n\n1. Stand with your feet wider apart than the width of your shoulders.\n\n2. Place your hands on your thighs above your bent knees and straighten your arms. Take several deep breaths. The position feels like sitting on an invisible chair.\n\n3. With your head forward and your bottom back, adjust your feet so your knees are directly above your ankles and your back is straight. You are creating a kind of suspension bridge.\n\n4. Slowly stretch one shoulder across and down toward the opposite knee (see Figure 2-14). Repeat with the other shoulder. It is a crossover exercise. You will feel the stretch across your back. You can do this stretch more than once.\n\n5. Very slowly raise yourself with your arms hanging down, until you are in a standing position.\n\n_Figure 2-14_ SUSPENSION BRIDGE\n\n**Fortifying Your Aura and Crossover Energies**   \n**with the Celtic Weave** (time\u2014about a minute)\n\nYour aura is a multilayered sphere of energy that emanates from your body and interacts with the atmosphere of the earth. It is itself a protective atmosphere that surrounds you, filtering out many of the energies you encounter, and drawing in others that you need. The health of your aura reflects the health of your body; the health of your body reflects the health of your aura.\n\nWhen you feel happy, charismatic, and spirited, your aura may fill an entire room. When you are sad, despondent, and somber, your aura crashes in on you, forming an energetic shell that isolates you from the world. Sometimes if I feel closed in by too many intruding energies, I will very slowly place my open palms against the inside band of my auric field and push it away. If I go slowly enough, I can feel the pressure of the field as I push on it. Try it sometime when you are having difficulty claiming your space in the world, or are feeling sad, small, or squashed. Imagine yourself surrounded by an eggshell-like energy and exhale slowly as you push it away from you, beginning at about two inches from your body. It is easier to do than you might think. You may be able to feel an energetic force against your hands, and in any case, it will give you more breathing room, energetically and psychologically.\n\nThe Celtic Weave exercise keeps the energies within your body crossing over and also helps keep your aura strong and healthy. When these two energy systems are in good repair and working in harmony, your overall health is greatly supported. To do the Celtic Weave:\n\n1. Stand tall, with your hands on your thighs. Throughout the exercise, breathe slowly and deeply, in through your nose and out through your mouth.\n\n2. Swing your arms around and bring them into a prayer position in front of your chest.\n\n3. Rub your hands together, shake them off, face your palms toward each other, and see if you can feel the energy between them. Do not be concerned if you cannot feel it. Sensitivity will grow if you keep working with energy.\n\n4. Rub your hands together again, shake them off, put them up next to your ears, about three inches out, and take a deep breath (see Figure 2-15a). Exhale.\n\n5. With your next in-breath, bring your elbows together.\n\n6. On the out-breath, cross your arms (see Figure 2-15b) and swing them out (see Figure 2-15c).\n\n7. Cross them again in front of your body and swing them out again.\n\n8. Do this again, but as you swing out, bend down and cross your arms over the top part of your legs (see Figure 2-15d).\n\n9. Stay bent. Swing out again, in front of your ankles.\n\n10. Bend your knees slightly. Turn your hands toward the front, scoop up that energy, stand, raise your arms, and pour that energy down the front, sides, and back of your body.\n\n_Figure 2-15_ CELTIC WEAVE\n\n#  REPROGRAMMING DEEP-SEATED ENERGY PATTERNS IN BODY AND MIND\n\nEnergy runs in habits. Like flowing water that forms a riverbed, energy patterns that are repeated day after day tend to become entrenched. They are difficult to change even if they are no longer needed or useful. If one of your ancestors was born during a famine and meals were scarce and inadequate, her body would learn to turn protein that was optimally meant to build muscle into fat storage. When the famine ended, and food was plentiful, the energy systems that control metabolism would still be storing fat, now unnecessarily. This dynamic is true of the energies that govern your mind as well as those that govern your body. Your famine-born ancestor might be obsessed with food in ways her sibling, born four years later and after the end of the famine, might not be.\n\nFortunately, it is possible to change deep-seated energy habits that affect our bodies and minds. David, Gary Craig, and I wrote an entire book on this topic, _The Promise of Energy Psychology,_ and we hear amazing stories about its effectiveness (see Gary's Web site, www.emofree.com). But I do not want to be glib about this topic. As Western medicine has slowly embraced the concept of the mind-body connection, many well-meaning health practitioners tell people to change their attitudes, beliefs, or self-concept as if it is like changing a pair of shoes. Too often, however, they don't give them the tools to do it, even though those tools exist. While I am a fan of the power of intention, simply willing yourself to change a belief or other mental pattern that has become deeply embedded in your energy system often has the effect of just giving you another reason to feel bad about yourself, beyond having recognized the self-defeating mental pattern in the first place. You can't think yourself out of a dark room, and you can't usually think yourself out of a major depression, the desire for a cigarette, or the longing for love from someone who is not available to give it. Shifting your energies, however, creates a new internal atmosphere that supports new ways of feeling and thinking.\n\nI offer here three powerful techniques you can experiment with and integrate with the other methods presented throughout this book. The Blow-out\/Zip-up\/Hook-in, the Temporal Tap, and the Meridian Energy Tap each combine your focused intention with a way of shifting the energies that maintain the mental habit you wish to change.\n\n**Blow-out\/Zip-up\/Hook-in** (time\u2014one to two minutes)\n\nWhen you are troubled, upset, or simply frustrated that you have not been able to change a pattern in your life, tune in to the feeling and:\n\n1. Stand up straight. Put your arms out in front of you, bend your elbows slightly, make fists with the insides of your wrists facing up, and take a very full breath (see Figure 2-16).\n\n2. Swing your arms behind you and up over your head. Hold here for a moment.\n\n3. Reach way up, turn your fists so your fisted fingers are facing each other, and rush your arms down the front of your body as you emphatically open your hands. Let out your breath and your emotions with a \"whooooosh\" sound or any other sounds that come naturally.\n\n4. Repeat several times. This will feel good. The last time, bring your arms down in a slow and controlled manner, blowing your breath out of your mouth as you go. Stand tall and breathe deeply.\n\n5. In this moment of emptiness and release, formulate a simple statement about yourself that you wish were true. \"I am calm and at peace.\" \"I am feeling strong and confident.\" \"I am a force to be reckoned with.\" Affirmations of this nature are in the first person, present tense, stated as if they are already true.\n\n6. Zip the affirmation into body and mind by stating it slowly and deliberately while doing a modified Zip-up: Place your hands at your pubic bone, take a deep in-breath, and while stating your affirmation, move your hands, slowly and deliberately, straight up the center of your body to your lower lip. Do this three times.\n\n7. Lock it in with a Hook-up: Breathing in through your nose and out through your mouth, place the middle finger of one hand between your eyebrows above the bridge of your nose, the middle finger of the other hand in your navel, and gently press each finger into your skin (see page 62, Figure 2-11). Pull your fingers upward and hold for about 15 seconds.\n\n_Figure 2-16_ BLOW-OUT\n\n**Temporal Tap** (time\u2014about a minute)\n\nThe Temporal Tap was used for pain control in ancient China, but it is now being recognized as a surprisingly effective method for breaking harmful habits and simultaneously establishing desired ones. Tapping around the temporal bone\u2014beginning at the temples and traveling around the back sides of the ears\u2014makes the brain more receptive to learning while temporarily suspending other sensory input. It also relaxes triple warmer, the energy system that maintains physiological habits. This allows you to more easily slip in a new habit.\n\nIn the 1970s, the founder of Applied Kinesiology, George Goodheart, discovered that by tapping along the cranial suture line that starts between the temporal and sphenoidal bones, you can temporarily shift the mechanisms that filter sensory input. If you introduce a self-suggestion or spoken affirmation while you are tapping, your mind will be particularly receptive to it. The Temporal Tap takes advantage of the differences between the left and right hemispheres of the cerebral cortex. In most people, the brain's left hemisphere is more critical and skeptical than the right hemisphere. So statements made with a negative wording are in closer accord with the way the left hemisphere functions and are more likely to be assimilated. Such statements are tapped into the left side of the head. Likewise, because the right hemisphere is highly receptive to favorable input, statements that are worded positively are tapped into the right side. This pattern is reversed in some left-handed people, and you can energy test to see which works for you. If you stay strong while tapping a negative statement into the left side, stay with the instructions; if you become weak, swap the words \"right\" and \"left\" in the instructions.\n\nBegin by identifying a habit, an attitude, an automatic emotional response, or a health condition you would like to change. Describe the change you would like to bring about in a single sentence, stated as an affirmation in present time, that is, as if the desired condition already exists. It does not have to be a truth yet, but rather a statement that you would like to be a truth in the future. For instance, you might say, \"Under pressure, I stay calm and centered.\" It usually helps to write out your statement.\n\nThen restate it, keeping the same meaning, but with a negative wording (\"no,\" \"not,\" \"never,\" \"don't,\" \"won't,\" etc.). Thus the \"calm and centered\" statement may be worded negatively as \"I no longer get stressed under pressure.\" Notice that the meaning is still positive, even with the negative wording. Another example: \"I eat for health and fitness\" can be negatively worded as \"I don't eat from anxiety or compulsion.\" \"My fingernails are growing long and healthy\" can be negatively worded as \"I no longer bite my nails.\"\n\n1. Starting at the temple, tap the left side of your head from front to back with the three middle fingers of your left hand (see Figure 2-17). State the negatively worded version of your statement in rhythm as you tap. Tap hard enough to feel a firm contact and a bit of a bounce. Tap from the front to the back about five times, making your statement with each pass.\n\n2. Repeat the technique on the right side, tapping with your right hand, but this time using your statement in its positively worded form.\n\n3. Repeat the procedure several times per day. The more you tap in the affirmation, the quicker and stronger the effect on your nervous system and your field of habit.\n\n_Figure 2-17_ TEMPORAL TAP\n\nThe Temporal Tap combines a variety of powerful elements, including repetition, autosuggestion, and neurological reprogramming. It affects not only the brain but also each meridian, so the message of your intention is carried to every system in your body. It is a disarmingly simple way to change many patterns that cannot be overcome by willpower alone.\n\nThe parallels between our physical and our emotional difficulties are often striking. Sometimes an organ will literally begin to behave in a way that reflects the way _you_ behave. I'll use myself as an example. I sometimes have difficulty setting clear boundaries with people. Given the nature of my work, this can be an overwhelming problem. At one point some years ago, I became very vulnerable to infections.It was as if my thymus gland, which is responsible for protecting the body from infection, was modeling _its_ behavior after mine. I began to tap into my left temple the statement, \"My thymus gland no longer allows foreign invaders to enter my system,\" and into the right, \"My thymus gland vigilantly maintains strong boundaries that keep out foreign invaders.\" Not only did my susceptibility to infections decrease, my ability to set clear boundaries improved as well. Part of the beauty of the Temporal Tap is that in funneling your intention down to two short statements, you can sometimes get to the nub of a complex and puzzling relationship among body, mind, and spirit.\n\nThe Temporal Tap is such an important tool for taking greater control of your life that I want to relate a number of stories where _it didn't work,_ as these stories are instructive for getting it to work. I have found that when the Temporal Tap fails, it is often because of how the statements are worded. The words need to be in a language that is comfortable for you. Sometimes it just means translating the words into the simplest words possible. It is also important that the words be aligned with your values and congruent with your feelings, that you not be saying one thing while thinking another, and that you not be attempting to negate a primal need.\n\nA woman who wanted to lose twelve pounds gained eighteen using this approach. On the left side, she tapped in the words, \"I no longer hold on to extra weight.\" On the right side: \"My set point is dropping to 134.\" These wordings seemed reasonable, but the tapping was having the reverse effect. I asked her to notice if her thoughts were following her words as she tapped, if her mind was wandering, or if any images were entering her awareness. It turned out that what came into her mind _every_ time she tapped was, \"Oh, hell, I've got a Slavic body, I'm always going to have a Slavic body, and I'm going to end up looking just like my [fat] Aunt Sophie.\" She was tapping this in five times every day. And it was working! The thought and image were far more primal self-suggestions than her carefully worded statements. She did, by the way, lose a good deal of weight after tapping into her left side, \"I did not inherit Aunt Sophie's body\" and \"I inherited a body that can be thin and lithesome\" into her right side.\n\nA man whose job was eliminated after twenty-four years with a company was sent back to college for a year, at the firm's expense, to upgrade his skills. Five weeks into the first quarter, he had achieved Cs and Ds on the midterms of several of his courses, threatening his future reemployment. He was trying very hard, but he found himself unable to concentrate during any sustained period of study. Having heard that I had helped people with learning disabilities, he scheduled a session with me. I could not find a learning disability, but I did sense that after working for a quarter of a century in a people-oriented position, his habit field had no juice for inwardly oriented, concentrated study. I explained this to him, and he was highly enthusiastic after I taught him how to use the Temporal Tap for changing the habit field that was affecting his ability to study. But he came back a week later deeply discouraged. He had followed my instructions to the letter and devoted all the time required, but he could detect no change in his study habits.\n\nHe told me the statements he had been tapping in, and I watched him do the tapping. The words sounded good to me, but his energies weren't receiving them very well. As I explained how important it was that the words must sound right to him, I learned that he had come to the United States at age six and that English was his second language. After doing the Temporal Tap for another week, this time using his native language to make the same statements, he reported respectable improvements in his ability to read and concentrate.\n\nA friend wanted to quit smoking after fighting a chronic cough for more than a year. I taught her the Temporal Tap. As she tapped in her statements\u2014\"I no longer smoke\" on the left side and \"I enjoy my cigarette-free existence\" on the right\u2014she started to smoke like crazy, and she felt miserable. Something about the tapping was making the habit stronger, and it was also making her extremely anxious. But since the tapping was having an undeniable effect, even if it was the opposite of the desired one, she knew it was potent and decided to figure out how to use it to her benefit. She sensed that her anxiety was a key to the problem. She figured out that saying \"I no longer smoke\" was triggering her anxiety, in part because smoking was her major method of relaxation. Playing her intuition, she found new words. She began to tap into the left side \"Anxiety no longer causes me to smoke\" and into the right side \"I only smoke now for my highest health and pleasure.\" The anxiety dramatically receded, and she went down from a pack per day to about three cigarettes per week. She kept it at that level for several months, using her occasional cigarette as a mantra, an act of meditation. Her cough diminished. Eventually she quit smoking completely.\n\nAs with many energy medicine techniques, the Temporal Tap often requires more finesse than simply following a pat formula. To summarize some of the considerations to keep in mind, your Temporal Tap statements will be more effective if: (1) the wording of the statement tapped into the left side of your head contains a negative word and the statement tapped into the right side contains only positive words, (2) the statements are in harmony with the way you naturally talk and think, (3) as you say the statements, you keep your attention on the words and their meaning, and (4) the statements do not instruct you to do something that contradicts a core value or a more basic need.\n\nI have seen the Temporal Tap be effective for an enormously wide range of problems. It has helped people stop smoking, drinking, overeating, and scratching compulsively. It has assisted them in building confidence, optimism, and self-esteem. It has aided them in stimulating their immune systems when fighting a serious illness, improving their metabolisms when trying to lose weight, and enhancing their coordination when trying to learn a new skill. It has been a factor in reducing tumors, clearing eczema, and lowering blood pressure.\n\nSelect a target behavior, emotion, or physiological condition that you want to change. Craft a negatively worded statement to tap around your left ear and a positively worded statement to tap around your right ear. You can test whether your statement is in harmony with your energies by tapping with one hand while making your statement and using the other hand as a friend energy tests you. Temporal Tap your statements four or five times a day for at least a week. With deeply entrenched habits, I have seen it take up to thirty days before the results became evident. Even if you are skeptical, judge for yourself based on the results you attain.\n\n**Meridian Energy Tap** (time\u2014two minutes)\n\nTapping on various sets of acupressure points while bringing to mind a psychological problem seems to rapidly shift the neurology that maintains that problem. Using the same approach, improvements in physical problems are also increasingly being reported. While energy psychology, as it is called, can be easily learned and our book on the topic is readily available, it is beyond the scope of this book to do it justice. But I present one simple technique that is very powerful and well worth knowing.\n\nIf you are able to stimulate a series of acupressure points distributed on different meridian pathways, you can pump your meridian system into action. Where pumping adrenaline through the body arouses the defense system, pumping energy through the meridians brings new vitality to body and mind, often bringing about unanticipated positive effects. It is always good for the immune system. Sometimes it corrects minor physical problems without ever having focused on them. It can also decrease anxiety and enhance your sense of well-being. It is an all-purpose intervention. And it is easy to do.\n\nIf you do it while mentally focusing on a problem, it has an added benefit. Focusing on a problem causes the body to go into whatever stress reactions are associated with that problem. If you simultaneously energize your meridian system, you can often neutralize that stress reaction, and in many cases prevent it from occurring the next time that same problem is encountered. So this is a good one to do when you are distraught, thinking about something that bothers you, or simply wanting a boost.\n\n1. Stretch your forehead to get blood flowing by placing your thumbs at your temples on the sides of your head, curling your fingers, and resting your fingertips on the middle of your forehead. Slowly, and with some pressure, pull your fingers apart so that you are stretching the skin. Breathe deeply.\n\n2. Tap each of the following points for the length of one deep vigorous breath (see Figure 2-18). It is a firm brisk tap, but of course not so hard as to hurt or bruise yourself:\n\na. The points on the inside edges of your eyebrows\n\nb. The points on the outsides of your eyes\n\nc. Tap with all your fingers at your temples and continue tapping up, over, around the back, and to the bottoms of your ears. Do this three or four times.\n\nd. The points under your eyes at the tops of your cheekbones\n\ne. The point between your top lip and your nose\n\nf. The point between your bottom lip and your chin\n\ng. The three thumps (pages 53-55) in the order of K-27, thymus, and spleen\n\nh. The points on the outsides of your legs between the knees and the hips (hang your arms, bend your middle fingers toward your legs, and start tapping)\n\n_Figure 2-18_ MERIDIAN ENERGY TAP\n\nThis chapter has been a primer of basic energy medicine techniques that enhance the flow of the energies in your body. The following chapter is the first to present energy techniques that specifically address managing your hormones, focusing on those having to do with your immune system and with stress. The chapter begins, however, by examining the chemistry of hormones, their role in your health, and how medical politics influence the ways we understand them.\n\n_**Chapter 3**_\n\nDancing with Your Hormones\n\nHORMONES  \n_More Unique Than Your Thumbprint!_   \n_More Troublesome Than Your Thighs!_   \n_More Potent Than Your Therapist!_\n\n_Every woman I know wants to enjoy strength, vitality, and vigor._   \n_She wants to maintain optimum health, radiant skin, and luxurious hair._   \n_She also wants to stay physically active and vital, and still be able to think_   \n_clearly and remember facts and events. That's where hormones come in._\n\n\u2014 SUSAN LARK, M.D. _Hormone Revolution_\n\n_Hormones play our lives like instruments._\n\n\u2014 D . LINDSEY BERKSON\n\n_Hormone Deception_\n\n**Y** our brain is not your only source of intelligence. If you think of your _entire_ _body_ as a large brain imbued with incredible intelligence, you are closer to the truth. And your _hormones_ are the chemical messengers that keep the whole system adapting to your world, moment by moment. Being able to influence your hormones with understanding and wisdom gives you a strong edge for adapting to a world that is radically different from the one in which our bodies evolved.\n\n#  YOUR BODY'S SIGNALING SYSTEM\n\nEnergy medicine is a major key for bringing that understanding and wisdom to the functioning of your hormones. Your body is a latticework of subtle energy fields, all of which impact your physiology, and hormone production can be directly influenced _by moving the energies in your body._ This demonstrable fact allows you to have control over your health and mood in ways that other cultures have understood but ours has not. Hormones are messengers that orchestrate the activity of your organs, tissue, and cells. They govern every aspect of human functioning, from digestion to reproduction, from thought to action. They rush from cell to cell, providing millions of instructions every second. But their messages have to deliver the right information for everything in your body and mind to work in balance and harmony. If your hormonal system begins to malfunction, your life can suddenly plummet into illness, despair, and misery. If hormonal disturbances are corrected, your health and mood can soar.\n\nThe two most important ideas in this book are:\n\n1. For almost every health condition a woman faces, hormonal imbalances are in the foreground or in the background, and\n\n2. You can influence your hormones and improve your health, happiness, and vitality by using energy techniques that are accessible and easy to learn.\n\nFemale hormones remain, in many ways, a mystery to the medical community. The biochemistry of hormones has been mapped, but in the critical area of individual differences, Western medicine and science are stumped. The same medication for the same symptoms will help one woman magnificently, not help another at all, and for a third will produce side effects that are more harmful than the original problem. In addition, enormous contradictions among sophisticated research studies on the dangers of hormone replacement therapy, as well as on many other aspects of a woman's biochemistry, have become a tremendous embarrassment to the medical community and underscore the fact that one woman's cure is another's calamity.\n\n#  ONE SIZE FITS ALL: NOT!\n\nA strength of energy medicine is that its methods can be tailored to the unique energies and needs of each person. Routinely giving two people the same medication or other treatment simply because they have the same symptoms is medicalmadness. We are too different from one another genetically, temperamentally, and energetically to base health care decisions on broad generalizations or on research showing that a percentage of people improve with a specific treatment. You deserve better, and you saw in Chapter 2 how it is possible to assess the compatibility between your body's energies and a treatment you are considering. Individual differences must be taken into account in any well-considered health care decision.\n\nI know this too well because my body does not follow the rules. Medical protocols that make most people better often make me worse. During the summer of 1974, I participated in an experiment conducted by Dr. Dorian Paskowitz, the resident surfer and physician at that time at Palomar College, just north of San Diego. A garden with a variety of vegetables and herbs had been planted on the campus and, for one month, the two dozen of us in the experiment ate only what we picked, raw and uncooked, fresh from the ground. Most of the other participants lost weight and reported that they were thinking more clearly and feeling energized. But though I was exercising regularly and vigorously several times per week and ate no more than the others, I gained 12 pounds and felt dreadful.\n\nDr. Paskowitz was stumped. He was trying to prove how healthy a natural diet of live foods is for everyone (his challenge to conventional wisdom was, \"How can you be fully alive when you are eating dead foods?\"), and I was not fitting his theory. Why? I was caught in his enthusiasm for a natural diet and expected the experiment to work. I didn't cheat and sneak bags of Fritos at night. I didn't appear to be genetically different from everyone else. What caused me to respond so differently?\n\nOver the years I've come to realize that the riddle has a one-word answer: hormones. My hormones _really_ don't follow the rules. I was once brought to a hospital, passed out, and the treatment I was given almost killed me. I also had the worst PMS of everyone I'd ever known, an outcome of low progesterone and high estrogen levels, yet all the remedies I could find at the health food store made me worse. No doctor was ever able to figure out my fluctuating thyroid. I didn't match the Western models about hormones on much of anything. In fact, it is in part because my own endocrine system is so screwy that I've had to learn enough about hormones to be able to write and teach about them. But in my teens, twenties, and thirties, when such information would have been invaluable, I didn't have access to it.\n\n#  BRINGING AN ENERGY APPROACH TO HORMONAL CHAOS\n\nAlthough Telisha sometimes felt challenged by the demands of raising four small children, she loved being a mom. However, like clockwork, for one week out of each month, she hated the role and herself in it. For the seven days prior to the start of her period, she found herself screaming at her kids with little or no provocation, retreating into bed crying regardless of what the children needed, and generally living in a dark gloom of despair and debilitation.\n\nThen, again like clockwork, her period would start, her mood would lift, her competence would return, and she would sail through the next three weeks, though under a cloud of guilt for what had occurred during the previous bout of PMS and dread of the next one.\n\nTelisha scheduled a session with me, hoping for help with her monthly ordeal but not really knowing what to expect. The session happened to coincide with the start of her PMS time. I found that every system in her body was being compromised by energy disturbances. One of Telisha's major meridians was overenergized and several were underenergized. Because you can very quickly correct such imbalances, I was able to experiment with her by shifting the flow of the energies and asking her how she felt after each intervention. Her body relaxed as each pathway was balanced, and she felt an immediate change in her mood.\n\nShe went home with several exercises I taught her for maintaining the new balance. She did them faithfully until her period started, and the days that would normally have been torture for her were mercifully unusual in that she did not experience the dark gloom or mood swings that were characteristic of her PMS experience. She thought she was cured. As the next cycle descended, however, she forgot about the exercises and was taken by surprise into the depths of another PMS nightmare. She was soon on my doorstep. The treatment was straightforward. We used the same procedures we'd used in the first session, but with more emphasis on the importance of continuing to practice the methods at home every month. She came to appreciate that her monthly cycle is a natural rhythm. She also learned that when she was feeling good, not in PMS, she could easily apply energy methods that would make the coming of her period less extreme and disturbing. The difference this made in her life was enormous. Two years later, I was pleased to learn that she was teaching other women how to manage their PMS at our local community clinic.\n\n#  YOUR HORMONES IN ACTION\n\nIf you think of each cell in your body as a theater with a thousand stages, hormones raise and lower the curtains. A hormone may be sent from a distant neighborhood to raise the curtain and start the action on a stage that boosts your immune system or on another that regulates your metabolism. When the job is finished, another hormone lowers the curtain and the chemical actors nap until that curtain rises again. A hormone is a molecule, produced by a cell, which tells other cells, tissues, and organs what to do. Each hormone gives a precise instruction for raising or lowering a specific curtain, and in coordination with millions of other hormones, may turn you on sexually, create a hot flash, or increase the size of your breasts in preparation for pregnancy.\n\nIt was once thought that the brain and nervous system somehow regulated many of the processes now understood to be governed by the secretion of hormones. Rather than a top-down communication system, we have with our hormones a lively community inside with trillions of potent and seemingly independent decision makers. Hormones operate from three levels of distance: (1) those produced within a cell to regulate activities in that cell or in nearby cells, (2) those that travel a very short distance through an established duct, and (3) those that are produced by the endocrine glands, which are then secreted into the bloodstream to control events in cells and organs that may be far away. Energy fields are vital in broadcasting the signals that make this enormous community of decision makers work in harmony.\n\nIt is difficult to grasp how potent your hormones actually are. The estrogen that regulates your menstrual cycle, influences your sex drive, and prepares you for pregnancy\u2014secreted continually during your thirty or so childbearing years\u2014will, according to some estimates, have a total weight less than that of an olive. A tiny fraction of that, at the time of puberty, transforms a girl's body into that of a woman. We are speaking of a very potent drug.\n\nRemove the body's ability to produce certain hormones and death or debilitating illness follows. Even minuscule imbalances can inflict a range of serious maladies.Describing the pervasive role of hormones and the glands that produce them, and writing in the 1950s when their place in human functioning was just beginning to be fully appreciated, physician J. D. Ratcliff observed: \"Our amazing endocrine glands take some part in virtually everything we do. Lift an eyelid\u2014it was hormones that made sure sugar was in the blood to provide muscle power. Cut a finger and hormones will be there to help control inflammation and wall off infections. Hormones are our common denominators with animal life. Some of the movie starlet's sex hormones are identical with those of a whale; some of a prize-fighter's pituitary products match those of a mouse.\"\n\nAny plant or animal that has more than one cell produces hormones\u2014that is how its cells communicate. While nearly every type of human organ and tissue secretes hormones, the most familiar hormones are those released by the endocrine glands, including the adrenals, hypothalamus, ovaries, pancreas, parathyroid, pineal, pituitary, testes, thyroid, and thymus. As a group, these glands weigh only a few ounces, yet they are so decisive in our functioning that Ratcliff noted that \"this tiny amount of tissue acts as a kind of council of ministers for the body . . . the body's master chemists.\" The endocrine glands dispatch their hormone messengers directly into the bloodstream with clarity of purpose and precision in destination. The mission of a single hormone might serve in the stimulation or suppression of growth, the activation or inhibition of the immune system, the regulation of metabolism, the preparation for activity (e.g., fighting, fleeing, or mating), or the management of a phase-of-life change (e.g., puberty, caring for offspring, menopause).\n\nMost people do not normally think of hormones as a force in shaping their brains. We think of our brain as a given of our inheritance, with its wonderful capacities and any limitations built in and relatively fixed. But daily experiences literally change the physical structure of the brain. Learning builds new neurons. Our brains are continually being reconfigured, and our hormones, according to Mona Lisa Shulz, M.D., even serve to \"mold how our right and left brains process feelings and thoughts.\" In her groundbreaking _The New Feminine Brain,_ 4 Schulz, who is both a neuropsychiatrist and a medical intuitive, explains how our brains adapt to our world, so your brain is literally a different kind of brain from your grand-mother's brain. And we are at the cusp of the evolution of a \"new feminine brain\" that takes advantage of our deep intuition and the unique wiring of the female cortex, with its many more connections than men have between the right- and left-brainhemispheres. Shulz shows how understanding your hormones can help you develop your brain to new possibilities. Meanwhile, understanding the energies that govern your hormones gives you a special advantage.\n\n#  YOUR ENERGIES TELL YOUR HORMONES WHAT TO DO\n\nHormones, in infinitesimal quantities, are able to pull off epic achievements because they are catalysts that _excite_ your body's energies (the word \"hormone\" is, in fact, derived from the Greek _hormon,_ which means \"I excite\" or \"set in motion\"). Hormones and energy impact each other in a mutual feedback loop. Hormones tell your cells, tissues, organs, and their energies what to do; your body's energies tell your hormones what to do. In each of the subsequent chapters of this book, you will learn to direct your energies to better manage the hormones that shape your journey through life.\n\nEnergy medicine is the art and science of working with the relationships between energy and chemistry to promote health, vitality, and well-being. The important part of the formula is not _energy_ or _chemistry._ It is both. While changes in your chemistry (such as those you experience in your monthly cycle) change your energy, shifts in your energy also change your chemistry. This has been evident time and again in my practice. Often simple energy interventions have changed a woman's measurable hormone counts, restoring balance. While I don't know of any formal studies, I have worked with many nurses who had easy access to laboratory tests. I often encouraged them to take a reading prior to a session and another right after the session. Changes brought about during a session appeared in their white blood counts, blood sugar levels, thyroid levels, and estrogen levels.\n\n#  FABRICATING THE MYTH THAT NATURE'S WAY IS THE WRONG WAY\n\nThere has been a great deal of confusion about hormones within science and medicine. When combined with deeply accepted cultural myths that devalue some of the distinguishing qualities of the female body and psyche, a fog of misinformation prevents most women from fully understanding how their bodies work. If you are like me, so far from the norm that it is critical to know how to step in and manage your own hormones, this is a terrible handicap. But even if your hormones operate \"normally,\" you can be enormously empowered in managing your moods, your health, and your vitality if you understand your hormones. So accurate information is essential.\n\nA 1963 article by New York gynecologist Robert A. Wilson and his wife, Thelma, a nurse, began: \"The unpalatable truth must be faced that all postmenopausal women are castrates.\" The article, published in the _Journal of the American Geriatrics Society,_ was followed three years later by Dr. Wilson's _Feminine Forever,_ 5 a best-selling book that was widely serialized and excerpted in magazines and newspapers across the country. Wilson argued that menopause was a \"deficiency disease,\" much like diabetes. The estrogen depletion associated with menopause, according to Wilson, led to \"a horror of living decay.\" The menopause in this rendition stole not only a woman's health, but also her femininity and her sexuality, leading to debilitating depression, or at least a \"vapid cow-like feeling.\" Postmenopausal women, he said, \"exist rather than live,\" seeing the world \"through a gray veil\" and moving through their days \"as docile harmless creatures missing most of life's values.\"\n\nWhile this characterization would not have flown in any of the thousands of cultures throughout history that have venerated their female elders, America bought it. Wilson's solution was to insist that women take estrogen pills to replace what their deficient bodies could no longer produce, and he founded a private trust for promoting estrogen therapy (with $1.3 million funding from the pharmaceutical industry). Soon \"hormone replacement therapy pills,\" as the _Boston Globe_ would later report, \"sat like pastel candies inside the medicine cabinets of middle-aged women from coast to coast. They were easy to use, widely available, and touted as the pharmaceutical embodiment of Ponce de Le\u00f3n's fountain of youth, promising youthful vigor, sharper thinking, and softer skin for women confronting menopause and the years after.\"\n\nBy 1993, the science backing this development included many studies published in reputable medical journals such as _The Lancet_ and the _Journal of the American Medical Association_ ( _JAMA_ ). They suggested that hormone replacement therapy in menopausal and postmenopausal women enhanced memory and reduced the risk of breast cancer, genital cancer, heart disease, stroke, and bone loss. Yet many women began to discontinue HRT within months of starting it because of side effects such as depression, mood swings, bleeding, fatigue, headaches, increased blood pressure, chest pains, varicose veins, water retention, decreased bladder control, and weight gain. By 2002, the only benefit that had not been contradicted by larger and better-designed studies involved the reduction of bone loss. That same year, an NIH study of 16,608 women, published in _JAMA,_ reported that taking HRT for five years increased the risk of breast cancer by 26 percent, of stroke by 41 percent, and of heart disease by 29 percent. A subsequent report from the same study found that HRT doubled the risk of dementia.\n\n#  HOW COULD SCIENCE GET IT SO WRONG?\n\nThe _Boston Globe_ article \"Hormone Therapy's Rise and Fall: Science Lost Its Way, and Women Lost Out\" suggests: \"The stellar rise and spectacular fall of hormone replacement therapy ranks among the biggest medical mistakes in history, fueled by a combination of weak science, relentless hype, the herd mentality of doctors, and women's dawning redefinition of menopause from an inevitable 'change of life' to a manageable condition.\"\n\nYep, women lost out. Many of the early studies were based on old and sometimes unreliable medical records that often did not indicate the strength of the medication or even the exact medication. Another error in the research was the failure to track individual differences. The same symptoms might stem from a variety of causes. For instance, you may not be producing the estrogen your body needs, but your neighbor may have similar symptoms caused by excess estrogen. Giving both of you the same medication because the research shows that a percentage of people improve with a specific treatment is indeed madness.\n\nWilson's crusade to rescue women from what he called the \"tragedy\" of menopause was given a huge platform by the pharmaceutical industry. After all, if menopause were just considered a natural occurrence instead of a disease, hundreds of millions of women throughout the world would remain an untapped market for the pharmaceuticals used to treat this \"disease.\" Wilson's doctrine that menopause was a disease needing treatment was the perfect message for the pharmaceutical industry to spread in order to open \"the largest commercial market in the world.\"\n\nMenopause is every girl child's future. Leslie Kenton points out, in her classic book on menopause, _Passage to Power,_ that drug manufacturers \"spend a fortune each year in an attempt to convince governments, doctors, and the public that . . . using their products is the only choice available to women\" for battling the dreaded disease known as menopause.\n\n#  IS SCIENCE GETTING IT RIGHT YET? THE POLITICS OF MEDICINE\n\nYou may or may not care about the intricacies and politics of scientific research, but if you go to your doctor, describe your symptoms, and are given one of the more than 3 billion prescriptions that will be written in the United States this year, there is good reason to care. _You are betting your well-being, if not your life, on the accuracy of scientific research._ More than 100,000 people die each year from taking those pills _as instructed._ You cannot just stare at the pills to decide whether they will actually help your unique body with your unique symptoms, whether the recommended dosage is too weak or too strong, or what unintended effects they might produce. You rely on the research findings to have answered these essential questions _before_ your doctor settles on the prescription.\n\nScientific research has been considered the pillar of truth in our culture, the knowledge you can count on. Whereas so much of the information that comes your way is trying to get you to buy or believe something according to someone else's agenda, science has the authority of objectivity. Or does it? Well, to rephrase the old adage, people who love sausage and trust scientific findings shouldn't watch either of them being made. This is not to say that the basic scientific method is not one of humanity's greatest achievements\u2014it is! But the tail of politics and economics is now wagging the dog of health care research. With those three billion prescriptions per year, depending on where that sacred canine points, some fierce infighting is involved.\n\nThe Boston Women's Health Book Collective (of _Our Bodies, Ourselves_ fame) has summarized the \"quiet but radical transformation\" that has occurred in medical research over the past three decades. Before 1970, government sources funded the vast majority of clinical research. Today pharmaceutical companies fund three out of four of the clinical trials published in the most highly respected medical journals,such as _JAMA,_ the _New England Journal of Medicine,_ and _The Lancet._ About 80 percent of the research the drug companies were doing back in 1991 was at least being conducted in universities, where various types of reviews and balances supported the researchers' independence and objectivity. By 2000, two-thirds of this research was being conducted by for-profit research centers.\n\n\"So what!\" you may think. Research is research. To be published in a reputable journal, it must abide by scientific standards. Well, think again. Scientists employed by commercial enterprises know a great deal about how to get the results their organization wants to see. A common strategy is that if a study does not appear to be producing the findings the drug company wants the world to know, the study is discontinued so the unfavorable findings do not have to be reported. Another deceptive practice is to test the safety of a medication that will be taken mostly by older people on younger healthy subjects who are not as likely to experience side effects and who are not as likely to be taking other medications that might negatively interact with the drug being tested. The Women's Collective summarizes: \"By controlling the design of the research, the criteria by which patients are selected, the analysis of the data, and the selection of results to be published, drug companies shape medical knowledge.\"\n\nDo these technical points make _that much_ difference? Two separate investigations of just that question were published in 2003, one in _JAMA,_ the other in the _British Medical Journal._ Guess what? When a study was funded by the company that sold the product being studied, it was between 3.6 and 4 times more likely to show positive results for the product than if the researchers did not have a stake in the outcome. A subsequent study in _JAMA_ looked to see if this ignoble distortion was still the case if only the highest-quality journals were considered. Yes, and more so. In the most prestigious studies, the sponsor's product was 5.3 times more likely to show positive results than in studies of the same product where the researchers were objective. These biased studies, however, when they appear in the medical journal, are not labeled with neon signs that say \"biased material\" or \"Warning, the Surgeon General has determined that what you are about to read is crap.\" They are, rather, among the primary sources with which your doctor makes decisions about what will help you and whether it will put you in harm's way. And it doesn't stop with corrupted research practices. The Woman's Collective goes on to show how the agencies we count on to protect the interests of health consumers are increasingly compromised:\n\n\u2022 The guidelines that inform and direct clinical practices are formed by committees dominated by experts who may have active financial ties to the very companies that produce the drugs and other products under consideration.\n\n\u2022 About 70 percent of the continuing medical-education courses that doctors attend are paid for by drug and other medical-product manufacturers.\n\n\u2022 \"Objective\" medical journals rely on revenues from pharmaceutical advertising and from selling reprints of published articles to corporate sponsors, who then distribute them to physicians.\n\n\u2022 More than half the funding for the division of the U.S. Food and Drug Administration (FDA) that approves new drugs and oversees drug safety comes from user fees paid by drug companies.\n\nIn brief, financial gain is getting to be the health industry's new \"objectivity.\"\n\nHave you filled a prescription lately and been shocked by the cost? Consider 3 billion such prescriptions each year in the United States. While the research doesn't tell us with confidence whether they work or whether they are safe, the revenues to keep them being written exceed your wildest dreams. Spending on drug marketing, according to congressional investigators, jumped from $11 billion in 1997 to nearly $30 billion in 2005. Profit margins among the leading pharmaceutical companies are typically three to four times higher than for other Fortune 500 industries.\n\nAs for natural treatments, drug companies can make a profit only on what they can patent, and treatments based on natural plants cannot be patented. Therefore, a huge market has, for instance, been developed for synthetic progesterone (progestin). While progestin is generally less effective and has many more side effects than natural progesterone, it can be patented. Since natural progesterone cannot be patented, it is not advertised or promoted by the medical industry. You, along with your doctor, are less apt to know much about it (more about progesterone in subsequent chapters), nor for that matter are you likely to hear about energy medicine from your doctor. After all, the drug companies have not yet figured out how to patent energy. The missing information, misinformation, and contradictory information is not limited to HRT, but extends to some of the most vital areas where Western medicine meets a woman's health challenges, from mammograms to osteoporosisto PMS. In brief, the information generated by the complex web of drugmakers, researchers, and government regulators does not necessarily have your best interests at heart.\n\nMost of the doctors I know bemoan the fact that their profession is controlled by a web of social and economic forces that are unrelated to their patients' well-being. The point of this little diatribe through the health care propaganda industry is not really to point fingers. I must admit, however, that I don't mind suggesting that a teeny bit of institutional bias might be entering the system some of those 3 billion times this year that a doctor reaches for a prescription pad instead of suggesting a more natural remedy. Or that the 100,000 to 300,000 people who will die this year from taking their medications exactly as prescribed might have been better protected. But mainly, because this is a self-help book, I am emphasizing this parody of science to encourage you to remain vigilant. Do not let your own perceptions and well-considered judgments about your body and its needs simply dissolve in the opinions of authorities. No matter how contradictory the information or how smooth the propaganda, you can counter the disempowering impact of such messages. Becoming the chief scientist in the laboratory of your own body's energies allows you to assess possible remedies for your ills by observing the subtle effects they have on your energy system. These, in turn, predict the likely long-term impact the remedies will have on your body.\n\n#  THE MYTHS THAT KEEP US TRAPPED\n\nThe medical myths that impact our health choices work in concert with another set of myths that impact us even more personally. We live in a culture where beauty and youth are given more status and resources than wisdom, compassion, and ability combined. We live in a culture where we are continually assaulted with (and compare ourselves with) media images of the \"perfect\" woman: tall, willowy, weighing at least 20 percent less than her height requires, rarely looking older than 25, and with all flaws on her skin airbrushed away. We live in a culture where an exemplary woman, in a failed sexual-harassment case, was denied partnership in a top-ten accounting firm \"because she needed to learn to walk, talk, and dress more femininely.\" We live in a culture where you know exactly what I am talking about.\n\n_When Tina first came into my office, her very life force seemed depleted. She was exhausted and depressed. In her early fifties, she had spent her life trying to please men, trying to be pretty, trying to do everything she was \"supposed\" to do in order to be loved, and now it felt too late. Three men had left her for younger women. She was very angry. She had \"done it right,\" she had followed the rules, she had sacrificed her own needs for others, she had gone for her collagen treatments, and none of it had gotten her what she wanted. She wasn't pretty enough, she wasn't thin enough, and now she wasn't young enough. And none of this was ever going to change. When menopause hit, it seemed the last blow. She believed it would all be downhill from here. And she was left with the inescapable fact that she hadn't lived her life for herself. Her own instincts and life force had been subjugated for . . . for what? Now in my office she presented a list of health problems that included hypothyroidism, compromised adrenals, menopausal symptoms, chronic fatigue, terrible depression, and thoughts of suicide._\n\n_Her physical ailments were clearly tied to her emotional problems. I began by focusing directly on the energy imbalances that were associated with the physical symptoms. A lifetime of denying your legitimate needs can leave your energies badly scrambled. Tina's life force (specifically, the vital energy that flows through the kidney meridian) had been dangerously depleted. As we nurtured her life force so that it pumped more vigorously through her body, she began to meet the world with a new sense of empowerment. As the energies shifted, so did her consciousness. She came to recognize that rather than enforced isolation, this was the first time in her life when she was not taking care of others and had an opportunity to really take care of herself. This in itself was exhilarating. Beyond that, the kidney meridian governs fear. She found a new courage to be herself without fear of rejection, and she blossomed in an internal atmosphere that didn't long for or seek anyone else's approval._\n\nTina's story isn't unusual. Pretty by most conventional standards, much of her life force as a young woman still went into working toward an airbrushed image that is not attainable in real life. Today, in the United States, more money is spent on cosmetics than on education or social services. Each minute, 2,055 jars of skin-care products and 1,484 tubes of lipstick are sold. Each week, tens of thousands of women voluntarily have their bodies mutilated in deference to an image of beauty that defies nature. Women pay dearly to look thinner, younger, or bustier, only to find out that while cosmetic surgery cannot usually implant genuinely higher self-esteem, it often creates unanticipated problems. And it reflects a sad but widespread failure in self-affirmation. A vital and lonely challenge for women as they mature, according to Susun Weed, is that \"they must recast their own opinion of beauty so that it includes old women.\" The impact on our self-concept of the cultural messages around beauty and age are so pervasive that it is hard to get our minds beyond the assumptions of the \"beauty myth.\" This affects not only how you feel about yourself and what you do to influence your appearance, it also affects the way you care for your health. This was tragically brought home to me by my experiences in Fiji. When I lived there in the mid-1970s, I had never met a happier people. The Fijian women were robust, voluptuous, and joyful. Television was not introduced to the islands until 1995. Within three years, 11 percent of Fijian girls, comparing themselves with the slender actresses and models that they saw showcased, were bulimic, regularly purging themselves.\n\nOur culture's pathological standards regarding beauty and weight infest other cultures as they gain the technology to find out how we think. It is not a pretty picture. If we had a department of mental health that was given the same power and resources as our war department, we would as a society vigorously and unambiguously teach girls how to appreciate and value their bodies as the unique gifts of nature that they are. Instead, we have created powerful industries that forcefully teach girls to negatively compare themselves with the unattainable images that Hollywood and Madison Avenue have somehow decided are the ideal look. We hypnotize our men with those images as well, and everyone loses.\n\nChristiane Northrup, M.D., reflects that the \"devaluation\" of real female bodies within our society, combined with the cultural legacy that \"male is superior to female, young is superior to old,\" provided fertile soil for Robert Wilson's misogynistic campaign: \"Our better-living-through-chemistry society was poised to help us control our unruly female physiology through birth control pills during our reproductive years and estrogen during menopause.\"\n\nAs women, we bought the message from the medical industry that redemption from the horror of menopause was to be found in estrogen replacement, and we continued to buy it even with mounting evidence that for many women the solution would create far worse problems than the condition it was meant to address. Embedded in our acceptance of this message is a series of other cultural myths that affect us all. We have learned that our natural rhythms and cycles are to be fought and controlled rather than valued and honored.\n\n#  HOW RITUAL AND KNOWLEDGE CAN REEMPOWER US\n\nThe epoch passage from little girl into woman is not celebrated in our culture. It is not commemorated in a special ceremony like a confirmation, a bat mitzvah, or a wedding. At a time in our lives when we are, really, still little girls, needing guidance and nurturance and assurance about what is happening to us and where we are going and what it all means, we get very little of any of that unless we have extraordinary parents. And even for very wise parents, it is not easy to pluck a ritual out of the air when your culture is so ambivalent and confused about the passage between being a little girl and becoming a woman.\n\nI, however, was part of a changing era. Women were rising to collectively celebrate who we were, perhaps for the first time in thousands of years, since before the patriarchy became the dominant social force. And we wanted rituals to celebrate our daughters. We were waking up to what it means to be a woman, and we wanted to pass these sweetest of revelations along to the next generation.\n\nThe Woman's Liberation Movement carried a consciousness that for some of us was like a flower blossoming. Latent dimensions of our power and creativity and potentials were being rediscovered in the most exciting ways. The pendulum was swinging away from blindly accepting the status of a woman as depending upon being a stay-at-home mom, subservient to her husband, getting dinner on the table in time, and trying not to make trouble. (I remember a bumper sticker that captured our dawning realization about our \"place\" in society: \"Well-behaved women never make history.\" )\n\nThis shift included exuberant talk about how to ritualize a young girl's period. I personally was excited, wanting to truly commemorate this great passage and help my daughters know it was a wonderful event happening in their bodies. But we had no role models. We could only imagine how it might have been wonderful for ourselves if our moms and dads had given us a ritual. But such wisdom had not been passed down to us. Our memories, instead, held visions of all the girls in seventh grade being marched off to see _the_ film on menstruation.\n\nIn mine, we walked in a straight line to the room where the film would be shown, past the boys, all pointing, laughing, and taunting us. Somehow they all knew. There was humiliation and even crying among some of the girls. Then we watched a stupid film that portrayed the narrowest of facts, and it left us all afraid.\n\nAll the teachers were embarrassed and didn't know how to talk about it. Mrs. McDonald was younger but wanted, nonetheless, to give some advice and wisdom from her 24 years: \"Just remember this: it is completely up to you not to get pregnant. Boys can't help it, so you must never 'make out.' If you get pregnant, you will miss the prom and be sent away to have a baby alone.\" Menstruation was still given the name \"the curse\" by many mothers along with these same dire warnings that you could get pregnant. There was no pill yet.\n\nThe Woman's Liberation Movement was a massive energized effort to change all this and bring everything about our cycles out into the open. We were going to empower our little girls with the joy of their cycles. Even though our society was not ritualizing a young girl's first period, I certainly would.\n\nUnfortunately, my husband stepped in first. This was before either girl had even been out on a date. He had decided that he was going to have \"the talk\" with them. I was immediately concerned. \"Are you sure you know what to say, Ray?\" Ray was 12 years older than me, from a generation preceding mine, and he was decidedly not into Women's Lib. He assured me that he knew exactly what to say. And he did. In fact, what he said had a lot to do with my deciding to divorce him before the girls became full-fledged teenagers. In their talk, he told them calmly and firmly, \"If you ever get pregnant, don't come home.\" End of discussion.\n\nWhen my turn came, however, I didn't quite create the example for all parents to aspire to, either. For instance, Dondi's first period came earlier than I expected. She was only 11 and the day sneaked up on me. In my inept hunger to make her first period beautiful, I lovingly held her and spoke of how she was moving into womanhood and what that meant. I didn't know what I did wrong, but after a horrified look came over her, she started crying. She whimpered, \"I want to die!\" I had so held the dream of doing this perfectly and beautifully for her that I couldn't bear her response. I slapped her, like you might in panic slap a sleepwalker heading toward a ledge. It was the first and only time I ever hit her. As we talked further to try to recover the mess I'd made of this precious moment, I learned that she didn't want her childhood gone. She was hearing what I was telling her in black and white. That morning she was a little girl who could play and have fun, and now she thought she no longer was and no longer could. Now she was supposed to step into the shoes of a grown woman. But she still felt like a little girl, and her mind didn't want to go where I was seeming to lead her.\n\nWe all need to develop rituals and practices that are based on a solid understanding of our own nature and place in life. And medicine needs to develop procedures that are based on a solid understanding of our bodies and their not-so-solid energies. So far, it has not. But _you_ can. As Judith Duerk ponders:\n\n_How might it have been different for you, if, on your first menstrual day, your mother had given you a bouquet of flowers and taken you to lunch, and then the two of you had gone to meet your father at the jeweler, where your ears were pierced, and your father bought you your first pair of earrings, and then you went with a few of your friends and your mother's friends to get your first lip coloring, and then you went, for the very first time, to the Women's Lodge, to learn the wisdom of women? How might your life be different?_ 17\n\n#  ENERGY MEDICINE FOR YOUR HORMONES\n\nFor every human body, the cascading chemical dance that determines health and well-being rests with the proper interplay of three major hormones: insulin, adrenaline, and cortisol. Because of the stresses, pollutants, and diet that are part of modern life, all three of these hormones tend to be oversecreted. Diana Schwarzbein, M.D., points out that well-informed changes in nutrition and lifestyle habits \"bring the major hormones back _down_ into balance,\" and by balancing a few essential hormones, \"you will balance all your hormones.\"\n\nThree ways to promote the optimal functioning of insulin, adrenaline, and cortisol are proper diet and exercise, and enhancing your energy system through a practice such as the Five-Minute Daily Energy Routine (page 51). While energy medicine has much to offer if these three major hormones get out of balance, the primary focus of this book is on hormones that are specific to a woman's body and her journey into the miracles and mysteries of menstruation, fertility, pregnancy, and menopause. One energy system, however, triple warmer, governs the production of adrenaline and cortisol and, in conjunction with spleen meridian, influences the production of insulin. The management of triple warmer will be our focus for the remainder of this chapter.\n\n**Triple Warmer.** Triple warmer is one of the body's most potent and least understood energy systems. While the hypothalamus is considered the master gland because it helps regulate breathing, heart rate, body temperature, blood pressure, and hormone production, it teams with triple warmer in carrying out all these functions. Triple warmer, among evolution's great success stories, is charged with keeping you alive in much the way your local police force and fire department are charged with keeping you safe. It governs three of the body's most extraordinary mechanisms:\n\n1. The immune system\n\n2. The emergency response to threat (fight, flee, or freeze)\n\n3. The ability to form physiological and behavioral habits for managing stress or threat.\n\nTriple warmer's energies move in two ways. They follow along the path of triple warmer meridian, and they also are able to jump from this pathway and instantly go to any spot in the body where triple warmer's help is needed. This natural hyperlinking ability is a property of the radiant circuits (see page 32). Triple warmer simultaneously functions as a meridian and as a radiant circuit, and it carries the supreme responsibility of keeping you alive regardless of how hostile circumstances might become.\n\nWhen you think about the brilliant design of the immune system (able to recognize and ward off menacing invaders) and the fight-or-flight response (setting you into life-preserving action, no contemplation required), you begin to appreciate how the force that manages these survival modes must certainly be one of evolution's greatest achievements. However, the fly in the evolutionary ointment is that the immune system and the fight-or-flight response are masterful adaptations to a world that no longer exists. The strategies used by triple warmer were designed to help you survive in the brutal world of your ancestors.\n\n**Adrenal Fatigue: The Stress Syndrome of the Twenty-first Century.** We live in what has been referred to as \"the most agonizing century in history,\" the vast improvements in quality of life and opportunities for personal fulfillment notwithstanding. The stresses and information overload you face every day could not have been imagined by your great-grandparents when they were your age. The sense of one's world and one's place in it, which once remained stable for generationafter generation, changes every decade for us, with our sense of security dwindling in the process. A child psychologist told me recently that the _average_ child today carries more anxiety than did children who were psychiatric patients a few decades ago. Our unmanaged stress, according to one study, is a greater risk factor for cancer and heart disease than cigarette smoking.\n\nAdrenal fatigue has been called \"the stress syndrome of the twenty-first century.\"  Your adrenals are two walnut-size glands that sit on top of your kidneys. They produce or contribute to the production of some 150 vital hormones that influence every major physiological process in your body, including the release of two of the three \"major\" hormones, adrenaline and cortisol. Both of these hormones help the body cope with stress. Cortisol also fosters immune-system function.\n\n**Triple Warmer and the Adrenals.** Triple warmer, which governs the adrenals and thus the production of adrenaline and cortisol, is always on alert for a foreign invader. It triggers both the fight-or-flight response in the face of external threat and the immune response when the body takes in unfamiliar substances in food, water, or air. It was a brilliant strategy when our food, water, and air were pure and when external threats were limited to attacks by predators or crises such as low food supplies. But today, your immune system has to sort through any of tens of thousands of artificial chemicals that appear in our food. Not recognizing them on the evolutionary list of what is familiar and safe, it has to decide whether to initiate a biologically costly immune response to fight them off. Triple warmer also has to distinguish between real threats and stresses of modern life that do not require a full-fledged fight-or-flight reaction. When your computer dies just before you hit Save on an inspired, middle-of-the-night outpouring of glorious words for your speech next week, that is certainly a stress, but it does not warrant the same biologically intensive response as when a saber-toothed tiger walked into your ancestors' cave.\n\nThe physiological and psychological cost of these false alarms is substantial, invisibly draining your life force. The physiological changes of the fight-or-flight response include: heart rate may double or triple; blood pressure increases as the coronary arteries dilate; respiratory rate increases; muscle tension increases; hormones such as adrenaline, noradrenaline, cortisol, oxytocin, and vasopressin are released into the bloodstream; hydrochloric acid is secreted into the stomach; glucoseis released from the liver; basal metabolic rate increases; blood leaves the forebrain and digestive tract and moves into the muscles and limbs; pupils dilate, improving eyesight; systems not essential for fighting or escaping, such as the immune, digestive, and sexual systems, virtually shut down. That's a lot of biological capital to spend each time your computer crashes, a friend is upset with you, your husband is late getting home from work, or your daughter uses a swear word. But it is the same circuitry and physiological response for any of these relatively minor disturbances as it was when the saber-toothed tiger walked into the cave. That's what I mean by saying that your body evolved for a world that no longer exists.\n\nIn addition to the physiological effects of the stress-response syndrome there are also psychological consequences. The ability to maintain perspective diminishes significantly, as do other logical functions. The tendency is to rely on instinctive or habitual stress-induced behavioral patterns rather than forming a creative response to the situation. In addition, anger or rage tends to accompany and support the fight response, while fear or panic generally accompanies and supports the flight response. Hysteria, feeling overwhelmed, or numbness tend to result when the fight or the flight response is activated but then inhibited or otherwise not acted upon.\n\nBeyond the immediate costs, if the stress response is continually activated, the physiological consequences can lead to disorders of the immune and autonomic nervous systems. These include susceptibility to infection, autoimmune diseases, chronic anxiety, chronic fatigue, and depression. The body's ability to produce adrenaline and cortisol, two of its master hormones, also becomes exhausted.\n\n**Retraining Triple Warmer.** In short, each of us landed in a world that tends to send triple warmer into overdrive, and many of our health problems result from, or are at least exacerbated by, this single evolutionary twist. The good news is that you can literally retrain triple warmer to not be on overdrive. This is, in fact, one of the most important gifts energy medicine has to offer.\n\nFor instance, when a stress (whatever its nature) sends you into the fight-or-flight response, and up to 80 percent of the blood leaves your forebrain to go into your arms and chest to fight, or into your into legs to run, the whole cascade of ancient chemicals prepares you for action you probably will not take. A simple technique from energy medicine called the Stress Dissolver lets you interrupt the process whenever you feel stressed or threatened. Lightly place the pads of your fingers on your forehead and your thumbs on your temples (see Figure 3-1) and hold firmly but with no pressure for a minute or two. An alternative version is to lightly place the palm of one hand on your forehead and the palm of the other hand on the back of your head, just above your neck. Both versions activate points, called neurovasculars, which impact circulation and bring the blood back to your head, shifting you out of the stress response. Breathe deeply as you do the Stress Dissolver. It is a physiological contradiction to stay in this position and to be in fight-or-flight mode for more than a couple of minutes, so the fight-or-flight response subsides.\n\n_Figure 3-1_ THE STRESS DISSOLVER\n\n**Partnering with Triple Warmer in an Emergency.** While I was writing this chapter, I had a powerful experience using this and related procedures for working with triple warmer. We had retreated to Baja to write. I was driving early one evening, alone, and about to turn into a market, when I came upon a gruesome scene. A bus had just hit a young woman and one of its back tires had stopped directly on her pelvis, hip, and upper legs. Mine was the first car to arrive. I pulled over, jumped out, and went directly over to her. The bus driver, who looked like he was in shock himself, said, \"No, no,\" when I got to her and gestured for me not to touch her, fearing, I'm sure, that an untrained person might inadvertently cause further harm. I don't speak Spanish, but I heard the words \"Yo soy doctor\" (I am a doctor) tumble out of my mouth. Everyone who was on the bus or who was now coming onto the scene cleared a space for me to be with her. Time stopped; everything seemed surreal. The girl was unconscious. I saw that her life force was leaving her. Her aura had collapsed completely into her body and the energies were draining out at her feet. I focused on what could be done to keep her alive. The tire sitting on her became secondary. I was on the right side of her body and immediately began to hold triple warmer points that would stabilize her energies and take her out of shock. These points are on both sides of the body. I signaled to the nearest person, a Mexican boy probably about 13, to hold the same points on the left side of her body, and he promptly mimicked what I was doing, somehow maneuvering around the tire.\n\n**The Triple Warmer Smoothie** (time\u201420 seconds)\n\nTriple warmer energy starts at the tip of the ring finger, travels up to the neck, behind the ears, and ends at the temples. You can use the electromagnetic energies of your hands to readily calm overactive triple warmer energy by tracing part of this pathway backward. Here are the steps:\n\n1. Place your fingers at your temples. Hold for one deep breath, again breathing in through your nose and out through your mouth (see Figure 3-2a).\n\n2. On another deep in-breath, slowly slide your fingers up and around your ears, smoothing the skin while maintaining some pressure (see Figure 3-2b).\n\n3. On the out-breath, slide your fingers down and behind your ears, press them down the sides of your neck, and hang them on your shoulders.\n\n4. Push your fingers into your shoulders and then, when you are ready, firmly drag them over the tops of your shoulders, and smooth them to the middle of your chest, with one arm resting on top of the other (see Figure 3-2c). This is the heart chakra. It brings you home to yourself.\n\n5. Hold here for several deep breaths.\n\n_Figure 3-2_ THE TRIPLE WARMER SMOOTHIE\n\n**The Triple Warmer Tap** (time\u2014about a   \nminute or as needed)\n\nThis is a great \"sedative\" to use _while_ you are feeling fear. It is simple, quick, and effective, sending impulses that turn off the fear or stress response.\n\n1. Place the hand to be tapped over the center of your chest.\n\n2. With your other hand, tap it on the valley between the ring finger and little finger, above the knuckle (toward the wrist). Tap firmly, about 10 times, breathing deeply (see Figure 3-3).\n\n3. Pause and take a deep breath.\n\n4. Tap about 30 more times.\n\n5. Repeat, switching hands.\n\n_Figure 3-3_ THE TRIPLE WARMER TAP\n\nWe held these points until I felt the shock start to subside. While I was mostly on automatic pilot, I realize in retrospect that stabilizing her and starting to bring her out of shock was a necessary first step. Then other triple warmer points could be held. These would support her life force and get the energies that were draining out of her to remobilize and literally start to fight for her life. I suspect that we held the first set of points for about three minutes before it felt right to move to the second set of points. We just sat there on the ground with our fingers firmly touching these points. It felt like we were holding her life force in our hands. The boy was solid and steady in his task, but I noticed that he had tears in his eyes. Finally, I started to feel a power under my hands, as if her life force was coming back. As soon as this began to stabilize (I would guess that we held these points about three or four minutes), I was drawn to hold her neurovascular points (described on page 106). I showed the boy how to hold the end points of the heart meridian to give her heart all the support we could, and I went to the neurovascular points on her forehead. Besides interrupting the fight-or-flight response, which clearly was not adaptive in this situation, these points stabilize the circulatory system, helping balance the flow of blood throughout the body.\n\nAt about this time, an ambulance arrived with two paramedics. The bus driver immediately explained that I was a doctor. They approached to take over, but I felt it was essential to continue to stabilize her blood flow. I was able to indicate to them that they should let me continue, and they did. Somehow between the few Spanish words that were coming to me and whatever understanding they had of English, I was able to explain that the boy was holding points that stabilize the heart and I was holding points that could help with blood flow and internal bleeding. The paramedics turned their attention to how to get the bus off the woman. About five men had a brief discussion with the bus driver. Since the tire was directly over the woman's body, they needed to decide whether to roll forward or backward. I sensed that if they rolled forward they would damage more internal organs and indicated that they needed to roll backward. I also sensed that having the bus move over her again could put her back into shock and that the neurovascular points needed to further stabilize her first. At the moment that I could feel a pulse on her forehead, she opened her eyes and looked at me. Then she closed them again. I signaled to the paramedics and the driver that it was okay now to move the bus. I continued to hold the points on her forehead as the 32-passenger bus rolled back, off her pelvis, left hip, and leg. Almost instantly it seemed, she was on a stretcher, in the ambulance, and rushed off. The bus driver hugged me.\n\nI was in a bit of shock myself. I went back to my car and cried. I then held my own neurovascular points, breathing deeply, until I felt I could drive. The next morning, I went back to the market to find out the hospital to which the woman was taken. When I got to the hospital, a woman who spoke broken English was able to identify the woman. She called a doctor, who came to greet me. The paramedics had apparently described the strange behavior of this blond gringo mystery woman. The doctor smiled, took my hands in his, and said, in good English, \"You are the one who helped Maria at the accident. Something wonderful happened there. Somehow her blood coagulated and stopped her internal bleeding. Did it have something to do with what you did? It probably saved her life. What did you do?\" I put my fingers on his forehead and said, \"This is what I did.\" I've never seen appreciation turn more rapidly into bewilderment.\n\n**Ways to Partner with Triple Warmer.** Hopefully, you will never need to use these techniques in such extreme circumstances, but you can use them whenever you feel fear that is out of proportion to the actual danger or whenever life triggers any other kind of stress response. By using one or more of the following triple warmer techniques at such times, you are giving a direct command to triple warmer to not pump unnecessary adrenaline and cortisol into your system. You are partnering with it, training triple warmer to respond more appropriately to the stresses of modern life and literally helping it to evolve. If you use these procedures regularly, incorporating one or more of them into your Daily Energy Routine, for instance, you can help free yourself from the epidemic of triple warmer overwhelm facing all of us and the problems it causes with adrenaline, cortisol, and insulin, the hormones that regulate almost all the other hormones in your body. Two of the techniques have already been presented, the Meridian Energy Tap (page 80) and the Stress Dissolver (which is also described in greater detail on page 105). Here are four additional methods to consider for balancing triple warmer:\n\n_Figure 3-4_ TRIPLE WARMER\/ SPLEEN HUG\n\n**Triple Warmer\/Spleen Hug**   \n(time\u2014about a minute)\n\nThis comforting position simultaneously calms triple warmer while energizing spleen meridian. Use it anytime you are upset or need comfort:\n\n1. Wrap your left hand around your right arm, just above the elbow.\n\n2. Wrap your right arm around the left side of your body, underneath your breast (see Figure 3-4).\n\n3. Hold this position for at least three deep breaths.\n\n4. Reverse sides.\n\n**Holding the Triple Warmer Sedating Points** (time\u20146 minutes)\n\nHolding your fingers on certain combinations of acupressure points can replenish depleted energies (these are called \"strengthening points\") or release jammed or stagnant energies (\"sedating points\"). The strengthening points strengthen meridians by _adding energy._ The sedating points strengthen meridians by _releasing excess energy._ Triple warmer is almost always overcharged in our stress-filled culture.\n\nSedating triple warmer not only allows it to function more effectively, but also keeps it from draining other meridians of their energies. Holding the sedating points accomplishes what the Triple Warmer Smoothie, Tap, and Hug accomplish, but with a deeper impact. Be sure to find a very comfortable position while holding the points. Holding acupressure points to sedate or strengthen your meridians is among the most time-consuming procedures presented in this book. But I love the results I get, as do my students and clients. And this is one procedure where you don't have to be mindful. You can hold the points while watching TV, talking to a friend, or relaxing in the tub. To sedate triple warmer meridian (the numbers in the meridian-strengthening and -sedating instructions throughout the book, as in \"Stomach 36,\" refer to the name of the acupressure points being held):\n\n1. Place the middle finger of one hand on Stomach 36 and simultaneously place the middle finger of your other hand on Triple Warmer 10 (see Figure 3-5a), on either side of your body.\n\n2. Hold for about two minutes.\n\n3. Repeat on the other side.\n\n4. Then place the finger of one hand on Bladder 66 and the finger of the other hand on Triple Warmer 2 (see Figure 3-5b) and hold for about a minute.\n\n5. Repeat on the other side.\n\n_Figure 3-5_ TRIPLE WARMER SEDATING POINTS\n\nEnergy medicine can help you understand your body in new ways and empower you with natural and highly effective methods to care for yourself. After examining some of the culture's distorted and insidious medical myths, this chapter has presented several basic methods for managing your body's master hormones, particularly in relationship to the stresses of daily life. In the following chapters you will learn how to use energy medicine methods with a variety of issues that specifically impact a woman's body, such as PMS, sexuality, fertility, pregnancy, and menopause. And you will be invited to learn how to combine energy interventions with energy tests to help you and your doctor determine the kinds and amounts of natural hormones or other supplements you may need. An energy approach does not necessarily replace the best of what medical science has to offer. But it can refine it, and it can go beyond mere chemical interventions. With knowledge about your body's energies, you can responsibly insist on being your doctor's collaborator in your treatment and, together, you will both be more effective.\n\n_**Chapter 4**_\n\nReclaiming the Wisdom of Your Menstrual Time\n\n_We can reclaim the wisdom of the menstrual cycle by tuning in to our cyclic nature and celebrating it as a source of our female power._\n\n\u2014 CHRISTIANE NORTHRUP, M.D. _Women's Bodies, Women's Wisdom_\n\n**P** remenstrual syndrome (PMS) affects sixty percent of the women in the United States. It is a medically diagnosable disorder and, with extreme PMS, a woman's very sense of sanity may feel assaulted. Beyond the physical symptoms, it is for many women no less than a ferocious wrestling match with forces from deep within. It is as if the soul is trying to burst out at the seams while the body's disorienting biochemical fluctuations are squashing it down. Our culture has not been particularly adept in preparing us for this challenge. Working with your body's energies not only may help you feel better, immediately and palpably, but also may shift your body's chemistry so that your journey through your monthly cycle is an adventure of the female soul rather than a fight with the female body.\n\nAfter discussing the chemical, emotional, and energetic dynamics of a woman's cycle, this chapter coaches you on measures you can take throughout the month that will make your premenstrual time less troubling. It then presents energy techniques to do at the first signs of the onset of PMS and outlines a Premenstrual Module you can add to your Daily Energy Routine for bringing your body into better harmony with the chemical changes that occur prior to your period. It also offers specific energy techniques that can help with the physical discomfort and emotional challenges that may accompany PMS. Finally, it describes ways to energy test for hormonal and other supplements that may augment the energy approach.\n\n#  IS PREMENSTRUAL SYNDROME A SYNDROME?\n\n_Hysteria_ and _uterus_ have the same derivation ( _hystera_ means uterus in ancient Greek). Wide variations in the moods of many women, moods that correspond with their monthly cycle, have led to the use of terms like hysterical (or worse) to characterize some of the more extreme behavior resulting from monthly hormonal imbalances. Ranging from anxious to depressed, aggressive to weepy, and reactive to numb, these mood swings are described as part of premenstrual syndrome (PMS). Along with severe shifts in mood that are independent of events, conspicuous physiological changes may also accompany the time prior to menstruation, including bloating, swelling, fatigue, constipation, and skin changes.\n\nAt the same time, our attunement to the ebb and flow of natural cycles is a deep source of a woman's power. But when the medical establishment bestows a name like PMS on a natural process, that process is set on a slippery slope toward becoming pathologized and something to treat. We begin to move away from self-acceptance, away from understanding nature's plan, away from time-honored wisdom. Here we see the medical industry reshaping reality. First of all, the term \"premenstrual syndrome\" has a unique meaning for every woman. Some never experience it. For others, the symptoms last much of the month. For some, it seems related to lower levels of progesterone, in which case supplemental progesterone greatly reduces the most unpleasant features of their monthly cycle. For others, adding progesterone doesn't touch the misery they endure each month. PMS is a single term for many hormonal landscapes.\n\nFor all who experience it, however, PMS potentially holds a positive, creative, rehabilitative function. Like the sacred plants used in shamanic healing ceremonies to open a doorway to the world of spirit, PMS changes the body's chemistry to clear a path for consciously journeying into the domain of the soul. It unleashes the profound power of your inner regions, often in ways that women find overwhelming, bringing them face-to-face with great forces that usually reside in the netherworld. Viewing PMS as only a medical condition misses half the fun. Tribal cultures knew that the premenstrual days were a sacred time, a time to honor and respect, a time to sanctify in customs such as the moon hut.\n\nAnyone who knows me, however, knows that I am not one to romanticize or become overly sentimental about the spiritual potentials of PMS. I know it in its darkness as well as its light, and in its darkness, PMS is a doozie.\n\n#  YOUR MONTHLY CYCLE: A PLAY OF TWO HORMONES\n\nThe seemingly tangled biochemistry of a woman's monthly cycle is actually among nature's most remarkable designs. While you are going about your daily business of eating, sleeping, working, shopping, playing, loving, and parenting, your body is busy choreographing a most amazing and time-honored pageant. One of the two million or so immature eggs stationed in your ovaries when you were born is selected as Queen-apparent and parades down the runway of the fallopian tubes toward the reproductive palace that is your uterus. If you have made the choices of nature's urging, she will be met by millions of suitors, choose one who has proven himself a most robust and healthy fellow, and they will join to grow into the newest achievement in humanity's mission to perpetuate itself.\n\nThe Queen-apparent has only about a day to find her suitor. If not, she is recycled (perhaps it was her hairdo), and preparations for a new Queen-apparent are set in motion. Nature seems harsh. But she wants only the best for you. And guess what? The complex stage directions for the pageant are coded largely in the release of two chemical messengers\u2014yup, the hormones estrogen and progesterone.\n\nHowever, other chemical messengers are also involved. In fact, the whole cycle begins when your hypothalamus tells your pituitary gland to release into your bloodstream follicle-stimulating hormone (FSH) and other hormones related to ovulation. Follicles are tiny sacs within your ovaries that house immature eggs. The hormones sent from your pituitary gland prompt the follicle holding the egg that is most ready to mature (the winner of the Queen-for-a-day competition) to get on with it. Showtime!\n\nAs the Queen-apparent puffs out her chest and steps toward the mike, the follicle splits, and she is suddenly free. Meanwhile, the blood supply to the ovaries increasesand its ligaments contract, pulling the ovaries closer to the fallopian tubes, paving the way for the Queen-apparent's journey down the runway. The follicle's job, however, is not done. Not content to be an idle empty nest, it continues to support its offspring by rapidly taking on new cells so it becomes the ovaries' largest inhabitant and, now called the corpus luteum, begins to produce the estrogen and progesterone that orchestrate the pageant.\n\nEstrogen directs the body to prepare a nursery for the much-anticipated child of the blessed excursion down the runway. The uterine lining is thickened so the nursery can contain the rapid growth of new life. Acting in tandem, progesterone causes glands within the uterus to produce secretions that stabilize the surface of its lining (the endometrium). Meanwhile, estrogen also gives the guys a hand by changing the consistency of the cervical mucus, making the juicy atmosphere more hospitable. \"Fertile\" cervical mucus draws the sperm into the fallopian tubes (where the festivities will take place) and helps them survive as they await the appearance of the lady of the hour.\n\nBut if, after all that, by the end of the cycle none of the guys have come through, or if they never showed up, the corpus luteum realizes that its chances for becoming a grandparent have passed, and it stops contributing its estrogen and progesterone to the family trust. Without these, blood vessels in the uterine wall contract and spasm, the uterine lining is shed, and the menstrual blood cleans all, initiating the start of a new cycle.\n\nNot surprisingly, this internal royal life-and-death drama can have some effect on your mood. Even if you are oblivious to the plight of the Queen and her suitors, the hormones that manage the pageant may be harder to ignore. For some women the hormonal changes over a month are extreme; for others they are relatively minimal. Some women are also neurologically wired to have strong reactions to the slightest hormonal changes; others hardly notice massive fluctuations. The changes in estrogen and progesterone also trigger neurotransmitters, most specifically serotonin and gamma-aminobutyric acid (GABA), which can impact mood dramatically. If hormonal levels rise and fall too quickly, the adrenal glands become burdened and begin to secrete abnormal amounts of the stress hormone cortisol. This strains the entire endocrine system, making it ever more difficult to achieve hormonal balance as estrogen and progesterone are telling cells to multiply. Then, if the egg has not been fertilized and there is no baby for the uterus to carry, enzymes liquefy the uterine lining, the flow of blood begins, and the month's surplus uterine lining is shed. The entire body is involved. Even the bones are receptors for estrogen and progesterone. Is it possible to manage the ever-changing ingredients of this divinely inspired biochemical soup? I will share my journey.\n\n#  MONTHLY VISITS TO HORMONAL HELL\n\nIn reflective moments, I like to believe that my many health challenges were a necessary part of my training as a healer. The concept of the \"wounded healer\"\u2014that terrible illnesses and woundings establish a healer's expertise\u2014resonates with me. Multiple sclerosis, tuberculosis, heart problems, a malignancy, and asthma have all been personal teachers for me. But beyond any of these, few people have been better prepared by their own life's journey to write a book on working with hormonal challenges.\n\nMy menses began when I was ten. Because I was so young, my mother had not yet had \"the talk\" with me. My panties were red with blood. I was dumbfounded. I thought I must have done something terrible, but I did not have a clue what it was. It was so disturbing that I decided to hide my panties in the attic so nobody would see. Mostly, I didn't want Mama to be scared. But with a nose for such deception, Mama quickly found them. When she saw me next, I received a knowing and compassionate, \"Oh, Donna!\" Oh, Mama, bless you!\n\nEvery month from that point onward, two days before the start of my period, terrible cramps, backache, and groin pain were accompanied by severe shifts in my perceptions and emotional reactions. The day before my period, the pain and pressure in my abdomen, groin, and at the top of my legs was so intense that I could hardly stand.\n\nThe experience is intensely biological. Women who suffer extreme PMS are, in fact, experiencing something akin to drug withdrawal, with a sharp decline in the body's natural opiates in the days prior to menstruation. It felt like I was being pulled low to the ground, toward the earth, and deep into myself. It was beyond choice. My legs felt heavy. The chemistry that fueled my usual exuberance completely waned. Energies from others and from the world now hurt my body as well as my psyche.\n\nThen there was a day each month, just before my period started, when I was so awake that I would stay up all night, seeming to accomplish everything. The house would be made spotless, letters would be answered, papers filed, clutter organized. Nothing could tire me. I was in full joy and energy. Then, right before my period, I felt full of lead, pulled down, compelled to rest. This would actually sometimes knock me out, and I would go into long slumbers. They always felt to me like an ancient hibernating response. This was a time that my body needed to do major internal work, making a thousand adjustments as it readied itself to discard another month's preparations for carrying a baby and enter a new cycle.\n\nDavid describes a day in my PMS life if I don't get to be moon-hut internal: \"It is like I went to bed with Mrs. Jekyll and woke up with Mrs. Hyde. It is like the happiest, most joyful person I know becomes a ranting, tormented soul, sure she is responsible for healing the pain of every pitiful soul who passes her way. Any missteps of the past month are put under a microscope, magnified a dozen times, and dwelled upon endlessly, though it is a very democratic microscope\u2014it is equally sharp with my missteps as it is with hers. Words meant to comfort are heard as evidence that I will never be able to understand her. Her misery is extreme, and my usual first-response tools\u2014reason and repair\u2014make things even worse. Was there nothing in my training as a psychologist, I would ask myself, that prepared me for each new installment that, like Groundhog Day, presented itself like clockwork, giving me yet another slim chance of getting it right?\" I would like to add, David, that I was far more tormented than you were by not getting it right!\n\nThe role of hormones, though I didn't yet think in those terms, became patently clear during my first pregnancy. No monthly descent into physical and emotional agony! Also, for the first time in my life, I didn't have to watch my weight. It was as if my metabolism allowed me to process food like other people. Before, I could fast for days and not lose a pound, and it almost seemed that I could gain weight by merely breathing the aromas from the kitchen. Now I could eat whatever I wanted and, except for supporting the growing baby, be losing weight. I gained zero pounds during that pregnancy! And I ate and felt better than ever before. My body and my spirit loved being pregnant. And after my daughter Tanya was born totally healthy, I weighed 20 pounds less than I did when I conceived her.\n\nI now understand that a woman produces a great deal of progesterone during pregnancy, which is necessary for building up the wall around the uterus and preparing the womb to carry a baby. Progesterone would later turn out to be my drug of choice. The elevated progesterone levels were, in part, why I loved being pregnant. I also loved being a mom. My progesterone production, however, went back to its former level, and PMS again became my private terror. After Tanya was born, my hormone-induced cramps and cyclic swings uncharitably returned, along with distortions of reality that could be dangerous. Our apartment was upstairs. Sometimes it would seem perfectly logical in my happy reveries as a young mother that if I dropped Tanya, sending her and her basket over the railing, she would gently float down to the bottom and I could then walk down the stairs, pick her up still comfortably snuggled in her basket, and continue into our day. Thankfully, a small voice from deep within was able to persuade me that this might not be a good idea. But I am not exaggerating to say that I had to fight the impulse, and I wonder how many mothers sit in jail for not having been able to sort such reveries from reality during the postpartum period. It had nothing to do with wanting to harm her or with postpartum blues, but rather a temporary inability to distinguish fantasy from reality.\n\nWhen I was pregnant again with my second daughter, Dondi, I was equally happy and joyous, full of vigor, cheer, and good spirits. No cycles. No distortions. No pain. And after the birth, again the magic faded as abruptly as it had begun. It took some self-restraint not to have 15 or 20 children so that I could live as much of my life as possible enjoying the delights of pregnancy.\n\nAs the years went on, my sensitivities in the week before my period became even more extreme. As PMS approached each month, I would begin to feel trepidation. What was I going to do if I had a client scheduled or had to teach a class? I am a very positive person with lots of energy but, when PMS came, the bottom dropped out chemically, energetically, and emotionally. At this time of the month, other people's energies could fry my circuits. Especially irritating were positive, outgoing people, the kind everybody loves to be around. It was excruciating to feel my compulsion to respond to other people's coaxing or needs as my body and soul were pulling me inward. Yet I still tried to connect with them.\n\nI never knew anyone who had PMS as bad as I did. Women who don't go through severe PMS often cannot even identify with my PMS experiences. Once, when one of my closest friends was furious at me for \"losing it\" during a highly inconvenient moment, she shouted, \"Shouldn't you be held responsible for what you did?\" I considered this for a moment and then replied with the certainty of fresh and profound insight, \"No, I absolutely am not accountable!\" Nothing even gave me a clue that I might be about to blow up. I might be perfectly normal, even happy and serene in one moment, and then something would trigger me, and I would feel nuts, finding myself screaming or in tears. I could be shielded from those triggers when I was able to take \"moon hut\" time. But with all the demands of parenting and career, that was infrequent, so I was often vulnerable.\n\nMine was certainly not a politically correct PMS. It was bad news for women to hear about severe PMS. Even my female gynecologist wanted me to conceal my experience, saying \"Donna, I don't think I'd speak of this out in public and let people know just how bad it can be for a woman. After all, we want to get a woman into the presidency.\" Women were coming into their power at that time, and it did not seem that a person who had this condition could be head of her own affairs, much less of her country. They didn't want women showcasing their monthly impaired judgment and deep psychological struggles. They felt PMS gave women a bad rap. There was already a glass ceiling, and they didn't want anything to lower it. I, on the other hand, felt that these were the very qualities that the country needed\u2014 leaders who went within once a month and wrestled with their souls.\n\nNonetheless, I was terrified as PMS approached each month, and so were those who loved me. But not my clients. They were thrilled. People in my hometown would mark on their calendars when I was going to be premenstrual and try to schedule sessions at that time. Personal boundaries, which have not been my strong suit under any circumstances, were virtually nonexistent when I had PMS. It was impossible for me _not_ to feel what was going on with anyone in my presence or not to offer every ounce of my life force. I knew their pains, the dynamics of their illnesses, and their personal sorrows as if they were mine, and my empathy and desire to help were far beyond reasonable. But what had people fighting to be given appointments during my premenstrual week was that I was extremely psychic during that time. The information I would receive about them was uncannily accurate. Having no boundaries does have its rewards.\n\nBut for me, working during this time was terrible. It threw my hormones off even further. At exactly the time everything in my being needed to turn inward, to retreat from the world, my sensitivities became so acute that I suffered for others to the degree that I would find a way to meet with them no matter what the personal cost or how late the hour. And there was no shortage of clients. Yet it was precisely at this time that I was not meant to be out in the world healing anyone. It was time to care for myself, to drop in deep, and see what I wasn't seeing. \"Healer, heal thyself,\" the concept that had gotten me through so much in my life, got blanked from my screen while I was premenstrual.\n\n#  THE UPSIDE OF THE DOWNSIDE\n\nBut now, a surprising revelation. I loved PMS! Not while it was happening, of course. But I loved it when it was over. And I love it now when I look back. Beyond the many horrendous parts of PMS, if I listened to my soul and my body, I found an upside to the downside. I cherish that I had such an uncontrollable force hit my body each month that humbled me and put me on a monthly spiritual path. My soul beckoned to me. This was not my persona. It was not a trick of my psyche or a chemical imbalance. I was wrestling with the deepest forces of my being. My truest nature was peering out through the cracks of my hormonally altered reality. This was a great gift, even if presented in frightening wrappings.\n\nMy body's insistence that I go deep within was not easily denied. It felt painful and wrong to come up from that place to interact with people. Positive energy felt negative. But the deep internal energy that drew me also felt scary as I began the descent. I felt anxiety and fear about all the things I couldn't control, most especially my own emotions. It felt as if I was falling down an elevator shaft, and I didn't know how far I was going to go.\n\nBut there was no escape. The wisdom of the feminine soul doesn't allow it. Difficult truths I'd been able to ignore or deny throughout the month came up large in my heart, often exaggerated into negatively tinged caricatures of themselves. The polarity from being external and out in the world to being internal and deep within myself was paralleled with a swing from light to dark. From this immersion into the dark night of the soul, I came back transformed. Hellish as it is, at least I didn't have to pay thousands of dollars to go on a vision quest. And if I had embraced the process\u2014left worldly time and entered moon-hut time\u2014I came back transformed for the better. As I began my ascent back up each month, I knew I was changed, touched by profound truths and a larger perspective. The journey was like meditating, but on steroids. I have always been a bit ambivalent about disciplined meditation practices since my body plummeted me into the depths this way. While recognizing the profound places meditation can take people, it seemed a genteel substitute, something perhaps for men or for women who didn't have the privilege of extreme PMS.\n\nWhen I got to the other side of my period, I was always deeply grateful for it because I understood things about myself and my life that I don't believe I could have otherwise. Beyond its role in renewing our monthly biological ovulation cycle, we _need_ PMS for its role in renewing our contact with the deep forces at the very heart of life. However, in our culture, which doesn't support women being in their own rhythm at that time of the month, we don't get to follow our soul's calling. Because we try to be out in the world and \"normal,\" our PMS intensifies and sometimes becomes debilitating.\n\nA woman who can fully appreciate her cycle, and the depths it can lead her to, develops wisdom that is emotionally rich and intuitive. She becomes more sensitive to her own intimate rhythms and the greater rhythms of life. She learns her body and its energetic ways as well as its hormonal ways. She is forced to surrender every month, and so she learns to embrace the quality of surrendering. The best part for me was how my soul _insisted_ that I be in conscious contact with it. I could not ignore it, and I learned to surrender to its beckoning. The wisdom of my inner impulse was further supported when David got with the program by building me a \"moon hut\" (does it count if it has propane gas, a top-of-the-line Simmons bed, and a stereo system?) right in our backyard.\n\nDespite the ways PMS yanked me off center, it was also my truth serum. Anything I had been able to deny or repress or not see would suddenly appear large on the screen of my inner life, with sharp tones and vivid emotions. Feelings that had been stuffed, truths that had been disregarded, and injustices that had been ignored throughout the month were in my face. Brutally in my face. I could not put up with a lie or arrogance or someone devaluing another. I had no censor in me. No tolerance for the nonsense we all accept as part of daily life. No way to hold back feelings that discretion might have kept restrained. I could not go along with anyone else's agenda. I often did not even know if my marriage would survive each month when I was premenstrual. PMS is nature's design for making you reexamine your life intensively for a few days every month, and few institutions can withstand such scrutiny (self-defense, by the way, was the _real_ reason David built my moon hut!).\n\nAn elemental life force is always at work in a woman's body. While we are fertile, we enter deeply into this space every month. The cycle is both intimately personaland universal. The physical journey through the days before and during menstruation is miraculous, but PMS is also the passage through a doorway to a holy place. It is a sacred, intimate journey with the self and the soul.\n\nDuring my PMS time, my dreams were often lucid and powerful. I sometimes couldn't tell in the daytime whether I was daydreaming or having psychic visions. In terms of yin and yang, it was extreme yin, extreme feminine. When I got to the other side of my period, I knew I was fortunate among women to know this extreme of being woman. While it seems on the surface that women who do not experience PMS are very lucky, I have secretly felt that missing out on the sacred journey of the worst kind of PMS is a great loss. Evolution seemed to speed up for me during premenstrual time, and my soul gave me guidance in a way that just wasn't possible in my normal busy days. The voice of my soul would call, and I _had to_ answer. PMS is a path of power and wisdom but, in a world without \"moon time,\" it can also be a path of distress and despair.\n\n#  SEARCHING FOR THE MAGIC PILL\n\nDuring clearheaded, upbeat times of the month, I began to experiment with ways that I might help myself during the challenging times. Doctors didn't have remedies for PMS beyond drugs designed to help with headaches and cramps. These did not touch my symptoms. I went to health food stores in search of relief. They offered several different herbs that supposedly helped with PMS. These consistently made my symptoms worse. I later learned that the active ingredient in all of these was natural estrogen. The popular belief back in the 1970s was that estrogen levels needed to be increased to help with PMS. While this did help relieve cramps for some women, it often did not help with the other symptoms and, for many women, it was actually harmful because their PMS symptoms were caused by already having too much estrogen in relation to progesterone. The pharmaceutical companies were not doing research on natural substances because they couldn't patent them. So the herb companies were operating with good intentions but little accurate information.\n\nI had so many bad experiences with herbs that were supposed to be helpful for PMS or women's hormone problems that I became afraid to try the new ones that became available. But then\u2014and this is before I knew about energy testing\u2014I beganto notice how my body reacted energetically to the different herbs on the shelf. My energy would literally separate from an herb that was not good for me; it would pull back. It was as if my energies and the energies of the substance didn't want to merge. That's how I energy tested before I knew energy testing. I would sense the energetic impact of the herb on me so that I wouldn't keep buying and ingesting substances that were harming me. Then came that fortuitous day when a young Mexican woman I knew was chewing on the root of a plant. I could see that there was an elegant energy between her and the plant. I asked her what it was. She explained to me that she did not feel well (\"the time of the month,\" she confided) and that chewing on this root helped her immensely. The energy of that root felt so good to me that I asked her how I could get some. She generously cut off some of the root and gave it to me.\n\nThe next time I felt PMS symptoms, I began to chew the plant, Mexican yam root, which is very high in natural progesterone (Mexican and Siberian yam root are the best of the various forms of yam root I know about). It was so good for me that it was almost like a magic potion. But it tasted _so_ bad. I talked a local health food store into grinding it into a powder so I could put it into little capsules that could be swallowed without having to taste it. I began checking each woman I knew who was having difficulty with PMS. I would put the yam root into her energy field to see if it made her energies look better. If it did, her PMS symptoms were related to low progesterone. Pretty soon, my friends and clients were getting the health food store to grind the root for them, and they were reporting such good results that the people at the health food store began asking their suppliers if they could get Mexican yam root in capsules. No one was distributing that at the time, but the Solaray rep took the idea back to his company and the people there decided to add it to their product line. Solaray sent me a year's supply with a letter thanking me for my contribution.\n\nA few years later, in the late 1970s, Kathryn Dalton, a physician in England, did research with female prisoners who had been convicted of impulsive acts of violence. She suspected that these women were taken over by extreme PMS caused by a progesterone deficiency. After she provided them with progesterone supplements while they were premenstrual, their emotional imbalances and outbursts ceased. This was so obvious to everyone involved that she actually was able to get several women released from jail on the promise that they would faithfully take progesterone at the appropriate times.\n\nWhen my father was dying, in 1987, a time of intense emotion for me, I found that yam root was no match for the extremes of my PMS. Taking a cue from Kathryn Dalton, I made it my quest to find pure, natural progesterone. As luck would have it, one of the very few pharmacies in the country that was able to supply natural progesterone was in my hometown of Ashland, Oregon. David wanted to nominate the pharmacist, Jack Sabin, for the Nobel Peace Prize.\n\nHowever, as with all good things, the pendulum can swing too far. While I was viewing progesterone as God's gift to women, I assumed that if a little was good, more was better. I began to rely on progesterone any time I felt a little off emotionally. The result was that my uterus began to grow and thicken. That is progesterone's job, fortifying the uterus so it is able to carry a baby. While this started imperceptibly, it had become quite a problem before I made the connection between overdosing on progesterone and the changes in my uterus. I have been told that my uterus is the size it would be if I were two and a half months into a pregnancy. So here I am, stuck to this day with a tummy that won't go flat. The lesson is, everything in balance. And that is the great value of energy testing, to be able to assess what your body needs at any point in time. Meanwhile, I have been alarmed to see the culture's pendulum also swinging toward progesterone as a wonder drug. Credible health providers are advocating that everyone, with no assessment of their own natural progesterone\/estrogen balance, should take progesterone in doses that are simply, according to my experiences, over the top.\n\nTaking too much progesterone also creates a different ratio between your progesterone levels and all your other hormones, and this can have further unintended effects. I am someone who rarely gets depressed\u2014I get hysterical\u2014but suddenly everything was upside down. My levels of estrogen became too low in relationship to my progesterone, and I felt a deadness I'd never known before. When I backed off from taking too much progesterone, and particularly from taking it when I wasn't premenstrual, the deadness went away. In short, the progesterone panacea had its limitations. If I used it to keep all my emotions in control, it hurt my body, and eventually my spirit as well.\n\nOver time, I learned that by combining energy techniques with the use of progesterone, each became more effective. PMS involves a _chemical_ imbalance in hormones that can be corrected chemically _or_ energetically, and it involves an _energetic_ imbalance that also can be corrected using either approach. But working directly with your body's energies does not lead to unwanted side effects! And it can produce the same kinds of chemical changes for good feelings that ingesting hormones can produce.\n\nFor myself, when I had PMS, I could suddenly become hysterical. Most women know the signs that they are chemically, hormonally, or energetically off, and they can take steps to correct the imbalances using energy interventions. Family members can be supportive as well. David quickly learned his job. If he saw me become hysterical or anxious or extremely emotional, instead of derisively asking, \"Oh, God, is it that time of the month again?!!\" he would lovingly say: \"Here, let me hold your triple warmer points.\" This was a form of first aid (not to mention self-defense) that would immediately calm me. Other steps could follow that would further balance and stabilize.\n\nWhile the biochemistry of PMS is extraordinarily complex, if you approach it energetically, instead of somehow trying to control it chemical reaction by chemical reaction, it becomes manageable. And while the energies in your body can seem as complex as its biochemistry, what you can do to influence your energies is much simpler. Here is how it looks to me.\n\n#  YOUR MONTHLY CYCLE: A PLAY OF MANY ENERGIES\n\nWhen a new female client walks through the door, my eyes see the same things your eyes would see. In addition, I see a luminance surrounding her body, known in many healing traditions as the _aura_ (page 30). If the woman is healthy and hormonally balanced, the energies surrounding her body seem full and billowy, and they go out a long way. If the woman is having a difficult premenstrual time, the aura will be collapsed, dense, and hovering close to her body. The energies that normally move in an easy exchange with the environment have a compressed look. Colors in the aura that normally move from one band to the next are splotchy. They do not flow smoothly. Beyond the surrounding aura, the energies that move _within_ the body also change in their appearance at different times in her cycle.\n\nWhen I look at the energies _in_ a woman as she lies on my table, I might see any of many variations of how her energies are flowing. Emotions show up in the energies, and different emotions show up differently. With depression, the energies may appear to be sinking; with panic, it may seem they are rising, sometimes lookinglike they want to explode. If the woman is exhausted or ill, I might see meridian energy traveling backward, chakras (vortexes of energy) hardly spinning, or perhaps all the energy will be very slow. I may see blockages that need to be freed. Before the treatment is through, I hope to see the meridians streaming in a forward manner, the chakras spinning at a vibrant tempo, and the obstructions unblocked.\n\nHormonal imbalances also reveal themselves in the energies. The energies of a woman who is having difficulties with PMS often appear sluggish. There may be places where there is almost no movement, where energies are not reaching their natural destination. Sometimes still energies will move suddenly or change direction or appear chaotic. They may seem compressed and wanting to burst.\n\nSeeing such patterns in a woman's energies guides me. If we can bring her energies into balance and harmony, the woman is going to have a much easier premenstrual time. The energies will not, of course, look like her energies at non-premenstrual times, but they will not be as collapsed, compressed, or splotchy, and the flow among the colors will be smoother. Bringing about these changes in the energies will adjust the hormones. _Chemistry follows energy!_ Simple energy procedures harmonize hormones!\n\nThe techniques I use are physical and they are simple. You do not need to be able to see the energies for the techniques to be helpful. They work whether you see the energies or not, and even whether or not you believe the techniques will work. That you do not need to be able to see energy for energy techniques to be effective is, in fact, a core premise on which my approach to energy medicine is built. And I've seen this premise demonstrated again and again over the past 30 years.\n\n#  ENERGY TECHNIQUES FOR MANAGING PMS\n\nAny of the individual exercises in the Five-Minute Daily Energy Routine can serve as a kind of first aid to give you a boost during PMS. If you suddenly find yourself coming unglued, you can scream, you can hide, or you can do a Hook-up (page 62). If you feel vulnerable but are required to be with other people, do the Zip-up (page 61). The Three Thumps (page 53) instantly recharges your batteries. I am certain that the Wayne Cook Posture (page 56) helped save my marriage a number of times by bringing me back to myself and my senses _before_ I called the lawyer. It is one of my all-time favorite unscramblers. Use it whenever you are feeling stressed or overwhelmed. Another technique that is invaluable when you can't think clearly, your physical coordination is off, or you find yourself spiraling into depression (any of which is a sign that you are homolateral) is the Homolateral Crossover (page 67). In fact, I know of nothing else that works as well when your energies are not crossing over from one brain hemisphere to the other. If you can't get yourself into gear, if nothing else seems to be working, assume that you need it.\n\nThe energy techniques for managing PMS are grouped as follows: (1) Maintaining good energies throughout the month; (2) Calming steps to take at the first signs of PMS; (3) Adding a \"PMS Module\" to your Daily Energy Routine; (4) Techniques for working with physical discomfort and pain; (5) First aid for the emotional crazies; and (6) Energy testing for hormonal and other supplements.\n\n##  **1. MAINTAINING GOOD ENERGIES THROUGHOUT THE MONTH**\n\nEnergy habits are the established ways your energies flow, and they may be for better or for worse. If you can institute positive energy habits in your body during times that are less stressful, your body will be able to cope more effectively when stress hits. Keep these principles from Chapter 2 in mind:\n\n\u2022 Stretching makes space in the body so your energies can move in their most natural way.\n\n\u2022 Clearing toxins supports the healthy flow of your body's energies.\n\n\u2022 Energies need to move in crossover patterns.\n\n\u2022 A set of simple exercises done on a daily basis can stimulate all your energy systems, which in turn brings a better vibration to your organs, your immune system, and your spirit.\n\nBy doing the Five-Minute Daily Energy Routine daily and combining it as needed with other stretching and crossover exercises from Chapter 2, you will be helping your body establish healthy energy habits that will better support you throughout your monthly cycle. Beyond that, time will show that these healthy energy habits become embedded\u2014so you are giving yourself a gift that keeps on giving.\n\nMost women intuitively know that they are best served by going internal during the premenstrual part of their cycle. Cooperate! Do not schedule nonessential social activities. Plan for moon-hut time. Keep life as calm and simple as possible. And be prepared. If the miseries of PMS occur, be ready to swiftly shift your energies.The following techniques show you how. I suggest that you familiarize yourself with them during the time of the month when you are not premenstrual so that you can use them with a minimum of effort. If you learn them during calmer times, you are much more likely to use them when you need them.\n\nThe techniques in the remainder of this chapter, and in the remainder of this book for that matter, can seem like so many more chores added to your day. The thing is, they work. And they do not take long. They take less time than a brisk walk or most people's driving time to the gym. In these ways, they are very efficient. Also, these exercises are much easier to do than they seem at first when read from the printed page. And you need to do only a portion of them to get helped. I try to give you good guidance in determining which ones are going to be the most useful for you.\n\nIf you treat this book as a guide, you will have at your disposal powerful, simple tools for being more in charge of your life, more at peace, and more able to deal with life's challenges. I hope you will come along with me on this journey and give the techniques a fair try. So begin with the routines in Chapter 2, and then when you notice the first symptoms of PMS, take the following definitive steps.\n\n##  **2. CALMING STEPS TO TAKE AT THE FIRST SIGNS OF PMS**\n\nAt the first bout of irritability, lethargy, bloating, or other sign of PMS, carve out some time and make yourself very comfortable. My preference is to be in a bathtub or hot tub. Immersing our bodies in warm water may be the closest experience we have to returning to the womb. It shifts your energy field so that your body is more receptive to healing and energy exercises that calm your hormones. Once you have relaxed a bit (lovely soothing music in the background enhances the experience), lie back in the tub, get completely comfortable, and do the following:\n\n**A. Sedating the Triple Warmer with the Smoothie** (time\u201420 seconds)\n\nYou were introduced to this procedure in the previous chapter. When you are stressed, whether due to external circumstances or bodily disruptions such as those that are part of PMS, triple warmer steps into action. Triple warmer, you may recall, is designed to keep you safe by invoking the immune or fight-or-flight response and by maintaining survival habits. But because it evolved for a world that no longer exists, it is often on overdrive. During PMS, it is almost always valuable to calm the triple warmer energies that move through your body. This relaxes your entire energy system, releases tension, and reduces fear and anxiety. Fortunately, the Triple Warmer Smoothie (page 107) is a quick and simple way to calm triple warmer, and sometimes it is all that is needed. The Triple Warmer Tap (page 108) and the Triple Warmer\/Spleen Hug (page 110) are also simple and effective. A more decisive though more time-consuming approach is to hold the Triple Warmer Sedating Points (page 110).\n\n**B. Holding the Spleen Meridian Strengthening Points**   \n(time\u2014about 5 minutes)\n\nSpleen and triple warmer meridians exchange energy in a teeter-totter kind of relationship. When triple warmer is activated in PMS (or in any other situation), it springs into action and drains energy from spleen to nourish itself. Spleen meridian often gets so depleted that your immune system and your ability to adapt become compromised. When spleen meridian energy is low, your body insists that you slow down. If you go with this slower flow, you can do much to restore balance. By _deliberately_ slowing down in the bathtub or another comfortable setting, and then calming triple warmer and strengthening spleen meridian, you are telling your body that you are attuning yourself to the monthly shift that is about to occur. You are speaking to your body in its native language\u2014 _energy._ This prepares you to usher in this period attuned to the needs of your body, mind, and spirit. Strengthening spleen meridian also happens to help with the heavy, weak, and aching legs that are often a symptom of PMS.\n\nMeridians can be strengthened in several ways. One of the most powerful is by holding acupressure points that affect the meridian. The procedure is similar to the way you sedated triple warmer in the previous chapter. You simply place your fingers on the points, hold for a couple of minutes, and then hold a second set of points. By finding a comfortable position in the tub, on the couch, or in another setting, this can be an enjoyable, centering, meditative time. While that is my favorite way to do it, you can also multitask. Holding the points also works while you are watching television, talking on the speakerphone, or chatting with a friend. You can even get your friend to hold the points for you (adjust the instructions accordingly).\n\n_Figure 4-1_ SPLEEN MERIDIAN STRENGTHENING POINTS\n\nTo strengthen spleen meridian, simultaneously place your fingers on Speen 2 and Heart 8 on either side of your body and hold for about two minutes (see Figure 4-1a). You can do this by placing the middle finger of one hand on Heart 8 and the middle finger of the other hand on Spleen 2. Repeat on the other side. For the second set of points (Figure 4-1b), use the thumb and pointer fingers to hold Liver 1 and Spleen 1. You can hold these points on both feet at the same time. Hold for about a minute.\n\n##  **3. ADDING A \"PMS MODULE\" TO YOUR DAILY ENERGY ROUTINE**\n\nTo keep your energies humming during your premenstrual time, you might want to do the Five-Minute Daily Energy Routine twice, once early in your day and again in the afternoon or evening, particularly when you are dragging. I suggest adding the following techniques to the routine when you are premenstrual and continuing to include them until two or three days after your period begins. Insert the following five techniques right after the Lymphatic Massage (page 59), and then end the routine as usual, with the Zip-up and Hook-up.\n\n**A. Connecting Heaven and Earth** (time\u2014about 2 minutes)\n\nThis is an ancient exercise that women have used in many cultures throughout time. See instructions on page 44.\n\n_Figure 4-2_ THE TRIPLE AXIS HOLD\n\n**B. The Triple Axis Hold**   \n(time\u2014about 30 seconds)\n\nThe Triple Axis Hold is a quick and simple technique that brings immediate comfort. It calms and balances your hormones, and benefits the pituitary and hypothalamus glands in particular:\n\n1. Place the heel of either hand on your forehead, then take your middle finger and rest it on the top of your head.\n\n2. Make a three-finger cluster with the thumb, index, and third fingers of your other hand and place it just beyond the curve of your head (see Figure 4-2).\n\n3. Hold during three deep breaths.\n\n**C. Abdominal Stretch** (time\u2014about 30 seconds)\n\nYour abdomen, with all the organs in it, can become full and tight before your period, interfering with the normal movement of energy. This bloating causes a feeling in the middle of your torso like a dam being backed up. The Abdominal Stretch opens the spaces, relieves some of the pressure, and allows fresh energy to move through the area, giving you a reprieve from that unpleasant stuffed-bird feeling.\n\n1. Stand facing the back of a chair or other piece of furniture that is a similar height, placing your hands on top of it. Straighten your arms. Alternatively, reach to a ledge about the height of your shoulders and straighten your arms, standing now an arms distance from the ledge.\n\n2. Look up, and while stretching your neck gently backward, swing your right leg backward, keeping it straight at the knee. Feel the stretches in your neck and your abdomen (see Figure 4-3).\n\n3. Return to center.\n\n4. Repeat with your left leg.\n\n5. This will feel good. Do as many as you wish.\n\n_Figure 4-3_ THE ABDOMINAL STRETCH\n\nD. The Sideways Stretch   \n(time\u2014less than a minute)\n\nThis exercise also stretches the ligaments in your abdomen and is particularly beneficial for the energetic functioning of your heart, liver, spleen, and gallbladder, each of which may become stressed during your premenstrual time. Like the Abdominal Stretch, it also helps counteract bloating. To do the Sideways Stretch:\n\n1. Stand with your hands on your thighs and take a deep breath, inhaling through your nose and exhaling through your mouth.\n\n2. Swing your arms out to the side and above your head on a deep in-breath.\n\n3. Grab your right wrist with your left hand, and on an out-breath, bend to the left while pulling your right arm with your left hand (see Figure 4-4).\n\n4. Return to center on an in-breath. Repeat on the other side. Do at least three rounds.\n\n5. Finally, with palms facing outward and hands over your head, circle your arms down to your sides.\n\n_Figure 4-4_ THE SIDEWAYS STRETCH\n\n**E. Spleen Meridian Flush and Tap** (time\u2014less than a minute)\n\nIf you feel weak during your premenstrual time, spleen meridian almost always needs to be strengthened. A very good way to strengthen a meridian is by flushing it, which is to trace it backward one time and forward three times, kind of like backwashing a filter, pulling out stagnant energies. You can brush the meridian lightly with your hands or keep your hands a couple of inches above the meridian, using a slow, deliberate motion. You will be following the meridian pathway shown in Figure 4-5. Throughout the exercise, move your hands slowly and deliberately. To flush and tap spleen meridian:\n\n1. Stand with your hands flat on the sides of your body at your waist, fingers pointing straight down.\n\n2. Take a deep in-breath, and draw your fingers up to your armpits.\n\n3. Let your breath out as you bring your hands down your sides, again flattening them. When you reach your waist, move your hands to the front of your hipbones, then continue down the inside of your legs with fingers spread, hands flat.\n\n4. Go over your anklebones and continue forward along the insides of the feet and off the big toes.\n\n5. Take a deep in-breath and retrace these steps in the other direction, starting at the insides of the feet at the big toes and, with your hands flat, trace straight up the insides of the legs, flaring out at the hips, to the sides of the body, up to the armpits, and then down to the bottom of the rib cage. Retrace in this direction two more times.\n\n6. Tap the spleen neurolymphatic reflex points beneath the breasts, in line with your nipples, and down one rib for about ten seconds as you breathe deeply (see Figure 2-5, page 54).\n\n7. Slide your fingers to the spleen acupressure points on the side of your body and tap for ten more seconds (see Figure 4-5).\n\n_Figure 4-5_ SPLEEN FLUSH AND TAP\n\n##  **4. TECHNIQUES FOR WORKING WITH PHYSICAL DISCOMFORT AND PAIN**\n\nAmong the most common physical symptoms of PMS are bloating, breast tenderness, cramps, back pain, and groin pain. For cramps and many types of pain, people often instinctively circle their hand over the area of discomfort. There is good reason to do this. Your hand can pull blocked or stagnant energy out of an area. Circling in a counterclockwise motion tends to draw out disruptive energy; circling in a clockwise motion tends to stabilize the energies that are there. Your hand does not even need to make physical contact with your body. Circle a couple of inches above the area of discomfort in a counterclockwise motion and notice if you feel your hand pulling energy off your body. Shake your hand off from time to time, as if sending the excess energy into the ground. Once you are feeling even a little bit better, you can stabilize the improvement by circling in a clockwise direction.\n\n**A. Stomach Meridian Flush** (time\u2014about a minute)\n\nAnother good technique for PMS discomfort is to flush the stomach meridian. Rather than taking energy out of the specific area of discomfort (as with circling), tracing the stomach meridian backward (the first part of flushing the meridian) is like cleansing a pathway that keeps the energies involved with PMS flowing. It immediately relaxes everything in the area of your stomach, so it is a natural response to cramps or stomachaches. But it can also help with difficult emotions, particularly heartache or other emotional pain whose physical counterpart is in your stomach or chest.\n\nAs with flushing the spleen meridian, you can brush the meridian lightly with your hands or keep your hands a couple of inches above the meridian, using a slow, deliberate motion. Begin by tracing the stomach meridian backward, using both hands to follow the path of the meridian in the opposite direction of its normal flow (see Figure 4-6):\n\n1. Starting with your fingers at the second toe of each foot, travel to your ankles and straight up your body, slightly toward the outsides of your legs.\n\n2. Come straight up the front of your body, over your ovaries, over your breasts, straight up your neck, over your jawbone, and up through your eyes, to your hairline.\n\n_Figure 4-6_ STOMACH MERIDIAN FLUSH\n\n3. Then circle down, following the outline of your face, down to your jawbone, and finally come straight up to the points beneath your eyes.\n\n4. Shake the energies off your hands. Trace this backward pathway again one or two times if you wish.\n\n5. Now trace the stomach meridian in the opposite (forward) direction (again, see Figure 4-6).\n\n6. Trace the meridian forward a total of three times. End by shaking the energies off your hands.\n\n**B. Additional procedures to address specific types of PMS discomfort**\n\n_Figure 4-7_ MASSAGING SPLEEN 9\n\n**Bloating.** The best exercises I know for bloating are the Abdominal Stretch and the Sideways Stretch, both suggested as additions to your daily routine. Any stretch that involves the stomach opens space for energy to move. You can also massage an acupressure point called Spleen 9 on the side of the leg beneath the knee, in the depression behind the tibia (see Figure 4-7) for about ten seconds in a counterclockwise direction (as if your knee were the clock), and then tap it for five seconds. This is a traditional treatment for edema, though it is not suggested for women who are pregnant.\n\n**Breast Swelling and Tenderness.** PMS pain and swelling are signs of hormonal changes that block the movement of energy, causing excess energy to accumulate and become dense and sluggish. Your stomach meridian passes over your breasts, down through your abdomen, and over your ovaries. During PMS, the stomach meridian is often overtaxed, trying to manage the complex changes occurring in these areas. It often accumulates excess energy. When your breasts swell or are tender, releasing some of that energy can provide relief. Begin by tracing the stomach meridian backward as described in the discussion of the Stomach MeridianFlush. The sedating stomach meridian can further reduce the pressure in these areas, giving relief to cramps as well as breast tenderness, and pain in the ovaries. Sedating the meridian is like opening a tap that allows this energy to drain and allows healthy circulation to be restored. Instructions for sedating the stomach meridian, and several other meridians involved with PMS, are presented after the following discussion of cramps, back pain, groin pain, late periods, and sedating the meridians.\n\n**Cramps.** Sedating the stomach meridian also helps with cramps. A procedure that specifically focuses on cramps is to massage the fourth acupressure point on the spleen meridian. Relax, position your feet so you can comfortably reach both, and simultaneously massage the points that are on the inside of each foot, halfway between your toe and your heel (Spleen 4, see Figure 4-8). Push up against the bone and massage in a counterclockwise direction for at least ten seconds and up to a minute. Then tap the points while taking two deep breaths. Meanwhile, massaging Liver 4 (also shown in Figure 4-8) relieves pressure from the uterus and may also provide relief from cramps. Again, massage in a counterclockwise direction and then tap the points during a couple of deep breaths.\n\n_Figure 4-8_ CRAMP RELIEVER\n\n**Back Pain.** In energy medicine, pain relief is based on addressing the underlying problem rather than masking the pain, as with \"painkillers.\" Instead of just blocking out the symptoms, energy medicine techniques for alleviating pain correct, at the level of the body's energies, the reasons for the pain. PMS can cause pain in a sturdy back because the kidneys have to filter more blood and hormones during this time and can become full and stressed, leading to pain in the surrounding area. Sedating the kidney meridian removes tense energy from the kidneys, relaxes them, and provides relief. If you have upper back pain together with cramps, sedating the small intestine meridian, which governs the entire abdominal cavity, can be helpful. This can relieve cramps as well as back tension. When back pain is located in and around the spine at the waist, the large intestine meridian is often involved, and sedating the large intestine meridian can provide relief.\n\n**Groin Pain.** During PMS, as the uterine lining thickens, the weight of the uterus increases. This places tension on the groin ligaments, which may tighten in order to give the uterus better support but can result in groin pain. The simplest technique is to lie down and place your fingertips right on the groin lines where your legs attach to your body. Relax your hands so they are lying flat. Your hands are bringing good energy to your ovaries, the sides of your uterus, and two valves (the ileocecal and Houston's valves) that often lose their rhythm during PMS, causing constipation. Hold this position for about two minutes.\n\nAnother approach for relieving groin pain is to work with the liver meridian. This meridian controls the tension in the ligaments of the body, and the liver itself has the job of dispelling the hormones that have been released into the system and need to be eliminated. So the liver meridian is also working overtime during PMS. Sedating the liver meridian (described on page 146) can relax unnecessary tension in the ligaments, help the system restabilize itself, and help the liver process hormones that have served their function.\n\n**If Your Period Is Late.** You can often get your period to start by relaxing the gluteal muscles and the entire pelvic area. Sedating a meridian that controls the circulation system, the pericardium, and the gluteal muscles\u2014called the circulation-sex meridian\u2014can bring on a late period with no risk of aborting a fetus. Another reason keeping this meridian in a good flow is so important is that it works synergistically with the spleen, stomach, and triple warmer meridians.\n\n##  **Sedating the Meridians Involved with PMS**\n\nMany of the uncomfortable symptoms associated with PMS are based on excess energy rather than not enough energy. This may seem counterintuitive because you may be feeling that you need more energy, which is also true. Here is how it works. When too much energy accumulates in certain areas of the body, the normal flow of energy is blocked, causing weariness and pain. When there is too much energy in one place, the flow stops, so there is too little energy getting to other areas. The reason too much energy may accumulate in an area is that when a meridian is being overworked (as happens with PMS), the meridian may overcompensate, or other meridians may feed it extra energy in an effort to support it.\n\nA powerful way to help get blocked energy flowing again is to \"sedate\" the meridian pathway in which the clogged energy has accumulated. This makes it possible for the energy to start moving again and can provide tremendous relief while allowing your whole system to get back into balance. The procedure is similar for each of the meridians, though different acupressure points are involved with each. In Chapter 3, you were shown how to sedate the triple warmer. Again, when you sedate a meridian, you place your fingers on the designated acupressure points, hold for a couple of minutes (it is a firm touch but does not involve deep pressure or massage), and then hold a second set of points for about a minute. The total time involved is three to six minutes, depending on whether it is physically possible to hold both sides simultaneously. Be sure when holding the points to find a comfortable position so you are not taxing your body. The meridians most involved with the _physical_ symptoms of PMS include the stomach, small intestine, kidney, large intestine, liver, and circulation-sex meridians. The earlier discussion of specific PMS symptoms tells you when it might be valuable to sedate each.\n\n**Sedating the Stomach Meridian** (breast tenderness and cramps; time\u2014 about five minutes)\n\nYou can work from the figures, holding the first set of points (Figure 4-9a) for two minutes on one side, then the other side; and the second set of points (Figure 4-9b) simultaneously, for about a minute. Here are the instructions spelled out in greater detail:\n\n1. Take the thumb and middle finger of your right hand and hold the end of your right second toe (this covers the Stomach 45 acupressure point).\n\n2. At the same time, hold the sides of the end of the index finger of your right hand with the thumb and index finger of your left hand\u2014this covers Large Intestine 1 (see Figure 4-9a).\n\n3. Hold for about two minutes. Be sure you are physically comfortable.\n\n4. Repeat on the other side.\n\n5. Bring your feet into a position where you can comfortably reach both with your hands.\n\n6. Move your thumbs along the valley between the ligaments of the second toe and third toe of each foot until you are about an inch back from where your toes begin (Stomach 43).\n\n7. Take the index fingers of each hand and place them on the ligament that goes to your fourth toes, about an inch farther back (toward your ankle) from the position of your thumb (Gallbadder 41) [see Figure 4-9b].\n\n8. Hold for about 60 seconds.\n\n_Figure 4-9_ STOMACH MERIDIAN SEDATING POINTS\n\n**Sedating the Small Intestine Meridian** (pain in upper back combined with cramps; time\u20146 minutes)\n\nSimultaneously place the middle finger of one hand on Stomach 36 (a hand's width beneath your knee, in line with your second toe) and the middle finger of the other hand on Small Intestine 8 (about three fingers' width beneath your elbow, in line with the little finger), on either side of your body, and hold for about two minutes (see Figure 4-10a). Repeat on the other side. Then place the middle finger of one hand on Small Intestine 2 (beneath the little finger, between the base of the finger and the wrist) and the middle finger of the other hand on Bladder 66 (in the indentation at the outside bottom of the little toe) and hold for about a minute (see Figure 4-10b). Repeat on the other side.\n\n**Sedating the Kidney Meridian** (back pain; time\u2014about 3 minutes)\n\nIt is possible to hold both sets of points simultaneously with this one.\n\n1. To do both sides at once, sit in a chair and pull your feet up onto the chair. Place the middle finger of each hand between the second and third toes and bring these fingers over the ball of the foot, resting them on the edge of the ball that is away from your toes (Kidney 1). You can hold either foot with your right or left hand, whichever is easier.\n\n2. Place your thumbs on the base of the opposite big toenail, in the corner closest to the second toe (Liver 1) [see Figure 4-11a].\n\n3. Be sure you are comfortable. Hold for about two minutes. If you are pregnant, however, hold for only about 30 seconds.\n\n4. Place the middle finger of your right hand on the outside of your right big toe and slide just over the \"bunion bone\" to the indentation (Spleen 3). Find the same point on your left foot.\n\n5. Place your thumbs in the indent just behind your inner anklebones (Kidney 3) [see Figure 4-11b].\n\n6. Hold about 60 seconds.\n\n_Figure 4-10_ SMALL INTESTINE MERIDIAN SEDATING POINTS\n\n**Sedating the Large Intestine Meridian** (pain in lower back; time\u2014 about 6 minutes)\n\nFigure 4-12a shows the first pair of points, Large Intestine 2 (inside the base of the index finger) and Bladder 66 (outside the base of the little toe, in the indentation). Hold for about two minutes on each side. Figure 4-12b shows the second pair, Large Intestine 5 (on top the of wrist in line with the index finger) and Small Intestine 5 (on top of the wrist, in line with the little finger). Hold for about one minute on each side.\n\n**Sedating the Liver Meridian** (groin pain; time\u2014about 6 minutes)\n\nSimultaneously place the middle finger of one hand on Liver 2 and the middle finger of the other hand on Heart 8 (see Figure 4-13a), on either side of your body, and hold for about two minutes. Repeat on the other side. Then place the middle finger of one hand on Liver 4 and the middle finger of the other hand on Lung 8 (see Figure 4-13b) and hold for about a minute. Repeat on the other side.\n\n_Figure 4-11_ KIDNEY MERIDIAN SEDATING POINTS\n\n_Figure 4-12_ LARGE INTESTINE MERIDIAN SEDATING POINTS\n\n_Figure 4-13_ LIVER MERIDIAN SEDATING POINTS\n\n**Sedating the Circulation-Sex Meridian** (late period; time\u2014about 6 minutes)\n\nSimultaneously place the middle finger of one hand on Spleen 3 (inside of the foot, just before the lump that leads to the big toe) and the middle finger of the other hand on Circulation-Sex 7 (middle of the wrist, in line with the middle finger), on either side of your body, and hold for about two minutes (see Figure 4-14a). Repeat on the other side. Then place the middle finger or thumb of one hand on Kidney 10 (on the inside of the leg where the knee bends) and the middle finger of the other hand on Circulation-Sex 3 (on the inside of the elbow, in the middle) and hold for about a minute (see Figure 4-14b). Repeat on the other side.\n\n##   **5. FIRST AID FOR THE EMOTIONAL CRAZIES**\n\nEnergy medicine offers a number of techniques worth having at your fingertips during any emotional roller coaster, and PMS can be the Giant Dipper of emotionalroller coasters. I can assure you of several things about energy medicine methods for emotional turmoil. They are quick. They are powerful. You can learn them in advance so you have instant access. You will feel a shift in the energy. And you will not remember to use them when you need them. Okay, prove me wrong!\n\n_Figure 4-14_ CIRCULATION-SEX MERIDIAN SEDATING POINTS\n\nThree principles to keep in mind are stretch, tap, and cross and hold. If you don't remember anything else, breathe deeply as you: (1) stretch your midsection, (2) tap anywhere on your face, head, and body, and (3) cross your arms and hold your shoulders. You have also already learned several great emotional first-aid techniques: the Hook-up (page 62), Zip-up (page 61), Triple Warmer Smoothie (page 107), and Wayne Cook Posture (page 56). Four others you can experiment with are Charging Your Batteries, Expelling the Venom, the Neurovascular Hold, and the Darth Vader Breath.\n\n**A. Charging Your Batteries** (time\u2014a little more than a minute)\n\nWhen you feel exhausted, on edge, or ready to burst into tears, this technique can bring relaxation, support, and strength to the liver, spleen, bladder, and kidney meridians, each of which can be highly stressed when you have PMS. If done regularly, it will help keep your hormones balanced and bring greater ease to your cycle.\n\n1. Sitting down with your left foot bent in your lap, squeeze the sides of your left foot (see Figure 4-15a).\n\n2. Begin to drag the fingers of your left hand with pressure up the inside of your leg.\n\n3. Draw your hands up your torso, with pressure, your left hand moving up the left side of your body and your right hand moving up the left side of the midline at the front of your body.\n\n4. When your right hand reaches underneath your collar bone (K-27) and your left hand reaches to the side of your breasts on your rib cage (Spleen 21), tap both areas 10 or 12 times while breathing deeply (see Figure 4-15b).\n\n5. Repeat on the other side.\n\n_Figure 4-15_ CHARGING YOUR BATTERIES\n\n**B. Expelling the Venom** (time\u2014less than a minute)\n\nThe first two parts of the Blow-out\/Zip-up\/Hook-in technique from Chapter 2 (page 74) are so powerful that in tandem, I call them Expelling the Venom. They move stress and rage out of your body, unblock congested energies, and free the spirit:\n\n1. Stand up straight. Put your arms out in front of you, bend your elbows slightly, make fists with the insides of your wrists facing up, and take a very full breath (see Figure 2-16, page 75).\n\n2. Swing your arms behind you and up over your head. Hold here for a moment.\n\n3. Reach way up, turn your fists so your fisted fingers are facing each other, and rush your arms down the front of your body as you emphatically release your fists. Let out your breath and your emotions with a \"whooooosh\" sound or any other sounds that come naturally.\n\n4. Repeat several times. This will feel good. The last time, bring your arms down in a slow and controlled manner, blowing your breath out of your mouth as you go.\n\n5. End with a Zip-up. With a deep breath, place your hands at your pubic bone (see Figure 2-10, page 61) and pull them straight up the midline of your body, up in front of your face. Stretch high above your head, and turn your palms outward. Slowly releasing your breath, extend your arms out to the sides as far as they will go and back down to your pubic bone.\n\n**C. The Neurovascular Hold** (time\u20141 to 3 minutes)\n\nWhen you are stressed or feeling blue, this technique releases tension from your body as well as from your mind. Your body still reacts to threat and stress as our ancestors' bodies did millions of years ago. Running away or fighting required that the blood rush away from the brain and into the body to support quick action. You are designed to _act_ in an emergency, not think. The Neurovascular Hold, already mentioned in the previous chapter, interrupts the fight-or-flight response, bringing blood back to your forebrain so you can think more clearly.\n\n1. Sitting or lying down, tune in to a stress you are feeling or focus on a stressful thought, memory, or situation.\n\n2. Place your thumbs at your temples and the pads of your fingers on your forehead, on the bony area above your eyebrows (see Figure 3-1, page 106).\n\n3. Hold here softly for up to three minutes, breathing deeply, in through your nose and out through your mouth.\n\n**Alternative:** Place the palm of one hand across your forehead and the palm of your other hand across the back of your head. Again, hold softly for up to three minutes, breathing deeply.\n\n**D. The Darth Vader Breath** (time\u2014less than a minute)\n\nThis technique calms triple warmer, balances hormones, releases stress, and brings a steadiness to body and mind. The breathing for this technique is very slow and controlled.\n\n1. Stand and empty your breath completely.\n\n2. Breathe in through your mouth, pulling your breath very slowly through the back of your throat, making a raspy \"Darth Vader\" type of sound.\n\n3. Breathe out making the same sound, again very slowly.\n\nWhile some of these methods will work better for you than others, each can be effective. Experiment.\n\n **6. ENERGY TESTING FOR HORMONAL AND OTHER SUPPLEMENTS**\n\nWhile our emphasis has been on techniques for shifting your body's energies, a definition from Chapter 3 provides the context: Energy medicine is the art and science of working with the relationships between energy _and chemistry_ to promote health, vitality, and well-being. Our focus is on energy because, in addition to that being my specialty, it is the area that has received less concrete attention in conventional as well as alternative approaches to helping women with their health concerns. But in practice, energy medicine is very attuned to chemical as well as energetic balances and imbalances. All herbs and medicines, in fact, have energeticproperties. And energy testing (see pages 63-65 and the Appendix) is a tool for determining the types of hormonal, nutritional, or other supplements that might best complement hands-on energy interventions.\n\nIf you know that you always feel better after taking a particular herb to relieve your cramps, there is no need to energy test it. But energy testing can help you decide which of the eight herbs at the health food store is most likely to work for your unique body, complaints, and energy system. Every food, pill, or capsule carries an energy field. Energy testing allows you to determine, before ingesting it, whether the unique vibration of that substance resonates and is compatible with your own body's energies and needs. Since chemistry follows energy, that will also give you an indication of how your body will respond to the substance.\n\nFor myself and my approach with clients, I first try to handle monthly fluctuations using the energy techniques alone because they are less invasive. If you rely on outside sources for a chemical your body produces naturally, your body's ability to make that chemical is often diminished (an important exception is progesterone).\n\nEnergy techniques certainly work independently of any other interventions. But there are times where herbs and other supplements that are used for PMS can complement energy work, and I fully support that combination when it is necessary.\n\nEven if you do not take it upon yourself to learn energy testing, you can still experiment with the substances described here on a trial-and-error basis. I have personally learned a great deal about my own body and what it needs through such experimentation, particularly before I was introduced to energy testing in the late 1970s. Please check the Internet for contraindications and discuss with a competent practitioner before ingesting any substance about which you are not well informed.\n\nIf you have cultivated an ability with energy testing, or have access to a practitioner who can reliably energy test substances you are considering, you will not only depend less on trial and error, you will also be able to figure out in advance the amount you need and when you need it. My body may need a tad of progesterone, but three tads may throw off my progesterone-estrogen balance. The next day I may not need any, and the day after that I may need three tads in the morning and one tad in the afternoon. I know of no better tool than energy testing for establishing a harmony between your body's fluctuating chemistry and what you offer it to maintain peak functioning.\n\nMonthly changes in estrogen and progesterone balances are at the core of the most severe PMS symptoms. Estrogen levels are at their peak right up to ovulation, when the hormone is telling you to find a suitor and get on with it. With ovulation, progesterone levels begin to rise. If you did the first part right, according to nature's calling, then your body needs to start preparing for a baby. Progesterone remodels the uterus to carry the baby. Whereas estrogen excites you and makes you more sexually active and interested, progesterone calms you. It makes you more tranquil. In fact, one of its primary functions besides strengthening the uterus is to relax the central nervous system. But if you are in the postovulation part of your cycle and not producing enough progesterone, there is a deep-body anxiety that your uterus is not able to build the lining it needs for the anticipated baby\u2014something is not right! Beyond this, your entire nervous system is out of balance because the expected progesterone is not being produced.\n\nImproper progesterone levels may be due to many causes. Nature's design lottery simply edged some women toward more estrogen and others toward more progesterone. Many women begin to have a decline in their natural progesterone levels by their mid- to late thirties. If PMS is, for you, a time of increased anxiousness, hysteria, or reactivity, you may not be producing as much progesterone as your body needs. While some of the energy exercises in this chapter may lead to a better balance, you might also consider supplementing what your body is producing. Other signs of low progesterone for menstruating women, beyond irritability and extreme mood swings, include irregular menstrual cycles, heavy menstrual bleeding, bloating, back pain, edema, fatigue, excessive food cravings, endometriosis, uterine fibroids, and tender breasts.\n\nWhile the most terrible parts of my own PMS were emotional, my younger daughter, Dondi, was physically debilitated every month, going into intense sweats and shakes, with terrible cramps and pain. And she never knew when her period would begin, so she could not plan for it, which proved costly for her in her career as an actress. Whereas progesterone was my wonder drug, it did nothing for her. The surprise solution was when she started taking birth control pills, which increased her estrogen levels. Suddenly and decisively, her period was regular and her symptoms disappeared.\n\nFor supplemental progesterone, the simplest way to introduce it into your body is to take Mexican or Siberian **yam root** capsules. Some people also use yam root or other progesterone creams. All of this is available over the counter. The strongest forms of progesterone are by prescription only (be sure you get natural progesterone rather than progestin, which is synthetic and more likely to be harmful). Natural progesterone may be taken as a suppository or orally in capsules, or you can punch a hole in the capsule and spread small amounts on your inner wrist or abdomen, allowing you to completely control the dosage. Your skin will absorb it directly into your bloodstream, whereas if you take it orally, your liver needs to process it. From yam root to prescription progesterone, you can experiment and see what happens, or you can energy test to find out which form and how much your body needs day by day. Sometimes that need fluctuates hour by hour. And in any case, please be careful not to make the error I made of seeing progesterone as a panacea and using it so much that it caused harm (see page 127). Possible symptoms of excess progesterone include irritability, dullness, depression, diminished libido, uterine cramping, headaches, and insomnia.\n\nBefore I found yam root, **vitamin B** **6** was the only substance that really helped regulate my extreme PMS mood swings. B6 calms the nervous system. I took it in higher doses than are generally recommended, but again you can energy test. I use **B complex,** which helps process the hormonal fallout of the stresses of daily life, along with extra B6. **Potassium** , or sometimes a full-spectrum mineral capsule, helps normalize blood pressure and maintain electrolyte balances, which are also important for the nervous system. **Magnesium** is a mineral that is often deficient in women who have PMS. Magnesium relaxes the muscles, so it can help with numerous symptoms, from abdominal and leg cramps to headaches to constipation. Another favorite of mine was **chromium,** which helps balance blood sugar levels. Many women have sugar and chocolate cravings when they have PMS. Chocolate cravings often reflect the body's need for magnesium; sweet cravings, its need for chromium.\n\nMany women experience a weakness in their legs during their premenstrual time. An herb called **butcher's broom** was a godsend to me when my legs would get weak. I could pop it before I danced or climbed in our Oregon mountains. It allowed me to use my legs in ways I couldn't have imagined while they were feeling weak and achy. The derivation of the name butcher's broom is interesting. During the plague in Europe, butchers were not afflicted. It turned out that butchers made special brooms of a shrubby evergreen plant, which absorbed into their hands (particularly where they had cuts), and this plant turned out to have strong medicinal properties. Butcher's broom works by attaching to the collagen and strengthening the blood vessels, which helps keep blood circulating throughout the body.\n\nOther herbs that are frequently recommended for PMS include motherwort, dandelion root, dong quai, and ginkgo biloba. You can experiment with these or energy test, and you can learn a lot more about each of them on the Internet. Briefly, **motherwort** is said to calm the emotions and bring inner security. **Dandelion root** is used to strengthen the spleen and liver meridians as well as to help with water retention. **Dong quai** balances estrogen and progesterone levels for some women, though it always made me worse, another reason for energy testing. **Ginkgo biloba** is used to help with memory, brain fog, and blood circulation\u2014I can't recall if it worked for me. Increasing the **fiber** in your food can help keep your digestive system moving during sluggish times. **Full-spectrum lights** give some women an energy boost and that uplifts their mood.\n\nYour monthly cycle, however challenging, is a truly miraculous process. My hope is that the simple energy techniques described in this chapter will give you tools for meeting those challenges more effectively, and that they will help you tap more readily into the sources of deep wisdom that can be accessed with every cycle of the moon.\n\n_Chapter 5_\n\nSexuality, Fertility, Pregnancy, and Birth\n\n_For centuries, women stood in their caves stirring the soup with one hand, bouncing a baby on the hip, and kicking the woolly mammoth out the door with the other foot. Women have, over the centuries, developed a brain structure and style of thinking that is characteristically different from men._\n\n\u2014 JEAN HOUSTON, PH.D.\n\nA human, according to a somewhat reductionistic accounting, is \"a gene's way of making another gene.\" We experience the process more lavishly. Our sexuality is juicy, ecstatic, and compelling; our fertility, the pinnacle of potential; pregnancy, the journey from a microdot to a miracle; birth, the exemplar of manifestation.\n\nOur bodies know the plan. Our deepest biological programming ushers us through every stage of reproduction. Yet even here, the modern world has turned the playing field upside down. The sweetness and passion of sexuality\u2014once a private matter between two people involving innocent responses of soul, heart, and hormones\u2014have been distilled and injected into the ever-present media images that shape our self-concept and our images of love and sex. Our fertility is challenged by the preponderance of artificial hormones in our foods and electromagnetic pollution in the air. The effects of whatever may be unhealthy in our complex lifestyles are magnified during pregnancy. Even our ways of giving birth have been divorced from our natural rhythms and instincts.\n\nEnergy medicine has much to offer in each of these arenas. By bringing you into better communication with your energies, your body\u2014the world's foremost expert about your sexuality, fertility, pregnancy, and birthing\u2014is able to talk to you in its own language. Besides helping you re-embrace your natural instincts, energy medicine provides tools for discovering what may have gone awry in any of these areas and supplying effective corrections. These tools can also be used for creating an atmosphere\u2014an energetic environment within your body\u2014that will help you flourish in every aspect of reproduction. This chapter offers basic concepts and methods in each area.\n\n#  **Sexuality**\n\nI am part Cherokee. Cherokee grandmothers teach their granddaughters how to pleasure themselves without a man. This is not a self-indulgent practice, but a vital skill for maintaining their independence and self-reliance until they are ready to be with a mate. The original meaning of the word _virgin,_ in fact, meant \"a woman who owns herself.\" Healthy sexuality for a woman begins by early on getting to know her body, revering it, becoming adept at pleasuring it, cherishing it as a holy vessel that may one day carry a baby, and developing a fierce intent not to violate it. Owning your body as a young woman also means that you do not casually give away your virginity or your sacred sexual involvement as a barter for being liked, accepted, or popular; as a way of feeding a hunger for love or affirmation; or as a collapse into serving another's desires at the expense of your own integrity. All this is as relevant today as it ever was; I am not suggesting a return to prudish times. I am saying, however, that a young woman may pay a great price for having multiple partners. Aside from sexually transmitted diseases and the decreased self-esteem that is often found in girls who do have multiple sexual partners, the body reacts. The risk of cervical cancer in women who have had multiple partners is vastly increased. Some women who took on multiple partners when they were young lose the capacity to derive deep pleasure from sex even after having found a partner they deeply love. This sad fallout from the exuberance of the free-love era has been reported to me so frequently by my female clients that I am surprised it is not spoken of more openly in these times when we are still coming to terms with the freedoms birth control has opened to us.\n\n#  THE CHEROKEE GRANDMOTHER'S SELF-EMPOWERMENT SECRET\n\nWhile I did not grow up in the Cherokee tradition, I was shown the grandmother's self-empowerment secret from an old Cherokee woman when I was in my 30s. I was astonished because I had spontaneously \"discovered\" it when I was 11, as if it were passed along to me as an echo from my ancestors. Back then, I never thought of this as sexual, nor is it particularly focused on the genitals. Rather it was a sense of immense pleasure and wonderful invigoration. I would not feel free to share it had I not discovered it on my own, but since I did, gather round, ladies.\n\nThe Cherokee Grandmother's Self-Empowerment Secret can be approached in two ways, one that is done in total stillness and one that involves movement. The first is the way I discovered it when I was 11. I was lying in the sun on a large boulder behind our house in Ramona, California. It was my favorite place. A deep stillness quieted my body and mind. From this stillness, I began to feel strong pulsations coming up into my body from the earth. It was as if my breath was no longer powered by my own lungs, and instead the earth was breathing me. After a while, the pulsations took the form of a figure-eight energy moving in waves lengthwise through the trunk of my body. Up and over, around and under they went, again and again, in a perfect rhythm. I felt like I was an instrument being played by an exquisite force I couldn't explain. The sensations throughout my body brought a pleasure that satisfied at the deepest level\u2014physically, emotionally, spiritually. It was pure joy, indescribably wonderful to engage these deepest energies that activate sensual pleasure, peace, and restoration.\n\nTo this day, when I want to be nurtured and restored, I go into that deep stillness, and the \"energy eight\" begins to travel. It is not necessary to be on a rock or grass or earth to do this. A bed or floor works just fine. I believe these vital figure-eight energies are always there waiting to be engaged. I just become still, set my intention, and my body seems to do the rest. It does require a degree of surrender. The version that involves movement is a good way to learn the method. My clients have told me that even if it is difficult to attune themselves to these figure-eight energies, or to surrender to them, the physical movements help them experience their latent energies. Eventually the form that does not involve movement may become available to them.\n\nThe old woman taught me the physical movement, but then she made a commentthat the girls eventually find that they don't even have to move their bodies. They discover that the energy wave begins on its own, just as I had initially experienced it. Here is how she taught it.\n\n##  GRANDMOTHER'S SELF-EMPOWERMENT SECRET\n\n1. Begin by lying down on your back and take two deep breaths, in through the nose and out through the mouth. You may prefer to close your eyes. Then let your breath become natural and easy. Force and effort do not work with this method. Surrender does the trick.\n\n2. Focus your attention on your root chakra, the lower part of your pelvis. Sense into the energies that are always there. Imagine energy and oxygen moving in and out of every cell.\n\n3. Circle your hips in an undulating up-and-down figure-eight movement.\n\n4. Bend your legs at your knees, and place your hands on your inner thighs, with the sides of your thumbs lying on the creases where your legs and body join (see Figure 5-1). Move your whole body in an undulating figure-eight motion. The liver, spleen, and kidney meridians (all yin, or female, energies) come up the insides of the legs and cross the thighs. The points your hands touch when you take this position can open sexual energy.\n\n5. Allow the energies that will naturally build to flow with these movements.\n\n6. After a time, lay your legs flat, place your hands at the sides of your body, slow your breath, and let go. The sensation of the figure-eight movement may continue on its own and even build.\n\n_Figure 5-1_ GRANDMOTHER'S SELF-EMPOWERMENT SECRET\n\nThis obviously is not limited to the Cherokee tradition. It is aligned with the natural architecture of a woman's body. When I shared the method in an energy medicine course for women, a belly dancer in the class said, \"That comes out of the Middle East. Traditional belly dancers know this method.\" Another woman said, \"It originated in China. It is a Triple Warmer\/Pericardium technique to get heat and pleasure pumping through your body. It even predates Tantric yoga.\"\n\n#  EVOLUTION'S JOKE, SOCIETY'S DILEMMA\n\nIn our culture, many of us were brought up believing that we should not enjoy our sexuality outside a loving relationship. And there are physiological as well as emotional consequences for women having casual sex with multiple partners. Unfair as it might in some ways seem, our hormones tend to support the traditional model that confines intercourse to a loving, committed relationship. Just to be clear, I am not saying, \"Wait for Mr. Right before engaging\"\u2014this isn't necessarily even my opinion about what is best for all women at all times\u2014but let's look at the hormonal landscape that underlies such choices.\n\nYour sexual and romantic decisions are based less on clear thinking and free will than on your hormones. It is not that rational thought and well-considered choices cannot counter the urges of our hormones. They can, and they need to. But we should also not underestimate the strong magnetic pull our hormones can exert beneath our radar.\n\nFree will is, in terms of evolution, a recent development. Other species do not enjoy anything on the order of our awareness of choice and freedom to choose. Long before nature began its perilous experiment of granting us free will, however, hormones and brain chemicals were doing the job of ensuring that we would perform the behaviors necessary for survival and procreation. Those programs still supplement and sometimes dominate our choices. The fight-or-flight response to threat is a classic example, as are the urgencies to seek food, water, and shelter. These behaviors are biologically programmed, with biology talking to us through the language of pleasure and pain. Most primal here are behaviors that help us survive. Next are behaviors ensuring that we create the next generation.\n\nIn that arena, men and women are wired so differently that we surely must be witnessing nature's sense of humor, penchant for tragic drama, or indifference (though who knows which). Frequent intercourse with multiple partners increases the self-esteem of young men, while withholding intercourse correlates with higher self-esteem in young women. Men's fantasies are more about sex, women's more about romance. Men and women have markedly different biologically driven sexual agendas. Remember the Queen-apparent, the month's single fertile egg, poised to carry the woman's genes forward through impregnation, gestation, and years of nurture and protection. The opportunity might appear some 300 times during the woman's lifetime, with impregnation transforming the drama from a monthly festival into a long-term commitment. For the man to fulfill the minimum requirements in ensuring his contribution to the gene pool, impregnation is the only act in the play that really matters, and he has billions of sperm that are at the ready at any time the opportunity might arise. And his hormones are telling him to seek those opportunities frequently.\n\nEnergetically, arousal for a man quickly ignites the genitals, and the energy moves up toward the heart and warms it. For a woman, arousal starts in the mind or heart and then travels to the genitals. The hormones testosterone, vasopressin, and dopamine are also well recognized as driving male sexual pursuits; however, the role of oxytocin is especially entertaining. Oxytocin, the \"tend and befriend\" hormone, is associated with women bonding with one another and nurturing their families. And indeed, it is often ten times as concentrated in women's bodies as in men's. Except at one particular time\u2014during orgasm. That's when men get to have a deep bonding moment. Their oxytocin jumps to five times normal levels. For a brief instant, nature achieves something close to emotional parity between the sexes. Beyond release, beyond pursuit, attaining this state is part of men's obsession with copulation. The common wisdom that men give love in order to get sex is only half the story\u2014men also want sex in order to feel love. Quickly, however, while the woman is basking in the shared intimacy, the man's oxytocin returns to its normal low levels and she is somewhat deflated when his romantic murmurings are replaced with Monday-night football. While nature hopes that if you can entice him into 20 or 30 of those bonding moments it will begin to involve other parts of his brain and become a habit so he starts to feel love that transcends the exhilaration of the pursuit and stays around to protect your babies, the hormone-driven differences in male and female sexual behavior should not be underestimated. Most specifically, do not assume that the rules generated by his brain are the rules your body should follow.\n\nThis is far more than a mere personal matter, and it extends far beyond sexuality. We live in a culture whose governance conforms much more to the influence of testosterone than of oxytocin. When a woman faces stress, her body produces oxytocin, which buffers the fight-or-flight response and causes her to seek ways of connecting with others. After any immediate physical threat, her focus goes to conferring with other women and tending to the welfare of the children. The more she does this, the more oxytocin she produces. The behavior is self-reinforcing. Beyond that, estrogen enhances the bonding effects of oxytocin. Testosterone, on the other hand, which men produce in high levels when stressed, tends to inhibit the bonding effects of oxytocin. When relationships become strained, most women want to talk; most men turn inward and seek solutions in isolation. Ease of communication was not nature's first priority in fashioning our differences. Culturally, we are challenged to advance oxytocin-based strategies to our problems by mediating the fight-or-flight response and elevating \"tend and befriend\" so it becomes a much stronger possibility for both men and women.\n\nThe ongoing suppression of feminine values goes to the heart of whether humanity is going to alter its current rapid course toward extinction, and it impacts our physical health as well as our mental health. While the stifling of the very values that could save us all takes many forms, subtle and blatant, its primal foundation traces to physical domination. The world's tolerance of violence toward women and the resulting suppression of the feminine principle is not just another bothersome problem affecting undeveloped countries. Describing her own waking to the pivotal role of this issue, Sister Mary Eve recounts: \"Slowly, it dawned on me that nothing was more important than stopping violence toward women\u2014that the desecration of women indicated the failure of human beings to honor and protect life and that this failing would, if we did not correct it, be the end of us all. I do not think I am being extreme. When you rape, beat, maim, mutilate, burn, bury, and terrorize women, you destroy the essential life energy on the planet. You force what is meant to be open, trusting, nurturing, creative, and alive to be bent, infertile, and broken.\"\n\nWhy is this detour into social commentary relevant for a health book? Because if you are dancing to a drummer that is seriously out of phase with evolution's best strategies, as coded in your hormones, your body pays a price. Most specifically, your energies become conflicted within themselves, and your health, your happiness, and your sexuality may all be compromised. When clients lie down on my table, not only do their energies reflect their genetic predispositions and physical circumstances, they also reflect their life choices. This is just as true for men as it is for women, and nature's trick of providing us with such different hormone-based strategies for bonding, resolving problems, and reproducing is the fundamental puzzle whose solution our collective survival depends on.\n\n#  OPENING SEXUALLY IS MORE THAN PUSHING POINTS\n\nEnergy medicine shows you which points to hold, massage, or tap in order to awaken your sexual and other energies. More than holding points, however, the first order of business is to attend to yourself as an energetic being. My very first client was a well-known personality who was in his late 80s. I hadn't actually set up a practice yet, but I had just taught my first energy medicine class, and one of the participants told him about me. While he had a number of health problems, he had only one goal in seeing me. He had lost the ability to have an erection. He wanted a girlfriend but didn't feel it was fair to a woman to pursue her if he couldn't perform sexually. This was definitely not my area of expertise. But I did believe that if I got his body's energies flowing, it could only help. So that is what I set about doing. Wherever his energies were blocked, I opened the channels.\n\nWhile every health problem in his body showed improvement within just a few sessions, there was only one he wished to focus upon. A successful man, he was blunt and direct about his priorities. I received a triumphant phone call from him one morning announcing that he was holding it and it was hard. He told me I was worth my weight in gold and that he was going to get himself a girlfriend. He did. That was a very instructive experience for me. I didn't know the points that are used for enhancing sexual performance. I didn't know cures for anything. I only knew how to balance the energies in the body.\n\nSo all of the methods in this book for balancing your energies contribute to your becoming a more robust sexual being. However, a few techniques can open sexual energy that has been chronically blocked or inhibited.\n\n##  OPENING YOUR SEXUAL ENERGIES\n\nAside from childbirth, sex is the ultimate physically intimate act, usually more so for a woman than for a man because she takes her partner _into_ her body. It is far beyond our scope here to address all the emotional, interpersonal, and spiritual dimensionsof your sexuality, but I can share some techniques for sparking the energies that affect it.\n\n**Getting Your Sexual Feelings to Flow.** In energy medicine, the meridians most involved in healthy sexuality are the kidney, spleen, and liver meridians.\n\n\u2022 The kidney meridian, in some traditions, is considered the storage area of sexual energy, and while I don't quite see it that way, it is a strong force in keeping our sexual energy flowing.\n\n\u2022 The spleen meridian bolsters the kidney meridian in this function, and also supports all the organs involved in female sexuality.\n\n\u2022 The liver meridian is able to open blocked energies, and supports the muscles and ligaments that are part of sexuality.\n\nTo maintain a healthy sexuality, or to open blocked sexual energy, keep your kidney, spleen, and liver meridians humming. Use any combination of the following methods:\n\na. In the Cherokee Grandmother's Self-Empowerment Secret, placing your hands on the insides of your thighs as your body undulates stimulates all three of these meridians. For some women, simply keeping their hands there gets energy flowing. While this can initiate a warm, liquid-like feeling and you may not want to be any more intrusive than that, gently massaging the points on the insides of your thighs in a circular motion will also stimulate these energies.\n\nb. You can use your hands to keep the kidney, spleen, and liver meridians open and flowing. Your hands are electromagnetic, and moving them along segments of the meridian lines can invigorate these meridians. Place your open hands at the insides of your feet. With your hands spread, draw them very slowly and deliberately up the insides of your legs to the tops of your inner thighs.\n\nc. Acupressure points for the kidney, spleen, and liver meridians are also situated on the creases where your legs and body join. Gently press or massage along the creases to stimulate the energies in these meridians.\n\nd. Massaging any tender points on your pelvic bone also keeps the pathways clear so sexual energy will be able to flow.\n\nDeliberately propelling the kidney, spleen, and liver energies through your thighs once or twice each day changes the energetic habits in your entire pelvic area so that sensuality and sexual excitement are more readily accessible. Regularly pressing the points on your inner thighs, leg-body creases, and pubic bone also has the effect of keeping your consciousness attuned to yourself as a sexual being.\n\n**The Back Side of Sexuality.** The energies held in your buttocks are ruled by the circulation-sex meridian. If these energies become stagnant or blocked (descriptions of personality characterizations like \"tight-assed\" often have a physiological counterpart), your capacity for sexual enjoyment is inhibited. You want to release these energies so they can flow forward and become fully engaged during sex. One way to know if your circulation-sex energies are blocked is by pushing in on your buttocks and seeing if there are areas that are sore. Sedating the circulation-sex meridian frees congested energies and helps keep the energies moving throughout your root chakra, an energy center that is strongly involved in your sexuality. Sedating the circulation-sex meridian (see page 148) not only opens a door to sexuality, it can also help stabilize an irregular menstrual cycle.\n\n_Figure 5-2_ CENTRAL SEXUAL CHANNEL (HAND POSITIONS)\n\n**Opening the Central Sexual Channel.** Another set of points, which affect the central and governing meridians and a radiant circuit called the penetrating flow, open the energies involved in deep, whole-body sexual experiences.\n\na. Lying down, cup your hands and bring the three middle fingers of each hand together, back to back, fingernails touching. Place them about an inch above your pelvic bone (see Figure 5-2).\n\nb. Push your fingers in. Lie this way quietly for about three minutes. Notice the movement of energy.\n\n**The Vortex Revival.** There is an energetic parallel to the way a woman's genitalia differ from those of a man. At the root or sexual chakra, the energies pull in, like a magnetic vortex, in a woman, while in a man there is a spiraling force that pushes outward. One receives, one projects. It is because energies spiral into our bodies that we, as women, feel so deeply. Men, who express pride in being more objective than us, are processing their emotions in the spiraling chakra energies _outside_ their bodies. To play off lyrics by Janis Joplin, \"Objective's just another word for nothing gets inside.\" Of course all men and all women have both yin and yang energies in varying combinations, so the generalizations do not always fit and are not really fair, but for the most part, for women more than for men, our relationships penetrate us, not just physically but energetically. We take our partners into our bodies physically, emotionally, and spiritually, and if we are hurt too much or too often, that spiraling inward force, which draws a partner's energies into our own, begins to diminish and lose its magnetic pull. Whether you have been wounded or just want to give that magnetic pull a boost, the following exercise energizes and revives the natural vortex energies in your root chakra.\n\n**The Vortex Revival Preliminaries**   \n(time\u2014about 90 seconds)\n\n1. Sitting in a chair, place your hands on the top of your hip bones and slowly and firmly move them down the fronts and sides of your legs and massage the energies off your feet, like squeezing a tube of toothpaste. Then slowly and firmly bring your hands back up on the insides of your legs and out to your hips. Do this two or three times.\n\n2. On a deep in-breath, draw your hands straight up the middle of your body (along the central meridian and over the penetrating flow) from your pubic bone to your chin. Release your breath.\n\n3. On another deep in-breath, push your fingers up over your chin and cheekbones with some pressure until you are cradling your face in your hands, with the middle fingers at your temples (see Figure 5-3). Release your breath.\n\n4. Pushing your fingers up from your chin and onto your face opens the flow of the stomach, small intestine, large intestine, gallbladder, and triple warmer meridians.\n\n5. Do a Triple Warmer Smoothie (page 107) and a Hook-up (page 62).\n\n_Figure 5-3_   \nTHE VORTEX REVIVAL   \nPRELIMINARIES\n\nThis in itself opens the meridians involved with sexuality, and you can stop here. But it is also a great preliminary to the Vortex Revival, which is more like a meditation.\n\n**The Vortex Revival** (time\u2014five to ten minutes)\n\n1. Lie down on a bed or on the floor with your palms up and your hands open.\n\n2. Imagine a swirling, spiraling energy at the end of your feet.\n\n3. Imagine it now very slowly and deliberately being drawn into your body, coming in through your feet. Energy follows imagination!\n\n4. Imagine it coming to your pelvis and climbing up through the center of your body so there is a swirling energy within and above your pelvic area.\n\n5. Imagine this as a spiraling force that literally sucks energy from outside you into your body. This is a vortex revival moment! Spend some time playing with these energies.\n\n6. Linger here, or take it further if you like, by experimenting, in your imagination, and expanding the size of the swirls until, with the spiral centered at your pelvis, it grows to encompass your heart. This connects the heart and genitals in a full-body experience.\n\n7. End by placing one hand on your second chakra (just below your navel) and the other on your heart chakra. Within one to three minutes, you are likely to feel a strong connection between these chakras (see Figure 5-4).\n\n_Figure 5-4_ THE VORTEX REVIVAL (HAND POSITIONS)\n\nSometimes in my practice, a woman wails and sobs just from feeling the energies moving again. This energy is very real, and it originates within you, far different from the physical connection with a partner. A variation of the Vortex Revival with a partner or even good friend can, however, be a wonderful experience.\n\n**The Vortex Revival with a Partner** (time\u2014five to ten minutes)\n\n1. As you lie on your back, your partner puts both hands on your hips and gently, slowly, firmly, pulls her\/his hands down the outsides of your legs.\n\n2. Then with more pressure, have your partner pull the energy off each foot, with both hands on one foot and then the other.\n\n3. When you do the Vortex Revival on yourself, you use your imagination to engage the spiraling energies. When you have a partner, the electromagnetic energies of his or her hands can interact with your energies to make the exercise even more powerful. Your partner sits or stands about 24 inches away from the base of your feet and, with a flat open hand, begins to make slow, counterclockwise circles in the energy field of your feet (see Figure 5-5).\n\n4. If he or she moves slowly enough, you will begin to feel a spiraling force at your feet that begins to move up your legs toward your pelvis. Your partner can support this movement by coming closer and closer to your feet with the spiraling motion.\n\n5. Focus on the way this spiraling energy engages your pelvis or even moves up to your heart.\n\n6. You may experiment with using your imagination to expand or intensify the spiral.\n\n7. End by placing one hand on your second chakra (just below your navel) and the other on your heart chakra (see Figure 5-4). Within one to three minutes, you are likely to feel a strong connection between these chakras. Alternatively, your partner's hands can be placed over these chakras.\n\n_Figure 5-5_ THE VORTEX REVIVAL WITH A PARTNER\n\n##  ENERGY TECHNIQUES FOR ENHANCING SEXUALITY WITH A PARTNER\n\nPlayfully thinking outside the box does not require techniques. You follow the energy and the energy follows you. Start with creative kissing. Here are Three Great Kisses:\n\n1. Slowly, softly, mindfully, let your lips linger behind one of your partner's knees. The neurovascular points that sit there are some of the sexiest points you can stimulate, and just before the tickle zone is an exhilarating space.\n\n2. One of the most romantic spots for a kiss is on the top of the upper eyelid. Gently and fully attuned to the connection you are making, soften your lips and know you are stimulating the meridian that helps two people cozy up with each other.\n\n3. A long held kiss on the side of the cheek stimulates several energy points that help awaken the flow of love.\n\nThe most frequently reported reason couples who have been together for a long time seek sex therapy is because one or both have lost interest in their sexual relationship, or as the professionals label it when they bill insurance companies, the patient has an \"Inhibited Desire Disorder.\" While it is, of course, important to work with communication, resentments, and failed expectations, a behavioral approach is often what is most effective in successfully helping a loving couple respark their sexual desire for each other. Even as simple a technique as the Three Great Kisses can get the energy into a habit of flowing between the two partners.\n\nMargot Anand's classic _The Art of Sexual Ecstasy_ 6 is a wonderful guide to heightening the experience of making love, grounded in knowledge of ancient wisdom such as Tantric and kundalini yoga, as well as insights from contemporary sex therapy. Making love can be elevated into a true art form, yet at its core is energy. Making love can become a shared journey in discovering the magic of the body's energies and how the energies of two bodies can interact, blend, dance, and merge. It is an epic adventure. By working with your body's energies and flows, you will already be enhancing your sexuality, but it is also possible to focus energy techniques specifically for sexual enhancement. This, however, like many of our topics, could be the basis of another book. Fortunately, Anand's book is very attuned to the dimension of energy within sexuality, so I'm taking the afternoon off.\n\n##  HEALING WOUNDS\/FEELING SAFE\n\nWhen we are young, the passion of the moment may fuel great sex. If we are innocent and undamaged, great sex opens our souls to our partners. But let's face it, in most of the scenarios likely to follow when we are young, we get wounded. Not fair to us. Not fair to the partner, who was just looking for an entertaining way to spend the evening and is genuinely surprised when we start to stalk him. Okay, gross exaggeration, but the hormonal aftermath of great sex may be very different for men than for women. For us, great sex _is_ risky business. In the wild, if you become pregnant and he leaves you, your literal survival is threatened. But even in a world where we can control pregnancy and have many levels of social support to fall back on, abandonment after great sex is still a primal loss and our wound may be soul deep. Both sexuality and survival depend on the energies of the root chakra, and therein lies the rub. Your root chakra doesn't register the subtleties. If such abandonments occur a number of times, or even if you have one whopper, the root chakra energy may shut down your sexual juices. The passion of the moment may become very difficult to enter.\n\nAs we mature, great sex grows from having confidence in who we are and in the truth of our own experience. If you are guided primarily by a focus on your partner's needs, desires, and judgments about you, your sense of self, and soon your enjoyment, get lost in your sweetness. Sustained great sex also grows from having confidence in your partner's love and caring for you. Feeling victim to your mate's expectations isn't particularly sexy; succumbing to pressure closes down your pleasure circuits. Establishing a deep foundation of trust leads to emotional safety and the intimacy that allows you to laugh, or cry, from the depths of your soul. It is a romantic container that is fashioned both within and beyond the bedroom.\n\nIf we have been deeply wounded, sometimes the wound does not even fully reveal itself until we are graced by profound love. Women who have been sexually or physically abused may not be able to open fully to their own sexual pleasure without revisiting the traumatic past and healing emotional wounds that may trace back to childhood. This sometimes may require a substantial course of psychotherapy, but energy therapy is proving itself to be able to markedly speed the healing process. Sometimes if you can heal the emotional wound at an energetic level, the psychological healing follows rapidly. The approach, applying energy techniques for healing emotional wounds, is called energy psychology. While it is beyond the scope of this book to go into that complex territory, here is a case history that is typical of what can be rapidly accomplished. It is from the book on energy psychology that David and I wrote with Gary Craig.\n\n_Sandy and her partner came to one of our colleagues_ _9_ _for premarital counseling. Among the issues they were concerned about was their sexual relationship. Although Sandy had been married before, she found herself reacting with uncontrollable negative feelings when her fianc\u00e9 initiated sexual play. He was willing to be patient, kind, and understanding, and he seemed genuinely interested that sex be a shared experience. While she freely acknowledged that she had no problems with his attitude, she still would usually became upset and turned off by his overtures. They asked for help with this problem, and a private session with Sandy was arranged._\n\n_When she came in, the therapist gently asked, \"Is there something in your earlier years that you could talk about?\" She immediately burst into tears. Red blotches appeared on her skin, and her words were punctuated with heavy sobbing and gasping as she began to relate her story: \"When I was seven years old, we lived in [a small rural town]. One day my stepfather took me for a walk down a country road. It was in the summer. We hiked up the side of a hill. Then we stopped. Then he took off all my clothes. Then he took off all his clothes.\"_\n\n_At this point she was scarcely able to breathe. The therapist stopped her and said that it was not necessary to go any further. He had her state her distress rating_ _about the memory on a 0-to-10 scale. Not surprisingly, it was a 10. He then led her through a tapping sequence similar to the Meridian Energy Tap_ [page 80]. _Her intensity dropped from 10 to 6. After another round of tapping and the use of affirmations such as \"Even though I still have some of this upset, I deeply love and accept myself,\" the intensity fell to 2. Then another round of tapping was completed, along with another simple affirmation._\n\n_By this time, Sandy was breathing quietly. Her skin was free of blotches, her eyes were clear, and she was looking at her hands, lying folded in her lap. The therapist said, \"Sandy, as you sit there now, think back to that hot summer day when your stepfather took you for that walk down that country road. Think about how you hiked up the side of that hill until you stopped. Think about how he took off all your clothes. Think of how he took off all his clothes. Now, what do you get?\"_\n\n_She sat there without moving for maybe five seconds, then looked up calmly and said, without strong emotion, \"Well, I still hate him.\" The therapist, after agreeing that hating him seemed a perfectly human response, then asked, \"But what about the distress you were feeling?\"_\n\n_Again she paused before answering. This time she laughed as she said, \"I don't know. I just can't get there. Well, that was 20 years ago. I was just a little girl. I couldn't protect myself then the way I can now. What's the point in getting upset about something like that? . . . I never let that man touch me again, and my kids have never been allowed to be near him. I don't know, it just doesn't seem to bother me like it did.\"_\n\nAfter this single session, Sandy no longer experienced negative feelings in response to her partner's sexual advances. In a two-year follow-up, she reported that the problem was \"good and gone,\" and her partner, now her husband, confirmed that there was no sign of the former difficulties. Such shifts in relationship to a traumatic memory that has been emotionally cleared using an energy intervention are surprisingly common. While therapy around past abuse may require skilled assistance, the Meridian Energy Tap (page 80) can be adapted and helpful in many contexts, and it is quite harmless. If you combine the tapping with a memory or a worry that plagues you, doing the tapping sequence again and again, the memory or worry will often lose its grip.\n\n#  **Fertility**\n\nYou are designed to have an egg poised for fertilization every month. You are designed to have hormones raging through your bloodstream that cause your brain chemistry to urge you to find a mate. You are designed to become pregnant, carry, and birth a baby, and to quickly become pregnant again and again and again, from your early teens into your forties, or until your body says \"enough\" and you die in childbirth. Meanwhile, nearly every human culture has been trying to find ways to put the brakes on nature's prolific plan for you. Sometimes the brakes, however, are permanently applied, through culture or biology, in a woman who desperately wants to conceive a baby.\n\nAt least 100 women or couples have come to me over the past 30 years asking for help after having difficulty conceiving. All but three of them became pregnant. Usually if there was serious and obvious physical damage interfering with the ability to become pregnant, I would not be consulted, which admittedly did up my percentages. But unless there is a decisive physical obstacle, conception is usually possible. There are many factors involved in infertility, and the circumstances of the three women who did not conceive are instructive. One of them eventually discovered that her fallopian tubes were totally closed off, although this had not been detected in an earlier medical exam. Another, after substantial individual work with me, learned that her apparently virile young husband had an extremely low sperm count. A third couple was already headed toward divorce and hoping at some level that having a baby would fix their marital problems. The sad truth was that their energies did not easily dance together, and the tension between them worsened their incompatibilities. While energetic differences form the spark of romance, the topic of another program, stress and personality incompatibilities exaggerate the energetic differences at all levels. But for everyone else, baby pictures were not far off.\n\nPregnancy occurs when a woman releases an egg from her ovary, it travels into the fallopian tube, it is fertilized by a sperm, and it continues down the tube to the uterus, where it implants itself and begins to develop. Infertility means that this sequence is not able to be completed. Medical exams and laboratory tests can often determine the physical reasons for infertility. Among the most common are blockages in the fallopian tubes or damage to the ovaries. Pelvic inflammatory disease, often caused by a sexually transmitted infection, may damage the fallopian tubes. Endometriosis, where tissue from the uterine lining grows outside the uterus, may also damage the fallopian tubes as well as the ovaries. Fibroids may crowd the uterus so that an egg cannot implant or a baby is unable to grow. The sperm or egg may be damaged, sometimes from environmental causes such as radiation or pollution or from the use of tobacco or other drugs or simply from age. Attempts to intervene have led to the growth of a $2 billion fertility industry. But, as an article in _Time_ points out, \"the more doctors have to intervene with drugs, needles and surgery to get sperm to meet egg, the greater the chance that something will go wrong.\" In addition, according to Randine Lewis in her excellent book _The Infertility Cure,_ hormonal imbalances \"contribute to 40 percent of infertility, yet are considered untreatable by conventional Western Medicine.\" A woman's body can, however, in my experience, be brought into a hormonal balance that supports pregnancy by using energy medicine, with larger amounts of the hormones needed for conception being produced naturally.\n\nEnergy work can change chemistry, open pathways, get the body to expel toxins, enhance the health of the blood, and help the body cycle regularly. It can keep the uterus flushed and healthy and even repair it. And of course, it can change energy habits (an energy habit might, for instance, promote the growth of fibroids or keep your cycle at 23 days instead of 28). It is actually surprising how many ways energy medicine can help a woman become fertile.\n\nEven physical obstructions were often energy obstructions first. By correcting imbalances in the energy field, physical blockages can literally be dissolved. Dietary changes are a natural way to shift the body's energies, and they can be used to increase the chances of conceiving. Commonsense notions that it is probably not a good idea to mess too much with the way Mother Nature produces our food are supported by a study of the impact diet has on fertility. The findings, based on the experiences of 18,000 women who wanted to conceive, were striking and surprising. Whole milk and rich ice cream promoted pregnancy, while skim milk and other \"light\" dairy products such as low-fat yogurt actually worked against fertility. Natural, unsaturated fats increased the chances of pregnancy, turning on genes that enhanced fertility, while trans fats had a detrimental effect. A book by researchers from Harvard Medical School, _The Fertility Diet,_ presents practical guidelines based on the study.\n\nAn \"energy habit\" that often impacts fertility is what I might call \"chronic fatigue of the reproductive organs.\" Everything seems slowed, from the movement of the egg to the delivery system for the sperm. I suspect that this is a factor in those infertility cases where all the standard tests show up as normal. Stagnation in the liver meridian or diminished levels in the energies affecting the thyroid can also interfere with the body's ability to conceive, as can a dominance of a woman's yang energies (hot, bright, quick, aggressive) over her yin energies (cool, dark, slow, receptive). Lewis calls conception \"a fragile miracle that can be affected by any one of a thousand factors.\" By counteracting energy habits that interfere with conception and bringing the body's energies back into their natural balance, the body's remembered wholeness and script for making a baby can be restored.\n\nEmotions\u2014fears or ambivalence or interpersonal difficulties with one's partner\u2014can also cause energy patterns that interfere with pregnancy. There is a saying in traditional Chinese medicine that \"fear scatters _chi,_ \" and in my experience, it is indeed harder to retain the embryo in the uterus when anxiety and worry are too strong. Confronting emotional concerns energetically, using techniques from energy psychology, can remove some of their charge, making it easier to directly and consciously address their causes.\n\n#  AN ENERGY APPROACH TO FERTILITY\n\nA woman's body can be gently nourished with energy methods, like tending a tree in an orchard, and encouraged to bear fruit. Many women, however, crumble under the sorrow and fear of believing that they are unable to have a child. It has been a remarkable privilege to first demonstrate, through energy tests, that the door is not closed, and then to help these women systematically shift the energies that are interfering with pregnancy, and to finally hear the triumphant announcement that they have conceived.\n\nWhen infertility is an issue, unless it involves collapsed tubes or other substantial physical damage, the first steps are to regularly do the Five-Minute Daily Energy Routine (page 51) and to be sure your energies are not running in a homolateral pattern (page 67). With infertility, spleen meridian is also almost always involved. Spleen meridian is considered the \"mother\" of all the body's energies. It tries to nurture you whenever and wherever there is threat. It will create inflammation if you get a splinter in an attempt to remove the invader and initiate a healing process for the tissue. Its primary strategy of protecting you, however, is to mobilize your entire body to stay strong. In the sense that a lion will attack the weakest wildebeest in the pack, and the weakest individuals, constitutionally, are the most vulnerable to invasion by harmful microorganisms, a strong energy system is one of the best ways to protect yourself from outside invaders or internal breakdown. Spleen meridian works to keep your whole system strong. It is a community builder, orchestrating your organs, blood, cardiovascular, and other systems.\n\nTriple warmer, on the other hand, which also protects you in the face of threat, tends to use attack as its primary strategy. If spleen mobilizes your \"inner mom,\" triple warmer meridian mobilizes your \"inner militia.\" In our culture, just as patriarchal values and militaristic strategies dominate in the outer world, triple warmer tends to dominate in our inner world. In the balance between triple warmer and spleen, triple warmer can conscript energy from spleen to fight its battles. This diminishes your overall strength, vitality, and joy. Triple warmer perceives constant threat in the modern world, with its pollutants, stresses, and chemicals that were not there while it was evolving. So for many people, triple warmer is in perpetual reactivity and overcharge, while spleen meridian is in perpetual depletion. Among the many problems this causes, if spleen meridian is depleted, less support is available\u2014in keeping with its nature as the \"mother\" energy\u2014for becoming pregnant or for carrying a baby.\n\nSpleen meridian, in fact, governs the sea of blood and the cycle of menstruation. It governs whether the blood expels completely or coagulates and stays in the body, where it can grow as endometriosis or fibroids. It governs metabolism as well as how quickly the body can restore and replace natural hormones. It governs the body's ability to renew itself. And it governs the vitality of the sexual organs and the uterus, keeping them in a state that is inviting for a baby. Strengthening spleen meridian is often the first order of business when working with infertility. It is also a way of keeping your entire body strong.\n\nTo nurture spleen meridian, you can strengthen it directly, but it is often more effective to calm triple warmer so that it is not perpetually pulling energy from spleen meridian. An overactive triple warmer can, in fact, undo your efforts to strengthen spleen directly. I personally had an extreme imbalance between triple warmer and spleen meridian that was the underlying energetic cause of an assortment of medical conditions, though they seemed unrelated. Strengthening spleen meridian and calming triple warmer began to restore my health. The epidemic imbalance between triple warmer and spleen creates energy patterns that are dysfunctional, and sometimes you need to be diligent to shift these habits. You've already been introduced to each of the techniques in the following sequence. Using it two or three times each day will do a great deal to keep your spleen meridian strong:\n\n**Spleen\/Triple Warmer Balancing Sequence** (time\u2014about 3 minutes)\n\n1. The Triple Warmer Smoothie (time\u201420 seconds, see page 107).\n\n2. The Triple Warmer Tap (time\u2014about a minute, see page 108).\n\n3. Spleen Meridian Flush and Tap (time\u2014less than a minute, see page 136).\n\n4. Neurovascular Hold (time\u20141 to 3 minutes, see page 151).\n\n5. Triple Warmer\/Spleen Hug (page 110) Each of these supports a strong working partnership between your spleen and triple warmer meridians. The \"Hug\" is particularly convenient because you can do it while you are talking with someone or watching television, and many other times throughout the day. It is something you don't have to think about.\n\n**Fertility Stimulation Points** (time\u2014less than 2 minutes)\n\nIn addition to the vital role of spleen meridian, attention to your kidney, liver, and circulation-sex meridians also supports fertility. Stimulating the following sequence of points will help optimize all three meridians.\n\n1. Use two or three fingers to tap the points directly in front of the inside anklebones for three deep breaths. You can tap both sides simultaneously (Liver 4).\n\n2. Tap in back of your inside anklebones, again on both sides for three deep breaths (Kidney 3).\n\n3. Tap two thumb widths above the inner anklebones for about three deep breaths (Kidney 7) [see Figure 5-6 for all three points].\n\n4. Push your breasts up and find the points directly behind your nipples. Massage as firmly as you can tolerate for several seconds (circulation-sex neurolymphatic reflex points).\n\n5. Do the Triple Axis Hold from the PMS chapter (page 134) to stimulate the endocrine glands related to fertility.\n\n_Figure 5-6_ FERTILITY STIMULATION POINTS\n\nOne additional concern, even after doing all of the above, is that if the energy in the circulation-sex meridian is not flowing well, it gets trapped in the gluteal muscles, interfering with all the organs and processes related to conception. You can know if this is occurring for you by simply pushing in with your fingers or thumb on your buttocks to see if there is tenderness. If there is, you can \"loosen up\" the circulation-sex meridian by sedating it (page 148). You will probably find that there is less tenderness in your glutes immediately after this procedure.\n\n##  PHYSICAL CONDITIONS THAT MAY PREVENT CONCEPTION.\n\nFour physical conditions that may prevent conception are problems with the ovaries, fibroids, endometriosis, and the impact of age on fertility.\n\n**Energies Not Flowing through the Ovaries.** Your ovaries are situated directly on the stomach meridian line. If energies are not flowing well through the ovaries, you are less likely to become pregnant. This flow can be enhanced by sedating the stomach meridian (page 143), followed by tapping the points at the top of your cheekbones, directly beneath your eyes (Stomach 1 and 2).\n\n**Fibroids.** Fibroids are noncancerous tumors composed of blood and disorganized tissue. If blood doesn't completely leave the uterus, it can coagulate, adhere to the uterus, and grow. You are wise to take measures to prevent this. As a fibroid grows in the uterus, in a tragic case of mistaken identity, the body begins to fiercely protect the fibroid like a baby and to marshal all available resources to help it grow. One of the simplest ways to keep the blood from staying trapped in the uterus is to place a heating pad over your uterus for 10 to 15 minutes on the days your period is winding down and on the day that follows. The heat will facilitate release of the menstrual blood. The same meridians that are involved with fertility\u2014the spleen, triple warmer, kidney, liver, and circulation-sex meridians\u2014may contribute to fibroid growth if they are compromised. So each of the procedures mentioned earlier for reversing infertility also addresses fibroid growth. Particularly important is the liver, which processes hormones. Sedating the liver meridian (page 146) restores balance among your hormones and aids in the prevention and shrinking of fibroids.\n\nDietary supplements can also be helpful. Essential fatty acids such as those found in flaxseed, fish oils, and evening primrose oil may help prevent and dissolve fibroids, as can nettles, kelp, motherwort, and a combination of Chinese herbs called the \"blood movers,\" including cinnamon twig, peach kernel, and red peony.  Fibroids are discussed later in the chapter in the personal story about my daughter: \"Energy Medicine and Conventional Medicine Together in Concert.\"\n\n**Endometriosis.** The endometrium\u2014the mucous membrane that lines the uterus\u2014can grow outside the uterus, with its cells somehow migrating and becoming embedded in sites such as the cervix, ovaries, fallopian tubes, vagina, colon, rectum, or bladder wall. Misplaced endometrial cells have also been found in the abdominal wall muscles, lungs, nose, and even the brain. Wherever they are, however, they respond to fluctuations in estrogen and progesterone just like the endometrium, so they cause bleeding during menstruation. This blood often then becomes trapped. Severe inflammation, scarring, terrible cramps, painful intercourse, and infertility may result. Energetically, the same problems with the spleen, triple warmer, kidney, liver, and circulation-sex meridians that play into infertility and fibroids may also be the cause of or a large contributor to endometriosis. Both fibroids and endometriosis trace to blood that gets trapped and stagnates. The challenge is that since menstruation does not clean out the sites where the stray endometrial cells have lodged, the body must be helped to clear them out. Supplements that can assist include flaxseed oil, baby primrose oil, Pycnogenol, and anti-inflammatory herbs such as bromelain. The energy medicine approach to preventing and reducing endometriosis, beyond a Daily Energy Routine, involves restoring balance to triple warmer (page 110), strengthening the spleen meridian (page 132), flushing the stomach meridian (page 138), massaging the liver meridian points on the hands and feet (page 47), and sedating the kidney (page 145) and circulation-sex meridians (page 148).\n\n**The Impact of Age on Fertility.** Of the women who have had an impulse to kill me, up at the top of the list were those who came for help as they entered menopause and wound up getting pregnant. I was balancing their hormones and energies so they would feel better as they went through menopause but, in regulating these energies, they became fertile again, sometimes after years of sexual activity without birth control. More than the other meridians, the spleen and kidney meridians tend to weaken with age. Since meridians are energies and not organs, however, it is possible to keep these important energies youthful and vital. The Daily Energy Routine, as well as the procedures listed earlier in this chapter, is valuable, not just for fertility, but for, as the saying goes, dying young as late as possible.\n\nWherever the hands on your biological clock, fertility depends on healthy eggs. If you are racing against time, you can extend your fertility finish line by keeping yourself youthful and vital through keeping the spleen and kidney meridians, in particular, strong and balanced. Healthy body, healthy eggs. While many herbal remedies may also be helpful, you might consider in particular antioxidants, such as coenzyme Q10, which eliminate the free radicals that inhibit healthy cell duplication as we grow older.\n\nDespite all the ways it is possible to increase the likelihood of having a child, using both energy medicine and modern technology, motherhood simply is not the fate of every woman. Many of my closest friends were unable to have or chose not to have children. But they bring a quality of incredible love to my girls, to other people's children, and to the world. I am often humbled by the love and wisdom of these \"mothers of the heart.\" As Randine Lewis says in the closing of her powerful treatise on overcoming infertility: \"You never know what place you will fill in the universal plan, but I do believe with all my heart that the love that makes us want to be parents is not meant to go to waste. . . . If there is a divine plan and we are placed on this earth to learn and grow, then perhaps the lessons of our souls are taught through those who are put and who are not put in our lives. Ultimately, however, we must recognize [that] the children we wanted so much and have done so much to bear are not really ours to begin with.\"\n\n#  ENERGY MEDICINE AND CONVENTIONAL MEDICINE TOGETHER IN CONCERT: A PERSONAL STORY\n\nMy daughter Tanya began feeling pain in the area over her ovaries when she was in her midthirties, and the early stages of fibroid growth were detected. Her symptoms included long periods with heavy bleeding, painful intercourse, bladder incontinence, blurry vision, and pain in her pelvis, back, legs, and arms. She also had many symptoms suggesting hormonal imbalances, including extreme PMS, terrible headaches, depression, anxiety, exhaustion, weight gain, dry skin, and hair loss. Poor baby! She was living in Boulder, and my work had me traveling almost constantly. When I was able to visit and work with her, it was obvious which energies were most problematic: the second chakra, the stomach meridian (which travels right over the ovaries), and the small intestine meridian (which governs the abdominals). The spleen\/triple warmer balance is also important in almost any illness, with both also playing a significant role in hormone regulation. Each also needed attention. I'd get these meridians back into a healthy flow, Tanya's pain would be relieved almost instantly, her hormone balance would seem restored, and I'd give her homework to foster further healing. I'd leave for another three months, feeling quite sure the fibroid would now shrink and the pain would not return, as this had always been my experience when I treated clients who presented with similar symptoms.\n\nFollowing each visit, I was surprised to learn that Tanya's pain had returned. It was clear to me that the recurrent pain, headaches, and the fact that the fibroids were not shrinking were all related to a hormonal imbalance, but it was not clear why the energy treatments were not reversing that imbalance. I was again learning what I'd learned many times before, which is that when I am close to someone, I can't be objective. And it's not just about wanting to help too much or having too much stake in the outcome. It is a literal change in how I perceive the energies. The best way I can explain it is that I couldn't tell where I ended and she began. This was very confusing because I usually know exactly whose energies I am looking at when I am interacting with a client. I was traveling around the world being represented as this great healer, yet I wasn't having success helping my nearest and dearest with a condition that had been relatively easy to turn around with many others.\n\nEver my daughter, Tanya had been trying to manage her condition exclusively with holistic remedies. Some of the best complementary health practitioners in Boulder had worked on her, but even they could not crack the code. She felt much better after energy sessions. They always helped. But the benefits did not last. In retrospect, taking medical tests that would have resulted in her being given bioidentical hormones would have served her better.\n\nEven with all the care she was receiving, the fibroids did not shrink. In fact, after years of not increasing in size, they grew wildly during a nine-month period following an emotionally significant event that underscored the fact that she was now forty years old, had not had a child, yet desperately wanted one. It was, in fact, as if she were pregnant. All the healers who worked with her during this period commented that it was like a baby was inside her and her body was protecting that baby. The feeling they all reported was that it was as if the fibroid was defiant. No interventions could slow its growth.\n\nShe now looked pregnant, and many other symptoms began to appear. Because so much blood was feeding the fibroid, she became anemic and extremely weak. She had difficulty lifting anything, and had to literally climb on all fours to get up the stairs in her home. Her eyesight became bad. She was having heavy periods with blood clots. Her doctors at Kaiser recommended a blood transfusion and a hysterectomy, but as her medical problems become more apparent, Kaiser (coincidentally?) dropped her insurance based on an administrative technicality. This turned out to be a fortunate injustice. She still wanted to have a baby, so she fiercely resisted the recommendation of a hysterectomy, and a deep instinct told her not to have a blood transfusion despite strong medical advice that her life was in danger if she didn't. We found an herbal combination, a powerful iron supplement called Floradix. It went into her cells as if it were being injected, and she brought her extreme anemia to a normal state. Her doctor expressed amazement.\n\nBut even as the Floradix was helping her anemia, and the energy workers were making her feel better, her fibroid continued to grow. Strangers would make conversations about her baby, and airport security would direct her away from X-ray devices. We were all in denial, hoping for that magical natural cure that would shrink the fibroid without her having to undergo a hysterectomy.\n\nThe final wake-up call was during a long-planned trip to Hawaii. After landing, Tanya noticed a hard swelling near her left ankle. That leg was also in terrible pain. She called me, and I offered some suggestions to her and her partner, including energy exercises to reduce the swelling and help with the pain. These did give some relief, but the pain continued over the next three days, when they called again. Not only was she still battling the pain, the lump she had noticed on her ankle was now traveling up her leg through a vein she could see on the outside of her leg. It was above her knee when she called on the third day. I was alarmed. I told her that it sounded like a blood clot, a thrombosis, and that she had to go right away to a doctor to find out. It was evening, and she was not sure where she could go. We decided that in the meantime, I would do a long-distance healing. Tanya lay down to receive it. I realized after I got off the phone with her that I didn't know to which leg I should direct the healing. But the moment I began to attune to her, the inside of my left leg began to hurt terribly. It turned red and began to swell. There was no doubt now which leg it was. I continued to send the healing energy to her until my own pain diminished and the swelling and redness in my leg had disappeared. I knew something major had been accomplished. She called me right afterward and said that the swelling had gone way down and the pain was gone. Meanwhile, her partner had been on the phone, and he had located a walk-in clinic. They went and the doctor immediately identified it as a blood clot, which she confirmed with an X-ray. She was very interested to hear from Tanya how the energy work and distant healing had apparently caused the clot to shrink and stopped the pain. When Tanya was not enthused about the medication being recommended, the doctor said, \"Okay, continue what you're doing and see if you can make it go away.\" But she warned that Tanya still wasn't out of danger. She said it would be life-threatening if it reached her groin. Tanya also would not be able to get on the plane to return home unless the clot was gone completely. By the time of Tanya's planned return, the deep thrombosis had become a superficial thrombosis and the doctor assured her that she was out of harm's way.\n\nWith this close call behind us, we realized that we needed to consider having her fibroid surgically removed, even though the idea terrified me. Besides my bias that many doctors routinely put people at far more risk than is necessary, both my parents had died of medical mistakes made in hospitals. Fibroid tumors affect at least half of all women. The primary aggressive treatments (though not all need invasive procedures) are hysterectomy, myomectomy, and embolization. While myomectomy and embolization look encouraging on the Web sites of the clinics that perform these procedures, each has risks we were not willing to entertain. As to hysterectomy\u2014well, not only did Tanya still want a child, my desire for grandchildren trumped any objectivity on my part about that choice.\n\nDespite all the measures we had taken, Tanya's fibroid had caused a dangerous blood clot and was implicated in her anemia, blurry eyesight, general weakness, terrible pain, and other symptoms. It was stealing blood and energy from other parts of her body. It was also pushing against her liver, so her liver wasn't processing hormones correctly, creating emotional havoc, and she was looking yellowish much of the time. Not a pretty sight for a mother who was\u2014wisely or not\u2014still actively involved in her daughter's treatment.\n\nI was starting to think of the fibroid as an independent force with a mind of its own. It wanted food, energy, blood, life force. It was out of control. The plant in the movie _Little Shop of Horrors_ came to mind. It wanted to be fed. It was growing ever faster. There seemed no way to stop it. Its power was awesome. We reluctantly decided it was time to bring in the cavalry.\n\nAs fate would have it, Tanya was at one of my classes when we were coming to this decision, and one of my advanced students gave her a session. She told us about a doctor who she believed had saved her own life. This was no ordinary doctor. This was a woman who had gotten laws passed in three states regarding stricter requirements for informed consent prior to hysterectomies. This was a woman who had strongly advocated for more sensible practices in gynecology, invented less invasive surgeries for a number of female problems, and had been a thorn in the side of many of her more traditional colleagues. This was also a woman who had lost her license to practice medicine, perhaps for being a courageous pioneer, willing to utilize methods that were ahead of their time. But still, a revoked medical license is not the first qualification one usually insists upon when choosing a surgeon.\n\nWe met in a restaurant: Tanya, the doctor, and I. She quickly won our confidence. Her depth of knowledge permeated her answers to every question we asked. While I had learned very little I didn't already know from the other doctors and healers who had been helping with Tanya's fibroid, I learned a dazzling amount in a very short time about the issues and controversies in gynecology as she knowledgeably addressed each of our doubts and concerns. I was impressed by her humanity and caring for Tanya, for all her patients, for all women. She tuned in to Tanya's fibroid energetically, over the french fries, and she knew it as I knew it. With Tanya's permission, she laid her right hand over the area of the fibroid, and at one point turned to me so Tanya wouldn't hear and said, \"She's really at risk!\" She knew all our options, was able to quickly go through each of the alternatives our research had revealed, and discussed the reasons for and the liabilities of each. For instance, with embolization, what they tell you is that it is an ingenious procedure where silicone pellets are injected up through the femoral artery to cut off the blood supply to the uterus so the fibroid can't grow and ultimately shrinks because its blood supply has been cut. What they don't tell you is that cutting off the blood supply to the uterus can shrink and mummify it, so that within two years women often have to have a hysterectomy anyway. Beyond this, it is possible for the silicone pellets to go astray and permanently compromise the blood supply to other parts of the body.\n\nAfter going through the available alternatives, each quite undesirable, the doctor described a surgery she had developed in which the fibroid is delicately separated from the uterine lining bit by bit and the uterus is then repaired. Because her license had been revoked, she told us about the hospital in Tijuana where she was currently doing her surgeries. A few days later, Tanya checked in. While part of me was concerned about a doctor who had been disbarred from practice performing an unknown procedure in a Tijuana hospital, I was feeling deeply blessed that I had stumbled upon this maverick of a physician who seemed to have examined conditions such as Tanya's from every conceivable angle and developed the most potent and compassionate ways of approaching them.\n\nIt was a tiny hospital with a capacity of only seven patients. Each private room had two beds, and a family member was expected to stay with the patient as advocate and support system. That honor went to me. Unlike American hospitals, the food was fabulous, cooked to order, and could be anything from gourmet dining to wheat grass juice. I prepared Tanya for the surgery by giving her extensive energy balancings several times per day in the days between the decision to proceed and the actual surgery. David had been giving her energy psychology sessions to prepare for the event emotionally. Besides addressing her natural fears about a highly invasive procedure and bolstering her confidence and positive expectations about the surgery, one of the themes of their work together was Tanya's lifelong reluctance to act decisively when someone, particularly a boyfriend, was hurting her. Cutting out a growth that was clearly doing great damage was a perfect metaphor for this issue. They focused on the part of her that did not want to interfere, that was racked with guilt about the thought of destroying this autonomous being growing inside her, no matter what harm it was causing. By the time of the surgery, she understood the metaphor perfectly and was feeling a warrior's determination to ward off anything attacking her or her body. Xena, television's beautiful warrior princess, had, in fact, been one of the figures she used in her mental imagery. I received from David my own energy psychology sessions as well, with one theme being that I not faint while watching my daughter being cut open.\n\nOn the morning of the surgery, besides balancing Tanya, I insisted on balancing the doctor as well. When the surgery began, I was there, scrubbed in with mask and gown, and there I remained through the whole four-and-a-half-hour procedure. The surgeon, a second doctor who was also the owner of the hospital, a nurse, and the anesthesiologist were also all in the room. The opening event\u2014well, this was actually before the _opening_ event\u2014was conducted by the anesthesiologist. I was given the opportunity to energy test Tanya on the sedatives and local anesthesia that were going to be used. The choice and the dosages tested out well, though in other surgeries where I have been invited to participate, even when the choice of medication was correct, the amount was often too great. Not so here; the dosage was perfect for Tanya's body. I considered this a good sign about the medical team's intuition and attunement. To lend energetic support during the incision, I held my fingers about an inch and a half above Tanya's forehead, over the main neurovascular stress points, to keep her body from going into any kind of shock.\n\nI thought for a moment that I wouldn't be able to watch the knife slice into my daughter's flesh, but I decided that I must force myself to witness the entire procedure. To my surprise, I wasn't a bit woozy. While I knew that the energy psychology had helped me prepare emotionally, a healing energy beyond anything I might have expected, and larger than any of us there, entered the room. We were suddenly in a loving, healing, sacred space. It was like an aura surrounding us, perhaps a product of the good doctors, the humble hospital, my love, and Tanya's spirit. It was palpable. I felt strong and glad to be there.\n\nThe cut was made. A large chunk of flesh had to be cut out in order to free the uterus. Directly beneath Tanya's skin, going four inches downward, was a thick layer of fat that had to be cut out. It was a creamy yellow and salmon color, with a healthy glow. I'd never thought of fat as I did in that moment. You saw instantly the necessity for fat in the body, to buffer your organs and bones, and why a woman is designed to carry more fat around her middle than a man. That fat protects the baby, as well as all the female organs. The shock to me was how beautiful the fat was, and how beautiful was the energy it emitted. It is a buffer that is there to comfort and protect. Society has given a bad name to this magnificent cushion. I've been much more accepting of my lust for Dairy Queens ever since.\n\nThe surgeon's hands disappeared into Tanya's body and reemerged with her uterus, now a hard, reddish-pink, glistening sphere slightly larger than a basketball. It was shiny and so taut it looked as if it were ready to pop. I instantly understood that there had been no time to waste\u2014the need for this surgery had been urgent. The doctor lifted Tanya's uterus and placed it on her abdomen, attached now only by ligaments. Its energy was filled with a light that was so beautiful, it took my breath away.\n\nWhen the knife cut into Tanya's uterus to begin to remove the fibroid, something shocking happened. With the first slit into the uterus, an energy sprayed out that was exactly the opposite of what I had been seeing. Rather than a beautiful healthy glow, it looked like an evil dark force. I didn't have any visual cues from looking at her uncut uterus, even its energies, that such a force was held inside it. Not only was the energy ugly, so was the fibroid that was to be removed. It was one of the ugliest things I'd ever seen. I immediately had the thought that it held years of psychological trauma and pain. It was a terrible energy; it looked absolutely malevolent, both as a red, disorganized blob and in the energies it was emitting.\n\nTanya's energy, by contrast, looked so pure. She was an innocent lying there, while a negativity that seemed absolutely evil had lodged and taken a strong hold in her body. It came to me in a flood that the surgery was the most holistic thing we could do, to cut out this darkness from Tanya's body, from her life. The surgeon, on first seeing the fibroid, said, in memorable understatement, \"I'll bet Tanya's been in a bad mood for a long time.\"\n\nTo remove the fibroid in one piece would have required a cut that the uterus could not have survived, so the fibroid was cut in two. The first part to be removed was as big as a huge melon. It adhered to the inside of the uterus, so the surgeons had to very delicately cut it away. The care that went into this\u2014small snippet by small snippet\u2014was astounding. While Tanya's uterus was sitting on her stomach, the doctors had their instruments inside it, like working in a tiny cave. Their painstaking steps were ensuring that Tanya's uterus would be preserved. The whole process was amazingly respectful of the integrity of her body.\n\nWhen the last of the fibroid had finally been cut away from the uterus, along with some 25 fibroid \"seeds,\" everyone in the room took a big sigh. Hours of care and precision had succeeded in separating and removing the invader. Throughout the surgery, clamps had been used, so there was virtually no bleeding. I was awed by the skill and care I was witnessing. Except for brief, soft communications, the operating room had been in a holy silence throughout this battle to excise the fibroid without destroying Tanya's uterus. Now lively, light classical music was brought in as the process began of stitching the uterus back together. It was not just a matter of closing the incision. Wounds remained wherever the fibroid had adhered to the inside lining of the uterus. So the doctors were now sewing the inside of the uterus, determined to leave it strong enough to carry my grandchildren. At various points along the way, the surgeon explained to me why she was choosing a particular procedure rather than an alternative for the purpose of preserving Tanya's fertility.\n\nThe two doctors stitched and stitched, reminding me of a seamstress and a tailor. My appreciation was immense. I myself am a seamstress. Sewing is my relaxation and my meditation, and here I was watching my daughter's womb being ever so skillfully and patiently sewed, like a beautiful multilayered quilt being hand-stitched. After an hour of rapt attention and awe, I said, \"You must have put 300 stitches in there.\" They laughed. \"Closer to 500!\"\n\nWhen the uterus was completely sewn up, the lead surgeon stood back, looked at the other doctor, nodded her head, and smiled. This was his cue to lift the uterus from Tanya's stomach and cup it in his hands. His arms were outstretched and he placed his head in his forearms. All was silent. The music had been cut off. No one spoke. He continued in this posture for what I would guess was at least three minutes. Then he stood up, took a breath, and placed the uterus back on Tanya's stomach.\n\nAt this point, the surgeon told me, \"He is helping the cells in her womb remember what it was like before they were harmed.\" She explained that \"cells hold memories!\" (as if this weren't one of the core premises on which my career is built). Then she nodded to him, and he again lifted and held the uterus for what seemed to be another three minutes. When he did this, everyone else in the room was in a quiet reverence. I took photos of the entire operation, and one of the most beautiful is of the doctor holding Tanya's uterus. It had a beautiful glow within and around it that everyone could see and that is evidenced in the photo.\n\nThis was a most sacred experience, like being at a holy altar during a high spiritual ceremony. I told the doctor that it looked as if he was praying. Taken aback, he said, \"Well . . . I was.\" Then they placed Tanya's uterus back in her body and did the final stitching. When the surgery was completed, Tanya opened her eyes and smiled at me. As we had planned, I began to give her another energy session to help her body restore after the trauma of surgery. One of the givens after a surgery is that the patient's energies will be running in a homolateral pattern. This occurs, actually, any time a person's health is compromised, a way that the body conserves its energies. But it is harder to function and harder to heal while the energies are in that pattern. So one of the first things to do after a surgery is to begin to redirect the energies from a homolateral to a crossover pattern. I've never seen anyone right after a surgery who was not homolateral. To my amazement, Tanya was not! Her body was so prepared for the surgery that her energies just kept on flowing! I was elated! During Tanya's recovery, a variety of medications were prescribed for pain and swelling, and when some of them energy tested weak, the doctors were good enough to change the prescriptions until medications and herbs that were compatible with Tanya's energies were found.\n\nHad we not been led to this unusual surgeon, Tanya almost certainly would have been forced to have a hysterectomy, and in any case, would have lost the possibility of carrying a baby. Now, at the time of this writing, it is still a choice she can make. But whether or not she ultimately carries a child, I am thankful her uterus was preserved. The full consequences of a hysterectomy are often not disclosed to women. For instance, the female organs are tightly arranged so that removing the uterus, even if it is prolapsed, may itself cause a prolapse, a falling out of place, of another organ. One of my friends wound up in a wheelchair, unable to walk, due to complications from a hysterectomy. While the procedure should never be done casually, hysterectomies are performed almost routinely in the United States. They are carried out at nearly double the rate of that in England or Europe, suggesting that social and economic forces within the medical system are a strong factor in determining whether a hysterectomy is to be performed.\n\nNonetheless, my indebtedness and gratitude for the tools of modern medicine were immense around the time of Tanya's surgery, and they will never be forgotten. Even though my professional role seems to have become one that challenges the medical profession to expand its paradigm and soften its arrogance and rigidity, when surgery or pharmaceuticals are required and skillfully administered, I can only watch with awed respect that humanity has devised such marvels of compassion and healing. Tanya was, of course, very fortunate to have found a physician who also thinks outside the box, but ultimately it was her surgical skills that saved the day. I, in fact, have always viewed energy medicine and conventional medicine as complements to each other rather than competing approaches. Many of my cries of protest have been pleas for conventional medicine to come to that view as well. For instance, if you prepare your body energetically prior to surgery, and attend to it energetically following surgery, as I did with Tanya, the chances that your body will respond to the surgery as intended are enhanced, and your chances for a robust and speedy recovery greatly increased. When I revise _Energy Medicine_ for its tenth-anniversary edition (how time flies!) as soon as I complete this book, I plan to add a section on how to apply energy methods to support necessary but invasive medical procedures such as surgery. I'm thinking of calling it: \"If surgery becomes necessary: The cutting edge as a last resort.\"\n\n#  **Pregnancy**\n\nPregnancy is the miracle that occurs after two microdots of protoplasm, each the product of an independent universe, implausibly find their way to each other and merge into a new microdot. This new microbeing carries all the necessary informationfor creating a trillion-celled organism that hosts the most sophisticated computing system known, capable of building skyscrapers, writing poetry, exuding love, and transcending itself in mystical rapture. Much of this information had been carried in the energy fields of the egg and the sperm, and it is now in the _energy field_ of the new microbeing (called a zygote), rather than in its _genes,_ but that is another story, laid out in Chapter 1. Discussed here is the growth of the microbeing into a human baby, a play that unfolds in the theater of your womb. The stage management you provide, both physically and energetically, will impact the production in ways you already know and in some ways you might not. Following is a guide to the energetic aspects of that stage management.\n\nPregnant women came into my office for a myriad of reasons. Some came with concerns about the health of their baby. Some came because they were fearful of a Cesarean section and wanted to work with someone who would support a natural birth. Some came because they were swelling or suffering with morning sickness or other symptoms. Some wanted me to tell them the color of their baby's aura. But all desired to stay very healthy during their pregnancy so their baby would have an auspicious entry into the world.\n\n**Supporting the Health of the Baby.** Whatever the mother does to energetically stay in balance helps the baby. Do the basics laid out in Chapter 2, particularly the Daily Energy Routine and all the techniques that stretch your body and that keep your energies crossing over. As with everything else, but particularly with pregnancy, keeping spleen meridian strong is vital to your health. When a woman becomes pregnant, one of the earliest energetic changes to occur is the intensification of her spleen energies. Each meridian has a pulse that can be felt on the wrist, and one of the ways I know that a woman is pregnant is by checking her spleen pulse for otherwise unexplained increases in her spleen meridian energy. This increase is not surprising. Spleen meridian takes on a host of new tasks oriented toward nurturing the baby. So you want to be sure it has lots of reserve energy. The spleen\/triple warmer balancing sequence suggested to promote fertility (page 180) can do much to keep spleen meridian strong and balanced. The other meridians that are most intimately involved with fertility\u2014the kidney, liver, stomach, and circulation-sex meridians\u2014also have important roles in fostering a healthy pregnancy. Methods for supporting them are suggested below. One of my favorite ways to keep mother and baby connected follows.\n\n**Heart-to-Womb Connection** (time\u2014about 9 months)\n\nOne of my favorite ways to support the mother- baby bond is to bolster the energetic connection between heart and womb. Standing or lying down, take a few deep breaths in through your nose and out through your mouth. Place one hand over the middle of your chest and the other over your womb (Figure 5-7). After a few more deep breaths in through your nose and out through your mouth, breathe naturally. The breathing instructions help ensure a healthy flow of energy, blood, and oxygen between the lungs and the womb. And the power of your love toward your baby is laser-focused.\n\n**Paving the Way for a Natural Birth.** A pregnant woman whose first two children were delivered by Cesarean section came to me with the hopes of being able to deliver her third child naturally. Her obstetrician was amused by the techniques I gave her. He told her these were good to do to keep her relaxed, but that she should not fool herself into thinking she would not have to have another Cesarean section after already having had two. He insisted, in fact, that she could not continue to be his patient if she did not abandon her plan for a home delivery. A home delivery in her case felt foolhardy and dangerous to him. I, however, was encouraged by her progress, by the way the energies were no longer blocked in the areas where she had scar tissue from the first two Cesareans, and by how her whole abdominal area looked healthy and strong. I told her that I felt confident she could deliver naturally. She, understandably, bowed to her doctor's recommendation and dropped the plans for a home birth. When the time arrived, however, there was no choice to be made. The baby came about three days before the doctor expected, and with almost no warning. Labor came on suddenly. She was in bed. Her husband was home. They called the doctor and began preparing to go to the hospital. She made it as far as the livingroom. She squatted, and the baby came out. It was a remarkably easy birth. No complications. The doctor, to his credit, put substantial effort into finding out what had happened and wound up learning a great deal about energy medicine.\n\n_Figure 5-7_ HEART-TO-WOMB CONNECTION\n\n**Healthy Pregnancy Sequence** (time\u2014less than a minute)\n\nTo support a natural birth and decrease the chances of needing a Cesarean section, begin with everything recommended in the discussion under Supporting the Health of the Baby. In addition, to support the energies most involved with pregnancy, do the Healthy Pregnancy Sequence on a daily basis. Breathe deeply throughout the entire sequence.\n\n1. Push your thumbs into the points at the top of your cheekbones, directly beneath your eyes (Stomach 1 and 2). At the same time, lightly place the pads of your fingers on your forehead, on the bony area about two inches above your eyebrows (see Figure 5-8). Hold this position for at least a minute while breathing deeply. This part of the sequence can be done independently of the other two. Do-ing it several times per day reduces stress, relaxes the abdomen and uterus, and gives energy to the legs to support your growing body.\n\n2. Place your fingertips on the arch of your pubic bone and massage deeply the areas under each finger (these are neurolymphatic reflex points that open the energies in your entire pelvic region).\n\n3. Start with the middle finger of your left hand on the center of the pubic bone; move it left about two inches and drop it about one inch, staying in the groin crease. Press and gently massage the point there on your groin (Liver 12). With your right hand, massage the K-27 points (page 53). Repeat on the other side, massaging the liver point on the right side with your right hand and K-27 with your left hand.\n\n_Figure 5-8_ HEALTHY PREGNANCY SEQUENCE\n\n**Calming the Circulation-Sex Meridian.** During the last trimester of pregnancy, calm the circulation-sex meridian by having a friend or partner hold its sedating points (page 148). This helps your circulation, supports your own heart and your baby's heart, and reduces the lower back pain and exhaustion that can result from carrying the baby. If no one is available, and if it is difficult for you to reach the sedating points, gently hold the \"devil's horn points\" (about three inches above the ears) for two to three minutes. These are circulation-sex neurovascular points.\n\n**Balancing Your Root and Womb Chakras.** All of these procedures will help your energies flow and your hormones stay balanced so that your body's natural, organic rhythms take over in preparing you for the birth. Another extremely valuable procedure is to clear your chakras, particularly your root and womb chakras, the spiraling energies directly over your pubic bone and over your womb. An entire chapter of _Energy Medicine_ (Chapter 5) is devoted to understanding and working with your chakras, but here is a brief version of the procedure. While you can do this on yourself, it is much nicer to relax into it as another person does it for you. The Five-Minute Daily Energy Routine (page 51), or at least the Neurolymphatics, Crown Pull, and Hook-up, are good preliminaries. Then, circle either hand in a counterclockwise direction two to three inches above your pubic bone. Circle slowly for at least a couple of minutes, breathing deeply. Shake your hand off and circle over the same area in a clockwise direction for a minute or two. Repeat the same counterclockwise and clockwise circling over the area of your womb, just below your navel. End with figure-eight patterns over the entire area.\n\n**Goddess Belly Hold.** This technique keeps the energetic connection between you and your baby strong and clear by keeping your belt circuit in a good flow. The belt circuit, or belt flow, is a radiant circuit that surrounds your waist and, when you are pregnant, also moves below your belly, supporting the growth of the baby. It connects the energies of the top and bottom parts of your body, and allows a strong flow of blood, lymph, and oxygen to move through the womb and assist a healthy birth and baby. This vertical distribution of the energies is critical to the belt flow when you are pregnant. Place your hands at the sides of your waist with your thumbs on your back. Slowly and firmly pull your hands forward and bring them downward until they are cupping the bottom of your womb (see Figure 5-9). Take a deep breath and then slowly let go. Repeat at least one more time. This procedure feels wonderful for most people, and it reestablishes a top-to-bottom harmony in your body. If you can get another person to do this for you, it is a joy. You can lie down on a bed so the other person can place each hand at the center of your back and pull you up from behind your waist. As the person's hands pull out to the sides of your body, you can relax again into the bed.\n\n_Figure 5-9_ GODDESS BELLY HOLD\n\n**Breech Position.** A Cesarean section is often performed when the fetus is in a breech (feetfirst) or transverse (sideways) position. But energy work can sometimes get the baby back into the headfirst position that nature intended. The energies of the unborn child are distinct from those of the mother, and when you do a chakra balancing on the mother, you can feel the mother and the child responding independently. Feeling the child's response always makes me happy. Sometimes a baby is in no rush to get out, and if you relax the womb area with a chakra clearing, the baby will actually begin to follow your hand. In one case, a woman was overdue for delivery, and her child was in a breech position. The mother was paralyzed with fear, and did not want to have a Cesarean section. She asked me if I could find anything in her energy that was keeping her expected daughter, already named Melissa, from turning to the right position. Melissa's energy appeared healthy, but she was in no rush to get out. The only problem I could find was that the mother's abdominal muscles were tightly constricted from stress. I relaxed this area with a chakra clearing over her womb. After about two minutes of counterclockwise motion, it looked and felt to me as though Melissa's energy was getting in sync with mine and picking up the rhythm of my hand. After about fifteen minutes, she began to turn, as if following the energy of my hand, until she was upside down, ready for birth. Another person in the room was jumping up and down and screaming, \"I can see her turning!\" The mother went directly from my office to the hospital, where she gave birth to a beautiful baby girl. While you cannot always count on this technique to prevent a breech delivery, I have had nearly identical experiences several times. Also, since first describing this method in the original edition of _Energy Medicine,_ a number of midwives and nurses have told me that the technique led to the repositioning of a fetus, making a natural delivery possible.\n\n**Miscarriages.** Again, everything mentioned to this point supports a healthy birth: the Daily Energy Routine; enhancing crossover patterns; stimulating and balancing the spleen, triple warmer, stomach, kidney, liver, and circulation-sex meridians; balancing your root and womb chakras; and keeping your belt circuit flowing. Also of particular value in preventing things from going awry are techniques that keep the energies connected, such as the Hook-up (page 62) and Connecting Heaven and Earth (page 44). It is also important to accept that sometimes a miscarriage is nature's way of releasing a baby who is damaged or otherwise not right to live in this world and that nothing you could have done was going to change that.\n\nThree common problems during pregnancy are morning sickness, back pain, and toxemia or preeclampsia. Each can be addressed by energy medicine, often in partnership with other approaches.\n\n**Morning Sickness.** Bouts of nausea and vomiting affect between 50 and 95 percent of women during the first trimester of pregnancy. While not harmful to mother or child (unless they lead to dehydration or malnutrition), they are no fun either. Folk remedies that are helpful to some women include eating six smaller meals instead of three large ones, getting plenty of rest and good exercise, drinking fluids 30 minutes before or after meals but not with meals, sniffing lemons or ginger, and taking about 200 mg of vitamin B6 daily. A variety of energy techniques can also help bring morning freedom.\n\n1. The quickest and most direct step you can take for morning sickness is the Hook-up (page 62), placing your middle fingers at your navel and third eye, pushing in, and pulling up. Hold for at least three deep breaths. This hooks up the central and governing meridians so that energies can realign.\n\n2. A second intervention is to squeeze the bottom corners of one big toenail while your other hand pushes up under your cheekbone beneath your eye. Sit this way for a minute or two, breathing consciously. Repeat on the other side. These two techniques will alleviate morning sickness for many women. If they do not work for you, the following helps with the most common patterns. None of them will hurt you, so experiment and see which works best for you.\n\n3. Rub your hands together until you feel warmth. Place one hand where you feel discomfort and the other hand on the back of your neck. Breathe deeply and hold for a couple of minutes.\n\n4. Become as comfortable as you can in a chair. With one hand, grasp with your thumb and middle finger on the back of your foot, just behind your ankle. Squeeze the area behind your ankle on either side of the Achilles tendon. At the same time, with your other hand, place your middle finger beneath your cheekbone and push upward (see Figure 5-10). Hold for a couple of minutes. Repeat on the other ankle and cheekbone.\n\n5. Hold the first set of small intestine meridian sedating points (page 145) for about two minutes, and then lay the heels of your hands against your cheekbones, with your thumbs on your temples and your fingers on your forehead, for about a minute.\n\n_Figure 5-10_ GOOD MORNINGS POSITION\n\nAt least one or two of these should work quite well, and that is all you need to use. Experiment to find out what works for you. The feedback will be almost instantaneous.\n\n**Back Pain.** The weight of the growing baby places tremendous physical pressure on the expectant mother's back.\n\n1. Having your partner or a friend do a Spinal Flush daily (page 60) prevents stagnant energies from accumulating as the back does double duty, and this keeps energies moving through and strengthening your back.\n\n2. If you are feeling pain around your kidneys, sedating the kidney meridian (page 145, following the special instruction there for pregnant women) releases tension in the kidney itself. If you are more than a few months pregnant, you probably will not be able to reach the points on your feet and will need to ask your partner or a friend to hold the points. If sedating the kidney meridian did not alleviate the pain, sedate the large intestine meridian (page 146), which governs the muscles at the waist in the back.\n\n3. If the pain is in your gluteal muscles or sacrum, sedate the circulation-sex meridian (page 148). Again, you will need your partner or a friend to hold the points.\n\n**Toxemia.** Toxemia, also called preeclampsia, is characterized by high blood pressure, swelling that will not subside, and large amounts of protein in the urine. All the techniques suggested above, such as maintaining a balance between the spleen and triple warmer meridians and keeping the liver and kidney meridians in a good flow, help prevent or reduce toxemia. The Five-Minute Daily Energy Routine is usually enough to maintain these balances, though sedating triple warmer, liver, and kidney meridians, and strengthening spleen, gives an extra boost. Massaging the lymphatics (page 59) can also further reduce toxemia.\n\n#  **Delivery**\n\nMy dear friend Sandy Wand, one of Ashland's legendary midwives, tells me that at about eight months into pregnancy, it occurs to the expectant mother that the \"microdot\" (she had just read a draft of the pregnancy section above), which has now become a beach ball, is going to have to find a way out through an opening that suited the microdot but seems quite inadequate for the beach ball. Where all the awareness had been focused on the pregnancy and growing baby, now there is a shift of awareness that this pregnancy is about to lead to a birth. During the last weeks of pregnancy, a natural energy comes in to prepare for delivery, changing physiology and consciousness. The pelvic floor is spreading. The baby is dropping into it. The baby's head becomes engaged in the pelvic floor, and the baby settles into the position in which it will be birthed. There is no escape, and biology bypasses all theories about how this is going to happen.\n\nFortunately, the body _knows_ how to give birth. And it can be supported in many ways. Simple techniques such as the Hook-up (page 62) or having your partner or friend press in for about a minute at the power point (the center indentation where the head joins the back of the neck) help calm the nervous system and reorient the body's energies.\n\nDuring labor, contractions create tension. Energy can be pulled off the body between contractions by simply smoothing with flat hands down the body and off the feet. Because labor is such an intense physical and emotional experience, gently placing your hands on the neurovascular points on the forehead or behind the knees (some women do not want to be touched on the head during labor) can relax the body, release anxiety, and calm the mind so that the body can do what it knows how to do naturally. This is a simple intervention that may yield major benefits. Reducing tension and panic allows the body's organic wisdom to operate unimpeded. Gently smoothing the forehead with a damp cloth stimulates the same neurovasculars.\n\nPlacing your hands on the woman's sacrum with pressure during contractions helps keep a strong current moving in the radiant circuit called the penetrating flow. This taps into and supports the womb's powerful energies, aligns the mother with the energies of the baby, reduces pain in the lower back, and harnesses all the radiant circuits, the natural resource energies for delivery. Holding the sides of the woman's feet early during labor connects all the body's energy systems and helps them coordinate with one another. A method called the Brazilian Toe Technique can be enormously comforting during labor, bringing greater efficiency to all the body's energy systems. When you do this for someone, the recipient is lying on her back. Be sure to also find a comfortable position for yourself.\n\n**Brazilian Toe Technique** (time\u201410 to 15 minutes)\n\n1. Position yourself at the feet of the mother-to-be. Place your thumbs beneath her middle toes and your middle fingers on top of the same toes, on the toenails (see Figure 5-11). Hold lightly for two to three minutes. Breathe in through your nose and out through your mouth.\n\n2. Slide your thumbs to beneath her fourth toes, then slide your fourth fingers on top of the fourth toes. Hold lightly for two to three minutes. Continue the same breathing.\n\n3. Slide your thumbs beneath her little toes, then slide your little fingers on top of the toenails of her little toes. Hold lightly for two to three minutes.\n\n4. Slide your thumbs beneath her second toes, then slide your index fingers on top of the toenails of her second toes. Hold lightly for two to three minutes.\n\n5. Slide your thumb beneath her big toes, then place your index fingers and your middle fingers so that one is on each side of the base of the toenails of her big toes. Hold lightly for two to three minutes.\n\n_Figure 5-11_ BRAZILIAN TOE TECHNIQUE\n\nEvery physical and mental resource a woman has is pushed to and beyond Olympian proportions during labor. In giving us a skull large enough to contain a human brain while streamlining our pelvis to allow us to walk on two legs, nature was pressed to make some ingenious compromises as well as some sacrifices in evolving the modern human, and easy childbirth was one of those sacrifices. A culture's practices in counteracting this big-brain\/small-pelvis dilemma tells a great deal about its compassion, wisdom, and attunement to the feminine principle, a topic I will skip before I go into a diatribe of anguished bewilderment. Suffice it to say that the midwife tradition has much to teach us, and a few Western hospitals are making great positive strides. And whatever the birthing environment,wherever nature has made significant biological compromises, things can go wrong.\n\nIt is beyond the scope of this book to go into the emergency procedures that may be required during a birth, and Western medicine excels at saving lives in desperate situations. Yet even here, less invasive energy techniques are available as a first response, and they can often prevent the need for more Draconian measures. For instance, if there is weakness or too much variability in the fetal heartbeat, holding the acupressure strengthening points on the mother's heart and spleen meridians, or tracing her heart meridian, will often resonate between mother and child, strengthening and stabilizing the baby's heartbeat and decreasing the likelihood that medication will be needed. If, after a certain level of dilation and contraction activity, the labor stalls, techniques for getting the energies grounded and moving\u2014such as squeezing and massaging the sides and tops of the feet, holding and swaying the mother in figure-eight movements, or simply getting her to move her body\u2014will often spontaneously restore progress.\n\nWhen I went into labor with my daughter Dondi, my doctor, Paul Brenner, M.D. (a legend in our family about what a doctor ought to be), was with another patient in a different location and couldn't get to the hospital for a couple more hours. He was able to contact one of his friends, and he got him to agree to immediately go to the hospital to be with me. He gave him two directives, to hold me when contractions came and to keep the nurses away from me until Paul arrived. This man not only wasn't a doctor, he showed up in a football uniform, cleats and all. Whenever a contraction would come, he would hold me as instructed, and he also instinctively rocked me in a figure-eight manner. It was incredibly comforting. My body would relax in these big arms. I felt totally taken care of. When Paul arrived, he took over these duties, also instinctively rocking me in a figure-eight motion. He asked me what would make this the most wonderful birth experience possible. I said that I would really like to be able to watch the birth. A couple of hours later, a giant mirror was installed on the wall of the room where I would be giving birth. With a beaming smile, Paul said, \"Take a look, Donna! Happy 'birth' day!\" After the delivery of a beautiful baby girl, Paul was crying openly and hugging me and my perplexed husband. A few years later, after no contact, Paul called me at home about an unrelated matter and three-year-old Dondi answered. He said, \"This is Paul Brenner. Is your mommy there?\" Dondi squealed, \"Paul Brenner! I'm your baby!\"\n\nFemale sexuality, fertility, pregnancy, and delivery are a woman's distinctive birthrights. The culture in which a woman lives maps a route through each of these profound journeys. The maps provided by our culture have become increasingly estranged from our most fundamental energies, natural rhythms, and biological foundations. Outside our cultural frameworks, however, are maps that are more life-affirming, more keyed to our highest potentials. As we fully grasp the situation, we are pressed to become cartographers, mapping old wisdom into new territory that holds both opportunities and threats our grandmothers could not have imagined. We are, through the very lives we are living in these challenging times, changing the maps and fueling hope for the next generation about what is possible for a woman. I hope the tools presented in this chapter support you in these grand adventures.\n\n_Chapter 6_\n\nMenopause\u2014Gateway to Your _Second_ Prime of Life\n\n_Menopause is not just the end of fertility and childbearing years that are so often seen as women's prime. It is the time for a woman to take charge of her health and make the changes that will carry her through her_ second _prime of life._\n\n\u2014 BERNADINE HEALY, M.D. _Former Director, National Institutes of Health_\n\nMenopause is a journey of mythic proportions. It is the gateway into your second biological prime! My intention with this chapter is to help you navigate through that gateway into a _richer_ second prime. Let me know if I succeeded: www.ed-em.com\/2ndprime-survey.htm.\n\nUntil you get there, it is hard to imagine how strange this territory can be. Since I knew PMS at its worst, I thought I would sail through menopause. You sometimes get off easy on one if the other is dreadful, much as I had enjoyed easy pregnancies after monthly PMS agony. Women like me, who are rich in estrogen and thus have difficult periods, often have an easier time of pregnancy and menopause because the ratio of progesterone, which we are low on, increases in relationship to estrogen. I, however, turned out to be a victim of my own expertise. Natural progesterone had so firmly become my drug of choice by serving me magnificently during the PMS times that I kept taking it into menopause, and I threw off my progesterone\/estrogen balance. So I wound up with the biochemistry of a more typical menopause instead of taking advantage of my natural excess estrogen. I had the experience for the first time of drenched sheets, insomnia, and a feeling of deadness, none of which would have occurred if I had not overdosed on progesterone. Sometimes you just can't win. But you can learn from my mistakes. Read on.\n\nDozens of books on menopause can be found in any major bookstore, but to my knowledge this is the first whose approach is based primarily on energy medicine. Several, however, are excellent compilations of conventional medical as well as folk remedies for common symptoms (e.g., Wingert and Kantrowitz's _Is It Hot in Here?_ 2). I have also loved five masterpieces that address the topic from their own unique angles, written by Leslie Kenton, Ph.D., Susun Weed, Christiane Northrup, M.D., John Lee, M.D.,and Susan Lark, M.D., respectively. While it is beyond the scope of this chapter to attempt to cover all the material in these superb resources, some of the essential ideas that provide a context for an energy-based approach to menopause follow.\n\nComing into your second prime requires a major rearrangement of the perspectives and values that served you earlier in your life. Many of the books about menopause focus on the psychological challenges, which are substantial. Our focus, however, is on energy tools that can help you move through menopause with greater ease and in better health. This is the best concrete preparation I know for the psychological and spiritual as well as the physical journey of menopause. Energy medicine helps you at all levels.\n\n#  **PREPARING BODY, MIND, AND SPIRIT FOR THE JOURNEY**\n\nEvery emotional challenge is a physiological experience. Trying to think your way out of depression or anxiety usually does not work. Our culture may tell us that will and determination are all that are needed, but it would do much better teaching us how to move our energies when they are stuck rather than teaching us to blame ourselves for _being_ stuck. It is not \"all in your mind\"; it is all in your energies. The emotions of menopause reside in the chemistries of our bodies, and they are powerful. At the same time, massive, cataclysmic shifts are retooling your body from a baby-making machine to an independence you never before could have imagined. While nothing is going to stop you from going through this tumultuous process, energy medicine can deeply support you during the transition and help you come out of it healthier, happier, and more independent. By becoming adept at balancing your energies, you are better able to navigate your way into the gifts of menopause: empowerment, self-reliance, and freedom from the compulsion to cater to others' needs or agendas for you.\n\nMy own passage through menopause was great training for being attuned to the women who consulted me in my practice. After you have served as your own laboratory, you get much better at guiding others. As it happened, most of my closest friends started through menopause about four years after me. Still in the experience myself, but a little further along, I was able to resonate deeply with their complaints, terror, and confusion.\n\n_One day I dropped in on a friend who was known for being extremely fiery and opinionated, a powerhouse personality. I hadn't heard from her in a while and wanted to check in. She casually told me, in a dry humdrum voice, that she no longer cared about me. It wasn't personal at all, she assured me. She didn't care any longer about anybody or anything. The well of her human passion, along with her compassion, had dried up. She said she didn't desire a thing. Her hearty sexual appetite was gone. She felt dead to all of life. And she hated her itchy, flaky, dry skin. When I offered to give her a session to help her deal with her dead emotions, she was eye-roll irritated and felt patronized. She was, however, willing to let me help her with the torture of her itchy skin._\n\n_So I had her lie down on the floor, and we got started. I energy tested every meridian in her body that might be involved with skin conditions and corrected each imbalance. By the end, her skin wasn't itching anymore. This got her attention. The relief was so great, in fact, that she wanted more. We went to work on her hormones. While several energy systems were involved\u2014such as her chakras and radiant circuits\u2014her liver meridian turned out to be the key. Deep energies that had seemed lost to her were again available. At one point, to her surprise, she began to sob. She had believed she would never feel enough to cry again, and her tears seemed to open a floodgate for all the liquids in her body. She could cry again. She could feel sexy again. Her dead mood began to lift, and all of life began to seem juicy. Over a number of sessions, these shifts became more stable. She no longer felt \"dried up.\" While her body would continue for some time to go in_ _and out of chemical imbalances, the fluctuations became far less extreme, and with new energy tools in her back pocket, menopause became a bothersome challenge rather than a hopeless descent into the void._\n\nIt is striking to me how the decade between the years of approximately 48 and 58 often determines whether a woman finds her way into a vibrant second prime\u2014 strong, healthy, and with higher self-esteem than before\u2014or a period of rapid aging, as if the woman's life is winding down. While many psychological and physiological factors are involved, the common denominator seems to be that the women who find their way into a second prime actively engage their body's energies (whether or not they are specifically thinking in those terms), not accepting at all that this is a downward spiral. Those who have come to me for help presented symptoms associated with menopause, but their journeys included hearing what their bodies were really trying to tell them\u2014in their native language, energy\u2014and aligning themselves with their bodies' requests.\n\n##  Menopause as Nature's Plan for You\n\nFrom maiden to mother to crone: the archetypal phases of a woman's life. The spirit of the crone arises in the postmenopausal woman, according to Leslie Kenton, and becomes \"deeply and spontaneously sexual, assertive, straight, incorruptible, prophetic, intuitive, and free . . . the qualities that most terrify the patriarchal culture in which we live.\"\n\nIn the midst of my own menopause, with its multiple emotional and physical challenges, a new energy slowly began to emerge, a force that seemed to come up from the earth and permeate my cells. It was a power I'd not known before. I was new to this realm. I was in undeniably unfamiliar territory with novel demands and newfound empowerment. While I wasn't exactly thinking in these terms, I was entering the world of the crone. I liked it. Despite the emerging challenges, I felt more whole, and certainly more assertive. I was free in ways I'd not known before. A silly incident stands out in my memory as an emblem of this new assertiveness. I was crossing the street, lost in my thoughts, and an impatient young man, behind the wheel of a shiny new convertible, \"flipped me off.\" I guess I was taking too long to get across. While in the past I would have felt terrible for having inconvenienced him, I was now totally amused. How stupid it seemed for him to spend his energy and upset on something so trivial. I turned to him and said with calm assertiveness, \"Hey, I'm fifty years old. I don't have time to go into that space. And you don't, either!\" As his eyes met mine, his hostile expression turned into a quizzical look, and then it was as if he really saw me for the first time. He said, \"No, we don't, do we? Thank you!\" By fully embracing the way of her own heart, the crone invokes the power to change the way others think about their choices.\n\nAfter half a lifetime of subduing your more independent qualities to serve husband, children, and society, you are ready to be subdued no more. Your goals become clearer, your mind more focused, and your power unleashed. The wisdom of innumerable experiences congeals, and you find yourself to be a teacher and leader in your extended family and in your community. That is nature's plan for you. Are you ready?\n\nThis plan, for a second prime, is supported by the hormonal changes that correspond with menopause. It is programmed into our genes. Our sexuality is another remarkable evolutionary achievement. No other creature on Earth, male or female, comes close to having our experience as sexual beings.\n\nAccording to Leonard Shlain, \"a yawning chasm separates the reproductive life [of human females] from that of the females of the other three million sexually reproducing species.\" We are the only female species capable of sexual union year-round rather than only when we are \"in heat.\" We are the only female species with such dramatic menses or with periods that are synchronized with the moon or that discharge so much blood that we are, in the wild, all but dependent on animal meat to replace the iron we lose. We are the only species with a brain that is capable of overriding the circuitry that demands obedience to our sexual urges, allowing us to refrain from sexuality for the purpose of choosing when and if we become pregnant, and also allowing us to be highly selective in our choice of a mate. In contrast, a female chimp mates, on average, 138 times with thirteen different males for each infant she births. We are the only female species capable of prolonged and multiple orgasms of such intense and sustained pleasure. We are the only female species whose architecture contributes to extreme pain during childbirth. And we are the only females to experience menopause so early in relationship to our potential life span.\n\nShlain's rather extraordinary hypothesis is that these distinctive qualities of human female sexuality combined to drive human evolution forward so we would thrive by intellect rather than brawn. We can't outrun or outclaw a lioness, but we can outplan her! Foresight\u2014\"the ability to maneuver conceptually in the dimension of time\"\u2014ultimately provided us with control over no less than the earth's destiny. Women's sexuality, according to Shlain, \"taught our species how to tell time.\" Connecting the lunar and menstrual cycles, noticing the relationship between intercourse and the cessation of menses, and observing that babies appear nine moons after the last menses were all training grounds for beginning to understand the workings of time. Many of these reproductive oddities were actually compromises. They did not, in themselves, advance our biological fitness. But they combined to show us how present events relate to the past and to the future. Mortal risk at childbirth, for instance, increased exponentially because of the biological compromises that were necessary when our skull size increased to accommodate a large brain and our pelvis was redesigned to allow us to walk on two legs. There was, in fact, a period of history when so many women were dying in childbirth that our entire species was at risk. What was the evolutionary advantage in that? But we survived because women figured out the relationship between sex and subsequent pregnancy, and began to take control of their reproductive lives.\n\nDiscussing menopause, Shlain reaches back to Hecate, the prototypical crone of Greek mythology, recognized by both men and women for her freedom, courage, strength, wisdom, and magical powers. Although the crone certainly earned the reverence she received by adeptly distilling a lifetime of experience, another major factor contributes to her potency. This, according to Shlain, involves \"the dramatic realignment of the serum concentrations of her estrogen, progesterone, and testosterone.\"  While menopause launches a steep drop in all three hormones, the proportion of testosterone in relation to the other two jumps sharply. Estrogen production often decreases 70 to 80 percent following menopause; progesterone production decreases by more than 99 percent; but testosterone production decreases by only 50 percent or less. This means that the amount of testosterone relative to estrogen often doubles (in some cases, it increases twentyfold) and, relative to progesterone, it increases at least fiftyfold. With these radical changes in the balances among the hormones that control personality come dramatic shifts in a woman's psyche. The indecision and pliability that often mark a woman's youth \"are replaced by clearheaded assertiveness.\" And when mature women, \"brimming with testosterone (relatively speaking),\" are freed from child-rearing duties and reenter the world on their own terms, they \"begin to exert a wider influence on the welfare of the society.\"\n\nAs Margaret Mead once said, \"There is no greater power in the world then the zest of a postmenopausal woman.\" During our first prime, we are wired to bear children and nurture a family. Then, in our second prime, we are wired to lead our communities with passion and zest\u2014using all the pragmatic ways of loving and caring we learned during our first prime as training for the job. This is, in my humble opinion, much better training for community leadership than is a political science degree.\n\n##  Menopause: New Chemistry\/New Energy\n\nThe transformation from a biochemistry that is oriented toward hosting the monthly drama of releasing an egg and preparing it for conception to a biochemistry supporting newfound independence is indeed jarring. In addition, estrogen and progesterone serve many functions beyond promoting sexual desire and supporting pregnancy. The effect of these hormones on secondary characteristics\u2014from skin texture to hair luster to bone strength\u2014also goes into turmoil at menopause. Seventy-five percent of the 40 million menopausal women in the United States complain of symptoms associated with these changes.\n\nThe appeal of hormone replacement therapy (HRT) is not surprising. And on closer analysis, some of the dangers revealed in the 2002 NIH study of 16,608 women on estrogen therapy (page 93) are not as definitive as originally assumed. When examined according to age groups, for instance, women who received HRT within the first decade after the onset of menopause did not show the increased risk of heart disease found in the overall study. On the other hand, the study has been criticized for underrepresenting the dangers of HRT. Potential participants were screened out of the study if they had some of the risk factors for heart disease, diabetes, stroke, or breast cancer, while a percentage of the actual population being prescribed HRT in the real world has those risk factors. In addition, 40 percent of the women in the study dropped out, usually due to side effects. Increased incidences of cancer and heart disease in the women who had the most adverse initial reactions to HRT were, therefore, not tracked.\n\nThe current advice from the NIH and the Food and Drug Administration (FDA) is that women should discuss with their doctor the risks and benefits before embarking on hormone therapy treatment (the standard fallback when science knows it doesn't have the answers). They also suggest that the dosage and duration of therapy should be the least amounts possible for achieving the treatment goals. The tone of caution in these recommendations is well founded. At least 25 percent of the women who do start HRT discontinue it due to side effects. The official recommendations, however, assume that doctors can make good choices based on bad information, with the great error in the original study being that rather than natural progesterone, the synthetic and far-inferior progestin was used.\n\nWe don't know from the NIH study what impact natural progesterone might have had. The molecules in natural progesterone fit the receptor sites in a woman's body like lock and key. The molecules in the synthetic progestins have been altered so that they could be patented, and the body often does not respond in the desired ways, with adverse reactions being common.\n\nThe late John Lee, M.D., investigated the effects of natural progesterone on thousands of menopausal women. He suggested that many of the symptoms that are treated with estrogen are actually caused by progesterone deficiency, not estrogen deficiency. Too much estrogen, in relation to progesterone, becomes toxic to the body. The truth is, none of us can avoid inadvertent \"estrogen replacement therapy\" because our environment bathes us in estrogen. The earlier onset of puberty, which came on average at about 14 years a century ago and is now closer to 12 (with a girl's first period at eight or nine not being unusual), has been linked to the hormones indiscriminately used to puff up meat and dairy products, and to the powerful estrogens in the sprays used on grains fed to animals. Our estrogen levels can actually be raised by car exhaust, liquids we drink from plastic containers, and the chemicals found in nail polish, glue, soaps, cosmetics, and paint remover.\n\nTo support this thesis, Lee pointed out that for women in nonindustrial cultures, menopausal complaints are minor or unknown. In many agrarian cultures, there isn't even a word for \"hot flash,\" and symptoms such as vaginal dryness, osteoporosis, and mood swings are not associated with the time when a woman stops menstruating. With the introduction of industrialization to Third World countries, resulting in less exercise, greater calorie intake, and greater exposure to external estrogens, dramatic increases in the incidence of menopausal symptoms have occurred. Lee laments that \"we have managed to make menopause, a perfectly natural part of a woman's life cycle, into a disease.\" Implicated here are environmental pollutants, poor diet, unhealthy lifestyle, cultural attitudes, and the incorrect use of synthetic hormones.\n\nLee's thesis is that few Western women are truly deficient in estrogen. Rather, at menopause, most become deficient in progesterone. When symptoms such as breast tenderness, weight gain, bloating, and mood swings occur in adolescent girls, they are attributed to increases in estrogen; when these same symptoms occur in women during the transition to menopause, they are attributed to \"estrogen deficiency.\" Rather than estrogen deficiency, \"estrogen dominance\" is the problem, according to Lee. While estrogen is certainly necessary for good health throughout a woman's life, it becomes toxic in quantities that are too high or that are not balanced by sufficient progesterone. Lee acknowledged that some women may need estrogen supplements at menopause (and estrogen deficiency can be confirmed with a salivary hormone-level test). Most estrogen is made in body fat after the ovaries stop producing it, so these women tend to be thin and petite. But for most women, Lee was suggesting that the pharmaceutical and medical establishments have it upside down when it comes to estrogen replacement.\n\nLee was not alone. In 1991, feminist Germaine Greer impugned the proponents of HRT for not having established that estrogen deficiency is the cause of what they are attempting to treat, a situation that still exists. Sandra Coney's 1994 _The Menopause Industry_ argued that women were making disastrous health choices after being subjected to \"a demoralizing propaganda campaign aimed at brainwashing them into accepting that they are 'estrogen deficient,' in other words, defective in their normal state.\" Jerilyn Prior, M.D., professor of endocrinology at the University of British Columbia, points out that it is \"backward science\" to assume that the symptoms of menopause are tied to decreased estrogen levels, kind of like \"calling a headache an aspirin deficiency disease.\"\n\nLee summarized: \"The estrogen 'deficiency' hypothesis is not supported by the facts of estrogen blood levels, by worldwide ecologic surveys, or by endocrinology experts.\" His remedies include simple lifestyle choices such as using natural progesterone, adequate exercise, and good diet, including organic meat and dairy products that have not been filled with artificial hormones. Natural progesterone addresses many of the problems caused by the \"estrogen dominance\" that Lee believed is almost epidemic. Progesterone cream frequently reduces or eliminates symptoms such as hot flashes, night sweats, lowered libido, and \"middle-age\" weight gain; helps protect against breast cancer; and helps stop and even reverse osteoporosis.\n\nWomen to Women, the first medical clinic devoted to health care for women, by women, is now also a source of Internet information and services (www .womentowomen.com). They recommend that any hormone replacement therapy use bioidentical hormones, which are manufactured to have the same molecular structure as the hormones made by your own body. Synthetic hormones are intentionally different, not for health reasons, but because the pharmaceutical companies can't patent bioidentical structures, even though they are proving to be more effective and less likely to produce side effects. However, while I agree in theory about bioidentical hormones, I have also seen bioidentical hormones cause side effects. This is why I rely so strongly on energy testing, not only for the choice of supplement, but also for when and how much is needed from one time to the next.\n\nDifferent women metabolize different hormones in different ways, and their hormonal balances are ever-changing. If the supplement, whether bioidentical or synthetic, throws you into hormonal imbalance, it is doing harm. In addition, so-called bioidentical hormones made by compounding pharmacies are not always truly bioidentical, and they are not as carefully regulated by the FDA as those made by pharmaceutical companies. One lot may have a different concentration in its hormonal level from the next. Any drug carries risks! Women to Women's guideline is to get symptom relief at the lowest possible dosage.\n\nYour doctor, however, is more likely to have been informed by the pharmaceutical industry, which promotes a combination of synthetic estrogens and progestins. Such contradictory information from authorities is mirrored not only in HRT, but also in nearly every issue women face at menopause, from keeping our skin soft to keeping our bones strong. Where do we turn amid the conflicting information?\n\n#  **AN ENERGY APPROACH TO THE CHEMISTRY OF MENOPAUSE**\n\nShort of moving to a nonindustrialized culture, many basics of diet, exercise, and lifestyle are within your control and are not matters of controversy. Start there. Meanwhile, the energies that are the living presence of the hormonal changes during menopause can be managed directly.\n\nIn my practice, I found that helping a woman come to her optimal hormonal balance during perimenopause, menopause, and afterward would generally proceed along three basic steps. More often than you might think, the first step was in itself sufficient. Simply balancing a woman's energies and teaching her how to keep them balanced led to tremendous improvements in how she felt. Many times the woman initially believed she needed estrogen or progesterone, and balancing her energies caused her body to begin producing the hormones she was lacking. But if this was not enough, the second step was to use energy testing to determine which herbs or hormones would be beneficial. The third step was to figure out quantities and timing for using these substances in harmony with her body while minimizing side effects or overdependence.\n\n**First Step: Energy Balancing.** Before considering more invasive measures, such as taking hormones that your body didn't produce, a first step is to get your system into optimal energetic balance, which in turn balances your chemistry naturally. You learned the Five-Minute Daily Energy Routine in Chapter 2, and that is the ground floor for keeping your energies strong and balanced during menopause. In this chapter, you will be introduced to (1) a special Menopause Module to add to your Daily Energy Routine, designed to promote hormonal balance for women who are entering or are in menopause, (2) specific exercises for increasing your body's production of estrogen, progesterone, and testosterone, along with methods for figuring out which your body needs, and (3) techniques to address specific symptoms of menopause. Energy exercises that correct a specific problem also help harmonize your entire energy system.\n\n**Second Step: Energy Testing Vitamins, Herbs, and Hormones.** \"The healthy body,\" according to Christiane Northrup, \"is equipped to produce all the hormones a woman needs throughout her life.\" Using an energy approach supports this natural capacity. However, if you are lacking a vital ingredient necessary for health\u2014and it is not easy today to ensure a truly balanced diet attuned to your unique physiological needs\u2014your body has to work hard to compensate. If you have gotten your energies into good balance and symptoms persist, the next step is to evaluate whether you need to take natural hormones or other substances. Lab tests can give clues, but hormone levels fluctuate markedly on a daily and even hourly basis. Because estrogen is so potent and is produced only in minute quantities compared to progesterone and testosterone, \"estrogen dominance\" (see page 215) is often at play even with relatively low estrogen levels. Thus, many women get prescriptions for estrogen that their bodies do not need and that may actually be harmful. Meanwhile, many over-the-counter remedies are also available. Some may be exactly right for your body. The challenge, of course, is figuring out what you need. This requires educating yourself, finding conventional medical and pharmaceutical information about the choices you are considering, and consulting sources that favor a more natural perspective. The People's Pharmacy (www.peoplespharmacy.com\/index.asp) is a superb source for such information. Once you have identified promising vitamins, minerals, herbs, or hormones, energy testing is the best way I know to determine, day by day, whether what you are considering is what you need.\n\n**Third Step: Minimize Dependence, Maximize Balance.** Natural hormones and botanical products are readily available. Why not just routinely take such supplements? There are reasons for caution when interfering with nature's way. Adding chemicals that are already produced naturally may result in your body producing less of those chemicals. Balances may be thrown off, such as when my overreliance on progesterone caused a permanent thickening of my uterus. What happens, for women who take HRT, to the natural and desirable increase of testosterone, relative to the estrogen and progesterone, that fosters the power of the crone? This is highly complex territory, and it is not surprising that one expert or another advocates virtually every solution imaginable. We are all so different that almost any avenue will work for some women. So-called experts often develop a career by advocating for everyone an approach that works only for some. The menopause industry\u2014on both the pharmaceutical and the natural-healing sides\u2014 is a multibillion-dollar enterprise that lacks good road maps. Again, this is where energy testing becomes invaluable. My needs for progesterone and estrogen change on a daily basis, and energy testing gives me a gauge to learn what my body requires and doesn't require during a particular time frame. As you use energy testing, you also begin to notice patterns, so you become less reliant on the energy test. If, for instance, I am feeling a bit hysterical, it is often a sign that my progesterone level has gotten too low, and I may do energy exercises to raise it or may put a drop or two from a natural progesterone liquid capsule onto my wrist. In the rare cases that I feel deadness, depression, or a top-of-the-head headache, it is usually because I have used too much progesterone and my balance of estrogen relative to the progesterone has been thrown off.\n\nBalance is, again, the essential concept and, again, energy testing can be a great gauge. You will also find that once you are regularly balancing your energies, your need for supplemental hormones will decrease. The remainder of this chapter details the first of these three basic strategies, energy balancing. It presents an add-on module to the Daily Energy Routine for women who are dealing with menopause. It shows you how to use energy techniques to increase the production of estrogen, progesterone, thyroid, or testosterone, as needed. And it discusses specific symptoms that are often associated with menopause, such as osteoporosis, hot flashes, and depression, suggesting ways of working with your energies when such symptoms occur.\n\n##  Adding a Menopause Module to Your Daily Energy Routine\n\nThe Daily Energy Routine (page 51) is a cornerstone of how I teach energy medicine. One exercise, however, if you need it, even trumps the daily routine, and that is the Homolateral Crossover (page 67). If you are short on energy, brain power, motivation, or spirit, or if you are depressed, your energies are probably moving in a homolateral pattern. You don't need an energy test to tell you that something is off. You can, however, verify the homolateral pattern with the energy test described on page 68. If your energies are homolateral, start with the Homolateral Crossover, even before the Daily Energy Routine. If your energies are not crossing over, little else you do\u2014exercise, massage, or energy techniques\u2014will yield its full benefit.\n\nIf you are in the midst of menopause, or just noticing the first signs of menopause, I suggest that you add the following six techniques to your Daily Energy Routine. As a group they will require less than five additional minutes, and they will have a balancing effect on your hormones. Three of them have already been introduced as part of the Premenstrual Module: Connecting Heaven and Earth (about two minutes, page 44), the Spleen Meridian Flush and Tap (about 30 seconds, page 136), and the Abdominal Stretch (about 30 seconds, page 134). To these, add the Diaphragm Breath, the Hormone Hook-up, and Sunrise\/Sunset. Insert all six techniques right after the Lymphatic Massage (page 59). Then end the routine as usual, with the Blow-out\/ Zip-up\/Hook-in.\n\n**Diaphragm Breath** (time\u2014about 30 seconds)\n\nThe diaphragm is a strong, thin shelf of muscle partitioning the chest from the abdomen. It fans oxygen throughout the body. The following exercise not only helps distribute oxygen to every cell, gland, and organ, it also enhances the coordination and effectiveness of the hormone system.\n\n_Figure 6-1_ DIAPHRAGM BREATH\n\n1. Firmly place your left hand under the center of your rib cage and place your right hand on top of it. With your hands flat, pull your elbows close to your body so you are hugging your midsection (see Figure 6-1).\n\n2. Inhale deeply and push your body toward your hands while your hands push back against your body. Hold your breath and push hard. Although there is no set amount of time, the longer you hold your breath and push (without becoming light-headed), the better.\n\n3. Release your breath naturally, along with your hands.\n\n4. Relax. Do two more times.\n\n**Hormone Hook-up** (time\u201440 seconds)\n\nThe pineal, pituitary, and hypothalamus glands form a complex axis. Through direct feedback to one another they control the release of many of the hormones that govern your body. Points on the scalp and skin called neurovascular reflex points, when touched lightly, increase the flow of blood in the general area of the point being touched. By holding more than one neurovascular reflex point, you can increase the flow of blood between the areas being held. Because the bloodstream is a primary mode of transportation for the hormones, touching the specific neurovascular reflex points described below coordinates the hormones regulated by the pineal-pituitary-hypothalamus axis, three of the glands that most strongly influence the body's hormones. Doing the Hormone Hook-up daily is like taking an energy vitamin that supports harmony among your hormones. It has the added benefit of reducing stress and helping clear the body of stress chemicals. It feels good and is good to do on a routine basis as well as any time you sense you might benefit from it.\n\n1. Bring the thumb, index, and middle fingers of your right hand together and place this \"three-finger cluster\" at the top of your head.\n\n2. Simultaneously make a three-finger cluster with your left hand and place it just over the curve behind the top of your head. Hold both points for the length of three deep breaths.\n\n3. Flatten your right hand so your palm is on your forehead and your middle finger is at the point on the top of your head (see Figure 6-2). Relax into this hold for another three deep breaths.\n\n_Figure 6-2_ HORMONE HOOK-UP\n\n**Sunrise\/Sunset** (time\u201430 seconds)\n\nThe following exercise can help stabilize blood pressure (it lowers high blood pressure and raises low blood pressure) and is generally very good for the heart. In addition, it calms body and mind and tends to bring equilibrium to hormones that are out of balance.\n\n1. Begin with your hands at the sides of your body, palms facing outward. Slowly with an in-breath, circle your arms above your head, as if directing the sun to rise (see Figure 6-3a).\n\n2. With both arms now extended upward and palms facing each other, reach for the risen sun. Stretch one arm up high toward the sun. Imagine grabbing a rope with this hand. Close your fist around it and pull it down, drawing the sunlight onto you. As this hand comes down, reach up to grab the rope with your other hand and pull it down. Keep drawing the sun, alternating hands for several more pulls (see Figure 6-3b).\n\n3. Turn your palms outward. Release your breath slowly as you circle your arms far out to the sides and down to the sides of your legs, as if directing the sun to set and your body to calm (see Figure 6-3c).\n\n_Figure 6-3_ SUNRISE\/SUNSET\n\n**Daily Energy Routine with Menopause Module.** The entire expanded routine still requires under ten minutes per day once you have learned each technique. Begin with the Homolateral Crossover if required. Then, while the order is not overly important, one good sequence is:\n\n1. Three Thumps (30 seconds, page 53)\n\n2. Cross Crawl (30 seconds, page 56)\n\n3. Wayne Cook Posture (90 seconds, page 56)\n\n4. Crown Pull (30 seconds, page 58)\n\n5. Lymphatic Massage (60 seconds, page 59)\n\n6. Connecting Heaven and Earth (2 minutes, page 44)\n\n7. Abdominal Stretch (30 seconds, page 134)\n\n8. Spleen Meridian Flush and Tap (30 seconds, page 136)\n\n9. Diaphragm Breath (30 seconds, page 220)\n\n10. Hormone Hook-up (30 seconds, page 220)\n\n11. Sunrise\/Sunset (30 seconds, page 221)\n\n12. Blow-out\/Zip-up\/Hook-in (30 seconds, page 74)\n\n#  **ENERGY TECHNIQUES FOR INCREASING THE PRODUCTION OF ESTROGEN, PROGESTERONE, THYROID, AND TESTOSTERONE**\n\nThe above methods organically help keep your hormones in a better balance. The following methods can be used to increase the production of specific hormones. You can use lab tests to find out if you have hormone deficiencies or excesses, but reliability is a problem due to daily fluctuations. More reliable once you master the art of energy testing is to place a natural form of the hormone in your energy field and energy test it. But even without any tests, certain symptoms are signs of specific imbalances. While only rough indicators (each person's body is different, the same symptom may have numerous causes), and you might want to verify with lab tests, you can also experiment with the energy techniques for increasing production of hormones that seem to be deficient and observe the effects.\n\n##  Stimulating Estrogen and\/or Progesterone Production\n\nThe fact that you are no longer wired to bring forth a baby after menopause doesn't mean that estrogen production has stopped, though it is at only one-tenth its previous level (produced now primarily in the adrenal glands and fat cells). Progesterone production, on the other hand, is nearly absent, though tiny amounts are still made in the adrenals and fat cells, and are actually necessary for the continued production of estrogen.\n\nProgesterone production can, however, be inhibited by the hormones found in meats, dairy products, and pesticides. In addition, because everything from soap to nail polish leaks unnatural estrogens and estrogen-mimics into our bodies (which bind them to our estrogen receptors), we may need to balance this estrogen invasionby stimulating the natural production of progesterone or by adding it as a supplement through gels or creams. At the same time, the estrogen absorbed from improbable sources such as car exhaust is not the estrogen that wards off osteoporosis. You may need to produce or introduce natural estrogen, as well. Maintaining a balance between estrogen and progesterone is as important a challenge in postmenopause as at any time before.\n\nHow do you know if you need more estrogen, more progesterone, neither, or both? I wish I had an easy answer or guideline. The truth is that what you need at one time may be altogether different from what you need at another. These balances are always fluctuating, and even the symptoms of a deficiency or an excess of progesterone or estrogen may be different from one woman to another. I once put the following note on my refrigerator to remind myself of the patterns in my own body: \"Dragging or no motivation\u2014estrogen down! Panic or fury, progesterone down!\"\n\nThese generalizations probably hold for most women, as do those in Table 1. However, they are not an infallible guide. For most women, estrogen deficiency causes hot flashes, and that is the prevailing belief. Yet it is when my own balance goes even slightly toward _too much_ estrogen that I get hot flashes. Even from eating tofu, or any other substance with estrogen in it, I get not only hot flashes but also such night sweats that I may wake up drenched. My own experience has led me to a theory that hot flashes are not caused by estrogen deficiency per se. I believe they are caused by rapid shifts in estrogen-progesterone balances. It is also the case that for most women, the imbalance that causes hot flashes is on the side of estrogen deficiency, but for some women the imbalance can be in the other direction.\n\nIn addition to individual differences, if a woman takes excess amounts of either hormone, unexpected effects may result. For instance, if you realize you are low on progesterone because you are feeling overly emotional, but then apply so much progesterone cream that you create a relative estrogen deficiency, you may have brought on feelings of depression. Because I periodically use progesterone cream, I have inadvertently done just that. Another way I know I need estrogen is that I get a headache on the top of my head. But if I try to correct it with the amount of estrogen in the lowest-dose bioidentical pill available, I will soon be experiencing the symptoms of progesterone deficiency. Instead, I will do an energy exercise to bring about the balance. This usually works. If it doesn't, I cut the lowest-dose estrogen tablet available to a fraction of its size, and the headache will very quickly be gone. Regardless of how I handle it, I usually do not need external estrogen again for a good while.\n\n**Table 1: Possible Symptoms of Progesterone and Estrogen Imbalances**\n\nThe way I suggest you approach this complex territory is that you begin by making some educated guesses regarding your own progesterone and estrogen balances based on the above discussion and any energy testing you have done. Then apply the Daily Energy Routine with the Menopause Module (page 219). This in itself is designed to bring your hormones into a better balance. There is a high probability that you will be feeling better immediately after doing the exercises. If you still sense that you are not in a hormonal comfort zone after doing the Menopause Module for about three days, do one of the following to produce more progesterone or estrogen, based on your best guess of what you need. If you guess wrong, you will not get the relief you are seeking, but no real harm will be done since the body tends to self-regulate when hands-on energy interventions are used.\n\n**Preliminaries.** Prior to stimulating the production of estrogen or progesterone using energy methods:\n\n1. Help balance the endocrine system with the Triple Warmer Smoothie (page 107).\n\n2. Sedate the liver meridian (page 146), which will prepare the liver to process the hormone increase.\n\n3. Do the Triple Axis Hold (page 134) to optimize the balance in your hormonal system.\n\n4. Then use the appropriate techniques from Table 2.\n\n**Table 2: Stimulating Progesterone and Estrogen Production**\n\n##  Details on the Specific Procedures in Table 2\n\n**To Stimulate Progesterone Production** (time\u2014about 10 minutes)\n\n1. After the preliminaries above, sedate the kidney meridian (time\u20143 minutes, see page 145).\n\n2. Tap the kidney meridian strengthening points (time\u2014about 30 seconds): with a firm tap and a steady rhythm of about one tap per second, tap about two inches above the inside anklebone of each foot for about three deep breaths.\n\n3. Strengthen the spleen meridian (time\u20146 minutes, see page 132).\n\n**To Stimulate Estrogen Production** (time\u2014a bit more than a minute)\n\n1. To locate the adrenal reflex points, touch an inch above the navel and then go out an inch to each side (see Figure 6-4). Massage these points firmly for about 15 seconds.\n\n2. With your hand open, lightly place the second, third, and fourth fingers of your left hand on your left temple, just outside the eye socket. Place the index and middle fingers of your right hand in the crease behind your right knee. Relax as you hold this position for four deep breaths. Repeat on the other side. These are the neurovascular points for the triple warmer and gallbladder meridians, respectively, and this combination stimulates the adrenals to produce estrogen. Recall that neurovascular points are always held gently.\n\n_Figure 6-4_ STIMULATING ESTROGEN PRODUCTION\n\nHave these methods been scientifically confirmed? They have not. They grow out of my practice and experimentation with thousands of women, as well as my own and their observations about subsequent shifts in mood, energy patterns, and physical symptoms. While they are based on the empirically supported theory that chemistry follows energy, I look forward to laboratory research measuring hormonal changes after the various techniques. This will beyond doubt further refine the methods and also provide better information on how best to match the procedure with the person.\n\nMeanwhile, register how you feel after each technique in the Menopause Module for stimulating progesterone or estrogen. Do you feel calmer? Do you feel like you have your old self back? These are signs that your hormones are in better balance and more effectively reaching their mark. Some women, however, may find that even though these exercises make them feel better, they don't fully counteract the plummeting hormones that may be involved with menopause. In either case, you may still want to use the exercises because they build energetic habits that can help you go through menopause and postmenopause more easily. And you can also consider a range of supplements that may help your body find an optimum adjustment to the changes it is navigating.\n\n##  Vitamins and Herbs That Enhance Estrogen and Progesterone Balance\n\nI recommend here supplements that I, or my clients, have personally found particularly helpful as they moved into and through menopause. I have seen so many benefits from flaxseed and flaxseed oil that if they were illegal, I would owe it to my clients to become an underworld flaxseed dealer. Beyond its many well-established health-promoting and anticancer properties, **flaxseed** keeps your skin moist, bones stronger, and body juicier through menopause. Interestingly, according to Susan Lark, M.D., the lignans found in flaxseed also help eliminate estrogen deficiency by binding with unoccupied estrogen-receptor sites and mimicking the body's own estrogen. **Essential fatty acids** (omega-3 and omega-6) are also great supplements. They help in the production of female sex hormones, and they also help keep the body moist and supple. **Black cohosh** is an herb that contains phytoestrogens (chemicals that plants produce that act like gentle forms of the estrogen produced in our own bodies), and for some women, it eliminates their hot flashes. I have loved **Mexican yam root** for the gentle form of progesterone it provides. **Ginkgo biloba** helps with memory and other mental capacities as well as circulation. **Ginseng** , a medicinal plant used in China for thousands of years, can counter fatigue, build stamina, and produce estrogen and progesterone, as well as support all the endocrine glands. Dong quai is used for a range of menopausal symptoms, from hot flashes to insomnia.\n\nI also recommend taking a good-quality multiple vitamin and a separate full-spectrum mineral supplement. The processed food we generally consume does not usually supply us with the range of nutrients that we need. However, when it comes to nutrition, herbs, and vitamins, we are all very different, so a great deal of personal research is called for. Use energy testing to the best of your ability to determine in advance what might be helpful and what might not be. Inform yourself as much as possible through books and the Internet about any supplement you are considering. Discuss such choices with health or dietary professionals, and remember that introducing any substance into your body is an experiment, so observe carefully. Be alert for unwanted effects, which may indicate that you are taking too much, mixing too many supplements, or simply taking something your body can't assimilate. And proceed with optimism that the attention you are giving to finding out what your body needs is going to pay off big.\n\n##  Supplemental Estrogen and Progesterone\n\nI have at my disposal all of the above advice and methods, apply them as needed, and still take and love natural progesterone and, on rare occasions, natural estrogen. Am I a fraud? We live in extraordinary times. We have never been more challenged. And we have never had more resources for meeting our challenges.\n\nThe environment strains our natural chemical balances. The many stresses of modern life strain our natural chemical balances. Living longer and in less harmony with nature stresses our natural chemical balances. Energy medicine can do much to restore them. So can the judicious use of the various supplements that are readily available. Better living through energy. Better living through chemistry.\n\nMy first advice is to not accept advice from anyone. At least not uncritically. Cookie-cutter recommendations and prescriptions are for indistinguishable cookies. You are unique. But the pharmaceutical industry spends untold millions telling you that moving forward naturally and according to your body's plan is the path to becoming old and decrepit, and that the solution is to take their drugs, which are designed to reverse time and give you the chemistry that you had when you were 25. It is a lie based on a myth. On the other hand, if you take the time to experiment with energy techniques and natural supplements, you can keep your body healthy, supple, and vibrant well into old age.\n\nIf you have determined through the various means discussed earlier that your body is low in progesterone and needs more than the methods presented can produce, my suggestion is to begin by experimenting with progesterone creams that the staff at a good health food store recommends (also do a critical search on the Internet). Regardless of the dosages recommended, energy test and experiment.\n\nRecommended dosages are averages based on many women. You need to find the dose needed for _this_ woman, and at a particular point in time. If you find that you do not get enough relief, you may need to get a prescription. While your doctor may have other up-to-the-minute suggestions, Prometrium is an excellent form of progesterone, made from yam and encapsulated with peanut oil (and thus contraindicated if you are allergic to peanuts). While it comes in a capsule that is meant to be taken orally, you can instead pierce the capsule with a pin and rub the liquid onto the inside of your wrist or your stomach. This sends the progesterone directly into your bloodstream, bypassing your liver and your entire digestive system, and it also allows you to easily modulate the amount. Another form that bypasses the digestive system is the vaginal suppository available by prescription, though not all pharmacies are able to make it. As discussed earlier (see page 127), it is very easy to tip the scales too far to the other side, so be very alert to the symptoms of too much progesterone, described on the right side of Table 1 (see page 225).\n\nYour body needs a balance of progesterone and estrogen to support your heart, keep your bones strong, and help your brain stay sharp. If you have determined that you need supplemental estrogen, the choices include phytoestrogens, herbal compounds, bioidentical estrogen, and standard pharmaceutical hormone replacement therapies such as Premarin. The same principles apply in terms of informing yourself; energy testing to the extent of your abilities, starting with the more natural compounds (though I have sometimes even found standard pharmaceuticals to energy test stronger than bioidenticals\u2014again, individual differences trump everyone's theories); and considering taking less than the recommended dosages. For instance, the lowest-dose Premarin tablet can be cut in half and cut in half again with a pill cutter, and even a third time, and meets the needs of many women. Premarin is also generally taken on a daily basis, yet unlike an antibiotic, there is no compelling logic for this if you are able to carefully monitor your day-by-day needs using self-observation or energy testing. With estrogen, it is also now standard treatment to supplement it with progesterone, as the harmful effects of \"unopposed\" estrogen replacement therapy are well documented. In any case, after taking a hormonal supplement, tap the spleen neurolymphatic reflex points on both sides of the body (page 136) to help you metabolize it.\n\n##  Thyroid\n\nUntil 2002, all thyroid tests were developed and interpreted on the basis of lab data from men, even though nearly 90 percent of the complaints doctors receive that might involve thyroid dysfunction are made by women. Because of this little oversight, many women with symptoms of thyroid dysfunction were showing up as normal on lab tests and not receiving needed treatment. The problem was so extensive that there was actually a \"hypothyroid patient movement,\" supported by knowledgeable endocrinologists, demanding that tests be made more sensitive and that supplemental thyroid be more readily prescribed.\n\nMeanwhile, others have decried this movement, not because they object to symptoms being alleviated, but because supplemental thyroid hormones are, according to Diana Schwarzbein, M.D., \"Band-Aids to cover up symptoms. They do nothing to restore balance to your body.\" She counsels: \"If you take thyroid hormone replacement therapy for a lifestyle-based endocrine disorder instead of improving your nutrition and lifestyle habits, you will actually create more hormonal imbalances and further destroy your metabolism.\" She insists that proper nutrition is the first order of business. Since hormones are made primarily from proteins, cholesterol, and essential fats, a balanced diet is critical for maintaining the production of your hormones. Adequate amounts of iodine are required. Herbs that can help stimulate your own production of thyroid hormones include nettle leaf, kelp, parsley, mullein, selenium, and Irish moss. A full-spectrum mineral supplement is generally valuable and is specifically helpful for the thyroid. Progesterone also supports the thyroid.\n\nSchwarzbein points out that because the body makes hormones in minuscule and fluctuating amounts, it is very hard to answer many hormonal questions using blood or urine tests. While I cannot advise you on whether to take thyroid replacement therapy, using energy medicine to help balance your thyroid is a noninvasive approach, more akin to nutrition and lifestyle interventions, and it may prevent the need for more extensive measures. As we age, our thyroid gland tends to produce fewer of the thyroid hormones that are essential for metabolism, vitality, and good mood. Symptoms and signs that your thyroid hormone production is off are shown in Table 3.\n\n**Table 3: Some of the Major Symptoms and Signs of Thyroid Imbalance**\n\nPhysical exercise and adequate sleep are essential for proper thyroid function. Simply stretching the skin above, below, and to the sides of the Adam's apple instantly brings good energy to the thyroid. In addition, the following techniques, alone or in combination, reduce stress in the thyroid and stimulate the production of thyroid hormones.\n\n_Figure 6-5_ THYROID BOOSTER\n\n**Thyroid Booster**   \n(time\u2014about a minute)\n\n1. Do the Triple Warmer Smoothie (page 107) and then tap the spleen neurolymphatic reflex points (page 136).\n\n2. Place the middle finger of one hand just above the Adam's apple, the middle finger of the other hand just below the Adam's apple, and separate your fingers so you are stretching the skin on your neck. Do the same with your fingers on the sides of your Adam's apple. Also stretch diagonally.\n\n3. Rest the thumb, index finger, and middle finger of one hand in the hollow below your Adam's apple and rest the pads of the fingers of your other hand at your temple (see Figure 6-5). Both are triple warmer neurovascular reflex points. Hold gently for four or five deep breaths and then repeat on the other side.\n\n##  Testosterone\n\nWomen produce about 10 percent less testosterone as men, primarily through their ovaries and adrenal glands. The production of testosterone declines substantially just prior to menopause, at which time a woman produces about half the amount she was producing in her midtwenties. However, after menopause, the ovaries continue to make testosterone in quantities such that testosterone is proportionately higher than estrogen as compared with earlier in life, supporting some of the increased assertiveness and independence seen in postmenopausal women.\n\nThe doctor who gave me my annual exams in the early 1980s was a feminist who said to me one day: \"You know, Donna, you'd like your life a lot better if you weren't _so_ nice. A lot of women are experimenting with testosterone, and they are loving it. I think it would be really good for you. You'd strut around owning your space, and you'd have clear 'no's' for people.\" She talked me into trying it. She had some samples in her office, and she gave me the lowest dose available. She hoped I would take it and tell her what I experienced. It did make my life different. I was furious all the time. And not the kind of juicy anger that engages you with the people you love; it was more like a marine about to attack an enemy squadron that had just wiped out an innocent village. While I stopped taking it almost immediately, it did give me much greater respect for men who are naturally juiced up with that stuff all the time and are still able to behave in a civilized manner.\n\nSo I am not going to offer suggestions about taking testosterone replacement\u2014 you can weigh the risks\u2014plus there are other forms besides the synthetic macho type that I took. **Muira puama** , for instance, is a plant from Brazil whose bark and roots have been shown to increase libido in menopausal women by upping testosterone levels. **Maca** is a plant from Peru whose root has a similar effect. **Damiana** is a shrub, also from South America, that has been used as an aphrodisiac since Mayan times. While I do not have substantial experience with any of these, I can offer some energy interventions if you think your testosterone levels may be lacking. Table 4 describes symptoms and signs of testosterone imbalances.\n\nIn the early 1980s, I did volunteer work at a women's shelter. These women had been abused and had come to the shelter for safety and refuge. They were so deeply exhausted and demoralized that they were terrified of leaving, even if they didn't particularly like being there. I remember one little boy who saw me giving a session to his mother. She began to sob as she relaxed on my table, and he must have thought I was hurting her like his dad had hurt her. He started beating on my leg with all the fury his little three-year-old body would allow. It was actually quite poignant. Testosterone in action! His mom, however, seemed totally deficient of anything like such a life force, as were most of the women when they got to the shelter. No fight was left in them. They had no reserve power. We all have yin and yang, female and male energies within us, and we need each. These women were depleted of both. Triple warmer, which governs the fight\/flight\/freeze responses, had been on continual alert for a long time, and they had used all they had to fight and flee their way to the shelter. Now their energies were in freeze mode. I'm sure their testosterone levels and corresponding yang energies were as low as at any time in their lives. Yet they needed those yang energies if they were going to survive in the world and not return to their abusive partners. So I began sedating triple warmer meridian and strengthening its counterpart, spleen meridian. You might think that sedating triple warmer would take away power, but it had the opposite effect. I would calm it and calm it and calm it, and I could see the yin forces in the body gathering, gaining strength. Then the yang forces. In the process, testosterone production gets back on track.\n\nThe other meridian involved in producing testosterone, in addition to triple warmer, is the stomach meridian. While a yang or \"male\" energy, the stomach meridian in a woman carries the force of the mother tiger. During her mothering years, this energy is focused on children and family. After menopause, it is available for establishing the power of the crone. It is a grounding force that helps a woman hold her own space. So to set the stage for more testosterone production, you want to be sure your stomach meridian is strong, steady, and uncompromised. Steps include regularly doing the Stomach Meridian Flush (page 138). Precede it with the Blow-out (page 74), the Zip-up (page 61, with an affirmation about stepping into the power of the crone), and the Three Thumps (page 53). Then connect the energies between the triple warmer and stomach meridians by placing your thumbs on your temples and your fingers directly above your eyes. These points get the energies of both meridians in sync with each other. Finally, having created a readiness in your body to utilize more testosterone, the Testosterone Triple Massage is designed to activate testosterone production. While I have never evaluated this sequence with lab tests, women have reported that it gave them more power, stronger boundaries, greater mental clarity, emotional balance, and renewed libido. To naturally increase testosterone, do the Testosterone Triple Massage two or three times per day:\n\n**Table 4: Some of the Major Symptoms and Signs of Testosterone Imbalance**\n\n**Testosterone Triple Massage**   \n(time\u2014less than a minute)\n\n1. Place your middle fingers in your navel, move them up one inch, and then out one inch to either side. Take a deep breath and massage these points deeply for about 15 seconds. This stimulates the adrenal points, strengthening the adrenals and enhancing testosterone production.\n\n2. Place your middle fingers at the knob on the top inside edges of your hip bone. Move your fingers two inches toward the center, breathe in deeply, and massage for about 15 seconds (see Figure 6-6). This stimulates and strengthens the ovaries and enhances testosterone production.\n\n3. Deeply massage the points on the outsides of your legs between your knees and hips for about 15 seconds (to find these points, hang your arms and bend your middle fingers toward your legs). These are gallbladder meridian points that stimulate the yang forces in the body.\n\n_Figure 6-6_   \nTESTOSTERONE TRIPLE   \nMASSAGE\n\n##  Osteoporosis\n\nA woman who had broken a rib by simply sneezing sought me out as a last resort. The only indicated medication at the time for reversing bone loss was causing severe side effects. She was only 49, and she did not want to see her body deteriorate further. But she was also a physician, well educated in Western medicine, and did not believe she could reverse her condition, citing research in our opening minutes together to support this view. She hoped instead that if she could build the muscles supporting her skeletal structure by using physical exercise and could strengthen the energies in her body (she knew very little about energy work, but I had helped a few of her patients and she was open to giving it a try), she would be less vulnerable to the effects of her \"genetically destined\" loss of bone density. I felt that more was possible, but I decided to start the session by showing her how energy work could both help lift her sense of despair and leave her feeling better in her body. This gained me enough credibility that I could talk to her about how energy feeds the bones and can literally reverse bone loss. She was skeptical but willing to let me try. Bone density is regulated by the kidney meridian. But for her, a simple formulistic approach was not enough. The kidney meridian is fed by the lung meridian, and it was a deficiency in her lung meridian that was keeping her kidney meridian weak. So we wound up giving as much attention to her lung meridian as to her kidney meridian. The lung meridian governs grief, and to her surprise, one of the keys in helping get her lung meridian back to an optimal flow involved emotional work around a loss she had not fully processed. After four or five sessions and faithful compliance with back-home assignments on her part, we both knew her bones were growing stronger. After a bone density test showed dramatic improvement, she spoke to me about doing a research project, one of those opportunities I never quite had time for. But it is a project I hope someone reading this book will carry forth.\n\nOsteoporosis afflicts more women than stroke, diabetes, breast cancer, or arthritis. It is a substantial challenge that causes a progressive loss of minerals, bone mass, and bone density. It may have increased as much as sixfold in recent decades. It is also surrounded by misinformation. Estrogen, for instance\u2014whose decrease during menopause has been identified as the culprit in osteoporosis\u2014plays a far less significant role in protecting against the disease than does progesterone.\n\nYour bones are in a constant process of breakdown and regeneration, a miracle of renewal that depends on two kinds of cells. The first are called osteoclasts, which are modulated by estrogen, and take away old, worn-out bone. The second are called osteoblasts, which are modulated by progesterone. They go to those same sites and create new bone. It is critical for keeping your bones strong to maintain an adequate progesterone level.\n\nAs we age, osteoclast activity becomes more prominent, and bone mass is lost. But women in their 70s and 80s can still slow their bone loss, strengthen their bones, and even build new bone. Long before that, however, by their 40s or 50s, they can be taking steps to stop the loss of minerals in their bones, particularly calcium and magnesium. Getting these minerals into your system, however, can be tricky.\n\nCalcium pills, widely touted as a preventive for osteoporosis, may interfere with the metabolism of magnesium and zinc, thus ultimately harming rather than promoting bone strength. Calcium is best obtained from your diet and, even then, cannot be adequately absorbed unless your body has the proper amounts of magnesium, hydrochloric acid, and various vitamins. Magnesium may, in fact, be as important as calcium in preventing osteoporosis. Meanwhile, many prescription medications, as well as over-the-counter medications such as antacids, disrupt calcium absorption. The calcium that is naturally found in herbs, such as nettles, is an excellent way to supplement your calcium intake. If you take calcium in a pill form, it is most likely to be absorbed if it is part of a multimineral that contains more magnesium than calcium.\n\nAn enormously unfair contributor to osteoporosis is that our bones absorb toxic lead from the environment, which displaces calcium and results in bone loss. Adding further injury, at menopause and after (and also, alarmingly, during pregnancy and breast-feeding), lead in the bones is released into the bloodstream, where its poisonous effects may contribute to hypertension, kidney disease, dementia, heart disease, and nerve damage. Because lead can leach into the bloodstream whenever calcium is released from the bones, one way to keep lead out of your bloodstream is to be sure you have adequate amounts of calcium available. If the body isn't pulling calcium from your bones, it won't be leaking lead into your bloodstream. In addition, several supplements have been documented for helping rid the body of lead. **NAC (N-acetyl cysteine)** enhances urinary excretion of heavy metals, including lead. **SAM-e (S-adenosylmethionine)** reduces the toxic effects of lead and binds lead to the bile for elimination. **Epsom salts (magnesium sulfate)** also reduce lead toxicity. Other nutrients that help with bone strength include **vitamin K,** chondroitin, MSM, and vitamin D (particularly D3). **Vitamin D** is provided most naturally by the sun, but those of us who live most of our lives indoors are often deficient in D. It is also blocked every time we use sunscreen. Don't ever let yourself get a sunburn, but do let your skin soak the sun's rays in moderation.\n\nSimplistic solutions are, however, particularly misleading with osteoporosis. Proper diet, regular _weight-bearing_ exercise, and maintaining a healthy estrogen\/ progesterone balance form a triangle of prevention and bone regeneration. Energy exercises can strengthen the foundation of that triangle. Just as magnets and electromagnetic therapies have been shown to speed the healing of broken bones, subtle energy work as taught in this book helps maintain bone strength and density.\n\nAn excellent and very simple exercise that engages your energies to help prevent osteoporosis is the Shoulder Lift. It lubricates the shoulders, releases lymph, helps your upper back stay strong and straight, and even helps prevent dowager's hump. It is valuable not only in the physical results it brings, it also helps you establish a more conscious and positive relationship with your shoulders and upper back. Lift your shoulders to your ears and slowly roll them backward, around, and up again several times while breathing deeply. Then roll them forward, either simultaneously or one shoulder at a time.\n\n##  Isometrics for Preventing Osteoporosis\n\nIsometric exercise is a type of strength training that has proven to be extremely valuable in preventing and correcting osteoporosis. Loss of bone density had become a problem for astronauts, who might spend months with little physical movement. Just ten minutes of isometric exercises each day can reverse bone loss. Isometric exercise, which involves the static contraction of a muscle without any visible movement, is also great for your energies. Getting some instruction or a book on isometrics is a superb way to address or prevent bone loss. Here are two of my favorite isometric exercises:\n\n**Torso Stretch** (time\u2014about 20 seconds)\n\n1. Sit tall on your hands with your fingers facing inward and straighten your arms.\n\n2. Breathe deeply while holding this position for several breaths.\n\nYou will feel the stretch in your back, neck, and arms.\n\n**Bone Saver** (time\u2014less than 2 minutes; also a great tummy trimmer)\n\n1. Place your arms out in front of you, take a deep breath in, hold your breath, and push the heels of your hands hard against each other (see Figure 6-7a).\n\n2. At the same time, squeeze your stomach in toward your spine, tighten your buttocks, and tilt your pelvis forward and up.\n\n3. When you want to exhale, instead take three more rapid breaths in through your nose while holding the above posture.\n\n4. Then exhale slowly, as if blowing through a straw (your stomach will tighten naturally).\n\n5. When it feels like your exhalation is complete, blow three more rapid breaths out through your mouth.\n\n6. Relax after each round. Do several rounds.\n\n7. Repeat the entire sequence, except this time bend the fingers of each hand, hook them into one another, and try to pull your arms apart so there is isometrc tension at your fingers and in your arms and shoulders (see Figure 6-7b).\n\n_Figure 6-7_ BONE SAVER\n\nIn my practice, when someone's bones were weak or fragile, giving them the homework assignment of sedating and strengthening the kidney meridian two or three times per day began to rebuild bone strength in ways that the person could readily recognize within two or three weeks. When my father had bone cancer, he was told that his fifth lumbar vertebra (L-5, a spinal vertebra located at the waist) was \"hanging on by a thread.\" He had to wear a very uncomfortable contraption to keep it stable because if he were to so much as sneeze, they said, the bone could snap and he would be paralyzed for life. One of my frequent extended visits came at about the time he was fitted for the contraption. With a father's unconditional confidence in his daughter's magical powers, he said, \"Donna, you can fix this!\" I didn't have any inkling that I could fix it, but I applied the tools I had available. I tested all his meridians several times each day to look for patterns. The kidney meridian kept showing weak.\n\nMy father's original diagnosis had been prostate cancer. The prostate is governed by the kidney meridian, and the kidney meridian also governs bone strength and the building of bone. I will take a little detour here, as this was one of the experiences that has made me so insistent in advising that you not squash your own judgment in deference to medical authorities. My father's PSA counts had been somewhat elevated. His doctor was not particularly concerned. But my father loved life and decided to get a second opinion from an oncologist. The oncologist insisted that he have surgery, and as the situation unfolded, it was clear that the two physicians strongly disagreed with each other. Fear won, the surgery was performed, and bone cancer was diagnosed soon after. In reviewing the course of the illness, my father's primary care physician was furious. He advised us to sue the oncologist, explaining that during the surgery, cancer cells had \"spilled\" into his bloodstream and then gone into his bones. He told our family, after my father died, that the bungled surgery had been a \"death sentence.\"\n\nIt was. But we did not know that at the time, and Daddy had asked me to fix his back. Since the kidney meridian governs both the prostate and bone strength and was the meridian that was consistently weak, it was my focus. Every day, three times each day, I would sedate and then strengthen his kidney meridian. This was very reinforcing because immediately following each mini-session, he would report that the pain had diminished. Within three weeks, he was feeling no pain in the area of his lower back. He persuaded his doctor to do another X-ray of the area. L-5 was now _normal._ It had grown back completely. This is not supposed to happen! The medical team was so flummoxed that the first thing they did was check to be sure that the original X-ray was not that of another patient. With the kind of cancer my father had, they expected to see only further deterioration, yet the uncomfortable back brace went into permanent storage.\n\nIf you have osteoporosis or are concerned about the condition, you want to be sure to keep your kidney meridian strong. You might want to get an assessment from an energy medicine practitioner (www.innersource.net\/links\/links_practitioners.htm) or from a good acupuncturist. The energy test required is beyond the instructions given in the Appendix. If the kidney meridian is weak, you can sedate it (page 145), which relaxes the kidneys themselves, allowing toxins to move more easily through them. Then follow by tapping the kidney meridian strengthening points (page 226) to infuse it with more power to help build new bone cells. If you do this at least twice per day for a month, and then go in for another bone density test, you are likely to see improvement. Sedating and strengthening the kidney meridian does no harm, even if the meridian is already strong, and in fact is always valuable, kind of like exercising a muscle.\n\n#  **STIFFNESS AND JOINT PAIN**\n\nOsteoporosis, in its early stages, is a \"silent\" disease. Joint pain is not. It screams at you. Many postmenopausal women suffer with joint pain, stiffness, or aches in the back, knees, shoulders, hips, and neck\u2014both on a temporary and chronic basis.Despite decades of wear and tear on our joints, and even injuries, we can counter and help prevent such pain and stiffness by keeping our joints lubricated.\n\nInflammation is a primary culprit in joint problems. Inflammation is the normal and wholesome process by which the body protects itself from irritation, injury, or foreign substances such as bacteria and viruses, sending chemicals and white blood cells to affected areas. But with unceasing stress, inadequate or excess exercise, or poor diet, inflammation may be triggered on a false-alarm basis, so it is produced when it is not needed and becomes harmful. High cortisol levels are endemic for many people experiencing the stresses of modern life, and sustained high levels of cortisol damage bone tissue and impair its regeneration. While inflammation is the common denominator in numerous complex diseases, from arthritis to diabetes to cancer, it has the straightforward physical effect of interfering in the exquisite dance at the joints where bones meet.\n\nTo keep your joints well lubricated, a simple and very basic step you can take is to be sure you have enough essential fatty acids in your diet. Don't let the \"fat\" in the name put you off. You need them, and your body cannot produce them. You have to get them from what you eat. Unless you live in a fishing village, however, you probably aren't getting enough essential fatty acids if you aren't conscious about your need for them. Fortunately, foods with both types of essential fatty acids\u2014omega-3 and omega-6\u2014are readily available. Natural sources include fish and shellfish, flaxseed and flaxseed oil, hemp oil, soy oil, canola oil, evening primrose oil, chia seeds, pumpkin seeds, sunflower seeds, leafy vegetables, avocados, and walnuts. The health benefits of taking adequate amounts of essential fatty acids are enormous, ranging from preventing inflammation to promoting cardiovascular health. Flaxseed and flaxseed oil are great sources of essential fatty acids, and their multiple health benefits include preventing cancer, helping dry eyes, and keeping the body's tissues moist. You can buy flaxseed oil, mixed with peppers and garlic, as a delicious salad dressing, or make your own. I add lemons and basil leaf as well (which is also good for keeping your body alkaline as you age). But you can also take it in capsules. You can energy test dosage.\n\nAnother straightforward way of keeping your joints strong and healthy involves keeping the energies moving freely through them. A number of stretching exercises and energy techniques can help ensure this. Connecting Heaven and Earth (page 44) is excellent. Three more include:\n\n**Circular Joint Rub** (time\u2014about 30 seconds)\n\nSimply rub, in a counterclockwise circle, the joint pain with the heel of your hand or with your fingers.\n\n**Tape the Magnet on the Pain** (time\u201410 to 20 minutes)\n\nTape the north side of a weak flat magnet (Radio Shack's round donut magnets work well) against the skin over the bone or joint that hurts for about ten minutes. You can use a compass to determine which is the north side. The needle that points to north will point to the side of the magnet that goes on your skin. If the magnet hasn't made a significant difference in the first ten minutes, remove it. If it has made a difference, keep it on as long as you feel it is continuing to work. Do not leave it on overnight. Do not tape the south side against an inflamed joint, and do not place either side over a vein.\n\n**Wrist Twist** (time\u2014less than a minute)\n\nStretching creates space that helps lymph and lubricating fluids move through the joints. When trapped energy is freed, it can become a positive healing force. While yoga and other stretching exercises are great to do on a regular basis, the Wrist Twist is specifically for stiffness in the hands, shoulders, and arms, and it is a method you can use in the moment.\n\n1. Let your arms hang and then lift them out to the side about 30 degrees.\n\n2. With your hands open and your fingers leading, rotate your hands inward until you feel a stretch in your forearms and shoulders (see Figure 6-8). Hold that stretch for about ten to fifteen seconds.\n\n3. Relax. Repeat several times.\n\n_Figure 6-8_ WRIST TWIST\n\n#  **OTHER SYMPTOMS ASSOCIATED WITH MENOPAUSE**\n\nThe remainder of the chapter presents energy interventions that may be helpful in working with other common challenges women face during and after menopause. Because the same symptom may have a totally different cause for one woman than for another, no list of complaints and remedies is going to be completely reliable. But I can describe the procedures that women in my practice found most helpful for each of the following conditions. If several methods are presented for a condition that is of concern to you, experiment to find which, or which combination, is most effective.\n\n##  Hot Flashes\n\nIn addition to the following energy methods and the Menopause Module, several herbal remedies are widely reported to help with hot flashes. Those that seem to have been most effective with my clients include flaxseed or flaxseed oil, evening primrose oil, and for some women, black cohosh. Many of my clients stopped HRT with no adverse effects using energy exercises with flaxseed oil as the only supplement. If the flaxseed and the Menopause Module are not enough, additional energy methods that are useful with hot flashes include:\n\n**The \"Cool - Down\" Acupoint.** While in a hot flash, press the middle fingers of each hand into the point about two inches below your navel (this is the point of intersection of the central and kidney meridians). Take a deep breath and hold the point as you then breathe normally. This quickly redistributes energy in a manner that reduces body temperature during a hot flash.\n\n**Triple Warmer Smoothie.** Because triple warmer manages body temperature as well as the stress and immune responses, calming it with a method such as the Triple Warmer Smoothie often provides instant relief (see page 107).\n\n**The Darth Vader Breath.** Another technique that can quickly calm triple warmer and may provide quick relief during a hot flash is the Darth Vader Breath (page 152).\n\n**Dowsing the Fires.** A fourth technique not only provides relief, it also prevents future hot flashes. I found that when women did the following procedure on a daily basis, their hot flashes became less intense and, in some cases, subsided altogether. This technique is also good for maintaining thyroid health.\n\n1. Cross your hands, placing your middle fingers on K-27 (page 53) and massage both K-27 points with some pressure.\n\n2. Remove your right hand and slide the middle finger of your left hand above the corner of your clavicle.\n\n3. Turning your head to the left as far as you can (see Figure 6-7), slide the middle finger of your left hand out toward the outer edge of your clavicle while you breathe out.\n\n4. Repeat with your right hand on your left clavicle.\n\n_Figure 6-7_ DOWSING THE FIRES\n\n##  Depression\n\nFor people who have suffered with depression, the deadness that may be experienced during menopause is at least familiar territory. However, if you are prone more toward hysteria, like me, and less toward depression, it can be new and terrifying. I didn't know what to do with the unfamiliar sense of emptiness\u2014the strange lack of feeling, motivation, or caring. Though I always felt compassion, my understanding for people who experience depression increased profoundly by going through it myself. Depression is not only in the mind. Every cell of the body, every organ, every body function becomes sluggish. Your reflexes and thinking are slowed. Too little estrogen in relation to progesterone can cause depression. The Daily Energy Routine with the Menopause Module, along with the other steps already discussed for bringing estrogen and progesterone into balance, will help ward off menopausal depression. Following are more targeted steps. When you are depressed, it is hard to muster the energy to turn the depression around, and you lose incentive. So be alert to the fact that you probably won't feel like doing these exercises, but when you are through, you are likely to know that something has begun to shift.\n\n**Stretch, Stretch, Stretch** (time\u2014about 3 minutes)\n\nIf you are depressed, energy is not moving. You can instantly activate the energies in your body by simply stretching the skin on your face, scalp, and neck. In the process you will be stimulating the endpoints of your stomach, small intestine, large intestine, bladder, gallbladder, triple warmer, central, and governing meridians. Breathe deeply throughout all these stretches.\n\n1. Begin with the Crown Pull (page 58). Then give yourself a scalp massage.\n\n2. Continue to stretch every part of your face.\n\n3. Push in at your cheekbones and move your fingers to the opening of your ears, stretching your skin.\n\n4. Place your middle fingers above your upper lip, push in, and stretch the skin to the outside of your mouth.\n\n5. Repeat, this time starting below your bottom lip.\n\n6. Place your fingers on your jawbone and push up along the line of your jaw.\n\n7. Pull your ears, starting at the bottom lobe and traveling up about half an inch at a time, until you have stretched the entire outer lobes.\n\n8. With your head bent back, stretch the skin on your neck in every way you can.\n\n9. Place the fingers of one hand on the opposite shoulder, push in, and pull forward. Repeat on the other side.\n\n10. Experiment with stretching any other area of skin that might feel good or doing yoga-like stretches of your muscles and joints.\n\n11. Optional power boost: Tap vigorously with several fingers on all the bones throughout your face and head.\n\n**Homolateral Crossover** (time\u20143 to 5 minutes)\n\nAny time you are depressed, your energies are running in a homolateral pattern. Period! Do all you can do to get them crossing over (page 65), particularly figure-eight patterns. The Homolateral Crossover (page 67) is also one of the most powerful methods you can use, though you may need to do it two or three times per day for a while to shift the energetic pattern and get the new pattern to stick. Then other methods for countering depression are more likely to be effective. Getting your energies back into a crossover pattern also helps bring your hormones out of the chaos that homolateral energies create.\n\n**Mellow Mudra** (time\u20143 minutes)\n\nBy combining the main neurovascular points with a hand posture called a mudra in yoga, you are able to stimulate the flow of blood to your head while calming and aligning energies in the body that are reacting against one another, a property of depression. Moving the blood also gets energies moving, and moving energy is the antithesis of depression. Do another Crown Pull (page 58), followed by rubbing your head all over, giving yourself a pleasant massage. Then put your thumbs over the nails of the index fingers of each hand (see Figure 6-10a). Keeping this position or mudra, place your third and fourth fingers gently on your forehead, directly above your eyes and equidistant between your eyebrows and your hairline. Rest the thumb and index finger circles gently on the temples and, with a slight pull to the sides from your third and fourth fingers, begin to hum. Hold these points for two to three minutes as you continue to hum (see Figure 6-10b). Breathe deeply.\n\n_Figure 6-10_ MELLOW MUDRA\n\n**Heart-Womb Connection** (time\u20143 minutes)\n\nStanding, sitting, or lying down, place one hand over your heart chakra (middle of your chest) and the other over your womb chakra (between your navel and your pu-bic bone\u2014see Figure 6-11). Notice the energies between these two chakras beginning to connect as you stay in this position for about three minutes, breathing deeply. People have told me that feeling the energies moving between these places attunes them to an aliveness within that counteracts depression.\n\n_Figure 6-11_ HEART-WOMB CONNECTION\n\n##  Mental Sharpness\n\nThe same techniques that help overcome depression also help in overcoming the diminished mental acuity that is sometimes experienced by women going through menopause. The Daily Energy Routine, the Menopause Module, and the stretching, crossover, and neurovascular techniques described above are particularly effective in keeping your mind sharp. **Ginkgo biloba** is proving to have brain-enhancing properties. While early research led to mixed findings, more recent studies, based on higher dosages, showed substantial improvement for a range of mental functions, including reversing some symptoms of Alzheimer's. Many of my clients have reported that it helped them with memory and mental clarity. Established biological effects of ginkgo biloba are that it improves blood flow to most tissues and organs, protects against cell damage from free radicals, and blocks many of the effects of blood clotting associated with various cardiovascular, renal, respiratory, and central nervous system disorders. It is contraindicated for certain individuals with blood-circulation disorders. In summary, the exercises mentioned above, possibly combined with brain-enhancing supplements such as ginkgo biloba, can help your mind stay vibrant and sharp.\n\n##  Spongy Tummy and Thickening Waist, Oh My\n\nThink oxygen. While Chapter 7 is all about weight management, start by increasing your oxygen levels. A few facts:\n\n\u2022 Oxygen breaks down fat molecules, turning them into carbon dioxide (CO2) and water (H2O).\n\n\u2022 Your body is continually processing toxins\u2014your health depends on it\u2014 and you are designed to discharge, with your breathing, more than two-thirds of the toxins your body processes.\n\n\u2022 If oxygen is in short supply, the ability of the villi in the intestinal tract to absorb nutrients and digest food effectively plunges by as much as 72 percent.\n\n\u2022 When people use only the top part of their lungs to breathe, their metabolism is slowed and other problems ensue.\n\n\u2022 The metabolic rate can instantly increase by up to 30 percent when oxygen intake is increased.\n\n\u2022 Deep breathing exercises can triple the fat-burning oxygen available to us and increase by up to 70 percent the digestive system's ability to convert toxins into gases, which are then exhaled.\n\nResearch has shown that introducing three five-minute segments of daily breathing exercises can substantially increase metabolic rate. It also improves muscular strength, toning, and emotional outlook. Diaphragm Breath (page 220), part of the Menopause Module, is a 30-second daily breathing exercise designed to remind your body how to breathe well. You can supplement it with the following isometric exercise, which I call the Metabolic Breath, or the Menopausal Tummy Trimmer. The basic principle is to breathe deeply in the manner described while tensing several muscles.\n\n**Metabolic Breath** (time\u2014about a minute, with a nod of appreciation to Greer Childers)\n\n1. Bend forward at the waist and place your hands on your knees.\n\n2. Take a fast, deep in-breath through your nose. As you do, round your back and squeeze your stomach in toward your spine.\n\n3. Force your breath out through your mouth (see Figure 6-12a).\n\n4. When all your breath is out, suck your stomach in tight and hold without breathing until you want to catch your breath (see Figure 6-12b).\n\n5. Release. Your lungs will fill with air.\n\n_Figure 6-12_ METABOLIC BREATH\n\n##  VAGINAL DRYNESS\n\nWomen I've worked with who complained of vaginal dryness reported substantial improvement after increasing their _daily_ intake of essential fatty acids. Again, sources include fish oils, flaxseed and flaxseed oil, hemp oil, soy oil, canola oil, evening primrose oil, chia seeds, pumpkin seeds, sunflower seeds, leafy vegetables, walnuts, and avocados. While avocados are always included on the lists of good oils to take, I can give them a personal plug from my own experience. When I moved to an area where avocados were plentiful, I noticed within a couple of months that my skin and rough feet became markedly softer, moister, and suppler, even though my new home was in a drier climate. In figuring out why this might be after years of leathery feet, I realized that the only real change was a substantial increase in my intake of avocados, usually as guacamole (well, maybe it could have been the margaritas I had _with_ the guacamole?). The Menopause Module also helps promote moist skin and prevent vaginal dryness. Occasionally, additional energy work is needed, and this focuses on keeping the kidney meridian balanced by sedating it (see page 145) and then tapping its strengthening points (see page 227).\n\n#  **HEADACHES**\n\nHeadaches that come with menopause may have many causes. Your energies may be off. Your life may be full of stress. Your sleep patterns may be disturbed. Your hormones may be out of balance. I found that yam roots could help me since I tended, like many women, to be low in progesterone. For other women, bioidentical estrogen might restore the balance. Exercises that free energy that gets trapped in the skull can sometimes provide immediate relief. Start with a vigorous Crown Pull (page 58). That may be enough. If not, the Pendulum Head Swing is an excellent next step.\n\n**Pendulum Head Swing** (time\u2014about two minutes)\n\n_Figure 6-13_ PENDULUM HEAD SWING\n\n1. Take a deep in-breath.\n\n2. Letting your breath out, drop your head forward to your chest, and relax your shoulders.\n\n3. Breathe in as you let your right ear pull your head over your right shoulder and feel the stretch on the left side of your neck (Figure 6-13). Take two or three deep breaths in this position.\n\n4. Slowly as you let your breath out, let your head complete the circle, with your left ear leading toward your left shoulder. Feel the stretch on the right side of your neck as you take two or three deep breaths.\n\n5. Come back to center, dropping your head forward to your chest.\n\n6. Repeat at least two more times.\n\n#  **INSOMNIA**\n\nWhile insomnia may have many causes, physiological as well as psychological, calming the overactive or chaotic energies that contribute to insomnia often works at a deeper level and brings peaceful sleep. You may have to experiment with a variety of methods and find which works best for you. Beyond energy methods, sleep labs (www.sleepcenters.org) have studied insomnia and innovated various techniques you might want to investigate. They are described in a number of popular books as well as various Web sites. One of the most common causes of insomnia for menopausal women is low magnesium levels. You can energy test magnesium capsules to determine whether you need additional magnesium, which you can take just prior to going to bed.\n\nNow is also a good time to introduce you to the energy medicine question-and-answer area of my Web site (access from www.innersource.net and put \"insomnia\" in the search engine). One or two of the techniques described there is likely to be a good resource for you. Several of the techniques that help with insomnia are general-purpose exercises, and you have already been introduced to them, such as Expelling the Venom (page 151), Crown Pull (page 58), Triple Warmer Smoothie (page 107), Connecting Heaven and Earth (page 44), and Hook-up (page 62). Because hormonal imbalances may be involved in the insomnia that is often reported with menopause, doing the Daily Energy Routine with the Menopause Module shortly before going to bed may also help you get a good night's sleep.\n\nGoing through menopause in our culture can be a soul-shaking experience. But it is also a mythic journey into a deeper engagement with life, designed to lead to the wisdom of the crone and the profound ways she is called upon to influence her extended family and community. Knowing how to evoke your energies to support you in this journey is like going on a safari with hiking boots and proper gear instead of sandals and a parasol.\n\n_Chapter 7_\n\nWeight Management Tips\n\n_Fifty percent of [women in the U.S.] diet two or more times each year with little or no long-term success at reaching or maintaining their desired weight._\n\n\u2014WEIGHT WATCHERS\n\nEvery day the media confront us with an embarrassment of anorexic-looking heroines. They coagulate in our psyches as an unworkable yet coercive ideal that is hurtful to our health, our sense of well-being, and our pride in ourselves. Before using energy medicine to address my own weight challenges, my life was permeated with the familiar self-criticism, self-denial, weight loss\/weight gain\/weight loss\/weight gain, and bottled-up hysteria that is entwined in this cultural insanity. Like many women, I was tyrannized by the ideal of slenderness.\n\nBut beyond those who feel fat at a healthy weight, it is also true that more than 150 million Americans weigh substantially more than what is optimal for their health, with at least 60 million of them being clinically obese. And these gloomy numbers have been skyrocketing in recent decades. How do we deal with this epidemic of obesity without swinging back to anorexic ideals that carry their own evils?\n\nIn addition to the blow to our vanity, excess weight leads to numerous health problems that could be avoided, from diabetes to heart attacks to strokes to liver failure. The vast majority of the 150 million-plus overweight individuals have tried many strategies to lose those pounds. But they run into a number of paradoxes. For instance, your body takes comfort in excess fat. It wants to survive the next famine, and extra fat is like cash in its emergency bank account. In fact, if you allow yourself to become extremely hungry, you throw off your blood sugar and metabolism, triggering the ancient response of storing whatever you eat as fat. Even sleep enters the picture. Your body tends to store more weight if you are sleep-deprived. So giving yourself permission to get enough sleep isn't just something you do so you can be more rested and alert. Think of it as part of dieting!\n\nHundreds of diet books come and go! Clearly, dieting is not the whole answer for most of us. In fact, dieting often has a paradoxical effect, turning on mechanisms that cause your body to produce fat cells and preserve the fat it already has. Dieting signals to your body that the famine that was universally dreaded by our ancestors has begun. Turn every morsel of nutrition into fat storage! Now!!!\n\nShelves of different diet books also reflect the fact that each of us is biochemically unique. The same food combinations that cause one woman to lose weight are stored on another's hips. While the \"calories-in minus calories-burned equals calories-stored\" theory is true for some people, it totally misses the complex genetic and biochemical realities of weight storage for many others. Thin people often assume that excess eating is the only cause of excess weight. End of story. The rest is lies and excuses. Their more portly companions, noticing that their own food intake is far less than that of their judgmental beanpole friends, are not convinced that this is the scientifically accurate conclusion. The fact is that a large proportion of people who are heavy eat less and exercise more than their thinner friends. Yes, totally unfair! No weight management plan fits more than a minority of people. And even those programs that attempt to account for individual differences provide general guidelines that often are simply inaccurate. You are too unique, and thus such generalities are not reliable.\n\nIn my experience, energy medicine can be extremely effective in weight control. Several energy systems are involved in metabolism, in the creation of muscle cells, and in the storage or burning of fat cells. Because these energy systems tell your chemistry what to do, you can manage your weight by managing these energies. In the sense that it is easier to control the temperature of your refrigerator by adjusting the temperature knob than by dumping in ice cubes, working with the _energies_ that control weight is like having your hand on the thermostat.\n\nI once led an ongoing weight and energy medicine workshop for 16 people. The group included six women who were referred to me by local physicians because their weight was posing a serious hazard to their health. Each was more than 300 pounds. Because I was still experimenting with using energy medicine for weight management, I offered to do the group for free under one condition. And that was that they not diet (I'm known locally as a very tough negotiator). We worked together for about 16 months, two sessions per week. Each of the six women referred by their doctors lost more than 100 pounds. And they kept the weight off.\n\nI wish it were easy to put into a book the methods that work for weight management. I have found, instead, that there are few requests that come through my door that require a more individualized program or more careful monitoring than controlling weight. Successful treatment combines an understanding of the woman's unique biochemistry, energies, self-image, internal conflicts, and weight history. It is easy to underestimate the physiological pressure to keep fat on the body. Willpower may have little impact if your strategies do not take into account the energetic dimension of weight control. Plus, the approach needs to be continually monitored, as the strategies that work one week may need to be revised the next. Your body's energies are always shifting. Perhaps one day I will attempt to put all of that into a book, but it will not, alas, be a simple book. What I can offer in this chapter, however, are some of the techniques that seem to be effective for a reasonable number of people.\n\nRather than laying out a systematic program that pretends to address the unique biochemistry of each reader, I simply present them as \"Weight Management Tips\" in three categories: Tips for making healthy choices when your body is screaming \"FEED ME,\" tips for reprogramming your mind to support healthy eating habits, and tips for reprogramming your body to maintain a healthy weight. While starting here with full disclosure that effective weight management using energy medicine often requires a highly sophisticated individualized approach, you may experiment with these tips and find that some are genuinely useful for you. For instance, if the energies moving through your spleen meridian are low, it is very hard to metabolize food properly and therefore to burn calories efficiently. So several of the \"tips\" offer ways of promoting a healthy energy flow in the spleen meridian. Simply tapping your spleen points (Figure 2-5, page 54) for about ten seconds before and after eating improves your metabolism. How cool is that! Consider each tip an experiment and discover the ones that are helpful to you. I hope you will find them useful and empowering.\n\n#  WHEN YOUR BODY IS SCREAMING, \"FEED ME\"\n\nThe desire to place tasty food in your mouth is nearly as primal a biological urge as breathing itself. Your body is designed, in one of nature's striking examples of the body-mind connection, to cause you to crave the foods it needs when it needs them. Not to be outdone by nature, however, the food industry attempts to manipulate your body's natural preferences with artificial sweeteners and other miracles of modern chemistry, fooling your taste buds into responding to substances that are not good for you as if they were good for you. In addition, because of imbalances in the body or the psyche\u2014often caused by stress, habit, or biochemistry\u2014 the signals sent to your brain regarding the need to eat do not match your body's actual need for sustenance. The result may be compulsive or anxious eating that not only forces you to have to process material you do not need, but also taxes your body and hurts your health.\n\nEach of the Weight Management Tips addresses such dilemmas with energy interventions, from restoring physiological and psychological balances to what to do at the moment of temptation. The tips in this first set present straightforward steps you can take when your body is screaming, \"FEED ME.\" If uncontrolled eating is an issue for you, experiment with these and keep the ones that work best in your back pocket. The techniques presented here address:\n\n1. Stressed Eating\n\n2. Comfort Eating\n\n3. Anxious Eating\n\n4. Compulsive Eating\n\n5. That Hollow Hungry Feeling\n\n6. Specific Cravings in the Moment\n\n7. Changing a Pattern of Cravings\n\n8. Overcoming Hunger\n\n9. Subclinical Allergies\n\n##  TIP 1: STRESSED EATING\n\nDon't eat when you are stressed! First unstress! Then enjoy your meal. When your body is in fight-or-flight mode, the resources are not available for proper digestion, making it hard to metabolize food or acquire the nutrients it has for you. The Five-Minute Daily Energy Routine (page 51) is a good way to bring yourself out of stress, at least enough to enjoy a meal. A quicker and simpler way to bring your body out of stress is the Neurovascular Hold (page 151). Another is the following (time\u2014less than a minute):\n\n1. Raise your hands high above your head, make fists, and on a slow, controlled exhalation, bring your arms down slowly and deliberately, fists in front of your body (see Figure 2-16, page 75). Open your hands when they reach the bottom.\n\n2. Place your hands back-to-back and slowly and deliberately slide both hands up the center of your body (see Figure 7-1).\n\n3. Lift your hands in front of your face and above your head on another deep in-breath.\n\n4. Turn your palms outward, and circle them down to the sides of your legs with another out-breath.\n\n_Figure 7-1_ EAT IN PEACE\n\n_Taking your arms down the central meridian releases tension buildup and calms the body. The upward motion of your hands \"zips\" this calmness into your central meridian. Lifting your arms above your head and then taking them down your sides is calming to your nervous system._\n\n##  TIP 2: COMFORT EATING\n\nSometimes we eat simply because we want to be comforted. Hunger is not even a factor. We may take in food because we are feeling lonely, depressed, lost, or overwhelmed. An alternative way to comfort yourself is to calm the energies that are involved in these emotions, all of which ultimately trace to triple warmer. A simple yet powerful way to calm these emotions is the Triple Warmer Smoothie (page 107).\n\n##  TIP 3: ANXIOUS EATING\n\nOther times we eat to quell feelings of anxiety. Again, hunger is not a factor. And again, guess which meridian is directing the energies: triple warmer! The Triple Warmer Smoothie can help manage the urge for anxious eating as well as for comfort eating. While other methods for dealing with causes of the anxiety may be called for, the following technique is another way of calming triple warmer, turning off anxiety's alarm response. This method also sends a signal up the central meridian, calming the central nervous system and providing immediate relief without the calories:\n\n1. Place your middle finger on a point about two inches below your belly button.\n\n2. Take a deep in-breath, and as you release the breath, pull your finger up to your navel, slowly and with pressure.\n\n3. Repeat two more times.\n\n4. Flatten one hand on your womb chakra, below your navel, and the other on your heart chakra, over the middle of your chest. Rest them there for several deep breaths.\n\n##  TIP 4: COMPULSIVE EATING\n\nHere are three simple techniques that can be used, individually or in combination, to interrupt the compulsion to eat what you don't need:\n\n1. Place either thumb in your mouth and suck it while bending your index finger over the area between your lip and your nose (as a baby does) and rub this area up and down slowly with your knuckle. _This hooks up the central and governing meridians to create a force field that is comforting and also provides a minicranial adjustment that moves oxygen and cerebrospinal fluid through the head, shifting the focus from food to the comforting feelings that are elicited._\n\n2. Place one hand on the middle of your chest. With your other hand, tap the point that is between the fourth and fifth fingers, just above the knuckles, toward the wrist (see Figure 3-3, page 108). Breathe deeply as you tap and think about the food you are obsessing about. _This is an acupressure power point on the triple warmer meridian that turns off the ancient, primal response to grab any available food out of a fear of scarcity._\n\n3. Read the instructions for Expelling the Venom (page 151). Instead of releasing anger or frustration on the outbreath, release the panic or urgency to eat. On the Zip-up at the end, use an affirmation such as \"I am feeling peaceful inside\" or \"I am satisfied and pleased with myself.\"\n\n##  TIP 5: THAT HOLLOW HUNGRY FEELING\n\nThe Sunrise\/Sunset technique (page 221) not only stabilizes blood pressure, it also, like a sedative, impacts your chemistry in a way that produces an inner peace. As you lift your arms to the heavens, you also open to larger forces, allowing beneficial surrounding energies to come in and fill you. Another quick technique when you are tempted to use food to fill that hollow hungry space inside is called Mad-die's Belly Scoop (time\u2014about 20 seconds):\n\n1. Place your hands at your waist with your thumbs on your back and your fingers on your front.\n\n2. Take a deep breath in, and then pull your thumbs forward with pressure (Figure 7-2a).\n\n3. As you release your breath, let your thumbs continue to slide forward toward your navel.\n\n4. When they are about two inches from your navel, let your hands and arms swoosh forward with power, scooping out the energy and sending it outward (see Figure 7-2b). Repeat several times.\n\nA third technique for countering the hollow hungry feeling is rooted in a 6,000-year-old Taoist practice that takes your attention to the part of you that feels empty.\n\n1. Lie on your back and place the palm of either hand on your navel. Move your hand slowly in small clockwise circles, at first only brushing the skin. Slowly make larger and larger circles. Feel the warmth. As you continue, allow your mind to experience all the sensations that are produced in your body. Once the circles are surrounding your entire stomach area, begin moving inward again, toward the center, continuing in a clockwise direction.\n\n2. Repeat the above sequence, this time using more pressure, and have your mind's eye watch the energies as they move.\n\n3. Do the sequence one last time, now pushing in quite deeply, literally massaging your intestines. As you follow the sensations, imagine that excess fat is dissolving with each circular motion. _The circular motion quiets hunger, removes the \"hollow\" feeling, and stimulates peristaltic activity. Other benefits include tightening the flesh in the areas being massaged, strengthening the blood vessels, and helping the digestive system work more efficiently._\n\n_Figure 7-2_ MADDIE'S BELLY SCOOP\n\n##  TIP 6: SPECIFIC CRAVINGS IN THE MOMENT\n\nA quick diversion when you are craving a snack your body does not need is to place one middle finger in your navel, push in, and pull up several inches with pressure.\n\nThen pull down, back to the navel. Repeat, this time pulling out to the left side and back to the navel. Continue around until you have made a five-pointed star. _The technique frees blocked energies in your abdomen, providing its own kind of satisfaction, and breaking the focus that was going toward the craved delicacy._\n\nAnother way to quell a craving is to do the variation of Expelling the Venom described in Tip 4 (step 3, page 261), but for the affirmation, say something along the lines of: \"I feel _so_ content and satisfied. Losing weight feels better than eating that pie.\"\n\n**Caution:** _Watch the labels on what you ingest. For instance, some foods are manufactured with addictive sweeteners, such as leptin and ghrelin, that manipulate hormones to tell your brain you are hungry no matter how much you have eaten._\n\n##  TIP 7: CHANGING A PATTERN OF CRAVINGS\n\nMany forms of hypnosis and self-suggestion have been applied for helping people manage their eating habits. Two of the best that I know are the Zip-up, with an affirmation (page 75), and the Temporal Tap (page 75).\n\nIf potato chips are your downfall, an affirmation you might use with the Zip-up might be \"I feel empowered saying no to potato chips.\" With the Temporal Tap, you might state as you tap on the left side (negative phrasing), \"I don't even like that processed taste of potato chips anymore,\" and on the right side (positive phrasing) tapping in, \"I love being freed from the tyranny of the potato chip.\"\n\nAnother way to break an addictive craving is to completely change your body chemistry and then to use more psychologically oriented interventions, such as the Zip-up or Temporal Tap. For instance, doing a juice fast for a couple of days can have many benefits, including interrupting physiological habits so new ones can be put into their place. Get instruction from a knowledgeable health care provider or good health food store so you will do the fast in a manner that gives your body the least disruption and greatest benefit.\n\n##  TIP 8: OVERCOMING HUNGER\n\nIn the best of all possible worlds, we would eat when we are hungry, stop when we are wonderfully satisfied, and maintain the ideal weight for feeling great and staying healthy. That is how we are designed. Sometimes in today's world, however, you may want to refrain from eating even though your body is legitimately hungry. Here are three simple techniques to make this easier:\n\n1. Directly in front of the opening to your ear is a flap. Grasp it with your thumb and index finger and begin to twist and massage it. _Acupressure points that have been used for thousands of years to turn off hunger when food was not available are situated on that flap._\n\n2. I find what I call \"hypothalamus imagery\" to be a great focus when I am dealing with hunger. The hypothalamus is a gland that, among other things, regulates metabolism and the desire for food intake. I imagine a little guy in the back of my head who is at the controls of my hypothalamus and whose job is to take care of me by inducing me to want food to stave off the famine and store all the fat I could possibly need. I just talk to him. I reassure him, telling him I am safe, putting him at ease in every way I can. When he is at ease, I am at ease, my body is at ease, and I am in control of how I respond to my hungers and cravings.\n\n3. Do the Abdominal Stretch (page 134), Connecting Heaven and Earth (page 44), Diaphragm Breath (page 220), or any other exercise that stretches your abdomen. _Stretching the abdomen opens space for energy to flow, providing energetic nourishment._\n\n##  TIP 9: SUBCLINICAL ALLERGIES\n\nSubclinical allergies are responses to certain foods that do not easily metabolize for you. The symptoms are much subtler than a skin condition or indigestion. These foods can cause weight gain that is disproportionate to their calories. Identifying such foods is yet another reason to become adept in energy testing so you can \"test before you ingest.\" If you know your body has difficulty with a specific food and want to experiment to see if you can help your body adapt to it, tapping on your spleen points (page 55) as you chew it can help your body metabolize it more easily.\n\n#  REPROGRAMMING YOUR MIND\n\nAt more than 300 pounds and just starting to work with me, Joan sent this letter:\n\n_Well, I know I should feel ashamed of myself, and sometimes I do. Sometimes I feel rotten about other people's assumptions about me. That I'm sloppy and dumb and I eat lots of starch and have no self-respect and I just don't care. But I have my moments. After I get home and I undress, I look in the mirror. I see rings of flesh, massive planes, mountainous plateaus. Suddenly, I'm in a glory of fleshy existence. I see a power in it, in my own body. Fertility. Abundance. A large unrestricted sense of life. I'm actually proud of the way I look. How dare people presume I'd rather be thin!_\n\nIt is a great paradox that accepting your body just as it is can be the start of a wonderful transformation. Eighteen months after writing the above, Joan was at 125 pounds following a twice-weekly weight support group that focused on energy medicine methods and doing daily homework, but no dieting for the purpose of losing weight. Her doctor expressed amazement that the energy work reversed a serious metabolic disorder that other medical interventions had not been able to correct. The last time I saw Joan, 16 years later, she was still at 125 pounds. Her set point (discussed later) had gone south for good.\n\nThis achievement, interestingly, was at first hard on Joan's emotions. After attaining such a wonderful relationship with her body in its fullness, she became depressed as the energy work changed her metabolism and she began shedding weight faster than anyone else in the group. She lost respect for the people who liked her so much more as a slim woman than as a round one. She left her marriage after a similar dynamic produced disdain for her husband. Going from large to small was not an easy journey. Had I been more experienced, I would have addressed her self-image and her acceptance of the transformation.\n\nYour mind is a powerful player not only in how you feel about your weight, but also in how much weight your body stores and in the behaviors that affect your weight. The following seven tips present energy techniques that can help establish a better partnership with your body in relation to weight.\n\n **10.** I Love Me Just the Way I Am\n\n **11.** Maintaining an Upbeat Feeling While Dieting\n\n **12.** Changing a Present Truth to the Truth That You Want\n\n **13.** Countering the Fear of Fat\n\n **14.** Countering the Fear of the Bathroom Scale\n\n **15.** Countering Depression About Weight Gain\n\n **16.** Appreciating Plateaus\n\n##  TIP 10: I LOVE ME JUST THE WAY I AM\n\nAs the great American psychologist Carl Rogers used to say, \"The curious paradox is that when I accept myself just as I am, then I can change.\" Loving and accepting your body is the best way to get it to cooperate with any intentions you might have for changing it. Trust me. In my weight classes, each woman had the homework assignment of looking at herself in the mirror and expressing appreciations toward each part of her body. It began, however, with focusing on a body part about which she had judgments or bad feelings. While looking and thinking about this body part, she went through the following instructions:\n\n1. Face a mirror, place your thumbs on your temples, in the indentation just outside your eyes, and rest the other fingers on your forehead. Take three or four deep breaths while holding this position. _The points you are holding reduce stress levels. Holding them helps blow the negative charge you have been harboring about this part of your body._\n\n2. Take a moment to recognize this part of your body as a miracle of nature, a sacred achievement of evolution. Ask your body for forgiveness about your judgments, something along the lines of: \"Forgive me for having been so mad at you and so judgmental. You have loved me unconditionally, doing the very best you could to serve me.\"\n\n3. Now place one hand over the middle of your chest (heart chakra) and the other just beneath your navel (womb chakra), mindfully connect with the part of your body you are looking at in the mirror, and state a sincere appreciation for it.\n\nYou may have to reflect a bit to identify the kinds of gratitude that emerge when you recognize what this part of your body is doing for you, but the process will engage you with your body, and its parts, in new ways. Here are some appreciations I have stated:\n\n\"Thank you, arms, for reaching for what I want, for expressing my feelings, for your strength, for your ability to embrace.\"\n\n\"Thank you, legs, for walking me to wherever I want to go.\"\n\n\"Thank you, tummy, for digesting my food and for keeping me warm and safe.\"\n\n\"Thank you, breasts, for making milk for my girls when they were little, for making me feel sexy, and for bringing me pleasure.\"\n\nWhisper loving-kindness to your body as you would to any other intimate.\n\n##  TIP 11: MAINTAINING AN UPBEAT FEELING WHILE DIETING\n\nThe concept of dieting is connected with the concept of deprivation. It is more accurate, however, to recognize that smart dieting is a way of _giving_ your body what it truly needs. You are not fighting it, you are loving it. Any time you begin to feel sorry for yourself or cheated or deprived of a food you might crave in the moment, this exercise, the Three-Heart Hook-up, creates a sense that all is well.\n\nYou will be tracing three hearts: one over your face, one over your torso, and one in the larger field that encompasses your entire body:\n\n1. Cradle your face in your hands, with the bottom of your palms touching at your chin and your fingers reaching up over your cheeks. Draw your middle fingers over to the center, between your eyebrows (the \"third eye\"), push up, and with a deep breath, trace a heart with your middle fingers, ending at your chin. Trace this heart over your face three times, breathing deeply.\n\n2. Then draw your fingers from your chin down to your chest. Trace a large heart up and around your rib cage, ending by flattening and moving your hands in front of your hips until your fingers touch. Then move your fingers straight up to the center of your chest and create two more of these hearts, again breathing deeply.\n\n3. At the bottom of that heart, place your hands so they are back-to-back, thumbs against your body, and bring your thumbs up the center of your body and over the center of your face, separating as they move over your head, and creating as large a heart as your arms can comfortably reach in the space that surrounds you, ending with your hands touching again at the bottom of your torso. Again place your hands back-to-back and bring them up the center of your body, breathing deeply while creating two more hearts.\n\n##  TIP 12: CHANGING A PRESENT TRUTH TO THE TRUTH THAT YOU WANT\n\nThe Temporal Tap (page 75) is a powerful method for changing self-limiting mental programming. You often have to do it every day for about a month before it locks in, but for a long-standing, deep belief, you should expect that some persistence might be required. In relation to weight, these may be very specific and in-the-moment beliefs, such as \"I must have chocolate right now,\" or quite far-reaching ones, such as \"all the women in my family are fat by the time they reach 50, and fat is my fate as well.\" The wording used with the Temporal Tap is very important, and it is also important that you find wording that fits for you. The following are simply possible wordings for the two examples just mentioned:\n\nFreeing yourself from chocolate:\n\nLeft (negative phrasing): I don't even like chocolate anymore.\n\nRight (positive phrasing): I love feeling free from that yearning for chocolate.\n\nChanging your expectations about the future:\n\nLeft (negative phrasing): It isn't my fate to get fat as I grow older.\n\nRight (positive phrasing): I am going to be slim and fit at 50.\n\n##  TIP 13: COUNTERING THE FEAR OF FAT\n\nThe culture's messages about being overweight create a tyranny of fear that literally permeates into your cells. That tyranny can keep you from seeing yourself accurately, while creating a panic to diet in unhealthy ways. If and when it grips you, the following, in any combination, can help quell it.\n\n1. While tuning in to the feeling, place either hand over the center of your chest. With your other hand, tap the hand on your chest in the valley between the ring finger and little finger, above the knuckle (toward the wrist). Tap on each hand for several deep breaths.\n\n2. While tuning in to the feeling, hold your neurovascular points (page 105).\n\n3. While tuning in to the feeling, do the Blow-out\/Zip-up\/Hook-in (page 74).\n\n4. While tuning in to the feeling, do the Meridian Energy Tap (page 80)\n\n##  TIP 14: COUNTERING THE FEAR OF THE BATHROOM SCALE\n\nMore women have told me of their terror about getting on the bathroom scale than you might imagine. Their scales have become triggers for self-doubt, emblems of emotional discouragement. For some women, this has become so extreme that they haven't weighed themselves for years. I've even been told, \"That's why I don't get regular medical checkups\u2014I don't want to see how much I weigh, and I certainly don't want anyone else to see it and record it.\" If thinking about weighing yourself sends you into fear or despair, then your relationship to the bathroom scale deserves attention. The following sequence can help you break the tyranny of that seemingly innocuous piece of equipment.\n\n1. Stand in front of the scale. Do a Blow-out (see page 74). Blow out feelings of fear or despair.\n\n2. Step onto the scale. Take a good look at your weight. Continuing to look at the numbers, breathe in deeply through your nose.\n\n3. With your breath still held, touch the thumb of each hand with the index finger of that hand and press them together hard. Hold for several seconds. Let your breath out through your mouth very slowly as your fingers release.\n\n4. Whatever the news from the scale, step off it and take another deep breath.\n\n5. Place your left hand over the center of your chest and, with your right hand, tap between the ring finger and the little finger of your left hand, just above the knuckles, toward the wrist (see Figure 3-3). Tap for several deep breaths and then repeat on the other hand.\n\n6. Do the Zip-up (page 75) with an affirmation such as, \"I am grateful I have this gauge to give me good and helpful information.\"\n\n##  TIP 15: COUNTERING DEPRESSION ABOUT WEIGHT GAIN\n\nSometimes it is very hard to be unattached about your weight. We may have done a great deal to free ourselves from the culture's tyranny about staying absurdly thin, yet when we step on that scale and find that it has been climbing upward, we may feel discouraged and depressed. The methods described for menopausal depression (pages 245-248), including stretching techniques, the Homolateral Crossover, the Mellow Mudra, and the Heart-Womb Connection, in any combination, can be helpful for countering any form of depression.\n\n##  TIP 16: APPRECIATING PLATEAUS\n\nHowever effective your efforts, you are going to hit plateaus\u2014periods where progress seems to stop or even go backward. While these can be discouraging, plateaus are inevitable aspects of the ebb and flow of change. They are necessary to help your body regroup. One way not to be thrown off by plateaus is to anticipate and reframe them. You can recognize a plateau as a good sign, a reason to celebrate, and a reason to rest. It is a time to drop into gratitude for the progress that has been made. You can't push here. Your body's intelligence knows how long you need to stay at an interim spot before it can embrace further change. By accepting the plateau, you don't stress your body. Stressing your body triggers the triple warmer and can reverse progress. When you do reach a plateau, one energy technique that is always helpful is to strengthen the spleen meridian (see page 132). The spleen meridian governs metabolism, and keeping this meridian strong can, along with time, help your body get used to the weight loss rather than bouncing back to the earlier weight. Strong spleen energies may also shorten the amount of time you need to be in the plateau.\n\n#  REPROGRAMMING YOUR BODY\n\nWhile the techniques for reprogramming your mind may be vital for weight management, making shifts in physiological processes that impact weight, without the use of drugs, is one of the great strengths of an energy approach. The areas of focus in this section include:\n\n **17.** Lowering Your Set Point\n\n **18.** Improving Your Metabolism\n\n **19.** Assimilating Food More Effectively\n\n **20.** Activating a Sluggish Digestive Tract\n\n **21.** Slimming Your Waist and Belly\n\n **22.** Oxygenating Your Body\n\n **23.** Providing First Aid for Low-Blood-Sugar Attacks\n\n **24.** Countering Hormone-Induced Weight Gain\n\n **25.** Keeping Your Thyroid Healthy\n\n **26.** Eliminating Toxins\n\n##  TIP 17: LOWERING YOUR SET POINT\n\nThe set point is a set weight the body attempts to maintain. Like a thermostat turning on and off the heat or AC to maintain a predetermined temperature in a room, the body tends to keep bouncing back to a given weight. If you manage to get your weight to be less than your set point, a host of physiological mechanisms kicks in to store fat, such as changes in metabolism that reduce the rate at which calories are burned. It doesn't matter if your set point is higher than you wish it to be. When your weight goes below your set point, you can feel starved, with images of food flooding your mind. Your body chemistry might compel you to eat a whole bag of cookies, and even then you might not feel satiated. This is nature in action, trying to prepare you for the next famine with absolutely no regard about your personal desire for that anorexic look.\n\nMany weight-loss approaches recognize that lowering the set point would be an excellent strategy for managing weight. How to do this, however, is a matter of debate. While aerobic exercise, for instance, is known to have a positive effect, it isn't exactly a magic bullet. The set point is _set_ by many factors. Some, such as heredity, body structure, and age, are fixed\u2014you cannot control them. Others, such as health and stress level, fluctuate. What is not usually discussed is how decisively the set point is controlled by your body's energies. Triple warmer, the energy system that maintains survival habits, manages the set point. If chronic stress pushes triple warmer into a condition of perpetual threat, for instance, raising the body's weight may be a misguided protective response. Reducing chronic stress can cause triple warmer to lower the set point. The techniques for balancing your overall energies presented in this book can have the happy side effect of lowering your set point. When I learned how to strengthen my spleen meridian, which was always my Achilles heel, I lost 17 pounds in seven weeks without changing my diet. This wasn't even my goal, yet it happened. Maintaining a good balance between your spleen and triple warmer meridians, as emphasized throughout this book, is among the most important steps I know for keeping your set point stable and sometimes lowering it as well. You can also reset your set point more directly by using powerful self-suggestion techniques bolstered by stimulating energy points.\n\nThe Temporal Tap (page 75) is a powerful approach. You need to find wording that fits you and your situation. Take into account internal obstacles to losing weight. If the comfort brought by eating has become a substitute for other rewards in life, you must address this directly. Remember, use a negative wording when tapping on the left side and a positive wording when tapping on the right side. So while tapping on the left side, you might say, \"I no longer eat to curb my sense of emptiness, and my set point does not resist going down.\" When tapping on the right side, you might say, \"I am savoring every bite and my set point is dropping to [five pounds less than your current weight].\" If having extra weight is at some level a way of protecting yourself against unwanted sexual advances, your wording on the left side might be, \"I do not need to keep extra fat to set clear boundaries.\" On the right side, you might say, \"I know how to assert a clear 'No!' as my set point drops to [five pounds less than your current weight].\" Thousands of variations are possible, but 30 days of tapping with wording that fits for you can make a real difference. I've seen this happen over and over. Being able to lower your set point through self-suggestion and energy work is highly empowering.\n\n##  TIP 18: IMPROVING YOUR METABOLISM\n\nAnother fast track to reducing weight is to get your body to break down food and burn calories more rapidly and effectively. Oxygen is a key to healthy metabolism (see page 249), and the Metabolic Breath (page 250) is a one-minute technique for helping your body metabolize food more efficiently. If your energy is also dragging, which often goes along with slow metabolism, chances are that you are in a homolateral pattern and will benefit from the Homolateral Crossover (page 67). Then do the following two or three times each day:\n\n1. Do the Five-Minute Daily Energy Routine (page 51) and follow it with Connecting Heaven and Earth (page 44).\n\n2. Place your thumbs on the triple warmer neurovascular points at your temples and the pads of your fingers on your forehead, on the bony area above your eyebrows (see Figure 3-1, page 106). Hold here softly for up to three minutes, breathing deeply in through your nose and out through your mouth. _Triple warmer's default setting is to keep fat on your body so you will survive if food gets scarce. It did not read your diet books. And the more stress you are facing, or the greater your sleep deprivation, the more fat it wants to store. Calming triple warmer is a way of telling it that it can relax its extreme survival strategies, including extra fat storage._\n\n3. Do the Spleen Meridian Flush and Tap (page 136) to speed your metabolism. _The spleen meridian governs metabolism._\n\nWhile these energy techniques address metabolism directly and effectively, there is also no substitute for regular aerobic exercise and vigorous stretching. Build them into your daily routines.\n\n##  TIP 19: ASSIMILATING FOOD MORE EFFECTIVELY\n\nYour small intestine's job is to extract the nutrients from food and assimilate them into your body. If the small intestine does not accomplish this effectively, you are required to take in greater quantities of food to get the same nutritional value. Your body literally \"hungers\" for the missing nutrients. It compels you to eat additional quantities because it was not able to absorb the nutrition that was available in the\n\n_Figure 7-3_ ASSIMILATION MASSAGE\n\nmeal you just had. This adds unnecessary weight. In addition to taking the steps discussed above for optimizing your metabolism, you can assist your small intestine in assimilating food more effectively by massaging its neurolymphatic reflex points two or three times each day. You can massage the two major sets of small intestine points in about 20 seconds:\n\n1. Cup your fingers and massage along the bottom of your rib cage (see Figure 7-3a).\n\n2. Grasp your thighs with your hands and massage your inner thighs with your thumbs (see Figure 7-3b).\n\n##  TIP 20: ACTIVATING A SLUGGISH DIGESTIVE TRACT\n\nIf your body is not eliminating properly, it is often because the ileocecal valve\u2014the valve that connects the small and large intestines\u2014is not opening and closing as it should. When this occurs, and it is quite common, the digestive system's peristaltic rhythm is not fully supported, digestion becomes sluggish, and material that is meant to be eliminated can even back up into the small intestine. A series of folds in the far end of the large intestine, called the Houston's valve, may also become involved. While a variety of physical problems may lead to malfunctions in the ileocecal and Houston's valves, stress and tension are often the causes. If you are feeling bloated or experiencing difficulties in eliminating, massaging the ileocecal and Houston's valves as follows is likely to be beneficial. Resetting both valves creates a symmetry between them.\n\n**Massaging the Ileocecal and Houston's**   \n**Valves** (time\u2014about 20 seconds)\n\n1. Place your right hand on your right hip bone with your little finger at its inside edge (your hand is over your ileocecal valve).\n\n2. Place your left hand at the corresponding spot at the inside edge of your left hip bone (your hand is over the Houston's valve) [see Figure 7-4]. Exert pressure as you slowly drag the fingers of each hand up six to seven inches with a deep inhalation.\n\n3. Shake the energy off your fingers with the out-breath and return to the original position. Repeat about four times.\n\n4. End by dragging your thumbs from top to bottom, backward along the same paths, one time with pressure.\n\n5. Close with the Three Thumps (page 53).\n\n_Figure 7-4_ VALVE MASSAGE\n\n##  MASSAGING THE LARGE AND SMALL INTESTINE NEUROLYMPHATIC REFLEX POINTS\n\nAnother way to support your digestive system when it is sluggish is to massage the large intestine and small intestine neurolymphatic reflex points. Massage with\n\n_Figure 7-5_ INTESTINE POINTS\n\npressure using circular movements. Proceed down the legs, staying two or three seconds at each point and then moving about half an inch farther toward the knees.\n\n1. The small intestine points follow the inside seam, starting about a hand's width below the groin and ending about a hand's width above the knee (see Figure 7-3b, page 274).\n\n2. The large intestine points extend from your hip to your knee and follow the outside seam of a pair of pants (see Figure 7-5).\n\n##  TIP 21. SLIMMING YOUR WAIST AND BELLY\n\nAnything that dissolves tension and allows energy to move through the middle of your body clears toxins and helps your metabolism. The following four techniques, in any combination, accomplish this.\n\n1. Massaging the Large and Small Intestine Neurolymphatic Reflex Points (as described above).\n\n2. Connecting Heaven and Earth (see page 44).\n\n3. Abdominal Stretch (see page 134).\n\n4. Sideways Stretch (see page 135).\n\nAnother very simple technique for slimming the waist is Tanya's Spiraling Swing and Slap (time\u2014about a minute).\n\nWith your feet planted shoulder-width apart and your knees relaxed and slightly bent, twist to the right and allow your left arm to swing in front of your body, your right arm in back, both slapping your body with the twist (see Figure 7-6a). Swing back to center (Figure 7-6b) and to the other side with the same motion, swing, and slaps. If you have a tendency to get dizzy, looking straight ahead will prevent this.\n\n_Figure 7-6_ TANYA'S SPIRALING SWING AND SLAP\n\n##  TIP 22. OXYGENATING YOUR BODY\n\nThe diaphragm is like a bellows that opens and shuts as you breathe. With each inhalation, it creates suction that draws oxygen into the lungs. Stress can interfere with the diaphragm's operation, keeping the cells in your body from receiving optimal levels of oxygen. We actually have little conscious control over its actions, yet we can exercise and strengthen it with the Diaphragm Breath (page 220). This results in a better distribution of oxygen throughout the body. Among the many health benefits are better metabolism and reduced levels of fat storage.\n\n##  TIP 23. PROVIDING FIRST AID FOR LOW-BLOOD-SUGAR ATTACKS\n\nAny diet plan that subjects you to low blood sugar is not a good plan. When blood sugar drops, the body's energies become weak and disorganized. The pancreas governs blood sugar levels, and the pancreas is governed by the spleen meridian. If food is not immediately available, doing the Triple Warmer Smoothie (page 107), followed by tapping the spleen neurolymphatic reflex points (page 55), helps reorganize your energies. The Wayne Cook Posture (page 56) can then further stabilize them.\n\n##  TIP 24. COUNTERING HORMONE-INDUCED WEIGHT GAIN\n\nHormones may be involved in a sudden weight increase or may be keeping weight on that you cannot shed. Medications may also disrupt your hormones in ways that affect your weight. Techniques from throughout this book, such as those in the Daily Energy Routine (page 51) or the Hormone Hook-up (page 220) or others in the Menopause Module (page 219), may help balance your hormones so food is metabolized more effectively. Such methods always bring some benefit to the body's energy systems. You can experiment to see which have the strongest impact. A quick way to stimulate the organs that process hormones is to firmly tap up the insides of your legs several times, from above your ankle to your groin, for about 30 seconds. This stimulates the liver, kidney, and spleen meridians. Then firmly tap down the outsides of your legs several times, again for about 30 seconds. This stimulates the stomach and gallbladder meridians. As you tap, breathe deeply in through your nose and out through your mouth.\n\n##  TIP 25. KEEPING YOUR THYROID HEALTHY\n\nYour thyroid impacts all your hormones, and it can become less effective as you age. It is often involved in the weight gain that may begin in the middle years. See the discussion on pages 231-233 for ways to keep your thyroid functioning at its best.\n\n##  TIP 26. ELIMINATING TOXINS\n\nThe body is continually processing the air you breathe, the food you eat, and the stresses you encounter. Toxins are harmful by-products of these activities, and they may accumulate in your body. Toxins can hinder your health in numerous ways, including interfering with your metabolism. The techniques presented for moving toxins out of your body (pages 45-51) are generally valuable, and they can be of particular importance when holding excess weight is an issue.\n\n##  FARE WELL\n\nJust as energy methods can empower you to be master instead of victim of your own weight, I hope this book and the methods and perspective it offers serve you in becoming more the master of your own fate. An approach that attunes you to your body's energies does empower you. It allows for deeper levels of involvement\u2014conscious and purposeful\u2014with the master control centers of your body and mind. I have time and again witnessed how a shift in the energy systems that govern health resulted in renewed vitality, strength, and joie de vivre. These energies are a bridge between your physical body and your spirit, so health maintenance becomes a journey into the foundation of your being.\n\nMy intention with this book has been to lead you on a path of discovery and empowerment by guiding you through the energy systems that form the infrastructure of your physical body. On this journey:\n\n\u2022 We began with an exploration of the ways that energy fields control every biological mechanism.\n\n\u2022 You were introduced to the concepts that by shifting those energy fields, you can bring about desired changes in any physical condition on which you wish to focus.\n\n\u2022 You learned a variety of techniques for stimulating your body's overall energy system to promote health, clear thinking, and well-being, packaged as a Daily Energy Routine, the energy medicine counterpart of a physical workout.\n\n\u2022 You learned to apply energy methods to specific concerns, from managing stress and emotional distress to reprogramming deep-seated energetic patterns in body and mind, and then to issues that are of particular concern to women, including menstruation, sexuality, fertility, pregnancy, delivery, menopause, and weight management.\n\n\u2022 In the process, you learned about the web of connections among your body's hormones, your energies, and your well-being.\n\n\u2022 You learned techniques for managing those energies that can help your hormones function at their best, and I hope you have felt the benefits that good management of your energies and hormones can have on every aspect of your life.\n\nAll this was in the larger context of a culture that has been evolving and changing at a rate that by any measure is out of control. As the requirements of daily life pull us further and further from the natural rhythms of our body, energy medicine becomes an antidote, reminding us of who we are and reconnecting us with our deepest nature. And there, we find, lies the true gift of creation. We who give birth know in our hearts and bones and wombs how energy creates, acclimates, adjusts, and builds healthy bodies. As we step into a future where the feminine principle walks hand in hand with the physician, the statesman, and the educator, the new world we so deeply long for is giving birth. This book opened and now closes with an emphasis on the happy fact that every woman is an exquisite vibration, a spark of the divine archetypal feminine principle. As you embrace your deepest nature more fully, the energies at the foundation of your being shine forth into your body, your mind, and your world.\n\nAppendix\n\n**Talking with Your Body's Energies** **1** **\u2014 The Art of Energy Testing**\n\n_The Chinese physician can detect imbalances in meridians by feeling the pulses, but this is a sensitive touch, and it may take ten to twenty years to develop proficiency with it._\n\n\u2014 JOHN THIE _Touch for Health_\n\nEnergy testing, on the other hand, can be learned quickly. You don't need any\n\ngreat intuitive powers or an ability to \"see\" energies. In fact, when I learned energy testing, what excited me the most was that it gave me a way to demonstrate to others what I was seeing in their energy field. With energy testing, you can reliably assess, in any given moment, the body's unique energies and energy fields and their ever-oscillating fluctuations.\n\nYour body is, in fact, a cascading fountain of energy systems, remarkably complex, exquisitely coordinated, and entirely unique. That is why no book can tell you exactly what you must do to thrive. Not only does every person carry a distinguishing energy, so too does every cell, every organ, and every system of the body. Just as your thumbprint, your heart, and your brain are unique, so are the energies of each. These local energies speak their own language, and they also share a universal language. Energy testing is a precise tool for translating these languages, phrase by phrase. It is also the case, and all my students eventually experience this, that the more you work with your own energies and the energies of others, the more strongly you develop your ability to intuitively sense them.\n\n#  THE ART AND SCIENCE OF ENERGY TESTING\n\nAs you develop your sensitivities to the subtle energies swirling within and around you, it will be extremely useful to have a tool that is tangible and palpable rather than to be left to rely totally on intuition. Enter energy testing. Energy testing, developed as \"muscle testing\" by George Goodheart, the founder of Applied Kinesiology, and meticulously refined by his prot\u00e9g\u00e9 Alan Beardall, is a concrete and very tangible procedure. It allows you to determine whether an energy pathway is flowing or blocked, whether an organ is getting the energy it needs to function properly, or whether an outside energy (such as the energy of a particular food or a suspected toxin) is harmful to your system. I call the procedure \"energy testing\" instead of \"muscle testing\" because the test really assesses the _flow of energy_ through the body's meridians rather than the _strength_ of the muscles used in the test.\n\nWhen I was first learning energy testing, a problem whose solution had eluded me all my adult life was solved. After years of experimentation, I had not been able to figure out why I couldn't manage my hypoglycemia or my weight, even though I was diligently taking the steps that should have controlled both. My spleen meridian consistently tested weak. The pancreas, which controls hypoglycemia and often weight, is on the spleen meridian. By working with the spleen meridian, I was able to counter my lifelong hypoglycemia. I was impressed. I also lost seventeen pounds without changing my diet. I was _really_ impressed. Over the years, energy testing has been priceless within my personal as well as professional life.\n\nWhile MRIs, EEGs, and CAT scans provide vital and often lifesaving information about the body and its energies, I have yet to find a medical device that can reliably make the _subtle_ determinations afforded by energy testing. Built into each of us, however, is all the equipment we need to ascertain which energies are good for a person and which are not. Energy testing is always available, night or day, and requires no instrumentation. If you practice it regularly, it can begin to feel almost instinctive. It is a tool that lets you ask your body for information about what it needs. And it lets your body answer in a language with a small enough vocabulary that you can master it.\n\nIn fact, energy testing is quite easy to learn. Deceptively easy. As a result, many people apply and misapply it casually and often inaccurately. Misapplied, it may reveal more about the beliefs of the tester, the fears or hopes of the person being tested, or other factors that have nothing to do with the information being sought. Many people have encountered energy testing by an inexperienced practitioner, have been given flamboyant explanations of its capabilities, or have seen it used more like a parlor trick than a tool for self-knowledge. I want to lift energy testing in your mind to its rightful place, which is between science and art. It is an art to learn how to energy test reliably. Once you have, it becomes a dependable barometer of your body, your energies, and your environment.\n\nIf I had the power to impact the medical profession in just one way, it would be that physicians add energy testing to their diagnostic tool kit for determining the choice and dosage of medications. Iatrogenic illnesses\u2014disorders induced by medical treatment\u2014are among the most serious problems in medical care today. Energy testing could significantly reduce their incidence. And if I could choose just a few ways for this book to impact your lifestyle, one of them would be that you use energy testing to determine what foods, vitamins, and supplements you should, and should not, take into your body. In maintaining health, knowledge is power, and because each of us is unique, energy testing is vital in my work.\n\nEnergy testing allows you to assess the state of your own or another's energies, identify imbalances, and tailor the procedures presented throughout this book to suit your own unique needs or those of someone you care about. A woman with multiple chemical sensitivities and a history of life-threatening allergic reactions needed an antibiotic for a strep infection. When she went to the drugstore to get the prescription filled, she asked if she could bring it back if she proved to be allergic to it. The pharmacist of course said that would not be possible, but she was on a limited budget, and she was adamant that she not wind up buying medicine she couldn't use. She knew how to energy test, and she convinced one of the drugstore staff to energy test the medication on her. She tested strong and purchased the medication.\n\nThe pharmacist was so amused that the story got back to me. The story had an unexpected but instructive ending. The medication seemed to be effective, with her symptoms subsiding within a few days, but it was supposed to be taken several times per day for a full course of ten days. At about day six, the woman's neck began to itch, her stomach and ankles swelled, and her heart began to fibrillate. She recognized this as a severe allergic reaction and had her granddaughter energy test everything she had recently ingested. The antibiotic now tested weak. After she stopped taking it, the symptoms soon subsided. As often happens for people with hyperallergenic tendencies, she had built up an intolerance to the medication. Does this mean that the original energy test was inaccurate? No. The body is a dynamic system that is always in flux. Intolerances can and do develop. Because energy testing is quick, free, and always available, it is an extraordinarily useful tool for obtaining current information about your body's changing needs.\n\n**How Reliable Is Energy Testing?** A 1984 article in the journal _Perceptual and Motor Skills_ is one of the first published laboratory studies that supports the potential value of muscle testing, or energy testing. The investigator, Dean Radin, later commented: \"To my surprise I found that in double-blind tests, people were slightly weaker when holding an unmarked bottle of sugar than they were when holding an unmarked bottle of sand of the same weight.\" A literature review in 2007 of more than 100 subsequent studies, including 12 randomized controlled trials, led the reviewers to conclude that there is scientific support for the method. Some of the studies were quite impressive. An objective instrument called a \"force transducer\" has been developed. The force transducer has demonstrated a correlation between measured muscle resistance and the practitioner's assessment, and it has also been used to show that experienced practitioners using muscle testing had reliability and reproducibility when their outcomes were compared with one another's.  Another persuasive and well-controlled study compared muscle testing with measurements on computerized instruments and showed the difference in muscle firmness as the subject made congruent or noncongruent statements (truths or lies) to be highly significant (at the .001 level of confidence). Some studies have yielded contradictory results. One showed significant agreement among three examiners when the test used the piriformis or the pectoralis muscles but not when the test used the tensor fascia lata muscle. Other studies did not find the method to be reliable. Because there are so many ways that energy testing is conducted, and so many nuances within a single energy test, much more study is needed to determine how the method works and under what conditions.\n\nWhile research confirmation always lags behind clinical experience, some nuances that must be incorporated into both clinical work and research studies have been established. For instance, different testers consistently reported the same resultson the same clients when one set of instructions was given the client regarding resisting the tester's pressure, but not when another set of apparently valid instructions was given. Other studies have investigated the physiology of the test. For instance, muscles that show weakness in an energy test register different voltage levels than muscles that are simply tired, so the test is measuring an internal shift that is different from fatigue. Moreover, energy testing correlates with electrical activity in the central nervous system, so the information gathered during an energy test reflects brain activity, not just the state of the indicator muscle.\n\nAlthough more definitive research is still emerging, I have introduced energy testing to tens of thousands of individuals during the past three decades, and I have feedback from hundreds of them\u2014who have returned to classes, written letters, and scheduled private sessions\u2014that energy testing has proven a useful and accurate way of obtaining information about the body's needs. My experience has consistently been that the energy test corresponds with what I see in a person's energies. Information from the test also suggests where to work on the body, and the subsequent results have again and again confirmed for me the validity of the tests. The client frequently confirms the test as well, as when the test shows a weakness on the bladder meridian and the client immediately responds, \"I'm just recovering from a bladder infection!\" Some caveats apply, however. While this information is enormously useful, there are enough variables in an energy test that it should always be interpreted in the context of other sources of information. And unless a practitioner is highly experienced and proficient, decisions about medication should not be made on the basis of an energy test alone, and even then, only in consultation with the physician prescribing the medication.\n\n**A Biological Basis for Energy Testing.** Your nervous system is a phenomenally sensitive thirty-seven-mile-long antenna, and your body's energy systems reverberate to the external energies that come into their range. Everything from the food you eat to the people you encounter carries its own frequency and affects you. While most of these vibrations exist below the threshold of your awareness, your body resonates to some and tenses against others. As a result, you will embrace the energies of some foods or some people while rejecting the energies of others.\n\nYour sensitivity to these external energies is reflected in an energy test. The frequency of the substance being tested affects your nervous system, and this is reflected by the resistance in the muscle used in the energy test. Many kinds of energy can be measured scientifically, and I believe that energy testing can also discriminate among subtle energies that cannot be detected by existing scientific instruments.\n\nBecause energy testing detects the vibrational impact of a substance on your nervous system, subtle distinctions may be identified that could not, for instance, be revealed by a blood test. Foods that may appear identical, such as two apples\u2014 one organic, one not\u2014may have very different vibrations and impact your energy system differently. I personally test strong for raw milk, weak for pasteurized milk, and very weak for skim milk. Natural foods are balanced within themselves, and when we remove a portion of that food, this balance may be corrupted, and the food's vibration altered. Your body then has to assimilate a vibration that has been skewed, which may upset its own balance or at least prevent it from gaining the full nutritional value of the food. It is very hard to be smarter than Mother Nature as we tamper with foods that sustain us, but energy testing can tell you if your body's vibration is in harmony with the vibration of a food or vitamin.\n\n#  LEARNING TO ENERGY TEST\n\nYou can learn energy testing in a few minutes, but to master it, you need to incorporate it into your kinesthetic skill bank, like learning to ride a bicycle. Even as you begin to master the simple steps of energy testing, please understand that it is only with practice that you will be able to control for extraneous influences and attune yourself to the subtle distinctions that make an energy test accurate. The pressure you use in energy testing a child, for instance, is different from the pressure you use to energy test your sister, which is yet again different from working with a football player from your local college. But when your energy aligns properly with the energy of the person you are testing, the test will work regardless of the other person's strength. I became a bit of a legend in the Ashland High School athletic department when the coach brought me in to demonstrate energy medicine to the football team. These guys were not going to let this middle-aged blond lady budge their arm. But none of them could hold his arm firm after I fluttered the meridian that governed the muscle I was pushing against. Then they all wanted to know how to do this to the other team.\n\nBeing energy tested reinforces the link between your brain and subtle energies in your body, establishing new levels of internal communication. New areas of self-awarenessbegin to unfold. Many people find that they intuitively know what the result of an energy test will be before they apply pressure to the other person's arm. It's not like guessing before the test but rather launching a communication in which your awareness is working in tandem with subtle energies.\n\nWhile energy testing is usually done with a partner, you can perform an energy test on yourself, and I describe that procedure as an alternative. For learning the process, however, it is better to have a partner. I cannot state too strongly how valuable it will be for you to push through any shyness or reluctance to involve another person in your learning here. Whether it is someone already close to you or just a casual acquaintance, it will be a gift for you both. We tend to touch people for affection, for sexual gratification, or in anger. Another very important reason to become comfortable with touching one another is for healing. It is an altogether different kind of touching. Beyond opening a door to new perceptions, feelings, and understanding, healing touch can save your life!\n\n**Energy Testing with a Partner.** Every muscle, every meridian, and every organ in your body can be energy tested. A meridian is a fixed energy pathway that distributes energy to and from at least one organ. We will begin with a single test that you will be able to use in many contexts. This test determines the way the energy is flowing along the spleen meridian\u2014the energy pathway that passes through the spleen and the pancreas. The spleen is involved with the immune system. It also determines whether the body will be able to metabolize a particular food, emotion, thought, energy, or other external influence. The spleen and the pancreas are both involved with food metabolism, blood sugar levels, and the mood swings associated with them. Both organs and their shared meridian influence your general energy level, and are extremely responsive to stress. For all these reasons, testing the spleen-pancreas energies can answer many questions you may have about the way your body will respond to something you are considering ingesting or otherwise bringing into your life. This is also an excellent gauge for assessing your body's overall health.\n\nBecause preconceptions can affect an energy test, you don't try to guess what the results will be. Subtle energies are responsive to your thoughts, so clear your mind as well as you can prior to conducting an energy test. If either of you is thirsty, begin with a drink of water. Water conducts electricity, and dehydration interferes with the flow of energy in your body. Avoid eye contact during the test, as this can make it more a test of interpersonal dynamics than what you are intending to test. Remove cell phones or other electronic devices, crystals, and heavy jewelry. Also be sure to ask if the person being tested has any injuries that might be aggravated by having pressure applied to the arm being tested, and if so, use the other arm. Having the person place his or her thumb and second finger on the points where the neck meets the head, about an inch from each side of center, stops the influence of thoughts and beliefs on the outcome. To energy test the spleen meridian:\n\n1. Both of you take a deep breath. With your exhalation, release your expectations.\n\n2. The person being tested places either arm straight down the side of the body with the thumb touching the side of the leg, fingers pointing down.\n\n3. The tester slips an open hand between the body and the arm, just above the wrist, and rests the other hand on the person's shoulder.\n\n4. The tester then asks the other person to hold his or her arm firm, elbow straight. I often use words such as, \"Hold your arm secure next to your body\" or simply \"Hold.\"\n\n5. With an open hand and with pressure from one to two seconds, the tester slowly pulls on the arm (see Figure Ap-1).\n\n_Figure Ap-1_ SPLEEN ENERGY TEXT\n\nWith neither person straining, the arm will either pull away relatively easily or it will feel locked in place. When pressure is applied, a muscle with energy flowing through it may also move a fraction of an inch, but it will bounce right back. Do not struggle so much to hold your arm firm that you involve other muscles. If you are the tester, do not struggle to pull the other's arm away. This is not a competition, nor is it about muscle strength. If the energy is flowing freely, the arm will stay locked in place or give a bit but bounce right back when you pull on it.\n\nIf the Spleen Energy Test shows the meridian to be weak, strengthen it by vigorously tapping or deeply massaging the spleen points that are illustrated in Figure 2-5 (page 54) and then retest. In fact, if you find yourself overly susceptible to exhaustion, infection, or illness, one way to help keep your spleen meridian and your immune system strong is by routinely doing the Triple Warmer Smoothie (page 107) and then tapping your spleen points. Immediately after this sequence, the Spleen Energy Test will probably indicate that the meridian is strong. If the test still shows it to be weak, your energies are probably quite scrambled. Don't be alarmed. Do the Daily Energy Routine (plus the Homolateral Crossover) presented in Chapter 2 to unscramble your energy field, and then return here and continue. It is also important that the person conducting the test be relatively balanced, so some practitioners routinely do the Three Thumps (page 53), Cross Crawl (page 56), or Wayne Cook Posture (page 56) along with their client prior to performing an energy test.\n\nIf the muscle holds firm, you can challenge it, and this is particularly useful while you are learning the procedure. This is a way of determining the optimal amount of pressure to apply to distinguish whether the meridian's energy is flowing strongly. One way to calibrate this is to make a statement such as \"I am wearing a blue shirt,\" where the truth of the statement is obvious to both parties. Immediately after the statement has been made, perform the energy test. If it is a true statement, the meridian will test strong, and you have a gauge of the pressure that can be applied when the energy is flowing. If it is a false statement, the meridian will not hold and you have a gauge of the pressure that indicates a weak meridian. Similarly, if you have the person think of something pleasant or joyful and immediately do the test, the meridian will generally stay firm. While holding an embarrassing or frightening thought, the meridian will generally lose its firmness.\n\nIf the meridian holds firm no matter what you do, or if it is chronically weak and will not be strengthened by tapping the spleen points or by the Daily Energy Routine, use the General Indicator Test described below. Sometimes, however, the body's energies are running in \"irregular\" patterns that need to be corrected before an energy test will be accurate. The energy moving through a muscle may be \"frozen\" or \"submerged,\" or its polarity may be reversed. While more advanced than the scope of this book, instructions for working with such energies are available on the Web site of the Energy Medicine Institute (www.energymed.org).\n\n**The General Indicator Test.** The spleen meridian test you just learned is particularly useful for determining your response to foods, supplements, and environmental conditions. If, however, the spleen meridian is chronically weak, or if irregular energies are moving through it, it will not serve well in a general-purpose test. The General Indicator Test is a second energy test that may be used in virtually any situation because it does not isolate a specific meridian but rather indicates a more general disturbance in the energy field.\n\nThe basic principles for both tests are the same. The only difference is in how you hold your arm. To do the General Indicator Test:\n\n_Figure Ap-2_ GENERAL INDICATOR TEST\n\n1. Place either arm straight out in front of you, parallel with the ground, and then move it 45 degrees to the side. Your elbow should be straight and your hand open, palm facing the floor, or thumb pointing down, as in Figure Ap-2.\n\n2. Have your partner place the fingers of an open hand on your arm, just above your wrist, resting the other hand on the other shoulder (see Figure Ap-2). As you hold firm, your partner pushes down slowly for up to two seconds and only hard enough to determine whether there is a \"bounce.\"\n\nAs with the Spleen Meridian Test, you need the muscle to initially stay firm to get an accurate test, and you may strengthen the flow of energy through the muscle by using the Daily Energy Routine. You may then determine the amount of pressure to use by testing a true statement and a false statement. And if the muscle will not weaken or will not strengthen, more advanced methods may be needed.\n\n**Energy Testing Without a Partner.** Because I sometimes wanted someone to energy test me when no one was available, I hit upon a solution while I was first experimenting with energy testing. I went to a sporting-goods store and found a dumbbell that I could lift if I held it straight out in front of me when I was holding a comforting thought but could not lift when I was holding a depressing thought (thoughts generate subtle energies that impact our bodies, and comforting as well as depressing thoughts affect the muscles).\n\nI placed the dumbbell on a dresser that was the height of my shoulders. I would put my arm out in front of me, grab the weight, and try to lift it. If I wanted to energy test for a specific food or vitamin, I would hold the substance in one hand and try to lift the weight with the other. The energy of the food would affect my energies as decisively as an uplifting or depressing thought. I could find out whether the substance was having a positive or negative impact on my energies by my ability or inability to lift the weight. Because dumbbells exert a steady pressure downward, they can provide a reasonably objective measure of what is being tested. The critical physical factor for this self-test is finding a dumbbell with the correct weight\u2014a weight you can lift off the desk or other surface while thinking a positive thought but not while thinking a negative thought. You can get the exact weight you need for this test by using a one-gallon water jug. Experiment with the amount of water needed to just be able to lift it off a surface with your arm straight out while keeping a comforting thought active but not a disturbing thought. Alternatively, hang your arm straight down while holding the weight and, with your elbow straight, lift it to the side, along the path you'd use if a partner was testing you.\n\nNumerous other ways for energy testing without a partner have been devised. Because most of them require the person to simultaneously exert and resist pressure, to be both the tester and the tested, it is very difficult to assure an unbiased result unless the person is highly experienced. The most effective method I know has you use your body as a pendulum rather than exerting pressure on a muscle. While standing, place the item being tested at your stomach, holding it evenly with both hands, and bring your elbows in so the sides of your arms touch the sides of your body. Then bring your feet together, facing straight ahead. Become still, centered, take a deep breath, and release. If after a few moments you feel yourself pulled toward the item, that is, falling forward, it indicates that the item is in harmony with your energies (see Figure Ap-3). If you find yourself falling backward, away from the substance being tested, it indicates that the item is in conflict with your energies. It is as if, at a subtle level to which you must attune yourself, a substance that is in harmony with your energies has a magnetic attraction pulling you toward it while a substance that is in disharmony repels you. In this test, you need to be particularly careful about your wishes or preconceived ideas influencing the results, and the instructions for fine-tuning your energy-testing abilities with a partner, beginning on page 299, can be adapted for this test as well.\n\n_Figure Ap-3_ SELF-TEST\n\n**Practice Energy Testing at Your Next Meal.** Although there is a different energy test for each of the fourteen meridians, the test for the spleen-pancreas meridian can serve you well in innumerable contexts. The test is based on the latissimus dorsi muscle, and it is particularly attuned to food metabolism. You can, however, use the Spleen Meridian Test as a _general indicator test_ to find out about almost anything that is going on in your body.\n\nYour next meal can become a personal workshop for practicing energy testing. Energy test each of the foods you are planning to eat. Touch the food and allow your other arm to be energy tested. If you lose your strength upon touching the food, your body's vibration is not in harmony with the food's vibration. This may mean the food is never good for your body, the food is not good for your body right now, or you have an allergy to the food.\n\nTesting the food at various points in time will give you information about whether it is always, sometimes, or never good for you. Because everyone's chemistry is unique, nutrition is an entirely individual matter. One person's vitamin is another's poison. If the energy of a food, vitamin, or supplement doesn't match the energy of your body, you will not absorb and metabolize it even if all the experts in the world say you need it. Even good food is toxic if its vibration triggers your immune system to tie itself up in a defensive reaction. Such food allergies often go undetected, but not without accumulating damage. Energy testing can help you know what your body needs at a given moment, and it can help you develop a superb nutrition program for your unique body.\n\nOne caveat is that if your adrenals are exhausted, sugar and caffeine will often test strong. Even if a substance is generally harmful for you, if your body is needing the burst of energy it can supply, the test may be misleading.\n\n**Energy Testing Your Way to a Better Diet.** For many people, learning to energy test for food changes their relationship to what they eat. You can test to see if your body is going to like a particular food. You can also get children involved with picking out foods you buy at the grocery store or energy test them before you cook a meal. It can be a game. They will enjoy it, at least until the first time they energy test weak on something they really want. But often, even if they are begging for something, seeing their arm lose its strength on an energy test of a junk food will bring laughter and a reduction in the tension.\n\nOf course they may want to eat it anyway, but I have seen kids become less interested in foods that consistently cause them to lose their energy. The best results happen if they make the connection that losing strength on an energy test may mean that the food being tested may reduce their energy level and leave them feeling bad. But you have to be honest. A healthy body may be able to metabolize a certain amount of junk food, and it would undermine the process to overpower the child with your strength or your mind. Use the guidelines presented here to be sure the energy test is accurate, and particularly that it isn't being influenced by your beliefs or your child's desires.\n\nFinally, let children draw their own conclusions about how eating food that energy tested weak makes them feel. When they can authentically relate energy testing to their life\u2014that it isn't just a trick or a frivolous game\u2014energy testing becomes a source of empowerment, a way to get useful information and biofeedback. They might also like turning the tables on Mom and Dad and energy testing you on that cheesecake or cigarette. And why not? You may be interested to find out for yourself whether that daily glass of red wine really is good for your heart. You can also use energy testing to assess the likely reaction of your own body or your children's bodies to vitamins and other food supplements.\n\n**How to Energy Test an Infant, a Pet, or Someone in a Coma.** \"Surrogate testing\" allows you to test someone who is not able to offer resistance in an energy test. If someone is too sick to use his or her own strength or is mentally impaired or too young to follow your instructions, surrogate testing can provide valid information. This will even work with a pet.\n\nIt can also be used when you cannot get an accurate test on someone. For instance, because the energetic links with family members are so complex, you may find that you can get accurate results on everyone but your spouse or your child.\n\nOr sometimes a person is muscular and macho, and recruits auxiliary muscles so that he or she will not look weak on the test. Surrogate testing bypasses these difficulties. Along with the person (or animal) being tested, it requires two others (the surrogate and the tester). If you are conducting the test:\n\n_Figure Ap-4_ SURROGATE TEST\n\n1. Have the two individuals hold hands or make other physical contact.\n\n2. Energy test the arm of the \"surrogate,\" the person touching the one you want information about. The results of the test on the surrogate tell you what is happening with the other (see Figure Ap-4).\n\nThis seems bizarre at first, but it is simply based on the way energy flows. If you want to know, for instance, if a particular food is causing an adverse reaction in an infant, you can be a surrogate for the infant. With the food on the infant's skin, place your hand over the infant's hand or abdomen and have someone energy test your other arm. If the test shows weak, the energy of the food is probably not compatible with the infant. If this seems too strange, please don't take my word for it. You can demonstrate its validity for yourself by first having yourself energy tested on a number of substances and then having someone surrogate test you on the same items.\n\n**What You Can't Energy Test.** An energy test tells you whether the energy is flowing through or inhibited in the muscle being tested. Some people use energy tests to ask the body questions that go far beyond the scope of the procedure. \"Is this condo a good investment?\" \"Will I enjoy going to Machu Picchu for my next vacation?\" \"Should I take the next Eden Energy Medicine Five-Day Basic Workshop?\" I have seen more nonsense emerge from the misuse of energy testing than I like to think about, and it is an embarrassment to the field how many practitioners casually use energy testing to answer all sorts of bizarre questions.\n\nEfforts to ask the body questions about the future seem particularly ludicrous to me: \"Will I be cured of this illness by next month?\" While I may have a lot of information to make an educated guess about such a question, an energy test is not a source I rely on for that information. I believe there is fate, there is free will, and there is circumstance. An energy test about the future assumes that it is all fated. But free will, unpredictable circumstances, and shifting relationships with other people all converge with whatever may be fated. This is why readings from even the most talented psychics are only a percentage game. Many factors influence the single question being asked.\n\nBut even questions about treatment are very tricky. \"Should this problem be treated with the meridians? The chakras?\" Energy tests can be influenced by many factors, even for people of the highest integrity who do everything they can to get their beliefs, hopes, or expectations out of the way. It is challenge enough not to exert an influence based on your beliefs, desires, or unconscious expectations. This is true for any energy test, but much more so when it is based on a verbal question rather than letting the body respond in its own language.\n\nAs I write this, I am having images of colleagues who ask questions before an intervention such as, \"Is this for the body's highest good?\" or \"Do I have permission to proceed?\" I ask these questions intuitively, but I do not energy test for them. However, practitioners I really admire do ask these questions and test for the answers so respectfully, almost as a prayer, that the questions themselves set an energy field of honor and respect. So the energy test itself serves a very powerful purpose of creating a shared energy field for healing.\n\nSo while emphasizing the pitfalls of asking verbal questions of the body, I do not want to dismiss the practices of colleagues who have found ways to make this work for them. Some practitioners have learned to use verbal questions followed by an energy test as a way of tuning in to a higher source of information, so it becomes for them a way of channeling in to their intuition, a special bridge to the truth of the situation. For people who have developed this to a refined art, it can be quite trustworthy. But for our purposes here, I simply want to present energy testing within the scope that I use it, which is much more traditional\u2014checking, under specific conditions, the relative strength of an indicator muscle to determine the energetic state of a meridian or other energy system. While there are many truths and many ways to get at them, I believe that energy testing after asking the body a verbal question is one of the trickier of the bunch.\n\n#  TESTING THE IMPACT OF THE PHYSICAL ENVIRONMENT ON YOUR ENERGY FIELD\n\nThe moment you come into contact with an external energy, before you are even dimly aware of its impact on you, your own energy field is already responding and adjusting to it. By simply bringing something near you and doing an energy test, you can discern its immediate impact on your subtle energies. I have many times arranged to meet a client at a supermarket for the purpose of energy testing to see which foods elicit a positive response and which do not. Even different brands of the same food or vitamin may affect you in different ways. It is also possible for our dietary needs to change from one stage of our lives to the next, but a tremendous amount of useful information can be garnered from a trip to the grocery store.\n\nYour first energy tests are experiments for you and a partner, and you will both become more proficient in your ability to perform accurate energy tests as you proceed. For now, play with energy testing. Have your friend test you when you walk into your favorite part of the house and then when you walk into your least-favorite spot. Have a third person think \"negative\" thoughts while in your proximity. Find out how that affects your energy. Then have someone give you a genuine smile. Test again. Explore how the television affects you when you are two feet from it, then eight feet. Experiment with different setups of your computer or your office. Discover how your energies are affected when you are exposed to a particular piece of art or music.\n\nAs you explore your environment using energy testing, do not be concerned about temporarily bringing something that weakens you into your own energy field. Because we are continually entering fields of energy that affect us, the body\u2014 at least when it is relatively healthy\u2014adjusts to the initial impact of a disturbing energy field by quickly rebalancing itself. In fact, one definition of health is how readily your body can adapt to a spectrum of environmental conditions. You can use energy testing to explore any element of your environment that stirs your curiosity. By having fun with it, you will be developing a strong aptitude with this invaluable tool.\n\n#  FINE-TUNE YOUR ENERGY-TESTING ABILITIES\n\nI have long sensed that energy testing does more than simply provide information. In an energy healing session, it feels like the early part of the treatment, as if performing the test somehow engages my energies and the client's energies within a healing context. Energy testing immediately directs healing forces to the area being tested. It is as if the energy test focuses your subtle energies in a way that prepares them for the work to follow. I have found that the identical procedure yields better results if I have energy tested it first. How can this be?\n\nBecause subtle energies are affected by thoughts and intentions, and because they are nonlocal, the energies of healer and client start mingling as soon as an energy test is even contemplated. We cannot be truly objective in any test. Not only does the act of observing influence the observed in quantum physics, subtle energies are ultrasensitive to many influences. Yet we can learn to control for our hopes, fears, and expectations, and create a container within which an energy test provides information that is very useful and quite precise. In fact, energy testing with a partner can reveal a deeper level of information than can any inanimate instrument. The complex interactions of tester and tested evoke two-way feedback that can make the energy test an incomparable dance of exploration for the purpose of obtaining trustworthy information. Having now trained hundreds of students whom I know to be capable of providing reliable energy tests, I can tell you with confidence that this skill can be yours, and that it is a skill worth developing. The following tips will increase your accuracy. Practice them now, or proceed with the earlier sections of this book and return here when the time is right.\n\n1. **Maintain a Beginner's Mind.** Your mind affects your energy field, instantly and potently. To factor out preconceived ideas about what the test will reveal, approach energy testing as a contemplative practice, coming into a centered, meditative \"beginner's mind.\" If you are concerned that your hopes or preconceived ideas, or your partner's, may be getting in the way of the test, the following procedure can energetically disengage them.\n\n2. **Neurologically Disengage Expectations.** This approach for neurologically disconnecting preconceived ideas from an energy test, taught to me by Gordon Stokes, my first Touch for Health instructor, may seem implausible, but over the years I have found that it works. It can be used by the tester, the tested, or both.\n\na. Take the thumb and middle finger of one hand and place them in the two indentations where the back of your neck meets your head.\n\nb. With your other hand, proceed with the energy test (see Figure Ap-1, page 290).\n\nThese indentations are primary headache points in traditional Chinese medicine, but holding them does more than relieve headaches. The energy that travels up your nervous system along your spine passes through these points. When you are the tester, holding the points on yourself breaks the energetic circuitry with the other person and brings you back into your own energy loop, disengaging you and your beliefs from the other person. When you are being tested and place your hands on these points, it disengages you from your own thoughts about the matter and seems to allow the test to reveal a deeper level of information. Keeping your own beliefs and expectations from interfering is the most important single step you can take to ensure an accurate energy test.\n\n3. **Focus Your Intention.** Your intention can influence a test's outcome. Rather than having to work around the effects of intention, however, you can use the fact that expectations influence energy testing, and resolutely set your intention for valid results. To demonstrate, find a food or other substance that tests strong on a friend. Repeat the test on the same substance without telling your friend what you are up to. On the second test, decide in advance that the test will show weak. Do not pull harder. Just see if this shift in your intention influences the results. It often does. So set your intention for an accurate test.\n\n4. **Establish a Resonance between Tester and Tested.** You can create an energetic resonance with your partner:\n\na. Both take a deep breath in tandem.\n\nb. Together release the breath along with all thoughts.\n\nc. Test once the breath is released.\n\n5. **Stay Alert.** Be sure the person is ready before beginning the test. Pull the arm smoothly. Apply pressure only long enough to determine whether the arm's resistance holds (generally between one and two seconds). If a muscle is locked in place or gives a fraction of an inch and then bounces back, the test has shown that the energy is flowing through the muscle. Do not overwhelm the muscle \"just to be sure.\" Along with learning the mechanics of energy testing, I am also interested in helping you cultivate a strong and reliable intuition. Energy testing is a superb means to that end because it provides tangible information about subtle energies. With each energy test you perform, you receive feedback that attunes your intuition to the flow of energies within you and around you as well as to the subtleties of energy testing itself.\n\n6. **Practice under Double-Blind Conditions.** You can develop confidence in your ability to do an accurate energy test by practicing under conditions where you will get immediate feedback. You will need two other people. One will supervise the test, and one will be tested. Find some substances that the person being tested _knows_ are healthy and agreeable to his or her system, such as organic apples or mint tea. Take other substances that you _know_ would be bad to ingest, such as a bottle of ammonia. The person supervising the experiment takes a substance and puts it into the energy field of the person being tested in a way that neither you as the energy tester nor the person being tested can see, perhaps against the back of the person's shirt. Energy test. Do not be surprised if you do not get a perfect score at first, while you are learning the subtleties of the process. The double-blind test can be a great forum for learning the intricacies of energy testing, and it provides a benchmark as you continue to build your proficiency.\n\nEnergy testing is a method for assessing the body's energies and for evaluating how the environment, including the food you eat, is affecting them. You can use it to augment and refine the instructions provided throughout this book.\nNotes\n\n#  **Introduction EMBRACING YOUR EXQUISITE VIBRATION**\n\n \"String theory\" is a recent though controversial idea in physics that attempts to reconcile the \"ill-concealed skeleton in the closet of physics\" that what scientists understand about the behavior of the universe on extremely large scales (general relativity theory) and on extremely small scales (quantum mechanics) cannot both be right. If confirmed, string theory would resolve this contradiction and also provide a way of understanding how the material world is simultaneously both energy and matter. A good introduction to string theory is Brian Greene's _The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory_ (New York: W.W. Norton, 2003).\n\n James Oschman, _Energy Medicine: The Scientific Basis_ (New York: Harcourt, 2000).\n\n Rephrased from Shakespeare, _Hamlet,_ Act 2, Scene II.\n\n Gary Null, Carolyn Dean, Martin Feldman, Debora Rasio, and Dorothy Smith, _Death by Medicine_ (New York: Nutrition Institute of America, 2003).\n\n##  **Chapter 1 A MEDICINE CALLED ENERGY**\n\n Leopold Dorfer et al., \"A Medical Report from the Stone Age?\" _Lancet,_ 1999; _354:_ 1023-1025.\n\n National Center for Complementary and Alternative Medicine. (2002). \"What Is Complementary and Alternative Medicine?\" Bethesda, MD: NCCAM. Retrieved December 3, 2006, from www.nccam.nih.gov\/health\/whatiscam.\n\n Donna Eden and David Feinstein, \"Energy Medicine: Uses in Medical Settings,\" 2006, paper posted at www.energymed.org\/hbank\/handouts\/ener-med_in_medical_set.htm\n\n Dawson Church, _The Genie in Your Genes: Epigenetic Medicine and the New Biology of Intention_ (Santa Rosa, CA: Elite, 2007), pp. 137-138.\n\n Church, _The Genie in Your Genes._\n\n Richard Gerber, _Vibrational Medicine,_ 3rd ed. (Rochester, VT: Bear & Co.), p. 428.\n\n William A. Tiller, _Psychoenergetic Science_ (Walnut Creek, CA: Pavior, 2007).\n\n Valerie Hunt, _Infinite Mind: The Science of Human Vibrations_ (Malibu, CA: Malibu Publishing, 1995).\n\n David Feinstein and Donna Eden, \"Six Pillars of Energy Medicine,\" _Alternative Therapies in Health and Medicine_ 2007, 14, 44-54. Available online from www.EnergyMedicinePrinciples.com.\n\n Beverly Rubik, \"The Biofield Hypothesis: Its Biophysical Basis and Role in Medicine,\" _Journal of Alternative and Complementary Medicine_ 2002, 8, 703-717.\n\n Valerie Hunt, _Infinite Mind._\n\n Timothy Ferris, _Coming of Age in the Milky Way_ (New York: William Morrow, 1998).\n\n Bill Bryson, _A Short History of Nearly Everything_ (New York: Broadway, 2003), p. 141.\n\n The Human Genome Project expected to identify some 120,000 genes but found, to everyone's astonishment, that fewer than 24,000 do the job (a simple roundworm contains 20,000 genes, so you can imagine the surprise).\n\n Robert O. Becker, _Cross Currents: The Perils of Electropollution_ (New York: Tarcher, 1990).\n\n Lynn McTaggart, _The Field: The Quest for the Secret Force of the Universe_ (New York: Harper, 2003), p. 45.\n\n Cited in Bruce Lipton, \"Mind Over Genes: The New Biology.\" Available from www.brucelipton.com\/article\/mind-over-genes-the-new-biology.\n\n Harold S. Burr, _The Fields of Life_ (New York: Ballantine, 1972).\n\n Abraham R. Liboff, \"Toward an Electromagnetic Paradigm for Biology and Medicine,\" _Journal of Alternative and Complementary Medicine_ 2004, 10(1), 41-47.\n\n Patricia Ellen Winstead-Fry and Jean Kijek, \"An Integrative Review and Meta-Analysis of Therapeutic Touch Research,\" _Alternative Therapies in Health and Medicine,_ 1999, 5(6), 58-67.\n\n Melinda H. Conner, Genevieve Tau, and Gary E. Schwartz, \"Oscillation of Amplitude as Measured by an Extra Low Magnetic Field Meter as a Physical Measure of Intentionality. Poster presentation, Toward a Science of Consciousness,\" University of Arizona, Tucson, AZ, May 2006.\n\n Rolland McCraty, \"The Energetic Heart: Bioelectromagnetic Communication within and between People,\" in Paul J. Rosch and Marko S. Markovs (eds.), _Clinical Applications of Bioelectromagnetic Medicine_ (New York: Marcel Dekker, 2004), 541-562.\n\n Reported in Paul Pearsall, _The Heart's Code_ (New York: Broadway Books, 1998), p. 7.\n\n James L. Ochsman, _Energy Medicine: The Scientific Basis_ (London: Churchill Living-stone, 2000).\n\n Bruce H. Lipton, _The Biology of Belief_ (Santa Rosa, CA: Elite, 2005).\n\n Gary Null, Carolyn Dean, Martin Feldman, Debora Rasio, and Dorothy Smith, _Death by Medicine_ (New York: Nutrition Institute of America, 2003).\n\n Ibid. p. 33.\n\n Lipton, _The Biology of Belief,_ p. 112.\n\n Ibid., p. 115.\n\n Ibid., p. 119.\n\n##  **Chapter 2 ENERGY TECHNIQUES FOR HEALTH AND VITALITY**\n\n Sections of this chapter are adapted from Chapters 3, 6, and 9 of _Energy Medicine._\n\n Certain points are called \"forbidden points\" in traditional acupressure, and activating the Large Intestine 4 point in a pregnant woman is not recommended. Forbidden points during pregnancy are discussed further in note 1 of Chapter 4.\n\n Robert Frost, _Applied Kinesiology: A Training Manual and Reference Book of Basic Principles and Practices_ (Berkeley, CA: North Atlantic Books, 2002).\n\n Marcello Caso, \"Evaluation of Chapman's Neurolymphatic Reflexes via Applied Kinesiology: A Case Report of Low Back Pain and Congenital Intestinal Abnormality.\" _Journal of Manipulative and Physiological Therapeutics_ 2004, 27(1), 66-72.\n\n Daniel G. Amen, _Making a Good Brain Great: The Amen Clinic Program for Achieving and Sustaining Optimal Mental Performance_ (New York: Three Rivers Press, 2006).\n\n Ibid., pp. 131-133.\n\n Florence Peterson Kendall (ed.), _Muscles: Testing and Function, with Posture and Pain,_ Fifth Edition (Baltimore: Lippincott, Williams & Wilkins, 2005).\n\n Scott C. Cuthbert and George J. Goodheart, \"On the Reliability and Validity of Manual Muscle Testing: A Literature Review,\" _Chiropractic & Osteopathy_ (online journal) 2007, _15_ , 4.\n\n Henry Pollard, Bronwyn Lakay, Frances Tucker, Brett Watson, and Peter Bablis, \"InterexaminerReliability of the Deltoid and Psoas Muscle Test,\" _Journal of Manipulative and Physiological Therapeutics_ 2005, 28(1), 52-56.\n\n Educational applications of these methods are known as \"Brain Gym.\" For further information, visit www.braingym.com and www.braingym.org.\n\n David Feinstein, \"Energy Psychology: A Review of the Preliminary Evidence,\" _Psychotherapy: Theory, Research, Practice, Training_ (in press). Available from www.EnergyPsychologyResearch.com.\n\n David Feinstein, Donna Eden, and Gary Craig, _The Promise of Energy Psychology: Revolutionary Tools for Dramatic Personal Change_ (New York: Tarcher\/Penguin, 2005). www.EnergyPsychEd.com.\n\n##  **Chapter 3 DANCING WITH YOUR HORMONES**\n\n Barry Sears, _The Age-Free Zone_ (New York: HarperCollins, 2000).\n\n J. D. Ratcliff, \"The Endocrine Glands: Centers of Control,\" _Our Human Body_ (Pleasantville, NY: The Reader's Digest Association, 1962), 271-276.\n\n Norman Doidge, _The Brain That Changes Itself_ (New York: Viking, 2007).\n\n Mona Lisa Shulz, _The New Feminine Brain: How Women Can Develop Their Inner Strengths, Genius, and Intuition_ (New York: Free Press, 1995).\n\n Robert A. Wilson, _Feminine Forever_ (New York: Evans, 1966).\n\n See Judith A. Houck's \"'What Do These Women Want?': Feminist Responses to _Feminine Forever,_ 1963-1980\" ( _Bulletin of the History of Medicine,_ 2003, _77:_ 103-132) for a superb discussion of the juxtaposition and aftermath of Wilson's original article and Betty Friedan's _The Feminine Mystique,_ both published in 1963.\n\n Stephen Smith, \"Hormone Therapy's Rise and Fall: Science Lost Its Way, and Women Lost Out\" ( _The Boston Globe,_ July 20, 2003).\n\n Leslie Kenton, _Passage to Power: Natural Menopause Revolution_ (Carlsbad, CA: Hay House, 1998), p. 4.\n\n Ibid.\n\n The Boston Women's Health Book Collective, _Our Bodies, Ourselves: Menopause_ (New York: Simon & Schuster, 2006), pp. 24-25.\n\n Naomi Wolf, _The Beauty Myth_ (Toronto: Vintage Books, 2007).\n\n Nancy Etcoff, _Survival of the Prettiest: The Science of Beauty_ (New York: Anchor, 2000).\n\n Susun Weed, _New Menopausal Years: The Wise Woman Way_ (Woodstock, NY: Ash Tree, 2002), p. 104.\n\n Wolf, _The Beauty Myth._\n\n Christiane Northrup, _The Wisdom of Menopause: Creating Physical and Emotional Health and Healing During the Chang_ e (New York: Bantam, 2001), p. 135.\n\n Merlin Stone, _When God Was a Woman_ (New York: Harcourt, 1978).\n\n Judith Duerk, _Circle of Stones: Woman's Journey to Herself_ (New York: New World Library, 2004).\n\n Diana Schwarzbein, _The Schwarzbein Principle II: The Transition_ (Deerfield Beach: Health Communications, 2002), p. 54.\n\n James Wilson, _Adrenal Fatigue: The 21st Century Stress Syndrome_ (Petaluma, CA: Smart Publications, 2002).\n\n##  **Chapter 4 RECLAIMING THE WISDOM OF YOUR MENSTRUAL TIME**\n\n Acupuncture points may be activated with needles, moxibustion (the burning of the herb _artemesia vulgaris_ at the acupoint), electrical impulses, lasers, and manual stimulation such as massaging, tapping, or holding the points. I find acupuncture needles, moxibustion, and the electronic and laser stimulation of acupuncture points to be relatively intrusive compared with manual stimulation, and not necessarily more effective, though each method has its place. In the practice of acupuncture, certain points\u2014including Spleen 9, Kidney 1, and Large Intestine 4\u2014are not to be penetrated with a needle on a woman who is pregnant. My experience, and the experiences of many other hands-on energy medicine practitioners, has been that holding these points when sedating or strengthening a meridian does not cause adverse reactions. In fact, points that are considered unsafe to stimulate for pregnant women with the use of needles or moxibustion have, in clinical practice, been found to be safe when activated by electronic or laser stimulation. While I have no direct experience in using electronic or laser stimulation with pregnant women, these reports seem to support my experience that the even less invasive, hands-only methods taught in this book are safe. Nonetheless, since this is a self-help book and a practitioner is not there to monitor the procedures, the amount of time for holding the Kidney 1 point if you are pregnant has been, as a precautionary measure, reduced from two minutes to thirty seconds, long enough to get the process started but so short as to have no chance of causing an adverse effect. For a discussion of clinical findings related to acupuncture points that may have adverse effects, see Dr. John Amaro's \"The 'Forbidden Points' of Acupuncture\" at www.chiroweb.com\/archives\/18\/10\/01.html.\n\n##  **Chapter 5 SEXUALITY, FERTILITY, PREGNANCY, AND BIRTH**\n\n Michael Gurian, _What Could He Be Thinking: How a Man's Mind Really Works_ (New York: St. Martin's Griffin, 2003).\n\n Ibid.\n\n Ellen Eatough's Web site and free \"Love Tips\" expands upon these and many other important ideas about sexuality. www.extatica.com.\n\n Ibid.\n\n Mary Eve, \"Remembering V-Day,\" www.truthout.org\/docs_2006\/021407H.shtml.\n\n Margot Anand, _The Art of Sexual Ecstasy_ (New York: Tarcher\/Penguin, 1989).\n\n Kenneth Ray Stubbs, _The Essential Tantra: A Modern Guide to Sacred Sexuality_ (New York: Tarcher\/Putnam, 1999).\n\n David Feinstein, Donna Eden, and Gary Craig, _The Promise of Energy Psychology: Revolutionary Tools for Dramatic Personal Change_ (New York: Tarcher\/Penguin, 2005).\n\n Alan Batchelder was the therapist.\n\n \"The Energies of Love\" has been one of our most popular classes. An introduction to this topic is on a DVD of that name, available from www.innersource.net\n\n Christine Gorman, \"The Limits of Science,\" _Time,_ April 15, 2002, 159 (15), p. 52.\n\n Randine Lewis, _The Infertility Cure_ (New York: Little, Brown & Co., 2004), p. 15.\n\n Jorge E. Chavarro, Walter C. Willett, and Patrick J. Skerrett, _The Fertility Diet_ (New York: McGraw-Hill, 2008).\n\n Ibid., pp. 12-13.\n\n Ibid., pp. 149-151.\n\n Ibid.\n\n Ibid., p. 282.\n\n Vicki Hufnagel, _No More Hysterectomies,_ rev. ed. (New York: Penguin, 1989).\n\n Ibid., p. 60.\n\n##  **Chapter 6 MENOPAUSE\u2014GATEWAY TO YOUR** _**SECOND**_ **PRIME OF LIFE**\n\n To examine your life as a mythic journey, see David's award-winning _The Mythic Path: Discovering the Guiding Stories of Your Past: Creating a Vision for Your Future_ (Book of the Year\u20142007 U.S. Book News Awards, Psychology\/Mental Health Category). For an overview, visit www.innersource.net.\n\n Pat Wingert and Barbara Kantrowitz, _Is It Hot in Here? Or Is It Just Me? The Complete Guide to Menopause_ (New York: Workman Publishing Company, 2006).\n\n Leslie Kenton's _Passage to Power: Natural Menopause Revolution_ (Carlsbad, CA: Hay House, 1995) is a wonderful synthesis of the biological and spiritual dimensions of menopause.\n\n Susun Weed's _New Menopausal Years: The Wise Woman Way_ (Woodstock, NY: Ash Tree, 2002) is a distillation of what _works_ after having interviewed some 50,000 menopausal women over a 13-year period.\n\n Christiane Northrup's _The Wisdom of Menopause: Creating Physical and Emotional Health and Healing During the Change_ (New York: Bantam, 2001) is a gynecologist's highly personal yet information-packed guide for menopausal women.\n\n John Lee's _What Your Doctor May Not Tell You about Menopause: The Breakthrough Book on_ Natural _Hormone Balance,_ rev. ed. (New York: Warner, 2004), is an analysis of the good science and the bad science behind hormone replacement therapy by a gynecologist who was a pioneer in the use of natural hormones.\n\n Susan Lark's _Hormone Revolution: Yes, You Can Naturally Restore & Balance Your Own Hormones_ (Los Altos, CA: Portola Press, 2008) is rich with information and strategies for hormone balance, menopause, perimenopause, fatigue, hypertension, and many other areas that affect your health and well-being. Based on Dr. Lark's 30 years of medical practice and innovative teaching, it focuses on five hormones\u2014estrogen, progesterone, testosterone, pregnenolone, and DHEA\u2014and includes a section on energy medicine.\n\n Kenton, _Passage to Power,_ p. 205.\n\n Leonard Shlain, _Sex, Time and Power: How Women's Sexuality Shaped Human Evolution_ (New York: Penguin, 2004), p. 183.\n\n Ibid., p. 159.\n\n Ibid., p. 184.\n\n Ibid., p. 95.\n\n Ibid.\n\n Ibid., p. 96.\n\n Jacques E. Rossouw, et al., \"Postmenopausal Hormone Therapy and Risk of Cardiovascular Disease by Age and Years Since Menopause.\" _JAMA_ 2007, 297(13), 1,465-1,477.\n\n Lee, _What Your Doctor May Not Tell You About Menopause._\n\n Ibid., pp. 3-4.\n\n Germaine Greer, _The Change: Women, Aging and Menopause_ (New York: Fawcett Columbine, 1991).\n\n Sandra Coney, _The Menopause Industry_ (Alameda, CA: Hunter House, 1994), p. 25.\n\n Cited in Lee, _What Your Doctor May Not Tell You About Menopause,_ p. 52.\n\n Ibid., p. 57.\n\n Bethany Hays, \"Solving the Puzzle of Hormone Replacement,\" _Alternative Therapies in Health and Medicine_ 2007, 13(3), 50-57.\n\n Christiane Northrup, _The Wisdom of Menopause,_ PBS Television Special, 2001.\n\n Susan Lark, _A Woman's Guide to Embracing \"The Change,\" Special Report of The Lark Letter_ (Forrester Center, WV: Healthy Directions: 2005).\n\n Diana Schwarzbein, _The Schwarzbein Principle II: The Transition\u2014A Regeneration Program to Prevent and Reverse Accelerated Aging_ (Deerfield Beach, FL: HIC, 2002), pp. 44-46.\n\n Leslie Kenton, _Passage to Power,_ p. 105.\n\n Ibid., p. 110.\n\n##  **Appendix TALKING WITH YOUR BODY'S ENERGIES\u2014THE ART OF ENERGY TESTING**\n\n Adapted from Chapter 2 of _Energy Medicine_ (2nd ed., New York: Tarcher\/Penguin, 2008).\n\n Dean I. Radin, \"A Possible Proximity Effect on Human Grip Strength,\" _Perceptual and Motor Skills_ 1984 (58), 887-888.\n\n Personal communication, January 16, 1998.\n\n Scott C. Cuthbert and George J. Goodheart, \"On the Reliability and Validity of Manual Muscle Testing: A Literature Review.\" _Chiropractic & Osteopathy_ (online journal) 2007, 15:4.\n\n William Caruso and Gerald Leisman, \"A Force\/Displacement Analysis of Muscle Testing,\" _Perceptual and Motor Skills_ 2000, 91, 683-692.\n\n D. Monti, J. Sinnott, M. Marchese, E. Kunkel, and J. Greeson, \"Muscle Test Comparisons of Congruent and Incongruent Self-Referential Statements.\" _Perceptual and Motor Skills_ 1999, 88, 1019-1028.\n\n Arden Lawson and Lawrence Caleron, \"Interexaminer Agreement for Applied Kinesiology Manual Muscle Testing,\" _Perceptual and Motor Skills_ 1997, 84, 539-546.\n\n Chang-Yu Hsieh and Reed B. Phillips, \"Reliability of Manual Muscle Testing with a Computerized Dynamometer,\" _Journal of Manipulative and Physiological Therapeutics_ 1990, 13, 72-82.\n\n Gerald Leisman, Robert Zenhausern, Avery Ferentz, Tesfaye Tefera, and Alexander Zemcov, \"Electromyographic Effects of Fatigue and Task Repetition on the Validity of Estimates of Strong and Weak Muscles in Applied Kinesiological Muscle-Testing Procedures,\" _Perceptual and Motor Skills_ 1995, 80, 963-977.\n\n Gerald Leisman, Philip Shambaugh, and Avery Ferentz, \"Somatosensory Evoked Potential Changes During Muscle Testing,\" _International Journal of Neuroscience_ 1989 (45), 143-151.\n\n Stephanie Eldringhoff and Victoria H. Matthews, \"Frozen and Irregular Energies: Hidden Energy Stumbling Blocks,\" 2006, www.energymed.org\/hbank\/handouts\/frozen_irregular_energies.htm.\nResources\n\n_Most people have no idea of the giant capacity we can immediately command when we focus all of our resources on mastering a single area of our lives._\n\n\u2014TONY ROBBINS\n\n#  **Join or Start a Local Study Group**\n\nA great way to learn energy medicine is to find study buddies with whom you can practice. This may be one other person or a small group, and it may be as simple as discussing this book or practicing techniques together after watching them on a DVD. Many communities have local energy medicine study groups. You can find them, along with other individuals interested in energy medicine, at www.EnergyMedicineDirectory.com.\n\n#  **Find an Energy Medicine Practitioner**\n\nEvery local community is enjoying a rapid increase in the number of health practitioners who incorporate an energy medicine perspective. Practitioners may be found in all of the healing professions, from physicians and chiropractors to nurses to personal trainers to massage therapists. Practitioners of acupuncture, qi gong, Reiki, Ayurveda, Applied Kinesiology, homeopathy, Touch for Health, Healing Touch, and Therapeutic Touch, among many others, are all working directly with the body's subtle energies. An excellent guide to finding a qualified energy practitioner in your own local community, as well as a listing of practitioners trained in my approach to energy medicine, can be found in the \"Practitioners and Links\" area at www.innersource.net.\n\n#  **Find Classes and Trainings**\n\nLocal communities are continually offering classes in various aspects of energy medicine. In larger cities, widely distributed free monthly local magazines announce dozens every month. Courses with David or me can be found on the \"Classes\" area at www.innersource.net.\n\n#  **Eden Energy Medicine Certification Program**\n\nTaught by gifted practitioners who have been studying with me for many years, and who are still overseen by me, this two-year program has earned rave reviews from hundreds of graduates. Energy medicine is a career for the future, and this program is a route into it. Learn more on the \"Certification Program\" area at www.innersource.net.\n\n#  **Additional Books, Videos, and Other Learning Resources**\n\nBooks by David and\/or me have collectively won four national awards, including two Book of the Year designations in their respective categories. The six-hour _Energy Medicine: The Essential Techniques_ video program takes most of the exercises from _Energy Medicine_ and shows me personally instructing you in how to use them. It is like bringing me into your living room. See all of our books, DVDs, and CDs, and the superb Energy Medicine Kit by Sounds True, at www.innersource.net.\n\n#  **Home Study Resources**\n\nMany of our books and videos can form the basis of home study programs with exams, certificates of completion, and professional continuing education credit available. Learn more at www.EnergyHomeStudy.com.\n\n#  **The Energy e-Letter**\n\nLearn about numerous topics of interest; new books, DVDs, and other resources; and upcoming classes and training events, and generally stay informed about\n\nDavid's and my approach to energy healing. You can sign up free from nearly any page at www.innersource.net. Your name will never be given to others.\n\n#  **The Energy Medicine Handout Bank**\n\nAs more and more of my students have been teaching energy medicine, they have put great amounts of energy into developing handouts for their classes. They have often developed material that was similar to handouts used by others. We decided to take the best of these handouts and place them on our \"Handout Bank.\" The Handout Bank is a free resource designed to (1) help make energy medicine more widely accessible, (2) aid those who are teaching classes or providing services in energy medicine, and (3) create a high-quality archive of principles and methods. It is designed for the energy medicine practitioner, but others interested in the field may also find it a valuable resource. The Handout Bank is posted on the site of our sister organization, the nonprofit Energy Medicine Institute, www.EnergyMed.org.\n\n#  **Getting an Energy Medicine Perspective on Health Questions**\n\nEnergy medicine does not _diagnose_ or _treat_ illness or disease. Instead, it corrects energetic imbalances that are at the foundation of health and vibrancy. But physical symptoms often provide clues about the types of energy imbalances the body needs to have addressed. I have received thousands of inquiries about how to apply energy medicine with various health-related concerns. While the numbers now make it impossible to respond to people personally, for many years I did just that, so I have hundreds of responses to such questions. Our staff has selected questions and answers that may apply to others with similar concerns, concealed the writers' identities, and edited and posted them so the information might serve many. These are well indexed in the \"Questions and Answers\" section at www.innersource.net.\n\n#  **Learn About Energy Psychology**\n\nApplying the principles of energy medicine to emotional problems and to promoting peak vitality is proving to be one of the most exciting developments in the field of psychology. Learn more at www.EnergyPsychEd.com.\nIndex\n\n_Page numbers in italics refer to photographs._\n\nAbdominal Stretch exercise __\n\nacceptance, for weight loss\n\nacupuncture (acupressure) points\n\nin energy psychology\n\nholding\n\nK-points\n\nMeridian Energy Tap __\n\nstimulating\n\nadrenal glands\n\nadrenaline\n\naffirmations\n\nTemporal Tap exercise __\n\nfor weight management\n\naging, cultural messages concerning\n\nAmen, Daniel\n\nAnand, Margo\n\nanger\n\nappetite\n\nanxious eating\n\ncomfort eating\n\ncompulsive eating\n\ncravings\n\nEat in Peace exercise __\n\nhollow hunger\n\nMaddie's Belly Scoop exercise __\n\nmind-body connection\n\novercoming hunger\n\nsubclinical allergies\n\napplied kinesiology\n\n_Art of Sexual Ecstasy, The_ (Anand)\n\nAssimilation Massage exercise __\n\nattitude\n\nacceptance, for weight loss\n\ncreative potential of PMS\n\neffect of thoughts on brain chemistry\n\nonset of menstruation\n\nself-ownership of body\n\ntransition to menopause\n\n_See also_ cultural values\n\naura\n\nCeltic Weave exercise __\n\ncolors\n\nNewton's observation of\n\noverview\n\nduring premenstrual time\n\nas protective atmosphere\n\nautoimmune disorders\n\nback pain\n\nduring PMS\n\nduring pregnancy\n\nSedating Kidney Meridian exercise __\n\nback pain ( _cont._ )\n\nSedating Large Intestine Meridian exercise __\n\nSedating Small Intestine Meridian exercise __\n\nbasic grid\n\nbaths, calming effect of\n\nBeardall, Alan\n\nbelt flow\n\nBerkson . Lindsey\n\nbirth. _See_ labor and delivery\n\nblack cohosh supplements\n\nbloating __\n\nblood pressure, Sunrise\/Sunset exercise for __\n\nblood sugar\n\nlow-blood-sugar attacks\n\nSpleen Points Thump exercise __\n\nBlow-out\/Zip-up\/Hook-in exercise __\n\nbone mass loss. _See_ osteoporosis\n\nBone Saver exercise __\n\nBoston Woman's Health Book Collective\n\nbrain function\n\nDaily Energy Routine benefits\n\neffect of hormones on\n\nginkgo biloba supplements\n\ninteraction between right and left hemispheres\n\nlearning during temporal bone tapping\n\nmaintaining mental sharpness\n\nduring postpartum period\n\nbreach birth position\n\nbreast swelling and tenderness\n\nSedating Stomach Meridian exercise __\n\nstomach meridian\n\nBrenner, Paul\n\nBurr, Harold\n\nbutcher's broom supplements\n\ncalcium supplements\n\ncalming techniques\n\nbathing\n\nCrown Pull exercise __\n\nat onset of PMS symptoms\n\nStress Dissolver exercise __\n\nSunrise\/Sunset exercise __\n\nTriple Warmer Smoothie exercise __\n\nTriple Warmer Tap exercise __\n\nWayne Cook Posture exercise __\n\nCeltic weave energy system\n\nCeltic Weave exercise __\n\ncrossover energy patterns\n\nHomolateral Crossover exercise __\n\noverview\n\nRhythmic Eights exercise __\n\nSuspension Bridge exercise __\n\ncentral meridian\n\nchanneling of negativity\n\npenetrating flow and sexual feelings\n\nZip-up exercise __\n\nchakras\n\ninformation in\n\noverview\n\nroot, in men versus women\n\nroot, sexual feelings and\n\nroot, Vortex Revival exercises for __\n\nroot and womb balance during pregnancy\n\nwomb clearing to reposition fetus\n\nChapman, Frank\n\nCharging Your Batteries exercise __\n\nCherokee Grandmother's Self-Empowerment Secret exercise __\n\nchondroitin supplements\n\nchromium supplements\n\nChurch, Dawson\n\nCircular Joint Rub exercise\n\ncirculation-sex meridian\n\nendometriosis\n\nFertility Stimulation Points exercise __\n\nflow of sexual feelings\n\nlate or irregular periods\n\nduring pregnancy\n\nSedating Circulation-Sex Meridian exercise __\n\nuterine fibroids\n\ncoenzyme Qsupplements\n\ncolors of energy\n\ncomfort eating\n\ncompulsive eating\n\nconception. _See_ fertility\n\nConey, Sandra\n\nConnecting Heaven and Earth exercise __\n\nCook, Wayne\n\n\"Cool-Down\" Acupoint exercise\n\ncortisol\n\ncramps\n\ncircling hand over\n\nCramp Reliever exercise __\n\nSedating Small Intestine Meridian exercise __\n\nSedating Stomach Meridian exercise __\n\nsmall intestine meridian\n\nstomach meridian\n\nCross Crawl exercise __\n\ncrossover energy pattern. _See also_ homolateral energy pattern\n\nCeltic weave energy system\n\nCeltic Weave exercise __\n\nfigure-eight energies\n\nGrandmother's Self-Empowerment Secret exercise __\n\nHomolateral Crossover exercise __\n\nRhythmic Eights exercise __\n\nSuspension Bridge exercise __\n\nCrown Pull exercise __\n\ncultural values\n\nlove and sex\n\nmale-orientation of\n\nmenopause as disease\n\nslenderness\n\nviolence toward women\n\nyouth and beauty\n\nDaily Energy Routine\n\nCross Crawl __\n\nCrown Pull __\n\nHook-up __\n\nincorporation into daily activity\n\nLymphatic Massage\n\nmenopause module\n\nPMS module\n\nThree Thumps __\n\nWayne Cook Posture __\n\nZip-up __\n\nDalton, Kathryn\n\ndandelion root supplements\n\nDarth Vader Breath exercise\n\ndelivery. _See_ labor and delivery\n\ndepression\n\nbody's energies during\n\nconcerning weight gain\n\nHeart-Womb Connection exercise __\n\nMellow Mudra exercise __\n\nduring menopause\n\nprogesterone\/estrogen balance\n\nStretch, Stretch, Stretch exercise\n\nDiaphragm Breath exercise __\n\ndiet\n\nenergy testing\n\nessential fatty acids\n\nfertility and\n\nhormone balance and\n\nsubclinical food allergies\n\n_See also_ supplements and herbal remedies; weight management\n\ndong quai supplements\n\nDowsing the Fires exercise __\n\nDuerk, Judith\n\neating. _See_ appetite\n\nEat in Peace exercise __\n\neducational kinesiology\n\nelectrics\n\nelectromagnetic fields\n\nemotions\n\nCharging Your Batteries exercise __\n\nenergy techniques for\n\nExpelling the Venom exercise\n\nimprint of, in energies\n\ninfertility and\n\nduring menopause\n\nmood swings\n\nNeurovascular Hold exercise\n\nphysical symptoms and\n\nprogesterone\/estrogen imbalance and\n\nfrom sexual abandonment or abuse\n\nstomach meridian and\n\n_See also_ depression\n\nendocrine system. _See_ hormones\n\nendometriosis\n\nenergies\n\nin body\n\ncoordination of genes and cell behaviors\n\nand hormones, interactions between\n\ninformation transmission by\n\nintelligence of\n\nin material world\n\npersonal history encoded in\n\nsensory perceptions of\n\nyin and yang energies\n\nenergy habits\n\nBlow-out\/Zip-up\/Hook-in exercise __\n\nfertility and\n\nMeridian Energy Tap exercise __\n\nreprogramming, difficulty of\n\nfor stress management\n\nTemporal Tap exercise __\n\n_See also_ Daily Energy Routine\n\nenergy medicine\n\nand conventional treatments, combined\n\nfocus on overall health and functioning\n\nmethods used\n\npotential of\n\nprinciples\n\nshift of body's evolutionary strategies\n\nside effects\n\ntestimonials\n\ntraditional Chinese medicine\n\nenergy psychology\n\nenergy routine. _See_ Daily Energy Routine\n\nenergy testing\n\nconnection to wider dimensions\n\nas diagnostic tool\n\nfor environmental impact on personal energy field\n\nfine-tuning\n\nof foods and diet\n\nGeneral Indicator Test __\n\nof herbs and supplements\n\nfor homolateral energy pattern\n\nlimitations\n\nreliability\n\nscientific basis for\n\nself-test __\n\nSpleen Energy Test __\n\nsurrogate testing __\n\nenvironment and modern lifestyle\n\nadaptation to\n\nenvironmental estrogen sources\n\nenvironment and modern lifestyle ( _cont._ )\n\nimpact on personal energy field\n\nimpact on reproductive processes\n\nmenopause symptoms and\n\noutmoded stress response to\n\nepsom salts supplements\n\nessential fatty acids\n\nestrogen\n\nbone density and\n\ndeficiency hypothesis\n\nenergy methods to stimulate production\n\nenvironmental sources\n\nfunctions\n\nimproper levels, signs of\n\nin menstrual cycle\n\nprogesterone\/estrogen balance\n\nStimulating Estrogen Production exercise __\n\nsupplements\n\nevening primrose oil supplements\n\nExpelling the Venom exercise\n\n_Feminine Forever_ (Wilson)\n\nfertility\n\nenergy approach\n\nFertility Stimulation Points exercise __\n\nmedical and energy approaches combined\n\nmeridians involved\n\nobstacles\n\nSpleen\/Triple Warmer Balancing Sequence\n\n_Fertility Diet, The_ (Chavarro, Willett, and Skerrett)\n\nfibroids\n\naggressive treatments\n\nenergy of\n\nherbal treatment\n\nto prevent or shrink\n\nsymptoms\n\nunconventional surgical treatment\n\nfight-or-flight response. _See_ stress and stress response\n\nfigure-eight energies. _See_ crossover energy pattern\n\nFive-Minute Daily Energy Routine. _See_ Daily Energy Routine\n\nfive rhythms\n\nflaxseed and flaxseed oil\n\nfood cravings\n\nfoods. _See_ diet\n\nFoot Massage exercise __\n\nfull-spectrum mineral supplements\n\nGerber, Richard\n\nGeronilla, Linda\n\nginkgo biloba supplements\n\nginseng supplements\n\nglands. _See_ hormones\n\nGoddess Belly Hold exercise __\n\nGoodheart, George\n\nGood Morning Position exercise __\n\ngoverning meridian, sexual feelings and\n\nGraham, Martha\n\nGrandmother's Self-Empowerment Secret exercise __\n\nGreer, Germaine\n\ngroin pain\n\nto relieve\n\nSedating Liver Meridian exercise __\n\nhabits. _See_ energy habits\n\nHand Massage exercise __\n\nheadaches\n\nPendulum Head Swing exercise __\n\nfrom progesterone\/estrogen imbalance\n\nhealers\n\ncontact with another's divine nature\n\nexperience of another's trauma\n\nlong-distance healing\n\nhealth. _See_ energy medicine\n\nhealth problems. _See_ illness\n\nHealthy Pregnancy Sequence __\n\nHealy, Bernadine\n\nheart, energy field of\n\nHeart-Womb Connection exercise __\n\nheavenly realm. _See_ spirituality\n\nHenka, Sammy\n\nherbal remedies. _See_ supplements and herbal remedies\n\nhome birth\n\nhomolateral energy pattern\n\ncompromise of healing ability\n\nfollowing surgery\n\nHomolateral Crossover exercise __\n\nsigns of and testing for\n\n_See also_ crossover energy pattern\n\nHook-up exercise __\n\n_Hormone Deception_ (Berkson)\n\nHormone Hook-up exercise __\n\nhormone replacement therapy (HRT)\n\ncultural values and\n\nestrogen deficiency hypothesis\n\nhealth risks associated with\n\nwith natural progesterone\n\npharmaceutical industry promotion of\n\nrecommendations\n\n_Hormone Revolution_ (Lark)\n\nhormones\n\nadrenal glands\n\nadrenaline\n\nbrain structure and\n\ncortisol\n\nDiaphragm Breath exercise __\n\nendocrine glands\n\nin feedback loop with energies\n\nfunction\n\nHormone Hook-up exercise __\n\nhypothalamus gland\n\nimbalance, energy patterns of\n\nimbalance, physical symptoms of\n\nimpact on health\n\ninfluence on sexual and romantic choices\n\ninsulin\n\nliver meridian and\n\nlymphatic massage for\n\nmood swings and\n\nnutrition for production of\n\npineal-pituitary-hypothalamus axis\n\nspleen meridian and\n\nSunrise\/Sunset exercise __\n\ntestosterone\n\nTestosterone Triple Massage exercise __\n\nThymus Thump exercise __\n\nThyroid Booster exercise __\n\nthyroid gland\n\ntriple warmer and\n\nweight gain and\n\n_See also_ estrogen; progesterone\n\nhot flashes\n\nHouston, Jean\n\nhunger. _See_ appetite\n\nHunt, Valerie\n\nhypothalamus gland\n\nhysterectomy\n\nhysteria\n\nprogesterone level and\n\nas stress response\n\nWayne Cook Posture exercise __\n\nillness\n\ndisturbed energies preceding\n\nenergy patterns during\n\nfrom hormone imbalances\n\nfrom hormone replacement therapy\n\ninterrelationship between physical and emotional problems\n\nfrom unmanaged stress\n\nimmune system\n\ncortisol and\n\nLymphatic Massage exercise __\n\nlymph system and\n\nMeridian Energy Tap exercise __\n\nThree Thumps exercise __\n\ntriple warmer and\n\ninfertility. _See_ fertility\n\n_Infertility Cure, The_ (Lewis)\n\ninsomnia\n\ninsulin\n\nIntestine Neurolymphatic Reflex Points exercise __\n\niron supplements\n\nisometric exercises\n\nBone Saver exercise __\n\nfor osteoporosis prevention and correction\n\nTorso Stretch exercise\n\njoint pain and stiffness\n\nCircular Joint Rub exercise\n\nin postmenopausal women\n\nTape the Magnet on the Pain exercise\n\nWrist Twist exercise __\n\nKenton, Leslie\n\nkidney meridian\n\nback pain\n\nbone strength\n\nCharging Your Batteries exercise __\n\nendometriosis\n\nfear\n\nfertility and health of eggs\n\nFertility Stimulation Points exercise __\n\nlife force in\n\nduring pregnancy\n\nSedating Kidney Meridian exercise __\n\nsexuality\n\nuterine fibroids\n\nkinesiology\n\nK-Points exercise __\n\nlabor and delivery\n\nbody's preparation for delivery\n\nBrazilian Toe Technique __\n\nemergency procedures\n\nenergy flow during\n\nnatural birth after Cesarean section\n\nlarge intestine meridian\n\nback pain\n\nSedating Large Intestine Meridian exercise __\n\nLark, Susan\n\nLee, John\n\nLewis, Randine\n\nlife force\n\ndepletion by stress\n\ndepletion through self-denial\n\nin kidney meridian\n\nmobilization of\n\nLipton, Bruce\n\nliver function\n\nFoot Massage for Liver exercise __\n\nHand Massage for Liver exercise __\n\ntoxin removal\n\nliver meridian\n\nCharging Your Batteries exercise __\n\ndandelion root supplements\n\nendometriosis\n\nFertility Stimulation Points exercise __\n\ngroin pain\n\nduring pregnancy\n\nSedating Liver Meridian exercise __\n\nSedating Liver Meridian ( _cont._ )\n\nsexuality\n\nuterine fibroids\n\nlymphatic points\n\nLymphatic Massage exercise __\n\nmassage\n\nSpinal Flush exercise __\n\nlymph system\n\nMaddie's Belly Scoop exercise __\n\nmagnesium supplements\n\nmagnet therapy, Tape the Magnet on the Pain exercise\n\nMary Eve, Sister\n\nMassaging Spleen exercise __\n\nMassaging the Ileocecal and Houston's Valves exercise __\n\nMcTaggart, Lynne\n\nMead, Margaret\n\nmedical practice, Western\n\nfocus on drugs and surgery\n\nas industry\n\npharmaceutical company control of\n\nunderstanding of female hormones\n\nview of menopause as disease\n\nview of PMS as medical condition\n\nweaknesses and failures\n\nmedical treatments\n\ndeaths from\n\nand energy approaches, combined\n\nenergy testing of\n\nfor fibroids\n\nhormone replacement therapy (HRT)\n\nprotocols\n\nMellow Mudra exercise __\n\nmenopause\n\nBone Saver exercise __\n\nbooks on\n\nDaily Energy Routine module\n\ndepression\n\nDiaphragm Breath exercise __\n\nas disease\n\nDowsing the Fires exercise __\n\nemotions and attitude\n\nenergy balance\n\nestrogen\/progesterone balance\n\nheadaches\n\nHeart-Womb Connection exercise __\n\nHormone Hook-up exercise __\n\nhormone replacement therapy (HRT)\n\nhot flashes\n\nas industry\n\ninsomnia\n\nlifestyle and environmental factors\n\nMellow Mudra exercise __\n\nmental sharpness\n\nMetabolic Breath exercise __\n\nnewfound assertiveness and influence\n\nosteoporosis\n\nPendulum Head Swing exercise __\n\nstiffness and joint pain\n\nStimulating Estrogen Production exercise __\n\nStretch, Stretch, Stretch exercise\n\nSunrise\/Sunset exercise __\n\nsupplements\n\ntestosterone imbalance\n\nTestosterone Triple Massage exercise __\n\nThyroid Booster exercise __\n\nthyroid imbalance\n\nTorso Stretch exercise\n\nvaginal dryness\n\nweight gain\n\n_Menopause Industry, The_ (Coney)\n\nmenstrual cycle\n\ncelebration of\n\nearly onset\n\nlate or irregular periods\n\nmechanics of\n\nspleen meridian and\n\n_See also_ PMS\n\nmental sharpness. _See_ brain function\n\nMeridian Energy Tap exercise __\n\nmeridians. _See also specific meridians_\n\nmetabolic rate\n\nMexican yam root\n\nmind. _See_ attitude; brain function\n\nmineral supplements\n\nmiscarriage\n\nmood swings\n\nenergy rebalancing\n\nextreme behaviors\n\nhormonal fluctuations\n\nvitamin B\n\nmorning sickness\n\nMortifee, Devon\n\nmotherwort supplements\n\nMSM supplements\n\nmultiple vitamin supplements\n\nmuscle testing. _See_ energy testing\n\nNAC (N-acetyl cysteine) supplements\n\nNational Institutes of Health (NIH)\n\nnatural birth\n\nneurolymphatic reflex points\n\nLymphatic Massage exercise __\n\nmassaging\n\nSpinal Flush exercise __\n\nNeurovascular Hold exercise\n\nneurovascular points\n\n_New Feminine Brain, The_ (Shulz)\n\nNorthrup, Christiane\n\nOpening the Central Sexual Channel exercise __\n\nOrloff, Judith\n\nosteoporosis\n\nBone Saver exercise __\n\ncauses\n\nisometrics\n\nmeridians involved\n\nsupplements\n\nTorso Stretch exercise\n\novaries, energy flow through\n\noxygen, metabolism and\n\noxytocin\n\nOz, Mehmet\n\npain control. _See specific type of pain_\n\nPaskowitz, Dorian\n\n_Passage to Power_ (Kenton)\n\nPendulum Head Swing exercise __\n\nPeople's Pharmacy\n\nperiods. _See_ menstrual cycle; PMS\n\npharmaceutical industry\n\ncontrol of research and clinical practices\n\ndrug patenting and marketing\n\nplateaus in weight loss\n\nPMS\n\nAbdominal Stretch exercise __\n\ncalming techniques\n\nCharging Your Batteries exercise __\n\nDaily Energy Routine module\n\nDarth Vader Breath exercise\n\nenergy approach\n\nenergy patterns during\n\nestrogen\/progesterone imbalance\n\nExpelling the Venom exercise\n\nextreme experience of\n\nherbal remedies and supplements\n\nMassaging Spleen exercise __\n\nmeridians involved\n\nmood swings\n\nNeurovascular Hold exercise\n\npain and discomfort management\n\npositive potential of\n\nSedating Kidney Meridian exercise __\n\nSedating Large Intestine Meridian exercise __\n\nSedating Liver Meridian exercise __\n\nSedating Small Intestine Meridian exercise __\n\nSedating Stomach Meridian exercise __\n\nSideways Stretch exercise __\n\nSpleen Meridian Flush and Tap exercise __\n\nSpleen Meridian Strengthening Points exercise __\n\nStomach Meridian Flush exercise __\n\nTriple Axis Hold exercise __\n\n_Positive Energy_ (Orloff)\n\npotassium supplements\n\npreeclampsia\n\npregnancy\n\nback pain\n\nbreach position, to correct\n\nchakra balancing\n\nGoddess Belly Hold exercise __\n\nGood Morning Position exercise __\n\nHealthy Pregnancy Sequence __\n\nHeart-to-Womb Connection exercise __\n\nmeridians involved\n\nmiscarriage\n\nmorning sickness\n\nmother-baby connection\n\nnatural birth after Cesarean section\n\nprogesterone levels\n\ntoxemia (preeclampsia)\n\npremenstrual syndrome. _See_ PMS\n\nPrior, Jerilyn\n\nprogesterone\n\nbone density and\n\nenergy methods to stimulate production\n\nestrogen\/progesterone balance\n\nimproper levels, signs of\n\nin menstrual cycle\n\nduring pregnancy\n\nsupplements\n\nsynthetic form of (progestin)\n\npsychic opening. _See_ spirituality\n\nradiant circuits\n\nbelt flow\n\nenergy hyperlinking\n\noverview\n\npenetrating flow\n\nRadin, Dean\n\nRatcliff, J. .\n\nRhythmic Eights exercise __\n\nrhythms\n\nritual celebration for onset of menstruation\n\nRogers, Carl\n\nroot chakra\n\nof men versus women\n\nroot chakra ( _cont._ )\n\nduring pregnancy sexual feelings and\n\nVortex Revival exercises __\n\nSabin, Jack\n\nSAM-e (S-adenosylmethionine) supplements\n\nSchwarzbein, Diana\n\nscientific research\n\nSedating Circulation-Sex Meridian exercise __\n\nSedating Kidney Meridian exercise __\n\nSedating Large Intestine Meridian exercise __\n\nSedating Liver Meridian exercise __\n\nSedating Small Intestine Meridian exercise __\n\nSedating Stomach Meridian exercise __\n\nsedation. _See_ calming techniques\n\nself-suggestion. _See_ affirmations\n\nset point for body weight\n\nsexuality\n\nenergy in buttocks\n\nenhancing\n\nfemale qualities\n\nGrandmother's Self-Empowerment Secret exercise __\n\nhealing from trauma\n\nhormonal influence on choices\n\nfor men versus women\n\nmeridians involved\n\nOpening the Central Sexual Channel exercise __\n\nself-owning of body\n\nThree Great Kisses techniques\n\nVortex Revival exercises __\n\nShealy, Norm\n\nShlain, Leonard\n\nShulz, Mona Lisa\n\nsickness. _See_ illness\n\nSideways Stretch exercise __\n\nsleep patterns\n\nsmall intestine meridian\n\nback pain\n\nSedating Small Intestine Meridian exercise __\n\nSpinal Flush exercise __\n\nspirituality\n\nCrown Pull exercise __\n\nhealer's contact with another's divine nature\n\nPMS as sacred journey\n\nthrough energy testing\n\nspleen meridian\n\nCharging Your Batteries exercise __\n\nCramp Reliever exercise __\n\ndandelion root supplements\n\nendometriosis\n\nfertility and health of eggs\n\ninsulin production\n\nMassaging Spleen exercise __\n\nmetabolism\n\nduring pregnancy\n\nsexuality\n\nSpleen Energy Test __\n\nSpleen Meridian Flush and Tap exercise __\n\nSpleen Meridian Strengthening Points exercise __\n\nSpleen Points thump exercise __\n\nSpleen\/Triple Warmer Balancing Sequence\n\ntriple warmer, interaction with\n\nTriple Warmer\/Spleen Hug exercise __\n\nuterine fibroids\n\nin weight management\n\nstiffness. _See_ joint pain and stiffness\n\nStimulating Estrogen Production exercise __\n\nStokes, Gordon\n\nstomach meridian\n\nendometriosis\n\novaries and\n\nduring PMS\n\nduring pregnancy\n\nSedating Stomach Meridian exercise __\n\nStomach Meridian Flush exercise __\n\ntestosterone production\n\nStomach Meridian Flush exercise __\n\nstress and stress response\n\nadrenal fatigue\n\nDarth Vader Breath exercise\n\nEat in Peace exercise __\n\nenergy habits and\n\nExpelling the Venom exercise\n\nhormone-driven responses\n\nMeridian Energy Tap exercise __\n\nneurolymphatic reflex point massage\n\nNeurovascular Hold exercise\n\nas outmoded evolutionary adaptation\n\npsychological effects\n\nStress Dissolver exercise __\n\ntriple warmer and\n\nTriple Warmer Smoothie exercise __\n\nTriple Warmer Tap exercise __\n\nWayne Cook Posture exercise\n\nstretching\n\nConnecting Heaven and Earth exercise __\n\nspace for energy flow\n\nStretch, Stretch, Stretch exercise\n\nsubclinical allergies, energy testing for\n\nsubtle energies\n\nSunrise\/Sunset exercise __\n\nsupplements, conditions alleviated by\n\nbone mass loss\n\nendometriosis\n\nfibroids\n\nhot flashes\n\ninfertility due to aging eggs\n\ninsomnia\n\nlead toxicity\n\nmental decline\n\nPMS symptoms\n\nprogesterone\/estrogen imbalance\n\nthyroid deficiency\n\nvaginal dryness\n\nsupplements, energy testing of\n\nsupplements and herbal remedies\n\nblack cohosh\n\nbutcher's broom\n\ncalcium\n\nchondroitin\n\nchromium\n\ncoenzyme Q\n\ndandelion root\n\ndong quai\n\nepsom salts\n\nessential fatty acids\n\nestrogen\n\nevening primrose oil\n\nflaxseed oil\n\nfull-spectrum mineral supplements\n\nginkgo biloba\n\nginseng\n\niron\n\nmagnesium\n\nmotherwort\n\nMSM\n\nmultiple vitamin supplements\n\nNAC (N-acetyl cysteine)\n\npotassium\n\nprogesterone\n\nSAM-e (S-adenosylmethionine)\n\ntestosterone\n\nthyroid hormones\n\nvitamin B complex\n\nvitamin B\n\nvitamin\n\nvitamin K\n\nyam root\n\nSuspension Bridge exercise __\n\nSzent-Gy\u00f6rgyi, Albert\n\nTanya's Spiraling Swing and Slap exercise __\n\nTape the Magnet on the Pain exercise\n\nTarzan Thump exercise __\n\nTemporal Tap exercise __\n\ntestosterone\n\nimbalance\n\nin male stress response\n\nTestosterone Triple Massage exercise __\n\nThie, John\n\nthinking. _See_ brain function\n\nThree Great Kisses techniques\n\nThree-Heart Hook-up exercise\n\nThree Thumps exercise __\n\nThymus Thump exercise __\n\nthyroid\n\nhormone imbalance\n\nThyroid Booster exercise __\n\nweight management\n\n\"Tibetan Energy Rings and the Celtic Weave\" (Eden)\n\nTorso Stretch exercise\n\n_Touch for Health_ (Thie)\n\ntoxemia\n\ntoxins\n\nclearing from body\n\nenergy toxins\n\nFoot Massage for Liver __\n\nHand Massage for Liver __\n\nlymphatic system and\n\nweight management and\n\ntrauma\n\ndamage to basic grid\n\nimprint in chakra energies\n\nsexual abandonment and abuse\n\ntreatments, medical. _See_ medical treatments\n\nTriple Axis Hold exercise __\n\ntriple warmer meridian\n\nDarth Vader Breath exercise\n\nemergency mobilization of\n\nendometriosis\n\nfibroids\n\nhormone production governed by\n\nin immune and stress responses\n\noverview\n\nPMS symptoms\n\nas radiant circuit\n\nretraining of\n\nsedation, benefits of\n\nspleen meridian, interaction with\n\nSpleen\/Triple Warmer Balancing Sequence\n\nStress Dissolver exercise __\n\ntoxemia\n\nTriple Warmer Sedating Points exercise __\n\nTriple Warmer Smoothie exercise __\n\nTriple Warmer\/Spleen Hug exercise __\n\nTriple Warmer Tap exercise __\n\nuncontrolled eating\n\nweight set point\n\nuterine fibroids. _See_ fibroids\n\nvaginal dryness\n\nvitamin B complex supplements\n\nvitamin Bsupplements\n\nvitamin supplements\n\nvitamin K supplements\n\nVortex Revival exercises __\n\nWand, Sandy\n\nWayne Cook Posture exercise __\n\nWeb resources\n\nadvanced energy techniques\n\narticle on Celtic weave\n\narticle on energy medicine fundamentals\n\nenergy techniques program\n\ninsomnia information\n\nto locate energy medicine practitioner\n\nmenopause survey\n\nPeople's Pharmacy\n\nWomen to Women clinic\n\nWeed, Susun\n\nweight management\n\naffirmations\n\nanxious eating\n\nassimilation of food\n\nAssimilation Massage exercise __\n\nchallenges\n\ncomfort eating\n\ncompulsive eating\n\ncravings\n\ndigestive tract invigoration\n\nEat in Peace exercise __\n\nenergy work for\n\nfears\n\nhollow hunger\n\nhormone-induced weight gain\n\nhunger, overcoming\n\nIntestine Neurolymphatic Reflex Points exercise __\n\nlove of body\n\nlow blood sugar\n\nMaddie's Belly Scoop exercise __\n\nMassaging the Ileocecal and Houston's Valves exercise __\n\noxygen and metabolic rate\n\nplateaus in weight loss\n\nset point\n\nslimming waist and belly\n\nsubclinical allergies, energy testing for\n\nTanya's Spiraling Swing and Slap exercise __\n\nThree-Heart Hook-up exercise\n\nthyroid and\n\ntoxins, eliminating\n\nupbeat attitude\n\nWestern medicine. _See_ medical practice, Western\n\nWilson, Robert A. and Thelma\n\nwomb chakra\n\n_Women's Bodies, Women's Wisdom_ (Northrup)\n\nWomen to Women\n\nWrist Twist exercise __\n\nyam root\n\nyin and yang energies\n\nZip-up exercise __\nOther Books and Programs of Interest\n\nThe _Energy Medicine for Women_ DVD program, a highly recommended companion to this book, shows Donna Eden demonstrating most of the procedures the book presents.\n\n_Energy Medicine_ by Donna Eden (with David Feinstein, Ph.D.), the bestselling classic, is available now in a thoroughly revised and up-to-date tenth anniversary edition. It also has a companion DVD program, _Energy Medicine: The Essential Techniques,_ which features six hours of Donna demonstrating the most fundamental procedures of energy medicine.\n\nAn energizing first look at energy medicine is _Introduction to Energy Medicine,_ a DVD based on a two-hour live seminar with Donna. Another introductory resource, the _Energy Medicine Kit,_ was a One Spirit featured selection. A variety of other basic to advanced DVD programs is also available for bringing energy medicine into your living room.\n\n_The Promise of Energy Psychology,_ by David Feinstein, Ph.D., Donna Eden, and Gary Craig, provides simple step-by-step instructions for creating constructive changes in your energy system _and brain chemistry_. These changes can help you alter behavioral patterns; overcome psychological challenges such as inappropriate fear, guilt, shame, jealousy, and anger; and succeed with challenging goals. A companion DVD program, _Introduction to Energy Psychology,_ explains the essential concepts, and demonstrates the basic procedures emerging from this revolutionary development within psychology. The book was a finalist in the Self-Help category at the 2007 PubInsider National Book Awards (the Indies).\n\n_Personal Mythology: Using Ritual, Dreams, and Imagination to Discover Your Inner Story_ and a companion CD present a 12-week program for transforming the deep guiding myths that shape our lives. Its new third edition, which adds an energy psychology module, was named a _USA Book News'_ 2007 Book of the Year (Psychology \/Mental Health category) and 2007 Book of the Year (New Age Nonfiction category) at the Indies Awards. The book provides a compass for these troubling times.\n\nLive classes, ranging from introductory workshops to a two-year professional certification program in energy medicine, are available. A catalog describing other materials and training programs will be sent upon request (contact energy@ innersource.net). Professional continuing education credit is available for many of these programs.\n\n_www.innersource.net_\n\n24-hour Order Line: 800-835-8332\n\nInnersource, 777 East Main Street, Ashland, OR 97520\nAbout the Authors\n\nDonna Eden is a pioneer in the field of energy medicine. She has addressed audiences around the world and is consulted frequently within both traditional and alternative health care settings.\n\nDavid Feinstein is a clinical psychologist who has served on the faculties of The Johns Hopkins University School of Medicine and Antioch College. His award-winning books include _Personal Mythology, The Promise of Energy Psychology_ , and _Energy Psychology Interactive._\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \nAmalie Howard\n\nTHE ALMOST GIRL\n\nTo Valerie, because she couldn't be here.\nPROLOGUE\n\nTHREE YEARS EARLIER\n\nThe slight figure is lithe and quick, a shadow of a shadow in the darkness. It runs along the edgy gloom of the halogen-lit streets, flying over electric fences and scaling walls with the practiced ease of a skilled athlete. One would never suspect that it was being chased by an entire army of soldiers, but it was, several hundred of them.\n\nIn a fluid twist to gauge the remaining distance from its pursuers, the runner's profile is visible for a brief second. It is the face of young girl, barely fourteen, as she glides into a narrow alley. Blood drips from a self-inflicted gash in her arm, the silver implant she'd dug from the wound slipping from her fingers to the oily ground before it is crushed beneath her boot.\n\nGlancing at the gauge on her wrist, she sees a red flash that tells her that she's nearly at the eversion checkpoint. Her timing and positioning must be exact for the universe transition. She ducks into a crouch as the first of the small army reaches the dark alleyway; he is faster than the others.\n\nThey're always fast.\n\n\"Surrender yourself,\" a voice says. The soldier stands, weapon at the ready. He knows that she is there. The girl steps out from behind a crate. There is no fear on her face, just a silent calmness, an acceptance of the situation. The soldiers are programmed to obey and to subdue hostiles, but she tries to divert them anyway. She knows that nothing she says will deter the soldier \u2013 after all, she's been their leader for the better part of a year.\n\nAnd now, she is the traitor... the fugitive.\n\n\"Stand down, Lieutenant,\" she says firmly in a husky voice far too mature for her years. The soldier doesn't even acknowledge her words. \"That is an order.\"\n\n\"Surrender,\" he repeats, raising the electro-rod slightly. \"General.\"\n\nHis voice is dead, just like the rest of him, but he understands exactly who she is. He's half-alive but still far from a mindless drone. She sees a glimmer of blue sparks at the rod's three-pronged tip. He'd have it set to stun she knew, but she wasn't going back alive. She couldn't go back.\n\n\"OK. Have it your way,\" she says.\n\nThe girl lunges at him, barely half his size, to slide on her knees beneath the blunt edge of the metal rod swinging toward her head. Her hand snakes out, a fist thumping into the hard, cold flesh beneath his ribs lightning-fast, and the soldier grunts, doubling over at the dull crack. In a reverse motion, her fist slices past his Achilles tendon, the blade between her fingers a blur, and he crumples to the floor.\n\nThey may be immune to pain, but they're still made of flesh and bone.\n\nNot losing momentum, she jabs him in the back of the neck just above the top of his spine with the point of her knife. The strike is snake-like and true. A spark and the sharp smell of singed flesh, and in a matter of seconds, the soldier is lying prostrate on the ground, twitching slightly, disabled for the moment.\n\nGlancing around, the girl listens for sounds of the others before emptying the soldier's pockets quickly. As well as the rod, she sticks a communication earpiece, a long-handled knife, six packets of dried food dust, and two pen-like instruments into her own black knapsack.\n\nIt is more than she could have hoped for.\n\nThe soldier stirs with a whining noise, and the girl grasps his face between her hands, pulling open his eyelids with her thumbs and forefingers. His skin is cold and clammy but he's not dead; far from it. Her blow to his cortex chip would only have caused it to reboot, but the nanocells in his retina would still be relaying real-time to her pursuers. She wants the message to be clear and stares directly into his eyes, straddling his chest with her knees.\n\n\"Don't try to find me,\" she growls. \"Don't send anyone. If you do, they will end up like this one; that I can promise you.\"\n\nHer hands twist, tugging the soldier's chin upward and jamming his knife into the back of his neck. It is such a smooth motion of her hands that the soldier's body barely twitches as she severs his spinal cord, the critical connector between the brain and the body. Her face will be the last thing her pursuers see. The light in his eyes fades but it's only a trick of the shadowed gloom around them. There's no life in these creatures... only death.\n\nThe sensor on her wrist flashes to blue. Without a backward glance, she is away in an instant, swallowed up by the inky darkness, punching in a sequence on a flat computer-like device connected to the sensor. After a moment, all that's left in her wake is a brief shimmer in the fabric of space and air. She's gone.\nPART ONE\n\nTHE OTHERWORLD\nPRESENT DAY\n\nCOLORADO\n\nMy thoughts rain like spatters of blood against the colorless landscape of drab walls and wooden faces. A bell rings, and it is a mad rush as chairs are pushed back loudly. A tall woman with a no-nonsense face calls for silence.\n\n\"The class roster for the end-of-year projects has been posted in the hallway. You have been paired in groups of four with a different assignment based on what we have covered this semester. If you don't know your partners, I suggest you meet them quickly, as these projects will count for half of your final grade.\"\n\nA collective groan rolls its way across the classroom.\n\n\"But Mrs Taylor,\" a girl three rows across stands and complains loudly, \"why can't we pick our own groups? Wouldn't that be better for everyone?\"\n\n\"Miss Hall, in the future, if you'd like to say something, please refrain from yelling it across the classroom. The groups have been allocated according to last year's class standings.\"\n\n\"But\u2013\"\n\n\"The groups are final, Miss Hall.\" Mrs Taylor's voice brooks no argument, and the girl falls silent, although her face remains puckered with frustration as she exits the classroom.\n\nI sit huddled at the back, waiting until the classroom is almost empty before gathering my things and walking noiselessly to the front.\n\n\"Mrs Taylor?\" I ask. My voice is slightly roughened from a lack of sleep, and the teacher jumps, looking up questioningly. I paste a suitably contrite look on my face. \"Sorry to startle you, I'm... er... Riven. I transferred in last week. About the groups...\"\n\n\"Ah, yes, Riven, I do have a note about you, as a matter of fact,\" Mrs Taylor says, shuffling through a pile of papers on the desk. \"You have already been assigned. It's on the board along with the others. If you run into any trouble, let me know.\" Mrs Taylor pushes the wire-rimmed glasses up her nose, her dark eyes sharp. \"Anything else?\"\n\n\"No, that's it,\" I mumble, unable to hold back the yawn that overtakes my facial muscles.\n\n\"Are you alright? You look quite pale.\"\n\n\"I'm fine, just tired. Jet lag,\" I smile and hoist my backpack over my arm. \"Thanks, Mrs Taylor.\"\n\n\"Riven?\" I freeze at the door and turn my head in her direction. Her black eyes are still piercing, unsettling as if they can see right through me. I feel an odd, unwelcome shiver take hold at the base of my spine. \"Welcome to Horrow.\"\n\n\"Thanks,\" I mumble and shift away from her impaling gaze. She's looking at me as if she knows who I am... an imposter, a stranger.\n\nA killer.\n\nI sneak a glance into the classroom once I'm in the hallway, and Mrs Taylor is back to studying the papers on her desk. I must have been wrong. I yawn again as exhaustion consumes me. In my tired delirium, I'm starting to imagine things. I've been pushing myself way too hard without enough rest intervals between jumps. It's foolish and reckless.\n\nBlack dots fill my vision. I'm disoriented as if the ground is tilting beneath my soles. I glance down, only to see the checkered tile floor undulating like a breaking wave. Gasping for breath, I haul open the first door I see.\n\nA janitor's closet.\n\nLeaning against the cool plastic of a recycling bin, I breathe in huge gulps of stale closet air. The fatigue is becoming worse, ever since the last jump. My fingers begin to shake uncontrollably as I smooth open the crumpled paper with my new class schedule.\n\nThirty class schedules in as many months, with time slowly running out. Trying to locate the boy here has been like looking for a drop in a bucket of water \u2013 near impossible. But I can't give up. I won't. Because in my gut, I can sense that there are already others here... others looking for him.\n\nAnd I have to find him first.\n\nSwallowing, I blink back the grit from my burning eyes and squint in the dim light at my schedule. I have Phys Ed and then lunch. I won't make it to lunch, it's an absolute certainty. My body slides down the side of the bin and I sit in the muted shadows as light filters from the cracks underneath the door. Maybe I'll just sit here for a second to catch my breath.\n\nMy eyelids droop heavily, and then there is only sweet aching darkness.\n\nWhen I open my eyes again, there is no longer any light seeping through the door and there is only silence beyond it. I must have slept through the entire day on the floor of this tiny closet. I inch my way up, hearing my joints creak painfully, and crack open the door. The hallway is deserted, the clock on the wall showing 4 o'clock in the morning. A chill sweeps along my skin as the fluorescent light flickers eerily. School hallways just aren't the same without kids in them. Suppressing a shudder, I exit through a side entrance onto one of the practice football fields as the door locks behind me with a soft click.\n\nEarly fall, and the night is dark and cold. I tug my black sweater down along my arms. It's only when I reach the empty parking lot where the Ducati is parked that I feel the first painful rumble in my stomach. I haven't eaten anything today. Swearing at my own carelessness, I unwrap a snack bar from my backpack with clammy hands and shove it into my mouth.\n\nIt's one of my few rules of survival \u2013 always eat. In my condition, hunger can bring on far worse things, things that you can't come back from, not in this world anyway, and my body is unstable enough already. The food slides down like hard cardboard along the soft sides of my throat and I gag, but force myself to swallow. There's a bottle of water in my backpack and I drink it so quickly that half of it spills down my sweater. I'm hoping that it isn't already too late. I throw on my helmet and take a deep breath. I have to make it back to the motel. It's only a few miles.\n\nTires squealing in protest against the cold asphalt, I pull out onto the main road and ride as fast as I dare over the speed limit. The last thing I need is to get pulled over. It happened once before when the sickness started. The cop ended up in the hospital that night, and I had to leave town quickly, trusting that what I'd been looking for hadn't been there. I couldn't risk anything similar happening, not again and not here.\n\nI stop at a red light, concentrating on taking slow mechanical breaths. But the pain in my belly only deepens as if in silent mockery of my efforts.\n\nYou can make it, I tell myself firmly, accelerating across the intersection.\n\nThe panic recedes but then returns in a wave so violent that I am gasping as my back arches like a bow. There's no way I can make it. How could I have been so foolish to think that I could beat the odds... beat time?\n\nToo late, too late, too late.\n\nA brutal wave of nausea drives me to jerk roughly on the Ducati's handlebars, the motorcycle's wheels protesting angrily on the asphalt, just as a lance-like pain stabs through me. My fingers jam reflexively against the throttle, twisting it. The bike lurches forward and careens across the two opposite lanes, my thighs burning from gripping the sides of the tank to steady it.\n\nThat's when the shakes start. Within seconds, I can feel my hands curl into hardened claws, my body spasming uncontrollably. My eyes roll back and I barely see the oncoming lights, as the bike swings precariously once more to the left, grinding off the road and spinning into gravel. My body is flung like a sack of rocks as the Ducati skids to a shattered halt on its side.\n\nThe sky above me is dark and wide with nothing in it. No stars, no moon, nothing. Just blackness. I suck in a shallow breath, keeping my jaw tightly closed, knowing how easily I can bite my own tongue off if I'm not careful. My chest aches with the strained intake of air, but I already know from years of training that it's mostly bruises, and nothing's broken. Hot white dots cloud my vision and I focus myself, searching for my backpack. It was flung from me upon impact with the ground, but it's just a foot away.\n\nReach out slowly, the bag is right there, I tell myself, but my body refuses to cooperate. Inside I know that it is too late, I can feel myself shutting down. I should have rested today, stayed in bed and given myself a chance to recuperate from the jump, but I'd been stupid, arrogant. I hadn't wanted to lose any time, and now I'm going to pay the price. My eyes slip shut.\n\nAs if from afar, I hear a rustling and then a loud banging. Someone yelling. Shadows flit across my closed eyelids. \"Help,\" I whisper. \"Help.\"\n\n\"Oh god! She came off a bike. Don't move her; she could have a concussion.\"\n\n\"Hey! Hey, you OK?\"\n\nThe voices are dull as if coming from far away. My thoughts won't even turn toward them. Noises followed by a dull thud as someone stoops beside me. Gentle fingers slip across my arm, moving upward to open my visor.\n\nBackpack. I try to say my single thought but my tongue is thick against my teeth. I can only open and close sticky lips that taste like metal.\n\n\"She's alive! Help me get this helmet off. Careful with her.\"\n\nMy head lolls backward as the helmet slips off, but I'm caught by strong hands and cradled gently. A bottle is placed against my lips and I feel cool water trickle into my mouth, washing away the coppery taste. It hurts to swallow, but I ignore the pain. The water moistens my gums and loosens my tongue.\n\n\"Injector... backpack...\" It won't be long before I go into shock. \"Have to... stick...\"\n\n\"Don't worry, I got it. I'm allergic, too,\" one of the voices says. I hear a rustle and feel the rough jab of the needle piercing into my skin through my jeans, and then soft fingers are brushing against my forehead. \"Hang on, it's going to be OK.\"\n\n\"Should we call 911?\" the other voice asks. \"What's with the needle?\"\n\n\"No hospital, please. Be OK...\" I direct my plea to the one who'd administered the auto-injector. \"Please, can't afford...\"\n\n\"Rest,\" the voice says. \"It's OK, Jake, looks like an allergic reaction. Could be peanuts, bees, anything, don't know.\" I hear the rustling of a wrapper. \"My aunt's off tonight. I'll take her home with me and see what she says. If she says to go to the hospital, I'll take her.\"\n\n\"What about her bike? We shouldn't just leave it, right? We can probably get it in the back of my truck,\" the voice belonging to Jake says. \"I can take a better look at it tomorrow.\"\n\n\"OK. Help me get her inside first. Careful, she may be hurt from when it went off the road.\"\n\n\"Thank you,\" I murmur as they lift me gently into the backseat of the truck. They are the only words I can manage before my brain shuts down. I can feel the serum making its way through my body, stopping my cells from going into anaphylactic shock.\n\nThe boy's right \u2013 I am having an allergic reaction, just not to any food.\n\nIn some dark corner of my mind, I know that I should be worried or be afraid that I have fallen into the wrong hands, but somehow I know... I trust that I am safe. The thing, is I can't remember the last time I felt safe. Oblivion sweeps my remaining consciousness away.\n\nWhen I open my eyes again, I'm lying in a bed in an airy room. It's quiet and peaceful. A fan on the ceiling wafts cool air into my face, and for a second it feels as if I'm in some kind of dream. Then I see the boy slumped in the armchair in front of the window and instantly know that this is reality. He seems asleep, although I can't really tell from the way his hair is curling into his face. I search for my backpack. It's sitting next to him on the floor. Sitting up gingerly, I swing my left leg over the side of the bed and wince at the pain now radiating up my back and around my ribs.\n\n\"You shouldn't really move, you know.\" The boy is awake now and I can feel him watching me carefully. I ignore him and shift my other leg to the floor. The pain is excruciating, echoing along every nerve ending like fire.\n\n\"My aunt says you need to keep that leg up,\" he says and moves to stand next to me, his hand pressing onto my shoulder. With his free hand, he carelessly shoves the hair out of his face and sits beside me on the edge of the bed. \"You're pretty banged up.\"\n\nOur eyes collide and it is like I am being sucked into a vortex that I can't control.\n\nIt's him.\n\nThe boy I'm supposed to find.\n\nHis hair is lighter, almost golden brown, and swept to the side around his face, but his nose and chin are the spitting image of the one I know. And his eyes... those impossibly green eyes, filled with vibrant life. I'd prepared myself that he would look like him but they're so alike that it leaves me speechless.\n\nAnd he found me. He saved me.\n\nI shake myself hard. What are the odds? Searching for someone for nearly three years only to find them via an accident of fate? The questions make my head pound, and I blink, disoriented.\n\n\"Where am I... What happened?\" I croak. My voice is unfamiliar. Weak.\n\n\"Don't you remember? You crashed your bike and had some kind of crazy allergic reaction. You're at my house now. You didn't want me to take you to the hospital because you said something about money, so I brought you here,\" he says in a rush and then clarifies, \"My aunt's a doctor.\"\n\n\"How long have I been here?\" I say and try to stand, gasping at the soreness of my ribs.\n\nHis nearness is overwhelming, confusing me as thoughts of Cale race through my clouded brain. My throat is raw, and the effort to swallow makes my head pound. A wave of dizziness overcomes me and I fall back to the bed. A knife-like pain slices through my leg.\n\nThe boy leans forward to grasp my shoulder gently. \"Look, you really should\u2013\"\n\n\"Don't touch me,\" I snap, flinching away from the warmth in his fingers. My body may be beaten, but it's still poised to attack. The boy's offended expression throws me, and my anger fades as my brain struggles to keep up. \"Sorry, I'm still a little freaked out, and I don't like people touching me,\" I say by way of apology. He still looks miffed so I force a tiny smile to my lips. \"You go to Horrow, right?\"\n\n\"Yes, we're in the same Physics class,\" he says, the hurt look draining away slowly, \"and in the same project group. I only knew who you were because Mrs Taylor asked me to help you out if you needed a hand since you're new. You started last week, right?\"\n\n\"Yes,\" I say, remembering the profile of a boy I'd barely given a second glance to. I grind my teeth together \u2013 that had been sheer carelessness on my part. Or maybe all those jumps are finally catching up to me; otherwise, why else would I be lying here in this bed, weak as a newborn kitten?\n\n\"I'm Caden, by the way,\" he says, sticking his hand out. Staring at his fingers as if they're snakes, I raise my hand in an awkward half-wave. My smile feels forced. His hand falls away, and the weird look returns to his face. \"You're not too friendly, are you?\"\n\nI breathe out the pent-up air in my lungs and feel the rush of adrenaline recede. I stare at the boy through the corner of my eye who could be Cale. No, not Cale. They may look the same, but they're entirely different people underneath their doppelganger skins.\n\n\"Sorry. I mean, I know who you are,\" I whisper under my breath.\n\nIt's not Cale, I remind myself for good measure.\n\nMy head still feels wobbly like some kind of horrible hangover. Only, I wouldn't know what that would be like \u2013 the only time I'd tasted spirits had been with Cale, celebrating the Winter Solstice when I was ten. It was an experience I never want to repeat. But I'd seen other people drunk enough to guess what a hangover would feel like.\n\nA tremor runs through my hands and I flex them automatically. My veins are blue against my skin, the tendons still corded and raised along the backs of my hands. Black and blue bruises mottle the length of both arms. My torso probably looks worse. A hollow feeling fills my stomach as I realize just how close the shakes had brought me to an irreversible outcome last night. Too close... and now that I'd found the boy, I needed to have all my wits functioning. Others would be close too. The ones who would also come for him.\n\n\"I like your tattoo,\" Caden says, interrupting the turn of my thoughts. Instinctively, my fingers touch the gold circular seal and the three black lines \u2013 two whole and one broken \u2013 beneath it on my neck. \"Does it mean anything?\"\n\nI almost want to laugh. A filial brand and a line for each traitor I'd killed? He'd be running away as fast as he could or calling the police if he even guessed what it meant.\n\n\"No.\"\n\n\"So, what's your name?\" Caden asks, shoving his hands in his pockets. I had to give him credit for trying. In that, he was just like Cale \u2013 neither of them took \"no\" for an answer.\n\n\"Riven.\"\n\n\"I thought that was your last name?\"\n\n\"Riven is my last name,\" I say, and bite back a grin at his immediate frown. \"I only have one name. Where I come from we don't have two names, just one.\" I see his frown deepen, and kick myself for my telling choice of words.\n\n\"Where you come from,\" he repeats slowly. \"Everyone has two names here, unless you're like Usher or Madonna.\" At my blank stare, he clarifies, \"You know, the singers?\"\n\nI nod quickly. I've seen them on the television. \"Just Riven,\" I say.\n\n\"Just Riven.\" He draws my name out slowly like he's trying to taste it or something. \"That's a weird name. I mean, unique,\" he says hastily. \"Does it mean anything?\"\n\n\"It means ripped apart.\"\n\n\"Oh.\" I can see that he's at a loss for words. I don't blame him. Back home, my name strikes fear into anyone who hears it \u2013 but that's more a factor of the reputation that precedes me than anything else.\n\nFrom his expression, I can see him wondering why someone would name a child with such an odd, violent name. I feel my lips curling in a smile \u2013 as far as names go, I like the fierceness of it, the simplicity. In a weird way, it fits me.\n\nAfter a couple minutes, Caden speaks. \"No idea what mine means. So, is that from Asia or Africa, then? You know, where people have one last name? Is that where you grew up?\"\n\nI can only manage a terse nod. At Caden's questions, I wish I could pull out the notebook in my backpack and leaf through it. Even after three years of blending in\u2014appearance, accent and behavior-wise\u2014I'm still not familiar with the exact geographical topography of this world. His questions are making my head spin, and I can't afford to make any more mistakes, not when I am almost home... now that I've finally found him.\n\nI shake myself mentally once more. If my body were stronger, I'd grab him and go, but in my weakened condition, that would be sure suicide for us both. I'd die, and he'd never make it without me. Not there.\n\nMy eyes fall to the glass of water sitting on the bedside table next to an alarm clock, and I take a slow sip. It's almost 11 on Saturday morning. I need to make some kind of exit and compose myself for travel. And the travel I'm talking about is not as simple as buying an airplane ticket and showing up at a mass-transit airport; it's way more complicated. Any number of things could go wrong, especially when there is more than one traveler \u2013 one of them a fugitive, the other a target.\n\n\"You don't look Asian,\" Caden continues his monologue, considering I'm barely participating in the conversation. \"I mean, you look like me, well, except the hair. Yours has green and blue in it,\" he points out. I touch the strands and remember that I'd dyed it four schools before, after the incident with the police. It was haphazardly chopped around my face except for a single braid that wound down one side.\n\n\"It's cool, your hair,\" Caden adds and then reddens. \"For a punk look, I mean.\"\n\nI'd butchered it myself when I'd been short on time, leaving only the slim blue and gold braid. I hadn't been able to let it go \u2013 the only reminder of my position, my rank. But overall, it was an edgy, fierce look that tended to make people stay away, which I'd liked.\n\nIt wasn't doing much to shut Caden up, though. \"You definitely stand out, especially at Horrow,\" he remarks. \"The girls are all pretty much vanilla. You meet any of them yet?\"\n\n\"No. I keep to myself.\"\n\nA wry smile. \"I get it. You don't like being touched, you want to be alone, and you're not looking for any friends.\"\n\nCaden moves to stand near the window and moves my backpack from the floor to the chair. He doesn't open it but just stares at it thoughtfully. It's a brief respite from the conversation, so I use the silence to figure out how to tactfully say thank you and leave.\n\nHe eyes me. \"What exactly happened to you last night?\"\n\nBut I'm saved from having to respond to Caden or tell him rudely to shut up, when a neatly dressed woman enters the room. She is no taller than I am but sturdily built; she looks like a strong woman. Her dark hair is pulled off of her face into a tidy bun at the base of her neck. She has kind eyes with lines at the corners, but there's something else in them, too... warning that her kindness shouldn't be mistaken for weakness.\n\n\"How's our patient doing this morning?\"\n\nShe glances at Caden, who is still flushed, and then back to me where I'm sitting on the edge of the bed with a frown on my face. A strange expression curls the corners of her lips upward, and I can feel my brows snapping together even more tightly. I don't recognize or like the amused look on her face, as if she thinks there's something going on between the two of us.\n\n\"I'm Caden's aunt,\" she says to me. \"He's been in here constantly. I've never seen him so solicitous of anyone.\"\n\n\"What? I wasn't.\" Caden flushes and stares at the ground.\n\n\"I hope you haven't been keeping her from resting, Caden. She needs to keep that foot elevated.\"\n\n\"It's fine,\" I say, and then more clearly, \"My foot?\" For the first time, I notice that I am wearing some sort of cotton pants, and I wonder whether Caden's aunt had removed my own clothes. Curiously, I don't feel any embarrassment, because I'm more worried about whether the injury will slow me down.\n\n\"Lay back,\" she tells me gently and places a hand against my forehead. \"That's good.\"\n\n\"What happened?\" I repeat, trying to pull the pajama material up to see. She stalls my hand.\n\n\"Try not to move, you have some badly bruised ribs, too. It's your ankle, nothing too serious. You must have torn a ligament from the convulsions or when you fell, but you do need to keep pressure off of it for now. I iced it and wrapped it last night. Let's have a look.\"\n\nCarefully unwrapping the bandage, I see that my ankle is a blotchy greenish purple and twice the size of my other foot. I am sure that it looks far worse than it is. I wiggle my toes slowly and I know from experience it's a good sign. It means nothing's broken.\n\n\"A lot of the swelling has gone down, which is good,\" Caden's aunt says. I can't imagine my ankle being any fatter, but it must have been because even Caden is nodding.\n\n\"It matches your hair,\" Caden remarks. I ignore him, more concerned with trying to calculate how much this injury will set me back.\n\n\"How long?\" I ask.\n\n\"A few weeks.\"\n\n\"A few weeks!\" I gasp. \"Can't you do anything to speed it up?\"\n\nA gentle smile while deftly re-wrapping the bandage across my ankle. \"No, honey. Best you can do to recover quickly is rest, ice, compression, and elevation. R. I. C. E. Simple enough to remember, right? If the pain gets any worse or it doesn't get better, you'll need to get it checked out. For now, I can give you some ibuprofen to help with the pain and the swelling.\"\n\n\"No meds. I can manage the pain,\" I say. \"I'm allergic to most medications,\" I add at her curious look. The truth is that anything that inhibits the functions of the brain is a risk, especially during eversion. I need to be clear.\n\n\"I guess that explains why you had such a high-tech injector in your bag,\" Caden chimes in, pulling the pen-like instrument from the front pocket of the backpack where he'd replaced it the night before. \"I've never seen anything like it. My emergency one is like a plastic piece of crap compared to yours. Bees are my nemesis,\" he reminds me, twisting the silver cylinder between his fingers.\n\nI smile, a cheap attempt at reassurance and normalcy even though my heart is pounding. I've never wanted to lurch forward and grab anything more than at that moment. Like the teacher earlier, I feel that Caden's aunt can see right through me. Her blue eyes are as sharp as Mrs Taylor's had been, and although there's no mistrust in them, I feel uncomfortable just the same.\n\nIt's one of the reasons that I don't like getting close to people. Too many questions. And too many that can't be answered. But I know that I owe them both some kind of explanation for my bizarre behavior... and for the injector that looks like it comes from some kind of super advanced robotics lab.\n\n\"Mine is a little more complicated,\" I say. \"I'm not allergic to bees or food. It's a... a genetic brain thing. If I don't take my medication regularly, like yesterday, things can go south pretty quickly, especially with the seizures. Sometimes something as simple as hunger can set it off.\" I glance up to test the waters. They are both watching me, but with more concern than any kind of disbelief on their faces. My lies are getting more convincing. \"The injector is custom-made for my condition. You couldn't use it,\" I say in Caden's direction. \"And it's really expensive so... \"\n\nI don't have to finish my sentence before Caden carefully replaces the injector in the backpack.\n\n\"Sorry,\" he says stuffing his hands in his pockets. \"So are you OK now?\"\n\nI nod slowly. I haven't had to use the injector before but it has definitely come in handy to say the least. I am alive. Each cylinder has six doses, so I have five remaining. I hope fervently that I don't have to use them. Even thinking about the pain makes my head spin. Caden's aunt pulls the sheet up and pats my forehead.\n\n\"You can stay here as long as you need to, Riven. Can I call someone for you? Your parents? They must be worried.\"\n\n\"No thanks,\" I say quickly. \"My father is out of town on business. He usually calls me to check in. You can talk to him then.\"\n\nShe frowns for a second but nods. \"You're welcome to stay as long as you like.\"\n\n\"I will. Thanks for taking care of me, Mrs...?\" I trail off realizing that I don't even know their family name.\n\n\"Just call me June.\"\n\n\"Thank you, June,\" I say.\n\nI'm overwhelmed at her generosity, letting some stranger into her home. I could have easily been one of the others looking for the boy. How easy would it be to kill him? One swipe of a knife, a pillow over the face, a twist of a finger? They're so trusting, these people. Back home, getting within an arm's length of another person is virtually impossible, much less getting into someone else's home. It's astonishing that the boy has survived for so long.\n\nThe odds weren't in his favor, yet here he was, unhurt and obviously thriving... hidden in plain sight. And I'd found him quite by accident \u2013 this town hadn't been on my list. I'd just stopped here on my way to Wyoming and randomly decided to stay for a few days to recuperate after the last eversion. It had been a spur of the moment decision.\n\nI glance at Caden, chewing on his thumb and staring at me out of the corner of his eye. He seems to be just like all the other kids of this world, so oblivious to everything but their immediate sphere of existence. Watching him, I know that he has been well protected, but he is clearly unprepared.\n\nHe thinks he's just a normal boy. But I know better.\n\nHe has no idea about anything \u2013 no idea of who is after him or what's coming for him. I frown. So how has he survived? How has he been able to stay here undetected and in the dark about who he really is for this long?\n\nThere is only one answer that I can think of. It is one that chills me to my bones.\n\nSomeone has to be helping him.\n\nSomeone who knows that I would be coming.\nBLACKOUT\n\nCaden and his aunt insisted I stay with them until late Sunday afternoon. Despite June's protests, I took a taxi back to the cash-only cheap motel on the outskirts of town as soon as I could. Due to her expert care, my injury is healing well, albeit slowly. In my world, muscle and tissue would be repaired in minutes in a laser lab. Still, I'm surprised that after just two days, I can bear weight on it. I sigh, frustrated. It couldn't have come at a worse time. Despite racking my brain for alternatives, I am a sitting duck. Attempting to evert with any kind of physical weakness is a death sentence. Eversion doesn't just mean physical stress \u2013 any kind of strain that sends mixed messages to the brain could upset the timing and the result. And no one wants to end up inside out with a jump gone wrong.\n\nHauling myself out of bed, I clear my mind and perform a series of meditational exercises that send energy flowing through my body. Despite the hollow ache in my ribs, it feels good. I stretch each muscle carefully until my movements are fluid and limber, taking care with my ankle, then move into a series of simple calisthenics that has a fine sheen of sweat coating my skin when I'm finished. It's a process that I repeat every morning without fail, with the exception of Saturday. I frown, redoing the exercises once more, a compensation of sorts for the missed interval. Even impaired, I can take on a couple of Vectors, but probably no more than three. I have to be prepared for the worst.\n\nI unfold the leather case lying tucked inside the back of my bag. Shiny silver knives and an array of weapons greet me, and I finger one of their edges carefully. They've never failed me. Without glancing behind me, I flick two toward the back of the motel door and they lodge with thick precision into the wood of the narrow doorjamb. Not much of a target, but I shrug and retrieve the blades. I repeat the knife throws, managing to get both in the same incision points as before. Better.\n\nGrabbing the crutches I'd borrowed from June, I hobble to the door, swearing under my breath. Having to move this slowly is worse than the pain. I hitch a ride to school in the back of a pickup truck, and before I can lose my nerve, I grit my teeth and awkwardly shuffle my way up the stairs to the doors. I'd like nothing more than to not have to attend another day of high school now that I've found Caden, but I also don't want anything to happen to him, either. I still have that feeling of things not being quite right, and vigilance and caution are two things that have kept me alive all these years.\n\nSo another day of Horrow High it is.\n\nTrudging to my class, I realize that I don't know anything about Caden. The little I do know tells me that he is nothing like Cale. It confuses me. Still, what did I expect? They're not exactly the same people \u2013 made even more dissimilar by the whole nature-versus-nurture thing. But the truth is, I don't need to know anything about him. Why should I care? He's a target, and one that I need to get back as quickly as possible.\n\n\"Hey, Riven! How's the leg?\"\n\n\"Thanks for the general announcement,\" I growl sourly just as Caden walks alongside me with two of his football friends in tow. \"It's fine.\"\n\n\"Guys, this is Riven. New girl,\" Caden says to his friends with a wide grin. \"But be warned\u2013\"\n\n\"Hey, I'm Jake,\" a redheaded boy interrupts with a smile. \"I was there the other night with Cade when you trashed your bike...\" Jake trails off at the dark scowl on my face.\n\nCaden laughs out loud. \"As I was saying, just don't mention her riding skills or ask her about anything personal; she gets a little touchy about that. And she's not interested in making friends, so forget I introduced her and move along.\"\n\nI shoot him a withering glare just in time to see a willowy blonde swing her arms around Caden from the back. Her demeanor is not friendly, nor is the acid warning look she launches in my direction. My body tenses immediately, and already my brain is calculating the distance to exits and casualty ratios of the dozens of kids swirling around me. I force myself to relax.\n\nShe's a kid, not a threat.\n\nThe adrenaline seeps from my system as the girl tosses an icy smile in my direction, her designer white pants like a second skin and a pink shirt unbuttoned enough to show a lacy pink bra, leaving little to the imagination.\n\n\"Who's your friend?\" the blonde says to Caden, her tone dripping venom. My hand hovers over the blade wedged into my belt. No metal detectors in this school makes it a hell of a lot easier to deal with threats, unlike the public schools in New York, which had been an eye-opening experience. I'd received detention for a week because of a concealed knife in my boot. Forcing my hand to my side, I try to act normal.\n\n\"Hey Sadie, this is Riven. She's new,\" Caden says and turns to embrace Sadie, who jumps up to wrap her very long cheerleader legs around his waist. I want to laugh at her overt territorial display, but something inside tells me that this will probably not be the best thing to do. Still, I can't quite help myself, and the side of my lips twitches into a smirk. Sadie's eyes narrow and I bend my head, biting my lip to stifle my amusement.\n\n\"Nice name,\" she drawls after a minute, her tone indicating that my name is anything but nice. A cutting response rises to my lips, but provoking this girl won't accomplish much, other than to serve my own ego. And I need to keep a low profile.\n\n\"Thanks,\" I say instead and quicken my step. \"Catch you later, Caden.\"\n\n\"It's just Cade, remember?\"\n\nI shoot a hand in the air and keep walking. Cade, Caden, it makes no difference to me.\n\n\"Where'd that one get dragged in from, juvie rehab?\"\n\nSadie's scornful words reach me as I walk into the classroom, but I don't hear Caden's response. It wouldn't surprise me one bit if he said that I was a foreign transfer. This time, I can't hold back my laugh, and it comes until I feel tears running down my cheeks and the sides of my stomach ache from it. They couldn't imagine just how foreign I actually am. I am still snorting even when class begins and Mrs Taylor's eyes laser me with a quelling look.\n\n\"Class, please sit in your designated project groups. We will be working on them during the second half of class today.\"\n\nGroans mix in with the noisy screech of chairs as students move around, shuffling to other tables, and for the first time I look around the classroom, staring at faces with interest instead of my usual detached assessment of potential danger. I see them as people instead of targets or threats, and I am surprised by how young they all are. Not that I am much older, but truth is, I feel older.\n\nHarder.\n\nHalf of these kids haven't felt the sharp edge of hunger or had to fight for anything in their lives. They are plump, satisfied, and ignorant. Where I'm from, our training begins the minute we are born, and we face survival tests far worse than a quiz on Shakespeare before the age of five. Education is mandatory, but so are other things \u2013 physical education, weapons education, life education.\n\nI realize that I'm judging once more and give myself a shake. Be fair, I think. It's not their fault that they are the way they are, and have evolved in a different world under a different set of rules and circumstances. They are people, the same flesh and blood as I am.\n\nWell, maybe not Sadie, I think with a grin, she's pure venomous angst.\n\nThe ones in my group seem likable enough: a girl, Charisma, who is quiet but friendly; Caden, of course; and another jumpy thin boy, Philip, with an overbite and fingernails bitten to the quick. His head is already buried in his physics textbook. Leafing mentally through my slang file, he is what most in this world would call a nerd or a geek, but where I'm from, Philip would be a coveted asset. Someone with his skills would be selected in a heartbeat. My father would have loved him.\n\nCaden opens his books on the table and leans back in his chair, grinning at me. \"Hey, Crutches, let me know if you need any help.\" He's joking, of course, knowing what my response will be. I shoot him a look communicating exactly what I will do with my crutches if he offers to help me again. His grin widens and I can't help smiling back at him, the acid thought of my father melting away.\n\nI feel eyes lasering into my back, but I refuse to give their owner the satisfaction of a response. Sadie is a nuisance, nothing more, and truth be told, I'd rather pass the next few weeks in anonymity instead of some full-on teenage feud with a hormonally-challenged seventeen year-old who thinks I'm after her boyfriend.\n\nAt seventeen, where I come from, you're a full grown contributor, responsible for a heck of a lot more than picking out a prom dress or fighting over a boy. If you have disagreements with another, you duel it out. It's that simple. If you need intervention for bigger disputes or feuds, you get it in front of the Royal Tribunal. Here, in the capricious world of high school, it's an entirely different story. I sigh. Needless to say, I've had enough of this version of high school to last me a lifetime.\n\n\"So what's the verdict?\" a low voice says. Caden is staring at me with a raised eyebrow and a half-smile.\n\n\"What?\"\n\n\"Your analysis of us? Of me? Come on, be honest, what do you really think?\"\n\nIt is one of my few unguarded moments, and for a second, I see a flash of Cale's perceptiveness in him. My face composes itself into its normal emotionless mask, but Caden's knowing expression is so hauntingly familiar that it throws me. I feel an uncomfortable warmth seeping up into my neck and across my ear lobes. I don't like this feeling. It makes my response far sharper than it should be.\n\n\"You like being the center of attention, you're smart but lazy, you want to get out of this town as soon as you graduate, and your girlfriend is a bitch. How's that?\" The words snap like rubber bands to my lips, but as soon as I say them, I find myself wanting to take them back. My anger is directed at myself, not anyone else.\n\nCharisma is staring at me slack-jawed, and geek-boy has even raised his head from his book, looking nervously from Caden to me and back again as if expecting a full-on brawl right there in the middle of physics. But Caden's expression is measured, and he holds my challenging stare easily.\n\nA slow smile. \"Three out of four, so not bad.\"\n\nAgain, spoken so much like Cale. I know I shouldn't be surprised at the similarities, but nonetheless, I am. My teeth are ground together so tightly that my jaw aches. I don't want to feel anything for Caden other than as a means to an end. He isn't Cale.\n\n\"Your girlfriend isn't a bitch?\"\n\nA laugh. \"No, she is, but Sadie's harmless. Remind me to tell you about it later,\" Caden says just as Mrs Taylor raps on her desk. There won't be a \"later\" if I have anything to say about it. If our brief exchange is any indication of what the next few days will bring, I'll watch over him from a distance. I can't risk any rapport with him \u2013 my loyalty is to Cale, and this boy is nothing more than a target. I look away, pretending to listen to Mrs Taylor.\n\nNewton's laws of motion are a familiar subject, and while it's something that I learned long ago, that doesn't mean that I don't have to pay attention. I learned that lesson the hard way in Boston when I carelessly answered a question with an analysis worthy of a doctorate-level dissertation. They treated me like some kind of second-coming prodigy. Needless to say, I didn't stay at that school too much longer. Drawing attention tended to draw other bad things... way worse things.\n\nLike Vectors.\n\nI suppress an automatic shudder, and not even the workbook equations of motion can distract me. The Vectors are one of the most-feared and worst creations of my world. They are engineered creatures, made from human corpses... reanimated dead beings with one purpose. To hunt and to kill.\n\nOur technology is advanced, but we learned a hard lesson with artificial intelligence centuries before. While this world was embroiled in the French Revolution, my world was facing its worst crisis in history \u2013 the Tech War \u2013 a war that had devastated us to the point that topographical boundaries had been unrecognizable and continents reshaped. Whole oceans were destroyed. What had once been lush forests were now dusty plains of gray acid ash. Parts of my world are still black and oozing with toxic matter. But people survived, eventually reclaiming and rebuilding what they could.\n\nThe Vectors were a new kind of soldier, created after the laws were put in place against any kind of self-evolving robotic intelligence. Nanogen technology became the perfect combination of programmable robotics and human genetics, and the Vectors were bred to supplement human loss, to protect and defend those who survived. They had been the brilliantly sick creation of a madman.\n\nMy father.\n\n\"You OK?\" Caden whispers, distracting me. \"You look ill.\"\n\n\"I'm fine.\"\n\nBut I'm not fine. Everything inside of me feels like it's on fire. My ribs are splintering with alternating bands of hot and cold surging like rotating tides. There's a strange buzzing sensation in my head, and I can feel myself becoming light-headed and fuzzy with each passing second. What the hell is happening to me?\n\n\"Riven, can you hear me? Riven!\"\n\nCaden's face swims blurrily into my vision. And then it's Cale's. I reach toward him but he fades away, and I am left alone, terrified. There is nothing but darkness and the faces of the Vectors, inexorably closing in.\n\nWhen I awaken this time, I am lying in a sterile room with metal leads stuck to my chest and plastic tubing stretching across my face. White light stabs into my eyes and I lurch upright. Panicked, I pull against the tubes and wiring only to cause a frantic beeping. Someone in a white coat rushes into the room, and I shrink back, hands protectively curled in front of my chest.\n\n\"It's OK, Riven, it's OK. Calm down, you're safe. It's me, June.\" Caden's aunt is standing next to me, a soft smile on her face. \"We really need to stop meeting like this,\" she says as she checks the monitor and replaces the oxygen tubing under my nose.\n\n\"Where am I? What happened?\"\n\n\"You went into circulatory shock and you fainted.\" I eye the wires and she smiles again. \"They're just monitoring your heart rate. I took a look at your foot, and it's healing quickly, faster than I expected.\"\n\n\"I can't be here,\" I choke out as June uncaps a syringe and deftly injects it into my arm. \"What's that? I don't want any drugs!\"\n\n\"It's just a sedative. Riven\u2013\"\n\n\"No! You don't understand. I need to leave. They'll find me. They'll find all of you.\" My heart rate escalates with every breath, and concern crosses June's face.\n\n\"Calm down. You're not making any sense. Who'll find you? Your parents? Well, that's good. I meant to ask you. For some reason, none of the numbers listed on your file at school seem to work. We need to get in contact with them and let them know you've been admitted to the clinic.\"\n\n\"My dad is out of town.\" It's a programmed response.\n\n\"Cell phone?\"\n\n\"No. He calls me, remember?\" My mind is racing trying to assess the situation. Somehow, my body is failing. I know that something isn't right \u2013 I can feel it inside of me, the holes, and I clutch my chest. I need to get out of there. \"June, hospitals... I can't be here.\" I'm gasping for air with each word. \"Please. My mother... she died in a room just like this one... please!\"\n\nI am not even pretending at this point. White spots are exploding like clouds of mist behind my eyelids. All I can see are the memories I do have of my mother tied down in a white room with tubes embedded in every part of her, screaming her head off.\n\nWaiting to die.\n\nIt's all I can do not to rip out the ones attached to me, and then tears are running down my face, and I can't stop them, there's so much pain like everything is dying inside of me. Where did it all come from? What's happening? What is wrong with me?\n\nAnd then I am screaming, staring at the red dot on my arm. \"What did you do? What did you do to me? What kind of poison did you put in me? Get it out, get it out!\"\n\n\"Help me restrain her arms,\" I hear June say. \"Gentle with her.\"\n\nI fight like I have never fought before, as if I am fighting to keep my very last breath, scratching and scraping, kicking and punching. The full weight of a body collapses against me, and my arms thud, pinned to my sides against the bed. I am drowning in a sea of my tears, the salt of them covering my cheeks and my lips. My body stills and the world grows unnaturally quiet. Now I'm the one waiting to die.\n\nIn my mind, I see the Vectors. They're coming.\n\n\"Please,\" I whisper. \"Please, June. Don't let me die here.\"\n\n\"Don't worry, darling. You're safe now.\"\n\nSomething cool swirls around my veins, and then the world goes dark once more.\n\nVoices and colors fade in and out.\n\n\"Is she going to be OK?\" That one I recognize. It's Caden's. I want to tell him that I'm fine, but my lips won't cooperate. My bones feel like syrup, like there's nothing inside of the skin that's holding my body together. Darkness takes me again.\n\n\"... no consent for testing... can't draw blood...\"\n\n\"... levels stabilized...\"\n\n\"... take her home... haven't been able to locate her father...\" That last one is June's lilting voice.\n\nI sleep again.\n\nImages move along in an endless collage, bits of white and gray intermingling. Later on, there are bits of sky and swatches of green. The sound disappears. There's only the quiet of gentle voices murmuring around me, like the sound of rain.\n\nMy mind is quieter now, no longer so manic, allowing me to think and feel and process. A soft voice tells me that everything is going to be OK. Is it June or is it Caden? But I listen, and once more feel that sense of safety, of trust. And again, I know that I am breaking one of my own hard and fast rules.\n\nNever trust anyone. Especially the Otherworlders.\n\nBut do I really have a choice? Other than Caden and June, I've thought of them as nothing but a means to an end. Did I think of them as people? They aren't real to me. They're a parallel species that has nothing to do with me. But if trusting them means that maybe I can stay alive, then I'll have a sliver of a chance to get back home. A full-body internal scan will detect anything that is wrong with me in seconds.\n\nIf I can make it back.\n\nStrong arms are carrying me. My eyelids hurt but they open and close like heavy drapes. Within moments, I am back in a familiar room... the airy one with the fan and gauzy white curtains.\n\n\"Caden, don't stay too long. She needs to get some rest,\" June says as she props the pillows behind my head. I manage a weak smile. I can't even begin to express my gratitude, but something in her eyes tells me that she knows more than she's saying. Or is it just my constant sense of paranoia?\n\nCaden brushes the hair out of my face. His green eyes are soft and comforting.\n\n\"You're awake. How are you feeling?\" he asks.\n\n\"I've been better.\" My voice feels like I haven't used it in years and rubs against the inside of my throat like gritty sandpaper. Caden pours me a glass of water from the pitcher, and I sip gratefully.\n\n\"Has this ever happened to you before?\"\n\n\"No.\" Nothing like this has ever happened to me before. I've never been sick or fainted a day in my life. I wonder if it has to do with the pills or the injector or being here this long. I stare at Caden and then say the words that are playing on the tip of my tongue. \"Thanks, by the way. That's twice now you've saved me. I don't know if I like it.\"\n\n\"Why?\"\n\n\"I don't like owing people.\"\n\n\"You don't owe me anything, Riven. You needed help and I was in the right place at the right time, that's all.\"\n\nWe are quiet for a moment. The edge of the bed dips under Caden's weight as he sits next to me and rests his head on his hands propped on his knees. A silky lock of hair curls into his face.\n\n\"I like your hair long,\" I murmur. \"I mean, I like it short, too.\"\n\n\"You've never seen me with short hair.\" Caden's voice is quiet, but I can see his eyes narrow. I want to kick myself. In the next moment, I decide that I am done with conversation. All it will lead to is confusion, and questions and doubt when the time comes. Caden is staring at me, waiting for an explanation. I shrug and take a big gulp of water.\n\n\"You look like someone I know. He has short hair.\"\n\n\"You mean Cale?\"\n\nI choke and almost spit water all over the room as a wave of shocked coughing overcomes me. \"What did you say?\" I whisper after several painful seconds.\n\n\"Cale. You said his name while you were unconscious in the hospital. I think you thought I was him.\" He pauses, watching me carefully. \"Now it all makes sense.\"\n\nI am overwhelmed by my own stupid carelessness. What else did I say? Did I talk about what I was doing here? About who Caden really was? I can't process the questions fast enough as ten more pop up in their place. What have I done? I should have just risked it and everted the minute I'd found Caden, and dealt with the consequences later. He was the important one, and he was healthy enough. That was all that mattered.\n\nYou're thinking crazy, Riven, I tell myself. If you everted from here, without you Caden wouldn't have half a chance in the Outers before he got to the city. The Outers would have swallowed him whole.\n\nCaden's voice interrupts the ominous chaos of my thoughts. \"So, who is he?\"\n\n\"A friend,\" I say.\n\nCale is far more than that \u2013 he's my best friend, my brother, my liege \u2013 but I know that explaining any of it won't make much difference. Caden won't understand our politics, or any of the intricacies of who I am to Cale. I'm bound to Cale in a way that these people could never fathom... one bred of steel and blood and undying loyalty.\n\n\"In the hospital, it sounded like he was a lot more than that, more like a boyfriend.\"\n\n\"Not that it's any of your business, but no, he's my best friend.\" I cough and take another sip of water. \"Did I talk about anything else? You know, while I was in there?\"\n\n\"Only about your mother and how she died. I'm sorry.\"\n\nI frown but I do have a vague recollection of begging June to let me leave. \"It was a long time ago. She had a brain tumor.\" I pause, hesitating. \"Did I say anything else? About... Cale?\"\n\nCaden shakes his head, and my relief is so great I can feel my entire body relax in my next breath. But it is too soon. He is struggling with the lie, I can see it in his eyes. The dread fills me again.\n\n\"Tell me,\" I say. \"I won't get mad, I promise.\"\n\n\"You said, 'I'm sorry I failed you. Sorry I didn't come back right away.' Or something like that.\" A pause, and then Caden's green eyes meet and hold mine. \"Is he OK?\"\n\n\"No.\"\n\nMy voice is so quiet that Caden strains forward to hear it. Unthinkingly, I tuck the lock of hair back over his ear. Something in his eyes flares at my touch. I hardly notice what I've done, but it doesn't matter. Suddenly everything is so clear that it feels like I can see for days... the clarity that comes after a fever breaks. I take a deep breath and things become even clearer.\n\nCaden isn't Cale.\n\nI know what I have to do.\n\nMy sense of conviction is so strong that Caden shies away instinctively, a far cry from the warmth I'd felt from him just before. My mask is back, my chilling words matching the icy hardness he must see on my face. It is a reflex action almost, the sudden shape of my purpose here. I don't question the calmness that settles into me; in fact, it feels like old, comfortable clothing. It's as if everything inside of me has somehow been magically reset... who I am, my duty, my mission.\n\nAnd my mission is to secure the target.\n\nI shake my head slowly. \"I was wrong before. I haven't failed him. I've found what he needs, and I will get it to him.\" The words are hopping on my tongue, burning to get out. \"There is nothing, dead or alive, that will stop me.\"\nSECRETS\n\n\"Riven, you look wonderful. I can't believe that brace is off your ankle already!\"\n\nJune's face is welcoming and warm at the door, but I steel myself against it. I am not here to make friends or have any further doubts as to why I'm here. Still, I don't want to be rude when she opens the door wide and invites me inside.\n\n\"Good genes,\" I acknowledge with a small smile, hovering near the entrance. \"I can't stay, but wanted to come by to say thank you, and also to get my bike. I'm OK to ride now,\" I assure her hurriedly as her brows begin to knit together. \"I'm really alright. I can walk, run, everything's back to normal.\"\n\n\"I'm not comfortable with you getting on a motorcycle this soon. Especially one that was in an accident barely a week ago. Not to mention what else happened \u2013 the shock and fainting.\"\n\n\"June, I'm fine. It was only stress, and I've been taking better care of myself. Plus, Cade told me that Jake checked out the bike already,\" I interject hastily. \"Really, I've been riding that thing forever, and I'm safer on that than in anything on four wheels. And my ankle's fine, I swear.\"\n\nJune shoots me a skeptical look. \"Just let me have a quick look to be sure. It will make me feel better.\" Knowing that getting out of there will be a lot easier if I just let her look for herself, I nod and sit on the chair closest to me. Her fingers are warm against my skin as she gently feels along the bones and then twists my ankle to the left and right. \"Any pain?\"\n\nI shake my head. \"It's been fine for a few days now.\"\n\n\"Good genes,\" she says repeating my earlier words and then frowns. \"Maybe I was wrong about the torn ligament; could have been just a mild sprain that looked worse than it was.\" She checks my eyes, heart rate, and blood pressure. \"All good, too.\"\n\nI can see June second-guessing her own doubt. She's too good of a doctor to have been wrong, and she knows it. Quickly, I say, \"I'm just glad I'm better now. Crutches are a pain. Being the local fainting gimp is even worse.\"\n\nThe truth is, I feel better than I have in years. Ever. I can't explain it, but it's as if a switch has been turned on inside of me \u2013 my body feels limber, my brain crystal-clear. Maybe I'm finally getting used to this place.\n\nA smile. \"I still want you to take it easy for a couple more days. There's no swelling, but with this kind of recovery, I wouldn't want you to overdo it just because it looks and feels OK.\"\n\n\"Got it.\"\n\nShe's silent for a while, then says quietly, \"Amazing,\" her fingers still resting against my foot.\n\n\"June? Caden said that my keys were in the kitchen,\" I suggest helpfully.\n\nAs if in a trance, June blinks and stands, but I can see that her brain is still furiously ticking. \"Oh, yes, of course. Caden isn't here; he's at a fencing meet at the school, I think,\" she says while she's looking for the keys in the kitchen. \"But I think he would have put them in one of these drawers.\"\n\n\"Fencing?\"\n\nJune shrugs. \"Don't look at me. I like chess myself, but he has a natural affinity for it, and while I'm not a fan of any combat-weapon sport, he does seem to enjoy it. I've only been to a few of his meets, but he's pretty good. Or so I see.\" She grins. \"Have you seen the trophy shelves in his room?\"\n\nI shake my head. No. \"Fencing,\" I repeat softly to myself.\n\nThe sport itself is beautiful to watch, as elaborate as dancing, with elongated parries and delicate thrusts. I know it well because we are all trained in the art of most hand-to-hand combat techniques by the tender age of five, and all manner of weaponry by seven. Swords, bows, knives, spears, axes, and guns... everything you could conceivably use to dispatch an enemy. The sword has always been my favorite. Cale always favored the crossbow. For a second, the memory of one of our first training sessions together flashes through my brain, and June fidgeting through the drawers in the kitchen fades into the background.\n\nWe had been assigned to one another for formal training, and had to face a mock obstacle course with various threats. I'd just turned eight. Though I was small for my age, I was lightning-fast and held the advantage of having held a sword before the age of two. Already, I was at the top of my age group in any kind of martial arts training.\n\nWhen we were paired up for the test before the final assignments on specialized-weapon training were made, we automatically sized the other up. His shock of glossy brown hair made him look impish, but the expression on his face was boldly confident.\n\n\"You're small,\" he said, his voice matching the arrogant expression. \"Looks like I'll have to pick up the slack.\"\n\n\"How come you're not paired with someone your own level?\" I blurted out.\n\n\"Guess they think you need babysitting.\"\n\nI'd felt like slapping him. My scowl was fierce, but he'd just laughed in my face as if I were nothing more than a kitten defending its toy. I found this boy's arrogance to be so grating and his overconfidence so irritating that I vowed then and there to teach him a lesson.\n\n\"Try to keep up,\" I snarled, and took off just as the whistle blew, jumping over fences and scaling walls. I didn't even look behind me to see if he was keeping up, even though part of the test was to protect your partner at all times. I was too angry. Irrationally, I wanted to show this rude boy exactly what I could do.\n\nTwo hologen targets jumped up in front of me, one a wild jaguar and the other some kind of bird. I shot them with the rifle slung across my back, both easy hits. It was then that I heard the shout of pain, and in that brief second, the red clouding my brain cleared. The boy was hurt.\n\nI could go back and get him, or I could leave him.\n\nJust as I was deciding to press forward \u2013 let him tough it out on his own \u2013 something niggled at the base of my neck. I couldn't leave him, it wouldn't be right. Resigned, I turned around and backtracked, only to find him leaning against a tree calmly chewing on an apple.\n\nI frowned. \"I heard someone cry out.\"\n\n\"Yeah, it was a new kind of hologen. It made human noises. I killed it.\" His nonchalant words were almost enough to make me miss the blood dripping from his sleeve. Almost.\n\n\"You're hurt?\"\n\n\"That's mostly the thing's blood. But it did break my leg. I can't walk.\" He finished the apple, tossing the core near the body of the dead creature. \"You should go on, I'll tell the trainers that I tripped. You should finish.\"\n\nI could feel the guilt at leaving him earlier ripping through me like acid. \"No, it's my fault. I shouldn't have left. I broke the rules and you got hurt.\"\n\n\"We could default.\"\n\n\"No!\" I shouted. Defaulting meant quitting, and a strike that would go on your training record, even if something happened that was no fault of your own. Failure was not an option, not for any of us. If you couldn't defend yourself in the field, you were as good as dead. It was as simple as that. \"We'll figure something else out,\" I said confidently despite the hole of anxiety in my stomach.\n\n\"I guess I shouldn't have teased you.\" It was the smallest offer of truce and barely an apology, but I took it anyway. He had been hurt because of me... because I deserted him.\n\n\"I'm Riven,\" I said, sealing the fragile truce. \"Can you walk if you hold on to me?\"\n\n\"You're half my size; we'll never make it. Just leave me.\"\n\n\"Shut up,\" I told him. \"I'm stronger than I look.\"\n\nAnd so, with me half dragging, half pushing him along, we inched through the rest of the course. When we ran out of bullets in my rifle, we stood back-to-back, sword and crossbow in hand to take out our enemies. I admired the boy's skill with his weapons, switching fluidly from longbow to crossbow and back before I could blink. Anything that got too close, I fought off capably with my sword and, occasionally, a short knife.\n\nIn the end, it wasn't the most graceful completion of the course, but we came out together panting, winded, and utterly exhausted. The medical team rushed out of nowhere so quickly that I didn't have a chance to say anything more to the boy before he was hover-ported away, surrounded by a team of elite guards. It was surprising but I didn't think too much more about it.\n\nIt was only afterwards that I learned that my partner was the son of the Monarch of Neospes. A prince. Mortified, I kept waiting for the soldiers who would surely come to arrest me for my heedless words and thoughtless actions in the holo dome, but I needn't have worried. From that point on, the boy who I would come to know as Cale always asked for me, and only me, to partner with him.\n\nOne day, when I asked him why he'd always chosen me, he responded flippantly, \"Someone needs to protect the little people.\"\n\nBut later in a rare moment of unguarded honesty, Cale told me that it was because I'd come back for him without knowing who he was, even after his intentionally provoking words. He'd liked that I had never even considered giving up or leaving him behind. I remember telling him that quitting would never be an option for me. It wasn't then, and it would never be now.\n\nAnd just like back then, Cale knows that he can count on me. And that can't change, not now, not ever.\n\n\"Riven?\"\n\nI jerk out of my thoughts so quickly that my fingers are on the hilt of a knife tucked into a side pocket of my pants before I can exhale the breath hitched in my lungs. I relax, my brain belatedly recognizing June's voice and release my hot fingers.\n\n\"Sorry,\" I breathe. \"Homework on the brain.\"\n\n\"The hospital just paged me and I need to go,\" June says pulling on her jacket. Her face is apologetic. \"Your keys aren't in here; they must be on Caden's desk somewhere. Can you have a look and lock the door on your way out? Take the stairs just past the room you were in. Second door on the left. You take care of yourself, OK?\"\n\nShe's out the door before I can even nod. But I am too grateful to speak \u2013 she's just given me the opportunity I've been looking for. An engine turns over, and I wait until I see her car pulling out of the driveway before I make my way upstairs.\n\nPictures line the hallway walls in a variety of colors and shapes with faces and landscapes filling their matte frames. They weave a story of vibrant life as I climb the staircase. Some of the frames are new and some antique, with some photos in black and white, while others are in color. Despite the hodge-podge, there's a beautiful artistry and a love beneath it all connecting it together. It's breathtaking.\n\nIt's only seconds later that I realize that Caden isn't in a single one.\n\nPushing open the door to the first room, I tell myself that I'm not snooping, merely familiarizing myself with the layout, but I still feel uncomfortable anyway. This room is obviously June's room, with cherry furniture and a large four-poster bed covered with a hand-quilted floral bedspread. The room is airy and overlooks the street. A thickly bound book and a pair of glasses are resting on the bedside table. The room is feminine yet strong, just as she is.\n\nCrossing the few feet to the right side of the room, I perform a cursory search. I don't know what I'm looking for... clues, weapons, anything that will tell me who June really is, because I know without a doubt that she isn't Caden's aunt, if only because of the photos in the hallway.\n\nWhy is she helping him?\n\nA fluttery feeling tingles along the back of my neck, and I spin around in attack mode. But there's nothing there. I can't shake the feeling that has now sunk into my bones, that feeling of being watched. I wonder if half of it is my own imagination combined with the events of the last few days.\n\nI give myself a mental shake. The house is empty; I would know instantly if it weren't. Turning my attention back to the job at hand, I make my way to June's dresser, sliding my hands along the walls and along its backside. The drawers are filled with clothes, nothing exceptional. Something snatches my attention on the bedside table. The heavy gold letters on the book are like a neon warning.\n\nQuantum Mechanics: Intuition or Theory?\n\nThis time, I can't control the realization that rushes through me. June knows a lot more than she's letting on. How much does she know? Did she know about me?\n\nNo. I've been careful. Haven't I? The self-doubt crawls in, cold and relentless. I haven't really been myself lately, getting injured, fainting, and ending up in a hospital. Being all too careless. I could have let something unintentional slip over the past few weeks.\n\nNo, my inner voice argues. You are meticulous. She doesn't know anything other than what she knew anyway. Otherwise, why would she have left you here alone if she thought you were dangerous? There's no way.\n\nMollified somewhat, I run my finger along the edge of the book. Quantum mechanics isn't exactly bedtime reading material. Hefting it up, I flip open the cover, skimming through the first quarter, and almost drop it to the floor. Instead, I sink to the bed and hold the book carefully on my lap. In a cutout hidden in its pages, in a bed of soft chamois, lies an innocuous-looking gun, barely palm-sized. I know instantly that it is loaded and it is lethal.\n\nThere's a magazine of bullets in a slim brown box next to the gun in the book. I examine them carefully. Custom hollow-points, meant to shred the inside of a target. The blue markings on the side of the box indicate that there's some kind of modified burst mechanism within the bullet. These have been specifically designed to annihilate whatever or whoever they come into contact with. I place the bullets back into the case and replace the book.\n\nBefore closing the cover, I stare for a long second. The gun is new. From the minimal residue and shiny oiled insides, it's probably only been fired a few times. If June, who for all intents and purposes is a civilian, is anticipating this level of danger, then I've been miscalculating things all along.\n\nShe's expecting someone. Or something.\n\nAfter I've smoothed the bed and verified that everything is back in its place, I exit the room quickly. Across the hall is another room. This room, unlike the rooms I've been in, is completely sparse, with a single bed with a metal frame and a slim desk sitting under the window. It appears to be unused, but still, instinct propels my feet to cross over to the nearest closet door. Empty. I release the breath I've unconsciously been holding. Maybe it's a spare room that June hasn't gotten to yet. Still, something about its sparse efficiency strikes a familiar chord inside of me. The clock on the wall is five minutes fast, which is curious because it matches the time on my own watch. Time can be your own worst enemy. I, too, prefer to always be ahead of it. Retracing my footsteps, I close the door behind me and make my way to the next door. A decent-sized bathroom.\n\nThe third door is Caden's room.\n\nI know it instantly but something holds me back, my fingers hovering on the doorknob. Why am I so afraid to open it? Caden means nothing to me. A shiver sweeps through me, tingling along the undersides of my arms and up my neck, and a forgotten sense of anxiety hits me full-force. Instinctively, my fingers draw back.\n\nDrawing a shaky breath, I check the other doors down the hallway. One leads to another bathroom, the next to a linen closet, and the third to a stairway to a shadowy attic that's filled with old furniture. On the attic steps, I sit staring at Caden's door as if the devil himself is on the other side. Why is my imagination suddenly running wild? Why am I afraid? There's nothing there, nothing that can harm me.\n\n\"Get it together, Riv!\" I tell myself harshly. \"He's just a kid like everyone else here, nothing more than that. Now find your keys and let's go.\" I catch a glimpse of myself in the long mirror at the end of the hallway, and I have to laugh.\n\n\"Keep talking to yourself and you know where you'll end up,\" I say to the fierce-looking girl, and watch her pitiful attempt to stare me down. I step closer. \"The loony bin,\" I inform her threateningly and then roll my eyes as she shakes her head and grins at me.\n\nI wonder briefly if losing your mind is a part of the eversion sickness that afflicts about fifty percent of the people who attempt it, because not only am I talking to no one in particular, but my hair is sticking out like a prickly bush, and my light gray eyes have a slightly desperate quality to their shadowy dark-circled edges. I look like a homeless runaway.\n\nI tug on the second-hand Grateful Dead T-shirt, some obscure music band that I'd never heard of, and hike my jeans out of the beat-up black combat boots. Not much I can do about the hair, but I try anyway, fingering the choppy locks lying on either side of the blue-and-silver braid hanging to my shoulder. Better, but not much. I may feel like a million bucks inside, but I definitely don't look it. I shrug. Once I get back, I can work on my appearance. Right now, I have a job to do.\n\nFind my stupid keys.\n\nWith a hiss of exasperation, I stride to Caden's door and shove it open. The room is painted in rich intertwining hues of blue creating the illusion of being submerged underwater. A large bed occupies most of the space, leaving room for little else, but I expect that's the point. It's an undersea sanctuary of sorts, and one that is only truly appreciated lying down. Complete immersion. It's beautiful and serene, not at all what I'd expected.\n\nAs I am crossing over to Caden's desk, my brain registers other details, like the shelves in the window alcove above it, covered in trophies. The majority of them are for fencing, but some, again not surprisingly, are for archery. A small sound escapes my lips, half gasp, half cry of some sort, and a knot immediately forms in my belly. With the bow, Cale had been an expert marksman. It shouldn't be so strange that they have so much in common, given what they are, but the similarities are still overwhelming.\n\nA tiny amateurish landscape painting above the bed catches my eye, and I lean against the mattress for a closer look. It is all I can do not to fall backwards as my weight dips into the bed in a very unnatural way, as if I am on some kind of strange floating device. Instead, I spring backward to compensate and bang my still-healing ankle into the desk chair next to the bed.\n\n\"Mother of...\" I mutter, launching the offending chair across the room, as a cloud of pain threatens to suffocate me. \"Ouch!\"\n\n\"I'd hate to know what that poor chair did to deserve such treatment.\"\n\nI blink the stars in my vision away. Caden is leaning nonchalantly against the door, his mouth twisted in a grin, and shaking his head in mock consternation.\n\n\"Hey,\" I blurt out. \"Sorry, June told me to look for my keys up here. She couldn't find them where you said in the kitchen. I hope you don't mind.\" The words blend together in a rush, and I'm not entirely sure why they sound a trifle defensive. I can't believe I hadn't heard him come in. \"She told me to come up.\"\n\n\"Yeah, I forgot that I'd moved them. They're in my desk drawer. In the back.\"\n\n\"What?\" Why would he have put my keys in there? I know I'm frowning.\n\nA shrug as he walks toward me. \"I thought you were going to do something stupid like try to ride right after the accident, so I hid them.\"\n\n\"I wasn't\u2013\"\n\n\"Sure,\" Caden says, his grin widening, and reaches around me. I feel my entire body freeze as his arms graze against mine, and suddenly I am holding my breath. Every second feels elongated as the smell of the sweat on his skin from his fencing meet seeps into my nostrils. He smells so much like Cale that my knees buckle... but that's impossible. There's no conceivable way that they should even smell alike. Is there? My confusion must be apparent because Cale \u2013 I mean Caden \u2013 grasps my arms.\n\n\"Riven? What's wrong? You're staring at me like I'm a ghost.\"\n\n\"I need to sit,\" I rasp, ignoring the keychain he's holding in one hand. \"Not on your bed,\" I say hastily. \"Something's wrong with it. It's... broken or something. Soggy.\"\n\nCaden's laugh rings through the room. \"It's just a waterbed. They're supposed to be soggy.\"\n\n\"A water what?\" The thought is inconceivable to me... a bed with water in it. When I think of the scarcity and the high cost of water where I come from, the thought of the overindulgence of Caden's bed makes me physically sick. I shake my head to cover my discomfort. \"I don't get it. Why don't you have a real bed?\"\n\n\"It is a real bed. Don't you know what a waterbed is?\" I shake my head, still mute. \"They're pretty common. I like the feel of it, and it's good for my back. Something about it is calming, and when I lie on it, in the silence, I really feel like I'm in the middle of the ocean. Come on, try it.\"\n\n\"No.\"\n\nBut I have to admit that I am intrigued. The whole notion of the sea and the ocean is as foreign to me as my entire existence probably is to Caden. He reads my hesitation \u2013 and my curiosity \u2013 easily.\n\n\"Here,\" he says, and turns my shoulders so that my back is facing the bed. \"Don't jerk down; just sit gently. Good. Now lie back.\"\n\nI comply until the top of the bed is literally cupping my entire body. \"It feels so weird,\" I say.\n\n\"One sec, check this out.\"\n\nI barely notice when Caden pulls the shades over his windows and presses a switch on a light in one corner of the room, so taken I am with the gently sloshing motion of the waterbed. But in the next moment, I'm transported to another world as white bands of light radiate against the blue mosaic of the walls, and the deep sound of marine life thrums into my ears from a box on the bedside table. I can't even speak, far less breathe, when I feel Caden lie on his back next to me, the movement from his weight sending a slow, rolling wave into my right side.\n\n\"This is...\"\n\nBut I can't find the right words for the magnitude of the feelings inside of me. I have never ever seen a real ocean other than in pictures, and this is as close to that as I have ever gotten, even if it is just an illusion of light and sense.\n\n\"Incredible, right?\"\n\n\"Amazing,\" I whisper in a childlike voice. \"Does the ocean really look like this?\"\n\n\"In the right spot, if the sun is shining down through the water, this comes pretty close. I think I've loved the ocean ever since I can remember,\" Caden says in a quiet voice. \"I don't remember much about when I was really little, but I do remember my mom taking me to this seaside village when I was eight years old, and she couldn't get me out of the water, even when it was so dark that I couldn't see two feet in front of me. Back then, I had to have my bedroom painted blue, too.\" Caden laughs, a sound halfway between humor and pain.\n\n\"Your mom? Is she here?\"\n\nCaden turns to face me, a shadow crossing his features, and shakes his head slowly. \"No. She died.\"\n\n\"How?\"\n\n\"When I was seven. Seizure, they said.\" His mouth twists. \"Some kind of brain or nervous system infection, but the doctors weren't really sure.\"\n\n\"I'm sorry.\"\n\n\"Thanks. It's OK. I miss her but it was a long time ago.\"\n\nI turn back to stare at the ceiling, the light and sound doing nothing to dissipate the sudden weight in my chest, but I remain silent. There's nothing I can say \u2013 death is a natural part of life for me, but knowing that Caden's mother had died from eversion sickness leaves me cold. I couldn't imagine how painful it would have been, or how hard it would have been for Caden to watch his mother die.\n\nCale's mother.\n\nA shiver runs through me, and warm fingers slide against my wrist. The shiver deepens. The waterbed shifts, rolling me upward as Caden turns on his side to face me. I can feel him staring at me, but I keep my eyes glued to the ceiling. His fingers skim downward to cover my closed fist in his hand.\n\nI can't move. My entire body is rigid at the light touch.\n\n\"Who are you really, Riven?\" he whispers, his right hand shadowing the blue swirls for a second before lifting to move the braid out of my face. He holds it for a second, studying it before releasing it. My breath catches. The sheer force of him imprisons me, as his fingers trail down my face, turning my chin toward his. \"You seem so tough on the outside, but you're not. Not really.\"\n\nMy eyes meet his. They are warm but unreadable. His thumb stirs against my temple.\n\n\"You don't know anything about me.\" The words are sticky on my tongue, clumsy. For some reason, I feel inexplicably awkward.\n\n\"I know you're not like other girls, but I know you aren't as hard as you pretend to be,\" Caden says, propping himself up on one shoulder and cupping the right side of my face in his palm. Caden's eyes are liquid like the imaginary water wonderland surrounding us, his irises mirroring shades of hazy blue. They are mesmerizing. His head bends toward mine, and all the breath steals out of me.\n\n\"I can't stop thinking about you,\" he murmurs. \"You're so different.\" His words slice through me like ice shards as I pull away. What the hell am I doing? I am different... more different than he knows. I jerk sideways and upwards, causing the bed to undulate violently, and wrench my hand out of his.\n\nHe's a mark, for heaven's sake. A mark!\n\n\"What's wrong? You OK?\" Caden asks quickly. An embarrassed look flits across his face for a second, but it's gone as quickly as it came. He hasn't done anything wrong, and I hadn't done anything to dissuade him. I'd been idiotic to ignore the obvious signals \u2013 the bed, the lights, his gentle touch \u2013 but my senses had been muddled by the magic of the ambient lights and sounds.\n\nFor the hundredth time since I've been here, I curse myself, but the truth is, I'm far better at fighting than I am at flirting... or clearly, even recognizing it. I glare at him.\n\n\"I'm fine,\" I say, snapping the words through my teeth, struggling to compose myself as self-disgust rages through me \u2013 I'd been stupid to let myself go like that. But my self-loathing still boils over. It's poisoning my throat, the inside of my eyes, and I want to scream. My fingers curl into fists, but my voice when I speak is calm. It is inflectionless, emotionless.\n\n\"You are right about one thing. I'm not like other girls.\" I meet his gaze and hold it ruthlessly until his drops away. I grab my keys off his desk and walk to the door, glancing once more over my shoulder. \"I'm worse. Don't for a second delude yourself otherwise.\"\nPREPARATION\n\nIt is a tide of moving bodies, all flowing toward the door. Friday afternoon, and they all can't escape the confines of the classroom fast enough. I don't know why I even bother to continue going to Horrow anymore. I've already set the plan in motion \u2013 we're heading for Denver this weekend to see some play that June had gotten Caden tickets for. The timing will be perfect, and I need him to be willing, at least until we evert. There's really no reason to be at school, but I tell myself that it is for Caden's own safety.\n\nThe truth is that I'm enjoying high school for the first time in months, and in particular, this physics class. Something about Mrs Taylor's no-nonsense confidence reminds me of my teachers back home. Not surprisingly, given who my father is, physics has always been one of my strong suits. I like this class even if it is rudimentary.\n\n\"Riven, can you stay back a minute, please?\" Mrs Taylor asks just as I walk past her desk.\n\nI nod to Caden who's walking ahead of me. \"I'll meet you in the quad.\"\n\nSince the other day in his room, neither of us has spoken about what happened. But sometimes, I see an odd look in his eyes whenever he thinks I'm not looking, and he is quick to conceal it when I do. I don't know what to make of it, but it's not like it has any bearing on the job I'm there to do.\n\nOne thing I've learned about high school here is that it is a roiling mass of boys, girls, frenemies, and insta-crushes... in love one day and at loggerheads the next. I didn't expect Caden to be exempt \u2013 this was his world, after all \u2013 but for my part, what had happened was already forgotten. I rule my emotions. They do not rule me.\n\n\"Make that the gym,\" Caden tells me, hefting a large bag with his fencing gear. \"I have a meet, remember?\"\n\n\"OK, I'll come by when I'm done.\" I nod again and walk over to Mrs Taylor with a sense of foreboding born of following years of pure gut instinct. Did I repeat the mistake I'd made in Boston? Said something that is way beyond my supposed educational level? Written about some theory that doesn't already exist in this universe?\n\n\"Yes, Mrs Taylor?\"\n\nShe glances up, her eyes as dark and piercing as ever over her wire-rimmed glasses. \"Sit down, please. I want to discuss the last quiz.\"\n\n\"I had help,\" I blurt out before she can say anything else, but she stares at me with those obsidian eyes until I sit down. Stupidly, I realize that insinuating that I'd had help for a quiz meant that I'd been cheating. Still, Mrs Taylor doesn't say anything, and even with all my training I find that I can't read her at all, and my palms are clammy with sweat. She shuffles through the pile of papers and moves mine to the top. Even from where I am sitting, the huge circled letter \"D\" is glaring. D? That couldn't possibly be right. But still, I couldn't have done better if I'd planned it. I can't help the smile that sweeps across my face.\n\nMrs Taylor glares at me over her glasses. \"Obviously, you didn't have help,\" she remarks, her sarcasm stinging like a wet slap. \"And this is no laughing matter, young lady. May I remind you that if you fail my class, you will have to take it again in summer school?\"\n\nI almost laugh out loud. I definitely won't be around by summer. I compose myself. \"I'm sorry, Mrs Taylor. I will try to do better.\" I'm about to rise when I realize that Mrs Taylor hasn't quite finished with me.\n\n\"The thing that confuses me, Riven, is that your transcripts from previous schools are more than satisfactory, and you also seem to have an excellent grasp of the material during class-time and in discussion group, both of which suggest to me that you either weren't prepared for the quiz or, more likely, that you deliberately answered incorrectly.\"\n\nI'm at a loss for words. \"I wasn't prepared,\" I begin but the look on her face freezes any more lies from leaving my lips. My ploy, it seems, has drawn more attention than if I'd aced the test. Squirming inside at my gauche stupidity, I wait for her to continue.\n\n\"I also see from your transcripts from your last five schools that you have moved around quite a bit, more than usual for a girl your age.\"\n\n\"My father's job requires him to travel.\"\n\n\"Seems excessive. What field, if I may ask?\" Mrs. Taylor's mild expression suggests that she is merely curious, but I take nothing for granted, especially if it is something that can compromise my safety. Or Caden's.\n\nTrust no one. They were the last words that Cale said to me.\n\nI shrug and smile. \"Sales. He doesn't really talk about it.\" My smile turns calculating. \"Kind of like the mob.\" But Mrs Taylor doesn't take the bait, and instead regards me with an unreadable smile of her own. Something uncomfortable slides along my spine; apart from June, she's the first person to make me uneasy the whole time I've been looking for Caden, and I don't like the feeling at all. \"Can I go now?\" I say, more testily than I'd intended.\n\n\"In a minute. I want to ask you about one more thing. Your discussion group's project is the law of universal gravitation, correct?\"\n\n\"Yes.\" The uncomfortable feeling digging into my spine spreads its fingers along my ribs and across my chest. It's Boston all over again. I can see it in Mrs Taylor's slightly fixated expression.\n\n\"Mr Perkins... Philip,\" she amends at my blank face, \"your group partner, mentioned the other day that you had an interesting contention regarding the laws of gravity.\"\n\nMy mind is racing now, trying to recall every bit of the offhand discussion I'd had with Philip. Bored out of my mind during one of the group sessions, I'd wanted to have a little fun, poking holes into Philip's vast amount of book knowledge and his theories. What's to say that this scenario couldn't exist? Or what about this principle? Have you ever thought about if this could happen? And the killer, what about sub-quantum gravitational distortion? Little did I know that he would have gone back to Mrs Taylor. I grit my teeth to keep from kicking myself.\n\nStupid, stupid, stupid!\n\nI paste a vacuous look on my face and twirl a strand of hair around my finger just as I'd seen Sadie doing earlier. It makes me sick to my stomach to be imitating someone that vapid, but I grit my teeth and twirl as if my life depends on it.\n\n\"Philip,\" I repeat in what I hope is a dreamy voice. \"I think he really likes me. I was only trying to impress him, Mrs Taylor. The thing is, I don't know the first thing about gravity except what they say on that television show, Star Trek. That's where I got the ideas. Did he say something bad about me?\"\n\nBy the end of my mini-tirade, my voice has degenerated into an irritating whine. I am sickened at the empty-headed sound of it, but know that I have no choice. Hopefully, Mrs Taylor will believe me, but the truth is I have no idea about Star Trek other than a couple reruns I'd seen at a motel in Philadelphia, which I'd thought hilarious. I can only hope that my impersonation of a vacuous valley girl will work.\n\n\"Which episode?\" she asks without batting an eye.\n\n\"I think it was called 'Gravity,' it was about some kind of gravitational distortion.\" Mrs Taylor's eyes are relentless but I force myself to look as clueless as possible. My relief is palpable when I sense rather than see her shoulders relax and her body tilts away from me.\n\n\"Sometimes the writers of those television shows deserve more credit than they're given,\" she says after a long moment.\n\n\"I wouldn't know,\" I say. \"Most of the time I have help... even at the other schools. People tend to feel sorry for me. Boys, in particular.\" Something tells me I'm pushing it, but I can't seem to stop the excessive overcompensation for my slip with Philip, even though I'm obviously in the clear.\n\nThe thoughtfulness in Mrs Taylor's eyes wanes to actual distaste, and I squirm in my seat. In a different world, I'm sure she and I could have had a scintillating conversation about sub-quantum theory and gravitational distortion.\n\nIt is the reason I am even able to come here, after all.\n\nI stare at the floor chewing on my lip until Mrs Taylor says briskly, \"Well, thanks for clearing that up. Star Trek aside, I will expect you to perform better on the next quiz. And try to take on your share of the work, will you, Riven?\"\n\nI'm almost home free, but for some reason, I stop at the door. Even though it shouldn't matter, it bothers me that she thinks I'm some sort of vapid idiot who would use others to get ahead. Failing is just not a part of who I am.\n\n\"I'm sorry. I'll do better next time. I just want you to know that this isn't who I am. I pull my own weight and I don't cheat.\"\n\nAware that I'm babbling for no reason, I'm already out the door, so I nearly miss the speculative glance she sends in my direction, but I've had enough interrogation for one day. Mrs Taylor will be no more than a distant memory in a few days. Maybe as a goodbye present, I'll leave her a paper on sub-quantum string theory and its practical application to move between universes. Then again, altering the course of history is a big no-no, as in strictly \u2013 we're talking punishable by imprisonment \u2013 forbidden.\n\nOutside, the day has waned to a cool, clear evening. I check my watch. Caden's meet will be in full swing... and full of more people. I have the biggest urge to race back to my motel room and lay on the bed in the dark for a while where it's quiet and I am alone, and where I can think. Instead, I sit on a nearby bench and close my eyes just for a moment.\n\nAll of this interaction is tiresome. Remembering what to say and what not to say takes a huge toll after a while, and I'm mentally exhausted, especially after the confrontation with Mrs Taylor. Before, I'd shift in and out, looking for Caden and then move on. Now that I've found him, coupled with my hindrance of an injury, I've had more interaction with these people than I'd ever intended. And it's literally draining.\n\nPlus, too much contact means bad things could happen. It means that my presence could unknowingly set something into motion... a disturbance in the natural course of events. It means that other people \u2013 not just the Vectors \u2013 could find me, but also the Guardians, who monitor such disturbances.\n\nI've never met a Guardian, but Cale's father told me that they were there to make sure that people on both sides stayed where they were supposed to be. For centuries, the Guardians have been an elite group bound to the same code on both sides, preventing people from shifting, under the Laws of Eversion after the Great Infection of 1927. They answer only to the Faction, a trio of leaders supposedly older than the monarchy of my world.\n\nYou evert, you die. It's as simple as that. If the environmental differences don't get you, the Guardians will.\n\nOnly with Caden, the Guardians had failed. Until recently, everyone thought that Caden was dead. But obviously, he isn't dead... he is very much alive, a secret that Cale only revealed to me a few months ago when he mysteriously became sick. So, somehow, Caden has managed to outwit them and survive all this time. I can't quite shake the feeling that there's something more, something I am missing that's right in front of my face... something essential to his survival here.\n\nHe told me that his mother had died here, from a seizure. She probably had the pills that my father had given her, but they didn't help. My father warned that the pills with their brain stabilization agents wouldn't work for everyone, and everting would only put more pressure on the body's central nervous system. But the plain truth is that we don't belong here, and the universe has its own way of righting wrongs, of fixing inconsistencies. Her seizure was just that... nature's way of dealing with cheaters.\n\nI take in a few deep breaths and complete a set of mental exercises to clear my head. A quick glance at my watch suggests that the meet should nearly be over, so I start to make my way across the quad. A part of me doesn't want to watch Caden fence \u2013 I don't want to see what a natural he is, just like Cale.\n\nCale.\n\nFor a second, I wonder how he's doing. Whether he's surviving in what has become a sea of snakes and traitors. They won't kill him, that I am sure of, because they need his name too much to control the people of Neospes; loyalty to the monarchy was too hard-won to be usurped by a single coup. We were too careful, too suspicious of sudden changes.\n\nPeople trusted Cale's family. They trusted his father, and now he was dead... assassinated in cold blood by his half-brother, a bloodthirsty man greedy for power. Without a doubt, I know that Cale was next. His life is collateral for the moment... collateral for support. His uncle will keep him alive for as long as it suits his purpose. I have to trust that he is somehow holding on; otherwise, everything I'm doing will be for nothing.\n\nFear for Cale's safety clouds my mind so much that I almost crash into a group of kids standing in a shadowy corner near the gym.\n\n\"Watch where you're going!\" shouts a slurred voice. A foul breath blows into my face, and I almost gag.\n\n\"Sorry,\" I say, and then belatedly recognize one of the faces in the crowd: Charisma, the other girl from my physics discussion group. The slurred voice belongs to a dark-haired guy she's leaning against... the one with the foul breath. He's obviously drunk or high on something, and she looks dazed but doesn't say anything.\n\n\"Hey, Charisma,\" I say, but she won't even meet my eyes, as if she's staring right through me. Something about the way the guy's arm is wrapped around her shoulders rubs me the wrong way and I hesitate.\n\nIt's not your problem, my inner voice hisses. None of these people, other than Caden, are. Keep walking.\n\nI listen, take two steps, and halt. Even though I've only had a few classes with Charisma, she's grown on me with her upbeat personality and her willingness to help others. She's one of those types of people I wish I could be more like \u2013 selfless and caring \u2013 but I am so far beyond that person, it's not even funny. Hardness and cynicism drives me. With so much loss in my world, allowing myself to care about anything other than my own survival is a death sentence. I guess a part of me feels drawn to Charisma for that reason. She seems untouched by anything ugly.\n\n\"Hey, you OK?\" I ask her.\n\n\"She's fine,\" the boy says, pulling her away from me in the opposite direction.\n\n\"I wasn't talking to you,\" I say to him, and grab Charisma's shoulders so that she's facing me. Her eyes are dilated, and she's looking at me as if she's trying to focus but can't. \"Charisma, are you OK?\"\n\n\"I told you she was fine!\" the boy snaps, pushing me backward with one hand. My brain registers two things in immediate succession. One, Charisma is drugged, and two, this boy is lucid enough to shove me backward. My body kicks into battle mode, and everything slows to the point where I can sense the movements of his friends behind me.\n\n\"She's not fine, and I am going to take her home. Back off; I don't want to hurt you,\" I say quietly. I figure I should prepare him for what's about to happen.\n\n\"You and what army?\" he jeers in a loud voice. Instantly, he has the attention of everyone within ten feet of us. \"Look, guys, we have a late addition to the party. Get her a drink before she hurts herself.\"\n\nHe laughs, and his friends join in. Someone thrusts a cup in my face, and even without tasting it, I know that there's something wrong with it. I can smell it. My eyes narrow and I bat the cup away with the back of one hand.\n\n\"You guys don't go here, do you?\" More slurred laughter. They must have come for the meet and then decided to take advantage of girls while they were at their drink fest. \"Don't you know it's a crime to drug people's drinks?\"\n\n\"Lookit, we got ourselves a deputy,\" one of the boys giggles. \"You gonna arrest us?\"\n\n\"Arrest me, arrest me, Ociffer. I'm underage!\" another says.\n\nI glare him into silence. Where I come from, there's no drinking age. Consuming spirits is a rite of passage, and considering it's cheaper and more accessible than water, people don't make that much of a big deal over it. It's mostly consumed in toasts and celebrations. And frankly, people are too busy to risk the effects on their day-to-day responsibilities.\n\n\"I don't want any trouble. I just want Charisma. Just pass her over, and you can go back to getting yourselves drunk.\"\n\n\"Charisma wants to stay,\" the dark-haired boy says and turns to her. \"Don't you, baby?\"\n\n\"Mm hmm...\" Charisma murmurs incoherently. A line of drool has made its way down her chin, and she's starting to teeter on her feet. The boy glares at me with a vile expression in his eyes and then kisses her while I am watching, his tongue slithering over her mouth. I don't flinch, not even when he grabs the front of her chest. \"You're my girl, aren't you baby?\"\n\n\"Touch her again, and it will be the last thing you do.\"\n\n\"You mean like this\u2013\"\n\nI break his fingers with a single flex of my own before he can even touch the front of her dress, and then I'm on the move, spinning backward and knocking two of the guys behind me off their feet. The fourth boy takes one look at me and flees in the opposite direction.\n\nI turn back to where Charisma is still standing near the dark-haired boy, who is screaming on his knees and clutching his mauled fingers. With his uninjured hand, he removes a switchblade from his pocket and brandishes it, weaving unsteadily to his feet. Out of the corner of my eye, I notice that Charisma has teetered her way to a tree and has slumped down against its trunk. At least she's out of harm's way, but I know that I don't have a lot of time. She can't fall asleep before I can get her some help.\n\n\"You're going to be sorry,\" the boy snarls, pointing the knife at me.\n\nThis time, I can't stop from laughing. \"Someone once told me that if you point a knife, you'd better be prepared to use it,\" I inform him. His answer is to swipe at my face, an attack that I dodge easily. \"You should know that where I come from, I graduated the top of my class in hand-to-hand combat.\"\n\n\"What are you? Some kind of army grunt?\"\n\n\"Something like that. Bet you're wishing that you'd just let her go when I'd asked you, right?\" I know my lack of fear must be aggravating him, but honestly, it's like fighting an uncoordinated toddler. Given the odds, I could quite conceivably fight him blindfolded.\n\n\"Shut up. Who has the knife, anyway?\" he taunts lunging blindly at me.\n\nI spin again and clip the knife out of his hand with the heel of my boot so that it flies upward and lands in my own fist. \"This knife?\"\n\n\"Doesn't matter. I don't need it,\" he says with forced bravado. His eyes dart to the motionless forms of his friends. I can see the fear in his eyes and a dawning understanding of what he has gotten himself into, underscored by sheer disbelief that a girl is somehow getting the best of all of them. It is the same fear that makes him charge toward me in a football-type tackle.\n\nI dance out of the way and laugh again. I'm exhilarated. It's the first real combat exercise I've had in weeks, this coming from someone who typically trains three hours a day in a rigorous simulation mode and then another hour in actual combat. I should be sluggish, but I'm wired. Ever since the clinic, it's as if I can feel the neurons firing inside of me, getting stronger. And now, my body feels wired, like it's plugged in to a giant electrical outlet, every move charged with lethal fire. I am invincible.\n\nMy last move has brought me near to where Charisma is sitting, and I notice that her head is slumping forward onto her knees. She's nearly unconscious. A surge of anger rips through me and I advance on the boy. His eyes widen because now I am no longer laughing. My face is dead, emotionless. It is a look that has been partially responsible for the rank I hold in my own world.\n\n\"You like to take advantage of defenseless girls?\" His head snaps back as my fist makes contact with his right cheek. It's barely a touch, but he stumbles backward. \"You put something in their drinks, and then what do you do? Pretend to care about them? Then you hurt them?\"\n\nEach word is a staccato of fury. Fury at what girls here had to put up with over and over again. I've seen it at almost every school I've been to, and until now, I'd always walked away, telling myself that there was nothing I could do.\n\nWhere I come from, girls \u2013 women \u2013 know how to defend themselves from everything and everyone: human, animal, or machine. Drugged or not, any girl from my world would have had this guy, or one three times his size, on his backside before he could even lay a finger on her.\n\nIn this world, in neighborhood high schools, others like this boy prey on innocent girls, and more often than not get away with it because the girls are too ashamed or humiliated or aren't able to remember to do anything about it. It sickens me. Drugging another person in my world for something as revolting as sexual gratification is an offense punishable by exile \u2013 a fate more feared than death. Let's just say it doesn't happen too often. Exile is not a gentle end.\n\nSomeone needs to teach this boy a lesson. For Charisma, I'd be that person.\n\nI grab the boy by the front of his shirt and pull him close to me. He's a fair head taller than I am, but I am practically holding him off the ground. I press the butt of his knife against his crotch so deeply that I can see the water spring into his eyes. My voice is a low snarl. \"I ever see you near her, I will end you. Got it?\"\n\nWithout waiting for any acknowledgement, I spear my knee into his groin, feeling the immediate grunt radiating up through his entire body as he collapses against me, crying. I shove him away, a whimper from Charisma drawing my attention. The boy is curled into a fetal ball on the ground, but I still send his knife spinning behind me without a backward glance. I know without looking that it thumps right into the sliver of space on the ground between his stomach and his thighs. The sudden sour odor of urine permeates the night air.\n\nI lift Charisma easily. \"It's OK; you're safe now,\" I tell her. \"But you need to stay awake for just a few more minutes, OK?\"\n\n\"Mmm... OK.\"\n\nAfter I file a report with school security with a condensed version of the events and accompany Charisma in the ambulance to the local hospital, where she will stay overnight \u2013 apparently, the boy had used some kind of hypnotic sedative in an excessive amount \u2013 I catch a cab back to Horrow. But the parking lot is empty, and it looks like I've missed the whole meet, and Caden, too.\n\nA sense of exhaustion overcomes me, and I rest my head against the handlebars of my bike. I want to leave this place as fast as I can. Everything about it unnerves me. I want to go back to where I belong, where I feel whole. Here, I am starting to feel broken, the natural result of living in a broken world. Although they have more landmass, water, and people than we do, I have no doubt that this world is far more lost than mine.\n\nGritting my teeth, I rev the bike with one thing on my mind. Come hell or high water, we are leaving tomorrow. With a sense of rejuvenated intent, I ride to Caden's house. I don't let the fact that his car isn't in the driveway or that there aren't lights on in the house deter my new sense of purpose. I'll wait. Throwing a jacket across my shoulders, I make my way to the front porch, but there's already someone there.\n\nMy heart plummets to my stomach in a free fall that is magnified by the fact that time has slowed to abnormal proportions. My blood thunders in my ears like a solid force.\n\n\"Shae,\" I breathe.\n\nMy sister. My family. My enemy.\nCONFLICT ARISING\n\n\"I knew they'd send you sooner or later.\"\n\nThe breath that leaves my lips in response to the husky familiarity of her voice is deflating and harsh, taking with it every bone in my body.\n\n\"No one... sent me. I came alone,\" I manage in a shaky voice.\n\nShae looks more or less the same as when I saw her last, right before she caught me off-guard, armed with a double electro-rod, except that there's an oozing red gash across her face. Hair in blond dreadlocks, tanned face, if thinner, and eyes the color of a glacial sky. Those cold eyes were the last thing I'd seen before she'd everted.\n\nSeeing her now is like being dunked in a bucket of ice until my entire body feels like it's going to peel out of my skin. I want to run to her so badly it hurts, but underneath it all her betrayal is as fresh as it was thirteen years before, and the pain just as sharp. She left me with no regrets and no explanations. The monarchy had branded her a traitor, and I had to live with her shame until I built myself into something large and powerful enough to eclipse it.\n\nI hate her. That isn't going to change. Not now, not ever.\n\n\"So you're the one helping Caden,\" I say. \"I should have guessed. I learned everything I know about covering up the marks of eversion from you.\"\n\n\"And yet you found me.\"\n\nI laugh, a hollow sound made harsh with a coil of emotions I can't begin to unravel. \"It wasn't easy to track you, trust me. You were careful, I'll give you that... everting and then traveling by their transportation. Smart. But my coming here was just pure luck.\"\n\n\"Luck,\" Shae repeats, a small frown creasing her brow. I notice that there are more bruises and cuts along her arms, all of them fresh.\n\n\"Maybe I sensed you in my subconscious?\" I offer snidely. I can't keep the sarcasm from my voice. \"So who's June, really? Does she know who you are?\"\n\n\"No.\" Shae shrugs her shoulders, not giving up much. I raise a skeptical eyebrow, and Shae continues. \"She's part of an organization here that helps with supervised independent living until Caden turns eighteen.\"\n\nI return her shrug, thinking of the gun I found. It makes no difference to me whether Shae's lying about June or not. She's not my target. \"Is Caden here?\" I ask abruptly.\n\nShae's face is expressionless but her body is poised, anticipating an attack. \"I know why you're here, and I can't let you take him back,\" she says.\n\n\"You don't know anything,\" I snap, bristling. \"And I'm not a little girl anymore. You don't get to make decisions for me or tell me what I can and can't do.\"\n\n\"You don't understand\u2013\"\n\n\"That's what you said when you left. I'm older now. Try me,\" I say, folding my arms across my stomach and tapping a booted foot against the flagstone path. My fingers close comfortingly around the handle of a blade that's tucked into the waistband of my second-hand black fatigues. I don't trust anyone, far less Shae, who's deceived more than her share of people.\n\nShe stands, both hands in the air, and I back away a couple steps in instinctive response. \"If you take him back, they're going to kill him. The Vectors looking for him have drawn the attention of the Guardians on this side. We don't have a lot of time. I have to take him somewhere safe. All I'm asking is for you to trust me.\"\n\n\"Trust you?\" I sputter and laugh at the same time. \"Like that's going to happen.\"\n\n\"You think you can stop the Vectors? There's a dozen of them within hours, less even, of finding us. Where do you think I was? I was fighting them, trying to lead them away from Caden. They're coming here, Riven. For him. And for you.\"\n\n\"I can take care of myself.\"\n\n\"And Caden? Can you take care of him?\" Shae says staring at me as if she's trying to anticipate what I'm going to say. \"He's a prince, unless you've forgotten.\"\n\nI stare back. My voice is cold. \"No, he isn't. He's more than that, Shae. You know that. He has a purpose. And you must know why I have to take him back. Cale is dying.\"\n\n\"Then they will both die. What you're doing is suicide.\"\n\n\"Don't you even care that your home is about to be at the mercy of a madman if Caden doesn't go back? What's the alternative? That he stays here and lives out a life he wasn't meant for, while everyone we know, everyone we love, dies or is enslaved by Murek?\"\n\nShae sighs, the movement rippling wearily through her whole body, and interrupts my tirade. \"Riven, he doesn't know.\"\n\n\"What?\"\n\n\"Caden doesn't know anything about who he is or where he came from, none of it. He had very little memory of his life in Neospes and what little he did remember, Leila erased. She never meant for him to go back, you know. They were going to kill him.\"\n\n\"It doesn't matter,\" I snap. \"He's needed there now.\"\n\nShae sits on the step again, her entire body slumping forward. Pain spasms across her face from the movement.\n\n\"Not that I care, but what happened?\" I ask, nodding at the gashes on her arms.\n\n\"Two Vectors. I took care of it.\" She sighs and leans against the railing. \"Did our father send them?\"\n\nI spit on the ground and nod. \"Yes. He's with them, but you knew that already, didn't you?\" I see the flare of pity in her eyes \u2013 meant for me \u2013 but I ignore it. \"Look, you haven't been back to the city in years. Neospes is on the brink of war, and more people will die if we allow Murek to take control of the monarchy. He has an army of Vectors at his back and he means to control our world, and this one too, I expect. It's why I have to help Cale. He's got no one he can trust, and he has to live, or we all die.\"\n\n\"I know more than you think,\" Shae says. \"But sacrificing Caden isn't the answer, Riven. I took an oath to protect him. What makes his life worth any less than Cale's? Or yours? Or mine? This is the only place he's safe.\"\n\nI take a deep breath, trying to calm the sudden rush of anger swirling inside of me at her blindness. \"Safe for what, Shae?\" I growl. \"Caden has a purpose, one which he is bound to fulfill. I took an oath too, to protect our king. You and I both know that this is what he was born for.\"\n\n\"You're wrong. Leila left because she loved her son too much to let him die.\"\n\n\"Cale's her son!\"\n\n\"Caden is, too.\"\n\n\"He's not her son,\" I mutter doggedly.\n\nWe're at an impasse, staring at each other with stubborn fury. I'm not above using force to get my way, but Shae's just taken out two Vectors. Alone. I'm not about to make any rash and stupid decisions about her combat skills or apparent exhaustion. The silence hangs between us like an impenetrable wall as we stare at each other across the five feet that could well be an abyss between us. After a few tense minutes, Shae clears her throat.\n\n\"Do you remember that day? When I left?\" she asks, and I give the barest of nods. It's a day etched into my memory so deeply that I couldn't forget it if I tried. \"I wanted to take you with me, but you wouldn't go,\" she says. Her laugh is empty. \"You remember what you called me?\"\n\nDefector.\n\nShe doesn't have to say it; we both know what I said.\n\n\"That was the day I realized that you were your father's daughter. He robbed you of anything close to love,\" Shae says. \"You chose to stay with him out of fear of him than out of love for me. He owned you then, and he owns you now.\"\n\n\"I am nothing like him,\" I grit out. \"No one owns me.\"\n\n\"He used to say that I could never be a killer, because I was too emotional. Said I loved too much. How can someone love too much? He said I was too much like Mom. Soft. I proved him wrong, even though I died a little bit inside every time I took a life.\" At Shae's words, my eyes feel like there's sand behind them, and a boulder settles in the pit of my stomach. \"But he was right about you, wasn't he? The stone-cold sister? The one who wouldn't be torn by emotion, the one who kills without feeling a thing? Servant to the monarchy... obeying orders without question... He's made you into their killing machine. So what? You're going to kill me now, Riven?\"\n\nHer words are like daggers, hardening every bit of resolve I'd lost over the last few weeks. Living in this world has softened me, made me forget who I am. But Shae's right. I am a killer. And I obey. It's what I do.\n\nShe wanted stone-cold? I step forward and pull a bone-handled sword from the underside of my backpack so that it lies flush down the length of my thigh. Her eyes narrow at my blatant challenge.\n\n\"If I have to,\" I say. \"I'm taking him whether you like it or not, Shae. He is nothing more than a target to me; you said it yourself.\" I stare at her with cold eyes, feeling nothing but icy purpose. \"You chose your path and I chose mine. And you are nothing but an obstacle that I am more than willing to remove.\"\n\nShae pulls herself to her feet, a slender double-edged sword materializing in her left hand. \"So be it,\" she agrees. \"But you know I can't let you do that.\"\n\n\"Can't let her do what?\" a familiar voice calls out. Caden is striding up the driveway with his bag of gear slung over one shoulder and carrying a trophy. I turn around, twisting my hand in a backwards motion so that my blade slides under my backpack into its sheath. His timing couldn't have been worse. Or maybe it's a blessing in disguise.\n\n\"Nice trophy,\" Shae says, discreetly pocketing her own blade. \"You win again?\"\n\nCaden dumps his bag on the grass and tosses the trophy on top of it with a grin and a thumbs-up. \"You're back,\" he says to Shae with raised eyebrows. \"Visiting?\"\n\n\"Checking in.\"\n\n\"My warden,\" Caden says to me still grinning. The cocky sideways smile reminds me so much of Cale that my breath hitches in my throat. I am beginning to hate these moments when I'm caught off guard, when the two of them start to merge into one, or worse, when Caden starts to become someone real on his own.\n\n\"Hey,\" I say tightly.\n\n\"Thought you were going to the meet?\" he asks, his shoulder brushing mine.\n\n\"I know, but I got caught up, something with Charisma. I'll fill you in later,\" I say, flicking an eye over to Shae, who seems to be watching us both with more than a little curiosity. Without thinking, I step away, and the immediate quirk of Shae's eyebrow irritates me. I know exactly what it's meant to say, mocking my earlier words.\n\nJust a target?\n\nI return her look evenly.\n\nCaden's gaze pans slowly from Shae to me, and back again. There's very little family resemblance between us, considering we're only half sisters, but we've been trained by the same people, a fact that's evident in the similar set of our shoulders and the stance of our feet. It's an environmental similarity, and obviously one that Shae hasn't lost, despite all her years here.\n\nBelatedly noticing Shae's appearance, Caden walks past me and sits next to her. \"You look like hell. What happened to you?\"\n\nShae smiles, and it is a smile that I haven't seen in a very long time. \"You should see the other guy,\" she jokes, and then says, \"It's no big deal. I tripped over my own two feet. You know how clumsy I can be.\" She says those last words with a fleeting look toward me.\n\nBut they have already drawn me into a near-forgotten memory.\n\nWhen we were little, Shae used to be constantly teased about how clumsy she was, tripping over furniture or nothing in particular. But the minute she got on the combat field, it was as if she transformed into the most graceful fighter. In the arena, no one could match Shae's skill. Rising quickly into the ranks of the elite, Shae's name became known by many, something that made her fall to disgrace \u2013 and our family's shame \u2013 even more noticeable.\n\nI had to bear the brunt of it after she left, but it had only made me stronger and compelled me to outperform every other kid in my training classes. By the time I was twelve, I was put into elite training \u2013 the youngest ever \u2013 and at fourteen, the coveted rank of general was mine. Of course, that made no difference now. At Cale's request, barely two months after my inauguration, I'd left to find Caden.\n\nThree years later, I'd finally found him. And her.\n\n\"That cut on your face looks nasty,\" Caden says, interrupting my thoughts. \"June has some antibiotic cream that you should put on it.\"\n\nGoing into the house with Shae is not part of my plan. It's confined and I'm not as familiar with the layout as she would be. In hindsight, I realize that the bare and spartan room would have been hers. I should have known... its austerity and the clock set ahead like mine should have been dead giveaways. Instead, I'd been blind.\n\nCareless.\n\nCaden pokes his head around the front door. \"You coming or what?\"\n\nI hesitate. The truth is, I feel much safer outside, even with the deepening twilight shadows already dappling the front lawn. Night isn't too far behind, but at least it's out in the open. But I don't trust Shae either, and I don't know whether she'd try to escape with Caden. Making a decision, I nod and walk up the porch steps.\n\nInside, while Caden cleans the cut on Shae's face in the kitchen, I sit at a stool beside the granite island separating us.\n\n\"So, Shae,\" I say casually, picking up an apple from a bowl on the countertop and biting into it. \"You don't live here?\"\n\nShae's eyes laser onto mine, but I keep my expression blank. \"No. I go to college a couple hours away.\"\n\n\"Funny that Caden never mentioned you.\"\n\nCaden laughs, pouring antiseptic cleaner onto a gauze cloth. \"What's to mention? She's sour, unfriendly, falls down all the time, and gets very pissed when I talk about her to anyone, like she's some kind of secret agent. So I just pretend she doesn't exist. Works for everyone.\"\n\n\"Shut up, Caden,\" Shae says grimacing. \"Ouch, ease up that stuff, will you?\"\n\n\"Stop being a baby,\" he shoots back, and winks at me. \"Shae's my unofficial warden. I swear she's got spies everywhere telling her my every move like I'm in witness protection.\"\n\n\"Caden...\" Shae growls in warning.\n\nI know that voice, but I draw the fire in my direction. \"Seems pretty safe here from what I can tell.\" I grin at Caden. \"Something you're not telling me? Like you're some kind of Princess Diaries royalty?\" If looks could kill, I'd be incinerated, but I deliberately don't look at Shae.\n\nCaden snorts and rolls his eyes. \"I wish. No, Shae's just Shae. Super protective ever since my mom died.\" Shae looks like she's going to explode and starts to get up, but Caden pushes her back down onto the stool. \"Look, you can't move while I do this, or it'll get all messed up and won't heal properly.\" I take another bite of my apple and watch as he deftly places tiny strips of surgical tape across the cleaned and medicated wound.\n\n\"How long have you guys lived here? With June?\"\n\n\"Ten years next February.\"\n\n\"You've been here this whole time?\" He shoots me an odd look, and I rush to clarify. \"I mean, I thought you'd lived on the East Coast and moved around.\"\n\n\"We moved here after my mom died. I told you, remember?\"\n\nNow I'm confused. I'd tracked them across fifteen states in the past three years. And then it hits me. I've tracked Shae... the decoy, not them. He's been here all along living a normal life while she's been everting back and forth to throw anyone looking for them off the scent. Our eyes meet and I know that my guess is right.\n\nI frown. Everting so many times at what cost? Genetically, human bodies aren't built to jump back and forth. Our cells start to break down, even with the pills reinforcing them.\n\n\"Cade, do you know if June's got any more of those painkillers in her medicine cabinet? My leg's killing me,\" Shae asks, glancing at Caden and breaking our connection.\n\n\"The ones for your migraines?\" he asks. I shoot Shae a sharp glance but she ignores me.\n\n\"Yeah.\"\n\n\"Think so. Be right back.\"\n\nAs Caden leaves the room, Shae and I stare at each other across the granite divider. The seconds stretch into silent agonized minutes. Migraines are the first sign of brain degeneration. Despite her betrayal, I wouldn't want anyone to suffer that kind of pain.\n\n\"Shae...\" My voice almost fails me. \"How long?\"\n\n\"A few months ago. Don't worry. I'm not going anywhere just yet.\" She leans forward, pressing her palms on the counter and correctly interpreting my frown. \"And I can still take you out if I have to,\" she warns.\n\n\"If you can't protect him, then at least give him a chance with me.\"\n\n\"I can protect him,\" Shae says. \"It could be years before anything more happens.\"\n\n\"But it could be less.\"\n\nThe onset of migraines is the beginning of the end; the only unknown is exactly how long it takes before the appearance of brain tumors and the fatal seizures, like the one that had taken Leila, Caden's mother. Sometimes it could be years, other times weeks. It depends on physiology.\n\n\"He doesn't belong there, Riv,\" Shae says.\n\n\"Don't call me that,\" I snap, ignoring the stab of hollowness in my belly. \"Once I get him to Cale, we will be safe.\"\n\n\"Safe?\" she hisses. \"You're taking him to his death.\"\n\n\"Shae, we've been through this; it's what he was born for. If you die, he's not going to survive much longer. Neither will Cale. At least let Caden fulfill his purpose.\" I move slowly around the granite island until I'm standing right in front of her. I grasp her shoulders gently. \"I know what you promised Leila,\" I say. \"But she made a promise too, seventeen years ago. If you don't let me do this, they're both going to die. Caden here and Cale there. At least if I can get Caden back, one of them has a fighting chance to live. Can't you see that?\"\n\nShae's eyes are wet with tears. \"What if that chance belongs to Caden?\" she whispers. \"What gives us the right to choose who lives and who dies?\"\n\n\"Shae, that choice was made years ago.\"\n\n\"But he's different. I know you've seen it, felt it. I saw the way you looked at each other today.\"\n\nMy fingers tighten of their own volition in denial, digging into the flesh of her arms. \"No, he isn't. He's nothing but a ghost.\"\n\nI can feel her body flinch under my hands at my biting words. She squares her shoulders. \"You and I both know that he's more than that. Caden is strong. He'll be fine with or without me,\" she says stubbornly.\n\nI step back, releasing her, and walk to the other side of the island. I pick up my half-eaten apple and study the browning parts of the white flesh. My voice is almost a whisper.\n\n\"And if you die, how are you going to tell him? That he's from another universe? That you've lied to him for his whole life? What's he going to do if he comes up against a Guardian? Or worse... Vectors? His blood and Cale's blood will be on your hands.\"\n\n\"He doesn't have to know about\u2013\"\n\nA loud crash and a scream from upstairs have us both bolting like a gunshot for the staircase. The fear on Shae's face is mirrored by the fear on mine, but her fear is rooted in losing Caden. Mine is tied to losing Cale. I take the steps up two at a time with Shae hot on my heels, weapons already in hand and prepared for the worse.\n\nBut what awaits us is more deadly than either of us can ever imagine. The odor is unmistakable; it's the smell of death.\n\nBlood, breath, and bones inside of me fall prey to an instant crippling fear.\n\nThe Vectors are here.\nDEAD MAN WALKING\n\n\"How many?\" Shae asks me.\n\nWe eye each other in silent truce for the moment, our only objective to see Caden safe despite our polar opposite endgames.\n\n\"At least two, I think,\" I say. Her eyes widen and I can see that she's thinking about running in there no matter the cost. \"Wait,\" I whisper urgently. \"Can you get around to the bathroom that connects your room to Caden's?\" Shae nods. \"OK, on my mark, in three.\" She nods but hesitates. I place a hand on her fingers and squeeze. \"Don't worry, Shae; they won't kill him. They have orders to take him back to Murek.\"\n\nShae shoots me a defeated smile, and I stare at the raw cuts on her hands for a second. Her fingers twist to squeeze mine, and a strange sensation chokes my chest. We both know that she's not at her fighting best, and if things take a turn for the worse, there's a very real chance that she'll die. The choking feeling spreads to my neck and paralyzes every muscle in my face. Time slows between us, and suddenly it's as if the years \u2013 and the betrayal \u2013 separating us no longer exist. My numb fingers tighten around hers.\n\n\"Wind at your back, sister,\" Shae says. My eyes are burning so fiercely that they feel like they will catch fire at any moment.\n\n\"And at yours,\" I choke out.\n\nBut she's already gone in a whisper of movement. I suffocate the useless emotion inside of me, knowing it will only help my enemy, and eye the time carefully on my watch. I know that Shae's doing the same. The seconds count down, and with a final short breath, I shove open the door. My just-drawn breath hitches at the scene before me.\n\nI was wrong.\n\nThere are four of them. One's holding Caden in a chokehold, the others are staring at us with merciless dead eyes. I haven't faced a Vector in years, and I'd forgotten how normal they seem. They look human: extra-large military-type people, but still people. But if you look more carefully, you notice things like the unnatural pasty pallor of their skin and the bluish tinge of their eyes... their very dead eyes. The Vectors aren't human. Not anymore.\n\n\"Riven,\" Caden wheezes. \"Look out!\"\n\nI duck and spin just as one of the Vectors slides in my direction, swinging an electro-rod to my head, and skewer him with one of my blades. For a dead creature, it's incredibly agile for its size. It barely twitches from the impact, lunging toward me again. I spin and scissor my blades across its waist, seeing fabric and skin split apart. But there's no blood, only a brackish gray-blue fluid.\n\nOut of the corner of my eye, I see that Shae has been cornered by the other two, but she's wielding her double-edged saber like it's a lethal piece of ribbon, wheeling and ducking with mesmerizing speed. I turn back to the Vector lurching toward me, its gashes healing before my eyes, but I follow Shae's lead and keep slicing, opening new wounds as quickly as the old ones regenerate.\n\nDespite how easily I'd fought the boys earlier, with this Vector it's only a matter of minutes before I'm out of breath and my newfound confidence wavers. The training exercises I've been doing are woefully inadequate, child's play compared to fighting these things. Three years is a long time to lose my edge.\n\nI grit my teeth \u2013 this is what I've worked for, what I was born for. Sparing a glance toward Caden, his body held paralyzed by the well-placed pincer grip of the Vector beside him, there's no way I'm going to give up. Not now.\n\nPain ricochets through me as the Vector's electro-rod catches me on the thigh, and I slam backward into another of the Vectors behind me. Without thinking, I clip my elbow backward, slamming into its head, before diving toward the first aggressor. Though I am far from out of shape, they are faster and smarter than I remember.\n\nThe one facing me seems to anticipate my every move, forestalling every turn and bend as if it can read my mind. I need to trick it somehow, take back the advantage. I swing my swords as if I'm possessed, spinning under the Vector's outraised arms until I see my opening.\n\n\"Game over,\" I hiss, slamming my heel against the back of its leg. The creature crumples to its knees, and I waste no time in crossing my arms together as hard as I can, the swords in each hand meeting to cleave the Vector's head completely off its shoulders. I watch, huge gulps of searing breath filling my chest, as it keels over with a sick clump.\n\nCaden is watching us with horrified eyes, his gaze swinging from Shae back to me in disbelief. But he's the least of my worries right now. The other Vector is now holding him immobile on the shoulder with one hand while tapping into a handheld computer with the other. The Vector isn't concerned with any of us, its mission different from the others. I gasp, recognizing the device in its hands. It sets the coordinates and parameters for eversion.\n\n\"Shae, he's going to eve\u2013\" I begin turning toward Shae and freeze. The two Vectors have Shae up against the wall, barely a hair's breadth from killing her. Her face is covered in blood, old wounds reopened and new ones oozing red. Her eyes lock with mine and I can see the regret in the curl of her lips. My head's already shaking before I dart forward, arms raised and weapons out.\n\n\"No!\" I scream. \"Cease! That is an order.\"\n\nTo my utter disbelief, both Vectors stop, turning in dumb submission toward me. Shae's shock reflects my own. My brain is spinning. These must have been sent here before I defected, and are still somehow programmed to obey my orders.\n\n\"Release her,\" I say quickly, not wasting a second. Shae slumps to the floor but still manages to swing her saber into the Achilles tendons of the one closest to her. Taking no chances, even before its body collapses, her sword is already buried in the back of its neck severing its spinal column. She's hurt badly. I can hear it in her labored breathing and see it in her eyes.\n\n\"Riven.\"\n\nThe soft voice behind me is a whisper of a warning, but the tiny hairs on the back of my neck are already standing at stiff attention. I turn to meet the eyes of the Vector restraining Caden whose attention had been on the eversion device. That attention is now riveted on me like a laser. Its eyes are a lightless black with the familiar blue halo surrounding the pupils.\n\nBut the similarities end there.\n\nThere's something different about this one, I can feel it deep in my bones. Like the others, its uniform is black, but there's a jagged swatch of red cloth across his chest in the shape of a crescent moon. Every part of me knows that it means something terrible.\n\n\"Kill them,\" it says in a guttural growl. \"That is an order, soldier.\"\n\nI blink. It's some kind of leader, then, a Commander. The command is directed at the remaining Vector, but its stare remains focused on me. It's impossible. Vectors don't speak. Speaking infers functioning brain capability, and they're dead. A shiver of cold dread runs down my spine.\n\nAren't they?\n\n\"Now!\" it growls again.\n\nBut the remaining Vector is motionless, staring from me to him as if confused. Its programming must not have been overridden to counter a direct order from me, its last leader, even with an order from its new one. The window of opportunity is no bigger than a sliver, but I grasp it without a second thought.\n\n\"Kill it,\" I shout, jerking my head in the big Commander Vector's direction.\n\nIt's all I can do to get out of the way as the Vector launches its considerable bulk toward its commanding officer. But within the blink of an eye, its body is flung back in our direction and crashes into a bookcase. Splintered wood peppers the air like wooden darts, and I shield my eyes instinctively, covering Shae's body with mine.\n\nThe big Vector is motionless, still holding Caden. It flicked its attacker off like a bothersome gnat with one finger.\n\n\"Again!\" I scream at the fallen Vector. \"Get up. Don't stop until it's destroyed.\"\n\nEven as the words drop from my lips and the creature launches itself once more in silent submission, I pull a silver instrument shaped like a four-leafed clover from my boot and fling it toward Caden. He's not the target; the fingers pinching into his shoulder are. My aim is true and the star clips off the Vector's fingers cleanly.\n\n\"Caden, get down!\" I yell, but he doesn't move even though thick grayish blue fluid is spraying into his face from the Vector's severed fingers. I gnash my teeth.\n\n\"Go...\" Shae wheezes as if sensing my hesitation to leave her unprotected, \"...be fine for a minute.\"\n\nThere's no time to think as I take Caden down in a football tackle that would rival any in this world's Super Bowl, rolling underneath the desk just as the two Vectors smash into the floor beside us. Even though the commanding Vector is pummeling the one below it, its stare is still fixated on me as if I'm the one it's punishing. I rip my eyes away with effort and kick it in the side so that the one fighting for us gets some leverage to twist over and above it.\n\n\"You hurt?\" I ask Caden urgently.\n\n\"No,\" he rasps. \"But my legs feel funny. What did that thing to do me? What is that thing? Are they going to kill us?\" His voice is rising with every second and I can see the terrified panic building in his eyes as they dart toward the two grappling on the other side of the desk.\n\n\"No. You're going to be fine,\" I say. \"Can you get over to Shae?\" He nods and I squeeze his hand. \"Pull her into the bathroom if you can and close the door. Here, take this.\" I don't look at him. I don't want him to see the panic in my eyes. Instead, I shove one of the swords that had fallen to the ground into his hands and push him toward Shae. \"Don't be afraid to use it.\"\n\nI take a deep breath and turn to the two remaining Vectors. The one that obeyed me is not going to last much longer. It doesn't have much of a face and its ribs are concave in a way that suggests imminent fatality. It's only a matter of time before the nanobes inside of it stop communicating, and I need it alive for questioning. If it's loyal to me, there's no way I can let such a windfall go.\n\nWith all the strength I can summon, I kick the Vector Commander in the face, hearing the crunch of bone as my blow dislocates its jaw. Blue liquid seeps down the side of its face as my boot tears away skin and tissue from its chin, exposing filed, jagged teeth. It turns toward me, a sick grin tugging the exposed tissue upward, and digs a heavily booted foot into the other Vector's chest until gray-blue fluid pools around his sole.\n\nEmotion? Impossible.\n\nBut it is a grin... a horrible mockery of a grin. Vectors are inanimate, robot-cell controlled hosts. They don't think for themselves, and they certainly don't smile.\n\nBut this one does.\n\n\"What are you?\" I whisper as it twists its head in both hands to realign its neck, staring at me with a knowing expression. I have never felt such fear, not even when I was running for my life to escape Murek's guard.\n\n\"A general,\" it answers. \"Like you.\"\n\n\"I am no general,\" I snap.\n\n\"Yes, you were a colossal failure, weren't you?\"\n\n\"You don't know what you're talking about. You're a machine. You're dead.\" I don't know why I feel the need to defend myself against the poisonous words of this thing, but something about it reminds me of my father... judging me, even now.\n\n\"That's where you're wrong,\" it says. \"Come home, Riven. Bring the boy. All will be well.\" The Vector bends its head in a conciliatory way, which only makes it seem more macabre without half its face, but I am mesmerized by its last words. And terrified.\n\n\"You don't want to kill him?\" The thing wavers as if reading something on my face and I deaden my expression, but it's too late.\n\n\"No,\" it says. The word is a lie. There's no way they'd keep Caden alive. Murek wants Cale dead. It made sense to reason that they wanted Caden dead, too. Unless... \"Why do you think I didn't kill this boy?\" the Vector says, distracting my ugly train of thought. It holds a black-gloved hand up that's easily twice the size of mine. \"It would be so easy,\" it says squeezing his fist.\n\n\"So why didn't you?\" I hide the fear sliding around inside of me with bravado, but I know without a doubt that I can't trust anything this creature says. But I need to buy time... time to think.\n\n\"Orders are orders. I don't ask questions.\"\n\n\"Why are you different from the others?\" I say. \"How can you talk?\"\n\n\"That is the question,\" the Vector says taking a step toward me. It's not aggressive, but I step back anyway and feel the bed frame against the back of my knees. I'm nearly trapped. The only way around the creature will be over the bed to the bathroom door or the window. \"Your father created me after you left.\"\n\n\"Are you alive?\"\n\nThat gruesome smile again. \"More than the others. Less than you.\" Its cryptic words irritate me. It's as if the thing is playing some kind of game, one that I'm sure has no rules.\n\n\"Why would they make you?\" I say. The Vector smiles again, and I can feel the bottom of my stomach drop even before it says the words. The sick pleasure on its face makes me want to retch.\n\n\"Because the Lord King is dead.\"\n\n\"You're lying.\"\n\nEverything inside of me feels like it's disappearing \u2013 bones, blood, air \u2013 until I'm nothing but a shell collapsing upon itself. I can't even breathe. In slow motion, I fall back against the side of the bed, legs buckling, but my senses haven't completely deserted me, and out of the corner of my eye, I see the Vector reach for a long-handled spiked weapon. On autopilot, I scramble across the bed and shove myself to the other side just as it lifts the entire bed frame with one hand and smashes it against the wall. Chunks of wood and steel explode into the exposed parts of my body.\n\nBefore I can move, the Vector's spike swings toward my head, and I dive forward, my swords cleaving into its calves before rolling to its left. It barely deters the creature, and I fend off another attack, sparks flying as our weapons meet in midair. The vibration echoes painfully down my arm, and I can see a detached metal spike protruding from the flesh of my upper arm. My shirt is sticky with blood. I don't want to bleed to death by trying to pull it out, so I grit my teeth and leave it, hefting my other arm high while protecting my body with my injured arm.\n\nScanning the room, I notice that the bed frame is blocking the bathroom door and the nearby window is locked, which means precious extra seconds lost trying to unlock it. The Vector's bulk shields the bedroom door. The window is the only choice I've got, and if worse comes to worse, I can go through it headfirst and hope for the best. Either way, it seems that I'm facing the possibility of broken bones. I need to distract the creature to buy some time.\n\nI lower myself into a crouch and sweep my leg out, but the Vector moves out of the way, fast for something of such size. I switch to words, hoping beyond hope that making it think will help slow it down.\n\n\"You know how I know you're lying? About the king?\" It watches me like a bird toying with a worm half-submerged in the dirt. \"Because people, important people, know about Caden. Cale's alive; otherwise, Murek would just forget about Caden and rule Neospes as he's always wanted.\"\n\nI pause again, snaking my uninjured arm out to catch it across his left flank. Blue liquid seeps through its clothing and drips to the floor. Now we're even.\n\n\"Did he teach you to lie?\" I continue my one-sided conversation, gaining confidence with each breath. \"My father? He's very good at lying. After all, he convinced me to lead your kind. But he had a hidden agenda, didn't he?\"\n\nI spin and jab at my opponent's body, but it anticipates my movements this time and dodges, only to return a blow that stuns me senseless. Something wet and warm plasters my hair to my scalp, but I can barely feel it beneath the hot welt flowering against the side of my face. I spit a mouthful of blood to the floor and lean against the wall. My vision begins to blur as the Vector morphs into three separate beings, each wavering like smoke.\n\n\"That all you got?\" I grit out, holding my sword across my body and praying that my shaking legs don't give out. The Vector pauses with another grin, as if sensing impending victory. My only comforting thought is that Shae and Caden are safe. She'll get what she wanted \u2013 Caden will never return to Neospes.\n\nAnd I would have failed... in my promise to Cale. But if the Vector is right, then it won't matter either way. I stare into its dead blue eyes, and smile. \"We will never let you take him.\"\n\n\"You have no choice, General,\" the thing says finally, removing the pocket device from his vest. \"The boy will go back, and so will you, dead or alive. Your father wants you alive, of course. But Lord Murek has no preference. Regardless, you cannot stop me.\"\n\n\"But I can,\" a voice says, just as the sound of a cannon tears through the room. The Vector pitches forward as gunfire rips through its bulk, June's hollow-points doing what they're designed to do. It's a volley of bullets as Caden holds June's semiautomatic gun with shaky calm.\n\n\"Aim for the head, Caden,\" I try to shout, but my voice is barely a whisper as I feel myself sliding downward against the wall. \"It'll only regenerate anywhere else.\"\n\nBut my words are lost beneath the sound of the exploding shells as the acrid smell of gunpowder fills my nose. I can feel my cells desperately trying to re-engage, when the incongruity of the situation hits me. Caden's the one protecting me. I want to laugh, but only a choked gurgle takes shape in my mouth as Caden empties round after round into the monster.\n\nAfter what seems like an eternity, Caden flings the spent gun to the floor and brandishes the sword I'd handed him earlier. My eyes are on fire, but I have to see if one of the bullets has miraculously hit the Vector in the head or in the spine. It's the only way to stop them. But instead, I watch in horrified slow motion as the Vector pushes off the wall, provoked to the point of rage, and hurls its bullet-ridden body toward Caden.\n\n\"Caden, run!\" It's all I can manage as black stars cloud my vision, unconsciousness threatening to sweep me away. But Caden ignores my warning, darting to the left and sliding to his knees, before reaching upward and back to pierce the sword's tip into the Vector's exposed back. The thing stumbles forward toward me, gurgling, as the sword lodges in its spine. Game over.\n\nFor a second, our eyes meet, and before I can even blink, the air in the room shimmers for a second, and without warning, the Vector disappears. The only memory of it is the red-hot end of Caden's sword, neatly lasered to half its size, and a blackened patch on the carpet. I'd forgotten about the eversion device.\n\nThe monster is gone. For now.\n\nBut it knows where we are, and it's only a matter of time before it comes back with more. It's my last thought before I slip into an unwelcome oblivion.\nTRUTH BE TOLD\n\nMy vision is swimming when I awake. The room is dark, lit only by a single flickering candle. It hurts to focus, and I am confused because Cale and Caden are both in the room, staring at me with wide frightened eyes.\n\n\"You OK?\" they ask me simultaneously. I lift my hand toward their faces.\n\n\"How is this possible?\" I rasp. \"Where... am I?\"\n\n\"You're safe, Riven,\" they both say. \"Drink this.\"\n\nA cold rim touches my lips and I sip the liquid gratefully. My throat feels like it is on fire when the liquid touches it, but I feel better and less woozy as it goes down. A small silver flask dances at the edge of my vision. \"What is that?\"\n\n\"Shae said to give it to you.\"\n\n\"Shae's here, too?\" My head is ringing, and the feeling that something isn't quite right slips around inside of it. \"Cale?\"\n\n\"No. Riven, it's me. It's Caden. Here, drink some more.\"\n\nI sip obediently, the liquid tearing a path again into my insides. It's bitter but warming. I sit up, pushing my elbows back against the pillows. Surprisingly, it takes very little effort to move, despite the pain in my head that would suggest otherwise. The room starts to take shape, and as I grow more and more awake, I realize that nothing else hurts.\n\n\"Where am I?\" I ask again after a couple minutes. \"What happened?\"\n\n\"Don't you remember?\" Caden says. \"Those things that attacked us?\"\n\nAnd then it's like a tidal wave as the events from earlier come rushing back. My fingers curl into the scratchy blankets on the sides of my legs.\n\n\"How long have I been out?\"\n\n\"Only a couple hours.\"\n\n\"Where are we?\"\n\nCaden comes closer, and the metal cot dips as he sits next to me. \"We're in the basement. It used to be a tornado shelter back in the early fifties. It's why Shae picked this house out of all the others. She's a bit of a Miss Doomsday, but I guess she was right.\" He nods over to the far side of the room that's still shrouded in darkness. \"She's pretty hurt, but I gave her some stuff that June uses for head injuries. It's a mild sedative too, so she's sleeping now. She didn't want to call 911.\"\n\nNo, Shae wouldn't; too many questions. I hobble over to where she's lying on a cot similar to mine and stare at her bloodied face. Caden has cleaned off some of the blood, but her injuries are starting to blacken and swell. She looks far worse for wear than I. My fingers drift to her neck, and I can feel a faint but steady pulse. Her breathing is shallow and wheezy. Caden has cut off the legs of her pants to bandage some of her wounds, but his efforts are amateur at best. It won't be long before her cuts become infected. And the migraines... Those are the beginning of the end. The injector in my bag would help, but unless she gets real help from our doctors, it would only provide temporary relief at best.\n\n\"She doesn't look so good,\" I say.\n\n\"I used what we had.\" Caden's voice is apologetic. \"Riven, we need to get her to a hospital.\"\n\n\"No.\" I shake my head emphatically. \"No hospitals; too many questions that we can't afford to answer. They wouldn't be able to help her, anyway. I need to get my backpack. Does June have a medical kit upstairs?\"\n\n\"Yes, but I don't even know how to use half the stuff June has in there. It's hospital-grade stuff.\"\n\n\"Then we're going to need to figure it out,\" I say flatly, resting my hand against Shae's hot forehead. \"And fast.\" Infection has already begun to set in. I walk back to my cot, where Caden is still sitting, and squat to retrieve my boots.\n\n\"Riven,\" Caden asks quietly, \"what were those things?\"\n\nI stare at him, wondering how after all these years Shae could have singlehandedly protected him from ever coming up against them. I don't even know what she's told him, if anything at all. My guess is nothing. She's tried to protect him the only way she knew how \u2013 by keeping him in the dark, letting him have as normal a life as possible here with some kind of chance to be happy. Glancing over my shoulder at her sleeping form, I am unsure of what to say, but Caden is far from stupid, and he certainly isn't blind. I settle for something near the truth.\n\n\"They're called Vectors, a government experiment. Reanimated corpses.\"\n\n\"Reanimated? Like zombies?\"\n\nI shake my head, a faint smile at his childlike response curling my lips. \"Zombies are dead, period. And they aren't real. Vectors are very real dead bodies, controlled by nanobes. Tiny little microscopic robots that operate inside the hosts.\"\n\n\"Microscopic robots?\" His expression is skeptical. \"You're kidding, right?\" I shoot him a look and raise an eyebrow. \"Is that even possible?\" he asks.\n\nA dozen mocking responses slip to my lips, but I stifle them. I lace up my left boot and start on my right. \"Not everything's impossible. Remember the blue fluid?\" Caden nods. \"That's nanoplasm... the robots.\"\n\n\"I don't get it; why dead bodies?\"\n\n\"Easier to control than live ones, I expect,\" I say bluntly, and grab my weapons, walking over to the steel door. \"How do you open this thing?\"\n\nCaden grabs my arm. \"Where are you going? Those things, the Vectors could be up there. What do they want, anyway?\"\n\nI try to keep the fear slinking around deep inside my belly out of my eyes.\n\nThey want you.\n\n\"I need to check the bodies to see if there's anything we can use. And Shae needs something I have in my backpack. I'll be back; just sit tight.\" I watch as he unbolts the heavy door. \"Lock it behind me. When I come back, ask me who our physics teacher is, OK?\"\n\n\"OK,\" he says, squeezing his fingers, his hand still on my shoulder. \"Be safe, Riven.\"\n\nI climb the basement stairs carefully, hearing the heavy steel bolts fall into place behind me. The entire entrance has been reinforced with some kind of thick metal, and I trail my fingers across the shiny, cool surface. Shae has definitely made sure to be prepared for something. The door at the top leads into the kitchen. It's a narrow trapdoor-like entrance that I'd never noticed before, not any of the previous times I'd been in their kitchen. It, too, is heavily reinforced, with special seals and gaskets. There are no visible handles for re-entry, so I stick a nearby cookbook in the gap. I have no idea if it will hold or not, but it's the best I can come up with.\n\nIt's quiet, which isn't necessarily a good thing, so I'm cautious when I make my way back upstairs. The room is a shambles, furniture tossed and broken, blood and blue fluid spattered everywhere, with three dead creatures in various stages of decay gracing the floor. The smell is putrid, like a wall of rotting human compost curling against me, and I feel the answering bile rise in my throat. That's the thing with Vectors \u2013 when the nanoplasm dies, the bodies decompose rapidly. My father had once said that it was a disgusting but necessary element of control. As a society, we'd learned that the hard way.\n\nTrying not to breathe and careful not to touch any of the fluid, I methodically check each of the Vectors for weapons and anything else of use. I pocket an electro-gun, some rods, a couple metal golf balls that I'm sure are some kind of high-tech explosive devices, as well as any wireless communications headgear I can find. I'm onto the third in less than five minutes when I hear a faint sound. My weapons are at the ready before I'm even in a standing position. I tiptoe to the bedroom door, ears straining, but everything is quiet. I must have imagined the sound.\n\nThe low whine behind me catches me off guard and I swing around to an empty room until I realize that the sound is coming from the third Vector. It's not dead! I pull what's left of its head to face me, wincing at the stench of its wounds. If it's not dead, it will be soon.\n\n\"Soldier,\" I say urgently. \"Can you hear me?\" No response. I tug on its jacket and its head lolls forward. \"Answer me. That's a direct order.\"\n\nIts uninjured eye cracks open and the entire pupil is covered in pale bluish ooze. I doubt it can even see me, but somehow it's registering my voice.\n\n\"Who sent you?\"\n\nThe Vector's eyes roll back in its head. \"Is Cale dead?\" There's nothing, and I rephrase, desperate now. \"Is the Lord King dead?\"\n\nThe Vector's head moves slightly from left to right. It's a no! My relief is tangible, and I sink back onto my haunches. It's more than I could have hoped for. \"What does my father want?\"\n\nA single outstretched finger points to me. The Vector's eye rolls back into its head, and its mouth opens and closes haphazardly, as if choking. The hand thumps to the floor. Within seconds, its head lolls to the side, and the pungent smell intensifies as its internal organs degrade and liquefy. Swallowing past the sourness in my mouth, I release the jacket and finish my search of its body, pocketing a pair of infrared glasses and a silver pearl-like earpiece communications device.\n\nI move to leave but pause at the door, thinking ahead. I don't have a plan in place, but if any of us are to make it back to Neospes, we will need clothing. The Vector's uniforms are designed to keep their bodies protected and are made from a rare type of engineered fabric-like armor, which also provides warmth and heat depending on weather conditions, both of which are unpredictable in Neospes. It would be stupid to leave them.\n\nI frown at the task at hand but move quickly before I have time to think about what I'm doing. In no time at all, I have three sets of uniforms peeled off of the Vectors' bodies. They stink, but I can't help that. I put them along with the weapons in Caden's fencing bag and sling it over my back.\n\nNow for the medical kit.\n\nAt the door, I glance back into the room. Looking at their naked, decaying flesh is far more repulsive than seeing them clothed. Curved ribs and sharp hipbones protrude against their milky, opaque skin with grotesque prominence: the stuff of nightmares. Blue veins traverse their near-transparent skin to route the nanoplasm from their artificial central nervous systems to the rest of their bodies, like a ghostly blue spider web. They barely look human now. Instead, they look like rotting, dead wraiths. I shake my head, swallowing thickly \u2013 the Vectors are true abominations of my culture.\n\nThe sound of the front door jerks me out of my thoughts.\n\n\"Hello? Caden? Anyone home?\" It's June's voice. She must have come home early. I glance down at my filthy shirt and grab one of Caden's clean T-shirts off the dresser, shrugging into it. \"What is that horrific smell?\"\n\n\"Hey, June,\" I call out, taking the steps down three at a time. \"Sorry, we were doing an experiment for bio. Went bad. I wouldn't go up there if I were you for at least ten minutes.\" The last things I need her seeing are the three dead bodies in her house that look like something out of a science fiction movie. I fake an embarrassed grin and offer her an apologetic look.\n\n\"Why am I not surprised?\" she says slowly, after glancing with narrowed eyes to the stairs before putting her keys and bag on the counter. I hesitate \u2013 I still need to get the medical supplies.\n\n\"June, we were looking for your... medical bag?\" I ask in as casual a manner as I can manage.\n\n\"Why?\" So much for putting anything past her as her eyes meet mine, immediately full of concern. \"Are you hurt?\"\n\n\"Nothing major,\" I say quickly. \"I hurt my leg fooling around with Caden's foils earlier. I'm worried that it will get infected.\" It's not an outright lie, as one of the Vectors caught the back of my calf, but it's not like I've paid much attention to it with everything else going on.\n\n\"Well, let me just wash up, and I'll take a quick look. My bag's in my office.\"\n\n\"I'll get it,\" I say, and all but sprint to June's office. I grab the bag and a couple of the blankets lying on her couch, and go back to the kitchen where she's still washing her hands. June stares quizzically at the blankets and the medical bag in my arms, and her eyes flick to mine. She dries her hands slowly, her gaze drifting between Caden's gear bag, the blankets, and me. Then her eyes flit to the staircase.\n\n\"What's going on, Riven?\" Her voice is quiet, but there's something in it that raises the hairs on the back of my neck. It's an instinct that has kept me alive all these years. Her gaze settles on some fluid spattered on my collar peeking out over Caden's shirt. My stomach sinks. I can see something dawning in her eyes. Mistrust. Fear.\n\nGently placing the bag on the floor, I shift my balance from toe to heel and back again. There is no easy way to explain what I'm about to do, no lie that will make my actions any less terrible. She has to go down below, willing or not. And the fact is, I don't know June, which means I can't trust her. I edge closer and place my hands in the air in a non-threatening motion.\n\nNot missing a beat, June edges nearer to the kitchen island so that it stands between us. \"Where is Caden?\" she asks carefully.\n\n\"Caden's fine.\" My voice is inflectionless and slow. \"You have to trust me, June. But I need your help. Shae's hurt.\"\n\n\"Shae?\" A small furrow of worry shadows her brow, but she steels her expression almost immediately. \"Shae's not home. She'd have called to let me know.\"\n\n\"She came back today,\" I say. \"She had an accident.\"\n\nA sharp glance. \"And the dead Vectors upstairs?\"\n\n\"What?\" This time it's my eyes that rivet on hers. \"What do you know about Vectors?\"\n\nShe has taken me by surprise, and just as I'm considering leaping across the island and knocking her unconscious, a small voice has us both spinning around. Shae's leaning against the wall, her face a mottled collage of purples in the fluorescent lighting. Climbing the stairs from the secret room has her wheezing.\n\n\"June's a... Guardian, Riven,\" she gasps, besieged by a round of ugly-sounding coughs. A trail of bloody spit runs down her chin as her body slumps down against the wall. I stare at June's impassive face, incredulous.\n\nA Guardian! My hands grasp the hilts of the blades tucked into my waistband.\n\n\"Was a Guardian,\" June corrects, this time placing both her own hands in the air. She turns her head toward Shae, and I understand what she wants to do. I nod but don't release the handles of my weapons lying flat against my back. She cradles Shae's head against her. \"Can you pass me the bag?\" she asks me. Her eyes, so warm before, are now cold and expressionless.\n\nUnconsciously, I steel my expression to equal hers. \"You can't help her. She's everted too much. She needs more than the help you can give her.\"\n\n\"I can try.\"\n\nWith a glance at Shae, I push the bag across with the toe of my boot, ever cautious. I am the enemy here, the one who has come to take Caden back. I can't trust either of them, even after what happened with the Vectors.\n\n\"Was a Guardian?\" I ask, after a couple minutes watching her take out several glass bottles from her bag. \"I didn't think someone could stop being a Guardian.\"\n\n\"Well, I did.\"\n\n\"Why?\"\n\n\"Look at you; you're just a kid,\" June says softly, not answering my question.\n\n\"I'm not a child,\" I snap back.\n\nJune's eyes are gentle. \"But you are, Riven. Look around you, look at the children in your school: they're kids. The same age as you are. You're babies trained to kill.\" I can't stand the pity in her voice, and I bristle.\n\n\"They're useless and wouldn't last a minute in Neospes. Answer the question, June.\"\n\nA long, searching look as if she's trying to see inside my head. \"I didn't believe in executing innocent people... innocent kids.\" Now it's my turn to stare at her. \"The Guardians honor a code to protect the fabric of the universe,\" June continues. \"You know what would happen if people were to jump back and forth, don't you?\" It's a rhetorical question, so I remain silent. \"The threat of infection, of disease, is of course the worst, not to mention altering the course of a civilization's future. We honor an agreement between the worlds to protect each side from the other... more so to protect this world from the greed of yours. Eversion was never meant to be permanent. It was a mistake to let it go this far, to create an algorithm that allows abominations like the Vectors to come here.\"\n\n\"What do you mean, it was a mistake?\"\n\nJune answers my question with an equally blunt one of her own. \"Why do you think Murek wants Caden so badly?\"\n\n\"I don't know.\" It's not a lie. I have many theories but none of them strike me as accurate. The truth is I have no idea why he wants Caden, especially if Murek wants to rule Neospes. Getting rid of him would be the easiest thing to do, after Cale is out of the way. It makes no sense that he would want him so badly. \"So why does he?\"\n\n\"Pass me the blankets,\" she tells me, and I comply automatically. She makes Shae, who keeps slipping in and out of consciousness, more comfortable on the floor. I glance at my watch, knowing that each second we remain here becomes more and more risky. June sends a sidelong glance in my direction and continues speaking while sticking a thermometer into Shae's mouth. \"It is a secret that many would kill to protect.\" She pauses as if assessing whether to tell me or not, and I wait, silent. Nothing prepares me for the next words that come out of her mouth. \"Caden, like Cale, is a hybrid. A product of both universes.\"\n\n\"That's impossible,\" I shoot back. \"I may not know Murek's endgame, but I do know what happens to any progeny that comes out of any union between universes. They are abominations and are all to be disposed of... by you, the Guardians, and the Vectors.\" I can hardly keep the vitriol out of my voice. \"It's the law. You track them, and the Vectors eliminate them.\"\n\nJune is calm. \"It's true. Caden's mother was from this world. She never returned, because of her children. It was only when Caden was in danger that she came back, but she couldn't survive. Her immune system had become too weakened to protect her. And that is the sole reason I stopped being one of them. Caden was an innocent child. And Leila, too...\" She trails off.\n\n\"I don't get it. Why do you care about either of them?\"\n\nAfter another searching look, June sighs. \"We grew up together. She was like my sister. My first mistake was to tell her what I was, and from then on, she couldn't let it go. We were barely your age, but it consumed her to the point of obsession. My second mistake was that she everted there because of me... all because I was careless and told her in the first place.\n\n\"She went so far as to major in quantum mechanics at school, and even though I wouldn't tell her anything I knew \u2013 I was terrified of the consequences \u2013 she was determined to find a way. And she did. That was the night she almost got herself killed trying to evert using some home-designed calculation that she must have stolen from my notes somehow. She almost succeeded too, but in the end, her body couldn't take the force and started to collapse on itself, half stuck in this world, half of it in yours. I panicked, and instead of going to my father as I should have for help, I everted us both to Neospes.\" She glances at me, breaking off to place a cold compress on Shae's head after cleaning off the remaining blood on her face. I keep my face composed despite my racing thoughts.\n\n\"Your father saved her. Her injuries were too great for us to return, and by the time she was well enough to make the jump back, it was too late. The Lord King was fascinated by her, and then she got pregnant. That was the last time I saw her until she came to me ten years ago with Caden.\" June shrugs. \"How could I say no to what she was asking? For help. For protection. It was my fault she went there in the first place. I broke the law, and she was the one who paid the price. I owed her.\"\n\n\"But she's from here,\" I say.\n\n\"The Lord King of Neospes doesn't answer to the law. He forbade her to return.\"\n\nI frown to cover my sense of shock at what she is telling me about Cale, about Cale's father... about who his mother is. I can't get my mind around it.\n\n\"That's a large debt,\" I say for lack of anything else. June shrugs again, her lips twisting in a sad, wry smile.\n\n\"It is what it is.\"\n\nDespite my shock, her story rings true as I think back to all of the times I'd seen Cale's mother. She always seemed so odd to me, as if her mind was always somewhere else, like she didn't quite fit in with everyone else in Neospes. She used to wear these long, flowing, brightly colored dresses \u2013 custom-tailored, Cale had once told me \u2013 instead of the standard black or gray tunic and leggings that most of us wore. I'd always thought the dresses fanciful and strange. And now I know \u2013 she had never belonged there at all.\n\n\"Did Shae tell you anything about me?\" I ask June abruptly.\n\n\"No,\" she says, checking Shae's eyes with a thin instrument. \"She didn't have to. I realized what you were after the clinic.\"\n\nMy eyes narrow. I voice the words pounding in my head. \"What I was?\"\n\n\"A soldier of Neospes.\"\n\n\"And yet you still trusted me with Caden?\" I couldn't help the derision in my voice.\n\n\"Not at first \u2013 I wanted to keep you close \u2013 but then I saw something there... something about the way you were with him. And he with you. I thought you cared about him. But I was wrong, wasn't I?\"\n\nMy teeth grind together, and what escapes my lips is little more than a snarl despite the unfamiliar tug in my chest her words provoke. \"You are wrong. I don't give a damn about him. Caden is a target, nothing more.\"\n\n\"Riven?\"\n\nWe both turn at the quiet voice behind us. The betrayal on Caden's face hits me like a slap. I meet his eyes and drop them just as quickly. I don't know how long he's been standing there, but I know it's been long enough for him to hear my last few words. I sling my backpack across my chest as if it's some kind of shield, a distraction maybe, and rifle through its contents until I find what I am looking for. I slide the silver case toward June. I won't need it anymore \u2013 when I return to Neospes, I won't be coming back.\n\n\"Give her this. It will help.\" I stand, slowly stretching my legs. I nod toward the stairs and grab the bags of gear I've piled together before leaning over the gas stove in the middle of the kitchen island to tuck one of the metal golf balls that I'd found on one of the dead Vectors in the middle of the grate. I'm business now, emotion tucked deep. \"More of them will come, if they're not here already. We need to move and seal the door. Either you come down with me or you can stay here to greet them. One way or another, there's not going to be much left up here. It's your call.\"\n\nI don't look at Caden as I push past him to the trapdoor above the basement stairs. Truth is, I can't even look at him. My curiously burning eyes won't allow it.\nUNDERGROUND\n\nBy the time I've carried Shae down the stairs along with a few extra supplies that June's thrown in, I've almost forgotten that Caden's even there. But I feel him staring at me, with heavy thoughtful glances that make me far more unsettled than if they were filled with anger. June has gone quiet as well, but I expected that. Knowing what she knows, I'd be the last person she would ever fully trust, but still, there's an uneasy understanding between us that at the moment we both need each other.\n\n\"Where does this lead?\" I ask her, noticing another steel door that opens to a dark tunnel behind it.\n\nShe stares at me before answering and throwing me a ratty map. \"Couple miles underground. This tunnel forks to the hospital and to an abandoned building near Horrow.\" She jabs at the map I've opened. \"See all the tunnels? There's an entire web of them down here, most of them collapsed and unusable. Used to be a safe-house for an old underground military base back in the Forties,\" June adds, noticing my expression as I peruse the piece of paper. \"It's why Shae chose it.\" She moves over to check on Shae. \"She's looking better,\" she murmurs more to herself than to me.\n\n\"It won't last,\" I blurt out before I can stop myself. I tuck the map into my back pocket. \"She's everted too much already. Her brain can't take the pressure.\"\n\n\"What pressure?\" The low voice belongs to Caden. He's sitting on the cot I was lying on earlier, pretending to sort through the gear in his fencing bag. \"What does 'everted' mean?\"\n\nI pause for a beat before answering him. \"Ever heard of the bends?\"\n\n\"Decompression sickness? Like when you come up too fast from a deep depth and pressurized gases are released into the body too quickly?\"\n\nI have to fight the instant urge to eyeroll. Caden's so technical even with the little things. \"Exactly. Well, it's like the bends, only it starts in the brain. Then it becomes physical because humans aren't built to evert\" \u2013 I spare a glance at Shae's twitching form, knowing she can still hear me \u2013 \"to jump between universes. Our bodies are too frail, and when they start to break down, they become susceptible to infection and disease.\" June's fists are clenched at her side, her eyes unreadable. \"It's why the Guardians were put in place. To stop any contamination.\"\n\n\"That makes no sense. Guardians? Contamination?\" Caden says, lurching to his feet, interrupting my quiet words. \"Listen to yourself. People don't jump between universes!\"\n\nI shake my head and amend my earlier thought. He may be good with the little things, but when it comes to the big picture, he can be pretty obtuse. \"Where did you think those things came from, Caden? From the zoo? They're from somewhere else, a world just like this one, only far, far worse.\"\n\n\"No,\" he says. \"How is that even possible?\"\n\n\"A lot of things are possible.\" I glare at June. \"Didn't you tell him anything?\"\n\n\"We didn't have to,\" June says. \"Until you got here.\"\n\nI stiffen at her tone, but Caden moves to stand in front of me. \"What are you talking about? What haven't they told me, Riven?\"\n\n\"Get out of my face, Caden. I mean it.\" I can hear the desperation in his voice even in the face of his bravado, but there's nothing I can say. Telling him anything at all means that I'd have to tell him why I'm there in the first place... that I'm as bad as the Vectors... that June is right about me. I push past him, pretending to study the crates of food along the wall. \"I thought you were into all of this scifi stuff? All those DVDs in your room about stargates and whatnot? You're the genius; you figure it out.\"\n\n\"Those are movies.\" Caden's words are slow and deliberate. \"They're made up, you know. Science fiction?\"\n\n\"More like science fact.\"\n\nThe only sound in the room is the shallow hiss of Shae's breathing. Caden is staring at me, disbelief, confusion, and anger written all over his face. I'm not surprised. When Cale first told me about the existence of this world, I thought he was been playing me. But in the end, I understood that technology and physics theory had made it not only plausible, but also possible. And the universe was far wider than any of us really knew. Cale speculated that hundreds of other universes existed, but ours was one of the few to come into parallel contact with another.\n\nI throw my palms into the air and raise an eyebrow. Caden faces my challenge with narrowed eyes, and I can see his mind ticking through the probabilities. \"Even if it were possible,\" he says grudgingly, \"are you saying that Shae \u2013 my cousin \u2013 is sick because she jumped from this world to another universe and back?\"\n\n\"Yes. That's exactly what I'm saying.\" I gesture at myself. \"She everted, just as I did. And just like June, once upon a time.\" Caden rocks back onto his heels, his face as white as a sheet, staring from June back to me as if we're ghosts. I know I'm being blunt, but I don't have time to sugarcoat secrets that Shae and June had concealed from him. \"And she's not your cousin. She's your warden. Your word, not mine.\"\n\n\"Are you serious, right now? I was kidding when I said that.\"\n\n\"Well, I'm not. Look, I don't care if you believe me, or think we're on the USS Enterprise, or think you're dreaming. More of those things are going to come, and I need to think for a second without having to explain the nuts and bolts of quantum physics theory to you. Think about it for half a second and you'll see that it's not as improbable as it seems.\" I open the duffel bag and lay out all of the devices I took off the Vectors. I'm so rattled that I can't help myself when I pick up one of the golf balls and thrust it into Caden's face. \"Does this look like any technology you've ever seen? No? It's not from here. None of us are, except for June.\" I stare him in the eyes, my words like bullets.\n\n\"What are you saying? That I'm not?\"\n\nI turn to June, who's staring at me with a clenched jaw. \"Tell him.\"\n\nJune sighs but doesn't shy away from the task. \"She's right, Caden. What she says is true.\"\n\n\"No. No, that's impossible.\"\n\n\"It's true, Caden,\" June says. \"Shae would tell you the same.\"\n\n\"So you lied to me? All this time?\" Caden hisses to her before spinning to walk away and then twisting back around in the same step. \"Were you ever going to tell me?\" June doesn't answer, but her expression clearly says that they haven't planned to. \"I don't believe this,\" Caden mutters. \"This is insane.\"\n\n\"Believe it,\" I say flatly just as Shae turns heavily on the bed to face us, gasping. June was right. She's looking better after the injector, but she's clearly still in a lot of pain.\n\n\"I'm sorry, Cade,\" she wheezes, \"...my fault.\"\n\nCaden turns toward me, with a measured glance at Shae's tortured expression. His eyes are gentle. \"Still, it doesn't make sense. Even if I believed you, then why would Shae endanger herself, knowing the risks of doing it over and over? That she'd... die?\"\n\nI can't help the twist of my lips nor the snarl that slips from them. \"To protect you.\"\n\nJune is already on her feet at my tone, her body bridling and ready to defend Caden. I unclench my jaw and try to breathe the spiraling rage out of my body. June feels no such self-control and she's in my face before I take two breaths.\n\n\"It's not his fault! That was Shae's cho\u2013\"\n\nThe explosion takes us by surprise, even though I was the one who'd left the gas stove burning in the kitchen, and we're all slammed to the ground in different directions. Pain rockets through my head and along my sides as I thump against the steel door I've been standing next to. Despite the intense throbbing in my head, I jump to shaky feet. Years of training force me to do an automatic check of myself for injuries.\n\nThe Vectors are back.\n\nThey'd be the only things that could have triggered the gas. Everting generates minute pockets of electricity, but for some reason when the Vectors do it, the electrical fields are bigger... big enough to ignite a gas-filled room. The golf balls would have done the rest, and no doubt there won't be much left of the bodies, or anything else above ground, for that matter.\n\nA hazy memory drifts through my head \u2013 now I remember why the golf balls are called cleaners. Hot enough to incinerate bones and liquefy metal, such that anything in their path would be completely vaporized. The heat from the fire diffuses through the heavy trapdoor despite its thickness.\n\n\"What the hell was that?\" Caden grunts, following my lead to stand on shaky legs.\n\n\"A cleaner. One of those silver balls.\" I dust the grit from my clothes and blink the soreness from my eyes. A glance in Shae's direction confirms that she's unhurt; I can hear her labored breathing over the ringing in my ears. \"We need to move. It won't be long before they find that door. And they will. Murek won't stop now.\"\n\n\"Who's Murek?\" Caden says.\n\n\"A dictator.\" I toss a pack toward him, hard. \"Get this on. Take only what you need.\"\n\n\"Where are we going? We can't leave. What about school?\" The inane question throws me for a second and I stare at him. He reddens and adds, \"Shae said\u2013\"\n\n\"School's out, Caden. And I'm in charge, not Shae.\"\n\nI know he's confused, but school is probably a comforting constant. I bring myself back to the task at hand, a part of my brain belatedly realizing that June hasn't gotten up.\n\n\"June, you OK?\" In the seconds that it takes to turn around, the quiet sense of knowing is already like a shiver across my neck. Shae, for her part, is sitting up and staring at June's inert body a few feet from where she's now sitting. The antidote injector has done its job \u2013 despite her bloody clothing and the unexpected force of the explosion, she looks nearly back to normal.\n\n\"June's dead,\" she says.\n\nHer voice breaks the silence and my sudden inability to move. Within seconds, I am at June's side with Caden not far behind me, and I gently pull her inert body toward me. She's been thrown against something sharp and her death was instant. The gash on her head is bloody, her sightless eyes wide open and looking right through me. Questioning... judging even in death. Hastily, I close them and turn to Shae with a deep breath.\n\n\"Are you OK? OK to go?\" I ask her, not hiding the urgency in my voice. We don't have a lot of time.\n\nShe nods, distracted, and I can see that her attention is on Caden. The broken look in his eyes reminds me of someone with little experience with death, but my words fade before I can speak them. In my world, death is an expected companion \u2013 whether in our brutal history or a foray gone wrong outside the city wall \u2013 and I've seen more than my fair share of it. Instead, my fingers find Caden's and I squeeze them, suddenly conscious of Shae's stare that is fluttering like a moth between our hands and my face. I wrench my hand away as if his fingers are on fire.\n\n\"We need to go,\" I growl, removing June's map from my pocket and opening it next to Shae on a small crate beside the bed. \"Have you been down all of these?\"\n\nShae ignores me with a glare to pull the blanket off her bed and tuck it carefully over June. I watch as she and Caden lift the body up to place it gently on the bed. Apart from the blood, June looks like she could be sleeping. Caden stands next to the cot as if he's in some kind of trance, and doesn't move until Shae grasps his shoulders with both hands, turning him to face her and shaking him.\n\n\"Caden, remember what I taught you \u2013 we take them with us in our hearts. Let her go, OK? There's nothing you could have done; it was just her time.\" Her voice is thready but grows stronger by the second as she pulls him into a tight hug. \"We will always carry her with us.\"\n\nThe pressure behind my nose and eyes is sudden, like a blow to the head at the sound of Shae's words, so achingly familiar. She told me the same thing when our mother died. The emotion flooding my body is hot and eviscerating. I swallow past the solid lump in my throat and meet Caden's wet eyes. He's staring at me over Shae's shoulder, and the moment is unending, the mirrored empathy in them acting like a salve on my ridiculous emotions. It is all I can do to tear myself away, grateful for the moment when Shae moves to break the silent and unexpected raw connection stretching between us.\n\nI compose myself, digging my nails into clammy palms so hard that it stings. \"Moving on,\" I repeat stonily. \"The tunnels?\"\n\n\"Haven't changed a bit, have you, Riv? Still as cold as ice.\"\n\n\"Occupational hazard,\" I toss back, smoothing the map on the crate. I'm clenching my teeth so tightly, it feels like they will shatter at any moment. I can't look at Caden even though I know he, unlike Shae, is looking at me and seeing right through my bluster. Thankfully, he says nothing.\n\nShae kneels beside me and jabs at a spot on the map. \"We're here.\" She traces her finger along a faded brown line. \"We need to get to here. It's a long way, about twenty miles.\"\n\n\"That's not too bad,\" I say. On foot and injured, that distance would take about six hours, give or take some rest time for Shae. On my own, it'd be a couple hours max, but I've trained hard, running large distances across unfriendly terrain for years. Shae's still hurt, despite her brave rallying, and Caden... well, there's no way he can maintain my speed without any training.\n\nShae disrupts my thought process. \"That's just the exit point of the tunnel. We still need to get to the Denver airport, which is at least another forty miles away. That's the closest eversion point. Above ground, I mean,\" she adds. \"There's a bus station not too far from where we get out. It's straightforward.\" I stare at her with narrowed eyes. It's the closest she's come to admitting that she'll trust me with Caden. She shrugs, understanding my skeptical glance, and jerks her head down toward her body, still wrapped in bloodstained bandages.\n\nShe doesn't have much choice. Without her, Caden only has me. And without me, he would be a sure thing for the Vectors. I can see the question in her eyes \u2013 whether I'll protect him \u2013 and there's only one answer I can give. I nod.\n\n\"Why can't we... evert from here?\" Caden says, interrupting our wordless exchange. \"I mean, those things, the Vectors did, like Riven said...\"\n\n\"They're dead, remember?\" Shae answers. \"They're designed to evert when and where as necessary. We're not. It doesn't hurt them because their cells are already dead. We have to find certain areas where there's a zero point gravitational field so we can pass through with the least amount of physical and mental aftereffects.\"\n\n\"But what about you? You're sick already. Won't that be bad? I won't leave you, Shae. I can't. Not after...\" Caden trails off to stare at June's body.\n\n\"Let's cross that bridge when we come to it,\" Shae says gently. \"Right now, we need to get out of here with everything we can carry, and fast. Can you get the food packs off that shelf and the gear from the trunks over in the corner?\"\n\n\"Where are we going?\"\n\n\"Somewhere safe. I promise I will tell you everything, but for now, we need to get our gear together and get out of here, or we're not going to have that chance. Remember what I told you back in New York? We have to be ready to leave at a second's notice. Nothing's changed. It's the same.\"\n\n\"But that's back when I thought we were minors not wanting to get separated,\" he argues. \"This is different. Isn't it?\"\n\n\"Yes, and I promise I will explain when I can. Right now, we need to move.\"\n\nShae's become the leader I used to know; the confidence in her voice has Caden in automatic reaction mode. She pulls June's medical bag toward her, tugging two dark turtlenecks and two pairs of black cargos from a nearby box, which she places on the bed, and begins to undress. I'm startled at the mottled colors of her skin, the bruises from her earlier fight fading into a shocking kaleidoscope of purple, yellow, and black, interspaced with bands of red, dirty bandages. Her body, despite its damages, is wiry with lean, corded muscle. I watch as she deftly changes the dressings, smearing antiseptic cream across any open wounds.\n\n\"I don't heal as quickly as you do, remember?\" Shae says with a wry smile, noticing my look. We'd always used to joke about that, my ability to heal quickly, and we'd always put it down to different fathers and the luck of genetics. I used to feel like a freak, but Shae was the one to help me see it for what it was \u2013 a gift, and one that I'd used to my every advantage in battle. \"Here, these are for you,\" she adds, throwing one each of the turtlenecks and pants in my direction. \"We're still close to the same size.\"\n\nI catch it one-handed, looking away as she tugs on her clothes, irritated by her thoughtfulness. Instead, I concentrate on cleaning the blood and pale blue gore from my twin ninjata blades with gun oil until they're as spotless as mirrored glass. Both deadly Artok weapons, they were a gift from Cale. Skilled assassins, the Artok are a tribe from the East, and what's left of them lives in sector seven in Neospes, one of the few areas on the periphery of the core. My mother's grandfather had been Artok. Cale liked the symbolism of it, and I liked his unexpected kindness.\n\nI put the ninjatas carefully to the side and line up all the other items from my backpack on the floor \u2013 sleeping bag, rope, tools, blanket, a collapsible tent, emergency food packets, a survival kit with various first-aid items including the silver case with the anti-eversion injector, which I open; only three left out of the five slots on the cylinder.\n\nDesigned to counter the physical effects of jumping between universes, they were yet another concoction invented by my father's medical team. I frown, staring at the syringes. They are effective, but they're only meant to be temporary fixes to combat cell degradation. The weekly pills are supposed to be taken during the jumps, and the injectors are meant for emergency purposes if anything goes wrong.\n\nI fish deeper into my backpack and pull out a circular silver case. Diligent about taking the stabilization pills \u2013 well, at least until lately \u2013 I've only had to use one of the injectors since I've been here. I have several years' worth of pills left \u2013 I've been prepared to be in this world for as long as it took to find Caden.\n\nBut now, Shae would need both cases, at least until we get back to Neospes, where most of the damage to her nervous system could be reversed with our medical technology. I slide the cases over to her with one hand. If she makes it...\n\n\"You keep these, in case we get separated or anything,\" I say gruffly.\n\n\"Won't you need them?\"\n\n\"I don't plan on coming back here.\"\n\n\"Riv\u2013\" she begins.\n\n\"Don't,\" I say. \"Just take them. You need them more than I do. And like you said, I heal quickly.\" I pause, and stare at the ground. \"Plus, I haven't taken the pills in weeks, and I don't feel any different. No headaches, nothing. It's like I've adapted or something.\" I shrug. \"After I crashed my bike, my body went supernova on me. Figure I nearly died and it wanted to live. Or something like that.\"\n\nShae's eyes narrow. \"You always did recover fast. That's weird, though.\"\n\n\"Don't you think I know that?\"\n\n\"Strong survival instinct.\" Her mouth opens and closes like she wants to say something more, but can't find the words. Her hesitation is grating.\n\n\"What?\" I snap.\n\n\"You ever wonder why... your body can do those things?\"\n\nI stare at her. \"No. Everyone's wired differently. I heal fast. You've known that for years. What's the big deal?\"\n\n\"Nothing.\" Her lips twist, her eyes dropping away from mine as she puts the cases into her own pack. \"Thanks.\"\n\nThe silence is like a web between us, sticky with so many things left unsaid over all the years. It's suffocating, the way she's been watching me when she thinks I'm not looking, studying me like I'm some kind of pariah. I can't stand it.\n\n\"No problem,\" I grit out, and pull the gear bag I'd packed upstairs toward me, removing the Vectors' uniforms. Luckily, they don't stink as much. They're designed with a special self-cleaning technology that eliminates body odors, meant for hours of prolonged use. I dab at some of the stains that look fresh with a rag and smooth them out in front of me just in case \u2013 who knows how long the Vectors had been wearing them?\n\nThey don't look like much, but are engineered for the rough terrain outside of the city wall, where the days are hot and the nights freezing cold. The solar panels on the back and shoulders store and diffuse heat through the suit as needed, and the ventilation pockets recycle sweat for cooling during hot weather. A multitude of pockets and utility latches hold anything from food to gear to weapons. I run my fingers from shoulder to cuff on one of the suits to skim lightly over the recessed keypad in the wristband indented with various symbols. The keypad controls the suit's special programming.\n\n\"What are those?\" I jump in surprise when Caden squats down beside me. \"Wait a sec! Are these the Vectors' uniforms? The dead things?\" I almost laugh at the unbridled disgust on his face. I nod. \"Nasty,\" he adds, and moves away to lug a couple more trunks over to Shae.\n\n\"Maybe, but you won't be saying that when your bones think they're shattering inside of you from the cold,\" I mutter under my breath, rolling up the suits into their reversible pouches sewn into the leg cuffs and tucking them into my backpack. I repack all of the other items, including the ones I also took off the Vectors, and refill the water pouches that fit along the sides of my pack with some of the jugs of drinking water lining the wall.\n\n\"Is there a bathroom down here?\" I ask Shae, who nods toward a door on the left side of the room.\n\nInside the cramped space that's little more than a closet with a toilet and a tiny sink, I remove my filthy torn shirt and wipe the blood from my neck and chest. I don't even bother to smooth the mess that's my hair, but I wash my face with the trickle from the tap, dabbing the cool water under my arms and along my sides. I remove my torn pants and twist over the toilet to examine the wound on my leg where the Vector had caught me with the electro-rod, but I'm surprised to see that it's barely a thin, blackened welt under the crusty blood. I frown \u2013 I've never healed this fast before. Must have felt worse than it looks. I clean it off and pull on Shae's clean clothes before walking out. The pants are a little snug, but they're clean and not ripped.\n\nIn the outer room, Shae and Caden are leaning over a case lined with all manner of weapons \u2013 guns, knives, chains, maces, spears, bows, and swords.\n\n\"Guns won't kill Vectors,\" Shae is explaining to Caden. \"They're programmed to dodge the trajectory of bullets. Something about the sound of the metal, I think.\" Brandishing a curved knife, she adds, \"The only way to kill them, as you saw, is a sharp blow to the head or severing the spinal column.\"\n\n\"But people are much slower than bullets,\" Caden argues.\n\n\"But we're less noticeable,\" I interject, heading over to them. \"That's our advantage over them. By the first bullet, they know where you are. If you miss, you're dead. It's a small window, but usable.\" I pull on the worn black leather harness over my shoulders and slip my short swords into their sheaths flat against my back. Choosing two short knives from the pile, I tuck those into my backpack along with a handful of four-pointed steel throwing stars. \"It's all about speed, flexibility and unpredictability. With a knife or a sword, you have to get in real close, but once you strike true, they go down.\"\n\nShae hefts a mini-crossbow in her hand. \"This is my favorite.\"\n\n\"I don't get it. What's the difference between that and a gun?\" Caden says. \"Plus, I got the one with bullets upstairs, remember?\"\n\n\"Like I said, Vectors can hear bullets coming a mile away. Arrows are a lot quieter. Half the time, taking out a Vector quietly is the biggest challenge, because who knows what else you can attract, or how many of them?\" Shae says. \"And you didn't kill the one upstairs. They can withstand a lot of physical damage as long as you don't touch their vulnerable spots. It's a waste.\"\n\n\"Which weapon should I use, then?\" Caden asks.\n\n\"The sabre,\" I say quickly. \"It's the perfect weapon for you. It'd be like fencing, only you're fighting for your life, not points.\"\n\n\"Super,\" Caden's droll response elicits a short laugh from Shae, and I, too, fight to keep the smile from my mouth, but the moment of levity is gone like a breath in the wind. They share a look that I ignore.\n\n\"We need to move,\" I tell them. \"Shae, how're you feeling?\"\n\n\"Good,\" she responds, hoisting her backpack on her shoulders. Her color is back, and she's standing in fighting stance. That's one thing about the anti-eversion injectors \u2013 they pack a combined punch of pain inhibitors, counteractive pressure meds, and a low dose of epinephrine to get the nervous system functional. Shae tosses a headlamp in my direction and looks at me expectantly.\n\n\"OK, we follow you. Let's go,\" I say, pulling the lamp over my forehead. \"Caden's in the middle, and I'll be behind him.\"\n\nWe make our way through the steel door, dropping the heavy bar across the back once it's shut. If the Vectors make it into the basement room, it won't be much of a deterrent to them, but it will still take considerable force to open it from the inside.\n\nThe tunnel smells musty, and it's dark, with the three beams from our lamps barely piercing the blackness. No one speaks as we make our way, walking as fast as possible. There's enough room for us to probably walk three abreast, but we remain in single file with Shae at the lead and me at the rear. After nearly forty minutes of hard walking, the tunnel forks. Shae takes the one on the left without missing a beat, and I have to force myself to not pull out the map in my pocket. I am flying blind and it's not a feeling I like. Nor do I like depending on a sister that lied to me in the first place, even if we are now working against a common enemy.\n\n\"Are you sure this is the right way?\" I can't stop myself from asking.\n\n\"Yes. I've done this dozens of times before,\" Shae shouts back.\n\nHer words do little to reassure. The tunnel walls are starting to close in, and all I want to do is sprint as fast as I can to the other end. I've never been good with being underground. Performing routine security checks in the belowground shelters back home used to have me breaking out in cold sweats. I practice my old trick, counting softly in my head backwards from one hundred and breathing in through my nose and out through my mouth. It helps a little.\n\nOur pace is grueling, and the calming breaths I'm trying to take in are becoming more and more like shallow pants. I slow for a second to pull an ultrathin device from my pocket, one that I'd taken from one of the Vectors. It tells me how far we've already travelled \u2013 we're just past the halfway point of twenty miles. I pocket the electronic tablet and almost crash into Caden's back.\n\nIt's only at that moment that I realize there's a loud rushing noise coming from a ragged-edged vent in the ceiling, and we're in a wider band of rock. The space is a roughly hewn cave, its walls dotted with smaller caves and darker tunnels. Both Shae and Caden are standing in front of a three-fork tunnel at the far end, their heads together, staring at a piece of paper. They both glance at me.\n\n\"What's the matter?\" I ask. \"We're almost there; why are we stopped?\"\n\n\"The right tunnel may be faster, according to this map,\" Shae says. I nod impatiently. All I want to do is get out of here as fast as possible. It's only after I follow Caden's shadowy form into the tunnel that the thought occurs to me that Shae had said she'd done this trek dozens of times before.\n\nWhy would she suggest a new route? What had they been talking about in the seconds before I caught up with them in the noisy area? My brain jumps into overdrive as the pieces come together enough to make me freeze in my tracks.\n\nBut I'm a half-second too late.\n\nThere's no one ahead of me. In that exact moment, the silence drums into my ears and it's so dead quiet that I don't even hear footsteps. The only sound is that of my breathing. I curse my stupidity.\n\nMy eyes strain forward into the darkness, the beam from my penlight offering little clarity. There's no one there; they're gone. But they have to be close. Shae knows these tunnels inside out \u2013 there has to be some kind of alcove nearby, somewhere they're hiding. I close my eyes and exhale silently, letting my other senses do the work. But Shae knows me too well. There's nothing, no movement at all. I could be alone, even though every instinct inside of me screams that I'm not.\n\nClenching my teeth, I delve into my pack, searching futilely for the pair of infrared glasses I'd taken off the dead Vector in Caden's house. I hear an indistinct noise like the sound of some tiny animal rustling around behind me, and my hands grasp the hilts of my blades as I swing around. Nothing. The darkness surrounds me, heavy and dry, as I inch back the way we'd come. I release my grip on the swords, leaving them in their sheaths for the moment \u2013 I can't risk hitting Caden. Shae, I don't really care about one way or another. She has proven that she is still my enemy. I resort to words instead of blades.\n\n\"Shae? What are you doing?\" I whisper furiously into the dark. \"We need to stay together. You're hurt and you need me. Shae! Where are you?\"\n\nI'm still inching along when I sense the shift in the air and I swing around into a crouched stance, protecting my body instinctively with my forearms. The cold tip of an electro-rod presses against the soft spot just beneath my ear, and my body freezes. I forgot how quiet Shae can be, catching me unawares for years during training when we were little. I gnash my teeth in frustration. Her voice in my ear is soft with bittersweet notes of regret.\n\n\"I'm sorry, Riv. I can't let you take him. Don't try to find us.\"\n\n\"Shae, don't\u2013\"\n\n\"And I'm sorry for this, too.\"\n\nI feel a sharp zap against my skin, and then the darkness blinds me.\nTHE PREY\n\nLights blink in and out. Warm sun flickers against my eyelids. I can hear someone laughing, and I turn toward the voice from where I'm lying, hefting myself onto my elbows. Lights flash again, this time like popping light bulbs exploding behind my eyelids, as a searing pain lances through my shoulder.\n\nCoughing, I taste burnt blood.\n\nIt all comes back to me in a rush. Caden, the Vectors, the dark tunnel... Shae. There is no sunlight, no laughter. Instead, I'm breathing in the rank, dusty air of the tunnel where my sister has left me. Pulling myself to my knees, I'm still groggy from the electro-shock. Obviously, Shae had set it to stun, but it still hurts something fierce.\n\nGingerly, I touch the welt along the side of my neck and wince. She's been generous; this stun is more or less mild. I would have taken out an enemy \u2013 even if it were my sister \u2013 without a second thought. Those rods have a kill setting that could liquefy the insides of anything human, and melt the internal wiring of anything not human. But even so, one of the stun settings could knock a live person out for days at a time.\n\nI look at my watch quickly, noting that I've only been out for a few hours. A quick check of my body, other than the welt on my neck and a cut on my lip from where I'd fallen, tells me that I'm otherwise unhurt. My pack lies off to the side, hanging drunkenly off one arm, and I hoist it onto my lap. They haven't touched it. My blades are still snug against my back under my jacket. At least Shae hasn't left me with nothing to protect myself, even though I wouldn't have done the same.\n\nI haul myself up against the wall and stand, trying to get my bearings and ignoring the dizziness that threatens to make my knees buckle.\n\n\"Get a grip, Riven,\" I snarl to myself. \"It's not like you've never been on the wrong end of an electro-rod before.\"\n\nGrabbing my pack, I remove the first-aid kit and pour some cold liquid from a slim bottle onto a piece of gauze, careful not to let it touch my fingers. I dab it onto my neck, a shiver snaking through me as an icy sensation immediately dulls the raw ache of the welt. The liquid anesthetic hardens into a thin, flexible shell over the sore area, its under-layer seeping into my skin to deaden raw nerve endings and rebuild cells. Within seconds, the pain is gone, and within an hour I know my neck will be as good as new.\n\nNormally, I'd just leave my wounds to heal on their own, but now I have no time to lose. The cell-regeneration remedy is yet another of my father's inventions... and one that I'd steadfastly refused to use. Using anything of his makes me sick to my stomach, but now it's a necessity to find Caden quickly. It's a brutal reminder of what is at stake \u2013 I can't let my hatred for my father affect my decisions and actions now.\n\nPocketing the bottle, I try to reorient myself. I shine my flashlight down one end of the tunnel, and it's soon swallowed up by the blackness. I do the same down the other end. Eyesight isn't going to help me, so I close my eyes, engaging my other senses and letting the flow of the stale air in the tunnel waft around me. The changes are subtle, but they're there \u2013 the ones that tell me which direction has more movement in the airflow.\n\nWithout hesitation, I sprint down the tunnel on confident feet. Recalling the treacherous, veiny patchwork of the tunnel map, I know I can get lost with a single wrong turn, so I'm careful not to veer off the pathway. If I can make my way back to the place where the tunnels fork into three, I'll be able to figure out which way they've gone and track them from there.\n\nI run past several other tunnels and alcoves that I haven't noticed before, refusing to let any fear enter my mind. But it does, inexorably. And I know I've made the wrong choice.\n\nJust backtrack, I tell myself silently. Follow your feet, and trust your instincts.\n\nI can hardly help the next thought that follows that one \u2013 as my instincts had told me to trust Shae? But they hadn't; my emotions had. I grit my teeth and press on, clearing my mind of any thought but getting to the end of the tunnel. And within minutes, I do.\n\nOnly it's a dead end.\n\nI punch my fist into the wall and a shower of pebbles scatters at my feet. How could I have missed a turn? I stayed straight, didn't I? Could I have missed it somewhere?\n\nThink! I urge myself. I retrace the path in my head, then backtrack about half a mile before I see it \u2013 a barely discernable twist in the path. I had veered in the wrong direction into what was now clearly an offshoot from the main tunnel. I take a deep breath to calm my racing nerves. Things could be far worse. I could have ended up running in circles or gotten even more lost.\n\nBack on the right track, it's no time at all before I am in the area with the four tunnels \u2013 the one that we'd come from, and the three we'd chosen between. Squatting down, I notice faint scuffmarks in the dirt in front of the tunnel on the far left. A slight color change in the ground suggests that this tunnel has been used more than the others. I check my watch. Shae and Caden are probably near the other end, if not out already, but the window of opportunity isn't completely closed for me to track them. Still, I have to move, and fast.\n\nI'm just about to enter the leftmost tunnel when something stops me dead in my tracks \u2013 the sound of something heavy moving, something coming from behind me... something big. The hairs on the back of my neck stand at stiff attention, because I know that it can't be Caden and Shae. They would never have gone back.\n\nIt has to be Vectors. They've found us. Or more precisely, found me.\n\nI deliberate between making a beeline down the tunnel and facing them head-on. But I have no idea how many they are. For a second, my body feels like it is splitting down the middle with equal urges to fight and flee pulling me in opposite directions. It's not in my nature to run, but fighting an unknown number of Vectors in such an enclosed space will not be to my advantage, despite my skills.\n\nI decide to wedge myself into one of the many alcoves lining the walls of the cave. I'll get some idea of their numbers and assess potential attack options. And, at the very least, their tracking technology is far better than mine, and we are looking for the same thing.\n\nThe enemy of my enemy is my friend.\n\nNot that I'll ever align myself with Vectors, but I will use them however I need to, and then get rid of them when I don't. Hoisting myself up the cave wall, I find a recessed nook and crawl inside to wait, pressing my body back into the dark space until rocks are digging painfully into my flesh.\n\nMy eyes adjust slowly to the muted dark of the outer cave. I've covered my scuffmarks in the dirt and sprinkled myself with anti-tracking dust from the bottom of my pack. They probably aren't even looking for me, but I have to play it safe on the off chance that they are. My father has his own reasons for wanting me back in Neospes.\n\nIt isn't long before they enter the big cave: three of them, with one a familiar face, the ruthless commander from before. My teeth clench. The smell of them hits me like a rolling wave, the pungent scent of formaldehyde. Even though I'm used to it, it's something that automatically raises the hairs on my whole body. In Neospes, we cremate our dead, except for the Vector soldiers, who are put through an unnatural rigorous embalming-like process. They carry the smell of death like armor.\n\nHalogen lights on their uniforms illuminate the cave. I watch the commander carefully. It was a tough fight earlier, and its ability to speak had been unnerving. I can't help noticing that its bullet-ridden body has been completely repaired in a matter of hours. It's nothing for our reconstructive technology \u2013 the technological differences between my world and this one are like night and day. But then my mind flashes back to the abundance of water in this world and extravagances like Caden's waterbed. Limitless water over advanced robotics is a no-brainer. So is a world without creatures like these, without the Vectors.\n\nWith inhuman stealth, the Vectors move purposefully, examining the ground in front of the three tunnels. The big one turns to study the rest of the cave, and I imagine his eyes slowing and stopping at my alcove. I can hear the blood rushing in my ears and the shakiness of my breath in the dead silence. I'm barely breathing, and even though I know he can't possibly see me, for a split second, it feels like our eyes connect. Adrenaline rushes through my bloodstream, but then the moment is gone as he turns back to his subordinates, pressing a series of buttons on the wrist-pad of his suit to initiate the tracking device.\n\n\"Two trails,\" he says. His voice is guttural, and as before, it chills me to the bone.\n\nA red light streams from his wrist as it scans the entrances to both tunnels. Yellow markings spin through the red, more on the right than the left: heat readings. The commander moves over to that tunnel pressing another series of buttons, and the light switches to a pale laser-like blue. A long noise bleeps on his console, loud in the silence and almost making me jump.\n\n\"The tracks return,\" he growls and studies the device before stepping over to the far left tunnel to repeat the sequence. This time there are three short beeps, and he nods, satisfied. \"The boy went this way. Move out,\" he commands.\n\nThe other two nod and immediately obey, disappearing into the tunnel. For a moment, the Vector commander turns around and re-scans the room. His gaze doesn't stop at my hiding place this time, but I can feel his suspicion, and this, too, is unlike the traditional programming of the Vectors. They operate by computer rules and programming algorithms, not instinct. Can he somehow sense me? He punches another sequence of numbers into the wrist-pad, different from the ones he used before. Unconsciously, I press myself deeper against the wall and ignore the searing slice of stone against my skin.\n\nMy heart jumps into my throat as he flashes the same blue tracker in a slow clockwise circle. I can only hope that the anti-tracker dust will do its job. As the blue light filters into my cave, I can see the analysis data in the light discoloring its surface, and I hold my breath. These scanners are built to analyze individual scent \u2013 yet another advanced biometric profile weapon incorporated into the suits \u2013 almost like electronic sniffer dogs. Terrified, I wait for the beeps that will surely come if he's looking for me, but there's no sound.\n\nThe light moves past my cave, but I don't exhale until the Vector Commander has completed his scan and followed the other two into the tunnel. My muscles remain clenched until enough time has passed by, when I scramble out of my cramped hiding space. It takes only a couple minutes for my blood to recirculate through my body, but I am already moving, digging through the bottom of my pack and locating the infrared glasses I couldn't find earlier. There's no way I can see in the dark or use the penlight \u2013 they'll see me coming from a mile away.\n\nI head into the tunnel, walking as quickly as I dare. I am tracking on scent alone, the unique smell of their dead flesh wafting on the stale air in the tunnel. The sharp odor of it makes me remember the first time I saw one of them.\n\nIt'd been during a time when my whole family was still together \u2013 my father, my mother, Shae, and I \u2013 a time before betrayal and lies ripped us apart. At only six, and one of the youngest recruits, I'd been released from training early and called to my father's underground experimental lab. At the time, he was the head scientist in the advanced robotics and genetic testing facility, and already navigating the waters of reanimating the dead with cyborg technology.\n\nUsing my unrestricted passkey \u2013 being his daughter had its privileges \u2013 I found him on one of the lower levels in one of the test labs. Unnoticed in a corner of the outer office, I stared fascinated through the glass wall into the room beyond where my father had just finished decontaminating a corpse on a long silver table. Decontamination, my father once explained to me, meant getting rid of all internal bodily fluid and unnecessary organs, and preserving the remaining husk and heart with an electro-chemical solution.\n\n\"It's all biological,\" he said to me. \"The body is a capable host, even though it's no longer alive. With the nanoplasm, we can use and program these shells to operate almost as well as a fully functioning live person would. And they would make even better soldiers, as there's no emotion, just programming.\" He paused then to look at me with dark, narrowed eyes. \"And the beauty of them is that they're expendable. One command, and the nanoplasm shuts down. No loss, nothing compromised. Think of it as a type of recycling.\"\n\n\"How do they go to the bathroom?\" my perfectly logical, then six year-old self asked in all seriousness. My father smiled widely and lifted me onto his desk.\n\n\"Smart question,\" he answered. \"The simple answer is that they don't. We remove all the parts that we don't need, like the kidneys or the liver or the stomach. We keep the heart because it moves the nanoplasm around the body, and we keep part of the brain and spinal cord to process the information we give it.\"\n\n\"What makes it go, then? Like how does it work? What do they eat?\" I cocked my head and frowned. \"It's not a robot, is it?\" Even then I couldn't keep the trepidation out of my voice, having learned about our violent history in my civilization lessons. The Tech War had obliterated our world, leaving the sparse little we had now as a harsh lesson of the perils of artificial intelligence.\n\n\"No, princess. They're safe. This nanoplasm responds to programming only. It's not self-aware.\" He patted my head reassuringly, as only he could. \"And they don't eat anything; they have a special lithia core that keeps the heart pumping and the brain ticking. Once you compromise the spinal cord or the brain, it's an automatic kill switch. Don't worry, sweet; it's all under control this time.\"\n\n\"What are they called?\"\n\n\"Vectors.\"\n\nThe entire process disgusted and fascinated me at the same time. I couldn't imagine, even then, how a dead person could be used as a soldier, and watching my father at work was the first time I was able to see the process up close and personal. I remained crouched in my corner, thrilled and terrified of being caught, watching as my father and his team worked the corpse from top to bottom.\n\nMachines around the body beeped constantly, with long tubes of various sizes connected to the table. They were filled with metallic-colored fluids. Slowly, in succession, two of the tubes were emptied, and then combined into a larger tube before being injected into the dead man's body. Two of my father's assistants exited to another adjoining secure room filled with flashing computer screens. My father followed them just as the third tube with the silvery blue fluid slowly started emptying like the others into the body beneath it.\n\nHe punched some numbers into the keypad on one of the desktops. Something was starting to happen in the room. The corpse on the table bucked and began convulsing against the metal shackles connecting it to the table. I could see the bunched muscles of its arms and legs cording as if it were in terrible pain, even though my father had said that they couldn't feel anything. And then suddenly, it stopped moving, and the only noise was the flatline sound of a long beep.\n\n\"Dammit!\" I heard my father swear and punch the wall before stalking out of the room into his office, where I was hiding. He raked his hands through his hair as he studied some papers lining his desk, grinding his teeth in frustration. Not wanting to get caught in the crossfire of his anger \u2013 which could be nasty at times \u2013 I curled my body into as tight a ball as I could manage, pressing myself into the wall behind me, and prayed that he wouldn't notice me.\n\n\"We did the sequencing right. What did I miss? What did I miss?\" he muttered to himself before punching a button on the intercom and hissing to his assistants, \"Get me another prepped body. We go again in thirty minutes!\" Without even looking in my corner, he left his office through the outer door, slamming it behind him. A shaky breath left my mouth but I stayed curled tightly for several seconds before standing very carefully.\n\nThe area was empty... except for me and the Vector lying on the table. I couldn't help myself, knowing that I only had a few minutes before my father or his assistants returned, and without hesitation, I punched in the code on the inner door. My birthday. In hindsight, I always wondered how my brilliant father could be so clueless or predictable as to use his daughter's birth date as a code on one of the most dangerous areas in the facility. But the thought of it made me feel warm inside \u2013 he may not be the most demonstrative of fathers, but this was a sign, a sign that I mattered. Ignoring the warning clench of my stomach muscles, I pushed open the door.\n\nInside the room, I approached the body with trepidation, even though I knew it was not operational. The test had failed. My nose wrinkled against the suffocating chemical smell that caused my eyes to sting. Still, I inched nearer. Up close, the thing was huge \u2013 a dead giant of a man. His skin was a dull gray color, bleached out by all the compounds stopping it from decaying, but the metallic fluids now inside of him still gave his body muscular definition. Clad only in a pair of medical undergarments, his bare chest and arms were sleek and hairless, his head shaved. I moved closer to the table and placed my hand carefully alongside the hand caught by the metal shackle on the table. They were ten times the size of mine!\n\nA long red scar on the giant's side caught my attention and I bent closer. This was where the lithia cell my father had told me about would have been inserted to connect to the lower spine. I ran my finger along the cold line of flesh and shivered. It felt like clammy linoleum. The man also had a tattoo running along the top of his thigh that read, \"Test Specimen 74.\" Seventy-four of them they'd already burned through... no wonder my father had been so upset. It wasn't like we had dozens of dead people lying around; as a society, we valued life too much.\n\nI was so intent on staring at the number and thinking about how little of us were left in our tiny pocket of the world, that I didn't feel the yank on my head until it was too late, and the tears were already springing to my eyes in agony as something hauled on my braid with brute, uncompromising force.\n\nPanicked, I wondered whether my hair had gotten caught on the table somehow, and twisted despite the tearing sensation on my scalp to see what I was caught on. Instead, I found myself face-to-face with a milky-blue stare that was so devoid of any life that my terror made me freeze completely.\n\nThe giant was alive! But it wasn't possible. The experiment had failed.\n\nAt the exact moment that I realized that the experiment hadn't failed, the alarms in the room went off, and things galvanized into motion as the dead man broke through the steel shackle like it was butter \u2013 oblivious to the silvery-blue fluid that poured down its forearm from where the sharp steel edges had razored through flesh \u2013 and taking my head with it until I was half spread-eagled over its body. It was as if my head were on fire, my skin tearing off of my skull and my hair loosening in whole clumping handfuls.\n\nClose up, its eyes were even more terrifying, sucking the life out of me with their blank deadness, completely devoid of any soul. The thing opened its mouth, and I started screaming. I didn't stop screaming, not until hands pulled me back, and I felt someone cutting away its grip on my hair. Voices fluttered in and out. I could see huge steel needles being jammed into the thing's side as people tried to hold the giant in place, until it finally shut down. I glanced at its face once more, and even though it was completely lifeless; its colorless blue eyes still stared at me as if the creature wanted to swallow me whole.\n\nFor years afterward, I couldn't even look at them \u2013 their gray faces and milky-blue eyes, terrifying \u2013 a constant reminder of that moment when I'd been caught unawares. I'd never trusted them, not ever, not even when I'd led them. They were a means to an end, and Murek, the Legion Commander at the time, had loved them because of their absolute loyalty. Despite my own reservations, my father had been right. The Vectors had made excellent soldiers.\n\nFinding myself in a new open cave area underground, I bring myself back to the present, clearing my head from my memories to focus on the task at hand. They are great soldiers, but their allegiance is and will always be to Murek and the crown \u2013 a loyalty that is programmed into the very fluid that runs in their dead veins. Even though the one in Caden's house somehow recognized me as its general, I know that it had been a glitch \u2013 and one that will be quickly rectified. Nothing will stand in the way of getting Caden... not even me.\n\nThere's only one exit out of the space at the far end, and I make my way there on silent swift feet. I am so intent on getting into the tunnel that I don't sense the attack until it is too late, as something large and heavy tackles me to the ground. My night-vision glasses are jerked from my head in the collision, and I'm suddenly flying blind. Without losing a beat, I roll and kick upward with all the force I can muster, feeling my boots connect with a thick body and pushing it off of me. In a flash, I'm in a crouch, taking three steps back. I loosen my backpack and remove the swords from their sheaths against my back.\n\nI can't see it, but I know it's one of the Vectors. It has a slight advantage because it can see me. I say slight because part of our training means fighting blindfolded. All of our senses are honed to the point of razor-sleek sharpness, and facing an opponent in darkness is par for the course. I take a deep breath and pull it into my center, letting it fill me and closing my eyes. I exhale and wait.\n\nThe air shifts to my left and instead of spinning away, I move toward it, at the last minute grinding to my knees and slashing outward with my blades. They strike into something thick and heavy \u2013 a leg or upper thigh. The thing makes no sound, but now I can hear it moving as it drags one foot slightly. It lunges toward me once more, and this time I wait and take the hit on my left flank, moving out slightly so that I don't receive the full weight of the strike.\n\nWith Vectors, hesitation is the difference between life and death. Ignoring the pain, I whirl alongside the Vector's body until I am facing its back. My swords don't falter, swinging up and scissoring across the back of its neck in a smooth decisive motion born from years of honing the same move in training and in actual combat. I hear the thump of its head followed by the second thump of its body, and pull the thin penlight from my pocket, trying to filter the sharp light through my fingers. I'm still cautious. The creature had been waiting for me, but there's no other movement or noise.\n\nI retrieve my glasses quickly \u2013 they aren't broken, thankfully \u2013 and find my pack. In the muted green glare of the glasses, the Vector's shape is a dull outline. I quickly divest him of any of his gear that I can carry, including a wireless communications headpiece, which I tuck into my ear, and an electro-coil, a flexible strand of wire attached to a thick short handle, which I slip into my boot. I haul his body to the side of the area, and move toward the tunnel.\n\nThat's when the screaming starts.\n\nREVERSAL OF FORTUNE\n\nI bolt down the narrow passageway without any thought for my own well-being. All I can hear is Shae's bloodcurdling scream, and it chills me to the bone. It sounds as if she's being gutted while still alive. I'm flat-out running because I know the Vectors have no use for her, and while I am still upset at her betrayal, there's no way I'd want my sister to suffer or die at the hands of those things... and especially not the big one.\n\nBut as I draw closer, I can see that he's not there. And neither is Caden.\n\nI refuse to panic, or even imagine that somehow the big one has already taken Caden back. I take a deep breath \u2013 first, I have to help Shae. The infrared green vision of my goggles identifies thrashing movement down the tunnel in front of me, and I click a button on the side of the arm that magnifies the image \u2013 there are only two of them and I recognize Shae's wiry thinness immediately.\n\nThe Vector lifts her body and slams it into the side of the wall like a sack of potatoes. I dart forward, ducking my head and full-on tackling him to the ground. There's a dull thud as our bodies crash to the dirt, and I'm rolling, sword in hand, before I can even take another breath.\n\nThe Vector is just as quick and already on his feet facing me. I spare a glance at Shae. I can smell the rusty odor of blood in the cramped space. Dread makes me dash forward, but I misjudge the Vector's movement, as his fist catches me in the solar plexus and knocks all the air out of me.\n\nGasping, my knees buckle and I swing my blade weakly across its side, but not before its elbow catches me in the back of the head. My face smashes against the rough rock of the tunnel, and I can feel the sharp sting of my skin peel off against the stone even as stars blind my vision. The Vector is on top of me before I can move, bashing my weapons out of my hands. I curl into a ball to protect myself, because I can't get a strike in edgewise.\n\nShae claws at its body from behind, and distracted, it slams a heel in her direction. I hear a vicious thump \u2013 presumably her body against the tunnel wall behind us \u2013 before it swings back to me, blows crunching into my ribs like a jackhammer. The pain is excruciating, but I force myself to stay focused.\n\nSomehow, I have to get out of this, if only to find Caden. If I don't, we are all lost. The thought gives me a boost of strength as my hand slides down the side of my leg into my boot to grasp the handle of the electro-coil I took off the other Vector earlier. With a lurch, I flip my body around and jam my legs against the wall with all the force I can muster, wrapping one knee around the thing's head. In quick succession, I twist to the side and jerk my knee backward, but I'm not quick enough.\n\nI'm thwarted as the Vector's knee smashes into my skull, disabling my glasses... and suddenly, I'm blinking as my eyes adjust from infrared vision to shadowy darkness. In that second, it's on top of me, crushing the breath from my chest with its weight. Fingers close around my windpipe and I can only kick wildly, my strength seeping out of me, clawing at its face and head in desperation. But it's too strong and I can't find the leverage to get out from under it.\n\nSuddenly, the Vector crumples like a dead weight against me, its steel fingers loosening around my neck, and I'm wheezing long breaths of the stale tunnel air like it's the freshest I've ever breathed. A flashlight flicks on and Shae's there, a dripping electro-coil in her hand and the severed Vector's head a few feet away from where she's standing.\n\n\"Don't you think we're even, because we aren't,\" I rasp, squeezing out from beneath the creature. Shae smiles weakly and helps to kick the rest of the Vector's body off of me. \"Where's Caden?\" I ask her, and then say in the same breath, \"How badly are you hurt?\" Even in the dim light of the flashlight, I can see that her shirt is soaked with blood and she's holding her middle with one arm.\n\n\"Just a scratch,\" she says, but I know that she's lying \u2013 there's too much blood and a heavy, foul odor of charred flesh. \"Don't worry; Caden's safe. He's outside. The big one shot me... he got too close, and I couldn't slow Caden down, so I told him to go, and then I backtracked.\"\n\n\"Outside where?\"\n\n\"He's safe,\" Shae repeats resolutely, and I glare at her. There's nothing but mistrust left between us. My eyes narrow.\n\n\"Are you sure? What about the big one? Did you get it?\"\n\nShae's eyes darken with frustration. \"Just before you got here, it everted. I almost had it. One minute my blade was at its neck, and the next it was gone. It talked, too. About you.\"\n\nI stare at her sharply. \"What did it say?\"\n\n\"That you were a fugitive, that you would betray Caden, and me, that you weren't to be trusted.\" I keep my eyes fixed on hers, and her voice continues softly. \"But I knew that already. Then it said that it was sorry... sorry that it hadn't killed me when it had the chance.\" She pauses, and I know what she's going to say even before she finishes her story. \"That's what got me, the knowing in its voice, like it somehow knew me. I hesitated with the kill strike and then it was gone.\"\n\n\"It's father's latest creation,\" I say flatly. \"Thinking Vectors, as if they aren't terrible enough already. It's some new prototype, one with its own memories. Did you hear its voice? It's the same as his.\" Softly to myself, I say, \"A tribute to his vanity.\" My eyes connect with Shae's, and for a second, it's like we're trainees again, but the moment is gone in a breath, suffocated by everything since between us. \"We need to get out of here.\" I know that she isn't going to tell me where Caden is, but I can track him easily enough on my own once we get outside. \"Which way is out?\"\n\nShae nods back in the direction from where I'd come. \"That way. Look, Riv, I'm sorry... for back there.\"\n\nI don't look at her, and instead pocket the dead Vector's special terrain glasses, which are way more advanced than the night-vision ones I have. \"Forget it. You did what you had to. I would have done the same. Or probably worse.\" I shrug. It's no secret between us what I am or what I'm capable of. \"You're OK to walk at least until we get outside?\"\n\nShae nods and falls into step with me. \"It doesn't mean I'm not sorry. I just can't let anything happen to him.\"\n\n\"Well, you did a good job of that, didn't you?\" As soon as the words are out of my mouth, I regret them. She did the only thing she could have, short of killing Caden herself. \"I just mean he has no training and he's on his own.\"\n\n\"He has training,\" Shae says softly. \"Haven't you seen him fence?\"\n\n\"No.\" I'd missed his meet the one time I'd promised to go, helping to save Charisma from those predators. \"Why?\"\n\n\"I taught him. All the techniques I know. It's why he's so good; he's ranked number one in the state.\" Before Shae defected, she'd been a master swordswoman. Though her personal preference was the crossbow, she'd been chosen to instruct others, myself included, in the intricacies of sword martial arts. She'd been the best of the best, until she'd trained me.\n\n\"What did you fight against? Dummies filled with straw?\" I snicker.\n\n\"He fought me. He's good, Riv.\"\n\n\"Good enough to fight a Vector?\" I shoot back.\n\n\"Good enough to fight you.\" I remain silent, feeling the threat still thick in the air. I pick up the pace a bit, knowing that it won't be easy for Shae, but she doesn't complain, despite her slightly labored breathing. We've both been trained to withstand near-fatal injuries, and I don't feel any sympathy despite her being my sister. She'd simply refuse any offer of my help, anyway. \"The Vector also said there's a bounty on your head. They want you alive.\"\n\n\"I know. He wants me back.\"\n\n\"Why?\" Shae asks.\n\nI shrug. \"To punish me, I suppose. I did torch their biggest genetic research lab before I left.\"\n\n\"You didn't!\" I can hear the smile in Shae's voice, and I stifle the brief whisper of pride that flutters inside of me.\n\n\"That was years ago,\" I say flatly. \"Who knows what he's planning now?\"\n\nWe make our way out of the underground after a few more miles, and we end up in a deserted warehouse on the far outskirts of a neighboring town. I pull a couple bales of hay over the trapdoor that we just exited. It won't stop anything from getting out, but we'll at least have some notice if the hay starts moving.\n\nShae fumbles in her pack and jams one of my injectors into her leg before collapsing to the ground and closing her eyes. Two in the space of a few hours is not exactly what they're designed for, but I can see that our pace has cost her. Despite my reticence, I move to her side. I owe her my life, not that I would ever tell her that. She doesn't protest when I unbuckle her vest, only to see a gaping hole on her left side with singed, blackened edges. A shard of something shiny glimmers on the inside of the wound.\n\n\"Part of the electro-rod,\" she rasps, wincing as my fingers gently touch the sides of it.\n\n\"Why didn't you say something before?\" I grit out. My voice is angry, but it's directed at myself, not her. I can't think about how painful it would have been, and I feel even guiltier for not checking in the tunnel or at least offering some help.\n\n\"No time before. Just need to... get it out now.\"\n\nI nod and spray my hands with an alcohol solution from my med-kit. I can only imagine what had happened before I got there, when I'd first heard her scream. The Vectors are known for operating their electro-rods in open wounds. Even on the stun setting, the agony is brutal, like a laser on skinless tissue. It's one of their well-used torture techniques. The Vector must have wedged it so hard into her that the silver tip of it had shattered.\n\nMy body cringes with a phantom pain that isn't mine. The agony would have been excruciating. I shiver and assess the damage carefully. The good thing is that the electric shock somewhat cauterized the gash, which means that at least Shae won't bleed to death. Waiting for the adrenaline from the injector to kick in, as gently as I can, I pull out the sliver of silver and check carefully for any other stray pieces. By the time I'm certain there aren't any left, I've removed twelve shards.\n\nI glance at Shae. Her eyes are closed and her breathing is shallow but even. I use the rest of my father's numbing repair liquid to patch up the gaping wound, deftly taping a square of thick padded gauze from the med-kit over it. Tearing a strip from around the base of my shirt, I wrap the material as tightly as I can around her waist. I'm not entirely sure that the liquid and the bandage will do the trick, but it's all I can do.\n\n\"Thank you,\" Shae whispers, staring at me as I buckle her vest closed. \"Why are you even helping me? You should just leave me. You want me dead, remember?\"\n\n\"I don't want you dead,\" I say dully. Despite my anger at what she's done, I don't want her to be hurt at my hands. She is my sister, after all. Me wanting to kill her and someone else doing it are two totally different things. \"Shae, I don't want you dead,\" I repeat firmly, as if to convince her and myself at the same time, and slide down the wall to sit by her side. \"Look, I can't promise you anything, but trust me, I don't want to hurt Caden. The Vectors do \u2013 they want to kill him. For all I know, Cale could already be dead and Caden is going to be the last hope in his line. Either way, I need him. We need him. Don't you get it?\"\n\n\"He's a person, too. He's real. Surely you can see that?\" she says, countering my question with one of her own, and for a second, I'm afraid to answer. Because the truth is, for some reason when I think about him, my chest tightens in jerky response. And I know that Caden isn't just a target anymore. I don't answer Shae, but she sees the reaction in my eyes. \"I see you do,\" she says softly.\n\n\"What I feel doesn't matter.\"\n\n\"Yes, it does,\" she pauses, her voice quiet. \"Because it means I can trust you with him. And trust that you'll do the right thing. I thought I saw it before, but you were so cold, I just didn't know. You've always been so good at keeping your feelings hidden. But now, I see it. You do care about him. Don't you, Riv?\"\n\nI hate the way her words are making my feelings about Caden even more real than they already are. And I'm not ready to deal with them yet. I'm not ready to open myself up to anyone, especially not Shae. I don't look at her.\n\n\"They're not going to stop until he's dead,\" I say softly. \"Until we're all dead or taken.\"\n\n\"He's back at Horrow,\" Shae says after a while. She digs into her pack and takes out the circular case of stabilization pills. She pops a couple into her mouth. \"That's where he went.\"\n\n\"Horrow?\" I reply, ignoring the stab of immediate worry that twists through me at the sight of her taking more pills on top of the injector. The meds can turn toxic in the body with overuse. She must be more desperate than I thought, to risk dying. \"But that's the first place they'll look.\"\n\nShae shakes her head and smiles. \"That's the beauty of it. They won't. Vectors don't understand high school, or the concept of school in this world, because their programmers don't understand it.\"\n\n\"What?\" I say, confused.\n\n\"The idea of high school for kids this age doesn't exist in Neospes. It's a foreign concept to us. So, unless the Vectors were ordered specifically to look there, they won't. Make sense?\"\n\n\"Not really,\" I say. \"Won't they still track him there?\"\n\n\"Eventually, but we have a day or two at least. Caden won't lead them directly back there.\"\n\nBut the more I think about it, what she's saying makes an uncanny sort of sense. The Vectors follow orders \u2013 they don't have the intuitive sense to think for themselves. They tracked Shae back to their house. I think back to when I came here on my own years ago, and the concept of school was been so utterly alien to me that it'd been a huge adjustment to even try to pretend to be a high-school student. In fact, I still am not good at it, which is why more often than not I usually get the \"most likely to be a sociopath\" label.\n\n\"I don't get it. Why do we even have to go back there? Caden should meet us. We should stick to the plan and get to the eversion point. That's what we're\u2013\"\n\nShae's expression freezes the rest of my words on my lips. \"That's not an option anymore. I got a text before. It's crawling with Vectors. The minute they knew where we were heading, they swarmed it. They're waiting for us. It's a trap, Riven.\"\n\n\"So what are you saying? We can't get out?\"\n\n\"Not there, not anymore. We need to regroup and rethink.\" I stare at her, my eyes narrowed. Her last betrayal is still an open wound, and despite her earlier words about trusting me, I can't trust her for a second where Caden is concerned. \"What do you mean, you got a text? From who?\"\n\n\"A Guardian.\" Shae doesn't offer any more explanation other than those two words, but I continue to press. It's not enough for me. I need to know what her plans are and who her friends are... and whether they, too, will toss me over to the Vectors the first chance they get. I have no doubt they would know exactly who I am. Plus, Guardians are solitary in nature and spread out in this world. More than one in any one area is an anomaly.\n\n\"Another Guardian? At Horrow?\"\n\nA glare in response to my doggedness. \"Yes. Let's go.\"\n\nShae stands wearily. If she's in pain, she certainly doesn't show it, but I know that she's operating on pure will right now. The stabilization pills on top of the injector would have helped slow her heartbeat and numb most of the pain. People always used to say I was the tough one, \"the ice queen,\" they'd called me. But Shae's got a core of solid steel \u2013 she could probably outlast me ten times over on sheer will alone. Only, I remember now that Shae isn't operating just on will; she's got a body packed full of an explosive cocktail of meds.\n\n\"Shae, the pills\u2013\"\n\n\"I know what I'm doing,\" she growls.\n\n\"Do you?\" I say gently, and follow as she moves to the back of the warehouse to throw open a door to a small wooden shed that I had noticed upon my first look around. She pulls out a grungy half-rusted dirt bike and then another. Then she stops to look at me.\n\n\"I'm dying, Riven. My body is dying. I can feel it. It's been through too much.\" Her words are no surprise, but I'm shaken all the same. They echo in the sudden dead silence between us.\n\n\"We can go to a clinic, the one where June worked. They'll help you.\"\n\nA strained laugh. \"Are you kidding? And what will they do when they find strange fluids and medicine from the injector inside of me that don't even exist here? That's only going to draw more attention to us... to Caden. As far as I see, I've got about one more solid fight in me, and I'm not going to go down without it.\" Her eyes meet mine. \"Actually, I don't intend to go down at all if I can help it, and if by some miracle, I make it back to Neospes, maybe... he... Father or someone else can help...\"\n\nPerhaps it's the blunt admission that she's dying, but I have no idea what comes over me as I grab Shae so tightly that we both almost fall over. She stinks of blood and gore and Vector fluid, but all I can smell is the scent of lilacs and sunshine and horse-riding leather \u2013 the scent of our childhood \u2013 and it's nearly my undoing. Shoving her away roughly without daring to meet her eyes, I hoist myself onto one of the bikes and grunt over my shoulder, \"You OK to ride?\"\n\n\"Yes,\" she says. I keep my eyes downcast. My thoughts are heavy and spinning. Wanting to kill an enemy, even if the enemy is my sister, is far different from knowing that I will lose her regardless... knowing that she's going to die and there's nothing I can do about it, even if I wanted to. Which I don't.\n\nDo I?\n\nA dull ache takes hold of my forehead and nose, and then my eyes are stinging with an unfamiliar soreness. I thought I had purged all emotion after the betrayal of my father, but the hot sadness snaking through my veins right now screams otherwise. I gun the engine, letting the roar of it dull my thoughts and the vibrations beneath me soothe my knife-edged anguish.\n\nEmotion is weakness, my inner voice says. She's just a person, nothing more.\n\nShe's my sister.\n\nOne who betrayed you.\n\nI would have done the same.\n\nShe'll betray you again.\n\nI don't have an answer. The argument bounces off the walls of my brain... the logical side of me saying that the odds are Shae will deceive me again. But this previously dormant side, this quiet feeling side, argues otherwise. Still, despite my unexpected compassion, deep down I know that it's too late for either of us. She's dying and I'll have to let her go when the time comes.\n\nShae's bike growls past me and I follow her down the dirt road that leads to a main road. It's deserted. I glance down at my watch. It's early, just before 6 in the morning on a Wednesday. Shae doesn't stay on the main road, and soon we're off on another circuitous gravelly road that winds up into the base of the Rockies. Even though there are barely any other people around, she isn't taking any chances of being tracked.\n\nThe view is overwhelming, and my breath stops in my throat at the sight of a deep lake-filled gorge on my left. The water is like glass, so reflective with the barest tint of the rising sun dancing across its surface that it's like looking at heaven and earth wrapped up in each other's arms. I have never seen anything so incredibly beautiful in my life. I haven't even realized that I've come to a stop on my bike until Shae backtracks toward me.\n\n\"What are you doing?\"\n\nIt takes me a minute to find my voice, and even when I do, it's raspy and hollow with longing. \"I've never seen anything like this. Ever. Can you imagine having water like this back home?\"\n\n\"It's called the Horsetooth Reservoir, and we used to,\" Shae says softly. \"Before the war.\"\n\n\"It's so pure, so untouched... so beautiful.\" My voice trails off and I'm lost once more in a silent reverence that's so sweet, I am consumed by it. Watching the sun's rays dance across the surface as they shimmer over the tops of the mountains it feels like I'm in another world, one so perfect that all I have to do is breathe and believe. There's no war or hate or pain, just beauty. Soothed by the water's glittering depths, somehow I feel restored.\n\nAfter a few minutes, I gun the bike's engine and nod to Shae. We ride in silence until we reach the edges of Fort Collins. Time seemed to go a lot more quickly to get back there than when we had left, but in all fairness, we had been on foot in the underground tunnels. Things are starting to get a little busier. Ever careful, we stick to the lesser-known roads off the main street running through the town, but nothing seems different from the hustle and bustle of any other standard weekday morning. Still, I am diligent in assessing anything at all that looks out of place. We are very careful not to go anywhere near their old house, although by now it's probably razed to the ground. No doubt, someone will be watching.\n\nAfter a while, Shae pulls off to the side and I follow suit. \"What's up?\" I ask her. On both sides of us are quiet horse farms, and the only living things are two ravens circling overhead and a few grazing animals that look like horses off in the distance on the left. At the top of a small hill on the right side of the road, is a rambling farmhouse.\n\n\"Did you notice anything coming into town?\"\n\nI shake my head. \"No.\"\n\n\"I didn't see anything either,\" she says. \"Doesn't mean they aren't here, so we need to be extra cautious, just in case.\" Shae looks almost back to normal. The fresh air has put some color back into her cheeks, but I know that it's probably all still a fa\u00e7ade. I can tell by the slight hitch of her left side and from the way she's holding herself.\n\n\"How's your wound?\" I ask.\n\n\"I'll live,\" she says. \"We should split up.\"\n\nMy eyes narrow immediately. \"No way,\" I snap. \"You can't make it on your own, and the truth is, I don't trust you.\"\n\n\"Riv, we don't have a choice. We have a better chance of getting to him separately.\" Her voice wavers. \"And the truth is, do you really think I could outrun anyone \u2013 even you \u2013 in this condition? And you know that,\" she adds gently. \"Go to Horrow. You're still a student there. I'll meet you afterward in the parking lot next to the gym.\"\n\nA thought occurs to me, one that I hadn't followed up on earlier in the conversation, about the Horrow connection. A hazy recollection of Caden asking about school in the basement of his house flits through my brain. Maybe it'd been more than just a comfort. Is it some kind of safe house?\n\nI frown at Shae. \"We are running for our lives. Why is Caden in school, of all places?\"\n\n\"Like I said, it's the safest place he can be right now. Gym. After school. OK?\"\n\nI nod, my eyes burning into her face, daring her to betray me. But I know she's right \u2013 there's no way she could outrun me, not in her condition. \"What then? What's your plan? What's to stop me from just taking Caden and leaving the minute I get there?\"\n\n\"Because you need my help, and he won't go anywhere without me. We have allies, Riven. They will protect him.\" She moves over to stand near me, where I'm still straddling my bike. \"There's a lot you don't know, Riv. I didn't know if I could trust you, but now\" \u2013 Shae pauses, gesturing to her body \u2013 \"it seems like I have no choice. I'll tell you everything. 3 o'clock at the gym, I promise.\" I don't even notice as she slides her fingers along mine, hooking her little finger into the crook of mine. \"Sister-swear,\" she whispers, and all I can feel is the hot imprint of her skin against my own with the force of a hundred memories behind it.\n\nI may be a fool, but I trust her.\nREVELATIONS\n\nI stare down at my filthy shirt and the dried blue remnants of dead Vector under my fingernails. Shae's already left, and I'm standing on the road facing the farmhouse that I saw before. I can't see my face, but I can feel the dirt from the tunnels caking my skin. There's no way I can go to Horrow looking like I've just been at the wrong end of a rodeo bullfight gone bad. What I wouldn't give for a long hot shower!\n\nFor a brief second, I toy with the idea of heading back to the motel where I'd stayed before, but it's on the other side of town. If I waste any more time, I'll be far later than I already am. They take tardiness very seriously in high schools, I've noticed, as if it's some odd measure of teenage responsibility.\n\nAlthough I have a healthy respect for punctuality, being late has saved my life more than a few times. In Neospes, predictability is something that can get you killed. After a few more minutes of hemming and hawing, I decide to head up to the farmhouse to see if I can get cleaned up and swipe a fresh shirt. On the porch, I'm torn between breaking and entering, or just asking for help, which is another utterly foreign concept to me.\n\nI glance at the watch on my wrist \u2013 I can do it in five minutes. In and out. I try the handle of the door, and it opens inward with a loud creak. I sigh inwardly... looks like I'm going to have to play the damsel-in-distress role, and I almost laugh aloud. The thought of me being any kind of damsel is hysterical.\n\n\"Hello?\" I call out.\n\nNo one answers. Not wasting any time, I check the doors along the front hallway, encountering a closet and a kitchen before I find a small bathroom. My reflection greets me like a still clip from a horror movie. It's worse than I've even imagined. Blood, blue-gray ichor, and dark brown and black smudges streak my face like an artist's palette. My hair is clumped into a tuft on one side of my head and plastered to my scalp on the other. My clothes are disgusting, and my pants are ripped all the way down one leg. I'm probably going to need some other clothes, but first things first.\n\nI peel off my leather jacket, outer shirt, and ninjata harness, remaining in a thin tank, and dunk my entire head under the faucet, washing the grime from my face and shoulders with water as hot as I can manage. A couple minutes later, with a twinge of guilt, I'm toweling myself dry with one of the two pristine white embroidered towels from the towel rack, when I hear a sound behind me. Without thinking I swing around in fighting stance before I can blink, to see the face of a small boy. His blue eyes are merely curious, so I relax slightly.\n\n\"Hello,\" he says. \"Are you one of my sister's friends?\" I nod slowly and peer around his small frame to see if anyone else is there. The boy continues to chatter. \"She already went to school. I'm sick today. Don't tell that I'm out of bed, OK? I only wanted to get a drink. Mommy's out checking on one of the horses. It's having a baby!\" He starts coughing then until his eyes start to water. I follow him to the kitchen, where he helps himself to a cup of water. \"You're really dirty,\" he remarks. \"And you smell a little.\"\n\n\"What's your name?\" I say in a soft voice, glancing nervously to the back porch. A seven year-old I can handle, not a parent... especially one who would quickly recognize me as not one of her daughter's friends.\n\n\"Josh.\"\n\n\"You're right,\" I say. \"I was on my way to school, and I fell off my bike. Do you think you could help me? I don't want to be late. I need to borrow a sweater from your sister. Could you show me her room, and we can get you back into bed?\" My voice drops to a stage whisper. \"Don't worry; I won't tell if you won't tell. I don't want anyone to know that I fell into horse poop. You promise?\"\n\nJosh giggles and wrinkles his nose at my confession, but his face warms with a conspiratorial smile. \"Promise.\"\n\nAfter grabbing my pack and dirty clothes, we head up the stairs. Josh points to his sister's room and heads into his own across the hall without a backward glance. Looks like I've lost his interest for the moment, so I move quickly to the room he'd pointed to.\n\nIt is painted completely in shades of pink with pristine white furniture, and is the most delicate room I have ever seen. A canopied bed dominates most of the room. Pink and white floral curtains flutter at the window. The carpet is plush, and also pink.\n\nI'll be better off taking clothes from Josh!\n\nThe sound of a door closing from downstairs startles me into action and I open the owner's closet door. It smells like vanilla and roses, and everything inside it is also pink and white or pastel-colored.\n\nI sigh, resisting the urge to gag, and methodically search through the racks... full of dresses, skirts, and sweaters, all color coordinated. Doesn't this girl own any pants? More noises from downstairs have me moving faster, but starting to panic. After a second, I see a pair of dark-wash jeans on a hanger, and I pull them on only to find that they can't button and are six inches too long. This girl is not only a princess, she's a tall skinny princess.\n\nSo that eliminates pants, I think to myself. Someone is moving around near the stairs, and I grab the only dress I can find that isn't vile and fling it over my head. It's a bottle-green color with a lighter green-and-yellow pattern of vines that comes to my knees, but at least it isn't pink. There's a floor-length mirror in the adjoining bathroom.\n\nThe soft cotton feels good against my skin. The dress doesn't look half bad, and I'm spared from looking too girly, especially as it's paired with my worn black combat boots. I pat down my wet hair and grimace. I look good but still edgy... a motorcycle-riding badass in a dress. It's not that I haven't worn dresses before; they're just impractical.\n\n\"You look pretty,\" Josh says standing in the doorway of the bathroom. \"That's one of Sadie's favorites. It's from Grandma in New York.\"\n\nA moment of pure mirth bubbles against my lip. \"Sadie? Your sister is Sadie?\" Josh frowns and nods as if I've asked the dumbest question, and I bite my lips. Sadie? As in Caden's girlfriend? I shake my head rolling my eyes skyward... someone has a sick sense of humor. I think briefly about changing, but a voice coming up the stairs stops me.\n\n\"You better be in bed, young man,\" a lilting voice warns.\n\n\"Pinkie swear, I won't tell if you won't,\" I whisper with a wink, kneeling and locking my finger with his before Josh bolts to his room. \"See you around, Josh.\"\n\nAs Josh disappears across the hall, I swing my harness on, tuck in my blades, and pull on my jacket. I grab my backpack, stuffing my pants into it before stepping out of the window onto a narrow balcony. I'll get rid of them later. I shimmy down the apple tree leaning against the side. A few minutes later, I'm back on the bike and heading toward Horrow.\n\nBreathless at my sneaky entrance to the school through the locker rooms, I slide into my seat long after the last bell chimed down the hallway, but it must be my lucky day because Mrs Taylor is nowhere to be seen. A momentary panic grips me as I don't see Caden in his usual place in Physics. Shae had said that he'd be here. Then I notice that about half the class is missing.\n\n\"Where's Caden?\" I say.\n\n\"You're late. Nice dress, by the way,\" Philip says, and I glare him into silence.\n\n\"Where's everyone else?\" I snap.\n\n\"They're in lab. Group lab experiment today, remember?\" I can only stare dumbly at him. \"We go in for the second half of the class when they come back. It was the homework from last week?\"\n\n\"I was sick. What homework?\" Philip doesn't answer, but Charisma looks over at me shyly and slides over her notebook to my side of the table.\n\n\"It's about refraction and diffraction. Don't worry; I've got you covered,\" she whispers. In an even softer voice, she says, \"I never got to thank you properly. Thank you, Riven. I don't know what I would have done if you hadn't been there.\"\n\n\"It's OK,\" I say, stunned into momentary silence by her gratitude. \"They had it coming.\"\n\n\"Where'd you learn to fight like that?\"\n\nBefore I can answer, Mrs Taylor walks into the room, her dark eyes piercing. They land on me immediately, and I feel myself shrinking down in my seat. The last thing I need is any kind of confrontation, to draw even more attention to myself. Her brows snap together as her eyes dart to the clock and back to me, but she doesn't say anything, just walks over to her desk for a folder and leaves the classroom. I'm grateful for the reprieve, as small as it is, and stare down at my textbook even though I feel her eyes flutter like a moth on me once more at the door.\n\n\"Did you hear about Caden?\" Philip whispers, confirming that she'd left.\n\n\"What about Caden?\" I say, suddenly alert.\n\n\"His house burned down. Electrical fire, they're saying. His aunt was in there. Everyone's talking about it. Where were you this weekend? Under a rock?\"\n\n\"I was sick,\" I repeat. \"Did they find her aunt?\"\n\n\"No,\" Charisma says. \"But she's missing. They're assuming the worst.\"\n\nI say nothing. It's likely that June's body will never be found in the hidden basement room. My guess is that the Vectors who tracked us into the tunnels torched it completely once they figured out where we went. \"How's Caden doing?\" I say.\n\n\"He seems really out of it, but I guess that's to be expected. He was at a travel meet when it happened, I think. I don't even know where he's staying. Maybe at Sadie's.\"\n\nI try to resist even thinking about Caden being in that frosting-colored room or lying on that pink ruffled bed covered in frivolous throw pillows, but it's useless as a sour feeling invades my stomach. I crush it with a hiss and make my mind blank. What Caden does with Sadie makes no difference whatsoever to me.\n\nMrs Taylor walks back in and announces that the rest of us should make our way to the lab. In the hallway, I see Caden a split second before he sees me. His eyes widen as they glance from my face to my boots, and I can hardly hide the answering scowl overtaking my face.\n\n\"Nice dress,\" he says, echoing Philip's earlier words as we brush shoulders.\n\n\"Yeah, if you're homeless,\" Sadie says, standing next to him with narrowed eyes. She's dressed in a pair of white pants and a pink sweater. I'm actually breathless with anticipation to see if she will recognize the dress as hers, but of course, she doesn't. Still, I can see her brain ticking over, considering the fact that somehow \u2013 as impossible as it seems \u2013 we own exactly the same dress. \"It's hideous.\"\n\nI can barely hold back the truth that stings my lips. Instead, I smile sweetly at her and say, \"Like you would know anything about fashion. I think it's from New York. It's designer.\"\n\nShe shoots me a look that would incinerate a building. \"Well, you should stick to pants. Dresses don't really suit dykes,\" she says nastily.\n\nI bite back a grin and shrug. \"Takes one to know one. And I'd rather be a dyke any day than look like a frosted cupcake without any imagination.\"\n\nI won't admit it to anyone, but it actually feels good to have a pissing contest with Sadie \u2013 if anything, to take the edge off everything that's happened during the last few days. In a weird way, it feels normal, and just the feeling of being able to flay Sadie with my tongue instead of my blade gives me great satisfaction.\n\nCaden's eyes are burning through me, even though he's remained quiet during the entire exchange. I feel his arm brush against the sleeve of my jacket, and then their group is past us. As we walk into the lab, I pull a piece of paper from my pocket. Shae must have given him a heads-up that I'd be back.\n\nMeet me in the boys' bathroom in ten minutes, it reads. I crumple it and tuck the paper back into my pocket.\n\nI fumble my way through the experiment, but Charisma is as good as her word, picking up my considerable slack. Philip doesn't volunteer anything, but I can see from his expression that Mrs Taylor will undoubtedly hear about it at some point. Although I'm familiar with the subject matter, I am too distracted to pay attention, which is probably for the best. I'd rather be noticed for being brainless than for knowing far more than I should. Refraction is something we learn as part of our weapons training, since some of our more advanced laser weapons harness refraction technology. Philip's eyes would likely bug out of his head if he knew that our technology could bend lasers around corners.\n\nAfter eight minutes, I excuse myself and bolt to the bathroom. It's empty but I wait in the last stall until I hear the outer door open and close. Caden is dragging the trashcan against the door as I step out of the stall. His eyes are clouded, and his face is tired. The last two days must have been a whirlwind for him.\n\n\"You OK?\" I ask.\n\n\"Is Shae OK?\" Our words merge into each other's at the same time.\n\nI nod. \"She's fine. She told me to meet her in the gym after school.\"\n\n\"Yeah, she texted me the same thing. Is she hurt? That thing almost got us. She told me to go, and the last thing I saw was her running toward it.\"\n\n\"She took care of it,\" I say.\n\nCaden stares at me awkwardly. \"Look, Riven, I'm sorry for what Shae did. I don't know why she did it. She didn't explain anything to me. But I wanted to say that I was sorry.\"\n\n\"It's OK. She did what I would have done.\" I tug with strangely clammy hands at the skirt of the dress. For some reason, being in it with Caden looking at me is making me feel edgy and uncomfortable. When I'm wearing my own clothing, it's like armor. I'm a soldier first and a girl second. Now the reverse feels true. I don't like it.\n\n\"I'm still sorry.\" Caden's voice is quiet and my heart is thumping like it's going to jump out of my chest as he steps closer until he is so near that I can see the downy fuzz on his cheekbones. I stop breathing as goose bumps break out along my arms. Thank the stars my jacket is covering them. \"I was really worried that one of those things would have gotten you.\"\n\nI clear my throat, the rough sound gritty in the short silence between us, and swallow. \"Don't you know? I can take care of myself,\" I say, desperately trying to sound nonchalant and only succeeding in sounding breathlessly weak.\n\n\"I know.\" Caden's fingers are soft as they trail down the side of my head, and it's all I can do to not lean into them. The soft caress is more than I've felt in a long time, and pressure builds behind my eyes and across my cheekbones.\n\nGet it together, I tell myself angrily, and step backward, only there's a sink in the way and there's nowhere else I can go. My eyes slide to Caden's chest and focus on a button, as I will myself to toughen up. Don't even think about crying!\n\nBut it's like trying to stop a dam that has too many cracks, and my tears are no match for Caden's aching gentleness. And so they come, hot and earnest, as I lean my head against him and feel his arms wrap around me. For the first time, I forget what I am and give in to just being a girl... not a soldier, not a grown-up, just a girl.\n\nI cry for myself, I cry for Shae, and I cry most of all for Caden. Deep down inside I know that no matter how much I love Cale and where my loyalty lies, there's no way that I could ever be the one to take Caden's life from him... even to save Cale. It's a realization that leaves a part of me numb. It's also one that terrifies me, because for the first time I'm confronted with some of the feelings I know I've been hiding. Shae's right \u2013 I do care about him.\n\nAnd it will be a problem.\n\nCaden's arms are tight, holding me so close it's as if we are one body. I hadn't realized that my arms are twisted around his, my fingers gripping his shirt. There's not even room for breath between us, and even though my tears stop, we still hold each other close. Truth is, I don't want to move, because I know that the moment I stir, things will be different. They have to be. There's no way I can let this thing blossoming between us get in the way of what I have to do... of who I am. But for one moment, I can pretend that we are somehow more than we are, that we are in a universe of our own making, that we are in love and the world is ours.\n\nCaden's breath brushes against my skin, and I feel his lips press into the hollow of my temple. It feels as if my entire body is disappearing into the imprint of his touch against me. My body tenses automatically, rejecting the light caress.\n\n\"Just let go,\" he whispers against my hair. \"You don't have to be strong every single second. Let go, Riv.\"\n\nFor one millisecond, it's the only thing I want to do, as I let my eyes find the deep green vulnerability of his. I can't even think around the knot in my chest at what I see there, and I know he's waiting... waiting for me to say something... to agree, to do something to make it all real. But I can't.\n\nLove is weakness, my father's memory hisses in my head. It infects your will.\n\nI close my eyes as the lessons of my childhood overwhelm me with barbed cruelty. We're taught early on to curb our emotions and to make decisions with sound logic. Love isn't something that's valued in Neospes. Families are engineered based on survival abilities and genetic compatibility, not love. It's a useless emotion that causes people to do stupid things in its name. Emotion is the seed of weakness, and making any more of this thing between Caden and I will lead to nothing but loss or pain.\n\n\"Riv?\" His voice is gentle and so beautiful that every cell in me bends toward it. But I can feel the cold chill of reason battling within me. Nothing good can ever come of this.\n\n\"I can't,\" I whisper, and push myself away with all the strength I can muster, the cold edge of the sink grazing against the small of my back as I twist past it. It hurts more than I can imagine to even look at him. \"I just can't... not with you. Not with anyone. I'm not built like that. You'll only get hurt, Caden. I don't know how...\"\n\nMy words trail off and I press my hot palms to the cold porcelain of the neighboring sink, my eyes falling away from his. My face is unrecognizable in the mirror above it \u2013 puffy eyelids and flushed cheeks. My hair is mussed, and I fix it automatically, sweeping away any evidence of Caden's fingers. I splash cold water on my face and feel better, more like myself. But I still can't face him, so I dart a look at his reflection.\n\nHe's standing quiet, his face open and vulnerable. The fleeting thought that Cale would never be this way flits through my brain. He's always been unreadable, the master of all emotion. Some call him cold, but I see it as a gift. Hardly anyone ever knows what Cale is thinking. In the same breath, I understand that maybe this is what gets to me about Caden \u2013 he's so different, so unguarded.\n\nOur eyes meet and he smiles, a deep gentle smile that unhinges any resolve I have. \"Don't shut down, Riven. Please. I know what you've been taught to be, but there is something real between us. I know you feel it, too.\"\n\n\"You're wrong.\"\n\n\"Am I?\" He's standing right behind me again, his gaze impaling mine in the mirror, daring me to lie. I don't answer. Instead, I swing around and shove my hands into his chest so hard that he crashes into the metal doors of the stalls behind us. I resort to the only response I know: combat.\n\n\"You're going to fight me, Riv?\" he gasps, hugging his chest with one arm.\n\n\"If I have to,\" I say. \"Just stop. Stop right there.\"\n\n\"What if I don't?\" he says, advancing once more toward me, one step, then two.\n\n\"You won't like what you get, Caden. I don't want to hit you.\"\n\nAnother step. \"That's a risk I'm willing to take.\"\n\n\"I'm not,\" I say, and drive a closed fist into his shoulder. But either I'm not committed to the blow or Caden's gotten faster than I'd given him credit for, because he deflects my strike with an easy dart to the right and grabs my arm, my own momentum spinning me with considerable force into the wall at the end of the bathroom. Caden's right there with me, his breath warm in my face. \"Let me go,\" I seethe through clenched teeth. I twist my knee up, but he presses his body into mine so hard that my limbs are glued to the wall behind me.\n\nIrritation courses through me in violent waves. But something else flows hotly too, and I feel my breath falter. The truth is, I don't want to move. The only thing holding us in place is the length of his body against every inch of mine. Something unfamiliar stirs in my chest. I wrangle my free arm up under my jacket to pull out one of my weapons, but Caden grabs it with his hand. My breaths get shorter and shorter, and suddenly his mouth is crushed against my lips.\n\nEvery thought in my brain fizzles into shocked silence. And then I'm kissing him back, following his lead, with an intensity that has our teeth grinding together. I don't even know where he begins and I end \u2013 my lips are part of his mouth, and his part of mine. My hands, now free, are twining in his hair, and his arms are locked around me. Every living part of me is caught up in him.\n\nThe grating sound of the trash bin grinding across the floor pierces through the fog encasing us both, and I open languorous eyes to see Sadie, her body shaking with rage.\n\n\"What the fuck is going on?\"\nSCARS\n\n\"What the hell happened to you?\" Shae blurts out as we meet her in the gym parking lot after school. Shae, for her part, is looking a lot better. She's clean and has on a pair of dark jeans and a green sweater. Most of the blood is gone from her face and she looks semi-normal.\n\nI, on the other hand, am a completely different story. After a trip to the school nurse and principal, the rest of the day had passed in relative quiet, despite the occasional censuring look from students and teachers alike. I touch the three claw marks on my face and grin. Caden stares at the ground, but I can see that the corner of his lips is twitching.\n\n\"Sadie happened,\" I say.\n\nShae's eyes narrow, jumping from me to Caden and back again. \"Caden's Sadie?\"\n\nEven though I bristle inside at the comment that Sadie was Caden's anything, I shrug. \"She's a little high-strung, if you hadn't noticed. Anyway, she attacked me for no reason,\" I say. Caden launches a hot look in my direction, but I ignore it. \"I took care of it; don't worry.\"\n\n\"What do you mean, you took care of it?\" The suspicion and patronizing edge of Shae's voice irritate me. I know that this isn't Neospes, and apart from what she may think about me, I do have half a brain in my head. I throw my hands into the air in exasperation.\n\n\"I drowned her in a toilet,\" I say, and then after seeing Shae's expression, \"Kidding! Look, Princess Pinkalicious is fine. She'll have to go to the salon to fix those curls, but a little toilet water never killed anyone.\"\n\n\"Don't look at me!\" Caden says as Shae turns the force of her glare on him. \"I got punched in the face. And I think she broke up with me.\"\n\nThe brief flutter of elation I feel at Caden's words disappears under the heavy weight of reality as Shae herds us toward a green car parked in the lot. Deep down, I know that Caden will probably be a hell of a lot better off with Sadie than with someone like me. Despite letting my guard down in the bathroom, I don't know the first thing about being anyone's girlfriend or letting someone get close to me.\n\n\"You're better off without her,\" Shae says, catching my attention. \"There's something about her that rubs me the wrong way, like she's too perfect. I never got why you liked her. She just didn't seem like your type.\"\n\n\"And yet you let me date her for a year?\"\n\nShae shrugs. \"I only protect you. I don't smother. If Sadie was your idea of a girlfriend, then that was your choice. Out of curiosity, though, why'd you go out with her?\"\n\nCaden flushes, embarrassed at the direct question, but he answers Shae. \"Because it was easy. I didn't have to think about anything. And you're right, she isn't my type. I didn't even know I had a type until recently.\"\n\n\"So what is it?\" Shae asks. She sounds as if she's only making conversation, but I'm listening very carefully.\n\n\"Not her,\" Caden says. His glance slips to me, and then falls to my lips before looking away. An immediate response sears through me, but I refuse to think about the kiss and the way it'd made me feel like a live electrical wire. I'd never even kissed anyone before, and now I can't look at him without my ears flaming or my pulse racing. I hate it. I hate how unpredictable it makes me feel, like I can't depend on myself anymore, like a part of me is caught up in someone else. It's unfamiliar and unwelcome territory.\n\n\"Where are we going?\" I snap gruffly to Shae.\n\n\"Somewhere safe.\"\n\nI nod to the dirt bike I left parked on the right side of the green car. \"I'll follow you guys.\"\n\nNot that I don't want to go with Shae, but I like having my options open, especially if I need a getaway ride. I also don't want to be in such close proximity to Caden. He makes me agitated for obvious reasons, and I need to have all my wits about me going into this blind. Shae's expression remains calm, but she flicks a raised eyebrow in my direction.\n\nAfter a couple seconds, Shae jumps into the car, and nods for Caden to get in. Instead, he walks over to me sitting on the bike. I tense immediately at his soft touch on my shoulder. \"What's wrong?\"\n\nI can feel Shae's heavy stare through the car window. Sighing, I look Caden full in the face, steeling my voice and my heart. \"It never happened, OK? This thing with you and me. It can't happen. Do you understand what I'm saying?\"\n\n\"Why? What do you mean?\" His voice is wounded, and my eyes fall away from his, unable to bear the pooling hurt there. \"I thought\u2013\"\n\n\"No, you were wrong, Caden. I didn't feel anything. I was testing to see how far you would go,\" I say without inflection. I know the words are like daggers to him, but I don't care. I have to end it before it grows into something worse, something that could endanger the both of us. So instead, I deaden any emotion inside of me with practiced ease. \"I'm sorry you thought it was something more. It wasn't. I just don't feel that way about you.\"\n\nThe pain in his voice is worse than I could imagine. \"You were testing me? For what? To lead me on? To see me make a fool of myself?\"\n\n\"Yes.\" The word is like a gunshot, but it fulfills its purpose. Caden backs away, his eyes wide and angry. I can feel my betrayal eating away at him. And even though my insides feel like a pulverized mess, I know what I'm doing is right. Maybe in a different time, we could have been more, but not now... not with everything at stake. Wearing Sadie's dress has made me weak. It has made me just as selfish and as self-indulgent as she is.\n\n\"Shae was right, wasn't she?\" he whispers. \"I thought she was out of her mind when she called you the ice queen. But you are, aren't you? You're heartless, Riven.\"\n\nI flinch at his words, but I shrug, nonchalant. \"I've been called far worse. You'd do best to keep your distance from me, Caden. I'm no good for anyone,\" I say quietly. My voice is barely a whisper now. \"Shae knows me better than most. I don't know how to love. Just ask her. She'll tell you.\"\n\n\"How would she know anything about you? You just met.\"\n\nI stare at him sadly. \"Because she's my sister,\" I say, and gun the engine on the bike, pulling away as fast as I can from the shattered look on his face. At the end of the parking lot, I wait, watching as he gets into the car with Shae. He doesn't even look at me as they drive past, and I'm grateful for that even though it cuts through me like a hot blade.\n\nAs I follow Shae's car, my thoughts inexorably return to the kiss. It had been so unexpected and so fiery that even now my chest heats up just from thinking about it. I don't know what I feel about Caden and whether those feelings are indeed separate from Cale. The thing is that Cale is like a brother to me, but with Caden, there's something more. I was primed to love him just from knowing Cale, but I'm falling in love with him because he's Caden. I'm so confused that it's making me crazy, my thoughts whirling into a blurring jumble in my head. The only thing standing out from everything else is whether I'm being disloyal to Cale.\n\nBecause I can't help but feel like I am betraying him somehow.\n\nAhead of me, Shae slows down in front of a wide iron gate. I can't see the house from where we are, but I expect that there is one behind the imposing wall of metal. The gate is attached to a high stone wall, and for a second I wonder what whoever lived in there wanted to keep out. Or in.\n\nThere is a small camera on the right side of the gate, and I can feel its eye centered on me. After a couple seconds, the gate slides open. My pulse is racing, but I follow Shae inside onto a long and narrow driveway flanked by slender pine trees. The gates close noiselessly behind us, and the creepy silence of it is unnerving. The thought of an asylum drifts through my head and I shiver. Something isn't right. I don't know if it's nerves or adrenaline, but I'm on edge and my instincts are screaming.\n\nBefore I can accelerate to cut in front of Shae to voice my misgivings, we pull up in front of a looming stone house. It's as forbidding as the iron gate and stone wall surrounding it. Removing my helmet, I dismount and check to make sure all my weapons are in place, just in case. Even if they're Shae's friends, I still don't trust easily. Everyone is an enemy until they prove otherwise.\n\nThere's a person on the stairs in front of the giant double door, and Shae's shaking hands with him. I follow, intentionally tucking Caden behind me. His feelings toward me are obvious, but I ignore them. His safety is paramount. The person turns, and for a second, my breath halts in my chest as a pair of familiar obsidian eyes freeze my body in its tracks.\n\n\"Mrs Taylor?\" I say, dumbfounded. \"You're the Guardian?\"\n\nA tight smile. \"Riven,\" she says with a nod. \"Welcome to my home. Any friend of Shae's is a friend of ours. Please call me Era.\"\n\nCaden's eyes are as wide as my own, but he remains silent. I hide my own surprise with indifference as I walk past her through the door. Her home is dark on the inside with heavy shutters covering all the windows. My tension mounts a notch, and the hairs on the back of my neck are at stiff attention. It takes everything inside of me not to draw my weapons.\n\n\"Drinks? Food? Can I get you anything?\" Era asks as we follow her down a few steps into a long hallway that has no windows.\n\n\"No, thanks,\" Shae says. Caden nods that he didn't want anything either, and I stay silent. There's no way I'm eating anything in this house.\n\nShae and Era are at the front with Caden behind them. I'm bringing up the rear. In the narrow hallway, I feel my instincts kicking in, and not in a good way. I catch up to Caden and squeeze his arm. His eyes meet mine. For a minute, it seems like he's going to ignore me, but then he falls back alongside me.\n\n\"What's up?\"\n\n\"Look,\" I whisper. \"If anything happens, get behind me, OK?\"\n\n\"Why? What's wrong?\"\n\nI purse my lips and shake my head, any awkwardness between us forgotten for the moment. I pause, checking that Era and Shae are still far enough ahead of us. \"Something doesn't feel right. If Mrs Taylor knew about us, why didn't she say anything? It doesn't make sense.\"\n\n\"Maybe she didn't know,\" he says. I feel his eyes studying me. \"I didn't even know you and Shae were related, remember?\"\n\n\"Stop,\" I hiss at his potshot. \"Even if she didn't know about me, she had to know about you. And if she is a Guardian, why wouldn't Shae have warned her about me?\"\n\nThere's a long pause before Caden answers. \"I see your point.\"\n\nUp ahead, Shae and Era have stopped at a gray metal door at the end of the hallway. It reminds me of the door that June had in her house, the one leading to the secret room in the basement. Only on either side of this door, there are two recessed keypads with LED biometric pads. The lights in the corridor flicker and I frown. Era turns to me with a tight smile.\n\n\"It's nothing; don't worry. A power surge,\" she says, and nods to a white table in the corner. \"Please leave any weapons you're carrying over there.\"\n\n\"What's behind that door?\" I counter flatly, ignoring her reassurance that does nothing to reassure me one bit. Neither does the command to leave my weapons out here. The muscles in my neck remain as tight as coiled springs. A power surge? What's in that room that would affect all the power in the house? Some kind of electrical torture chamber? My frown deepens.\n\nEra smiles a smile that goes nowhere near her eyes. But it doesn't surprise me \u2013 she's a Guardian, and an active Guardian at that. They're chosen for their complete lack of reaction. They make decisions based on logic and reason, and they don't deviate from the jobs they're supposed to do.\n\nIt's hard to reconcile the two Mrs Taylors \u2013 the teacher at school and the uncompromising Guardian standing in front of me. Without a doubt, I know that she's a Guardian first and a teacher second. She isn't my friend \u2013 after all, I'd flaunted the Guardian's cardinal law by defecting to this universe. The only reason she's giving me the time of day is because she thinks I'm an ally of Shae's. I wonder just how much Shae has told her about me. Would she have told Era the truth? I take a deep breath to calm my panicked thoughts.\n\n\"I mean it. I'm not going in there,\" I repeat. \"Especially without my weapons.\"\n\n\"It's OK, Riven,\" Shae says. Then she grasps my arm, and leans in. \"You have to trust me. We'll be safe. Just do as she says.\"\n\nI shoot her a scathing look but remove my jacket and harness, tossing my backpack into the corner. Caden does the same. Era nods with satisfaction before she and Shae punch in some kind of simultaneous code on either side of the door and lean in for a retina scan. My unease spirals.\n\nWhat in hell is behind that door?\n\nAs the door swings open on noiseless hinges, I don't realize I'm gripping Caden's wrist so tightly that he winces and pulls away. But I can't even think to apologize. Instead, as pale blue fluorescent light and the nauseatingly familiar smell of formaldehyde rush out to greet me, I'm frozen into immobility.\n\nThe room is circular. The floor is metal. There are men in blue head-to-toe suits walking past us with steel trays and tablet computers. There are computer flat-screens everywhere full of trending data that I can't even begin to process, but I'm guessing that it's some kind of laboratory. My eyes take in the details, categorizing them in my head and assessing for danger. And there's a lot of it... that I know for certain.\n\nCaden pushes past me to follow Shae and Era inside, and I do the same only to be assaulted by what awaits me. I forget the men in the blue suits immediately. A row of Vectors stand in circular man-sized specimen tubes strapped vertically to the wall, a handful of them to the right of where I'm standing. My jaw drops to the floor and stays there.\n\n\"What is this place?\" I hear Caden say.\n\n\"It's a research facility,\" Shae responds. \"Era's been studying them for years, trying to recreate the parameters to evert as they do.\"\n\n\"But you have to be dead,\" I say automatically.\n\nEra impales me with an unnerving, piercing stare. I hold it this time and raise an uncowed eyebrow. \"Not necessarily,\" she says. \"We can reduce certain things to mimic the state of the Vectors, to put a body at rest if you will.\"\n\nSkepticism threads my voice. \"No, the bodies have to be dead to withstand eversion from anywhere. Anything less than a zero gravity point will pulverize us.\"\n\nA slow smile. \"Come now, Riven. Put that physics knowledge of yours to good use,\" she says in a patronizing voice that sets my teeth on edge. \"Why do you think the Vectors generate so much electricity when they evert?\" She waits, and I shrug. I never got into the technicalities with my father. All I knew was that they were dead things that could jump from anywhere and we could not.\n\nEra smiles. \"The electrical energy generated by the nanoplasm inside of them acts as a force field to protect the human tissue. Dead or alive, it doesn't matter. We can increase the electricity inside our neurons to jump as they do.\"\n\n\"How?\"\n\n\"We've developed a serum,\" Era says, her hand gesturing to the men walking around us. \"Come, let me show you.\"\n\nWe follow her past the Vectors, but I can't help staring at the darkly veined, translucent bodies on display. On close inspection, I can see the outline of their bones as well as the metal wires from the lithia core connecting the spine to the skull. I step closer. Numbers are inked into their skin. One reads 104. The one closest to me twitches, and I almost jump out of my skin.\n\n\"Are they active?\" I blurt out.\n\n\"Of course not,\" Era says. \"Don't worry, pet; you're safe. They can't hurt anyone.\"\n\nIgnoring her supercilious tone, my gaze falls back to the Vectors. They're all still functioning, but somehow they've been neutralized. I wonder for a second how they'd been able to deactivate them. It isn't like they have a kill switch, and all of my father's programming is strongly encrypted. But then I remember. Shae would have provided some, if not all, of that information.\n\n\"How did you get them?\" I ask, nodding at the five Vectors standing inside the glass half-tubes against the wall. I can't help the edge to my voice. \"And how did you deactivate them?\"\n\n\"With great difficulty.\"\n\nShae turns toward me. \"They were older, first generation. When we... left, I took any of Father's research I could get my hands on. I knew he'd send the Vectors after us, and I was hoping I could use it to avoid them. Era discovered that they were programmed with some kind of kill code.\"\n\n\"The Vectors all have self-destruct codes,\" I say slowly. \"It's in their operating parameters. If their objectives are compromised, the nanoplasm shuts down.\"\n\n\"That's where I come in.\" I dimly recognize the voice but not the face behind the blue mask.\n\n\"Philip?\" Caden asks, and his voice startles me. I'd forgotten that he was even there. But he's staring at the person removing the gaskets connecting a wide helmet to a blue safety suit.\n\n\"Wait, Philip?\" I echo.\n\n\"Extra credit,\" he says in a bland voice. \"And training. I'm next in line.\" He jerks his head toward Era. My brows snap together, eyes darting between Philip and Era. The resemblance had not been obvious before, but now I see the same long noses and the wide, angular cheekbones, even though Philip's hair is blond and not dark like Era's. I feel a flush redden my cheeks as I recall the strange look Era had given me when I'd lied saying that Philip seemed to like me. Cringing embarrassment hardens my voice.\n\n\"Wait, what? You're in training to be a Guardian?\"\n\nPhilip nods briefly and gets back to business, consulting an tablet he's holding in his left hand. He types in a sequence. \"I embedded a code to override the self-destruct programming. These three on the left,\" he says gesturing to the Vectors closest to us, \"have been completely deprogrammed.\"\n\n\"And the other two?\"\n\n\"They were the last ones we got. Still working on them. Their security has an additional layer that we haven't been able to get past, some kind of eleven-digit code. They're slightly more advanced. I've run different algorithms, and nothing.\"\n\nMy mind is racing. I still don't quite understand what they want to achieve by everting from any point like the Vectors, especially using first-generation Vectors from twelve years ago. My legion had been third-gen, and I'd bet anything that Vector Commander we had fought before was far more advanced than any I'd ever had under my command.\n\n\"I'm sorry. I don't get it. Why would you want to evert from anywhere? What's the point?\"\n\nPhilip doesn't answer and Shae avoids my eyes. Of course; I see it now. It's an exit strategy if anyone pursing them \u2013 like me \u2013 somehow manages to get too close.\n\nShae nods, confirming my guess. \"It was our way out if they sent anyone after us.\"\n\nHer voice is quiet, and I notice she doesn't imply that I'm one of the people chasing them. It occurs to me right at that moment that she hasn't told Era the truth about me. It seems odd that Shae would protect me, even after everything. \"I just didn't think it was going to take this long to get a workable serum.\"\n\n\"So how does the jump work?\" Caden's voice is small. He's been quietly listening all along and trying to put together the pieces, from what Shae and I had told him earlier to what he was hearing now.\n\nPhilip brings up a controller on his tablet and keys in some numbers. A picture forms on the flat-screens in the center of the room. It looks like an hourglass broken up into small squares. \"That is a two-dimensional drawing of a traversable wormhole. It's basically a bridge in space with two different end points. Think of the universe as made up of an infinite number of universes. Some of these universes are coupled by a gravitational field, which means that we can communicate between them. OK so far?\" Caden nods, and Philip continues, pointing to one side of the diagram. \"In this case, this is our world, and that is Neospes, where Shae \u2013 and the Vectors \u2013 are from.\"\n\nPhilip pauses to bring up another image on the computers, this time a series of numbers and symbols on a graph with moving waves. \"How it works is a whole other story. We're talking string theory and sub-quantum mechanics, basically the relationship between space-time, gravity, energy, and matter. OK?\"\n\n\"Not really. You lost me at string,\" Caden says, dazed. I bite back a smile.\n\n\"What Philip is trying to say,\" Shae explains, \"is that our physicists figured out a way to manipulate electrons and gravity to jump between one universe to the other. Like through a kind of passage.\"\n\n\"You mean like a stargate?\" Caden offers, remembering my earlier jibe. \"You know, the movie?\"\n\nPhilip sighs and rolls his eyes. \"Sort of, only there's no gate, but the transference is similar.\"\n\n\"OK, so why doesn't everyone know about this? I mean, it's amazing,\" Caden says. \"Can you imagine if everyone could do that?\"\n\n\"They can't.\" Era moves to the front screen and taps in some more commands that clear them all. Her voice is hard. \"Though the universes exist in parallel, we're not meant to go between them. We were breaking the laws of evolution and nature. And nature has a tough way of evening things out. Where do you think the bubonic plague came from that wiped out millions of people in Europe? We couldn't control it, so we closed the wormholes and founded the Guardians on this side.\" She nods toward Shae. \"They created the Vectors on theirs to police and deal with illegal jumpers.\"\n\n\"What about them?\" My voice is harsh in the vacuum of silence following Era's words. \"Why wasn't Leila or Shae dealt with for jumping illegally?\" I'm more curious than anything \u2013 the Guardians have such rigid laws that it doesn't make sense as to why they would allow Shae to remain here.\n\n\"Caden's mother was seeking asylum to return home with her son from the Lord King at the time,\" Era says. \"The Faction could not deny it.\"\n\nAt her words, Caden is already backing away, his face confused and betrayed, staring from Shae to Era to me. His anger finds an easy mark.\n\n\"I'm the target you were talking about in my house that day?\" he hisses in my direction. \"I'm what you were looking for? The next in line to your stupid monarchy?\"\n\n\"It's not what you think \u2013\" I begin, but he cuts me off.\n\n\"No. It's not what you think,\" he says. \"I don't belong there. I belong here. I'm not going with you. Not now, not ever. I could care less if your idiot boyfriend is dying.\"\n\nI'm flinching inside at the words he's throwing so carelessly at me, but the truth is, I am there to take him back. My voice is cold when I respond. My words are for everyone.\n\n\"That idiot boyfriend is your brother and the reason you're even alive. And you're going whether you like it or not.\" I turn to Philip, who's staring at me like I've grown a pair of horns on my forehead. \"Try these numbers.\" I reel off a series of numbers and he hastily enters them onto his tablet.\n\n\"What are they?\"\n\n\"My name and my birth date. Trust me, that's the code you're looking for.\"\n\nPhilip punches in the numbers and for a second, nothing happens. My father had been anything if predictable with the earlier versions of his pet project. Maybe I've got it wrong.\n\n\"I'm sorry\u2013\"\n\nBut the wailing of a loud siren \u2013 the sound of security being breached \u2013 cuts off my words. Philip's eyes are wide and horrified, focused on something behind me. Era is already diving for some kind of control pad on the computer desk, slamming her left hand down so that more alarms are shrieking. In her right hand, she's holding an electro-rod. In that millisecond, I notice that it's not set to stun.\n\nIt's set to kill.\n\nShae screams and pushes Caden to the floor as people in blue suits streak past us. Doors start closing, separating the room into smaller quarters. I turn in slow motion, only to meet the cold dead eyes of one of the Vectors that I had clearly just activated.\nTRACK OR TRAP\n\nFather had been predictable, all right.\n\nPredictable enough to know that one day I'd be the one to initiate the Vector spies that had gone missing in this world. I curse myself again, reaching for my ninjatas, but there's nothing there but bare skin. This time, I curse Era for making me leave them outside of the room that is now sealed behind a giant metal door. Scanning the space, I see another white door on the other side of the room that has a small window in the center of it. Caden's face is peering at me from the other side. His eyes are terrified, jumping from the Vector to me. I nod reassuringly \u2013 at least he's safe for the moment.\n\n\"Open that door,\" I shout.\n\n\"We can't; standard security protocol,\" Era says from behind me. Her voice is controlled and calm. She's holding two electro-rods in each hand, and one of them is pointed right at me. \"They'll stay closed until the situation is resolved. Who are you? How did you activate them?\"\n\n\"I didn't\u2013\"\n\n\"What the hell did you do?\" Philip says angrily. \"And why is it just standing there?\"\n\nPhilip is right. The Vector across from us is not moving, even though its eyes are open and staring at the three of us. Something's not right. Vectors don't hesitate \u2013 they follow a program, a very specific program. This one is waiting for a reason.\n\n\"Era,\" I say urgently. \"There were two of them, right?\" She nods, the electro-rod dropping slowly downward. \"Two. Where's the other one?\"\n\nIn slow motion, we both turn to the white door in the center of the wall at the same time that a bright swath of blood splashes across the window. My heart leaps into my throat, but neither of us can get to the door, as the one in our room darts to stand between us and the computer panel. I've never seen a Vector move that quickly. This one is different. For a second, I wonder if the other one in Caden's room is as quick. Shae will protect Caden, I tell myself. She has to. That thought is the only thing keeping me together.\n\nThe Vector moves again, this time delivering a blow to Philip's head and spinning behind Era. Her cry is loud even as she lunges toward it, but it's too fast, darting out of her way with incredible speed. I skid over to Philip. He's still breathing, and I shove him under the console in the middle of the room. \"Stay here,\" I tell him, and he nods woozily.\n\nEra has gotten in a couple good hits. I can see the charred skin from where she's shocked the Vector, but he's too quick for her to strike where it will disable him.\n\n\"Why isn't it fighting back?\" Era shouts, edging back to where I'm standing next to Philip. \"Do you see that? It's toying with us. What's it waiting for?\"\n\nIts movements seem vaguely familiar to me. I've seen one of these kinds of Vectors before. They're wiry and fast, and aren't programmed to fight. They're programmed for other things.\n\nI think back to the moment when Murek had planned the coup and killed his brother. He'd been with a Vector like this one. I remember thinking at the time that it didn't seem like much protection, but Murek had insisted that the creature accompany them. Then they'd been flash-attacked. Murek had miraculously survived, but his brother and most of the group had been killed, including the strange Vector. Whatever had attacked them had known exactly where they would be, because the attack had been strategic and swift. By the time the royal guards had gotten there, the attackers were already gone. Murek had said it had been doomed timing, but I'd known better.\n\n\"It's a shadow,\" I breathe.\n\n\"A what?\" Era's voice is suspicious. \"What the hell is a shadow?\"\n\n\"They're spies. They send out some kind of signal to call for more Vectors.\" Era doesn't answer but Philip does. He's holding his tablet in his hand and staring at it in horror.\n\n\"She's right,\" he whispers. \"There's some kind of radio wave that wasn't here before.\"\n\n\"You can track signals?\" I ask.\n\n\"We need to deactivate it right now,\" Era says, veering toward the thing before I can stop her. Just then an explosion rocks the wall in the back of the room, and I see chunks of colored plaster fly into the small window. But I can't even think about Caden and Shae. I have way worse things to worry about, staring from Era to the shadow Vector. She's going to make it worse.\n\n\"Era, no!\" I tackle her from behind, and we both roll crashing into the unresponsive Vectors on the far side of the room. She punches me in the face and I feel my head snap backward from the blow. Stars spin in my vision, but I manage to block her second strike. \"Stop!\" I scream. \"It's a trap; if you hurt it, it will explode. That's what just happened in the other room. I don't even know if the others are alive.\"\n\n\"But if we don't disarm it, we're dead either way,\" Philip yells.\n\nBut there is another way. That is... if the thing will listen to me as the other one in Caden's bedroom had. \"Let me try.\" I approach the Vector slowly, my hands in the air. It watches me but doesn't move out of my way. I glance back at Era and then Philip, both staring at me with wary expressions. \"Stand down, soldier,\" I say firmly. I can feel the hot lasers of Era's eyes on my back, but I have no other option. \"That is a direct order from your commanding officer.\"\n\nIts answer is to lurch forward and grab my neck in its cold hands, so fast that I can barely get my own hands up to block it. Black dots dance in my vision. And then everything goes eerily quiet. I don't know if I'm dead or dreaming when the big metal doors swing open, and Caden rushes in, sword in hand to cleave the Vector's head clean off its body.\n\nSomeone's screaming for us to get out, and I feel rough hands pulling me. Vaguely, I see Shae's face at the doorway holding it open with painted red arms. I have no voice. The metal door closes behind us and an explosion rocks the entire house, throwing us to the floor and plunging the entire hallway into darkness.\n\n\"Riven, get up. GET UP!\"\n\nMaybe I am dreaming. And we're all already dead. But Caden's voice in my ear is not a figment of my imagination. I feel his hands on my face, and then under my arms, lifting me. I stand on shaky feet \u2013 that thing had almost killed me. My breaths are long pants, but I'm starting to feel like maybe I'm not actually dead.\n\n\"Caden?\" I croak. \"You're alive. How\u2013\"\n\n\"No time, Riv. They're already here.\" Shae's voice comes from the other side of me. \"We have to move.\" She shoves something in my arms, and I realize that it's my backpack and my jacket. \"Let's go.\"\n\n\"This way,\" Era says. We navigate the dark hallway, holding on to each other, until we're at a narrow staircase. \"This will take us to the office on the third floor.\" Noiselessly, we make our way up, listening for sounds that we're being followed or something's waiting for us at the other end.\n\nBut we're safe for the moment. The office is empty, but the electricity works. There are only six of us now: Shae and Caden, Era and Philip, one of the research guys who looks like he's in some kind of shock, and me.\n\n\"Everyone OK?\" Shae whispers. She's once more covered in blood and gore, as is Caden.\n\n\"She's one of them,\" Era says glaring at me. \"An officer.\"\n\n\"I know,\" Shae says, surprising them, and me most of all.\n\n\"And you brought her here?\" Philip hisses. I throw a scowl in his direction and he scowls back just as fiercely. I am the enemy.\n\n\"She's with us now,\" Shae says firmly. \"Right, Riven?\" I notice Era says nothing, but I can feel her eyes on me, studying. I nod \u2013 I need their help if we have any hope of escaping the Vectors. For now, \"the enemy of my enemy is my friend\" is truer than ever.\n\n\"How many?\" I ask, glancing toward the office door.\n\n\"We got the one in the room,\" Caden whispers. \"Jim overrode the system so we could get out through an emergency hatch, but we lost two of the other guys in the explosion. We had to circle around to get to the door to your room. Shae got one of the other Vectors, and I took one out, but there's more. Those things told them exactly where we were.\"\n\n\"You mean she did,\" Philip says with an accusing look in my direction.\n\n\"Shut up, Philip,\" Shae says, but I still feel an unwelcome stab of guilt, knowing that I'd been the one to reactivate them. \"It's not your fault, Riv,\" Shae says softly in my ear. \"I was going to suggest the same thing. He played us both.\"\n\nI nod, silent. \"What now, then?\"\n\n\"We try to get out of here.\" She turns to Era. \"The exit tunnels, are they still usable?\"\n\nEra shakes her head. \"The explosion in the lab would have compromised them. We can't go back down there. The only way out is up to the third floor and then off the balcony to the back.\"\n\n\"We have to be very quiet, then,\" Shae says. \"There's too many of them for us to take them head-on.\"\n\n\"OK, let's go,\" Caden says, and I'm surprised at this side of him... a leader. In the last few hours, he looks older and harder. He said that he'd taken out one of the other Vectors \u2013 not bad for someone whose skill is based on sport fencing. My lip twitches, and then I remember that Shae had told me that she'd trained Caden, and that he was good enough to fight me. The beginnings of the smile fade away and I stare at him with a grudging respect.\n\nI follow them on silent feet as we make our way to the staircase leading to the third floor. There are crashing noises on the lower level, and it won't be long before they come upstairs. But we have some time. They are nothing if not methodical and will strip each floor completely before moving on to the next. Their programming gives us a tiny sliver of opportunity to escape.\n\nCaden's shoulder brushes mine. \"You OK?\" I whisper to him, and he nods. He looks uncomfortable for a second and then leans in.\n\n\"I'm sorry for what I said earlier,\" he whispers back. \"You know, in the room. About Cale. I didn't know.\" The sound of Cale's name on his lips is like a painful dart in my chest. \"Shae explained that he was sick and that I'm the only one who can help him.\"\n\n\"Yes.\" It's the only word I can manage. I can only imagine what kind of lies \u2013 or truths \u2013 Shae had told him about Cale or why I was there. But nothing prepares me for Caden's next words.\n\n\"She said it was up to me,\" he says quietly. \"She told me that she's dying and can't protect me anymore. But that I would be safest with you no matter what.\" He pauses to stare at me searchingly. \"So I'm in.\"\n\n\"Next door on the left.\" Era's voice drifts back to us, but we're both standing still staring at the other until we're alone in the hallway.\n\n\"Are you sure?\" I grit out. \"I can't promise you anything, Caden. And I don't even know that Cale is still alive. Maybe you're better off here with Shae and Era. Shae's going to be fine. Trust me, you don't know her like I do.\"\n\nAnother crash from downstairs makes us both jump. Caden links his hand into mine, and squeezes. \"All I know is that those things want to kill me, and with you I have a shot of survival, whether that's here or there. So I'm in, whatever you decide. I just wanted you to know that.\"\n\n\"You guys, come on!\"\n\nEra's hiss has us both moving, and we follow the others silently into the room. Shae's already got a rope tied to the balcony. Her gaze flutters to our hands, and she nods just once before hopping over the side. Jim, the research guy, has already gone, and so has Philip.\n\n\"You go after Shae, Caden. Then her,\" she says, nodding at me. Her glance slips to our fingers and something in it makes me pull away, self-conscious. \"I'll bring up the rear in case any of them make it up here. Go!\"\n\nCaden leaps over and shimmies down the rope. It's three stories up but he slides down easily and gives me a thumbs-up once he's on the ground. I turn around to let Era know that I'm going, but am completely blindsided from the blunt metal that crunches into my face. Something snakes around my wrist and pulls it tight against a hard edge.\n\nDazed, I feel warm wet blood trickling down my face and blink through the blood to see Era jumping over the side. Her black lifeless eyes meet mine for a brief second and I see nothing there, not even regret for her betrayal.\n\nI am still her enemy.\n\n\"Era, you're making a mistake,\" I gasp. \"Shae can't make it without me.\"\n\n\"You're with them,\" she shoots back. \"You were the one who brought them here.\"\n\n\"No,\" I say, desperate. \"I didn't. I'm against them; you don't understand.\"\n\nEra slides me a circumspect look. \"So you're not their general?\" I want to lie, to say anything that would make her release me, but I know that it would only backfire in my face. I had to take a chance that she would believe the truth.\n\n\"Yes; I mean I was but not anymore. I defected to find Caden. He's next in line for the monarchy.\" But it was the wrong thing to say. Her face hardens, and I gnash my teeth in frustration at my foolish mistake.\n\n\"Yes, for parts, correct?\" Her voice is acidic. \"You deserve to die with your kind,\" she spits toward me.\n\n\"No! You're wrong. I'll figure something out, please, Era!\" Pulling against the plastic tie tethering my hand to the metal screen door, I try to reach for her, but she's already gone.\n\nIn the room, I can hear noises of movement on the stairs. They're coming to this level. I know that I only have a couple minutes, if that, before the Vectors swarm. I swing myself onto my stomach, pulling against the plastic tie so hard that I can feel its edges slicing into my skin. But I take the pain for the moment, twisting my free arm up and back to get one of my swords. My fingers touch the bottom hilt and peel their way upward, but they're slick with sweat.\n\nI tilt my body and my fingers find a grip. The sword is in my hand just as the first Vector enters the room. The curtains to the balcony billow with the wind, but I know that it will search routinely unless I attract its attention. As quietly as possible, I slide my blade's edge toward the plastic shackling my wrist, wincing at the pain, and with a quiet snap, I'm free.\n\nBut I'm afraid to even take a breath as another Vector joins the first, and then a third. The smell of them is gagging. The rope is still tied to the top of the stone balustrade, but it's on the other side of the balcony from where I am. There's no way I could be quick enough to get to it and get safely down without drawing their attention, especially with the only cover being the sheer curtains and two double glass doors.\n\nSliding my blade soundlessly into its sheath, I step quietly to the edge and without hesitation fling myself over the side just as one of them comes through the doors. Without daring to breathe, I hang by the tips of my fingers on the lip of the stone's edge, hoping beyond hope that the Vector doesn't do any kind of heat scan. But luckily, it doesn't, and after a grunt, I hear its footsteps recede into the room.\n\nMy fingers feel like they're going to break off even though it's only been about thirty seconds, and I dig the points of my boots into the wall behind me to get some leverage. But it's no use; I feel my fingers start to slip from their precarious hold. And then I'm sliding down, panicked, until my feet slam into a three-inch decorative ledge of stone running around the side of the house. I hang on for dear life even though the skin on the pads of my fingers is red-raw from the sharp rock.\n\nThe scrabbling noises of my fall have two of the Vectors back out on the balcony. If they look over the side, they'll see me dangling like some kind of weird spider in plain sight, but fortunately for me they're distracted by the rope. One of their suits emits a series of loud, short beeps, and they all march back inside.\n\nI'll have to find another way down, and fast. I start inching toward the right corner of the house. I cling to the shadows along the wall and keep moving, hoping for a miracle. I squint in the darkness \u2013 there's something long and dark in front of me. A gutter pipe! I almost lose my hold in my haste to reach it, and then I'm moving downward without hesitation like a human-sized monkey, hand over fist.\n\nFloodlights illuminate the lawn just as I reach the bottom, and I flatten myself against the wall behind some shrubs. They're prickly and scratch my face and legs, but I don't feel any of it. My attention is riveted on the couple dozen Vectors flowing out of the house. Where had they all come from? The others wouldn't stand a chance against so many!\n\n\"Find them,\" I hear a guttural voice command, and my blood turns to ice. It's the same Commander, the one that had been tracking us, my father's number one. \"Bio sensors on.\"\n\nOne of the Vectors makes its way toward me, and I crawl backward, mindless of the sharp branches. Against the wall, I slide my swords out without a sound. I stop breathing as it comes within inches of me, and leap up to sever its spinal cord silently. Its body slumps backward, and I shove it under the bushes before swiping its bio sensor and communicator, and then I'm running into the cover of the trees as fast as my feet can take me.\n\nI have to find the others before they do.\n\nThe biometric readings on the device show Caden about half a mile away from my present location. I run hard, knowing that the horde of Vectors is right behind me, until my lungs are burning and my breath is coming in shallow, desperate pants. And then I hear them, crashing through the underbrush all around me. I'm too late.\n\n\"Riv! In here.\" I look around but all I can see are three big rocks and some bushes. \"Up here.\"\n\nThey're in some kind of cave on the top of the rock ledge. Without a second thought, I hoist myself up and climb into the brush, not looking back. I can hear the sounds of movement on my heels, but it only makes me climb faster. After a couple minutes, I see an arm and then Caden's face. Shae is leaning against him and her eyes are closed. My relief is tangible, even though Shae looks like she's on her last legs.\n\n\"Where are the others? Era and Philip?\" I whisper.\n\nCaden shakes his head. \"Philip lost it and took off. And then Era went after him, and that was the last we saw of either of them.\"\n\n\"How's she doing?\" I whisper with a nod at Shae.\n\n\"I'm fine, little sister.\" Her voice is barely a wheeze, but her grip is strong as she grabs my hand. \"It's time.\"\n\n\"Time for what?\"\n\n\"You know what: to go,\" Shae says. \"We can't last in here. They're already out there. There's too many, and the only way is to evert. Now.\"\n\n\"But we can't. We need an eversion point.\"\n\nShae wheezes again. \"Hand me my bag.\" She pulls out the silver case I'd given her. There's one injector dose left. She stabs it into her leg, wincing, and places the empty cartridge next to her on the ground. She pulls out a white bag and removes two more injectors. The liquid is a deep jewel blue color. \"It's the serum we've been working on.\"\n\n\"What if it doesn't work?\"\n\n\"It does. I've used it,\" Shae says with a smile. \"To get away from you. We were trying to refine it. The aftereffects were bad on the nervous system.\" Her smile turns sad, and something inside of me crumples. \"As you can clearly see. Someone had to be the test rabbit. I'm just glad it was me.\"\n\n\"Shae\u2013\"\n\n\"Riv, you promised, remember?\" A thin dribble of blood leaks out of the corner of her mouth. Her voice is urgent. \"When you inject it, evert right away. Run as fast as you can on the other side, because you'll be in the Outers. You have thirty minutes to evert back before the serum becomes compromised. That should be enough time to get you away from them in any direction. After thirty minutes, you'll need to find an eversion point to get back here. Whatever you decide, keep Caden safe.\" Her fingers clutch mine, and I feel her knuckles slide against my face. \"I believed in you, you know. I never stopped.\"\n\nAnd then before I can say anything, she shoves her backpack and an eversion device into my hands and pushes past us, jumping down to the clearing below. I'm already twisting to leap after her when Caden's hands haul me back. I can feel the tears on my face, hot and violent. I want to hurt them, the Vectors \u2013 him \u2013 for ripping us apart again. My rage is all-consuming until I'm nothing but fury.\n\n\"Riven, no,\" Caden says, grasping my shoulders. \"Don't make her sacrifice be in vain. We have to go.\"\n\n\"She's my family,\" I growl.\n\n\"She's mine, too. It's what she wanted. Please, I can't do this without you. There's nothing for me here anymore. My life is in your hands now.\"\n\nHis words are like a bucket of cold water. Inside I know he's right, but still, the agony is scalding my insides like acid. Going out there would be suicide, just as she'd known it would be. But she'd done it for him... and for me. Caden is the priority. He'd always been the priority, even if the rules had shifted and the playing fields flipped. I nod once and pick up the syringes.\n\n\"For Shae,\" Caden says.\n\n\"For Shae,\" I agree, swallowing past the grapefruit-sized knot in my throat. \"On the count of three,\" I say. \"One, two, three.\"\n\nWe stick the injectors into our legs at the same time. Caden's eyes are wide, and I smile reassuringly at him, even though I'm falling apart inside.\n\n\"I'm scared,\" he whispers. \"Will it hurt?\"\n\n\"Just hold on to me,\" I say dully, my voice breaking. Truth is, I don't know if it will hurt. But at the moment, it feels like nothing could hurt as much as the gaping hole inside of me. \"It'll be OK. Shae said we'll be OK. And she's never steered us wrong before.\"\n\nAnd then I pull Caden close, and we're both crying against each other for the sister we're leaving behind... the one who had surrendered herself to save us. I hit enter on the eversion device.\n\nHot white light glows around us in the small cave, and within seconds, we are gone.\nPART TWO\n\nNEOSPES\nOVER THERE\n\nWe're on a training ground, running back to back, the wind in our hair and glee on our faces. Our enemies fall gracefully to our weapons, because together we are invincible. I turn to Shae, the wind lifting her braids off her face, and laugh out loud. We are breathless with victory. But suddenly, something in her face shifts, and she backs away, her hands outstretched, warding off something horrible.\n\nI call her name over and over, but my feet are leaden and stuck to the ground, and she doesn't stop moving. In a few seconds, I am alone and the orange sunshine has disappeared behind a dark, ominous cloud. The ground crumbles beneath my feet, and I'm lying in a desert with a mouth full of sand. Something crawls into my eye and I can feel it moving inside me, feeding off me. I can't stop screaming, but no one hears me.\n\nMy mouth tastes like metal, dirty, sour metal. I spit and it's an odd blue color, dead Vector's blood. The caked, parched ground sucks it up greedily, like it's something precious. There's some kind of creature crawling toward me. It looks like a black scorpion, only it's shiny and metallic, and its eyes are glowing white orbs. It crawls up onto my arm and then digs its sharp forward claws deep into my flesh and starts to feed on me. I am disappearing into this thing's mouth until it's gigantic and sated on my flesh, and I am nothing but a speck.\n\nIt has eaten me, absorbed me. I have become the monster. And I feel drunk with it, exhilarated. Alive.\n\nAnd that's when I see Shae, running toward me and throwing herself into my arms as if she hadn't seen me in weeks. She smells so good that I can't help myself. She is sunlight in a world of darkness. My mouth opens and I take her into me without a second thought. She must die to feed the monstrosity I have become.\n\nI can't stop screaming.\n\n\"Shae, no. I'm sorry.\" The words feel like knives, tearing into the roof of my mouth, and I gag reflexively. Everything hurts. I don't know if I'm awake or asleep, or if I'm dead or alive. Wetness soaks my cheeks.\n\n\"Riven, wake up.\" A hand is shaking me. \"It's OK, it's OK. We're OK.\"\n\nI pry open my sticky eyelids, and Caden's worried face swims into focus. I'm shivering so hard I can almost feel my bones rattling. \"Whererewe?\" I mumble unintelligently.\n\n\"Neospes, I think,\" Caden says. \"But it's so desolate. I can't see a city. There's nothing for miles.\"\n\n\"How long have we been here?\" I try to sit up. I've never blacked out during eversion before. I don't even want to think about what that means. My blood feels hot, like it's on fire, but when I touch my skin it's icy cold. I rub my hands up and down my arms. The movement hurts.\n\n\"I don't know. A half hour, maybe more. I tried to wake you, but you weren't even blinking. Your eyelids were moving, so I knew you were alive, but your heartbeat was so slow. I wasn't sure whether you'd had some kind of reaction to the serum. But I felt OK,\" Caden rushes out. \"Then you started murmuring, and a couple minutes after, you woke up. Where are we?\"\n\nI look around quickly. We're in a barren stretch of land. There's not a tree in sight, but I know it's all an illusion. The scavengers would smell us. \"We're in the Outers.\"\n\n\"That doesn't sound good.\"\n\n\"It's not.\"\n\n\"So should we run a bit and then evert back?\" Caden asks. I shake my head and check my watch.\n\n\"We couldn't do it and be safe. If I've been out as long as you say, we're way past the thirty-minute mark that Shae was talking about. It'd be too risky. Plus, my blood feels like it's on fire; I don't think I'd survive another jump.\" I pause, staring at my skin. \"Maybe you're right and I did have some weird reaction to the serum.\"\n\n\"So what do we do?\"\n\n\"We make our way back to the city, but first I need to figure out exactly where we are.\" I open my backpack and pull out two of the Vectors' uniforms I'd stuffed in there. I toss one to Caden. \"Get undressed and put this on.\" I glance up at the sky that's still covered in a reddish haze. \"It gets really hot here during the day. We're going to have to find a spot to rest, quickly.\"\n\nCaden looks confused as he shrugs out of his clothes and into the Vector's uniform. \"Wouldn't it be safer to travel by day?\" His face reddens as he notices me stripping out of my clothes at the same time, and he turns away. I shrug. There's no room for propriety.\n\nMy reply is short. \"Not here.\"\n\nI zip up the second uniform, thankful that it doesn't smell, and throw Sadie's dress into my pack \u2013 cotton is rare here, and it would be unwise to leave it. My ninjata harness I slip on over the uniform, tightening the straps across my chest. I roll the leather jacket and tuck it in the top. The red power button on the suit's computer is blinking, indicating that it's already charging. Nearly all of the suit's power comes from solar energy. I tap the keypad on the forearm of the suit and type in a sequence of numbers, wincing at the pinch at the back of my neck. I repeat the same on Caden's. \"What are you doing?\"\n\n\"Resetting the suit to calibrate to our bodies. Hang on a sec; this may hurt a bit.\" His eyes widen at the sting.\n\n\"What was that?\" Caden hisses, jerking away.\n\n\"It's OK. It won't hurt anymore. It's a wire that the suit connects to your nervous system. Works in tune with your biometrics.\"\n\n\"Oh.\"\n\n\"It may look like a piece of nylon, but it's really pretty advanced technology. This computer controls the whole suit; it records your heart rate, the amount of fluid in your body, and basically makes sure you stay alive. It's thermo-responsive, too.\"\n\n\"Oh.\" I smile this time at Caden's dazed response. \"What's thermo-responsive?\"\n\n\"Our daytime temperatures are very different from yours. It's boiling hot during the day, upward of a hundred and fifty degrees, and very cold at night, like negative sixty. The suit adjusts to each, keeping your body at a constant temp.\"\n\n\"It can do that?\" Caden's voice is still full of amazement. \"Wow, no wonder you wanted to take them off the Vectors.\"\n\nI raise my eyebrows. \"That's just the beginning of what they can do, but let's get moving. I'll fill you in later. The Outers isn't exactly the most friendly place, and we need to find somewhere safe to rest for the day before we fry,\" I say.\n\nOff to the right, there's nothing but more flat land, but on my left I can see the vague outline of some mountains. I type in a command on the suit keypad, and a hologram of a compass hovers in the air. Underneath the 3-D rendition is a map. Neospes is to the north. The map says that it's over sixty miles away. That's three days of hard hiking in very unfriendly terrain... extremely unfriendly terrain.\n\nI glance at Caden. He's poking through all the pockets on his suit.\n\n\"Here, take this,\" I tell him, handing over one of the two handheld crossbows from Shae's bag. I strap the other to a special hitch on my suit that has a retractable lead. \"Your sabre isn't any good against flying things. And they come from all sides here.\"\n\nI haven't gone through everything she has in there yet. I'm a little terrified of what I may find or not find. I have never been this unprepared for an eversion \u2013 no food, no supplies, no medical gear... nothing. All I have are the weapons \u2013 better than nothing! \u2013 and half-used supplies in the bottom of my backpack. But Shae was always prepared for any eventuality, and I can't help hoping that she somehow planned for this. I'll wait until I've found us somewhere safe to hide before doing a full tally of what we had, and pray that it will be enough.\n\n\"If anything moves, shoot it. Don't think, just shoot,\" I say to Caden. \"OK?\"\n\nCaden's eyes are dark, but there's no fear in them. It lifts me up a tiny bit. \"Are there Vectors out here?\"\n\n\"No, they don't come out this far, unless it's a raid or a search party. Out here\" \u2013 I gesture to the barren landscape around us \u2013 \"there are worse things than Vectors.\" I glance upward. \"And the worst thing of all is that sun, so let's move.\"\n\nWe jog in silence for a while, tracking north on the compass. Our pace is hard and it's already sweltering. The ground is covered in an oily red haze that makes it look as if it's shimmering, and I can feel the sweat slicking across my forehead. Glancing at Caden, his face is determined. I check the computer on the suit. The temperature gauge reads one hundred and ten degrees. The worst part is that it's not even 7 in the morning.\n\nI quicken my pace, and Caden follows without complaint. By my calculation we have about two hours before we get to the base of the hills on the map, and hopefully some cover.\n\nIf we make it...\n\nA noise overhead has me twisting around with my crossbow in hand. It looks like some kind of bird, but I know instinctively that it's not a bird. Sunlight glints off its wings as it drops lower, making a beeline in our direction. I wonder if it's some kind of tracking spy, and I toy between killing it midflight or figuring out exactly what it is. The things that exist out here live by no rules \u2013 they live to survive any which way they can.\n\n\"Caden, drop!\" I shout and Caden rolls to the ground just as the thing dives past his head, talons outstretched. It's far bigger than it looked at first, smaller now as it climbs into the sky. It's not going to attack again, I notice, but there's no way I can let it attract any more attention to us. I place the bolt and shoot. My aim is true and the thing drops like lead to the ground.\n\nMy crossbow remains at the ready, although blue sparks from my first arrow ripple across the bird's body, rendering it powerless. Up close, the bird's wingspan is about three feet across, and it reeks of rotten flesh. Metal wires crisscrossing and woven into bands of tissue make up the bird-thing's body. I can see the faint outline of a skeleton underneath between the gaping holes. Where the eyes should be are only two glowing white orbs. I feel the bile rising.\n\n\"Stay back,\" I warn Caden, but he ignores my words of caution.\n\n\"I want to see what I'm up against out here,\" he says pushing past me, and then freezes, his body like rock. \"What the hell is that?\"\n\n\"It's a hybroid. Half-android, half-something alive.\" I shrug. \"It's just one of the many things out here.\"\n\n\"Android?\" Caden repeats. \"Like a robot?\"\n\nI shoot him a glare. \"You know, like the Vectors? Only, they take whatever parts they can get out here. It's slim pickings in the Outers, and we are like Mecca, so let's go. I have no idea if there are any more of these floating around.\"\n\nI grind the heel of my boot into the thing's head, and it makes a sizzling sound. The eyes dull, but I'm not taking any chance. Sliding the short knife from my boot, I sever the wires connecting the head, cringing from the rot of decay.\n\n\"Come on,\" I tell Caden. \"We wasted ten minutes on this. Here, this will be better.\" I click a button on the neck of his suit and a hood with a thin transparent mesh unravels. I tuck it over his head, and the suit seals the closure. I do the same on mine, and at his look, I say, \"It's so your skin doesn't burn off. Let's run.\"\n\nBy the time we get to the base of the hill \u2013 which is more like a rock cliff than a hill \u2013 the sun has climbed high into the sky and is beating down on top of us with the force of a hammer. The edge of the hill is shrouded in pale shadow, but with every passing second, the line of sun moves inexorably toward it. If the sun catches us climbing that face, it will be a struggle.\n\nI'm wheezing, and I can hear Caden's labored breath behind me. My legs are burning and my heart is pounding. My mouth is so dry that every time I swallow, my tongue sticks to the roof of my mouth and I have to peel them apart. One of the side pockets on my pack holds a thin root, and I break it in half, offering part to Caden.\n\n\"What is it?\" he asks.\n\n\"A root; chew it.\"\n\nThe root provides some reprieve, filling my mouth with spit. The more you chew it, the more saliva it releases. I look upward along the cliff face, shielding my eyes from the unrelenting sun, searching. And then I spot what I'm looking for, a dark dot under a ledge. \"Up there. You see it?\" Caden nods, and then we are climbing faster than we've ever climbed. We're racing against the line of sun burning a fiery path up the rock beneath us.\n\n\"Won't the suits protect us?\" Caden grunts.\n\nI don't answer because I'm not really sure. I haven't been back to Neospes in three years, and these suits are newer than the ones I'm used to. The earlier prototypes used to become unstable in really hot temperatures. I glance at the heat reading on my forearm: one hundred and nineteen degrees and mounting. I can't take the chance of any malfunctions, especially when the suit is directly connected to our bodies.\n\nAs we get nearer to the small cave mouth, I pause, signaling Caden to wait before grabbing a flare from my pack and throwing it inside. Other things would be looking for shelter, too. The flare hits the back wall and flames brightly for a second. Nothing happens, and then some kind of huge lizard-like creature shuffles out. My long blade is embedded in its head before it could even spit venom in our direction.\n\n\"Is that a hybroid?\" Caden whispers.\n\n\"No, it's like a komodo dragon,\" I say, kicking the carcass to the edge of the ledge. \"Only\u2013\"\n\nCaden interrupts me with a wry look. \"I know, worse.\"\n\nI grin. \"You're getting it. This one's pure. No metal. They're survivors through and through. And they're carnivores, so keep your eyes open for any others.\" I nudge its mouth, shuddering at the jagged shark-like, rust-colored teeth. \"They're poisonous and can spit their venom to paralyze their prey.\"\n\n\"Sick!\" We share a laugh that quickly evaporates to silence. Levity is a luxury right now. I squeeze his shoulder and shine a flashlight into the shallow cave. It's not bad for shelter, with the top ledge dipping down like a lip. The komodo must have just been resting there, because there's no sign of a nest or excrement. We should be safe enough from the sun or other predators.\n\nThe dead komodo is already starting to stink. Leaving it visible would be like putting up a \"free buffet dinner\" sign. I don't say anything to Caden, but if we run into a bind on the food front, it would be foolish to waste such a windfall, especially if Shae hasn't been as prepared as I hope. But I can't even look at it and think about eating it without gagging.\n\nIn Neospes, most of our food is powdered or in gel form and engineered. We don't grow crops or have farms. Our food is processed in factories underground in one of the sectors responsible for food production. Half the time, we don't even know what we're eating; all we know is that it has exactly the right combination of calories and nutrition to keep our bodies at top functioning capacity. The only time I've ever had the luxury of organic food is during the Solstice Games.\n\nOccasionally when we were younger, on raids to the Outers, Shae and I would trap some of the animals. For some of the poorer people on the outskirts of the city, that's all they had to eat. I remember one old Artok woman telling Shae and me that some of the Outers komodos used to be delicacies in the old days. She'd offered us some, and while Shae had gamely eaten the roasted beast, I hadn't been able to even stomach the abnormal, charred flesh. I'd thrown up immediately afterward.\n\nA smile twists my lips. My revulsion for cooked animals had shifted pretty quickly when I everted to the Otherworld. It was either eat or starve, or become what they called a vegetarian. In Neospes, food was not a pleasure; it was a means of survival. But that didn't mean anything in Caden's world. Food was celebrated and revered to the point of excess. It shocked me.\n\nI stare at the carcass and kick it with all the force I can manage off the ledge. It makes a wet thud near the bottom, and from where I am, I can see a thin line of black ant-like creatures mobbing it in seconds. Only thing is... they're not ants, and they're not small. I duck back into the cave and almost crash into Caden standing beside me.\n\n\"Was it poisonous?\" Caden asks me. \"You know, to eat? I mean, we have to eat, right?\"\n\n\"No. I just couldn't...\" But I have no explanation other than my own stupid disgust. I sigh. Caden's right.\n\n\"It's just meat, Riven.\" I stare at him, and after a couple seconds, Caden makes to leave the cave. \"I'll go get it. We don't know how long we're going to be out here.\"\n\nBut I stop him. \"You can't go down there. It's already gone.\"\n\nSure enough, the carcass has all but disappeared, picked clean to the bones, and even half of those, too, are gone.\n\n\"What the... \" Caden's voice is small.\n\n\"They're carrion eaters, scavengers. Those\" \u2013 I jerk my head to the ground below \u2013 \"are hand-sized burying beetles. They'll scavenge anything. Dead or alive. They can smell dead things from miles away.\" I turn to him. \"Which is why we're probably better off with the komodo down there.\"\n\nIt's boiling now. The temperature reading is a hundred and thirty-two, and the sun's arms are about halfway down the rock face. Soon, nothing will be out in the noon sun, not even the metals. The shade of the rock doesn't provide much relief from the heat, but it's a lot cooler and the suits do the rest. I unroll a flat black shade from my pack and run it along the cave's mouth. The powerful magnetic edges curve and seal to the rock.\n\n\"What's that? More fancy cloth?\"\n\n\"Yep, a holo-tube,\" I agree. \"Sort of. It's like a body tent. If I couldn't find any shelter, as an emergency, I could roll up in this, but it can't hold two people.\" For some reason, I'm blushing. The tube could hold two people but they'd be literally sandwiched together. Shae and I had done it once, and we'd been much smaller than Caden and I would be. I shake the unnerving image out of my head. \"It blocks out all UV light and heat. And it's holographic.\" I point to a tiny control pad on one of the edges. \"The outer side mimics the surrounding area. Like camouflage. So anything out there would just see rock face.\"\n\n\"Cool.\" Caden inspects the edges of the material and rubs it carefully between his fingers, frowning at the evenly spaced bars beneath it. \"What are these? Magnets?\"\n\n\"Yes. They stick to the lodestone.\" I raise an eyebrow. \"How's your geology?\"\n\n\"Naturally occurring magnetite in igneous rock,\" Caden says primly. \"Supposedly magnetized by lightning.\"\n\nI grin, impressed. \"Well, here it's from all the electrical fields during the War.\"\n\n\"Wait, what about our gear?\" Caden asks. \"Wouldn't the fields affect them, too?\"\n\n\"Nope, shielded.\" I say, leaving Caden to ponder the geological and technological wonders of my world, and head to the back of the cave where I empty out both Shae's pack and mine.\n\nI sort everything into three piles: food, medicine, and gear. Shae was prepared. There's a lot of food from the other world, including dried strips of some kind of meat and packets of dried fruit and grain, but Shae must have stockpiled a ton of our powdered food whenever she'd everted back here. There are a few dozen thin tubes, more of the water-root, and both water bladders on the inside of each of the packs are full.\n\nThe water we'll have to use sparingly, more so than the food. The suits will keep us from losing too much of our body water, but we will die faster without water than food. My guess is that the komodo would have had a water source not too far away. If we're lucky, maybe it will be on the way to the city.\n\nMost of Shae's bag contains more food than anything, but I still have my med-kit from before, so that will have to do. I remove all of the various weapons I stole from each of the Vectors and line them up in a row. Three night-vision goggles, two coiled electro-whips, one rod, my ninjata swords, the two crossbows with bolts, Caden's sabre, and a handful of throwing stars and knives. On top of that, I also have two communications headsets and the eversion device that brought us here.\n\nAll in all, we aren't in terrible shape. My fingers catch against something in the front pocket, and I pull out a tiny device that looks like an electronic thumb drive. I frown. There's a piece of paper on it that reads \"For Riven.\" I shove it back into the pack. I'm not ready to process whatever it is that Shae has prerecorded on there. I'm also not ready to admit to myself that she's really gone or to hear her voice and see her face.\n\n\"You OK?\" Caden says, watching me from where he's still sitting near the front of the cave. \"Need any help with anything?\"\n\n\"No, I'm OK. Just doing inventory.\"\n\n\"Then we should add my stuff in there, too.\" Caden crawls over and tosses the contents of his backpack into the mix, and I frown.\n\n\"What's that?\" I ask, pointing to a slim leather-bound case the size of a watch box that had fallen out next to June's gun, and a pile of miscellaneous junk looking vaguely like random Eagle Scout supplies. I don't say anything about the gun, because it could come in handy, but the box looks elegant and out of place.\n\n\"Something my mother left me.\"\n\n\"Your mother?\" I ask, startled. \"May I?\"\n\nCaden smiles sadly. \"Go ahead. It's some kind of chip. We couldn't read it, no matter what we did. I put it in there just in case.\"\n\nI open the box carefully. Inside, there's a silver-colored ring with a dark blue circular stone. The Neospes royal crest is emblazoned on it. The chip Caden's talking about is a pill-sized silver cylinder. I stare at Caden. \"Shae didn't tell you what's on it?\" I ask.\n\n\"No, she said she didn't know.\"\n\nBut Shae did know. We all did. She must have had some reason for not wanting to tell Caden what the chip meant. I slide it and his ring into an inside pocket on my vest. Caden doesn't object, and together we repack everything, including Caden's thermal blankets, fire-sticks, water packets, emergency food bars, flashlights, and snare wire.\n\n\"Where'd you get all this stuff?\" I ask, fingering an odd-looking multi-tool that will undoubtedly come in handy. I'm unwillingly impressed by his foresight.\n\n\"Online,\" Caden says with a grin. \"All I had to look up was survival gear for the zombie apocalypse.\"\n\n\"The zombie apocalypse?\"\n\n\"What can I say? We like to be prepared. Hurricanes, tornados, and zombies.\"\n\nI can't help the laugh that rolls its way out of my stomach, and I'm laughing until my sides feel like they're about to split. Go figure for a world that has instructions on how to survive a zombie apocalypse. Thinking about it sets me off again. I don't stop laughing, not even when Caden throws a water packet at my head.\n\nWe eat a couple strips of meat and dried fruit, saving the longer-life food items for later, and lie alongside each other on the rocky floor of our little cave. I haven't forgotten about the chip sitting inside my vest pocket. It's almost burning a hole into my skin.\n\nIt's time for Caden to tell me everything he knows.\nIN THE OUTERS\n\nWhen I awake, my throat feels gritty like I've chewed on sandpaper. Despite the hard ground, we've managed to both get some sleep. Caden is curled up on his side, something on a silver chain curled in his fist. He showed me the locket last night when he recounted everything he knew from his childhood. It was tarnished platinum with a photo of his mother inside wearing one of the flowing dresses that she'd loved. On the other side was a photo of Caden as a baby. Caden said that he never took the locket off.\n\nIt's boiling in the cave, but the temperature is already changing, lowering. Sometime during the day, I shrugged out of the suit and just slept under one of the thermal blankets. I stretch as quietly as I can, soothing the aches out of my muscles, and get dressed after taking care of minor bodily functions in a back corner of the cave. I smile, remembering that Caden asked whether the suit also took care of those kinds of needs.\n\n\"The human ones can inhibit waste,\" I told him, \"but these are designed for the Vectors. They don't need to use the bathroom,\" I explained. He stared at me like I'd grown two heads.\n\n\"They can do that?\" His eyes were wide. \"Do they process your pee so it's drinkable, you know, like in Dune?\"\n\nThen it was my turn to stare at him. I shook my head. \"That's gross. No, our suits are designed to inhibit those needs, at least until we take the suits off.\" I pointed to the neuron connector at the back of the neck. \"Remember? They're not designed for long-term use, although I'm sure they've probably figured that out by now. As you can probably guess.\"\n\n\"How do you know all this stuff? Like how it works?\"\n\nMy answer was a toss-up, but I chose the least complicated answer. \"My father designed them.\"\n\nI glance over at Caden as he turns in his sleep. He, for his part, had told me everything that he could remember, but it still wasn't a lot. His memories were sporadic, mostly centered around his life in the other world. His mother rarely smiled, Caden said, and when she did, it was always short-lived. It was as if a part of her had been missing, like she was living lost in a daydream somewhere else half the time.\n\nCaden hadn't known about Cale. Neither his mother nor Shae had told him anything of Neospes or where he was truly from. And yet, he faced the Vectors that had come in search of him, with innate courage and bravery. He accepted the near-impossible that I was from a parallel universe, just as Shae was. And I know he has more questions, but it seems like he is waiting... waiting for me to tell him the truth of who he is.\n\nMy mouth twists. I wouldn't even know where to start.\n\nHey, Caden, you're the son of a king in this world. Your father was murdered by his brother, your uncle, by the way, who now wants to kill the brother you never knew you had, and kill you before you can somehow save him so that he can assume the throne. Oh, and we live in a giant glass dome of a city, because everywhere else is contaminated because of an android war that pretty much killed off everything in the process.\n\nAnd by the way, I'm somehow falling for you.\n\nMaybe I'll leave out that last part. Love is the seed of weakness, as my father had always said. I'd be better off burying any feelings I have for Caden, and he'd be wise to do the same. I touch my fingers to my suddenly warm lips, thinking of our unexpected kiss in the bathroom at Horrow. It seems like eons ago instead of mere days. For a brief second, I allow myself to savor the feelings blooming inside my chest, if just for that moment.\n\n\"Why are you staring at me?\" Caden murmurs sleepily. An embarrassed flush invades my face and neck.\n\n\"I wasn't. I mean, I was but I was looking through you,\" I mumble.\n\n\"You were smiling.\"\n\n\"Yeah, well I was thinking of ice cream and donuts,\" I snap, irritated. \"Come on, we have to get moving. The sun's just setting and I want to get a head start.\"\n\nWe gather up what we've used, and tuck them into the packs. Caden has Shae's pack, since it's more lightweight and durable like mine. His sabre is in its sheath lying along his back underneath the pack. Caden stares at me and grins.\n\n\"We look like ninjas.\"\n\nMy mouth twitches. The description is appropriate. \"We'll need to fight like ninjas to get through the next two days.\"\n\n\"What about the wrappers and garbage?\" Caden asks looking around. \"Shouldn't we...\"\n\n\"Forget it. It's extra weight. Just leave it,\" I say over my shoulder as I'm repacking the holo-tube at the entrance. \"Don't worry; it'll get used by something out here. The things that live here use everything they can get their hands on. Think of it as natural recycling.\"\n\nThe landscape is still red, but a different kind of red with tinges of gray along its edges. It's seventy-nine on the temperature monitor. Here and there, we spot movement. I try not to let my tension show, but it's a guarantee that we'll run into some kind of trouble. Predators in the Outers are vicious. I wasn't kidding when I told Caden earlier that the creatures out here use everything they could get their hands on. I meant it \u2013 they'd slaughter each other for hair, teeth, and bone. They'd rip us apart for less. A shiver races across my back.\n\n\"Here, let's put these to work,\" I say, grabbing Caden's wrist and pressing one of the buttons on the keypad. \"It's a security protocol that identifies any metal, I mean, hybroids. You'll feel a vibration and this console will flash.\"\n\n\"How does that work?\"\n\n\"Sound waves.\"\n\nWe climb down the mountain quickly, moving across the terrain at a good pace. We keep our hoods off, as the warmth of the open air is comfortable for the moment. The sky is an odd mixture of crimson and gray and black. There's no blue in the Neospes sky, but it has its own unique beauty. I realize that I've missed it, but a part of me misses the blue, too.\n\nOur pace is grueling, but I'm relishing the chance to push my muscles hard, to run so that my heart is pumping like a piston in my chest. Caden keeps pace with me easily, and the realization that he's fitter and stronger than I thought is a delayed one. He's not as lean as Cale, even with their identical build, but I misjudged him. I wonder if he fights as easily as he runs... just as Shae boasted. She said that he could take me. I doubt that, but it doesn't mean I'm not curious. Cale was good, but he could never take me in a one-on-one fight, not even with all his years of training.\n\nI've always had a good sense of where people are going to be before they strike, like a fighting sixth sense. It gave me that extra edge that I put to good use in my unprecedented rise to the rank of general. No, Caden wouldn't be able to best me, especially not having been trained in our ways from the beginning.\n\nThe landscape starts to change as the sun disappears and the moon rises into the night sky, covering everything in an oily silver veil. Normally, moonlight is beautiful, but out here it has different implications. Moonlight is ominous, insidious. It means hunters are on the prowl and predators out for prey. And so we have to be extra vigilant.\n\n\"What is that?\" Caden huffs, his gaze drawn by something off to our right. I don't want to stop, but I look over and my breath hitches just as the console of my wrist-pad vibrates and the light flashes red.\n\n\"Stop,\" I hiss, and drop to the ground. Caden follows unbidden. Removing the infrared goggles from my pack, I stare silently at the group of six or seven raptor-sized creatures pawing and shoving into each other. They're about a mile away from where we are.\n\n\"What are they? More hybroids or real things?\" Caden whispers.\n\n\"Definitely hybroids. They're pack hunters. Reptiles.\" I'm hoping that they haven't seen us... or smelled us. They're machines that have taken on the most savage aspects of territorial beasts, and have fused themselves with horns and tusks, teeth and scales. Reptiles are notoriously hard to kill and even harder to outrun.\n\nI turn the goggles to the left of where we are crouched. There's a dip where the landscape shifts, a gorge of some sort. I hadn't planned on going in that direction \u2013 as much as the inhabitants of the Outers live aboveground, the worst of them prefer the underground. Or so it's rumored.\n\nIn the city, scary tales of the Outers filled our ears from birth. They were the stories that people whispered into children's ears: \"the monsters from the Outers will get you!\" or \"break the law and risk exile to the Outers.\" That one had been the worst one \u2013 to be banished from the city and forced to survive was a fate worse than death. Many traitors had killed themselves to avoid the outcome of the Outers. No one exiled there had ever returned.\n\n\"Follow me,\" I say to Caden, making a decision. \"We're going to crawl over there to where that drop is, got it?\" Caden nods. \"Just try to move slowly, without any jerky movements.\"\n\nCommando-style, we inch our way across the dusty ground. I can feel the dirt like grit against my teeth, clogging my windpipe. The fine dust is everywhere. I cough softly and spit to the side, and my saliva is rust-colored. My elbows and knees are burning from the scraping against the hard earth, but we're almost there. I can see that the cracks along the ground are starting to widen. Soon they'll be big enough for us to drop into and at least be safe if the hybroids do see us. Once we're out of sight, we'll be out of mind.\n\nI look back triumphantly to Caden, and he's a few feet away, his face screwed up and frozen. He looks as if he's just stuffed an entire lemon into his mouth.\n\n\"Don't\u2013\" I warn. But it's too late. The sneeze echoes across the open ground like thunder. In the next second, all we hear is the rumbling of distant hooves and screeching that could rival an oncoming hover-train. \"Run, Caden, run! To the chasm! Run!\"\n\nAnd we are sprinting for our lives as the thunderous rumbling draws closer. I don't want to look back, but I do. I have to see how close they are... whether we have a shot in hell of getting away from them. One we can probably take, maybe two, but definitely not six. With their breath hot on our backs, now I understand why anyone exiled to the Outers would prefer death. They'll capture us and take us back to whoever their tribal leader is, and then we'll be stripped for parts \u2013 skin, bones, organs, blood.\n\n\"Don't look back,\" I gasp to Caden, but his face is already a mix of pain and terror. Ignoring my own advice, I glance back again. One of them is faster than the others and gaining on us with every leap. It's the runner. All the pack reptiles have a runner \u2013 the quick one that snares the prey, and then the others follow to immobilize it. They're all fast, but the runners are faster.\n\nI run out in a wide arc veering away from Caden. \"Keep running to that gap, OK,\" I scream. \"No matter what.\"\n\n\"What are you doing?\"\n\n\"Trying to keep us alive.\"\n\nAnd then I can't think as I'm running into the reptile runner head on, ninjatas in both hands. I have thirty seconds before the others close the space between them and the runner. It's smaller than the ones I've seen before, about horse-sized, but I know it's no less lethal with its heavily muscled and metaled body. Its eyes glow white as its pointy snout gapes open, full of sharp, cracked teeth. Angled plates curve down its back and tail into some kind of pike.\n\nWe are seconds from head-on impact, and in full sprint I roll head over heels, slashing out with my blades at the same time, snapping through the intertwined wires and tissue at the base of its hooves. I don't stop. Momentum keeps me going, and I'm on my feet and running back toward Caden, but I hear the thump and screech of agony as it bowls forward onto its face. A grim triumphant smile graces my face for a second... It won't be running for a while, that one.\n\nI've earned us a sliver of time; the rules of existence in the Outers mean that the others will fight to pick the fallen apart. It's survival of the fittest and strongest. Sparing a look behind me, I see they're already converging on their fallen pack-mate. I dash harder, pumping my legs like pistons until I'm almost flying. I have no idea where my extra strength comes from; I just go, taking advantage of my second wind. The pounding of hooves behind me echoes the pounding in my chest.\n\nAhead of me, Caden barely squeezes into a tight gap, and I throw myself down behind him, crashing into his back and sending us both spiraling into the rock walls of the narrow grotto. We're both hauling stale air into our lungs and clutching each other with numb fingers, even as dirt, rocks, and sour saliva fall on us from the creatures already snapping their mouths above us, trying to get in. We crouch farther down into the shaft but we're safe. They're too big to follow.\n\n\"You OK?\" I gasp.\n\n\"What is wrong with you? You could have killed yourself!\" I'm startled at the reprimand and his snarky tone, but I smile a tired smile.\n\n\"Better me than you. Come on.\"\n\nCaden doesn't return my smile. He stands in front of me, staring at me with fiery eyes. \"Riv, you can't do that anymore, OK? I know you're way better than I am navigating this terrain, but moving forward, it's both of us or neither of us. Got it? We do it together.\"\n\nI nod, an involuntary smile curving my lips again. Something in his voice makes me feel funny deep down inside, like he wants to protect me. People don't tend to question my orders, especially here in Neospes. I lead and they follow. The smile grows into a full-on grin as I imagine what Caden will do once he finds out that I've been a general, commanding an entire legion here, or even worse that I'm now some kind of marked-for-death, dangerous renegade.\n\n\"It's not funny, Riven,\" Caden hisses, misunderstanding my expression. \"I already lost Shae. I can't lose you too.\"\n\n\"I know,\" I agree, his quiet comment slapping the grin from my face. \"Together, from here on out.\"\n\nAbove us, the reptiles are still scrabbling \u2013 they'll do anything to get in here as long as we remain this visible, including digging out a bigger hole, which they've already started to do. The foul steam of their breath swirls around us. I stare at Caden, and we both nod at the same time. We need to move. They'll give up eventually and move on to some other target once we're out of sight.\n\nThe gap opens out to a deep gorge, above which the reptiles are furiously digging, but underneath the crack to our right, the tunnel burrows downward, disappearing into darkness. We don't have much choice but to follow it.\n\n\"Let's go. Be vigilant,\" I whisper to him, switching on the halogen lights on our suits. I'm torn between taking the lead and having to worry about him behind me every step of the way, or letting him go ahead and having to face any dangers head-on. In the end, I decide to take the lead. \"Stay close, and don't stop for anything. Got it?\" Caden agrees and then we are off, moving as quickly as we can deeper into the gritty tunnel.\n\nI sigh inaudibly, feeling the weight of the dark earth on all sides of me. Why does it always have to be tunnels? They're everywhere \u2013 in Neospes, in the Otherworld. I'm no longer claustrophobic, but tunnels still irk me. I was locked in a box underground for hours at a time to learn to face and harness my fear of enclosed spaces. It sounds cruel, but it worked. A key part of our mental training means I had to face and understand all of my fears.\n\nShae had had a fear of heights, so she'd had to jump off the tallest point in Neospes every day for four weeks. The day she did it without crying was the day she overcame that fear. She was still be wary of heights, but it no longer weakened her. One of the trainees in my group had a fear of snakes. He ended up dropping out, unable to take the counter-fear measures.\n\nThe system is brutal, but it's effective. An initiate can always lie about their fears, but it's counterproductive. Facing fear in a controlled environment and trying to overcome it without any preparation in a hostile environment are two vastly different things. In Neospes, fear is the paper-thin difference between life and death.\n\nThe tunnel widens a bit and I drag my gloved hand along the oddly-hewn edges of the rock. It's curiously smooth in areas and roughly chopped in others. I frown. My instincts warn that it's probably because some combination of living tissue and metal had grated against it, but a part of me wonders whether I'm being overly paranoid after the run-in with the reptiles above ground.\n\n\"Cade, you OK?\" I whisper back. He's tucked his hood around his neck, and even though the temperature is dropping consistently, there's a thin sheen of sweat on his forehead. His eyes are dark in the stark halogen lighting of our suits.\n\n\"Yeah. Where are we now?\"\n\nI consult the holographic compass and map, which works just as well underground. We're still making good time and heading in the direction of the city, about a quarter of the way there, but it's far slower now that we're not running. I squint at the map, noticing that it has now placed us below ground. The technology of the suit has dynamic virtual properties that allow its various operations to self-adjust, depending on external stimuli.\n\nCurious, I touch a spot on the base of the hologram that is highlighted with an \"S.\" The map is interactive and immediately shifts to show the entire network of the subterranean tunnels around us. Caden's gasp is soft beside me, but I'm still staring at the intricate web of tunnels that connect nearly to the edges of the city so many miles away.\n\nHave those always been there?\n\n\"Wow,\" Caden breathes. \"So I guess this means we can pretty much stay down here and not have to deal with those things up there.\"\n\n\"We don't know that things exactly like that aren't down here,\" I say flatly. \"Or worse.\"\n\n\"You're right,\" Caden agrees. \"But we haven't seen anything even close to any kind of life for the last couple hours.\"\n\n\"Doesn't mean they aren't there.\"\n\nI move the map with a swish of my thumb and forefinger, opening a wider subsection of the part nearest the city. It's not connected, but it's definitely close enough to get us to the east side \u2013 the least-policed side of the city. I trail my finger back toward the dot that indicates my current position, noticing several other tunnels below us that go deeper underground, but there's no way I'm going to risk heading down one of those.\n\n\"See here,\" I tell Caden. \"Here's what we're going to do. See this path, nearest the surface?\" He nods. \"We'll take that. It may take longer, but it could be safer. And that way, if we do run into trouble, we can always get out.\" I point to a few thin white lines marked at the surface by some odd red dots. \"I think these are cracks like the one we came in. It looks like this runs along the base of the Peaks.\"\n\n\"The Peaks?\" Caden repeats.\n\nI flip the map back to an over ground view, and show him the ridge of cliff-like mountains. \"We're running along the base of that.\" I frown. \"Actually, I think our path may even go under in parts. The inside of that mountain is literally pure volcanic glass. No metals can live there for some reason. Some kind of electromagnetic pulse.\"\n\nCaden echoes my frown. \"Volcanic glass.\"\n\n\"Ever see a mountain that has no caves, that looks like a sheath of sheer black glass? Well, get a good look, because that's one.\" I tap the map, switching the view to what the cliff mountains would look like at that moment on the hologram. \"Used to be an active volcano thousands of years ago that the metals pulverized. Now it's dead and impenetrable.\"\n\nI close the map and pull a food bar out of my pack, handing it to Caden. \"Now's as good a time as any. Let's rest for a second.\" We eat in silence and drink a water packet each.\n\n\"Can my suit do all that, too?\" Caden asks, and I flinch at the sound of his voice in the quiet between us. I nod and show him the control panels on his wrist. The overview is short, but Caden picks it up rapidly and is soon flicking through all the versions of the map. Just in case, I also quickly run through the security parameters of the suit that I initiated for both of us earlier.\n\n\"The suit is intuitive and attuned to you. It stores your data. Technically, we're not really supposed to interchange the suits \u2013 they're designed for each person \u2013 but Shae and I used to steal the Vectors' suits because they had way cooler tech. I developed an algorithm to erase the programmed data so we could use them.\"\n\nI blush and trail off, realizing that I'm showing just how much of a geek I am, but as Shae used to say, the apple doesn't fall far from the tree. My father knew what I was capable of at an early age and fed my brain a steady diet of bioengineering, physics, and advanced robotics.\n\nIt was a conscious act of rebellion on my part when I opted to train with the soldiers and become a part of Cale's personal guard. My father ranted and raved for days that I belonged in his lab with him. It incensed me to the point that I requested living quarters in the castle under the guise of protecting Cale, when it was only to escape my father's manic rages. In the end, he twisted it to suit him, because I ended up leading his greatest creations \u2013 the Vectors. I'd never understood why he was so pleased about that, but it got him off my back, and that was reward enough for me.\n\nCaden's voice makes me jump again. \"I think it's cool that you know all that.\"\n\n\"We learn different things than you do,\" I say, shrugging. \"But it doesn't make what you know any less important. We just evolve differently based on where we live.\"\n\nAn indistinct sound brushes gently across my ears, and for a second I think that I've imagined it. But then I hear it again, like a single note of something. It's some kind of bell-like sound. \"Did you hear that?\" I whisper to Caden.\n\n\"No, what?\"\n\n\"It's a chime or something. Listen.\"\n\nWe both sit in silence, our ears straining, waiting until the sound comes again. \"There it is! It sounds like music,\" Caden says, his eyes wide. \"Count it; it'll come back in six beats. Listen.\"\n\nCaden's right. The sound repeats, nearly inaudible as if it's coming from a long distance away. It's barely an echo of an echo, but we're both standing and staring at each other, our eyes shifting around us. I glance down at the security pad on my suit and check Caden's for good measure.\n\nNothing.\n\nIt doesn't beep, not even when the shadows materialize from the walls before us, with weapons pointing directly at our hearts.\n\nTHE OTHERS\n\n\"What the\u2013\"\n\n\"Shut up, Caden,\" I hiss, staring at the five men in dirty brown tunics. Despite the weapons pointed toward me, I shift slightly to the right so that Caden is standing behind me. My eyes slide down to the keypad at my wrist \u2013 the alarm is still silent, confirming that none of them are hybroids. One of the men steps forward, and I tense automatically, wishing that the empty water packet in my hand were one of my blades.\n\nHe holds a wand-like device in his hand, which he waves up and down our bodies. I realize quickly what it is \u2013 some kind of metal detection tool. But unlike our suits, it makes a harsh static-like sound, as if it's malfunctioning. The man bangs it against his side, staring at it and then our uniforms as if confused.\n\nDid he think we're Vectors?\n\n\"Come,\" he says, and I frown. His voice is non-threatening but firm. I have no idea who these people are, but it's clear that they want us to go with them. The fact is, I could take them all out easily without endangering Caden, but I'm curious. I've never heard of people living in the Outers. I stare at the men. They all look physically fit, if a little thin.\n\n\"Where?\" I ask.\n\n\"No questions,\" he says, and then jerks his head to two of the men behind him. They remove our packs and our weapons, and I let them. Nodding for Caden to do the same, I study the leader carefully. There are no marks on his face or any other distinctive characteristics giving any clue as to who he is, but it is clear that he's in charge. His body is lean with muscle, his eyes clear and bright. I need to know who these people are and what they're doing out here.\n\nThe men bind our wrists behind our backs, and we walk in single file down a tunnel offshoot that is completely concealed behind a bit of rock face right in front of where we had been sitting. I kick myself mentally, knowing that I should have investigated the tunnel maps in more detail when I found them earlier.\n\nThey don't blindfold us to conceal where they are going, and I understand the reason for that relatively quickly. I try to keep track of where we're heading, but it's difficult with all the twists and turns, and after a while, I realize that I have no idea where we are. Everything looks the same. Rock and more rock. Dirt floor. Dirt ceiling. The floor tilts slightly, and I wonder for a second how long we'd been walking downward. I hadn't even noticed.\n\nAfter a while, the tunnel widens into a large cave that's lit with some kind of sconces. There's a guard at the end of the space who eyes us ferociously. Caden tenses beside me, but I throw my shoulder into his and force a reassuring smile to my face. When I pass the guard, I glare so hard that I see the shock in his face. He's barely a boy, I notice with a grin. Good to see that my General mojo still works.\n\nIn the cave beyond the one we're in, the walls are black and shiny. The air feels cooler, as if there's some kind of draft coming from outside, but I know that that's impossible if we're as far down underground as I'm guessing we are.\n\nThere are a few more people now, dressed in the same brown garb. They stare at us with anxious, scared expressions as if we are somehow the enemy, and some of them even scurry away. A small face peeps from behind one of the people, and I feel my heart lurch. A child, barely three years old, stares curiously at me.\n\n\"What is this place?\" I ask, but the only answer I get is a burlap sack over my head. I struggle against the ties at my wrists and fling my head back so hard that it crunches wetly into bone.\n\n\"Stop!\" a voice yells, and then lowers at my ear. \"No one's going to hurt you, but we do not know who you are, and so we cannot trust you. If you are judged to be a friend, then we will remove the bag, but until then you must keep it on. Please do not struggle. There are ways for us to restrain you, or worse, sedate you.\"\n\n\"Where's the boy?\" I grit out, knowing I'm indeed at the voice's mercy. I don't know what kind of sedation techniques they use here, but I don't want either Caden or me to find out the hard way. \"I need to know he's with me, and you have my word that I will not struggle.\"\n\nSomething heavy and warm is thrust into my left side. \"Caden? Is that you?\" I say urgently through the bag. \"You OK?\"\n\n\"Yeah. I'm OK. Caught a hook to my chin, but can't say I blame them. I kicked out when I felt the bag on my head.\" He pauses and leans closer, led only by instinct. We're standing back to back at that point, and I can feel his head pressing into my shoulder. \"Where are we?\"\n\n\"I don't know,\" I answer honestly. \"I've never even heard of people, as in real live people, living out here.\"\n\nA prod, and we're walking again. I stay close to Caden, making sure that our arms are always touching at every step. His voice is nearly silent. \"Are they going to kill us?\"\n\n\"I don't think so,\" I whisper back. \"At least not right now.\"\n\nEngaging my other senses, I can feel the air change against my body. The space feels far more open than any of the others before. If I didn't know better, I would say that we were once more outside. It even smells different, but my sense of smell is a little undermined by the musty odor of the bag over my head. Once in a while, I'll also hear a gasp or a voice. More people? It baffles me that there's a whole community of humans living in the Outers, obviously by choice. The Outers is a place devoid of life, devoid of anything but metals left over from the war, and hybroids that scavenge to survive.\n\nA few minutes later, we stop walking and the bags are removed. I blink against the sudden light. Only three of the initial men are with us. We're standing in some kind of holding room, with a rough-hewn door barred with metal on one end and two thin cots on the other. They thrust us into it and swing the door shut behind them. Our packs are placed along the wall near the outer passageway. Our weapons are nowhere in sight.\n\n\"Wait!\" I shout, but they've faded into the darkness before I can take a breath.\n\n\"What do we do now?\" Caden asks. He looks concerned but no worse for wear, other than a reddening bruise on his chin.\n\n\"That looks like it hurts,\" I say.\n\nHe grins wryly. \"The one who hit me was my age, I think. I saw it coming, but my brain said freeze instead of duck. Don't worry; I have my eye on him for a little payback when we get out of here.\"\n\n\"Cade,\" I begin. \"I don't know who these people are or what they're doing here, but you need to know that they may not want us here.\" I don't say that they wouldn't want me there, especially if they know anything about who I used to be. But if they are from Neospes, sooner or later one of them will recognize me. \"And if they don't, we need to do anything we can to get out, OK, even if they look like kids your age.\"\n\n\"I get it.\"\n\n\"No, you don't. In this world, we are taught how to kill before we even learn how to talk. They are not kids like you are. If they see you as the enemy, they will take you out without blinking.\"\n\n\"Riv, I said I got it,\" he snaps.\n\nI stare at him but he looks away. I can tell that it's bothering him. That kid had probably hit him out of pure gut instinct, and he maybe hadn't expected it to hurt quite as much as it did. I, for my part, was happy that it had only been a fist and not something worse.\n\nMy head feels fuzzy, and I'm not sure it's because we're underground. Something feels off but I can't quite put my finger on it. I lean against one side of the room and close my eyes for a second before taking stock of the cell we're trapped in.\n\nThe walls of our prison are the same dark, shiny rock from the first cave, and it's oddly warm and smooth to the touch. There are no openings, but I can feel airflow coming from somewhere against my legs. I push off the wall to follow the changes in the air to an inch-long vent carved into the floor. The air is cool and smells fresh, as if it's being piped in from the outside. I find that odd and am intrigued, because it seems like this whole place has somehow been constructed. Such an elaborate venting system hasn't happened by accident.\n\nSomeone enters the hallway beyond our door with a tray. It's a young girl. Her face is covered with a veil, but she doesn't look at us. She's been instructed to deliver whatever is on that tray and leave without any eye contact.\n\n\"What's your name?\" Caden asks, his hands wrapped around the bars of the door as she slides the tray underneath along the floor.\n\nStartled, she answers automatically. \"Sela, sir.\"\n\n\"Thank you, Sela, for the food.\"\n\nShe smiles despite herself at his gentle words, and then without warning, her eyes grow wide and terrified as she glances up into the far corner of the room. Realizing that I'm following her gaze, her eyes drop hastily to the floor. Sela shuffles out far more quickly than she'd entered, and is gone without another word.\n\n\"You scared her,\" Caden says reproachfully to me.\n\nHe pulls the tray to the middle of the room, and I squat next to him. There are two wooden cups of water and some kind of ground meal in a bowl. He stares at me.\n\n\"You think it's safe?\" he says. I frown. \"You said if they wanted us dead, they would have killed us by now, right?\"\n\n\"Yes, but killing by poisoning is far less messy than an arrow in the stomach.\"\n\nCaden rolls his eyes at me and stares glassily at the food. \"I'm starving. It feels like hours since I ate that food bar. And I'm so thirsty.\" He looks away from the tray with effort. \"The smell of it is killing me.\"\n\nFor a second, I stare at him mutely, wanting to answer but unable to. The outline of his body shimmers into two people as my vision flickers. Forming the words in my head takes energy. \"You can eat if you want to, Cade.\"\n\n\"Are you going to?\"\n\nI shake my head thickly. \"No, not until I find out where we are and who these people are. Force of habit.\"\n\n\"Are you OK?\" Caden asks me, frowning. \"You look a little woozy.\"\n\n\"I feel really tired. You?\" I ask. Caden shakes his head. \"Must just be me then.\"\n\nI stretch, circling and pumping my arms to get the blood flowing in my body. It helps a little. I study the tiny black spot on the corner of the wall. If Sela hadn't looked directly there, the spot would have been unnoticeable, as had obviously been intended, but now it bothers me. What exactly had she been so afraid of when she'd looked up there? The way her eyes had dropped to the floor right afterward makes me think twice about it.\n\nSquinting out of the corner of my eye, I study the dot. Something flashes \u2013 it's the barest hint of a reflection in a lens \u2013 and I belatedly realize that I am looking at a minuscule camera. Someone is looking at us right at that moment, and has been looking at us all along.\n\n\"Caden,\" I hiss. \"Don't even think about touching that food.\"\n\n\"But you just said\u2013\"\n\n\"I know what I said,\" I say exasperated. \"But I didn't know two minutes ago that we were being watched like rats in a cage.\"\n\nCaden's hand drops so quickly I almost laugh, but the shaken look on his face makes my humor fade instantly. He shoves the tray under the door.\n\nHis voice is a worried rasp. \"Where?\"\n\n\"Upper left corner of the room. Don't look now,\" I warn, grimacing as his eyes flick towards it.\n\n\"I see it.\" After several moments, he mouths. \"What's the plan?\"\n\n\"No plan. We wait.\"\n\nCaden frowns but I shake my head imperceptibly. If they can see us, I'm pretty sure that they can hear us. My guess is that they're waiting for the direction of some leader, some person who will decide what to do with us.\n\nI want \u2013 no \u2013 I need to know who that person is and what they're doing here outside the boundaries of the city, and building some kind of secret sector in the Outers. And most of all, how do they stay safe from the reptiles and the metals? It's baffling. Neospes, like other city pods around the rest of the world, was built on the knowledge that the Outers were uninhabitable. And yet, real people are living here.\n\nHave things changed that much in the three years I've been gone?\n\nThere's no doubt in my mind that the tiny camera has a full view of the entire room, and while they've taken our weapons, we still have the suits. I can at least try to understand the layout of where we are being held, and where we are on the map.\n\nI hold my arm up and press the map command. Nothing happens. I press it again but there's no response. Frowning, I tap in the security sequences to power up the suit but the entire thing remains unresponsive. I swear under my breath and try again. But it's no use. The suit has somehow been deactivated.\n\n\"What's wrong?\" Caden asks.\n\n\"It's not working.\"\n\n\"What do you mean, it's not working? I thought you said the suits were automatic?\" Caden whispers. He averts his eyes hastily as I unzip and roll down my suit, separating the connection from my body and studying the microchips around the neural connector. But there's no light whatsoever.\n\n\"They've definitely been deactivated,\" I say quietly. \"Let me see yours.\"\n\nUncaring of the cameras or the fact that I'm clad in just a sports bra, I detach Caden's suit and roll it down to his waist, but it's the same as mine \u2013 completely unresponsive. He moves to shrug it back on, but I put a hand on his arm.\n\n\"Don't,\" I say. \"I don't know if the neural connectors are safe if the suit's been disabled.\" My fingers are still resting on his arm and I'm startled at the unexpected warmth of his skin. My eyes flutter to his leanly muscled chest, and my hand falls away along with my eyes. I step back, irritated at myself for being distracted. I've seen Cale shirtless countless times, but seeing Caden without a shirt bothers me far more than it should.\n\nThe buzzing in my head grows louder and my vision distorts once more, so much so that the floor starts to undulate beneath my feet. I shake my head roughly, trying to clear the static that seems to be crackling and coming from the inside my skull.\n\nI don't realize that I've slumped to the floor until I feel Caden's hands on my shoulders, shaking gently. \"Riven? Are you OK? You're freezing.\"\n\n\"Get off me,\" I say, shrugging him off. \"I'm fine.\"\n\nBut Caden is right. My body is cold, as if it can no longer regulate my internal temperature. My teeth are chattering so hard that it feels like they'll break at any second. And claustrophobia overtakes me \u2013 a feeling I'd conquered years ago. As if on cue, my heartbeat elevates rapidly and suddenly; I can't breathe.\n\nCaden whips one of the scratchy wool blankets off one of the cots at the end of the room and throws it around my shoulders. The fuzziness in my head gets worse, like some kind of high-pitched drill. The walls seem like they're closing in, amplifying the sound tenfold. I clap my hands against my ears, but if anything, the sound grows louder. Caden's shouting something but I can't hear him over the noise in my brain.\n\nFocus, I tell myself. Focus. Breathe. Focus. Breathe.\n\nIt's an old mantra that Shae had taught me when I'd been held underground during fear training. It calms the noise as my breathing evens out. My skin feels warmer to the touch already. I have no idea what caused the episode, but I'm sure it has to do with some kind of post-traumatic eversion stress or some weird extended reaction to the serum.\n\n\"Something's wrong,\" Caden is saying to the dot in the wall. \"Can't you see that? We need help!\"\n\nI'm about to tell him that his cries for help are useless when I see movement out of the corner of my eye coming down the passageway.\n\n\"It's about goddamn time,\" Caden says, and then stands back as one of the same men from before opens the metal-barred gate. He walks to stand next to me, his arms folded against his chest. \"I don't know what you guys did to her, but you better fix her fast.\"\n\nCaden's expression is stony like his voice, the realization that I could be in real danger hard-hitting. All traces of the boy Caden are now gone.\n\n\"I'm fine, Cade,\" I say weakly. \"I feel better.\" I try to smile reassuringly at him, but it fades as one of the men steps forward. As before, they are all dressed in the same brown clothing, their faces stoic and expressionless. Caden's hand comes to rest on my shoulder, squeezing gently. His fingers knead the muscles , releasing some of the built-up tension, and I let my eyes drift close, despite the risk.\n\n\"Who are you?\" The voice is soft and musical... a woman, then. My gaze snaps open. \"Who sent you?\"\n\n\"No one sent us,\" I say, keeping my voice modulated and my hands flat against my sides. But the woman squints at Caden and steps closer. I see the recognition dawn in her eyes, and my heart sinks. She stares from Caden to me, and again, there's an odd familiarity in her expression. Does she think he's Cale, as I'd done at first?\n\nStill, she's not too sure about me and steps close enough to touch the faded blue braid wound into my hair. My rank. Her fingers drift down to touch the inked seal on the side of my neck \u2013 the one that marks me as a general of Neospes \u2013 and lingers against the black lines beneath it.\n\n\"You're the general,\" she says. \"The one who defected three years ago.\" Her eyes narrow, but something in her voice tugs at me... a familiar tone, perhaps. Maybe I'm still fuzzy. I've never seen \u2013 or heard \u2013 this woman before in my life. \"Have you been here all this time?\"\n\n\"No,\" I say. I don't confirm or deny that I am whom she guesses. Caden is staring at me with wide eyes, but the understanding swirling in them is indisputable. Cat's out of the bag now, and there's no use pretending I'm not who I am.\n\n\"Who are you?\" I toss my own question in response back to the woman, who's studying me with an odd expression that makes me feel open and uncomfortable. Naked. I tug the blanket across my shoulders and stand. All the men behind the woman rest their hands on the hilts of their weapons, but I ignore them. \"Who are you?\" I repeat, my tone sharper.\n\nShe laughs at my posturing. It is a sound devoid of any humor. \"We're nobody. Exiles. Traitors. Enemies of Neospes. Defectors.\" She says the last word with the hint of a smile, but I am growing tired of this game. They're toying with us. I can take the five of them out blindfolded, even in my unfamiliar state. Without looking at Caden, I tighten my body into a state of readiness but freeze at her next words. \"You really don't know me, do you, Riven?\"\n\nAs before, I'm certain I've never seen this woman before, but then again, her hair is covered in a brown wrap. Her face is the same color as her clothing. Her eyes look dark in the light. She could be anyone.\n\nI shrug, arrogant, raising my palms upward in expectation.\n\nShe lifts a hand and removes the wrap, and her hair falls loose, so long that the silken waves reach past her back. It is so blond that it's nearly white, and I know without even seeing them that her eyes in the daylight are light gray. Silver.\n\nLike mine.\n\nThe only thing I inherited from her.\nA WEB OF LIES\n\n\"You're not her. You can't be her,\" I say in an emotionless voice. \"She's dead. My mother died years ago, strapped down to a bed in a lab. I saw her with my own eyes.\"\n\nThe words sting like poison barbs against my lips, scorching my insides on the way out, and I can't even look at this woman who makes my heart pound and my eyes burn. No one in Neospes has hair that color, but where she came from \u2013 one of the cities on the other side of the burned oceans \u2013 that hair color is common.\n\nIt's not her.\n\nMy heart argues otherwise, but my brain sees it for what it must be... a trick, some kind of lie to disable me. I cross my arms over my chest. My eyes narrow. \"Don't push me. Who are you?\"\n\n\"I did die that day,\" the woman says. Her voice is soft so only I can hear it. \"Just not in the way that you think.\"\n\n\"I don't know what game you're playing, and I don't care. You know who I am, and you know that if I choose, none of you will be left standing.\" The men once more bristle at my words, but I ignore them. All I can feel is the hot emotion welling up inside me like a tide, uncontrollable and violent. It's at odds with the coldness of my words. \"My mother is dead.\"\n\n\"Look again, Riven. Trust yourself, not what you've been told.\"\n\n\"I know what I saw,\" I say flatly.\n\n\"Look again.\"\n\nAt her gentle insistence, I struggle to keep myself in check, but it's too late. There isn't a shred of doubt on her face, no sign of untruth in her eyes, but I refuse to give in. I can't... because if what she says is true, then my whole life has been a lie. She has to be lying.\n\nHow dare this woman presume to be my mother?\n\nMy rage erupts like a volcano, burning my mind with lost memories and thoughts I want forgotten. Without thinking, I drop to my knees in a crouch and spin behind her. I clip one of the men with my boot and take another out with a jab to the jaw before anyone can blink. The other two lurch toward me with their weapons, but I'm spinning again, my feet and hands darting out with swift, lethal purpose. And they too join their brothers on the floor.\n\nThe woman hasn't moved or drawn any weapons, but I still let my fury pin her up against the wall, my forearm under her slim neck.\n\n\"Riv, no,\" Caden says from behind me. \"What are you doing?\"\n\n\"Shut up,\" I growl to him. \"She's an imposter. An exile.\" I turn back to her. The compassion in her clear eyes irritates me even more. \"Otherwise, why would she and all her followers be hiding out here instead of in the city? And she's a liar. A fucking liar!\"\n\nMy voice breaks on the last word, but still the woman doesn't move, staring at me with those brilliant eyes and daring me to believe the impossible. I flinch against the light palm pressing along the base of my spine. \"Riven,\" Caden says again, his voice soft, \"Stop. You're not a soldier anymore. Calm down. What these people are doing here is anyone's guess, but it's not our place to judge. We need their help. Remember why we're here?\"\n\nHis words are tentative, but I can't see past the emotion storming inside of me. The fact that some stranger could elicit such a frightening response out of me because of the mere color of her hair or her eyes makes me furious... furious at myself, and furious at her for claiming to be someone that she isn't. Caden's foolishly timed words nudge me over the edge of my tenuous control on my anger.\n\nMy voice is calm. Deadly calm. \"It became my place to judge when they imprisoned us here.\" I turn to face him, my expression echoing my voice, releasing my hold on the woman so that she slumps back against the wall, clutching her neck but still silent. \"And don't you ever talk about me being a soldier, ever again. You know nothing of it. I will be a soldier until the day that I die. So don't presume to think that you know me or what I am, ever. Understand?\" I jab at the seal on my neck. \"You asked me once what this was. It's a seal, a brand. It marks me as theirs and I can never escape it as long as I'm alive. And the black lines\" \u2013 I'm spitting now, advancing on him, my words merging together \u2013 \"mark the lives I've taken, those I've killed. Still think I'm not so tough?\" I say, mocking his words from his bedroom, a lifetime ago.\n\nCaden is up against the other wall and I'm so close that I can feel his shallow breaths on my skin. But there's no real fear on his face; instead his eyes are wide and worried, focused on me. His hands are flat against my shoulders but non-threatening.\n\n\"I may not know who you were, Riven,\" he says quietly. \"But I know who you are now.\"\n\n\"Lady Aurela!\"\n\nI jerk and spin around at the shouts behind me, but it's the name that knocks the wind out of me. My mother's name. This has to be some kind of intricate plot. They're all in on it, trying to weave some convoluted web around me, but for the life of me I can't seem to figure out all the pieces. Why does my head feel so clouded, as if I can't think properly, like there's something blocking me, confusing my thoughts and weakening my resolve?\n\n\"Get off, Caden,\" I mutter, my anger at him draining away, and shoving his hands off my shoulders. \"You don't know anything about Neospes and you don't know anything about me.\"\n\nMy eyes are drawn to the blur of movement in the outer passageway, but the woman holds up a silent palm, and the men milling there stop. \"You're not angry with him. You're angry with me,\" she says, walking forward. \"Riven, it is me. I didn't die that day.\"\n\n\"I saw her die,\" I insist. \"On that table.\"\n\nAurela shook her head, her eyes darkening with pain and regret. \"You shouldn't have been there. He knew you were there, too. He wanted you to see it so you'd know what challenging him would cost.\"\n\n\"It's common knowledge that I was in that room, and what my father was capable of,\" I say softly, but my words have less conviction. \"It still doesn't make you her.\"\n\n\"Riven, please,\" she says. \"I know you've changed. You've had to, but I know that you know it's me. Listen to what your heart is telling you and not your reason. You and Shae\u2013\"\n\n\"Don't you talk about her,\" I hiss, pain seething through my voice, but Aurela continues despite my interruption.\n\n\" \u2013were everything to me. She asked you to come with her, but you didn't. She couldn't tell you about me, because he would have found out, and as you said, you know what he was capable of.\"\n\nDespite my mistrust, my mind is reeling. \"Shae never asked me...\" But my voice trails off as the memory snaps into my brain, followed by Shae's answer to why I had chosen not to go with her: He owned you then, and he owns you now. I stare at Aurela. \"She never said anything about you.\"\n\n\"She couldn't. The quarters were monitored. He would have killed her.\"\n\nI push past the growing knot in the pit of my stomach. My voice is cold and hard. \"So you left, and she left. You left me with him.\"\n\nAurela is in front of me now, her face constricted. \"If I could change that night, I would. I would go back to get you whatever the cost; you have to believe that, Riven. No matter what.\"\n\nEverything inside me wants to argue, to scream, to rail against this woman who had always told me she loved me in the dead of night when only my nightmares kept me company, but suddenly I am tired of fighting. I am tired of pushing away everyone around me. I am tired of loss. I'm wary of letting logic rule my every action when it's obvious that my heart knows it's her. The emotion of it is overwhelming... splintering all of my carefully constructed walls.\n\nCaden is watching me, and as I meet his eyes, he nods just once. He will support whichever path I decide. I shake my head tiredly and sit on the edge of one of the cots. The woman is quiet, waiting and knowing that my next words will decide how we leave this room. Truth and anguish and regret are written all over her face. I close my eyes for a long time.\n\n\"No, it would have been a suicide mission for anyone to come back,\" I say eventually. \"That was the night he activated the Vector program... because of Shae. There were kill orders on sight for all defectors of the realm.\" I pause, the bitter question burning a hole in my mouth. \"How did you survive?\"\n\nAurela's face is wet with silent tears but she answers. \"Annis.\"\n\nAnnis was my mother's research assistant. Like my father, my mother was a brilliant advanced genetics scientist. They were paired to lead the bioengineering and robotics programs, but when my father grew interested in reanimating the dead, my mother disagreed. She fought him tooth and nail on it, saying that it was a gateway mistake heading straight back to what they were just rebuilding from as a society \u2013 the Tech War. But my father was headstrong and arrogant.\n\nTheir intellectual arguments were epic, their fights shattering. Shae used to take me into our sleeping cell with her fingers covering my ears. Then one day, it went too far. In a fit of rage, he'd shoved her and she'd hit her head. Hard. Everyone at the hospital had said that her odds of survival were excellent, but somehow, she'd still died the next day.\n\nThey'd never been able to prove anything, but I knew. I'd seen the security footage that he'd stolen and hidden \u2013 saw him screaming at her in the dead of night alone in that hospital room, trying to force her to stay with him. She'd wanted to leave him and take us with her. In the end, he'd poisoned her. I saw him inject the poison himself into her intravenous tubing. If he couldn't have her, then no one else would. That was the day that I lost both my parents.\n\n\"Annis?\" I say weakly. \"But he poisoned you. I saw.\"\n\nA soft smile. \"She was there, hiding. She bled the poison out and then she helped me to flee Neospes before she returned to fake my death.\"\n\n\"Yes. I remember. She handled your organ reassignment and cremation.\"\n\n\"Someone else's, but there had to be a record.\"\n\n\"So where did you go?\" I ask. \"Here?\"\n\n\"Not immediately. I was too weak to survive the Outers, despite what little we'd heard about others surviving. So I went to Sector Seven where my great-aunt still lived. She concealed us, and then when I was well enough, we made our way out here.\"\n\nI nod slowly, shaken by the onslaught of memories. \"It never made sense to me after you died why Father still branded you as an enemy of Neospes. You were dead. Why sully your memory and what you had contributed?\" The realization is slow like icy water. \"So he knows you're not dead?\"\n\n\"Maybe,\" Aurela concedes. \"He never believed Annis, even though the data was all there. He sent waves of soldiers out to all the sectors, saying that I had been named a traitor and had stolen valuable crown research. He offered wealth and food and weapons, but the Artok are a proud tribe. They protect their own. I was never found. So for all intents and purposes, I am dead.\"\n\n\"But does he think you're dead?\"\n\n\"I honestly don't know,\" Aurela says.\n\n\"And Shae?\"\n\n\"Unlike your father, Shae never stopped looking for me.\" Her words unlock another memory, one of Shae telling me at night that my mother would always care for us no matter what, and that I'd see her again one day. But I never believed her. I cried and nodded but never believed. Once more, Shae was right. Aurela smiles again; this time, it's a proud smile. \"She's so tenacious. She made Annis tell her, and two days later, we found her wandering out here on her own, nearly half-dead.\" Aurela stares at me expectantly, and I feel the answering dread unfurl in my stomach. \"She's not with you?\"\n\nI try to speak but I can't. The words are lodged in my throat like sharp stones, choking me with their broken edges. I open and shut my mouth like a flailing fish, gasping. Caden answers from behind me.\n\n\"She didn't make it,\" he says. \"Shae's dead.\"\n\nWithout warning, Aurela falls back like a rock, and I find myself lurching forward to catch her. Her skin is warm and she smells the same as she did ten years before, of vanilla and earth. If I wasn't convinced before, I am now. Her silvery eyes find mine and then I'm hugging her so tightly, I can't even breathe. I'm sobbing and breathless, and I pull even tighter.\n\n\"I'm sorry, I'm sorry, I'm sorry. She... died to save us. So stupid and so bullheaded.\"\n\nMy face is wet against hers. \"Shhh,\" she whispers, comforting me as if I'm still two years old, crying from a skinned knee. \"It's OK. Shae knew what she'd gotten into. Like you, she had a soldier's heart. I just... Nothing really prepares you to lose a child, does it? Don't worry; we'll get through this together.\"\n\nIt seems like forever that we've been lying on the ground, until I hear a suspicious sniff behind me and glance up to see Caden hastily wiping his face. I do the same and roll my eyes at him. Already, I can feel the cold numbness seeping through my body \u2013 my usual self-defense mechanism against emotion. Showing vulnerability is still not something I'm entirely comfortable with, despite my deepest tiny desire to stay in my mother's arms forever.\n\nI stand, pulling Aurela up with me, and step away as if it will erase the last few minutes. But this time, the deadening numbness is slower than usual. My regret cripples me, as do whatever feelings I have for this woman I haven't seen in a decade. My mother.\n\n\"This is Caden,\" I rasp.\n\n\"I know who he is,\" Aurela responds, pulling Caden into a long hug. She looks at me over his shoulder. \"Thank you for keeping him safe.\"\n\nI want to agree that that's exactly what I've done, but for some reason I can't hold her gaze. My eyes drop to the floor and I mumble something unintelligible. The truth is, I went to the other world to find Caden at Cale's request. Not to keep him safe at all. I haven't even been sure of what I'm going to do with Caden once we get to the city, but that's the thing that has been driving me. Complete the mission. Worry about everything else like feelings and casualties later.\n\nOnly, now we were stuck in some underground secret village in the Outers on the outskirts of the city \u2013 a flourishing city that isn't under the thumb of Neospes.\n\n\"How do you know who I am?\" Caden asks, quietly interrupting my thoughts, staring from Aurela and back to me.\n\n\"I've known you a very long time. Come,\" Aurela says, \"Let's go to my quarters, and we'll talk more.\"\n\nThe men in the passageway clear a path as we walk past in Aurela's wake. I grab our packs on the way out despite some hard stares from the men standing there. A quick search through the contents confirms that nothing of value has been removed, other than the missing weapons.\n\n\"Where are my blades?\" I ask.\n\n\"They're safe,\" Aurela responds. \"Come.\"\n\nAs we move away from the room, I notice that the rock composition of the wall shifts into something less glassy and more earthy, although still marbled with the black rock from the cave before. At the same moment, I feel something vibrate at my waist, where the shirtsleeves of my suit are wrapped underneath the blanket I still have slung over my shoulders. The lights on the keypad are blinking as if the suit's trying to reactivate.\n\nI frown, shrugging off the blanket and pulling the top half of the suit back over my shoulders, wincing at the slight sting of the neck connector plugging into my central nervous system. It's trying desperately to reboot, but something's still running interference.\n\nAurela turns glances at me out of the corner of her eye. \"It's not going to work,\" she says quietly.\n\n\"What?\"\n\n\"The suits. Do you know why they didn't work? Especially in that room?\" Aurela waves a hand at the black-veined walls of the tunnel. Her voice is still careful and quiet. \"It's electromagnetically charged volcanic glass. It disables all computers, all robotics.\" I frown at her words. Volcanic? Are we near the Peaks? I'm still frowning as she continues. \"It's how we're safe from the metals and the hybroids. Even in here, where the pulse is less strong, it still interferes with any computer signals. We're virtually undetectable.\"\n\nI power down the suit, and remove the neural connector to ask the question heavy in my mouth. \"Are we near the Peaks?\"\n\nA laugh. \"Honey, we're in the Peaks.\"\n\n\"But how is that even possible...\" I gasp. \"The Peaks are impenetrable.\"\n\n\"To androids, yes,\" Aurela agrees. \"But not to us.\"\n\nShe ushers us into another room, this one veined with light blue quartz-like glass. Oddly, I feel like a huge weight is lifted off me. I don't know if it's the color of the walls \u2013 it reminds me of the sky in Caden's world \u2013 but I take a long deep breath and feel less muddled. The remaining cobwebs in my head clear as if by magic, just as my suit boots up. Aurela is watching me closely, a slight frown marring her forehead, but she looks away when I make eye contact.\n\nI turn to look around. A huge table surrounded by chairs dominates the room. In one corner, there's a long desk with several flat-screen computers, data flashing across all their screens. I turn to Aurela but Caden beats me to the punch.\n\n\"I thought you said no computers worked here,\" Caden says. He too had been listening quietly and paying attention.\n\n\"They do in this room. We have to have some means of communication with people outside and to keep up to date on what's going on in Neospes.\" She nods to my clothes. \"Those will work now.\"\n\nBut I'm already powering up the suit. I can see that the three men who have accompanied us all have their hands on their weapons. Obviously, the technology of the Vectors' suits is something they are wary of.\n\nAnd rightly so.\n\nThe suit just doesn't just control temperature or create awesome topological holographs. It's an advanced bio-weapon itself. At the touch of a button, bladed spikes rise from the fabric of the suit. It can harden like armor in less than a second, or change color to fade into a background. The suit is designed for espionage, insinuation, and attack, and its defensive and offensive properties are legendary. But as I'd showed Caden, the best thing about them is the neural connector that taps into the brain's signals to the body. Regardless of programmed or natural human impetus, the suit responds to flight or fight signals in a millisecond. In some of the later models, the suit calibrates to your very thoughts.\n\nI've never been a fan of the automatic-pilot mode. It's too unpredictable when tapped into humans. That technique works best with the Vectors, because they're emotionless and run by computers. Humans are too subject to emotional decisions, especially under stress. If the suit gets mixed messages, it will opt for the first directive, even if there's a later counter-command. It'll then have to be manually overridden, which in many cases means it's already too late.\n\nStill, limitations aside, short of flying, it's a super-suit, unmatched in design or abilities.\n\nDefense, I command silently, and feel the programming engage as the suit tightens against my body. The movement is barely noticeable to human eyes, but the slight smile twitching along the corners of my mother's mouth irks me. It's no surprise \u2013 she knows the suits better than anyone. After all, before she defected, she invented bioprogramming of the early prototypes.\n\nI stare at her with narrowed eyes.\n\n\"You still don't trust easily, do you?\" Aurela remarks.\n\nI'm hard-pressed to wipe my standard frown off my face. \"Not really. Where are my weapons?\"\n\n\"Why do you need them? Surely you're protected enough already.\" Her meaningful glance dips to my uniform, but in the same breath, she gestures to a cot against the far wall, and with some relief I notice my harness and scabbard on top of it along with our two crossbows. I walk over to inspect them, but they seem the same as before. If anything, my swords look like they've been cleaned and oiled. I frown but sling the harness over my arm.\n\nAurela sits at the table and inclines her head for Caden and me to do the same. I place the ninjatas carefully on the table and take a seat. At a glance from their leader, the men who had accompanied us leave the room. The last one \u2013 a boy near my own age \u2013 shoots a glare at me that could melt rock, but I just grin, baring my teeth at him in a mockery of a smile. It's a look that has scattered crowds. His glare fades quickly.\n\n\"Haven't lost your touch, I see,\" Aurela says, noticing the exchange. At my stare, she continues, her voice soft. \"There are many stories of you, even ones as a mother I wish I'd never heard. But you did what you were commanded to do, and you did it extraordinarily well.\" I remain silent. \"At fourteen, your reputation preceded you. Even grown men were terrified to face you. What made you leave? Defect?\"\n\nI knew the question would come. I spare a brief look to Caden but his face is carefully expressionless. He wants to know the answer, too. \"Cale ordered me to find Caden.\" I pause, searching for the right words. \"I was to bring him back alive.\"\n\n\"Why?\"\n\n\"You know why,\" I snap, evasive.\n\nAurela's return stare is measured. \"We need to trust each other, Riven. I know you feel you can't trust me, and that I will have to earn that from you. Even though I don't fully know your motives, you're here and he\" \u2013 she says with a searching glance at Caden \u2013\"is safe.\" She reaches her fingers across the table where my hand is resting, but I pull it away at the last second. She lets hers rest where mine had been. \"I know that my daughter is in there somewhere. I felt it before. I don't know what happened with Shae, but she would have moved heaven and earth to keep Caden away from you if she thought you were a danger.\"\n\n\"She did,\" I grit out. \"She failed. She didn't trust me, either.\"\n\n\"She didn't at first.\"\n\n\"How do you know that? Did she tell you?\"\n\nAurela nods, folding her hands in her lap. \"Whenever she everted back here to evade the Vectors or you, she appraised me on what was happening in the other world.\"\n\nI slam to my feet, anger coloring the inside of my skin with dull red flame. \"And you let her? You let her evert over and over again, knowing what it would do to her? How could you do that? Knowing it would kill her?\" The words are rushing out of me like a river of pain from between clenched teeth. \"What kind of mother are you?\"\n\n\"A fighter, like you.\"\n\n\"You sent her to die!\" I scream, fist curling at my sides.\n\n\"She was there to protect him from you.\"\n\nAll I could see is her face behind the fire now burning in my brain, inflaming me. And then I'm lurching toward her. Out of the corner of my eye, Caden jerks out of his chair, but I don't let it distract me. For an older woman, my mother is fast, darting out of my way and spinning across the top of the table like a gymnast. I leap over it. I don't know whether it's my rage that blinds me, but I don't see her strike until it's too late. Her arm catches me across my side, knocking the wind out of me. In her other hand, I see the glinting blade of my ninjata that had been on the table. Its point is pressed against my neck.\n\nCaden flings himself on top of me before I can move. \"Riven, stop,\" he growls. \"Shae was there for me. You know that. She told you that. Stop blaming yourself or anyone else for her death. It was what she wanted.\"\n\nAll the fury seeps out of me like the air in a balloon, and I close my eyes. My voice is strangled and bitter. \"She died for nothing.\"\n\n\"That's not true,\" Caden says. \"She died for us.\"\n\n\"Then she died for nothing.\" I open my eyes to meet his, and then Aurela's. The pity in hers is suffocating. \"It is true. My mission\" \u2013 I spit the words \u2013 \"was to bring you to Cale. He's the king here. And he's sick. He needed me to find you\" \u2013 my heart twists, but Caden has to know all of it \u2013 \"for parts.\"\n\n\"For parts?\" he echoes vacantly. Caden's eyes are horrified but I have to finish. I have to tell him what he is. He'll find out sooner or later, and either way, he'll hate me for it.\n\n\"Body parts. You're a clone, Caden. You're not real.\" \nCONFESSIONS\n\nAurela's incredulous laugh is long and hollow and cold at my bombshell. I push Caden off and stand slowly, dragging him up beside me. Aurela is wiping tears from her eyes, waving away the people who had rushed into the room at the crashing sound of Caden's chair.\n\nCaden jerks away from me, shrugging off my arm. \"What do you mean, I'm a clone?\"\n\nBut I don't answer, my eyes still resting on Aurela, who is shaking her head with an expression of complete disbelief on her face. Caden is a clone, I'm sure of it. I'd heard it from Cale's own mouth.\n\n\"You're no clone,\" Aurela chokes and then looks at me, her stare discerning. \"Is that what Cale told you?\"\n\nI flinch inwardly at her words but I nod, wary, studying her face for deception. But her eyes are clear and her voice even more so. \"He's not. Caden is the real prince of Neospes.\"\n\n\"That's impossible.\" Though I think them, they're not my words. They're Caden's. He sinks into the chair behind us with a stupefied expression. \"I'm not... not a... prince.\"\n\n\"You're wrong,\" I say to Aurela. \"He's not. Cale is.\"\n\n\"Am I? Think back, Riven. You know I'm telling the truth.\"\n\nHad Cale lied to me? Or is Aurela the one lying? But as quickly as I ask myself the second question, I'm replaying the events in my head, searching for anything that could have told me Cale was lying. Why should I doubt him now, if I hadn't then?\n\nUnless I hadn't seen it. I hadn't wanted to see it.\n\nCloning is an old technology, used only by the royals, and forbidden everywhere else. It has undesirable side effects \u2013 we learned the hard way that clones had odd frailties \u2013 like weakened immune systems or psychosomatic disorders. In a world rebuilt on utopian principles, genetic purity was critical to survival, and cloning was outlawed shortly after the Tech War. But clones were still commissioned as safeguards for the monarchy, an extra layer to protect the royal line from insurgents. When Cale confided that his clone had been taken, I was suitably outraged on his behalf. He said that he was sick and would die if I weren't able to locate and bring the clone back to Neospes. I was the only one he could trust.\n\nAfter his father was murdered, I was so eager to do Cale's bidding, so eager to save him that I agreed without a second thought. But in hindsight, I still don't see any deception. Was I been so gullible? Am I still?\n\n\"I'd rather be a clone,\" Caden says dully to no one in particular. \"I always knew I was different, but this just takes the cake.\"\n\n\"So there's no clone?\" I ask slowly, Caden's inane words piercing the sudden fog of activity in my brain.\n\n\"No,\" Aurela says. \"There is. It's just not Caden.\"\n\nThe realization hits me through the fog like a ton of bricks to the face. I feel my feet stagger backward, and my hip braces against the side of the table. Grasping the edges with numb fingers, I hold myself from sliding down.\n\n\"Cale's the clone.\" My voice is a monotone. Aurela nods, her expression compassionate. The realizations come more quickly after the first. I put two and two together quickly. Cale's mother had left when he'd been four. \"Their mother left to protect the prince,\" I muse quietly, sparing a glance at Caden, who still looks like he's in some kind of waking dream. I don't blame him. My mind is spinning like an unstoppable top. I can't imagine the confusion he's feeling.\n\nAurela nods. \"She suspected Murek long before. She knew he was collaborating with your father to assassinate the king. It was only a matter of time before they came for her son.\" Aurela stops, lowering her voice. \"They wanted to use him. Leila knew she was in danger when she realized she was in the way. When she took him, she went to the Artok. They brought her to us, but the only real safe place for her and the prince was her world.\"\n\nJune wasn't lying after all. Leila had always been from over there, and she took her son to the only refuge she knew. And that's why June and Era broke all the oaths they'd taken as Guardians to protect her. The puzzle is far more intricate than I've ever imagined. I shake my head. \"Even if she did switch them before she\u2013\"\n\n\"She didn't switch them,\" Aurela interrupts. \"That was your father's idea after she everted. They needed a puppet, so they made up this story that his mother had died. But Cale\u2013\"\n\n\"\u2013told me the truth. That she'd everted with the clone. I mean, that's what he thought. Or what he'd been told.\" I pause and half stand. \"By Murek and my father.\"\n\nIt all made sense. My father knew that Cale was the only person who I ever had any loyalty to, so when Cale got sick, he took the opportunity to tell him about the clone, and it was natural that Cale asked me \u2013 his most trusted confidant \u2013 to track down the clone, even if by obeying his orders in secret, it made me appear to be a defector in the eyes of everyone else.\n\n\"I see you now understand.\"\n\n\"Why would they send me?\"\n\n\"Because you were the best,\" Aurela says gently. \"You are the best. The Vectors reported that Caden was with Shae in the Otherworld, and Cale knew you were the only one who could fight or beat her.\"\n\n\"So is Cale really sick, or is that all a lie, too?\"\n\nAurela pours something black that looks like coffee into three mugs, and sets a steaming cup in front of Caden and then in front of me. I take a gulp, and although the bitter taste overwhelms me at first, the aftertaste is thick and mellow like butterscotch.\n\n\"No, our reports confirm that he is sick.\" Her words cause the cracks inside of me to widen into furious chasms that I can feel splitting me apart. Relief seeps in to flood the fractures as I realize that Cale didn't lie to me. He didn't send me on some fool's mission. He didn't betray me to my father. He sent me to help him live, even if it were part of some misguided plot of Murek's. I'm sure Cale would never betray me. He is as much a victim of Murek and my father as we are.\n\nSighing heavily, I glance over at Caden, who hasn't moved, sitting with his head in his hands, staring into the mug as if it holds answers only he can see. I can't even imagine what this must be like for him.\n\n\"You OK, Caden?\" I ask.\n\n\"What do you care?\" he rasps without looking up. \"You brought me here to die, didn't you?\"\n\n\"No!\" I say, pounding my fist on the table so hard that the shock runs up my arm and through my back. \"I didn't. I mean, at first you were a target, but now...\" My voice trails off, caught in the turmoil of what exactly Caden has come to mean to me.\n\n\"Now what?\" he says, turning around with eyes so green, they're like the grass in his world. It's like they're seeing right into me, past all the flesh and bone, deep down where there's nothing else but truth. \"Now what, Riv?\" he whispers.\n\nI stare at Aurela, but she can't say the words for me. Instead, her face is compassionate, as if somehow she already knows. She knows what I feel... everything I've kept buried under my orders. I owe Caden the truth, don't I?\n\n\"But now...\" My throat is clogged and my eyes are smarting. \"Now it's different. You're my friend.\" I can see that those words are not enough. Caden's gaze drops from mine to stare once more into the coffee mug.\n\nI'm struck dumb. My mouth won't move. Nothing is moving. A single tear weeps out of the corner of my left eye. I leave it, feeling its hot path meandering down my cheek. My tears are the words I cannot say.\n\n\"Come,\" Aurela says gently, interrupting the heavy tension between Caden and me. \"Let's get you two settled for the night. We'll talk more tomorrow, once you've had some rest.\"\n\nThe minute we leave the room, I feel disoriented again, as if I'm suffering vertigo. I take deep breaths as Aurela escorts us down some more dark passageways. It helps. I notice that the men fall into silent step behind us, guarding their leader at every moment. Even though I know how important she is in this small community, for the first time I start to question just how important she is. She holds herself with a quiet confidence \u2013 the same self-assurance I remember as a child. But there's no arrogance in her words or her manner. She is one of them even as she leads them.\n\nCaden has a room all to himself with an armed guard. Now that I know who he is, I'm not surprised. Aurela is taking no chances that word has already gotten out about Caden's identity or that some zealous defector will try to get back into Murek's good graces by offering up the runaway prince.\n\n\"Cade,\" I say at the entrance. \"I'll see you tomorrow. OK?\"\n\n\"Yeah.\" But he doesn't look at me at all, not even when he lies back on the cot at the far end of the room and lays his head on his palms, staring up at the ceiling. I stand there for a moment, uncomfortable, before I hear Aurela gently calling my name. \"Night, Cade.\"\n\nHe doesn't answer as I turn to follow Aurela. She stares at me with a knowing expression. \"Don't worry; we're not far away. And he'll be different in the morning. It's a lot for him to have to take in; just give him some time.\" I know that she's right, but Caden's aloofness hurts more than I ever thought it would.\n\nThe thought of Cale pops into my brain, and I shove him away. I can't choose. I won't. I'll have to find some other way around all of this... some way to save them both. I'll give myself up to my father if I have to, if there's anything he can do to save Cale.\n\n\"This is where you are,\" Aurela says. She points to an adjoining room. \"My quarters are just over there.\" Glancing around, I see that people are taking notice of where I've been placed \u2013 a room adjoining hers. Aurela reads my expression easily. \"It's because you're the bigger threat,\" she says smoothly.\n\nBut inside, I know there's more to it than that. She could have left me in the cell we'd been detained in, surrounded by armed guards. Instead, I'm like some guest in her private quarters. It's a message.\n\nA message that I am important.\n\nFor some reason, I don't like it. I'm not sure that I've forgiven her for everything between us, for everything that Shae knew... that I did not. For leaving me behind with him. My voice is bitter. \"I'd rather stay in the first room.\"\n\n\"Riven, that is a holding cell. None of them trust you as I do.\"\n\nI narrow my eyes at her. \"Why? Why do you trust me? You know who I am. What I do. What I've done.\"\n\n\"Yes, I know all of that. But I also know that you are my daughter, and there's some of me still in there, no matter how much he's tried to weed it out of you all those years.\"\n\n\"What do you mean?\" I snap.\n\nHer voice is quiet. \"Nothing.\" Aurela stares at me for a second, her white-blonde hair curling around her shoulders. \"Get some rest. Things will look better in the morning,\" she says, and then, \"Riven?\"\n\n\"What?\"\n\n\"I lo\u2013\"\n\n\"Don't say it,\" I snap back, cutting her off midsentence. \"You don't even know me.\" I stare with dead calm into her silver eyes. \"And you're right. He did cut every last part of you out of me. Everything human, everything that should feel something. He made me emotionless just like him. And you know what? I like it.\"\n\nI am so proud of the strength and conviction of my voice, but her tiny smile is my undoing. She steps forward and I hold my ground. I don't even blink when she takes a strand of hair that is stuck to my cheek and tucks it behind my ear, nor when her fingers trail over the tattoos on my neck to rest on my shoulder with a reassuring squeeze. \"And yet you wouldn't tell me this if you weren't fighting it inside this very moment, would you? Sleep well, my little blackbird.\"\n\nAnd with that, she was gone.\n\nI stare at the wooden door between our rooms for a long moment. She called me little blackbird. The thought draws me backward, and I'm sitting in my room, crying my eyes out.\n\n\"Why'd you name me Riven? It's so horrible,\" I was wailing. \"It's not even pretty. It's ugly, like me.\"\n\n\"You're not ugly, darling,\" my mother soothed. \"Your name was supposed to be Raven, which is the name of a tiny blackbird that visits me from the gorge behind the house. It has the prettiest whistle. When you were born, you made that sound.\"\n\n\"Really?\" I asked, still sobbing but curious.\n\n\"Yes, but well, mistakes happen. And your name was recorded as Riven.\" She kissed me on the nose then. \"I have an idea. How about if I call you little blackbird, just you and me? It could be our little secret.\"\n\nAnd I nodded, thrilled with having a secret name that remained a secret between us until the day she died... I mean, until now. Even Shae didn't know. If I had any doubt that she was who she said she was \u2013 my mother \u2013 I didn't anymore. No one would have known about that name but the two of us. It was ours alone.\n\nI glance around at my quarters. From force of habit, I'd already taken inventory of the small, square-shaped room the minute we'd walked in, but it looks the same as all the others \u2013 spartan, with the exception of a small table and chair on one side, next to a cot. Nothing, except its position next to Aurela's, marks it as superior.\n\nThe flame of a small candle dances against the wall, illuminating the white quartz and onyx colors in the rock. I stare at the rock and tilt my head to one side. I don't feel as claustrophobic or as unbalanced in this room. It's odd how I feel more uncomfortable in some areas of the Outer underground than I do in others, almost as if the rock composition is tied to my ability to function, like the computers. I laugh \u2013 I must be more worn out than I think.\n\nI glance at the cot, but I can't sleep. Too much nervous energy is swirling inside of me. My mind still feels muddled, so I strip down to my underclothes, taking care to fold the suit over the chair. I sit cross-legged on the floor and pull energy into my center for a long period until my heartbeat is steady and my breathing full. I extend each arm forward, and then ease my legs out into a side-split, stretching my tight muscles. The sequence of calisthenics falls into place as I twist my torso over my left leg, and my mind goes blank, muscle memory kicking in.\n\nNearly an hour later, my body is dripping with sweat, but I haven't felt so alive since being underground. I'm wired, energy coursing through me and filling my cells with vibrant life. Without missing a beat, I grab my ninjata blades and start swinging them in a graceful arc, my legs extending outward at the same pace.\n\nThe exercise starts out slow and then gradually builds in speed until I'm gasping for breath and whirling the blades with incredible swiftness. I'm moving so quickly that the glossy blades are a blur in the room, the flicker of candlelight on them almost making them look like liquid flame between my fingers.\n\nThe swords are moving faster than I am, and my body strives to keep up, moving faster and faster and faster, until something hot nicks the back of my leg. I jerk to a halt, staring at the watery crimson trail that is welling against my skin. The voice at the corner of the room takes me by surprise.\n\n\"Getting a little rusty?\"\n\n\"What are you doing here?\" I pant, wiping the sweat off my face with my forearm. \"You should be sleeping.\"\n\n\"Like you are?\" Caden saunters into the room and pulls out one of the chairs at the table, straddling it with his legs on either side and his arms across the top. His dark hair is unruly as if he's been running his hands through it one too many times in aggravation. He rests his chin across his crossed forearms staring at me through squinty eyes.\n\n\"I need the exercise,\" I say.\n\n\"And I couldn't sleep,\" he tosses back. \"You know, a boy doesn't find out he's a prince from the magical land of Far Far Away every day.\" The sarcasm is heavy in his voice, and I bite back a smile at the reference to Shrek.\n\n\"At least you're not an ogre, and it's probably a lot less magical than the one you were in.\"\n\nHis stare is assessing, a lock of hair curling into one eye. \"So, tell me something. When you thought I was a clone, you were coming to get me to bring me back here, and Shae was protecting me from you?\" I nod, uncertain of the direction of his thoughts, but continue my movements, albeit more carefully now. \"So I was your target?\"\n\n\"Yes.\" I slow my pace further with the swords, lunging and stretching both my arms in an arc over my head before pulling them around to the front and twisting away from him.\n\n\"So am I still your target?\" His voice is louder than it was, and I whirl around. But his voice isn't louder, and Caden is no longer sitting. He's right in front of me. His hands grasp my wrists, halting them mid-motion.\n\nHe's so close that I can feel his warm breath feathering against my cheek. In a smooth motion, he removes the ninjata from my left hand, stepping back and swinging it in a slow circle. I take a slow breath.\n\n\"Don't hold a weapon\u2013\"\n\n\"\u2013that you're not prepared to use,\" Caden finishes. \"Shae told me.\"\n\n\"Cade...\" I begin.\n\n\"What? Are you so afraid to fight me?\" he asks softly. There's something in his voice that I can't identify, something painful and aching.\n\n\"I don't want to hurt you,\" I say. \"These are real swords, you know. Not fencing foils.\"\n\n\"I know.\"\n\nAnd before I can think, all I see is the flash of a blade curving toward me. The clash of steel in the small room is like thunder as my blade meets his in a shower of sparks, but Caden is already sidestepping and striking from the underside.\n\nIn a few seconds, I'm aware that Caden is more than good. He's really good.\n\nShae wasn't joking about how easily he holds a sword. I sense that most of it is instinct, but he has the basics of what we are all taught in Neospes. He's taken that a step further with his own fencing training. Despite the fact that I've spent the better part of two hours practicing, Caden has me on my toes. Even though my body wants to go into full attack mode, I restrain myself.\n\n\"Why are you holding back?\" he taunts, reading my slowed movements accurately.\n\n\"I don't want to hurt you.\"\n\n\"We'll see about that.\"\n\nI drop to a crouch and jerk upward, only to find the top end of my sword crashing into the bottom of his on its way down. His ability to read responses is uncanny, almost as uncanny as my own, and I grin widely at the unexpected challenge.\n\nCaden's bare foot catches my heel and I fall backward, only to catapult to my feet in a crouch, my sword at his back. He fends me off capably, and then we are spinning to the discordant tune of crashing metal, until I am against the wall with his sword upon my neck. Caden's eyes are triumphant.\n\nBut so are mine.\n\nI tap the point of my sword against the inside of his hip, and as he looks down, I grin. And then I'm laughing, and Caden is laughing, until his fingers slide against my cheek, and the laughter slips from my lips. His eyes are so green, it feels like I'm drowning in them. I want to move, but my body won't listen. My arms drop to my sides, and my tongue slips out to moisten suddenly dry lips. I pull my lower lip between my teeth.\n\n\"Don't do that.\" The harsh whisper is Caden's.\n\n\"Don't do what?\" I say chewing unconsciously on the corner of my bottom lip.\n\nHis eyes darken. A storm-tossed meadow. \"That thing with your mouth.\"\n\n\"I didn't\u2013\" But his lips silence mine in mid-sentence, the soft warm pressure of them hugging the curves of mine like they'd known them forever. Our breaths mingle as we draw apart, and Caden is staring at me with those impossibly green eyes. I can't help myself. I lean into him, parting my lips and slanting my mouth against his. His hands are on the back of my neck and around my back, drawing me against him so tightly I can barely breathe. But I clutch him tighter, lost.\n\nThe second kiss of my life.\n\nKissing is an anomaly in Neospes. Couples are paired by genetic compatibility, not by what they feel about each other. But humans are social creatures, and sometimes love blooms after the pairing, although that is incredibly rare. I remember one boy in my training group who developed an affinity for another trainee. It'd affected his performance so clearly that within a day, the girl had been transferred to another sector.\n\nLove made us vulnerable, made us weak. Those were our rules.\n\nBut Caden's kiss makes me weaker and stronger all at the same time. And the way it makes me feel \u2013 like I am flame on the outside and liquid on the inside. It makes me feel alive, as if I can take on anything. And the only time I ever feel like that is when I'm fighting, when the adrenaline takes over and I'm only fire and fight.\n\nNow I'm fire and something else entirely.\n\nMy hands tangle in his hair, into the soft mess of it, and I draw him closer. Not even the clatter of the swords on the floor tears us apart. Eventually, we come up for air, and as we pull apart, my mind drifts to our first kiss in the bathroom at Horrow, so similar to this one but so intoxicatingly different. My fingers slide against the square line of his jaw and across the sharp rise of his cheekbone.\n\nCaden presses his lips into my hair and stays there for what seems like an eternity. I can't move, not even when he leans into me and rests his head on my shoulder, turning his face into my damp neck. In fact, every part of me is motionless as his lips find the curve of my collarbone, winding their way up to my ear, fanning the fire once more unfurling in my chest. My legs are unsteady.\n\n\"I love you.\"\n\n\"I love...\"\n\nFor a second, I imagine that's what I started to say. And then I'm splintering into an abyss of darkness and cold and pain.\n\nDECEPTION'S DAUGHTER\n\n\"Is she going to be OK?\"\n\nThere are white, bright lights everywhere, flashing. They hurt my eyes, even closed. I try to move, but my arms are restrained. So are my legs. I'm lying on a cold, white surface in what appears to be an emergency medical bay. I crack open an eyelid, squinting at the wave of agony that threatens to send me back into an unconscious stupor.\n\nA hologram of a human body is suspended in the middle of the room and surrounded by all kinds of shifting miscellaneous data. I blink. Everything is so white. Even the medical garments barely covering my torso are white. My lips are cracked and sore. My tongue is glued to the roof of my mouth, and when I try to talk, my voice crumbles like dust. I blink again, opening both eyes and trying to focus. Pain stabs through my head.\n\n\"Water,\" I manage to gasp.\n\nA shadow looms and the rim of a cold cup is held gently to my lips. The water is like ice, soothing the dryness inside my mouth. I want more, but the cup is gently taken away.\n\n\"WhereamI?\" The words merge into one. I try to sit up, but forget that I am restrained. The panic is immediate. \"Where am I?\" I scream, my throat seared raw.\n\n\"Sector Seven,\" a voice says. \"You're safe.\"\n\nMy brain registers only two things. I'm back in the city, in a Sector on the outskirts of Neospes. And there are no bioengineering facilities in Sector Seven. Where am I being held?\n\nFlashes of Cale and Shae and my mother wind their way through my mind, and then I see another face. It's Cale's but not Cale's. And he's looking at me with gentle eyes full of something that terrifies me. It terrifies me because my heart understands what his eyes are saying, and it feels the same. Who was the boy? A dream? A figment?\n\nMy mind is angry, squelching the tiny spark of emotion. I am two separate things. Memory and present. Frailty and strength. I have to be strong. The boy is nothing to me.\n\nBut he's not a dream. The shadow forms into focus, and I see him. His name is Caden, my brain whispers. He's the target. You brought him here. You have to take him to Cale, the Lord King. But not yet.\n\nWhy not yet?\n\nThe questions are overwhelming, dueling inside of me.\n\n\"Hey,\" the boy says with a crooked smile. \"You OK?\" I nod, and close my eyes, turning away from him.\n\n\"Caden,\" another voice says, a woman's voice. \"You need to get some rest. We're through the worst, and she's awake now. You'll be able to talk to her soon.\"\n\n\"What happened back there?\" I hear Caden ask. \"No one told me anything.\"\n\n\"Get some rest. I promise I will explain later.\"\n\nI hear the sound of the door closing as Caden leaves. The woman checks my arms and the neural leads that run from my chest and temple to the base of the metal table. It connects below to the row of computers monitoring my vitals. I realize then that the suspended holo body is mine.\n\nHer fingers are gentle. I stare at her through my lashes, surreptitious. She has long silvery blond hair that has been tied into a row of braids across the top of her head, and hangs thickly over one shoulder. Her eyes are clear, her face youthful.\n\nShe is your mother, my mind tells me. Aurela. But it is a piece of information, nothing more. I don't feel anything overwhelming, not like the rush I'd had with the boy earlier. I open my eyes to meet hers.\n\n\"How long have I been here?\" I ask hoarsely.\n\n\"Three days.\" Her hand brushes across my forehead. Her touch feels odd against my skin, tingly and tender. I shy away automatically. \"Do you remember anything?\" she asks but I only stare at her silently. \"What's your name?\"\n\n\"Riven. Legion General.\" I watch her face slide from relaxed to anxious to pained.\n\n\"Do you remember the Otherworld? The Outers? Anything?\"\n\n\"Yes, of course.\"\n\nAurela studies me, her eyes narrowed as she comes to some understanding in her head. It worries me, and I feel my brow furrow, uncannily matching hers ridge for ridge. \"What is your mission?\" she says slowly.\n\n\"To secure the target.\" I blink. The response is automatic, programmed in my brain, but something doesn't feel right. The feelings in my head don't match the words in my mouth. They're coming from somewhere else... some alternate source of information feeding answers to my lips. The thoughts are there but something tells me that they are not my own.\n\nThey're orders. Someone else's orders.\n\n\"No... I don't know,\" I say, pulling my arms against the steel manacles. \"Someone is in my head. It's not me!\"\n\n\"Riven, calm down,\" Aurela says. \"Let me get the doctors. Everything will be OK.\"\n\nBut my panic is swift, and I'm wrenching my arms on the table until the metal starts to cut through my flesh. Oddly, I don't feel a thing. The veins in my arms and neck are corded so tightly that they are raised and navy against my skin, and I'm mesmerized by the intertwined deep red and pale blue fluid oozing out of the lacerations at my wrists.\n\nRed. And blue.\n\nThe machines are beeping so loudly that they sound like some kind of terrible security breach. One hand breaks free, the twisted wrist bracelet still attached to the table, and I'm clawing at my other hand. I don't know where I'm finding the strength, but I rip off the leads stuck to my chest without even blinking. More machines bleat in immediate succession, but they've dulled to a low hum in the back of my mind as everything goes strangely calm. My only objective is to escape. It has ahold of me like a starving dog defending a bone, relentless.\n\nA team of people rushes into the room, and Aurela's face is hanging over mine. Something cold slides into the back of my neck and spreads through the rest of my body. Aurela's eyes are clear, holding my own. I can see the answers in them. The last thing I think before sedation grabs hold of me is that she knows. She knows.\n\nShe's always known.\n\nWhen I awake once more, I'm no longer in the operating analysis room. I'm in a small four-walled gray cube, lying on a thin bed. There are cameras in the ceiling and a metal stool and table in the corner of the space. Someone has dressed me in soft flannel clothing.\n\nI try to sit up, and surprisingly do so with little pain. It feels weird, like I am somehow back to normal. I stretch my neck in slow circles, but everything feels fluid and strong. It almost seems that everything I remember must be some sort of bad dream, but of course, I know that it is not. I run through my mental checklist in my head, confirming all of the factual information I know about myself and who I am.\n\nMy name is Riven. I am a Legion General. My target is Caden. I have acquired the target. Caden is my friend.\n\nThe last thought shoots into my head like an errant arrow, and I analyze it carefully like a piece of forensic evidence. I feel nothing for the thought itself, but it's an oddity that intrigues me. In Neospes, friends are a luxury and oftentimes, in my opinion, more of a hindrance. Unlike family, they complicate things. Lines become blurred, and I like being able to make decisions objectively. As a result, I have no friends, so it's odd that I would have a memory that a target had become my friend. My only \"friends\" have been Cale, and Shae once upon a time.\n\nShae.\n\nThe name is a bullet exploding inside of my consciousness... a part of my brain that I know is wholly mine. A wave of agony ripples through me, and my body folds in on itself, the memories flooding my head like boiling lava. My arms grip around my torso so hard that I can barely breathe. I can hear someone screaming \u2013 keening \u2013 and it takes a second before I realize that the sound is coming from my own mouth.\n\nShae is dead. She died to save me. My sister is dead. She died to protect Caden.\n\nCaden is my friend.\n\nThe waterfall of memories assaults me anew, and it all comes back in a rush \u2013 Cale, the Otherworld, the Vectors, Shae, Caden, the Outers, Aurela.\n\nAll of it.\n\nThe keening sound continues, and I'm rocking back and forth, curled over. I feel like I am being torn down the middle, between an overly rational part of my brain that doesn't even feel like me and the tiny insistent part that nearly does. Neither feels like who I am. Because I don't know who I am.\n\n\"Who am I?\" The words spit themselves from my lips like acid. I stand in front of the camera in the corner of the room and scream the words again.\n\nThe door opens and a man clothed in a black uniform walks in, his hand on the butt of an electro-rod sitting at his hip. He is a Legion Commander; I can see the seal on his neck clearly. I feel my eyes narrow, and I unconsciously sink my weight back into my haunches. His hand hovers over the weapon, correctly interpreting my movements. But he has no chance.\n\nMy reaction is instantaneous, despite my lack of practice. I run at the side of the wall, kicking off of it and somersaulting to land directly behind the man, my elbow around his neck and his left arm up against the middle of his back before he can even breathe. I jam the heel of my foot into the back of his leg, and he sinks to his knees.\n\n\"Move and you die,\" I say. \"Where am I?\"\n\n\"You're in Sector Seven,\" the man gasps. \"Please...\"\n\nSo I'm still in Sector Seven. That surprises me. Are we still in the bio-research facility?\n\n\"Where in Sector Seven?\"\n\n\"Underground research lab,\" he answers. \"Off the grid. Look, I can explain if you just let me up.\" I can feel myself wavering between wanting to eliminate the threat and getting answers. A fully stocked bio-research plant that's off the grid? In all my years in Neospes I've never heard of one of those. My arms tighten involuntarily. \"I'm... I'm Aurela's first-in-command,\" the man gasps. \"She left me to watch you.\"\n\nReleasing my hold a fraction, I remove the electro-rod from its harness and toss it away. Against my better judgment, I let him go, and he falls back to his haunches, his hands to his neck.\n\n\"Aurela's first-in-command,\" I repeat slowly. \"What do you mean, she left you to watch me? Where is she? Who are you? Why do you have a Commander rank seal on your neck?\" My questions are fired in immediate succession. The man smiles and removes his black beanie to run his hands through his white-blond hair.\n\n\"I'm Commander Sauer, an active soldier in Sector Seven,\" he begins. \"But I'm also Artok and your mother's guard,\" he adds.\n\n\"She never mentioned you,\" I snap unmoved.\n\n\"It was part of our agreement,\" he says quietly, his stare meaningful. I understand where he's coming from. He's her agent on the inside. If he'd been connected to Aurela in any way, especially after her defection, he would have been targeted and executed as an example.\n\n\"So you're a spy.\"\n\n\"Yes,\" he says.\n\nI study the seal on his neck and the colors that mark him as Sector Seven. He is a Sector Commander, which would give him \u2013 and Aurela \u2013 unrestricted access to valuable information. Still, being an inside spy in a place like Neospes is incredibly risky, as in face-the-Vectors risky. I eye Sauer with a little more respect.\n\nAs the Legion General of the elite Vector task force, I never heard of him, but Neospes is a big place, with allotted sectors policed by different branches of the monarchy, so it isn't farfetched that we never met. Unless, of course, I had been the one to catch him being a traitorous spy. The old me would have fed him to the Vectors myself.\n\n\"How old are you?\"\n\n\"Twenty.\"\n\n\"Did you grow up here?\"\n\n\"I came here with my grandmother.\" Sauer is unfazed by my rapid-fire questions. \"Any more questions?\"\n\nHis cool attitude annoys me. \"Why is this place off the grid?\"\n\n\"Aurela started to build it before she left. She needed a base with the appropriate technology to face whatever your father was planning. We finished it. It's not a registered facility.\" Sauer smiles again. \"If Murek knew this existed, he'd probably torch the whole sector.\"\n\nI frown. \"So everyone in Sector Seven is a part of this?\"\n\nI think back to what I know of the area. Sector Seven has always been a peripheral colony and classified as non-confrontational. Their citizens have always lived on the outskirts, keeping to themselves. They'd been deemed artisans \u2013 despite some of their Artok roots \u2013 and weren't considered a threat to the monarchy. How wrong we'd been, I understand slowly. From the little I knew of the Artok, they'd been far from a placid tribe of people. There are obviously more of them than anyone had guessed living in Sector Seven, and my mother now leads them all.\n\nSauer nods. \"Not all but most of us. We're the liaison to the Outers, to Aurela. We keep tabs on Murek and the Lord King.\" He says the last words quietly, and I can feel his eyes on me. I keep my face expressionless. The doubt in his voice is easy to read. He doesn't trust me. And why should he? I don't even trust myself. I don't even know myself.\n\nSauer stands slowly and stretches his arms across his head. My body tenses, but I don't move. He hasn't made any offensive moves toward me, and even if he'd been lying, we would have been surrounded in seconds by scores of Vectors desperate to get their hands on their absconding general.\n\nI turn on my heel, giving him my back. It doesn't really mean anything, but I've learned that it implies a level of trust and vulnerability. It's a misconception, though \u2013 I've been trained to detect shifts of movement in the air by fighting blindfolded with steel weapons. If he pulls any kind of weapon to attack me, it will have the same outcome.\n\nI walk over to the cot and sit on it. Sauer, to his credit, hasn't moved an inch. \"Let's assume you've been telling me the truth all along,\" I begin. \"Why are we in Sector Seven now?\"\n\n\"That's for Aurela to explain.\"\n\n\"OK. So where is Aurela?\"\n\n\"She's... engaged at the moment.\" His evasive answer irritates me.\n\n\"OK. So where is Caden?\" My tone is patronizing, my anger under tight control but simmering just below the surface. I cock my head to one side and lean back on my arms, waiting. I am deliberately trying to provoke him, goading him into an answer. He is obviously under orders.\n\nSauer smiles tightly. \"In a room just like this one. Resting.\" His emphasis on the last word is clear, and I almost grin at his blatant suggestion that I should be doing the same.\n\n\"I want to see him.\"\n\n\"That's not possible.\"\n\nMy eyes narrow. \"You do know that I can leave this room at any point with or without your consent? And I can find Caden myself if I have to.\"\n\nSauer shrugs and nods. He knows exactly what I'm capable of. I showed him that clearly when he'd entered. He walks over to the chair on the side of the room and slides it out.\n\n\"May I?\" he asks politely, and then sits, crossing his ankle over the opposite knee. His eyes are penetrating, staring right through me. I don't say anything caustic at his expression, because I'm a little intrigued by him myself. The more I stare at him, the more I see the resemblance to Aurela's people, and the marks of the Artok tribe. Apart from his hair, which is the same distinctive silvery white, they have a similar angularity in their faces. Despite my caution, I'm fascinated by any connections to Aurela's past... to my past.\n\nSauer's voice is soft, interrupting my thoughts. \"I've always wanted to meet you, you know. My entire division recounted accounts of your fearlessness, of soldiers who cowered at the sight of you. A girl, far younger than I was, leading the Vectors. The Lord King's private guard.\" I hold Sauer's gaze without responding. \"Impressive record for one so young.\"\n\n\"If you say so,\" I return in an inflectionless voice.\n\n\"But of course, your father engineered you to be that way,\" Sauer muses. I frown, but remain silent. \"He built you to be the perfect soldier. What made you defect?\"\n\n\"I don't know that I have,\" I say carefully, my brain firing at his provocative words. I am no one's puppet, far less my father's. \"I was following orders from my king. My loyalty lies and will always lie with him.\" Sauer looks like he has something more to say, but doesn't. A sour expression crosses his face for an instant as he leans back in his chair. It fades after a moment, replaced by his former thoughtful expression.\n\n\"I knew Shae,\" he says. The mere mention of her name sends my stomach into a tailspin. I feel the tide of emotion surge inside of me. It's all still there, simmering. I shove the thoughts away, but my fingers clench into fists at my sides. \"She often told me that I would like you.\"\n\n\"Well, she was wrong. And she's dead.\"\n\nI'm unprepared for the naked ache that slashes across his eyes. My normally acute ability to read people has taken a beating over the last few weeks, but even Sauer can't conceal his feelings for my sister from me. Sauer has just gotten a lot more interesting.\n\n\"You were the reason she kept everting back here,\" I say slowly. Sauer doesn't answer, but his clear eyes are so pain-ridden that it's obvious. \"I knew it wasn't just to provide updates to Aurela or throw the Vectors off Caden's trail. He was safe. The Vectors tracked me there, not them. So she was coming back here for you?\"\n\n\"Yes. We were... in love.\"\n\nI laugh, and the sound in the room is ugly, echoing emptily against the stark walls. I can barely get my mind around it. My mother had been paired with my father because of their combined brilliance and what they could contribute together to the monarchy. People didn't fall in love. They didn't get to choose who they wanted to be with. Partnerships were allocated based on what was best for Neospes. I laugh again emptily. In the end, love had killed Shae.\n\n\"Love?\" I spit in his direction, launching to my feet. \"Do you know what you did to her? Did you see what you were doing to her? She was dead on her feet, but she still came back here for you!\" I'm in Sauer's face now, not even bothering to control the violence of my rage. \"You. Killed. Her.\" My finger jabs into his chest with each word. \"You made her weak.\"\n\nSauer doesn't even respond to my vitriolic words. Instead, he watches me with those same heavy eyes. I can see the regret \u2013 and his love for Shae \u2013 in them, but I don't want to see either. I don't want to see anything that reflects the feelings inside my own heart. I don't want to admit that somehow, somewhere, I'd let compassion or love weaken me, too. So I let my anger take over. I let my fury fuel me. They are the things I know, the mindsets I understand.\n\n\"Get up,\" I rage at him. I slap him across the face, and then a second time. \"Get up and face me like a man. She deserved that much, don't you think?\"\n\n\"I'm not going to fight you, Riven,\" Sauer says. His face is bright red from where I've struck him, but his voice is even. \"You're right. I failed her. I let her die. I deserve everything and more that you say, but she wouldn't want this. She loved you, too.\"\n\nSomething hot and responsive rushes inside of me at his soft words, and then something cold immediately floods my veins, suppressing it in seconds. I step back and then back again, until I feel the bed against the backs of my legs. My mind is clear one minute, and then fogged the next. Once more, the feeling of being two different people threatens to tear me apart.\n\n\"What's wrong with me?\" I say, clutching my head in my hands. \"What's happening? Every time I think about her, it hurts. I'm splitting in two.\"\n\n\"It's the programming,\" Sauer says gently, walking forward to fold me into his arms. I let him, shivering so hard that my teeth are rattling. Sauer's words sink in slowly.\n\n\"The what?\" I whisper.\n\n\"The programming,\" Aurela says from the doorway. I hadn't even noticed her there. \"I see you've meet Sauer. I'm sorry I wasn't here when you woke up, but I had to take care of something. How are you feeling?\" She tips my head back gently to peer at my face.\n\nMy brain is spinning, but somehow I already know. A hazy memory of the pale blue fluid combined with the blood on my wrist in the operating room fills my eyes. \"What programming?\"\n\n\"There's no easy way to\u2013\"\n\n\"Just say it,\" I whisper. \"Say it.\"\n\nAurela's face is tormented, her hands fluttering against my head like protective mother birds trying to safeguard their young. But she can't protect me anymore. The damage has already been done. She didn't protect me then... and she can't protect me now. Sauer's words fill my head: he built you to be the perfect soldier...\n\n\"I want to hear you say it,\" I repeat doggedly.\n\nShe nods, just once. \"Your programming. To control the nanoplasm inside of you.\"\nON THE BRINK\n\n\"So I'm a Vector?\"\n\nI'm amazed at how calm my voice is, but the truth is, somewhere deep down, I've always known that I was like them. There were too many inconsistencies, things that made me better than everyone else... things that made me less human. I could run faster, react more quickly, heal miraculously when injured. There's no way I could have been normal. I don't even feel any sense of violation that my own father experimented on me without my knowledge. Yet.\n\n\"I'm an android,\" I say.\n\n\"No,\" Aurela says. \"You're... a hybrid. Your father experimented with nanoplasm before the early success of the Vectors. He experimented on himself and then you when you were little.\" Aurela holds my face in her hands, but my eyes are unseeing. \"I never knew, but when I found out, that's when I knew I had to leave. I had to find a way to stop him.\"\n\nWhat kind of man experiments with an unproven technology on his own child? I feel the hate inside of me boil and metamorphose into something large and ugly. I'm not even human. No wonder he'd loved that I'd chosen to lead the Vectors. I'm one of them. Controlled, just as they are. Built. Engineered. A thing, like them.\n\nDead, just like them.\n\nThe self-disgust must be evident on my face, because Sauer's arms tighten around me. I've forgotten that I'm still caught in his grip. \"Let go of me,\" I tell him. \"I'm fine.\"\n\nBut he doesn't, so I shove him away easily. So easily, like his arms are nothing but string. I stare at my curled fists, for the first time aware of the tensile power in them... power that isn't mine. It's fake, engineered strength, driven by the robots in my blood. My hands drop to my sides in revulsion.\n\n\"Riven,\" Aurela says, and I jerk at the sound of my name. So apt, I think. My name means broken... it's a perfect name, after all. I raise burning eyes to hers. \"You're nothing like them,\" she says, correctly reading the thoughts rising to suffocate me with their poisonous intensity. \"You're different.\"\n\nI shake my head, struggling to reconcile everything I know about the Vectors and everything I know about myself. I'm not just different... what my father has done is impossible. I'm a live person made of flesh and blood and bone.\n\n\"I'm missing something,\" I say to Aurela. \"How is this even possible? I thought you couldn't combine nanoplasm with human DNA?\"\n\n\"I believe your father experimented on himself before you were conceived.\" Aurela pauses, as if thinking to herself. \"He planned it all from the beginning. His body eventually rejected the nanobes, but yours didn't, because your DNA had already transmutated as a result of his earlier testing. When you were conceived, the law of natural selection made your cells adapt. He'd hoped all along that they would have some sort of genetomorhic effect.\"\n\n\"Did you know?\"\n\n\"Not at first,\" she says, running a hand through her hair tiredly and sitting in the seat that Sauer vacated. \"But he would take you to the lab with him, even when I insisted that you didn't have to go. After a while, I started to suspect something untoward, but he denied it.\" Aurela stops, her face wet. \"Then I found contaminated blood samples he'd hidden. Yours. And I knew for sure. That was when I told him I was leaving with you and Shae. My mistake, of course, was to tell him at all.\"\n\n\"I thought you were fighting about the Vector tech?\"\n\n\"No, my darling, it was always about you. Then you had your first training op, and he was so proud that you, the youngest of all your peers, had finished way ahead of everyone else. Way ahead. You were a prodigy. His prodigy. After that, he never let you out of his sight.\"\n\nAurela is talking about the placement trials. At four, we are put through a rigorous series of mental and physical exercises to determine when we start our training and instructor assignments. I was placed in the elite section with children far older than I. My father was so proud. I bite my lip so hard that it bleeds, a metallic sourness filling my mouth.\n\nOf course he was proud \u2013 his creation, his abomination was a smashing success.\n\nBut even in his jubilation, he had to keep it all a secret, because what he'd done went against all our laws. After the War, any combination of human genetics and android technology had been forbidden. But my father had flaunted the laws, driven by his own pride.\n\n\"And the Vectors? Was it his plan that they'd be his own private army? Loyal to him because they were loyal to me? Was that in my programming?\" My sarcasm is acidic.\n\nAurela sighs. \"Only your father knows what he intended to create. Even now, we don't know where his loyalties truly lie.\" She glances at Sauer and then me, as if working out something in her head, something more that she has to tell me. \"So many things have changed since you left, Riven. The monarchy is unstable. Murek and your father have spies everywhere. The Vectors are stationed in every sector, even here.\"\n\nThe thought of the Vectors leaves a sour taste in my mouth. My stomach heaves nauseatedly. I don't want to be anything like them. The big Commander from the Otherworld fills my visions, and I remember the sound of my father's voice coming from its mouth.\n\n\"They're thinking now,\" I say, remembering the chilling words of the Vector that it was more alive than the others but less alive than me. Even it had known what I was... what I am. \"The Vectors. There was one in the Otherworld that spoke with his voice. He wants me back. Badly. I don't know for what but he does.\"\n\nAurela's face blanches at my quiet words. Understanding is in her eyes as she nods. \"He wants to replicate your genetic code.\"\n\nI feel a hot tear slide down my cheek, and I swipe it away viciously, furious at myself for even shedding a single tear for the monster that was my father. \"I'm just a thing to him. An experiment. A Vector.\"\n\n\"No,\" Aurela insists. \"You're not.\"\n\n\"Barely. I'm half-alive, and they're dead. But I'm the same... a thing.\" My voice is as lifeless as the words coming out of my mouth.\n\n\"No, Riven, you're alive. You control the nanobes, not the other way around.\"\n\nBut a troubling thought occurs to me as Aurela says those words. If I do control them, why is everything within me suddenly shutting down? Has something set them off within the parameters of their code? I frown, confused. \"So, what happened before? You know, with Caden, before Sector Seven, when I blacked out? What caused it?\"\n\n\"My guess is that it was some kind of failsafe in your father's programming, a phrase or something,\" Aurela says, and I feel myself flush, knowing instantly the phrase that had caused it all \u2013 Caden said he loved me and I returned the sentiment. It makes sense. My father despised weakness and anything that caused it, especially love. \"Whatever it was caused the nanoplasm to reboot, restoring a set of baseline defaults, which is why you were so confused about who you were and couldn't remember everything.\"\n\n\"Will it happen again?\" My question flies through gritted teeth. The last thing I need is anyone using some programming catch phrase that will stop me from doing what every bone in my body wants to do \u2013 make my father pay for what he did to me.\n\n\"I don't think so,\" Aurela says. \"I worked with my engineers to erase and rewrite the default programming. We backdoored all the code. It's clear.\"\n\n\"Thank you.\" It's the least I can offer. \"Did Shae know?\" I ask after several minutes of silence. So many things are falling into place... all of Shae's sidelong glances, her pointed questions about my healing ability, other things she wanted to say to me in the Otherworld and couldn't. She wanted to tell me; I see that now. But the sad truth was that I never would have believed her and she probably knew that, which was why she always stopped herself.\n\nAurela nods. \"That's why she tried so hard to get you to leave with her, but she knew that a part of you would want to stay with him. She was so afraid that you would tell him about her... about me, in spite of yourself. You were a risk, and one we didn't fully understand.\" She pauses, her face earnest. \"We didn't know how deep the programming went or whether it undermined your own thought.\"\n\n\"Does it?\" I blurt out.\n\n\"No, it's built to obey your commands.\"\n\nThe knowledge is overwhelming, but things are starting to come together in my head, like migrant puzzle pieces. \"So that's why I felt so sick in the Peaks. Because I have machines inside of me,\" I say. \"I should have known.\" I gesture at my body. \"You know why I used to feel so comfortable leading the Vectors? It was because deep down, I felt just like them. And I followed orders just like a good soldier, just like a good little reptile.\" I swing around to stare at Sauer who is standing near the far side of the room. \"Now you know why I was so good, because I was one of them.\"\n\nSauer shakes his head, a small smile darting around the corners of his mouth. \"No, Riven,\" he says in that soft drawl of his. \"You're better than they are. You're super strong, you heal quicker than any of us could ever hope to heal, you can think more quickly. And you're alive. You're still you. That has to count for something.\"\n\nHis words strike a chord inside of me, and realization dawns slowly but surely. As brilliant as my father is, he isn't a genetic scientist. I stare at Aurela. \"It was your genetic coding that made this possible, wasn't it?\" I say to her. \"You were the only one who could have found a way to string nanoplasm with live human DNA.\"\n\n\"Yes, you're right. I developed the bio-gen coding,\" she confesses sadly. \"He was working on a project to test the nanoplasm on live creatures and convinced me that he needed to test to see if it could operate within a live host.\" Aurela grabs my shoulders. \"I never would have done it had I known that he was going to use it on himself or you; you have to believe that!\"\n\nI nod, because I can't speak. My tongue is bonded to the roof of my mouth. I swallow painfully. \"So can you take them out? The nanobes?\"\n\n\"No,\" she says, \"they're part of you. Unlike the nanoplasm for the Vectors, which fire off a lithia core, yours are linked to your body. They fuel from food just like your blood does because they're tied in to your DNA. If we even tried to separate the strands, you would die. Your body has already adapted to coexist with them. You're unique, Riven. That's probably why he wants you so badly. You're the experiment that went viral. His biggest triumph.\"\n\nI'm at a loss. I don't even know what to think, far less say. I have live microscopic robots inside of me that can never be taken out. I can never be fully human, never be normal. Everything inside funnels into a tornado of fury against the man responsible for making me into a freak. The man who thought himself some sort of god. His arrogance would be his destruction.\n\nI would be his destruction.\n\n\"Hey, guys,\" a voice says, and I whirl around, only to collide with Caden. His hair is rumpled around his face and there are pillow lines creasing his skin as if he's only just awakened. He smells like soap and outdoors. I've never wanted more to fling myself into his arms and close my eyes, but I steel myself. \"Glad you're awake,\" he says to me, his eyes gentle. \"You scared me for a while.\"\n\n\"I'm fine,\" I snap more harshly than I intended. Hurt flashes in his expression as if I've slapped him, and his eyes pan slowly from me to Sauer to Aurela.\n\n\"What's going on?\" he asks.\n\n\"Nothing. We're talking logistics,\" I say dismissively, and turn to Aurela. \"Can we get out of here? This place is making me sick.\"\n\nThe rest of the bio-facility is the same as any I've been in with my father, a veritable maze of white walls and white doors. Once in a while, people dressed in white walk past us. They all nod or bow respectfully in Aurela's direction. She's more than a leader, I realize. She's their unofficial queen. Even Sauer walks a step behind her, I notice, in some kind of dutiful deference.\n\nWe enter an elevator at the far end of one of the corridors and make our way to the top. The elevator opens into a simple, nondescript two-story house. I look backward as the wall slides shut behind us, completely concealing the hidden elevator behind it. Aurela was right; there's so much I don't know about what has happened over the last three years.\n\n\"Make yourself comfortable. I'll be back,\" Aurela says, and walks downstairs.\n\nThe room we're in is simple, with a long dining table surrounded by wooden chairs. It's sparse but comfortable, and unlike the clinical austerity of the bio-facility beneath us, it's painted a peaceful yellow. Silvery moonlight filters through one side of the house, and the cool air and smell of civilization hits my face through the open windows. It's never smelled so good. Even in the outlying sectors, Neospes had a unique odor. It's not offensive or foul, just achingly familiar, like an odd combination of well-oiled machinery and bread. I breathe deeply, walking to one of the half-opened windows.\n\nIt's nighttime, but I can see the tip of the citadel rising in the distance. It has been my home for so long that a part of me twinges. I squash it as quickly as it fills me \u2013 it's been a cage, not a home, and all part of my father's plan. I push the shutters open and lean out slightly, watching the people bustle on the street below as the cool air rushes in. Even though it's night, it's nowhere near as cold as it was in the Outers. I look up at the giant glass perimeter of the dome spanning as far as the eye can see. It performs a double function. It protects us from any predators from the Outers, and it regulates the unpredictable temperatures of our atmosphere. So right now, the night air is crisp rather than freeze-blood-to-ice cold.\n\nBut any way I look at it, we are all still prisoners in a giant fish bowl, and a small part of me would rather be out there than in here. Sirens go off in the distance, and Sauer's voice from behind me jerks me from my thoughts.\n\n\"Be careful,\" Sauer warns. \"The Vectors are on high alert.\"\n\nMy eyes narrow. \"Why?\"\n\nHe shoots me a wry grin, putting some tableware and a pitcher of some liquid on the long table. \"Mostly rumors that you're back. Of course, since you removed your tracking chip, you're off the grid, too. But as much as the Artok are our allies, we can't stop people from talking, especially those loyal to the monarchy.\" He grabs my pack from a corner of the room and tosses it to me. \"Don't worry; I'm sure you're more than familiar with their search techniques. And if worse comes to worse, you can take care of yourself just fine.\"\n\n\"Thanks,\" I say drily, but I shut the window carefully behind me. The contents of my backpack look the same, but I know that Aurela's guards would have been meticulous and careful in their search. \"What about you? Are you off the grid?\"\n\n\"No, they know where I am. But I'm off duty, and technically this is my house, so we're good.\" Sauer adds three different-shaped bottles to the table and pours himself a glass from one of them. After a pointed glance from Caden, Sauer slides him a glass across the table. Caden knocks it back in one swift shot as if he's done it a hundred times before. I can smell the alcohol from where I am standing. Sauer nods at an empty glass and then in my direction.\n\n\"No, thanks,\" I say. I've never mixed well with spirits of any kind, even though it feels like I probably should have a glass like Caden to calm the storm still simmering inside of me. I look away and haul two deep breaths into my lungs. I'm pretty sure that giving in to the temptation would only have a worse outcome... like me running off to confront my father in a blind rage or something equally stupid.\n\n\"Where's my jacket and blades?\" I ask sifting through my bag.\n\n\"They're still in the Peaks,\" Sauer says. \"Caden grabbed your pack when you passed out. Don't worry; Aurela sent some men to retrieve them.\"\n\nMost of the other, smaller weapons are in the bag. I notice that Shae's thumb drive is resting on the top of everything else. I still haven't been able to bring myself to see whatever she has recorded on there. In hindsight, I probably should have listened to it the minute we left the Otherworld \u2013 no doubt she would have mentioned Aurela or even the thing that I am. A shudder rips through me at the thought, and I slap the flap closed.\n\nSauer stares at me over the rim of his glass as I stretch the corded muscles in my neck. He bangs the glass on the table, making both Caden and I jump. \"Come on, I have an idea.\"\n\n\"What about Aurela?\" Caden says.\n\n\"She'll be a while, and we're not leaving the house,\" Sauer throws over his shoulder. \"We're just going to get some tension out.\"\n\nCaden and I follow him down the stairs to the main floor and then down another set of stairs into a wide room that's lit with overhead recessed lighting. I recognize the layout immediately \u2013 it's a training room. The floor is padded and two opposite walls are lined from top to bottom with mirrors. There are a few combat dummies on either end of the room, all of them battered, with chunks torn out of their torsos and faces. A vast array of weapons, from knives to spears to lances to pikes, lines one of the other two opposite-facing walls, but what takes my breath away is the antique collection of intricately carved Artok bows lining the near wall.\n\n\"Wow,\" I hear myself say.\n\n\"Shae's,\" Sauer says softly. \"She used to say that it inspired her.\"\n\n\"It's beautiful,\" Caden agrees.\n\nI agree wholeheartedly but I can't speak. The thought that my sister fought \u2013 trained \u2013 in this room fills me with an odd choking sensation that leaves me empty for words. There was so much of her life that I missed, so much that I didn't understand. If only I'd made the choice to go with her all those years ago, how different everything could have been. I'd have had a sister and a mother, instead of an egomaniacal father who cared more for his zombie robots than he did his real flesh and blood.\n\nBut \"what if\" never did anyone any good, far less me. I made my choice. I didn't go with them, and now my sister is dead. I feel the unspent anger swelling within me again like a monstrous tide, and I understand why Sauer has brought us down here.\n\nTraining is what he knows. It's what I know. I throw off my outerwear and join Sauer where he and Caden are standing in the middle of the room.\n\n\"Standard warm-up drill?\" Sauer asks as I windmill my arms.\n\nI nod and glance at Caden, who like Sauer is only wearing black fatigues. He isn't half as chiseled as Sauer, but the sight of his lean chest makes my stomach waver. A faint flush fills my cheeks, and I cover it up with gruffness. \"You want to watch first and then join in?\"\n\n\"Shae taught me,\" he says.\n\n\"It's not like fencing practice,\" I toss back.\n\n\"You just try to keep up.\" I almost snort out loud at his overconfidence, and then I remember our sparring back in the Peaks and his skill. Bowing mockingly in his direction, I take a deep breath and focus, centering my energy.\n\nIn a line, we bend and twist through a complicated series of rhythmic calisthenics, moving and breathing in silent unison. The movements are slow and long, extending to the edges of my center of gravity, strengthening my core. In the mirror, I can see that Caden is keeping up easily, his body as flexible as ours, as if he's been doing the exercises for years. Obviously, Shae taught him well.\n\nCompleting the first stage, we move sinuously into the second phase, which incorporates more jumping, kicking, and thrusting. Our training is built on a dynamic combination of hard and soft martial arts that focuses on energy and core strength as well as defensive and offensive strategies. I'm covered in a sheen of sweat, but it feels so good \u2013 every jab and shove helping to dissipate the raw tension in my body.\n\nNearly two hours later, Sauer retrieves three long, slender black staffs from the side and hands them to Caden and me. We're moving into the soft-weapon stage of the training, comprised of delicate and quick sparring movements against each other, the staffs acting as extensions of our bodies. We spin and lunge in unison, and in the mirror we are a blur of graceful but lethal movement. The staffs click against each other in a precise, dangerous staccato as we meet in various steps of the exercise.\n\nThe final phase of the training is actual sparring. We pair off for a sequence of moves and then spin back to the third opponent. Sauer is strong but occasionally leaves his flanks unprotected. If we were in actual combat, he would be a worthy opponent, but I know that I would eventually beat him.\n\nCaden is another matter altogether. His movements are graceful and catlike. He reads my strikes almost in advance of them, like a mind-fighter, nor does he expose many weaknesses. He doesn't falter in his offensive strikes, and his defensive moves flow like yin and yang. He knows when to attack and when to withdraw, which is something only learned after years of experience.\n\nI'm slowly realizing that Caden has mastered most, if not all, of our elite training techniques. With a grin, I understand now that Shae wasn't kidding when she said he could probably take me. Caden had been her final and best trainee. Preoccupied, I spin out of the way of his staff at the last minute, but it still catches me on the back of the shoulder.\n\nWincing, I see that Caden is grinning at me. \"Almost had you that time.\"\n\n\"I was distracted, and almost doesn't get you any points,\" I shoot back, and then incline my head. Credit should always be given where it's due. \"But yeah, you were good.\"\n\n\"Feel better?\" Sauer asks.\n\nI nod. \"That was exactly what I needed. I forgot how good it felt to do that from beginning to end. In the Otherworld, I only got to practice it in pieces, and finding an adequate sparring partner was difficult enough.\" I squeeze Caden on the back, ignoring the spark of electricity that shoots up my fingers at the damp touch of his skin. \"Shae taught you well.\"\n\n\"Thank you. She made me do it every day before school, rain or shine,\" Caden says. \"She was relentless.\"\n\nSauer hands us each a cup of drinking water. \"You were both good.\" He glances at me knowingly. \"But you, you were holding back. Why?\"\n\n\"I wasn't,\" I begin, but realize that Sauer is right. Despite my fatigue, my body still feels like it could go for several more hours. I shrug. Before knowing the truth, going all out was the only thing I knew. Now, it feels weird. It feels fake because of what I know I am... as if I'm cheating somehow. I stare at the ground, tension hovering against my shoulders, eager to weigh me down once more.\n\n\"Try this.\" Sauer walks to the wall and removes a thin longsword. The scabbard is blood red with black markings, and the sword's handle is black interlaced with silver. Sauer's face is solemn as he stands in front of me, slowly removing the elegant silvery blade, and only then do I see the inscription of my name near the hilt. I gasp.\n\n\"Shae had this made for you in the Artok way,\" Sauer tells me. \"The sword's name, like yours, is Riven. It means 'to cleave asunder.'\"\n\n\"I know what it means,\" I say for lack of anything else, taking the sword reverently in my hands. The craftsmanship is extraordinary, the weight of it perfect.\n\n\"So,\" Sauer says. \"Let's see you try it out.\" I frown at him, but he is staring at me expectantly. I glance at Caden, who agrees and moves to sit with Sauer at the far end of the room, giving me enough space with the deadly weapon.\n\nHolding the blade vertically between my palms in front of my face, I take a deep breath and then lay the sword flat out in front of me on both palms, as I sink into sideways lunge and bow. I close my eyes, letting my breathing and the movement of my body guide me.\n\nThe training studio fades away and I am alone. Flicking the hilt of the sword up, I grab it easily and spin into a complicated series of thrusts, jumps, and parries. This time I don't hold back as the sword flows in and out of my movements like a binding thread, a fluid extension of me.\n\nI'm spinning faster and faster, letting the power heating up within me take over, letting the part of me that isn't human lend me strength, until I am nothing but shadow and flashes of light. And still I go faster. I can feel my heart burning, a vortex of flame and liquid exhilaration. Perhaps Sauer is right \u2013 maybe I am better than the Vectors. My chest pounding, I finish with a final slash and bend into a forward lunge with the blade held over my head in perfect symmetry.\n\nI open my eyes. There is dead silence in the room. Self-conscious, I stand, only to find that it's no longer just Caden and Sauer standing there. Aurela is there too, accompanied by six people in hooded cloaks. I hadn't even heard them come in.\n\nI recognize four of them immediately as the Sector Leaders of Sectors Three, Four, Six, and Eight. The other two I don't know, but they're all staring at me with identical expressions of astonishment, awe, and something else I'm not quick to identify, despite seeing it thousands of times before.\n\nFear. \nREVOLUTION\n\nI wipe the grime off my skin with a washcloth, using a special cleansing oil. What I wouldn't give for one of the decadent showers of Caden's world. That's one of the things I treasured \u2013 that and the beauty of a world that hasn't been ravaged by war. For a second, I think of the lake in the mountains that I saw with Shae, and my heart aches. I won't see either of them again. They have become synonymous with beauty and loss.\n\nDespite the unexpected audience at the end, the training exercise was exactly what I needed to feel normal... to feel like myself again. Tugging on an undershirt and a pair of leggings, I drag a brush through my hair. It's gotten longer, but I haven't bothered to cut it. Staring at myself in the mirror over the water basin, I hardly recognize myself. Faded chunks of green intertwine with darker strands. I've lost weight, and my cheekbones ridge prominently on the sides of my face, but my eyes are bright and cheeks rosy.\n\nMy thin blue-weaved braid hangs over my left ear to my shoulder. Carefully, I unravel the braid, using my knife to hack off the strands to match the length of the rest of my hair. I don't know why I left it \u2013 but it's been a part of me for so long, a status of my rank. Of course, all that has changed. I am no longer a Legion General. I am no one.\n\nNo, I am worse. I'm a fugitive.\n\nLike my mother. Like Shae.\n\nThinking of her, I glance at the long blade resting in its scabbard on my bed and feel a tear slide down my cheek. I was on her mind even though I barely spent a second of thought on her. She was always the more compassionate of the two of us, and no wonder; I'm half a real person with little empathy for anyone. A flash of bitter self-loathing surges through me... for the thing that I am.\n\nGrabbing my backpack roughly, I rifle through it to take my mind off that train of thought. My hand is drawn to a slim rectangular object. Shae's drive. It's the last remaining piece of the puzzle. Swallowing hard, I plug it into a holograph port resting on the small table on one side of the room. Shae's face swims into focus. My stomach swan dives to my feet but I force myself to double-tap the play button.\n\nHello, Riven. My entire body flinches at the sound of her voice and the soft expression on her face. She could be sitting across the table from me. My fingers curl into my sides, bloodless, as her words continue. If you're seeing this, then I'm gone. And hopefully you and Cade are still alive and you found your way back to Neospes. There's so much I have to say to you, so much I need to explain, but finding the right words is a challenge in itself. First of all, I love you. I'm pretty sure you don't think I ever did, but it's true. I've always loved you and I always believed in you. The real you.\n\nI want to throw the drive and the hologram against the wall. I can't even bear to look at her beautiful face. Every cell in me is shaking with regret and grief. It's nearly more than I can take, but I force myself to listen despite the gut-wrenching anguish spiraling through me.\n\nYou're probably going to find out a lot about yourself if you're back in Neospes, things about your father and our mother. She's alive, Riven! I wanted to tell you that day when I came back for you, but it would have been too risky for her safety. She's been here all along. She's the one who got Caden and his mother out. She's the one who sent me to protect him. If you haven't found her already, find her. Go to Sector Seven and ask for Commander Sauer. He will take you to her. If you do see Sauer, tell him I'll see him again and he's always with me.\n\nI know you're being extra careful about staying off the grid, but trust no one.\n\nTRUST NO ONE. Shae's face and words are emphatic.\n\nOne last thing, my sister. Thank you for looking out for Caden. He's the real deal. Cale's the clone. I'm sure you already know this by now, but if you don't, I'm sorry about him and that you had to find out the truth this way. I know you were \u2013 are \u2013 close. But there are things underway in Neospes that make it hard to see where the real betrayal lies... with Murek, your father, or even Cale himself.\n\nPlease be careful, and watch out for Caden. I tried to train him in our ways as best as I could without having to reveal everything. I think he will be ready for whatever comes your way, whether that's in the Otherworld or in Neospes. How I wish I could have been there with you, to protect and fight with both of you, but everything happens for a reason, doesn't it? You're meant to be Caden's protector now. He's the future of everything. Don't fail him, don't fail us.\n\nI know you won't, because you don't know how to fail.\n\nOne other last thing. I smile at the tone in her voice, watching her eyes rolling and the sideways smile that was unique to her. You're probably going to learn some troubling truths about yourself. But you know who you are, Riven. You've always known who you are. Don't let that change you. You're strong. Stronger than I could ever be. So don't fight it; instead, accept it and take it for all the good things that make you so incredibly unique.\n\nLove you, Riv; don't ever forget that. Wind at your back, my sister.\n\nThe voice spins into silence, and the hologram grows fuzzy and then disappears. But I don't need it to hear her words that are replaying over and over in my head. What did she mean that Caden was the future of everything? It sounded so ominous. There's so much I don't understand, and it's not that I don't trust Cale. I trust him with my life. He hasn't lied to me \u2013 he has been sick. I've seen it with my own eyes. He sent for Caden because he thought he needed replacement organs, and my mission had been of the utmost secrecy because of Cale's mistrust of Murek and my father.\n\nA soft knock on the door jerks me out of my chaotic thoughts.\n\n\"Who is it?\"\n\n\"It's Aurela.\" I eject and throw the drive into the pack, sweeping a hand through my hair and tucking the freshly shorn strands behind my ear self-consciously. Maybe she won't notice.\n\n\"Come in,\" I say. \"I'll be done in a sec.\"\n\nShe has changed into a gilded blue tunic and wide skirt. She looks feminine and delicate, two things that I can never seem to master. I'm no spring flower, that's for sure, but then looks have never mattered to me. Survival has. Sighing, I smooth my plain tunic and leggings, and sit to pull on a worn pair of combat boots before standing to face my mother.\n\nShe smiles gently, her eyes landing with the touch of a butterfly on the side of my head where the braid had been, but doesn't say anything. Instead, she just nods. I can feel her approval, and for some reason, it warms me.\n\n\"I'm sure you have questions,\" she says. \"Ones that Sauer or I didn't answer earlier. I want you to be prepared before you go into that room.\"\n\nShe's talking about the room where all the other Sector heads \u2013 the ones who had been watching my performance earlier \u2013 are waiting.\n\n\"She told me to find you, you know,\" I say softly. \"Shae.\"\n\n\"Yes, I know.\"\n\n\"She knew about me.\" It's not a question, but Aurela nods, her eyes compassionate. For a long second, we stare at each other. Shae would have looked just like her the older she got, with the same features... the same eyes. There is grace in Aurela's shoulders, but there's hard strength, too. I swallow. She's at the head of this whole revolution, this whole coup against the Neospes monarchy. She's more than strong. I realize the core of steel I'm seeing in her is the very same core in myself that men have grown to fear here in Neospes. I'm more like her than I care to admit.\n\nDistracting my thoughts, I grab ahold of the black and gold wrap lying on a chest and tug it over my shoulders. Dress golds. It's a plain shift with a high collar and shiny black buttons, one of the more formal items of clothing usually reserved for award ceremonies or funerals. I smile at the irony. We aren't exactly going to either.\n\n\"Are you going to war against the monarchy?\" I ask, deftly fastening the over-garment's ties around my waist. The soft material drapes nearly to mid-thigh.\n\nAurela shoots a glacial smile in my direction. \"It's complicated, Riven. War isn't the answer for anyone. But what Murek is planning will undermine everything we have built, not just in Neospes, but also in the Otherworld.\"\n\n\"What exactly is he planning?\"\n\n\"If my intelligence is correct, he's building an army,\" she says. \"An army of Vectors to take to the Otherworld. Guardians have been assassinated. We believe that he means to make the people there into slaves and to control their vast resources. Water, to name one of the most important.\"\n\n\"He who controls water controls the world,\" I murmur. It's an old Neospes saying. Water here is traded like gold. If Murek somehow manages to control the Otherworld's resources, he would become more than a king; he'd be a god. \"But even if he were able to create a bridge, we can't survive in their world,\" I blurt. \"I mean, you and the others. Your immune systems are too different. You'd die.\"\n\nI flush, knowing that it was only because of my unique nature that I'd been able to survive, thrive even. Aurela stares at me, and for a second I see something like a flash of sympathy in her eyes.\n\n\"That's where you come in,\" she says softly. \"The next phase. It's why your father wants you so badly. You're the only living person who has ever adapted to the nanobes.\"\n\nMy words come out in a rush as I collapse back onto the bed. \"He wants to clone me?\"\n\n\"More specifically, replicate your DNA.\"\n\nMy breath is coming in short bursts, but her words are no surprise to me. As much as I hate what I am, I know that she is right. I am the blueprint to universe domination, because the laws of natural selection do not apply to me. No wonder I stopped having any side effects in the Otherworld. My body \u2013 my nanoplasm \u2013 was forced to adapt once I stopped taking the pills.\n\n\"Aurela, I know Cale's not involved in any of this. Shae\" \u2013 I nearly gag on her name \u2013 \"said that I shouldn't trust him, but I do. I'd been at his side every day before I left to find Caden. He is dying. Murek killed his father, and his mother left to protect her other son.\"\n\nAurela nods. \"We will protect him if we can.\" She pauses. \"But Caden comes first. He is the true crown prince; do you understand that?\"\n\n\"Yes.\"\n\n\"Good, then let's go, unless you have any more questions.\" She stands and stares pointedly at me.\n\n\"Just one more question,\" I say, but my eyes drop to the ground as a rush of warmth races across and down my back. \"When you and father were paired together... did you ever feel anything for him?\"\n\nAurela's stare morphs into something pained. \"Your father was very different when he was younger. He was smart and vibrant, and I admired him very much.\" Her voice is at odds with the surge of painful emotion in her eyes. \"We were friends as children and often played together. Our partnership was a natural progression of that.\"\n\n\"But did you love him?\" I blurt out, and feel the heat of a thousand suns on my neck.\n\n\"I loved you,\" Aurela says gently. \"And Shae.\"\n\nI understand her restraint. How can one ever love someone who is a monster? And inside, I'm just like him. He's made sure of that.\n\nAurela moves to stand in front of me as if reading my thoughts and grasps my arms. \"Your father was driven by a desire to prove himself. He had no room in his life for love. You have me in you too, and we are Artok. Our choices, our lives are based on love.\"\n\n\"I'm not Artok,\" I say automatically.\n\n\"But you are.\"\n\nI ignore the tender look on her face and the gentle fingers that slide against my cheek. I've never felt connected to my Artok roots. The Artok do things differently than most other people in Neospes. Although they believe in the inherent value of genetic preselection, they also believe that emotions like love are important in relationships. Maybe I felt the disconnect because my mother had left, and the only love I'd ever known was from a father whose sole goal was to make me a monster.\n\n\"Did you love Shae's dad?\"\n\nAurela's eyes grow even sadder. \"Yes.\"\n\nShae's father \u2013 also Artok \u2013 was killed in the Outers when Shae was barely a year old. Shae told me the story when we were children. Her father was an atmospheric field technician who never returned from an Outer mission. His entire group was attacked by Reptiles, and everyone had survived but him. Shortly after, Aurela had been reassigned to my father.\n\n\"Did he do it?\"\n\n\"Did who do what, Riven?\" Aurela's voice is thick, and I know that she is lost in her own memories of the love she has lost... and the daughter she's lost.\n\n\"My father,\" I spit in disgust. \"Sounds just like him, so convenient. Did he arrange to have him accidentally left alone? Did the guards do it and blame it on Reptiles?\"\n\n\"It was an accident, Riven. The transport sensors were faulty, and there was nothing anyone could do.\" Her voice is calm but her words are monotone as if she's already thought through some other explanation herself and come to the same conclusion that I'd had. \"Your father would never\u2013\"\n\nI interrupt her. \"Never what? Destroy people to get what he wants? Right.\" Walking toward the door, I glance over my shoulder. Aurela is watching me carefully. \"The next time we meet face to face, I'm going to be the one to destroy him; that I promise you. He will pay for what he did to you, to me, and to Shae. Now let's go plan a war.\"\n\nThe room is deathly quiet, even though it's full of people. I stand like a shadow in the back, watching, listening, and waiting. The hood of my jacket obscures my face so that I look like half a dozen of the other soldiers dressed in similar uniforms in the room. Caden, too, is clad the same, with his hood covering much of his face.\n\nI stare at my mother, who has become this dynamic, tough-jawed person at the front of the room. She looks regal but stone cold, with eyes like slivers of ice. In a few minutes, I can see why they all defer to her. She's incredible.\n\nThey've already been over the plans and strategy, and the best way into the castle. Caden is standing to the left of me, half hidden in the darkness, and Sauer is on his left. We're all near the doors just in case things get disruptive \u2013 Caden has to be protected at all costs. But since he insisted on being present, Aurela wanted to keep him as inconspicuous as possible. I peep around the edge of my hood and Caden catches my eye.\n\n\"You look good in dress golds,\" he whispers.\n\nI roll my eyes at him. \"Too flashy for me.\" I glance at him, wearing similar colors to mine. \"You look good, too. Now shut up and pay attention. You asked to be here, remember?\"\n\n\"What do you think about what they want to do?\" Caden asks under his breath, but Sauer gives us both the eye of death and we fall silent.\n\nTruth is, I don't know that what they're suggesting will work. The Winter Solstice is around the corner, and it's one of the two times in Neospes that there are any festivities. The other is during the Summer Solstice. We celebrate Winter and Summer with Games \u2013 a combination of exhibition combat, dancing, and trading \u2013 when all the Sectors come together, even the Royals.\n\nFor the most part, each Sector operates independently, focused on its particular role. The scientists stick with scientists, the farmers with farmers, the builders with builders. The only common threads are the soldiers who are ensconced in every single Sector. It pains me to think it, but after experiencing the freedom of the Otherworld, Neospes is little more than a police state. Soldiers are everywhere, enforcing the rules \u2013 and the will \u2013 of the monarchy.\n\nBut even despite the fact that armed guards surround the celebrations, and more recently armed Vectors, people still love them. It gives all of the people from the different Sectors a reason to come together. They compete in friendly exhibition games like sword fighting and archery. They trade animals and goods. Parades of dancers and acrobats make their way to the castle for an all-night ball in the grand courtyard. But in spite of the revelry and the communal spirit of the Games, security is always tighter than normal.\n\nWhich makes what Aurela is suggesting so dangerous.\n\nShe is planning to stage simultaneous attacks \u2013 one to destroy my father's robotics facility and engage the kill switch for all the Vectors, and two to infiltrate the castle during the ball and reveal the corruption that has corroded the once-revered monarchy. I've added a third step \u2013 to save Cale from any fallout.\n\nCaden nudges me in the side, as if reading my thoughts. \"I don't get it,\" he whispers. \"Why would there be an actual kill code in the facility? That's like having a computer password written in a diary next to your bed.\"\n\n\"It's part of robotics security. During the Tech War, we couldn't shut them down, and the rogue droids executed the programmers who had all the kill codes. When we started developing the Vector technology, the king mandated a failsafe in case the unexpected happened again.\" My mouth twists in a rueful grin. \"So yes, dumb but necessary.\" My grin turns wider. \"Plus, it's not like it's going to be in a bedside diary... more like in a locked triple titanium-enforced room surrounded by around a hundred Vector guards.\"\n\n\"Oh.\"\n\n\"Shhh!\" Sauer's hiss is loud, drawing the censorious stares of two people at the end of the long table. Caden and I shut up.\n\nOne of Aurela's commanders is showing something on a hologram that's hanging in the middle of the room. He's making things move and appear with his fingers, shifting diagrams and plans to either side of the screen. His fingertips hold and drag an image from the center that explodes into a hologram of the castle blueprints across the middle of the table. I hear Caden's audible gasp and bite back a smile. Our technology, despite the ravaged world we live in, is remarkable.\n\n\"Pretty cool, huh?\" I say to Caden under my breath with an anxious glance at Sauer.\n\n\"It's the same tech of the suits, just way more advanced, right?\" I nod and stifle a snort at Caden's slack-jawed expression. Luckily, Sauer doesn't hear me and instead moves to the front of the room, where he takes charge of the presentation. As Sauer launches into an advanced schematic of the castle's security, I find my gaze wandering again, this time around the room.\n\nI recognize a few soldiers, although I am unable to put names to any of the faces. I'm not surprised. Soldiers don't have a long life expectancy in Neospes. They don't comfortably retire. The strong survive, and the weak do not. It's that simple. Most of the others are high-ranking officials, and they, with the exception of the four Sector heads I recognized before, likely came into power after I left in search of Caden.\n\nMy eyes collide with a pair of steely black ones. A willowy girl at the far end of the room is watching me and making no attempt to hide that she is. At first glance, she is pretty, but then I see the scars... three long, ugly scratch marks marring her face. They are nearly white against her dark skin, and not even her thick curly hair can conceal them. The girl's face tugs at my memory. She is dressed in camouflage gray, the color of Neospes dust, and wears the stripes of a Commander. I frown, trying to place her face in my head but the memory evades me.\n\nAs if sensing my stare, Caden nudges me in the ribs again. \"Who's that?\"\n\n\"What?\"\n\n\"The girl over there who looks like she wants to grind you into teeny tiny pieces with her eyes alone.\"\n\n\"I don't know,\" I answer automatically before glaring at him to stay quiet.\n\nI'm bored by the sudden political turn of the conversation, and I'm agitated by the offensive stare of the girl I can't quite recognize. I know I've seen her before, but I can't place where. Her face with her violent scars is making me edgy. And the fact that she's a Commander makes me crazy. I'm longing to ask Sauer, but he's still up near where Aurela is now sitting. Instead, I sink farther into the shadow of the wall and try to ignore the girl.\n\nWhat seems like hours later, Aurela finally announces the results of the vote. It is unanimous. Her plan will proceed as indicated. I'm still not convinced that they'll be able to infiltrate the labs and the castle, but I'm a spectator... a hired gun at this point, with no say in their martial strategy. I'm going along because any distraction they create will help me with my objective \u2013 to locate and kill my father.\n\nI turn to leave with Caden in tow, but Aurela's voice halts me in my tracks.\n\n\"General Riven will require some volunteers,\" she says, waving her arm in my direction. \"She's taking point on a side assault.\"\n\nAs all eyes converge upon me, the silence in the room is even worse than in the beginning. It is heavy with a dark undercurrent. They don't trust me, not because of what I am \u2013 no one really knows the truth other than a few \u2013 but because of who I am... my reputation.\n\n\"General Riven?\" someone hisses. \"We can't trust her.\"\n\n\"Where has she been all this time? How convenient that she's back.\"\n\nThe noise in the room escalates as pandemonium erupts. The dark-eyed girl is watching me with a smirk. My mouth curls down, and it's all I can do to tear my eyes away.\n\n\"Silence, please.\" Aurela's voice is quiet, but the voices recede. \"Our plans do not stop at removing Murek or exposing his endgame to the people of Neospes. It also extends to reinstating the true crown prince alongside his brother.\" The murmurs in the room rise again and once more regress at Aurela's hand. \"That is where General Riven has been, protecting said prince in the Otherworld.\"\n\n\"The Otherworld!\" someone else gasps.\n\n\"Still, she's the General,\" a voice argues. \"Loyal to the false prince. We know who she is.\"\n\n\"Yes, she is the General,\" Aurela agrees mildly. \"And no, she is bound to us and to Prince Caden. I trust her, and that should be enough.\"\n\nI feel Caden squirm next to me, at odds with all the sudden attention. But my mind is whirling at the fact that my mother and leader of the rebels has openly supported me \u2013 stating that anyone who doesn't accept me does not accept her. I'm at a loss for words, breath even, at her complete and utter trust.\n\nFor a brief second, I wonder whether that trust in me is misplaced.\n\nAnd in that same moment, I realize who the dark-eyed girl is. She's been someone I trusted. Once upon a time, I would have considered her a loyal teammate. But when she went against orders with a group of insurgents led by her brother, the King called for swift justice.\n\nI know exactly who she is. She's a Commander I exiled to the Outers four years ago; she is the single broken line tattooed on my neck.\n\nAnd now she is one of us.\nFATHER DEAREST\n\nDespite the fact that Loren, the soldier I'd exiled, was one of the volunteers, oddly enough, to be on my team, things were on track with Aurela's wishes. I was suspicious the minute Loren voiced her desire to come with me \u2013 she obviously had a bone to pick. After all, I executed her brother and exiled her.\n\nBut at the end of the day, I know that it doesn't matter. I don't trust anyone, and no one here trusts me either, for good reason. I'd been the worst of the worst, and a part of me is still that person. I was programmed to be emotionless, to obey orders, to kill without question. And because of the nanoplasm, I'm faster and fiercer than anyone else. I'm death in a girl's body.\n\nIt stands to reason that people will hate me.\n\nBut I have a job to do, and I can't let some soldier with a vendetta get in the way, even if she has every reason to want to stick an electro-rod through my guts the first chance she gets. We all have our demons.\n\nAnd the old saying about keeping your enemies close is true enough.\n\n\"Why are we breaking into your old house again?\" Caden asks, bringing me back to the present.\n\nWe are crouched behind a building, dressed in identical gray fatigues, the ones the soldiers on the streets wear. I stole them the day before once I realized what I needed to do as part of my plan to infiltrate the palace. I debated against bringing Caden, but leaving him behind wasn't an option. And it isn't exactly the most dangerous situation \u2013 my father will be at the lab \u2013 and we'll be in and out before anyone realizes that we've gone. We'll be safe as kittens. I hoped.\n\n\"I need something. Something important.\"\n\n\"Won't Aurela be pissed that we snuck out?\"\n\nExasperated, I turn to face him. \"Well, she's not going to find out, is she? If we hurry up, that is. Come on; we don't have all day. In and out, OK?\"\n\nMy father's home, the place where I grew up, is in Sector One. Naturally it's the closest sector to the castle, but, oddly enough, doesn't have that much security. People from other sectors don't voluntarily go to Sector One. Shae and I always joked that we needed guards to keep people in instead of keeping undesirables out. Sector One citizens didn't mingle with people from other sectors \u2013 they were the cream of the crop, the brains of Neospes.\n\nI borrowed Sauer's transport hover without his permission but I figured that's what they were for... emergencies. Caden and I had lugged the hover-bike near the side of a warehouse behind some crates and covered it with a tarp.\n\n\"Get changed,\" I snap to him, and toss him a silver oversuit.\n\n\"Why?\"\n\nI wave a hand around us and shrug into my own jumpsuit over my clothes. \"Notice that everything in Sector One is silver? It's a sign of privilege. And we need to blend in. The fatigues were only to get us here.\"\n\nThere aren't many people walking around. Most of them are at work, and the younger ones are training in their chosen vocation or in the fields. The elite training fields are between Sector One and Sector Two, and most of the soldiers' families are housed in Sector Three. So Sector One is quiet.\n\n\"Stay close,\" I warn Caden. \"And be vigilant. There are cameras everywhere. If they scan us without a chip, we're toast.\"\n\nIt's a fine line between appearing casual to any onlookers and staying out of sight of the many electronic eyes in each Sector. In Neospes, peace isn't a result of freedom. It's a result of swift military action. Break the law and face the punishment. Those electronic eyes watch everything all the time \u2013 and freedom is the price we have to pay for the security of life in the dome.\n\n\"I thought you said this was going to be easy.\"\n\n\"This is easy. Come on.\"\n\nWe make our way along the side of a row of identical white and light gray-patterned housing cubes, staying out of sight of the main street and weaving in and out of smaller alleys, until we come to a house with tiny purple flowers planted along the edges. My heart trembles in my chest. Shae and I planted them when we were little, in defiance of all the gray and white. Surely my father would have destroyed them. Why has he kept them? As a memory of what? How much we hated Sector One? How much we both hated him?\n\n\"This is it. There shouldn't be anyone here, but be alert,\" I say. \"Quick, around the back.\"\n\nI don't want to draw any attention by going to the front, street side, especially for a wanted fugitive. Located on the far side of Sector One, my father's house backs onto a deep, rocky gorge.\n\nThe gorge appears bottomless and beautiful in a raw-nature kind of way, with a sheer, reflective rock face on one side and broken rubble on the other. The sun glints off the different colored minerals in the rock so brightly that it's like being inside a prism. Free of security cameras and prying eyes, the gorge was one of my favorite places to hide.\n\n\"Wow, so this is where you lived?\" Caden says. \"It's amazing.\"\n\nI allow myself a smile. \"See that ledge over there? Across the gorge? That's actually a cave; it used to be my hideout. My secret place to get away.\"\n\nThe mouth of the cave is barely noticeable. I can see him squinting, so I move closer to show him, lining my face up against his and pointing. Caden turns into me, and it brings back memories I don't want to think about. Not now, especially given what happened the last time. Despite the electric zing in my belly and my sudden breathlessness, I step away. A wounded look slashes across Caden's expression, but he hides it quickly.\n\n\"So it was like your tree house. What'd you keep in there, dolls and stuff?\"\n\nI snort. \"More like a stash of knives and electro-grenades.\"\n\nCaden's green eyes flash with humor. \"Come on, every little girl has to have dolls somewhere.\"\n\n\"So then,\" I toss back, \"where was your stash of dolls?\"\n\nA wide grin. \"Are you calling me a girl?\"\n\n\"Hey, if the shoe fits...\"\n\n\"I'll admit I played with dolls, only they were called GI Joes, so technically they were action figures. But if you still think I'm a girl, I'd be happy to prove otherwise.\" Caden's grin turns wicked and makes a hundred butterflies go aloft in my belly. A flash of his sweaty, defined abs in the training room at Sauer's jumps into my head, and this time I can't control the crazy blush that invades my entire body. He is definitely all boy.\n\n\"No, thanks, I'm good.\" I manage to sound normal, even though my heartbeat is on a rampage following the shameless turn of my thoughts.\n\nGet a grip, Riven, I hiss to myself. This isn't the time or place to be lusting after a boy like a slutty Sadie clone. But now that that door is open, it's kind of hard to get the images out of my head.\n\n\"Well, if you change your mind,\" Caden says, wiggling his eyebrows suggestively and sashaying his hips.\n\n\"You are such a freak,\" I retort, blushing. \"Come on, let's go. And keep it together, will you? We could get arrested or die any second.\"\n\n\"So you keep saying.\"\n\nPulling a small laser device from my pocket, I hold it up to the keypad sensor on the back door. A whine, and the locks click back. Inside, the house is exactly the same as I remembered. Not a little; exactly the same. My father, as brilliant as he is, has never been one for any kind of change. The small but efficient kitchen is spotless. So is the high table with its benches. Off the kitchen and dining area is a larger living space, but it looks unused. The sleeping quarters are on either side of the main area.\n\nCaden is behind me on silent feet. His eyes are wide and he's taking it all in. I know he has to be curious about where Shae and I used to live. After all, he'd been with her for years in the Otherworld, only to find out that he didn't know the truth about who she actually was all along.\n\nOr me either, for that matter.\n\n\"Try not to touch anything, OK? He can't know that we \u2013 or anyone \u2013 has been here.\"\n\n\"Aren't there security cameras in here, too?\" Caden asks nervously.\n\nI shake my head. \"No, my father is exempt from many of Neospes' laws.\"\n\nMy fingers curl into knots at my side, knowing exactly why he hadn't wanted to be seen. I don't bother to go to my parents' old room. I don't even want to think about him more than I already have. Instead, I push open the door to the room that Shae and I shared, my gloved fingers cold on the thin metal door.\n\nCaden's gasp behind me is audible. \"What the...\"\n\n\"Now you understand what we're dealing with?\" I say quietly. \"A psychopath.\"\n\nThe room is a shambles, littered with broken furniture. The top bunk, soldered into the walls, is now chopped into white and black remnants with bits of mattress strewn all over the floor. The bottom bunk hangs drunkenly off its remaining hinge. It looks like he'd gone after the room with a blowtorch. Huge black gashes discolor the walls.\n\nIt's the work of a madman... one consumed by unimaginable rage.\n\nHe'd torn holes in the thick carpet and ripped any of our remaining clothing to shreds. Not much of Shae's was there after she left, but he shredded my old uniforms until they were nothing but tatters. The only thing left intact in the room is the glass case with all my trophies won during the different Games. They meant more to him than they ever did to me.\n\n\"Wow,\" Caden breathes, staring at the garish display. \"You won... like everything.\"\n\n\"Robot blood. What can I say?\" I reply drily. \"Technically, I cheated, so I don't deserve any of those.\"\n\n\"He cheated,\" Caden reminds me. \"Not you. You were innocent.\"\n\nA deep guttural laugh that I recognize as my own crawls from my mouth. \"Don't make me into some kind of martyr, Caden. I was never an innocent, not for a second. Not after all I've done. I did all those things.\" I thump my chest emphatically. \"Me. Not him, not anyone else. Me.\"\n\n\"Riv,\" Caden says slowly. \"I understand you did all those things, but you did them because you were following orders, and you didn't question them, because you were compromised by your father.\" Caden grasps my shoulders, tugging me around to face him. I can't look at him but I do. I meet his eyes and it's like I'm standing at the edge of the gorge outside, on the brink of freefalling. \"You have to stop punishing yourself for something that was out of your control. Can't you see that? You have to see that.\"\n\nBut it isn't his voice that convinces me. It's the complete faith in his eyes... those green eyes that I've somehow known my whole life. And for the first time, I don't think of Cale.\n\nIt's only Caden... this boy who believes in me more than I believe in myself. And I'm terrified. I've never been more afraid of anything in my life. I shift backward but his arms tighten, drawing me against his chest. His chin rests on top of my head, his breath warm against my hair.\n\n\"Let it go, Riv,\" he whispers, so close that I can see the soft pulse beneath his skin. \"Let go of the past. It can't hurt you anymore.\"\n\nAnd then I'm weeping, long silent tears that soak my face and his shirt front... tears for all the years of pain and anger and loss. I cry until there's nothing but dry sobs racking my body, and still I can't stop. I cry for myself and my inability to love a boy who's clearly in love with me. I'll only hurt him. I'll destroy him, just like everyone else in my life.\n\n\"I'm scared. I can't lose anyone else.\" My voice is so soft, it's more of a rasp than words against his shirt. But Caden raises his head. His hands move to hold my face on either side.\n\n\"I know. But fear is just fear. It can only hurt you if you let it.\" His thumbs rub gently against my cheekbones. \"And I won't let you let it.\" His voice is hypnotic, weaving through all the broken parts of me and somehow bringing them back together. \"Look at me, Riven,\" he says, his eyes capturing mine easily. \"You think I don't know anything about fear? I know that if you let it, it can consume you.\" His voice is so soft I can barely hear it. \"You can't be afraid all the time, afraid to lose everything. Shae told me how you did it in your training. Faced it, confronted it head on. This is the same thing.\"\n\n\"I didn't save her,\" I say, my words distraught.\n\n\"You couldn't save her.\" His voice gentles. \"Stop beating yourself up for something that was also out of your control. You only control what you do. Not the actions of others.\"\n\nCaden is right \u2013 he's seen right through me, right into the core of me. And he loves me. Just as I love him. I started to say it before my internal programming went berserk. I've never allowed myself to feel anything for anyone, not when everything I've been taught is the opposite; love precedes weakness. Then why do I like knowing that Caden feels something for me?\n\nI like it. I like him.\n\nHis eyes widen as if he's reading my thoughts, the internal battle I'm waging with myself... the war between want and reason, between girl and soldier. And I'm losing. I'm losing myself to him. My eyes speak the words that my mouth cannot.\n\nYes, they say. Yes.\n\nCaden leans in as his right hand slides into my hair to the nape of my neck and brings my face toward his, his eyes never letting mine go. At the last minute before his lips touch mine, I close my eyes. The kiss is soft and featherlight, but I still slide my hands up his arms, gripping tightly, pressing myself against his body and demanding more.\n\nHe smiles against my mouth, and our kiss deepens into something fierce and frantic. Suddenly, I can't even think, standing there in my father's house as the pain and fear fall away. All I can do is feel... the soft contours of his lips against mine, the taste of his mouth, the strength of his body. My fingers clutch his clothing and I am overwhelmed, breathless. Weightless.\n\nCaden trails kisses against my temple, my eyes, my nose. His eyes are a stormy jade, dark with passion. He kisses me again, and this time everything around us disappears \u2013 my father's house, my destroyed life, my broken self. All that remains is Caden.\n\nFingers tangle in clothing and zippers as we fall to the floor together. My palms are flat against the taut planes of Caden's chest, and my breath is unsteady at the sheer beauty of him above me. His body is not bulky with muscle; instead, he's sleekly toned, his stomach and sides hard against my wandering fingertips.\n\nHe is kissing me again, and we stop only to pull my tunic over my head. A sudden shyness envelops me and I feel my face flush. No one has ever seen me this unclothed. No one. Not ever. My embarrassment must be obvious because Caden grins and grasps my chin.\n\n\"You're perfect, Riven,\" he whispers.\n\n\"No. I'm not,\" I say, my thoughts burning, my hands fluttering to hide the scars peppering my ribcage and stomach. \"But you make me feel that way.\"\n\nHis smile transforms his entire face and he bends toward me again, pushing away my embarrassed hands. His kisses grace my ear, and then my neck, and wander lower, kissing each raised welt, each ugly wound that I tried to hide with aching gentleness. My fingers thread their way into his hair and down his neck. Caden growls low in his chest as my hands slide down his shoulders and his muscled back, and lower still.\n\nWe are combustible, electric.\n\nI'd never kissed anyone before Caden, far less gotten half-naked with anyone, but this feels more right than anything I've ever done. He's the one who makes me feel less splintered, less shattered. I feel whole when I'm with him. That has to count for something, doesn't it? I have to tell him the truth about how I feel.\n\n\"Caden,\" I begin.\n\n\"Mmm,\" he says, trailing more kisses up my arm. But before I can continue, the sound of a door slamming jerks me into action. I shove Caden to one side and press my fingers urgently against his lips, my eyes wide. Someone is outside.\n\nHow could I have been so reckless?\n\nShrugging my arms into the white overalls, I crawl along the floor to the thin window of glass against the far wall and slide up to the right of it. A quick glimpse confirms that there's a vehicle outside. It's black and sleek and official-looking.\n\n\"We have to hide,\" I mouth to Caden who has already dressed. His hair is tousled and his eyes are still glazed from what we'd shared seconds before. I'm sure I must look the same.\n\n\"Where?\" he mouths back.\n\nMy father ripped the room apart in rage, but he still didn't find the secret space that Shae and I painstakingly built. Half of me wants to leap toward it to get what I came for and make a run for it, but the other half warns that getting caught would be foolish. I glance at Caden. I can't risk putting him in danger, not with my father.\n\nSo instead, I jerk my head toward the twisted bottom bunk bed. \"Under,\" I say. \"Now.\"\n\nI slide in behind him until we are both jammed up against the wall. Every inch of Caden's body is glued to every inch of mine, but there's no fire between us now... only cold dread.\n\n\"Don't breathe,\" I warn.\n\nSounds reach us from the other room. A door clicking, a staccato of footsteps on the floor, voices. He's not alone.\n\n\"Sir,\" a voice says. \"Reports came in of trespassers in the back.\"\n\n\"Any confirmation of what they looked like?\" My father's voice is mellifluous and just as loathsome as the last time I'd heard it. My stomach curdles like spoiled milk and lurches unsteadily. A cold sweat pinpricks its way along my back.\n\n\"No, sir. There's no sign of them. And no sign of forced entry on the perimeter. It's clear.\"\n\nThe footsteps draw closer, and I press myself even harder against Caden. I'm not even breathing by the time the door to the room slides open and a shiny pair of black boots come into view. His smell assaults me... still the same, a dark musky scent of spice.\n\nIt spins me back into a whirlwind of memories \u2013 my father and I playing chess in his laboratory, my father glowing with praise at my first-place trophies for archery and swordplay, my father pinning my rank of general to my vest. It's hard to reconcile the man I'd known as a child to the monster I now realize he is.\n\nThe seconds drag by and still the black boots don't move, but I can sense heavy eyes roving slowly over every inch of the space. I scan the floor in a panic, but nothing appears noticeably different from the overall chaos of the room. My eyes swivel back to the black boots. It's so silent you could hear an eyelash fall. Neither Caden nor I are breathing.\n\nI'm so sure that he's going to peer beneath the bunk that my pulse thrums and my blood rushes madly in my ears. My body tenses, and then I feel Caden's fingers, quiet against the small of my back, stroking, soothing. They slide over and intertwine with mine. My pulse slows and calms.\n\n\"All clear, sir,\" the other voice says. \"There's no one here, now.\"\n\nAfter a minute, the boots retreat and the door slides closed.\n\nStill, I don't breathe properly until I hear the hum of the engine outside. Tense, I crawl silently out from under the bed and press my ear to the door. There's no sound in the rest of the house, and the car is gone.\n\nWe're safe for now.\n\nI cross the room swiftly and pull the trophy case gently from the wall, sliding my hands along the floor. I pull back a portion of the carpet, and press the bare tiles gently, this time in a certain sequence. A soft click, and one of the tiles glides back to reveal a hidden compartment.\n\nInside, there's a box. I don't have time to sift through all the contents, even though my heart stings at some of the treasures Shae and I had hidden there. Bits of ribbon, silver stars, letters written to our future selves.\n\n\"What's in there?\" Caden asks.\n\n\"Stuff we didn't want anyone to find.\" I pull out the thing I'd risked our lives to come here to get.\n\nA Vector's uniform.\n\nOnly this one is different. This one, my father recalibrated to work with my own DNA. A wry smirk twists my mouth. We were so shocked and proud that the suit responded to me \u2013 thinking my father brilliant \u2013 and little knowing that the suit responded to the nanoplasm inside of me instead of pure human DNA. The suit was designed for me.\n\nHe knew. And he created the perfect defense for his perfect weapon. The suit is unmatched in its capabilities, offensive and defensive. No wonder he tore the room apart trying to find it, and then assumed that I'd somehow hidden it or taken it with me.\n\n\"Let's go,\" I say to Caden. \"We're done here.\"\n\nWe sneak out the back after checking carefully to make sure we aren't being watched, and make our way to where we'd hidden the hover bike. The journey back to Sector Seven is far quicker than the time it took to leave, and my stomach sinks as we pull into Sauer's garage.\n\nAurela's face is blacker than a thundercloud.\nTHE WINTER SOLSTICE\n\nAurela is still furious, even though I've assured her a hundred times that we hadn't encountered any danger. Of course, I hadn't mentioned our near-discovery by the man himself. She would have flipped her lid and put both Caden and me on lockdown. To his credit, Caden had followed my lead like a pro, and after only an hour of intense interrogation Aurela had relented.\n\nWe're back in the meeting room, going over the plans one last time. I toss a casual glance at Caden, who's sitting next to Loren. Caden's eyes meet mine meaningfully over the table, and I'm glad I'm sitting, because my knees suddenly feel like water. Earlier, he had told me that he'd have given anything for a few more minutes before my father showed up.\n\nEven though I feel far older at times, Caden has more experience in the relationship area than I do. I have exactly zero skill when it comes to boys, the not-beating-them-to-a-pulp kind, I mean. Caden's been with Sadie, and she's not exactly the unadventurous type. Sadie is a girl who calls the shots, especially in the relationship department.\n\nDragging my eyes away, I focus instead on Loren's piercing glare. We haven't spoken since she volunteered to be on my team, and it's clear that she doesn't trust me. Then again, I don't trust her, either. I agreed for her to join us because before I exiled her, she'd been one of the best. And now, with her skills honed in the Outers, she's only become more so. I'm not about to begrudge someone a chance to change, not after everything I've learned about myself.\n\nI turn my attention to Sauer's computer tablet in front of me, showing a schematic of the castle. Security is near impenetrable. There are four layers of guards, two sets outside on the perimeter and two sets inside. They are only the human line of defense. There are also Vectors at each entry point. I sigh, steepling my fingers in front of me, and meet Sauer's eyes. Getting in undetected will be a challenge for one person, far less four.\n\n\"What are you thinking?\" I ask him.\n\n\"The Winter Games are two days away, but we still have a lot to do,\" he says. \"To even get past the castle guards, we'd have to register by identity chip, and technically, you're already off the grid. So is Caden.\"\n\nI removed my implanted birth chip the minute I decided to leave Neospes \u2013 it was incredibly painful but a necessity. At the time, I hadn't known if the Vectors could track an implant even in another dimension, but I didn't want to take any chances. Given that all citizens of Neospes have to be accounted for at all times, Sauer explained that both Caden and I would require new identity chips, which means fake ones, and they aren't that easy to get. It's tricky, because if someone dies, the system automatically records it. You have to use the chip of someone who is still alive. If someone's chip is forcibly removed, it releases an alarm that goes straight to the Vectors. After Leila disappeared with Caden, those precautions were deemed necessary.\n\nCaden is frowning with a confused look on his face. \"I don't get it,\" he says, staring at me. \"You took your chip out three years ago. How come they didn't get you?\"\n\n\"They did, but I killed the Vector who'd caught me. Then I everted.\"\n\n\"So basically, I need a chip, and you need a chip?\"\n\n\"Yes,\" Sauer says. \"And Aurela, but she already has a fake one she's been using for years.\"\n\nCaden's frown deepens. \"So what do we have to do, exactly?\"\n\nLoren answers. \"You get fitted with a new one. Aurela's people have been growing fake people for years. So technically, you'll get the chip of someone who's only alive electronically.\"\n\n\"Wow, fake people,\" Caden breathes. \"That's extreme.\"\n\n\"Extreme times call for extreme measures,\" Sauer says matter-of-factly.\n\nLoren leans forward. \"Aurela has been planning this for a long time, very, very carefully. She knew identity chips would be needed one day for those of us who'd been exiled, so she made sure we could be accounted for.\" Her voice is monotone, but I avoid her stare like it's the plague.\n\n\"So who do I get?\" Caden asks.\n\n\"They're all soldiers,\" Sauer says, and then looks at me with a weird expression, his face reddening. \"Well, most of them anyway.\"\n\nI don't hesitate. \"What do you mean?\"\n\n\"Your identity is an... um... danseuse.\" His face dulls to a dark red. \"Loren was assigned the last female soldier chip.\"\n\nMy chair flies back crashing into the wall. \"A what?\" I scream.\n\nDanseuses are nothing more than paid female escorts, paid to entertain wealthy citizens of Neospes. Aurela's calm voice breaks through my fury as she walks into the room. \"It's the best disguise for you, Riven, to infiltrate the palace. If you look like a soldier, people will recognize you. Don't forget who you were... who you still are here.\"\n\nBut the thought of it fills me with disgust. \"I'm not using the identity of a freaking floozy! Pick something else. I'll be a cleaner, a goat trader, anything.\"\n\n\"We don't have any other female identity chips, Riven, and this is the best for you. Trust me,\" Aurela says. She walks to stand in front of me and turns me toward her. My chest feels like it's going to explode. I've never been asked to do anything more ludicrous. I'm a general, not some dancer dressed in a handkerchief! I don't even know how to dance.\n\n\"Aurela, I can't.\"\n\nAurela's voice is harsh, harsher than I've ever heard it. \"You have no choice. If you don't, we abort. It's that simple. You want to save Cale. Well, you are the only person who can ensure his safety if we get you into the castle.\" Aurela pauses with a long stare at me. \"You decide.\"\n\nI know she's right, but I can't reconcile wearing some kind of revealing costume with any of it. It's a huge strike against me \u2013 Riven \u2013 the name that strikes fear into the hearts of all men will now cause ridicule. But I know that's only my pride talking. I should do whatever I have to do to save Cale and Caden without being selfish. Being a danseuse for ten minutes is only a means to an end. A crappy, ridiculous means, but it will get the job done. I'll get into the castle unnoticed.\n\n\"Fine,\" I snap, stalking to the far end of the room, where I pour myself a shot of spirits. I drink it quickly, ignoring its hot path searing my stomach. Loren is laughing at me, her dark eyes mocking.\n\nHours later, there are three of Aurela's lackeys in my chamber, fussing over my hair and my face and my clothes. I have never felt more conspicuous or more ridiculous. Strange dark-gold extensions have been applied and braided intricately with sweet-smelling blossoms into my own hair. My skin has been oiled to a gilded sheen and colored shimmery dust applied to my eyes and cheeks. The women gesture for me to step into my costume \u2013 a filmy white and silver getup that looks like some kind of confection instead of a dress.\n\nThere's a knock at the door, and one of the women answers it.\n\n\"I'd hardly recognize you, Riven,\" Aurela says. She's dressed head-to-toe in black stealth gear, pretty much what I want to be wearing.\n\nInstead, I'm a porcelain doll. An ugly scowling porcelain doll.\n\nI turn to glare at her. \"That was the point, wasn't it?\" I gesture down at myself. \"I couldn't attack a fly and win in this dress. In fact, the fly would probably try to eat me, since I look like a giant frosted cupcake.\"\n\n\"A what?\"\n\nBelatedly realizing I've mentioned a food from the Otherworld that doesn't exist here, I shrug. \"Never mind. It means a delicious but totally useless food... which is what I look like.\" One of the women draws me to the mirror, and my half-joking, self-deprecating words desert me completely. I don't recognize myself.\n\nThe girl staring at me in the mirror has glossy dark braids piled in rows along her head and draped artfully to one side. Her eyes are lined with kohl, making their normally dull gray depths vibrant and jewel-toned. Her cheeks are glittery and flushed with color, her lips glistening. The dress is impractical but undeniably lovely, falling in triangular swathes of chiffon to the ground.\n\nI am an imposter.\n\nIt can't possibly be me, but I blink and the girl does the same. I resist the urge to pick a fight with the first soldier I can find.\n\n\"You look beautiful.\" Caden's voice is hushed and awed, and I spin around only to find my swift retort cut off from my own mouth. He is resplendent in a silk moss-green shirt and black dress pants, complete with official-looking scabbard.\n\n\"So do you,\" I blurt out, and then immediately look to Aurela, who is busy addressing one of the women who had helped get me ready. Thank the stars she didn't notice my schoolgirl response to his appearance. I flush and Caden grins, reading me easily.\n\n\"Thank you, my lady. I'm supposed to be your... er... partner.\"\n\n\"My what?\" I splutter. \"But you were supposed to be a soldier.\" In a rush, I notice the lines of gold stars and royal stripes decorating his upper left breast pocket, marking his status as a lord of Neospes.\n\nAurela walks over with an approving smile at Caden's garb. \"Too obvious a disguise. This way, you can be near each other without too many questions. Caden's a wealthy lord and you're his entertainment escort for the ball.\" I can't help the blush that heats my face at the thought of being anyone's escort, far less Caden's.\n\n\"You know,\" Caden begins laughing at my obvious embarrassment, \"there's a name for that where I'm from\u2013\"\n\n\"Shut up,\" I say, swearing under my breath. \"Or you'll see exactly how entertaining I can be with two sets of blades.\"\n\n\"Oh, feisty. I like that in my women.\"\n\n\"Aurela!\" I protest weakly. \"Really? Can't he just be a soldier and go with Sauer? That way, I don't end up killing him by accident or something.\"\n\nBut Aurela rolls her eyes and throws me a sling pack for my gear. It's a form-fitting pack that I can wear around my waist under the dress. There isn't much room under there, but I'll be damned if I go anywhere without weapons. She also hands me some kind of frilly fan that I just stare at dumbfounded.\n\n\"It's a fan,\" she says helpfully. My scowl deepens as I grasp the offending object as if it's a snake. \"Ready to go in ten,\" she reminds us with a quickly concealed smile. I swear that she's laughing under her breath.\n\nSmoothing the ruffled skirt of the dress, I turn to Caden once Aurela is a few feet away and engaged in conversation with one of her maids. \"Joking aside, you don't think I look ridiculous?\"\n\n\"I hardly recognize you,\" Caden says honestly. \"I mean, I've never seen you out of cargo pants, unless you count the time when you wore Sadie's dress.\" He winks and I choke out a startled laugh.\n\n\"You knew about that?\"\n\nCaden smiles drily. \"I think Sadie knew about that, too, but she wasn't going to admit it in a million years.\" He shakes his head. \"You have no idea how pissed-off she was.\"\n\n\"Sorry, but she deserved way worse, and you know it.\"\n\n\"Well, I wasn't too upset. The last time you wore a dress, epic things happened, so win-win for me,\" Caden says with a meaningful look, clearly referring to our last day at Horrow. He leans in so only I can hear him. \"Can't wait to see what happens this time.\" At his whispered words, I can't control the tingle that weaves its way through my body, but I fight it just the same.\n\n\"Keep talking, my lord,\" I snap with a glare, \"and the only thing that's going to happen is your ass meeting the ground, dress or no dress. Now, go away.\"\n\nCaden backs away, hands in the air, grinning, and I stare at myself in the mirror again, blinking just to make sure it's still me. The top of the dress is lower than anything I've ever worn, and I tug at the bodice uncomfortably. \"Seriously Cade, I look like a stranger. This isn't me. And how the hell am I going to fight in this?\"\n\n\"Just because you're wearing a dress doesn't make you a different person, Riv. Plus, Sauer's got your suit and your sword for when we get inside. It's just clothes... wear a tunic underneath if it'll make you feel better. And you don't look like a stranger; your eyes are the same, and those I'd know anywhere.\"\n\n\"I am wearing a tunic and I still feel naked,\" I flash back. \"And these don't even look like my eyes. Mine are more dirt-colored than silver.\"\n\nCaden's glance is like velvet. \"Well, they always look like this to me.\"\n\nIt's all I can do not to melt inside, but I keep my face expressionless, then scowl slightly for good measure. I sneak one last look at the mirror and sigh.\n\nI have a job to do.\n\nSauer and Loren are ready and waiting by the time we get downstairs. Outside, the streets are already crowded with people heading to the training fields between Sector Two and One for the Games. When evening starts to fall, everyone will eventually find their way to the castle, the final event and celebration of the Winter Solstice.\n\nIt's a tradition dating back to the beginning of Neospes, one of the first decrees of the then newly crowned king. It used to be a celebration of our survival after the War. Now, its meaning has become far more mundane \u2013 merely a way for the monarchy to fawn over itself and its power. We've lost sight of what's important, as has our leadership.\n\nAurela is right \u2013 I see that now. And I understand why it has to change.\n\nI glance at Sauer, who nods curtly in my direction as we merge into the throng of people on the street. He's all business, a commander first and a friend second. His role as a double agent is about to end, once we wage our combined attack on the monarchy. Until today, I hadn't realized how deep the unrest reaches... and how many followers Aurela has, not just in Sector Seven but all the sectors. She has thousands of reserve soldiers on hand if things get ugly.\n\nBut that isn't Aurela's plan. She prefers a quiet, stealthy infiltration. She doesn't want hundreds of innocent people to die, especially at the hands of the Vectors. Our population is devastated enough without diminishing it even further. Which is why we split into three groups \u2013 me, Sauer, Caden, and Loren to safeguard Cale; Aurela's team to infiltrate the lab and locate the kill code for the Vectors; and finally, the third team to depose Murek.\n\nMy father will have to wait until Cale is safe. Then I have my own plans for him.\n\n\"You OK?\" Caden asks, jerking me out of my murderous thoughts.\n\n\"What?\"\n\n\"I said, are you OK?\"\n\n\"Of course,\" I snap. \"Why wouldn't I be? It's just a mission. Stay as close to me as possible. I don't want to draw undue attention.\"\n\n\"I'm pretty sure that's the point of how we're dressed, but don't worry; I won't embarrass you. I'll just think of it like going to prom.\"\n\nI half-choke, half-snort on my retort. \"Prom?\"\n\nCaden grins at me. \"Yeah, you know... the whole life-or-death school dance thing in my world when girls murder each other with their eyes, and guys are expected to fall in line or face certain death. Don't worry; got it down pat.\"\n\nWhich is why I am grinning from ear to ear when Sauer announces that we are splitting from the rest of the other groups. Just before we jump into Sauer's hovercraft, I feel Aurela's hand on my arm.\n\n\"Riven, be careful,\" she says quietly. \"And please don't do anything rash. I'm trusting you.\"\n\n\"I know,\" I say. \"Cale and Caden will be safe, I promise you.\" She frowns at my evasive response, but I climb into the hover. There's nothing that will stop me, once Cale is with Sauer, from doing what I have to do where my father is concerned. And if the opportunity presents itself during the raid, I'll take it. But I know what Aurela is worried about. I turn around. \"Don't worry; I won't jeopardize the mission,\" I say. \"The code in his lab is my birth date. At least, it was until I defected. And Mom, you be careful, too.\" The hover accelerates, but not before I see the startled look on her face that I'd called her something other than Aurela.\n\nDriving is slow going, especially around all the people, but after a while Sauer pulls off onto a less major road where there's far less pedestrian traffic. The Games are well underway, and the training fields are packed with huge colored tents full of participants and street vendors selling wares and trinkets. People are laughing and dancing, and cheering on the various combatants.\n\nWe ditch the hover and make our way to the main tent in the middle. It's where the exhibition fighting is and where anyone important will be. Falling into my role, Caden and I separate from Loren and Sauer, pretending to be interested in a couple battling it out with swords in the middle of the tent. It's an exhibition match, so the opponents are clothed in protective padding. Still, it is amazing to watch. Their technique is faultless and perfect in its form.\n\nOut of the corner of my eye, I see Loren taking up her position near the entrance of the tent and Sauer making his way to the far end, where there is a raised dais, likely for any royal attendees and their court. Although I'm straining my neck to tell who's there, I can't quite see over the heads of everyone, so my attention swings back to the exhibition fight. Four Vectors are stationed around the ring like silent, dead pillars. There seem to be a lot more of them than usual, I notice, frowning. Has someone tipped them off about Aurela's plans? Or maybe I've just been gone too long from Neospes. Shaking my head, I let go of my heightened paranoia \u2013 this stupid dress is making me edgy.\n\nA fat man in a gaudy gold outfit jostles me, and my fingers curl into fists at his lascivious leer. He's staring at my chest like it's a giant pile of precious stones. My fists tighten. I've never wanted to smash in someone's face more.\n\n\"Not here,\" Caden warns in my ear, moving to stand between me and the man who's just been spared having my fist in his face. He turns to the offending man. \"The lady is with me,\" he says quietly. His words aren't threatening, but the low growl underscoring his words is.\n\n\"I wasn't going to do anything,\" I grit out as the man moves away with a lingering glance at me. \"I'm fine.\"\n\nIgnoring the low burn that Caden's possessive tone had ignited in my belly, I turn my attention back to the exhibition match and notice that the combatants have switched to double-edged Artok weapons similar to mine. The one on the right is half a step slower than the one on the left, but they are evenly matched. Again, their combat technique is brilliant.\n\nI wish I could see who they are, but they're both wearing antique decorative facemasks. The truth is, I probably wouldn't know them, anyway. Our mortality rates aren't that great, and they're probably up-and-comers in the ranks. The fighter on the left drops into a low crouch, spinning to the left to catch the other unawares, and wins the point.\n\n\"He's good,\" Caden comments. \"The one on the left.\"\n\n\"Not that good,\" I say snarkily, still semi-peeved to be in a dress. To tell the truth, a part of me wants nothing more than to be in that ring, showing off and strutting my skills. Winning meant everything to my father, so I won everything I could, including all of my exhibition matches, which didn't technically have winners. His approval used to mean so much.\n\nI turn away, unable to watch, and catch Sauer's eye. He's gesturing to someone on the dais. Caden gasps, his eyes connecting with the four Vectors standing on either side of it. But I knew they'd be there. I want to see who they're protecting so vigilantly.\n\nWe inch our way forward, the throng of people thinning the closer we get, and I recognize the gaunt face and the bulbous, fish-like eyes immediately. Instinctively, my fan opens in front of my face, concealing my own features from view as a hollow feeling spreads in my stomach. My disguise may fool most people, but there's one person who could see right through it... a master manipulator and schemer himself.\n\nMurek.\n\nUsurper. Traitor. Murderer.\nTHE WINNER TAKES IT\n\nTwilight is falling as the last of the Games wrap up, and people are already starting to take the revelry to the castle courtyard for the Midnight Ball. The streets are lit with multicolored halogen lights with shiny decorations draped over storefronts and houses. Everyone is laughing and dancing... well, everyone except us.\n\nI want to leave but am at the mercy of Sauer's lead. I still haven't seen Cale. He's nowhere to be found in the main tent or the surroundings, and Murek did the presentation of the Games' trophies in his stead as if he were already acting in the position of king. I wonder for a moment whether Cale is already dead and hidden away somewhere deep in the castle where no one would ever find him. Murek is twisted like that.\n\nMy father is also mysteriously absent. Not that I expected that he would attend the Games, but he'd been present at every single match I'd ever won. His pride would keep him away, I realize. My only hope is that he isn't in the lab \u2013 he is my prize, not Aurela's. I have to be the one to make him pay for what he'd done to all of us.\n\n\"Riven,\" a voice rasps in my ear. It's Sauer, standing with his back against mine. \"Remember the strategy. We separate after entry and regroup thirty minutes later. You take point to extract Cale, and Loren and I watch for Vectors. Here's your comms.\" He palms two tiny skin-colored devices into my hands. \"Stick to the plan, and remember your cover. Code word for help is reptile.\" There's a clear warning in his voice. \"Clear?\"\n\n\"Clear.\" My own voice is terse, ready.\n\nThe wireless comms device is a tiny bean-shaped pod that tucks snugly into my ear. I hand the second one to Caden, who does the same. Together, we join the crowd walking toward the first security checkpoint. Loren and Sauer fall in a little way back behind us.\n\n\"Scan, please.\"\n\nCaden goes through first without any problem. I hold out my wrist underneath the scanner, and for a second my stomach lurches as the screen flickers before validating my identity as Tania, a danseuse. The electronic words are tiny, but they may as well have been on a neon banner over my head. Inside, I'm cringing at the appreciative look the soldier tosses my way, but instead of punching him in the face, I smile coquettishly, thinking of Sadie, and play the part I'm supposed to play.\n\n\"Bravo,\" Caden says. I purposefully stomp on his toe as I walk past him. \"Ow! What was that for? I was paying you a compliment.\"\n\n\"You know what for,\" I hiss back. \"And I was only channeling Sadie; so glad to see her ridiculously simpering ways still work on you.\"\n\n\"What\u2013?\"\n\n\"Come on,\" I snap, inexplicably angry. I'm a soldier, not a seductress. The pretense does not come easily.\n\nThe next security checkpoint is at the entrance of the courtyard beyond the tall and forbidding gilded gates. Cale and I used to scale them as children \u2013 a fair feat, given that they are over eight feet tall and spiked, but we were inventive and determined. I glance up at the parapets overlooking the grand courtyard that are decorated with brightly colored banners and flags, hoping to see a glimpse of Cale's face, but he's not there.\n\nMy body freezes at the sight of the two Vectors stationed with the guards at the checkpoint. Just seeing them standing there, their milky blue eyes staring at nothing, sends a chill deep into my bones. To think I used to command them without a second thought! Now I'm the enemy.\n\nFollowing in Caden's footsteps, I flick my wrist through the scanner, and the screen flickers and then fades to snow. The guard frowns and taps the scanner lightly. One of the Vectors swivels his eyes toward me, his gaze heavy. Feigning a bored look, I hold my wrist out even though my heart is racing. The guard scans my chip again. This time, it goes through.\n\nI feel the Vector's eyes on me all the way to the middle of the courtyard.\n\n\"What happened?\" Sauer's voice is tinny inside my ear. I can barely hear him over the loud music.\n\n\"I don't know,\" I reply, touching my ear gently. \"I think the nanobes are attacking the chip. They must think it's something foreign. There was a lot of electronic interference on the scanners. I'm not sure that it will work again.\"\n\n\"OK,\" Sauer says. \"Be careful. Those two were the main ones Aurela knew about so we'll figure out any others. And Riven?\" His voice is quiet, raising the hairs on the back of my neck. \"You have a Vector on your tail. He's been with you since the checkpoint.\"\n\nSauer's right. I can still feel the Vector's oily, wet gaze on me. So I do the only thing I can do. Half-dragging a bewildered Caden to the platform in the center, I throw my arms on his shoulders. I giggle loudly, a flirtatious sound that sounds grating and false to my own ears, and wave my fan teasingly in front of my face. Caden's eyes widen.\n\n\"Put your arms around my waist,\" I hiss urgently from behind the fan. \"Don't you know how to dance?\"\n\n\"I thought you didn't do dancing.\"\n\n\"I don't, but we have company, and I need it to not consider us a threat.\"\n\n\"Happy to oblige.\" Caden's hands slip around my waist to the small of my back, pulling me close. He has that smile on his lips again... the one that makes me breathless. He raises an eyebrow.\n\n\"Oh, get over it,\" I snap, watching as his smile turns into a full-fledged knowing grin. \"Really, Cade? We're on a mission, remember?\"\n\n\"I know, but we're dancing, so let's at least try to enjoy it.\"\n\n\"Why do you do that?\" I ask. \"Play around when you're supposed to be serious?\"\n\n\"Because life's too short. Even if one of those guys\" \u2013 he gestures to the Vector watching us \u2013 \"wants to kill me, I'd rather die with a smile in my heart. Life is for the living, and death is for the dead.\"\n\nThat shuts me up.\n\nWe make a few showy turns, not realizing that others on the dance floor have given us a wide berth. Apparently, hardcore martial arts and weapons training give a person an edge in the dancing arena. I feel as if there's wind beneath my soles, and although I know that most of it is probably due to Caden's expert partnering, I have no trouble keeping up.\n\n\"You're a good dancer,\" I tell him.\n\n\"Fencing,\" he says. \"Half dancing, half fighting. Oh, and Shae made me take ballroom. Don't laugh,\" he says at my incredulous expression, twirling me around him. \"It was either ballroom or singing, and since I can't hold a tune, well, I didn't have much choice.\"\n\n\"Did she say why?\" I ask, mirth in my eyes.\n\nHe shrugs. \"Something to do with being well-rounded. Honestly, I was seven at the time, so it was just something interesting to do. All old-school, too, like waltzes and the foxtrot.\" As if to demonstrate his point, Caden twirls us into several complicated loops, my toes barely grazing the floor.\n\n\"Well, you do it very well,\" I say.\n\nCaden's grin is wicked as he spins me outward and pulls me back toward him so quickly that my body snaps into his like a rubber band. \"Constantly surprising you, aren't I?\"\n\nI am breathless.\n\nBeing this close to him in front of so many people is sweet torture, flashes of two days before tormenting me. His dancing is as masterful as his... kissing. I realize that I'm staring at his finely-shaped lips and drop my eyes hastily.\n\n\"I want to, too,\" he whispers in my ear, lowering me into a graceful dip.\n\n\"Want to what?\"\n\n\"Kiss you right now.\"\n\nI almost lose my grasp on his shoulders, but Caden slips his free hand around my back and brings me back up. My cheeks are flaming and my heart is bursting. The music stops and Caden bows with a flourish to the loud sound of cheering. I sink into a dazed half-curtsy, and we move off the dance floor.\n\n\"The Vector's gone,\" Caden says, handing me a thin, fluted glass of something misty.\n\n\"What?\" Again, he has caught me unawares. I flush dully. I'm still thinking of him kissing me. \"Oh, right.\"\n\nThe ice-cold vapor in the glass helps to slow the burn in my chest, bringing me back to reality and some semblance of self-control. If I don't get it together, this is going to be an extraction gone horribly wrong. I shove any thoughts of Caden and kissing out of my head.\n\nMoving along the edge of the crowd, we walk to the stone terrace that branches off into the castle's lush rose gardens. I don't know why I never noticed it before: so many people without water in Neospes, and here it's indulgently wasted on roses. My time in the Otherworld has changed me, made me appreciate the little that we do have. We walk casually into the garden.\n\n\"Anyone see us?\" I ask.\n\n\"No,\" Caden says. \"And even if they did, they'll just think we came in here to finish what we started out there on the dance floor.\"\n\n\"You're disgusting,\" I say, but my cells are firing at the very thought.\n\nGritting my teeth and ignoring Caden, I locate the duffel bag that Sauer has managed to stash under a bench in a shadowed corner of the gardens. The sky is a deep bluish purple, and the faded moonlight barely illuminates the small square.\n\n\"Turn the other way,\" I say primly to Caden, who of course grins even more widely but complies. Stripping off the offending white gown, I pull on the black Vector suit I retrieved from my house. I don't link the neck connector to my nervous system just yet. Instead, I clip some weapons onto the suit's belt and throw my star-shaped sheath over my shoulder to fasten it over my suit, flush against my back. My ninjata blades scissor into the bottom of it, and the long sword Shae gave me slides in down the middle from the top. Finally, my pack goes over everything.\n\n\"OK, your turn,\" I say to Caden. \"I'll keep watch.\"\n\n\"You can watch.\"\n\n\"You're so annoying. I said keep watch. Just hurry up!\"\n\nBut Caden is already slipping off his shirt just as I'm finishing my last words, and my breath slams against my lips. Every curve is as chiseled as I remembered and looks even better in the moonlight. I drag my eyes away and stare at a point on the castle wall so hard that my eyes ache. He's a distraction... and one that could get us killed, get me killed.\n\nI slow my mind and focus on the task at hand, each breath hardening my resolve. Caden fades into the background as my body readies itself for action, my brain sharp and thoughts fluid. An icy sensation slips through my veins, and for the first time I recognize it for what it is \u2013 the android side of me, readying itself for battle, too.\n\n\"Time to go,\" I say, my voice rigid. \"You good?\"\n\n\"Yes.\"\n\nIt's as if Caden, too, has flicked a switch inside of him. Now the teasing smile is gone. In its place is a look of grim determination. He's also dressed in black gear similar to Aurela's and has stuffed the duffel out of sight. I pinch the collar of the suit against my skin and feel a small jolt as the suit powers up.\n\n\"Radio silence,\" I say to Sauer, pressing my earpiece. \"On my mark.\"\n\n\"Clear.\"\n\nAnd Sauer is gone.\n\nThere are guards patrolling, but security doesn't seem any tighter than usual. Like shadows, we cross the rest of the garden, keeping ourselves pressed against the bushes until we reach the far wall. There's a little-used passage in the back. All of the servants will be on full duty tonight, so we shouldn't encounter anyone.\n\nCaden and I split up, approaching the entrance from opposite sides. There's a guard to the right of Caden. He takes the guard out effortlessly with two jabs to the temple. We've all agreed to minimal casualties, so the guard is only unconscious.\n\nThe crumbling stone staircase is narrow and smells musty with disuse, but we're still careful not to step on anything that could draw unwanted attention to us. Noises and voices filter through as we pass a part of the kitchens, continuing upward until I've counted four floors and stopping at the back of a heavy wooden slab.\n\n\"Where exactly are we?\" Caden asks softly.\n\n\"East wing, fourth floor. It's the library.\"\n\n\"How do you know for sure that no one's on the other side of this door?\"\n\n\"Cale and I used to sneak out without anyone knowing, using this passage,\" I whisper to him. \"I don't think anyone knows that it exists anymore. Come on.\"\n\nThe slab cracks open, and as expected, there's no one in the room. We slip out and the other side of the door \u2013 an actual bookcase \u2013 slides back into place with a low click.\n\n\"This way,\" I say.\n\nAurela was right that everyone would be outside during the ball. We haven't encountered a single person, not even a guard. As soon as the thought crosses my mind, a shiver flutters across my neck. Why aren't there any guards? Where is everyone?\n\nSignaling to Caden to stop, I creep around a tall pillar and peer down into a section of the hall on the floor below. Not a body in sight.\n\n\"Something's wrong,\" I whisper. \"Something's not right. We have to find Cale now.\"\n\nMoving with urgent purpose toward the west wing with Caden close behind, I slip into Cale's old room. I'm unprepared for the onslaught of memories, especially of the last day I saw Cale... sick and weak. He begged me to find Caden so that I could save him.\n\n\"Cale?\"\n\n\"Riven, no,\" Caden warns in a low voice behind me, but I'm already moving toward the canopied bed. There's someone sitting behind the curtains \u2013 I can just see his outline.\n\n\"Cale? Is that you?\"\n\nThe figure turns moving into the light. \"No, but it is good to see you again, General.\" It's the Vector Commander, the one with the terrible voice. \"Your father said you had returned.\"\n\n\"My father?\" I blurt out.\n\n\"Of course,\" the Vector said. \"Your mother told him.\"\n\n\"You lie!\" I spit, but then compose myself in the next breath. \"My mother is dead. If you had your facts correct, you'd know that.\" The Vector stands, and I hear Caden's short hiss of indrawn breath behind me. I forgot how big the giant was, but I ignore the thrum of panic in my abdomen. \"Where's Cale? Where's the Lord King?\"\n\n\"Indisposed.\"\n\n\"What did Murek do with him?\" But even as I ask the question, I know I'm not going to get any answers. That thing is there for a sole purpose \u2013 for me. I shift into a battle stance, but the creature just watches me, its milky gaze fluttering to land on something just beyond my shoulder. In the next instant, I realize that I'm dead wrong.\n\nThat thing is there for Caden.\n\n\"Looks like they fixed you up good after the last time we met,\" I taunt, knowing it won't elicit any kind of response. It's what makes them the perfect killing machines \u2013 they don't feel, they just obey. \"There won't be much left of you this time, that I can promise you.\"\n\nMy ninjatas are in my hand and I'm springing toward the Vector with lethal precision before the last word leaves my lips. It meets me with a swift sidestep and an elbow to the back of the head that has me reeling. It moves fast, faster than the last time I'd seen it. I'm operating on old data \u2013 this thing has been recalibrated.\n\nCaden circles around the back to jab with his sword, but the Vector deflects the strike easily with its armored forearm. We attack it full-on from either side, slashing and weaving between offensive and defensive moves, but despite the double attack, we don't land any lethal blows. Frowning, I see that the Vector has been programmed to protect its vulnerable spots... or maybe it had somehow learned our tactics from our last fight.\n\nAs if reading my mind, the thing speaks. \"Let me have the boy.\"\n\n\"The boy's right here if you want him,\" Caden says defiantly, and I shoot him a glare. This is no time to be flippant. We could be stuck here in a never-ending fight with this thing preempting our every move while Cale inches closer to dying. The insidious way the Vector said \"indisposed\" makes my skin crawl.\n\nWe need backup.\n\n\"Reptile,\" I say urgently touching my ear. \"Reptile!\"\n\nBut there's no response. We're on our own.\n\nWithout wasting time, I tuck my body into a spin crouch, slashing at the Vector's sides. I connect, but the damage once more is just minor. The giant dances away, light on its feet. Its face slashes open in a grimace, baring broken, stained teeth. I can only imagine it's some kind of macabre grin. It's toying with us.\n\n\"Don't make this harder than you have to, General. You don't have a choice.\"\n\n\"What's it doing?\" Caden mouths, confused as I am by the Vector's cavalier attitude toward us. \"Why isn't it attacking?\"\n\n\"I don't know. Orders. I think it's waiting.\"\n\nI don't bother to hide what I'm saying. We're probably already surrounded. Pressing a button on my arm, I engage the suit's internal armor. Time to piss the big brute off while we still have a fighting chance. Caden and I attack simultaneously, slamming into the Vector from both sides. It only has two arms and we have four between us. Caden bears the brunt of the defense \u2013 slamming into the side of a bedpost like he's fluff \u2013 but it gives me the opening I need.\n\nPropelling my body toward the creature, I leap, my foot pressing off of the Vector's thigh, and I twist my torso in midair, slamming my blades forward. A row of spikes on the outside of my leg rips into its arms just as my ninjata blade sinks itself hilt-deep into the back of the Vector's thick neck. I wrench with all my might, ripping upward through the corded muscle. It grunts loudly, trying to shrug me off, but I'm hanging on to the hilt protruding from its neck with everything I have left.\n\nMy second blade, as sharp as it is, won't penetrate the Vector's uniform in armor mode. Instead, I swipe at its arms as they claw at me, trying to get me off its back. My strike is true as three of the creature's fingers fall like stones to the floor. Blue fluid sprays me in the face.\n\nI spare a glance to Caden, who is still lying on the floor. He's alive but dazed from the blow. He won't survive another hit like that, even if the Vector wants to keep him alive. Time to end this. Swinging my legs up, I literally crawl up the Vector's back and throw myself around and over its neck so that my thighs are straddling its head.\n\nAnd then I twist, throwing us both to the ground and hearing the sick crunch of bone as its neck dislocates. My attack won't kill it. It will only disorient it for a minute, if that. I land hard on my shoulder and the pain rockets through my bones, but I lurch to my knees, conscious not to waste the precious little advantage I have. The nanobes do what they're supposed to, assuaging the painful areas immediately and repairing any internal damage, so only seconds pass before I grab Shae's sword and swing it with both hands... right at its head.\n\nOur eyes meet in the nanosecond it takes for its android brain to recognize its imminent destruction. The Vector's hands still crunch into my sides and grasp at my neck. But it's already over \u2013 the blade slides through tendons and wiring like they're nothing but paper. Its body crashes to the ground.\n\n\"Try fixing that, asshole,\" I growl.\n\nVery carefully, I remove the ninjata that was spiked into its neck and jam it into the base of its cerebral cortex, where any thought processing function would likely be. Cutting open a flap of skin and ignoring the blue foul-smelling fluid oozing out, I dig my fingers in until I find the small square.\n\n\"Riven...\" Caden's voice is thready but I don't turn around until I've destroyed the chip. \"Riven!\" His voice is more insistent.\n\n\"What?\" I snap in irritation, spinning around and freezing.\n\nA dozen Vectors or more line the far perimeter of the room. They stand, unmoving, in a silent, deadly line. I glance back at the open balcony doors. Even if I grab Caden and escape, it's a four-story drop to the bottom, and there's no way I can take them all on my own, even if I am a super advanced hybrid humanoid.\n\nThe devil we do and the devil we don't \u2013 we're dead if we don't do something.\n\nI inch my way slowly toward Caden, who's only a couple of feet away. The Vectors watch me carefully but don't take any action. Unlike the commander I just pulverized, as soldiers, they're operating on direct orders only. I hope.\n\n\"You OK?\"\n\n\"Yeah, feel like I got hit with a sledgehammer,\" Caden whispers back, rubbing his head. The side of his face is covered in blood, but it looks like it's only from a long cut at his temple instead of a cracked skull.\n\n\"You did,\" I say with a grimace. \"Look, we have no chance against all of them, and it looks like they want to keep us here, not attack us. So we're going to make a run for those doors. We need to find Sauer.\"\n\n\"Riv, it's a long way down.\"\n\n\"Do you trust me?\"\n\nHis eyes are wide. \"Yes.\"\n\n\"Good. Now hold on to me and don't let go!\"\n\nAnd then we run like hell for the doors. The noise behind us is sudden and thunderous, but I don't look back. As I'm running, I unwind a length of black cord that's hooked to my belt. The end of the cord opens into a metal three-pronged device, and I throw it just before flinging myself over the balustrade and grabbing Caden with one hand.\n\nWe're freefalling in space for an eternity before the rope snaps tight and we swing to the parapet that's just below us. Releasing the rope, we roll to the ground, unhurt. My blood feels hot, but I'm energized with adrenaline, my heart pounding in my ears.\n\n\"Move,\" I yell to Caden. \"There's an entrance on the far side.\"\n\nOutside, the noise is deafening with music and shouting and singing below us. We run the length of the parapet, the same one I looked up to when we went through the security checkpoint. The courtyard is packed with people, but no one's looking up at us. Half of it is shrouded in shadow, and the other is brightly lit from the castle floodlights.\n\nI still can't shake the feeling that I've missed something... that despite our escape, we're somehow being herded.\n\nThe Vectors, including the commander, were prepared for us in advance, which explains why we aren't being treated as trespassers and why they aren't attacking us. Someone in control wants us alive. Murek? My father?\n\n\"Hey!\" a voice shouts from behind us. It's Loren, running as if her life depended on it. She doesn't slow down until she reaches Caden, who's a couple feet behind me, closest to her.\n\n\"Where's Sauer?\" I yell.\n\nThe smile that breaks across her face is full of malice as she slams an electro-rod into Caden's side. His eyes widen and I lurch forward instinctively, but I'm too late as she presses a button on the rod.\n\n\"Yes, you know what that is, and it's not set to stun. Weapons down, General.\" My hands lower, releasing the ninjatas to the ground.\n\n\"Loren, what're you doing? We're on the same side,\" Caden gasps weakly.\n\n\"I was never on your side,\" Loren laughs, spit flying from her mouth. \"The minute I found out who Aurela's daughter was, I knew I'd do anything to take her down. It was fate that you returned.\" Her eyes are manic, staring at me. \"You destroyed my family, as I will destroy yours.\"\n\n\"Caden has nothing to do with that,\" I say softly. \"That's between you and me.\"\n\nHer laugh is hollow, but something in it makes the hairs on the back of my neck stand at nervous attention. \"He has everything to do with it.\"\n\nI whip around at the sound of another familiar voice behind me. \"All in good time, Loren, all in good time.\"\n\nMurek.\n\nHe isn't alone. A squadron of Vectors and armed guards is fanned out behind him. We are hosed. I swing back to Loren. My instincts were correct about her, but it went far deeper than I ever thought. I thought she'd want to get even and it would come to blows at some point, but I never suspected that she was some kind of spy for Murek.\n\n\"Aurela trusted you.\"\n\nAnother satisfied smile. \"She did. I wanted her to. I was going to bring her down... all of it, her plans to infiltrate the palace. She deserves to die just as my brother died. And then you came back, and things couldn't get any sweeter. Two for the price of one.\"\n\nI need to antagonize her, to get her to drop Caden. \"What do get out of this? Vengeance? Your brother's still dead.\" Her eyes flash with anger, but I continue to push forward, sick inside at the vitriol spilling from my lips. \"You couldn't save him then, and you can't save him now, because you're weak.\"\n\nWith a cry of fury, Loren dives toward me, but I'm ready for her attack, sidestepping swiftly and swinging my elbow up and outward into the back of her head. She stumbles to the ground, but I'm already moving toward Caden. Together, we back away from Murek.\n\n\"Thank you,\" he says nastily. \"She was a loose end that needed tying up.\"\n\n\"But she's still alive!\" Caden blurts out.\n\n\"She isn't now,\" Murek says, watching as one of the Vectors electrocutes her with the butt of his weapon, the smell of singed flesh hot in the air. Loren's body arches upward twice and collapses. Caden's eyes are wide with horror. \"There isn't a way out, General. You know that. Surrender, and you'll live.\" Murek's voice is low and compelling, and as slick as I remembered. \"We have Commander Sauer. We have your mother. It's over. Hand over the boy.\"\n\n\"I can't do that,\" I say.\n\n\"Then we have no choice.\"\n\n\"Enough,\" a soft voice says from behind us. Caden stiffens as I turn around in slow motion, disbelieving. My heart beats with impossible hope that he is somehow still alive and that my mind isn't playing tricks on me. He was sick and near death when I'd left.\n\nBut it is he, and he looks better than before. Emotion clogs every part of me. Strong, healthy... and dressed in exhibition clothing.\n\nCale, the Lord King of Neospes.\nSIBLING OF FLESH AND BONE\n\n\"That was you in the ring?\" I demand through clenched teeth, recalling the Vectors who stood at stiff attention during the exhibition fight. I'm kicking myself \u2013 I should have recognized Cale's fluid style immediately. But in the grand scheme of things, it doesn't matter. There's nothing I could have done differently.\n\n\"I'm surprised you couldn't tell,\" Cale says.\n\n\"Are you better? Why didn't you send word? I would have come back.\"\n\nI'm ranting now, with Cale watching me from the corner of his private salon. Caden stands in silence by the window, his glance flitting from us to the two guards standing on either side of the door. For all intents and purposes we haven't been restrained or disarmed, and I still have my weapons. If push comes to shove, I can get both Caden and Cale out of there without breaking a sweat. But I'm too angry to consider alternatives. I continue my tirade.\n\n\"Do you have any idea what I've been through? I thought you were dead and Murek was in control.\"\n\n\"Riven,\" Cale begins gently. \"He is. It's the only reason I'm not dead yet.\"\n\n\"What? I don't get it.\" I'm stunned. \"But you're alive, Cale. You're strong again. How could you let someone like him take control?\" Cale remains quiet, his eyes more distant than I've ever seen them. He hasn't even given Caden more than a cursory look after we were ushered into his private room. In fact, he's doing his damnedest not to pay him any attention at all, which is odd, considering that they are mirror images of each other. Wouldn't he be curious? Frowning, I remember what Loren was saying.\n\n\"Did you know about Aurela? That my mother was alive?\"\n\n\"And that she was planning some kind of coup?\" I flinch at the acerbity in his voice. \"Of course I did. It's my job to know about those who want to usurp me.\"\n\n\"Usurp?\"\n\nHe ignores my outburst and leans back in his chair. His manner is indolent, scratching against my rattled senses like nettles. \"That's why I sent Loren in. When she first approached our soldiers in the Outers, I realized it was my chance to find out exactly what your mother was planning. And then when you came back with him, no less, it was perfect.\"\n\nA cold feeling makes its way through me. \"So did you know she was alive before you sent me to the Otherworld?\"\n\nHe doesn't answer, but I see the truth in his eyes. \"You were the only one, Riven. The only one who could find him and bring him back.\" Only then do Cale's eyes drift to Caden \u2013 his exact likeness \u2013 and remain there, cold and unflinching. \"He has something of mine. Something that my mother stole, and I need it back.\"\n\n\"What?\" Caden bursts out from behind us. \"I don't have anything of yours. I didn't even know about you or any of this\u2013\" he says gesturing into the air with his hands, \"\u2013until a few weeks ago!\"\n\nI'm shaking my head slowly as understanding filters through my brain. \"You lied to me?\" Cale shrugs, opening his hands wide, palms upward. \"You're not really dying, are you?\"\n\n\"Not exactly. Your father fixed me, gave me an electronic lung. But I needed you to believe that I was dying so you'd go.\" He cocks his head to one side like a bird, his gaze speculating, switching between Caden and myself. Calculating. \"Did they tell you what you are?\" Cale says slyly to Caden.\n\nCaden looks to me, confused. \"But Shae said...\"\n\n\"Let me guess,\" Cale says, ridicule tainting his words, \"Shae said that you're the heir, and I'm the byproduct? Well, you're wrong. I'm the only heir. Give me the chip, brother.\"\n\n\"Chip? What chip?\" Caden says, but I can hardly hear him. I completely forgot about the chip I pocketed from Caden in the Outers. That and the blue signet ring.\n\n\"Come now, don't play dumb. She would have given it to you \u2013 it's the only way for you to prove who either of you are. By the way, my condolences; I heard that she died, quite painfully as it were. It would have been something to see, I'm sure.\"\n\n\"Cale!\" I've never seen this cruel side of him, and every instinct in me is reacting to it. Caden's eyes are stormy, his fingers clenching into fists at his sides at Cale's ugly, provoking taunts.\n\nAurela was right. Cale isn't who I thought he was. He's changed. How could I have not seen the malevolence eating away at him earlier? Have I been so blind to Murek's evil influence? Or have I changed after my time in the Otherworld... after my time with Shae and Caden?\n\n\"I don't have it,\" Caden grits out. \"I lost it in the Outers.\"\n\n\"He doesn't have it, Cale,\" I say, stepping forward. Out of the corner of my eyes, I see the guards' hands drop to their weapons. I raise an eyebrow to Cale, who shoots me a look in return that clearly says his security comes first. \"I have it,\" I say. I feel Caden's panicked glance, but I can't even look at him. I need to be the general, not the girlfriend. \"What's so important about it?\" I toss back at Cale.\n\nI can see the wheels in Cale's head turning, as if he's trying to figure out how much he can tell me. For my part, I've never underestimated anyone more. \"Like hell you don't know, Riven. That chip is a record of my birth. It does not belong to him,\" he says, jerking his head toward Caden.\n\n\"And you need it to prove that you're the rightful king.\" And now I have something he wants desperately. \"Or the people will always question your right, especially if they know Caden has returned.\"\n\n\"Yes, your mother took care of that when she and her rebels sowed the seeds of dissent in Neospes.\" Cale is scowling now, but the dark expression vanishes from his face in seconds, replaced instead by something more conciliatory. \"Riv,\" he says, placating. \"You know me. You know who I am, don't you? Don't you trust me anymore? Can't you see they're lying?\"\n\n\"I don't know who to trust, Cale,\" I answer honestly. \"Everyone has lied, even you. The only one who hasn't lied to me is Caden. I don't know what's truth or not.\"\n\n\"I knew you'd take his side,\" Cale sneers, his face twisted in an ugly grimace. \"He told me, you know, your father. About what you were doing in his house. With him.\" I feel myself color, heat blooming in my cheeks, followed by horror that my father knew I was there, somehow saw what Caden and I did. I should have known that he'd have cameras in the house. He is far more paranoid than he used to be. I kick myself mentally for the tenth time for being so foolish and reckless.\n\nMy eyes narrow. \"Where is he?\"\n\n\"Taking care of some personal business at the lab,\" Cale responds in a sly tone. \"I am sure it will be a very sweet family reunion, don't you think? Now give me what I want, Riven, or I will take away what you want.\" He jerks his head in Caden's direction, nodding to the guards, who step forward obediently to restrain him.\n\nMy heart slides downhill at the betrayal. I've been such a fool.\n\n\"So Murek isn't after the throne?\" I say in a soft voice. Cale smiles widely, shaking his head. \"He's your tool,\" I breathe. \"You use him just like you used me.\"\n\n\"Well,\" Cale murmurs conversationally so only I can hear him. \"Father and I had different ideas about the monarchy. I thought he was far too soft with the people. When my lungs started failing, he started asking questions. Pointed questions. I think in the end, he knew that I wasn't his real son.\" He spreads his hands wide. \"So I took care of him.\"\n\n\"You murdered your own father?\"\n\n\"Technically, my uncle did it, but yes, it was at my behest.\" His eyes are cold and dead like the monster he had become. \"Don't worry, the reptiles took care of him quickly, I heard.\"\n\n\"You're insane,\" I whisper.\n\n\"Ah, the dreaded insanity,\" Cale mocks. \"That's another thing I need your friend for. As brilliant as your father is, it's not like he can create a whole new perfect brain for me.\"\n\nAppalled, I can only stare at him. \"What does he get out of helping you?\"\n\n\"Why, you, of course.\" Cale rolls his eyes skyward. \"Must be some skewed father-daughter thing.\"\n\nDespite his sordid insinuation, I breathe a sigh of relief. Cale doesn't know about me... about what my father had created. And of course my father didn't say anything, because he had something to hide... something he knew that others \u2013 like Cale or Murek \u2013 would desperately want.\n\nMe.\n\nA part of me prays that Aurela circumvented whatever confrontation awaited them, but I know it's a hope beyond hope. If he knew she was coming, he would have been prepared with a thousand Vectors. She's probably already dead. The only tangible thing left to negotiate is Caden's safety.\n\n\"Cale, if I give you the chip,\" I begin, \"will you promise to let Caden return unharmed to the Otherworld?\"\n\n\"No!\" Caden hisses, wrenching against the unbreakable grip of the Vectors.\n\n\"Yes,\" I say. \"You belong there. You never belonged here.\"\n\n\"Riven, I have nothing to go back to. I belong with y...\" He falters. \"Here.\"\n\nLoud clapping interrupts our exchange. \"Isn't that so touching? The captive fell in love with the captor.\" Cale turns to me. \"You've done your job well, General. So yes, you have my word that he will be allowed to leave unharmed as soon as I have the chip.\"\n\nSaccharine treachery drenches his words. If he can murder his own father without a second thought, he will never let Caden go once he gets his hands on the only proof of any claim to the throne. We're at an impasse \u2013 Cale staring at me with persuasive eyes and Caden staring at me with fury in his.\n\n\"I'll tell you where the chip is right now if you let him go.\"\n\n\"No, Riven!\" Caden shouts, but I ignore him. \"Don't do this. Don't let him do this.\" My heart feels like it's splintering inside my body but I know it's the only way. It's the only way for me to keep Caden safe from Cale's insanity.\n\nI'm so sorry, Cade.\n\n\"So where is it?\" Cale snaps waving his arm impatiently.\n\nI deflect his question with one of my own. There's no way I can give him the chip here in the castle with us under Vector arrest. Neither Caden or I would have a chance. \"What happened to you, Cale? Things used to be different. You believed in Neospes and rebuilding what your ancestors started.\"\n\nA hollow laugh. \"Didn't you get the memo? They aren't really my ancestors. I'm merely a by-product of him.\" Cale's hands are shaking with rage as he points to Caden. \"That doesn't mean that I'm any less than he is, but the sad truth is, everyone will think that way. As soon as I understood what I was, I knew that I had to get rid of him for good.\" His face is tortured, and my chest fills with an unfamiliar ache. He is my best friend \u2013 one for whom I'd sworn a blood oath to protect. \"Do you know,\" he continues bitterly, \"my father tried to banish me? Murek was the one who told me, who saved me. So my father had to die. It was him or me.\"\n\n\"Cale, you can't trust Murek. He's a snake. He's always been a snake. You know that. You know what he's planning with the Vectors and the Otherworld?\"\n\nA slow, cunning smile rolls over Cale's face. \"Of course I do. It was my idea.\"\n\n\"What?\"\n\n\"I will be greater than any king of Neospes, and when we take the resources from the Otherworld, I will rule them too.\"\n\n\"Listen to yourself, Cale. This is madness. The Guardians will never stand for it. You know what happened before, when people jumped back and forth centuries ago. We were decimated by disease. Our very existence was nearly derailed.\" I'm almost screaming now. \"There's a reason the anti-eversion laws are in place... a reason the Guardians exist.\"\n\n\"The Guardians?\" he mocks me, laughing cruelly. \"What Guardians? What do you think the Vectors have been doing all this time? Surely you didn't think they were only looking for either of you?\"\n\n\"What have you done, Cale?\" I say. I can't keep the horror from my voice as I remember what Aurela said about Guardians randomly dying. \"The Guardians are neutral. Untouchable.\"\n\n\"By whose authority? The Faction?\" Cale scoffs venomously. \"Some obscure group of universe overseers from a thousand years ago?\" His face is contorted with malice. \"They're just humans whom the Vectors have killed with no penalty whatsoever. The Faction or whatever they're called is a made-up entity. I'm the king. I call the shots.\"\n\nHis face is manic, and it's clear Cale has completely lost his mind. His thirst for power has consumed him. But even though I know it will be of no use, I still try one more time. \"Cale, what you are doesn't define you. It's who you are inside, who you make of yourself. Trust me, I know.\" I want to tell him the truth about me, but I know that I can't. \"Being the master of two universes won't make you happy. It won't make you whole or feel any less broken. You are what you are. The choices you make are the things that define you, and this isn't you. I know you.\"\n\n\"You know a ghost.\" Cale's jaw clenches and his eyes darken to a stormy green. \"We're done here. Where's the chip?\"\n\n\"I can't give it to you,\" I say. \"I know you'll never keep your word. I could always tell when you were lying, and you are now.\"\n\n\"So be it,\" Cale says, his face hardening even more. He nods to the guards, who escort Caden through the door into the main hall. \"They kill him if you so much as twitch toward me, Riven,\" he warns. He glares at me to follow, a twisted smile curving his lips in warning. He knows that I'm itching to take him out now that we're alone. But there's too much at stake, and with Caden's life hanging in the balance, I can't risk it. I grit my teeth and follow Caden through the doors.\n\nThe great hall is full of Vectors, and the chemical smell of them all is almost suffocating. Murek is standing next to a bloodied but alive Sauer, and staring at me with a self-satisfied smirk on his bulbous face. I send him a clear message of my own \u2013 if I get within an inch of him, his life will be mine. This time, I smile.\n\n\"You see,\" Cale whispers in my ear. \"I know you just as well, Riven. I knew you would say no to me, so I have something very special planned for you.\" Cale walks to the throne at the end of the hall, where the two Vectors are waiting with the restrained Caden. I don't meet his eyes. I can't meet them, because I need to be strong for whatever is coming next. Murek bows respectfully as Cale takes his place on the throne.\n\n\"Lord Murek, please bring out our special guest.\"\n\nI am prepared to see Aurela bound and gagged, because there's no one more important to me than Caden other than her, and for the moment, he's safe. It would be amusing if they had somehow put back together the commander I'd destroyed earlier. Folding my arms across my chest, I paste a bored expression on my face.\n\nBut nothing prepares me for what comes through those doors.\n\nMurek walks back into the hall with a malicious, triumphant smile. At his side is a Vector, but it's not just any Vector. This Vector is slim in stature and has the face of someone I love.\n\nSomeone I loved.\n\nShae, my heart breathes her name. What have they done to you?\n\nMy stomach plummets and the world spins out from beneath my feet. I'm backing away with an outstretched arm, my other hand clapped to my mouth. My eyes are burning. As if from a dense fog, I hear Sauer's pained shriek. I feel wetness on my cheeks \u2013 everything hurts, and I'm choking.\n\n\"No,\" I gasp.\n\n\"Yes,\" Cale shouts, clapping.\n\nMy sister is a Vector.\n\n\"Come on, Riven; let's see those infamous skills of yours,\" Cale crows.\n\n\"You are a monster. You'll never rule Neospes. I will kill you first. I'll kill you all.\" I'm screaming as I'm keening, my rage and pain making me berserk. Now I understand why he'd let me keep my weapons. I can hardly see through the cloud of fury that's flaying me, but somehow my eyes meet Caden's. Although his pain is as great as mine, maybe even greater, there's something else there... a silent glance that slivers through the haze in my brain.\n\nDo what you must.\n\nAnd everything disappears \u2013 all the anger, the hurt, the aching sadness \u2013 all of it, and it's just Caden anchoring me to him. Caden is my past, my present, and my future. He is my world. He's the only whole thing in all of this broken madness, reminding me of everything that's true... everything that is real.\n\nMy sister died in the Otherworld. I grieved. I mourned her.\n\nThis is not my sister.\n\nCale and my father have made a mockery of what she was. They'll pay for that too. They all will. I leave the war paint of tears on my face \u2013 they're for her.\n\nGrimly, I turn to face my opponent, ignoring the long dark-blonde braids and the curve of her cheekbones. Instead, I focus on the milky eyes that are nothing like her vibrant full-of-life ones. Her arms are long and slender, no longer full of caring. They wield a deadly staff with double-edged blades at both ends.\n\n\"Hello, Riven,\" it says in her voice.\n\nReeling, I don't see the first blow coming until the last second, and I duck, but the edge of the blade clips my temple and forehead, just missing my eye. Blood splatters along the floor following the arc of the staff flying upward, and drips down my neck.\n\nShe's a talking Vector, like the Commander earlier, nothing more.\n\n\"Miss me?\"\n\nThis is not my sister.\n\nGritting my teeth, I pull out the sword \u2013 the Artok sword my real sister had made for me \u2013 from its sheath, swinging it upward just in time to meet the blow of the staff coming toward me. The force of it ricochets up my arms. She's strong... stronger now. I have to tell myself again that this is not Shae. It's a Vector... a tough Vector that has her face and her voice, but one that will never fight with her heart, or her skill.\n\nSo we dance.\n\nMost of my moves are defensive. I'm watching her movements, learning what's been programmed. We circle each other, the violent sound of steel clashing echoing in the huge hall. We could be exhibition sparring in the training field. I need to change the game, to get her to slip up. I step in, ducking under one of her strikes, my foot colliding with her chest so hard that she staggers backward, but it doesn't stop her. Advancing, I whip the sword up to catch her across the chest, but her Vector suit is in armor mode and deflects the strike harmlessly. She lunges toward me as I spin and crouch, throwing my leg out to catch her in the back of her knee. She stumbles but rights herself with the inhuman grace only a Vector has. She smiles, cocking her head to one side.\n\n\"You left me to die,\" she says. \"You didn't even try to fight. You wanted me to die.\"\n\nHer words are worse than her blows. Guilt chokes me.\n\nI left my sister to die.\n\nGrinning, she comes at me again, swinging that staff of hers in a circular motion like a fan. I weave and bob, but I can feel the air of the blades whipping far too close to my face. I'm disoriented, disabled by her barbed taunts. At the last second, she stops, the weapon in one hand, her dead eyes holding mine.\n\n\"You killed me,\" she whispers, and then twists to kick me in the ribs. Payback. The sick crunch of bone snaps wetly and I double over, only to feel the flat of her weapon crashing against the back of my head.\n\nI drop like a sack of stones, pain rocketing through my body. The sword skids across the room. Everything is swimming through my blurred vision as I crawl to my knees. It feels like I'm moving through quicksand, and the booted foot comes far too quickly, smashing me in the side of the face. Color explodes like fireworks inside my eyelids, and blood fills my mouth as my entire body flips up and over from the force of her kick.\n\nI cough, spluttering blood through loose teeth.\n\nCacophonic laughter echoes around me. It's Murek, his face lit up with glee. Cale is sitting on the edge of his throne, something hungry in his expression \u2013 his desire to punish me is more than personal. All eyes are on us, even the rest of the dead Vectors standing silently around the hall's perimeter, but the only ones I care about are fixed on me. I can feel Caden's strength, willing me to get up.\n\nI can't.\n\nYou can, his eyes say.\n\nI shake my head. She's too fast, faster than any Vector I've ever fought. She's invulnerable.\n\nYou're faster than any human.\n\nIt feels like the words are Caden's words, but I know the conversation is entirely in my head. Still, he's right or I'm right. I'm faster than any human. I'm like them, only better, a reverse-engineered reptile.\n\nI turn over, spitting a mouthful of blood to the floor and pulling my arm underneath me. The suit's armor is already on, but I enter a sequence on the wrist-pad engaging synchronous mode. The suit will attune to me, and I with it.\n\nI'll fight fire with fire.\n\nSomersaulting to my feet, I feel myself powering up, the nanobes inside of me responding to the energy from the suit. I am reptile. The smile that opens across my face is the Vector's only warning before I launch myself toward her, running at full speed and drawing my ninjatas from the harness.\n\nOne I toss in Cale's direction, and I see the momentary shock on his face. But it's not meant for him. I don't even turn around to hear the satisfied thud of steel into flesh or the thump of Murek's dead body on the ground. Instead, I'm leaping, launching myself like a missile to the abomination in front of me.\n\nHer staff goes flying and we both fall to the ground, rolling. The smell of her and formaldehyde fills my nose as I straddle her chest. This time I stare into the milky blue eyes.\n\n\"You are not my sister,\" I hiss, and slam the remaining ninjata up through her chin into her head. Blue fluid spurts out.\n\nHer leg swings up and over my neck, thrusting me forward, but I jump easily to my feet. So does she, her staff back in hand. The blade is still stuck in her neck, the hilt of it protruding outward. She starts the spinning trick again, but this time I stand my ground. The blade whips so close to my hair that I see wisps of it falling to the ground. I bare my teeth in a grin.\n\n\"No,\" I say, grasping the staff in the middle and stopping it mid-motion, my gloved fingers closing on hers. The armored suit is a fluid extension of me, unbendable and unyielding. The Vector tries to pull it away, but now I am as strong \u2013 or stronger even \u2013 than she is. I move closer, looking deep into the Vector's eyes that once belonged to my sister, because I know he's watching remotely.\n\n\"I'm coming for you, Father.\"\n\nAnd then I lean back and kick the hilt of the ninjata sideways so that it rips across the Vector's spinal column and clatters to the floor in a spray of blue. There's dead silence in the hall for an instant as the Vector falls to the ground. Out of the corner of my eye, I see Caden flip up and over, twisting the arms of his captors. He slams their heads together, and they crumple to the ground. Grabbing the wounded Sauer and tossing him over his shoulder, he darts toward me in the center of the room.\n\n\"Restrain the prisoners!\" Cale screams to the Vectors surrounding us, and suddenly pandemonium erupts as we're rushed on all sides.\n\nThere's no way in hell we can win. But we won't go down without a fight. I'm going to take as many of them as I can with me. I feel Caden's fingers slip into mine, and I squeeze. We lock eyes, and I know that we're in this together. Sauer stands weakly next to us, the crossbow in his hand already firing into the oncoming horde. With a deep breath, I raise the staff I took off Vector Shae and prepare to do battle.\n\nBut then the unexpected happens.\n\nEvery single one of the Vectors stops in their tracks, their arms and weapons falling to their sides, eyes going blank and dull. The remaining human guards back away in confusion. There's no way they will take us on without the Vectors.\n\n\"What happened?\" Caden asks, his voice loud in the unnatural silence. Even Cale is staring at us in the middle of an unmoving mob with his jaw on the floor.\n\n\"Aurela,\" I breathe. \"She's done it. She's deactivated them.\"\n\nOnly then do I notice the row of red-clad masked fighters surrounding Cale's throne. Where did they come from? Who are they? As if in response, the doors to the great hall open, and a woman walks in, followed by more of the same ominously garbed warriors. I recognize her immediately, and this time, my jaw drops to the floor.\n\nEra Taylor.\nLONG LIVE THE KING\n\nWhy is my physics teacher from the Otherworld in Neospes? What is she doing here? Caden's eyes are as wide as I'm sure mine must be, so I know I'm not seeing ghosts. Era Taylor \u2013 a Guardian \u2013 is here in Neospes.\n\n\"Era?\" I say warily. \"What are you doing here?\"\n\nHer dark eyes are as piercing as I remembered. \"We are the Faction\"\n\n\"What?\" I repeat dumbly. The Faction is here? And Mrs Taylor is a part of it? Which means that the scarlet-clad warriors are the Faction's infamous Red Guard. \"But you're a Guardian.\"\n\nEra smiles a humorless smile. \"A Guardian and also the leader of the Faction, an order far older than your young king standing over there.\"\n\n\"How dare you?\" Cale shouts back. \"I am the King of Neospes! What have you done with my guards?\"\n\n\"He's not my king.\" I glance at him dispassionately and turn my attention back to Era, who is still assessing a murderous Cale with her eyes.\n\n\"You will be tried and judged for your crimes,\" Era says calmly to him. \"Your guards must obey the Faction first and the Neospes monarchy second. At least they, if not you, still understand the chain of rule.\" Cale falls silent.\n\n\"Era, how did you get here?\" Caden asks quietly. \"We thought you were dead. Did Phillip refine the serum you were working on?\"\n\n\"Yes, he did,\" Era answers. \"And I'm not dead, as you can clearly see. We tried to find you after we got separated but only discovered dead-end tracks. We found the cave later, but there was no trace of you.\"\n\n\"We everted,\" Caden gulps. \"Shae died.\"\n\n\"I gather that, and I'm sorry for her loss. She was a great asset.\"\n\nOf their own volition, my eyes fall on the nearly-decapitated body of my dead sister. It wasn't her before, but it's her now. It's her body. My rage resurfaces. My father has gone too far this time, defiling her for his and Cale's own nefarious ends. How he must have loved punishing her and pretending it was me.\n\nI walk over to where Cale is still sitting. The red ninja-clad warriors restraining him are silent, only their glittering eyes visible through their masked hoods. \"You are despicable,\" I say. \"I trusted you. How could you let him do that to Shae? You made her a monster just so that you could hurt me?\"\n\nCale's eyes are as dead as the metal lungs in his chest. \"You betrayed me.\"\n\n\"I did what you asked. You're the one who lied.\"\n\nCale's eyes flick to Caden. \"You were mine. You betrayed me with him.\"\n\n\"I was never yours,\" I say. \"As flawed as you are, I believed in you, and there was a time that I did love you. I would have done anything for you, and I did. I went to another universe for you... just so that you could find someone you wanted to kill. You betrayed me. You want to know the truth?\" I lean in so that my lips are nearly touching his ear and only Cale can hear my words. \"Caden is everything you are, and more. He's more of a king than you will ever be.\"\n\nRage reddens Cale's face. \"Shae got what\u2013\"\n\n\"Don't!\" My open palm collides with his face. \"That's for my sister. I hope you get everything you deserve for what you've done. Either way, you are dead to me.\"\n\nI walk back without a backward glance to Era, who nods to her guards to take Cale away. He'll be held until Era decides what his punishment will be. Caden's arm drops across my trembling shoulders, and he gathers me close. We both ignore Era's raised eyebrows.\n\n\"What happens now?\" he asks her.\n\nThis time her smile is real. \"Now the rightful Lord King of Neospes takes his place.\"\n\n\"What? No, I couldn't\u2013\"\n\n\"Yes, Cade,\" I say, pulling out of his hug. \"She's right. You have to. This is what you were meant to be, why your mother risked everything for you. Why... \" my voice breaks, and I glance over to where my sister's corpse is lying, \"...Shae protected you with her life. You were always the rightful heir.\"\n\n\"Why didn't my mother ever tell me who I was?\" Caden's voice is small.\n\nEra flicks her hand to two of her guards, signaling them to take Shae's body to the mortuary, and folds her arms across her chest. \"I'm sure she meant to someday, but she didn't get the chance; her immune system was too weak. You survived because you were young and strong.\" Her gaze flits to Murek's dead body still lying near the throne with my ninjata buried in his chest. \"She was right to take you away.\" Era's eyes have a faraway look in them. \"I was young then, newly part of the Faction. When she came to me, desperate for help, I agreed to let her evert. You would both be safe there, and when the time was right, you would return. But things changed. Murek grew more cunning \u2013 he knew what Cale was, and he was the one who told the King that your mother had taken his real son, not the clone, and that you were both dead. Your father was consumed by grief and refused to recognize Cale as anything but an imposter. It began to eat away at Cale, and he turned to the only person who seemed sympathetic: Murek.\"\n\n\"So Murek was playing both sides?\" I say, and Era nods.\n\n\"But he was still his son,\" Caden whispers. \"I mean, he was me, wasn't he? He couldn't help what he was.\"\n\n\"The King didn't see it that way,\" Era says gently. \"Royal clones are meant as a failsafe to the line, not meant to be recognized as part of it.\"\n\n\"But I saw him. He was still real. He's still a person.\"\n\n\"One who ordered the murder of your father after I left,\" I remind him quietly. \"He was my friend, remember? But Cale is twisted. He's not the same person I once knew. His bitterness killed him from the inside out. Clone or not, he's still broken inside.\"\n\n\"And Murek couldn't have planned it more perfectly \u2013 he had what he wanted nearly in his grasp. His own kingdom,\" Era says, walking out of the hall and indicating that we should follow her.\n\nCaden looks to me, and I nod. It's her show now. A few of the Red Guard fall in line with us, the majority remaining behind to sequester the Vectors. Before we leave, I retrieve Shae's sword and my ninjatas, wiping them clean and re-sheathing them.\n\nIn the hallway, Era continues speaking, her voice low. \"Even Cale didn't understand the complexity of Murek's cunning. He told Cale about the chip that would disprove his place as heir, and that Caden was indeed still alive. Cale's plan was to kill Caden and destroy the chip, but Murek had other plans. He wanted the chip to dethrone Cale, and he knew that Cale would send Riven.\" Her black eyes meet mine. \"You were the only one who could hinder his plans.\"\n\n\"And me? Why did you leave me for the Vectors?\"\n\nEra's glance turns into a thoughtful frown. \"I didn't know then that you'd turned. But Aurela vouched for you... her life for yours. I found it curious. I found it even more curious to learn that your father almost killed Murek for using you to find Caden.\"\n\n\"What do you mean?\" I am careful to hold Era's stare. I don't know if she knows the truth about me, but it wouldn't surprise me. The Faction seems to have eyes and ears everywhere, based on what she's telling us about the whole Murek mastermind show.\n\n\"Murek and Cale needed your father to build an army of Vectors, so they gave him free rein to experiment and play around with unsanctioned robotics.\"\n\n\"So the Faction knew?\" I toss back. \"Why didn't you stop him?\"\n\n\"It's not our duty to interfere in the politics of each universe. We protect the Guardians and maintain the separation of the two worlds,\" Era returns calmly. \"But once we learned about their plot to bridge the worlds, then we were oath-bound to act.\"\n\n\"So my father basically got carte blanche to do whatever he wanted?\" My voice is bitter. \"Hurt whomever he wanted, take whatever he wanted, destroy whole lives.\"\n\n\"Your father was driven by his creations... by his ego.\" The voice comes from behind us, and I jerk around to see Aurela entering through a door.\n\nI can't help myself; my feet are already racing toward her. She's dirty and weary and limping slightly, but her face is victorious. We embrace, her fingers grazing the oozing cut on my temple, and I search her eyes.\n\n\"Did you see him?\" She nods silently. \"Is he dead?\" She shakes her head. For a second, I wonder why she's avoiding my questions, and then I realize that all eyes are on us, including Era's and the Red Guard. Sighing, I say, \"I'm glad you found the kill switch for the Vectors. It got a little hairy in there.\"\n\nAurela squeezes my hand as we walk back toward the others, and gives Era a brief recap of what happened at the lab. Era nods in satisfaction and walks to a pair of heavy wooden doors. It's Cale's old bedchamber. We enter the room, where servants are already waiting to attend Caden.\n\n\"Time to look the part,\" Era says to him.\n\n\"I'll be waiting right outside, OK?\" I smile reassuringly, wishing we'd gotten the chance to speak privately. His green eyes meet and hold mine. This time, he's the one reassuring me. I smile and walk outside.\n\nTwo guards station themselves inside the door, and two outside of it. Era stops to give them instructions, and I use the opportunity to speak to Aurela.\n\n\"So he's not dead?\" I ask urgently.\n\n\"No,\" she says. \"He everted at the last minute.\"\n\n\"Everted?\" I gasp. \"To the Otherworld?\"\n\n\"I don't know.\"\n\n\"Did you talk to him?\" I ask.\n\nShe stops to squeeze my shoulder, her eyes pained. \"He's not going to let you go, Riven. He's never going to let you go.\" Her voice fades to a barely audible whisper. \"He's lost it, he's so consumed by what you are. In his eyes, you belong to him, and he'll kill you if he can't have you.\" Tears are falling down her face. \"I tried to reach him, to see if there was some part of the man I knew still in there, but he's gone. There's nothing in there now. He's as dead inside as the Vectors he creates.\"\n\n\"Then there's only one thing I can do; I have to find him first, before he finds me.\"\n\n\"Riven, there's something else. We need to let the Faction know about you,\" Aurela says. \"About what you are and why he wants you so badly. They can protect you. If he gets his hands on you, who knows what he's capable of? We can't risk it.\"\n\nI know that she's right, even though a part of me wants to hunt him down in silence on my own terms, but I also know that that's my ego talking. I still want to be the one to punish him, but I also know now more than ever that I cannot underestimate him. He is devious and brilliant... and utterly insane.\n\nEra joins us, and I excuse myself, nodding to my mother to do what she has to do. Outside, the revelry is loud. I can't imagine that they have any clue as to what has happened inside the castle \u2013 that a false king has been replaced by the true heir, that people have died, that Neospes is finally about to enter a new phase... one not undermined by a king with his own selfish interests.\n\nI hope that Caden will be a good king and that he'll be a true leader to our people. I have no doubt that he will be. After all, he is forgiving to a fault. He is smart and will take the time to listen and learn. Although he has been shaped in the Otherworld, his heart is here. He is everything that is the best of both worlds. I find myself smiling, watching all of the joyful faces below me. His mother would have been proud of what her son had become.\n\n\"Hey,\" Caden's voice is soft.\n\nI turn slowly, my eyes widening, butterflies causing havoc in the pit of my stomach. It's still not a feeling that I'm comfortable with... that any one person could make me feel so shattered and whole at the same time. Caden is magnificent in monarch purple and black, his long overcoat trimmed in sable. His hair has been combed back off his face, and his eyes are a sparkling green. He looks beautiful, perfect.\n\nI find my voice. \"You clean up good.\"\n\n\"Thanks.\"\n\n\"You look like a prince.\"\n\n\"I feel like a weirdo,\" Caden says with a grin. \"I mean, have you seen this cape?\"\n\nI grin back. \"Sadie would definitely approve.\" The look in Caden's eyes drives any other words about Sadie from my mouth.\n\n\"Do you approve?\"\n\n\"Yes,\" I say. \"I definitely approve. You belong here, Cade. This is where you were meant to be; this is who you are.\"\n\n\"I thought you wanted me to go back to my world,\" Caden says, stepping forward to brush a strand of hair out of my face.\n\nHis light touch makes the butterflies take flight, and unconsciously, I lean into his caress. \"At the time, it was the only place where you would have been safe from Cale. But everything worked out, didn't it?\"\n\n\"Riven,\" Caden says, moving to stand directly in front of me, both his hands pressed to the sides of my face. \"None of this could have happened without you, you know that, right? I mean, you're the reason I'm here and alive. You're the reason I even know who I am and what I'm meant to do.\" Caden pauses, staring out at the throng of people below us. \"The only reason I don't feel like I don't belong here is because of you. You made me fall in love with Neospes just like I've fallen\u2013\"\n\n\"Cade,\" I interrupt him, blushing furiously. \"You always had Neospes in your heart. It's in your blood.\"\n\n\"Will you let me finish?\" he says, bending his head to graze my lips so softly that it's barely a touch, but it silences me effectively.\n\n\"You know what happened the last time,\" I breathe, my eyes fluttering closed, and tilt my face up to his, pressing my lips against his mouth. His hands slide into the hair at the nape of my neck, and he deepens the kiss, opening his mouth against mine. My fingers curl into the fur of his collar, pulling him closer. I never want to let him go.\n\nThat kiss says everything that Caden wants to say and more, and I kiss him back, leaving him with no doubt of my own feelings. Because now, more than ever, he needs to know how much I do love him, especially with what I know I have to do.\n\nThe loud sound of cheering breaks us apart, and we glance down to see the crowd below staring up at the parapet with their hands in the air.\n\n\"Go do your thing, Cade,\" I say, drawing away from the edge of the stone balustrade. \"I believe in you. And I will always be here.\" I clutch my fist to my heart, the tears burning a pathway behind my eyelids and down my cheeks.\n\n\"Wait, why are you crying?\" he says.\n\n\"Because I love you, and I'm so proud of you. Now go, address your people,\" I say with a laugh, swiping my face with the back of my hand. \"And stop making me say such reckless feeling things.\"\n\nWatching Caden address the crowd, I have no doubt that he will be a great leader, and with people like Aurela and Sauer at his side, he will unite the people of Neospes. I walk back to where Era and Aurela are standing. Era stares at me and I shift uncomfortably, but her stare is curious instead of anxious.\n\nEven if she is Faction, I still don't quite trust her. Maybe it's the way her eyes seem to pierce right through me, or the fact that she is one of the most powerful women in our combined universes. No one person should have that much power. That said, it's not like she ever abused her power. She stepped in where she had to.\n\nAnd I know I'm going to need her help if I am to track down my father.\n\nI feel Aurela's arms encircle me from the side, and I lean into her, inhaling the scent of my childhood. My arms cross over my mother's arms and I squeeze.\n\n\"Take care of him,\" I tell her.\n\nI punch in a sequence on the arm of my suit and pull out an eversion device from the pack at my side. Time is of the essence, and every second I delay here means my father will get further away. I meet Era's eyes with a wry grin. I don't need her serum \u2013 I can evert from anywhere just as the Vectors do.\n\nJust another one of the perks of being a half-human, half-robot hybrid.\n\n\"Guess I'll see you in physics class,\" I say to her with a forced grin, even though it feels like my very bones are splintering inside of me.\n\nMy eyes flit from Aurela to Sauer to Caden... my family is no longer broken. It's beautiful and it's more real than anything I can imagine. I never thought it would be so hard to leave anyone, because I never had anyone to leave before.\n\nNo wonder people say that love begets weakness. It hurts like a beast. But I know that it isn't weakness; it's might. It takes far more strength to care. It makes you want to fight harder... to live for something worth believing in.\n\nIt makes you more.\n\nAs if he can sense what I'm about to do, Caden turns around, his eyes widening in delayed understanding. I smile to reassure him, but my lip quivers. I can't even lie to not hurt him... all I can see is the pain in his eyes, those intense green eyes asking me why, pleading for me to stay.\n\nBut there's no time for answers, no time for reasons. I can only smile and weep.\n\nEverything fades to black, the fabric of time and space shimmering around me, but I hold on to Caden's face. I hold on to his green eyes. I hold on to him and the weightless way he makes me feel. I hold on to the touch of his lips upon mine and the salt of his tears in our kiss. I hold on to love.\n\nI know I will see him again. \nACKNOWLEDGMENTS: THE ALMOST GIRL\n\nTo my fabulous and feisty Strange Chemistry editor, Amanda Rutter, who saw so much potential in The Almost Girl that she bought the duology, thank you so much for collaborating with me and letting me share this story with the world. It's such an honor to be inducted into the nefarious ranks of the Strange Chemists! This book could not have found a more perfect home at Strange Chemistry, and for that I have to thank my warrior princess agent, Liza Fleissig of the Liza Royce Agency. I honestly cannot thank you enough for everything you do\u2014your foresight, your enthusiasm, your drive and your passion are all astounding. I'm so grateful to have you in my corner.\n\nTo Kristi Cook\u2014critique partner, friend, therapist, sidekick and partner in crime\u2014thank you for being awesome. That is all. To my writing retreat group\u2014Cindy, Danielle, Angie and Kate\u2014who keep me full of inspiration and in good humor, thanks for your generous words of wisdom and your friendship. Without you, I'd be a hotter mess than usual. To my friend and fellow author, Kim Purcell, thanks for all the sanity checks and energetic conversations. To Jenny, thanks for the lunches, the brainstorming and the plotting\u2014I'm starting to think we may need some super-villain masks. A big shout out to Julie, Marissa, Sami and Grace at JKS for their superhuman publicity efforts\u2014thank you! To all the bloggers, booksellers and readers who spread the word about my books and humble me with their unwavering support, I wouldn't be here without you. Thanks for doing all that you do.\n\nTo my extended family, friends and fans, a heartfelt thank you for being such amazing champions. Much gratitude to my parents, who always encouraged me to fuel my voracious imagination and to push the boundaries of possibility. Thank you for never holding me back. To my brothers, thanks for always looking out no matter what. Lastly, to the loves of my life\u2014Cameron, Connor, Noah and Olivia\u2014thank you for making no other universe equal this one.\nABOUT THE AUTHOR\n\nSeventeen Magazine Summer Club author Amalie Howard grew up on a small Caribbean island where she spent most of her childhood with her nose buried in a book. Twenty-two countries, surfing with sharks and several tattoos later, she has traded in bungee jumping in China for writing the adventures she imagines instead. She isn't entirely convinced which takes more guts.\n\namaliehoward.com\n\ntwitter.com\/AmalieHoward\n\nStrange Chemistry\n\nAn Angry Robot imprint\n\nand a member of the Osprey Group\n\nLace Market House,\n\n54-56 High Pavement,\n\nNottingham\n\nNG1 1HW\n\nUK\n\nAngry Robot\/Osprey Publishing\n\nPO Box 3985\n\nNew York\n\nNY 10185-3985\n\nUSA\n\nAngry Robot\/Osprey Publishing,\n\nwww.strangechemistrybooks.com\n\nStrange Chemistry #22\n\nA Strange Chemistry paperback original 2014\n\nCopyright \u00a9 Amalie Howard 2014\n\nAmalie Howard asserts the moral right to be\n\nidentified as the author of this work.\n\nCover art by Steven Wood\n\nDistributed in the United States by Random House, Inc., New York.\n\nAll rights reserved.\n\nAngry Robot is a registered trademark and Strange Chemistry, the Strange Chemistry icon and the Angry Robot icon are trademarks of Angry Robot Ltd.\n\nThis is a work of fiction. Names, characters, places and incidents are the products of the author's imagination or are used fictitiously. Any resemblance to actual events, locales, organizations or persons, living or dead, is entirely coincidental.\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 and destroyed\" and neither the author nor the publisher may have received payment for it.\n\nUS ISBN: 978 1 90884 480 4\n\nUK ISBN: 978 1 90884 479 8\n\nEbook ISBN: 978 1 90884 481 1\n\nSet in Meridien and Dirty Headline by Argh! Oxford\n\nPrinted in the United States of America\n\n9 8 7 6 5 4 3 2 1\n\n# Contents\n\n  1. Title Page\n  2. Prologue: Three Years Earlier\n  3. Part One: The Otherworld\n  4. Present Day: Colorado\n  5. Blackout\n  6. Secrets\n  7. Preparation\n  8. Conflict Arising\n  9. Dead Man Walking\n  10. Truth Be Told\n  11. Underground\n  12. The Prey\n  13. Reversal of Fortune\n  14. Revelations\n  15. Scars\n  16. Track or Trap\n  17. Part Two: Neospes\n  18. Over There\n  19. In The Outers\n  20. The Others\n  21. A Web of Lies\n  22. Confessions\n  23. Deception's Daughter\n  24. On the Brink\n  25. Revolution\n  26. Father Dearest\n  27. The Winter Solstice\n  28. The Winner Takes It\n  29. Sibling of Flesh and Bone\n  30. Long Live the King\n  31. Acknowledgments\n  32. About the Author\n  33. Imprint Page\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\nWhat Others Are Saying About  \nREBOOT YOUR LIFE\n\n_\"Reboot Your Life_ is a wonderful guide for reconnecting with the dreams we gave up to become adults.\"\n\n\u2014 RON ANDREWS,  \nVice President, U.S. Businesses,  \nHead of Human Resources, Prudential\n\n\"While taking a sabbatical may sound deceptively simple, making time for oneself can be a difficult task in today's busy society. Some companies, including ours, have found however, that people thrive as both individuals and as employees when they are given a sabbatical opportunity and can set aside real time to reflect, refresh, and re-energize. _Reboot Your Life_ provides valuable advice, wisdom, and tips to help readers both make their sabbatical dreams become a reality and reap as much value and benefit from their sabbatical time as possible.\"\n\n\u2014 TAMI GRAHAM,  \nDirector of Global Benefits Design, Intel Corporation\n\n\"It's a tough lesson to learn: Time is more valuable than money\u2014because all the money in the world can't buy you time. And that's the lesson to be learned in these pages\u2014how to appreciate and make the most of that uncertain and precious commodity, the time you are given to live a fulfilled life.\"\n\n\u2014 TERRY SAVAGE,  \nNationally syndicated _Chicago Sun-Times_  \nfinancial columnist and author of _The New Savage Number:  \nHow Much Money Do You Really Need to Retire?_\n\n\"In a world that has gone from 9-5 to 24\/7 in a generation, our ability to lead a professionally productive yet reflective life is increasingly challenged. A time of professional disengagement offered by a sabbatical is invaluable. The chance to cultivate personal interests and family and friends free of the demands of the office and Blackberry leads to a richer life and thereby a refreshed perspective when returning to work. These four writers understand this and offer wise counsel. Take their advice and take a sabbatical.\"\n\n\u2014 PETER H. DARROW,  \nSenior Counsel, Cleary Gottlieb Steen and Hamilton\n\n\"During my 25 years at IMG, I witnessed professional athletes returning to competition after a needed break with re-honed skills and a fresh outlook. When i left IMG in 2006, i wanted to do the same but there were no roadmaps. _Reboot Your Life_ walks you through how to plan your sabbatical, what to expect and how to reenter the work place. it's a fun, practical guide to exploring this life-changing experience.\"\n\n\u2014 STEPHANIE TOLLESON,  \nFormer Senior Corporate VP, IMG and  \n(current) Chair of the Board, Women's Sports Foundation\n\n\"Talk about discovery! _Reboot Your Life_ gives the inspiration, insight, and practical tips to take a work break for a breather or a real life change.\"\n\n\u2014 WONYA LUCAS,  \nExecutive Vice President and Chief Marketing Officer,  \nDiscovery Communications\n\n\"This gem of a book is from four authors who are role models of lives well lived. They provide a motivational, yet practical, blueprint for that 're-boot' that any busy person can use and apply. A must read!\"\n\n\u2014 PHYLLIS J. CAMPBELL,  \nChairman, Pacific Northwest, JP Morgan Chase\n\n\"Taking a sabbatical to grow, expand and explore is so essential to your ability to create and innovate. But most of us don't think it's possible until we retire. _Reboot Your Life_ gives you a road map as well as tips and stories from those who have done it. I highly recommend it for anyone thinking about making themselves, as well as their organizations, more open to innovative and creative thinking.\"\n\n\u2014 CLAUDIA B KOTCHKA,  \nFormer Vice President, Design, Strategy and Innovation  \nProcter & Gamble\n\n\"I don't care how good you are at your work. All of us need time off to refresh and recharge. Until recently I had never taken more than one contiguous week off from work. I wish I had taken a Reboot Break earlier in my career. _Reboot Your Life_ gives you the courage and the tools to do so.\"\n\n\u2014 GREG JOSEFOWICZ,  \nFormer CEO, Borders, Inc.\n\n\"We all need time off from our careers to prepare for the next phase of our lives. Whether it is to regroup when we are \"burned out,\" take care of our families when they need us most, or to explore new passions and opportunities, taking a sabbatical allows us the journey. _Reboot Your Life_ is the roadmap!\"\n\n\u2014 SUSAN C. KEATING,  \nPresident & CEO, National Foundation for  \nCredit Counseling & former President & CEO  \nof Allfirst Financial\n\n\"Peter Drucker, the father of management, often encouraged readers to take the time to examine their accomplishments and objectives. In his seminal Harvard Business Review article, 'Managing Oneself,' he wrote, 'To stay mentally alert and engaged during a 50-year working life, one must know how and when to change the work one does.' _Reboot Your Life_ makes for a friendly, thought-provoking companion along the way.\"\n\n\u2014 ELIZABETH HAAS EDERSHEIM,  \nAuthor of _The Definitive Drucker_ and Founder  \nand Director, New York Consulting Partners\n\n\"The shocks of the last decade are reminders that there is more to life than livelihoods. In this new world, the more diverse our experiences and knowledge, the more connections our brains and hearts can make. _Reboot Your Life_ is a practical, relevant, inspiring easy-to-read guide on the journey for growth, rest, and renewal. The Sabbatical Sisters retreats are also an enjoyable and actionable way to give yourself the \"gift of time.\"\n\n\u2014 SUSAN SCHIFFER STAUTBERG,  \nPresident of PartnerCom and Co-Founder  \nof Women Corporate Directors (WCD)\n\n\"Taking a sabbatical was one of the best things I ever did for my life! _Reboot Your Life_ covers every aspect of that adventure\u2014the fears, the money, the big dreams, other people and everything else in between. Get the book.\"\n\n\u2014 ARIANE DE BONVOISIN,  \nAuthor of _The First 30 Days: Your Guide to Making Any Change Easier_\n\n# Reboot Your Life\n\nENERGIZE YOUR CAREER AND LIFE BY TAKING A BREAK\n\n| THE SABBATICAL SISTERS |\n\n_Catherine Allen_  \n _Nancy Bearg_  \n _Rita Foley_  \n _Jaye Smith_\n\nCopyright \u00a9 2011 by Catherine Allen, Nancy Bearg, Rita Foley, Jaye Smith\n\nReboot Your Life\u2122, Sabbatical Sisters\u2122, and Reboot Break\u2122, are used throughout this book, as US trademarks owned by Reboot Partners LLC and are used with Reboot Partners' permission.\n\nFIRST EDITION\n\nAll rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without permission in writing from the publisher, except by a reviewer who may quote brief passages in a review.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nReboot your life : energize your career and life by taking a break \/ The Sabbatical Sisters, Catherine Allen . . . [et al.].  \np. cm.  \nIncludes index. ISBN 978-0-8253-0564-1 (alk. paper)  \n1. Leave of absence. 2. Sabbatical leave. 3. Quality of work life. I. Allen,  \nCatherine A.  \nHD5255.R43 2011  \n650.1\u2014dc22\n\n2010050686\n\nFor inquiries about volume orders, please contact:\n\nBeaufort Books  \n27 West 20th Street, Suite 1102  \nNew York, NY 10011  \nsales@beaufortbooks.com\n\nPublished in the united States by Beaufort Books  \nwww.beaufortbooks.com\n\nDistributed by Midpoint Trade Books  \nwww.midpointtrade.com\n\nPrinted in the united States of America\n_We dedicate this book to our families._\n\n## **| CONTENTS |**\n\nIntroduction\n\n[ CHAPTER 1  \nGiving Yourself the Gift of Time](ch01.html#ch01)\n\n[ CHAPTER 2  \nTypes of Sabbaticals: The Why and What](ch02.html#ch02)\n\n[ CHAPTER 3  \nIt's All in the Planning](ch03.html#ch03)\n\n[ CHAPTER 4  \nFunding Your Freedom](ch04.html#ch04)\n\n[ CHAPTER 5  \nPhases and Navigating the First Thirty Days](ch05.html#ch05)\n\n[ CHAPTER 6  \nThe Heart of the Reboot Break: Reconnection and Exploration](ch06.html#ch06)\n\n[ CHAPTER 7  \nRenewed, Recharged, Now What?](ch07.html#ch07)\n\n[ CHAPTER 8  \nDeflecting Sabbatical Robbers](ch08.html#ch08)\n\n[ CHAPTER 9  \n\"You're Taking a _What_?\" Life with Someone on Sabbatical](ch09.html#ch09)\n\n[ CHAPTER 10  \nLiving the Lifelong Sabbatical](ch10.html#ch10)\n\nConclusion\n\n APPENDIX\n\nPlanning Checklist\n\nFunding Your Freedom\n\nOrganizations That Get It\n\nResources\n\nIndex\n\nAcknowledgments\n\nOur Sabbatical Stories\n\nAbout the Authors\n\n## **| INTRODUCTION |**\n\n\"I always knew that one day  \nI would take this road, but yesterday  \nI did not know today  \nwould be the day.\"\n\n\u2014Nagarjuna\n\nTIME OFF. TAKING A BREAK. Going on sabbatical. Being between gigs. These phrases describe a growing need among some of us to step out of the working world for a while. That need may be motivated by a desire to reset the balance in our lives, follow a creative dream, make a difference, give back to society, or explore something different. It may also be driven by less positive causes\u2014job burnout, disillusionment with a career, or a life-changing event like death, illness, or divorce.\n\nWhatever the reason, millions of us are ready for change\u2014for a \"Reboot Break\"\u2014for taking several months away from our everyday work in order to refresh and renew.\n\nUnfortunately, most of us don't know how to do it. Or we feel we can't. Or we think we need permission. Fears about not having enough money, losing the respect of colleagues, being out of the \"game,\" or altering that self-image keep many of us from expanding our horizons.\n\nWhen we had a farm economy, natural breaks provided time to renew. As recently as the 1980s, people spent more time at home than they do today\u2014time having dinner with their families and watching television, reading, or playing board games. There was more time for relaxation, exploring, thinking.\n\nToday we rarely have time for rest. We have lost even our short breaks. Technology\u2014pagers, PCs, laptops, the Internet, cell phones, BlackBerrys, and smartphones\u2014beckons us to be \"on\" 24\/7. Even when we're supposedly \"off\"\u2014on weekends or in the evenings\u2014we are \"on,\" and it is taking a toll.\n\n**We're a nation on the verge of burnout.** We need to take a break. We need to regroup, renew, and reinvigorate our lives. We need to give ourselves the \"gift of time.\"\n\nAccording to a 2009 Gallup Poll and statistics from Monster.com, the numbers underscore the stress people are feeling in the workplace and the desire for change. American workers are working more hours than they did twenty years ago, with men averaging 49.9 hours and women 44 hours per week. Eighty-six percent of workers are experiencing job stress, and half describe their stress as \"extreme fatigue\" or \"feeling out of control.\" Sixty percent of workers feel pressure to work too much, and 83 percent of employees want more time with their families. Over 50 percent of employees are either somewhat or completely dissatisfied with their jobs, and 83 percent of workers plan to look for a new job when the economy improves.\n\nWhat a testament to the need for taking time off!\n\nTo be competitive as a nation and thrive as individuals in the twenty-first century, we all need time to refresh and recharge. To be creative and productive, we must be nourished, too.\n\nIt is time to give ourselves a break and \"reboot our lives.\"\n\nMost would agree that Americans don't know how to take a real break, and many of us long to modify work to recapture our evenings, Saturdays and Sundays, and our vacations. Those who have taken sabbaticals have found that when they return to work, they are far more likely to take breaks in the future. They are able to acknowledge that they need them. They've seen the benefits of time that is truly \"off\" and the importance of pacing themselves. As one engineer told us after a six-month break, \"A sabbatical actually resets our 'beingness,' making us aware of the need to check in with our inner selves.\"\n\nThis book is about regaining that time and creating a more desirable cycle of work, relaxation, and personal growth, and about the self-discovery and acceptance that comes from exploration and rest.\n\nWe've talked to more than two hundred people who have taken a Reboot Break\u2014men and women from their twenties to their seventies, from a variety of ethnic, socio-economic, and professional backgrounds\u2014to learn why they decided to take time off from work, how they did it, and what they learned.\n\nThis includes, of course, those who have had to take unexpected sabbaticals because they were laid off in a down economy. Rather than taking the first job they're offered, many people spend three to six months or more stepping back and reassessing their goals and their opportunities.\n\n**Taking time out from work to reboot your life is not just a new and enduring trend; it's a necessity in our stress-ridden world.** This book will give you the guidance and resources to negotiate a Reboot Break, whatever your age and stage of life and even in tough economic times. Meaningful time off can be an important path, not only to personal development, but also to career advancement. What's more, you deserve it!\n\nIn addition to interviewing sabbatical takers about their experiences, we wanted to understand how employers are viewing sabbaticals. To that end, we have examined more than fifty corporations, law firms, non-profits, small businesses, and educational institutions that provide funding for sabbaticals, allow their employees to take unpaid leave, or support the concept in some other way. Many organizations see sabbaticals as a recruitment and retention tool that helps create a resilient and loyal workforce. Later in the book, we help you make the case to your employer for your Reboot Break and for instituting a Reboot Break policy.\n\n### **SOME COMMON THEMES**\n\nOne of the key themes that emerged in our interviews with men and women who took time off was the importance of planning. We spend two chapters on this (Chapters 3 and 4), one of which is entirely devoted to financial planning and the consideration of things like health insurance and retirement savings.\n\nA second major theme was that there are at least four phases to a sabbatical:\n\nI. Creating Space\u2014Putting your life in order\n\nII. Reconnection\u2014Revitalizing connections to people, places, activities, and self\n\nIII. Exploration\u2014Learning new things, especially through travel\n\nIV. Reentry\u2014Starting a new chapter of your life\n\nWe devote three chapters to these phases, Chapters 5, 6, and 7.\n\nA third major theme was that all the people we interviewed, no matter the experience, felt that their lives had improved after the sabbatical. They experienced better career opportunities, or enhanced personal relationships, or a new sense of self-respect. We use their stories throughout the book to illustrate this.\n\nOur mission is to empower overworked Americans and others to plan for and take much-needed career intermissions in order to rest, recharge, stimulate new thinking, and come back better prepared for the challenges and opportunities they face.\n\nWe hope to broaden your horizons by encouraging you to give yourself the \"gift of time\" to find your real interests and explore them. In the pages ahead, we address common hesitations and fears head-on and provide practical, easy-to-read, and actionable ways to plan, prepare for, and actualize the life-changing break from work that we call rebooting your life. Each chapter has exercises that will help you with your plan, and the Appendix is rich in worksheets and resources.\n\nThe book is designed to allow you to skip around, find topics and exercises of interest, and take what is relevant to you now. It is also designed to take you step by step through the planning and implementation of time off from work . . . what we call the Reboot Break. We hold retreats across the United States to help people think about, and plan for, their time off. The Appendix has more information about the Reboot Your Life Retreats.\n\nWe invite you to use this book as your companion, friend, advisor, and support group, all rolled into one. Our message to you is simple: _By taking time out to reboot, personally and professionally, you too can live the better, richer, fuller life you've been seeking._\n\n## |CHAPTER 1|\n\n## _Giving Yourself the Gift of Time_\n\n\"Whatever you can do, or dream you can, begin it.  \nBoldness has genius, power, and magic in it.  \nBegin it now.\"\n\n_\u2014Johann Wolfgang von Goethe_\n\nWhat would you do if you only had one year to live?\" That's the question Jaye asked Nancy, Cathy, and Rita the first time we talked. It was 2006, and we were seated around a table on a veranda in Vieques, Puerto Rico. We were at the annual conference of a group of women CEOs and senior business and public policy executives who network, support each other, and give back to the community\u2014as well as have fun. One of the ways we get to know each other better is through \"table topics\" at lunch.\n\nWe four had chosen the table with the topic \"Dreamweaving: Following Your Dreams.\" Each of us wondered as we sat there awkwardly how much we would reveal of our dreams and ourselves. Who were these other women? None of us thought of ourselves as \"touchy feely\" or \"new age,\" so we didn't know what to expect of the others.\n\nIt turned out we were all contemplating a sabbatical or integrating the benefits of one we'd taken. We began to realize that our dreams had everything to do with taking time off to discover what they really were or to make them happen. If we had a year to live, we each agreed, it might just be a sabbatical year.\n\nAt the end of the lunch, we agreed to check back during the year to see where we were in achieving our sabbatical dreams. \"I left feeling I had a support network that would cheer me on,\" said Rita. \"It was an unexpected result of the lunch and taught me again about how small risks are ways to open the mind and heart.\"\n\nThe four of us stayed in touch after the conference but did not see each other again until a few months later at a crowded New York City event. Cathy called across the room to Rita, \"Hey, Sabbatical Sister, I'm going to do it\u2014I'm going to take my sabbatical!\" Rita, with a huge smile, called back, \"Hey Sabbatical Sister, I just started mine!\" The name Sabbatical Sisters stuck.\n\nThe book was born the next year at our conference in Ecuador, and we went on to coin the term Reboot Break as an updated term for sabbatical.\n\n### **WHO TAKES REBOOT BREAKS**\n\nIf you bought this book on your own, you must be thinking about taking time off. If someone else bought it for you, it may be to nudge you in the direction of a needed break. In any case, as you contemplate your own dream of a Reboot Break, your age may have a great deal to do with the kind of break you're looking for and the way you decide to go about it.\n\n#### IF YOU'RE IN YOUR TWENTIES AND THIRTIES\n\nYou, who are just starting out, have multiple reasons for taking time off. You may find you need a \"course-correction\" in your career or in the way you've balanced your life. You may feel you chose the wrong career and want to continue your education in order to switch to something more suited to you. You may want to start a family or spend more time with loved ones. Or you may just be bored with your job. Because you're young, you are probably less encumbered and more able to travel, explore new interests, give back to your community, or just try something new.\n\nIn your twenties and thirties there are many real and perceived barriers to taking some time off, but there are ways to get around them.\n\nThe three greatest fears at this age are:\n\n\u2022 You can't afford it financially.\n\n\u2022 You won't be taken seriously in your career.\n\n\u2022 You won't be able to come back to your job or employer.\n\nLater in the book we go into more detail on the real and perceived barriers to taking time off, how to deal with these barriers, and how to make time off a possibility by planning for it.\n\nThe chapters on planning and ways to fund a Reboot Break address the financial concerns. In our chapter on planning and what organizations are doing to create programs to attract and retain employees, you'll see how to convince your employer that it's in his or her best interest, as well as your own, to allow you to take time off.\n\nMany people your age have \"set themselves apart from the pack\" by taking time to volunteer, upgrade skills or education, explore and travel, and come back refreshed.\n\n#### IF YOU'RE IN YOUR FORTIES AND FIFTIES\n\nPeople in their forties and fifties tend to be mid-career and may need a break to recover from job burnout or stressful events in their personal lives like illness, death of a family member, or divorce. Or they may want to explore a totally new career, or take time to volunteer and give back. In this age group, the need to renew, refresh, and become more creative is often paramount.\n\nYour key fears may be:\n\n\u2022 If you leave, a peer may get your promotion or job.\n\n\u2022 You have clients, employees, or partners who you think can't cope without you.\n\n\u2022 You need the benefits for your family.\n\n\u2022 You wonder if you step away whether you will ever find work again.\n\nThese fears and the others we talk about later in the book are all manageable. It just takes planning. There will always be peer competition in large organizations, but the stories from our interviewees show that taking the risk paid off. They came back to the same organizations refreshed and invigorated. They were often promoted. Or they were approached by other organizations because of their confidence and risk taking. **To be competitive, organizations are seeing that they need to retain the experienced and knowledgeable people in this age group.**\n\nWe have a discussion in Chapter 3 on how to take time off if you are an entrepreneur, a small organization employee, a partner in a law firm, or any other position where clients, customers, and employers may be significantly impacted. We show you how to do it. And we share the stories of those who have done it.\n\n#### IF YOU'RE IN YOUR SIXTIES AND SEVENTIES\n\nPeople in their sixties and seventies are showing interest in taking a Reboot Break as they increasingly reject traditional models of retirement. Many are starting new careers, exploring non-profit involvement, or launching their own businesses. They are spending time exploring their passions, from art to travel to grandparenting. Some are taking a \"pre-retirement break\" to explore what they want to do when they eventually leave work.\n\nAt this age, the fears may be more philosophical than just juggling finances and responsibilities. But there are still concerns that must be addressed:\n\n\u2022 You are stepping off the career track. Can you get back on if you want?\n\n\u2022 Have you saved enough for retirement, and will this impact finances and benefits negatively?\n\n\u2022 What will you do with all that time?\n\nBaby Boomers hate the \"R\" word (retire) and want to do something different, but don't know what. Taking time off at this juncture of your life gives you time to explore different options and plan for the next phase of your life. Baby Boomers are likely to live well into their eighties and nineties, and be far healthier than their parents. There is much you can do, and rebooting may help you figure out what you want to do for the next twenty to thirty years.\n\nWhy is it we often undervalue one of our most valuable assets\u2014time? Anyone over sixty will tell you how important it is, but at twenty or thirty, we think we have all the time in the world to do what we want. We willingly spend time working and planning our finances, our careers, or our vacations, but we rarely spend time planning how to step back for a while and decide what we really want to do. People contemplating sabbaticals worry, \"What will I say when people ask what I'm doing now?\" or \"What do I put on my business card?\"\n\n**We in the United States are so tied to what we do that leaving our jobs, even temporarily, is a scary proposition.**\n\n### **THE GIFT OF TIME**\n\nRemember the play and movie _Stop the World\u2014I Want to Get Off?_ We have all felt that way. Our days are full of obligations, demands, and details. From the time the alarm goes off to when we collapse in bed at night, we bounce from activity to activity.\n\nFor most of us, our busy schedules are what enrich our lives. We love coaching basketball while we tap at our BlackBerrys in between plays. We like pushing ourselves at our jobs and crossing things off our lists. Being busy is a way of life, and most of us like it that way\u2014at least until we don't.\n\nDo you ever find yourself thinking:\n\n_Why do I do this to myself all the time?_\n\n_What if I could clear my schedule for a week, a month ... a year?_\n\nInwardly, many of us are yearning to take some meaningful time off from work, to give ourselves the gift of time.\n\nThink forward. Imagine a period of richness and fulfillment. Imagine floating in your own time\u2014fast if you want, slow if you want. Imagine responding to your own internal rhythm more than to external stimuli and requirements. How would that feel?\n\nImagine some more: You are getting things done that you've left undone for years, trying new things, spending real time with friends, being with family, stretching yourself mentally and physically, and filling your mind with something other than cell phone messages and schedules.\n\nDream along with us some more. Here's the end game: You're back from your sabbatical. You are refreshed and reenergized. Your mind is clear and focused. So are your priorities. You've rebooted.\n\nNow, flash back to where the journey begins: deciding that the benefits of a sabbatical are right for you, and then going through the essentials that will get you there.\n\nThe decision to take a sabbatical\u2014to answer that call to oneself\u2014is a huge step. There's no question about it. The worries that can cloud people's minds and tighten their guts as they contemplate taking a break are real and practically universal. At the same time, **sabbaticals are as old as time and are a natural rhythm of life.** Knowing that and learning a little more about sabbaticals\u2014like their cultural context and great outcomes\u2014may make it easier to decide to take one. So let's get more comfortable with the concept and explore how taking a sabbatical is a natural human desire and can make a big difference in one's life.\n\n### **A LITTLE HISTORY**\n\nThe Bible's Old Testament tells the story of the Sabbath. It teaches that God created the world in six days and rested on the seventh. The Bible tells us that the work-rest balance, the fourth of the Ten Commandments, was handed down by God to Moses and from Moses to his people. This idea is probably familiar to you, even if you're not religious. The concept of the Sabbath recognizes the universal need for a renewal break.\n\nWe just call that \"rebooting.\"\n\nIn Biblical times, the practice that left fields fallow in the seventh year was called Shemitah. Also during Shemitah, individual debts were forgiven, and slaves were released from bondage. Similar to a sabbatical period, Shemitah was a time of release and freedom, symbolic of returning to oneself.\n\nLike taking a Reboot Break today, the decision to live freely was a difficult one. Some Hebrew slaves remained slaves even though they had the legal right to become free. For them, the known was more comfortable than the unknown. Perhaps they told themselves, \"Seven years from now, I'll take my freedom,\" postponing that challenge and its potential gifts, just as many of us now say, \"Someday I will take real time off from my job. Someday I'll take time for myself.\" Perhaps the indebted even listened to the inner voice so many of us have heard, whispering that they did not deserve the freedom.\n\nEuropean culture has managed to preserve a modified version of the sabbatical practice. It's not uncommon for some Europeans to get (and take) six to eight weeks of vacation each year. That's their rebooting period. Americans, on the other hand, have a worldwide reputation for their short vacations, or for taking no vacation at all. And most Americans tend to work longer hours than people in most other developed countries. So we start from a personal time deficit each year and keep building the time debt to ourselves year after year. It takes a sabbatical to step away and recharge.\n\nAmerican universities have been ahead of the rest of American society. They have long recognized (beginning back in 1880 with Harvard) that professors need to get away from teaching every few years to refresh and renew. Most academic institutions grant faculty members a sabbatical of several months to a year every six or seven years for research, study, or writing. The late computer science professor Randy Pausch, famed for his \"last lecture\" and book by the same name, took an academic sabbatical to fulfill his lifelong dream of working as an Imagineer at Disney.\n\n### THE NEW SABBATICAL (REBOOT BREAK): IT's NOT JUST FOR ACADEMICS\n\n**The new sabbatical\u2014what we call a Reboot Break\u2014is based on the same historical theories of rest and renewal as the academic sabbatical, but it offers _all_ working people the opportunity to refresh and come back to the working world stronger and more creative.**\n\nThe Reboot Break can be done in agreement with one's workplace, or by separating from it. It may include pay and benefits, or it may not, thus requiring a plan for financing and insurance. Taking a break entails a little thinking outside the box, but not more than most people can handle when armed with information, creativity, and determination.\n\nHow long is a Reboot Break? Ideally, it is a period of at least three months. People need at least that much time to truly benefit. With more time, the reward becomes deeper, so a year is even better. Whatever its length, a sabbatical is the gift of time, and it cannot be crammed into a night or weekend or even a one- or two-week vacation.\n\nWhat is the formula? The Reboot Break has no precise formula. It takes many forms, and anyone at any stage in his or her adult life can take one.\n\nWhat are some of the benefits? Time is precious, and time spent with yourself is especially precious because it allows you to examine your assumptions, habits, career choices, priorities, and purpose in life. For the more than two hundred people we interviewed for this book (what we like to call \"the Reboot Break chorus\"), the break provided the opportunity to think and reflect in a way they could never do in the rush of their everyday lives. All interviewees said they were personally and professionally better for their Reboot Break\u2014in their work, in relationships, mentally, and physically.\n\nOne member of our Reboot Break chorus, Kate, thirty-five, tells her story:\n\n_I now work forty to forty-five hours a week, see friends, eat dinner with my kids at least four times a week, have dates with my husband, and am happy\u2014and very productive\u2014in my work. I have a ton of energy, and my creativity is back._\n\n_A year or so ago, I was in a very different place. I was lucky if I ate one dinner a week with the kids. My cat couldn't even depend on eating before 10 pm if I was feeding him. The plants in my office were dying, and I hadn't had sex in six months. By the time I realized that I was too stressed even to like my job anymore\u2014much less enjoy my life\u2014my family already knew that I was unhappy and unfulfilled. But their disapproving looks barely registered on me._\n\n_Then my company\u2014a maturing startup\u2014instituted a sabbatical program and I qualified. Actually, I was sort of nudged toward it by management, and they were helpful in providing my full salary and benefits. Together we created a plan for covering my work while I was away, and it turned out to be great training for more junior members of the team._\n\n_I had fought tooth and nail for that job, and I wanted to keep it and to love it again. The sabbatical was the answer. On my four-month sabbatical, I cleared some of the clutter from my house and life, reassessed my priorities, traveled first with my family, then alone on an adventure, and worked on my relationships. When I returned to work, the company welcomed me back and\u2014armed with my sabbatical lessons\u2014I became a better, more focused employee who really does love her job._\n\n* * *\n\n### REBOOTING YOUR LIFE: TWO PATHS\n\nFor most people, the first and most important questions in preparing for a Reboot Break are: Do I want to take a break from my current job and then return there? Or will I leave my position and, when the sabbatical is over, look for something new?\n\nThese are the two sabbatical paths:\n\n\u2022 A **workplace sabbatical,** in which you leave your job for a time of renewal and then return to the same job.\n\n\u2022 A **between-gigs sabbatical,** in which you leave your current job and take a break before moving on to a new job or different line of work.\n\nWhich may be right for you?\n\nOf course, not everyone's time off fits neatly into one category or the other. Some people leave their jobs with every intention of going back, and then decide not to return. Some people arrange their next job before leaving their current one, scheduling the sabbatical for the months in between. Some even retire and then figure out that they want to go back to work, creating an unexpected sabbatical.\n\n#### THE WORKPLACE SABBATICAL\u2014FREEDOM WITH A SAFETY NET\n\nAre you ready for a break from your job, but committed to the organization and want to return to do bigger and better things there? Are you happy with your career choice and want to stay in the track you are on, but with time out for personal renewal, or more education, or an adventure? Are you a few years from retirement and want a break to explore what you'd love to do in retirement?\n\nIn all of those scenarios, it may be that a workplace sabbatical is right for you.\n\nEach year, thousands of Americans leave their jobs temporarily for a much-deserved break of months, a year, or longer. How do they do it?\n\n##### _Formal Programs_\n\nBecause relatively few of us work for companies with formal sabbatical policies or know someone who's taken substantial time off\u2014and because our culture tells us that two weeks of vacation is enough\u2014we tend to think it's impossible. **In fact, scores of organizations, from multinational corporations to tiny non-profits, offer sabbatical programs. And those that don't have official policies are making deals with their employees that allow them much-needed time to reboot their lives.**\n\nThe Families and Work Institute, a non-profit research group, surveyed 1,100 companies with fifty or more employees for its 2008 National Study of Employers and found that 24 percent of companies with fewer than one hundred workers and 14 percent of companies with one thousand or more workers allowed paid or unpaid sabbaticals of six months or more.\n\nIntel, based in Silicon Valley, California, has had a sabbatical program for fifteen years. More than 69,000 employees have taken time off to date. Everyone in the firm is eligible after seven years, regardless of level. It is a two-month program to which employees may add four weeks vacation time, so many end up with three months off. While people are on sabbatical, their colleagues take up or redistribute their work.\n\nKeith took two sabbaticals at Intel and benefitted greatly from both. The first gave him time to be with his family and travel. The second helped him make some decisions about the next phase of his life.\n\nHenry, also an Intel employee, said, \"It is a great break. No one checks email and even if you do, you don't have to solve the problem. It is a great way to free yourself from the responsibilities of work for reinvigoration.\" Both Keith and Henry feel the program keeps people coming back to Intel and broadens the skill set of the workforce. The process of delegating responsibilities and empowering others to do the job of the person on sabbatical has led to a much more diversified and effective employee group.\n\nIntel employees are no different from the rest of us. Their jobs are demanding. Their co-workers depend on them. And Intel is the same as any company: competition is intense; it can't afford to let projects go undone.\n\nSo how does it work? It's not as hard as it sounds. First, the employee has to plan. He tells his manager of his intention to take advantage of the program. Together, they decide on the length and timing of the sabbatical. He works with management and co-workers to ensure that his job gets done while he's out. He commits to a date when he'll return.\n\nThough sabbaticals certainly make employees happier, companies don't grant sabbaticals to be nice. They do it because sabbatical programs make good business sense. They make companies stronger. Employees come back refreshed and recharged, with new energy and creativity and gratitude to the company that gave them a sabbatical. Organizations get a periodic infusion of new blood while retaining the work experience and wisdom of their seasoned employees. And staff members who cover for those on leave have opportunities for meaningful professional growth.\n\nCommonfund, an asset management company based in Connecticut, has about 170 employees and a paid sabbatical program that allows for eight weeks off. The first sabbatical is given after ten years of service, and then one is given for every seven years after that. Vacation time can be added to the sabbatical time. Employees must complete forms that describe the purpose of the sabbatical, and they must create a corporate coverage plan for while they are gone.\n\n\"The employees are great at covering for each other,\" Alyssa Kraft, Managing Director of Human Resources, says \"because they work as teams.\" Commonfund is instituting changes to the program to make it mandatory for all US-based employees, from secretaries to the CEO, and to require that sabbaticals be taken within thirty-six months of earning them to encourage more employees to see the benefits of time off. \"All employees have come back,\" Alyssa says, \"They found out it was a great way to rejuvenate themselves, to refresh, and to return to work with new insights and ideas.\"\n\nSabbaticals are also great recruiting and retention tools. They breed employee loyalty, which begins with the attraction of the sabbatical program and is reinforced by the sabbatical experience.\n\nConsider Andy, a marketing manager at a mid-sized biotech company. He took a company-approved sabbatical to travel. After six months hiking the Himalayas, exploring Eastern cultures, eating exotic foods, and meeting new people, he was excited to return to work. He missed his co-workers, whom he considered friends.\n\nWhen he sat down at his workstation that first day back, Andy found that he could see the problems his team faced with fresh eyes. During his first few days, he was able to break through several challenges that had stumped his teammates and his boss. When a new project landed on his desk, he found his creativity was heightened. He worked more quickly and noticed that he was enjoying the intellectual stimulation. Work didn't feel like work.\n\nAndy's boss, Eleanor, loved his new energy and ideas. But most exciting was that Andy wasn't a new employee. He held all the valuable knowledge, work experience, and wisdom of a seasoned worker. When Eleanor commented to Andy on his improved performance, he said he thought the whole team had been improved. Eleanor agreed. While Andy was out, she had seen his co-workers take on new responsibilities, making the whole department stronger and more flexible. Now Andy could delegate more work, giving him more time to be creative and strategic.\n\nEveryone knew that Andy had been on a company-approved sabbatical. Because they were working with someone who had actually done it, new employees who were told about the program when they were hired became more interested in it. In his review, Eric, an administrative assistant in Andy's department, told Eleanor how important it was to him to work for a company who understood people's need for time off. He said he was committed to staying with the company long enough to do what Andy did.\n\n* * *\n\n##### _If There Is No Formal Program_\n\nBut what if your employer doesn't offer a formal sabbatical program? You're not alone: most companies don't. The majority of people who take sabbaticals are granted them informally.\n\nWhen there is no formal program, the onus is on you to convince your boss to let you go. The key is to be able to make the business case: articulate how your sabbatical will benefit the company rather than focusing on why you need time off. (For more on this, see Chapter 3, \"It's All in the Planning.\")\n\nOnce the time off is granted, you'll have to negotiate the details. Approach the conversation as carefully as you would the terms of a new job. Know how much time you want to take. If you would like to be paid and receive benefits while you're gone, ask for them up front. If you don't think you're getting the amount of time you need, consider taking some of the leave without pay. (Read more about this in Chapter 4, \"Funding Your Freedom.\")\n\nMary described her sabbatical in equally glowing terms. \"At forty-three and just after 9\/11, I was trying to juggle a high-pressure career in commodity trading with my roles as wife and mother. Twenty years in the white-hot intensity of the trading floor had exhausted me and burned me out. And I was on emotional overload. My father was dying. I knew I needed to take time off for my own good, personally and professionally.\"\n\nMary took the scary step of asking for a three-month sabbatical, knowing no one at her company had ever taken one. To ease her anxiety, she first approached the company's human resources department for advice and confirmation that what she wanted to do was possible. With their encouragement and support, she went to her manager, who immediately approved her request. She was relieved, though she had been prepared to quit if they had not granted her the time off. \"Otherwise, I would have had a breakdown,\" she said.\n\nShe took the sabbatical without pay and had to cover her own benefits at the company rate, but her job was guaranteed when she came back. A woman came down from the office to do her work on the trading floor during her absence. Planning and preparation were important to insure that her reentry would be as smooth as possible with her clients, so she let them know by newsletter about her impending sabbatical. They were surprised, but in a positive way. Some were even jealous.\n\n\"I came back energized and better prepared to deal with the stress and pressures of the job and environment. The job used to own me, but now I'm in control of it. I'm a better performer, and I enjoy my work more.\" Her career stayed on track, and she was thrilled to soon be promoted to senior vice president. On the personal front, too, the sabbatical was a resounding success, as Mary learned to balance her life better. Her experience inspired her husband to take a sabbatical the following year, and together they have created a much more satisfying and fulfilling family life, individually, as parents, and as a couple.\n\nUnlike Mary's case, it may take a bit of convincing to get your boss or co-workers to agree to let you take a sabbatical and to negotiate the details to your satisfaction. This can be a challenge, but it can be managed, as described further in the planning chapter.\n\nThe other category of sabbaticals is a bigger leap than the workplace sabbatical but no less popular and no less feasible, once you decide to do it.\n\n#### THE BETWEEN-GIGS SABBATICAL: DECLARING YOUR INDEPENDENCE\n\nWhen people ask Kim what she's doing these days, she just smiles and says, \"I'm between gigs.\" People who know her know this means she's left one job and is taking time out before starting another. She loves the freedom, the excitement, and the anticipation of having unhitched herself from what for so long has defined her: her job.\n\nYou may want a between-gigs sabbatical, too. Are you ready for a job change? Maybe you are burned out where you are, or dissatisfied with your career choice? Maybe you want to go back and fulfill an old dream or explore new career ideas. Have you retired and are now restless and starting to think about an encore career? Maybe you are out of a job\u2014not at your choosing\u2014and need time to figure out what to do next.\n\nBeing between gigs is about untethering from a job, either to figure out what to do next or to take a break before starting a new gig. Some people leave to search for an entirely different career that will satisfy them more deeply. The sabbatical provides the luxury of time to step back, relax, reassess, and figure out where and what you really want to be or do.\n\nA between-gigs sabbatical offers a real sense of freedom. The job has been left behind and there are new fields to find and adventures to discover. One can spend time on special personal projects and activities before turning attention to the next gig, whether it is in the same field or a whole new endeavor.\n\nHow much time, you ask?\n\nIt varies by person (and financial circumstances), but sabbatical veterans agree that it's sensible to wait at least several months or even a year before actively looking for the next gig.\n\nIf the sabbatical taker doesn't know what the next gig will be, or even what field she or he will be in, a between-gigs sabbatical can have more of an open-ended quality than a workplace sabbatical: \"Carry me away. Let the tide take me where it will. Let's see how long it takes to figure out my next career step.\"\n\nBeverly needed more than just a break from working. A California pediatrician, she was becoming more and more disillusioned with the healthcare system, which was preventing her and her colleagues from giving their patients the attention they needed. So she left her practice.\n\nIt was a wrenching decision that took years to make, but she finally did it.\n\nBeverly used the time off to get a master's degree in public health. She also traveled, visited friends, and connected with her grown children. She honed her medical skills and stayed current professionally through committee work at the local children's hospital.\n\nToday Beverly is back at work, consulting and working on committees. She's still looking for a full-time job, but she's awaiting the right opportunity that will allow her to truly make a difference in healthcare policy.\n\nBetween-gigs sabbaticals also can bring surprises. Jason, as the manager of an art-framing store, was uninspired in his work, and he was willing to sacrifice income for a job that made him happy. While he was between gigs, he spent time with his brother, who was very ill. One day when he was massaging his brother's shoulders, a light bulb went off. He could be creative with his hands through massage. Not long afterward Jason enrolled in massage school.\n\nToday, Jason makes less money as a massage therapist than he did as an art framer. But he knows his sabbatical led him to a happier and more satisfying life. \"Life's too short to work every day at a job that you don't like,\" he said. \"And you can't think about what you want to do next while you're working. Being between gigs without a set job to go to can be daunting. But it can be exhilarating, too.\"\n\n### THE UNEXPECTED SABBATICAL: LEARNING TO LOVE YOUR LAYOFF\n\nBecause of ups and downs in the economy and employment trends that have made it much less likely that a person will work forever (or for many years) for the same employer, many of us will experience some kind of unexpected layoff or furlough in the course of our careers.\n\nLosing a job can be like having a life raft snatched away. Your first reaction is survival: grab another job, quick! Within a few hours of getting the news, you're dusting off your resume, making calls, looking at the want ads. The search is on, and it is overlaid with extraordinary worry about what comes next.\n\nThousands of good, talented people find themselves in this situation. It is one that often raises a lot of questions: Should you get more schooling or training to enhance your credentials? Should you move to a city with more job opportunities in your field? Should you use this opportunity to change fields?\n\nThe best solution might be to take a short time out. You probably have been working since you were twenty-one or younger. If you build in a few months to reassess and figure out what to do before you start looking, you may be more likely to end up in a job you find rewarding.\n\nHow you use that time-off period will affect the longer-term success of your career and life. Many employers provide outplacement services for their employees to assist them with the sudden transition, but outplacement firms we interviewed say that many of the people entering into their programs and the job market are unfocused and not ready, and therefore they don't present themselves in a confident way. They agree that many are in need of an emotional break and a time to reflect on what they really want and need to do next. Jumping at any job is not the answer. It wastes time, tarnishes your resume if it doesn't work out, and reinforces any negative feelings you might have about being unexpectedly out of work.\n\nYou also may need time to heal wounds. On a Wednesday in June, Barbara was dismissed as president of a corporate division, leaving her feeling hurt and humiliated. In shock, she moved to her country house, deciding to put off any serious job hunting until the end of January.\n\n_The beginning period was very hard. I don't know where the time went. I got up early every day and went to the gym every morning. If I got one thing done in a day, it was a lot. I was very weepy. I had lost my identity, and I went through a people-controlling stage. I was in their faces. If I couldn't control what was around me, I was going to control others. I had such anger over the loss of my job._\n\n_Then I read a book on the five steps of transition, and it led to a lot of soul searching. The key question was, \"What would make you click for the next stage of your life?\" I needed to move from making my mark to doing something that made me happy._\n\n_Gradually, I got my ego back. I traveled to Italy and France for two weeks to see good friends. I reworked my bio and began using the services of an outplacement company. I formulated my strategy, talked to people, assessed my opportunities, and connected with retailheadhunters and women entrepreneurs. Almost a year after I lost my job, a recruiter who received my resume said that I would make a great recruiter. After a trial run in that company, I decided to take a risk and go into this totally new field._\n\n_I am comfortable in my own skin in this new place of self and work. My sabbatical was a gift. Here's my advice: Do it. Time goes by very quickly. It takes three to six months to decompress from the rat race. It then takes three months to feel normal, especially after a job loss._\n\nLisa was a computer programmer who had worked for the same company for seven years. Her job had become more and more stressful, and she knew the company was reorganizing and cutting costs.\n\nEven though she had seen it coming, Lisa felt angry when her manager told her they were cutting her position. She had worked hard and liked her job. She had been a good employee. It didn't seem fair.\n\nAfter a few weeks, the dark clouds that had been hanging over Lisa began to clear. For the first time in years, she stayed at home for hours at a stretch, giving much-needed attention to the fixer-upper she had bought five years before. She started to sleep more and eat healthier foods. She began to feel relaxed and more in touch with who she really was.\n\nLisa had always had a strong spiritual side, and she used her newfound freedom to study Buddhism with a well-known teacher. She traveled with him to Italy, Africa, and then Vietnam, deepening her practice.\n\nHer time off let Lisa detox from the bad feelings of being laid off, explore a long-held interest, and focus on a new goal: changing careers. Today, she's left programming behind and works in corporate giving. She continues to practice Buddhism and take occasional trips with her teacher. She says she's never been happier.\n\nIf you are laid off without an economic cushion\u2014such as a buyout or at least six months of severance pay\u2014the economic worries stemming from job loss can be frightening. How do I pay the bills, the college tuition? Do we cancel our family vacation? How long can I go on without a salary? Chapter 4, \"Funding Your Freedom,\" will show you ways to buy yourself a few months if you didn't receive adequate severance pay or a buyout.\n\nJoe's company was taken over, and the new company had its own person for his job. Joe had just bought a new home, and his son was about to start college. There were more bills to pay than ever. Naturally, Joe's first impulse was to grab for that life raft.\n\nBut two months into his job search, Joe had to admit that he had no interest in finding another job in human resources. He realized he wanted something totally different.\n\nInstead of sending out resumes, Joe shifted gears. He bought himself a few extra months by selling one of the family cars and making other adjustments. He used his time to learn how some of his skills might transfer to a different career, public relations. \"I was more scared than I've ever been,\" Joe said. \"But reinventing myself was exciting. I love my new job.\"\n\nLosing a job may turn out to be a blessing in disguise, providing you with the impetus to make the changes you may not have realized you wanted to make. The key is to shift your mindset from feeling like a victim to taking charge.\n\nJim had found himself with unexpected time off a couple of times in a short period. Each time his instinct was to find a new job as quickly as possible. He jumped into two different jobs, one after the other, with little thought about what he really wanted. After the second job change, he stepped back and decided to pursue his long-term dream of co-owning and running a bed-and-breakfast on Cape Cod.\n\nFinances were the biggest issue, but he was able to live on severance and accrued vacation pay while using a home equity loan to pay for the renovation of the bed-and-breakfast he bought with a friend.\n\nNot everyone is as certain about what he wants to do as Jim, but many, forced by a sudden change in their working environments, realize that taking a step back is the only way to move forward.\n\nIn summary, leaving a job unexpectedly can be the impetus to make the changes you have always wanted to make. Shifting your mindset from a victim of circumstances to taking charge and using the unexpected time to redirect your life can also be seen as an unexpected gift. The key is to take control of your life!\n\n### IS A SABBATICAL REALLY FOR ME?\n\nTaking a sabbatical sounds great, right? Why, then, do so many people stand at the edge of a divide between doing a sabbatical and not doing it, wondering if they should take the leap? They are so tempted as they stand there looking at the possibilities, but they are rooted to the ground in deep thought and a certain amount of turmoil. There are decisions to make and details to handle. They think it may be easier not to take the big step. They wonder if they will lose their identity while not working, and lose their edge when they return to work. Somewhat like the Hebrew slaves mentioned earlier, people today sometimes choose to continue working simply because it's easier than taking up the challenge of freedom.\n\nMaybe this sounds like you. It was certainly us.\n\nSome skeptical onlookers may question why someone would step out of a job, especially a job they like and value, for a few months or a year, for some loosely defined purpose, and then return.\n\nOne might ask the same of a parachutist: \"Why would someone jump out of a perfectly good airplane?!\" To the sport parachutist, the answer is obvious: he or she doesn't doubt that the experience is worth the leap from the security of the plane. Careful planning has preceded the jump. There is always some risk, but the exhilaration, confidence, and focused determination gained from this experience are larger and more important than the risk. In fact, without the risk, the meaningful experience sought cannot be found.\n\nNo one can tell you when the time is right to reboot your life. What we can tell you is this: **whenever you decide to do it, it will be worth it.** When you begin to live your life differently after your time off, spending more time doing what you love, you'll feel its power in your life. When you start a new career that you'd never dreamed of when you walked out of your old job, you'll know it was the greatest gift you ever gave yourself. When you return to the office brimming with energy and ideas, you will see what a difference rebooting your life can make.\n\nThe trick is to believe what the parachutist knows: it's worth the risk.\n\nAll you need is to give yourself permission to take the leap.\n\nYOU MAY SAY, \"My life doesn't need rebooting. It's fine. I'm a multitasker, and I enjoy my long days and the rush I get from pressure.\"\n\nYes, but ask yourself: \"Am I as strategic and successful as I could be in my work? Am I as thoughtful to my family and friends as I might be? Am I too focused on one area of my life?\" Taking time out to change and do new things can bring a whole new perspective on how to approach your work\u2014and your life.\n\nOur experience and research show that life can be better after a break to refresh, get some space, and recharge. Furthermore, in today's workplace, people who take a sabbatical have an edge. They have figured out how to balance their life and be better at what they do.\n\n###   EXERCISES\n\n##### _Exercise 1-1: Thinking It Through_\n\nUse a journal or notebook for answering these exercises and those at the end of each chapter. Try to answer quickly, almost stream-of-consciousness, to get your ideas down on paper.\n\n\u2022 What prompted you to read this book?\n\n\u2022 Why do you think you need to take time off? Write down what stresses, desires, or pressures are driving this decision.\n\n\u2022 Which kind of sabbatical (workplace or between-gigs) is right for you?\n\n\u2022 What are your fears, obstacles, and anxieties about taking time off?\n\n## | CHAPTER 2 |\n\n## _Types of Sabbaticals: The Why and What_\n\n'Don't be too timid or squeamish about your actions.  \nAll life is an experiment.  \nThe more experiments you make, the better.\"\n\n\u2014 _Ralph Waldo Emerson_\n\nSabbaticals, or Reboot Breaks, typically fall into seven types:\n\n\u2022 Career enhancement\n\n\u2022 Course-correction or reinvention\n\n\u2022 Family-related\n\n\u2022 Emotional healing\n\n\u2022 Volunteering\n\n\u2022 Life-enhancement\n\n\u2022 Pre-retirement\n\nThese categories are not hard and fast, though, because **behind every sabbatical decision are uniquely individual motivations, and the mixture of motivations for sabbaticals makes them hard to fit into tidy boxes. As you read, you'll probably recognize yourself in one or more ofthe types and find yourself thinking about your own situation and motivation.** The idea is not to choose a type\u2014though that may happen. The idea is to begin to understand why a sabbatical appeals to you and just how ready you are to reboot.\n\n### CAREER-ENHANCEMENT SABBATICAL\n\nA career-enhancement sabbatical may be right for you if:\n\n\u2022 _You need a graduate degree or special training to move ahead._\n\n\u2022 _You want to explore something new or a new aspect of your current specialty._\n\n\u2022 _You enjoy learning and want to take your knowledge to the \"next level\" or to learn something new and interesting._\n\nNancy's first Reboot Break was for career enhancement.\n\n_I was thirty and had been working for seven years. At that point, I was on Capitol Hill, where I first worked for the Senate Armed Services Committee, then moved to the Congressional Budget Office. I needed an advanced degree to move ahead in my career, and my boss suggested that I get a master's degree from the Kennedy School of Government at Harvard. I was thrilled to be accepted and to get a student loan. I resigned from my job and left Washington DC behind temporarily. I returned to find a job at the Pentagon directing an office and working on Middle East\/Persian Gulf issues. My Reboot Break was a wonderful ten months and a great career boost._\n\nA career-enhancement sabbatical can be taken from the workplace or between gigs, to burnish credentials for the next career move, like Nancy's. The motivation may be higher pay, more job satisfaction, or a climb up the ladder to a position with greater responsibility. It can also be an excuse to take time off to come back a better, recharged employee. Inevitably, people who take these Reboot Breaks add personal experiences to their time off, such as travel or more time with the family.\n\nSandra is a grade school teacher in the Bronx, New York, who took a one-year educational sabbatical at age forty-seven to increase her salary and spend more time with her young son. She had been working for seventeen years in the New York school system, which allows a one-year sabbatical with partial pay after fourteen years of service.\n\nTo obtain her master's degree, Sandra took three classes each semester, concentrating the classes all on the same day so that she would have the other days free. She wanted to be at home for her fourth-grader and be his class parent. Also, her husband had some health issues, and for several months he was at home out of work. She told us, \"I was so glad to be completely free to take him to the doctor and be there with him. Our family life and bonds benefited so much from this break.\"\n\nCareer-enhancement sabbaticals can be very satisfying, as they offer a real change of lifestyle\u2014away from work\u2014that encompasses more than just professional improvement. Instead of squeezing in night classes while they work during the day, people who take career-enhancement sabbaticals are able to engage fully in the academic experience.\n\nLauren's Reboot Break and MBA were fully paid by her corporation. She said of this time, \"It was wonderful being a full-time student and being able to separate who I am from the job I do. Not having a work identity while I was in school was very empowering.\"\n\nSome corporations let their up-and-coming executives spend time in Washington DC for a stint in the corporate affairs office. Others allow employees to spend time in the government, such as being a White House Fellow. John, a former executive with Bethlehem Steel, spent time in Washington DC when it was important for the steel industry to understand and influence policy in Congress. \"It was one of the best experiences I had in my career, enabling me to see how policy was developed and influenced. It was valuable not only in my Bethlehem Steel career, but later when I moved into other industries.\"\n\nSometimes a break is essential to creating the space to concentrate and make the career enrichment actually happen. As people learn and explore more, they often become more insightful about what they really want to do.\n\n### COURSE-CORRECTION OR REINVENTION SABBATICAL\n\nYou may be looking for a course-correction sabbatical if:\n\n\u2022 _You are bored with your job and dream of doing something else._\n\n\u2022 _You were discouraged when you were younger from following your passions and doing what you really wanted to do._\n\n\u2022 _You feel that you have a \"calling\" but you're not sure what it is._\n\nCathy's first Reboot Break was a course-correction from the academic to the corporate world in the 1970s.\n\n_I had always thought being a professor would be the ideal position for me. I would teach subjects I loved to students who were attentive in a bucolic setting with like-minded people._\n\n_The reality was much different. As an assistant professor, I did like my subject (management and marketing) and my students. But the pettiness and politics of academic life came as a huge surprise. So much time was spent on things of so little importance. So little was accomplished when there were so many major issues to address. Most disappointing was the lack of resources and time for research, which was what I really wanted to do._\n\n_My first Reboot Break, which included traveling and lecturing in Asia for a year, was planned as a way to get some time and space to decide whether I wanted to stay in the academic world and consult, or take the leap to the corporate world. I had sent out my resume to several corporations before I left for the year. Dun & Bradstreet, for whom I had consulted, was one of them. When I got back, they had a position waiting for me. They told me that my confidence in taking a year off demonstrated to them that I was exactly the kind of person they wanted to head a new entrepreneurial division of the company._\n\n**Many people go to work every day with a terrible secret: they know they've chosen the wrong career, but they don't know how to get out of it.**\n\nA course-correction or reinvention sabbatical can help you explore possibilities for change and lead you in a new or even unexpected direction. In some cases, the change is returning to an old dream. Maybe you always wanted to be a painter or a writer, but your parents steered you toward law because it was more practical. Or maybe you wanted to run your own small business, or be a school teacher, or use your foreign language talents, but ended up doing something else.\n\nBrian always loved music. He played drums in high school and in college, where he majored in music. After college, Brian discovered he had a talent for stocks. He became involved in finance and moved from Virginia to New York City. But he never forgot his dream. \"I was successful, but I wasn't fulfilled,\" he said. \"Music was calling me.\" So he took a Reboot Break to figure out how to bring music back into his life. Today, at twenty-nine, he is a substitute music teacher in the Virginia public schools, gives private percussion lessons, and does gigs with orchestras. The next step toward his dream of performing with world-class orchestras is a graduate degree in classical music.\n\nOf course, not everyone is like Brian. Many people aren't sure what they really want to do. They just know that it is not what they are doing. Time off for course-correction helps you leave behind the dull ache of doing a job that you just don't like and make room for a new passion. It creates the time and space to discover what kind of work feels right, and it can lead you in a direction you never expected.\n\nMarco, a young lawyer, knew law wasn't his passion. After just one year, he left his high-powered law firm for a Reboot Break. His time off allowed him to explore his interest in business. Not long after the break, he founded a tech company. But his work wasn't done. Two years later he took another course-correction sabbatical. This time he followed his new dream of working in foreign policy. Today he's a policy analyst in Washington DC.\n\nSome people try something new on their Reboot Break only to return to what they had been doing before. That doesn't mean the time was wasted. The opportunity to explore a passion can go a long way toward reducing the stress of feeling as though you've missed an opportunity or that you really should be doing something else.\n\nLinda, a therapist, used her Reboot Break to live overseas and write, then start a small business. The business didn't succeed, but she loved trying it. Today she's returned to her work as a therapist, and in her free time she writes plays and travels.\n\nIf you are feeling as if you've handed your life over to your job, or you are resentful because you think your job is keeping you from fulfilling your dreams, a course-correction sabbatical can help you feel more satisfied or find a new direction. You might find that what you needed wasn't a career change, but the freedom to expand your life outside of work. When a passion has been ignited, sometimes the old job doesn't seem so bad, especially if it helps pay for your new hobby.\n\n### FAMILY-RELATED SABBATICAL\n\nA family-related sabbatical may be right for you if:\n\n\u2022 _You are finding it more and more difficult to balance your work and home life._\n\n\u2022 _You wish you had more time to spend with a loved one who has special needs or is ill or dying._\n\n\u2022 _You have family goals you aren't able to meet while working full-time._\n\nNancy took a family-related sabbatical at thirty-five to have children.\n\n_My U.S. Army husband was being transferred from Washington DC to Europe, and I agreed to leave my White House national security job to accompany him. My number one goal was to start a family. Much toour delight, we had two daughters in the next three years. I was at home with them for their earliest years in Germany and Japan, so I had the wonderful gift of time to bond with them._\n\nA family-related sabbatical allows you to let go of the guilt and devote time to the people who need you most. It may not fix all the problems, but it can give you some relief when you're feeling pulled in too many directions. It can let you listen to your heart when it tells you that nothing is more important than being there for the ones you love when they truly need you. Unfortunately, our loved ones' timing isn't always ideal. Maybe you're up for a promotion when your father becomes ill, or your teenage daughter is failing school right after you win the big account.\n\nAlison took a workplace sabbatical in 1994 at age forty-eight from her financial services firm, where she was one of the highest-ranking officers. The primary purpose was to be more involved with her son during the summer before he started high school. John, a high-functioning autistic teen, was completing junior high. It had been a tough time for him. Alison and her husband wanted to help their son prepare to enter high school. She wanted to be with him to help work on his coordination, self-confidence, and self-sufficiency, which would in turn help his ability to make friends and fit in.\n\nShe took her Reboot Break as family leave under the Family and Medical Leave Act (See Chapter 3). Her bosses were reluctant to give her the time, but she knew they had no choice because of the law, which was new at the time, and they ultimately agreed. Alison's summer with her son was precious. She was with him every day, just spending time together or doing special projects. One interesting project they did involved sorting their closets and drawers, which hadn't been done in years. She turned the task into a fun, joint activity of 15-30 minutes each day to get rid of stuff as a team. She even taught him to cook.\n\nJohn blossomed into \"more of a guy\" as he became more coordinated and better at sports under the tutelage of a trainer. Alison worked with him in other ways too, such as his independence skills and how he dressed. They took a trip to visit her dad in Nebraska, but she had John stay longer and then navigate his way home alone through two flights and a plane change. He became more self-confident and knew that he could master the change ahead of him called high school.\n\nReturning to work, Alison felt refreshed and energized. She looks back on the Reboot Break as one of the highlights of her career and a key to a burst of energy and productivity. And, most importantly, it helped her son. Alison says, \"The Reboot Break was really the right thing at the right time.\"\n\nWhile family-related sabbaticals provide the flexibility to do what is necessary for a family member, they may also benefit the sabbatical taker. Many find the experience life changing, an impetus to reorder priorities.\n\nAfter Jimmy's wife died, he and his children took a Reboot Break so they all could heal, travel, and just be together to re-establish their lives.\n\nSometimes people don't take their Reboot Break for family reasons, but life circumstances bring the family more into focus. Leena was on a Reboot Break when her father suffered a stroke. It was a blessing that she had the flexibility to be able to crisscross the Atlantic Ocean several times to be with him and help manage his care.\n\n### EMOTIONAL HEALING SABBATICAL\n\nYou may need an emotional healing sabbatical if:\n\n\u2022 _You recently lost a spouse, partner, parent, child, or other loved one._\n\n\u2022 _You are divorced or separated from a spouse or partner._\n\n\u2022 _You are a survivor of an illness or situation that was traumatic or life-threatening._\n\n\u2022 _You recently experienced an upsetting job loss._\n\nDifficult emotional times are a natural part of life. Romantic relationships fall apart. People we love die. Life deals us a hand that we didn't expect, and we do our best to cope.\n\nAll the while, we're plugging away at our jobs, rushing to meetings, traveling, pushing papers. Sometimes the work is therapeutic, distracting us from our crisis. Other times we wonder how we can keep smiling at our co-workers, customers, and clients, acting as though things are okay. We need to take time off.\n\nAn emotional healing sabbatical is about nursing one's wounds and transitioning to a new \"place\" with lessons learned. The wounds may not be entirely gone at sabbatical's end, but time and a new understanding of life will have made them more bearable.\n\nIn her popular book\u2014now a movie\u2014 _Eat, Pray, Love,_ Elizabeth Gilbert set out to find balance in her life by taking a Reboot Break. Gilbert plumbed the depths of her soul and told the world about it. Readers are the beneficiaries of her struggles and insights, of her pain when she left her marriage and suffered through the stages of divorce. Also, she was seeking her own truth about her place in society and how she would live her life. She took refuge and sought answers in food, prayer, and love in three different countries, and eventually she found inner peace and a new understanding of herself.\n\nIn the classic book _Gift from the Sea_ (1955), Anne Morrow Lindbergh's insights from her time away at the beach to seek balance are about understanding how recognizing and embracing the emotional threads of life will weave in more strength, texture, richness, and clarity. Her lessons are strikingly similar to those we have gleaned from our own Reboot Breaks and from our many interviews with both women and men.\n\nLeena shared the story of her time on a Reboot Break. Her twenty-month sabbatical journey combined emotional healing, searching for her next job and place to live, and surprises along the way.\n\nLeena had not planned on taking a Reboot Break, but at thirty-six, when she sold her Internet-based food delivery business in London, she realized that she needed to take time off to heal from a broken heart. So, a professional circumstance and an emotional need came together and spelled \"Reboot Break.\" Now, she would have time to heal the wound and to figure out her next professional calling. She even called into question where she lived. Untethered from her business, she was moving into a period of being open to what could happen next, both personally and professionally.\n\nShe first returned for several months to Lebanon, where she had grown up. She had been away for fourteen years, except for short visits, and longed for the beautiful Lebanese mountains and exciting Beirut. Leena wanted to catch up with the family and friends she missed so much. She hadn't realized how low she had sunk emotionally and was thankful for the time away from daily routine and work pressures to heal. It helped so much to be back in the nurturing bosom of family and friends. She read self-help books, practiced yoga, and worked on moving past that \"dark place.\" She began to feel more hopeful about the future.\n\nLeena traveled to Madrid to stay with friends. There, she took a painting and drawing class, which she had always wanted to do. She worked on improving her Spanish, too, which enabled her to start meeting new people and building new relationships. The more time she spent in Madrid, the more she began to see that this would be a good place for her to live. An interesting work idea came along, and she began to make concrete plans to move to Madrid.\n\nWith this newfound direction and plan, Leena returned to London to sell her apartment and gather her things for the move. As luck or kismet would have it, she met someone while back in London. His name was Jos\u00e9. She found herself facing the kind of situation that happens to many a sabbatical taker. Because she had opened herself up to life in a new way, she now was presented with a new opportunity\u2014and the challenge of thinking deeply about what was truly important to her. She felt torn and had to decide what she was going to do: continue in an orderly fashion with her plan to move to Madrid, or explore her relationship with Jos\u00e9.\n\nLeena, the former \"strictly business\" businesswoman, followed her heart. It helped that she was still on her Reboot Break. She had the flexibility to shelve the Madrid plan and extend her time away from work. The relationship blossomed and, after a time, she and Jos\u00e9 married and now live in the Washington DC area.\n\nLeena's experience demonstrates that taking the time to reflect and allow for new thoughts, ideas, and experiences can enrich life with new possibilities. In more dramatic terms, one could say that taking time off can nudge the unfolding of one's destiny, one's future. Leena is certain that without her Reboot Break, she would not have been open to building a relationship with Jos\u00e9. First, her heart may not have healed. Second, she would have been tied to a job that would not have afforded her the time to explore and pursue the relationship and discover what she really wanted and needed.\n\nLeena also transformed her view of work and changed her career direction. She now works at a non-profit organization, and her life's work\u2014her found passion\u2014is promoting better understanding between Muslims and non-Muslims and being involved in peace projects in the Middle East.\n\n### VOLUNTEERING SABBATICAL\n\nA volunteering sabbatical may be right for you if:\n\n\u2022 _You long to make a difference in the lives of others._\n\n\u2022 _You have a particular skill you'd like to use to make the world a better place._\n\n\u2022 _You want to travel to distant lands, but traveling as a \"tourist\" doesn't appeal to you._\n\nRita says that volunteer work has always been one of her passions. It took a Reboot Break to satisfy her desire to become more deeply involved.\n\n_In the past several years when I was working eighteen-hour days and traveling constantly, something had to give, and it was my ability to commit time to volunteering. Though I contributed financially to several non-profit organizations and was on the board of one, I started to crave spending more time on work that seemed so important to me. I was quickly coming to the conclusion that I needed to make volunteering my work, rather than having it be the thing I did on top of a full day. Whenmy company announced that it was moving its headquarters to another part of the country, it was the perfect time to take a Reboot Break._\n\n_I used the time to immerse myself in microfinance, a field where the results are immediate, tangible, and scalable. I was soon asked to become chair of Pro Mujer, a non-profit organization that gives $170 loans to women in Latin America whose families earn under $2 a day. Pro Mujer also provides business and self-esteem training and health services for the whole family. The 99.9 percent repayment rate shows that there is a real need._\n\n_I felt good about my efforts and contributions. I was able to bring my business knowledge and experience to bear on several strategic areas for the organization. But I felt even better when I traveled to our banks. Before doing so, I enrolled in Spanish classes so that I could converse with our clients. I complemented the classes at home with Spanish immersion weeks in Antigua, Guatemala, and Oaxaca, Mexico. I traveled to our existing programs in Bolivia, Nicaragua, Peru, and Mexico and went with the founders to Argentina to set up a new program._\n\n_My life is so much more enriched by all the women I have had a chance to meet. I cried when hearing their stories, with tears for the hardship of their lives and tears of happiness for their successes._\n\nMany people long to spend time volunteering. They want to contribute in some personal way to making the world a better place, to making a difference in the face of relentless problems. Many of us take a weekday night or Saturday afternoon to mentor a child or clean litter from a highway or debris from a rundown part of town. But some of us want to spend more time and have a stronger engagement. A volunteering sabbatical can be the answer.\n\nFor us and many of our interviewees, the break was a time to give back, in small ways and large. Some of us served on non-profit boards, while others wanted the more hands-on experience of feeding the homeless in their own communities, providing medical care in Africa, or establishing schools at home or abroad.\n\nVolunteering sabbaticals can change people in profound ways. Sometimes the transition back to your old life can be jarring. When Nigel returned to his law firm in the UK after volunteering with his wife in Trinidad and Tobago and Ecuador, he deeply questioned his priorities and those of his colleagues. Sitting at his desk, he couldn't help but think, is all this really that important? Others were so changed by their volunteer experiences that they quit the corporate world altogether for a life of non-profit work and volunteering.\n\nA number of corporations encourage their employees to take volunteering sabbaticals, which gives the employees rewarding experiences and the company a way to contribute to social responsibility.\n\nThe \"new volunteerism\" is a rapidly growing industry of customized travel programs and accommodations in places around the world where you pay to volunteer. It combines cultural and learning experiences with making a positive impact. Many websites offer listings of meaningful volunteer opportunities for people who want to give back during their Reboot Break. (See Resources in the Appendix and further discussion of volunteering in Chapters 3 and 6.)\n\n### LIFE-ENHANCEMENT SABBATICAL\n\nYou may be ready for a life-enhancement sabbatical if you:\n\n\u2022 _Feel uninspired by your work or your life._\n\n\u2022 _Feel out of touch with your dreams and interests._\n\n\u2022 _Feel the need to connect to life's deeper meeting through spiritual exploration._\n\n\u2022 _Just crave meaningful time for yourself._\n\nAlthough Jaye's Reboot Break captures elements of several sabbatical types, life-enhancement was her primary objective.\n\n_I had started and built a company with three partners. Four years later, one of them retired, which caused me to take on roles and responsibilities that I didn't enjoy. And I enjoyed doing it twelve hours a day even less! After six years of forcing myself, I finally raised my hand to make amajor change. Not only was I unhappy with what I was doing, I had lost touch with those I cared most about, and especially myself. I needed a break. On sabbatical, I was able to have enriching experiences that were otherwise elusive: see long-lost friends, spend quality time with family, and travel a lot to reconnect and to rediscover myself._\n\nReboot Breaks don't have to be motivated by a deep need or noble purpose. There may be no trauma to recover from or ambition to follow. Sometimes a Reboot Break happens just because you want to take some time for yourself and see what it feels like to stop working for a while.\n\nGeoff told us about his experience.\n\n_My Reboot Break was a journey within myself. I had taken time off of work in my thirties to search for a better work\/life balance. I started to think about how I wanted to structure my life in the future. In my travels, I met up with an old friend who had moved to Australia. I proudly told this self-made man who had retired at thirty-five that I had just been made a vice president. He responded, \"That's great. So tell me, are you happy?\" I spent the rest of my break working on how I could be certain to answer that question \"Yes.\" I don't have all the answers, but I'm closer. I learned that you don't need to come out of a Reboot Break with all the perfect answers. It's okay to just recharge your batteries._\n\nLife-enhancement sabbaticals often have an important dose of spiritual enlightenment as people seek ways to nourish their souls and get to know themselves better. It can be in small doses or as a major focus. On Marco's two course-correction sabbaticals, he felt that introspection was the most important activity. \"Without logging a lot of time with myself on sabbatical, I never would have been able to figure out the career that will sustain me through my life. I learned about how I want to live my life and conduct my relationships by undertaking a search for my own spirituality. That was important and satisfying.\"\n\nMany sabbatical takers find their path to self-discovery through yoga, meditation, and participating in weekend or longer programs that guide them through a process of self-reflection and spiritual exploration. **Many people are hungry for a connection to a part of themselves that is untapped and under-celebrated.** Some also want to feel connected to a larger spiritual community of like-minded people to help support them.\n\n### PRE-RETIREMENT SABBATICAL\n\nA pre-retirement sabbatical may be right for you if:\n\n\u2022 _You want to \"test the waters\" of retired life._\n\n\u2022 _You know you'll need to work during retirement and want to figure out what kind of work you'd like to do._\n\n\u2022 _You know what you want to do during your retirement\u2014you're going to start a business or build a house or become a yoga instructor\u2014and you want to get a head start on your plans._\n\nCathy's sabbatical in 2007 was a pre-retirement sabbatical.\n\n_I, like many Baby Boomers, was not ready to retire. Rather, I see myself working well into my seventies, but with more time for other things. I wanted more control over my work life and to follow some of the passions and dreams I had left behind in my last career phase._\n\n_I used the eight months off to get more involved with philanthropy and work behind the scenes on fundraising and policy development for a presidential campaign. I continued to give presentations in the financial services and technology fields so that I kept up connections and continued to learn._\n\n_The most exciting parts of the time off were closer to my passions of travel and design. I was building a new guest house on my property in Santa Fe and loved every minute of the planning. I also traded houses with an acquaintance and spent a month in France with friends, some from my business connections. What gelled from those two experiences was how to use my guesthouse and house in Santa Fe as places for talented and engaged people to meet and work on some of the problemsfacing the United States in its need to be globally competitive. I'm back at the Santa Fe Group now, working on how to turn that thought into a business opportunity._\n\n**As people approach retirement age, they often begin to think about what they will do in retirement and want to position themselves for it.** A pre-retirement sabbatical in your fifties or sixties can help you explore what to do later. You may look forward to the extra time and freedom of retirement and want a taste of it in advance. Or you may contemplate retirement with some trepidation and want to lay some groundwork for it. A pre-retirement sabbatical can be used to try volunteering, take a class or an extended trip, start a hobby, or anything else that offers a taste of what may be in store after leaving a fulltime career. A pre-retirement sabbatical can be taken either from the workplace or between gigs.\n\n### PHASED RETIREMENT: THE NEW WAY\n\nBaby Boomers and others are foregoing old-fashioned retirement. Instead of ending their careers only to spend their time in leisure, they are increasingly spending their post-retirement years working part-time or volunteering. There are seventy-eight million Baby Boomers in their sixties now, at or near traditional retirement age, but polls indicate that only about thirty percent actually plan to quit work. And the number of those who plan to keep working at least part-time is growing, according to a 2008 Gallup poll. Some of this is financial necessity, but some of it is simply not wanting to retire.\n\nWe call this new lifestyle that pairs traditional retirement's relaxation and discovery with rewarding work \"phased retirement.\" It's a term we have borrowed from law firms, which have a long and enlightened tradition of allowing their older associates and partners, like Sam, to cut back their work without leaving the firm. An attorney in his sixties, Sam is still a partner but he spends most of his time working on pro bono cases and mentoring the firm's younger employees.\n\nFor some, phased retirement means leaving an old job behind for new, more interesting work. Dick \"retired\" from an active legal career at sixty-five. After some time off, he became a volunteer with the Baltimore Symphony Orchestra. Today he heads the orchestra's youth outreach program.\n\nThe new gig may be the realization of an old dream (even a reinvention), or a different incarnation of the previous career. Often, it fulfills a desire to use one's lifetime of personal and work skills in volunteering. Roberta, a retired banker, volunteers her financial skills, helping young entrepreneurs prepare their taxes. Frank, a retired government worker, donates his legal expertise at a nearby homeless shelter.\n\nMany people don't realize they want a phased retirement until after they retire. They have the idea that being on vacation\u2014for the rest of their life!\u2014will be nothing but bliss. And for some it is.\n\nBut after six months, a year, or maybe two, many think to themselves, _Is this all there is?_\n\nKevin, an Idaho entrepreneur and professor, knew he wasn't ready for traditional retirement. But he was ready to find a better balance between his work and personal life. At sixty, he resigned from his last full-time job and took time off to figure out the right balance of work and personal time, as well as the kind of work he could do that would be satisfying. Today Kevin spends three days a week counseling small business entrepreneurs and teaches one college-level business course. He also manages an \"angel fund\" that he created to help entrepreneurs who want to start new companies and serves as a governor of the Rotary Club.\n\n\"I'm doing these things because I have the time now,\" Kevin said. \"The angel fund keeps me connected to that community. Being the Rotary Club governor is a labor of love, and I have talent to bring to it. It's a gift that I have time to do all of this.\"\n\nPhased retirement can take many different forms. And it is truly changing the way people live their lives in their older years. Marc Freedman, in his book _Encore: Finding Work That Matters in the Second Half of Life,_ talks about what we all have to gain from people who seek active and purposeful lives long after our parents stepped off life's stage.\n\n###   EXERCISES\n\n##### _Exercise 2-1: Your Motivations_\n\n\u2022Which type of sabbatical(s) do you identify with most?\n\n\u2022Write down the issue or motivation driving you toward one or more of the types of sabbaticals.\n\n\u2022Think about the direct impact a Reboot Break break could have, and make lists of the pros and cons.\n\n## | CHAPTER 3 |\n\n## _It's All in the Planning_\n\n\"A goal without a plan is just a wish.\"\n\n_\u2014Antoine de Saint-Exup\u00e9ry_\n\nCongratulations! Now that you've decided to take a Reboot Break, it's time to put together a plan to help you to make the most of it. Whether you are a non-planner, a great planner, or an over-planner, this chapter will provide you with helpful insights on:\n\n\u2022 The crucial role that planning plays.\n\n\u2022 How to pre-empt potential emotional hurdles about planning.\n\n\u2022 How to prepare for conversations with bosses, spouses, and children.\n\n\u2022 The tactical steps you will need to consider in advance for a successful break.\n\n\u2022 How planning can also enable entrepreneurs, sole practitioners, and small business owners to take time off.\n\n\u2022 Resources that will help in planning and giving you the \"gift of time.\" (Understanding the ways in which you can fund your time off is so important that we have dedicated a full chapter\u2014Chapter 4\u2014to the multitude of options and resources available to anyone considering taking time off.)\n\nNo sabbatical dreams come true by happenstance. Each dream is unique and needs first to be imagined, then planned and implemented by the one person who can do it . . . you!\n\n### WHY PLANNING IS IMPORTANT AND WHY IT's SOMETIMES HARD\n\nPlanning is the key for everyone taking a Reboot Break. It enables you to fulfill your goals, expectations, and needs for this time-off period. It lays the groundwork, ensuring that you will get the most from the sabbatical experience. It helps determine the resources needed, the things that need to be put in place, and at what stage. It leads one to think about contingencies and to be better prepared.\n\nWe recommend starting six months to a year ahead of your Reboot Break, but many people plan successfully on shorter notice. Others choose to plan several years ahead. Every situation is unique. The trick is to leave enough flexibility to be spontaneous and open to learning lessons along the way, but at the same time to think through enough details in advance to make sure opportunities are not lost.\n\nEileen planned a year in advance of taking a twelve-month Reboot Break, starting with a list of her goals, which included travel and reconnecting with friends. Although she had a terrific time and enjoyed her trip to France and Switzerland, she later regretted that she hadn't planned more of the specifics. She had wanted to take cooking classes in France, for example, but missed the opportunity because it needed to be researched and reserved in advance.\n\nThere is creative tension between planning and not planning, being over-programmed and not being programmed enough. The objective is to have structure to achieve your goals but also leave lots of downtime and room for flexibility and opportunity. Planning also does not come easily to everyone. What keeps us from planning?\n\n\u2022 Many of us are procrastinators.\n\n\u2022 We have lingering emotions of fear or guilt about taking time off.\n\n\u2022 It feels easier to us to just let things happen.\n\n\u2022 We just don't know how to plan.\n\nAll of these circumstances are normal, but they must be overcome. Push past these inhibitors and get started.\n\n### PLANNING BASICS\n\nHere are some basics for anyone starting to plan for a time away:\n\n\u2022 Identify your major goals.\n\n\u2022 Decide how much time to take. If it is an unexpected sabbatical, figure out how flexible you can be with time off.\n\n\u2022 Figure out what financial and other resources you will need.\n\n\u2022 Decide what you will need to do ahead of time to be able to take time off.\n\n\u2022 Determine whom to tell, who will be impacted, and how you will deal with that.\n\n\u2022 If you are an entrepreneur or sole practitioner, decide what planning is needed well in advance.\n\n\u2022 Decide whether travel is included, and if so, what advance planning is needed.\n\n\u2022 Do a plan outline and timelines for the Reboot Break period.\n\n\u2022 Identify fears that might get in the way and how to address them.\n\n\u2022 Schedule down time for yourself. If your goal is to get off the racetrack and smell the roses, then carving out time for yourself to think, reflect, and feel can be enough of a plan, at least for the beginning of a Reboot Break.\n\n* * *\n\n### GETTING STARTED: IDENTIFYING GOALS\n\nHere are some preliminary steps you might take in identifying your major goals and outcomes:\n\n1. Think and daydream about your time off.\n\n2. Write about it, which makes it more real.\n\n3. Talk about it to others to get reinforcement.\n\n4. Write down steps to take toward those dreams.\n\nYou know yourself. Think carefully about what you need to do to feel afterward that you made the best use of your break and special time off. What are the categories of things you would like to do? Do you have a major, overarching goal, like getting an MBA, traveling to Asia, or running a marathon? If so, what do you need to do to accomplish that? What are some of your other goals, large and small, and how might you reach them?\n\nLaura needed a break from her high-stress job. She and her husband had dreamed about traveling internationally during the ten years of their marriage. He had found a summer project assignment in Australia, and now here was their chance to realize those dreams. She jumped at the opportunity to take a leave of absence and join him. Together they planned what they would do and how they would do it. That was half the fun. They got their kids involved and planned places they wanted to see, activities they wanted to do, and what their budget might be. The kids each had research assignments around those activities and places, and Laura and her husband used the time to broaden their kids' knowledge as well as their own.\n\nWe suggest several techniques for helping the creative process:\n\n**Visualize** \u2014Movement assists in thinking more creatively and boldly because it allows the mind to shift from the left side of the brain to the right side, which enables you to be open to new possibilities. We recommend taking a walk in a calm and peaceful place, such as a park or by the water\u2014somewhere where you won't be distracted\u2014to contemplate your time off. Spend at least thirty minutes permitting your mind to go to a place that allows it to stretch and imagine the ideal. Assume there are no barriers. Let yourself imagine what your ideal break would look and feel like. Make a point of not talking to anyone while on this walking\/visualization exercise. Later, when you return to a place where you can write, get those ideas down in a journal uncensored.\n\nJaye used the visualization technique to plan her Reboot Break. It included how she would tell her partner she was leaving the business, what she would say to clients, and, more importantly, what she wanted to do when she had the time off. This included everything from addressing health issues to exploring a new career path. It worked for her.\n\nOn the other hand, Ned didn't take the time to visualize before he began his six-month sabbatical. All he knew was that he was tired and overwhelmed and needed an emotional break. He began to realize when his time off was coming to an end that he never did the things he would have liked to do. He hadn't planned for them. He is now a strong advocate for planning and is, in fact, planning his next sabbatical.\n\n**Think Boldly** \u2014Zoe at age twenty-five left her job at an environmental organization after months of dreaming of doing the St. James pilgrimage in Spain. After discussion with colleagues, plus a lot of planning, she embarked on a complex ten-month travel sabbatical. Her planning included where she wanted to go, what it took to do the pilgrimage, whether her friends would be available to visit, what experiences she wanted to have, and a budget. She did the pilgrimage in Spain, hiking the Pyrenees Mountains along the famed 480-mile Camino del Compostela, also called the Way of St. James. Then she stayed on in Spain for a few months, taking Spanish classes. Next she traveled to Germany, living with friends for two months. Then she was off to northern Thailand before trekking in Nepal. She finished her travel living on a French farm for the last three months. The experience led her to the field of conflict resolution, a master's degree, and a new career.\n\nWithout a bold step in life, some things would never be. Surround yourself with pictures, articles, books, and inspiring quotes. Make a collage with images of all the places you want to go and things you want to do, and hang it by your desk or bed to remind you of your dreams.\n\n**Talk to Others** \u2014If you don't have a clear idea of your life goals or Reboot Break goals, you can talk to others. By verbalizing your thoughts and ideas, especially with someone who will give you encouragement, you make them more real and achievable. Beyond family and friends, blogs are one way to do this. Please join us at our blog on _www.reboot yourlifebook.com._ Another way is creating or joining a community, such as our Reboot Your Life groups on Facebook and LinkedIn. Another example is the community we have created through the retreats we hold, with the attendees staying in touch afterward.\n\n**Write It Down** \u2014Putting pen to paper is another way to start to turn dreams into reality. Journaling about your thoughts right from the start is an excellent way to capture ideas, fears, and wishes\u2014all a significant part of the planning process. Maybe you already write in a journal daily, or maybe you never have. Grab your journal or a handy notebook or go out and buy a blank book for writing, and then sit in a quiet place to think and write. Research has shown that there's a greater connection between the paper and the mind when writing by hand as opposed to using a computer.\n\n### DECIDING HOW LONG TO TAKE OFF\n\nIf you are taking a between-gigs sabbatical or are out of work on an unexpected sabbatical, you still need to plan a timeframe for your sabbatical and for returning to work. The length of time you stay on your Reboot Break most likely will relate to your financial resources. We advise making a conscious decision to plan a Reboot Break after losing a job and not to seek a new job right away. We recommend planning at least a three-month break, if possible, plus having an additional three months in financial reserves in case the job search takes longer than anticipated once you begin it. If you can take a longer break, such as six months to a year, go for it.\n\nIf you are working, your company's policies will influence how long you can take off. Does the company have a sabbatical or leave-of-absence policy, and would it work for you? How long does it allow? Is it paid or unpaid? If there is no policy, has anyone else ever asked to take a sabbatical? Has anyone taken time off without pay and been able to return to the company and his or her same job? These are questions your human resources department can help you to answer. You might also speak to longtime employees who may have taken advantage of some of these benefits.\n\nOther factors in deciding how long a Reboot Break to take may include:\n\n\u2022 How long you think it will take to achieve all you dream about doing.\n\n\u2022 How much time you need to accomplish a specific goal, such as an academic program.\n\n\u2022 How much time your company provides.\n\n\u2022 How much time you can afford to take off financially.\n\n\u2022 A season or specific period of time, such as summer when the kids are out of school.\n\n\u2022 Requirements of a specific program, such as the Family and Medical Leave Act, or any other program in which you might be participating.\n\nThe Family and Medical Leave Act (FMLA) is designed to assist people needing time to tend to medical issues for a family member. Should your Reboot Break be focused on caring for an ill family member, you can take up to three months off, intermittently or all at once. You must have proof of the illness and the relationship to get approval.\n\nDorothy's daughter, Shanna, gave birth to her first child. All had gone well but the baby was born prematurely. Shanna was living at home at the time, with Dorothy and her husband. While Dorothy was thrilled to have her first grandchild come into the world, she knew that her daughter was not prepared to care for her new baby alone. With Dorothy's very demanding job running a child welfare agency, there was little time for her own family and to support her daughter. Dorothy researched and discovered that she could take advantage of the FMLA to assist her daughter and new grandchild.\n\nDorothy was able to take two weeks at first and then three weeks intermittently over the next two months to spend time with her new grandchild. She was thrilled to have this special time with him, to support her daughter, and to get away from the distractions of her busy job.\n\nBen's mother was diagnosed with Alzheimer's. He wanted to help her to get her life in order and get her settled somewhere where she would feel well taken care of and comfortable. He asked for, and received, a three-month FMLA leave. He used the break to spend quality time with his mother but also to clean out her house, sell it, and get her settled somewhere safe. He loved being with her and was able to take care of her proactively while also giving himself the \"gift of time\" with her. It gave him a sense of relief and peace to know that she was where she needed to be.\n\nSometimes people end up extending their Reboot Break if they can, either because they aren't ready to return to a more structured routine, or because they feel they just haven't yet accomplished what they set out to do. For those with the flexibility, extending the time away is a great option.\n\nMary Pat had originally planned to take eight months off when she resigned from her senior-level marketing position. She planned a number of things to do during this time of healing and exploration. As she approached the eight-month mark, she realized that she had much more to do and that the doors had just started opening up for her. Going back to a full-time work routine felt counterproductive and premature; she had already invested considerable time to get where she was. While on her Reboot Break, Mary Pat was approached by a number of recruiters. Since she still wasn't ready to return to full-time work, she turned the opportunities into some consulting assignments to rebuild her financial resources. Then she joyfully continued her break. Today, Mary Pat is consulting and still leaving more time for herself. She integrated what she learned from her time off to move into a career that gave her more control over where she lived, what she did, and how much time she worked.\n\n#### WHAT IF YOU WERE TAKING A YEAR OFF?\n\nSusan started charting her course in her late twenties, twelve months ahead of the year-long sabbatical she planned to take from her job in banking. Susan knew that she was planning for the next stage of her life and wanted to be thoughtful about what she was doing. She applied to a six-month academic program in England, then requested and received permission from her employer to attend and extend the time off to a year. \"I knew that I needed a full year,\" she said, \"and I know that when I have things planned, I am more productive and get more out of my time away. If you don't plan, time escapes you and the things you had dreamed of fade away.\"\n\nMore and more people are taking a full year off in order to achieve a longer-term life shift. Many people travel for a year and leave their creature comforts behind. Some volunteer during their time away. Studying often requires a commitment of at least a year for career enhancement or to acquire enough knowledge for a career shift and to be more credible.\n\nWe recommend a year or more break to truly get the impact and lasting benefits of being away, but any amount of time away is nourishing and recharging. For those of you still uncertain about taking a longer break, mini-sabbaticals are a good way to test the waters.\n\n#### STARTING OUT SMALL: TESTING THE WATERS\n\nStarting with a long Reboot Break can be daunting, in terms of planning, financing, or facing fears. In a down economy, small steps may seem more palatable. Although we recommend that Reboot Breaks ideally last several months to a year, one suggestion is trying a shorter \"mini-sabbatical,\" to pave the way. You can try taking a break for a few days to a month. Structured the right way, a weekend, a week, or a month can give you a glimpse of what a real stretch of time can do.\n\nSome people try something or someplace new for a short time and then decide whether they want to pursue the activity on a more sustained basis or go back to the place for longer.\n\nFor some people, taking a long weekend to unplug from the BlackBerry and computer and be alone with no schedule offers a small test of what it would be like to take a Reboot Break. A mini-sabbatical can also test the logistical and emotional planning required with one's family or significant other, the financial resources needed, how the office will get along without you, whether you can handle simply being alone for a period of time, and ways of overcoming feelings of guilt.\n\nWill left his work as a photographer in New York City for the month of March to refresh his soul, mind, and spirit. \"It was an imperative for me. I had to do it. You have to do this kind of thing for yourself. You have to work it out financially and figure out how long you can take. I scheduled my photography jobs so I could be gone, and I went to Texas with my camera in hand just for pleasure. I drove for one month all over the state, loving not working, listening to music, taking pictures.\"\n\nWhen Cathy traded houses to spend a month in Italy in 2006, it went so smoothly with her staff in her absence that she realized her real sabbatical could be much longer than the three months she had planned.\n\nCathy and others offer some good advice about planning mini-sabbaticals:\n\n\u2022 **Unplugging can be the hardest part.** Tell your office and others ahead of time that you are not going to be available electronically. Then try to stick to that plan to get the most out of your time off.\n\n\u2022 **You can do a lot in a month.** It's twice the time of a two-week vacation and twice the opportunity to get away from it all and relax or try something new. Planning helps you make the most of it.\n\n\u2022 **Involve your family in the planning and\u2014if it fits the plan\u2014 in the mini-sabbatical itself.** It should be a time to get away from normal chores and routines.\n\n\u2022 **You can be creative without spending huge amounts of money.** For example, some people trade houses or even house-sit for a short period of time. That gets them away from their routine and into position to explore new places. The web can be helpful in identifying opportunities.\n\n\u2022 **Plan ahead, starting with making a list of things you have been wanting to do.** Figure out how to make it happen.\n\nIn Chapter 10, Living the Lifelong Sabbatical, we talk more about ideas and tips for mini-sabbaticals and how they can be a lifelong practice.\n\n### WHOM TO TELL AND HOW: PLANNING CONVERSATIONS\n\nMake a list of all the people who will be impacted by your taking time off. Start with your family members, friends, and others who count on you. Then list employers, colleagues, clients, and customers. Include any corporate or not-for-profit boards you may be on or community projects in which you are engaged. Plan what to tell them, and when.\n\n#### TALKING TO YOUR SPOUSE\/PARTNER, CHILDREN, AND FRIENDS\n\nYour transition will be smoother if you include, early in the planning process, the people who will be affected by your taking a Reboot Break. They can be sounding boards and challenge you to think more\u2014or less\u2014broadly. Fearing the reactions of others and thus avoiding those potentially difficult conversations will cause more problems than dealing with them up front in a calm and confident way. If you are convinced of your decision, others will get on board more quickly, especially if you ask for their help, support, and ideas. If you receive some pushback from naysayers, don't let them rain on your parade. (We talk more about reactions of loved ones in Chapter 9.)\n\nWhen a spouse or partner is a co-sabbatical taker, it is even more important to share each other's goals and expectations and to find alignment and build a plan that suits both of you.\n\nBarry wanted to take three months off from his position as leader of a synagogue. He felt it was important to take time to reflect and learn something new to bring back to his congregation. He started planning for the time off with the help and involvement of his wife, Sarah. She worked as a teacher and needed to plan long in advance if she was to participate in any way. Together they made a list of what they each wanted to accomplish during this time away. It was important for both of them to come back refreshed and armed with a fresh outlook and new perspectives. They each took courses that had been on their list and took three two-week vacations away together. Their unexpected gift from this time together was that it reinforced their own relationship and strengthened it.\n\nChildren, too, need to be a part of the conversation, once you are clear and have a plan. They will want to know what to expect and how it will affect them. At the same time, you can leave room for their additions and suggestions. From the mouths of children can come creative, unfettered thoughts, ideas, and possibilities.\n\nTalk to your friends. They may provide just the encouragement, support, and ideas you need.\n\n#### GETTING YOUR EMPLOYER ON BOARD\n\nNow that you have decided to take a Reboot Break, you need to get your employer on board. Employers want to have energized and productive staff members, so there is something in this for your boss as well as for you. You might approach it that way in your discussions.\n\n\"Are you crazy?\" you ask. \"You don't know my boss. This would be impossible at my company.\" Well, you may just be surprised, and you'll never know if you don't ask. If you are working for a small company or a start-up and you and your colleagues are in the throes of launching the company, new products, or a new line of business, then the timing may not be right for you. It's up to you to pick the right time, communicate it, and plan well in advance for it.\n\nBill was the VP of a major business line at a top computer company, and he was planning to start a two-month sabbatical within days of the most important product launch in the company's history. As the head of this major business, the lifeline of the company, how could he disconnect completely for two months when the product had to be flawless and there were major customers and analyst meetings to be had? Besides that, the company had no sabbatical program and no one had ever taken a sabbatical.\n\nBill explained to those who wondered how he could do it that he had made a personal resolution to take two months off to be with his family every three years in the summertime. He started communicating his resolve three years ahead of time, telling his boss, the CEO, and co-workers that \"in three years time I will be taking two months off.\" Of course, no one paid any attention to him for something that was so far off. Every six months he reminded everyone of his plans. As the time got closer, he reminded people every month and created a plan so that his employees were empowered and prepared to run the business. The launch went off flawlessly and catapulted the company. He had a wonderful time refreshing himself and reconnecting with his wife and his teenage children as they traveled and explored Europe.\n\nMaybe you work for a very small company and you and your co-workers are each doing the work of three people. Again, timing is everything. This may simply not be the time to ask, but that doesn't mean that you give up your dream. The people who get the most out of a Reboot Break dream about it and plan for it a year or more ahead of time. If you were suddenly taken ill or if you are a woman and became pregnant, even small companies would give you time off\u2014six weeks in the case of a maternity break\u2014and they survive. You may want to start with a mini break of two or three weeks, but share with your boss your desire and your resolve to be able to take a longer Reboot Break someday.\n\nWe've spoken to many lawyers who have said there is no way a lawyer can take a sabbatical or Reboot Break. They claim that their clients depend on the individual client-attorney relationship and would never accept a replacement. Others point to the partner revenue sharing as an obstacle, feeling that it wouldn't be fair to take off two or three months. But many well recognized and well run law firms successfully offer sabbatical programs. They have found a way to handle these objections and claim that their programs work well for lawyers and for the clients. For a list of some law firms and other entities offering programs, see the Appendix \"Organizations That Get It.\"\n\nBe sure to prepare for your conversation with your boss. It is important to be clear that this is something you want and need to do, that you have thought through how your work can be managed during your absence, and that you have a reentry plan that will enable you to transition back with little or no interruption in the business. Thus, the benefits will clearly outweigh any negative concerns.\n\n##### _Making the Business Case_\n\nWe suggest scripting the conversation so that you don't miss valuable points in presenting the business case, not just your case. As you approach your conversation, it might be helpful for you and your boss to know that many successful companies and organizations are offering sabbaticals. We have researched or talked to 150 corporations, law firms, non-profits, trade associations, government agencies, and small enterprises that offer sabbaticals or otherwise support employees who want to take time off. Some of these organizations have formal sabbatical programs with paid leave, and some just allow people to have a job waiting when they return. Some organizations cover their employees' healthcare benefits while they're not working or offer the insurance at the company rate.\n\nEmployers are more willing to help those desiring a Reboot Break because they see the value to the organization's culture, recruitment, retention, and capacity to innovate. Sabbaticals are also fundamental to building the breadth and depth of employees within the organization, both those who take them and those who fill in. The latter become more experienced and flexible by standing in for the absent colleague, and the organization builds resilience.\n\nIn 2009 there were nineteen companies on Fortune's \"100 Best Companies to Work For\" that sponsored sabbatical programs, a 36 percent increase on 2008. Our research suggests that the trend is in fact on the rise. Some companies\u2014such as AARP, American Express, Charles Schwab, Deloitte, eBay, Intel, McDonalds, Newsweek, Random House, Inc., Scholastic, and Text 100\u2014all offer one- to three-month paid sabbaticals. Others, like the consulting firm Accenture, help employees set aside part of their paychecks to finance a three-month leave, with a continuation of benefits.\n\nThe trend toward sabbaticals has remained true even during the economic downturn, with the added innovation of furloughing to keep employees but cut costs. Furloughing is somewhat common, with employees being asked to take a month or even a year off with reduced pay or even no pay. They get to refresh and come back reinvigorated and rededicated as even stronger employees. Some law firms are encouraging first-year associates to delay coming to the firm and instead do volunteer work for three to six months, and most firms are paying them partial salaries to do so.\n\nGiven the business advantages of granting sabbaticals to employees, you can make a strong business case to your employer. Here are some points you might want to make:\n\n\u2022 **Sabbaticals and taking time off are the state-of-the-art way to energize talent and build leadership skills.** Leaders and managers need to take a step back and away to refuel themselves to bring that renewed energy back to the organization.\n\n\u2022 **Sabbaticals enhance staff capabilities and succession planning for those who remain behind.** These employees get the opportunity to step up and stretch their capabilities. They can have more visibility and lay the groundwork for their own future advancement in the organization.\n\n\u2022 **Sabbaticals induce loyalty.** Employees who are allowed time off for rebooting are more likely to stay in the organization longer, lowering turnover and recruitment costs.\n\n\u2022 **Returning employees are usually healthier and happier,** improving company morale and lowering healthcare costs.\n\nWhen asking for the time off, be prepared to talk about why it is important for you to take the time off, what you plan to do, and then\u2014 more importantly for your employer\u2014the benefits to them. When going in to negotiate your time off, be prepared to discuss what your ideal break would include. Can you get salary continuation? How will health benefits be covered? If they are not paid for by your company, make sure that you personally contribute so that your health insurance is not interrupted. Check that this Reboot Break is treated as a leave of absence and does not create an interruption in years of service for things like future vacation accrual, pensions, 401(k), etc.\n\nHaving a conversation with employers and managers doesn't have to be difficult. Mary, who you may recall from Chapter 1 was desperate to take a break, approached the human resources department at her commodity-trading company first to test the waters. There was no sabbatical leave policy, and no one would dream that a trader would ask for time off. But, they were supportive and gave her some valuable advice about how to approach her manager. Following their suggestions, she offered to help train someone to step temporarily into her shoes, thereby increasing that employee's abilities. She explained why she needed a sabbatical and the benefits she hoped to gain from it. The manager wasn't thrilled, but considering the costs of finding someone of Mary's caliber to replace her, the manager decided she was worth waiting for and gave her the go-ahead. (Chapter 1 has other relevant examples and information on workplace sabbaticals.)\n\nYou can prepare your employer by making a strong argument for yourself and for the business case. The more in tune you are to what is in it for them to agree to your taking this time away, the more likely they are to agree. At the same time, once employers or business partners see your commitment, it is harder for them to turn down the idea. Think about whether you want to return to the same job in the organization or have flexibility to move to something new, and be prepared to discuss that.\n\nGo into the meeting with a well thought-out plan about the delegation of your work load, the timeframe, the benefits to you, and the potential benefits to the company. Including other managers in your decision and in the design of how to handle things during your sabbatical will help smooth the transition. Acknowledge their advice and make sure they feel heard, but never let them derail the plan. Reinforce the benefits to them, the employees, and clients.\n\nIf the business case doesn't work, there is always the business proposition approach. Dale says she got time off from her job in New Zealand for her sabbatical by \"striking a deal\" with her boss on how the work would get done:\n\n\"I had to produce two major reports before leaving (a huge task) and one when I got back. I basically worked a few seventy-hour weeks before going and on my return. I also arranged for a friend of mine to work part-time to cover me while I was away. But it was worth it in the end. The time off really energized me.\"\n\n#####  _Announcement_\n\nMake your boss's life easier. Here is a sample email announcement to your organization regarding your planned time off. You can decide with your boss if the letter comes from you or from your boss. There are a number of ways to approach this; this is one option:\n\nDear colleagues and staff,\n\nMany of you are aware that I will be taking advantage of the company's sabbatical program this summer. I will be away from xxxx to xxxx. While I am away you will be able to speak to xxxx, who has been fully brought up to speed on all the projects in which I am involved and can answer any of your questions to help move things forward in my absence. You will find him\/her insightful and well prepared. I appreciate your support of our mutual projects during my absence and look forward to returning in xxx, energized and ready to jump back in to carry on our work together. I am available for questions or concerns up until my leave begins on xxx.\n\nThank you.\n\n#### TALKING TO CO-WORKERS, PEERS, BOARD MEMBERS\n\nThis needs to be handled with care. Some co-workers may be resentful or jealous and might in some way try to undermine your plan. Being sensitive to their feelings is critical for the success of the plan. They will want to feel included and considered, and they will want to know what part they will play in your absence.\n\nReactions vary. Many people reported that although their co-workers were a little jealous, they also cheered them on and felt encouraged that they too could take time off later.\n\nMake a personal call to all of the critical people in groups with which you interact regularly to share the plan for your time off and to answer any questions they might have. You don't want to surprise anyone. A clear memo to everyone a week or two before your departure, outlining who will be handling what, how to reach key people, and reassuring them that the business will be well handled during this time, is very important for maintaining your reputation and keeping the organization and your responsibilities moving forward.\n\n#### TALKING TO CLIENTS AND CUSTOMERS\n\nPlanning ahead is critical in making a smooth transition both out of your role and then later back into it, whether you are working in a corporation or have your own business. Your clients and customers need to know with confidence that their needs will be met, so providing them with information in advance will ease their minds and help insure that they will be there when you return.\n\nMost people have had very positive reactions from clients and customers. Many have built even better relationships as a result because of how the sabbatical period was communicated, planned, and handled throughout the process. Unknowns breed fear and concern, so communicate in advance and make it clear when you will return and reenter their lives and businesses.\n\n### PLANNING FOR ENTREPRENEURS AND SOLE PRACTITIONERS\n\nDoctors, accountants, and other professional specialists face their own unique set of issues. Many we interviewed were daunted at first at even considering a sabbatical. \"How will customers, patients, and clients feel if I am not available?\" \"Will I lose my business if I am not there to captain the ship?\" \"No one will do it the way I would do it.\"\n\nA striking number of people we interviewed fit into this category of having no obvious replacement during a time of leave. They all overcame their fears, took successful Reboot Breaks, and were able to return to their business or practice.\n\nThe good and bad news is that someone _can_ do what you do, maybe even as well. Think of colleagues who could take your patients or customers temporarily in your full role or perhaps a reduced role. Consider hiring someone to do it as a freelance employee, asking people you trust for recommendations.\n\nTwo friends who each ran very small businesses made a deal with one another. They didn't know each other's business, but they were both successful entrepreneurs and knew how to service customers, listen, solve problems, and keep employee morale high. The two agreed to take a Reboot Break a year apart. Each would be the guardian\/mentor to the person or team left in charge while the boss was away. They did so by occasionally attending meetings, as a board member might, and then during their friend's break they talked to the manager in charge at least once a week. Good leaders know how to ask good questions regardless of industry and business. They made sure they each had a good number two who could step up to run the day-to-day part of the business.\n\nAs you consider who will carry out or support your functions while you are away, plan how to train and empower them as necessary. It will be a test of your succession planning\u2014and ability to take future Reboot Breaks\u2014if you can work this out.\n\nIf you cannot figure out someone to take over while you are away, think about whether your customers, clients, or patients can get along without your services for a while until you return. Notify them well in advance that you are closing the office for a specific period of time.\n\nOther things to think about to prepare you and your business while you are away:\n\n\u2022 Document your work flow and processes for those who will be supporting you during your break.\n\n\u2022 When possible, take care of unfinished business issues.\n\n\u2022 Prepare backup for anything that may come along, such as new business.\n\n\u2022 Keep up liability and other insurance.\n\n\u2022 Let everyone know important contact information, including accountants, lawyers, technical support, building maintenance, etc.\n\n\u2022 Communicate in advance to all of your customers and clients about your plan and back-up plan so they know what to expect and whom to call.\n\nBetsy, a hand surgeon in North Carolina, took a one-year break from her medical practice to volunteer in one of the poorest parts of South Africa. At first she doubted that she could leave her solo practice, but with her two employees leaving and a necessary move out of her medical building, she felt the time was right. She arranged for a new orthopedic doctor just moving to town to handle her patients for several months and was able to take the sabbatical of her dreams. Though she was in single practice, she was still part of a larger, twenty-three-physician corporation. They had to vote to keep her in the corporation. She said that she still would have gone on sabbatical even if they said no. They didn't, but some believed she would not return.\n\nOthers in the local medical community were interested and positive. After she sent out letters to the larger medical community announcing her sabbatical, she was surprised to hear back from several doctors saying what a novel idea a sabbatical was in medicine and that they were envious. Her patients were very supportive and believed that she was coming back, which she did, and today she still has a thriving practice.\n\nGlenn, an independent financial planner, worried that his clients would be unhappy when he told them about his plan to take three months off. Instead, they were impressed by his commitment and applauded his decision. Some even admitted they were jealous. His careful planning for covering their interests in his absence\u2014 offering to transfer accounts to another financial planner\u2014was thorough and much appreciated. When he returned, some clients told him he was a role model and inspiration for them. Glenn's practice flourishes today.\n\n### PLANNING TRAVEL\n\nMost people we interviewed included some form of travel as part of their sabbatical experience. Once you have identified where you want to go, you will need to research what is needed to enter the countries you would like to visit. Some have special visa requirements that might take from a week to three months to be processed. Although some people prefer to just head to a place and see what happens and make plans as they go, other types of travel require more advance planning. Reservations for travel, housing, rail, etc. might need to be made in advance. More importantly, make sure your passport is not only up to date, but will not expire within six months of entering any country.\n\nFor those looking to study while they are away, there is usually an application process with deadlines for enrollment. Researching everything you can and speaking to people who have done what you are planning to do will save time and aggravation and make the travel experience richer and more fulfilling. Plan early and ahead, when you can.\n\nThe Appendix includes a long and thorough list of things to consider and plan for before traveling as part of your Reboot Break, and Chapter 4 is also rich in travel planning information.\n\n### PLANNING TO VOLUNTEER\n\nA desire to \"give back\" drives people to volunteer, and it may you as well. Volunteering can be part of a company's sabbatical program or sponsored by a church or community group. Other times it is part of the \"new volunteerism,\" where you pay to volunteer in interesting places overseas or at home.\n\nWhen Larry Fish accepted his job as Chairman and CEO of Citizens Financial Group in 1991, his predecessor said he could come on board whenever he liked. He decided to take a six-month sabbatical before he started. \"I went to a social services housing agency and said, 'Look, I would like to help, but I don't want to do a strategic plan or raise money or fix the accounting department. I will scrub floors, mentor kids, work the food bank, and be here as early and late as you want.'\"\n\nSo that's what he did, literally, for six months. \"I got much more out of it than I gave,\" Fish said, \"And I took the experience and started the sabbatical program at my institution.\" Citizens offers its full-time employees who have been with the bank for at least three years and are in good standing the opportunity to volunteer at a non-profit of their choice. They get full pay and benefits and a guarantee that the same job or better will be available when they return.\n\nOther firms encourage volunteerism as well. Antonia talked of the experience of Shearman & Sterling LLP, a major New York law firm, giving its employees two to three months off to volunteer. \"It makes us a better firm and makes us better lawyers as well as people.\"\n\nTwo organizations that list opportunities for volunteering abroad are Working Abroad and Greenforce. Working Abroad provides a scheduled report on volunteer activities across the world for a small fee. Greenforce, a London-based organization, runs volunteer programs in ten countries. Other resources are listed in the Appendix. Confirm that the organization is a registered 501(c)(3) at _www.guidestar.org_.\n\nBefore you sign on with a volunteer program, research the operator, the prices, and the environment. Some tour operators charge for organizing trips that you could do for free directly with the not-for-profit.\n\n### THE PLAN OUTLINE\n\nThe task now is to take the elements of your Reboot Break and plan step by step for each part. At this point, you know your goals and timeframe, have consulted with important people, including your boss and\/or colleagues, and can now look into the details of what you will do on this well-deserved break. This is the fun part because it brings you closer to the reality of your coming adventure.\n\n#### PLANNING GRID FOR A SUCCESSFUL REBOOT Break\n\nUse the grid on the next page or make your own as a way to organize your thinking around each goal you want to achieve during your Reboot Break. First think about the bigger goals, i.e., improve your health, travel, spend time with the family, learn something new. As you drill down in any one of those larger sabbatical dreams, you will need to think through the various steps to make them happen. With each goal in mind, what will you need to do first? Build from there. Add who can help you realize your goals and by when.\n\nBe as detailed and specific as possible, giving yourself timelines to ensure that you actually get it done. You can continue to add to the grid and change things as you learn from each step you take.\n\nIn the Appendix of this book is a simple planning checklist, plus more detailed planning tips, organized by timeframe starting at one year out. The steps can be compressed into a shorter timeframe as necessary, and not all steps apply to everyone.\n\n### READY FOR TAKEOFF\n\nHaving a solid plan and taking steps day-by-day toward the sabbatical dream will calm the jitters and keep the path clear for takeoff. Keeping your goals in mind, you can explore ideas and make choices. If your plan isn't working, that's okay, because you can always change it.\n\nYou are now armed with some of the tools and steps that can lead to a successful Reboot Break. It is the planning that makes it so. Look within yourself for what you really want to do and think through what it will take to get there. Everyone needs a break; you wouldn't be reading this book if you didn't think you needed one too. Planning is important, it can actually be fun, and it will lead to a more satisfying and successful Reboot Break.\n\n###   EXERCISES\n\n##### _Exercise 3-1: Building Your Dream Reboot Break_\n\n\u2022 Make a list of all the things you would ideally like to do while on your Reboot Break. Don't leave anything out; this is the time to throw in everything you have always thought about doing but never had the time to do.\n\n\u2022 Now go through and prioritize the top 5, top 10, and top 15 things you would like to do.\n\n##### _Exercise 3-2: Taking Your \"Business Case\" to Your Employer_\n\n\u2022 Draft a sabbatical plan \"business case\" to take to your employer, including responses to questions the employer might ask and a list of the ways your time off would benefit the company or organization.\n\n##### _Exercise 3-3: Create Your Own Planning Checklist_\n\n\u2022 Review the Reboot Break Planning Checklist from the Appendix and the list of things to do when preparing ahead of time.\n\n\u2022 Compile a list of what you need to get done and when before starting your Reboot Break, with timeframes to keep you on track.\n\n## | CHAPTER 4 |\n\n## _Funding Your Freedom_\n\n\"Good fortune is what happens when  \nopportunity meets with planning.\"\n\n_\u2014 Thomas Edison_\n\nOne of the greatest challenges people face in taking time off is how to finance it. Maybe you're concerned about the current economy. Maybe the market crash has sucked your retirement funds dry. Maybe you are afraid to leave a job for fear of not finding another. Or maybe you are already on an \"unexpected sabbatical\" because you have lost your job. Maybe you fear that tapping into your savings will leave you vulnerable or that you need the health insurance that your employer provides. Or maybe you fear criticism from your family for being \"frivolous\" with hard-earned money. In this chapter, we offer creative ways of funding your freedom without putting your financial health at risk.\n\nFinding the money to buy your freedom and nourish your soul is often the single biggest roadblock to getting away. Your goal should be time off free of financial concerns so that you can focus on new learning and experiences. That means, ideally, untouched retirement funds and stable home finances.\n\nMarco, who took two course-correction Reboot Breaks in his twenties (Chapter 2), put it well: \"Financial worries\u2014fear of financial insecurity, uncertainty, and instability\u2014threaten one's identity. People have a fear of being in limbo, especially financial limbo. It takes an inner strength to set that period of uncertainty in place.\"\n\nIn this chapter, we'll show you:\n\n\u2022 How to separate the misconceptions about financing Reboot Breaks from the reality.\n\n\u2022 How to save for time off.\n\n\u2022 How to plan your finances, including anticipating unexpected expenses and special considerations for the period when you are returning to work.\n\n\u2022 Resources you may be able to draw on to help finance your break.\n\n### CLEARING UP THE MISCONCEPTIONS\n\nYou hear many misconceptions about funding and the ability to take time off from work, such as:\n\n\u2022 Sabbaticals are only for middle- and upper-income people.\n\n\u2022 A Reboot Break will bankrupt your savings.\n\n\u2022 Taking time off will ruin your career.\n\n\u2022 It's irresponsible to take a sabbatical, especially in bad economic times.\n\n**You don't have to be rich to take time off.** We interviewed people from every income group who have taken time off. How much money someone makes is not the best predictor of who takes a sabbatical. It has more to do with the stage of life people are in and their dreams and goals. Teachers, nurses, ministers, construction workers, and other individuals have found ways to take a Reboot Break.\n\nThree age groups tend to be the most interested in taking sabbaticals, and they fund them in different ways. The twenty- to thirty-year-olds we interviewed were able to pack up their apartments and leave stuff with their parents, if they had anything, and be free to take off on little money. The thirty- to fifty-year-olds had mortgages and other expenses, but were most likely to get some support from their employers or their own savings or find work while off work. The sixty-plus-year-olds tended to use savings for their time off.\n\nOne of the most important misconceptions is about the costs associated with taking time off. Yes, it does cost something to do what you'd like on sabbatical, but it doesn't have to break the bank.\n\nPeople fund their time off using their retirement, vacation, education or travel funds, or funds from an account created specifically for a Reboot Break. We call this the \"Reboot Break account,\" and it is an important vehicle for realizing your dreams. You can direct your own savings, as well as gifts from others, to help fund your time off in the future.\n\nSome may use full or partial pay from a company sabbatical program (if there is one) and\/or accumulated vacation time. Others tap severance packages, bonuses, family gifts, inheritances, or tax returns to fund their freedom. There are many financial possibilities and ways to begin imagining how you might finance your break.\n\n#### FINANCIAL IMPACT ON YOUR CAREER?\n\nOne of the most common fears we heard from people of all ages centered on the impact taking time off would have on their career. Many worried they would not be taken seriously or would be passed over for promotions and bonuses and thus suffer financial consequences. They wondered how to explain the employment gap in the future.\n\nSome were concerned about colleagues who were competitive with them, and what they might do to take over their positions. In a shaky economy, where people have seen much of their gains in the stock market evaporate, their companies downsize, and their neighbors out of work, there is an understandable reluctance to do something risky. Some were concerned that their jobs would not be there when they returned.\n\n**We found you do not have to sacrifice your career and financial stability to take time off.** Of the 200 people we interviewed, over half went back to their current employers, and every one of them ended up feeling that they were more valued by their company. Those who didn't go back or severed the relationship before going on sabbatical ended up in positions they liked as well. It turns out that career breaks can be good for your financial as well as your personal life.\n\nOver 60 percent of Generation X employees, born between 1964 and 1978, want to take an extended leave or sabbatical, according to a 2001 survey quoted by _American Demographics._ Many progressive employers see it as a sign of drive and initiative. Travel abroad builds interpersonal and cultural skills. Our interviewees learned new languages, talents, and work-related knowledge on their time off. Being in new cultures made them more flexible, adaptable, and tolerant.\n\nMany people prepared for their return before taking off by doing speeches before and during the sabbatical, going to industry conferences, staying current on the literature of their industry, and checking with recruiters on how they would be perceived on their return, even going so far as to revise their resumes to explain the time off before they left, and sending them out a month or two before their return.\n\n#### THE CONNECTION BETWEEN MONEY AND SABBATICAL GOALS\n\nResearch by psychologists and economists indicates that people see money as a source of security, freedom, prestige, power, and reward.\n\nTaking time off is viewed as a source of freedom\u2014the freedom to do what you want, enjoy a more balanced life and still feel secure about your future and that of your family. The burden of making a living often keeps us from stepping back and re-evaluating our lives. A Reboot Break allows us that time.\n\nInterestingly, sometimes the wealthier people are, the more they fear taking time off because they are so tied to a certain level of income and spending. The thought of breaking the tie between money and happiness, or at least questioning it through a sabbatical, is too frightening. All of us at one time or another _need_ time out to question our beliefs. We talk more about how to define success beyond money in Chapter 10, Living the Lifelong Sabbatical.\n\nNo one can predict what will happen with the economy or financial markets for the next few years. All of us have been impacted by living through the dramatic changes in home values, retirement accounts, and jobs in this economic downturn. But you can be proactive and take steps to get your finances under control. Hard-earned savings can go quickly if thought isn't given to what you want to do and how to do it.\n\n#####  _Living Light_\n\nOne new notion includes the concept of \"living light\"\u2014saving as much as you can, spending wisely, and letting go of material things in exchange for more true freedom. Let's walk through what this means.\n\nLeena, whose story you first read in Chapter 2, had a beautiful home in London but gave it up when she decided to take time off. \"I had to let go of certain attachments. I had an exquisite flat in London. I decorated it but only lived there for three years. In order to take time off, I had to rent it out. My friends were horrified that I would leave this place I had just designed and furnished. **By letting go of material attachments, though, I was able to have my freedom.\"**\n\nFor decades, America has been a \"more is better\" society: more food, more fashion, more money, and more spending. The global financial crisis has curbed our profligate tendencies and struck fear in consumers' hearts. With housing values plummeting and 401(k)s shrinking, saving for tomorrow makes sense.\n\nMore Americans today are \"living light\"\u2014lightening up on debt, forgoing excessive spending, and spurning showy, expensive brands. This is not only necessary for most; it's also appealing. More and more, we are looking beyond conspicuous consumption to find meaning in life's more prosaic pleasures, such as spending time with family and friends.\n\nLiving light is a fundamental of a Reboot Break. Few of us can afford to maintain an extravagant lifestyle while we're working, never mind while taking time off from work. Instead, many sabbatical takers free themselves in the same way Leena did: by forgoing material possessions in exchange for more authentic personal fulfillment.\n\nKevin, like many others we interviewed, learned to live on less and to become much less attached to material things. He found ways to travel more inexpensively and live more authentically. He consciously chose on his Reboot Breaks to find ways to simplify his life to give him the breathing room to think.\n\n#### CASH OUT YOUR RETIREMENT SAVINGS, HOME, OR BUSINESS?\n\nA sabbatical is meant to give you time away from work to refresh, renew, reflect, and then come back to your same employer or business or decide to do something else. Cashing in your retirement fund or selling your home or business may put undue financial or emotional pressures on you. If you're thinking of doing this, be sure you've given yourself time to think seriously about what you want to do next. You may be surprised that after your time off, you wish you had kept the house or business, and certainly want the security of your retirement savings. Few people taking a career break cash in everything and take off. Keeping a tie to your home, work, non-profit board, or family is important. Having a place to come back to at the end of the journey may be critical to your happiness on your Reboot Break.\n\n### CREATING A BUDGET\n\nNothing works better than creating and adhering to a budget when you need to plan or when money is tight. While some people think of budgets as a chore, they are powerful tools for realizing goals. So start with your goals and include the amount of time off you want to give yourself before you go back to work. Your career break may be voluntary or not, but in tough economic times, we all worry. Having a structure for your budget is critical to using this time between jobs or careers. The basics of budgeting for a sabbatical include:\n\n\u2022 Determining your current spending needs and cash flow.\n\n\u2022 Determining the approximate amount of time you want to take off.\n\n\u2022 Listing the bills that will need to be paid, such as mortgage, insurance, college tuition, etc., on an ongoing basis.\n\n\u2022 Determining specific financial needs while on sabbatical, such as travel funds.\n\n\u2022 Brainstorming ways to cut costs or earn money, such as renting your home or apartment during your period away.\n\nTo understand how you spend money, plot out the previous year's monthly expenditures. Include tax statements, credit card statements, check registers, and other documentation you have on hand. If you don't have all these records, get financial planning software, such as a spreadsheet, or get a notebook and record all expenditures each day, week, and month for a few months. Include credit purchases, as well as cash, debit, and direct payments. Use a spreadsheet or notebook or online software to create categories going forward for revenue and expenses. Start tracking now in preparation for your time off as well as during the sabbatical. Just the process of writing things down or recording them daily is enlightening.\n\nYou will begin to see ways to save automatically. We tend to spend more freely with credit cards, and often don't realize how much we spend on snacks, coffee, and other things we really don't need. Identify areas of spending where you can cut back before and during your time off\u2014from gym fees to gifts\u2014as well as those expenditures that remain constant, such as the mortgage or car payments.\n\nWhen Rita started planning for her most recent leave from work, she started keeping track of daily expenses and income to help understand better how to budget. \"I was shocked at how much I was spending daily on frivolous things. By recording them, I knew just where to cut back without much pain.\" She continues to monitor her spending in this way today.\n\nNext, draw up a monthly budget that allows for realizing your financial goals: saving for a sabbatical, spending on sabbatical, paying down all credit cards, saving money for a down payment on a house, putting money into the retirement fund, etc. Microsoft Money and Quicken have excellent budgeting and cash flow planning forms. Or search online for other budgeting software. (In the Appendix is a listing of some resources to help you develop and track a budget.) The budget process should include not only what income you have to draw on and expenses you can cut, but also other resources you can tap in an emergency. These might be savings, loans, rental income, or sale of a major asset.\n\nIn addition to being one of the most rewarding experiences in your life, taking a Reboot Break may also teach you financial planning skills and practices. The process of planning, saving, budgeting, and controlling spending before and during the time you take off will serve you the rest of your life. An added benefit may be learning that you can take risks and succeed.\n\n### WAYS TO FUND YOUR FREEDOM\n\nLet's start with some of the ways you might fund your freedom, categorized into four types:\n\n\u2022 Saving ahead of time\n\n\u2022 Getting your employer to pay\n\n\u2022 Getting paid for work done while on sabbatical\n\n\u2022 Using a windfall\n\n#### SAVING AHEAD OF TIME\n\n**The more you save ahead of time, the more stress-free your time off will be and the more options you will have.** Using \"found money,\" like a severance or inheritance or a large tax return, can be the initial basis for that savings, as can be regular contributions to a \"Reboot Break account.\" Saving is a commitment to yourself and to your goals. Putting even small sums away each week or month creates momentum.\n\nWays to Save:\n\n\u2022 **Automating savings from your paycheck** through direct deposit with your employer or via your bank. You pay your Reboot Break account first and adjust your spending accordingly. Once you start, you may find it's not as hard as you think, so review the amount you're saving every few months and consider saving even more.\n\n\u2022 **Asking your friends and relations to contribute** to a Reboot Break account in lieu of giving you holiday, birthday, and other types of gifts.\n\n\u2022 **Earning more** through a raise, bonus, overtime, extra job, or investment and putting those extra earnings into your Reboot Break account. This allows you to continue with your current spending habits while you build a reserve for the time off.\n\n\u2022 **Renting out your home or apartment,** your office space, your second home, or anything else that might be of value to others in order to offset your living expenses.\n\n\u2022 **\"Cashing out\" or liquidating any assets** you no longer want or need. Use eBay, consignment shops, garage sales, auctions, and other venues and reserve the returns for your time off.\n\n\u2022 **Putting your money out of easy reach** of your ATM card, debit card, or checkbook. Make it a savings account, CD, or other account where it is out of mind and less accessible.\n\n\u2022 **Changing jobs or building your business** to give yourself more earning power, and then saving the difference for your break.\n\n\u2022 **Asking your employer to defer your income or pay 75 percent of your salary now and the rest when you are \"rebooting your life.\"** Learning to live on less before the sabbatical is good practice for living light.\n\n\u2022 **Managing your credit cards** by trying to pay off your balances each month, which will save you interest and fees.\n\nWhile you are contributing to the Reboot Break account, don't forget to max out (contribute as much as is possible to) any 401(k)s, especially if they are matched by your employer. Many employers will match what you put into savings anywhere from 50 percent to 100 percent. You are leaving money on the table if you do not continue to fund any matching program your employer has while you are taking time off. If you are over fifty, you can contribute more than the stipulated $15,000 as a way of catching up, and it is prudent to do so. The 401(k) savings will continue to grow your retirement and back-up \"emergency fund\" as you take time off from your career.\n\nBuild a safety net or \"emergency fund\" by putting money into savings, like a high-yield money market fund or laddered short-term CD, with at least six months income to cover monthly expenses. This is your safety net to be used and replenished for emergencies, such as car breakdowns, repairs to the house, to cover unexpected loss of income, or if it takes you longer to find a position when you return from your between-gigs sabbatical. Always replenish it.\n\nSome other advice, depending on your age and financial situation, that helps in planning your finances for a sabbatical:\n\n\u2022 Save at least 15 to 20 percent of your income for long-term goals, such as owning a house, college tuition, and retirement.\n\n\u2022 Keep your debt-to-income ratio at no more than 30 percent.\n\n\u2022 Look at refinancing when rates are at least one percentage point lower than your existing rate.\n\n\u2022 Invest no more that 5 percent of your stock portfolio in your company stock, or any single stock.\n\n\u2022 Keep discretionary spending under 20 percent of your take-home pay.\n\n\u2022 Keep your FICO\/credit score high.\n\nSaving for time off gives you the freedom to do what you want without having to tap family resources or worry how you will impact your finances. Bruce, who worked for a financial analysis division of a large institution, spent several years planning for his time off, budgeting for expenses, living on his salary alone, and banking his bonuses in a savings account. His good saving habits allowed him to take an entire year off.\n\nRita consciously chose to leave her position as President of Mead-Westvaco's Consumer Division to take time off, then start a \"portfolio career.\" She worked with her financial advisor ahead of time to plan for the transition. She prepaid many expenses, including her mortgage, used stock options for savings, maximized her deferred income while she was working, and chose to live a more simple lifestyle after leaving her full-time job.\n\n#### WHAT IF I HAVEN'T SAVED FOR MY SABBATICAL?\n\nIf you need time off now and haven't saved enough, the next best thing is to \"borrow\" from your existing savings. Your home equity line of credit (HELOC) is a good place to start, as interest on a loan up to $100,000 is tax deductible, usually at a reasonable rate. However, HELOCs are harder to get if the value of your home has gone down. (The average American home lost 13 percent to 20 percent of its value between 2005 and 2009.) The ratio of equity required has gone up as well. Shop _www.bankrate.com_ for example rates for home equity lines of credit. While we don't advocate taking money early from your 401(k) or any account that might carry a penalty for withdrawal, some people have done just that with a plan to replenish the account.\n\nBorrow only when it makes good financial sense for you. By 2007, US households owed $1.33 for every $1 of disposable income. Look at your debt-to-income ratio (monthly debt payments divided by monthly pre-tax income) and stay under 30 percent.\n\nHere are some questions to ask when considering borrowing to finance your break:\n\n\u2022 Are you investing in yourself\/your career to enhance your value as an employee or entrepreneur, and ultimately add to your net worth?\n\n\u2022 Is your credit score high enough to qualify for the lowest rates?\n\n\u2022 Will you be able to afford the payments if you're out of work for six months?\n\n\u2022 Will your debt repayments remain under 30 percent of your pre-tax monthly income when you return to work?\n\nWhen your preparation timeline is short, there are still some things you can do to set the stage for time off. First, try to pay down your credit card debts as much as possible. Transfer credit card balances to lower-interest cards or call your issuer to lower the interest rate. (Call your credit card companies to negotiate, since more than half of them will reduce interest rates when a customer calls and makes the request.) If you can, pay off the cards with higher interest first, then the next higher. Mortgage and home equity debt (up to $100,000), however, are tax deductible. If your interest rate is low, you're better off paying the minimum amount due and paying out over time. Payments on student loans are also tax deductible, so pay them off last.\n\nTo get the best credit terms, you need high credit scores. The best rates go to those with scores of 760 or higher out of a possible 850. Boost your score by paying bills on time, reducing credit card balances to less than 20 percent of your limits, and correcting any errors on your reports. Order your credit report from _www.annualcreditreport.com._ You get one free from each credit bureau every twelve months. Buy the Equifax version of your FICO scores for $8 at their website.\n\nThe factors lenders look at in a credit report are:\n\n1. How many inquiries you have made. (It's better to shop for loans in a fourteen-day window to have them recorded as only one.)\n\n2. What you are juggling. (Having a mix of types of credit is considered an indicator of credit-worthiness.)\n\n3. How much credit you are using. (Aim to use less than 20 percent.)\n\n4. How timely you've been in paying bills: thirty to sixty days is yellow, which means they have the account on watch, and over ninety is red, which means in arrears and will impact your rating.\n\n5. Whether you've ever really messed up: foreclosures, bankruptcies, liens, and other major problems will stay on your report for seven to ten years.\n\nSome people borrow from family members or their regular savings. Most said they hoped to repay the amounts in the future, either to the family member they borrowed from, or to their savings. But many borrowed from their savings without worrying about paying it back right away because they understood the importance of the time off and that, regardless of finances, they had made an excellent return on their investment\u2014their investment in themselves.\n\n#### GETTING YOUR EMPLOYER TO Pay\n\nLooking to cut costs without laying off employees in 2009, the New York law firm Skadden, Arps, Slate, Meagher & Flom LLP began offering associates worldwide the option of accepting one-third of their base salary to take a year off, especially to volunteer and do pro bono work. As part of the offer, they were guaranteed job security: if there were layoffs while they were away, Skadden assured them, they would be immune.\n\nAlso as a cost-cutting measure, KPMG has offered employees paid sabbaticals at 20 percent of their salaries, keeping their benefits intact and guaranteeing their jobs upon their return.\n\nAcademic sabbaticals are the norm because colleges and universities recognize that professors need to be refreshed and renewed periodically, in order to return to the classroom excited and invigorated. Academics are paid during their time off by the university or college for research or teaching in another area, geographically or intellectually.\n\nTo apply for a sabbatical, academics are usually required to have taught at the institution for at least seven years. In their applications, instructors set out a plan for what they will do and usually make a presentation to the faculty upon their return. The time off may be a semester, half year, or year.\n\nIncreasingly, nonacademic organizations, such as corporations, churches, non-profits, and legal and consulting firms, are providing financial support of some type for their employees to take time off to renew as well, as described in Chapters 1 and 3. We offer several examples of the types of programs and the benefits to the organization.\n\nIf you plan to stay with the same employer, the ideal situation is to convince them to pay you your salary while you are taking the sabbatical, or work out a special deal. Geoff told us about a creative approach someone took of getting his employer to pay half his salary for six months and defer the income to the next year. Then he took the next year off at 70-80 percent salary, so that he had income all along and tax advantages as well. This is just another example of deferred income that many highly paid executives use. You have the organization spread out your salary over a longer period of time, especially into years where you might not have as much income normally.\n\nWhen planning for a break where you expect to return to the same company, it is important to negotiate explicitly that the time off is for \"retooling\" or \"renewal\" that will benefit the company and should contribute toward service, options, restricted stock, 401(k) matching eligibility, prior years' service, long-term disability, etc. Ask your employer how they will consider the time off in terms of vesting for retirement or stock options. We recommend negotiating with your employer for continued health insurance coverage paid by them, paying it yourself by reimbursing your employer, or finding some interim coverage. Even though health insurance can be expensive, you don't want to have it as a worry when you are taking time off.\n\nExplore with your human resources department the option of going back to school for undergraduate or graduate work and\/or coursework that would help you in your career. Often corporations will reimburse all or part of employee tuition costs. While this isn't exactly your employers paying for a sabbatical, it does pay for your education and expenses related to it when on your break. There are many programs that go unused because employees do not know of them.\n\n#### GETTING PAID WHILE TAKING TIME OFF\n\nWe found many creative ways people made money while on sabbatical by being inventive as well as persuasive. Here are some examples:\n\n\u2022 **Rent your house,** office space, second home, or other real estate so you will have income coming in despite your taking time off.\n\n\u2022 **Freelance your skills** in other environments\u2014graphic design, teaching, doing artwork, babysitting, doing bookkeeping, cutting hair, travel writing\u2014all while you are on sabbatical.\n\n\u2022 **Apply for a grant or fellowship,** as there are many in the arts, writing, government service, and corporate world that are underutilized. It takes some digging, but they are there.\n\n\u2022 **Apply for an internship** and learn a new field of work, industry, or set of skills. Many pay a small stipend or offer other benefits.\n\n\u2022 **Be a travel companion** and learn about compassion as well as a different perspective on travel.\n\n\u2022 **Explore opportunities to give lectures abroad** and represent your country, company, or industry. At least the expenses may be covered.\n\n\u2022 **Try out a passion by getting another job,** such as being a sous chef or getting part-time work in a field you want to enter. (Get a job that is not too demanding so that you can do other things.)\n\n\u2022 **Write your own blog** or articles for travel magazines, special interest magazines, or other publishers. Get a sponsor for your blog.\n\n\u2022 **Start a business that you can do minimally from home or while traveling.** Internet-based businesses are more amenable to this.\n\nRemember, Cathy and her husband paid for much of their eleven-month journey through Asia by lecturing for the U.S. Information Agency (USIA) on how Congress works and how to market to the United States. USIA paid for hotels, meals, transportation, and a small per diem when they were giving lectures sponsored by embassies, academic institutions, or newspapers. \"It was a fabulous experience and paid for much of our time in those countries.\"\n\nRick, a potter, found a two-month visiting artist program at Arizona State University, which allowed him to explore new techniques and types of clay work. While it did not provide a stipend, it did provide housing, a place to work, and access to resources at the university.\n\nJan used a National Endowment for the Arts fellowship to return to university for a master's program and upgrade her skill set. She had been a reporter before and returned to the journalism field after her sabbatical, this time as a newscaster. Her time off not only gave her new skills, but the confidence to be front and center with her audience.\n\nThe caveat, however, is not to get so focused on making money that you defeat the purpose of taking time off. While working, albeit differently, during a sabbatical may ease your financial concerns, it may also deter you from the reflection and breaking away time you need. Disconnecting from work is one of the most important things to do, our interviewees say. That's what allows you to get perspective. If you work during your time off, be sure to put enough boundaries around it to leave yourself free time. Make the work fit into your overall goal, rather than the other way around.\n\n#### USING A WINDFALL\n\nSome of the people we interviewed used windfalls that came their way to pay for their time off. They ranged from inheritances to severance payments to bonuses and large tax refunds. Sometimes they were annual gifts from family members.\n\n#### UNEMPLOYMENT BENEFITS\n\nUnemployment benefits can be used to fund sabbaticals too. Amy, an attorney who was let go from her law firm because of the collapse of the corporate litigation department, used her unemployment check while doing an internship at a winemaking organization. The unemployment benefit allowed her to do the internship for no pay and was granted by the state because she was improving her chances for a new position.\n\n### CUTTING BACK ON EXPENSES\n\nLooking closely at all your current spending patterns can help you find ways to cut costs and reallocate spending while on your Reboot Break. It's a matter of being open to changing your patterns of spending and saving. Acknowledge that there is flexibility in your budget, and you are off on the right foot! All of us have had to do this, but most of us only do so when forced.\n\nYour sabbatical need not be an expensive proposition. Being flexible in what you want to do and how you do it takes more of the pressure off. As Bruce, whom you met earlier in this chapter, said, \"Think about your expenses realistically and make sure you are comfortable so that stress about money does not interfere with your plan. Have good control over your expenses.\"\n\nIf you plan to travel for all or most of the time, think about what you won't need to spend money on during that time period: your home or apartment, your car, your office space, cable TV, housekeeper, car insurance, dry cleaners, gym membership, etc. Cancel or re-negotiate memberships until you return from your break. Try for a zero cost, or at least costs at one-third to one-half of normal living. One interviewee had great advice for the sabbatical period: \"Just stay out of the stores!\"\n\n#### CUT YOUR HOME EXPENSES\n\nCutting your home expenses while you're taking a Reboot Break can turn it into independent bliss. Sabbatical costs are, and should be, different from everyday life costs. Be inventive, imaginative, and thoughtful in what you do. Your savings will deplete quickly if your ongoing home expenses continue at the current level.\n\nBefore you take time off, prepay expenses such as taxes, insurance, contributions to retirement plans, and fees that may come up while you are off work. Do your regular medical and dental checkups before you leave employment. Ask a friend or your accountant or financial advisor to pay bills while you are gone, and give them power of attorney for any emergencies that come up. Housing costs and maintenance are often the largest expenditures per month. Some ways to cut housing costs are:\n\n\u2022 **Renting your apartment or home** at a rate that covers mortgage and utilities payments. Ideally, rent it furnished to save on storage and moving costs. Use a rental agent or do it yourself, but get references and credit checks and write out a contract for the renters, even if they are friends or family.\n\n\u2022 **Getting a house sitter** who can watch the house, take care of pets, pay bills, take care of repairs, etc. while you are away. Create a contract for this, to make the house sitter more like a professional. This eliminates the cost of paying someone to do these things for you.\n\n\u2022 **Trading or swapping your apartment or home** through an Internet business, such as _www.homeexchange.com ,_ or a network of your friends who have houses in desirable places. This at least mitigates the costs of a new place.\n\n\u2022 **Selling your home,** but only if you are looking to buy elsewhere or that is part of your sabbatical plan. There are costs to doing this, and the market should be right.\n\nWe'll expand on the concept of trading your house or apartment, because it is becoming a major trend and many opportunities are emerging. Are you planning to travel to a foreign country? Do you want to have some time to write your great American novel, but do it on the coast of California?\n\nMany people interested in living in a different area for a period of time are looking to trade or rent houses and apartments for that short period. There are websites that vet potential candidates for you and provide you with choices of homes and apartments elsewhere. Some of those sites are listed in the Appendix.\n\nThere are also listings at universities and colleges, where professors who take sabbaticals look for short-term rentals. If you live in the New York or Washington DC areas, there are diplomats who are always looking for short-term rentals or stays. The same goes for cities all over the world where there are large diplomatic communities.\n\nTalk to your accountant or financial advisor about possible tax implications of what you are doing, from renting your house to what can be deducted for education or career enhancement. Even if you don't earn money during your time off, if it results in the publication of a book or research for a project, there may be tax relief opportunities.\n\nIf the sabbatical involves a legitimate professional activity, everything (within limits, of course) may be deductible as a business expense if you leave your home. In case you're ever audited, keep excellent records so that you'll be able to demonstrate how the expenses contributed to your business or career enhancement. If you stay away for a year or more, up to $80,000 in foreign income can be excluded from your federal income taxes. Renting your house and any repairs that are made while it's being rented are tax deductible, too. Talk to your tax advisor!\n\n#### BUDGET TRAVEL\n\nWhile some of you may be accustomed to staying at nice hotels and eating at good restaurants as part of your business or regular life, traveling on a budget during your time off can open you to the real life and culture of a country.\n\nWhen Cathy and Gary traveled in Asia for a year in 1985, they tried to live on $50 a day. \"We often did so by saving on transport,\" Cathy says, \"and that is where we really had some fabulous experiences, such as riding from Karachi to Peshawar, Pakistan in a brightly colored, patterned bus with people, poultry, and potatoes. Another was riding on a junk around Hong Kong with a Chinese family, rather than taking the tour boat. We visited the outer islands of Japan by traveling on a mail boat. In all cases, they were safe modes of transportation and quite clean . . . but not the expected way for tourists to travel. A highlight was traveling through the jungles of Thailand on an elephant, by boat, and on foot.\"\n\nThe point is, open your mind to new adventures on your Reboot Break!\n\nHousing can include hostels, bed-and-breakfasts, guest houses, or a friend's apartment. For those more outdoor-oriented, there is camping or caravan hires. And then there's always the overnight train that combines transport with housing. The real treat of staying in places like these is the people you meet, and it is those people and experiences you tend to remember, not the fifteenth Hilton hotel room you've stayed in.\n\nAnother advantage of budget travel is that you spend more time walking, exploring free sites, just \"being\" rather than spending money on shows, events, paid museums, etc. You have more time for reading, writing, hiking, and reflection. Don't exchange busyness at work for busyness in travel.\n\nOne way Cathy saved money on her sabbatical in 2007 was to trade her house in Santa Fe for a house in Dordogna, France, owned by a friend. Both were big enough houses to accommodate friends or family for three weeks, and the costs were minimal\u2014cleaning people, tips for the overseers, and utilities. It gave Cathy and her friends three weeks in France for the cost of getting there. Saving on eating out by cooking in breakfasts and some dinners was a plus. In fact, they would have missed out on the fabulous food markets had they stayed in a hotel. She still dreams about the good breads and pure butter from the markets. The fact that both houses were comfortable, attractive, and located in key tourist areas was fundamental to the reason for choosing those places for travel centers.\n\nDavid, an entrepreneur who lost his business and was in mourning for the time and effort down the drain, took off for Prague, where it was less expensive to live. \"I lived for four months on $1,000. I had saved some money, rented my apartment, and scaled way back financially. It was one of the most interesting times of my life. I learned things I could never have learned the way I lived before. Being free of the 'material stuff' allowed me to explore what life had to offer.\"\n\nOne way to travel free is to save up your frequent flier miles and use them for travel during the sabbatical. Use credit cards that help you to earn points, and try to concentrate them on one or two airlines.\n\nOther considerations include how long you want to be gone and whether you are willing to rent or trade your home. Do you need rental income, or will a trade work for you? Who could look after your property while you are gone? Do you have a place to store valuables? What kind of a place are you looking for? What would you have to do to your house or apartment to make it attractive to trade?\n\nAt the very least, you need to give someone the responsibility for looking after your home or apartment, paying bills, picking up mail, and having power of attorney for emergencies. You can automate many financial payments through direct deposit and bill payment with your bank or through a bill payment service.\n\n#### OTHER WAYS TO SAVE MONEY\n\nGet your family in on the planning on ways to save money. A 2009 _USA Today_ survey on the economic downturn showed that 74 percent of children were worried about, and 67 percent wanted to talk to their parents about, the family finances.\n\nBeth, a vice president of a major manufacturing firm in New York, enlisted her husband and children in the planning a year ahead of their family sabbatical. They took a year off, home-schooled their four kids, and traveled all over the United States in a retrofitted bus with a car towed behind. They tapped in to savings and sold some stock options to fund the year off. The kids were told they would have no spending money unless they earned it themselves. Each was told to create a job or find ways to raise money. At the time, they were two, six, nine, and eleven. The three older kids got into it and had jobs, such as pet sitting and lawn care, to raise their spending money for the year they took off. The whole family went through things to sell and had a huge garage sale, plus cleaned up their house! Everyone agreed not to buy new clothes for the year except at secondhand stores. They agreed not to order but one drink at restaurants and to eat healthy snacks along the route. While they had originally planned to sell their house, they decided to rent it and were glad they had it to come back to at the end of the sabbatical.\n\nYou don't have to travel far or spend much money at all to enjoy your time off. Many of the people we interviewed treated their hometowns as places to explore and acted as if they were tourists in their area, visiting museums, shops, schools, theatres, farms, and other places on long weekends or daily jaunts. It's time to realize it has been years since you went to some of the places your houseguests love.\n\n#### THE UNEXPECTED SABBATICAL SURVIVAL PLAN\n\nIf you have been let go by your employer, can you still fund time off without being worried about finances and finding another job when you return? The answer is yes, if you take a breath and think about it.\n\nFirst, see what can be negotiated with your employer. Is there a severance package? Outplacement services? Continuation of benefits? Use of company facilities? Payment for attendance at conferences? Payment for financial planning?\n\nApply for unemployment benefits as soon as possible. While it is not in any way a replacement for your salary, it is something and will help you get through. Look at applying for social security or company pensions if you are over sixty. You may find it worthwhile to begin drawing them earlier.\n\nSit down with your family and go over what has happened and what resources you have. By making them part of the process, they will worry less and you will all face the challenges together. They can also be a resource for you to explore new job and career opportunities.\n\nIf you have never created a budget, now is the time to do so. Treat the unexpected sabbatical as you would a regular sabbatical. Look where expenses can be reduced and where assets you may have can be rented or traded or sold.\n\nTake at least a couple of weeks to a month to do nothing but mourn the job loss, get yourself organized, and take time to reconnect with your family, your friends, your body, and your mind. Chances are it has been stressful leading up to the job loss, and you need the time to recoup mentally and physically before you start a job hunt.\n\nGive yourself time even when you are in the job hunt process. Even those on an unexpected sabbatical found ways, after taking a month or two to reboot, to contain their job hunt to three or four days a week to give them downtime to think more strategically.\n\nTHIS CHAPTER IS just the beginning of thinking about funding your freedom . . . there are many stories and good ideas to come in the next few chapters. The Appendix has a list of resources and a sample financial planning list that will help you think through what you need to do. There are myriad ways to fund your time off and enjoy the insights that brings. View it as one of the best financial investments you will ever make!\n\n###   EXERCISES\n\n##### _Exercise 4-1: Facing Your Fears_\n\n\u2022 What are your biggest fears about money and taking time off from work? Write them down. Write potential solutions next to the fears.\n\n##### _Exercise 4-2: Figuring Out What You Need_\n\n\u2022 Write down how much money you think you will need to maintain your ongoing financial obligations and expenses during your Reboot Break.\n\n\u2022 Write down potential sources of funds you might tap in to to finance your time off.\n\n\u2022 What is the gap and how will you fill it?\n\n(Note: the Appendix has a sample budget checklist.)\n\n##### _Exercise 4-3: Cutting Back on Expenses_\n\n\u2022 Write down ten ways you can cut back on expenses before and during your sabbatical.\n\n1 \"The New Consumer Value: 'Living Light.'\" _www.santa-fe-group.com\/wp-content\/uploads\/2010\/07\/NewConsumerValueNov2008.pdf_\n\n## | CHAPTER 5 |\n\n## _Phases and Navigating the First Thirty Days_\n\n\"Go confidently in the directions of your dreams!  \nLive the life you've imagined.  \nAs you simplify your life,  \nthe laws of the universe will be simpler.\"\n\n_\u2014 Henry David Thoreau_\n\nEvery Reboot Break is unique, but our interviews revealed a common pattern of how sabbatical takers allocated their time. Most tended to divide time into major blocks, each with a different quality and feel to it. Many people think about their break in quarters. The character and length of the phases vary, but in general they describe the rhythms you can expect during your time off. The next three chapters explore these four phases of a Reboot Break. This chapter introduces the phases and covers the first phase in more depth, especially the first thirty days.\n\n### PHASES OF A SABBATICAL\n\nI. Creating Space\u2014Putting your life in order\n\nII. Reconnection\u2014Revitalizing connections to people, places, activities, and self\n\nIII. Exploration\u2014Learning new things, especially through travel\n\nIV. Reentry\u2014Starting a new chapter of your life\n\nPhase I, the \"Creating Space\" stage, often involves sorting, cleaning, and simplifying one's environment. This is a time of regrouping, of clearing and \"getting your house in order\" so that the time spent can be as free and uncluttered as possible. Many people don't know what to do with themselves in the beginning. It's not uncommon to experience work withdrawal or contact withdrawal.\n\nIn particular, **the first thirty days are crucial\u2014they set the stage for the rest of the time off.** They are also, for many, a shock to the system, going from a busy work life to time off.\n\nOnce the Creating Space period is done, sabbatical takers feel comfortable expanding their horizons. They tend toward activities in their town or city that they enjoyed but hadn't done for years. We call this stage \"Reconnection.\" Reconnection is a time of reaching out to others and enjoying personal growth. During this period, people broaden their horizons by trying new things, renewing old connections, and rediscovering passions long left behind.\n\nThis is a time to explore new or old or forgotten activities\u2014from new foods to taking a dance class to picking up a new sport. The adventure of Reconnection is often coupled with the comfort and fun of reaching out to friends and family, especially people sabbatical takers have not had time to see during their busy everyday lives. They are able to enjoy the little pleasures in life, like reading the newspaper cover to cover, cooking dinner for a friend, going to a play, taking a walk, or writing in a journal.\n\nAfter a period of fun and revitalization, sabbatical takers are ready to test the waters with more adventure and greater risks, often farther from home. We call this stage \"Exploration.\" Whether traveling to Asia, taking Spanish classes, exploring your spiritual side on a retreat to the American Southwest, or embarking on a volunteer project in Africa, external adventures can lead to internal journeys and explorations of the mind and self, giving birth to new and unexpected interests and passions.\n\nAfter a time of Exploration, people are refreshed and exhilarated, and ready to consider their next gig. Reflecting on their experiences, ideally over a period of months, they are able to \"reenter\" and begin to shape a new stage in their personal and work lives. This fourth stage of the sabbatical, \"Reentry,\" is where a new chapter of your life begins. This is one of the most exciting and satisfying parts of rebooting your life, as you assimilate what you have experienced and make time for reflection. It is a time of new ideas, new possibilities, and new perspectives.\n\nWhile not everyone goes through the four stages in the same way, this last phase, Reentry, is one that should not be rushed. Too often, people try to predict what they will do at the end of the sabbatical before they've even started. We encourage readers not to feel pressure to answer when your families and colleagues ask what you're going to do at the end. The answer will come when it's time.\n\nTime for reflection is so important at this stage because it is where the activities and experiences, the emotions and feelings, and the wants and needs come together to pave the path to post-reboot life. Reentry is usually about coming to terms with the end of a wonderful period in your life and figuring out how to incorporate all those wonderful experiences in the future, as you set about to return to work or to start a new profession.\n\nThe next three chapters describe how we and those we interviewed divided and navigated our time off. This chapter is dedicated to the first phase\u2014the first thirty days and how to think about a daily schedule. The next chapter will segue into the Reconnection and Exploration stages, and Chapter 7, on Reentry, will address the process and potential fears of seeking that new career or returning to work when you're moving to a new chapter in your life.\n\n### THE FIRST THIRTY DAYS\n\nOne would think that the first day of a Reboot Break would be the happiest of one's life. No work, no worries, and a whole bunch of free time. What could be better?\n\nFor many, it is wonderful from the beginning, but others experience anxiety about the change. And don't be surprised if you even feel unwell in the early days. You might feel nauseated, have headaches, or find you can't eat. Some find that their bodies need to \"detox\" as the stress dissipates. As Lori shared:\n\n_I had a physical reaction. I was literally dizzy. My head was spinning from having operated at such a pace that my body had a hard time slowing down. All that adrenaline had no place to go. This was the most dramatic time. My euphoria was muted by confusion about my physical reaction._\n\nMany report that they did not know what to do with themselves in the beginning. It's not uncommon to experience work withdrawal or contact withdrawal. Rita said: I found during my second break that I suffered from email withdrawal.\n\n_For me, email was like brushing my teeth; it was one of the first things I did each morning. After getting over 100 emails a day and needing to check email multiple times a day, it felt strange not having this ritual or habit in my daily life._\n\nOthers reported sleeping until noon. Fear was common, too. Susan said she woke up on the first Monday and was terrified. She was so used to being scheduled all her life. Jason felt guilty not going to work. Jerry also said that at first he had trouble not being in a routine. He told us, **\"Starting a Reboot Break is a shock. It's a loss of rhythm, rhyme, and reason.\"**\n\nDr. Roberta Lee knows a lot about decompressing. As the Medical Director of Beth Israel's Continuum Center for Health and Healing in New York City, she has seen hundreds of people suffering from stress and burnout from work. She has written a book called _The SuperStress Solution_ and is a frequent speaker on the topic of how to relieve stress in our lives. She believes that stress frequently implies that you have gotten out of touch with what is comfortable in work.\n\nAnd she thinks Reboot Breaks are a great idea. \"They afford time to immerse yourself, to step out of your work self, and to see the totality of how stress got you where you are. The first thirty days need to be a time of recovery. Each person has his or her own rhythm. Everyone goes through different phases; there is usually a period of regrouping, and a period of the body recovering and recouping.\"\n\nStress manifests itself in different ways. For some it's in stomach problems, while others have neck or back problems. And for others their immune systems are more vulnerable. Some people are much more affected by stress than others.\n\nThere is also a difference in how men and women handle stress. Often women in high-powered careers give up friendships. Tending and befriending can help them recover. A distinct pattern of recovery with women is the need to talk issues out. Some career women have become so masculine that they assume the way to reduce stress is by finding quiet time, when in reality they need relationships. Men, on the other hand, tend to seek quiet time, \"cave time.\" But they need relationships too.\n\nDr. Lee believes the first thirty days offer an important time and space to ask yourself: Are you living well, and what helps or hinders you? How many friends and loved ones do you have around? How many have you pushed away? This is particularly an issue for men.\n\nShe recommends staying in touch with your dreams. Every life that sparkles is a much more stress-resilient life. It's good to have five- and ten-year goals and dreams.\n\nDr. Lee also recommends keeping a journal so that you can get in touch with parts of you that you have tended to ignore while you were in the highly focused part of your job. \"There are lots of things going on in our heads that we don't even know about. Journaling is a great way to get in touch with the ticker tape in our head. If you have the option of a sabbatical, I would encourage you to do it. It's a way of valuing your life and yourself.\"\n\nIn addition to journaling, we recommend treating the first week or even month of your break as a time for indulgence. Get a massage. Read junk novels. Make a schedule and fill it with activities that you enjoy. Jason, who turned from art framing to massage therapy, spent the first month learning the natural rhythms of his body for sleeping and waking, which helped to clear his mind.\n\nOne UK couple chose to spend their time off mostly traveling. They spent the first three weeks of their first month in the United States \"doing nothing\" in a remote, secluded cabin. \"It was fantastic,\" said Nigel. \"We did a bit of skiing, a bit of walking, and chilled out. It was a nice, relaxing way to start our year off.\"\n\nVictor found he needed two or three months to decompress. \"Not having a preconceived time schedule is very smart. Letting go of all the tension from work is a great thing.\"\n\nBeverly said, \"I loved the joy of not having to be in any one place at a certain time.\"\n\nJerry agreed, adding, \"There were fewer rules and less fitting in required. I didn't feel I had to be in places. I could sleep or exercise or not.\"\n\nAriane de Bonvoisin, reboot veteran and author of _The First 30_ _Days,_ said:\n\n_The first thirty days can last 10 or 200 because it's related to the mindset with which you approach change. There is external pressure to have a plan, but for me it was best not to have one. There were 100 books that I wanted to read and 10 countries that I wanted to go visit. I said to myself, let's just go have fun._\n\nWe advise all sabbatical takers to give themselves a break if they don't get everything done on their list right away. Nancy did just this at the beginning of her break.\n\n_After two years of working until 10 pm most nights and just generally having no balance between work and life, I left my job as president and CEO of an international non-profit development organization. It had been teetering between life and death when I took the job and was now going to survive. I had given it my all enthusiastically and was ready for someone else to steer it into the future._\n\n_The first thing I did was sleep. In fact, I would sleep long into the morning every day and still be tired. I had various ambitions related to going through boxes from the office and organizing other things around the house, but I didn't have the motivation. I worried that my whole break would be wasted on sleep and doing odds and ends around the house. The only energy I expended was on planning a trip, which gave me something to focus on. I knew the trip would inspire and recharge me. After the first thirty days I was off to Argentina and already feeling better._\n\n### CLEAR\/ORGANIZE\n\nWhen leaving the hectic pace of work and before launching into something new, we and other sabbatical takers found the need to clear our heads, clear the clutter in our lives, organize our dwellings, and organize ourselves. The first stage is a time to put things in order. Some decided that they wanted to take care of doctors' appointments, clean out that cluttered room, organize three years of loose photos, paint the kitchen, repair a broken bike, or visit relatives. This is a period of letting go, part of a major transition in which one is leaving an old life behind and preparing to be open to new experiences, change, and renewal.\n\nMany people reported gleefully charging ahead with a list of all the things they wanted to organize, clean up, or do. It didn't faze them that the list covered ten years of neglected chores.\n\nNancy and others sold their homes and handled all the necessary work associated with moving. Nancy says she has never gotten out of this phase and feels like she is in the movie _Groundhog Day._\n\n_I'm not the best person to tell a clearing-out story. You won't hear: \"Here's how I did it, and now I'm done.\" I'm a pack rat, and it seems like my clearing-out never ends. Boxes from two previous non-profit organization jobs. Old financial files (what to throw away, what to keep?). They all cohabited in my previous house and somehow ended up in this one too when I downsized. I did downsize many things to move into mycurrent house, mostly furniture and old luggage, and did it pretty efficiently, but it just didn't all get done. I even hired an organizer to help clear out before the move, but I hired her too late in the process to finish. She came for a few hours at a time for several days. I didn't rehire her for the new house\u2014silly me. The moving-in process meant newly organized drawers and closets, which was great, but many boxes found their way into the extra room._\n\nSome people use part of this organizing time to prepare for the next phases, Reconnection and Exploration. As you have read, Cathy spent the first part of her first break planning, packing, and organizing for an eleven-month trip through Asia, including spending time reading about each country she planned to visit.\n\n#### EMPTYING THE EMOTIONAL CLOSET\n\nMany sabbatical takers reported needing time to heal from the breakup of a romance, a divorce, or another sad or stressful experience. Whether it was having time in their own place for the first time in a very long time or setting up a new home, people talked about \"getting settled.\"\n\nLeena, whose main story of change and new love was in Chapters 2 and 4, began her sabbatical after her heartbreak by spending a few months in her home country of Lebanon. She focused on relaxation, catching up with old friends, and exploring lost relationships. She did lots of yoga and read several self-help books about mind, body, and spirit.\n\nFor her third Reboot Break, Regan had just returned to the United States from Bosnia where she was in diplomatic service. \"I came home totally fried,\" she said. Regan returned to New York just after the 9\/11 attacks. \"I needed time to find my home and how I would relate to it. Besides, I was really tired.\" So she decided to build a Japanese garden.\n\n_It became a mental vacation. I was away from political and environmental stress. It was a project meant to create beauty rather than workto build a nation. Whereas Bosnia was about the macro, the garden was about the micro. It was one of the ways I was able to refresh myself. The garden brought me home in a very concrete way._\n\nMary, the woman in Chapter 1 who was the first person to take a sabbatical at her workplace, took four months off in the summer from June until September. Who would have known that this perpetually smiling and cheerful friend was being torn apart inside? As a trader on the floor on Wall Street, Mary had witnessed 9\/11 firsthand and had lost many friends. She had been grieving over them for the past eight months, and at the same time bearing the burden of other family issues. On top of that, juggling taking her children to school, working under the pressure of a trading floor, cooking dinner, and being a good wife, mother, and friend just got to her.\n\nJust before she started her break she went to a spa for a long weekend. \"I needed to start my detox,\" she said. Then she and her family rented a home in the Berkshires. Her kids went to summer camp while she explored the area and started to explore her inner self. She started to explore many of the deep issues inside that she had never addressed.\n\n_I used to drive around the countryside and just cry a lot. My sabbatical was about opening up all the things that I had inside. I was emotionally drained. I needed to do personal therapy, which I had never done before. I never had the time to do it. I was leading such a busy life that I never stopped to feel. I had a lot of issues to face. For years I kept stuffing feelings and emotions into my emotional closet. I kept adding and adding things, and then one day the closet door just wouldn't shut anymore._\n\nThe other analogy that Mary used is what she called \"plaque.\" Plaque builds and builds. You can floss, but until you go to the dentist for a deep cleaning, you don't address the issue. She said that for the first couple of days you may feel crazy. It hurts while you're getting rid of all those germs, but then you feel great afterwards.\n\nThis might be a good time to step back and acknowledge the things that have been on your mind. It often helps to write them down and make a list of steps you could take to empty your emotional closet. Don't force it. You might want to take small steps and set yourself up for success along the way. And remember to congratulate yourself. If you take steps forward and then backward, you are normal. Just keep at it. Remember that you deserve to be able to open space in your mind for new things. Emotional baggage does not have long-term residential rights.\n\n#### A DAILY APPROACH\n\nLet's turn to how you might approach a typical day. Time and time again people we interviewed said that one of the challenges was the tendency to over-schedule vs. just relaxing. We recommend starting the day with a reflection period. Greg said, \"The first two hours of the day are my thinking and reading time. I still get up at the same hour, but I haven't used an alarm clock since starting my sabbatical.\"\n\nSome people identified the beginning of their break and the lack of a daily schedule as their low points. Susan, the banker you met earlier, realized how structured her life had become by her workday.\n\n_On Monday, the first day of my break, I woke up and felt that I should go to the office. My dad had coincidently started his retirement on the same day. I called him up and asked him if he was struggling with that first day as much as I was. We met for breakfast._\n\nAn ultra-organized person, Susan found it hard not to have a schedule, so she made one for herself. Regan felt similarly. \"This amorphous state, without the requirement to structure my day, left me shiftless and disoriented. It took me three months to leave unstructured time for myself. Afterward it was an irreversible insight.\"\n\nA friend of Rita's and former CEO who had taken a Reboot Break recommended reading _The Artist's Way at Work,_ a twelve-week self-reflection and writing course.\n\n_The book introduced me to the concept of writing 'morning pages,' three of them the first thing in the morning. I have been writing morning pages\u2014some call it journaling\u2014ever since. For someone who likes to charge into the day, stopping and reflecting on what happened the previous day\u2014what was significant, how a certain situation was handled\u2014was important. It also caused me to think ahead about that day's activities, which in turn allowed me to mentally prepare for and relish the activities even more. By the simple act of writing the date at the top of the page of my journal, I suddenly remembered that it was someone's birthday. Had I done my normal charging ahead, I would have missed that wonderful opportunity to call that friend or drop them a note on their special day. Which room or outdoor space I chose to write in was also a treat each day. It allowed me to appreciate my environment so much more, to hear the birds at 6 am, to feel the breeze through the trees, to sit in bed as if it were my throne._\n\nMornings are always a challenge for Rita because they are her favorite time in the day, and there are so many activities that she wants to do in a given time. Writing, yoga, reading the newspaper, reading a novel, gardening, studying Spanish, and other things could all take up four to five hours. During her time off work, she tried to divide her days so that she had reflection time, time for emails or general paperwork and bills, cooking, and one sports activity a day, whether it was yoga, tennis, golf, swimming, kayaking, skiing, Pilates, or hiking. She even snuck in the occasional nap, as many others reported doing. Cathy said, \"Each day my goals were to write my morning pages, exercise, reach out to friends, do something career-related, and enjoy downtime.\"\n\nAnother concept in the book _The Artist's Way at Work_ is about taking an \"artist date\" once a week, usually alone. It could be walking in a beautiful field or looking up at the architecture in a city. The date could be drawing, visiting a museum, or going to a show, a dance performance, or a concert. It is anything that awakens your artistic sense. Even those who are not artists have come to love their \"artist dates\" and ensure that they have at least one a week.\n\nMark, who left his high-stress New York job for Brazil and working in non-profits, spent his days reading three newspapers first thing in the morning, one to three hours of physical exercise, cooking, exploring, and lying on a beach. In between, he did volunteer work for various NGOs. He liked keeping a schedule of activities, but he also made sure that he left spaces of free time for himself. \"The high point of the day was waking up every morning feeling as good as I did, remembering the day before and what I did, what I said, and to whom I said it.\"\n\nAnother common refrain from the people we interviewed is the enjoyment of things that they hadn't done in years. \"I read the paper and watched the news, which I had never done.\" \"I spent more time on the beach than I had in the twenty-five years I lived there and enjoyed it more than before.\" \"I could go to dinner without having to be on call.\" \"I had the joy of not having to be someplace at a certain time.\"\n\nEveryone's definition of what fun is or what they want to clear or organize in their life is different. There is, however, the common theme of taking time for oneself and rediscovering the joy that exists in doing little things. After an initial adjustment, people basked in their freedom. Several people we interviewed actually used the same phrase to sum up how they felt starting their Reboot Break: \"It was terrific!\" And it gets even better in the Reconnection and Exploration phases.\n\n###   EXERCISES\n\n##### _Exercise 5-1: What I'll Accomplish_\n\n\u2022 List five things that you hope to accomplish during the beginning phase of your Reboot Break. Whether it is cleaning out the closet, making overdue doctors' appointments, or whatever you may have been putting off due to your work schedule, tackling these will allow you time to organize yourself and free you up for the next stage of your time off.\n\n##### _Exercise 5-2: Clearing My Emotional Closet_\n\n\u2022 List the emotional areas that need to be addressed and cleared out in the first phase of the break so you can be open to new exploration and possibilities for your future.\n\n## | CHAPTER 6 |\n\n## _The Heart of the Reboot Break: Reconnection and Exploration_\n\n\"Twenty years from now you will be more disappointed by the things that you didn't do than by the ones you did do. So throw off the bowlines. Sail away from the safe harbor. Catch the trade winds in your sails.\n\nExplore. Dream. Discover.\"\n\n_\u2014 Mark Twain_\n\nYou have come through the first phase of the Reboot Break. You've stepped off the work treadmill and cleared away some of your clutter. Order may have been restored in closets or the garage, old emails have been purged, and you've worked on your online photo albums. (Or\u2014for those of us of a certain age\u2014precious photos have been moved from a stack of shoeboxes to family albums.)\n\nNow you are headed into the heart of the Reboot Break: the Reconnection and Exploration phases. This period is what you went on sabbatical to experience, and it is usually the most satisfying part of the process. Go in search of your dreams. During the Reconnection phase, sabbatical takers tend to stay closer to home, as they strengthen relationships and go back to old or forgotten but beloved activities. Reconnection is about getting more comfortable in your reboot skin. And once that happens, it's time for Exploration, which is often a time for new challenges and adventure away from home territory.\n\nKim's break unfolded in a way that clearly illustrates the two stages. At fifty-nine, she left her executive job at the Portland, Oregon, mass-transit system. She had logged eight years there and nearly thirty in other public roles, and she was ready for a break. When she reached the Reconnection stage of her between-gigs sabbatical, she had already organized the basement and done those photo albums. Then she hosted a family reunion, went skiing with friends, and took on new volunteer roles in public service. She and her husband also decided to downsize and move into the city, so she orchestrated the process of selling the house and the complicated move, while also fitting in regular trips to the gym and taking time for herself. Like many sabbatical takers, she wondered how she ever had time for a full-time job.\n\nMoving to Exploration, Kim and her husband traveled to Africa to go on safari and visit their son in the Peace Corps. It was a decided drain on the family savings, but she convinced her husband that it was a \"once in a lifetime\" chance. Then they ended up moving again, this time from their cramped temporary apartment to another house. Kim was in such a reboot rhythm at that point that she fit the move in around knee surgery, a trip to the Galapagos with her other son, and a jaunt to Washington DC to represent a non-profit at a political event.\n\n### RECONNECTION\n\nDuring her Reboot Break, Jaye was so excited to have time to herself that she was giddy with all the possibilities before her.\n\n_After I had settled into a new sleeping and exercise routine, I started calling and surprising old friends. What a thrill! We planned lunches and some weekend getaways to renew our relationships and catch up on years of our lives. Some were divorced, parents had died, illnesses had transformed us, and now we could finally support each other. I was quickly forgiven for not being there as much as they and I would have liked. I also tried to travel somewhere new each month._\n\nReconnection is a time of expansion too. What does it mean to expand? According to _Webster's Dictionary,_ expanding is unfolding, enlarging, opening, or spreading. When you expand yourself, you feed your mind, your heart, your soul, and your spirit.\n\nReconnection is a time for serious personal growth. Many rebooters described feeling like they were being revived during this period\u2014being brought back to life, to consciousness, or renewed strength.\n\nMost people find that they want to try something new during this phase, from taking up a new sport to art lessons to volunteering. But they also love doing the fun or satisfying things that were crowded out of their working lives. Maybe it's reading a historical novel or the latest magazines, seeing the Picasso exhibit at the local art museum, or spending a summer afternoon hiking with the kids.\n\nMarvin had a blast during the Reconnection phase. He did home renovation projects he'd put off for years, skied on weekdays, read, and learned French. Having the freedom to do nothing or a lot, depending on their mood that day, is a common pleasure of people we interviewed. **Having the freedom to set your own schedule is a gift of time,** and it is one of the most important parts of the Reboot Break, especially in the Reconnection and Exploration stages.\n\n#### FEEDING THE BODY\n\nIf you've been neglecting yourself physically, the Reconnection phase is a good time to establish new habits and even set goals for your health. That might mean jogging or swing dancing, or just getting into the fresh air each day. It can also be a time to test yourself physically by running a road race or trying rock climbing. Rita started her Reconnection phase by taking a yoga class.\n\n_Having spent so much time on airplanes, my back was killing me, and I was very stiff. To feel good mentally, I first needed to feel good physically. I soon added a twice-weekly tennis game to my yoga routine. I hadn't played tennis since I was a child. It was a little embarrassing picking upa racquet some thirty-five years later, but I had always loved the sport and the time was right._\n\nJan is the journalist you met in the funding chapter. Over the years, she had stopped paying attention to her health. She ate too much junk food, she hardly exercised, and she was lucky if she got six hours of sleep most nights. On her year-long Reboot Break, Jan's first goal was to build some form of exercise into her life. She chose swimming because she could tag along with a friend and use her friend's routine to help establish her own. Twenty years later, Jan is still a regular swimmer.\n\nFeeding the body can mean finally getting rid of a nagging health problem like backaches. It can also mean being kind to your body by having a massage, seeing an acupuncturist, or quitting smoking.\n\nFor some people, feeding the body simply means spending more time in the produce department and less in the junk food aisle. Just by changing your diet, getting enough sleep, and getting a little exercise, you'll be happier and more energetic, leaving you in great shape for everything else you'll be doing during the rest of your time off.\n\n#### FEEDING THE SOUL\n\nJust as feeding the body prepares your physical self for other adventures, feeding the soul gets your mind and spirit in shape. Food for the soul comes in three varieties: having choices, enjoying the moment, and clarifying goals.\n\n#####  _Having Choices_\n\nReconnection can be all about choices. You choose how to spend your time, and you can try new things\u2014or not. Rita says:\n\n_I visited at least ten museums during this period. It wasn't that I was an ardent art lover. Just the opposite. But given the time and freedom to choose, I wanted to know what I was missing during all those years of non-stop work when my friends were visiting art galleries. Learningabout art became a personal goal. As important as the activity itself was, the ability to choose to do it was more important._\n\nA new-found sense of spontaneity can be one of the best rewards during the Reconnection phase. For most of our everyday lives, what we do and how we do it are in large part determined by our bosses, customers, spouses, children, and other important people in our lives. A Reboot Break can provide the simple freedom to decide what you want to do and when and how you want to do it. It can take some getting used to, but we found, as others did, that it can lead to personal discovery and growth.\n\nGail took time off from her executive bank job to figure out how to move into not-for-profit work. During her Reconnection phase, she watched _The Today Show_ every morning and went to the gym every afternoon. But the rest of her time was open to whim. She often had lunch with friends on the spur of the moment. \"I was willing to be very spontaneous, and I was unscheduled most of the time.\"\n\nGail spent a lot of time on her own. She wouldn't hesitate to go alone to a movie in the afternoon or a Broadway matinee. She visited all the museums in Manhattan and thought nothing of hopping on the subway just to shop at a specialty food store downtown. Her friends said they had never seen her so happy. So many people we interviewed told us they had no idea before taking a sabbatical how much stress they carried all the time. Sabbatical takers are often told they look ten years younger.\n\nMany people choose to add community service or work with nonprofit organizations to their schedules during Reconnection.\n\nBetsy, the forty-five-year-old North Carolina orthopedic surgeon introduced in Chapter 3, always felt she had missed out on a key part of community living. As a successful doctor, she never had the time to do volunteer work on a regular basis between medical school, residency, and her long office and hospital hours. It was important to her to serve the community during her time off. \"I wanted to see what it was like to be part of my community without the conflicts of work all the time,\" she said.\n\nJust before taking time off, she had been elected president of the local medical society for a one-year term. During her Reboot Break, she led the organization in creating Project Access, in which each physician in the county agreed to see a certain number of uninsured patients each year. Also, as you'll read later, during the Exploration phase, Betsy deliberately chose to travel to faraway places where she could participate in deeply meaningful and important volunteer projects.\n\n#####  _Enjoying the Moment_\n\nBuddhists have been preaching mindfulness, or being in the moment, for thousands of years. It means being fully aware of what is going on inside and outside yourself at any given moment. Mindfulness is all about training yourself to put the brakes on all those thoughts running through your head, so you can enjoy what's going on right now\u2014to stop the past and future from stealing your present.\n\nIn his book, _The Power of Now: A Guide to Spiritual Enlightenment,_ Eckhart Tolle emphasizes the benefits of being present in your life now and creating greater consciousness. He talks of how to access our own power and inner resources to integrate more fully who we are, so we can create a more authentic future.\n\nAriane, the _First 30 Days_ author you met in Chapter 5, experienced significant internal growth during her sabbatical: **\"My sabbatical totally changed me\u2014physically and in terms of my energy,\"** she says. \"I developed a lightness. The biggest change was inner reliance. I have a deep, intimate relationship with myself now. The sabbatical made me softer, more flexible, more 'in the flow' of life, and less fearful.\"\n\nIn our time-starved environment, we often spend more time planning than doing and don't take the time to just be. Reboot Breaks allow the time and perspective for just being and not doing. It might be sitting on the beach enjoying the sunset or the sound of the waves. It might be listening to the conversations around the dinner table with no interruptions. Or it might be painting a still life and keeping your own thoughts still.\n\nNancy wrote about a time of being in the moment in April 2007.\n\n_I scheduled a trip to the Bay Area for my daughter Rachel's last college lacrosse home game and her twenty-second birthday. The best airfare meant traveling on a weekday and staying an additional day and a half before flying back home to Washington DC. Being on sabbatical, I had time flexibility, so I bought that flight. Once I was in California, I decided to use the extra hours to wander up the California coast._\n\n_I write this on a perch overlooking the Pacific Ocean, with waves crashing below me onto a Stonehenge-like rock formation, then subsiding into turquoise and white froth before pounding again. I've come from a wonderful time with Rachel and her teammates and friends, and am now deep into my own time._\n\n_Last night, I drove across the Golden Gate Bridge and continued north. The spectacular drive at sunset flooded me with unexpected emotions of release and thankfulness. This morning I've loved climbing across rocks, walking a flowered field, and now sitting here quietly. Within seconds, I will put my pen down to just \"be\" here. Later, I will drive through the peaceful wine country. What a joy. It is giving me such a peaceful moment to reflect._\n\n#####  _Clarifying Goals_\n\nWho says January 1st is the only day you can make a resolution? Reconnection can be a time to set goals for your sabbatical, or for your life. Think of it as an extended New Year's Day. It can be a period of stretching and testing yourself, physically or mentally.\n\nAlong with getting regular exercise, Jan, the journalist we met previously, decided to use her time away from work to explore the concept of \"enough.\"\n\nJan had always been driven. She was, in fact, a perfectionist. In her line of work, she had to be. Every day she dove into details and worked long hours, digging until she had the full scoop.\n\nJan had all the skills she needed to be a great reporter. But what she didn't know, especially after so many years of ambitious work, was when to stop. She decided the most important goal of her break would be to learn when she had done enough. \"I wanted to let go of my need to always push myself to do more,\" she said.\n\nJan applied and was accepted for a fellowship for journalists at Columbia University. Though she was away from work, she was immersed in classes and studying. So she set rules for herself: she felt herself wanting to study all night, but she forced herself to close the books and just go to sleep. Pretty soon she saw she could get a good night's rest and still get her work done.\n\nEvery day, Jan became more conscious of the choices she was making. Over time, she learned that \"enough\" was really about priorities. Is getting a little more work done more important than a good night's rest? By making everything a priority, Jan was leaving an important element out of her work: herself. When she was able to pull off the superhighway of her career and look around, Jan could see that a little more work definitely wasn't more important than sleep. Quite the opposite.\n\nWhen she went back to work, Jan became a national assignment editor at CBS, and then an off-air reporter at ABC. Her workload grew and grew. She was bombarded with more information than ever before.\n\nEvery day, though, she used what she had taught herself during her break. Instead of making everything a priority, she switched off her autopilot. She still worked quickly, but now she checked in with herself. After a careful evaluation, she pursued some stories and let other less compelling or interesting ones go. Over time, what had been prioritization grew into that all-important reporter skill: instinct.\n\nJan went on to become a TV anchor, one of the most demanding jobs in the business. \"The 'art of enough' is one that you have to learn and use all of the time,\" she said. \"It's not a matter of getting extra credit. It's all about doing something well, mastering it, and moving on.\"\n\n### EXPLORATION\n\nNancy tells an Exploration phase story that combines a dream and travel time with her younger daughter:\n\n_The phone rang in early June, 2008. \"Mom, I've got from June 23 to July 2 if you want to go on a trip. Did you say you wanted to go to Morocco?\" It was Rachel. She would graduate in a week, and we'd been talking about having a mother-daughter adventure before she started a new job at a solar power startup._\n\n_Within two weeks, we were having our own solar power experience on camelback in the hot Moroccan sand dunes. I had had one last major goal on my sabbatical during the Exploration phase before going back to work, and this was it: the African desert. Now I was looking for inspiration as a Berber guide was leading our camels across the impossibly beautiful desert sands at dusk. Our heads were wrapped in bright cloths against the wind and blowing sand. That night we slept outside under the stars before heading back across the dunes\u2014this time bathed in morning light and total stillness\u2014and on to Marrakech and the rest of our Moroccan adventure._\n\n_It was an exotic exploration trip, but it could have been a trip to a national park or Boston or Disneyland. The point is that I was on a Reboot Break and had the flexibility of schedule and mind to plan a spontaneous trip with my daughter._\n\nThe Exploration phase deepens the personal enrichment and growth of Reconnection. Choices, reflection time, personal challenges, family, and friends all remain part of the picture. This is a period of pushing yourself by trying or learning more new things and traveling to new places, and it is likely the time when you achieve major goals of your sabbatical.\n\nYou may find that you use the Exploration period to explore intensively new places in your own mind and self. Rita says, \"I am scared of heights, but in Belize a friend convinced me to go zip-lining, swinging through very high trees on a wire, connected by a harness. It was exhilarating. That day I bought a tee shirt that has become my favorite. It reads: **When was the last time you did something for the first time?**\n\n#### EXPLORING THE WORLD\n\nMany cultures encourage people to take time off for travel. Australians traditionally take a month off each year, often using that time to travel. Young Israelis who have finished their military duty often take a year off to see the world. In the United States, though, few people get more than two weeks of paid vacation a year, and not many can afford to use all of their vacation time for travel. In May 2009, _Digital Journal_ reported that the number of Americans who said that they were going to take a vacation in the next six months was at a thirty-year low, according to an April survey. Expedia also found in their 2009 survey that one-third of employees don't take all of their vacation time.\n\nGiven the statistics, it's hardly surprising that so many Americans use their Reboot Breaks to travel. Greg, a former CEO, echoed the sentiment of many when he described the first trip he took in his sabbatical: \"A month of walking in the English Cotswolds was the longest consecutive time off I had taken in twenty years. It was the only time I can remember when I didn't have to do something else.\"\n\nSome people hop in a car or rent an RV to crisscross the landscape of their own country that they've never explored. More often, sabbatical takers choose to go farther afield. Trips are as diverse as one can imagine\u2014walking tours in England, foreign language immersion classes in Germany, cooking classes in Thailand, windsurfing in Australia, skiing in Norway, becoming a registered tour guide in Brazil, teaching English as a second language in Guatemala, and visiting the African desert, to name just a few.\n\n#####  _Volunteering_\n\nSome people incorporate volunteer work into their travel. Betsy, the orthopedic surgeon you read about earlier, wanted to give back during her Reboot Break, even as she was traveling. Her parents had reinforced this value when she was growing up, and two close friends her age had just died. \"I felt my mortality,\" she said quietly. \"I felt I needed to do things that were important to me before I died.\"\n\nAt the top of the list was a month-long medical volunteer project in South Africa for Health Volunteers Overseas. Her orthopedics assignment was in an area with no water or electricity. Betsy was deeply struck by the poverty of the area, juxtaposed with the people's generosity and gratitude and the beauty of the countryside. She loved putting her medical skills to use with these people. \"The work of this trip was the reason I went into medicine.\"\n\nTravel often gives people a new perspective on life, and volunteering can bring that new perspective into sharper focus. As part of their year-long sabbatical, British lawyer Nigel and his wife Sarah worked in Trinidad for two months doing environmental conservation work and coaching village children in soccer. They found that their perspective was broadened by living in a new country where everything about daily life was different from back home.\n\nNigel felt this even more strongly later in his time off, volunteering in Ecuador. He and his fellow volunteers arrived to create a summer camp program for children in a remote village. The village had just been damaged from excessive rainfall, so the group spent the first three days clearing out a mudslide and creating fields and areas for sports.\n\nThe camp was a huge success, and for Nigel it offered an important window into village economic realities, fears, and hopes. A little girl, Isabella, especially caught Nigel's attention. She was usually off by herself and seemed awkward with the other children. When Nigel asked about her, the response was, \"Oh, she's that way because her family can only afford to send one child a year to school. Schools are free, but the books and other necessary materials add up to $20 a year. Her brother goes one year and then she goes on the alternate year.\"\n\nNigel was profoundly moved. His perspective has shifted forever. Back in his law firm, when someone demands a rush turnaround on something, Nigel frequently thinks of Isabella and the $20 annual school fee. \"Yes,\" he thinks, \"it's important to you and I'll do my best, but this is not the most important issue in life.\" Today, Nigel continues to financially support those needier than he and to find ways during his free time and vacations to volunteer. And the summer sports camp set up by a handful a people during their time off from work continues today.\n\nSometimes it's hard to explain these experiences and their profound effects to others. Mark and Margaret traveled to the Sudan to volunteer with victims of famine. They set up a medical and feeding clinic and later a project for homeless street kids. Clearly what they experienced firsthand was different from most people's everyday experience back in their hometown.\n\nWhen they returned home, they thought their friends and family would want to know all about their travels and what life was like in a country so different from their own. Instead they discovered that people gave them little time to describe their extensive experiences and volunteer work.\n\nWe and other sabbatical takers who traveled to faraway places had similar experiences. Somehow, there is just a little less enthusiasm for seeing those hundreds of photos or hearing minute details about a story that you still feel in your bones, but that is just too remote for others. In time, Mark and Margaret learned to value their experience for themselves and to accept that others wouldn't be able to understand it in the same way.\n\n#### EXPLORING THE MIND\n\nMany people devote the Exploration phase to learning. Some take language courses. Some enroll in degree programs. Some do it to change careers. Some want to learn new skills that will bring higher pay.\n\nThose who opted out of work to study for a higher degree or more skills training got a bonus: more pay. Some chose further education as a mid-course-correction. Kevin says that at forty-five he was done with the business world and wanted to teach. And he went off to Texas to get his PhD.\n\nLeslie, at thirty-five, says \"I'm done with being a secretary who gives all her friends fabulous home decorating advice. I'm going to get a degree and start my own business.\"\n\nSeveral people liked their work and planned to return to their positions but felt unchallenged and needed a change. Learning a new skill or language got them excited again.\n\nChauncey turned a traditional academic sabbatical into an international networking and learning opportunity. He went from an assistant professor of marketing at a business school to someone on a super-charged Reboot Break having exciting and diverse experiences. At the same time, he was enhancing his academic work and benefiting his university.\n\nHe began his time away teaching at Beijing University for the fall semester. It was the start of an enduring academic relationship that would enhance his school's China study tours and international curriculum development. In the spring, he taught at the university of Split in Croatia. While there, he initiated an effort to build a center for the multi-disciplinary study of sustainability on the Croatian island of Vis.\n\nThen he got involved with Starbucks to fulfill another of his reboot objectives, completing and publishing a corporate social responsibility (CSR) case study. He also used this opportunity to publish a manuscript on advertising fair trade coffee.\n\nChauncey summarizes, \"Having new and exciting experiences rejuvenated my teaching. Among the other benefits, I got to build a unique laboratory for students to discover and learn global aspects of sustainable management practices. And it provided me access to hard-to-find research data for meaningful academic publishing.\"\n\nIt is less common for teachers in elementary or secondary schools to take sabbaticals, yet the bold ones do, enhancing their skills and their pay.\n\nWyatt, a high school physics teacher, used his one-year sabbatical to go to graduate school to enhance and complement his existing knowledge. For him, the free time was as important as the time he spent earning his master's degree in environmental science at Brown University. During the winter recess, he traveled to Thailand for one month, a trip he would not have been able to take without his educational sabbatical. He also used some of that time to explore a career change. He went back to teaching for a year, and then moved to a job in environmental policy. This kind of exploration is not uncommon among sabbatical takers. It's a great time to test the waters of change.\n\nWyatt wasn't burned out. He just wanted to take a year to do something that he couldn't have done otherwise and enhance his skills and perspective as a teacher. And, of course, the pay increase for the new master's degree wasn't bad\u2014either as a teacher or in his new career!\n\nMany non-academics in the Reboot Break chorus used their breaks for intellectual exploration. The young banker Susan felt stale and unchallenged at work. At thirty, she was the youngest in her group of fifteen at the bank, and the only woman. It was depressing to her that her co-workers had spent thirty-five years at the same desks. That was not what she wanted, so she set out to enrich her credentials.\n\nSusan applied for and was accepted into a study program at the London School of Economics. She found the course material and discussions very stimulating, but equally stimulating was the travel that she was able to do every weekend, using London as her base. Then she traveled to new places for the rest of her year. Susan returned to her company refreshed and rebooted. She has since been promoted twice and is thrilled that her company relocated her back in London for her next assignment.\n\nSometimes the education on a Reboot Break is less formal. Gary was a senior public relations and communications coordinator who aspired to be a published author of children's books. During his time off, Gary wrote, took a writing course, attended writers' conferences, and searched for a publisher. He also joined a professional writers' organization to meet other aspiring writers and stay abreast of events and opportunities in his area.\n\nA surprise came in the form of photography and a course-correction. Gary had always had an interest in photography, but during this period it blossomed into a passion. He never expected to uncover another creative outlet, but there he was, taking photos all day long. Today he has created a website and has even sold some of his photos. What was a hobby is becoming a new career expression. \"It's hard to describe the lightness, the serenity, joy, and creativity I felt\u2014and still feel,\" said Gary about his Reboot Break.\n\n#### EXPLORING THE HURT\n\nRobert Frost once said that \"the best way out is always through.\" Sometimes the most important work of exploration is to acknowledge a painful situation, face up to it, and let it run its course. That's not as much fun as learning French or hiking in the Himalayas, but if you're hurting, sometimes it's the only way to heal.\n\nDavid, the entrepreneur in Chapter 4, found his once-very-successful business suffering from stiff competition from cheaper labor sources overseas. He spent all of his own money trying to stem the company's losses. Instead, he lost everything. Soon afterward, his partner died. When it was over, he was financially ruined and emotionally drained. David thought about finding another job, but in his heart he knew he needed a break, one that would take him to a world totally different from the one he had left behind. He went to Prague.\n\n_I spent my days waking up whenever I wanted to and hanging out with whomever. I just spent the time doing nothing but looking at life through conversations with people I would never have met under normal circumstances: people working for Radio Free Europe, bar owners from Benin and Nigeria who were trying to make it in the West, Czech people caught up in the transfer of the country into the European Union. In Prague, time stopped. Whenever anyone offered me an opportunity to do anything, I said 'yes,' whether it was touring the countryside, going to a beer fest, or visiting a castle. My trip felt like something I should have done when I was younger, but I never took time off after college. Not a day off. My time in Prague felt unnatural after working so hard for so long, but it was completely natural._\n\nDavid had no definite end to his break. He was very much between gigs. He took comedy classes back in the United States and kept working on a book he had started writing in Prague. He was still recovering from the feeling of failure about losing his business.\n\n_It takes as long to come out of something as it takes to go in. People give far too short a time to grieving. Same with my company. I had spent two to three years invested in making it happen, building it up, and watching it crumble. It took two to three years for me to come back. What finally ended this for me was the book. It gave me the external validation of a success, which allowed me to say, \"I really am good at something and I can now go to my grave with that.\" No one can take that away from me. And if I had never gone away, I never would have had that. I would have kept spinning my wheels._\n\nNow David is a freelance corporate consultant on technology-related topics, specifically business continuity and \"disaster recovery.\" He is writing his second book.\n\n#### EXPLORING THE SELF\n\nThe Exploration phase can sometimes bring isolation and loneliness. Whether your journey is physical, intellectual, emotional, or all three, being far away from your normal network or experiencing a major life change can be scary and lonely. But it can also be a gift, allowing for profound self-discovery and growth.\n\nDavid, as we mentioned, moved to Prague. \"Leaving the country was important,\" he said. \"I had to get as far away as possible so I wouldn't be able to reach into my old life and fall into old habits. The time zone difference helped, as did the physical isolation of the city.\" For David, being removed from everything familiar freed him to do the things that he had always wanted to do.\n\nSusan, the banker who took time off to study in London, felt similarly, saying, \"My time in London was the only time in my life when I had no connections and was so completely by myself. The first week I thought that I had put myself way too far out there. But once I started to meet people and to create a new social structure, I was fine. I started to travel, too. I had felt totally disconnected, but my new environment allowed me 'think time.'\"\n\nLike many of the people we interviewed, Susan is a doer. She's smart, efficient, hardworking, and usually takes on several activities at once. But all this \"doing\" hadn't allowed a lot of time for reflection and getting to know herself. When Susan was forced to be alone, she was able to discover the things that were truly important to her. Susan returned to the same company after her year off, but she switched from portfolio management to hedge funds because of the insights she gained about herself and her needs.\n\nFor some people, self-discovery means spiritual exploration. Many begin keeping a daily journal. Some decide to go to a place or participate in a process that will help them get more in touch with their spiritual side. We know people who took classes in meditation and built that into a daily practice. A few stayed in an ashram to learn the benefit of deeper thought in mediation, and others visited spiritual centers or listened to spiritual leaders. Lorraine, a conflict resolution expert, retreated into silent meditation for a week at a monastery in the beautiful Virginia countryside. Speaking with no one but herself and the universe, immersed in the simplicity of her surroundings, she got back in touch with herself and what is important to her.\n\n#### EXPLORING RELATIONSHIPS\n\nJaye was looking for an opportunity during her break to get to know her brother's two children better. They were already eight and twelve years old.\n\n_Fortunately my brother provided the perfect opportunity to spend a few weeks with his kids while he and his wife spent time together and had their own reconnection. I jumped at the chance. We had a wonderful time together getting to know each other in a way quick family visits would never allow. It was a gift of time for us all._\n\nIf alone time is the yin of exploration, relationships are the yang. A sabbatical is a perfect time to connect with the people you love, whether it's traveling across the globe to meet a long-lost parent or getting reacquainted with your spouse of twenty-five years. The extra time is precious, whether with children, siblings, parents, or friends. It can be planned or spontaneous\u2014a family reunion you host or a pickup softball game.\n\nMichael took his first Reboot Break of three months from corporate work at thirty-five. For three weeks he traveled through northern Europe with his new girlfriend. The trip cemented their relationship, and they ended up getting married a year later. When his girlfriend had to return to work, Michael invited his younger brother to join him. Michael explained that they had never really gotten along well because of their age difference, yet they both found that this time together created a great bonding experience.\n\nMichael's other objective for his time off was to meet his biological father. His mother had remarried when he was very young, and he had never gotten to know his biological father. So he traveled to Germany, where he knew his father still lived. He found his number through relatives and called. \"Hi, my name is Michael and I think I'm your son.\" After a long pause, the response was, \"Yes, you are.\" Michael and his father spent four days getting to know each other and sharing stories of their lives. They have become close friends. Michael now goes to Berlin every two years to visit his father and half-brother, another new friend.\n\nJerry, an executive recruiter in London, had hoped to use his time away from his job to build a second home and get back to competitive sailing. Then he found out that he and his wife were expecting. Sailing would have required spending twelve weekends away from his wife and newborn baby, so he changed course and spent his sabbatical doing things around the house and immersing himself in parenting.\n\nFor some, a Reboot Break brings the bittersweet gift of time with a dying parent or loved one. You have the time to say goodbye meaningfully, doing things you might not otherwise have had time for, such as making a scrapbook of your father's life or listening to your grandmother's favorite music with her.\n\nRod was let go as part of a corporate restructuring. When his father, who lived in a different town some distance away, became seriously ill, Rod was able to travel there often. When he was home, he found he had more time with his grandchildren.\n\nFriends can be an important part of exploration, too. Regan, the diplomat who returned from Bosnia, devoted her three-month break to traveling **\"Reconnecting with my friends energized me in ways I never expected,\"** she said. Her friends helped her understand what she wanted from her life and her work. \"They reminded me of my passions, my skills, my talents, and my dislikes,\" she said. \"My friends reminded me of who I was.\"\n\nBeing with family and friends can be the most reaffirming part of your Reboot Break. Its importance was mentioned by every person we interviewed. It solidifies the base of who we are and what we care about.\n\n#### KEEPING IT SIMPLE\u2014PACKING LIGHT\n\nYou cleared space and put things in order in the first phase, but this phase goes more deeply into simplifying your life. Clearing away more of life's clutter can open you up to new experiences and tune you in to the things that really matter.\n\nCathy's first Reboot Break was a time of freeing herself and packing light.\n\n_It was 1984 and my mother had died the year before. I had spent many trips back and forth to Missouri to care for her and her affairs. I was exhausted both mentally and physically, and felt I was carrying the weight of the world on my shoulders\u2014in fact my back started hurting, symbolic of that. I also had been working on my doctorate, having finished the comprehensives and coursework, and I was having difficulty with my dissertation advisor, who was always giving me veiled hints that if I slept with him, the process would go smoother, never mind that I was married._\n\n_Then there was my marriage and problems there that needed focused attention. Everywhere I turned, responsibility piled on. I dreamed of being free, of leaving problems behind until I could get my head cleared. I dreamed of freedom and packing light, both mentally and figuratively._\n\n_When my husband and I decided to take a year-long Reboot Break, using some of the money I inherited from my mother, that's just what I did, taking the time to mourn, reassess my career objectives, and be free. We packed one suitcase each\u2014a Kelty frame suitcase that turned into a backpack\u2014and one small backpack, and took off to travel in Asia for nearly a year. It was one of the most freeing times of my life, and I learned the value of \"packing light.\"_\n\nLeena, whom you met before, rented out her apartment in London to finance her break. In addition to the joy and enrichment of traveling, Leena discovered the pleasure of \"traveling light,\" telling us, \"I liked not having lots of baggage that could weigh me down and being free of the obligations of my home and belongings. It allowed me to focus more on the people in my life and the direction of my life.\"\n\nThis notion of \"packing light\" or \"traveling light\" can refer to emotional baggage, too: letting go of a schedule, going with the flow, or being unburdened. By packing a small \"emotional suitcase,\" Cathy, Leena, and others left room for more important travel essentials, such as confidence, belief in themselves, a sense of humor, happiness, and a complete openness to adventure.\n\nAriane was thirty years old when she left her corporate job to take a year-long Reboot Break.\n\n_The job I was in just wasn't what I wanted. I thought that being in charge of venture capital for a major media company would make me happy. I realized in the first thirty days that it wasn't right for me, but I stayed for two years anyway. I had a nudge from the universe when my company merged with another, and I was told I would have had to move to another state. Around that time, I woke up with a rash from head to toe that lasted for a couple of days. All the tests for allergies came backnegative. It was then that I first spoke the words to myself, \"I am allergic to my life.\"_\n\n_One of the first things I did was to go to Italy, which I thought of as a warm and friendly place. I took only a few things with me, mostly books. I also packed light mentally. I lowered my expectations. Everything in New York is big, important, and serious. I wanted to get away from all that._\n\nAriane didn't have a plan. Her attitude was that things would show up and that the universe would fill the void. For her, it was a real journey of stepping out of control and away from what was expected of her. Change can be scary and threatening. But when you face change head on\u2014as everyone must do on a Reboot Break\u2014something good comes from that change, even if the reward isn't immediate. In her suitcase, Ariane replaced material treasures with what she calls the five L's: Live, Laugh, Love, Learn, and Lighten up. She went on to found a company and write her book, _The First 30 Days,_ about how to navigate change. Both draw on the lessons she learned on her sabbatical.\n\nTime alone, time caring for one's physical and mental health, time with friends and family, spontaneity, being in the moment, reaching out to help others, exploring, learning\u2014these are all the pieces of the natural threads of a sabbatical quilt, the pieces of the sabbatical dream.\n\nAll these wonderful experiences are individual and yet universal, and they are only a few examples of the endless possibilities. Renewed and enriched, you are ready for the next phase. In our experience, as you turn to looking for a new job or preparing to return to your workplace, the activities of Reconnection and Exploration will continue as part of your sabbatical rhythm\u2014accompanied by a new sense of self-confidence and direction.\n\n* * *\n\n###   EXERCISES\n\n##### _Exercise 6-1: Reconnecting to Your Life and Yourself_\n\n\u2022 Make a list of four or five things you'd like to do during the Reconnection stage of your Reboot Break.\n\n\u2022 If one or two items had to come off the list, what would they be?\n\n##### _Exercise 6-2: Exploring_\n\n\u2022 Make a list of four or five things you'd like to try during the Exploration stage.\n\n\u2022 If one or two items had to come off the list, what would they be?\n\n##### _Exercise 6-3: Packing Light for Exploration_\n\n\u2022 If you plan to travel during the Exploration stage, what is the bare minimum that you need to take with you? List the things that you will absolutely need during your journey.\n\n\u2022 What are the things you would like to have on your journey within you? (By within you, we mean qualities such as courage, a sense of adventure, joy, curiosity, enthusiasm, and strength.)\n\n\u2022 Is any emotional baggage\u2014a romantic relationship gone bad or self-limiting ideas about your life\u2014holding you back from achieving your sabbatical or its goals? Write it down.\n\n\u2022 Review the \"emotional baggage\" list in a week. What might you be ready and willing to let go of?\n\n## | CHAPTER 7 |\n\n## _Renewed, Recharged, Now What?_\n\n\"The unexamined life is not worth living.\"\n\n\u2014 _Plato_\n\nThe fourth stage of the Reboot Break, what we call \"Reentry,\" marks the start of a new chapter of your life. This is one of the most exciting and satisfying parts of rebooting your life, as you assimilate all that you have experienced during your time off and now make time for reflection. It is a time of new ideas, new possibilities, and new perspectives.\n\nYour adventures have ended and reality has set in. What now? You've expanded your time to include things that bring you satisfaction. You've gotten out and explored, reconnected with old friends, and hopefully allowed plenty of time for rest and introspection. To others, it may seem that you have been on a vacation, especially if you traveled to your dream spot in the Exploration phase, but behind all this wonderful time off, there has been some inward caretaking. Now it's time to bring all those experiences together and determine what's next.\n\n**Don't rush yourself. Too often, people try to predict what they will do at the end of their time off before they've even started.** When your family asks, don't rush to answer. The answer will come when it's time.\n\nReentry is usually about coming to terms with the end of a wonderful period in one's life. Take the time to absorb the best parts of your break and figure out how to incorporate those experiences into your post-Reboot Break life. You may be getting a little antsy about your financial situation and what you are going to do next. Or you may be thinking about what it will be like to return to your job or begin a new one. This is a time to start to talk to others about work and to go on job interviews. In this chapter we will explore some ways to think about reentering the work world and give examples of how others handled the transition.\n\nRita had taken a between-gigs break and was now entering the Reentry stage. An old school acquaintance approached her about a full-time job. Rita didn't think she was interested in the opportunity, but the day before she was going to turn it down, she woke up in the middle of the night with the idea of turning the offer into a two-month consulting engagement. It provided her intellectually stimulating and challenging work while still allowing her the flexibility she had grown to love during her Reboot Break. It also gave her more time to figure out what she really wanted to do.\n\n### TRANSITIONS\n\nReentry is a transition. In fact, the whole reboot process can be understood as a transition between your former work and a new lifestyle. Even if you return to the same company, a transition takes place.\n\nThe author William Bridges has written extensively about transition. He made his own course-correction when he left his career as an English professor for the field of transition management. Then he experienced an even more profound transition with the death of his wife. His book, _The Way of Transition,_ is based on the deeper learning and understanding he forged during that second transition. Bridges describes a three-stage process, starting with the necessity of ending a chapter of your life and leaving something behind. No matter what the circumstances are\u2014the need for a change, the death of a loved one, the birth of a child\u2014the first step of a transition is \"letting go.\" It involves realizing a phase of life is over and that you are moving away from it and into something else.\n\nLetting go is followed by a \"neutral zone,\" an in-between time of self-examination and discovery. This can be an empty and lonely time, or a very creative time, or both. Bridges describes it as a time of reorientation, personal growth, (self) authentication, and creativity. It is similar to the Reconnection and Exploration phases in this book.\n\nThe third stage Bridges describes is \"beginning again,\" with all the attendant uncertainty and exhilaration. For us, it is Reentry.\n\nUltimately, transition results in renewal. Bridges talks about life's imperative to renew itself and says renewal is possible only by going through the process of transition. \"Just what causes the timing of these turnings is beyond our knowing,\" he writes. \"All we know is that periodically, some situation or event deflects us from the path that we thought we were on and, in so doing, ends the life-chapter we were in. In order to continue our journey, we are forced to let go of the way we got that far. Having let go, we find ourselves in the wilderness for a time, and not until we have lived out that time can we come back around to a new beginning.\"\n\n**Part of a successful sabbatical is recognizing, accepting, and celebrating transition and change.**\n\n### REFLECTION\n\nTime for reflection is so important at this stage because it is where the activities and experiences, the emotions and feelings, and the wants and needs come together to pave the path to post-sabbatical life. If you learn only one thing from your sabbatical, it should be to take time regularly and frequently for reflection. We as individuals all too often move from our head to our feet, from thought to action, without spending time contemplating how we really feel about something.\n\nAbout half of the people we interviewed returned to their same jobs. And of those, about half were burnt out, frustrated, and fed up with their jobs before taking their Reboot Break. It was only by taking time off and going deep inside that they realized that they actually liked their jobs but just needed more balance. Their Reboot Break gave them renewal and an awareness that there were ways to enjoy life and have balance while still being fully engaged in their work.\n\nSusan, the thirty-year-old banker you've read about in past chapters, never doubted that after her year-long break, she would return to the same company. She liked the company and its values but knew that she would quickly get bored doing her old work. She handled her reentry well and shortly after returning was able to negotiate a rotation into a new department where she could be more innovative. Susan credited her time by herself in London:\n\n_I had never been so alone in all my life. But that was good for me and allowed me to figure out what I wanted to do next. Without all of my reflection time I would have never been able to create a plan to return to my same company, but in a new city and in a whole new area that offered a lot more creativity._\n\n### EXPLORING NEW OPTIONS\n\nWhen Barbara was ready to think about work again after losing her job and taking a Reboot Break, she chose to go to an intensive three-day session with a life coach. First, she spent a week answering a questionnaire. The coach raised a lot of intense questions that caused her to think about what was important to her. He also gave her the William Bridges book on transitions.\n\n_We talked about the free fall, separation, anger, healing, etc. We talked about what made me click. I did a lot of soul searching. I asked myself,\"What would make me click for the next stage of my life?\" I needed to move from making my mark to doing something that made me happy. It had to encompass entrepreneurship and creativity, but I knew that I also needed structure._\n\nBarbara decided to kick off her Reentry phase by contacting people and asking them to write testimonials about her. She explained that she was going through a transition and period of self-exploration and that she would appreciate their points of view on what was special about her.\n\nBarbara sent out about forty emails and thirty people responded. That's an incredible response rate. Barbara found their responses not only very informative but also moving and emotional. By providing friends and previous business colleagues the time and space to give her honest feedback, she was able to hear what made her good at her job, what made her special.\n\nBarbara found her ego again. Now she was armed with the right ammunition and clarity to recreate her bio and start talking to recruiters. She spent time formulating her strategy and assessing various opportunities.\n\nShe connected with headhunters and women entrepreneurs. In the process of sending her resumes out to recruiters, one of them said that she would make a great recruiter. Barbara thought to herself, \"I am a negotiator and a deal closer, and I am flexible.\" She decided to take a risk and go into this totally new field. She said that the worst that could happen was that she would choose to leave. Plus, her Reboot Break had armed her with better clarity about her own strengths and where best to use them.\n\nElizabeth is another person who loved her work and lost it in a company restructuring. When she had been let go from a company several years prior to this time, she was scared of being in her home alone all day. She admitted to feeling a bit panicky each time her husband left in the morning. She therefore went straight to job searching without giving herself the gift of time. She quickly found employment, but not her dream job. This time, rather than jumping right back to work, she turned a crushing event into a gift of time. For the first couple of months, she exercised, traveled, reconnected with college roommates, and enjoyed sailing before breaking out to a new chapter in her life.\n\nAfter much soul searching, Elizabeth reaffirmed to herself that she really liked training sales forces, and it would therefore be a good idea to connect directly with other people who did similar work. Rather than making individual calls or appointments, she attended a Strategic Account Management Association conference.\n\nWhen she arrived, the name of her former company was printed on her badge. She had to cross it out and handwrite her name. At the beginning of the meeting, everyone was asked to introduce him\/herself. She said \"A few months ago, I was the VP of Global Accounts for my company. I am no longer with them, and I now contribute to the unemployment statistics. What you see on my badge is code for saying I am trying to figure out what to do next.\"\n\nEveryone laughed. Elizabeth said it was very extemporaneous and that she didn't know what was going to come out of her mouth. During the first coffee break and throughout the remainder of the meeting, several people approached her, offering ideas and brainstorming about next jobs. It probably helped that she exhibited a sense of humor and confidence as she handled that difficult introduction. She left the conference with several leads. Today she works with one of the firms that she met there.\n\nUpon returning to work, she looked great and was calmer and more confident. It was only after spending reflection and reconnection time during her break that she realized how many people thought that she now displayed a totally different personality. When she talked or even just thought about work, she had always tensed up. With newfound knowledge, she committed to leaving some of that intensity behind as she entered her next career phase.\n\nMany people worry about how they will explain their Reboot Break to prospective employers, especially in a down economy. Professional recruiters confirm our strong belief that it is not a problem, depending on how you describe your time off. **Remember, this is not \"time off\" but in fact \"time on\" to invest in yourself.** Employers might be turned off by negative explanations like \"I was totally burnt out and just needed time to be by myself and in nature.\" However, talking about how you bettered yourself, what you learned, and how you expect your new perspective to lead you to be more creative and innovative will work. One hiring manager said, **\"The person who has traveled or tried new things and who is open to different experiences and views is exactly the person we want in our company.\"**\n\nSome other tips for searching for a new career after your Reboot Break:\n\n1. Polish your resume and bio. There are professional services that can help you with this if needed.\n\n2. Remember to do a financial assessment of your current situation and what you need from your next job to help you negotiate.\n\n3. Put your best foot forward by selecting clothing that portrays the image of a confident, put-together person.\n\n4. Think about how you are going to explain your time off. Get your answers ready for other predictable questions, especially any that might worry you.\n\n5. Prepare questions for the interviewers. Remember, you can now choose a job that fits who you are and the quality of life you want to continue to lead.\n\n6. Send thank you notes after each interview.\n\n7. Google yourself. You may be surprised what is out there about you, and you should know.\n\n### NETWORKING\n\nAfter Rita fell into that first consulting gig, she went on to do two more short consulting engagements for other companies. She liked the flexibility it provided her as well as the mental stimulation, but still wasn't sure if it should be her next full-time career move. During her time off, she met with as many consultants as she could to ask what they liked and disliked about their work. How did she find those people to talk to? Networking the old-fashioned way\u2014picking up the phone. First, she called friends to ask them whom they might know. Then she called professional groups with whom she had been associated in the past and asked which members were consultants who might be willing to speak with her.\n\nHere are some tips for Networking:\n\n\u2022 Ask all the people you come in contact with if they can suggest an expert who works in a given field that you want to explore.\n\n\u2022 Tap in to your parents, friends, neighbors, fellow churchgoers, people you meet at the gym, etc.\n\n\u2022 Send emails to friends asking if they know someone in the field you've selected.\n\n\u2022 Make your request hassle-free by drafting a paragraph that they can send on your behalf. A sample is \"David, a very good friend of mine, is exploring changing careers and is very interested in knowing more about the consulting world. Since you too made the switch from working for a large corporation to consulting on your own, I was hoping that you might be willing to speak with him about the challenges and rewards of doing so.\"\n\n\u2022 Don't forget to tap in to the power of social media for networking. The one most used for work-related discussions is LinkedIn. Enter your profile and remember to add all the key words to cover your new areas of interest. You can say what you are looking for under \"status updates.\" You can read job opportunities there too. Ask former colleagues and bosses to write a recommendation for you. You can also see who has been reviewing your background and looking at your profile under \"who's viewed my profile.\"\n\n\u2022 You don't always need introductions to approach people. You can do research online and even write to people you don't know, asking for a brief conversation. Tools such as Google or LinkedIn may be of help. There is a wealth of knowledge out there.\n\n\u2022 Once a connection has been made, ask to schedule a brief phone call or meeting, and do not take too much of their time. We find that people are usually willing to give a helping hand.\n\n\u2022 Invite the person out for a meal or a cup of coffee. Let her choose and you pay. A face-to-face meeting is preferable.\n\n\u2022 Before the meeting, do online research into his or her background and organization. Come to the meeting prepared with knowledge of the field and questions to ask. Remember, you have asked for the meeting, so take the lead. Sample questions might include: How did you get into your field? What do you like about it? What don't you like about it? What are your biggest challenges? What additional research should I do if this were the next job for me? Are there others to whom I should speak? What have I not asked you that I should have asked?\n\n\u2022 Think what you can do for this person in exchange. Is there a book or piece of research that you have that you can send to him? Is there a contact you can set up for him to help him in his endeavors?\n\n\u2022 Immediately after the meeting, write or type a note to yourself on what you learned in your meeting. Later, after you have met with several people, it will be helpful to be able to refer back to all the notes and to pull out common themes.\n\n\u2022 Remember to send thank you notes.\n\nUtilizing this networking phase of her Reboot Break, Rita determined that consulting was not for her. \"It was only after I talked to several people that I better understood the challenges of scaling up a consulting business. And more importantly, I realized that at that stage in my life, I liked to be the implementer of recommendations and that I wanted to stay and see people grow and organizations morph.\"\n\nTo her friends' shock, after swearing that she would never work for a large organization again, Rita went to work for a Fortune 500 global company\u2014the one for which she did the first consulting gig. She would not have done so if it were not for her time off and the very deliberate time she took during the Reentry phase to network and then deeply reflect upon what she valued.\n\nVictor is a consummate networker. Victor had taken a between-gigs break, and it was time to focus on what he really wanted to do. He reflected on who he was, what he was good at, and what gave him the most satisfaction. He thought that he wanted something in the nonprofit field. He knew that the best way to do research was to go to the primary sources and get answers firsthand.\n\n_I reached out to people I hadn't seen in years. I reached out to some I never knew before. People were really helpful and very willing to talk and to give me their time. From each meeting I got a gem of an idea. Talking to all sorts of people was invaluable. Finding out what others valued in work and what they did was key. I listened hard._\n\nThen, when Victor was close to choosing a job with a not-for-profit group, a friend challenged him by asking, \"Is that really what you want to do?\"\n\n\"I had to admit that I wasn't ready to give up the money one can earn in the corporate world,\" Victor said. As they talked further, Victor realized that he could have a bigger impact volunteering for a not-for-profit and having money to give to them. Victor would not have come to this realization without reaching out to friends and asking for their advice.\n\n### MINI-TRIALS AND INTERNSHIPS\n\nAfter you have done your research and talked to several people, you may be perfectly clear about your next career move. However, if you still have some open questions, one of the best ways to \"try out\" a career is to ask one of those people you called if you could shadow someone at their company for a day or a week. If longer, perhaps it could be an unpaid internship.\n\nAmy is the young attorney who was let go from her law firm because of the collapse of the corporate litigation department during the recession. She decided that she wanted to explore something totally different from law. She took a two-month Reboot Break to live in Italy and study the language, as well as the foods. She also explored career opportunities via the Internet. One intriguing idea was doing an internship to segue into a different field. She approached a wine-making company about doing an internship with them. The experience included all aspects of winemaking, from crushing the grapes and blending the tastes to bottling to marketing. The internship was unpaid, but it allowed her to parlay her legal and business skills into a new career field. She was offered a full-time position at the end of the internship and is now happily working as head of compliance and the commercial division in the profession she sought during her time off. Also, she has created her own wine and labeling . . . soon to be commercialized!\n\n### ONCE YOU ARE READY TO START BACK\n\nNow that you have decided to reenter the workplace, we recommend that you spend some time thinking through how you are going to react to being back in a work environment. If you are returning to your previous job, call your colleagues to find out the highlights of what has happened while you were out. Start to look at emails and key reports that will bring you up to speed so that you do not feel totally out of it when you return. By the way, don't be surprised if you return after several months off feeling as if very little has changed. That happens frequently. Also, think about what you are going to share about your time off and what you don't want to say. Moderation is the key. You want people to be happy for you and to see your new level of energy and passion, but you don't want to sound like a broken record, describing every detail of your time off and making them jealous.\n\nIf you are starting a new job, learn what you can about the culture before starting, and set the tone for what people can and cannot expect from you right from the start. Focus on building new relationships and setting boundaries where you can to enable better balance. It is all too easy to get back into old bad habits, staying late every night, working on weekends, giving up personal time when the work can most likely be managed or staffed better. Don't let people begin to expect that you will always be available no matter what. Setting clear expectations is important to do right from the beginning, and perhaps you can be a good role model for others to do the same.\n\nIf you are an entrepreneur running a small company or the boss of a larger organization, think about how you might return to your old title but with a more strategic focus. Your employees or key managers have probably gotten along fine without you. After all, haven't you been preparing them for more responsibility all along? Now is the time to let them handle the day-to-day activities so that you can reflect more on what new strategic directions the business could take.\n\nWhether you are reentering your old place of work or beginning a new career, your sabbatical has made you a new person. Think through the ways your priorities have shifted. Take care to protect the insights you have gained. Take time for your friends. Take time for yourself. Take time for reflection.\n\nSusan is one of those people who has a natural tendency to charge into things. Upon returning to work she felt that she had to deliberately schedule \"reflection time\" as if it were an appointment in her day. So she asked her secretary to block off time weekly to do so. \"It made me a better leader and a more strategic thinker.\"\n\nDuring your break, you most likely chose to spend your time only with people you enjoy. We all know the workplace is not like that, and you may be reminded of that as you return. You may have an overly demanding boss, a cranky customer with a deadline, or a colleague who just does not seem supportive of you. The beauty of the workplace is that it brings together a mosaic of different personalities, backgrounds, and thinking styles. In fact, diversity among these elements can encourage a culture in which innovation can flourish.\n\nBut sometimes people's work styles go beyond a healthy difference in thinking and processing styles. Cathy Allen, in her research for _The Artist's Way at Work_ coined the phrase \"a crazy maker\" to describe a person in the workplace who is long on problems, short on solutions, and drains his co-workers' creative energy. Rather than getting sucked in, take a deep breath, think of one of those picturesque, peaceful places where you spent time during your Reboot Break, and set some clear boundaries.\n\nSABBATICALS ARE LIFE CHANGING. Make the best use of this last phase of your Reboot Break. Take time to consider how your priorities may have shifted, what you've learned, and how you will use your new knowledge and self-awareness in the next phase of your life.\n\n###   EXERCISE\n\n##### _Exercise 7-1: Exploring Career Paths_\n\n\u2022 Make a list of five possible career paths you might take, including staying with your current job or field. Prioritize the top three. How would you explore these?\n\n\u2022 Make a plan to learn more through networking. To whom might you reach out? Try to meet with at least two people a week, and afterward write down the key takeaways from each conversation.\n\n##### _Exercise 7-2: Assessing Opportunities_\n\n\u2022 When assessing a work option, first reflect upon your ideal work and make a list of what is important to you. Then make a list of the pros and cons. Under pros, list all of the things you perceive as the positives about a particular career\/work option, and under cons list all the negatives or potential drawbacks.\n\n\u2022 What is missing? Often people think they only have one choice. By doing this little exercise you can often approach the opportunity and ask, up front, for a few other things that are important to you.\n\n##### _Exercise 7-3: Returning Mindset_\n\n\u2022 As you prepare to return to work, write down at least three characteristics of your ideal mindset. For example, \"I will take time for reflection, friends and family, self, and patience.\" This is the time to think about how your priorities may have shifted.\n\n## | CHAPTER 8 |\n\n## _Deflecting Sabbatical Robbers_\n\n\"Knowledge is learning something every day.  \nWisdom is letting go of something every day.\"\n\n\u2014 _Zen Proverb_\n\nNow you've made a plan and know about the phases of a Reboot Break. You have goals, timetables, intentions, and you are in charge of it all. Experienced sabbatical takers know this is where the rubber meets the road. Will you really do all those things\u2014and do them on the schedule you laid out? What will interfere with your plans? For every good intention there are as many \"Sabbatical Robbers\" and traps to lure your attention away. This chapter will describe some and outline how to overcome them.\n\n### WHO AND WHAT ARE SABBATICAL ROBBERS?\n\nThe actors in this chapter are you and everyone else in your environment\u2014your family, friends, colleagues, and other professional acquaintances. They all can have a role in slowing your progress toward your goals\u2014usually unintentionally\u2014by making demands on you or causing you to use your time less effectively.\n\nSabbatical Robbers come in two categories: internal (you) and external (everyone else). Here are a few examples:\n\n#### ROBBERS THAT YOU CREATE:\n\n\u2022 You spread yourself too thin by taking on too many commitments.\n\n\u2022 You let day-to-day things (household, etc.) take up the time.\n\n\u2022 You can't say no.\n\n\u2022 You procrastinate.\n\n\u2022 You are having too much fun to do the more serious things you planned.\n\n#### EXTERNAL ROBBERS:\n\n\u2022 **Everyone wants a piece of your time because you aren't working.**\n\n\u2022 You are a target for helping out with charitable causes or your children's school.\n\n\u2022 Your family expects you to babysit because you have so much time on your hands.\n\n\u2022 You are the reliable relative in every emergency or need.\n\n\u2022 The house needs work, and you are in charge.\n\n\u2022 \"What about me? I'd like you to spend more time with me.\"\n\nRecognize these robbers and deal with them!\n\nIt should be clear, though, that **one person's Sabbatical Robber may be another person's sabbatical objective or joy.** For example, Jack loved going to the park every afternoon with his grandson while his daughter went to class. One of Jack's sabbatical goals had been to spend more time with his grandson, which was far more important to him than having that time to himself. If his daughter had pressured him to watch the child, and it really wasn't what he wanted to do with his time, Jack might have viewed it as a Sabbatical Robber rather than a sabbatical joy.\n\nSometimes, too, there are unintended intrusions\u2014the inevitable unplanned life impositions that simply happen, and you can't prevent them. It may relate to you, such as a time commitment that has to be rescheduled from before or after your Reboot Break to the middle of it. Or it might be something that happens to someone close to you, such as your spouse breaking a leg. These circumstances become not so much robbers as incidents to treat with flexibility and good humor.\n\n#### WHEN YOU'RE THE ROBBER\n\nMany of you have overcome much and accomplished much to reach this point. Most of you are having a positive experience, already benefiting from unfettered time to relax, enjoy nature, clear away clutter, explore and try new things, see friends, be spontaneous. Whatever stage you are in, some of you may be experiencing a gnawing continuation of some of the feelings you encountered when you were contemplating this Reboot Break, and those feelings may be interfering with it.\n\n**Guilt often is the biggest culprit, in particular, cultural and family guilt.** You may feel guilty not to be working, not to be earning money, not to be volunteering more, or not to be taking on a larger role in your family, since you now have time. This manifests itself in your spending more time than you envisioned on professional activities, volunteering, and family support. Some of the activity is self-initiated; some is in response to requests or expectations.\n\nIt is inevitable when you are on a break from work that people will think that you have unending free time and ask for more and more of it. Sometimes they just assume that you have the time and want you to spend it as they expect.\n\nYou are more vulnerable to others' requests for your time when you are carrying guilt about what you should be doing or simply feeling that you should be doing more. There is also the danger of inadvertently being flattered into what you really don't want to do.\n\nHere is our prime list of Sabbatical Robbers in this category:\n\n\u2022 _You,_ who can't say \"no\" to the people who think you have unending free time and keep asking you, even though their requests pull you off your reboot goals.\n\n\u2022 _You,_ who feel that since you aren't working and have more time, you should take on expanded volunteer roles, even though these roles conflict with your sabbatical plans.\n\n\u2022 _You,_ who keep working, even though you are supposed to be between gigs. During Martha's academic \"half sabbatical,\" she cut way back on her teaching time but remained at the university, and the space filled up with so much job-related activity that she says, \"The sabbatical was really dreadful.\"\n\n\u2022 _You_ and your inability to deal well with lack of structure, and as a result either waste time, procrastinate, or let time drift away.\n\n\u2022 _You,_ who feel that you shouldn't be paying that nanny or babysitter or caregiver or house cleaner or lawn mower since you are no longer working at a job outside the home.\n\n\u2022 _You,_ who take on the reliable relative role because you have the time to do it, even if it pulls you off the track you set.\n\n\u2022 _You,_ who are enjoying the luxuries of downtime so much that you don't get to the meat of your reboot goals.\n\n\u2022 _You,_ who treats your break like a job, rigidly holding to the schedule and asking, \"What did I accomplish today?\"\n\nSome people found that they traveled too much, which made it more challenging to find downtime and to get to other sabbatical goals. Nigel, a UK solicitor, and his wife scheduled a whole year of travel, but in retrospect would have cut that down, as it got too tiring and all-consuming. You need downtime. Schedule it. Leave enough time for yourself and what you enjoy.\n\n#### EXTERNAL ROBBERS\n\nAnd beware of those external robbers:\n\n\u2022 _Your children and\/or spouse or partner_ who think you aren't performing up to standard because they haven't adjusted their lenses to the fact that you didn't take time off to add on household chores and family responsibilities.\n\n\u2022 _Your grown children or others_ who expect you to be available for babysitting, driving kids around, and other such tasks. As Marie's retirement sabbatical approached, a relative noted multiple times how nice it would be that Marie would be able to look after her young nephew Jason when his mother had meetings in town. Laments Marie, \"I wish they would ask me if this is something I'd like to do rather than making the plan for my time. I want to make my own plans.\"\n\n\u2022 _Bosses or colleagues_ who want you to continue expending effort on your former job, or possibly want you to start early with a few tasks leading up to your new job.\n\n\u2022 _Acquaintances_ who ask you to apply your professional skills to their non-profit organization pro bono.\n\n\u2022 _Your child's school_ that wants you to head a committee.\n\n\u2022 _Your bowling league,_ which wants to put you in charge.\n\n\u2022 _Your friends, former colleagues, and former bosses_ who can't imagine what you do with all that time. Nancy tells the story of when she saw then-Vice President George H. W. Bush a few months after leaving her job as his National Security Advisor to move to Germany in hopes of having kids with her Army general husband. He asked her, \"Nance, what do you _do_ all day?\" \"Well, I study German, go to German-American discussion groups, explore Heidelberg, cook good meals, try to get pregnant . . . \" It can be hard to explain what you are doing every day to someone you admire so much, professionally and personally, who has always known you in a work context.\n\n### RESPONSES TO YOURSELF\n\nHere's where we tackle how flexible or rigid you want to be regarding your schedule, activities, and goals. It is so easy to get caught up in myriad day-to-day things or to keep saying yes.\n\nCathy observed ruefully partway through her Reboot Break, \"I still haven't done art lessons or painting, although I have my studio space cleaned up.\" The rest of the Sabbatical Sisters heartily congratulated her for cleaning up the studio and pointed out there was still time for art. Indeed, before the sabbatical ended, she had explored her own artwork fantasies and is still dabbling in art.\n\nIt is worth it at the beginning and at times during the sabbatical to consider how one stays on track, but said, \"Don't keep poking at it. You don't have to keep asking yourself where you are going.\"\n\nYou can congratulate yourself and think about where you are and what you have accomplished to get to this point, even in the face of daily distractions. Of course, you can make course-corrections too, if you wish. As a Latin proverb pronounces, \"It's an ill plan that cannot be changed.\"\n\nKim overcame her guilt regarding Sabbatical Robbers by realizing that \"if I served in all the causes that ask for some of my time, I couldn't have the psychological space to do what _I_ needed. If you get too many things on your plate, it's like being at work.\"\n\nOther helpful tips:\n\n\u2022 Remember the wise Japanese saying that you need to take 15 percent of your energy for yourself in whatever you do. If you run yourself dry, you will not have the reserves necessary to do what you need to do.\n\n\u2022 If you've come from a highly scheduled work life, you might need a plan (at least at the outset) for how you'll spend your time. To suddenly have totally unstructured time and more time alone than usual can be \"like being in a foreign country, and you don't speak the language,\" observed Margaret. Schedule only half the day and leave the other half open, allowing time for the unintended. There are ways to learn to become more \"in the moment.\"\n\n#### LOOK HOW FAR YOU'VE COME\n\nIf you are overly self-critical and restless about what you are accomplishing, you may want to perform this exercise to take stock. Some will never feel the need to do it, and others will need to do it more than once.\n\n\u2022 Title a blank paper or a journal page _Look How Far I've Come_ and date it.\n\n\u2022 Write the date you decided to take a Reboot Break, the reasons why, and your sabbatical goals.\n\n\u2022 Make a header called _What I've Done to Get Here_ and write the steps you took to make the sabbatical a reality. This includes financial preparation, steps to leave a job, living and lifestyle changes, etc.\n\n\u2022 Write the feelings or naysayers you confronted successfully to go on your break. Those two sets of things are _big_ accomplishments.\n\n\u2022 Now, make a header called _What I've Done on My Sabbatical Thus Far_ and write those things down, including everything, even the smaller things. Here are some examples:\n\n\u25aa Simple pleasures like reading the newspaper every day or calling someone special more often\n\n\u25aa Spending more time with your mother or children or friends\n\n\u25aa Cleaning out even one drawer\n\n\u25aa Trying something new like yoga, art, metalwork, or carpentry\n\n\u25aa Getting more exercise\n\n\u25aa Saying no to various demands on your time, or increasing your volunteer work\n\n\u25aa Finding yourself letting go of the work or situation you left behind\n\n\u25aa Traveling\n\nYou will be amazed at how long a list you can produce, and most likely your list will have some bigger things on it than the examples given. For example, you may already have decided to sell your house or moved or made other major decisions.\n\n\u2022 Now, look at your sabbatical goals again and consider whether you are being true to those goals. If you are lagging or off track or want to change the goals, be gentle with yourself as you consider how you are spending your time and whether you want to change it.\n\n\u2022 Write these changes in a section called _Adjustments_ or _Potential Adjustments._\n\n\u2022 Set it aside and go back to it in a few days to see if you still feel the same way and want to pursue the changes you have suggested to yourself. Be sure to read the entire document you have created.\n\n### RESPONSES TO OTHERS\n\nHere are suggested some responses to external Sabbatical Robbers. These have worked for others and may work for you.\n\n\u2022 **Have an elevator speech or cocktail party statement at the ready about your sabbatical.** Develop a great way of describing your time off that makes it more understandable to others and gets you off the hook of being called upon to help out. Take Rita's great line: \"You can't imagine all the projects I'm involved in.\" Or you can say, \"I'm on a sabbatical between work gigs (or jobs) and I'm taking time off to explore new things.\" Mark just likes to say, \"I'm an activist.\" We prefer the response: \"I'm taking a Reboot Break!\"\n\n\u2022 **Make up titles for chunks of time to explain how busy you are.** You can say you've got a conference call, which may be a call with yourself. We suggest being able to straightforwardly say what you are doing, but for some period of time or for some interlocutors, this approach might help.\n\n\u2022 **It can take time to fully separate from the job you are leaving, but you may want to give it a limit and stick to it so you can get on with your break.** Martha says stay away from your former workplace. On the other hand, Margaret, whose previous sabbatical took her and husband Mark to Sudan, left her job at the non-profit Snake River Alliance after fourteen years but joined the board to \"keep her finger in\" and remain supportive.\n\n\u2022 **Don't over-schedule your kids just because you have more time to drive them around.** One sabbatical mother we know only schedules two ongoing activities per child, which leaves more time for her, but it also leaves more time to be closer to the children and have more quality time with them.\n\n\u2022 **Learn to say \"no.\"** Have a response ready and practiced for the times you will want to say \"no.\" \"I'd love to join your board, but I'm really concentrating during this break on xyz, and I need to stay focused on that.\"\n\n### SABBATICAL ROBBER OR GIFT?\n\nMary Pat gave voice to a worry, saying, \"I live in fear that someone I know will get sick or will need me and will take me away from these lovely days and my peaceful pattern.\" But, as the next story shows, what may seem like a Sabbatical Robber can turn out to be a gift.\n\nLeena's father fell ill in England with a stroke while she was on sabbatical, and she was greatly relieved to have the flexibility to travel and spend precious time with him and help him contend with the labyrinth of the British medical system. Her siblings were working and couldn't do it. Also during this time, Leena met Jos\u00e9 as she was preparing to move to a new job in Madrid. She thought of not giving the relationship a chance because it could take her off the track she had already set for the next chapter in her life. Instead she said, \"Why not give it a chance?\" She turned down the job in Madrid, extended her sabbatical, and is now married to Jos\u00e9, the love of her life. For Leena, \"Both meeting a new man and tending to my father were gifts.\"\n\nKim spent a chunk of her break hosting big family events, but, she says, \"I was happy to get all these things scheduled when I wasn't working. I looked forward to hosting the family reunion without having to do it at a fever pitch. Some would call it a Sabbatical Robber, but it wasn't for me.\"\n\n### LESSONS LEARNED\n\n**Give yourself permission to say \"no.\"** Don't be flattered into accepting roles you don't really want. \"Keep your ego out of it,\" says Marvin. Kim's advice is to look for your gut reaction when the request comes. You may be flattered at first, but does your stomach tighten or your subconscious signal a warning about overload? Have a clear set of goals and stick to them.\n\nFor her Reboot Break, Kim left one non-profit board, joined another, and turned down several more. The most difficult conflict was being asked to play a major role in organizing an environmental conference that would have national impact. \"I knew they needed skills like mine to get it done,\" she says. \"But I didn't want to do it because it would be 24\/7. It is important work, and it would have been wonderful to be part of it, but at a different stage of my life. My time off has taught me that I don't want to go back into all that stress. I'm seeing this from the standpoint of how hard it is to do it, rather than the challenge and glory that I would have loved before.\"\n\n**Make time for yourself and realize that it's okay to take the time.** You've earned it. You are doing something important. It's part of a life cycle. In the book _How to Get What You Want and Want What You Have,_ author John Gray talks about having to be happy in all aspects of your life to be happy overall. You need a balance, which includes time for yourself, especially on a sabbatical. Absorb this and make it your own.\n\n**Keep your goals clearly in mind.** You may choose not to stick to those goals, but that should be a mindful decision.\n\n**Reframe feelings of cultural and family guilt.** One solution is to give family members a role in your sabbatical. Ask them to do something they are good at, such as researching travel on the Internet. Negotiate with your spouse about your time away. Cut a deal on chores. Share your time off with your family: take them with you to museums, on trips, or on other adventures.\n\n**Learn to do the rockwork first and fill in with the sandwork.** Linda, a therapist, tells her clients to envision a big jar into which you must put a pile of rocks and a pile of sand. If you start with the sand, there won't be room for the rocks. If the rocks go in first, the sand can be filled in around the rocks and there will be space for both. Brandon does his writing in the morning when he is fresh and saves routine tasks for the \"least desirable part of the day.\" He says, **\"I save the best parts of the day for myself.\"**\n\n###   EXERCISES\n\n##### _Exercise 8-1: Identify Your Potential Robbers_\n\n\u2022 Name the people or forces who might intrude on your sabbatical plans.\n\n\u2022 Think through what their requests might be and whether you want to agree.\n\n##### _Exercise 8-2: Practice Saying \"No\"_\n\n\u2022 Write down a gracious \"no\" response to each request you would refuse.\n\n\u2022 Practice saying the responses in front of a mirror.\n\n##### _Exercise 8-3: (Tell Yourself) Look How Far I've Come_\n\n\u2022 As described in this chapter, write down your sabbatical goals, then _What I've Done to Get Here,_ then _What I've Done on My Sabbatical Thus Far._\n\n\u2022 Assess whether you are being true to your goals. If you are lagging or off track, consider whether you want to change the goals or how you spend your time.\n\n\u2022 If you do want to change, write down the changes.\n\n## | CHAPTER 9 |\n\n## _\"You're Taking a What?\"  \nLife with Someone on Sabbatical_\n\n\"The person who says it cannot be done  \nshould not interrupt the person doing it.\"\n\n\u2014 _Chinese Proverb_\n\nRachel found her mother's Reboot Break a little stressful. Nancy started flying all over the country to attend Rachel's college lacrosse games with other dedicated parents and fans\u2014a treat for both mother and daughter. But Rachel soon found she had to remind her mom constantly that she, not Rachel, was the one on a Reboot Break.\n\nNancy was excited to spend time with her daughter and suggested a mother\/daughter trip every week. Meanwhile, Rachel's academic and athletic schedule left her no time for an adventure in Chicago, a drive up the New England coast, or a trip to visit Great-Aunt Ginny.\n\n\"Mom!\" Rachel would explain, \"I have to go back with the team, and I can't just take a day or two off with you just because you have time to spare.\"\n\nRachel did like having her mom at the games and hated to ask her to restrain her enthusiasm, but Nancy's free time sometimes evoked in Rachel a sense of uncomfortable responsibility. Nancy wanted Rachel's constant attention, and Rachel couldn't always comply. Rachel felt some of her twenty-year-old independence threatened as she tried to balance her time between her mother's newfound freedom and her own life as a college student.\n\n**It can be challenging to live with someone on sabbatical. It also can be delightful.** From the moment of the announcement of the break until the end, some of life's normal rhythms and patterns are changed, and adjustments are necessary by everyone in the family.\n\nThis chapter is for both the sabbatical takers and those closest to them. It contains observations, stories, and advice for all. We urge you to share it with your spouse or partner, children, and others who live with you or are very close to you.\n\n### FIRST REACTIONS BY LOVED ONES TO YOUR ANOUNCEMENT\n\nReactions when the intention to take a Reboot Break is first announced may be \"You're taking a _what?\"_ accompanied by surprise, jealousy, shock, worry, happiness and support, anger, or all of these. These emotions are normal, and in every case everyone wonders how the sabbatical will go\u2014both for them and their loved ones.\n\nDale, a New Zealander in her late twenties, planned to leave her office job for four months to travel to India. When she announced her plan at home, her parents were shocked. They could hardly disguise their discomfort and disapproval. Their professionally successful and normally responsible daughter was walking out on a small company that depended on her. She had a future there. What was she doing? \"That's not how we raised her,\" they lamented.\n\nDale tried to tell them that this time off from work was something she needed and longed for, and that she had cleared it with her boss and figured out with him how her work would get done. The company could manage. She would come back as a better, more committed employee. Her future was intact. Her father was unconvinced and quite upset.\n\nDale's boyfriend, on the other hand, was very supportive. \"Tama, my boyfriend at the time, was really keen on my going,\" she says, \"He thought that I worked too much and that my life was imbalanced. Perhaps he just wanted me to get a life!\"\n\nWhen Mike announced his Reboot Break to his wife Myra, she smiled broadly. She knew what it meant: they would be planning an extended community service trip to Costa Rica. As a biology teacher, she had summers free, but Mike worked in real estate. The only holdup in their plans had been Mike's reluctance to ask for time off. Now he had done it, and they could go.\n\nConstance lost her job and decided to take a meaningful break before looking for another. She and her husband were within ten years of retirement, and she wanted to explore possible ideas. She told her husband the news, somewhat proudly, certain that it was a mature approach. Her eyes shone as she talked about the fun she would have and the trips she would take sleuthing out places.\n\nHer joy met stony silence. He was jealous. He would be toiling while she played and worked on their future. Furthermore, he was hurt that she hadn't consulted more with him.\n\nWhat reactions do you think you will encounter when you announce your Reboot Break? How will you respond to these emotions?\n\n#### YOUR FIRST RESPONSES\n\nCommunicate with and include your family. **Early communication with loved ones can help set the Reboot Break on a positive course and determine its success.** One needs allies and supporters, not surprised naysayers or Sabbatical Robbers.\n\nTo prevent a rocky start, contemplate out loud taking a Reboot Break, so it's not a complete surprise. When you announce your decision, be as comprehensive and clear as you can about your plans and goals, and how your family or significant other will be included. Be open to ideas and changes you could make that still meet your goals.\n\nTalk about expectations on all sides. You will run up against a brick wall if you make it sound like the agenda is entirely yours. By the same token, you don't want to hear relief from your loved one that you can now take over childcare or another major task. At least you want to be able to talk it out when that is suggested. Whatever the initial reaction, you will want to be ready to talk about your ideas and get advice. You will need to plan well for this discussion. Do some of the planning in Chapters 3 and 4 so you can respond to initial concerns about finances, what you will do, what they will do, what you will do together, and your future after your time off. For example, you may need responses like these:\n\n\u2022 _Here's how we can do it financially._\n\n\u2022 _No, I'm not going off by myself for the whole time. There will be lots of family time._\n\n\u2022 _Yes, Mom, I'm being responsible in giving up my job. Lots of people do this and end up in jobs they like much better._\n\n\u2022 _Yes, I know what I'm doing. Even though I've lost my job, I am taking time to figure out how I can move into a more satisfying line of work._\n\n\u2022 _Yes, I'm going to take a sabbatical and I want you to do it too, so we can travel together. Now is the time._\n\n\u2022 _Yes, I'm going to take time off now to take a breather before my last years of work. It's time to lay the groundwork for the move to Florida we've discussed._\n\n### SETTING EXPECTATIONS\n\nYour assumptions about what a Reboot Break is and what it can include may differ from those of your partner, spouse, or family members\u2014particularly when it comes to his or her role.\n\nWhen Alexandra told her husband that she had been let go from her job in education administration, she was understandably hurt and angry. But when she told him, \"I'm leaving in two days for three weeks in St. Lucia in the Caribbean so I can do yoga and relax and figure out what to do next,\" he was shocked.\n\nAfter Roger recovered a bit, he started to get into the spirit of things and said, \"That sounds great. I'll make the arrangements to go with you. We can get someone to stay with the kids. I may join you a day or so after you get there, but I won't be far behind with my swim fins in hand. We'll have a wonderful time together.\"\n\n\"Not so fast,\" came the reply. \"I'm going alone. I need this time by myself to decompress and get over this blow. You need to stay home with the kids.\"\n\nSabbatical takers plan activities to fulfill _their_ needs. If there is no early and explicit communication about developing the plans and describing the role of the spouse or partner, the spouse or partner will naturally have his or her own ideas of what it means. Roger thought it meant he was going too.\n\nSudden job loss can upset normal communications. Alexandra usually consulted with Roger on travel and other issues, and they planned together. But, in this case there was less communication than usual.\n\nAlexandra ended up going to the Caribbean alone, but she extended her stay by a week and Roger joined her for the second part of her trip. They figured out that she could continue to take a break from work rather than start looking for a new job immediately, and he encouraged her to think about starting her own business. She did just that a few months later by establishing a consulting firm, and she has been thriving in the change and new challenge.\n\nWhen asked to describe the most stressful thing for him about his then-girlfriend's, now wife's, break, Paul replied, \"I wanted to be with Cathy to travel and explore new things with her, but I had a business to run. I also had my own routines, being one of the 'regular guys' playing tennis. My routines were disrupted, and I was conflicted. I had my own life to live. We had to work at keeping things in balance.\"\n\nCathy relates that Paul had those concerns about his time because he envisioned that Cathy would be totally free from other obligations and wanting to spend every day with him. The dilemma was soon solved when Cathy set forth a very busy schedule for her Reboot Break that had less \"Paul time\" than he expected. He was taken aback at first, but Cathy's written schedule set the expectations, and it worked out well for both of them. The highlight was a month-long trip to France that turned out to be a mini-sabbatical for Paul.\n\nSometimes there is an expectation\u2014as described in the Sabbatical Robbers chapter\u2014that the person taking a break will have plenty of time to take over childcare and extra household duties. The break taker often has an entirely different expectation of how his or her time will be used. They both see it as free time to be filled, but their agendas are far apart. The only way to deal with this is to get it out on the table upfront and talk it through.\n\nThen there's the expectation of travel and adventures together. We talked to Carol about her husband, Mark's, academic sabbaticals. As a professor and ordained minister, he is fortunate to have taken two sabbaticals and has another one coming up. Clearly, Mark and Carol had differing expectations. Here's a bit of the interview with Carol:\n\n**Question:** Tell me about when Mark is on sabbatical. How is it for you?\n\n**Answer:** Every time he takes a sabbatical, I can't wait for it to be over.\n\n**Question:** Why?\n\n**Answer:** He says, \"Where's my lunch?\" I don't do lunch. I have my own schedule. He is underfoot. Also, he never plans for us to go somewhere special. I think of being on sabbatical as being in a special place. We just stay home.\n\n**Question:** Does he involve you in planning the time off?\n\n**Answer:** Never.\n\nCarol's answers were somewhat tongue-in-cheek, but they do provide insight into some of the issues for family members that can be assuaged with planning and discussing expectations.\n\n#### RESPONSES\n\n**It is exceptionally important to discuss expectations on both sides.** The sabbatical taker must set forth his or her goals and what that means in terms of practical, everyday activities. He should talk about expectations for himself and family members. It should be a discussion, not an ultimatum. It is good to talk about what you will and won't do. For example, \"I will spend more time with the kids, but I will not take over their childcare. Please do not cancel those arrangements.\" Or, \"I expect to clean the attic and would love to have your help and advice on a couple Saturday mornings.\"\n\nThe family members should be clear about their expectations and assumptions as well, and ask as many questions as necessary. The idea is to avoid frustration by having open, specific communication from the beginning. This may sound like drudgery, but it can lay the groundwork for similar conversations as the Reboot Break develops. It will certainly help avoid misunderstanding and disappointment over differing expectations.\n\n### VOICES OF FAMILY MEMBERS\n\nIn our interviews, we heard stories with a range of scenarios\u2014both highs and lows\u2014that illustrate what those close to someone on a break may be feeling. The next story is a best-case scenario, in which assumptions about the Reboot Break matched and worked out well.\n\nSteve was downsized in March from a technology company in Boise, Idaho. His whole unit's jobs went to Taiwan. The workers were let go in groups, and he was the second to last group, so he and Teresa had known that he would soon be out of a job. In fact, Teresa had been looking forward to having Steve home for awhile. He had been in maintenance before he turned to microchip manufacturing, and he was very handy around the house. She wasn't anxious for him to find another job immediately.\n\nFortunately, Steve had the same idea. He had so many projects at home and at their small cabin that he couldn't wait to get started. He decided to take a break and not look for work right away. Also, he wanted to think about what kind of work to seek next. Teresa thought that was great.\n\nWe asked if Steve was underfoot at home, and Teresa said, \"Not at all. I loved having him at home. It was terrific that he could do all the little things and also the big projects we had been talking about for years.\" Steve was most appreciative of how supportive Teresa was of his not working. She kept telling him that he didn't need to rush back to work, that she was very comfortable with his work break. They had the income from her job and could get along, at least for awhile.\n\nIn the summer, he started a new job, but it was a false start. Teresa welcomed him back home full time with open arms. She still had the attitude that his being home was highly valuable to the family, not just for the handyman aspect, but also everything related to family logistics, including having family and young grandchildren nearby. \"I always thought he'd make a great house husband,\" said Teresa, smiling. Steve's view, also expressed with a smile, was, \"I liked the role for a time, but not that title.\"\n\nSteve figured out in the course of his Reboot Break that he wanted to return to maintenance, his original field. In November, he started a new job maintaining the dorms at the university. He counts both the layoff and the Reboot Break as positives in his life. Teresa agrees.\n\n#### YOUR LOVED ONES' WORRIES AND FEARS\u2014SOME COMMON THEMES\n\nIt is not uncommon for worries and fears to creep in during a Reboot Break, not to ruin it, but just to be there as a tension sometimes, often related to how the expectations are playing out. Worries and fears fall into two categories: family members may worry about their loved ones taking time off from work, and they may worry about themselves. Sometimes they express these feelings openly, sometimes they do not. So it is important to be sensitive to both spoken and unspoken worries. Some common ones follow.\n\n#####  _Concerns for the Sabbatical Taker_\n\n**Boredom.** _\"What will she DO all day?\"_ Angus agreed that it was time for Kim to leave her current job, but he wasn't so sure about her not taking another. They could manage without the income, but wouldn't she get frustrated or bored without the daily challenge? The boys were in college, so demands at home had diminished. Then they moved\u2014 downsized from a home and yard sized for two boys to a house with half the space and in need of serious remodeling. Now it seemed fortunate that Kim wasn't working. She coordinated the moves and the contractors over fifteen demanding months. Angus continued his job responsibilities and schedule as the founder of a non-profit energy business and foundation.\n\nThey agree it couldn't have been done if both had been working full time\u2014at least not while keeping their relationship and their collective sanity intact. \"We'll never have a leisurely marriage; we're neither of us wired that way,\" Angus says. \"Kim operates at full throttle, whether it's a paying job, a string of non-profit boards, the house remodels, managing a social calendar during the holidays, or all of it at once. I just relax\u2014well, tense up a little less\u2014and hang on.\"\n\nKim told him, \"This is a very deliberate decision for me to learn and have time to reflect. I have goals, and I intend to meet them. I'll be plenty busy. Don't worry about me. This is the gift of time I'm giving myself.\"\n\n**Career.** _\"Will she be okay? Will her career be ruined?\"_ Dale, the New Zealander who went to India on her break, had had a variety of reactions from her parents and boyfriend, both immediately and as the planning developed. Though she continued to receive pushback from her parents, they understood better after she explained, and even became supportive. On the eve of another Reboot Break to India from her job in Washington DC, Dale tells about the first one:\n\n_My Dad grumbled and groaned and gave me many long lectures about facing up to my responsibilities and letting people down at work. It was a sermon once a week for the four months I was planning my trip. Aaahhhhhh! Just quietly though, I know he was proud of my loyalty to the friend I was going over to India to spend time with. I guess he was worried that I would damage my career by taking time off at a critical juncture. He was also worried that I would get sick in India. He's a doctor.My Mum didn't say too much but was quietly supportive. They gave me some money toward my ticket for my birthday, so they couldn't have been too badly opposed._\n\nDale did a great job of easing her parents' concerns and staying in touch. She emailed them frequently, kept a blog about her activities, sent postcards from every city. She filled them in on communications from her office so they'd know that all was well there and that her boss looked forward to her return. She reassured her parents that she wasn't being irresponsible. And she never failed to remark on her continuing good health in communications to her doctor dad.\n\n**Happiness.** _\"Will he like it? Will it work out for him?\"_ Victor's son Benedict warily watched his father plan his time away from work. His father was a workaholic, and Benedict only knew him as such. Sure, they had family time, and he got attention from both of his parents, but his dad was a banker and worked seriously long hours. Benedict worried that his dad wouldn't be happy on his Reboot Break. What would he do with all that time? Would he be frustrated or short tempered? Would he plan major projects to reorganize everything in the house and make him help? Benedict just didn't know what to expect.\n\nBenedict was pleasantly surprised. He had underestimated the power of the gift of time, coupled with both planning and spontaneity. Victor really did stay away from the office and endless hours on the computer and on conference calls. And Benedict was the beneficiary! Victor was very happy, and one of his priorities was to spend time with his son. They had a great time skiing, watching special programs on television, and just being together. Also, Victor acknowledges that Benedict's early concerns reinforced his intention of making the Reboot Break truly a time away from work.\n\n**Fulfillment.** _\"Is he doing what he set out to do and taking advantage of this time?_ Another worry is whether the sabbatical taker is doing enough on sabbatical and living up to his or her own dreams.\n\nKurt had an unexpected but welcome Reboot Break. He had worked for over ten years as a psychiatrist specializing in medications. His division at a prominent university medical school closed, and he found himself with the time he craved to remodel the house he owned with his partner, Perry.\n\nPerry's days off were Monday and Tuesday. That time alone at home was precious to him, but now Kurt was there. It caused a little problem, in that he didn't have his time alone, and also he felt that Kurt was becoming too narrowly focused. Kurt had begun to focus on every detail of the house and the dogs, not\u2014as Perry thought he should\u2014visiting with friends and family, going to the museums he had long wanted to visit, or even planning a great trip.\n\nThe moment that Perry absolutely knew that Kurt needed to broaden his horizons was when Kurt said, \"I'm so glad you are home. The dogs have been following me around and need so much attention. I need a break.\" Of course, the dogs were acting the way they always did, but their needs had become magnified by Kurt as he felt more and more responsible for everything that went on in the house. Perry said that even his chores were taken over by Kurt. Yes, it was nice that he didn't have to clean on his days off and that Kurt always had dinner ready when he got home, but things had gotten out of balance.\n\nAnother story:\n\nPeter, a university professor, took a sabbatical, and his wife thought he was squandering it. He still went to the office and to various university meetings. She worried that he would wake up a few months after the sabbatical and say with unhappiness and regret, \"What did I do with all that time?\"\n\nThis can be delicate. The family member may be dealing with Sabbatical Robbers, mainly that inner voice saying it's not good to be too carefree when others are still working. Or maybe the sabbatical taker hasn't planned well enough and is procrastinating. It is frustrating for the family member, who wants the sabbatical taker to be happy. (Or of course, there's the possibility of simply wanting the loved one less underfoot.) In such cases, it's time to check in about expectations. Is it going as planned? Should there be changes? Is it time to plan something special together?\n\n#####  _Personal Concerns_\n\n**Change.** Life changes when someone is on a Reboot Break. It can impact normal roles and even household tasks, as with Perry and Kurt. It might take the form of a lot of attention all of a sudden, as in the Nancy and Rachel story at the beginning of this chapter. And it can create new demands.\n\nRachel and her sister Sarah thought it was good that their mother had shed the job that kept her at the office such long hours. They loved seeing her happier and loved the rewards. Both girls benefitted from several lengthy mother-daughter trips and lots of wonderful times in between.\n\nBut then there were the matters of Nancy's reboot goals and the roles of her daughters, especially Rachel. Nancy had a few clear goals for her time off, including establishing an exercise regimen and setting up new technological devices. Rachel had not realized that she would be called upon in this regard, but she became, as she tells it, her mother's fitness coach and technology consultant. (\"Rachel, how do I email pictures from my phone and computer?\" \"What are the best exercises for increasing strength in my legs and training for skiing?\")\n\n\"I had to learn how to manage that, \" Rachel says, \"All of a sudden I had to complete all these manual labor tasks that accomplished my mom's cleaning goals and provide what were essentially technology manuals and fitness regimens for her. Then I heard about all the successes and newfound joys: 'Look what I can do now!' At some point, getting hundreds of photos emailed to me after I taught her how to do it made her accomplishments start to seem less thrilling.\"\n\nThere was also the matter of the move. Nancy decided her break was a perfect time to sell the suburban family home, downsize, and move to the city. The downsizing process, which found Rachel making numerous trips to eBay sellers and the Salvation Army, and performing many feats of manual labor, like packing and carrying numerous boxes, convinced Rachel that she'll never buy another non-essential item in her life.\n\nBoth girls were great sports about it all, but a general rule is to be clear in advance about your goals and expectations. It normally doesn't work well to spring major responsibilities or time demands on the people in your life.\n\n**Self-confidence.** _\"Where do I stand?\"_ There was a bit of this in Paul's reaction earlier in this chapter. The partner wants to be supportive and keep up with time-off activities, and may be a bit anxious about how to accomplish that. Or there may be issues of inclusiveness. \"Is my loved one trying to get away from me or just have a growth experience alone? Will I be forgotten in the rush of new learning, new rhythms, and newfound passions? Can I keep up with my partner's new dimensions and explorations as I continue my job and daily routine?\"\n\nThe sabbatical taker needs to be inclusive. Plan special occasions. Call frequently when away. Email may not be sufficient for reassurance, as it can feel less personal.\n\n**Resentment.** Spouses and partners who still must work every day may be envious or angry. They may make more demands of the sabbatical taker to do the household chores, run errands, or take care of the kids. Friends may be envious too that they are not on a Reboot Break.\n\nThe sabbatical taker who gets this kind of pushback may need to have another conversation about expectations.\n\n#####  _A Word on Guilt_\n\nWe addressed guilt in the Sabbatical Robbers chapter, but it comes up here too. The emotional reaction of those at home can cause guilt and anguish for the person on a break, even to the extent of impacting the quality of the Reboot Break. Again, reassurance and communication are crucial so that the family reaction will be less likely to be emotional and engender guilt. If it involves children and a spouse, the spouse can play a key role in explaining the situation to the children.\n\n#####  _Unanticipated Events_\n\nJanie's husband, Scott, was grateful that his wife was on a Reboot Break when he was taken ill, but he was concerned that it would ruin her break. \"I was so worried, but Janie was wonderful about it. She postponed a special trip that was to have kicked off her leave, and it became something to look forward to as the scheduled time off wound down. It was so great to know that Janie was there for my initial hospitalization and all my doctors' appointments without her having to arrange to be away from work. It was always so uncomfortable for her when she was working to take time off for family matters. I know she would have done it, but it was just better this way. Though it was a down time for us with my uncertain health, we were close and less stressed than if she had been torn between work and being with me. Janie's focus, lack of stress, and time\u2014just the gift of time\u2014brought us through the crisis of my heart attack in beautiful shape.\"\n\nReboot Breaks\u2014just like life\u2014have surprises. You need to be able to make adjustments along the way and sometimes change priorities, while still holding on to your goals as much as possible. The best way to accommodate change is to be in open and honest communication with your loved ones and others who are affected.\n\n### ADVICE FOR THE SABBATICAL TAKER\n\nThese points summarize what you can do during the Reboot Break to make it a positive experience for your loved ones. Remember that, while your spirit is high, your loved ones are facing the same old pressures at work and home.\n\n\u2022 Keep talking, informing, sharing, and asking opinions. Respond to resentments or any other reactions. It's a time to be patient and generous.\n\n\u2022 Remember to check in with how your loved ones are feeling. They won't always express themselves openly without being asked.\n\n\u2022 Anticipate worries loved ones might have about you, about themselves and their role, or about your relationship.\n\n\u2022 Make a regular \"date\" with your significant other.\n\n\u2022 Call and email frequently if you are away.\n\n\u2022 Involve them in the learning you are experiencing, and share as much as you can. This can be a great time for family trips that are longer than normal vacations, if that is possible. Everyone grows and creates memories when time allows it.\n\n\u2022 Be considerate of their time and responsibilities. They don't have the flexibility you have, and they may not always want to do what you want to do. Let them know if you plan to lean on them for major or minor sabbatical goals.\n\n### COPING SKILLS FOR FAMILY MEMBERS\n\nAnd here are some thoughts from us that could be helpful for your loved ones.\n\n\u2022 Say how you are feeling, as positively as possible.\n\n\u2022 Don't lose self-confidence. Your loved one is not deserting you, but just going on a journey of self-discovery.\n\n\u2022 Discuss demands that you think are reasonable to make upon your sabbatical taker as spouse, fellow parent, or significant other, while respecting the reboot goals.\n\n\u2022 Ask to be part of the planning and the plan, but give breathing room.\n\n\u2022 Ask to be part of changes along the way.\n\n\u2022 Plan your own Reboot Break!\n\n\u2022 Be understanding and encouraging.\n\n\u2022 Enjoy the time together.\n\n\u2022 Enjoy the attention\u2014if your parent overly dotes on you and your activities, try to accept it. If you are ill and are able to be with your loved one more than expected, relish it. If you are able to develop a relationship that otherwise might have faltered or been squashed by the everyday rush, bask in it.\n\nTo summarize, the sabbatical experience should be positive for everyone involved. We have given lots of advice here, but the main thing to remember is to communicate, which means listening as well as talking. Remember, the other person has a life too. And you can both enjoy the Reboot Break.\n\n###   EXERCISES\n\n##### _Exercise 9-1: Important Others_\n\n\u2022 List family members, friends, colleagues, and others who might be affected by your sabbatical.\n\n##### _Exercise 9-2: Addressing Concerns_\n\n\u2022 Write about the concerns those people might have and how you would address them.\n\n## | CHAPTER 10 |\n\n## _Living the Lifelong Sabbatical_\n\n\"Learn as though you will live forever;  \nlive as though you will die tomorrow.\"\n\n\u2014 _Mahatma Gandhi_\n\nWe have come to the last chapter, and one we think is most important. **Why would we take time off unless we expected it to change us in some positive way, so that we could truly keep living the lifelong sabbatical,** or at least reap the benefits from it?\n\nIn this last chapter, we come back to our own stories\u2014how the four of us are using what we learned during our sabbaticals in our everyday life. We include other examples as well. We also give you ways to take mini-sabbaticals of a week or a month until you can plan your next Reboot Break.\n\nWe shared with you earlier some of the statistics about the U.S. workforce, but here are more that are relevant to why we need to incorporate permanent changes into our lives, not just a one-time leave or time off from work. The Conference Board, an independent economic research and membership organization, released an extensive report in January 2010 indicating that job satisfaction is at its lowest level in two decades. The report shows that only 45 percent of those surveyed are happy with their work, down from 61.1 percent in 1987, when the survey was first conducted. Through economic boom and bust, the statistics on satisfaction indicate a consistent downward trend.\n\nThese latest statistics on job satisfaction point to the need for us to renew and refresh our perspectives, even if it is in short doses. Most of us cannot just leave our positions or change our bosses, but we can change our attitudes. This chapter will focus on how we can build into our daily lives ways to keep perspective, to be grateful for what we have, and plan for what we need.\n\nClearly, you are not alone. We all are over-committed, over-worked and over-tired\u2014especially if we are trying to balance a career, a family, and our personal lives. And most of us seem to have lives that are unbalanced. This chapter is about keeping our lives in balance and finding ways to enjoy all aspects of life to the fullest.\n\n### TEN THINGS TO TAKE WITH YOU FOR LIVING THE LIFELONG SABBATICAL\n\nJohn had spent six months on a break after leaving a high-pressure corporate position in New York. When he started his Reboot Break, he was burned out and overweight. He vowed that he would change the way he lived his life, if only during the six-month period he was taking off, and he did that. He started playing tennis again, worked out, lost twenty pounds, started eating healthfully, and took time to get to know his family and friends better, as well as taking time to rethink what he wanted to do next. Before he began his next job hunt, he wrote down several things he wanted to keep in his life after he started back to work. One was to stay in shape and the other was to be home for dinner with his kids at least three nights a week and hold Saturdays as family days, no matter what.\n\n_I learned a great deal about myself, as well as my family, during my time off from work, and I wanted to keep that knowledge in my heart. I have changed the way I work and am as a person for the better. It is possible to do if you remember how important the lessons were when you had the time to reflect._\n\nSo what can we do to change our lives for the better? Following are ten things we found that can make a difference. We learned them from our own experiences as well as from the stories of others. These may work for you, or there may be ten other things that work better. **The important thing is to spend time thinking about what is important to you, and then schedule it into your life, just as you do business meetings or visits to the doctor.**\n\n1. **Start journaling** every morning to capture your thoughts, dreams, concerns. It will help you observe yourself on that \"inner journey\" and keep you focused on what's really important to you.\n\n2. **Use the Goals Circle Exercise** at the end of this chapter to identify goals you want to achieve in at least six areas of your life. Do the Goals Circle annually and spend time each year assessing where you are with those goals. Carry the short version of the Goals Circle with you.\n\n3. **Schedule time for yourself into every day, every week, every month.** It can be down time to just listen to music uninterrupted. It can be an \"artist date\" with yourself to explore a museum or park. It can be time to follow your passion. But put it on your calendar just as you would any meeting or appointment.\n\n4. **Practice saying \"no\" to things you really don't want to do.** It is a skill to learn. It is often hard to do so because of guilt or obligation or fear of being rude. Say \"I'll think about it\" or \"I'll get back to you\" if you can't say no right away. Don't say yes right away, even if you want to do something. Practice giving yourself the time to set your priorities, rather than having others set them for you.\n\n5. **Plan longer weekends and evenings out with friends and family.** Get the chores and errands done at other times so when you can take time off, you use it for yourself and doing things you want. Get your family to help out so you're not doing laundry, grocery shopping, or fixing the gate during those times. Alison told us how she set a timer for twenty minutes and got her daughter and son to help sort and organize a closet each day as a family project. Even if it means working late during the week or on other nights or one weekend a month, organize, consolidate, then delegate the tasks.\n\n6. **Plan longer vacations.** Take at least two weeks together and try to do a month every three to five years\u2014or better yet, every year, if you can. Structure it so you are using your time off like the mini-sabbaticals described in Chapter 3.\n\n7. **Simplify, simplify, simplify.** Your office, your closets, your house, your car, your lifestyle. We live at such a hurried pace. And we have too much stuff! Engage your family in this. You can work on one room at a time. Hold a garage sale, sell stuff on eBay. Put the proceeds in your Reboot Break fund. Give things to charity. Don't buy something if you don't need it. Clear the clutter. It will give you the space you need, physically and mentally, to enhance your life.\n\n8. **Pack \"light\"\u2014both physically and mentally.** Practice carrying on luggage for trips on planes. Do one thing at a time. Use less and carry less with you. Leave the baggage behind. This goes along with simplifying your life.\n\n9. **Try one new thing and take some small risk each week.** By taking baby steps, we build the courage to do more. It can be trying a new food or restaurant, taking a course, calling an acquaintance for lunch. It can be wearing something different, walking a new route to work, or exploring a new neighborhood. Try a new sport. Just do something that stretches you. Not everything will be great or even good, but that is part of the learning too.\n\n10. **Nourish yourself during stressful times.** Ask for help. Don't be afraid of what's ahead. Think more about creating your support network\u2014family, friends, colleagues, church members. Give to get. Be generous with yourself and reach out to others. They need you and you need them.\n\n### STORIES OF LIVING THE LIFELONG SABBATICAL\n\n#### OUR STORIES\n\nCathy still writes in her journal every morning to keep her focused and to get her ready for the day. She schedules \"artist dates\" every week to try something new or go to a new place, whether she is home in Santa Fe or on business travel. She carries a \"List Book\" with her goals and adds to or deletes from it to be sure she is doing some of those things she has wanted to do. The categories include Career, Health, Relationships, Creative, Friends and Family, Philanthropy, Finances, and House Design and Maintenance. She also carries her Goals Circle with her in her purse.\n\nCathy spends approximately two days a week on The Santa Fe Group, her strategic consulting company, one day a week on philanthropy and non-profit board work, one day a week on creative pursuits, such as writing, and the rest on fun and time with family and friends. She has empowered her employees to do the day-to-day administration and focuses her efforts on ideas, client generation, and strategic senior-executive-level consulting, which affords her more flexibility.\n\nShe is, of course, at a phase in her life where she can enjoy such freedom, but still continually has to make sure her work and nonprofit involvement doesn't encroach on her creative and relationships time. Learning to say \"no\" has to be practiced every day!\n\n_I plan to trade my house in Santa Fe each year for a house abroad to learn new cultures, languages, and food. Like my fellow Baby Boomers, Idon't really plan to retire. I just want more control over my time and the ability to continue to explore and learn, to spend time with my family and friends, to live life to the fullest._\n\nWhen Rita returned to work from her third sabbatical, it was to be president of a major division of MeadWestvaco, with over 6,000 employees working for her. The challenge was how to manage the responsibilities of the company and still have time for herself. On her time off, Rita had taken up tennis again, learned yoga, spent more time with her family, and reprioritized what she wanted to do with them.\n\nWhen she reentered everyday life, she had her assistant schedule a half hour between appointments to give her time to reflect, scheduled tennis on Friday mornings for an hour on a regular basis, and did yoga on planes and when she was on conference calls and no one could see her. She also put her family first, asking her boss if he was willing to put her on early in the board meeting agenda so she could make it to her son's teacher meeting, for example, which is rarely done in the corporate environment. In an effort to bring more fun into her workplace, she used the holidays as a reason to have everyone dress in costume for an employee luncheon and lots of laughs.\n\nToday, Rita has a portfolio career. One-third of her time is spent serving as a director on corporate boards and coaching clients on how to find their first board position. A second third is dedicated to her passion\u2014giving back. She is involved in numerous not-for-profit activities. The one for which she spends most time is as past chair and board member of Pro Mujer, a microfinance organization dedicated to helping women and children in South and Central America. The last third is for having fun, and that part includes new activities, such as learning Spanish and salsa dancing, as well as writing this book.\n\n_I am working hard at learning to \"be in the moment\" and enjoying the little things in life. I have gone to a lot more shows, plays, museums, and walks this past year. Most important has been the time I have had to spend with friends and family. Each day I think of my favoritequestion: \"When was the last time you did something for the first time?\" Yes, I've had incredible firsts, including trips to Ecuador, Jordan, Spain, and Cuba, attending a sound healing conference in Santa Fe, and snowshoeing and kayaking. But that quote has a much more profound meaning for me. Seeing or doing things for the first time includes how I look at a building that I may have walked by 100 times or how I look at the quality of the sky. Every day is an adventure and a new learning opportunity for me._\n\nIt isn't easy to change our lifelong work styles, and outside factors often intervene. Nancy went back to work with the intention of including important habits from her sabbatical in her new lifestyle. She vowed to take a lunch break every day to get outside in nature and do errands, and to leave the office by 6 pm every day so that she could have a full evening. She also set aside special events on the calendar with her family and friends to keep sacrosanct and looked forward to relaxing weekends. But she has struggled with keeping these promises.\n\nOne big work project derailed her from what was otherwise a successfully balanced lifestyle. Her workaholic tendencies have returned, but less strongly than before, and she has kept to her schedule of special events, like her college reunion and trips with family and friends. Weekends are sometimes relaxing and sometimes taken up with work or other time-consuming responsibilities\u2014the \"keeping up with everything else\" that we all have to do. She's relearning firsthand the importance of putting personal activities on her calendar, such as going to museums and concerts, exercising, and dinners out. She knows she needs a better balance between work, other obligations, and play, and she is striving to achieve it, starting with rereading this chapter periodically!\n\n_You have heard the old adage \"practice what you preach\"! Well, I am trying to do that. But it isn't easy to change habits when external situations make you revert to old ways. It takes courage, consistency of motivation, and a promise to yourself to keep giving yourself the gift of time. I do think it is important to have a clear set of priorities, and to knowhow to say \"no\" to requests for your time. I love doing my Circle Goals in January, then revisiting them from time to time during the year. I still keep my daughters, Sarah and Rachel, as my top priorities. I'm doing volunteer projects in Washington DC, and going to the theatre regularly. One of my main goals is to continue to get out West several times a year, where I experience renewal through the beauty and open sky._\n\nJaye was very concerned about not losing the calm and peace of mind she gained from her time off, along with a sense of connectedness to her full self. Deciding not to go back to a structured corporate job enabled her to have more flexibility to build in the things that were important to her. Jaye increased her involvement with the Harlem Dowling Westside Center, a foster care and adoption agency in New York City, and now serves as vice president. She spends more time with her family, especially her younger nieces and nephews, and sees them every Wednesday for dinner, to keep up with their busy lives. Jaye spent time on her Reboot Break thinking about what really makes her happy, and travel was at the top of the list. She now travels three to four times a year with friends. She schedules events with friends on a regular basis. Her work is intense, so she needs to balance her life with more \"fun\" things to do.\n\nJaye is incorporating the work we've done for this book\u2014the research, learning, forums, community building, websites, and blogs\u2014 into her consulting practice, Breakwater Consulting.\n\n_Seven years ago I took a five-day course on living the life you want. I was obviously ready in many ways to change my life, but I needed the time away to really look at my work and life and clarify what I wanted it to look like going forward. I learned through that process and then the subsequent year off that it is so critical for me to build in little breaks on a regular basis. I try to take some time off every month to renew and refresh myself. If I don't, I feel it and suffer the consequences. This has been one of the most important lessons so far. Seven years later, I am continuing to do it. I find that, for me to be the best person, consultant, and colleague I can be, I must take a step back to learn new things,refresh my thinking, and stimulate my creativity. It shows up obviously in the work I do._\n\n### MINI-SABBATICAL\n\nWe talked about the concept of mini-sabbaticals earlier in the book. While a week, or even a month, doesn't count as a Reboot Break from our point of view, these can serve as shorter refresher breaks to build up to taking more time off. The problem is that most of the week, two weeks, or month you take off can get eaten up by doing pragmatic things or stressful travel or too much activity. In a short time period it is also very hard to detach from work. You still carry your cell phone, PDA, and laptop, and invariably use them. The tips in Chapter 3 should be helpful in this regard.\n\nFollowing are some ideas of what to do with short periods of time off.\n\n#### THINGS TO DO IF YOU ONLY HAVE A WEEK\n\n1. **Don't travel anywhere\u2014vacation at home** and take day trips to explore the area around you so you do not have the added stress of travel involved.\n\n2. **Don't do errands or chores\u2014structure the time off to do just what you want to do:** read, take a class, go to a museum, host a dinner party, go on a hike, go to movies, write in your journal, have lunch with friends, but DON't DO CHORES.\n\n3. **Take an immersion course in a culture, language, or art technique, or go to a spa** \u2014ideally close to home with very little travel.\n\n4. **Take an \"inner journey\" and devote the week to exploring yourself** \u2014see a therapist, a palm reader, an astrologer; read self-help books, meditate, go on a retreat, write in a journal; do yoga, take long walks.\n\n5. **Plan your next Reboot Break** \u2014read this book again, do the exercises and checklists, explore your employer's policies, talk to people who've taken time off, plan what you will do when you take a Reboot Break, get your financial planning started.\n\n**Keep in mind that giving yourself the gift of time has no boundaries.** It can be that quiet moment in a park or a day at the beach or a week devoted to following your dreams. Time is our most precious commodity. Use it wisely.\n\n\"It was actually taking a month off to rent a house in Umbria with friends on my birthday that gave me the impetus to do a Reboot Break,\" said Cathy. She had never taken more than two weeks of vacation at a time since joining the corporate world in 1986. Even then, she always had her BlackBerry with her and called into the office regularly.\n\n_The most enlightened experience I had during the month off in Italy was how well my staff did, once empowered. I told them I wouldn't be calling in or checking emails. I prepared them for what issues might come up and I empowered them to make decisions. I said I would not second-guess them or punish them if I didn't agree with their decisions, and I didn't. They were fabulous and professional, and the experience did two things: It changed the way I managed to be even more collaborative, and it made me realize I would still have an effective, functioning business even when I wasn't actively involved for a while. It did wonders for the staff as well._\n\n#### THINGS TO DO IF YOU ONLY HAVE A MONTH\n\n1. **Rent or trade your house for a house in a different part of the country or world** \u2014anything to get you out of your known environment and the temptation or guilt to do chores or follow your regular routine.\n\n2. **Minimize your contact with work.** Arrange it so you do not have to check emails, call in, or go to meetings. Disconnect. If it is absolutely necessary to connect, make it only one time a week for one hour. Guess what? They can get along without you for that time period. If that is what you fear, reread Chapter 3!\n\n3. **Don't plan a stressful schedule at home or even traveling.** Make time for yourself. Schedule downtime and do different things. If you have to do chores, concentrate them on one day a week. Use the other time to explore, expand, learn\u2014about yourself and the things you've wanted to try.\n\n4. **Plan your time off, or you will end up frittering it away.** Yes, it is nice to sleep late, watch old movies, and read, but don't make that your primary activity. Set goals for what you want to experience and learn. Take risks to do things you've never done. Structure your time so that you get outside your comfort zone, yet give yourself downtime.\n\n5. **Think of taking a month off every year or at least every three to five years.** Make it part of your life.\n\n### THE GOALS CIRCLE EXERCISE\n\nHow many of us have talked about or written New Year's resolutions only to forget them about two weeks later? Then we go right back to our habits, whether they relate to our health, finances, relationships, or work, which results in lives that are not balanced and are therefore unfulfilling. The Goals Circle exercise is about changing your life and putting it back into harmony and balance.\n\nIn the exercise section of this chapter, we're going to ask you to complete a Goals Circle project that will serve as your guide for integrating all aspects of your life and being sure you create a more balanced approach to life. Cathy created this and wrote about it in her book, _The Artist's Way at Work,_ after having used it for several years. Now all the Sabbatical Sisters are creating Goals Circles and sharing them with each other at the beginning of each year. The purpose of the exercise is to balance your life around the most important aspects of it for the coming year. The simple chart is just a circle divided into six or eight pie slices of equal size to remind you to give equal time, or at least mental thought, to them.\n\nLabel the pie slices on the chart with six to eight aspects of your life. Examples might be: career, philanthropy, spirituality or religion, creativity, friends and family, romantic relationships, health and exercise, education, house renovation, or whatever you have on your horizon for the next year. The categories may be for that year only, such as renovating a house, or ongoing, such as health and exercise. Cathy says:\n\n_I usually do this on New Year's Eve or Day, or at least that week. I find a comfortable, quiet place to sit, build a fire, and get my cup of tea and journal or notebook. I first look at what I had said I wanted to do for the past year . . . and I usually have not looked at it for six months. I am always amazed by how much I have accomplished in many areas of my life . . . never 100 percent, but usually 60 to 80 percent, which is great. Somehow, \"lose fifteen pounds\" keeps reappearing on my lists and never quite makes it off! After reflecting on what I had done and what made me happy, I look to see if I want to include any of the existing categories or goals in the new Circle. I then draw my new Circle and begin to first flesh out the categories, then the goals. Some years I have had as many as twelve pie slices, but I havelearned that fewer are better and keep me more focused on what is really important. It doesn't mean that other things are not part of my life or responsibilities, but these six to eight areas should be my real focus for a happy and serene life._\n\nOnce you have decided on the aspects or categories, write five goals for each one that are measurable and obtainable in the next year. They may be as specific as \"redo my resume\" or directional as \"travel to a new country,\" but you should be able to sit down and know at the end of the year whether you obtained them or not.\n\nHere are some example categories and specific goals:\n\n#### FINANCES:\n\n\u2022 Get all financial records entered into a software program like Quicken or Money Management.\n\n\u2022 Read three books on financial management.\n\n\u2022 Meet with a financial planner.\n\n\u2022 Set up a Reboot Break fund and contribute at least $200 to it each month.\n\n\u2022 Pay off credit card balances each month.\n\n#### TIME FOR SELF:\n\n\u2022 Schedule an \"artist date\" each week for an hour.\n\n\u2022 Play basketball or tennis once a week.\n\n\u2022 Go to a concert or music event twice a month.\n\n\u2022 Have a massage once a month.\n\n\u2022 Take a golf weekend to learn or improve my game.\n\nIt may take you a couple of hours, or even days, to come up with five goals for each category, but it is important to do so to stretch yourself. When you are finished, go back and pick out the one goal from each category that is most important for you to do this coming year and put those on another circle. This circle is one you might carry with you and refer to every once in awhile.\n\n\"I recommend putting the larger chart, with five goals each, away for six months, then reviewing it,\" Cathy says. \"Don't try to update or change it . . . just look at it and see where you are against your goals. It is at the end of the year that you want to take the time to reflect.\" What we have learned from doing this for several years in a row is that most people accomplish more of their goals because they have taken the time to think about them and write them down. The process makes one think about all aspects of life and consider how to keep the parts in better balance. Having five goals in each category makes people think about what it would take to achieve their goals. By spending time planning what we want from life, we are more in control of how we realize those goals. We call it \"being present and open\" to what comes along. As we mentioned, the Sabbatical Sisters have taken this a step further by sharing our goals with each other at the beginning of the year. The extra step of verbalizing the goals in a supportive environment makes the goals even more attainable. You might try sharing your Goals Circle with a group of friends or family members whom you trust. You can help each other achieve them!\n\n###   EXERCISES\n\n##### _Exercise 10-1: Ten Things to Do_\n\n\u2022 Write out ten things you want to include in your life on a regular basis.\n\n\u2022 Schedule them into your calendar each week or month.\n\n\u2022 Keep the list in your wallet or purse.\n\n##### _Exercise 10-2: Goals Circle Exercise_\n\n\u2022 Draw your circle with six to eight equal pie slices and categories.\n\n\u2022 Put five goals under each category.\n\n\u2022 Review it every six months.\n\n\u2022 Create a second circle with just the categories and one key goal in each.\n\n\u2022 Put that in your wallet or purse and look at it every once in awhile.\n\n## _Conclusion_\n\nWe have learned a great deal not only from our own sabbaticals but from talking to so many others who have given themselves the gift of time. Below we share some of the best lessons we've learned about planning for and taking time off. To reboot your life, you need perspective, time for growth, rest, and renewal. You need to be in concert with the natural cycles of life.\n\n**Here are some favorite lessons learned by us and those we have interviewed:**\n\n\u2022 Give yourself permission to take time off\u2014don't be afraid.\n\n\u2022 You are not alone; this is emerging as a major trend.\n\n\u2022 You will receive many personal and professional benefits.\n\n\u2022 There are many benefits to your employers and society as a whole.\n\n\u2022 You will want to do this several times.\n\n\u2022 You can incorporate what you learn into your everyday life.\n\n\u2022 Financial planning is critically important.\n\n\u2022 It is important to plan how to handle naysayers, the unexpected, and time robbers.\n\n\u2022 Time is precious; don't waste it.\n\n\u2022 You will know yourself better after your Reboot Break. The inner journey continues.\n\nReading this book is just the beginning of the journey. Join us and other sabbatical takers on our blog to share your thoughts and experiences. You can reach us by going to _www.rebootbreak.com_ to blog and learn about our upcoming retreats held across the United States. Our email address is on the website as well.\n\nGood luck on your journey!\n\nThe Sabbatical Sisters:  \n _Catherine Allen_  \n _Nancy Bearg_  \n _Rita Foley_  \n _Jaye Smith_\n\n## | APPENDIX |\n\n### _Planning Checklist_\n\n#### **PRE-PLANNING**\n\n__ Make the decision to take time off to reboot your life.\n\n__ Begin to keep a journal of your thoughts, ideas, and dreams; write what you want to accomplish.\n\n__ Identify and prioritize your sabbatical dreams and goals.\n\n__ Determine how much time you need on the sabbatical.\n\n__ Tell your spouse, partner, significant other, or friends what you are thinking and include them in the process.\n\n__ Select dates to start and end your sabbatical.\n\n__ Plan when and how to tell your manager, business partner, staff; prepare a script.\n\n__ Have the conversation with everyone who needs to know your plans and discuss the arrangements for the time away.\n\n__ Train staff and make sure there are backups and alternative arrangements for all of your responsibilities.\n\n__ Begin to save and to prepare a budget for your time off.\n\n__ Speak to a financial advisor.\n\n__ Cut unnecessary expenses.\n\n__ Explore alternative ways to generate income (rental properties, selling things on eBay, part-time job or consulting opportunities, etc.).\n\n__ Prepare your house or apartment for rental if leaving on an extended trip.\n\n__ Have all medical exams and doctors' visits before you leave.\n\n__ Sign up for classes that you will want to take.\n\n__ Research places to volunteer, visit, travel.\n\n__ Make sure your passport is up to date if you will travel outside the country.\n\n__ Make flights and other travel arrangements.\n\nYou can set up a section in your notebook for plans that you develop. Include a timetable. For example, you might include the category of Reconnecting with Family and Friends. Part of that may be phone calls as well as dinners together. Part of it may involve travel. List these things separately with rough dates.\n\nCreate two calendars, starting with a wall calendar to write in your main events, such as travel, classes, and big projects, so you can visualize the main events. The second calendar is the planning calendar, which will be full of planning reminders and event activities. It can be online, on a white board, or whatever works for you. Just develop a system that helps in your process and doesn't end up stressing you out. You need to be able to organize tasks and information for the outcome you seek.\n\n#### **DETAILED PLANNING CHECKLIST**\n\nHere are more detailed planning tips that have been taken from scores of sabbatical takers. We recommend, if possible, that you start planning one year in advance of a Reboot Break, but it can be compressed as time dictates. Create your own list that meets your needs. This is just an example.\n\n##### **Planning: One Year Before**\n\n  _Goal Setting and Research_\n\n\u2022 Build a wish list of all the things you would ideally like to do.\n\n\u2022 Begin to log and journal your plans, hopes, and dreams. Capture them so that you do not forget. This will be fun to look back over as you get closer and after you have returned.\n\n\u2022 Include your spouse\/partner in the decision-making process by sharing ideas, hopes, and dreams.\n\n\u2022 Involve family and friends in the research of aspects of your sabbatical to share the experience and help to ease their and your anxiety.\n\n\u2022 Speak to others who have taken time off or done some of the things you would like to do.\n\n\u2022 Map out a plan of what needs to get done and break it into manageable chunks over a year's time.\n\n  _Travel and Educational_\n\n\u2022 Develop criteria for picking where you want to travel, if you want to travel. This can be to see nearby friends and family, explore a new place in your country, or travel abroad.\n\n\u2022 If traveling abroad, research all you can about the country, the culture, and the people.\n\n\u2022 Look at websites that offer options for volunteering or working abroad in countries of your choice. (See Resources in this Appendix.)\n\n\u2022 Family travel planning is important to start early\u2014 _www.suite101.com_ can help you with a wide range of options and information to help you think through the many steps of preparing yourself and your family and selecting places to go.\n\n\u2022 Learning about all the possible travel options in itself is a fun project. A good place to start is _www.thetravellerslounge.co.uk._\n\n\u2022 Begin studying a foreign language at home or build that into your plan. There are many options where you can live with a family in country or in reasonable accommodations that the language schools will provide. To get started, you can check out _www .languagelearningabroad.com,www.abroadlanguages.com, www.cesalanguages.com, and www.language-learning.net._\n\n  _Work_\n\n\u2022 Find out what your company's sabbatical or personal\/family leave policies are.\n\n\u2022 Talk to colleagues who have taken a sabbatical from your company.\n\n\u2022 Begin to build a plan for transitioning your work while you are away.\n\n\u2022 Identify who will step into your shoes.\n\n\u2022 Document what you are currently working on and what needs to be tracked during your absence.\n\n\u2022 Plan what you will say to your manager and when.\n\n\u2022 Prepare an email (or draft one for your boss) communicating your plans to your team, your organization, and your clients.\n\n  _Financing_\n\n\u2022 Explore options to finance your time off if you are not getting paid while away.\n\n\u2022 Speak to your financial advisor and begin putting things in place.\n\n\u2022 Begin a Reboot Break fund.\n\n\u2022 Consider what costs can be cut now.\n\n\u2022 See the Funding Your Freedom section in this Appendix.\n\n  _Home_\n\n\u2022 If traveling for an extended period of time, prepare your house for rental or sale. Plan longer-term projects and repairs. Consider house-swapping for a while, as another option. (See Resources for home-swap websites.)\n\n##### **Eight Months Before**\n\n  _Travel and Educational_\n\n\u2022 Contact and apply to academic programs.\n\n\u2022 Apply to special volunteer programs at home or abroad that require an application process and advance deposits to hold your place.\n\n\u2022 Look at career sites to get you thinking more about how you will be using your time while away. Some fun career sites include _www.escapefromcubiclenation.com_ and _www.gapyearforgrownups.co.uk._\n\n\u2022 Research travel sites to help you book inexpensive housing, such _aswww.hostelbookers.com._\n\n\u2022 Teaching English as a Foreign Language (TEFL) is a popular way to see the world and earn as you go. You will need to get certified; _www.online-education.net_ can help, as can _www.transitionsabroad.com._\n\n  _Work_\n\n\u2022 Cleaning can begin at work. It might be helpful to get rid of unnecessary piles and old project folders to clear the way for others to use your office during your absence. It will also make your office more welcoming when you return, paving the way for new ideas, projects, and energy.\n\n  _Financing_\n\n\u2022 Contribute to your Reboot Break fund.\n\n\u2022 Continue cutting expenses.\n\n  _Home_\n\n\u2022 Clear clutter and get your house ready for others or identify projects that you may want to do during your sabbatical.\n\n\u2022 If renting or swapping your home, there may be things that need to be repaired or adapted for your temporary house guests. It's a good time to do some of the things that had been waiting to be done anyway.\n\n  _Other_\n\n\u2022 Practice taking a mini-sabbatical for a week or even just a weekend. Disconnect from your computer and BlackBerry.\n\n##### **Six Months Before**\n\n  _Travel and Educational_\n\n\u2022 Update all passports and necessary travel documents. Most countries will require that your passport be valid for six months past your entry into their countries.\n\n\u2022 Check to see if a visa is needed.\n\n\u2022 Make extra copies of your passport; carry one with you and ask someone else to hold a copy as a backup.\n\n\u2022 If needed, arrange for your children's schooling when traveling abroad or elsewhere in the United States.\n\n  _Work_\n\n\u2022 Speak to your boss\/manager about your desire to take a sabbatical. Present a well-thought-out, positive plan covering what you want to do, why you want to do it, and the advantages and opportunities for the company. Describe your ideas and suggestions about how you will make sure that your work is covered and how others will benefit from the new experiences and responsibilities. Get buy-in from your boss, and after the meeting send an email to both of you that summarizes all that was discussed.\n\n\u2022 Begin to implement a training plan for those who will be covering for you while you are away.\n\n\u2022 Keep a printed copy of the company's sabbatical policy in case you need to refer to it while you are away.\n\n  _Financing_\n\n\u2022 Meet with your insurance broker to make sure you have the insurance you need to protect you during your time away.\n\n\u2022 Continue adding to your Reboot Break fund.\n\n\u2022 Research and prepare to have your bills paid online or through a trusted friend or professional.\n\n  _Home_\n\n\u2022 If needed, begin your research on storing options for furniture or other personal items you would feel better about storing for safety. While away, you don't want to have to worry about your personal valuables.\n\n##### **Three Months Before**\n\n  _Travel and Educational_\n\n\u2022 Get in touch with people you want to visit if you have not yet done so.\n\n\u2022 Begin to think about what you want to pack to take with you on any travels. (Remember to \"pack light.\")\n\n\u2022 Register for any courses or programs that you want to participate in while on your sabbatical.\n\n\u2022 Get an international driver's license.\n\n\u2022 Buy a railway card for the countries you are visiting.\n\n  _Work_\n\n\u2022 Train staff who will be taking over during your absence.\n\n\u2022 Plan for how your work will be handled and by whom, arrange for and train any temporary help you need while you are away, and make sure your liability insurance will be in effect during your absence.\n\n  _Especially For Business Owners, Entrepreneurs, and Sole Practitioners_\n\n\u2022 Figure out who will be running the show and how the work will be handled.\n\n\u2022 Arrange for and train any temporary help you need while you are away.\n\n\u2022 Make sure your liability insurance is in effect while you are away.\n\n  _Financing_\n\n\u2022 Make a will or update your existing one, and let people know where you will be and when.\n\n\u2022 Set up systems for personal finances and for paying monthly bills.\n\n  _Home_\n\n\u2022 Work on identifying ways to rent your house or apartment. Start by checking with local real estate agents. Also see listings in the Resources section of this Appendix.\n\n\u2022 Prepare your house or apartment for rental by identifying what should be packed and what stays.\n\n\u2022 Make doctor and dentist appointments, make sure you are up-to-date on your checkups to avoid any unnecessary health issues.\n\n\u2022 Clarify your health coverage for traveling locally and abroad.\n\n\u2022 Identify a place to store your car.\n\n\u2022 Identify someone to care for pets, and perhaps plants, while you are away.\n\n##### **Two Months Before**\n\n  _Travel and Educational_\n\n\u2022 Arrange for an international calling card. (Once there, get in-country calling cards.)\n\n  _Work_\n\n\u2022 Build and implement a communication plan for staff and customers\/clients.\n\n\u2022 Begin to transition things to those who will be working on your behalf and test out the new approach and process.\n\n  _Financing_\n\n\u2022 Speak to the bank about how to access your money while abroad.\n\n\u2022 Automate paying your bills.\n\n\u2022 Identify friends, family members, an accountant, or small business services to pay your bills if they cannot be automated, and to handle affairs when you are away. Bills can be sent directly to them.\n\n\u2022 Sign and designate a power of attorney.\n\n\u2022 Make arrangements for paying taxes if you will be away during tax time.\n\n  _Home_\n\n\u2022 Pack the house.\n\n\u2022 Look into voicemail services so you can retrieve calls while away or arrange for an international cell phone.\n\n\u2022 International calling options also might include SKYPE or resources like MagicJack or Vonage. Research and secure a service that is easily accessible and works with your budget.\n\n\u2022 Speak to your doctor about getting a six-month supply or more of your medications, and research sources for getting it while away from home, should that become necessary.\n\n\u2022 Find out about getting an absentee ballot for voting, should you plan on being away during any election.\n\n\u2022 Set up a blog and learn to post pictures and stories there or on sites like _www.snapfish.com or www.flickr.com._\n\n\u2022 Get your camera in working order and get extra memory cards and batteries.\n\n##### **One Month Before**\n\n  _Travel and Educational_\n\n\u2022 Look into and sign up for local classes and activities, such as yoga.\n\n\u2022 Get a library card!\n\n\u2022 Pack light. You want to be free to move around and not be encumbered by heavy bags and items you don't really need.\n\n\u2022 Make sure you have a list of phone numbers and email addresses for all your friends, family, and important contacts, including your lawyer, accountant, plumber, electrician, and anyone you might need to speak to while you are away.\n\n\u2022 Check that all of your camera, computer, and telephone equipment is ready to go and charged for action.\n\n\u2022 Make sure you have the right adaptors for different electric power if you are traveling.\n\n  _Work_\n\n\u2022 Introduce the person taking on your role or working in your company to clients and staff to ease the transition.\n\n  _Financing_\n\n\u2022 Cancel or suspend your newspaper or magazine subscriptions, gym memberships, and any other unnecessary expenses while you are going to be away. Suspend your phone service.\n\n  _Home_\n\n\u2022 Arrange for mail hold or redirection to a friend's or family member's address.\n\n\u2022 Arrange for plumbers, electricians, and handymen to be available while you are gone.\n\n\u2022 Notify your utility company if services should be suspended for a period of time.\n\n\u2022 Do all the grooming you typically like to do (hair, nails, etc.).\n\n## _Funding Your Freedom_\n\nFollowing are some financial checklists that can help you plan for, and fund, your Reboot Break. Use them to capture where you are today financially and where you want to be while on your break.\n\n  Monthly Fixed Expenses: _(Create columns for current spending and planned changes)_\n\nMortgage #1\n\nMortgage #2\n\nAuto Loan\/Lease #1\n\nAuto Loan\/Lease #2\n\nOther Auto\/Truck Loans\n\nHome Equity Loan\n\nDebt Consolidation Loan\/Payment\n\nStudent Loan(s)\n\nRent\n\nCondo or Homeowner Association Fee\n\nElectricity Oil or Gas Heat Water\n\nGarbage Collection Sewer\n\nPhone (land line)\n\nCell Phones, Pagers, PDA\n\nCable\/Satellite TV\/TiVo Boxes\n\nInternet Access\n\nCar Insurance\n\nHealth Insurance\n\nChild Support\n\nAlimony\n\nMedical\/Dental Payments\n\nRetirement Savings\n\nEmergency Fund Savings\n\nOther\n\n**Total Monthly Fixed expenses:**\n\n  **Money to Set Aside for Recurring expenses:** _(Create columns for total amount, monthly savings amount, and planned changes)_\n\nProperty Taxes\n\nSchool Taxes\n\nState and Local Taxes\n\nQuarterly Income Taxes (Local, State and Federal)\n\nHome Insurance\n\nLong-Term Care Insurance\n\nLife Insurance\n\nCar License Renewal\n\nCar Maintenance\n\nHome Repair\n\nVeterinarian\n\nGifts\n\nVacation\/Travel\n\nTuition and School Costs\n\nTithing\n\nMemberships\n\nCharitable Donations\n\nOther\n\n**Total Monthly Savings for Periodic Expenses:**\n\n  **Monthly Variable expenses:** _(Create columns for current spending and planned changes)_\n\nCredit Card #1\n\nCredit Card #2\n\nCredit Card #3\n\nCredit Card #4\n\nOther Credit Cards\n\nStore Cards\n\nGas Cards\n\nOther Credit Lines\n\nGroceries\n\nEating Out\n\nSchool Lunches\n\nHousehold Supplies\n\nGas\/Tolls\/Parking\n\nPublic Transportation\n\nHealth Club Membership\n\nDaily Coffee\/Snacks\n\nLaundry\/Dry Cleaning\n\nPet Care and Supplies\n\nBaby Items\n\nChildrens' Allowances\n\nHaircuts\/Grooming\/Manicures, etc\n\nCosmetics\n\nClothes\n\nEntertainment (movies, rentals, on-demand, sports events, theatre, concerts, day trips)\n\nCDs\n\nClub Dues\n\nDaycare\/Babysitters\n\nLessons\n\nField Trips\n\nATM Fees\n\nComputer\/Online Expenses\n\nDonations\/Tithes\n\nLawn Service\n\nHousekeepers\n\nEmergency Savings\n\nTobacco\/Alcohol\n\nSubscriptions\n\nPostage\n\nOther\n\n**Total Monthly Variable expenses:**\n\n  **Monthly Income Sources:**\n\nNet Income #1 (income less taxes, Social Security, Medicare)\n\nNet Income #2\n\nNet Income #3\n\nRental Property\n\nAlimony\n\nChild Support\n\nPensions\n\nRetirement Income\n\nSocial Security\n\nInvestment Income\n\nOther Income\n\n**Total Take-Home Income:**\n\n## _Organizations That Get It_\n\nSabbaticals have long been associated with universities and religious organizations. Over the last decade there have been a rising number of corporations, law firms, service organizations, and not-for-profit organizations that embrace the benefits of sabbaticals or Reboot Breaks.\n\nBelow is a list of companies and firms that have implemented some form of a sabbatical program for their employees. The majority of programs were created to offer a fully paid Reboot Break after five, seven, or ten years of service and with an average duration of one to three months. Some programs offer six months to a year. Many other organizations do not have formal programs, but, when approached by employees in good standing, will allow them to customize a personal Reboot Break\u2014while holding their jobs for their return. Hewlett Packard (HP), for example, does not have a formal program, but employees can create their own. At Citibank, one can request up to one year unpaid leave.\n\nIn times of economic difficulty, some firms implement temporary programs to help them manage costs while still retaining key talent. Furlough programs offer employees the opportunity to take time off, unpaid or partially paid, but with a guarantee that their jobs will be waiting for them upon their return. Accenture offered such a one-time program. Workers who agreed to an 80 percent pay cut were allowed a six to twelve month sabbatical with benefits. Cisco staffers who accepted a two-thirds pay cut were offered the chance to volunteer at one of twenty-nine pre-selected not-for-profit organizations. Cisco management expected thirty people to take advantage of this program; 300 signed up. In the recent economic downturn, Skadden, Arps, Slate, Meagher & Flom, along with other law firms, paid newly hired lawyers one-third of their starting salaries not to show up for a year. They retained talent while giving their new employees a gift of time. Some were expected to work pro bono for nonprofit organizations, build experience, and meet potential contacts and future clients.\n\nFor more on organizations that offer formal sabbatical programs, visit our website, _www.rebootbreak.com._\n\nHere is a list of some companies and firms that have implemented some form of a sabbatical program for their employees:\n\nAA Appointments\u2014www.aaappointments.com\n\nAARP\u2014www.aarp.org\n\nAbacus Planning Group\u2014www.abacusplanninggroup.com\n\nAccenture\u2014www.accenture.com\n\nActel\u2014www.actel.com\n\nAddleshaw Goddard\u2014www.addleshawgoddard.com\n\nAdobe Systems\u2014www.adobe.com\n\nAdvanced Micro Devices (AMD)\u2014www.amd.com\n\nAlston & Bird\u2014www.alston.com\n\nAmerican Century Investments \u2014www.americancentury.com\n\nAmerican Express\u2014www.home.americanexpress.com\n\nAmmex\u2014www.ammex.com\n\nAnderson ZurMuehlen\u2014www.azworld.com\n\nApple\u2014www.apple.com\n\nAppRiver\u2014www.appriver.com\n\nArco Construction\u2014www.arcoconstruction.com\n\nArrow Electronics\u2014www.arrow.com\n\nAutodesk\u2014www.autodesk.com\n\nAvalonBAY Communities\u2014www.avalonbay.com\n\nBAIN & Co.\u2014www.bain.com\n\nBarfield, Murphy, Shank & Smith\u2014www.bmss.com\n\nBingham McCutchen\u2014www.bingham.com\n\nBKD\u2014www.bkd.com\n\nBlast Radius\u2014www.blastradius.com\n\nBoston Consulting Group\u2014www.bcg.com\n\nBrogan & Partners Consulting Group\u2014www.brogan.com\n\nBureau of National Affairs\u2014www.bna.com\n\nCapital One\u2014www.capitalone.com\n\nCDW\u2014www.cdw.com\n\nCharles Schwab\u2014www.schwab.com\n\nCirque du Soleil\u2014www.cirquedusoleil.com\n\nCisco\u2014www.cisco.com\n\nCitibank\u2014www.citibank.com\n\nCitizens Financial Group\u2014www.citizensbank.com\n\nCitrin Cooperman\u2014www.citrincooperman.com\n\nClark Nuber\u2014www.clarknuber.com\n\nCleary Gottlieb Steen & Hamilton\u2014www.cgsh.com\n\nClif Bar & Company\u2014www.clifbar.com\n\nCMS Cameron McKenna\u2014www.cms-cmck.com\n\nCoblentz, Patch, Duffy & Bass\u2014www.coblentzlaw.com\n\nCommonfund\u2014www.commonfund.org\n\nContainer Store\u2014www.containerstore.com\n\nCredit Suisse First Boston (CSFB)\u2014www.credit-suisse.com\n\nDaxko\u2014www.daxko.com\n\nDeloitte\u2014www.deloitte.com\n\nDeutsche Bank\u2014www.db.com\n\nDoubleday Broadway Publishing Group\u2014doubleday.knopfdoubleday.com\n\nDrivers Jonas Deloitte\u2014www.djdeloitte.co.uk\n\nDurfee Foundation (finances sabbaticals for non-profit executives)\u2014www.durfee.org\n\neBay\u2014www.ebay.com\n\nEdelman\u2014www.edelman.com\n\nEdward Jones\u2014www.edwardjones.com\n\nEhrhardt Keefe Steiner & Hoffman\u2014www.eksh.com\n\nEide Bailly\u2014www.eidebailly.com\n\nEpic Systems\u2014www.epic.com\n\nFarella Braun & Martel\u2014www.fbm.com\n\nFedEx\u2014www.fedex.com\n\nFleishman-Hillard\u2014www.fleishmanhillard.com\n\nFoley and Mansfield\u2014www.foleymansfield.com\n\nFoster Pepper\u2014www.foster.com\n\nGuardian Media Group\u2014www.gmgplc.co.uk\n\nGenentech\u2014www.gene.com\n\nGeneral Mills\u2014www.generalmills.com\n\nGoldman Sachs\u2014www2.goldmansachs.com\n\nHallmark\u2014www.hallmark.com\n\nHammonds\u2014www.hammonds.com\n\nHewlett Packard\u2014www.hp.com\n\nHLB Tautges Redpath\u2014www.hlbtr.com\n\nHopping Green and Sams\u2014www.hgslaw.com\n\nHotel Equities\u2014www.hotelequities.com\n\nHutchinson Black and Cook\u2014www.hbcboulder.com\n\nInfosys\u2014www.infosys.com\n\nIntel\u2014www.intel.com\n\nJohn Lewis Partnership\u2014www.johnlewispartnership.co.uk\n\nJoie de Vivre Hospitality\u2014www.jdvhotels.com\n\nJones Hall\u2014www.joneshall.com\n\nKatten Muchin Rosenman LLP\u2014www.kattenlaw.com\n\nKetchum\u2014www.ketchum.com\n\nLinchris Hotel Corporation\u2014www.linchris.com\n\nLinear Technology\u2014www.linear.com\n\nLinklaters\u2014www.linklaters.com\n\nLogos Research Systems\u2014www.logos.com\n\nMacdonald and Company\u2014www.macdonaldandcompany.com\n\nMcDonald Jacobs\u2014www.mcdonaldjacobs.com\n\nMcDonald's\u2014www.mcdonalds.com\n\nMcGladrey & Pullen\u2014www.mcgladrey.com\n\nMen's Warehouse\u2014www.menswearhouse.com\n\nMenlo Innovations\u2014www.beta.menloinnovations.com\n\nMicrosoft\u2014www.microsoft.com\n\nMITRE\u2014www.mitre.org\n\nMortenson Construction\u2014www.mortenson.com\n\nNet Atlantic\u2014www.netatlantic.com\n\nNew Leaf Community Markets\u2014www.newleaf.com\n\nNewsweek\u2014www.newsweek.com\n\nNike\u2014www.nike.com\n\nNorton Rose\u2014www.nortonrose.com\n\nNRG\u2014www.nrgenergy.com\n\nPatagonia\u2014www.patagonia.com\n\nPeace Over Violence\u2014www.peaceoverviolence.org\n\nPerkins Coie\u2014www.perkinscoie.com\n\nPlante & Moran\u2014www.plantemoran.com\n\nPricewaterhouseCoopers\u2014www.pwc.com\n\nProcter & Gamble\u2014www.pg.com\n\nQuad\/Graphics\u2014www.qg.com\n\nQuikTrip\u2014www.quiktrip.com\n\nRalston-Purina\u2014www.purina.com\n\nRandom House\u2014www.randomhouse.com\n\nRecreational Equipment (REI)\u2014www.rei.com\n\nReznick Group\u2014www.reznickgroup.com\n\nRobert W. Baird\u2014www.rwbaird.com\n\nRossetti\u2014www.rossetti.com\n\nRSM McGladrey\u2014www.mcgladrey.com\n\nRussell Investments\u2014www.russell.com\n\nSatyam Technologies\u2014www.satyamtechnologies.net\n\nS.C. Johnson & Son\u2014www.scjohnson.com\n\nScholastic\u2014www.scholastic.com\n\nSchroder Investment Management\u2014www.schroders.com\n\nSeattle-Northwest Securities\u2014www.snwsc.com\n\nSegal\u2014www.segalco.com\n\nSeventh Generation\u2014www.seventhgeneration.com\n\nShearman & Sterling\u2014www.shearman.com\n\nSilicon Graphics\u2014www.sgi.com\n\nSkadden, Arps, Slate, Meagher & Flom\u2014www.skadden.com\n\nStanding Partnership\u2014www.standingpr.com\n\nStayner Bates & Jensen\u2014www.stayner.com\n\nStrategic Actions for a Just Economy (SAJE)\u2014www.saje.net\n\nThe Sun\u2014www.thesun.co.uk\n\nSybase\u2014www.sybase.com\n\nText 100\u2014www.text100.com\n\nTimberland\u2014www.timberland.com\n\nUnicef UK\u2014www.unicef.org.uk\n\nUnited States Navy\u2014www.navy.mil\n\nVauxhall\u2014www.vauxhall.co.uk\n\nWaggener Edstrom\u2014www.waggeneredstrom.com\n\nWilliam Mills Agency\u2014 www.williammills.com\n\n## _Resources_\n\n#####   _Books:_\n\nBach, David. _Smart Women Finish Rich: 7 Steps to Achieving Financial Security and Funding Your Dreams._ New York, NY: Broadway Books, 1999.\n\nBishop, Kimberly, Dale Burg and Ginendolyn Penner. _Get Down to Business and You'll get the Job!_ Canada: Gemma B. Publishing, 2010.\n\nBonvoisin, Ariane de. _The First 30 Days: Your Guide to Any Change._ New York, NY: HarperCollins Publishers, 2008.\n\nBridges, William. _The Way of Transition: Embracing Life's Most Difficult Moments._ Jackson, TN: De Capo Press, 2001.\n\nCameron, Julia. _The Artist's Way._ New York, NY: Jeremy P. Tarcher\/Putnam, 1992.\n\nCameron, Julia, Mark Bryan, and Catherine Allen. _The Artist's Way at Work._ New York, NY: William Morrow and Company, 1998.\n\nClements, Dan and Tara Gignac. _Escape 101: Sabbaticals Made Simple._ Canada: Brainranch, 2007.\n\nDlugozima, Hope, James Scott, and David Sharp. _Six Months Off: How to Plan, Negotiate and Take the Break You Need Without Burning Bridges or Going Broke._ Gordonsville, VA: Henry Holt, 1996.\n\nFerriss, Timothy. _The 4-Hour Workweek._ New York, NY: Crown Publishers, 2007.\n\nFreedman, Marc. _Encore: Finding Work That Matters in the Second Half of Life._ New York, NY: Public Affairs (Perseus Group), 2007.\n\nGilbert, Elizabeth. _Eat, Pray, Love: One Woman's Search for Everything Across Italy, India and Indonesia._ New York, NY: Viking Penguin, 2006.\n\nGray, John. _How to Get What You Want and Want What You Have: A Practical and Spiritual Guide to Personal Success._ Scranton, PA: HarperCollins Publishers, 2000.\n\nLee, Roberta. _The Superstress Solution._ New York, NY: Random House, 2010.\n\nLevine, Robert. _Power Sabbatical: The Break that Makes a Difference._ Scotland: Findhorn Press, 2007.\n\nLindbergh, Anne Morrow. _Gift from the Sea._ New York, NY: Knopf Doubleday, 1955.\n\nSavage, Terry. _The Savage Number: How Much Money Do You Need to Retire?_ New York, NY: John Wiley & Sons, Inc., 2005.\n\nSedlar, Jeri, and Rick Miners. _Don't Retire, Rewire: 5 Steps to Fulfilling Work That Fuels Your Passion, Suits Your Personality, or Fills Your Pocket._ New York, NY: Alpha Books, 2003.\n\nSmith, Jaye, and Dina von Zweck. _Venus Unbound: A Guide to Actualizing the Power of Being Female._ New York, NY: Simon & Schuster, 1989.\n\nStanny, Barbara. _Secrets of Six Figure Women: Surprising Strategies to Up Your Earnings and Change Your Life._ New York, NY: HarperCollins, 2002.\n\nTolle, Eckhart. _The Power of Now: A Guide to Spiritual Enlightenment._ Canada: Namaste Publishing, Inc., 1997.\n\nTracy, Brian. _Eat That Frog!: 21 Great Ways to Stop Procrastinating and Get More Done in Less Time._ San Francisco, CA: Berrett-Koehler, 2006.\n\n##### **White Papers:**\n\nAllen, Catherine, Leslie Mitchell, and Janey Place. \"The New Consumer Value: Living Light.\" White paper by The Santa Fe Group, 2008. See _www.santa-fe-group.com\/wp-content\/uploads\/\/2010\/07\/NewConsumerValueNove2008.pdf._\n\nAllen, Catherine, Nancy Bearg, Rita Foley, and Jaye Smith. \"Job Loss: Turning Downtime into Your Time.\" White paper by The Santa Fe Group, 2009. See _www.santa-fe-group.com\/wp-content\/uploads\/\/2010\/07\/SFG-WhitePaper-Sabbatical-Mar2009.pdf._\n\n#####   _Sources of Information on Financial Planning and Budgeting_\n\n#### **Magazines and Newsletters:**\n\n_Consumer Reports Money Advisor_  \n _Smart Money_  \n _Money Magazine_\n\n#### **Websites:**\n\n_www.bankrate.com_\n\n_www.betterbudgeting.com_\n\n_www.billshrink.com_\n\n_www.consumerreports.org\/cro\/money\/index_\n\n_www.mint.com_\n\n_www.money.cnn.com\/tools_\n\n_www.myfico.com_\n\n_www.nfcc.org_\n\n_www.savingforcollege.com_\n\n_www.smartmoney.com_\n\n_www.terrysavage.com_\n\n#####   _Sources of Information on Planning for Time Off:_\n\n_www.encore.org_\n\n_www.first30days.com_\n\n_www.gapyearforgrownups.co.uk_\n\n_www.rebootbreak.com_\n\n_www.retiredbrains.com_\n\n_www.thecareerbreaksite.com_\n\n#####   _Sources of Information on Renting\/Trading Homes and Apartments:_\n\n_www.cyberrentals.com_\n\n_www.homeaway.com_\n\n_www.homeexchange.com_\n\n_www.rentvillas.com_\n\n_www.sabbaticalhomes.com_\n\n#####   _Sources of Information on Volunteering:_\n\n_www.abroaderview.org_\n\n_www.crossculturalsolutions.org_\n\n_www.enkosiniecoexperience.com_\n\n_www.vocationvacations.com_\n\n_www.volunteervacations.com_\n\n## | INDEX |\n\nacademic sabbaticals, ,\n\nage at time of Reboot Break, \u20135\n\nAllen, Catherine, \u2013122, \u2013172, \u2013214\n\nannualcreditreport.com,\n\nartist dates, ,\n\n_Artist's Way at Work, The_ (Cameron, et al.), \u2013100, \u2013137\n\nart of enough, developing, \u2013110\n\nbankrate.com,\n\nbarriers to planning, \u201343\n\nbarriers to time off, \u20135,\n\nBearg, Nancy\n\nAfrican desert exploration,\n\non being in the moment,\n\ncareer-enhancement break,\n\non clearing clutter, \u201397\n\nfirst 30 days, \u201396\n\nlifelong sabbatical journey, \u2013174\n\nsabbatical story, \u2013216\n\nbetween-gigs sabbaticals, \u201317\n\nblogging,\n\nBonvoisin, Ariane de, , , \u2013123\n\nBridges, William, \u2013127\n\nbudgets and budgeting, \u201373, \u201386, \u201388. _See also_ living light\n\nburnout, xii, , \u201399,\n\nCameron, Julia, \u2013100\n\ncareer-enhancement sabbaticals, \u201326, ,\n\nCircle Goals, , \u2013181\n\nCitizens Financial Group,\n\nclients, talking to about your plans, \u201359\n\nCommonfund,\n\ncommunicating about your plans\n\nwith clients and customers, \u201359\n\nwith co-workers, peers, and board members,\n\nwith employers, \u201358\n\nwith family and friends, , \u201352, , \u2013157,\n\noverview,\n\ncourse correction sabbaticals, \u201324, \u201328\n\nco-workers, telling your plans to,\n\nCreating Space phase\n\nclearing and organizing, \u201397, ,\n\ndaily approach, \u2013101\n\nemptying the emotional closet, \u201399\n\nexercises, \u2013102\n\nfirst 30 days, \u201397\n\noverview, \u201392\n\ncredit cards\n\npaying down, , , ,\n\ntravel rewards from,\n\ncredit reports, \u201378\n\ncustomers, talking to about your plans, \u201359\n\ndebts, paying down, , , ,\n\ndebt-to-income ratio, \u201377\n\ndeciding on a sabbatical, \u201322\n\ndeferred income,\n\ndreams and dreaming\n\njournaling and, ,\n\noverview, \u20132,\n\nreturning to an old dream, \u201328\n\nstaying in touch with,\n\nwalking\/visualization exercise, \u201345\n\n_See also_ goals; journaling\n\n_Eat, Pray, Love_ (Gilbert),\n\neducational sabbaticals, \u201326, \u201337, \u2013117, \u2013188,\n\nemotional closet, clearing your, \u201399, \u2013118\n\nemotional healing sabbaticals, \u201324, \u201333\n\nemployment. _See_ workplace sabbaticals\n\n_Encore: Finding Work That Matters in the Second Half of Life_ (Freedman), \u201340\n\nentrepreneurs, sabbaticals for, \u201361, ,\n\nexpansion,\n\nexpectations, \u2013157,\n\nExploration phase\n\nexercises,\n\nexploring relationships, \u2013121\n\nexploring the hurt, \u2013118\n\nexploring the mind, \u2013117\n\nexploring the self, \u2013119\n\nexploring the world,\n\noverview, \u201391, , , \u2013112\n\npacking light, \u2013123,\n\nvolunteering, \u2013114\n\nexploring new options, \u2013131\n\nexternal sabbatical robbers, , , \u2013150\n\nFamilies and Work Institute,\n\nfamily\n\ncoping skills for, \u2013166\n\nexercises,\n\nof laid-off workers, \u201388\n\nmoney saving and, \u201387\n\npersonal concerns of, \u2013164\n\nreactions to your sabbatical, \u2013153, \u2013162\n\nsabbaticals related to needs of, \u201324, \u201330, \u201348, \u2013121\n\nsupporting your, \u2013165\n\ntalking about your plans with, , \u201352, , \u2013157,\n\n_See also_ relationships\n\nFamily and Medical Leave Act (FMLA), , \u201348\n\nFICO scores, ,\n\nfinancing your sabbatical\n\nbudgeting, \u201373, \u201388\n\nbudget travel, \u201386\n\nemployer support, \u201379\n\nexercises, \u201389\n\nfinancial checklists, \u2013196\n\nfinancial impact on your career, \u201369\n\ngoals and, \u201370\n\nfrom home equity, \u201378\n\nliving light, \u201371, \u201384, \u201387\n\noverview, \u201347, \u201368, \u201381\n\nplanning checklist, , , , , ,\n\nfrom savings, , \u201376\n\ntraveling light, \u201386\n\nunemployment, \u201382\n\nunexpected sabbaticals, \u201388\n\nwith windfalls, \u201374,\n\n_First 30 Days, The_ (Bonvoisin), , ,\n\nFoley, Rita, \u201334, \u2013173, \u2013218\n\nformal programs in the workplace, \u201313,\n\nFortune's \"100 Best Companies to Work For,\"\n\nFreedman, Marc, \u201340\n\nfrequent flyer miles,\n\nfurloughs, . _See also_ unexpected sabbaticals\n\n_Gift From the Sea_ (Lindbergh),\n\nGilbert, Elizabeth,\n\ngoals\n\nclarifying goals, \u2013110, , \u2013180,\n\nfinancing your sabbatical and, \u201370\n\nidentifying, \u201346,\n\n_See also_ dreams and dreaming; journaling\n\nGoals Circle exercise, , \u2013181\n\nguilt, \u2013142, , \u2013164\n\nhome\n\ncutting expenses, ,\n\nrenting out or trading, , , \u201384, \u201387,\n\nrenting out or trading planning checklist, , \u2013190, , , ,\n\nselling your, , , \u201397, , \u2013163\n\nhome equity line of credit (HELOC), \u201377\n\n_How to Get What You Want and Want What You Have_ (Gray), \u2013149\n\nindulging yourself, \u201395\n\ninheritance, \u201374,\n\nIntel, \u201312\n\ninternal sabbatical robbers, \u2013142, \u2013146\n\ninternships and mini-trials, \u2013135\n\ninteroffice announcement, \u201358\n\njob stress, xii, \u2013168\n\njournaling\n\nfor connecting with self,\n\nas lifelong sabbatical,\n\noverview, , ,\n\nquieting self-criticism with, \u2013146\n\nlaid-off employees. _See_ unexpected sabbaticals\n\nlawyers and sabbaticals,\n\nlearning as goal, \u201324, \u201337, \u2013117. _See also_ Exploration phase\n\nLee, Roberta, \u201394\n\nlife-enhancement sabbaticals, \u201324, \u201337, \u2013117\n\nlifelong sabbaticals\n\nauthors' examples, \u2013175\n\nexercises,\n\nGoals Circle exercise, , \u2013181\n\noverview, \u2013171\n\nLindbergh, Anne Morrow,\n\nliving light, \u201371, \u201384, \u201387. _See also_ budgets and budgeting\n\nmindfulness, \u2013109\n\nmini-sabbaticals, \u201351, , \u2013177,\n\nmini-trials and internships, \u2013135\n\nmotivations for sabbatical decision, \u201324, . _See also_ sabbaticals, types of\n\nmultitasking, ,\n\nnetworking, \u2013134\n\noutplacement services,\n\npacking light, \u2013123,\n\npeers, talking to about your plans,\n\npersonal regrowth\n\nallowing time for, , , ,\n\nauthor's stories, \u2013220\n\nclearing your emotional closet, \u201399, \u2013118\n\ndeveloping mindfulness, \u2013109, \u2013171\n\nemotional healing sabbaticals, \u201324, \u201333\n\nguilt vs., \u2013142, , \u2013164\n\noverview,\n\nreflection during Reentry, , \u2013131\n\nself- exploration, \u2013119\n\n_See also_ goals; journaling; lifelong sabbaticals; Reconnection phase\n\nphases of a sabbatical, \u201391. _See also_\n\nCreating Space phase\n\nExploration phase\n\nlifelong sabbaticals\n\nReconnection phase\n\nReentry phase\n\nphysical healing, \u2013106\n\nplanning\n\nand age at time of Reboot Break, \u20135\n\ncommunicating your plans, \u201359\n\nfor entrepreneurs, \u201361,\n\nexercises,\n\nidentifying goals, \u201346\n\noverview, xiii\u2013xiv, \u201343, \u2013177, \u2013183\n\nplan outline, \u201364\n\npre-planning checklist, \u2013186\n\nresponses to questions, \u2013147,\n\ntimeframe choice, \u201351\n\nvolunteering plans, \u201363\n\n_See also_ travel plans\n\nPlanning Checklist, \u2013194\n\nPlanning Grid, \u201364\n\n_Power of Now, The_ (Tolle),\n\npre-retirement sabbaticals, \u201324, \u201338\n\npriority clarification, \u2013110\n\nReconnection phase\n\nclarifying goals, \u2013110\n\nenjoying the moment, \u2013109\n\nexercises,\n\nfeeding the body, \u2013106\n\nfeeding the soul, \u2013108\n\noverview, \u201391, , \u2013105\n\nrecruiting and retention tool, sabbatical as, xiii, ,\n\nReentry phase\n\nexercises, \u2013138\n\nexploring new options, \u2013131\n\nmini-trials and internships, \u2013135,\n\nnetworking, \u2013134\n\noverview, \u201391,\n\nplanning for, , , \u2013137\n\nreflection time, \u2013128\n\nreflection during Reentry, , \u2013131\n\nreinvention sabbaticals, \u201324, \u201328\n\nrelationships\n\nexploring new, \u2013121\n\nrejuvenating, , , , , \u2013213\n\n_See also_ family\n\nresources, \u2013204\n\nretirement\n\nBaby Boomers and, \u20135\n\nphased retirement, \u201340\n\npre-retirement sabbaticals, \u201324, \u201338\n\nsavings for,\n\nrockwork and sandwork,\n\nsabbatical gifts, , \u2013148\n\nsabbatical robbers\n\nexercises, \u2013150\n\nexternal robbers, , , \u2013147\n\ninternal robbers, \u2013142, \u2013146\n\noverview, \u2013140, \u2013149\n\nsabbaticals\n\nage at time of, \u20135\n\nhistory of, \u20138\n\nas necessity, xiii\n\nsabbaticals, types of\n\ncareer-enhancement sabbaticals, \u201326, ,\n\ncourse correction sabbaticals, \u201328\n\neducational sabbaticals, \u201326, \u201337, \u2013117, \u2013188,\n\nemotional healing sabbaticals, \u201333\n\nfamily-related sabbaticals, \u201330\n\nlife-enhancement sabbaticals, \u201337, \u2013117\n\noverview, \u201324\n\npre-retirement sabbaticals, \u201338\n\nreinvention sabbaticals, \u201328\n\nvolunteering sabbaticals, \u201335\n\n_See also_ unexpected sabbaticals\n\nSanta Fe Group, \u201338\n\nsaving for your sabbatical, , \u201377\n\nsaying \"no,\" , , ,\n\nself-discovery paths, \u201337\n\nself- exploration, \u2013119\n\nself-healing. _See_ personal regrowth\n\nShearman & Sterling LLP, \u201363\n\nSmith, Jaye, \u2013175, \u2013220\n\nsole practitioners, sabbaticals for, \u201361, ,\n\nstress, xii, , \u2013168,\n\n_SuperStress Solution, The_ (Lee), \u201394\n\ntaxes\n\nbenefits from sabbaticals,\n\ndeduction for HELOC interest, \u201377\n\ndeduction for student loans,\n\ndeferred income and,\n\ntimeframe choice, , \u201351\n\nTolle, Eckhart,\n\ntransition, \u2013127\n\ntravel plans\n\nbudget travel, \u201386\n\noverview, \u201362,\n\npacking light, \u2013123,\n\nplanning checklist, \u2013188, , , , ,\n\ntuition costs, employer reimbursement of,\n\nunemployment benefits, \u201382,\n\nunexpected sabbaticals\n\nfinancing your sabbatical, \u201388\n\noverview, \u201320\n\nreassessing goals and opportunities, xiii\n\nunemployment benefits and, \u201382,\n\nvolunteerism\n\nguilt-driven,\n\nplanning to volunteer, \u201363\n\nvolunteering sabbaticals, \u201324, \u201335, \u2013114\n\nwalking\/visualization exercise, \u201345\n\n_Way of Transition, The_ (Bridges), \u2013127\n\nwindfalls, \u201374,\n\nworkplace sabbaticals\n\nbusiness proposition approach to getting time off,\n\nconversation with the boss, \u201357\n\ndeciding to take time off, \u201354\n\nemployer's financial support, , \u201379\n\nformal programs, \u201313, , \u2013201\n\ninteroffice announcement, \u201358\n\nno formal program, \u201315\n\noverview, xiii, \u201311\n\nplanning checklist, , , , \u2013192, ,\n\nReentry phase, \u2013137\n\nwork-rest balance,\n\nyear-long sabbaticals, \n\n## | ACKNOWLEDGMENTS |\n\nWE HAVE GAINED SO MUCH from the help of others in creating this book as well as our Reboot Your Life Retreats. We are grateful for the support, insights, editing, advice, introductions, and research that our friends, colleagues, and relatives have provided us. We are particularly grateful to our interviewees, who shared with us their experiences as well as lessons learned, and the retreat attendees, who have shared their dreams and their fears.\n\nA special thank you to Susanna Space, Julie Kline, Penny Peters, Margot Atwell, Eric Kampmann, Candace Ishmael, Julie Koch-Beinke, Lynn Coppotelli, Sandra Poirier Diaz, Linda Lowenthal, Linda Cashdan, Amy Friedman, Carol Carlisle, Robin Slade, Paul Rooker, Sarah Born, Rachel Dyke, Michael and Nathaniel Buckley, Susan Stautberg, and the wonderful members of the Belizean Grove.\n\n## | OUR SABBATICAL STORIES |\n\n#### CATHY'S STORY\n\n_Cathy has taken two sabbaticals, one for eleven months when she was in her thirties, and the most recent one when she was in her late fifties._\n\nThe earlier sabbatical was a travel tour of Asia with my then-husband, giving lectures for the United States Information Agency (USIA) and traveling for pleasure. That sabbatical was motivated by my mother's death, a strained marital relationship, and disenchantment with the academic world, where I was an assistant professor of business administration. I had been taking care of my mom from a distance. I lived in Washington DC, and she was in Missouri. She died in September 1983, and I was exhausted and overwhelmed from taking care of her. My husband and I had married in 1980, separated in 1982, gotten back together in 1983, and were dealing with financial and infertility challenges. I was teaching at American University and working on my doctorate. I was growing increasingly disenchanted with the academic world because of the pettiness of the politics. And then there was the implied threat by my dissertation lead advisor that if I slept with him, I would get my dissertation accepted, and if not, well . . .\n\nMy first sabbatical was harder to justify to others than the later one. In 1985 sabbaticals were mostly academic-driven, and the go-go corporate world couldn't conceive of the concept. I was mid-career, as was my husband, and people said we were crazy. (\"You'll never get corporate jobs.\" \"You are wasting money.\" \"You will end up divorced.\")\n\nAs it turned out, the sabbatical was one of escape, relief, and renewed focus on our relationship. We planned to take nine to twelve months off, tour Asia, and come back to new jobs. We traveled in Asia for eleven months, lecturing for the USIA on trade with Asia and the United States, how Congress works, and how to market to the US. Between lectures, we traveled throughout fifteen Asian countries, absorbing the culture, arts, scenery, history, and lifestyles.\n\nFor each country, we read fiction, history, economic development, and cultural publications and met with people in the diplomatic, corporate, academic, and arts worlds. We also traveled by every mode, spent time in villages and small towns, visited tourist sites, and ate native food. We challenged ourselves to learn new skills, such as eating with chopsticks the whole time we were in Japan. We tried new foods, learned local languages (at least some basics), and studied cultural protocols.\n\nThat first 1985 sabbatical ended in November. I didn't know what would be waiting when we returned. I had sent my resume to a number of contacts before I left on the sabbatical and told them approximately when I would be back. I had a job interview with Dun & Bradstreet waiting for me\u2014a job I took and which propelled us to move to the New York area from Washington DC, and from the academic world to the corporate. From the 1985 sabbatical I learned that taking calculated risks was good, that taking a sabbatical mid-career was not hurtful to my career, that Asia was going to be a substantial economic power in the future, and that I had confidence from the experience that served me well in the corporate world.\n\nThe most recent sabbatical was planned as well. I had taken early retirement from my position as founding CEO of a major financial services consortium. I didn't know what would be next, except that I would be managing my consulting firm, The Santa Fe Group, when I returned from my time off.\n\nThis time around, it was a natural break. People understood the reasons I wanted time off\u2014personal and professional\u2014and wished me well. There were no naysayers, except my internal voice saying, \"Can you really forego the income for a year?\"\n\nThe planning for both sabbaticals included setting goals of what I wanted to accomplish, how I might finance it, what steps were needed to take care of the house, finances, work, etc. I am a planner by nature, and I found it almost as much fun to plan and dream about what I would do as it was to go on sabbatical.\n\nFor the earlier sabbatical, I spend 10 percent of the time planning, packing, and organizing for the trip, 20 percent reading about the country, 20 percent speaking at USIA events, and 50 percent traveling, touring, and enjoying the culture. The 2007 sabbatical was less specific in time. I spent most of the first month and a half sorting and organizing stuff in Santa Fe, working 20 percent of the time on Santa Fe Group\/career issues, exercising, reading, and traveling and being with my soon-to-be husband. We spent the month of July in France sightseeing with friends. Each day my goals were to write, exercise, reach out to friends, do something career-related, and enjoy downtime and reflection.\n\nIn this recent sabbatical, I was mostly relieved, happy to be free and not in Washington DC, and excited about the journey. I had twinges of worry about finances\u2014mostly because I was building an addition to my house\u2014but I felt confident I'd find a way to earn more when I returned to work.\n\nHad I not taken the time off to reflect, learn, travel, and renew, I would not be where I am today, spiritually, emotionally, and in terms of health. The sabbatical helped me focus my career goals for the next phase of my life.\n\nFrom the recent sabbatical I learned that planning is important, and having goals and checking in on them makes you prioritize your time. I learned that the hardest thing to do is say \"no\" and maintain time to just reflect. I am busier than I've ever been, albeit doing things I want to do. I just need to remember to make time for reflection.\n\n#### NANCY'S STORY\n\n_Nancy has taken several sabbaticals during her career in international security and foreign policy. In each case, she resigned from a job to take a break without knowing what the next job would be, and it always turned out well._\n\nAfter marrying at age thirty-three, when I was in a Pentagon job during the Carter Administration, and then going to the White House to serve as then-Vice President George H. W. Bush's national security advisor, I left Washington to accompany my Army general husband on an overseas assignment. I was thirty-five and wanted to have kids, and it meant leaving the best job in the world after only eighteen months. I missed my career, but I knew I could go back to it. As the two-year Army assignment lengthened into three years in Germany and another three in Japan, so did my sabbatical. It was a time full of discovery and fulfillment: new adventures, travel, language learning, many new international and American friends, public service using my professional skills\u2014and two beautiful baby daughters.\n\nUpon returning to the United States, I took one of those more-than-full-time White House positions at the National Security Council for four years. The girls were then in grade school, and the impetus to take a sabbatical to spend more time with my family was my daughter Sarah's plaintive, \"We want you to walk us to and from school like the other mothers do.\" I did just that, along with volunteering, doing a bit of professional writing, and actually having the time to send out Christmas cards.\n\nThen I mentioned to a friend that I was ready to take on a part-time job, and the non-profit world beckoned with a job that quickly became full-time, as they always seem to. Seven years later, I was still directing the program I founded on International Peace, Security and Prosperity at the Aspen Institute but badly needed a break as a result of the pressures of a divorce, coupled with trying to make it to all the kids' sports games plus everything else. There wasn't enough time and energy (physical and emotional) for work, kids, and a divorce, so I took another sabbatical.\n\nIn retrospect, the break from work was too short to get enough recharge and renewal because, after only a few months, I took on a new challenge as president and CEO of a struggling international development non-profit organization. After two years, the organization was on an even keel and I had promises to keep to my family, specifically overseas trips with my daughters, who were now in college, and I wanted to spend time with my mother and with a childhood friend who was struggling with breast cancer. And I wanted to return to the policy world in conflict prevention after taking some time off for myself. So I began another career break at age fifty-seven.\n\nThis break was truly one of giving myself the gift of time. There were so many possibilities for family time, friends, travel, some consulting work, non-profit boards, and teaching a university course. I dove into the smorgasbord, though I was living mostly on savings. A major highlight was being with my daughter Sarah in the Balkans where she was studying. Amid it all, the biggest project was moving the family from the suburbs to urban Washington DC, which I believe couldn't have been done if I were working. The time flexibility was fabulous, and it even led to writing this book to share sabbaticals with others.\n\n#### RITA'S STORY\n\n_Rita has taken four sabbaticals over her career. Here is one of her stories._\n\nFour days on the West Coast, three on the East, that is, when I wasn't in Europe or Asia. That is how I had spent the better part of the one and a half years before my second sabbatical. I lived on planes. It was exhilarating, and it was exhausting.\n\nI had been working for a computer company for seventeen years when an opportunity came along to take a software company public. The problem was that I was living in Boston, had a home in Brooklyn, and the job was in California. My husband, children, and I determined that it was a too good a job to pass up. They moved back to Brooklyn and I flew home on as many weekends as I could, except for the summer when one son went to camp and the other came out with a friend to live and work with me in California.\n\nI asked myself, could I handle that type of crazy schedule? Yes, for a while. My life was so segmented between work and family that I threw myself into both. In California my work days started at 7 am after a walk on the beach. They usually ended around 11 pm after a working dinner. I read and did my strategic thinking on planes. Back in Brooklyn, it was cooking and doing family activities.\n\nIt was an exciting time to see a young software company grow, to work with a very energized workforce, to be able to conduct business globally, and to be admired by your customers. Yet my life felt schizophrenic. I had a reputation for enormous work capacity and being able to multitask. But inside, my two worlds were tearing me apart. As time went on, I discovered that the founders were having trouble with growing pains and they were learning what it took to be a public company. Our management and strategic views collided. We agreed to part ways.\n\nThat was the beginning of October. I knew that once word leaked out that I had left, the phone would start ringing. And it did. But I had determined ahead of time that I needed at least three months to refresh and to reconnect with my life in New York. I politely told headhunters that I was taking a break and would be glad to speak with them after January 1st.\n\nSo what did I do? Nothing and everything. My goal was to enjoy the simple things in life, all those little things that my hectic schedule never allowed. First on my list was reading the _New York Times_ from cover to cover in the relaxing environment of Starbucks. After that, I either played tennis or went to a yoga class. I did lots of walking, cooking, visiting museums, and reconnecting with friends and family.\n\nIn January, the calls began and I did meet with several people. I thought before my sabbatical that my next position would most likely be as a CEO of a technology company or an Internet start-up. But I had lots of reflection time during my time off, and I found that inside I was rebelling from the insidiousness of the Internet and software boom. Make your millions, sell, and get out was not a work model that I had grown up with. It offended my sense of responsibility and business ethics.\n\nI fell into consulting when an old school acquaintance approached me for help with a specific business need. I turned our discussion into a two-month consulting engagement, which gave me more time to reflect and network. I enjoyed consulting and went on to work with other companies. But I soon realized that if I wanted to keep doing what I was doing I would have to scale up my practice and hire help. I met with as many consultants as I could to ask what they liked and disliked about their work.\n\nThese discussions helped me realize that consulting was not for me. I like to be the implementer of recommendations. I enjoy seeing people grow and organizations morph. To my friends' shock, after swearing that I would never work for a large organization again, I went to work for a Fortune 500 global company\u2014the one for which I had done the first consulting gig. I would not have done so, were it not for my sabbatical and the very deliberate time that I took to reflect upon what I valued.\n\n#### JAYE'S STORY\n\n_After seven years as an independent consultant, Jaye was unfulfilled and unchallenged. She was in a rut, and she didn't know how to get out._\n\nMy sister was gravely ill and terminal. I knew I needed to find something that was fulfilling but would also enable me to support her three children. After much deliberation, self-analysis, and networking with colleagues who knew me well, I created a new vision of a consulting firm focused on corporate consulting, called Partners in Human Resources International. I quickly found collaborators in that vision. As we built the business plan, we began writing our own ideal job descriptions. Because of the diversity of our skills, we were able to design the company structure around each of our strengths, interests, and goals. I was head of quality control in the delivery of services, consultant selection, and project management, and also in charge of new business.\n\nThe company grew quickly. By year five, we had reached $7 million in revenue. One of my two partners decided to retire. That's when things started to change for me. I was thrust into the role of president of the firm, a role more focused on operations, which was not my passion, but where I had capability. As a good partner, I took on the role enthusiastically. I was determined to succeed. We continued to grow.\n\nAfter four more years, the needs of the company had changed. I began to feel I was no longer the right person for the job. I found myself frustrated and unsatisfied. It seemed unthinkable that I would leave a company I had created, yet I was increasingly unhappy. Then a catalyst occurred that began to move me into a new direction: I was diagnosed with an immune-related illness that was potentially serious. Reducing stress was critical to my living a long and healthy life. I faced my own self-limiting beliefs and fears. I needed to leave the firm, and my role as president, to find the space to reconnect and take care of myself. My sister's children were now grown and more independent. I was able to take more risks. I took my own coaching advice and began an internal journey, and at age forty-eight, I began my first sabbatical. I had just purchased my first home, so I had a place to rest and recuperate. My lifestyle required a steady income, so I negotiated a buyout that gave me a year.\n\nFirst I did all of the things that I had longed to do over the last ten years that I never had time for. I spent leisurely time alone and with my family. I visited friends I had neglected. I traveled a lot, which opened me up to new possibilities. Everything I did stimulated ideas, and I matched those against the backdrop of what I knew about myself. I found a new inner peace that was, however, mixed with the anxieties and pressures of the unknown.\n\nI began reading and attending conferences on emotional intelligence. I thought about how I could incorporate these new ideas into my work as an executive coach and team facilitator. I also did other fun things I had always wanted to do, like learning to play the Conga drums. As time passed, the ideas for using all of these new elements in my work came together. I designed Team Beat, a team-building facilitation technique, using drumming and rhythm as a metaphor for teams working in sync. Little by little, a new direction began to emerge for the consulting practice I would build.\n\nI was surprised at how long the process took. I wanted answers quickly. I wanted a plan. I wanted to take action. After one year, I still felt that I had more work to do on self-reflection. I realized that I would carry this continued self-analysis and learning with me, probably for the rest of my life. I learned that I had changed over the past ten years, and I got to know my new self better. As I emerged from uncertainty and began to understand what I had and what I wanted to become, I felt more alive and connected to myself, and certainly more healthy.\n\nNow I'm doing what I want, where I want, and how I want. I rebuilt my expertise in corporate coaching and team building, integrating new techniques and ideas I developed during my time off. I have my own practice and can manage my work\/life balance more successfully. My work is incredibly rewarding, and my clients are diverse. I've maintained a good relationship with my former partner and company, so I haven't had to lose what I built there.\n\nMy work is focused on coaching senior executives. I design and facilitate team processes and events, very often using music. I'm growing every day. I make a point of surrounding myself with supportive friends and family to keep me on track with this new way of living. Without my sabbatical, and the time and space it provided, I would not be where I am today.\n\n## | ABOUT THE AUTHORS |\n\nTHE SABBATICAL SISTERS are four successful senior executives with careers in diverse fields. When they met, they discovered that they were all sabbatical veterans, joining thousands of men and women who have taken time out from their careers to reflect and renew. The four started calling that a Reboot Break and dubbed themselves the Sabbatical Sisters. Because they found it so important to their own professional careers and personal lives, they wanted to share the concept of \"rebooting your life\" with others, so they decided to write a book. They talked to over 200 people who have taken time off and researched well over fifty firms that help their employees take sabbaticals in some way. In 2009, they began holding retreats to help people plan and create support systems for taking a sabbatical. Their mission is to help others give themselves the \"gift of time.\" For more information, see their website, _www.rebootbreak.com ,_ and find them on Facebook, and LinkedIn.\n\nCATHERINE ALLEN is a co-author of _The Artist's Way at Work_ (William Morrow) with Julia Cameron and Mark Bryan and _Smart Cards: Seizing Strategic Opportunities_ (McGraw-Hill Inc) with William Barr. She is known as an innovator and expert in financial services and technology. Her career includes executive positions at Dun & Bradstreet and Citibank, and assistant professorships at several universities. Most recently, she was the founding CEO of BITS, the industry consortium made up of the CEOs of 100 of the largest financial services institutions. Today she owns and manages The Santa Fe Group, a strategic consulting firm of financial industry and technology experts. She sits on the boards of Stewart Title Guaranty, El Paso Electric Company, and Citibank Global Transaction Services, plus the boards of two foundations, the Los Alamos National Laboratory and Museum of New Mexico, and several non-profits. Recently, she was appointed by Governor Bill Richardson to the New Mexico State Investment Council. Catherine has taken two Reboot Breaks. She lives in Santa Fe, New Mexico.\n\nNANCY BEARG has had a forty-year career in international security policy. Her early career was in government at the Department of Defense, National Security Council (White House), Senate Armed Services Committee, and Congressional Budget Office. She was the National Security Advisor to then-Vice President George H.W. Bush and later served on the National Security Council staff when he was president. After one of her Reboot Breaks, Nancy changed gears and moved into the not-for-profit world, directing the International Peace, Security and Prosperity program for the Aspen Institute and then as President and CEO of EnterpriseWorks\/VITA, an international development organization. Today Nancy works in the area of conflict prevention and post-conflict peacebuilding, consults on national security, and teaches a university course on leadership. She published five books on national security while at the Aspen Institute. Nancy has taken five Reboot Breaks. She lives in Washington DC.\n\nRITA FOLEY is a corporate director, retired Fortune 500 Global President, and a committed leader in numerous organizations dedicated to improving the health and lives of individuals. Rita sits on the boards of the publicly traded PetSmart and Dresser-Rand, plus two not-for-profit boards: Pro Mujer, a not-for-profit microfinance and health organization operating in Latin America, and The HealthCare Chaplaincy, a leader in palliative care. Rita is an advisor with Crenshaw Associates, leading their board services practice. These appointments follow a very successful business career, which culminated at MeadWestvaco as Global President of the $1.1B Consumer Packaging Group. Prior to that, Rita held various leadership positions at QAD, Digital Equipment Corp, and Harris Lanier. Rita began her career at Polaroid in St. Albans, England. She has taken four Reboot Breaks. She lives in Brooklyn, NY.\n\nJAYE SMITH is the resident expert on the topic of navigating careers, with more than twenty-five years of experience in the field. Jaye is co-author of _Venus Unbound_ (Simon & Schuster, Inc.), a guide for building balance in women . Jaye took a Reboot Break after being the President of Partners in Human Resources International, a consulting firm she co-founded. Today, Jaye continues to work closely with her former firm and is also CEO of her own firm, Breakwater Consulting, where she coaches executives on renewal and leadership as well as designs and facilitates team and organizational effectiveness programs for corporations, not-for-profits, and universities. She is an honored adjunct faculty member at New York University's Center for Career Planning and on the board of Harlem Dowling Westside Center, a foster care and adoption agency. She lives in New York City.\n\nLeft to right: Jaye Smith, Catherine Allen, Nancy Bearg, Rita Foley\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nAustralian Screen Classics\n\n# Wake in Fright\n\n# Tina Kaufman\n\nCurrency Press \u2022 Sydney\n\n# Table of Contents\n\nTitle Page  \nCopyright Page  \nAuthor's Biography  \nAustralian Screen Classics  \nAcknowledgements  \nIntroduction\n\n1 - How the Film Came About  \n2 - How the Film Got Made  \n3 - How the Film Got Lost\u2014and Found  \n4 - How the Film was Received\n\nThen...  \nOut of sight, but not out of mind  \n... and now\n\nNotes  \nBibliography  \nFilmography  \nCredits\n\nwww.currencypress.com.au\nFirst published by Currency Press Pty Ltd and the National Film and Sound Archive in 2010.\n\nCurrency Press Pty Ltd   \nPO Box 2287   \nStrawberry Hills NSW 2012   \nwww.currencypress.com.au   \nenquiries@currency.com.au\n\nNational Film and Sound Archive   \nGPO Box 2002, Canberra   \nACT 2601   \nwww.nfsa.gov.au\n\nCopyright \u00a9 Tina Kaufman, 2010\n\nFirst electronic edition published in 2012 by Currency Press Pty Ltd.\n\nCopying for Educational Purposes\n\nThe Australian Copyright Act 1968 (Act) allows a maximum of one chapter or 10% of this book, whichever is the greater, to be reproduced and\/or communicated by any educational institution for its educational purposes provided that educational institution (or the body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL) under the Act. For details of the CAL licence for educational institutions contact CAL, Level 15, 233 Castlereagh Street, Sydney NSW, 2000. Tel: (02) 9394 7600; email: info@copyright.com.au\n\nCopying for Other Purposes\n\nExcept as permitted under the Act, for example a fair dealing for the purposes of study, research, criticism or review, all rights are reserved. This publication (or any part of it) may not be reproduced or transmitted, copied, stored, distributed or otherwise made available by any person or entity (including Google, Amazon or similar organisations), in any form (electronic, digital, optical, mechanical) or by any means (photocopying, recording, scanning or otherwise) without prior written permission from the publisher.\n\nPrinted book ISBN: 9780868198644\n\nePub ISBN: 9781921429071\n\nTypeset by Currency Press. Cover design by Emma Vine, Currency Press. Front cover shows Gary Bond as John Grant. Back cover shows Chips Rafferty as Jock Crawford and Gary Bond as John Grant.\n\n# TINA KAUFMAN\n\nTina Kaufman is a freelance journalist who writes on film, media and communication issues. She was editor of the independent film newspaper, Filmnews, for seventeen years, until its much-lamented demise in 1994. Since then she has been writing for various film, media and art publications; regularly for Metro magazine, for which she is a contributing editor, occasionally for RealTime, and more recently and more regularly for Screen Hub, the online journal for film professionals. A board member of the Sydney Film Festival for twenty-five years, and a founding member of the Film Critics' Circle of Australia, she is now an honorary life member of both organisations.\n\n# AUSTRALIAN SCREEN CLASSICS\n\nJANE MILLS\n\nSeries Editor\n\nOur national cinema plays a vital role in our cultural heritage and in showing us at least something of what it is to be Australian. But the picture can get blurred by unruly forces such as competing artistic aims, inconstant personal tastes, political vagaries, constantly changing priorities in screen education and training, technological innovation and the market.\n\nWhen these forces remain unconnected, the result can be an artistically impoverished cinema and audiences who are disinclined to seek out and derive pleasure from a diverse range of films, including Australian ones.\n\nThis series is a part of screen culture which is the glue needed to stick these forces together. It's the plankton in the moving image food chain that feeds the imagination of our filmmakers and their audiences. It's what makes sense of the opinions, memories, responses, knowledge and exchange of ideas about film.\n\nAbove all, screen culture is informed by a love of cinema. And it has to be carefully nurtured if we are to understand and appreciate the aesthetic, moral, intellectual and sentient value of our national cinema.\n\nAustralian Screen Classics will match some of our best-loved films with some of our most distinguished writers and thinkers, drawn from the worlds of culture, criticism and politics. All we ask of our writers is that they feel passionate about the films they choose. Through these thoughtful, elegantly-written books, we hope that screen culture will work its sticky magic and introduce more audiences to Australian cinema.\n\nDr Jane Mills is an Associate Professor at the Journalism and Media Research Centre, University of NSW. She is the Series Editor of Australian Screen Classics, a former Head of Screen Studies at AFTRS (1995\u20132000) and a founder\u2013member of 'Watch on Censorship'. She is the author of several books including _The Money Shot: Cinema, Sin and Censorship_ (Pluto Press) and _Loving and Hating Hollywood: Reframing Global and Local Cinemas_ (Allen & Unwin).\n\n# ACKNOWLEDGEMENTS\n\nThanks to Jane Mills for pushing me to do something I was initially so reluctant to take on, to all the people who talked to me about their early filmmaking and filmgoing, and especially to Howard Rubie and Richard Brennan, for being so generous with their memories and their time. The series editor and the author would like to thank Graham Shirley for his valuable comments on the manuscript.\n\nTina Kaufman\nFor Silvia, Max, Viveka and Alfie,   \nwith the hope that the love of movies carries on.\n\n# INTRODUCTION\n\nWake in Fright is a film that plays an important role in Australia's recent cinema history; it's a film that people remember vividly no matter how long ago they saw it. And yet it's a film that was made at a time when Australian audiences were indifferent to local production, a film that disappeared from view for many years and was only restored to its proper place by the amazing diligence and sheer determination of one man. For someone like me, who loves films and who has written about the ups and downs of the Australian film scene for many years, the story of Wake in Fright is one that is both intriguing and mysterious, a story that demands to be told.\n\nFilms have always been part of my life. I grew up at Bayview, on Pittwater in Sydney's northern beaches, where I was one of a gang of local kids. It was a great place to grow up; we spent most of our leisure time outside, swimming, riding and climbing trees. But we did go to Saturday matinees at the local cinemas at Narrabeen and Collaroy, and even as far afield as Manly when there was something particularly enticing. It was a good hour's bus ride to Manly, but there were four cinemas there, including the Metro by the beach which showed wonderful MGM musicals, such as Singin' in the Rain (1952) and all those ones with the diminutive all-singing, all-dancing Jane Powell, as well as swashbuckling films like Scaramouche (1952). But it was at another cinema, the one in the Manly Corso, where I remember seeing a thrilling double bill of Jacques Tourneur's Cat People (1942) and Gunther von Fritsch and Robert Wise's The Curse of the Cat People (1944), an early introduction to the joys of producer Val Lewton's subtle and creative take on the horror genre. Yet another Manly cinema, the one around near the Manly ferry wharf, was where my mother took me to see James Stewart in Anthony Mann's great western, Bend of the River (1952), although I wouldn't have realised then how stark and implacable a tale it was. These are films I've never forgotten\u2014and have seen many times since.\n\nIn my last years at high school I'd take the odd day off, especially if I was running late or hadn't done my homework, and I'd go into the city for a couple of sessions at the city cinemas. Although it was mainly Hollywood fare that enticed me, I did take in the occasional British film. Once I left school I started working in the city, where I not only kept up my diet of contemporary films but started attending screenings of a quite different kind\u2014at the Sydney University Film Group, the Realist Film Group, the National Film Theatre of Australia\u2014and very soon I started going to the Sydney Film Festival. All these screenings introduced me to a wide range of national cinemas, genres and filmmaking periods and really expanded my film knowledge and understanding.\n\nAustralian film, however, came a bit later. As is well known, there wasn't visible filmmaking activity at this time, in the early to mid-1960s. But I moved in a social milieu that included people involved in all sorts of creative activity, from art to publishing, from acting to music. Several friends were involved with showing American surfing movies in halls and surf clubs up and down the coast and soon started talking about making their own films; other friends were working at the ABC and the Commonwealth Film Unit (later to become Film Australia) and were also venturing into filmmaking.\n\nThen I moved into a big share house where a friend, Albie Thoms, not only screened experimental films in the front room, but shot footage on the stairs. Albie, together with David Perry, Aggy Read and John Clark, was setting up UBU Films, and I watched from a front-row seat as Sydney's small but active underground film movement began to grow and become more visible. Taking its name from Alfred Jarry's absurdist 1896 play, Ubu Roi, UBU Films became a Sydney-based independent filmmaking co-operative dedicated to making, exhibiting and distributing experimental films. From the mid-1960s the UBU circle took in many young film-makers who in later years became prominent mainstream film-industry figures, including Peter Weir, Phillip Noyce and Bruce Beresford, as well as artists who experimented with filmmaking, like Garry Shead, Mick Glasheen and Peter Kingston. These filmmakers, and many others, would become members of the Sydney Filmmakers Co-op when it was established in 1969.\n\nThe films that UBU produced were extraordinary, highly inventive and always a challenge to the mainstream. I spent a lot of time driving around the city with Aggy in his battered Mini Minor, delivering bundles of Oz, the soon-to-be-notorious libertarian magazine produced by Richard Neville, Martin Sharp, Richard Walsh and others. Aggy had used lots of magazines to make his very short and for a time quite famous film, Boobs a Lot (1970), which simply, but startlingly, cuts together numerous short shots of women's breasts to The Fugs' pop song of that title. Produced by Albie and photographed by David Perry was Tobias Icarus Age Four (1968), a short but moving film about the mother-and-child relationship made by my friend Clem Weight, the first of us to have a child. It actually travelled successfully to several overseas film festivals. In fact, a number of UBU films screened at overseas festivals; as Albie Thoms later observed, in those early years they were taken much more seriously outside Australia.\n\nBut filmmaking for many at that time was more like painting or acting or playing in a band: it was another expression of a rich and lively local art scene. Painter Garry Shead put together lots of home-movie footage and some especially-shot sequences to make Ding a Ding Day (1966), which chronicles his life as an art student, including his early involvement with Oz magazine, and I'm actually in it, along with many friends. Budding artists Peter Kingston, Johnny Allen and Mick Glasheen made what can be looked back on as a really early underground movie, Who Plus Live Equals Home (1963). Friends lived in the now legendary Yellow House in Kings Cross, where film was just one of the many art forms that were being wildly experimented with, and film screenings were held spontaneously. And practically everyone I knew was involved with Chris McCullough's Vision for a New World (1968), a rather enigmatic tale of a suicide that might not have happened, which went on to win an Australian Film Institute (AFI) Award.\n\nAnother friend, David Price, constructed a sort-of documentary called Surfing Roundabout (1965), featuring a group of surfies who, driving around to various surfie hangouts in their old bomb, rarely managed to get wet, let alone catch a wave. This sly look at the surfing scene starred Jenny Kee (later a celebrated designer), my gorgeous friend Anou Kiisler, and our favourite local band, The Missing Links, with a title song by another friend, Stephen Little. It had a breathless narration written and spoken by Oz editor Richard Neville and, as Albie Thoms says in his book Surfmovies: the history of the surf film in Australia,'it purported to be a serious examination of the surfing craze, but took every opportunity to ridicule the claims... made for the sport'.\n\nAnou also featured in another surfing documentary, The Surfing Years, made by Peter Thompson (brother of actor Jack, and who would later become the resident film critic on Sunday, a long-running current affairs program on the Nine network). The film followed a group of young surfers up the coast where Anou and Peter's soon-to-be wife Victoria, who went along for the trip, spent a lot of time running up and down deserted beaches with rather a lot of dogs. (It was going to movies with Anou that made an unexpected but dramatic difference to me; she was short-sighted, so we had to sit in the front stalls, and I suddenly discovered how much more immersed in the film I would become sitting there. I've been sitting there ever since, and strongly recommend it to all serious film-lovers; an added advantage is that being less crowded, it's easier to sit in the centre of the row, the ideal viewing position.)\n\nAs this all suggests, filmmaking for most of us at this time seemed mainly to be about having fun, something you or your friends did as a group, as others might go dancing or support a footy team. Most of the films were influenced by what we knew of the largely amateur European and US avant-garde and surrealist cinema, using techniques that included hand-drawn animation and images and shapes produced by directly scratching onto the celluloid itself. And until UBU Films, and then the Sydney Filmmakers Co-op, started distributing and exhibiting in the late-1960s (and by the early 1970s there were a number of coops Australia-wide, making over 600 films available around the country), you could really only ever see them in the living room, or at the occasional group screening in a hired cinema.\n\nWhile many of the filmmakers would go on to play important roles in the development of cinema in Australia, at the time my feeling was still that 'real' filmmaking, either the mainstream Hollywood fare or the exciting work we were seeing at festivals and film societies, was something that was done elsewhere. Albie Thoms, however, thought differently; he told me recently that the UBU mob always thought they could make a successful cottage industry out of their sort of filmmaking, completely separated from the hierarchical Hollywood model.\n\nGetting married, having a child and spending some time in England diverted my attention somewhat from cinema, but in the mid-1970s my love of film and my work as a journalist suddenly came together when I was offered the job of editor of Filmnews. For seventeen years in that role, and since then as a freelance journalist, I've written about many aspects of film in Australia. Filmnews was the ideal starting point for me: coming out of the Sydney Filmmakers Co-op, it initially concentrated on Film Co-op issues, representing the voice of the newly active independent Australian film and video maker. As the local industry grew, Filmnews widened its brief, covering the areas of media production and practice, media policy and film culture and theory, always endeavouring to make links between these areas. Through its pages we interviewed filmmakers, policymakers, film bureaucrats and broadcasters, and wrote about the changing institutions, funding regimes and media policies that were the backbone of the emerging industry. Over this period one of my particular interests, both in Filmnews and in other publications since its demise, has been writing about the discovery and restoration of missing films, about the search for lost films, and about both the work and the complicated history of the National Film and Sound Archive.\n\nI wouldn't have seen Wake in Fright when it was first released. I was still in London, only returning late in 1972, just in time for the election of the Whitlam government. So I probably first saw it at the Sydney Film Festival in 1975, when it was screened as part of the festival's Salute to Australian Film, a retrospective that was quite an eye opener, I have to admit, as it introduced me to a range of Australian filmmaking that I'd never really known about. But while I don't remember clearly when I first saw it, I do remember the impact it had. When I saw it for the second time, at a conference in 1999, I was surprised not only by how well the film held up, nearly thirty years after it was made, but by how vividly I remembered it, how many striking sequences had stayed with me. Seeing it again in 2009, and hearing more about how it was lost and found, only added to my interest.\n\nYes, Wake in Fright is a powerful, challenging film that held up what many saw as an unwelcome mirror to Australian society at a time when it just wasn't ready for it. I'm intrigued by how well people remember it, even if they haven't seen it for over thirty years; it's as if scenes have been imprinted in their memories. And not just the bloody, messy, kangaroo shoot, but the claustrophobic scenes in the two-up school, in the RSL, and in the pubs, all that relentless drinking, and that initially playful, friendly wrestle between the two drinking and shooting buddies Dick and Joe, that goes on and on and on...\n\nWhat a film is 'about' and what its plot is can be quite different. I don't think I'll be spoiling it for those who've never seen it if I briefly outline the plot of Wake in Fright, because what the film is 'about' is far more complex and far more interesting than its storyline. It's also shocking, savage, and amazingly contemporary, but to get the full measure of that you need to see it. If you really don't want to know the plot, please skip the next paragraph which I quote from the Australian Screen website:\n\nJohn Grant (Gary Bond) is the bored teacher at a one-room school inTiboonda, a tiny railway junction on the far western plains of New South Wales. On his way to Sydney for Christmas, he stops overnight in Bundanyabba, a frontier mining town. The local policeman, Jock Crawford (Chips Rafferty), introduces him to after-hours drinking and a two-up game where he loses all his money. At the pub next morning, businessman Tim Hynes (Al Thomas) takes pity and invites him home for lunch. After a disastrous sexual encounter with the man's voracious daughter Janette (Sylvia Kay), Grant gets drunk again with the enigmatic'Doc'Tydon (Donald Pleasence) and his two roughneck pals, Dick (JackThompson) and Joe (Peter Whittle). He wakes up in Tydon's dishevelled shackto a breakfast of kangaroo, hash and pills. Tydon is a disbarred doctor, describing himself as an alcoholic tramp who lives without money or pretence. He taunts Grant over his sexual failure with Janette Hynes. That night, Grant joins Tydon and his friends on a kangaroo-shooting trip. He learns how to kill a wounded kangaroo with a knife. They trash an outback pub and he is sexually assaulted by the drunken Tydon. He tries to get away the next day by hitchhiking, but the truckie just brings him back to 'The Yabba'. In a fury, he goes to Tydon's shack to kill him, but succeeds only in wounding himself. After discharge from the hospital, Grant takes the train back to Tiboonda for another year of teaching.\n\nAs I've said, what the film is 'about' extends far beyond its plot. Critics occasionally reach for hyperbole when discussing it, but this could be an indication of just how powerful a film it is. Here's Paul Byrnes, a former Sydney Film Festival Director and a regular critic for the Sydney Morning Herald, and his Curator's note from Australian Screen:\n\nWake in Fright is Australian cinema's Paradise Lost. It is probably the most unflattering depiction of the country that has ever been filmed, but it remains a profoundly ambiguous work, suspended between disgust and a kind of admiration for the honest depravity of what it shows... John Milton's epic poem Paradise Lost, published in 1667, tells the story of the fall of Man, and his temptation by Satan. Wake in Fright is virtually the same story. At the core of both is a question of free will, the freedom to choose. No-one forces John Grant into depravity; all they do is offer him the chance to choose it.\n\nFor me, Wake in Fright is a film that, despite its initial rejection by Australian audiences, still managed to give encouragement to young and emerging Australian filmmakers at a time when both they and the nascent industry needed it; it's a film that now can be seen and appreciated both for its qualities as a film and for the role it played in the development of that industry. And it's a film with a fascinating history. As a journalist who has written about the ups and downs of most aspects of this growing and very complex screen community, what I find I want to write about Wake in Fright is the story behind the film: how it got made, how it got lost, how it got found, and how it was restored. And, importantly, how it was received both on its first release, and on its much more triumphant re-release.\n\n# 1\n\n# HOW THE FILM CAME ABOUT\n\nThe late 1960s and early 1970s were the years of what has come to be called the 'rebirth' of the Australian film industry, although at the time nobody could ever have imagined how much that industry would grow over the next forty years and what films would come to be made. It's fascinating that in 1971 both Wake in Fright and Walkabout screened at the Cannes Film Festival: two films made by outsiders\u2014a Canadian and a Brit\u2014that have come to be seen as enormously important in that rebirthing process. Both films are unlike anything that had previously been made in Australia, but of the two, Wake in Fright is perhaps the stronger, more savage and harder-hitting film. The more I discover about it, the more intrigued I am by how such a film got made, at that time and in such an unlikely fashion.\n\nIn his book on Nicolas (Nic) Roeg's film Walkabout, Louis Nowra talks about the barren cultural landscape out of which the two films emerged. But even if it appeared barren, there must have been something there to provide the fertility for such an energetic, striving plant to germinate and eventually to thrive. Was that barrenness an illusion? Were there were little patches of vegetation and, underneath, many young shoots getting ready to sprout? Was there dormant and emerging filmmaking life in what would appear to future commentators as a wasteland?\n\nAustralia had been making films (it could only occasionally have been called an industry), since almost the beginning of cinema in the 1890s, but it had always been a very stop-start affair. In the heyday of production in the 1920s and 1930s, between ten and fifteen features were made a year, but in the years leading up to 1970 production had slowed to something less than a trickle\u2014more an occasional splutter. Of course, there were the overseas productions. In fact, 1959 saw a veritable flurry of activity, with Stanley Kramer making On the Beach in Melbourne, Harry Watt directing The Siege of Pinchgut in Sydney, and Leslie Norman making Summer of the Seventeenth Doll also in Sydney. And in 1960 Fred Zinneman made The Sundowners with Deborah Kerr and Robert Mitchum. But not one of these directors was Australian; indeed, as critic and former Sydney Film Festival Director David Stratton says in his Afterword to the new edition of Ken Cook's original novel Wake in Fright:\n\nthe high-profile 'Australian' films made during this period weren't Australian productions at all. Most of them, including The Overlanders, Eureka Stockade, Bush Christmas, The Shiralee, Smiley, The Siege of Pinchg ut and the 1957 version of Robbery under Arms, were British films, often shot in studios in the UK with Australia used only for the location work; others, including Kangaroo, On the Beach and The Sundowners, were mainstream Hollywood productions made on location here.\n\nAnd as Stratton also comments in his book, The Last New Wave: 'Australian stories were being filtered through foreign eyes, and a strange variety of foreign actors were [sic] pretending to be Australians.'\n\nBut if very few Australian features were made, by the mid-1960s there was an active production sector making newsreels, television programs, and television commercials, while government-funded films were being produced at the Commonwealth Film Unit. Newer, lighter cameras helped the filmmakers who were making surfing, travel and adventure documentaries to go up the coast and into the bush and the outback in search of stories. There were six small studios with sound stages in Sydney, another in Melbourne, and there were about a dozen laboratories to do the processing and post-production. In Australian Cinema: the First Eighty Years, film historians Graham Shirley and Brian Adams refer to 'more than fifty film production companies employing from five to over a hundred people each' and comment that 'among these various film production bodies, along with the television studios, underground filmmakers, filmmakers returning from abroad, and individual film critics, the desire for a reborn film industry grew.'\n\nIn fact, as legendary filmmaker Ken G. Hall wrote in 1967 in the short-lived theatre and film journal Masque:\n\nthere is an Australian film industry at this moment and it is keeping more people in regular employment than ever before. There are available to prospective producers ten times the facilities\u2014studio space, modern equipment\u2014than was available to, say, Charles Chauvel and myself in the 30s and 40s. The days of one camera and, at best, two microphones, together with some equipment made out of Meccano parts and literally tied up with wire, are no more. Much more footage is being shot than in that production heyday and there are more laboratories, including many excellent colour laboratories, than were ever dreamed of then. The trouble is that these people, this studio space and facilities, are not being used to make what most interested people would want them to be used for\u2014feature film production.\n\nOf course, this supposes that the production of feature films is the ultimate goal of any film industry, an issue raised by Susan Dermody and Elizabeth Jacka in The Screening of Australia (Vol 1). 'Why is feature film production assumed to be the real point of an industry?'they ask, and continue:\n\nThe assumption goes back to the phase of film history, from the silent period to the peak audiences of 1946, when movies were the dominant popular art, when people 'went', on average, two to three times a week, when the industry was the third or fourth biggest revenue-earner in the United States, and there was no television. That's part of the answer. Another part is that the marketing of films (to paying audiences, rather than to client groups, as is the case with educational films or broadcasting interests) is organised around the event of the feature, which must be large, emphatic and powerful enough to warrant travel, ticket-price and several hours of voluntary attendance in the dark.\n\nWhile it is the feature film that attracts that paying audience, its production is also what many filmmakers see as their ultimate goal, and even in these bleak years some determined filmmakers did manage to make features. One was Tim Burstall, who made Two Thousand Weeks in 1969, a semi-autobiographical tale about the frustration and isolation of an artist in the Australian wasteland. Actor and writer Graeme Blundell, who was in the film, writes about the experience in his very entertaining memoir of those early years, The Naked Truth: A Life in Parts, describing it as 'a subjective view of a writer's crisis when he calculates he only has 2000 weeks in which to express himself.' It was, as David Stratton (who programmed it in that year's Sydney Film Festival) says in The Last New Wave, 'a remarkably ambitious film'.\n\nTim Burstall had made a short children's film, The Prize, which won a prize at the Venice Film Festival in 1960. He'd then made some documentaries and a children's TV series, had been working in the US film industry for two years on a Harkness Fellowship, and thought he was ready for his first feature. Released in Melbourne in March 1969 with high (probably too high) expectations, it was reviewed badly and taken off after eleven days, while the Sydney Film Festival screening some months later was a disaster. 'Burstall was devastated', reported Graeme Blundell, 'and developed a deep hatred for what he called \"the intelligentsia\". They were \"highbrows and ABC types\" and they seemed affronted by the simple decency of the film.' In 1971 Burstall made Stork and he had to pay for the first screening in Melbourne. It did so well that it got picked up by mainstream distributors Roadshow, becoming not only the new Australian cinema's first box office success, but reinvigorating Burstall's career, and scooping the pool at the 1973 AFI Awards, the year he made the highly successful Alvin Purple. In that film Blundell let everything hang out, shocking some of the more straight-laced critics but nevertheless delighting the cinema-going public and delivering a very large profit. But Burstall never forgot or forgave the treatment given to Two Thousand Weeks.\n\nBehind the scenes, many of those working in film and television had been lobbying the government for years for some form of government support for production; broadcaster Phillip Adams, then both a successful advertising man and prospective filmmaker, and his friend, historian and Labor Party stalwart Barry Jones (who had been unofficially advising then Liberal Prime Minister John Gorton on cultural issues), proposed to Gorton the setting up of a national film school. In 1969 a screening for federal parliamentarians of Anthony (Tony) Buckley's documentary, Forgotten Cinema, an account of the rise and fall of the Australian feature film pioneers, was given some credit for pursuading Gorton to finally give some assistance to the industry. (Tony, we'll discover, will play an important role in the story of Wake in Fright.)\n\nIn 1969 the commonwealth government announced a three-part program of assistance: an investment corporation to support feature films and television programs, a national film school, and an experimental film fund to assist in the making of low-budget films and encourage emerging filmmakers. While the film school didn't come into being until 1973, the Australian Film Development Corporation was set up in 1970, as was the Experimental Film and TV Fund, initially administered by the Australian Film Institute.\n\nFor many of those who have since written about the period prior to government support, in the numerous takes on how the industry got going again, it may have seemed a very insular, bleak and unrewarding time. For those who were there, however, it was different: their memories reveal a screen culture buzzing with activity and ideas, infused by commercial enterprise, by television and commercials, and energised by popular, classic and foreign films. Nor was it a one-way street: just as the Australian industry was open to ideas from overseas, Australian ideas, energies and talent also flowed abroad.\n\nTake Producer Richard Brennan, for example. As he wrote in early 2009 just after his retirement as a project officer from the federal government funding body Screen Australia, he has been in love with cinema since he was ten: 'I have read about films, watched them, and studied them. And since I was seventeen I have worked on them.' That all started when he met Bruce Beresford at Sydney University in 1960, 'both dreaming of being filmmakers', and they made a short film called The Devil to Pay. Richard then made several other short films and, upon graduating, went on to work as, first, a production assistant at the ABC, and then at the Commonwealth Film Unit (later Film Australia). In 1970 he produced and directed a documentary in support of the May moratorium which launched the anti-Vietnam war movement in Australia, Or Forever Hold Your Peace, which was the first film financed by the newly created Experimental Film and TV Fund. He then produced Peter Weir's short drama Homesdale, which won the rarely-given Grand Prix at the Australian Film Institute Awards.\n\nSo, for Richard and others like him, it was a time of terrific optimism. He recalls vividly how cinema opened up the rest of the world for him, telling me:\n\nWe'd just seen two locally-made features in release, The Naked Bunyip and Nickel Queen, and at Film Australia we were working on two more, Brian Hannant's Flashpoint and Cecil Holmes' Gentle Strangers. We actually saw a future in filmmaking. And we were inspired by the films we were seeing; that year, at the Sydney Film Festival, I'd seen Truffaut's Bed and Board, Peter Watkins' Punishment Park, Pasolini's Teorema, Bunuel's Tristana, Eric Rohmer's Claire's Knee, and Bernardo Bertolucci's The Conformist.\n\nRichard's immaculately-kept diaries, which have become a great source of information for friends whose memories or record-keeping aren't nearly as reliable, record that he thought Wake in Fright 'outstanding, Donald Pleasence's performance very impressive, and he notedwhat a star Jack Thompson is going to be'. They also record what novelist Frank Moorhouse told him on 3 November 1971\u2014that Wake in Fright'gave him a thirst, an interest in homosexuality and an itchy \"trigger finger'\".\n\nThe next month Richard was off to London to work as production manager on The Adventures of Barry McKenzie, written by Barry Humphries and directed by his old friend Bruce Beresford.\" Thirty or so films later, some as production manager, many more as producer (and occasionally as actor), Richard remembers this as 'one of the highlights of my life; it was my first feature and my first time overseas'.\n\nHoward Rubie, who would be the first assistant director on Wake in Fright, is another who remembers the early years of the Australian film renaissance\u2014and even those before, in the 1950s and 1960s\u2014as a highly productive period. In 1954, aged sixteen, he had started working at Cinesound as a camera assistant, working on newsreels, short documentaries for cinemas, corporate documentaries and, later, on news footage for Channel Nine. He started about the same time as a young editing assistant, Tony Buckley, and by 1967 they were both working for Ajax Films. 'We were making a lot of TV commercials, especially cigarette commercials', Rubie told me, continuing:\n\nThere was an incredible amount of activity in the non-feature area, and commercial production was the cornerstone of that activity. Even in the mid-sixties everyone was aspiring to features. We talked about the UK productions that were being made here by Harry Watt and Ealing Studios. And we talked about Charles Chauvel's Sons of Matthew and about Lee Robinson and Chips Rafferty's films as showing the way. There was a feeling of disappointment that the Australian industry had declined so much, but there was always a hope that it could be revived.\n\nInvolved in that possible revival was the company that would co-produce Wake in Fright, NLT Productions, so named after its entrepreneurial partners, Jack Neary, Bobby Limb and Les Tinker. The company had been making 'family' shows for television since the early 1960s, centred around the successful weekly variety show. The Bobby Limb Show featuring Limb, who was at the time a popular personality, a singer and comedian who'd made his name and reputation on radio and in the clubs. Jack Neary, his manager and agent, also owned nightclubs, while Les Tinker was a Leagues Club manager who had financed the company.\n\nIn 1968 this small Australian company got involved with the US company Group W, a division of the major US multinational Westinghouse, which at the time operated a small US television network, and signed a co-production deal to make ten features in five years. This was an incredibly ambitious undertaking even for an experienced production team. There is some uncertainty about how these two companies got into bed with each other but Howard Rubie thinks it had something to do with the Channel Nine connection. As explained by film historians Andrew Pike and Ross Cooper in their book Australian Film 1900-1977:\n\nNLT was supported in the venture by Motion Picture Investments, a company directed by prominent Australian businessmen, including Sir Reginald Ansett. Investment was on a major scale and the budget for the first film was $750,000.\n\nThat first feature, Squeeze a Flower, was a comedy about a secret recipe for a liqueur known only to an Italian monk, played by Italian actor Walter Chiari, who comes to Australia to work at a Hunter Valley vineyard. The film was made quickly, with the principal members of cast and crew imported and everyone else Australian. The director, Mark Daniels, was American. Released in February 1970, the reviews weren't good and it flopped.\n\nBut by this time NLT was getting ready to make their second feature\u2014Wake in Fright.\n\n# 2\n\n# HOW THE FILM GOT MADE\n\nAustralian painters, writers and filmmakers have long been fascinated by the outback, the never-never, the back of beyond; that great Australian emptiness has been a continuing theme. Over the years white Australians have invented many names for the arid country that few ever visit and even fewer really understand. It has become much more than a location; whether it's hostile, indifferent or even, very occasionally, welcoming, it has an identity that is palpable. And it's been a part of our national cinema almost from the start; myths have vied with reality in stories that have been told and retold in ever more dramatic ways.\n\nFrom the early silents, through the films of the 1920s and 1930s and into the lean years of the 1940s and 1950s, there had been stories about white settlement, about farmers and settlers taming and being shaped by the land, of desperation and the never-ending struggle against the natural world, of the physical and mental hardship of their lives, of the ravages of drought and fire; there are films of explorers, convicts and bushrangers challenging or taking refuge in the desert or the bush. In all of these the great Australian emptiness has played a major role. From the postman with his long and isolated mail run on the Birdsville Track in John Heyer's wonderful documentary Back of Beyond (1954), to the pioneering families of Charles Chauvel's Sons of Matthew (1949) and the cattle drovers of Harry Watt's The Overlanders (1946), those who take on the outback bears the scars of isolated living on their character and behaviour.\n\nWhen Ted Kotcheff, an outgoing, energetic, and naturally positive man, was given the task of portraying the outback in Wake in Fright, it was a challenge he took on with mixed feelings.\n\nI was trepidatious about depicting a culture I didn't know\u2014but I did recognise some coincidences. Canada and Australia are both ex-British colonies, both big countries with vast empty spaces where life is hard. Remote area mining, forestry, and construction industries serve as a powerful backdrop to very masculine societies where there is lots of drinking. Women, when present, are marginalised by the hard drinking culture.\n\nKotcheff believed that the experience of people in the Australian outback would be similar to those in the Canadian North. 'Our north is vaster than your outback', he explained, 'but you have that same sense of brooding, empty landscapes that don't liberate you, they scare you and entrap you. That whole masculine ethos was also very true of the Canadian north, so I had some sense of that.'\n\nBroken Hill is a mining town near the NSW-SA border, whose tourism brochures even today describe as 'in the breathtaking Australian Outback... and an extreme amount of pubs!' Researching and shooting the film there, Kotcheff came to understand better why violence was such a part of life in a town where men outnumbered women by three to one. He came to believe that the roughness and violence was, for men so deprived of female company, a way of getting some sort of physical contact. Later he speculated whether it needed an outsider to see the harshness of life in this remote mining town; indeed after the Sydney Film Festival screening in the State Theatre in 2009 a member of the audience said to him that no Australian would have dared to make the film in the 1970s.\n\nWake in Fright was Kenneth Cook's first successfully published novel. It drew on his time as a journalist in Broken Hill and was first released in 1961 when he was 32. It was published in the UK and in the USA, translated into several languages, and was added to school reading lists in Australia. The story tells of an indentured schoolteacher John Grant (played by Gary Bond in the film) who, on his way to Sydney for his summer holiday with his girlfriend, gets trapped over one weekend in a nightmare of gambling, never-ending drinking, kangaroo shooting and dark sexual encounters in the outback mining town of Bundanyabba. The novel's bleak atmosphere of isolation and menace, of hellish immediacy, has been translated into the film.\n\nHow and why NLT Productions acquired such a tough and challenging project is something of a mystery. The English actor Dirk Bogarde had purchased the film rights in 1963 and wanted to make it with his chosen director Joseph Losey, with whom, by the late 1960s, he had already established a strong working relationship in Sleeping Tiger (1954), The Servant (1963), King and Country (1964),and Accident (1967). Ken Cook actually told the Australian media that the Bogarde-Losey production would start shooting in December 1963, but it didn't eventuate. British screenwriter Evan Jones, who worked with Losey on The Damned in 1961 and wrote the original script for Wake in Fright, surmised that it was never really a prospect as the money was not there. Australian writer Morris West, a friend of Cook's, bought the rights from Bogarde in 1966, worked on it again with Evan Jones, then sold the project to NLT in a deal reported at the time to be worth $49,000. As a project for NLT it was completely different from their previous feature, the light comedy Squeeze a Flower, or from anything they had done for television.\n\nKotcheff believed he could make the film of the book and the script (which followed the book closely). When he arrived in Australia he consulted with Ken Cook who filled him in on a lot of the background to the novel. He then went to Broken Hill and spent much time with the locals, doing his research and learning about the ways of the bush and the outback. He admits to spending a lot of time in the pubs and bars. He later recalled one incident from a location survey:\n\nWe came across the pub (that appears in the film) with the giant beer bottle on top. It's real! There were about thirty cars outside, each with a woman sitting in it, and inside the pub the men were having a great time. First I have to explain that at the time I had a handlebar moustache and hair down to here. I said, 'Let's go in'\u2014and location manager John Shaw, who's built like a rugger player, is afraid to go in with me\u2014but we do. One guy looked at my hair and said 'Shee-it', looked at my moustache, and said 'Sheee-it', and then said, 'Hello, Stalin'. I didn't answer, and he repeated it. I said, 'I'd love to talk to you, but I'm dead'. There's was silence for a bit, and then he burst out laughing, saying, 'I love a bloke with a sense of humour.'\n\nIt's a sentence that appears word for word in the film, although from the lips of Chips Rafferty it has a certain malicious twist. But for Kotcheff it set up a relationship in Broken Hill that would prove useful. When he wandered around the pubs at night he'd often encounter people who would want to pick fights with him, but luckily, 'some of the blokes from the beer-bottle pub followed me around and protected me'.\n\nBroken Hill proved hospitable to almost all of the film people. While the town would usually be closed up tight by eight o'clock, the dining room in the RSL club was kept open especially so they could get something to eat late, without having to request it in advance. (Of course, behind closed doors the drinking and pokies were going on illegally long into the night, as in the film.) The film used many of the Broken Hill locals as extras and the school children in the opening scene of the film were local kids from farms around the town. There was only one minor incident: a camera assistant taking cans of film to the plane was set upon by some locals and a can rolled into the gutter. We got it back and it was OK, but the thought of what could have happened to that precious can of film was a bit concerning', remembers assistant director Rubie.\n\nKotcheff, now a vigorous 78, has shed light on why he, a Canadian who knew nothing about the Australian outback, got to direct the film. He grew up in Toronto, but with no film industry in Canada, his only chance to make feature films was to go somewhere else. So in 1958 he gravitated towards England, eventually directing stage plays and some television as well as feature films. 'I didn't start making films in Canada, I only directed television there', he says. 'In London, there's theatre, film and television all in the same city, so I went to England and that was where my film career really started.'\n\nAfter making Tiara Tahiti in 1962, Kotcheff directed the much admired Life at the Top (1965) with Lawrence Harvey and Jean Simmons. This, he recalls, was followed by\n\nanother good film about the racial situation in London in the 1960s, Two Gentleman Sharing (1969). The writer of that film, Evan Jones, said to me, 'Ted, there is this great book you would be perfect for. It's by Kenneth Cook and it's called Wake in Fright; it's a lost weekend of self discovery. You should read it. I'm going to start an adaptation and, if you like it, you can go to Westinghouse (who were half financing the film and had an office in London) to see if they'll hire you and we'll work on it together'. I read the book and loved it, and I went to Peter Katz, who ran Westinghouse's Group W Films, and said,'Look, Evan and I would like to work together on this project. He'll write it and I'll direct it'. Peter Katz gave me the job.\n\nFor Kotcheff, 'the best part of directing is creating characters. To create interesting characters is the fun for me in making films'. He had empathy for the main characters in Wake in Fright, whom he thought of as musical instruments: 'Most of the time there's only four characters\u2014Doc, John and the kangaroo men (Dick and Joe)\u2014so you had to pay attention to every little detail in their behaviour. It wasn't a symphony. It's a piece of chamber music.' He was particularly taken with a line early in the script, when John Grant is talking about the aggressive hospitality in the town, and Doc (Donald Pleasence) says, 'It's death to farm out here, it's worse than death in the mines; you want them to sing opera as well?'\n\nThe film was shot on location in and around Broken Hill for eight weeks, following a four-week studio shoot in Sydney. NLT appointed another company, Ajax Films, to set up the production and, as with Squeeze a Flower, brought in an overseas director, Kotcheff, and the UK cinematographer who had shot Squeeze a Flower, Brian West. The rest of the crew was Australian, most of whom had worked on at least some of the features made locally in recent years. Kotcheff was impressed with this young, enthusiastic crew; even at the time, he recalls, there was a feeling that 'they were making something special, something important'.\n\nHoward Rubie was working at Ajax Films as a television director, and he enthusiastically joined the production as assistant director. 'Wake in Fright was one of the few screenplays I'd read, and I'd already read many, that I couldn't stop reading. It was a real page turner', he recalls. He was concerned that the $800,000 budget was inadequate (and was duly chastised by producer George Willoughby for questioning it) because\n\nwe had a slightly bigger crew [than usual], and there was a lot of location shooting, a lot of extras. Some sequences had to be dropped, [so] there was a bit of cutting and rewriting of the original shooting script. For logistical and financial reasons all the pub and house interiors were shot on location or in the studio in Sydney, all exteriors in Broken Hill. But the cinematographer Brian West was excellent; he seemed to adapt very well to Australian conditions and like most overseas cinematographers he was amazed at the quality and clarity of the light\u2014he used to bounce a lot of light off white card to get the glare.\n\nThe shoot had its ups and downs, but on the whole it went smoothly. Film is a collaborative medium and one advantage for Kotcheff was that many of the crew had worked together before, and at a faster pace than US or UK crews. Rubie, however, recalls one problem they encountered on location:\n\nWe did have to reshoot the fight scene, because Donald Pleasence decided he'd act better if he was a little drunk and badgered the props boy into giving him real beer. While Ted would rarely go beyond four takes for a scene, we had to do so many takes for that scene that Peter Whittle and Jack Thompson were physically buggered\u2014they'd have to drag themselves into another take.\n\n'There were 47 takes', recalls editor Tony Buckley, 'although we ended up using take eight, with a bit of take ten.'\n\nWhile Kotcheff believes Australians are natural actors because they are so exuberant and extroverted,the production company insisted on casting an English actor as John Grant. It had been hard: every eligible British actor, it seemed, had turned down the role. Kotcheff had initially wanted Michael York, who was intense, intelligent and blonde (and who told Kotcheff years later how he came to regret not doing the film), but finally chose Gary Bond, a strong stage and television actor with some film credits who had something of the same look (and who would be promoted as the'new Peter O'Toole'). While the casting of a British actor could be seen as unnecessary, Bond's Englishness actually highlighted his difference and contributed a naivety that strengthened his fine performance.\n\nKotcheff had seen Chips Rafferty in several of the Ealing films about the Australian outback, (The Overlanders, Eureka Stockade and Ralph Smart's Bitter Springs (1950)), and says:\n\nEven back then I had thought he was a wonderful type. It was when we were casting that I was told about his iconic status in the Australian film industry. But I had always liked him and thought he would be perfect for the role of the policeman.\n\nKotcheff saw this character as a local petty tyrant, the king of this particular circle of hell. 'I always felt there was a sinister side to him when, at the end, he takes a certain sadistic pleasure from John's mental and emotional demise.' And Chips Rafferty shows an edge, adding an undertone of menace to his portrait of the outwardly genial and friendly local cop.\n\nBroken Hill was Chips Rafferty's birthplace, and Wake in Fright would be his last feature film. In his biography, Chips:The Life and Times of Chips Rafferty, Bob Larkins says,'Chips gave the finest performance of his career, displaying a subtlety of characterisation rarely evident before. But then, he had never had a better part to play, or, with the exception of Fred Zinneman and Lewis Milestone, a better director...'\n\nRafferty was able to add his own element of authenticity to this very authentic looking film: there was a lot of beer drinking in the film, and of course the production provided a non-alcoholic beverage for the actors, but there was one actor who would never drink it\u2014Chips Rafferty.\n\nJack Thompson says he learned much from the film and from Kotcheff in particular.\n\nHe'd say, 'Don't act, just think!' It was my first major film role and it was a wonderful experience because it was an extraordinary film, and a very disturbing film. For me it was a great thing to be involved in, and what I learned making it I carried for the next couple of years of my career.'\n\nDonald Pleasence was another mentor. 'We were discussing very earnestly the difference between stage and film acting, and he said, sotto voce, \"Feed the camera, Jack.\" It's something I've followed ever since.' Thompson also talks about\n\nthe incredible energy that Ted brought to the whole shoot. In those bad days before health and safety issues brought in strict regulations, when we did all that driving madly about, there was an unseen companion. Ted was lying (in the ute) down at our feet, yelling directions!\n\nKotcheff had gone down into the mines when he was on the location search and found it'a nightmare down there'. He particularly wanted to include a scene showing the two roughnecks, Dick (Jack Thompson) and Joe (PeterWhittle), at work in the mines but the budget couldn't didn't stretch to that. It's something he regrets to this day:\n\nUnfortunately, it became a victim of the budget because it was very complex to shoot down in the mines and it was going to take another whole day. Afterwards I was always sorry I didn't shoot it, because you would have seen them working in this God-damned hellish hole and you would have understood why they went crazy and drank and gambled and shot.\n\nKotcheff grew to love the outback, what he saw as'its unearthly colours, extraordinary shapes and totally exotic vegetation,' and for years looked for a subject that would allow him to return to film it, but neverfound one. His major concern was to recreate in the studio what he had found in the outback. He'd compiled many notes on the colours of the landscape and told art director Dennis Gentle to use only the hot colours of red, orange and brown to convey the oppressive heat. He'd also noted that every surface and object was covered in a layer of fine red dust.'What I did was to bring barrels of earth that was the same colour as the outback, put it into fly pumps and before every take I would have it sprayed into the air so that it would hang there for a while and then create a film of dust over everything.' And, of course, they needed flies. The science department at Sydney University produced thousands of sterilised houseflies and a swarm was released before each take for the scenes in Doc's shack. 'I went to infinite pains to never let the audience be free of the heat, the dust, the flies', Kotcheff says.\n\nThen there were the scenes that have attracted so much attention and outrage:the kangaroo shoot in which kangaroo after kangaroo is chased, spotlit and shot in an orgy of driving, drinking and slaughter. The footage was shot on a professional cull and brilliantly edited into the scenes of the four men driving and spotlighting kangaroos. It was, and still is, important for Kotcheff that not one kangaroo was killed specifically for the purpose of the film; he has never forgotten the brutality so disturbingly captured for the film.\n\nJackThompson believes the controversy over the kangaroo-shooting scenes belied the truth of Australian life in the outback, which he, for one, had already experienced. 'When I was sixteen I wenton a kangaroo shoot. What really disturbed me about it was the adrenalin. There was a sense of danger and a sense of excitement about it that I tried to portray in those scenes.' Kotcheff seems to agree; despite shooting images to disturb even many years later, he describes that day in the back of the ute, bumping and bouncing around with camera operator John McLean, as 'the most exhilarating day of my life!'\n\nThere is another kangaroo scene Kotcheff is much happier about, however. When they filmed the scene in which Joe (Peter Whittle) boxes with a kangaroo, they first tried the kangaroos they had been using for other scenes in the film, but these, as Kotcheff explains\n\nwere mainly pacifists. We tried everything to get them to fight, but nothing was happening; I thoughtto myself, 'This is the nightmare I thought it was going to be. How am I going to do this sequence?' Then Howard Rubie came back with the biggest roo I'd ever seen, about eight feet tall. Someone had shot his eye out, so we called him Lord Nelson, and he hated humans, wanted to smash, kick, kill any human being he saw. When we put him upto fight, he just went for Peter, and what guts he had to face this demon! Nelson wanted to destroy him, grab him in a hug, raise his hind legs and smash every bone in Peter's body. Peter was trying to lift his tail to put him off balance, but Nelson would not let him and would swing around and go in for another attack. We finished the whole sequence within three hours, and somewhere I have a great photograph of Peter and Nelson, exhausted and embracing each other. At some point during those three hours, Nelson had understood that Peter wasn't actually trying to hurt him, and at the end of it they were leaning against each other's shoulders for support.\n\nWhen he was sure that they had all the footage they needed, Kotcheff turned to Nelson and told him:\n\n'You were great!'The whole crew applauded him. He looked around, puzzled. I said,'Open the gates! Alright Lord Nelson, you can go.' But he looked at me still puzzled and again I said,'You can go. Leave!' He hopped away about five steps, turned around and looked backat me. 'I mean it', I said.'You did a great job and now you can go', and then he hopped off into the darkness.\n\nNelson was a wild kangaroo but he got his own credit as 'Nelson, the fighting kangaroo'. Every film needs a bit of luck sometimes and to have the one kangaroo in the whole of Australia who was like Moby Dick was my amazing bit of luck.\n\nWhen the shoot was finished Kotcheff cut the footage with editor Tony Buckley at Ajax Films, where, he says, the relatively inexperienced Buckley 'did an absolutely superb job in editing'. He recalls that producer George Willoughby,\n\nlike all old Brits, thought that they do it better in Britain and said to me, 'Ted, do you mind if I bring in a British editor called Tom Noble?' (Tom, by the way, won an Oscar for Witness, and he has edited a lot of my films, including The Apprenticeship of Duddy Kravitzand First Blood.) I said to George, 'It's cut brilliantly, what you want him to do with it?'George said, 'I only want him to lookat it with a professional editor's eye. Humour me, Ted, humour me.' So I screened the film for Tom and when the lights went up I looked over and said, 'Well, Mr Noble?' He said, 'Ted, I wouldn't touch a frame of it. This editor is brilliant and he's done a superb job for you'.\n\nKotcheff also especially remembers sound editor Tim Wellburn doing a fantastic job:\n\nHe told me that he didn't like the tracks we had, and so what he did was to take the vehicle we'd used, and I think he also took Peter Whittle with him, and went back into the out back and re-recorded the car turning and speeding and screeching and spinning, accelerating, decelerating, crunching and banging into things. They spent days out there. I was present at the mix after they got back and I thought it was the most superb sound track I ever heard.\n\nAll in all, the production was a good experience for everyone involved, especially the director. As JackThompson said, 'at the end of the shoot I asked Ted if there was anything he'd like to take back with him, and he said, \"the crew'\".\n\n# 3\n\n# HOW THE FILM GOT LOST\u2014AND FOUND\n\nFilms disappear all the time. Distributors dispose of unwanted prints, filmmakers leave prints on top of cupboards, producers forget or fail to lodge their original material with the National Film and Sound Archive, companies change hands or get taken over, and stuff just gets lost or thrown away. To an extent this might be expected with short films, or unsuccessful films, or films that people have forgotten about, but it happens with many more. As film archivist and historian Ken Berryman wrote in 1999 in his entry on 'Lost Films' in the Oxford Companion to Australian Film,more than ninety percent of Australia's silent films, and a large slice of our film production since then, has probably not survived'.\n\nThere is an International Federation of Film Archives (FIAF) whose members are national institutions dedicated to the rescue, collection, preservation and screening of moving images, valued both as works of art and culture and as historical documents; it now comprises more than 140 institutions in over 77 countries\u2014a reflection of the extent to which the preservation of moving image heritage has become a world-wide concern. Australia's National Film and Sound Archive (NFSA)\u2014a co-publisher of this book\u2014was established in 1984 (although its antecedents go back to 1935) and, after years of being attached, in one form or another, to other institutions, in 2008 it finally became a permanent, autonomous national institution with its own statutory base. Despite its complex history it has grown into one of the world's larger and better-recognised audiovisual archives; its role is to collect, store, preserve and make available screen and sound material relevant to Australia's culture, and it is the repository of much of Australia's film heritage. But, while films made with government support are obliged to lodge preservation material with the NFSA, it is not obligatory for the many other films made every year. And certainly films made in earlier periods were not required to lodge such material.\n\nIn the 1980s the NFSA initiated the Last Film Search in response to the urgent need to recover missing nitrate films. Nitrate film stock was used in all pre-1951 professional productions, but not only was it highly flammable and sometimes self-igniting, it also decomposed if not stored in optimum conditions. So if old films were to be preserved they had to be transferred to safety film. While by the 1980s many early films were already irretrievably lost, it was believed that many were still out there waiting to be found. And they were.\n\nThe Last Film Search, well-supported by the media, encouraged people to contact the NFSA with information about the location of any film reels they happened to know about. And happily for Australian film culture citizens from all around Australia responded; in addition to much nitrate feature film footage, home movies, newsreels, actuality footage, cinema advertisements and trailers were also discovered. This resulted in the NFSA acquiring another ten percent of pre-1930s cinema and much more of our more recent film history. Highlights from the search included a 1926 two-hour documentary on New Guinea (this 16mm actuality footage was recorded during the filming of two back-to-back feature films\u2014The Hound of the Deep on Thursday Island and The Jungle Woman at Merauke in Dutch New Guinea), a very good original print of Charles Chauvel's 40,000 Horsemen, home movies from as early as 1910 and, most excitingly, three hundred feet of Australia's first feature, The Story of the Kelly Gang (1906), found by teenagers on a rubbish tip.\n\nWhile filmmakers are advised to locate their original negatives and lodge them with the NFSA, many don't. They might leave it at the lab thinking it will be safe there, but labs close or change hands. Films that have been sold into foreign territories sometimes have their original negative or intermediates sent overseas to make additional prints; that material can be very hard to locate and recover. Despite the Last Film Search campaign and the automatic lodgement of government-supported films, many films remain missing from the national collection, including the eighty percent of pre-1930s production, most of which will never been found as many silent films were deliberately destroyed with the coming of sound. Many later films are also missing, while much of the commercial television production from 1956 to 1962 was never preserved.\n\nWhile films continue to disappear all the time, some are also found. After its release in 1971, Wake in Fright screened in Australia in only a few city cinemas, never reaching the suburbs before it gradually faded from view. In 1996 Bobby Limb, the middle initial of NLT Productions, became increasingly concerned over the fate of the film that his company had made over twenty-five years earlier, and to which he believed it still owned the rights. Where, he wanted to know, was the negative and soundtrack of Wake in Fright? While a handful of increasingly damaged 35mm and 16mm prints were still around, no-one seemed to know where the original material was.\n\nWake in Fright was made as a co-production; despite a warm reception at the Cannes Film Festival and by critics in Australia and overseas, its release, both domestically and overseas, was unsuccessful. This, combined with a history of rights-holding companies changing hands, parent companies being taken over and smaller companies being deregistered, made it more than possible the film would be mislaid. When Bobby Limb contacted the film's editor, Tony Buckley, by now an eminent producer and a leading member of Australia's screen community, and asked him to help locate the missing negative and soundtrack, Buckley proved to be exactly the right person. In Ted Kotcheff's words, Buckley was'the relentless lead detective on this hunt'.\n\nIn his detailed description of his detective work in his memoir, Behind a Velvet Light Trap: A Filmmaker's Journey from Cinesound to Cannes, Buckley describes how initially he didn't think it would be at all difficult to trace the negative of Wake in Fright: 'I confidently tell Bobby that it would be in London, at one of the laboratories... where I know a lot of Australian films had been sent for their overseas sales requirements to be serviced'.\n\nBut when he contacted the laboratories, remembering to say the film may be labelled by its international title Outback, it couldn't be found. And so began the hunt that took him from the UK to France, Germany and Spain (where a Spanish-dubbed print was found, but with no supporting material). In 1998 he visited the vault in London where the reels were thought to be stored, only to learn they had been sent to the US the previous week. For a while the trail went cold. Then, in 1999, the Dublin Film Festival found the only Technicolour print in existence. This was sent back to Australia and screened at the 1999 Australian Screen Directors' conference, to much excitement and acclaim. But Buckley was still not satisfied: 'It's not a bad print, but it had some very nasty joins'. The hunt was still on.\n\nHe finally succeeded in tracking down Harvey Rappaport at the US TV network CBS (which had inherited much moving-image material when it merged with Westinghouse, the original parent company of the film's co-producer, Group W). Rappaport eventually found some 16mm prints of the film, but this wasn't enough for super-sleuth Buckley: his holy grail was the film negative from which clear, clean prints could be struck. He implored Rappaport to have one final search for the negative. In his book, Buckley tells the rest of the story:\n\nRappaport got so fed up with my pestering him about the film that he agreed to go to Pittsburgh to look for it. And he found it in this big container marked 'For destruction'. There were more than 200 reels there, virtually a complete set. According to the foreman at the warehouse, the container had been about to go to the tip. We don't know how close we were to losing it but it might have been a matter of days.\n\nWith the help of Ausfilm (a government and industry partnership), all these reels were then repatriated to the NFSA in Australia. They arrived in September 2004, more than eight years after Limb first asked Buckley for his assistance. As Kotcheff said, it is only because of Buckley's 'commitment and years of dogged pursuit that this film has survived'.\n\nWhat kept him going? Buckley thinks he 'got a bug in my head that it had to be found, although I nearly gave up in 2002; of course I didn't think it would be the trial that it became, or that it would be discovered so close to being destroyed'. Recovering the film after such a long search was 'a relief more than anything. Like any national treasure, once it's gone, it's gone; and it would have been a terrible thing had it been lost. We were so close that I didn't dare ask what the dates were for that crate's destruction!'\n\nLike many a good film narrative, this story has a subplot, and in this case it's one that reveals just how important the history of film is to a nation's screen heritage, almost as important as the existence of the film itself.\n\nWhile Buckley was searching for the film, entertainment lawyer Raena Lea-Shannon was conducting her own search. Buckley had rung and asked whether she'd be interested in clarifying title to the film on a pro-bono basis. As NLT Productions Pty Ltd was deregistered in September 1990, she had to make an application to the Australian Securities and Investments Commission (ASIC), seeking their agreement to transfer the rights in the film on behalf of the de-registered company. This involved locating the documentation that proved NLT Productions was the maker of Wake in Fright, and had done all the right things in terms of legal contracts. Upon learning from film historian Nick Herd, who was researching the history of Australian commercial television, that he had discovered about twenty boxes of papers in the Mitchell Library relating to NLT Productions, and other related companies, and that they included two boxes connected with Wake in Fright, she spent much spare time going through all the (very dusty and dirty) papers looking for the documentation. In a bizarre echo of the search for the film's negative, all these boxes had been found in a building slated for destruction, and the builder, who thought they could be important, decided to save them at the last minute.\n\nMost of the papers relating to Wake in Fright are photo and carbon copies, not originals and not signed. However, because they are the only ones known to exist, they have been relied on by the ASIC to establish that NLT Productions made the film. Because NLT Productions was deregistered, ASIC became owner of the copyright. It was decided on advice from Lea-Shannon that a nonprofit Trust should be established to receive the film rights from ASIC. Lea-Shannon set up the Wake in Fright Trust in January 2009 and ASIC transferred the copyright to the Trust, whose aim is:\n\nto preserve and promote public awareness of the motion picture Wake in Fright and the general public awareness of the significance of Australian-produced feature films and to support the cultural development of the Australian Film Industry as the Trustee sees fit.\n\nIn time those boxes of papers may also provide more information on how the Australian company, NLT productions, got together with its co-producer, the US company Group W owned by Westinghouse. They include copies of letters from NLT's Jack Neary to the celebrated Australian dancer, actor, director and choreographer Robert Helpmann (later Sir Robert), who was very interested in playing Doc; there are also copies of letters from Helpmann giving Neary contacts for possible co-production partners. When Helpmann learned the part had gone to Donald Pleasence, Lea-Shannon reports that he was not at all happy. Jack Neary was no longer considered a friend.\n\nThis paper search, which has already taken over five years so far, isn't finished; the details of the co-production agreement with CBS (inherited from Westinghouse) are in the process of being retrieved by CBS. Once those rights are settled Wake in Fright will be available for distribution worldwide.\n\nThis legal history of its loss and recovery offers the sort of detailed information that film historians need in order to understand how and why films do or don't get made, why and where audiences go to see them or not, and why a film is or isn't a success.\n\nAnd so the next stage of the process began: the restoration of the film. Restoration is another part of the story of our national screen heritage: the expertise and dedication needed to restore a film, the costs involved in restoration and preservation, and the priorities that have to be established. After the original negative and other material arrived at the NFSA in Canberra, an initial inspection confirmed that the images and sound negatives were virtually complete and reasonably intact. The NFSA, together with Deluxe Australia (formerly Atlab), the film laboratory with which it has an ongoing print restoration program, then had to determine how to preserve the film and the method that would best restore its image and sound closest to its original state.\n\nWhen Atlab made a print from the negative, it was discovered that, as well as the expected dirt and scratches, the colours were badly faded, and that conventional film grading methods would probably not restore them. Anthos Simon, General Manager Creative Services for Deluxe Australia, thought they could do a better job through their recently launched digital facility, EFILM. 'We did some tests on some of the worst shots', Simon has said, 'and Tony Buckley and the [NFSA] were blown away by the improvements. The colour was back, it was clean and there were details that simply weren't visible before\u2014such as a dartboard over Chips Rafferty's shoulder in a pub scene'.\n\nAlthough the cost for such a process was much higher than had been anticipated for the film's restoration, it was determined that Wake in Fright was a crucial part of Australia's screen heritage, and Deluxe and the NFSA formed a partnership to do the full digital restoration. The work proceeded slowly, utilizing laboratory down-time as far as possible. Painstakingly, often manually, often frame by frame, the colour was corrected, the dirt and scratches removed, the image stabilised. Buckley was able to direct much of the process; as the film's editor he knew exactly how it had looked at its first screening in 1970. Digital technology meant that alterations made for the US version were easily corrected to the original, and even a couple of problems from the original edit were fixed. Several shots with nudity had been censored for the US release, and replaced with alternative, clothed takes; now copies of the original shots from the Australian version were inserted back into the negative and digitally colour-graded to match.\n\nAfter more than a year's work, the completed digital master was transferred to 35mm camera negative stock, and new prints made from this negative. For the soundtrack, Danny Roberts from NFSA Audio Services, working with then NFSA Film Specialist David Noakes, spent much time transferring the many different components that were used to remaster the soundtrack. These were pieced together digitally by Soundfirm, the sound post-production company, to create the restored and remastered Dolby soundtrack. Finally, a pristine durable 35mm negative was struck to be preserved by the NFSA.\n\nAfter the restored film's first public screening in Australia at the Sydney Film Festival 2009, with some of the crew and a delegation from Broken Hill in attendance, Ted Kotcheff said:\n\nI thought the film was pretty damn good\u2014I haven't seen it for so many years, and of course even if you've got something ninety-eight percent right, you only see the two percent wrong\u2014but I thought the restoration was amazing. I kept looking at the texture. And the performances\u2014they were so good!\n\nSo, one film, a very important film in Australia's cinema history, has been found and restored. At this point it feels important to say that many films are still missing; the NFSA is currently searching for the original materials for two other important films from the same period, Jim Sharman's strange first feature, Shirley Thompson vs the Aliens (1972), featuring bodgies and widgies, visitors from space, madness, and Sydney's Luna Park, and the Margaret Fink-produced film of David Williamson's play and screenplay, The Removalists (Tom Jeffrey, 1975). There are prints of both, but nothing more. As Ken Berryman writes,\n\nwhile Australian film culture and our heritage awareness have come a long way in recent decades, it is still possible for more recent films to slip through the preservation net. Without effective compulsory-deposit legislation, final responsibility for lodging original negatives and release prints with the NFSA rested with producers who often lack[ed] the time, energy, finance, or motivation to attend to the proper storage of a just-completed film.\n\nIn 1980 UNESCO adopted a Recommendation for the Safeguarding and Preservation of Moving Images, a document of historic importance, in which UNESCO called upon its member states and the international public to treat moving images, because of their educational, cultural, artistic, scientific and historical value, as an element of national culture. It furthermore called upon nations and their citizens to protect and preserve their films for transmission to future generations. Audiovisual archives all around the world, some at vastly different stages of development and funding, all strive to carry out this work for their own national screen collections. In recent years, however, with the enormous changes in technology and the emergence of digital media, the responsibilities of an audiovisual archive have become more complex and demanding. The concept of national cinema is harder to define in an age of globalisation and of international co-productions, while the issues of copyright, the diversification of delivery technologies, and the increased marketing opportunities for archival material, all need to be taken into account.\n\nIn Australia, the NFSA has an obligation to collect, store, preserve and provide access to Australia's ever-growing heritage of moving images and recorded sound, and it's an obligation that carries with it much responsibility. I've been writing about the NFSA for years, sometimes very critically, but I believe that the role it has played in the discovery and restoration of Wake in Fright is one that provides guidance for the future.\n\n# 4\n\n# HOW THE FILM WAS RECEIVED\n\n# Then...\n\nWake in Fright opened in Sydney on Saturday 9 October 1971 at the Embassy (the dark and luxurious cinema where I used to see British films when I bunked off from school; now gone, like so many cinemas from the Sydney CBD\u2014and from towns and cities across Australia). The reviews were good, although they pulled no punches. 'It's a harsh, jotting movie with an almost physical impact, though its vitality and rough humour prevent it from descending into resigned despair', wrote Beverly Tivey in the Sunday Australian, adding that it was'a startlingly accurate visual translation of Ken Cook's novel; andabsolutely wonderful cinema'. Martha Dubose, an American critic living and working for a time in Sydney as the Sydney Morning Herald reviewer, said that when she left the cinema she felt somehow 'blood-spattered and dirty,' and described the film's achievement as 'characterising a borderless human condition in particularly Australian terms. It succeeds in going beyond the Aussie types it utilises to portray the senseless codes that all men concoct to justify themselves'. 'Bawdy, brutal, cruel\u2014and a masterpiece', ran the headline in the Daily Mirror. Colin Bennett in the Age called it'the most savage comment on Australia ever put on film'. (Bennett, who David Stratton describes in his book The Last New Wave, as a'highly respected contributor to the Age, and just about the only writer earning his living in Australia as film critic', was the reviewer who had so savaged Tim Burstall's 2,000 Weeks two years before .)\n\nPerhaps the most perceptive comment came from Sylvia Lawson, writing in the Australian, who did not see it as either caricature or realism: 'It is more in the nature of one man's bad dream, told with a dream's intensity and a dream's apparent illogic.' She continued, 'Ken Cook's tight, terrifying little story is of course absurd, but when you take it as an imaginative concentration of horrific real-life elements, it makes full sense and impact'.\n\nAnd impact it most certainly had. I've talked to many friends about their memories of seeing the film on its original release, and I've had widely differing responses but, as I've mentioned earlier, none could forget it. One confessed that the film was such an awful, traumatic experience that he couldn't even consider seeing it again. Another went with her mother, who had grown up in the country, and who found its portrayal of the treatment of women bore out what she remembered from her own experience. For me, what is most interesting is the clarity with which people remember the film, retaining vivid images not only of the kangaroo shoot, but many other scenes.\n\nJack Thompson, speaking at the premiere screening at the Sydney Film Festival in June 2009, recalled hearing reports that during its first release, people were apparently yelling out in the cinema, 'That's not us. That's not Australia. That's not how we are'. But, he said, 'it was, and still is. I think it was more that we hadn't made any films about us for a long time. We'd not seen that image of ourselves on the screen, and Australians were terribly confronted by what they saw'.\n\nMichael Thornhill was writing about films and filmmaking for the Australian at that time. He'd made a short film, The American Poet's Visit, in 1969, would make his first feature, Between Wars, in 1974 and go on to enjoy the same sort of bumpy career that many Australian filmmakers have, all the while remaining an active, critical, and very vocal member of the screen community. On 27 January 1970, reporting on his visit to the set of Wake in Fright when they were shooting the two-up sequence, he described'the euphoric experience' of watching a film being made, but concluded rather ominously:'The producers of Wake in Fright seem to be aiming somewhere in between Easy Rider and If on the one hand and Hello, Dolly! on the other. They may well do it. Wake in Fright will have to gross about $3 million at the box office to break even'.\n\nHe was even more concerned in March the next year, reporting on 23 March that\n\nUnited Artists claim they are making top plans for the release of Wake in Fright in Australia. This was contradicted by another United Artists source, who said that Wake in Fright would have to compete for theatre space with Fiddler on the Roof and the new James Bond film, Diamonds Are Forever.\n\nWhen the film was released in October he was quite impressed. 'If Wake in Fright is not the definitive film about Australia', he wrote on 12 October in the Australian, 'it will do as thought-provoking entertainment, warts and all, until some such masterpiece is forthcoming.' He was less positive about its chances at the box office, saying,'it will be interesting to see if this relatively expensive international potpourri is commercially successful... it is unusual to find an Australian company involving itself in a film that purports to treat seriously an aspect of the Australian social ethos'. In this he was prescient. Despite the strong reviews the film did not perform well at the box office, closing in Brisbane after only a week. And in Sydney, although Tony Buckley recalled that word of mouth was working and audiences were slowly growing, six weeks into the run the film was taken off to make way for more mainstream fare.\n\nGraham Shirley and Brian Adams wondered whether the film 'was perhaps too uncomfortably direct and uncompromising to draw large Australian audiences', and they quote one of the film's two executive producers, Bill Harmon, as saying the film had been treated badly by its local distributor, United Artists' Australian branch. Harmon had said in the Australian, 'It almost seems nobody wants anything to succeed here. Here we are, wanting to do pictures here, and it's terribly important for the investor to get his money back. But if a good movie can't get any money back here, who the hell's interested?'\n\nWhen Wake in Fright opened in Australia it had already screened as the official Australian entry in the 1971 Cannes Film Festival in May, where it gained what Ted Kotcheff remembers as 'ecstatic reviews... One said that there was so much heat and dust (on screen) he had to go home and have a shower afterwards.' It was released in Paris as Le Reveil dans la Terroir (Waking in Terror), received great reviews from French critics, and ran for nearly five months. Kotcheff remembers\n\nOne French critic thought it was a masterpiece and interviewed me two or three times, on television and on radio. The French really loved that film. They loved the idea of people under great existential stress, and were attracted to the dust and heat and the powerful rawness of it.\n\nUnited Artists had completed a deal with NLT Productions and Group W for global distribution, but they changed the title of the film to Outback for its international release. This version of the film was slightly different: both the opening and end titles were changed, and a scene in which Gary Bond appeared naked now had him in underpants. In October 1971 Outback was released in Britain, and four months later in the United States.\n\nThe overseas advertising played the originality of the film to the hilt by dwelling on the strangeness of the outback location, declaring: 'Outback. Drama of Despair. Filmed in Australia' and went on,\n\nOutback derives its title from the setting of its story, the 'outback' area of Australia. It is a violent section of the down-under continent: violent in color, violent climate, violent gambling and violent drinking. The 'Outback' cages [John Grant], strips him bare, and tortures him, but in exchange it gives him, on his brief, destructive stay, a new understanding of his own humanity.\n\nDespite its relatively brief releases and lack of success in the UK and in the US, the film did get written about. In Britain it was reviewed in The Times, the Telegraph and the film buffs' bible, the Monthly Film Bulletin, while in the US it was mentioned quite a few times in Variety, and reviewed in the Village Voice and Vanity Fair. Roger Greenspun wrote in the New York Times of 'a sense of general foreboding that crystalizes often enough into particular terror and that is not quite like anything else I can remember feeling at the movies'. He went on,\n\nscarier for me is a scene in one of the city's great sprawling saloons, a moment of hushed, terrifying solemnity when in a kind of lunatic praise of long-departed soldiers, sailors and airmen, the drinkers are admonished overa spectral loudspeaker, 'Lest We Forget!' I suspect that these scenes are real, too, and to have discovered them is a filmmaker's coup.\n\nThe film critic once described as 'a national treasure', Pauline Kael, treated the film very seriously in the New Yorker: 'There's talent and intelligence in the Australian film Outback. More remarkably, it has a subject\u2014the crude comradeship among the white men in the vast desert areas, and their erratic destructiveness... it records the same kind of senseless destruction that in recent American movies has been blamed on the corruptiveness of American racism and capitalist exploitation and the Vietnam war'. The film's strength, she continued,\n\nis that, unlike the American films in which vengeful, hypocritical (usually middle-class) whites destroy mercilessly, slaughtering Indians, or animals, or both, Outback is fair-minded. And because the mindless violence you're shown isn't attributed to what you know damned well didn't cause it, you can't shrug it off; you're stuck with it, trying to understand it, trying to figure out what, if anything, can be done about it.\n\nKnowle wrote that you come out of the cinema 'with a sense of epic horror'. Not all overseas reviewers, however, were so impressed by the film. Writing in the British tabloid the Sun, Fergus Cashin commented 'anybody who emigrates to Australia after seeing this must need treatment'.\n\nIn 1975 the Sydney Film Festival presented a 'Salute to Australian Film', an 'Australian Feature Film Retrospective 1911-1971'. The special program explained that,\n\nnearly seventy years since the release of the first Australian feature film, The Story of the Kelly Gang, in 1906... and now that local production is at a higher level than it has been for many years, it seemed a very appropriate time to take a long view at some of the most important films produced here in the intervening period.\n\nThe Salute included almost seventy productions from 1911 on, culminating in Wake in Fright, which was described in the program by the Festival's then Director, David Stratton, as 'the most uncompromising film on the Australian mateship ethos'. The retrospective, which screened throughout the festival, was enthusiastically attended and Stratton remembers Wake in Fright receiving a very positive response. And this was only four years after that first disappointing release.\n\nIn the years after its theatrical release Wake in Fright gradually disappeared from view. It did, however, screen on Channel 10 at 8.30 pm one night in 1988, where it was presented with some passion by film presenter Bill Collins, who said that it held a mirror up to life, and that 'it should be considered one of the best films ever made in this country'. Advising readers about this screening, Walter Sullivan in the Daily Telegraph wrote that no-one could describe Wake in Fright 'as an attractive film, but there is no denying it is effective', and went on to say that some of the men set out to show John Grant's character what Australian mateship is about. 'What it is, in fact, about, the film suggests, is booze. Booze in quantities beyond belief, booze whether you want it or not, booze while you are still conscious and more of it as soon as you wake up.'\n\n# Out of sight, but not out of mind\n\nIn the intervening years, as the film itself gradually slipped from sight, its reputation and importance grew. In his book Australian National Cinema film academic Tom O'Regan credits Wake in Fright and director Ted Kotcheff as one of the films and directors to which the post-1970s revival is particularly indebted, preparinglocal and international audiences for the Australian films that followed'. What he says, it seems to me, is insightful and undeniable:\n\nWake in Fright's prototypical middle-class male school teacher experiencing a vernacular working-class male regional culture fashioned the male ensemble film. With its dystopian view of mateship and misogyny, Wake introduced the idea of endemic and structural evil to Australian cinema... The preparedness to accept, exploit, entertain and at times exaggerate this possibility provided an important maturity to Australian film-making and undoubtedly aided its circulation in the international festival market. It helped create... an unsavoury Australian past and present centred largely on the deeds, misogyny, limited horizons, and xenophobia of white [Anglo] males. Kotcheff's film prepared the way for that mix of hyperrealism, excessive masculinity, ambiguous sexuality, and misogyny so insistently present in subsequent Australian cinema.\n\nIn The Screening of Australia Volume 2, the first serious analysis of the Australian film revival, Susan Dermody and Elizabeth Jacka are also astute, describing the film as:\n\nan outsider's essay, a kind of Australian heart of darkness, but at moments a deeply perceptive and unadorned one, without either false politeness or undue reversion to gross stereotype... The adaptation of Kenneth Cook's novel helps to anchor this co-production in this culture; it is also worth speculating that Ted Kotcheff's Canadianness presented less of a problem of cultural difference than would have been the case if an American or British director had been used. Dominion provincialism, a geography of extremes and great isolation, and a similar sense of a cultural coming of age in the seventies: Canada and Australia had a lot of common ground.\n\nAs it happens, this prefigures much of what Ted Kotcheff has said on how he felt about making the film. Indeed, as he said at Wake in Fright's screening at the Toronto Film Festival in September 2009, 'The truth is, I felt I knew these people. Australia, like Canada, is huge, open and claustrophobic. The vastness of both places is imprisoning. And I'd seen men doing bad things in my own country. They weren't strangers to me'.\n\nThe critic and cultural commentator Meaghan Morris wrote something similar back in 1980 in The New Australian Cinema:\n\nWake in Fright places its harrowing vision of ruthless sexual segregation and violent repression in the context of an industrialized desert hell. The environment is a brutalizing one; but there is a difference between the lethal monotony of the tiny settlement at the beginning of the film , and the crazed violence of the mining town, which is purely man-made\u2014the cruel conditions of labor which underlie the town's sense of amusement.\n\nShe went on to point out that Grant, the young teacher in the film, 'is marked by his outsider status, which brings upon him a fearful education in the ways of the \"here\" of the mining town. Otherness can be a source of both attraction\u2014offering something which cannot be realized in a here and now\u2014or a source of repulsion, an occasion for persecution'.\n\nAndrew Zielinski pursues this further. Writing about four films directed by overseas filmmakers, in an article entitled 'Australia from the Outside In' in Issue 49 of Screen Education, a journal for media teachers, he comments:\n\nWake in Fright can be taken as a journey through hell. The film populates the outback with aggressive males who unleash violence on the landscape, interrogating the male myth of mateship. Any positive aspect of landscape and the settler or battler is totally subverted. Instead of harmony between the landscape of the bush and the inhabitants we have a strong dislocation. The inhabitants willingly abuse themselves, women, the bush and kangaroos. As intruders are welcomed to this male clique, the hospitality becomes an aggressive beer-swilling odyssey. Survival here is to dominate and lower any standards to a basic instinctual drive.\n\nIn 1999 I attended my favourite conference, held annually by the Australian Screen Directors Association (now the Australian Directors Guild). That year, the print recently discovered by the Dublin Film Festival was screened to highlight the theme of the conference: 'Australian Stories in a Global Industry'. Ted Kotcheff gave a video introduction from New York and Tony Buckley was on hand to discuss the film; it was received enthusiastically and un-nostalgically, and recognised for its important place in the ongoing battle for a sustainable and strong local industry. Seeing it again, and especially in this context, made both its historical significance and its contemporary relevance equally clear.\n\nIn a prescient article in Quadrant in 2006, entitled 'The Best Australian Film You've Never Seen', Simon Caterson describes Wake in Fright as 'a treasure of the Australian cinema, a film which like all classics in any art form has dated only in ways that don't matter'. Having somehow heard of the recovery of the film's negative, and recognising its high critical reputation, he tracked down a (apparently pretty awful) DVDcopyfrom a Canadian media company. Despite its inferior quality, he wrote that the film's reputation is'richly deserved'. Commenting that 'no other Australian film prepares you for Wake in Fright', he goes on to speculate 'that a kind of unconscious self-censorship\u2014a sort of cultural amnesia\u2014has allowed Wake in Fright to all but disappear, while other, \"nicer\" films of the period... have been cared for and canonised'.\n\n# ... and now\n\nIn April 2009, the film's rediscovery and restoration was announced in a flurry of press and radio comment, with the ABC's 7.30 Report doing a detailed story, which featured a brief clip of Bill Collins' impassioned introduction to that single TV screening over twenty years earlier.\n\nA month later it was once again screened at the Cannes Film Festival, this time in the Cinema Classics section, and was once again acclaimed. In June, when Ted Kotcheff arrived for the Sydney Film Festival screening and to record (with Tony Buckley) the commentary for the DVD, he embarked on a short but very effective publicity tour. A number of follow-up stories and interviews appeared, the most detailed and thoughtful of which appeared in the online cinema journal Senses of Cinema's May '09 issue, conducted by Raffaele Caputo (although, having just read so many of the reviews of the film, both then and now, I take issue with his comment that it was originallyunkindly received' by critics). Caputo makes the following interesting comment:\n\nThe main criticism levelled at the film when it was released in Australia was its uncompromisingly scurrilous view of Australians, exposing a dark side to the mateship myth that, as JackThompson had said to you, people felt was unrealistic. Yet, a question I have always asked myself is why John Grant, near the end of the film when he boards the train that's taking him back to Tiboonda, and after all that he has been through, graciously accepts a beer that's offered to him? It tells me that what we have seen prior to this moment\u2014the brutality, the menace, the aggressive hospitality of the Australian character\u2014was not meant to betaken realistically, but that it's John's heightened, surreal, distorted point of view of that world.\n\nKotcheff agrees, responding that'this guy is an outsider and does not really understand what is going on in that world and has a kind of distorted vision of it'.\n\nThe film made international waves: after the Sydney screening, Toni O'Loughlin reported in the British paper, the Guardian,\n\nlast month the 1971 Australian horror film Wake in Fright was screened at the Cannes Film Festival, and now it is finally being accepted at home in Australia after years of neglect... amid the crowd of long-time fans who attended the film's screening at the Sydney Film Festival on Saturday were some from Broken Hill, the New South Wales town where much of the film was shot. One couple, too young to have seen the film when it was released, were curious to see the film that had riled so many in their town. They laughed along with the rest of the audience, who guffawed at the laconic dialogue and brutal behaviour as if they were watching a caricature.\n\nThe restored film was given a limited but well-attended and profitable Australian release from the end of June, and reviews this time around were if anything even stronger and more positive, with many critics referring to the film's (and often their own) history. Evan Williams in the Weekend Australian, after asking himself why he hadn't remembered to include Wake in Fright in his list of the five best Australian films some months earlier, then gave a glowing commendation:\n\nAt a technical level, the film is masterly. The dusty, heat-throbbing outback landscapes are beautifully caught in Brian West's cinematography,and (Tony) Buckley's editing has wonderful fluidity and power... in more ways than one, this is Buckley's film.\n\nSeveral critics and commentators recalled the first time they had seen the film. Paul Byrnes in the Sydney Morning Herald commented,\n\nrediscovering Wake in Fright is less like running into an old friend than someone you feared as a child. There has never been a more savage and scabrous film about Australia. Unfortunately, it was also uncomfortably true, which was one reason Australians didn't go to see it... in 1971. It was just too confronting. Its power has hardly diminished in the years since.\n\nAnd in the Age Brigid Delaney wrote that she first saw Wake in Fright\n\nas a child, and fragments have never left my consciousness\u2014the bright, slanting light, the frenzied gambling and drinking, the roo shoot. The film was assumed by some to be a dystopian, exaggerated and nightmarish version of Australia, almost like a horror film. Last week I saw the re-release at Nova. It was just as horrible and brilliant as I had remembered. But the horror and brilliance resided not in the violence\u2014but in something else\u2014the truth. The truth ofTed Kotcheff's 1971 film is even more startling for its endurance. What it says about some isolated parts of Australia is as true now as it was then. The film also throws a bright light on this nation's great disconnect; mainstream or urban Australia's lack of engagement with its more isolated parts\u2014that myth that we are one nation.\n\nThe Age's reviewer, Jake Wilson, young enough not to remember the film from an earlier screening, was less overwhelmed, but he did think\n\nthe casting of Pleasence as the Yabba's resident pervert turns out to be Kotcheff's masterstroke. Long typecast as a whiney runt, he couldn't be further from the stereotype of the bronzed working-class Aussie. Yet he's wholly persuasive as a demonic variant of a figure beloved of Australian storytellers from Henry Lawson to David Williamson: the ratbag intellectual who stands both inside and outside a dominant male group. Giving his high, nasal voice an Australian twang, Pleasence makes Tydon a recognisable, strangely pathetic monster. Perhaps only Graham Kennedy could have matched his ability to convey abject need through overdone mateyness and a confiding gleam in the eye.\n\nPerhaps younger reviewers have a slightly different take. Dave Hoskin, in Metro magazine, while believing that it's a'classic and unparalleled movie', was more interested in the fact that for him, at least, 'the film positively screams that it's the product of a bygone era'. But, as he says, compared to the 'safeness' of contemporary films, this iswild filmmaking, scary filmmaking, classic filmmaking'.\n\nIn 2009 Sylvia Lawson revisited the film in the online journal Inside Story where she described the film as\n\na version of the Australian inland, about the injuries of isolation, and the gulf between the country and the city. It operates a particular way of seeing the desert, opening with a high-level pan around flat, sandy emptiness, before coming down into the one-room schooihouse\u2014and the room full of kids, mixed ages, under orders to keep silent until the clock ticks three o'clock and they can race off for the holidays. (I haven't seen anyone remarking on the way John Grant, being a bitterly discontented teacher, is also a pretty bad one.)\n\nBut, as she goes on to say,\n\nsince Wake in Fright was made thirty-eight years ago, there have been massive shifts in general consciousness on the nation's deeply bicultural nature, on Aboriginal people, the inheritance of dispossession, the crucial idea of country in the Indigenous sense\u2014and also on country towns, complicated places as they are, and now places which, more than ever, understand themselves historically. Watching now, but notthen, it's possible to think about the Bundanyabba milieu as a pack of blokes who, simply, didn't know where they were.\n\nThe Sydney Morning Herald even used the film in its editorial about local film distribution, saying 'The re-release of Wake in Fright is a reminder of just how powerful Australian film can be, and also how fleeting', before going on to say 'we are still producing good films... and they are still not finding a large audience', and making an argument for putting more effort into'ensuring our good films find an audience'. This is an aim with which I entirely agree, while recognising that it's not something to which there are any easy answers.\n\nAfter making Wake in Fright, Ted Kotcheff directed The Apprenticeship of Duddy Kravitz (1974), which, based on Mordecai Richler's important Canadian novel, is probably his favourite film. He made a number of other films in the 1970s and 1980s, then went on to direct a lot of TV in the 1990s. Now well in his seventies, he's an ebullient, active man, who shows no sign of slowing down; since 2000 he has been executive producer and occasional director of the USTV series Law & Order: Special Victims Unit. When in September 2009 he introduced a special screening of Wake in Fright at the Toronto International Film Festival, Greg Quill, writing about the event in the Toronto Star, reported that New Zealand-born actor Sam Neill, who was there, remarked, 'I was stunned when I first saw it. It confirmed my worst fears about Australians'. Quill grasped the reason for the film's original reception in Australia, writing,\n\nWhen Wake in Fright so cruelly exposed the slimy underbelly of the nation's hidden psychoses\u2014brutish misogyny, racism, bloodlust, alcoholism, a rampant gambling streak, xenophobia and a fear of the emptiness of their own landscape\u2014it was a shocking revelation most Australians weren't prepared for, not a family portrait Australians were happy to hand around.\n\nHe further reported that Rachel Ward, in Toronto for the screening of her feature film Beautiful Kate, sawWake in Fright soon after moving to Australia from England in the early 1980s, and acknowledged its role in the development of a defining vernacular among her peers. As he says,\n\nIndeed, Beautiful Kate, a Gothic mystery about incest, suicide and misplaced love in a decaying family in the South Australian bush, owes much to the harsh visual language and uncompromising honesty Kotcheff introduced. 'If I had seen it before I decided to settle there, I might not have gone', [Rachel Ward] said. 'It was challenging... and uncomfortable. And people don't want to go to the movies to be made uncomfortable... But I like stories that are morally ambiguous, where there's no clear idea of what's good and what's evil, or of what's forgivable and what's not.'\n\nQuill also reported Kotcheff as saying, 'I just know I'm very pleased with Wake in Fright. It shone a light into the dark corners of the human psyche. And it's one film that I never thought, \"If only I'd done this or that another way'\".\n\nIn its two commercial releases, thirty-eight years apart, Wake in Fright has attracted many reviews, and while most have been favourable, they haven't tried to gloss over the film's almost visceral impact. In the years between, its reputation has grown and what has been written about it has multiplied. Although I've barely scratched the surface, I've tried to give a sense of how critics, reviewers, film historians and commentators have regarded this strong, tough film, this film about survival that has managed to survive.\n\nThe film now seems set to become a permanent part of our screen culture heritage. Just before I finished writing this I had dinner with an old friend, writer and editor Marsha Rowe, who has lived in London for many years. Interestingly, the dinner was also attended by some friends of those early, light-hearted filmmaking days, including Albie Thoms and Richard Neville. I told them about the book, and we remembered those times, and we talked about Wake in Fright, which Marsha, so far away, hadn't heard about. A few days later I got an email from London. Amazingly, she had seen the film on the flight back'in the middle of the worst last leg of the flight, from Bangkok to London, and was transfixed. It's fantastic that the film is not lost.'\n\nIn the last weeks as I was finishing this book, ABC Radio National, by another amazing bit of timing, was repeating the book reading of the novel of Wake in Fright. Every morning, as I sat at my computer, I heard another chapter; it was bleak and spare and almost cold, and frightening in its very matter-of-factness. And as I listened to the words I could see the images that were crafted by the people and through the processes that I've tried to convey in this book. And I'm very glad to have had the chance to do my own piece of investigation, in a story filled with searches and detection, into just how such an important film was made and lost and found again.\n\n# NOTES\n\nMore information about these and many other such films of this period can be found in Peter Mudie's book, Ubu Films: Sydney Underground Movies, 1965-1970, Kensington: University of New South Wales Press, 1971.\n\nThe Fugs werea satirical-political New York rock band named from a euphemism for 'fuck' used in Norman Mailer's novel The Naked and the Dead.\n\nWomen, of course, didn't actually ride surfboards in those days, and rarely even got wet (in the movies, at least). It wasn't until Puberty Blues that this situation was revised, cinematically. For more about this, see Nell I Schofield Puberty Blues, Currency Press & NFSA, Redfern, 2004.\n\nYounger and newer Australians and other readers who have little idea what 'two up' is will find an excellent history of this very Aussie gambling game at the Australian War Memorial website at http:\/\/www.awm.gov.au\/encyclopedia\/two_up\/index.asp. Retrieved on 14 September 2009.\n\nhttp:\/\/aso.gov.au\/titles\/features\/wake-in-fright\/notes.Retrieved on 31 October 2009.\n\nhttp:\/\/aso.gov.au\/titles\/features\/wake-in-fright\/notes.Retrieved on 31 October 2009\n\nSee Catherine Lumby, Alvin Purple, Currency Press & NFSA, Redfern, 2008.\n\nScreen Hub, 16 February 2009.\n\nThe first Australian moratorium protesting the Vietnam War was held in May 1970 when over 200,000 people across Australia marched in peaceful demonstrations. The movement's aims were to force a withdrawal of Australian and other foreign troops from Vietnam and to repeal the National Service Act 1964. Six months after the third moratorium was held in June 1971, the newly-elected Labor Party abolished conscription, freed imprisoned conscientious objectors and announced the withdrawal of all Australian troops from Vietnam.\n\nAll quotes from Richard Brennan are from interviews conducted by the author.\n\nSee also Tony Moore, The Barry McKenzie Movies, Currency Press & NFSA, Redfern, 2005.\n\nAll quotes from Howard Rube are from telephone interviews conducted by the author. England's Ealing Studios produced several films in Australia in the 1950s including The Overlanders (1946) and Eureka Stockade (1949) both directed by the British Harry Watt and starring Chips Rafferty. Lee Robinson directed the iconic Rafferty, his partner in the production company Southern International, in a series of films in the 1950s, using exotic locations to sell Australia to a mainly overseas market. The films, which included The Phantom Stockman (1953), King of the Coral Sea (1954), and Walk into Paradise (1956), did quite well, but a number of factors, including the difficulty of getting financing and problems with local distribution, contributed to the company getting out of feature production. The arrival of television didn't help.\n\nAdrian Martin's monograph on the Mad Max movies in this series is a good starting point for more information on how the outback has been treated cinematically in the years since 1971.\n\nQuotes by Ted Kotcheff throughout this book have come from his appearances at the Sydney Film Festival screening and discussion, and from a number of informal interviews conducted with the author.\n\nIn 1952-54 Cook worked for the Australian Broadcasting Commission at Broken Hill\u2014a town he apparently loathed\u2014and then at Rockhampton, Queensland. While in Rockhampton he wrote a novel that was subsequently considered libellous and pulped. He moved to Brisbane and then moved back to Sydney in 1954. In 1961 his novel Wake in Fright was published.\n\nQuotes from Jack Thompson are from the discussion following the screening of the film at the Sydney Film Festival in 2009.\n\nThe exception was France, where it ran for five months at a Paris cinema.\n\nLimb died in 1999, never knowing whether the film would be found.\n\nAll the reviews quoted here were published at the time of the film's release in 1971.\n\nAnother 'lost and found'story is attached to the history ofThe Story of the Kelly Gang, which is now available on DVD. For more information on the film seethe NFSA webpage at http:\/\/australianscreen.com.au\/titles\/story-kelly-gang\/\n\n# BIBLIOGRAPHY\n\nBlundell, Graeme. The Naked Truth: A Life in Parts. Sydney: Hachette Australia, 2008.\n\nBuckley, Anthony. Behind a Velvet Lighttrap: A Filmmaker's Journey from Cinesound to Cannes. Prahran: Hardie Grant Books, 2009.\n\nDermody, Susan and Jacka, Elizabeth. The Screening of Australia (Vol 1): Anatomy of a Film Industry. Sydney: Currency Press, 1987.\n\nDermody, Susan and Jacka, Elizabeth. The Screening of Australia (Vol 2): Anatomy of a National Cinema. Sydney: Currency Press, 1988.\n\nLarkins, Bob. Chips: The Life and Films of Chips Rafferty. South Melbourne: Macmillan Company of Australia, 1986.\n\nMcFarlane, Brian, Mayer, Geoff, and Bertrand, Ina (eds). The Oxford Companion to Australian Film. Oxford University Press, 2000\n\nMoran, Albert and O'Regan, Tom (eds). Australian Film Reader. Sydney: Currency Press, 1985\n\nMoran, Albert and O'Regan, Tom (eds). The Australian Screen. Ringwood: Penguin Books Australia, 1989.\n\nMudie, Peter. Ubu Films: Sydney Underground Movies, 1965-1970. Kensington: University of New South Wales Press, 1971.\n\nMurray, Scott (ed). The New Australian Cinema. West Melbourne: Nelson, 1980\n\nO'Regan, Tom. Australian National Cinema. London: Routledge, 1996.\n\nPike, Andrew, and Cooper, Ross. Australian Film 1900-1977. Oxford University Press, 1980.\n\nShirley, Graham, and Adams, Brian. Australian Cinema:The First Eighty Years. Sydney: Angus and Robertson in association with Currency Press, 1983 (revised edition 1989).\n\nStratton, David. The Last New Wave: The Australian Film Revival. Australia: Angus and Robertson Publishers, 1980.\n\nThoms, Albie. Surfmovies: The History of the Surf Film in Australia. Noosa Heads: Shore Thing Publishing, 2000.\n\n# FILMOGRAPHY\n\nAccident, Joseph Losey, 1967\n\nThe Adventures of Barry McKenzie, Bruce Beresford, 1972\n\nAlvin Purple, Tim Burstall, 1973\n\nThe American Poet's Visit, Michael Thornhill, 1969\n\nThe Apprenticeship of Duddy Kravitz, Ted Kotcheff, 1974\n\nThe Back of Beyond, John Heyer, 1954\n\nBeautiful Kate, Rachel Ward, 2009\n\nBed and Board, Fran\u00e7ois Truffaut, 1970\n\nBend of the River, Anthony Mann, 1952\n\nBetween Wars, Michael Thornhill, 1974\n\nBitter Springs, Ralph Smart, 1950\n\nBoobs a Lot, Aggy Read, 1970\n\nBush Christmas, Ralph Smart, 1947\n\nCat People, Jacques Tourneur, 1942\n\nClaire's Knee, Eric Rohmer, 1970\n\nThe Conformist, Bernardo Bertolucci, 1971\n\nThe Curse of the Cat People, Gunther von Fritsch and Robert Wise, 1944\n\nThe Damned, Joseph Losey, 1961\n\nThe Devil to Pay, Bruce Beresford, 1962\n\nDiamonds Are Forever, Guy Hamilton, 1971\n\nDing a Ding Day, Garry Shead, 1966\n\nEasy Rider, Dennis Hopper, 1969\n\nEureka Stockade, Harry Watt, 1949;\n\nFiddler on the Roof, Norman Jewison, 1971\n\nFirst Blood, Ted Kotcheff, 1982\n\nFlashpoint, Brian Hannant, 1972\n\nForgotten Cinema, Tony Buckley, 1969\n\n40,000 Horsemen, Charles Chauvel, 1941\n\nGentle Strangers, Cecil Holmes, 1972\n\nHello, Dolly!, Gene Kelly, 1969\n\nThe Hound of the Deep, Frank Hurley, 1926\n\nIf... Lindsay Anderson, 1968\n\nThe Jungle Woman, Frank Hurley, 1926\n\nKangaroo, Lewis Milestone, 1952\n\nKing & Country, Joseph Losey, 1964\n\nLaw & Order: Special Victims Unit, US TV series, 1999-\n\nLife at the Top, Ted Kotcheff, 1965\n\nThe Naked Bunyip, John B Murray, 1970\n\nNickel Queen, John McCallum, 1971\n\nOn the Beach, Stanley Kramer, 1959\n\nOr Forever Hold Your Peace, Richard Brennan, 1970\n\nThe Overlanders, Harry Watt, 1946\n\nThe Prize, Tim Burstall, 1960\n\nPunishment Park, Peter Watkins, 1971\n\nRobbery Under Arms, Jack Lee, 1957\n\nScaramouche, George Sidney, 1952\n\nThe Servant, Joseph Losey, 1963\n\nThe Shiralee, Leslie Norman, 1957\n\nThe Siege of Pinchgut, Harry Watt, 1959\n\nSingin' in the Rain, Gene Kelly, Stanley Donen, 1952\n\nThe Sleeping Tiger, Joseph Losey, 1954\n\nSmiley, Anthony Kimmins, 1956\n\nSons of Matthew, Charles Chauvel, 1949\n\nSqueeze a Flower, Marc Daniels, 1970\n\nThe Story of the Kelly Gang, Charles Tait, 1906\n\nStork, Tim Burstall, 1971\n\nSummer of the Seventeenth Doll, Leslie Norman, 1959\n\nSurfing Roundabout, David Price, 1965\n\nThe Sundowners, Fred Zinneman, 1960\n\nThe Surfing Years, Peter Thompson, 1966\n\nTeorema, Pier Paolo Pasolini, 1968\n\nTiara Tahiti, Ted Kotcheff, 1962\n\nTobias Icarus Age Four, Clemency Weight, 1968\n\nTristana, Luis Bu\u00f1uel, 1970\n\nTwo Gentleman Sharing, Ted Kotcheff, 1969\n\nTwo Thousand Weeks, Tim Burstall, 1969\n\nVision for a New World, Chris McCullough, 1968\n\nWalkabout, Nicholas Roeg, 1971\n\nWho Plus Live Equals Home, Peter Kingston, John Allen, 1962\n\nWitness, Peter Weir, 1985\n\n# CREDITS\n\n# Key Cast\n\nJock Crawford\n\nChips Rafferty\n\n'Doc'Tydon\n\nDonald Pleasence\n\nJohn Grant\n\nGary Bond\n\nJanette Hynes\n\nSylvia Kay\n\nDick\n\nJack Thompson\n\nJoe\n\nPeter Whittle\n\nTim Hynes\n\nAl Thomas\n\nCharlie\n\nJohn Meillon\n\nAtkins\n\nJohn Armstrong\n\nJarvis\n\nSlim DeGrey\n\nReceptionist\n\nMaggie Dence\n\nJoe the cook\n\nNorm Erskine\n\nFirst controller\n\nOwen Moase\n\nSecond controller\n\nJohn Dalleen\n\nCharlie Jones\n\nBuster Fiddess\n\nStubbs\n\nTex Foote\n\nStockman\n\nColin Hughes\n\nVan Driver\n\nJacko Jackson\n\nRobyn\n\nNancy Knudsen\n\nJoyce\n\nDawn Lake\n\nHiggins\n\nHarry Lawrence\n\nPig Eyes (as Bob McDarra)\n\nRobert McDarra\n\nPoker player\n\nCarlo Manchini\n\nMiner\n\nLiam Reynolds\n\n# Key Crew\n\nProduction company\n\nNLT Productions Pty Ltd, Group W\n\nFilms\n\nDirector\n\nTed Kotcheff\n\nScreenplay\n\nEvan Jones\n\nbased on the novel\n\nWake in Fright by Kenneth Cook\n\nProducer\n\nGeorge Willoughby\n\nExecutive Producers\n\nBill Harmon, Howard G. Barnes\n\nAssociate Producer\n\nMaurice Singer\n\nComposer\n\nJohn Scott\n\nDirector of Photography\n\nBrian West\n\nEditor\n\nAnthony Buckley\n\nArt Director\n\nDennis Gentle\n\nFirst Assistant Director\n\nHoward Rubie\n\nCamera Operator\n\nJohn McLean\n\nPost-Production Coordinator\n\nThom Noble\n\nLocation Manager\n\nJohn Shaw\n\nSound Mixer\n\nHugh Strain\n\nSound Recording\n\nJohn Appleton\n\nSound Editors\n\nKeith Palmer, Tim Wellburn\n\nContinuity\n\nRita Cavill\n\nProduction Secretary\n\nVivian Holmes\n\nProduction accountant\n\nPhilip Moylan\n\nGaffer\n\nTony Tegg\n\nMakeup Artist\n\nMonica Dawkins\n\nHairdresser\n\nRobert Hynard\n\nWardrobe\n\nRon Williams\n\nCasting Directors\n\nJill Dempster, John Merrick\n\nTitles by\n\nEllis-Wright\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nTable of Contents\n\nAbout the Author\n\nTitle Page\n\nCopyright Page\n\nDedication\n\nCHAPTER 1\n\nCHAPTER 2\n\nCHAPTER 3\n\nCHAPTER 4\n\nCHAPTER 5\n\nCHAPTER 6\n\nCHAPTER 7\n\nCHAPTER 8\n\nCHAPTER 9\n\nCHAPTER 10\n\nCHAPTER 11\n\nCHAPTER 12\n\nCHAPTER 13\n\nCHAPTER 14\n\nCHAPTER 15\n\nCHAPTER 16\n\nCHAPTER 17\n\nCHAPTER 18\n\nPraise for _The Crow Road:_\n\n'Brilliantly done' _Daily Mail_\n\n'Tight with detail and close observation and creates a strong sense of a particular period of growing up' _Independent_\n\n'Banks has woven a warm and funny story, rich with characters and adventures ... an utterly enchanting piece of fiction' _New Woman_\n\n'Magnificent... a poignant, very funny study of life growing up in Banks's native Scotland. At times as wonderfully light and colourful as its setting on the west coast of Scotland, and as darkly comic as _The Wasp Factory ...' For Him_\n\n'This substantial novel indicates a restless author very firmly in the driver's seat, back on what appears to be a Scottish route with intriguing potential destinations' _The Scotsman_\n\n'What makes Banks a significant novelist is the love and effort that go into his works, and his acute sense of the ways in which people can suffer ... this is Banks's finest novel yet' _Independent on Sunday_\n\n'Prentice is a most engaging narrator, self-deprecating, funny and hopelessly self-deceiving' _Daily Telegraph_\nIain Banks came to widespread and controversial public notice with the publication of his first novel, _The Wasp Factory_ , in 1984. He has since gained enormous popular and critical acclaim for both his mainstream and his science fiction novels.\n**_Also by lain Banks_**\n\nTHE WASP FACTORY\n\nWALKING ON GLASS\n\nTHE BRIDGE\n\nESPEDAIR STREET\n\nCANAL DREAMS\n\nTHE CROW ROAD\n\nCOMPLICITY\n\nWHIT\n\nA SONG OF STONE\n\nTHE BUSINESS\n\nDEAD AIR\n\nTHE STEEP APPROACH TO GARBADALE\n\nTRANSITION\n\nSTONEMOUTH\n\nTHE QUARRY\n\n**_And as Iain M. Banks_**\n\nCONSIDER PHLEBAS\n\nTHE PLAYER OF GAMES\n\nUSE OF WEAPONS\n\nTHE STATE OF THE ART\n\nAGAINST A DARK BACKGROUND\n\nFEERSUM ENDJINN\n\nEXCESSION\n\nINVERSIONS\n\nLOOK TO WINDWARD\n\nTHE ALGEBRAIST\n\nMATTER\n\nSURFACE DETAIL\n\nTHE HYDROGEN SONATA\n\nThe Crow Road\n\nIAIN BANKS\n\nHachette Digital\n\nwww.littlebrown.co.uk\n\nPublished by Hachette Digital 2008\n\nCopyright \u00a9 lain Banks 1992\n\nThe right of lain Banks to be identified as author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.\n\n_All characters in this publication are fictitious_   \n_and any resemblance to real persons, living or dead,_   \n_is purely coincidental._   \nAll rights reserved.   \nNo part of this publication may be reproduced,   \nstored in a retrieval system, or transmitted, in any   \nform or by any means, without the prior   \npermission in writing of the publisher, nor be   \notherwise circulated in any form of binding or   \ncover other than that in which it is published and   \nwithout a similar condition including this   \ncondition being imposed on the subsequent purchaser.\n\nA CIP catalogue record for this book   \nis available from the British Library.\n\nISBN 978 0 7481 0993 7\n\nThis ebook produced by JOUVE, FRANCE\n\nHachette Digital  \nAn imprint of  \nLittle, Brown Book Group  \n100 Victoria Embankment  \nLondon EC4Y 0DY\n\nAn Hachette UK Company\n**Again, for Ann,**   \n**And with thanks to:**   \nJames Hale,   \nMic Cheetham,   \nAndy Watson   \nand Steve Hatton\nCHAPTER 1\n\nIt was the day my grandmother exploded. I sat in the crematorium, listening to my Uncle Hamish quietly snoring in harmony to Bach's Mass in B Minor, and I reflected that it always seemed to be death that drew me back to Gallanach.\n\nI looked at my father, sitting two rows away in the front line of seats in the cold, echoing chapel. His broad, greying-brown head was massive above his tweed jacket (a black arm-band was his concession to the solemnity of the occasion). His ears were moving in a slow, oscillatory manner, rather in the way John Wayne's shoulders moved when he walked; my father was grinding his teeth. Probably he was annoyed that my grandmother had chosen religious music for her funeral ceremony. I didn't think she had done it to upset him; doubtless she had simply liked the tune, and had not anticipated the effect its non-secular nature might have on her eldest son.\n\nMy younger brother, James, sat to my father's left. It was the first time in years I'd seen him without his Walkman, and he looked distinctly uncomfortable, fiddling with his single earring. To my father's right my mother sat, upright and trim, neatly filling a black coat and sporting a dramatic black hat shaped like a flying saucer. The UFO dipped briefly to one side as she whispered something to my father. In that movement and that moment, I felt a pang of loss that did not entirely belong to my recently departed grandmother, yet was connected with her memory. How her moles would be itching today if she was somehow suddenly reborn!\n\n'Prentice!' My Aunt Antonia, sitting next to me, with Uncle Hamish snoring mellifluously on her other side, tapped my sleeve and pointed at my feet as she murmured my name. I looked down.\n\nI had dressed in black that morning, in the cold high room of my aunt and uncle's house. The floorboards had creaked and my breath had smoked. There had been ice inside the small dormer window, obscuring the view over Gallanach in a crystalline mist. I'd pulled on a pair of black underpants I'd brought especially from Glasgow, a white shirt (fresh from Marks and Sparks, the pack-lines still ridging the cold crisp cotton) and my black 501s. I'd shivered, and sat on the bed, looking at two pairs of socks; one black, one white. I'd intended to wear the black pair under my nine-eye Docs with the twin ankle buckles, but suddenly I had felt that the boots were wrong. Maybe it was because they were matt finish ...\n\nThe last funeral I'd been to here \u2013 also the first funeral I'd ever been to \u2013 this gear had all seemed pretty appropriate, but now I was pondering the propriety of the Docs, the 501s and the black biker's jacket. I'd hauled my white trainers out of the bag, tried one Nike on and one boot (unlaced); I'd stood in front of the tilted full-length mirror, shivering, my breath going out in clouds, while the floorboards creaked and a smell of cooking bacon and burned toast insinuated its way up from the kitchen.\n\nThe trainers, I'd decided.\n\nSo I peered down at them in the crematorium; they looked crumpled and tea-stained on the severe black granite of the chapel floor. Oh-oh; one black sock, one white. I wriggled in my seat, pulled my jeans down to cover my oddly-packaged ankles. 'Hell's teeth,' I whispered. 'Sorry, Aunt Tone.'\n\nMy Aunt Antonia \u2013 a ball of pink-rinse hair above the bulk of her black coat, like candy floss stuck upon a hearse \u2013 patted my leather jacket. 'Never mind, dear,' she sighed. 'I doubt old Margot would have minded.'\n\n'No,' I nodded. My gaze fell back to the trainers. It struck me that on the toe of the right one there was still discernible the tyre mark from Grandma Margot's wheelchair. I lifted the left trainer onto the right, and rubbed without enthusiasm at the black herring-bone pattern the oily wheel had left. I remembered the day, six months earlier, when I had pushed old Margot out of the house and through the courtyard, past the outhouses and down the drive under the trees towards the loch and the sea.\n\n'Prentice, what is going on between you and Kenneth?'\n\nThe courtyard was cobbled; her wheelchair wobbled and jerked under my hands as I pushed her. 'We've fallen out, gran,' I told her.\n\n'I'm not stupid, Prentice, I can see that.' She looked up at me. Her eyes were fierce and grey, as they always had been. Her hair was grey now, too, and thinning. The summer sun cleared the surrounding oaks and I could see her pale scalp through the wisps of white.\n\n'No, gran, I know you're not stupid.'\n\n'Well, then?' She waved her stick towards the outhouses. 'Let's see if that damn car's still there.' She glanced back at me again as I wheeled the chair round on its new heading, towards the green double doors of one of the courtyard garages. 'Well, then?' she repeated.\n\nI sighed. 'It's a matter of principle, gran.' Stopping at the garage doors, she used her stick to knock the hasp off its staple, pushed at one door till its planks bowed slightly, then, wedging her stick into the resulting gap, levered the other door open, a bolt at one corner scraping and tinkling through a groove worn in the cobbles. I pulled the chair back to let the garage door swing. Inside it was dark. Motes swirled in the sunlight falling across the black entrance. I could just make out the corner of a thin green tarpaulin, draped angularly about level with my waist. Grandma Margot lifted the edge of the tarp with her stick, and flicked it away with surprising strength. The cover fell away from the front of the car and I pushed her further into the garage.\n\n'Principle?' she said, leaning forward in the chair to inspect the long dark bonnet of the car, and pushing the tarp back still further until she had revealed the auto up to its windscreen. The wheels had no tyres; the car rested on blocks of wood. 'What principle? The principle of not entering your father's house? Your own family home?' Another flick of the cane and the covering moved up the screen, then fell back again.\n\n'Let me do that, gran.' I stepped to the side of the car and pulled the tarpaulin back until it lay crumpled on the boot, revealing that the car had a missing rear window. More dust revolved in the light from outside, turning Grandma Margot into a seated silhouette, her almost transparent hair shining like a halo.\n\nShe sighed. I looked at the car. It was long and quite beautiful, in a recently-old-fashioned way. Beneath the patina of dust it was a very dark green. The roof above the missing rear window was battered and dented, as was the exposed part of the boot lid.\n\n'Poor old thing,' I whispered, shaking my head.\n\nGrandma Margot sat upright. 'It or me?' she said sharply.\n\n'Gran ...' I said, tutting. I was aware that she could see me very well, sunlit from behind her, while all I could see of her was a dark shape, a subtraction of the light.\n\n'Anyway,' she said, relaxing and poking at one of the car's wire wheels with her walking stick. 'What's all this nonsense about a matter of principle?'\n\nI turned away, rubbing my fingers along the chrome guttering over a rear door. 'Well ... dad's angry at me because I told him I believed in ... God, or in something, anyway.' I shrugged, not daring to look at her. 'He won't ... well, I won't ... We're not talking to each other, so I won't come into the house.'\n\nGrandma Margot made a clucking noise with her mouth. 'That's it?'\n\nI nodded, glancing at her. 'That's it, gran.'\n\n'And your father's money; your allowance?'\n\n'I \u2013' I began, then didn't know how to put it.\n\n'Prentice; how are you managing to survive?'\n\n'I'm managing fine,' (I lied.) 'On my grant.' (Another lie.) 'And my student loan.' (Yet another lie.) 'And I'm doing some bar work.' (Four in a row!) I couldn't get a bar job. Instead I'd sold Fraud Siesta, my car. It had been a small Ford and kind of lazy about starting. People used to imply it looked battered, but I just told them it came from a broken garage. Anyway, that money was almost gone now, too.\n\nGrandma Margot let out a long sigh, shook her head. 'Principles,' she breathed.\n\nShe pulled herself forward a little, but the wheelchair was caught on part of the tarp. 'Help me here, will you?'\n\nI went behind her, pushed the chair over the ruffled canvas. She hauled open the offside rear door and looked into the dull interior. A smell of musty leather wafted out, reminding me of my childhood and the time when there was still magic in the world.\n\n'The last time I had sex was on that back seat,' she said wistfully. She looked up at me. 'Don't look so shocked, Prentice.'\n\n'I wasn't \u2013' I started to protest.\n\n'It's all right; it was your grandfather.' She patted the wing of the car with one thin hand. 'After a dance,' she said quietly, smiling. She looked up at me again, her lined, delicate face amused, eyes glittering. 'Prentice,' she laughed. 'You're blushing!'\n\n'Sorry, gran,' I said. 'It's just ... well, you don't ... well, when you're young and somebody's ...'\n\n'Past it,' she said, and slammed the door shut; dust duly danced. 'Well, we're all young once, Prentice, and those that are lucky get to be old.' She pushed the wheelchair back, over the toe of my new trainers. I lifted the chair clear and helped complete the manoeuvre, then pushed her to the door. I left her there while I put the tarpaulin back over the car.\n\n'In fact some of us get to be young twice,' she said from the doorway. 'When we go senile: toothless, incontinent, babbling like a baby ...' Her voice trailed off.\n\n'Grandma, please.'\n\n'Och, stop being so sensitive, Prentice; it isn't much fun getting old. One of the few pleasures that do come your way is to speak your mind ... Certainly annoying your relatives is enjoyable too, but I expected better of you.'\n\n'I'm sorry, Grandma.' I closed the garage door, dusted off my hands, and took up my position at the back of the wheelchair again. There was an oily tyre print on my trainer. Crows raucoused in the surrounding trees above as I pushed my gran towards the drive.\n\n'Lagonda.'\n\n'Sorry, Gran?'\n\n'The car; it's a Lagonda Rapide Saloon.'\n\n'Yes,' I said, smiling a little ruefully to myself. 'Yes, I know.'\n\nWe left the courtyard and went crunchily down the gravel drive towards the sparkling waters of the loch. Grandma Margot was humming to herself; she sounded happy. I wondered if she was recalling her tryst in the Lagonda's back seat. Certainly I was recalling mine; it was on the same piece of cracked and creaking, buttoned and fragrant upholstery \u2013 some years after my gran's last full sexual experience \u2013 that I had had my first.\n\nThis sort of thing keeps happening in my family.\n\n'Ladies and Gentlemen of the family; on the one hand, as I don't doubt you may well imagine, it gives me no great pleasure to stand here before you at this time ... yet on the other hand I am proud, and indeed honoured, to have been asked to speak at the funeral of my dear old client, the late and greatly loved Margot McHoan ...'\n\nMy grandmother had asked the family lawyer, Lawrence L. Blawke, to say the traditional few words. Pencil-thin and nearly as leaden, the tall and still dramatically black-haired Mr Blawke was dressed somewhere in the high nines, sporting a dark grey double-breasted suit over a memorable purple waistcoat that took its inspiration from what looked like Mandelbrot but might more charitably have been Paisley. A glittering gold fob watch the size of a small frying pan was anchored in the shallows of one waistcoat pocket by a bulk-carrier grade chain.\n\nMr Blawke always reminded me of a heron; I'm not sure why. Something to do with a sense of rapacious stillness perhaps, and also the aura of one who knows that time is on his side. I _thought_ he had looked oddly comfortable in the presence of the undertakers.\n\nI sat and listened to the lawyer and in short order wondered (a) why Grandma Margot had chosen a lawyer to make the address, (b) whether he'd be charging us for his time, and (c) how many others of my family were wondering the same things.\n\n'... long history of the McHoan family in the town of Gallanach, of which she was so proud, and to which she so ... usefully and, and industriously contributed throughout her long life. It was my privilege to know and serve both Margot and her late husband Matthew well, in Matthew's case first as a school friend, back in the twenties. I well remember ...'\n\n'Grandma, I mean; good grief.'\n\n'What?'\n\nMy grandmother drew deeply on the Dunhill, flicked her wrist to close the brass Zippo, then put the lighter back in her cardigan.\n\n'Grandma, you're smoking.'\n\nMargot coughed a little and blew the smoke towards me, a grey screen for those ash-coloured eyes. 'Well, so I am.' She inspected the cigarette closely, then took another drag. 'I always wanted to, you know,' she told me, and looked away, over the loch towards the hills and trees on the far side. I'd wheeled her down to the shore path at Pointhouse near the old cairns. I sat on the grass. A soft breeze disturbed the water; seagulls flew stiff-winged, and in the distance the occasional car or truck disturbed the air, making a lazy throat-clearing noise as they emerged from or disappeared into the channel the main road drove between the trees. 'Hilda used to smoke,' she said quietly, not looking at me. 'My elder sister; she used to smoke. And I always wanted to.' I picked up a handful of pebbles from the path-side and started throwing them at the waves, lapping against the rocks a metre below us, almost at high tide. 'But your grandfather wouldn't let me.' My grandmother sighed.\n\n'But gran,' I protested. 'It's bad for you.'\n\n'I know.' She smiled broadly. 'That was another reason I didn't ever take it up, after your grandfather died; they'd found it was unhealthy by then.' She laughed. 'But I'm seventy-two years old now, and I don't give a damn.'\n\nI chucked a few more pebbles. 'Well, it isn't a very good example to us youngsters, is it?'\n\n'What's that got to do with the price of sliced bread?'\n\n'Eh?' I looked at her. 'Pardon?'\n\n'You're not really trying to tell me that young people today look to their elders for an example, are you, Prentice?'\n\nI grimaced. 'Well ...' I said.\n\n'You'd be the first generation that did.' She pulled on the cigarette, a look of convincing derision on her face. 'Best do everything they don't. That's what tends to happen anyway, like it or lump it.' She nodded to herself and ground the cigarette out on her cast, near the knee; flicked the butt into the water. I tutted under my breath.\n\n'People react more than they act, Prentice,' she said eventually. 'Like you are with your dad; he raises you to be a good little atheist and then you go and get religion. Well, that's just the way of things.' I could almost hear her shrug. 'Things can get imbalanced in families, over the generations. Sometimes a new one has to ... adjust things.' She tapped me on the shoulder. I turned. Her hair was very white against the rich summer green of the Argyllshire hills and the brilliant blue of the sky beyond. 'D'you feel for this family, Prentice?'\n\n'Feel for it, gran?'\n\n'Does it mean anything to you?' She looked cross. 'Anything beyond the obvious, like giving you a place to stay ... well, when you aren't falling out with your father? Does it?'\n\n'Of course, gran.' I felt awkward.\n\nShe leaned closer to me, eyes narrowing. 'I have this theory, Prentice.'\n\nMy heart foundered. 'Yes, gran?'\n\n'In every generation, there's a pivot. Somebody everybody else revolves around, understand?'\n\n'Up to a point,' I said, non-committally, I hoped.\n\n'It was old Hugh, then your grandfather, then it was me, and then it got all confused with Kenneth and Rory and Hamish; they each seem to think they were it, but...'\n\n'Dad certainly seems to think he's paterfamilias.'\n\n'Aye, and maybe Kenneth has the strongest claim, though I still think Rory was more clever. Your Uncle Hamish ...' She looked troubled. 'He's a bit off the beaten track, that boy.' She frowned. (This 'boy' was nearly fifty, of course, and himself a grandfather. It was Uncle Hamish who'd invented Newton's Religion, and who had taken me in when my father and I had fallen out.)\n\n'I wonder where Uncle Rory is,' I said, hoping to divert my gran from areas that sounded portentous and daft with the familiar game that anybody in our family can play; making up stories, conjectures, lies and hopes about Uncle Rory, our one-time golden boy, professional traveller and some-time magician, whose most successful act had been his own disappearance.\n\n'Who knows?' My gran sighed. 'Might be dead, for all we know.'\n\nI shook my head. 'No, I don't think so.'\n\n'You sound certain, Prentice. What do you know we don't?'\n\n'I just feel it.' I shrugged, threw a handful of pebbles into the waves. 'He'll be back.'\n\n'Your father thinks he will,' Margot agreed, sounding thoughtful. 'He always talks about him as though he's still around.'\n\n'He'll be back,' I nodded, and lay back in the grass, hands under my head.\n\n'I don't know, though,' Grandma Margot said. 'I think he might be dead.'\n\n'Dead? Why?' The sky was deep, shining blue.\n\n'You wouldn't believe me.'\n\n'What?' I sat up again, swivelled to face her, looking over the much-scribbled-upon grey-white cast (as well as signatures, get-well-soon messages and silly drawings, there were at least two shopping lists, a recipe copied down from the radio and detailed instructions on how to get by car to the flat I shared in Glasgow).\n\nGrandma Margot pulled up her sleeve to expose her white, darkly spotted right forearm. 'I have my moles, Prentice. They tell me things.'\n\nI laughed. She looked inscrutable. 'Sorry, gran?'\n\nShe tapped her wrist with one long pale finger; there was a large brown mole there. Her eyes were narrowed. She leaned closer still and tapped the mole again. 'Not a sausage, Prentice.'\n\n'Well,' I said, not sure whether to try another laugh. 'No.'\n\n'Not for eight years, not a hint, not a sensation.' Her voice was low, almost husky. She looked as though she was enjoying herself.\n\n'I give in, gran; what are you talking about?'\n\n'My moles, Prentice.' She arched one eyebrow, then sat back with a sigh in her wheelchair. 'I can tell what's going on in this family by my moles. They itch when people are talking about me, or when something ... remarkable is happening to the person.' She frowned. 'Well, usually.' She glared at me, prodded me in the shoulder with her stick. 'Don't tell your father about this; he'd have me committed.'\n\n'Gran! Of course not! And he wouldn't, anyway!'\n\n'I wouldn't be too sure of that.' Her eyes narrowed again.\n\nI leant on one of the chair's wheels. 'Let me get this right; your moles itch when one of us is talking about you?'\n\nShe nodded, grim. 'Sometimes they hurt, sometimes they tickle. And they can itch in different ways, too.'\n\n'And that mole's Uncle Rory's?' I nodded incredulously at the big mole on her right wrist.\n\n'That's right,' she said, tapping the stick on one footrest of the wheelchair. She held up her wrist and fixed the raised brown spot with an accusatory glare. 'Not a sausage, for eight years.'\n\nI stared at the dormant eruption with a sort of nervous respect, mingled with outright disbelief. 'Wow,' I said at last.\n\n'... survived by her daughter Ilsa, and sons Kenneth, Hamish and Roderick.' The good lawyer Blawke had helpfully nodded at my dad and my uncle when he mentioned them. Dad kept on grinding his teeth; Uncle Hamish stopped snoring and gave a little start at the mention of his name; he opened his eyes and looked round \u2013 a little wildly, I thought \u2013 before relaxing once more. His eyelids started to droop again almost immediately. At the mention of Uncle Rory's name Mr Blawke looked about the crowded chapel as though expecting Uncle Rory to make a sudden and dramatic appearance. 'And, sharing, I'm sure, in the family's grief, the husband of her dear late daughter, Fiona.' Here Mr Blawke looked very serious, and did indeed grasp his lapels for a moment, as he nodded, gravely, at Uncle Fergus. 'Mr Urvill,' Mr Blawke said, completing the nod that had developed pretensions to a bow, I thought, and then clearing his throat. This genuflection completed, the reference to past tragedy duly made, most of the people who had turned to look at Uncle Fergus turned away again.\n\nMy head stayed turned.\n\nUncle Fergus is an interesting enough fellow in himself, and (of course) as Mr Blawke knew to his benefit, probably Gallanach's richest and certainly its most powerful man. But I wasn't looking at him.\n\nBeside the thick-necked bulk of the Urvill of Urvill (soberly resplendent in what I assumed was the family's mourning tartan \u2013 blackish purple, blackish green and fairly dark black) sat neither of his two daughters, Diana and Helen \u2013 those long-legged visions of money-creamed, honey-skinned, globetrotting loveliness \u2013 but instead his niece, the stunning, the fabulous, the golden-haired, vellus-faced, diamond-eyed Verity, upwardly nubile scionette of the house of Urvill, the jewel beside the jowls; the girl who, for me, had put the lectual in intellectual, and phany in epiphany and the ibid in libidinous!\n\nSuch bliss to look. I feasted my eyes on that gracefully angular form, just this side of her uncle and sitting quietly in black. She had worn a white quilted skiing jacket outside, but now had taken it off in the unfittingly chilly crematorium, and sat in a black blouse and black skirt, black ... tights? Stockings? My God, the sheer force of joy in just imagining! and black shoes. And shivering! The slick material of the blouse trembling in the light from the translucent panes overhead, black silk hanging in folds of shade from her breasts, quivering! I felt my chest expand and my eyes widen. I was just about to look away, reckoning that I had gazed to the limits of decency, when that shaven-sided, crop-haired head swivelled and lowered, her calm face turning this way. I saw those eyes, shaded by her thick and shockingly black brows, blink slowly; she looked at me.\n\nSmall smile, and those diamond eyes piercing, marking me.\n\nThen the gaze removed, refixed, directed somewhere else, once more facing the front. My neck felt un-oiled as I turned away, blasted and raddled by the urge of that directed consideration.\n\nVerity Walker. Eating my heart out. Consuming my soul.\n\n'And dad's mole?'\n\n'Here,' Grandma Margot said, tapping her left shoulder. She laughed a little as we went along the path between the shore and the trees. 'That one itches fairly often.'\n\n'And mine?' I asked, plodding after the wheelchair. I'd taken my biker's jacket off and it lay now on my gran's lap.\n\nShe looked up at me, her expression unreadable. 'Here.' She patted her tummy, looked forward again. 'Pivotal, wouldn't you say, Prentice?'\n\n'Ha,' I said, still trying to sound non-committal. 'Could be. What about Uncle Hamish? Where's he at?'\n\n'Knee,' she said, tapping the plaster on her leg.\n\n'How is your leg, gran?'\n\n'Fine,' she said tetchily. 'Plaster comes off next week. Can't happen soon enough.'\n\nThe wheels of the chair sighed through the grass on either side of the narrow path. I remembered something I'd been meaning to ask.\n\n'What were you doing up that tree anyway, gran?'\n\n'Trying to saw a branch off.'\n\n'What for?'\n\n'To stop those damn squirrels using it as a diving board to get to my bird table, that's what for.' She used her stick to whack a crumpled drinking-yoghurt bottle off the path and into the water.\n\n'You could have asked somebody else.'\n\n'I'm not totally incapable, Prentice. I'd have been all right if that hoodie hadn't started dive-bombing me; ungrateful wretch.'\n\n'Oh, it was a bird's fault, was it?' I had a mental picture of some beetle-eyed carrion crow swooping on my gran, knocking her off her ladder. Maybe it had seen _The Omen._\n\n'Yes, it was.' Grandma Margot twisted in her wheelchair and raised both her stick and her voice. 'And a few years ago I'd only have been bruised, as well. Brittle bones are one of the things that make getting old such a damn nuisance, too, especially if you're a woman.' She nodded brusquely. 'So think yourself lucky.'\n\n'Okay,' I smiled.\n\n'Damn birds,' she muttered, glaring at a stand of ash trees on the edge of the plantation with such severity that I half expected to hear a parliament of crows cry out in answer. 'Ach well,' she shrugged. 'Let's head back to the house; I need to go.'\n\n'Right you are,' I said, and wheeled the chair around. Grandma Margot lit another cigarette.\n\n'That branch is still there, by the way.'\n\n'I'll deal with it.'\n\n'Good lad.'\n\nA lark trilled, high overhead.\n\nI wheeled my gran back along the path by the water, over the main road and up the gravel drive, through the sunlit cobbled courtyard towards the tall house with the crow-stepped gables.\n\nI cut the offending branch down that afternoon, before I went back to Gallanach, to my Uncle Hamish's house, for tea. My dad arrived while I was up the ladder, sawing away at the sappy oak and swatting at flies. He stopped and looked at me when he got out of the Audi, then he disappeared into the house. I kept on sawing.\n\nMy great-great-great grandfather, Stewart McHoan, was buried in a coffin made from black glass by the craftsmen he had commanded in his capacity as manager of the Gallanach Glass Works (a post now filled by my Uncle Hamish). Grandma Margot had gone for the more conventional wooden model; it slid away into the wall as Bach's Mass reached one of its choral climaxes. A wood-fronted door slid back up to block the hole the coffin had disappeared into, then a little purple curtain lowered itself over the doorway.\n\nThe head honcho of the undertakers supervised us as we all formed up for what was obviously the important and formal business of Leaving The Chapel. My father and mother left first. 'I told you we sat in the wrong place, Tone,' I heard my Uncle Hamish whisper behind me. (Aunt Tone just went 'Ssh!')\n\nOutside it was a calmly sombre day, chill and a little damp. I could smell leaves being burned somewhere. The view down the crematorium's birch-lined drive led towards the town and the ocean. In the distance, through the haze, North Jura was dark pastel and flat-looking on the unruffled grey blanket of sea. I looked around; dark-dressed people were everywhere amongst the parked cars, talking quietly. Their breath rose in clouds through the still air. Uncle Hamish was talking to the lawyer Blawke; Aunt Antonia to my mother. Dad was with the Urvills. The wonderful Verity was mostly hidden by my father, her snow-white ski jacket in eclipse behind the old man's tweed coat. I considered shifting my position so I could see her properly, but decided against it; somebody might notice.\n\nAt least, I thought brightly, she was here alone. For the last two years that I'd been worshipping Verity from afar she'd been going out with a gorm-free creature called Rodney Ritchie; his parents owned Ritchie's Reliable Removals in Edinburgh and were keen on alliteration. My father had met them once and coined a new collective noun: an embarrassment of Ritchies.\n\nAnyway, Urvill family gossip had it that Verity might be coming to her senses regarding Rodney's removal, and it was a positive and encouraging sign that she had turned up here without the geek in tow. I thought about approaching her. Maybe when we got back to the castle.\n\nI also thought about talking to James, but little brother was leaning against the crematorium wall looking bored but cool in his borrowed great-coat, earplugs in, getting his Walkman fix at last. Still mainlining The Doors, probably. For a moment I almost missed my elder brother, Lewis, who hadn't been able to make it back for the funeral. Lewis is better-looking, smarter and wittier than I am, so I don't miss him often.\n\nI was standing beside Uncle Hamish's Jaguar. Maybe I should just get into the car. Or find somebody else to talk to. I could feel that an attack of awkwardness \u2013 the kind of episode I am unhappily prone to \u2013 was imminent.\n\n'Hi, Prentice. You okay?'\n\nThe voice was deep and throaty but female. Ashley Watt strolled up, put her hand on the side of my shoulder, patting. Her brother Dean was just behind. I nodded.\n\n'Yeah. Yeah; fine. How's yourself? Hi, Dean.'\n\n'Hi, man.'\n\n'You just back for this?' Ash asked, nodding her head at the low grey granite of the crematorium buildings. Her long fawn hair was gathered up; her strong, angular face, dominated by a blade of a nose and a pair of large round-lensed glasses, was concerned and sad. Ash was my age, but she always made me feel younger.\n\n'Yeah; back to Glasgow on Monday.' I looked down. 'Wow, Ash; I don't think I've ever seen you in a skirt before.' Ash always wore jeans. We'd known each other since we'd used to crawl around on the same carpets together, but I couldn't remember seeing her in anything else but jeans. Yet there were her legs all right; pretty good-looking ones too, under a knee-length black skirt. She wore a big naval-looking jacket with the cuffs turned over, and black gloves; medium-high heels made her the same height as me.\n\nShe grinned. 'Short memory, Prentice. Recall school?'\n\n'Oh, yeah,' I nodded, still looking at the legs. 'Apart from then, though.' I shrugged, smiled warily at her. I'd gone through a protracted Unbearable stage while I'd been at high school \u2013 it had lasted from my first day through to about fourth year \u2013 and the most vivid memory I had of Ash from that time was when I and her two brothers had carried out a highly successful snowball ambush on her, her sister and their pals as they'd walked back from school one dark evening. _Somebody's_ snowball had broken that long sharp nose of Ashley's, and I suspected it had been one of mine if for no other reason than because as far as I knew nobody else had been deploying snowballs whose ballistic properties had been enhanced by the judicious reinforcement of their cores with moderately sizeable chuckie stones.\n\nHer nose had been reset, of course, and we'd got on better since we'd each left school. Ash frowned a little, her slightly magnified grey eyes searching mine.\n\n'I was sorry to hear about the old lady. All of us were.' She swivelled briefly to Dean, standing lighting up a Regal behind her. He nodded; black jeans and a dark blue crombie that looked like it had seen better decades.\n\nI wasn't sure what to say. 'I'll miss her,' I said eventually. I'd been trying not to think about it, ever since I'd heard the news.\n\n'Was it a heart attack, aye, Prentice?' Dean inquired through his cloud of smoke.\n\n'No,' I said. 'She fell off a ladder.'\n\n'I thought she did that last year,' Ash said.\n\n'She did; off a tree. This time she was clearing the gutters. The ladder slipped and she went through the conservatory roof. She was dead by the time they got her to the hospital. Shock from blood-loss, apparently.'\n\n'Oh, Prentice, I'm sorry,' Ash said, and put her hand on my arm.\n\nDean shook his head and looked mystified. 'Ah thought she had a heart attack.'\n\n'She did have one,' I nodded. 'About five years ago; got a pacemaker fitted.'\n\n'Maybe she had a heart attack while she was up the ladder,' Dean suggested. Ash kicked his shin. 'Oo-ya!' he said.\n\n'Excuse Mr Sensitivity here,' Ash said. 'But like I said: we were all really sorry to hear, Prentice.' She looked around. 'Haven't seen Lewis here; could he not make it?'\n\n'He's in Australia,' I sighed. 'Being funny.'\n\n'Ah.' Ash nodded, smiling faintly. 'Well, that's a shame.'\n\n'For the Australians, perhaps,' I said.\n\nAsh looked sad, even pitying. 'Aw, Prentice \u2013'\n\nDean prodded his sister in the back with the hand he wasn't rubbing his shin with. 'Hoi; what was that about yon guy ye bumped into in that jacuzzi in Berlin? Said ye were goantae tell \u2013'\n\n'Oh yeah ...' Ash turned from frowning at her brother to frowning at me, took a breath, then let it out. 'Hey; you fancy a pint later, Prentice?'\n\n'Well, maybe,' I said. 'I think we're ordered up to the castle for drinks and a bite to eat.' I shrugged. 'This evening?'\n\n'Okie-dokie,' Ash nodded.\n\n'A _jacuzzi?'_ I asked, looking at Dean and Ash in turn. _'Berlin?'_\n\nDean grinned broadly and nodded.\n\nAsh said, 'Aye, Prentice; watchin the wa' come doon. And a shocking and decadent tale it is, too, let me tell you. See you in the Jacobite about eight?'\n\n'Right you are,' I said. I leaned close and nudged her. 'What jacuzzi?'\n\nI saw the expression on Dean's face, then heard the noise, then watched Ashley's gaze rise from my face to fasten somewhere over my left shoulder. I turned slowly.\n\nThe car came screaming up the crematorium drive, leaves swirling into the air behind. It was a green Rover, and it had to be doing sixty. Probably exceeding the previous speed record within the crematorium grounds by a factor of at least three. It was heading more or less straight for us, and braking distance was running out fast.\n\n'That no Doctor Fyfe's car?' Dean said, as Ash grabbed my sleeve and started to pull me back, at the same time as the Rover's engine note fell from its wail, its nose dipped and the rear end wavered as the tyres tried to bite the moist tarmac.\n\n'I thought he had an Orion,' I said, as Ashley pulled Dean and me past the rear of Uncle Hamish's car and onto the grass. Everybody in the crowd outside the crematorium was watching the green 216 as it skidded to a stop, avoiding a head-on collision with the Urvill's Bentley Eight by only a few centimetres. The tyres rasped on the tarmac. Doctor Fyfe \u2013 for indeed, that was who it was \u2013 jumped out of the driver's seat. He was as small, rotund and be-whiskered as ever, but today his face was red and his eyes were staring.\n\n'Stop!' he yelled, slamming the door and running for the chapel entrance as fast as his little legs would carry him. 'Stop!' he shouted again; a little unnecessarily, I thought, as everybody had quite entirely stopped whatever they'd been doing some time before his car had even begun braking. 'Stop!'\n\nI still insist that I heard a muffled crump at this point, but nobody believes me. That was when it happened, though.\n\nThe sensitive morticians who run the Gallanach Corporation Crematorium usually wait until night before they burn the bodies, to avoid the possibility of resulting smoke-plumes sending overwrought relations into unsightly paroxysms of grief, but Grandma Margot had specified that she wanted to be incinerated immediately; her cremation was therefore genuinely under way as we stood there.\n\n'Ah!' said Doctor Fyfe, stumbling just before he was intercepted before the door of the chapel by a concerned undertaker. 'Ah!' he said again, and crumpled, first into the undertaker's arms and then to the ground. He was on his knees briefly, then turned and sat down, clutched at his chest, stared at the granite flagstones outside the chapel, and to the assembled, still stunned and quieted crowd of us announced, 'I'm sorry, folks, but I believe I'm having a coronary ...' and keeled over on his back.\n\nThere was an instant when nothing much seemed to happen. Then Dean Watt nudged me with the hand holding his Regal and said quietly, 'There's a funny thing, eh?'\n\n'Dean!' hissed Ashley, as people crowded round the doctor.\n\n'Oo-ya!'\n\n'Call an ambulance!' somebody shouted.\n\n'Use the hearse!' yelled my dad.\n\n'Och, it's only a bruise,' Dean muttered, rubbing vigorously at his shin. 'Oo-ya! Will ye quit that!'\n\nThey used the hearse, and got Doctor Fyfe to the local hospital in ample time to save his life if not his professional reputation.\n\nThe muffled crump \u2013 which I still maintain that I heard \u2013 was my grandmother exploding; Doctor Fyfe had neglected to ask the hospital to remove her pacemaker before she was cremated.\n\nLike I say, this sort of thing keeps happening in my family.\nCHAPTER 2\n\nThese were the days of fond promise, when the world was very small and there was still magic in it. He told them stories of the Secret Mountain and the Sound that could be Seen, of the Forest drowned by Sand and the trees that were time-stilled waters; he told them about the Slow Children and the Magic Duvet and the Well-Travelled Country, and they believed all of it. They learned of distant times and long-ago places, of who they were and what they weren't, and of what had and what had never been.\n\nThen, every day was a week, each month a year. A season was a decade, and every year a life.\n\n'But dad, Mrs McBeath says there is so a God, and you'll go to a bad place.'\n\n'Mrs McBeath is an idiot.'\n\n'No she's no, dad! She's a teacher!'\n\n'No she's not, or better still, no she _isn't._ Don't use the word \"no\" when you mean \"not\".'\n\n'But she's no a niddyott, dad! She is a teacher. Honest.'\n\nHe stopped on the path, turned to look at the boy. The other children stopped too, grinning and giggling. They were almost at the top of the hill, just above the Forestry Commission's arbitrary tree line. The cairn was visible, a lump on the sky-line. 'Prentice,' he said. 'People can be teachers and idiots; they can be philosophers and idiots; they can be politicians and idiots ... in fact I think they have to be ... a genius can be an idiot. The world is largely run for and by idiots; it is no great handicap in life and in certain areas is actually a distinct advantage and even a prerequisite for advancement.'\n\nSeveral of the children giggled.\n\n'Uncle Kenneth,' Helen Urvill sang out. 'Our daddy said you were a commie.' Her sister, alongside her on the path and holding her hand, gave a little squeal and put her free hand up to her mouth.\n\n'Your father is absolutely correct, Helen,' he smiled. 'But only in the pejorative sense, and not the practical one, unfortunately.'\n\nDiana squealed again and hid her face, giggling. Helen looked puzzled.\n\n'But dad,' Prentice said, pulling at his sleeve. 'Dad, Mrs McBeath _is a_ teacher, really she is, and she said there is so a God.'\n\n'And so did Mr Ainstie, too, dad,' Lewis added.\n\n'Yes, I've talked to Mr Ainstie,' McHoan told the older boy. 'He thinks we should send troops to help the Americans in Viet Nam.'\n\n'He an idiot too, dad?' Lewis hazarded, decoding the sour expression on his father's face.\n\n'Definitely.'\n\n'So there isnae a God, eh no Mr McHoan?'\n\n'No, Ashley, there isn't.'\n\n'Whit aboot Wombles, Mr McHoan?'\n\n'What's that, Darren?'\n\nThe Wombles, Mr McHoan. Of Wimbledon Common.' Darren Watt was holding the hand of his little brother, Dean, who was staring up at McHoan and looking like he was about to burst into tears. 'Are they real, Mr McHoan?'\n\n'Of course they are,' he nodded. 'You've seen them on television, haven't you?'\n\n'Aye.'\n\n'Aye. Well then, of course they're real; real puppets.'\n\n'But they're no _really_ real, naw?'\n\n'No, Darren, they're not really real; the real creatures on the real Wimbledon Common are mice and birds and maybe foxes and badgers, and none of them wear clothes and live in nice well-lit burrows with furniture. A lady made up the Wombles, and made up stories about them, and then people made the stories into television programmes. That's what's real.'\n\n'See, ah told ye,' Darren said, shaking his little brother's hand. 'They're no real.'\n\nDean started to cry, face screwing up, eyes closing.\n\n'Oh, good grief,' McHoan breathed. It never ceased to amaze him how quickly a small child's face could turn from peach to beetroot. His own youngest, James, was just leaving that stage, thank goodness. 'Come on, Dean; up you come up here and we'll see if we can get to the top of this hill, eh?' He lifted the howling child up \u2013 after he'd been persuaded to let go of his brother's hand \u2013 and put him on his shoulders. He looked at the little up-turned faces. 'We're nearly there, aren't we? See the cairn?'\n\nThere was a general noise of agreement from the assembled children.\n\n'Come on, then! Last one there's a Tory!'\n\nHe started off up the path; Dean was crying more quietly now. The other children ran round and past him, laughing and shouting and scrambling straight up the hillside, over the grass towards the cairn. He quit the path and started after them, then \u2013 holding Dean's legs \u2013 turned to look back at Diana and Helen, who were still standing quietly, hand-in-hand, on the path. 'You two not playing?'\n\nHelen, identically dressed to her sister in little new green dungarees and staring out from under her precisely-trimmed black fringe, shook her head, frowned. 'We better go last, Uncle Kenneth.'\n\n'Oh? Why?'\n\n'I think we're Tories.'\n\n'You might well turn out to be,' he laughed. 'But we'll give you the benefit of the doubt for now, eh? On you go.'\n\nThe twins looked at each other, then, still hand-in-hand, started up the grassy slope after the rest, earnestly concentrating on the business of clumping one foot in front of the other through the long rough grass.\n\nDean was starting to cry loudly again, probably because he thought his brother and sister were leaving him. McHoan sighed and jogged up the hill after the kids, shouting encouragement and making sure he trailed the last of them to the top and the cairn. He made a great show of being out of breath, and wobbled as he sat down, collapsing dramatically on the grass after setting Dean to one side.\n\n'Oh! You're all too fit for me!'\n\n'Ha, Mr McHoan!' Darren laughed, pointing at him. 'You're the toerag, so ye are!'\n\nHe was mystified for a moment, then said, 'Oh. Right. Toe-rag, Tuareg, Tory.' He made a funny face. 'Tora! Tora! Tory!' he laughed, and so did they. He lay in the grass. A warm wind blew.\n\n'What for are all these stones, Mr McHoan?' Ashley Watt asked. She had climbed half-way up the squat cairn, which was about five feet high. She picked up one of the smaller rocks and looked at it.\n\nKenneth rolled over, letting Prentice and Lewis climb onto his back and kick at his sides, pretending he was a horse. The Watt girl, perched on the cairn, bashed one rock against another, then inspected the struck, whitened surface of the stone she held. He grinned. She was a tyke; dressed in grubby hand-me-downs like the rest of the Watt tribe, she always seemed to have a runny nose, but he liked her. He still thought Ashley was a boy's name (wasn't it from _Gone With The Wind?),_ but then if the Watts wanted to call their children Dean and Darren and Ashley, he supposed that was up to them. Could have been Elvis and Tarquin and Marilyn.\n\n'D'you remember the story of the goose that swallowed the diamond?'\n\n'Aye.'\n\nIt was one of his stories, one he'd tried out on the children. Market research, his wife called it.\n\n'Why did the goose eat the diamond?'\n\n'Please, Uncle Kenneth!' Diana Urvill said, holding up one hand and trying to click her fingers.\n\n'Yes, Diana.'\n\n'It was hungry.'\n\n'Naw!' Ashley said scornfully from the cairn. She blinked furiously. 'It wiz fur teeth!'\n\n'It _swallowed_ it, smarty-pants, so there!' Diana said, leaning towards Ashley and shaking her head.\n\n'Hey!' McHoan said. 'You're both ... sort of right. The goose swallowed the diamond because that's what geese do with things like pebbles that they find; they swallow them so that they go into their ... anybody know?' He looked round them all as best he could without disturbing Lewis and Prentice.\n\n_'Gizzltrd,_ Mr McHoan!' Ashley shouted, waving the stone she held.\n\nDiana squealed and put her hand to her mouth again.\n\n'Well, a gizzard is part of a bird, too, that's right Ashley,' he said. 'But the diamond actually went into the goose's crop, because, like lots of animals and birds, geese need to keep some wee stones, like pebbles or gravel, in their crop, down here,' he pointed. 'So that they can grind their food up small and digest it better when it goes into their tummy.'\n\n'Please, Mr McHoan, Ah remember!' Ashley shouted. She clutched the stone to her chest, getting her ragged, thin grey jumper a little dirtier.\n\n'Me too, dad!' Prentice shouted.\n\n'And me!'\n\n'Me too!'\n\n'Well,' he said, rolling slowly over and letting Lewis and Prentice slide off his back. He sat up; they sat down. 'Way back, a long long time ago, there were these big enormous animals that used to live in Scotland, and they \u2013'\n\n'What did they look like, dad?' Prentice asked.\n\n'Ah.' McHoan scratched his head through his brown curls. 'Like ... like big hairy elephants ... with long necks. And these big huge animals \u2014 '\n\n'What were they called, please, Uncle Kenneth?'\n\n'They were called ... mythosaurs, Helen, and they would swallow rocks ... big rocks, way down into their crops, and they used these rocks to help crunch up their food. They were very very big animals, and very heavy because of all the rocks they carried around inside them, and they usually stayed down in the glens because they were so heavy, and didn't go into the sea or the lochs because they didn't float, and they stayed away from marshes, too, in case they sank. But \u2013'\n\n'Please, Mr McHoan, did they up climb trees, naw?'\n\n'No, Ashley.'\n\n'Naw, ad didnae think so, Mr McHoan.'\n\n'Right. Anyway, when they were very very old and they were going to die, the mythosaurs would come to the tops of hills ... hills just like this one, and they'd lie down, and they would die peacefully, and then after they were dead, their fur and their skin would disappear, and then their insides would disappear too af \u2013'\n\n'Where aboots did their fur and their skin go, please, Mr McHoan?'\n\n'Well, Ashley ... they turned into earth and plants and insects and other wee animals.'\n\n'Oh.'\n\n'And eventually there would just be a skeleton left \u2013'\n\n'Eek,' said Diana, and put her hand over her mouth again.\n\n'Until even that crumbled away and became dust, and \u2013'\n\n'And their tusks, Mr McHoan?'\n\n'Pardon, Ashley?'\n\n'Their tusks. Did they go intae dust as well?'\n\n'Umm ... yes. Yes, they did. So after a while everything was dust ... except for the stones that the big animals had carried in their crops; those lay in a big pile where the mythosaurs had laid down to die, and that,' he turned and slapped one of the larger stones protruding from the base of the rock pile behind him. 'That,' he grinned, because he liked the story he had just thought up and told, 'is where cairns come from.'\n\n'Ah! Ashley! You're standing on stuff that's been in a animal's gizzurd!' Darren shouted, pointing.\n\n'Eaurgh!' Ashley laughed and jumped down, throwing the stones away and rolling on the grass.\n\nThere was a deal of general tomfoolery and wee high squealing voices for a while. Kenneth McHoan looked at his watch, and wound it up as he said, 'All right, kids. Time for your dinner. Anybody hungry?'\n\n'Me!'\n\n'Me, dad!'\n\n'We are, Uncle Kenneth.'\n\n'Ah could eat a missasore, so ah could, Mr McHoan!'\n\nHe laughed. 'Well, I don't think they're on the menu, Ashley, but not to worry.' He took his pipe out and stood up, filled the bowl and tamped it down. 'Come on, you horrible rabble. Your Aunt Mary's probably got your dinner ready for you by now.'\n\n'Will Uncle Rory be doing tricks, Uncle Kenneth?'\n\n'If you're good, and eat up your vegetables, Helen, aye, he might.'\n\n'Oh good.'\n\nThey trooped down. Dean had to be carried because he was tired.\n\n'Dad,' Prentice said, falling back to talk to him while the rest whooped and yelled and capered on the slope. 'Are miffasores real?'\n\n'As real as Wombles, kiddo.'\n\n'As real as Dougal in _The Magic Roundabout?'_\n\n'Every bit. Well, almost.' He drew on the pipe. 'No; just as real. Because the only place anything is ever real is inside your head, Prentice. And the mythosaur exists inside your head, now.'\n\n'Does it, dad?'\n\n'Yes; it used to just exist in my head but now it exists in your head too, and the others'.'\n\n'So is God in Mrs McBeath's head, then?'\n\n'Yep, that's right. He's an idea in her head. Like Father Christmas and the Tooth Fairy.' He looked down at the child. 'Did you like the story about the mythosaur and the cairns?'\n\n'Was it just a story then, dad?'\n\n'Of course it was, Prentice.' He frowned. 'What did you think it was?'\n\n'I don't know, dad. History?'\n\n'Histoire, seulement.'\n\n'What, dad?'\n\n'Nothing, Prentice. No, it was just a story.'\n\n'I think the story about you meeting mum's more better, dad.'\n\n'Just \"better\" will do, Prentice; the \"more\" isn't required.'\n\n'Still a better story, dad.'\n\n'Glad you think so, son.'\n\nThe children were entering the forest, funnelling into the path between the pines. He looked away then, across the rough geography of bough and leaf, to the village and the station, just visible through the trees.\n\nThe train chuffed off into the evening, the red light on the final carriage disappearing round the bend in the cutting through the forest; the steam and smoke climbed into the sunset skies beyond. He let the feeling of return wash over and through him, looking across the deserted platform on the far side of the tracks, down across the few lights of Lochgair village to the long electric-blue reflection that was the loch, its gleaming acres imprisoned between the dark masses of the land.\n\nThe noise of the train faded slowly, and the quiet susurration of the falls seemed to swell in recompense. He left his bags where they lay and walked to the far end of the platform. The very edge of the platform dropped away there, angling down to the deck of the viaduct over the rushing water beneath. A chest-high wall formed the furthest extent of the rest of the platform.\n\nHe rested his arms on the top of the wall and looked down the fifty feet or so to the tumbling white waters. Just upstream, the river Loran piled down from the forest in a compactly furious cataract. The spray was a taste. Beneath, the river surged round the piers of the viaduct that carried the railway on towards Lochgilphead and Gallanach.\n\nA grey shape flitted silently across the view, from falls to bridge, then zoomed, turned in the air and swept into the cutting on the far bank of the river, as though it was a soft fragment of the train's steam that had momentarily lost its way and was now hurrying to catch up. He waited a moment, and the owl hooted once, from inside the dark constituency of forest. He smiled, took a deep breath that tasted of steam and the sweet sharpness of pine resin, and then turned away, went back to pick up his bags.\n\n'Mr Kenneth,' the station master said, taking his ticket at the gate. 'It's yourself. Back from the varsity, are you?'\n\n'Aye, Mr Calder; that's me done with it.'\n\n'You'll be coming back then, will you?'\n\n'Aye, maybe. We'll see.'\n\n'Indeed. Well, I'll tell you now; your sister was here earlier, but wi' the train bein late an that ...'\n\n'Ach, it's not far to walk.'\n\n'Indeed not, though I'll be shutting up shop very soon now, and I could offer you a lift on the back of my bike if you liked.'\n\n'I'll just walk, thank you.'\n\n'As you will, Kenneth. It's good to see you back.'\n\n'Thank you.'\n\n'Ah ... that might be her, actually ...' Mr Calder said, looking down the curve of the station approach. Kenneth heard a car engine, and then headlights swung white light across the iron railings holding the rhododendrons back from the tarmac road.\n\nThe big Super Snipe growled into the car park, heeling as it turned and stopping with the passenger's door opposite Kenneth. 'Hello again, Mr Calder!' a voice called out from the driver's seat.\n\n'Evening, Miss Fiona.'\n\nKenneth threw his bags onto the back, settled into the passenger seat and accepted a kiss from his sister. He was pressed back into the seat as the Humber accelerated off down the road.\n\n'Okay, big brother?'\n\n'Just grand, sis.' The car skidded briefly as it swung onto the main road. He clutched at the grab handle on the door pillar, looked at his sister, sitting hunched over the big steering wheel, dressed in slacks and blouse, her fair hair tied back. 'You have passed your test, haven't you, Fi?'\n\n'Course I have.' A car, coming in the opposite direction, honked at them and flashed its lights. 'Hmm,' she said, frowning.\n\n'Try the dip switch.'\n\n'Ah hah.'\n\nThey swept off the main road and into the house drive, roared up between the dark masses of the oaks. Fiona took the car grinding over the gravel, past the old stable block and round the side of the house. He looked back over his shoulder. 'Is that a wall?'\n\nFiona nodded as she brought the car to a halt in front of the house. 'Dad wants a courtyard, so he's building a wall by the stables,' she said, turning off the engine. 'We're going to have a conservatory overlooking the garden, if mum has her way, which I dare say she will. I think your room's all right, but Hamish's is being redecorated.'\n\n'Heard from him?'\n\n'Getting on famously with the piccaninnies, apparently.'\n\n'Fi; really. They're Rhodesians.'\n\n'They're little black Rhodesians and I shall always think of them as piccaninnies. Blame Enid Blyton, say I. Come on, Uncle Joe; you're just in time for supper.'\n\nThey got out; there were lights on in the house, and a couple of bikes lying against the steps curving up to the front door. 'Whose are those?' he asked, taking his bags from the back of the car.\n\n'Couple of lassies camping over there,' Fiona pointed, and he could just make out a dim orange shape, lit from inside, under the elms on the west side of the lawn.\n\n'Friends of yours?'\n\nFiona shook her head. 'No; just turned up, asked to camp; think they thought we were a farm. They're from Glasgow, I think.' She took his briefcase from him and bounded up the steps to the opened double doors of the porch. He hesitated, reached into the car and took the keys out of the ignition, then glanced at the tent. 'Ken?' Fiona called from the door.\n\nHe made a tutting noise and put the keys back, then shook his head and pulled them out again. Not because there were strangers around, and certainly not just because they were from Glasgow, but just because it was irresponsible to leave keys in the car like that; Fiona had to learn. He pocketed the keys and picked up his bags. He glanced over at the tent, just as it flared with light.\n\n'Oh!' he heard Fiona say.\n\nAnd that was when he first saw Mary Lewis, running out of a tent in her pyjamas with her hair on fire, screaming.\n\n'Christ!' He dropped the bags, ran across the gravel drive towards the girl haring across the grass, hands beating at the blue and orange flames crackling round her head. He leapt down to the lawn, pulling off his jacket as he went. The girl tried to run past him; he tackled her, bringing her down with a ragged thump; he had the jacket over her head before she properly started struggling. After a few seconds, while she whimpered, and the stink of burning hair filled his nostrils, he pulled the jacket away. Fiona came running; another girl, dressed in too-big pyjamas and a fawn duffle coat, and holding a small flat kettle, followed her from the house, wailing.\n\n'Mary! Oh, Mary!'\n\n'Nice tackle, Ken,' Fiona said, kneeling by the girl with the burned hair, who was sitting quivering. He put one arm round her shoulders. The second girl fell to her knees and put both arms round the girl she'd called Mary.\n\n'Oh, hen! Are you all right?'\n\n'I think so,' the girl said, feeling what was left of her hair, and then burst into tears.\n\nHe extracted his arm from between the two girls. He brushed his jacket free of grass and burned hair, and put it round the shoulders of the crying girl.\n\nFiona was pulling bits of hair away and peering at her scalp in the gloom. 'Think you've been lucky, lassie. But we'll call the doctor anyway.'\n\n'Oh no!' the girl wailed, as though this was the worst thing in the world.\n\n'Now, now, Mary,' the other girl said, her voice shaking.\n\n'Come on, let's get into the house,' Kenneth said, rising. 'Take a look at you.' He helped the two girls to their feet. 'Maybe get you a cup of tea, eh?'\n\n'Oh, that's what caused all this in the first place!' Mary said, standing pale and shaking, eyes bright with tears. She gave a sort of desperate laugh. The other girl, still hugging her, laughed too. He smiled, shaking his head. He looked into the girl's face, finally seeing it properly, and thought how bizarrely beautiful she looked, even with half a head of frizzy, whitened hair, and eyes red raw with crying.\n\nThen he realised he was seeing her \u2013 and seeing her better all the time \u2013 in the light of a flickering glow that was blooming in the west of the garden, under the elms. Her eyes widened as she looked past him. 'The tent!' she howled. 'Oh _no!'_\n\n'And I missed it! Damn damn damn! I hate going to bed this early!'\n\n'Shush. I've told you; now go to sleep.'\n\n'No! What happened next? Did you have to take all her clothes off and put her to bed?'\n\n'No! Don't be ridiculous! Of course not!'\n\n'Oh. That's what happened in this book I read. 'Cept the girl was wet from being in the sea ... she's fallen in the water!' Rory completed the latter part of this sentence in his Bluebottle voice. 'She's fallen in the water!' the wee voice said again, in the darkness of the room.\n\nKenneth wanted to laugh, but stopped himself. 'Please shut up, Rory.'\n\n'Go on; tell me what happened next.'\n\n'That's it. We all came into the house; mum and dad hadn't even heard anything. I got the hose going eventually but by that time it was too late to save much of the stuff in the tent; and anyway then the primus really blew up, and \u2013'\n\n'What? In an explosion?'\n\n'That's the way things normally blow up, yes.'\n\n'Holy smoke! Oh damn, hell and shite! I missed it.'\n\n'Rory; mind your language!'\n\n'Weeeellll.' Rory turned over in the bed, his feet prodding Kenneth in the back.\n\n'And mind your feet, too.'\n\n'Sorry. So did the doctor come or not?'\n\n'No; she didn't want us to call him, and she wasn't badly hurt; just her hair, really.'\n\n'Waa!' Rory gave a squeal of excitement. 'She's not bald, is she?'\n\n'No, she isn't bald. But she'll probably have to wear a scarf or something for a while, I expect.'\n\n'So they're staying in the house, are they? These two lassies from Glasgow? They're in the house?'\n\n'Yes, Mary and Sheena are staying in my room, which is why I've got to sleep with you.'\n\n'Ffworr!'\n\n'Rory, shut up. Go to sleep, for Pete's sake.'\n\n'Okay.' Rory made a great bouncing movement, turning over in bed. Kenneth could feel his brother lying still and tense beside him. He sighed.\n\nHe remembered when this had been his room. Before his dad had unblocked the fireplace and put a grate in it, the only heating during the winter had been that ancient paraffin heater they hadn't used since the old house, back in Gallanach. How nostalgic he had felt then, and how distant and separated from Gallanach at first, even though it was only eight miles away over the hills, and just a couple of stops on the train. That heater had been the same height as him, at first, and he'd been told very seriously never ever to touch it, and been slightly frightened of it at the start, but after a while he had grown to love the old enamelled heater.\n\nWhen it was cold his parents would put it in his room to heat it up before he went to bed, and they would leave it on for a while after they'd said good-night to him, and he'd lie awake, listening to the quiet, puttering, hissing noise it made, and watching the swirling pattern of flame-yellow and shadow-dark it cast on the high ceiling, while the room filled with a delicious warm smell he could never experience after that without a sense of remembered drowsiness.\n\nIt had been a precious light, back then; must have been during the war at first, when his dad was using the probably illegal stockpile of paraffin he'd built up before rationing began.\n\nRory nudged him with one foot. He ignored this.\n\nHe ignored another, slightly stronger nudge, and started snoring quietly.\n\nAnother nudge.\n\n'What?'\n\n'Ken,' Rory whispered. 'Does your tassel get big sometimes?'\n\n'Eh?'\n\n'You know; your tassel; your willy. Does it get big?'\n\n'Oh, good grief,' he groaned.\n\n'Mine does. It's gone big now. Do you want to feel it?'\n\n'No!' he sat up in the bed, looking down at the vague shape of his brother's head on the pillow at the other end of the bed. 'No, I do not!'\n\n'Only asking. Does it, though?'\n\n'What?'\n\n'Your willy; get big?'\n\n'Rory, I'm tired; it's been a long day, and this isn't the time or the place \u2013'\n\nRory sat up suddenly. 'Bob Watt can make stuff come out of his; and so can Jamie McVean. I've seen them do it. You have to rub it a lot; I've tried but I can't get any stuff to come out, but twice now I've got this funny feeling where it's like heat; like heat coming up as if you're getting into a bath, sort of. Do you get that?'\n\nKenneth sighed, rubbed his eyes, rested his back against the low brass rail at the foot of the bed. He drew his legs up. 'I don't think it's really up to me to have to go into all this, Rory. You should talk to dad about it.'\n\n'Rab Watt says it makes you go blind.' Rory hesitated. 'And he wears glasses.'\n\nKenneth stifled a laugh. He looked up at the dim roof, where dozens of model aircraft hung on threads and whole squadrons of Spitfires and Hurricanes and ME 109s attacked Wellingtons, Lancasters, Flying Fortresses and Heinkels. 'No, it doesn't make you go blind.'\n\nRory sat back, legs drawn up too. Kenneth couldn't make out his brother's expression; there was a soft glow from the small nightlight candle on Rory's desk, near the door, but it was too weak to let him see the boy's face clearly.\n\n'Ha; I told him he was wrong.'\n\nKenneth lay back down. Rory said nothing for a while. Then Rory said, 'I think I'm going to fart.'\n\n'Well, you'd better make damn sure it goes out the way.'\n\n'Can't; got to keep it under the covers or it might ignite on the nightlight and blow the whole house up.'\n\n'Rory; shut up. I'm serious.'\n\n'... 'sail right.' Rory turned over, settled down. 'It went away.' There was silence for some time. Ken fitted his legs round Rory's back, closed his eyes, and wished that his father had concentrated on restoring more rooms in the old house rather than building courtyard walls.\n\nAfter a while, Rory stirred again and said sleepily, 'Ken?'\n\n'Rory; please go to sleep. Or I'll kick you unconscious.'\n\n'No, but Ken?'\n\n'Whaaat?' he breathed. _I should have beaten him up when we were younger; he isn't scared of me at all._\n\n'Have you ever shagged a woman?'\n\n'That's none of your business.'\n\n'Go on; tell us.'\n\n'I'm not going to.'\n\n'Please. I won't tell anybody else. Promise. Cross my heart and hope to die I won't.'\n\n'No; go to sleep.'\n\n'If you tell me, I'll tell you something.'\n\n'Oh, I'm sure.'\n\n'No, really; something dead important that nobody else knows.'\n\n'I'm not buying it, Rory. Sleep or die.'\n\n'Honest; I've never told anybody, and if I do tell you you mustn't tell anybody else, or I might get put in the jail.'\n\nKenneth opened his eyes. _What's the kid talking about?_ He turned over, looked to the head of the bed. Rory was still lying ' down. 'Don't be melodramatic, Rory. I'm not impressed.'\n\n'It's true; they'd put me in jail.'\n\n'Rubbish.'\n\n'I'll tell you what I did if you tell me about shagging.'\n\nHe lay there, thought about this. Apart from anything else, the horrible and ghastly truth was that at the ripe old age of practically twenty-two, he had never made love to a woman. But of course he knew what to do.\n\nHe wondered what Rory's secret was, what he thought he had done, or what story he had made up. They were both good at making up stories.\n\n'You tell me first,' Kenneth said, and felt like a child again.\n\nTo his surprise, Rory said, 'All right.' He sat up in bed, and so did Kenneth. They waggled closer until their heads were almost touching, and Rory whispered, 'You remember last summer, when the big barn burned down on the estate?'\n\nKenneth remembered; it had been the last week of his vacation, and he had seen the smoke rising from the farm, a mile away along the road towards Lochgilphead. He and his dad had heard the bell sound in the ruined estate chapel, and had jumped into the car, to go and help old Mr Ralston and his sons. They'd tried to fight the fire with buckets and a couple of hoses, but by the time the fire engines arrived from Lochgilphead and Gallanach the old hay barn was burning from end to end. It stood not far from the railway line, and they'd all assumed it had been a spark from an engine.\n\n'You're not going to tell me \u2013'\n\n'That was me.'\n\n'You're joking.'\n\n'Promise you won't tell, please? Please please please? I've never told anybody and I don't want to go to jail, Ken.'\n\nRory sounded too frightened to be lying. Kenneth hugged his young brother. The boy shivered. He smelled of Palmolive.\n\n'I didn't mean to do it, Ken, honest I didn't; I was experimenting with a magnifying glass; there was this wee hole in the roof, and this beam of sunlight, and it was like a sort of searchlight falling on the straw, and I was playing with my Beaufighter; not the Airfix one, the other one, and I was melting holes in the wings and fuselage 'cos they look dead like bullet holes and you can melt a big long line of them and they look like twenty millimetre cannon holes, and I pretended the sunshine really was a sort of searchlight, and the plane crashed, and I'd thought I'd see if I could make the straw go on fire, just a little bit, round where the plane had crashed, but I didn't think it would all burn down, really I didn't; it just all went up dead sudden. You won't tell, will you, Ken?'\n\nRory pulled back, and Kenneth could just make out the boy's eyes, shining in the gloom.\n\nHe hugged him again. 'I swear; on my life. I'll never tell anybody. Ever.'\n\n'The farmer won't have to sell his car to buy a new barn, will he?'\n\n'No,' he laughed quietly. 'It's old Urvill's farm anyway, really, and being a good capitalist, I'm sure he had it well insured.'\n\n'Oh... okay. It was an accident, honest it was, Ken. You won't tell Mr Urvill, will you?'\n\n'Don't worry; I won't. It was only a barn; nobody hurt.'\n\n'It was an accident.'\n\n'Sssh.' He held the boy, rocked him.\n\n'I was that frightened afterwards, Ken; I was going to run away, so I was.'\n\n'There now; sssh.'\n\nAfter a while, Rory said, groggily, 'Going to tell me about shagging, Ken, eh?'\n\n'Tomorrow, all right?' he whispered. 'Don't want you getting all excited again.'\n\n'You promise?'\n\n'I promise. Lie back; go to sleep.'\n\n'Mmmm. Okay.'\n\nHe tucked the boy in, then looked up at the dull crosses of the planes, poised overhead. Young rascal, he thought.\n\nHe lay back himself, toyed briefly with his own erection, then felt guilty and stopped. He closed his eyes and tried to sleep, but couldn't stop thinking of the girl whose hair had gone on fire. He'd seen quite far down her pyjama top when he'd put his arm round her shoulders.\n\nHe forced himself to stop thinking about her. He reviewed the day, the way he often had since childhood, trying to fill the time between the light going out and his brain finally relaxing, letting him go to sleep.\n\nWell, so much for his plan to tell his parents as soon as he got home that he too wanted to travel, that he didn't want to stay here, or get a job at the factory, managerial or not, or become a teacher like Hamish. Maybe something settled and bourgeois like that could come later, but he wanted to taste the world first; there was more to it than this wee corner of Scotland, more to it than Glasgow and even Britain. The world and his life were opening up before him and he wanted to take full advantage of both (apart from anything else, there was always the Bomb, that lurking presence forever threatening to close it all back down again with one final, filthy splash of light that heralded the long darkness, and made a nonsense of any human plan, any dream of the future. Eat, drink and be merry, because tomorrow we blow up the world).\n\nHe had intended to tell his parents all this as soon as he got in, but the incident with the girls and their tent and that poor, shocked, bonny lassie with her hair on fire had made it impossible. It would have to wait until tomorrow. There would be time. There was always time.\n\nHe wondered what her skin would feel like. It had been the colour of pale honey. He wondered what it would feel like to hold her. He had touched her \u2013 he had been sprawled on top of her, dammit \u2013 but that wasn't the same thing, not the same thing at all. She had been slim, but her breasts, soft globes within the shadows of those silly pyjamas, had looked full and firm. There had been something fit and limber about the way she'd moved, even when she'd been shivering after her ordeal. He would have believed she was an athlete, not a student of \u2013 what had she said? \u2013 geography. He smiled in the darkness, touching himself again. He'd like to study her geography, all right; the contours of her body, the swelling hills and deep dales, her dark forest and mysterious, moist caves ...\n\nThe girls stayed at Lochgair for another six days. The McHoans were used to keeping open house, and wouldn't hear of the girls just packing up what was left of their possessions and cycling or taking a train back to Glasgow.\n\n'Och, no; you must stay,' Margot McHoan said, at breakfast the next morning. They were all sat round the big table; Mary with a towel round her head, looking prettily embarrassed, her friend Sheena, big-boned, blonde and apple-cheeked, happily wolfing down sausage and eggs, Fiona and Kenneth finishing their porridge, Rory searching for the plastic toy concealed somewhere in the Sugar Smacks packet. Dad had left for the glass factory earlier.\n\n'Oh, Mrs McHoan, we couldn't,' Mary said, looking down at the table. She had only nibbled at her toast.\n\n'Nonsense, child,' Margot said, pouring Rory another glass of milk and smoothing the _Herald_ on the table in front of her. 'You're both very welcome to stay, aren't they?' She looked round her three children.\n\n'Certainly,' Fiona said. She had already found Sheena to be a kindred spirit when it came to Rock 'n Roll, which might provide her with a valuable ally when it came to displacing dad's folk songs and Kenneth's jazz on the turntable of the family radiogram.\n\n'Of course.' Kenneth smiled at Mary, and at Sheena. 'I'll show you around, if you like; much better to have a local guide, and my rates are very reasonable.'\n\n'Muuuum, they've forgotten to put the wee boat in this box,' Rory complained, arm deep in the Sugar Smacks packet, face dark with frustration and ire.\n\n'Just keep looking, dear,' Margot said patiently, then looked back at the two girls. 'Aye; stay by all means, the two of you. This big house needs filling up, and if you feel guilty you can always help with a bit of decorating, if there's any wet days, and if my husband gets round to it. Fair enough?'\n\nKenneth glanced at his mum. Margot McHoan was still a striking-looking woman, though her thick brown hair was starting to go grey over her forehead (she had dyed it at first, but found it not worth the bother). He admired her, he realised, and felt proud that she should be so matter-of-factly generous, even if it might mean that he had to keep sleeping in the same bed as his young brother.\n\n'That's awful kind, Mrs McHoan,' Sheena said, wiping her plate with a bit of fried bread. 'Are you sure?'\n\n'Totally,' Margot said. 'Your parents on the phone?'\n\n'Mine are, Mrs McHoan,' Mary said, glancing up.\n\n'Good,' Margot said. 'We'll call them, tell them you'll be here, all right?'\n\n'Oh, that's awfully nice of you, Mrs McHoan,' Mary said, and flickered a wee nervous smile at the older woman. Kenneth watched her and the smile ended up, albeit briefly, directed at him, before Mary looked down, and crunched into her toast and marmalade.\n\nHe drove the two girls round the area in the Humber when his dad wasn't using it; sometimes Fiona came too. The summer days were long and warm; they walked in the forests south of Gallanach, and in the hills above Lochgair. A puffer captain let them travel through the Crinan canal on his boat, and they took the family dory puttering over to Otter Ferry for lunch one day, over the smooth waters of Lower Loch Fyne, one windless day when the smoke rose straight, and cormorants stood on exposed rocks, wings held open like cloaks to the warm air, and seals popped up, black cones of blubber with surprised-looking faces, as the old open boat droned slowly past.\n\nThere was a dance on in Gallanach Town Hall that Saturday, the day before the two girls were due to return to Glasgow; Kenneth asked Mary to go with him. She borrowed one of Fiona's dresses, and a pair of his mother's shoes. They danced, they kissed, they walked by the quiet harbour where the boats lay still on water like black oil, and they sauntered hand-in-hand along the esplanade beneath a moon-devoid sky full of bright stars. They each talked about their dreams, and about travelling to far-away places. He asked if she had given any thought of maybe coming back here some time? Like next weekend, for example?\n\nThere is a loch in the hills above Lochgair; Loch Glashan, reservoir for the small hydro power station in the village. Matthew McHoan's friend, Hector Cardle, a Forestry Commission manager, kept a rowing boat on the loch, and the McHoans had permission to use the boat, to fish the waters.\n\nRory was bored. He was so bored he was actually looking forward to school starting again next week. Back in the spring, he had hoped that Ken being back home would make the summer holidays fun, but it hadn't worked out that way; Ken was either up in Glasgow seeing that Mary girl, or she was here, and they were together all the time and didn't want him around.\n\nHe had been in the garden, throwing dry clods of earth at some old model tanks; the clouds of dust the clods made when they hit the hard, baked earth looked just like proper explosions. But then his mum had chased him out because the dust was getting the washing dirty. He hadn't found anybody else around to play with in the village, so he'd watched a couple of trains pass on the railway line. One was a diesel, which was quite exciting, but he'd soon got bored there, too; he walked up the track by the river, up to the dam. It was very warm and still. The waters of the loch were like a mirror.\n\nHe walked along the path between the plantation and the shore of the loch, looking for interesting stuff. But you usually only found that sort of thing down at the big loch. There was a rowing boat out in the middle of the little loch, but he couldn't see anybody in it. He was banned from making rafts or taking boats out. Just because he'd got a bit wet a few times. It was unfair.\n\nHe sat down in the grass, took out a little die-cast model of a Gloster Javelin, and played with it for a while, pretending he was a camera, tracking the plane through the grass and over the pebbles and rocks by the loch side. He lay back in the grass, looked at the blue sky, and closed his eyes for a long time, soaking up the pink-ness behind his eyelids and pretending he was a lion lying tawny and sated under the African sun, or a sleepy-eyed tiger basking on some rock high over a wide Indian plain. Then he opened his eyes again and looked around, at a world gone grey, until that effect wore off. He looked down at the shore; little waves were lapping rhythmically at the stones.\n\nHe watched the wavelets for a while. They were very regular. He looked along the nearby stretch of shore. The waves \u2013 hardly noticeable, but there if you looked \u2013 were coming ashore all along the lochside. He followed the line they seemed to indicate, out to the little rowing boat near the middle of the loch. Now he thought about it, it was _very_ odd that there was nobody in the boat. It was moored; he could see the wee white buoy it was tied to. But there was nobody visible in the boat.\n\nThe more carefully he looked, the more certain he became that it was the rowing boat that all these little, rhythmic waves were coming from. Hadn't Ken and Mary been going fishing today? He had thought they'd meant sea-fishing, in Loch Fyne, but maybe he hadn't been paying attention. What if they had been fishing from the rowing boat and fallen overboard and both been drowned? Maybe that was why the boat was empty! He scanned the surface of the loch. No sign of bobbing bodies or any clothing. Perhaps they'd sunk.\n\nAnyway, what was making the boat make those waves?\n\nHe wasn't sure, but he thought he could see the boat moving, very slightly; rocking to and fro. Maybe it was a fish, flopping about in the bottom.\n\nThen he thought he heard a cry, like a bird, or maybe a woman. It made him shiver, despite the heat. The boat seemed to stop rocking, then moved quite a lot, and then went totally still. The little waves went on, then a few slightly bigger, less regular ones lapped ashore, then the water went still, and was as flat as a pane of glass.\n\nA gull, a white scrap across the calm sky, flapped lazily just above the blue loch; it made to land on the prow of the little rowing boat, then at the last second, even as its feet were about to touch, it suddenly burst up into the sky again, all panic and white feathers, and its calls sounded over the flat water as it flapped away.\n\nWhat sounded very like laughter came from the little rowing boat.\n\nRory shrugged, put the model plane in the pocket of his shorts and decided to go back down to the village and see if there was anybody around to play with yet.\n\nKenneth and Mary held hands at tea that evening, and said they wanted to get married. Mum and dad seemed quite happy. Fiona didn't seem in the least surprised. Rory was nonplussed.\n\nIt was years before he made the connection between those tiny, rhythmically lapping waves, and that blushing, excited announcement.\nCHAPTER 3\n\nGaineamh Castle, home of the Urvills once again, stands amongst the alders, rowans and oaks that cover the northern flanks of the Cnoc na Moine, due south of the carbuncular outcrop that supports the First Millennium fort of Dunadd, and a little north-west of the farm rejoicing in the name of Dunamuck. The castle, a moderately large example of the Scottish Z-plan type, with cannon-shaped stone waterspouts, has a fine view through the trees and across the parkland and fields to the town of Gallanach, which spreads round the deep waters of Inner Loch Crinan like some slow but determined beach-head of architecture somehow landed from the sea.\n\nThe sound of gravel crunching beneath a car tyre has always meant something special to me; at once comforting and exciting. Of course the one time I tried to explain this to my father he suggested that what it really signified was the easy rolling pressure the middle and upper classes thought it was their right to exert upon the multitudinous base of the workers. I have to confess that the entire counter-revolution in world affairs has come as something of a personal relief to me, making my dad seem no longer quite so remorselessly well-clued-up, but rather \u2013 if anything, any more \u2013 just quaint. It would have been sweet to tackle him on that subject at the time, especially given that Gorby's unleashed restructuring had just resulted in the spectacular and literal deconstruction of one of the age's most resonantly symbolic icons, but at the time we weren't talking.\n\n'Prentice,' rumbled the slightly bloated Urvill of Urvill, taking my hand and briefly shaking it, as if weighing my mitt. I felt for a moment the way a young bull ought to feel when the man from McDonalds slaps its haunch ... but then probably doesn't. 'So very sorry.' Fergus Urvill said. I wondered whether he was referring to Grandma Margot's death itself, her detonation, or Doctor Fyfe's apparent attempt to up-stage the old girl. Uncle Fergus let my hand go. 'And how are your studies going?'\n\n'Oh, just fine,' I said.\n\n'Good, good.'\n\n'And the twins; are they both well?' I asked.\n\n'Fine, fine,' Fergus nodded, presumably allocating his two daughters a word each in his reply. Ferg's gaze went smoothly to my Aunt Antonia; I took the hint, and (like Margot) passed on. 'Antonia,' I heard behind me. 'So very sorry ...'\n\nHelen and Diana, Uncle Ferg's two lusciously lissom daughters, sadly couldn't be here; Diana spent most of her time either in Cambridge or the least touristy part of Hawaii, which is the bit thirty kilometres away from the beaches \u2013 four of them vertically \u2013 at the Mauna Kea observatory, studying the infra-red. Helen, on the other hand, worked for a bank in Switzerland, dealing with the ultra-rich.\n\n'Prentice, are you all right?' My mother took me in her arms, held me to her black coat. Still splashing on the _No. 5,_ by the smell of it. Her green eyes looked bright. My father had been at the head of the reception line; I had ignored him and the compliment had been returned.\n\n'I'm fine,' I told her.\n\n'No, but are you really?' She squeezed my hands.\n\n'Yes; I'm really really fine.'\n\n'Come and see us, please.' She hugged me again, said quietly, 'Prentice, this is silly. Make it up with your father. For me.'\n\n'Mum, please,' I said, feeling like everybody was looking at us. 'I'll see you later, okay?' I said, and pulled away.\n\nI walked into the hall, taking off my jacket, blinking hard and sniffing. Coming from cold into warmth always does this to me.\n\nThe entrance hall of Gaineamh Castle sports the business end of a dozen or so beheaded male red deers, perched so high up on the oak-panelled walls that attempting to utilise them for their only conceivable practical purpose in such a location \u2013 hanging coats, scarves, jackets, etc. on their impressively branched antlers \u2013 only exposes them as the venue for a kind of non-returnable sport rather than a sensible amenity. Rather more prosaic brass hooks, like smooth unsuitable claws beneath the glass-eyed stares of the stags, accepted our garments in their stead. My much be-zippered black leather pretend-biker's jacket seemed a little out of place amongst the sober wools and furs; Verity's snow-white skiing jacket looked ... well, just sublime. I stood and stared at it for a second or two longer than was probably fit; but it really did seem to glow in the dark company. I sighed, and decided to keep my white silk M\u00f6bius scarf on.\n\nI entered the hammer-beamed Solar of the castle; the great hall was filled with a quietly chattering crowd of McHoans, Urvills and others, all nibbling canapes and vol-au-vants, and sipping whisky and sherries. I suspect my grandmother would have preferred pan-loaf sarnies and maybe a few slices of ham-and-egg pie, but it had, I suppose, been a kind gesture of the Urvill to ask us back here, and one should not carp. Somehow the McHoan home, still bearing the scars of grandma's sudden, unorthodox and vertical re-entry into the conservatory following her abortive attempt to de-moss the gutters, seemed unfitting as our post-cremation retreat.\n\nThere! I caught sight of Verity, standing looking out of one of the Solar's tall mullioned windows, the wide grey light of this chill November day soft upon her skin. I stopped and looked at her, a hollowness in my chest as though my heart had become a vacuum pump.\n\nVerity: conceived beneath a tree two millennia old and born to the flare and snap of human lightning. Emerging to emergency, making her entrance, and duly entrancing.\n\nWhistling or humming the first phrase of Deacon Blue's _Born In A Storm_ whenever I saw her had become a sort of ritual with me, a little personal theme in the life lived as movie, existence as opera. See Verity; play them tunes. It was in itself a way of possessing her.\n\nI hesitated, thought about going over to her, then decided I'd best get a drink first, and started towards the sideboard with the glasses and bottles, before I realised that offering to refresh Verity's glass would be as good a way as any of getting talking to her. I turned again. And almost collided with my Uncle Hamish.\n\n'Prentice,' he said, in tones of great import and sobriety. He put one hand on my shoulder and we turned away from the window where Verity stood, and away from the drinks, to walk up the length of the hall towards the stained-glass height of the gable-end window. 'Your grandmother has gone to a better place, Prentice,' Uncle Hamish told me. I looked back at the vision of wonderful-ness that was Verity, then glanced at my uncle.\n\n'Yes, Uncle Hamish.'\n\nDad called Uncle Hamish 'The Tree' because he was very tall, moved in a rather awkward way \u2013 as though made out of something less flexible than the standard issue of bone, sinew, muscle and flesh \u2013 and (so he claimed, at any rate) because he had seen him act in a school play once, and he had been very, well, wooden. 'Anyway,' my dad had insisted when he'd originally confided this private piece of nomenclature, only half a decade earlier, on the occasion of my sixteenth birthday, when we'd got drunk together for the first time, 'he just lumbers about!'\n\n'She was a good woman, and did little that was bad and much that was good, so I'm sure she has gone to a reward rather than a punishment, living amongst our anti-creates.'\n\nI nodded, and as we strolled amongst them, looked around at the various members of my family, the McGuskies (Grandma Margot's maiden-family), the Urvill clan, and sundry worthies from Gallanach, Lochgilphead and Lochgair, and pondered, not for the first time, what on Earth (or anywhere else for that matter) had given Uncle Hamish the idea for his bizarre, home-made religion. I really didn't want to go into all this right now, and anyway found the whole subject a little awkward, because I wasn't actually quite as gung-ho for Hamish's personal theology as he seemed to think I was.\n\n'She was always very kind to me,' I told him.\n\n'And therefore your anti-create will be kind to her,' Uncle Hamish said, still with one hand on my shoulder, as we stopped and looked up at the stained-glass monstrosity at the far end of the hall. This showed in graphic form the story of the Urvills from about the time of the Norman conquest, when the family of Urveille, from Octeville in Cotentin, had crossed into England, percolated northwards, swirled briefly around Dunfermline and Edinburgh, and finally come to rest \u2013 perhaps afflicted by some maritime memory of their ancestral lands on the seam of the Manche \u2013 in what had been the very epicentre of the ancient Scots kingdom of Dalriada, losing only a few relatives and a couple of letters on the way. Swearing allegiance to David I, here they have stayed, to mingle their blood with that of the Picts, the Scots, the Angles, the Britons and the Vikings who have all variously settled, colonised, raided and exploited this part of Argyll, or maybe just arrived at one time and forgotten to leave again.\n\nThe peregrinations and subsequent local achievements of the clan Urvill make interesting history, and would make fascinating viewing if the giant window telling the tale wasn't so badly done. The fashionable but untalented son of one of the previous head Urvill's school pals had been commissioned to execute the work, and had taken the brief all too literally. Deadly dull and eye-squintingly garish at the same time, the stained glass window made me want to grit my teeth.\n\n'Yes, I'm sure you're right, uncle,' I lied.\n\n'Of course I am, Prentice,' he nodded slowly. Uncle Hamish is balding, but of the school that believes long wisps of hair grown on one side of the head and then combed delicately across the pate to the other edge look better than naked skin exposed to the elements. I watched the coloured light from the stained glass window slide over shiny skin and hardly less luminescent oiled hair, and thought what a prat he looked. I inadvertently found myself humming the appropriate piece of music from the Hamlet cigar adds and thinking of Gregor Fisher.\n\n'Will you join me in worship this evening, Prentice?'\n\nOh shit, I thought. 'Perhaps not, actually, uncle,' I said, in tones I hoped sounded regretful. 'Have to pop down the Jac to talk to a girl about a jacuzzi. Probably go straight from here.' Another lie.\n\nUncle Hamish looked at me, the grain-like lines on his forehead bunching and tangling, his brown eyes like knots. 'A _jacuzzi,_ Prentice?' He pronounced the word the way the lead in a Jacobean tragedy might pronounce the name of the character who has been his nemesis.\n\n'Yes. A jacuzzi.'\n\n'That's a form of bath, isn't it?'\n\n'It is.'\n\n'Not meeting this young lady _in_ a bath, are you, Prentice?' Uncle Hamish's lips twisted slowly into what was probably meant to be a smile.\n\n'I don't believe the facilities of the Jacobite Bar run to such a thing, uncle,' I told him. 'They've only recently got round to installing hot water in the gents. The relevant jacuzzi is in Berlin.'\n\n'The German city?'\n\nI thought about this. Could I have mis-heard Ash and she have been talking about the briefly famous chart-topping band of the same name? I thought not. 'Yes, uncle; the city. Where the wall was.'\n\n'I see,' Uncle Hamish nodded. 'Berlin.' He stared up at the violently clashing leaden imagery of the great stained-glass window. 'Isn't that where Ilsa is?'\n\nI frowned. 'Aunt Ilsa? No, she's in Patagonia, isn't she? Incommunicado.'\n\nUncle Hamish looked suitably confused as he contemplated the garish gable glass. Then he nodded. 'Ah yes. Of course.' He looked back down at me. 'However. Shall we see you for supper, Prentice?'\n\n'I don't know,' I admitted. 'Just as likely to end up with a kebab, I imagine. Or a fish supper.'\n\n'Well, you have your key with you?'\n\n'Oh yes. Thanks. And I'll be ... you know; quiet, when I come in.'\n\n'Right.' Uncle Hamish gazed back up at the crass glass. 'Right. We'll probably be off in a half-hour or so; let us know if you do want a lift.'\n\n'Surely.'\n\n'Right you are, then.' Uncle Hamish nodded, turned, then looked back with an intensely puzzled expression. 'Did I hear somebody say mother _exploded?'_\n\nI nodded. 'Pacemaker. That's what Doctor Fyfe was rushing to tell us; told dad in the ambulance. But it was too late by then, of course.'\n\nUncle Hamish looked more baffled than ever, but nodded eventually and said, 'Of course,' and walked off over the parquet with a startlingly tree-like creaking noise which I realised \u2013 with a small but welcome surprise \u2013 was issuing from his black brogues.\n\nI made straight for the sideboard with the drinks, but a quick inspection of the casement of the relevant window on my way there revealed that Verity the Comely had gone.\n\nFortingall is a modest hamlet in the hills north of Loch Tay, and it was there in the winter of 1969 that my Aunt Charlotte was determined to consummate her marriage. Specifically, she wanted to be impregnated beneath the ancient yew tree that lies in an enclosure within the graveyard of the small church there; she was convinced that the tree \u2013 two thousand years old, according to reliable estimates \u2013 must be suffused with a magical Life Force.\n\nIt was a dark and stormy night (no; really), the grass under the ancient, straggling, gnarled yew was sodden, and so she and her husband, Steve, had to settle for a knee-trembler while Charlotte held onto one of the overhanging boughs, but it was there and then \u2013 despite the effects of gravity \u2013 that the gracile and quiveringly prepossessing Verity was conceived, one loud night under an ink black sky obscuring a white full moon, at an hour when all decent folk were in their beds and even the indecent ones were in somebody's, in a quaint little Perthshire village, back in the fag end of the dear old daft old hippy days.\n\nSo my aunt says, and frankly I believe her; anybody wacko enough ever to have bought the idea that there was some sort of weird cosmic energy beaming out of a geriatric shrub in a back-end-of-nowhere Scottish graveyard on a wet Monday night probably hasn't the wit to lie about it.\n\n'Naw, she's great, I mean really _really_ great. I'm in love. I love her; I'm hers. Verity; take me; put me out of my misery. O God ...'\n\nI was drunk. It was getting on towards midnight in the Jacobite bar and at my normal rate of drinking that meant I'd had about ten pints of export. Ash and Dean Watt, and another couple of old pals, Andy Langton and Lizzie Polland, had all drunk about the same as I had, but then they'd been home for their tea and they hadn't been swilling back the Urvill's whisky for a significant part of the afternoon.\n\n'So have you told her, Prentice?' Ash said, putting down another set of pints on the pocked copper table we were hunched around.\n\n'Ah, Ash,' I said, slapping the table. 'I admire a woman who can carry three pints at the same time.'\n\n'I said, have you told this lassie you love her, Prentice?' Ash said, sitting down. She took a bottle of strong cider from one breast pocket of her navy shirt, and a glass of whisky from the other.\n\n'Wow!' I said. 'Ash! I mean, like; wow! Wicked.' I shook my head, took up my old pint and finished it.\n\n'Answer the lassie,' Dean said, nudging me.\n\n'No, I haven't,' I confessed.\n\n'Ya coward,' said Lizzie.\n\n'I'll tell her for you if you like,' Droid offered (there is an entire generation of Andrews with the shared nickname of Droid, post _Star Wars)._\n\n'Na,' I said. 'But she is just fabulous. I mean \u2013'\n\n'Why not tell her?' Liz asked.\n\n'I'm shy,' I sighed, hand on heart, eyes heaven-ward, lashes fluttering.\n\n'Get out a here.'\n\n'So tell her,' Ash said.\n\n'Also,' I sighed. 'She's got a boyfriend.'\n\n'Ah-ha,' Ash said, looking at her pint.\n\nI waved one hand dismissively. 'But he's a wanker.'\n\n'That's all right, then,' Liz said.\n\nI frowned. 'Actually, that's the only flaw Verity seems to have; her lousy taste in men.'\n\n'So you _are_ in with a chance then?' Liz said brightly.\n\n'Yeah,' I said. 'I think she's going to chuck him.'\n\n'Prentice,' Ash insisted, tapping the table. _' Tell_ her.'\n\n'I can't.'\n\n'Why not?'\n\n'Because I wouldn't know how to,' I protested. 'I've never told anybody I love them before. I mean, how do you? The words sound so corny, so devalued. It's so ... it's just such a cliche.'\n\nAsh looked scornful. 'What rubbish.'\n\n'Well, smarty-pants,' I said, leaning over to her. 'Have _you_ ever told anybody you love them?'\n\n'Hundreds of times, darling.' Ash said in a deep voice, pouting. Dean guffawed. Ash drank from her pint, then shook her head. 'Well, actually, no.'\n\n'Ha!' I said.\n\nAsh leaned over to me, her long nose almost touching mine. 'Tell the girl, you idiot.'\n\n'I can't,' I said, sitting back. 'I just can't. She's too perfect.'\n\n_' What?'_ Ash frowned.\n\n'Infallible. Too perfect; ideal.'\n\n'Sounds like misogynist romantic shite to me,' snorted Liz, who's always taken a hard line on such things.\n\n'It is,' I admitted. 'But she's just incredible. D'you know where she was conceived?'\n\nDean and Ash exchanged looks; Andy spluttered into his beer while Lizzie rolled her eyes. 'Aw yeah,' Dean said, nodding and looking quite serious. 'Doesn't everybody?'\n\nI was shocked, and almost cut short my next gulp of beer. 'You don't really, do you?'\n\n'Course not, Prentice,' Ash said shaking her head. Her long fair hair spilled from over one shoulder. 'What diff \u2013'\n\n'Aw, it's just incredible,' I told them. 'Her mum told me; Aunt Charlotte. Bit of a nutter, but okay. I mean totally aff her heid really, but anyway \u2013' I took another gulp of beer, '\u2013 she had this thing about psychic energy or some crap like that ... and about Scottish history \u2013'\n\n'Aw; runs in the family, does it, Prentice?' Dean asked.\n\n'Naw; she's not a McHoan ... anyway; she'd married this English guy called Walker and they hadn't consummated the marriage, right, not on their wedding night; she wanted to wait, and when they did get it together she made sure it was in this wee village called Fortingall, right? Near Loch Tay. Thing is, she'd heard something about Fortingall being where Pontius Pilate \u2013'\n\n'Wait a minute,' Dean said. 'How long was it between them getting married and them humping?'\n\n'Eh?' I scratched my head. 'I don't know; a day or two. Oh! I mean, they'd done it before, like. It wasn't their first time or anything. It was just Aunt Charlotte's idea that it'd be more special if they hadn't done it for a while, and then did it under this tree. But they had been fucking before. I mean; good grief, this is the love generation we're talking about here.'\n\n'Right,' Dean said, apparently mollified.\n\n'Anyway; Fortingall is where some people say Pontius Pilate was born, and \u2013'\n\n'Whit?' Andy said, wiping his beard. 'Away ye go.'\n\n'So they say,' I insisted. 'His dad was in the ... shit ... the seventh legion? The ninth? Damn ...' I scratched my head again, looked down at my trainers (and thought with some relief that at least tonight I would not have the long struggle to undo the buckles and untie and then loosen the laces on the Docs, which were my usual drinking gear these days). 'Or _was_ it the seventh legion?' I pondered, still staring at my Nikes.\n\n'Never mind if it was the fuckin' foreign legion,' Droid said, exasperated. 'You're no trying to tell us Pontius fucking Pilate was born in Scotland!'\n\n'Well maybe!' I said, spreading my arms wide and almost spilling Ash's whisky. 'His dad was in the legion stationed there! Apparently! I mean, the Romans had a military camp and Pontius Pilate's pa was stationed there, maybe, and so young Pontius could have been born there! Why not?'\n\n'You're making this up,' laughed Ash. 'You're just like your dad; I remember those stories on a Sunday afternoon.'\n\n'I am not like ma dad!' I yelled.\n\n'Hey, shoosh,' Lizzie said.\n\n'Well, I'm not! I'm telling the truth!'\n\n'Aye, well,' Ash said. 'Maybe. People get born in funny places. David Byrne was born in Dumbarton.'\n\n'Anyway; Pontius Pi \u2013'\n\n'Whit?' Dean grimaced. 'The guy that wrote _Tutti Frutti?'_\n\n'Listen; Pontius \u2013'\n\n'Na; that was John Byrne,' Lizzie said. _'David_ Byrne; the guy in Talking Heads, ya heidbanger.'\n\n'Look, anyway, forget Ponti \u2013'\n\n'Anyway, it was Little Richard.'\n\n'Will you shut up? This isn't about Pon \u2013'\n\n'What? In Talking Heads?'\n\n'Shut up! I'm telling you; Po \u2013'\n\n'Na; that wrote _Tutti Frutti.'_\n\n'I give in,' I said, sitting back. I sighed, supped my export.\n\n'Aye, the song; but no the film.'\n\n'It wasnae a fillum; it was a series.'\n\n'Ah _know;_ you knew what ah meant.'\n\n'I hate these drunken, rambling conversations,' I breathed.\n\n'Aye, but I've heard worse.' Ash nodded.\n\n'Anyway, it wasnae fillum at all; it was video.'\n\n'It was naawwwt!' Dean drawled scornfully. 'Ye could _see_ it was fillum! What sort a telly have _you_ got?'\n\nI crossed my legs, crossed my arms and swivelled to look at Ash. I rubbed my rather greasy face and focused on her. 'Hi. Come here often?'\n\nAshley pursed her lips and studied the ceiling. 'Just the once,' she said, frowning at me. 'In the toilets.' She gathered my shirt lapels in her fist and pulled me close to her face. 'So who talked?'\n\n'Fnarr fnarr,' I breathed over her. Ash's face wrinkled, quite attractively, actually. But then it was late.\n\n'Hi youse,' a deep voice said, bending over us. 'Yer oan.'\n\n'On what?' I asked the very large fellow with very long hair who had spoken.\n\n'The pool table; PM and AW; that's youse, is it no?'\n\n'Shit, aye, right enough.'\n\nAsh and I went to play pool.\n\nI'd been just about to ask her about the jacuzzi in Berlin, but now didn't seem like the right time.\n\nUncle Fergus had the observatory built back in 1974 (when the heavenly Verity was four). The idea was two-fold. First of all \u2013 according to my father \u2013 Fergus wanted a bigger and better telescope than he had. Dad had a three-inch refractor in a shed in the garden at Lochgair. Fergus ordered a six-inch reflector. Also, it was a business sample. The lenses and mirror were to be made in the new Specialist Glass Division of the Gallanach Glass Works, the Urvill-owned factory which even yet provides the town with a significant proportion of its employment. Not only, therefore, would Uncle Fergus have a fascinating and unique additional feature for his not-long restored castle, it would be both an advertisement for his Glass Works _and_ tax-deductible!\n\nThe fact the telescope was a wee bit close to Gallanach itself, and so possibly prone to light pollution from the town's sodium vapour lamps, was less of a problem than it might appear; with Uncle Fergus's connections he could have the offending lamps shaded at the council's expense. So Uncle Fergus was prepared if necessary \u2013 and only selectively, of course \u2013 to dim his home town.\n\n(His niece had already bettered that; when the diminutive, bloody and bawling form of Verity Walker had appeared on the scene, the lights had actually gone out.)\n\nI'd met the sublime Verity for the first time in some years in the observatory, one coal-sack-black moonless night in 1986, a few days before I left to go to University, when I was already full of the exhilaration and fear of departure and independence, and the whole huge world seemed to be opening up before me, like some infinite blossom of opportunity and glamour. The twins had taken to having star-gazing parties in the cold, cramped hemisphere which protruded from the summit of the compact castle, and I'd arrived late after being out on the hill with little brother James during the afternoon and then suffering a delayed tea because some friends of dad's had showed up unannounced and had to be catered for.\n\n'Aye, it's yourself, Prentice,' boomed Mrs McSpadden, informatively. 'And how are you?' Mrs McSpadden was the Urvill's housekeeper; a rotundly buxom lady of perpetual middle-age with a big baw-face that gave the impression of being freshly scrubbed. She had a very loud voice and dad always told people that she hailed from Fife. A ringing noise in one's ears after a close encounter with the lady tended to enforce the impression this was literally true. 'The rest are up there. Will you take this tray up? There's coffee in these pots; you just turn the wee spot to the front here, ken, and \u2013' She lifted the corner of a heavy napkin smothering a very large plate. '\u2013 there's hot sausage rolls under here.'\n\n'Right, thanks,' I said, lifting the tray. I'd come in through the castle kitchen; entering through the main door after it had been shut for the night could be a performance. I made for the stairs.\n\n'Here, Prentice; take this scarf up to Miss Helen,' Mrs McSpadden said, flourishing the article. 'That lassie'll catch her death of cold up there one night, so she will.'\n\nI bowed my head so that Mrs S could put the scarf over my neck.\n\n'And mind them there's plenty of bread, and some chicken in the fridge, and cheese, and plenty of soup forbye, if you get hungry again.'\n\n'Right, thanks,' I repeated, and jogged carefully upstairs.\n\n'Anybody got any roach paper?'\n\nI squeezed into the brightly-lit dome of the observatory; it was about three metres in diameter, made from aluminium, the telescope took up a lot of it, and it was cold, despite a wee two-bar electric heater. A modestly proportioned ghetto-blaster was playing something by the Cocteau Twins. Diana and Helen, bundled in enormous Mongolian quilted jackets, were crouched round a small table with Darren Watt, playing cards. My elder brother, Lewis, was at the telescope. We all said our hellos. 'This is cousin Verity. Remember her?' Helen said, as she draped the scarf I'd brought her over Darren's head. Helen pointed at a cloud of smoke, and as it blew towards me and cleared I saw her.\n\nThere was a sort of cubby-hole in the non-rotating part of the observatory, built into the attic of the castle's main block. It was just a long cupboard really, but you could coorie down into it to make more space in the dome proper. Verity Walker was lying in a sleeping bag there, only her upper half protruding into the dome; she was smoking one joint and rolling another, on the cover of a pictorial atlas of the universe. 'Evening,' she said. 'Got any roach paper?'\n\n'Yeah; hi,' I said. I put the tray down, searched my pockets, pulled out some stuff. The last time I'd seen Verity Walker, maybe five or six years earlier, she'd been a scrawny tyke with a mouth full of orthodontic brace-work and a serious Shakin' Stevens habit. Now \u2013 once seen through the smoke \u2013 she had short, pure blonde hair, and a delicate, almost elfin face which tapered to an exquisite chin that looked like it had been made to be grasped lightly in three fingers and pulled closer to your lips ... well, to my lips, anyway. Her eyes were the blue of old sea-ice, and when I saw her complexion all I could think was: _Wow; Lloyd Cole city!_ Because she had perfect skin.\n\n'That'll do.' She took something from my hand. 'Thanks.'\n\n'Hey! That's a library ticket!' I grabbed it back. 'Here.' I handed her half a book token my mother had given me.\n\n'Thanks.' She started cutting it with a little pair of scissors.\n\n'It's just a tokin' token,' I told her, squatting down beside her.\n\nShe grunted with laughter, and my heart performed manoeuvres that the connecting plumbing makes topologically impossible.\n\n'All set for the big move, bro?' Lewis grinned down from the wee seat under the eye-piece of the telescope. He reached over to the table where I'd set the tray down and started pouring coffee into the mugs. My big brother has always seemed more than two years older than me; a little taller than my 1.85, and a little more thick-set, he looked bigger still at the time thanks to a beard of the burst-sofa persuasion. Back then, it was his turn to be in disgrace with my father, because he'd just dropped out of University.\n\n'Yeah, all set,' I told him. 'Found a place to stay.' I nodded at the telescope. 'Anything interesting tonight?'\n\n'Got it on the Pleiades just now. Take a look.'\n\nWe took turns star-gazing, playing cards, crouching round the little electric heater, and constructing joints. I'd brought a half bottle of whisky, and the twins had some brandy, which we used to beef up the coffee. The munchies struck again an hour or so after we'd polished off the last of the sausage rolls; the twins mounted an expedition into the depths of the castle in search of the mythical Soup Dragon (we spoke in Clanger while they were gone) and returned with a steaming tureen and a half-dozen bowls.\n\n'Where're you staying in Glasgow, Prentice?' Darren Watt asked.\n\n'Hyndland,' I said, slurping my soup. 'Lauderdale Gardens.'\n\n'Ah, that's no far from us. Going to be around on the thirtieth? We're having a party.'\n\n'Oh, ah, yeah; probably.' (Actually, I'd been going to come home that weekend, but I could juggle things.)\n\n'Ah well, come along; should be fun.'\n\n'Thanks.'\n\nDarren Watt was in his last year at Art School and \u2013 for me, at least \u2013 had been the epitome of cool since New Year two years earlier. After the bells, mum had driven Lewis and me into Gallanach; we went to a party Droid and his chums were giving. Darren had been there; blond, lean, drop-dead bone structure, and exuding style. I'd admired the looped silk scarf he'd worn over a red velvet jacket that would have looked silly on most people but in which he looked totally poised. He'd given me the scarf, and \u2013 when I'd tried to demur \u2013 explained he was growing bored with it; better it went to somebody who would appreciate it, though he hoped I'd hand it on too, if I ever tired of it.\n\nSo I took it. It was just an ordinary silk scarf, given a half twist and the ends carefully sewn together, but that, of course, made it a M\u00f6bius scarf, the very idea of which I just thought was wonderful. I thought Darren was pretty wonderful, too, and for a while wondered if maybe I was gay, too, but decided against it. In fact, a large part of the attraction of an invite to a party at Darren's place was due to the fact his flat-mates were three salivatingly attractive and reputedly enthusiastically heterosexual female arts students (I'd met them when he'd brought them to Gallanach on a day trip the previous year).\n\n'You still making models of these wave-powered hoodjie-ma-flips?' I asked him, finishing my soup. Darren was wiping his plate with a bit of bread, and I found myself copying him.\n\n'Aye,' he said, looking thoughtful. 'Looks like I've found a sponsor for the real thing, too.'\n\n'What? Really?'\n\nDarren grinned. 'Big cement company's interested; talking about a serious money grant.'\n\n'Wow! Congratulations.'\n\nFor the last eighteen months or so, Darren had been making these tenth-scale wood and plastic models of sculptures he wanted to build full size in concrete and steel one day. The idea was to construct these things on a beach; he'd need planning permission, lots of money, and waves. The sculptures were wave-powered mobiles and fountains. When a wave struck them a giant wheel would revolve, or air would be forced through pipes, producing weird, chest-shaking, cathedral-demolishing bass notes and uncanny howls and moans, or the water in the waves themselves would be channelled, funnelled, and emerge in a whale-like spout of spray, bursting from the top or sides of the sculpture. They sounded great, perfectly feasible, and I wanted to see one work, so this was good news.\n\nI went downstairs for a pee, and came back to a good-natured but confused argument. 'What do you mean, no it doesn't?' Verity said from her sleeping-bagged cubby-hole.\n\n'I mean, what is sound?' Lewis said. 'The definition is; what we hear. So if there's nobody there to hear it ...'\n\n'Sounds a bit anthro-thingy to me,' Helen Urvill said, from the card table.\n\n'But how can it fall without making a sound?' Verity protested. 'That's crazy.'\n\nI leaned over to Darren, who was sitting looking amused. 'We talking trees falling in forests?' I asked. He nodded.\n\n'You're not listening \u2013' Lewis told Verity.\n\n'Maybe you're not making a sound.'\n\n'Shut up, Prentice,' Lewis said, without bothering to look at me. 'What I'm saying is, What is a sound? If you define it as \u2013'\n\n'Yeah,' interrupted Verity. 'But if the tree hits the ground that must make the air move. I've stood near a tree when it's felled; you feel the ground shake. Doesn't the ground shake either, when there's nobody there? The air has to move; there must be ... movement, in the air; its molecules, I mean ...'\n\n'Compression waves,' I provided, nodding to Verity, and thinking about Darren's wave-powered organ-pipe coast sculptures.\n\n'Yeah; producing compression waves,' Verity said, with an acknowledging wave at me (oh, my heart leapt!). 'Which birds and animals and insects can hear \u2013'\n\n'Ah!' Lewis said. 'Supposing there aren't \u2013'\n\nWell, it got silly after that, dissolving into the polemical equivalent of white noise, but I liked the robustly common-sensical line Verity was taking. And when she was talking, of course, I got to stare at her without anybody thinking it odd. It was wonderful. I was falling in love with her. Beauty _and_ brains. Wow!\n\nMore sounds, more spliffs, more star-gazing. Lewis did his impression of a radio being tuned through various wavelengths; fingers at his lips to produce the impressively authentic between-stations noises, then suddenly putting on silly voices to impersonate a news reader, compere, quiz contestant, singer ... 'ttttrrrrsssshhhh ... reports that the London chapter of the Zoroastrians have fire-bombed the offices of the Sun newspaper for blasphemy ... zzzoooowwwaaanngggg ... athangyou, athangyou, laze an ge'men, andenow, please put your hands together for the Siamese Twins ... kkkkrrrraaasshhhwwwaaaassshhhaaa ... uh, can you eat it, Bob? Ah, no, you can't. I'm afraid the answer is; a Pot Nooddle ... bllbllbllbl ... Hey hey, we're the junkies! ... zpt!'\n\nAnd so on. We laughed, we drank more coffee, and we smoked.\n\nThe gear was black and powerful like the night; the hollow aluminium skull of the observatory tracked the 'scope's single eye slowly over the rolling web of stars, or \u2013 hand-cranked \u2013 swivelled the universe about our one fixed point. Soon my head was spinning, too. The music machine played away \u2013 far away \u2013 and when I started to understand the lyrics of a Cocteau Twins song, I knew I was wrecked. The stars shone on in mysterious galactic harmonies, constellations like symphonies of ancient, trembling light; Lewis told weird and creepy stories and bizarrely apposite jokes, and the twins \u2013 hunkered over the little card-table in their quilted jackets, their night-black hair straight and shining and framing their broad-boned beautiful faces \u2013 looked like proud Mongolian princesses, calmly contemplating creation from the ribbed dome of some fume-filled yurt, midnight-pitched on the endless rolling Asian steppe.\n\nVerity Walker \u2013 professed sceptic though she was \u2013 read my palm, her touch like warm velvet, her voice like the spoken ocean, and in her eyes each iris like a blue-white sun stationed a billion light years off. She told me I'd be sad and I'd be happy and I'd be bad and I'd be good, and I believed all of it and why not, and she told me the last part in Clanger, the tin-whistle pretend language from one of the children's programmes we'd all watched as youngsters, and she was trying to keep a straight face, and Lew and Dar and Di and Hel were snorting with laughter and even I was grinning, but I'd been singing happily along to the Cocteau Twins' other-worldly words for the past hour, and I knew exactly what she said even though she might not have known herself, and fell completely in love with her iris-blue eyes and her wheat-crop hair and her peat-dark voice and the peach-skin fuzz of infinitesimally fine hairs on her creamy skin.\n\n'What was all that stuff about Pontius Pilate, anyway?' Ash said.\n\n'Aw ...' I waved my hand. 'Too complicated.'\n\nAsh and I stood on a low little mound overlooking what had been the Slate Mine wharf, at the north-west limit of Gallanach where the Kilmartin Burn flows out of the hills, meanders without conviction, then widens to form part of Gallanach Bay before finally decanting into the deeper waters of Inner Loch Crinan. Here was where the docks had been, when the settlement had exported first coal then slate then sand and glass, before the railway arrived and a subtle Victorian form of gentrification had set in the shape of the railway pier, the Steam Packet Hotel and the clutch of sea-facing villas (only the fishing fleet had remained constant, sporadically crowded amongst its inner harbour in the stony lap of the old town, swelling, dying, burgeoning again, then falling away once more, shrinking like the holes in its nets).\n\nAshley had dragged me out here, now in the wee small hours of what had become a clear night with the stars steady and sharp in the grip of this November darkness, after the Jacobite Bar and after we'd trooped (victorious at pool, by the way) back to Lizzie and Droid's flat via McGreedy's (actually McCreadie's Fast Food Emporium), and after consuming our fish\/pie\/black pudding suppers and after a cup of tea and a J or two, and after we'd got back to the Watt family home in the Rowanfield council estate only to discover that Mrs Watt was still up, watching all-night TV (does Casey Casen _never_ sit down in that chair?), and made us more tea, and after a last wee numbrero sombrero in Dean's room.\n\n'I'm going for a walk, guys, okay?' Ash had announced, coming back from the toilet, cistern flushing somewhere in the background, pulling her coat back on.\n\nI'd suddenly got paranoid that I had over-stayed my welcome and \u2013 in some dopey, drunken excess of stupidity \u2013 missed lots of hints. I looked at my watch, handed the remains of the J to Dean. 'Aye, I'd better be off too.'\n\n'I wasn't trying to get rid of you,' Ash said, as she closed the front door after us. I'd said goodbye to Mrs Watt; Ash had said she would be back in quarter of an hour or so.\n\n'Shit. I thought maybe I was being thick-skinned,' I said as we walked the short path to a wee garden gate in the low hedge.\n\n'That'll be the day, Prentice,' Ash laughed.\n\n'You really going to walk at this time of night?' I looked up; the night was clear now, and colder. I pulled on my gloves. My breath was the only cloud.\n\n'Nostalgia,' Ash said, stopping on the pavement. 'Last visit to somewhere I used to go a lot when I was a wean.'\n\n'Wow, really? How far is it? Can I come?' I have a fascination with places people think powerful or important. If I hadn't been still fairly drunk I'd have been a lot more subtle about asking to accompany Ash, but, well, there you are.\n\nHappily, she just laughed quietly, turned on her heel and said, 'Aye; come on; isn't far.'\n\nSo here we stood, on the wee mound only five minutes from the Watt house, down Bruce Street, through a snicket, across the Oban road and over the weedy waste ground where the dock buildings stood, long ago.\n\nThe dock-side was maybe ten metres away; the skeletal remains of a crane stood lop-sided a little way along the cancered tarmac, its foundations betrayed by rotten wooden piling splaying out from the side of the wharf like broken black bones. Mud glistened in the moonlight. The sea was a taste, and a distant glittering that all but disappeared if you looked at it straight. Ash seemed lost in thought, staring away to the west. I shivered, un-studded the wide lapels of the fake biker's jacket and pulled the zip up to my right shoulder so that my chin was encased.\n\n'Mind if I ask what we're doing here?' I asked. Behind and to our left, the lights of Gallanach were steady orange, like all British towns, forever warning the inhabitants to proceed with caution.\n\nAsh sighed, her head dropped a little. She nodded down, at the ground we stood upon. 'Thought you might know what this is, Prentice.'\n\nI looked down. 'It's a wee lump of ground,' I said. Ash looked at me. 'All right,' I said, making a flapping action with my elbows (I'd have spread my hands out wide, but I wanted to keep them in my pockets, even with my gloves on). 'I don't know. What is it?'\n\nAsh bent down, and I saw one pale hand at first stroke the grass, and then dig down, delving into the soil itself. She squatted like that for a moment, then pulled her hand free, rose, brushing earth from her long white fingers.\n\n'This is the Ballast-Mound, the World-Hill, Prentice,' she said, and I could just make out her small thin smile by the light of the gibbous moon. 'When the ships came here, from all over the world, for whatever it was they were shipping from here at the time, they would sometimes arrive unladen, just ballast in them; you know?'\n\nShe looked at me. I nodded. 'Ballast; yeah, I know what ballast is; stops ships doing a _Herald of Free Enterprise.'_\n\n'Just rocks, picked up from wherever the ship last set sail from,' Ash said, looking to the west again. 'But when it got here they didn't need it, so they dumped it \u2013'\n\n'Here?' I breathed, looking at the modest mound with new respect. 'Always here?'\n\n'That's what my grampa told me, when I was a bairn,' Ash said. 'He used to work in the docks. Rolling barrels, catching slings, loading sacks and crates in the holds; drove a crane, later.' (Ashley pronounced the word 'cran', in the appropriate Clyde-side manner.) I stood amazed; I wasn't supposed to be getting ashamed at my lack of historical knowledge until Monday, back at Uni.\n\n' \"Hen,\" he'd say, \"There's aw ra wurld unner yon tarp a grass.\"'\n\nI watched from one side as Ashley smiled, remembering. 'I never forgot that; I'd come out here by myself when I was a kid, just to sit here and think I was sitting on rocks that had once been a bit of China, or Brazil, or Australia or America ...'\n\nAsh squatted down, resting on her heels, but I was whispering, '... Or India,' to myself just then, and for one long, swim-headed instant my veins seemed to run with ocean-blood, dark and carrying as the black water sucking at the edges of the tumbledown wharf beneath us. I thought, _God, how we are connected to the world!,_ and suddenly found myself thinking about Uncle Rory again; our family connection to the rest of the globe, our wanderer on the planet. I stared up at the broken face of moon, dizzy with wonder and a hunger to know.\n\nWhen he was younger than I am now, my Uncle Rory went on what was supposed to be a World Trip. He got as far as India. Fell in love with the place; went walk-about, circulating; to Kashmir from Delhi, then along the hem of the Himalayas, crossing the Ganga at Patna \u2013 asleep on the train \u2013 then zig-zagging from country to coast and back again, but always heading or trying to head south, collecting names and steam trains and friends and horrors and adventures, then at the very hanging tip of the subcontinent, from the last stone at low tide on Cape Comorin one slack dog-day; reversing; heading north and west, still swinging from interior to coast, writing it all down in a series of school exercise books, rejoicing in the wild civility of that ocean of people, the vast ruins and fierce geography of the place, its accrescent layers of antiquity and bureaucracy, the bizarre images and boggling scale of it; recording his passage through the cities and the towns and villages, over the mountains and across the plains and the rivers, through places I had heard of, like Srinagar and Lucknow, through places whose names had become almost banal through their association with curries, like Madras and Bombay, but also through places he cheerfully confessed he'd visited for their names as much as anything else: Alleppey and Deolali, Cuttack and Calicut, Vadodara and Trivandrum, Surendranagar and Tonk ... but all the while looking and listening and questioning and arguing and reeling with it all, making crazed comparisons with Britain and Scotland; hitching and riding and swimming and walking and when he was beyond the reach of money, doing tricks with cards and rupees for his supper, and then reaching Delhi again, then Agra, and a trek from an ashram to the great Ganga, head fuddled by sun and strangeness to see the great river at last, and then the long drift on a barge down to the Farakka Barrage a train to Calcutta and a plane to Heathrow, half dead with hepatitis and incipient malnutrition.\n\nIn London, after a month in hospital, he typed it all out, got his friends in the squat where he lived to read it, called it _The Deccan Traps And Other Unlikely Destinations,_ and sent it to a publisher.\n\nIt very nearly sank without trace, but then it was serialised in a Sunday newspaper, and suddenly, with no more warning or apparent cause than that, _Traps_ just was the rage, and he was there.\n\nI read the book when I was thirteen, and again four years later, when I understood it better. It was hard to be objective \u2013 still is \u2013 but I think it is a good book; gauche and naive in places, but startling; vivacious. He went with his eyes open, and, not having taken a camera, just tried to record everything on the pages of those cheap exercise books, straining to make it real for himself, as though he could not believe he had seen and heard and experienced what he had until it was fixed somewhere other than in his stunned brain, and so he could describe walking towards the Taj Mahal \u2013 ho-hum, thinks the reader, immediately in the realm of the tacky postcard \u2013 and still give you a wholly fresh impression of the exact scale and actual _presence_ of that white tomb; delicate but powerful, compact and yet boundlessly imposing.\n\nEpic grace. With those two words he encapsulated it, and you knew exactly what he meant.\n\nAnd so our Rory became famous, at that moment on the very lip of the escarpment of his fame, the rosy cliffs forever at his back as he wandered on.\n\nAsh squatted down, resting on her heels. She tore a piece of grass from the mound, ran it through her fingers. 'And I'd come here when my daddy-paddy was beating the living shit out of my mum, and sometimes us too.' She looked up at me. 'Stop me if you've heard this one before, Prentice.'\n\nI hunkered down too, shaking my head as much to clear it as to deny. 'Well, not exactly, but I knew it wasn't all sweetness and light, _chez_ Watt.'\n\n'Fuckin right it wasn't,' Ash said, and sounded bitter. The blade of grass ran through her fingers, was turned round, passed through again. She looked up, shrugged. 'Anyway, sometimes I came out here just because the house smelled of chip-fat or the telly was too loud, just to remind myself there was more to the world than 47 Bruce Street and endless arguments about fag money and which one of us got a new pair of shoes.'\n\n'Aye, well,' I said, at a loss really to know what to say. Maybe I get uncomfortable being reminded there are worse backgrounds than coming from a family of mostly amiable over-achievers.\n\n'Anyway,' she said again. 'They're levelling the lot tomorrow.' Ash looked back over her shoulder. I followed her gaze. 'That's what all that plant's for.'\n\nI remembered the Triffid jokes we used to make about Heavy Plant Crossing, and only then saw the dim outlines of a couple of bulldozers and a JCB, a little way off down the piece of waste ground.\n\n'Aw, shit,' I said, eloquently.\n\n'An exclusive marina development with attractive fishing-village-style one\u2013 and two-bedroom flats with dedicated moorings, double garages and free membership of the private health club,' Ash said, in a Kelvinside accent.\n\n'Fffuck,' I shook my head.\n\n'What the hell,' Ashley said, rising. 'I suppose the Glasgow middle classes have to go somewhere after they've braved the treacherous waters of the Crinan canal.' She gave her hands one final dust. 'Hope they're happy there.'\n\nWe turned to leave the mound, me and the Ash, then I grabbed her arm. 'Hi.' She turned to me. 'Berlin,' I said. 'The jacuzzi; I just remembered.'\n\n'Oh yeah.' She started walking down the slope, back to the weeds, the junk and the ankle-high remains of old brick walls. I followed her. 'I was in Frankfurt,' she said. 'Seeing this friend from college? We heard things were happening in Berlin so we hitched and trained it; met up with ... Well, it's a long story, but I ended up in this fancy hotel, in the swimming pool; and had a big whirl-pool bath in a wee sort of island at one end, and this drunken English guy was trying hard to chat me up, and making fun of my accent and \u2013'\n\n'Cheeky basturt,' I said as we got to the main road.\n\nWe waited while a couple of cars sped north out of town.\n\n'That's what I thought,' Ashley nodded, as we crossed the road. 'Anyway, when I told him where I came from he started saying he knew the place well and he'd been shooting here, and fishing, and knew the laird and \u2013'\n\n'Do we have a laird? I didn't know. Perhaps he meant Uncle Fergus.'\n\n'Maybe, though when I asked him that he got cagey and said no ... but the point is he was acting all mysterious about something, and he'd already said there was somebody here who was having the wool pulled over their eyes, and had been for a long time, and he thought their name was ...' Ash stopped at the snicket that led up to Bruce Street. My route back to Uncle Hamish's house went straight along the main road.\n\nI looked up the wee path, lit by a single yellow street lamp, half way up. Then I looked back into Ashley Watt's eyes.\n\n'Not McHoan, by any chance?' I asked.\n\n'Yep,' Ash nodded.\n\n'Hmm,' I said. Because McHoans are fairly thin on the ground around here. Or anywhere, for that matter.\n\n'Who was this guy?'\n\n'Journalist. There to cover the big knock-down.'\n\n'What was his name?'\n\n'Rudolph something, I think somebody called him. He wouldn't say.'\n\n'You might have used your feminine wiles.'\n\n'Well, at the time they were more or less fully employed on a systems analyst from Texas with shoulders wide as the prairie sky and a gold company Amex card, to be perfectly honest, Prentice.' Ash smiled sweetly.\n\nI shook my head. 'Saucy bitch.'\n\nAsh grabbed my balls through my 501s and squeezed gently. My breath baled out.\n\n'Language, Prentice,' she said, then released, covered my mouth with hers, wiped my teeth with her tongue, then swivelled, walked away.\n\n'Wow,' I said. The old testes were complaining, but only slightly. I cleared my throat. 'Night, Ashley,' I said, cool as I could.\n\nAsh turned, grinning, then reached into her big, naval-looking jacket with the brass buttons and fished something out; threw it.\n\nI caught the projectile; a little lump of grey concrete, smooth and dark on one side.\n\n'Die Mauer,' she said, walking backwards. 'Actually from the section near the Brandenburg Gate where it said, \"Viele viele bunte Smarties!\". The red paint on one surface used to be in the middle of the dot in the last \"i\". Bit of the world that used to be between Germanies.' She waved. 'Night, Prentice.'\n\nI looked at the grainy chunk of concrete in my hand. 'Wow,' I breathed. Ash's fair hair flared briefly under the street light, then dimmed as she walked away. 'Wow!'\nCHAPTER 4\n\nHe looked round the Solar of the castle. The big new window at the gable end of the hall was still covered with a translucent plastic sheet which rustled in the wind and crackled as the rain blew onto it. A shifting grid of dark lines was the shadow of the scaffolding outside. The high-ceilinged hall smelled of paint, varnish, new wood and drying plaster. He walked over to one of the mullioned windows, and stood there, looking out at the low clouds as they drifted over Gallanach, soaking the dull town with the curving veils of rain they dragged beneath them, like the train of some vast grey gown.\n\n'Daddy, daddy! Uncle Fergus says we can go up on the roof with mum if we're careful! Can we? Please can we? Promise we won't jump off!' Lewis skipped into the hall, dragging little Prentice behind him. Lewis had his anorak back on, and Prentice was dragging his behind him over the shining parquet floor.\n\n'Aye son, I suppose so,' Kenneth said, sitting on his ankles to pull the younger boy's jacket on and zipping it up. Lewis went leaping and whooping round the hall while this was going on. 'Not so loud, Lewis,' Kenneth said, without much conviction.\n\nPrentice smiled at his dad. 'Daddy,' he said in his slow, croaky voice. 'Need the toilet.'\n\nKenneth sighed, pulled the child's hood up, then pushed it back down again. 'Aye well; your mum will take you. Lewis!' he shouted. Lewis darted guiltily away from the paint pots he'd been examining at the other end of the hall, and came running over.\n\n'This is great, daddy! Can we get a castle too, aye?'\n\n'No. We can't afford it. Take your brother back to his mum; he needs the toilet.'\n\n'Aww,' whined Lewis, staring accusingly at his young brother, who just grinned at him and wiped his nose on one cuff of his anorak. Lewis prodded Prentice in the back. 'You're always spoiling things!'\n\n'Do as you're told, Lewis,' McHoan said, straightening. His knees complained as he did so. 'On you go. And be careful on that roof.' He waved them both towards the double doors they'd entered through.\n\nLewis made a show of plodding off, clumping one foot in front of another, body swaying exaggeratedly. He was pulling Prentice by one toggle of his anorak hood.\n\n'By the _hand,_ Lewis,' Kenneth said wearily.\n\n'You're a pest, boy,' Lewis told his younger brother as they reached the doorway.\n\nPrentice turned and waved to his dad with his free hand. 'Bye, daddy,' the wee voice said. Then he was pulled out of the room.\n\n'Bye, son,' McHoan said, and smiled. Then he turned back to the window and the rain.\n\n'It's a bit damp still.'\n\n'Ach, yer no afraid of a bit a wet, ur ye? Yer no a girrul ur ye?'\n\n'No I'm _not_ a girl. But if I get my clothes mucky \u2013'\n\n'Your dad's rich; he can buy you new clothes.'\n\n'Aye; yer paw's rich. You could probably have new claes every day if you wantit.'\n\n'Don't be ridiculous. All I'm saying \u2013'\n\nKenneth could see both points of view; Lachy, in a grimy shirt held together by odd buttons and a safety pin, and tattered, patched short trousers that drooped below his knees and had probably belonged to at least two elder brothers, was already grubby (and sporting the vivid remains of a black eye no one had mentioned because it had probably been his dad who gave it him). Fergus had nice, well-fitting clothes on: grey serge short trousers, a new blue jersey and a tweed jacket with leather patches sewn on the elbows. Even Kenneth felt a little dowdy in comparison. His shorts had been darned at the back, though he was getting a new pair when the next clothes rations came through. The girls all wore skirts, blouses and jerseys; their socks were white, not grey. Emma Urvill had a coat with a little hood that made her look like a pixie.\n\n'Are we playing this game or not?' she asked.\n\n'Patience,' Lachy said, turning to the girl, still standing holding her bike. 'Patience, lassie.'\n\nEmma looked skywards and made a tutting noise. Beside her, Kenneth's sister, Ilsa, also on her bike, shook her head.\n\nThe castle stood on the side of the hill. The tall trees around it were still dripping, and its rough, uneven stones were dark and wet from the rain that had not long stopped. A watery sun gleamed on the dark leaves of the ivy that clung to one side of the ruin, and in the forest behind, a wood pigeon cooed softly.\n\n'Oh, what the heck,' Fergus Urvill said, and rested his bike against a tree.\n\nLachy Watt let his bike fall to the ground. Kenneth lowered his to the damp grass alongside. The girls propped theirs against the wooden rails at the start of the bridge. The short wooden bridge, about wide enough for a cart, crossed a steep, bush-choked gully about thirty feet deep. At the bottom of this tiny, dank glen a burn splashed and foamed; it rushed out of the woods, curled round three sides of the rock and grass knoll the roofless castle stood upon, fell over a small waterfall, then progressed gently afterwards, joining the River Add near the main road, so that eventually its waters flowed through and beneath the town of Gallanach and into the bay near the railway pier.\n\nSun came suddenly, making the grass bright and the ivy leaves sparkle; the wind pushed through the forest with a quiet roar, releasing drops of water all around. Kenneth watched a train on the viaduct at Bridgend, about a mile away; the west wind was keeping its noise from them, but he could see the steam rising quickly from the dark locomotive and whipping back over the half-dozen burgundy coaches in little white clouds that spread and were torn apart and flung away by the wind.\n\n'Right,' Lachy said. 'Who's het?'\n\n'Het?' Fergus said. 'You mean \"it\"?'\n\n'You know what ah mean; who's goin het first?'\n\n'Do One potato, Two potato,' suggested Emma.\n\n'Oh Goad, all right,' Lachy said, shaking his head.\n\n'And you shouldn't take the Lord's name in vain,' Emma told him.\n\n'Christ, Ah'm sorry,' Lachy said.\n\n'You did it again.'\n\n'You a Tim or sumhin?'\n\n'I'm a Christian,' Emma said primly. 'And I thought you were, too, Lachlan Watt.'\n\n'Ah'm a Protestant,' Lachy said. 'That's what am are.'\n\n'Can we get on with this, please?' Ilsa said.\n\nThey all lined up, fists clenched; Lachy ended up being it, much to his own annoyance.\n\nKenneth had never been inside the old castle; you could just see it from the house, if you knew what you were looking for, and you could see it quite well if you used dad's binoculars, but it was on the Urvill estate, and even though their families had been friends for years \u2013 generations, dad said, which meant even longer \u2013 Mr Robb, on whose farm the castle stood, didn't like children, and chased them off his fields and out of his woods whenever he could, threatening them with his shotgun. He couldn't chase Fergus and Emma Urvill off though, so they were all safe. Kenneth had wondered if Mr Robb was secretly a fifth columnist or even a Nazi, and was hiding men washed up from a sunken U-boat, or preparing a place for paratroopers to land, but despite him and some of the other children watching Mr Robb very carefully from the woods a few times, they had never been able to prove anything. But they had explored the hidden garden a bit, and decided the castle looked worth investigating.\n\nThe castle had dark, intact dungeons at ground level, and a stone stair-case in a circular tower that rose to the open heart of the ruin, where a few jumbled stones and a floor of earth and weed looked up to the sky. The stairs wound further up inside the corner tower, pausing at each long-collapsed floor above, where a doorway looked out onto the central well. Another stairway pierced the walls themselves on the far side of the shell of the keep, rising through their thickness past another three doorways hanging like internal balconies, to a couple of small rooms at the top of dark chimneys which led to the base of the walls outside.\n\nThe castle held a variety of other dark nooks and shadowy crannies you could hide in, as well as windows and fireplaces set high in the thick walls, where you could climb if you were good at climbing, and if you were really good you could climb up from the circular stairs to the very summit of the ruin, where you could walk, if you dared, right round the thick tops of the walls, over the weeds and the ivy, sixty feet or more above the ground. From there you could look out to sea, over Gallanach, or into the mountains to the north and the forested hills to the south. Closer, there was the overgrown walled garden, across another bridge behind the castle, where tangles of rhodies crowded under monkey-puzzle trees and a riot of exotic flowers attracted buzzing clouds of insects in the summer.\n\nThe rules were that you could hide anywhere in the castle; Kenneth and the others left Lachy at the track side of the bridge, counting slowly. They laughed and squealed, bumping into each other and shushing each other as they tried not to shout too loudly while they dithered and giggled over where to hide.\n\nKenneth climbed up into a high window and crouched down.\n\nEventually, Lachy came into the open hall of the castle, looking around. Kenneth watched him for a moment, then ducked back in, flattened himself down as close to the stone sill of the window as he could.\n\nHe was the last to be found, and for a while delighted in the fact that none of them could find him, even after Lachy had caught the rest and they were all shouting at him to come out so they could have another go. He lay there, feeling the damp breeze coming through the window and tickling the hairs on his bare legs. He listened to the shouts of the others, echoing in the castle's emptied shell, and to the voices of the crows and the wood pigeons in the trees, and he smelled the dark, wet smell of the moss and weeds that had found a foothold amongst the ruin's grey stones. He kept his eyes tightly closed, and as he listened to them search for him and call on him, there came a strange, tight, quivery feeling in his tummy which made him want to clench his teeth and bring his knees together and made him worry about wetting his pants.\n\nI love this here, he thought to himself. I don't care if there is a war on and Fergus's uncle got killed in North Africa, and Wullie Watt got killed in the North Atlantic and Lachy gets hit by his dad and we might have to move to another house because Mr Urvill wants ours back and I don't understand trigonometry and the Germans do invade us; I love this. If I died right now I wouldn't care; wouldn't care at all.\n\nLachy climbed right up onto the top of the walls eventually, and only then did he see Kenneth. Kenneth came down, yawning widely and rubbing his eyes and claiming he had fallen asleep. He'd won, had he? Oh, jolly good.\n\nThey played some more, and made fun of Fergus after he'd won a game because Kenneth had worked out what the two little rooms at the top of the second set of stairs were; they were toilets, and that was why the chimneys led down and went out of the castle; it was so all the number one and numbers twos could fall down there. Fergus had hidden down a latrine! And him worried about getting his clothes dirty, too! Fergus denied they were toilets; they were completely clean and didn't smell at all and they must be chimneys.\n\n'Chimneys, ma arse!' Lachy laughed. 'They're shite-holes!'\n\n(Emma tutted, but couldn't help smiling.)\n\n'Chimneys!' Fergus insisted desperately, blinking hard. He looked at Kenneth as though expecting him to agree. Kenneth looked down at the tramped-down earth under his feet.\n\n'They're shite-holes, so they are,' laughed Lachy. 'And you're just a big jobbie!'\n\n'Chimneys,' Fergus protested, his voice rising, his face going red.\n\n'Big jobbie, big jobbie; big smelly jobbie!' Lachy chanted.\n\nKenneth watched Fergus shake with anger while Lachy danced round the interior of the keep, singing out, 'Big jobbie, big jobbie; big smelly jobbie!'\n\nFergus stared angrily at his sister and at Kenneth, as though betrayed, then just stood and waited for Lachy to get bored with his taunting, and as Kenneth watched, a blank, emotionless expression gradually replaced the anger on Fergus's face.\n\nKenneth had the fleeting, extraordinary impression of seeing something buried alive, and felt himself shake suddenly, almost spastically, shivering.\n\n'... jobbie, jobbie; big smelly jobbie!'\n\nIn the last game, Kenneth hid with Emma Urvill in one of the dungeons, showing her how to turn her back to the light and put the hood of her coat up to hide her face, and sure enough when Ilsa came to the door of the dungeon \u2013 and he felt that quivering, scary, glorious feeling in his tummy again \u2013 she didn't see them, and they hugged each other once she had gone, and the hug was warm and tight and he liked it and she didn't let go, and after a while they put their mouths together and kissed. He felt a strange echo of that terrifyingly wonderful sensation in his belly and his heart, and he and Emma Urvill held onto each other for ages, until all the others were caught.\n\nLater, they played in the tangling undergrowth of the walled garden, and found an old over-grown fountain with the stone statue of a naked lady in it, and an old shed at one corner where there were ancient tins and jars and bottles with Victorian-looking writing on them. The rain came on for a while and they all stayed in there, Fergus complaining about his bike rusting, his sister and Kenneth exchanging the occasional sly look, Ilsa staring out at the rain and saying there were places in South America where it hadn't stopped raining for hundreds of years, and Lachy mixing various sticky, treacley subtances together from the shelves of old bottles and tins, trying to find a combination that would explode, or at least burn, while the rain hammered then whispered then dripped on the tarred roof overhead, and plopped through holes onto the springy wooden floor of the shed.\n\n'Of course, we haven't moved all the bottles yet,' Fergus said, pointing with his pipe at the still unfilled racks that covered the wall of the cellar. The cellar was painted white, and lit by naked bulbs; wires hung and there were un-plastered holes for cables and plumbing leading through the walls and up to other floors. The wood and metal wine racks gleamed, as did the two hundred or so bottles that had already been stored.\n\n'Should keep you going for a bit, eh, Fergus?' he grinned. 'Once you've filled this lot up.'\n\n'Mmm. We were thinking of touring a few vineyards next summer,' Urvill said, scratching his thick chin with his pipe. 'Bordeaux; the Loire, that sort of thing. Don't know if you and Mary fancy making a foursome or not, hmm?'\n\nFergus blinked. Kenneth nodded. 'Well, perhaps. Depends on holidays and that sort of thing. And the kids, of course.'\n\n'Oh,' Fergus said, frowning as he picked a little sliver of tobacco off his Pringle sweater. 'We weren't thinking of taking the children.'\n\n'Ah, well, no; of course not,' Kenneth said, as they went to the door. Fergus switched the lights out in the various cellars and they went up the stone-flagged steps towards the utility room and kitchen.\n\nIt was that cellar, he thought to himself as he followed Fergus's Hush Puppies up the steps. That was where I hid with Emma Urvill, and kissed her. That cellar; I'm sure it was that one. And that window I was looking out earlier; that was the one I hid in that day, nearly thirty years ago; I'm certain.\n\nHe felt a terrible weight of time and loss settle on him then, and a slight feeling of resentment at the Urvills in general and Fergus in particular, for having \u2013 with so little thought \u2013 stolen part of his memories from him. At least malice might have acknowledged the value of his nostalgia.\n\n'Ferg, this dishwasher's like a Chinese Puzzle,' Fiona stood up from the recalcitrant machine, then saw her brother and smiled broadly, came towards him, hugging.\n\n'Hiya, Ken. Been getting the guided tour, have you?'\n\n'Yes; very impressive.' Kenneth kissed his sister's cheek. How old had she been when he'd come out here with Fergus and the rest? About two, he guessed. Not old enough to come all this way on a bike. He must have been eight or nine. He wondered where Hamish had been; ill, maybe. He'd always been taking colds.\n\nFiona Urvill, n\u00e9e McHoan, wore old flared Levis and a loose green blouse knotted over a white T-shirt. Her copper-coloured hair was tied back. 'How're you?'\n\n'Oh, I'm well,' Kenneth nodded; he kept an arm round her waist as they walked over to the dishwasher, where Fergus crouched, consulting the instruction booklet. The door of the dishwasher was hinged open like a drawbridge.\n\n'Appears to be written in code, my dear,' Fergus said, scratching the side of his head with his pipe. Kenneth felt a smile form on his face as he looked down at the man. Fergus seemed old before his time: the Pringle jumper, the Hush Puppies, even the pipe. Of course, Kenneth could remember when he used to smoke a pipe; but that had been different. Looked like Fergus was losing his hair already, too.\n\n'How's school?' Fiona asked her brother.\n\n'Och, getting on,' he said. 'Getting on.' He had been promoted to Principal Teacher in English the previous autumn. His sister always wanted to know how things were going at the high school, but he usually felt reluctant to talk about work around her and Fergus. He wasn't sure why, and he suspected he probably wouldn't like acknowledging the reason, if he ever did work it out. He was even more chary about revealing he was writing down some of the stories he'd told the kids over the years, hoping to publish them some day. He was worried people might think he was trying to out-do Rory, or \u2013 worse still \u2013 think he hoped to use him as a contact, an easy way in.\n\n'No, I tell a lie,' Fergus admitted. 'Here's the English bit. Well, American, anyway.' He sighed, then looked round. 'Talking about English-speaking furriners, McHoan; you still all right for the International next Saturday?'\n\n'Oh aye,' Kenneth nodded. They were meant to be going to the Scotland-England rugby game in a week's time. 'Who's driving?'\n\n'Umm, thought we'd take the Morgan, actually.'\n\n'Oh God, Fergus, must we? I'm not sure I can find my bobblehat.'\n\n'Oh, come on man,' Fergus chuckled. 'Thought we'd try a new route: down to Kintyre; across to Arran, Lochranza to Brodick; Land Ardrossan and then the A71 to the A of the N. Strikes and power cuts permitting, of course.'\n\n'Fergus,' Kenneth said, putting one hand to his brow. 'It sounds enormously complicated.' He refused to rise to the bait about strikes and power cuts. He guessed that 'A of the N' meant Athens of the North. 'Are you sure the Lochranza ferry runs outside the high season, anyway?'\n\nFergus looked troubled, stood up. 'Oh, it must, mustn't it? Well, I think it does.'\n\n'Might be best to check.'\n\n'Righty-oh, will do.'\n\n'Anyway, couldn't we take the Rover?' Kenneth wasn't keen on the Morgan; its stiff ride hurt his back and gave him a headache, and Fergus drove too fast in the ancient open-top. Maybe it was the sight of all that British Racing Green paint and the leather strap across the bonnet. The Rover, 3.5 though it was, seemed to calm Fergus a little.\n\n'Oh, come on man, where's your sense of occasion?' Fergus chided. 'The hotel won't let us into the car park if we show up in the Rover,'\n\n'Oh God,' Kenneth sighed, and squeezed his sister's waist. 'The Morgan it is then.' he looked at Fiona. Those green eyes sparkling. 'I'm getting old, sis. Do you think I'm getting old?'\n\n'Positively ancient, Ken.'\n\n'Thanks. How're the twins?'\n\n'Oh, glowing.'\n\n'Still taking them to Windscale for their hols then, are you?'\n\n'Ha! Oh, Ken, you're still so comparatively witty.'\n\n'Have you tried switching it on?' Fergus suggested, squatting on the floor in front of the dishwasher again. His voice echoed inside the machine as he tried to stick his head inside amongst the racks.\n\n'Don't be catty, Ferg,' Fiona told him. She smiled at her brother. 'Haven't seen young Rory out here for a while, and he never calls us; he okay?'\n\n'Still in that squat in Camden, last we heard, living off his ill-gotten sub-continental gains.'\n\n'A _squat?'_ Fergus said, words muffled. 'Thought he made a packet on that ... travel book thingy.'\n\n'He did,' Ken nodded.\n\n'About India, wasn't it?'\n\n'Yep.'\n\n'Ferg,' Fiona said, exasperated. 'You bought the book, remember?'\n\n'Of course I remember,' Fergus said, reaching into the dishwasher to fiddle with something. 'Just haven't read it, that's all. Who needs to read a book to find out about India? Just go to bloody Bradford ... What's he doing living in a _squat?'_\n\nKen ground his teeth for a second, looking appraisingly at Fergus's ample rear. He shrugged. 'He just likes living with the people there. He's a social animal, Ferg.'\n\n'Have to be a bloody animal to live in a squat,' Fergus muttered, echoing.\n\n'Hoi, don't be horrible about my brother,' Fiona said, and tapped Fergus's backside with her foot.\n\nFergus glanced quickly round and glared at her, his plump, slightly reddened face suddenly grim. Kenneth felt his sister stiffen next to him. Then Fergus gave a little wavering smile, and with a quiet grunt turned back to the opened machine and its instruction booklet. Fiona relaxed again.\n\nKenneth wondered if things were really all right with the couple. He thought he sensed a tension between them sometimes, and a couple of years earlier, not long after the twins had been born, he'd thought Fergus and Fiona had seemed distinctly cold towards each other. He had worried for them, and he and Mary had discussed it, wondering what might have caused this unhappiness, and if there was anything they could do (they had decided there wasn't, not unless they were asked). Still, he had tried broaching the subject with Fergus once, after a dinner party, while they nursed whiskies in the conservatory of the old Urvill house and watched the lights of the navigation buoys and lighthouses scattered around and through the Sound of Jura as they winked on and off.\n\nFergus hadn't wanted to talk. Mary had had no more success with Fiona. And anyway it had all seemed to come gradually right again.\n\nMaybe I'm just jealous, he thought to himself, as Fiona pulled away from him and went to the big new Aga, sitting squat, cream and gleaming against one wall of whitewashed stone. She put a hand over part of the cooker's surface, gauging the heat. The silence in the kitchen went on.\n\nKenneth had never given Freud much credence; mainly because he had looked as honestly into himself as he could, found much that was not to his taste, found a little that was even just plain bad, but nothing much that fitted with what Freud's teachings said he ought to find. Still, he wondered if he did resent Fergus, at least partly because he had taken his sister away, made her his.\n\nWell, you never knew, he supposed. Maybe everybody's theories were right, maybe the whole world and every person, and all their relationships within it were utterly bound up with one another in an intricate, entangled web of cause and effect and underlying motive and hidden principle. Maybe all the philosophers and all the psychologists and all the theoreticians were right ... but he wasn't entirely sure that any of it made much difference.\n\n'Mary and the kids with you?' Fiona said, turning from the Aga to look at him.\n\n'Taking in the view from the battlements,' Kenneth told her.\n\n'Good,' she nodded. She glanced at her husband. 'We're getting an observatory, did Ferg tell you?'\n\n'No.' He looked, surprised, at the other man, who didn't turn round. 'No, I didn't know. You mean a ... a telescope; an astronomical observatory?'\n\n'Bloody astronomically expensive,' Fergus said, voice echoing in the dishwasher.\n\n'Yes,' Fiona said. 'So Ferg can spend his nights star-gazing.' Mrs Urvill looked at her husband, still squatting in front of the opened machine, with an expression Kenneth thought might have been scorn.\n\n'What's that, my dear?' Fergus asked, looking over at his wife, an open, innocent expression on his face.\n\n'Nothing,' his wife said brightly, voice oddly high.\n\n'Hmm,' Fergus adjusted something inside the dishwasher, scratched above his ear with his pipe again. 'Jolly good.'\n\nKenneth looked away then, to the windows, where the rain spattered and ran.\n\nConceived in a howling gale, Verity was born \u2013 howling \u2013 in one, too. She came into the world a month before she was due, one windy evening in August 1970, by the shores of Loch Awe \u2013 a birth-place whose title, Prentice at least had always thought, could hardly have been more apt.\n\nHer mother and father had been staying at Fergus and Fiona Urvill's house in Gallanach for the previous two weeks, on holiday from their Edinburgh home. For the last night of their holiday the young couple decided to visit a hotel at Kilchrenan, an hour's drive away to the north east up the side of the loch. They borrowed Fergus's Rover to make the journey. The bulging Charlotte had that week developed a craving for salmon, and duly dined on salmon steaks, preceded by strips of smoked salmon and followed by smoked salmon mousse, which she chose in preference to a sweet. She complained of indigestion.\n\nWell \u2013 if in Charlotte's case rather monotonously \u2013 fed, they began the return journey. The evening was dull, and although there was no rain a strong warm wind was blowing, waving the tops of the trees and stroking lines of white breakers up the length of the narrow loch. The gale increased to storm force as they drove south west into it, down the single-track road on the western shore.\n\nThe narrow road was littered with fallen branches; it was probably one of those that produced the puncture.\n\nAnd so, while her husband struggled with over-enthusiastically-tightened wheel-nuts, Charlotte went into labour.\n\nBarely half an hour later a stunning blue flash \u2013 the colour of the moon and brighter than the sun \u2013 burst over the scene from the hill above.\n\nThe noise was thunderous.\n\nCharlotte screamed.\n\nAbove, on the hillside, stood the lattice forms of two electricity pylons, straddling the heather like grey gigantic skeletons wreathed in darkness. The black wind howled and there was another blinding flash and a titanic concussion; a line of violet incandescence split the night mid-way between the two huge pylons as energy short-circuited through the air between the wind-whipped power-lines.\n\nCharlotte screamed again, and the child was born.\n\nThe tail end of Hurricane Verity passed over the British Isles that night; it had been born in the doldrums, cut its teeth flooding bits of the Bahamas, flirted with the coast of North Carolina, and then swept off across the North Atlantic, gradually losing energy; a brief encounter with the angle between a cold front and a warm front just off Ireland refreshed it unexpectedly, and it trashed numerous pleasure boats, rattled a few acres of windows, played frisbee with a multitude of slates and broke many a bough as it passed over Scotland.\n\nThe stretch of the national electricity grid down the western shore of Loch Awe towards Gallanach was one of the storm's more spectacular victims, and Charlotte always claimed that it was right on the stroke of the final massive arc between the thrashing cables \u2013 which tripped circuit breakers in the grid to the north and plunged all of Gallanach into darkness \u2013 that her child (wrinkled, blood-flecked and salmon pink) finally slid out into her father's hands.\n\nThey named her Verity, after the hurricane.\n\nWhen she was eighteen, Fergus Urvill gave his niece Verity a very special present made from one of the exhibits in the museum attached to his glass factory. For the child born to the blaze and crack of human lightning, her entry into this world marked by the same brilliant arcs of short-circuited energy that plunged Gallanach into powerless gloom, he had a necklace fashioned which was made from fulgurite.\n\nFulgurite is a natural glass, like another of the museum's minor treasures, obsidian. But while obsidian is born purely of the earth, formed in the baking heat and furious pressure of volcanic eruptions, fulgurite is of the earth and of the air, too; it is made when lightning strikes unconsolidated sand, and fuses it, vitrifies it in long, zig-zag tubes. God's glass, Hamish McHoan called it.\n\nThe Gallanach Glass Works Museum contained a collection of tubular fulgurites, plucked from the sands of Syria by Walter Urvill \u2013 Fergus's grandfather \u2013 on a visit there in 1890, and transported back to Scotland with great care and not a little luck so that they arrived intact. One of the crinkled, gnarled little tubes was over a metre long; another just a fraction shorter. Fergus had the smaller of the two sent to a jeweller in Edinburgh, to be broken, the pieces graded and ground and polished and threaded together like dark little pearls, to create a unique necklace for his niece.\n\nHe presented the result to the lightning-child during her birthday party, at her parents' house in Merchiston, in Edinburgh, in August 1988 (it was, perhaps unfittingly, a perfectly fine, warm, clear and calm night, on that anniversary). Fergus \u2013 always a rather dour, prematurely elderly figure, characterised by those collar-contacting jowls \u2013 improved immensely in the eyes of both Kenneth and Prentice McHoan with that single, elegant, and rather unexpectedly poetic act.\n\nVerity had the grace to accept the necklace with a particular gratitude that acknowledged the thought behind the gift, and the taste to make it a regular, even habitual, part of her wardrobe.\n\nThe upholstery of Fergus's Rover was cleansed of the debris and stains associated with Verity's birth and the car continued to serve the Urvill family for another five years or so until 1975, when it was traded in (for what Prentice would thereafter maintain was a scandalously small sum, considering that the thing ought to have been preserved as some sort of internationally-recognised shrine to Beauty) for an Aston Martin DB6.\n\nIt was once Prentice's dream, shortly after he'd passed his driving test, to find that old Rover \u2013 lying in a field somewhere, perhaps \u2013 and to buy it; to own the car his beloved had been born in; to drive it and to cherish it. He realised, of course, that it had almost certainly been scrapped long before, but that had not prevented him harbouring the perhaps irrational notion that somehow a little of its recycled metal must have found its way into at least one of the three old bangers he'd owned.\n\nThe defiantly thunderous and lightning-fast Aston Martin DB6 was the car that Fergus and Fiona Urvill were travelling in on the night they were involved in a crash at Achnaba, just south of Lochgair, in 1980.\nCHAPTER 5\n\nRight, now this isn't as bad as it sounds, but ... I was in bed with my Aunty Janice.\n\nWell, actually, in one sense it's exactly as bad as it sounds because when I say I was in bed with her, I don't mean I was in bed with her because we'd gone hill-walking together and been caught out in a snow storm and eventually found shelter in some exceptionally well-appointed bothy that just so happened to have only one bed and we had to get into it together to keep warm; nothing like that. We were fucking.\n\nBut (phew), she wasn't a real aunt; not a blood relation, not even an aunt by marriage. Janice Rae had been Uncle Rory's girlfriend, and I just called her Aunty. However she had been my father's brother's lover, and \u2013 perhaps more embarrassingly \u2013 it had been her daughter, Marion, who had initiated me into the whole sticky, smelly, noisy, potentially fatal, potentially natal, sordid and sublime act in the first place, on the dry, cracked green leather surface of the garaged Lagonda Rapide Saloon's back seat, one hot and musty summer's afternoon, eight years earlier.\n\n(We brought the house down.)\n\nBlame Lewis.\n\nThe voice has gone quiet, deep, almost gravelly now. A light \u2013 harsh and white \u2013 shines from one side, so that his lean, clean-shaven face looks hard and angular, even cruel.\n\n'I have this door in my house,' he breathes, then pauses. 'It's a very special door.' He looks to one side. The way he does it, you get the urge to look that way too, but you don't. 'Do you know what I keep on the other side of that door?' He raises one eyebrow, but there is silence in the darkness. You wait. 'Behind the door I keep ...' (He leans forward now, towards us, somehow confiding and threatening together.) '... the rest of the Universe.' A wintery smile, and if you were prone to that sort of thing, your skin might crawl.\n\nThere is a little nervous laughter. He waits patiently for it to subside. 'I have a special name for that door,' he says, eyes narrowing. 'Do you know what I call it?' (This is the dangerous bit, where it could all end in disaster, but he holds the pause, and the silence is eloquent.) 'I call it ...' he pauses again, looks into the darkness to one side, then towards the light again. '... my Front Door.'\n\nThere is more laughter, like relief. He smiles for the first time; a thin, unimpressed expression. 'Perhaps you have one like it, in your house.' He steps back, the lights go up, and he makes a sort of half nod, half bow. 'My name is Lewis McHoan. Good night.'\n\nHe walks off to loud applause; cheers, even.\n\nI look from the television to my flatmates.\n\n'Aye, he's no bad,' Gav says, pulling open another can of cider.\n\n'He's okay,' agrees Norris, and drinks from his. 'That last bit was a bit weird but. He really your brother, aye?'\n\nI glare at the screen as the MC appears, signing off. Lewis had been the last act. 'Yes.' I say, taking my empty Export can between both hands, and crushing it. 'Yes, he is.' The credits roll. I throw the squashed can at the litter bin, but it misses, hits the wall, rolls across the floor and dribbles flat beer onto the threadbare carpet.\n\nI stood in the bookshop, reading the story about the magic dressing gown, tears in my eyes.\n\nA hand tapped me on the shoulder. I put the book down quickly on the pile and hauled my hanky from my pocket, bringing it up to my face as I turned. I blew my nose.\n\n'Come on, slow-coach,' mum said, smiling down at me. Her gaze flicked to the book-pile. 'Reading your dad's stories at last, eh? What's brought this on?' Not waiting for an answer, she put one arm round my shoulders and guided me out onto the Departures concourse. 'Come on; let's go and wish your Uncle Rory _bon_ voyage, shall we?'\n\n'All right,' I said, sniffing.\n\nMum frowned down. 'Prentice, have you been crying?'\n\n'No!' I said vehemently, shaking my head and stuffing the hanky back into my trousers. Mum just smiled. I felt the tears try to come again, prickling behind my eyes.\n\n'Prentice!' Uncle Rory said, picking me up. 'God, you're getting big. I'll soon not be able to lift you.'\n\nGood, I thought; this is embarrassing. I hugged him, as much to get my face out of sight as to express any regret at his leaving.\n\n'Aye,' I heard my mum saying. 'I think we had a wee tear or two, there.'\n\n'We didn't, did we?' Uncle Rory laughed, bringing me back round in front of him, holding me there. His big face, entirely framed by curly auburn hair, looked happy and kindly. I wanted to hit him and my mum, or maybe burst into tears and hug them; either would do. 'Ah, dinnae greet, laddy,' he laughed, lapsing into the working-class Scots I had grown ashamed of because my beautiful cousins Diana and Helen didn't speak like that, and those coarse Watt children did.\n\nStop it! I beamed at him (I was trying to develop a technique for aiming my thoughts at people to get them to do things for me; there were promising developments, but it was early days still, and I was suffering a lot of teething problems. That bastard George Lucas hadn't had the decency to reply to my letter about The Force yet, either).\n\n'I have not been crying, honest I haven't, Uncle Rory,' I said, sniffing.\n\n'Of course you haven't,' Uncle Rory grinned, winking at my mum.\n\n'That's right,' I said. Now put me down!\n\nUncle Rory put me down with a grunt. 'That's better,' he said, roughing up my hair. 'Ah; a wee smile!'\n\n_Of course I'm smiling, you big fool; you are prey for my thoughts!_\n\n'Will you be away awful long, Uncle Rory?' I asked.\n\n'Yes, I dare say I will, Prentice,' Uncle Rory said. The PA system shouted that the Heathrow flight was boarding. The voice mentioned something about a gate, but I doubted it would be anything as interesting as a stargate. Uncle Rory picked up his shoulder bag and the three of us started to walk towards a big crowd of people. A loud roar outside the glass expanse of one wall sounded excitingly like a crash ... but it was only a plane landing.\n\n'If you're in Hollywood and bump into George Lucas \u2013'\n\nUncle Rory laughed mightily, and exchanged one of those infuriatingly knowing adult looks with my mum. 'I don't think that's very likely, Prentice, but if I do ...'\n\n'Will you ask him if he got my letter?' I said. We reached the place where everybody was standing around and hugging, and we stopped. 'He'll know what it's about.'\n\n'I certainly will.' Uncle Rory laughed, squatting down. He made a worse mess of my hair and gripped both shoulders of my blazer. 'Now you be a good boy and I'll see you all in a few months.' He stood up. Him and mum had a brief cuddle, and she kissed him on the cheek. I turned my face away. I was glad my father wasn't here to see this. How could they do that sort of thing in public? I had a look round to see if my dad was watching from behind a potted palm or through holes cut in a newspaper, but he didn't seem to be.\n\n'Bye, Rory; safe journey.'\n\n'Bye, Mary. Tell Ken I'll call when I can.'\n\n'Will do. Take care now.'\n\nUncle Rory grinned. 'Yeah.' He squeezed one of her shoulders and winked at her _again!_ 'Bye love; see you.'\n\n'Bye.' We watched him show his ticket to the man at the gate, then with one last wave he was gone.\n\nI turned to mum. 'Mum, can I have some more money for the Star Wars machine?' I pointed at the video games. 'I got through three stages last time and I almost got to the fourth; I think I know how to deal with the big towers now and I'm getting really good at \u2013'\n\n'I think you've had quite enough of that machine, Prentice,' mum said, as we walked away through the people. We were heading for the stairs. I tried to pull her towards the row of video games.\n\n'Aw, mum, please; come on; I'll let you watch if you like.'\n\n_You will let me play the machine. You will let me play the machine._\n\nShe had the nerve to laugh. 'That's very kind of you, Prentice, but I'll pass on that. We have to get back home.'\n\n'Can I go home on the train mum, please can I?'\n\n_You will let your son take the train home. You will let your son Prentice take the train home._\n\n'Something wrong with my driving, you wee rascal?'\n\n'No mum, but can I please?'\n\n'No, Prentice; we'll take the car.'\n\n'Aww, but mum ...'\n\n'Will I buy you a book?' Mum stopped near the bookshop. 'Would you like that?'\n\n'There's a Judge Dredd annual out,' I said helpfully.\n\nShe _tssked._ 'Oh, I suppose, if it'll keep you quiet...'\n\nWhile she paid for it, I went to the pile of dad's books, and when nobody was looking I tore a couple of pages in one book, then put a load of somebody else's books over the top of dad's, so that nobody could see them.\n\nHow _dare_ he take the stories he'd told me and Lewis and James and the others and tell them to other people, to strangers? They were ours; they were _mine!_\n\n'Come on, terror,' mum said.\n\nA hand between my shoulder blades propelled me from the shop. But at least it wasn't the Vulcan Death Grip.\n\n_You will change your mind about letting your son take the train._\n\n_Mrs Mary McHoan, you will change your mind about letting your son Prentice take the train home ... and about playing the Star Wars machirie ..._\n\n'I mean, nobody tells you sex is going to be so noisy, do they? I mean, they can be quite specific about the actual act itself; there is no gory detail, no technical nuance that is not gone into, by teachers or parents or books about sex or the Joy of LURVE or television programmes or just the boys or girls in the year above you at school telling you behind the bike sheds, BUT NOBODY TELLS YOU ABOUT THE NOISE!\n\n'They don't! The first time I ever got laid it was the summer, it was hot, we were doing it naked in the old missionary position, and there I was, trying to pretend I'd been doing this for years, and thinking am I doing this right? Was that enough foreplay, did I devote sufficient time to going down on her or did it look like I was doing it because I read you ought to in _Cosmopolitan..._ and I did want to spend more time down there, but my neck was getting sore ... and I'm thinking should I start chewing the other earlobe now, and should I sort of pull back so I can get my mouth to her nipples, because I'd like to suck them; I would, but my neck's still sore, and just as I'm thinking about all this, and _still_ trying to think about putting this MFI kitchen unit together to stop myself from coming too soon but it isn't working any more more because I keep thinking of screws and pre-drilled holes and male and female parts and I'm stroking her and it's great and she's panting and I'm panting and then, just then, from in between our two naked, heaving bodies, THERE IS A NOISE LIKE A RHINOCEROS FARTING!\n\n'There is the noise of a fart the like of which you have never heard in your life before; it echoes off nearby tall buildings; it leaves your ears ringing; little old half-deaf ladies three streets away run to the broom cupboard and start hammering on the ceiling and threatening their upstairs neighbours with the Noise Abatement Society. I mean, a Loud Fart, okay?\n\n'And she is laughing and you don't know what to do; you try to keep going but it happens again and she's in hysterics and it is all deeply, deeply, _deeply_ embarrassing, and you keep going but there's this constant farting noise caused by all the sweat and it just isn't the same any more and you're thinking why didn't they tell me about this? Why wasn't I told? I mean, do other people put a towel in between them, or what?\n\n'... And you come eventually and after a cuddle and you've whispered a few sweet somethings, you withdraw, holding the old johnny on because that's what it says on the packet after all, and you go to the loo to dispose of the horrible dangly greasy thing and you have a very full bladder by now and you think you'll have a pee ... Ha ha ha ha ha; WRONG! You _think_ you'll have a pee, but you _can't!_ ...'\n\nI shook my head, remembering the times Lewis had ranted away like this in the past; in pubs, amongst friends, at parties. I'd enjoyed it usually, back then; I'd felt almost privileged to witness these chaotic fulminating tirades, and even been proud that Lewis was my brother ... But then I'd come to my senses and decided that my elder sibling was in fact a vainglorious egomaniac with a runaway sarcasm-gland problem. Now he was taking what had been relatively amusing examples of a private wit and exposing them to everybody, to make money and amass praise. My family are always doing this sort of thing to me.\n\nI looked at Gav. Gav was standing at my side, clutching his pint glass up near his shoulder and howling with laughter. He was sweating. He had tears in his eyes and his nose was running. He was having a great time. Gavin \u2013 one of my two flatmates \u2014 is a chap of the world; he has been there, he has done all this, he has had everything that Lewis was describing happen to him, too, and he didn't mind who knew it; this was the comedy of recognition; it was mature, it was happening, it was ideologically correct in terms of sexual politics, but it was also extremely _rude,_ and Gav just thought it was all totally hilarious. He was spilling what was left of his pint down his coat, but I suspected he wouldn't have cared even if he had noticed.\n\nI shook my head again and looked back at the low stage, where Lewis was still stalking back and forth like a caged hyena, grinning and sweating and gleaming under the lights and shouting into the microphone and flinging one arm about and smiling wickedly and striding side to side, side to side, talking to individuals at the front, to the people at the side and in the middle of the crowded audience, talking to us standing here at the back, talking to everybody.\n\nLewis was dressed in black jeans and a white tuxedo over a white T-shirt which had three enormous black letters on it; FTT. In much smaller letters underneath, it read: (have carnal knowledge of the conservative and unionist party and their supporters). You could buy these T-shirts at the door. Gav had one, wrapped in polythene and stuffed in one pocket of his coat.\n\nWe were upstairs in Randan's, the latest incarnation of a bar that had previously traded under the name Byre's Market, and before that had been called Paddy Jones's; premises forever apostrophised. That original appellation was before my time, and I confess to a degree of yearning for an age when bars had, in the main, sensible names, and did not pride themselves on serving their own creakingly-titled cocktails, a Choyce Selection of Our Eftimable Home-Made Pies, Hotpottes And Other Fyne Dishes, and twenty different designer lagers, all of which taste identical, cost the earth and are advertised on the tellingly desperate Unique Selling Points of having a neat logo, a top that is difficult to open or a bottle neck whose appearance is apparently mysteriously enhanced by having a slice of citrus fruit rammed down it.\n\nBut if this is the price we have to pay for all-day opening and letting women into public bars, then I admit it may well be churlish to carp. I used to think dad was kidding about bars closing in the afternoon, and at ten in the evening (TEN, for Christ's sake; I don't go out until midnight sometimes!), and about some not having women's toilets at all ... but apparently it's all true, and scarcely a decade and a half gone.\n\nI looked at my watch, wondering how long Lewis was going to keep this up. Telling conventionally-structured jokes uses up material appallingly quickly and if that had been what Lewis was up to I might not have had the prospect of enduring too much more challenging, non-sexist, politically aware, near-the-bone (well, near the bone-head, at any rate) alternative humour, but this observational stuff \u2013 telling people things they already know and getting them to pay you for the privilege (sort of the light entertainment equivalent of psychoanalysis) \u2013 can go on virtually indefinitely. Indeed, I felt like it already had.\n\nLewis was moderately big all of a sudden, after a series of appearances on that late-night TV show. The programmes had been recorded at a Comedy Festival in Melbourne, Australia, which Lewis had been invited to (hence his inability to make old Margot's funeral). Tonight was the premiere date on his first solo UK tour, and it looked depressingly likely that it would be totally sold out, thanks to the advertising power of television. If he hadn't given me the complimentary tickets I doubted that Gavin and myself would have stood any chance of getting in (but then if he hadn't given me the complimentary tickets a troop of wild Clydesdales on speed wouldn't have dragged me here).\n\nI looked at my watch again. Half an hour gone. So far he had said exactly one thing I found even slightly amusing, and that was right at the start: 'At one stage I thought I was a complete asshole.' (There followed the inevitable pause for effect). 'But I passed through that.'\n\nLaugh? I almost.\n\n'... about my family, ladies and gents, because I come from this very strange family, you know; very strange family indeed ...' Lewis said.\n\nGav turned, big red face beaming; he nudged me. I didn't turn to look at him. I was staring \u2013 glaring \u2013 at the stage. My mouth felt dry. He wouldn't dare, would he?\n\n'There's my Uncle Alfred \u2013'\n\nI started to relax. We do not have an uncle Alfred. Still, maybe he was going to use some true or embroidered slice of family history and just disguise it with a false name.\n\n'Uncle Alfred was a very unlucky man. He was so unlucky we actually called him Unlucky Uncle Alfred. We did. Unlucky Uncle Alfred was so unlucky, he's the only man in history ever to have been killed by an avalanche on a dry ski-slope.'\n\nI relaxed a bit more. He hadn't dared. This was just a joke.\n\n'No, really. He was skiing down when it sort of started to come undone at the top and roll down ... crushed to death by three hundred tons of nylon tufting. Haven't been able to look at a Swiss Roll the same way since.'\n\nAnother nudge from a highly amused Gavin. 'That true, Prentice, aye?'\n\nI gave what I hoped was a suitably withering look, then turned back to the stage. I drank my heavy and shook my head.\n\n'Prentice,' Gav insisted from my side, missing the first part of Lewis's next mirth-infused effusion. 'Zat true, aye?'\n\nObviously my withering look needed more work in front of the mirror. I turned to Gavin. 'Every word,' I told him. 'Except his real name was Uncle Ethelred.'\n\n'Aw aye.' Gav nodded wisely, took a sip from his beer without significantly moving the glass from near his right shoulder, and frowned as he tried to catch up with what Lewis was saying, only to succeed in catching the predictably below-the-belt punch-line. Everybody else laughed, so so did Gav, no less enthusiastically than anybody else, and, interestingly, no less enthusiastically than he had at any other part of Lewis's act, when he'd heard every word. Remarkable. I watched Gav for a while from the corner of my eye, wondering, not for the first and \u2013 barring serious accidents and justifiable homicide \u2013 almost certainly not for the last time, what I was doing sharing a flat with somebody whose cogitative powers I had last had cause to ponder only a few hours earlier, when I had discovered \u2013 while watching the news with Gav \u2013 that he had believed up until then that the Intifada was an Italian sports car.\n\nIn a way I envied Gav, just because he found life such a hoot. He also seemed to think that it was \u2013 like himself, perhaps \u2013 comparatively uncomplicated. As is the way with such things, these subjectively positive qualities tend to have precisely the opposite effect on the temperaments of those in close proximity to the person concerned.\n\nThis was a man, after all, who had not yet mastered something as fundamental and as linear in its properties (for the most part) as running a bath at the correct temperature. How many times had I gone into the bathroom in our flat to find that the bath was full almost to the brim of hot, steaming water? This was an indication that Gav was planning to bathe in an hour or so. Gavin was of the opinion that the way to draw a bath was to fill it entirely from the tap that had the little 'H' on it (thereby reducing the flat's supplies of immediately available hot water to zero), then leaving the resulting body of liquid to cool to something approaching a state in which a human body could enter it without turning instantly the colour of a just-boiled lobster. This normally took about thirty minutes in the depths of winter, and sometimes well over an hour in high summer, during which time Gav was inclined to amuse himself watching television \u2013 soap operas and the less intellectually taxing game shows, preferably \u2013 or eating, say, banana and Marmite sandwiches (just one example from Gavin's extensive repertoire of unique snackettes that entirely substituted culinary originality for anything as boring as tasting pleasant).\n\nMy attempts to explain the subtle dialectics of utilising both hot and cold taps \u2013 consecutively or concurrently \u2013 to produce a bath that could be used immediately without recourse to the Western General's burns unit (with the resulting benefits of freeing the bath for the use of others earlier and in the process using a great deal less electric power, which both we and the planet could ill afford), fell not so much on deaf ears as on open-plan ones. In automotive terms, if Lewis was a motor-mouth, then Gavin was a cross-flow head.\n\nI drained my glass, studied the flattening dregs of foam at the bottom.\n\n'Nuther beer, big yin?'\n\n'No thanks, Gav; I'll buy my own.'\n\nGavin, I had long ago concluded, believed that life revolved around rugby and beer, and that \u2013 especially under the influence of too much of the latter \u2013 sometimes it just revolved. Perhaps it might be a mistake to match him pint for pint.\n\n'Ah; go on. Heavy, aye?' He grabbed my empty glass, and with that he was gone, shouldering his way through the pack of bodies for the distant dream that was the bar. He was still grinning inanely. Probably a good point for him to mount an expedition to the bar. Lewis was in the middle of a long, right-on, faux-na\u00efve spiel about post-isms which Gav probably found a little bewildering. ('I mean, what is post-feminism? Eh? Answer me that? What do they mean? Or have I missed something? I mean, was there a general election last week and nobody told me about it and half the MPs are now women? Are fifty per cent of the directors of all major industries female? Is it no longer the case that the only way to hold on to your genitals if you're brought up in Sudan is to be born a boy? Don't Saudi Arabian driving licences still have a section that says Title: Mr, Mr or Sheik, please delete?')\n\nI really had been going to buy my own drink; anybody who has ever been hard-up will tell you it's the easiest way to regulate one's finances while still remaining nominally sociable, but Gav, profligate though he may have been with the heat plumes from his baths (and kettles; Gavin's determination to wreck the ecosphere through the generation of copious volumes of unnecessary hot water extended to never boiling a kettle that was less than brim-full, even if only a single cup was required), was equally generous when it came to buying drink. At such moments it was almost possible to forget he was also the inventor of custard and thousand-island dressing pudding.\n\nMy brother seemed to be thinking along the same epicurean lines. However, to my horror (emulsified with a small amount of schadenfreudian delight), he appeared to be proposing to sing.\n\nI closed my eyes and looked down, ashamed not just for Lewis but for my whole family. So this was the cutting edge of British alternative humour. Finishing with a song. Good grief.\n\nI shall draw a veil over this performance, but let history record that this pretended paean of praise for Mrs Thatcher \u2013 comparing her to various foods, with only a hint of sarcasm most of the way through ('as English as Blueberry pie') \u2013 ended with the couplet 'Maggie, you're a Spanish omelette, like an egg you just can't be beaten, \/ Maggie, you're all the food that I eat ... twenty-four hours after it's eaten.'\n\nThe puzzled patrons of Randan's, who had been worriedly thinking that perhaps Lewis wasn't quite so right-on after all, and had had his head turned by a sniff of fame and a glimpse of the flexible stuff, suddenly realised their man was still okay (phew), and it had all been an elaborate joke (ha!) as well as a knowing dig at more conventional comedians (nudge), and so duly erupted with applause (hurrah!).\n\nI breathed a sigh of relief that at last it was all over \u2013 barring encores, of course \u2013 clapped lightly, looking at my watch as I did so. A glance revealed that the besieged bar was under further pressure now that the attacking forces had been reinforced following the end of Lewis's act. I suspected that for all my scorn I might yet be grateful for Gav's rugbying skills that evening, not to mention his Neanderthal build (perhaps that was why he found rugby so attractive; he was a throw-back!).\n\nI looked at my watch again, wondering if Lewis would be unduly insulted, and Gav overly disappointed, if we didn't go back-stage to see the great performer afterwards. Things had gone so appallingly well that Lewis would undoubtedly be on a high and hence unbearable.\n\nPerhaps I could plead a headache, if that wasn't too un-butch for Gav to accept. ('Ach, have another few beers and a whisky or two and it'll soon go away, ya big poof,' would be the sort of reply my flat-mate would favour, as I knew to my cost.)\n\n'Excuse me, are you Prentice? Prentice McHoan?'\n\nI'd noticed the woman sidling through the crowd in my direction a few seconds earlier, but paid no real attention, assuming I just happened to be on her route.\n\n'Yes?' I said, frowning. I thought I recognised her. She was short, maybe early forties; curly brown hair and a round, attractive face that looked run-in without being worn out. I coveted her leather jacket immediately, but it wouldn't have fitted me. A glint in her eyes could have been animal lust but was more likely to be contact lenses. I tried to remember where I'd seen her before.\n\n'Janice Rae,' she said, offering her hand. 'Remember?'\n\n'Aunty Janice!' I said, shaking her hand. I suspected I was blushing. 'Of course; you used to go out with Uncle Rory. I'm sorry I knew I recognised you. Of course. Aunt Janice.'\n\nShe smiled, 'Yeah, Aunt Janice. How are you? What are you doing?'\n\n'Fine,' I told her. 'At Uni; last year. History. And yourself?'\n\n'Oh, keeping all right,' she said. 'How are your parents, are they well?'\n\n'Fine. Just great,' I nodded. I looked round to see if Gav was on his way back; he wasn't. 'They're fine. Umm ... Grandma Margot died last month, but apart from that \u2013'\n\n'Oh no!' she said. 'Margot? Oh, I'm sorry.'\n\n'Yes,' I said. 'Yes, well, we all were.'\n\n'I feel terrible; if only I'd kept in touch ... Do you think it would be all right if I, if I wrote ... to your mum and dad?'\n\n'Oh, sure; yeah; fine. They'd be delighted.'\n\n'Even if I'd just made the funeral ...' she said, downcast.\n\n'Yes ... Big turn-out. Went ... not with a whimper.' I nodded at the empty stage. 'Lewis couldn't make it, but everybody else was there.'\n\nHer eyes widened; it was like a light went on beneath her skin, then started to go out even as she said, 'Rory, was he \u2013?'\n\n'Oh,' I said, shaking my hand quickly in front of her, as though rubbing something embarrassing out on an invisible blackboard. 'No; not Uncle Rory.'\n\n'Oh,' she said, looking down at her glass. 'No.'\n\n''Fraid we haven't heard anything for, well, years.' I hesitated 'Don't suppose he ever got in touch with you, did he?'\n\nShe was still looking at her glass. She shook her head. 'No; there's been nothing. No word.'\n\nI nodded my head, looked around for Gav again. Janice Rae was still inspecting her glass. Broke or not I'd have offered to buy her another drink, but her glass was full. I was aware that I was sucking in my lips, trapping them between my teeth. This is something I do when I'm feeling awkward. I wished she would say something more or just go away.\n\n'I always felt,' she said, looking up at last, 'that your dad knew more than he was letting on.'\n\nI looked into her bright eyes. 'Did you?'\n\n'Yes. I wondered if Rory was still in touch with him, somehow.'\n\n'Well, I don't know,' I said. I shrugged. 'He does still talk about him as though ...' I had been going to say as though he were still alive, but that might have hurt her. 'As though he knows where Uncle Rory is.'\n\nShe looked thoughtful. 'That was the way I felt, when I was down there, after Rory ... left. There was one time when ...' She shook her head again. 'I thought he was going to tell me how he knew; let me in on his secret, but ... well, at any rate, he never did.' She smiled at me. 'And how is Lochgair? Your parents still in that big house?'\n\n'Still there,' I confirmed, catching sight of Gav making his way through the scrum of bodies, concentrating on the two full beer glasses in front of him.\n\nJanice Rae looked warm and happy for a moment, and her eyes narrowed a little, her gaze shifting away to one side. 'It was a good place,' she said softly. 'I have a lot of happy memories of that house.'\n\n'I guess we all do.'\n\nUncle Rory had met Janice Rae at some literary do in Glasgow. She was ten years older than him, a librarian, divorced, and had a ten-year-old daughter called Marion. She lived with her mother, who looked after Marion while Aunt Janice was at work. I could remember the two of them coming to the house for the first time. Uncle Rory had brought various women to the house before; I'd ended up calling them all aunty, and I was calling Janice that by the end of the first weekend they spent at Lochgair.\n\nDespite the fact that Marion was a girl and a couple of years older than me, I got on well enough with her. Lewis \u2013 also two years older than me \u2013 was going through an awkward stage during which he wasn't sure whether to treat girls with scorn and contempt, or sweeties. James, born the year after me, liked what and who I liked, so he liked Marion. She became one of The Rabble, the generic and roughly affectionate term my father applied to the various kids he would tell stories to on a Family Sunday.\n\nA Family Sunday was one when either the McHoans or the Urvills played host to the other family, plus that of Bob and Louise Watt. Aunt Louise had been born a McHoan; her father was the brother of Matthew, my paternal grandfather and husband of Grandma Margot, she of the heart that broke only after she was safely dead. Bob Watt was brother of Lachlan, whose taunting of Uncle Fergus concerning the matter of hiding inside a medieval lavatory led to the unfortunate incident with the display case and resulted in Lachlan becoming the man with four eyes, but who did not wear glasses.\n\nBob Watt never turned up for Family Sundays, though Aunt Louise did, often wearing thick make-up and sometimes dark glasses. Sometimes the bruises showed through, all the same. Now and again there'd be something she didn't even try to hide, and I can recall at least two occasions when she turned up with her arm in a sling. I didn't think very much about this at the time, just assuming that my Aunt Louise was somehow more fragile than the average person, or perhaps excessively clumsy.\n\nIt was Lewis who eventually told me that Bob Watt beat up his wife. I didn't believe him at first, but Lewis was adamant. I puzzled over this for a while, but at length just accepted it as one of those inexplicable things that other people did \u2013 like going to the opera or watching gardening programmes \u2013 which seemed crazy to oneself but made perfect sense to the individuals concerned. Maybe, I thought, it was a Watt family tradition, just as Family Sundays and at least one person in each generation of our family managing the Gallanach Glass Works seemed to be two of our traditions.\n\nMum and Aunt Janice became friends; she and dad were much closer in age to Janice than Rory, and they were parents, too, so perhaps it was no surprise they got on. Whatever; after Uncle Rory disappeared, Aunt Janice and Marion still came down to the house every now and again. It was the year after Rory vanished that Marion, then about fifteen, got me into the garage where the car was. We'd been out on our bikes, riding round some of the forestry tracks one hot and dusty September day; everybody else was in Gallanach, shopping, or \u2013 in Lewis's case \u2013 playing football.\n\nMarion Rae had the same curly brown hair her mother did. She had a round, healthy-looking face which even I could see was quite pretty, and was about the same height as I was, though a little heavier (I was of that age and body-type concerning which adults help to ease the difficult journey through the age of puberty by making remarks about disappearing if you turned sideways, and running around in the shower to get wet). We'd seen some old burnt-out wreck of a car abandoned in a ditch, up in the hills; I'd said something about the sports car under the covers in the courtyard garage back at the house; Marion wanted to see it.\n\nI still maintain I was seduced, but I suppose I was inquisitive as well. Girls were still less interesting to me than models of the Millennium Falcon and my Scalextric set, but I had conducted a couple of masturbatory experiments which had set me thinking, and when Marion, exploring the warm, dim, tarpaulin-green gloom of the old car with me, said, Phew she was hot, wasn't I? and started unbuttoning her blouse, I didn't say No, or run away, or suggest we get out of the stuffy garage.\n\nInstead I blew on her.\n\nWell, she was sweaty, and I could see moisture on the top of her chest, above the little white bra she was wearing, trickling between the white swells of her breasts. She seemed to appreciate the gesture, and lay back and closed her eyes.\n\nI remember her asking if I wasn't hot, and feeling my leg, and her hand running up to my thigh, then there was some silly line like, 'Oh, what's this?' as she felt inside my shorts, expressing what even then I thought was probably fake surprise at what she discovered there. My own words were no less inane, but something \u2013 either the heat of the moment or just retrospective embarrassment \u2013 seems to have wiped them and most of the subsequent relevant details from my memory. Still, I recall being pleased that everything seemed to fit, and work as well, and if our (now I think about it, ridiculously fast) mutual thrusts hadn't unsettled the car on its blocks, that sense of having successfully risen to the occasion and worked out what to do with relatively little guidance would have been my abiding impression of the proceedings.\n\nInstead, just as I was both coming and going (going; 'Wow!'), and Marion was making some extremely interesting noises, the car collapsed under us.\n\nIt shuddered and fell onto the concrete floor of the garage with an apocalyptic crash. We'd shaken it off its blocks. Some bizarre sense of symmetry had made me insist that we should not lie across the back seat, but that I should instead squat on the transmission tunnel, with Marion half on the rear seat, and half on me. As a result, the Rapide fell backwards off its wooden supports and its boot rammed into a load of drums and cans stored behind it, crushing them in turn against an old Welsh dresser that had been consigned to the garage years earlier; this \u2013 loaded up with tins and tools and spare parts and junk until it was top heavy \u2013 proceeded to over-balance. It leant, creaking, towards the car, and \u2013 although it did not actually fall over \u2013 distributed most of its load of paint, spanners, plugs, bolts, spare bulbs, bits of trim, hammers, wrenches and assorted boxes and tins all over the tarpaulin-covered boot, rear window and roof of the Lagonda.\n\nThe noise was appalling, and seemed to go on forever; I was dead still, my orgasm \u2013 more quality than quantity \u2013 completed, and my mouth hanging open as the cacophony reverberated through the garage, the car and my body. Dust filled the car's interior; Marion sneezed mightily and almost squeezed me out of her. Something heavy hit the rear window, and it went white all over, crazed into a micro-jigsaw of tiny glass fragments.\n\nEventually the noise stopped, and I was about to suggest that we ran away very soon and to some considerable distance before anybody discovered what had happened, when Marion grabbed both my buttocks with a grip like steel, stuck her panting, sweat-streaked face against mine, and snarled those words with which I \u2013 in common with most men, I suspect \u2013 would eventually become relatively familiar, in similar, if rather less dramatic situations: 'Don't Stop.'\n\nIt seemed only right to comply, but my mind wasn't really on what I was doing. Another precedent, perhaps.\n\nMarion seemed to have some sort of fit; it coincided with \u2013 or perhaps was the cause of \u2013 the rear window falling in. It showered us both with little jagged lumps of glass, green under the tarpaulin-light, like dull emeralds. We both stayed like that for a bit, breathing heavily and brushing crystalline fragments out of each other's hair and laughing nervously, then started the delicate business of disengaging and trying to dress in the back of a tarpaulin-covered car full of gravelly glass.\n\nWe completed dressing outside the car, in the garage, shaking bits of glass out of our clothes as we did so. I had the presence of mind to put these fragments back into the car, and spread the glass more evenly over the seat, removing the shard-shadow of Marion from the cracked green leather (there was, I noticed with a little pride and considerable horror, a small stain there \u2013 probably more Marion than me, to be honest \u2013 but there was nothing I could do about that beyond wiping it with my hanky). We closed the garage, grabbed our bikes and headed for the hills.\n\nIt was a week before dad discovered the disaster scene in the garage. He never did work it out.\n\nLewis threatened to tell him, but that was only because I'd been stupid enough to blab to my brother, and then been incensed to discover he'd screwed Marion too, twice; on the two previous weekends she'd been down. I immediately threatened to tell the police because Lewis was older than she was and that made it Statue-Tory Rape (I'd heard of this on TV); he said if I did that he'd tell dad about the car ... and so there we were, me barely a teenager and already arguing over a woman with my brother.\n\n'It was good to meet you again, Janice,' Lewis said, shaking Aunt Janice's hand, then taking her elbow in the other hand, kissing her on the cheek. 'You should get in touch with Mary and Ken again; I'm sure they'd love to hear from you.'\n\n'I will,' she said, smiling, then fastened the collar of her glove-leather jacket.\n\nLewis turned to me. 'Bro; sure we can't tempt you?'\n\n'Positive,' I said. 'Got a lot of work to do. Enjoy yourselves.'\n\n'Aw, come on, ya big poof,' Gav said, breathing beer. He put one arm round my shoulders and hugged. From the amount of pressure involved, I gathered he was trying to fold me in half. ''Sno even wan yet!'\n\n'Yes, Gavin, the night is yet senile; but I have to go. You have fun, all right?'\n\n'Aye, okay.'\n\n'Taxi!' shouted Lewis.\n\nWe were standing on Byres Road, outside Randan's, which would be closing soon. Lewis, some guy he'd been friendly with at Uni, a girl who may or may not have been Lewis's girlfriend, and Gav had all decided to head for some bar in the centre of town. I had demurred, as had Janice.\n\n'Prentice; see you at the weekend.' Lewis hesitated as he pulled the taxi door open for the maybe-girlfriend, then came up to me, hugged me. 'Good to see you, little brother.'\n\n'Yeah; you take care,' I said, patting his back. 'All the best.'\n\n'Thanks.'\n\nThey left in the taxi; Janice and I walked up Byres Road to where she'd left her car. It started to rain. 'Maybe I will take that lift,' I told her.\n\n'Good,' she said. She pulled a small umbrella from her shoulder bag, opened it as the rain came on heavier. She handed it to me. 'Here; you'd better hold this; you're taller.' She took my arm and we had to lean towards each other to keep even our heads dry under the little flimsy umbrella.\n\nShe smelled of Obsession and smoke. She, Gav and I had gone to meet Lewis, holding court in the small dressing room. Later we had all gone to the downstairs bar, then Lewis had announced he wanted to keep on drinking after they called time. Janice had had a couple of fizzy waters, and seemed totally sober, so I reckoned it was safe to accept a lift.\n\n'You don't really like your brother that much, do you?' she asked.\n\n'Yes, I do,' I told her. The traffic hissed by, heading up Byres Road. 'He just ... annoys me sometimes.'\n\n'I thought you seemed a bit reluctant when he suggested going back home this weekend.'\n\nI shrugged. 'Oh, that's not Lewis; that's dad. We aren't speaking.'\n\n'Not speaking?' She sounded surprised; maybe amused. 'Why not?'\n\n'Religious differences,' I said. It had become my stock reply.\n\n'Oh dear.' We turned onto Ruthven Street, away from the bright shop fronts and traffic. 'Still a bit further to go,' she said.\n\n'Where are you parked?'\n\n'Athole Gardens.'\n\n'Really? Not a good place to live if you had a lisp.'\n\nShe laughed, squeezed my arm.\n\nHello, I thought. I switched the umbrella from one hand to the other and put my arm lightly round her waist. 'I hope I'm not taking you out of your way. I mean, I could walk. It isn't far.'\n\n'No problem, Prentice,' she said, and put her arm round my waist. Hmm. I thought. She gave a small laugh. 'You were always thoughtful.' But somehow, the way she said it, I thought, No, she's just being _friendly._\n\nWe got into the Fiesta; she dumped the brolly in the back. She put both hands on the wheel, then turned to me. 'Listen, I've got some ... some papers Rory left with me. I did mean to send them to your father, but to be honest I lost track of them, and then didn't find them again until mum died and I was clearing stuff out ... I don't suppose it's anything ... you know, that the family needs, is it?'\n\nI scratched my head. 'Dad has all Rory's papers, I think.'\n\n'It's just old poems and notes; that sort of thing.' She started the car; we put our belts on. She took a pair of glasses from her shoulder bag. 'All a bit confusing, really.'\n\n'Hmm,' I said. 'I suppose dad might want a look at them. Wouldn't mind looking at them myself, come to think of it.'\n\n'Do you want to pick them up now?' She looked at me, her round face soft-looking in the orange blush of the sodium vapour. Her hair was like a curly halo. 'It isn't far.'\n\n'Yeah, okay. I guess so.'\n\nI watched her face. She smiled as we pulled away. 'You sound just like Rory sometimes.'\n\nJanice Rae was the last person known to have seen Uncle Rory, one evening in Glasgow. Rory had been staying with friends in London for the previous fortnight. He had talked to his agent and seen some television people about doing some travel series, but whatever deal he'd been trying to set up with the BBC, it had fallen through.\n\nAt the time Rory was still \u2013 just \u2013 living off Traps, which was attracting a trickle of money even then, when he'd spent everything he'd got for later travel books and occasional articles. He was sharing a flat with an old pal called Andy Nichol who worked in local government; according to Andy, Rory had moped around their flat for a couple of days, shut in his room mostly, supposedly writing, then when Andy had come back from work one day, Rory had asked if he could borrow Andy's motorbike for the night. Andy had given him the keys, and Rory had set off; he'd stopped briefly at Janice Rae's mum's place, and said something about having an idea; some way of saving the project he'd been working on; adding some new ingredient.\n\nHe'd given Janice the folder that she now wanted to give me, eight years later, and then rode off into the sunset, never to be seen again.\n\nHer flat was on Crow Road, not all that far away, down near Jordanhill. As she showed me into the place, down a hall lined with old movie posters, I asked her if she'd ever heard Grandma Margot use the saying: away the Crow Road (or the Craw Rod, if she was being especially broad-accented that day). It meant dying; being dead. 'Aye, he's away the crow road,' meant 'He's dead.'\n\nJanice looked away from me when I said those words, mumbled about the papers and went to get them.\n\nIdiot, I told myself. I stood in the living room; it was full of heavy old furniture that looked as though it belonged somewhere else, and some limited edition modern prints. On a sideboard, there was a photograph of Janice Rae's dead mother, and another of her daughter Marion and her husband. Marion was a police-woman in Aberdeen. I shook my head, grinning and feeling very old and very young at once.\n\n'Here,' Aunt Janice said. She handed me a cardboard folder stuffed with loose papers. On the spine it said CR in black felt-tip. The folder was burgundy but the spine was faded to grey.\n\n'CR?' I said.\n\n_'Crow Road,'_ Janice said quietly, looking down at the folder in my hands.\n\nI wasn't sure what to say. While I was still thinking, she looked up, bright-eyed, glanced around at the walls of the flat and shrugged. 'Yeah; I know. Sentimental of me, eh?' She smiled.\n\n'No,' I said. 'It's ... it's \u2013' The words sweet and nice suggested themselves, but didn't seem right. '\u2013 fitting. I guess.' I stuck the folder under my arm, cleared my throat. 'Well ...' I said.\n\nShe had taken off her jacket; she wore a blouse and cords. She shrugged. 'Would you like some coffee? Something stronger?'\n\n'Umm ...' I said, taking a deep breath. 'Well ... aren't you tired?'\n\n'No,' she said, folding her arms. 'I usually read way past this time of night. Stay; have some whisky.'\n\nShe took my jacket, poured me a whisky.\n\nI sat down on a huge, surprisingly firm old couch. It looked like brown leather, but any smell it had had was gone. I held the whisky glass up. 'Won't you?' I said. This is like playing chess, I thought.\n\n'Well, not if I have to drive you home, Prentice.'\n\n'Oh ... I could ... walk,' I smiled bravely. 'Can't be more than three or four miles. Less than an hour. You'd lend me a brolly, wouldn't you? Or there might be a night bus. Please; have a whisky; sit down, make yourself at home.'\n\nShe laughed. 'Okay, okay.' She went to the table where the bottles were, poured herself a whisky. Somewhere in the distance, that sound of the city: a siren warbling.\n\n'Stay here, if you like,' she said, slowly putting the top back on the bottle. She turned, leaning back against the table, drinking from her glass, looking down at me. 'That's if you want to ... I don't want you to think I'm seducing you or anything.'\n\n'Shit,' I said, putting my glass down on a rather over-designed coffee table. I put my hands on my hips (which is rather an unnatural thing to do when you're sitting down, but what the hell). 'I was kinda hoping you were, actually.'\n\nShe looked at me, then gave a single convulsive laugh, and right until then I think it might still have gone either way, but she stood there, her back to the table, set her glass down upon its polished surface, put her hands behind her back, and looked down, her head forward and a little to the left. Her weight was on her left leg; her right leg was relaxed, knee bent in slightly towards the left. I could see she was smiling.\n\nI knew I'd seen that stance before, and even as I was getting up from the couch to go over to her I realised she was standing just the way Garbo does in _Queen Christina,_ during the Inn sequence, when she's sharing the best room with John Gilbert, playing the Spanish ambassador who doesn't realise until that point the disguised Garbo is a woman, not a man. She starts to take her clothes off eventually, and gets down to her shirt; then Gilbert looks round, does a double take and looks back; and she's standing just like that, and he knows.\n\nIt had \u2013 I recalled, even as I went over to her \u2013 been one of Uncle Rory's favourite old films.\n\nIt was one of those wonderful first nights when you never really do more than drowse between bouts of love-making, and even when you do think no more; that's it, finito ... you still have to say good-night, which itself means a kiss, and a hug; and each touch begets another touch more sweet, and the kiss on the cheek or neck moves to the lips, the lips open, the tongues meet ... so every touch becomes a caress, each caress an embrace, and every embrace another coupling.\n\nShe turned to me, during that night, and said, 'Prentice?'\n\n'Mm-hmm?'\n\n'Do you think Rory's ... away the crow road? Do you think he's dead?'\n\nI turned on my side, stroked her flank, smoothing my hand from thigh to shoulder, then back. 'I really don't know,' I admitted.\n\nShe took my hand, kissed it. 'I used to think, sometimes, that he must be dead, because otherwise he'd have been in touch. But I don't know.' There was just enough light seeping in past the curtains to let me see her head shaking. 'I don't know, because people sometimes do things you'd never have thought they would ever do.' Her voice broke, and her head turned suddenly; she pushed her face into the bedclothes; I moved over to hold her, just to comfort her; but she kissed, hard, and climbed on top of me.\n\nI had, up until that point, been performing an agonising re-appraisal of the indignant signals of total, quivering, painful exhaustion flooding in from every major muscle I possessed. My body's equivalent of the Chief Engineer was screaming down the intercom that the system just wouldn't take any more punishment, Jim, and there was no doubt that I really should have been pulling out and powering down just then ...\n\nBut, on the other hand, what the heck.\n\n' \"... all your nonsenses and truths, your finery and squal-adoptions, combine and coalesce, to one noise including laugh and whimper, scream and sigh, forever and forever repeating, in any tongue we care to choose, whatever lessened, separated message we want to hear. It all boils down to nothing, and where we have the means and will to fix our reference within that flux; there we are. If it has any final signal, the universe says simply, but with every possible complication, 'Existence,' and it neither pressures us, nor draws us out, except as we allow. Let me be part of that outrageous chaos ... and I am.\" '\n\nHer voice was sleepy; the hand that had been quietly ruffling my hair had now gone limp. The litany subsided, the quiet words not echoing in the dark room.\n\nUncle Rory's words, apparently. At first just thought; a mantra to delay ejaculation \u2013 a slightly more civilised, if narcissistic, alternative to brother Lewis's thoughts about constructing MFI kitchen units. Then, once, she had asked him what he thought of when they made love (and smoothed over his protestations of eternal in-head fidelity) to discover that \u2013 purely to prolong her pleasure \u2013 he sometimes recited a piece of his own poetry to himself. He was persuaded to repeat it, for her, and it became a shared ritual.\n\n'Always ... always liked that,' she said quietly, shifting a little to fit her body to mine. 'Always...'\n\n'Hmm,' I said, and felt her breathing alter. 'Good-night, Janice,' I whispered.\n\n'Night, Rore,' she murmured.\n\nI wasn't sure what to feel. Eventually I yawned, pulled the duvet over the two of us, and smiled into the darkness.\n\nI went to sleep wondering what on earth had possessed Uncle Rory to write a miserable, incomprehensible line 'your finery and squal-adoptions'.\n\nWhat in the name of hell was a squal-adoption, for goodness' sake?\n\nThere was something else nagging me; my conscience. The embarrassing truth was that despite having taken a sort of policy decision years ago, the gist of which was: no condom, no sex, Janice and I had not been using one. She'd emplaced a cap, but that, as the leaflets will tell you, don't provide no AIDS protection. So here I was indulging in casual \u2013 if intensive \u2013 sex with a woman I hadn't even heard anything about for eight years; hell, she could have been up to anything! But she had claimed the opposite, and I'd believed her. It was probably the truth, but it was exactly such instances of casually misplaced trust that were undoubtedly going to kill better men and women than me over the next decade or so.\n\nStill, it was done. I drifted away.\n\nI swear I was asleep when my eyes flicked on their own and in a burst of dark certainty I thought: _squalid options!_ that's what he wrote: _Squalid options,_ before going instantly back to sleep again.\nCHAPTER 6\n\nThey sat, stood or lay within the shattered cone-stump of the old broch, looking out over the more recent, but just as empty, equally abandoned, and even more forlorn square crater of the never-used production-platform yard. Above, a lark \u2013 just a speck against the blue \u2013 sang, its shrill voice jetting fluid bursts of song.\n\n'Aw, tell us, Mr McHoan; please.'\n\n'Yeah, dad; what is it?'\n\n'Please, Uncle Ken. Pleeease.'\n\n'Yeah, come on, Mistur McHoan. Tell us. Whit is it?'\n\n'What's what?'\n\n'The sound you can see!' Prentice shouted, jumping down from the broken wall of the broch; Ashley was climbing higher.\n\n'The sound you can see?' he said thoughtfully. He leant back on the sun-warmed stones, looking across the grass circle inside the old ruin, over the spray of grey stones downhill where the broch had fallen or been torn away, over the sharp green tops of the pines to the waters of Loch Fyne. A white-hulled yacht ran gull-winged before the wind, heading north-east up the loch towards the railway bridge at Minard point; perhaps heading for Inveraray. In the distance, a few miles behind, he could see another boat, its spinnaker a tiny bright bulb of pure yellow, like a flower on a gorse bush.\n\n'Well,' he said. 'You can't see it from here.'\n\n'Aw naw!'\n\n'Where can you see it from then, uncle?'\n\n'Well, where we were when I told you about it; we could see it from there.'\n\n'In the Old House?' Diana said, looking puzzled.\n\n'That's right.'\n\n'It isn't the wind, then,' Helen Urvill said, and sat down beside him.\n\nLewis snorted derisively. 'The wind!' he said. 'Don't be so stupid.'\n\n'Aunt Ilsa said it might be breeze block, but I wasn't to say anything until ... aw ... _heck!'_ Prentice flattened his hand and struck it off his forehead with a loud slap; he fell over backwards into the long grass.\n\n'Very amusing, Prentice,' Kenneth sighed.\n\n'Hi, Mr McHoan; look at where I am!'\n\n'Good grief, Ashley; be careful.' She was at the top of the wrecked broch wall, rising into the sky like a grey sine on a sheet of blue paper; Ashley a point.\n\n'I'm no scairt, Mr McHoan!'\n\n'I bet you aren't, but I didn't _ask_ you whether you were scared or not, Ashley; I told you to be careful. Now get down here.'\n\n'I'll come down if ye tell us whit the sound ye can see is, so ah will, Mr McHoan.'\n\n'Get down here, you wee monkey!' he laughed. 'I was about to tell you, before you started hollering. Down; now.'\n\n'Aw, dinnae get yer knickers in a twist, Mr McHoan,' Ashley said, shaking her blonde-haired head and starting to climb down the curved edge of the wall.\n\n'I won't, young lady,' he said. Diana and Helen looked shocked, then giggled. Lewis and Prentice sniggered quietly.\n\n'She said knickers, Mr McHoan,' Dean Watt said.\n\n'Ah'm tellin mum,' Darren told his sister as she made her way, feet and bum first, down the slope of stone.\n\n'Ach, away and bugger yourself, Darren Watt,' the girl said, checking on her next footstep.\n\n'Haaaw!' gasped Diana.\n\n'Ashley!' Kenneth said, exasperated.\n\n'Oh, Mr McHoan, did you hear whit she said! Did ye! Yur a wee bissum, so ye are, Ashley.'\n\n'Yes, I did, and \u2013'\n\n'That's very rude you know, young lady,' Prentice said, wagging his finger at the girl. ('Oh shut up, Prentice,' said Lewis.)\n\n'Ah'm no a bissum \u2013'\n\n'Uncle Ken: what's a bu \u2013'\n\n'Waa! She said \u2013'\n\n'Knickers knickers kni \u2013'\n\n' \u2013buggerlugs.'\n\n'All right, all right!' Kenneth said, raising his voice over a high-pitched babble of Childish. 'That's enough! Do you want to hear the answer or not, you horrible rabble?'\n\n'But-'\n\n'She \u2013'\n\n'Ah'm \u2013'\n\n'Stop it!' he roared. He jumped to his feet and shook one fist in the air, dramatically pirouetting so that the gesture included each of them. 'You're all acting like children! If I'd wanted this sort of treatment I'd have stayed a teacher!'\n\n'But dad, we _are_ children,' Prentice said, rolling his eyes and shaking his head and falling over in the grass again, sighing loudly.\n\n'Innocence is no excuse, Prentice McHoan!' he roared, shaking one finger at the prone child. 'That was the motto of my old school and you'd all better remember it!'\n\nLewis was the only one not amused by the performance. He played with a bit of grass. The others were either laughing outright or sat, bunched up, heads down between shoulders, arms tense by sides, making snorting, guffawing noises and exchanging nodding, wide-eyed looks.\n\n'Oh dear lord!' Kenneth shouted to the bare blue sky, his arms wide, head thrown back. 'Look down upon this awful stupid bairn of mine and teach him some common sense before the world gets him!'\n\n'Ha, Mr McHoan! You dinnae believe in the lord!' Ashley roared from half-way up the wall, almost level with his head.\n\nHe swivelled to her. 'And that's enough of your old buck, Ashley Watt! I don't believe in Santa Claus but Prentice still gets presents at Christmas, doesn't he?'\n\n'Ah!' Ashley said, pointing at him. 'That's different, Mr McHoan; there's hunners of them!'\n\nHe took a step back, looked shocked. 'You little barrack-room barrister; what sort of extra-mural comment is that?' He threw his arms out wide again. To his shock, Ashley jumped right into them, yelling,\n\n'Jurmonimo!'\n\nThe girl slammed into his chest, clunked her head off his chin, little arms whacking round his neck, knees hoofing into his stomach. He put his arms out to hold her, staggered back, almost fell, aware that behind him were the twins, sitting on the grass.\n\nHe bent his knees, flexed his back, and did not collide with or fall over the twins. He wobbled upright with Ashley still clinging to him, legs round his waist now. She smelled ... sweaty, was probably kindest. 'Well,' he wheezed, winded. 'Thank you for that contribution, Ashley.' The others were relatively quiet. Ashley was rubbing her forehead vigorously with one hand. He frowned, lifted the girl away from his chest so he could look at her face. Apart from being grubby, it looked all right. _'What_ did you shout, Ashley?'\n\n'Please, Mr McHoan,' the wee rough voice said, 'Ah said JURMONIMO!'\n\nHe started to laugh and had to put her down. He went to his knees, then sat down and rolled over. All the rest joined in except Ashley, who stood, arms folded, bottom lip stuck out fiercely.\n\n'This isnae funny,' she said, turning away. 'Ah'm away tae get fed.'\n\n'Ha ha ha ha,' Kenneth McHoan said, holding his tummy. 'Ha ha.'\n\n'Were your classes this hilarious?'\n\nKenneth opened his eyes.\n\n'Uncle Rory!' Prentice said, and ran for the man; the boy jumped up onto him rather the way Ashley had leapt down onto his father. Rory laughed and caught him, swung him around, let go an arm and grabbed a leg, whirling the boy round once. 'Wheee!' Prentice yelled. Rory landed him one-handed.\n\nKenneth went to Rory, hugged him. 'God, man, it's good to see you.'\n\n'You too, Ken.'\n\n'You just get back?' Kenneth laughed.\n\n'Ten minutes ago.'\n\nThe two men separated; Kenneth looked his brother up and down.\n\n'Uncle Rory! Uncle Rory! Do some magic; do a trick!'\n\nRory's chestnut curls had been cut back to what was almost a crew-cut; his face was tanned, clean-shaven. Rory pursed his lips, took a coin from his pocket, bent down to the children, made the coin progress across the knuckles of one hand, then disappear into his fist; he waved over it, and when the fist became hand, the coin was gone. Squeals.\n\nRory looked lean and a little tired; his jeans were white with wear, and frayed at one knee. He wore a cheesecloth shirt and smelled vaguely of patchouli.\n\nThe coin re-appeared behind Diana's ear. She put one hand to her mouth, eyes wide. The others went, 'Yaaayy!'\n\nKenneth grinned, shook his head, as Rory straightened, a little stiffly. 'More! More! Do it _again!'_\n\n'Later,' Rory said, looking serious, mysterious, and winking.\n\n'So,' Kenneth said. 'How's the world?'\n\nRory shrugged, 'Still there.'\n\n'Back for long?'\n\nAnother shrug, and an easy smile. 'Dunno. Maybe.'\n\n'Well,' Ken said, putting one arm round his brother's shoulders and starting to walk towards the path, where the still-frowning form of Ashley Watt stood, arms crossed tight as her brows. Ken smiled broadly at her, glanced at Rory. 'Better get all the family in the one place before you start answering questions; otherwise you'll get fed up telling the same stories all the time.' Kenneth turned round, waved to the rest of the children. 'Come on, rabble; your Uncle Rory's back from exotic places and he's got much better stories than me!'\n\nThe children started after them. The two men came up to Ashley; Rory ruffled her hair. She frowned. Kenneth lifted her up with a grunt, held her dangle-legged in front of him. 'Sorry if I upset you, Ashley,' he told her.\n\n'Huh, okay, Mr McHoan,' she said. 'Ah'm sorry ah swore.'\n\n'Okay,' he set her down.\n\nShe looked down the hillside to the forestry track that led back to Lochgair, glanced up at him, then back at the other children, and said loudly, 'Ah bet ah can be back at the hoose first, though but.' She turned and ran.\n\nThe rest raced after her, whooping and hollering past Kenneth and Rory.\n\nKenneth shook his head. 'Preprandial stampede; traditional,' he told his brother. He made a show of squeezing Rory's boney shoulder. 'Woa; feels like you could do with a bit of feeding up yourself.'\n\n'Yeah,' Rory said, looking down at the heather. 'Well, my stories might be a bit thin, too; maybe I should tell them to you first. Let you re-tell the kids.' He gave a small laugh. 'You're the professional fictioneer in the family. I'm just a glorified hack.'\n\n'Hey, is that false modesty or even a note of jealousy there, young Rore?' Kenneth laughed, squeezing his brother's shoulder again. 'Come on, man; I stayed here and had weans and taught weans and you were off getting famous; consorting with tigers and wandering through the Taj Mahal and then wowing us all; fucking celebrity; toast of the town and plenty of bread; literary festivals, awards \u2013'\n\n'Travel writing awards,' Rory sighed.\n\n'Nothing wrong with that. Jeez; last time I _saw_ you, you were on TV. What was that line? \"Better lionised than mauled.\"?' Ken laughed as they walked down the hill.\n\nRory made an exasperated noise, shook his head. 'Ken, don't you remember anything?'\n\nKen looked nonplussed. 'What? Did I get it wrong?'\n\n'No, but that was your line. _You_ said that. Years ago. One night. We were drunk; I don't know ... but you said it, not me.'\n\n'Did I?'\n\n'Yes.'\n\nKen frowned. 'You sure?'\n\n'Positive,' Rory snapped.\n\n'Good grief. I'm wittier than I thought.' Ken shrugged. 'Well; you're welcome to it. But anyway; let your poor old brother have his turn. Don't begrudge me for being able to distract the odd pre-adolescent from the TV for the odd half-hour.'\n\nRory shook his head. 'I don't, Ken,' he said, and sighed again. 'I'm not jealous.' He looked at his brother; bearded, hair still dark, face cheerfully lined but still young-looking. 'Just those end-of-ramble blues.' Rory shrugged, the thin shoulders moving under Kenneth's arm. 'But it's good to be back.'\n\nKen smiled. They saw Prentice walking back up through the grass and fern towards them, panting. The others were kicking up a cloud of dust on the forestry track; a small and highly noisy storm heading Lochgair-ward.\n\n'What is it, Prentice?' Kenneth called.\n\n'Dad!' the boy gasped from some distance off.\n\n'What?'\n\n'What _was_ the sound ...' He took a deep breath. 'You can see?'\n\n'The Sound of Jura!' he yelled. 'Now keep running or you'll get no dinner!'\n\n'Okay!' Prentice called. He jogged off, shaking his head.\n\nThe rain fell with that impression of gentle remorselessness west coast rain sometimes appears to possess when it has already been raining for some days and might well go on raining for several more. It dissolved the sky-line, obliterated the view of the distant trees, and continually roughened the flat surface of the loch with a thousand tiny impacts each moment, every spreading circle intersecting, interfering and disappearing in the noise and clutter of their successors. It sounded most loud as it pattered on the hoods of their jackets.\n\n'Ken, are you sure fish are going to bite in this weather?'\n\n'Course they will, Prentice. Have some faith.'\n\n'Well that's good, coming from you.'\n\nKenneth McHoan looked at his son, sitting looking suitably miserable in waterproofs in the bows of the little boat. 'Just a phrase. I could have said, \"Trust me,\" I suppose.'\n\n'Huh.' Prentice said. 'That's no better. Who was it used to say \"If someone says 'Trust me' ... don't\"?'\n\n'Na,' Kenneth said, shaking his head. 'That was Rory. I never said that.'\n\n'You did!' Prentice said, then seemed to realise he was sounding petulant, and looked away again. He plonked the rear end of the fishing rod down in the bottom of the boat, watched the thin end waggle up and down for a while. He folded his arms, leaned forward, hunching up. 'God, I'm depressed.'\n\n'Cheer up,' Kenneth said, falsely hearty. 'Have some more coffee.'\n\n'I don't want coffee.'\n\n'Well, you forced me into it; I was saving this for later, but ...' Kenneth opened the poppers on the Berghaus jacket, unzipped and dug into the deep internal pocket, pulled out a hip-flask. He offered it to Prentice.\n\nPrentice looked at it, looked away. 'I don't think that's going to solve anything.'\n\nKenneth sighed, put the flask away again, completed reeling in, cast again, and slowly wound the lure in once more. 'Prentice; look, we're all sorry about \u2013'\n\nDarren Watt was dead.\n\nHe'd been on his motorbike, driving to Glasgow one bright day. He was overtaking a truck on the long straight at the start of Glen Kinglas; a car pulled out onto it from the Cowal Road. Darren had assumed the driver had seen him, but the driver had only looked one way; hadn't thought to check there was nothing overtaking on his side of the road. Darren's bike hit the wing of the car doing eighty; he might have survived being thrown into the open road or the heather and grass at the road-side, but he had started to turn as he saw the car coming out in front of him, so hit it at a slight angle; he was catapulted across the road and into a lay-by; he hit the big concrete litter bin full on, and was dead by the time the ambulance arrived.\n\n'It's not just Darren,' Prentice said. 'It's everything; it's ... it's Uncle Rory; Aunt Fiona, and ... shit, it's even doing History, dad. Jesus; do human beings ever just get on with each other? Why are we always at each other's throats?'\n\n'Well, I wouldn't worry about Rory,' Kenneth said quietly.\n\n'Why shouldn't I? He's dead. He must be; it's been six years; we could probably have him legally declared dead.' Prentice kicked the rod. 'Good excuse for a wake; and we wouldn't even have the expense of a coffin or anything.'\n\n'Prentice ...' Kenneth said.\n\n'Well!' Prentice shouted. 'You're always so fucking smug about Rory being alive! What do you know? What makes you so smart?'\n\n'Prentice, calm down.'\n\n'I will not! Christ, dad, do you realise how insufferable you can be? Mr Omniscience. Jeez.' Prentice looked away at the grey landscape of water, cloud and dripping trees.\n\n'Prentice, I don't know for certain Rory's alive, but I'm fairly sure. In a round-about sort of way, he keeps in touch. I think. That's all I can say.' He started to say something else, then stopped himself. 'Oh, I don't know what to say. I want to say, \"Trust me,\" but ... looks like Rory himself has ruled that out. Can't say he isn't right about that ... It's true, most of the time. But I'm not lying to you.'\n\n'Maybe not,' Prentice said. He looked back at Kenneth. 'But you might be wrong about the things you're so busy telling us the truth about.'\n\n'I did say I wasn't certain.'\n\n'Yeah? What about Darren?'\n\nKenneth looked puzzled. He shook his head. 'No, you've lost me; what do you \u2013'\n\n'I can't believe he's just ... gone, like that, Ken. I can't believe there isn't something left, some sort of continuity. What was the point of it all, otherwise?'\n\nKenneth put the rod down, clasped his hands. 'You think Darren's ... personality is still around, somewhere?'\n\n'Why not? How can he be such a great guy, and clever and just ... just a good friend, and some fuckwit forgetting to look both ways cancels out all that ... probably not even a fuckwit; probably some ordinary guy thinking about something else ... How ...' Prentice shoved his hands under his oxters, rocked forward, head down. 'God, I hate getting inarticulate.'\n\n'Prentice, I'm sorry. Maybe it sounds brutal, but that's just the way it is. Consciousness ... goodness, whatever; they haven't got any momentum. They can stop in an instant, just snuffed out. It happens all the time; it's happening right now, all over the world; and Darren was hardly an extreme example of life's injustice, death's injustice.'\n\n'I know!' Prentice put his hands up to the jacket hood, over his ears. 'I know all that! I know it's happening all the time; I know the death squads are torturing children and the Israelis are behaving like Nazis and Pol Pot's preparing his come-back tour; you keep telling us; you always told us! And people just scream and die; get tortured to death because they're poor or they help the poor or they wrote a pamphlet or they were just in the wrong place at the wrong time; and nobody comes to rescue them, and the torturers never get punished; they retire, they even survive revolutions sometimes because they have such fucking useful skills, and no super-hero comes to save the people being tortured, no Rambo bursts in; no retribution; no justice; nothing ... and _that's just it!_ There _has_ to be something more than that!'\n\n'Why?' Kenneth said, trying not to sound angry. 'Just because we feel that way? One wee daft species, on one wee daft planet circling one wee daft star in one wee daft galaxy; us? Barely capable of crawling into space yet; capable of feeding everybody but ... nyaa, can't be bothered? Just because we think there must be something more and a few crazy desert cults infect the world with their cruel ideas; _that's_ what makes the soul a certainty and heaven a must?' Kenneth sat back, shaking his head. 'Prentice, I'm sorry, but I expected better of you. I thought you were smart. Shit; Darren dies and you miss Rory, so you think, \"Bugger me; must be a geezer with the long flowing white beard after all.\"'\n\n'I didn't say \u2013'\n\n'What about your Aunt Kay?' Kenneth said. 'Your mum's friend; she did believe; must be a God; prayed every night, went to church, practically claimed she had a vision once, and then she gets married, her husband dies of cancer within a year and the baby just stops breathing in its cot one night. So she stops believing. Told me that herself; said she couldn't believe in a God that would do that! What sort of faith is _that?_ What sort of blinkered outlook on the world is it? Didn't she _believe_ anybody ever died \"tragically\" before? Didn't she ever read her precious fucking Bible with its catalogue of atrocities? Didn't she believe the Holocaust had happened, the death camps ever existed? Or did none of that matter because it had all happened to somebody else?'\n\n'That's all you can do, isn't it?' Prentice shouted back. 'Shout people down; skim a few useful anecdotes and bite-sized facts and always find something different to what they've said!'\n\n'Oh I'm sorry! I thought it was called argument.'\n\n'No, it's called being over-bearing!'\n\n'Okay!' Kenneth spread his arms out wide. 'Okay.' He sat still for a time, while Prentice remained hunched and tense-looking in the bows. When Prentice didn't say anything, Kenneth sighed. 'Prentice; you have to make up your own mind about these things. I ... both your mother and I have always tried to bring you up to think for yourself. I admit it pains me to think you ... you might be contemplating letting other people, or some ... some doctrine start thinking for you, even for comfort's sake, because \u2013'\n\n'Dad,' Prentice said loudly, looking up at the grey clouds.'I just don't want to talk about it, okay?'\n\n'I'm just trying \u2013'\n\n'Well, stop!' Prentice whirled round, and Kenneth could have wept to see the expression on the face of his son: pained and desperate and close to tears if he wasn't crying already; the rain made it hard to tell. 'Just leave me alone!'\n\nKenneth looked down, massaged the sides of his nose with his fingers, then took a deep breath. Prentice turned away from him again.\n\nKenneth stowed the fishing rod, looked round the flat, rain-battered waters of the small loch, and remembered that hot, calm day, thirty years earlier, on another fishing trip that had ended quite differently.\n\nHe took up the oars. 'Let's head back in, all right?'\n\nPrentice didn't say anything.\n\n'Fergus, darling! You're soaked! Oh; you've brought some little friends with you, have you?'\n\n'Yes, mother.'\n\n'Good afternoon, Mrs Urvill.'\n\n'Oh, it's young Kenneth McHoan. Didn't see you under that hood. Well, jolly good; come in. Take off your coats. Fergus, darling; close that door.'\n\nFergus closed the door. 'This is Lachlan Watt. His dad works in our factory.'\n\n'Oh, really? Yes. Well ... You've all been out playing, have you?'\n\nMrs Urvill took their coats, handling Lachy's tattered and greasy-looking jacket with some distaste. She hung the dripping garments up on hooks. The rear porch of the Urvill's rambling house, at the foot of Barsloisnoch hill, beyond the north-west limits of Gallanach, smelled somehow cosy and damp at the same time.\n\n'Now, I dare say you young men could do with some tea, am I right?'\n\nMrs Urvill was a tall, aristocratic-looking lady Kenneth always remembered as wearing a head-scarf. She wasn't that day; she wore a tweed skirt, sweater, and a pearl necklace which she kept fingering.\n\nShe made them tea, accompanied by some slices of bread and bramble jelly. This was served at a small table in Fergus's room, on the first floor.\n\nFergus had one slice of bread, and Kenneth managed two before Lachy wolfed all the rest. The war was only over a few months, and rationing was still in force. Lachy sat back, belched. 'That was rerr,' he said. He wiped his mouth on the frayed sleeve of his jumper. 'See the breed in our hoose; it's green, so it is.'\n\n'What?' said Kenneth.\n\n'What rot,' Fergus said, sipping his tea.\n\n'Aye it is,' Lachy said, pointing one grubby finger at Fergus.\n\n'Green bread?' Kenneth said, grinning.\n\n'Aye, an' ah'll tell ye why, tae, but ye've goat tae promise no tae tell anybudy.'\n\n'Okay,' Kenneth said, sitting forwards, head in hands.\n\n'Hmm, I suppose so,' Fergus agreed unenthusiastically.\n\nLachy glanced from side to side. 'It's the petrol,' he said, voice low.\n\n'The petrol?' Kenneth didn't understand.\n\n'Load of absolute rot, if you ask me,' Fergus sneered.\n\n'Na; it's true,' Lachlan said. 'See the Navy boys, oot oan the flyin boat base?'\n\n'Aye,' said Kenneth, frowning.\n\n'They pit this green dye in thur petrol, an if yer foun wi that in the tank uv yer motor car, ye get the jile. But if ye pit the petrol through breed, the dye comes oot, an ye can use the petrol an naebudy kens a thing. It's true.' He sat back. 'An that's why we huv green breed in oor hoose, sometimes.'\n\n'Woof,' Kenneth said, fascinated. 'Bet it tastes horrible!'\n\n'That's illegal,' Fergus said. 'My mother knows the C.O. at the base; if I told her she'd tell him and you'd probably all be arrested and you _would_ get the jail.'\n\n'Aye,' Lachy said. 'But you promised no tae tell, didn't ye?' He smiled thinly over at Fergus, sitting on the other side of the small table. 'Your maw always call ye \"Darlin\", aye?' .\n\n'No,' Fergus said, sitting straight and drawing a hand across his forehead, moving some hair away from his eyes. 'Only sometimes.'\n\nKenneth got up and went to stare at a big model ship in a glass case on the far side of the room. It was an ordinary steamer, not a warship, unfortunately, but it looked magnificent, like one of the ones he'd seen in the big museum in Glasgow when his dad had taken him there. The ship was wonderfully detailed; every stanchion and rail was there; every tiny port-hole, even the oars in the tiny shore-boats behind the tall funnel, their seats and internal ribs thinner than match-sticks.\n\n'You her darlin, ur ye?' Lachy said, wiping some crumbs from the plate. 'You her wee darlin, that right, Fergus?'\n\n'Well, what if I am?' Fergus said sniffily.\n\n'Weyl, whort if a eym?' Lachy mimicked. Kenneth looked round from the gleaming, perfect model.\n\nFergus's face looked pinched. 'At least my mum and dad don't hit me, _Master_ Watt.'\n\nLachy sneered, stirred in his seat. 'Aye, great fur some,' he said, standing up. He walked round the room, looking at some wooden aircraft models on a desk, tapping them. 'Very fancy carpet, Fergus darling,' he said, going up and down on his heels on the thick pile of the intricately-patterned rug. Fergus said nothing. Lachy picked up some lead soldiers from a couple of trays ranked full of them, then stood inspecting some maps on the wall, of Scotland, the British Isles, Europe and The World. 'They red bits aw ours, are they?'\n\n'No, they're the King's actually,' Fergus said. 'That's the Empire. They're not red because they're _commie_ or anything.'\n\n'Ach,' Lachy said, 'Ah ken that; but ah mean they're British; they're ours.'\n\n'Well, I don't know about \"ours\", but they belong to Britain.'\n\n'Well,' Lachy said indignantly. 'Ah'm British, am ah no?'\n\n'Hmm. I suppose so,' Fergus conceded. 'But I don't see how you can call it yours; you don't even own your own house.'\n\n'So whit?' Lachy said angrily.\n\n'Yes, but, Fergus,' Kenneth said. 'It is the British Empire and we're all British, and when we're older we can vote for MPs to go to parliament, and they're in power, not the King; that's what the Magna Carta says; and we elect them, don't we? So it is our Empire, really, isn't it? I mean you when think about it.'\n\nKenneth walked into the middle of the room, smiling at the other two boys. Fergus looked unconvinced. Lachlan rolled his eyes, looked at the small single bed, then at a couch in one corner. 'You got this room all tae yerself?' Lachy said, voice high.\n\n'Yes, so?' Fergus replied.\n\n'Bi Christ, it's all right for some, eh, Ken?' Lachy said, winking at Kenneth and walking over to the model ship in the glass case. 'Aye,' he said, tapping the glass, then twisting a little key in a lock at one end of the case; the side panel of the case opened. 'Ah bet ye can get up tae all sorts aw things in here by yourself at nights.' He started trying to haul the model out of the case.\n\n'Stop that!' Fergus shouted, standing up.\n\nLachy shifted the whole glass on its stand, reached in and lifted the model out of its two wood and brass cradles. Kenneth saw the rear mast bend against the top of the case. The black threads of the radio wires sagged.\n\n'How can ye play with it in here?' Lachlan protested, straining to pull the model out.\n\n'Lachy \u2013' Kenneth said, starting over to him.\n\n'It's not a toy!' Fergus said, running over. He swatted Lachy's arm. 'Stop it! You'll break it!'\n\n'Ach, all right,' Lachy said. He slid the model ship back in. Kenneth noticed with some relief that the mast flexed back into shape, hauling the radio antennae taut again. 'Keep yer hair on, _darling.'_\n\nFergus locked the door of the case and pocketed the key. 'And don't call me that!'\n\n'Sorry, _darling.'_\n\n'I said stop it!' Fergus shrieked.\n\n'Ach, dinnae wet yer knickers, ya big lassie.'\n\n'You disgusting little \u2013'\n\n'Oh, come on, you two; act grown-up,' Kenneth said. 'Fergus,' he pointed over to the window, and a slope-topped display case standing under it. 'What's all this stuff?'\n\n'That's my museum,' Fergus said, glaring at Lachy and walking to the window.\n\n'Oo, a museum,' Lachy said in a pretend posh voice, but came over too.\n\n'Things I've found, locally,' Fergus explained. He stood over the case, pointing. 'That's a Roman coin, I think. And that's an arrowhead.'\n\n'Whit's that green thing?' Lachy said, pointing to one corner.\n\n'That,' Fergus told him, 'is a fossilized pear.'\n\nLachy guffawed. 'It's a bit aw bone, ya daft bugger. Where'd ye get yon? Back a the butcher's shop? Find it in the dug's bowl, aye?'\n\n'No I did not,' Fergus said indignantly. 'It's a fossilized pear; I found it on the beach.' He turned to Kenneth. 'You've got some education, Kenneth; you tell him. It's a fossilized pear, isn't it?'\n\nKenneth looked closer. 'Hmm. Umm, I don't know, actually.'\n\n'Fuckin bit a bone,' Lachy muttered.\n\n'You filthy-mouthed little wretch!' Fergus shouted. 'Get out of my house!'\n\nLachy ignored this, bent down, face over the cabinet.\n\n'Go on; get out!' Fergus screamed, pointing to the door.\n\nLachy looked sourly at the pitted, vaguely green exhibit labelled 'Fossilized Pear, Duntrunne Beach, 14th of May 1945.'\n\n'I'm not kidding! Out!'\n\n'Fergus \u2013' began Kenneth. He put a hand on the other boy's arm. Fergus hit it away, face white with fury.\n\nLachy wrinkled his nose, which was almost touching the glass of the cabinet. 'Still, whit dae ye expect frae a laddy that hides in a lavvy?'\n\n'You _pig!'_ Fergus screamed, and brought both fists thudding down on the back of Lachy's head. Lachy's face crashed through the glass, into the display case.\n\n'Fergus!' Kenneth yelled, pulling him away as Fergus kicked at Lachy's legs. Lachy screamed, jerked back, spilling glass, arms flailing, face covered in blood.\n\n'Aah, ya basturt!' he wailed, staggering. 'Ah canny see!'\n\n'Lachy!' Kenneth shouted, hauling his hanky out of his pocket. He went to Lachy, grabbed his shoulders. 'Lachy; stand still! Stand still!' He tried to wipe the blood from the other boy's eyes; it was all over his jumper, dripping onto the carpet.\n\n'But ah canny see! Ah canny see!'\n\n'What on _earth_ is going on in he \u2013 Oh my God!' Mrs Urvill said, from the doorway. 'Fergus! What have you been letting him do? And get him off that carpet; it's Persian!'\n\nLachlan lost an eye. The Gallanach Glass Works, Ornaments Division, made him an artificial one. Fergus was soundly beaten by his father, and not allowed out for a fortnight. The Urvills granted the Watt family the sum of one thousand guineas in full and final settlement of the matter, the papers drawn up by the firm of Blawke, Blawke and Blawke.\n\nLachlan was still growing, and perhaps because of that during his mid-teens the eye kept falling out, so another, slightly larger, was made; Lachlan was allowed to keep the old one. He had a third glass eye, which he'd got from the hospital when the first one had been lost for a week (it was eventually discovered, months later, under a chest of drawers in Lachy and Rab's bedroom, where presumably it had rolled during the night), but it was of inferior quality; duller and less lifelike, and he kept it as a spare.\n\nHe was the boy with four eyes, and he didn't even need glasses. Or rather a monocle.\n\n'Keep an eye out for us, Lachy!' and variations thereof became a popular phrase amongst his school-mates, though not to his face after the first boy to say it within Lachy's earshot, if not sight, was held down by a half-dozen powerful young Watts and forced to swallow the brown-irised orb, and then to bring it back up.\n\nMary McHoan sniffed the air. 'Prentice, you smell of petrol.'\n\nPrentice collapsed into a seat in the living room. 'Sorry,' he said.\n\nHis mother looked over the top of the _Guardian_ at him. On the television, a game of snooker was proceeding silently. Prentice sat and looked at it. Mary put the paper down, took off her reading glasses.\n\n'Where's Ken?' Prentice asked. He still had his black leather jacket on.\n\n'In bed, reading,' Mary told him. She folded the paper, went over to her son, and sniffed the air above him. 'And smoke! You smell of ... of non-pub smoke,' she said, going back to her seat. 'What have you been up to?'\n\nPrentice leaned towards her. 'Promise you won't tell dad?'\n\n'No, Prentice,' she said, smoothing her skirt. She took a coffee mug from the small table at her side and sipped from it. 'You know I'm terrible with secrets; not like your father.'\n\n'Hell's teeth. Oh, well,' Prentice said. 'Whatever; we got let off, so \u2013'\n\n'Let off what?' Mary said, alarmed.\n\n'We were in the Jac and Bill Gray said he'd heard the Watts saying \u2013 well, it was Ashley, he said, which was why I didn't believe him at first \u2013 but he'd heard them ... they were all sitting, all the young ones; the Watts, anyway, sitting there being antisocial and morose, cause of Darren getting killed, and anyway, Bill heard Ash saying there was only one way to deal with it, or they'd never get over it properly, and they should all get sledgehammers and stuff \u2013'\n\n'Sledgehammers!' Mary said, clutching at her elbows.\n\n'That's what I said!' Prentice said, sitting forward, unzipping his jacket. 'Sledgehammers? And Bill said yeah, he was sure; and crowbars, that sort of stuff; they were going to get it out of their system, and I believed Bill because he's so straight; no side at all, and I looked over and they were all standing up and putting their coats on and drinking up, and I tried to talk to Ashley, but they were out the door, and Ash said something about coming along too, and it was Bill had the car, and he'd dashed for a pee, and by the time we got out to the car park they were tearing off in Dean's Cortina, and then Bill couldn't get his car started and we headed for the Watts' house, but by then they'd been there and they passed us; we turned round, followed their lights, caught up with them at those new houses out by Dalvore, but they were just throwing stuff in the boot. I shouted to them, but they got back in and screamed off again, so we followed.\n\n'Jeez, I thought they knew where that guy lived that hit Darren, but Bill said he was from East Kilbride, and I said but we're heading that way! And they just kept going; past here and up to Inveraray, and then I thought, God, I hope I know what they're really going to do, and I told Bill, and he said Shit, let's hope so.'\n\n'Prentice.'\n\n'Sorry. Anyway, I was right. They drove to Kinglas; Glen Kinglas, with us following, and they got parked in the lay-by, and we did too, and we all got out, and we all stood there for a while, and nobody said anything. Then they got the sledgehammers and the crowbars out, and we turned the cars and left the engines running so we had plenty of light, then Bill and I sat on the bank and watched them ... Oh, wow! Mum; you should have seen them! They smashed that fucking litter bin \u2013'\n\n'Prentice!'\n\n'Sorry. But they did; they pulverised the mother. They whacked and smashed and blasted the damn thing to smithereens then tore them to shreds too; hammered the metal bins inside flat, turned the concrete shell of the thing to dust, and I'd asked Bill if it was okay, and he'd said, In the circumstances ... so I went to his car and got his spare can of petrol, because Bill's really organised that way, and said, Was it all right? And they were all standing there, sweating and panting and looking just so _drained,_ and Ash just sort of nodded, and I emptied the petrol all over the remains of the bin and Dean threw a match at it and Whumph! up it went, and we just stood there.\n\n'And then this cop stopped! I couldn't believe it! What were the chances? And like only a couple of cars had passed; hadn't stopped, though one had slowed down, certainly, but it had gone off again. And this enormous fu \u2013 great sergeant got out and he was, like _incandescent!_ The bin was nothing on this guy! And we all just stood there, and I thought, Oh no, this really could end badly, because there was just him by himself, and he was cursing us up and down and the Watts weren't taking it too well and I thought I could hear Dean starting to growl, and I finally managed to get a word in edgeways when he said who'd set it on fire and I said me and stepped forward, showing him the petrol can, and told him what it was all about; about Darren hitting the thing and it being like \u2013 well I tried not to use too many long words, but like, expiation ... and he listened, and I was like in that way when you're really nervous where once you've started you can't stop, and I was probably repeating myself all over the place and rambling and not making much sense, but I just kept on going, and he just stood there with this look like thunder on his face, all lit by the fire, and I stopped and said we knew it was wrong and we'd accept having to be punished for it \u2014 even though I heard Dean growling when I said it \u2014 but even so, although we might be sorry we'd done it, we were glad too, and that was just the way it was, and if we didn't normally have respect for public property, it wouldn't mean so much to us to destroy it like we had.'\n\nPrentice swallowed. 'And I shut up at last, and nobody said anything, and the fire was nearly out by this time, and the big sergeant just says, 'Get on your way and pray I never haul any of you up for anything else.' And I'm like, Yessuh, massa, and kicking dirt over the wee bits of the fire that's left and the Watts are still surly but they're putting all the stuff back in the boot of the Cortina and the big guy's just standing with his arms folded watching us, and I'm thinking; Guildford Four, Birmingham Six, hell; there's still a few good apples left, and we just got into our cars and drove away, with the big sergeant still standing there glowering like Colossus in our tail lights.' Prentice spread his hands. 'That's it.'\n\n'Well,' Mary said. 'Good grief.' She shook her head, glanced at the snooker, then put her glasses on and took up her paper again. 'Hmm, well, I probably won't tell your father that. Away and wash your hands, try to get rid of that smell. There's plenty of milk in the fridge if you want cereal.'\n\n'Right-oh, mum.' He came over to her, kissed her hair.\n\n'Yuk; what a stink. Go and wash, you vandal.'\n\n'Thanks for listening, mum,' he said, on his way to the door.\n\n'Oh, I had a choice, did I?' she said, pretending primness.\n\nPrentice laughed.\nCHAPTER 7\n\nWe passed the lay-by near the Cowal Road junction doing about ninety. I watched as we went by. Nothing; it was just a damp, deserted parking place with a big new concrete litter bin (replaced with unusual alacrity, in less than six months). We swept past, trailing light spray. It was a dim, grey day; light drizzle from the overcast, mountains hidden past about a thousand feet. We were on dipped-beam; the instruments glowed orange in front of the delicious, straight-armed, black-skirted, Doc-shoed, crop-blonde, purse-lipped Verity; my angelic bird of paradise, driving like a bat out of hell.\n\n'Yo; Prentice. Get you out of bed?'\n\n'Oh, you guessed.'\n\n'It's a gift. Pick you up at one?'\n\n'Umm ... Yeah. Where are you, Lewis?'\n\n'At the Walkers', in Edinburgh.'\n\n'Oh ... Is Verity there?'\n\n'Yeah; she's coming.'\n\n'Eh?'\n\n'She's coming; to Lochgair. Charlotte and Steve are off to the States this morning, skiing, and Verity \u2013'\n\n'Skiing, to the States? Sheesh, that pack-ice gets \u2013'\n\n'Shut up, Prentice. The upshot is Verity's going to be Festival Perioding with the Urvills. She's going to drive us there.'\n\nAnd me insane, I thought.\n\n'Great,' I said. 'No Rodney?'\n\nLewis laughed. 'No Rodney. Verity is finally a Rod-free zone.'\n\n'Couldn't have happened to a nicer chap.'\n\n'Agree grade and comments. See you thirteen hundred hours.'\n\n'Yeah; see you then.' I put the phone down.\n\nThere was a dartboard above the phone with a picture of Thatcher taped over it. I kissed it. 'Yeeeeee-HA!' I shouted, leaping back into the bedroom.\n\n'Shut up, Prentice,' Gav moaned, muffled, from his bed. He was invisible under a heap of duvet. My bed was on the other side of the room, away from the window, and so not quite as cold as Gav's in the winter. I fell into it, bounced. (Technically I should have Norris's solo room because I've been in the flat longest, but that room's small and noisy; also, Gav doesn't snore and he's quite happy to retreat to the living room couch if I have female company ... That's another thing; there's only room for a single bed in Norris's room). 'Put the heater on, ya bastard,' Gav mumbled.\n\nI leapt up, ninja'd over to Gav's bed and wheeched the duvet off.\n\n'Aw ya \u2013!' He grabbed the duvet back, cocooned himself again. '\u2013 bastard!'\n\n'Gavin,' I told him. 'You are a skid-mark on the lavatory bowl of life. But I respect you for it.' I turned, grabbed my dressing gown and made for the door; with one mighty ninja kick, the side of my right foot connected with all three switches of the fan heater at the same time and it hummed into life. 'I shall make some tea.'\n\n'Dunno about tea; fuckin good at makin a noise.'\n\n'Thank you for sharing that with us, Gav. I shall return.'\n\n'What's the weather like?'\n\n'Hmm,' I said, staring at the ceiling, one finger to my lips. 'Good question,' I said. 'The weather's like, a manifestation of the energy-transfer effected between volumes of the planet's gaseous envelope due to differential warming of the atmosphere at various latitudes by solar radiation. Surprised you didn't know that, actually, Gavin.'\n\nGavin stuck his head out from under the duvet, giving me cause once more to marvel at the impressive way the lad's shoulders merged into his head with no apparent narrowing in between (this appeared to be the principal physical benefit bestowed by the game of rugby; the acquisition of an extremely thick neck, just as the most important thing one could take to the sport was a thick skull, and from it an intact one still in satisfactory two-way communication with one's spinal cord).\n\nGav \u2013 who probably epitomised thick-skulledness, though admittedly would not be amongst one's first fifteen when it came to offering proof of heavy traffic within the central nervous system \u2013 opened one bleary eye and focused on me with the same accuracy one has grown to expect from security forces aiming baton rounds at protesters' legs. 'What the fuck's made you so unbearable this morning?'\n\nI clasped my hands, smiled broadly. 'Gavin, I am in a transport of delight, or at least shall be shortly after one o'clock this afternoon.'\n\nThere was a pause while Gavin's duty-neuron struggled to assimilate this information.\n\nThe intense processing involved obviously exhausted too much of Gav's thinly stretched grey matter to allow speech in the near future, so he contented himself with a grunt and submerged again.\n\nI boogied to the kitchen, singing, 'Walking On Sunshine'.\n\nI watched the orange-white needles swing across their calibrated arcs. Ninety. Jeez. I was sitting behind Lewis, who was in the front passenger seat. I kind of wished I'd sat behind Verity; I wouldn't have seen so much of her \u2013 not even a hint of that slim, smooth face, frowning in concentration as she barrelled the big black Beemer towards the next corner \u2013 but I wouldn't have been able to see the speedometer, either. Lewis seemed unperturbed.\n\nI shifted in my seat, a little uncomfortable. I pulled the seat belt tight again. I checked Verity wasn't watching and adjusted my jeans a little. The folder containing Rory's work lay on the seat by my side; I lifted the file onto my lap, concealing a bulge. There was a reason for this.\n\nWe'd been on the bit of fast dual carriageway between Dumbarton and Alexandria, not long after Verity and Lewis had picked me up. Verity made a sort of wriggling motion a couple of times, straining back against her seat. This force was applied by those long, black-nyloned legs, and though most of the pressure was provided by her left limb, some residual effort pushed her right foot down as well, and on each occasion we speeded up, just momentarily, as her amply-soled Doc Marten pressed against the accelerator.\n\n'You okay?' Lewis had asked, sounding amused.\n\nShe'd made a funny face. 'Yup,' she'd said, shifting down to fourth as a car she'd been waiting to pass pulled back into the slow lane. We were all pressed back into our seats. 'Problem of wearing sussies, sometimes; they sort of pull a bit, you know?' She flashed a smile at Lewis, then me, then looked forward again.\n\nLewis laughed, 'Well, no, can't claim I do know, but I'll take your word for it.'\n\nVerity nodded. 'Just getting things sorted out here.' She strained against the back-rest again, her bum lifting right off the seat. The car, already doing eight-five, roared up to over a hundred. The rear of a truck was approaching rapidly. Verity wiggled her bottom, plonked it back down, calmly braked and shifted up to fifth, dawdling along behind the green Parceline truck while she waited for it to overtake an Esso tanker. 'Parceline, parceline ...' she breathed, tapping her fingers on the thick steering wheel. She made it sound French, pronouncing the word so that it rhymed with 'Vaseline'.\n\n'That better?' Lewis inquired.\n\n'Mm-hum,' Verity nodded.\n\nMeanwhile I was fainting in the back seat, just thinking of what that tight black mid-thigh skirt concealed.\n\nIt had taken until the long, open left-hander that leads down into Glen Kinglas before my erection had finally subsided, and that had been mostly naked fear; Verity had lost it just for a second, the rear of the car nudging out towards the wrong side of the road as we whanged round the bend. Sitting in the rear, maybe it had felt worse, but I'd been petrified. Thankfully, there'd been no traffic coming; the concept of striking up an intimate \u2013 indeed potentially penetrative \u2013 relationship with the rocks on the far side of the road had been bad enough; but even the prospect of a head-on with another lump of metal travelling at anything remotely like the sort of speed we were sustaining might have resulted in me making my mark in the most embarrassing fashion on the leather upholstery of the Bavarian _macht-wagen._\n\nVerity just went 'Whoa-yeah!' like she'd accomplished something, jiggled the steering wheel once and accelerated cleanly away.\n\nAnyway, it's one of the minor unfortunate facts of life that a detumescing willy is prone to trap stray pube hairs under the foreskin as it scrolls forward again, and that was why I was adjusting my clothing as we braked for the bend above Cairndow.\n\nI opened the _Crow Road_ folder lying on my lap and leafed through some of the papers. I'd read the various bits and pieces a couple of times now, looking for something deep and mysterious in it all but not finding anything; I'd even done a little research of my own, and discovered through mum that dad had some more of Rory's papers in his study; she'd promised she'd try and look them out for me. I took a sheet of paper out of the folder and held the page of scribbled, multi-coloured notes up, resting it on one raised knee, gazing at it with a critical look, wondering if Verity could see what I was doing. I cleared my throat. I'd rather been hoping Lewis or Verity might have asked me what the file contained by now, and what I was doing, but \u2013 annoyingly \u2013 neither of them had.\n\n'Sounds?' Lewis asked.\n\n'Sounds.' Verity nodded.\n\nI sighed. I put the sheet back in the folder and the folder back on the other rear seat.\n\nWe rounded the top of Upper Loch Fyne listening to an old Madonna tape, the Material Girl singing 'Papa Don't Preach,' which raised a smile from me, at least.\n\n... Back to Gallanach, for Christmas and Hogmanay. I felt a strange mixture of hope and melancholy. The lights of on-coming cars glared in the dull day. I watched the lights and the drizzle and the grey, pervasive clouds, remembering another car journey, the year before.\n\n'Sounds daft to me, Prentice,' Ashley said, lighting another cigarette.\n\n'It sounds daft to _me,'_ I agreed. I watched the red tip of her cigarette glow; white headlights streamed by on the other side of the motorway, as we headed north in the darkness.\n\nDarren had been dead a couple of months; I had fallen out with my father and I'd been in London for most of the summer, staying with Aunt Ilsa and her long-term companion, whose only name appeared to be Mr Gibbon, which I thought made him sound like a cat for some reason ... Anyway, I'd been staying with them in darkest Kensington, at Mr Gibbon's very grand, three-storeyed town-house in Ascot Square, just off Addison Road, and working at a branch of Mondo-Food on Victoria Street (they were trying a new line in Haggisburgers at the time and the manager thought my accent would help shift them. Only trouble was, when people said, 'Gee, what's in these?' I kept telling them. I don't believe they're on the menu any more). I'd saved some money, grown heartily sick of London, fast food and maybe people, too, and I was getting out.\n\nAsh had been in London for a programming interview with some big insurance company and had offered me a lift back home, or to Gallanach anyway, as I'd exiled myself from Lochgair. Her battered, motley-panelled 2CV had looked out of place in Ascot Square, where I think that anything less than a two-year old Golf GTi, Peugeot 209 or Renault 5 was considered to be only just above banger status, even as a third car, let alone a second.\n\n'Sorry I'm late, Prentice,' she'd said, and kissed my cheek. She and Lewis had been out for a meal the night before. Big brother was staying in Islington, making a living from TV comedy shows by being one of the twenty or so names that zip up the screen under where it says Additional Material By:, and trying to be a stand-up comic. I'd been invited to dinner too, but declined.\n\nI'd hoped she'd just pick me up and we'd be on our way, but Ash hadn't seen Aunt Ilsa for a long time and insisted on exchanging more than just pleasantries with her and Mr G.\n\nAunt Ilsa was a large, loud woman of forbiddingly intense _bonhomie;_ I always thought of her as being the most remote outpost of the McHoan clan (unless you counted the still purportedly peripatetic Uncle Rory); a stout bulwark of a woman who \u2013 for me at least \u2013 had always personified the dishevelled ramifications of our family. A couple of years older than dad, she had lived in London for three decades, on and off. Mostly, she was off; travelling the world with Mr Gibbon, her constant companion for twenty-nine of those thirty years. Mr Gibbon had been an industrialist whose firm had employed the ad agency which Aunt Ilsa had worked for when she'd first moved to London.\n\nThey met; he found her company agreeable, she found his a new slogan. Within a year they were living together and he had sold his factory to devote more time to the rather more demanding business of keeping Aunt Ilsa company on her peregrinations; they had been on the move more or less ever since.\n\nMr Gibbon was a grey-haired pixie of a man, ten years older than Aunt Ilsa, and as tiny and delicate as she was tall and big-boned. Apparently he was quite charming, but as the basis of his charm seemed to rest upon the un-startling stratagem of addressing every female he encountered by the fullest possible version of her name (so that every Julie became a Juliana, every Dot extended to a Dorothea, all Marys became Mariana, Sues Susanna, etc. Sorry; etcetera) as well as the slightly perverse habit of calling all young girls 'madam' and all old women 'girls,' it was a charm to which I at least was quite prophylactically immune.\n\n'And you are ...?' he asked Ashley as he welcomed her in the hallway.\n\n'Ash,' she said. 'Pleased to meet you.'\n\nI grinned, thinking Mr Gibbon would have a hard job finding a convincing embellishment for Ash's uncommon monicker.\n\n'Ashkenazia! Come in! Come in!' He led the way to the library.\n\nAsh turned back to me as we followed, and muttered, 'He's a pianist, isn't he?'\n\nTotally misunderstanding what she meant, I sneered slightly at Mr Gibbon's back, and nodded. 'Yeah; isn't he just.'\n\nAunt Ilsa was in the library; she had a heavy cold at the time and I am tempted to say we discovered her poring over a map, but the inelegant truth is that she was searching the shelves for a misplaced book when we entered.\n\nShe spent most of the next half hour or so talking about the extended holiday to Patagonia she was planning, in an extremely loud voice and with an enthusiasm that would probably have embarrassed the Argentinian Tourist Board.\n\nI sat fretting, wanting to be away.\n\nBy some miracle, the 2CV hadn't been towed away when I'd finally dragged Ash out; we'd made it to the M1, picked up a hitcher and \u2013 rather beyond the call of duty, I'd have said \u2013 dropped him where he was going, in Coventry. We got lost in Nuneaton trying to get back on the M6, and were now heading through Lancashire at dusk, still an hour or more from the border.\n\n'Prentice, there are a lot of better reasons for not talkin to your dad, believe me.'\n\n'I believe you,' I said.\n\n'What about your mother?'\n\n'No, she's still talking to him.'\n\nShe tutted. 'You know what I mean. You're still seeing her, I hope.'\n\n'Yeah; she came to Uncle Hamish's a couple of times, and she drove me back to Glasgow once.'\n\n'I mean, what's the big argument? Can't you just agree to disagree?'\n\n'No; we disagree about that.' I shook my head. 'Seriously; it doesn't work that way; neither of us can leave it alone. There's almost nothing either of us can say that can't be taken the wrong way, with a bit of imagination. It's like being married.'\n\nAsh laughed. 'What would you know? I thought your mum and dad were pretty happy.'\n\n'Yeah, I suppose. But you know what I mean; when a marriage or relationship is going wrong and it's like everything that one person says or doesn't say, or does or doesn't do, seems to rub the other one up the wrong way. Like that.'\n\n'Hmm,' Ash said.\n\nI watched the red tail lights. I felt very tired. 'I think he's angry that having given me the freedom to think for myself, I've not followed him all down the line.'\n\n'But, Prentice, it's not as though you even believe in Christianity or anything like that. Shit, I can't work out what it is you do believe in ... God?'\n\nI shifted uncomfortably in the thin seat. 'I don't know; not God, not as such, not as a man, something in human form, or even in an actual thing, just ... just a field ... a force \u2013'\n\n' \"Follow the Force, Luke,\" eh?' Ash grinned. 'I remember you and your Star Wars. Didn't you write to Steven Spielberg?' She laughed.\n\n'George Lucas.' I nodded miserably. 'But I don't even mean anything like that; that was just background for the film. I mean a sort of interconnectedness; a field effect. I keep getting this feeling it's already there, like in quantum physics, where matter is mostly space, and space, even the vacuum, seethes with creation and annihilation all the time, and nothing is absolute, and two particles at opposite ends of the universe react together as soon as one's interfered with; all that stuff. It's like it's there and it's staring us in the face but I just can't ... can't access it.'\n\n'Maybe it isn't accessible,' Ash said, fag in mouth, holding the steering wheel with her knees and making a stretching, circling motion with her shoulders (we were on a quiet stretch of motorway, thankfully). She took her cigarette from her mouth again, put her hands back on the wheel. I hoped she wasn't getting sleepy; the drone of the wee Citro\u00ebn's engine was cataleptically monotonous.\n\n'How not?' I said. 'Why shouldn't it be accessible?'\n\n'Maybe it's like your particle; inevitably uncertain. Soon as you understand one part of what it means, you lose any chance of understanding the rest.' She looked over at me, brows furrowed. 'What was that routine Lewis used to do? About Heisenberg?'\n\n'Oh,' I said, annoyed now. 'I can't remember.'\n\n'Something about being at school and bursting into this office and saying, look, are you Principal here or not, Heisenberg? And him going, weellll ...' She gave a small laugh. 'Mind, it was funnier the way Lewis told it.'\n\n'A little,' I conceded. 'But \u2013'\n\n'Lewis seems to be making it in the old alternative comedy scene, doesn't he?' Ash said.\n\n'So we're told,' I said, looking away. 'I don't imagine Ben Elton or Robin Williams have considered early retiral quite yet, though.'\n\n'Aye, but good for him, though, eh?'\n\nI looked at Ash. She was watching the road as we roared down a slight incline at all of seventy. Her face was expressionless; that long, Modigliani nose like a knife against the darkness. 'Yeah,' I said, and felt small and mean-spirited. 'Aye, good for him.'\n\n'It true you've not seen much of him in London?'\n\n'Well, he has his own friends, and I was usually too tired after work.' (A lie; I wandered art galleries and went to films, mostly.) 'And I couldn't have paid my way, either.'\n\n'Ach, Prentice,' Ashley said, chiding. She shook her head (the long mane of fair hair was tied up, so it did not swish and fall over her shoulders). 'He'd have liked to have seen you more often. He's missed you.'\n\n'Oh, well,' I said.\n\nI watched the lights again for a while. Ashley drove and smoked. I felt myself nodding off, and shook myself awake. 'Ah, dear ...' I rubbed my face with both hands, asked, 'How do you keep awake?'\n\n'I play games,' she told me.\n\n'Oh yeah?'\n\n'Yeah,' she nodded, licking her lips. 'Like Name That Tail-Light.'\n\n'What?' I laughed.\n\n'True,' she said. 'See that car up ahead?'\n\nI looked at the two red lights. 'Yeah.'\n\n'See how high up the lights are, not too far apart?'\n\n'Yo.'\n\n'Renault 5.'\n\n'No kidding!'\n\n'Mm-hmm. One it's over-taking?'\n\n'Yeah?'\n\n'Horizontally divided lights; that's an old Cortina; mark 3.'\n\n'Good grief.'\n\n'Here's a Beemer. New five series, I think ... about to pass us; should have lights that slant in slightly at the bottom.'\n\nThe BMW passed us; its rear lights were slanted in slightly, at the bottom. We overtook the old Ford and the 5 a little later.\n\n'Course,' Ash said. 'It's more fun in a fast car when you're doing all the overtaking, but even just sitting at seventy you'd be surprised how much you pass, sometimes. Now.' She held up one finger. 'Listen and feel as we pull back into the slow lane.'\n\nAsh swung the ancient 2CV to the left, then straightened.\n\n'What?' I said.\n\n'Nothing.' She grinned. 'Missed all the cats' eyes. Bump-free lane-changing. A great skill, you know.' She glanced at me, mock-serious. 'Not so easy in a Ferrari, or whatever; the tyres are too wide. But skinny wee tyres like this thing's got are just about ideal.'\n\n'Allow me to sit back in amazement, young Ashley,' I said, crossing my arms and twisting in my seat to face her. 'I had no idea it was possible to extract such multifarious enjoyment from a simple night-time car journey.'\n\nAshley laughed. 'Cobbled streets are even more fun, if you're a girly.'\n\n'Huh. Trust you to lower the tone of the whole conversation _and_ introduce a note of clitoris envy at the same time.'\n\nAsh laughed louder, ground the cigarette butt out in the ashtray, flipped it closed. 'Och, it's a gift; I'd be ashamed of myself if I wasn't just so fucking nice with it.' She put her head back and roared with laughter at this, before shaking her head and restoring her attention to the road. I laughed a little too, then stared out of the side window, wondering suddenly if Ash had slept with Lewis last night.\n\nShe clicked the indicator on. 'Ye Olde Motorway Services. Come on; yer Aunty Ashley'll buy you a coffee and a sticky bun.'\n\n'Gee, you sure know how to show a boy a good time.'\n\nAsh just smirked.\n\nWhen I woke, about mid-day in the flat on Crow Road, Janice Rae had gone. To work, I assume. There was a note, on a small blue sheet of writing paper: 'You're the better stand-up. Call me, sometime, if you want. J.'\n\nI looked at that qualified second sentence with an odd feeling of sadness and relief.\n\nDrying off after a shower, I stood looking at two framed movie posters on the bathroom wall. _Paris, Texas_ and _Dangerous Liaisons._\n\nI had a coffee and some toast, washed up and let myself out. I'd put the _Crow Road_ folder in a Tesco bag, and walked back to our flat under grey skies and through a mild and swirling wind, swinging the carrier to and fro, and whistling.\n\nOur flat was in Grant Street, near St George's Cross (and just off Ashley Street, funnily enough). My flat-mates were out when I got back, which was fine by me; I did not relish the prospect of facing the single-entendres that were Gav's best approximation of wit, and which inevitably followed any sexual adventure of mine or Norris's \u2013 real or imagined \u2014 Gav ever found out about. If I was lucky, Gav would be so shocked at the very idea I had had carnal knowledge of an aunt \u2013 even one of the not-really-an-aunt variety \u2013 that he would just pretend it hadn't happened. Hell, if I was _really_ lucky he might stop talking to me altogether, I thought ... but that didn't seem likely. Or preferable, to be honest; part of me rather looked forward to such taunting. I'd caught a glimpse of my face in the hall mirror once, when Gav was berating me for such rakish tendencies, and I'd been smiling.\n\nI made myself another coffee, extended myself on the sofa \u2013 my legs quivery with fatigue \u2013 opened the folder, pulled out the sheets of paper and started to read.\n\n_Crow Road_ seemed to be the title of Uncle Rory's Big Idea. From the notes, he seemed unsure whether its final form would be a novel, a film, or an epic poem. There were even some pages discussing the possibility of it being a concept album. I lay there on the couch and shuddered at the very thought. So _seventies._\n\nThe material in the folder seemed to fall into three basic categories : notes, bits of descriptive prose, and poems. A few of the notes were dated, all between the early and late seventies. The notes were on a mixture of papers, mostly loose-leaf; ruled, plain, squared, graph. Some were on cartridge paper, some on pages torn from what looked like school exercise books, and some on folded, green-lined computer print-out. Napkins and old cigarette packets did not, sadly, put in an appearance. The notes were scribbled in a no-less motley variety of different-coloured pens (ball, felt and micro-liner) and used a lot of abbreviations and compressions: H _crshd twn carige & tr? Erlier proph. by Sr: 'kld by t. livng &_ t. _ded((?)) H Chrst-lk figr (chng nm to start with T!!???); fml Chrst fr new times? Scot mrtyr? Or Birnam wd idea \u2013 disgsd army??? (2 silly?) ..._ and that was one of the more comprehensible bits.\n\n2 silly, indeed.\n\nThe prose was mostly about places Rory had been; they read like out-takes from his travel pieces. _San Jos\u00e9, Ca: Suddenly, the Winchester House itself seemed like a emblem for the restless American soul...._ about some weird house Rory wanted to use in his story, judging by some cryptic notes at the end of the passage.\n\nThen there was the poetry:\n\n... We know this life   \nis merely a succession   \nof endless brutal images,   \npunctuated,   \nfor effect,   \nby relative troughs   \nwhose gutsy heaves   \nat first disguised   \nbut power us to the next disgrace.\n\n'Not applying for a job with Hallmark cards, then,' I muttered to myself, sipping at my coffee.\n\nBut I kept on reading.\n\nMy head wasn't really in the right state for assimilating all this stuff, but as far as I could gather, Uncle Rory had been trying for years to come up with something _Creative_ (his capital, his italics). Something that would establish him as a Writer: script-writer, poet, lyricist for a rock band, novelist, playwright ... it didn't matter. Being recognised for having kept a glorified diary while wandering through India when he was young and naive wasn't enough for him. It wasn't _serious._ This work, _Crow Road,_ would be Serious. It would be about Life and Death and Treachery and Betrayal and Love and Death and Imperialism and Colonialism and Capitalism. It would be about Scotland, (or India, or an 'Erewhon???') and the Working Class and Exploitation and Action, and there would be characters in the work who would represent all of these things, and the working out of the story _would itself prove the Subjectivity of Truth._\n\n... There were pages of that sort of stuff.\n\nThere were also pages of poems forced into some sort of rhyming structure so that they might conceivably have worked as songs, several paragraphs of references to critical works (Barthes, especially; _Death of the Author!_ shouted what looked like a headline over one entire page of notes devoted to ideas about a _loose-leaf novel\/poem??_ There were location notes for a film and sheets about the physical appearance of the characters and the sort of actors who might play them, these surrounded by doodles, mazes and uninspired drawings of faces. There was a list of bands that might be interested in doing an album (a musical tone-scale running all the way from Yes to Genesis), and a sheaf of sketches for the sets in a stage presentation. What there wasn't was any indication whatsoever that Rory had actually written any part of this great work. The only things that might have been classed as narrative were the poems, and they didn't seem to have anything to do with each other, apart from the fact a lot of them seemed to be vaguely about Death, or Love. Tenuous, was the word that came to mind.\n\nI looked in the folder again to see if I'd missed anything.\n\nI had. There was another small sheet of blue writing paper, in Janice Rae's hand. 'Prentice \u2013 had a look at this \u2013' (then the word 'while', crossed out heavily, followed by the word 'before', also nearly obliterated) '\u2013 Can't find any more; R had another folder. (?) If you find it and work out what it's all about, let me know; he said there was something secret buried in it. (Gallanach)?'\n\nI bobbled my head from side to side. 'Gallanach?' I said, in a silly high-pitched voice, as though quoting. I stretched, grunting with pain as my leg muscles extracted their revenge for having been ignored twelve hours earlier.\n\nI reached for my coffee, but it was cold.\n\n'Dear God, we beseech ye, visit the reactive wrath of their own foulness upon those nasty wee buggers in the Khmer Rouge in general, and upon their torturers, and their leader Pol Pot, in particular; may each iota of pain they have inflicted on the people of their country \u2013 heathen or not \u2013 rebound upon their central nervous system with all the agony they originally inflicted upon their victims. Also, Lord God, we ask that you remember the dark deeds of any communistic so-called-interrogators, in this time of great upheaval in eastern Europe; we know that you will not forget their crimes when their day of reckoning comes, and their guttural, Slavic voices cry out to ye for mercy, and ye reward them with all the compassion they ever showed to those unfortunate souls delivered unto them. Prentice?'\n\nI jumped. I'd almost fallen asleep while Uncle Hamish had been droning on. I opened my eyes. The Tree was looking expectantly at me.\n\n'Oh,' I said. 'Umm ... I'd just like to put in a word for Salman Rushdie. Or at least take one out for old Ayatollah Ruhollah Khomeini ...' I looked at Uncle Hamish, who was making quiet signals that I should clasp my hands and close my eyes. We were in the front lounge of Uncle Hamish and Aunt Tone's Victorian villa in the attractive Gallanach suburbette of Ballymeanoch, facing each other over a card table. I closed my eyes.\n\n'Ah,' I said. 'Dear God, we pray that as well as suffering whatever part of the general physical unpleasantness involved in the Iran-Iraq war you may judge to be rightly his, you can find a spare area in his suffering, er, anti-create, for Mr R. Khomeini, late of Tehran and Qom, to experience at least some of the, umm, despair and continual worry currently being undergone by the novelist Mr S Rushdie, of Bombay and London, heathen and smart-alec though he may well be. Amen.'\n\n'Amen,' echoed Uncle H. I opened my eyes. Uncle Hamish was already rising from his seat, looking positively twinkly with health and good cheer. He rubbed his hands. 'Very good,' he said, moving in that oddly stiff and creaky way of his for the door. 'Let us repair for some repast,' he chuckled as he held the door open for me. 'I believe Antonia has prepared something called Cod Creole.' He sniffed the fishy air in the hall; we crossed to the dining room.\n\n'Not Lobster Creole? Or Kid?' I inquired.\n\nBut I don't think Uncle Hamish heard me. He was humming something sombre and looking pleased with himself.\n\nUncle H has developed a fascinating heresy based on the idea that exactly what you did to other people while you were alive gets done right back to you once you're dead. Torturers die \u2013 in agony \u2013 hundreds, maybe thousands of times, before their ravaged souls are finally dropped from the jaws of a fearsome and vengeful God. Those who authorise the dreadful deeds carried out by the torturers (or whoever) also share whatever proportion of this retrospective agony the deity \u2013 or his angelic cost-benefit-calculating representatives \u2013 deem they deserve. Having quizzed The Tree on the details of this scheme, it would appear that said burden of transferred pain is debited from the account of the guy at \u2013 or rather wielding \u2013 the sharp end of the original action, which seems only fair, I suppose.\n\nApparently Uncle Hamish is awaiting divine inspiration on the knotty problem of whether the good things one has accomplished in one's life are also re-lived from the other side (as it were), or simply subtracted from the nasty stuff. At the moment he seems to be veering towards the idea that if you did more good than bad during your life you go straight to Heaven, an arrangement which at least processes the merit of simplicity; the rest sounds like something dreamt up by a vindictive bureaucrat on acid while closely inspecting something Hieronymus Bosch painted on one of his bleak but imaginative-days.\n\nStill, it has its attractions.\n\nAunt Tone and the family's two children, Josh and Becky, and Becky's infant daughter, Iona, were already in the dining room, filling it with bustle and chat.\n\n'Said your prayers?' Aunt Tone said brightly, depositing a steaming dish of potatoes on the table.\n\n'Thank you, yes,' Hamish replied. My uncle worships alone these days, and has done ever since his son left home to become a devout Capitalist (neither his wife nor his daughter had ever bothered with my Uncle's unique brand of condemnationist Christianity; as a rule, the McHoan women, whether so by blood or marriage, have displayed a marked reluctance to take their men-folk's passions seriously, at least outside the bedroom). I think that was why Uncle Hamish had been so delighted when I'd come to stay with the family, and also \u2013 perhaps \u2013 why he was in no hurry to help effect a reconciliation between me and my father.\n\nWe dined on spicy fish which repeated on me for most of the evening in the Jac, meeting pals, until I drowned it in an ocean of beer.\n\n'Happy New Year!' Ashley yelled, flourishing a bottle of generic whisky with more enthusiasm than care; she cracked the bottle off the oak-panelled wall of the castle's crowded entrance hall, but without, apparently, causing damage to either. Clad in a sparkly jacket and a long black skirt, wreathed in silly string and clumps and strands of paper streamers from party poppers, her long hair bunned, she enveloped me in a very friendly kiss, breathing whisky and wine fumes. I kissed right back and she pushed away, laughing. 'Wo, Prentice!' she shouted over the noise. The hall was packed with people; music spilled out from the main hall beyond; pipes and fiddles, tabors and accordions, guitars and a piano, several of them playing the same tune.\n\n'I thought you gave up,' I said, pointing at the cigarette she had stuck behind one ear. Josh and Becky were still at the doors, greeting people they knew.\n\n'I did,' she said, taking the fag from behind her ear and putting it in her mouth. She left it there for a few seconds, then restored it to its previous position. 'See? Still given up; no temptation at all.'\n\nAsh and I levered our way through the press of people while I undid my jacket and struggled to extricate my half-bottle of whisky from a side pocket. We made it into the hall, which was actually less crowded, though still full. A huge fire roared in the grate; people balanced on the fire-seat which ran around the hearth, and on every other available perch, including the stairs and the piano. A few enthusiasts within the midst of the crowd were trying to dance the Eightsome Reel, which in the circumstances was a little like trying to stage a boxing match in a telephone box; not totally impossible, just pointless.\n\nAsh and I found a space over near the piano. She reached over the piano to a pile of little plastic cups, grabbed one and shoved it into my hand. 'Here; have a drink.' She sloshed some whisky into the cup. 'How've you been?'\n\n'Fine,' I said. 'Broke, and I can see that 2.1 disappearing over the event horizon, but fuck it; I've still got my integrity and my M\u00f6bius scarf, and a boy can go a long way with those things. You got a job yet?'\n\n'What?'\n\n'Let's stand away from this fucking piano.'\n\n'What?'\n\n_'Have you got a job yet?_\n\n'Na. Hey.' She put one hand on my shoulder. 'Heard what David Bowie's latest film's called?'\n\n'This sounds Lewisian,' I shouted.\n\n'No,' she shook her head. ' \"Merry Christmas, Mister Ceausescu\"!' Ashley laughed like a drain; a teetotaller might have said her breath smelled like one.\n\n'Very funny,' I yelled into her ear. 'Haven't laughed so much since General Zia got blown up. Where is Lewis, anyway? We were waiting for them to turn up at Hamish and Tone's but they never showed. He and James here?'\n\nAsh looked concerned for a second, then her smile returned. She put her arm round my shoulders. 'Saw James over by the accordion earlier. Hey; you want to take a stroll round the battlements?' She pulled a spliff half out of her breast pocket, let it fall back. 'Got a number here, but Mrs McSpadden keeps wandering through, and I seem to remember she took inordinate and _extremely_ loud interest in one of these last year when wee Jimmy Calder stoked up. You comin?'\n\n'Not right now,' I said, looking around the crowd, acknowledging a few waves and some distant mouthings that were probably shouts. I stood on tip-toes to look round the hall; a paper-plane battle seemed to be taking place at one end. 'You seen Verity?'\n\n'Not for a bit,' Ash said, pouring herself more whisky. I refused. 'Hey.' Ash nudged me. 'There's dancing upstairs.'\n\n'Verity there?'\n\n'Maybe,' Ash said, raising her eyebrows.\n\n'Let's check it out.'\n\n'Way to go, Prent.'\n\n... No Verity in the Solar, loud with sounds and dark with light, and less crowded still. Ash and I danced, then cousin Josh asked her, and I sat watching the people dance for a while \u2013 the best way to extract any real enjoyment from dancing, I've always thought, but I seem to be unusual in not gaining any real pleasure from performing the movements \u2013 and then saw Helen Urvill, entering the hall holding a lager can. I went over to her, through the dancers.\n\n'Happy New Year!'\n\n'Hey, Prentice. Same to you ...'\n\nI kissed her, then lifted her up and spun her round; she whooped.\n\n'How are _you?'_ I yelled. Helen Urvill, elegantly tall and judiciously lean, straight thick hair obsidian black, dress combat-casual, back on holiday from Switzerland and looking as thoroughly kempt as ever, passed the lager can to me.\n\n'I'm fine,' she said.\n\nI looked at the tin she'd handed me. _'Carling Black Label?'_ I said, incredulous. Somehow this did not quite seem Helen's style.\n\nShe smirked. 'Try some.'\n\nI tried some; the stuff foamed, went up my nose. I spluttered, stepping back, dripping, while Helen took the can back and stood grinning. 'Champagne?' I said wiping my chin.\n\n'Lanson.'\n\n'What else? Oh you're so _stylish,_ Helen,' I said. 'Wanna dance?'\n\nWe danced, and shared the can of champagne. 'How's Diana?' I shouted above the music.\n\n'Couldn't get back,' Helen yelled. 'Still out in Hawaii.'\n\n'Poor thing.'\n\n'Yeah.'\n\nHelen continued to circulate; I decided it was time for a pee and then maybe some food, which took me via the garden (there was a queue for the downstairs loo, and the upper part of the castle was locked) to the kitchen.\n\nMrs McSpadden was in command, over-seeing a production line of sandwiches, sausage rolls, bowls of soup and chilli, slices of black bun and Christmas cake and accompanying slices of cheese.\n\n'Prentice!' Mrs McSpadden said.\n\n'Mthth MnThpndn!' I replied, mouth full of cake.\n\nShe shoved a set of keys into my hand. 'Will ye pop down to the cellar, for us?' Mrs S shouted. 'Get another litre of whisky; it's the second archway on the left. Dinnae let anybody down with you, mind; keep that door locked.' The microwave chimed and she hauled a still half-frozen block of chilli out on a big plate; she started breaking it up with a large wooden spoon.\n\nI swallowed. 'Okay,' I said.\n\nI went through to the utility room, cool and dark after the noise and chaos of the kitchen. I turned the light on, sorted through the keys for one that looked like it might match the door to the cellar. A movement outside caught my eye and I peered through the window; looked like I'd put on an outside light, too.\n\nVerity Walker, clad in a short black dress, was dancing sinuously on the roof of Uncle Fergus's Range Rover. Lewis sat cross-legged on the bonnet of the car, watching her. He glanced over, shading his eyes, and seemed to see me, looking through the window from the utility room. Verity pirouetted. Holding her shoes in one hand, she ran the other down over her body to one thigh, then back to her head and through her cropped blonde hair. The floodlight outside \u2013 harsh and white \u2013 lit her like she was on stage. Her hair glowed like pale flame.\n\nLewis jumped off the Range Rover (Verity wobbled a little as the car bounced on its springs, but recovered); he stood at the side of the car, between me and it, and held one hand up to Verity. She danced on, oblivious, then he must have said something, and she danced seductively, fluidly, to the edge of the roof, hips moving slow, a big smile on her face as she looked down at Lewis, then she threw herself off the roof. Lewis caught her, staggered back a couple of steps, then forward, as Verity wrapped her arms round his neck and her legs round his waist; white glances of thigh against the black. Lewis put his arms round her as he pitched forward.\n\nThey thumped together into the Range Rover. I thought the impact must have hurt her back, but it didn't look like it had. Her arms and legs stayed where they were, and Lewis's head bent down to hers. Her hands started to stroke and caress the nape of his neck and the back and sides of his head.\n\nAfter a while, one of Lewis's arms disengaged, waving behind him. One finger pointed up to the bright flood-light that was showing me all this. His hand made a cutting, chopping motion.\n\nWhen he did it a second time, I put the light out.\n\nI let myself into the cellar, locked the door behind me. The cellar was cold. I found the whisky, let myself out of the cellar and locked it, turned all the lights out, gave Mrs McSpadden the bottle, accepted a belated new-year kiss from her, then made my way out through the kitchen and the corridor and the crowded hall where the music sounded loud and people were laughing, and out through the now almost empty entrance hall and down the steps of the castle and down the driveway and down to Gallanach, where I walked along the esplanade \u2013 occasionally having to wave or say 'Happy New Year' to various people I didn't know \u2013 until I got to the old railway pier and then the harbour, where I sat on the quay-side, legs dangling, drinking my whisky and watching a couple of swans glide on black, still water, to the distant sound of highland jigs coming from the Steam Packet Hotel, and singing and happy-new-year shouts echoing in the streets of the town, and the occasional sniff as my nose watered in sympathy with my eyes.\nCHAPTER 8\n\nRory stood on the dunes, facing the sea. Lewis stomped away along the tide-line, kicking at the odd piece of driftwood and the occasional plastic bottle. His hands were stuffed into the pockets of his camouflage jacket; his head \u2013 short-haired, these days \u2013 was down.\n\nSouth Uist. Lewis seemed to be taking it as a personal insult that the family had come to the Hebrides for their summer holiday. People kept asking him what he was doing on Uist; Lewis was further north, ha ha.\n\n'He's awful moody, isn't he, Uncle Rory?'\n\nRory watched Lewis walk away along the beach. 'Yeah.' He shrugged.\n\n'Why do you think he doesn't want to walk with us?' Prentice's thin face looked genuinely puzzled. Rory smiled, looked once more at Lewis's retreating back, then started down the far side of the dune heading for the narrow road. Prentice followed. 'I think,' Rory said, 'it's called being at an awkward age.'\n\nKenneth, Mary and the boys had come holidaying to the Hebrides, as they did most years. Rory had been invited along too, as he usually was, and for a change had accepted. So far, they'd been lucky; the Atlantic weather systems had been kind, the days bright and warm, the nights calm and never completely dark. The big rollers boomed in, the wide beaches lay mostly empty, and the machair \u2013 between dunes and cultivation \u2013 was a waving ocean of bright flowers thrown across the rich green waves of grass. Rory loved it, somewhat to his surprise; a holiday from holidays. A place to stay where he didn't have to take notes about flights and ferries and hotels and restaurants and sights. No travel book to think about, no articles, no pressure. He could laze.\n\nHe volunteered to take the boys on a walk after breakfast that Sunday. James had stayed behind and Lewis had been sullen for the half-hour or so they'd been walking before suddenly announcing he wanted to be alone.\n\nRory and Prentice walked on together, their short shadows preceding them. The road would be turning east soon, and taking them back to the main road so that they could turn south and walk back to the house. Lewis knew his way about the area, so Rory was happy to let him wander off alone.\n\nA car passed them on the single track road, heading north; they stood aside to let it pass, waving at the single occupant when he waved at them. The surf was a distant wash of noise, rolling over the sparkling machair in invisible waves. Larks warbled, points of sound in the sweep of blue sky and small puffy clouds.\n\n'Is it all right to walk on a Sunday, Uncle Rory?'\n\n'All right?' Rory said, glancing at the boy. In shorts and a short-sleeved shirt, he looked almost painfully thin. Rory wore an old cheesecloth shirt and cut-off jeans.\n\n'Aye; dad was saying you're not even allowed to walk in some islands on a Sunday!' Prentice rolled his eyes and puffed his cheeks out.\n\n'Well, yeah,' Rory said. 'I think they're like that in Lewis and Harris. But that's the hard-line prods up there. Down here they're Catholics; bit more relaxed about that sort of thing.'\n\n'But not being able to _walk!'_ Prentice protested, shaking his head at his shadow on the grey-black tarmac.\n\n'I think you're allowed to walk to church and back.'\n\n'Ho! Big deal!' Prentice didn't sound impressed. He was silent for a while. 'Mind you,' he said, sounding sly. 'I suppose you could always take a very long way round.'\n\nRory laughed, just as his attention was caught by a little white blossom lying on the road surface in front of them. Prentice looked up, at first surprised, then smiling, when Rory laughed. Prentice stood on the flower, then jumped, shrieking with pain.\n\n'Ah; my foot! My foot! Oh! Oh!'\n\nRory stood, open mouthed for a second, watching Prentice hop around on the tarmac, clutching at one ankle, his face contorted. Rory thought for a second Prentice was pretending, but the boy's expression convinced him he was in real pain. Prentice hopped onto the grass and fell over, still clutching at his foot; Rory could see something white stuck to the sole of the boy's sandshoe.\n\n'What is it?' he said, crouching down by Prentice's side. The boy was shaking, and when he looked up at Rory there were tears in his eyes.\n\n'I don't know,' he sobbed. 'Stepped on something.'\n\n'Let me see.' Rory sat on the grass in front of Prentice and held his foot. The little white blossom he'd seen on the road's surface was stuck to the boy's sandshoe; it wasn't a flower, it was a little paper charity flag for the Royal National Lifeboat Institution, the sort you secured to your lapel with a pin. The flag was still attached to its pin, which was buried in the sole of Prentice's shoe. Rory sucked his breath in when he saw it; most of the pin must be inside the boy's foot, near the middle of the broadest part of the sole.\n\nPrentice's foot and leg shuddered as he rolled on the grass. 'It's awful sore, Uncle Rory,' he said, voice trembling.\n\n'It's just a wee pin,' Rory said, trying to sound encouraging. 'I'll have it out in a second.'\n\nHe licked his lips, rubbed his right index finger and thumb together for a couple of seconds and held Prentice's foot steady with his left hand. He used the nails of his finger and thumb to find the head of the pin, itself almost buried in the tan rubber sole of the sandshoe. He grasped it. Prentice whimpered, foot trembling in Rory's grip. Rory gritted his teeth, pulled.\n\nThe pin slid out; an inch of it, shining in the sunlight. Prentice cried out, then relaxed. Rory put the boy's foot down gently.\n\nPrentice sat up, face quivering. 'That's better,' he said. He used one shirt sleeve to wipe at his face. 'What was it?'\n\n'This.' Rory showed him the pin.\n\nPrentice grimaced. 'Ouch.'\n\n'You're probably going to need a tetanus injection,' Rory told him.\n\n'Aw no! More needles!'\n\nThey took his shoe and sock off. Rory sucked at the tiny wound and spat, trying to remove any dirt. Prentice, eyes still watering, laughed nervously. 'Is that not a horrible smell, no, Uncle Rory?'\n\nRory threw the boy's white sock at him, grinning. 'I've been to India, kid; that ain't nuthin.'\n\nPrentice put his shoe and sock back on and got to his feet, obviously in some pain when he stood. 'Here; I'll give you a carry-coal-bag,' Rory said, turning his back to the boy and putting his arms out from his sides as he crouched.\n\n'Really, Uncle Rory? You sure? Will I not be awful heavy?'\n\n'Hop on; you're a bean-pole, laddie. I'll probably go faster with you on my back; you walk too slow. Come on.'\n\nPrentice put his arms round Rory's neck and got up onto his back; Rory set off at a run. Prentice whooped.\n\n'See?' Rory said, slowing to a fast walk.\n\n'I'm not too heavy, honest, Uncle Rory?'\n\n'What? A skelf like you? Never.'\n\n'Do you think this is a punishment from God for talking about walking on a Sunday, Uncle Rory?'\n\nRory laughed. 'Certainly not.'\n\n'Do you not believe in God either, Uncle Rory?'\n\n'No. Well; not in the Christian God. Maybe something else.' He shrugged his shoulders and shifted Prentice into a more comfortable position on his back. 'When I was in India, I thought then I knew what it was I might believe in. But when I came back it all seemed to go away again. I think it was something to do with the place.' He looked to one side, at the dazzling expanse of machair; endless emerald green scattered thick with flowers so bright they seemed lit from inside. 'Places have an effect on people. They alter your thoughts. India does, anyway.'\n\n'What about when you went to America? Did that affect what you thought?'\n\nRory laughed gently. 'Yeah; it did that all right. Kind of in the opposite way, though.'\n\n'Are you going to go away again?'\n\n'I expect so.'\n\nPrentice elapsed his hands in front of Rory's chin. Rory glanced at his wrists; thin and fragile looking. Prentice was still holding the little Lifeboat flag, twirling the pin between his fingers.\n\n'When did you stop believing in God?' Prentice asked.\n\nRory shrugged. 'Hard to say; I think I started to think for myself when I was about your age, maybe a bit younger.'\n\n'Oh.'\n\n'I tried to imagine how the world had been created, and I imagined Sooty \u2013 you know; the glove puppet \u2013'\n\n'I know; they still have him. Sooty and Sweep.' Prentice giggled.\n\n'Well, I imagined him standing on a wee planet about the size of a football \u2013'\n\n'But he hasn't got any legs!'\n\n'Ah, but he did in the annuals I got for Christmas. Anyway, I imagined him waving a wand, and the world came into existence. Like, I'd been to church, been to Sunday School, so I knew all the stuff in the Bible, but I guess I needed to envisage it ... see it, in my own terms.'\n\n'Uh-huh.'\n\n'But then I thought; wait a minute; where does the planet Sooty's standing on come from? I thought Sooty could have waved his wand and made that appear too, but where would he stand while he was doing it? I mean, I didn't think, Well, he could float in space, and it never occurred to me to ask where Sooty himself had come from, or the wand, but I was already heading towards not believing, I suppose. It was like the dragons.'\n\n'Dragons?' Prentice said, sounding excited and wary at once. Rory felt the boy tremble.\n\n'Yeah,' Rory said. 'I used to hide under the covers of my bed at night, imagining there were dragons out there; in the room when the light was out, when there was nobody else there. I'd hunch down under the covers with just an air-hole to breath through, and shelter there. The dragons couldn't get you through the air-hole; they could only get you if you put out a foot or a hand, or worst of all your head; that was when they struck; bit it off, or pulled you right out and ate all of you.'\n\n'Waa! Alien!' Prentice said. His arms squeezed Rory's neck.\n\n'Yeah,' Rory said. 'Well, I guess a lot of horror films come from that sort of background. Anyway; I used to be petrified of these dragons, even though I knew they probably didn't exist; I mean I knew there was no Santa Claus, and no fairies and elves, but still thought ghosts and dragons were a possibility, and it only took one to kill you ... I mean how did I really know I could trust adults? Even mum and dad? There were so many things I didn't really understand about people, about life. Most of the time you could just ignore a lot of the stuff you didn't know; it'd come in time, you'd be told when you needed to know... But how did you know that there wasn't some big secret, some big, evil deal going down that involves you but had been kept secret from you?\n\n'Like, maybe your parents were just fattening you up until you would make a decent meal for these dragons, or it was an intelligence test; the kids smart enough to have sussed out the fact there were dragons around were the ones that would survive, and the ones that just lay there, trusting, each night, deserved to die, and their parents couldn't tell them or the dragons would eat them, and stories about dragons were the only clues you were ever given; that was all the adults could do to warn you ... I was pretty paranoid about it. I used to be frightened to fall asleep at night sometimes, afraid I'd stick my head out from under the clothes while I was asleep and wake up to find my head in a dragon's mouth, before I died.'\n\n'Wow!'\n\nRory grunted, shifting Prentice's weight again. Kid wasn't so feather-light after all. 'But then one night, under the covers \u2013 I was just getting older, I guess, but anyway \u2013 I was sort of reviewing the day, and I was thinking about school, and what we'd learned, and we'd been doing the Second World War, and I hadn't liked the sound of this Hitler guy at all; and I'd asked dad, just to double-check, and \u2013'\n\n'So he was still alive? When you were ten?'\n\n'Oh yeah; didn't die until I was twelve. Anyway; he brought down this book; history of the War in pictures, and it had like all these photos of the death camps, where the Nazis murdered millions of Jews, and communists, and homosexuals, and gypsies and anybody else they didn't like ... but mostly Jews, and there were like just piles of bodies; incredibly thin bodies, like bones; skeletons wrapped with tissue paper, and piled higher than a house ... and pits; long pits full of bodies, and the metal stretchers they were put onto to be shoved into the ovens, and the piles of wedding rings and spectacles; glasses, and even artificial legs and weird stuff like that ...\n\n'Anyway, that night they put a night-light in my room, in case I had nightmares, but the shadows were even worse than the darkness, and so I just lay there, under the covers, quivering with fear thanks to these damn dragons, and I wished Ken was back from University because sometimes I was allowed to sleep in his room, and I wished I was allowed a torch in my room, but I wasn't, and I was wondering about crying really loudly, because that would bring mum and dad in to see me, but then what did I say was wrong? And then I suddenly thought ...\n\n'The dragons might be there; they might be real and they might be every bit as vicious as I'd imagined, but I'm a human being; so was Adolf Hitler and he killed millions of people!\n\n'And I threw back the bedclothes before I had any more time to think about it and burst out of the bed; threw myself into the middle of the bedroom, screaming and roaring and thrashing about.'\n\n'Ha!' Prentice said, squirming.\n\n'That brought mum and dad through; thought I was having a fit or something. But I just looked up from the carpet with this great big reassuring smile and said there was nothing to worry about.' Rory smiled at the memory, bringing his head up to look around. A break in the dunes let the sound of surf grow louder. There was a car in the distance coming towards them.\n\n'Brilliant!' Prentice said.\n\nRory grunted, shifting Prentice's weight once more. 'Never had any trouble with dragons after that.'\n\n'I'll bet you didn't!'\n\nThe car hummed nearer as the view to one side slowly opened up through the dunes to reveal the shining beach and blue-green ocean.\n\n'Let's see if we can get a lift off this car, eh?' Rory said. 'You okay to get down?\n\n'Yeah!' Prentice slid off onto the grass and stood there, favouring his good leg, while Rory stretched and rubbed at his lower back. He stuck one thumb out when the car was still a few hundred yards away. Prentice reached up and put something on the thin collar of Rory's shirt. It was the little paper Lifeboat flag. Rory held his collar out so that he could look at it. He looked down at the boy's grinning face. 'Thanks,' he said.\n\n'That's your medal, Uncle Rory,' Prentice told him. 'For being a brilliant uncle.'\n\nRory ruffled the boy's hair. 'Thanks, Prentice.' He looked back at the car. Was it slowing?\n\n'I used to worry about Darth Vader,' Prentice confessed, putting his arm round Rory's waist and lifting his foot to massage it with one hand. 'I'd lie under the covers and make the noise he makes when he's breathing, and then I'd stop, but sometimes it would go on after I'd stopped!' Prentice shook his head, and slapped one hand off his forehead. 'Crazy, eh?'\n\nRory laughed, as the approaching car started to slow down. 'Yeah, well, that's what stories do to you, sometimes. Your dad's always tried never to tell you lies, or stories that would scare you or make you superstitious, but \u2013'\n\n'Ha!' Prentice said, as the battered Cortina II drew to a stop just past them. 'I remember he tried to tell us clouds came from the Steam Packet Hotel, in the town. That's what they were: packets of steam from the Steam Packet Hotel. Ha!'\n\nRory smiled as they walked towards the car, him supporting the limping boy. Rory looked away for a second, towards the beach, where the long Atlantic rollers crashed against the broad expanse of gold.\n\nHe sniffed the glass; the whisky was amber, and there wasn't much of it. The smell stung. He put it to his lips, hesitated, then knocked it back in one go. The drink made his lips and tongue tingle; his throat felt sore and the fumes went up his nose and down into his lungs. He tried very hard not to cough like he'd seen people cough in westerns when they tried whisky for the first time, and got away with just clearing his throat rather loudly (he looked round at the curtains, afraid somebody might have heard). His eyes and nose were watering, so he pulled his hanky from his trousers, blew his nose.\n\nThe whisky tasted horrible. And people drank this stuff for pleasure? He had hoped that by trying some whisky he'd understand adults a bit better; instead they made even less sense.\n\nHe was standing between the curtains and the windows of the ballroom of the Steam Packet Hotel, on the railway pier at Gallanach. Outside, the afternoon was wet and miserable-looking, and what little light there had been \u2013 watery and grey \u2013 was going now. Sheets of rain hauled in off the bay, blew around the steamers and ferries moored round the windswept quay, then collapsed upon the dark grey buildings of the town. The street lamps were already lit, and a few cars crawled through the rough-mirror streets with their lights on and their wipers flapping to and fro.\n\nMusic played behind Rory. He balanced the empty whisky glass on the window-sill and gave his nose a last wipe, pocketing his hanky. He supposed he'd better go back into the ballroom. Ballroom; he hated the word. He hated the music they were playing \u2013 Highland stuff, mostly \u2013 he hated being here in this dull, wet town, with these dull people listening to their dull music at their dull wedding. They should be playing the Beatles or the Rolling Stones, and they shouldn't be getting married in the first place \u2013 modern people didn't.\n\n_'Heeee-yooch!'_ a voice shouted, startlingly nearby, making Rory jump. The curtains bowed in a few yards away, almost touching the window-sill, the movement like a wave. Rory could hear the stamping, slapping feet move in time to the fiddles and accordions as they played a jig. People were clapping, shouting out. God, it was all so provincial.\n\nRory straightened his tie, and with his whisky still burning in his throat, and now his stomach too, he moved along to the gap in the curtains and slid through, back into the ballroom, where people sat drinking at long wooden tables and groups of dancers went whirling round in complicated, ever-changing patterns, all flowing dresses and clasping hands and big red sweaty faces and white shirts and ties and narrow trousers or \u2013 even worse \u2013 kilts.\n\nRory moved near the stage, behind the tables where Kenneth and Mary sat, talking to mum. Boring Hamish and the horse-resembling Antonia were on the floor, him in a kilt, her still in her white bridal gown, both dancing badly and out of time, but seemingly thoroughly enjoying themselves.\n\n'Well,' he heard his mum saying, 'you two had better get a move on, or Hamish and Antonia will beat you to it.' She laughed and drank from her glass. She wore a hat. Rory hated his mother in a hat. He thought she sounded drunk. Kenneth and Mary smiled uncertainly at each other.\n\n'Well, mum,' Kenneth said, sitting back, filling his pipe. 'We have been practising.'\n\n'Kenneth!' his wife said quietly.\n\nMum shook her head. 'Ah, don't mind me; plenty of time yet, I dare say.' She looked into her empty glass. 'I wouldn't be missing grandchildren so much, but ...' She shrugged. There was an awkward silence between the three people then, while the music played and the dancers whooped and shouted and clapped and stamped. Rory saw his mother's shoulders move once, and she put her head down for a second, sniffed. She reached down for her handbag on the floor. Kenneth handed her his hanky. He put his arm round his mum's shoulders. Mary moved her seat closer, reached out and took one of the older woman's hands in hers.\n\n'God, I miss that old devil,' mum said, and blew her nose. Eyes bright with tears, she looked at Mary, and then saw Rory standing behind and to one side of them. 'Rory,' she said, trying to sound all right. 'We wondered where you were. Are you enjoying yourself, darling?'\n\n'Yes,' he lied. He hated her calling him 'darling'. He stayed where he was because he didn't want to get close enough for them to smell his breath. His mother smiled.\n\n'Good lad. See if you can find your cousin Sheila; you said you'd ask her to dance, remember?'\n\n'Yeah, all right,' he said, turning away.\n\nHe didn't like boring cousin Sheila, either. She was about the only girl here who was his own age. It was horrible being this age when nobody else was; they were all either adults or children. He blamed his parents. Mostly he blamed his dad. If he'd looked after himself, not had a heart attack, he'd still be around. That was how thoughtless he'd been. Rory supposed it was the same thoughtlessness that had made dad and mum have him so much later than the rest of their children. People just didn't think, that was the trouble.\n\nHe didn't go looking for Sheila. He decided to go wandering. He would slip away. He had always liked slipping away from things. At parties he would just quietly leave when nobody was watching him, so that only much later would anybody wonder where he was. When he was out with a group of other kids, playing kick-the-can or soldiers, he would often sneak away, so that they would never find him, or think he had fallen down a hole or into a burn or a loch. It was a wonderful feeling, to disappear like that; it made him feel different and special. He gloried in the cunningness of it, the feeling of having outwitted the others, of knowing what they did not; that he was out and away and they were back there where he'd left them, ignorantly worrying where he was, searching; wondering.\n\nHe slunk out through the doors while they were clapping the band after finishing one of their noisy, interminable Highland dances.\n\nIt was cooler in the lobby. He drew himself upright and walked confidently through the bit of the lobby that gave onto the Cocktail Lounge, where ruddy-faced men stood panting and laughing, sleeves rolled up, ties loose, queuing for drinks or holding trays of them, laughing loudly in deep voices.\n\nHe went through another set of doors, down some steps, round a corner, and found the hotel's single small lift. He pulled both sets of gates open with an effort, entered, then closed them again. The lift was a little bigger than a phone box. He pressed the brass button for the top floor. The lift jerked into motion and set off, humming. The white-washed walls of the lift shaft moved smoothly downwards as the lift ascended. Stencilled letters painted inside the shaft said 1st Floor ... 2nd Floor ... God, he thought, Americans must think they're in the Stone Age when they come to stay in a place like this.\n\nHe felt ashamed.\n\nThe top floor was boring. He went from one end to the other of the U-shaped hotel, up and down steps that marked the boundaries of the three separate buildings that made up the Steam Packet Hotel. There were no windows; only skylights, each spattered with rain drops and lined with little rivulets of running water. He'd been hoping for windows, and a view over the bay or the town.\n\nHe trod the corridors again, looking for an unlocked door. Maybe the maids would have left some of the rooms open, if there was nobody staying in them just now. He tried a few handles. The only open door led to a broom cupboard.\n\nThen at the next door he heard giggling. He looked at the number. It was room 48. 48 was a good number; not as good as 32 or 64, but better than, say, 49, and much better than 47 (though that was interesting too because it was a prime). The very best numbers were numbers like 20, 23, 30, 40, 57, 75, 105 and 155. Calibre numbers; gun numbers. Those were luckiest. But 48 was all right.\n\nMore giggling. He looked back down the corridor, then crouched and looked through the key-hole. It was a bit clich\u00e9d, but what did people expect in a boring hotel like this in a boring town like this in a boring country like this? It was all you could do.\n\nThere was no key in the lock, so he could see in through the big old-fashioned key-hole. He saw a large dressing table sitting in a broad bay window. The dressing-table held a big, tippable mirror, and most of the rest of the room was visible in it. In the mirror Rory saw his sister Fiona, and then Fergus Urvill. They were making the big double bed.\n\nFiona still wore her peach-coloured bridesmaid's dress, very long and smooth-looking. There were flowers in her hair, which made her look quite good. Rory suspected she looked so good because she didn't live here any more; she lived in London, and Aunt Ilsa had got her a job working for a television company. Fiona sold time to people. That was how she put it. She sold advertising space. She sold time. Rory thought that sounded pretty interesting.\n\nFergus Urvill was on the other side of the bed, dressed in a kilt, shirt and waistcoat. Rory knew Fergus was ages with Kenneth, but somehow he always seemed older. Maybe it was because he had gone to a private high school. Rory didn't really know Fergus Urvill very well; although he did sometimes visit Lochgair, he spoke differently \u2013 posher \u2013 and seemed to spend a lot of his time shooting at birds and animals with other rich people.\n\nRory had always found Fergus Urvill to be a little frightening. Kenneth had told him the story, years ago, about when Fergus put Lachy Watt's eye out; he'd stuck a fossil bone in it, or something. Rory thought now that his brother must have exaggerated the story, made it more horrific than it really had been, and he _certainly_ didn't believe that Lachy had run away to sea just so that he could wear an eye-patch and pretend he was a pirate. He had joined the merchant navy \u2013 Rory had asked dad about that \u2013 but he had an artificial eye, not a patch. Rory knew because he'd been with mum once when they'd met Lachy and a woman in the street in Lochgilphead. Rory had looked very hard but hadn't been able to decide which was the false eye.\n\nHis own eye smarted, exposed to the draft coming through the key-hole. He blinked, then used his other eye.\n\nFiona and Fergus were making the bed, but doing it in a funny sort of way; the bottom sheet had been doubled up half-way down the bed. They were both chuckling to themselves, and talking in quiet, urgent whispers. Fiona glanced off to one side a couple of times. Rory worked out she was looking at the door he was crouched behind.\n\nThey made the bed up, so that it looked ordinary. Rory got ready to run away down the corridor. But they didn't leave the room; instead, Fiona and Fergus, still breathless with giggles, still chattering excitedly away, started to turn the furniture in the room upside down. They left the bed, of course, but they turned a table, a chest of drawers, two bedside cabinets, two chairs and an easy chair upside down. They carefully replaced lights and vases and other bits and pieces as they went along. They stood before the dressing table for a while, looking at it and discussing it, apparently, but eventually just turned it round so that it faced the wrong way, rather than turn it upside down.\n\nFiona leant back against the rear of the dressing table, breathing hard, and waved one hand, wafting air over her face. Her cheeks were pink, and a couple of coils of copper hair had fallen from her hairdo, one on each side of her head. She pulled at her bodice, blew down, went 'Whoo!' Rory couldn't see Fergus Urvill. Then he reappeared, stood by Fiona. He was holding a key and a couple of toilet rolls; he said something Rory didn't catch. 'Oh no,' Fiona said, touching Fergus's arm. Her face looked amused but concerned. 'No, that's naughty ...'\n\nFergus stood there for a moment. Rory couldn't see his face, but Fiona's looked glowing and bright. 'I like being naughty,' he heard Fergus say, and then he stepped forward and took Fiona in his arms, still holding the key and the toilet rolls.\n\n_What?_ thought Rory. This really was something. Sister Fiona and big Fergus Urvill? Stupid girl; probably only after her body.\n\n'Ferg!' Fiona said, breaking away. Her face looked surprised, cheeks even redder. She smiled broadly, held Fergus's elbows. 'Well, this is ... unexpected.'\n\n'I've always ...' Fergus lowered his voice as he bent to kiss her again, face in her hair and then his mouth on hers. Rory missed the exact words.\n\nGo on, thought Rory. Go on. Do it. Let me see!\n\nFergus's hands dropped the key and the toilet rolls, grabbed Fiona's bum. She pushed away from him. 'Ferg ...' she said, breathless, lip-stick smeared.\n\n'Fiona,' Fergus moaned, clutching her. 'I want you! I need you!'\n\n'Well,' Fiona said, gulping. 'That's very, ah ... but not here, eh?'\n\nFergus pulled her close again. 'Let me drive you home tonight.'\n\n'Umm, well, I think we were getting a taxi.'\n\n'Please; let me. Please. Fiona. You don't know ...' Fergus stuck his nose into her hair again, made a sort of moaning noise. 'Feel me.' And he guided one of Fiona's hands to the front of his kilt.\n\nGood God, thought Rory. He took another quick glance down the hall, then looked back through the key-hole.\n\nFiona took her hand away. 'Hmm. Yes; actually I already could, Fergus.'\n\n'I need you!' He pulled her close again.\n\n_'Not here,_ Fergus.'\n\n'Fiona; please ...'\n\n'All right; all right, Fergus. I'll try. We'll see, okay?'\n\n'Yes; yes, thank you!' Fergus gathered Fiona's hands in his.\n\n'Right,' she laughed. 'Well, come on; let's get out of here before the happy couple arrive. Put those back in the loo.' She pointed at the toilet rolls. Fergus retrieved them. She busied herself with her hair, restoring it. Fergus turned and disappeared from Rory's view. 'And put some cold water on _that,'_ Fiona said, grinning. 'Looks like your sporran's trying to levitate.'\n\nShe came towards the door. Rory leapt back, staggered on legs that had gone half to sleep, and only just scrambled into the broom cupboard and got the door shut before the bedroom door opened. The broom cupboard key-hole didn't let him see anything. He heard muffled conversation but no footsteps.\n\nHe waited, breathless, heart hammering in the darkness, one hand in his trouser pocket, stroking himself.\n\n_'Do you know where the twins were conceived?'_\n\n_'No idea,' he said, and belched._\n\n_'Fucking McCaig's Folly, that's where.'_\n\n_'What, Oban?'_\n\n_'The very place.'_\n\n_'Good grief.'_\n\n_'You don't mind me saying this, I mean talking about Fiona like this, do you?'_\n\n_'No, no.' He waved one hand. 'Your wife; you talk about her. No, no, that's bad, that sounds bad. I'm all for women's lib.'_\n\n_'Might have bloody known. Might have bloody known you would be. Bloody typical, if you ask me. You're a Bolshie bastard, McHoan.'_\n\n_'And you are the unacceptable face of Capitalism, Ferg.'_\n\n_'Don't quote that fairy at me, you Bolshie bastard. And don't call me Ferg.'_\n\n_'Beg your pardon. Some more whisky?'_\n\n_'Don't mind if I do.'_\n\n_Rory got up out of the creaking wooden seat and walked unsteadily over to where Fergus lay on the bare wooden floorboards, head against the ancient, burst couch. The fire crackled in the grate, its light competing with that of the little gas lamp. Rory unscrewed the top from the bottle of Bells carefully and topped up Fergus's little silver cup. Fergus had brought a leather case with him; it held three of the silver cups and a big hip flask. Rory had brought the bottle in his rucksack._\n\n_'There you go.'_\n\n_'Ta much. You're a decent fellow for a Bolshie bastard.'_\n\n_'One tries, old bean,' Rory said. He walked carefully to his seat,_ _picked his little cup up from the floor and went to the room's single window. It was black outside. There had been a moon when they'd first arrived, but the clouds had come while they were chopping wood, and the rain while they'd cooked dinner on the two little primus stoves._\n\n_He turned from the darkness. Fergus looked like he was almost asleep. He was dressed in plus fours, tweed waistcoat (the jacket, and his waxed Barbour were hanging behind the door of the bothy), thick socks, brogues, and a fawn country shirt with a button-down collar. God, he even had his tie on still. Rory wore cords, mountain-hiking boots and a plain M &S shirt. His nylon waterproofs were draped over a chair._\n\n_What an odd pair we make, he thought._\n\n_He had been back from his travels for a while, staying first in London then at Lochgair, while he tried to work out what to do with his life. He had the impression things were sliding past him somehow. He'd made a good start but now he was faltering, and the focus of attention was drifting slowly away from him._\n\n_He had returned to discover that \u2014 like his brother before him \u2014 Ken had given up being a teacher. Hamish had taken up the managerial place at the factory that everyone had expected would be Kenneth's, when Kenneth had decided to teach. Now Ken too was quitting the profession to try something else: writing children's stories. Rory had always thought of Hamish as a sort of ponderously eccentric fool, and Ken a kind of failure because he had so much wanted to travel, and instead had settled down with Mary, stayed in the same wee corner of the world as he'd been born and raised in, and not only raised his own children, but chosen to teach others', too. Rory had felt slightly sorry for his elder brother, then. Now he felt envious. Ken seemed happy; happy with his wife, with his children, and now with his work; not rich, but doing what he wanted to do._\n\n_And why hadn't Ken told him he was writing too? He might have been able to help him, but even if Ken had wanted to do it all_ _without any assistance from his younger brother, he might at least have told him what he was doing. Instead Rory had found out only when Ken had had his first story published, and now it was as though they were passing each other travelling in opposite directions; Ken slowly but surely building up a reputation as a children's story-teller while his own supposed career as a professional recounter of traveller's tales sank gradually in the west. Books people forgot about and articles in Sunday supplements that were only one notch above the sort of shit tourist boards put out._\n\n_And so he'd left London, to come here, hoping to lick the closing wise wound of whatever talent it was he had._\n\n_He'd spent a lot of time just wandering in the hills. Sometimes Ken came too, or one of the boys if they were in the mood, but mostly he went by himself, trying to sort himself out. What it boiled down to was: there was here, where he had friends and family, or there was London where he had a few friends and a lot of contacts, and it felt like things were happening, and where you could fill time with something no matter how mixed up and fraudulent you felt... or there was abroad, of course; the rest of the world; India (to take the most extreme example he'd found so far), where you felt like an alien, lumbering and self-conscious, materially far more rich and spiritually far more poor than the people who thronged the place, where just by that intensity of touching, that very sweating crowdedness, you felt more apart, more consigned to a different, echoing place inside yourself._\n\n_One day, on a long walk, he'd almost literally bumped into Fergus Urvill, crouching in a hide up amongst the folds in the hills, waiting with telescope and .303 for a wounded Sika deer. Fergus had motioned him to sit down with him behind the hide, and to keep quiet. Rory bad waited with the older man \u2013 silent for quarter of an hour apart from a whispered hello and a quick explanation of what was going on \u2013 until the herd of deer appeared, brown shapes on the brown hill. One animal was holding the rest back; limping heavily. Fergus waited until the herd was as close as it looked like it was_ _going to come, then sighted on the limping beast, still two hundred yards away._\n\n_The sound of the shot left Rory's ears ringing. The Sika's head jerked; it dropped to its knees and keeled over. The rest raced off, bouncing across the heather._\n\n_He helped Fergus drag the small corpse down the slope to the track, where the Land Rover was parked, and accepted a lift back to the road._\n\n_'Hardly recognised you, Roderick,' Fergus said, as he drove. 'Not seen you since Fi and I got shackled. Must be at least that long.'_\n\n_'I've been away.'_\n\n_'Of course; your travels. I've got that India book of yours, you know.'_\n\n_'Ah.' Rory watched the trees slide past the Land Rover's windows._\n\n_'Done any others?'_\n\n_'There was one about the States and Mexico. Last year.'_\n\n_'Really?' Fergus looked over at him briefly. 'I didn't hear about that.'_\n\n_Rory smiled thinly. 'No,'he said._\n\n_Fergus made a grunting noise, changed gear as they bumped down the track towards the main road. 'Ken said something about you living in a squat in London ... or something ridiculous like that. That right?'_\n\n_'Housing cooperative.'_\n\n_'Ah-ha.' Fergus drove on for a while. 'Always wanted to take a look at India myself, you know,'he said suddenly. 'Keep meaning to go; never quite get around to it, know what I mean?'_\n\n_'Well, it isn't the sort of place you can just take a look at.'_\n\n_'No?'_\n\n_'Not really.'_\n\n_The Land Rover came down to the main road between Lochgilphead and Lochgair: 'Look, we've got a do on this evening, in the town \u2013' Fergus glanced at his watch. '\u2013 bit late already, to tell the_ truth. But how about coming round tomorrow _for ... In_ fact, d'you fish?'\n\n_'Fish? Yeah, I used_ to.'\n\n_'Not against your vegetarian principles, is it?'_\n\n_'No. India didn't change me that much.'_\n\n_'Well, then; come fishing with me tomorrow. Pool on the Add with a monster trout in it; been after the swine for months. Plenty of smaller stuff too, though. Fancy it? Course, I'll never talk to you again if you catch the big feller, but might make a fun afternoon. What do you say?'_\n\n_'Okay,' he said._\n\n_So they became friends, after a fashion. Most of Rory's pals in London were in the International Marxist Group, but here he was; wandering the hills with an upper class dingbat who just happened to be married to his sister and who lived for huntin', shootin' and fishin' (and seemed to spend the absolute minimum amount of time in his castle with his wife), and who had just last year rationalised half the work force in the glass factory out of a job. Still, they got on together, somehow, and Fergus was an undemanding companion; company of a sort, but not taxing; none of Ken's garrulousness, Lewis's moodiness or Prentice or James's ceaseless questioning. It was almost like walking the hills on your own._\n\n_And a couple of days ago Fergus had suggested they go for a longer hike, up into the trackless hills where the Landy couldn't reach. They would take collapsible rods, a couple of guns, and have to fish and shoot to eat. They could stay in the old lodge; it would save taking a tent._\n\n_So here they were, on the first floor of the old lodge, which was now used just as a bothy. The room they were in contained a single big dormer window, a fireplace, a couch, a table and two seats, and two bunkbeds. There were other rooms with more beds, but keeping to one room meant only lighting one fire; the autumn weather had turned chilly early._\n\n_'No,'Fergus said, looking up from where he lay, slumped against_ _the couch. 'But you don't mind me talking about Fiona like this, do you? I mean, your sister. My wife. You sure you don't mind, do you?'_\n\n_'Positive.'_\n\n_'Good man.'_\n\n_'McCaig's Folly, eh?'_\n\n_'Hmm? Oh; well yes... at least I think so. Got the idea from Charlotte, actually.'_\n\n_'What, your sister?'_\n\n_'Mmm. The one that married that chap Walker, from Edinburgh.'_\n\n_'Oh yeah; I remember.' Rory went over to the seat that held his jacket._\n\n_'Funny girl, Charlie; had this thing about ... antiquity. Got Walker_ to deflower _her under this ancient fucking_ yew _tree in Perthshire_. So she told _me, anyway.'_\n\n_'Uh-huh.'Rory rummaged in his jacket pockets._\n\n_'Fiona and I thought we'd try something like that, one time we were in Oban, for some do. You know; put a bit of sparkle back in ... You sure you don't mind me talking about your sister like this?'_\n\n_'Yeah.' Rory took his tobacco tin from the jacket. He held the tin up. 'As long as you don't mind me having a little smoke?'_\n\n_'Not at all, not at all. Bloody cold it was, in that damn folly. Had to sit on a \u2013 Oh,' Fergus said, suddenly realising. 'You mean the old wacky baccy.'_\n\n_Rory smiled, sat down. 'That's the stuff.'_\n\n_'Not at all,' Fergus said, waving one hand. 'Go ahead.' He watched carefully as Rory set out the papers. 'Mmm, go ahead.'_\n\n_Rory looked up, saw Fergus's fascinated expression. 'Do you want any of this, Fergus?'_\n\n_'Umm,' Fergus said, sitting back, blinking. 'Could do, I suppose. Never really tried it, to be honest. Couple of chaps at the school got booted out for that stuff and I never did get round to it.'_\n\n_'Well, I'm not forcing you.'_\n\n_'Not at all. Not at all.'_\n\n_They smoked the joint. Fergus, used to the occasional cigar with his brandy now that he'd given up his pipe, pronounced the smoke quite cool, and objected more to the sweet taste of the Old Holborn than to the scent of resin._\n\n_'This any good for hanky-panky?' he said, passing the roach back to Rory, who took a last hot toke then flicked the remains into the heart of the fire._\n\n_'Can be,'he said._\n\n_'Might try it some time. God knows we could do with something to \u2013 Look, you absolutely sure you don't mind me talking about your sister this way?'_\n\n_'Positive.'_\n\n_'Good man \u2013 hey! Did you hear that?'_\n\n_Rory looked up at the ceiling. Fergus was staring at the plasterboard expanse above them. Rory listened. Then, above the crackling of the fire, he did hear something; a quiet, scrabbling noise in the roof-space above them._\n\n_'Rats, I'll bet!' Fergus said, and rolled over to his pack._\n\n_Rory thought about it. They were here in a deserted old house in the middle of nowhere on a black and starless night in one of the more mysterious bits of Scotland, and there was a scrabbling, clawy sort of noise coming from the ceiling above him and this other drunk, stoned man. He shrugged. Yeah; probably rats. Or mice. Or birds._\n\n_Fergus pulled his pack gently to him, scraping over the floorboards. He lifted the rucksack up. The .303 and the shotgun were in a waterproof bag strapped to the side of the pack. Fergus undid the straps. 'Ssh,' he said to Rory. Rory had started building another joint. He waved. He drank some more whisky._\n\n_He was just inserting the roach when Fergus rolled over to him and held the shotgun out to him. 'Here!' he whispered urgently._\n\n_'Hmm,' Rory said, nodding thanks. He heard some clicks._\n\n_Fergus held the ancient Lee Enfield at his side. He knelt close by Rory. 'Think the little bastard's over there.' He pointed. He reached_ _up, touched the gun Rory held. It was hard doing the roach one handed. 'Put that down, man!' Fergus hissed. He took the tin from Rory's lap and put all the makings down on the floor. Rory felt peeved._\n\n_'There,' Fergus said. 'Safety's off. When I fire, aim where I do, all right?'_\n\n_'Yup,' Rory said, forgetting about the J. He took the shotgun. Fergus walked on his heels, still hunkered down, across the room, eyes and gun pointed towards the plasterboard ceiling. He stopped. There was a noise like a spider running across a very sensitive microphone._\n\n_Bang! went the rifle. Rory almost dropped the shotgun. 'There!' yelled Fergus. Plaster was falling from a small hole in the ceiling; there_ was _smoke in the air._ Rory _aimed at the small hole, pulled the trigger. The gun struck back against his shoulder, sending him falling back off his seat. He clattered to the floor._\n\n_'Well, pump it, man, pump it!' he heard Fergus shouting from somewhere._\n\n_Awful lot of smoke around. Ears seemed to be ringing. He pumped the gun. (Funny; he'd have thought Fergus would have been a side-by-side man.) There was another sharp crack of sound from the .303. He saw the hole appear in the plaster almost right above him. Great; he could get the little bastard without having to get up from the floor. The floorboards ought to provide extra firing stability, too. He pulled the trigger again. The gun went Blam! with a little less sonic enthusiasm than before, though it hurt his shoulder a little more._\n\n_A white waterfall of plaster burst down from the ceiling and slapped and pattered all over him. Rory spat bits out of his mouth, blinked the white dust out of his eyes. He heard Fergus colliding with something in the room. He pumped the gun, looked round. Fergus was lying on the couch, aiming at the centre of the ceiling. He fired the Lee Enfield again; Rory was getting the hang of this now, and aimed the shotgun at the same place and fired it, almost before_ _the noise of Fergus's shot had stopped echoing. The room was getting a bit hazy, and there was probably blood coming from his ears, but what the fuck. Rory readied the gun again._\n\n_He tried to follow where Fergus was pointing his rifle. As he did so, still lying there with his legs up on the chair he'd fallen over, he started-to over-balance to one side, towards Fergus._\n\n_'Aah!' Rory said. He tried to put one hand out to stop himself, but the gun was still in his grip. The long, blue-black barrel arced towards Fergus. Fergus looked, as Rory fell helplessly over, the gun barrel falling like some felled tree, wide muzzle pointed straight towards him._\n\n_Rory could tell exactly what was going to happen, and couldn't stop it._\n\n_Fergus's eyes widened. He jumped; fell over the back of the couch._\n\n_Rory fell onto his side; the shotgun roared and the rear of the couch blew open in a dusty horsehair explosion._\n\n_Rory let the gun down to the floor. The noise still rang in his ears. The room stank of smoke and the fire had gone strangely quiet. 'Ferg?' he said, tentatively. Couldn't hear himself speak._ 'Ferg!' _he shouted._\n\n_He sat upright, leaving the gun on the floorboards. Plaster tumbled off his body in clouds of dust._\n\n_'Hello?'Fergus said, appearing over the top of the couch, gunless._\n\n_Rory looked at him. They both blinked, eyes watering. 'Did we get it?' Rory asked._\n\n_'Don't know,' Fergus said. He staggered round the rear of the couch, feet crunching in plaster, and sat down. He looked at the still slightly smoking hole in the couch, just beside where he'd sat, then up at the holes in the ceiling._\n\n_He stayed looking at the holes in the ceiling for a while. Then he started crying._\n\n_Rory watched for a while, befuddled. 'What's the matter, man?' he said._\n\n_Fergus took no notice; he kept on crying, still staring up at the holes in the ceiling. He took big lungfuls of air and then let them out in great racking sobs that shook his whole body. After a while, he put his head in his hands and sat there, rocking back and forward, clutching his hair just above his ears. The tears flowed, trickling off his nose and spotting the white plaster dust on the floorboards at his feet._\n\n_'Ferg,' Rory said, going over to him. He hesitated, then put his arm on the man's shoulders. 'Fergus; for God's sake man, what's wrong?'_\n\n_Fergus looked up and suddenly Rory felt older than him. Fergus's heavy, ruddy face was puffed and bloated, and tears had streaked through the dust on his cheeks, disappearing in the bristles on his jaw-line and chin. When he spoke it was in the voice of a small, hurt boy._\n\n_'Oh God, Rory, I've got to tell somebody, but you must promise; you must give me your word you won't breathe a word to anybody else. On your life.'_\n\n_'Hey, you haven't killed anybody or anything, have you?'_\n\n_'No,' Fergus shook his head, screwed his eyes up. 'No! Nothing like that! It's not something I did.'_\n\n_'Okay; my word. All that stuff. '_\n\n_Fergus looked at him and Rory shivered. 'You swear?' Fergus said, voice hollow._\n\n_Rory nodded. 'I swear.' He felt dizzy. The smoke-filled room seemed to tip and waver. He wondered if they put something trippy in shotgun or rifle cartridges. And why did I mention killing somebody? That wasn't too sensible, way out here on this moonless night, etc., with a couple of lethal fire-arms lying around._\n\n_'All right,' Fergus said, sitting back, breathing deeply. He looked almost soberly at Rory. 'You sure you don't mind me talking about your sister?' he said slowly, with what might have been some sort of smile on his face._\n\n_Oh god, thought Rory, and felt sick._\n\n_But it was too late to go back now._\n\n_The way he told it, it took maybe five minutes. Fergus Urvill was crying like a baby again at the end of it. Rory cuddled him. And after as many tender words as he could think of, to try and lighten the load, to try and make it seem less of a confession, even to try and compensate for the shared and shaming confidence, he told Fergus that he had been responsible for the fire that had burned down the barn near Port Ann, fifteen years earlier._\n\n_They ended up laughing about that, but it was the uneasy laughter of desperation and displacement, and all they could do after that was finish the whisky and have the joint Rory had been working on, and it was almost a relief when Fergus was sick as a dog out of the window, hanging out barfing onto the slates and into the guttering while Rory tried to clean the plaster off the top bunk and stowed the guns out of harm's way._\n\n_They woke with raging hangovers to a wrecked room and the smell of black powder and vomit. There was a dead rat, blown almost in two, resting on the hearth of the fire._\n\n_They left the place as it was, picked up their gear and walked away. Neither of them mentioned anything that had been said during the night; they just agreed to head back to civilisation and not to mix whisky and cannabis like that again._\n\n_There were no more huntin' shootin' and fishin' trips. Rory went back to live in London that winter, and ended up \u2014 funnily enough \u2014 living in a squat._\n\n_He wrote poems._\nCHAPTER 9\n\nThe train sat, wrapped in rain and rocked by gusts, waiting to join the main line. Sidelined again, I watched the cold wind flatten the grubby-looking grass of a weedy field outside Springburn. A man walked across the field, some mongrel dog padding ahead of him. Two paths crossed the rectangular field, forming a neat St Andrew's Cross of down-trodden grass. The dog stopped to sniff at something in the grass, then squatted, urinating. The man following behind was dressed in cheap looking jeans and a donkey jacket, there was a bonnet perched on his head, and his hands were stuffed in his pockets. He walked up behind the dog and kicked its arse. The animal loped away, putting more distance between them, then resumed its casual, padding walk along the path. It was getting dark. Street lights were starting to come on in the distance, crimson slivers slowly brightening to orange.\n\nI looked at my watch. We'd been stuck here, waiting to join the main line into Queen Street, for about ten minutes. You often had to wait here while the Edinburgh trains came and went, but the delay didn't usually last this long. The station was only five minutes away; more importantly, _food_ was only five minutes away. I'd forgone breakfast because I hadn't got to bed till four in the morning, lunch because I had a hangover and anyway I was late for the train, and due to the fact that it was \u2013 according to British Rail at any rate \u2013 still part of the extended Festive Period, there had been no buffet trolley on the train. I was starving. I was so hungry I'd have eaten pork scratchings. Queen Street station, a scant mile and a half away, had burgers, sandwiches, shell pies, french fries and french sticks, bridies and pasties and patties. My God, if all they had were Haggisburgers, I'd eat those.\n\n'Ladies and gentulmun ...' crackled a gruff Glaswegian voice from the carriage loudspeakers. My heart sank. The perfect end to a perfect holiday. 'Due to a signalling failure ...'\n\nI looked out of the wind-shaken carriage, where people were moaning and cursing and making vows to start going by bus, or take the car next time, or buy a car, or learn to drive ... looked out through the rain-spattered sheets of glass, watching the cold January day leach out of the grey skies above the drenched city, and witnessed the rain fall upon the tramped-on, pissed-on, shat-on grass of the narrow path in the scrubby field with a feeling of wry but nevertheless wretched empathy.\n\nGod, what did any of it matter, in the end? You lived; you died. You were as indistinguishable from a distance as one of these blades of grass, and who was to say more important? Growing, surrounded by your kin, you out-living some, some out-living you. You didn't have to adjust the scale much, either, to reduce us to the sort of distant irrelevance of this bedraggled field. The grass was lucky if it grew, was shone upon and rained upon, and was not burned, and was not pulled up by the roots, or poisoned, or buried when the ground was turned over, and some bits just happened to be on a line that humans wanted to walk on, and so got trampled, broken, pressed flat, with no malice; just effect.\n\nAnd intelligence? Control? There were things that we had no more control over yet than the grass did over the developer who chose to plough it all under and build a factory on top. Perhaps some asteroid, nudged out of its place in the great gravitational gavotte, would fall to Earth; a bullet into a face, obliterating. Unwitnessed, for what would be visible, from even a nearby star? A blink of flame, like a match struck beside a search-light ... And then nothing.\n\nBut didn't there have to be something out there, just to witness, just to _know?_ Hell, it didn't even have to do anything; it didn't have to act on prayers or have us singled out as a special species, or play any part in our history and development; it didn't even necessarily have to have created us, or created anything, all it had to do was exist and have existed and go on existing, to record, to _encompass._\n\nI watched as the rain battered the grass and the wind pummelled it, quick gusts flattening patches of the field like sudden bruises beneath the dull sky. I could just imagine my father jumping up and down on this argument, this need for meaning, for faith.\n\nThe train jerked. I started, too, shaken from my reverie. Then the train went into reverse, motors growling, occupants groaning, and trundled slowly back through the squalls of rain, passing Maryhill and looping down through Anniesland and over Great Western Road.\n\nWe paralleled Crow Road for a bit, and stopped, waiting for signals, outside Jordanhill station; I looked up at the rear of the flats which fronted Crow Road, trying to work out which was Janice Rae's.\n\nI thought of Uncle Rory, then remembered that I had some more of his papers with me, and a load of his poems. Mum had found them for me in the house at Lochgair. I got my bag down from the rack. Uncle Rory could not be more depressing than reality was, just now.\n\nAny hope I might have entertained that Lewis and Verity's little Hogmanay hug had been an aberration, something they would fail to follow through, or feel for some reason embarrassed about, was comprehensively quashed the next evening when they turned up together at Uncle Hamish and Aunt Tone's, bearing all the signs of new lovers (literally so in the case of Lewis's neck, which displayed a line of passion purpurae worthy of an industrial vacuum cleaner, and which were ill-concealed by Lewis's longish dark curls and a white shirt fastened with a bootlace tie).\n\nLewis and Verity kept exchanging looks, laughing at anything even remotely amusing each other said, sitting close together, finding a hundred small excuses to touch each other ... I wanted to throw up. We had all gathered for Hamish and Tone's traditional Ne'erday partyette; a necessarily quietish affair during which people exchanged tales of drunkenness, broken resolutions and recipes for hangover cures, as well as taking advantage of the opportunity to compare notes regarding blank spots in the memories of any of the assembled penitents.\n\nI was helping Aunt Tone prepare stuff in the kitchen but had to give up when Lewis and Verity volunteered to assist as well, and then spent most of the time feeding one another little bits of food, goosing each other and going into sardinely-intimate huddles punctuated by low whispers, bursts of baboon-like giggles and convincingly porcine snorts. I went through to the dining-room and helped myself to a pint of the neuron-friendly punch Uncle Hamish always made for the event.\n\nMum and dad turned up later. There were about twenty of us, all told; mostly McHoans but with a smattering of civilians. We sipped \u2013 or in my case gulped \u2013 the weak but tasty punch, nibbled on Aunt Tone's buffet-bits, and played Alternative Charades; an invention of my father's in which one first has to guess the category of the thing one is being asked to decipher. When it was my turn to mime, I tended to concentrate on Popular Communicable Diseases, Well-Known Poisons, Famous Mass Murderers and Great Natural Disasters.\n\nMy last memory is of trying to mime Rare Gynaecological Disorders, preparatory to attempting Toxic Shock Syndrome. But apparently people insisted that one stand up to do one's piece, and I \u2013 successfully acclimatised to the horizontal by this time \u2013 refused to pander to this sort of nit-picking, and so passed my turn on to Cousin Josh with as much good grace as I could muster.\n\n'The congenitally odd-jeaned person to my left will take my place,' I mumbled, waving one hand in his direction before letting my head resume its communion with Hamish and Tone's lounge carpet.\n\nThe bit about odd jeans was totally accurate, by the way; Cousin Josh made his fortune firstly by dealing in cars, then by risking all on a jeans company which at the time was tottering on the very hem of bankruptcy; under Josh's regime, their jeans weren't any better or any cheaper than anybody else's, but he had the garments made in odd sizes; waists of 29, 31, 33 inches, and so on, as opposed to the products from all other companies, domestic and foreign, which tended to favour the even numbers.\n\nIt was one of those brilliantly simple ideas people always wish they had had themselves, and believe that somehow they could have had; no need to incur any extra expense or make any more sizes than anybody else, or necessarily to distinguish one's product in other way, yet just by the idea one has a potential market of half the jeans-buying public, or at least that proportion of it which has always felt that they are somehow perpetually between the usual sizes.\n\nI vaguely remember dreaming about Verity's jeans that night; how graphically, geographically tight they were and how wonderful it must be to take them off her. Then I imagined Lewis, boots tied round his neck, for some reason suddenly resembling Shane MacGowan, skinning her jeans off, not me, and he turned into Rodney Ritchie, at home with his parents, unpicking the individual stitches of her jeans with a tiny knife, and the Ritchies all wore badly-fitting jeans and had denim curtains and denim carpets and denim light shades and denim wallpaper with the little rivets left on like poppers so you could just press paintings and photos onto the wall ... except that Mr Ritchie looked like Claude Levi-Strauss, which is when I think I started to get confused.\n\nEither I had been put to bed, I thought, as I woke up next morning, in the wee cold room at the top of the house, or my standard drunk-person's on-board auto-pilot facility was improving with experience.\n\nI bathed, dressed, and broke my fast with some left-overs from the fridge, a pint of water and a couple of brace of Paracetamol, all without encountering anybody else in the house. It was only eight o'clock; obviously I'd conked out some time before everybody else, and they were still asleep (I had heard appropriate log-sawing-like noises coming from Hamish and Tone's room on my way back from the bathroom). The day looked bright but cold; I laced up the Docs and went for a walk in the hills behind Gallanach.\n\nI felt like shit and I was trying so hard not to think of Lewis and Verity that I couldn't think about anything else, but the day was fabulous; clear and cold, the sky crystal blue and reflecting in the waters of hill-cupped lochans and the glinting length of Loch Add. On such days the hills hold a mixture of azure and gold never seen at any other time of year; the cobalt sky is more intense than it ever is in summer, and the straw-coloured hills shine strong in the light from the low winter sun. Against the shifting mirror that is the surface of a loch, the colours shimmer and dance; they take your breath away, and \u2013 for a brief, relieving while \u2013 they can even take your thoughts away.\n\nUp in the hills, at the place of marching water, I found Ashley Watt and one of her more exotic cousins.\n\nThe concrete spillway below the Loch Add reservoir comes down to a stepped slope above the confluence of several small burns draining nearby slopes. A short bridge carries the track over the spillway, and that was where Ashley and Aline were sitting, legs dangling over the stream in the concrete gully, arms resting on the lower bar of the bridge rails.\n\nThey were sitting side by side, watching the marching water.\n\nWhat happened was that the water first backed up behind the lipped edge of the top step, then over-flowed, and spilled with increasing force, in a sort of hydro-chain-reaction, down each subsequent step to the bottom of the channel. There followed a period of comparative quiet, while the water built up again behind the top step and those beneath. You might guess it was my dad who first pointed out this odd (and classically Chaotic) phenomenon and brought it to the attention of us kids. None of us had ever been able to discover whether it was a deliberate effect, or the result of pure chance. Whatever, it was wonderfully restful, unpredictable and therapeutic.\n\n'Hey, Prentice,' Ash said. She looked a little worn and bleary-eyed, though her long, lion-coloured hair shone like health itself in the brassy sunlight of mid-day.\n\n'Hi.' I nodded to her and to Aline, who was Franco-Vietnamese and engaged to Hugh Watt, one of Ashley's multitudinous cousins from the branch of the family that seemed to favour consorts of an exotic provenance (Hugh's brother Craig was going out with a stunning, lanky Nigerian called Noor). Aline looked even smaller and blacker-haired than usual, beside Ashley. 'Aline; \u00e7a va?'\n\n'Magic, Prentice,' Aline replied in fluent Glaswegian.\n\n'Have some skoosh,' Ash said as I sat down next to her. She reached between her and Aline and handed me a half-finished bottle of Irn-Bru. I had, over the course of the morning, already gulped down about a gallon of teeth-achingly cold stream-water at various points up in the hills, but the traditional Scottish hangover treatment was probably just what I needed. I took a couple of mouthfuls, handed the bottle back, wiping my lips.\n\n'You look terrible,' Ash said.\n\n'Feel worse,' I said glumly, watching the water cascade down the concrete stair-case of the spillway.\n\n'Lost track of you at the Urvills' party, Prentice,' Ashley said. 'You just slope off, or did you get a lumber?'\n\n'Oh God,' I moaned, and lowered my head to the cool steel pipe of the bridge rail.\n\n'Hey ...' Ash said gently, putting her hand on my head and patting me. 'There there, Prentice ma man. What's the matter?'\n\n'Oh, nothing much,' I sighed, slowly raising my head again and gazing at the water. 'I saw the woman I love wrap herself round my older, smarter and wittier brother like clingfilm round a sandwich, and it looks like they're enjoying each other the way ... Oh, God, I'm so pissed off I can't even think of a decent comparison. Or even an indecent one, which would probably \u2013 certainly \u2013 be more to the point.'\n\n'Part from that; everything okay, aye?' Ash said, putting her arm round my shoulders.\n\n'Help me, Ashley,' I said, closing my eyes and putting my head on her shoulder. 'What am I to do?'\n\n'You must think of her on the toilet,' Aline said, and giggled.\n\n'Off-white woman speak truth,' Ash said, lowering her head to rest it on mine. 'The hots rarely survive an intense course of imagining the beloved on the cludgie.'\n\n'No,' I sighed, opening my eyes as a series of splashes announced another chaotic event on the spillway. 'I'd probably only develop a fetish for coprophagy.'\n\n_'Pardon?'_\n\n'That as unpleasant as it sounds?'\n\n'Unpleasanter.'\n\n_'Merde!'_\n\n'Yup.'\n\n'You're a hopeless case, Prentice, so you are. Have you contemplated suicide?'\n\n'Yeah; soon as it's finished, I'm going to throw myself off the Channel Tunnel.'\n\nAshley's shoulders moved once under my head. 'Plenty of time to set your affairs in order, then.'\n\n'It's not my affairs I'm concerned with.'\n\n'Ach, she wasn't your sort, anyway, Prentice.'\n\n'What; you mean not good enough for me?'\n\n'No, Prentice; I mean too much taste. You never stood a chance with a woman that choosy.'\n\nI pulled away and looked dubiously at Ashley, who smiled sweetly. 'What is this?' I said. 'You auditioning for the Exit chapter of the Samaritans, or what?'\n\nAshley took my hands in hers. 'Ah, Prentice. Dinnae worry; maybe it's just an infatuation; hers, or Lewis's ... or yours. Whatever. Maybe she'll come to her senses. Maybe she wants to work her way through all the McHoan brothers in order of age \u2013'\n\n'Or weight.'\n\n' \u2013 or weight. Maybe she'll get married to Lewis but have a lifelong affair with you.'\n\n'Oh, great.'\n\n'See? You don't know what might happen,' Ashley said happily, spreading her hands.\n\n'Anyway, Prentice,' Aline said in her sing-song voice. 'There are plenty more fishes in the sea, yes?'\n\nI looked over at Aline. 'Hey, can I quote you on that?'\n\nAline winked at me, tapped the side of her nose. 'The toilet,' she said conspiratorially.\n\nI started to get up. 'It's no good,' I sighed. 'You two are cheering me up too much and I can't stand the excitement.' I got wearily to my feet, muscles aching from the effects of drink and walk.\n\n'See you down the Jac tonight?' Ash said.\n\n'Maybe,' I said. 'I keep trying to drown my sorrows but they appear to be marginally more buoyant than expanded polystyrene.' The water cascaded down the face of the spillway again, the noise like a million stamping feet heard from a long way off. I shrugged. 'Fuck it, though; worth another try. Gotta start working some time.'\n\n'That's my boy.'\n\n'See you, gals.'\n\n'Bye-bye, Prentice.'\n\n'Try not to fall in love with anybody else before tonight.'\n\n'Yo.'\n\nAn hour or so later I saw my mother's green Metro, just about to turn out of the drive-way of Hamish and Tone's house. She stopped when she saw me, wound the window down. 'Here you are,' she said.\n\n'Here I am,' I agreed.\n\n'I was waiting for ages there.' She glanced at her watch. 'Oh well. Getting in?'\n\nI got into the car; we started to reverse the fifty yards back up the drive. Actually, my legs were so tired I was quite grateful for the lift. 'I brought what I could find of Rory's stuff.' Mum nodded. 'Your dad thinks there's more, but it's buried in the filing.' I looked at the back seat, where a folder lay. 'Not that you deserve it,' she added.\n\n'Oh, thanks,' I said. I picked the folder up; _CRII_ said the lettering on the spine. It looked similar to the folder I already had, but perhaps a little thicker. I vaguely remembered reminding mum last night that I was looking for the rest of Uncle Rory's papers.\n\n'Well?' she said.\n\nI looked over, yawning. 'Well?' I repeated.\n\nWe drew to a stop outside the door of the house. 'You don't remember last night, do you?' mum said, turning the ignition off. She was dressed in angora and chunky cords; new perfume. She looked slightly unamused and not a little worried.\n\n'Not... in its entirety, no,' I confessed.\n\nShe shook her head. 'God, you were drunk, Prentice.'\n\n'Umm,' I said, weighing the folder in my hands. '... Yes.' I smiled my best 'but I'm still your wee laddy' smile.\n\nShe raised those delicate brown brows. 'My God, you don't remember embarrassing Lewis and Verity last night, do you?'\n\nI looked at her.\n\n'I mean, apart from embarrassing your father and me,' she added.\n\nI felt the blood draining from my face like somebody had opened a valve in my ankle. Oh-oh.\n\nI swallowed. 'I wasn't doing my impression of the Bradford City supporter, the King's Cross Disaster victim and the guy from Piper Alpha meeting up in Hell, was I?' (Requires three cigarettes; offends everybody.)\n\n'It's not funny, Prentice; poor Verity was nearly in tears. You're lucky Lewis didn't throttle you.'\n\n'Oh my God,' I said, feeling cold. 'What did I say?'\n\n(Duck, and cover.)\n\n'Told her \u2013 told everybody \u2013 you were madly in love with her!' she said, eyes flashing. 'Then, having declared undying worship of the poor girl, you proceeded to slag her off for taking up with Lewis.' Mum shook her head angrily, tears in her eyes. 'Prentice! What were you _thinking_ of?'\n\n'Oh my God,' I moaned. KYAG. I put the folder down in my lap and put my forehead on the folder.\n\n'Then you followed that up with some fairly off-colour remarks about Lapland, and what you referred to, I believe, as \"the old earth-moving equipment\".'\n\n'Oh my God.'\n\n'And I think we all successfully worked out what \"doing the Delta Foxtrot\" was, as well, before you became totally incoherent.'\n\n'Oh my God!'\n\n'I don't think saying \"Oh my God\" will make it any better, Prentice. I think you should apologise to Verity and Lewis as soon as you can. They're up at the castle.' My mother brought her voice under control with an effort. 'Though you might also think about saying sorry to Hamish and Antonia, too, as you were their guest and it was their party you brought grinding to an embarrassing halt. Just as well you agreed to go quietly when Kenneth suggested it was time you went to bed; though apparently he and Hamish practically had to carry you upstairs, and the whole way up you were muttering something vile about Lewis being thrown naked into a tub of starving Elephant Leeches.'\n\nAnd dad put me to bed! Oh no! Dad and the Tree! The shame of it!\n\n'Mum, I want to die,' I mumbled into the folder.\n\n'Just at the moment, Prentice, I don't think there'd be any shortage of volunteers to help you on your way, if you were serious.'\n\n'I am.'\n\n'Stop being melodramatic, Prentice; it doesn't suit you. Sarcasm's more your forte.'\n\n'Oh my God.'\n\n'Prentice,' mum said, putting her hand on my head and running her fingers through my hair. 'Prentice ...'\n\nI looked up, straightened. Mum's eyes looked red. She shook her head. 'Prentice, why are you so stupid with your cleverness sometimes?'\n\nI took a deep breath. 'Wish I knew, mum,' I said, and sniffed, eyes smarting. Best not to say anything about it running in the family.\n\nShe took me in her arms, hugged me. I was surprised, as I always was at such moments, how slim and small she felt.\n\nAfter a bit we let go of each other. She glanced in the mirror and declared I had wrecked her eyes for the rest of the day. Then we went in to Hamish and Tone's for tea and apologies, and later drove to the castle for what would have been the most excruciating interval of my life if Verity and Lewis had still been there, but they weren't; they had taken off in the car to visit some friends of Verity's who lived in Ardnamurchan, and wouldn't be back until late tomorrow at the earliest.\n\nMum took me back chez Hamish and Tone; she agreed to pass on my expressions of contrition to my father. She'd wanted me to come to Lochgair and say sorry to him there, but I had begged for mercy, and \u2013 rather to my surprise \u2013 been granted it.\n\nI had already decided that tomorrow I would take the train back to what was now your official European City of Culture for the following twelve months. In theory, Verity and Lewis were meant to be giving me a lift in four days time, but that was obviously out, now.\n\nI had to promise mum I'd write to each of them, and apologise in person at the first possible opportunity, and also that I'd stop off at Lochgair before I returned to Glasgow, to see dad.\n\nAshley met me in the Jac that night, listened to my woes, bought me drink when I ran out of money (I'm sure I was short-changed at the bar) even though she probably had less dosh than I did, and listened to my woes all over again when we went back to her mum's and sat up till God knows when, talking low so we wouldn't wake Dean in the next room. She made me coffee, gave me hugs, and at one point I fell asleep, and was at peace for a while, and woke up sprawled on the floor, my head on her lap, one gentle hand stroking my head. 'Ash,' I croaked, 'you're a saint.'\n\nShe just smiled.\n\nA last cup of coffee and I left; back to H and T's in time for a few hours' fitful sleep; then up and away, run to the station by Aunt Antonia. I only just caught the train, and when, a quarter of an hour later, we pulled in to Lochgair, and I should have got my bag and quit the Sprinter and walked to the house and finally have talked \u2013 sober, and not in the context of a game of Alternative Charades \u2013 to my father, and apologised, and spent the three hours until the next Glasgow train with my mother and father in some longed-for spirit of reconciliation, I did nothing of the sort.\n\nInstead I put my head to one side so that it rested against the cold glass of the window, closed my eyes and let my mouth hang open a little. I stayed like that for the minute or so we waited at the Lochgair station platform, and didn't stir again \u2013 yawning convincingly for any other passengers who might be watching \u2013 until we were crossing the viaduct at Succothmore.\n\nStill stuck on the track within sight of Janice Rae's flat, I got up out of my seat, took down my bag and fished out the file mum had brought from the house. I found some much-Tippexed poems typed on foolscap, plus about twenty or so printed A4 pages which looked like they were part of a play or film script. I selected a page at random and started reading.\n\n_Lord:_ ... And I see them as they will be, dead and torn; shocked, mutilated and alone, on battlefields or by long roads, in ditches or against high walls, in echoing white corridors and misty woods, in fields, by rivers; dumped in holes, thrown in piles; neglected and absolved. Or, if living on, filled with petty, bitter memories, and a longing for the war they fought to end. Oh captain, I see in this my ending, what I think you didn't start to glimpse with your most cunning intuition; the soldiers are always the real refugees. Their first victim is themselves, their life taken from them well before \u2014 as though seeking a replacement from another freed \u2014\n\nBut I couldn't take any more. I put the papers in the folder and the folder in my bag, then stuffed that under my seat.\n\nI looked out at the rain instead; it was cheerier.\n\nI'd avoided stopping off to see mum and dad. It made my eyes close, every time I thought about it. What was wrong with me?\n\nWell, I thought; they made me. They produced me; their genes. And they brought me up. School and university still hadn't changed me as much as they had; maybe even the rest of my life could never compensate for their formative effect. If I was too embarrassed, too full of shame to go and see them, it wasn't just my fault; it was theirs too, because of the way they'd brought me up (God, I thought I'd stopped using that excuse when I left Lochgair Primary School). But there _was_ a grain of truth in there.\n\nWasn't there?\n\nAnd hell, I thought; I _had_ been tired; I _was_ tired still, and I would phone that evening \u2013 definitely \u2013 and say I'd fallen asleep, and nobody would be too bothered, and after all a chap could only cope with so much sorrow-saying in one day ... of course I'd phone. A bit of soft soap, a bit of flannel, like dad would say.\n\nNo sweat; I could charm them. I'd make everything all right.\n\nStill, it was the hangover of that piece of moral cowardice at Lochgair station, along with everything else, that led to me feeling so profoundly awful with myself that evening (after the train finally did get into Queen Street and I walked back, soaked and somehow no longer hungry, in the rain to the empty flat in Grant Street), that mum had to call me there, because I hadn't been able to bring myself to phone her and dad ... and I still managed to feign sleep and a little shame and a smattering of sorrow and reassure her as best I could that really I was all right, yes of course, not to worry, I was fine, thanks for calling ... and so of course after that felt even worse.\n\nI made a cup of coffee. I was feeling so bad that I treated it as a kind of moral victory that I was able to empty most of the water out of the obviously Gav-filled kettle and leave the level at the minimum mark. I stood in the kitchen waiting for the water to heat up with a distinct feeling of eco-smugness.\n\nIt was just as I was sitting down in the living room with my cup of coffee that I realised I'd left my bag on the train.\n\nI couldn't believe it. I remembered getting out of my seat, putting on my jacket, wondering about trying to get something to eat, deciding I didn't feel hungry, glancing at the empty luggage rack, and then heading through the station and up the road. With no bag.\n\nHow could I? I put the coffee down, leapt out of my chair and over the couch, ran to the phone, and got through, ten minutes later, to the station. Lost Property was closed; call tomorrow.\n\nI lay in bed that night, trying to remember what had been in the bag. Clothes, toiletries, one or two books, a couple of presents ... and the folder with Uncle Rory's papers in it; both folders, including the one I hadn't read yet.\n\nNo, I told myself, as panic tried to set in. It was inconceivable that I'd lost the bag forever. It would turn up. I had always been lucky that way. People were generally good. Even if somebody had picked it up, maybe they had done so by mistake. But probably a guard had spotted it and it was right now sitting in some staff-room in Queen Street station, or Gallanach. Or maybe \u2013 in a siding only a mile or two from where I lay \u2013 a cleaner's brush was at this moment encountering the bag, wedged back under the seat ... But I'd get it back. It couldn't just disappear; it had to find its way back to me. It had to.\n\nI got to sleep eventually.\n\nI dreamt of Uncle Rory coming home, driving the old Rover Verity had been born in, the window open, his arm sticking out, him smiling and holding the missing folder in his hand; waving it. In the dream, he had a funny looking white towel wrapped round his neck, and that was when I woke up and remembered.\n\nMy white silk scarf; the irreplaceable M\u00f6bius scarf, the gift of Darren Watt, had been in the missing bag as well.\n\n'Noooo!' I wailed into the pillow.\n\nWaking up was a process of gradually remembering all the things I had to feel bad about. I rang Lost Property first thing. No bag. I got them to give me the number for the cleaners' mess-room and asked there. No bag. I tried Gallanach, in case the train had got back there before the bag had been discovered under the seat by some honest person. No bag.\n\nI tried both stations again in the afternoon; guess what?\n\nI did the only thing I could think of, and retired to bed; if I was to be a blade of grass doomed to be trampled flat, then I might as well accept it and lie down. I stayed in bed for the next twenty-four hours, sleeping, drinking a little water, not eating at all, and only rousing myself when Gav arrived back (from his parents', I wrongly assumed), loudly declaring himself to be of unsound liver but totally in love.\n\nOh, lucky ewe, I said, does she come from a respectable flock?\n\nHa ha, it's your au \u2013 fr ... parents' friend, Janice, Gav beamed, radiating unrepentant guilt; came round here the other day looking for you we got talking went for a curry had a few drinks ended up back here one thing led to another know how it is always liked older women they're more experienced know what I mean arf arf anyway spent an extremely enjoyable New Year at her place apart from the usual visit to my folks' of course oh by the way she's coming round here tonight I'm cooking lasagne can you swap rooms seeing Norris won't be back until tomorrow it's just I didn't expect you back until then either, that okay?\n\nI stared at Gav from my bed, blinking and trying to take in this torrent of exponentially catastrophic information. I attempted desperately to convince myself that what I was experiencing was just a particularly cruel and hateful dream concocted by some part of my mind determined to exact due penalty from my conscience for my having behaved with such despicable lack of grace during the holidays ... but failed utterly; my sub-conscious' stock of nightmare-paradigms includes nothing so banally twisted as Gav.\n\nFinally, scraping together the last microscopic filaments of my tattered pride to produce a quorum fit for emergency ego-resuscitation if not actual wit, I managed: 'Gav, I'm shocked.' (Gav looked defensive for all of a micro-second, a concession my lacerated self-respect fell upon with all the pathetic desperation of a humiliatingly defeated politician pointing out that well, things can only get better.) 'You never told me you could cook lasagne.'\nCHAPTER 10\n\nOnce upon a time, long ago, there was a rich merchant who thought that the city where he lived was full of bad people, and especially bad children.'\n\n'Were they Slow Children?'\n\n'Some of them were, as a matter of fact, but at the time they didn't have the signs to tell them so.'\n\n'Are the Slow Children only in Lochgair, dad?'\n\n'No; there are Slow Children in various places; watch out for the road-signs. Now; back to the story. The rich merchant thought the children should always salute him and call him \"sir\" when they passed him in the street. He hated beggars and old people who couldn't work any more. He hated untidiness and waste; he thought that babies who threw things from their cradles should be punished, and children who wouldn't eat their food should be starved until they ate what they had been given in the first place.'\n\n'Dad, what if it had gone rotten?'\n\n'Even if it had gone rotten.'\n\n'Aw, dad! Even if it had maggots and things in it and it was all horrible?'\n\n'Yes; that would teach them, he thought.'\n\n'Awwrr! Yuk!'\n\n'Well, the rich merchant was very powerful, and he came to control things in the city, and he made everybody do as he thought they ought to do; snowball-throwing was made illegal, and children had to eat up all their food. Leaves were forbidden to fall from the trees because they made a mess, and when the trees took no notice of this they had their leaves glued onto their branches ... but that didn't work, so they were fined; every time they dropped leaves, they had twigs and then branches sawn off. And so eventually, of course, they had no twigs left, then no branches left, and in the end the trees were cut right down. The same happened with flowers and bushes too.\n\n'Some people kept little trees in secret courtyards, and flowers in their houses, but they weren't supposed to, and if their neighbours reported them to the police the people would have their trees chopped down and the flowers taken away and they would be fined or put in prison, where they had to work very hard, rubbing out writing on bits of paper so they could be used again.'\n\n'Is this story pretend, dad?'\n\n'Yes. It's not real; I made it up.'\n\n'Who makes up real things, dad?'\n\n'Nobody and everybody; they make themselves up. The thing is that because the real stories just happen, they don't always tell you very much. Sometimes they do, but usually they're too ... messy.'\n\n'So the rich merchant wouldn't like them?'\n\n'That's right. In the city, nobody was allowed to tell stories. Nobody was allowed to hum, or whistle or listen to music, either, because the merchant thought that people should save their breath the way they saved their money.\n\n'But people didn't like living the way the merchant wanted them to; most mums and dads wouldn't serve their children rotten food, and hated having to pretend that they did. People missed the trees and flowers ... and having to walk around with one eye covered by an eye-patch.'\n\n'Why was that, dad? Why did they have \u2013'\n\n'Because the merchant thought it was a waste of light to have both eyes open; why not save the light the way you save money?'\n\n'Were they like Mr Lachy, dad?'\n\n'Well, not exactly, no; Lachlan Watt only has one eye; the other one looks like a real one but it's glass. The people in the city could change from one eye to the other on different days, but Lachlan \u2013'\n\n'Aye, dad, but they're like him sort of, aren't they?'\n\n'Well, sort of.'\n\n'Why has Mr Lachy only got one eye, dad?'\n\n'Uncle Fergus punched him! Eh, dad?'\n\n'No, Prentice. Uncle Fergus didn't punch him. It was an accident. Fergus and Lachy were fighting and Fergus meant to hit Lachy but he didn't mean to put his eye out. Now; do you two want to hear this story or not?'\n\n'Aye, dad.'\n\n'Aye, dad.'\n\n'Right. Well, the city wasn't a nice place to live because of all the silly laws the merchant had passed, and people started to leave it and go to other towns and other countries, and the merchant was spending so much time passing new laws and trying to make people obey the ones he'd already passed that his own business started to fail, and eventually the city was almost deserted, and the merchant found that he owed people much more money than he had in the bank, and even though he sold his house and everything he owned he was still broke; he was thrown out of his house and out of the city too, because he had become a beggar, and beggars weren't allowed in the city.\n\n'So he wandered the countryside for a long time, starving and having to beg for food, and sleeping in barns and under trees, and eventually he found a little town where all the beggars and old people he'd had thrown out of the city had gone; they were very poor, of course, but by all helping each other they had more than the merchant had. He asked if he could stay with them, and eventually they agreed that he could, but only if he worked. So they gave him a special job.'\n\n'What, dad?'\n\n'What was the job, dad?'\n\n'He had to make brooms.'\n\n'Brooms?'\n\n'Old fashioned brushes made from bundles of twigs tied to a wooden handle. You know up in the forest you sometimes see those things for beating out fires?'\n\n'The big flappy things?'\n\n'Yes; they're big bits of rubber \u2013 old tyres \u2013 attached to wooden handles, for beating out fires on the ground. Well, in the old days, those used to be made from twigs, and even longer ago people used to use brooms like that to sweep the streets and even to sweep their houses. Not all that long ago, either; I can remember seeing a man sweeping the paths in the park in Gallanach with a broom like that, when I was older than either of you are now.'\n\n'Ah, but dad, you're ancient!'\n\n'Ha-ha ha ha!'\n\n'That's enough. Now listen; about these brooms, right?'\n\n'What?'\n\nWhat, dad?'\n\n'The man who had been a rich merchant, and who was now a beggar, had to make brooms for the town. He had a little hut with a stone floor, and a supply of handles and twigs. But to teach the man a lesson they had given him a supply of twigs that were old and weak; poor twigs for making brooms with.\n\n'So, by the time he had made one broom the floor of the hut was covered in bits of twigs, and he had to use the broom he'd just made to sweep the floor of his hut clean before he could start making the next broom. But by the time he'd cleaned the floor to his satisfaction, the broom had worn right away, right down to the handle. So he had to start on another one. And the same thing happened with that broom, too. And the next, and the next; the mess made making each broom had to be cleared up with that same broom, and wore it away. So at the end of the day there was a great big pile of twigs outside the hut, but not one broom left.'\n\n'That's silly!'\n\n'That's a waste, sure it is, dad?'\n\n'Both. But the people had done it to teach the man a lesson.'\n\n'What lesson, dad?'\n\n'Ah-hah. You'll have to work that out for yourselves.'\n\n'Aw, dad!'\n\n'Dad, I know!'\n\n'What?' Kenneth asked Prentice.\n\n'Not to be so damn silly!'\n\nKenneth laughed. He reached up and ruffled Prentice's hair in the semi-darkness; the boy's head was hanging out over the top bunk. 'Well, maybe,' he said.\n\n'Dad,' James said from the lower bunk. 'What happened to the merchant?'\n\nKenneth sighed, scratched his bearded chin. 'Well, some people say he died in the town, always trying to make a broom that would last; others say he just gave up and wasted away, others that he got somebody else to make the brooms and found somebody to provide better twigs, and got people to sell the brooms in other towns and cities, and hired more people to make more brooms, and built a broom-making factory, and made lots of money and had a splendid house made ... And other people say he just lived quietly in the town after learning his lesson. That's a thing about stories, sometimes; they have different endings according to who you listen to, and some have sort of open endings, and some don't actually have proper endings yet.'\n\n'Aw, but dad ...'\n\n'But one thing's definite.'\n\n'What, dad?'\n\n'It's light-out time.'\n\n'Aw ...'\n\n'Night-night.'\n\n'Night, dad.'\n\n'Yeah; night.'\n\n'Sleep tight.'\n\n'Don't let the bugs bite.'\n\n'Right. Now lie down properly; noddles on pillows.'\n\nHe made sure they were both tucked in and went to the door. The night-light glowed softly on the top of the chest of drawers.\n\n'Okay ... Dad?'\n\n'What?'\n\n'Did the man not have any family, dad?' Prentice asked. 'In the story: the merchant. Did he not have any family?'\n\n'No,' Kenneth said, holding the door open. 'He did, once, but he threw them out of his house; he thought he wasted too much time telling his two youngest sons bed-time stories.'\n\n'Aww ...'\n\n'Aww ...'\n\nHe smiled, padded back into the room, kissed the boys' foreheads. 'But then he was a silly man, wasn't he?'\n\nThey left Margot to look after the children and set off in the car, heading for Gallanach. Kenneth smiled when he saw the hand-painted sign at the outskirts of the village that said, 'Thank You.'\n\n'What are you grinning at?' Mary asked him. She was bending down in her seat, staring into the little mirror that hinged up from the glove-box flap, inspecting her lip-stick.\n\n'Just that sign,' he said. 'The one that goes with the Slow Children sign at the other end of the village.'\n\n'Huh,' Mary said. 'Slow children, indeed. I hope you weren't telling my bairns horrible stories that'll keep them awake all night.'\n\n'Na,' he said. The Volvo estate accelerated down the straight through the forest towards Port Ann. 'Though maggoty meat and people with one eye did come into it at one point.\n\n'Hmm,' Mary said. She snapped the glove-box closed. 'I heard Lachy Watt's back in the town; is that true?'\n\n'Apparently.' Kenneth rotated his shoulders as he drove, trying to ease the nagging pain in them that too much drink the night before always seemed to give him these days.\n\nThey had spent Hogmanay at home, welcoming the groups of people roaming the village as they came round. The last revellers had finally been seen off at nine in the morning; they and Margot had done some cleaning up before going to bed, though Ken had anyway had a couple of hours' sleep between three and five, when he'd fallen into a deep slumber on the wicker couch in the conservatory. The boys had gone out to play on the forestry tracks with their new bikes on what had proved a bright but cold day; Mary had got three hours' sleep before they came back, noisily demanding to be fed.\n\n'Haven't seem him for ... what? Ten years?' Mary said. 'Has he been away at sea all that time?'\n\n'Well, hardly,' Ken said. 'He was in Australia, wasn't he? Settled down there for a while. Had some sort of job in Sydney, I heard.'\n\n'What was he doing?'\n\n'Don't know; you could ask him yourself. Supposed to be coming to Hamish and Tone's shindig tonight.'\n\n'Is he?' Mary said. The Volvo hissed along the dark road; a couple of cars went past, holes of white light in the night, scattering spray which the water jets and wipers of the Volvo swept away again. Mary took a perfume spray from her handbag, applied the scent to wrists and neck. 'Fergus and Fiona are coming tonight, aren't they?'\n\n'Should be,' Ken nodded.\n\n'Do you know if Lachy and Fergus still talk to each other?'\n\n'No idea.' He laughed. 'Don't even know what they'd talk about; a member of the factory-owning Scottish gentry and a second mate \u2013 or whatever Lachy is these days \u2013 who's spent the last few years in Oz. What is there to say; aye-aye, captain of light industry?'\n\n'Fergus isn't gentry, anyway,' Mary said.\n\n'Well, good as. Might not have a title, but he acts like he does sometimes. Got a castle; what more do you want?' Kenneth laughed lightly again. 'Aye-aye. Ha ha.'\n\nThe lights of Lochgilphead swung into view ahead, just as rain started to spot the windscreen. Kenneth put the wipers on. 'Aye-aye!' he sniggered.\n\nMary shook her head.\n\n'Going to the dogs, if you ask me.'\n\n'Fergus, people like you have been saying that since somebody invented the wheel. Things get better. They're always looking up.'\n\n'Yes, Kenneth, but you're basically Bolshie, so you would think so.'\n\nKenneth grinned, took a drink of his whisky and water. 'It's been a good year,' he nodded. Fergus looked suitably disgusted, and threw back the remains of his own whisky and soda in one gulp.\n\nThey stood in the lounge of Hamish and Antonia's house, watching the others help themselves to the buffet Antonia had prepared. Neither of the two men had felt hungry.\n\n'You might not be saying that when the refugees come back from Australia,' Fergus said sourly. Kenneth glanced at him, then looked round for Lachlan Watt; he was sitting on a distant chair, a plate of food balanced on his knees, talking to Shona Watt, his sister-in-law.\n\nKenneth laughed as Fergus refilled his glass from one of the whisky bottles on the drinks trolley behind them. 'Fergus, you're not talking about the Domino Theory by any chance, are you?'\n\n'Don't care what you call it, McHoan; not saying it'll be next, either, but you just watch.'\n\n'Fergus, for God's sake; not even that asshole Kissinger believes in the Domino Theory any more. The Vietnamese have finally got control of their own country after forty years of war; defeated the Japs, the French, us, and the most powerful nation in the history of the planet in succession, with bicycles, guns and guts, been bombed back into the bronze age in the process and all you can do is spout some tired nonsense about little yellow men infiltrating the steaming jungles of the Nullarbor Plain and turning the Aussies into Commies; I think a Highland League side winning the European Cup is marginally more likely.'\n\n'I'm not saying they won't pause to draw breath, Kenneth, but I can't help feeling the future looks black for those of us interested in freedom.'\n\n'Fergus, you're a Tory. When Tories say freedom they mean money; the freedom to send your child to a private school means the money to send your child to a private school. The freedom to invest in South Africa means the money to invest there so you can make even more. And don't tell me you're interested in freedom unless you support the freedom of blacks to come here from abroad, which I know you don't, so there.' Kenneth clinked his glass against Fergus's. 'Cheers. To the future.'\n\n'Huh,' said Fergus. 'The future. You know, I'm not saying your lot won't win, but I hope it doesn't happen in my lifetime. But things really are going to the dogs.' Fergus sounded genuinely morose, Kenneth thought.\n\n'Ah, you're just peeved your lot have elected a woman leader. Even that's good news ... even if she is the milk snatcher.'\n\n'We got rid of an old woman and replaced him with a younger one,' Fergus said, mouth turned down at the corners, staring over his whisky tumbler and across the room to where his wife was talking to Antonia. 'That's not progress.'\n\n'It is, Fergus. Even the Tories are subject to change. You should be proud.'\n\nFergus looked at Kenneth, a wealth of sombre disdain in his slightly watery-eyed look. Kenneth gave him a big smile. Fergus turned away again. Kenneth looked at the other man's heavily jowled, prematurely aged face and shook his head. Chiang-Kai-Shek and Franco dead, Angola independent, Vietnam free at last ... Kenneth thought it had been a great year. The whole tide of history seemed to be quickening as it moved remorselessly leftwards. He felt vaguely sorry for Fergus. His shower had had their reign, he thought, and grinned to himself.\n\nIt had been a good year for Kenneth personally, too. The BBC, bless its cotton socks, had taken some of the stories from his first collection; a whole week of _Jackanory_ to himself, just six weeks before Christmas! At this rate he could start thinking about giving up teaching in a year or two.\n\n'I wish I shared your enthusiasm for change.' Fergus sighed, and drank deeply.\n\n'Change is what it's all about, Ferg. Shuffling the genes; trying new ideas. Jeez, where would your damn factory be if you didn't try new processes?'\n\n'Better off,' Fergus said. He looked sourly at Kenneth. 'We're just about making enough from traditional paperweights to keep the Specialist Division afloat. All this hi-tech stuff just loses us money.'\n\n'Well, it must be making money for somebody; maybe you weren't able to invest enough. Maybe the big boys'll take over. That's the way things go; capitalists all want to have a monopoly. Only natural. Don't get depressed about it.'\n\n'You won't be saying that if we have to close the factory and put everybody out of work.'\n\n'God, Ferg, it isn't that bad, is it?'\n\nFergus shrugged heavily. 'Yes, it is. We've told them it might come to that; the shop-stewards, anyway. Another strike, or too big a pay rise, and we might go under.'\n\n'Hmm,' Kenneth said, sipping at his whisky. He wondered how serious the other man was. Industrialists often made that sort of threat, but they rarely seemed to be carried out. Kenneth was a little surprised that Hamish hadn't said anything about the factory being in such dire straits, but then his brother did seem to put the church and the factory above family and friends.\n\n'I don't know.' Fergus shook his head. 'If we weren't tied to this bit of the country, I'd almost think about chucking it all in and heading off somewhere different \u2013 Canada, or Australia, or South Africa.'\n\nIt was Kenneth's turn to look sour. 'Yes,' he said. 'Well, you'd probably get on fine in the RSA, Ferg. Though that's the one place I _wouldn't_ recommend if you want to keep well away from the red tide.'\n\n'Hmm,' Fergus nodded, still watching his wife, now talking to Shona Watt. 'Yes, you may be right.' He knocked back his drink, turned to the bottle-loaded table behind and poured himself another large whisky.\n\nAntonia clapped her hands, singing out: 'Come on, you boring lot; let's all play charades!'\n\nKenneth drained his glass, murmured. 'God, I hate charades.'\n\n_'Heniiss ... never liked him either; fat lipped beggar ... queer, y'know; thass wha he's singing you know; d'you know that? \"Scuse me while I kiss this guy... disgussin ... absluley disgussin ...'_\n\n_'Fergus, do shut up.'_\n\n_' \"Scuse me, while I kiss this guy\" ... bloody poofter coon.'_\n\n_'I'm sorry about this, Lachy.'_\n\n_'That's okay, Mrs U. You no goin to put your seat belt on, no?'_\n\n_'No; not for short journeys \u2013'_\n\n_'Lachy? Lachy ... Lachy! Lachy; I'm sorry about your eye ... really really sorry; never forgave myself, never... here, shake ...'_\n\n_Fergus tried to lever himself up from the rear bench seat of the old Rover, but failed. He got as far as lifting his head and getting one shoulder off the seat, but then collapsed back onto the leather, and let his eyes close._\n\n_The car rumbled about him ... even more restful than the noise of train wheels in the old days; he tried to remember the old days ..._\n\n_'You sure you don't mind doing this, Lachy?' Fiona said, swinging the car off the'main road and onto the drive that led to the castle. The headlights made a tunnel of the trees and rhododendrons._\n\n_'Na, it's okay.'_\n\n_Lachlan Watt had been about to leave Hamish and Antonia's party when Fergus had fallen over and Fiona had decided it was time to take her husband home; she had offered Lachy a lift back to his brother's house, but when they'd got there Fergus had seemed fast asleep, snoring loudly and taking no apparent notice of Fiona shaking him and shouting at him; Lachy had volunteered to come back to the castle to help get Fergus out of the car and upstairs to bed; Fiona would run Lachy back afterwards._\n\n_'God that man's a nuisance,' Fiona said, as they turned the corner in the drive and the lights of the castle came into view against the coal-dark_ night. _'Like I_ say; _I could have_ got _the baby-sitter to help me with him, but she's just a skelf... not our regular girl. She's built like a rugby player, could probably put Ferg over her shoulder, but not this girl. Leanne's her name ... that's her car there; doesn't look old enough to drive if you ask me ...'_\n\n_Fiona brought the Rover to a halt behind a beaten-up mini, standing on the gravel in front of the castle's main entrance._\n\n_'This really is awful good of you, Lachy._ '\n\n_'Aye, it's no problem, Mrs U.'_\n\n_Fiona turned to him. She smiled. 'Lacby; it's Fiona. You make me feel old when you call me Mrs U.'_\n\n_'Sorry; Fiona.' Lachy grinned._\n\n_He had been a thin, light-framed boy, and he had grown to become a lean, wiry man; the years of life on merchant ships, and then in Australia, had left his skin looking well-used, like soft and fine-grained \u2013 but slightly distressed \u2013 leather. His hair was unfashionably short, and both eyes glittered. It was a spare, uncluttered, characterful face, especially compared to Fergus's._\n\n_'That's better.' Fiona smiled. She turned and looked in disgust at the body in the back seat, just as Fergus started to snore again. 'Well; better get this lump out of the car, I suppose.'_\n\n_Fergus had gone back into a deep sleep. They couldn't wake him. Fiona went in to tell the baby sitter she was free to go, while Lachy tried to rouse Fergus._\n\n_'Hoi you; Fergus. Ferg; wake up, man.'_\n\n_'Aarg ... Henriss, bassard.'_\n\n_'Fergus; wake up, Fergus.' Lachlan tried slapping the man's cheeks; his heavy jowls wobbled like jellies._\n\n_'Hhnn...'_\n\n_'Wake up,' Lachlan said, slapping Fergus's cheeks again, harder. 'Wake up,' he said quietly. 'Ye upper class cunt ye.' He fairly walloped Ferg on one chop._\n\n_Fergus awoke suddenly; arms waving about, eyes wild and bright, making no sound other than a faint gurgling noise. Then he rolled off the seat into the footwell and immediately started snoring again._\n\n_'Any luck with the sleeping beauty?' Fiona said, coming down the steps alongside a slim, blonde-haired girl who was zipping up an anorak._\n\n_Lachlan turned round. 'Na; be's sound.'_\n\n_'That'll be the day,' muttered Fiona. She glanced in at Fergus, then turned to the girl. 'Thanks, Leanne, dear; now drive carefully, won't you?'_\n\n_'Aye, Mrs Urvill,' the girl said, taking out some keys and heading for the mini. 'Night-night.'_\n\n_'Bye now.'_\n\n_Fiona and Lachy took an end of Fergus each; Lachy held him under the shoulders, Fiona by the ankles. They struggled up the steps, through the entrance hall, rested in the main hall, then took him up to the first floor._\n\n_'In here,' Fiona said, nodding._\n\n_Lachy supported Fergus's shoulders with one knee while he twisted the handle of a darkly-stained wooden door. It swung open to darkness._\n\n_'There's a light, aye?'_\n\n_'Just there; down a bit.'_\n\n_The room was small and bright; there was a single bed, a dressing table and chair, and a wardrobe. There was a print of a hunting scene on one wall, opposite a small window._\n\n_'Guest room's good enough for him tonight,' Fiona grunted_ as _they swung him onto the bed and dropped him._\n\n_'Shooch!' Fiona said, collapsing onto the floor. Lachy sat down on the pillow at the head of the bed, breathing hard. Fiona wiped her brow. She got up shakily._\n\n_'That was hard work,' she said. She pulled Fergus's shoes off and nodded to the door. 'Come_ on; _let's break into the old bugger's best malt before we run you back. You deserve it.'_\n\n_'Fair enough,' Lachy said, smiling. 'No takin his clothes off, no?'_\n\n_'Ugh. Certainly not,' Fiona said. She drew back a little against the door to let Lachy go past her into the hallway. 'He's lucky we didn't leave him in the car.' She turned out the light._\n\n_Fergus woke in utter darkness, wondering where he was; he felt as though he was falling backwards forever into darkness. For an instant he thought perhaps he was dead, consigned to perdition and gloom until the end of time, his only sensation that of falling back and back and back, head over heels forever. He heard himself moan, and felt with his hands: bedclothes. He was still wearing his own_ _clothes, too. Here was his shirt on his wrist; there his trousers, sweater... shoes off. He flexed his feet, feeling his toes in his socks. His hands found the sides of the bed; it was a single, then._\n\n_It was still totally dark. He tried to remember where he'd been last._\n\n_The party; Hamish and Antonia McHoan's. Of course. He must still be there, as this wasn't his own bed. Put to bed. Bit bad, that; probably in the dog-house as far as the lady wife was concerned, too, but then what was new?_\n\n_He put one hand out, feeling for a table; he found what felt like one, and then a long cold metal stem. Reaching up, he felt a switch._\n\n_The light clicked on and suddenly everything was white and horribly bright. He shielded his eyes. God, his head felt fuzzy, and sore. He needed a drink very badly; water would do._\n\n_He looked round the white-painted room, thinking that it looked somehow familiar. Perhaps he had slept here before. Or maybe he'd given the McHoans some bits and pieces of furniture._\n\n_He listened but couldn't hear anything. The door of the room looked familiar, too. Odd to find a door so comforting, somehow._\n\n_He got up, wobbled across to the door. He was quite cold. He opened the door; a dark hall. Funny; the place didn't smell like the McHoans' house did. It smelled of wood and a sort of quite pleasant mustiness. This place smelled of stone and polish. Bit like the castle._\n\n_He went out into the hallway, felt along the wall for a light switch; he found one, switched it on. Stairs led up; the wood-panelled hall led to another set of stairs going down. There were old paintings on the walls. He felt very dizzy, and sat down on the bottom step of the stairs. He was home. This was the castle._\n\n_He got up, walked up the stairs. The door to the short flight of stairs that led to the two topmost floors was locked. He didn't understand. He searched his pockets but could find no key._\n\n_He pushed at the door again. He gathered a chestful of air to_ shout at Fiona \u2013 dozy bitch had locked him out of his own fucking castle, his own _bedroom_ \u2013 but then thought of the children. Might _wake the little beggars up. Didn't want that._\n\n_He went down through the lower hall to the kitchen, drank some water. His watch said it was two o'clock; so did the kitchen clock. There ought to have been keys hanging by the door to the utility room, but they weren't there. Bloody fishy. Had Fiona hidden them? Did she think he was dangerous, was that it?_\n\n_Maybe she thought he would get up in some drunken stupor and ravish her. 'Huh, that'll be right,' he said to himself. His voice sounded rough in the quiet kitchen; he cleared his throat, coughed, and felt the dull pounding of his headache suddenly sharpen._\n\n_Damn it all. Perhaps he was being punished. Maybe she was punishing him for getting drunk. Had he done anything disgraceful? He couldn't remember, but he doubted it. He usually held his drink well, and behaved like a gentleman even when he did have one too many._\n\n_He looked at his reflection in the window over the sink. He pulled one splayed hand through his hair. Maybe he ought to have a shower or something. There was always the bathroom on the first floor ..._\n\n_He felt bloody annoyed, Fi locking him out of their apartments like that._\n\n_Then he remembered the observatory._\n\n_You could get up to it by the stairs to the roof. He'd been up there, in the roof_ space _when the men had been installing the dome. For that matter, he'd seen that loft being put together, knew it almost as well as that self-opinionated young architect had. He'd crawled around in there, him and the builder, with a torch, discussing where the observatory could be built; what joists and supports would have to go, what extra bracing would be needed._\n\n_He chuckled to himself, put down the cup he'd been drinking the water from, wiped his lips._\n\n_He padded through the hall, up the four flights of stairs to the little landing where you either went straight ahead and out onto the battlements, or ducked through the wee door into the observatory._\n\n_It was bitterly cold inside the aluminium hemisphere. He wished_ _he'd thought to put shoes on before he'd started on this piece of nonsense; feet felt like blocks of ice. Still._\n\n_He opened the door that gave into the extended cupboard under the roof. Dark. Damn; should have thought to bring a torch, too._\n\n_'Sloppy, Urvill, bloody sloppy,' he breathed to himself._\n\n_He squeezed inside the little cupboard. Really must lose some weight. Well, festive period well and truly over now; time to go on a diet, or do a bit more exercise. He wriggled to the rear of the cupboard; felt for the wooden battens on the panel at the end of the dark space. The panel came away after a little while; he put it on the floor in front of him, and wriggled through on his elbows and knees into the darkness beyond._\n\n_'Getting too old for this sort of thing, 'he told himself. It was very nearly totally dark in the roof space; only a little light came from behind him, through the cupboard from the dome of the observatory. He felt his way across the joists in front of him, got his legs free from the cupboard and was able to get up into a crouch, balancing on a joist, hands just above his head, holding on to rough, undressed wood._\n\n_He swung one foot out, to the next joist, then put out one hand and felt for the next rafter; he transferred his weight carefully. There; did it. He was aware of the lath and plaster clinging to the bottom of the joists; put a foot through that and you'd be right through the ceiling below; chap could fall slap into the bath from here, probably; or into the twins' room, maybe; perish the thought; daddy coming crashing through the ceiling, give the little perishers nightmares for the rest of their lives._\n\n_He swivelled from joist to joist, rafter to rafter, feeling horribly like a monkey and getting very cold feet in the process even though he was breaking out in a sweat at the same time. His knees and his neck were making ghastly creaking noises and protesting like hell._\n\n_He looked back at the light coming from the observatory cupboard, now a good twenty feet away, and thought about going_ _back; this whole prank was becoming a bit much, really. But he'd started, 'so he'd finish._\n\n_He saw the faintest of glows ahead, from between two of the joists. He smiled. 'That's the ticket,' he breathed to himself. With the next joist it came closer; then he could see one edge of the little hatch; then he was over it. A soft light gave away the outline of the door. He heard voices. God, the silly woman had probably left the radio on._\n\n_He got down on his knees again, feet supported on the joist behind; his knees gave sharp twinges of pain, taking almost all his weight._\n\n_He felt for the edges of the square door, found them and lifted it gently. What a locked-room mystery this would present the old girl with, if he could get in without her hearing him, and get undressed and slide in beside her! She'd never be able to work it out. Of course, he thought, as he levered the hatch door open slowly, letting more soft light spill out from underneath, he'd have to cover his tracks in the morning; damn silly to have left the cupboard back there open, and the light on in the dome. But never mind. Fi hardly ever went up there anyway._\n\n_He'd lifted the near edge of the door up about three inches above the top of the joist in front of him. He held it there, lowered his head, peeped into the_ room, _smiling, wondering if he could see Fi from this angle. Voices. Warm air and voices._\n\n_'Oh ... God, God, God, God; yes, yes, yes, yes...'_\n\n_It took him a moment to work out what was going on._\n\n_But then he realised._\n\n_That was Fiona, in the bed, on the bed, covers half off, the only light in the room coming from a little candle by the bedside, her hair spilled on the pillow (the other pillow was on the floor) ... and that was Lachlan Watt, wrapped round her, body bucking like some horse, his hands at her neck, at one breast, in her hair, cupping her neck; the covers sliding off, Fiona putting her arms wide, clutching at the bottom sheet of the bed at one side, clutching the edge of the_ _bedside table with the other. Her head beat from side to side and she said, 'Yes, yes, yes, yes,' again, then Lachlan \u2013 wiry, athletic-looking, skinny shanks ramming back and forth like some skinny bull \u2013 reached under her, pulled her up, his legs spreading, kneeling; she hung onto him, arms round his neck, then after a few vertical stabs he threw her down, back onto the bed; she grunted, arms still tight round his back, then she brought her legs up, right up over his thin, plunging, globe-buttocked behind, until her ankles were in the small of his back, rocking to and fro, feet crossed one over the other, locked there; with one splayed hand she held onto his back, pressing him to her, and with the other hand she felt down the length of his body, over ribs and waist and hips, and with another grunt reached round and under, taking his balls in her hand, pressing them and kneading them and squeezing them._\n\n_'Aw Christ!' he heard Lachlan Watt say, body arching. Fiona shuddered, her voice almost a squeal as she took a series of sudden, deep in-rushing breaths, and buried her head in the hollow between Lachlan Watt's shoulder and neck._\n\n_Fergus let the little door down without making a noise._\n\n_He felt very cold, and he had pissed himself. The urine was warm around his balls and tepid down his leg, but it was cold at his knees. He knelt there in the darkness, listening to the sounds of the subsiding passion in the room below, then swivelled silently and with even greater care than before, and feeling far more sober, moved back towards the thin, escaping light at the far end of the chill, cramped roof space._\nCHAPTER 11\n\nIf the year of our folly 1990 had started inauspiciously for me, then the Fates, Lady Luck, Lord Chance, God, Life, Evolution \u2013 whoever or whatever \u2013 immediately thereafter set about the business of proving that the entangled disasters distinguishing the year's first few days were but a mild and modest prelude to the more thorough-going catastrophes planned for the weeks and months ahead ... and this with a rapidity and even an apparent relish which was impressive \u2013 if also bowel-looseningly terrifying \u2013 to behold.\n\nGav and my Aunt Janice got on like a house on fire, a combined location and fate I occasionally wished on them as I lay awake listening to the sounds of their love-making, a pastime I sometimes suspected I shared with people in a large part of the surrounding community, not to say northern Europe.\n\nI had made the mistake of volunteering to sleep on the couch in the living room on the nights that Janice stayed at our flat; this offer was made with what I thought was obvious sarcasm one evening while Gav and Norris were attempting to develop a technique for cooking poppadoms in the microwave. They were having an intense and appropriately heated discussion on the problems of cold-spots (as evinced by the fact that their first attempts came out looking like braille roundels), and on the unfortunate instability of three poppadoms balanced together \u2013 caused not so much by the jerk they received when the turntable started up as by their movements while they cooked and swelled \u2013 but eventually my flatmates settled on the concept of standing the things up individually on the glass turntable, and so instigated what they termed a 'brain-storming session' in an attempt to find a suitable support mechanism. (I suppressed the urge to point out that the chances of two such patently zephyr-grade minds producing anything remotely resembling a storm was roughly equivalent to the likelihood of somebody called Cohen landing a pork scratching concession in Mecca during Ramadan.)\n\n'An alligator clip with the chrome bits removed.'\n\n'Naw; still metal.'\n\n'Maybe we could shield it.'\n\n'Na; has to be plastic. Yer non thermosetting stuff, for preference.'\n\n'Well, look, Gav,' I said from the kitchen doorway. 'I only overhang the couch by a foot or so at each end; why don't I attempt to curl up there when you and Janice are in residence, if not flagrante, in the bedroom?'\n\n'Eh?' Gav said, swivelling that thick neck of his to look at me, his massive brows furrowing. He scratched at one rugby-shirt shrouded armpit, then nodded. 'Aw; aye.' He looked pleased. 'Thanks very much, Prentice; aye, that'd be grand.' He turned back to the microwave.\n\n'Maybe we could suspend them from this bit in the middle with a length of thread,' Norris grunted, sticking his head almost right inside the appliance. Norris, still clad in his white lab coat, was one of those medical students whom fate has seemingly marked out to spend the bulk of their studies and initial training suffering from quite stupendous hangovers incurred through the intake of near-fatal levels of alcohol the night before, and their subsequent professional careers sternly finger-wagging at any member of the general public who dares to consume over the course of a week what they themselves had been perfectly happy to sink during the average evening.\n\n'I mean, don't let the fact I'm the longest serving flat-dweller put you off; the last thing I want to do is embarrass you, Gav,' I said (just a tad tetchily).\n\n'Na, it's all right, Prentice; ta,' Gav said, then crouched down by Norris and squinted into the lit interior of the microwave. 'Nowhere to attach it,' he told Norris. 'Anyway; wouldnae turn, would it?'\n\nThey both looked thoughtful, heads side by side at the open oven door, while I wondered what the chances were of both heads fitting \u2013 and jamming \u2013 inside and the door safety-catch somehow short-circuiting.\n\n'Na,' Norris said. 'We're looking at some form of support from below, know what ah mean? Come on, Gav, you're the engineer ...'\n\n'I mean, that old duvet's bound to cover most of the important parts of my body, and the chances of the pilot on the fire blowing out again and gassing me in my sleep can't really be _that_ high,' I said.\n\n'Hmm,' Gav said. He straightened, then bent forward and tapped at the white plastic strip on the kitchen window ledge which retained the cheaply horrid secondary double-glazing the flat's owners had fitted.\n\n'Just a block of wood, maybe,' Norris said.\n\n'Get hot,' Gav said, looking more closely at the white plastic strip. 'Depending on how much water there is in the wood; could warp. Still think plastic's your best bet.'\n\n'After all, Gav, I can just stay up till your drinking pals have decided to head home, or Norris's card school chums finally drag themselves away, or crash out and snore on the Richter scale, whatever; the fun rarely extends beyond three or four o'clock in the morning ... why, that would leave me a good four or five hours' sleep before an early lecture.'\n\n'Aye, that's great, Prentice,' Gav said, still closely inspecting the window sill. Then he stood up suddenly and snapped his fingers. 'Got it!' he said.\n\nWhat, I thought? Had my tone of reason in the face of monstrosity finally registered? But no.\n\n'Blu-tack!'\n\n'What?'\n\n'Blu-tack!'\n\n'Blu-tack?'\n\n'Aye; Blu-tack. You know: Blu-tack!'\n\nNorris thought about this. Then said excitedly. 'Aye; Blu-tack!' 'Blu-tack!' Gav said again, looking wide-eyed and pleased with himself.\n\n'The very thing!' Norris nodded vigorously.\n\nI shook my head, quitting the kitchen doorway for the comparative sanity of the dark and empty hallway. 'You crack the Bollinger,' I muttered. 'I'll just phone the Nobel Prize Committee and tell them their search is over for another year.'\n\n'Blu-tack, ya beauty!' I heard from the white-glowing crucible of cutting-edge technological advancement that our humble kitchen had become.\n\n'You mean you haven't read them all?'\n\n'I went off the idea,' I said. I was sitting in what had effectively become my boudoir; our living room. Aunt Janice seemed to prefer staying here with Gavin to travelling out to Crow Road most nights.\n\nGav and Janice sat on the couch, loosely attired in dressing gowns and watching a video.\n\nI had been sitting at the table housed in the living room bay window, trying to write a paper for a tutorial the next day, but Gavin and Janice had chosen to punctuate their highly audible coupling sessions (in what the more tenacious core-areas of my long-term memory still sporadically insisted had once been my bedroom) with an almost equally noisy episode of tortilla chip eating. The corny raucousness which ensued of course meant that the television volume had to be turned up to window-shaking levels so that the happy couple could savour the exquisitely enunciated phrasing of Arnold Schwarzenegger's lines over the noise of their munching.\n\nI had admitted defeat on the subject of the links between agricultural and industrial revolution and British Imperialism, and sat down to watch the video. Perhaps appropriately, given the inflammatory nature of the effect Gav and Janice seemed to have on each other's glands, it was called _Red Heat._\n\n'Oh,' I'd said. 'A Hollywood movie about two cops who don't get along at first but are thrown together on a case involving drugs, foreigners, lots of fights and guns and which ends up with them respecting each other and winning. Sheech.' I shook my head. 'Makes you wonder where these script-writer guys get their weird and zany ideas from, doesn't it?'\n\nGav had nodded in agreement, without taking his eyes off the screen. Janice Rae had smiled over at me, her hair fetchingly disarrayed, her cheeks flushed. 'Oh yes, Prentice,' she'd said. 'What did you think of Rory's work, in that folder?'\n\nHence the exchange above.\n\nJanice looked back at the telly and stretched one leg out over Gavin's lap. I glanced over, thinking that she had much better legs than a woman of her age deserved. Come to that, she had much better legs than a man of Gav's mental age deserved.\n\n'So you haven't found any hints about what it was Rory had hidden in there?' she said.\n\n'I've no idea what he wanted to hide,' I said, wishing that Janice would hide a little more of her legs.\n\nI was uncomfortable talking about the poems and Rory's papers; the bag lost on the train coming back from Lochgair at the start of the year had stayed lost, and \u2013 stuck with just the memory of the half-finished stuff that Janice had given me originally \u2013 I'd given up on any idea I'd ever had of trying to rescue Uncle Rory's name from artistic oblivion, or discovering some great revelation in the texts. Still, it haunted me. Even now, months later, I had dreams about reading a book that ended half-way through, or watching a film which ended abruptly, screen whiting-out ... Usually I woke breathless, imagining there was a scarf \u2013 shining white silk looped in a half-twist \u2013 tightening round my neck.\n\n'It was something he'd seen, I think.' Janice watched the distant screen. 'Something ...' she said slowly, pulling her dressing gown closed. 'Something ... over-seen, if you know what I mean.'\n\n'Vaguely,' I said. I watched Gavin's hand move \u2013 apparently unconsciously, though of course with Gav that could still mean it was fully willed \u2013 to Janice's polyester-and-cotton covered thigh. 'Something,' (I suggested, watching this,) 'seen voyeuristically, perhaps?'\n\n'Mmm,' Janice nodded. Her right hand went up to Gav's short, brownish hair, and started to play with it, twirling it round her fingers. 'He put it in ... whatever he was working on.' She nodded. 'Something he'd seen, or somebody had seen; whatever. Some big secret.'\n\n'Really?' I said. Gavin's hand rubbed up and down on Janice's lap. Gav's face gave no sign he was aware of doing this. I pondered the possibility that the lad possessed some dinosaur-like secondary brain which was controlling the movements of his hand. Palaeobiological precedent dictated such an organ be housed in Gavin's ample rear, and have responsibility for his lower limbs \u2013 not to say urges \u2013 rather than his arms, but then one never knew, and I reckoned Gavin's modest forebrain \u2013 doubtless fully occupied with the post-modernist sub-texts and tertiary structuralist imagery of _Red Heat_ \u2014 could probably do with all the help it could get. 'Really?' I repeated.\n\n'Mmm,' Janice nodded. 'So he said.' She bit her lip.\n\nGavin had a look of concentration on his face now, as though two parts of his brain were attempting the tricky and little-practised operation of communicating with each other.\n\n'Something about \u2013' Janice moved her hips, and seemed to catch her breath. '\u2013 the castle.' She clutched at Gav's hair.\n\nI looked at her. 'The castle?' I said. But too late.\n\nPerhaps lent the final impetus necessary for successful reception by the proximity of the area of stimulus to that of cognition, this hair-pulling signal finally seemed to awaken Gavin to the perception that there might be something else going on in his immediate area other than the video, undeniably captivating though it was. He looked round, first at his hand, then at Janice, who was smiling radiantly at him, and finally at me. He grinned guiltily.\n\nHe yawned, glanced at Janice again. 'Bit tired,' he said to her, yawning unconvincingly once more. 'Fancy goin' to \u2013?'\n\n_'What'_ Janice said brightly, slapping her hand down on Gavin's bulky shoulder, '\u2013 a good idea!'\n\n'Tell us how it ends, eh Prentice?' Gav said, nodding back at the television as he was half hauled out of the room by Aunt Janice, en route to the land of nod after a lengthy detour through the territories of bonk.\n\n'With you going \"Uh-uh-uh-uh-uh!\"' I muttered to the closed door. I glared at the screen. '\"How it ends,\"' I muttered to myself. 'It's a video, you cretin!'\n\nI returned to the changes in British society required to bring about the Empire on which the light of reason rarely shone. It was going to be a long night, as I also had to finish an already over-due essay on Swedish expansion in the seventeenth century (it would have to be a goodish one, too; an earlier remark \u2013 made in an unguarded moment during a methodically boring tutorial \u2013 ascribing Swedish territorial gains in the Baltic to the invention of the Smorgasbord with its take-what-you-want ethic, had not endeared me to the professor concerned; nor had my subsequent discourse on the innate frivolity of the Swedes, despite what I thought was the irrefutable argument that no nation capable of giving a Peace Prize to Henry Kissinger could possibly be accused of lacking a sense of humour. Pity it was actually the Norwegians.\n\nI remembered a joke about Kissinger ('no; fucking her.') and found myself listening to Gav and Janice. They were still at that stage of their coital symphony where only the brass section was engaged, as the old metal bed creaked to and fro. The wind section \u2013 essentially vox humana \u2013 would join in later. I shook my head and bent back to my work, but every now and again, as I was writing or just thinking, a niggling little side-track thought would distract me, and I'd find myself remembering Janice's words, and wondering what exactly Uncle Rory might have hidden within his later work (if he really had hidden anything). Not, of course, that there was much point in me wondering about it.\n\nFor about the hundredth time, I cursed whatever kleptomaniac curmudgeon had walked off the train with my bag. May the scarf unravel and do an Isadora Duncan on the wretch.\n\n'Uh-uh-uh-uh-uh!' came faintly from what had been my bedroom. I ground my teeth.\n\n'Married?' I gasped, aghast.\n\n'Well, they're talking about it,' my mother said, dipping her head towards the table and holding her Paisley-pattern scarf to her throat as she nibbled tentatively at a large cream cake.\n\nWe were in Mrs Mackintosh's Tea Roomes, just off West Nile Street, surrounded by straightly pendulous light fitments, graph-paper pierced wooden screens, and ladder-back seats which turned my usual procedure of hanging my coat or jacket on the rear of the seat into an operation that resembled hoisting a flag up a tall mast.\n\n'But they can't!' I said. I could feel the blood draining from my face. They couldn't do this to me!\n\nMy mother, neat and slim as ever, ploughed crunchingly into the loaf-sized meringue cream cake like a polar bear breaking into a seal's den. She gave a tiny giggle as a little dollop of cream adhered to the tip of her nose; she removed it with one finger, licked the pinky, then wiped her nose with her napkin, glancing round the restaurant through the confusing topography of slats and uprights of the seats and screens, apparently worried that this minor lapse in hand-mouth coordination was being critically observed by any of the surrounding middle-class matrons, perhaps with a view to passing on the scandalous morsel to their opposite numbers in Gallanach and having mother black-balled from the local bridge club. She needn't have worried; from what I had seen, getting a little bit of cream on your nose was practically compulsory, like getting nicked on the cheek in a ritualised duel before being allowed to enter a Prussian drinking sodality. The atmosphere of middle-aged ladies enjoying something wicked and nostalgic was quite palpable.\n\n'Don't be silly, Prentice; of course they can. They're both adults.' Mother licked cream from the ice-cave interior of the meringue, then broke off part of the superstructure with her fingers and popped it into her mouth.\n\nI shook my head, appalled. Lewis and Verity! Married? No!\n\n'But isn't this ...' My voice had risen a good half-octave and my hands were waggling around on the end of my arms as though I was trying to shake off bits of Sellotape. '... rather soon?' I finished, lamely.\n\n'Well, yes,' mum said, sipping her cappuccino. 'It is.' She smiled brightly. 'I mean, not that she's pregnant or anything, but \u2013'\n\n'Pregnant!' I screeched. The very idea! The thought of the two of them fucking was bad enough; Lewis impregnating that gorgeous creature was infinitely worse.\n\n'Prentice!' Mother whispered urgently, leaning closer and glancing round again. This time we were getting a few funny looks from other customers. My mother smiled insincerely at a couple of Burberried biddies smirking from the table across the aisle; they turned sniffily away.\n\nMy mother giggled again, hand to mouth, then delved into the meringue. She sat back, munching, face red but eyes twinkling, and with those eyes indicated the two women who'd been looking at us; then she raised one finger and pointed first at me, then at her. Her giggle turned into a snort. I rolled my eyes. She dabbed at hers with a clean corner of napkin, laughing.\n\n'Mother, this is not funny.' I drank my tea, and attacked another chocolate eclair. It was my fourth and my belly was still growling. 'Not _at all_ funny.' I knew I was sounding prissy and ridiculous but I couldn't help it. This was a very trying time for me, and the people who ought to be offering support were offering only insults.\n\n'Well,' mother said, sipping at her coffee again. 'Like I say, there's no question of that. I mean, not that it makes much difference these days anyway, but yes, you're right; it is a bit soon. Your father and I have talked to Lewis and he's said they aren't going to actually rush into anything, but they just feel so ... right together that it's ... just come up, you know? Arisen naturally between them.'\n\nI couldn't help it. My obsessed, starveling brain was conjuring up all sorts of ghastly images to accompany this sort of talk; things arising, coming up ... Oh God ...\n\n'They've talked about it,' mother said, in tones of utmost reason, with a small shrug. 'And I just thought you ought to know.'\n\n'Oh, thanks,' I said, sarcastically. I felt like I'd been kicked by a camel but I still needed food, so I polished off the eclair, belched with all the decorum I could, and started eyeing up a Danish pastry.\n\n'They're in the States right now,' mother said, licking her fingers. 'For all we know they might come back married. At least if that happens it won't come as quite such a shock now, will it?'\n\n'No,' I said miserably, and took the pastry. It tasted like sweetened cardboard.\n\nIt was April. I hadn't been back to Gallanach yet this year, hadn't spoken to dad. My studies weren't going so well; a 2.2 was probably the best I could hope for. Money was a problem because I'd spent all the dosh I'd got for the car, and I needed my grant to pay off the overdraft I'd built up. There was about a grand in the old account \u2013 my dad's money came by standing order \u2013 but I wouldn't use it, and what I regarded as my own finances were \u2013 judging from the tone of the bank's increasingly frequent letters \u2013 somewhere in the deep infrared and in serious danger of vanishing from the electromagnetic spectrum altogether.\n\nI had paid my rent early on with the last inelastic cheque I'd written, hadn't paid my Poll Tax, had tried to find bar work but been unsuccessful, and was borrowing off Norris, Gav and a few other pals to buy food, which comprised mostly bread and beans and the odd black pudding supper, plus a cider or two when I could be persuaded to squander my meagre resources on contributing to the funds required for a raid on the local off-licence.\n\nI spent a lot of time lying on the couch in the living room, watching day-time television with a sneer on my face and my books on my lap, making snide remarks at the soaps and quizzes, chat shows and audience participation fora, skimming the scummy surface of our effervescent present in preference to plumbing the adumbrate depths of the underlying past. I had taken to finishing off the flat beer left in cans by the members of Norris's itinerant card school after its frequent visits _chez nous,_ and was seriously considering starting to steal from bookshops in an attempt to raise some cash.\n\nFor a while I had been ringing the Lost Property office at Queen Street station each week, still pathetically hoping that the bag with Uncle Rory's poems and Darren Watt's M\u00f6bius scarf would somehow miraculously turn up again. But even they weren't having anything to do with me any more, after I'd _definitely_ detected an edge of sarcasm in the person's voice and lost my temper and started shouting and swearing.\n\nRejected by Lost Property; it seemed like the ultimate insult.\n\nAnd Aunt Janice never did remember any more about whatever Rory had hidden in his later work.\n\nMum sipped her coffee. I tore the Danish to bits, imagining it was Lewis's flesh. Or Verity's underclothes \u2013 I was a little confused at the time.\n\nWell, let them get married. The earlier the better; it would end in tears. Let them rush into it, let them repent at leisure. They weren't right for each other and maybe a marriage would last a shorter time than a more informal, less intense liaison; brief and bitter, both of them on proximity fuses with things coming rapidly to a crunch, rather than something more drawn out, where they might spend long periods apart and so forget how much they hated being together, and enjoy the fleeting, passionate moments of reunion ...\n\nI fumed and bittered away while my mother finished her coffee and made concerned remarks about how thin and pale I was looking. I ate another Danish; mother told me everybody else was fine, back home.\n\n'Come back, Prentice,' she said, putting one hand out across the table to me. Her brown eyes looked hurt. 'This weekend, come back and stay with us. Your father misses you terribly. He's too proud to \u2013'\n\n'I can't,' I said, pulling my hand away from hers, shaking my head. 'I need to work this weekend. Got a lot to do. Finals coming up.'\n\n'Prentice,' my mother whispered. I was looking down at my plate, licking my finger and picking off the last few crumbs, transferring them to my mouth. I could tell mum was leaning forward, trying to get me to meet her eyes, but I just frowned, and with my moistened finger-tip cleared my plate. 'Prentice; please. For me, if not for your dad.'\n\nI looked up at her for a moment. I blinked quickly. 'Maybe,' I said. 'I don't know. Let me think about it.'\n\n'Prentice,' my mother said quietly, 'say you will.'\n\n'All right,' I said, not looking at her. I knew I was lying but there wasn't anything I could do about it. I couldn't send her away thinking I could be so heartless and horrible, but I also knew that I wasn't going to go home that weekend; I'd find an excuse. It wasn't that this dispute between my dad and me about whether there was a God or not really meant anything any more, but rather the fact of the history of the dispute \u2013 the reality of its course, not the substance of the original disagreement \u2013 was what prevented me from ending it. It was less that I was too proud, more that I was too embarrassed.\n\n'You promise?' mother said, a slight stitching of her brows as she sat back in the ladder-backed seat the only indication that she might not entirely believe me.\n\n'I promise,' I nodded. I felt, wretchedly, that I was such a moral coward, such a sickening liar, that making a promise I knew I had no intention whatever of keeping was hardly any worse than what I had already done. 'I promise,' I repeated, blinking again, and set my mouth in a firm, determined way. Let there be no way out of it; let me really _make_ this promise. I was so disgusted with myself that I wanted to make myself suffer even more when I did \u2013 as I knew I would \u2013 break my word. I nodded fiercely and smiled bravely, utterly insincerely, at my mother. 'I really do promise. Really.'\n\nWe said goodbye outside, in the street. I told her the flat was in too disgusting a state for her to come and visit. She hoisted her umbrella to ward off the light drizzle that had started to fall, gave me a couple of twenty-pound notes, said she'd look forward to seeing me on Friday, kissed my cheek, then went off to do her shopping.\n\nI had dressed as well as I could that morning, in more or less the same stuff I'd worn for Grandma Margot's funeral. Minus the lost M\u00f6bius scarf, of course. I turned up the collar of my fake biker's jacket and walked off.\n\nI gave the money to a thankfully dumb-struck fiddle-player on Sauchiehall Street and walked away feeling like some sort of martyred saint. As I walked, this mood was gradually but smoothly replaced by one of utmost depression, while my body \u2013 as though jealous of all the obsessive regard my emotions were receiving \u2013 came up with its own demands for attention, evidenced by an unsteady, fluid shifting in my guts, and a cold sweat on my brow.\n\nI felt fainter and fainter and worse and worse and more and more nauseous, unsure whether it was the bitterness of sibling-thwarted love, or just too much starch and refined sugar. It felt like my stomach had decided to take a sabbatical; all that food was just sitting there, unprocessed, locked in, slopping around and making me feel horrible.\n\nAfter a while I stopped telling myself I wasn't going to be sick, and \u2013 resigned to the fact that I was going to have to throw up at some point \u2013 kept telling myself instead that I'd manage to hold it in until I was back in the flat, and so do it in private, rather than into the gutter in front of people.\n\nEventually I threw up into a litter bin attached to a crowded bus shelter on St George's Road.\n\nI was still gagging up the last few dregs when somebody punched me on the cheek, sending the other side of my head banging against the metal wall of the shelter. I spun round and sat down on the pavement, a ringing noise in my head.\n\nA tramp dressed in tattered, shiny trousers and a couple of greasy-looking, buttonless coats bent down, looking at me. He smelled of last year's sweat. He gestured angrily up at the litter bin. 'Ye wee basturt; there might a been somethin good in there!' He shook his head in obvious disgust and stalked off, muttering.\n\nI got to my feet, supporting myself on the side of the shelter. A wee grey woman wearing a headscarf peered out at me from the end of the bus queue. 'You all right, sonny?' she said.\n\n'Aye,' I said, grimacing. 'Missus,' I added, because it seemed appropriate. I nodded at the bin. 'Sorry about that; my stomach's on strike and my food's coming out in sympathy.'\n\nShe smiled uncomprehendingly at me, looking round. 'Here's ma bus son; you look after yoursel, okay?'\n\nI felt the side of my head where it had hit the bus shelter; a bruise was forming and my eye felt sore. The wee woman got on her bus and went away.\n\n'Oh, Prentice!' Ash said, more in despair than with disgust. 'You're kidding.' She looked at me in the candle-light. I was past caring about feeling guilt and shame and everything was collapsing anyway, so I just looked straight back at her, resigned, and after a while I shook my head. Then I picked up a bit of naan bread and mopped up my curry sauce.\n\nThe naan bread was big; we'd both stuffed ourselves with it during the meal but it was still big. When it had arrived it had needed a separate table just to accommodate it; luckily the restaurant wasn't busy. 'Not so much a naan bread, more a toasted duvet,' I'd said. Ash had laughed.\n\nDuring the course of the meal we'd reduced the blighter to the proportions of a couple of pillows, not to mention disposing of portions of chicken kalija and fish pakora to start, followed by garlic chilli chicken, lamb pasanda, a single portion of pulao rice, and side dishes of Bombay potato and sag panir to accompany. Two dry sherries and a couple of bottles of Nuit St Georges had washed it all down and now we were onto the coffee and brandy. It was Ashley Watt's treat, of course; I still couldn't afford to eat out unless it was in the street and out of a paper poke. Ash was passing through Glasgow and staying with us on her way to a new job down in London.\n\nIt was mid-summer, and unseasonably warm for Glasgow; Ash wore a long, rough silk shirt, and leggings. A light cotton jacket hung over the back of her chair. I was still wearing out the regulation Docs and the thick black jeans. I had borrowed one of Norris's big paramilitary-style fawn shirts to wear as a jacket over my anti Poll Tax T-shirt. I'd left it to the end of the meal before I said anything about being arrested.\n\n'Aw, man,' Ash said, sitting back slackly in her seat. The candle-light reflected in her glasses. _'Why,_ Prentice?' The Anarkali was dark and quiet and a lot of the light was coming from the candle between us. She looked sad; concerned for me, I thought.\n\nI rather liked it. I liked the idea of other people feeling sorry for me, even though I also despised them for it, because I wasn't worth their sympathy and that made them fools.\n\nOf course, I despised myself for despising them for showing such genuine and unselfish emotions, but that's just one of the things you have to get used to when you're in a serious self-destruction spiral. Mine was feeling rather like a power-dive right now. I shrugged. 'Why not? I needed the money.'\n\n'But your family's rich!'\n\n'No, they're ... Well, they might be fairly well ...' I smiled, sat closer, took up my brandy and cradled it in front of the candle flame. 'Actually, there's quite a good exchange on those lines in _Catch-22,_ the movie \u2013 much underrated film \u2013 which isn't in the book, so Buck Henry must have written it, where Nately's been killed and Yossarian's been to Milo's whorehouse to see Nately's whore and Milo's picked him up in the half-track and he's saying Nately died a rich man; he had such-and-such a number of shares in M&M enterprises, and Yossarian says \u2013'\n\nAshley was glaring at me over the candle flame the way a hawk must glare at a field mouse the instant before it parts mouse from field forever. I saw this predatory, outraged expression building on Ash's face like a line of dark clouds on the horizon, and stopped talking, though entirely out of inquisitiveness, not trepidation.\n\n'Shut the fuck up about _Catch-22,_ ya cretin;' Ash said, storming forward and planting both fore-arms on the table cloth. 'What the fucking hell are you doin stealing books for money when you've no need to, eh? Just what sort of dick-head are you, Prentice? I mean, what the fuck are your parents going to think if they hear? How are they goin to feel? Or is that it? Are they supposed to feel bad? Are you tryin to get back at your dad because of this stupit religious thing? Well, come on; are you?'\n\nI sat back, amused.\n\nI played with the dumpy stem of the brandy glass, smirking at Ashley through the candle flame. Ashley's long hair was tied back and she looked rather attractive, now I thought about it. I wondered what the chances were of bedding the girl. A little recreational fornication would go down quite well just now. I wondered if Ash was into rough sex. I had no idea whether I was into it myself, but for some reason just then the idea seemed rather intriguing. I smiled at her, gave a small laugh. 'Really, Ashley, I didn't think you'd take it all so melodramatically. It's only shop-lifting, after all. Just one silly book, too; worse things happen at C & A's.' I sat back, still smiling; legs crossed, arms crossed.\n\nAsh's face was close to the flame, its yellow oval glowing like some magical caste-mark on her forehead. Much closer and she'll melt her glasses, I thought. She appeared to be trying to out-stare me, but actually I'm rather good at that sort of thing when I want to be, and I didn't let my eyes flicker.\n\nA waiter was approaching from behind her, I noticed, without taking my eyes off hers; I felt the grin broaden on my lips. The waiter would distract her, especially as she had ordered the meal and was obviously paying, and anyway she almost certainly hadn't heard the waiter approaching.\n\nAsh reached one hand out across the table and spilled my brandy into my lap.\n\nJust as I was reacting, going 'Wha \u2013!' and jerking forward, Ash turned smoothly to the waiter and with a broad smile said,\n\n'The bill, please.'\n\n'It does look like I've pissed myself!' I protested as we walked back to the flat. 'Those people were definitely laughing at me.'\n\n'Oh, shut up, Prentice.'\n\n'You're telling _me_ to shut up!' I laughed. The July night was warm and muggy and the traffic rumbled like thunder down Great Western Road. 'You throw drink all over me, expect to sleep in my flat tonight and _you_ tell _me_ to shut up!'\n\nAsh paced purposefully on, long flinging strides I was having difficulty in keeping up with. She was still glaring, though straight ahead now. I noticed people coming towards us weren't getting in her way.\n\n'I didn't throw the drink, I tipped it,' she told me. 'And I'm only coming back to the flat to get my bag, if that's the way you feel. I'll sleep in the car. Or find a hotel.'\n\n'I didn't!' I protested, waving my arms and running after her as I saw the possibility of getting into Ashley's increasingly attractive body slipping away from me. 'I didn't say that! I just don't like being told to shut up! I'm sorry! I mean, I'm really sorry I'm annoyed that you spilled \u2013 or tipped \u2013 drink all over me!'\n\nAsh stopped so suddenly I wondered where she'd gone for a moment. I turned, looked, and went back to her, standing looking furious in the light of a Spud-U-Like.\n\n'Prentice,' she said calmly. 'You've practically exiled yourself from your family and your home and your friends, you think you've failed your finals but you say you've no intention of sitting your re-sits even if you have; you've no money and you haven't even been looking for a job; you're getting done for shop-lifting and you're acting like such a fucking dick-head you seem determined to get shot of the last few pals you do have left ... and all you can do is make smart-ass remarks.'\n\nI looked through her bright red glasses into her light grey eyes and said, 'Well, so far so good, certainly, but let's not count our \u2013'\n\nShe stamped on my right big toe, forcing me to produce an involuntary and appallingly undignified yelp. She stormed off; I half limped, half hopped after her.\n\n'Let's not count our vultures before they're hatched, eh?' I laughed. She powered on, ignoring me. I hopped after her. 'Spare a shekel for a healthy beggar?' I cackled. 'Able was I ere I saw Michael; where can you land a Palin? And in what?'\n\nAsh kicked my other shin. Wonderful girl; didn't even seem to break stride.\n\nShe disappeared into an off-licence. I waited outside, rubbing my shin and inspecting the damage to my Docs; luckily the scuff on the right toe didn't show up the way it would have with polished boots.\n\nAsh reappeared with a bag; she swept past me, briefly showing me the bottle of Grouse it contained. I skipped after her down the street. 'After trying the fluid on a small unnoticeable area, you now wish to wash all of my trousers the spirited way, am I right, madam? Now; will you swap these two bottles of warm urine for that one bottle of our product?'\n\nShe shook her head, not looking at me. 'You and I are going to get filthily drunk, Prentice, and if by the time we get to the bottom of this bottle I haven't got some sort of sense out of you I'm going to break it over your thick fucking skull.' She turned, beamed a toothy non-smile at me for about a micro-second, then strode determinedly on.\n\nI tried to keep up. I looked at the bottle in the bag. 'Couldn't you just leave the whisky, I'll drink it all, wake up in the morning \u2013 no, make that the afternoon \u2013 with a head that _feels_ like you hit me over the skull with the bottle, and you sleep in the car ready for that long and demanding journey down the notoriously dangerous A74 tomorrow?'\n\nAsh shook her head.\n\nWe got back to Grant Street. I looked up, saw some lights on in the flat. Maybe, I thought, Ash would be so turned on by the sounds of frantic coupling emanating from Gav and Aunt Janice in the bedroom that she'd tear my clothes off. Or maybe Norris and his pals would distract her from this crazed idea of getting air-locked drunk by suggesting a friendly game of cards.\n\nAsh followed my gaze. She held the bottle up in front of my eyes. 'Ready for this, Prentice?'\n\n'Drink doesn't solve anything, you know,' I told her. 'Just dissolves brain cells.'\n\n'I know,' she said. 'I'm working on the principle that most people are okay unless they get muroculous with drink, when they become arse-holes; you're behaving like an arse-hole now, so maybe drink'll make you okay.'\n\nI tried to look as sceptical as I could. 'I bet you believe in crop circles, too.'\n\n'Prentice, I believe you seem determined to fuck your life up, and I just want to know why.'\n\n'Oh,' I said brightly. 'That's easy; my affections have been rejected by the one I love and her carnality is being most thoroughly investigated by my elder and smarter brother on a more or less hourly basis, so I am spurned and she is spermed; my father believes his children should be free to make up their own minds, but preferably only out of the spare-parts that he provides ... And apart from that ... I mean the exams and getting nicked and stuff ... Well,' I sighed, looking up to the night sky, where the clouds were starting to blot out the few stars that the city lights did not obscure. I spread my arms wide. '... I'm just a waster.'\n\nAsh looked at me. I could see her chest move in and out inside the light cotton jacket. 'Naw, Prentice,' she said quietly, after a while. 'You're just a bairn.'\n\nI shrugged. 'Maybe. Come on.' I indicated the close. 'Let's get as drunk as you think we have to, and you can tell me all the reasons I'm so childish.' I glanced at my watch as we headed for the stairs. 'Better get started, though; we've only got all night.'\n\nWe climbed the stairs, reached the flat.\n\n'You know,' Ash was saying, breathing hard and looking down the stair-well as I opened the door. 'I don't know anybody who lives in a flat who doesn't live on the top floor.'\n\n'Friends in high places,' I said, opening the door to Janice Rae.\n\nAunt Janice was clothed (shirt and jeans), which made rather a refreshing change, and standing in the hallway. She looked distraught. Her eyes were red and her mascara had left what appeared to be a diagram of the Los Angeles freeway system down her cheeks. Beyond her Gav stood looking awkward and sheepish. I glanced from Janice to Gav and back again, while Janice looked at me, lip trembling.\n\nLet me guess, I thought; they've finally done it; they've broken the bed.\n\n'Oh, Prentice!' Janice said suddenly, throwing herself at me and enveloping my upper torso in a hug that would have done credit to a grizzly. I wondered what had brought this on, and how to peel Aunt Janice off. What must Ashley be making of all this? (She'd be getting jealous, with any luck.)\n\nJanice pulled away; I could breathe again, and promptly did so.\n\n'Oh, Prentice,' she said again, holding my head in both hands and shaking her own. Her eyes closed, she turned her face away, released her hold on my cheekbones and let me go on into the hall. Gav stood by the hall table, shifting his weight from side to side and glancing nervously down at the phone now and again.\n\nHe avoided my eyes.\n\nI took a couple of steps forward, then heard something whispered from behind me, and looked back to see Janice hugging Ash, almost violently.\n\nThey'd never met before. How shocking, I thought. Where was that traditional British reserve only abandoned for cloying camaraderie under the influence of injuriously vast quantities of alcohol? I wondered, if nervously.\n\nAsh was looking over Janice Rae's shoulder at me, those grey eyes behind the bright red glasses filling with tears.\n\n'Um; you've to phone home,' Gav mumbled, apparently addressing his trainers.\n\n'ET or BT?' I heard myself say to him, though the different sections of my brain seemed to have slipped out of synch somehow, and I was aware of all sorts of different things at once, and time seemed to have slowed down and at the same time some part of my brain was racing, trying to come up with some logical explanation for what was going on that didn't involve calamity ... and failing.\n\n'It's \u2013' Gav said, this time seemingly directing his remarks to his rugby-shirted chest. 'It's your dad,' he whispered, and suddenly started to cry.\nCHAPTER 12\n\nThis is the Specialist Glass Division,' Hamish said, opening a door. They found themselves in a long corridor with one glass wall that looked out into a bright, modern, open-plan and spacious area. Everything gleamed and the few people visible wore white coats; apart from the exposed brickwork of a couple of rotund furnaces, linked to the ceiling by shining metal ductwork, the place looked more like a laboratory than a factory.\n\nThere was a silence none of the three brothers seemed inclined to fill. Hamish, an immaculate white coat over his three piece suit, gazed with a rapt expression at the almost static panorama on the far side of the glass. Kenneth looked bored. Rory stood at Janice Rae's side, humming something monotonous, one arm round Janice's waist and attempting to tickle her, just above her right hip.\n\n'Very clean,' Janice said eventually.\n\n'Yes,' Hamish said gravely. He nodded slowly, still observing the scene beyond the glass. 'It has to be, of course.' He turned to the tables against the wall behind them, on which lay various glassy-looking objects, some in display cabinets, most loose, all with explanatory notes stuck to the wall above them. From a wooden plinth on one table, Hamish picked up a dull black cone that looked a little like a Viking helmet without the horns.\n\n'This is a missile nose-cone,' he said, turning the cone over in his hands. He held it out to Janice. She took it.\n\n'Hmm. Quite heavy,' she said. Rory tickled her again and she nudged him.\n\n'Yes, heavy,' Hamish said gravely, taking it back and carefully replacing it on its wooden block. 'Strictly speaking, this is a glass ceramic rather than ordinary glass,' he said, adjusting the precise position of the nose cone on the plinth. 'The basis is lithium aluminosilicate, which withstands heat very well. Cooker hobs are made from this sort of thing ... and obviously missiles need to withstand a lot of heat from friction with the air.'\n\n'Obviously,' Kenneth said. He and Rory exchanged looks.\n\nHamish turned to another exhibit; a broad bowl, also dull and dark, and over half a metre across, it was like a gigantic plate with no lip. He lifted an edge so that they could look underneath, where it was criss-crossed with a lattice of deep ribs.\n\n'Satellite aerial?' Kenneth said.\n\n'No,' Hamish said, though a hint of a smile crossed his dour face. 'No, this is a substrate for an astronomical telescope mirror.'\n\n'Like the one Fergus has in the castle?' Rory asked.\n\n'That's right. All the substrates and optics for Mr Urvill's telescope were made here. Though of course they were on a smaller scale than this piece.' Hamish lowered the edge of the bowl and flicked a bit of dust off one edge. 'This is made from the same type of material as the nose cone there. It resists distortion under thermal shock.'\n\n'Hmm,' Janice in a tone that suggested that she was really trying to be interested as well as sound it.\n\n'Over here,' Hamish said, plodding towards another table, 'we have what are called the passivation glasses, related to the Borate glasses but made from zinc-silicoborate ...'\n\n'All I said was I'd like to see the factory,' Janice whispered to Rory as they moved to follow Hamish. 'The outside would have done.'\n\n'Tough shit,' Rory said, and tickled her with both hands this time, producing a yelp.\n\nAnother man in a white coat came up to Hamish from the far end of the corridor. 'Excuse me a moment,' Hamish said to the others, and turned to talk to him.\n\nKenneth turned to Rory and Janice. He tugged on Rory's sleeve and in a low monotone said, 'Dad, I'm bored, dad; dad, are we nearly finished yet, dad? Dad, want to go home, dad.' He leant one hand against the glass wall, glanced back at Hamish \u2013 still deep in conversation, and nodding \u2013 and rolled his eyes. He looked at Janice. 'My elder brother,' he said quietly. 'The man who put the Bore in Boro-silicate.'\n\n'You don't have to stay.' Rory grinned. 'We could get a train home.'\n\nKenneth shook his head. 'No; it's okay.' He glanced at his watch. 'Maybe we can drag the Tree out for lunch soon.'\n\n'Sorry about that,' Hamish said, coming up behind them.\n\nThey all smiled at him. Hamish moved one arm up to indicate they should move down the corridor to where they could see the exciting zinc-silicoborates. He took a pristine white handkerchief out of his pocket and rubbed at the faint hand-print Kenneth had left on the glass partition as he said, 'These passivation glasses are of much use in the semi-conductor industry, and we have high hopes that with the burgeoning of the Scottish computer industry \u2013 Silicon Glen as it is sometimes jocularly called \u2013 we shall shortly be supplying ...'\n\n'And to think, all that could have been mine.' Kenneth sighed with pretended regret, putting his feet up on the low wall of the terrace and rocking his seat back on its rear legs as he shaded his eyes with one hand. He brought his drink up to his lips with the other.\n\nJanice and Rory were tucking into their salads; the terrace of the Achnaba Hotel was crowded with tourists, and on the road in front of the hotel cars, caravans and coaches hummed past, heading for Lochgilphead, Gallanach, or Kintyre. A brisk warm wind blew from the south west, laden with the vanilla smell of gorse blossom, mixed with pine off the forests and a salt hint from the sea.\n\n'Well, that's just the way it goes, Ken,' Rory said. 'Hamish got to be manager of the factory and you didn't. No use crying over spilled boro-silicate ...'\n\nKenneth grinned, staring out over the balustrade of the terrace towards the hills on the far side of Loch Fyne. 'I wonder where that saying comes from. I mean, why milk? If it means something not very valuable, why not water? Or \u2013'\n\n'Maybe crying over milk was unlucky,' Rory suggested.\n\n'It was years before I realised it was even common parlance,' Kenneth said, still staring out to the loch. 'I used to think it was something only mum came out with. Like \"I couldn't draw a herring off a plate.\" I mean, what the hell does that mean? Or, \"Och aye; that's him away the Crow Road.\" Jeez. Opaque or what?'\n\n'But they might all have some ... some basis in reality,' Rory insisted. 'Like crying over milk was bad news; spoiled it.'\n\n'Maybe it spoiled un-spilled milk,' Kenneth nodded. 'Some chemical reaction. Like they say thunder can curdle milk; ions or something.'\n\n'Ah,' Rory said. 'Then maybe you were _supposed_ to cry over milk, because it helped preserve it, or made it easier to turn into cheese. And so it was a waste crying over spilled milk.'\n\n'I think this is where we came in,' Kenneth said. He squinted at a car on the road as it hurried north. 'Isn't that Fergus?' he said, nodding.\n\n'Where?'\n\n'Racing green Jag; heading north.'\n\n'Is that what Ferg's driving these days?' Rory said, rising up in his seat a little to watch the car pass. It swept round the long bend that carried the road towards the forest. He sat back down and took up his fork again. 'Yeah, looked like Ferg.'\n\n'This is Fergus Urvill, who owns the factory?' Janice asked. She sat back in the white plastic chair, fanning herself with her napkin.\n\nKenneth looked at her. 'Yep, that Fergus,' he said. 'Of course, you haven't had the dubious pleasure yet, have you?' He put his glass down on the circular table, and inspected the rolled up sun-shade that protruded from the centre of the table like an unopened flower.\n\n'No,' Janice said. 'What's he like?'\n\nKenneth and Rory exchanged glances. 'Bearing up remarkably well,' Kenneth said.\n\nJanice looked puzzled for a second, then said, 'Oh; yes, of course; Fiona ...' she looked embarrassed. Rory patted her hand on the table.\n\nKenneth looked away for a moment, then cleared his throat. 'Yeah; anyway.' He stretched his shoulders, sat back. 'Fergus ...\n\nUpper-class; huntin'-shootin'-fishin' type ... Could be worse, I suppose.'\n\n'Still,' Rory said. 'Not what you'd call a happy man.'\n\n'Well, of course,' Janice said quietly, and bit her lip.\n\nKenneth frowned. 'His precious factory's making a profit,' he said briskly, draining his glass. 'The Greedy Party's in power. What more does he want?'\n\n'A wife?' Rory suggested, and then sucked on one finger.\n\nKenneth looked down, studying his glass. There was silence.\n\nRory rubbed a mark off the white table's surface. Janice lifted the scooped neck of her bright print dress and blew down.\n\n'Want some shade?' Kenneth asked Janice. She nodded.\n\nKenneth stood, lifted the stalk of the sun-shade and opened the big parasol, casting a shadow over Janice and Rory.\n\n'Did you know,' Janice said to Rory, squeezing his hand. 'In the Dewey Decimal System, glass-making comes under the code six six six?'\n\n'Woo,' Rory whistled. 'Number of the beast! Spooky, eh?'\n\n'Not many people know that,' Janice said. She smiled.\n\nKenneth laughed. He sat back in his chair again, dragging it round so he was under the shade too. 'Shame Ferg isn't superstitious.' He chuckled. 'Mind you, Hamish is. Maybe we should tell him that. The Tree has some pretty weird ideas about religion; he might just swallow the idea he's been working for the devil all this time. Renounce the whole business, start going round smashing windows.'\n\n'Really?' Janice said. 'What is he? I mean what religion?'\n\nKenneth shrugged. 'Oh, just Church of Scotland; but if they had a Provisional Wing, I think he'd be on it.'\n\n'He's always had a soft spot for the royal family \u2013' Rory began.\n\n'Yes; his head,' Kenneth said.\n\n'\u2013 Maybe he could start the Royal Church of Scotland.'\n\n'Maybe he could start thinking like a rational human being instead of a cave-man frightened by lightning,' Kenneth said tartly.\n\n'Oh, you're so cruel,' Rory told him.\n\n'I know,' Kenneth sighed, rolling the base of his glass around on the table top. 'Time for another drink, I think.'\n\n'My round,' Janice said, rising.\n\n'No,' Kenneth said, 'Let \u2013'\n\n'Sit down,' Janice told him, taking his glass from his hand. 'Same again?'\n\nKenneth looked glum. 'No; Virgin Mary this time. Gotta drive.'\n\nThe two men watched Janice head for the bar.\n\n'What _did_ Fergus ever say to you?' Kenneth asked Rory.\n\n'What?' Rory said, blinking. 'What about?'\n\n'God, I hate it when you're mysterious!' Kenneth shook his head. 'You know damn well. Before the crash; way before. What did Fergus ever tell you? Was it after you came back from India that second time; before you went back to London? You two went hill-walking a lot then, didn't you? Old Ferg spill some beans up in them there hills?'\n\n'We talked,' Rory said awkwardly, using his fork to push bits of lettuce around his plate. 'He told me things, but ... I don't want to go into it, Ken, it would only complicate matters. It's nothing that directly touches you.'\n\n'What about Fiona?' Kenneth said, voice low, staring at his brother. 'Did it touch her?'\n\nRory looked away, across the loch. He shrugged. 'Look, Ken, it isn't something you'd benefit by knowing, all right? Just leave it at that.' The fork continued to shift the lettuce leaves around the plate.\n\nKenneth watched his brother for a moment, then sat back. 'Oh well, serves me right for being nosey. Let's change the subject. How's this new project thing coming along?'\n\n'Oh, I'm still working on it.'\n\n'I wish you'd let me look at it.'\n\n'It isn't finished yet.'\n\n'When will it be?'\n\n'When it is,' Rory said, frowning. He put the fork down. 'I don't know. Look; it's sort of a personal story ...\n\n'Ah,' Kenneth said.\n\nRory leaned forward over the table, closer to his brother. 'Look,' he said, glancing round towards the french windows that led to the bar. 'I've had a few more ideas ... well, I've thought about ... areas I didn't think I could use that I now think I can, and I want to develop that stuff, and \u2013'\n\n_'What_ stuff?' Kenneth said, laughing in exasperation and throwing his arms wide. 'Just tell me what _sort_ of stuff!'\n\nRory sat back, shaking his head. 'I can't say. Really.' He glanced up at Kenneth. 'But things ... things might start to happen soon, anyway. I can't say any more for now.'\n\nKenneth shook his head sadly. 'They might have happened by this ... opera, TV series, pop-up-book, whatever the hell it is; _and_ if you'd let me talk to a few people. I mean, if it's just that you're too close to it and you don't want _me_ to look at it, there are people I know who're good at that sort of thing; they can see the wood from the trees; they could \u2013'\n\n'Aw, come on, Ken,' Rory said, a pained expression crossing his face. He ran a hand through his short, straight hair. 'This is my show; this is the way I want to do it. Just let me, all right?'\n\n'I don't know, Rore,' Kenneth said, sitting back. 'Sometimes you play your cards so damn close to your chest I don't think you can see them yourself. You should open up a bit more, share your problems. Share some secrets.'\n\n'I do,' Rory said, biting his lip and looking down at his glass.\n\n'Rory,' Kenneth said, sitting forward and lowering his voice to conspiratorial levels, 'the last secret I remember you telling me was that it was you who set fire to that barn on the Urvill's estate.'\n\nRory grinned, stirring his finger through a little patch of moisture on the side of his glass. 'Hey, I'm still waiting to see if you tell anyone.'\n\nKen laughed. 'Well, I haven't. Have you?'\n\nRory smiled, sucking air through his teeth at the same time, clinked one thumb-nail against his glass. He glanced at his brother. 'Don't worry; my secret is safe with us.' He shook his head, then shrugged. 'Okay,' Rory sighed, trying to suppress a smile, looking away. 'There might be a job with Aunty in the offing, okay?'\n\n'What?' Kenneth laughed. 'The Beeb? You going to be a TV star?'\n\n'It's not definite yet,' Rory shrugged. 'And it's ...' he frowned at his brother. 'Shit, Ken; it's just more hack-work. It's better paid, is all.'\n\n'What is it though?'\n\n'Oh, a fucking travel programme, what else?' Rory rolled his eyes. 'But anyway; we'll see, okay? It's not definite, like I say, and I don't want to get anybody's hopes up, so keep it quiet; but things might start to happen.'\n\n'But that's _great_ news, man,' Kenneth said, sitting back.\n\n'Talking about me, I hope, boys,' Janice said, returning with their drinks on a tray.\n\n'... said, \"My _God,_ Rory, I've never seen one that big!\" and I said \u2013 oh; hello dear,' Rory grinned, pretending only then to notice Janice.\n\nShe sat down, smiling. 'Talking about the size of your overdraft, are we, dear?'\n\n'Gosh-darn,' Rory said, snapping his fingers, looking at Kenneth. 'Caught telling tales again.'\n\n'Runs in the family,' Kenneth said, taking up his glass. 'Cheers, Janice.'\n\n'Your health.'\n\n'Slange.'\n\nThey left after that drink and went back to the house at Lochgair; Rory and Kenneth cleared a tangled choke of bushes and shrubs at the rear of the garden, where Mary wanted the lawn extended. They sweated through the insect-loud afternoon, while the sun shone. Janice sunbathed, and later helped Mary and Margot prepare the evening meal.\n\nJanice had taken that day off from the library. She and Rory left on the last train back to Glasgow that night.\n\nIt was the last time Kenneth ever saw Rory.\n\n_Fiona sat in the passenger seat of the car, watching the red roadside reflectors drift out of the night towards her. She was thrown against one side of the seat as Fergus powered the Aston round the right-hander that took the road out of the forest, down, into and through the little village of Furnace. She was pressed back against the seat as_ _Fergus accelerated again. They swung out and past some small, slower car, over-taking it as though it was stationary; headlights ahead of them glared, the on-coming car flashed its lights and she heard its horn sound as they passed, a few seconds later. The sound was quickly lost in the snarl of the Aston's engine._\n\n_'If you're driving like this to try and prove something, don't bother on my account,' she said._\n\n_Fergus was silent for a while, then, in a very controlled and even voice said, 'Don't worry. Look, I just want to get home as soon as possible. All right?'_\n\n_'Everything'll suddenly get better once we're home, will it?' Fiona said. 'Kiss the kids on the head and get Mrs S to make some tea; stiff whisky for you, G and T for me. Maybe we should call up the McKeans to say we got back safely; you can ask after Julie ...'_\n\n_'For Christ's sake, Fiona \u2013'_\n\n_' \"For Christ's sake, Fiona\",' Fiona sneered, imitating Fergus's voice. 'Is that all you can say? You've had half an hour to think up another excuse, and \u2013'_\n\n_'I don't need,' Fergus sighed, 'any excuses. Look; I thought we had agreed to just leave this \u2014 '_\n\n_'Yes, that would suit you fine, wouldn't it, Ferg? That's your way of dealing with everything, isn't it? Pretend it hasn't happened, maybe it'll go away. If we're all terribly polite and decorous and discreet, maybe the whole horrid thing will just ...' She made a little fluttering motion with her hands, and in a high-pitched, girlish voice, said, 'Disappear!'_\n\n_She looked at him; his broad, soft-jowled face looked hard and set in the dim light shining from the car's instruments. 'Well,' she told him, leaning over as far as she could towards him. 'They won't just go away, Ferg.' She tried to make him look at her. He frowned, put his head slightly to one side and lifted it, trying to look round and over her head. 'Nothing ever goes away, Fergus,' she told him. 'Nothing ever doesn't matter.' She strained over a little more._ 'Fergus _-' she said._\n\n_He pushed her away with his left hand, back into her seat._\n\n_She sat there, mouth open. He seemed to understand the silence and glanced over, a weak smile flickering on his face. 'Sorry,' he said. 'Getting in the way a bit there. Sorry.'_\n\n_'Don't you push me!' she said, slapping his shoulder. She hit him again. 'Don't you ever dare push me again!'_\n\n_'Oh stop it, Fiona,' he said, more exasperated than angry. 'One minute I'm in the dog-house because ... well, because I'm not all over you all the time; next second \u2013'_\n\n_' \"Not all over you all the time\"?' Fiona said. 'You mean not fucking me, Fergus, is that what you mean?'_\n\n_'Fiona, please \u2013'_\n\n_'Oh.' Fiona slapped one palm off her forehead, then crossed her arms, looked away, out of the dark side window. 'Fuck; did I swear? Oh fuck. Oh what a silly fucking cow I must fucking be.'_\n\n_'Fiona \u2013'_\n\n_'I said something straight. I'm so sorry. I actually said what I meant, used the sort of word you'd normally only hear from your golfing chums or your rugby pals. Or does Julie use that sort of language? Does she? Do you like her to talk dirty? Does that get you going, Ferg?'_\n\n_'Fiona, I'm getting rather tired of this,' Fergus said through his teeth, his fingers gripping the wheel harder, rubbing round it. 'I'm sorry you think what you do about Julie. As I have tried to tell you, she was the wife of an old friend and I've kept in touch since she got divorced \u2013'_\n\n_'Still stuck on that, Fergus?' Fiona said, impersonating concern._\n\n_'Oh dear; we had that line back at Arrochar, I seem to recall. And what was the rest of it? Oh yes, one of her sons has leukaemia, poor little kid, hasn't he? And you've helped her and the little darling with BUPA out of the goodness of your heart \u2013'_\n\n_'Yes I have, and I'm sorry you choose to sneer about it, Fiona.'_\n\n_'Sneer!' laughed Fiona. 'It's a joke, Fergus. Jesus, she was practically taking your zip down.'_\n\n_'Oh, don't be ridiculous. It's not my fault Julie got a bit tipsy.'_\n\n_'She was smashed out of her brains, Fergus, and about the only thing she remembered was that she wanted to get your trousers off. God knows why, but she seemed to associate that with pleasure.' Fiona gave a sort of strangled laugh, then put one hand up suddenly to her nose, and looked away, and sobbed once._\n\n_Fergus drove quickly on, trees flicking past like green ghosts to the right, the waters of the loch just a dark absence on the left._\n\n_Fiona sniffed. 'Trying the great silence again, eh, Ferg?' She pulled a handkerchief from her handbag on her lap, dabbed at her nose. 'Still pretending it'll all go away. Still sticking your head in your precious fucking optical-quality sand.'_\n\n_'Look, can't we talk about this in the morning? I mean, when you're ...'_\n\n_'Sober, Fergus?' she said, looking over at him. 'That what you were going to say? Blaming it on drink again? Is that all it was? Of course, silly me. I should have realised. Dear Julie gets drunk and for some bizarre reason suddenly starts feeling you up under the table while we're nibbling our cheese and biscuits, and making pathetic double-entendres, and attacks you outside the bathroom; totally unprovoked, of course, and it's all just the drink talking. And I'm just being hysterical, I suppose, because I've had too many of John's terribly strong G and Ts and it'll all look different in the morning and I'll come to you and say sorry and wasn't I being a silly girl last night, and you can pat me on the head and say yes, wasn't I? And we can still go for cocktails at the Frasers' and bridge at the McAlpines and tee off with the Gordons and cruise with the Hamiltons with a united front, a respectable face, can't we, Fergus?'_\n\n_'Fiona,' Fergus said, face set and teeth clenched. 'I don't know,' he breathed, 'why you're making such a big thing of this. It's just one of those things that happens at parties; people do get drunk and they do do things they wouldn't normally think of. Maybe Julie has ... or has had, in the past, a crush on me or something. I don't know. Maybe \u2013'_\n\n_'A crush on you,' said Fiona. 'Jesus. Well, that's a better try, Ferg. But I don't think you're quite as good a liar as you think you are. And she's not that good an actress.' Fiona looked down, twisting the handkerchief in her fingers. 'Oh God, Ferg, it was so fucking obvious. I mean. I knew there was something going on; all those trips away, and getting drunk and not being able to come home, staying at one of your chums' delightful little pied-\u00e0-terres. Oh, sorry, no, you can't phone back, he's only just got it and it hasn't had a phone put in yet. Or coming back with bruises; how you suddenly became so very clumsy or so easily marked. But at least I could still kid myself, at least I didn't have my nose rubbed in it.'_\n\n_'Fiona!' Fergus shouted, knuckles white on the steering wheel. 'For God's sake, there's nothing to have your nose rubbed in! Julie's just a friend. I haven't touched her!'_\n\n_'You didn't have to, she was touching you,' Fiona said, voice quiet, looking away from Fergus, out to the darkness of the loch. A few weak lights shone on the far side, and headlights on the Otter Ferry road, two miles away across the black expanse of waves, swung out briefly, like a lighthouse beam ... and then dimmed and disappeared. The car roared through another small village before the trees hid the view again._\n\n_Fiona kept her face away from him, looking out into the night, watching the vertical bright line of light the car threw onto the serried mass of dark conifers. Even there she could not escape him; she could see his distorted image in the slanted glass of the car's windows, dim in the background, still lit by his instruments._\n\n_She wondered how she could ever have thought that she loved him, and why she had stayed with him for so long after she'd realised that if she ever had, she did not love him now._\n\n_Of course she could say it was for the children, as people always did ... It was true, up to a point. How terrible it was to have those easy phrases, trotted out so often in the course of gossip, or heart-to-hearts, or in magazine articles, or even court cases, become so real._ _It was never the sort of thing you thought about when you were young, when you were \u2013 or thought you were \u2014 in love, and all the future shone with promise._\n\n_Problems belonged to other people. You might imagine supporting them, talking with them when they needed to talk, trying to help, but you didn't imagine that you would be the one desperate to talk (or the one too embarrassed to talk, too ashamed or too proud to talk); you didn't imagine you would be the one who needed help, not even when you told friends that of course there might be problems, or agreed with your beloved that you would always talk about things ..._\n\n_Staying together for the children._\n\n_And for the adults, she thought. For the sake of appearances. God, she had thought she was above that sort of thing, once. She had been bright and free and determined and she had decided she was going to make her own way in the world, just as well as any of her brothers might. She'd been a sort of feminist before it became fashionable; never had much time for all that sisterly stuff, but she was positive she was as good as any man and she'd prove it ... And marrying Ferg had seemed like an extra boost to her life-plan. London had been exciting, but she had not shone out there, she felt, the way she had here. She had never felt any affection for the place and had made no friends there she would miss; and anyway, she would find fields to conquer up here, coming home triumphant to wed the lord of the manor._\n\n_But it had not been as she had imagined. She had expected to be the centre of things in Gallanach, but the McHoans as a family had so many other things happening to them; she had felt peripheral. The Urvills' own history, too, made her feel like something unimportant on the family tree, for all that Fergus talked of responsibility and duty and one's debt to the next generation._\n\n_She was a leaf, expendable. A twig \u2013 maybe \u2013 at best._\n\n_Somehow all her dreams had disappeared. It seemed to her now that all she had ever had had been the dream of having dreams; the_ _goal of having goals one day, once she had made her mind up what it was she wanted._\n\n_But that had never happened. First Fergus, then the twins, then her own small part in the society of the town and the people there, and in the wider, still circumferential concerns of this wee country's middle-to-ruling classes, and in the more dissipated commonwealth of mildly powerful people who were their peers beyond that \u2013 in England, on the continent, from the States and elsewhere \u2013 took up her time, sapped her will and replaced her own concerns with theirs._\n\n_So now, she thought, I am married to a man whose touch disgusts me, and who anyway does not seem to want to touch me. She looked at Fergus's dim reflection, distorted in the glass, then tried to re-focus on her own image. Can he find me as repellent as I find him? I can't look that bad, can I? A few grey hairs, but you don't notice them; still a size twelve, and I've looked after myself. I look good in this, your standard little black number, and I still get into a tight pair of jeans ... What's wrong with me? What did I do? Why does he have to spend half his time with that drunken, brassy bitch?_\n\n_God, the best time I've had in the past five years was one night with Lachy Watt, angry at Ferg, and more surprised than anything else. They way he just took my hair in one hand, while we were standing looking up at that God-awful window in the great hall, and turned my head to him, and pulled me close; tongue down my throat before I knew what was happening, and there was something adolescent and desperate beneath all that working-class directness, but Jesus, I felt_ wanted ...\n\n_She shook her head. That was best left out of it. Once was once; dismissible. Ever again would set a pattern. Lachy had been back one time afterwards that she knew of, a year later, and he had called, but she'd told him she wouldn't be able to see him, and put the phone down on him. No, that didn't matter._\n\n_She looked at the reflection of Fergus again, as he pulled the wheel; the car tunnelled into the forest, the wall of trees on either_ _side a blur, their greenness more remembered than seen._\n\n_I could leave him, she thought. I could always have left him. But mother's too close for comfort; there'd be too many nearby friends, too many chances of bumping into people I'd rather not bump into; too much mitigating against the clean break; new start. God, I'm pathetic, though, that's so petty. Why haven't I the sheer drive to just get up and go, take the twins and emigrate to Oz or Canada? Or live in wild eccentricity in London or Paris?_\n\n_Or I can stay, as I know I probably will. Muddle through. Look after the twins and try to make sure they negotiate the reefs of puberty and adolescence, set them up to make their way in the world, and do so without becoming just like me ..._\n\n_She looked out, into the grey sweep of road ever rushing towards them. Fergus powered the car down out of the forest, through some more houses and a few lights. The car lurched. Fergus looked over, smiled at her. She didn't know whether to smile back or not, and she wondered what that expression had meant, and what had been going through his head for the last few miles._\n\n_The car jiggled on its springs, lurched again and settled. She clutched at her seat, looking forward. The engine roared._\n\n_She looked back at Ferg, saw tears in his eyes. 'Ferg?' she said._\n\n_The car skidded a little, came straight; she glanced forward at the road, saw the corner and the trees. She clutched at the dashboard with both hands. 'Ferg!' she screamed. 'Look \u2013!'_\nCHAPTER 13\n\nI was eleven when Aunt Fiona died; I remember feeling both peeved and cheated that I was thought too young to go to the funeral. It would have been my chance to show how mature I had become, and anyway from what I'd seen on television and films, funerals looked like rather dramatic and romantic events; people dressed in black and looked sombre. They had thin, tight lips, and they sometimes wept, and there was a lot of grim clutching of other people's shoulders, and low mutterings about how so-and-so had been a good person, and that sort of thing. But under it all was the simple, joyous fact: they were dead and you weren't yet!\n\nI hadn't got to see Aunt Fiona being buried, but I did see Uncle Fergus in hospital. I was in, too, getting my appendix out, and I went along from my ward to his room just to say how sorry I was.\n\nHe had a broken arm, some cracked ribs, and his whole face was bruised; kids with face-paints couldn't have matched all those colours. I'd never seen anything like it.\n\nThere wasn't much to say; I can't remember what I did say. He kept talking about not being able to remember anything after passing Lochgair, no matter how hard he tried. He couldn't understand why she hadn't been wearing her seatbelt. He'd thought she had been, but they said she hadn't. She hadn't. He started to cry.\n\nI sat on the giant, corroded lump of concrete and steel, legs crossed, arms folded, watching the waves break on the sands below and listening to the strange, whooping, hooting sounds and hollow clanging noises produced by the fluted pipes and iron doors embedded in the fractionally tilted concrete mass.\n\nIt was a little after sunset, three days after my father's death. The sun had dipped behind North Jura, and abandoned the sky to a skeined mass of glowing clouds, sinking through the spectrum from gold towards blood-red, all against a wash of deepening blue. The wind was still warm, coming in from the south west, sharp with salt as the remnants of the rolling Atlantic swell hit the rocks nearby and sent up spray, but maybe also \u2013 well, you could imagine it, at least \u2013 containing a hint of grasses, too; something directed over the distant greenery of Ireland, or swept round from the Welsh hills along the circling wind.\n\nThe concrete block was more or less a cube, about four metres to a side, though it looked more squat than that, its lower metre buried in the sand of the small beach a few miles west of Gallanach, about level with the southern tip of Island Macaskin. The concrete and pipe-work block \u2013 four years old now, and streaked with rust and seagull droppings \u2013 was the only full-size work Darren Watt ever completed.\n\nDarren had got his sponsorship from a cement company, which agreed to provide materials and a grant, but finding a place to put the finished piece had been tricky, and it had been Uncle Fergus, no less, who had finally come to the rescue with a site for the work; the town council hadn't liked the idea of a gigantic concrete object the size of four garages being stationed anywhere near the town itself, and for a while it had looked like Darren was going to have real problems finding anywhere to put his concrete edifice (especially after a couple of the more pygmy-brained newspapers had taken up the story and started fuming about a ridiculous waste of public money and the outrageous despoiling of our fragile landscape with queer, arty-farty, loony-left monstrosities).\n\nDarren had thought about playing up to this drivel by giving the thing some wonderfully pretentious title, and I recall him at a party discussing the merits of calling it The Lusitanian Coast Dialectical Kinetic\/Static Object Alpha. In the end, though, he just called it Block One.\n\nIt was a three-kilometre hike from the nearest path, and even the odd yachtsperson, passing close enough to catch sight of the block, would probably have dismissed it as some old war-time ruin. Not exactly as public as Sauchiehall Street, then, but Darren had been happy. It worked; when the tide was at the right level, it produced noises like a ghost trapped in badly tuned organ-pipes, sonorous slammings as waves opened and slammed shut heavy doors like hinged manhole covers within the set tonnes of the block's hollow insides, and \u2013 depending on the waves \u2013 impressive spouts of water, bursting into the air from its rusted throats as though from some stranded cubist whale. He'd learned a lot from it, he'd said; just you wait till the next one, and the ones after that ...\n\nI was thinking about Aunt Fiona because death and dying were on my mind, and I was going back through all the people I'd known who'd had the nerve to pop their clogs before they should have, while I was still around to miss them. Aunt Fiona was a vague memory, even though I'd been eleven when she'd died and I'd known her for so many years. It was as though by her early death the memories had lost the chance of being renewed every now and again, and instead were somehow built over, the spaces that should have been hers recycled and used-up by those of the family who were still alive.\n\nShe'd been okay; I'd liked her, from what I could remember. She'd let us play in the castle and its grounds, and she'd taken us on walks round the coast sometimes. She'd seemed young and old at once to me; of a different generation to Fergus and Lachlan, and even my father. She had seemed younger than them, never mind the real elders, like Grandma Margot; closer to us when we were children. It was a quality she'd shared with Uncle Rory.\n\nThe still absent Uncle Rory. We'd thought that \u2013 as dad's death had gone reported in a few papers, partly because of his modest fame, and partly because of the bizarre nature of his demise \u2013 Rory might hear, and finally get in touch ... but nothing had happened yet, and the funeral was tomorrow. The romantic in me wanted him to reappear at the ceremony, in the grounds of the house at Lochgair, but I doubted that he would. Too pat, too neat, too kind a thing for fate to throw up now.\n\nI looked up at the violet sky, feeling the wind move my hair across my forehead and the nape of my neck. I could see a few stars. I stared at the heavens until my neck got sore, then said, aloud and loud,\n\n'Well?'\n\nNothing.\n\nThe waves shushed across the sands. I lowered my head. Out to sea, a couple of birds flew wing-tip low across the sky-reflecting waters. I shook my head, wondering at it all.\n\nDad died \u2013 my Uncle Hamish seemed to be maintaining \u2013 in suspicious circumstances; God killed him.\n\nUncle Hamish appeared to be almost perversely upset and appalled by the implications of this supposed act; his own part in the bizarre and fatal episode troubled him less, I guessed, than the terrifying idea that there really might be, after all, a God who listens, thinks, decides and acts, just like an ordinary mortal, except more powerful. It rather indicated, I suspected, that all this time my uncle had just been playing a game, and his retributive proto-heresy was exactly as frivolous as my dad had been given to claiming. Whatever, Uncle Hamish was, in short, under sedation.\n\nAnd dad was under the care of the undertaker, and would soon be under the roses at the rear of the garden in Lochgair, un-christened at the start of his life, and joined to unconsecrated ground after its end.\n\nSome generation, I thought. If Uncle Rory was dead (and who was to say he wasn't) then Hamish, my uncle, The Tree, at that moment lying in a darkened room, moaning about a jealous God and being his brother's keeper and the divine and blinding light come from the skies and the smell of the devil and all his works and popping Valium every few hours and muttering about anti-creates and asking his wife to tell my mother that for all his atheism \u2013 so powerfully and dramatically disproved \u2013 he was sure Kenneth had been a mostly good man, and would not suffer unduly in the after-life, even though the gates of heaven were irredeemably closed to him ... This prattling wreck, this bed-bound, hide-bound bag of gibbering nonsense was all that remained of that generation's one-time promise.\n\nRory gone from us for a decade, at least as good as dead; Fiona gone for want of a seat belt; and my father, drunk and angry, furiously determining to prove ... _something_ by a prank barely worthy of some over-privileged Oxbridge undergraduate.\n\nJust Hamish left, and him half-mad with an amalgamated fever of grief, guilt, and re-inoculated faith.\n\nSome result.\n\nI'd surprised myself, when Gav broke down like that, and I knew that dad was dead. I believe I actually came close to fainting. I stood, watching Gav greet, hearing Janice Rae sob into Ashley Watt's shoulder behind me, and gradually I started to feel I was no more attached to or in control of my own body than I was Gavin's. I don't mean that I stood or floated outside myself, just that I was somewhere inside me that wasn't connected with the usual channels of communication, let alone action.\n\nI heard a noise like continual surf, and the view went sort of grey, and tunnel-like for a bit. I was suddenly aware of how delicately balanced we are on our two skinny legs, and my skin seemed to be contracting, pressing in all about me, and going cold, leaving sweat.\n\nI wobbled, apparently; Ash took me by the shoulders and sat me down on the little chair by the table. She got Janice to make some sweet tea. I said thank you, drank the tea, shivered a bit, and then Ash dialled Lochgair for me.\n\nThe phone was engaged, but Ashley kept trying. It was a friend of mum's from the village who answered initially.\n\nI didn't think I was crying, while I was on the phone; I felt calm and in control and I spoke quietly to my mum, who sounded trembly and yet flat-toned, and told me what had happened, but after I'd put the phone down I found that my eyes were full of tears and my cheeks wet with them. They'd dribbled round my chin and onto my chest, inside the open shirt.\n\n'Oh dear,' I said, feeling that I ought to feel embarrassed. Ash handed me a clean tissue, and I dabbed myself dry.\n\n'I'll drive you back,' Ashley said, squatting in front of me in the hall, my hands gathered in hers, her long face serious, eyes shining.\n\n'You've drunk too much. We've both drunk too much,' I said. 'Anyway, you've got to get to London, start your new job.' I took a deep breath. 'Thanks, though.' I bent forward, kissed her nose.\n\nShe put her head down. I sat back in the seat again and gazed over her head at the white-painted wallpaper on the far side of the hallway.\n\nShe looked up into my eyes.\n\n'What happened, Prentice?'\n\nI shrugged. 'Crazy,' I said, my gaze sliding away from those sternly concerned eyes, to look at the worn hall carpet and an old red wine stain from a party two years ago. 'Just crazy.'\n\nAsh patted my hands. 'I'll take you down in the morning, then. I can get them to hold the job. There was no rush. Only if you want.'\n\n'I don't know,' I said, and really didn't. I bent forward, put my head between my knees and stared at the black-taped edge of the carpet under the seat and the rough floorboards beyond. I felt Ash stroke my head, her hands soft and gentle through my hair.\n\nI didn't want to go to bed and anyway could not have slept. She stayed up with me, and we finished the real coffee and then the instant. I talked about the family, about Rory and Fiona and mum and dad. Thunder rolled over the city, just before sunrise, and I found myself laughing, sitting there on the couch, in the living room with Ashley; laughing at the thunder. She held me, shushed me.\n\nThe dawn came up dull at first, then the clouds cleared from the west and a bright blue day was there. Ashley left a note for Gav and Janice, helped me pack a bag \u2013 I couldn't decide on anything \u2013 then we left. The old 2CV, freshly pillar-box red after its latest re-spray, puttered through the near-empty streets of the bright and silent city, and rocked and rolled its way back down towards Gallanach.\n\nThe weather was perfect, the new day glorious. I talked incessantly and Ash listened, sometimes smiled and seemed always to have a kind word.\n\nWe arrived at Lochgair about breakfast-time, with the sun shining through the trees and the birds loud in the garden. Ashley stopped the car at the opened gates at the end of the drive where it entered the courtyard. 'I'll drop you here, okay?' she said.\n\n'Oh, come in,' I told her.\n\nShe shook her head, yawned. Her long fawn hair shone in a beam of sunlight coming through the car's open side window. 'I don't think so, Prentice. I'll get home, get some sleep. Give me a call if there's anything I can do, okay?'\n\nI nodded. 'Okay.'\n\n'Promise?' She smiled.\n\n'Promise,' I said.\n\nShe leaned over, put one hand behind my head and kissed my forehead. I heard her take a breath, like she was about to speak, but then she exhaled, just patted my head. I put one arm round her, held her for a moment, then pulled away, reached into the back and got my bag, opened the door and got out. 'Thanks,' I said.\n\n'It's okay, Prentice,' she said.\n\nI closed the flimsy door. The car revved up and turned round, one skinny front wheel poking out alarmingly from its wheel arch. The little Citro\u00ebn clattered off down the drive. Ashley stuck one hand out of the window and waved; I raised my arm, and held it there as I watched the car head away under the trees through the dappling light. It paused at the main road, then turned away, its noise soon lost in the background of bird-song and wind-ruffled leaves.\n\nThe cool morning air smelled clean and fresh; I took a deep breath and rubbed my smarting eyes, feeling spaced-out from lack of sleep.\n\nThen I picked up my bag and turned to the house.\n\nIt was a well-travelled country, dad told us. Within the oceanic depths of time that lay beneath the surface of the present, there had been an age when, appropriately, an entire ocean had separated the rocks that would one day be called Scotland from the rocks that would one day be called England and Wales. That first union came half a billion years ago. Some of those rocks were ancient even then; two billion years and counting, and shifting and moving across the face of the planet while that primaeval ocean shrank and closed and all that would become the British Isles still lay south of the equator. Compressed and folded, the rocks that would be Scotland \u2013 by then part of the continent of Euramerica \u2013 held within their crumpled, tortuously layered cores the future shape of the land.\n\nBy a third of a billion years ago, that part of Euramerica lay on the equator, covered by great fern forests that would be buried and folded and pressed and heated and so turn to oil and coal, in the future that was yet to come. Meanwhile the mass of rocks, afloat on the molten stone beneath, were heading slowly northwards, and sundering; the climate became hot and the rains sparse; the great dinosaurs, tree-tall and house-heavy, tramped slowly through a semi-desert while a new ocean opened to the west. After the dinosaurs had gone, and while the Atlantic still grew, the volcanoes erupted, smothering the old rock on the surface under their own vast, deep oceans of lava.\n\nThe land then held mountains higher than Everest, but they were worn down eventually by nothing harder than wind and water, until, much later still \u2013 now that Scotland was level with Canada and Siberia and the earth cooler \u2013 the glaciers came, covering the rocks with their own chill, inverted image of the old and weathered lava plains. The sheer mass of that frozen water etched the mountain rock like steel engraving glass, and pressed the roots of those fire-floating hills deeper into the dense sea of magma beneath.\n\nThen the climate changed again; the glaciers retreated and the water they had held filled the oceans, so that the waters rose and cut what would eventually be called the British Isles off from main-land Europe, while the scoured, abraded hills to the north, set free at last from that compressing weight of ice, rose slowly back out of the earth, to be colonised again by plants and animals, and people.\n\nOn walks, on day trips and holidays, he found and pointed out the signs that told of the past, deciphering the symbols written into the fabric of the land. In Gallanach, we saw the bright seam of white cretaceous sandstone that had provided the Gallanach Glass Works with raw material for a century and a half. On Arran, he showed us rocks folded like toffee, ribboned and split; on Staffa, the even, keyboard-regular columns of cooled lava; in Edinburgh, the rubble-tailed stumps of ancient volcanoes; in Glasgow, the black, petrified remains of trees three hundred million years old; in Lochaber, the parallel roads that marked the shores of lochs dammed and un-dammed by glaciers, millennia earlier; throughout Scotland we saw hanging valleys, drumlins and corries; and in the Hebrides we walked the raised beaches where the ocean swells had crashed until the land rose, and touched rocks two and a half billion years old; half as old as Earth itself; a sixth of the age of the entire universe.\n\nHere was magic, I remember thinking, as we drove north towards Benbecula one day, looking out at the machair, gaudy with flowers. I was just old enough to grasp what dad had been telling us, but still young enough to have to think about it in childish terms. Magic. Time was Magic; and geology. Physics, chemistry; all the big, important words dad used. They were all Magic.\n\nI sat listening to the car's engine, as we drove; mum at the wheel, dad in the passenger seat, shirt-sleeved arm out of the Volvo's window, Lewis, James and I in the back.\n\nThe car engine made a steady growling noise, and I remember thinking it was funny that those long-dead plants had been turned into the oil that had been turned into the petrol that made the car growl. I chose to forget the absence of reptiles in those carboniferous forests, and imagined that they had been populated by great dinosaurs, and that they too had fallen into the ooze, and made up part of the oil, and that the noise the car made was like the angry, bellowing growls they would have made while they were alive, as though their last dying breath, their last sound on this planet, had been saved all these millions and millions of years, to be exhaled along a little road on a little island, pushing the McHoan family north, one summer, on our holidays.\n\nI looked out of the open window; the machair lay dazzling under the midsummer sunshine to our left.\n\n'Prentice! Prentice! Oh, Prentice; pray for your father!'\n\n'Hello, Uncle Hamish,' I said, as Aunt Tone ushered mother and me into the bedroom where my uncle lay, propped up, splendid but demented in a pair of blue cotton pyjamas and a red silk dressing-gown decorated with blue dragons. The room was behind dim closed curtains, and smelled of apples.\n\n'Mary! Oh, Mary,' Uncle Hamish said, seeing my mother. He clasped his hands together, holding a black handkerchief. His hair was a bit mussed and he had a stubble shadow; I'd never seen him look so disarrayed. In front of him there was a huge tray with short legs, partly covered by a quarter-completed jigsaw puzzle. I walked up to the bed and put my hand out. I clutched Uncle Hamish's still clasped hands, held them briefly, squeezed and let go.\n\nCloser inspection revealed that Hamish was putting the jigsaw puzzle together upside-down; every cardboard flake was grey, turned the wrong way up.\n\nMum gave Hamish a brief hug and we sat down on a couple of chairs on either side of the bed. 'I'll make some tea,' Aunt Tone said, and quietly closed the door.\n\n'And biscuits!' shouted Uncle Hamish at the closed door, and smiled broadly at first mum and then me. After a moment, though, his face seemed to collapse and he looked like he was about to weep.\n\nThe door opened again. 'What's that, my dear?' Aunt Tone asked.\n\n'Nothing,' Uncle Hamish said, the mouth-only smile suddenly there again, then fading just as quickly. The door closed. Hamish peered down at the jigsaw puzzle, toyed with a couple of the pieces, looking for a place to fit them into what he had already completed. The squint bottom edge of the puzzle, some small spaces between joined pieces, a few tiny flecks of cardboard \u2013 half grey, half coloured \u2013 gathered like dust along the raised edges of the tray, and a small pair of collapsible scissors lying on the bedspread near the pillows, indicated that Uncle Hamish had \u2013 not to put too fine a point on it \u2013 been cheating.\n\n'Thank you, both, for coming,' he said, absently, still fiddling with the grey pieces. He sounded bored, like he was talking to a couple of factory workers summoned to his office for some formality of business. 'I appreciate it.' I exchanged looks with my mother, who appeared close to tears again.\n\nMum had done pretty well till now; we'd both cried a bit when Ashley had deposited me at the gates of the house at Lochgair, but since then she had coped pretty well. We'd visited the good lawyer Blawke that first day, and the next day he'd actually made a house-call, a concession which, extrapolating from the attitude of his secretary when she rang us up to tell us the sacred presence was on his way, we ought to have treated with the sort of awe and respect the average person reserves for royalty and major religious figures. I was a little surprised he didn't kneel and kiss the door-step when he unfolded himself from his Merc.\n\nThe undertaker had been dealt with, a few reporters fended off, Lewis \u2013 in London \u2013 reassured that there was nothing he could do up here for now, and told not to cancel his gig dates, and James, on a school trip in Austria, finally contacted. He would arrive the day of the funeral; one of his teachers would come back with him.\n\nDad's study proved to be a wilderness of papers, disorganised files, chaotic filing cabinets, and an impressive-looking computer that neither mum nor I knew how to operate. The afternoon I got back mum and I had stood looking at the machine, knowing there might be stuff in it we'd need to look at, but unable to work out what to do with the damn thing after switching it on; the relevant manual had disappeared, mum had never touched a keyboard in her life and my computer expertise was confined to having a sound tactical sense of which alien to zap first and a leechlike grip on continuous-fire buttons.\n\n'I know just the person,' I said, and rang the Watts' house.\n\nTwenty-four hours before the funeral, Aunt Tone had rung and said could we possibly come and see Uncle Hamish? He'd asked to see us.\n\nAnd so here we were. Mum sat on the far side of the bed, her eyes bright.\n\nI cleared my throat. 'How are you, Uncle Hamish?' I asked.\n\nHe looked at my mother, as if he thought she'd talked, not me. He shrugged. 'Sorry to drag you out here,' he said. His voice was flat, emotionless. 'I just wanted to say how, how sorry I am, and I want you all to forgive me, even though I didn't ... didn't encourage him. He insisted. I told him not to do it.' He sighed and tried to press one of the cardboard pieces into place on the puzzle without success. 'We were both a little the worse for wear and,' he said. 'I did try. I tried to stop him, tried to talk to him, but ... but ...' He stopped talking, tutted in apparent exasperation and took up the little scissors. He trimmed a couple of finger-nail sized bits of cardboard off the piece and forced it into place. 'Don't make the damn things right any more,' he muttered.\n\nI began to wonder at the wisdom of leaving Uncle H with a pair of scissors, even small ones.\n\nHe looked at me. 'Headstrong,' he said brightly, then looked down at the puzzle. 'Always was. Good; liked him; brother after all, but ... there was no sense of God in him, was there?' Hamish looked at mum, then me. 'No sense of something greater than him, was there, Mary?' he said, turning back to mum. 'Proof all round us; goodness and power, but he wouldn't believe. I tried to tell him; saw the minister yesterday; told him he hadn't tried hard enough. He said he couldn't force people to go to church. I said, why not? Did in the old days. Why not?' Uncle Hamish took up another piece of grey cardboard, turned it this way and that. 'Good enough then, good enough now; that's what I told him. For their own good.' He grunted, looked displeased. 'Idiot told me not to blame myself,' he said, staring grimly at the puzzle-piece, as though trying to pare bits off it with just the sharpness of his stare. 'I said I don't, I blame God. Or Kenneth for ... for goading ... inciting Him.' Uncle Hamish started to cry, his bottom lip quivering like a child's.\n\n'There, Hamish,' mum said, reaching out and stroking one of his hands.\n\n'What exactly happened, Uncle Hamish?' I asked. Sounded to me like the man had cracked completely, but I still wanted to see if he could come up with more details.\n\n'Sorry,' sniffed Hamish, wiping his eyes then blowing his nose into the black hanky. He put the hanky in his breast pocket, clasped his hands on the edge of the tray holding the jigsaw, and lowered his head a little, seeming to address the centre of the puzzle. His thumbs started to circle each other, going round and round.\n\n'We had a few drinks; we'd met in the town. I'd been at the Steam Packet, meeting with some people. Showed them round the factory in the morning. Just paperweights. Man from Harrods. Nice lunch. Thought I'd look for a present for Antonia's birthday, bumped into Kenneth coming out of the stationer's. Went for a pint; bit like the old days, really.'\n\n'Here we are,' Aunt Antonia announced from the door, appearing with a tray full of crockery. There was a pause while tea was poured, biscuits dispensed. 'Shall I stay here, dear?' Aunt Tone asked Hamish.\n\nI thought she looked worse than mum did. Her face was drawn, there were dark shadows under her eyes; even her brown, bunned hair looked greyer than I remembered.\n\nHer husband ignored her, talking on as before, though now having apparently shifted his attention to the cup of tea Aunt Tone had placed in front of him on the puzzle tray. His thumbs were still circling each other.\n\n'Went to the Argyll Lounge; good view of the harbour from there. Drank pints. It was like when we were younger. Had a cigar. Good chat, really. Rang the office, said I was playing truant. He rang Lochgair. We were going to go for a Chinese meal, just for old time's sake, but we never got round to it. Thought it would be fun to go on a bit of a pub-crawl, so we went on to the Gallery bar, in the Steam Packet. That was where we started talking about faith.'\n\nUncle Hamish stopped talking, took up his cup of tea, sipped quickly from it without raising his gaze from the tray, then replaced the cup in the saucer. 'He called me a crack-pot,' Hamish said. His eyebrows rose up his forehead; his voice rose too. Then it fell again as he said, 'I called him a fool.'\n\nHamish looked quickly, furtively, at my mother. 'Sorry,' he mumbled, and looked forward at the tray and the puzzle again. He sighed; his thumbs kept going round. 'I told him Christ loved him and he just laughed,' Hamish complained. 'He refused to see; he refused to understand. I told him he was like a blind man, like somebody who would not open their eyes; all he had to do was accept Christ into his life and suddenly everything would fall into place. The world would look a different place; a whole new plane of existence would open up. I explained that all we did here was merely a preparation for the next life, where we would be judged, punished and rewarded.' Hamish shook his head, face radiating dismay. 'He went all snide, asked me when exactly I'd had the brain by-pass operation.'\n\n(God \u2013 or whatever \u2013 help me; at that point, despite it all, I had to stifle a guffaw. I coughed, and dabbed at my suddenly brimming eyes with a tissue.)\n\nHamish rattled on. 'I told him that only religion gave any meaning to life; only God, as an absolute, gave us a ... peg to hang our philosophies on. What was the meaning of life, otherwise? He said, What meaning? He said, How long is a piece of string? and, What colour is the wind?' Uncle Hamish shook his head again. 'I told him faith was love, the most beautiful thing in the world. He said it was nonsense, surrendering our humanity. Humanity!' Hamish scoffed. 'Religion gives us rules; it can keep people from doing wrong; it helps us be good. But he wasn't having it, would not listen. \"Religion is politics,\" he told me, several times. As though repeating it made it true. \"Religion is politics! Religion is politics!\" Blasphemed. We left the last bar \u2013 can't even remember which one it was, to be honest \u2013 and we were walking back here, for a nightcap, I think, coming along Shore Road \u2013 I left the car in the Steam Packet Hotel car park, of course \u2013 and we had some argument about the Shore Street Church. He said he liked it, liked the architecture, but it was really a testament to the skill of humans, not to the glory of God, and just a symbol. I said it was the house of God, and he'd better not trespass.' Hamish looked up at mum for a moment. 'He was walking along the wall, you see.'\n\nMum nodded. Hamish was already staring at the tray again.\n\n'He said what was any church or temple but a giant, hollow idol? I told him he was sick; he said he was infected with reason. I said Reason was his God, and it was false; it was the true idol.' Hamish sighed. 'The street was wet; there had been rain. I remember noticing that ... Kenneth shouted at me, told me...' Hamish shook his head. '... he said; \"Hamish; all the gods are false. Faith itself is idolatry.\"'\n\nUncle Hamish swivelled his big, grey head and gazed gloomily at me. His eyes looked cold and jelly-like; they reminded me of frog-spawn discovered in some ditch. '\"All the gods are false. Faith itself is idolatry,\"' Uncle Hamish breathed, staring at me. I shivered. 'Can you credit that, Prentice?' He looked down, away from me, shaking his head.\n\nHamish returned his gaze to the puzzle tray. His thumbs kept circling. 'I can't remember exactly what he said,' Hamish whispered, and then sighed. 'But he jumped off the wall and ran over to the church. He started climbing.'\n\nI heard my mother sob once, very quietly.\n\n'I had to climb over the wall,' Hamish breathed, 'Gate was locked. By the time I got there he was out of reach. I thought he was shinning up a drainpipe. Just assumed. Heard rumbles, I think, but ... didn't think anything of it. No flashes, that I can remember. Kenneth was yelling and swearing and shouting imprecations; calling down all sorts of punishment; I was trying to get him to come down; told him he'd fall; told him the police were coming; told him to think of his family. But he kept climbing.'\n\nI studied my hands in the pink-tinged light, turning them over and looking at the lines on my palm, the veins on the back. I tried to imagine dad, climbing up that tower, hauling himself up, hand over hand, sweating and straining in the darkness, trusting to his own strength and the cool metal strip beneath his hands.\n\nThe block beneath me was silent now; the last of the waves had retreated from it and were breaking further down the beach as the tide went out. The sky was still gaudy with crimson clouds, though much of the brightness had gone. I glanced at my watch. I ought to be jumping down off this thing and heading back to the road; it was a rough hike over the headland, and dangerous in the dark. But the red streaks of the clouds were dissolving as the sunset went on, leaving the sky clear above me. This near the centre of the year, on a clear night, it would never get totally dark. I had a while yet, but I wouldn't leave it too late; mum would worry. That would just be the cherry on it, me taking the Crow Road too.\n\nUncle Hamish took another sip of his tea, frowned at the cup and spat the tea back into it. 'Cold,' he said apologetically to his wife. He dabbed at his lips with his handkerchief. I realised only then I hadn't touched the cup that Aunt Tone had poured for me.\n\nHamish went on: 'There was a very strange noise, a sort of humming noise seemed to come from under my feet, from the stones of the church. Couldn't work out what it was, thought it was the drink or just the effect of looking up like that, craning my neck. But it wouldn't go away, and it got louder and I felt my hair stand on end. I shouted up to Kenneth; he was about half-way up, still climbing. Then there was a flash, a blinding flash.\n\n'Saw a glowing red line in front of me, like a vein of burning blood, like lava, in front of me. Noise terrific. Smell of sulphur; something of that nature; smell of the devil, though I think that was just coincidence. Fell down. Half blind, thought a bomb had gone off. Heard ringing, like the church bells all going on at once.' Uncle Hamish went to sip from his tea again, then thought the better of it and put the cup back on the saucer. 'Realised it had been lightning. I still couldn't believe it; found Kenneth behind me, lying on the grass and a sort of slab thing, over a grave. Hands burned. Been climbing the lightning conductor, blew him off. Don't know if that would have killed him, but he'd landed on the stone. Dead. Blood from his head.' Hamish looked slowly over at mum, who was crying silently. 'Sorry,' he told her.\n\nShe didn't say anything.\n\n'Idiot,' I whispered, sitting there on Darren's great grey concrete block. 'Idiot,' I said, and for once I wasn't talking to myself. _'Idiot!'_ I shouted at the sky. 'IDIOT!' I bellowed, hands clawing at the pitted concrete surface beneath me. 'IDIOT!' I screamed, emptying my lungs to the soft sea airs. Coughing and choking, I sat there, tears in my eyes, breathing hard. Eventually I wiped my nose on my shirt sleeve, feeling like a little kid again, and then sniffed, swallowed, and breathed slower, clenching my teeth to stop my jaw trembling.\n\nI sat back, shivering, legs out straight in front, arms behind, hands splayed on the rough concrete. I thought about them all. Dad, falling; Grandma Margot, falling. Darren, broken against the tomb-white concrete of a council litter bin; Aunt Fiona, through the windscreen of the Aston Martin, neck snapped, into the young trees by the roadside ... and who knew what had happened to Rory? Well, in a day or two I was going to start trying to find out. So far mum and I \u2013 with Ashley's help \u2013 had only dealt with the papers and files we had to, to deal with the legal formalities. But there was a lot more stuff to go through, and somewhere in all that bumf there might be something that would tell us about Uncle Rory, and why dad had always been so sure his brother was still alive.\n\nBut for all we knew he'd died a roadside death, too.\n\nUncle Hamish turned to me. 'Swear he was still alive.' He nodded, frowning at me. I raised my eyebrows, feeling very cold inside. Hamish nodded again. 'Still alive; he said something to me. I swear Kenneth said, \"See?\"' Hamish shook his head. 'Said that to me; said, \"See?\" without opening his eyes.' He looked down at his rotating thumbs. His frown seemed to stop them. 'That was what he said; and it was so ... wrong; such a silly, silly thing to say, that I thought I must have only thought I heard it, but I'm sure, that's what he said. \"See?\"' Uncle Hamish shook his head. '\"See?\"' He kept shaking his head. '\"See?\"' He turned to me. 'Can you credit that, Prentice?'\n\nHe looked away again before I could think of what to say. '\"See?\"' he repeated to the tray with the ruined puzzle, and shook his head again. '\"See?\"'\n\n'Excuse me.' Mum got up and left the room, crying.\n\nHamish stared at the cardboard puzzle. Aunt Antonia sat at the end of the bed, staring hollow-eyed at her silent husband. The tray over Uncle Hamish's legs started to vibrate. I could see the duvet over Uncle Hamish's thighs shaking. The bed began to squeak. My uncle stared, appalled, at the tray on his lap, as the little grey pieces of the up-turned puzzle migrated across the vibrating surface of the tray, gradually collecting against one edge.\n\nThe spasms in Uncle Hamish's legs seemed to grow more severe; the cup of tea I'd put on the bedside table near my right elbow showed a concentric pattern of standing ripples. I suddenly thought of the scene in The Unbelievable _Prevalence_ of Bonking, when the tanks enter Prague. Uncle Hamish made a strange keening noise; Aunt Tone patted his feet under the duvet and rose from the end of the bed.\n\n'I'll get your pills, dear.'\n\nShe left the room. Hamish turned to me, his whole body shaking now, the puzzle on the tray starting to break up as the tray bounced up and down beneath it. 'Jealous,' Hamish croaked through clenched teeth. 'Jealous, Prentice; jealous! Jealous! Jealous God! Jealous!'\n\nI got up slowly, patted his trembling hands and smiled.\n\nI've always had this fantasy that, after Uncle Rory borrowed his flat-mate Andy's motorbike and headed off into the sunset, he crashed somewhere, maybe coming down to Gallanach; came off the road and fell down some gully nobody's looked in for the last ten years, or \u2013 rather more likely, I suppose \u2013 crashed into the water, and there's a Suzuki 185 GT lying just under the waves of Loch Lomond, or Loch Long, or Loch Fyne, its rider somehow entangled in it, reduced by now to a skeleton in borrowed leathers, somewhere underwater, perhaps between here and Glasgow; and we all pass it every time we make the journey, maybe only a few tens of metres away from him, and very possibly will never know.\n\nI know that dad \u2013 who had indeed assumed that Rory had been on his way here \u2013 drove the Glasgow road a few times, immediately after Andy and then Janice raised the alarm, looking for some sign of an accident, a skid mark, a damaged fence or wall, always wondering if maybe his brother was lying unconscious or paralysed in a field or a ditch somewhere, invisible from the road ... But all he ever found were road cones, assorted litter and the occasional dead sheep or deer.\n\nWhatever; neither dad nor the police ever found any trace of Rory or the bike. No unidentified bodies turned up that could have been his, and no hospitals received any unknown coma victims fitting his description.\n\nI don't think any of us ever mentioned suicide, but I at least considered the possibility that he had killed himself. Rory had been depressed, after all; his one success had been a travel book written a decade earlier, and everything else he'd tried since had failed to live up to that; he had recently failed to become a TV presenter \u2013 a job he'd thought beneath him but which he needed for the money (and so had been all the more galled when he hadn't been chosen) \u2013 and maybe, too, he'd finally admitted to himself he was never going to write his magnum opus ...\n\nHell, his life just wasn't going anywhere special; people kill themselves for poorer reasons.\n\nI reckoned the chances of him being under the waves somewhere improved significantly if he had committed suicide; he could have picked his spot to drive straight at a wall or a crash barrier, maybe on top of a cliff. Could be anywhere. I could think of a few places, further north in the Highlands, which would be perfect. If he'd tied himself to the bike somehow ...\n\nBut why go to the effort of doing that in the first place? It wasn't as though there was some big insurance sum involved, or any funny business with wills or family money. Rory had inherited some capital when grandad died, held in trust until he was eighteen; he'd used that up travelling round India the first time, then lived off the success of Traps and \u2013 later \u2013 the declining advances and journalistic commissions he'd received after that. When he'd disappeared he'd had a small overdraft.\n\nMaybe he'd been murdered. I'd thought of that years ago, even on the evening we'd heard he was missing. I had been playing down on the shore of Loch Gair with Helen and Diana Urvill, and when we came back for our tea there was a police car in the courtyard of the house.\n\nA police car! I recall thinking, getting all excited.\n\nOf course, in my fantasy I was the one who discovered Rory's evil murderer and brought him to justice, or fought with him and watched him fall off a cliff or into a combine harvester or under a steam-roller or whatever.\n\nOnly I couldn't see that anybody had had much of a motive. It had crossed my mind that it might have something to do with _Crow Road;_ somebody wanted to steal the idea and keep Rory out of the way, but it wasn't even as though there was much to steal. Notes and poems; wow.\n\nI stood up on the silent concrete block and dusted my hands off. The disappearing clouds were the colour of dried blood in a sky gone close to purple. More stars were coming out. A contrail blazed pink overhead, as a plane headed for America. I looked at my watch; I had to go. I'd told mum I'd be back for supper in an hour or so. We were expecting Lewis and Verity that evening; they were flying up from London, where Lewis had been working, and they would hire a car at Glasgow. They might be back when I returned.\n\n'Shouldn't have mentioned you,' Uncle Hamish said, as I walked to the door of the dim bedroom. I turned back. He was still trembling. It hurt me to look at him, the way it hurts to hear nails scraped down a blackboard. 'Shouldn't have said anything about you, Prentice,' he said, the words whistling out between his clenched teeth. I could hear Aunt Tone's footsteps coming up the stairs in the hall outside. 'Shouldn't have said, Prentice; shouldn't have said.'\n\n'Said what, uncle?' I said, hand on the door knob.\n\n'That you were closer to me; that I'd won you, saved you from his heathen faith!' Uncle Hamish's eyes stared at me from a shaking, ash-grey face.\n\nI nodded and smiled at him. 'Oh well,' I said. The door opened and I got out of the way of Aunt Tone, bearing pills and a glass of water. 'See you tomorrow, Prentice,' she whispered to me. She patted my arm. 'Thank you.'\n\n'It's all right. See you tomorrow, Aunt Tone.'\n\nOutside, on the landing, I looked down the stairs to where my mother was standing by the front door, putting on her jacket. I leant back against the closed bedroom door for just a second, and \u2013 looking at nothing in particular \u2013 said very quietly to myself, 'See?'\n\nI went to the land-side edge of the concrete cube, and faced back at the remains of the sunset, trying to work out how I was going to feel seeing Lewis and Verity again, after the way I'd behaved at New Year. But search as I tried, I could find no trace of dread or jealousy; I was even looking forward to seeing them again. Something of the coldness that had settled over me in the last few days seemed to have spread to the way I felt about Verity. It felt like all my jealous passion had dissipated like the clouds overhead.\n\nI thought about jumping down onto the beach, but that might have been asking for another family tragedy, so I climbed down, walked to the end of the shallow scoop of bay and set off through the grass by the side of the burn, heading back to Gallanach through the calm summer gloaming.\n\n... He told us about the plants on the islands, too; how the open, glorious machair, between the dunes and the farmed land, was so dizzily sumptuous with flowers because it was the place where the acidic peat and the alkali sands produced a neutral ground where more plants could flourish in the sunlight. And just the names of those plants were a delight, almost a litany; marsh samphire, procumbent pearlwort, sand-spurrey, autumnal hawkbit, cathartic flax, kidney vetch, germander speedwell, hastate orache, sea spleenwort; eyebright.\n\nWe learned about the people who had made Scotland their home: the hunter-gatherers of eight or nine thousand years ago, nomads wandering the single great wood and stalking deer, or camping by the edge of the sea and leaving only piles of shells for us to find; the first farmers, just beginning to clear the land of the blanket of thick forest a few millennia later; the neolithic people who had built the tomb of Maes Howe before the pyramids were constructed, and the stone circle at Callanish before Stonehenge, in the thousand-year summer of the third millenium; then came the Bronze Age and Iron Age people, the Vikings and Picts, Romans and Celts and Scots and Angles and Saxons who had all found their way to this oceanically marginal little corner of northern Europe, and left on the place their own marks; the treeless slopes themselves, the roads and walls, cairns and forts, tombs, standing stones, souterrains, crannogs and farms and houses and churches; and the oil refineries, nuclear power stations and missile ranges, too.\n\nHe made up stories, about the secret mountain, and the sand-drowned forest, the flood that turned to wood, the zombie peat, and the stone-beings that drilled for air. Sometimes the location for, or the subject of, a story would have some basis in fact; the secret mountain was a real hill on which grew a flower that grew nowhere else in the world. There had indeed been great storms that had moved whole ranges of sand dunes inland, drowning forests, and villages ... And peat _was_ un-dead, the surrounding rocks' acidity, the chill Atlantic airs and ever-likely rain conspiring to prevent the corpses of the dead plants from decomposing.\n\nOther stories were pure fantasy, the result of a kind of child-like quality in him, I think. If you.looked at certain stands of trees from a distance, especially in a glen, and when in full leaf, they did look like great bulging torrents of green water, bursting from the depths of the earth and somehow frozen. There was a sort of visual naivety at work there that verged on the hallucinogenic, but it did, I'd argue, make a warped sort of visual sense. Magmites \u2013 the people who lived in the mantle of the earth, beneath the crust, and who were drilling up for air the way we were drilling down for oil \u2013 must just have appealed to that part of him that loved turning things around. Opposites and images fascinated him, excited him; magicked inspired absurdity from him.\n\nI think Uncle Rory would have given almost anything to have tapped the lush gravidity of that source as well.\n\nTelling us straight or through his stories, my father taught us that there was, generally, a fire at the core of things, and that change was the only constant, and that we \u2013 like everybody else \u2013 were both the most important people in the universe, and utterly without significance, depending, and that individuals mattered before their institutions, and that people were people, much the same everywhere, and when they appeared to do things that were stupid or evil, often you hadn't been told the whole story, but that sometimes people did behave badly, usually because some idea had taken hold of them and given them an excuse to regard other people as expendable (or bad), and that was part of who we were too, as a species, and it wasn't always possible to know that you were right and they were wrong, but the important thing was to keep trying to find out, and always to face the truth. Because truth mattered.\n\nI suppose we all want to pass on our beliefs; they seem even more our own than the genes we transmit ... but maybe they are largely inherited too, even if sometimes what you inherit is the exact opposite \u2013 the reversed image of what was intended.\n\nSometimes I felt he was trying to brain-wash us; that he wanted us to be images of himself, thinking the way he thought, doing what he would have done, as if that would help him cheat death, make him less mortal somehow. Then all his parables and laws seemed like.megalomania, and his reasoned certainties like dogma.\n\nOther times he seemed genuinely altruistic, and on occasion I thought I could sense something like desperation in him, trying so hard to equip us as best he could for the vicissitudes of life, while the world changed all around us so fast that some of his ideas and theories \u2013 which had seemed so important to him in his life, and so crucial for us to know in turn \u2013 became irrelevant; were proved wrong, or just shown to be not so important after all.\n\nMy mother was different, and always had been. I don't think she ever really laid down the law like that, not even once; she just got on with things. We knew we were loved, and we knew when something we'd done was disapproved of, but she trained us by example, and let us make mistakes. The only idea I think she could ever be accused of trying to put into our heads was the welcome realisation that whatever happened to us, she'd be there.\n\nI'm not sure that it wasn't the more effective method in the end, and \u2013 in its own way \u2013 more confident, too.\n\nHalf an hour after I'd left Darren's post-post-modernist concrete block I stood in the dusk light beneath the dun on the hill of Bac Chrom, within sight of the track at last, the lights of Slockavullin village beneath me, the eastern edge of Gallanach a thin grid of orange sparks to my right, the main road to Oban and the north busy with lights of white and orange and red, and the dark landscape below full of soft undulations, littered with chambered cairns, cup and ring marked rocks, standing stones, tumuli and ancient forts.\n\nAll the gods are false, I thought. Faith itself is idolatry.\n\nI looked into that ancient, cluttered darkness, wondering.\nCHAPTER 14\n\nSo anyway, bro, how are things?'\n\nLewis shook his head slowly and deliberately. He held up his whisky glass and studied it from close range, focusing with explicit care, one eye at a time. I formed the impression he was attempting to fix the tumbler's image in his memory so he'd know the identity of the receptacle to blame come the following morning. I was so drunk at the time this actually seemed like quite a smart idea, and I would probably have attempted to do the same thing myself if I'd thought I was remotely capable of coordinating my hand, eyes and brain to that degree. The only reason I could get my drinking hand and my mouth in roughly the same place at approximately the same time at this stage in the evening was because I'd had so much recent practice at it. And even that comparatively simple system wasn't a hundred per cent any more; I'd missed my mouth twice already and spilled small amounts of whisky onto my chin and shirt. I'd carried it off with dignity, though.\n\nLewis looked like he was going to sleep. Either that or the superior intellect of the whisky glass had hypnotised him. I knew the problem.\n\n'Lewis?' I said.\n\n'Wha \u2013 what?' he looked at me, confused.\n\n'I was saying,' I said. 'How are things?'\n\n'Oh,' he said, and sighed. 'I don't know.' He frowned. 'Verity said to me just yesterday ... she said, \"Lewis, I don't think we understand each other any more.\"'\n\n'What did you say?' I sipped my whisky carefully.\n\n'I said, \"Whadaya mean?\",' Lewis snarled.\n\nThen he burst out laughing. It must have been infectious laughter because I started laughing too, and then we were both laughing, but we couldn't have been that drunk because we didn't over-do it. Five minutes later \u2013 well, maybe ten, absolute max \u2013 we' d stopped laughing almost entirely.\n\n'Really?' I said, wiping my eyes.\n\nLewis shook his head. 'Na, course not. Everything's ... was pure dead brilliant, actually.'\n\n'Good,' I said, and drank. I meant it too, but even as I realised that I meant it, I thought: ah, it's just the drink. I'll be worse in the morning. Still, I looked up at Lewis and said, 'I think I'm better.'\n\n'Better ...?' Lewis began, giggling.\n\n'Better than ... yesterday, Mr Creosote?' I started to laugh.\n\n'Better get a bucket \u2013' Lewis howled, but couldn't manage the rest of the line, because by then we were on the floor. I laughed until my ears hurt.\n\nI stood beneath the larches in the rain, holding an umbrella, wearing a kilt and feeling a little self-conscious. The stand of dripping trees had gone yellow and dropped their needles during the last few weeks, turning the ground beneath them a dully shining blond that seemed like a tinted mirror to the ash-bright expanse of overcast sky. I touched the plain black obelisk, slick and cold in the chill October rain. Behind me, the noise from the marquee \u2013 an increasing choir of chattering voices \u2013 was slowly drowning the patter of the drizzle as it fell through the twigs and branches above onto the sodden ground; a busy, buzzy, shared excitement displacing what the solitary soul perceived as a sort of tranquil gloom.\n\nWhat guy? I thought. What is Ash going to show me; who? (And already thought I might have guessed.) Shit, I didn't like the sound of this.\n\nThe rain came on harder and I listened to it drumming on the taut black skin of the umbrella, remembering remembering.\n\n'Remember the River Game?'\n\n'Remember the _Black_ River Game?'\n\n'Ha!'\n\nWe were digging dad's grave, waist deep in the rich black earth of Lochgair, partially shielded from the house by the dense mass of rhodie bushes and tall tangles of wild roses. Jimmy Turrock, the council workman sent from the municipal cemetery to dig the grave officially, and who'd been in the same class as Lewis at school, was sitting against the wheel of his miniature earth-mover, arms folded, head back, mouth open, snoring. That morning over breakfast, Lewis and I had decided we'd dig the hole ourselves. If nothing else, it would take our minds off our hangovers, which were industrial strength.\n\nThe River Game was something dad made up himself. He did it for Lewis and me. The first version was roughed out in a big sketch book, while he tinkered with the rules. When he was happy with it all, he got a big bit of white cardboard from a display company in Glasgow, drew out the playing surface, painted it, sprayed it with lacquer and edged the board with black tape. He'd bought various Lego packs and made the ships and the cargoes out of those. The rules were typed, the cards were printed on labels and they were stuck onto the back of ordinary playing cards. We were presented with the result as a sort of extra present to be shared between Lewis and me for Christmas 1981. James was still a bit young; he'd only have chewed the ships and choked on the cargoes.\n\nLewis \u2013 who had asked for and got a television for his room, and a new Walkman \u2013 had the good grace to express gratitude. I was still celebrating having finally worn down dad's resistance to having a computer in the house, and was therefore far too busy kicking pixel and re-staging the attack of the Imperial AT-ATs on the rebel snow trenches to be bothered sparing more than the most cursory glance at what was, when all was said and done, a lump of amateurishly painted cardboard, a handful of non-motorised and very basic Lego bits, a few adulterated cards and what looked suspiciously like an exam paper. 'Yeah; great, dad. Got any more PP9 batteries for this wee car? The one out your calculator didn't last long,' was about as enthusiastic as I got about the game for most of the festive period.\n\nLater, I deigned to play.\n\nThe River Game was based on trade; dad had wanted something that would distract us from all the war games Lewis and I played: soldiers, with our friends in the woods, battles with our toys, wars on friends' computers. He really wanted something non-capitalistic as well as non-military, but the River Game was going to be just his first effort; he would \u2013 he told us \u2013 be working on something much more right-on, once he had the time to spare. He'd see if we liked the River Game first.\n\nYou had two or three ships; you sailed them from a port on one side of the board to a port on the other side through what was either a big loch or lake choked with islands, or a piece of territory with an awful lot of waterways snaking through it, depending how you chose to look at it. You picked up cargo at the second port and sailed back. The cargo was worth a certain amount when you got back to your home port, and with the money you could buy more ships, configured for speed or capacity. There were at least half a dozen major routes from one port to the other, and basically, the shorter the route you took, the more hazardous it was; there were whirlpools, channels prone to rock falls, stretches of river where the sand-banks changed all the time, and so on. The weather had a chance to change every few moves, and how much the different types of cargo were worth depended on ... Oh, what your opponents had chosen to carry, what the weather was, whether the month had an 'r' in it; I can't remember it all.\n\nIt was quite a fun game, mildly addictive, with a reasonable balance of skill and luck, and Lewis and I eventually got quite a few of our friends playing it, but the truth is it improved dramatically when Lewis \u2013 with my help \u2013 drew up an extra set of rules which let you build _warships!_\n\nWe played that game for weeks before dad caught us at it in the conservatory, one rainy May day, and asked how come there were all these ships with funny-coloured cargoes clustered so close together and surrounded by wrecks where there were no hazards.\n\nOops.\n\nWe called it the Black River Game (Dad even objected to the title). He had been working on a new improved version of the original game that involved using some of the money to build railways across the board; you laid track, you built bridges, dug tunnels, coped with rock falls and marshes and recalcitrant land owners, and the first one to finish his or her railway was, in effect, the winner. But he stopped work on this sophistication when he found us acting out furiously destructive naval engagements on his painstakingly crafted board. He didn't take it away, though. I think for a while he was trying to develop another non-combative game that he'd defy us to turn martial, but it stayed at the development stage and never did see the light of day.\n\nI stopped digging for a moment, wiped some sweat from my brow with the hem of my T-shirt, which was lying on the ground at the head of the grave. I leant on my shovel, looking at Jimmy Turrock's up-ended face while he snored. Lewis stopped digging for a moment too, breathing hard.\n\nI said, 'We disappointed him, though, didn't we?'\n\nLewis shrugged. He took a handkerchief from his back pocket and wiped his face. 'Oh, Prentice, come on; boys will be boys. Dad knew that.'\n\n'Yeah, but he expected better of us.'\n\n'Dads always do, it's traditional. We turned out not too bad.'\n\n'Neither of us did as well as he expected at Uni,' I said. I'd told Lewis \u2013 though not my mother \u2013 that I was fairly certain I'd failed my finals.\n\n'Well, for a start, he didn't know about you,' Lewis said, scraping some earth off the blade of his spade. 'And he was smart enough to know degrees aren't everything. Come on, we're not in prison, we're not junkies and we're not Young Tories.' He waggled his eyebrows. 'It's no small achievement.'\n\n'I suppose,' I said, and started digging again. (Pity he'd mentioned prison; another thing I hadn't told Lewis about was that I'd been nicked for shop-lifting. Not that I'd be going to prison, but it's the thought that counts.)\n\nLewis kept on digging. 'We could have done worse,' he insisted.\n\n'We could have done better,' I said, shovelling another load of earth out of the pit.\n\nLewis was silent for a while, then said, quietly, 'Better than ... yesterday?'\n\nI laughed in spite of myself (and in spite of the grave, and my aching head and still bruised heart). 'Shut up,' I said, 'please.'\n\nLewis shut up. I encountered another rhodie root and attacked it with the hacksaw, then took up the spade again, blinking sweat out of my eyes and waving a couple of flies away.\n\nLewis muttered almost inaudibly as he dug; 'It's only _waffer_ thin ...'\n\nWe snorted and guffawed for a while, then took a break for yet more Irn-Bru, sitting at the edge of the grave, legs dangling into it, with Jimmy Turrock still blissfully \u2013 and vocally \u2013 in the land of nod across the grave in front of us.\n\nI drank deeply from the bottle, passed it to Lewis. He finished it, grimaced, looked at the bottle. 'You know, I've finally realised what this stuff reminds me of,' he said, and belched heavily. I followed suit, trumping his sonorous burp with one that disturbed a few drowsy crows from nearby trees and even had Jimmy Turrock stir in his sleep.\n\n'What?' I said.\n\n'Chewing gum,' Lewis said, screwing the cap back on the bottle and chucking it into the grass near the council earth-digger.\n\nI nodded wisely. 'Yeah, right enough.'\n\nWe sat there, silent for a while. I looked at Jimmy Turrock's spotty, open-mouthed face and his wispy red hair. His snores sounded like somebody forever trying to start a badly-tuned buzz-saw. I listened to it for a while, and watched a couple of flies buzzing around in a tight but complicated holding pattern in front of his mouth, as though daring each other to be the first to investigate inside. After a while they broke off, though, and settled for exploring the rough landscape of Jimmy's checked shirt. My head hurt. Come to that, almost everything hurt. Ah well, self-inflicted wounds.\n\nJimmy Turrock snored on, oblivious.\n\nLewis and Verity had arrived the night before, an hour after I'd got back from my sunset visit to Darren's sea-side sculpture. Their plane had been late and they'd had problems hiring the car, so they arrived nearly two hours later than we'd expected. Rather than phone from the airport, Lewis had hired a mobile along with the car but then when they'd tried to use it, it hadn't worked. The upshot was that mum and I had been getting into a fine panic, and I'd been dreading watching the news: '... and we're just getting reports of an incident at Glasgow airport ... details still coming in ...'\n\nI mean, statistics tell you family tragedies oughtn't to come in quite such close succession, but Jeez; it gets to you, when somebody dies as unexpectedly as dad. Suddenly everybody you know seems vulnerable, and you fear for them all. Every phone-call sends your heart racing, every car journey anyone takes you want to say, Oh God be careful don't go above second gear have you thought of fitting air-bags is your journey really necessary be careful be careful be careful ... So there we were; mum and I sitting watching the television, on the couch together, side by side, holding hands tightly without even realising it and watching the television but not taking in what we were watching, and dreading the sound of the phone and waiting waiting waiting for the sound of a car coming up the drive.\n\nUntil I heard it, and leapt over the couch and hauled open the curtains and the car drew up and Lewis waved at me as he got out and I whooped, 'It's them!' to my mum, who smiled and relaxed and looked suddenly beautiful again.\n\nThere was a big three-cornered hug in the hall; then mum saw Verity standing by the door, taking a very long deliberate time to take her jacket off and hang it up; and so she was brought into the scrum too, and that was the first time, I realised, that I'd ever actually embraced her, even if it was just one arm round her slim shoulders. It was all right.\n\nThen the phone rang. Mum and I jumped.\n\nI got it. Mum took Lewis and Verity into the lounge.\n\n'Hello?'\n\n'Hello!' shouted a voice of immodestly robust proportions. 'To whom am I talking?' the booming voice demanded. It was Aunt Ilsa. We'd left a message at the only contact address we had for her, two days earlier. She was in Ladakh, a place so out of the way it would take several international airports, a major rail terminus and substantial investment in a network of eight-lane highways to promote it to the status of being in the middle of nowhere.\n\n'It's Prentice, Aunt Ilsa.' There was a satellite delay. I was talking to what I suspected was the only satellite ground station between Islamabad and Ulaan Baatar. There was a lot of noise in the background; it sounded like people shouting, and a mule or something.\n\n'Hello there, Prentice,' Aunt Ilsa bellowed. 'How are you? Why did you want me to call?' Perhaps, I thought, she'd been taking steroids and they'd all gone to her vocal chords.\n\n'I'm ... there's \u2013'\n\n'\u2013 ello?'\n\n'\u2013 some bad news, I'm afraid.'\n\n'What? You'll have to speak up, my dear; the hotelier is proving refractory.'\n\n'It's dad,' I said, thinking I might as well get this over with as quickly as possible. 'Kenneth; your brother. I'm afraid he's dead. He died three days ago.'\n\n'Good God! What on earth happened?' Aunt Ilsa rumbled. I could hear shouting. The thing that sounded like a mule went into what appeared to be a fit of coughing. 'Mr Gibbon!' roared Aunt Ilsa. 'Will you control that fellow!'\n\n'He was struck by lightning,' I said.\n\n'Lightning?' Aunt Ilsa thundered.\n\n'Yes.'\n\n'Good God. Where was he? Was he on a boat? Or \u2013'\n\n'He was \u2013'\n\n' \u2013golf course? Mr ... hello? Mr Gibbon had a friend once who was struck by lightning on a golf course, in Marbella. Right at the top of his back-swing. Bu \u2013'\n\n'No; he was \u2013'\n\n' \u2013course it was an iron.'\n\n' \u2013climbing,' I said.\n\n' \u2013number seven, I think. What?'\n\n'He was climbing,' I shouted. I could hear what sounded like a fight going on at the other end of the phone. 'Climbing a church.'\n\n'A _church?'_ Aunt Ilsa demanded.\n\n'I'm afraid so. Listen, Aunt Ilsa \u2013'\n\n'But he wouldn't be seen dead near a church!'\n\nI bared my teeth at the phone and growled. My aunt, the unconscious humorist.\n\n'I'm afraid that's what happened,' I said as evenly as I could. 'The funeral is tomorrow. I don't suppose you can make it, can you?'\n\nThere was a noise of some Ladakhian confusion for a while, then, fortissimo; 'I'll have to leave you now, Lewis \u2013'\n\n'Prentice,' I breathed through gritted teeth.\n\n' \u2013Our yak has escaped. Tell your mother our thoughts are with her at \u2013'\n\nAnd it was goodbye downlink.\n\nI looked at the phone. 'I'm not sure you have any to spare, aunt,' I said, and put the phone down with a feeling of relief.\n\n'I need a drink,' I said to myself. I strode purposefully towards the lounge.\n\nLewis had been marginally more sensible than me, later on, that night before the funeral; he'd gone to bed one whisky before I had, leaving me in the lounge alone, at about three in the morning.\n\nI should have gone then too, but I didn't, so I was left to get morose and self-pitying, re-living another evening in this room, another whisky-connected two-some over a year earlier.\n\n'But it's not _fair!'_\n\n'Prentice, \u2013'\n\n'And don't tell me life isn't fair!'\n\n'Aw, _think,_ son,' dad said, sitting forward in his seat, clutching his glass with both hands. His eyes fixed on mine; I looked down, glaring at his reflection on the glass-topped coffee table between us. 'Fairness is something we made up,' he said. 'It's an idea. The universe isn't fair or unfair; it works by mathematics, physics, chemistry, biochemistry ... Things happen; it takes a mind to come along and call them fair or not.'\n\n'And that's it, is it?' I said bitterly. 'He just dies and there's nothing else?' I could feel myself quivering with emotion. I was trying hard not to cry.\n\n'There's whatever he left behind; art, in Darren's case. That's more than most get. And there's how people remember him. And there might have been children \u2013'\n\n'Not very likely in Darren's case, was it?' I sneered, grabbing at any opportunity to score even the smallest rhetorical point over my father.\n\nDad shrugged, staring into his whisky. 'Even so.' He drank, looked at me over the top of the tumbler. 'But the rest,' he said, 'is just cells, molecules, atoms. Once the electricity, the chemistry, stops working in your brain, that's it; no more. You're history.'\n\n'That's defeatist! That's small-minded!'\n\nHe shook his head. 'No. What you're proposing is,' he said, slurring his words a little. He pointed one finger at me. 'You're too frightened to admit how big everything else is, what the scales of the universe are, compared to ours; distance and time. You can't accept that individually, we're microscopic; here for an eye-blink. Might be heading for better things, but no guarantees. Trouble is, people can't believe they're not the centre of things, so they come up with all these pathetic stories about God and life after death and life before birth, but that's cowardice. Sheer cowardice. And because it's the product of cowardice, it promotes it; \"The Lord is my shepherd\". Thanks a fucking lot. So we've to live like sheep. Cowardice and cruelty. But everything's okay, because we're doing the Lord's work. Fuck the silicosis, get down that mine and work, nigger; Aw shucks; sure we skinned her alive and threw her in the salt pans, but we were only doing it to save her soul. Lordy lordy, gimme that old time religion and original sin. Another baby for perdition ... Shit; original sin? What sick fuckwit thought that one up?'\n\nDad drained his glass and put it down on the glass-topped table between us. 'Feel sorry for yourself because your friend's dead if you want, Prentice,' he said, suddenly calm and sober. 'But don't try to dignify it with what's supposed to be metaphysical angst; it's also known as superstitious shit, and you weren't brought up to speak that language.'\n\n'Well, thanks for the fucking censorship, dad!' I yelled. I jumped up and slammed my own glass down. The table top cracked; a single big flaw crossed, deep and green and not quite straight, like a dull ribbon of silk somehow suddenly embedded in the thick glass, from one edge of the table to the other, almost underneath our tumblers.\n\nDad stared at it then snorted, chuckling. 'Hey, yeah! A symbol.' He shook his head, glum, muttering as he sat back: 'Hate the fuckers.'\n\nI hesitated, looking at the cracked glass, instinct \u2013 or training \u2013 telling me to apologise, but then did what I'd intended to do, and set about storming out of the room.\n\n'Just fuck off, dad,' I said before I slammed the door.\n\nHe looked up, pursed his lips and nodded, as though I'd asked him to remember and put the lights out before he went to bed. 'Yup; okay.' He waved one hand. 'Night.'\n\nI lay in bed seething, thinking of all the smart things I should have said, until I fell into a troubled sleep. I woke early and left before anybody else was up, driving my hangover back to Glasgow and shouting at caravans that got in my way, and that was the last meaningful, full and frank exchange of views with my dad that I ever had.\n\n'I wish he hadn't died right now,' I said. I didn't look at Lewis. I was still looking at Jimmy Turrock, asleep against the wheel of his council digger. 'I wish I could \u2013 I wish we could have started talking again.' One of the two flies exploring the cotton landscape of Jimmy's shirt suddenly buzzed up to his forehead. His snoring hesitated, then went on. 'It was so stupid.' I shook my head. 'I was so stupid.'\n\n'Yeah,' Lewis said after a bit. 'Well, that's just the way it is, Prentice. You weren't to know.' I heard Lewis sigh. 'There was something I wish I'd told him, too. Could have said, over the phone, end of last week.'\n\nI looked at Lewis. 'Oh yeah?'\n\nLewis looked awkward. He crossed his arms and sucked at his bottom lip. He glanced at me. 'Were you really that ... you know; keen on Verity? I mean; are you?'\n\nI kicked my heels against the sides of the grave, checked out a couple of tree roots we'd have to tackle before we could dig much deeper. I shrugged. 'Ah, it was just infatuation, I suppose. I mean, you know, I'll always like her, but... all that stuff at New Year ... that was ... well, partly the drink, but ... mostly just sibling rivalry; sibling jealousy,' I said. We both grinned. He still looked awkward. This time, instead of sucking his bottom lip, he bit his top one.\n\nI knew, just like that.\n\n'You are getting married,' I said, gulping.\n\nLewis looked at me with wide eyes. \u2013 'She's _pregnant?'_ I spluttered, contralto.\n\nLewis's mouth was hanging open. He shut it quickly. He wiped his face with the hanky; his eyebrows and eyes registered surprise.\n\n'Um, both,' he said. 'Almost certainly.' He wrung the hanky out over the hole, but it didn't drip (still, though, we would leave a fair amount of sweat in our father's grave).\n\nLewis nodded and his smile was flickering, uncertain. I hadn't seen him look so unsure of himself since the time when he was sixteen and I _almost_ had him convinced the Boxer Rebellion had been about underpants.\n\n'Fooof,' I said.\n\nSeemed as appropriate as anything. I stared over at Jimmy Turrock, blinking.\n\nLewis was making a clicking noise with his mouth. He cleared his throat. 'Wasn't exactly planned, to tell the truth, but... well; I mean, we both, you know; want it, so ... And, well, you know how I feel about marriage and all that stuff, but ... Fuck it, it just keeps things simple.'\n\nHe sounded almost apologetic.\n\nI shook my head and, turning to him with a big smile, I said, 'You total bastard.' I put my hands on my hips. He looked concerned, but I guess my grin must have looked sincere. 'You total, complete and utter bastard; I hate you,' I told him. 'But I hope you're disgustingly happy.' I hesitated, just a little, then I hugged him. 'Obscenely happy,' I said. Probably have cried but I was pretty cried out by that stage.\n\n'Man.' He breathed into my shoulder. 'I didn't know how you'd take it.'\n\n'In the neck,' I said, pushing him away. 'Told mum?'\n\n'Wanted to wait till after the funeral. Mind you, I was going to wait till then to tell you, too, so maybe Verity's spilling the beans right now.'\n\n'So when's the big event?'\n\n'Which one?' Lewis smiled; embarrassed, I do believe. He shrugged. 'We thought October, and the sprog thinks March.'\n\nI let out a long, shuddering sigh, head feeling a bit swimmy. 'Marriage, eh?' I said, shaking my head again. I looked him down and up, hoisted one brow. 'Think you'll take to it?'\n\nLewis grinned. 'Like a lemming to water.'\n\nI laughed. Eventually I laughed so loudly I woke Jimmy Turrock, who looked at me \u2013 sitting on the edge of my father's grave on the day of his burial, guffawing away fit to wake the living \u2013 with undisguised horror.\n\nLike a lemming to water. Lewis knew as well as I did the maligned little buggers are perfectly good swimmers.\n\nJames arrived back about mid-day. He was ... well, pretty distressed, and all the fragile defences mum, Lewis and I had been constructing for the past few days \u2013 Lewis and I joking, mum staying quiet and keeping busy \u2013 crumbled. James seemed to blame dad, blame us; blame everybody. He was ugly with anger and he was like a racing outboard in the calm little pond we'd been trying to create; the house felt hellish and we all started snapping at each other. Outside, at the back of the garden, we could hear the council digger, excavating the rest of the hole. The engine revved up and down; it sounded like a machine snoring. James wished us all dead and ran up to his room and slammed the door. It was a relief to get back out to the grave and help Jimmy Turrock apply the finishing touches.\n\nThen it was time to get showered and changed and wait for the hearse and the mourners. The funeral was suitably grim, despite the sunshine and the warm breeze. The words Lewis said over the grave sounded awkward and forced. Mum looked white as paper. James stood, mouth twisted, furious; he stalked off the instant the coffin touched the bottom of the grave. I threw some earth down onto the pale wood of the lid, putting back a little of what I'd helped dig out.\n\nBut it passed, and the people who came \u2013 a good hundred or more \u2013 were kind. We were busy in the house afterwards, feeding and watering them, and then that passed too.\n\nMy big brother and his intended asked me to be their best man the day after dad's funeral. I'd slept, fitfully, on the idea, but finally said yes. It had already been agreed between the two families that the wedding would be held at Lochgair. Lewis and Verity stayed another day after that, then left to go back to London so that Lewis could resume his gigs. He was almost ashamed when I saw him next, when he confessed that nobody thought his delivery had altered a bit; he was just the same on stage after dad's death as he had been before. The only thing he changed was that he stopped telling the joke about the uncle that dies in an avalanche on a dry ski-slope.\n\nI told him not to worry about it; you had to be a different person on stage. The person he was up there would only change if he told a story about dad dying. Maybe a routine based on the idea of an atheist getting struck by lightning while climbing a church tower would be therapeutic for him, one day.\n\nLewis had the decency to be appalled at the idea.\n\nMum and I went through dad's papers, and were able, after Ashley's tuition, to work the computer and access the information it held.\n\nDad's will, which had been written at the time of Grandma Margot's death, had turned up in the strongbox hidden under the study floorboards. The strongbox had been no big secret; we all knew about it. It was just something to make any burglar's job more difficult. Mum had already seen the will when she had opened the strongbox the morning after dad's death, in the company of one of her friends from the village. She had only looked at the first paragraph, which confirmed that dad wanted to be buried in the grounds of the house. She'd been too upset to look at any more of it, and had put the will back under the floor.\n\nSo we opened the strongbox again, divided the papers, took a desk each, and looked at what we had. Mum had given the pile with the will in it to me. I read it first, and my heart sank after I'd scanned quickly through it and got to the end.\n\n'Oh no,' I said.\n\n'What's wrong?' she asked from the main desk in front of the window.\n\n'It's the will,' I said, turning it over, looking at the last part again, looking over the page but still failing to find what I was looking for. 'It hasn't been witnessed or anything.'\n\nMum came over and stood behind me. She took the four hand-written sheets from me, frowning. Her skin was pale and her eyes looked dark. She wore black jeans and a dark blue shirt and her hair was tied back with a piece of blue ribbon. She handed the will back to me. 'I think it's all right,' she said slowly. She nodded. 'I'll call Blawke to make sure. He'll need to look at it anyway.' She nodded again, walked back to sit in her seat and started reading through the papers she had in front of her. Then she looked up at me. 'You phone him, would you?'\n\n'All right,' I said and watched her bend to the papers again. She appeared to read for a few moments; I almost wanted to laugh, she seemed so unconcerned. She looked up again after a few seconds and just sat there, looking out through the open velvet curtains at the back lawn.\n\nShe sat like that for a full two minutes, unmoving, face unreadable. I smiled; I wanted to weep, to laugh. Eventually I said softly, 'Mum?'\n\n'Hmm?' She turned to me, a hesitant smile on her thin face.\n\nI held the will up from where it lay on the desk. 'This is dad's will.' I managed a smile. 'Don't you want to know what it says?'\n\nShe looked confused, then embarrassed, and put her hand to her mouth. 'Oh, of course. Yes. What does it say? Let's see.'\n\nI pulled my seat over alongside hers.\n\nThe good lawyer Blawke opined that the will was perfectly legal; under Scottish law, a hand-written will did not have to be witnessed. He even came out and looked at it personally, which made two visits in one week. Truly our cup of honour ranneth over.\n\n'Yes,' the lawyer Blawke said, reading the will as he sat in the front lounge. 'Well, I can't see anything wrong with it.' He looked unhappy. 'Unarguably his writing ...'\n\nHe studied it again.\n\n'Yes,' he nodded, finally. 'I actually warned him against doing this, some time ago, but he seems to have got away with it.' The heron-like lawyer seemed sad that the will was litigation-proof. He smiled weakly, and mum offered to re-fill his whisky glass.\n\nThe will \u2013 expressed with a brevity and a lack of ambiguity the best lawyers would have been proud of, and the rest alarmed at \u2013 left the house, grounds and so on to mum, along with a two-fifths share in both the residue of dad's savings and any money made after his death. Lewis, James and I had one-fifth shares each. There were specified amounts to an almost archetypal spread of right-on causes: CND, Amnesty International and Greenpeace. Ten grand each. Ten grand! I was initially stunned, fleetingly annoyed, then ashamed, and later vaguely impressed. Mum just sighed, like she'd been expecting something like that.\n\nI confess to having experienced a sensation of relief on discovering I had not been written out of dad's will; I wouldn't have blamed him. I think and hope that that feeling was engendered more by a desire to feel I'd still been loved \u2013 despite everything \u2013 than by avarice. I didn't think there would be all that much to go round after those donations, anyway.\n\nDad's agent, his accountant and the lawyer Blawke worked it out between them (though I checked their figures later). The good lawyer summoned us all to his office a fortnight after dad's death. Only James wouldn't come. Lewis flew up specially.\n\nIt had all, indeed, been just about as simple as it had looked. Blawke told us the sums involved and I was pleasantly surprised. The donations to right-onnery seemed much more in proportion now; I can only claim that I had spent (what at least seemed like) so long living on bread and cottage cheese and fish suppers in Glasgow \u2013 measuring my money in pennies and reluctantly-parted-with pounds \u2013 that I had an excuse for not being able to imagine that the thirty K dad had salved his conscience with when he'd written the will had actually been quite a small part of the modest fortune he'd built up over the years.\n\nDad had left over a quarter of a million pounds, after the government had taken its cut.\n\nMy share came to well over forty thousand smackeroos. The likelihood was that for the next few years at least, I'd bank about fifteen grand per annum, which might or might not tail off, abruptly or gradually, depending on how well dad's stories held up against the tests of time ... not to mention the likes of Thomas the Tank Engine, the Teenage Mutant Ninja Turtles, and whatever other delights the future of the children's fiction market held.\n\nAnyway, suddenly I was, if not quite within range of the mountains of Rich, certainly well into the foothills of Comfortable. It entirely made up for the discovery, a few days earlier, that the estimate I had made of my chances of passing my final exams had been considerably more accurate than any of the conclusions I had drawn in the course of them. I had distinguished myself by failing, a result the department prided itself on happening only rarely.\n\nMy initial reaction had been to cut my losses on the honours front and see if I could take an MA instead; a re-sit would mean a whole extra year at university. But that sentiment had only lasted for a day. In the turmoil of feelings and fortunes dad's death had produced, the prospect of another year's study, with the framework and time scale that would provide \u2013 especially if I applied myself, as I thought I would now be able to \u2013 seemed suddenly a relief rather than a chore.\n\nAt any rate I still had a little time to decide what to do, and the money would make the choice easier. A return to Glasgow need not now also mean a return to the joys of sharing a flat with Gav, Aunt Janice, and their sonically extrovert passions.\n\nWe stood, the three of us, mum, Lewis and I, on the pavement outside the Main Street offices of the lawyer Blawke, in Gallanach. I was still thinking, Forty grand? and trying not to look too stunned. Mum was slowly putting on her black leather gloves.\n\nLewis and I looked at each other.\n\nLewis wasn't doing too badly himself, down in London, but he too had looked pretty surprised when Blawke told us the sums that were heading our way. Mum hadn't really shown any reaction; she'd just thanked the good lawyer politely and asked after his wife and family.\n\n'Fancy a drink?' Lewis said to me. I nodded. I felt slightly faint. 'Mum?' Lewis said.\n\nShe looked round at him, small and neat in her dark blue coat. It was a bright, warm day and I could see the silver in her dark brown hair. She looked so delicate. I felt like I was in my early teens again, mum seemingly getting shorter and shorter with each season that passed.\n\n'What?' she said. I found myself sniffing the air; I was downwind, but all I could smell was Pear's soap and Lewis's Aramis. Mum seemed to have stopped wearing perfume.\n\n'I think a drink, to steady our nerves,' Lewis said to her.\n\n'Aye,' mum said, looking thoughtful. She gave a thin wee smile and nodded at us. 'Aye, he'd have liked that.'\n\nAnd so we went to the Lounge of the Steam Packet Hotel, looking out over the tourist-crowded pier and the packed car park. The water was bright amongst the hulls of the moored yachts, and the Mull ferry was a black shape in the distance, heading away.\n\nWe drank vintage champagne and fifteen-year-old malt. I suspected dad would have approved.\n\nLewis had to head back to London that night. Mum and I had been busy for a week, tidying up all the loose ends an unexpected death leaves, especially when the deceased is somebody as socially and professionally entangled as dad.\n\nThen mum had gardened while I'd sorted through less urgent papers; printing out everything on the disks, searching out all the rest of the stuff on stories, and sending it \u2013 or copies \u2013 to dad's literary executor, his editor in London. I had become modestly PC-COMPUTERATE (Ashley had given me a grounding in the basics, though PCs were not really her field). I'd even learned how to change the toner cartridge in the photocopier without making a mess.\n\nOn one of the earliest computer disks dad had used, dated shortly after he'd finally joined the computer age and bought the Compaq, in 1986, I found copies of some of Rory's poems; dad must have been putting them onto the system from the drafts Rory had left. I printed those out. It didn't look like dad had been very impressed with the poems, or he'd presumably have transcribed them all onto the computer (they weren't on the hard disk or backed up onto another floppy either \u2013 another indication my father had regarded them as relatively unimportant), but at least I again had something Rory had written. I was still hoping more of Rory's papers would surface somewhere.\n\nDad's old diaries turned up in a cardboard box at the back of a cupboard. Mum glanced at them, handed them to me. They looked pretty boring, frankly: 'M&I to Gal; shops, prom walk, back; did VAT.' and 'me Glasg car 1040 LHR 1315 F'furt; late, missed others. tel. L'gair Din in room, TV' were two of the more exciting and informative entries for last year. Dad's ideas books \u2013 A4 pads, usually \u2013 were where the interesting stuff was. We'd look at the diaries later.\n\nThen, one day, at the back of dad's oldest and most decrepit filing cabinet, I'd discovered treasure. It was in the form of three tatty, falling-apart Woolworth folders stuffed with old exercise books and shorthand-pads, bulging manila envelopes stuffed with old tickets, timetables and assorted scraps of paper, as well as assorted sheets of paper of various sizes, some stapled together, most loose, some typed and some hand-written, and all the work of Uncle Rory. There was one sealed envelope, too.\n\nHere were all the poems I'd seen before and more, typescripts of all the travel pieces, and the progenitor of Traps; Rory's India journal; tattered, battered, stained and torn and littered with doodles and little hand-drawn maps and sketches. A fold-out map of India was stapled to the inside back cover of the first exercise book, and on it Rory's spastically erratic route round the country was picked out in blue Biro. The back cover of the second book was covered with little faded train and bus tickets, attached to the cheap, fibrous blue paper with rusting staples. The last exercise book had only one ticket stapled to the rear cover: Rory's Air India ticket home. Some of the pages were stained with what looked like saffron, and I swear one book still smelled of curry.\n\nI'd sat down there and then and begun to read, flicking through the thin, brittle pages of the journal, smiling at the spelling mistakes and the awkward, amateurish drawings, looking for passages I remembered.\n\nI'd looked at the other stuff too, and found one play \u2013 another martial yarn about death and betrayal, and apparently nameless \u2013 which contained not only the passage about the fate of soldiers which I'd read in the delayed train back in January, in the rain on the line at the back of Crow Road, but which also ended with the lines I'd heard first a few weeks before that, in Janice Rae's flat. In Janice Rae's bed, in fact.\n\n_And all your nonsenses and truths,_ I'd read.\n\n'Your finery and squalid options,' I'd said quietly, to myself.\n\nRory's climax-delaying mantra was all there, right down to the last, three-word line. But, given the situation the narrator was in at that point, the lines took on an extra resonance, and an irony I had not been able to appreciate before. The section was circled with red ink and under that last line was written a note in large letters:\n\n_USE FOR END CR._\n\nGradually though, as I'd looked at it all, my feeling of quiet elation faded as I realised there seemed to be nothing else in any of the folders that seemed to relate to Crow Road. All I found was one cryptic note scribbled in pencil on the inside flap of what looked like the most recent of the three tatty files. It said:\n\n_\"CR: !B killsH!!? (save)_   \n_(jlsy? stil drwnd)\"_\n\nB and H. I vaguely remembered these abbreviations from the notes I'd lost. I shook my head, cursing my own idiot negligence, and Uncle Rory's frustrating delight in abstract abbreviation.\n\n... Jlsy. Well, _that_ was a recurring theme in Rory's work.\n\n... Stil drwnd. But Hell, I thought H got crshd btwen crge & tr!\n\n'Fuck it,' I'd said, and closed the file. I'd turned the bulky, heavy, sealed manila A4 envelope over in my hands for a while, then opened it. Computer disks. (That was a surprise. As far as I knew Uncle Rory had never possessed a computer.) Eight big floppy computer disks each in their own brown paper envelopes. Hewlett-Packard Double-sided Flexible Disks, Recorder # 92195A (Package of 10 disks). Well, yes; of _course_ there would have to be two missing. They were numbered 1 to 8 in black felt-tip, and that was the only indication they weren't brand new and unused. The write-protect holes were still taped over.\n\nI'd looked over at the Compaq, sitting on dad's desk, but the big, somehow already old-fashioned looking disks wouldn't even have fitted into the Compaq's drive if you'd folded them in half.\n\nMaking a mental note to call Ashley in London about the disks some time, I put them back in their manila envelope and the envelope back in its faded folder, and spent a fair while after that just leafing through Uncle Rory's India journal, smiling sadly at it all and becoming almost as willingly lost in it as it seemed Rory had in the pungent, teeming wastes of India itself ... until mum called me from the foot of the stairs, and it was time for tea.\n\nA few days later, I'd travelled back up to Glasgow by train; we'd got all the immediate matters regarding my father's death sorted out. It had been a perfect day; summer-warm and spring-fresh, the air winter-clear, the colours more vivid than in autumn.\n\nI'd felt a sort of shocked calm settle over me as I'd travelled, and been able to forget about death and its consequences for a while.\n\nThe familiar route had looked new and startling that day. The train had travelled from Lochgair north along the lower loch, crossed the narrows at Minard, and stopped at Garbhallt, Strachur, Lochgoilhead and Portincaple Junction, where it joined the West Highland line and took the north shore of the Clyde towards Glasgow. The waters and the skies blazed blue, the fields and forests waved luxuriously in a soft, flower-scented breeze and the high hill summits shimmered purple and brown in the distance.\n\nMy spirits had been raised just watching the summer countryside go past \u2014 even the sight of the burgeoning obscenity of the new Trident submarine base at Faslane hadn't depressed me \u2014 and when the train had approached Queen Street (and I'd been making very sure I had all my luggage with me) I'd seen something sublime, even magical.\n\nIt had been no more than that same scrubby, irregularly rectangular field of coarse grass I'd sat looking at so glumly from the delayed train in the rain that January. Then, the field's sodden, down-trodden paths had provided an image of desolation I had fastened onto, in my self-pity, like a blood-starved leech onto bruised flesh.\n\nAnd now the field had burned. Recently, too, because there was no new growth on the brown-black earth. And yet the field was not fully dark. All the grass had been consumed save for a giant green X that lay printed, vivid and alive, on the black flag of the scorched ground. It was the two criss-crossing paths through the wedged-in scrap of field that still shone emerald in the sunlight. The flames had passed over those foot-flattened blades and consumed their healthier neighbours on either side while they themselves had remained, made proof against the blaze and guaranteed their stark survival just by their earlier oppression.\n\nI'd stood there, in the act of taking my bags down from the luggage rack. And smiling to myself, I'd said, 'See?'\n\nDad hadn't specified any memorial; all his will had said was that he wanted to be buried somewhere in the grounds of the house. There was some discussion, and eventually mum decided on a plain black marble obelisk with his name and the relevant dates on it.\n\nI stood there, dressed in my slightly preposterous Highland finery, half-way through this wedding in the rain and remembering the funeral in the sunlight a season earlier, and I thought again how damn ugly that dark monument had turned out, then I shook my head and turned, and walked back to the lawn and the marquee. The ground was squelchy and I had to tread carefully to avoid getting mud on my thick white socks. The kilt swung against my knees.\n\nI wondered if Ash was back yet.\n\n'What what what? Come on, Prentice! My first chance to snog tongues with your brother as a married man and you're dragging me away waving ... ha! ha! Where did you get these?'\n\nThe hall\u2013 of the Lochgair house was swarming with people, crowding in, laughing, brandishing presents, shaking hands, demanding drinks, slapping Lewis on the back, hugging Verity, talking quietly to my mother, wandering through the press of people greeting each other and bumping and smiling and talking away and generally making me feel I might have arranged the reception line a little better; it had been a relief to spot Ashley struggling through the crowd at the front door, remember the computer disks, dash upstairs to get them and then down to intercept her and haul her into the lounge.\n\nFound them in dad's study,' I told Ashley, holding the disks out to her. She put a gaily wrapped package down on a chair, took one of the big disks from me and slid it out of its paper wrapper, grinning.\n\nThen she looked up, frowned and stepped back, arms wide. 'Prentice,' she said, voice deep with censure. 'You haven't said how stunning I look yet. I mean, come now.'\n\nAsh wore loose black pants and a shimmery silver top; hair back-swept and piled up. The glasses had been replaced by contacts. 'You look great,' I told her. I nodded at the disk in her hand. 'Think you can do anything with that?'\n\nAsh sighed and shook her head. 'I don't know. Haven't you got the machine they ran on?'\n\nI shook my head. 'I asked my mum about it; she thinks they might have been Rory's.'\n\n'That long ago, eh?' Ash tapped the disk sleeve dubiously, as though expecting it to crumble to dust at any second.\n\n'I didn't know until today he even had one; I mentioned to mum I'd ask you about these, and she said Rory did have a computer, or a word processor or whatever. Got it out in Hong Kong about a year before he disappeared.'\n\n'Hong Kong?' Ash looked even more dubious.\n\n'Some sort of ... copy; clone? Of an ... well, mum said an Orange, but I guess she means Apple. She remembers him complaining that it \u2013 or the program or whatever \u2013 didn't come with proper instructions, but he got it to work eventually.'\n\n'...Uh-huh.'\n\n'Dad left it in the flat Rory shared in Glasgow when he took Rory's papers away. Wouldn't have a computer in the house, at the time.'\n\n'Wise man.'\n\n'I'm going to try and track down the guy Rory shared the flat with, but I reckon the machine's been chucked out or whatever long ago, and I just thought, could you ... you know ... you might know somebody who perhaps could be able to ... to decode what's on there?' I shrugged, suddenly feeling awkward. Ash was now looking at the disk as though fully anticipating that creepy crawlies were about to start emerging from it. 'I mean,' I said, clearing my throat, suddenly feeling hot and sweaty. 'There might not be anything on them at all, but ... I just thought ...'\n\n'So,' Ash said slowly. 'Let me get this straight: you don't know the machine, but it's probably some ancient nameless Apple clone from the dark grey end of the market, almost certainly using reject chips; it probably had a production run that lasted until the first month's rent fell due on the shed the child-labourers were assembling them in, it used an _eight-inch drive_ and ran. what sounds like dodgy proprietorial software with more bugs than the Natural History Museum?'\n\n'Umm... Yeah. That about sums it up.'\n\nAsh nodded a few times, lips tight, weighing the disk in her hands. 'Right.' She nodded at the ones I still held. 'Okay. Can I take these?'\n\n'Sure.' I handed them to her and she turned for the door.\n\n'See you later,' she said, heading into the crowded hall. I went after her; she was excusing her way to the front door.\n\n'Ash!' I said, squeezing through after her. 'Not now! Come and enjoy the party!'\n\n'Don't worry,' she said, glancing back. 'I shall return. I'll drop these at home so I don't forget to take them back to London; I know people who might be able to help ... but I just remembered I forgot something; something for you. Left it at mum's.' She looked out the door; it was starting to rain. 'Shit.'\n\nThere was an old giant brass cartridge case by the hall hat stand which held our assorted umbrellas and walking sticks; I lifted a brolly from it. Ash turned to me, a worried expression on her face as she said, 'I saw that guy again. I'll show you. Give my present to the happy couple!'\n\n'What guy \u2013?' I said, but she was already sprinting through the still-arriving guests for the little red 2CV, parked a good fifty metres down the car-crowded drive, the disks held tight to her chest. I watched her high-heels flashing over the gravel, and the other guests turning to look at her, then there were more people to greet and hands to shake.\n\nI took the brolly myself eventually and went for a walk up the garden to dad's grave, just to get away from the crowds for a bit.\n\nBack in the house, I dodged one of the waitresses from the Lochgair Hotel, carrying a huge tray with champagne flutes out of the kitchen towards the marquee; I waved at mum, splendid in black with white stripes and standing talking to Helen Urvill, and went through to the hall. I put the umbrella in the old cartridge case. Then I thought maybe I should open it out and dry it, like you're supposed to, so I hauled it out again and left it opened in the hall.\n\n'Prentice,' Verity said, coming down from upstairs.\n\nShe was enfolded in white silk; a creation of some clothes-designer, friend of hers in Edinburgh. Technically it was a blouse, medium-length skirt and jacket, but when she wore it it looked like a single piece, and handsome it was too. She was hardly showing yet, but the outfit would anyway have disguised an almost full-term pregnancy. She wore white leggings, and high-heels that made her taller than me. She also wore the fulgurite necklace; mum had guessed both that Verity would want to wear it, and that she might think it best not to, in case the association hurt, so she'd made a point of telling Lewis she thought Verity ought to wear it, if it suited the outfit she had chosen. Verity's hair was as short-cropped as ever, but she looked none the worse for that, and the little white micro-hat she wore, complete with thrown-back, white fish-net veil, sat well on her too. She came up to me, took me by the shoulders and kissed me on the cheek.\n\n'That was a great speech; thanks,' she said. She was still holding my shoulders, and squeezed them. She looked the way you're supposed to look, both when you're pregnant and on the day of your marriage; glowing, radiant, suffused with joy. Still had perfect skin. She put on a convincingly upper-class English accent as she said, 'You've been en ebserloortly soopah byest men, my deah.'\n\nI put my hands lightly on her still slim waist and made a small bowing motion. 'Any time,' I said, and grinned.\n\nShe laughed, shaking her head. She stepped back, folded her arms, looked me up and down and said, 'And so smart.'\n\nI curtsied, fluttering my eyelashes.\n\nShe laughed again and held out her hand. 'Come on; let's find my husband. He's probably flirting with the bridesmaids by now.'\n\n'I thought that was my job,' I said, taking her arm in mine as we went towards the rear of the hall. I heard the front door open behind us. I turned and looked, stopped, then turned back to Verity. 'I'll take a rain check on that, shweetheart.'\n\nVerity smiled at Ashley Watt, shaking a glistening waterproof she'd just taken off, and nodded. 'Well, there's appropriate, today.' She winked at me and walked off.\n\nAshley met me at the foot of the stairs, brandishing a VHS cassette. 'Got it. Take me to your video.'\n\n'Walk this way,' I told her, heading up the stairs two at a time.\n\n'Do I get to look up your kilt?' she said from behind.\n\n'Not if you're lucky.'\n\nI switched the lights on in the study; we tended to keep the curtains closed. There was a TV and video in the study. I switched it all on and put the cassette in the machine.\n\n'Cool,' Ashley said, standing hands on hips in the middle of the study, heels neatly over the centre of dad's Persian rug, bunned hair directly beneath the big brass and stained-glass light fixture, hanging extravagantly beneath an ornate plaster ceiling rose. She swivelled, surveying the book-case walls, the maps, the prints and paintings and various interesting bits and pieces scattered around shelves, tables, desks and the floor.\n\n'Bit cluttered for my taste,' I said, starting the tape and watching some end-credits. 'Dad found it conducive enough.'\n\n'Fast forward,' Ash said. The screen scrolled quickly, then the BBC Nine O'Clock news started flashing before us. Ash turned away, so I let it roll.\n\nAshley crossed to an over-crowded book case; there was an empty crystal bowl perched on a pile of loose papers on top of the book case, and Ashley tapped the bowl very gently with one finger. She took her hand away, held it in the air near the ice-coloured ornament, and clicked her fingers. She bent her head towards the bowl, seemingly listening for something. I frowned, wondered what she was up to.\n\nShe turned and faced the bowl, went 'Ah,' in a high-pitched voice, then listened again, head tilted, smiling this time.\n\n'Ashley, what exactly are you doing?'\n\nShe nodded at the bowl. 'Crystal; you can make it ring by producing the right noise.' She grinned like a little girl. 'Good, eh?' She looked behind me. 'That's you,' she said, nodding at the screen.\n\nI hit Play. We stood, watched.\n\n'... talked to Rupert Paxton-Marr of the _Inquirer,_ one of the journalists held by the Iraqis, and asked him how he'd felt,' said the BBC man in Amman.\n\nI couldn't resist a thin smile, one journalist asking another how he felt.\n\nRupert Paxton-Marr was a tall, blond, blue-eyed man with exactly the jaw-line I'd have chosen for myself, given the opportunity ; sickeningly handsome, he had an accent to match. 'Well, Michael,' he said. ('Air, hair lair,' I said to myself.) 'I don't think we were really in much danger; clearly international attention has fixed on Iraq, and I think they knew we knew that, and accepted we were ... weren't a threat to them. Umm ... our driver had taken a wrong turning, and that was that. Of course, one does remember what happened to, ah, Farhzad Bazhoft, but I don't think you can let that stop you; in the end one has a job to do.'\n\n'Thank you, Rupert. And now, reporting fr \u2013'\n\nI hit Stop and turned to Ashley, standing beside me. She was still looking at the blank screen where the little green zero symbol sat in one corner, wobbling almost imperceptibly. She had sucked her cheeks in and her lips were pursed. There was a whoop of laughter from somewhere downstairs. Ash nodded slowly, looked at me. 'That's ma boy,' she said.\n\n'You're sure about that?' I said.\n\n'I'm sure.' She looked serious. She looked pretty good, too, now I looked properly; I couldn't remember ever seeing Ashley wearing make-up, and you'd have thought that not having had the practice she'd be crap at it, but she looked great; maybe a little overenthusiastic with the dark stuff round the eyes, but why quibble? She nodded. 'Don't look at me like that; I'm really sure.'\n\n'Sorry. I believe you,' I said. I spun the tape back, to play it again. Ashley put one hand on my arm and rested her chin against the shoulder of my Prince Charlie jacket.\n\n'Turn the sound down,' she said. 'That guy's voice is like chewing on silver paper.'\n\nI turned the sound down. The noise of people laughing and talking in the marquee came through the double glazing and the heavy burgundy of the velvet curtains. I heard an amplified voice outside say, 'Testing.' It was probably Dean Watt; he and his band had been hired by Lewis and Verity to play during the afternoon (for the evening they'd booked a more traditional wheech-your-partner fiddle and accordion band).\n\nI ran the clip again. 'Definitely, officer,' Ashley said, tapping the top corner of the TV. 'Recognise him anywhere, even with his clothes on.'\n\nI switched the TV off and ejected the cassette. I stood for a moment, rubbing my chin.\n\n'Whoops,' Ashley said, and delicately rubbed a little of that transferred make-up from the black shoulder of my jacket.\n\nI waited till she'd finished, then went to dad's desk, unlocked a lower drawer and took out a slide tray; one of those plastic things that holds a few hundred transparencies. 'So, when you saw this guy, Paxton-Marr,' I said, opening the tray and putting the lid on the desk, 'in Berlin, in this hotel, in the jacuzzi ...' I looked up at Ashley, standing sceptically by the TV, one elbow resting on it as she watched me. 'What was the hotel called again?'\n\n'I told you,' she said. 'I can't remember. I called June, and neither could she. It's probably the only place she ever stayed and forgot to nick a towel or yet another emergency sewing kit or whatever.' Ash shrugged. 'Frankly, Prentice, I was stoned out of my brains most of the time I was there. All I can remember is it had a big pool in the basement with a jacuzzi at one end, and they did really good breakfasts.' She sighed. 'Excellent hopple-popple.' Her eyebrows flicked once.\n\n'Hopple-popple?' I grimaced.\n\n'Scrambled eggs,' Ash smiled. 'Take me to Berlin and I'll find it for you. It was somewhere near the zoo.'\n\nI put the tray down on the desk, went over to Ashley, holding out a little piece of cardboard; it was the front cover off a book of matches, torn off the piece that held the matches.\n\n'Wasn't the Schweizerhof, was it?' I asked her.\n\nShe looked steadily into my eyes for a little while, then took the piece of card, looked at it and turned it over.\n\n'Twenty-seven eleven eighty-nine,' she muttered. She nodded and handed me the coyer back. 'Yeah,' she said, frowning. 'Yeah; it was. That was the place.'\n\nI put the little torn bit of cardboard back in the slide tray. It was the second last one, out of about forty of them.\n\n'What's the significance of the date?' Ashley asked, coming over to the desk. Outside, I could hear the sound of an electric guitar chord and a few drum beats.\n\n'I think that was when dad received it.' I picked the latest torn cover out of the tray. 'This one arrived just after he died.' We both sat on the edge of the desk; Ashley looked at the little piece of glossy cardboard.\n\n'Woo,' she whistled. 'Amman Hilton. Spooky, or what?'\n\n'Yeah. Spooky as fuck,' I said, tapping the cover with one finger-nail. 'And I'm sure I recognise that guy Paxton-Marr, too. From Glasgow, or Edinburgh, or here. I've met him. In the flesh, I think.'\n\nAsh put her elbow on my shoulder. 'And damn firm, tanned flesh it was too, let me tell you,' she said.\n\nI looked into those grey eyes, smiling. 'But not as firm and tanned as your programmer from Texas.'\n\nAsh laughed, skipped off the desk. 'Systems Analyst. And you're right; they breed them bigger and better in Texas.'\n\nMusic started up in the marquee. _Kiss The Bride._ Ash stood on the Persian rug again, putting one hand to her ear. 'Hark; it's young brother and his pals.' She frowned. 'Doesn't sound like a Mark E Smith or Morrissey track to me.' She shook her head. 'Tsk. How are the mighty fallen.' She put her head down so that, if she'd been wearing glasses, she'd have been looking over them at me. 'Want my advice?'\n\n'Mm-hmm,' I nodded.\n\n'Come on and dance. We can sort \u2013 or you can sort \u2013 this out later, when you've had time to think.' She struck a dramatic, arguably dance-inspired pose and held out one hand. 'Hey baby, let's boogie!'\n\nI laughed, shut the match-book covers away and locked the desk drawer.\n\n'That's it, laddy,' Ashley said, holding my arm as we went to the door. 'You put that key in yer sporran.'\n\n'At least I know down there it's safe from interference,' I told her. She smiled. I locked the study door too.\n\n'By the way, by the way,' Ash muttered into my ear as we headed along the landing for the stairs, 'got a gramme of the old Bogot\u00e1 talcum powder about my person. Fancy a toot, later?'\n\n'What, the real thing?' I grinned. 'I thought speed was your poison.'\n\n'Normally,' she nodded. 'But this is a special occasion; I splashed out.'\n\n'You wee tyke,' I said. I pulled her closer as we walked, held her tight. 'You just stick with me, kid, all right?'\n\n'Whatever you say, ma man.'\n\nWe did kick-steps down the stairs. Risky, when you're wearing a kilt as it is meant to be worn, but invigorating.\n\nI danced with Ashley, and with Verity, and with Helen Urvill, and with mum, cutting in on Lewis after he'd persuaded her onto the floor. Most of the time though she just sat, surrounded by family and friends, watching us all with an expression that, to me at least, spoke of a kind of stricken joy; a surprise that such pleasure could still exist \u2013 and she feel even remotely a part of it \u2013 when dad was not here to share in it all.\n\nI am not a natural dancer but I made an exception for Verity's wedding. I grooved and sweated and drank and made a point of doing the old red blood cell impression, circulating; bathing in, soaking up and transmitting onwards the oxygen of family news and gossip from cell to cell ...\n\n'Where are you off to next, Aunt Ilsa?' I asked the lady in question, during our waltz. Aunt Ilsa \u2013 even larger than I remembered her, and dressed in something which looked like a cross between a Persian rug and a multi-occupancy poncho \u2013 moved with the determined grace of an elephant, and a curious stiffness that made the experience a little like dancing with a garden shed.\n\n'Canada, I think, Prentice. Churchill, on Hudson's Bay. To observe the arctic bear.'\n\nI confess I had to re-process that sentence a couple of times as we danced, before working out that she did not intend to study the region naked (an image I found rather alarming), but was merely using a more pedantically accurate term for a polar bear.\n\n'Super.' I smiled.\n\nUncle Hamish sat at the table with the rest of the family and got slowly drunk. I danced with Aunt Tone, and asked after her husband's health.\n\n'Oh, he's getting better all the time,' Aunt Tone said, glancing at him. 'He hasn't had the nightmares for weeks now. I think going back to work helped him. Fergus was very understanding. And he's had a lot of long chats with the minister. People have been very kind, altogether. You haven't talked to him?' Aunt Tone looked at me critically.\n\n'Not for a bit.' I gave her a big smile. 'I will, though.'\n\nUncle Hamish watched the dancing. He lifted his whisky to his lips, sipped at it, then shook his head with such slow deliberation I caught myself listening for the creak. 'No, Prentice. I have been foolish, and even vain. I did not pay sufficient heed to the scriptures. I thought that I knew better.' He sipped his whisky, shook his head. 'It was vanity; my theories, my beliefs about the hereafter; vanity. I have renounced them.'\n\n'Oh,' I said, disappointed. 'No more anti-creates?'\n\nHe shook, as though a chill had passed through him. 'No, that was my mistake.' He looked at me straight for the first time. 'He punished both of us, Prentice.' Uncle Hamish flicked his gaze towards the roof of the marquee. 'Both of us,' he repeated. He looked away again. 'God knows we are all his children, but he is a strict father, sometimes. Terribly, terribly strict.'\n\nI put my head on my hand and looked at my uncle as I considered this idea of God as child-abuser. Hamish started to shake his head again before he'd sipped his whisky, and I experienced a brief feeling of excited horror, waiting for the resulting catastrophe; but he just stopped in mid-shake, sipped, then shook his head slowly again. 'Aye; a strict father.'\n\nI patted his arm. 'Never mind.' I said, helpfully.\n\nI danced with Aunt Charlotte, Verity's still-handsome and determinedly superstitious mother, who told me that the newly-weds would surely be happy, because their stars were well-matched.\n\nExhibiting a generosity of spirit I rather surprised myself with, I agreed that certainly the stars in their eyes seemed to augur well.\n\nI bumped into the smaller than life-size Mr Gibbon near the bar at one point; I was in such a gregarious, clubbable mood I actually enjoyed talking to him. We agreed Aunt Ilsa was a wonderful woman, but that she had itchy feet. Mr Gibbon looked over at Aunt Ilsa, who had \u2013 I could only imagine by force \u2013 got Uncle Hamish up for a dance. Together they were having the same effect on the dance floor as a loose cannon manned by hippos.\n\n'Yes,' Mr Gibbon said, sighing, his eyelids fluttering. 'I am her kentledge.' He smiled at me with a sort of apologetic self-satisfaction, as though he was the luckiest man alive, and tip-toed off through the crowds with his two glasses of sherry.\n\n'Kentledge?' I said to myself. I'd have scratched my head but my hands were full of glasses.\n\n'Prentice. Taking a breather too, eh?'\n\nI had stepped outside the marquee for a breather, late on, after the hoochter-choochter music started and the place got even warmer. I looked round in the shadows and saw Fergus Urvill, Scottishly resplendent in his Urvill dress tartan. Fergus came into the light spilling from the open flap of the marquee. He was smoking a cigar. The rain had ceased at last and the garden smelled of earth and wet leaves.\n\nFergus glittered; crystal buttons sparkled on his jacket; black pearls of obsidian decorated his sporran, and the skean dhu stuck into the top of his right sock \u2013 a rather more impressive and business-like example of the traditional Highland-dress knife than mine, which looked like a glorified letter opener \u2013 was crowned with a large ruby, glinting in the light against the hairs of his leg like some grotesquely faceted bulb of blood.\n\n'Yes,' I said. 'Yes, getting my breath back.'\n\nFergus looked into the marquee. 'They're a handsome couple, eh?'\n\nI glanced in, to see Lewis and Verity, hand in hand, talking to some of Verity's relations. Lewis had changed into a dark suit and a bootlace tie; Verity wore a dark skirt and long, gold-coloured jacket.\n\nI nodded. 'Yes,' I said. I cleared my throat.\n\n'Cigar?' Fergus said, digging an aluminium tube from one pocket of his jacket. I shook my head. 'No,' Fergus said, looking at me tolerantly. 'Of course you don't, do you.'\n\n'No,' I said. I grinned inanely.\n\nI was surprised at just how uncomfortable I felt in his presence, and how hard it was both to work out precisely why I felt that way, and to disguise the fact. We talked for a little while. About my studies; going better now, thank you. And about flying. Fergus was learning to fly; up at Connel, the air field a few miles north of Oban. Oh, really? Yes. Hoped to be going solo by the end of the year, if all went according to plan. He asked me what I thought of the Gulf crisis and I, quailing, said it all kind of depended how you looked at it.\n\nI think I made him feel as awkward eventually as I had from the start of the conversation, and I took the opportunity of a new reel beginning to head back into the marquee, to join in another swirling, riotous dance.\n\nAshley, Dean and I retired to my room in the house during the supper interval, while people got their breath back and the band \u2013 four oldish guys mysteriously called the Dougie McTee Trio \u2013 tried to get drunk.\n\nWe snorted some coke, we had a couple of Js, and in response to a single question from Dean, I told them both all about the River Game; its history, every rule and feature, a thorough description of the board, an analysis of the differing playing styles of myself, Lewis, James, dad, mum and Helen Urvill, some handy tips and useful warnings, and a few interesting excerpts from certain classic games we'd played. It took about ten minutes. I don't think I repeated myself once or left anything out, and I finished by saying that all of that, of course, wasn't to mention the secret, banned version; the Black River Game.\n\nThey both stared at me. Dean looked like he hadn't believed a word I'd said. Ashley just seemed amused.\n\n'Aye. Good coke, isn't it?' she said.\n\n'Yep,' Dean said, busy with mirror and blade again. He glanced at his sister and nodded at me. 'For God's sake, Ash, stick that number in his mouth and shut him up.'\n\nI accepted the J with a smile.\n\nThe three of us kick-stepped down the stairs.\n\n'Hoy, all that stuff about that game,' Dean shouted as we three swung into the marquee, where an Eightsome Reel of extravagant proportions and high decibel-count was in its Dervish phase. 'That gospel, aye?'\n\nI frowned deeply as I looked at him. 'Oh no.' I shook my head earnestly. 'It's _true.'_\n\nLater, I sat alone at a table, quietly drinking whisky, watching them all. I'd lowered my head; one hand lay flat, palm-down, on the table. I felt very calm and deadly and in control; shit, I felt like I was Michael Corleone. The tunes and laughs and shouts washed through me, and the people, for that moment, seemed to be dancing about me, for me. I felt ... pivotal, and drank a silent toast to Grandma Margot. I drank to my late father. I thought of Uncle Rory, wherever he was, and drank to him. I even drank to James, also absent.\n\nJames was coming down only slowly from his peak of anger. Even now, he was still so sullen and difficult to get on with that it had almost been a relief when he'd said he didn't want to be involved in the wedding. He'd gone to stay with some school pals at Kilmartin, a little north of Gallanach, for the weekend. I think mum was unhappy he wasn't here today, but Lewis and I weren't.\n\nI drank some more whisky, thinking.\n\nA marriage.\n\nAnd a little information.\n\nNot to mention more than a little suspicion.\n\nAll it had taken was one blurred face, glimpsed far away by somebody else, seen soundless for a second on a fuzzed TV in a noisy, crowded, smoky pub in Soho, one Friday evening \u2013 just one tiny example of all the inevitable, peripheral results of a confrontation in a distant desert \u2013 and suddenly, despite all our efforts, we were back amongst the bad stuff again; shrapnel from the coming war. Although, of course, I couldn't be sure.\n\nMum went past, dancing with Fergus Urvill, who was sweating. Mum looked small, next to him. Her expression was unreadable. Jlsy, I thought, and drank to Uncle Rory.\n\nLewis and Verity left at midnight in a taxi. None of that let's-make-a-mess-of-the-car nonsense for them. The taxi was supposedly heading for Gallanach; only mum and I knew they were actually booked into the Columba in Oban, and heading for Glasgow and the airport tomorrow.\n\nThe four-man trio played; the dancing continued. Mum left with Hamish and Tone; she was staying with them tonight. I was in charge of the house. I danced until my legs ached. I talked until my throat hurt. The band, and the bagpipe players who'd joined in with them, stopped playing at about two. Dean and I fed some home-made compilation tapes through the PA, and the dancing went on.\n\nLater, after everybody had either left or crashed out in the house, Ash and I walked out along the shore, by the calmly lapping waters of Loch Fyne, in a clear, cool dawn.\n\nI remember babbling, high and spacey and danced-out all at once. We sat and stared out over the satin grey stretch of water, watching low-flying seagulls flapping lazily to and fro. I treated Ash to bits of Uncle Rory's poetry; I knew some of it by heart, now.\n\nAsh suggested heading back and to the house, and either having some coffee or getting some sleep. Her wide eyes looked tired. I agreed coffee might be an idea. The last thing I remember is insisting I had whisky in my coffee, then falling asleep in the kitchen, my head on Ash's shoulder, mumbling about how I'd loved dad, and how I'd loved Verity, too, and I'd never find another one like her, but she was a heartless bitch. No she wasn't, yes she was, no she wasn't, it was just she wasn't for me, and if I had any sense I'd go for somebody who was a kind and gentle friend and who I got on well with; like Ash. I should take up with Ash; I should fall in love with her, that's what I ought to do. Only if I did, I muttered into her shoulder, she'd be sure to fall for somebody else, or die, or get a job in New Zealand, but that's what I ought to do, if only things worked that way ... Why do we always love the wrong people?\n\nAsh, silent beneath me, above me, just patted my shoulder and laid her head on mine.\n\nMum woke me in the late afternoon. I moaned and she put a pint glass of water and two sachets of Resolve down on the table near my head. I tried to focus on the water. Mum sighed, tore the sachets open and tipped the powder fizzing into the glass.\n\nI checked things out with the one eye that would open. I was in my room at Lochgair, on my bed, still mostly clothed in shirt and kilt and socks. My head felt like it had been recently used for a very long and closely contested game of basketball. Somebody had stolen my real body and replaced it with a Prentice-shaped jelly mould packed full of enhanced-capacity pain receptors firing away like they were auditioning for a Duracell commercial. Mum was dressed in faded jeans and an old holed sweater. Her hair was tied back and she wore violently yellow rubber kitchen gloves which were doing horrible things to my visual cortex. A yellow duster dangled from her hip pocket. I couldn't think what else to do, so I moaned again.\n\nMum sat down on the bed, put a hand on my head and ran my curls through her rubber-clad fingers.\n\n'What's that you've got in your hair?' she said.\n\nMy brain cells? I wondered. Certainly it felt like they'd been squeezed out of my ears. Damn rim-shots. Not that I could share this insight with my mother, for the simple reason that I couldn't talk.\n\n'What is it?' mum said. 'This black stuff?' She rubbed her fingers together in my hair, the rubber gloves squeaking horribly. 'Oh, stop moaning, Prentice. Drink your water.' She sniffed at her fingers. 'Hmm,' she said, rising and heading for the door. 'Mascara, eh?'\n\nI looked up, monocular, at the closing door, grimacing.\n\nMassacre?\nCHAPTER 15\n\nI sipped my Bloody Mary, looking down at huge, white, piled-up clouds so bright in the mid-day sunshine they looked yellow. The plane had just levelled out and there was a smell of food; they were serving lunch further forward in the cabin. I watched the clouds for a moment, then looked at my magazine. I was on my way to London, a couple of torn-off match-book covers in my pocket, hoping to confront Mr Rupert Paxton-Marr.\n\n'Thanks mum ... Ash?'\n\n'Yo, Prentice. How's it hanging?'\n\n'Oh, plum.'\n\n'Still wearing the kilt, eh? Look, I've had some word from \u2013'\n\n'How about you?'\n\n'Eh?'\n\n'How are you?'\n\n'Oh, rude health. Verging on the obscene. Listen; my computer wizard's been in touch.'\n\n'What? About the disks?'\n\n'Cor-rect.'\n\n'What's on them? What do they say? Is there anyth \u2013'\n\n'Hey ... hold your horses. Had to get the stuff to him first.'\n\n'Oh. Where is he?'\n\n'Denver.'\n\n_'Denver?'_\n\n'Yup.'\n\n'Denver _Colorado?'_\n\n'... Yes.'\n\n'What, in America?'\n\n'Yeah, Northern Hemisphere, The World, The Solar System ...'\n\n'Okay, okay, so he's ... hey, is this your Texan programmer? Has he moved states?'\n\n'Systems Analyst, for the last fucking time, Prentice, and no, it isn't him; just a guy I exchange E-mail with sometimes.'\n\n'Right. And he's got the disks?'\n\n'No, of course he hasn't got the disks.'\n\n'What? Then \u2013'\n\n'He has the information that was held on them. Well, on the one that held anything. Seven were blank; not even formatted.'\n\n'Ah, right. I see ... so what does it say? What is on it? Was it all Rory's \u2013'\n\n'It's a little more complicated than that, Prentice.'\n\n'Oh.'\n\n'I've got a message on my screen here from him. Thought you might be interested in it.'\n\n'Oh; you're at work. Hey, have you seen the time? You're working late, aren't you?'\n\n'Yes ..., Prentice. Do you want to hear the message?'\n\n'Will I understand it?'\n\n'You'll get the gist of it.'\n\n'Okay.'\n\n'Right. I quote: \"I thought your man up there in the misty glens might like to know \u2013\"'\n\n' \"Misty glens\"? That's sounds a bit patronising.'\n\n'Prentice; shut up.'\n\n'Sorry.'\n\n' \"... might like to know what our game plan is with respect to your word-processed file(s). As we don't yet know what geek program this mutant _No-namo-brand_ clone was running, we have had to resort to extreme measures to access the data. Dr Claire Simmons of London University, who picked up the disks, will use a vintage Hewlett Packard TouchScreen (which has compatible eight-inch drives) in the establishment's Museum of Computing to extract the raw binaries, _sector by sector,_ praying all the while that somebody has posted an ediger to Usenet that she can use to strip off the physical addressing; she will then attack the content one word at a time, swapping bytes as needed and inverting bits if none of it looks like ASCII, stripping the eighth bits if they're in the way or un-encoding the lot if we can't do without them, and unload the result to a Prime mini-computer (another indestructible antique) somewhere on the campus network. She moves all this to her Iris, double-encrypts it and E-mails it via Internet (off JANUS or BITNET to nsfnet-relay.ac.uk, probably) via Cornell to an account I'm not supposed to have on the Minnesota Supercomputer Center's Cray-2 (currently the biggest and quickest compute-server short of a Connection Machine at the high end, so I might as well use it to do the decryptions and perhaps take my own first whack at demangling before moving the data along). From there I download via a dedicated T3 line to an SGI 380SX-VGX at one of AT&T's Bell Labs (the one in Boulder, I think \u2013 another unofficial account) from where I can further download \u2013 and filter out certain offending control characters \u2013 to a Mac II at my office. Then I dump the results onto a floppy and bike them home to tinker with in my basement, which is where the _hard_ work starts.\" ... Get all that, Prentice?'\n\n'Yeah. Basically what he's saying is, it's a piece of piss.'\n\n'Absolutely. A doddle.'\n\n'Great. So when can we expect to see some results?'\n\n'No idea. Don't forget the guy's doing it for fun, and he's a busy man. No promises, but he sounds confident. I'll call him in a week if he doesn't get in touch first.'\n\n'Tell him I'll fax him a crate of champagne or something.'\n\n'Certainly. So, when ...? Ah shit. Fucking decollator's jammed again. Gotta go attend the print, Prent.'\n\n'Okay. Bye. Oh, and thanks.'\n\nI now had a better idea of what Rory had been doing in the days before his disappearance. It looked like he had been working on _Crow Road_ between the time he'd come back from London after seeing his friends and the evening he disappeared, on the motor bike he'd borrowed from his flat-mate. That was what he'd been doing, stuck in his room in the flat in Glasgow; finally actually writing something on his bizarre contraption of a computer.\n\nHe'd done it, he'd stopped writing notes and started on the work itself.\n\nI'd talked to a retired policeman who at the time had looked \u2013 briefly \u2013 into what had happened to Rory. The police hadn't come up with anything; they'd interviewed Janice Rae, and Rory's flat-mate Andy Nichol, and looked at the papers Rory had left with Janice. There was no suicide note, so they'd decided the papers weren't relevant. Apart from checking the hospitals and eventually listing Rory as a Missing Person, that had been that.\n\nThe only useful information I'd got from the police was that Rory's flat-mate had left local government and joined the civil service a few months after Rory had disappeared. I'd tracked Andy Nichol down at a tax office in Plymouth and called him there, but apart from saying he'd heard a lot of keyboard-clattering noises coming from Rory's room during the days before Rory had borrowed his bike and disappeared, he'd only been able to confirm what I already knew. He did say he'd tried working Rory's Neanderthal computer after dad had said he could have it, but he couldn't make the beast work; he'd sold the machine and the two blank disks that had come with it to a friend in Strathclyde University. It had been chucked out years ago.\n\n... Whatever; after those few days work, Rory had suddenly upped and offed, and never came back. Maybe the stuff on the disks would give me a clue why he'd suddenly done that. If there _was_ anything useful there; that clattering noise didn't prove anything ... I'd seen _The Shining._\n\nThe cloud cover started to break up over the midlands; I chomped through my lunch. The starter was smoked salmon. I thought of Verity and Lewis, on honeymoon in the Bahamas, and \u2013 with just a tinge of sadness \u2013 silently wished them well.\n\nI saw Ashley come into the pub. She stood near the door, looking round, that strong-boned head swivelling, those grey eyes scanning. She didn't see me on the first sweep; I was mostly hidden by other people. I watched her take a couple of steps forward, look round again. She was dressed in a dark, skirted suit, under the old but still good-looking jacket I remembered her wearing at Grandma Margot's funeral. Her hair was gathered up and tied; she wasn't wearing her glasses. Her face looked tense and forbidding. She seemed harder, more capable and more self-contained than I recalled her being in Scotland.\n\nIn those few moments, in the noise and smoke of a pub by the river, a quarter mile from the Tower, in the great, cruel, headless monster that was London after a decade of Hyaena rule, I wondered again at my own feelings for Ashley Watt. I knew I didn't love her; she didn't make me feel anything like the way I had about Verity, and yet I'd been \u2013 I realised \u2013 looking forward to seeing her, and now that I had seen her, just felt, well ... happier, I guess. It was all puzzlingly simple. Maybe \u2013 to lapse into the humdrum continuum for a moment \u2013 she was the sister I'd never had. I remembered the mascara mum had discovered in my hair after the wedding, and wondered if the position of honorary sibling was one Ashley would entirely welcome.\n\nI tried to remember Ashley's tone when I'd rung her, a couple of days ago, to say that I was coming down (this about a week after I'd had my own personal info-dump on the workings of the world computer network). I had already called Aunt Ilsa and arranged to stay with her and Kentledge Man, and I'd wondered at the time if I'd detected the merest hint of reproach in Ashley's voice when I'd told her I would be staying in deepest Kensington. At any rate, she'd told me there was a sofa-bed and a spare duvet of indeterminate tog value at the flat she shared in Clapham, in case Aunt Ilsa went on some sudden expedition to Antarctica and forgot to tell her Filipino maid, or whatever. She'd added that the two girls she shared with really wanted to meet me (I felt pretty sure the person they really wanted to meet was Lewis).\n\nI raised my hand as Ashley's gaze passed again over where I stood; she caught the movement, and that city-hard expression changed instantly, relaxing and softening as she smiled broadly and walked over.\n\n'Hiya, babe.' She punched my shoulder, then gave me a big hug. I hugged right back. She smelled of _Poison._\n\n'How are you?' I asked her.\n\nShe put one fist on her hip and held her other hand up in front of my face, fingers spread. 'Drinkless,' she grinned.\n\n'You got _off?'_\n\n'Yeah,' I said, swirling the remains of my pint round in my glass.\n\nAsh shook her head. 'I thought you were going to plead guilty.'\n\n'I was,' I confessed. I shrugged, looked down. 'I got a smart lawyer. She said it was worth fighting. Ended up in a jury trial, eventually.'\n\nAsh laughed. 'Well done,' she said. She lowered her head until she could look into my eyes. 'Hey, what's the matter?'\n\n'Well,' I said, trying not to smile. 'I _did_ do it after all; it seems wrong I got off because I dressed in a suit and I could afford an expensive advocate and people in the jury had heard of dad and felt sorry for me because he'd died. I mean if I'd come from Maryhill and I wasn't reasonably articulate and didn't have any money, even if I _had_ just forgotten I hadn't paid for the book, I bet everybody would have told me to plead guilty. Instead, thanks to the money, I had an advocate who'd probably make God look just a little lacking in gravitas, and discovered a talent for lying through my teeth that promises a glittering career as a _Sun_ journalist.'\n\nAsh leaned conspiratorially forward over the small table we were crouched round, and quietly said, 'Easy, boy, you're on their turf.'\n\n'Yeah,' I sighed. 'And don't drink the tap water.' I looked around the place, all crowds and smoke. The English accents still sounded oddly foreign. 'No sign?' I asked.\n\nAsh looked round too, then shook her head. 'No sign.'\n\n'You sure he drinks here?'\n\n'Positive.'\n\n'Maybe he's been sent away, back to the Gulf.'\n\nAsh shook her head. 'I spoke to his secretary. He's having some root canal work done; he's here till the end of next week.'\n\n'Maybe I should have just arranged to see him.' I sighed. 'My new-found talent as a con-man might have come in useful. I could have said I had pictures of Saddam Hussein torturing a donkey or something.'\n\n'Maybe,' Ash said.\n\nWe had discussed this sort of thing. Ash's first idea was simply that she should ring him up, tell him she'd seen him on television and heard he worked in London; she was here too, now, and did he fancy a drink sometime? But I wasn't sure about this. If he'd been reluctant to give Ash his name in Berlin, and thought even there that he'd already said too much, he might be suspicious when she rang up. So I felt; so my \u2013 by now rather paranoid \u2013 feelings suggested. A chance meeting seemed more plausible, or at least it had when I'd been talking to Ash from dad's study in Lochgair. Now I wasn't so sure.\n\n'How's your wizard?' I asked her.\n\n'Eh?' Ash looked confused for a moment. 'Oh; Doctor Gonzo? Still working on the files. They weren't just weird shit, they were corrupted weird shit; where did your dad keep those things; inside a TV? But anyway; he's still hopeful.'\n\n'Doctor _Gonzo?'_ I said, tartly.\n\n'Don't look like that, Prentice,' Ash chided. 'This guy's knocking his pan in for you for nothing. And he has got a doctorate.'\n\nI smiled. 'Sorry.'\n\n'Oh, and supposing the good Doctor can decipher all that corrupted crap you presented him with, what format do you want these files in eventually anyway, you ungrateful wretch?'\n\n'How d'you mean?'\n\n'I mean what program do you use on the Compaq?'\n\n'Oh, Wordstar,' I nodded knowledgeably.\n\n'Version? Number?'\n\n'Ah ... I'll have to come back to you on that one. Look; just ask him to print it out and send it to me. Would that be okay?'\n\nShe shrugged. 'If you want. Or you could get a modem; E-mail's about a zillion times faster.'\n\n'Look, I'm still not all that comfortable around computers that don't come with a joystick and a \"fire\" button; just ... just ordinary airmail and real paper will be fine.'\n\nAsh grinned, shook her head. 'As you wish.' She stood up. 'Same again?' she asked, clinking my glass.\n\n'No,' I said. 'I'll have a half.'\n\n'Any particular sort?'\n\n'Na, anything.'\n\nI was alternating pints and whiskies on principle; they keep giving you your old glass back down here.\n\nI watched Ash weave her way to the bar.\n\nI still felt nervous about meeting this guy Paxton-Marr, but all-in-all, I told myself, things weren't so bad. Those of us most affected by dad's death were \u2013 with the possible exception of Uncle Hamish \u2014 bearing up pretty well, I might yet find out what Uncle Rory had written, I didn't have to worry about money, I had no criminal record, and I was being a good young(ish) adult again, attending diligently to my studies. Mostly I stayed in Glasgow during the week, and went back to Lochgair at weekends, unless mum \u2013 sometimes accompanied by James \u2013 came to stay with me. I had got filthy drunk just once since dad had died, and then with good reason; it had been the day Thatcher resigned. Bliss was it, etc., even if the Tapeworm Party was still in power.\n\nThe lawyer Blawke had found me a place to rent for the year I needed to be in Glasgow. It was part of the property of a Mrs Ippot, who'd died rich but intestate at a sourly ripe old age, having throughout her life promised part, or all, of her sizeable fortune to various individual relations and combinations of relations within her extensively and antagonistically divided family, in a blizzard of contradictory letters, and with what appeared to be a profound lack of consideration for the litigious chaos that was bound to ensue. Mrs Ippot, in short, had been the sort of client probate lawyers have wet dreams about.\n\nMy own theory was that Mrs I had actually thoroughly detested every single one of her relatives, and had hit on a nicely appropriate way of confounding all of them. By Jarndyce out of Petard, Mrs Ippot's lawyer-infested legacy had ensured that her rebarbatively consistent family would suffer years if not decades of self-inflicted hatred and frustration as the increasing legal fees gradually corroded the monies she had left; a tortuously slow method of telling your relatives from beyond the grave exactly what you thought of them that makes giving all the loot to a cats' home look positively benign in comparison.\n\nAnd so I stayed in the late Mrs Ippot's enormous town house in Park Terrace, overlooking Kelvingrove Park and the River Kelvin running through it. The museum and art gallery sat red, huge and stately to the left, its sandstone bulk crammed with the silt of time and human effort, while on the hill to its right, skirted by the black outlines of trees, the university soared with self-impressed Victorian fussiness into the grey autumnal skies, positively exuding half a millennium's experience in the collation and dissemination of knowledge.\n\nThe high ceilings and vast windows of Mrs Ippot's former home appeared to have been the work of an architect anticipating the design of aircraft hangars; the interior was cluttered with paintings, rugs, chandeliers, life-sized ceramics of the smaller big cats, small statues, large statues and objets d'art of every imaginable description, all interspersed with heavy, dark, intricately gnarled wooden furniture that gave the appearance of being volcanic in origin. The house's inventory \u2013 drawn grimly to my attention by a spotty clerk who obviously resented the fact I was younger than he was \u2013 came in three volumes.\n\nI christened the place Xanadu, but never did find any sleds.\n\nMy friends, of whom I saw less these days, suggested parties when they first heard about the place. On seeing it, they usually agreed with me that to mount a serious whoopee on the premises would be to invite cultural catastrophe on a scale usually only witnessed during major wars and James Last concerts.\n\nOne of my pals \u2013 graduated, employed; moving on to better things \u2013 sold me his old VW Golf, and I drove down to Lochgair most weekends, usually on a Thursday night as I didn't have any classes on a Friday. James and I helped mum, who was redecorating the house. She was talking about knocking down the old conservatory and putting in a new one, perhaps covering a small swimming pool. She had also formed the idea of building a harpsichord, and then learning to play it. We took tea at the Steam Packet Hotel on occasion, and James kept an Ordnance Survey map on which he inked in all the walks we undertook, on the hills and through the forests around Gallanach.\n\nMum and I had started going through dad's diaries. Some were pocket size, some were desk diaries; a couple of early ones were effectively home made. They went back to when he'd been sixteen. I'd suggested Mum read them first in case there was anything embarrassing in them, though I think in the end she just skimmed them. They weren't the stuff of scandal, anyway; the entries we'd sampled when we first discovered them in the box at the back of the cupboard were about as revealing as they ever got; really just appointments, notes on what had happened that day, where dad had been, who he'd met. If there was a single indiscretion recorded there, I never found it. The same went for any but the most basic observation or idea; he'd kept those in the A4 pads.\n\nIt was at the bottom of the box containing dad's diaries, in an old presentation tin which had held a bottle of fifteen-year-old Laphroaig, that I found Rory's diaries; little pocket books, usually a week-per-two-pages. Dad must have filed them separately from the other papers.\n\nI got very excited at first, but then discovered that Rory's diaries were even more sparse \u2013 and considerably more cryptic \u2013 than my father's, with too many initials and acronyms to be easily understood, and too full of week\u2013 and even month-long gaps to form a reliable impression of Rory's life. There was no diary for the year he disappeared. I'd tried to make sense of Rory's diaries, but it was uphill work. The entry for the day of my birth (when Rory had been in London) read:\n\nK r; boy 8\u00a3. _Prentis?!?_ M ok Eve, pub.\n\nThe entry for the next day read: \"vho\" in shaky writing, and that was all. \"ho\" and \"vho\" (or sometimes h.o. and v.h.o.) often followed entries regarding pubs or parties the night before, and I strongly suspected they stood for hungover and very hungover. K meant Kenneth and M Mary, pretty obviously. ok was itself (its opposite was nsg, which stood for Not So Good; he'd spelled it out the first time he'd used it, following a \"48hr h.o.\" after Hogmanay the previous year). A small r meant \"rang\"; a telephone call. And I had indeed weighed in at eight pounds.\n\nI found a few mentions of \"CR\" \u2013 I even recognised some of the notes I'd read the previous year; he must have jotted them down in his diary first before transferring them to his other papers. But there was nothing to provide any new answers.\n\nThe one thing that stayed with me as a result was not a solution to anything, but rather another mystery. It was on a page at the back of the last diary, the diary for 1980; a page headlined by the mysterious message:\n\nJUST USE IT!\n\n... a page covered with notes, some in pencil, some in ball-point, some in very thin felt-tip, but a page which held the only instance anywhere in all the papers I had where Rory had made an effort not just to alter or score out some words or letters, but to obliterate them. It read:\n\nshow Hlvng pty wi C?\" (whoops): 2 close??\n\nThe symbols just before the H and C had been obliterated by a heavy black felt-tip marker, but the original note had been written with a ball-point, and by holding the page up to the light at just the right angle, I could see that the first letter had been an F and the second an L.\n\nF and L. Those abbrevations didn't turn up anywhere else in Rory's notes for either _Crow Road_ or anything else that I knew of. Rory _never_ crossed stuff right out; he only ever put a line through it. Why the big deal with the felt-tip? And who were F and L? And why that \"whoops\"? And what was too close to what?\n\nI found myself cursing Uncle Rory's inconsistency. F in the diaries sometimes meant Fergus (aka Fe), sometimes Fiona (also Fi), and sometimes Felicity, a girl Rory had known in London, also recorded as Fls, Fl or Fy (I guessed). The only L in the diaries seemed to be Lachlan Watt, though he \u2013 mentioned on the rare occasions when he came back to visit from Oz \u2013 was LW, more usually.\n\nSome nights at Lochgair, after long evenings spent poring over those little, thin-paged diaries on the broad desk in dad's study, trying to make sense of it all, and failing, I'd fall asleep in my bed with the symbols and acronyms, the letters and numbers and lines and boxes and doodles and smudges all swirling round in front of me even after I'd put the light out and closed my eyes, as though each scribbled sign had become a mote of dust and \u2013 by my reading \u2013 been disturbed; lifted from the page and blown around me in a vortex of microscopic info-debris, chaotic witnesses of a past that I could not comprehend.\n\nI found one thing which \u2013 after a little puzzled thought \u2014 I could comprehend, but which I hadn't been expecting, in Uncle Rory's 1979 diary. Stuck to the inside back cover with a yellowing stamp hinge was an old, faded, slightly grubby paper Lifeboat flag, without its pin.\n\nThe sentimentalist in me was reduced almost to tears.\n\nIn Glasgow I had taken to sitting in churches. It was mostly just for the atmosphere. Catholic churches were best because they felt more like temples, more involved with the business of religious observance. There was always stuff going on; candles burning, people going to confession, the smell of incense in the air ... I'd just sit there for a while, listening but not listening, seeing but not seeing, there but not there, and finding solace in the hushed commerce of other people's belief, absorbed in the comings and goings of the public and the priests, and their respective professions of faith. A father would approach me, now and again ... but I'd tell him I was just browsing.\n\nI walked a lot, dressed in my Docs and jeans and a long tweed coat that had been my father's. Uncle Hamish sent me thick letters full of original insights into the sacred scriptures, which I dipped into sometimes when I couldn't sleep. I never got further than page two of any of them. I frequented the Glasgow Film Theatre, and installed a video and a TV in the lounge. I bought a ghetto-blaster which usually lived in the flat's kitchen (and so became known as the gateaux-blaster) but which I would take walkabout with me sometimes, at least partly for the weight-training which transporting the brute from room to room provided. I'd stand and look at time-dark paintings, or run a finger over the line of some cold, marble animal, while the tall, glittering rooms resounded to the Pixies, REM, Goodbye Mr Mackenzie, The Fall and Faith No More.\n\n'He's here,' Ash said, coming back with the drinks. She sat down.\n\nI looked around. I saw him after a while. A little shorter and a little younger-looking than I'd expected, from the tape I'd seen. He was talking to a couple of other guys; they were all dressed in grey trench coats, and one had put a hat down on the bar that at least looked like it ought to be called a fedora. I wondered if the other two were also journalists.\n\nRupert Paxton-Marr; a foreign correspondent, his meticulously-trained, razor-sharp mind ready in an instant to describe a place as 'war-torn' and bring home to us all events and disasters in far away places, to talk of people tearing at the rubble with their bare hands, to reveal that only with dawn did the full extent of the devastation become apparent, and even \u2013 in the very best traditions of British popular journalism \u2013 to ask people who'd just seen their entire family duly butchered, burned, crushed or drowned, How do you feel?\n\nAsh seemed contemplative, eyeing me with a steady gaze. 'Well ...' I said, feeling my heart beat faster and my palms start to sweat. I took the two torn match-book covers out of my pocket. 'Think I'll go see what he has to say for himself.'\n\n'Want me to come?' Ash started to move in her seat.\n\nI shook my head. Then bit my lip. 'Shit, I don't know. All the way down here, I was just going to go up to him and say, \"You send these to my dad?\", but now I don't know. It feels a bit weird.' I looked over at the three men. 'I mean,' I laughed. 'They're even wearing trench-coats!'\n\nAsh looked briefly over too. 'Hey,' she said, smiling. 'They're on wine; they're not just knocking back whiskies and heading off. They'll be here a while yet. Sit and think for a moment.'\n\nI nodded, took a deep breath and drank some whisky.\n\nI thought about it some more. Then I said, 'Okay. Maybe we should go together. You could sort of introduce ... I could go out and pretend to just come in ... Hell; I could just tell him the truth ... I don't know.' I closed my eyes, appalled at my own lack of gumption.\n\nAsh got up, putting a hand on my shoulder. 'Sit here. I'll tell him I've just recognised him. You come over later; just mention the match-books. Don't show them, not at first. How does that sound?'\n\nI opened my eyes. I shook my head and said, 'Oh, I don't know, good as anything.'\n\n'Right.' Ash went over to the men. She pulled something from the back of her head as she went, and shook her long fawn hair down. It was the length of the jacket. I smiled to myself. That's my girl, I thought.\n\nI saw them look her up and down. Rupert smiled, then looked mystified as she talked, animated, hands waving. Then he laughed, his tanned, handsome face smiled and he looked her up and down again. The expression changed just a little, though, after that, as though something else had occurred to him. He looked a little more wary. So it appeared to me, anyway. He held out one hand, seeming to make introductions. Ash nodded. He pointed to the bar; she shook her head, then nodded back at me.\n\nRupert Paxton-Marr gazed above me, then dropped his gaze. He looked at me then back at Ashley. She was talking to him. His expression went through puzzlement, maybe concern, then went wary again, finally cold, studiously expressionless. He nodded, leaning back against a post supporting the front of the bar. Ash glanced back at me, her eyes opening wide for an instant, then she turned back to the men.\n\nI started to get to my feet.\n\nRupert's expression didn't change as I walked over. Two couples passed in front of me, weaving their way between the tables. When they'd passed, Rupert was already on his way to the door, mouth smiling broadly, one hand alternately waving and pointing at his watch as he backed off. By the time I got to where Ash and the two guys in the trench coats were standing, he'd made it out to the street.\n\nI stood there, frowning at the door Rupert Paxton-Marr had exited through. Something about the way he'd moved as he'd backed off had left me with an uncanny feeling of d\u00e9j\u00e0 vu.\n\nAsh looked surprised. So did the two guys. One of them looked me up and down. 'Jesus Christ,' he said. 'How'd you do that? Usually only women with toddlers screaming \"Daddy!\" in tow have that sort of effect on Rupe.'\n\n_Remember, remember,_ I thought to myself, and smiled. I turned to the man and shrugged. 'It's a gift,' I told him.\n\n'He owe you money or somefink?' the second man said. They were both about thirty, lean and clean-cut. Both were smoking.\n\nI shook my head.\n\nAsh laughed loudly. 'No,' she said, holding her hands out to the two men. 'It's just that the last time we all met up, we all got filthy drunk \u2013 didn't we, Presley? \u2013 and Rupert thinks Presley here \u2014 '\n\nPresley? Ash was indicating me when she said the name. _Presley?_ I thought.\n\n'... thinks that Rupert tried to proposition him. Which he didn't, of course, but it was all a little embarrassing, wasn't it, dear?' Her happy, smiling face looked demandingly at me.\n\nI nodded dumbly as the two men looked at me as well.\n\n'Embarrassing,' I confirmed.\n\nAsh was beaming smiles all over the place like a laser gone berserk. 'I mean,' she said, tossing her hair. 'Rupert isn't gay, is he? And Presley ...' She looked suddenly sultry, voice slowing, going a little deeper. 'Here ...' She took an extra breath, her gaze flickering down from my face to my crotch and back, '... _certainly_ isn't.'\n\nThen she seemed to collect herself and directed a broad smile to the two men. They looked suitably confused.\n\n_'Presley?_ PRESLEY?' I yelled as we walked rapidly along Thomas More Street. 'How _could_ you?' I waved my hands about. A light drizzle was falling out of the orange-black sky.\n\nAshley strode on, grinning. She held a small umbrella; her heels clicked. 'Sorry, Prentice; it was just the first thing I thought of.'\n\n'But it isn't even very _different_ from Prentice!' I shouted.\n\nShe shrugged. 'Well then, that's probably why it was the first thing I thought of.' Ash laughed.\n\n'It's not funny,' I told her, sticking my hands into my pockets, stepping over some empty pizza containers.\n\n'It wasn't funny,' Ash agreed, almost prim. 'It's your reaction that is.' She nodded.\n\n'Great,' I said. 'There are two guys going around now who think my name is Presley, but to you it's just a hoot.' I stepped on a wobbly paving stone and jetted dirty water up my chinos. 'Jeez,' I muttered.\n\n'Look,' said Ash, sounding serious at last. 'More to the point, I'm sorry I fucked that up. I don't know _why_ he dashed off like that. All I said was I'd a friend with me. I didn't even say you wanted to meet him or anything. It was weird.' She shook her head. 'Weird.'\n\nWe had escaped from the pub after finishing our drinks and chatting \u2013 awkwardly on my part, easily on Ashley's \u2013 with Rupert's two friends (Howard and Jules); a stilted conversation whose most useful result seemed to have been a general agreement that old Rupe was a lad, eh?\n\n'Doesn't matter,' I told her. I saw a taxi coming with its light on and suddenly remembered I was rich. 'I know where I saw him, now.'\n\nI stepped into the road and waved.\n\n'You do?' Ash said from the kerb.\n\n'Yep.' The cab pulled in. Things were looking up; my usual Klingon Cloaking Device \u2013 which has tended to engage automatically on the rare occasions I have felt rich enough in the past to afford a taxi \u2013 seemed to have been de-activated. I held the door open for Ashley.\n\n'So; you going to tell me, or be all mysterious?' she said as she got in.\n\n'I'll tell you over dinner.' I sat beside her and closed the door. 'Dean Street, Soho, please,' I told the driver. I smiled at Ashley.\n\n'Dean Street?' she said, eyebrow arching.\n\n'Amongst many other things, I owe you a curry.'\n\nWhen I was fifteen I had my first really bad hangover. On Friday nights I and some of my school pals used to meet at the Droid family house in Gallanach; we'd sit in Droid's bedroom, watching TV and playing computer games. And we'd drink cider, which Droid's big brother purchased for us \u2013 for a small commission \u2013 from the local off-licence. And smoke dope, which my cousin Josh McHoan, Uncle Hamish's son, purchased for us \u2013 at an exorbitant commission \u2013 in the Jacobite Bar. And sometimes do speed, which came from the latter source as well. Then one night Dave McGaw turned up with a litre of Bacardi and he and I finished it between the two of us, and the next morning I was woken up by my dad to a strange and horrible new feeling.\n\nvho, as Rory would have written. nsg at all.\n\nThere had been a phone-call for me; Hugh Robb, from the farm near the castle, reminding me I'd agreed to come and help with making the bonfire for Guy Fawkes' night. He was coming out to pick me up.\n\nThis, of course, was not really what I needed (any more than I needed dad lecturing me on how unsound a custom it was to build bonfires on November the fifth and so celebrate religious bigotry; didn't I know it had been an anti-catholic ceremony, and the effigy burned on the fire used to be the Pope?), but I couldn't admit to mum and dad I'd been drinking and had a hangover, so I had to get dressed with my head pounding and my insides feeling distinctly unwell. I waited outside on the porch steps, taking deep breaths in the cool clear air and wishing the hangover would just go away. Then I suddenly thought maybe it wasn't a hangover; maybe this pounding in my head was the first symptom of a brain tumour ... and so I ended up praying that I _did_ have a hangover.\n\nHugh Robb was a big, amiable Scotch Broth of a lad; he was a full year older than I was but we were in the same class at school because he'd been kept back a year. He arrived in a tractor hauling a trailer full of branches and old wood and I rode with him in the cab, wishing that the tractor had better suspension and that Hugh could have thought of something else to talk about other than the prolapsed uterus of one of the farm's cows.\n\nRound the hill from the castle there was a big east-facing field; it was surrounded by trees on all sides but the slope gave it a view towards Bridgend. I still thought of it as the ponies' field because it was where Helen and Diana's ponies had been stabled originally before they'd been moved to a more level paddock west of the castle.\n\nHugh and I unloaded the broken planks and the great bare grey branches from the trailer. We worked together for a bit, then I continued to stack the wood while Hugh went to collect some more. He made a couple of trips, dumping what looked like about a tonne of wood each time before announcing he was off to another farm where they had even more wood.\n\nI let the tractor disappear, bumping along the track towards the castle, then collapsed back in the huge pile of branches awaiting my attention. I lay, spread-eagled and half-submerged on the springy mass of grey, leaf-nude wood and stared up at the wide blue November sky, hoping the bass drum inside my head would hit a few thousand rest-bars reasonably soon.\n\nThe sky seemed to beat in time to the throbbing inside my head, the whole blue vault pulsing like some living membrane. I thought about Uncle Rory and his discovery that it was not possible to influence TV screens from afar by humming. I wondered \u2013 as ever \u2014 where he was; he'd been gone a couple of years by that time.\n\nA bird swung into view over the trees behind my head, and I lay there and I watched it; broad, flat-winged, flight feathers at the square wing-tip ruffling like soft fingers, the small, quick head flicking this way and that, the brown-grey body between the soft density of wings tilting and turning as it glided the cool air, tail feathers like a rich brown fan.\n\n'Beauty,' I whispered to myself, smiling despite the pain in my head.\n\nThen suddenly the buzzard burst and sprayed across the sky; it fell plummeting, limp and trailing feathers, to the ground. A double crack of sound snapped across the field.\n\nThe bird fell out of sight behind me. I blinked, not believing what had happened, then rolled over, looking through the mask of branches at the trees edging the field where the sound of the shots had come from. I saw a man holding a shotgun, just inside the trees, looking to one side then the other, then running out into the field. He wore green strapped wellies, thick brown cords, a waxed jacket with a corduroy collar, and a cloth bunnet. One more prick in a Barbour jacket, but this one had just shot a buzzard.\n\nHe gazed down at something in the grass, then smiled. He was tall and blond and he looked like a male model; enviable jaw line. He stamped down on the thing in the grass, looked around again then backed off, finally turning and walking smartly back into the woods.\n\nI should have shouted, or taken the bird to the police as evidence \u2013 buzzards are a protected species, after all \u2013 but I didn't. I just watched the Barbour disappear into the trees, then rolled over and breathed, 'Fuckwit.'\n\nHe was at the firework party the following night, laughing and talking and sharing a dram from Fergus's hip flask. I watched him, and he saw me, and we looked at each other for a few moments before he looked away, all in the furious, writhing light of the pyre that I had put together, and which contained \u2013 pushed in near its now blazing centre \u2013 the corpse of the bird he had killed.\nCHAPTER 16\n\nWe stood beside the observatory dome, on the battlements of Castle Gaineamh, facing into a cool westerly breeze. Lewis, in cords and a grease-brown stockman's coat, looked through the binoculars, his black hair moving slightly in the wind. Verity stood at his side, face raised shining to the winter blue sky, bulky in her thermal jacket, her ski-gloved hands clasped thickly under the bulge of her belly. The plain beyond the woods below, holding Gallanach and cupping the inner bay, was bathed in the deep-shadowed sunlight of late afternoon. Wisps of cirrus moved high above, tails trailing up, promising clear weather. A two-coach sprinter moved in the distance, on the viaduct at Bridgend, windows glinting in the sunlight. I took a deep breath and could smell the sea.\n\nThe unopened air-mail packet from Colorado, lodged next to my chest between shirt and jacket, make a crinkling, flexing noise, giving me a funny feeling in my belly.\n\n'No sign?' Verity asked.\n\nLewis shook his head. 'Mm-mm.'\n\nVerity shivered. She hunched her shoulders, bringing them up and in towards her neck. 'Brr,' she went. She linked arms with Lewis.\n\n'Ah,' he said, protesting, still looking through the field glasses, though now at a slight angle.\n\nVerity tutted, and with a gorgeously pretended scowl moved away from her husband and stepped over to me. She slid an arm round my waist, snuggling. I put an arm round her shoulders. She rested her head against my arm; I looked down at her. She was growing her hair a little. The sides of her head weren't actually shaved any more. She smelled of baby oil; Lewis had what sounded like the enviable job of smoothing it over The Bulge, in an attempt to fend off stretch marks later. I smiled, unseen, and looked back to the north.\n\n'Is that what-do-you-call-it?' Verity said, nodding.\n\n'No, that's thingy-ma-bob,' Lewis said, just as I said,\n\n'Hey; well remembered.'\n\n'Dunadd,' Verity said patiently, ignoring both of us. She was looking at the small, rocky hill a kilometre to the north. 'Where the footprint is.'\n\n'Correct,' I said.\n\nLewis glanced at us, grinned. He lowered the binoculars a little. 'Can't see it from here, but that's where it is.'\n\nDunadd Rock had been the capital of Dalriada, one of the early and formative kingdoms in Scotland. The footprint \u2013 looks more like a bootprint, actually, just a smooth hollow in the stone \u2013 was where the new king had to place his foot when he made his vows, symbolically \u2014 I suppose \u2014 to join him to the land.\n\n'Can I have a look?' Verity said. Lewis handed her the glasses, and she leant against the stone battlements, supporting her belly. Lewis stood behind her, chin lowered onto her shoulder.\n\n'Right at the summit, isn't it?' Verity asked.\n\n'Yep,' Lewis said.\n\nShe looked at Dunadd for a bit. 'I wonder,' she said, 'if you had one of your feet planted there, when you gave birth ...'\n\nI laughed. Lewis went wide-eyed, drawing up and back from his wife. She turned round, grinning wickedly at Lewis and then me. She patted Lewis's elbow. 'Joke,' she said. 'I want to be in a nice warm birthing pool in a nice big hospital.' She turned back to the view. Lewis looked at me.\n\n'Had me fooled,' I shrugged. 'Runs in the family, after all.'\n\n'Can you see that stone circle, too?' Verity said, lifting the binoculars to gaze further north.\n\nEarlier that day, Helen Urvill, Verity and Lewis and I had been behaving like tourists. The land around Gallanach is thick with ancient monuments; burial sites, standing stones, henges and strangely carved rocks; you can hardly put a foot down without stepping on something that had religious significance to somebody sometime. Verity had heard of all this ancient stoneware but she'd never really seen it properly; her visits to Gallanach in the past had been busy with other things, and about the only place she had been to before was Dunadd, because it was an easy walk from the castle. And of course, because we had lived here most of our lives, none of the rest of us had bothered to visit half the places either.\n\nSo we borrowed Fergus's Range Rover and went site-seeing; tramping through muddy fields to the hummocks that were funeral barrows, looking up at moss-covered standing stones, plodding round stone circles and chambered cairns, and leaning on fences staring at the great flat faces of cup-and-ring marked rocks, their grainy surfaces covered in the concentric circular symbols that looked like ripples from something fallen in a pond, frozen in stone.\n\n'Did I ever tell you about the time I used to be able to make televisions go wonky, from far away?'\n\nIt was a bright and warm day, back in that same summer Rory had come out to the Hebrides with us. Rory and I were walking near Gallanach, going from the marked rocks in one field to the stone circle in another. I remember I had a pain in my side that day and I was worrying that it was appendicitis (one of the boys in my class that year had almost died when his appendix had ruptured). It was just a stitch, though. Uncle Rory was a fast walker and I'd been intent on keeping up with him; my appendix waited another year before it needed taking out.\n\nWe had been visiting some of the ancient monuments in the area, and had started talking about what the people who'd built the cairns and stone circles had believed in, and that had led us on to astrology. Then suddenly he mentioned this thing about televisions.\n\n'Making them go _wonky?'_ I said. 'No.'\n\n'Well,' Rory said, then turned and looked behind us. We stood up on the verge as a couple of cars passed us. It was hot; I took off my jacket. 'Well,' Rory repeated, 'I was ... a few years older than you are now, I guess. I was over at a friend's house, and there was a bunch of us watching _Top of the Pops_ or something, and I was humming along with a record. I hit a certain deep note, and the TV screen went wavy. Nobody else said anything, and I wondered if it was just coincidence, so I tried to do it again, and after a bit of adjusting I hit the right note and sure enough, the screen went wavy again. Still nobody said anything.' Rory laughed at the memory. He was wearing jeans and T-shirt and carried a light jacket over his shoulder.\n\n'Well, I didn't want to make a fool of myself, so I didn't say anything. I thought maybe it just worked on that one particular television set, so I tried it at home; and it still happened. The effect seemed to work from quite a distance, too. When I stood out in the hall and looked into the lounge, it was still there, stronger than ever.\n\n'Then we were going up to Glasgow, mum and I, and we were passing a shop window full of TVs, and so I tried this new gift for messing up TV screens on them, and hummed away to myself, and all the screens went wild! And I was thinking Great, I really can do magic! The effect is getting stronger! I could appear on TV and do this! Maybe it would make everybody's screens go weird!'\n\n'Wow,' I said, wanting to get home and try this myself.\n\n'So,' Rory said. 'I stopped in my tracks and I asked mum. I said, \"Mum; watch this. Watch those screens.\" And I hummed for all I was worth, and the pictures on the screens went wavy. And mum just looked at me and said, \"What?\" And I did it again, but no matter how hard I tried I couldn't get her to see the effect. Eventually she got fed up with me and told me to stop being silly. I had screens going mental in every TV shop we passed in Glasgow that day, but nobody else seemed to be able to see it.'\n\nRory grimaced, looking across the edge of the plain beyond Gallanach to the little rocky hill that stuck up from the flat fields.\n\n'Now, I wish I could remember just what it was that made the penny drop, but I can't. I mean, usually a beautiful assistant says something stupid and the clever scientist says, \"Say that again!\" and then comes up with the brilliant plan that's going to save the world as we know it ... but as far as I remember it just came to me.'\n\n'What?' I said.\n\nRory grinned down at me. 'Vibrations,' he said.\n\n'Vibrations?'\n\n'Yeah. The vibrations I was setting up in my own skull \u2013 actually in the eyeball, I suppose \u2013 were making my eyes vibrate at about the same frequency as the TV screen flickers. So the screen looked funny, _but only to me,_ that was the point. And it made sense that the further away you were from the screen \u2013 as long as you could still make it out, of course \u2013 the more pronounced the effect would appear.' He looked down at me. 'You see?'\n\n'Yeah,' I said, 'I think so.' I studied the road for a bit, then looked up, disappointed. 'So it doesn't really work after all?'\n\nRory shook his head. 'Not the way I thought it did, no,' he said.\n\nI frowned, trying to remember how we'd got onto this. 'What's that got to do with what we were talking about?' I asked.\n\nRory looked at me. 'Ah-ha,' he said, and winked. He nodded at a gate set in the low wall facing the road; beyond were the standing stones. 'Here we are.'\n\n'Here we are.' The stair-door creaked open; I went to help Helen. She handed me a tray with four pewter mugs. The mulled wine steamed; it smelled wonderful. 'Mmm, great,' I said. Helen took my hand and stepped out through the little half-size door onto the battlements. Her broad face was tanned and her body looked lean and fit after some early-season skiing in Switzerland. She wore Meindl boots, an old pair of leather trousers that had belonged to her mother, a cashmere sweater and a flying jacket that looked distressed enough to have seen action over Korea, if not Greater Germany. Her hair was shining black and shoulder length.\n\nShe took a mug. 'Help yourself,' she said. 'Any sign of him?'\n\n'Nope,' I said. I held the tray out to Lewis and Verity, who made the appropriate noises and took a mug each.\n\nHelen nodded at the corner of the airmail package inside my jacket. 'Still incubating that, Prentice?'\n\nI grinned. 'Yeah.'\n\nWe stood there sipping the hot, spicy wine, looking north.\n\n'Prentice? Ash. He's done it.'\n\n'Who? Doctor Gonzo?'\n\n'Yeah; printing it out now; mail you the hard copy tomorrow morning; he'll E-mail it to me too, I'll download the files and stick them on a disk your Compaq'll accept and bring it with me next week when I come up for Hogmanay ... unless you got a modem yet, did you?'\n\n'No, no I didn't.'\n\n'Okay; that'll be the arrangement, then. Sound all right to you?'\n\n'Yeah; great. I'll write back to say thanks. He, ah ... say what the files actually were in the end?'\n\n'Text.'\n\n'That all he said?'\n\n'Yep. Prentice, he hasn't read whatever he found; well, probably not more than the first few lines to check they were in English, not gibberish. Once he'd cracked it I don't think he was really interested in what was actually written there. But it is text.'\n\n'Right. Text.'\n\n'Should get to you in a few days, airmail.'\n\nThe big envelope from the States had arrived in the mail this morning, five days after Ash had called; the return address was Dr G, Computing Science Faculty, University of Denver, Co. I'd stared at the thing as it lay there on the front door mat, my mouth gone oddly dry. I had a slight hangover, and had decided \u2013 as I gingerly picked the disappointingly slim package up \u2013 that I'd open it after breakfast. Then after breakfast I'd thought maybe I should leave it until later, especially when Verity rang and invited me over to the castle.\n\nIt was the last day of 1990; a full twelve months after that fateful party when Mrs McSpadden had sent me down to the cellar for some more whisky. We were all back here for the usual round of parties and visits and hangovers. I was mostly looking forward to it all, even though I was still trying to find reasons not to ask Fergus the things I knew I ought to ask him about Mr Rupert Paxton-Marr. But then, maybe I oughtn't to ask them at all. Maybe what was in this airmail package would relieve me of the need to ask any questions (I told myself). I kept coming back to the distinct possibility that maybe I was making something out of nothing, treating our recent, local history like some past age, and looking too assiduously, too imaginatively for links and patterns and connections, and so turning myself into some sort of small-scale conspiracy theorist.\n\nI had been immersed in my studies all term, and everything seemed to be going well. My professor, reading my tutorial papers and essays, had gone beyond noises of encouragement and vicarious complacency to a sort of uncomprehending peevishness that I'd contrived to fail so spectacularly the year before.\n\nMeanwhile, in the history that we were currently living through, it looked like a war was going to start in just over two weeks, but \u2013 apart from a kind of low background radiation of species-ashamed despair because of that \u2014 personally I felt not too bad. Mum appeared to be holding out, despite Christmas and New Year traditionally being a bad time for the bereaved. She had actually started building the much talked-about harpsichord, turning a spare bedroom at Lochgair into a workshop which at the moment looked suitably chaotic; James had mostly rejoined humanity, to the extent that several times over the last few weekends before Christmas he'd taken his Walkman phones off long enough to have what could, with only a little generosity, be described as a conversation. Lewis was doing well, Verity was almost disgustingly healthy (apart from the occasional sore back and a bladder that appeared to have become inordinately susceptible to the sound of running water), and I actually looked forward to seeing Lewis and Verity now, much to my own amazement; there was still a distant pang when Verity smiled at me ... but it was more remembered than real.\n\nI looked at my watch. In four hours I'd be setting off for Glasgow Airport, for Ashley. She was booked on a late flight and I'd volunteered to go pick her up. She'd be working until half-five in London this evening, and it would have been pushing the old 2CV a bit to get to Scotland \u2013 let alone here \u2013 in time for the bells.\n\nI scanned the skies to the north, watching for movement. I was looking forward to the drive, even if it was at night and I wouldn't be able to see the scenery.\n\nThe wind gusted a little and I supped my mulled wine. The woods \u2014 evergreen and deciduous-bare \u2013 swept down to the fields and then the town; forests rose on the hills to the east.\n\n'Anybody heard the news today?' Lewis asked.\n\n'Nothing special happening,' Helen said.\n\nI guessed she'd got an up-date from Mrs McSpadden, who tended to keep the TV on in the kitchen these days.\n\n'All quiet on the desert front,' breathed Lewis, taking up the glasses again and looking north towards Kilmartin.\n\n'You sure he'll come that way?' Verity asked.\n\nLewis shrugged. 'Think so.'\n\n'Said he would,' Helen confirmed.\n\nVerity stamped her feet.\n\n'Hey,' Helen said. 'I never asked you, Lewis; you got any Gulf jokes?'\n\nLewis made an exasperated noise, still looking through the binoculars. 'Na. I heard a couple of crap Irish ones, and the usual suspects in different disguises, but there hasn't been anything good. I was trying to work on a routine about if the Stealth bomber worked as well as it did in Panama, the B-52 as it did in Viet Nam and the marines as effectively as they did in Lebanon, then Saddam had nothing too much to worry about, but it wasn't funny enough.' He brought the glasses down from his eyes for a moment. 'In fact, it wasn't funny at all.'\n\n'I know a girl from school who's out there,' Helen said. 'Nurse.'\n\n'Yeah?' Verity said, stamping her feet again.\n\n'Ha!' Lewis said suddenly.\n\n'You seen him?' Verity said, clutching Lewis's arm.\n\nHe laughed, glancing down at her. 'No,' he said, and grinned at me. 'But guess who got called up as a reservist?'\n\nI shrugged. 'I give in.' I didn't think I knew anybody in the forces.\n\nLewis smiled sourly. 'Jimmy Turrock. Used to be a bandsman. They're the stretcher-bearers in war time.'\n\nI frowned, not recognising the name. 'Jimmy \u2013?' I began. Then I remembered.\n\n'The grave-digger!' I laughed.\n\n'Yeah,' Lewis said, turning away again, raising the glasses. 'The grave-digger.'\n\nI felt cold inside and my smile faded. 'Wow,' I said. 'Some sense of humour the army has.'\n\n'Work experience,' Lewis muttered.\n\n'Who's this you're talking about?' Verity asked.\n\n'Guy helped us bury dad,' Lewis said.\n\n'Oh,' she said. She hugged Lewis.\n\n'You going to go if they call you up, Prentice?' Helen Urvill said, not looking at me.\n\n'Hell no,' I said. 'Which way's Canada?'\n\n'Yeah,' muttered Lewis. 'Shame dad's dead; maybe he could have got us into the equivalent of the National Guard, if we have one.'\n\n'Traffic wardens?' I suggested. Lewis's shoulders shook once.\n\n'You really wouldn't go?' Helen said to me, one eyebrow raised.\n\n'I might send them some blood if they ask me nicely,' I told her. 'In an oil can.'\n\n'I suppose we can't use the telescope, can we?' Verity said suddenly, nodding at the white dome to our right. 'As well as the binoculars I mean.' She looked at each of us.\n\n'Nup,' Helen said.\n\n'Too narrow a field of view,' Lewis said.\n\n'And upside down,' I added.\n\nVerity looked over at the dome. There was a smile on her face. 'Do you remember that night we met in the dome?' she said, looking up at Lewis. 'We hadn't seen each other since we were kids ...'\n\nLewis handed the glasses to Helen, who held them one-handed, straps dangling. Lewis hugged his wife. 'Of course I do,' he said, and kissed her nose. She buried her face in his coat. I looked away, thinking about the drive up to the airport this evening.\n\nMaybe I should allow another half-hour or so for the journey, just in case of hold-ups. And of course they were building new bits onto the airport at the moment; could be a problem parking, and tonight was bound to be busy. I'd leave early, no sense in leaving late and having to hurry. I had taken to driving a bit slower and more carefully these days. Mum still worried, but at least I could reassure her with a clear conscience.\n\nI sighed, and the package against my chest flexed again. I looked down at it. Hell, this was silly; I ought to read the stuff. Waiting until I got back to the house and was sitting at the desk in the study was just putting it off.\n\n'How's the wonderful world of Swiss banking these days anyway, Hel?' Lewis asked.\n\n'Oh, wacky and transparent as ever,' Helen said. They started talking about Z\u00fcrich and London, and I sat down on the slope of slated roof, behind them. I pulled the airmail package out and opened it carefully. Verity looked back at me and smiled briefly, before turning back to Helen and Lewis to share some joke about the Hard Rock Caf\u00e9. My hands felt clammy as I slid the sheets of paper out of the thick white envelope. This is daft, I thought. It probably is gibberish, or Rory's job application for that travel programme presenter's job; a CV for the TV. Nothing important, nothing revelatory.\n\nThe first sheet was a letter from the good Doctor, arcane with acronyms and abbreviations, telling me how he'd deciphered the binary mush he'd been sent and turned it into what I held in my hands. He sounded like a likeable guy, but I kind of just glanced at the letter. I went on to the print-out.\n\nThere were about fifty or sixty pages of single-space laser print. The first twenty or so pages were taken up with pieces I recognised: articles and poems and the nameless play Rory had apparently decided to cannibalise for the end of Crow Road. Then came three passages of prose.\n\nI glanced up at the others; Helen and Verity were still talking, Lewis was looking through the binoculars towards Gallanach. I started reading, and my mouth went dry.\n\nI raced through each of the passages, my eyes bulging, hands shaking. The voices of the others, the cool December air and the chill slates under my backside seemed like they were all a million miles away, as I read what Uncle Rory had written.\n\n_'D'you know where the twins were conceived?'_\n\n_'No idea,' he said, and belched._\n\n_'Fucking McCaig's Folly, that's where.'_\n\n_'What, Oban?'_\n\n_'The very place.'_\n\n_'Good grief.'_\n\n_'You don't. mind me saying this, I mean talking about Fiona like this, do you?'_\n\n_'No, no.' He waved one hand. 'Your wife; you talk about her. No, no, that's bad, that sounds bad. I'm all for women's lib.'_\n\n_'Might have bloody known. Might have bloody known you would be. Bloody typical, if you ask me. You're a Bolshie bastard, McHoan.'_\n\n_'And you are the unacceptable face of Capitalism, Ferg.'_\n\n... That was how the first passage began. I finished it and realised my mouth was hanging open. I closed it and started, dazed, the next passage:\n\n_'Henriss ... never liked him either; fat lipped beggar... queer, y'know; thass wha he's singing you know; d'you know that? \"Scuse me while I kiss this guy ...\" disgussin ... absluley disgussin ...'_\n\n_'Fergus, do shut up.'_\n\n_' \"Scuse me, while I kiss this guy\"... bloody poofter coon.'_\n\n_'I'm sorry about this, Lachy.'_\n\n_'That's okay, Mrs U. You no going to put your seat belt on, no?'_\n\n_'No; not for short journeys \u2014 '_\n\n_'Lachy? Lachy ... Lachy! Lachy; I'm sorry about your eye ... really really sorry; never forgave myself, never... here, shake...'_\n\n'Holy fucking shit,' I whispered, when I finished it. Suddenly my hands felt very cold. I looked at the slates I was sitting on, then over at the dome of the observatory, gleaming in the low winter sun.\n\n'You okay, Prentice?' Verity said, frowning at me from the battlements.\n\nI nodded, tried to smile. 'Fine,' I gulped. I turned to the third and last passage.\n\n_Fiona sat in the passenger seat of the car, watching the red roadside reflectors as they drifted out of the night towards her; she was thrown against one side of the seat as Fergus powered the Aston around the right-hander ..._\n\n... And on through to the end:\n\n... _'Look \u2014 !'_\n\nAnd that was all. I looked up, brain reeling.\n\n'Yo,' Helen said, looking through the binoculars. She bent at the knees and put her mug down on the stones under her feet, then rose smoothly again.\n\n'You see him?' Verity said, turning, still hugged within Lewis' arms, to look out over the battlements.\n\n'Could be,' Helen said. She handed the field glasses over to Lewis.\n\n'Yeah, might be,' he said. It was Verity's turn next with the binoculars.\n\nI swallowed a few times, put the sheets of paper back in their envelope. I stood up and walked over to the others, in a kind of trance.\n\nVerity shook her head. 'Na, I can't see the damn thing.' She handed the glasses to me. 'You're looking pale, Prentice. You sure you're okay?'\n\n'Fine,' I croaked, not looking at her. I took the binoculars. 'Thanks.'\n\nI'd seen the speck unaided by that time. Once I'd found it again the binoculars enlarged the dot into the frontal silhouette of a high-winged light aircraft, flying more or less straight towards us, its body pointed a little to the south west to compensate for the wind. It waggled in the air a little as it flew down the glen, encountering a gust high above Kilmartin.\n\n'Christ,' Lewis said. 'It's a Mig on a bombing run; everybody down!'\n\nI handed the glasses back to Helen, who didn't look particularly amused. She frowned at me. 'You okay, Prentice?'\n\n'Fine,' I said.\n\n'You should have loaned your dad your jacket,' Lewis told Helen.\n\n'Doesn't fit him,' Helen said, binoculars at her eyes. I watched the dot of the plane drift closer towards us through the northern sky.\n\n'You were in a sleeping bag,' I heard Lewis say softly to Verity. He was holding her from behind, chin on the crown of her head. I must have missed what they'd said earlier. I felt weird; I was glad the battlements were too high to fall over if I fainted.\n\nVerity smiled. 'I remember. We were all smoking and playing cards and taking turns to look at the stars, and we got the munchies.' She frowned. 'There was Diana and Helen, and ... what was that guy's name?' She glanced round and up at Lewis. 'Wayne somebody?'\n\n'Darren somebody,' Lewis said. He accepted the glasses from Helen, held them with one hand and balanced them on Verity's head. 'Hoy, stand still.'\n\n'Sorry, _sir,'_ she said.\n\n'Darren Watt,' I said. The plane was closer now but harder to see; it had dropped below the level of the hills behind and was no longer silhouetted against the sky. You could still see it with the naked eye, though. It glinted, once.\n\nVerity nodded. Lewis tutted in exasperation. 'He was the gay guy, wasn't he?' Verity said.\n\n'Yup,' Lewis said. 'Sculptor. Good, too; fucking shame, that was.'\n\n'Oh God,' Verity said. 'Of course, he died.'\n\n'Bike crash,' Helen said, scooping up her mug of cooled wine from the flagstones, and draining it.\n\nThe plane was flying over Gallanach now. I thought I could hear its engine. I remembered standing here once with mum, years ago. Fergus and dad were shooting at clay pigeons in a field to our right somewhere, and I remembered hearing the flat Crack ... Crack noise of the guns, and thinking they sounded just like one plank falling on top of another. Blam! indeed. _Remember, remember ..._\n\nVerity laughed, making Lewis tut again. 'You were doing your radio impressions,' she said, 'that night. Remember?'\n\n'Of course,' Lewis said.\n\n'Why was I in a sleeping bag?' Verity said, frowning at the approaching plane.\n\n'You were in the cupboard.' Helen smiled. She waved out across the chill afternoon air above Gallanach. I looked back at the plane, which was switching its lights on and off.\n\n'Oh,' Verity said. 'Yeah; the wee cubby hole.'\n\n'Ah ha,' Helen said, as Lewis waved too, still watching through the binoculars, now elevated above Verity's head. 'But it was really a secret passage.'\n\n'Was it?' Verity asked, glancing at Helen.\n\n'Yeah. Di and I used to take the bit of wood off at the back and get into the attic. Wander all over.'\n\n'Anything interesting in there?' Lewis asked. The plane was in a shallow dive, angling towards us a few hundred metres away.\n\n'Just pipes and tanks,' Helen shrugged. 'There was a loft door into mum and dad's room.' She smiled. 'When we started getting interested in sex, we used to pretend we'd get up there one night and see if we could catch them at it, but we were too frightened.' Helen laughed lightly. 'Had us giggling ourselves to sleep a few nights, though. And anyway, Ferg had put a bolt on it.'\n\nThe little white Cessna roared overhead, waggling its wings. Lewis and Verity and Helen all waved. I stared up, seeing the single tiny figure waving in the cockpit. The plane banked, circled round the hill the castle stood on and came back over, lower, engine loud and echoing in the woods beneath.\n\nI made myself wave.\n\nOh dear fucking holy shit, I thought.\n\nThe plane waggled its wings again, then straightened out over Dunadd as Fergus took the Cessna \u2013 his Christmas present to himself \u2013 back north to its home at Connel.\n\n'That it?' said Verity.\n\n'Yup,' Helen said.\n\n'What did you expect?' Lewis asked. 'A crash?'\n\n'Oh ...' said Verity, heading for the door to the stairs. 'Let's get back in the warm.'\n\nBlam! _Remember, remember._ Amman Hilton. Look \u2013! JUST USE IT Kiss the sky, you idiot ...\n\n'Prentice?' Lewis said, from the little door. I looked over at him. 'Prentice?' he said again. 'Wake up, Prentice.'\n\nI'd been staring after the departing plane.\n\n'Oh,' I said. 'Yeah.' On still shaky legs, I followed the others down from the wind-blown battlements and into the warm bulk of the great stone building.\n\n'So the televisions weren't going wonky at all,' I said, still struggling to understand.\n\n'That's right,' Rory said. 'It just looked like it, to me only.' He plucked a long piece of grass from beside one of the standing stones and sucked on the yellow stalk.\n\nI followed suit. 'So it was in your head; not real?'\n\n'Well ...' Rory frowned, turning away a little and leaning back on the great stone. He folded his arms and looked out towards the steep little hill that was Dunadd. I stood to one side, watching him. His eyes looked old.\n\n'Things in your head can be real,' he said, not looking at me. 'And even when they aren't, sometimes they ...' he looked down at me, and I thought he looked troubled. 'Somebody told me something once,' he said. 'And it sounded like it had really hurt him; he'd seen something that made him feel betrayed and hurt by somebody he was very close to, and I felt really sorry for this person, and I'm sure it's affected them ever since ... but when I thought about it, he'd been asleep before this thing had happened, and asleep again afterwards, and it occurred to me that maybe he'd dreamed it all, and I still wonder.'\n\n'Why don't you tell him that?'\n\nRory looked at me for a while, his eyes searching mine, making me feel awkward. He spat the blade of grass out. 'Maybe I should,' he said. He nodded, looking out across the fields. 'Maybe. I don't know.' He shrugged.\n\nI stood there, back at the same stone my Uncle Rory had rested against, a decade earlier. I'd left the castle and driven here to the stone circle shortly after we'd come down from the battlements. There was still plenty of time to get back to Lochgair for dinner before I had to set off for Glasgow, and Ash.\n\nI leant against the great stone, the way Rory had when he'd talked about the man betrayed, the man who'd seen \u2013 or thought he'd seen \u2013 something that had hurt him. I looked ahead, out over the walls and fields and stands of trees. I shivered, though it wasn't especially cold.\n\n'See?' I said, quietly, to myself.\n\nMaybe Rory had been looking at Dunadd that day, as I'd assumed at the time. But beyond Dunadd, just a little to the right on this line of sight, I could see the hill where Gaineamh castle stood, its walls showing blunt and steel grey through the naked trees.\n\n'Prentice!'\n\n'... Yeah?'\n\n'Food! Come on, it's getting cold!'\n\nMum had been calling from the bottom of the stairs. I was sitting at the desk in the study, curtains open to the darkness, just the little desk light on, its brass stalk gleaming, its green shade glowing. I looked back down from my reflection in the dark computer screen, first to my watch \u2013 still half an hour before I had to leave to pick up Ashley \u2013 and then to the thin, battered-looking pocket diary lying opened on the desk.\n\n_Fri_ F @ Cas, L.Rvr, trak, hills. Bothy;   \nfire, fd, dnk, js. (F stnd) rt in clng!   \nguns. F nsg. trs & scrts. F barfd   \nWELCOME TO ARGYLL!\n\nI saw her hair first, shining tight-tied in a spotlight somewhere down the domestic arrivals concourse. I hadn't seen Ashley Watt for about six weeks, after that night in London when I'd seen but not talked to Rupert Paxton-Marr. Ashley was dressed in the same business-like suit she'd worn that night, and carried a big shoulder bag. Her smile was broad.\n\n'Ash. Great to see you.' I hugged her, lifting her off her feet.\n\n'Woo!' she laughed throatily. 'How ya doin, Presley?'\n\nI winced, dramatically, but still offered to carry her bag.\n\n'Prentice; you read a couple of things your uncle wrote and suddenly you're accusing people of murder? Come on.'\n\n'Haven't you looked at the files Doctor Gonzo sent over?'\n\n'Of course not; not my business, Prentice.' Ashley sounded indignant. 'Oh; before I forget,' she said, reaching for her jacket on the back seat and digging into a pocket. She took out a little three-inch Sony disk and handed it to me. 'Present from Colorado. Yours to tinker with.'\n\n'Thanks,' I said, putting the disk in my shirt pocket. 'I might, too; the spelling mistakes have been annoying me.' I moved my head. 'The stuff's in that envelope on the back seat.'\n\n'You don't want me to read it _now,_ do you?'\n\n'There's a torch.'\n\n'Am I allowed to finish building the spliff first?'\n\n'Okay, but then read.'\n\nI'd waited till we were out of Glasgow before I'd told Ashley about the horrible ideas concerning Fergus that I just couldn't get out of my head.\n\nMost of the journey from Lochgair up to Glasgow I'd spent thinking, trying to work out what might be true and what false in the fragments of writing that Rory had left on disk. The rat in the ceiling and the confession of something over-seen; that was what had taken me back to stand amongst the standing stones that afternoon, after I'd left the castle.\n\nAnd remembering what Rory had said to me there had taken me back to that 1976 diary entry.\n\nrt in clng! F nsg.   \ntrs & scrts\n\nAnd the 1980 diary with the words JUST USE IT!, and the L that had been changed to a C; the L _must_ stand for Lachlan Watt and the F for Fiona. That was the secret Fergus had told Rory, that night in the bothy; the story of Fergus waking up after being brought home from Hamish and Tone's party and crawling through the castle roof-space to see his wife in bed with Lachy Watt. That was the party that Fiona and Lachlan had left together.\n\nOf course, all I had was Rory's fictionalised word for any of it.\n\nSo I'd asked my mum, over dinner.\n\n'Did Fiona ... leave a party with somebody else?' she repeated, looking mystified.\n\n'It's just something in one of Rory's poems,' I said. '... Not earth-shakingly important or anything, but there's an odd sort of note that ... well, I just wondered if you knew, or had heard ...' I shrugged, sipping my glass of water.\n\nMum shook her head, helping herself to some more peas. 'The only time I ever saw Fiona leave a party with somebody else, Fergus was there too. In body, at least.'\n\n'Uh-huh?' I said.\n\n... scrts ...\n\nI owed the last, absurdly simple part of the theory to a stag that had suddenly run onto the road while I was zapping down Glen Croe, between the Rest-and-be-Thankful and Ardgartan. One moment the road ahead was clear in the headlights, next second Wha! Something dark brown looking big as a horse with huge antlers like some twisted aerial array came belting out of the forest across the road and leapt the downhill crash barrier. I slammed the brakes on, nearly locking the wheels. The beast disappeared into the darkness and the car swept through the single cloud of steamy breath it had left behind.\n\nI'd come off the brakes and accelerated again almost immediately, shaking my head and muttering curses at all kamikaze deer, and feeling my heart-beat start to slow again after my fright. I'd adjusted my seat belt and looked over at the passenger's seat. Something had moved there, when I'd braked.\n\nI'd left the airmail envelope holding the print-out of Rory's pieces sitting on the passenger seat, because I wanted Ashley to read them. The envelope had slipped forward under the deer-induced sharp braking, plonking down into the passenger footwell. I'd tutted, waited for the straight along the side of Loch Long, checked for traffic, then reached over, retrieved the package from the footwell and put it back on the seat.\n\nAnd that had set me pondering.\n\nI'd passed through Arrochar in a daze, thinking, of course!\n\nAshley read the relevant passage while we travelled the new, fast stretch of the Loch Lomond road.\n\n'Yeah,' she said slowly. 'Mm-hmm.' She put the sheaf of papers down, switched off the torch, looked at me, then lit the J. 'So this is Rory's idea of what happened just before Fergus and Fiona crashed?'\n\n'Yeah.'\n\n'And is this Rory indicating your Aunt Fiona fucked my Uncle Lachy?' she sounded almost amused.\n\n'Right,' I said. I glanced at her.\n\n'Kind of fanciful, isn't it?' she said. 'Jeez, he was hardly ever here, and they didn't really move in the same social circles.'\n\n'Damn,' I breathed. 'Maybe you should have read the other two bits first.'\n\n'Hmm.' Ashley drew smoke in, handed the J to me.\n\nI took a small toke. 'Yuk; what's this?'\n\n'Herbal mixture,' Ash said. 'No point giving up fags and then smoking tobacco in Js.'\n\n'Hmm,' I said, handing the number back.\n\n'So what are you saying, Prentice? Did I miss something?'\n\n'Maybe.' I shook my head, letting the car slow as we approached Tarbet. 'Or maybe I'm reading too much into it ... want to read the other two bits?\n\nAsh sighed, accepted the J back and switched the torch back on.\n\nWe passed Tarbet, accelerating over the shallow neck of land to Arrochar, pottered through the village at less than forty, then gathered speed again as we curved round the head of Loch Long, passing the place where I'd retrieved the airmail package from the footwell a couple of hours earlier.\n\n'Yeah, but what _did_ Fergus tell Rory?' Ash said, finishing that part.\n\n'Read the next bit,' I said. I waved my hand when Ash offered me the joint.\n\nThe road started to climb along the dark shoulder of the hillside towards the Rest-and-be-Thankful, leaving the old road still down in the floor of the glen. I kept a careful look out for Mad Stags From Hell crashing across the road, but none appeared.\n\n'Woof,' Ash said, closing the last page. 'Horny stuff towards the end there.' She switched the torch off again. 'You think that last bit is what Fergus told Rory in the bothy, if it really happened?'\n\n'Yeah,' I said. 'There's a diary entry to back it up, and there is a way through the castle's attic from the observatory to the master bedroom, and a loft door. Helen mentioned it just today.'\n\n'But Prentice!' Ash laughed, coughing. 'All you've got is Rory's ... written word for it!'\n\n'It's all circumstantial, I know. Although mum does remember the party at Hamish and Tone's, and Lachy did help Fiona take Fergus home.'\n\n'Wow,' Ash said, tartly.\n\n'So anyway, what's happened is: Fergus has spilled these beans to Rory, who's spent years trying to come up with some creative ideas for his big project and failed dismally, then decided Just Use It; use the one spectacular piece of real-life drama only he and Fergus know about; he's written this sort of diary piece about the time they were in the bothy together; another, more fictionalised bit about what Fergus actually saw; and then a third passage that ... well, that's the point.' I glanced over at her. 'I was hoping you might see the same thing I did in that last bit, the bit in the car. I think that was what Rory was writing just before he borrowed the bike and went to see Fergus, because of what he had started to suspect, when he was writing that.'\n\n'Went to see Fergus?'\n\n'Yes.' I looked over at her. 'And Fergus killed him.'\n\n_'What?_ And Fergus _killed_ him?' Ash said, voice high. 'Why, Prentice?' She opened the window a crack and threw the roach out.\n\n'I'll come to that,' I said, holding up one finger. We were passing Loch Restil now; I was still watching out for stags.\n\nAshley shook her head. 'Prentice, have you been reading crime novels instead of your history books?'\n\nI gave a small laugh. 'No. The worst crimes are always in the history books, anyway.'\n\nAsh undid her hair, reached into her bag and started to brush her hair with a long-tooth comb. 'Hmm,' she said. 'Okay. So keep going.'\n\n'Right,' I said. 'That guy Paxton-Marr. He'd been sending dad those match-book covers ... I mean _match-book_ covers, right?'\n\n'Yeah, so?'\n\n'So he knows Fergus; Fergus was getting the guy to send them, making dad think Rory was still alive, farting around all over the world. Why should Fergus want to do that?'\n\n'I don't know, but what's so special about match-book covers?'\n\nI looked over for a moment; her face was pale in the lights of an on-coming car. 'That bit in the bothy,' I told her. 'Rory tells Fergus he accidentally set fire to a barn on the estate when he was a kid. I think the only other person Rory'd ever told about that was dad, who thought that nobody else knew. So when these match-book covers came from all over the world, he thought it was a secret sign from Rory.'\n\nAshley was silent for a while, then sighed. 'What fertile imaginations you have in your family.'\n\n'Yeah,' I said. 'I'm afraid so.'\n\nWe rounded the long left-hander into Glen Kinglas, where Verity had almost lost the back-end of the Beemer a year earlier. The long straight disappeared into the darkness. A few tiny red sparks in the distance were tail lights. I had another shivery feeling of d\u00e9j\u00e0-vu.\n\nAshley tapped her fingers on the dashboard, then ran them through her hair. After a while she said, 'And what did Rory suspect?'\n\n'Murder. His sister's murder.'\n\nAsh took her time before answering. 'You think Fergus killed your Aunt Fiona _as well?\"_\n\nI nodded. 'You guessed it.'\n\n'She was already dead when they had the crash?'\n\n'Hmm, I hadn't thought of that,' I admitted. I came off the power. I checked the mirror; there was nothing following us, and no headlights in front. 'No, I believe Rory got it right in that bit he wrote, and she was alive when they crashed; I was thinking of something else.'\n\n'What?' Ash said.\n\nI braked smoothly as though we were approaching a sharp corner, not on a long straight. From the corner of my eye, I could see Ash looking at me. I changed down to second gear, let the engine brake the car. I reached over and hit the little red release button on Ashley's seat belt, then I slammed the brakes on. The Golf skidded briefly along the road on locked wheels. I heard Ashley shout something. Her hands went out in front of her. She shot forward, harder than I'd intended and went 'Oof!' against the dashboard, blonde hair flying. Her head hit the screen.\n\nThe car juddered to a stop.\n\nI stared in horror.\n\nAsh sat rubbing her forehead. She glared at me. She was holding her chest just underneath her breasts with her other hand. She glared at me. 'What the _fucking_ hell was that for, Prentice?'\n\n'Oh shit,' I said, hand to mouth. 'Oh God, are you all right?' I checked the mirror, put both hands to my mouth. 'I didn't mean to actually hurt you.'\n\n'Well you actually did, you idiot.' She looked down at her seatbelt anchorage, then at the buckle. One side of the belt was still wrapped round her. I sat staring at her, my back against the driver's door, my heart pounding. Ashley patted her forehead, studied her fingers, then scowled at me and sat back in her seat, re-fixing her seatbelt. She waggled her shoulders, sticking her chest out a little and grimacing through the screen at the dark grey length of road exposed to the headlights. 'You complete fool, Prentice; I may never dance the rhumba again.' She looked at me, then pointed forwards. 'Drive.'\n\n'God, I'm sorry,' I said. I got the car moving again.\n\nAsh patted her chest and inspected her forehead in the mirror on the back of the sun-visor, using the torch she'd been reading with. 'No lasting damage done, I think,' she said, snapping the torch off and the visor shut.\n\n'I'm really sorry,' I said. I rubbed my hands on my trousers, one at a time. 'I didn't mean \u2013'\n\n'Enough,' Ash said. 'I promise I won't sue, okay?'\n\n'Yeah,' I said, shaking my head. 'But I'm really \u2013'\n\n'You think,' Ash interrupted, 'that your Uncle Fergus killed his wife by driving off the road and undoing her seat belt just before they hit?'\n\nI took a deep breath. 'Yes.'\n\n'Slow down, will you?' Ash said.\n\n'Eh?' I said, slowing. We hadn't been going particularly fast.\n\nThen I realised. 'Oh. Yeah,' I said, feeling even worse. 'I pick my places, don't I?'\n\nAsh didn't reply; we both watched, silent, as the Golf dawdled past the parking place at the Cowal road junction where Darren Watt had died.\n\n'Shit,' I said. 'Oh God, I'm doing an awful lot of apol \u2013'\n\n'Forget it,' Ash said. 'Let's get home.'\n\nI shook my head. 'Oh shit,' I said miserably.\n\nThe lights of Inveraray were off to our right, steady across the dark waters of Loch Shira as we rounded Strone Point, when Ash spoke again. 'Bit of a risky way to top your wife, isn't it?'\n\n'Convincing, though. And maybe ... Don't laugh,' I said, glancing over at her. 'Maybe the perfect crime.'\n\nAsh looked at me dubiously. 'Oh dear, Prentice. Really.'\n\n'I'm serious,' I said. 'He banged his head; he doesn't remember the last few miles of the drive. He even asked to be hypnotised, though they never did. Short-term memory gone, see? Hell, if he did it just on the spur of the moment, maybe even he isn't sure he meant it. He told me himself that he thought Fiona had been wearing a seat belt. I saw him just after the crash, while I was in hospital too, getting my appendix out. So nobody \u2013 maybe not even him \u2013 will ever know. It's fucking perfect. Risky but perfect, if it does work.'\n\nWe stopped at traffic lights by the ornate, hump-backed bridge that took the road over the Aray. I sat staring at the red light; Inveraray sat ahead, round the side of the little bay, white buildings glowing in the sodium twilight of the street lights.\n\n'But if he doesn't know he did it,' Ash said, putting the sun visor down again and checking her forehead in the mirror in the lights of the on-coming stream of traffic, 'why would he kill Rory anyway?'\n\nI shrugged. 'Maybe he _does_ know he did it; but even if he doesn't, he might guess that he did. Maybe he was afraid Rory would publish something too close to the truth, maybe Rory was threatening to tell people about his theory; the police, for a start. Maybe neither murder was premeditated; maybe Fergus just reacted, both times. I don't know.'\n\n'Hmm,' Ash said. She sat looking baffled for a bit, then shook her head. The lights changed and we crossed the bridge.\n\n'If I'm right,' I said, 'Fergus probably had thought about killing her before he did. Maybe he only actually decided then and there, on the road that night; but he must have thought about it. Like I say; even if he isn't sure he did it himself, he knows he might have. I mean how hard do you have to think about something, how seriously, before it becomes something you could do, in the heat of the moment?'\n\n'I give in,' said Ash. 'You tell me.'\n\n'Jeez,' I said. 'When I was feeling really bad last year I used to lie awake at night thinking that if there was some way of killing Lewis, quickly, painlessly, with no way of being found out, I might just do it, especially if I knew somehow that Verity would turn to me afterwards \u2013'\n\n'Oh for God's sake, Prentice,' Ash said, turning her head to watch downtown Inveraray slide past. A minute later we were out, accelerating down the darkness of the loch side.\n\n'Look,' I said. 'I was pretty fucked-up. I mean, I'm not saying it wasn't my own fault, Ash; I know it was. I'm not looking for sympathy. I'm just trying to explain that some crazy stuff can go through your head sometimes through love, or jealousy, and maybe, if it's triggered by something ... I mean if somebody had actually given me a method of killing Lewis like that I'd probably have been horrified. I hope I couldn't even have thought about doing it any more once I knew it was possible. It was just a fantasy, a kind of warped internal therapy, something I day-dreamed about to make me feel better.' I shrugged. 'Anyway, that's the case for the prosecution.'\n\nAsh sat, mulling, for a while.\n\n'So,' she said. 'Have you checked out whether Fergus was alone the night Rory may or may not have gone to see him? I mean this whole thing falls to pieces if your uncle \u2013'\n\n'He was alone, Ash,' I told her. 'Mrs McSpadden had gone to visit relatives in Fife that weekend. Mum and dad had suggested the twins came and stayed with us. Fergus brought them over about tea time; I remember talking to him. He had a couple of drinks and then he left. So he was alone in the castle.'\n\nAsh looked at me. I just shrugged.\n\n'Okay,' she said eventually. She rested her elbow on the door, and tapped at her teeth with one set of nails. Her skirt had ridden up a little, and I stole the occasional glance at her long, blackly shining legs.\n\n'So,' she said later when we were in the forest, away from the loch side and a few kilometres out of Furnace. 'What is to be done, Prentice?'\n\n'I don't know,' I confessed. 'There's no body ... Well, there is Aunt Fiona's, but that's neither here nor there. But Rory's still missing, in theory. I suppose I could go to the boys in blue with what I've got, but Jeez, can you imagine? Right, sonny, so you think this wee story that ye've read means yer uncle wiz kilt ... Ah see. Would you mind just putting on this nice white jaikit? Aye, the sleeves are a wee bitty on the long side, but you won't be needing yer hands much in this braw wee room we've got for you with the very soft wallpaper.'\n\nWe curved down into Furnace, the road finding the loch shore again. I could sense Ash looking at me, and chose not to look back, concentrating on checking the mirrors and the instruments. Eventually she took a breath. 'Okay. Supposing Fergus did kill Rory, what did he do with the body?'\n\n'Probably hid it,' I said. 'Not too near the castle ... He had plenty of time; all night. He had a Land Rover; he could have got the bike in the back. Bit of a struggle maybe, but Fergus is a biggish lad, and a 185 Suzi isn't that heavy. It did occur to me he could have driven the bike himself with the body lashed to his back looking like a pillion. It's a bit Mezentian, but possible. But then he'd have had to have walked back from wherever he left it ...' I looked over at Ashley, who was staring at me with a worried, even frightened expression. I shrugged. 'But I think he took it up to one of the lochs in the hills, in the Landy; used the forestry tracks and dumped body and bike together into the water. There are plenty of places. The forest to the south of the castle, on the other side of the canal ... It's just full of little lochs up there, and there are tracks to most of them; it's the obvious ... What's the matter?' I asked.\n\n'You're right into all this, aren't you?'\n\n'What do you expect?' I laughed, a strange, tight feeling in my belly. 'What if I'm right? Jeez, this guy might just have killed two of my close relations; wouldn't _you_ be kind of interested?'\n\nAsh breathed out. 'Oh dear, Prentice,' she sighed, shaking her head and staring out of the window at the night as we swept through the forest towards Lochgair. 'Oh dear, oh dear ...'\n\nWe pulled up outside the Watt house in Bruce Street before eleven. Ash looked in the visor mirror again. She frowned and held her hair away from her face, turning her head from side to side. 'Can't see a bruise,' she said.\n\nI looked over. 'No, I think you're all right there.' I spread my hands. 'Look, I'm really sorry \u2013'\n\n'Oh, shush,' Ash said. She nodded at the house. 'Coming in?'\n\n'Just for a minute. I'd like to ask a favour of your mum.'\n\n'Yeah?' Ash said, reaching into the back for her flight bag. 'Let me guess; you want to get in touch with Uncle Lachy.'\n\nI turned the engine off and killed the lights. 'Aye; I wondered if she might let me have his phone number in Australia. I'd like a wee word with him.'\n\n'Yeah, I bet you would.'\n\nWe got out of the car and walked up the path towards the door.\n\nI had a brief chat with Mrs Watt, gracefully refused a dram, and left after five minutes. A shower scattered raindrops in bright cones under the street lights as I drove away. I went up Bruce Street then took a couple of lefts onto the Oban road where it ran along the side of what had been the Slate Mine wharf.\n\nWhen I saw the building site, I pulled in and stopped the car.\n\nThe site was lit with a sort of hollow orange dimness by the nearby sodium lamps. It was here I'd come with Ashley that night after Margot's percussive cremation; we sat here on the Ballast Mound, the World Hill. It was the night she'd told me about Berlin, the Jacuzzi, and the man who'd hinted there was some trick being played on somebody in Gallanach. She'd given me that piece of the Berlin Wall, shortly after we'd sat together here. The developers had been going to level the mound the following day, preparatory to putting up some new houses.\n\nBut it looked like they hadn't got very far.\n\nThe old wharf was derelict again; levelled all right, and with foundation trenches dug, but no more. Little wooden stakes were stuck into the ground near a few of the trenches; loose bits of wet string tied to them lay straggled across the ploughed-up ground. There were no earth movers or dumper trucks on the site any more, just a couple of loose piles of bricks, the bottom few layers already overgrown by weeds. A picket fence round the site had been knocked flat almost all the way round, and the developer's signboard hung flapping in the breeze, secured at only one corner to a rickety, lop-sided framework.\n\nGone bust, I supposed and, with a look at where the Ballast Mound had been, drove away.\nCHAPTER 17\n\nThe line went dead. Twenty thousand kilometres away \u2013 and a lot more than that if you took the satellite route my words had \u2013 a man put the phone down on me. I listened to the electronic buzz for a while, then replaced the onyx handset in its gold cradle.\n\nI put my hands between my knees, looked out through my own reflection in the study windows to the darkness of the park and the string of orange lights along Kelvin Way, and felt a cold, sick feeling coiling in my belly. I was running out of excuses for doing nothing.\n\nIf Lachlan Watt had said _'What?'_ or _'How dare you!'_ or something like that; even if he'd just denied it \u2013 indignant or amused \u2013 and perhaps especially if he'd asked me to repeat what I'd just said, I'd have had some doubt. But to put the phone down ... Did that make sense? I mean, you're living quietly in Australia, the phone goes, and somebody you last remember as a kid in Scotland has the nerve to ask if you ever slept with his aunt in her marital bed. Do you put the phone down without another word if the answer's No?\n\nMaybe you do. Everybody's different. Maybe I still didn't know enough. I lowered my head to the green leather surface of the antique desk and banged my head softly a couple of times, my hands still clasped between my knees.\n\nI'd been putting this off for days. And anyway weeks had passed. First, Ashley's mum hadn't had Lachy's number, then she got it off somebody else in the family, then it turned out it was an old number (I hadn't tried it anyway) and he'd moved, then there was a delay getting the new number, and when Mrs Watt did phone up with it, I'd dithered. What was I supposed to say? How did I broach the subject? Come right out with it? Talk round it? Hint? Accuse? Make up some story about a just-discovered will, with a bequest to the one man she'd been unfaithful with? Or the one she'd most enjoyed being unfaithful with? Should I pretend to be a lawyer? A journalist? Offer money? I fretted for days and could have gone on doing so for months.\n\nI'd stayed in Glasgow that Thursday night, completing a paper on the effect of industrial growth on the drive towards the unification of Germany in the eighteenth century; it wasn't actually due in until the following Friday, but I reckoned that slamming the blighter in a whole seven days early would keep the Prof. happy.\n\nI'd turned one of the late Mrs Ippot's first-floor reception rooms into a study, moving a giant oak and leather desk over to the window with the help of Gav and Norris; I'd bought a PC similar to but faster than the machine at Lochgair and plonked it roughly in the middle of the mega-desk, where it looked a bit lost, but clashed nicely. For the essay on German unity, I'd surrounded the computer with a dozen delicately beautiful pieces of Meissen pottery. Whether they had any positive effect on the worth of the paper I don't know, but they were a lot more soothing to look at while I was searching for inspiration than a blinking cursor.\n\nI'd finished the paper about 2am and printed it out. I thought about getting in the car there and then, dropping the paper through the letter-box of a pal who'd take it in to the department for me tomorrow, and then heading for Lochgair. But I was tired, and I'd already told mum I'd be down in the morning; I didn't want to wake her by arriving in the middle of the night.\n\nSo I'd had a whisky and gone to bed.\n\nThe main bedroom in Mrs Ippot's expansive town house contained a canopied four-poster about the size of a double garage, the sleeping surface of which was about the same height as a mini's roof. The posts were telegraph-pole thick; highly polished mahogany carved into representations of fairies, elves and gnomes, all stacked like little caryatids. I liked to imagine they were the work of an Amerindian totem-pole maker who'd read too much Tolkien.\n\nThe centre-piece of the bedroom was a vast chandelier cut from ruby-coloured Murano glass; it hung like a glistening spray of frozen blood from the centre of a gilt-smothered ceiling whose few flat patches were covered in paintings of cherubs and fawns cavorting in a sylvan landscape that appeared to be equal parts Rubens and Disney.\n\nThe walls of the room, when not hidden from view by the bed's luxurious (but Islamically abstract) brocade side curtains, were covered with huge Rococo canvases of Venus in various guises, settings and ages, though all shared the same state of deshabille and a rotundity of figure that must have required the painterly equivalent of soft-focus to appear so leniently attractive.\n\nWhere the walls did not glow with acres of flesh, they reflected that golden voluptuousness with great gilt-frame mirrors which almost visibly strained the walls they hung on, and which, I couldn't help but notice, also provided rather a good view from and of the silk-sheeted bed. I'd understood Mrs Ippot had been elderly and frail when she died, but I rather hoped she'd had more fun in the bed than just lying there contemplating the condign punishment I'd decided she had devised for her immediate family (certainly _I_ had yet to share the space between those sheets with anybody, though the bedroom's sheer scale and stateliness did lend masturbation an air of solemnity and arguable dignity the apprehension of which had previously quite passed me by). Even the bedside tables were Chippendale; one of them was topped with a large cut-crystal Venetian vase which I kept fruit in, when I remembered. Otherwise it played host to the little lump of concrete that had been part of the Berlin Wall, which Ashley had given to me over a year earlier.\n\nThe bedroom also contained the greater part of Mrs I's collection of camphor-wood chests; a few too many, perhaps. Despite the visual and tactile splendour, olfactorily it was like sleeping in a chemist's.\n\nHowever, the sad truth is that being surrounded by art treasures designed to excite the eye, gladden the gland and animate the avarice does not guarantee a full night's kip. I'd woken at about half-six, lain there restlessly for a bit, then given up trying to get back to sleep and got up to have some toast and a cup of tea.\n\nI'd put the TV on in the kitchen and found we were at war.\n\nI sat and watched it for a while; heard the CNN guys in Baghdad, saw the reporters report from Saudi airfields, listened to the studio pundits gibber about surgical strikes and pinpoint accuracy, and discovered that, these days, war is prosecuted, not waged. Actually, both words struck me as possessing greasily appropriate connotations in the circumstances.\n\n'Fuck it,' I said to myself. What was telephoning somebody you hardly know on the other side of the planet and asking them impertinent questions about their sex life, compared to this gratuitous malfeasance? I strode up the stairs to the reception-room study, determined to make the phone call.\n\nI settled on the direct approach and the truth about myself.\n\nAnd Lachy Watt put the phone down on me.\n\nMaybe he just wanted to get back to the TV and watch our exciting Third World War for a bit.\n\nI'd stayed in the Lochgair house over Hogmanay itself. We had plenty of drink in, and mum and I had prepared loads of food, but not many people actually visited after the bells. Verity went to bed about ten past midnight after struggling to stay awake from about ten o'clock. She had a very small glass of whisky at the bells. Some people from the village came in about one, Aunt Tone and Uncle Hamish arrived about two for half an hour of strained conversation, and some of James's pals called in after four, but mostly it was just mum, Lewis, James and I together. James conked out about six, but Lewis and I were determined to see the dawn come up just on principle.\n\nWe sat in the conservatory, talking and listening to CDs on the gateaux blaster, which I'd brought down with me from Glasgow because it sounded better than the Golfs own sound system (which anyway didn't include a CD player). We were drinking whisky, chasing it with pints of mineral water; pacing ourselves. Lewis felt we were both starting to nod off at one point and so suggested a game of chess. I mooted for the River Game, but we'd have had to have dug the board and everything out and read through all the rules, so we decided chess would be simpler.\n\n'I've been too sensible,' I told him, while pondering a pawn exchange.\n\n'Sensible?' Lewis sounded surprised. _'You?'_\n\nI grinned. 'Well ... Look at me; I'm studying, I'm living quietly, I'm coming home to mother each weekend ... I even bought a sensible, reasonably cheap car. All that money I got...' I shook my head. 'I'm twenty-two; I should have blown it all on floozies or dangerous drugs, or just took off round the world, or bought a Ferrari.'\n\n'You can't buy a Ferrari for forty grand,' Lewis said, chin in hands, studying the board.\n\n'I didn't say it had to be a new one.'\n\nLewis shrugged. 'Well, you've still got most of the dosh. Go ahead; go do some of that stuff.'\n\n'Yeah, but I sort of promised mum I'd get this degree.'\n\n'Okay, so wait till the summer and _then_ do it.'\n\n'But mum'll just worry if I get a fast car.'\n\n'So take off round the world.'\n\n'Yeah. Maybe. I might.'\n\nLewis looked up at me. 'What are you intending to do, anyway, Prentice?' He grimaced, stretched, rubbed his face. 'I mean, are you still just going to wait and see who's recruiting graduates and then take what sounds like the best job, or have you settled on something yet? Something you actually want to do?'\n\nI shook my head. 'Still open, that one,' I said. I took the pawn Lewis had offered. He looked vaguely surprised. 'I still like the idea of just _being_ a historian,' I told him. 'You know, ideally. But that means staying in academia, and I don't know if that's what I want. Somehow I don't think they let you go straight from graduation onto prime-time TV with a twenty-six part dramatised history of the world.'\n\n'Sounds a little unlikely,' Lewis agreed, taking my pawn. 'You given up on the diplomatic service?'\n\nI smiled, thinking back a year to Uncle Hamish's party. 'Well, I'm not sure I'm cut out for that. I've met some of those people, they're bright ... But in the end you have to do as you're damn well told by fuckwit politicians.'\n\n'Ah! Politics, then?' Lewis said.\n\nI bit my lip, looking the length and breadth of the board, trying to work out if the bishop I wanted to move next was going to cause any problems in its new position. 'Na, I should have started by now anyway, but ... shit; you have to make deals. You have to lie, or come so damn close to lying it makes little difference. It's all so fucking expedient, Lewis; they all have this thing about my enemy's enemy is my friend. \"He may be a son-of-a-bitch, but he's our son-of-a-bitch.\" I mean; good grief. What a crock of shit that is. I despair for our species.'\n\n'Not politics, then.'\n\n'I wonder if Noam Chomsky needs an assistant,' I said.\n\n'Probably got one,' Lewis said.\n\n'Yeah,' I sighed. 'Probably.'\n\nLewis looked quizzical. 'Everything else all right?'\n\n'Yeah,' I said, feeling awkward. 'Why?'\n\nHe shrugged, studied the board again. 'I don't know. I just wondered if there was anything ...'\n\n'Hi guys.'\n\nWe both looked over to see Verity, hair in spiky disarray, face soft with sleep, wrapped in a long white towelling dressing gown, padding into the conservatory holding a glass of milk.\n\n'Morning,' I said.\n\n'Hi there, darlin,' Lewis said, swivelling so she could sit on his lap. She put her head on his shoulder and he kissed her forehead. 'You okay?'\n\nShe nodded sleepily. Then she straightened, drank some milk and ruffled Lewis's hair. 'Might get dressed,' she said, yawning. 'Been having nightmares.'\n\n'Aw,' Lewis said tenderly. 'You poor thing. Want me to come to bed?'\n\nVerity sat on Lewis's lap, rocking back and forth a little, her bottom lip pouting. She frowned and said, 'No.' She smoothed Lewis's hair again. 'I'll get up. You finish your game.' She smiled at me, then looked up. 'Nearly dawn.'\n\n'Why, so it is,' I said. Beyond the glass of the conservatory there was just the faintest hint of grey in the sky over the house.\n\nVerity waved bye-bye and went off, head down, rubbing her eyes, back into the house.\n\nI moved the bishop. Lewis sat and thought.\n\nI had won a knight and another pawn for the bishop when Verity came back. She was washed and dressed; she looked fabulous in leggings and a black maternity dress with a black leather jacket over the top. She stood at the doors, clapped her hands together and \u2013 when we appeared quizzical \u2013 waved some keys at us and said, 'Fancy a drive?'\n\nWe looked at each other and both shrugged at the same time.\n\nWe took Lewis and Verity's new soft-top XR3i \u2013 roof down, heater up full \u2013 out into the grey-pink dawn and drove through Lochgilphead and then into Gallanach and just cruised about the town, waving at the people still walking around the place and shouting Happy New Year! at one and all. Lewis and I had brought the whisky, just in case we met anybody we felt we ought to offer a dram. So we started with each other, and all that water during the night must have done us the power of good because the whisky tasted great.\n\n(I'd looked back at the castle, as we'd passed the hill on the outskirts of Gallanach, feeling guilty and ashamed and nervous because I still hadn't done anything about my suspicions, but telling myself that I _still didn't have any real_ evidence, and anyway I was off-duty now; this was the season to have fun, after all. Hogmanay; let's-get-oot-oor-brains time. And, naturally, an end-of-year truce. Hell, it was traditional.)\n\n'Let's go down Shore Road and drop some whisky on that grave dad hit!' Lewis shouted suddenly. 'Mr Andrew McDobbie 1823\u2013 1875 and his wife Moira 1821-1903 deserve to be thought of at this time!'\n\n'Ugh, you ghoul,' said Verity.\n\n'No,' I said. 'It's a great idea. Verity; to the church!'\n\nWhich is how we came to find Helen Urvill and Dean Watt wandering through Gallanach along Shore Road, arm in arm. Dean was playing \u2013 necessarily softly \u2013 on his Stratocaster, while Helen held a bottle of Jack Daniels. They were being followed by a bemused-looking dog.\n\n'Happy New Year!' shouted Dean Watt, and struck a chord. There ensued a great deal of Happy New Years; the mongrel that had been following Helen and Dean joined in by barking.\n\nThere were lots of hugs and handshakes and kisses too, before Helen Urvill yelled, 'Yo Verity!' as she hung on Dean's shoulder and breathed bourbon fumes at us. 'You sober, girl?'\n\n'Yep,' Verity nodded briskly. 'Want a lift anywhere?'\n\nHelen swung woozily round to look at Dean, who was fiddling with a machine-head. 'Well, we were heading back for the castle ...' She frowned deeply, and her eyes flicked around a bit. 'I think ...' She shrugged; her thick black eyebrows waggled. 'But if you're going somewhere ...'\n\n'Let's go somewhere,' Verity said to Lewis, who was in the passenger's seat. 'Somewhere further.' She nudged Lewis.\n\n'Okay,' Lewis said. 'Got a full tank; where we going to go?'\n\n'Oban!'\n\n'Boring!'\n\n'Glasgow!'\n\n'What _for?'_\n\n'How about,' I suggested, over the noise of the barking hound. 'That bit north of Tighnabruaich, where you can look out over the Kyles of Bute? That's a nice bit of scenery.'\n\n'Brilliant!' Lewis said.\n\n'Great idea!'\n\n'Let's go!'\n\n'Get in, then.'\n\n'And let's take the dog.'\n\n'Is it car-trained?'\n\n'Who cares? We can point it over the side if it comes to it.'\n\n'Fuck it, yeah, let's take the mutt.'\n\n'Might not want to come,' Dean said, and handed the Fender to Lewis, who put it at his feet with the neck by the door, while Dean knelt down by the side of the dog, which was sniffing at the rear wheel of the Escort.\n\n'Course it wants to come,' Helen said, with the conviction only the truly drunk can muster. 'Not a dog been born doesn't like sticking its nose out a car window.'\n\n'Here you go,' grunted Dean, lifting a puzzled-looking canine of medium build, indeterminate breed and brownly brindled coat into the car and onto my lap.\n\n'Hey, thanks,' I said, as Helen clambered in beside me and Dean squeezed in on her far side. 'So it's me that gets to find out if this beast's shit-scared of driving.'\n\n'Ah, stop whining,' Helen said, and pulled the fishy-smelling dog away from me to plonk it in Dean's lap.\n\n'All set?' asked Verity.\n\n'I wonder if its wee eyes'll light up when the brakes go on?' Dean said, trying to look into them.\n\n'All set!' Helen yelled, then yodelled lustily as we performed a U-turn and went smoothly back through the town. Helen offered me some Jack Daniels, which I accepted. We still shouted Happy New Year! at people, and the dog barked enthusiastically in accompaniment; it didn't seem in the least discomfited when we left Gallanach and headed through Lochgilphead and away.\n\nWe stopped briefly at Lochgair. I ran into the house. Mum was up, washing dishes. I kissed and hugged her and said we'd be a few hours. Not to worry; Verity was bright as a button and so sober it ought to constitute a crime in Scotland at this time on a Hogmanay morning. She told me to make sure nobody else drove, then, and be careful. She made me take a load of sandwiches, dips and God-knows what, two bottles of mineral water and a flask of coffee she'd just made, and I staggered out the house and had to put most of it in the convertible's rather small boot, but then that was that and off we went through the calm, brightening day, playing lots of very loud music and munching through the various bits and pieces of food I hadn't stashed in the boot. Dog liked the garlic dip best.\n\n'I don't give a fuck what colour he is; a man who can't pronounce his own name shouldn't be in charge of the most destructive military machine the world has ever seen,' I heard Lewis say, while I sat looking at Dean Watt, and thought, Shit, not again.\n\n'She did, did she?' I said, trying to look pleased. 'Well. Good for her. Nice chap, is he?'\n\nDean shrugged. 'Okay, ah suppose.'\n\nWe were sitting on the rocks beyond the car-park crash barrier at the viewpoint above West Glen, overlooking the Kyles of Bute. The island itself stretched away to the south, all pastel and shade in the slightly watery light of this New Year's morning. The waters of the sound looked calmly ruffled, reflecting milky stretches of the lightly clouded sky.\n\nDamn, I thought.\n\nAshley had got off with somebody at Liz and Droid's party. Dancin and winchin, as Dean had put it. Then gone off together. And suddenly I felt like it had happened again. Maybe not quite as stylish as jumping off your uncle's Range Rover into your future husband's arms, but just as effective. My heart didn't exactly go melt-down this time, but it still wasn't too pleasant a feeling.\n\nDean seemed happy to adjust his Strat and pick out the occasional tiny, tinny-sounded phrase; Lewis and Verity and Helen were arguing about the coming war. Or at least Lewis was ranting and they were having to listen.\n\n'Aw, Hell,' Lewis said. 'I'm not arguing he isn't an evil bastard ...'\n\nAshley, I thought, staring out into the view. Ashley, what was I thinking of? Why had I taken it so slow? What had I been frightened of? Why hadn't I said anything?\n\nHadn't I known what it was I wanted to say?\n\n'\u2013 democracy and freedom, what Our Brave Boys are actually going to be fighting for is to restore the nineteenth century to Kuwait and defend the seventeenth century in Saudi Arabia.'\n\nNow I thought I knew what I wanted to say, but it might already be too late. The knowledge and the provenance of its uselessness were the same; a feeling of loss I couldn't deny. Did that mean I was in love with her? If I was, it felt quite different from what I'd felt for Verity. (Verity sat at Lewis's side, huddled in her leggings and leathers and wearing Lewis's startlingly bright skiing jacket, all orange and purple and lime; she looked like a little psychedelic blonde Buddha perched on the tartan car rug.) Something calmer than that, something slower.\n\n'\u2013 ternational law is only so goddman sacrosanct when it isn't something awkward like the World Court telling America to quit mining Nicaraguan harbours.'\n\nBut perhaps I was wrong about Ash being interested in me, anyway. Ashley was the one I remembered talking to in the Jac that evening after Grandma Margot's cremation; she was the one who kept telling me to tell Verity I loved her. If you love her, tell her. Wasn't that what she'd said? So if Ashley felt anything for me beyond friendship, why hadn't she said anything to me? And if she did feel anything, what was she doing going off with this friend of Droid's?\n\n'\u2013 next time the US wants to invade somewhere and see what happens; out'll come that good old veto again. Heck, we got _lots_ of practice using that. _We'll_ do it if the Yanks don't. Panama? This place with the ditch? You don't like the guy in power any more after paying him all that CIA drug money over the years? Ah, why not? On you go. Seven thousand dead? Never mind, we can hush that up.'\n\nCould I finally be right, and a woman was taking up with somebody else to make me feel jealous? I doubted it. Maybe she had been patiently waiting for me to tell her how I felt, or make some sort of move, and now she was fed up waiting, so all bets were off. But why should she have been so passive? Was Ashley that old-fashioned ? Didn't sound like it; from what she'd told me, it was her who went after that Texan systems analyst, not the other way round. If she'd fancied me at all she'd have said or done something about it before now, wouldn't she?\n\n'\u2013 resolutions are fine, unless they're against Israel, of course, in which case, Aw sheeit; you guys just stay in them Golan Heights, and that Gaza Strip. Shoot; them Palestinians probably weren't \u2013 aw, gosh-darn; did I say shoot them Palestinians? Well, hell no, we won't mention that. Twenty-three years the Israelis have been ignoring UN resolutions and occupying foreign territory; south, east and north. Hell's teeth, they'd probably invade the Mediterranean if you told them the fish were Palestinian. But does the US lay siege to them? Impose sanctions? Like fuck, they _bank-roll_ the place!'\n\nMaybe she did think of me as a brother. All those times I'd rambled drunkenly away to her about how much I loved Verity and what a hard time I was getting from everybody, and how wonderful Verity was, and what a poor, hard-done-by kid I was, and how much I loved Verity and how nobody understood me, and how wonderful Verity was ... How could you expect anybody to listen to all that moronic, self-pitying, self-deluding crap for so long and not think. Poor jerk?\n\n'\u2013 we paid him to fight the Iranians for us, but now the scumbag's getting uppity, so we'll pay other scumbags like Assad to help fight _him,_ and it'll all happen \u2013'\n\nUnloading all that stuff on Ash; most people would have told me to fuck off, but she listened, or at least didn't interrupt ... but what must she have been thinking? The response just couldn't be, Oh, he's so sensitive, or Oh, what a deep capacity for lurve this young fellow has ... _Poor jerk._ That about covered it. Or just, _Jerk._\n\n'\u2013 a modern day Hitler it's Pol Pot; even Saddam Hussein hasn't obliterated two million of his own people. But does the West mount a crusade against _that_ genocidal mother-fucker? No! We're _supporting_ the vicious scumbag! The United fucking States of America and the United fucking Kingdom think he's just the bee's knees because he's fighting those pesky Vietnamese who had the _nerve_ to beat Uncle Sam \u2013'\n\nBut maybe she hadn't really got off with this guy. Maybe it was all a mistake, maybe there was still a chance. Oh shit, I thought, and watched a seagull glide smoothly through the air below us, over the tops of the trees and the bundled rocks that led down to the distant shore.\n\n'Oh,' said Verity suddenly, and clutched her belly, and looked wide-eyed at Lewis, who was in full flight over the vexed sands of Kuwait, and apparently quite beyond verbal interception.\n\n'\u2013 Sabra and Chatila; ask the Kurds in Halabja \u2013' He stopped dead, looked at his wife, who was still clutching her belly, looking pleadingly at him.\n\nLewis's jaw dropped and his face went white.\n\nVerity hugged herself, put her head between her knees and started to rock back and forth. 'Oh-oh,' she said.\n\nLewis staggered to his feet, hands flailing, while Verity's shoulders started to quiver. The dog, which had been snoozing at Lewis's feet, jumped up too.\n\n'Verity, what's wrong? Is it \u2013?' began Helen, leaning over and putting an arm on Verity's shoulders.\n\n'Who's the least drunk?' Lewis hollered, gaze oscillating rapidly between the car parked a few metres behind us and his wife, sitting rocking back and forth and shaking. The dog barked, bouncing up on its front feet, then sneezing.\n\n'Oh! Oh! Oh!' said Verity, as Helen hugged her.\n\n'Aw Christ,' said Dean. 'Verity, you're no about to drop, are ye?'\n\nLewis stood with his hands out, fingers splayed, eyes closed, on the rock. 'I don't believe this is happening!' he yelled. The dog barked loudly in what sounded like agreement.\n\nHelen Urvill, her face down at Verity's knees \u2013 where Verity's head was still wedged \u2013 suddenly slapped Verity across the back and rolled away, laughing.\n\nDean looked confused. I felt the same way, then realised.\n\nLewis opened his eyes and stared at Helen lying laughing on the rock.\n\nVerity rose quickly and gracefully, her face pink and smiling. She stepped up to Lewis and hugged him, rocking him, her face tipped up to his as she giggled. 'Joke,' she told him. 'It isn't happening. I keep telling you, this baby's going to be born in a nice warm birthing pool in a nice big hospital. Nowhere else.'\n\nLewis sagged. He might have fallen if Verity hadn't held him. He slapped both hands over his face. 'You unutterable ... minx!' he roared, and put a hand to each side of Verity's grinning face, holding her head and shaking it. She just giggled.\n\nSo we sat and had some coffee and sandwiches.\n\n_'Damn_ fine coffee,' muttered Lewis.\n\nWell, he had a tartan shirt on.\n\nWe drove back later; I watched buzzards and crows and gulls stoop and wheel and glide across the under-surface of thickening grey cloud. We were all very tired save Verity, and I must have fallen asleep because it came as a surprise when we had to stop to put the top up, in Inveraray, when the rain came on. It was a cramped, claustrophobic journey after that, and the dog whined a lot and smelled.\n\nWe got to Lochgair; I staggered into the house, collapsed into my bed and slept for the rest of the day.\n\nI kept missing Ashley after that. Whenever I rang the Watt house she was out, or asleep. She rang me once, but I'd been out walking. Next time I called she had caught the train for Glasgow, en route for the airport and London.\n\nTone and Hamish's usual post-Hogmanay soiree had been even more subdued than usual. Hamish had given up drink, but apparently found his heretical ideas on retribution more difficult to jettison, and so spent most of the evening telling me \u2013 with a kind of baleful enthusiasm \u2013 about a Commentary he was writing on the Bible, which cast new light on punishment and reward in the hereafter, and which had great contemporary relevance.\n\nI drove back to Glasgow on the fifth of January. After New Year's Eve, watching Fergus show off his new plane, I hadn't visited the castle again.\n\nTwo weeks later, after I had had my abbreviated conversation with Lachlan Watt in sunny Sydney, I set off for Lochgair at nine that Friday morning, listening to the war on the radio for as long as I could, until the mountains blocked out the signal.\n\nWar breaks out amongst the oilfields and the price of crude plummets. From being an ally so staunch he can missile American ships and it passes as an understandable mistake, and gas thousands of Kurds with barely a gesture of censure (Thatcher promptly increased his export credits, and within three weeks Britain was talking about all the lovely marketing opportunities Iraq represented; for chemicals, presumably), Saddam Hussein had suddenly become Adolf Hitler, despite more or less being invited to walk into Kuwait.\n\nIt was a war scripted by Heller from a story by Orwell, and somebody would be bombing their own airfield before too long, no doubt.\n\nFrom Glasgow to Lochgair is a hundred and thirty-five kilometres by road; less as the crow flies, or as the missile cruises. The journey took about an hour and a half, which is about normal when the roads aren't packed with tourists and caravans. I spent most of the time shaking my head in disbelief at the news on the radio, and telling myself that I mustn't allow this to distract me from confronting Fergus, or at the very least sharing my suspicions with somebody other than Ash.\n\nBut I think I already knew that was exactly what would happen.\n\nAnd Ash ... God, the damn thing may be just muscle, merely a pump, but my heart really did seem to ache whenever I thought of her.\n\nSo I tried not to think about Ashley Watt at all, utterly unsure whether by doing so I was being very strong, or extremely stupid. I chose not to make an informed guess which; my track record didn't encourage such honesty.\n\nMum dropped her laser-guided bombshell over lunch that day. We were sitting in the kitchen, watching the war on television, dutifully listening to the same reports and watching the same sparse bits of footage time after time. I was already starting to get bored with the twin blue-pink glowing cones of RAF Tornadoes' afterburners as they took off into the night, and even the slo-mo footage of the exciting Brit-made JP-233 runway-cratering package scattering bomblets and mines with the demented glee of some Satanic Santa was already inducing feelings of weary familiarity.\n\nOn the other hand, such repetition left one free to appreciate the subtler points in these reports that might otherwise have gone unnoticed, such as the fact that the English _could_ pronounce the soft _ch_ sound, after all. The little rascals had only been teasing us all these years, saying 'Lock' Lomond and 'Lock' Ness! Why, it must be something genetic, we'd all thought. But no! Places like Bah'rain and Dah'ran were rolled confidently off the tongue by newsreader after newsreader and correspondent after correspondent as though they'd been using the technique for years.\n\nUnfortunately, rather like a super-gun, there appeared to be a problem traversing such a sophisticated phonetic delivery system, and while the Arabian peninsula obviously lay in the favoured direction, nowhere unfortunate enough to be located to the north of London seemed able to benefit from this new-found facility.\n\n'Oh,' mum said, passing the milk across the kitchen table to me, 'assuming we're all still alive next Friday, Fergus has asked me to the opera in Glasgow. Is it all right if we stay with you?'\n\nI watched the lines of tracer climb above Baghdad, impotent spirals of light twisting to and fro. I felt frozen. Had I heard right? I looked at my mother.\n\nShe frowned. 'Prentice, are you okay?'\n\n'Wha-?' I said. I could feel the blood draining from my face. I put the jug down, feeling as white as the low-fat it contained. I tried to swallow. I couldn't talk, so I settled for clearing my throat and looking at mum with a interrogatory expression.\n\n'Fergus,' mum said tolerantly. 'Invited me to the opera in Glasgow, next Friday. May we stay with you? I assume there's room ... I do mean separate rooms, Prentice.' She smiled. 'Are you all right? You're not worried about the war, are you? You look white as a sheet.'\n\n'I'm fine,' I waved one hand weakly. Actually I felt sick.\n\n'You look sick,' mum said.\n\nI tried to swallow again. She shook her head. 'Don't worry, Prentice. They won't conscript you; you're far too Bolshie. I really wouldn't worry.'\n\n'Hg,' I said, almost gagging.\n\n'Is that all right? Are we allowed to stay with you? Does your lease, or whatever, cover that?'\n\n'Ah,' I said at last. 'Yeah.' I nodded, finally swallowing successfully. 'Yeah, I think so. I mean, of course. Yes. Why not? Loads of room. What opera? What are you going to see?'\n\n'Macbeth.'\n\n_Macbeth!_ 'Oh,' I said, trying to smile. 'That's Verdi, isn't it?'\n\n'Yes, I think so,' mum said, still frowning. 'Would you like to come? It's a box, so there should be room.'\n\n'Um, no thanks,' I said. I didn't know what to do with my hands, which seemed to want to shake. Finally I shoved them in the pockets of my jeans.\n\n'You sure you're all right, Prentice?'\n\n'Of course!'\n\nMum tipped her head to one side. 'You're not upset because I'm going out with Fergus, are you?'\n\n'No!' I laughed. 'Why, are you?'\n\n'We've partnered each other at bridge a couple of times. He's a friend, Prentice, that's all.' Mum looked puzzled.\n\n'Right. Well,' I said. 'Yes, of course there's room. I'll ... no problem.'\n\n'Good,' mum said, and clicked a couple of sweeteners into her tea. She was still looking at me strangely. I turned and watched the war for a while. Jumping Jesus, now what?\n\nI sat at dad's desk. It took longer to write down what I suspected than I'd thought it would. I started with pen and paper, but my writing looked funny and I kept having to dry my hand. Finally I used the computer and printed out what I'd typed. I put the sheet of paper in an envelope and left it lying in the top right drawer of the desk. I wished dad had had a gun, but he hadn't. I settled for the old Bowie knife I'd had since my Scouting days, sticking the leather sheath down the back of my jeans. I changed into a T-shirt and a shortish jumper so that I could get at the knife quickly, feeling frightened and embarrassed as I did so.\n\nMum was in what had been a spare bedroom, constructing the harpsichord. When I stuck my head round the door, the room stank of varnish and the sort of old-fashioned glue you'd rather not know the original source of. 'I'm just going up to the castle, to see Uncle Fergus,' I said. 'You reminded me: there are some pieces of Lalique in the house I'm staying in. I thought I'd have a talk to Fergus about them, see if he fancied bidding for them when the contents are eventually auctioned.'\n\nMum was standing at the work-bench, dressed in overalls, her hair tied back. She was polishing a piece of veneer with a cloth. 'Pieces of what?' she said, blowing from the side of her mouth to dislodge a wisp of hair that had escaped the hair clasp.\n\n'Lalique. Ren\u00e9 Lalique. Glass; you know.'\n\n'Oh, yes.' She looked surprised. 'Fergus'll see them on Friday, won't he?'\n\n'Well, they're in storage in the cellar,' I said. 'I haven't actually seen them. They're in the inventory. I took a note of them. But I thought if he did want to look at them, maybe I could look them out in time for Friday.'\n\n'Oh.' Mum shrugged, tipped oil from a bottle onto the brown-stained cloth. 'Okay, then. Say hello from me.'\n\n'Yeah,' I said. I closed the door.\n\nI walked away thinking I should have said more, should have said ... well, the conventional things you tell people when you're going in fear of your life. But I couldn't think of a way to say them that wouldn't sound ridiculous and melodramatic. I'd closed off the letter I'd left in the desk with quite enough of that sort of thing, I thought.\n\nI took the Golf out of Lochgair, along the Gallanach road. The Bowie knife was an uncomfortable lump down and across the small of my back, its wood and brass handle cold on my back at first, then warming.\n\nI stopped and made a phone call in Lochgilphead.\n\n'Mr Blawke, sorry to trouble you at home \u2013'\n\nOstensibly I was just checking out whether it was all right for me to mention the Lalique to Fergus, before the expensive French glass-ware was included in any auction, but really I was making sure the lawyer Blawke knew where I was going.\n\nIt wasn't until I was at the foot of the castle driveway that I realised all this time I'd just been assuming Fergus would be there. As I hesitated, hands shaking on the wheel, it occurred to me there was probably a good chance he wasn't. I hadn't checked, after all, and Fergus frequently went away for the weekend; maybe he wasn't at the castle. Relief coursed through me, along with an annoying current of shame that I felt so relieved.\n\nI took the Golf up the drive.\n\nThe gravel circle in front of the castle held five cars, including Fergus's Range Rover. 'Oh God,' I said to myself.\n\nI parked the Golf behind a Bristol Brigand which sat half on the gravel and half on the grass. I walked up to the doors and rang the bell.\n\n'Prentice!' Mrs McSpadden roared. 'Happy New Year to you.'\n\n'Happy New Year,' I said, realising only then that I hadn't seen Mrs McS since the turn of the year. I was permitted to kiss the formidable ramparts of one of Mrs McS's cheeks. 'Is Uncle Fergus in?' I asked. _Say,_ No, I thought, _Say,_ No!\n\n'Aye, he is that,' she said, letting me into the castle. 'I think they're playing billiards. I'll take you up.' She stood aside to let me into the entrance hall with its glassy-eyed audience of stags' heads.\n\n'Actually, it's sort of personal,' I said, smiling faintly, aware I was blinking a lot.\n\nMrs McS looked at me oddly. 'Is that a fact? Well, then, would you wait in the library?'\n\n'Ah ... all right,' I said.\n\nWe walked through the hall. 'Isn't this Gulf thing terrible?' Mrs McSpadden shouted, as if trying to be heard there. I agreed it was terrible. She showed me into the library, on the other side of the lower hall from the kitchen entrance. I stood in there nervously, trying to breathe normally, letting my gaze flick over the ranked rows of impressive, dark leather spines. I wished my own was half so noble and upright. The room smelled of leather and old, musty paper. I went to look out one of the room's two small windows, at the garden and the wood beyond. I adjusted the knife down the back of my jeans so that I could get at it easily.\n\n'Prentice?' Fergus Urvill said, entering the library. He closed the door behind him. He was dressed in tweed britches and a Pringle jumper over a checked country shirt, with thick socks and brogues. He brushed some grey-black hair away from his face. His jowls flexed as he smiled at me, lifting a little from the collar of his shirt.\n\nI cleared my throat.\n\nFergus stood there, his arms folded. After a moment he said, 'What can I do for you, young man?'\n\nI moved from the window to the large wooden table that filled the centre of the room, and put my hands lightly on its surface to stop them shaking. A seat back pressed into my thighs.\n\n'Fergus ...' I began. 'I wondered ... I wondered if you knew where ... where my Uncle Rory might be.'\n\nFergus frowned, then one eye closed and he sort of cocked his head. Still with his arms folded, he leaned forward a little. 'Sorry? Your uncle \u2013'\n\n'Uncle Rory,' I said. Maybe a little too loudly, but at least my voice didn't sound as shaky as I'd expected. I lowered it a little to say, 'I thought you might have an idea where he is.'\n\nFergus stood straight again. The frown was still there around his eyes, but his lips were smiling. 'You mean Rory, who disappeared ...?'\n\n'Yes,' I nodded. My mouth felt dry and I had to fight to swallow.\n\n'I've no idea, Prentice.' Fergus scratched behind one ear with one hand. He looked mystified. 'Why do you think I might know?'\n\nI felt myself blinking too much again, and tried to stop it. I took a breath.\n\n'Because you got a man called Rupert Paxton-Marr to send match-book covers to my dad.' My hands were shaking even though they were planted on the surface of the table. I pressed down harder.\n\nFergus rocked back a little on his brogues. His frown-smile intensified. 'Rupert? Sending your dad ... what?' He looked a little amused, a little confused, and not nervous in the least. _Oh God, what am I doing?_ I thought.\n\nOf course, I hadn't thought to bring any of the match-book covers with me. 'Match-book covers,' I said, my dry throat rasping. 'From all over the world, so that dad would think Rory was still alive.'\n\nFergus looked to one side and unfolded his arms, sticking his hands in his pockets. He looked up at me. 'Hmm. Would you like a drink?' he said.\n\n'No,' I told him.\n\nHe moved to the other end of the table, where there was a small wooden desk like the top of a lectern. He opened it and took out a squat decanter and a crystal glass. He took the glittering, faceted stopper out of the decanter and poured some of the brown liquid into the glass, frowning all the time. 'Prentice,' he said, shaking his head and mating stopper and decanter again. 'I'm sorry, you've lost me. What are you ... what is ... what do you think is going on? Rupert's sending, or was sending Kenneth ...?'\n\n'Match-book covers, from hotels and restaurants and bars in various parts of the world,' I told him, as he stood, relaxed, one hand in pocket, one hand holding the glass, his face scrunched up in the manner of one trying hard and with some sympathy to understand what another is saying. 'Somehow,' I struggled on, 'they were meant to convince dad that Rory was still alive. But I think he's dead.'\n\n'Dead?' Fergus said, drinking. He nodded at the seat I was standing over. 'Aren't you going to take a seat?'\n\n'No thanks,' I said.\n\nFergus shrugged, sighed. 'Well, I can't imagine ...' The frown came back again. 'Has Rupert told you he was doing this?'\n\n'No,' I said.\n\n'And are you sure it wasn't Rory?' Fergus shrugged. 'I mean, was it his handwriting?'\n\n'There wasn't any handwriting.'\n\n'There wasn't ...' Fergus shook his head. He smiled, an expression that looked to be half sympathy and half incomprehension. 'Prentice, I'm lost. I don't see ...' His voice trailed off. The frown returned. 'Now, wait a moment,' he said. 'You said you thought I might know where Rory is. But if he's dead ...?' He stared, looking shocked, into my eyes. I tried hard not to look away, but in the end I had to. I looked down at the table-top, biting my lip.\n\n'Prentice,' Fergus said softly, putting his glass down on the table. 'I've no idea where your uncle is.' There was silence for a while. 'Rupert is an old school friend of mine. He's a journalist who goes all over the world; he's out in Iraq at this moment, in fact. I haven't seen him for a couple of years, though he used to come and shoot on occasion. He is a bit of a practical joker at times, but ...' Fergus looked thoughtful. He shrugged. 'Rory did tell me something once about setting fire to a barn on the estate once; accidentally, when he was very young. That might tie in with these match boxes ...' He shook his head, inspected the contents of his glass. 'But I don't _think_ I ever mentioned that to Rupert.'\n\nI felt sick. 'Nothing about ... some pieces of writing makes any sense, does it?'\n\n'Writing?' Fergus said, tilting his head, one eye narrowing. He shook his head. 'No. Whose writing?'\n\n'Rory's. Based on something that you saw here; up in the roof-space of the castle, and which you told Rory when you were in that bothy together. The night you shot the rat.'\n\nFergus had leaned forward again. He looked totally bemused. Finally he jerked upright and laughed. He looked at the glass he held. 'Maybe I should lay off this stuff. You're making less and less sense as you go along here, Prentice. Rory and I did spend a night in a bothy once, on the estate. But there wasn't any ... rat.' He smiled and frowned at the same time. 'Or any shooting. I don't think we even had guns with us; we were fishing some of the out-of-the-way lochans and streams.' He sighed, giving the impression of patient weariness. 'Is this something you've read?'\n\n'Yes,' I conceded.\n\n'What, in your father's papers, since his death?' Fergus looked as though he felt pity for me.\n\nI nodded, trying not to look down from his gaze. 'Sort of,' I breathed.\n\n'And who is meant to have seen what?' He raised one finger to his mouth, bit briefly at a nail and examined it.\n\n'None of that makes any sense to you, does it?' I said. 'No ... confession, revelation? Nothing to do with Lachy Watt?'\n\nFergus looked hurt. He swirled the glass, drained it. 'That was a very long time ago, Prentice,' he said quietly.\n\nHe looked at me more sorrowfully than accusatorily. 'We were only children. We don't always appreciate the seriousness of what we do ...' He glanced at his empty glass ... 'when we're younger.'\n\nHe put the glass on the table.\n\nI couldn't match his gaze, and lowered mine again. I felt dizzy.\n\nI heard Fergus take in a breath. 'Prentice,' he said, eventually. 'I was quite close to Kenneth. He was a friend. I don't think we saw eye-to-eye on anything really, but we ... we got on, you know? He was a gifted man, and a good friend, and I know I feel the loss. I can imagine how you feel. I ... I've had my own ... What I mean is, it isn't an easy thing to cope with, when somebody that close dies so suddenly. Everything can look ... Well, everything can look very black, you know? Nothing seems right. You even resent other people their happiness, and, well, it just all seems very unfair. It is a terrible strain to be under; don't think I don't appreciate that. And just now, when the world seems ...' He took another deep breath. 'Look, old son \u2013'\n\n'I'm sorry,' I said, stopping him. I smiled shakily. 'Uncle Fergus; I'm very sorry I came here. I've been silly. I don't know what I was ...' I shook my head, looked briefly down. 'I don't know what I was thinking. I've not been getting much sleep recently.' I smiled bravely. 'Watching too much television, maybe.' I waved one hand round a little, as though flailing out for something just beyond reach, then shrugged. 'I'm sorry,' I concluded.\n\nFergus looked serious for a moment. Then he gave a small smile. He crossed his arms again. 'Oh well. I think everything looks a bit sort of mad, really, at the moment, doesn't it?'\n\n'A bit,' I agreed. I sniffed, wiped my nose with a paper hanky.\n\n'Sure you won't have that drink?' Fergus said.\n\nI nodded, stuffed the hanky back into my jeans. 'No thanks, I have to drive. Better be getting back.'\n\n'Right you are,' Fergus said.\n\nHe saw me to the door. He patted me on the shoulder as I stood in the doorway. 'Don't worry, Prentice, all right?'\n\n'Yeah,' I said.\n\n'Oh, and I don't know if your mother's mentioned it \u2013'\n\n'Opera; Friday.' I smiled.\n\nFergus smiled too, jowls wobbling. 'Ah, she has.'\n\n'Yes. No problem,' I said.\n\n'Jolly good. Well, that's all right then.' He offered me his hand.\n\nWe shook. 'Thanks, uncle,' I said. He nodded, and I walked down the steps and across the gravel to the Golf.\n\nHe waved goodbye from the steps, looking concerned but encouraging.\n\nI let the Golf trundle down to the bottom of the hill, where the drive levelled out and joined the tarmac single-track which swept round the base of the hill towards the main road between Gallanach and Lochgilphead. At the junction I stopped. I just sat there for a while. I raised my right hand and looked at the palm for a while, then spat on it and rubbed it hard on the side of my thigh. I tore the knife and its sheath out from my jeans and threw them down into the passenger footwell. I looked in the rear-view mirror, where I could just see the reflection of the top of the castle \u2013 its battlements and silver observatory dome \u2013 through the limbs of the leafless trees.\n\n'Guilty as charged, you bastard,' I heard myself say. Then with a quick look either way, I revved up, slipped the clutch and sent the VW screaming along the road away from the castle.\n\nThe courtyard was empty and the house storm doors were shut when I got back to Lochgair. I parked the Golf in the yard and got out; my hands were shaking. I felt like getting furiously drunk. I stood there, breathing hard in the calm air, listening to gulls crying above the drive down towards the loch, while crows crackled in the trees around the house like some drunken chorus, scornful. My heart was thudding now and my trembling hands were slick with sweat. I had to rest back against the side of the car. I closed my eyes. The cries of the birds were replaced by a roaring noise in my ears.\n\nJesus, I thought, if this was how I felt, how must Fergus be reacting, if I was right, and he was guilty? Now would be the time to watch him, study him. But I could barely have walked just then, let alone drive back to the castle, even if I had been able to summon up the courage to return.\n\nEventually I felt better again, and instead of going into the house, went for a walk through the woods and the forest and up into the hills, and sat on an old ruined wall on the hill topped by the cairn where dad had told us about the mythosaurs, all those years ago. I looked down to the trees and the loch in the pastel light cast by the bright, gauzy overcast, while the mild wind freshened. I replayed that scene in the castle library time after time after time, imagining that I remembered every word, every movement, every nuance of tone and phrasing, every millimetric increment of body language, trying to work out whether I was being terribly sensitive and acute, or just insanely fanciful and paranoid.\n\nSometimes I thought it was perfectly obvious that Fergus was utterly genuine, and all my ideas, all my suspicions were demonstrably ludicrous. Of course the man was innocent; I was insane. Guilty as charged, indeed; who was I to judge?\n\nOther times it was as though his every inflection and gesture shrieked artifice, lies, deception. Very good deception, cunningly deployed lies and artful artifice, but everything false all the same.\n\nHe had reacted just as you would expect somebody to react. But was that the way somebody actually would react? I didn't know, and could not decide.\n\nI got so angry and confused at it all I threw my head back and screamed at the grey sky, roaring full force, all noise and no meaning till my lungs emptied and my throat ached. I doubled up, coughing and spluttering, eyes watering, feeling marginally better but looking round guiltily, hoping nobody had heard or seen. Only a couple of crows answered, harsh voices calling from the trees beneath.\n\nI'd chosen a vantage point from which I could watch the road and the house, and only went back down there when I saw mum's Metro turn off the loch road from Gallanach and flicker like a green ghost as it moved up the drive, half-obscured by the trunks and branches of the bare, grey oaks.\n\nI suppose I was uncommunicative with James and my mother that evening; I spent most of the time in dad's study, reading and rereading the three pieces Rory had written about himself, Fergus, Aunt Fiona and Lachlan Watt. I looked through some of Rory's diaries, gritting my teeth at the impenetrable paucity of their desiccated information. I turned on the Compaq and looked at the letter I'd written that morning. Damn; found a spelling mistake that had got through the spell-checker; 'saw' where I'd meant to type 'was'.\n\nI started drinking whisky after dinner, sitting at the desk in the study at first, craning over its leather surface, sifting through the various papers and diaries, my eyes getting sore. I nearly spilled my whisky into the Compaq at one point, so I turned off the little green-shaded light on the desk and went over to the couch, taking all the bits and pieces with me. I switched on the standard lamp behind me and lay lengthwise on the couch, surrounded by paper. I had the TV on with the sound turned down most of the time, using the remote to turn it up whenever it looked like there was something interesting coming in from the Gulf. I heard James go to bed about eleven-thirty. Mum looked in to say good-night about twelve. I waved, wished her pleasant dreams and kept on reading.\n\nI woke up just after two with the whisky glass balanced on my chest and my eyes feeling gritty. I finished the whisky even though I didn't really feel like it, then went to bed. I drank some water before I fell asleep.\n\nThe clock said 4:14 when I woke up, my bladder just at that point where it might or might not be possible to fall asleep again without having to go for a pee (it didn't usually wake me with so poor an excuse). I lay there for a bit, listening to soft rain hitting the bedroom window. Maybe that was what had woken me. I turned over to go to sleep again, then suddenly started to wonder if I'd turned the computer off. I had the feeling that I had, but I couldn't actually remember doing so. Fuck it, I thought; it would be safe enough. I rolled over onto my other side.\n\nBut my bladder had woken up properly in the meantime and was demanding attention. I sighed, swung out of bed, not bothering with my dressing gown even though the house had grown a little chilly by now. There was an orange night-light plug in a socket in the corridor; I decided to save my eyes from the shock of putting on any more powerful illumination and navigated the anyway familiar route to the bathroom by the plug's pale orange glow.\n\nI sat in the darkness, peeing. A sort of quarter-erection had made it advisable to sit down. I smiled, remembering Lewis's spiel about trying to pee when you had a full bladder and a full erection at the same time. I flushed the toilet, washed my hands and drank some water from the tap. Mum must have been varnishing some part of the harpsichord earlier, judging by the smell in the corridor. I padded along to the study.\n\nI could just make out the dim shape of the desk and computer on the far side of the room when I opened the study door. I couldn't hear the Compaq's fan running, or see a light on, but I went over to it anyway. I stood with my thighs against the wood and leather back of the desk's chair, and leant forward, pressing the computer's disk eject button in case I had switched it off but had left a disk in it. No disk. I yawned, straightened, and rubbed the inside of my right fore-arm where it had brushed against the glass shade of the little desk light. The shade had been hot.\n\nThere was a little red dot glowing on the dark screen of the computer monitor; must be the reflection of the TV on the other side of the room. Ha; so I had left it \u2014\n\nI froze, suddenly wide awake.\n\nWhy was the light shade hot?\n\nThe little red light reflected on the screen winked out, as though suddenly obscured.\n\nI threw myself back from the desk, just starting to sense movement behind me; I fell backwards as something dark scythed past in front of my face and a noise like the wind terminated in a splintering crash. Somebody \u2013 just a silhouette in the dim vague shadows of the room, lit only by the feeble light spilling from the hall night-light \u2013 stumbled forward, just behind where I had stood, arms reaching in front of them, pulling something long and dark and thin out of the wrecked back of the seat. The figure started to turn as I landed heavily on my back on the rug; I kicked out at their nearest knee, wishing I was wearing my Docs. Or anything, come to that.\n\nI felt my heel hit their leg. 'Huh!'\n\nSounded male; he staggered a little, then came forward at me, one arm raised as I started to roll, suddenly feeling very vulnerable and naked. A smashing noise sounded from overhead; metal and glass. I kept rolling, pushing up with my hands and leaping to my feet. Glass was falling from the ceiling as something thudded into the floor where I'd lain. I was at the man's side as he staggered forward, raising the bar or jemmy or whatever the hell it was from where it had struck the carpet. I kicked him in what I hoped was the kidneys and watched him stumble to one side, then something banged into the top of my head and hit my shoulder, confusing me. My feet crunched over something hard on the rug as I staggered. More light from the hall, as I stood swaying, dazed, and the attacker recovered. I could see him better now; all in black. Gloves, balaclava. His build ...\n\n'Uncle Ferg?' I heard somebody whisper. It sounded like me.\n\n'Prentice?' said a woman's voice, distantly, worriedly, from the corridor.\n\nI watched the man in front of me seem to hesitate, arm raised. I was falling. I staggered backwards, trying not to fall, crashing into a filing cabinet.\n\n'Prentice!' mum screamed, somewhere. Then; 'James! Get back!'\n\nThe dark figure looked towards the hallway, where the light was. I nearly fell round the side of the filing cabinet, then pulled myself up on some shelves, staring back at the black-dressed man in the middle of the room. There was movement at the study door; sparks flashed in the middle of the ceiling. I clutched at something on the bookshelf; graspable, heavy enough; an ashtray or bowl. I threw it, heard it hit his body and clunk to the floor. He still stood there, maybe only for a second or so, but it seemed like an eternal hesitation, while he glanced from me to the hallway again. I thought I heard a door slam. I roared, shouting incoherently the way I had on the hillside that afternoon as I stumbled from the shelves, past the filing cabinet and nearly fell over the desk while he came forward at me, arm raised again; I picked up the computer's keyboard from the desk, hauling it bursting free and swinging it as hard as I could at him as he brought his arm down.\n\nThere was a terrific, bone-ringing crash that seemed to infect the whole world, like an electric shock and a thunder-clap and an earthquake all at once. There was an odd pattering and clinking noise from every part of the room. I stood, holding nothing, blinking in the darkness while somebody moved stumbling away, obscuring light.\n\nI felt weird. My feet and arms and head felt buzzy and sore, but when I felt my head I couldn't feel any blood. Feet felt slippy. I heard the phone on the desk make a noise, and picked it up, still dazed.\n\n'Which service?' said a man's voice.\n\n'Police!' I heard my mother shout.\n\n'Sorry,' I mumbled. I put the phone down, pushing myself away from the desk. I tripped on the pale remains of the keyboard. Its lettered keys lay scattered about the floor like teeth. I stubbed my toe on something, bent down and picked up a long steel bar. I limped to the top of the stairs in time to see the front door slam shut.\n\nMy head felt buzzy again; I went into the kitchen, found the broken door lock and two full red plastic petrol cans sitting on the kitchen table, then got back out into the hall, still holding the steel bar even though it was beginning to feel very heavy, and shouted, 'Mum? Mum; it's all right! I think ...' before I had to sit down at the kitchen table, because my tongue had suddenly become a clapper in the bell of my skull, and my head was ringing. I put my arms on the table and rested my head on them while I waited for the echoes in my head to go away.\n\n'Welcome to Argyll,' I told myself.\n\nThe kitchen light was painfully bright when it went on. Mum brought me my dressing gown and put a blanket over my shoulders and made me drink heavily sugared tea, and I remember thinking, Sugared tea; dad must have died again, and mumbling something about having a flag in my foot when mum washed them and put bandages on them, and wondering why she was looking so upset and James so frightened; then police came. They seemed very large and official and asked me lots of questions. Later, Doctor Fyfe appeared looking slightly dishevelled, and I recall asking him what he was doing up at this time in the morning, and how he was these days. Old ticker holding out all right, was it?\nCHAPTER 18\n\nWe were on the battlements; I faced into the cool north wind. I waited to feel the dizziness of d\u00e9j\u00e0-vu, but didn't. Maybe too much had happened, or not enough time had passed.\n\n'Well, whatever the heathen equivalent is,' Lewis said. 'Will you?'\n\n'Of course,' I said. I looked down into the small pink face bundled inside the old family shawl; Kenneth McHoan had his eyes tightly closed and wore an expression of concentration on his features that implied sleep was a business of some deliberation. One of his hands \u2013 the thumb so small it could have fitted on just the nail of one of my own thumbs \u2013 was held up near his chin; the tiny fingers made a slow waving motion, like a sea anemone in a stray current, and I jiggled up and down a little, cradling the sleeping child and going, 'Shh, shh.'\n\nI glanced at Verity, sitting beside Lewis, her arm round his waist. She looked up from her son's face for a moment.\n\n'Uncle Prentice, the Godfather.' She smiled.\n\n'An offer only a churl could refuse.'\n\n'People have their own absorption spectra, Prentice,' said Diana Urvill, as she took a Corning turn-of-the-century cut glass plate out of the display case in the castle Solar and \u2013 after wiping the plate with a lint-free cloth \u2013 handed it carefully to me. We both wore white gloves. I took the plate \u2013 like an immense ice crystal with too many angles of symmetry \u2013 and placed it on the table, on the topmost sheet of foam. I folded the translucent padding over \u2013 thinking how much it looked like prawn crackers \u2013 secured it with tape, then found a suitably sized box and placed the plate in the centre, on a bed of small white expanded-polystyrene wafers that looked like flattened infinity symbols.\n\nI lifted one of the giant sacks of the wafers and filled the box to the brim with them, covering the wrapped-up plate, then closed the box and took the little card Diana had left on the table and taped it to the side of the box where it could be read. Then I put the box on a five-high pile near the door; the stacking limit was six, so it completed that column.\n\n'Absorption spectra?' I said sceptically, as we started to repeat the whole process with a Fritsche rock crystal ewer.\n\nDiana, dressed in baseball boots, black tracksuit bottoms and a UCLA sweatshirt, her black hair tied in a pony tail, nodded, and breathed on the ewer before polishing it. 'Things they get absorbed in. Interests, that sort of thing. If you could take a sort of life-spectrum for everybody, of all the things they believed in and took an interest in and became involved in \u2013 all that sort of stuff \u2013 then they'd look like stellar spectra; a smooth band of colour from violet to red, with black lines where the things that meant something to those people had been absorbed.'\n\n'What an astronomical imagination you have, Diana,' I said. 'Getting enough oxygen up on Mauna Kea, yeah?' I grinned.\n\n'Just a pet theory, Prentice.' She finished polishing the ewer. 'Better than believing in,' she said, and handed me the elaborately carved jug, 'crystals.'\n\n'Well, that's true, in a very un-Californian way, isn't it?' I filled the inside of the ewer up with little polystyrene beads from another giant sack, a broad smile on my face as I remembered.\n\nShe cried out and the crystal sang in reply.\n\nLater, we exchanged signals.\n\n'Help me fold these sheets, will you?'\n\nThe day after all the excitement at Lochgair, I sat at the dining table with what looked like a turban on my head. It was a towel wrapped round one of those sealed liquid containers you freeze and put in cool boxes.\n\nI signed the statement.\n\n'Thank you, sir.'\n\n'Davey, stop calling me \"sir\", for God's sake,' I breathed. Constable David McChrom had been in my class at school and I couldn't bring myself to call him \"officer\". His nickname had been Plooky, but that might have been carrying informality a little too far.\n\n'Ach, second nature these days, Prent,' he said, folding the papers and standing up. He looked depressingly fresh and well-scrubbed; joining the police force seemed to have done wonders for his skin condition. He lifted his cap from the table top, turning to my mother. 'Right. That's all for now, Mrs McHoan. I'll be getting back, but if you think of anything else, just tell one of the other officers. We'll be in touch if we hear anything. You all right now, Mrs McHoan?'\n\n'Fine, thanks, Davey,' mum smiled. Dressed in jeans and a thick jumper, she looked a little dark around the eyes, but otherwise okay.\n\n'Right you are, then. You look after that heid of yours, okay, Prentice?'\n\n'As though it were my own,' I breathed, adjusting my towel.\n\nMum saw him out.\n\nThe CID were still in the study, looking for fingerprints. They'd be lucky. I looked out of the dining-room window to where a couple of policemen were searching the bushes near the kitchen door.\n\nMy, we were being well looked after. I doubted a roughly equivalent fracas in one of the poorer council estates would have attracted quite such diligent and comprehensive investigation. But maybe that was just me being cynical.\n\nMy head hurt, my feet hurt, my fingers hurt. All the extremities. Well, save one, thankfully. Most of the damage came from the central light fixture in the study ceiling. It was part of that \u2013 a large, heavy, brass part of it \u2013 which had hit me on the head, and it was the shattered glass of its shades which had cut my feet as I'd stumbled around the study. My fingers hurt from the impact of computer keyboard and steel tyre-iron.\n\nThe desk drawers had been levered open. The back of the desk's matching chair had taken the full force of a blow with the tyre-iron, the light fixture had been hit accidentally by the same implement and the ceiling rose damaged, the Compaq's keyboard was wrecked and the kitchen door needed a new lock. I felt I could use a new head.\n\nNothing had been stolen, though I'd noticed that all the papers I'd been looking at earlier that night \u2013 and which I'd left scattered round the couch \u2013 had been neatly gathered together and piled on one end of the desk, under a paperweight. The envelope I'd left in the desk's top right drawer that morning was still here. The police didn't open it. Apart from the damage, and that one contrary act of tidiness, it looked like our attacker had taken nothing, and left behind him only the petrol and the tyre-iron.\n\nI wanted to phone Fergus; ask him how he was. Good night's sleep? Any aches and pains? But mum had been fussing over me after Doctor Fyfe had said I'd need watching for a day or two and I wasn't being allowed to do very much. Somehow I lacked the will, anyway.\n\nThey'd asked me if I had any idea who it might have been, and I'd said No. I didn't say anything to my mother, or anybody else, either.\n\nWhat could I say?\n\nI was certain it had been Fergus \u2013 his build had been right, and even though I'd been dazed, I swear he did hesitate when I spoke his name \u2013 but how was I supposed to convince anybody else? I shook my head, then grimaced, because it hurt. I couldn't believe I'd been so stupid, not even thinking that he might try and steal or destroy whatever evidence he thought I had. 'Is this something you've read?' I whispered to myself, remembering what Fergus had asked me. 'In your father's papers, after his death?'\n\nJeez. I felt myself blush at my na\u00efvety.\n\nMum continued to fuss, but I got better through the day.\n\nAfter the CID boys finished in the study, I photocopied all Rory's papers \u2013 though I had to drag a chair over to the photocopier and sit down to do it \u2013 then, before the police left, and after much pleading, got mum to drive into Gallanach and deposit the parcelled originals in the bank. She came back with a new lock for the kitchen door. I hadn't been able to persuade her that a little holiday \u2013 in Glasgow, maybe \u2013 would be a good idea, so while she was away I rang Dean Watt and asked if he and Tank Thomas fancied coming to stay at Lochgair for a few days. Tank was a quiet and normally docile friend of the Watts', two metres tall and one across; I'd once seen him carry a couple of railway sleepers, one over each shoulder, without even breaking sweat.\n\nJames \u2013 who'd earlier been appalled that he'd only missed the first two periods of school while the police interviewed him \u2013 arrived back at four, glowing with glory. Apparently his part in the night's events \u2013 which I'd thought consisted largely of sticking his head round his bedroom door and being told to get back in again (and doing as he was told, for once) \u2013 had gained something in the translation at school; I suspected the gains involved the single-handed beating-off an attack by an entire gang of ninja assassins while mum and I slept.\n\nI told mum about Dean and Tank, but she wasn't having it, and rang Dean up to cancel the protection I'd arranged. The police had promised to keep an eye on the house over the next few nights, after all; a patrol car would check up the drive. This didn't sound like much good to me, but mum seemed reassured.\n\nOld Mr Docherty, a leathery-faced octogenarian with wispy white hair who was one of our neighbours in the village, arrived at tea-time and offered to come over with his shot-gun and sit up all night. 'Ah've nuthin tae steal maself, Mrs McHoan, and Ah'd rather make sure you and the bairns were all right. Canny have this sort aw thing going on in Lochgair, ye know. Be Glasgow people, Ah tell ye. Be Glasgow boys.'\n\nMum thanked him, but refused. He seemed happy when we asked him to help us fit the new lock on the kitchen door. Lewis was all set to come up from London when we told him what had happened, but mum persuaded him we were fine, really.\n\nFretting for something else to do, I rang up Mrs McSpadden at the castle and related all that had happened, and twice told her how I suspected the raider had been after Rory's papers, which I'd copied and deposited in the bank. 'In the bank, Prentice,' she repeated, and I could hear her voice echoing. 'Good idea.'\n\nI asked after Fergus and Mrs McSpadden said he was fine. He and his friends had been out fishing that day.\n\nTo my own amazement, I slept soundly that night. James said lights came up to the drive twice. I had to go and see Doctor Fyfe that day, and mum insisted on driving me into Gallanach, despite the fact I felt fine. Doctor Fyfe gave me permission to go back to Glasgow that evening, providing I took the train and stayed with friends.\n\nI stayed the extra night instead, and left by car in the early hours, taking Rory's diaries and the copies of his papers with me. I phoned Mrs McSpadden from Glasgow and told her that, too, and discovered that Fergus had gone to Edinburgh for a couple of days. On impulse, I told her I'd remembered something more from the attack, and I'd be going to the police in a day or two, once I'd checked on something.\n\nBack at university, I attended lectures \u2013 hobbling a little on my cut feet \u2013 and I studied, though I had headaches on the Monday and the Tuesday night. I made sure Mrs Ippot's house was securely locked each night, and closed all the shutters. I rang mum three or four times each day. Mum said Fergus had sent a huge bouquet of flowers to the house, when he'd heard what had happened. He'd phoned from Edinburgh and advised getting an alarm system fitted, and knew a firm in Glasgow who'd do it cost price, as a favour to him. Wasn't that sweet of him? Oh, and I hadn't forgotten she and Fergus would be coming to Glasgow for the opera at the end of the week, had I?\n\nI said of course not.\n\nI put the phone down, numb, my thoughts racing in a kind of aimless short-circuit as I wondered what on earth I was going to do.\n\nAnd, naturally, I followed the war like a good little media-consumer.\n\nThe clich\u00e9s were starting to come out. It was hardly possible to open a newspaper, turn on a television or listen to a radio programme without having rammed down the relevant orifice some witless variation on the facile adage concerning truth being the first casualty of war; a truism that is arguably a neat piece of propaganda itself, implying as it does that the majority of the military, politicians and media have any interest in, respect for or experience with disseminating the truth even in times of profoundest peace.\n\nI started inventing reasons for not putting mum and Fergus up on the Friday. I would be ill. I would have a bad cold. I would discover that the tenancy agreement specified I couldn't have anybody else to stay over-night at the Ippot house. The electricity had been cut off due to a computer error. A gas leak. Serious structural deficiencies caused by the weight of mirrors and chandeliers. Anything.\n\nI stopped watching the war at Tuesday lunch-time because if I'd carried on the way I had been, the history we were living through was going to stop me getting my degree for the history that had been and gone.\n\nAsh rang on the Tuesday evening. I told her everything that had happened, at the castle and Lochgair. She didn't seem to know what to make of it all; she said maybe I ought to go to the police. She sounded low, and said things weren't too good at work, though she wouldn't be more specific.\n\nMeanwhile, the sound of her voice was pulling me apart; it filled me with elation at the same time as it plunged me into despair. I wanted to shout Look, woman, I think I'm falling in love with you! I am! I do! I love you! Honest! I'm sure! Well, almost certain! ... but you couldn't; I couldn't. It wasn't the sort of equivocal thing to shout at any time, and even if I had been completely sure how I felt, I probably couldn't have told her, not just then. I got the impression it wasn't the sort of thing she wanted to hear anyway. She sounded like she just wanted to keep her head down for the moment; keep things quiet, uncomplicated; just cool out. Recently banged-on-the-head nutters raving down the phone at her suddenly declaring undying passionate love for no apparent reason was probably the last thing she needed. I was sure about that. Well, fairly certain.\n\nSo it was a desultory kind of phone call. I felt pretty depressed myself at the end of it. I didn't ask her about her love-life.\n\nI put the phone down feeling the same way I had a year earlier, the day I'd been travelling from Gallanach to Glasgow after Hogmanay, and I'd pretended to be asleep when the train stopped at Lochgair. Remembering that cowardice and that shame, I almost picked the phone up again to call Ash back, and my hand reached out a couple of times, and I debated with myself, muttering, my face contorting with silly expressions, and I told myself I was acting like a madman, and I really wanted to make that call and I really ought to, but I was terrified to do it as well, even though I knew that I should ... shouldn't I? Yes; yes I should; yes I definitely ought to, it was obvious, clear definite. I should.\n\nBut in the end I didn't.\n\nAt least there was always work to be done. I'd submerged myself in my studies with a feeling of almost orgasmic relief. The very fact the past can be taken or left made me want to accept it; the sheer demanding immediacy of the present made it repulsive.\n\nAnd so everything returned to a sort of normality, which didn't last, of course.\n\nOn Wednesday, the 23rd of January 1991, shortly after noon, Fergus Walter Cruden Urvill left Gaineamh Castle in his Range Rover and travelled north through the town of Gallanach and the village of Kilmartin, passing Carnasserie Castle and the cairn and standing stone at Kintraw, crossed the thin flood plain of the Barbreck River above Loch Craignish, travelled inland again to rejoin the shore at the cut-off for the Craobh Haven marina development, and then curved past the village of Arduaine, skirting Loch Melfort before passing through Kilmelford and entering the forest that led to Glen Gallain and then down to the shore of Loch Feochan and the twisting road heading for Oban. The Range Rover passed through the town a little before one o'clock and continued north to Connel, waited for the traffic lights to change at the old bridge over the Falls of Lora, then crossed, negotiated some road-works and finally turned left off the road a little further on, entering the thin strip of level coastal ground that was the Connel airstrip.\n\nFergus Urvill parked the Range Rover in the airfield car park. He talked to one Michael Kerr, from the village of Benderloch a couple of kilometres up the road from the field. Kerr was repairing the car-park fence; Mr Urvill said he wanted to use the telephone in the Portakabin that served as the airfield office. Michael Kerr said that Mr Urvill seemed in a good mood, and told him that he would be flying out to one of the Outer Hebrides ('the Utter HeBrides,' were his exact words), where an old school friend lived. He was going to surprise this friend and take him a bottle of whisky for a belated Hogmanay. He showed Michael Kerr the bottle of Bowmore whisky he was taking with him, in a small leather suitcase which also contained some clothes and toiletries. The only thing Kerr noticed that was out of the ordinary was that Mr Urvill grimaced a couple of times, and flexed his shoulders oddly. Kerr asked the older man if he was all right, and Fergus said yes, but it felt like a couple of ribs were acting up a little. An old injury; nothing to worry about.\n\nMrs Eliza McSpadden, the housekeeper at the castle, had confirmed that Mr Urvill had complained of chest pains the night before, and had taken some Paracetamol painkillers. He had taken a box of the tablets with him that morning, when he drove to Connel. He had said he would be away for a couple of days, and \u2013 apparently on impulse as he was about to get into the car \u2013 asked Mrs McSpadden to prepare some of her Cullen Skink soup for his lunch on the Friday. He wouldn't need more as he would be dining with Mrs Mary McHoan in Glasgow that evening, before the opera. The Colonial restaurant in Glasgow later confirmed that they had a booking for two for the Friday evening in Mr Urvill's name.\n\nWhen Mr Urvill came back out of the airfield office, it was about one-thirty. Michael Kerr helped him check the Cessna aircraft. The plane taxied to the end of the runway, faced into a wind and then took off into a five-knot south-westerly breeze, in good visibility under a five-thousand-foot cloud-base of light overcast. The forecast said the breeze would freshen and veer to the south east that evening, and the following few days would be bright and clear with a steady southerly wind of force three or four.\n\nThe Cessna was spotted by the British Army radar base on the island of St Kilda flying into an area that was restricted for missile testing. The light aircraft was flying at an altitude of two thousand feet on a bearing of 320\u00b0, which would take it towards Iceland. There was no radio response from the plane, and an RAF Nimrod, on patrol over the North Atlantic, was diverted to intercept.\n\nThe Nimrod rendezvoused with the light aircraft at 1516 GMT. It decreased speed and flew almost alongside, a little above and ahead of the Cessna for twenty-five minutes, attempting to make radio and visual contact. The Nimrod crew reported that the single occupant of the plane seemed to be unconscious, slumped back in his seat.\n\nAt 1541 GMT the Cessna's engine started to cut out and the plane \u2013 presumably out of fuel \u2013 began to lose altitude. The engine stopped altogether less than a minute later. The plane pitched forward, causing the pilot's body to slump over the controls, whereupon the aircraft went into a steep dive and started to spin. It fell into the sea, impacting at 1543.\n\nThe Nimrod circled, dropping a life raft and reporting the position of the wreck to nearby shipping. The plane sank twenty minutes later, as the sun was setting. There was little visible wreckage. An East German trawler picked up the Nimrod's liferaft during the following morning.\n\nThe crew of the Nimrod reported that at no time had the figure on board the light aircraft shown any sign of consciousness.\n\n'Hello?'\n\n'Prentice?'\n\n'Speaking. Is \u2013?\n\n'It's Ashley. I just heard about Fergus.'\n\n'Ashley! Ah ... Yeah. I heard this afternoon. I was going to call; I don't have your work number.'\n\n'Well?'\n\n'Well, what?'\n\n'Do you know any more than what's been on the news?'\n\n'Well, mum went up to the castle to see if Mrs McSpadden needed a hand, and she said she seemed kind of shell-shocked; kept talking about soup.'\n\n'Soup?'\n\n'Soup. Cullen Skink, specifically.'\n\n'Oh.'\n\n'Yeah, well, apparently Fergus seemed in good spirits, but he'd had some chest pains, the night before. Anyway, he drove up to Connel to fly out to the Hebrides to see some chum of his out there, and next thing we know he's dive-bombing the Atlantic and forgetting to pull up. Unconscious, apparently.'\n\n'Hmm ... so what do you think?'\n\n'Well, I don't know. Mum said she asked Mrs McSpadden who he was going to see, and she said she didn't know who it could have been. The police had already asked her that, apparently; they said they would make enquiries.'\n\n'Right. You think it was a heart attack?'\n\n'I don't know. Umm ...'\n\n'What?'\n\n'Well, apparently Mrs McSpadden said Fergus had a phone call the night before. She took it initially, then handed the phone to him.'\n\n'Yeah? And?'\n\n'Whoever it was, they were Scottish, but it was an international phone call; a satellite call. Mrs McSpadden thought she recognised the voice but she wasn't sure.'\n\n'Hmm. Recognised the voice.'\n\n'Yeah. Did ... I mean, did she know Lachy?'\n\n'Yes. Yes, she did. They both worked behind the bar in the Jac, about ... twenty years ago, maybe.'\n\n'Ah-ha.'\n\n'Ah-ha indeed.'\n\nI took a deep breath. 'Look, Ash, I've been mean \u2013' I heard a noise in the background.\n\n'Shit, that's the door. What?'\n\nThe breath sighed out of me. 'Ah ... nothing. Take care, Ash.'\n\n'Yeah, you too, bye.'\n\nI put the phone down, put my head back, looked up at the plaster stalactites that formed the ceiling frieze in the study of the Ippot house, and howled like a dog.\n\nThe Strathclyde Police received a telephone tip-off at their head-quarters in Glasgow that a drug ring was using Loch Coille Bharr \u2013 just south of the Argyllshire village of Crinan \u2013 as a hiding place for cocaine, at 1325 on January the 23rd. The tip-off was quite specific; talking of weighted, water-tight plastic cylinders towed behind yachts coming from the Continent and transferred to the loch to await pick-up by dealers from Glasgow. The loch was cordoned off that day and police divers started searching the south end of the loch the following morning, while policemen in small boats used grappling hooks to drag the rest.\n\nNo drug-packed cylinders were ever found, but on the second day one of the boats snagged something heavy. A diver went down to free the line from what was expected to be a water-logged tree.\n\nHe surfaced to report that the line had hooked onto the rear wheel of a motor-bike which had, tied to it, the remains of a body.\n\nThe bike and the body were brought to the surface that evening. The corpse had decomposed and been eaten by fish, to the point of being a skeleton held together more by the clothes it still wore than by the few pieces of connective tissue left. The clothes suggested the deceased had been a male, but the police weren't sure of the skeleton's sex until the body was examined in Glasgow the following day.\n\nWhat they did know was that the bike \u2013 a Suzuki 185 GT registered in 1977 \u2013 had been reported stolen by its owner in Glasgow in 1981, after it had been loaned to a friend and never returned. Probably that alone would have led to the police coming to Lochgair to see us, but one of the local policemen with a long memory had already put two and two together when he'd heard the make and model of the bike.\n\nThe corpse carried no identifying papers, but dental records matched. We knew then it was Rory.\n\nThe skeleton had been found wearing a crash-helmet, but it must have been put back on after Rory had been murdered; according to the pathologist's report, he'd been killed by a series of blows to the back of the head with a smooth, hard, spherical or nearly spherical object, approximately nine centimetres in diameter. He was probably unconscious after the first blow.\n\nAnd so, after the coroner had released the remains following the inquest in late February, Uncle Rory's bones came back to Lochgair at last, and were laid to rest at the back of the garden, under the larches, between the rhododendrons and the wild roses, at the side of his brother. The stone-mason added Rory's name and dates to the black marble obelisk, and we held a small ceremony just for the immediate family and Janice Rae. It fell to me to read out the words Rory had, apparently, intended to close Crow Road with, by way of a funeral oration.\n\nThe passage came from Rory's nameless play, and began: 'And all your nonsenses and truths ...'\n\nJanice cried.\n\nI remarked to Lewis that the way things were going in our family it might work out cheaper in the long run if we bought our own hearse.\n\nI do believe he was shocked. Or maybe he just wished he'd said it.\n\nTechnically the case remained open and Rory's murderer was still being sought, but beyond briefly interviewing mum, Janice and Rory's old flat-mate Andy Nichol, the police took no further action. I never did find out just how good at adding-up that policeman was.\n\nThe firm Ashley Watt was working for in London went into receivership in the last week of January. She was made redundant, but remained in the city looking for another job.\n\nThe war ended, in a famous victory. Only their young men died like cattle, and there was even talk of the US making a modest profit on the operation.\n\nVerity's baby was born \u2013 bang on time \u2013 on March the 2nd, in London, in a warm birthing pool in a big hospital. The boy was registered as Kenneth Walker McHoan; he weighed three and a half kilos and looked like his father.\n\nLewis, Verity and young Kenneth travelled up to Lochgair two weeks later.\n\nThe lawyer Blawke read Fergus Urvill's will in Gaineamh Castle on the 8th of March. I had been asked to be present, and travelled down by train \u2013 the Golf was in for a service \u2013 with feelings of bitterness and dread.\n\nHelen and Diana, solemnly beautiful in black, both looking tanned \u2013 Helen from Switzerland, Diana from Hawaii \u2013 sat together in the tall-ceilinged Solar and heard that they were to inherit the estate, with the exception of various pieces of glass held in the castle, which \u2013 as the twins had already known \u2013 were to be donated to the Glass Museum attached to the factory. Mrs McSpadden \u2013 sitting hunched and crying with what was, in retrospect, a quite baffling quietness \u2013 received the sum of twenty-five thousand pounds, and the right either to live on in the castle, or receive a similar amount if the property was sold or if she was asked to vacate her apartments by the twins or their heirs. Fergus had asked to be buried in the old castle garden, but as they never did recover the body a monument was decided on instead. A memorial service would he held in Gallanach at a later date.\n\nThe Range Rover was part of the estate, but the Bentley Eight had been willed to my father. Fergus had changed his will after dad's death \u2013 following promptings by the good lawyer Blawke \u2013 and so the car and its contents passed to me instead, which came as something of a surprise.\n\nThere were various other bits and pieces \u2013 bequests to charities and so on \u2013 but that was the gist of it.\n\nThe lawyer Blawke handed me the keys to the Bentley after the reading, while we were standing around awkwardly drinking small sherries dispensed by a quietly tearful Mrs McSpadden and I was still in a slight daze, thinking, What? Why? Why did he give me the car?\n\nI talked to the twins. Helen just wanted to get away, but Diana had decided to stay on for a while; I agreed to come and help her pack stuff away in a few days time. Fergus's personal effects were going to be stored in the cellar, and of course the glass had to be packed up to be taken to the museum. The twins said they still hadn't decided what to do with the castle long-term, and I got the impression it depended on what Mrs McSpadden chose to do.\n\nI said my good-byes as soon as I decently could. I had intended to take mum's Metro straight back to Lochgair; I'd told Helen and Diana that I'd probably come back that afternoon with mum, to take the Bentley away. But for some reason, when I got out of the castle doors, I didn't go crunching over the gravel to the little hatchback but turned and went back into the Solar and asked if I could take the Bentley to Lochgair instead, and come back for the Metro later.\n\nDiana told me the garage was open, so I walked round to the rear of the castle where the garage and outhouses were. The Bentley sat inside the opened double garage, burgundy bodywork gleaming like frozen wine. I opened the car, wondering why the will had mentioned the contents of the Eight as well as the vehicle itself.\n\nI got in and sat in that high armchair of a driver's seat, smiling at the walnut and the chrome and breathing in the smell of Connelly hide. The car looked showroom-clean; un-lived in. Nothing in the door pockets, on the back seats or the rear shelf; not even maps. I hesitated before opening the glove box. I was just paranoid enough to think maybe there was a bomb wired to that or the ignition, but, well, that didn't seem very Fergus-like, despite it all. So I opened the glove box.\n\nIt contained the car's manual \u2013 I'd never seen one bound in leather before \u2013 the registration documents, and a cardboard presentation box I recognised as coming from the factory gift shop.\n\nI took it out and opened it. There was a paperweight inside, which was what the box was meant to contain, but the big lump of multi-coloured glass was a little too large for the cardboard insert that went with the box. When I looked at the base it was an old limited edition Perthshire weight, not a Gallanach Glass Works product at all.\n\nI left the paperweight lying on the seat and got out, checked the car's boot \u2014 carefully, thinking of the end of _Charley Varrick_ \u2014 but that was in concourse condition too.\n\nI went back to the driver's seat and sat there for a while, holding the paperweight and gazing into its convexly complicated depths, wondering why Fergus had left this lump of glass \u2013 not even from his own factory \u2013 in the car.\n\nThen I weighed the glassy mass in my hand, and clutched it as you might a weapon, and took another, evaluating look at it, and realised. It was spherical, or nearly spherical, and probably pretty well exactly nine centimetres in diameter.\n\nI almost dropped it.\n\nI shivered, and put the paperweight back in the presentation case, put that in the glove-box, and \u2013 after the car did not blow up when I turned the ignition \u2013 drove its quietly ponderous bulk back to Lochgair.\n\nFergus's memorial service was held a week later, at the Church of Scotland, on Shore Street in Gallanach, mid-Argyll. Kind of a traumatic location for the McHoans, and I wouldn't have gone myself \u2013 it would have felt too much like either hypocrisy or gloating \u2013 but mum wanted to attend, and I could hardly not offer to escort her.\n\nWe put some flowers on the McDobbies' grave, where dad had died, then went in to the church, each kissing the sombrely beautiful twins.\n\nI stood listening to the pious words, the ill-sung hymns and the plodding reminiscences of the good lawyer Blawke \u2013 who must be becoming Gallanach's most sought-after after-death speaker \u2013 and felt a furious anger build up in me.\n\nIt was all I could do to stand there, moving my mouth when people sang, and looking down at my feet when they prayed, and not shout out some profanity, some blasphemy, or, even worse, the truth. I actually gathered the breath in my lungs at one point, hardly able to bear the pressure of fury inside me any longer. I tensed my belly for the shout: Killer! Fucking MURDERER!\n\nI felt dizzy. I could almost hear the echoes of my scream reflecting back off the high walls and arched ceiling of the church ... but the singing went on undisturbed. I relaxed after that, and looked around at the trappings of religion and the gathered suits and worthies of Gallanach and beyond, and \u2013 if I felt anything \u2013 felt only sorrow for us all.\n\nI looked up towards the tower. All the gods are false, I thought to myself, and smiled without pleasure.\n\nI talked to a red-eyed Mrs McSpadden after the service, walking down through the gravestones towards the road and sea, under a sky of scudding cloud; the wind tasted of salt. 'Aye,' Mrs McSpadden said, in what was for her almost a whisper. 'You never think it's going to happen, do you? We all have our little aches and pains, but when I think about it, if I'd just said something when he mentioned a sore chest that night to go to the doctor ...'\n\n'Everybody hurts, Mrs McSpadden,' I said. 'And he had broken those ribs, in the crash. Anybody would have assumed it was just those.'\n\n'Aye, maybe.'\n\nI hesitated. 'Mum said he'd had a phone call from abroad, the night before?'\n\n'Hmm? Oh, yes. Yes, he did. I thought I ... Well, yes.'\n\n'You don't know who it was?'\n\n'No,' she said slowly, though I saw her frown.\n\n'It's just that a friend of mine from university who's abroad at the moment had been going to call Fergus, to ask permission to visit the factory \u2013 he's writing a dissertation on the history of glass making \u2013 and I haven't heard from him for a while; I wondered if it might have been him, that's all.' (All lies of course, but I'd tried to ring Lachy Watt in Sydney and found that the phone had been disconnected. Ashley's mum didn't know where he was now, and I did still want to know what had finally driven Fergus to do what he had.)\n\n'Oh, I don't know,' Mrs McSpadden said, shaking her large, florid head. A big black bead of glass glittered at the end of her hatpin; a stray strand of white hair blew in the gusting wind.\n\n'You didn't hear anything that was said,' I prompted.\n\n'Och, just something about putting somebody up. I was on my way out the door.'\n\n'Putting somebody up?'\n\n'Aye. He said he hadn't put anybody up, and that was all I heard. I suppose he must have been talking about people who'd stayed at the castle, or hadn't stayed; whatever.'\n\n'Yes,' I said, nodding thoughtfully. 'I suppose so.' I shrugged. 'Ah well. Perhaps it wasn't who I was thinking of after all.'\n\nOr maybe it was. Maybe if Mrs McS had heard one more word before she'd closed that door, it would have been the word 'to'.\n\n'Come to think of it,' Mrs McSpadden said, 'I'd just been talking about you, Prentice, when the phone went.'\n\n'Had you?'\n\n'Aye; just mentioning to Mr Urvill what you'd said about remembering more details of when your house was burgled.'\n\n'Really?' I nodded, putting my gloved hands behind my back and smiling faintly at the grey and restless sea beyond the low church wall.\n\n_'Canada?'_ I said, aghast.\n\n'I've got an uncle there. He knows somebody working in a firm installing a system I know a bit about; they swung the work permit.'\n\n'My God, when do you go?'\n\n'Next Monday.'\n\n'Next _Monday?'_\n\n'I'll be going up to Gallanach tomorrow, to say goodbye to mum.'\n\n'Flying?'\n\n'Driving. Leaving the car there. Dean can use it.'\n\n'Jesus. How long are you going to Canada for?'\n\n'I don't know. We'll see. Maybe I'll like it.'\n\n'You mean you might stay?'\n\n'I don't know, Prentice. I'm not making any plans beyond getting there and seeing what the job's like and what the people are like.'\n\n'Shee-it. Well, can I see you? I mean; I'd like to say goodbye.'\n\n'Well, you going to Gallanach this weekend?'\n\n'Umm ... Would you believe that this weekend I was intending to drive a Bentley to Ullapool, get a ferry to the island of Lewis, drive to the most north-westerly point on the island I could find and throw a paperweight into the sea? But ...'\n\n'Well, don't let me stop you. I've got plenty of family to see, goodness knows.'\n\n'But-'\n\n'But I'm flying out from Glasgow on the Monday morning. You can put me up in this palace you're living in, if you like.'\n\n'Sunday? Yeah. Let me think; can't get a ferry on a Sunday, but I can get to Ullapool on Friday, travel over; back Saturday. Yeah. Sunday's fine. What time do you think you'll get here?'\n\n'Six all right?'\n\n'Six is perfect. My turn to take you for a curry.'\n\n'No it isn't, but I accept anyway. I promise not to throw brandy all over you.'\n\n'Okay. I promise not to act like an asshole.'\n\n'You have to act?'\n\n'Gosh, you know how to hurt a chap.'\n\n'Years of practice. See you Sunday, Prentice.'\n\n'Yeah. Then. Drive carefully.'\n\n'You too. Bye.'\n\nI put the phone down, looked up at the ceiling, and didn't know whether to whoop with joy because I was going to see her, or scream in despair because she was going to Canada. Caught between these two extremes, I experienced an odd calmness, and settled for a low moan.\n\nI was starting to think that maybe the Bentley wasn't really me. People gave me funny looks when I drove it, and I had already been stopped by some traffic cops on Great Western Road the day I drove the beast back from Lochgair to Glasgow. Is this your car, sir? they'd asked.\n\nWith hindsight, perhaps saying, Gosh, I thought you only did this to black people! wasn't the most politic reply to have made, but they only kept me waiting for an hour while they checked up on me and scrutinised the car. I spent the time sitting in the back of the police car thinking of all the worthy causes I could give the proceeds of the Bentley's sale to (I certainly wasn't going to keep Fergus's blood-money). The African National Congress and the League Against Cruel Sports were two names that suggested themselves as fit to spin Ferg's remains up to near turbo-charger speeds in his watery grave. Thankfully the Bentley's tyres were nearly new and the lights, like everything else, were all in perfect working order, so the boys in blue had to let me go.\n\nAnyway, it felt right that it was the monstrous burgundy-coloured Eight I took to the Hebrides rather than the Golf.\n\nI started out on the Friday morning and took the A82 to Inverness, then crossed to the west coast and Ullapool. The drive confirmed that the Bentley would have to go. It hadn't been as unwieldy as I'd imagined it might be, but I just felt embarrassed in the thing. There hadn't been anything in Fergus's will to say I couldn't do what I wanted with the car, so what the hell, I'd sell it.\n\nI caught the afternoon ferry to Stornoway. I stayed in the Royal Hotel that night, read history books about ancient wars and long-gone empires, and dipped into our currently interesting times via the television. I stationed the paperweight on the bedside table, as though to guard me through the night.\n\nAt ten o'clock the next morning I stood in a strong wind and light drizzle, wrapped in my dad's old coat, near the lighthouse at the Butt of Lewis \u2013 trying to think of a good joke about that to tell my brother \u2013 and wishing I'd brought a brolly. I hadn't been able to decide whether this really was the most north-westerly point of the island \u2013 there was a place with the appropriate name of Gallan Head that might have done as well \u2013 but in the end I thought maybe it didn't really matter that much, and anyway this headland was easier to get to.\n\nThere were some cliffs, not especially high. I had the paperweight in my pocket, and I took it out, feeling suddenly self-conscious and foolish even though there was nobody else around. The wind tugged at the coat and threw light, soaking spray into my eyes. The sea was tarnished rolling silver and seemed to go on forever into the light grey watery expanse of spray and air and cloud.\n\nI hefted the glass ball, then threw it with all my might out to sea. I don't think it would have mattered especially to me if it had hit the rocks and shattered, but it didn't; it just disappeared into the greyness, heading towards the piling, restless waves. I think I saw it splash, but I'm not sure.\n\nI had been thinking about saying something, when I threw the paperweight into the sea; 'You forgot something,' had been the line I'd been toying with on the drive up, through the peat-smoke smell. But it seemed trite; in the end I didn't say anything.\n\nInstead I stood there for a while, getting wet and cold, and looking out at the waves and thinking of that wreckage, lying out there on the floor of the Atlantic, a few hundred kilometres to the northwest, far beneath the surface of that grey receiving sea.\n\nWas Fergus Urvill anywhere, still? Apart from the body \u2013 whatever was left of him physically, down there in that dark, cold pressure \u2013 was there anything else? Was his personality intact somehow, somewhere?\n\nI found that I couldn't believe that it was. Neither was dad's, neither was Rory's, nor Aunt Fiona's, nor Darren Watt's. There was no such continuation; it just didn't work that way, and there should even be a sort of relief in the comprehension that it didn't. We continue in our children, and in our works and in the memories of others; we continue in our dust and ash. To want more was not just childish, but cowardly, and somehow constipatory, too. Death was change; it led to new chances, new vacancies, new niches and opportunities; it was not all loss.\n\nThe belief that we somehow moved on to something else \u2013 whether still recognisably ourselves, or quite thoroughly changed \u2013 might be a tribute to our evolutionary tenacity and our animal thirst for life, but not to our wisdom. That saw a value beyond itself; in intelligence, knowledge and wit as concepts \u2013 wherever and by whoever expressed \u2013 not just in its own personal manifestation of those qualities, and so could contemplate its own annihilation with equanimity, and suffer it with grace; it was only a sort of sad selfishness that demanded the continuation of the individual spirit in the vanity and frivolity of a heaven.\n\nThe waves surged against the cliffs, thudding into the rock and being reflected. The shapes of their energy charged back into that wild, disturbed water, obliterated and conserved at once.\n\nIt seemed to me then that it was this simple; individual life has no momentum, and \u2013 just as dad had said \u2013 the world is neither fair nor unfair. Those words are our inventions, and apply only to the results of thought. To die as Darren had, and as my father had, and perhaps as Rory had, with what might have been great things still to do, and much to give and to receive, was to make our human grief the greater, but could not form part of any argument. They were here, and then they weren't, and that was all there was.\n\nMy father had had the right of it, when I'd been so upset at Darren Watt's death; it had been a sort of petulance I had felt towards the world, an anger as well as a sadness that Darren had died so soon (and so uglily, so sordidly; a litter bin, for fuck's sake). How dare the world not behave as I expected it to? How dare it just rub out one of my friends? It wasn't fair! And, of course, indeed it was not fair. But that was beside the point.\n\nWell, the old man had been right and I had been wrong, and I just hoped that he'd known somehow that I would come to my senses eventually.\n\nBut if he had gone to his grave \u2013 via the McDobbies' \u2013 thinking that his middle son was a credulous fool, and likely to stay that way, well, that hurt me; hurt me more than I could say, but there was no fixing that now. It was over.\n\nI turned and left and caught the ferry back to Ullapool from Stornoway that afternoon, drinking cups of styrofoam coffee and eating greasy pies while I stood out on deck watching the beating waves.\n\nWe'd seen dolphins following the ship once, coming back this way past the Summer Isles after a holiday, one day many years ago; mum and dad and Lewis and James and me.\n\nBut that was then.\n\nI was back in Glasgow six hours later. I slept well.\n\nAnd so we went back to the Anarkali restaurant on that Sunday night, Ashley Watt and I, and we had a meal that was almost identical to the one we'd had before, on the summer night when dad had died, except we got along just fine this time, and Ashley didn't throw any brandy over me, and I didn't act like a complete asshole, and as I sat there, talking about all the old times and about the future, again I didn't know whether to laugh or cry, because it was so good to see her, but she was going away tomorrow, flying off across that wide grey ocean I'd stood looking at just the day before, flying away to Canada and maybe going to stay there, and I didn't know whether to ask about any men in her life or not \u2013 even though I knew from Dean that the guy she'd gone off with at Hogmanay had only been a one-night thing \u2013 and I still didn't feel I could tell her how I felt about her because she was going to go away now, and how could I suddenly say I love you when I'd never said it to anybody in my life before? How could I say it now especially, the night before she was due to leave? It would look like I was trying to make her stay, or just get her into bed. It would probably wreck this one precious evening that we did have, and upset her, confuse her, even hurt her, and I didn't want to do any of that. And through it all I knew there must have been a moment when I could have told her, some time in the past, some time over the last few months, when it would have been the right time and the right place, and it would have felt like the most natural thing in the world to say and do, but somehow, in the heat of things, just during the complexity of events \u2013 and thanks to my own stupidity, my hesitation, my indecision; my negligence \u2013 I'd missed it, and that, too, was gone from me; over.\n\nSo I just sat there, across from her, looking into her soft-skinned face all glowing in the candle-light, that long, thin nose rising straight above her small, smiling red mouth as if together they made an exclamation mark, and I felt lost in the grey sparkle of those eyes.\n\nWe walked out into the cool March night. It was fair but it had been wet and the pavements shone. Ashley stood on the steps as I put on the old tweed coat that had been my dad's. She wore a black dress and the old naval jacket with the turned-over cuffs I remembered from Grandma Margot's funeral. She leant against some railings, watching me button my coat up, and with her left foot she clicked her toe and heel as if in accompaniment to some song I couldn't hear.\n\nI looked down at her tapping black shoe as I adjusted my collar. 'Morse code?'\n\nShe shook her head, long fawn hair spilling over her dark shoulders.\n\nWe went arm in arm down the steps. 'What was that film that had a dancer tapping out insults at somebody?' I said.\n\n'Dunno,' Ash said, click-clicking her feet as we walked.\n\n'Was it Dead Men _Don't_ Wear Plaid?' I scratched my head. I wasn't wearing gloves and I could feel Ashley's warmth through her jacket. She smelled of _Samsara,_ which was a departure for her, I thought.\n\n'Maybe,' she said, and then she laughed.\n\n'What?'\n\n'I was just remembering,' she said, squeezing my waist. 'Mrs Phimister's class. Remember? The French teacher? We were in the same class.'\n\n'Oh yeah,' I said. We turned onto Woodlands Road.\n\n'You hated her because she'd confiscated a radio or something, and you used to tap out insults in morse code.' Ash laughed loud.\n\n'God, yeah,' I said. 'That's right.'\n\n' \"Fuck off you old cow\", was the witticism I recall best,\" Ash said, still snorting with laughter.\n\n'Jeez,' I said, pulling away from her a little to look into her eyes. 'You mean you could decipher it?'\n\n'Yeah,' Ash said, with a sort of friendly scorn.\n\n'You rotter!' I laughed. 'You absolute cad-ess. You cad-ette; I thought that was my secret. I only told people later, after I'd left school, and then nobody believed me.'\n\n'Yeah,' Ash said, grinning at me. 'I knew. A couple of times I almost got detention because I was giggling so much. Nearly wet my knickers trying not to laugh. Got some very stern looks from Mrs Phimister.' She laughed again, throwing her head back.\n\n'I didn't even know you knew morse code,' I said. 'I learned it in the scouts. Where did you learn it?'\n\n'My grandad taught me,' Ash said, nodding. 'We used to sit and pass messages at meal times by clinking our cutlery off the plates. Mum and dad and the others always wondered what we found so hilarious about yet another helping of shepherd's pie and chips.'\n\n'And you never said!' I shook my head. 'You rascal!'\n\nShe shrugged, looked down at her black, medium-high heels as she did a little tap-dance. 'You didn't like me; what was the point?'\n\n'I didn't like any girls,' I told her. 'In fact I wasn't that keen on any of the boys either. Come to think of it, I felt mostly contempt even for my friends.'\n\n'Yeah,' Ash said, leaning over towards me so that her grinning face was almost on my chest. 'But you didn't break their noses with a boulder disguised as a snowball, did you?'\n\nI stopped in my tracks.\n\nAsh gave a little squeal as she staggered, suddenly losing support on one side. She steadied and turned. She faced me, looking puzzled, from a metre or so away. I just stood there open-mouthed.\n\n'You knew that was me?'\n\n'Course I did.' She frowned and smiled at the same time.\n\n'Another secret gone!' I exclaimed, waving my arms. 'I've felt guilty about that for years!'\n\nAsh tipped her head to one side.\n\n'Well, not all the time,' I said. 'I mean, on and off.'\n\nShe raised one eyebrow.\n\n'Okay,' I said, slumping a little. 'Mostly off. But I did feel bad about it. I really did. I always felt bad about that.'\n\nAshley shook her head gently and came forward, took my arm and led me along the street. 'Never mind,' she said. 'I never told anybody. And I forgave you.'\n\n'Really?' I said, putting my arm round her again, 'When?'\n\n'At the time. Well, after it stopped hurting, anyway.'\n\nWe turned the corner into Woodlands Gate. I shook my head. 'Why didn't you ever say you knew it had been me?' I asked her.\n\nShe shrugged. 'The subject never really arose before.'\n\nI shook my head again. 'Good grief,' I said. 'All this time. Good grief.'\n\nAshley had been ravenous when she'd arrived at the house in Park Terrace a little after seven that Sunday evening, so she'd just dumped her bags and we'd gone straight out to the restaurant. When we got back after the meal, I showed her round the place. We opened a bottle of Graves I had in the kitchen \u2013 after first agreeing that of course we shouldn't \u2013 and then walked from room to room while I did my guided tour bit and pointed out the more interesting or valuable works of art, while we sipped our wine and the statues gleamed and the chandeliers glittered and the paintings glowed and the carpets spread before us like gigantic blow-ups of oddly symmetrical printed circuits.\n\nAshley shook her head a lot. When she saw the main bedroom she laughed.\n\nWe went back to the kitchen. She demurred when I offered to top her glass up. 'I should go to bed now,' she said, pulling a hand through her hair. She put her glass down on an oak working surface. 'Take some water in a big glass and get to me bed ...' she said. 'Do you mind?' She looked at me.\n\nI shrugged. 'No, of course not. There's glasses in the bathroom, beside your room.' A terrible sadness settled on me then, and I had to swallow hard a couple of times. I drank, to hide it, then said, as matter-of-factly as I could, 'What time do you want up tomorrow?'\n\n'About seven should do.'\n\n'Right,' I said, looking at my glass. 'Right. Seven. I'll bring you tea and toast, all right?'\n\n'Fine.'\n\n'Okay then,' I said.\n\nI looked up and she was smiling. She looked at her watch. 'Well,' she said, and flexed her brows. 'Night-night.'\n\nShe came forward, put one hand on my shoulder, kissed my cheek.\n\nI put my hand on her hip, let my head nuzzle towards hers a little. She put her arm round my waist and I turned to her, hugged her, my lips at her neck, kissing delicately. She pushed her head against mine, and we started to turn to each other at the same moment, as she put her arms round me; the kiss just seemed natural after that.\n\nIt went on for some time. Ashley seemed to loosen and grow more tense at the same time; her mouth appeared to want to swallow mine, her hands grabbed my curls, nails scratching at my scalp. I pulled on her hair, kissed and licked her neck. She dug her nails into the small of my back through my shirt. We kissed again and I kneaded her backside, then pulled the dress up while she wriggled a little to make it easier, and I found skin, stockings, her knickers, and pushed my hands inside, gripping her smooth, warm bum. She pulled herself up against me.\n\n'This,' she said, breaking off, breathing hard, while her hands stroked the nape of my neck and her gaze flicked from my mouth to my eyes and back again, 'this might be better suited to that ridiculous bedroom, what do you think?'\n\nI nodded. 'Good idea.'\n\n'Bring the wine.'\n\n'Better yet.'\n\nIt was something. On that monumentally ostentatious bed of the late Mrs Ippot's, Ashley and I made love like we'd done it for years and then been apart for years and just met up and hadn't forgotten a thing.\n\nA couple of times, lying there panting afterwards while we trickled with sweat and licked at each other, or were stroking and caressing and thinking about starting all over again, she laughed.\n\n'The room?' I said, first time.\n\n'No,' she said, shaking her gorgeous head, all tawny hair and flushed face. 'It's just you and me; I never thought this was going to happen.'\n\nAnd, later, when she cried out, I heard the crystal bowl on the table by the side of the bed ring, pure and faint, as if in reply.\n\nIt was later still, when we'd put the lights out and had agreed just to cuddle, exhausted and drained, but had not been able to merely cuddle, and so had coupled once more, and I still lay on top of her, inside her, while she breathed and I breathed and our hearts gradually slowed down again, that I did what I'd done before in that situation, flexing whatever muscle it is in the male genitals or the associated support systems that briefly fills the slowly detumescing penis with blood again, sending a small pulse of socketed touch into Ashley's body. She gave a little exhalation half-way between a sigh and a laugh, and then squeezed back with her vaginal muscles, like a hand round me.\n\nThere was a pause, and I thought I felt her go very still for a second, and then she squeezed me again; two quick grippings in succession. There was a pause, and I responded, but she dug her fingers into the small of my back as though to stop me, and so I relaxed.\n\nShe squeezed again, four times, the second pulse longer than the other three. Another pause, during which I realised \u2013 it was morse! Then another four pulses, the second one short and the others long.\n\nI. L. Y.\n\nI had raised my head away from her shoulder while I concentrated on what she was doing in there; now I lowered my face to her skin again. I laughed, very lightly, and after a moment so did she, and then I sent the same signal back, with a single long pulse at the end: I.L.Y.T.\n\nAnd I swear the sending made the signal all the truer.\n\nAnd that falling was followed by two more shared fallings, as we fell apart, and then asleep.\n\nI woke and she was dressed, standing by the bed, a beatific smile across her face, which was washed and glowing and framed by neatly combed hair. I struggled to get up on one elbow.\n\n'Ash?'\n\nShe put one hand to the back of my head and kissed my lips. 'I have to go,' she said.\n\n'What? But \u2013 you mean to _Canada?'_\n\n'Prentice, I promised. I have to.'\n\nI felt my jaw drop. I rolled onto my back for a second, then sat bolt upright. 'But last night!' I said, spreading my arms wide.\n\nAshley smiled even more broadly and climbed half onto the bed, one black-stockinged knee on the crumpled sheets. She kissed me. 'Was wonderful,' she said, 'but I have to go.'\n\n'You can't!' I slapped myself on the forehead with one palm. 'This can't be happening! It's a dream! Stay!' I reached out to her, held her face between my hands. 'Ashley! Please! Stay!'\n\n'I can't, Prentice. I said I'd go. I promised.'\n\n'I'm serious!' I said. 'I don't \u2013'\n\nShe put one soft hand gently to my mouth, shushing me, then kissed me long and tenderly. 'I'm going, Prentice,' she said, 'but it doesn't have to be for ever.'\n\n'Well, how long?' I wailed.\n\nShe shrugged, stroked my shoulders with her hands. 'You get this degree, okay? If you still want me then, well ...'\n\n'Promise?' I said, in what was meant to be a terminally sarcastic manner, but came out pathetically.\n\nShe smiled. 'I promise.'\n\n'Oh my God!' I said, looking at the clock by the crystal bowl. 'I don't believe this!' Maybe, if I could just stall her ...\n\n'There's a taxi waiting,' she told me. 'It's all right.' She smoothed some hair away from my eyes, her touch like silk.\n\n'But I was going to drive \u2013'\n\n'You rest,' she said. 'You probably had too much wine last night, anyway. The taxi really is waiting.' She slipped her hand under the covers, held my penis as she kissed me, then slipped away as I fell forward, trying to embrace her, hold her, keep her.\n\n'Ashley!' I said desperately. She was at the door.\n\n'Yes?' she said.\n\n'I didn't dream that... signal last night, did I?'\n\nShe laughed. 'Nope. Meant every letter; every word. With all my heart.' One brow flicked. 'Amongst other organs.' She tipped her head to one side, eyebrows raised. 'And you?'\n\n'The same,' I gulped.\n\nShe looked down at the floor, then back at me, still smiling. 'Good. Well, we can take it from there, okay?'\n\n'I'll write every day!' I told her.\n\n'Don't be ridiculous,' she laughed, with one shake of her head. 'Just pass those exams.'\n\n'They'll be over by mid-June,' I said, more to keep her there in my sight for a few seconds longer that for any other reason.\n\n'Then I'll be back in mid-June,' she said.\n\nShe pulled her black gloves from her jacket pockets and put them on. 'Bye, Prentice.' She blew me a kiss.\n\n'Bye,' I gulped. She closed the door. I flopped back, stunned, staring at the glittering red chandelier.\n\nI jumped out of bed as the front door banged closed; I tore downstairs bollock-naked and waved to her from one of the drawing room windows, which went from about human knee level to giraffe's head level.\n\nShe saw me; I could see her laughing. She pushed the window down and waved, and pointed to my groin and made a shocked expression as the cab started away. The driver saw me too and looked amused and shook his head. The cab drove off around the curbed terrace. I opened the window and leaned out, waving, and Ashley pushed the cab's window right down and stuck her head and arms out and blew me kisses through her wildly waving, slip-streamed hair all the way until the cab rounded the corner and disappeared.\n\nI sat down on the parquet, staring at the white gauziness of the huge net curtains, all my muscles complaining, my head pounding, my penis tingling, my flesh goose-pimpling against the cool wood of the floor. I shook my head. I collapsed back, banging my already internally abused head on a Persian rug. The carpet's pile was luxuriously deep however, so it didn't hurt as much as it might.\n\nI looked up at the ornately carved wooden ceiling, not entirely sure what to think. Then I started to laugh, lying there in the enormous room, naked, tummy wobbling, laughing like an idiot and hoping the resemblance ended there.\n\n'Oh well,' I said, laughing, to the ceiling. 'Here's hoping.'\n\n'Good; you're getting sensible,' mum said. She walked carefully towards me, the big blue sheet folding and drooping between us. She took the sheet's other two corners from me.\n\n_'Getting?'_ I said indignantly.\n\nMum smiled, folded the sheet over twice more and put it on top of the tumble drier. I pulled another sheet down off the old clothes pulley that hung under the ceiling of the utility room. We took an end each, stood apart, pulled the sheet taut.\n\n'Mm-hmm,' she said, tugging at the sheet again. 'I think selling the Bentley is very sensible.' She folded the sheet over, hand to hand; I did the same. We pulled it taut again. Mum looked thoughtful. 'Maybe we should sell that ancient thing sitting in the garage out there, as well.'\n\n'The Lagonda?' I said. We folded the sheet over again.\n\n'Yes,' mum said, walking towards me again. 'It's just a waste of space at the moment.'\n\n'You mean you weren't thinking of going in for classic car restoration after you've finished the harpischord?'\n\nMum smiled as she took the sheet from me. 'Well, actually that had occurred to me, but ...' She wrinkled her nose. 'No; I don't think so.'\n\n'Well, we won't get much for it in the state it's in at the moment.' I pulled another sheet down.\n\n'I'm not bothered about the money,' mum said. She folded the sheet away, shot me a mischievous look. 'And besides, whose fault is it the car's in the state it is, anyway?'\n\n'What?' I said. I stood looking at her.\n\nMum took the sheet from me and put two of its corners in my hands as she backed off, pulling it tight. She smiled. 'It was you who tipped the big dresser down onto it in the garage that time, wasn't it?'\n\nShe pulled the sheet; it flew out of my fingers, billowing over the floor of the room like some slow motion wave. I ran after it, catching it. I retrieved the corners, untwisted the sheet and studied the amused expression on my mother's face. She tugged the sheet again and I held onto it this time.\n\nI'm ashamed to admit that it even occurred to me to deny it, albeit briefly. I grinned sheepishly as we folded the sheet over. 'Yeah, guilty as charged, but it was an accident.' I shook my head. 'How did you work that out?'\n\nShe walked towards me, took the sheet from me. 'Found a bit of broken glass in your underpants when I was washing them,' she said, and gave a tiny laugh as she turned away to place the sheet on the drier.\n\nI looked up at the ceiling. 'Oh dear,' I said.\n\nMum turned round, standing there in her jeans and blouse, glowing with what might well have been self-satisfaction. She reached up and pulled a last sheet down off the pulley, handing one end to me. 'Yes. Well, we'll draw a discreet veil over that little incident, shall we?'\n\nI nodded, pursed my lips. 'Might be best,' I agreed. I coughed, pulled the sheet taut with her, and with a textbook expression of interested interrogation, asked, 'And how is the harpsichord-construction project going, anyway?'\n\n'Well \u2013'\n\nIt didn't end there, either. Nobody had thought to tell me, but obviously it was open season on Prentice's ignorance. If you were female, anyway.\n\n'Well,' I said. 'I think my absorption spectrum must be hazy.'\n\n'No,' Diana said. 'I think it's much like anybody else's.' She took a Waldglas beaker out of the display cabinet and glanced at me. She may have seen a hurt expression because she shrugged and smiled and said, 'Okay, maybe yours has a few more black lines. You were always interested in all sorts of stuff, weren't you?'\n\nI shrugged. 'It runs in the family.'\n\n'Fact is,' Diana said, breathing on the knobby green glass, 'it's probably thanks to you I spend so much of my life fourteen thousand feet above Hawaii looking for I-R stars.'\n\n'It is?' I said. \/\n\n'Yeah,' Diana said, smiling at the glass as she polished it. 'You remember the night there was Helen, me, you, Lewis and Verity and ... Darren? We were up in the observatory?'\n\n'I remember,' I said.\n\n'You got really stoned and started gibbering about how fantastic the universe was?'\n\nI shook my head. 'I don't remember that,' I confessed.\n\n'Well, you were pretty ripped,' Diana said. She handed me the beaker. 'But you were coherent, mostly, and you were really enthusiastic. I mean you even shut Lewis up; you just raved about how amazing astronomy was. You meant cosmology, but what the heck. You were just bubbling with it.' She brought a second Waldglas beaker out of the cabinet.\n\n'Huh.' I filled the beaker with polystyrene beads, found a box big enough to hold the two beakers and put the first one carefully into its bed of little white infinity symbols. 'Well, I'll take your word for it.'\n\n'Oh, you were just so fascinated with it all. Especially with stellar evolution. That had obviously really blown your mind. \"We are made of bits of stars!\" you shouted.' Diana laughed a little. 'You'd been reading about all that stuff and it just tickled you pink. You told us about how the sun and the solar system were made out of the remnants of older stars that had blown up; how the elements that made up the world had been made in those ancient stars, and that meant our bodies, too, every atom. Jeez, I thought you were going to explode.' She handed me the second beaker.\n\n'Hmm,' I said. 'Well, I sort of remember that, I think.'\n\nActually, I wasn't sure I really did at all. My recollection of that evening got very hazy after the bit where Verity had pretended to tell me my fortune.\n\n' \"We are made from bits of stars! We are made from bits of stars!\" you kept yelling, and went through it all: super novae scattering heavy atoms; the debris swirling through space, other novae and supers sending shock-waves through the debris, compressing it; stars forming, planets; geology, chemistry; life.' Diana shook her head. She extracted a thin, delicate, old-looking flute of a wine glass from the display case. 'And Jeez, you made me feel ashamed. I mean, dad had built the observatory for us; it was a present, in a way. And we hardly used it. We went up there to smoke dope. And here you were, knew all about this stuff, and actually made it sound interesting. You were really gone on the idea that we were stuck down here on this one little planet and still just savages really, but we'd glimpsed the workings of the universe, worked out from light and radiation what had happened over the last fifteen billion years and could talk sensibly about the first few seconds after the big bang \u2013 even if the jury has gone back out on that idea nowadays \u2013 and could predict what would happen to the universe over the next few billion, and understand it ...' Diana held the wine glass up to the light, and cleaned it with the cloth. 'You were pretty scathing about religion, too; tawdry and pathetic in comparison, you said.' She shrugged. 'I didn't necessarily buy that, but you made me ashamed not to have used the telescope more. And so I did, and then I got some books on astronomy, and found out a lot of it was about maths, which I was good at anyway, though somehow the fact astronomy was about numbers and equations as well as stars and telescopes hadn't occurred to me. But anyway, that was the start of it, I was hooked. Been a star-junkie ever since, Prent, and it's all your fault.'\n\nShe flashed a shocked expression at me and handed me the glass.\n\nI shook my head. 'You as well, eh?'\n\n'Hmm?' Diana said.\n\n'Nothing,' I breathed, running a hand through my hair. 'Shit, I never knew.' I looked mockly serious. 'This is something of a responsibility, Diana. I trust you haven't had cause to regret your decision.'\n\n'Not at all, Prentice.' She closed the now empty cabinet, and took off her white gloves. 'I mean, maybe I'd have settled on astronomy anyway, without your one-man show. Whatever; it's been fun. Cold at nights and a long way from the beach, and the air's a bit thin ... but it's the skies that really take your breath away.' She nodded. 'You should visit, come see it all some time.'\n\n'I'd like to,' I said. 'People allowed to come and look round?'\n\nDiana folded her arms and rested her back against the display cabinet. 'It can be arranged.'\n\n'There's somebody I'd like to take there.'\n\nDiana smirked. 'Yeah? Somebody special? Who's that?'\n\n'Oh ... friend of mine. In Canada at the moment.'\n\n'Ashley, huh?'\n\nI felt myself blush. 'Well, yeah,' I said, trying not to grin too much.\n\nDiana nodded, still smiling. 'It'd be great to see you both out there. You two sort of an item these days?'\n\nI shrugged, felt myself blush again. 'Sort of. I hope so. I think so.'\n\nDiana laughed, which was good to hear; I didn't think she had laughed since Fergus died. 'Yeah, I think so.'\n\nVerity and Lewis brought young Kenneth to the castle that day, so that Mrs McSpadden could go all gooey over him. She did. Diana seemed equally charmed. Kenneth just slept.\n\nDiana broke open a bottle of twenty-five-year-old Macallen, which was older than any of us (well, except Mrs McS, but she'd gone back to the kitchen by then), and an awful lot older than Kenneth.\n\n'Let's wet his head,' Diana said.\n\n'Can we go up on the roof?' I said. It just seemed like a good idea.\n\nSo we climbed up there, into a bright March afternoon with a keen blue sky and a smell of wood-smoke on the westerly breeze. We sat on the slates and drank our whiskies and took turns holding the baby, who was still fast asleep.\n\n'You having him christened?' Diana asked Verity softly, peering down at the infant's tiny scrunched-up face. She rocked him to and fro.\n\n'Well, I think mum and dad would rather he was, but I'm not bothered one way or the other. Lewis isn't too keen, are you, my love?'\n\nLewis showed his teeth. 'Over my dead body, actually.' he said.\n\n'See?' Verity said to Diana, who was smiling broadly and holding the boy close, sniffing him. She just nodded.\n\nVerity glanced at Lewis, then said, 'Prentice?'\n\n'Yo?'\n\n'We'd like you to be his godfather. Would you be?' She actually looked as though she thought I might refuse. Lewis was grinning at me.\n\nI cleared my throat. 'Well ... in terms of the actual title, I'm sort of taking a long hard look at my previous statements about the existence or non-existence of a supreme being at this moment in time, re-appraisal-wise,' I said, a suitably pained expression on my face as Diana handed the baby to me.\n\nLewis laughed.\n\nAnyway, it was agreed, and then we thought the little blighter ought to have at least a semblance of a christening, so Lewis dabbed his finger in his whisky and reached over and put a tiny drop of the spirit on his son's head, and said, 'There; that's all he's going to get.'\n\n'Kenneth Walker McHoan,' I said, cradling him with one arm and raising my glass in the other hand.\n\nWe drank the lad's health. Then Diana threw her glass away over the battlements towards the woods. Lewis, Verity and I all looked at each other, then followed suit, and heard a couple of the tumblers smash somewhere in the trees beneath. Young Kenneth opened his eyes at that point, looked woozily up at me and let out a small, plaintive cry. I laughed and kissed his tiny nose, then handed him back to his mother so she could feed him.\n\nI stood up then and went to the battlements, and held the ancient rough stones beneath my hands. I looked out over the woods and the plain and the fields; to Gallanach, with its quays and spires and serried streets, and out to the crumpled hills beyond, the brindle of forests to the east and the glitter of waves to the west, where the ocean was. I thought of Ashley, on the other side of that ocean, and wondered what she was doing right now, and hoped that she was well, and happy, and maybe thinking of me, and then I just stood there, grinning like a fool, and took a deep, deep breath of that sharp, smoke-scented air and raised my arms to the open sky, and said, _'Ha!'_\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n##### Copyright\n\nThe 7 Habits of Highly Effective People\n\nCopyright \u00a9 1989, 2004 by Stephen R. Covey\n\nCover art to the electronic edition copyright \u00a9 2012 by RosettaBooks, LLC\n\nAll rights reserved. No part of this book may be used or reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without permission in writing from the publisher, except by a reviewer who may quote brief passages in a review.\n\nElectronic edition published 2012 by RosettaBooks LLC, New York.\n\nePub ISBN: 978-0-7953-2342-3\nTo my colleagues,  \nempowered  \nand empowering\n\n### Acknowledgments\n\nInterdependence is a higher value than independence.\n\nThis work is a synergistic product of many minds. It began in the middle seventies as I was reviewing 200 years of success literature as part of a doctoral program. I am grateful for the inspiration and wisdom of many thinkers and for the trans-generational sources and roots of this wisdom.\n\nI am also grateful for many students, friends, and colleagues at Brigham Young University and the Covey Leadership Center and for thousands of adults, parents, youth, executives, teachers, and other clients who have tested this material and have given feedback and encouragement. The material and arrangement has slowly evolved and has imbued those who have been sincerely and deeply immersed in it with the conviction that the Seven Habits represent a holistic, integrated approach to personal and interpersonal effectiveness, and that, more than in the individual habits themselves, the real key lies in the relationship among them and in how they are sequenced.\n\nFor the development and production of the book itself I feel a deep sense of gratitude:\n\n\u2014to Sandra and to each of our children and their spouses for living lives of integrity and service and for supporting my many travels and involve\u00adments outside the home. It's easy to teach principles loved ones live.\n\n\u2014to my brother John for his constant love, interest, insights and purity of soul.\n\n\u2014to the happy memory of my father.\n\n\u2014to my mother for her devotion to her more than 87 living descendants and for her constant demonstrations of love.\n\n\u2014to my dear friends and colleagues in the business, especially:\n\n\u2014to Bill Marre, Ron McMillan, and Lex Watterson for feedback, encour\u00adagement, editorial suggestions, and production help.\n\n\u2014to Brad Anderson, who at great personal sacrifice for over a year, developed a Seven Habits video-based development program. Under his leadership this material has been tested and refined and is being implemented by thousands of people across a broad range of organiza\u00adtions. Almost without exception, after initial exposure to this material, our clients desire to make it available to greater numbers of employees, underscoring our confidence that it \"works.\"\n\n\u2014to Bob Thele for helping to create a system for our firm that gave me the peace of mind to enable me to really focus on the book.\n\n\u2014to David Conley for communicating the value and power of the Seven Habits to hundreds of business organizations so that my colleagues, Blaine Lee, Roice Krueger, Roger Merrill and Al Switzler, and I have the constant opportunity to share ideas in a large variety of settings.\n\n\u2014to my proactive literary agent Jan Miller, and my \"can do\" associate Greg Link and his assistant Stephanni Smith and Raleen Beckham Wahlin for their creative and courageous marketing leadership.\n\n\u2014to my Simon and Schuster editor Bob Asahina for his professional competence and project leadership, for his many excellent suggestions and for helping me to better understand the difference between writing and speaking.\n\n\u2014to my earlier devoted assistants Shirley and Heather Smith and to my present assistant Marilyn Andrews for a level of loyalty which is truly uncommon.\n\n\u2014to our Executive Excellence magazine editor Ken Shelton for his editing of the first manuscript years ago, for helping refine and test the material in several contexts, and for his integrity and sense of quality.\n\n\u2014to Rebecca Merrill for her invaluable editing and production assistance, for her inner commitment to the material, and for her skill, sensitivity, and carefulness in fulfilling that commitment, and to her husband, Roger, for his wise, synergistic help.\n\n\u2014and to Kay Swim and her son, Gaylord, for their much appreciated vision which contributed to our organization's rapid growth.\n\n### Contents\n\nFOREWORD\n\n**PART ONE: PARADIGMS AND PRINCIPLES**\n\nInside-Out\n\nThe Seven Habits\u2014An Overview\n\n**PART TWO: PRIVATE VICTORY**\n\nHABIT 1 Be Proactive\n\nPrinciples of Personal Vision\n\nHABIT 2 Begin with the End in Mind\n\nPrinciples of Personal Leadership\n\nHABIT 3 Put First Things First\n\nPrinciples of Personal Management\n\n**PART THREE: PUBLIC VICTORY**\n\nParadigms of Interdependence\n\nHABIT 4 Think Win\/Win\n\nPrinciples of Interpersonal Leadership\n\nHABIT 5 Seek First to Understand, Then to Be Understood\n\nPrinciples of Empathetic Communication\n\nHABIT 6 Synergize\n\nPrinciples of Creative Cooperation\n\n**PART FOUR: RENEWAL**\n\nHABIT 7 Sharpen the Saw\n\nPrinciples of Balanced Self-Renewal\n\nInside-Out Again\n\nA Personal Note\n\nAFTERWORD: Questions I am Often Asked\n\nAPPENDIX A: Possible Perceptions Flowing out of Various Centers\n\nAPPENDIX B: A Quadrant II Day at the Office\n\nPROBLEM\/OPPORTUNITY INDEX\n\nINDEX\n\n### Foreword\n\nWhen I wrote this book in 1989, I had no idea how the world would change and that people would be able to read The 7 Habits of Highly Effective People in this amazing way on these amazing devices.\n\nSince then, this book has been called \"the most influential business book of the century\" (by The Wall Street Journal). It stayed on The New York Times best-seller list for five years. A \"must-read\" translated into 38 languages, with 20 million copies in print (and is arguably the most pirated business book in the world). Google 7 Habits and you'll get more than 12 million hits.\n\nI've been humbled and gratified by the worldwide audience this book has reached. Presidents, prime ministers, and kings read the book, but so do college students, construction workers, and kitchen help. I've heard from literally thousands of people who say things like this:\n\n  * \"I learned to focus on truly important things, not just urgent things.\"\n  * I listen\u2014really listen\u2014to other people for the first time.\"\n  * \"Since I started thinking win-win in a job that was killing me, I've found a new mission and purpose in my professional life.\"\n\nHow will The 7 Habits impact you? My hope is that you will find new hope, a greater sense of purpose, more peace of mind, and far more rewarding relationships in both your personal and professional life.\n\nStephen R. Covey\n\nDecember 2009\n\nFranklinCovey Co.\n\n## _PART ONE_\n\n# PARADIGMS  \nand PRINCIPLES\n\n### Inside-Out\n\n_There is no real excellence in all this world  \nwhich can be separated from right living._\n\nDavid Starr Jordan\n\nIN MORE THAN 25 YEARS of working with people in business, university, and marriage and family settings, I have come in contact with many individuals who have achieved an incredible degree of outward success, but have found themselves struggling with an inner hunger, a deep need for personal congruency and effectiveness and for healthy, growing relationships with other people.\n\nI suspect some of the problems they have shared with me may be familiar to you.\n\n_I've set and met my career goals and I'm having tremendous professional success. But it's cost me my personal and family life. I don't know my wife and children any more. I'm not even sure I know myself and what's really important to me. I've had to ask myself\u2014is it worth it?_\n\n_I've started a new diet\u2014for the fifth time this year. I know I'm overweight, and I really want to change. I read all the new information, I set goals, I get myself all psyched up with a positive mental attitude and tell myself I can do it. But I don't. After a few weeks, I fizzle. I just can't seem to keep a promise I make to myself._\n\n_I've taken course after course on effective management training. I expect a lot out of my employees and I work hard to be friendly toward them and to treat them right. But I don't feel any loyalty from them. I think if I were home sick for a day, they'd spend most of their time gabbing at the water fountain. Why can't I train them to be independent and responsible\u2014or find employees who can be?_\n\n_My teenage son is rebellious and on drugs. No matter what I try, he won't listen to me. What can I do?_\n\n_There's so much to do. And there's never enough time. I feel pressured and hassled all day, every day, seven days a week. I've attended time management seminars and I've tried half a dozen different planning systems. They've helped some, but I still don't feel I'm living the happy, productive, peaceful life I want to live._\n\n_I want to teach my children the value of work. But to get them to do anything, I have to supervise every move... and put up with complaining every step of the way. It's so much easier to do it myself. Why can't children do their work cheerfully and without being reminded?_\n\n_I'm busy\u2014really busy. But sometimes I wonder if what I'm doing will make any difference in the long run. I'd really like to think there was meaning in my life, that somehow things were different because I was here._\n\n_I see my friends or relatives achieve some degree of success or receive some recognition, and I smile and congratulate them enthusiastically. But inside, I'm eating my heart out. Why do I feel this way?_\n\n_I have a forceful personality. I know, in almost any interaction, I can control the outcome. Most of the time, I can even do it by influencing others to come up with the solution I want. I think through each situation and I really feel the ideas I come up with are usually the best for everyone. But I feel uneasy. I always wonder what other people really think of me and my ideas._\n\n_My marriage has gone flat. We don't fight or anything; we just don't love each other anymore. We've gone to counseling; we've tried a number of things, but we just can't seem to rekindle the feeling we used to have._\n\nThese are deep problems, painful problems\u2014problems that quick fix approaches can't solve.\n\nA few years ago, my wife Sandra and I were struggling with this kind of concern. One of our sons was having a very difficult time in school. He was doing poorly academically; he didn't even know how to follow the instructions on the tests, let alone do well on them. Socially he was immature, often embarrassing those closest to him. Athletically, he was small, skinny, and uncoordinated\u2014swinging his baseball bat, for example, almost before the ball was even pitched. Others would laugh at him.\n\nSandra and I were consumed with a desire to help him. We felt that if \"success\" were important in any area of life, it was supremely important in our role as parents. So we worked on our attitudes and behavior toward him and we tried to work on his. We attempted to psych him up using positive mental attitude techniques. \"Come on, son! You can do it! We know you can. Put your hands a little higher on the bat and keep your eye on the ball. Don't swing till it gets close to you.\" And if he did a little better, we would go to great lengths to reinforce him. \"That's good, son, keep it up.\"\n\nWhen others laughed, we reprimanded them. \"Leave him alone. Get off his back. He's just learning.\" And our son would cry and insist that he'd never be any good and that he didn't like baseball anyway.\n\nNothing we did seemed to help, and we were really worried. We could see the effect this was having on his self-esteem. We tried to be encouraging and helpful and positive, but after repeated failure, we finally drew back and tried to look at the situation on a different level.\n\nAt this time in my professional role I was involved in leadership development work with various clients throughout the country. In that capacity I was preparing bimonthly programs on the subject of communication and perception for IBM's Executive Development Program participants.\n\nAs I researched and prepared these presentations, I became particularly interested in how perceptions are formed, how they govern the way we see, and how the way we see governs how we behave. This led me to a study of expectancy theory and self-fulfilling prophecies or the \"Pygmalion effect,\" and to a realization of how deeply imbedded our perceptions are. It taught me that we must look at the lens through which we see the world, as well as at the world we see, and that the lens itself shapes how we interpret the world.\n\nAs Sandra and I talked about the concepts I was teaching at IBM and about our own situation, we began to realize that what we were doing to help our son was not in harmony with the way we really saw him. When we honestly examined our deepest feelings, we realized that our perception was that he was basically inadequate, somehow \"behind.\" No matter how much we worked on our attitude and behavior, our efforts were ineffective because, despite our actions and our words, what we really communicated to him was, \"You aren't capable. You have to be protected.\"\n\nWe began to realize that if we wanted to change the situation, we first had to change ourselves. And to change ourselves effectively, we first had to change our perceptions.\n\nTHE PERSONALITY AND CHARACTER ETHICS\n\nAt the same time, in addition to my research on perception, I was also deeply immersed in an in-depth study of the success literature published in the United States since 1776. I was reading or scanning literally hundreds of books, articles, and essays in fields such as self-improvement, popular psychology, and self-help. At my fingertips was the sum and substance of what a free and democratic people considered to be the keys to successful living.\n\nAs my study took me back through 200 years of writing about success, I noticed a startling pattern emerging in the content of the literature. Because of our own pain, and because of similar pain I had seen in the lives and relationships of many people I had worked with through the years, I began to feel more and more that much of the success literature of the past 50 years was superficial. It was filled with social image consciousness, techniques and quick fixes\u2014with social band-aids and aspirin that addressed acute problems and sometimes even appeared to solve them temporarily, but left the underlying chronic problems untouched to fester and resurface time and again.\n\nIn stark contrast, almost all the literature in the first 150 years or so focused on what could be called the Character Ethic as the foundation of success\u2014things like integrity, humility, fidelity, temperance, courage, justice, patience, industry, simplicity, modesty, and the Golden Rule. Benjamin Franklin's autobiography is representative of that literature. It is, basically, the story of one man's effort to integrate certain principles and habits deep within his nature.\n\nThe Character Ethic taught that there are basic principles of effective living, and that people can only experience true success and enduring happiness as they learn and integrate these principles into their basic character.\n\nBut shortly after World War 1 the basic view of success shifted from the Character Ethic to what we might call the Personality Ethic. Success became more a function of personality, of public image, of attitudes and behaviors, skills and techniques, that lubricate the processes of human interaction. This Personality Ethic essentially took two paths: one was human and public relations techniques, and the other was positive mental attitude (PMA). Some of this philosophy was expressed in inspiring and sometimes valid maxims such as \"Your attitude determines your altitude,\" \"Smiling wins more friends than frowning,\" and \"Whatever the mind of man can conceive and believe it can achieve.\"\n\nOther parts of the personality approach were clearly manipulative, even deceptive, encouraging people to use techniques to get other people to like them, or to fake interest in the hobbies of others to get out of them what they wanted, or to use the \"power look,\" or to intimidate their way through life.\n\nSome of this literature acknowledged character as an ingredient of success, but tended to compartmentalize it rather than recognize it as foundational and catalytic. Reference to the Character Ethic became mostly lip service; the basic thrust was quick-fix influence techniques, power strategies, communication skills, and positive attitudes.\n\nThis Personality Ethic, I began to realize, was the subconscious source of the solutions Sandra and I were attempting to use with our son. As I thought more deeply about the difference between the Personality and Character Ethics, I realized that Sandra and I had been getting social mileage out of our children's good behavior, and, in our eyes, this son simply didn't measure up. Our image of ourselves, and our role as good, caring parents was even deeper than our image of our son and perhaps influenced it. There was a lot more wrapped up in the way we were seeing and handling the problem than our concern for our son's welfare.\n\nAs Sandra and I talked, we became painfully aware of the powerful influence of our own character and motives and of our perception of him. We knew that social comparison motives were out of harmony with our deeper values and could lead to conditional love and eventually to our son's lessened sense of self-worth. So we determined to focus our efforts on us\u2014not on our techniques, but on our deepest motives and our perception of him. Instead of trying to change him, we tried to stand apart\u2014to separate us from him\u2014and to sense his identity, individuality, separateness, and worth.\n\nThrough deep thought and the exercise of faith and prayer, we began to see our son in terms of his own uniqueness. We saw within him layers and layers of potential that would be realized at his own pace and speed. We decided to relax and get out of his way and let his own personality emerge. We saw our natural role as being to affirm, enjoy, and value him. We also conscientiously worked on our motives and cultivated internal sources of security so that our own feelings of worth were not dependent on our children's \"acceptable\" behavior.\n\nAs we loosened up our old perception of our son and developed value-based motives, new feelings began to emerge. We found ourselves enjoying him instead of comparing or judging him. We stopped trying to clone him in our own image or measure him against social expectations. We stopped trying to kindly, positively manipulate him into an acceptable social mold. Because we saw him as fundamentally adequate and able to cope with life, we stopped protecting him against the ridicule of others.\n\nHe had been nurtured on this protection, so he went through some withdrawal pains, which he expressed and which we accepted, but did not necessarily respond to. \"We don't need to protect you,\" was the unspoken message. \"You're fundamentally okay.\"\n\nAs the weeks and months passed, he began to feel a quiet confidence and affirmed himself. He began to blossom, at his own pace and speed. He became outstanding as measured by standard social criteria\u2014academically, socially and athletically\u2014at a rapid clip, far beyond the so-called natural developmental process. As the years passed, he was elected to several student body leadership positions, developed into an all-state athlete and started bringing home straight A report cards. He developed an engaging and guileless personality that has enabled him to relate in nonthreatening ways to all kinds of people.\n\nSandra and I believe that our son's \"socially impressive\" accomplishments were more a serendipitous expression of the feelings he had about himself than merely a response to social reward. This was an amazing experience for Sandra and me, and a very instructional one in dealing with our other children and in other roles as well. It brought to our awareness on a very personal level the vital difference between the Personality Ethic and the Character Ethic of success. The Psalmist expressed our conviction well: \"Search your own heart with all diligence for out of it flow the issues of life.\"\n\nPRIMARY AND SECONDARY GREATNESS\n\nMy experience with my son, my study of perception and my reading of the success literature coalesced to create one of those \"Aha!\" experiences in life when suddenly things click into place. I was suddenly able to see the powerful impact of the Personality Ethic and to clearly understand those subtle, often consciously unidentified discrepancies between what I knew to be true\u2014some things I had been taught many years ago as a child and things that were deep in my own inner sense of value\u2014and the quick fix philosophies that surrounded me every day. I understood at a deeper level why, as I had worked through the years with people from all walks of life, I had found that the things I was teaching and knew to be effective were often at variance with these popular voices.\n\nI am not suggesting that elements of the Personality Ethic\u2014personality growth, communication skill training, and education in the field of influence strategies and positive thinking\u2014are not beneficial, in fact sometimes essential for success. I believe they are. But these are secondary, not primary traits. Perhaps, in utilizing our human capacity to build on the foundation of generations before us, we have inadvertently become so focused on our own building that we have forgotten the foundation that holds it up; or in reaping for so long where we have not sown, perhaps we have forgotten the need to sow.\n\nIf I try to use human influence strategies and tactics of how to get other people to do what I want, to work better, to be more motivated, to like me and each other\u2014while my character is fundamentally flawed, marked by duplicity and insincerity\u2014then, in the long run, I cannot be successful. My duplicity will breed distrust, and everything I do\u2014even using so-called good human relations techniques\u2014will be perceived as manipulative. It simply makes no difference how good the rhetoric is or even how good the intentions are; if there is little or no trust, there is no foundation for permanent success. Only basic goodness gives life to technique.\n\nTo focus on technique is like cramming your way through school. You sometimes get by, perhaps even get good grades, but if you don't pay the price day in and day out, you never achieve true mastery of the subjects you study or develop an educated mind.\n\nDid you ever consider how ridiculous it would be to try to cram on a farm\u2014to forget to plant in the spring, play all summer and then cram in the fall to bring in the harvest? The farm is a natural system. The price must be paid and the process followed. You always reap what you sow; there is no shortcut.\n\nThis principle is also true, ultimately, in human behavior, in human relationships. They, too, are natural systems based on the law of the harvest. In the short run, in an artificial social system such as school, you may be able to get by if you learn how to manipulate the man-made rules, to \"play the game.\" In most one-shot or short-lived human interactions, you can use the Personality Ethic to get by and to make favorable impressions through charm and skill and pretending to be interested in other people's hobbies. You can pick up quick, easy techniques that may work in short-term situations. But secondary traits alone have no permanent worth in long-term relationships. Eventually, if there isn't deep integrity and fundamental character strength, the challenges of life will cause true motives to surface and human relationship failure will replace short-term success.\n\nMany people with secondary greatness\u2014that is, social recognition for their talents\u2014lack primary greatness or goodness in their character. Sooner or later, you'll see this in every long-term relationship they have, whether it is with a business associate, a spouse, a friend, or a teenage child going through an identity crisis. It is character that communicates most eloquently. As Emerson once put it, \"What you are shouts so loudly in my ears I cannot hear what you say.\"\n\nThere are, of course, situations where people have character strength but they lack communication skills, and that undoubtedly affects the quality of relationships as well. But the effects are still secondary.\n\nIn the last analysis, what we are communicates far more eloquently than anything we say or do. We all know it. There are people we trust absolutely because we know their character. Whether they're eloquent or not, whether they have the human relations techniques or not, we trust them, and we work successfully with them.\n\nIn the words of William George Jordan, \"Into the hands of every individual is given a marvelous power for good or evil\u2014the silent, unconscious, unseen influence of his life. This is simply the constant radiation of what man really is, not what he pretends to be.\"\n\nTHE POWER OF A PARADIGM\n\nThe Seven Habits of Highly Effective People embody many of the fundamental principles of human effectiveness. These habits are basic; they are primary. They represent the internalization of correct principles upon which enduring happiness and success are based.\n\nBut before we can really understand these Seven Habits, we need to understand our own \"paradigms\" and how to make a \"paradigm shift.\"\n\nBoth the Character Ethic and the Personality Ethic are examples of social paradigms. The word paradigm comes from the Greek. It was originally a scientific term, and is more commonly used today to mean a model, theory, perception, assumption, or frame of reference. In the more general sense, it's the way we \"see\" the world\u2014not in terms of our visual sense of sight, but in terms of perceiving, understanding, interpreting.\n\nFor our purposes, a simple way to understand paradigms is to see them as maps. We all know that \"the map is not the territory.\" A map is simply an explanation of certain aspects of the territory. That's exactly what a paradigm is. It is a theory, an explanation, or model of something else.\n\nSuppose you wanted to arrive at a specific location in central Chicago. A street map of the city would be a great help to you in reaching your destination. But suppose you were given the wrong map. Through a printing error, the map labeled \"Chicago\" was actually a map of Detroit. Can you imagine the frustration, the ineffectiveness of trying to reach your destination?\n\nYou might work on your behavior\u2014you could try harder, be more diligent, double your speed. But your efforts would only succeed in getting you to the wrong place faster.\n\nYou might work on your attitude\u2014you could think more positively. You still wouldn't get to the right place, but perhaps you wouldn't care. Your attitude would be so positive, you'd be happy wherever you were.\n\nThe point is, you'd still be lost. The fundamental problem has nothing to do with your behavior or your attitude. It has everything to do with having a wrong map.\n\nIf you have the right map of Chicago, then diligence becomes important, and when you encounter frustrating obstacles along the way, then attitude can make a real difference. But the first and most important requirement is the accuracy of the map.\n\nEach of us has many, many maps in our head, which can be divided into two main categories: maps of the way things are, or realities, and maps of the way things should be, or values. We interpret everything we experience through these mental maps. We seldom question their accuracy; we're usually even unaware that we have them. We simply assume that the way we see things is the way they really are or the way they should be.\n\nAnd our attitudes and behaviors grow out of those assumptions. The way we see things is the source of the way we think and the way we act.\n\nBefore going any further, I invite you to have an intellectual and emotional experience. Take a few seconds and just look at the picture Figure 1.\n\nNow look at the picture Figure 2 and carefully describe what you see.\n\nDo you see a woman? How old would you say she is? What does she look like? What is she wearing? In what kind of roles do you see her?\n\nYou probably would describe the woman in the second picture to be about 25 years old\u2014very lovely, rather fashionable with a petite nose and a demure presence. If you were a single man you might like to take her out. If you were in retailing, you might hire her as a fashion model.\n\nBut what if I were to tell you that you're wrong? What if I said this picture is of a woman in her 60's or 70's who looks sad, has a huge nose, and is certainly no model. She's someone you probably would help across the street.\n\nWho's right? Look at the picture again. Can you see the old woman? If you can't, keep trying. Can you see her big hook nose? Her shawl?\n\nIf you and I were talking face to face, we could discuss the picture. You could describe what you see to me, and I could talk to you about what I see. We could continue to communicate until you clearly showed me what you see in the picture and I clearly showed you what I see.\n\nBecause we can't do that, turn to Figure 3 and study the picture there and then look at this picture again. Can you see the old woman now? It's important that you see her before you continue reading.\n\nI first encountered this exercise many years ago at the Harvard Business School. The instructor was using it to demonstrate clearly and eloquently that two people can see the same thing, disagree, and yet both be right. It's not logical; its psychological.\n\nHe brought into the room a stack of large cards, half of which had the image of the young woman you saw in Figure 1, and the other half of which had the image of the old woman in Figure 3.\n\nHe passed them out to the class, the picture of the young woman to one side of the room and the picture of the old woman to the other. He asked us to look at the cards, concentrate on them for about ten seconds and then pass them back in. He then projected upon the screen the picture you saw in Figure 2 combining both images and asked the class to describe what they saw. Almost every person in that class who had first seen the young woman's image on a card saw the young woman in the picture. And almost every person who had first seen the old woman's image on a card saw an old woman in the picture.\n\nThe professor then asked one student to explain what he saw to a student on the opposite side of the room. As they talked back and forth, communication problems flared up.\n\n\"What do you mean, 'old lady'? She couldn't be more than 20 or 22 years old!\"\n\n\"Oh, come on. You have to be joking. She's 70\u2014could be pushing 80!\"\n\n\"What's the matter with you? Are you blind? This lady is young, good looking. I'd like to take her out. She's lovely.\"\n\n\"Lovely? She's an old hag.\"\n\nThe arguments went back and forth, each person sure of, and adamant in, his or her position. All of this occurred in spite of one exceedingly important advantage the students had\u2014most of them knew early in the demonstration that another point of view did, in fact, exist\u2014something many of us would never admit. Nevertheless, at first, only a few students really tried to see this picture from another frame of reference.\n\nAfter a period of futile communication, one student went up to the screen and pointed to a line on the drawing. \"There is the young woman's necklace.\" The other one said, \"No, that is the old woman's mouth.\" Gradually, they began to calmly discuss specific points of difference, and finally one student, and then another, experienced sudden recognition when the images of both came into focus. Through continued calm, respectful, and specific communication, each of us in the room was finally able to see the other point of view. But when we looked away and then back, most of us would immediately see the image we had been conditioned to see in the ten-second period of time.\n\nI frequently use this perception demonstration in working with people and organizations because it yields so many deep insights into both personal and interpersonal effectiveness. It shows, first of all, how powerfully conditioning affects our perceptions, our paradigms. If ten seconds can have that kind of impact on the way we see things, what about the conditioning of a lifetime? The influences in our lives\u2014family, school, church, work environment, friends, associates, and current social paradigms such as the Personality Ethic\u2014all have made their silent unconscious impact on us and help shape our frame of reference, our paradigms, our maps.\n\nIt also shows that these paradigms are the source of our attitudes and behaviors. We cannot act with integrity outside of them. We simply cannot maintain wholeness if we talk and walk differently than we see. If you were among the 90 percent who typically see the young woman in the composite picture when conditioned to do so, you undoubtedly found it difficult to think in terms of having to help her cross the street. Both your attitude about her and your behavior toward her had to be congruent with the way you saw her.\n\nThis brings into focus one of the basic flaws of the Personality Ethic. To try to change outward attitudes and behaviors does very little good in the long run if we fail to examine the basic paradigms from which those attitudes and behaviors flow.\n\nThis perception demonstration also shows how powerfully our paradigms affect the way we interact with other people. As clearly and objectively as we think we see things, we begin to realize that others see them differently from their own apparently equally clear and objective point of view. \"Where we stand depends on where we sit.\"\n\nEach of us tends to think we see things as they are, that we are objective. But this is not the case. We see the world, not as it is, but as we are\u2014or, as we are conditioned to see it. When we open our mouths to describe what we see, we in effect describe ourselves, our perceptions, our paradigms. When other people disagree with us, we immediately think something is wrong with them. But, as the demonstration shows, sincere, clearheaded people see things differently, each looking through the unique lens of experience.\n\nThis does not mean that there are no facts. In the demonstration, two individuals who initially have been influenced by different conditioning pictures look at the third picture together. They are now both looking at the same identical facts\u2014black lines and white spaces\u2014and they would both acknowledge these as facts. But each person's interpretation of these facts represents prior experiences, and the facts have no meaning whatsoever apart from the interpretation.\n\nThe more aware we are of our basic paradigms, maps, or assumptions, and the extent to which we have been influenced by our experience, the more we can take responsibility for those paradigms, examine them, test them against reality, listen to others and be open to their perceptions, thereby getting a larger picture and a far more objective view.\n\nTHE POWER OF A PARADIGM SHIFT\n\nPerhaps the most important insight to be gained from the perception demonstration is in the area of paradigm shifting, what we might call the \"Aha!\" experience when someone finally \"sees\" the composite picture in another way. The more bound a person is by the initial perception, the more powerful the \"Aha!\" experience is. It's as though a light were suddenly turned on inside.\n\nThe term paradigm shift was introduced by Thomas Kuhn in his highly influential landmark book, The Structure of Scientific Revolutions. Kuhn shows how almost every significant break\u00adthrough in the field of scientific endeavor is first a break with tradition, with old ways of thinking, with old paradigms.\n\nFor Ptolemy, the great Egyptian astronomer, the earth was the center of the universe. But Copernicus created a paradigm shift, and a great deal of resistance and persecution as well, by placing the sun at the center. Suddenly, everything took on a different interpretation.\n\nThe Newtonian model of physics was a clockwork paradigm and is still the basis of modern engineering. But it was partial, incomplete. The scientific world was revolutionized by the Einsteinian paradigm, the relativity paradigm, which had much higher predictive and explanatory value.\n\nUntil the germ theory was developed, a high percentage of women and children died during childbirth, and no one could understand why. In military skirmishes, more men were dying from small wounds and diseases than from the major traumas on the front lines. But as soon as the germ theory was developed, a whole new paradigm, a better, improved way of understanding what was happening made dramatic, significant medical improvement possible.\n\nThe United States today is the fruit of a paradigm shift. The traditional concept of government for centuries had been a monarchy, the divine right of kings. Then a different paradigm was developed\u2014government of the people, by the people, and for the people. And a constitutional democracy was born, unleashing tremendous human energy and ingenuity, and creating a standard of living, of freedom and liberty, of influence and hope unequaled in the history of the world.\n\nNot all paradigm shifts are in positive directions. As we have observed, the shift from the Character Ethic to the Personality Ethic has drawn us away from the very roots that nourish true success and happiness.\n\nBut whether they shift us in positive or negative directions, whether they are instantaneous or developmental, paradigm shifts move us from one way of seeing the world to another. And those shifts create powerful change. Our paradigms, correct or incorrect, are the sources of our attitudes and behaviors, and ultimately our relationships with others.\n\nI remember a mini-paradigm shift I experienced one Sunday morning on a subway in New York. People were sitting quietly\u2014some reading newspapers, some lost in thought, some resting with their eyes closed. It was a calm, peaceful scene.\n\nThen suddenly, a man and his children entered the subway car. The children were so loud and rambunctious that instantly the whole climate changed.\n\nThe man sat down next to me and closed his eyes, apparently oblivious to the situation. The children were yelling back and forth, throwing things, even grabbing people's papers. It was very disturbing. And yet, the man sitting next to me did nothing.\n\nIt was difficult not to feel irritated. I could not believe that he could be so insensitive as to let his children run wild like that and do nothing about it, taking no responsibility at all. It was easy to see that everyone else on the subway felt irritated, too. So finally, with what I felt was unusual patience and restraint, I turned to him and said, \"Sir, your children are really disturbing a lot of people. I wonder if you couldn't control them a little more?\"\n\nThe man lifted his gaze as if to come to a consciousness of the situation for the first time and said softly, \"Oh, you're right. I guess I should do something about it. We just came from the hospital where their mother died about an hour ago. I don't know what to think, and I guess they don't know how to handle it either.\"\n\nCan you imagine what I felt at that moment? My paradigm shifted. Suddenly I saw things differently, and because I saw differently, I thought differently, I felt differently, I behaved differently. My irritation vanished. I didn't have to worry about controlling my attitude or my behavior; my heart was filled with the man's pain. Feelings of sympathy and compassion flowed freely. \"Your wife just died? Oh, I'm so sorry! Can you tell me about it? What can I do to help?\" Everything changed in an instant.\n\nMany people experience a similar fundamental shift in thinking when they face a life-threatening crisis and suddenly see their priorities in a different light, or when they suddenly step into a new role, such as that of husband or wife, parent or grandparent, manager or leader.\n\nWe could spend weeks, months, even years laboring with the Personality Ethic trying to change our attitudes and behaviors and not even begin to approach the phenomenon of change that occurs spontaneously when we see things differently.\n\nIt becomes obvious that if we want to make relatively minor changes in our lives, we can perhaps appropriately focus on our attitudes and behaviors. But if we want to make significant, quantum change, we need to work on our basic paradigms.\n\nIn the words of Thoreau, \"For every thousand hacking at the leaves of evil, there is one striking at the root.\" We can only achieve quantum improvements in our lives as we quit hacking at the leaves of attitude and behavior and get to work on the root, the paradigms from which our attitudes and behaviors flow.\n\nSEEING AND BEING\n\nOf course, not all paradigm shifts are instantaneous. Unlike my instant insight on the subway, the paradigm-shifting experience Sandra and I had with our son was a slow, difficult, and deliberate process. The approach we had first taken with him was the outgrowth of years of conditioning and experience in the Personality Ethic. It was the result of deeper paradigms we held about our own success as parents as well as the measure of success of our children. And it was not until we changed those basic paradigms, until we saw things differently, that we were able to create quantum change in ourselves and in the situation.\n\nIn order to see our son differently, Sandra and I had to be differently. Our new paradigm was created as we invested in the growth and development of our own character.\n\nParadigms are inseparable from character. Being is seeing in the human dimension. And what we see is highly interrelated to what we are. We can't go very far to change our seeing without simultaneously changing our being, and vice versa.\n\nEven in my apparently instantaneous paradigm-shifting experience that morning on the subway, my change of vision was a result of\u2014and limited by\u2014my basic character.\n\nI'm sure there are people who, even suddenly understanding the true situation, would have felt no more than a twinge of regret or vague guilt as they continued to sit in embarrassed silence beside the grieving, confused man. On the other hand, I am equally certain there are people who would have been far more sensitive in the first place, who may have recognized that a deeper problem existed and reached out to understand and help before I did.\n\nParadigms are powerful because they create the lens through which we see the world. The power of a paradigm shift is the essential power of quantum change, whether that shift is an instantaneous or a slow and deliberate process.\n\nTHE PRINCIPLE-CENTERED PARADIGM\n\nThe Character Ethic is based on the fundamental idea that there are principles that govern human effectiveness\u2014natural laws in the human dimension that are just as real, just as unchanging and unarguably \"there\" as laws such as gravity are in the physical dimension.\n\nAn idea of the reality\u2014and the impact\u2014of these principles can be captured in another paradigm-shifting experience as told by Frank Koch in Proceedings, the magazine of the Naval Institute.\n\nTwo battleships assigned to the training squadron had been at sea on maneuvers in heavy weather for several days. I was serving on the lead battleship and was on watch on the bridge as night fell. The visibility was poor with patchy fog, so the captain remained on the bridge keeping an eye on all activities.\n\nShortly after dark, the lookout on the wing of the bridge reported, \"Light, bearing on the starboard bow.\"\n\n\"Is it steady or moving astern?\" the captain called out.\n\nLookout replied, \"Steady, captain,\" which meant we were on a dangerous collision course with that ship.\n\nThe captain then called to the signalman, \"Signal that ship: We are on a collision course, advise you change course 20 degrees.\"\n\nBack came a signal, \"Advisable for you to change course 20 degrees.\"\n\nThe captain said, \"Send, I'm a captain, change course 20 degrees.\"\n\n\"I'm a seaman second class,\" came the reply. \"You had better change course 20 degrees.\"\n\nBy that time, the captain was furious. He spat out, \"Send, I'm a battleship. Change course 20 degrees.\"\n\nBack came the flashing light, \"I'm a lighthouse.\"\n\nWe changed course.\n\nThe paradigm shift experienced by the captain\u2014and by us as we read this account\u2014puts the situation in a totally different light. We can see a reality that is superceded by his limited perception\u2014a reality that is as critical for us to understand in our daily lives as it was for the captain in the fog.\n\nPrinciples are like lighthouses. They are natural laws that cannot be broken. As Cecil B. deMille observed of the principles contained in his monumental movie, The Ten Commandments, \"It is impossible for us to break the law. We can only break ourselves against the law.\"\n\nWhile individuals may look at their own lives and interactions in terms of paradigms or maps emerging out of their experience and conditioning, these maps are not the territory. They are a \"subjective reality,\" only an attempt to describe the territory.\n\nThe \"objective reality,\" or the territory itself, is composed of \"lighthouse\" principles that govern human growth and happiness\u2014natural laws that are woven into the fabric of every civilized society throughout history and comprise the roots of every family and institution that has endured and prospered. The degree to which our mental maps accurately describe the territory does not alter its existence.\n\nThe reality of such principles or natural laws becomes obvious to anyone who thinks deeply and examines the cycles of social history. These principles surface time and time again, and the degree to which people in a society recognize and live in harmony with them moves them toward either survival and stability or disintegration and destruction.\n\nThe principles I am referring to are not esoteric, mysterious, or \"religious\" ideas. There is not one principle taught in this book that is unique to any specific faith or religion, including my own. These principles are a part of most every major enduring religion, as well as enduring social philosophies and ethical systems. They are self-evident and can easily be validated by any individual. It's almost as if these principles or natural laws are part of the human condition, part of the human consciousness, part of the human conscience. They seem to exist in all human beings, regardless of social conditioning and loyalty to them, even though they might be submerged or numbed by such conditions or disloyalty.\n\nI am referring, for example, to the principle of fairness, out of which our whole concept of equity and justice is developed. Little children seem to have an innate sense of the idea of fairness even apart from opposite conditioning experiences. There are vast differences in how fairness is defined and achieved, but there is almost universal awareness of the idea.\n\nOther examples would include integrity and honesty. They create the foundation of trust which is essential to cooperation and long-term personal and interpersonal growth.\n\nAnother principle is human dignity. The basic concept in the United States Declaration of Independence bespeaks this value or principle. \"We hold these truths to be self-evident: that all men are created equal and endowed by their Creator with certain inalien\u00adable rights, that among these are life, liberty and the pursuit of happiness.\"\n\nAnother principle is service, or the idea of making a contribution. Another is quality or excellence.\n\nThere is the principle of potential, the idea that we are embryonic and can grow and develop and release more and more potential, develop more and more talents. Highly related to potential is the principle of growth\u2014the process of releasing potential and developing talents, with the accompanying need for principles such as patience, nurturance, and encouragement.\n\nPrinciples are not practices. A practice is a specific activity or action. A practice that works in one circumstance will not necessarily work in another, as parents who have tried to raise a second child exactly like they did the first can readily attest.\n\nWhile practices are situationally specific, principles are deep, fundamental truths that have universal application. They apply to individuals, to marriages, to families, to private and public organizations of every kind. When these truths are internalized into habits, they empower people to create a wide variety of practices to deal with different situations.\n\nPrinciples are not values. A gang of thieves can share values, but they are in violation of the fundamental principles we're talking about. Principles are the territory. Values are maps. When we value correct principles, we have truth\u2014a knowledge of things as they are.\n\nPrinciples are guidelines for human conduct that are proven to have enduring, permanent value. They're fundamental. They're essentially unarguable because they are self-evident. One way to quickly grasp the self-evident nature of principles is to simply consider the absurdity of attempting to live an effective life based on their opposites. I doubt that anyone would seriously consider unfairness, deceit, baseness, uselessness, mediocrity, or degeneration to be a solid foundation for lasting happiness and success. Although people may argue about how these principles are defined or manifested or achieved, there seems to be an innate consciousness and awareness that they exist.\n\nThe more closely our maps or paradigms are aligned with these principles or natural laws, the more accurate and functional they will be. Correct maps will infinitely impact our personal and interpersonal effectiveness far more than any amount of effort expended on changing our attitudes and behaviors.\n\nPRINCIPLES OF GROWTH AND CHANGE\n\nThe glitter of the Personality Ethic, the massive appeal, is that there is some quick and easy way to achieve quality of life\u2014personal effectiveness and rich, deep relationships with other people\u2014without going through the natural process of work and growth that makes it possible.\n\nIt's symbol without substance. It's the \"get rich quick\" scheme promising \"wealth without work.\" And it might even appear to succeed\u2014but the schemer remains.\n\nThe Personality Ethic is illusory and deceptive. And trying to get high quality results with its techniques and quick fixes is just about as effective as trying to get to some place in Chicago using a map of Detroit.\n\nIn the words of Erich Fromm, an astute observer of the roots and fruits of the Personality Ethic:\n\nToday we come across an individual who behaves like an autom\u00adaton, who does not know or understand himself, and the only person that he knows is the person that he is supposed to be, whose meaningless chatter has replaced communicative speech, whose synthetic smile has replaced genuine laughter, and whose sense of dull despair has taken the place of genuine pain. Two statements may be said concerning this individual. One is that he suffers from defects of spontaneity and individuality which may seem to be incurable. At the same time it may be said of him he does not differ essentially from the millions of the rest of us who walk upon this earth.\n\nIn all of life, there are sequential stages of growth and development. A child learns to turn over, to sit up, to crawl, and then to walk and run. Each step is important and each one takes time. No step can be skipped.\n\nThis is true in all phases of life, in all areas of development, whether it be learning to play the piano or communicate effectively with a working associate. It is true with individuals, with marriages, with families, and with organizations.\n\nWe know and accept this fact or principle of process in the area of physical things, but to understand it in emotional areas, in human relations, and even in the area of personal character is less common and more difficult. And even if we understand it, to accept it and to live in harmony with it are even less common and more difficult. Consequently, we sometimes look for a shortcut, expecting to be able to skip some of these vital steps in order to save time and effort and still reap the desired result.\n\nBut what happens when we attempt to shortcut a natural process in our growth and development? If you are only an average tennis player but decide to play at a higher level in order to make a better impression, what will result? Would positive thinking alone enable you to compete effectively against a professional?\n\nWhat if you were to lead your friends to believe you could play the piano at concert hall level while your actual present skill was that of a beginner?\n\nThe answers are obvious. It is simply impossible to violate, ignore, or shortcut this development process. It is contrary to nature, and attempting to seek such a shortcut only results in disappointment and frustration.\n\nOn a ten-point scale, if I am at level two in any field, and desire to move to level five, I must first take the step toward level three. \"A thousand-mile journey begins with the first step\" and can only be taken one step at a time.\n\nIf you don't let a teacher know at what level you are\u2014by asking a question, or revealing your ignorance\u2014you will not learn or grow. You cannot pretend for long, for you will eventually be found out. Admission of ignorance is often the first step in our education. Thoreau taught, \"How can we remember our ignorance, which our growth requires, when we are using our knowledge all the time?\"\n\nI recall one occasion when two young women, daughters of a friend of mine, came to me tearfully, complaining about their father's harshness and lack of understanding. They were afraid to open up with their parents for fear of the consequences. And yet they desperately needed their parents' love, understanding, and guidance.\n\nI talked with the father and found that he was intellectually aware of what was happening. But while he admitted he had a temper problem, he refused to take responsibility for it and to honestly accept the fact that his emotional development level was low. It was more than his pride could swallow to take the first step toward change.\n\nTo relate effectively with a wife, a husband, children, friends, or working associates, we must learn to listen. And this requires emotional strength. Listening involves patience, openness, and the desire to understand\u2014highly developed qualities of character. It's so much easier to operate from a low emotional level and to give high-level advice.\n\nOur level of development is fairly obvious with tennis or piano playing, where it is impossible to pretend. But it is not so obvious in the areas of character and emotional development. We can \"pose\" and \"put on\" for a stranger or an associate. We can pretend. And for a while we can get by with it\u2014at least in public. We might even deceive ourselves. Yet I believe that most of us know the truth of what we really are inside; and I think many of those we live with and work with do as well.\n\nI have seen the consequences of attempting to shortcut this natural process of growth often in the business world, where executives attempt to \"buy\" a new culture of improved productivity, quality, morale, and customer service with strong speeches, smile training, and external interventions, or through mergers, acquisitions, and friendly or unfriendly takeovers. But they ignore the low-trust climate produced by such manipulations. When these methods don't work, they look for other Personality Ethic techniques that will\u2014all the time ignoring and violating the natural principles and processes on which a high-trust culture is based.\n\nI remember violating this principle myself as a father many years ago. One day I returned home to my little girl's third-year birthday party to find her in the corner of the front room, defiantly clutching all of her presents, unwilling to let the other children play with them. The first thing I noticed was several parents in the room witnessing this selfish display. I was embarrassed, and doubly so because at the time I was teaching university classes in human relations. And I knew, or at least felt, the expectation of these parents.\n\nThe atmosphere in the room was really charged\u2014the children were crowding around my little daughter with their hands out, asking to play with the presents they had just given, and my daughter was adamantly refusing. I said to myself, \"Certainly I should teach my daughter to share. The value of sharing is one of the most basic things we believe in.\"\n\nSo I first tried a simple request. \"Honey, would you please share with your friends the toys they've given you?\"\n\n\"No,\" she replied flatly.\n\nMy second method was to use a little reasoning. \"Honey, if you learn to share your toys with them when they are at your home, then when you go to their homes they will share their toys with you.\"\n\nAgain, the immediate reply was \"No!\"\n\nI was becoming a little more embarrassed, for it was evident I was having no influence. The third method was bribery. Very softly I said, \"Honey, if you share, I've got a special surprise for you. I'll give you a piece of gum.\"\n\n\"I don't want gum!\" she exploded.\n\nNow I was becoming exasperated. For my fourth attempt, I resorted to fear and threat. \"Unless you share, you will be in real trouble!\"\n\n\"I don't care!\" she cried. \"These are my things. I don't have to share!\"\n\nFinally, I resorted to force. I merely took some of the toys and gave them to the other kids. \"Here, kids, play with these.\"\n\nPerhaps my daughter needed the experience of possessing the things before she could give them. (In fact, unless I possess something, can I ever really give it?) She needed me as her father to have a higher level of emotional maturity to give her that experience.\n\nBut at that moment, I valued the opinion those parents had of me more than the growth and development of my child and our relationship together. I simply made an initial judgment that I was right; she should share, and she was wrong in not doing so.\n\nPerhaps I superimposed a higher-level expectation on her simply because on my own scale I was at a lower level. I was unable or unwilling to give patience or understanding, so I expected her to give things. In an attempt to compensate for my deficiency, I borrowed strength from my position and authority and forced her to do what I wanted her to do.\n\nBut borrowing strength builds weakness. It builds weakness in the borrower because it reinforces dependence on external factors to get things done. It builds weakness in the person forced to acquiesce, stunting the development of independent reasoning, growth, and internal discipline. And finally, it builds weakness in the relationship. Fear replaces cooperation, and both people involved become more arbitrary and defensive.\n\nAnd what happens when the source of borrowed strength\u2014be it superior size or physical strength, position, authority, credentials, status symbols, appearance, or past achievements\u2014changes or is no longer there?\n\nHad I been more mature, I could have relied on my own intrinsic strength\u2014my understanding of sharing and of growth and my capacity to love and nurture\u2014and allowed my daughter to make a free choice as to whether she wanted to share or not to share. Perhaps after attempting to reason with her, I could have turned the attention of the children to an interesting game, taking all that emotional pressure off my child. I've learned that once children gain a sense of real possession, they share very naturally, freely, and spontaneously.\n\nMy experience has been that there are times to teach and times not to teach. When relationships are strained and the air charged with emotion, an attempt to teach is often perceived as a form of judgment and rejection. But to take the child alone, quietly, when the relationship is good and to discuss the teaching or the value seems to have much greater impact. It may have been that the emotional maturity to do that was beyond my level of patience and internal control at the time.\n\nPerhaps a sense of possessing needs to come before a sense of genuine sharing. Many people who give mechanically or refuse to give and share in their marriages and families may never have experienced what it means to possess themselves, their own sense of identity and self-worth. Really helping our children grow may involve being patient enough to allow them the sense of possession as well as being wise enough to teach them the value of giving and providing the example ourselves.\n\nTHE WAY WE SEE THE PROBLEM IS THE PROBLEM\n\nPeople are intrigued when they see good things happening in the lives of individuals, families, and organizations that are based on solid principles. They admire such personal strength and maturity, such family unity and teamwork, such adaptive synergistic organizational culture.\n\nAnd their immediate request is very revealing of their basic paradigm. \"How do you do it? Teach me the techniques.\" What they're really saying is, \"Give me some quick fix advice or solution that will relieve the pain in my own situation.\"\n\nThey will find people who will meet their wants and teach these things; and for a short time, skills and techniques may appear to work. They may eliminate some of the cosmetic or acute problems through social aspirin and band-aids.\n\nBut the underlying chronic condition remains, and eventually new acute symptoms will appear. The more people are into quick fix and focus on the acute problems and pain, the more that very approach contributes to the underlying chronic condition.\n\nThe way we see the problem is the problem.\n\nLook again at some of the concerns that introduced this chapter, and at the impact of Personality Ethic thinking.\n\n_I've taken course after course on effective management training. I expect a lot out of my employees and I work hard to be friendly toward them and to treat them right. But I don't feel any loyalty from them. I think if I were home sick for a day, they'd spend most of their time gabbing at the water fountain. Why can't I train them to be independent and responsible\u2014or find employees who can be?_\n\nThe Personality Ethic tells me I could take some kind of dramatic action\u2014shake things up, make heads roll\u2014that would make my employees shape up and appreciate what they have. Or that I could find some motivational training program that would get them committed. Or even that I could hire new people that would do a better job.\n\nBut is it possible that under that apparently disloyal behavior, these employees question whether I really act in their best interest? Do they feel like I'm treating them as mechanical objects? Is there some truth to that?\n\nDeep inside, is that really the way I see them? Is there a chance the way I look at the people who work for me is part of the problem?\n\n_There's so much to do. And there's never enough time. I feel pressured and hassled all day, every day, seven days a week. I've attended time management seminars and I've tried half a dozen different planning systems. They've helped some, but I still don't feel I'm living the happy, productive, peaceful life I want to live._\n\nThe Personality Ethic tells me there must be something out there\u2014some new planner or seminar that will help me handle all these pressures in a more efficient way.\n\nBut is there a chance that efficiency is not the answer? Is getting more things done in less time going to make a difference\u2014or will it just increase the pace at which I react to the people and circumstances that seem to control my life?\n\nCould there be something I need to see in a deeper, more fundamental way\u2014some paradigm within myself that affects the way I see my time, my life, and my own nature?\n\n_My marriage has gone flat. We don't fight or anything; we just don't love each other anymore. We've gone to counseling; we've tried a number of things, but we just can't seem to rekindle the feeling we used to have._\n\nThe Personality Ethic tells me there must be some new book or some seminar where people get all their feelings out that would help my wife understand me better. Or maybe that it's useless, and only a new relationship will provide the love I need.\n\nBut is it possible that my spouse isn't the real problem? Could I be empowering my spouse's weaknesses and making my life a function of the way I'm treated?\n\nDo I have some basic paradigm about my spouse, about marriage, about what love really is, that is feeding the problem?\n\nCan you see how fundamentally the paradigms of the Personality Ethic affect the very way we see our problems as well as the way we attempt to solve them?\n\nWhether people see it or not, many are becoming disillusioned with the empty promises of the Personality Ethic. As I travel around the country and work with organizations, I find that long-term thinking executives are simply turned off by psych up psychology and \"motivational\" speakers who have nothing more to share than entertaining stories mingled with platitudes.\n\nThey want substance; they want process. They want more than aspirin and band-aids. They want to solve the chronic underlying problems and focus on the principles that bring long-term results.\n\nA NEW LEVEL OF THINKING\n\nAlbert Einstein observed, \"The significant problems we face cannot be solved at the same level of thinking we were at when we created them.\"\n\nAs we look around us and within us and recognize the problems created as we live and interact within the Personality Ethic, we begin to realize that these are deep, fundamental problems that cannot be solved on the superficial level on which they were created.\n\nWe need a new level, a deeper level of thinking\u2014a paradigm based on the principles that accurately describe the territory of effective human being and interacting\u2014to solve these deep concerns.\n\nThis new level of thinking is what Seven Habits of Highly Effective People is about. It's a principle-centered, character-based, \"inside-out\" approach to personal and interpersonal effectiveness.\n\n\"Inside-out\" means to start first with self; even more fundamentally, to start with the most inside part of self\u2014with your paradigms, your character, and your motives.\n\nIt says if you want to have a happy marriage, be the kind of person who generates positive energy and sidesteps negative energy rather than empowering it. If you want to have a more pleasant, cooperative teenager, be a more understanding, empathic, consistent, loving parent. If you want to have more freedom, more latitude in your job, be a more responsible, a more helpful, a more contributing employee. If you want to be trusted, be trustworthy. If you want the secondary greatness of recognized talent, focus first on primary greatness of character.\n\nThe inside-out approach says that private victories precede public victories, that making and keeping promises to ourselves precedes making and keeping promises to others. It says it is futile to put personality ahead of character, to try to improve relationships with others before improving ourselves.\n\nInside-out is a process\u2014a continuing process of renewal based on the natural laws that govern human growth and progress. It's an upward spiral of growth that leads to progressively higher forms of responsible independence and effective interdependence.\n\nI have had the opportunity to work with many people\u2014wonderful people, talented people, people who deeply want to achieve happiness and success, people who are searching, people who are hurting. I've worked with business executives, college students, church and civic groups, families and marriage partners. And in all of my experience, I have never seen lasting solutions to problems, lasting happiness and success, that came from the outside in.\n\nWhat I have seen result from the outside-in paradigm is unhappy people who feel victimized and immobilized, who focus on the weaknesses of other people and the circumstances they feel are responsible for their own stagnant situation. I've seen unhappy marriages where each spouse wants the other to change, where each is confessing the other's \"sins,\" where each is trying to shape up the other. I've seen labor management disputes where people spend tremendous amounts of time and energy trying to create legislation that would force people to act as though the foundation of trust were really there.\n\nMembers of our family have lived in three of the \"hottest\" spots on earth\u2014South Africa, Israel, and Ireland\u2014and I believe the source of the continuing problems in each of these places has been the dominant social paradigm of outside-in. Each involved group is convinced the problem is \"out there\" and if \"they\" (meaning others) would \"shape up\" or suddenly \"ship out\" of existence, the problem would be solved.\n\nInside-out is a dramatic paradigm shift for most people, largely because of the powerful impact of conditioning and the current social paradigm of the Personality Ethic.\n\nBut from my own experience\u2014both personal and in working with thousands of other people\u2014and from careful examination of successful individuals and societies throughout history, I am persuaded that many of the principles embodied in the Seven Habits are already deep within us, in our conscience and our common sense. To recognize and develop them and to use them in meeting our deepest concerns, we need to think differently, to shift our paradigms to a new, deeper, \"inside-out\" level.\n\nAs we sincerely seek to understand and integrate these principles into our lives, I am convinced we will discover and rediscover the truth of T. S. Eliot's observation:\n\n_We must not cease from exploration and the end of all our exploring will be to arrive where we began and to know the place for the first time._\n\n### The Seven Habits\u2014An Overview\n\nWe are what we repeatedly do. Excellence, then, is not an act, but a habit.\n\nAristotle\n\nOUR CHARACTER, BASICALLY, is a composite of our habits. \"Sow a thought, reap an action; sow an action, reap a habit; sow a habit, reap a character; sow a character, reap a destiny,\" the maxim goes.\n\nHabits are powerful factors in our lives. Because they are consistent, often unconscious patterns, they constantly, daily, express our character and produce our effectiveness... or ineffectiveness.\n\nAs Horace Mann, the great educator, once said, \"Habits are like a cable. We weave a strand of it everyday and soon it cannot be broken.\" I personally do not agree with the last part of his expression. I know they can be broken. Habits can be learned and unlearned. But I also know it isn't a quick fix. It involves a process and a tremendous commitment.\n\nThose of us who watched the lunar voyage of Apollo 11 were transfixed as we saw the first men walk on the moon and return to earth. Superlatives such as \"fantastic\" and \"incredible\" were inadequate to describe those eventful days. But to get there, those astronauts literally had to break out of the tremendous gravity pull of the earth. More energy was spent in the first few minutes of lift-off, in the first few miles of travel, than was used over the next several days to travel half a million miles.\n\nHabits, too, have tremendous gravity pull\u2014more than most people realize or would admit. Breaking deeply imbedded habitual tendencies such as procrastination, impatience, criticalness, or selfishness that violate basic principles of human effectiveness involves more than a little willpower and a few minor changes in our lives. \"Lift off\" takes a tremendous effort, but once we break out of the gravity pull, our freedom takes on a whole new dimension.\n\nLike any natural force, gravity pull can work with us or against us. The gravity pull of some of our habits may currently be keeping us from going where we want to go. But it is also gravity pull that keeps our world together, that keeps the planets in their orbits and our universe in order. It is a powerful force, and if we use it effectively, we can use the gravity pull of habit to create the cohesiveness and order necessary to establish effectiveness in our lives.\n\n\"HABITS\" DEFINED\n\nFor our purposes, we will define a habit as the intersection of knowledge, skill, and desire.\n\nKnowledge is the theoretical paradigm, the what to do and the why. Skill is the how to do. And desire is the motivation, the want to do. In order to make something a habit in our lives, we have to have all three.\n\nI may be ineffective in my interactions with my work associates, my spouse, or my children because I constantly tell them what I think, but I never really listen to them. Unless I search out correct principles of human interaction, I may not even know I need to listen.\n\nEven if I do know that in order to interact effectively with others I really need to listen to them, I may not have the skill. I may not know how to really listen deeply to another human being.\n\nBut knowing I need to listen and knowing how to listen is not enough. Unless I want to listen, unless I have the desire, it won't be a habit in my life. Creating a habit requires work in all three dimensions.\n\nThe being\/seeing change is an upward process\u2014being changing seeing, which in turn changes being, and so forth, as we move in an upward spiral of growth. By working on knowledge, skill, and desire, we can break through to new levels of personal and interpersonal effectiveness as we break with old paradigms that may have been a source of pseudo-security for years.\n\nIt's sometimes a painful process. It's a change that has to be motivated by a higher purpose, by the willingness to subordinate what you think you want now for what you want later. But this process produces happiness, \"the object and design of our existence.\" Happiness can be defined, in part at least, as the fruit of the desire and ability to sacrifice what we want now for what we want eventually.\n\nEFFECTIVE HABITS\n\nInternalized principles and patterns of behavior\n\nTHE MATURITY CONTINUUM\n\nThe Seven Habits are not a set of separate or piecemeal psych-up formulas. In harmony with the natural laws of growth, they provide an incremental, sequential, highly integrated approach to the development of personal and interpersonal effectiveness. They move us progressively on a Maturity Continuum from dependence to independence to interdependence.\n\nWe each begin life as an infant, totally dependent on others. We are directed, nurtured, and sustained by others. Without this nurturing, we would only live for a few hours or a few days at the most.\n\nThen gradually, over the ensuing months and years, we become more and more independent\u2014physically, mentally, emotionally, and financially\u2014until eventually we can essentially take care of ourselves, becoming inner-directed and self-reliant.\n\nAs we continue to grow and mature, we become increasingly aware that all of nature is interdependent, that there is an ecological system that governs nature, including society. We further discover that the higher reaches of our nature have to do with our relationships with others\u2014that human life also is interdependent.\n\nOur growth from infancy to adulthood is in accordance with natural law. And there are many dimensions to growth. Reaching our full physical maturity, for example, does not necessarily assure us of simultaneous emotional or mental maturity. On the other hand, a person's physical dependence does not mean that he or she is mentally or emotionally immature.\n\nOn the maturity continuum, dependence is the paradigm of you\u2014you take care of me; you come through for me; you didn't come through; I blame you for the results.\n\nIndependence is the paradigm of I\u2014I can do it; I am responsible; I am self-reliant; I can choose.\n\nInterdependence is the paradigm of we\u2014we can do it; we can cooperate; we can combine our talents and abilities and create something greater together.\n\nDependent people need others to get what they want. Indepen\u00addent people can get what they want through their own effort. Interdependent people combine their own efforts with the efforts of others to achieve their greatest success.\n\nIf I were physically dependent\u2014paralyzed or disabled or limited in some physical way\u2014I would need you to help me. If I were emotionally dependent, my sense of worth and security would come from your opinion of me. If you didn't like me, it could be devastating. If I were intellectually dependent, I would count on you to do my thinking for me, to think through the issues and problems of my life.\n\nIf I were independent, physically, I could pretty well make it on my own. Mentally, I could think my own thoughts, I could move from one level of abstraction to another. I could think creatively and analytically and organize and express my thoughts in under\u00adstandable ways. Emotionally, I would be validated from within. I would be inner directed. My sense of worth would not be a function of being liked or treated well.\n\nIt's easy to see that independence is much more mature than dependence. Independence is a major achievement in and of itself. But independence is not supreme.\n\nNevertheless, the current social paradigm enthrones indepen\u00addence. It is the avowed goal of many individuals and social movements. Most of the self-improvement material puts indepen\u00addence on a pedestal, as though communication, teamwork, and cooperation were lesser values.\n\nBut much of our current emphasis on independence is a reaction to dependence\u2014to having others control us, define us, use us, and manipulate us.\n\nThe little understood concept of interdependence appears to many to smack of dependence, and therefore, we find people, often for selfish reasons, leaving their marriages, abandoning their children, and forsaking all kinds of social responsibility\u2014all in the name of independence.\n\nThe kind of reaction that results in people \"throwing off their shackles,\" becoming \"liberated,\" \"asserting themselves,\" and \"doing their own thing\" often reveals more fundamental depen\u00addencies that cannot be run away from because they are internal rather than external\u2014dependencies such as letting the weaknesses of other people ruin our emotional lives or feeling victimized by people and events out of our control.\n\nOf course, we may need to change our circumstances. But the dependence problem is a personal maturity issue that has little to do with circumstances. Even with better circumstances, immaturity and dependence often persist.\n\nTrue independence of character empowers us to act rather than be acted upon. It frees us from our dependence on circumstances and other people and is a worthy, liberating goal. But it is not the ultimate goal in effective living.\n\nIndependent thinking alone is not suited to interdependent reality. Independent people who do not have the maturity to think and act interdependently may be good individual producers, but they won't be good leaders or team players. They're not coming from the paradigm of interdependence necessary to succeed in marriage, family, or organizational reality.\n\nLife is, by nature, highly interdependent. To try to achieve maximum effectiveness through independence is like trying to play tennis with a golf club\u2014the tool is not suited to the reality.\n\nInterdependence is a far more mature, more advanced concept. If I am physically interdependent, I am self-reliant and capable, but I also realize that you and I working together can accomplish far more than, even at my best, I could accomplish alone. If I am emotionally interdependent, I derive a great sense of worth within myself, but I also recognize the need for love, for giving, and for receiving love from others. If I am intellectually interdependent, I realize that I need the best thinking of other people to join with my own.\n\nAs an interdependent person, I have the opportunity to share myself deeply, meaningfully, with others, and I have access to the vast resources and potential of other human beings.\n\nInterdependence is a choice only independent people can make. Dependent people cannot choose to become interdependent. They don't have the character to do it; they don't own enough of themselves.\n\nThat's why Habits 1, 2, and 3 in the following chapters deal with self-mastery. They move a person from dependence to indepen\u00addence. They are the \"Private Victories,\" the essence of character growth. Private victories precede public victories. You can't invert that process anymore than you can harvest a crop before you plant it. It's inside-out.\n\nAs you become truly independent, you have the foundation for effective interdependence. You have the character base from which you can effectively work on the more personality-oriented \"Public Victories\" of teamwork, cooperation, and communication in Habits 4, 5, and 6.\n\nThat does not mean you have to be perfect in Habits 1, 2, and 3 before working on Habits 4, 5, and 6. Understanding the sequence will help you manage your growth more effectively, but I'm not suggesting that you put yourself in isolation for several years until you fully develop Habits 1, 2, and 3.\n\nAs part of an interdependent world, you have to relate to that world every day. But the acute problems of that world can easily obscure the chronic character causes. Understanding how what you are impacts every interdependent interaction will help you to focus your efforts sequentially, in harmony with the natural laws of growth.\n\nHabit 7 is the habit of renewal\u2014a regular, balanced renewal of the four basic dimensions of life. It circles and embodies all the other habits. It is the habit of continuous improvement that creates the upward spiral of growth that lifts you to new levels of understanding and living each of the habits as you come around to them on a progressively higher plane.*\n\nThe diagram in the image below is a visual representation of the sequence and the interdependence of the Seven Habits, and will be used throughout this book as we explore both the sequential relationship between the habits and also their synergy\u2014how, in relating to each other, they create bold new forms of each other that add even more to their value. Each concept or habit will be highlighted as it is introduced.\n\nTHE SEVEN HABITS PARADIGM\n\nEFFECTIVENESS DEFINED\n\nThe Seven Habits are habits of effectiveness. Because they are based on principles, they bring the maximum long-term beneficial results possible. They become the basis of a person's character, creating an empowering center of correct maps from which an individual can effectively solve problems, maximize opportunities, and continually learn and integrate other principles in an upward spiral of growth.\n\nThey are also habits of effectiveness because they are based on a paradigm of effectiveness that is in harmony with a natural law, a principle I call the \"P\/PC Balance,\" which many people break themselves against. This principle can be easily understood by remembering Aesop's fable of the goose and the golden egg.\n\nThis fable is the story of a poor farmer who one day discovers in the nest of his pet goose a glittering golden egg. At first, he thinks it must be some kind of trick. But as he starts to throw the egg aside, he has second thoughts and takes it in to be appraised instead.\n\nThe egg is pure gold! The farmer can't believe his good fortune. He becomes even more incredulous the following day when the experience is repeated. Day after day, he awakens to rush to the nest and find another golden egg. He becomes fabulously wealthy; it all seems too good to be true.\n\nBut with his increasing wealth comes greed and impatience. Unable to wait day after day for the golden eggs, the farmer decides he will kill the goose and get them all at once. But when he opens the goose, he finds it empty. There are no golden eggs\u2014and now there is no way to get any more. The farmer has destroyed the goose that produced them.\n\nI suggest that within this fable is a natural law, a principle\u2014the basic definition of effectiveness. Most people see effectiveness from the golden egg paradigm: the more you produce, the more you do, the more effective you are.\n\nBut as the story shows, true effectiveness is a function of two things: what is produced (the golden eggs) and the producing asset or capacity to produce (the goose).\n\nIf you adopt a pattern of life that focuses on golden eggs and neglects the goose, you will soon be without the asset that produces golden eggs. On the other hand, if you only take care of the goose with no aim toward the golden eggs, you soon won't have the wherewithal to feed yourself or the goose.\n\nEffectiveness lies in the balance\u2014what I call the P\/PC Balance. P stands for production of desired results, the golden eggs. PC stands for production capability, the ability or asset that produces the golden eggs.\n\nTHREE KINDS OF ASSETS\n\nBasically, there are three kinds of assets: physical, financial, and human. Let's look at each one in turn.\n\nA few years ago, I purchased a physical asset\u2014a power lawnmower. I used it over and over again without doing anything to maintain it. The mower worked well for two seasons, but then it began to break down. When I tried to revive it with service and sharpening, I discovered the engine had lost over half its original power capacity. It was essentially worthless.\n\nHad I invested in PC\u2014in preserving and maintaining the asset\u2014I would still be enjoying its P\u2014the mowed lawn. As it was, I had to spend far more time and money replacing the mower than I ever would have spent, had I maintained it. It simply wasn't effective.\n\nIn our quest for short-term returns, or results, we often ruin a prized physical asset\u2014a car, a computer, a washer or dryer, even our body or our environment. Keeping P and PC in balance makes a tremendous difference in the effective use of physical assets.\n\nIt also powerfully impacts the effective use of financial assets. How often do people confuse principal with interest? Have you ever invaded principal to increase your standard of living, to get more golden eggs? The decreasing principal has decreasing power to produce interest or income. And the dwindling capital becomes smaller and smaller until it no longer supplies even basic needs.\n\nOur most important financial asset is our own capacity to earn. If we don't continually invest in improving our own PC, we severely limit our options. We're locked into our present situation, running scared of our corporation or our boss's opinion of us, economically dependent and defensive. Again, it simply isn't effective.\n\nIn the human area, the P\/PC Balance is equally fundamental, but even more important, because people control physical and financial assets.\n\nWhen two people in a marriage are more concerned about getting the golden eggs, the benefits, than they are in preserving the relationship that makes them possible, they often become insensitive and inconsiderate, neglecting the little kindnesses and courtesies so important to a deep relationship. They begin to use control levers to manipulate each other, to focus on their own needs, to justify their own position and look for evidence to show the wrongness of the other person. The love, the richness, the softness and spontaneity begin to deteriorate. The goose gets sicker day by day.\n\nAnd what about a parent's relationship with a child? When children are little, they are very dependent, very vulnerable. It becomes so easy to neglect the PC work\u2014the training, the communicating, the relating, the listening. It's easy to take advantage, to manipulate, to get what you want the way you want it\u2014right now! You're bigger, you're smarter, and you're right! So why not just tell them what to do? If necessary, yell at them, intimidate them, insist on your way.\n\nOr you can indulge them. You can go for the golden egg of popularity, of pleasing them, giving them their way all the time. Then they grow up without any internal sense of standards or expectations, without a personal commitment to being disciplined or responsible.\n\nEither way\u2014authoritarian or permissive\u2014you have the golden egg mentality. You want to have your way or you want to be liked. But what happens, meantime, to the goose? What sense of responsibility, of self-discipline, of confidence in the ability to make good choices or achieve important goals is a child going to have a few years down the road? And what about your relationship? When he reaches those critical teenage years, the identity crises, will he know from his experience with you that you will listen without judging, that you really, deeply care about him as a person, that you can be trusted, no matter what? Will the relation\u00adship be strong enough for you to reach him, to communicate with him, to influence him?\n\nSuppose you want your daughter to have a clean room\u2014that's P, production, the golden egg. And suppose you want her to clean it\u2014that's PC, production capability. Your daughter is the goose, the asset, that produces the golden egg.\n\nIf you have P and PC in balance, she cleans the room cheerfully, without being reminded, because she is committed and has the discipline to stay with the commitment. She is a valuable asset, a goose that can produce golden eggs.\n\nBut if your paradigm is focused on production, on getting the room clean, you might find yourself nagging her to do it. You might even escalate your efforts to threatening or yelling, and in your desire to get the golden egg, you undermine the health and welfare of the goose.\n\nLet me share with you an interesting PC experience I had with one of my daughters. We were planning a private date, which is something I enjoy regularly with each of my children. We find that the anticipation of the date is as satisfying as the realization.\n\nSo I approached my daughter and said, \"Honey, tonight's your night. What do you want to do?\"\n\n\"Oh, Dad, that's okay,\" she replied.\n\n\"No, really,\" I said. \"What would you like to do?\"\n\n\"Well,\" she finally said, \"what I want to do, you don't really want to do.\"\n\n\"Really, honey,\" I said earnestly, \"I want to do it. No matter what, it's your choice.\"\n\n\"I want to go see Star Wars,\" she replied. \"But I know you don't like Star Wars. You slept through it before. You don't like these fantasy movies. That's okay, Dad.\"\n\n\"No, honey, if that's what you'd like to do, I'd like to do it.\"\n\n\"Dad, don't worry about it. We don't always have to have this date.\" She paused and then added, \"But you know why you don't like Star Wars? It's because you don't understand the philosophy and training of a Jedi Knight.\"\n\n\"What?\"\n\n\"You know the things you teach, Dad? Those are the same things that go into the training of a Jedi Knight.\"\n\n\"Really? Let's go to Star Wars!\"\n\nAnd we did. She sat next to me and gave me the paradigm. I became her student, her learner. It was totally fascinating. I could begin to see out of a new paradigm the whole way a Jedi Knight's basic philosophy in training is manifested in different circum\u00adstances.\n\nThat experience was not a planned P experience; it was the serendipitous fruit of a PC investment. It was bonding and very satisfying. But we enjoyed golden eggs, too, as the goose\u2014the quality of the relationship\u2014was significantly fed.\n\nORGANIZATIONAL PC\n\nOne of the immensely valuable aspects of any correct principle is that it is valid and applicable in a wide variety of circumstances. Throughout this book, I would like to share with you some of the ways in which these principles apply to organizations, including families, as well as to individuals.\n\nWhen people fail to respect the P\/PC Balance in their use of physical assets in organizations, they decrease organizational effectiveness and often leave others with dying geese.\n\nFor example, a person in charge of a physical asset, such as a machine, may be eager to make a good impression on his superiors. Perhaps the company is in a rapid growth stage and promotions are coming fast. So he produces at optimum levels\u2014no downtime, no maintenance. He runs the machine day and night. The production is phenomenal, costs are down, and profits skyrocket. Within a short time, he's promoted. Golden eggs!\n\nBut suppose you are his successor on the job. You inherit a very sick goose, a machine that, by this time, is rusted and starts to break down. You have to invest heavily in downtime and maintenance. Costs skyrocket; profits nose-dive. And who gets blamed for the loss of golden eggs? You do. Your predecessor liquidated the asset, but the accounting system only reported unit production, costs, and profit.\n\nThe P\/PC Balance is particularly important as it applies to the human assets of an organization\u2014the customers and the employees.\n\nI know of a restaurant that served a fantastic clam chowder and was packed with customers every day at lunchtime. Then the business was sold, and the new owner focused on golden eggs\u2014he decided to water down the chowder. For about a month, with costs down and revenues constant, profits zoomed. But little by little, the customers began to disappear. Trust was gone, and business dwindled to almost nothing. The new owner tried desperately to reclaim it, but he had neglected the customers, violated their trust, and lost the asset of customer loyalty. There was no more goose to produce the golden egg.\n\nThere are organizations that talk a lot about the customer and then completely neglect the people that deal with the customer\u2014the employees. The PC principle is to always treat your employees exactly as you want them to treat your best customers.\n\nYou can buy a person's hand, but you can't buy his heart. His heart is where his enthusiasm, his loyalty is. You can buy his back, but you can't buy his brain. That's where his creativity is, his ingenuity, his resourcefulness.\n\nPC work is treating employees as volunteers just as you treat customers as volunteers, because that's what they are. They volunteer the best part\u2014their hearts and minds.\n\nI was in a group once where someone asked, \"How do you shape up lazy and incompetent employees?\" One man responded, \"Drop hand grenades!\" Several others cheered that kind of macho management talk, that \"shape up or ship out\" supervision approach.\n\nBut another person in the group asked, \"Who picks up the pieces?\"\n\n\"No pieces.\"\n\n\"Well, why don't you do that to your customers?\" the other man replied. \"Just say, 'Listen, if you're not interested in buying, you can just ship out of this place.'\"\n\nHe said, \"You can't do that to customers.\"\n\n\"Well, how come you can do it to employees?\"\n\n\"Because they're in your employ.\"\n\n\"I see. Are your employees devoted to you? Do they work hard? How's the turnover?\"\n\n\"Are you kidding? You can't find good people these days. There's too much turnover, absenteeism, moonlighting. People just don't care anymore.\"\n\nThat focus on golden eggs\u2014that attitude, that paradigm\u2014is totally inadequate to tap into the powerful energies of the mind and heart of another person. A short-term bottom line is important, but it isn't all-important.\n\nEffectiveness lies in the balance. Excessive focus on P results in ruined health, worn-out machines, depleted bank accounts, and broken relationships. Too much focus on PC is like a person who runs three or four hours a day, bragging about the extra ten years of life it creates, unaware he's spending them running. Or a person endlessly going to school, never producing, living on other peo\u00adple's golden eggs\u2014the eternal student syndrome.\n\nTo maintain the P\/PC Balance, the balance between the golden egg (production) and the health and welfare of the goose (production capability) is often a difficult judgment call. But I suggest it is the very essence of effectiveness. It balances short term with long term. It balances going for the grade and paying the price to get an education. It balances the desire to have a room clean and the building of a relationship in which the child is internally committed to do it\u2014cheerfully, willingly, without external supervision.\n\nIt's a principle you can see validated in your own life when you burn the candle at both ends to get more golden eggs and wind up sick or exhausted, unable to produce any at all; or when you get a good night's sleep and wake up ready to produce throughout the day.\n\nYou can see it when you press to get your own way with someone and somehow feel an emptiness in the relationship; or when you really take time to invest in a relationship and you find the desire and ability to work together, to communicate, takes a quantum leap.\n\nThe P\/PC Balance is the very essence of effectiveness. It's validated in every arena of life. We can work with it or against it, but it's there. It's a lighthouse. It's the definition and paradigm of effectiveness upon which the Seven Habits in this book are based.\n\nHOW TO USE THIS BOOK\n\nBefore we begin work on the Seven Habits of Highly Effective People, I would like to suggest two paradigm shifts that will greatly increase the value you will receive from this material.\n\nFirst, I would recommend that you not \"see\" this material as a book, in the sense that it is something to read once and put on a shelf.\n\nYou may choose to read it completely through once for a sense of the whole. But the material is designed to be a companion in the continual process of change and growth. It is organized incrementally and with suggestions for application at the end of each habit so that you can study and focus on any particular habit as you are ready.\n\nAs you progress to deeper levels of understanding and implementation, you can go back time and again to the principles contained in each habit and work to expand your knowledge, skill, and desire.\n\nSecond, I would suggest that you shift your paradigm of your own involvement in this material from the role of learner to that of teacher. Take an inside-out approach, and read with the purpose in mind of sharing or discussing what you learn with someone else within 48 hours after you learn it.\n\nIf you had known, for example, that you would be teaching the material on the P\/PC Balance principle to someone else within 48 hours, would it have made a difference in your reading experience? Try it now as you read the final section in this chapter. Read as though you are going to teach it to your spouse, your child, a business associate, or a friend today or tomorrow, while it is still fresh, and notice the difference in your mental and emotional process.\n\nI guarantee that if you approach the material in each of the following chapters in this way, you will not only better remember what you read, but your perspective will be expanded, your understanding deepened, and your motivation to apply the mate\u00adrial increased.\n\nIn addition, as you openly, honestly share what you're learning with others, you may be surprised to find that negative labels or perceptions others may have of you tend to disappear. Those you teach will see you as a changing, growing person, and will be more inclined to be helpful and supportive as you work, perhaps together, to integrate the Seven Habits into your lives.\n\nWHAT YOU CAN EXPECT\n\nIn the last analysis, as Marilyn Ferguson observed, \"No one can persuade another to change. Each of us guards a gate of change that can only be opened from the inside. We cannot open the gate of another, either by argument or by emotional appeal.\"\n\nIf you decide to open your \"gate of change\" to really understand and live the principles embodied in the Seven Habits, I feel comfortable in assuring you several positive things will happen.\n\nFirst, your growth will be evolutionary, but the net effect will be revolutionary. Would you not agree that the P\/PC Balance principle alone, if fully lived, would transform most individuals and organizations?\n\nThe net effect of opening the \"gate of change\" to the first three habits\u2014the habits of Private Victory\u2014will be significantly increased self-confidence. You will come to know yourself in a deeper, more meaningful way\u2014your nature, your deepest values and your unique contribution capacity. As you live your values, your sense of identity, integrity, control, and inner-directedness will infuse you with both exhilaration and peace. You will define yourself from within, rather than by people's opinions or by comparisons to others. \"Wrong\" and \"right\" will have little to do with being found out.\n\nIronically, you'll find that as you care less about what others think of you, you will care more about what others think of themselves and their worlds, including their relationship with you. You'll no longer build your emotional life on other people's weaknesses. In addition, you'll find it easier and more desirable to change because there is something\u2014some core deep within\u2014that is essentially changeless.\n\nAs you open yourself to the next three habits\u2014the habits of Public Victory\u2014you will discover and unleash both the desire and the resources to heal and rebuild important relationships that have deteriorated, or even broken. Good relationships will improve\u2014become deeper, more solid, more creative, and more adventuresome.\n\nThe seventh habit, if deeply internalized, will renew the first six and will make you truly independent and capable of effective interdependence. Through it, you can charge your own batteries.\n\nWhatever your present situation, I assure you that you are not your habits. You can replace old patterns of self-defeating behavior with new patterns, new habits of effectiveness, happiness, and trust-based relationships.\n\nWith genuine caring, I encourage you to open the gate of change and growth as you study these habits. Be patient with yourself. Self-growth is tender; it's holy ground. There's no greater investment.\n\nIt's obviously not a quick fix. But I assure you, you will feel benefits and see immediate payoffs that will be encouraging. In the words of Thomas Paine, \"That which we obtain too easily, we esteem too lightly. It is dearness only which gives everything its value. Heaven knows how to put a proper price on its goods.\"\n\n## _PART TWO_\n\n# PRIVATE VICTORY\n\n### HABIT 1  \nBE PROACTIVE\n\n### PRINCIPLES OF PERSONAL VISION\n\nI know of no more encouraging fact  \nthan the unquestionable ability of man  \nto elevate his life by conscious endeavor.\n\nHenry David Thoreau\n\nAs YOU READ THIS BOOK, try to stand apart from yourself. Try to project your consciousness upward into a corner of the room and see yourself, in your mind's eye, reading. Can you look at yourself almost as though you were someone else?\n\nNow try something else. Think about the mood you are now in. Can you identify it? What are you feeling? How would you describe your present mental state?\n\nNow think for a minute about how your mind is working. Is it quick and alert? Do you sense that you are torn between doing this mental exercise and evaluating the point to be made out of it?\n\nYour ability to do what you just did is uniquely human. Animals do not possess this ability. We call it \"self-awareness\" or the ability to think about your very thought process. This is the reason why man has dominion over all things in the world and why he can make significant advances from generation to generation.\n\nThis is why we can evaluate and learn from others' experiences as well as our own. This is also why we can make and break our habits.\n\nWe are not our feelings. We are not our moods. We are not even our thoughts. The very fact that we can think about these things separates us from them and from the animal world. Self-awareness enables us to stand apart and examine even the way we \"see\" ourselves\u2014our self-paradigm, the most fundamental paradigm of effectiveness. It affects not only our attitudes and behaviors, but also how we see other people. It becomes our map of the basic nature of mankind.\n\nIn fact, until we take how we see ourselves (and how we see others) into account, we will be unable to understand how others see and feel about themselves and their world. Unaware, we will project our intentions on their behavior and call ourselves objective.\n\nThis significantly limits our personal potential and our ability to relate to others as well. But because of the unique human capacity of self-awareness, we can examine our paradigms to determine whether they are reality- or principle-based or if they are a function of conditioning and conditions.\n\nTHE SOCIAL MIRROR\n\nIf the only vision we have of ourselves comes from the social mirror\u2014from the current social paradigm and from the opinions, perceptions, and paradigms of the people around us\u2014our view of ourselves is like the reflection in the crazy mirror room at the carnival.\n\n\"You're never on time.\"\n\n\"Why can't you ever keep things in order?\"\n\n\"You must be an artist!\"\n\n\"You eat like a horse!\"\n\n\"I can't believe you won!\"\n\n\"This is so simple. Why can't you understand?\"\n\nThese visions are disjointed and out of proportion. They are often more projections than reflections, projecting the concerns and character weaknesses of people giving the input rather than accurately reflecting what we are.\n\nThe reflection of the current social paradigm tells us we are largely determined by conditioning and conditions. While we have acknowledged the tremendous power of conditioning in our lives, to say that we are determined by it, that we have no control over that influence, creates quite a different map.\n\nThere are actually three social maps\u2014three theories of determinism widely accepted, independently or in combination, to explain the nature of man. Genetic determinism basically says your grandparents did it to you. That's why you have such a temper. Your grandparents had short tempers and it's in your DNA. It just goes through the generations and you inherited it. In addition, you're Irish, and that's the nature of Irish people.\n\nPsychic determinism basically says your parents did it to you. Your upbringing, your childhood experience essentially laid out your personal tendencies and your character structure. That's why you're afraid to be in front of a group. It's the way your parents brought you up. You feel terribly guilty if you make a mistake because you \"remember\" deep inside the emotional scripting when you were very vulnerable and tender and dependent. You \"remember\" the emotional punishment, the rejection, the comparison with somebody else when you didn't perform as well as expected.\n\nEnvironmental determinism basically says your boss is doing it to you\u2014or your spouse, or that bratty teenager, or your economic situation, or national policies. Someone or something in your environment is responsible for your situation.\n\nEach of these maps is based on the stimulus\/response theory we most often think of in connection with Pavlov's experiments with dogs. The basic idea is that we are conditioned to respond in a particular way to a particular stimulus.\n\nHow accurately and functionally do these deterministic maps describe the territory? How clearly do these mirrors reflect the true nature of man? Do they become self-fulfilling prophecies? Are they based on principles we can validate within ourselves?\n\nBETWEEN STIMULUS AND RESPONSE\n\nIn answer to those questions, let me share with you the catalytic story of Viktor Frankl.\n\nFrankl was a determinist raised in the tradition of Freudian psychology, which postulates that whatever happens to you as a child shapes your character and personality and basically governs your whole life. The limits and parameters of your life are set, and, basically, you can't do much about it.\n\nFrankl was also a psychiatrist and a Jew. He was imprisoned in the death camps of Nazi Germany, where he experienced things that were so repugnant to our sense of decency that we shudder to even repeat them.\n\nHis parents, his brother, and his wife died in the camps or were sent to the gas ovens. Except for his sister, his entire family perished. Frankl himself suffered torture and innumerable indignities, never knowing from one moment to the next if his path would lead to the ovens or if he would be among the \"saved\" who would remove the bodies or shovel out the ashes of those so fated.\n\nOne day, naked and alone in a small room, he began to become aware of what he later called \"the last of the human freedoms\"\u2014the freedom his Nazi captors could not take away. They could control his entire environment, they could do what they wanted to his body, but Viktor Frankl himself was a self-aware being who could look as an observer at his very involvement. His basic identity was intact. He could decide within himself how all of this was going to affect him. Between what happened to him, or the stimulus, and his response to it, was his freedom or power to choose that response.\n\nIn the midst of his experiences, Frankl would project himself into different circumstances, such as lecturing to his students after his release from the death camps. He would describe himself in the classroom, in his mind's eye, and give his students the lessons he was learning during his very torture.\n\nThrough a series of such disciplines\u2014mental, emotional, and moral, principally using memory and imagination\u2014he exercised his small, embryonic freedom until it grew larger and larger, until he had more freedom than his Nazi captors. They had more liberty, more options to choose from in their environment; but he had more freedom, more internal power to exercise his options. He became an inspiration to those around him, even to some of the guards. He helped others find meaning in their suffering and dignity in their prison existence.\n\nIn the midst of the most degrading circumstances imaginable, Frankl used the human endowment of self-awareness to discover a fundamental principle about the nature of man: Between stimulus and response, man has the freedom to choose.\n\nWithin the freedom to choose are those endowments that make us uniquely human. In addition to self-awareness, we have imagination\u2014the ability to create in our minds beyond our present reality. We have conscience\u2014a deep inner awareness of right and wrong, of the principles that govern our behavior, and a sense of the degree to which our thoughts and actions are in harmony with them. And we have independent will\u2014the ability to act based on our self-awareness, free of all other influences.\n\nEven the most intelligent animals have none of these endowments. To use a computer metaphor, they are programmed by instinct and\/or training. They can be trained to be responsible, but they can't take responsibility for that training; in other words, they can't direct it. They can't change the programming. They're not even aware of it.\n\nBut because of our unique human endowments, we can write new programs for ourselves totally apart from our instincts and training. This is why an animal's capacity is relatively limited and man's is unlimited. But if we live like animals, out of our own instincts and conditioning and conditions, out of our collective memory, we too will be limited.\n\nThe deterministic paradigm comes primarily from the study of animals\u2014rats, monkeys, pigeons, dogs\u2014and neurotic and psychotic people. While this may meet certain criteria of some researchers because it seems measurable and predictable, the history of mankind and our own self-awareness tell us that this map doesn't describe the territory at all!\n\nOur unique human endowments lift us above the animal world. The extent to which we exercise and develop these endowments empowers us to fulfill our uniquely human potential. Between stimulus and response is our greatest power\u2014the freedom to choose.\n\n\"PROACTIVITY\" DEFINED\n\nIn discovering the basic principle of the nature of man, Frankl described an accurate self-map from which he began to develop the first and most basic habit of a highly effective person in any environment, the habit of proactivity.\n\nWhile the word proactivity is now fairly common in management literature, it is a word you won't find in most dictionaries. It means more than merely taking initiative. It means that as human beings, we are responsible for our own lives. Our behavior is a function of our decisions, not our conditions. We can subordinate feelings to values. We have the initiative and the responsibility to make things happen.\n\nPROACTIVE MODEL\n\nLook at the word responsibility\u2014\"response-ability\"\u2014the ability to choose your response. Highly proactive people recognize that responsibility. They do not blame circumstances, conditions, or conditioning for their behavior. Their behavior is a product of their own conscious choice, based on values, rather than a product of their conditions, based on feeling.\n\nBecause we are, by nature, proactive, if our lives are a function of conditioning and conditions, it is because we have, by conscious decision or by default, chosen to empower those things to control us.\n\nIn making such a choice, we become reactive. Reactive people are often affected by their physical environment. If the weather is good, they feel good. If it isn't, it affects their attitude and their performance. Proactive people can carry their own weather with them. Whether it rains or shines makes no difference to them. They are value driven; and if their value is to produce good quality work, it isn't a function of whether the weather is conducive to it or not.\n\nReactive people are also affected by their social environment, by the \"social weather.\" When people treat them well, they feel well; when people don't, they become defensive or protective. Reactive people build their emotional lives around the behavior of others, empowering the weaknesses of other people to control them.\n\nThe ability to subordinate an impulse to a value is the essence of the proactive person. Reactive people are driven by feelings, by circumstances, by conditions, by their environment. Proactive people are driven by values\u2014carefully thought about, selected and internalized values.\n\nProactive people are still influenced by external stimuli, whether physical, social, or psychological. But their response to the stimuli, conscious or unconscious, is a value-based choice or response.\n\nAs Eleanor Roosevelt observed, \"No one can hurt you without your consent.\" In the words of Gandhi, \"They cannot take away our self respect if we do not give it to them.\" It is our willing permission, our consent to what happens to us, that hurts us far more than what happens to us in the first place.\n\nI admit this is very hard to accept emotionally, especially if we have had years and years of explaining our misery in the name of circumstance or someone else's behavior. But until a person can say deeply and honestly, \"I am what I am today because of the choices I made yesterday,\" that person cannot say, \"I choose otherwise.\"\n\nOnce in Sacramento when I was speaking on the subject of proactivity, a woman in the audience stood up in the middle of my presentation and started talking excitedly. It was a large audience, and as a number of people turned to look at her, she suddenly became aware of what she was doing, grew embarrassed and sat back down. But she seemed to find it difficult to restrain herself and started talking to the people around her. She seemed so happy.\n\nI could hardly wait for a break to find out what had happened. When it finally came, I immediately went to her and asked if she would be willing to share her experience.\n\n\"You just can't imagine what's happened to me!\" she exclaimed. \"I'm a full-time nurse to the most miserable, ungrateful man you can possibly imagine. Nothing I do is good enough for him. He never expresses appreciation; he hardly even acknowledges me. He constantly harps at me and finds fault with everything I do. This man has made my life miserable and I often take my frustration out on my family. The other nurses feel the same way. We almost pray for his demise.\n\n\"And for you to have the gall to stand up there and suggest that nothing can hurt me, that no one can hurt me without my consent, and that I have chosen my own emotional life of being miserable\u2014well, there was just no way I could buy into that.\n\n\"But I kept thinking about it. I really went inside myself and began to ask, 'Do I have the power to choose my response?'\n\n\"When I finally realized that I do have that power, when I swallowed that bitter pill and realized that I had chosen to be miserable, I also realized that I could choose not to be miserable.\n\n\"At that moment I stood up. I felt as though I was being let out of San Quentin. I wanted to yell to the whole world, 'I am free! I am let out of prison! No longer am I going to be controlled by the treatment of some person.'\"\n\nIt's not what happens to us, but our response to what happens to us that hurts us. Of course, things can hurt us physically or economically and can cause sorrow. But our character, our basic identity, does not have to be hurt at all. In fact, our most difficult experiences become the crucibles that forge our character and develop the internal powers, the freedom to handle difficult circumstances in the future and to inspire others to do so as well.\n\nFrankl is one of many who have been able to develop the personal freedom in difficult circumstances to lift and inspire others. The autobiographical accounts of Vietnam prisoners of war provide additional persuasive testimony of the transforming power of such personal freedom and the effect of the responsible use of that freedom on the prison culture and on the prisoners, both then and now.\n\nWe have all known individuals in very difficult circumstances, perhaps with a terminal illness or a severe physical handicap, who maintain magnificent emotional strength. How inspired we are by their integrity! Nothing has a greater, longer lasting impression upon another person than the awareness that someone has transcended suffering, has transcended circumstance, and is embodying and expressing a value that inspires and ennobles and lifts life.\n\nOne of the most inspiring times Sandra and I have ever had took place over a four-year period with a dear friend of ours named Carol, who had a wasting cancer disease. She had been one of Sandra's bridesmaids, and they had been best friends for over 25 years.\n\nWhen Carol was in the very last stages of the disease, Sandra spent time at her bedside helping her write her personal history. She returned from those protracted and difficult sessions almost transfixed by admiration for her friend's courage and her desire to write special messages to be given to her children at different stages in their lives.\n\nCarol would take as little pain-killing medication as possible, so that she had full access to her mental and emotional faculties. Then she would whisper into a tape recorder or to Sandra directly as she took notes. Carol was so proactive, so brave, and so concerned about others that she became an enormous source of inspiration to many people around her.\n\nI'll never forget the experience of looking deeply into Carol's eyes the day before she passed away and sensing out of that deep hollowed agony a person of tremendous intrinsic worth. I could see in her eyes a life of character, contribution, and service as well as love and concern and appreciation.\n\nMany times over the years, I have asked groups of people how many have ever experienced being in the presence of a dying individual who had a magnificent attitude and communicated love and compassion and served in unmatchable ways to the very end. Usually, about one-fourth of the audience respond in the affirmative. I then ask how many of them will never forget these individuals\u2014how many were transformed, at least temporarily, by the inspiration of such courage, and were deeply moved and motivated to more noble acts of service and compassion. The same people respond again, almost inevitably.\n\nViktor Frankl suggests that there are three central values in life\u2014the experiential, or that which happens to us; the creative, or that which we bring into existence; and the attitudinal, or our response in difficult circumstances such as terminal illness.\n\nMy own experience with people confirms the point Frankl makes\u2014that the highest of the three values is attitudinal, in the paradigm or reframing sense. In other words, what matters most is how we respond to what we experience in life.\n\nDifficult circumstances often create paradigm shifts, whole new frames of reference by which people see the world and themselves and others in it, and what life is asking of them. Their larger perspective reflects the attitudinal values that lift and inspire us all.\n\nTAKING THE INITIATIVE\n\nOur basic nature is to act, and not be acted upon. As well as enabling us to choose our response to particular circumstances, this empowers us to create circumstances.\n\nTaking initiative does not mean being pushy, obnoxious, or aggressive. It does mean recognizing our responsibility to make things happen.\n\nOver the years, I have frequently counseled people who wanted better jobs to show more initiative\u2014to take interest and aptitude tests, to study the industry, even the specific problems the organizations they are interested in are facing, and then to develop an effective presentation showing how their abilities can help solve the organization's problem. It's called \"solution selling,\" and is a key paradigm in business success.\n\nThe response is usually agreement\u2014most people can see how powerfully such an approach would affect their opportunities for employment or advancement. But many of them fail to take the necessary steps, the initiative, to make it happen.\n\n\"I don't know where to go to take the interest and aptitude tests.\"\n\n\"How do I study industry and organizational problems? No one wants to help me.\"\n\n\"I don't have any idea how to make an effective presentation.\"\n\nMany people wait for something to happen or someone to take care of them. But people who end up with the good jobs are the proactive ones who are solutions to problems, not problems themselves, who seize the initiative to do whatever is necessary, consistent with correct principles, to get the job done.\n\nWhenever someone in our family, even one of the younger children, takes an irresponsible position and waits for someone else to make things happen or provide a solution, we tell them, \"Use your R and I!\" (resourcefulness and initiative). In fact, often before we can say it, they answer their own complaints, \"I know\u2014use my R and I!\"\n\nHolding people to the responsible course is not demeaning; it is affirming. Proactivity is part of human nature, and, although the proactive muscles may be dormant, they are there. By respecting the proactive nature of other people, we provide them with at least one clear, undistorted reflection from the social mirror.\n\nOf course, the maturity level of the individual has to be taken into account. We can't expect high creative cooperation from those who are deep into emotional dependence. But we can, at least, affirm their basic nature and create an atmosphere where people can seize opportunities and solve problems in an increasingly self-reliant way.\n\nACT OR BE ACTED UPON\n\nThe difference between people who exercise initiative and those who don't is literally the difference between night and day. I'm not talking about a 25 to 50 percent difference in effectiveness; I'm talking about a 5000-plus percent difference, particularly if they are smart, aware, and sensitive to others.\n\nIt takes initiative to create the P\/PC Balance of effectiveness in your life. It takes initiative to develop the Seven Habits. As you study the other six habits, you will see that each depends on the development of your proactive muscles. Each puts the responsibility on you to act. If you wait to be acted upon, you will be acted upon. And growth and opportunity consequences attend either road.\n\nAt one time I worked with a group of people in the home improvement industry, representatives from twenty different organizations who met quarterly to share their numbers and problems in an uninhibited way.\n\nThis was during a time of heavy recession, and the negative impact on this particular industry was even heavier than on the economy in general. These people were fairly discouraged as we began.\n\nThe first day, our discussion question was \"What's happening to us? What's the stimulus?\" Many things were happening. The environmental pressures were powerful. There was widespread unemployment, and many of these people were laying off friends just to maintain the viability of their enterprises. By the end of the day, everyone was even more discouraged.\n\nThe second day, we addressed the question, \"What's going to happen in the future?\" We studied environmental trends with the underlying reactive assumption that those things would create their future. By the end of the second day, we were even more depressed. Things were going to get worse before they got better, and everyone knew it.\n\nSo on the third day, we decided to focus on the proactive question, \"What is our response? What are we going to do? How can we exercise initiative in this situation?\" In the morning we talked about managing and reducing costs. In the afternoon we discussed increasing market share. We brainstormed both areas, then concentrated on several very practical, very doable things. A new spirit of excitement, hope, and proactive awareness concluded the meetings.\n\nAt the very end of the third day, we summarized the results of the conference in a three-part answer to the question, \"How's business?\"\n\nPart one: What's happening to us is not good, and the trends suggest that it will get worse before it gets better.\n\nPart two: But what we are causing to happen is very good, for we are better managing and reducing our costs and increasing our market share.\n\nPart three: Therefore, business is better than ever.\n\nNow what would a reactive mind say to that? \"Oh, come on. Face facts. You can only carry this positive thinking and self-psych approach so far. Sooner or later you have to face reality.\"\n\nBut that's the difference between positive thinking and proactivity. We did face reality. We faced the reality of the current circumstance and of future projections. But we also faced the reality that we had the power to choose a positive response to those circumstances and projections. Not facing reality would have been to accept the idea that what's happening in our environment had to determine us.\n\nBusinesses, community groups, organizations of every kind\u2014including families\u2014can be proactive. They can combine the cre\u00adativity and resourcefulness of proactive individuals to create a proactive culture within the organization. The organization does not have to be at the mercy of the environment; it can take the initiative to accomplish the shared values and purposes of the individuals involved.\n\nLISTENING TO OUR LANGUAGE\n\nBecause our attitudes and behaviors flow out of our paradigms, if we use our self-awareness to examine them, we can often see in them the nature of our underlying maps. Our language, for example, is a very real indicator of the degree to which we see ourselves as proactive people.\n\nThe language of reactive people absolves them of responsibility.\n\n\"That's me. That's just the way I am.\" I am determined. There's nothing I can do about it.\n\n\"He makes me so mad!\" I'm not responsible. My emotional life is governed by something outside my control.\n\n\"I can't do that. I just don't have the time.\" Something outside me\u2014limited time\u2014is controlling me.\n\n\"If only my wife were more patient.\" Someone else's behavior is limiting my effectiveness.\n\n\"I have to do it.\" Circumstances or other people are forcing me to do what I do. I'm not free to choose my own actions.\n\n**Reactive Language**\n\nThere's nothing I can do.\n\nThat's just the way I am.\n\nHe makes me so mad.\n\nThey won't allow that.\n\nI have to do that.\n\nI can't.\n\nI must.\n\nIf only.\n\n**Proactive Language**\n\nLet's look at our alternatives.\n\nI can choose a different approach.\n\nI control my own feelings.\n\nI can create an effective presentation.\n\nI will choose an appropriate response.\n\nI choose.\n\nI prefer.\n\nI will.\n\nThat language comes from a basic paradigm of determinism. And the whole spirit of it is the transfer of responsibility. I am not responsible, not able to choose my response.\n\nOne time a student asked me, \"Will you excuse me from class? I have to go on a tennis trip.\"\n\n\"You have to go, or you choose to go?\" I asked.\n\n\"I really have to,\" he exclaimed.\n\n\"What will happen if you don't?\"\n\n\"Why, they'll kick me off the team.\"\n\n\"How would you like that consequence?\"\n\n\"I wouldn't.\"\n\n\"In other words, you choose to go because you want the conse\u00adquence of staying on the team. What will happen if you miss my class?\"\n\n\"I don't know.\"\n\n\"Think hard. What do you think would be the natural conse\u00adquence of not coming to class?\"\n\n\"You wouldn't kick me out, would you?\"\n\n\"That would be a social consequence. That would be artificial. If you don't participate on the tennis team, you don't play. That's natural. But if you don't come to class, what would be the natural consequence?\"\n\n\"I guess I'll miss the learning.\"\n\n\"That's right. So you have to weigh that consequence against the other consequence and make a choice. I know if it were me, I'd choose to go on the tennis trip. But never say you have to do anything.\"\n\n\"I choose to go on the tennis trip,\" he meekly replied.\n\n\"And miss my class?\" I replied in mock disbelief.\n\nA serious problem with reactive language is that it becomes a self-fulfilling prophecy. People become reinforced in the paradigm that they are determined, and they produce evidence to support the belief. They feel increasingly victimized and out of control, not in charge of their life or their destiny. They blame outside forces\u2014other people, circumstances, even the stars\u2014for their own situation.\n\nAt one seminar where I was speaking on the concept of proactivity, a man came up and said, \"Stephen, I like what you're saying. But every situation is so different. Look at my marriage. I'm really worried. My wife and I just don't have the same feelings for each other we used to have. I guess I just don't love her anymore and she doesn't love me. What can I do?\"\n\n\"The feeling isn't there anymore?\" I asked.\n\n\"That's right,\" he reaffirmed. \"And we have three children we're really concerned about. What do you suggest?\"\n\n\"Love her,\" I replied.\n\n\"I told you, the feeling just isn't there anymore.\"\n\n\"Love her.\"\n\n\"You don't understand. The feeling of love just isn't there.\"\n\n\"Then love her. If the feeling isn't there, that's a good reason to love her.\"\n\n\"But how do you love when you don't love?\"\n\n\"My friend, love is a verb. Love\u2014the feeling\u2014is a fruit of love, the verb. So love her. Serve her. Sacrifice. Listen to her. Empathize. Appreciate. Affirm her. Are you willing to do that?\"\n\nIn the great literature of all progressive societies, love is a verb. Reactive people make it a feeling. They're driven by feelings. Hollywood has generally scripted us to believe that we are not responsible, that we are a product of our feelings. But the Hollywood script does not describe the reality. If our feelings control our actions, it is because we have abdicated our responsibility and empowered them to do so.\n\nProactive people make love a verb. Love is something you do: the sacrifices you make, the giving of self, like a mother bringing a newborn into the world. If you want to study love, study those who sacrifice for others, even for people who offend or do not love in return. If you are a parent, look at the love you have for the children you sacrificed for. Love is a value that is actualized through loving actions. Proactive people subordinate feelings to values. Love, the feeling, can be recaptured.\n\nCIRCLE OF CONCERN\/CIRCLE OF INFLUENCE\n\nAnother excellent way to become more self-aware regarding our own degree of proactivity is to look at where we focus our time and energy. We each have a wide range of concerns\u2014our health, our children, problems at work, the national debt, nuclear war. We could separate those from things in which we have no particular mental or emotional involvement by creating a \"Circle of Con\u00adcern.\"\n\nAs we look at those things within our Circle of Concern, it becomes apparent that there are some things over which we have no real control and others that we can do something about. We could identify those concerns in the latter group by circumscribing them within a smaller Circle of Influence.\n\nBy determining which of these two circles is the focus of most of our time and energy, we can discover much about the degree of our proactivity.\n\nPROACTIVE FOCUS\n\n(Positive energy enlarges the Circle of Influence)\n\nProactive people focus their efforts in the Circle of Influence. They work on the things they can do something about. The nature of their energy is positive, enlarging and magnifying, causing their Circle of Influence to increase.\n\nReactive people, on the other hand, focus their efforts in the Circle of Concern. They focus on the weakness of other people, the problems in the environment, and circumstances over which they have no control. Their focus results in blaming and accusing attitudes, reactive language, and increased feelings of victimiza\u00adtion. The negative energy generated by that focus, combined with neglect in areas they could do something about, causes their Circle of Influence to shrink.\n\nREACTIVE FOCUS\n\n(Negative energy reduces the Circle of Influence)\n\nAs long as we are working in our Circle of Concern, we empower the things within it to control us. We aren't taking the proactive initiative necessary to effect positive change.\n\nEarlier, I shared with you the story of my son who was having serious problems in school. Sandra and I were deeply concerned about his apparent weaknesses and about the way other people were treating him.\n\nBut those things were in our Circle of Concern. As long as we focused our efforts on those things, we accomplished nothing, except to increase our own feelings of inadequacy and helplessness and to reinforce our son's dependence.\n\nIt was only when we went to work in our Circle of Influence, when we focused on our own paradigms, that we began to create a positive energy that changed ourselves and eventually influenced our son as well. By working on ourselves instead of worrying about conditions, we were able to influence the conditions.\n\nBecause of position, wealth, role, or relationships, there are some circumstances in which a person's Circle of Influence is larger than his or her Circle of Concern.\n\nThis situation reflects a self-inflicted emotional myopia\u2014another reactive selfish life-style focused in the Circle of Concern.\n\nThough they may have to prioritize the use of their influence, proactive people have a Circle of Concern that is at least as big as their Circle of Influence, accepting the responsibility to use their influence effectively.\n\nDIRECT, INDIRECT, AND NO CONTROL\n\nThe problems we face fall in one of three areas: direct control (problems involving our own behavior); indirect control (problems involving other people's behavior); or no control (problems we can do nothing about, such as our past or situational realities). The proactive approach puts the first step in the solution of all three kinds of problems within our present Circle of Influence.\n\nDirect control problems are solved by working on our habits. They are obviously within our Circle of Influence. These are the \"Private Victories\" of Habits 1, 2, and 3.\n\nIndirect control problems are solved by changing our methods of influence. These are the \"Public Victories\" of Habits 4, 5, and 6. I have personally identified over 30 separate methods of human influence\u2014as separate as empathy is from confrontation, as sepa\u00adrate as example is from persuasion. Most people have only three or four of these methods in their repertoire, starting usually with reasoning, and, if that doesn't work, moving to flight or fight. How liberating it is to accept the idea that I can learn new methods of human influence instead of constantly trying to use old ineffective methods to \"shape up\" someone else!\n\nNo control problems involve taking the responsibility to change the line on the bottom on our face\u2014to smile, to genuinely and peacefully accept these problems and learn to live with them, even though we don't like them. In this way, we do not empower these problems to control us. We share in the spirit embodied in the Alcoholics Anonymous prayer, \"Lord, give me the courage to change the things which can and ought to be changed, the serenity to accept the things which cannot be changed, and the wisdom to know the difference.\"\n\nWhether a problem is direct, indirect, or no control, we have in our hands the first step to the solution. Changing our habits, changing our methods of influence and changing the way we see our no control problems are all within our Circle of Influence.\n\nEXPANDING THE CIRCLE OF INFLUENCE\n\nIt is inspiring to realize that in choosing our response to circumstance, we powerfully affect our circumstance. When we change one part of the chemical formula, we change the nature of the results.\n\nI worked with one organization for several years that was headed by a very dynamic person. He could read trends. He was creative, talented, capable, and brilliant\u2014and everyone knew it. But he had a very dictatorial style of management. He tended to treat people like \"gofers,\" as if they didn't have any judgment. His manner of speaking to those who worked in the organization was, \"Go for this... go for that... now do this... now do that\u2014I'll make the decisions.\"\n\nThe net effect was that he alienated almost the entire executive team surrounding him. They would gather in the corridors and complain to each other about him. Their discussion was all very sophisticated, very articulate, as if they were trying to help the situation. But they did it endlessly, absolving themselves of responsibility in the name of the president's weaknesses.\n\n\"You can't imagine what's happened this time,\" someone would say. \"The other day he went into my department. I had everything all laid out. But he came in and gave totally different signals. Everything I'd done for months was shot, just like that. I don't know how I'm supposed to keep working for him. How long will it be until he retires?\"\n\n\"He's only fifty-nine,\" someone else would respond. \"Do you think you can survive for six more years?\"\n\n\"I don't know. He's the kind of person they probably won't retire anyway.\"\n\nBut one of the executives was proactive. He was driven by values, not feelings. He took the initiative\u2014he anticipated, he empathized, he read the situation. He was not blind to the president's weaknesses; but instead of criticizing them, he would compensate for them. Where the president was weak in his style, he'd try to buffer his own people and make such weaknesses irrelevant. And he'd work with the president's strengths\u2014his vision, talent, creativity.\n\nThis man focused on his Circle of Influence. He was treated like a gofer, also. But he would do more than what was expected. He anticipated the president's need. He read with empathy the president's underlying concern, so when he presented informa\u00adtion, he also gave his analysis and his recommendations based on that analysis.\n\nAs I sat one day with the president in an advisory capacity, he said, \"Stephen, I just can't believe what this man has done. He's not only given me the information I requested, but he's provided additional information that's exactly what we needed. He even gave me his analysis of it in terms of my deepest concerns, and a list of his recommendations.\n\n\"The recommendations are consistent with the analysis, and the analysis is consistent with the data. He's remarkable! What a relief not to have to worry about this part of the business.\"\n\nAt the next meeting, it was \"go for this\" and \"go for that\" to all the executives... but one. To this man, it was \"What's your opinion?\" His Circle of Influence had grown.\n\nThis caused quite a stir in the organization. The reactive minds in the executive corridors began shooting their vindictive ammunition at this proactive man.\n\nIt's the nature of reactive people to absolve themselves of responsibility. It's so much safer to say, \"I am not responsible.\" If I say \"I am responsible,\" I might have to say, \"I am irresponsible.\" It would be very hard for me to say that I have the power to choose my response and that the response I have chosen has resulted in my involvement in a negative, collusive environment, especially if for years I have absolved myself of responsibility for results in the name of someone else's weaknesses.\n\nSo these executives focused on finding more information, more ammunition, more evidence as to why they weren't responsible.\n\nBut this man was proactive toward them, too. Little by little, his Circle of Influence toward them grew also. It continued to expand to the extent that eventually no one made any significant moves in the organization without that man's involvement and approval, including the president. But the president did not feel threatened because this man's strength complemented his strength and com\u00adpensated for his weaknesses. So he had the strength of two people, a complementary team.\n\nThis man's success was not dependent on his circumstances. Many others were in the same situation. It was his chosen response to those circumstances, his focus on his Circle of Influence, that made the difference.\n\nThere are some people who interpret \"proactive\" to mean pushy, aggressive, or insensitive; but that isn't the case at all. Proactive people aren't pushy. They're smart, they're value driven, they read reality, and they know what's needed.\n\nLook at Gandhi. While his accusers were in the legislative chambers criticizing him because he wouldn't join in their Circle of Concern Rhetoric condemning the British Empire for their subju\u00adgation of the Indian people, Gandhi was out in the rice paddies, quietly, slowly, imperceptibly expanding his Circle of Influence with the field laborers. A ground swell of support, of trust, of confidence followed him through the countryside. Though he held no office or political position, through compassion, courage, fast\u00ading, and moral persuasion he eventually brought England to its knees, breaking political domination of three hundred million people with the power of his greatly expanded Circle of Influence.\n\nTHE \"HAVE'S\" AND THE \"BE'S\"\n\nOne way to determine which circle our concern is in is to distinguish between the have's and the be's. The Circle of Concern is filled with the have's:\n\n\"I'll be happy when I have my house paid off.\"\n\n\"If only I had a boss who wasn't such a dictator....\"\n\n\"If only I had a more patient husband....\"\n\n\"If I had more obedient kids....\"\n\n\"If I had my degree....\"\n\n\"If I could just have more time to myself....\"\n\nThe Circle of Influence is filled with the be's\u2014I can be more patient, be wise, be loving. It's the character focus.\n\nAnytime we think the problem is \"out there,\" that thought is the problem. We empower what's out there to control us. The change paradigm is \"outside-in\"\u2014what's out there has to change before we can change.\n\nThe proactive approach is to change from the inside-out: to be different, and by being different, to effect positive change in what's out there\u2014I can be more resourceful, I can be more diligent, I can be more creative, I can be more cooperative.\n\nOne of my favorite stories is one in the Old Testament, part of the fundamental fabric of the Judeo-Christian tradition. It's the story of Joseph, who was sold into slavery in Egypt by his brothers at the age of seventeen. Can you imagine how easy it would have been for him to languish in self-pity as a servant of Potiphar, to focus on the weaknesses of his brothers and his captors and on all he didn't have? But Joseph was proactive. He worked on be. And within a short period of time, he was running Potiphar's household. He was in charge of all that Potiphar had because the trust was so high.\n\nThen the day came when Joseph was caught in a difficult situation and refused to compromise his integrity. As a result, he was unjustly imprisoned for thirteen years. But again he was proactive. He worked on the inner circle, on being instead of having, and soon he was running the prison and eventually the entire nation of Egypt, second only to the Pharaoh.\n\nI know this idea is a dramatic paradigm shift for many people. It is so much easier to blame other people, conditioning, or conditions for our own stagnant situation. But we are responsible\u2014\"response-able\"\u2014to control our lives and to powerfully influence our circum\u00adstances by working on be, on what we are.\n\nIf I have a problem in my marriage, what do I really gain by continually confessing my wife's sins? By saying I'm not responsible, I make myself a powerless victim; I immobilize myself in a negative situation. I also diminish my ability to influence her\u2014my nagging, accusing, critical attitude only makes her feel validated in her own weakness. My criticism is worse than the conduct I want to correct. My ability to positively impact the situation withers and dies.\n\nIf I really want to improve my situation, I can work on the one thing over which I have control\u2014myself. I can stop trying to shape up my wife and work on my own weaknesses. I can focus on being a great marriage partner, a source of unconditional love and support. Hopefully, my wife will feel the power of proactive example and respond in kind. But whether she does or doesn't, the most positive way I can influence my situation is to work on myself, on my being.\n\nThere are so many ways to work in the Circle of Influence\u2014to be a better listener, to be a more loving marriage partner, to be a better student, to be a more cooperative and dedicated employee. Some\u00adtimes the most proactive thing we can do is to be happy, just to genuinely smile. Happiness, like unhappiness, is a proactive choice. There are things, like the weather, that our Circle of Influence will never include. But as proactive people, we can carry our own physical or social weather with us. We can be happy and accept those things that at present we can't control, while we focus our efforts on the things that we can.\n\nTHE OTHER END OF THE STICK\n\nBefore we totally shift our life focus to our Circle of Influence, we need to consider two things in our Circle of Concern that merit deeper thought\u2014consequences and mistakes.\n\nWhile we are free to choose our actions, we are not free to choose the consequences of those actions. Consequences are governed by natural law. They are out in the Circle of Concern. We can decide to step in front of a fast-moving train, but we cannot decide what will happen when the train hits us.\n\nWe can decide to be dishonest in our business dealings. While the social consequences of that decision may vary depending on whether or not we are found out, the natural consequences to our basic character are a fixed result.\n\nOur behavior is governed by principles. Living in harmony with them brings positive consequences; violating them brings negative consequences. We are free to choose our response in any situation, but in doing so, we choose the attendant consequence. \"When we pick up one end of the stick, we pick up the other.\"\n\nUndoubtedly, there have been times in each of our lives when we have picked up what we later felt was the wrong stick. Our choices have brought consequences we would rather have lived without. If we had the choice to make over again, we would make it differently. We call these choices mistakes, and they are the second thing that merits our deeper thought.\n\nFor those filled with regret, perhaps the most needful exercise of proactivity is to realize that past mistakes are also out there in the Circle of Concern. We can't recall them, we can't undo them, we can't control the consequences that came as a result.\n\nAs a college quarterback, one of my sons learned to snap his wristband between plays as a kind of mental checkoff whenever he or anyone made a \"setting back\" mistake, so the last mistake wouldn't affect the resolve and execution of the next play.\n\nThe proactive approach to a mistake is to acknowledge it instantly, correct and learn from it. This literally turns a failure into a success. \"Success,\" said IBM founder T. J. Watson, \"is on the far side of failure.\"\n\nBut not to acknowledge a mistake, not to correct it and learn from it, is a mistake of a different order. It usually puts a person on a self-deceiving, self-justifying path, often involving rationalization (rational lies) to self and to others. This second mistake, this cover-up, empowers the first, giving it disproportionate impor\u00adtance, and causes far deeper injury to self.\n\nIt is not what others do or even our own mistakes that hurt us the most; it is our response to those things. Chasing after the poisonous snake that bites us will only drive the poison through our entire system. It is far better to take measures immediately to get the poison out.\n\nOur response to any mistake affects the quality of the next moment. It is important to immediately admit and correct our mistakes so that they have no power over that next moment and we are empowered again.\n\nMAKING AND KEEPING COMMITMENTS\n\nAt the very heart of our Circle of Influence is our ability to make and keep commitments and promises. The commitments we make to ourselves and to others, and our integrity to those commitments, is the essence and clearest manifestation of our proactivity.\n\nIt is also the essence of our growth. Through our human endowments of self-awareness and conscience, we become conscious of areas of weakness, areas for improvement, areas of talent that could be developed, areas that need to be changed or eliminated from our lives. Then, as we recognize and use our imagination and independent will to act on that awareness\u2014making promises, setting goals, and being true to them\u2014we build the strength of character, the being, that makes possible every other positive thing in our lives.\n\nIt is here that we find two ways to put ourselves in control of our lives immediately. We can make a promise\u2014and keep it. Or we can set a goal\u2014and work to achieve it. As we make and keep commit\u00adments, even small commitments, we begin to establish an inner integrity that gives us the awareness of self-control and the courage and strength to accept more of the responsibility for our own lives. By making and keeping promises to ourselves and others, little by little, our honor becomes greater than our moods.\n\nThe power to make and keep commitments to ourselves is the essence of developing the basic habits of effectiveness. Knowledge, skill, and desire are all within our control. We can work on any one to improve the balance of the three. As the area of intersection becomes larger, we more deeply internalize the principles upon which the habits are based and create the strength of character to move us in a balanced way toward increasing effectiveness in our lives.\n\nPROACTIVITY: THE THIRTY-DAY TEST\n\nWe don't have to go through the death camp experience of Frankl to recognize and develop our own proactivity. It is in the ordinary events of every day that we develop the proactive capacity to handle the extraordinary pressures of life. It's how we make and keep commitments, how we handle a traffic jam, how we respond to an irate customer or a disobedient child. It's how we view our problems and where we focus our energies. It's the language we use.\n\nI would challenge you to test the principle of proactivity for thirty days. Simply try it and see what happens. For thirty days work only in your Circle of Influence. Make small commitments and keep them. Be a light, not a judge. Be a model, not a critic. Be part of the solution, not part of the problem.\n\nTry it in your marriage, in your family, in your job. Don't argue for other people's weaknesses. Don't argue for your own. When you make a mistake, admit it, correct it, and learn from it\u2014immediately. Don't get into a blaming, accusing mode. Work on things you have control over. Work on you. On be.\n\nLook at the weaknesses of others with compassion, not accusation. It's not what they're not doing or should be doing that's the issue. The issue is your own chosen response to the situation and what you should be doing. If you start to think the problem is \"out there,\" stop yourself. That thought is the problem.\n\nPeople who exercise their embryonic freedom day after day will, little by little, expand that freedom. People who do not will find that it withers until they are literally \"being lived.\" They are acting out the scripts written by parents, associates, and society.\n\nWe are responsible for our own effectiveness, for our own happiness, and ultimately, I would say, for most of our circumstances.\n\nSamuel Johnson observed: \"The fountain of content must spring up in the mind, and he who hath so little knowledge of human nature as to seek happiness by changing anything but his own disposition, will waste his life in fruitless efforts and multiply the grief he proposes to remove.\"\n\nKnowing that we are responsible\u2014\"response-able\"\u2014is fundamental to effectiveness and to every other habit of effectiveness we will discuss.\n\nAPPLICATION SUGGESTIONS\n\n  1. For a full day, listen to your language and to the language of the people around you. How often do you use and hear reactive phrases such as \"If only,\" \"I can't,\" or \"I have to\"?\n  2. Identify an experience you might encounter in the near future where, based on past experience, you would probably behave reactively. Review the situation in the context of your Circle of Influence. How could you respond proactively? Take several moments and create the experience vividly in your mind, picturing yourself responding in a proactive manner. Remind yourself of the gap between stimulus and response. Make a commitment to yourself to exercise your freedom to choose.\n  3. Select a problem from your work or personal life that is frustrating to you. Determine whether it is a direct, indirect, or no control problem. Identify the first step you can take in your Circle of Influence to solve it and then take that step.\n  4. Try the thirty-day test of proactivity. Be aware of the change in your Circle of Influence.\n\n### HABIT 2  \nBEGIN WITH THE END IN MIND\n\n### PRINCIPLES OF PERSONAL LEADERSHIP\n\nWhat lies behind us and what lies before us  \nare tiny matters compared to what lies within us.\n\nOliver Wendell Holmes\n\nPLEASE FIND A PLACE TO READ THESE NEXT FEW PAGES where you can be alone and uninterrupted. Clear your mind of everything except what you will read and what I will invite you to do. Don't worry about your schedule, your business, your family, or your friends. Just focus with me and really open your mind.\n\nIn your mind's eye, see yourself going to the funeral of a loved one. Picture yourself driving to the funeral parlor or chapel, parking the car, and getting out. As you walk inside the building, you notice the flowers, the soft organ music. You see the faces of friends and family you pass along the way. You feel the shared sorrow of losing, the joy of having known, that radiates from the hearts of the people there.\n\nAs you walk down to the front of the room and look inside the casket, you suddenly come face to face with yourself. This is your funeral, three years from today. All these people have come to honor you, to express feelings of love and appreciation for your life.\n\nAs you take a seat and wait for the services to begin, you look at the program in your hand. There are to be four speakers. The first is from your family, immediate and also extended\u2014children, brothers, sisters, nephews, nieces, aunts, uncles, cousins, and grandparents who have come from all over the country to attend. The second speaker is one of your friends, someone who can give a sense of what you were as a person. The third speaker is from your work or profession. And the fourth is from your church or some community organization where you've been involved in service.\n\nNow think deeply. What would you like each of these speakers to say about you and your life? What kind of husband, wife, father, or mother would you like their words to reflect? What kind of son or daughter or cousin? What kind of friend? What kind of working associate?\n\nWhat character would you like them to have seen in you? What contributions, what achievements would you want them to remem\u00adber? Look carefully at the people around you. What difference would you like to have made in their lives?\n\nBefore you read further, take a few minutes to jot down your impressions. It will greatly increase your personal understanding of Habit 2.\n\nWHAT IT MEANS TO \"BEGIN WITH THE END IN MIND\"\n\nIf you participated seriously in this visualization experience, you touched for a moment some of your deep, fundamental values. You established brief contact with that inner guidance system at the heart of your Circle of Influence.\n\nConsider the words of Joseph Addison:\n\nWhen I look upon the tombs of the great, every emotion of envy dies in me; when I read the epitaphs of the beautiful, every inordinate desire goes out; when I meet with the grief of parents upon a tombstone, my heart melts with compassion; when I see the tomb of the parents themselves, I consider the vanity of grieving for those whom we must quickly follow: when I see kings lying by those who deposed them, when I consider rival wits placed side by side, or the holy men that divided the world with their contests and disputes, I reflect with sorrow and astonishment on the little competitions, factions, and debates of mankind. When I read the several dates of the tombs, of some that died yesterday, and some six hundred years ago, I consider that great Day when we shall all of us be Contemporaries, and make our appearance together.\n\nAlthough Habit 2 applies to many different circumstances and levels of life, the most fundamental application of \"begin with the end in mind\" is to begin today with the image, picture, or paradigm of the end of your life as your frame of reference or the criterion by which everything else is examined. Each part of your life\u2014today's behavior, tomorrow's behavior, next week's behavior, next month's behavior\u2014can be examined in the context of the whole, of what really matters most to you. By keeping that end clearly in mind, you can make certain that whatever you do on any particular day does not violate the criteria you have defined as supremely important, and that each day of your life contributes in a meaningful way to the vision you have of your life as a whole.\n\nTo begin with the end in mind means to start with a clear understanding of your destination. It means to know where you're going so that you better understand where you are now and so that the steps you take are always in the right direction.\n\nIt's incredibly easy to get caught up in an activity trap, in the busy-ness of life, to work harder and harder at climbing the ladder of success only to discover it's leaning against the wrong wall. It is possible to be busy\u2014very busy\u2014without being very effective.\n\nPeople often find themselves achieving victories that are empty, successes that have come at the expense of things they suddenly realize were far more valuable to them. People from every walk of life\u2014doctors, academicians, actors, politicians, business professionals, athletes, and plumbers\u2014often struggle to achieve a higher income, more recognition or a certain degree of professional competence, only to find that their drive to achieve their goal blinded them to the things that really mattered most and now are gone.\n\nHow different our lives are when we really know what is deeply important to us, and, keeping that picture in mind, we manage ourselves each day to be and to do what really matters most. If the ladder is not leaning against the right wall, every step we take just gets us to the wrong place faster. We may be very busy, we may be very efficient, but we will also be truly effective only when we begin with the end in mind.\n\nIf you carefully consider what you wanted to be said of you in the funeral experience, you will find your definition of success. It may be very different from the definition you thought you had in mind. Perhaps fame, achievement, money, or some of the other things we strive for are not even part of the right wall.\n\nWhen you begin with the end in mind, you gain a different perspective. One man asked another on the death of a mutual friend, \"How much did he leave?\" His friend responded, \"He left it all.\"\n\nALL THINGS ARE CREATED TWICE\n\n\"Begin with the end in mind\" is based on the principle that all things are created twice. There's a mental or first creation, and a physical or second creation to all things.\n\nTake the construction of a home, for example. You create it in every detail before you ever hammer the first nail into place. You try to get a very clear sense of what kind of house you want. If you want a family-centered home, you plan to put a family room where it would be a natural gathering place. You plan sliding doors and a patio for children to play outside. You work with ideas. You work with your mind until you get a clear image of what you want to build.\n\nThen you reduce it to blueprint and develop construction plans. All of this is done before the earth is touched. If not, then in the second creation, the physical creation, you will have to make expensive changes that may double the cost of your home.\n\nThe carpenter's rule is \"measure twice, cut once.\" You have to make sure that the blueprint, the first creation, is really what you want, that you've thought everything through. Then you put it into bricks and mortar. Each day you go to the construction shed and pull out the blueprint to get marching orders for the day. You begin with the end in mind.\n\nFor another example, look at a business. If you want to have a successful enterprise, you clearly define what you're trying to accomplish. You carefully think through the product or service you want to provide in terms of your market target, then you organize all the elements\u2014financial, research and development, operations, marketing, personnel, physical facilities, and so on\u2014to meet that objective. The extent to which you begin with the end in mind often determines whether or not you are able to create a successful enterprise. Most business failures begin in the first creation, with problems such as undercapitalization, misunderstanding of the market, or lack of a business plan.\n\nThe same is true with parenting. If you want to raise responsible, self-disciplined children, you have to keep that end clearly in mind as you interact with your children on a daily basis. You can't behave toward them in ways that undermine their self-discipline or self-esteem.\n\nTo varying degrees, people use this principle in many different areas of life. Before you go on a trip, you determine your destination and plan out the best route. Before you plant a garden, you plan it out in your mind, possibly on paper. You create speeches on paper before you give them, you envision the landscaping in your yard before you landscape it, you design the clothes you make before you thread the needle.\n\nTo the extent to which we understand the principle of two creations and accept the responsibility for both, we act within and enlarge the borders of our Circle of Influence. To the extent to which we do not operate in harmony with this principle and take charge of the first creation, we diminish it.\n\nBY DESIGN OR DEFAULT\n\nIt's a principle that all things are created twice, but not all first creations are by conscious design. In our personal lives, if we do not develop our own self-awareness and become responsible for first creations, we empower other people and circumstances outside our Circle of Influence to shape much of our lives by default. We reactively live the scripts handed to us by family, associates, other people's agendas, the pressures of circum\u00adstance\u2014scripts from our earlier years, from our training, our conditioning.\n\nThese scripts come from people, not principles. And they rise out of our deep vulnerabilities, our deep dependency on others and our needs for acceptance and love, for belonging, for a sense of importance and worth, for a feeling that we matter.\n\nWhether we are aware of it or not, whether we are in control of it or not, there is a first creation to every part of our lives. We are either the second creation of our own proactive design, or we are the second creation of other people's agendas, of circumstances, or of past habits.\n\nThe unique human capacities of self-awareness, imagination, and conscience enable us to examine first creations and make it possible for us to take charge of our own first creation, to write our own script. Put another way, Habit 1 says, \"You are the creator.\" Habit 2 is the first creation.\n\nLEADERSHIP AND MANAGEMENT\u2014THE TWO CREATIONS\n\nHabit 2 is based on principles of personal leadership, which means that leadership is the first creation. Leadership is not management. Management is the second creation, which we'll discuss in the chapter on Habit 3. But leadership has to come first.\n\nManagement is a bottom line focus: How can I best accomplish certain things? Leadership deals with the top line: What are the things I want to accomplish? In the words of both Peter Drucker and Warren Bennis, \"Management is doing things right; leadership is doing the right things.\" Management is efficiency in climbing the ladder of success; leadership determines whether the ladder is leaning against the right wall.\n\nYou can quickly grasp the important difference between the two if you envision a group of producers cutting their way through the jungle with machetes. They're the producers, the problem solvers. They're cutting through the undergrowth, clearing it out.\n\nThe managers are behind them, sharpening their machetes, writing policy and procedure manuals, holding muscle development programs, bringing in improved technologies and setting up working schedules and compensation programs for machete wielders.\n\nThe leader is the one who climbs the tallest tree, surveys the entire situation, and yells, \"Wrong jungle!\"\n\nBut how do the busy, efficient producers and managers often respond? \"Shut up! We're making progress.\"\n\nAs individuals, groups, and businesses, we're often so busy cutting through the undergrowth we don't even realize we're in the wrong jungle. And the rapidly changing environment in which we live makes effective leadership more critical than it has ever been\u2014in every aspect of independent and interdependent life.\n\nWe are more in need of a vision or destination and a compass (a set of principles or directions) and less in need of a road map. We often don't know what the terrain ahead will be like or what we will need to go through it; much will depend on our judgment at the time. But an inner compass will always give us direction.\n\nEffectiveness\u2014often even survival\u2014does not depend solely on how much effort we expend, but on whether or not the effort we expend is in the right jungle. And the metamorphosis taking place in most every industry and profession demands leadership first and management second.\n\nIn business, the market is changing so rapidly that many products and services that successfully met consumer tastes and needs a few years ago are obsolete today. Proactive powerful leadership must constantly monitor environmental change, particularly customer buying habits and motives, and provide the force necessary to organize resources in the right direction.\n\nSuch changes as deregulation of the airline industry, skyrocket\u00ading costs of health care, and the greater quality and quantity of imported cars impact the environment in significant ways. If industries do not monitor the environment, including their own work teams, and exercise the creative leadership to keep headed in the right direction, no amount of management expertise can keep them from failing.\n\nEfficient management without effective leadership is, as one individual has phrased it, \"like straightening deck chairs on the Titanic.\" No management success can compensate for failure in leadership. But leadership is hard because we're often caught in a management paradigm.\n\nAt the final session of a year-long executive development program in Seattle, the president of an oil company came up to me and said, \"Stephen, when you pointed out the difference between leadership and management in the second month, I looked at my role as the president of this company and realized that I had never been into leadership. I was deep into management, buried by pressing challenges and the details of day-to-day logistics. So I decided to withdraw from management. I could get other people to do that. I wanted to really lead my organization.\n\n\"It was hard. I went through withdrawal pains because I stopped dealing with a lot of the pressing, urgent matters that were right in front of me and which gave me a sense of immediate accomplishment. I didn't receive much satisfaction as I started wrestling with the direction issues, the culture building issues, the deep analysis of problems, the seizing of new opportunities. Others also went through withdrawal pains from their working style comfort zones. They missed the easy accessibility I had given them before. They still wanted me to be available to them, to respond, to help solve their problems on a day-to-day basis.\n\n\"But I persisted. I was absolutely convinced that I needed to provide leadership. And I did. Today our whole business is different. We're more in line with our environment. We have doubled our revenues and quadrupled our profits. I'm into leadership.\"\n\nI'm convinced that too often parents are also trapped in the management paradigm, thinking of control, efficiency, and rules instead of direction, purpose, and family feeling.\n\nAnd leadership is even more lacking in our personal lives. We're into managing with efficiency, setting and achieving goals before we have even clarified our values.\n\nRESCRIPTING: BECOMING YOUR OWN FIRST CREATOR\n\nAs we previously observed, proactivity is based on the unique human endowment of self-awareness. The two additional unique human endowments that enable us to expand our proactivity and to exercise personal leadership in our lives are imagination and conscience.\n\nThrough imagination, we can visualize the uncreated worlds of potential that lie within us. Through conscience, we can come in contact with universal laws or principles with our own singular talents and avenues of contribution, and with the personal guidelines within which we can most effectively develop them. Combined with self-awareness, these two endowments empower us to write our own script.\n\nBecause we already live with many scripts that have been handed to us, the process of writing our own script is actually more a process of \"rescripting,\" or paradigm shifting\u2014of changing some of the basic paradigms that we already have. As we recognize the ineffective scripts, the incorrect or incomplete paradigms within us, we can proactively begin to rescript ourselves.\n\nI think one of the most inspiring accounts of the rescripting process comes from the autobiography of Anwar Sadat, past president of Egypt. Sadat had been reared, nurtured, and deeply scripted in a hatred for Israel. He would make the statement on national television, \"I will never shake the hand of an Israeli as long as they occupy one inch of Arab soil. Never, never, never!\" And huge crowds all around the country would chant, \"Never, never, never!\" He marshalled the energy and unified the will of the whole country in that script.\n\nThe script was very independent and nationalistic, and it aroused deep emotions in the people. But it was also very foolish, and Sadat knew it. It ignored the perilous, highly interdependent reality of the situation.\n\nSo he rescripted himself. It was a process he had learned when he was a young man imprisoned in Cell 54, a solitary cell in Cairo Central Prison, as a result of his involvement in a conspiracy plot against King Farouk. He learned to withdraw from his own mind and look at it to see if the scripts were appropriate and wise. He learned how to vacate his own mind and, through a deep personal process of meditation, to work with his own scriptures, his own form of prayer, and rescript himself.\n\nHe records that he was almost loathe to leave his prison cell because it was there that he realized that real success is success with self. It's not in having things, but in having mastery, having victory over self.\n\nFor a period of time during Nasser's administration Sadat was relegated to a position of relative insignificance. Everyone felt that his spirit was broken, but it wasn't. They were projecting their own home movies onto him. They didn't understand him. He was biding his time.\n\nAnd when that time came, when he became president of Egypt and confronted the political realities, he rescripted himself toward Israel. He visited the Knesset in Jerusalem and opened up one of the most precedent-breaking peace movements in the history of the world, a bold initiative that eventually brought about the Camp David Accord.\n\nSadat was able to use his self-awareness, his imagination and his conscience to exercise personal leadership, to change an essential paradigm, to change the way he saw the situation. He worked in the center of his Circle of Influence. And from that rescripting, that change in paradigm, flowed changes in behavior and attitude that affected millions of lives in the wider Circle of Concern.\n\nIn developing our own self-awareness many of us discover ineffective scripts, deeply embedded habits that are totally unworthy of us, totally incongruent with the things we really value in life. Habit 2 says we don't have to live with those scripts. We are response-able to use our imagination and creativity to write new ones that are more effective, more congruent with our deepest values and with the correct principles that give our values meaning.\n\nSuppose, for example, that I am highly overreactive to my children. Suppose that whenever they begin to do something I feel is inappropriate, I sense an immediate tensing in the pit of my stomach. I feel defensive walls go up; I prepare for battle. My focus is not on the long-term growth and understanding but on the short-term behavior. I'm trying to win the battle, not the war.\n\nI pull out my ammunition\u2014my superior size, my position of authority\u2014and I yell or intimidate or I threaten or punish. And I win. I stand there, victorious, in the middle of the debris of a shattered relationship while my children are outwardly submissive and inwardly rebellious, suppressing feelings that will come out later in uglier ways.\n\nNow if I were sitting at that funeral we visualized earlier, and one of my children was about to speak, I would want his life to represent the victory of teaching, training, and disciplining with love over a period of years rather than the battle scars of quick fix skirmishes. I would want his heart and mind to be filled with the pleasant memories of deep, meaningful times together. I would want him to remember me as a loving father who shared the fun and the pain of growing up. I would want him to remember the times he came to me with his problems and concerns. I would want to have listened and loved and helped. I would want him to know I wasn't perfect, but that I had tried with everything I had. And that, perhaps more than anybody in the world, I loved him.\n\nThe reason I would want those things is because, deep down, I value my children. I love them, I want to help them. I value my role as their father.\n\nBut I don't always see those values. I get caught up in the \"thick of thin things.\" What matters most gets buried under layers of pressing problems, immediate concerns, and outward behaviors. I become reactive. And the way I interact with my children every day often bears little resemblance to the way I deeply feel about them.\n\nBecause I am self-aware, because I have imagination and conscience, I can examine my deepest values. I can realize that the script I'm living is not in harmony with those values, that my life is not the product of my own proactive design, but the result of the first creation I have deferred to circumstances and other people. And I can change. I can live out of my imagination instead of my memory. I can tie myself to my limitless potential instead of my limiting past. I can become my own first creator.\n\nTo begin with the end in mind means to approach my role as a parent, as well as my other roles in life, with my values and directions clear. It means to be responsible for my own first creation, to rescript myself so that the paradigms from which my behavior and attitude flow are congruent with my deepest values and in harmony with correct principles.\n\nIt also means to begin each day with those values firmly in mind. Then as the vicissitudes, as the challenges come, I can make my decisions based on those values. I can act with integrity. I don't have to react to the emotion, the circumstance. I can be truly proactive, value driven, because my values are clear.\n\nA PERSONAL MISSION STATEMENT\n\nThe most effective way I know to begin with the end in mind is to develop a personal mission statement or philosophy or creed. It focuses on what you want to be (character) and to do (contributions and achievements) and on the values or principles upon which being and doing are based.\n\nBecause each individual is unique, a personal mission statement will reflect that uniqueness, both in content and form. My friend, Rolfe Kerr, has expressed his personal creed in this way:\n\n  * Succeed at home first.\n  * Seek and merit divine help.\n  * Never compromise with honesty.\n  * Remember the people involved.\n  * Hear both sides before judging.\n  * Obtain counsel of others.\n  * Defend those who are absent.\n  * Be sincere yet decisive.\n  * Develop one new proficiency a year.\n  * Plan tomorrow's work today.\n  * Hustle while you wait.\n  * Maintain a positive attitude.\n  * Keep a sense of humor.\n  * Be orderly in person and in work.\n  * Do not fear mistakes\u2014fear only the absence of creative, constructive, and corrective responses to those mistakes.\n  * Facilitate the success of subordinates.\n  * Listen twice as much as you speak.\n  * Concentrate all abilities and efforts on the task at hand, not worrying about the next job or promotion.\n\nA woman seeking to balance family and work values has expressed her sense of personal mission differently:\n\nI will seek to balance career and family as best I can since both are important to me.\n\nMy home will be a place where I and my family, friends, and guests find joy, comfort, peace, and happiness. Still I will seek to create a clean and orderly environment, yet livable and comfortable. I will exercise wisdom in what we choose to eat, read, see, and do at home. I especially want to teach my children to love, to learn, and to laugh\u2014and to work and develop their unique talents.\n\nI value the rights, freedoms, and responsibilities of our democratic society. I will be a concerned and informed citizen, involved in the political process to ensure my voice is heard and my vote is counted.\n\nI will be a self-starting individual who exercises initiative in accomplishing my life's goals. I will act on situations and opportunities, rather than to be acted upon.\n\nI will always try to keep myself free from addictive and destructive habits. I will develop habits that free me from old labels and limits and expand my capabilities and choices.\n\nMy money will be my servant, not my master. I will seek financial independence over time. My wants will be subject to my needs and my means. Except for long-term home and car loans, I will seek to keep myself free from consumer debt. I will spend less than I earn and regularly save or invest part of my income.\n\nMoreover, I will use what money and talents I have to make life more enjoyable for others through service and charitable giving.\n\nYou could call a personal mission statement a personal constitution. Like the United States Constitution, it's fundamentally changeless. In over two hundred years, there have been only twenty-six amendments, ten of which were in the original Bill of Rights.\n\nThe United States Constitution is the standard by which every law in the country is evaluated. It is the document the president agrees to defend and support when he takes the Oath of Alle\u00adgiance. It is the criterion by which people are admitted into citizenship. It is the foundation and the center that enables people to ride through such major traumas as the Civil War, Vietnam, or Watergate. It is the written standard, the key criterion by which everything else is evaluated and directed.\n\nThe Constitution has endured and serves its vital function today because it is based on correct principles, on the self-evident truths contained in the Declaration of Independence. These principles empower the Constitution with a timeless strength, even in the midst of social ambiguity and change. \"Our peculiar security,\" said Thomas Jefferson, \"is in the possession of a written Constitution.\"\n\nA personal mission statement based on correct principles becomes the same kind of standard for an individual. It becomes a personal constitution, the basis for making major, life-directing decisions, the basis for making daily decisions in the midst of the circumstances and emotions that affect our lives. It empowers individuals with the same timeless strength in the midst of change.\n\nPeople can't live with change if there's not a changeless core inside them. The key to the ability to change is a changeless sense of who you are, what you are about and what you value.\n\nWith a mission statement, we can flow with changes. We don't need prejudgments or prejudices. We don't need to figure out everything else in life, to stereotype and categorize everything and everybody in order to accommodate reality.\n\nOur personal environment is also changing at an ever-increasing pace. Such rapid change burns out a large number of people who feel they can hardly handle it, can hardly cope with life. They become reactive and essentially give up, hoping that the things that happen to them will be good.\n\nBut it doesn't have to be that way. In the Nazi death camps where Viktor Frankl learned the principle of proactivity, he also learned the importance of purpose, of meaning in life. The essence of \"logotherapy,\" the philosophy he later developed and taught, is that many so-called mental and emotional illnesses are really symptoms of an underlying sense of meaninglessness or emptiness. Logotherapy eliminates that emptiness by helping the individual to detect his unique meaning, his mission in life.\n\nOnce you have that sense of mission, you have the essence of your own proactivity. You have the vision and the values which direct your life. You have the basic direction from which you set your long- and short-term goals. You have the power of a written constitution based on correct principles, against which every decision concerning the most effective use of your time, your talents, and your energies can be effectively measured.\n\nAT THE CENTER\n\nIn order to write a personal mission statement, we must begin at the very center of our Circle of Influence, that center comprised of our most basic paradigms, the lens through which we see the world.\n\nIt is here that we deal with our vision and our values. It is here that we use our endowment of self-awareness to examine our maps and, if we value correct principles, to make certain that our maps accurately describe the territory, that our paradigms are based on principles and reality. It is here that we use our endowment of conscience as a compass to help us detect our own unique talents and areas of contribution. It is here that we use our endowment of imagination to mentally create the end we desire, giving direction and purpose to our beginnings and providing the substance of a written personal constitution.\n\nIt is also here that our focused efforts achieve the greatest results. As we work within the very center of our Circle of Influence, we expand it. This is highest leverage PC work, significantly impacting the effectiveness of every aspect of our lives.\n\nWhatever is at the center of our life will be the source of our security, guidance, wisdom, and power.\n\nSecurity represents your sense of worth, your identity, your emotional anchorage, your self-esteem, your basic personal strength or lack of it.\n\nGuidance means your source of direction in life. Encompassed by your map, your internal frame of reference that interprets for you what is happening out there, are standards or principles or implicit criteria that govern moment by moment decision-making and doing.\n\nWisdom is your perspective on life, your sense of balance, your understanding of how the various parts and principles apply and relate to each other. It embraces judgment, discernment, comprehension. It is a gestalt or oneness, an integrated wholeness.\n\nPower is the faculty or capacity to act, the strength and potency to accomplish something. It is the vital energy to make choices and decisions. It also includes the capacity to overcome deeply embedded habits and to cultivate higher, more effective ones.\n\nThese four factors\u2014security, guidance, wisdom, and power\u2014are interdependent. Security and clear guidance bring true wisdom, and wisdom becomes the spark or catalyst to release and direct power. When these four factors are present together, harmonized and enlivened by each other, they create the great force of a noble personality, a balanced character, a beautifully integrated individual.\n\nThese life-support factors also undergird every other dimension of life. And none of them is an all-or-nothing matter. The degree to which you have developed each one could be charted somewhere on a continuum, much like the maturity continuum described earlier. At the bottom end, the four factors are weak. You are basically dependent on circumstances or other people, things over which you have no direct control. At the top end you are in control. You have independent strength and the foundation for rich, interdependent relationships.\n\nYour security lies somewhere on the continuum between extreme insecurity on one end, wherein your life is buffeted by all the fickle forces that play upon it, and a deep sense of high intrinsic worth and personal security on the other end. Your guidance ranges on the continuum from dependence on the social mirror or other unstable, fluctuating sources to strong inner direction. Your wisdom falls somewhere between a totally inaccurate map where everything is distorted and nothing seems to fit, and a complete and accurate map of life wherein all the parts and principles are properly related to each other. Your power lies somewhere between immobilization or being a puppet pulled by someone else's strings to high proactivity, the power to act according to your own values instead of being acted upon by other people and circumstances.\n\nThe location of these factors on the continuum, the resulting degree of their integration, harmony, and balance, and their positive impact on every aspect of your life is a function of your center, the basic paradigms at your very core.\n\nALTERNATIVE CENTERS\n\nEach of us has a center, though we usually don't recognize it as such. Neither do we recognize the all-encompassing effects of that center on every aspect of our lives.\n\nLet's briefly examine several centers or core paradigms people typically have for a better understanding of how they affect these four fundamental dimensions and, ultimately, the sum of life that flows from them.\n\nSPOUSE CENTEREDNESS. Marriage can be the most intimate, the most satisfying, the most enduring, growth-producing of human relationships. It might seem natural and proper to be centered on one's husband or wife.\n\nBut experience and observation tell a different story. Over the years, I have been involved in working with many troubled marriages, and I have observed a certain thread weaving itself through almost every spouse-centered relationship I have encountered. That thread is strong emotional dependence.\n\nIf our sense of emotional worth comes primarily from our marriage, then we become highly dependent upon that relationship. We become vulnerable to the moods and feelings, the behavior and treatment of our spouse, or to any external event that may impinge on the relationship\u2014a new child, in-laws, economic setbacks, social successes, and so forth.\n\nWhen responsibilities increase and stresses come in the marriage, we tend to revert to the scripts we were given as we were growing up. But so does our spouse. And those scripts are usually different. Different ways of handling financial, child discipline, or in-law issues come to the surface. When these deep-seated tendencies combine with the emotional dependency in the marriage, the spouse-centered relationship reveals all its vulnerability.\n\nWhen we are dependent on the person with whom we are in conflict, both need and conflict are compounded. Love-hate over-reactions, fight-or-flight tendencies, withdrawal, aggressiveness, bitterness, resentment, and cold competition are some of the usual results. When these occur, we tend to fall even further back on background tendencies and habits in an effort to justify and defend our own behavior and we attack our spouse's.\n\nInevitably, anytime we are too vulnerable we feel the need to protect ourselves from further wounds. So we resort to sarcasm, cutting humor, criticism\u2014anything that will keep from exposing the tenderness within. Each partner tends to wait on the initiative of the other for love, only to be disappointed but also confirmed as to the rightness of the accusations made.\n\nThere is only phantom security in such a relationship when all appears to be going well. Guidance is based on the emotion of the moment. Wisdom and power are lost in the counterdependent negative interactions.\n\nFAMILY CENTEREDNESS. Another common center is the family. This, too, may seem to be natural and proper. As an area of focus and deep investment, it provides great opportunities for deep relationships, for loving, for sharing, for much that makes life worthwhile. But as a center, it ironically destroys the very elements necessary to family success.\n\nPeople who are family-centered get their sense of security or personal worth from the family tradition and culture or the family reputation. Thus, they become vulnerable to any changes in that tradition or culture and to any influences that would affect that reputation.\n\nFamily-centered parents do not have the emotional freedom, the power, to raise their children with their ultimate welfare truly in mind. If they derive their own security from the family, their need to be popular with their children may override the importance of a long-term investment in their children's growth and development. Or they may be focused on the proper and correct behavior of the moment. Any behavior that they consider improper threatens their security. They become upset, guided by the emotions of the moment, spontaneously reacting to the immediate concern rather than the long-term growth and development of the child. They may yell or scream. They may overreact and punish out of bad temper. They tend to love their children conditionally, making them emotionally dependent or counterdependent and rebellious.\n\nMONEY CENTEREDNESS. Another logical and extremely common center to people's lives is making money. Economic security is basic to one's opportunity to do much in any other dimension. In a hierarchy or continuum of needs, physical survival and financial security comes first. Other needs are not even activated until that basic need is satisfied, at least minimally.\n\nMost of us face economic worries. Many forces in the wider culture can and do act upon our economic situation, causing or threatening such disruption that we often experience concern and worry that may not always rise to the conscious surface.\n\nSometimes there are apparently noble reasons given for making money, such as the desire to take care of one's family. And these things are important. But to focus on money-making as a center will bring about its own undoing.\n\nConsider again the four life-support factors\u2014security, guidance, wisdom, and power. Suppose I derive much of my security from my employment or from my income or net worth. Since many factors affect these economic foundations, I become anxious and uneasy, protective and defensive, about anything that may affect them. When my sense of personal worth comes from my net worth, I am vulnerable to anything that will affect that net worth. But work and money, per se, provide no wisdom, no guidance, and only a limited degree of power and security. All it takes to show the limitations of a money center is a crisis in my life or in the life of a loved one.\n\nMoney-centered people often put aside family or other priorities, assuming everyone will understand that economic demands come first. I know one father who was leaving with his children for a promised trip to the circus when a phone call came for him to come to work instead. He declined. When his wife suggested that perhaps he should have gone to work, he responded, \"The work will come again, but childhood won't.\" For the rest of their lives his children remembered this little act of priority setting, not only as an object lesson in their minds but as an expression of love in their hearts.\n\nWORK CENTEREDNESS. Work-centered people may become \"workaholics,\" driving themselves to produce at the sacrifice of health, relationships, and other important areas of their lives. Their fundamental identity comes from their work\u2014\"I'm a doctor,\" \"I'm a writer,\" \"I'm an actor.\"\n\nBecause their identity and sense of self-worth are wrapped up in their work, their security is vulnerable to anything that happens to prevent them from continuing in it. Their guidance is a function of the demands of the work. Their wisdom and power come in the limited areas of their work, rendering them ineffective in other areas of life.\n\nPOSSESSION CENTEREDNESS. A driving force of many people is possessions\u2014not only tangible, material possessions such as fashionable clothes, homes, cars, boats, and jewelry, but also the intangible possessions of fame, glory, or social prominence. Most of us are aware, through our own experience, how singularly flawed such a center is, simply because it can vanish rapidly and it is influenced by so many forces.\n\nIf my sense of security lies in my reputation or in the things I have, my life will be in a constant state of threat and jeopardy that these possessions may be lost or stolen or devalued. If I'm in the presence of someone of greater net worth or fame or status, I feel inferior. If I'm in the presence of someone of lesser net worth or fame or status, I feel superior. My sense of self-worth constantly fluctuates. I don't have any sense of constancy or anchorage or persistent selfhood. I am constantly trying to protect and insure my assets, properties, securities, position, or reputation. We have all heard stories of people committing suicide after losing their fortunes in a significant stock decline or their fame in a political reversal.\n\nPLEASURE CENTEREDNESS. Another common center, closely allied with possessions, is that of fun and pleasure. We live in a world where instant gratification is available and encouraged. Television and movies are major influences in increasing people's expectations. They graphically portray what other people have and can do in living the life of ease and \"fun.\"\n\nBut while the glitter of pleasure-centered life-styles is graphically portrayed, the natural result of such life-styles\u2014the impact on the inner person, on productivity, on relationships\u2014is seldom accurately seen.\n\nInnocent pleasures in moderation can provide relaxation for the body and mind and can foster family and other relationships. But pleasure, per se, offers no deep, lasting satisfaction or sense of fulfillment. The pleasure-centered person, too soon bored with each succeeding level of \"fun,\" constantly cries for more and more. So the next new pleasure has to be bigger and better, more exciting, with a bigger \"high.\" A person in this state becomes almost entirely narcissistic, interpreting all of life in terms of the pleasure it provides to the self here and now.\n\nToo many vacations that last too long, too many movies, too much TV, too much video game playing\u2014too much undisciplined leisure time in which a person continually takes the course of least resistance gradually wastes a life. It ensures that a person's capacities stay dormant, that talents remain undeveloped, that the mind and spirit become lethargic and that the heart is unfulfilled. Where is the security, the guidance, the wisdom, and the power? At the low end of the continuum, in the pleasure of a fleeting moment.\n\nMalcolm Muggeridge writes \"A Twentieth-Century Testimony\":\n\nWhen I look back on my life nowadays, which I sometimes do, what strikes me most forcibly about it is that what seemed at the time most significant and seductive, seems now most futile and absurd. For instance, success in all of its various guises; being known and being praised; ostensible pleasures, like acquiring money or seducing women, or traveling, going to and fro in the world and up and down in it like Satan, explaining and experiencing whatever Vanity Fair has to offer.\n\nIn retrospect, all these exercises in self-gratification seem pure fantasy, what Pascal called, \"licking the earth.\"\n\nFRIEND\/ENEMY CENTEREDNESS. Young people are particularly, though certainly not exclusively, susceptible to becoming friend-centered. Acceptance and belonging to a peer group can become almost supremely important. The distorted and ever-changing social mirror becomes the source for the four life-support factors, creating a high degree of dependence on the fluctuating moods, feelings, attitudes, and behavior of others.\n\nFriend centeredness can also focus exclusively on one person, taking on some of the dimensions of marriage. The emotional dependence on one individual, the escalating need\/conflict spiral, and the resulting negative interactions can grow out of friend centeredness.\n\nAnd what about putting an enemy at the center of one's life? Most people would never think of it, and probably no one would ever do it consciously. Nevertheless, enemy centering is very common, particularly when there is frequent interaction between people who are in real conflict. When someone feels he has been unjustly dealt with by an emotionally or socially significant person, it is very easy for him to become preoccupied with the injustice and make the other person the center of his life. Rather than proactively leading his own life, the enemy-centered person is counterdependently reacting to the behavior and attitudes of a perceived enemy.\n\nOne friend of mine who taught at a university became very distraught because of the weaknesses of a particular administrator with whom he had a negative relationship. He allowed himself to think about the man constantly until eventually it became an obsession. It so preoccupied him that it affected the quality of his relationships with his family, his church, and his working associates. He finally came to the conclusion that he had to leave the university and accept a teaching appointment somewhere else.\n\n\"Wouldn't you really prefer to teach at this university, if the man were not here?\" I asked him.\n\n\"Yes, I would,\" he responded. \"But as long as he is here, then my staying is too disruptive to everything in life. I have to go.\"\n\n\"Why have you made this administrator the center of your life?\" I asked him.\n\nHe was shocked by the question. He denied it. But I pointed out to him that he was allowing one individual and his weaknesses to distort his entire map of life, to undermine his faith and the quality of his relationships with his loved ones.\n\nHe finally admitted that this individual had had such an impact on him, but he denied that he himself had made all these choices. He attributed the responsibility for the unhappy situation to the administrator. He, himself, he declared, was not responsible.\n\nAs we talked, little by little, he came to realize that he was indeed responsible, but that because he did not handle this responsibility well, he was being irresponsible.\n\nMany divorced people fall into a similar pattern. They are still consumed with anger and bitterness and self-justification regarding an ex-spouse. In a negative sense, psychologically they are still married\u2014they each need the weaknesses of the former partner to justify their accusations.\n\nMany \"older\" children go through life either secretly or openly hating their parents. They blame them for past abuses, neglect, or favoritism and they center their adult life on that hatred, living out the reactive, justifying script that accompanies it.\n\nThe individual who is friend- or enemy-centered has no intrinsic security. Feelings of self-worth are volatile, a function of the emotional state or behavior of other people. Guidance comes from the person's perception of how others will respond, and wisdom is limited by the social lens or by an enemy-centered paranoia. The individual has no power. Other people are pulling the strings.\n\nCHURCH CENTEREDNESS. I believe that almost anyone who is seriously involved in any church will recognize that churchgoing is not synonymous with personal spirituality. There are some people who get so busy in church worship and projects that they become insensitive to the pressing human needs that surround them, contradicting the very precepts they profess to believe deeply. There are others who attend church less frequently or not at all but whose attitudes and behavior reflect a more genuine centering in the principles of the basic Judeo-Christian ethic.\n\nHaving participated throughout my life in organized church and community service groups, I have found that attending church does not necessarily mean living the principles taught in those meetings. You can be active in a church but inactive in its gospel.\n\nIn the church-centered life, image or appearance can become a person's dominant consideration, leading to hypocrisy that undermines personal security and intrinsic worth. Guidance comes from a social conscience, and the church-centered person tends to label others artificially in terms of \"active,\" \"inactive,\" \"liberal,\" \"orthodox,\" or \"conservative.\"\n\nBecause the church is a formal organization made up of policies, programs, practices, and people, it cannot by itself give a person any deep, permanent security or sense of intrinsic worth. Living the principles taught by the church can do this, but the organization alone cannot.\n\nNor can the church give a person a constant sense of guidance. Church-centered people often tend to live in compartments, acting and thinking and feeling in certain ways on the Sabbath and in totally different ways on weekdays. Such a lack of wholeness or unity or integrity is a further threat to security, creating the need for increased labeling and self-justifying.\n\nSeeing the church as an end rather than as a means to an end undermines a person's wisdom and sense of balance. Although the church claims to teach people about the source of power, it does not claim to be that power itself. It claims to be one vehicle through which divine power can be channeled into man's nature.\n\nSELF-CENTEREDNESS. Perhaps the most common center today is the self. The most obvious form is selfishness, which violates the values of most people. But if we look closely at many of the popular approaches to growth and self-fulfillment, we often find self-centering at their core.\n\nThere is little security, guidance, wisdom, or power in the limited center of self. Like the Dead Sea in Israel, it accepts but never gives. It becomes stagnant.\n\nOn the other hand, paying attention to the development of self in the greater perspective of improving one's ability to serve, to produce, to contribute in meaningful ways, gives context for dramatic increase in the four life-support factors.\n\nThese are some of the more common centers from which people approach life. It is often much easier to recognize the center in someone else's life than to see it in your own. You probably know someone who puts making money ahead of everything else. You probably know someone whose energy is devoted to justifying his or her position in an ongoing negative relationship. If you look, you can sometimes see beyond behavior into the center that creates it.\n\nIDENTIFYING YOUR CENTER\n\nBut where do you stand? What is at the center of your own life? Sometimes that isn't easy to see.\n\nPerhaps the best way to identify your own center is to look closely at your life-support factors. If you can identify with one or more of the descriptions below, you can trace it back to the center from which it flows, a center which may be limiting your personal effectiveness.\n\n* * *\n\nIf you are... Spouse Centered\n\nSecurity\n\n  * Your feelings of security are based on the way your spouse treats you.\n  * You are highly vulnerable to the moods and feelings of your spouse.\n  * There is deep disappointment resulting in withdrawal or conflict when your spouse disagrees with you or does not meet your expectations.\n  * Anything that may impinge on the relationship is perceived as a threat.\n\nGuidance\n\n  * Your direction comes from your own needs and wants and from those of your spouse.\n  * Your decision-making criterion is limited to what you think is best for your marriage or your mate, or to the preferences and opinions of your spouse.\n\nWisdom\n\n  * Your life perspective surrounds things which may positively or negatively influence your spouse or your relationship.\n\nPower\n\n  * Your power to act is limited by weaknesses in your spouse and in yourself.\n\n* * *\n\nIf you are... Family Centered\n\nSecurity\n\n  * Your security is founded on family acceptance and fulfilling family expectations.\n  * Your sense of personal security is as volatile as the family.\n  * Your feelings of self-worth are based on the family reputation.\n\nGuidance\n\n  * Family scripting is your source of correct attitudes and behaviors.\n  * Your decision-making criterion is what is good for the family, or what family members want.\n\nWisdom\n\n  * You interpret all of life in terms of your family, creating a partial understanding and family narcissism.\n\nPower\n\n  * Your actions are limited by family models and traditions.\n\n* * *\n\nIf you are... Money Centered\n\nSecurity\n\n  * Your personal worth is determined by your net worth.\n  * You are vulnerable to anything that threatens your economic security.\n\nGuidance\n\n  * Profit is your decision-\u00admaking criterion.\n\nWisdom\n\n  * Money-making is the lens through which life is seen and understood, creating imbalanced judgment.\n\nPower\n\n  * You are restricted to what you can accomplish with your money and your limited vision.\n\n* * *\n\nIf you are... Work Centered\n\nSecurity\n\n  * You tend to define yourself by your occupational role.\n  * You are only comfortable when you are working.\n\nGuidance\n\n  * You make your decisions based on the needs and expectations of your work.\n\nWisdom\n\n  * You tend to be limited to your work role.\n  * You see your work as your life.\n\nPower\n\n  * Your actions are limited by work role models, occupational opportunities, organizational constraints, your boss's perceptions, and your possible inability at some point in your life to do that particular work.\n\n* * *\n\nIf you are... Possession Centered\n\nSecurity\n\n  * Your security is based on your reputation, your social status, or the tangible things you possess.\n  * You tend to compare what you have to what others have.\n\nGuidance\n\n  * You make your decisions based on what will protect, increase, or better display your possessions.\n\nWisdom\n\n  * You see the world in terms of comparative economic and social relationships.\n\nPower\n\n  * You function within the limits of what you can buy or the social prominence you can achieve.\n\n* * *\n\nIf you are... Pleasure Centered\n\nSecurity\n\n  * You feel secure only when you're on a pleasure \"high.\"\n  * Your security is short-lived, anesthetizing, and dependent on your environment.\n\nGuidance\n\n  * You make your decisions based on what will give you the most pleasure.\n\nWisdom\n\n  * You see the world in based on what what's in it for you.\n\nPower\n\n  * Your power is almost negligible.\n\n* * *\n\nIf you are... Friend Centered\n\nSecurity\n\n  * Your security is a function of the social mirror.\n  * You are highly dependent on the opinions of others.\n\nGuidance\n\n  * Your decision-making criterion is \"What will they think?\"\n  * You are easily embarrassed.\n\nWisdom\n\n  * You see the world through a social lens.\n\nPower\n\n  * You are limited by your social comfort zone.\n  * Your actions are as fickle as opinion.\n\n* * *\n\nIf you are... Enemy Centered\n\nSecurity\n\n  * Your security is volatile, based on the movements of your enemy.\n  * You are always wondering what he is up to.\n  * You seek self-justification and validation from the like-minded.\n\nGuidance\n\n  * You are counter-dependently guided by your enemy's actions.\n  * You make your decisions based on what will thwart your enemy.\n\nWisdom\n\n  * Your judgment is narrow and distorted.\n  * You are defensive, overreactive, and often paranoid.\n\nPower\n\n  * The little power you do have comes from anger, envy, resentment and vengeance\u2014negative energy that shrivels and destroys, leaving energy for little else.\n\n* * *\n\nIf you are... Church Centered\n\nSecurity\n\n  * Your security is based on church activity and on the esteem in which you are held by those in authority or influence in the church.\n  * You find identity and security in religious labels and comparisons.\n\nGuidance\n\n  * You are guided by how others will evaluate your actions in the context of church teachings and expectations.\n\nWisdom\n\n  * You see the world in terms of \"believers\" and \"nonbelievers,\" \"belongers\" and \"non-belongers.\"\n\nPower\n\n  * Perceived power comes from your church position or role.\n\n* * *\n\nIf you are... Self-Centered\n\nSecurity\n\n  * Your security is constantly changing and shifting.\n\nGuidance\n\n  * Your judgment criteria are: \"If it feels good... \"What I want.\" \"What I need.\" \"What's in it for me?\"\n\nWisdom\n\n  * You view the world by how decisions, events, or circumstances will af\u00adfect you.\n\nPower\n\n  * Your ability to act is lim\u00adited to your own resources, without the benefits of interdependency.\n\n* * *\n\nMore often than not, a person's center is some combination of these and\/or other centers. Most people are very much a function of a variety of influences that play upon their lives. Depending on external or internal conditions, one particular center may be activated until the underlying needs are satisfied. Then another center becomes the compelling force.\n\nAs a person fluctuates from one center to another, the resulting relativism is like roller coasting through life. One moment you're high, the next moment you're low, making efforts to compensate for one weakness by borrowing strength from another weakness. There is no consistent sense of direction, no persistent wisdom, no steady power supply or sense of personal, intrinsic worth and identity.\n\nThe ideal, of course, is to create one clear center from which you consistently derive a high degree of security, guidance, wisdom, and power, empowering your proactivity and giving congruency and harmony to every part of your life.\n\nA PRINCIPLE CENTER\n\nBy centering our lives on correct principles, we create a solid foundation for development of the four life-support factors.\n\nOur security comes from knowing that, unlike other centers based on people or things which are subject to frequent and immediate change, correct principles do not change. We can depend on them.\n\nPrinciples don't react to anything. They don't get mad and treat us differently. They won't divorce us or run away with our best friend. They aren't out to get us. They can't pave our way with shortcuts and quick fixes. They don't depend on the behavior of others, the environment, or the current fad for their validity. Principles don't die. They aren't here one day and gone the next. They can't be destroyed by fire, earthquake or theft.\n\nPrinciples are deep, fundamental truths, classic truths, generic common denominators. They are tightly interwoven threads running with exactness, consistency, beauty, and strength through the fabric of life.\n\nEven in the midst of people or circumstances that seem to ignore the principles, we can be secure in the knowledge that principles are bigger than people or circumstances, and that thousands of years of history have seen them triumph, time and time again. Even more important, we can be secure in the knowledge that we can validate them in our own lives, by our own experience.\n\nAdmittedly, we're not omniscient. Our knowledge and understanding of correct principles is limited by our own lack of awareness of our true nature and the world around us and by the flood of trendy philosophies and theories that are not in harmony with correct principles. These ideas will have their season of acceptance, but, like many before them, they won't endure because they're built on false foundations.\n\nWe are limited, but we can push back the borders of our limitations. An understanding of the principle of our own growth enables us to search out correct principles with the confidence that the more we learn, the more clearly we can focus the lens through which we see the world. The principles don't change; our understanding of them does.\n\nThe wisdom and guidance that accompany principle-centered living come from correct maps, from the way things really are, have been, and will be. Correct maps enable us to clearly see where we want to go and how to get there. We can make our decisions using the correct data that will make their implementation possible and meaningful.\n\nThe personal power that comes from principle-centered living is the power of a self-aware, knowledgeable, proactive individual, unrestricted by the attitudes, behaviors, and actions of others or by many of the circumstances and environmental influences that limit other people.\n\nThe only real limitation of power is the natural consequences of the principles themselves. We are free to choose our actions, based on our knowledge of correct principles, but we are not free to choose the consequences of those actions. Remember, \"If you pick up one end of the stick, you pick up the other.\"\n\nPrinciples always have natural consequences attached to them. There are positive consequences when we live in harmony with the principles. There are negative consequences when we ignore them. But because these principles apply to everyone, whether or not they are aware, this limitation is universal. And the more we know of correct principles, the greater is our personal freedom to act wisely.\n\nBy centering our lives on timeless, unchanging principles, we create a fundamental paradigm of effective living. It is the center that puts all other centers in perspective.\n\n* * *\n\nIf you are... Principle Centered\n\nSecurity\n\n  * Your security is based on correct principles that do not change, regardless of external conditions or circumstances.\n  * You know that true principles can repeatedly be validated in your own life, through your own experiences.\n  * As a measurement of self-improvement, correct principles function with exactness, consistency, beauty, and strength.\n  * Correct principles help you understand your own development, endowing you with the confidence to learn more, thereby increasing your knowledge and understanding.\n  * Your source of security provides you with an immovable, unchanging, unfailing core enabling you to see change as an exciting adventure and opportunity to make significant contributions.\n\nGuidance\n\n  * You are guided by a compass which enables you to see where you want to go and how you will get there.\n  * You use accurate data which makes your decisions both implementable and meaningful.\n  * You stand apart from life's situations, emotions, and circumstances, and look at the balanced whole. Your decisions and actions reflect both short- and long-term considerations and implications.\n  * In every situation, you consciously, proactively determine the best alternative, basing decisions on conscience educated by principles.\n\nWisdom\n\n  * Your judgment encompasses a broad spectrum of long-term consequences and reflects a wise balance and quiet assurance.\n  * You see things differently and thus you think and act differently from the largely reactive world.\n  * You view the world through a fundamental paradigm for effective, provident living.\n  * You see the world in terms of what you can do for the world and its people.\n  * You adopt a proactive lifestyle, seeking to serve and build others.\n  * You interpret all of life's experiences in terms of opportunities for learning and contribution.\n\nPower\n\n  * Your power is limited only by your understanding and observance of natural law and correct principles and by the natural consequences of the principles themselves.\n  * You become a self-aware, knowledgeable, proactive individual, largely unrestricted by the attitudes, behaviors, or actions of others.\n  * Your ability to act reaches far beyond your own resources and encourages highly developed levels of interdependency.\n  * Your decisions and actions are not driven by your current financial or circumstantial limitations. You experience an interdependent freedom.\n\n* * *\n\nRemember that your paradigm is the source from which your attitudes and behaviors flow. A paradigm is like a pair of glasses; it affects the way you see everything in your life. If you look at things through the paradigm of correct principles, what you see in life is dramatically different from what you see through any other centered paradigm.\n\nI have included in the Appendix section of this book a detailed chart which shows how each center we've discussed might possibly affect the way you see everything else*. But for a quick understanding of the difference your center makes, let's look at just one example of a specific problem as seen through the different paradigms. As you read, try to put on each pair of glasses. Try to feel the response that flows from the different centers.\n\nSuppose tonight you have invited your wife to go to a concert. You have the tickets; she's excited about going. It's four o'clock in the afternoon.\n\nAll of a sudden, your boss calls you into his office and says he needs your help through the evening to get ready for an important meeting at 9 A.M. tomorrow.\n\nIf you're looking through spouse-centered or family-centered glasses, your main concern will be your wife. You may tell the boss you can't stay and you take her to the concert in an effort to please her. You may feel you have to stay to protect your job, but you'll do so grudgingly, anxious about her response, trying to justify your decision and protect yourself from her disappointment or anger.\n\nIf you're looking through a money-centered lens, your main thought will be of the overtime you'll get or the influence working late will have on a potential raise. You may call your wife and simply tell her you have to stay, assuming she'll understand that economic demands come first.\n\nIf you're work-centered, you may be thinking of the opportunity. You can learn more about the job. You can make some points with the boss and further your career. You may give yourself a pat on the back for putting in hours well beyond what is required, evidence of what a hard worker you are. Your wife should be proud of you!\n\nIf you're possession-centered, you might be thinking of the things the overtime income could buy. Or you might consider what an asset to your reputation at the office it would be if you stayed. Everyone would hear tomorrow how noble, how sacrificing and dedicated you are.\n\nIf you're pleasured-centered, you'll probably can the work and go to the concert, even if your wife would be happy for you to work late. You deserve a night out!\n\nIf you're friend-centered, your decision would be influenced by whether or not you had invited friends to attend the concert with you. Or whether your friends at work were going to stay late, too.\n\nIf you're enemy-centered, you may stay late because you know it will give you a big edge over that person in the office who thinks he's the company's greatest asset. While he's off having fun, you'll be working and slaving, doing his work and yours, sacrificing your personal pleasure for the good of the company he can so blithely ignore.\n\nIf you're church-centered, you might be influenced by plans other church members have to attend the concert, by whether or not any church members work at your office, or by the nature of the concert\u2014Handel's Messiah might rate higher priority than a rock concert. Your decision might also be affected by what you think a \"good church member\" would do and by whether you view the extra work as \"service\" or \"seeking after material wealth.\"\n\nIf you're self-centered, you'll be focused on what will do you the most good. Would it be better for you to go out for the evening? Or would it be better for you to make a few points with the boss? How the different options affect you will be your main concern.\n\nAs we consider various ways of looking at a single event, is it any wonder that we have \"young lady\/old lady\" perception problems in our interactions with each other? Can you see how fundamentally our centers affect us? Right down to our motivations, our daily decisions, our actions (or, in too many cases, our reactions), our interpretations of events? That's why understanding your own center is so important. And if that center does not empower you as a proactive person, it becomes fundamental to your effectiveness to make the necessary paradigm shifts to create a center that will.\n\nAs a principle-centered person, you try to stand apart from the emotion of the situation and from other factors that would act on you, and evaluate the options. Looking at the balanced whole\u2014the work needs, the family needs, other needs that may be involved and the possible implications of the various alternative decisions\u2014you'll try to come up with the best solution, taking all factors into consideration.\n\nWhether you go to the concert or stay and work is really a small part of an effective decision. You might make the same choice with a number of other centers. But there are several important differences when you are coming from a principle-centered paradigm.\n\nFirst, you are not being acted upon by other people or circumstances. You are proactively choosing what you determine to be the best alternative. You make your decision consciously and knowledgeably.\n\nSecond, you know your decision is most effective because it is based on principles with predictable long-term results.\n\nThird, what you choose to do contributes to your ultimate values in life. Staying at work to get the edge on someone at the office is an entirely different evening in your life from staying because you value your boss's effectiveness and you genuinely want to contribute to the company's welfare. The experiences you have as you carry out your decisions take on quality and meaning in the context of your life as a whole.\n\nFourth, you can communicate to your wife and your boss within the strong networks you've created in your interdependent relationships. Because you are independent, you can be effectively interdependent. You might decide to delegate what is delegable and come in early the next morning to do the rest.\n\nAnd finally, you'll feel comfortable about your decision. Whatever you choose to do, you can focus on it and enjoy it.\n\nAs a principle-centered person, you see things differently. And because you see things differently, you think differently, you act differently. Because you have a high degree of security, guidance, wisdom, and power that flows from a solid, unchanging core, you have the foundation of a highly proactive and highly effective life.\n\nWRITING AND USING A PERSONAL MISSION STATEMENT\n\nAs we go deeply within ourselves, as we understand and realign our basic paradigms to bring them in harmony with correct principles, we create both an effective, empowering center and a clear lens through which we can see the world. We can then focus that lens on how we, as unique individuals, relate to that world.\n\nFrankl says we detect rather than invent our missions in life. I like that choice of words. I think each of us has an internal monitor or sense, a conscience, that gives us an awareness of our own uniqueness and the singular contributions that we can make. In Frankl's words, \"Everyone has his own specific vocation or mission in life... . Therein he cannot be replaced, nor can his life be repeated. Thus, everyone's task is as unique as is his specific opportunity to implement it.\"\n\nIn seeking to give verbal expression to that uniqueness, we are again reminded of the fundamental importance of proactivity and of working within our Circle of Influence. To seek some abstract meaning to our lives out in our Circle of Concern is to abdicate our proactive responsibility, to place our own first creation in the hands of circumstance and other people.\n\nOur meaning comes from within. Again, in the words of Frankl, \"Ultimately, man should not ask what the meaning of his life is, but rather must recognize that it is he who is asked. In a word, each man is questioned by life; and he can only answer to life by answering for his own life; to life he can only respond by being responsible.\"\n\nPersonal responsibility, or proactivity, is fundamental to the first creation. Returning to the computer metaphor, Habit 1 says \"You are the programmer.\" Habit 2, then, says, \"Write the program.\" Until you accept the idea that you are responsible, that you are the programmer, you won't really invest in writing the program.\n\nAs proactive people, we can begin to give expression to what we want to be and to do in our lives. We can write a personal mission statement, a personal constitution.\n\nA mission statement is not something you write overnight. It takes deep introspection, careful analysis, thoughtful expression, and often many rewrites to produce it in final form. It may take you several weeks or even months before you feel really comfortable with it, before you feel it is a complete and concise expression of your innermost values and directions. Even then, you will want to review it regularly and make minor changes as the years bring additional insights or changing circumstances.\n\nBut fundamentally, your mission statement becomes your constitution, the solid expression of your vision and values. It becomes the criterion by which you measure everything else in your life.\n\nI recently finished reviewing my own mission statement, which I do fairly regularly. Sitting on the edge of a beach, alone, at the end of a bicycle ride, I took out my organizer and hammered it out. It took several hours, but I felt a sense of clarity, a sense of organization and commitment, a sense of exhilaration and freedom.\n\nI find the process is as important as the product. Writing or reviewing a mission statement changes you because it forces you to think through your priorities deeply, carefully, and to align your behavior with your beliefs. As you do, other people begin to sense that you're not being driven by everything that happens to you. You have a sense of mission about what you're trying to do and you are excited about it.\n\nUSING YOUR WHOLE BRAIN\n\nOur self-awareness empowers us to examine our own thoughts. This is particularly helpful in creating a personal mission statement because the two unique human endowments that enable us to practice Habit 2\u2014imagination and conscience\u2014are primarily functions of the right side of the brain. Understanding how to tap into that right brain capacity greatly increases our first creation ability.\n\nA great deal of research has been conducted for decades on what has come to be called brain dominance theory. The findings basically indicate that each hemisphere of the brain\u2014left and right\u2014tends to specialize in and preside over different functions, process different kinds of information, and deal with different kinds of problems.\n\nEssentially, the left hemisphere is the more logical\/verbal one and the right hemisphere the more intuitive, creative one. The left deals with words, the right with pictures; the left with parts and specifics, the right with wholes and the relationship between the parts. The left deals with analysis, which means to break apart; the right with synthesis, which means to put together. The left deals with sequential thinking; the right with simultaneous and holistic thinking. The left is time bound; the right is time free.\n\nAlthough people use both sides of the brain, one side or the other generally tends to be dominant in each individual. Of course, the ideal would be to cultivate and develop the ability to have good crossover between both sides of the brain so that a person could first sense what the situation called for and then use the appropriate tool to deal with it. But people tend to stay in the \"comfort zone\" of their dominant hemisphere and process every situation according to either a right or left brain preference.\n\nIn the words of Abraham Maslow, \"He that is good with a hammer tends to think everything is a nail.\" This is another factor that affects the \"young lady\/old lady\" perception difference. Right brain and left brain people tend to look at things in different ways.\n\nWe live in a primarily left brain-dominant world, where words and measurement and logic are enthroned, and the more creative, intuitive, sensing, artistic aspect of our nature is often subordinated. Many of us find it more difficult to tap into our right brain capacity.\n\nAdmittedly this description is oversimplified and new studies will undoubtedly throw more light on brain functioning. But the point here is that we are capable of performing many different kinds of thought processes and we barely tap our potential. As we become aware of its different capacities, we can consciously use our minds to meet specific needs in more effective ways.\n\nTWO WAYS TO TAP THE RIGHT BRAIN\n\nIf we use the brain dominance theory as a model, it becomes evident that the quality of our first creation is significantly impacted by our ability to use our creative right brain. The more we are able to draw upon our right brain capacity, the more fully we will be able to visualize, to synthesize, to transcend time and present circumstances, to project a holistic picture of what we want to do and to be in life.\n\nExpand Perspective\n\nSometimes we are knocked out of our left brain environment and thought patterns and into the right brain by an unplanned experience. The death of a loved one, a severe illness, a financial setback, or extreme adversity can cause us to stand back, look at our lives, and ask ourselves some hard questions: \"What's really important? Why am I doing what I'm doing?\"\n\nBut if you're proactive, you don't have to wait for circumstances or other people to create perspective expanding experiences. You can consciously create your own.\n\nThere are a number of ways to do this. Through the powers of your imagination, you can visualize your own funeral, as we did at the beginning of this chapter. Write your own eulogy. Actually write it out. Be specific.\n\nYou can visualize your twenty-fifth and then your fiftieth wedding anniversary. Have your spouse visualize this with you. Try to capture the essence of the family relationship you want to have created through your day-by-day investment over a period of that many years.\n\nYou can visualize your retirement from your present occupation. What contributions, what achievements will you want to have made in your field? What plans will you have after retirement? Will you enter a second career?\n\nExpand your mind. Visualize in rich detail. Involve as many emotions and feelings as possible. Involve as many of the senses as you can.\n\nI have done similar visualization exercises with some of my university classes. \"Assume you only have this one semester to live,\" I tell my students, \"and that during this semester you are to stay in school as a good student. Visualize how you would spend your semester.\"\n\nThings are suddenly placed in a different perspective. Values quickly surface that before weren't even recognized.\n\nI have also asked students to live with that expanded perspective for a week and keep a diary of their experiences.\n\nThe results are very revealing. They start writing to parents to tell them how much they love and appreciate them. They reconcile with a brother, a sister, a friend where the relationship has deteriorated.\n\nThe dominant, central theme of their activities, the underlying principle, is love. The futility of bad-mouthing, bad thinking, put-downs, and accusation becomes very evident when they think in terms of having only a short time to live. Principles and values become more evident to everybody.\n\nThere are a number of techniques using your imagination that can put you in touch with your values. But the net effect of every one I have ever used is the same. When people seriously undertake to identify what really matters most to them in their lives, what they really want to be and to do, they become very reverent. They start to think in larger terms than today and tomorrow.\n\nVisualization and Affirmation\n\nPersonal leadership is not a singular experience. It doesn't begin and end with the writing of a personal mission statement. It is, rather, the ongoing process of keeping your vision and values before you and aligning your life to be congruent with those most important things. And in that effort, your powerful right brain capacity can be a great help to you on a daily basis as you work to integrate your personal mission statement into your life. It's another application of \"begin with the end in mind.\"\n\nLet's go back to an example we mentioned before. Suppose I am a parent who really deeply loves my children. Suppose I identify that as one of my fundamental values in my personal mission statement. But suppose, on a daily basis, I have trouble overreact\u00ading.\n\nI can use my right brain power of visualization to write an \"affirmation\" that will help me become more congruent with my deeper values in my daily life.\n\nA good affirmation has five basic ingredients: it's personal, it's positive, it's present tense, it's visual, and it's emotional. So I might write something like this: \"It is deeply satisfying (emotional) that I (personal) respond (present tense) with wisdom, love, firmness, and self-control (positive) when my children misbehave.\"\n\nThen I can visualize it. I can spend a few minutes each day and totally relax my mind and body. I can think about situations in which my children might misbehave. I can visualize them in rich detail. I can feel the texture of the chair I might be sitting on, the floor under my feet, the sweater I'm wearing. I can see the dress my daughter has on, the expression on her face. The more clearly and vividly I can imagine the detail, the more deeply I will experience it, the less I will see it as a spectator.\n\nThen I can see her do something very specific which normally makes my heart pound and my temper start to flare. But instead of seeing my normal response, I can see myself handle the situation with all the love, the power, the self-control I have captured in my affirmation. I can write the program, write the script, in harmony with my values, with my personal mission statement.\n\nAnd if I do this, day after day my behavior will change. Instead of living out of the scripts given to me by my own parents or by society or by genetics or my environment, I will be living out of the script I have written from my own self-selected value system.\n\nI have helped and encouraged my son, Sean, to use this affirmation process extensively throughout his football career. We started when he played quarterback in high school, and eventually, I taught him how to do it on his own.\n\nWe would try to get him in a very relaxed state of mind through deep breathing and a progressive muscle relaxation technique so that he became very quiet inside. Then I would help him visualize himself right in the heat of the toughest situations imaginable.\n\nHe would imagine a big blitz coming at him fast. He had to read the blitz and respond. He would imagine giving audibles at the line after reading defenses. He would imagine quick reads with his first receiver, his second receiver, his third receiver. He would imagine options that he normally wouldn't do.\n\nAt one point in his football career, he told me he was constantly getting uptight. As we talked, I realized that he was visualizing uptightness. So we worked on visualizing relaxation in the middle of the big pressure circumstance. We discovered that the nature of the visualization is very important. If you visualize the wrong thing, you'll produce the wrong thing.\n\nDr. Charles Garfield has done extensive research on peak performers, both in athletics and in business. He became fascinated with peak performance in his work with the NASA program, watching the astronauts rehearse everything on earth, again and again in a simulated environment before they went to space. Although he had a doctorate in mathematics, he decided to go back and get another Ph.D. in the field of psychology and study the characteristics of peak performers.\n\nOne of the main things his research showed was that almost all of the world-class athletes and other peak performers are visualizers. They see it; they feel it; they experience it before they actually do it. They begin with the end in mind.\n\nYou can do it in every area of your life. Before a performance, a sales presentation, a difficult confrontation, or the daily challenge of meeting a goal, see it clearly, vividly, relentlessly, over and over again. Create an internal \"comfort zone.\" Then, when you get into the situation, it isn't foreign. It doesn't scare you.\n\nYour creative, visual right brain is one of your most important assets, both in creating your personal mission statement and in integrating it into your life.\n\nThere is an entire body of literature and audio and video tapes that deals with this process of visualization and affirmation. Some of the more recent developments in this field include such things as subliminal programming, neurolinguistic programming, and new forms of relaxation and self-talk processes. These all involve explanation, elaboration and different packaging of the fundamen\u00adtal principles of the first creation.\n\nMy review of the success literature brought me in contact with hundreds of books on this subject. Although some made extravagant claims and relied on anecdotal rather than scientific evidence, I think that most of the material is fundamentally sound. The majority of it appears to have originally come out of the study of the Bible by many individuals.\n\nIn effective personal leadership, visualization and affirmation techniques emerge naturally out of a foundation of well thought through purposes and principles that become the center of a person's life. They are extremely powerful in rescripting and reprogramming, into writing deeply committed-to purposes and principles into one's heart and mind. I believe that central to all enduring religions in society are the same principles and practices clothed in different language\u2014meditation, prayer, covenants, or\u00addinances, scripture study, empathy, compassion, and many different forms of the use of both conscience and imagination.\n\nBut if these techniques become part of the Personality Ethic and are severed from a base of character and principles, they can be misused and abused in serving other centers, primarily the center of self.\n\nAffirmation and visualization are forms of programming, and we must be certain that we do not submit ourselves to any program\u00adming that is not in harmony with our basic center or that comes from sources centered on money-making, self interest, or anything other than correct principles.\n\nThe imagination can be used to achieve the fleeting success that comes when a person is focused on material gain or on \"what's in it for me.\" But I believe the higher use of imagination is in harmony with the use of conscience to transcend self and create a life of contribution based on unique purpose and on the principles that govern interdependent reality.\n\nIDENTIFYING ROLES AND GOALS\n\nOf course, the logical\/verbal left brain becomes important also as you attempt to capture your right brain images, feelings, and pictures in the words of a written mission statement. Just as breathing exercises help integrate body and mind, writing is a kind of psycho-neural muscular activity which helps bridge and inte\u00adgrate the conscious and subconscious minds. Writing distills, crystallizes, and clarifies thought and helps break the whole into parts.\n\nWe each have a number of different roles in our lives\u2014different areas or capacities in which we have responsibility. I may, for example, have a role as an individual, a husband, a father, a teacher, a church member, and a businessman. And each of these roles is important.\n\nOne of the major problems that arises when people work to become more effective in life is that they don't think broadly enough. They lose the sense of proportion, the balance, the natural ecology necessary to effective living. They may get consumed by work and neglect personal health. In the name of professional success, they may neglect the most precious relationships in their lives.\n\nYou may find that your mission statement will be much more balanced, much easier to work with, if you break it down into the specific role areas of your life and the goals you want to accomplish in each area. Look at your professional role. You might be a salesperson, or a manager, or a product developer. What are you about in that area? What are the values that should guide you? Think of your personal roles\u2014husband, wife, father, mother, neighbor, friend. What are you about in those roles? What's important to you? Think of community roles\u2014the political area, public service, volunteer organizations.\n\nOne executive has used the idea of roles and goals to create the following mission statement:\n\nMy mission is to live with integrity and to make a difference in the lives of others.\n\nTo fulfill this mission:\n\nI have charity: I seek out and love the one\u2014each one\u2014regardless of his situation.\n\nI sacrifice: I devote my time, talents, and resources to my mission.\n\nI inspire: I teach by example that we are all children of a loving Heavenly Father and that every Goliath can be overcome.\n\nI am impactful: What I do makes a difference in the lives of others.\n\nThese roles take priority in achieving my mission:\n\nHusband\u2014My partner is the most important person in my life. Together we contribute the fruits of harmony, industry, charity, and thrift.\n\nFather\u2014I help my children experience progressively greater joy in their lives.\n\nSon\/Brother\u2014I am frequently \"there\" for support and love.\n\nChristian\u2014God can count on me to keep my covenants and to serve his other children.\n\nNeighbor\u2014The love of Christ is visible through my actions toward others.\n\nChange Agent\u2014I am a catalyst for developing high performance in large organizations.\n\nScholar\u2014I learn important new things every day.\n\nWriting your mission in terms of the important roles in your life gives you balance and harmony. It keeps each role clearly before you. You can review your roles frequently to make sure that you don't get totally absorbed by one role to the exclusion of others that are equally or even more important in your life.\n\nAfter you identify your various roles, then you can think about the long-term goals you want to accomplish in each of those roles. We're into the right brain again, using imagination, creativity, conscience, and inspiration. If these goals are the extension of a mission statement based on correct principles, they will be vitally different from the goals people normally set. They will be in harmony with correct principles, with natural laws, which gives you greater power to achieve them. They are not someone else's goals you have absorbed. They are your goals. They reflect your deepest values, your unique talent, your sense of mission. And they grow out of your chosen roles in life.\n\nAn effective goal focuses primarily on results rather than activity. It identifies where you want to be, and, in the process, helps you determine where you are. It gives you important information on how to get there, and it tells you when you have arrived. It unifies your efforts and energy. It gives meaning and purpose to all you do. And it can finally translate itself into daily activities so that you are proactive, you are in charge of your life, you are making happen each day the things that will enable you to fulfill your personal mission statement.\n\nRoles and goals give structure and organized direction to your personal mission. If you don't yet have a personal mission state\u00adment, it's a good place to begin. Just identifying the various areas of your life and the two or three important results you feel you should accomplish in each area to move ahead gives you an overall perspective of your life and a sense of direction.\n\nAs we move into Habit 3, we'll go into greater depth in the area of short-term goals. The important application at this point is to identify roles and long-term goals as they relate to your personal mission statement. These roles and goals will provide the founda\u00adtion for effective goal setting and achieving when we get to the Habit 3 day-to-day management of life and time.\n\nFAMILY MISSION STATEMENTS\n\nBecause Habit 2 is based on principle, it has broad application. In addition to individuals, families, service groups, and organizations of all kinds become significantly more effective as they begin with the end in mind.\n\nMany families are managed on the basis of crises, moods, quick fixes, and instant gratification\u2014not on sound principles. Symp\u00adtoms surface whenever stress and pressure mount: people become cynical, critical, or silent or they start yelling and overreacting. Children who observe these kinds of behavior grow up thinking the only way to solve problems is flight or fight.\n\nThe core of any family is what is changeless, what is always going to be there\u2014shared vision and values. By writing a family mission statement, you give expression to its true foundation.\n\nThis mission statement becomes its constitution, the standard, the criterion for evaluation and decision making. It gives continuity and unity to the family as well as direction. When individual values are harmonized with those of the family, members work together for common purposes that are deeply felt.\n\nAgain, the process is as important as the product. The very process of writing and refining a mission statement becomes a key way to improve the family. Working together to create a mission statement builds the PC capacity to live it.\n\nBy getting input from every family member, drafting a state\u00adment, getting feedback, revising it, and using wording from different family members, you get the family talking, communicat\u00ading, on things that really matter deeply. The best mission state\u00adments are the result of family members coming together in a spirit of mutual respect, expressing their different views, and working together to create something greater than any one individual could do alone. Periodic review to expand perspective, shift emphasis or direction, amend or give new meaning to time-worn phrases can keep the family united in common values and purposes.\n\nThe mission statement becomes the framework for thinking, for governing the family. When the problems and crises come, the constitution is there to remind family members of the things that matter most and to provide direction for problem solving and decision making based on correct principles.\n\nIn our home, we put our mission statement up on a wall in the family room so that we can look at it and monitor ourselves daily.\n\nWhen we read the phrases about the sounds of love in our home, order, responsible independence, cooperation, helpfulness, meeting needs, developing talents, showing interest in each other's talents, and giving service to others it gives us some criteria to know how we're doing in the things that matter most to us as a family.\n\nWhen we plan our family goals and activities, we say, \"In light of these principles, what are the goals we're going to work on? What are our action plans to accomplish our goals and actualize these values?\"\n\nWe review the statement frequently and rework goals and jobs twice a year, in September and June\u2014the beginning of school and the end of school\u2014to reflect the situation as it is, to improve it, to strengthen it. It renews us, it recommits us to what we believe in, what we stand for.\n\nORGANIZATIONAL MISSION STATEMENTS\n\nMission statements are also vital to successful organizations. One of the most important thrusts of my work with organizations is to assist them in developing effective mission statements. And to be effective, that statement has to come from within the bowels of the organization. Everyone should participate in a meaningful way\u2014not just the top strategy planners, but everyone. Once again, the involvement process is as important as the written product and is the key to its use.\n\nI am always intrigued whenever I go to IBM and watch the training process there. Time and time again, I see the leadership of the organization come into a group and say that IBM stands for three things: the dignity of the individual, excellence, and service.\n\nThese things represent the belief system of IBM. Everything else will change, but these three things will not change. Almost like osmosis, this belief system has spread throughout the entire organization, providing a tremendous base of shared values and personal security for everyone who works there.\n\nOnce I was training a group of people for IBM in New York. It was a small group, about twenty people, and one of them became ill. He called his wife in California, who expressed concern because his illness required special treatment. The IBM people responsible for the training session arranged to have him taken to an excellent hospital with medical specialists in the disease. But they could sense that his wife was uncertain and really wanted him home where their personal physician could handle the problem.\n\nSo they decided to get him home. Concerned about the time involved in driving him to the airport and waiting for a commercial plane, they brought in a helicopter, flew him to the airport, and hired a special plane just to take this man to California.\n\nI don't know what costs that involved; my guess would be many thousands of dollars. But IBM believes in the dignity of the individual. That's what the company stands for. To those present, that experience represented its belief system and was no surprise. I was impressed.\n\nAt another time, I was scheduled to train 175 shopping center managers at a particular hotel. I was amazed at the level of service there. It wasn't a cosmetic thing. It was evident at all levels, spontaneously, without supervision.\n\nI arrived quite late, checked in, and asked if room service were available. The man at the desk said, \"No, Mr. Covey, but if you're interested, I could go back and get a sandwich or a salad or whatever you'd like that we have in the kitchen.\" His attitude was one of total concern about my comfort and welfare. \"Would you like to see your convention room?\" he continued. \"Do you have everything you need? What can I do for you? I'm here to serve you.\"\n\nThere was no supervisor there checking up. This man was sincere.\n\nThe next day I was in the middle of a presentation when I discovered that I didn't have all the colored markers I needed. So I went out into the hall during the brief break and found a bellboy running to another convention. \"I've got a problem,\" I said. \"I'm here training a group of managers and I only have a short break. I need some more colored pens.\"\n\nHe whipped around and almost came to attention. He glanced at my name tag and said, \"Mr. Covey, I will solve your problem.\"\n\nHe didn't say, \"I don't know where to go\" or \"Well, go and check at the front desk.\" He just took care of it. And he made me feel like it was his privilege to do so.\n\nLater, I was in the side lobby, looking at some of the art objects. Someone from the hotel came up to me and said, \"Mr. Covey, would you like to see a book that describes the art objects in this hotel?\" How anticipatory! How service-oriented!\n\nI next observed one of the employees high up on a ladder cleaning windows in the lobby. From his vantage point he saw a woman having a little difficulty in the garden with a walker. She hadn't really fallen, and she was with other people. But he climbed down that ladder, went outside, helped the woman into the lobby and saw that she was properly taken care of. Then he went back and finished cleaning the windows.\n\nI wanted to find out how this organization had created a culture where people bought so deeply into the value of customer service. I interviewed housekeepers, waitresses, bellboys in that hotel and found that this attitude had impregnated the minds, hearts, and attitudes of every employee there.\n\nI went through the back door into the kitchen, where I saw the central value: \"Uncompromising personalized service.\" I finally went to the manager and said, \"My business is helping organiza\u00adtions develop a powerful team character, a team culture. I am amazed at what you have here.\"\n\n\"Do you want to know the real key?\" he inquired. He pulled out the mission statement for the hotel chain.\n\nAfter reading it, I acknowledged, \"That's an impressive state\u00adment. But I know many companies that have impressive mission statements.\"\n\n\"Do you want to see the one for this hotel?\" he asked.\n\n\"Do you mean you developed one just for this hotel?\"\n\n\"Yes.\"\n\n\"Different from the one for the hotel chain?\"\n\n\"Yes. It's in harmony with that statement, but this one pertains to our situation, our environment, our time.\" He handed me another paper.\n\n\"Who developed this mission statement?\" I asked.\n\n\"Everybody,\" he replied.\n\n\"Everybody? Really, everybody?\"\n\n\"Yes.\"\n\n\"Housekeepers?\"\n\n\"Yes.\"\n\n\"Waitresses?\"\n\n\"Yes.\"\n\n\"Desk clerks?\"\n\n\"Yes. Do you want to see the mission statement written by the people who greeted you last night?\" He pulled out a mission statement that they, themselves, had written that was interwoven with all the other mission statements. Everyone, at every level, was involved.\n\nThe mission statement for that hotel was the hub of a great wheel. It spawned the thoughtful, more specialized mission state\u00adments of particular groups of employees. It was used as the criterion for every decision that was made. It clarified what those people stood for\u2014how they related to the customer, how they related to each other. It affected the style of the managers and the leaders. It affected the compensation system. It affected the kind of people they recruited and how they trained and developed them. Every aspect of that organization, essentially, was a function of that hub, that mission statement.\n\nI later visited another hotel in the same chain, and the first thing I did when I checked in was to ask to see their mission statement, which they promptly gave me. At this hotel, I came to understand the motto \"Uncompromising personalized service\" a little more.\n\nFor a three-day period, I watched every conceivable situation where service was called for. I always found that service was delivered in a very impressive, excellent way. But it was always also very personalized. For instance, in the swimming area I asked the attendant where the drinking fountain was. He walked me to it.\n\nBut the thing that impressed me the very most was to see an employee, on his own, admit a mistake to his boss. We ordered room service, and were told when it would be delivered to the room. On the way to our room, the room service person spilled the hot chocolate, and it took a few extra minutes to go back and change the linen on the tray and replace the drink. So the room service was about fifteen minutes late, which was really not that important to us.\n\nNevertheless, the next morning the room service manager phoned us to apologize and invited us to have either the buffet breakfast or a room service breakfast, compliments of the hotel, to in some way compensate for the inconvenience.\n\nWhat does it say about the culture of an organization when an employee admits his own mistake, unknown to anyone else, to the manager so that customer or guest is better taken care of!\n\nAs I told the manager of the first hotel I visited, I know a lot of companies with impressive mission statements. But there is a real difference, all the difference in the world, in the effectiveness of a mission statement created by everyone involved in the organiza\u00adtion and one written by a few top executives behind a mahogany wall.\n\nOne of the fundamental problems in organizations, including families, is that people are not committed to the determinations of other people for their lives. They simply don't buy into them.\n\nMany times as I work with organizations, I find people whose goals are totally different from the goals of the enterprise. I commonly find reward systems completely out of alignment with stated value systems.\n\nWhen I begin work with companies that have already developed some kind of mission statement, I ask them, \"How many of the people here know that you have a mission statement? How many of you know what it contains? How many were involved in creating it? How many really buy into it and use it as your frame of reference in making decisions?\"\n\nWithout involvement, there is no commitment. Mark it down, asterisk it, circle it, underline it. No involvement, no commitment.\n\nNow, in the early stages\u2014when a person is new to an organi\u00adzation or when a child in the family is young\u2014you can pretty well give them a goal and they'll buy it, particularly if the relationship, orientation, and training are good.\n\nBut when people become more mature and their own lives take on a separate meaning, they want involvement, significant involve\u00adment. And if they don't have that involvement, they don't buy it. Then you have a significant motivational problem which cannot be solved at the same level of thinking that created it.\n\nThat's why creating an organizational mission statement takes time, patience, involvement, skill, and empathy. Again, it's not a quick fix. It takes time and sincerity, correct principles, and the courage and integrity to align systems, structure, and management style to the shared vision and values. But it's based on correct principles and it works.\n\nAn organizational mission statement\u2014one that truly reflects the deep shared vision and values of everyone within that organization\u2014creates a great unity and tremendous commitment. It creates in people's hearts and minds a frame of reference, a set of criteria or guidelines, by which they will govern themselves. They don't need someone else directing, controlling, criticizing, or taking cheap shots. They have bought into the changeless core of what the organization is about.\n\nAPPLICATION SUGGESTIONS\n\n  1. Take the time to record the impressions you had in the funeral visualization at the beginning of this chapter. You may want to use the chart below to organize your thoughts.\n  2. Take a few moments and write down your roles as you now see them. Are you satisfied with that mirror image of your life?\n  3. Set up time to completely separate yourself from daily activities and to begin work on your personal mission statement.\n  4. Go through the chart in Appendix A showing different centers and circle all those you can identify with. Do they form a pattern for the behavior in your life? Are you comfortable with the implications of your analysis?\n  5. Start a collection of notes, quotes, and ideas you may want to use as resource material in writing your personal mission statement.\n  6. Identify a project you will be facing in the near future and apply the principle of mental creation. Write down the results you desire and what steps will lead to those results.\n  7. Share the principles of Habit 2 with your family or work group and suggest that together you begin the process of developing a family or group mission statement.\n\n* * *\n\n##### Footnote\n\n*Please refer to Appendix A.\n\n### HABIT 3  \nPUT FIRST THINGS FIRST\n\n### PRINCIPLES OF PERSONAL MANAGEMENT\n\nThings which matter most must never be  \nat the mercy of things which matter least.\n\nGoethe\n\nWILL YOU TAKE JUST A MOMENT and write down a short answer to the following two questions? Your answers will be important to you as you begin work on Habit 3.\n\nQuestion 1: What one thing could you do (you aren't doing now) that if you did on a regular basis, would make a tremendous positive difference in your personal life?\n\nQuestion 2: What one thing in your business or professional life would bring similar results?\n\nWe'll come back to these answers later. But first, let's put Habit 3 in perspective.\n\nHabit 3 is the personal fruit, the practical fulfillment of Habits 1 and 2.\n\nHabit 1 says, \"You're the creator. You are in charge.\" It's based on the four unique human endowments of imagination, conscience, independent will, and, particularly, self-awareness. It empowers you to say, \"That's an unhealthy program I've been given from my childhood, from my social mirror. I don't like that ineffective script. I can change.\"\n\nHabit 2 is the first or mental creation. It's based on imagination\u2014the ability to envision, to see the potential, to create with our minds what we cannot at present see with our eyes; and conscience\u2014the ability to detect our own uniqueness and the personal, moral, and ethical guidelines within which we can most happily fulfill it. It's the deep contact with our basic paradigms and values and the vision of what we can become.\n\nHabit 3, then, is the second creation, the physical creation. It's the fulfillment, the actualization, the natural emergence of Habits 1 and 2. It's the exercise of independent will toward becoming principle-centered. It's the day-in, day-out, moment-by-moment doing it.\n\nHabits 1 and 2 are absolutely essential and prerequisite to Habit 3. You can't become principle-centered without first being aware of and developing your own proactive nature. You can't become principle-centered without first being aware of your paradigms and understanding how to shift them and align them with princi\u00adples. You can't become principle-centered without a vision of and a focus on the unique contribution that is yours to make.\n\nBut with that foundation, you can become principle-centered, day-in and day-out, moment-by-moment, by living Habit 3\u2014by practicing effective self-management.\n\nManagement, remember, is clearly different from leadership. Leadership is primarily a high-powered, right brain activity. It's more of an art; it's based on a philosophy. You have to ask the ultimate questions of life when you're dealing with personal leadership issues.\n\nBut once you have dealt with those issues, once you have resolved them, you then have to manage yourself effectively to create a life congruent with your answers. The ability to manage well doesn't make much difference if you're not even in the \"right jungle.\" But if you are in the right jungle, it makes all the difference. In fact, the ability to manage well determines the quality and even the existence of the second creation. Management is the breaking down, the analysis, the sequencing, the specific application, the time-bound left-brain aspect of effective self-government. My own maxim of personal effectiveness is this: Manage from the left; lead from the right.\n\nTHE POWER OF INDEPENDENT WILL\n\nIn addition to self-awareness, imagination, and conscience, it is the fourth human endowment\u2014independent will\u2014that really makes effective self-management possible. It is the ability to make deci\u00adsions and choices and to act in accordance with them. It is the ability to act rather than to be acted upon, to proactively carry out the program we have developed through the other three endow\u00adments.\n\nThe human will is an amazing thing. Time after time, it has triumphed against unbelievable odds. The Helen Kellers of this world give dramatic evidence to the value, the power of the independent will.\n\nBut as we examine this endowment in the context of effective self-management, we realize it's usually not the dramatic, the visible, the once-in-a-lifetime, up-by-the-bootstraps effort that brings enduring success. Empowerment comes from learning how to use this great endowment in the decisions we make every day.\n\nThe degree to which we have developed our independent will in our everyday lives is measured by our personal integrity. Integrity is, fundamentally, the value we place on ourselves. It's our ability to make and keep commitments to ourselves, to \"walk our talk.\" It's honor with self, a fundamental part of the Character Ethic, the essence of proactive growth.\n\nEffective management is putting first things first. While leadership decides what \"first things\" are, it is management that puts them first, day-by-day, moment-by-moment. Management is discipline, carrying it out.\n\nDiscipline derives from disciple\u2014disciple to a philosophy, disci\u00adple to a set of principles, disciple to a set of values, disciple to an overriding purpose, to a superordinate goal or a person who represents that goal.\n\nIn other words, if you are an effective manager of your self, your discipline comes from within; it is a function of your independent will. You are a disciple, a follower, of your own deep values and their source. And you have the will, the integrity, to subordinate your feelings, your impulses, your moods to those values.\n\nOne of my favorite essays is \"The Common Denominator of Success,\" written by E. M. Gray. He spent his life searching for the one denominator that all successful people share. He found it wasn't hard work, good luck, or astute human relations, though those were all important. The one factor that seemed to transcend all the rest embodies the essence of Habit 3\u2014putting first things first.\n\n\"The successful person has the habit of doing the things failures don't like to do,\" he observed. \"They don't like doing them either necessarily. But their disliking is subordinated to the strength of their purpose.\"\n\nThat subordination requires a purpose, a mission, a Habit 2 clear sense of direction and value, a burning \"yes!\" inside that makes it possible to say \"no\" to other things. It also requires independent will, the power to do something when you don't want to do it, to be a function of your values rather than a function of the impulse or desire of any given moment. It's the power to act with integrity to your proactive first creation.\n\nFOUR GENERATIONS OF TIME MANAGEMENT\n\nIn Habit 3 we are dealing with many of the questions addressed in the field of life and time management. As a longtime student of this fascinating field, I am personally persuaded that the essence of the best thinking in the area of time management can be captured in a single phrase: Organize and execute around priorities. That phrase represents the evolution of three generations of time management theory, and how to best do it is the focus of a wide variety of approaches and materials.\n\nPersonal management has evolved in a pattern similar to many other areas of human endeavor. Major developmental thrusts, or \"waves\" as Alvin Toffler calls them, follow each other in succes\u00adsion, each adding a vital new dimension. For example, in social development, the agricultural revolution was followed by the industrial revolution, which was followed by the informational revolution. Each succeeding wave created a surge of social and personal progress.\n\nLikewise, in the area of time management, each generation builds on the one before it\u2014each one moves us toward greater control of our lives. The first wave or generation could be charac\u00adterized by notes and checklists, an effort to give some semblance of recognition and inclusiveness to the many demands placed on our time and energy.\n\nThe second generation could be characterized by calendars and appointment books. This wave reflects an attempt to look ahead, to schedule events and activities in the future.\n\nThe third generation reflects the current time management field. It adds to those preceding generations the important idea of prioritization, of clarifying values, and of comparing the relative worth of activities based on their relationship to those values. In addition, it focuses on setting goals\u2014specific long-, intermediate-, and short-term targets toward which time and energy would be directed in harmony with values. It also includes the concept of daily planning, of making a specific plan to accomplish those goals and activities determined to be of greatest worth.\n\nWhile the third generation has made a significant contribution, people have begun to realize that \"efficient\" scheduling and control of time are often counterproductive. The efficiency focus creates expectations that clash with the opportunities to develop rich relationships, to meet human needs, and to enjoy spontaneous moments on a daily basis.\n\nAs a result, many people have become turned off by time management programs and planners that make them feel too scheduled, too restricted, and they \"throw the baby out with the bath \"water,\" reverting to first or second generation techniques to preserve relationships, spontaneity, and quality of life.\n\nBut there is an emerging fourth generation that is different in kind. It recognizes that \"time management\" is really a misnomer\u2014the challenge is not to manage time, but to manage ourselves. Satisfaction is a function of expectation as well as realization. And expectation (and satisfaction) lie in our Circle of Influence.\n\nRather than focusing on things and time, fourth generation expectations focus on preserving and enhancing relationships and on accomplishing results\u2014in short, on maintaining the P\/PC Bal\u00adance.\n\nQUADRANT II\n\nThe essential focus of the fourth generation of management can be captured in the time management matrix diagrammed on the next page. Basically, we spend time in one of four ways.\n\nAs you can see, the two factors that define an activity are urgent and important. Urgent means it requires immediate attention. It's \"Now!\" Urgent things act on us. A ringing phone is urgent. Most people can't stand the thought of just allowing the phone to ring.\n\nYou could spend hours preparing materials, you could get all dressed up and travel to a person's office to discuss a particular issue, but if the phone were to ring while you were there, it would generally take precedence over your personal visit.\n\nIf you were to phone someone, there aren't many people who would say, \"I'll get to you in 15 minutes; just hold.\" But those same people would probably let you wait in an office for at least that long while they completed a telephone conversation with someone else.\n\nTHE TIME MANAGEMENT MATRIX\n\nUrgent matters are usually visible. They press on us; they insist on action. They're often popular with others. They're usually right in front of us. And often they are pleasant, easy, fun to do. But so often they are unimportant!\n\nImportance, on the other hand, has to do with results. If some\u00adthing is important, it contributes to your mission, your values, your high priority goals.\n\nWe react to urgent matters. Important matters that are not urgent require more initiative, more proactivity. We must act to seize opportunity, to make things happen. If we don't practice Habit 2, if we don't have a clear idea of what is important, of the results we desire in our lives, we are easily diverted into responding to the urgent.\n\nLook for a moment at the four quadrants in the time management matrix. Quadrant I is both urgent and important. It deals with significant results that require immediate attention. We usually call the activities in Quadrant I \"crises\" or \"problems.\" We all have some Quadrant I activities in our lives. But Quadrant I consumes many people. They are crisis managers, problem-minded people, deadline-driven producers.\n\nAs long as you focus on Quadrant I, it keeps getting bigger and bigger until it dominates you. It's like the pounding surf. A huge problem comes and knocks you down and you're wiped out. You struggle back up only to face another one that knocks you down and slams you to the ground.\n\nSome people are literally beaten up by problems all day every day. The only relief they have is in escaping to the not important, not urgent activities of Quadrant IV. So when you look at their total matrix, 90 percent of their time is in Quadrant I and most of the remaining 10 percent is in Quadrant IV, with only negligible attention paid to Quadrants II and III. That's how people who manage their lives by crisis live.\n\nThere are other people who spend a great deal of time in \"urgent, but not important\" Quadrant III, thinking they're in Quadrant I. They spend most of their time reacting to things that are urgent, assuming they are also important. But the reality is that the urgency of these matters is often based on the priorities and expectations of others.\n\nPeople who spend time almost exclusively in Quadrants III and IV basically lead irresponsible lives.\n\nEffective people stay out of Quadrants III and IV because, urgent or not, they aren't important. They also shrink Quadrant I down to size by spending more time in Quadrant II.\n\nQuadrant II is the heart of effective personal management. It deals with things that are not urgent, but are important. It deals with things like building relationships, writing a personal mission statement, long-range planning, exercising, preventive mainte\u00adnance, preparation\u2014all those things we know we need to do, but somehow seldom get around to doing, because they aren't urgent.\n\nTo paraphrase Peter Drucker, effective people are not problem-minded; they're opportunity-minded. They feed opportunities and starve problems. They think preventively. They have genuine Quadrant I crises and emergencies that require their immediate attention, but the number is comparatively small. They keep P and PC in balance by focusing on the important, but not urgent, high leverage capacity-building activities of Quadrant II.\n\nWith the time management matrix in mind, take a moment now and consider how you answered the questions at the beginning of this chapter. What quadrant do they fit in? Are they important? Are they urgent?\n\nMy guess is that they probably fit into Quadrant II. They are obviously important, deeply important, but not urgent. And because they aren't urgent, you don't do them.\n\nNow look again at the nature of those questions: What one thing could you do in your personal and professional life that, if you did on a regular basis, would make a tremendous positive difference in your life? Quadrant II activities have that kind of impact. Our effective\u00adness takes quantum leaps when we do them.\n\nI asked a similar question to a group of shopping center mana\u00adgers. \"If you were to do one thing in your professional work that you know would have enormously positive effects on the results, what would it be?\" Their unanimous response was to build helpful personal relationships with the tenants, the owners of the stores inside the shopping center, which is a Quadrant II activity.\n\nWe did an analysis of the time they were spending on that activity. It was less than 5 percent. They had good reasons\u2014problems, one right after another. They had reports to make out, meetings to go to, correspondence to answer, phone calls to make, constant interruptions. Quadrant I had consumed them.\n\nThey were spending very little time with the store managers, and the time they did spend was filled with negative energy. The only reason they visited the store managers at all was to enforce the contract\u2014to collect the money or discuss advertising or other practices that were out of harmony with center guidelines, or some similar thing.\n\nThe store owners were struggling for survival, let alone prosper\u00adity. They had employment problems, cost problems, inventory problems, and a host of other problems. Most of them had no training in management at all. Some were fairly good merchandis\u00aders, but they needed help. The tenants didn't even want to see the shopping center owners; they were just one more problem to contend with.\n\nSo the owners decided to be proactive. They determined their purpose, their values, their priorities. In harmony with those priorities, they decided to spend about one-third of their time in helping relationships with the tenants.\n\nIn working with that organization for about a year and a half, I saw them climb to around 20 percent, which represented more than a fourfold increase. In addition, they changed their role. They became listeners, trainers, consultants to the tenants. Their inter\u00adchanges were filled with positive energy.\n\nThe effect was dramatic, profound. By focusing on relationships and results rather than time and methods, the numbers went up, the tenants were thrilled with the results created by new ideas and skills, and the shopping center managers were more effective and satisfied and increased their list of potential tenants and lease revenue based on increased sales by the tenant stores. They were no longer policemen or hovering supervisors. They were problem solvers, helpers.\n\nWhether you are a student at the university, a worker in an assembly line, a homemaker, fashion designer, or president of a company, I believe that if you were to ask what lies in Quadrant II and cultivate the proactivity to go after it, you would find the same results. Your effectiveness would increase dramatically. Your crises and problems would shrink to manageable proportions because you would be thinking ahead, working on the roots, doing the preventive things that keep situations from developing into crises in the first place. In time management jargon, this is called the Pareto Principle\u201480 percent of the results flow out of 20 percent of the activities.\n\nWHAT IT TAKES TO SAY \"NO\"\n\nThe only place to get time for Quadrant II in the beginning is from Quadrants III and IV. You can't ignore the urgent and important activities of Quadrant I, although it will shrink in size as you spend more time with prevention and preparation in Quadrant II. But the initial time for Quadrant II has to come out of III and IV.\n\nYou have to be proactive to work on Quadrant II because Quadrants I and III work on you. To say \"yes\" to important Quadrant II priorities, you have to learn to say \"no\" to other activities, sometimes apparently urgent things.\n\nSome time ago, my wife was invited to serve as chairman of a committee in a community endeavor. She had a number of truly important things she was trying to work on, and she really didn't want to do it. But she felt pressured into it and finally agreed.\n\nThen she called one of her dear friends to ask if she would serve on her committee. Her friend listened for a long time and then said, \"Sandra, that sounds like a wonderful project, a really worthy undertaking. I appreciate so much your inviting me to be a part of it. I feel honored by it. For a number of reasons, I won't be participating myself, but I want you to know how much I appre\u00adciate your invitation.\"\n\nSandra was ready for anything but a pleasant \"no.\" She turned to me and sighed, \"I wish I'd said that.\"\n\nI don't mean to imply that you shouldn't be involved in significant service projects. Those things are important. But you have to decide what your highest priorities are and have the courage\u2014pleasantly, smilingly, nonapologetically\u2014to say \"no\" to other things. And the way you do that is by having a bigger \"yes\" burning inside. The enemy of the \"best\" is often the \"good.\"\n\nKeep in mind that you are always saying \"no\" to something. If it isn't to the apparent, urgent things in your life, it is probably to the more fundamental, highly important things. Even when the urgent is good, the good can keep you from your best, keep you from your unique contribution, if you let it.\n\nWhen I was Director of University Relations at a large university, I hired a very talented, proactive, creative writer. One day, after he had been on the job for a few months, I went into his office and asked him to work on some urgent matters that were pressing on me.\n\nHe said, \"Stephen, I'll do whatever you want me to do. Just let me share with you my situation.\"\n\nThen he took me over to his wallboard, where he had listed over two dozen projects he was working on, together with performance criteria and deadline dates that had been clearly negotiated before. He was highly disciplined, which is why I went to see him in the first place. \"If you want to get something done, give it to a busy man.\"\n\nThen he said, \"Stephen, to do the jobs that you want done right would take several days. Which of these projects would you like me to delay or cancel to satisfy your request?\"\n\nWell, I didn't want to take the responsibility for that. I didn't want to put a cog in the wheel of one of the most productive people on the staff just because I happened to be managing by crisis at the time. The jobs I wanted done were urgent, but not important. So I went and found another crisis manager and gave the job to him.\n\nWe say \"yes\" or \"no\" to things daily, usually many times a day. A center of correct principles and a focus on our personal mission empowers us with wisdom to make those judgments effectively.\n\nAs I work with different groups, I tell them that the essence of effective time and life management is to organize and execute around balanced priorities. Then I ask this question: if you were to fault yourself in one of three areas, which would it be: (1) the inability to prioritize; (2) the inability or desire to organize around those priorities; or (3) the lack of discipline to execute around them, to stay with your priorities and organization?\n\nMost people say their main fault is a lack of discipline. On deeper thought, I believe that is not the case. The basic problem is that their priorities have not become deeply planted in their hearts and minds. They haven't really internalized Habit 2.\n\nThere are many people who recognize the value of Quadrant II activities in their lives, whether they identify them as such or not. And they attempt to give priority to those activities and integrate them into their lives through self-discipline alone. But without a principle center and a personal mission statement, they don't have the necessary foundation to sustain their efforts. They're working on the leaves, on the attitudes and the behaviors of discipline, without even thinking to examine the roots, the basic paradigms from which their natural attitudes and behaviors flow.\n\nA Quadrant II focus is a paradigm that grows out of a principle center. If you are centered on your spouse, your money, your friends, your pleasure, or any extrinsic factor, you will keep getting thrown back into Quadrants I and III, reacting to the outside forces your life is centered on. Even if you're centered on yourself, you'll end up in I and III reacting to the impulse of the moment. Your independent will alone cannot effectively discipline you against your center.\n\nIn the words of the architectural maxim, form follows function. Likewise, management follows leadership. The way you spend your time is a result of the way you see your time and the way you really see your priorities. If your priorities grow out of a principle center and a personal mission, if they are deeply planted in your heart and in your mind, you will see Quadrant II as a natural, exciting place to invest your time.\n\nIt's almost impossible to say \"no\" to the popularity of Quadrant III or to the pleasure of escape to Quadrant IV if you don't have a bigger \"yes\" burning inside. Only when you have the self-awareness to examine your program\u2014and the imagination and conscience to create a new, unique, principle-centered program to which you can say \"yes\"\u2014only then will you have sufficient independent will power to say \"no,\" with a genuine smile, to the unimportant.\n\nMOVING INTO QUADRANT II\n\nIf Quadrant II activities are clearly the heart of effective personal management\u2014the \"first things\" we need to put first\u2014then how do we organize and execute around those things?\n\nThe first generation of time management does not even recog\u00adnize the concept of priority. It gives us notes and \"to do\" lists that we can cross off, and we feel a temporary sense of accomplishment every time we check something off, but no priority is attached to items on the list. In addition, there is no correlation between what's on the list and our ultimate values and purposes in life. We simply respond to whatever penetrates our awareness and appar\u00adently needs to be done.\n\nMany people manage from this first-generation paradigm. It's the course of least resistance. There's no pain or strain; it's fun to \"go with the flow.\" Externally imposed disciplines and schedules give people the feeling that they aren't responsible for results.\n\nBut first-generation managers, by definition, are not effective people. They produce very little, and their life-style does nothing to build their production capability. Buffeted by outside forces, they are often seen as undependable and irresponsible, and they have very little sense of control and self-esteem.\n\nSecond-generation managers assume a little more control. They plan and schedule in advance and generally are seen as more responsible because they \"show up\" when they're supposed to.\n\nBut again, the activities they schedule have no priority or recognized correlation to deeper values and goals. They have few significant achievements and tend to be schedule oriented.\n\nThird-generation managers take a significant step forward. They clarify their values and set goals. They plan each day and prioritize their activities.\n\nAs I have said, this is where most of the time management field is today. But this third generation has some critical limitations. First, it limits vision\u2014daily planning often misses important things that can only be seen from a larger perspective. The very language \"daily planning\" focuses on the urgent\u2014the \"now.\" While third generation prioritization provides order to activity, it doesn't question the essential importance of the activity in the first place\u2014it doesn't place the activity in the context of principles, personal mission, roles, and goals. The third-generation value-driven daily planning approach basically prioritizes the Quadrant I and III problems and crises of the day.\n\nIn addition, the third generation makes no provision for man\u00adaging roles in a balanced way. It lacks realism, creating the tendency to over-schedule the day, resulting in frustration and the desire to occasionally throw away the plan and escape to Quadrant IV. And its efficiency, time management focus tends to strain relationships rather than build them.\n\nWhile each of the three generations has recognized the value of some kind of management tool, none has produced a tool that empowers a person to live a principle-centered, Quadrant II life-style. The first-generation notepads and \"to do\" lists give us no more than a place to capture those things that penetrate our awareness so we won't forget them. The second-generation ap\u00adpointment books and calendars merely provide a place to record our future commitments so that we can be where we have agreed to be at the appropriate time.\n\nEven the third generation, with its vast array of planners and materials, focuses primarily on helping people prioritize and plan their Quadrants I and III activities. Though many trainers and consultants recognize the value of Quadrant II activities, the actual planning tools of the third generation do not facilitate organizing and executing around them.\n\nAs each generation builds on those that have preceded it, the strengths and some of the tools of each of the first three generations provide elemental material for the fourth. But there is an added need for a new dimension, for the paradigm and the implementa\u00adtion that will empower us to move into Quadrant II, to become principle-centered and to manage ourselves to do what is truly most important.\n\nTHE QUADRANT II TOOL\n\nThe objective of Quadrant II management is to manage our lives effectively\u2014from a center of sound principles, from a knowledge of our personal mission, with a focus on the important as well as the urgent, and within the framework of maintaining a balance between increasing our production and increasing our production capability.\n\nThis is, admittedly, an ambitious objective for people caught in the thick of thin things in Quadrants III and IV. But striving to achieve it will have a phenomenal impact on personal effectiveness.\n\nA Quadrant II organizer will need to meet six important criteria.\n\nCOHERENCE. Coherence suggests that there is harmony, unity, and integrity between your vision and mission, your roles and goals, your priorities and plans, and your desires and discipline. In your planner, there should be a place for your personal mission statement so that you can constantly refer to it. There also needs to be a place for your roles and for both short- and long-term goals.\n\nBALANCE. Your tool should help you to keep balance in your life, to identify your various roles and keep them right in front of you, so that you don't neglect important areas such as your health, your family, professional preparation, or personal development.\n\nMany people seem to think that success in one area can compensate for failure in other areas of life. But can it really? Perhaps it can for a limited time in some areas. But can success in your profession compensate for a broken marriage, ruined health, or weakness in personal character? True effectiveness requires balance, and your tool needs to help you create and maintain it.\n\nQUADRANT II FOCUS. You need a tool that encourages you, motivates you, actually helps you spend the time you need in Quadrant II, so that you're dealing with prevention rather than prioritizing crises. In my opinion, the best way to do this is to organize your life on a weekly basis. You can still adapt and prioritize on a daily basis, but the fundamental thrust is organizing the week.\n\nOrganizing on a weekly basis provides much greater balance and context than daily planning. There seems to be implicit cultural recognition of the week as a single, complete unit of time. Business, education, and many other facets of society operate within the framework of the week, designating certain days for focused investment and others for relaxation or inspiration. The basic Judeo-Christian ethic honors the Sabbath, the one day out of every seven set aside for uplifting purposes.\n\nMost people think in terms of weeks. But most third-generation planning tools focus on daily planning. While they may help you prioritize your activities, they basically only help you organize crises and busywork. The key is not to prioritize what's on your schedule, but to schedule your priorities. And this can best be done in the context of the week.\n\nA \"PEOPLE\" DIMENSION. You also need a tool that deals with people, not just schedules. While you can think in terms of efficiency in dealing with time, a principle-centered person thinks in terms of effectiveness in dealing with people. There are times when principle-centered Quadrant II living requires the subordination of schedules to people. Your tool needs to reflect that value, to facilitate implementation rather than create guilt when a schedule is not followed.\n\nFLEXIBILITY. Your planning tool should be your servant, never your master. Since it has to work for you, it should be tailored to your style, your needs, your particular ways.\n\nPORTABILITY. Your tool should also be portable, so that you can carry it with you most of the time. You may want to review your personal mission statement while riding the bus. You may want to measure the value of a new opportunity against something you already have planned. If your organizer is portable, you will keep it with you so that important data is always within reach.\n\nSince Quadrant II is the heart of effective self-management, you need a tool that moves you into Quadrant II. My work with the fourth-generation concept has led to the creation of a tool specifi\u00adcally designed according to the criteria listed above. But many good third-generation tools can easily be adapted. Because the principles are sound, the practices or specific applications can vary from one individual to the next.\n\nBECOMING A QUADRANT II SELF-MANAGER\n\nAlthough my effort here is to teach principles, not practices, of effectiveness, I believe you can better understand the principles and the empowering nature of the fourth generation if you actually experience organizing a week from a principle-centered, Quadrant II base.\n\nQuadrant II organizing involves four key activities.\n\nIDENTIFYING ROLES. The first task is to write down your key roles. If you haven't really given serious thought to the roles in your life, you can write down what immediately comes to mind. You have a role as an individual. You may want to list one or more roles as a family member\u2014a husband or wife, mother or father, son or daughter, a member of the extended family of grandparents, aunts, uncles, and cousins. You may want to list a few roles in your work, indicating different areas in which you wish to invest time and energy on a regular basis. You may have roles in church or community affairs.\n\nYou don't need to worry about defining the roles in a way that you will live with for the rest of your life\u2014just consider the week and write down the areas you see yourself spending time in during the next seven days.\n\nHere are two examples of the way people might see their various roles.\n\n  1. Individual\n  2. Spouse\/Parent\n  3. Manager New Products\n  4. Manager Research\n  5. Manager Staff Dev.\n  6. Manager Administration\n  7. Chairman United Way |   1. Personal Development\n  2. Spouse\n  3. Parent\n  4. Real Estate Salesperson\n  5. Community Service\n  6. Symphony Board Member\n\n---|---\n\nSELECTING GOALS. The next step is to think of one or two important results you feel you should accomplish in each role during the next seven days. These would be recorded as goals. (See next page.)\n\nAt least some of these goals should reflect Quadrant II activities. Ideally, these weekly goals would be tied to the longer-term goals you have identified in conjunction with your personal mission statement. But even if you haven't written your mission statement, you can get a feeling, a sense, of what is important as you consider each of your roles and one or two goals for each role.\n\nSCHEDULING. Now you can look at the week ahead with your goals in mind and schedule time to achieve them. For example, if your goal is to produce the first draft of your personal mission statement, you may want to set aside a two-hour block of time on Sunday to work on it. Sunday (or some other day of the week that is special to you, your faith, or your circumstances) is often the ideal time to plan your more personally uplifting activities, including weekly organizing. It's a good time to draw back, to seek inspiration, to look at your life in the context of principles and values.\n\nIf you set a goal to become physically fit through exercise, you may want to set aside an hour three or four days during the week, or possibly every day during the week, to accomplish that goal. There are some goals that you may only be able to accomplish during business hours, or some that you can only do on Saturday when your children are home. Can you begin to see some of the advantages of organizing the week instead of the day?\n\nHaving identified roles and set goals, you can translate each goal to a specific day of the week, either as a priority item or, even better, as a specific appointment. You can also check your annual or monthly calendar for any appointments you may have previ\u00adously made and evaluate their importance in the context of your goals, transferring those you decide to keep to your schedule and making plans to reschedule or cancel others.\n\nAs you study the following weekly schedule, observe how each of the nineteen most important, often Quadrant II, goals has been scheduled or translated into a specific action plan. In addition, notice the box labeled \"Sharpen the Saw\" that provides a place to plan vital renewing Quadrant II activities in each of the four human dimensions that will be explained in Habit 7.\n\nEven with time set aside to accomplish 19 important goals during the week, look at the amount of remaining unscheduled space on the schedule! As well as empowering you to put first things first, Quadrant II weekly organizing gives you the freedom and the flexibility to handle unanticipated events, to shift appointments if you need to, to savor relationships and interactions with others, to deeply enjoy spontaneous experiences, knowing that you have proactively organized your week to accomplish key goals in every area of your life.\n\nDAILY ADAPTING. With Quadrant II weekly organizing, daily plan\u00adning becomes more a function of daily adapting, of prioritizing activities and responding to unanticipated events, relationships, and experiences in a meaningful way.\n\nTaking a few minutes each morning to review your schedule can put you in touch with the value-based decisions you made as you organized the week as well as unanticipated factors that may have come up. As you overview the day, you can see that your roles and goals provide a natural prioritization that grows out of your innate sense of balance. It is a softer, more right-brain prioritization that ultimately comes out of your sense of personal mission.\n\nYou may still find that the third-generation A, B, C or 1, 2, 3 prioritization gives needed order to daily activities. It would be a false dichotomy to say that activities are either important or they aren't. They are obviously on a continuum, and some important activities are more important than others. In the context of weekly organizing, third-generation prioritization gives order to daily focus.\n\nBut trying to prioritize activities before you even know how they relate to your sense of personal mission and how they fit into the balance of your life is not effective. You may be prioritizing and accomplishing things you don't want or need to be doing at all.\n\nCan you begin to see the difference between organizing your week as a principle-centered, Quadrant II manager and planning your days as an individual centered on something else? Can you begin to sense the tremendous difference the Quadrant II focus would make in your current level of effectiveness?\n\nHaving experienced the power of principle-centered Quadrant II organizing in my own life and having seen it transform the lives of hundreds of other people, I am persuaded it makes a difference\u2014a quantum positive difference. And the more completely weekly goals are tied into a wider framework of correct principles and into a personal mission statement, the greater the increase in effective\u00adness will be.\n\nLIVING IT\n\nReturning once more to the computer metaphor, if Habit 1 says \"You're the programmer\" and Habit 2 says \"Write the program,\" then Habit 3 says \"Run the program,\" \"Live the program.\" And living it is primarily a function of our independent will, our self-discipline, our integrity, and commitment\u2014not to short-term goals and schedules or to the impulse of the moment, but to the correct principles and our own deepest values, which give meaning and context to our goals, our schedules, and our lives.\n\nAs you go through your week, there will undoubtedly be times when your integrity will be placed on the line. The popularity of reacting to the urgent but unimportant priorities of other people in Quadrant III or the pleasure of escaping to Quadrant IV will threaten to overpower the important Quadrant II activities you have planned. Your principle center, your self-awareness, and your conscience can provide a high degree of intrinsic security, guidance, and wisdom to empower you to use your independent will and maintain integrity to the truly important.\n\nBut because you aren't omniscient, you can't always know in advance what is truly important. As carefully as you organize the week, there will be times when, as a principle-centered person, you will need to subordinate your schedule to a higher value. Because you are principle-centered, you can do that with an inner sense of peace.\n\nAt one point, one of my sons was deeply into scheduling and efficiency. One day he had a very tight schedule, which included down-to-the-minute time allocations for every activity, including picking up some books, washing his car, and \"dropping\" Carol, his girlfriend, among other things.\n\nEverything went according to schedule until it came to Carol. They had been dating for a long period of time, and he had finally come to the conclusion that a continued relationship would not work out. So, congruent with his efficiency model, he had sched\u00aduled a ten- to fifteen-minute telephone call to tell her.\n\nBut the news was very traumatic to her. One-and-a-half hours later, he was still deeply involved in a very intense conversation with her. Even then, the one visit was not enough. The situation was a very frustrating experience for them both.\n\nAgain, you simply can't think efficiency with people. You think effectiveness with people and efficiency with things. I've tried to be \"efficient\" with a disagreeing or disagreeable person and it simply doesn't work. I've tried to give ten minutes of \"quality time\" to a child or an employee to solve a problem, only to discover such \"efficiency\" creates new problems and seldom resolves the deepest concern.\n\nI see many parents, particularly mothers with small children, often frustrated in their desire to accomplish a lot because all they seem to do is meet the needs of little children all day. Remember, frustration is a function of our expectations, and our expectations are often a reflection of the social mirror rather than our own values and priorities.\n\nBut if you have Habit 2 deep inside your heart and mind, you have those higher values driving you. You can subordinate your schedule to those values with integrity. You can adapt; you can be flexible. You don't feel guilty when you don't meet your schedule or when you have to change it.\n\nADVANCES OF THE FOURTH GENERATION\n\nOne of the reasons why people resist using third-generation time management tools is because they lose spontaneity; they become rigid and inflexible. They subordinate people to schedules because the efficiency paradigm of the third generation of manage\u00adment is out of harmony with the principle that people are more important than things.\n\nThe fourth-generation tool recognizes that principle. It also recognizes that the first person you need to consider in terms of effectiveness rather than efficiency is yourself. It encourages you to spend time in Quadrant II, to understand and center your life on principles, to give clear expression to the purposes and values you want to direct your daily decisions. It helps you to create balance in your life. It helps you rise above the limitations of daily planning and organize and schedule in the context of the week. And when a higher value conflicts with what you have planned, it empowers you to use your self-awareness and your conscience to maintain integrity to the principles and purposes you have determined are most important. Instead of using a road map, you're using a compass.\n\nThe fourth generation of self-management is more advanced than the third in five important ways.\n\nFirst, it's principle-centered. More than giving lip service to Quad\u00adrant II, it creates the central paradigm that empowers you to see your time in the context of what is really important and effective.\n\nSecond, it's conscience-directed. It gives you the opportunity to organize your life to the best of your ability in harmony with your deepest values. But it also gives you the freedom to peacefully subordinate your schedule to higher values.\n\nThird, it defines your unique mission, including values and long-term goals. This gives direction and purpose to the way you spend each day.\n\nFourth, it helps you balance your life by identifying roles, and by setting goals and scheduling activities in each key role every week.\n\nAnd fifth, it gives greater context through weekly organizing (with daily adaptation as needed), rising above the limiting perspective of a single day and putting you in touch with your deepest values through review of your key roles.\n\nThe practical thread running through all five of these advances is a primary focus on relationships and results and a secondary focus on time.\n\nDELEGATION: INCREASING P AND PC\n\nWe accomplish all that we do through delegation\u2014either to time or to other people. If we delegate to time, we think efficiency. If we delegate to other people, we think effectiveness.\n\nMany people refuse to delegate to other people because they feel it takes too much time and effort and they could do the job better themselves. But effectively delegating to others is perhaps the single most powerful high-leverage activity there is.\n\nTransferring responsibility to other skilled and trained people enables you to give your energies to other high-leverage activities. Delegation means growth, both for individuals and for organiza\u00adtions. The late J. C. Penney was quoted as saying that the wisest decision he ever made was to \"let go\" after realizing that he couldn't do it all by himself any longer. That decision, made long ago, enabled the development and growth of hundreds of stores and thousands of people.\n\nBecause delegation involves other people, it is a Public Victory and could well be included in Habit 4. But because we are focusing here on principles of personal management, and the ability to delegate to others is the main difference between the role of manager and independent producer, I am approaching delegation from the standpoint of your personal managerial skills.\n\nA producer does whatever is necessary to accomplish desired results, to get the golden eggs. A parent who washes the dishes, an architect who draws up blueprints, or a secretary who types correspondence is a producer.\n\nBut when a person sets up and works with and through people and systems to produce golden eggs, that person becomes a manager in the interdependent sense. A parent who delegates washing the dishes to a child is a manager. An architect who heads a team of other architects is a manager. A secretary who supervises other secretaries and office personnel is an office manager.\n\nA producer can invest one hour of effort and produce one unit of results, assuming no loss of efficiency.\n\nA manager, on the other hand, can invest one hour of effort and produce ten or fifty or a hundred units through effective delegation.\n\nManagement is essentially moving the fulcrum over, and the key to effective management is delegation.\n\nGOFER DELEGATION\n\nThere are basically two kinds of delegation: \"gofer delegation\" and \"stewardship delegation.\" Gofer delegation means \"Go for this, go for that, do this, do that, and tell me when it's done.\" Most people who are producers have a gofer delegation paradigm. Remember the machete wielders in the jungle? They are the producers. They roll up their sleeves and get the job done. If they are given a position of supervision or management, they still think like producers. They don't know how to set up a full delegation so that another person is committed to achieve results. Because they are focused on methods, they become responsible for the results.\n\nI was involved in a gofer delegation once when our family went water skiing. My son, who is an excellent skier, was in the water being pulled and I was driving the boat. I handed the camera to Sandra and asked her to take some pictures.\n\nAt first, I told her to be selective in her picture taking because we didn't have much film left. Then I realized she was unfamiliar with the camera, so I became a little more specific. I told her to be sure to wait until the sun was ahead of the boat and until our son was jumping the wake or making a turn and touching his elbow.\n\nBut the more I thought about our limited footage and her inexperience with the camera, the more concerned I became. I finally said, \"Look, Sandra, just push the button when I tell you. Okay?\" And I spent the next few minutes yelling, \"Take it!\u2014Take it!\u2014Don't take it!\u2014Don't take it!\" I was afraid that if I didn't direct her every move every second, it wouldn't be done right.\n\nThat was true gofer delegation, one-on-one supervision of methods. Many people consistently delegate that way. But how much does it really accomplish? And how many people is it possible to supervise or manage when you have to be involved in every move they make?\n\nThere's a much better way, a more effective way to delegate to other people. And it's based on a paradigm of appreciation of the self-awareness, the imagination, the conscience, and the free will of other people.\n\nSTEWARDSHIP DELEGATION\n\nStewardship delegation is focused on results instead of methods. It gives people a choice of method and makes them responsible for results. It takes more time in the beginning, but it's time well invested. You can move the fulcrum over, you can increase your leverage, through stewardship delegation.\n\nStewardship delegation involves clear, up-front mutual under\u00adstanding and commitment regarding expectations in five areas.\n\nDESIRED RESULTS. Create a clear, mutual understanding of what needs to be accomplished, focusing on what, not how; results, not methods. Spend time. Be patient. Visualize the desired result. Have the person see it, describe it, make out a quality statement of what the results will look like, and by when they will be accomplished.\n\nGUIDELINES. Identify the parameters within which the individual should operate. These should be as few as possible to avoid methods delegation, but should include any formidable restric\u00adtions. You wouldn't want a person to think he had considerable latitude as long as he accomplished the objectives, only to violate some long-standing traditional practice or value. That kills initiative and sends people back to the gofer's creed: \"Just tell me what you want me to do, and I'll do it.\"\n\nIf you know the failure paths of the job, identify them. Be honest and open\u2014tell a person where the quicksand is and where the wild animals are. You don't want to have to reinvent the wheel every day. Let people learn from your mistakes or the mistakes of others. Point out the potential failure paths, what not to do, but don't tell them what to do. Keep the responsibility for results with them\u2014to do whatever is necessary within the guidelines.\n\nRESOURCES. Identify the human, financial, technical, or organiza\u00adtional resources the person can draw on to accomplish the desired results.\n\nACCOUNTABILITY. Set up the standards of performance that will be used in evaluating the results and the specific times when reporting and evaluation will take place.\n\nCONSEQUENCES. Specify what will happen, both good and bad, as a result of the evaluation. This could include such things as financial rewards, psychic rewards, different job assignments, and natural consequences tied into the overall mission of an organization.\n\nSome years ago, I had an interesting experience in delegation with one of my sons. We were having a family meeting, and we had our mission statement up on the wall to make sure our plans were in harmony with our values. Everybody was there.\n\nI set up a big blackboard and we wrote down our goals\u2014the key things we wanted to do\u2014and the jobs that flowed out of those goals. Then I asked for volunteers to do the job.\n\n\"Who wants to pay the mortgage?\" I asked. I noticed I was the only one with my hand up.\n\n\"Who wants to pay for the insurance? The food? The cars?\" I seemed to have a real monopoly on the opportunities.\n\n\"Who wants to feed the new baby?\" There was more interest here, but my wife was the only one with the right qualifications for the job.\n\nAs we went down the list, job by job, it was soon evident that Mom and Dad had more than sixty-hour work weeks. With that paradigm in mind, some of the other jobs took on a more proper perspective.\n\nMy seven-year-old son, Stephen, volunteered to take care of the yard. Before I actually gave him the job, I began a thorough training process. I wanted him to have a clear picture in his mind of what a well cared for yard was like, so I took him next door to our neighbor's.\n\n\"Look, son,\" I said. \"See how our neighbor's yard is green and clean? That's what we're after: green and clean. Now come look at our yard. See the mixed colors? That's not it; that's not green. Green and clean is what we want. Now how you get it green is up to you. You're free to do it any way you want, except paint it. But I'll tell you how I'd do it if it were up to me.\"\n\n\"How would you do it, Dad?\"\n\n\"I'd turn on the sprinklers. But you may want to use buckets or a hose. It makes no difference to me. All we care about is that the color is green. Okay?\"\n\n\"Okay.\"\n\n\"Now let's talk about 'clean,' Son. Clean means no messes around\u2014no paper, strings, bones, sticks, or anything that messes up the place. I'll tell you what let's do. Let's just clean up half the yard right now and look at the difference.\"\n\nSo we got out two paper sacks and picked up one side of the yard. \"Now look at this side. Look at the other side. See the difference? That's called clean.\"\n\n\"Wait!\" he called. \"I see some paper behind that bush!\"\n\n\"Oh, good! I didn't notice that newspaper back there. You have good eyes, Son.\n\n\"Now before you decide whether or not you're going to take the job, let me tell you a few more things. Because when you take the job, I don't do it anymore. It's your job. It's called a stewardship. Stewardship means 'a job with a trust.' I trust you to do the job, to get it done. Now who's going to be your boss?\"\n\n\"You, Dad?\"\n\n\"No, not me. You're the boss. You boss yourself. How do you like Mom and Dad nagging you all the time?\"\n\n\"I don't.\"\n\n\"We don't like doing it either. It sometimes causes a bad feeling, doesn't it? So you boss yourself. Now, guess who your helper is.\"\n\n\"Who?\"\n\n\"I am,\" I said. \"You boss me.\"\n\n\"I do?\"\n\n\"That's right. But my time to help is limited. Sometimes I'm away. But when I'm here, you tell me how I can help. I'll do anything you want me to do.\"\n\n\"Okay!\"\n\n\"Now guess who judges you.\"\n\n\"Who?\"\n\n\"You judge yourself.\"\n\n\"I do?\"\n\n\"That's right. Twice a week the two of us will walk around the yard, and you can show me how it's coming. How are you going to judge?\"\n\n\"Green and clean.\"\n\n\"Right!\"\n\nI trained him with those two words for two weeks before I felt he was ready to take the job. Finally, the big day came.\n\n\"Is it a deal, Son?\"\n\n\"It's a deal.\"\n\n\"What's the job?\"\n\n\"Green and clean.\"\n\n\"What's green?\"\n\nHe looked at our yard, which was beginning to look better. Then he pointed next door. \"That's the color of his yard.\"\n\n\"What's clean?\"\n\n\"No messes.\"\n\n\"Who's the boss?\"\n\n\"I am.\"\n\n\"Who's your helper?\"\n\n\"You are, when you have time.\"\n\n\"Who's the judge?\"\n\n\"I am. We'll walk around two times a week and I can show you how it's coming.\"\n\n\"And what will we look for?\"\n\n\"Green and clean.\"\n\nAt that time I didn't mention an allowance. But I wouldn't hesitate to attach an allowance to such a stewardship.\n\nTwo weeks and two words. I thought he was ready.\n\nIt was Saturday. And he did nothing. Sunday... nothing. Monday... nothing. As I pulled out of the driveway on my way to work on Tuesday, I looked at the yellow, cluttered yard and the hot July sun on its way up. \"Surely he'll do it today,\" I thought. I could rationalize Saturday because that was the day we made the agreement. I could rationalize Sunday; Sunday was for other things. But I couldn't rationalize Monday. And now it was Tues\u00adday. Certainly he'd do it today. It was summertime. What else did he have to do?\n\nAll day I could hardly wait to return home to see what happened. As I rounded the corner, I was met with the same picture I left that morning. And there was my son at the park across the street playing.\n\nThis was not acceptable. I was upset and disillusioned by his performance after two weeks of training and all those commit\u00adments. We had a lot of effort, pride, and money invested in the yard and I could see it going down the drain. Besides, my neighbor's yard was manicured and beautiful, and the situation was beginning to get embarrassing.\n\nI was ready to go back to gofer delegation. Son, you get over here and pick up this garbage right now or else! I knew I could get the golden egg that way. But what about the goose? What would happen to his internal commitment?\n\nSo I faked a smile and yelled across the street, \"Hi, Son. How's it going?\"\n\n\"Fine!\" he returned.\n\n\"How's the yard coming?\" I knew the minute I said it I had broken our agreement. That's not the way we had set up an accounting. That's not what we had agreed.\n\nSo he felt justified in breaking it, too. \"Fine, Dad.\"\n\nI bit my tongue and waited until after dinner. Then I said, \"Son, let's do as we agreed. Let's walk around the yard together and you can show me how it's going in your stewardship.\"\n\nAs we started out the door, his chin began to quiver. Tears welled up in his eyes and, by the time we got out to the middle of the yard, he was whimpering.\n\n\"It's so hard, Dad!\"\n\nWhat's so hard? I thought to myself. You haven't done a single thing! But I knew what was hard\u2014self-management, self-supervision. So I said, \"Is there anything I can do to help?\"\n\n\"Would you, Dad?\" he sniffed.\n\n\"What was our agreement?\"\n\n\"You said you'd help me if you had time.\"\n\n\"I have time.\"\n\nSo he ran into the house and came back with two sacks. He handed me one. \"Will you pick that stuff up?\" He pointed to the garbage from Saturday night's barbecue. \"It makes me sick!\"\n\nSo I did. I did exactly what he asked me to do. And that was when he signed the agreement in his heart. It became his yard, his stewardship.\n\nHe only asked for help two or three more times that entire summer. He took care of that yard. He kept it greener and cleaner than it had ever been under my stewardship. He even reprimanded his brothers and sisters if they left so much as a gum wrapper on the lawn.\n\nTrust is the highest form of human motivation. It brings out the very best in people. But it takes time and patience, and it doesn't preclude the necessity to train and develop people so that their competency can rise to the level of that trust.\n\nI am convinced that if stewardship delegation is done correctly, both parties will benefit and ultimately much more work will get done in much less time. I believe that a family that is well organized, whose time has been spent effectively delegating on a one-on-one basis, can organize the work so that everyone can do everything in about an hour a day. But that takes the internal capacity to want to manage, not just to produce. The focus is on effectiveness, not efficiency.\n\nCertainly you can pick up that room better than a child, but the key is that you want to empower the child to do it. It takes time. You have to get involved in the training and development. It takes time, but how valuable that time is downstream! It saves you so much in the long run.\n\nThis approach involves an entirely new paradigm of delegation. In effect, it changes the nature of the relationship: The steward becomes his own boss, governed by a conscience that contains the commitment to agreed upon desired results. But it also releases his creative energies toward doing whatever is necessary in harmony with correct principles to achieve those desired results.\n\nThe principles involved in stewardship delegation are correct and applicable to any kind of person or situation. With immature people, you specify fewer desired results and more guidelines, identify more resources, conduct more frequent accountability interviews, and apply more immediate consequences. With more mature people, you have more challenging desired results, fewer guidelines, less frequent accountability, and less measurable but more discernable criteria.\n\nEffective delegation is perhaps the best indicator of effective management simply because it is so basic to both personal and organizational growth.\n\nTHE QUADRANT II PARADIGM\n\nThe key to effective management of self, or of others through delegation, is not in any technique or tool or extrinsic factor. It is intrinsic\u2014in the Quadrant II paradigm that empowers you to see through the lens of importance rather than urgency.\n\nI have included in the Appendix an exercise called \"A Quadrant II Day at the Office\" which will enable you to see in a busi\u00adness setting how powerfully this paradigm can impact your effectiveness.*\n\nAs you work to develop a Quadrant II paradigm, you will increase your ability to organize and execute every week of your life around your deepest priorities, to walk your talk. You will not be dependent on any other person or thing for the effective management of your life.\n\nInterestingly, every one of the Seven Habits is in Quadrant II. Every one deals with fundamentally important things that, if done on a regular basis, would make a tremendous positive difference in our lives.\n\nAPPLICATION SUGGESTIONS:\n\n  1. Identify a Quadrant II activity you know has been neglected in your life\u2014one that, if done well, would have a significant impact in your life, either personally or professionally. Write it down and commit to implement it.\n  2. Draw a time management matrix and try to estimate what percentage of your time you spend in each quadrant. Then log your time for three days in fifteen-minute intervals. How accurate was your estimate? Are you satisfied with the way you spend your time? What do you need to change?\n  3. Make a list of responsibilities you could delegate and the people you could delegate to or train to be responsible in these areas. Determine what is needed to start the process of delegation or training.\n  4. Organize your next week.* Start by writing down your roles and goals for the week, then transfer the goals to a specific action plan. At the end of the week, evaluate how well your plan translated your deep values and purposes into your daily life and the degree of integrity you were able to maintain to those values and purposes.\n  5. Commit yourself to start organizing on a weekly basis and set up a regular time to do it.\n  6. Either convert your current planning tool into a fourth generation tool or secure such a tool.\n  7. Go through \"A Quadrant II Day at the Office\" (Appendix B) for a more in-depth understanding of the impact of a Quadrant II paradigm.\n\n* * *\n\n##### Footnote\n\n*Please refer to Appendix B.\n\n*On the previous pages is a sample weekly schedule from the Seven Habits Organizer. If you would like samples of these schedules (which you can adapt to your current system), please call 1-800-255-0777 or visit our Internet home page at http:\/\/www.franklincovey.com. These schedules are also a feature of Microsoft Schedule + with Seven Habits.\n\n## _PART THREE_\n\n# PUBLIC VICTORY\n\n### Paradigms of Interdependence\n\nThere can be no friendship without confidence, and no confidence without integrity.\n\nSamuel Johnson\n\nBEFORE MOVING INTO THE AREA OF PUBLIC VICTORY, we should remember that effective interdependence can only be built on a founda\u00adtion of true independence. Private Victory precedes Public Victory. Algebra comes before calculus.\n\nAs we look back and survey the terrain to determine where we've been and where we are in relationship to where we're going, we clearly see that we could not have gotten where we are without coming the way we came. There aren't any other roads; there aren't any shortcuts. There's no way to parachute into this terrain. The landscape ahead is covered with the fragments of broken relationships of people who have tried. They've tried to jump into effective relationships without the maturity, the strength of char\u00adacter, to maintain them.\n\nBut you just can't do it; you simply have to travel the road. You can't be successful with other people if you haven't paid the price of success with yourself.\n\nA few years ago when I was giving a seminar on the Oregon coast, a man came up to me and said, \"You know, Stephen, I really don't enjoy coming to these seminars.\" He had my attention.\n\n\"Look at everyone else here,\" he continued. \"Look at this beautiful coastline and the sea out there and all that's happening. And all I can do is sit and worry about the grilling I'm going to get from my wife tonight on the phone.\n\n\"She gives me the third degree every time I'm away. Where did I eat breakfast? Who else was there? Was I in meetings all morning? When did we stop for lunch? What did I do during lunch? How did I spend the afternoon? What did I do for entertainment in the evening? Who was with me? What did we talk about?\n\n\"And what she really wants to know, but never quite asks, is who she can call to verify everything I tell her. She just nags me and questions everything I do whenever I'm away. It's taken the bloom out of this whole experience. I really don't enjoy it at all.\"\n\nHe did look pretty miserable. We talked for a while, and then he made a very interesting comment. \"I guess she knows all the questions to ask,\" he said a little sheepishly. \"It was at a semi\u00adnar like this that I met her... when I was married to someone else!\"\n\nI considered the implications of his comment and then said, \"You're kind of into 'quick fix,' aren't you?\"\n\n\"What do you mean?\" he replied.\n\n\"Well, you'd like to take a screwdriver and just open up your wife's head and rewire that attitude of hers really fast, wouldn't you?\"\n\n\"Sure, I'd like her to change,\" he exclaimed. \"I don't think it's right for her to constantly grill me like she does.\"\n\n\"My friend,\" I said, \"you can't talk your way out of problems you behave yourself into.\"\n\nWe're dealing with a very dramatic and very fundamental paradigm shift here. You may try to lubricate your social interac\u00adtions with personality techniques and skills, but in the process, you may truncate the vital character base. You can't have the fruits without the roots. It's the principle of sequencing: Private Victory precedes Public Victory. Self-mastery and self-discipline are the foundation of good relationships with others.\n\nSome people say that you have to like yourself before you can like others. I think that idea has merit, but if you don't know yourself, if you don't control yourself, if you don't have mastery over yourself, it's very hard to like yourself, except in some short-term, psych-up, superficial way.\n\nReal self-respect comes from dominion over self, from true independence. And that's the focus of Habits 1, 2, and 3. Inde\u00adpendence is an achievement. Interdependence is a choice only independent people can make. Unless we are willing to achieve real independence, it's foolish to try to develop human relations skills. We might try. We might even have some degree of success when the sun is shining. But when the difficult times come\u2014and they will\u2014we won't have the foundation to keep things together.\n\nThe most important ingredient we put into any relationship is not what we say or what we do, but what we are. And if our words and our actions come from superficial human relations techniques (the Personality Ethic) rather than from our own inner core (the Character Ethic), others will sense that duplicity. We simply won't be able to create and sustain the foundation necessary for effective interdependence.\n\nThe techniques and skills that really make a difference in human interaction are the ones that almost naturally flow from a truly independent character. So the place to begin building any relation\u00adship is inside ourselves, inside our Circle of Influence, our own character. As we become independent\u2014proactive, centered in correct principles, value driven and able to organize and execute around the priorities in our life with integrity\u2014we then can choose to become interdependent\u2014capable of building rich, enduring, highly productive relationships with other people.\n\nAs we look at the terrain ahead, we see that we're entering a whole new dimension. Interdependence opens up worlds of possibilities for deep, rich, meaningful associations, for geometri\u00adcally increased productivity, for serving, for contributing, for learning, for growing. But it is also where we feel the greatest pain, the greatest frustration, the greatest roadblocks to happiness and success. And we're very aware of that pain because it is acute.\n\nWe can often live for years with the chronic pain of our lack of vision, leadership or management in our personal lives. We feel vaguely uneasy and uncomfortable and occasionally take steps to ease the pain, at least for a time. Because the pain is chronic, we get used to it, we learn to live with it.\n\nBut when we have problems in our interactions with other people, we're very aware of acute pain\u2014it's often intense, and we want it to go away.\n\nThat's when we try to treat the symptoms with quick fixes and techniques\u2014the band-aids of the Personality Ethic. We don't understand that the acute pain is an outgrowth of the deeper, chronic problem. And until we stop treating the symptoms and start treating the problem, our efforts will only bring counterpro\u00adductive results. We will only be successful at obscuring the chronic pain even more.\n\nNow, as we think of effective interaction with others, let's go back to our earlier definition of effectiveness. We've said it's the P\/PC balance, the fundamental concept in the story of the goose and the golden egg.\n\nIn an interdependent situation, the golden eggs are the effec\u00adtiveness, the wonderful synergy, the results created by open communication and positive interaction with others. And to get those eggs on a regular basis, we need to take care of the goose. We need to create and care for the relationships that make those results realities.\n\nSo before we descend from our point of reconnaissance and get into Habits 4, 5, and 6, I would like to introduce what I believe to be a very powerful metaphor in describing relationships and in defining the P\/PC balance in an interdependent reality.\n\nTHE EMOTIONAL BANK ACCOUNT\n\nWe all know what a financial bank account is. We make deposits into it and build up a reserve from which we can make withdrawals when we need to. An Emotional Bank Account is a metaphor that describes the amount of trust that's been built up in a relationship. It's the feeling of safeness you have with another human being.\n\nIf I make deposits into an Emotional Bank Account with you through courtesy, kindness, honesty, and keeping my commit\u00adments to you, I build up a reserve. Your trust toward me becomes higher, and I can call upon that trust many times if I need to. I can even make mistakes and that trust level, that emotional reserve, will compensate for it. My communication may not be clear, but you'll get my meaning anyway. You won't make me \"an offender for a word.\" When the trust account is high, communication is easy, instant, and effective.\n\nBut if I have a habit of showing discourtesy, disrespect, cutting you off, overreacting, ignoring you, becoming arbitrary, betraying your trust, threatening you, or playing little tin god in your life, eventually my Emotional Bank Account is overdrawn. The trust level gets very low. Then what flexibility do I have?\n\nNone. I'm walking on mine fields. I have to be very careful of everything I say. I measure every word. It's tension city, memo haven. It's protecting my backside, politicking. And many organi\u00adzations are filled with it. Many families are filled with it. Many marriages are filled with it.\n\nIf a large reserve of trust is not sustained by continuing deposits, a marriage will deteriorate. Instead of rich, spontaneous understanding and communication, the situation becomes one of accom\u00admodation, where two people simply attempt to live independent life-styles in a fairly respectful and tolerant way. The relationship may further deteriorate to one of hostility and defensiveness. The \"fight or flight\" response creates verbal battles, slammed doors, refusal to talk, emotional withdrawal and self-pity. It may end up in a cold war at home, sustained only by children, sex, social pressure, or image protection. Or it may end up in open warfare in the courts, where bitter ego decimating legal battles can be carried on for years as people endlessly confess the sins of a former spouse.\n\nAnd this is in the most intimate, the most potentially rich, joyful, satisfying and productive relationship possible between two peo\u00adple on this earth. The P\/PC lighthouse is there; we can either break ourselves against it or we can use it as a guiding light.\n\nOur most constant relationships, like marriage, require our most constant deposits. With continuing expectations, old deposits evaporate. If you suddenly run into an old high school friend you haven't seen for years, you can pick up right where you left off because the earlier deposits are still there. But your accounts with the people you interact with on a regular basis require more constant investment. There are sometimes automatic withdrawals in your daily interactions or in their perception of you that you don't even know about. This is especially true with teenagers in the home.\n\nSuppose you have a teenage son and your normal conversation is something like, \"Clean your room. Button your shirt. Turn down the radio. Go get a haircut. And don't forget to take out the garbage!\" Over a period of time, the withdrawals far exceed the deposits.\n\nNow, suppose this son is in the process of making some important decisions that will affect the rest of his life. But the trust level is so low and the communication process so closed, mechan\u00adical, and unsatisfying that he simply will not be open to your counsel. You may have the wisdom and the knowledge to help him, but because your account is so overdrawn, he will end up making his decisions from a short-range emotional perspective, which may well result in many negative long-range consequences.\n\nYou need a positive balance to communicate on these tender issues. What do you do?\n\nWhat would happen if you started making deposits into the relationship? Maybe the opportunity comes up to do him a little kindness\u2014to bring home a magazine on skateboarding, if that's his interest, or just to walk up to him when he's working on a project and offer to help. Perhaps you could invite him to go to a movie with you or take him out for some ice cream. Probably the most important deposit you could make would be just to listen, without judging or preaching or reading your own autobiography into what he says. Just listen and seek to understand. Let him feel your concern for him, your acceptance of him as a person.\n\nHe may not respond at first. He may even be suspicious. \"What's Dad up to now? What technique is Mom trying on me this time?\" But as those genuine deposits keep coming, they begin to add up. That overdrawn balance is shrinking.\n\nRemember that quick fix is a mirage. Building and repairing relationships takes time. If you become impatient with his apparent lack of response or his seeming ingratitude, you may make huge withdrawals and undo all the good you've done. \"After all we've done for you, the sacrifices we've made, how can you be so ungrateful? We try to be nice and you act like this. I can't believe it!\"\n\nIt's hard not to get impatient. It takes character to be proactive, to focus on your Circle of Influence, to nurture growing things, and not to \"pull up the flowers to see how the roots are coming.\"\n\nBut there really is no quick fix. Building and repairing relation\u00adships are long-term investments.\n\nSIX MAJOR DEPOSITS\n\nLet me suggest six major deposits that build the Emotional Bank Account.\n\nUnderstanding the Individual\n\nReally seeking to understand another person is probably one of the most important deposits you can make, and it is the key to every other deposit. You simply don't know what constitutes a deposit to another person until you understand that individual. What might be a deposit for you\u2014going for a walk to talk things over, going out for ice cream together, working on a common project\u2014might not be perceived by someone else as a deposit at all. It might even be perceived as a withdrawal, if it doesn't touch the person's deep interests or needs.\n\nOne person's mission is another person's minutia. To make a deposit, what is important to another person must be as important to you as the other person is to you. You may be working on a high priority project when your six-year-old child interrupts with some\u00adthing that seems trivial to you, but it may be very important from his point of view. It takes Habit 2 to recognize and recommit yourself to the value of that person and Habit 3 to subordinate your schedule to that human priority. By accepting the value he places on what he has to say, you show an understanding of him that makes a great deposit.\n\nI have a friend whose son developed an avid interest in baseball. My friend wasn't interested in baseball at all. But one summer, he took his son to see every major league team play one game. The trip took over six weeks and cost a great deal of money, but it became a powerful bonding experience in their relationship.\n\nMy friend was asked on his return, \"Do you like baseball that much?\"\n\n\"No,\" he replied, \"but I like my son that much.\"\n\nI have another friend, a college professor, who had a terrible relationship with his teenage son. This man's entire life was essentially academic, and he felt his son was totally wasting his life by working with his hands instead of working to develop his mind. As a result, he was almost constantly on the boy's back, and, in moments of regret, he would try to make deposits that just didn't work. The boy perceived the gestures as new forms of rejection, comparison, and judgment, and they precipitated huge withdraw\u00adals. The relationship was turning sour, and it was breaking the father's heart.\n\nOne day I shared with him this principle of making what is important to the other person as important to you as the other person is to you. He took it deeply to heart. He engaged his son in a project to build a miniature Wall of China around their home. It was a consuming project, and they worked side by side on it for over a year and a half.\n\nThrough that bonding experience, the son moved through that phase in his life and into an increased desire to develop his mind. But the real benefit was what happened to the relationship. Instead of a sore spot, it became a source of joy and strength to both father and son.\n\nOur tendency is to project out of our own autobiographies what we think other people want or need. We project our intentions on the behavior of others. We interpret what constitutes a deposit based on our own needs and desires, either now or when we were at a similar age or stage in life. If they don't interpret our effort as a deposit, our tendency is to take it as a rejection of our well intentioned effort and to give up.\n\nThe Golden Rule says to \"Do unto others as you would have others do unto you.\" While on the surface that could mean to do for them what you would like to have done for you, I think the more essential meaning is to understand them deeply as individ\u00aduals, the way you would want to be understood, and then to treat them in terms of that understanding. As one successful parent said about raising children, \"Treat them all the same by treating them differently.\"\n\nAttending to the Little Things\n\nThe little kindnesses and courtesies are so important. Small discourtesies, little unkindnesses, little forms of disrespect make large withdrawals. In relationships, the little things are the big things.\n\nI remember an evening I spent with two of my sons some years ago. It was an organized father and son outing, complete with gymnastics, wrestling matches, hotdogs, orangeade, and a movie\u2014the works.\n\nIn the middle of the movie, Sean, who was then four years old, fell asleep in his seat. His older brother, Stephen, who was six, stayed awake, and we watched the rest of the movie together. When it was over, I picked Sean up in my arms, carried him out to the car and laid him in the back seat. It was very cold that night, so I took off my coat and gently arranged it over and around him.\n\nWhen we arrived home, I quickly carried Sean in and tucked him into bed. After Stephen put on his \"jammies\" and brushed his teeth, I lay down next to him to talk about the night out together.\n\n\"How'd you like it, Stephen?\"\n\n\"Fine,\" he answered.\n\n\"Did you have fun?\"\n\n\"Yes.\"\n\n\"What did you like most?\"\n\n\"I don't know. The trampoline, I guess.\"\n\n\"That was quite a thing, wasn't it\u2014doing those somersaults and tricks in the air like that?\"\n\nThere wasn't much response on his part. I found myself making conversation. I wondered why Stephen wouldn't open up more. He usually did when exciting things happened. I was a little disappointed. I sensed something was wrong; he had been so quiet on the way home and getting ready for bed.\n\nSuddenly Stephen turned over on his side, facing the wall. I wondered why and lifted myself up just enough to see his eyes welling up with tears.\n\n\"What's wrong, honey? What is it?\"\n\nHe turned back, and I could sense he was feeling some embar\u00adrassment for the tears and his quivering lips and chin.\n\n\"Daddy, if I were cold, would you put your coat around me, too?\"\n\nOf all the events of that special night out together, the most important was a little act of kindness\u2014a momentary, unconscious showing of love to his little brother.\n\nWhat a powerful, personal lesson that experience was to me then and is even now. People are very tender, very sensitive inside. I don't believe age or experience makes much difference. Inside, even within the most toughened and calloused exteriors, are the tender feelings and emotions of the heart.\n\nKeeping Commitments\n\nKeeping a commitment or a promise is a major deposit; breaking one is a major withdrawal. In fact, there's probably not a more massive withdrawal than to make a promise that's important to someone and then not to come through. The next time a promise is made, they won't believe it. People tend to build their hopes around promises, particularly promises about their basic liveli\u00adhood.\n\nI've tried to adopt a philosophy as a parent never to make a promise I don't keep. I therefore try to make them very carefully, very sparingly, and to be aware of as many variables and contin\u00adgencies as possible so that something doesn't suddenly come up to keep me from fulfilling it.\n\nOccasionally, despite all my effort, the unexpected does come up, creating a situation where it would be unwise or impossible to keep a promise I've made. But I value that promise. I either keep it anyway, or explain the situation thoroughly to the person involved and ask to be released from the promise.\n\nI believe that if you cultivate the habit of always keeping the promises you make, you build bridges of trust that span the gaps of understanding between you and your child. Then, when your child wants to do something you don't want him to do, and out of your maturity you can see consequences that the child cannot see, you can say, \"Son, if you do this, I promise you that this will be the result.\" If that child has cultivated trust in your word, in your promises, he will act on your counsel.\n\nClarifying Expectations\n\nImagine the difficulty you might encounter if you and your boss had different assumptions regarding whose role it was to create your job description.\n\n\"When am I going to get my job description?\" you might ask.\n\n\"I've been waiting for you to bring one to me so that we could discuss it,\" your boss might reply.\n\n\"I thought defining my job was your role.\"\n\n\"That's not my role at all. Don't you remember? Right from the first, I said that how you do in the job largely depends on you.\"\n\n\"I thought you meant that the quality of my job depended on me. But I don't even know what my job really is.\"\n\nUnclear expectations in the area of goals also undermine com\u00admunication and trust.\n\n\"I did exactly what you asked me to do and here is the report.\"\n\n\"I don't want a report. The goal was to solve the problem\u2014not to analyze it and report on it.\"\n\n\"I thought the goal was to get a handle on the problem so we could delegate it to someone else.\"\n\nHow many times have we had these kinds of conversations?\n\n\"You said....\"\n\n\"No, you're wrong! I said....\"\n\n\"You did not! You never said I was supposed to....\"\n\n\"Oh, yes I did! I clearly said....\"\n\n\"You never even mentioned....\"\n\n\"But that was our agreement....\"\n\nThe cause of almost all relationship difficulties is rooted in conflicting or ambiguous expectations around roles and goals. Whether we are dealing with the question of who does what at work, how you communicate with your daughter when you tell her to clean her room, or who feeds the fish and takes out the garbage, we can be certain that unclear expectations will lead to misunderstanding, disappointment, and withdrawals of trust.\n\nMany expectations are implicit. They haven't been explicitly stated or announced, but people nevertheless bring them to a particular situation. In marriage, for example, a man and a woman have implicit expectations of each other in their marriage roles. Although these expectations have not been discussed, or some\u00adtimes even recognized by the person who has them, fulfilling them makes great deposits in the relationship and violating them makes withdrawals.\n\nThat's why it's so important whenever you come into a new situation to get all the expectations out on the table. People will begin to judge each other through those expectations. And if they feel like their basic expectations have been violated, the reserve of trust is diminished. We create many negative situations by simply assuming that our expectations are self-evident and that they are clearly understood and shared by other people.\n\nThe deposit is to make the expectations clear and explicit in the beginning. This takes a real investment of time and effort up front, but it saves great amounts of time and effort down the road. When expectations are not clear and shared, people begin to become emotionally involved and simple misunderstandings become compounded, turning into personality clashes and communication breakdowns.\n\nClarifying expectations sometimes takes a great deal of courage. It seems easier to act as though differences don't exist and to hope things will work out than it is to face the differences and work together to arrive at a mutually agreeable set of expectations.\n\nShowing Personal Integrity\n\nPersonal Integrity generates trust and is the basis of many different kinds of deposits.\n\nLack of integrity can undermine almost any other effort to create high trust accounts. People can seek to understand, remember the little things, keep their promises, clarify and fulfill expectations, and still fail to build reserves of trust if they are inwardly duplicitous.\n\nIntegrity includes but goes beyond honesty. Honesty is telling the truth\u2014in other words, conforming our words to reality. Integrity is conforming reality to our words\u2014in other words, keeping promises and fulfilling expectations. This requires an integrated character, a oneness, primarily with self but also with life.\n\nOne of the most important ways to manifest integrity is to be loyal to those who are not present. In doing so, we build the trust of those who are present. When you defend those who are absent, you retain the trust of those present.\n\nSuppose you and I were talking alone, and we were criticizing our supervisor in a way that we would not dare to do if he were present. Now what will happen when you and I have a falling out? You know I'm going to be discussing your weaknesses with someone else. That's what you and I did behind our supervisor's back. You know my nature. I'll sweet-talk you to your face and bad-mouth you behind your back. You've seen me do it.\n\nThat's the essence of duplicity. Does that build a reserve of trust in my account with you?\n\nOn the other hand, suppose you were to start criticizing our supervisor and I basically told you I agree with the content of some of the criticism and suggest that the two of us go directly to him and make an effective presentation on how things might be improved. Then what would you know I would do if someone were to criticize you to me behind your back?\n\nFor another example, suppose in my effort to build a relationship with you, I told you something someone else had shared with me in confidence. \"I really shouldn't tell you this,\" I might say, \"but since you're my friend....\" Would my betraying another person build my trust account with you? Or would you wonder if the things you had told me in confidence were being shared with others?\n\nSuch duplicity might appear to be making a deposit with the person you're with, but it is actually a withdrawal because you communicate your own lack of integrity. You may get the golden egg of temporary pleasure from putting someone down or sharing privileged information, but you're strangling the goose, weakening the relationship that provides enduring pleasure in association.\n\nIntegrity in an interdependent reality is simply this: you treat everyone by the same set of principles. As you do, people will come to trust you. They may not at first appreciate the honest confrontational experiences such integrity might generate. Con\u00adfrontation takes considerable courage, and many people would prefer to take the course of least resistance, belittling and criticiz\u00ading, betraying confidences, or participating in gossip about others behind their backs. But in the long run, people will trust and respect you if you are honest and open and kind with them. You care enough to confront. And to be trusted, it is said, is greater than to be loved. In the long run, I am convinced, to be trusted will be also to be loved.\n\nWhen my son Joshua was quite young, he would frequently ask me a soul-searching question. Whenever I overreacted to someone else or was the least bit impatient or unkind, he was so vulnerable and so honest and our relationship was so good that he would simply look me in the eye and say, \"Dad, do you love me?\" If he thought I was breaking a basic principle of life toward someone else, he wondered if I wouldn't break it with him.\n\nAs a teacher, as well as a parent, I have found that the key to the ninety-nine is the one\u2014particularly the one that is testing the patience and the good humor of the many. It is the love and the discipline of the one student, the one child, that communicates love for the others. It's how you treat the one that reveals how you regard the ninety-nine, because everyone is ultimately a one.\n\nIntegrity also means avoiding any communication that is decep\u00adtive, full of guile, or beneath the dignity of people. \"A lie is any communication with intent to deceive,\" according to one definition of the word. Whether we communicate with words or behavior, if we have integrity, our intent cannot be to deceive.\n\nApologizing Sincerely When You Make a Withdrawal\n\nWhen we make withdrawals from the Emotional Bank Account, we need to apologize and we need to do it sincerely. Great deposits come in the sincere words:\n\n\"I was wrong.\"\n\n\"That was unkind of me.\"\n\n\"I showed you no respect.\"\n\n\"I gave you no dignity, and I'm deeply sorry.\"\n\n\"I embarrassed you in front of your friends and I had no call to do that. Even though I wanted to make a point, I never should have done it. I apologize.\"\n\nIt takes a great deal of character strength to apologize quickly out of one's heart rather than out of pity. A person must possess himself and have a deep sense of security in fundamental principles and values in order to genuinely apologize.\n\nPeople with little internal security can't do it. It makes them too vulnerable. They feel it makes them appear soft and weak, and they fear that others will take advantage of their weakness. Their security is based on the opinions of other people, and they worry about what others might think. In addition, they usually feel justified in what they did. They rationalize their own wrong in the name of the other person's wrong, and if they apologize at all, it's superficial.\n\n\"If you're going to bow, bow low,\" says Eastern wisdom. \"Pay the uttermost farthing,\" says the Christian ethic. To be a deposit, an apology must be sincere. And it must be perceived as sincere.\n\nLeo Roskin taught, \"It is the weak who are cruel. Gentleness can only be expected from the strong.\"\n\nI was in my office at home one afternoon writing, of all things, on the subject of patience. I could hear the boys running up and down the hall making loud banging noises, and I could feel my own patience beginning to wane.\n\nSuddenly, my son David started pounding on the bathroom door, yelling at the top of his lungs, \"Let me in! Let me in!\"\n\nI rushed out of the office and spoke to him with great intensity. \"David, do you have any idea how disturbing that is to me? Do you know how hard it is to try to concentrate and write creatively? Now, you go into your room and stay in there until you can behave yourself.\" So in he went, dejected, and shut the door.\n\nAs I turned around, I became aware of another problem. The boys had been playing tackle football in the four-foot-wide hallway, and one of them had been elbowed in the mouth. He was lying there in the hall, bleeding from the mouth. David, I discovered, had gone to the bathroom to get a wet towel for him. But his sister, Maria, who was taking a shower, wouldn't open the door.\n\nWhen I realized that I had completely misinterpreted the situa\u00adtion and had overreacted, I immediately went in to apologize to David.\n\nAs I opened the door, the first thing he said to me was, \"I won't forgive you.\"\n\n\"Well, why not, honey?\" I replied. \"Honestly, I didn't realize you were trying to help your brother. Why won't you forgive me?\"\n\n\"Because you did the same thing last week,\" he replied. In other words, he was saying, \"Dad, you're overdrawn, and you're not going to talk your way out of a problem you behaved yourself into.\"\n\nSincere apologies make deposits; repeated apologies interpreted as insincere make withdrawals. And the quality of the relationship reflects it.\n\nIt is one thing to make a mistake, and quite another thing not to admit it. People will forgive mistakes, because mistakes are usually of the mind, mistakes of judgment. But people will not easily forgive the mistakes of the heart, the ill intention, the bad motives, the prideful justifying cover-up of the first mistake.\n\nThe Laws of Love and the Laws of Life\n\nWhen we make deposits of unconditional love, when we live the primary laws of love, we encourage others to live the primary laws of life. In other words, when we truly love others without condition, without strings, we help them feel secure and safe and validated and affirmed in their essential worth, identity, and integrity. Their natural growth process is encouraged. We make it easier for them to live the laws of life\u2014cooperation, contribution, self-discipline, integrity\u2014and to discover and live true to the highest and best within them. We give them the freedom to act on their own inner imperatives rather than react to our conditions and limitations. This does not mean we become permissive or soft. That itself is a massive withdrawal. We counsel, we plead, we set limits and consequences. But we love, regardless.\n\nWhen we violate the primary laws of love\u2014when we attach strings and conditions to that gift\u2014we actually encourage others to violate the primary laws of life. We put them in a reactive, defensive position where they feel they have to prove \"I matter as a person, independent of you.\"\n\nIn reality, they aren't independent. They are counter-dependent, which is another form of dependency and is at the lowest end of the Maturity Continuum. They become reactive, almost enemy-centered, more concerned about defending their \"rights\" and producing evidence of their individuality than they are about proactively listening to and honoring their own inner imperatives.\n\nRebellion is a knot of the heart, not of the mind. The key is to make deposits\u2014constant deposits of unconditional love.\n\nI once had a friend who was dean of a very prestigious school.* He planned and saved for years to provide his son the opportunity to attend that institution, but when the time came, the boy refused to go.\n\nThis deeply concerned his father. Graduating from that particu\u00adlar school would have been a great asset to the boy. Besides, it was a family tradition. Three generations of attendance preceded the boy. The father pleaded and urged and talked. He also tried to listen to the boy to understand him, all the while hoping that the son would change his mind.\n\nThe subtle message being communicated was one of conditional love. The son felt that in a sense the father's desire for him to attend the school outweighed the value he placed on him as a person and as a son, which was terribly threatening. Conse\u00adquently, he fought for and with his own identity and integrity, and he increased in his resolve and his efforts to rationalize his decision not to go.\n\nAfter some intense soul-searching, the father decided to make a sacrifice\u2014to renounce conditional love. He knew that his son might choose differently than he had wished; nevertheless, he and his wife resolved to love their son unconditionally, regardless of his choice. It was an extremely difficult thing to do because the value of his educational experience was so close to their hearts and because it was something they had planned and worked for since his birth.\n\nThe father and mother went through a very difficult rescripting process, struggling to really understand the nature of uncondi\u00adtional love. They communicated to the boy what they were doing and why, and told him that they had come to the point at which they could say in all honesty that his decision would not affect their complete feeling of unconditional love toward him. They didn't do this to manipulate him, to try to get him to \"shape up.\" They did it as the logical extension of their growth and character.\n\nThe boy didn't give much of a response at the time, but his parents had such a paradigm of unconditional love at that point that it would have made no difference in their feelings for him. About a week later, he told his parents that he had decided not to go. They were perfectly prepared for this response and continued to show unconditional love for him. Everything was settled and life went along normally.\n\nA short time later, an interesting thing happened. Now that the boy no longer felt he had to defend his position, he searched within himself more deeply and found that he really did want to have this educational experience. He applied for admission, and then he told his father, who again showed unconditional love by fully accepting his son's decision. My friend was happy, but not excessively so, because he had truly learned to love without condition.\n\nDag Hammarskjold, past Secretary-General of the United Na\u00adtions, once made a profound, far-reaching statement: \"It is more noble to give yourself completely to one individual than to labor diligently for the salvation of the masses.\"\n\nI take that to mean that I could devote eight, ten, or twelve hours a day, five, six, or seven days a week to the thousands of people and projects \"out there\" and still not have a deep, meaningful relationship with my own spouse, with my own teenage son, with my closest working associate. And it would take more nobility of character\u2014more humility, courage, and strength\u2014to rebuild that one relationship than it would to continue putting in all those hours for all those people and causes.\n\nIn twenty-five years of consulting with organizations, I have been impressed over and over again by the power of that state\u00adment. Many of the problems in organizations stem from relation\u00adship difficulties at the very top\u2014between two partners in a professional firm, between the owner and the president of a company, between the president and an executive vice-president. It truly takes more nobility of character to confront and resolve those issues than it does to continue to diligently work for the many projects and people \"out there.\"\n\nWhen I first came across Hammarskjold's statement, I was working in an organization where there were unclear expectations between the individual who was my right-hand man and myself. I simply did not have the courage to confront our differences regarding role and goal expectations and values, particularly in our methods of administration. So I worked for a number of months in a compromise mode to avoid what might turn out to be an ugly confrontation. All the while, bad feelings were developing inside both of us.\n\nAfter reading that it is more noble to give yourself completely to one individual than to labor diligently for the salvation of the masses, I was deeply affected by the idea of rebuilding that relationship.\n\nI had to steel myself for what lay ahead, because I knew it would be hard to really get the issues out and to achieve a deep, common understanding and commitment. I remember actually shaking in anticipation of the visit. He seemed like such a hard man, so set in his own ways and so right in his own eyes; yet I needed his strengths and abilities. I was afraid a confrontation might jeopar\u00addize the relationship and result in my losing those strengths.\n\nI went through a mental dress rehearsal of the anticipated visit, and I finally became settled within myself around the principles rather than the practices of what I was going to do and say. At last I felt peace of mind and the courage to have the communication.\n\nWhen we met together, to my total surprise, I discovered that this man had been going through the very same process and had been longing for such a conversation. He was anything but hard and defensive.\n\nNevertheless, our administrative styles were considerably different, and the entire organization was responding to these differ\u00adences. We both acknowledged the problems that our disunity had created. Over several visits, we were able to confront the deeper issues, to get them all out on the table, and to resolve them, one by one, with a spirit of high mutual respect. We were able to develop a powerful complementary team and a deep personal affection which added tremendously to our ability to work effectively together.\n\nCreating the unity necessary to run an effective business or a family or a marriage requires great personal strength and courage. No amount of technical administrative skill in laboring for the masses can make up for lack of nobility of personal character in developing relationships. It is at a very essential, one-on-one level, that we live the primary laws of love and life.\n\nP PROBLEMS ARE PC OPPORTUNITIES\n\nThis experience also taught me another powerful paradigm of interdependence. It deals with the way in which we see problems. I had lived for months trying to avoid the problem, seeing it as a source of irritation, a stumbling block, and wishing it would somehow go away. But, as it turned out, the very problem created the opportunity to build a deep relationship that empowered us to work together as a strong complementary team.\n\nI suggest that in an interdependent situation, every P problem is a PC opportunity\u2014a chance to build the Emotional Bank Accounts that significantly affect interdependent production.\n\nWhen parents see their children's problems as opportunities to build the relationship instead of as negative, burdensome irrita\u00adtions, it totally changes the nature of parent-child interaction. Parents become more willing, even excited, about deeply under\u00adstanding and helping their children. When a child comes to them with a problem, instead of thinking, \"Oh, no! Not another prob\u00adlem!\" their paradigm is, \"Here is a great opportunity for me to really help my child and to invest in our relationship.\" Many interactions change from transactional to transformational, and strong bonds of love and trust are created as children sense the value parents give to their problems and to them as individuals.\n\nThis paradigm is powerful in business as well. One department store chain that operates from this paradigm has created a great loyalty among its customers. Any time a customer comes into the store with a problem, no matter how small, the clerks immediately see it as an opportunity to build the relationship with the customer. They respond with a cheerful, positive desire to solve the problem in a way that will make the customer happy. They treat the customer with such grace and respect, giving such second-mile service, that many of the customers don't even think of going anywhere else.\n\nBy recognizing that the P\/PC balance is necessary to effectiveness in an interdependent reality, we can value our problems as opportunities to increase PC.\n\nTHE HABITS OF INTERDEPENDENCE\n\nWith the paradigm of the Emotional Bank Account in mind, we're ready to move into the habits of Public Victory, of success in working with other people. As we do, we can see how these habits work together to create effective interdependence. We can also see how powerfully scripted we are in other patterns of thought and behavior.\n\nIn addition, we can see on an even deeper level that effective interdependence can only be achieved by truly independent people. It is impossible to achieve Public Victory with popular \"Win\/Win negotiation\" techniques or \"reflective listening\" techniques or \"creative problem-solving\" techniques that focus on personality and truncate the vital character base.\n\nLet's now focus on each of the Public Victory habits in depth.\n\n* * *\n\n##### Foot Note\n\n* Some of the details of this story have been changed to protect the privacy of those involved.\n\n### HABIT 4  \nTHINK WIN\/WIN\n\n### PRINCIPLES OF INTERPERSONAL LEADERSHIP\n\nWe have committed the Golden Rule to memory; let us now commit it to life.\n\nEdwin Markham\n\nONE TIME I WAS ASKED TO WORK WITH A COMPANY whose president was very concerned about the lack of cooperation among his people.\n\n\"Our basic problem, Stephen, is that they're selfish,\" he said. \"They just won't cooperate. I know if they would cooperate, we could produce so much more. Can you help us develop a human relations program that will solve the problem?\"\n\n\"Is your problem the people or the paradigm?\" I asked.\n\n\"Look for yourself,\" he replied.\n\nSo I did. And I found that there was a real selfishness, an unwillingness to cooperate, a resistance to authority, defensive communication. I could see that overdrawn Emotional Bank Ac\u00adcounts had created a culture of low trust. But I pressed the question.\n\n\"Let's look at it deeper,\" I suggested. \"Why don't your people cooperate? What is the reward for not cooperating?\"\n\n\"There's no reward for not cooperating,\" he assured me. \"The rewards are much greater if they do cooperate.\"\n\n\"Are they?\" I asked. Behind a curtain on one wall of this man's office was a chart. On the chart were a number of racehorses all lined up on a track. Superimposed on the face of each horse was the face of one of his managers. At the end of the track was a beautiful travel poster of Bermuda, an idyllic picture of blue skies and fleecy clouds and a romantic couple walking hand in hand down a white sandy beach.\n\nOnce a week, this man would bring all his people into this office and talk cooperation. \"Let's all work together. We'll all make more money if we do.\" Then he would pull the curtain and show them the chart. \"Now which of you is going to win the trip to Bermuda?\"\n\nIt was like telling one flower to grow and watering another, like saying \"firings will continue until morale improves.\" He wanted cooperation. He wanted his people to work together, to share ideas, to all benefit from the effort. But he was setting them up in competition with each other. One manager's success meant failure for the other managers.\n\nAs with many, many problems between people in business, family, and other relationships, the problem in this company was the result of a flawed paradigm. The president was trying to get the fruits of cooperation from a paradigm of competition. And when it didn't work, he wanted a technique, a program, a quick fix antidote to make his people cooperate.\n\nBut you can't change the fruit without changing the root. Working on the attitudes and behaviors would have been hacking at the leaves. So we focused instead on producing personal and organizational excellence in an entirely different way by developing information and reward systems which reinforced the value of cooperation.\n\nWhether you are the president of a company or the janitor, the moment you step from independence into interdependence in any capacity, you step into a leadership role. You are in a position of influencing other people. And the habit of effective interpersonal leadership is Think Win\/Win.\n\nSIX PARADIGMS OF HUMAN INTERACTION\n\nWin\/Win is not a technique; it's a total philosophy of human interaction. In fact, it is one of six paradigms of interaction. The alternative paradigms are Win\/Lose, Lose\/Win, Lose\/Lose, Win, and Win\/Win or No Deal.\n\n  * Win\/Win\n  * Win\/Lose\n  * Lose\/Win |   * Lose\/Lose\n  * Win\n  * Win\/Win or No Deal\n\n---|---\n\nWin\/Win\n\nWin\/Win is a frame of mind and heart that constantly seeks mutual benefit in all human interactions. Win\/Win means that agreements or solutions are mutually beneficial, mutually satisfy\u00ading. With a Win\/Win solution, all parties feel good about the decision and feel committed to the action plan. Win\/Win sees life as a cooperative, not a competitive arena. Most people tend to think in terms of dichotomies: strong or weak, hardball or softball, win or lose. But that kind of thinking is fundamentally flawed. It's based on power and position rather than on principle. Win\/Win is based on the paradigm that there is plenty for everybody, that one person's success is not achieved at the expense or exclusion of the success of others.\n\nWin\/Win is a belief in the Third Alternative. It's not your way or my way; it's a better way, a higher way.\n\nWin\/Lose\n\nOne alternative to Win\/Win is Win\/Lose, the paradigm of the race to Bermuda. It says \"If I win, you lose.\"\n\nIn leadership style, Win\/Lose is the authoritarian approach: \"I get my way; you don't get yours.\" Win\/Lose people are prone to use position, power, credentials, possessions, or personality to get their way.\n\nMost people have been deeply scripted in the Win\/Lose mental\u00adity since birth. First and most important of the powerful forces at work is the family. When one child is compared with another\u2014when patience, understanding or love is given or withdrawn on the basis of such comparisons\u2014people are into Win\/Lose thinking. Whenever love is given on a conditional basis, when someone has to earn love, what's being communicated to them is that they are not intrinsically valuable or lovable. Value does not lie inside them, it lies outside. It's in comparison with somebody else or against some expectation.\n\nAnd what happens to a young mind and heart, highly vulnera\u00adble, highly dependent upon the support and emotional affirmation of the parents, in the face of conditional love? The child is molded, shaped, and programmed in the Win\/Lose mentality.\n\n\"If I'm better than my brother, my parents will love me more.\"\n\n\"My parents don't love me as much as they love my sister. I must not be as valuable.\"\n\nAnother powerful scripting agency is the peer group. A child first wants acceptance from his parents and then from his peers, whether they be siblings or friends. And we all know how cruel peers sometimes can be. They often accept or reject totally on the basis of conformity to their expectations and norms, providing additional scripting toward Win\/Lose.\n\nThe academic world reinforces Win\/Lose scripting. The \"normal distribution curve\" basically says that you got an \"A\" because someone else got a \"C.\" It interprets an individual's value by comparing him or her to everyone else. No recognition is given to intrinsic value; everyone is extrinsically defined.\n\n\"Oh, how nice to see you here at our PTA meeting. You ought to be really proud of your daughter, Caroline. She's in the upper 10 percent.\"\n\n\"That makes me feel good.\"\n\n\"But your son, Johnny, is in trouble. He's in the lower quartile.\"\n\n\"Really? Oh, that's terrible! What can we do about it?\"\n\nWhat this kind of comparative information doesn't tell you is that perhaps Johnny is going on all eight cylinders while Caroline is coasting on four of her eight. But people are not graded against their potential or against the full use of their present capacity. They are graded in relation to other people. And grades are carriers of social value; they open doors of opportunity or they close them. Competition, not cooperation, lies at the core of the educational process. Cooperation, in fact, is usually associated with cheating.\n\nAnother powerful programming agent is athletics, particularly for young men in their high school or college years. Often they develop the basic paradigm that life is a big game, a zero sum game where some win and some lose. \"Winning\" is \"beating\" in the athletic arena.\n\nAnother agent is law. We live in a litigious society. The first thing many people think about when they get into trouble is suing someone, taking them to court, \"winning\" at someone else's expense. But defensive minds are neither creative nor cooperative.\n\nCertainly we need law or else society will deteriorate. It provides survival, but it doesn't create synergy. At best it results in compromise. Law is based on an adversarial concept. The recent trend of encouraging lawyers and law schools to focus on peaceable negotiation, the techniques of Win\/Win, and the use of private courts, may not provide the ultimate solution, but it does reflect a growing awareness of the problem.\n\nCertainly there is a place for Win\/Lose thinking in truly compet\u00aditive and low-trust situations. But most of life is not a competition. We don't have to live each day competing with our spouse, our children, our coworkers, our neighbors, and our friends. \"Who's winning in your marriage?\" is a ridiculous question. If both people aren't winning, both are losing.\n\nMost of life is an interdependent, not an independent, reality. Most results you want depend on cooperation between you and others. And the Win\/Lose mentality is dysfunctional to that cooperation.\n\nLose\/Win\n\nSome people are programmed the other way\u2014Lose\/Win.\n\n\"I lose, you win.\"\n\n\"Go ahead. Have your way with me.\"\n\n\"Step on me again. Everyone does.\"\n\n\"I'm a loser. I've always been a loser.\"\n\n\"I'm a peacemaker. I'll do anything to keep peace.\"\n\nLose\/Win is worse than Win\/Lose because it has no standards\u2014no demands, no expectations, no vision. People who think Lose\/Win are usually quick to please or appease. They seek strength from popularity or acceptance. They have little courage to express their own feelings and convictions and are easily intimidated by the ego strength of others.\n\nIn negotiation, Lose\/Win is seen as capitulation\u2014giving in or giving up. In leadership style, it's permissiveness or indulgence. Lose\/Win means being a nice guy, even if \"nice guys finish last.\"\n\nWin\/Lose people love Lose\/Win people because they can feed on them. They love their weaknesses\u2014they take advantage of them. Such weaknesses complement their strengths.\n\nBut the problem is that Lose\/Win people bury a lot of feelings. And unexpressed feelings never die: they're buried alive and come forth later in uglier ways. Psychosomatic illnesses, particularly of the respiratory, nervous, and circulatory systems often are the reincarnation of cumulative resentment, deep disappointment and disillusionment repressed by the Lose\/Win mentality. Dispropor\u00adtionate rage or anger, overreaction to minor provocation, and cynicism are other embodiments of suppressed emotion.\n\nPeople who are constantly repressing, not transcending feelings towards a higher meaning find that it affects the quality of their self-esteem and eventually the quality of their relationships with others.\n\nBoth Win\/Lose and Lose\/Win are weak positions, based in personal insecurities. In the short run, Win\/Lose will produce more results because it draws on the often considerable strengths and talents of the people at the top. Lose\/Win is weak and chaotic from the outset.\n\nMany executives, managers, and parents swing back and forth, as if on a pendulum, from Win\/Lose inconsideration to Lose\/Win indulgence. When they can't stand confusion and lack of structure, direction, expectation, and discipline any longer, they swing back to Win\/Lose\u2014until guilt undermines their resolve and drives them back to Lose\/Win\u2014until anger and frustration drive them back to Win\/Lose again.\n\nLose\/Lose\n\nWhen two Win\/Lose people get together\u2014that is, when two determined, stubborn, ego-invested individuals interact\u2014the re\u00adsult will be Lose\/Lose. Both will lose. Both will become vindictive and want to \"get back\" or \"get even,\" blind to the fact that murder is suicide, that revenge is a two-edged sword.\n\nI know of a divorce in which the husband was directed by the judge to sell the assets and turn over half the proceeds to his ex-wife. In compliance, he sold a car worth over $10,000 for $50 and gave $25 to the wife. When the wife protested, the court clerk checked on the situation and discovered that the husband was proceeding in the same manner systematically through all of the assets.\n\nSome people become so centered on an enemy, so totally obsessed with the behavior of another person that they become blind to everything except their desire for that person to lose, even if it means losing themselves. Lose\/Lose is the philosophy of adversarial conflict, the philosophy of war.\n\nLose\/Lose is also the philosophy of the highly dependent person without inner direction who is miserable and thinks everyone else should be, too. \"If nobody ever wins, perhaps being a loser isn't so bad.\"\n\nWin\n\nAnother common alternative is simply to think Win. People with the Win mentality don't necessarily want someone else to lose. That's irrelevant. What matters is that they get what they want.\n\nWhen there is no sense of contest or competition, Win is probably the most common approach in everyday negotiation. A person with the Win mentality thinks in terms of securing his own ends\u2014and leaving it to others to secure theirs.\n\nWhich Option Is Best?\n\nOf these five philosophies discussed so far\u2014Win\/Win, Win\/Lose, Lose\/Win, Lose\/Lose, and Win\u2014which is the most effective? The answer is, \"It depends.\" If you win a football game, that means the other team loses. If you work in a regional office that is miles away from another regional office, and you don't have any functional relationship between the offices, you may want to compete in a Win\/Lose situation to stimulate business. However, you would not want to set up a Win\/Lose situation like the \"Race to Bermuda\" contest within a company or in a situation where you need cooperation among people or groups of people to achieve maximum success.\n\nIf you value a relationship and the issue isn't really that important, you may want to go for Lose\/Win in some circumstances to genuinely affirm the other person. \"What I want isn't as important to me as my relationship with you. Let's do it your way this time.\" You might also go for Lose\/Win if you feel the expense of time and effort to achieve a win of any kind would violate other higher values. Maybe it just isn't worth it.\n\nThere are circumstances in which you would want to Win, and you wouldn't be highly concerned with the relationship of that win to others. If your child's life were in danger, for example, you might be peripherally concerned about other people and circum\u00adstances. But saving that life would be supremely important.\n\nThe best choice, then, depends on reality. The challenge is to read that reality accurately and not to translate Win\/Lose or other scripting into every situation.\n\nMost situations, in fact, are part of an interdependent reality, and then Win\/Win is really the only viable alternative of the five.\n\nWin\/Lose is not viable because, although I appear to win in a confrontation with you, your feelings, your attitudes toward me and our relationship have been affected. If I am a supplier to your company, for example, and I win on my terms in a particular negotiation, I may get what I want now. But will you come to me again? My short-term Win will really be a long-term Lose if I don't get your repeat business. So an interdependent Win\/Lose is really Lose\/Lose in the long run.\n\nIf we come up with a Lose\/Win, you may appear to get what you want for the moment. But how will that affect my attitude about working with you, about fulfilling the contract? I may not feel as anxious to please you. I may carry battle scars with me into any future negotiations. My attitude about you and your company may be spread as I associate with others in the industry. So we're into Lose\/Lose again. Lose\/Lose obviously isn't viable in any context.\n\nAnd if I focus on my own Win and don't even consider your point of view, there's no basis for any kind of productive relation\u00adship.\n\nIn the long run, if it isn't a win for both of us, we both lose. That's why Win\/Win is the only real alternative in interdependent realities.\n\nI worked with a client once, the president of a large chain of retail stores, who said, \"Stephen, this Win\/Win idea sounds good, but it is so idealistic. The tough, realistic business world isn't like that. There's Win\/Lose everywhere, and if you're not out there playing the game, you just can't make it.\"\n\n\"All right,\" I said, \"try going for Win\/Lose with your customers. Is that realistic?\"\n\n\"Well, no,\" he replied.\n\n\"Why not?\"\n\n\"I'd lose my customers.\"\n\n\"Then, go for Lose\/Win\u2014give the store away. Is that realistic?\"\n\n\"No. No margin, no mission.\"\n\nAs we considered the various alternatives, Win\/Win appeared to be the only truly realistic approach.\n\n\"I guess that's true with customers,\" he admitted, \"but not with suppliers.\"\n\n\"You are the customer of the supplier,\" I said. \"Why doesn't the same principle apply?\"\n\n\"Well, we recently renegotiated our lease agreements with the mall operators and owners,\" he said. \"We went in with a Win\/Win attitude. We were open, reasonable, conciliatory. But they saw that position as being soft and weak, and they took us to the cleaners.\"\n\n\"Well, why did you go for Lose\/Win?\" I asked.\n\n\"We didn't. We went for Win\/Win.\"\n\n\"I thought you said they took you to the cleaners.\"\n\n\"They did.\"\n\n\"In other words, you lost.\"\n\n\"That's right.\"\n\n\"And they won.\"\n\n\"That's right.\"\n\n\"So what's that called?\"\n\nWhen he realized that what he had called Win\/Win was really Lose\/Win, he was shocked. And as we examined the long-term impact of that Lose\/Win, the suppressed feelings, the trampled values, the resentment that seethed under the surface of the relationship, we agreed that it was really a loss for both parties in the end.\n\nIf this man had had a real Win\/Win attitude, he would have stayed longer in the communication process, listened to the mall owner more, then expressed his point of view with more courage. He would have continued in the Win\/Win spirit until a solution was reached they both felt good about. And that solution, that Third Alternative, would have been synergistic\u2014probably some\u00adthing neither of them had thought of on his own.\n\nWin\/Win or No Deal\n\nIf these individuals had not come up with a synergistic solution\u2014one that was agreeable to both\u2014they could have gone for an even higher expression of Win\/Win\u2014Win\/Win or No Deal.\n\nNo Deal basically means that if we can't find a solution that would benefit us both, we agree to disagree agreeably\u2014No Deal. No expectations have been created, no performance contracts established. I don't hire you or we don't take on a particular assignment together because it's obvious that our values or our goals are going in opposite directions. It is so much better to realize this up front instead of downstream when expectations have been created and both parties have been disillusioned.\n\nWhen you have No Deal as an option in your mind, you feel liberated because you have no need to manipulate people, to push your own agenda, to drive for what you want. You can be open. You can really try to understand the deeper issues underlying the positions.\n\nWith No Deal as an option, you can honestly say, \"I only want to go for Win\/Win. I want to win, and I want you to win. I wouldn't want to get my way and have you not feel good about it, because downstream it would eventually surface and create a withdrawal. On the other hand, I don't think you would feel good if you got your way and I gave in. So let's work for a Win\/Win. Let's really hammer it out. And if we can't find it, then let's agree that we won't make a deal at all. It would be better not to deal than to live with a decision that wasn't right for us both. Then maybe another time we might be able to get together.\"\n\nSome time after learning the concept of Win\/Win or No Deal, the president of a small computer software company shared with me the following experience.\n\n\"We had developed new software which we sold on a five-year contract to a particular bank. The bank president was excited about it, but his people weren't really behind the decision.\n\n\"About a month later, that bank changed presidents. The new president came to me and said, 'I am uncomfortable with these software conversions. I have a mess on my hands. My people are all saying that they can't go through this and I really feel I just can't push it at this point in time.'\n\n\"My own company was in deep financial trouble. I knew I had every legal right to enforce the contract. But I had become convinced of the value of the principle of Win\/Win.\n\n\"So I told him 'We have a contract. Your bank has secured our products and our services to convert you to this program. But we understand that you're not happy about it. So what we'd like to do is give you back the contract, give you back your deposit, and if you are ever looking for a software solution in the future, come back and see us.'\n\n\"I literally walked away from an $84,000 contract. It was close to financial suicide. But I felt that, in the long run, if the principle were true, it would come back and pay dividends.\n\n\"Three months later, the new president called me. 'I'm now going to make changes in my data processing,' he said, 'and I want to do business with you.' He signed a contract for $240,000.\"\n\nAnything less than Win\/Win in an interdependent reality is a poor second best that will have impact in the long-term relation\u00adship. The cost of that impact needs to be carefully considered. If you can't reach a true Win\/Win, you're very often better off to go for No Deal.\n\nWin\/Win or No Deal provides tremendous emotional freedom in the family relationship. If family members can't agree on a video that everyone will enjoy, they can simply decide to do something else\u2014No Deal\u2014rather than having some enjoy the evening at the expense of others.\n\nI have a friend whose family has been involved in singing together for several years. When they were young, she arranged the music, made the costumes, accompanied them on the piano and directed the performances.\n\nAs the children grew older, their taste in music began to change and they wanted to have more say in what they performed and what they wore. They became less responsive to direction.\n\nBecause she had years of experience in performing herself and felt closer to the needs of the older people at the rest homes where they planned to perform, she didn't feel that many of the ideas they were suggesting would be appropriate. At the same time, however, she recognized their need to express themselves and to be part of the decision-making process.\n\nSo she set up a Win\/Win or No Deal. She told them she wanted to arrive at an agreement that everyone felt good about\u2014or they would simply find other ways to enjoy their talents. As a result, everyone felt free to express his or her feelings and ideas as they worked to set up a Win\/Win agreement, knowing that whether or not they could agree, there would be no emotional strings.\n\nThe Win\/Win or No Deal approach is most realistic at the beginning of a business relationship or enterprise. In a continuing business relationship, No Deal may not be a viable option, which can create serious problems, especially for family businesses or businesses that are begun initially on the basis of friendship.\n\nIn an effort to preserve the relationship, people sometimes go on for years making one compromise after another, thinking Win\/Lose or Lose\/Win even while talking Win\/Win. This creates serious problems for the people and for the business, particularly if the competition operates on Win\/Win and synergy.\n\nWithout No Deal, many such businesses simply deteriorate and either fail or have to be turned over to professional managers. Experience shows that it is often better in setting up a family business or a business between friends to acknowledge the possi\u00adbility of No Deal downstream and to establish some kind of buy\/sell agreement so that the business can prosper without permanently damaging the relationship.\n\nOf course there are some relationships where No Deal is not viable. I wouldn't abandon my child or my spouse and go for No Deal (it would be better, if necessary, to go for compromise\u2014a low form of Win\/Win). But in many cases, it is possible to go into negotiation with a full Win\/Win or No Deal attitude. And the freedom in that attitude is incredible.\n\nFIVE DIMENSIONS OF WIN\/WIN\n\nThink Win\/Win is the habit of interpersonal leadership. It involves the exercise of each of the unique human endowments\u2014self-awareness, imagination, conscience, and independent will\u2014in our relationships with others. It involves mutual learning, mutual influence, mutual benefits.\n\nIt takes great courage as well as consideration to create these mutual benefits, particularly if we're interacting with others who are deeply scripted in Win\/Lose.\n\nThat is why this habit involves principles of interpersonal leadership. Effective interpersonal leadership requires the vision, the proactive initiative and the security, guidance, wisdom, and power that come from principle-centered personal leadership.\n\nThe principle of Win\/Win is fundamental to success in all our interactions, and it embraces five interdependent dimensions of life. It begins with character and moves toward relationships, out of which flow agreements. It is nurtured in an environment where structure and systems are based on Win\/Win. And it involves process; we cannot achieve Win\/Win ends with Win\/Lose or Lose\/Win means.\n\nThe following diagram shows how these five dimensions relate to each other.\n\nNow let's consider each of the five dimensions in turn.\n\nCharacter\n\nCharacter is the foundation of Win\/Win, and everything else builds on that foundation. There are three character traits essential to the Win\/Win paradigm.\n\nINTEGRITY. We've already defined integrity as the value we place on ourselves. Habits 1, 2, and 3 help us develop and maintain integrity. As we clearly identify our values and proactively orga\u00adnize and execute around those values on a daily basis, we develop self-awareness and independent will by making and keeping meaningful promises and commitments.\n\nThere's no way to go for a Win in our own lives if we don't even know, in a deep sense, what constitutes a Win\u2014what is, in fact, harmonious with our innermost values. And if we can't make and keep commitments to ourselves as well as to others, our commit\u00adments become meaningless. We know it; others know it. They sense duplicity and become guarded. There's no foundation of trust and Win\/Win becomes an ineffective superficial technique. Integrity is the cornerstone in the foundation.\n\nMATURITY. Maturity is the balance between courage and consideration. I first learned this definition of maturity in the fall of 1955 from a marvelous professor, Hrand Saxenian, who instructed my Control class at the Harvard Business School. He taught the finest, simplest, most practical, yet profound, definition of emotional maturity I've ever come across\u2014\"the ability to express one's own feelings and convictions balanced with consideration for the thoughts and feel\u00adings of others.\" As a part of his doctoral research, Hrand Saxenian had developed this criterion over years of historical and direct field research. He later wrote up his original research format in its com\u00adpleteness with supportive reasoning and application suggestions in a Harvard Business Review article (January-February 1958). Even though it is complementary and also developmental, Hrand's use of the word \"maturity\" is different from its use in the 7 Habits \"Ma\u00adturity Continuum,\" which focuses on a growth and development process from dependency through independency to interdependency.\n\nIf you examine many of the psychological tests used for hiring, promoting, and training purposes, you will find that they are designed to evaluate this kind of maturity. Whether it's called the ego strength\/empathy balance, the self confidence\/respect for others balance, the concern for people\/concern for tasks balance, \"I'm okay, you're okay\" in transactional analysis language, or 9.1, 1.9, 5.5, 9.9, in management grid language\u2014the quality sought for is the balance of what I call courage and consideration.\n\nRespect for this quality is deeply ingrained in the theory of human interaction, management, and leadership. It is a deep embodiment of the P\/PC balance. While courage may focus on getting the golden egg, consideration deals with the long-term welfare of the other stakeholders. The basic task of leadership is to increase the standard of living and the quality of life for all stakeholders.\n\nMany people think in dichotomies, in either\/or terms. They think if you're nice, you're not tough. But Win\/Win is nice... and tough. It's twice as tough as Win\/Lose. To go for Win\/Win, you not only have to be nice, you have to be courageous. You not only have to be empathic, you have to be confident. You not only have to be considerate and sensitive, you have to be brave. To do that, to achieve that balance between courage and consideration, is the essence of real maturity and is fundamental to Win\/Win.\n\nIf I'm high on courage and low on consideration, how will I think? Win\/Lose. I'll be strong and ego bound. I'll have the courage of my convictions, but I won't be very considerate of yours.\n\nTo compensate for my lack of internal maturity and emotional strength, I might borrow strength from my position and power, or from my credentials, my seniority, my affiliations.\n\nIf I'm high on consideration and low on courage, I'll think Lose\/Win. I'll be so considerate of your convictions and desires that I won't have the courage to express and actualize my own.\n\nHigh courage and consideration are both essential to Win\/Win. It is the balance that is the mark of real maturity. If I have it, I can listen, I can empathically understand, but I can also courageously confront.\n\nABUNDANCE MENTALITY. The third character trait essential to Win\/Win is the Abundance Mentality, the paradigm that there is plenty out there for everybody.\n\nMost people are deeply scripted in what I call the Scarcity Mentality. They see life as having only so much, as though there were only one pie out there. And if someone were to get a big piece of the pie, it would mean less for everybody else. The Scarcity Mentality is the zero-sum paradigm of life.\n\nPeople with a Scarcity Mentality have a very difficult time sharing recognition and credit, power or profit\u2014even with those who help in the production. They also have a very hard time being genuinely happy for the successes of other people\u2014even, and sometimes especially, members of their own family or close friends and associates. It's almost as if something is being taken from them when someone else receives special recognition or windfall gain or has remarkable success or achievement.\n\nAlthough they might verbally express happiness for others' success, inwardly they are eating their hearts out. Their sense of worth comes from being compared, and someone else's success, to some degree, means their failure. Only so many people can be \"A\" students; only one person can be \"number one.\" To \"win\" simply means to \"beat.\"\n\nOften, people with a Scarcity Mentality harbor secret hopes that others might suffer misfortune\u2014not terrible misfortune, but ac\u00adceptable misfortune that would keep them \"in their place.\" They're always comparing, always competing. They give their energies to possessing things or other people in order to increase their sense of worth.\n\nThey want other people to be the way they want them to be. They often want to clone them, and they surround themselves with \"yes\" people\u2014people who won't challenge them, people who are weaker than they.\n\nIt's difficult for people with a Scarcity Mentality to be members of a complementary team. They look on differences as signs of insubordination and disloyalty.\n\nThe Abundance Mentality, on the other hand, flows out of a deep inner sense of personal worth and security. It is the paradigm that there is plenty out there and enough to spare for everybody. It results in sharing of prestige, of recognition, of profits, of decision making. It opens possibilities, options, alternatives, and creativity.\n\nThe Abundance Mentality takes the personal joy, satisfaction, and fulfillment of Habits 1, 2, and 3 and turns it outward, appreciating the uniqueness, the inner direction, the proactive nature of others. It recognizes the unlimited possibilities for positive interactive growth and development, creating new Third Alternatives.\n\nPublic Victory does not mean victory over other people. It means success in effective interaction that brings mutually beneficial results to everyone involved. Public Victory means working to\u00adgether, communicating together, making things happen together that even the same people couldn't make happen by working independently. And Public Victory is an outgrowth of the Abun\u00addance Mentality paradigm.\n\nA character rich in integrity, maturity, and the Abundance Mentality has a genuineness that goes far beyond technique, or lack of it, in human interaction.\n\nOne thing I have found particularly helpful to Win\/Lose people in developing a Win\/Win character is to associate with some model or mentor who really thinks Win\/Win. When people are deeply scripted in Win\/Lose or other philosophies and regularly associate with others who are likewise scripted, they don't have much opportunity to see and experience the Win\/Win philosophy in action. So I recommend reading literature, such as the inspiring biography of Anwar Sadat, In Search of Identity, and seeing movies like Chariots of Fire or plays like Les Miserables that expose you to models of Win\/Win.\n\nBut remember: If we search deeply enough within ourselves\u2014beyond the scripting, beyond the learned attitudes and behaviors\u2014the real validation of Win\/Win, as well as every other correct principle, is in our own lives.\n\nRelationships\n\nFrom the foundation of character, we build and maintain Win\/Win relationships. The trust, the Emotional Bank Account, is the essence of Win\/Win. Without trust, the best we can do is compro\u00admise; without trust, we lack the credibility for open, mutual learning and communication and real creativity.\n\nBut if our Emotional Bank Account is high, credibility is no longer an issue. Enough deposits have been made so that you know and I know that we deeply respect each other. We're focused on the issues, not on personalities or positions.\n\nBecause we trust each other, we're open. We put our cards on the table. Even though we see things differently, I know that you're willing to listen with respect while I describe the young woman to you, and you know that I'll treat your description of the old woman with the same respect. We're both committed to try to understand each other's point of view deeply and to work together for the Third Alternative, the synergistic solution, that will be a better answer for both of us.\n\nA relationship where bank accounts are high and both parties are deeply committed to Win\/Win is the ideal springboard for tremendous synergy (Habit 6). That relationship neither makes the issues any less real or important, nor eliminates the differences in perspective. But it does eliminate the negative energy normally focused on differences in personality and position and creates a positive, cooperative energy focused on thoroughly understanding the issues and resolving them in a mutually beneficial way.\n\nBut what if that kind of relationship isn't there? What if you have to work out an agreement with someone who hasn't even heard of Win\/Win and is deeply scripted in Win\/Lose or some other philosophy?\n\nDealing with Win\/Lose is the real test of Win\/Win. Rarely is Win\/Win easily achieved in any circumstance. Deep issues and fundamental differences have to be dealt with. But it is much easier when both parties are aware of and committed to it and where there is a high Emotional Bank Account in the relationship.\n\nWhen you're dealing with a person who is coming from a paradigm of Win\/Lose, the relationship is still the key. The place to focus is on your Circle of Influence. You make deposits into the Emotional Bank Account through genuine courtesy, respect, and appreciation for that person and for the other point of view. You stay longer in the communication process. You listen more, you listen in greater depth. You express yourself with greater courage. You aren't reactive. You go deeper inside yourself for strength of character to be proactive. You keep hammering it out until the other person begins to realize that you genuinely want the reso\u00adlution to be a real win for both of you. That very process is a tremendous deposit in the Emotional Bank Account.\n\nAnd the stronger you are\u2014the more genuine your character, the higher your level of proactivity, the more committed you really are to Win\/Win\u2014the more powerful your influence will be with that other person. This is the real test of interpersonal leadership. It goes beyond transactional leadership into transformational leader\u00adship, transforming the individuals involved as well as the relation\u00adship.\n\nBecause Win\/Win is a principle people can validate in their own lives, you will be able to bring most people to a realization that they will win more of what they want by going for what you both want. But there will be a few who are so deeply embedded in the Win\/Lose mentality that they just won't think Win\/Win. So remem\u00adber that No Deal is always an option. Or you may occasionally choose to go for the low form of Win\/Win\u2014compromise.\n\nIt's important to realize that not all decisions need to be Win\/Win, even when the Emotional Bank Account is high. Again, the key is the relationship. If you and I worked together, for example, and you were to come to me and say, \"Stephen, I know you won't like this decision. I don't have time to explain it to you, let alone get you involved. There's a good possibility you'll think it's wrong. But will you support it?\"\n\nIf you had a positive Emotional Bank Account with me, of course I'd support it. I'd hope you were right and I was wrong. I'd work to make your decision work.\n\nBut if the Emotional Bank Account weren't there, and if I were reactive, I wouldn't really support it. I might say I would to your face, but behind your back I wouldn't be very enthusiastic. I wouldn't make the investment necessary to make it succeed. \"It didn't work,\" I'd say. \"So what do you want me to do now?\"\n\nIf I were overreactive, I might even torpedo your decision and do what I could to make sure others did too. Or I might become \"maliciously obedient\" and do exactly and only what you tell me to do, accepting no responsibility for results.\n\nDuring the five years I lived in Great Britain, I saw that country brought twice to its knees because the train conductors were maliciously obedient in following all the rules and procedures written on paper.\n\nAn agreement means very little in letter without the character and relationship base to sustain it in spirit. So we need to approach Win\/Win from a genuine desire to invest in the relationships that make it possible.\n\nAgreements\n\nFrom relationships flow the agreements that give definition and direction to Win\/Win. They are sometimes called performance agreements or partnership agreements, shifting the paradigm of pro\u00adductive interaction from vertical to horizontal, from hovering supervision to self-supervision, from positioning to being partners in success.\n\nWin\/Win agreements cover a wide scope of interdependent interaction. We discussed one important application when we talked about delegation in the \"Green and Clean\" story in Habit 3. The same five elements we listed there provide the structure for Win\/Win agreements between employers and employees, between independent people working together on projects, between groups of people cooperatively focused on a common objective, between companies and suppliers\u2014between any people who need to interact to accomplish. They create an effective way to clarify and manage expectations between people involved in any interdepen\u00addent endeavor.\n\nIn the Win\/Win agreement, the following five elements are made very explicit:\n\nDesired results (not methods) identify what is to be done and when.\n\nGuidelines specify the parameters (principles, policies, etc.) within which results are to be accomplished.\n\nResources identify the human, financial, technical, or organizational support available to help accomplish the results.\n\nAccountability sets up the standards of performance and the time of evaluation.\n\nConsequences specify\u2014good and bad, natural and logical\u2014what does and will happen as a result of the evaluation.\n\nThese five elements give Win\/Win agreements a life of their own. A clear mutual understanding and agreement up front in these areas creates a standard against which people can measure their own success.\n\nTraditional authoritarian supervision is a Win\/Lose paradigm. It's also the result of an overdrawn Emotional Bank Account. If you don't have trust or a common vision of desired results, you tend to hover over, check up on, and direct. Trust isn't there, so you feel as though you have to control people.\n\nBut if the trust account is high, what is your method? Get out of their way. As long as you have an up-front Win\/Win agreement and they know exactly what is expected, your role is to be a source of help and to receive their accountability reports.\n\nIt is much more ennobling to the human spirit to let people judge themselves than to judge them. And in a high trust culture, it's much more accurate. In many cases people know in their hearts how things are going much better than the records show. Discern\u00adment is often far more accurate than either observation or mea\u00adsurement.\n\nWin\/Win Management Training\n\nSeveral years ago, I was indirectly involved in a consulting project with a very large banking institution that had scores of branches. They wanted us to evaluate and improve their manage\u00adment training program, which was supported by an annual budget of $750,000. The program involved selecting college graduates and putting them through twelve two-week assignments in various departments over a six-month period of time so that they could get a general sense of the industry. They spent two weeks in commer\u00adcial loans, two weeks in industrial loans, two weeks in marketing, two weeks in operations, and so forth. At the end of the six-month period, they were assigned as assistant managers in the various branch banks.\n\nOur assignment was to evaluate the six-month formal training period. As we began, we discovered that the most difficult part of the assignment was to get a clear picture of the desired results. We asked the top executives the key hard question: \"What should these people be able to do when they finish the program?\" And the answers we got were vague and often contradictory.\n\nThe training program dealt with methods, not results; so we suggested that they set up a pilot training program based on a different paradigm called \"learner-controlled instruction.\" This was a Win\/Win agreement that involved identifying specific objec\u00adtives and criteria that would demonstrate their accomplishment and identifying the guidelines, resources, accountability, and consequences that would result when the objectives were met. The consequences in this case were promotion to assistant manager, where they would receive the on-the-job part of their training, and a significant increase in salary.\n\nWe had to really press to get the objectives hammered out. \"What is it you want them to understand about accounting? What about marketing? What about real estate loans?\" And we went down the list. They finally came up with over one hundred objectives, which we simplified, reduced, and consolidated until we came down to 39 specific behavioral objectives with criteria attached to them.\n\nThe trainees were highly motivated by both the opportunity and the increased salary to meet the criteria as soon as possible. There was a big win in it for them, and there was also a big win for the company because they would have assistant branch managers who met results-oriented criteria instead of just showing up for twelve different activity traps.\n\nSo we explained the difference between learner-controlled in\u00adstruction and system-controlled instruction to the trainees. We basically said, \"Here are the objectives and the criteria. Here are the resources, including learning from each other. So go to it. As soon as you meet the criteria, you will be promoted to assistant managers.\"\n\nThey were finished in three-and-a-half weeks. Shifting the training paradigm had released unbelievable motivation and cre\u00adativity.\n\nAs with many paradigm shifts, there was resistance. Almost all of the top executives simply wouldn't believe it. When they were shown the evidence that the criteria had been met, they basically said, \"These trainees don't have the experience. They lack the seasoning necessary to give them the kind of judgment we want them to have as assistant branch managers.\"\n\nIn talking with them later, we found that what many of them were really saying was, \"We went through goat week; how come these guys don't have to?\" But of course they couldn't put it that way. \"They lack seasoning\" was a much more acceptable expres\u00adsion.\n\nIn addition, for obvious reasons (including the $750,000 budget for a six-month program), the personnel department was upset.\n\nSo we responded, \"Fair enough. Let's develop some more objectives and attach criteria to them. But let's stay with the paradigm of learner-controlled instruction.\" We hammered out eight more objectives with very tough criteria in order to give the executives the assurance that the people were adequately prepared to be assistant branch managers and continue the on-the-job part of the training program. After participating in some of the sessions where these criteria were developed, several of the executives remarked that if the trainees could meet these tough criteria, they would be better prepared than almost any who had gone through the six-month program.\n\nWe had prepared the trainees to expect resistance. We took the additional objectives and criteria back to them and said, \"Just as we expected, management wants you to accomplish some additional objectives with even tougher criteria than before. They have assured us this time that if you meet these criteria, they will make you assistant managers.\"\n\nThey went to work in unbelievable ways. They went to the executives in departments such as accounting and basically said, \"Sir, I am a member of this new pilot program called learner-controlled instruction, and it is my understanding that you partic\u00adipated in developing the objectives and the criteria.\n\n\"I have six criteria to meet in this particular department. I was able to pass three of them off with skills I gained in college; I was able to get another one out of a book; I learned the fifth one from Tom, the fellow you trained last week. I only have one criterion left to meet, and I wonder if you or someone else in the department might be able to spend a few hours with me to show me how.\" So they spent half a day in a department instead of two weeks.\n\nThese trainees cooperated with each other, brainstormed with each other, and they accomplished the additional objectives in a week and a half. The six-month program was reduced to five weeks, and the results were significantly increased.\n\nThis kind of thinking can similarly affect every area of organiza\u00adtional life if people have the courage to explore their paradigms and to concentrate on Win\/Win. I am always amazed at the results that happen, both to individuals and to organizations, when responsible, proactive, self-directing individuals are turned loose on a task.\n\nWin\/Win Performance Agreements\n\nCreating Win\/Win performance agreements requires vital para\u00addigm shifts. The focus is on results; not methods. Most of us tend to supervise methods. We use the gofer delegation discussed in Habit 3, the methods management I used with Sandra when I asked her to take pictures of our son as he was waterskiing. But Win\/Win agreements focus on results, releasing tremendous indi\u00advidual human potential and creating greater synergy, building PC in the process instead of focusing exclusively on P.\n\nWith Win\/Win accountability, people evaluate themselves. The traditional evaluation games people play are awkward and emo\u00adtionally exhausting. In Win\/Win, people evaluate themselves, using the criteria that they themselves helped to create up front. And if you set it up correctly, people can do that. With a Win\/Win delegation agreement, even a seven-year-old boy can tell for himself how well he's keeping the yard \"green and clean.\"\n\nMy best experiences in teaching university classes have come when I have created a Win\/Win shared understanding of the goal up front. \"This is what we're trying to accomplish. Here are the basic requirements for an A, B, or C grade. My goal is to help every one of you get an A. Now you take what we've talked about and analyze it and come up with your own understanding of what you want to accomplish that is unique to you. Then let's get together and agree on the grade you want and what you plan to do to get it.\"\n\nManagement philosopher and consultant Peter Drucker recom\u00admends the use of a \"manager's letter\" to capture the essence of performance agreements between managers and their employees. Following a deep and thorough discussion of expectations, guide\u00adlines and resources to make sure they are in harmony with organizational goals, the employee writes a letter to the manager that summarizes the discussion and indicates when the next performance plan or review discussion will take place.\n\nDeveloping such a Win\/Win performance agreement is the central activity of management. With an agreement in place, employees can manage themselves within the framework of that agreement. The manager then can serve like a pace car in a race. He can get things going and then get out of the way. His job from then on is to remove the oil spills.\n\nWhen a boss becomes the first assistant to each of his subordi\u00adnates, he can greatly increase his span of control. Entire levels of administration and overhead can be eliminated. Instead of super\u00advising six or eight, such a manager can supervise twenty, thirty, fifty, or more.\n\nIn Win\/Win performance agreements, consequences become the natural or logical result of performance rather than a reward or punishment arbitrarily handed out by the person in charge.\n\nThere are basically four kinds of consequences (rewards and penalties) that management or parents can control\u2014financial, psychic, opportunity, and responsibility. Financial consequences include such things as income, stock options, allowances, or penalties. Psychic or psychological consequences include recogni\u00adtion, approval, respect, credibility, or the loss of them. Unless people are in a survival mode, psychic compensation is often more motivating than financial compensation. Opportunity includes training, development, perks, and other benefits. Responsibility has to do with scope and authority, either of which can be enlarged or diminished. Win\/Win agreements specify consequences in one or more of those areas and the people involved know it up front. So you don't play games. Everything is clear from the beginning.\n\nIn addition to these logical, personal consequences, it is also important to clearly identify what the natural organizational con\u00adsequences are. For example, what will happen if I'm late to work, if I refuse to cooperate with others, if I don't develop good Win\/Win performance agreements with my subordinates, if I don't hold them accountable for desired results, or if I don't promote their professional growth and career development?\n\nWhen my daughter turned 16, we set up a Win\/Win agreement regarding use of the family car. We agreed that she would obey the laws of the land and that she would keep the car clean and properly maintained. We agreed that she would use the car only for responsible purposes and would serve as a cab driver for her mother and me within reason. And we also agreed that she would do all her other jobs cheerfully without being reminded. These were our wins.\n\nWe also agreed that I would provide some resources\u2014the car, gas, and insurance. And we agreed that she would meet weekly with me, usually on Sunday afternoon, to evaluate how she was doing based on our agreement. The consequences were clear. As long as she kept her part of the agreement, she could use the car. If she didn't keep it, she would lose the privilege until she decided to.\n\nThis Win\/Win agreement set up clear expectations from the beginning on both our parts. It was a win for her\u2014she got to use the car\u2014and it was certainly a win for Sandra and me. Now she could handle her own transportation needs and even some of ours. We didn't have to worry about maintaining the car or keeping it clean. And we had a built-in accountability, which meant I didn't have to hover over her or manage her methods. Her integrity, her conscience, her power of discernment and our high Emotional Bank Account managed her infinitely better. We didn't have to get emotionally strung out, trying to supervise her every move and coming up with punishments or rewards on the spot if she didn't do things the way we thought she should. We had a Win\/Win agreement, and it liberated us all.\n\nWin\/Win agreements are tremendously liberating. But as the product of isolated techniques, they won't hold up. Even if you set them up in the beginning, there is no way to maintain them without personal integrity and a relationship of trust.\n\nA true Win\/Win agreement is the product of the paradigm, the character, and the relationships out of which it grows. In that context, it defines and directs the interdependent interaction for which it was created.\n\nSystems\n\nWin\/Win can only survive in an organization when the systems support it. If you talk Win\/Win but reward Win\/Lose, you've got a losing program on your hands.\n\nYou basically get what you reward. If you want to achieve the goals and reflect the values in your mission statement, then you need to align the reward system with these goals and values. If it isn't aligned systemically, you won't be walking your talk. You'll be in the situation of the manager I mentioned earlier who talked cooperation but practiced competition by creating a \"Race to Bermuda\" contest.\n\nI worked for several years with a very large real estate organization in the Middle West. My first experience with this organiza\u00adtion was at a large sales rally where over 800 sales associates gathered for the annual reward program. It was a psych-up cheerleading session, complete with high school bands and a great deal of frenzied screaming.\n\nOut of the 800 people there, around forty received awards for top performance, such as \"Most Sales,\" \"Greatest Volume,\" \"Highest Earned Commissions,\" and \"Most Listings.\" There was a lot of hoopla\u2014excitement, cheering, applause\u2014around the pre\u00adsentation of these awards. There was no doubt that those forty people had won; but there was also the underlying awareness that 760 people had lost.\n\nWe immediately began educational and organizational develop\u00adment work to align the systems and structures of the organization toward the Win\/Win paradigm. We involved people at a grass roots level to develop the kinds of systems that would motivate them. We also encouraged them to cooperate and synergize with each other so that as many as possible could achieve the desired results of their individually tailored performance agreements.\n\nAt the next rally one year later, there were over 1,000 sales associates present, and about 800 of them received awards. There were a few individual winners based on comparisons, but the program primarily focused on people achieving self-selected performance objectives and on groups achieving team objectives. There was no need to bring in the high school bands to artificially contrive the fanfare, the cheerleading, and the psych up. There was tremendous natural interest and excitement because people could share in each other's happiness, and teams of sales associates could experience rewards together, including a vacation trip for the entire office.\n\nThe remarkable thing was that almost all of the 800 who received the awards that year had produced as much per person in terms of volume and profit as the previous year's forty. The spirit of Win\/Win had significantly increased the number of golden eggs and had fed the goose as well, releasing enormous human energy and talent. The resulting synergy was astounding to almost everyone involved.\n\nCompetition has its place in the marketplace or against last year's performance\u2014perhaps even against another office or indi\u00advidual where there is no particular interdependence, no need to cooperate. But cooperation in the workplace is as important to free enterprise as competition in the marketplace. The spirit of Win\/Win cannot survive in an environment of competition and contests.\n\nFor Win\/Win to work, the systems have to support it. The training system, the planning system, the communication system, the budgeting system, the information system, the compensation system\u2014all have to be based on the principle of Win\/Win.\n\nI did some consulting for another company that wanted training for their people in human relations. The underlying assumption was that the problem was the people.\n\nThe president said, \"Go into any store you want and see how they treat you. They're just order takers. They don't understand how to get close to the customers. They don't know the product, and they don't have the knowledge and the skill in the sales process necessary to create a marriage between the product and the need.\"\n\nSo I went to the various stores. And he was right. But that still didn't answer the question in my mind: What caused the attitude?\n\n\"Look, we're on top of the problem,\" the president said. \"We have department heads out there setting a great example. We've told them their job is two-thirds selling and one-third management, and they're outselling everybody. We just want you to provide some training for the salespeople.\"\n\nThose words raised a red flag. \"Let's get some more data,\" I said.\n\nHe didn't like that. He \"knew\" what the problem was, and he wanted to get on with training. But I persisted, and within two days we uncovered the real problem. Because of the job definition and the compensation system, the managers were \"creaming.\" They'd stand behind the cash register and cream all the business during the slow times. Half the time in retail is slow and the other half is frantic. So the managers would give all the dirty jobs\u2014inventory control, stock work, and cleaning\u2014to the salespeople. And they would stand behind the registers and cream. That's why the department heads were tops in sales.\n\nSo we changed one system\u2014the compensation system\u2014and the problem was corrected overnight. We set up a system whereby the managers only made money when their salespeople made money. We overlapped the needs and goals of the managers with the needs and goals of the salespeople. And the need for human relations training suddenly disappeared. The key was developing a true Win\/Win reward system.\n\nIn another instance, I worked with a manager in a company that required formal performance evaluations. He was frustrated over the evaluation rating he had given a particular manager. \"He deserved a three,\" he said, \"but I had to give him a one\" (which meant superior, promotable).\n\n\"What did you give him a one for?\" I asked.\n\n\"He gets the numbers,\" was his reply.\n\n\"So why do you think he deserves a three?\"\n\n\"It's the way he gets them. He neglects people; he runs over them. He's a troublemaker.\"\n\n\"It sounds like he's totally focused on P\u2014on production. And that's what he's being rewarded for. But what would happen if you talked with him about the problem, if you helped him understand the importance of PC?\"\n\nHe said he had done so, with no effect.\n\n\"Then what if you set up a Win\/Win contract with him where you both agreed that two-thirds of his compensation would come from P\u2014from the numbers\u2014and the other one-third would come from PC\u2014how other people perceive him, what kind of leader, people builder, team builder he is?\"\n\n\"Now that would get his attention,\" he replied.\n\nSo often the problem is in the system, not in the people. If you put good people in bad systems, you get bad results. You have to water the flowers you want to grow.\n\nAs people really learn to think Win\/Win, they can set up the systems to create and reinforce it. They can transform unnecessar\u00adily competitive situations to cooperative ones and can powerfully impact their effectiveness by building both P and PC.\n\nIn business, executives can align their systems to create teams of highly productive people working together to compete against external standards of performance. In education, teachers can set up grading systems based on an individual's performance in the context of agreed upon criteria and can encourage students to cooperate in productive ways to help each other learn and achieve. In families, parents can shift the focus from competition with each other to cooperation. In activities such as bowling, for example, they can keep a family score and try to beat a previous one. They can set up home responsibilities with Win\/Win agreements that eliminate constant nagging and enable parents to do the things only they can do.\n\nA friend once shared with me a cartoon he'd seen of two children talking to each other. \"If mommy doesn't get us up soon,\" one was saying, \"we're going to be late for school.\" These words brought forcibly to his attention the nature of the problems created when families are not organized on a responsible Win\/Win basis.\n\nWin\/Win puts the responsibility on the individual for accom\u00adplishing specified results within clear guidelines and available resources. It makes a person accountable to perform and evaluate the results and provides consequences as a natural result of performance. And Win\/Win systems create the environment which supports and reinforces the Win\/Win performance agreements.\n\nProcesses\n\nThere's no way to achieve Win\/Win ends with Win\/Lose or Lose\/Win means. You can't say, \"You're going to think Win\/Win, whether you like it or not.\" So the question becomes how to arrive at a Win\/Win solution.\n\nRoger Fisher and William Ury, two Harvard law professors, have done some outstanding work in what they call the \"principled\" approach versus the \"positional\" approach to bargaining in their tremendously useful and insightful book, Getting to Yes. Although the words Win\/Win are not used, the spirit and underlying philosophy of the book are in harmony with the Win\/Win ap\u00adproach.\n\nThey suggest that the essence of principled negotiation is to separate the person from the problem, to focus on interests and not on positions, to invent options for mutual gain, and to insist on objective criteria\u2014some external standard or principle that both parties can buy into.\n\nIn my own work with various people and organizations seeking Win\/Win solutions, I suggest that they become involved in the following four-step process:\n\nFirst, see the problem from the other point of view. Really seek to understand and to give expression to the needs and concerns of the other party as well as or better than they can themselves.\n\nSecond, identify the key issues and concerns (not positions) involved.\n\nThird, determine what results would constitute a fully acceptable solution.\n\nAnd fourth, identify possible new options to achieve those results.\n\nHabits 5 and 6 deal directly with two of the elements of this process, and we will go into those in depth in the next two chapters.\n\nBut at this juncture, let me point out the highly interrelated nature of the process of Win\/Win with the essence of Win\/Win itself. You can only achieve Win\/Win solutions with Win\/Win processes\u2014the end and the means are the same.\n\nWin\/Win is not a personality technique. It's a total paradigm of human interaction. It comes from a character of integrity, maturity, and the Abundance Mentality. It grows out of high-trust relation\u00adships. It is embodied in agreements that effectively clarify and manage expectations as well as accomplishment. It thrives in supportive systems. And it is achieved through the process we are now prepared to more fully examine in Habits 5 and 6.\n\nAPPLICATION SUGGESTIONS:\n\n  1. Think about an upcoming interaction wherein you will be at\u00adtempting to reach an agreement or negotiate a solution. Commit to maintain a balance between courage and consideration.\n  2. Make a list of obstacles that keep you from applying the Win\/Win paradigm more frequently. Determine what could be done within your Circle of Influence to eliminate some of those obstacles.\n  3. Select a specific relationship where you would like to develop a Win\/Win agreement. Try to put yourself in the other person's place, and write down explicitly how you think that person sees the solution. Then list, from your own perspective, what results would constitute a Win for you. Approach the other person and ask if he or she would be willing to communicate until you reach a point of agreement and mutually beneficial solution.\n  4. Identify three key relationships in your life. Give some indication of what you feel the balance is in each of the Emotional Bank Accounts. Write down some specific ways you could make deposits in each account.\n  5. Deeply consider your own scripting. Is it Win\/Lose? How does that scripting affect your interactions with other people? Can you identify the main source of that script? Determine whether or not those scripts serve well in your current reality.\n  6. Try to identify a model of Win\/Win thinking who, even in hard situations, really seeks mutual benefit. Determine now to more closely watch and learn from this person's example.\n\n### HABIT 5  \nSEEK FIRST TO UNDERSTAND,  \nTHEN TO BE UNDERSTOOD\n\n### PRINCIPLES OF EMPATHIC  \nCOMMUNICATION\n\nThe heart has its reasons which reason knows not of.\n\nPascal\n\nSUPPOSE YOU'VE BEEN HAVING TROUBLE WITH YOUR EYES and you decide to go to an optometrist for help. After briefly listening to your complaint, he takes off his glasses and hands them to you.\n\n\"Put these on,\" he says. \"I've worn this pair of glasses for ten years now and they've really helped me. I have an extra pair at home; you can wear these.\"\n\nSo you put them on, but it only makes the problem worse.\n\n\"This is terrible!\" you exclaim. \"I can't see a thing!\"\n\n\"Well, what's wrong?\" he asks. \"They work great for me. Try harder.\"\n\n\"I am trying,\" you insist. \"Everything is a blur.\"\n\n\"Well, what's the matter with you? Think positively.\"\n\n\"Okay. I positively can't see a thing.\"\n\n\"Boy, are you ungrateful!\" he chides. \"And after all I've done to help you!\"\n\nWhat are the chances you'd go back to that optometrist the next time you needed help? Not very good, I would imagine. You don't have much confidence in someone who doesn't diagnose before he or she prescribes.\n\nBut how often do we diagnose before we prescribe in commu\u00adnication?\n\n\"Come on, honey, tell me how you feel. I know it's hard, but I'll try to understand.\"\n\n\"Oh, I don't know, Mom. You'd think it was stupid.\"\n\n\"Of course I wouldn't! You can tell me. Honey, no one cares for you as much as I do. I'm only interested in your welfare. What's making you so unhappy?\"\n\n\"Oh, I don't know.\"\n\n\"Come on, honey. What is it?\"\n\n\"Well, to tell you the truth, I just don't like school anymore.\"\n\n\"What?\" you respond incredulously. \"What do you mean you don't like school? And after all the sacrifices we've made for your education! Education is the foundation of your future. If you'd apply yourself like your older sister does, you'd do better and then you'd like school. Time and time again, we've told you to settle down. You've got the ability, but you just don't apply yourself. Try harder. Get a positive attitude about it.\"\n\nPause.\n\n\"Now go ahead. Tell me how you feel.\"\n\nWe have such a tendency to rush in, to fix things up with good advice. But we often fail to take the time to diagnose, to really, deeply understand the problem first.\n\nIf I were to summarize in one sentence the single most important principle I have learned in the field of interpersonal relations, it would be this: Seek first to understand, then to be understood. This principle is the key to effective interpersonal communication.\n\nCHARACTER AND COMMUNICATION\n\nRight now, you're reading a book I've written. Reading and writing are both forms of communication. So are speaking and listening. In fact, those are the four basic types of communication. And think of all the hours you spend doing at least one of those four things. The ability to do them well is absolutely critical to your effectiveness.\n\nCommunication is the most important skill in life. We spend most of our waking hours communicating. But consider this: You've spent years learning how to read and write, years learning how to speak. But what about listening? What training or education have you had that enables you to listen so that you really, deeply understand another human being from that individual's own frame of reference?\n\nComparatively few people have had any training in listening at all. And, for the most part, their training has been in the Person\u00adality Ethic of technique, truncated from the character base and the relationship base absolutely vital to authentic understanding of another person.\n\nIf you want to interact effectively with me, to influence me\u2014your spouse, your child, your neighbor, your boss, your coworker, your friend\u2014you first need to understand me. And you can't do that with technique alone. If I sense you're using some technique, I sense duplicity, manipulation. I wonder why you're doing it, what your motives are. And I don't feel safe enough to open myself up to you.\n\nThe real key to your influence with me is your example, your actual conduct. Your example flows naturally out of your character, or the kind of person you truly are\u2014not what others say you are or what you may want me to think you are. It is evident in how I actually experience you.\n\nYour character is constantly radiating, communicating. From it, in the long run, I come to instinctively trust or distrust you and your efforts with me.\n\nIf your life runs hot and cold, if you're both caustic and kind, and, above all, if your private performance doesn't square with your public performance, it's very hard for me to open up with you. Then, as much as I may want and even need to receive your love and influence, I don't feel safe enough to expose my opinions and experiences and my tender feelings. Who knows what will happen?\n\nBut unless I open up with you, unless you understand me and my unique situation and feelings, you won't know how to advise or counsel me. What you say is good and fine, but it doesn't quite pertain to me.\n\nYou may say you care about and appreciate me. I desperately want to believe that. But how can you appreciate me when you don't even understand me? All I have are your words, and I can't trust words.\n\nI'm too angry and defensive\u2014perhaps too guilty and afraid\u2014to be influenced, even though inside I know I need what you could tell me.\n\nUnless you're influenced by my uniqueness, I'm not going to be influenced by your advice. So if you want to be really effective in the habit of interpersonal communication, you cannot do it with technique alone. You have to build the skills of empathic listening on a base of character that inspires openness and trust. And you have to build the Emotional Bank Accounts that create a commerce between hearts.\n\nEMPATHIC LISTENING\n\n\"Seek first to understand\" involves a very deep shift in para\u00addigm. We typically seek first to be understood. Most people do not listen with the intent to understand; they listen with the intent to reply. They're either speaking or preparing to speak. They're filtering everything through their own paradigms, reading their autobiography into other people's lives.\n\n\"Oh, I know exactly how you feel!\"\n\n\"I went through the very same thing. Let me tell you about my experience.\"\n\nThey're constantly projecting their own home movies onto other people's behavior. They prescribe their own glasses for everyone with whom they interact.\n\nIf they have a problem with someone\u2014a son, a daughter, a spouse, an employee\u2014their attitude is, \"That person just doesn't understand.\"\n\nA father once told me, \"I can't understand my kid. He just won't listen to me at all.\"\n\n\"Let me restate what you just said,\" I replied. \"You don't understand your son because he won't listen to you?\"\n\n\"That's right,\" he replied.\n\n\"Let me try again,\" I said. \"You don't understand your son because he won't listen to you?\"\n\n\"That's what I said,\" he impatiently replied.\n\n\"I thought that to understand another person, you needed to listen to him,\" I suggested.\n\n\"Oh!\" he said. There was a long pause. \"Oh!\" he said again, as the light began to dawn. \"Oh, yeah! But I do understand him. I know what he's going through. I went through the same thing myself. I guess what I don't understand is why he won't listen to me.\"\n\nThis man didn't have the vaguest idea of what was really going on inside his boy's head. He looked into his own head and thought he saw the world, including his boy.\n\nThat's the case with so many of us. We're filled with our own rightness, our own autobiography. We want to be understood. Our conversations become collective monologues, and we never really understand what's going on inside another human being.\n\nWhen another person speaks, we're usually \"listening\" at one of four levels. We may be ignoring another person, not really listening at all. We may practice pretending. \"Yeah. Uh-huh. Right.\" We may practice selective listening, hearing only certain parts of the con\u00adversation. We often do this when we're listening to the constant chatter of a preschool child. Or we may even practice attentive listening, paying attention and focusing energy on the words that are being said. But very few of us ever practice the fifth level, the highest form of listening, empathic listening.\n\nWhen I say empathic listening, I am not referring to the techniques of \"active\" listening or \"reflective\" listening, which basically involve mimicking what another person says. That kind of listening is skill-based, truncated from character and relation\u00adships, and often insults those \"listened\" to in such a way. It is also essentially autobiographical. If you practice those techniques, you may not project your autobiography in the actual interaction, but your motive in listening is autobiographical. You listen with reflective skills, but you listen with intent to reply, to control, to manipulate.\n\nWhen I say empathic listening, I mean listening with intent to understand. I mean seeking first to understand, to really understand. It's an entirely different paradigm.\n\nEmpathic (from empathy) listening gets inside another person's frame of reference. You look out through it, you see the world the way they see the world, you understand their paradigm, you understand how they feel.\n\nEmpathy is not sympathy. Sympathy is a form of agreement, a form of judgment. And it is sometimes the more appropriate emotion and response. But people often feed on sympathy. It makes them dependent. The essence of empathic listening is not that you agree with someone; it's that you fully, deeply, under\u00adstand that person, emotionally as well as intellectually.\n\nEmpathic listening involves much more than registering, reflect\u00ading, or even understanding the words that are said. Communica\u00adtions experts estimate, in fact, that only 10 percent of our communication is represented by the words we say. Another 30 percent is represented by our sounds, and 60 percent by our body language. In empathic listening, you listen with your ears, but you also, and more importantly, listen with your eyes and with your heart. You listen for feeling, for meaning. You listen for behavior. You use your right brain as well as your left. You sense, you intuit, you feel.\n\nEmpathic listening is so powerful because it gives you accurate data to work with. Instead of projecting your own autobiography and assuming thoughts, feelings, motives and interpretation, you're dealing with the reality inside another person's head and heart. You're listening to understand. You're focused on receiving the deep communication of another human soul.\n\nIn addition, empathic listening is the key to making deposits in Emotional Bank Accounts, because nothing you do is a deposit unless the other person perceives it as such. You can work your fingers to the bone to make a deposit, only to have it turn into a withdrawal when a person regards your efforts as manipulative, self-serving, intimidating, or condescending because you don't understand what really matters to him.\n\nEmpathic listening is, in and of itself, a tremendous deposit in the Emotional Bank Account. It's deeply therapeutic and healing because it gives a person \"psychological air.\"\n\nIf all the air were suddenly sucked out of the room you're in right now, what would happen to your interest in this book? You wouldn't care about the book; you wouldn't care about anything except getting air. Survival would be your only motivation.\n\nBut now that you have air, it doesn't motivate you. This is one of the greatest insights in the field of human motivation: Satisfied needs do not motivate. It's only the unsatisfied need that motivates. Next to physical survival, the greatest need of a human being is psychological survival\u2014to be understood, to be affirmed, to be validated, to be appreciated.\n\nWhen you listen with empathy to another person, you give that person psychological air. And after that vital need is met, you can then focus on influencing or problem solving.\n\nThis need for psychological air impacts communication in every area of life.\n\nI taught this concept at a seminar in Chicago one time, and I instructed the participants to practice empathic listening during the evening. The next morning, a man came up to me almost bursting with news.\n\n\"Let me tell you what happened last night,\" he said. \"I was trying to close a big commercial real estate deal while I was here in Chicago. I met with the principals, their attorneys, and another real estate agent who had just been brought in with an alternative proposal.\n\n\"It looked as if I were going to lose the deal. I had been working on this deal for over six months and, in a very real sense, all my eggs were in this one basket. All of them. I panicked. I did everything I could\u2014I pulled out all the stops\u2014I used every sales technique I could. The final stop was to say, 'Could we delay this decision just a little longer?' But the momentum was so strong and they were so disgusted by having this thing go on so long, it was obvious they were going to close.\n\n\"So I said to myself, 'Well, why not try it? Why not practice what I learned today and seek first to understand, then to be under\u00adstood? I've got nothing to lose.'\n\n\"I just said to the man, 'Let me see if I really understand what your position is and what your concerns about my recommenda\u00adtions really are. When you feel I understand them, then we'll see whether my proposal has any relevance or not.'\n\n\"I really tried to put myself in his shoes. I tried to verbalize his needs and concerns, and he began to open up.\n\n\"The more I sensed and expressed the things he was worried about, the results he anticipated, the more he opened up.\n\n\"Finally, in the middle of our conversation, he stood up, walked over to the phone, and dialed his wife. Putting his hand over the mouthpiece, he said, 'You've got the deal.'\n\n\"I was totally dumbfounded,\" he told me. \"I still am this morning.\"\n\nHe had made a huge deposit in the Emotional Bank Account by giving the man psychological air. When it comes right down to it, other things being relatively equal, the human dynamic is more important than the technical dimensions of the deal.\n\nSeeking first to understand, diagnosing before you prescribe, is hard. It's so much easier in the short run to hand someone a pair of glasses that have fit you so well these many years.\n\nBut in the long run, it severely depletes both P and PC. You can't achieve maximum interdependent production from an inaccurate understanding of where other people are coming from. And you can't have interpersonal PC\u2014high Emotional Bank Accounts\u2014if the people you relate with don't really feel understood.\n\nEmpathic listening is also risky. It takes a great deal of security to go into a deep listening experience because you open yourself up to be influenced. You become vulnerable. It's a paradox, in a sense, because in order to have influence, you have to be influenced. That means you have to really understand.\n\nThat's why Habits 1, 2, and 3 are so foundational. They give you the changeless inner core, the principle center, from which you can handle the more outward vulnerability with peace and strength.\n\nDIAGNOSE BEFORE YOU PRESCRIBE\n\nAlthough it's risky and hard, seek first to understand, or diagnose before you prescribe, is a correct principle manifest in many areas of life. It's the mark of all true professionals. It's critical for the optometrist, it's critical for the physician. You wouldn't have any confidence in a doctor's prescription unless you had confidence in the diagnosis.\n\nWhen our daughter Jenny was only two months old, she was sick one Saturday, the day of a football game in our community that dominated the consciousness of almost everyone. It was an important game\u2014some 60,000 people were there. Sandra and I would like to have gone, but we didn't want to leave little Jenny. Her vomiting and diarrhea had us concerned.\n\nThe doctor was at that game. He wasn't our personal physician, but he was the one on call. When Jenny's situation got worse, we decided we needed some medical advice.\n\nSandra dialed the stadium and had him paged. It was right at a critical time in the game, and she could sense an officious tone in his voice. \"Yes?\" he said briskly. \"What is it?\"\n\n\"This is Mrs. Covey, Doctor, and we're concerned about our daughter, Jenny.\"\n\n\"What's the situation?\" he asked.\n\nSandra described the symptoms, and he said, \"Okay. I'll call in a prescription. Which is your pharmacy?\"\n\nWhen she hung up, Sandra felt that in her rush she hadn't really given him full data, but that what she had told him was adequate.\n\n\"Do you think he realizes that Jenny is just a newborn?\" I asked her.\n\n\"I'm sure he does,\" Sandra replied.\n\n\"But he's not our doctor. He's never even treated her.\"\n\n\"Well, I'm pretty sure he knows.\"\n\n\"Are you willing to give her the medicine unless you're abso\u00adlutely sure he knows?\"\n\nSandra was silent. \"What are we going to do?\" she finally said.\n\n\"Call him back,\" I said.\n\n\"You call him back,\" Sandra replied.\n\nSo I did. He was paged out of the game once again. \"Doctor,\" I said, \"when you called in that prescription, did you realize that Jenny is just two months old?\"\n\n\"No!\" he exclaimed. \"I didn't realize that. It's good you called me back. I'll change the prescription immediately.\"\n\nIf you don't have confidence in the diagnosis, you won't have confidence in the prescription.\n\nThis principle is also true in sales. An effective sales person first seeks to understand the needs, the concerns, the situation of the customer. The amateur salesman sells products; the professional sells solutions to needs and problems. It's a totally different approach. The professional learns how to diagnose, how to under\u00adstand. He also learns how to relate people's needs to his products and services. And, he has to have the integrity to say, \"My product or service will not meet that need\" if it will not.\n\nDiagnosing before you prescribe is also fundamental to law. The professional lawyer first gathers the facts to understand the situation, to understand the laws and precedents, before preparing a case. A good lawyer almost writes the opposing attorney's case before he writes his own.\n\nIt's also true in product design. Can you imagine someone in a company saying, \"This consumer research stuff is for the birds. Let's design products.\" In other words, forget understanding the consumer's buying habits and motives\u2014just design products. It would never work.\n\nA good engineer will understand the forces, the stresses at work, before designing the bridge. A good teacher will assess the class before teaching. A good student will understand before he applies. A good parent will understand before evaluating or judging. The key to good judgment is understanding. By judging first, a person will never fully understand.\n\nSeek first to understand is a correct principle evident in all areas of life. It's a generic, common denominator principle, but it has its greatest power in the area of interpersonal relations.\n\nFOUR AUTOBIOGRAPHICAL RESPONSES\n\nBecause we listen autobiographically, we tend to respond in one of four ways. We evaluate\u2014we either agree or disagree; we probe\u2014we ask questions from our own frame of reference; we advise\u2014we give counsel based on our own experience; or we interpret\u2014we try to figure people out, to explain their motives, their behavior, based on our own motives and behavior.\n\nThese responses come naturally to us. We are deeply scripted in them; we live around models of them all the time. But how do they affect our ability to really understand?\n\nIf I'm trying to communicate with my son, can he feel free to open himself up to me when I evaluate everything he says before he really explains it? Am I giving him psychological air?\n\nAnd how does he feel when I probe? Probing is playing twenty questions. It's autobiographical, it controls, and it invades. It's also logical, and the language of logic is different from the language of sentiment and emotion. You can play twenty questions all day and not find out what's important to someone. Constant probing is one of the main reasons parents do not get close to their children.\n\n\"How's it going, son?\"\n\n\"Fine.\"\n\n\"Well, what's been happening lately?\"\n\n\"Nothing.\"\n\n\"So what's exciting in school?\"\n\n\"Not much.\"\n\n\"And what are your plans for the weekend?\"\n\n\"I don't know.\"\n\nYou can't get him off the phone talking with his friends, but all he gives you is one- and two-word answers. Your house is a motel where he eats and sleeps, but he never shares, never opens up.\n\nAnd when you think about it, honestly, why should he, if every time he does open up his soft underbelly, you elephant stomp it with autobiographical advice and \"I told you so's.\"\n\nWe are so deeply scripted in these responses that we don't even realize when we use them. I have taught this concept to thousands of people in seminars across the country, and it never fails to shock them deeply as we role-play empathic listening situations and they finally begin to listen to their own typical responses. But as they begin to see how they normally respond and learn how to listen with empathy, they can see the dramatic results in communication. To many, seek first to understand becomes the most exciting, the most immediately applicable, of all the Seven Habits.\n\nLet's take a look at what well might be a typical communication between a father and his teenage son. Look at the father's words in terms of the four different responses we have just described.\n\n\"Boy, Dad, I've had it! School is for the birds!\"\n\n\"What's the matter, Son?\" (probing).\n\n\"It's totally impractical. I don't get a thing out of it.\"\n\n\"Well, you just can't see the benefits yet, Son. I felt the same way when I was your age. I remember thinking what a waste some of the classes were. But those classes turned out to be the most helpful to me later on. Just hang in there. Give it some time\" (advising).\n\n\"I've given it ten years of my life! Can you tell me what good 'x plus y' is going to be to me as an auto mechanic?\"\n\n\"An auto mechanic? You've got to be kidding\" (evaluating).\n\n\"No, I'm not. Look at Joe. He's quit school. He's working on cars. And he's making lots of money. Now that's practical.\"\n\n\"It may look that way now. But several years down the road, Joe's going to wish he'd stayed in school. You don't want to be an auto mechanic. You need an education to prepare you for some\u00adthing better than that\" (advising).\n\n\"I don't know. Joe's got a pretty good set up.\"\n\n\"Look, Son, have you really tried?\" (probing, evaluating).\n\n\"I've been in high school two years now. Sure I've tried. It's just a waste.\"\n\n\"That's a highly respected school, Son. Give them a little credit\" (advising, evaluating).\n\n\"Well, the other guys feel the same way I do.\"\n\n\"Do you realize how many sacrifices your mother and I have made to get you where you are? You can't quit when you've come this far\" (evaluating).\n\n\"I know you've sacrificed, Dad. But it's just not worth it.\"\n\n\"Look, maybe if you spent more time doing your homework and less time in front of TV....\" (advising, evaluating).\n\n\"Look, Dad. It's just no good. Oh... never mind! I don't want to talk about this anyway.\"\n\nObviously, his father was well intended. Obviously, he wanted to help. But did he even begin to really understand?\n\nLet's look more carefully at the son\u2014not just his words, but his thoughts and feelings (expressed parenthetically below) and the possible effect of some of his dad's autobiographical responses.\n\n\"Boy, Dad, I've had it! School is for the birds!\" (I want to talk with you, to get your attention.)\n\n\"What's the matter, Son?\" (You're interested! Good!)\n\n\"It's totally impractical. I don't get a thing out of it.\" (I've got a problem with school, and I feel just terrible.)\n\n\"Well, you just can't see the benefits yet, Son. I felt the same way when I was your age.\" (Oh, no! Here comes Chapter three of Dad's autobiography. This isn't what I want to talk about. I don't really care how many miles he had to trudge through the snow to school without any boots. I want to get to the problem.) \"I remember thinking what a waste some of the classes were. But those classes turned out to be the most helpful to me later on. Just hang in there. Give it some time.\" (Time won't solve my problem. I wish I could tell you. I wish I could just spit it out.)\n\n\"I've given it ten years of my life! Can you tell me what good 'x plus y' is going to do me as an auto mechanic?\"\n\n\"An auto mechanic? You've got to be kidding.\" (He wouldn't like me if I were an auto mechanic. He wouldn't like me if I didn't finish school. I have to justify what I said.)\n\n\"No, I'm not. Look at Joe. He's quit school. He's working on cars. And he's making lots of money. Now that's practical.\"\n\n\"It may look that way now. But several years down the road, Joe's going to wish he'd stayed in school.\" (Oh, boy! Here comes lecture number sixteen on the value of an education.) \"You don't want to be an auto mechanic.\" (How do you know that, Dad? Do you really have any idea what I want?) \"You need an education to prepare you for something better than that.\"\n\n\"I don't know. Joe's got a pretty good set up.\" (He's not a failure. He didn't finish school and he's not a failure.)\n\n\"Look, Son, have you really tried?\" (We're beating around the bush, Dad. If you'd just listen, I really need to talk to you about something important.)\n\n\"I've been in high school two years now. Sure I've tried. It's just a waste.\"\n\n\"That's a highly respected school, Son. Give them a little credit.\" (Oh, great. Now we're talking credibility. I wish I could talk about what I want to talk about.)\n\n\"Well, the other guys feel the same way I do.\" (I have some credibility, too. I'm not a moron.)\n\n\"Do you realize how many sacrifices your mother and I have made to get you where you are?\" (Uh-oh, here comes the guilt trip. Maybe I am a moron. The school's great, Mom and Dad are great, and I'm a moron.) \"You can't quit when you've come this far.\"\n\n\"I know you've sacrificed, Dad. But it's just not worth it.\" (You just don't understand.)\n\n\"Look, maybe if you spent more time doing your homework and less time in front of TV... .\" (That's not the problem, Dad! That's not it at all! I'll never be able to tell you. I was dumb to try.)\n\n\"Look, Dad. It's just no good. Oh... never mind! I don't want to talk about this anyway.\"\n\nCan you see how limited we are when we try to understand another person on the basis of words alone, especially when we're looking at that person through our own glasses? Can you see how limiting our autobiographical responses are to a person who is genuinely trying to get us to understand his auto\u00adbiography?\n\nYou will never be able to truly step inside another person, to see the world as he sees it, until you develop the pure desire, the strength of personal character, and the positive Emotional Bank Account, as well as the empathic listening skills to do it.\n\nThe skills, the tip of the iceberg of empathic listening, involve four developmental stages.\n\nThe first and least effective is to mimic content. This is the skill taught in \"active\" or \"reflective\" listening. Without the character and relationship base, it is often insulting to people and causes them to close up. It is, however, a first stage skill because it at least causes you to listen to what's being said.\n\nMimicking content is easy. You just listen to the words that come out of someone's mouth and you repeat them. You're hardly even using your brain at all.\n\n\"Boy, Dad, I've had it! School is for the birds!\"\n\n\"You've had it. You think school is for the birds.\"\n\nYou have essentially repeated back the content of what was being said. You haven't evaluated or probed or advised or inter\u00adpreted. You've at least showed you're paying attention to his words. But to understand, you want to do more.\n\nThe second stage of empathic listening is to rephrase the content. It's a little more effective, but it's still limited to the verbal communication.\n\n\"Boy, Dad, I've had it! School is for the birds!\"\n\n\"You don't want to go to school anymore.\"\n\nThis time, you've put his meaning into your own words. Now you're thinking about what he said, mostly with the left side, the reasoning, logical side of the brain.\n\nThe third stage brings your right brain into operation. You reflect feeling.\n\n\"Boy, Dad, I've had it! School is for the birds!\"\n\n\"You're feeling really frustrated.\"\n\nNow you're not paying as much attention to what he's saying as you are to the way he feels about what he's saying. The fourth stage includes both the second and the third. You rephrase the content and reflect the feeling.\n\n\"Boy, Dad, I've had it! School is for the birds!\"\n\n\"You're really frustrated about school.\"\n\nFrustration is the feeling; school is the content. You're using both sides of your brain to understand both sides of his commu\u00adnication.\n\nNow, what happens when you use fourth stage empathic listening skills is really incredible. As you authentically seek to understand, as you rephrase content and reflect feeling, you give him psychological air. You also help him work through his own thoughts and feelings. As he grows in his confidence of your sincere desire to really listen and understand, the barrier between what's going on inside him and what's actually being communi\u00adcated to you disappears. It opens a soul to soul flow. He's not thinking and feeling one thing and communicating another. He begins to trust you with his innermost tender feelings and thoughts.\n\n\"Boy, Dad, I've had it! School is for the birds!\" (I want to talk with you, to get your attention.)\n\n\"You're really frustrated about school.\" (That's right! That's how I feel.)\n\n\"I sure am. It's totally impractical. I'm not getting a thing out of it.\"\n\n\"You feel like school's not doing you any good.\" (Let me think\u2014is that what I mean?)\n\n\"Well, yeah. I'm just not learning anything that's going to help me. I mean, look at Joe. He's dropped out of school and he's working on cars. He's making money. Now that's practical.\"\n\n\"You feel that Joe really has the right idea.\" (Well....)\n\n\"Well, I guess he does in a way. He's really making money now. But in a few years, I bet he'll probably be ticked off at him\u00adself.\"\n\n\"You think Joe's going to feel he made the wrong decision.\"\n\n\"He's got to. Just look at what he's giving up. I mean, if you don't have an education, you just can't make it in this world.\"\n\n\"Education is really important.\"\n\n\"Oh, yeah! I mean, if you don't have a diploma, if you can't get jobs or go to college, what are you going to do? You've just got to get an education.\"\n\n\"It's important to your future.\"\n\n\"It is. And... you know what? I'm really worried. Listen, you won't tell Mom, will you?\"\n\n\"You don't want your mother to find out.\"\n\n\"Well, not really. Oh, I guess you can tell her. She'll probably find out anyway. Look, I took this test today, this reading test. And, Dad, they said I'm reading on a fourth-grade level. Fourth grade! And I'm a junior in high school!\"\n\nWhat a difference real understanding can make! All the well-meaning advice in the world won't amount to a hill of beans if we're not even addressing the real problem. And we'll never get to the problem if we're so caught up in our own autobiography, our own paradigms, that we don't take off our glasses long enough to see the world from another point of view.\n\n\"I'm going to flunk, Dad. I guess I figure if I'm going to flunk, I might as well quit. But I don't want to quit.\"\n\n\"You feel torn. You're in the middle of a dilemma.\"\n\n\"What do you think I should do, Dad?\"\n\nBy seeking first to understand, this father has just turned a transactional opportunity into a transformational opportunity. Instead of interacting on a surface, get-the-job-done level of communication, he has created a situation in which he can now have transforming impact, not only on his son but also on the relationship. By setting aside his own autobiography and really seeking to understand, he has made a tremendous deposit in the Emotional Bank Account and has empowered his son to open, layer upon layer, and to get to the real issue.\n\nNow father and son are on the same side of the table looking at the problem, instead of on opposite sides looking across at each other. The son is opening his father's autobiography and asking for advice.\n\nEven as the father begins to counsel, however, he needs to be sensitive to his son's communication. As long as the response is logical, the father can effectively ask questions and give counsel. But the moment the response becomes emotional, he needs to go back to empathic listening.\n\n\"Well, I can see some things you might want to consider.\"\n\n\"Like what, Dad?\"\n\n\"Like getting some special help with your reading. Maybe they have some kind of tutoring program over at the tech school.\"\n\n\"I've already checked into that. It takes two nights and all day Saturday. That would take so much time!\"\n\nSensing emotion in that reply, the father moves back to empathy.\n\n\"That's too much of a price to pay.\"\n\n\"Besides, Dad, I told the sixth graders I'd be their coach.\"\n\n\"You don't want to let them down.\"\n\n\"But I'll tell you this, Dad. If I really thought that tutoring course would help, I'd be down there every night. I'd get someone else to coach those kids.\"\n\n\"You really want the help, but you doubt if the course will make a difference.\"\n\n\"Do you think it would, Dad?\"\n\nThe son is once more open and logical. He's opening his father's autobiography again. Now the father has another opportunity to influence and transform.\n\nThere are times when transformation requires no outside coun\u00adsel. Often when people are really given the chance to open up, they unravel their own problems and the solutions become clear to them in the process.\n\nAt other times, they really need additional perspective and help. The key is to genuinely seek the welfare of the individual, to listen with empathy, to let the person get to the problem and the solution at his own pace and time. Layer upon layer\u2014it's like peeling an onion until you get to the soft inner core.\n\nWhen people are really hurting and you really listen with a pure desire to understand, you'll be amazed how fast they will open up. They want to open up. Children desperately want to open up, even more to their parents than to their peers. And they will, if they feel their parents will love them unconditionally and will be faithful to them afterwards and not judge or ridicule them.\n\nIf you really seek to understand, without hypocrisy and without guile, there will be times when you will be literally stunned with the pure knowledge and understanding that will flow to you from another human being. It isn't even always necessary to talk in order to empathize. In fact, sometimes words may just get in your way. That's one very important reason why technique alone will not work. That kind of understanding transcends technique. Isolated technique only gets in the way.\n\nI have gone through the skills of empathic listening because skill is an important part of any habit. We need to have the skills. But let me reiterate that the skills will not be effective unless they come from a sincere desire to understand. People resent any attempt to manipulate them. In fact, if you're dealing with people you're close to, it's helpful to tell them what you're doing.\n\n\"I read this book about listening and empathy and I thought about my relationship with you. I realized I haven't listened to you like I should. But I want to. It's hard for me. I may blow it at times, but I'm going to work at it. I really care about you and I want to understand. I hope you'll help me.\" Affirming your motive is a huge deposit.\n\nBut if you're not sincere, I wouldn't even try it. It may create an openness and a vulnerability that will later turn to your harm when a person discovers that you really didn't care, you really didn't want to listen, and he's left open, exposed, and hurt. The tech\u00adnique, the tip of the iceberg, has to come out of the massive base of character underneath.\n\nNow there are people who protest that empathic listening takes too much time. It may take a little more time initially but it saves so much time downstream. The most efficient thing you can do if you're a doctor and want to prescribe a wise treatment is to make an accurate diagnosis. You can't say, \"I'm in too much of a hurry. I don't have time to make a diagnosis. Just take this treatment.\"\n\nI remember writing one time in a room on the north shore of Oahu, Hawaii. There was a soft breeze blowing, and so I had opened two windows\u2014one at the front and one at the side\u2014to keep the room cool. I had a number of papers laid out, chapter by chapter, on a large table.\n\nSuddenly, the breeze started picking up and blowing my papers about. I remember the frantic sense of loss I felt because things were no longer in order, including unnumbered pages, and I began rushing around the room trying desperately to put them back. Finally, I realized it would be better to take ten seconds and close one of the windows.\n\nEmpathic listening takes time, but it doesn't take anywhere near as much time as it takes to back up and correct misunderstandings when you're already miles down the road, to redo, to live with unexpressed and unsolved problems, to deal with the results of not giving people psychological air.\n\nA discerning empathic listener can read what's happening down deep fast, and can show such acceptance, such understanding, that other people feel safe to open up layer after layer until they get to that soft inner core where the problem really lies.\n\nPeople want to be understood. And whatever investment of time it takes to do that will bring much greater returns of time as you work from an accurate understanding of the problems and issues and from the high Emotional Bank Account that results when a person feels deeply understood.\n\nUNDERSTANDING AND PERCEPTION\n\nAs you learn to listen deeply to other people, you will discover tremendous differences in perception. You will also begin to appreciate the impact that these differences can have as people try to work together in interdependent situations.\n\nYou see the young woman; I see the old lady. And both of us can be right.\n\nYou may look at the world through spouse-centered glasses; I see it through the money-centered lens of economic concern.\n\nYou may be scripted in the abundance mentality; I may be scripted in the scarcity mentality.\n\nYou may approach problems from a highly visual, intuitive, holistic right brain paradigm; I may be very left brain, very sequential, analytical, and verbal in my approach.\n\nOur perceptions can be vastly different. And yet we both have lived with our paradigms for years, thinking they are \"facts,\" and questioning the character or the mental competence of anyone who can't \"see the facts.\"\n\nNow, with all our differences, we're trying to work together\u2014in a marriage, in a job, in a community service project\u2014to manage resources and accomplish results. So how do we do it? How do we transcend the limits of our individual perceptions so that we can deeply communicate, so that we can cooperatively deal with the issues and come up with Win\/Win solutions?\n\nThe answer is Habit 5. It's the first step in the process of Win\/Win. Even if (and especially when) the other person is not coming from that paradigm, seek first to understand.\n\nThis principle worked powerfully for one executive who shared with me the following experience:\n\n\"I was working with a small company that was in the process of negotiating a contract with a large national banking institution. This institution flew in their lawyers from San Francisco, their negotiator from Ohio, and presidents of two of their large banks to create an eight-person negotiating team. The company I worked with had decided to go for Win\/Win or No Deal. They wanted to significantly increase the level of service and the cost, but they had been almost overwhelmed with the demands of this large financial institution.\n\n\"The president of our company sat across the negotiating table and told them, 'We would like for you to write the contract the way you want it so that we can make sure we understand your needs and your concerns. We will respond to those needs and concerns. Then we can talk about pricing.'\n\n\"The members of the negotiating team were overwhelmed. They were astounded that they were going to have the opportunity to write the contract. They took three days to come up with the deal.\n\n\"When they presented it, the president said, 'Now let's make sure we understand what you want.' And he went down the contract, rephrasing the content, reflecting the feeling, until he was sure and they were sure he understood what was important to them. 'Yes. That's right. No, that's not exactly what we meant here... yes, you've got it now.'\n\n\"When he thoroughly understood their perspective, he proceeded to explain some concerns from his perspective... and they listened. They were ready to listen. They weren't fighting for air. What had started out as a very formal, low-trust, almost hostile atmosphere had turned into a fertile environment for synergy.\n\n\"At the conclusion of the discussions, the members of the negotiating team basically said, 'We want to work with you. We want to do this deal. Just let us know what the price is and we'll sign.'\"\n\nTHEN SEEK TO BE UNDERSTOOD\n\nSeek first to understand... then to be understood. Knowing how to be understood is the other half of Habit 5, and is equally critical in reaching Win\/Win solutions.\n\nEarlier we defined maturity as the balance between courage and consideration. Seeking to understand requires consideration; seeking to be understood takes courage. Win\/Win requires a high degree of both. So it becomes important in interdependent situations for us to be understood.\n\nThe early Greeks had a magnificent philosophy which is embodied in three sequentially arranged words: ethos, pathos, and logos. I suggest these three words contain the essence of seeking first to understand and making effective presentations.\n\nEthos is your personal credibility, the faith people have in your integrity and competency. It's the trust that you inspire, your Emotional Bank Account. Pathos is the empathic side\u2014it's the feeling. It means that you are in alignment with the emotional thrust of another person's communication. Logos is the logic, the reasoning part of the presentation.\n\nNotice the sequence: ethos, pathos, logos\u2014your character, and your relationships, and then the logic of your presentation. This represents another major paradigm shift. Most people, in making presentations, go straight to the logos, the left brain logic, of their ideas. They try to convince other people of the validity of that logic without first taking ethos and pathos into consideration.\n\nI had an acquaintance who was very frustrated because his boss was locked into what he felt was an unproductive leadership style.\n\n\"Why doesn't he do anything?\" he asked me. \"I've talked to him about it, he's aware of it, but he does nothing.\"\n\n\"Well, why don't you make an effective presentation?\" I asked.\n\n\"I did,\" was the reply.\n\n\"How do you define 'effective'? Who do they send back to school when the salesman doesn't sell\u2014the buyer? Effective means it works; it means P\/PC. Did you create the change you wanted? Did you build the relationship in the process? What were the results of your presentation?\"\n\n\"I told you, he didn't do anything. He wouldn't listen.\"\n\n\"Then make an effective presentation. You've got to empathize with his head. You've got to get into his frame of mind. You've got to make your point simply and visually and describe the alternative he is in favor of better than he can himself. That will take some homework. Are you willing to do that?\"\n\n\"Why do I have to go through all that?\" he asked.\n\n\"In other words, you want him to change his whole leadership style and you're not willing to change your method of presenta\u00adtion?\"\n\n\"I guess so,\" he replied.\n\n\"Well, then,\" I said, \"just smile about it and learn to live with it.\"\n\n\"I can't live with it,\" he said. \"It compromises my integrity.\"\n\n\"Okay, then get to work on an effective presentation. That's in your Circle of Influence.\"\n\nIn the end, he wouldn't do it. The investment seemed too great.\n\nAnother acquaintance, a university professor, was willing to pay the price. He approached me one day and said, \"Stephen, I can't get to first base in getting the funding I need for my research because my research is really not in the mainstream of this department's interests.\"\n\nAfter discussing his situation at some length, I suggested that he develop an effective presentation using ethos, pathos, and logos. \"I know you're sincere and the research you want to do would bring great benefits. Describe the alternative they are in favor of better than they can themselves. Show that you understand them in depth. Then carefully explain the logic behind your request.\"\n\n\"Well, I'll try,\" he said.\n\n\"Do you want to practice with me?\" I asked. He was willing, and so we dress rehearsed his approach.\n\nWhen he went in to make his presentation, he started by saying, \"Now let me see if I first understand what your objectives are, and what your concerns are about this presentation and my rec\u00adommendation.\"\n\nHe took the time to do it slowly, gradually. In the middle of his presentation, demonstrating his depth of understanding and re\u00adspect for their point of view, a senior professor turned to another professor, nodded, turned back to him, and said, \"You've got your money.\"\n\nWhen you can present your own ideas clearly, specifically, visually, and most important, contextually\u2014in the context of a deep understanding of their paradigms and concerns\u2014you signif\u00adicantly increase the credibility of your ideas.\n\nYou're not wrapped up in your \"own thing,\" delivering gran\u00addiose rhetoric from a soapbox. You really understand. What you're presenting may even be different from what you had originally thought because in your effort to understand, you learned.\n\nHabit 5 lifts you to greater accuracy, greater integrity, in your presentations. And people know that. They know you're present\u00ading the ideas which you genuinely believe, taking all known facts and perceptions into consideration, that will benefit everyone.\n\nONE ON ONE\n\nHabit 5 is powerful because it is right in the middle of your Circle of Influence. Many factors in interdependent situations are in your Circle of Concern\u2014problems, disagreements, circumstances, other people's behavior. And if you focus your energies out there, you deplete them with little positive results.\n\nBut you can always seek first to understand. That's something that's within your control. And as you do that, as you focus on your Circle of Influence, you really, deeply understand other people. You have accurate information to work with, you get to the heart of matters quickly, you build Emotional Bank Accounts, and you give people the psychological air they need so you can work together effectively.\n\nIt's the inside-out approach. And as you do it, watch what happens to your Circle of Influence. Because you really listen, you become influenceable. And being influenceable is the key to influencing others. Your circle begins to expand. You increase your ability to influence many of the things in your Circle of Concern.\n\nAnd watch what happens to you. The more deeply you under\u00adstand other people, the more you will appreciate them, the more reverent you will feel about them. To touch the soul of another human being is to walk on holy ground.\n\nHabit 5 is something you can practice right now. The next time you communicate with anyone, you can put aside your own autobiography and genuinely seek to understand. Even when people don't want to open up about their problems, you can be empathic. You can sense their hearts, you can sense the hurt, and you can respond, \"You seem down today.\" They may say nothing. That's all right. You've shown understanding and respect.\n\nDon't push; be patient; be respectful. People don't have to open up verbally before you can empathize. You can empathize all the time with their behavior. You can be discerning, sensitive, and aware and you can live outside your autobiography when that is needed.\n\nAnd if you're highly proactive, you can create opportunities to do preventive work. You don't have to wait until your son or daughter has a problem with school or you have your next business negotiation to seek first to understand.\n\nSpend time with your children now, one on one. Listen to them; understand them. Look at your home, at school life, at the challenges and the problems they're facing, through their eyes. Build the Emotional Bank Account. Give them air.\n\nGo out with your spouse on a regular basis. Have dinner or do something together you both enjoy. Listen to each other; seek to understand. See life through each other's eyes.\n\nMy daily time with Sandra is something I wouldn't trade for anything. As well as seeking to understand each other, we often take time to actually practice empathic listening skills to help us in communicating with our children.\n\nWe often share our different perceptions of the situation, and we role-play more effective approaches to difficult interpersonal family problems.\n\nI may act as if I am a son or daughter requesting a special privilege even though I haven't fulfilled a basic family responsibil\u00adity, and Sandra plays herself.\n\nWe interact back and forth and try to visualize the situation in a very real way so that we can train ourselves to be consistent in modeling and teaching correct principles to our children. Some of our most helpful role-plays come from redoing a past difficult or stressful scene in which one of us \"blew it.\"\n\nThe time you invest to deeply understand the people you love brings tremendous dividends in open communication. Many of the problems that plague families and marriages simply don't have time to fester and develop. The communication becomes so open that potential problems can be nipped in the bud. And there are great reserves of trust in the Emotional Bank Account to handle the problems that do arise.\n\nIn business, you can set up one-on-one time with your employ\u00adees. Listen to them, understand them. Set up human resource accounting or stakeholder information systems in your business to get honest, accurate feedback at every level: from customers, suppliers, and employees. Make the human element as important as the financial or the technical element. You save tremendous amounts of time, energy, and money when you tap into the human resources of a business at every level. When you listen, you learn. And you also give the people who work for you and with you psychological air. You inspire loyalty that goes well beyond the eight-to-five physical demands of the job.\n\nSeek first to understand. Before the problems come up, before you try to evaluate and prescribe, before you try to present your own ideas\u2014seek to understand. It's a powerful habit of effective interdependence.\n\nWhen we really, deeply understand each other, we open the door to creative solutions and third alternatives. Our differences are no longer stumbling blocks to communication and progress. Instead, they become the stepping stones to synergy.\n\nAPPLICATION SUGGESTIONS:\n\n  1. Select a relationship in which you sense the Emotional Bank Account is in the red. Try to understand and write down the situation from the other person's point of view. In your next interaction, listen for understanding, comparing what you are hearing with what you wrote down. How valid were your assumptions? Did you really understand that individual's per\u00adspective?\n  2. Share the concept of empathy with someone close to you. Tell him or her you want to work on really listening to others and ask for feedback in a week. How did you do? How did it make that person feel?\n  3. The next time you have an opportunity to watch people commu\u00adnicate, cover your ears for a few minutes and just watch. What emotions are being communicated that may not come across in words alone?\n  4. Next time you catch yourself inappropriately using one of the autobiographical responses\u2014probing, evaluating, advising, or interpreting\u2014try to turn the situation into a deposit by acknowl\u00adedgment and apology. (\"I'm sorry, I just realized I'm not really trying to understand. Could we start again?\")\n  5. Base your next presentation on empathy. Describe the other point of view as well as or better than its proponents; then seek to have your point understood from their frame of reference.\n\n### HABIT 6  \nSYNERGIZE\n\n### PRINCIPLES OF CREATIVE COOPERATION\n\nI take as my guide the hope of a saint:  \nin crucial things, unity\u2014  \nin important things, diversity\u2014  \nin all things, generosity.\n\nInaugural address of President George Bush\n\nW HEN SIR WINSTON CHURCHILL WAS CALLED to head up the war effort for Great Britain, he remarked that all his life had prepared him for this hour. In a similar sense, the exercise of all of the other habits prepares us for the habit of synergy.\n\nWhen properly understood, synergy is the highest activity in all life\u2014the true test and manifestation of all of the other habits put together.\n\nThe highest forms of synergy focus the four unique human endowments, the motive of Win\/Win, and the skills of empathic communication on the toughest challenges we face in life. What results is almost miraculous. We create new alternatives\u2014something that wasn't there before.\n\nSynergy is the essence of principle-centered leadership. It is the essence of principle-centered parenting. It catalyzes, unifies, and unleashes the greatest powers within people. All the habits we have covered prepare us to create the miracle of synergy.\n\nWhat is synergy? Simply defined, it means that the whole is greater than the sum of its parts. It means that the relationship which the parts have to each other is a part in and of itself. It is not only a part, but the most catalytic, the most empowering, the most unifying, and the most exciting part.\n\nThe creative process is also the most terrifying part because you don't know exactly what's going to happen or where it is going to lead. You don't know what new dangers and challenges you'll find. It takes an enormous amount of internal security to begin with the spirit of adventure, the spirit of discovery, the spirit of creativity. Without doubt, you have to leave the comfort zone of base camp and confront an entirely new and unknown wilderness. You become a trailblazer, a pathfinder. You open new possibilities, new territories, new continents, so that others can follow.\n\nSynergy is everywhere in nature. If you plant two plants close together, the roots comingle and improve the quality of the soil so that both plants will grow better than if they were separated. If you put two pieces of wood together, they will hold much more than the total of the weight held by each separately. The whole is greater than the sum of its parts. One plus one equals three or more.\n\nThe challenge is to apply the principles of creative cooperation, which we learn from nature, in our social interactions. Family life provides many opportunities to observe synergy and to practice it.\n\nThe very way that a man and a woman bring a child into the world is synergistic. The essence of synergy is to value differences\u2014to respect them, to build on strengths, to compensate for weaknesses.\n\nWe obviously value the physical differences between men and women, husbands and wives. But what about the social, mental, and emotional differences? Could these differences not also be sources of creating new, exciting forms of life\u2014creating an environment that is truly fulfilling for each person, that nurtures the self-esteem and self-worth of each, that creates opportunities for each to mature into independence and then gradually into inter\u00addependence? Could synergy not create a new script for the next generation\u2014one that is more geared to service and contribution, and is less protective, less adversarial, less selfish; one that is more open, more trusting, more giving, and is less defensive, protective, and political; one that is more loving, more caring, and is less possessive and judgmental?\n\nSYNERGISTIC COMMUNICATION\n\nWhen you communicate synergistically, you are simply opening your mind and heart and expressions to new possibilities, new alternatives, new options. It may seem as if you are casting aside Habit 2 (to begin with the end in mind); but, in fact, you're doing the opposite\u2014you're fulfilling it. You're not sure when you engage in synergistic communication how things will work out or what the end will look like, but you do have an inward sense of excitement and security and adventure, believing that it will be significantly better than it was before. And that is the end that you have in mind.\n\nYou begin with the belief that parties involved will gain more insight, and that the excitement of that mutual learning and insight will create a momentum toward more and more insights, learnings, and growth.\n\nMany people have not really experienced even a moderate degree of synergy in their family life or in other interactions. They've been trained and scripted into defensive and protective communications or into believing that life or other people can't be trusted. As a result, they are never really open to Habit 6 and to these principles.\n\nThis represents one of the great tragedies and wastes in life, because so much potential remains untapped\u2014completely undeveloped and unused. Ineffective people live day after day with unused potential. They experience synergy only in small, peripheral ways in their lives.\n\nThey may have memories of some unusual creative experiences, perhaps in athletics, where they were involved in a real team spirit for a period of time. Or perhaps they were in an emergency situation where people cooperated to an unusually high degree and submerged ego and pride in an effort to save someone's life or to produce a solution to a crisis.\n\nTo many, such events may seem unusual, almost out of character with life, even miraculous. But this is not so. These things can be produced regularly, consistently, almost daily in people's lives. But it requires enormous personal security and openness and a spirit of adventure.\n\nMost all creative endeavors are somewhat unpredictable. They often seem ambiguous, hit-or-miss, trial and error. And unless people have a high tolerance for ambiguity and get their security from integrity to principles and inner values they find it unnerving and unpleasant to be involved in highly creative enterprises. Their need for structure, certainty, and predictability is too high.\n\nSYNERGY IN THE CLASSROOM\n\nAs a teacher, I have come to believe that many truly great classes teeter on the very edge of chaos. Synergy tests whether teachers and students are really open to the principle of the whole being greater than the sum of its parts.\n\nThere are times when neither the teacher nor the student knows for sure what's going to happen. In the beginning, there's a safe environment that enables people to be really open and to learn and to listen to each other's ideas. Then comes brainstorming, where the spirit of evaluation is subordinated to the spirit of creativity, imagining, and intellectual networking. Then an absolutely unusual phenomenon begins to take place. The entire class is transformed with the excitement of a new thrust, a new idea, a new direction that's hard to define, yet it's almost palpable to the people involved.\n\nSynergy is almost as if a group collectively agrees to subordinate old scripts and to write a new one.\n\nI'll never forget a university class I taught in leadership philosophy and style. We were about three weeks into a semester when, in the middle of a presentation, one person started to relate some very powerful personal experiences which were both emotional and insightful. A spirit of humility and reverence fell upon the class\u2014reverence toward this individual and appreciation for his courage.\n\nThis spirit became fertile soil for a synergistic and creative endeavor. Others began to pick up on it, sharing some of their experiences and insights and even some of their self-doubts. The spirit of trust and safety prompted many to become extremely open. Rather than present what they prepared, they fed on each other's insights and ideas and started to create a whole new scenario as to what that class could mean.\n\nI was deeply involved in the process. In fact, I was almost mesmerized by it because it seemed so magical and creative. And I found myself gradually loosening up my commitment to the structure of the class and sensing entirely new possibilities. It wasn't just a flight of fancy; there was a sense of maturity and stability and substance which transcended by far the old structure and plan.\n\nWe abandoned the old syllabus, the purchased textbooks and all the presentation plans, and we set up new purposes and projects and assignments. We became so excited about what was happening that in about three more weeks, we all sensed an overwhelming desire to share what was happening with others.\n\nWe decided to write a book containing our learnings and insights on the subject of our study\u2014principles of leadership. Assignments were changed, new projects undertaken, new teams formed. People worked much harder than they ever would have in the original class structure, and for an entirely different set of reasons.\n\nOut of this experience emerged an extremely unique, cohesive, and synergistic culture that did not end with the semester. For years, alumni meetings were held among members of that class. Even today, many years later, when we see each other, we talk about it and often attempt to describe what happened and why.\n\nOne of the interesting things to me was how little time had transpired before there was sufficient trust to create such synergy. I think it was largely because the people were relatively mature. They were in the final semester of their senior year, and I think they wanted more than just another good classroom experience. They were hungry for something new and exciting, something that they could create that was truly meaningful. It was \"an idea whose time had come\" for them.\n\nIn addition, the chemistry was right. I felt that experiencing synergy was more powerful than talking about it, that producing something new was more meaningful than simply reading something old.\n\nI've also experienced, as I believe most people have, times that were almost synergistic, times that hung on the edge of chaos and for some reason descended into it. Sadly, people who are burned by such experiences often begin their next new experience with that failure in mind. They defend themselves against it and cut themselves off from synergy.\n\nIt's like administrators who set up new rules and regulations based on the abuses of a few people inside an organization, thus limiting the freedom and creative possibilities for many\u2014or business partners who imagine the worst scenarios possible and write them up in legal language, killing the whole spirit of creativity, enterprise, and synergistic possibility.\n\nAs I think back on many consulting and executive education experiences, I can say that the highlights were almost always synergistic. There was usually an early moment that required considerable courage, perhaps in becoming extremely authentic, in confronting some inside truth about the individual or the organization or the family which really needed to be said, but took a combination of considerable courage and genuine love to say it. Then others became more authentic, open, and honest, and the synergistic communication process began. It usually became more and more creative, and ended up in insights and plans that no one had anticipated initially.\n\nAs Carl Rogers taught, \"That which is most personal is most general.\" The more authentic you become, the more genuine in your expression, particularly regarding personal experiences and even self-doubts, the more people can relate to your expression and the safer it makes them feel to express themselves. That expression in turn feeds back on the other person's spirit, and genuine creative empathy takes place, producing new insights and learnings and a sense of excitement and adventure that keeps the process going.\n\nPeople then begin to interact with each other almost in half sentences, sometimes incoherent, but they get each other's mean\u00adings very rapidly. Then whole new worlds of insights, new perspectives, new paradigms that insure options, new alternatives are opened up and thought about. Though occasionally these new ideas are left up in the air, they usually come to some kind of closure that is practical and useful.\n\nSYNERGY IN BUSINESS\n\nI enjoyed one particularly meaningful synergistic experience as I worked with my associates to create the corporate mission state\u00adment for our business. Almost all members of the company went high up into the mountains where, surrounded by the magnifi\u00adcence of nature, we began with a first draft of what some of us considered to be an excellent mission statement.\n\nAt first the communication was respectful, careful and predict\u00adable. But as we began to talk about the various alternatives, possibilities and opportunities ahead, people became very open and authentic and simply started to think out loud. The mission statement agenda gave way to a collective free association, a spontaneous piggybacking of ideas. People were genuinely em\u00adpathic as well as courageous, and we moved from mutual respect and understanding to creative synergistic communication.\n\nEveryone could sense it. It was exciting. As it matured, we returned to the task of putting the evolved collective vision into words, each of which contains specific and committed-to meaning for each participant.\n\nThe resulting corporate mission statement reads:\n\nOur mission is to empower people and organizations to signifi\u00adcantly increase their performance capability in order to achieve worthwhile purposes through understanding and living principle-centered leadership.\n\nThe synergistic process that led to the creation of our mission statement engraved it in the hearts and minds of everyone there, and it has served us well as a frame of reference of what we are about, as well as what we are not about.\n\nAnother high level synergy experience took place when I ac\u00adcepted an invitation to serve as the resource and discussion catalyst at the annual planning meeting of a large insurance company. Several months ahead, I met with the committee responsible to prepare for and stage the two-day meeting which was to involve all the top executives. They informed me that the traditional pattern was to identify four or five major issues through questionnaires and interviews, and to have alternative proposals presented by the executives. Past meetings had been generally respectful exchanges, occasionally deteriorating into defensive Win\/Lose ego battles. They were usually predictable, uncreative, and boring.\n\nAs I talked with the committee members about the power of synergy, they could sense its potential. With considerable trepida\u00adtion, they agreed to change the pattern. They requested various executives to prepare anonymous \"white papers\" on each of the high priority issues, and then asked all the executives to immerse themselves in these papers ahead of time in order to understand the issues and the differing points of view. They were to come to the meeting prepared to listen rather than to present, prepared to create and synergize rather than to defend and protect.\n\nWe spent the first half-day in the meeting teaching the principles and practicing the skills of Habits 4, 5, and 6. The rest of the time was spent in creative synergy.\n\nThe release of creative energy was incredible. Excitement re\u00adplaced boredom. People became very open to each other's influ\u00adence and generated new insights and options. By the end of the meeting an entirely new understanding of the nature of the central company challenge evolved. The white paper proposals became obsolete. Differences were valued and transcended. A new com\u00admon vision began to form.\n\nOnce people have experienced real synergy, they are never quite the same again. They know the possibility of having other such mind-expanding adventures in the future.\n\nOften attempts are made to recreate a particular synergistic experience, but this seldom can be done. However, the essential purpose behind creative work can be recaptured. Like the Far Eastern philosophy, \"We seek not to imitate the masters, rather we seek what they sought,\" we seek not to imitate past creative synergistic experiences, rather we seek new ones around new and different and sometimes higher purposes.\n\nSYNERGY AND COMMUNICATION\n\nSynergy is exciting. Creativity is exciting. It's phenomenal what openness and communication can produce. The possibilities of truly significant gain, of significant improvement are so real that it's worth the risk such openness entails.\n\nAfter World War II, the United States commissioned David Lilienthal to head the new Atomic Energy Commission. Lilienthal brought together a group of people who were highly influential\u2014celebrities in their own right\u2014disciples, as it were, of their own frames of reference.\n\nThis very diverse group of individuals had an extremely heavy agenda, and they were impatient to get at it. In addition, the press was pushing them.\n\nBut Lilienthal took several weeks to create a high Emotional Bank Account. He had these people get to know each other\u2014their interests, their hopes, their goals, their concerns, their backgrounds, their frames of reference, their paradigms. He facilitated the kind of human interaction that creates a great bonding between people, and he was heavily criticized for taking the time to do it because it wasn't \"efficient.\"\n\nBut the net result was that this group became closely knit together, very open with each other, very creative, and synergistic. The respect among the members of the commission was so high that if there was disagreement, instead of opposition and defense, there was a genuine effort to understand. The attitude was \"If a person of your intelligence and competence and commitment disagrees with me, then there must be something to your disagreement that I don't understand, and I need to understand it. You have a perspective, a frame of reference I need to look at.\" Nonprotective interaction developed, and an unusual culture was born.\n\nThe following diagram illustrates how closely trust is related to different levels of communication.\n\nThe lowest level of communication coming out of low-trust situations would be characterized by defensiveness, protectiveness, and often legalistic language, which covers all the bases and spells out qualifiers and the escape clauses in the event things go sour. Such communication produces only Win\/Lose or Lose\/Lose. It isn't effective\u2014there's no P\/PC balance\u2014and it creates further reasons to defend and protect.\n\nThe middle position is respectful communication. This is the level where fairly mature people interact. They have respect for each other, but they want to avoid the possibility of ugly confrontations, so they communicate politely but not empathically. They might understand each other intellectually, but they really don't deeply look at the paradigms and assumptions underlying their own positions and become open to new possibilities.\n\nRespectful communication works in independent situations and even in interdependent situations, but the creative possibilities are not opened up. In interdependent situations compromise is the position usually taken. Compromise means that 1 + 1 = 1\u00bd. Both give and take. The communication isn't defensive or protective or angry or manipulative; it is honest and genuine and respectful. But it isn't creative or synergistic. It produces a low form of Win\/Win.\n\nSynergy means that 1 + 1 may equal 8, 16, or even 1,600. The synergistic position of high trust produces solutions better than any originally proposed, and all parties know it. Furthermore, they genuinely enjoy the creative enterprise. A miniculture is formed to satisfy in and of itself. Even if it is short lived, the P\/PC balance is there.\n\nThere are some circumstances in which synergy may not be achievable and No Deal isn't viable. But even in these circumstances, the spirit of sincere trying will usually result in a more effective compromise.\n\nFISHING FOR THE THIRD ALTERNATIVE\n\nTo get a better idea of how our level of communication affects our interdependent effectiveness, envision the following scenario:\n\nIt's vacation time, and a husband wants to take his family out to the lake country to enjoy camping and fishing. This is important to him; he's been planning it all year. He's made reservations at a cottage on the lake and arranged to rent a boat, and his sons are really excited about going.\n\nHis wife, however, wants to use the vacation time to visit her ailing mother some 250 miles away. She doesn't have the oppor\u00adtunity to see her very often, and this is important to her.\n\nTheir differences could be the cause of a major negative experience.\n\n\"The plans are set. The boys are excited. We should go on the fishing trip,\" he says.\n\n\"But we don't know how much longer my mother will be around, and I want to be by her,\" she replies. \"This is our only opportunity to have enough time to do that.\"\n\n\"All year long we've looked forward to this one-week vacation. The boys would be miserable sitting around grandmother's house for a week. They'd drive everybody crazy. Besides, your mother's not that sick. And she has your sister less than a mile away to take care of her.\"\n\n\"She's my mother, too. I want to be with her.\"\n\n\"You could phone her every night. And we're planning to spend time with her at the Christmas family reunion. Remember?\"\n\n\"That's not for five more months. We don't even know if she'll still be here by then. Besides, she needs me, and she wants me.\"\n\n\"She's being well taken care of. Besides, the boys and I need you, too.\"\n\n\"My mother is more important than fishing.\"\n\n\"Your husband and sons are more important than your mother.\"\n\nAs they disagree, back and forth, they finally may come up with some kind of compromise. They may decide to split up\u2014he takes the boys fishing at the lake while she visits her mother. And they both feel guilty and unhappy. The boys sense it, and it affects their enjoyment of the vacation.\n\nThe husband may give in to his wife, but he does it grudgingly. And consciously or unconsciously, he produces evidence to fulfill his prophecy of how miserable the week will be for everyone.\n\nThe wife may give in to her husband, but she's withdrawn and overreactive to any new developments in her mother's health situation. If her mother were to become seriously ill and die, the husband could never forgive himself, and she couldn't forgive him either.\n\nWhatever compromise they finally agree on, it could be re\u00adhearsed over the years as evidence of insensitivity, neglect, or a bad priority decision on either part. It could be a source of contention for years and could even polarize the family. Many marriages that once were beautiful and soft and spontaneous and loving have deteriorated to the level of a hostility through a series of incidents just like this.\n\nThe husband and wife see the situation differently. And that difference can polarize them, separate them, create wedges in the relationship. Or it can bring them closer together on a higher level. If they have cultivated the habits of effective interdependence, they approach their differences from an entirely different paradigm. Their communication is on a higher level.\n\nBecause they have a high Emotional Bank Account, they have trust and open communication in their marriage. Because they think Win\/Win, they believe in a third alternative, a solution that is mutually beneficial and is better than what either of them originally proposed. Because they listen empathically and seek first to understand, they create within themselves and between them a comprehensive picture of the values and the concerns that need to be taken into account in making a decision.\n\nAnd the combination of those ingredients\u2014the high Emotional Bank Account, thinking Win\/Win, and seeking first to understand\u2014creates the ideal environment for synergy.\n\nBuddhism calls this \"the middle way.\" Middle in this sense does not mean compromise; it means higher, like the apex of the triangle.\n\nIn searching for the \"middle\" or higher way, this husband and wife realize that their love, their relationship, is part of their synergy.\n\nAs they communicate, the husband really, deeply feels his wife's desire, her need to be with her mother. He understands how she wants to relieve her sister, who has had the primary responsibility for their mother's care. He understands that they really don't know how long she will be with them, and that she certainly is more important than fishing.\n\nAnd the wife deeply understands her husband's desire to have the family together and to provide a great experience for the boys. She realizes the investment that has been made in lessons and equipment to prepare for this fishing vacation, and she feels the importance of creating good memories with them.\n\nSo they pool those desires. And they're not on opposite sides of the problem. They're together on one side, looking at the problem, understanding the needs, and working to create a third alternative that will meet them.\n\n\"Maybe we could arrange another time within the month for you to visit with your mother,\" he suggests. \"I could take over the home responsibilities for the weekend and arrange for some help at the first of the week so that you could go. I know it's important to you to have that time.\n\n\"Or maybe we could locate a place to camp and fish that would be close to your mother. The area wouldn't be as nice, but we could still be outdoors and meet other needs as well. And the boys wouldn't be climbing the walls. We could even plan some recre\u00adational activities with the cousins, aunts, and uncles, which would be an added benefit.\"\n\nThey synergize. They communicate back and forth until they come up with a solution they both feel good about. It's better than the solutions either of them originally proposed. It's better than compromise. It's a synergistic solution that builds P and PC.\n\nInstead of a transaction, it's a transformation. They get what they both really want and build their relationship in the process.\n\nNEGATIVE SYNERGY\n\nSeeking the third alternative is a major paradigm shift from the dichotomous, either\/or mentality. But look at the difference in results!\n\nHow much negative energy is typically expended when people try to solve problems or make decisions in an interdependent reality? How much time is spent in confessing other people's sins, politicking, rivalry, interpersonal conflict, protecting one's back\u00adside, masterminding, and second guessing? It's like trying to drive down the road with one foot on the gas and the other foot on the brake!\n\nAnd instead of getting a foot off the brake, most people give it more gas. They try to apply more pressure, more eloquence, more logical information to strengthen their position.\n\nThe problem is that highly dependent people are trying to succeed in an interdependent reality. They're either dependent on borrowing strength from position power and they go for Win\/Lose, or they're dependent on being popular with others and they go for Lose\/Win. They may talk Win\/Win technique, but they don't really want to listen; they want to manipulate. And synergy can't thrive in that environment.\n\nInsecure people think that all reality should be amenable to their paradigms. They have a high need to clone others, to mold them over into their own thinking. They don't realize that the very strength of the relationship is in having another point of view. Sameness is not oneness; uniformity is not unity. Unity, or oneness, is complementariness, not sameness. Sameness is uncreative... and boring. The essence of synergy is to value the differences.\n\nI've come to believe that the key to interpersonal synergy is intrapersonal synergy, that is synergy within ourselves. The heart of intrapersonal synergy is embodied in the principles in the first three habits, which give the internal security sufficient to handle the risks of being open and vulnerable. By internalizing those principles, we develop the abundance mentality of Win\/Win and the authenticity of Habit 5.\n\nOne of the very practical results of being principle-centered is that it makes us whole\u2014truly integrated. People who are scripted deeply in logical, verbal, left-brain thinking will discover how totally inadequate that thinking is in solving problems which require a great deal of creativity. They become aware and begin to open up a new script inside their right brain. It's not that the right brain wasn't there; it just lay dormant. The muscles had not been developed, or perhaps they had atrophied after early childhood because of the heavy left-brain emphasis of formal education or social scripting.\n\nWhen a person has access to both the intuitive, creative, and visual right brain, and the analytical, logical, verbal left brain, then the whole brain is working. In other words, there is psychic synergy taking place in our own head. And this tool is best suited to the real\u00adity of what life is, because life is not just logical\u2014it is also emotional.\n\nOne day I was presenting a seminar which I titled, \"Manage from the Left, Lead from the Right\" to a company in Orlando, Florida. During the break, the president of the company came up to me and said, \"Stephen, this is intriguing. But I have been thinking about this material more in terms of its application to my marriage than to my business. My wife and I have a real communication problem. I wonder if you would have lunch with the two of us and just kind of watch how we talk to each other?\"\n\n\"Let's do it,\" I replied.\n\nAs we sat down together, we exchanged a few pleasantries. Then this man turned to his wife and said, \"Now, honey, I've invited Stephen to have lunch with us to see if he could help us in our communication with each other. I know you feel I should be a more sensitive, considerate husband. Could you give me something specific you think I ought to do?\" His dominant left brain wanted facts, figures, specifics, parts.\n\n\"Well, as I've told you before, it's nothing specific. It's more of a general sense I have about priorities.\" Her dominant right brain was dealing with sensing and with the gestalt, the whole, the relationship between the parts.\n\n\"What do you mean, 'a general feeling about priorities'? What is it you want me to do? Give me something specific I can get a handle on.\"\n\n\"Well, it's just a feeling.\" Her right brain was dealing in images, intuitive feelings. \"I just don't think our marriage is as important to you as you tell me it is.\"\n\n\"Well, what can I do to make it more important? Give me something concrete and specific to go on.\"\n\n\"It's hard to put into words.\"\n\nAt that point, he just rolled his eyes and looked at me as if to say, \"Stephen, could you endure this kind of dumbness in your marriage?\"\n\n\"It's just a feeling,\" she said, \"a very strong feeling.\"\n\n\"Honey,\" he said to her, \"that's your problem. And that's the problem with your mother. In fact, it's the problem with every woman I know.\"\n\nThen he began to interrogate her as though it were some kind of legal deposition.\n\n\"Do you live where you want to live?\"\n\n\"That's not it,\" she sighed. \"That's not it at all.\"\n\n\"I know,\" he replied with a forced patience. \"But since you won't tell me exactly what it is, I figure the best way to find out what it is is to find out what it is not. Do you live where you want to live?\"\n\n\"I guess.\"\n\n\"Honey, Stephen's here for just a few minutes to try to help us. Just give a quick 'yes' or 'no' answer. Do you live where you want to live?\"\n\n\"Yes.\"\n\n\"Okay. That's settled. Do you have the things you want to have?\"\n\n\"Yes.\"\n\n\"All right. Do you do the things you want to do?\"\n\nThis went on for a little while, and I could see I wasn't helping at all. So I intervened and said, \"Is this kind of how it goes in your relationship?\"\n\n\"Every day, Stephen,\" he replied.\n\n\"It's the story of our marriage,\" she sighed.\n\nI looked at the two of them and the thought crossed my mind that they were two half-brained people living together. \"Do you have any children?\" I asked.\n\n\"Yes, two.\"\n\n\"Really?\" I asked incredulously. \"How did you do it?\"\n\n\"What do you mean how did we do it?\"\n\n\"You were synergistic!\" I said. \"One plus one usually equals two. But you made one plus one equal four. Now that's synergy. The whole is greater than the sum of the parts. So how did you do it?\"\n\n\"You know how we did it,\" he replied.\n\n\"You must have valued the differences!\" I exclaimed.\n\nVALUING THE DIFFERENCES\n\nValuing the differences is the essence of synergy\u2014the mental, the emotional, the psychological differences between people. And the key to valuing those differences is to realize that all people see the world, not as it is, but as they are.\n\nIf I think I see the world as it is, why would I want to value the differences? Why would I even want to bother with someone who's \"off track\"? My paradigm is that I am objective; I see the world as it is. Everyone else is buried by the minutia, but I see the larger picture. That's why they call me a supervisor\u2014I have super vision.\n\nIf that's my paradigm, then I will never be effectively interde\u00adpendent, or even effectively independent, for that matter. I will be limited by the paradigms of my own conditioning.\n\nThe person who is truly effective has the humility and reverence to recognize his own perceptual limitations and to appreciate the rich resources available through interaction with the hearts and minds of other human beings. That person values the differences because those differences add to his knowledge, to his understanding of reality. When we're left to our own experiences, we constantly suffer from a shortage of data.\n\nIs it logical that two people can disagree and that both can be right? It's not logical: it's psychological. And it's very real. You see the young lady; I see the old woman. We're both looking at the same picture, and both of us are right. We see the same black lines, the same white spaces. But we interpret them differently because we've been conditioned to interpret them differently.\n\nAnd unless we value the differences in our perceptions, unless we value each other and give credence to the possibility that we're both right, that life is not always a dichotomous either\/or, that there are almost always third alternatives, we will never be able to transcend the limits of that conditioning.\n\nAll I may see is the old woman. But I realize that you see something else. And I value you. I value your perception. I want to understand.\n\nSo when I become aware of the difference in our perceptions, I say, \"Good! You see it differently! Help me see what you see.\"\n\nIf two people have the same opinion, one is unnecessary. It's not going to do me any good at all to communicate with someone else who sees only the old woman also. I don't want to talk, to communicate, with someone who agrees with me; I want to communicate with you because you see it differently. I value that difference.\n\nBy doing that, I not only increase my own awareness; I also affirm you. I give you psychological air. I take my foot off the brake and release the negative energy you may have invested in defend\u00ading a particular position. I create an environment for synergy.\n\nThe importance of valuing the difference is captured in an often quoted fable called \"The Animal School,\" written by educator Dr. R. H. Reeves:\n\nOnce upon a time, the animals decided they must do something heroic to meet the problems of a \"New World,\" so they organized a school. They adopted an activity curriculum consisting of running, climbing, swimming and flying. To make it easier to administer, all animals took all the subjects.\n\nThe duck was excellent in swimming, better in fact than his instructor, and made excellent grades in flying, but he was very poor in running. Since he was low in running he had to stay after school and also drop swimming to practice running. This was kept up until his web feet were badly worn and he was only average in swimming. But average was acceptable in school, so nobody worried about that except the duck.\n\nThe rabbit started at the top of the class in running, but had a nervous breakdown because of so much makeup in swimming.\n\nThe squirrel was excellent in climbing until he developed frustrations in the flying class where his teacher made him start from the ground up instead of from the tree-top down. He also developed charley horses from over-exertion and he got a C in climbing and a D in running.\n\nThe eagle was a problem child and had to be disciplined severely. In climbing class he beat all the others to the top of the tree, but insisted on using his own way of getting there.\n\nAt the end of the year, an abnormal eel that could swim exceedingly well and also could run, climb and fly a little had the highest average and was valedictorian.\n\nThe prairie dogs stayed out of school and fought the tax levy because the administration would not add digging and burrowing to the curriculum. They apprenticed their children to the badger and later joined the groundhogs and gophers to start a successful private school.\n\nFORCE FIELD ANALYSIS\n\nIn an interdependent situation, synergy is particularly powerful in dealing with negative forces that work against growth and change.\n\nSociologist Kurt Lewin developed a \"Force Field Analysis\" model in which he described any current level of performance or being as a state of equilibrium between the driving forces that encourage upward movement and the restraining forces that discourage it.\n\nDriving forces generally are positive, reasonable, logical, con\u00adscious, and economic. In juxtaposition, restraining forces are often negative, emotional, illogical, unconscious, and social\/psychological. Both sets of forces are very real and must be taken into account in dealing with change.\n\nIn a family, for example, you have a certain \"climate\" in the home\u2014a certain level of positive or negative interaction, of feeling safe or unsafe in expressing feelings or talking about concerns, of respect or disrespect in communication among family members.\n\nYou may really want to change that level. You may want to create a climate that is more positive, more respectful, more open and trusting. Your logical reasons for doing that are the driving forces that act to raise the level.\n\nBut increasing those driving forces is not enough. Your efforts are opposed by restraining forces\u2014by the competitive spirit be\u00adtween children in the family, by the different scripting of home life you and your spouse have brought to the relationship, by habits that have developed in the family, by work or other demands on your time and energies.\n\nIncreasing the driving forces may bring results\u2014for a while. But as long as the restraining forces are there, it becomes increasingly harder. It's like pushing against a spring: the harder you push, the harder it is to push until the force of the spring suddenly thrusts the level back down.\n\nThe resulting up and down, yo-yo effect causes you to feel, after several attempts, that people are \"just the way they are\" and that \"it's too difficult to change.\"\n\nBut when you introduce synergy, you use the motive of Habit 4, the skill of Habit 5, and the interaction of Habit 6 to work directly on the restraining forces. You create an atmosphere in which it is safe to talk about these forces. You unfreeze them, loosen them up, and create new insights that actually transform those restraining forces into driving ones. You involve people in the problem, immerse them in it, so that they soak it in and feel it is their problem and they tend to become an important part of the solution.\n\nAs a result, new goals, shared goals, are created, and the whole enterprise moves upward, often in ways that no one could have anticipated. And the excitement contained within that movement creates a new culture. The people involved in it are enmeshed in each other's humanity and empowered by new, fresh thinking, by new creative alternatives and opportunities.\n\nI've been involved several times in negotiations between people who were angry at each other and hired lawyers to defend their positions. And all that did was to exacerbate the problem because the interpersonal communication deteriorated as it went through the legal process. But the trust level was so low that the parties felt they had no other alternative than to take the issues to court.\n\n\"Would you be interested in going for a Win\/Win solution that both parties feel really good about?\" I would ask.\n\nThe response was usually affirmative, but most people didn't really think it was possible.\n\n\"If I can get the other party to agree, would you be willing to start the process of really communicating with each other?\"\n\nAgain, the answer was usually \"yes.\"\n\nThe results in almost every case have been astounding. Problems that had been legally and psychologically wrangled about for months have been settled in a matter of a few hours or days. Most of the solutions weren't the courthouse compromise solutions, either; they were synergistic, better than the solutions proposed independently by either party. And, in most cases, the relation\u00adships continued even though it had appeared in the beginning that the trust level was so low and the rupture in the relationship so large as to be almost irreparable.\n\nAt one of our development programs, an executive reported a situation where a manufacturer was being sued by a longtime industrial customer for lack of performance. Both parties felt totally justified in the rightness of their position and perceived each other as unethical and completely untrustworthy.\n\nAs they began to practice Habit 5, two things became clear. First, early communication problems resulted in a misunderstanding which was later exacerbated by accusations and counteraccusations. Second, both were initially acting in good faith and didn't like the cost and hassle of a legal fight, but saw no other way out.\n\nOnce these two things became clear, the spirit of Habits 4, 5, and 6 took over, the problem was rapidly resolved, and the relationship continues to prosper.\n\nIn another circumstance, I received an early morning phone call from a land developer desperately searching for help. The bank wanted to foreclose because he was not complying with the principal and interest payment schedule, and he was suing the bank to avoid the foreclosure. He needed additional funding to finish and market the land so that he could repay the bank, but the bank refused to provide additional funds until scheduled payments were met. It was a chicken and egg problem with undercapitaliza\u00adtion.\n\nIn the meantime, the project was languishing. The streets were beginning to look like weed fields, and the owners of the few homes that had been built were up in arms as they saw their property values drop. The city was also upset over the \"prime land\" project falling behind schedule and becoming an eyesore. Tens of thousands of dollars in legal costs had already been spent by the bank and the developer and the case wasn't scheduled to come to court for several months.\n\nIn desperation, this developer reluctantly agreed to try the principles of Habits 4, 5, and 6. He arranged a meeting with even more reluctant bank officials.\n\nThe meeting started at 8 A.M. in one of the bank conference rooms. The tension and mistrust were palpable. The attorney for the bank had committed the bank officials to say nothing. They were only to listen and he alone would speak. He wanted nothing to happen that would compromise the bank's position in court.\n\nFor the first hour and a half, I taught Habits 4, 5, and 6. At 9:30 I went to the blackboard and wrote down the bank's concerns based on our prior understanding. Initially the bank officials said nothing, but the more we communicated Win\/Win intentions and sought first to understand, the more they opened up to explain and clarify.\n\nAs they began to feel understood, the whole atmosphere changed and a sense of momentum, of excitement over the prospect of peacefully settling the problem was clearly evident. Over the attorney's objections the bank officials opened up even more, even about personal concerns. \"When we walk out of here the first thing the bank president will say is, 'Did we get our money?' What are we going to say?\"\n\nBy 11:00, the bank officers were still convinced of their rightness, but they felt understood and were no longer defensive and officious. At that point, they were sufficiently open to listen to the developer's concerns, which we wrote down on the other side of the blackboard. This resulted in deeper mutual understanding and a collective awareness of how poor early communication had resulted in misunderstanding and unrealistic expectations, and how continuous communication in a Win\/Win spirit could have prevented the subsequent major problems from developing.\n\nThe shared sense of both chronic and acute pain combined with a sense of genuine progress kept everyone communicating. By noon, when the meeting was scheduled to end, the people were positive, creative, and synergistic and wanted to keep talking.\n\nThe very first recommendation made by the developer was seen as a beginning Win\/Win approach by all. It was synergized on and improved, and at 12:45 P.M. the developer and the two bank officers left with a plan to present together to the Home Owners Associa\u00adtion and the city. Despite subsequent complicating developments, the legal fight was aborted and the building project continued to a successful conclusion.\n\nI am not suggesting that people should not use legal processes. Some situations absolutely require it. But I see it as a court of last, not first, resort. If it is used too early, even in a preventive sense, sometimes fear and the legal paradigm create subsequent thought and action processes that are not synergistic.\n\nALL NATURE IS SYNERGISTIC\n\nEcology is a word which basically describes the synergism in nature\u2014everything is related to everything else. It's in the rela\u00adtionship that creative powers are maximized, just as the real power in these Seven Habits is in their relationship to each other, not just in the individual habits themselves.\n\nThe relationship of the parts is also the power in creating a synergistic culture inside a family or an organization. The more genuine the involvement, the more sincere and sustained the participation in analyzing and solving problems, the greater the release of everyone's creativity, and of their commitment to what they create. This, I'm convinced, is the essence of the power in the Japanese approach to business, which has changed the world marketplace.\n\nSynergy works; it's a correct principle. It is the crowning achievement of all the previous habits. It is effectiveness in an interdependent reality\u2014it is teamwork, team building, the devel\u00adopment of unity and creativity with other human beings.\n\nAlthough you cannot control the paradigms of others in an interdependent interaction or the synergistic process itself, a great deal of synergy is within your Circle of Influence.\n\nYour own internal synergy is completely within the circle. You can respect both sides of your own nature\u2014the analytical side and the creative side. You can value the difference between them and use that difference to catalyze creativity.\n\nYou can be synergistic within yourself even in the midst of a very adversarial environment. You don't have to take insults personally. You can sidestep negative energy; you can look for the good in others and utilize that good, as different as it may be, to improve your point of view and to enlarge your perspective.\n\nYou can exercise the courage in interdependent situations to be open, to express your ideas, your feelings, and your experiences in a way that will encourage other people to be open also.\n\nYou can value the difference in other people. When someone disagrees with you, you can say, \"Good! You see it differently.\" You don't have to agree with them; you can simply affirm them. And you can seek to understand.\n\nWhen you see only two alternatives\u2014yours and the \"wrong\" one\u2014you can look for a synergistic third alternative. There's almost always a third alternative, and if you work with a Win\/Win philosophy and really seek to understand, you usually can find a solution that will be better for everyone concerned.\n\nAPPLICATION SUGGESTIONS:\n\n  1. Think about a person who typically sees things differently than you do. Consider ways in which those differences might be used as stepping-stones to third alternative solutions. Perhaps you could seek out his or her views on a current project or problem, valuing the different views you are likely to hear.\n  2. Make a list of people who irritate you. Do they represent different views that could lead to 'synergy if you had greater intrinsic security and valued the difference?\n  3. Identify a situation in which you desire greater teamwork and synergy. What conditions would need to exist to support syn\u00adergy? What can you do to create those conditions?\n  4. The next time you have a disagreement or confrontation with someone, attempt to understand the concerns underlying that person's position. Address those concerns in a creative and mutually beneficial way.\n\n## _PART FOUR_\n\n# RENEWAL\n\n### HABIT 7  \nSHARPEN THE SAW\n\n### PRINCIPLES OF BALANCED SELF-RENEWAL\n\nSometimes when I consider what tremendous consequences come from little things... I am tempted to think... there are no little things.\n\nBruce Barton\n\nSUPPOSE YOU WERE TO COME UPON SOMEONE in the woods working feverishly to saw down a tree.\n\n\"What are you doing?\" you ask.\n\n\"Can't you see?\" comes the impatient reply. \"I'm sawing down this tree.\"\n\n\"You look exhausted!\" you exclaim. \"How long have you been at it?\"\n\n\"Over five hours,\" he returns, \"and I'm beat! This is hard work.\"\n\n\"Well, why don't you take a break for a few minutes and sharpen that saw?\" you inquire. \"I'm sure it would go a lot faster.\"\n\n\"I don't have time to sharpen the saw,\" the man says emphat\u00adically. \"I'm too busy sawing!\"\n\nHabit 7 is taking time to sharpen the saw. It surrounds the other habits on the Seven Habits paradigm because it is the habit that makes all the others possible.\n\nFOUR DIMENSIONS OF RENEWAL\n\nHabit 7 is personal PC. It's preserving and enhancing the greatest asset you have\u2014you. It's renewing the four dimensions of your nature\u2014physical, spiritual, mental, and social\/emotional.\n\nAlthough different words are used, most philosophies of life deal either explicitly or implicitly with these four dimensions. Philosopher Herb Shepherd describes the healthy balanced life around four values: perspective (spiritual), autonomy (mental), connectedness (social), and tone (physical). George Sheehan, the running guru, describes four roles: being a good animal (physical), a good craftsman (mental), a good friend (social), and a saint (spiritual). Sound motivation and organization theory embrace these four dimensions or motivations\u2014the economic (physical); how people are treated (social); how people are developed and used (mental); and the service, the job, the contribution the organization gives (spiritual).\n\n\"Sharpen the saw\" basically means expressing all four motiva\u00adtions. It means exercising all four dimensions of our nature, regularly and consistently in wise and balanced ways.\n\nTo do this, we must be proactive. Taking time to sharpen the saw is a definite Quadrant II activity, and Quadrant II must be acted on. Quadrant I, because of its urgency, acts on us; it presses upon us constantly. Personal P\/C must be pressed upon until it becomes second nature, until it becomes a kind of healthy addic\u00adtion. Because it's at the center of our Circle of Influence, no one else can do it for us. We must do it for ourselves.\n\nThis is the single most powerful investment we can ever make in life\u2014investment in ourselves, in the only instrument we have with which to deal with life and to contribute. We are the instruments of our own performance, and to be effective, we need to recognize the importance of taking time regularly to sharpen the saw in all four ways.\n\nThe Physical Dimension\n\nThe physical dimension involves caring effectively for our phys\u00adical body\u2014eating the right kinds of foods, getting sufficient rest and relaxation, and exercising on a regular basis.\n\nExercise is one of those Quadrant II, high-leverage activities that most of us don't do consistently because it isn't urgent. And because we don't do it, sooner or later we find ourselves in Quadrant I, dealing with the health problems and crises that come as a natural result of our neglect.\n\nMost of us think we don't have enough time to exercise. What a distorted paradigm! We don't have time not to. We're talking about three to six hours a week\u2014or a minimum of thirty minutes a day, every other day. That hardly seems an inordinate amount of time considering the tremendous benefits in terms of the impact on the other 162-165 hours of the week.\n\nAnd you don't need any special equipment to do it. If you want to go to a gym or a spa to use the equipment or enjoy some skill sports such as tennis or racquetball, that's an added opportunity. But it isn't necessary to sharpen the saw.\n\nA good exercise program is one that you can do in your own home and one that will build your body in three areas: endurance, flexibility, and strength.\n\nEndurance comes from aerobic exercise, from cardiovascular efficiency\u2014the ability of your heart to pump blood through your body.\n\nAlthough the heart is a muscle, it cannot be exercised directly. It can only be exercised through the large muscle groups, particularly the leg muscles. That's why exercises like rapid walking, running, biking, swimming, cross-country skiing, and jogging are so bene\u00adficial.\n\nYou are considered minimally fit if you can increase your heart rate to at least one hundred beats per minute and keep it at that level for thirty minutes.\n\nIdeally you should try to raise your heart rate to at least sixty percent of your maximum pulse rate, the top speed your heart can beat and still pump blood through your body. Your maximum heart rate is generally accepted to be 220 less your age. So, if you are 40, you should aim for an exercise heart rate of 108 (220 \u2013 40 = 180 x .6 = 108). The \"training effect\" is generally considered to be between 72 and 87 percent of your personal maximum rate.\n\nFlexibility comes through stretching. Most experts recommend warming up before and cooling down\/stretching after aerobic exercise. Before, it helps loosen and warm the muscles to prepare for more vigorous exercise. After, it helps to dissipate the lactic acid so that you don't feel sore and stiff.\n\nStrength comes from muscle resistance exercises\u2014like simple cal\u00adisthenics, push-ups, pull-ups, and sit-ups, and from working with weights. How much emphasis you put on developing strength depends on your situation. If you're involved in physical labor or athletic activities, increased strength will improve your skill. If you have a basically sedentary job and success in your life-style does not require a lot of strength, a little toning through calisthenics in addition to your aerobic and stretching exercises might be suffi\u00adcient.\n\nI was in a gym one time with a friend of mine who has a Ph.D. in exercise physiology. He was focusing on building strength. He asked me to \"spot\" him while he did some bench presses and told me at a certain point he'd ask me to take the weight. \"But don't take it until I tell you,\" he said firmly.\n\nSo I watched and waited and prepared to take the weight. The weight went up and down, up and down. And I could see it begin to get harder. But he kept going. He would start to push it up and I'd think, \"There's no way he's going to make it.\" But he'd make it. Then he'd slowly bring it back down and start back up again. Up and down, up and down.\n\nFinally, as I looked at his face, straining with the effort, his blood vessels practically jumping out of his skin, I thought, \"This is going to fall and collapse his chest. Maybe I should take the weight. Maybe he's lost control and he doesn't even know what he's doing.\" But he'd get it safely down. Then he'd start back up again. I couldn't believe it.\n\nWhen he finally told me to take the weight, I said, \"Why did you wait so long?\"\n\n\"Almost all the benefit of the exercise comes at the very end, Stephen,\" he replied. \"I'm trying to build strength. And that doesn't happen until the muscle fiber ruptures and the nerve fiber registers the pain. Then nature overcompensates and within 48 hours, the fiber is made stronger.\"\n\nI could see his point. It's the same principle that works with emotional muscles as well, such as patience. When you exercise your patience beyond your past limits, the emotional fiber is bro\u00adken, nature overcompensates, and next time the fiber is stronger.\n\nNow my friend wanted to build muscular strength. And he knew how to do it. But not all of us need to develop that kind of strength to be effective. \"No pain, no gain\" has validity in some circumstances, but it is not the essence of an effective exercise program.\n\nThe essence of renewing the physical dimension is to sharpen the saw, to exercise our bodies on a regular basis in a way that will preserve and enhance our capacity to work and adapt and enjoy.\n\nAnd we need to be wise in developing an exercise program. There's a tendency, especially if you haven't been exercising at all, to overdo. And that can create unnecessary pain, injury, and even permanent damage. It's best to start slowly. Any exercise program should be in harmony with the latest research findings, with your doctor's recommendations and with your own self-awareness.\n\nIf you haven't been exercising, your body will undoubtedly protest this change in its comfortable downhill direction. You won't like it at first. You may even hate it. But be proactive. Do it anyway. Even if it's raining on the morning you've scheduled to jog, do it anyway. \"Oh good! It's raining! I get to develop my willpower as well as my body!\"\n\nYou're not dealing with quick fix; you're dealing with a Quadrant II activity that will bring phenomenal long-term results. Ask anyone who has done it consistently. Little by little, your resting pulse rate will go down as your heart and oxygen processing system becomes more efficient. As you increase your body's ability to do more demanding things, you'll find your normal activities much more comfortable and pleasant. You'll have more afternoon energy, and the fatigue you've felt that's made you \"too tired\" to exercise in the past will be replaced by an energy that will invigorate everything you do.\n\nProbably the greatest benefit you will experience from exercising will be the development of your Habit 1 muscles of proactivity. As you act based on the value of physical well-being instead of reacting to all the forces that keep you from exercising, your paradigm of yourself, your self-esteem, your self-confidence, and your integrity will be profoundly affected.\n\nThe Spiritual Dimension\n\nRenewing the spiritual dimension provides leadership to your life. It's highly related to Habit 2.\n\nThe spiritual dimension is your core, your center, your commit\u00adment to your value system. It's a very private area of life and a supremely important one. It draws upon the sources that inspire and uplift you and tie you to the timeless truths of all humanity. And people do it very, very differently.\n\nI find renewal in daily prayerful meditation on the scriptures because they represent my value system. As I read and meditate, I feel renewed, strengthened, centered and recommitted to serve.\n\nImmersion in great literature or great music can provide a similar renewal of the spirit for some. There are others who find it in the way they communicate with nature. Nature bequeaths its own blessing on those who immerse themselves in it. When you're able to leave the noise and the discord of the city and give yourself up to the harmony and rhythm of nature, you come back renewed. For a time, you're undisturbable, almost unflappable, until grad\u00adually the noise and the discord from outside start to invade that sense of inner peace.\n\nArthur Gordon shares a wonderful, intimate story of his own spiritual renewal in a little story called \"The Turn of the Tide.\" It tells of a time in his life when he began to feel that everything was stale and flat. His enthusiasm waned; his writing efforts were fruitless. And the situation was growing worse day by day.\n\nFinally, he determined to get help from a medical doctor. Observing nothing physically wrong, the doctor asked him if he would be able to follow his instructions for one day.\n\nWhen Gordon replied that he could, the doctor told him to spend the following day in the place where he was happiest as a child. He could take food, but he was not to talk to anyone or to read or write or listen to the radio. He then wrote out four prescriptions and told him to open one at nine, twelve, three, and six o'clock.\n\n\"Are you serious?\" Gordon asked him.\n\n\"You won't think I'm joking when you get my bill!\" was the reply.\n\nSo the next morning, Gordon went to the beach. As he opened the first prescription, he read \"Listen carefully.\" He thought the doctor was insane. How could he listen for three hours? But he had agreed to follow the doctor's orders, so he listened. He heard the usual sounds of the sea and the birds. After a while, he could hear the other sounds that weren't so apparent at first. As he listened, he began to think of lessons the sea had taught him as a child\u2014patience, respect, an awareness of the interdependence of things. He began to listen to the sounds\u2014and the silence\u2014and to feel a growing peace.\n\nAt noon, he opened the second slip of paper and read \"Try reaching back.\" \"Reaching back to what?\" he wondered. Perhaps to childhood, perhaps to memories of happy times. He thought about his past, about the many little moments of joy. He tried to remember them with exactness. And in remembering, he found a growing warmth inside.\n\nAt three o'clock, he opened the third piece of paper. Until now, the prescriptions had been easy to take. But this one was different; it said \"Examine your motives.\" At first he was defensive. He thought about what he wanted\u2014success, recognition, security\u2014and he justified them all. But then the thought occurred to him that those motives weren't good enough, and that perhaps therein was the answer to his stagnant situation.\n\nHe considered his motives deeply. He thought about past happiness. And at last, the answer came to him.\n\n\"In a flash of certainty,\" he wrote, \"I saw that if one's motives are wrong, nothing can be right. It makes no difference whether you are a mailman, a hairdresser, an insurance salesman, a housewife\u2014whatever. As long as you feel you are serving others, you do the job well. When you are concerned only with helping yourself, you do it less well\u2014a law as inexorable as gravity.\"\n\nWhen six o'clock came, the final prescription didn't take long to fill. \"Write your worries on the sand,\" it said. He knelt and wrote several words with a piece of broken shell; then he turned and walked away. He didn't look back; he knew the tide would come in.\n\nSpiritual renewal takes an investment of time. But it's a Quadrant II activity we don't really have time to neglect.\n\nThe great reformer Martin Luther is quoted as saying, \"I have so much to do today, I'll need to spend another hour on my knees.\" To him, prayer was not a mechanical duty but rather a source of power in releasing and multiplying his energies.\n\nSomeone once inquired of a Far Eastern Zen master, who had a great serenity and peace about him no matter what pressures he faced, \"How do you maintain that serenity and peace?\" He replied, \"I never leave my place of meditation.\" He meditated early in the morning and for the rest of the day, he carried the peace of those moments with him in his mind and heart.\n\nThe idea is that when we take time to draw on the leadership center of our lives, what life is ultimately all about, it spreads like an umbrella over everything else. It renews us, it refreshes us, particularly if we recommit to it.\n\nThis is why I believe a personal mission statement is so impor\u00adtant. If we have a deep understanding of our center and our purpose, we can review and recommit to it frequently. In our daily spiritual renewal, we can visualize and \"live out\" the events of the day in harmony with those values.\n\nReligious leader David O. McKay taught, \"The greatest battles of life are fought out daily in the silent chambers of the soul.\" If you win the battles there, if you settle the issues that inwardly conflict, you feel a sense of peace, a sense of knowing what you're about. And you'll find that the public victories\u2014where you tend to think cooperatively, to promote the welfare and good of other people, and to be genuinely happy for other people's successes\u2014will follow naturally.\n\nThe Mental Dimension\n\nMost of our mental development and study discipline comes through formal education. But as soon as we leave the external discipline of school, many of us let our minds atrophy. We don't do any more serious reading, we don't explore new subjects in any real depth outside our action fields, we don't think analytically, we don't write\u2014at least not critically or in a way that tests our ability to express ourselves in distilled, clear, and concise language. Instead, we spend our time watching TV.\n\nContinuing surveys indicate that television is on in most homes some thirty-five to forty-five hours a week. That's as much time as many people put into their jobs, more than most put into school. It's the most powerful socializing influence there is. And when we watch, we're subject to all the values that are being taught through it. That can powerfully influence us in very subtle and impercep\u00adtible ways.\n\nWisdom in watching television requires the effective self-management of Habit 3, which enables you to discriminate and to select the informing, inspiring, and entertaining programs which best serve and express your purpose and values.\n\nIn our family, we limit television watching to around seven hours a week, an average of about an hour a day. We had a family council at which we talked about it and looked at some of the data regarding what's happening in homes because of television. We found that by discussing it as a family when no one was defensive or argumentative, people started to realize the dependent sickness of becoming addicted to soap operas or to a steady diet of a particular program.\n\nI'm grateful for television and for the many high quality educa\u00adtional and entertainment programs. They can enrich our lives and contribute meaningfully to our purposes and goals. But there are many programs that simply waste our time and minds and many that influence us in negative ways if we let them. Like the body, television is a good servant but a poor master. We need to practice Habit 3 and manage ourselves effectively to maximize the use of any resource in accomplishing our missions.\n\nEducation\u2014continuing education, continually honing and ex\u00adpanding the mind\u2014is vital mental renewal. Sometimes that in\u00advolves the external discipline of the classroom or systematized study programs; more often it does not. Proactive people can figure out many, many ways to educate themselves.\n\nIt is extremely valuable to train the mind to stand apart and examine its own program. That, to me, is the definition of a liberal education\u2014the ability to examine the programs of life against larger questions and purposes and other paradigms. Training, without such education, narrows and closes the mind so that the assump\u00adtions underlying the training are never examined. That's why it is so valuable to read broadly and to expose yourself to great minds.\n\nThere's no better way to inform and expand your mind on a regular basis than to get into the habit of reading good literature. That's another high leverage Quadrant II activity. You can get into the best minds that are now or that have ever been in the world. I highly recommend starting with a goal of a book a month, then a book every two weeks, then a book a week. \"The person who doesn't read is no better off than the person who can't read.\"\n\nQuality literature, such as the Great Books, the Harvard Classics, autobiographies, National Geographic and other publications that expand our cultural awareness, and current literature in various fields can expand our paradigms and sharpen our mental saw, particularly if we practice Habit 5 as we read and seek first to understand. If we use our own autobiography to make early judgments before we really understand what an author has to say, we limit the benefits of the reading experience.\n\nWriting is another powerful way to sharpen the mental saw. Keeping a journal of our thoughts, experiences, insights, and learnings promotes mental clarity, exactness, and context. Writing good letters\u2014communicating on the deeper level of thoughts, feelings, and ideas rather than on the shallow, superficial level of events\u2014also affects our ability to think clearly, to reason accu\u00adrately, and to be understood effectively.\n\nOrganizing and planning represent other forms of mental re\u00adnewal associated with Habits 2 and 3. It's beginning with the end in mind and being able mentally to organize to accomplish that end. It's exercising the visualizing, imagining power of your mind to see the end from the beginning and to see the entire journey, at least in principles, if not in steps.\n\nIt is said that wars are won in the general's tent. Sharpening the saw in the first three dimensions\u2014the physical, the spiritual, and the mental\u2014is a practice I call the \"Daily Private Victory.\" And I commend to you the simple practice of spending one hour a day every day doing it\u2014one hour a day for the rest of your life.\n\nThere's no other way you could spend an hour that would begin to compare with the Daily Private Victory in terms of value and results. It will affect every decision, every relationship. It will greatly improve the quality, the effectiveness, of every other hour of the day, including the depth and restfulness of your sleep. It will build the long-term physical, spiritual, and mental strength to enable you to handle difficult challenges in life.\n\nIn the words of Phillips Brooks:\n\nSome day, in the years to come, you will be wrestling with the great temptation, or trembling under the great sorrow of your life. But the real struggle is here, now... Now it is being decided whether, in the day of your supreme sorrow or temptation, you shall miserably fail or gloriously conquer. Character cannot be made except by a steady, long continued process.\n\nThe Social\/Emotional Dimension\n\nWhile the physical, spiritual, and mental dimensions are closely related to Habits 1, 2, and 3\u2014centered on the principles of personal vision, leadership, and management\u2014the social\/emotional dimen\u00adsion focuses on Habits 4, 5, and 6\u2014centered on the principles of interpersonal leadership, empathic communication, and creative cooperation.\n\nThe social and the emotional dimensions of our lives are tied to\u00adgether because our emotional life is primarily, but not exclusively, developed out of and manifested in our relationships with others.\n\nRenewing our social\/emotional dimension does not take time in the same sense that renewing the other dimensions does. We can do it in our normal everyday interactions with other people. But it definitely requires exercise. We may have to push ourselves because many of us have not achieved the level of Private Victory and the skills of Public Victory necessary for Habits 4, 5, and 6 to come naturally to us in all our interactions.\n\nSuppose that you are a key person in my life. You might be my boss, my subordinate, my coworker, my friend, my neighbor, my spouse, my child, a member of my extended family\u2014anyone with whom I want or need to interact. Suppose we need to communicate together, to work together, to discuss a jugular issue, to accomplish a purpose or solve a problem. But we see things differently; we're looking through different glasses. You see the young lady, and I see the old woman.\n\nSo I practice Habit 4. I come to you and I say, \"I can see that we're approaching this situation differently. Why don't we agree to communicate until we can find a solution we both feel good about. Would you be willing to do that?\" Most people would be willing to say \"yes\" to that.\n\nThen I move to Habit 5. \"Let me listen to you first.\" Instead of listening with intent to reply, I listen empathically in order to deeply, thoroughly understand your paradigm. When I can explain your point of view as well as you can, then I focus on communicating my point of view to you so that you can understand it as well.\n\nBased on the commitment to search for a solution that we both feel good about and a deep understanding of each other's points of view, we move to Habit 6. We work together to produce third alternative solutions to our differences that we both recognize are better than the ones either you or I proposed initially.\n\nSuccess in Habits 4, 5, and 6 is not primarily a matter of intellect\u2014it's primarily a matter of emotion. It's highly related to our sense of personal security.\n\nIf our personal security comes from sources within ourselves, then we have the strength to practice the habits of Public Victory. If we are emotionally insecure, even though we may be intellectu\u00adally very advanced, practicing Habits 4, 5, and 6 with people who think differently on jugular issues of life can be terribly threatening.\n\nWhere does intrinsic security come from? It doesn't come from what other people think of us or how they treat us. It doesn't come from the scripts they've handed us. It doesn't come from our circumstances or our position.\n\nIt comes from within. It comes from accurate paradigms and correct principles deep in our own mind and heart. It comes from inside-out congruence, from living a life of integrity in which our daily habits reflect our deepest values.\n\nI believe that a life of integrity is the most fundamental source of personal worth. I do not agree with the popular success literature that says that self-esteem is primarily a matter of mind set, of attitude\u2014that you can psych yourself into peace of mind.\n\nPeace of mind comes when your life is in harmony with true principles and values and in no other way.\n\nThere is also the intrinsic security that comes as a result of effective interdependent living. There is security in knowing that Win\/Win solutions do exist, that life is not always \"either\/or,\" that there are almost always mutually beneficial Third Alternatives. There is security in knowing that you can step out of your own frame of reference without giving it up, that you can really, deeply understand another human being. There is security that comes when you authentically, creatively and cooperatively interact with other people and really experience these interdependent habits.\n\nThere is intrinsic security that comes from service, from helping other people in a meaningful way. One important source is your work, when you see yourself in a contributive and creative mode, really making a difference. Another source is anonymous service\u2014no one knows it and no one necessarily ever will. And that's not the concern; the concern is blessing the lives of other people. Influence, not recognition, becomes the motive.\n\nViktor Frankl focused on the need for meaning and purpose in our lives, something that transcends our own lives and taps the best energies within us. The late Dr. Hans Selye, in his monumental research on stress, basically says that a long, healthy, and happy life is the result of making contributions, of having meaningful projects that are personally exciting and contribute to and bless the lives of others. His ethic was \"earn thy neighbor's love.\"\n\nIn the words of George Bernard Shaw,\n\nThis is the true joy in life\u2014that being used for a purpose recognized by yourself as a mighty one. That being a force of nature, instead of a feverish, selfish little clod of ailments and grievances complaining that the world will not devote itself to making you happy. I am of the opinion that my life belongs to the whole community and as long as I live it is my privilege to do for it whatever I can. I want to be thoroughly used up when I die. For the harder I work the more I live. I rejoice in life for its own sake. Life is no brief candle to me. It's a sort of splendid torch which I've got to hold up for the moment and I want to make it burn as brightly as possible before handing it on to future generations.\n\nN. Eldon Tanner has said, \"Service is the rent we pay for the privilege of living on this earth.\" And there are so many ways to serve. Whether or not we belong to a church or service organization or have a job that provides meaningful service opportunities, not a day goes by that we can't at least serve one other human being by making deposits of unconditional love.\n\nSCRIPTING OTHERS\n\nMost people are a function of the social mirror, scripted by the opinions, the perceptions, the paradigms of the people around them. As interdependent people, you and I come from a paradigm which includes the realization that we are a part of that social mirror.\n\nWe can choose to reflect back to others a clear, undistorted vision of themselves. We can affirm their proactive nature and treat them as responsible people. We can help script them as principle-centered, value-based, independent, worthwhile individuals. And, with the Abundance Mentality, we realize that giving a positive reflection to others in no way diminishes us. It increases us because it increases the opportunities for effective interaction with other proactive people.\n\nAt some time in your life, you probably had someone believe in you when you didn't believe in yourself. They scripted you. Did that make a difference in your life?\n\nWhat if you were a positive scripter, an affirmer, of other people? When they're being directed by the social mirror to take the lower path, you inspire them toward a higher path because you believe in them. You listen to them and empathize with them. You don't absolve them of responsibility; you encourage them to be proac\u00adtive.\n\nPerhaps you are familiar with the musical, Man of La Mancha. It's a beautiful story about a medieval knight who meets a woman of the street, a prostitute. She's being validated in her life-style by all of the people in her life.\n\nBut this poet knight sees something else in her, something beautiful and lovely. He also sees her virtue, and he affirms it, over and over again. He gives her a new name\u2014Dulcinea\u2014a new name associated with a new paradigm.\n\nAt first, she utterly denies it; her old scripts are overpowering. She writes him off as a wild-eyed fantasizer. But he is persistent. He makes continual deposits of unconditional love and gradually it penetrates her scripting. It goes down into her true nature, her potential, and she starts to respond. Little by little, she begins to change her life-style. She believes it and she acts from her new paradigm, to the initial dismay of everyone else in her life.\n\nLater, when she begins to revert to her old paradigm, he calls her to his deathbed and sings that beautiful song, \"The Impossible Dream,\" looks her in the eyes, and whispers, \"Never forget, you're Dulcinea.\"\n\nOne of the classic stories in the field of self-fulfilling prophecies is of a computer in England that was accidently programmed incorrectly. In academic terms, it labeled a class of \"bright\" kids \"dumb\" kids and a class of supposedly \"dumb\" kids \"bright.\" And that computer report was the primary criterion that created the teachers' paradigms about their students at the beginning of the year.\n\nWhen the administration finally discovered the mistake five and a half months later, they decided to test the kids again without telling anyone what had happened. And the results were amazing. The \"bright\" kids had gone down significantly in IQ test points. They had been seen and treated as mentally limited, uncoopera\u00adtive, and difficult to teach. The teachers' paradigms had become a self-fulfilling prophecy.\n\nBut scores in the supposedly \"dumb\" group had gone up. The teachers had treated them as though they were bright, and their energy, their hope, their optimism, their excitement had reflected high individual expectations and worth for those kids.\n\nThese teachers were asked what it was like during the first few weeks of the term. \"For some reason, our methods weren't working,\" they replied. \"So we had to change our methods.\" The information showed that the kids were bright. If things weren't working well, they figured it had to be the teaching methods. So they worked on methods. They were proactive; they worked in their Circle of Influence. Apparent learner disability was nothing more or less than teacher inflexibility.\n\nWhat do we reflect to others about themselves? And how much does that reflection influence their lives? We have so much we can invest in the Emotional Bank Accounts of other people. The more we can see people in terms of their unseen potential, the more we can use our imagination rather than our memory, with our spouse, our children, our coworkers or employees. We can refuse to label them\u2014we can \"see\" them in new fresh ways each time we're with them. We can help them become independent, fulfilled people capable of deeply satisfying, enriching, and productive relation\u00adships with others.\n\nGoethe taught, \"Treat a man as he is and he will remain as he is. Treat a man as he can and should be and he will become as he can and should be.\"\n\nBALANCE IN RENEWAL\n\nThe self-renewal process must include balanced renewal in all four dimensions of our nature: the physical, the spiritual, the mental, and the social\/emotional.\n\nAlthough renewal in each dimension is important, it only becomes optimally effective as we deal with all four dimensions in a wise and balanced way. To neglect any one area negatively impacts the rest.\n\nI have found this to be true in organizations as well as in individual lives. In an organization, the physical dimension is expressed in economic terms. The mental or psychological dimen\u00adsion deals with the recognition, development, and use of talent. The social\/emotional dimension has to do with human relations, with how people are treated. And the spiritual dimension deals with finding meaning through purpose or contribution and through organizational integrity.\n\nWhen an organization neglects any one or more of these areas, it negatively impacts the entire organization. The creative energies that could result in tremendous, positive synergy are instead used to fight against the organization and become restraining forces to growth and productivity.\n\nI have found organizations whose only thrust is economic\u2014to make money. They usually don't publicize that purpose. They sometimes even publicize something else. But in their hearts, their only desire is to make money.\n\nWhenever I find this, I also find a great deal of negative synergy in the culture, generating such things as interdepartmental rival\u00adries, defensive and protective communication, politicking, and masterminding. We can't effectively thrive without making money, but that's not sufficient reason for organizational existence. We can't live without eating, but we don't live to eat.\n\nAt the other end of the spectrum, I've seen organizations that focused almost exclusively on the social\/emotional dimension. They are, in a sense, some kind of social experiment and they have no economic criteria to their value system. They have no measure or gauge of their effectiveness, and as a result, they lose all kinds of efficiencies and eventually their viability in the marketplace.\n\nI have found many organizations that develop as many as three of the dimensions\u2014they may have good service criteria, good economic criteria, and good human relations criteria, but they are not really committed to identifying, developing, utilizing, and recognizing the talent of people. And if these psychological forces are missing, the style will be a benevolent autocracy and the resulting culture will reflect different forms of collective resistance, adversarialism, excessive turnover, and other deep, chronic, cul\u00adtural problems.\n\nOrganizational as well as individual effectiveness requires devel\u00adopment and renewal of all four dimensions in a wise and balanced way. Any dimension that is neglected will create negative force field resistance that pushes against effectiveness and growth. Organizations and individuals that give recognition to each of these four dimensions in their mission statement provide a pow\u00aderful framework for balanced renewal.\n\nThis process of continuous improvement is the hallmark of the Total Quality Movement and a key to Japan's economic ascen\u00addency.\n\nSYNERGY IN RENEWAL\n\nBalanced renewal is optimally synergetic. The things you do to sharpen the saw in any one dimension have positive impact in other dimensions because they are so highly interrelated. Your physical health affects your mental health; your spiritual strength affects your social\/emotional strength. As you improve in one dimension, you increase your ability in other dimensions as well.\n\nThe Seven Habits of Highly Effective People create optimum synergy among these dimensions. Renewal in any dimension increases your ability to live at least one of the Seven Habits. And although the habits are sequential, improvement in one habit synergetically increases your ability to live the rest.\n\nThe more proactive you are (Habit 1), the more effectively you can exercise personal leadership (Habit 2) and management (Habit 3) in your life. The more effectively you manage your life (Habit 3), the more Quadrant II renewing activities you can do (Habit 7). The more you seek first to understand (Habit 5), the more effectively you can go for synergetic Win\/Win solutions (Habits 4 and 6). The more you improve in any of the habits that lead to independence (Habits 1, 2, and 3), the more effective you will be in interdepen\u00addent situations (Habits 4, 5, and 6). And renewal (Habit 7) is the process of renewing all the habits.\n\nAs you renew your physical dimension, you reinforce your personal vision (Habit 1), the paradigm of your own self-awareness and free will, of proactivity, of knowing that you are free to act instead of being acted upon, to choose your own response to any stimulus. This is probably the greatest benefit of physical exercise. Each Daily Private Victory makes a deposit in your personal intrinsic security account.\n\nAs you renew your spiritual dimension, you reinforce your personal leadership (Habit 2). You increase your ability to live out of your imagination and conscience instead of only your memory, to deeply understand your innermost paradigms and values, to create within yourself a center of correct principles, to define your own unique mission in life, to rescript yourself to live your life in harmony with correct principles and to draw upon your personal sources of strength. The rich private life you create in spiritual renewal makes tremendous deposits in your personal security account.\n\nAs you renew your mental dimension, you reinforce your personal management (Habit 3). As you plan, you force your mind to recognize high leverage Quadrant II activities, priority goals, and activities to maximize the use of your time and energy, and you organize and execute your activities around your priorities. As you become involved in continuing education, you increase your knowledge base and you increase your options. Your economic security does not lie in your job; it lies in your own power to produce\u2014to think, to learn, to create, to adapt. That's true financial independence. It's not having wealth; it's having the power to produce wealth. It's intrinsic.\n\nThe Daily Private Victory\u2014a minimum of one hour a day in renewal of the physical, spiritual, and mental dimensions\u2014is the key to the development of the Seven Habits and it's completely within your Circle of Influence. It is the Quadrant II focus time necessary to integrate these habits into your life, to become principle-centered.\n\nIt's also the foundation for the Daily Public Victory. It's the source of intrinsic security you need to sharpen the saw in the social\/emotional dimension. It gives you the personal strength to focus on your Circle of Influence in interdependent situations\u2014to look at others through the Abundance Mentality paradigm, to genuinely value their differences and to be happy for their success. It gives you the foundation to work for genuine understanding and for synergetic Win\/Win solutions, to practice Habits 4, 5, and 6 in an interdependent reality.\n\nTHE UPWARD SPIRAL\n\nRenewal is the principle\u2014and the process\u2014that empowers us to move on an upward spiral of growth and change, of continuous improvement.\n\nTo make meaningful and consistent progress along that spiral, we need to consider one other aspect of renewal as it applies to the unique human endowment that directs this upward movement\u2014our conscience. In the words of Madame de Sta\u00ebl, \"The voice of conscience is so delicate that it is easy to stifle it: but it is also so clear that it is impossible to mistake it.\"\n\nConscience is the endowment that senses our congruence or disparity with correct principles and lifts us toward them\u2014when it's in shape.\n\nJust as the education of nerve and sinew is vital to the excellent athlete and education of the mind is vital to the scholar, education of the conscience is vital to the truly proactive, highly effective person. Training and educating the conscience, however, requires even greater concentration, more balanced discipline, more con\u00adsistently honest living. It requires regular feasting on inspiring literature, thinking noble thoughts and, above all, living in har\u00admony with its still small voice.\n\nJust as junk food and lack of exercise can ruin an athlete's condition, those things that are obscene, crude, or pornographic can breed an inner darkness that numbs our higher sensibilities and substitutes the social conscience of \"Will I be found out?\" for the natural or divine conscience of \"What is right and wrong?\"\n\nIn the words of Dag Hammarskjold,\n\nYou cannot play with the animal in you without becoming wholly animal, play with falsehood without forfeiting your right to truth, play with cruelty without losing your sensitivity of mind. He who wants to keep his garden tidy doesn't reserve a plot for weeds.\n\nOnce we are self-aware, we must choose purposes and principles to live by; otherwise the vacuum will be filled, and we will lose our self-awareness and become like groveling animals who live prima\u00adrily for survival and propagation. People who exist on that level aren't living; they are \"being lived.\" They are reacting, unaware of the unique endowments that lie dormant and undeveloped within.\n\nAnd there is no shortcut in developing them. The law of the harvest governs; we will always reap what we sow\u2014no more, no less. The law of justice is immutable, and the closer we align ourselves with correct principles, the better our judgment will be about how the world operates and the more accurate our paradigms\u2014our maps of the territory\u2014will be.\n\nI believe that as we grow and develop on this upward spiral, we must show diligence in the process of renewal by educating and obeying our conscience. An increasingly educated conscience will propel us along the path of personal freedom, security, wisdom, and power.\n\nMoving along the upward spiral requires us to learn, commit, and do on increasingly higher planes. We deceive ourselves if we think that any one of these is sufficient. To keep progressing, we must learn, commit, and do\u2014learn, commit, and do\u2014and learn, commit, and do again.\n\nTHE UPWARD SPIRAL\n\nAPPLICATION SUGGESTIONS:\n\n  1. Make a list of activities that would help you keep in good physical shape, that would fit your life-style and that you could enjoy over time.\n  2. Select one of the activities and list it as a goal in your personal role area for the coming week. At the end of the week evaluate your performance. If you didn't make your goal, was it because you subordinated it to a genuinely higher value? Or did you fail to act with integrity to your values?\n  3. Make a similar list of renewing activities in your spiritual and mental dimensions. In your social-emotional area, list relationships you would like to improve or specific circumstances in which Public Victory would bring greater effectiveness. Select one item in each area to list as a goal for the week. Implement and evaluate.\n  4. Commit to write down specific \"sharpen the saw\" activities in all four dimensions every week, to do them, and to evaluate your performance and results.\n\n### Inside-Out Again\n\nThe Lord works from the inside out. The world works from the outside in. The world would take people out of the slums. Christ takes the slums out of people, and then they take themselves out of the slums. The world would mold men by changing their environment. Christ changes men, who then change their environment. The world would shape human behavior, but Christ can change human nature.\n\nEzra Taft Benson\n\nI WOULD LIKE TO SHARE WITH YOU a personal story which I feel contains the essence of this book. In doing so, it is my hope that you will relate to the underlying principles it contains.\n\nSome years ago, our family took a sabbatical leave from the university where I taught so that I could write. We lived for a full year in Laie on the north shore of Oahu, Hawaii.\n\nShortly after getting settled, we developed a living and working routine which was not only very productive but extremely pleas\u00adant.\n\nAfter an early morning run on the beach, we would send two of our children, barefoot and in shorts, to school. I went to an isolated building next to the canefields where I had an office to do my writing. It was very quiet, very beautiful, very serene\u2014no phone, no meetings, no pressing engagements.\n\nMy office was on the outside edge of a college, and one day as I was wandering between stacks of books in the back of the college library, I came across a book that drew my interest. As I opened it, my eyes fell upon a single paragraph that powerfully influenced the rest of my life.\n\nI read the paragraph over and over again. It basically contained the simple idea that there is a gap or a space between stimulus and response, and that the key to both our growth and happiness is how we use that space.\n\nI can hardly describe the effect that idea had on my mind. Though I had been nurtured in the philosophy of self-determinism, the way the idea was phrased\u2014\"a gap between stimulus and response\"\u2014hit me with fresh, almost unbelievable force. It was almost like \"knowing it for the first time,\" like an inward revolu\u00adtion, \"an idea whose time had come.\"\n\nI reflected on it again and again, and it began to have a powerful effect on my paradigm of life. It was as if I had become an observer of my own participation. I began to stand in that gap and to look outside at the stimuli. I reveled in the inward sense of freedom to choose my response\u2014even to become the stimulus, or at least to influence it\u2014even to reverse it.\n\nShortly thereafter, and partly as a result of this \"revolutionary\" idea, Sandra and I began a practice of deep communication. I would pick her up a little before noon on an old red Honda 90 trail cycle, and we would take our two preschool children with us\u2014one between us and the other on my left knee\u2014as we rode out in the canefields by my office. We rode slowly along for about an hour, just talking.\n\nThe children looked forward to the ride and hardly ever made any noise. We seldom saw another vehicle, and the cycle was so quiet we could easily hear each other. We usually ended up on an isolated beach where we parked the Honda and walked about 200 yards to a secluded spot where we ate a picnic lunch.\n\nThe sandy beach and a freshwater river coming off the island totally absorbed the interest of the children, so Sandra and I were able to continue our talks uninterrupted. Perhaps it doesn't take too much imagination to envision the level of understanding and trust we were able to reach by spending at least two hours a day, every day, for a full year in deep communication.\n\nAt the very first of the year, we talked about all kinds of interesting topics\u2014people, ideas, events, the children, my writing, our family at home, future plans, and so forth. But little by little, our communication deepened and we began to talk more and more about our internal worlds\u2014about our upbringing, our scripting, our feelings and self-doubts. As we were deeply immersed in these communications, we also observed them and observed ourselves in them. We began to use that space between stimulus and response in some new and interesting ways which caused us to think about how we were programmed and how those programs shaped how we saw the world.\n\nWe began an exciting adventure into our interior worlds and found it to be more exciting, more fascinating, more absorbing, more compelling, more filled with discovery and insight than anything we'd ever known in the outside world.\n\nIt wasn't all \"sweetness and light.\" We occasionally hit some raw nerves and had some painful experiences, embarrassing experiences, self-revealing experiences\u2014experiences that made us extremely open and vulnerable to each other. And yet we found we had been wanting to go into those things for years. When we did go into the deeper, more tender issues and then came out of them, we felt in some way healed.\n\nWe were so initially supportive and helpful, so encouraging and empathic to each other, that we nurtured and facilitated these internal discoveries in each other.\n\nWe gradually evolved two unspoken ground rules. The first was \"no probing.\" As soon as we unfolded the inner layers of vulner\u00adability, we were not to question each other, only to empathize. Probing was simply too invasive. It was also too controlling and too logical. We were covering new, difficult terrain that was scary and uncertain, and it stirred up fears and doubts. We wanted to cover more and more of it, but we grew to respect the need to let each other open up in our own time.\n\nThe second ground rule was that when it hurt too much, when it was painful, we would simply quit for the day. Then we would either begin the next day where we left off or wait until the person who was sharing felt ready to continue. We carried around the loose ends, knowing that we wanted to deal with them. But because we had the time and the environment conducive to it, and because we were so excited to observe our own involvement and to grow within our marriage, we simply knew that sooner or later we would deal with all those loose ends and bring them to some kind of closure.\n\nThe most difficult, and eventually the most fruitful part of this kind of communication came when my vulnerability and Sandra's vulnerability touched. Then, because of our subjective involve\u00adment, we found that the space between stimulus and response was no longer there. A few bad feelings surfaced. But our deep desire and our implicit agreement was to prepare ourselves to start where we left off and deal with those feelings until we resolved them.\n\nOne of those difficult times had to do with a basic tendency in my personality. My father was a very private individual\u2014very controlled and very careful. My mother was and is very public, very open, very spontaneous. I find both sets of tendencies in me, and when I feel insecure, I tend to become private, like my father. I live inside myself and safely observe.\n\nSandra is more like my mother\u2014social, authentic, and sponta\u00adneous. We had gone through many experiences over the years in which I felt her openness was inappropriate, and she felt my constraint was dysfunctional, both socially and to me as an individual because I would become insensitive to the feelings of others. All of this and much more came out during those deep visits. I came to value Sandra's insight and wisdom and the way she helped me to be a more open, giving, sensitive, social person.\n\nAnother of those difficult times had to do with what I perceived to be a \"hang up\" Sandra had which had bothered me for years. She seemed to have an obsession about Frigidaire appliances which I was at an absolute loss to understand. She would not even consider buying another brand of appliance. Even when we were just starting out and on a very tight budget, she insisted that we drive the fifty miles to the \"big city\" where Frigidaire appliances were sold, simply because no dealer in our small university town carried them at that time.\n\nThis was a matter of considerable agitation to me. Fortunately, the situation came up only when we purchased an appliance. But when it did come up, it was like a stimulus that triggered off a hot button response. This single issue seemed to be symbolic of all irrational thinking, and it generated a whole range of negative feelings within me.\n\nI usually resorted to my dysfunctional private behavior. I sup\u00adpose I figured that the only way I could deal with it was not to deal with it; otherwise, I felt I would lose control and say things I shouldn't say. There were times when I did slip and say something negative, and I had to go back and apologize.\n\nWhat bothered me the most was not that she liked Frigidaire, but that she persisted in making what I considered utterly illogical and indefensible statements to defend Frigidaire which had no basis in fact whatsoever. If she had only agreed that her response was irrational and purely emotional, I think I could have handled it. But her justification was upsetting.\n\nIt was sometime in early spring when the Frigidaire issue came up. All our prior communication had prepared us. The ground rules had been deeply established\u2014not to probe and to leave it alone if it got to be too painful for either or both.\n\nI will never forget the day we talked it through. We didn't end up on the beach that day; we just continued to ride through the canefields, perhaps because we didn't want to look each other in the eye. There had been so much psychic history and so many bad feelings associated with the issue, and it had been submerged for so long. It had never been so critical as to rupture the relationship, but when you're trying to cultivate a beautiful unified relationship, any divisive issue is important.\n\nSandra and I were amazed at what we learned through the interaction. It was truly synergistic. It was as if Sandra were learning, almost for the first time herself, the reason for her so-called hang-up. She started to talk about her father, about how he had worked as a high school history teacher and coach for years, and how, to help make ends meet, he had gone into the appliance business. During an economic downturn, he had experienced serious financial difficulties, and the only thing that enabled him to stay in business during that time was the fact that Frigidaire would finance his inventory.\n\nSandra had an unusually deep and sweet relationship with her father. When he returned home at the end of a very tiring day, he would lie on the couch, and Sandra would rub his feet and sing to him. It was a beautiful time they enjoyed together almost daily for years. He would also open up and talk through his worries and concerns about the business, and he shared with Sandra his deep appreciation for Frigidaire financing his inventory so that he could make it through the difficult times.\n\nThis communication between father and daughter had taken place in a spontaneous way during very natural times, when the most powerful kind of scripting takes place. During those relaxed times guards are down and all kinds of images and thoughts are planted deep in the subconscious mind. Perhaps Sandra had forgotten about all of this until the safety of that year of commu\u00adnication when it could come out also in very natural and sponta\u00adneous ways.\n\nSandra gained tremendous insight into herself and into the emotional root of her feelings about Frigidaire. I also gained insight and a whole new level of respect. I came to realize that Sandra wasn't talking about appliances; she was talking about her father, and about loyalty\u2014about loyalty to his needs.\n\nI remember both of us becoming tearful on that day, not so much because of the insights, but because of the increased sense of reverence we had for each other. We discovered that even seem\u00adingly trivial things often have roots in deep emotional experiences. To deal only with the superficial trivia without seeing the deeper, more tender issues is to trample on the sacred ground of another's heart.\n\nThere were many rich fruits of those months. Our communica\u00adtion became so powerful that we could almost instantly connect with each other's thoughts. When we left Hawaii, we resolved to continue the practice. During the many years since, we have continued to go regularly on our Honda trail cycle, or in the car if the weather's bad, just to talk. We feel the key to staying in love is to talk, particularly about feelings. We try to communicate with each other several times every day, even when I'm travelling. It's like touching in to home base, which accesses all the happiness, security, and values it represents.\n\nThomas Wolfe was wrong. You can go home again\u2014if your home is a treasured relationship, a precious companionship.\n\nINTERGENERATIONAL LIVING\n\nAs Sandra and I discovered that wonderful year, the ability to use wisely the gap between stimulus and response, to exercise the four unique endowments of our human nature, empowered us from the inside out.\n\nWe had tried the outside-in approach. We loved each other, and we had attempted to work through our differences by controlling our attitudes and our behaviors, by practicing useful techniques of human interaction. But our band-aids and aspirin only lasted so long. Until we worked and communicated on the level of our essential paradigms, the chronic underlying problems were still there.\n\nWhen we began to work from the inside out, we were able to build a relationship of trust and openness and to resolve dysfunc\u00adtional differences in a deep and lasting way that never could have come by working from the outside in. The delicious fruits\u2014a rich Win\/Win relationship, a deep understanding of each other, and a marvelous synergy\u2014grew out of the roots we nurtured as we examined our programs, rescripted ourselves, and managed our lives so that we could create time for the important Quadrant II activity of communicating deeply with each other.\n\nAnd there were other fruits. We were able to see on a much deeper level that, just as powerfully as our own lives had been affected by our parents, the lives of our children were being influenced and shaped by us, often in ways we didn't even begin to realize. Understanding the power of scripting in our own lives, we felt a renewed desire to do everything we could to make certain that what we passed on to future generations, by both precept and example, was based on correct principles.\n\nI have drawn particular attention in this book to those scripts we have been given which we proactively want to change. But as we examine our scripting carefully, many of us will also begin to see beautiful scripts, positive scripts that have been passed down to us which we have blindly taken for granted. Real self-awareness helps us to appreciate those scripts and to appreciate those who have gone before us and nurtured us in principle-based living, mirroring back to us not only what we are, but what we can become.\n\nThere is transcendent power in a strong intergenerational family. An effectively interdependent family of children, parents, grand\u00adparents, aunts, uncles, and cousins can be a powerful force in helping people have a sense of who they are and where they came from and what they stand for.\n\nIt's great for children to be able to identify themselves with the \"tribe,\" to feel that many people know them and care about them, even though they're spread all over the country. And that can be a tremendous benefit as you nurture your family. If one of your children is having difficulty and doesn't really relate with you at a particular time in his life, maybe he can relate to your brother or sister who can become a surrogate father or mother, a mentor or a hero for a period of time.\n\nGrandparents who show a great interest in their grandchildren are among the most precious people on this earth. What a marvelous positive social mirror they can be! My mother is like that. Even now, in her late 80's, she takes a deep personal interest in every one of her descendants. She writes us love letters. I was reading one the other day on a plane with tears streaming down my cheeks. I could call her up tonight and I know she'd say, \"Stephen, I want you to know how much I love you and how wonderful I think you are.\" She's constantly reaffirming.\n\nA strong intergenerational family is potentially one of the most fruitful, rewarding, and satisfying interdependent relationships. And many people feel the importance of that relationship. Look at the fascination we all had with Roots some years ago. Each of us has roots and the ability to trace those roots, to identify our ancestors.\n\nThe highest and most powerful motivation in doing that is not for ourselves only, but for our posterity, for the posterity of all mankind. As someone once observed, \"There are only two lasting bequests we can give our children\u2014one is roots, the other wings.\"\n\nBECOMING A TRANSITION PERSON\n\nAmong other things, I believe that giving \"wings\" to our children and to others means empowering them with the freedom to rise above negative scripting that had been passed down to us. I believe it means becoming what my friend and associate, Dr. Terry Warner, calls a \"transition\" person. Instead of transferring those scripts to the next generation, we can change them. And we can do it in a way that will build relationships in the process.\n\nIf your parents abused you as a child, that does not mean that you have to abuse your own children. Yet there's plenty of evidence to indicate that you will tend to live out that script. But because you're proactive, you can rewrite the script. You can choose not only not to abuse your children, but to affirm them, to script them in positive ways.\n\nYou can write it in your personal mission statement and into your mind and heart. You can visualize yourself living in harmony with that mission statement in your Daily Private Victory. You can take steps to love and forgive your own parents, and if they are still living, to build a positive relationship with them by seeking to understand.\n\nA tendency that's run through your family for generations can stop with you. You're a transition person\u2014a link between the past and the future. And your own change can affect many, many lives downstream.\n\nOne powerful transition person of the twentieth century, Anwar Sadat, left us as part of his legacy a profound understanding of the nature of change. Sadat stood between a past that had created a \"huge wall of suspicion, fear, hate and misunderstanding\" between Arabs and Israelis, and a future in which increased conflict and isolation seemed inevitable. Efforts at negotiation had been met with objections on every scale\u2014even to formalities and procedural points, to an insignificant comma or period in the text of proposed agreements.\n\nWhile others attempted to resolve the tense situation by hacking at the leaves, Sadat drew upon his earlier centering experience in a lonely prison cell and went to work on the root. And in doing so, he changed the course of history for millions of people.\n\nHe records in his autobiography:\n\nIt was then that I drew, almost unconsciously, on the inner strength I had developed in Cell 54 of Cairo Central Prison\u2014a strength, call it a talent or capacity, for change. I found that I faced a highly complex situation, and that I couldn't hope to change it until I had armed myself with the necessary psychological and intellectual capacity. My contemplation of life and human nature in that secluded place had taught me that he who cannot change the very fabric of his thought will never be able to change reality, and will never, therefore, make any progress.\n\nChange\u2014real change\u2014comes from the inside out. It doesn't come from hacking at the leaves of attitude and behavior with quick fix personality ethic techniques. It comes from striking at the root\u2014the fabric of our thought, the fundamental, essential para\u00addigms, which give definition to our character and create the lens through which we see the world. In the words of Amiel,\n\nMoral truth can be conceived in thought. One can have feelings about it. One can will to live it. But moral truth may have been penetrated and possessed in all these ways, and escape us still. Deeper even than consciousness there is our being itself\u2014our very substance, our nature. Only those truths which have entered into this last region, which have become ourselves, become spontaneous and involuntary as well as voluntary, unconscious as well as conscious, are really our life\u2014that is to say, something more than property. So long as we are able to distinguish any space whatever between Truth and us we remain outside it. The thought, the feeling, the desire or the consciousness of life may not be quite life. To become divine is then the aim of life. Then only can truth be said to be ours beyond the possibility of loss. It is no longer outside us, nor in a sense even in us, but we are it, and it is we.\n\nAchieving unity\u2014oneness\u2014with ourselves, with our loved ones, with our friends and working associates, is the highest and best and most delicious fruit of the Seven Habits. Most of us have tasted this fruit of true unity from time to time in the past, as we have also tasted the bitter, lonely fruit of disunity\u2014and we know how precious and fragile unity is.\n\nObviously building a character of total integrity and living the life of love and service that creates such unity isn't easy. It isn't quick fix.\n\nBut it's possible. It begins with the desire to center our lives on correct principles, to break out of the paradigms created by other centers and the comfort zones of unworthy habits.\n\nSometimes we make mistakes, we feel awkward. But if we start with the Daily Private Victory and work from the inside out, the results will surely come. As we plant the seed and patiently weed and nourish it, we begin to feel the excitement of real growth and eventually taste the incomparably delicious fruits of a congruent, effective life.\n\nAgain, I quote Emerson: \"That which we persist in doing becomes easier\u2014not that the nature of the task has changed, but our ability to do has increased.\"\n\nBy centering our lives on correct principles and creating a balanced focus between doing and increasing our ability to do, we become empowered in the task of creating effective, useful, and peaceful lives... for ourselves, and for our posterity.\n\n### A Personal Note\n\nAs I conclude this book, I would like to share my own personal conviction concerning what I believe to be the source of correct principles. I believe that correct principles are natural laws, and that God, the Creator and Father of us all, is the source of them, and also the source of our conscience. I believe that to the degree people live by this inspired conscience, they will grow to fulfill their natures; to the degree that they do not, they will not rise above the animal plane.\n\nI believe that there are parts to human nature that cannot be reached by either legislation or education, but require the power of God to deal with. I believe that as human beings, we cannot perfect ourselves. To the degree to which we align ourselves with correct principles, divine endowments will be released within our nature in enabling us to fulfill the measure of our creation. In the words of Teilhard de Chardin, \"We are not human beings having a spiritual experience. We are spiritual beings having a human experience.\"\n\nI personally struggle with much of what I have shared in this book. But the struggle is worthwhile and fulfilling. It gives meaning to my life and enables me to love, to serve, and to try again.\n\nAgain, T. S. Eliot expresses so beautifully my own personal discovery and conviction: \"We must not cease from exploration. And the end of all our exploring will be to arrive where we began and to know the place for the first time.\"\n\n### Afterword:  \nQuestions I am Often Asked\n\nFrankly, I've always been embarrassed by personal questions like some in this afterword. But I am asked them so often and with such interest that I've gone ahead and included them here. Many of these questions and answers were also included in Living the 7 Habits.\n\nThe 7 Habits was published in 1989. Given your experiences in the many years that have followed, what would you change, add, or subtract?\n\nI'm not responding lightly, but frankly I wouldn't change any\u00adthing. I might go deeper and apply wider but I have had the opportunity to do that in some of the books released since then.\n\nFor example, over 250,000 individuals were profiled showing Habit 3, Put First Things First, as the habit most neglected. So, the First Things First book (published 1996) went deeper into Habits 2 and 3 but also added more substance and illustrations for all the other habits.\n\nThe 7 Habits of Highly Effective Families applied the 7 Habits framework of thinking into building strong, happy, highly effec\u00adtive families.\n\nAlso, my son, Sean, applied the framework to the unique needs, interests and challenges of teens in a very visually attractive, entertaining, and edifying way in The 7 Habits of Highly Effective Teens.\n\nWe have also had tens of thousands of people tell us of the significant impact of becoming the creative force of their own lives through internalizing the 7 Habits. Seventy six of them shared the details of their fascinating stories of courage and inspiration in Living the 7 Habits\u2014showing the transforming power of the principles in all kinds of personal, family, and organizational settings regardless of their circumstances, organizational position, or prior life experiences.\n\nWhat have you learned about the 7 Habits since the book's release?\n\nI have learned or had reinforced many things. I'll briefly mention ten learnings.\n\n1. The importance of understanding the difference between principles and values. Principles are natural laws that are external to us and that ultimately control the consequences of our actions. Values are internal and subjective and represent that which we feel strongest about in guiding our behavior. Hopefully we will come to value principles, so that we get the results we want now in a way that enables us to get even greater results in the future, which is how I define effectiveness. Everyone has values; even criminal gangs have values. Values govern people's behavior but principles govern the consequences of those behaviors. Principles are independent of us. They operate regardless of our awareness of them, acceptance of them, liking of them, belief in them, or obeying of them. I have come to believe that humility is the mother of all virtues. Humility says we are not in control, principles are in control, therefore we submit ourselves to principles. Pride says that we are in control, and since our values govern our behavior, we can simply do life our way. We may do so but the consequences of our behavior flow from principles not our values. Therefore we should value principles.\n\n2. From experiences all over the world with this material I have come to see the universal nature of the principles undergirding this material. Illustrations and practices may vary and are cultur\u00adally specific, but the principles are the same. I have found the principles contained in the 7 Habits in all six major world reli\u00adgions and have actually drawn upon quotations from sacred writings of those religions when teaching in those cultures. I have done this in the Middle East, India, Asia, Australia and the South Pacific, South America, Europe, North America, Africa, and among Native Americans and other indigenous peoples. All of us, men and women alike, face similar problems, have similar needs, and internally resonate with the underlying principles. There is an internal sense of the principle of justice or win\/win. There is an internal moral sense of the principle of responsibility, of the principle of purpose, of integrity, of respect, of cooperation, of communication, of renewal. These are universal. But practices are not. They are situationally specific. Every culture interprets universal principles in unique ways.\n\n3. I have come to see the organizational implications of the 7 Habits, although, in the strict technical sense, an organization does not have habits. Its culture has norms or mores or social codes, which repre\u00adsent habits. An organization also has established systems, processes, and procedures. These represent habits. In fact, in the last analysis, all behavior is personal. It is individual even though it often is part of collective behavior in the form of decisions made by management regarding structure and systems, processes and practices. We have worked with thousands of organizations in most every industry and profession and have found that the same basic principles contained in the 7 Habits apply and define effectiveness.\n\n4. You can teach all 7 Habits by starting with any one habit. And you can also teach one habit in a way that leads to the teaching of the other six. It's like a hologram where the whole is contained in the part and the part is contained in the whole.\n\n5. Even though the 7 Habits represents an inside-out approach, it works most successfully when you start with the outside challenge and then take the inside-out approach. In other words, if you are having a relationship challenge, say a breakdown of communication and trust, this will define the nature of the needed inside-out ap\u00adproach in winning the kind of private victory that enables the pub\u00adlic victory meeting that challenge. This is the reason I often teach Habits 4, 5, and 6 before I teach Habits 1, 2, and 3.\n\n6. Interdependence is ten times more difficult than independence. It demands so much more mental and emotional independence to think win\/win when another person is into win\/lose, to seek to understand first when everything inside you cries out for under\u00adstanding, and to search for a better third alternative when compromise is so much easier. In other words, to work successfully with others in creative cooperative ways requires an enormous amount of independence, internal security, and self-mastery. Oth\u00aderwise, what we call interdependency is really counter-dependency where people do the opposite to assert their independence, or codependency where they literally need the other person's weakness to fulfill their need and to justify their own weakness.\n\n7. You can pretty well summarize the first three habits with the expression \"make and keep a promise.\" And you can pretty well summarize the next three habits with the expression \"involve oth\u00aders in the problem and work out the solution together.\"\n\n8. The 7 Habits represents a new language even though there are fewer than a dozen unique words or phrases. This new language be\u00adcomes a code, a shorthand way of saying a great deal. When you say to another \"was that a deposit or a withdrawal?\" \"Is that reactive or proactive?\" \"Is that synergistic or a compromise?\" \"Is that win\/win or win\/lose or lose\/win?\" \"Is that putting first things first or second things first?\" \"Is that beginning with the means in mind or the end in mind?\" I've seen entire cultures transformed by a wide under\u00adstanding of and commitment to the principles and concepts sym\u00adbolized by these very special code words.\n\n9. Integrity is a higher value than loyalty. Or better put, integrity is the highest form of loyalty. Integrity means being integrated or centered on principles not on people, organizations, or even fam\u00adily. You will find that the root of most issues that people are deal\u00ading with is \"is it popular (acceptable, political), or is it right?\" When we prioritize being loyal to a person or group over doing what we feel to be right, we lose integrity. We may temporarily gain popularity or build loyalty, but, downstream, this loss of integrity will undermine even those relationships. It's like badmouthing someone behind their back. The person you are temporarily united with through badmouthing someone else knows you would bad mouth them under different pressures and cir\u00adcumstances. In a sense, the first three habits represent integrity and the next three loyalty; but they are totally interwoven. Over time, integrity produces loyalty. If you attempt to reverse them and go for loyalty first, you will find yourself temporizing and compromising integrity. It's better to be trusted than to be liked. Ultimately, trust and respect will generally produce love.\n\n10. Living the 7 Habits is a constant struggle for everyone. Every\u00adone falters from time to time on each of the seven and sometimes all seven simultaneously. They really are simple to understand but difficult to consistently practice. They are common sense but what is common sense is not always common practice.\n\nWhich habit do you personally have the greatest difficulty with?\n\nHabit 5. When I am really tired and already convinced I'm right, I really don't want to listen. I may even pretend to listen. Basically I am guilty of the same thing I talk about, listening with the intent to reply, not to understand. In fact, in some sense, I strug\u00adgle almost daily with all 7 Habits. I have conquered none of them. I see them more as life principles that we never really mas\u00adter and that the closer we come to their mastery, the more aware we become of how far we really have yet to go. It's like the more you know the more you know you don't know.\n\nThis is why I often gave my university students 50 percent of the grade for the quality of their questions and the other 50 per\u00adcent for the quality of their answer to their questions. Their true level of knowledge is better revealed that way.\n\nSimilarly, the 7 Habits represents an upward cycle.\n\nHabit 1 at a high level is vastly different from Habit 1 at a lower level. To be proactive at the beginning level may only be awareness of the space between stimulus and response. At the next level it may involve a choice such as, not to get back at or to get even. At the next level, to give feedback. At the next level, to ask forgiveness. At the next level, to forgive. At the next, to for\u00adgive parents. At the next level, to forgive dead parents. And the next level, to simply not take offense.\n\nYou're the vice-chairman of FranklinCovey Company. Does FranklinCovey live the 7 Habits?\n\nWe try to. Continually trying to live what we teach is one of our most fundamental values. But we don't do it perfectly. Like any other business, we're challenged by changing market realities and by integrating the two cultures of the former Covey Leader\u00adship Center and Franklin Quest. The merger took place in the summer of 1997. It takes time, patience, and persistence in apply\u00ading the principles and the true test of our success will be in the long run. No snapshot will give an accurate picture.\n\nAny airplane is off track much of the time but just keeps com\u00ading back to the flight plan. Eventually, it arrives at its destination. This is true with all of us as individuals, families, or organiza\u00adtions. The key is to have an \"End in Mind\" and a shared commit\u00adment to constant feedback and constant course correction.\n\nWhy seven? Why not six or eight or ten or fifteen? What is so sacred about seven?\n\nNothing is sacred about seven, it just so happens that the three private victory habits (freedom to choose, choice, action) precede the three public victory habits (respect, understanding, creation) and then there is one to renew the rest and that equals seven.\n\nWhen asked this question, I've always said if there were some other desirable characteristic you would like make into a habit, you would simply put that under Habit 2 as one of the values you are trying to live by. In other words, if punctuality is a desir\u00adable trait you want to make a habit, that would be one of the val\u00adues of Habit 2. So no matter what else you came up with you would put it under Habit 2, your value system. Habit one is the idea that you can have a value system, that you can choose your own value system. Habit 2 is what those choices or values are and Habit 3 is to live by them. So they are very basic, generic, and interconnected.\n\nIt so happens that at the writing of this afterword for this new edition of the Seven Habits, I have just completed a new book entitled The 8th Habit: From Effectiveness to Greatness. To some, calling it the 8th Habit may appear to be a departure from my standard answer. But you see, as I say in the opening chapter of this new book, the world has profoundly changed since The 7 Habits of Highly Effective People was published in 1989. The chal\u00adlenges and complexity we face in our personal lives and relation\u00adships, in our families, in our professional lives, and in our organizations are of a different order of magnitude. In fact, many mark 1989\u2014the year we witnessed the fall of the Berlin Wall\u2014as the beginning of the Information Age, the birth of a new reality, a sea change of incredible significance... truly a new era.\n\nBeing highly effective as individuals and organizations is no longer optional in today's world\u2014it's the price of entry to the playing field. But surviving, thriving, innovating, excelling and leading in this new reality will require us to build on and reach beyond effectiveness. The call and need of a new era is for fulfillment. It's for passionate optimization, for significant contribution and greatness. These are on a different plane or dimension. They are different in kind\u2014just as significance is different in kind, not in degree, from success. Tapping into the higher reaches of human genius and motivation\u2014what we could call voice\u2014requires a new mindset, a new skill-set, a new tool-set... a new habit.\n\nThe 8th Habit, then, is not about adding one more habit to the 7th\u2014one that somehow got forgotten. It's about seeing and har\u00adnessing the power of a third dimension to the 7 Habits that meets the central challenge of the new Knowledge Worker Age.\n\nHow does notoriety affect you?\n\nIt affects me in different ways. From an ego standpoint, it's flat\u00adtering. From a teaching standpoint it is humbling, but I must strongly acknowledge that I am not the author of any of these principles and deserve absolutely no recognition. I am not saying this because of a desire to be modest and humble. I am saying this because I believe it\u2014that I, myself, believe it. I see myself like most of you\u2014as a seeker of truth, of understanding. I am not a guru; I disdain being called a guru. I want no disciples. I am only trying to promote a discipleship toward principles that are already in people's hearts, that people will live true to their con\u00adscience.\n\nIf you had it to do over again, what is the one thing you would do differently as a businessperson?\n\nI would do more strategic, proactive recruiting and selecting. When you are buried by the urgent and have a thousand balls in the air, it is so easy to put people that appear to have solutions into key positions. The tendency is to not look deeply into their backgrounds and patterns, to do \"due diligence,\" nor is it to carefully develop the criteria that needs to be met in the particular roles or assignments. I am convinced that when recruiting and selecting is done strategically, that is, thinking long-term and proactively, not based upon the pressures of the moment, it pays enormous long-term dividends. Someone once said, \"That which we desire most earnestly we believe most easily.\" You really have to look deeply into both character and competence because even\u00adtually, downstream, flaws in either area will manifest themselves in both areas. I am convinced that although training and devel\u00adopment is important, recruiting and selection are much more important.\n\nIf you had it to do over again, what is the one thing you would do differently as a parent?\n\nAs a parent, I wish I had spent more time in carefully developing soft, informal win\/win agreements with each of my children in the different phases of their lives. Because of business and travels I often indulged my children and went for lose\/win too much instead of paying the price in relationship building sufficient to really develop thorough, sound win\/win agreements more con\u00adsistently.\n\nHow is technology going to change business in the future?\n\nI believe in Stan Davis's statement that \"When the infrastructure changes, everything rumbles,\" and I think the technical infrastruc\u00adture is central to everything. It will accelerate all good and bad trends. I'm also convinced that it is for these very reasons that the human element becomes even more important. High tech without high touch does not work, and, the more influential technology be\u00adcomes, the more important is the human factor which controls that technology becomes, particularly in developing a cultural commit\u00adment to the criteria in the use of that technology.\n\nAre you surprised or astounded at the universal popularity of the 7 Habits (with other countries\/cultures\/ages\/gender)?\n\nYes and no. Yes, in that I had no idea it would become a world\u00adwide phenomenon and that a few of the words would become part of Americana. No, in the sense that the material had been tested for over twenty five years and I knew that it would work primarily because it is based upon principles I did not invent and therefore take no credit for.\n\nHow would you begin to teach the 7 Habits to very young children?\n\nI think I would live by Albert Schweitzer's three basic rules for raising children: First, example; second, example; third, example. But I wouldn't go quite that far. I would say, first, example; sec\u00adond, build a caring and affirming relationship; and third, teach some of the simple ideas underlying the habits in the language of children\u2014help them gain a basic understanding and vocabulary of the 7 Habits and show them how to process their own experi\u00adences through the principles; let them identify what particular principles and habits are being illustrated in their lives.\n\nMy boss (spouse, child, friend, etc.) really needs the 7 Habits. How would you recommend I get them to read it?\n\nPeople don't care how much you know until they know how much you care. Build a relationship of trust and openness based upon a character example of trustworthiness and then share how the 7 Habits have helped you. Simply let them see the 7 Habits in action through your life. Then, at the appropriate time, you might invite them to participate in a training program or share your book as a gift or teach some of the basic ideas when the occasion calls for it.\n\nWhat is your background and how did you come to write the 7 Habits?\n\nIt was implicitly understood that I would follow in my father's footsteps and go into the family business. However, I found that I enjoyed teaching and training leaders even more than business. I became deeply interested and involved in the human side of organizations when I was at Harvard Business School. Later I taught business subjects at Brigham Young University and did consulting, advising, and training on the side for several years. During that time, I became interested in creating integrated lead\u00adership and management development programs around a sequential and balanced set of principles. These eventually evolved into the 7 Habits and then while applying it to organiza\u00adtions it evolved into the concept of principle-centered leadership. I decided to leave the university and go full-time into training executives from all different kinds of organizations. After a year of following a very carefully developed curriculum came the development of a business that has enabled us to take the mate\u00adrial to people throughout the world.\n\nWhat is your response to the people who claim to have the true formula for success?\n\nI would say two things. First, if what they are saying is based on principles or natural laws, I want to learn from them and I com\u00admend them. Second, I would say we are probably using different words to describe the same basic principles or natural laws.\n\nAre you really bald or do you shave your head for efficiency's sake?\n\nHey, listen, while you're busy blow-drying your hair, I'm out serving the customers. In fact, the first time I heard the expression, \"Bald is beautiful,\" I kicked the slats out of my crib!\n\n### APPENDIX A\n\n### POSSIBLE PERCEPTIONS FLOWING  \nout of VARIOUS CENTERS\n\n**  \n**\n\nIf your center is: Spouse\n\nThese are alternative ways you may tend to perceive other areas of your life:\n\nSpouse\n\n  * The main source of need satisfaction\n\nFamily\n\n  * Good in its place\n  * Less important\n  * A common project\n\nMoney\n\n  * Necessary to properly take care of spouse\n\nWork\n\n  * Necessary to earn money to care for spouse\n\nPossessions\n\n  * Means to bless, impress, or manipulate\n\nPleasure\n\n  * Mutual, unifying activity or unimportant\n\nA Friend or Friends\n\n  * Spouse is best or only friend\n  * Only friends are \"our\" friends\n\nEnemy or Enemies\n\n  * Spouse is my defender, or common enemy provides source of marriage definition\n\nChurch\n\n  * Activity to enjoy together\n  * Subordinate to relationship\n\nSelf\n\n  * Self-worth is spouse based\n  * Highly vulnerable to spouse attitudes and behaviors\n\nPrinciples\n\n  * Ideas which create and maintain relationship with spouse\n\n* * *\n\nIf your center is: Family\n\nThese are alternative ways you may tend to perceive other areas of your life:\n\nSpouse\n\n  * Part of the family\n\nFamily\n\n  * The highest priority\n\nMoney\n\n  * Family economic support\n\nWork\n\n  * A means to an end\n\nPossessions\n\n  * Family comfort and opportunities\n\nPleasure\n\n  * Family activities or relatively unimportant\n\nA Friend or Friends\n\n  * Friends of the family, or competition\n  * Threat to strong family life\n\nEnemy or Enemies\n\n  * Defined by family\n  * Source of family strength and unity\n  * Possible threat to family strength\n\nChurch\n\n  * Source of help\n\nSelf\n\n  * Vital part of but subordinate to family\n\nPrinciples\n\n  * Rules which keep family unified and strong\n  * Subordinate to family\n\n* * *\n\nIf your center is: Money\n\nThese are alternative ways you may tend to perceive other areas of your life:\n\nSpouse\n\n  * Asset or liability in acquiring money\n\nFamily\n\n  * Economic drain\n\nMoney\n\n  * Source of security and fulfillment\n\nWork\n\n  * Necessary to the acquisition of money\n\nPossessions\n\n  * Evidence of economic success\n\nPleasure\n\n  * Economic drain or evidence of economic stress\n\nA Friend or Friends\n\n  * Chosen because of economic status or influence\n\nEnemy or Enemies\n\n  * Economic competitors\n  * Threat to economic security\n\nChurch\n\n  * Tax write-off\n  * Hand in your pocket\n\nSelf\n\n  * Self-worth is determined by net worth\n\nPrinciples\n\n  * Ways that work in making and managing money\n\n* * *\n\nIf your center is: Work\n\nThese are alternative ways you may tend to perceive other areas of your life:\n\nSpouse\n\n  * Help or hindrance in work\n\nFamily\n\n  * Help or interruption to work\n  * People to instruct in work ethic\n\nMoney\n\n  * Of secondary importance\n  * Evidence of hard work\n\nWork\n\n  * Main source of fulfillment and satisfaction\n  * Highest ethic\n\nPossessions\n\n  * Tools to increase work effectiveness\n  * Fruits, badge of work\n\nPleasure\n\n  * Waste of time\n  * Interferes with work\n\nA Friend or Friends\n\n  * Developed from work setting or shared interest\n  * Basically unnecessary\n\nEnemy or Enemies\n\n  * Obstacles to work productivity\n\nChurch\n\n  * Important to corporate image\n  * Imposition on your time\n  * Opportunity to network\n\nSelf\n\n  * Defined by job role\n\nPrinciples\n\n  * Ideas that make you successful in your work\n  * Need to adapt to work conditions\n\n* * *\n\nIf your center is: Possessions\n\nThese are alternative ways you may tend to perceive other areas of your life:\n\nSpouse\n\n  * Main possession\n  * Assistant in acquiring possessions\n\nFamily\n\n  * Possession to use, exploit, dominate, smother, control\n  * Showcase\n\nMoney\n\n  * Key to increasing possessions\n  * Another possession to control\n\nWork\n\n  * Opportunity to possess status, authority, recognition\n\nPossessions\n\n  * Status symbols\n\nPleasure\n\n  * Buying, shopping, joining clubs\n\nA Friend or Friends\n\n  * Personal objects\n  * Usable\n\nEnemy or Enemies\n\n  * Takers, thieves\n  * Others with more possessions or recognition\n\nChurch\n\n  * \"My\" church, a status symbol\n  * Source of unfair criticism or good things in life\n\nSelf\n\n  * Defined by the things I own\n  * Defined by social status, recognition\n\nPrinciples\n\n  * Concepts which enable you to acquire and enhance possessions\n\n* * *\n\nIf your center is: Pleasure\n\nThese are alternative ways you may tend to perceive other areas of your life:\n\nSpouse\n\n  * Companion in fun and pleasure or obstacle to it\n\nFamily\n\n  * Vehicle or interference\n\nMoney\n\n  * Means to increase opportunities for pleasure\n\nWork\n\n  * Means to an end\n  * \"Fun\" work OK\n\nPossessions\n\n  * Objects of fun\n  * Means to more fun\n\nPleasure\n\n  * Supreme end in life\n  * Main source of satisfaction\n\nA Friend or Friends\n\n  * Companions in fun\n\nEnemy or Enemies\n\n  * Take life too seriously\n  * Guilt trippers, destroyers\n\nChurch\n\n  * Inconvenient, obstacle to recreation\n  * Guilt trip\n\nSelf\n\n  * Instrument for pleasure\n\nPrinciples\n\n  * Natural drives and instincts which need to be satisfied\n\n* * *\n\nIf your center is: A Friend or Friends\n\nThese are alternative ways you may tend to perceive other areas of your life:\n\nSpouse\n\n  * Possible friend or possible competitor\n  * Social status symbol\n\nFamily\n\n  * Friends or obstacle to developing friendships\n  * Social status symbol\n\nMoney\n\n  * Source of economic and social good\n\nWork\n\n  * Social opportunity\n\nPossessions\n\n  * Means of buying friendship\n  * Means of entertaining or providing social pleasure\n\nPleasure\n\n  * Enjoyed always with friends\n  * Primarily social events\n\nA Friend or Friends\n\n  * Critical to personal happiness\n  * Belonging, acceptance, popularity is crucial\n\nEnemy or Enemies\n\n  * Outside the social circle\n  * Common enemies provide unity or definition for friendship\n\nChurch\n\n  * Place for social gathering\n\nSelf\n\n  * Socially defined\n  * Afraid of embarrassment or rejection\n\nPrinciples\n\n  * Basic laws which enable you to get along with others\n\n* * *\n\nIf your center is: An Enemy or Enemies\n\nThese are alternative ways you may tend to perceive other areas of your life:\n\nSpouse\n\n  * Sympathizer or scapegoat\n\nFamily\n\n  * Refuge (emotional support) or scapegoat\n\nMoney\n\n  * Means to fight with or prove superiority\n\nWork\n\n  * Escape or opportunity to vent feelings\n\nPossessions\n\n  * Fighting tools\n  * Means to secure allies\n  * Escape, refuge\n\nPleasure\n\n  * Rest and relaxation time before the next battle\n\nA Friend or Friends\n\n  * Emotional supporters and sympathizers\n  * Possibly defined by common enemy\n\nEnemy or Enemies\n\n  * Objects of hate\n  * Source of personal problems\n  * Stimuli to self-protection and self-justification\n\nChurch\n\n  * Source of self-justification\n\nSelf\n\n  * Victimized\n  * Immobilized by enemy\n\nPrinciples\n\n  * Justification for labeling enemies\n  * Source of your enemy's wrongness\n\n* * *\n\nIf your center is: Church\n\nThese are alternative ways you may tend to perceive other areas of your life:\n\nSpouse\n\n  * A companion in or facilitator for church service, or a trial of faith\n\nFamily\n\n  * Models to exemplify adherence to church teachings, or trials of faith\n\nMoney\n\n  * Means to support church and family\n  * Evil, if greater in priority than church service or teachings\n\nWork\n\n  * Necessary for temporal support\n\nPossessions\n\n  * Temporal possessions of minimal importance\n  * Reputation and image of great worth\n\nPleasure\n\n  * \"Innocent\" pleasures as an opportunity to gather with other church members\n  * Others as sinful or time wasters, to be self-righteously denied\n\nA Friend or Friends\n\n  * Other members of the church\n\nEnemy or Enemies\n\n  * Nonbelievers; those who disagree with church teachings or whose lives are in blatant opposition to them\n\nChurch\n\n  * Highest priority source of guidance\n\nSelf\n\n  * Self-worth is determined by activity in the church, contributions to the church, or performance of deeds that reflect the church ethic\n\nPrinciples\n\n  * Doctrines taught by the church\n  * Subordinate to the church\n\n* * *\n\nIf your center is: Self\n\nThese are alternative ways you may tend to perceive other areas of your life:\n\nSpouse\n\n  * Satisfier and pleaser\n\nFamily\n\n  * Possession\n  * Need satisfier\n\nMoney\n\n  * Source of needs satisfaction\n\nWork\n\n  * Opportunity to \"do my own thing\"\n\nPossessions\n\n  * Source of self-definition, protection, enhancement\n\nPleasure\n\n  * Deserved sensate satisfactions\n  * \"My rights\"\n  * \"My needs\"\n\nA Friend or Friends\n\n  * Supporter, provider for \"me\"\n\nEnemy or Enemies\n\n  * Source of self-definition, self-justification\n\nChurch\n\n  * Vehicle to serve self-interests\n\nSelf\n\n  * Better, smarter, more right\n  * Justified in focusing all resources on personal gratification\n\nPrinciples\n\n  * Source of justification\n  * Those ideas that serve my best interests; can be adapted to need\n\n* * *\n\nIf your center is: Principles\n\nThese are alternative ways you may tend to perceive other areas of your life:\n\nSpouse\n\n  * Equal partner in a mutually beneficial interdependent relationship\n\nFamily\n\n  * Friends\n  * Opportunity for service, contribution, and fulfillment\n  * Opportunity for intergenerational rescripting and change\n\nMoney\n\n  * Enabling resource in the accomplishment of important priorities and goals\n\nWork\n\n  * Opportunity to use talents and abilities in a meaningful way\n  * Means to provide economic resource\n  * Time investment to be kept in balance with other time investments and in harmony with priorities and values in life\n\nPossessions\n\n  * Enabling resources\n  * Responsibilities to be properly cared for\n  * Secondary to people in importance\n\nPleasure\n\n  * Joy that comes from almost any activity in a focused life\n  * True re-creation as an important part of a balanced, integrated life-style\n\nA Friend or Friends\n\n  * Companions in interdependent livings\n  * Confidants\u2014those to share with, serve, and support\n\nEnemy or Enemies\n\n  * No real perceived \"enemies\"; just people with different paradigms and agendas to be understood and cared about\n\nChurch\n\n  * Vehicle for true principles\n  * Opportunity for service and contribution\n\nSelf\n\n  * One unique, talented, creative individual in the midst of many unique, talented, creative individuals who, working independently and interdependently, can accomplish great things\n\nPrinciples\n\n  * Immutable natural laws which cannot be violated with impunity\n  * When honored, preserve integrity and thus lead to true growth and happiness\n\n* * *\n\n### APPENDIX B\n\n### A QUADRANT II DAY  \nat the OFFICE\n\nTHE FOLLOWING EXERCISE AND ANALYSIS is designed to help you see the impact of a Quadrant II paradigm in a business setting on a very practical level.\n\nSuppose that you are the director of marketing for a major pharmaceutical firm. You are about to begin an average day at the office, and as you look over the items to attend to that day, you estimate the amount of time each one will take.\n\nYour unprioritized list includes the following:\n\n  1. You'd like to have lunch with the general manager (1-\u00bd hours).\n  2. You were instructed the day before to prepare your media budget for the following year (2 or 3 days).\n  3. Your \"IN\" basket is overflowing into your \"OUT\" basket (1-1\u00bd hours).\n  4. You need to talk to the sales manager about last month's sales; his office is down the hall (4 hours).\n  5. You have several items of correspondence that your secretary says are urgent (1 hour).\n  6. You'd like to catch up on the medical journals piled upon your desk (\u00bd hour).\n  7. You need to prepare a presentation for a sales meeting slated for next month (2 hours).\n  8. There's a rumor that the last batch of product X didn't pass quality control.\n  9. Someone from the FDA wants you to return his call about product \"X\" (\u00bd hour).\n  10. There is a meeting at two P.M. for the executive board, but you don't know what it is about (1 hour).\n\nTake a few minutes now and use what you have learned from Habits 1, 2, and 3 that might help you to effectively schedule your day.\n\n8-to-5 Schedule\n\n8 |\n\n* * *\n\n---|---  \n9 |\n\n* * *\n\n10 |\n\n* * *\n\n11 |\n\n* * *\n\n12 |\n\n* * *\n\n1 |\n\n* * *\n\n2 |\n\n* * *\n\n3 |\n\n* * *\n\n4 |\n\n* * *\n\n5 |\n\n* * *\n\nBy asking you to plan only one day, I have automatically eliminated the wider context of the week so fundamental to fourth generation time management. But you will be able to see the power of a Quadrant II, principle-centered paradigm even in the context of one nine-hour period of time.\n\nIt is fairly obvious that most of the items on the list are Quadrant I activities. With the exception of item number six\u2014catching up on medical journals\u2014everything else is seemingly both important and urgent.\n\nIf you were a third generation time manager, using prioritized values and goals, you would have a framework for making such scheduling decisions and would perhaps assign a letter such as A, B, or C next to each item and then number 1, 2, 3 under each A, B, and C. You would also consider the circumstances, such as the availability of other people involved, and the logical amount of time required to eat lunch. Finally, based on all of these factors, you would schedule the day.\n\nMany third generation time managers who have done this exercise do exactly what I have described. They schedule when they will do what, and based on various assumptions which are made and explicitly identified, they would accomplish or at least begin most of the items in that day and push the remainder onto the next day or to some other time.\n\nFor instance, most people indicate that they would use the time between eight and nine A.M. to find out exactly what was on the agenda for the executive board meeting so that they could prepare for it, to set up lunch with the general manager around noon, and to return the call from the FDA. They usually plan to spend the next hour or two talking to the sales manager, handling those correspondence items which are most important and urgent, and checking out the rumor regarding the last batch of product \"X\" which apparently didn't pass quality control. The rest of that morning is spent in preparing for the luncheon visit with the general manager and\/or for the two P.M. executive board meeting, or dealing with whatever problems were uncovered regarding product \"X\" and last month's sales.\n\nAfter lunch, the afternoon is usually spent attending to the unfinished matters just mentioned and\/or attempting to finish the other most important and urgent correspondence, making some headway into the overflowing \"IN\" basket, and handling other important and urgent items that may have come up during the course of the day.\n\nMost people feel the media budget preparations for the following year and the preparation for the next month's sales meeting could probably be put off until another day, which may not have as many Quadrant I items in it. Both of those are obviously more Quadrant II activities, having to do with long-term thinking and planning. The medical journals continue to be set aside because they are clearly Quadrant II and are probably less important than the other two Quadrant II matters just mentioned.\n\nThis is the kind of thinking which third generation time managers generally go through, even though they may vary as to when they will do what.\n\nWhat approach did you take as you scheduled those items? Was it similar to the third generation approach? Or did you take a Quadrant II, fourth generation approach? (Refer to the Time Management Matrix in Habit 3 - Put first things first.)\n\nThe Quadrant II Approach\n\nLet's go through the items on the list using a Quadrant II approach. This is only one possible scenario; others could be created, which may also be consistent with the Quadrant II paradigm, but this is illustrative of the kind of thinking it embodies.\n\nAs a Quadrant II manager, you would recognize that most P activities are in Quadrant I and most PC activities are in Quadrant II. You would know that the only way to make Quadrant I manageable is to give considerable attention to Quadrant II, primarily by working on prevention and opportunity and by having the courage to say \"no\" to Quadrants III and IV.\n\nTHE TWO P.M. BOARD MEETING. We will assume the two P.M. executive board meeting did not have an agenda for the attending executives, or perhaps you would not see the agenda until you arrived at the meeting. This is not uncommon. As a result, people tend to come unprepared and to \"shoot from the hip.\" Such meetings are usually disorganized and focus primarily on Quadrant I issues, which are both important and urgent, and around which there is often a great deal of sharing of ignorance. These meetings generally result in wasted time and inferior results and are often little more than an ego trip for the executive in charge.\n\nIn most meetings, Quadrant II items are usually categorized as \"other business.\" Because \"work expands to fill the time allotted for its completion\" in accordance with Parkinson's Law, there usually isn't time to discuss them. If there is, people have been so beaten and smashed by Quadrant I, they have little or no energy left to address them.\n\nSo you might move into Quadrant II by first attempting to get yourself on the agenda so that you can make a presentation regarding how to optimize the value of executive board meetings. You might also spend an hour or two in the morning preparing for that presentation, even if you are only allowed a few minutes to stimulate everyone's interest in hearing a more extended prepara\u00adtion at the next board meeting. This presentation would focus on the importance of always having a clearly specified purpose for each meeting and a well thought out agenda to which each person at the meeting has had the opportunity to contribute. The final agenda would be developed by the chairman of the executive board and would focus first on Quadrant II issues that usually require more creative thinking rather than Quadrant I issues that generally involve more mechanical thinking.\n\nThe presentation would also stress the importance of having minutes sent out immediately following the meeting, specifying assignments given and dates of accountability. These items would then be placed on appropriate future agendas which would be sent out in plenty of time for others to prepare to discuss them.\n\nNow this is what might be done by looking at one item on the schedule\u2014the two P.M. executive board meeting\u2014through a Quad\u00adrant II frame of reference. This requires a high level of proactivity, including the courage to challenge the assumption that you even need to schedule the items in the first place. It also requires consideration in order to avoid the kind of crisis atmosphere that often surrounds a board meeting.\n\nAlmost every other item on the list can be approached with the same Quadrant II thinking, with perhaps the exception of the FDA call.\n\nRETURNING THE FDA CALL. Based on the background of the quality of the relationship with the FDA, you make that call in the morning so that whatever it reveals can be dealt with appropriately. This might be difficult to delegate, since another organization is in\u00advolved that may have a Quadrant I culture and an individual who wants you, and not some delegate, to respond.\n\nWhile you may attempt to directly influence the culture of your own organization as a member of the executive board, your Circle of Influence is probably not large enough to really influence the culture of the FDA, so you simply comply with the request. If you find the nature of the problem uncovered in the phone call is persistent or chronic, then you may approach it from a Quadrant II mentality in an effort to prevent such problems in the future. This again would require considerable proactivity to seize the opportu\u00adnity to transform the quality of the relationship with the FDA or to work on the problems in a preventive way.\n\nLUNCH WITH THE GENERAL MANAGER. You might see having lunch with the general manager as a rare opportunity to discuss some longer-range, Quadrant II matters in a fairly informal atmosphere. This may also take thirty to sixty minutes in the morning to adequately prepare for, or you may simply decide to have a good social interaction and listen carefully, perhaps without any plan at all. Either possibility may present a good opportunity to build your relationship with the general manager.\n\nPREPARING THE MEDIA BUDGET. Regarding item number two, you might call in two or three of your associates most directly connected to media budget preparation and ask them to bring their recommen\u00addations in the form of \"completed staff work\" (which may only require your initials to finally approve) or perhaps to outline two or three well-thought-out options you can choose from and identify the consequences of each option. This may take a full hour sometime during the day\u2014to go over desired results, guidelines, resources, accountability, and consequences. But by investing this one hour, you tap the best thinking of concerned people who may have different points of view. If you haven't taken this approach before, you may need to spend more time to train them in what this approach involves, what \"completed staff work\" means, how to synergize around differences and what identifying alternative options and consequences involves.\n\nTHE \"IN\" BASKET AND CORRESPONDENCE. Instead of diving into the \"IN\" basket, you would spend some time, perhaps thirty to sixty minutes, beginning a training process with your secretary so that he or she could gradually become empowered to handle the \"IN\" basket as well as the correspondence under item number five. This training program might go on for several weeks, even months, until your secretary or assistant is really capable of being results-minded rather than methods-minded.\n\nYour secretary could be trained to go through all correspondence items and all \"IN\" basket items, to analyze them and to handle as many as possible. Items that could not be handled with confidence could be carefully organized, prioritized, and brought to you with a recommendation or a note for your own action. In this way, within a few months your secretary or executive assistant could handle 80 to 90 percent of all of the \"IN\" basket items and correspondence, often much better than you could handle them yourself, simply because your mind is so focused on Quadrant II opportunities instead of buried in Quadrant I problems.\n\nTHE SALES MANAGER AND LAST MONTH'S SALES. A possible Quadrant II approach to item number four would be to think through the entire relationship and performance agreement with that sales manager to see if the Quadrant II approach is being used. The exercise doesn't indicate what you need to talk to the sales manager about, but assuming it's a Quadrant I item, you could take the Quadrant II approach and work on the chronic nature of the problem as well as the Quadrant I approach to solve the immediate need.\n\nPossibly you could train your secretary to handle the matter without your involvement and bring to your attention only that which you need to be aware of. This may involve some Quadrant II activity with your sales manager and others reporting to you so they understand that your primary function is leadership rather than management. They can begin to understand that they can actually solve the problem better with your secretary than with you, and free you for Quadrant II leadership activity.\n\nIf you feel that the sales manager might be offended by having your secretary make the contact, then you could begin the process of building that relationship so that you can eventually win the confidence of the sales manager toward your both taking a more beneficial Quadrant II approach.\n\nCATCHING UP ON MEDICAL JOURNALS. Reading medical journals is a Quadrant II item you may want to procrastinate. But your own long-term professional competence and confidence may largely be a function of staying abreast of this literature. So, you may decide to put the subject on the agenda for your own staff meeting, where you could suggest that a systematic approach to reading the medical journals be set up among your staff. Members of the staff could study different journals and teach the rest the essence of what they learn at future staff meetings. In addition, they could supply others with key articles or excerpts which everyone really needs to read and understand.\n\nPREPARING FOR NEXT MONTH'S SALES MEETING. Regarding item number seven, a possible Quadrant II approach might be to call together a small group of the people who report to you and charge them to make a thorough analysis of the needs of the salespeople. You could assign them to bring a completed staff work recommendation to you by a specified date within a week or ten days, giving you enough time to adapt it and have it implemented. This may involve their interviewing each of the salespeople to discover their real concerns and needs, or it might involve sampling the sales group so that the sales meeting agenda is relevant and is sent out in plenty of time so that the salespeople can prepare and get involved in it in appropriate ways.\n\nRather than prepare the sales meeting yourself, you could delegate that task to a small group of people who represent different points of view and different kinds of sales problems. Let them interact constructively and creatively and bring to you a finished recommendation. If they are not used to this kind of assignment, you may spend some of that meeting challenging and training them, teaching them why you are using this approach and how it will benefit them as well. In doing so, you are beginning to train your people to think long-term, to be responsible for completing staff work or other desired results, to creatively interact with each other in interdependent ways, and to do a quality job within specified deadlines.\n\nPRODUCT \"X\" AND QUALITY CONTROL. Now let's look at item number eight regarding product \"X,\" which didn't pass quality control. The Quadrant II approach would be to study that problem to see if it has a chronic or persistent dimension to it. If so, you could delegate to others the careful analysis of that chronic problem with instructions to bring to you a recommendation, or perhaps simply to implement what they come up with and inform you of the results.\n\nThe net effect of this Quadrant II day at the office is that you are spending most of your time delegating, training, preparing a board presentation, making one phone call, and having a productive lunch. By taking a long-term PC approach, hopefully in a matter of a few weeks, perhaps months, you won't face such a Quadrant I scheduling problem again.\n\nAs you go through this analysis, you may be thinking this approach seems idealistic. You may be wondering if Quadrant II managers ever work in Quadrant I.\n\nI admit it is idealistic. This book is not about the habits of highly ineffective people; it's about habits of highly effective people. And to be highly effective is an ideal to work toward.\n\nOf course you'll need to spend time in Quadrant I. Even the best laid plans in Quadrant II sometimes aren't realized. But Quadrant I can be significantly reduced into more manageable proportions so that you're not always into the stressful crisis atmosphere that negatively affects your judgment as well as your health.\n\nUndoubtedly it will take considerable patience and persistence, and you may not be able to take a Quadrant II approach to all or even most of these items at this time. But if you can begin to make some headway on a few of them and help create more of a Quadrant II mind-set in other people as well as yourself, then downstream there will be quantum improvements in performance.\n\nAgain, I acknowledge that in a family setting or a small business setting, such delegation may not be possible. But this does not preclude a Quadrant II mind-set which would produce interesting and creative ways within your Circle of Influence to reduce the size of Quadrant I crises through the exercise of Quadrant II initiative.\n\n### Problem\/Opportunity Index\n\nThis index should not be interpreted as an attempt to provide \"quick fix\" solutions to deep concerns. It is, rather, a reference to material in this book that specifically deals with these concerns and, hopefully, will provide insight and help in working on them. These references reflect parts of the holistic, integrated approach of the Seven Habits and are most effective when understood and used in context.\n\nPERSONAL EFFECTIVENESS\n\nGrowth and Change\n\nCan you really create change in your life?\n\nIf you can change seeing, you can change being, 15-45\n\nGravity to pull to you, 46-48\n\nFrom \"you\" to \"I\" to \"we,\" 48-52\n\nYour companion in change, 59-60\n\nOpening your \"gate of change,\" 60-62\n\nBeing in charge of change, 65-94\n\nBecoming your own first creator, 103-106\n\nUsing your head, 132-135\n\nOnward and upward, 304-306\n\nWhen change is unsettling\n\nCreating a changeless core,\n\nWhen you've been handed a bad script\n\nChanging it, 103-106, 132-145\n\nPassing a new script on to others, 316-318\n\n\"But it's too hard to change!\"\n\nThe power of independent will, 147-149\n\nWhen to say \"yes!\" and what it takes to say \"no!,\" 150-158\n\nTrying to be something you're not\n\nYou can't get the fruits without the roots, 21-23,\n\nSymbol without substance, 35-38\n\nWhen you're burned-out on setting goals\n\nPutting yourself in control immediately, 91-92\n\nGoals that grow out of mission and roles, 135-137\n\nMaking it happen each week,\n\nPersonal Leadership\n\nWhen you don't know if you're being effective or not\n\nWhat is \"effectiveness\" anyway?, 52-59\n\nYou're \"really busy\"\u2014so what?, 98-99\n\nIf you're \"successful\"\u2014and miserable\n\nPrimary and secondary greatness, 21-23\n\nSymbol without substance, 35-36\n\nOutside-in, 43-44\n\nWhen your \"ladder of success\" is leaning against the wrong wall, 98-99\n\nWhen what you're doing doesn't work\n\nInside-Out, 15-45\n\nLiving a facade\n\nYou can only pretend for so long, 36-38\n\nWhen you don't have a solid core\n\nYou can't make a lighthouse move, 32-35\n\nCreating a \"principle center,\" 122-128\n\nWhen you can't find purpose or meaning in your life\n\nTranscending adversity, 74-75\n\nHabit 2: Begin With the End in Mind, 95-144\n\nMoving beyond self,\n\nRoots and wings, 314-318\n\nWhen you feel controlled by circumstances or other people\n\nYour companion in change,\n\nHabit 1: Be Proactive, 65-94\n\nHabit 2: Begin with the End in Mind, 95-144\n\nHabit 3: Put First Things First, 142-82\n\nUnderstanding your scripting\n\nThe power of a ten-second experience, 24-29\n\nHabit 1: Be Proactive, 65-94\n\nHabit 2: Begin with the End in Mind, 95-144\n\nInside-Out Again, 314-318\n\nAppendix A - Possible Perceptions Rowing out of Various Centers\n\nPersonal Management\n\nPlanning systems that don't work\n\n\"Efficiency\" may not be the answer,\n\nFour generations of time management, 149-150\n\nQuadrant II\u2014the new generation, 158-171\n\nLosing sight of what really matters\n\nHabit 2: Begin with the End in\n\nMind, 95-144\n\nHabit 3: Put First Things First, 145-182\n\nWhen you can't say \"no!\"\n\nWhat it takes to say \"no!,\" 156-158\n\nWhen you want to run away\n\nThe pleasure-centered life, 114-115\n\nEscape from the pounding surf, 152-153\n\nMaintaining Balance and Peace\n\nBeing great at something\u2014at the sacrifice of everything else\n\nHaving a solid center for balanced success, 109-128\n\nIdentifying roles and goals, 135-137\n\nThe fourth generation makes it possible, , 170-171\n\nWhen family and work demands conflict\n\nCreating a personal constitution, 106-109\n\nKeeping priorities straight, 113-114\n\nIdentifying roles and goals, 135-137\n\nIf you suffer from feelings of insecurity\n\nWhere do you get your security?, 109-128\n\nLike yourself before you can like others, 186-187\n\nSources of intrinsic security, 298-299\n\nHandling moods\n\nYou are greater than your moods, 66-67\n\nWhen your honor becomes greater than your moods,\n\nHandling tough problems\n\nTurning adversity into opportunity, 74-75\n\nActing instead of being acted upon, 76-77\n\nDefining your problem: direct, indirect or no control, 85-86\n\nMaking decisions\n\nThe value of a Personal Mission Statement, 108-109\n\nWhen \"PMA\" doesn't work\n\nFacing reality,\n\nAchieving success... and killing yourself in the process\n\nEffectiveness lies in the balance, 52-59\n\nHabit 7: Sharpen the Saw, 287-307\n\nWhen you're unhappy\n\nWhat is happiness, 47-48\n\nThe have's and the be's, 89-90\n\nBeing responsible for your own happiness,\n\nWhen you make a mistake\n\nThe other end of the stick, 90-91\n\nAsking forgiveness when you \"blow it,\" 197-199\n\nINTERPERSONAL EFFECTIVENESS\n\nFamily\n\nCreating strong families\n\nParents caught in management paradigm,\n\nCreating a family mission statement, 137-139\n\nUsing a force field analysis, 279-280\n\nRoots and wings, 314-318\n\n\"But I never have time for my family!\"\n\nHabit 2: Begin with the End in\n\nMind, 95-144\n\nHabit 3: Put First Things First, 145-182\n\nWho's more important\u2014the individual or the masses?,\n\nMaking one-on-one time, 258-259\n\nTaking time\u2014together, 309-314\n\nWhen the family climate is wrong\n\nParents caught in management paradigm,\n\nChanging the climate, 279-281\n\nBecoming a transition person, 316-318\n\nWhen your happiness is based on your family\n\nBeing family-centered, 112-113\n\nMarriage\n\nFalling out of love\n\nAre you empowering your spouse's weaknesses?, 41-42\n\nIf you want to have... be!,\n\nLove her!, 79-80\n\nWhen your spouse is the problem\n\nIf you want to have... be!,\n\nStay in your Circle of Influence, 81-90\n\nInfluencing your spouse\n\nWorking on the \"be's,\"\n\nThe thirty-day test, 92-93\n\nBuilding an Emotional Bank Account, 185-203\n\nThe key to your influence, 238-239\n\nWalking out on a marriage\n\nForsaking responsibility in the name of independence,\n\nDeteriorating or broken relationships\n\nWhen the greed for golden eggs kills the goose,\n\nBuilding the Emotional Bank Account, 188-203\n\nWhen your happiness is based on your spouse's\n\nBeing spouse-centered, 111-112\n\nDealing with differences\n\nWhen you're spouse-centered, \u2014112\n\nHabit 5: Seek First to Understand, 235-260\n\nHabit 6: Synergize, 261-284\n\nThe Frigidaire hang-up, 309-314\n\nWhen there's no trust in the relationship\n\nBuilding an Emotional Bank Account, 185-203\n\n\"But we never have time for each other!\"\n\nSpending one-on-one time, 257-259\n\nTaking time\u2014together, 309-314\n\nWhen there's no communication\n\nCommunication, cooperation and synergy, 270-274\n\nTaking time to talk things out, 309-314\n\nConfessing each other's \"sins\"\n\nBeing enemy-centered,\n\nNegative synergy, 274-277\n\nParenting\n\nOverreacting to a child's negative behavior\n\nWriting a new script, 104-106\n\nBeing family-centered, 112-113\n\nSeeing and being differently, 132-133\n\nWhen you \"blow it,\" 198-199\n\nWhen you don't like the kind of parent you are\n\nRewriting your script, 104-106\n\nSeeing and being differently, 132-133\n\nRole-playing your concerns, 258-259\n\nIf you or your spouse is emotionally immature\n\nBeing unwilling to change,\n\nParental \"image\" can overpower parental integrity, 38-40\n\nIf you have a need to be popular with your child\n\nBeing family-centered, 112-113\n\nGoing for the golden eggs, 55-56\n\nInfluencing a child\n\nFrom \"please\" to power, 38-40\n\nOverpowering children when they're young, 55-56\n\nBuilding an Emotional Bank Account, 185-203\n\nHabit 5: Seek First to Understand, 235-260\n\nGetting children to handle their jobs\n\ncheerfully and well\n\nGetting the golden eggs\u2014and preserving the goose,\n\n\"Green and clean,\" 174-179\n\nCreating a Win\/Win with a child\n\n\"No Deal\"\u2014a liberating option,\n\nLicensed to drive\u2014a Win\/Win opportunity,\n\nRaising a responsible child\n\nGetting golden eggs and preserving the goose,\n\n\"Green and clean,\" 174-179\n\nCreating a Win\/Win,\n\n\"If Mommy doesn't get us up in time...,\"\n\nEncouraging a child: \"Use your R&I!,\"\n\nBuilding a relationship with a child\n\nStar Wars, 56-57\n\nProactivity: the thirty-day test, 92-93\n\nBuilding an Emotional Bank Account, 185-203\n\nHabit 5: Seek First to Understand, 235-260\n\nWhen getting your own way weakens the relationship\n\nParental \"image\" can overpower parental integrity, 38-40\n\nOverpowering children when they're young, 55-56\n\nGetting the golden eggs\u2014and preserving the goose,\n\nRescripting over-reactive behavior, 104-106\n\nWhen one wins and one loses, both lose, 209-213\n\nWhen your child sees things differently\n\n\"I don't want to share!,\" 38-40\n\nStar Wars, 56-57\n\nHandling a \"rebellious\" son, 199-201\n\nHabit 5: Seek First to Understand, 235-260\n\nWhen a child has difficulty in school\n\nLooking at your lens as well as through it, 16-21\n\n\"Look at my child\u2014I'm a failure!\"\n\nGetting social mileage out of your child's behavior, 16-21\n\nParental \"image\" can overpower parental integrity, 38-40\n\nBeing family-centered, 112-113\n\nWhen your child's self-esteem is low\n\nChanging the way you see your child, 16-21\n\nPossessing self precedes sharing with others,\n\nReflections from the social mirror, 67-68\n\nHabit 5: Seek First to Understand, 235-260\n\nThe image you reflect back to others, 299-301\n\nWhen you can't understand your child\n\nHabit 5: Seek First to Understand, 235-260\n\nMotivating children\n\n\"Green and clean,\" 173-179\n\nWhen your child won't communicate\n\nLearning how to listen, 245-253\n\nRelationships with Others\n\nIf it's difficult to build good relationships\n\nYou can't get the fruits without the roots, 21-23\n\nFrom \"you\" to \"I\" to \"we,\" 48-52\n\nBuilding Emotional Bank Accounts, 185-203\n\nWhen you don't understand others\n\nHabit 5: Seek First to Understand, 235-260\n\nWhen others don't understand you\n\nThen seek to be understood, 255-257\n\nHandling blaming, accusing, or ugly feelings towards others\n\nA dependent response,\n\nListening to your language, 78-80\n\nUgly feelings are buried alive,\n\nCreating an abundance mentality, 219-220\n\nIf you don't have a Win\/Win character\n\nFinding a mentor or model,\n\nWhen you need to give feedback\u2014but it's going to hurt\n\nCourage and consideration: creating the balance, 218-219\n\nDealing with differences\n\nTwo people see differently; both can be right, 24-29\n\nStar Wars, 56-57\n\nThe glasses you wear make a difference, 126-128\n\nThe difference is all in your head, 130-131\n\nUnderstanding: one person's mission is another's minutia, 190-191\n\nHabit 5: Seek First to Understand, 235-260\n\n\"But you said...,\" 194-195\n\nThe laws of love and the laws of life, 199-201\n\nHabit 6: Synergize, 261-284\n\nWhen there's no communication\n\nTrying to see the \"old lady,\" 24-29\n\nBuilding bank accounts of trust, 185-203\n\nHabit 5: Seek First to Understand, 235-260\n\nSynergy and communication, 269-271\n\nWhen there's no cooperation\n\nCooperation, communication and trust,\n\nInfluencing others\n\n\"Search your own heart...,\" 16-21\n\nIf you want to have, be...,\n\nP\/PC in the human dimension, 55-56\n\nWorking in your Circle of Influence, 81-90\n\nBuilding Emotional Bank Accounts, 185-203\n\nOpen communication in Win\/Win, 220-222\n\nThe key to influence, 238-239\n\nTo influence, you have to be influenced,\n\nIf you can't reach a Win\/Win\n\n\"No deal\"\u2014a liberating option, 213-216\n\nNot all decisions need to be Win\/Win,\n\nCompetition has its place,\n\n\"Irreconcilable\" differences\n\nYou don't have to take them to court, 280-283\n\nWhen there's no trust\n\nBuilding Emotional Bank Accounts, 185-203\n\nBuilding strong relationships\n\nIf you want to have, be...,\n\nCreating a changeless core,\n\nA tool that makes it possible, 161-162\n\nBuilding Emotional Bank Accounts, 185-203\n\nRelationships\u2014a key to Win\/Win, 220-222\n\nBonding in the AEC, 269-270\n\nSocial\/emotional renewal, 297-299\n\nRebuilding broken relationships\n\nBuilding Emotional Bank Accounts, 185-203\n\nThinking Win\/Win when you're dealing with a Win\/Lose person\n\nDealing with Win\/Lose\u2014the real test of Win\/Win,\n\nScripting others\n\nThe power of perception, 16-21\n\nReflections from the social mirror, 67-68\n\nThe image you reflect back to others, 299-301\n\nLoving\u2014with no strings attached\n\nThe laws of love and the laws of life, 199-201\n\n\"But I know how to listen!\"\n\nAre you really listening?, 245-253\n\nWhen communication becomes emotional\n\nDon't unload\u2014listen!,\n\nRelationships on the Job\n\nWorking for a dictator\n\nIf you want to have, be...,\n\nBeing a proactive gopher, 86-88\n\nHandling peer resentment\n\nProactivity\u2014the key, 86-88\n\nWhen the job isn't clear\n\n\"But you said...,\" 194-195\n\nWhen you're not getting through to your boss\n\nMaking effective presentations, 255-257\n\nDealing with difficult relationships on the job\n\nGiving yourself to the one, 201-202\n\nWhen management resists change\n\nBeing a proactive gopher, 86-88\n\nFocusing on results, 224-226\n\nRunning scared of your boss\n\nThe capacity to earn\u2014your most important financial asset,\n\nGetting a job\n\nUsing your \"R&I,\"\n\nORGANIZATIONAL EFFECTIVENESS\n\nCreating top quality organizations\n\nOrganizational PC, 57-59\n\nMeasure once, cut twice,\n\nCreating organizational mission\n\nstatements, 139-143\n\nCreating synergy in business, 267-269\n\nStewardship delegation\u2014a focus\n\non results, 173-174\n\nRelationships at the top\u2014the key, 201-202\n\nHabit 4: Think Win\/Win, 204-234\n\nHabit 5: Seek First to Understand, 235-260\n\nHabit 6: Synergize, 261-284\n\nBalanced renewal in business, 302-303\n\nLack of teamwork in the organization\n\nIndependence precedes effective interdependence, 48-52, 185-188\n\nCreating organizational mission\n\nstatements, 139-143\n\nHabit 4: Think Win\/Win, 204-234\n\nHabit 6: Synergize, 261-284\n\nRivalry, politicking, conflict in the organization\n\nNegative synergy, 274-275\n\nWhen the environment is Win\/Lose\n\nYou get what you reward, 229-232\n\nPoor customer service\n\nWatering down the chowder,\n\n\"P\" problems are \"PC\" opportunities,\n\n\"My salespeople are just order takers!,\" 230-231\n\nGetting crunched by the changing economy\n\nAct or be acted upon,\n\nWrong jungle!, 101-103\n\n\"Creaming\"\n\n\"My salespeople are just order takers!,\" 230-231\n\nHigh producers who are interpersonal disasters\n\nWatering what you want to grow, 231-232\n\nMANAGERIAL EFFECTIVENESS\n\nWhen you're managing instead of leading\n\nWrong jungle!, 101-103\n\nWhen you have too much to do\n\n\"Efficiency\" may not be the answer,\n\nHabit 3: Put First Things First, 145-182\n\nA Quadrant II Day at the Office, 345-352\n\nIneffective meetings\n\nThe two P.M. board meeting, 336-337\n\nPreparing for next month's sales meeting, 351-352\n\nHandling correspondence\n\nThe \"in\" basket and correspondence,\n\nKeeping up on trade publications\n\nCatching up on medical journals,\n\nBudgets\n\nPreparing the media budget, 349-350\n\nIrresponsible employees\n\nIs it \"them\" or \"me?,\"\n\nTreating employees like your best customers, 58-59\n\n\"But you said...,\" 194-195\n\nNegative feelings in business relationships\n\nMoving from 5 to 20 percent Quadrant II time, 155-156\n\nNegative feelings are buried alive, 209-213\n\nLack of creativity, innovation\n\nStewardship delegation\u2014bringing out the best, 171-179\n\nHabit 4: Think Win\/Win, 204-234\n\nHabit 6: Synergize, 261-284\n\nDealing with a low trust culture\n\nThe price of a high-trust culture,\n\nTrying to legislate trust,\n\nBuilding Emotional Bank Accounts, 185-203\n\nHabit 4: Think Win\/Win, 204-234\n\nIf you can't reach a Win\/Win\n\n\"No deal\"\u2014a liberating option, 213-216\n\nNot all decisions need to be Win\/Win,\n\nCompetition has its place,\n\nExpensive, time-consuming training programs\n\nLearner-control and results focus, 224-226\n\nUnmotivated employees\n\n\"Green and clean,\" 174-179\n\nWin\/Win management training, 224-226\n\nWin\/Win performance agreements, 226-229\n\nToo many people to supervise\n\nIncreasing the span of control with Win\/Win, 226-229\n\nThe sales manager\u2014it can be delegated, 350-351\n\nIn difficult negotiations\n\nThe most viable alternative\u2014Win\/Win, 212-213\n\n\"No deal\"\u2014a viable option, 213-216\n\n\"We'd like you to write the contract...,\" 254-255\n\nYou don't have to take it to court, 280-283\n\nPerformance evaluation problems\n\nHelping people evaluate themselves, 226-229\n\nWatering what you want to grow, 231-232\n\n### INDEX\n\nAbundance Mentality, 219-21, ,\n\naccountability, , ,\n\nAddison, Joseph,\n\nadvisement response,\n\naffirmation, 132-35\n\n\"Aha!\" experience,\n\nAlcoholics Anonymous prayer,\n\nAmiel, 317-18\n\n\"Animal School, The\" (Reeves), 278-279\n\napologizing sincerely, 197-99\n\napplication suggestions:\n\nfor begin with end in mind,\n\nfor first thing first, 179-82\n\nfor proactivity, 93-94\n\nfor renewal, 306-7\n\nfor synergy,\n\nfor understanding, 259-60\n\nfor Win\/Win,\n\nAristotle,\n\nattitudes, 23-24\n\ndefinition of,\n\nsources of, , , , , ,\n\nattitudinal values, 74-75\n\nautobiographical responses, four, 245-53\n\nbalance, , 301-3\n\nbalanced self-renewal, principles of, 287-307\n\nBarton, Bruce,\n\nbegin with end in mind, 95-144\n\nall things are created twice, 99-100\n\nalternative centers, 111-18\n\napplication of, 146-47, , , , ,\n\napplication suggestions for,\n\nfamily mission statements, 137-39\n\nidentifying center, 118-22\n\nidentifying roles and goals, 135-37\n\nleadership and management in, 101-3\n\norganizational mission statements, 139-43\n\npersonal mission statement, 106-9, 128-29\n\nprinciple center, 122-28\n\nrescripting (becoming your own first creator), 103-6\n\nbehaviors, 23-24,\n\npatterns of,\n\nreactive vs. proactive, 71-72, , 78-80, , , 87-88\n\nsources of, , , , , ,\n\nthree areas of problems in, 85-86\n\nbeing, 31-32,\n\nBennis, Warren,\n\nBenson, Ezra Taft,\n\nbe's, 89-90\n\nbrain, left, functions of, , , , ,\n\nbrain, right, 130-35,\n\nexpanding perspective of, 131-32\n\nfunctions of, , ,\n\ntwo ways to tap into, 131-35\n\nvisualization by, 132-35\n\nbrain dominance theory, ,\n\nBrooks, Phillips, 296-97\n\nBush, George,\n\nbusiness, synergy in, 267-69\n\nCamp David Accord,\n\ncenter(s), 109-28,\n\nalternative, 111-18\n\nidentifying of, 118-22\n\nlife-support factors for, 109-11\n\npossible perceptions flowing out of,\n\nprinciples as, 32-35, 122-128, , ,\n\nchange:\n\nability to,\n\nForce Field Analysis in, 279-83\n\nfrom inside out, 35-40, 316-18\n\nprinciples of, 35-40, 47-48\n\nfrom renewal, 279-83, 304-6\n\nsee also inside-out process\n\ncharacter:\n\nand communication, 237-39\n\nthree traits of, 217-21\n\nWin\/Win, , 217-21\n\nCharacter Ethic, 18-23,\n\nvs. Personality Ethic, 18-21, , , , 238-39\n\nprimary greatness trait in, 21-23\n\nprinciples and values of, ,\n\n_Chariots of Fire_ ,\n\nchurch centeredness, 117-18, ,\n\nChurchill, Winston,\n\nCircle of Concern, 81-85, 90-91, ,\n\nCircle of Influence, 81-91\n\ncommitments and promises in,\n\nexpanding of, 86-88, 89-90\n\nnegative energy and, ,\n\none on one in, 257-59\n\npositive energy and,\n\nin relationships,\n\nrenewal in, ,\n\nclassroom, synergy in, 265-67\n\ncoherence, 160-61\n\ncommitment(s):\n\ninvolvement and,\n\nmaking and keeping, 91-92, , 193-94,\n\nin upward spiral,\n\n\"Common Denominator of Success,\" The (Gray), 148-49\n\ncommunication, ,\n\ncharacter and, 237-39\n\nand Emotional Bank Account, 188-190\n\nempathic, principles of, 235-60\n\nfour basic types of,\n\nfrom inside-out, 310-14\n\nintegrity and,\n\nlevels of,\n\nin relationships, 220-23\n\nsynergy and, 264-65, 269-71\n\nthird alternative in, 271-74\n\nsee also listening; understanding\n\ncompensation system, 231-32\n\ncompetition, , 208-9, , , ,\n\ncompromise, , , 270-73,\n\ncomputer metaphor, , , ,\n\nconditioning, 67-68, , ,\n\nconfrontation, 196-97\n\nconscience, , , ,\n\neducation and development of, 305-6\n\nas human endowment, , , , , 103-6, , , ,\n\nin proactive model,\n\nin time management,\n\nconsequences:\n\nof actions, ,\n\nas expectation, , , ,\n\nfour kinds of, 227-28\n\nconsideration, , 218-19,\n\nConstitution, U.S., 107-8\n\ncontribution, as law of life,\n\ncooperation:\n\nvs. competition, 205-6, , , ,\n\ncreative, principles of, 261-84\n\nas law of life,\n\nand levels of communication,\n\nCopernicus,\n\ncourage, 201-2, , 217-19, ,\n\ncreation:\n\nby design or default,\n\nfirst, 99-106, 129-31, , 146-47\n\nleadership and management, 101-3\n\nmental and physical, 99-100, 146-147\n\nrescripting as, 103-6\n\nsecond, 99-103,\n\ncreative process, see synergy\n\ncreative values,\n\ncredibility, 221-22,\n\ncrises, , , , , ,\n\nDaily Private Victory, , , , ,\n\ndaily vs. weekly scheduling, , 165-68,\n\nDeclaration of Independence, U.S., ,\n\ndefensive communication,\n\ndelegation:\n\nagreements, 223-24,\n\nfive elements in, , 223-24\n\n\"gofer,\" 173,\n\nincreasing P and PC by, 171-72\n\nstewardship, 173-79, 223-24,\n\nin weekly organizing,\n\ndeMille, Cecil B.,\n\ndependence, 49-51, ,\n\ndesire, , ,\n\ndeterminism:\n\nparadigm of, , ,\n\nthree theories of, 67-68\n\ndevelopment, see growth\n\ndiagnosis before prescription, 243-45\n\ndifferences, valuing, , , 277-79,\n\ndignity,\n\ndirect control, 85-86\n\ndiscipline, ,\n\nas law of life,\n\nin time management, 157-58\n\ndoing, in upward spiral,\n\ndriving forces, 279-83\n\nDrucker, Peter, , ,\n\nduplicity, , ,\n\necology,\n\neducation, 265-67, 294-97\n\neffectiveness, 52-4, , ,\n\nefficiency vs., , , 161-62, 169-170,\n\nhabits and paradigm of, , , 52-54, , , ,\n\nby organizational PC, 57-59\n\npersonal, ,\n\nEinstein, Albert,\n\nEinsteinian relativity paradigm,\n\nEliot, T. S, ,\n\nEmerson, Ralph Waldo, ,\n\nEmotional Bank Account, 188-90\n\nrelationships and, 221-22\n\nsix major deposits in, 190-202\n\nunderstanding and, , 241-43, , 251-59\n\nemotional deposits, 190-202\n\napologizing sincerely, 197-99\n\nattending to little things, 192-93\n\nclarifying expectations, 194-95\n\nkeeping commitments, 193-94\n\nlaws of love and laws of life in, 199-202\n\nshowing personal integrity, 195-97\n\nunderstanding individual, 190-92\n\nemotional dimension, in renewal, , 297-99, ,\n\nemotional responses,\n\nempathic communication, principles of, 235-60\n\nempathic listening, 239-43\n\nsee also listening\n\nendurance, 289-90\n\nenemy centeredness, 115-17, ,\n\nenergy, positive and negative, ,\n\nenvironmental determinism,\n\nethos,\n\nevaluation response,\n\nexcellence,\n\nexercise, 289-92\n\nexpectations:\n\nclarifying, 194-95\n\nin five areas, , ,\n\nand time management, , , ,\n\nexperiential values,\n\nfairness,\n\nfamily centeredness, 122-23, ,\n\nfamily mission statements, 137-39\n\nFarouk I, King of Egypt,\n\nFerguson, Marilyn, 60-61\n\nfinancial assets, ,\n\nfirst (mental) creation, 99-106, 129-131, , 146-47\n\nfirst things first, 145-82,\n\napplication suggestions for, 179-82\n\nbecoming Quadrant II self-manager, 162-68\n\ndelegation, 171-79\n\nand four generations of time man\u00adagement, 149-50, 159-60, 170-71\n\nliving it, 169-70\n\nmoving into Quadrant II, 158-60\n\nQuadrant II, 150-56,\n\nQuadrant II tool, 160-62\n\nwhat it takes to say \"no,\" 156-58\n\nFisher, Roger,\n\nflexibility:\n\nphysical,\n\nin planning,\n\nfocus:\n\nproactive, ,\n\nQuadrant II, ,\n\nreactive, ,\n\nForce Field Analysis, 279-83\n\nFrankl, Viktor, 68-70, , , , , 128-29,\n\nFranklin, Benjamin,\n\nfreedom to choose, 68-71,\n\nFreudian psychology,\n\nfriend centeredness, 115-17, ,\n\nFromm, Erich,\n\nGandhi, Mohandas K., ,\n\nGarfield, Charles,\n\ngenerations, in time management, 149-50, 159-60, 170-71\n\ngenetic determinism, 67-68\n\nGetting to Yes (Fisher and Ury),\n\ngoals:\n\nand expectations, 194-95\n\nlong-term, , 135-37\n\nselecting of,\n\nin weekly organizing, ,\n\nGod,\n\nGoethe, Johann Wolfgang von, ,\n\nGolden Rule,\n\ngoodness, ,\n\ngoose and golden egg fable,\n\nGordon, Arthur, 292-93\n\nGray, E. M. 148-49\n\ngreatness, primary and secondary, 21-23\n\ngrowth,\n\nForce Field Analysis in, 279-83\n\nmaturity continuum as, 48-52, 110-11,\n\nprinciples of, , , 35-40, 47-48\n\nfrom renewal, 279-83, 304-6\n\nguidance, 109-28,\n\nguidelines, as expectation, , , ,\n\nhabits, 47-48\n\none, 65-94\n\ntwo, 95-144\n\nthree, 145-82\n\nfour, 204-34\n\nfive, 235-60\n\nsix, 261-84\n\nseven, 287-307\n\nbreaking of, 46-47\n\nof effectiveness, , , 52-54,\n\nof interdependence,\n\nmaturity continuum and, 48-52\n\norganizational PC as, 57-59\n\nreaders' application of, 59-60\n\nand synergy,\n\nworking on,\n\nHammarskjold, Dag, ,\n\nHandel, George Frederick,\n\nhappiness, , , ,\n\nHarvard Business School,\n\nhave's, 89-90\n\nheart, exercising of, , 291-92\n\nHolmes, Oliver Wendell,\n\nhonesty, , 195-97\n\nhuman assets, , 55-56,\n\nhuman endowments, , , , , 103-6, , , 147-48, , ,\n\nhuman interaction:\n\nsix paradigms of, 206-16,\n\nwhich option is best in, 211-13\n\nsee also interdependence\n\nhuman nature,\n\nbalance in renewal of, 301-3\n\nfour dimensions of, 288-301\n\nhuman relations, see Personality Ethic\n\nhumility,\n\nIBM, , , 139-40\n\nignoring vs. listening,\n\nimage, , ,\n\nimagination, , 146-17\n\nas human endowment, , , , , 103-6, , ,\n\nin proactive model,\n\nvisualization and, 131-35,\n\nimportance, in time management, 150-54, , , ,\n\nindependence:\n\nand interdependence, 186-87, , ,\n\nparadigm of, 49-51\n\nindependent will,\n\nas human endowment, , , ,\n\npower of, 147-49\n\nin proactive model,\n\nindirect control, 85-86\n\nindividual, understanding of, 190-92\n\ninfluence, human:\n\nby listening, 238-43\n\nthirty separate methods of,\n\nsee also Circle of Influence\n\ninitiative, , 75-76,\n\nIn Search of Identity (Sadat), 103-4, ,\n\ninside-out process, 15-44, 309-18\n\nnew level of thinking in, 42-44\n\nPersonality and Character Ethics in, 18-21\n\npower of paradigm in, 23-31\n\nprimary and secondary greatness in, 21-23\n\nprinciple-centered paradigm in, 31-35\n\nprinciples of growth and change in, 35-40\n\nway we see problem is problem in, 40-42,\n\nintegrity, , , , , 195-97, ,\n\nas character trait,\n\nas law of life,\n\ninteraction, see human interaction\n\ninterdependence, 185-203\n\nEmotional Bank Account and, 188-190\n\nhabits of,\n\nand independence, 186-87, , ,\n\nparadigm of, 49-51, 185-203\n\nP problems as PC opportunities in, 202-3\n\nintergenerational living, 314-16\n\ninterpersonal communication, key to, ,\n\ninterpersonal effectiveness,\n\ninterpersonal leadership, 204-34\n\ninterpretation response,\n\ninvolvement, commitment and,\n\nJapan, economic success of, ,\n\nJefferson, Thomas,\n\nJohnson, Samuel, ,\n\nJordon, David Starr,\n\nJordan, William George, 22-23\n\nJoseph, story of,\n\nKerr, Rolfe, 106-7\n\nkindnesses, 192-93\n\nknowledge, , ,\n\nKoch, Frank,\n\nKuhn, Thomas,\n\nlanguage, listening to, 78-80\n\nlaws:\n\nof love and life, 199-202\n\nnatural, see principle(s)\n\nleadership:\n\nbasic task of,\n\ninterpersonal, principles of, 204-34\n\nvs. management, 101-3, , ,\n\npersonal, 95-144\n\ntransactional vs. transformational,\n\nlearning:\n\nmutual, ,\n\nin upward spiral,\n\nLewin, Kurt,\n\nliberty vs. freedom,\n\nlife, laws of, 199-202\n\nlife-support factors, 109-28\n\ncenters' affect on, 111-18\n\ndevelopment of, 122-28\n\nidentifying center by, 118-22\n\nmaturity of, 110-11\n\nLilienthal, David, 269-70\n\nlistening:\n\nempathic, , , 221-22, 239-43\n\nfour levels of,\n\nlogical or emotional responses in,\n\nskills for, 248-49,\n\ntime investment in, 252-53\n\ntraining in,\n\nsee also understanding\n\nliterature, ,\n\nlittle things, attending to, 192-93\n\nlogic, , , ,\n\nlogical responses,\n\nlogos,\n\nlogotherapy,\n\nlong-term organizing,\n\nLose\/Lose,\n\nLose\/Win, 209-10,\n\nlove, , 199-202\n\nloyalty, ,\n\nLuther, Martin,\n\nMcKay, David O.,\n\nmanagement:\n\nvs. leadership, 101-3, , ,\n\npersonal, 145-82\n\ntime, 149-50, , 159-60, 170-71\n\nmanagement training, Win\/Win, 224-227\n\nmanager, and input\/output,\n\nmanager's letter,\n\nMann, Horace,\n\nMan of La Mancha,\n\nMarkham, Edwin,\n\nMaslow, Abraham,\n\nmaturity, 217-19,\n\nmaturity continuum, 48-52, ,\n\nlife-support factors on, 110-11\n\nmeditation, ,\n\nmental (first) creation, 99-100, 129-131, , 146-47\n\nmental dimension, in renewal, , 294-95, ,\n\nMessiah (Handel),\n\nmethods vs. results, 173-74,\n\nmiddle way,\n\nmimic content, in listening, , 248-249\n\nmirror, social, 67-68\n\nMiserables, Les (Hugo),\n\nmission statements:\n\nfamily, 137-39\n\norganizational, 139-43\n\npersonal, 106-9, 128-29,\n\nin time management, ,\n\nmistakes, 90-91,\n\nmodels:\n\nproactive,\n\nreactive,\n\nmoney centeredness, 113-14, ,\n\nmotivation:\n\nsatisfied needs and,\n\nsee also desire\n\nmotives, , , ,\n\nMuggeridge, Malcolm,\n\nNasser, Gamal,\n\nnature, as synergistic, , 283-84\n\nneeds, satisfied,\n\nnegative energy, ,\n\nnegative synergy, 274-77,\n\nnegotiation, 206-16, ,\n\nsee also Win\/Win\n\nNewtonian model of physics,\n\n\"no,\" saying, 156-58\n\nNo Deal, Win\/Win or, 213-16\n\nobjective reality,\n\nobjectivity,\n\none on one, 257-59\n\nopportunities, , , 202-3, ,\n\nopportunity consequences, ,\n\norganizational mission statements, 139-43\n\norganizational PC, 57-59\n\n\"Organize and execute around priori\u00adties,\" ,\n\norganizing,\n\nlong-term and weekly, ,\n\nsee also first things first; Quadrant II\n\noutside-in paradigm, 43-14, ,\n\nPaine, Thomas,\n\nparadigms, 23-35\n\nattitudes and behaviors from, , , , , ,\n\nbasic, and paradigm shifts, 29-31,\n\ncore, 111-18\n\ndifferences and, 277-79\n\nnew level of thinking and, 43-44\n\npower of, 23-29,\n\nprinciple-centered, 32-35, 123-28, ,\n\nproactivity and, , ,\n\nrenewal and,\n\nof self,\n\nof seven habits,\n\nsix, of human interaction, 206-216\n\nparadigm shifts, ,\n\nnegative vs. positive,\n\npower of, 29-31,\n\nprocess of, 31-32\n\nby readers, 59-60\n\nas rescripting self, 103-6\n\nPareto Principle,\n\npartnership agreements, 223-24\n\nPascal, Blaise,\n\npathos,\n\npatience, ,\n\nPavlov, Ivan Petrovich,\n\npeace of mind,\n\npenalties, see consequences\n\nPenney, J. C.,\n\npeople, effectiveness vs. efficiency and, 161-62, 169-70,\n\nperception demonstration, 23-29, ,\n\nperceptions:\n\nchanging of, 17-18, 29-31\n\ndifferences in, 277-79\n\npossible, flowing out of various centers, 321-29\n\nunderstanding and, 253-55\n\nsee also paradigms\n\nperformance agreements, 223-24, 227-29\n\nPersonality Ethic, 18-44\n\nbasic flaws of, , , , 35-36\n\nvs. Character Ethic, 18-21, , , , 238-39\n\nimpact of, 40-42,\n\ntechniques in, , 21-22, ,\n\npersonal leadership, 95-144\n\npersonal management, 145-82\n\npersonal mission statement, 106-9,\n\nwriting and using, 128-29\n\npersonal vision, 65-94,\n\nperspective, expanding of, 131-32\n\nphysical assets, 54-55,\n\nphysical (second) creation, 99-100,\n\nphysical dimension, in renewal, , 289-92, ,\n\nplanning, in time management, , , , 160-62, ,\n\npleasure centeredness, 114-15, ,\n\npositive energy,\n\npositive mental attitude (PMA),\n\npositive thinking vs. proactivity,\n\npossession centeredness, , ,\n\nposterity,\n\npotential,\n\nPotiphar,\n\npower, 109-28,\n\nof independent will, 147-49\n\nof paradigm, 23-31\n\nP\/PC Balance:\n\nassets and, 54-57\n\ncourage and consideration as, 218-219\n\nin interdependence, ,\n\nin levels of communication, 270-71\n\nopportunities and, 202-3\n\nin organizational effectiveness, 57-59\n\nprinciple of, 52-54, ,\n\nin time management, ,\n\npractices vs. principles, 34-35\n\nprescription, diagnosis before, 243-45\n\npresentation process, 255-57\n\nprevention of crises, , , , ,\n\nprimary greatness, 21-23\n\nprinciple(s), 32-44, , ,\n\nof balance self-renewal, 287-307\n\ncentered paradigm, 32-35, 122-28, , ,\n\nas center of life, 122-28\n\nof creative cooperation, 261-84\n\nof empathic communication, 235-260\n\nexamples of,\n\nof growth and change, 35-40\n\nof interpersonal leadership, 204-34\n\nnew level of thinking and, 42-44\n\nof personal leadership, 95-144\n\nof personal management, 145-82\n\nof personal vision, 65-94\n\nof P\/PC Balance, 52-54, ,\n\nvs. practices, 34-35\n\nproblems and, 40-42\n\nvs. values,\n\nprincipled negotiation,\n\npriorities:\n\norganizing and executing around, 149-50,\n\nsee also first things first; Quadrant II\n\nPrivate Victory, 63-182, ,\n\nDaily, , , , ,\n\nproactivity, 65-94,\n\nact or be acted upon, 76-78\n\napplication of, , , , ,\n\napplication suggestions for, 93-94\n\nCircle of Concern\/Circle of Influ\u00adence in, 81-85\n\nconsequences and mistakes in, 90-91,\n\ndirect, indirect, and no control in, 85-86\n\nexpanding Circle of Influence in, 86-88\n\nfreedom of choice in, 68-70, ,\n\nhave's and be's in, 89-90\n\nlistening to language in, 78-80\n\nmaking and keeping commitments in, 91-92\n\nmodel,\n\nvs. reactivity, 71-72, , 78-80, , , 87-88, 104-6\n\nsocial mirror in, 67-68\n\ntaking initiative in, 75-76\n\nthirty-day test in, 92-93\n\nvalue-based, , , ,\n\nwriting personal mission statement by, 128-29\n\nprobe response,\n\nproblems:\n\nnew level of thinking for, 42-44\n\nP, as PC opportunities, 202-3\n\nthree areas of, 85-86\n\nin time management, 151-56,\n\nway we see problem as, 40-42\n\nand Win\/Win solutions,\n\nprocess(es), , 233-34\n\nhabits and, 46-48\n\nprinciple of, 36-38\n\nproducers:\n\nand input\/output, ,\n\nin time management, 150-56\n\nproduction (P), ,\n\nassets and, 54-57\n\nincreasing, , 171-72, ,\n\nproblems, as PC opportunities, 202-3\n\nproduction capability (PC), , , ,\n\nassets and, 54-57\n\nincreasing, , 171-72, ,\n\nopportunities in, P problems as, 202-3\n\norganizational, 57-59\n\npersonal, renewal as, 288-289\n\npromises, 91-92\n\nsee also commitments\n\npsychic determinism,\n\npsychological consequences,\n\npsychological dimension, in renewal, , 294-95, ,\n\npsychology, ,\n\nPtolemy,\n\nPublic Victory, 183-284, ,\n\nPygmalion effect,\n\nQuadrant I, 151-55, , , ,\n\nQuadrant II:\n\nadvances of fourth generation and, 170-71\n\nday at office in, ,\n\ndelegation in, 171-79\n\nexplanation of, 150-56\n\nfocus, ,\n\nliving, 169-70\n\nmoving into, 158-60\n\nparadigm,\n\nrenewal in, , , , , ,\n\nsaying \"no\" for, 156-58\n\nself-manager, 162-68\n\nsix important criteria in, 160-62\n\ntool, 160-62\n\nQuadrant III, 151-53, , , ,\n\nQuadrant IV, 151-53, , , 159-160,\n\nquality,\n\nreactive focus, ,\n\nreactive language,\n\nreactive model,\n\nreactive vs. proactive, 71-72, , 78-80, , , 87-88, 104-6\n\nreaders:\n\napplication by,\n\nexpectations from, 60-62\n\nsummary for, 51-52,\n\ntwo paradigm shifts by, 59-60\n\nreading,\n\nreality,\n\ninterdependent, , , ,\n\nreactive vs. proactive, , ,\n\nsubjective vs. objective,\n\nReeves, R. H., 278-79\n\nreflecting feeling, in listening,\n\nreflective listening, ,\n\nrelationships, , 221-23\n\nin organizations, 201-2; see also Public Victory\n\nin time management, , , , , , 169-70,\n\nsee also interdependence; Win\/Win\n\nrenewal, 285-318\n\napplication suggestions for, 306-7\n\nbalance in, 301-3\n\nfour dimensions of, 288-303\n\ngrowth and change process from, 279-83, 304-6\n\nmental dimension in, , 294-97\n\nphysical dimension in, , 289-92, ,\n\nscripting others in, 299-301\n\nsocial\/emotional dimension in, , 297-99, ,\n\nspiritual dimension in, , 292-94, , 303-4\n\nsynergy in, 303-4\n\nupward spiral in, 304-6\n\nrephrasing content, in listening,\n\nrescripting self, 103-6, ,\n\nresources, , , ,\n\nrespectful communication, ,\n\nresponse(s), 68-70,\n\nto consequences and mistakes, 90-91\n\nfour autobiographical, 245-53\n\ngap between stimulus and, 69-70, , , ,\n\nlogical or emotional,\n\nin proactive model,\n\nin reactive model,\n\nresponsibility, , ,\n\ndelegating of,\n\nproactive people and, , , , 89-90, , ,\n\nreactive people and, , , ,\n\nin Win\/Win,\n\nsee also proactivity\n\nresponsibility consequences,\n\nrestraining forces, 279-83\n\nresults:\n\ndesired, 173-74, , ,\n\nvs. methods, 173-74,\n\nin time management, 151-56,\n\nin Win\/Win, solutions,\n\nrewards, see consequences\n\nRogers, Carl,\n\nroles:\n\nand expectations, 194-95\n\nidentifying of, 135-37, 162-63,\n\nin long-term and weekly organiz\u00ading, ,\n\nRoosevelt, Eleanor,\n\nRoots,\n\nRoskin, Leo,\n\nSadat, Anwar, 103-4, , 316-17\n\nsameness, unity vs.,\n\nsatisfaction,\n\nof needs,\n\nsaw, sharpening of, , , , 287-307\n\nsee also renewal\n\nSaxenian, Hrand,\n\nsaying \"no,\" 156-58\n\nScarcity Mentality,\n\nscheduling, , 161-68\n\nscripting others, 299-301\n\nscripting self, 103-6, ,\n\nsecondary greatness, 21-23\n\nsecond (physical) creation, 99-103,\n\nsecurity, 109-28, , 298-99, ,\n\nseeing, 31-32,\n\nseek first to understand, then to be understood, 235-60\n\nselective listening,\n\nself-awareness, 66-67\n\nand freedom to choose, 69-70\n\nas human endowment, , , , , 103-6, , ,\n\nand language, 78-80\n\nin proactive model,\n\nsee also proactivity\n\nself-centeredness, , ,\n\nself-fulfilling prophecies,\n\nself-manager, Quadrant II, 162-68\n\nself-mastery, see Private Victory\n\nSelye, Hans,\n\nsequencing,\n\nservice, , 298-99\n\nSeven Habits, overview of, 46-62\n\nShaw, George Bernard,\n\nSheehan, George,\n\nShepherd, Herb,\n\nsincerity, 197-99\n\nskills,\n\nas effective habit, , ,\n\nlistening, 248-49,\n\nsocial dimension, in renewal, , 297-99, ,\n\nsocial mirror, 67-68\n\nsolutions, , ,\n\nsolution selling,\n\nspiral, upward, in renewal, 304-6\n\nspiritual dimension, in renewal, , 292-94, , 303-4\n\nspouse centeredness, 111-12, ,\n\nStael, Madame de,\n\nStar Wars, 56-57\n\nstewardship delegation, 173-79, 223-224,\n\nstimulus:\n\ngap between response and, 69-70, , , ,\n\nin proactive model,\n\nin reactive model,\n\nand response, 68-70\n\nstrength:\n\npersonal, , 201-2, 221-22,\n\nphysical, 290-91\n\nstructure, , 229-32\n\nStructure of Scientific Revolutions, The (Kuhn),\n\nstudents, 265-67, 300-301\n\nsubjective reality, , ,\n\nPersonality vs. Character Ethic of, 18-21, ,\n\nprimary and secondary traits of, 21-23\n\nprinciples and foundation of,\n\nWin\/Win in,\n\nsuccess literature, 18-19, , ,\n\nsupportive systems, , 229-32\n\nsympathy vs. empathy,\n\nsynergy, , , , , , 261-284\n\nall nature as, , 283-84\n\napplication suggestions for,\n\nin business, 267-69\n\nin classroom, 265-67\n\ncommunication and, 264-65, 269-71\n\nfishing for third alternative in, 271-274,\n\nForce Field Analysis in, 279-83\n\nintrapersonal, 274-75\n\nnegative, 274-77\n\nin renewal, 303-4\n\nvaluing differences in, , , 277-79,\n\nsystems, supportive, , 229-32\n\nTanner, N. Eldon,\n\nteachers, 265-67, 300-301\n\nteamwork, see interdependence, para\u00addigm of\n\nTeilhard de Chardin, Pierre,\n\nTen Commandments, The,\n\nthinking, new level of, 42-44\n\nthirty-day test, 92-93\n\nThoreau, Henry David, , ,\n\ntime management:\n\nfour generations of, 149-50, 159-60\n\nfourth-generation, five advance\u00adments in, 170-71\n\nmatrix,\n\nthird-generation limitations of, , ,\n\nsee also first things first; Quadrant II\n\nToffler, Alvin,\n\nTotal Quality Movement,\n\ntransaction vs. transformation, , 251-52,\n\ntransition person, 316-18\n\ntrust:\n\nas highest form of human motiva\u00adtion,\n\nand levels of communication, , ,\n\nin relationships, , 188-190, , 196-97, , 220-22, ,\n\ntruth, definition of,\n\ntruths, see principle(s),\n\n\"Turn of the Tide, The\" (Gordon), 292-93\n\n\"Twentieth-Century Testimony, A,\"\n\nunconditional love, 199-202\n\nunderstanding,\n\napplication suggestions for, 259-60\n\ncharacter and communication in, 237-39\n\ndiagnosis before prescription in, 243-45\n\nempathic listening in, 239-43\n\nfour autobiographical responses in, 245-53\n\nof individual, 190-92\n\none on one in, 257-59\n\nperception and, 253-55\n\nas primary function, 235-60\n\nseek to be understood after, 255-57\n\nUnited States, as paradigm shift,\n\nunity, , ,\n\nupward spiral, in renewal, 304-6\n\nurgency, in time management, 150-154, , , ,\n\nUry, William,\n\nvalues,\n\nvs. principles,\n\nproactivity based on, , , ,\n\nthree central, 74-75\n\nin time management,\n\nVietnam prisoners of war,\n\nvision, personal, 65-94,\n\nvisualization, 96-97, 131-35\n\nWarner, Terry,\n\nWatson, T. J.,\n\nweekly vs. daily scheduling, , 165-68,\n\nweekly worksheet, , 166-67, 181-182\n\nwill, see independent will\n\nWin, 210-11\n\nWin\/Lose, 207-9,\n\nWin\/Win, 204-34\n\nagreements, 223-24, 227-29\n\napplication suggestions for,\n\ncharacter, 217-21\n\ncooperation vs. competition in, 205-6, 208-9, , , ,\n\nfour-step process in, 233-34\n\nfive dimensions of, 216-34\n\nmanagement training, 224-27\n\nor No Deal, 213-16\n\nas paradigm,\n\nperformance agreements, 227-29\n\nprocesses, , 233-34\n\nrelationships, , 221-23\n\nsix paradigms of human interaction in, 206-16\n\nsolutions, , ,\n\nsupportive systems, 229-32\n\nsynergy and, ,\n\nthree character traits in, 217-21\n\nwisdom, 109-28,\n\nWolfe, Thomas,\n\nwork centeredness, , ,\n\nworksheet, weekly, , 166-67, 181-182\n\nWorld War I, success after,\n\nwriting, \n\n### About FranklinCovey\n\nMISSION STATEMENT\n\nWe enable greatness in people and organizations everywhere.\n\n### FOUNDATIONAL BELIEFS\n\nWe believe:\n\n  1. People are inherently capable, aspire to greatness and have the power to choose.\n  2. Principles are timeless and universal, and are the foundation for lasting effectiveness.\n  3. Leadership is a choice, built inside out on a foundation of char\u00adacter. Great leaders unleash the collective talent and passion of people toward the right goal.\n  4. Habits of effectiveness come only from the committed use of integrated processes and tools.\n  5. Sustained superior performance requires P\/PC Balance\u00ae\u2014a focus on achieving results and on building capability.\n\nVALUES\n\n  1. Commitment to Principles. We are passionate about our con\u00adtent, and strive to be models of the principles and practices we teach.\n  2. Lasting Customer Impact. We are relentless about delivering on our promises to our customers. Our success comes only with their success.\n  3. Respect for the Whole Person. We value each other and treat each person with whom we work as true partners.\n  4. Profitable Growth. We embrace profitability and growth as the lifeblood of our organization; they give us the freedom to fulfill our mission and vision.\n\nFranklinCovey (NYSE:FC) is the global leader in effectiveness train\u00ading, productivity tools and assessment services for organizations, teams and individuals. Clients include 90 percent of the Fortune 100, more than 75 percent of the Fortune 500, thousands of small and midsized busi\u00adnesses, as well as numerous government entities and educational institu\u00adtions. Organizations and individuals access FranklinCovey products and services through corporate training, licensed client facilitators, one-on-one coaching, public workshops, catalogs, more than 140 retail stores and www.franklincovey.com.\n\nFranklinCovey has 2,000 associates providing professional services, products and materials in twenty-eight languages, in thirty-nine offices and in ninety-five countries worldwide.\n\nPrograms and Services\n\n  * xQ Survey and Debrief (to help leaders assess their organization's \"Ex\u00adecution Quotient\")\n  * The 7 Habits of Highly Effective People workshop\n  * The 4 Disciplines of Execution worksession\n  * FOCUS: Achieving Your Highest Priorities workshop\n  * The 4 Roles of Leadership workshop\n  * The FranklinCovey Planning System\n\nTo learn more about FranklinCovey products and services,  \nplease call 1-888-868-1776 or 1-801-817-1776, or go to  \nwww.franklincovey.com.\n\n### About the Author\n\nStephen R. Covey is an internationally respected leadership authority, family expert, teacher, organizational consultant and author who has dedicated his life to teaching principle-centered living and leadership to build both families and organizations. He holds an M.B.A. from Harvard University and a doctorate from Brigham Young University, where he was a professor of organizational behavior and business management and also served as director of university relations and assistant to the president.\n\nDr. Covey is the author of several acclaimed books, including the in\u00adternational bestseller, The 7 Habits of Highly Effective People, which was named the #1 Most Influential Business Book of the Twentieth Century and one of the top-ten most influential management books ever. It has sold more than 15 million copies in thirty-eight languages throughout the world. Other bestsellers include First Things First, Principle-Centered Leadership and The 7 Habits of Highly Effective Families, bringing the com\u00adbined total to more than 20 million books sold.\n\nAs a father of nine and grandfather of forty-three, he received the 2003 Fatherhood Award from the National Fatherhood Initiative, which he says is the most meaningful award he has ever received. Other awards given to Dr. Covey include the Thomas More College Medallion for con\u00adtinuing service to humanity, Speaker of the Year in 1999, the Sikh's 1998 International Man of Peace Award, the 1994 International Entrepreneur of the Year Award and the National Entrepreneur of the Year Lifetime Achievement Award for Entrepreneurial Leadership. Dr. Covey has also been recognized as one of Time magazine's 25 Most Influential Ameri\u00adcans and has received seven honorary doctorate degrees.\n\nDr. Covey is the cofounder and vice chairman of FranklinCovey Com\u00adpany, the leading global professional services firm with offices in 123 countries. They share Dr. Covey's vision, discipline and passion to in\u00adspire, lift and provide tools for change and growth of individuals and organizations throughout the world.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":" \nTHE PLEASURE OF THE CROWN  \nAnthropology, Law and First Nations\n\nDara Culhane\n\nTalonbooks  \n1997\n_for feral travelers in the vast emptiness_\n\n_in memory of my mother_  \n_Claire Eglin Culhane_  \n_1918-1996_  \n_who never failed_  \n_to speak truth to power_\n**T ABLE OF CONTENTS**\n\n Generalized Map of British Columbia\n\nPART I: JUSTITIA OMNIBUS (JUSTICE FOR ALL)\n\n Chapter 1: How Did the Crown Acquire Title?\n\n ~ The Long and Short of It\n\n ~ Human Being, Citizen, Anthropologist\n\n Chapter 2: In Search of Reason\n\n ~ The Law in its Majesty\n\n ~ Late One Night at the Legion\n\nPART II: TERRA NULLIUS (UNOCCUPIED, EMPTY LAND)\n\n Chapter 3: Cultures Similar and Different\n\n ~ Creating Race in the Interests of Empire\n\n ~ Mutiny and Desertion as Common Sense\n\n Chapter 4: Beginning at the Beginning\n\n ~ Aboriginal Peoples Were Here\n\n ~ Oh, What a Tangled Web We Weave When First We Practice to Deceive\n\n ~ Eclecticism\n\n ~ Imperialist Competitions in Terra Nullius\n\n ~ Imperialist Nostalgia in Terra Nullius\n\n ~ Enough and As Good\n\n ~ Nation to Nation. Words to Paper\n\nPART III: TERRA INCOGNITA (UNKNOWN LAND)\n\n Chapter 5: The Great Chain of Precedent\n\n ~ Honour Among Thieves. South of the Border\n\n ~ Honour Among Thieves. North of the Border\n\n ~ Honour Among Thieves. In Africa\n\n Chapter 6: White Papers and Legal Tests: the 1970's\n\n ~ Law, Power and Resistance\n\n ~ Logic, Legal Hysteria and Rifle Shots on Vancouver Island\n\n ~ Civilization and the Whimsical Destruction of Property\n\n ~ From Law to Politics and Back Again\n\n ~ The Honour of the Crown Below Par\n\n Chapter 7: Conflicts and Constitutions: The 1980s\n\n ~ The Master's Tools: The Comprehensive Claims Policy\n\n ~ Title on the Baker Lake Tundra: Lower Peaks than Mexico\n\n ~ The Evolution of the Dedicated White People Band at Bear Island\n\n ~ The Master's House: The Repatriation of the Constitution\n\n ~ Testing, Testing: From \"Use and Occupancy\" to \"Culture\" in the Sparrow Decision\n\n ~ Ping Pong\n\nPART IV: ECCE SIGNUM (BEHOLD THE PROOF)\n\n Chapter 8: Delgamuukw versus The Queen\n\n ~ All the World's a Stage\n\n ~ Do Aboriginal Peoples Really Exist?\n\n Chapter 9: The Gitksan's and Wet'suwet'en's Case\n\n ~ A Native Point of View\n\n ~ Experts in Living Memory\n\n ~ The Ancestors are Watching\n\n Chapter 10: Experts on Behalf of the Plaintiffs\n\n ~ Anthropology: Interpretation and Understanding?\n\n ~ Here Come the Anthros\n\n ~ Law and Masculine Hysteria\n\n ~ Academic Freedom\n\n Chapter 11: The Crown's Case\n\n ~ The Political Economy of Litigation\n\n ~ Anthropology: The Science of \"Man\"?\n\n Chapter 12: Experts on Behalf of the Defendants\n\n ~ Her Majesty's Loyal Anthropologist\n\n ~ Malignant Mythologies\n\n ~ Hegemonic Moments\n\n Chapter 13: Jewels in the Crown\n\n ~ Temporal Purgatory: The Protohistoric Period\n\n ~ Who Created Private Property?\n\n ~ The Magic of Commodity Fetishism\n\n Chapter 14: From Fur Trading Frontier to White Man's Province\n\n ~ Believing What You See. Seeing What You Believe\n\n ~ On the One Hand. And On the Other\n\n Chapter 15: Old Questions. New Century.\n\n ~ The First Definite Hint\n\n ~ The Era of Permanent Penetration\n\n ~ Resistance versus Assimilation\n\nPART V: DIES IRAE (JUDGMENT DAY)\n\n Chapter 16: Reasons for Judgment\n\n ~ Nasty, British (Columbian) and Long\n\n ~ The Vast Emptiness\n\n ~ Might Makes Right\n\n ~ No Writing, Horses or Wheeled Wagons\n\n ~ Finders Keepers\n\n Chapter 17: Reasons in the Public Eye\n\n ~ Judgment Day\n\n ~ Music is Lost on Me. I Have a Tin Ear\n\n ~ Delgamuukw and the People Without Culture\n\n ~ Insult and Injury\n\n ~ Is There Intelligent Life in the Galaxy?\n\nPART VI: TABULA RASA (BLANK PAGE)\n\n Chapter 18: Judging the Experts\n\n ~ Experts v. Advocates\n\n ~ Ethics v. License\n\n ~ Revision v. Reversal\n\n ~ Politics v. Poetics\n\n ~ Chiefs v. Commoners\n\n Chapter 19: How Do You Know Whose Story is True?\n\n ~ Sense v. Nonsense\n\n ~ Fact v. Fiction\n\n ~ Truth v. Lies\n\nPART VII: AD ARBITRIUM (AT WILL)\n\n Chapter 20: Moonlighting as an Anthropologist and an Historian\n\n ~ Drive-by Ethnography\n\n ~ Civilization by Tautology\n\n ~ Reasons to Travel\n\n ~ Legal Rights and Social Wrongs\n\n ~ Communicating Deeds and Legislating Greed\n\n ~ Driven to Drink\n\n ~ Infantile Disorders\n\n ~ Camp Potlatch\n\n ~ The Battle of Britain\n\nPART VIII: AD INFINITUM (GOING ON FOREVER, INTO INFINITY)\n\n Chapter 21: _Delgamuukw_ at the British Columbia Court of Appeal, 1993\n\n ~ We Own Therefore We Are\n\n ~ We Buy Therefore We Are\n\n Chapter 22: The British Columbia Treaty Process and the Supreme Court of Canada, 1996\n\n ~ From Negotiation to Litigation to Negotiation to Litigation to\n\n ~ Appealing Culture\n\nPART IX: IN FUTURO (IN THE FUTURE)\n\n Chapter 23: The Past In the Present, In the Future\n\n ~ Just Another Day in Lotus Land\n\n ~ Justice. Then Peace.\n\n Chapter 24: Back to the Future\n\n ~ Colouring Outside the Lines\n\n Postscript: December 11, 1997\n\n ~ The Dawn of a New Day?\n\n ~ Patience is a Virtue\n\n ~ The Way Forward\n\n ~ The Irony of Ironies\n\n ~ Tomorrow Today\n\nBibliography\n\nTable of Cases\n\nAcknowledgements\n\n_Generalized map of British Columbia showing Territories claimed by Gitksan and Wet'suwet'en Peoples (not to scale)._\n**P ART I**\n\n**J USTITIA OMNIBUS**\n\n**(J USTICE FOR ALL)**\n**Chapter 1: How Did the Crown Acquire Title?**\n\n_The Indian elders in British Columbia question why they must subject their relationship to the land to a non-Indian court's strict scrutiny: why they must explain their use of the land to obtain 'rights' abstractly defined by others._\n\n_They believe that the Indians have rights to their land because their people go back with the land for thousands of years. What they do not understand is how the Crown acquired its 'rights' to their land._\n\n\u2014Lawyer Louise Mandell, 1987, _Native Culture on Trial_.\n\n**The Long and the Short of It**\n\nLaw, we are taught to believe by our educational institutions, embodies justice. Our courts' formal rules of evidence, and their procedures, enable the discovery of truth through the application of reason. \"Judicial neutrality\"\u2014the absence of bias among judges\u2014guarantees that fair and equitable resolutions to disputes between any and all people(s) will be arrived at. Equality before the law, we are told, is the linchpin of the Canadian judicial system; and justice, we are assured, is the outcome of legal processes. It follows from these official premises, and by this reasoning, that if contemporary Canadian sovereignty and Crown title and rights to land are confirmed by law, they must have been justly acquired.\n\nOf course, in these cynical times, few claim the Canadian legal system is infallible. Charges that interests other than the pursuit of justice may be at play in legal processes do not usually generate widespread alarm. Increasingly, we understand that what we call \"facts\" are frequently matters of interpretation that reflect particular points of view rather than unequivocal certainty. We have come to appreciate that something called \"culture,\" and more specifically, \"cultural differences,\" come between Aboriginal peoples and Canadian law, disrupting communication and mutual understanding. Suggestions that judges' decisions often reflect prejudices common in contemporary society at large, rather than being strictly determined by exclusively legal concerns, are hardly shocking revelations.\n\nHowever, periodically a particular judicial decision comes to public attention that shocks even the most complacent or embittered cynic. Allan McEachern, the former Chief Justice of the Supreme Court of British Columbia, created just such a controversy when he handed down his 1991 judgment against the Aboriginal claimants in the high profile Gitksan and Wet'suwet'en land rights case, also known as the case of _Delgamuukw v. The Queen_. Judge McEachern ruled that, before Europeans arrived in the late eighteenth century, the First Nations had been too \"primitive\" to have had property laws or institutions of governance. Today, he declared, they have no Aboriginal rights whatsoever. Charges of racism were hurled at the judge. He was accused of shamelessly favouring the interests of big forestry companies and non-Aboriginal elites against the rights of First Nations. The Chief Justice defended himself, using the letter of the law as his shield. He protested that he was compelled by historical precedents to repeat the rulings of previous judges that dated back hundreds of years. Chief Justice McEachern admitted that his decision might not be _just_ , but he was convinced that it was definitely _legal_.\n\nIndeed, law and justice have not always walked hand in hand through British Columbia's history, and nowhere is the distance between them greater than on the question of Aboriginal title and rights. How _did_ the British Crown acquire its rights to the territory we now call British Columbia? Many Canadians would respond, with pride, that unlike Americans, we gained sovereignty over the lands we live on, and established political jurisdiction over its Indigenous inhabitants, through the rule of law, rather than by military force.\n\nIn fact, the Crown acquired its title to First Nations' lands and resources by simply asserting sovereignty and ignoring Aboriginal title _in contravention of British colonial law_. British and then Euro-Canadian rule was enacted by sheer force of numbers as settlers overwhelmed fragments of Aboriginal populations who had survived the first waves of epidemic diseases brought by European fur traders during the eighteenth century. Aboriginal peoples were simply deemed, by law, not to exist. The colonists then established their own legal regime that validated their self-proclaimed sovereignty and jurisdiction. For over 100 years, the governments and the courts of the Province of British Columbia have defended their predecessors' initial trespass on Aboriginal lands, and their transgression of British law, by whatever means necessary: coercion, deceit, compromise, seduction, force. Law.\n\nAboriginal peoples have steadfastly insisted that they surrendered neither ownership of their lands, nor their political autonomy. Nor did they cease to exist. However, it was not until the late 1960s that First Nations\u2014after over a century of petitioning\u2014succeeded in compelling Canadian courts to begin adjudicating their disputes with the governments about who owns the land in British Columbia.\n\nWhen called upon to justify its actions in law, the Province of British Columbia retreats to a legal fortress, the cornerstone of which is the mystical \"original moment\" when Britain asserted sovereignty over First Nations in British Columbia, often without their knowledge, and always without their consent. The Crown's defense begins by giving legal sanction to this crude act of aggression, and then proceeds from this starting point to call upon a range of social theories, historical fictions, and popular \"common sense\" ignorance and prejudice to justify its actions. Crown lawyers present as evidence stereotypical caricatures of Aboriginal peoples as \"backward\" and \"primitive\" when Europeans arrived. Grisly tales of \"war-like savages\" are juxtaposed with an heroic epic of \"advanced\" and \"civilized\" white settlers who, it is claimed, brought Christianity, reason, and the rule of law to the untamed wilderness of British Columbia. The stark contrast in \"development\" between the two \"races,\" the Crown argues, made the superimposition of British sovereignty over Aboriginal sovereignty a \"natural\" outcome of the \"progress\" of history, and rendered inevitable the subordination of Aboriginal peoples to non-Aboriginal governance. A short answer to the Aboriginal elders' question is that the Crown acquired its 'rights' to their land by simply seizing it: through the force of law.\n\nHow could this have happened in this way? Here, in British Columbia, in Canada? More importantly, how can this violation of values that many Canadians hold dear\u2014like respect for the rule of law, a belief in equality, and a commitment to human rights\u2014continue to be legitimized in the 1990s? These are the questions that shape this book. By looking behind, around, and through the interconnections between law, history, culture, and power, I offer a long answer to the elders' question.\n\n**Human Being, Citizen, Anthropologist**\n\nThe landmark case of _Delgamuukw v. The Queen_ heard in British Columbia during 1987-1991 represented a crystallizing moment in the history of Aboriginal peoples and the law in British Columbia. All the variously interested parties to the dispute came together in one forum to present their positions, their arguments, and their supporting evidence. History followed them into the courtroom, and they carried the future out with them when they left.\n\nI wrote this book as a contribution to the ongoing discussions that are taking place in British Columbia and Canada about the possibilities for justice in Aboriginal\/non-Aboriginal relations; as a commentary on a particular dispute in one corner of the globe that is linked to more general struggles for social justice around the world; and, as an intervention in debates in the academic fields of anthropology and law. I write as a human being, a citizen, and an anthropologist. I present an explanation of issues and events that is necessarily one interpretation among many possibilities. Like any account, mine arises from and is shaped by some fundamental assumptions, principles, experiences and positions that I will set out as honestly as I can.\n\nMy inquiry begins and ends with the basic claim that all human beings are fundamentally equal and worthy of respect, and the simple assertion that Canadian law should reflect these principles, both in theory and in practice. My argument is that, in relation to Aboriginal peoples, this has not historically been, nor is it now, the case. I have come to this conclusion by way of a number of different paths.\n\nLong before I began this study of the _Delgamuukw v. R._ case, (and former Chief Justice Allan McEachern's _Reasons for Judgment_ ); and long before I had ever heard of the Gitksan or Wet'suwet'en First Nations, I was imbued as a child with certain understandings of both the danger of the law when it is abused as a weapon of terror, and of the promise of the law when it is employed to achieve and protect justice. Members of my Irish father's family were assassinated by English soldiers in the early years of this century: I was raised with few illusions about British justice. My mother's Jewish family fled pogroms in Czarist Russia and witnessed the Holocaust: I was taught to respect law as an alternative to brutality, and to value scholarship in the service of justice. These general understandings represent, to me, who I am and, as such, they shape how I interpret the world I live in.\n\nI began thinking seriously about the particular questions that constitute the subject matter of this book in the early 1970s, when I married a First Nations man, lived in his reserve community on the Central Coast of British Columbia, and became involved in the movement for recognition of Aboriginal title. I cannot remember when I first began to \"believe\" the simple truth that British Columbia First Nations hold unsurrendered Aboriginal title to their lands. If I ever thought differently, and there indeed was such a \"moment of revelation\" when I changed my mind, it happened so long ago that I cannot recall it now. For as long as Is can remember, this position has appeared to me as simple common sense: Aboriginal peoples were here, Europeans arrived. No wars of conquest were fought, no treaties were entered into. Hence, no Aboriginal title or rights were surrendered. These issues remain unresolved, so mutually-respectful negotiations should begin. These fundamental factual and moral \"truths\" are, to me, obvious. Nothing I have heard, seen, read, learned or experienced in the last twenty-five years has caused me to think or feel differently.\n\nI have brought up two Aboriginal children in contemporary British Columbia. I know something of the depths of racism in this province and in this country. I recognize, in the erudite language of the law, the same cruelty I have seen and heard directed to Aboriginal people in Canadian schools, hospitals, legislatures and many other \"public\" spaces. As a human being, I consider racism a violation of human rights. This is the first premise from which I write.\n\nSometime during the last twenty or so years, I also began to understand how the denial of Aboriginal title and rights in this context was simultaneously a denial of human equality. This understanding emerged first from learning how British and Canadian property law differentiates between categories of citizens according to their relationship to private property: fully \"matured\" citizens may own land, and others may not. These legal classifications mirror, and are mirrored by, cultural beliefs that accord greater value and worth to persons with property, than to those without. Ownership of private property stands as both criteria for, and evidence of, successful citizenship. Knowing this, when I hear Aboriginal people say that they are deeply hurt when governments and courts repeatedly refuse to recognize Aboriginal title and acknowledge Aboriginal rights, I interpret the injury they express as reflecting their very perceptive understanding of the meaning and value of property in the dominant Canadian culture, and the law's reflection of these meanings. That is to say, regardless of how diverse Aboriginal peoples may feel about property, they know how Canadian law feels about it, and they therefore rightly ascertain that they are being insulted\u2014treated as second class citizens\u2014by the law.\n\nAs a citizen, I wish to live in a just and democratic nation, and I object to the fact that Canadian sovereignty relies for its legitimacy on assumptions of human inequality: of European superiority and Aboriginal inferiority. The second premise I write from takes the form of an assertion of the democratic authority of a citizen to critique the law, and to insist that its practices accord with its avowed principles in ways that are comprehensible and transparent to the average person. I am neither a lawyer nor a legal scholar. I am, in this professional way, an \"outsider\" looking at the law. But as a citizen I am inevitably and irrevocably \"inside\" the law. The law claims to speak in the name of all people, and therefore the people should be able to discern the law's reasoning and the factual foundation it claims to rest on. All this I learned from my parents and other wise people. I brought these understandings and points of view with me when I began studying anthropology in 1982.\n\nAnthropology, historically, has been about western intellectuals investigating the ways of life\u2014the \"cultures\"\u2014of those classified as \"others\" in dominant European social theory: non-western, non-industrial, Indigenous peoples; and, to a lesser extent, marginalized groups within the borders of western nations. One of the central tasks of this traditional anthropology was identified as that of cross-cultural translation. Anthropologists sought to immerse themselves in the lives of the people they studied and, in this way, to achieve an \"insider's\" point of view. The next step in the anthropological enterprise was then to describe and explain\u2014to translate and represent\u2014these \"Native points of view,\" and their different cultural perspectives on the human condition, to western audiences.\n\nContemporary anthropology questions many of the foundations of this academic discipline. Most importantly, classical anthropology is undergoing a thorough re-examination by those \"subjects\" who were constituted as \"objects\" of study by earlier generations of ethnographers. Formerly colonized peoples, members of minority communities in the west, women, and other \"Others\" who were historically written about and analyzed by anthropologists, are turning the microscope around and scrutinizing those who originally examined them. They are reading and critiquing traditional ethnographic representations of themselves; conducting their own research into the cultures of both colonizers and colonized, oppressors and oppressed; producing a critical \"anthropology of anthropology.\" More and more, western and non-western anthropologists are studying their own societies: exploring the cultures they are always already immersed in, re-examining the histories that have produced them, and questioning their own taken-for-granted assumptions about the past, the present, and the future. This process is exciting and profoundly challenging. Errors in fact are being corrected. Alternative explanations of world history are being considered. Diverse visions of possible futures are being contemplated. We now struggle to read the \"great works\" of the European intellectual tradition through the eyes of these \"Others.\" It is now marginalized peoples' analyses of the centres, as well as of the margins, that we must try to grasp.\n\nTraditionally, an anthropologist might be expected to study the cultural beliefs and practices that First Nations draw on to support their appeals for legal recognition of Aboriginal title. I recognize and respect the fact that it is a consequence of persistent Aboriginal resistance that these issues are before the courts at all. Aboriginal litigants perpetually challenge the law, and occasionally jurists acknowledge some legitimacy to their claims. In these ways, First Nations have influenced Canadian law significantly. But I did not undertake this study for the purpose of learning what the evidence and testimony presented in _Delgamuukw v. R._ might tell us about Gitksan and Wet'suwet'en cultures and histories, or to unravel how legal strategies have shaped the representation of such issues in the courtroom. These are interesting questions, of course. But they are not the questions that have guided my inquiry. Rather, I take as given\u2014as \"common sense\"\u2014that the Gitksan's and Wet'suwet'en's fundamental position is valid. That is, I take for granted that regardless of what forms their social structures and cultural beliefs took at the time, they owned the territory in dispute when the British arrived in the eighteenth century, and they have not ceded, sold, lost or surrendered their title or rights to these lands and their resources under the terms of either Aboriginal, British or Canadian law. The Gitksan and Wet'suwet'en, and other First Nations in British Columbia, have sought recognition by Canadian courts of these facts in order that provincial and federal governments could be compelled to negotiate a mutually respectful relationship with them. First Nations have sought peaceful co-existence with newcomers who, like my own predecessors, came from many parts of the world\u2014for myriad reasons\u2014to live here, in Canada. I believe this is a just goal, and it is one I share. I take this position to constitute \"good sense.\"\n\nWhat I consider _not_ readily accessible to common sense, and not a reflection of good sense, and therefore in need of explanation and criticism, are the Crown's positions and the evidence and theories relied upon to support them. This book is therefore a project in the anthropology of European colonialism: a study of power and of the powerful. I turn my anthropologist's spyglass on the law, an institution that quintessentially embodies and reproduces Western power. This is the third premise from which I write.\n\nAs an anthropologist, I have tried to make sense of the law on Aboriginal title in British Columbia by placing the Gitksan and Wet'suwet'en case within the historical and contemporary political context of British and Canadian relations with First Nations, and considering the ways that law reflects and reinforces Euro-Canadian cultural beliefs, practices and diverse interests. I have explored how law shapes relations between people outside the courtroom as well as inside. Contrary to the image law holds of itself as a world unto itself, I understand law as inextricably enmeshed in society.\n\nThe first obligation of any responsible critic is to thoroughly investigate the object of critique, and I have tried to do so. In order to understand the story of _Delgamuukw v. R._ , I read, watched, listened to and discussed commentaries on the case by a wide range of people. I conducted a \"close reading\" of the various texts of this case: expert witness reports; transcripts of the trial; Reasons for Judgment; academic and popular commentaries on the case itself and the issues it raised. By \"close reading,\" I mean studying the texts not only for their literal or \"factual\" content, but also to understand the various ways they communicate meanings directly and subtly, by using language in particular ways; by writing and speaking in rhetorical styles; by deploying metaphor and evoking certain images and emotions; by using grammar, and constructing each text as a whole along specific lines. That is to say, I read these documents as cultural texts, using the tools of anthropological and cultural criticism. And, since particular texts do not make sense outside the broader context in which they are written and read, I have also studied related work in the fields of anthropology, history, law, and northwest coast ethnography.\n\nHowever, Chief Justice Allan McEachern's _Reasons for Judgment_ in _Delgamuukw v. R.,_ was not written as a work of fiction or ethnography. Judicial rulings carry with them a great deal of power and authority, based largely on assumptions that they represent reasonable and coherent conclusions drawn from a basis in empirical fact. I therefore considered it important to try to understand and critique this text on its own terms first. I began by analyzing the judge's ruling within its own context: the law on Aboriginal title in British Columbia. I considered it as a recent link in a long historical chain of Aboriginal title and rights cases that began when the British Empire began, and gave rise to the legal precedents that continue to shape contemporary judgments. My first task, then, was to trace the history of the law that Chief Justice McEachern claimed forced him to arrive at the ruling that he did.\n\nI conducted my critique from three positions, or points of view. First, I tried to step inside the law to understand it. I asked whether the law has followed its own rules and met its own criteria. As an anthropologist, the question that I sought to answer was whether or not the judge's rulings that relied on history and anthropology reflected reliable findings based on reputable research in these fields. Second, I stepped back outside the law, and read these texts from a position informed by a conventional anthropological critique of ethnocentrism that argues that all of humanity's cultures are worthy of equal respect, and should be understood on their own terms. The rules or values of one culture should not be applied to the evaluation of another. From this perspective, the dominant western culture is but one among many, neither universal nor superior. I asked, \"What cultural beliefs and practices did the Crown rely on to make sense of their arguments in _Delgamuukw v. R._?\" Interpreted from this position, the Crown's arguments and the Chief Justice's _Reasons for Judgment_ read like archaic, eurocentric, colonial texts, with a uniquely local, British Columbian flavour. Finally, I read and wrote from a third position, that of a critic of the dominant order. From this position, I argue that a critique of ethnocentrism must necessarily be the beginning, but should not be the conclusion, of a project aimed at re-imagining law and justice, and re-thinking how Aboriginal and non-Aboriginal people may live together. From this location on the political margins, I asked questions that address the future, as well as the past: regardless of the _cultural traditions_ these texts emerge from, what are the _cultural prescriptions_ embedded in the Crown's legal position and Chief Justice McEachern's law. Particularly, what visions of nature and society, what models of human relationships, did the Crown's defense of their position reflect, assert and defend? How are their images represented in the world they would create? What is the way of life they are empowered to command an army to defend? What kind of world\/country\/ province will we all, Aboriginal and non-Aboriginal, live in if desires like his continue to govern?\n\nMy interpretation and analysis therefore emerged from this critical \"ethnographic reading\" of the texts through which the law, and _Delgamuukw v. R._ , has made itself known to the public. I did not observe the trial, and I have not interviewed or consulted with any of the participants. My writing has been a solitary project, not carried out in collaboration with any representatives of any parties to the dispute. I claim to represent no one but myself. I retell the story of the Gitksan and Wet'suwet'en case, from my point of view, as a detailed account of an important moment in our recent history: a snapshot that artificially freezes time and records the instant so that we may return to it and re-view it, see it from different perspectives, read it in different ways, and learn different lessons from it. Many people have engaged in similar studies and arrived at similar\u2014and different\u2014conclusions, and I have learned a great deal from their work. Believing that what people say is inseparable from how they say\u2014or write\u2014it, I have tried to bring some of these other observers and commentators into conversation with each other on the pages of my text. Of course, it is I who have chosen which of their words to select, and directed at what moments they should speak and to whom. In so doing, I have interrupted, interpreted and appropriated their meanings to my own ends. My sources are not responsible for the uses I have made of their words. Believing too that humour is one of the strongest forms of cultural criticism, when particular moments in this story have struck me as funny or ironic, satirical or sardonic, I have presented them as such. Learning to laugh at ourselves and with others must surely be a worthy goal in the pursuit of justice.\n\nAny account of a process like the trial of _Delgamuukw v. R._ enacted over four years, that involved many people presenting complex evidence, debating obscure points of law, contradicting each other's interpretations of history, disputing understandings of esoteric cross-cultural translations, must necessarily be selective and partial. I have tried to provide enough detail and sufficient references that interested readers may take my account as an entry into, rather than the final word on, or an exit from, the many complex issues this story brings up.\n\nThis is not, however, a \"dialogue with texts.\" This is a \"dispute with texts,\" particularly with the texts of the Province of British Columbia's legal arguments in _Delgamuukw v. R._ , and with Chief Justice McEachern's _Reasons for Judgment_. At the same time, this book is not intended to be a polemic expressing an idiosyncratic or unique perspective. I have tried to present a meticulously documented, well-supported, and passionately reasoned, argument. I hope to persuade readers of the soundness of my position, and the weakness of the Crown's; of the injustices of the Crown's actions; and, of the desirability of an alternative vision for the future.\n\nThe story of litigation on Aboriginal title and rights in Canada, and particularly in British Columbia, is at one and the same time as remarkably simple as it is extraordinarily complex, as obtuse as it is simple-minded, and as familiar as it is strange. Truth and lies, fact and fiction, noble intentions and shameful motivations crisscross, undercut, and override each other as they dance across the pages of legal texts and treatises, creating a bewildering maze of claims and counter claims, accusations and denials. But when all is said and done, and many possible interpretations have been explored, the important questions remain: Has truth, by anyone's criteria, been told? Has justice, by anyone's measure, been done? Most importantly: how can justice be achieved in the present and protected in the future?\n\nStories like that of _Delgamuukw v. R._ illustrate how complex, and often contradictory, our feelings about law are. On the one hand, there is widespread cynicism in the public at large about the tenuous relationship between law and justice in Canada, especially in relation to Aboriginal peoples. And, there are hundreds of government reports, and volumes of academic research that support these criticisms. On the other hand, we continue to believe in the _possibility_ that the law will be just. Most importantly, we continue to _demand_ that the law _should_ be just, in both its theories and its practices. We can never cease to insist on this. How could we? Law, after all, is supposed to represent the principles and rules by which we have all agreed to live together in this nation-state, to constitute our particular social and cultural world. To cease trying to hold the law accountable to justice, no less than to condone law answering only to itself, would be to surrender to nihilism, despair and totalitarianism. The consequences of legal rulings are complex and far-reaching. Law's decisions are accompanied by powers of enforcement, including the exclusive right to legitimately use weapons and violence to resolve disputes. None of us can ignore the force of the law: it insists itself upon us in myriad ways every day. When it comes to law, then, critical skepticism, passionate outrage, perpetual hope, sardonic amusement, sheer terror, and utopian dreams of a moral social order are inevitably interwoven in our thinking, feeling, writing and conversation.\n\nMy initial goal was to make sense of the law. My next goal was to critique it. My long term goal is to help make the law sensible, and the society it reflects, just. I hope this book will challenge how you think about Aboriginal title and Canadian law, and how you choose to participate in shaping our collective future.\n\n* * *\n\n. Language is important. Throughout this text I use a variety of terms to refer to the peoples whose ancestors lived on the North American continent before Europeans. Each term has diverse connotations. \"Indigenous\" is the most all-encompassing and is the term of global representation chosen by the United Nations. Some people feel it erases the specificity of particular Nations, and suffers from an imprecise time frame, i.e. some people who others consider \"settlers\" identify themselves as \"indigenous\" because their families have lived in a region for many generations. \"Indian\" began with Christopher Columbus' error: landing in the Caribbean, he believed he had reached his destination of India, and so he called the people he encountered \"Indians.\" \"Indian\" is a term used in many legal documents, and in historical records. Some people find it offensive and feel its continued use reproduces its colonial legacy. \"First Nation\" is a term of recent emergence that is particularly popular in British Columbia. It is the language used in the _Constitution Act 1982_. Some people, particularly Metis representatives, interpret the \"first\" in \"First Nation\" as an implicitly hierarchical term that renders them \"Second Nations.\" \"Aboriginal\" is also a recent term that encompasses First Nations, Metis, Inuit, and Non-Status people. It is also used in legal documents, including the _Constitution Act 1982_. Some people feel that it is too broad and general and blurs important differences and erases diversity. I use all these various terms either because they are appropriate to the topic or time frame being discussed, or to the particular context. In unselfconscious contemporary everyday language, I hear all these terms deployed by Aboriginal and non-Aboriginal people and I use all these terms. I have reproduced this normal usage in the text. I regret if any readers are offended by any of these words.\n\n. In recent years, many Aboriginal nations have adopted spellings of their names that more accurately reflect their own correct pronunciation, replacing those spellings previously developed by Anglophone and Francophone colonial officials to facilitate English or French language and accent versions of Indigenous names. The Gixsan First Nation has adopted a new spelling of their name since this court case. I use the spelling as it appears in documents concerning the case.\n\n. Similarly, the Witsuwit'en have changed the spelling of their name.\n\n. The _Indian Act_ , as it was when I was married in 1975, conferred \"Indian status\" on non-Indian women when they married Indian men holding legal status. Conversely, when Indian women married non-Indian men, or Indian men who did not hold legal status, they and their children were dispossessed of their legal status as Indians. According to Canadian law, my marriage transformed me, in the eyes of the law, from a Canadian citizen of Jewish and Irish descent into a \"Registered Indian,\" member of the Nimpkish Band, and ward of the Crown. (The law was changed in 1985 so that today, marriage can neither confer nor take away legal Indian status.)\n\nI understand that, on the basis of my legal classification, some legal analysts would consider me an \"interested party\" to Aboriginal title and rights litigation. My children would hold entitlements under any treaty or settlement that might result. I do not consider myself an \"interested party\" in the manner of a potential beneficiary, nor do I think this legalistic aspect of my position significantly influences my analysis of the issues at stake. However, I offer this detailed explanation by way of respecting an obligation to \"put all my cards on the table\" when engaging in critique and arguments about morality and \"interests.\"\n\n. The late Antonio Gramsci, an Italian theorist, differentiated between \"common sense\" and \"good sense.\" \"Common sense,\" he said, consists of widely-held assumptions that are seen as \"natural,\" or in some other sense immutable, and taken for granted. There appears to be no need to question or challenge \"common sense\" since its \"facts\" are considered \"as obvious as the nose on my face.\" \"Good sense,\" on the other hand, refers to systematic reflection and conscientious planning based on practical and moral values: the best ideas of an era or populace. I am indebted to Richard Daly for bringing Gramsci's work on this to my attention. See Gramsci 1971.\n**Chapter 2: In Search of Reason**\n\n_It is the law that aboriginal rights exist at the \"pleasure of the Crown,\" and they may be extinguished whenever the intention of the Crown to do so is clear and plain.... The plaintiffs' claims for aboriginal rights are accordingly dismissed._\n\n\u2014Chief Justice Allan McEachern, 1991, _Reasons for Judgment, Delgamuukw v. R._\n\n**The Law in its Majesty**\n\nThus spoke Chief Justice Allan McEachern of the British Columbia Supreme Court as he rendered his long-awaited judgment in the most lengthy and costly Aboriginal land title litigation in Canadian history. The \"Gitksan and Wet'suwet'en case,\" also commonly referred to as \"the _Delgamuukw case_ ,\" or \" _Delgamuukw v. the Queen_ ,\" had been four years at trial, beginning in Smithers, British Columbia, on May 11, 1987, and concluding in Vancouver, B.C., on March 8, 1991. A total of 318 days of evidence from over 61 witnesses had been heard, additional evidence had been supplied by affidavit, and legal argument had taken up an additional 56 days in court. Verbatim transcripts of testimony now fill 23,503 pages of text, 82 binders of authorities now hold 9,200 pages of exhibits. An estimated 25 million dollars of public funds had been spent.\n\nThe issue before the court can be stated simply: The Gitksan and Wet'suwet'en had asked for legal recognition of their ownership and jurisdiction over 22,000 square miles of land and resources in the northwest region of the Province of British Columbia, Canada. They based their claim on the fact that they are descendants of people who lived there, in the same territory, since the beginning of time as they conceptualize it. When the Europeans arrived during what _they_ call the eighteenth century, the ancestors of contemporary Gitksan and Wet'suwet'en peoples were already living there. Neither the Gitksan nor the Wet'suwet'en ever entered into treaties with Britain or Canada, the governments representing the interests of the newcomers. Nor was there a war in which their territory was conquered militarily by the new colonial forces. Nor were any rights to land or resources ever sold by Indigenous peoples to any individual settlers, or to the British or Canadian governments. The Gitksan and Wet'suwet'en, therefore, continue to consider themselves the rightful owners and governors of the territory in question. They say they hold \"Aboriginal title and rights\" to this land.\n\nThe Province of British Columbia and the Government of Canada opposed the Gitksan's and the Wet'suwet'en's claims. The Crown asked the court for \"a declaration that the Plaintiffs have no right, title or interest in and to the Claim Area, and the resources thereon, thereunder or thereover.\" Since 1871, when British Columbia joined Canadian Confederation, all successive governments of that province had taken the position that no Aboriginal rights recognizable by \"civilized law\" existed prior to Britain declaring sovereignty over the territory. And even if those rights had existed, the Province of British Columbia's argument continued, the simple act of assertion of sovereignty by a European power over those lands was sufficient to extinguish any pre-existing Aboriginal title and rights. Legally, the Province of B.C. told First Nations, you do not exist. This position provided the rationale for provincial governments' consistent refusals, until 1990, to participate in any discussions or negotiations with federal government and Aboriginal representatives on Aboriginal title and rights issues.\n\nDuring the week preceding the release of Chief Justice McEachern's decision in the _Delgamuukw_ case, the _Vancouver Sun,_ British Columbia's largest circulation daily newspaper, ran a series of articles entitled \"Judgment Day\" in which spokespeople from government, industry, labour and the general public expressed the view that at least a partial court victory for the Aboriginal plaintiffs was anticipated by all concerned. Gitksan and Wet'suwet'en Tribal Council representative Herb George told the press that he expected to see \"the last little trace of honour in the Crown\" reflected in the judgment. \"We're not naive,\" George said, \"but we can still dream.\"\n\nThere seemed to be good cause for George's cautious optimism. Between 1987, when the _Delgamuukw v. R._ trial began, and 1991, when it concluded, some significant changes had occurred in the legal and political landscape of Aboriginal and non-Aboriginal relations in British Columbia. In 1989, the First Nations Congress, representing most of the 180 First Nations in B.C., had initiated discussions with representatives of the province's leading forestry, fishing and mining companies. The First Nations leaders' goals were to educate the business sector about the historical and legal bases of Aboriginal title, to calm industry fears about potential threats to economic stability in the event of recognition and settlement of Aboriginal claims, and to begin to build a foundation for direct negotiations between industry and First Nations governments on issues of economic development.\n\nFollowing these conferences, held at the exclusive Whistler Mountain Resort in the traditional territory of the Squamish First Nation, British Columbia Social Credit Premier Bill Vander Zalm appointed a \"native affairs advisory council\" to serve as a consultative committee. The advisory council held meetings with key players throughout the province during the spring and summer to discuss Aboriginal land claims. _Vancouver Sun_ columnist Vaughn\n\nPalmer observed that, \"The thinking in government circles is that the court will probably recognize aboriginal title...that it still exists today.\"\n\nOn May 31, 1990, less than a year before Chief Justice McEachern's judgment in the _Delgamuukw_ case was delivered, the Supreme Court of Canada had overturned earlier decisions of the British Columbia Supreme Court and the B.C. Court of Appeal in the case of _Regina v. Sparrow_. The Supreme Court concluded that certain Aboriginal rights\u2014in this case the fishing rights of the Musqueam First Nation\u2014had existed in British Columbia prior to the arrival of Europeans; had not been extinguished by the simple assertion of British sovereignty during the colonial era; and were now protected by section 35(1) of the _Constitution Act (1987)_ , the supreme law of Canada.\n\nAlthough the Supreme Court of Canada, in their decision in the _Sparrow_ case, upheld the position that, since Britain declared sovereignty, the \"underlying title to all land\" in Canada is vested in the Crown, they ruled that Aboriginal rights could not be _implicitly_ extinguished by the mere declaration that British sovereignty had been asserted. Rather, the Crown would have to _explicitly_ express their \"clear and plain intention\" in order to legally extinguish Aboriginal title to land. The first premise of the Province of British Columbia's position since 1871 had been that no legally-recognizable Aboriginal title existed at the time the British arrived in what is now British Columbia. Therefore, according to the Province of B.C., an _explicit_ expression of intent to extinguish title was neither required nor appropriate: what does not exist, could not be recognized. What could not be recognized, need not be extinguished.\n\nThe Supreme Court of Canada's rejection of the implicit extinguishment argument in the _Sparrow_ case, their reaffirmation of the requirement that the Crown make their intention to extinguish Aboriginal title _explicit_ , and their finding that this had not been done, and hence Aboriginal rights still existed in the province, was a significant victory for B.C. First Nations: the provincial government could now be legally required to acknowledge their existence.\n\nThe _Sparrow_ decision was hailed by many as the judgment that would bring an end to the long\u2014and increasingly contentious\u2014chapter in Canadian legal history in which colonial law, justified by archaic nineteenth-century notions of European superiority and Aboriginal inferiority, had dominated the courts. It was hoped that the _Sparrow_ decision would provide the required legal framework within which a new, more equitable relationship between Aboriginal and non-Aboriginal people in Canada could be developed, a relationship that would reflect contemporary commitments to social justice and equality in a multicultural Canada. Lower courts in British Columbia during 1990-1991 had granted several First Nations' applications for injunctions, halting development on their lands until the Aboriginal title question was settled by the courts.\n\nPublic opinion polls conducted during 1990 and 1991 had consistently shown that the majority of people in British Columbia thought the provincial government should reconsider its historic refusal to discuss land claims with the First Nations and should proceed, finally, to negotiate an agreement with them. Premier Vander Zalm\u2014heeding the courts, public opinion polls, and increasing pressure from corporate representatives seeking government guarantees that British Columbia could indeed boast a \"safe investment climate\"\u2014had allowed that the issue of land rights negotiations with First Nations should be revisited. For the first time in the province's 119-year history, a provincial Ministry of Aboriginal Affairs had been inaugurated in the Fall of 1990.\n\nAfter nearly 100 years of political lobbying for the right to even file such a petition in court, the Nisga'a First Nation\u2014neighbours of the Gitksan and Wet'suwet'en\u2014had been the first to launch a legal claim for recognition of their Aboriginal title in 1969. On March 7th, 1991, the day before Chief Justice McEachern's ruling in the Gitksan and Wet'suwet'en case was handed down, a landmark tripartite agreement between the Nisga'a Tribal Council and the federal and provincial governments was announced.\n\nNo one had anticipated what did happen on March 8, 1991.\n\nChief Justice Allan McEachern had not been swayed by public opinion polls or newspaper editorials, or by the Supreme Court of Canada's _Sparrow_ decision, or by the copious evidence in support of the Gitksan and Wet'suwet'en claim that he had listened to for four years. In his 400 page _Reasons for Judgment_ , the Chief Justice analyzed the testimony, reviewed the relevant points in law, and then dismissed the Gitksan and Wet'suwet'en claim. No Aboriginal title or rights had pre-existed European settlement, he ruled; and even if they had, they had been extinguished by the simple fact of Britain asserting sovereignty. Treaties had not been made, nor compensation paid, nor Aboriginal consent acquired. Nor were they required, he ruled.\n\n**Late One Night at the Legion**\n\nLike most everyone else interested in Aboriginal issues, especially in British Columbia, the judgment caught me off guard. The early morning CBC radio news on March 8, 1991, announced that the judgment had been released to the lawyers, who were in a \"lock up\" with the document until noon. I, too, had assumed that the outcome of this case would be relatively positive for the claimants: an historic turning point in the legal struggle for recognition of Aboriginal title and rights. When I turned on the CBC radio midday news it was not with bated breath or apprehension, but rather with curiosity to know the details of the judgment and what the implications were for all parties to this case. To say I was shocked by what I heard on the radio is an understatement.\n\nDuring the years 1988 to 1991, I had been a graduate student in Anthropology at the University of British Columbia and at Simon Fraser University. These were also the years that the Gitksan and Wet'suwet'en case was being heard in the British Columbia Supreme Court. While I was not directly involved in the trial in any way, I had followed developments in this case through media, attended public information sessions, special lectures, support demonstrations, benefit dances and fund-raising performances; and bought raffle tickets. Social movements in general, and Aboriginal politics in particular, form both the central focus of my work, and the personal and social milieu in which I live my everyday life. Therefore, the progress of this high-profile court case had been the subject of countless informal discussions among friends and relatives over the course of its four-year duration.\n\nFor days following the release of Chief Justice McEachern's ruling, the trial and the judgment were being talked about everywhere I went: at home, at the university, at social gatherings. Everyone I encountered had something to say about the Gitksan and Wet'suwet'en case and the Chief Justice's decision. Most were shocked. Many were outraged. A few were smug. Others were bitter. Some said the Gitksan and Wet'suwet'en should never have gone to court to begin with: what can you expect from the white man's courts? Since when could justice be found in law? Others felt betrayed by a legal system they had previously believed to be fundamentally fair. Some people were critical of the lawyers and the way they had conducted the case\u2014some said they had asked for too much and overwhelmed the judge with mountains of data. Others said they had not asked for enough, and that they should have called even more evidence. Some said the _Delgamuukw_ decision was an anomaly and didn't represent the thinking of the judiciary as a whole. Others said the opposite: McEachern had simply used unfashionable words and precedents to express what his colleagues, who had learned to obscure these ideas with more \"politically correct\" language, really thought.\n\nAnthropologists were insulted by the Chief Justice's wholesale rejection of ethnographic evidence. Many were dismayed that anthropologists like Hugh Brody, Richard Daly and Antonia Mills\u2014who had served as expert witnesses\u2014 had been summarily dismissed and reprimanded by the judge for being \"advocates\" whose participant observation research was not \"credible.\" Others were critical of the anthropology presented, and the anthropologists who had testified on behalf of the Gitksan and Wet'suwet'en: Hugh Brody had been too caustic. Richard Daly had been too obtuse. Antonia Mills had been too ethereal. They had presented too seamless a case. Others said they had indulged themselves in idealism and ideology: empiricism should have ruled the day.\n\nMost Aboriginal people I spoke with were less shocked by the judgment than their non-Aboriginal peers. They were disappointed, and deeply hurt by the Chief Justice's words. More than anything, they were angry at what they saw as the judge having insulted the Chiefs and Elders by his charge that their testimony had been \"untrue.\" Some Aboriginal people muttered resentfully about how much money the lawyers and the expert witnesses had made. Some asked why the anthropologists had ever been involved in the first place. Why hadn't the Gitksan and Wet'suwet'en stood by their own Chiefs and Elders and refused the need for \"representation and translation\" by white anthropologists?\n\nIn an unprecedented gesture, the government of British Columbia bound Chief Justice McEachern's _Reasons for Judgment_ in book form and distributed it widely throughout the province. Rumour had it that the Chief Justice hoped his text would form the basis of a public school curriculum. Shock turned to incredulity as the volume circulated among scholars and other people knowledgeable about British Columbia history and Aboriginal cultures. The Canadian Anthropology Society, representing 405 scholars, told the press that the judgment \"gratuitously dismisses scientific evidence, is laced with ethnocentric bias and is rooted in the colonial belief that white society is inherently superior.\" University of British Columbia anthropology Professor Robin Ridington added that \"if an Anthropology 100 student wrote anything like that in a paper, not only would you write a lot of red ink over it, you would say 'Look, please come in and talk to me. You have real problems.'\"\n\nWhen I read the _Reasons for Judgment_ , I found the text very familiar. I was immersed at university in the study of critiques of colonialism and western culture, written largely by colonized and formerly colonized people. This work focuses on how European social theories classify and represent non-European peoples as inferior \"Others,\" justifying colonial domination and exploitation by western powers. Within the borders of the west, similar processes of \"othering\" had long legitimated the subordination of women, minorities, gays, lesbians, the poor, the disabled, and political dissidents by dominant classes. Chief Justice McEachern's text read like a caricature of everything I was studying. The rendition the _Reasons for Judgment_ offered of British Columbia's history repeats what can be read in the blunt, unselfconscious language of Indian Agents' reports found in archival records, and in the memoirs of pioneers, missionaries and settlers that form the corpus of popular history in B.C.: what political scientist, Paul Tennant, has called \"the founding myth of White British Columbians.\" These local stories of Aboriginal primitivism, European superiority, and the historical inevitability of colonial domination are repeated throughout the colonized world. One can hear very similar stories in Australia, New Zealand, and South Africa. They are self-justifying accounts, told by a colonial people whose ongoing doubts and insecurities about the moral legitimacy of their occupation of Indigenous lands require that these stories be told, and retold. Having lived in a small coastal community in British Columbia for many years, I recognized in the Chief Justice's tome what I had long ago nicknamed the \"late one night at the Legion\" version of B.C.'s history.\n\nA month or so after the _Delgamuukw_ judgment was released, I attended a meeting of anthropologists and historians at the University of British Columbia to discuss how we might responsibly respond. We agreed that we would each write an article on a specific aspect of the anthropological and historical evidence that had been presented during the case. These papers would be compiled in a special edition of the journal, _B.C. Studies_. I chose to write a critique of the opinion report, testimony, and cross-examination of the expert witness who had provided anthropological evidence for the Crown, Dr. Sheila Robinson.\n\nI approached this project expecting to find in Dr. Robinson's evidence an interpretation of facts presented within a coherent, logical argument. I anticipated that I would likely disagree with Robinson's analysis, on a professional basis. Debate, after all, we like to think, is the foundation of scholarly integrity. I was shocked to find that her opinion report did not include an up-to-date review and analysis of relevant theoretical or substantive literature. Instead I found oxymoronic argument, and the substitution of theoretical speculations for empirical facts. I also read the reports and testimony of the anthropologists who testified for the Gitksan and Wet'suwet'en. I found issues to debate with Richard Daly, Antonia Mills and Hugh Brody, but the important difference between their submissions and those made on behalf of the Crown was that the former were meritorious, scholarly documents while the latter were not. The stark contrast between the expert opinion reports in anthropology and history submitted on behalf of the Gitksan and Wet'suwet'en, and those submitted on behalf of the Crown, was alarming in terms of their respective professional credibility. Furthermore, the Gitksan's and Wet'suwet'en's evidence and arguments accorded more obviously with conventional notions of what constitutes facts and logical argument than did the Crown's. Most importantly, the judge's crucial legal findings were based on particular interpretations of history and explanations of cultural differences. Clearly, the Crown's arguments and the testimony of their expert witnesses had appeared the more credible to Chief Justice McEachern. Why? What criteria had the judge used to evaluate the evidence he had heard?\n\nI read more of the transcripts and more of the legal analyses. I found in the texts of the _Delgamuukw v. R._ case an intense debate not only about Aboriginal land rights but also about moral visions, social theories and political strategies. In presenting their cases, the \"Plaintiffs\" and the \"Defendants\" disputed how stories would be told about the past, present and future of British Columbia; and about who could tell them. Above all, they competed for the judge's verdict: whose account would the law declare to be \"the truth\"? Whose story would have an army at its disposal?\n\nI decided to write my doctoral dissertation on the _Delgamuukw_ case. I began with law's history.\n\n* * *\n\n. Glavin, Terry 1991: B3.\n\n. Palmer, Vaughn 1990, quoted in Tennant, P. 1990: 237.\n\n. Previously, the First Nations of the Nass Valley of British Columbia were called \"Nishga.\" Currently, this First Nation spells its name as \"Nisga'a.\"\n\n. For a personal account of one lawyer's experience of this event, see Pinder, Leslie Hall 1991.\n\n. Empiricism is a theory of knowledge usually associated with the natural and physical sciences. Its basic tenets are that all \"factual knowledge\" originates in observation and experience, and can be falsified by observation and experience. \"Empiricism\" is usually understood as being opposed to other theories of knowledge that take greater account of the way social-cultural context and the subjectivity of the researcher and the reader can shape the meanings derived from observation and experience.\n\n. Canadian Press 1991: B3.\n\n. Ibid.\n\n. The literature on this subject is huge, multidisciplinary and growing rapidly. Texts that have become classics in the study of European constructions of \"others\" include: Asad (ed.)1973; Clifford and Marcus (eds.) 1986; Deloria 1969; Fabian 1991; Fanon 1963; McCrane 1989; Trinh T. Minh-ha 1989; Said 1978, 1992; Spivak 1988; Todorov 1982; West 1993; Wolf 1983.\n\n. Tennant 1991: 76.\n\n. See Miller (ed.) 1992.\n\n. An oxymoron is defined as \"a combination of contradictory or incongruous words (as _cruel kindness_ ),\" _Webster's New Collegiate Dictionary_ , 814.\n**P ART II**\n\n**T ERRA NULLIUS**\n\n**(U NOCCUPIED, EMPTY LAND)**\n**Chapter 3: Cultures Similar and Different**\n\n_Law is essentially historical, not just in the sense that the life histories of legal systems can be chronicled, but more importantly in the sense that it is characteristic of law to anchor justification to the past. Time is the soil of the lawyer's thinking._\n\n\u2014Philosopher Gerald Postema, 1991, _On the Moral Presence of Our Past._\n\n**Creating Race in the Interests of Empire**\n\n\"The point at which a storyteller chooses to begin,\" Edward Said wrote, \"is the first step in the intentional construction of meaning.\" The Gitksan and Wet'suwet'en begin their story\u2014locate their origins as sovereign peoples with title and rights to specific lands\u2014in \"time immemorial,\" when the Creator placed them on specific territories, and charged them with the responsibility of looking after the sentient beings who lived in and from and with these lands. They adopted the legal term, \"time immemorial,\" as a way of representing and translating\u2014communicating\u2014to English language courts their belief that they have been there since the beginning of time.\n\nChief Justice McEachern rejected what he considered the \"non-specific\" nature of Gitksan and Wet'suwet'en history, writing in his _Reasons for Judgment_ , \"I am not able to conclude on the evidence that the plaintiffs' ancestors used the territory since 'time immemorial' (the time when the memory of man 'runneth not to the contrary'). 'Time immemorial,' as everyone knows, is a legal expression referring to the year 1189 (the beginning of the reign of Richard II), as specified in the Statute of Westminster, 1275.\"\n\nAnother common starting point for legal historians to begin their rendition of the story of European\/Aboriginal relations in the Americas is the year Europeans call 1492: when Columbus sailed the ocean blue. The controversies that followed Columbus' arrival in the Caribbean concerning the moral and political legitimacy of conquest and colonialism culminated in the Conference of Valladolid. In 1550, King Charles V of Spain brought together clerics, lawyers and other scholars in the Spanish city of Valladolid and asked them to address the question of whether Indigenous peoples were part of the same human race, or species, as Europeans; and, depending on the answer to this fundamental question, how they should be treated by European colonial powers. The king asked a philosopher, Juan Gines de Sepulveda, and a Dominican monk, Bartolomeo de Las Casas, to answer the question: \"How can conquests, discoveries and settlements in the king's name be made to accord with justice and reason?\" Both Sepulveda and Las Casas agreed that all human beings, including Indigenous peoples, were of one species. Both also agreed that it was the duty of Europeans to convert all the world's peoples to Christianity, disagreeing only on method and rationale.\n\nSepulveda argued, relying on Aristotle, that some races are inferior to others, and that some people are born to slavery. By this reasoning, Europeans, a superior race, were justified in subjugating Indigenous peoples, an inferior race. Sepulveda used stories of cannibalism as evidence of the alleged inherent inferiority of the Indigenous peoples of South America. The technological achievements and complex social organization of the Aztecs and Incas, then as obviously sophisticated by prevailing European standards as they are universally acknowledged to be now, were absent from Sepulveda's analysis. Las Casas argued, on the other hand, that these \"Indians\" possessed an evolved culture, with social, economic and religious institutions. He claimed that Indigenous peoples were rational beings, fit to be compared to the Greeks and Romans. Las Casas did not argue that Spain should _not_ conquer the Indigenous Peoples of the Americas, but rather that its only justification for doing so should be to Christianize them. He was distraught that the cruelty of the conquistadors was inhibiting his mission of conversion.\n\nThe choice of the conference at Valladolid as the origin story of European\/ Indigenous relations in America, has been popularized in Canada by Judge Thomas Berger, a long time supporter of Aboriginal rights, and well-respected for having headed a public inquiry into the potential impacts of a proposed oil and gas pipeline on the Aboriginal peoples of the Canadian north. In his book, _A Long and Terrible Shadow: White Values, Native Rights in the Americas, 1492-_ _1992_ , Berger writes of the deliberations at Valladolid: \"Here was the very debate that I heard centuries later in the Mackenzie Valley Pipeline Inquiry.\" The \"Berger Inquiry\" broke with the traditional colonial reliance on non-Aboriginal experts that had historically characterized such processes, and encouraged and respected the participation of Aboriginal communities in its proceedings. The inquiry's findings and recommendations supported the Aboriginal peoples' opposition to the pipeline and their aspirations for local control. Berger proposed a ten-year moratorium on resource development to allow time for environmental and social impact studies to be completed, and for First Nations to consolidate their vision of self determination. (As it turned out, the oil and gas companies eventually abandoned their plans to build the pipeline for \"economic reasons\": it became an unprofitable proposition.)\n\nThe appeal of the Valladolid story, for liberal Canadians like Berger, may in part lie in its archetypal legal story form: a triangle consisting of a good learned person (Las Casas); a bad learned person (Sepulveda); and a benevolent sovereign (Charles), pondering a deeply important and complex issue in a quasi-judicial forum. Berger constructs the law's uninterrupted historical narrative from Columbus' encounters in Central and South America to contemporary Canada. This evokes notions of political conflict rooted in racial or cultural differences, often thought to be universal characteristics of all human societies in all historical periods, and thus to be immutable. The dominant image evoked is easily recognizable in contemporary liberal discourses on multiculturalism and identity-politics: white European colonizers, and brown Indigenous colonized.\n\nWhile the impact of Spanish colonization on the Americas is important, it is British imperialism and colonial law that is specifically of interest to this story, a story that differs from Spanish colonial history in some important ways. British imperial law traces its origins to the common law tradition that emerged in the fifth century A.D. when the Anglo-Saxons invaded what is now called England and conquered the indigenous Britons. The Anglo-Saxons went on to absorb immigrant Danes and Gauls, others of whom (having become, over the course of the ensuing five centuries, Normans) in turn conquered the Anglo-Saxons in 1066. The Norman conquest of 1066 brought to Great Britain a centralized state and church, and the arbitrary power of the king. Under the Norman-derived legal regime, the king, or sovereign, who mystically embodies the \"underlying title to all land\" and the implicit consent of all the people to his reign, is said to \"hover over the land.\" The origin of the sovereign's legal title is to be found in this abstract, imaginary vision, made concrete through the exercise of power as the sovereign became the symbolically omnipotent source of law, and his will its practical execution. A feudal regime, sometimes referred to as the \"Norman Yoke,\" was established, against which Anglo-Saxons, Britons, Danes, Gauls and some Normans (now having become English and Scots) waged civil war.\n\nAt the same time as Sepulveda and Las Casas were debating in Valladolid, English jurists were confronting similar problems regarding colonization in Ireland. Their dilemma was not precipitated by the sudden \"discovery\" of seemingly strange and alien peoples. Rather, the problem of justifying expropriations of lands and massacres of native populations that faced British imperial policy-makers was that of _recategorizing_ as radically \"different\" their Irish neighbours who had hitherto been similar and familiar.\n\nThat the Irish were Christian was never doubted by the Normans or their successors, but Christianity in Gaelic Ireland did not fully conform to Roman liturgical practice, and many pre-Christian traditions and customs had been only slightly veneered in these territories by Christianity. On this basis, the Irish were classified by British imperial law as atheists or infidels. Although, unlike \"certain savage tribes,\" the Irish were rarely accused of cannibalism, they were described as \"little better than Cannibals....\" In addition, the Englishtook the Irish practice of transhumance as proof that the Irish were nomads, hence barbarians. The English colonists thus developed a social theory that said the Irish had evolved to a level of cultural development analogous to the stage the ancient Britons existed at before they were civilized by the Romans. The Irish should therefore be made subservient to the colonizing English, (the true inheritors of Roman civilization) so that, through subjugation, they could come to appreciate civility and thus eventually achieve freedom as the former Britons had done. This belief that meting out punishment to subordinated peoples and individuals \"for their own good\" will result in their eventual emancipation, while an enduring one, is belied by the historical record which offers more support for the theory that cruelty breeds brutality.\n\nMy point is straightforward: the boundaries separating categories of people, and, more importantly, the significance of differences between and among them, are, like culture itself, constructed by people\u2014not given by nature or God\u2014and they change over the course of history as contexts and social relations change. Being created by people, these fluid boundaries can also be challenged, changed and recreated by people over time, as indeed, historically, they often have been.\n\nThe story of Aboriginal and non-Aboriginal relations in British Columbia that I am telling here begins with England's sixteenth-century colonization of Ireland, rather than with Columbus' landing in the Caribbean, for a number of reasons. First, there is the obvious historical continuity: the British colonized British Columbia, the Spaniards did not, although they \"discovered\" the territory before the British. Second, the story of the colonization of Ireland tells of recurring hostilities among people of the same \"race\" and \"cultural group.\" Rather than illustrating universal, ahistorical and immutable conflict based in \"natural\" racial or cultural difference, the history of England and Ireland is an example of contests for economic and political domination, and the historical process of \"racialization\" of the \"other\" so fundamental to colonial law and cultures. The image I wish to evoke is easily recognizable in contemporary debates about _critical_ multiculturalism and post-colonial politics: same-culture colonizers, and same-culture colonized.\n\nFinally, the appeal of Berger's choice of Valladolid as the preferred origin story for the history of Aboriginal legal rights in Canada may also be found in its acceptance of the historical inevitability, and hence justification, of European colonial conquest. Its insistence that colonial powers were and are motivated by good intentions and paternalistic\u2014yet humanitarian\u2014concerns for the best interests of the colonized, also reinforces familiar legitimations of colonial dominance.\n\n\"All this was done with the best intentions. We can only ask what would have happened had government's intentions _not_ been good?\" Phrases like this have become a common clich\u00e9 used by the Canadian media to wrap up stories about how another government program, or lack thereof, has been implicated in some tragic chain of events in an Aboriginal community.\n\nAssessing history solely on the basis of the intentions of the powerful, without reference to the consequences for the powerless, is a time-honoured tradition in western thought. Within this framework, the morality of a person's actions can only be judged on the basis of his or her intentions. This is based on the premise that people should not be held responsible for events beyond their control that contribute to their original actions having unintended consequences. At a very basic philosophical level, this is, of course, only fair. We as individuals are the only ones who know what our intentions are in the first place. Other people know our intentions only when we choose to communicate them. Our intentions motivate us to act in particular ways in relation to others, and our actions have consequences for ourselves and for those others. If our intentions, according to our own account, are honourable, but a person who experiences the consequences of our actions suffers, how is this to be resolved? The issue becomes more complex in real life where the questions \"what ought you to know before you act?\" and \"whose intentions will be acted upon and thus enabled to have consequences?\" must also be asked. \"Judgment by intention\" may be reasonable if \"the individual\" is considered as an isolated entity, but it becomes problematic when an individual is understood as a human being who lives in relationships with others in society.\n\nThe legal answer to this question in the liberal tradition is to grant rights to individuals, thereby hoping to ensure that the limit of one individual's rights is formed by the boundary at which another individual's liberty is inhibited. In order for _groups_ or _collectivities_ or _peoples_ to have similar protection within such a regime, they must be recognized in law as having legal rights. The history of Aboriginal\/non-Aboriginal relations in Canada has been one in which courts and governments have repeatedly refused to recognize Aboriginal rights, while continuing to insist that their actions have been, and continue to be, guided by their \"good intentions\" in relation to First Nations. This \"justification by intention\" argument has been repeated by governments for hundreds of years now, despite the fact that the _consequences_ of legally-sanctioned government policies have been, and continue to be, devastating for Aboriginal peoples. The federal government, in particular, has been repeatedly stymied by the unanticipated consequences of their good intentions. An option yet to be tried in the realization of governments' good intentions towards First Nations is the recognition of Aboriginal title to land, and rights to self-government.\n\n**Mutiny and Desertion as Common Sense**\n\n_The initial conditions for theorizing and reflecting on property rights in America are of a European people who arrive on a continent of roughly five hundred established Aboriginal nations and systems of property...and who do not wish to become citizens of the existing Aboriginal nations, but wish to establish their own nations and systems of property in accordance with their European institutions and traditions._\n\n\u2014Philosopher James Tully, 1993, _An Approach to Political Philosophy: Locke in Contexts_.\n\nBefore the arrival of Europeans, approximately 500 Aboriginal nations existed in North America. These nations were diverse in modes of living, language, religion, social organization and relationships to land and resources. Therefore, the first rational option available to early European explorers and settlers was to behave as guests or immigrants, to recognize the sovereignty of already existing Indigenous nations, to live by their laws and to seek acceptance by their people.\n\nBut many deeply ingrained assumptions and widely held beliefs limit and constrain the possibility of contemporary people immersed in the dominant western culture imagining this as a practical or viable choice. European theories arising from psychoanalytic traditions have argued that fear and hostility are instinctive human responses to encounters with cultural \"others.\" However, the weight of contemporary knowledge belies the proposition that such a reaction is either universal or natural. Oral historians and linguists have investigated the diverse ways in which non-Europeans have identified people other than themselves, including Europeans, at first contact. They found a variety of impressions and responses varying from adoration to repulsion, fear to amusement, hospitality to hostility and curiosity to indifference. Europeans have been represented in Indigenous thought by analogies as diverse as Gods, Devils, clowns and monkeys.\n\nEuropean images of Indigenous peoples have varied across time and space as well. Early visitors to \"new\" worlds, like explorers and traders, frequently expressed both interest in and admiration for the peoples they met on their travels. Later visitors wishing to claim these lands as their own, such as colonial governors and settlers, portrayed Indigenous peoples as wild savages. Most influential in western thinking, however, have been various theories about social and cultural evolution that postulate a hierarchical series of stages through which it is claimed all human societies must proceed. \"Primitive\" societies, in which people live from hunting, fishing and gathering, represent the \"first stage\" in this scheme; while \"civilized\" industrial societies exemplify the \"highest stage\" of human development. In the field of jurisprudence, Sir Henry S. Maine's 1861 volume, _Ancient Law_ , sets out a conjectural history of the evolution of law from \"primitive custom\" that \"mingled up religious, civil, and merely moral ordinances without any regard to differences in their essential character;\" to \"civilized law,\" which \"severs law from morality, and religion from law.\" The purported independence of law from the social conditions of its emergence, and law's professed autonomy from the society in which it is practiced, were evidence, for Maine, that western law \"belongs very distinctly to the later stages of mental progress.\"\n\nAnother common idea that arises from the same social evolutionary premises is the analogy drawn between the stages of individual psychological development hypothesized by western developmental psychology, and the social organization of non-western societies. In this framework, adults are to children as Europeans are to Aboriginals. Within historical analyses, contemporary Indigenous societies, imagined as being \"frozen\" at \"stages of development\" believed to be \"antecedent\" to the present, are seen as resembling historical European ones. Dutch anthropologist Johannes Fabian calls this notion that \"they\" are now, as \"we\" were then, a \"denial of coevalness\" that he identifies as the most enduring obstacle to achieving mutually respectful dialogue between Europeans and \"others.\" Fabian uses the term \"coeval\" to describe how colonized and colonizer share both geographic space (land); and temporal space (time). Social inequality between peoples of different cultures, Fabian argues, is shaped by the historically-specific political and economic relationships between them. It does not emerge because they live at different hypothesized stages of evolutionary development. Power, not time, separates people of different cultures.\n\nThese social evolutionary premises support the common belief that, during the eighteenth century, when they came into substantial and sustained contact with each other, Europeans and members of British Columbia First Nations were so radically different from each other in terms of material living conditions, social organization, political structures, religion, intellectual development and individual freedom, that for British settlers to have integrated into the Aboriginal societies would have necessitated them stepping \"backwards,\" both in historical time and in human psychological development. These beliefs are buttressed by images in paintings like those that decorate the Rotunda in the British Columbia Parliament buildings: luxuriously dressed and elaborately wigged British ship's captains benevolently bestowing gifts on scantily-clad and bewildered-looking Aboriginals.\n\nBut, consider what we know of the _real_ conditions of life for the majority of people in eighteenth-century Britain: feudalism\u2014dominated by the nobility, and the clergy living off the avails of peasant labour through tithing and maintaining power by force and threats of eternal spiritual damnation\u2014was being replaced by industrial capitalism as the real source of power in the nation state. Peasants, driven from the land, began to flock to the newly-emerging urban centres in search of subsistence from industry, rather than from hunting, gathering and agriculture. Accounts of factory conditions during early industrialization, and of the everyday life of ordinary citizens in cities like London, Manchester and Birmingham describe desperate urban poverty, squalor, alcoholism, prostitution and deprivation. It would be at least two centuries before women would be recognized as persons in British, and later Canadian, law, and we have yet to achieve mass literacy.\n\nCompare this to what we now know of life during the same historical epoch among, for example, the Gitksan and Wet'suwet'en peoples. Their eighteenth-century societies were characterized by a hierarchical social structure dominated by chiefs and shamans; divisions of labour based in clan, gender and generation; an economy dependent on the accumulation and unequal redistribution of wealth derived from abundant natural resources among different categories of persons; spiritual beliefs linking ancestors, the living and the land that prescribed reciprocal obligations; social life, including marriage, divorce and inheritance controlled by elders. An oral, rather than written, tradition of recording history and administering law prevailed. That European and Indigenous cultures differed significantly is obvious. But was one \"more advanced\" or \"less developed\" than the other? This is not at all obvious. Answers are complicated first by the problem of criteria. As we know, technological complexity is not necessarily correlated with spiritual satisfaction or emotional well-being. Second, different categories of persons in both societies would likely evaluate their own cultures differently, depending on their position within it. Establishing universal criteria upon which to base such evaluations is an extraordinarily complex process, if it is to represent any authentic consensus. And, the question may be asked if universal criteria of what constitutes a \"good society\" are possible, or desirable.\n\nWhat challenges the vision of unbridgeable cultural difference that has been codified in western law and in the popular imagination most fundamentally, however, are the actions of ordinary individuals in the everyday life of early colonial societies throughout history and around the world. Soldiers and sailors, who were often young unemployed men, were either pressed into service on trading and naval ships involuntarily (\"shanghaied\"), or signed on out of desperation in the absence of alternatives. Conditions at sea and in the armies were brutal, and troops frequently deserted their masters and escaped to live with the Aboriginal people they encountered on their voyages. Mutinies were common. Historical accounts, and popular culture, often ignore the fact that, for Britain and other colonial powers, there were two problematic populations that had to be managed and controlled in order for imperial designs to be realized: Indigenous peoples, and colonial troops and settlers. Most telling, perhaps, are the histories of fragments of non-European populations, like Hawaiians, and former African slaves who were fleeing the United States of America. Members of these groups, having no access to the protection of imperial power, frequently integrated into the Aboriginal societies whose acceptance their lives depended upon.\n\nThe belief reflected in law that Europeans arrived to lands without organized societies or laws is false. Legal scholar Brian Slattery has stated this flatly: \"All national myths involve a certain amount of distortion,\" Slattery writes, \"but some at least have the virtue of broad historical accuracy, roughly depicting the major forces at work. The myth that underlies much legal thinking about the history of Canada lacks that redeeming feature.\"\n\nTo answer the question of why the option of recognizing the sovereignty of already existing Aboriginal nations, behaving as guests, and attempting to live by Aboriginal laws does not appear to have been taken up by early European jurists and settlers, we have to look to British and Euro-Canadian law and culture, rather than to Aboriginal beliefs and practices.\n\n* * *\n\n. Said 1975: 1.\n\n. McEachern, Reasons 1991: 82. There are, however, debates about this date. Anthropologist Andrea Laforet commented: \"Although McEachern sees this as a fixed date, receding relentlessly into the past, there is an alternative interpretation, i.e. that in 1275 the British defined 86 years as the measure of a very long time\" (Laforet 1993:2). Lawyer William Henderson offers another correction, writing \"...1189 is the date of the accession of Richard I (the 'Lionheart')\" (Henderson, William B. 1991: 14 ftnt:45).\n\n. Hill 1958.\n\n. Canny 1973: 583, writes: \"The questions that we must pose are how, at the mid-sixteenth century, the Irish, a people with whom the English had always had some familiarity, came to be regarded as uncivilized, and what justifications were used for indiscriminate slaying and expropriation?\"\n\n. Ibid.,580.\n\n. \"Transhumance\" refers to the practice of owners and\/or herders moving livestock on a seasonal basis between mountains and lowland pastures. It is contrasted to agricultural settlement where pasture lands remained fixed in one location.\n\n. Historically, when British settlers, first in Ireland, then in the American colonies, and later in British Columbia, looked to history and law for moral and political legitimization in their struggles for independence from the British Crown, they constructed a rhetoric recalling a \"Golden Age\" of \"natural law\" that, their story went, existed prior to the Norman invasion of Britain. See Andrews, Canny, Hair (eds.) 1973; and Knafla 1986(b). This local, or common, law has come to be known by the phrase \"the fundamental laws of all Englishmen,\" which Robert Williams describes as \"the opinion that the Anglo-Saxons of England lived as free and equal citizens under a form of representative government that was inspired by divine principles of natural law and the common rights of all individuals\" (Williams, R. A. 1990:253).\n\n. \"Racialization\" refers to the ideological process whereby biological, genetic, or phenotypical characteristics are employed to classify categories of people. The most common example of the historically and socially constructed nature of racial categories is illustrated by the varying ways Jews have been classified throughout European history, where they have sometimes and in some places been classified as a distinct \"race\" of people, and at other times and in other places, not.\n\n. For an interesting examination of the process of racialization see Ignatiev, Noel 1995. The book examines the role played by identification with the dominant Caucasians and collusion in the oppression of Aboriginal and African-Americans in the historical ascendancy of Irish immigrants in the United States of America.\n\n. The notion that the morality of a person's actions can only be evaluated on the basis of their intentions is a central tenet of Kantian idealist philosophy. This privileges the subject's articulated interpretations and intentions, as compared to the interpretations and consequences experienced by others as a result of the subject's actions. This formulation assumes that individuals and their intentions and actions are not always already embedded in relationships, but can be decontextualized and considered independently. See Miles 1989 for a fuller discussion of \"judging by intentions\" in contexts of racial domination and subordination.\n\n. See Dauenhauer and Dauenhauer (eds.) 1994; Lips 1966; Wickwire 1994.\n\n. Maine (1861)1970: 15.\n\n. Fabian 1983.\n\n. For detailed discussions about mutinies in the early American colonies and the South Pacific, respectively, see Andrews et al (eds.)1973; and Obeysekere 1992.\n\n. See Koppel 1995.\n\n. See Alexander and Glaze 1996; Hudson 1997.\n\n. Slattery 1985: 114.\n**Chapter 4: Beginning at the Beginning**\n\n_Aboriginal peoples, of course, did not go around talking about their rights; mostly, they spoke in a discourse of responsibilities and respect. But that discourse was circulated among themselves. When others came and established\u2014or forced\u2014dominance, it became relevant to speak of rights as a way of negotiating relations._\n\n\u2014Historian Peter Kulchyski, 1994, _Unjust Relations: Aboriginal Rights in Canadian Courts_.\n\n_Regardless of how the inhabitants themselves perceived their connections with the land, in every case a physical and economic relationship necessarily existed. Quite simply, when the English arrived, these people were already there, using lands in accordance with their own needs and their own ways of life, as people everywhere do._\n\n\u2014Legal scholar Kent McNeil, 1989, _Common Law Aboriginal Title_.\n\n**Aboriginal Peoples Were Here**\n\nWhen Britain became engaged in the colonization of the Americas, British legal rules were already in place that addressed several major questions which arose whenever British settlers established themselves in another territory. The \"not-Christian enough\" rationale, developed to cope with the Irish situation, became codified in law in _Calvin's Case_ in 1608, when Britain's Chief Justice Coke articulated what has become known as the infidel rule: \"if a Christian King should conquer a kingdom of an infidel, and bring them under his subjection, there ipso facto the laws of the infidels are abrogated, for that they be not only against Christianity, but against the law of God and of nature.\"\n\nA memorandum of the Privy Council of Great Britain in 1722 consolidated imperial law by setting out rules for establishing British sovereignty in two possible situations. The first option, alternatively referred to as the doctrine of discovery, or the doctrine of occupation, or the doctrine of settlement, was to be applied in circumstances where the land discovered was _terra nullius_ \u2014 uninhabited by human beings. The second option, the doctrine of conquest, was to be applied where Indigenous populations were encountered.\n\nIn the case of _terra nullius_ , Britain simply proclaimed sovereignty by virtue of discovery and British law became, automatically, the law of the land. Where Indigenous populations were found inhabiting the desired land, the law required that British sovereignty had first to be won by military conquest, or achieved through the negotiation of treaties, before colonial law could be superimposed.\n\nOf course, Britain never had colonized and never would colonize an uninhabited land. Therefore, the doctrine of discovery\/occupation\/settlement based in the notion of _terra nullius_ was never concretely applied \"on the ground.\" Rather, already inhabited nations were simply legally _deemed to be uninhabited_ if the people were not Christian, not agricultural, not commercial, not \"sufficiently evolved\" or simply in the way. In British Columbia, the doctrine of _terra nullius_ has historically legitimized the colonial government's failure to enter into treaties with First Nations. The application of the doctrine of conquest to First Nations in British Columbia, which would have required recognition of the fact of their prior occupation, and their status as human beings, _was available within the confines of British imperial law_ but was rejected by colonial governments in British Columbia. When Aboriginal people say today that they have had to go to court to prove they exist, they are speaking not just poetically, but also _literally_.\n\n**Oh, What a Tangled Web We Weave When First We Practice to Deceive...**\n\n_Ordinarily, we think of language as describing a fact or a state of affairs...but a special capacity which is particularly inherent in the law makes things true simply by saying them.... This power is of course the attribute of judges and judicial decisions, among others. The texts of the law are thus quintessentially texts which produce their own effects._\n\n\u2014Sociologist Pierre Bourdieu, 1987, _The Force of Law: Towards a Sociology of the Juridical Field_.\n\nThe early history of British imperialism and its legal expressions constitute the beginning\u2014\"the first step in the intentional construction of meaning\"\u2014of the story which law itself tells about British and Canadian relations with First Nations in British Columbia. In law's imagination, a fundamental inequality was evident and established at the first moment of contact. This hierarchical relationship forms the cornerstone of the legal relations between Aboriginal peoples and Canadian governments, and is at the heart of each and every case of land rights litigation. It constitutes the foundational principles upon which the architecture of the Canadian state is built: the ultimate power of the British Crown to assert its will through simply _declaring_ its sovereignty over foreign lands and peoples, supported, if necessary, by armed force; the fundamental relationship of Euro-Canadian domination and Aboriginal subordination; and, the protection and advancement of the interests of the wealthy and the powerful classes of colonial society.\n\nIn these first moments in the story law tells us\u2014in its assertion of _terra nullius_ \u2014we see the central role played by abstraction and theory in western law and culture: the world is conceived, _and acted upon_ , as if reality can simply be conjured up in whatever form suits the desire of the powerful at the moment. Within this ideology, human beings can be considered, legally, not to exist, and can be treated accordingly. At this most fundamental, common sense level, a study of British and Canadian law in relation to Aboriginal title and rights therefore begins not \"on the ground,\" in concrete observations about different peoples' diverse ways of life, but rather \"in the air,\" in abstract, imagined theory. Hovering, like the sovereign, who embodies this abstraction, over the land. In the practices legitimized by this initiatory unleashing of the \"will to power\" we can see the antiquity of what continues to be a fundamental contradiction, paradox, or deceit in British and Canadian culture: an enduring abstract philosophical commitment to humanism\u2014defined at the most elementary level as the fundamental equality of all human beings\u2014co-exists with an enduring concrete material practice of inequality, and the domination of one group of people by another. It is within this space between the ideal and the real that ideologies of justification are constructed in law, government, imagination, and popular culture. This is the space wherein lies are legitimized and truths silenced. In the histories of colonial laws we can see both the mendacity and the crudeness of the original lie of European supremacy, and the shockingly unsophisticated nature of the edifice built upon it.\n\nThis same space between theory and practice, between avowed principles and lived experience, between the letter and the practice of the law, is one of the sites where Aboriginal peoples historically and contemporarily mount their resistance struggles. First Nations repeatedly expose both the failure of colonial law to obey itself in relation to Aboriginal peoples, and the presence of systemic racial and cultural bias in the justice system. When government policies and practices that systematically discriminate are juxtaposed with the Canadian state's formal commitment to democratic equality, hypocrisy is revealed. In these ways, Aboriginal peoples strike repeated blows to the heart of Canada's liberal self-image and international personality.\n\nSo begins the long dance we call Aboriginal\/nonAboriginal relations in Canada: a tango of domination and subordination, of resistance and repression, of compromise and intransigence, of accommodation and denial, of life and death.\n\n**Eclecticism**\n\n_The English, in fact, were eclectic in their choice of aims and methods; at one time or another they tried almost everything.... Late-comers to the New World, they had an abundance of precedents from which to choose. No other colonial empire employed so wide a range of legal devices in establishing settlements, or allowed so many diverse forms of social, religious, and economic organization._\n\n_Many factors contributed to this diversity: a habit of eclectic borrowing; differences in time, place, and circumstance; differences in personality and purpose; and the absence of sustained interest and continuous effective control by the central government._\n\n\u2014Historian K. R. Andrews, 1973, in _The Westward Enterprise: English Activities in Ireland, the Atlantic and America, 1480-1650_.\n\nIn Canada, France preceded England in settlement, and entered into treaties with the Micmac, Maliseets, Montagnaix-Naskapi, Huron and Abenaki to secure them as allies against both the Iroquois and the English. Throughout the seventeenth century, numerous agreements were entered into between and among Aboriginal peoples and the French and English. Many of these treaties were verbal agreements, solemnized through assembly and gift exchange, and symbolized by, for example, wampum belts. Other treaties were written in French, and later, English, by colonial representatives, and signed by themselves and by Aboriginal representatives.\n\nBefore the arrival of Europeans, different Aboriginal Nations had political and economic agreements with each other regarding trade and commerce, war and peace. These initially served as models for the treaties they entered into with Europeans. An important characteristic of early Indian-European treaty-making was that it conformed to Aboriginal political practices more than to European ones, reflecting the real balance of power that existed when Aboriginal peoples formed both the vast majority of the population, and possessed the necessary knowledge and skills to live in the North American environment. Current legal and political conflicts revolve around issues of competing interpretations of the intentions of the original treaty-makers, the obligations of governments that arise out of the treaties, and whether these obligations have been honoured or not.\n\nIn some cases, Aboriginal peoples, relying primarily on oral histories, argue that their ancestors entered into \"peace and friendship\" agreements with Europeans that allowed the newcomers certain rights to travel and harvest resources. These treaties, they say, were never understood to be final surrenders of lands, rights or political sovereignty. Rather, they were agreements that would be renegotiated as needed to respond to changing conditions, and renewed regularly through deliberations and ceremonies like the ones from which they had originally emerged.\n\nAgainst this, the Crown consistently argues that treaties should only be considered within the context of British law, without regard for Aboriginal legal practices at the time the treaties were made. It claims that the treaties are permanent and binding legal land cessions that also yield political sovereignty; and that the money paid to Indians constituted a trade of money for land, not lease or rental fees, or a toll, or a ceremonial exchange of gifts.\n\nA significant challenge to the Crown's position also emerges from within British and Canadian law and centres on debates about the nature of informed consent. Contractual agreements like treaties are only valid if both parties were fully aware of the terms and the consequences of the contract they entered into. In some cases, Aboriginal representatives are asking courts to consider whether or not Crown representatives made their interpretation of the meaning of treaties clear, verbally, to the Aboriginal peoples they were negotiating with. If not, and if the Aboriginal signatories to the treaties were not English speakers, readers, or writers, what is the legality of those treaties signed with an \"X\" scrawled next to an anglicized Aboriginal name printed by the governments' treaty negotiators?\n\n**Imperialist Competitions in Terra Nullius**\n\n_Englishman, although you have conquered the French, you have not yet conquered us. We are not your slaves. These lakes, these woods and mountains, were left to us by our ancestors...they are our inheritance; and we will part with them to none. Your nation supposes that we, like the white people, cannot live without bread, and pork and beef! But, you ought to know, that He, the Great Spirit and Master of Life, has provided food for us, in these spacious lakes, and on these woody mountains._\n\n\u2014Chippewa leader Minivavana, to an English trader, at Michilimackinac, Fall 1761.\n\nBy the Treaty of Utrecht in 1713, France ceded control over the Canadian Maritime regions to England, while retaining Cape Breton Island, Ile St. Jean and miscellaneous islands in the Gulf of St. Lawrence. In 1759 the British captured Quebec after seven years of war, and the subsequent Peace of Paris, signed in 1760, temporarily sorted out disputes between France, England and Spain. France ceded all its remaining territories in Canada and its territories east of the Mississippi River. Spain ceded Florida to Britain, but retained its territories west of the Mississippi captured from France in 1759. As conflicts between colonial powers intensified, a pattern developed that would continue into the present: European powers, and later federal and provincial governments, engaged in disputes with each other over lands and resources, that excluded Aboriginal peoples. As if they didn't exist.\n\n**Imperialist Nostalgia** **in Terra Nullius**\n\n_\"In the beginning all the world was America.\"_\n\n\u2014Political philosopher John Locke, 1672.\n\nWhile the Treaty of Utrecht and the Peace of Paris were being negotiated and signed, eighteenth-century British society was undergoing rapid industrialization and urbanization. Members of the new middle and professional classes became concerned about the human costs of this process, represented by the plight of former peasants who increasingly formed impoverished rural and urban populations. Europeans increasingly looked to science for guidance in social policy at home, and for rationalizations of colonial policies abroad. Various treatises emerged that claimed to explain the whole history of all humankind in systematic and coherent ways, and that all began with theories about human origins. Although available information about Aboriginal ways of life was principally limited to explorers' and traders' idiosyncratic and unsystematic records, the authors of the \"grand theories\" of Enlightenment Europe usually began with an evocation of so-called \"primitive life in the New World,\" meant to illustrate \"raw human nature\" and \"original human society.\" For romantics like Jean Jacques Rousseau, the \"state of nature\" was to be found in a simple Garden of Eden, and Indigenous peoples in America, imagined as \"noble savages,\" provided Europeans, steeped in the popular \"common sense\" of evolutionism, with appealing fantasies of their own primitive origins.\n\nAt the opposite extreme, for philosophers like Thomas Hobbes, the \"state of nature\" was an instinct-driven war of all against all. The establishment of society, and collective survival, depended upon the imposition of a sovereign's power and government to contain and control an essentially competitive and aggressive \"human nature.\" The Hobbesian \"state of nature\" that he, too, claimed was being lived in eighteenth century Indigenous societies in the Americas, was one of constant warfare, an absence of law or government, and a life that was \"nasty, brutish and short.\" Whatever the western rendition of \"primitive life,\" Aboriginal peoples occupied one or another of these \"savage slots\" in the European imagination: noble child of harmonious nature, or dangerous child of animal instinct. In whichever rendition, Aboriginal peoples were used as a _tabula rasa_ upon which Europeans could project whatever theoretical or cultural fashion seized the fantasy of the day. In both renditions, the reality of the ways of life, lived experiences and self representations of Aboriginal peoples are sacrificed to the European imagination. These \"grand theories\" competed with each other for acceptance amongst professional and popular audiences.\n\n**Enough and As Good**\n\nJohn Locke is usually identified as the most significant political theorist of this imperial era in Britain. Locke gathered together arguments circulating during the early seventeenth century and set them out in theories that would serve many of the later legal and political justifications of European seizure of property in North America. Locke, in the fashion of the times, argued that Aboriginal peoples lived in a pre-political state of nature representative of the first stage in universal human evolutionary development. Important characteristics of this early developmental phase included a hunting and gathering economy with no perception of established systems of property or government. Europe correspondingly represented the most advanced stage of evolutionary development. This was proven by the practice of agriculture; the existence of a legally codified system of property with written laws; a bureaucratic government; and a capitalist economy in which all exchanges of commodities in the market place between people are conducted through the common medium of money, the value of which is ultimately guaranteed by the state. The sovereign's image stamped on coinage symbolically represents the presence of the hovering sovereign mediating every monetary exchange between citizens.\n\nLocke went on to theorize that Aboriginal peoples therefore had property rights only \"in the products of their labour: the fruit they gather, the deer they catch and the corn they pick.\" In this they are governed by a \"natural law,\" Locke mused, that says each individual may appropriate what nature offers up without consultation with, or consent by, anyone else \"as long as there is enough and as good left in common for others.\" Locke reasoned that Europeans would increase the productivity of the land through agriculture and were therefore justified in appropriating Aboriginal lands without consent. Since this process was governed by the \"natural law\" of evolutionary development, if Aboriginal peoples inhibited European settlement, they would be in violation of natural law and could justifiably be eliminated. Philosopher James Tully concludes that, if Locke had recognized Aboriginal forms of property, and Aboriginal peoples as equal to Europeans, then \"settlement in America would have been illegitimate by his own criteria of 'enough and as good,' and consent would have been required.\"\n\n**Nation to Nation. Words to Paper.**\n\n_And whereas it is just and reasonable, and essential to Our Interest and the Security of Our Colonies, that the several Nations or Tribes of Indians, with whom We are connected, and who live under Our Protection, should not be molested or disturbed in the Possession of such Parts of Our Dominions and Territories as, not having been ceded to, or purchased by Us, are reserved to them, or any of them, as their Hunting Grounds._\n\n\u2014\"The Indian Provisions,\" _Royal Proclamation of 1763_.\n\nThe prevailing conditions in eastern Canada during the early 1760s were very complex, and formed the historical context in which the Royal Proclamation of 1763\u2014a document that continues to be the subject of legal and political debates today\u2014was issued. The British were faced with a number of problems. Their relationship with Indian nations who had been allies of the French was precarious. During the Seven Years War between Britain and France there had been active competition for Indian allies, and British army commanders in the field had been generous in their purchases of furs and exchanges of ammunition. After the war, London cut back these funds considerably and field commanders found themselves unable to honour commitments they had previously made. Entrepreneurial settlers and fur traders were making independent incursions on Indian lands and resources, impeding the development of the Crown monopoly. On the international front, Britain's hegemony in North America was still threatened by the Russians from the north, and the Spanish from the southwest. The Royal Proclamation of 1763 attempted to address all of these issues.\n\nCompeting interpretations of the historical, legal and political implications of the Royal Proclamation of 1763 have occupied a central place in Aboriginal title and rights discourse and litigation over the past two centuries. A key debate surrounds the question of whether or not the Royal Proclamation should be interpreted as having _recognized_ already existing Aboriginal rights, or as having created these rights. The former, which has since come to be known as the \"inherent rights\" position, is based on the assumption that Aboriginal rights today flow continuously from the sovereignty of nations that pre-existed European colonization. The latter\u2014the \"delegated rights\" position\u2014argues that, in law, no rights can exist except those created by the will of the sovereign. Hence, contemporary Aboriginal rights could only be those that a sovereign, or a court, or a parliament chooses to assign. Such rights would always be based in the desires of these bodies, and not in Indigenous history. And what the sovereign creates, the sovereign can also destroy or dismiss, at his pleasure.\n\nAnalysts agree, however, on some points. The Royal Proclamation differentiated Indian title to land in North America from non-Indian title in five significant ways. First, it reflected the fact that, in 1763, the Crown understood that it must at least formally recognize the legitimacy of, and negotiate on equal political grounds with, \"Indian _Nations_.\" The Royal Proclamation is guided by the doctrine of conquest set out in the Memorandum of the Privy Council in 1722.\n\nSecond, under the Proclamation, Indian title is defined as being collectively or communally, not individually, held. Furthermore, this title is limited to _use rights_ , like hunting and fishing, that are comparable to perpetual leases rather than to ownership. The uses in question must be only those practiced before European arrival. The Royal Proclamation therefore acknowledges First Nations as having some form of interest in their lands and resources. Whether, by legal interpretations, or cultural assumptions, this \"Indian title\" is of equal value to \"Crown title,\" or is some less valued form of property right that constitutes \"a burden\" on the Crown's title, is a subject of ongoing controversy. Third, the Proclamation dictates that Indian title can be only be transferred to the Crown. A number of debates have arisen concerning both the intent and the consequences of this clause. Many historians have argued that the insertion of this clause was primarily motivated by a humanitarian, paternalistic concern to protect Indians from unscrupulous frontier land speculators. Others argue that this clause reflects a power struggle between the Crown as state, and corporate and private interests, for monopoly over lands and resources. Still others focus on competition between international imperial interests and those of the emerging local, colonial governments.\n\nFourth, the Proclamation identifies Indians as \"Nations or Tribes\" and guarantees Indians the _protection_ of the Crown. This would later come to be expressed as a \"fiduciary duty.\" Ongoing debates involve whether or not this fiduciary duty and the obligations it entails should be interpreted as resulting from a negotiated, trust-like agreement between equal parties, or as reflecting a relationship of dependency analogous to the parent\/child relationship which is also classified, in law, as a \"fiduciary duty.\"\n\nFifth, the Royal Proclamation requires that Indigenous land rights can only be surrendered at a public assembly at which Indians give their consent. This issue comes up particularly in treaty litigation where some Aboriginal claimants argue that, whatever negotiations and agreements are claimed by the Crown to have been reached with their ancestors, the representatives who signed the treaties had not been mandated to do so by their constituencies, and there was insufficient knowledge of and\/or participation by the required majority of the Indigenous land holders to render the treaty legal.\n\nIn these interpretations, the possibility that Aboriginal peoples may not have wanted to sell, cede or by treaty give sovereignty or lands to anyone is not provided for in the written words of the Royal Proclamation of 1763. Nor is the possibility entertained that Aboriginal sovereignty could remain dominant, or could co-exist with Crown sovereignty.\n\nAnother significant debate about the Royal Proclamation of 1763 centres on the degree to which both British and Aboriginal intentions and understandings should inform contemporary interpretations. United States Native American legal scholar Robert Williams Jr. argues that the proclamation's \"two goals\u2014facilitating the profitable Indian trade and protecting Indian lands to prevent costly hostilities\u2014were viewed as complementary halves of a self-serving colonial policy put forward by mercantilist interests and their advocates in the British Ministry at Whitehall in the 1760s.\" \"Its discourse,\" Williams continues, \"was one of interest and expediency as articulated by armchair empire builders in the Old World, who viewed the honoring of promises made to savages in the New World as the cheapest, most 'expedient' means of containing both frontier defense costs and inland expansion by British American colonists.\"\n\nCanadian First Nations legal scholar, John Borrows, reiterates Williams' points about British intentions, but argues that interpretations of the Royal Proclamation limited to the words written in the document alone do not take into account the negotiations, gift-exchanges and other events that took place surrounding the production of the written text and recorded in First Nations oral traditions that reveal _their_ intentions and interpretations. Ignoring the oral history of the Royal Proclamation, Borrows charges, \"privileges one culture's practice over another.\" He goes on to argue that his inter-cultural interpretation of the Royal Proclamation suggests that \"The Proclamation uncomfortably straddled the contradictory aspirations of the Crown and First Nations when its wording recognized Aboriginal rights to land by outlining a policy that was designed to extinguish these rights.... The different objectives that First Nations and the Crown had in the formulation of the principles surrounding the Proclamation is the reason for the different visions embedded within its text. Britain was attempting to secure territory and jurisdiction through the Proclamation, while First Nations were concerned with preserving their lands and sovereignty.\"\n\n* * *\n\n. Calvin's Case 1608 cited in Walters 1993: 360.\n\n. For a thorough, and very readable, account of the application of _terra nullius_ in Canada, see Richardson 1993.\n\n. For a comprehensive analysis of this argument see Asch and Macklem 1991. The authors conclude: \"We believe it abhorrent that Canada was constituted in part by reliance on a belief in the inequality of peoples and that such a belief continues to inform political and legal practice in 1991\" (510).\n\n. For a very thorough articulation of this argument see Derrida 1992. See also Hunt 1993; and Sayer 1987.\n\n. Aboriginal legal scholar James Youngblood Henderson, for example, explains that \"Canadian law is not impersonal but racially biased; its legitimacy is threatened if not destroyed by its denial of order and freedom to Aboriginal people...\" Henderson and Henderson 1985: 186.\n\n. See, for example, Hamilton and Sinclair (eds.) 1991; and Turpel 1991(c).\n\n. See Dyck 1991.\n\n. See Venne 1997; and Chamberlin 1997.\n\n. Dickason 1992.\n\n. Henry 1809 quoted in Jones 1982; and Slattery 1985: 119.\n\n. The phrase, \"Imperialist Nostalgia,\" is taken from Rosaldo 1989, who defines it as the yearning for that which one has destroyed.\n\n. Laslett (ed.) 1964.\n\n. Francis 1992; and Trouhillot 1991.\n\n. Tully 1993(b).\n\n. Tully 1993(c): 10.\n\n. There have been a number of published versions of the Royal Proclamation of 1763, and wording varies in different publications. The quotation cited here is taken from Chief Justice Allan McEachern's 1991 _Reasons for Judgment_ , in _Delgamuukw v. R._ , 313.\n\n. For a survey of debates about both the historical context in which the Royal Proclamation was negotiated and the legal consequences for Canadian Aboriginal peoples in the present see Borrows 1992, 1997; Slattery 1991; Walters 1993. For analysis from a Native American (U.S.A.) perspective, see Williams, R. A. 1990(b).\n\n. The intended meaning of this language has also been hotly contested, with some arguing that the British Crown thereby recognized the sovereignty of Indian Nations, and others asserting that the words were used rhetorically or insincerely by the British to appease the Indians.\n\n. Williams, R. A. 1990: 237.\n\n. Borrows 1997: 160-161.\n**P ART III**\n\n**T ERRA INCOGNITA**\n\n(UNKNOWN LAND)\n**Chapter 5: The Great Chain of Precedent**\n\n_Suddenly, even the most hardened land-market capitalist assumed the mantle of zealous advocate of the Indians' natural-law right to engage in unregulated real estate transactions. Neither the King, nor the landed colonies \"owned\" the lands on the frontier, argued these speculators. The Indian tribes occupied these lands as free and sovereign peoples. By natural law, the Indians could therefore sell their rights to the land to whomever they pleased, the Proclamation of 1763 and the landed colonies' charter claims notwithstanding._\n\n\u2014Legal scholar Robert A. Williams, Jr., 1990, _The American Indian in Western Legal Thought: Discourses of Conquest_.\n\n**Honour Among Thieves. South of the Border**\n\nIn British law, one of the principal means that judges use to reach decisions about particular cases is through the use of precedent. Precedents are the decisions reached by previous judges in similar cases. A contemporary judge compares the facts before her or him, searches for cases that dealt with similar fact patterns, and interprets and applies the reasoning and findings of judges in those previous cases to the one at hand. This is referred to in legal parlance as the \"doctrine of _stare decisis_.\" It is based on the premise that fairness and equality before the law requires that like cases should be decided alike. Reliance on precedent is often pointed to as an inherently conservative characteristic of law, whereby justification must always be anchored in the past. However, contemporary critics argue that the notion that past precedents limit and determine present judgments is frequently overstated by representatives of the judiciary. Anthropologist Michael Asch and Law Professor Catherine Bell, for example, argue that \"it is not precedent itself that binds, but judicial interpretation of the past and its relevance to the present.... Adopting interpretive strategies, a judge chooses one precedent in favour of another, appearing to find, rather than create law. The appearance of finding is important because it deflects charges of result-oriented reasoning and judicial legislation.\" Legal scholars Gerald Torres and Kathryn Milun, point out that the rule of precedent serves more importantly to consolidate law's desire to define the future. They write, \"Law, by drawing constantly on precedent to develop itself, strives to collapse linear temporal sequence by bringing the past forward and, by creating a new precedent, drawing the future into itself.\" In a ritual practice judges call \"citing the authorities,\" _Reasons for Judgment_ are frequently written as narratives linking a chronology of precedents to the case at hand through descriptions of salient similarities of fact and argument. Contemporary Canadian judges ruling on Aboriginal title and rights cases usually begin their \"precedents narratives\" in eighteenth-century post-colonial U.S.A.\n\nIn 1776 the American colonies declared independence from Britain. In the ensuing years, three distinguishable factions emerged to dispute issues of land rights and lawful methods of acquisition of Indian lands. A faction of the American population that remained pro-British continued, after Independence, to argue for a literal interpretation of the Royal Proclamation of 1763, and asserted that the British Crown alone retained the prerogative to negotiate with and acquire land cessions from Indians. Another faction, consisting of legislators and political leaders of Virginia and the other colonies, argued that they held controlling rights to Indian lands on the basis of their Crown charters, having \"inherited\" the sovereign's prerogatives previously held by the British Crown and set out in the Royal Proclamation of 1763. Finally, a large group of frontier speculators claimed that, under natural law and natural right, the Indians themselves, as \"sovereign princes of the soil,\" could sell their land to whomever they wished. Philosopher John Locke became involved in the management and exploitation of the British colonies on the eastern seaboard of America, and was an influential theorist in post-Revolutionary United States. Processes and events there codified Locke's theories into laws that formed the basis of precedents still employed today. This was the context in which three decisive legal judgments were rendered by Chief Justice Marshall of the United States Supreme Court.\n\nThe first case, _Fletcher v. Peck_ , was heard in 1810. The State of Georgia had granted land to the New England Mississippi Land Company. The Company then divided and resold the land to a number of individuals, including the Plaintiff, Robert Fletcher. The Defendant, Peck, attempted to interfere with Fletcher's exercise of ownership, claiming to have acquired the same land directly from its original, Indigenous, owners. Chief Justice Marshall ruled, on the basis of the Royal Proclamation of 1763, that the State of Georgia's sale to Fletcher was illegal because the lands in question had never been surrendered by the Indians to either the British Crown, the government of the United States of America, or the State of Georgia. Marshall held that \"Indian title\" could only legitimately be extinguished by a European-derived sovereign, so neither Fletcher nor Peck had acquired lawful title. Another U.S. Supreme Court judge, Justice Johnson, dissented from Marshall's decision arguing that Indians \"retained absolute proprietorship of their soil\" which could be extinguished only by conquest or purchase. The significance of Chief Justice Marshall's ruling in _Fletcher v. Peck_ has manifested itself as the decision has been used as a precedent by contemporary Canadian judges in decisions that uphold the Royal Proclamation of 1763's dictum that Aboriginal title may only be extinguished by the Crown, and not by lower levels of government, corporations, or private citizens.\n\nThe second case, _Johnson v. McIntosh_ , was heard in 1823. Johnson claimed that he had inherited title to a tract of land from his father who had purchased it from the Piankeshaw and Illinois Indians. McIntosh said he had purchased the same lot from the U.S. federal government, who claimed they had acquired the land from the same Indians at a later date; that is, after Johnson's father said he had purchased it. Chief Justice Marshall found in favour of McIntosh, ruling that the federal government alone had the exclusive right to acquire Indian title, therefore neither Johnson's father nor the Piankeshaw and Illinois Indians could legally buy or sell Indian land. Marshall argued that his ruling was based in law, and not necessarily in justice. He wrote that his decision was determined by, \"History, and the decisions made and enforced by those Europeans who invaded America.\" Marshall relied on the doctrine of discovery\/occupation\/settlement, and the assumption of _terra nullius_ to defend his position, arguing that Crown title was grounded in the voyages of discovery made by the Cabots during the late fifteenth century. This second case in \"the Marshall trilogy,\" _Johnson v. McIntosh_ , is the decision most frequently selected as a precedent for application by contemporary Canadian judges to support judgments against Aboriginal claimants, on the basis that the Crown had an unfettered right to declare sovereignty over territories Britain deemed _terra nullius_.\n\nThe third and final case, _Worcester v. Georgia_ , was decided in 1832. The State of Georgia had attempted to enact jurisdiction over the Cherokee Nation by annexing its territory, annulling its constitution and laws, and requiring whites to obtain state permission before entering Cherokee territory. A white missionary, Samuel Worcester, after being arrested for refusing to comply with this statute, challenged the state's jurisdiction. These same lands, and jurisdictional arrangements, had been the subject of a treaty between the federal government and the Cherokee. Worcester argued therefore that the State of Georgia had no legal right to keep him off the Cherokee reservation.\n\nChief Justice Marshall ruled in favour of Worcester, saying that the doctrine of discovery\/occupation\/settlement was relevant only to governing relations between European nations, and not relations between states and the federal government. It yields to the Crown, Marshall wrote, only an exclusive right to acquire Aboriginal title as set out in the Royal Proclamation of 1763. It does not proscribe the terms of such acquisition, or what governmental powers may flow from the surrender of Aboriginal title to the Crown. In other words, Marshall ruled that the State of Georgia could not assume it had automatically inherited whatever powers the federal government may have acquired when it exercised its exclusive prerogative to extinguish Aboriginal title. More importantly, in his _Reasons for Judgment_ in _Worcester v. Georgia_ , Chief Justice Marshall argued that the doctrine of discovery\/occupation\/settlement did not, in itself, necessarily rule out the possibility that some form of negotiations between Aboriginal title holders and Crown representatives may still be required in order for the extinguishment of Aboriginal title to be legitimate. This decision provides a precedent that _could_ be interpreted to mean that Crown extinguishment of Native title and assumption of jurisdiction could require Aboriginal consent in order to take legal effect. Chief Justice Marshall's decision in _Worcester v. Georgia_ , however, has not been selected by Canadian judges for use as a precedent.\n\nNone of the Indigenous peoples whose lands and rights were at issue in the litigation described above were represented in court. As if they didn't exist.\n\n**Honour Among Thieves: North of the Border**\n\n_By the treaty of 1873 the Indian inhabitants ceded and released the territory in dispute, in order that it might be opened up for settlement, immigration, and such other purpose as to Her Majesty might seem fit, to the Government of the Dominion of Canada, for the Queen and Her successors forever.... The treaty leaves the Indians no right whatever to the timber growing upon the land which they gave up, which is now fully vested in the Crown, all revenues derivable from the sale of such portions of it as are situated within the boundaries of Ontario being the property of that Province...[that] possesses exclusive power to regulate the Indians' privilege of hunting and fishing...._\n\n\u2014Lord Watson, Judicial Committee of the Privy Council of the British House of Lords, 1888.\n\nIn 1888, the first significant legal decision involving issues of Aboriginal title and rights in Canada was heard by the Judicial Committee of the Privy Council of the British House of Lords, which was, until 1949, the highest court to which Canadian cases could be appealed. The _St. Catherine's Milling and Lumber_ case provided a precedent that would be cited in every subsequent Aboriginal title case in Canada up to and including the present.\n\nThe case involved a dispute between the federal government of Canada, the Province of Ontario, and a private corporation: St. Catherine's Milling and Lumber Co. The federal government had granted the company a permit to cut lumber on land they claimed had been surrendered to them by the Ojibway Nation under Treaty 3, signed in 1873. The Province of Ontario, however, charged St. Catherine's Milling and Lumber Co. with taking lumber without a valid permit. The federal government argued that the Ojibway had held full title to their lands until they surrendered their Aboriginal title to the Crown under the terms and conditions of Treaty 3, that included some monetary payments. To support their position, the federal government of Canada arguedthat Treaty 3 reflected the Royal Proclamation of 1763's recognition of full Aboriginal ownership and jurisdiction; which, having been surrendered to the Crown, now gave the federal government the right to issue permits to private companies like St. Catherine's Milling and Lumber. The Province of Ontario countered with the argument that, prior to the signing of Treaty 3, the underlying title to all the land at issue was not held by Aboriginal title, but was already owned by the hovering sovereign who acquired it by virtue of the doctrine of discovery\/occupation\/settlement set out in the Memorandum of the Privy Council in 1722. Treaty 3, according to the provincial argument, simply consolidated the extinguishment of whatever vague interests in the land the Ojibway _might_ have had prior to contact with Britain, that might have remained a \"burden on the Crown's title\" after the Royal Proclamation of 1763 was issued, rendering the lands in question unencumbered Crown land. The _British North America Act_ that consolidated Canadian Confederation in 1867, the Province of Ontario's argument continued, had transferred Crown land to provincial jurisdiction.\n\nThe Supreme Court of Canada agreed with the Province of Ontario, stating that \"the tenure of the Indians was a personal and usufructuary right dependent upon the goodwill of the sovereign.\" In other words, even if some form of Aboriginal title had pre-existed Britain's \"discovery\" of North America or survived the Royal Proclamation of 1763, it was a type of property ownership that was inferior to title in fee simple, recognized as paramount by British law. Regardless, Aboriginal rights of any sort, the Supreme Court of Canada ruled, were only those created\u2014not recognized\u2014by the Crown. The federal government appealed the court's decision to the Judicial Committee of the Privy Council; which, in turn, upheld the Supreme Court of Canada's ruling. Lord Watson cited the precedent set by Chief Justice Marshall's 1823 decision in _Johnson v. McIntosh_ upholding the doctrine of discovery\/occupation\/ settlement based in the concept of _terra nullius_ , as authority for the Lords' decision.\n\nMuch of the legal argument in the case of _St. Catherine's Milling and Lumber Co. v. R_. revolved around competing interpretations of what form of Aboriginal title or interest the Royal Proclamation of 1763 recognized or created. Lord Watson concluded that while \"there was a great deal of learned discussion at the Bar with respect to the precise quality of the Indian right...their Lordships do not consider it necessary to express any opinion upon the point.\" In effect, however, the Privy Council's decision affirmed that Aboriginal title was a mere \"burden\" on the hovering sovereign's underlying title to all the land.\n\nLord Watson's ruling that the only Aboriginal rights that could be recognized in law were those granted, or taken away, by the Crown, marked the rise to prominence of the theory of \"legal positivism\" in British and Canadian jurisprudence. Legal positivism is a term used to describe the \"tendency to treat jurisprudence as an exact science, a rational process that consists of identifiable data and rules,\" modelled on the natural or physical sciences. Positivism in social theory is based on the assumption that human social life can be studied using the same methods as those employed by the natural and physical sciences. Its adherents claim that \"objective\" knowledge about social reality that is free of any and all bias can be obtained by trained researchers. According to positivist theory, social scientists should first use the five senses recognized by western culture\u2014sight, sound, taste, touch and smell\u2014to capture data on human behaviour. The next step is to organize the data according to prescribed categories. Finally, research findings should be explained by reference to theoretical frameworks developed by previous generations of social scientists. The sense of sight and the practice of systematic observation is the most privileged source of knowledge in positivist social research: \"seeing is believing.\" Studies in behavioural psychology that place subjects in experimental environments and monitor how they respond to particular stimuli\u2014say, for example, fear\u2014represent the type of positivist social research that has been popularized, and that most people are familiar with. Conclusions are based on how a majority of research subjects respond to the particular stimuli. So, for example, positivist researchers conclude that \"humans respond to fear by either fight or flight.\" Of course, this does not tell us anything about what frightens who, where, when, or why. Nor does it tell us who can fight and who cannot, or where anyone flees to. We will never know how people who were not research subjects responded, or whether there are third or fourth alternatives that have escaped the research design.\n\nLegal positivism dispenses with the requirement that researchers must study phenomena that exist independently of the observer, _and substitutes law itself for \"objective\" reality_ , and judicial decision-making for scientific methodology. Simply put, the law creates reality that is real because it has been created by the law. Hence, regardless of what might actually exist, \"on the ground,\" under the doctrine of legal positivism, the Crown creates and extinguishes Aboriginal title and rights \"at its pleasure.\" Critics of legal positivism argue that jurisprudence is better understood as resulting from the accumulation of judges' interpretations of evidence and arguments over time in specific social contexts, and that the arts, literature and humanities provide more useful models for understanding law than the hard sciences.\n\nThe Ojibway, whose lands and histories were the subject of the dispute in the _St. Catherine's Milling and Lumber v. R._ case were neither consulted nor represented in court. As if they didn't exist.\n\n**Honour Among Thieves: in Africa**\n\n_The estimation of the rights of aboriginal tribes is always inherently difficult. Some tribes are so low in the scale of social organization that their usages and conceptions of rights and duties are not to be reconciled with the institutions or legal ideas of civilized society. Such a gulf cannot be bridged. It would be idle to impute such people some shadow of the rights known to our law and then to transmute it into the substance of transferable rights of property as we know them...._\n\n_On the other hand, there are indigenous peoples whose legal conceptions, though differently developed, are hardly less precise than our own. When once they have been studied and understood they are no less enforceable than rights arising under English law. Between the two there is a wide tract of much ethnological interest, but the position of the natives of Southern Rhodesia within it is very uncertain; clearly they approximate rather to the lower than to the higher limit._\n\n\u2014Lord Sumner, Judicial Committee of the Privy Council of the British House of Lords, 1919.\n\nWhen deciding Aboriginal title and rights cases, Canadian judges have available to them a wide range of historic precedents, diverse interpretations of these precedents, and variously reasoned arguments to select from in constructing their own rulings. They may draw on arguments waged, and decisions made, throughout the history and geography of the British Empire-cum-Commonwealth. The ruling quoted above in an African case\u2014 _Re: Southern Rhodesia_ \u2014along with Chief Justice Marshall's American cases, and the Canadian _St. Catherine's Milling and Lumber_ judgment, have come to constitute an unholy trinity of precedents repeatedly summoned by contemporary jurists.\n\nThe _Re: Southern Rhodesia_ judgment sets out the principle that, in order to determine the legitimacy of Indigenous peoples' land rights claims, imperial courts should first assess the claimants' position on the ladder of evolutionary progress hypothesized by nineteenth-century evolutionary social theory. Lord Sumner's first option for those found to be \"low on the scale,\" is a repetition of the Privy Council's Memorandum of 1722's doctrine of discovery\/occupation\/ settlement justified by the assumption of _terra nullius_ , itself a repetition of Lord Coke's judgement in _Calvin's Case_ in 1608. In 1722, whether or not British imperial law would classify Indigenous peoples as fully human beings was considered a religious question: were they Christians or infidels? In setting out his test in _Re: Southern Rhodesia_ , Lord Sumner relied on secular social theory, re-articulating and re-legitimating the now archaic seeming assumptions of 1722 in the pseudo-scientific language of Social Darwinist evolutionism that had become entrenched by 1919. Social Darwinism was an attempt to apply Charles Darwin's theories of evolution in the plant and animal worlds to human history. Simply put, some Indigenous peoples could be classified, according to the Privy Council, as belonging to a lower order of human being than the British. On the basis of this abstract act of classification they could be deemed, in law, not to exist. Their lands and resources, and their rights to govern themselves, could be eliminated: through the force of law.\n\nLord Sumner's option for those Indigenous peoples deemed more \"highly evolved\" reflects the second clause in the Privy Council's Memorandum of 1722. Under the doctrine of conquest, since Indigenous peoples were recognized as existing human beings, British domination had to be achieved through military conquest and\/or the negotiation of treaties. Indigenous legal codes could continue to be in force after the assertion of British sovereignty, until they were extinguished by the stroke of a pen, or by the cannons of the British navy. Following Sir Henry Maine, Lord Sumner elaborated evolutionary theory, applying its categories to legal codes and land tenure systems. He identified criteria by which courts could ascertain which Indigenous land tenure systems should be deemed worthy of being honoured by the British Crown. According to Lord Sumner, individual rather than tribal or communal ownership marked the difference between \"civilized\" and \"savage\" property law.\n\nNot all judges think alike. In 1921 Lord Sumner's 1919 decision in _Re: Southern Rhodesia_ was criticized and modified by Viscount Haldane, who issued a ruling in another African case before the Judicial Committee of the Privy Council. In _Amodu Tijani v. Southern Nigeria_ , Haldane argued that Indigenous land tenure systems should not be judged by British standards, but rather should be recognized and understood on their own terms: \"In interpreting the native title to land, not only in Southern Nigeria, but other parts of the British Empire, much caution is essential,\" Haldane wrote. \"There is a tendency, operating at times unconsciously, to render that title conceptually in terms which are appropriate only to systems which have grown up under English law. But this tendency has to be held in check closely.\"\n\nThe similarities and differences between Lord Sumner's ruling and Viscount Haldane's can be read as an historic illustration of the interconnections between law, history, anthropology and public opinion that have shaped Indigenous land rights litigation since the early days of British imperial expansion. Legal decisions in these cases, of necessity, rely on and reflect interpretations of history, culture, human nature and morality.\n\nBy the early years of the twentieth century, professional anthropologists for the most part had rejected the evolutionary theory of their nineteenth-century founding ancestors like Sir Henry Maine, Lewis Morgan, and Edward Tylor. The leading anthropologists of the 1900s-1930s, including Franz Boas, Bronislaw Malinowski, and H. R. Radcliffe-Brown, all argued for one version or another of positions based in cultural relativism, defined at its most essential level as respect for the fundamental equality of all human cultures, and the right of each culture to be judged on its own terms and not on the basis of another culture's evaluative criteria. This was not only a moral and a political stance, but reflected conclusions which had emerged as the discipline grew to include more professionally trained observers who spent longer periods of time living with non-European peoples. The empirical data collected, and the lived experience of fieldworkers, challenged the abstractions of the \"armchair academics,\" who, a generation before, had hypothesized the grand theories of universal human evolution from their ivory towers. Theories of scientific racism and Social Darwinist evolutionism could not sustain \"on the ground\" scrutiny. Many Indigenous peoples' ways of life and histories simply didn't conform to the evolutionary models. Aboriginal Australians, for example, valued little in the way of technology and material goods, but lived within highly complex social structures governed by intricate kinship relations, led rich spiritual lives guided by complex and sophisticated cosmologies, and produced abstract art. Evolutionary theory had hypothesized incorrectly that they should be completely consumed by satisfying survival needs and be incapable of abstract thought. Northwest Coast peoples in North America were neither agricultural nor industrial, yet they lived in a hierarchical social structure, and accumulated and stored surplus wealth which, evolutionary theory claimed, people classified as hunters and fishers should not be doing. Anthropologist Marshall Sahlins has challenged the notion that non-industrial peoples' lives were taken up with the daily struggle of wresting subsistence from nature. He argues that industrial capitalism requires the average person to expend more hours of labour per day to meet basic subsistence needs than is demanded of participants in hunting, gathering and fishing economies.\n\nIn his ruling in _Amodu Tijani v. Southern Nigeria_ Viscount Haldane adopted a position of cultural relativism, launching an explicit critique of Lord Sumner's ethnocentric evolutionism. Haldane argued that Aboriginal title, \"may not be that of the individual, as in this country it nearly always is in some form, but may be that of a community.... Such title...must be presumed to have continued to exist unless the contrary is established by the context or the circumstances,\" he concluded. While Haldane and Sumner differed on some points, neither questioned the fundamental premises that the British Crown had legitimate claims to sovereignty over the Americas; that the hovering sovereign held underlying title to all the land; and that colonial courts should determine what form of Aboriginal title could be recognized\u2014or created\u2014and by what criteria. The Royal Proclamation of 1763 defined Aboriginal title as communal and not individual. Lord Watson in _St. Catherine's Milling and Lumber_ in 1888 had declared that Aboriginal title was created by the Crown in whatever image the Crown chose. Lord Sumner had, in 1919, defined legally recognizable Aboriginal title as individual and not communal. Viscount Haldane in 1921 allowed that Aboriginal title could be communal and not individual. Both Sumner's and Haldane's rulings endorsed the dictum that legal recognition or non-recognition of Aboriginal title should be determined by the nature of Aboriginal societies and land tenure systems, _as these were understood by the court_. The microscope was resolutely trained on Aboriginal claimants who would be required to represent their cultures and laws in one or another framework determined by legal adaptations of European social theory. Questions about the moral, political or legal legitimacy of British colonialism were deemed by law to have been resolved and to not require further discussion. These issues were banished: silenced in legal conversations, and evicted from colonial courthouses. Their absence, however, continued to hover, over the sovereign.\n\nIn the legal disputes that followed Sumner's and Haldane's rulings, twentieth-century Canadian judges would select from any one of these precedents, or others, or some combination thereof, as their rationale of choice in rendering decisions in Aboriginal title and rights litigation. Various courts and judges would employ diverse and often contradictory understandings of a m\u00e9lange of anthropological and historical theories for analyzing Indigenous societies, It would not be until the closing decades of this century that the simple fact that \"when the British arrived, these people were already there, using lands in accordance with their own needs and their own ways of life, as people everywhere do,\" would figure prominently in debates about the grounds for legal recognition of Aboriginal or Crown sovereignty in Canada.\n\n* * *\n\n. Bell and Asch 1997:39-40. See also Postema 1991.\n\n. Torres and Milun 1990.\n\n. See Bell and Asch 1997; Doyle-Bedwell 1993; Macklem 1991; Slattery 1979; Williams R. A. 1990.\n\n. _Fletcher v. Peck_ , 10 U.S. (6 Cranch) 87 (1810) at 146.\n\n. _Johnson v. M'Intosh_ 21 U.S. (8 Wheat) 543 (1823) at 573.\n\n. Macklem 1991: 400.\n\n. _Worcester v. Georgia_ , 31 U.S. (6 Pet.) 515 (1832) at 543.\n\n. quoted in Kulchyski 1994: 30.\n\n. _St. Catherine's Milling and Lumber Co. v. R_. (1885) 10 or 196 (Ont.Ch.); (1886) 13 OAR 148 (Ont.CAO); (1887) SCR 577(SCC); (1888) 14 AC 46 (PC).\n\n. A \"usufructuary right\" means the right to use property owned by another, as long as that use is permitted by the rightful owner, and does not interfere with the owner's rights or desires.\n\n. \"Title in fee simple\" refers to the most privileged form of individual ownership of private property protected by Canadian law. Most Canadian homeowners hold \"title in fee simple\" to their property, subject of course to the good graces of their banks and mortgage companies.\n\n. quoted in Kulchyski 1994: 30.\n\n. Cotterrell 1984:10 quoted in Burtch 1992: 3.\n\n. For a more thorough discussion of the _St. Catherine's Milling and Lumber Company_ case, see Slattery 1987; Macklem 1991.\n\n. White 1985, 1990.\n\n. _Re Southern Rhodesia_ [1919] A.C. 211 at 233-4.\n\n. _Amodu Tijani v. Southern Nigeria_ [1921] 2 A.C. 399(p.c.) at 403.\n\n. See Stocking 1987, 1991, 1992, 1995.\n\n. See Vincent 1990.\n\n. Sahlins 1972. See also Lee and DeVore 1966.\n\n. McNeil 1989.\n**Chapter 6: White Papers and Legal Tests**\n\n_Legal orders may embody asymmetrical power relations, but power is always an interactional process. Dominant groups enjoy legally protected privileges, but they are also constrained by the law. And subordinated groups that suffer under particular legal systems may find that law offers them, the less powerful, a measure of protection from the powerful, just as it sometimes offers them resources for action._\n\n\u2014Anthropologists June Starr and Jane Collier, 1989, _History and Power in the Study of Law: New Directions in Legal Anthropology._\n\n**Law, Power and Resistance**\n\nTheoretical debates in the field of anthropology, politics and law center around questions of domination and resistance, justice and power: can marginalized groups, such as Aboriginal peoples, realize their aspirations for social change through legal processes?\n\nThe most fundamental and traditional forms of legal critique in the western world are arguments and stories that expose the absence of neutrality or autonomy in law and legal processes. Feminists critique male dominance and sexist bias; people of colour bring charges of systemic racism; colonized and formerly colonized peoples reveal the legacy of imperialism; advocates of labour and the poor unmask alliances between powerful business interests and members of the judiciary. In response to the perennial question of how autonomous the law is from economic and political interests in \"the last instance,\" the British theorist, E. P. Thompson, wrote: \"Well, for most of the time when I was watching, law was running quite free of economy, doing its errands, defending its property, preparing the way for it, and so on.... But...on several occasions, while I was actually watching, the lonely hour of the last instance actually came. The last instance, like an unholy ghost, actually grabbed hold of law, throttled it, and forced it to change its language and to will into existence forms appropriate to the mode of production, such as enclosure acts and new case-law excluding customary common rights. But was law 'relatively autonomous'? Oh, yes. Sometimes. _Relatively_. Of course (emphasis in the original).\"\n\nFor Aboriginal peoples, as for other disempowered groups, the legal arena is a complex and often contradictory one in which to pursue social change. On the one hand, entering into a legal battle requires that all parties accept the language and the rules of the court. Critics of this strategy argue that the extent of \"translation\" required, for example, to explain Aboriginal spiritual beliefs in the language of the law changes the meaning of those beliefs. On the other hand, limited gains and protection of some historical rights have been achieved through legal struggles.\n\nMany argue that Aboriginal peoples do not have the luxury of choosing _not_ to engage in litigation to secure and protect rights. As small minority communities\u2014geographically dispersed and politically and economically marginalized and dispossessed\u2014Aboriginal peoples in Canada have limited resources to draw upon in their struggles for economic justice, political rights, and cultural survival. There are many locations where resistance is mounted: maintaining and reproducing kinship-based communities and strong family bonds in everyday life; insuring personal and familial survival; continuing to hunt, fish, trap, and gather on the land; engaging in ritual and ceremony; negotiating with governments; litigating in the courts; and participating in civil disobedience. In British Columbia, struggles have been fought at each of these sites consistently and simultaneously since Europeans first arrived.\n\nCertainly, law is not monolithic. This is demonstrated by appeal courts overturning lower court rulings, and by the publication of majority and dissenting opinions by panels of judges who fail to reach consensus among themselves on key points. The history of resistance through law waged by First Nations in B.C. illustrates the \"use of law as a 'resource for action'\" that Starr and Collier refer to. However, this history also raises questions about whether, looked at over the long term, the legal victories of Aboriginal peoples aren't short-lived and quickly co-opted, leaving exhausted human and financial resources behind in their wake. The history of legal struggles over Aboriginal title in B.C. raises the question of whether law, after all is said and done, inevitably turns out to be a servant of the rich and powerful.\n\n**Logic, Legal Hysteria, and Rifle Shots on Vancouver Island**\n\n_There is no clear, evolutionary logic in the historical development of Aboriginal rights...(in the courts).... In spite of after-the-fact stories that have tried to imply a consistent logic...there was a basic incoherence, an instability and set of contradictions embodied in the original approach.... It is a history of sustained, often vicious struggle, a history of losses and gains, of shifting terrain, of strategic victories and defeats, a history where the losers often win and the winners often lose, where the rules of the game often change before the players can make their next move, where the players change while the logic remains the same, where the moves imply each other just as often as they cancel each other out. It is a complex history whose end has not been written and whose beginnings are multiple, fragmentary and undecidable._\n\n\u2014Historian Peter Kulchyski, 1994, _Unjust Relations: Aboriginal Rights in Canadian Courts._\n\nThe questions that have been before the courts in British Columbia and Canada can be summarized briefly: (1) Did First Nations have legally recognizable title to their lands before the assertion of British sovereignty? (2) If they did have such title, does it continue today, or has it been extinguished? (3) If Aboriginal title and rights have not been extinguished, how can federal or provincial governments lawfully contravene or extinguish them?\n\nDuring the years 1963-1991 a number of important cases were litigated in Canada concerning Aboriginal rights of various kinds. Those most relevant to the issues in dispute in the Gitksan and Wet'suwet'en case were: _R. v.White & Bob_; _Calder v. R_ ; _The Hamlet of Baker Lake v. Minister of Indian Affairs and Northern Development_ ; _Guerin v R_.; _R. v. Bear Island Foundation_ ; and _R. v. Sparrow_. Together, these six cases represent the legal context in which _Delgamuukw v. R._ was initiated, heard, and ruled on.\n\nIn the process of ruling on the six cases listed above, and others, different answers were given by the courts to the key questions; a series of legal tests that set out the criteria through which Aboriginal title and rights could be recognized in Canadian law emerged; and Aboriginal Elders, anthropologists, and other expert witnesses became an integral part of contemporary legal processes. Aboriginal and non-Aboriginal plaintiffs and defendants, lawyers and judges, witnesses and researchers involved in Aboriginal title and rights litigation since the 1960s have studied, created, defended, and opposed the evidence, arguments and rulings that emerged in these cases. In order to understand the strategies adopted by the parties to _Delgamuukw v. R._ , it is necessary to comprehend, at least in skeletal form, this legal and historical context.\n\nIn British Columbia, the first of the modern Aboriginal rights cases was initiated in the spring of 1963 when Clifford White and David Bob, two members of the Saalequn (Nanaimo) First Nation of central Vancouver Island in British Columbia, went deer hunting on the south slope of Mount Benson, a few miles inland from Nanaimo. They were arrested and charged under the _Game Act of British Columbia_ (R.S.B.C. 1960) for being \"in possession of deer during the closed season.\" White and Bob argued that their right to hunt and fish for food on unoccupied Crown lands was guaranteed to them under the terms of a treaty signed between Saanich Chief Whut-Say-Mullett, and Governor James Douglas in Fort Victoria on February 7, 1852. The case came to the attention of Thomas Berger, then leader of the British Columbia New Democratic Party and a practising lawyer in Vancouver. White and Bob's legal counsel argued that this treaty was protected by the _Indian Act_ which, as federal legislation, was paramount over provincial legislation such as the _Game Act_. The Province of B.C. argued that the treaty of 1852 did not constitute a treaty between the Crown and an Aboriginal nation but rather was a commercial conveyance between some individuals who were members of a group that were not a state and had no international personality, and a privateenterprise, the Hudson's Bay Company, represented by its Chief Factor, James Douglas, who doubled as Governor of the Colony of Vancouver Island at the time. In other words, the Crown argued that the document was not a treaty but a private exchange of cash for land. The ancestors of White and Bob, the Crown claimed, had sold their land, and therefore their descendants had no rights on it, or to it.\n\nThe _White & Bob_ case was the first to require a provincial government of British Columbia, and their lawyers, to develop legal arguments to defend their historical denial of Aboriginal title and rights. The province's solicitors adopted a position based in legal positivism and referred to the _St. Catherine's Milling_ precedent, arguing first that Aboriginal title could only exist if it was created by a sovereign. The king had chosen not to create such rights because the Aboriginal peoples of British Columbia were \"too primitive\" according to the _Re: Southern Rhodesia_ precedent. The province's legal argument relied on a particular interpretation of history that described the First Nations of British Columbia as minimally evolved nomads lacking law or government when Europeans first arrived. For the first time in court, the province articulated an interpretation of the Royal Proclamation of 1763 that was to become a hallmark of British Columbia's legal arguments in all subsequent cases. The province's lawyers claimed that, since British Columbia did not appear on most maps drawn by British cartographers (who had, in 1763, not yet travelled to the west coast), the Royal Proclamation of 1763 was not intended to apply to Aboriginal peoples here. What the sovereign (or, in this case, his mapmakers) did not see, did not exist.\n\nFurthermore, they referred to the use of the present tense by the writers of the Proclamation in the phrase \"the Indians with whom we are connected\" as evidence that the authors' intentions were to exclude any Aboriginal groups with whom the Crown had not yet made contact. Berger argued that the present tense is the grammar of legal statutes, proclamations, and constitutions which are usually considered, legally, to \"speak until they are repealed.\" The B.C. Crown's interpretation of the Royal Proclamation has been described by historians and legal scholars as \"implausible,\" \"indefensible\" and \"ridiculous.\" It has, however, been argued repeatedly by prestigious lawyers, and declared \"fact\" by eminent judges, for 30 years.\n\nThe next line of legal attack developed by the Province of B.C. in the _White & Bob_ case and in subsequent litigation, was that if there had, in fact, been some form of Aboriginal title before the arrival of Europeans in British Columbia, it was extinguished by the assertion of sovereignty by Britain, and by the enactment of land use legislation by the colonial legislature.\n\nThe absence of any First Nations consent to extinguishment of their land title was, within the framework of legal positivism, irrelevant: if Aboriginal rights are created and extinguished only by the sovereign, then Aboriginal consent is not required to do either. Anyway, concluded provincial lawyers, Indians \"implicitly\" consented to extinguishment of title. This, they argued, was \"proven\" by the fact that First Nations in British Columbia did not wage armed rebellions, appeared (according to non-Aboriginal interpretations) to acquiesce to colonial domination, were now believed to be assimilated into mainstream Canadian society, and were therefore no longer racially or culturally \"pure.\" Crown lawyers pointed to Aboriginal \"culture\" measured by language, religion, clothing, food, weapons, modes of earning a living, housing, and means of transportation, for evidence. Any cultural change since contact with Europeans was presented as the natural outcome of a passive, inferior culture coming into contact with an active, superior one. Only those activities whose origins could be traced to pre-contact Aboriginal cultures _and which continued to be practiced_ , could be classified under this rubric as \"truly Aboriginal.\" This is referred to in law as the \"frozen rights\" theory, wherein Aboriginal cultures are \"frozen\" at the moment of \"discovery,\" and any developments since that moment are interpreted as resulting from European influence. In other words, \"ignoring Indian title was...seen as extinguishing it. Put another way, the argument asserted that the ordinary operation of a British colonial government had the effect of wiping out the legitimacy of any preexisting aboriginal arrangements.... The argument of implicit extinguishment through ordinary legislation has not been raised in other parts of Canada.\"\n\nThis represents one of many \"double binds\" in legal and popular discourses on Aboriginal rights. A double bind is a \"damned if you do and damned if you don't\" situation in which the outcome is always pre-determined, and in which explanations are arrived at by reasoning backwards from consequences to causes. From a social evolutionary perspective, to the extent that Aboriginal cultures are understood as not having changed after contact with Europeans, they are analyzed as being \"arrested\" at a \"lower stage\" of development, and incapable of \"advancement.\" Such \"primitive\" peoples must not have had any concepts of property or law, and clearly cannot\u2014today\u2014be considered capable of being granted the same rights as those of \"civilized\" peoples. _They are too different to be considered equal_. To the extent, on the other hand, that Aboriginal cultures are understood as _having changed_ and adapted some European ways to their own, then they are said to have voluntarily \"assimilated\" into the colonial culture, and clearly then have no grounds on which to claim \"special\" rights different from everyone else's. _They are too equal to be considered different_. Heads, the Crown wins. Tails, Indians lose.\n\nWhen legal scholar, Patrick Macklem, tried to ascertain the way concepts of cultural similarity and cultural difference had impacted on \"how the law has contributed to the current status of First Nations in Canada,\" he found that \"Native difference is denied where its acceptance would result in the questioning of basic premises concerning the nature of property, contract, sovereignty or constitutional right. Native difference is acknowledged where its denial would achieve a similar result.\" Macklem concluded that this discourse on cultural similarity and difference constituted a \"rhetoric of justification\" for legal judgements.\n\nAnthropologists entered the legal fray in the _White & Bob_ case, led by Wilson Duff. Initially, anthropological research was principally concerned with documenting and describing various Aboriginal peoples' practices surrounding land and resource ownership and use, translating this data into language that lawyers and judges could understand, and considering whether or not these concepts of \"Aboriginal title\" were commensurable with concepts of property recognized by Canadian law. During the _R. v. White & Bob_ trial, Wilson Duff responded to Berger's question concerning the meaning of the term \"tribal territories\" for the Saalequn as follows:\n\nBerger: When you say tribal territories, can you tell us what you mean by that? What use would the Indians have made of their tribal territories?\n\nDuff: This could be a very complicated statement, because they used different kinds of territories...with different intensity. They would use the rivers, of course, for fish with great intensity, and the beaches with great intensity, and the mountains and forest with somewhat less intensity, yet they would go at least that far back, not only to hunt the land mammal, deer, and also other land mammals, but to get bark and roots for basketry and matting and such things. These territories would be definitely used by them and would be recognized by other tribes as belonging to them.\n\nThe lower court found both defendants guilty and fined Clifford White and David Bob $100.00 (or 40 days in jail in default) each. Berger was successful in obtaining a new trial for White and Bob and went on to win a decision by the County Court of Nanaimo, which was then upheld by the B.C. Court of Appeal and the Supreme Court of Canada. The document signed by Whut-Say-Mullett and Governor James Douglas was in fact a treaty. White and Bob indeed had the right to hunt and fish for food on unoccupied Crown land. Mr. Justice Tom Norris of the B.C. Court of Appeal further ruled that the treaty was, like other treaties signed between the Crown and Canadian Aboriginal peoples, consistent with the direction of the Royal Proclamation of 1763. He wrote: \"the aboriginal rights as to hunting and fishing affirmed by the Proclamation of 1763 and recognized by the Treaty...still exist.\"\n\nThe legal significance of this decision was that \"for the first time a judge, and a well-respected British Columbia judge of conservative leanings at that, had presented a comprehensive opinion endorsing both the pre-existence and the continuing existence of aboriginal rights in British Columbia.\"\n\n**Civilization and the Whimsical Destruction of Property**\n\n_The assessment and interpretation of the historical documents and enactments tendered in evidence must be approached in the light of present-day research and knowledge disregarding ancient concepts formulated when understanding of the customs and culture of our original people was rudimentary and incomplete and when they were thought to be wholly without cohesion, laws or culture, in effect a subhuman species. This concept of the original inhabitants of America led Chief Justice Marshall in his otherwise enlightened judgment in Johnson v. McIntosh...to say 'the tribes of Indians inhabiting this country were fierce savages, whose occupation was war.... 'We now know that that assessment was ill-founded...Chief Justice Marshall was, of course, speaking with the knowledge available to him in 1823. Chief Justice Davey...[of the B.C. Court of Appeal] with all the historical research and material available since 1823 and notwithstanding the evidence in the record...said of the Indians of the mainland of British Columbia: '...they were undoubtedly at the time of settlement a very primitive people with few of the institutions of civilized society, and none at all of our notions of private property.' In so saying this in 1970, he was assessing the Indian culture of 1858 by the same standards the Europeans applied to the Indians of North America two or more centuries ago...._\n\n\u2014Justice John Hall, Supreme Court of Canada, 1973, _Dissenting Opinion, Reasons for Judgment in Calder v. R._\n\nThere is another aspect to the relationship between law, culture and power that is interwoven with the role law plays in enforcing particular rules in the interests of specific sectors of society. Law also \"maintains power relations by defining categories and systems of meaning.\" This defining and productive activity of law\u2014setting the agenda, and constructing the parameters of what may be included and what must be excluded\u2014is exemplified in the legal tests for Aboriginal title and rights. The tests are based on criteria the courts have established for evaluating claims brought before them by Aboriginal litigants.\n\nLegal tests for Aboriginal rights reveal the power relations that are constitutive of legal encounters: judges construct the tests; claimants sit them; judges mark them: pass, fail or defer. Of course, the arguments and evidence presented by Aboriginal claimants, and political support for them from outside the courtroom, have also influenced the tests. That is to say, it is as a result of persistent Aboriginal resistance that these questions continue to be placed before the judiciary. To this extent, the evolution of the tests represents the outcome of contestation and negotiation. However, the Crown is the dominant partner in a hierarchical relationship with Aboriginal peoples in Canadian society; and this social inequality follows the parties into the courtroom in myriad ways. The Crown has greater access to more resources in every part of the legal process, and in every aspect of it. The playing field, as the popular expression goes, is not only theirs, it is far from level.\n\nThe history of legal tests for Aboriginal rights could start in 1550 with Valladolid. It could start in 1608 with Calvin's Case and the infidel rule. It could start with the memorandum of the British Privy Council in 1722 that set out the doctrine of conquest and the doctrine of discovery\/occupation\/settlement. It could start with Lord Sumner's dictum in _Re: Southern Rhodesia_ based in an evolutionary framework. This is the history that followed First Nations, lawyers, and judges into court in British Columbia where the story of legal tests for Aboriginal title began in 1969 with what is now called \"the _Calder_ case,\" or \"the Nishga case.\" Chief Frank Calder, on behalf of the Nishga Tribal Council, asked the Supreme Court of British Columbia for a declaration that:\n\n(1) the Nishga held title to their territory prior to the assertion of British sovereignty;  \n(2) that this title had never been lawfully extinguished; and,  \n(3) that this title is a legal right.\n\nIn support of their claim that their Aboriginal title had never been ceded, sold, surrendered or lost in war, the Nishga relied upon written archival evidence that showed they had begun petitioning the Queen and Colonial officials for recognition of their title since first contact with Europeans in the eighteenth century. Five Nishga hereditary chiefs, Frank Calder, James Gosnell, William McKay, Harry Nyce, and Anthony Robinson; one provincial archivist, Willard Ireland; and one anthropologist, Wilson Duff, testified as expert witnesses. Both the chiefs and the expert witnesses focused on demonstrating extensive Nishga use and occupation of the Nass Valley. Wilson Duff, acting as cultural translator, explained that the Nishga system of property ownership is different from, but analogous to, English common law property ownership. The skeleton of a legal test for Aboriginal title emerged in the following exchange between B.C. Supreme Court Judge Gould and Wilson Duff:\n\nThe Court: I want to discuss with you the short descriptive concept of your modern ownership of land in British Columbia, and I am going to suggest to you three characteristics (1) specific delineation of the land, we understand is the lot.\n\nDuff: Yes.\n\nThe Court: Specifically delineated down to the lot, and the concept of the survey; (2) exclusive possession against the whole world, including your own family. Your own family, you know that, you want to keep them off or kick them off and one can do so; (3) to keep the fruits of the barter or to leave it or to have your heirs inherit it, which is the concept of wills. Now, those three characteristics...are you with me?\n\nDuff: Yes The Court: Specific delineation, exclusive possession, the right of alienation, have you found in your anthropological studies any evidence of that concept being in the consciousness of the Nishgas and having them executing such a concept?\n\nDuff: My lord, there are three concepts.\n\nThe Court: Yes, or a combination of them.\n\nDuff: Specific delineation...Physical landmarks, physical characteristics. The exclusive occupation did not reside in an individual. It rested in a group of people who were a sub-group of the tribe...The owners in this sense had certain rights of alienation. They could give up the tract of land, lose it in warfare, but in practice it would not go to anybody outside of the tribe, that is, a tract of Nishga land might change hands but it wouldn't go to other than a Nishga family.\n\nThe Court: So am I correct in assuming that there are similarities in the Nishga civilization in the first two characteristics, but not the third? All that alienation means, of course, is that you can sell it to anybody you like?\n\nDuff: Yes... The Court: I will give two more characteristics of ownership, the right to destroy it at your own whim, if you like, and the other, that the exclusive possession should be of indeterminable time, that is, cannot be terminated by a person's life, that is, can be passed on to one's heirs. That makes five. Now, you have dealt with three. Now the right to destroy at whim, set fire to your own house; these matters you have been dealing with, would a group within the Nishga have the right, if the buildings at the mouth of a certain river had been in their exclusive use some time and they will say, 'Let's set fire to it,' would the tribe prohibit that?\n\nDuff: I would think that they would have that right.\n\nThe Court: You would think they would have that right?\n\nDuff: Yes.\n\nThe Court: Now, what about the duration of the right, not to destroy, but the right of exclusive ownership, would it go to their heirs?\n\nDuff: Yes.\n\nThe Court: Or go back to the tribe for distribution?\n\nDuff: In theory it belongs within that kinship group through time, with no duration in theory. It always remains with that same kinship group.\n\nThe Court: There is the matrilineal line?\n\nDuff: Yes.\n\nThe Supreme Court of British Columbia ruled that the Nishga were too \"primitive\" in the nineteenth century to have held concepts of property that could be considered on an evolutionary developmental par with the concept of property upheld by British law.\n\nEuropean colonial cultural beliefs, or ideologies, set up relationships between colonizer and colonized as a hierarchical set of binary oppositions in which one member of the pair is always symbolically superior to the other. Colonizers are rational, the colonized are irrational. Colonizers are guided by reason, the colonized are driven by instinct. Colonizers are industrious, the colonized are lazy. Colonizers are active agents, the colonized are passive objects. But binary oppositions are two-way streets: when colonizers describe the characteristics of the colonized, they are simultaneously describing themselves, through defining the differences between themselves and their \"Other.\" Each description of the other is at the same time a description of self. Interpreted from this perspective, Justice Gould's ruling can be read as defining Euro-Canadian property law, and presumably the more highly evolved cultural values reflected in that law as: individually owning land, building fences around it and kicking your family off it, and burning down your house on a whim.\n\nWhen the Nishga case was appealed, the Justices of the B.C. Court of Appeal declared that if any form of Aboriginal title had existed it had been explicitly extinguished by Britain's assertion of sovereignty and \"implicitly\" extinguished by provincial land legislation prior to Confederation. The Nishga, on this basis, had no legal rights.\n\nThe Supreme Court of Canada, however, found differently when the case was appealed to them. In their 1973 decision, of the seven judges, one dismissed the case on a technicality and did not comment on the issues at trial. The remaining six found unanimously that Aboriginal title had existed prior to European arrival, based on long term occupancy. On the other important questions before them, the six judges split into two groups: Justices Ritchie and Martland supported the main judgment written by Justice Judson; while Justices Laskin and Spence supported the dissenting judgment written by Hall. Judson, Ritchie and Martland agreed with the Province of British Columbia's argument that the Royal Proclamation of 1763 was not meant to apply to the then-undiscovered British Columbia. Relying on the _St. Catherine's Milling and Lumber Co._ precedent, they found that Aboriginal title was a \"mere burden\" on Crown title. From Chief Justice Marshall's ruling in _Johnson v. McIntosh_ they drew the conclusion that the Crown had the exclusive right to extinguish Aboriginal title; and that whatever Aboriginal rights might have existed were lawfully extinguished by Britain's declaration of sovereignty. Judson, however, differed with the Crown on the nature of Aboriginal title. In what was to become an oft-quoted statement, he wrote: \"...the fact is that when the settlers came, the Indians were there, organized in societies and occupying the land as their forefathers had done for centuries. This is what Indian title means and it does not help one in the solution of this problem to call it a 'personal or usufructuary right'.\"\n\nOn the other hand, Justice Hall, supported by Laskin and Spence, argued that the Royal Proclamation was meant to apply to British Columbia; that while the Crown did have the right to extinguish Aboriginal title it must state its intention to do so in \"clear and plain language,\" and it could not be concluded that \"implicit\" extinguishment had taken place by virtue of colonial authorities having simply ignored Aboriginal title. On the issue of the nature of Aboriginal title, Hall cited Lord Sumner's decision in _Re: Southern Rhodesia_ stressing the possibility that some Indigenous property systems were sufficiently well developed to be recognized by British law, and Lord Haldane's caution in _Amodu Tijani v. Secretary of Southern Nigeria_ , that non-British systems should be understood on their own terms and not only in relation to British law. Hall argued that, \"to ascertain how far this latter development of right has progressed involves the study of the history of the particular community and its usages in each case. Abstract principles fashioned _a priori_ are of but little assistance, and are as often as not misleading.\" The Supreme Court of Canada appeal panel judges relied extensively on both the written historical record and the testimony of the expert witnesses, reserving their highest praise for the contribution of Wilson Duff. Hall wrote: \"What emerges from the...evidence is the following: the Nishgas in fact are, and were from time immemorial a distinctive cultural entity with concepts of ownership indigenous to their culture and capable of articulation under the common law, having, in the words of Dr. Duff, 'developed their culture to higher peaks in many respects than in any other part of the continent north of Mexico.'\"\n\nThe _Calder_ decision therefore marks a significant departure from the positivist theory of Aboriginal rights which gave legal standing only to rights recognized by a sovereign. The Supreme Court of Canada's ruling in the _Calder_ case was a victory for British Columbia First Nations and their supporters. Six Supreme Court of Canada judges agreed that Aboriginal title in fact existed, and three allowed that it may continue to exist.\n\nNevertheless, certain aspects of this landmark decision would haunt future Aboriginal land rights litigation in other parts of Canada. Justice Hall did not reject the test Lord Sumner set out in 1919 in _Re: Southern Rhodesia_ , but affirmed it by using its criteria to distinguish the Nishga culture as qualifying for respect and recognition on the grounds of evolutionary theory. Hall's reasoning allowed for the possibility that other Aboriginal peoples might fail Sumner's and Haldane's tests. While both rejected crude and archaic concepts and language, neither Duff's anthropology, nor Hall's law, rejected the philosophical or theoretical premises of \"neo-evolutionary\" theory that in 1973 remained current in academic anthropology and embedded in popular culture. Various iterations of neo-evolutionary theory have emerged throughout this century. Most recent versions eschew ethnocentric judgments, while still retaining beliefs in the determining role of the economy and technology in shaping culture. While rejecting the notion that all people progress through pre-determined \"stages of development\" from hunting and gathering to industrial capitalism, some _neo_ -evolutionary theorists argue that European and European-derived cultures have brought most Indigenous peoples under western economic and cultural domination. This trajectory therefore now reflects historical experience, rather than theoretical hypotheses, they claim. Anthropologists like Wilson Duff, while they respected and supported First Nations peoples and were devoted to preserving their cultures, applied class-based western notions of \"high\" and \"low\" culture to their analyses of Indigenous cultures. Peoples like the Nishga, therefore, with their abstract art, ornate carving, sophisticated architecture, and elaborate ceremonial life were judged within this framework to be \"highly cultured.\" Like rural people and urban working classes whose cultures are classified as \"low,\" some Aboriginal peoples without material cultural objects like totem poles, masks and carvings desired by Europeans, or elaborate ceremonial rituals like potlatches, are classified as \"simple.\" Unexamined evolutionary assumptions continue to incipiently and insidiously influence literature and the arts, and to inform \"common sense\" and popular culture.\n\n**From Law to Politics and Back Again**\n\n_I feel like a man who has been told he is going to die and is now being asked how he would like to be killed._\n\n\u2014First Nations leader George Manuel, responding to the _White Paper Policy_ , Kamloops, 1969.\n\nOn August 8, 1969, at a press conference in Vancouver, B.C., newly-elected Canadian Prime Minister Pierre Elliot Trudeau was asked whether his government would recognize Aboriginal title and rights in British Columbia. He replied: \"Our answer is no. We can't recognize aboriginal rights because no society can be built on historical 'might have beens'.\" Trudeau's Minister of Indian and Northern Affairs, Jean Chretien, oversaw the drafting and publication of a new Canadian Indian Policy that was released later in 1969. The Liberal government's infamous \"White Paper Policy\" began with the statement, \"...aboriginal claims to land...are so general and undefined that it is not realistic to think of them as specific claims capable of remedy except through a policy and program that will end injustice to Indians as members of the Canadian community.\"\n\nThe White Paper Policy advocated the dissolution of any distinct legal or political status for Indigenous peoples, or their lands, and the rapid assimilation of Aboriginal peoples into the mainstream of Canadian society. Three B.C. provincial Indian organizations met in Kamloops, B.C. in 1969 to plan strategies for responding to the White Paper Policy. Unanimity was as elusive as ever, but, with equal consistency, was finally found in the fundamental premise that Aboriginal title had existed prior to European contact and still did. Several months later an \"All Bands Assembly\" in Vancouver agreed to have non-Aboriginal experts prepare an historical-legal argument in support of the recognition of Aboriginal title, to be presented to a parliamentary committee. Opposition to the White Paper Policy's assimilationist goals was voiced by Aboriginal representatives across the country, and for the first time in Canadian history, First Nations organized themselves on a Canada-wide basis, mobilizing their members to present a united front in defence of their cultural and political survival.\n\nFour years later, in 1973, when the Supreme Court of Canada handed down their decision in _Calder v. R_., ruling that Aboriginal rights had in fact existed in British Columbia before contact with Europeans, and perhaps had not been extinguished, Prime Minister Trudeau was moved to allow that First Nations without treaties \"May have more rights than we thought.\" The federal government revisited its position and instituted the \"Comprehensive Claims Policy\" whose terms of reference mirrored the Supreme Court's judgment in _Calder_. Aboriginal nations were invited to bring forth claims to land that had been used and occupied by their ancestors, and that they themselves were still using. The story of the _Calder_ decision and the Comprehensive Claims Policy is an archetype of the relationship between law and government policy in Aboriginal matters. The legal tail wags the political dog, and man's best friend rarely strays outside the yard the law fences in. Law is a force that politicians, bureaucrats, and Aboriginal peoples alike reckon with.\n\n**The Honour of the Crown Below Par**\n\n_...Indians have a legal right to occupy and possess certain lands, the ultimate title to which is in the Crown.... The nature of the Indians' interest is therefore best characterized by its general inalienability, coupled with the fact that the Crown is under an obligation to deal with the land on the Indians' behalf when the interest is surrendered.... Any description of Indian title which goes beyond these two features is both unnecessary and potentially misleading._\n\n\u2014Chief Justice Dickson, 1984, _Reasons for Judgment, Guerin v. R_.\n\nThe next significant court case was fought in the 1970s in British Columbia by the Musqueam First Nation. In 1950, federal officials of the Department of Indian Affairs and Northern Development (DIAND) arranged for the Musqueam Indian Band\u2014whose reserve forms a small enclave located in the elite urban neighbourhood of south-west Vancouver\u2014to lease a portion of their land to the exclusive Shaughnessy Golf and Country Club. Land values were assessed, and a lease agreement was ratified by the Musqueam Chief and Council by way of a band referendum. Since, under the _Indian Act_ , First Nations lands are held in trust for them as wards of the Crown who, like children, are not able to enter into legal contracts, representatives of the Department of Indian and Northern Affairs signed the final contract with the Shaughnessy Golf and Country Club on behalf of the Musqueam. In 1970, Musqueam Chief Delbert Guerin learned that before they had signed the final contract, DIAND officials had reduced the value of the land and agreed to terms significantly more favourable to the Shaughnessy Golf and Country Club than to the Musqueam First Nation. These were _not_ the terms that the band membership had agreed to in their referendum, and DIAND officials did not advise the Chief and Council that they had renegotiated the contract before they signed it.\n\nThe Musqueam sued the federal government for breach of trust and the federal court awarded them $10 million in damages. The Federal Court of Appeal overturned this decision. The Musqueam then appealed to the Supreme Court of Canada who found that, indeed, the Crown and its agents have a fiduciary obligation to act in the best interests of Indians, and that this had not been done in the case before them. While \"failing to uphold their fiduciary obligations\" may seem an undeservedly polite way to describe the deceit and duplicity engaged in by government representatives in this case, the _Guerin_ decision did mark a recognition by the courts of at least once instance of these all too common practices in the history of government relations with Aboriginal peoples.\n\nIt is the sections of the Supreme Court's decision that bear directly on the issue of Aboriginal title that are important to the story being told here. First, Chief Justice Brian Dickson's ruling in _Guerin_ reiterates these fundamental points: that the Crown in the form of the hovering sovereign holds underlying title to all land, and that Aboriginal title is not proprietary and can only be surrendered to the Crown. However, the significance of Dickson's judgment is that he ruled that Aboriginal rights can apply to off-reserve lands. Second, the Supreme Court of Canada in _Guerin_ ruled that the Crown's fiduciary duty is legally rooted in Aboriginal title, and not, as was argued by the Crown, in the discretionary benevolence of the Crown or the Department of Indian Affairs. Third, the _Guerin_ decision provides a precedent that future judges could adopt to recognize that Aboriginal rights pre-existed European arrival and are inherent: recognized, and not created, by the British sovereign. Fourth, in the _Guerin_ decision the Supreme Court of Canada confirmed that Aboriginal title and rights are _sui generis_ : \"of their own kind; constituting a class alone; unique; peculiar.\" The Courts, the ruling continued, have almost \"inevitably found themselves applying somewhat inappropriate property law\" to the determination of Aboriginal title and rights. The characterization of the legal nature of Aboriginal title as _sui generis_ provided Canadian courts with the opportunity to explore beyond the confines of English property law to determine the scope and content of Aboriginal title and its relationship with the Crown's interest.\"\n\nFinally, Madame Justice Bertha Wilson, the first woman to be appointed to the Supreme Court of Canada and one of the judges sitting on the bench when the case was heard, wrote in her contribution to the _Guerin_ judgment that: \"...the bands do not have the fee in the lands; their interest is a limited one. But it is an interest which cannot be derogated from or interfered with by the Crown's utilization of the land for purposes incompatible with the Indian title, _unless, of course, the Indians agree._ \" (Emphasis added.) This legitimated the position that Native consent should be required in order for Aboriginal title to be legally extinguished or contravened. Through this ruling, Justice Wilson created a precedent which was available to be adopted by judges when ascertaining whether or not particular Aboriginal peoples have consented to extinguishment of their Aboriginal title. Not only did the _Guerin_ decision mark an instance of legal recognition that Aboriginal peoples in fact exist, but it suggested that it might be appropriate, morally just, _and legal_ for their consent to be required before they could be \"extinguished.\"\n\nThe Supreme Court of Canada's _Calder_ decision marked a significant departure from the archaic legal positivism that had preceded it. And the _Guerin_ decision went several steps further. However, in both cases, all the Supreme Court judges were unanimous in upholding the legitimacy of Britain's assertion of sovereignty, and the hovering sovereign's possession of the \"underlying title to all the land.\" If extinguishment of Aboriginal title had not been properly effected historically, then the judges said it should be now, and they set out ways for extinguishment to be recognized and\/or accomplished.\n\nThe final goal that courts and governments have shared\u2014the desired outcome of litigation, as well as land claims and treaty negotiations\u2014is the extinguishment of Aboriginal title, absolutely and forever; and the confirmation of the singular sovereignty of Crown title, absolutely and forever. Legal and political recognition of Aboriginal title and extinguishment of Aboriginal title have been inextricably interdependent and mutually defining. The assertion of dominance and the surrender of autonomy must occur at the same instant. This demand for a forced coupling of _recognition_ of Aboriginal title with _extinguishment_ of Aboriginal title was upheld by every legal decision up to 1997 in Canada, and is reflected in the Comprehensive Claims Policy's requirement that the settlement of any claim and the signing of any modern-day treaty is dependent on the Aboriginal peoples' surrender of Aboriginal title, and agreement to its extinguishment. To Aboriginal peoples this demand is unjust and humiliating. Those First Nations who have entered into modern-day treaties\u2014often under tremendous pressures in the face of potential environmental crises\u2014have agreed to legal extinguishment of Aboriginal title for the purposes of resolving contemporary claims. In each set of negotiations, however, compliance with the demand for extinguishment has been resisted by First Nations representatives to the last moment, and it has only been with tremendous reluctance that extinguishment has been agreed to. It is a question that divides Aboriginal communities, and many continue to refuse to take this final step.\n\nThe 1970s marked the beginning of the contemporary period of B.C. Aboriginal politics in which debates concerning the relative merits and effectiveness of civil disobedience, political negotiation, and litigation are ongoing. The preferred tactic, as evidenced by the resolutions of provincial and national Aboriginal conferences, has always been political negotiation. The strength of their legal position, and the refusal of successive British Columbia governments until 1990 to recognize the legal existence of Aboriginal peoples in the province, has led British Columbia First Nations into the courts primarily as a means to achieve a strong enough bargaining position to \"bring the government to the table.\" First Nations have principally used the courts, not to _settle_ the dispute, but to acknowledge its _validity_ , after which nation-to-nation negotiations could honourably begin.\n\n* * *\n\n. Clifford 1988(b); Just 1992; Lazarus-Black and Hirsch (eds.) 1994; Merry 1992; Mertz 1988. See also Danielson and Engle (eds.)1995; Hart and Bauman (eds.)1996; Leonard 1995.\n\n. The literature on this topic is vast. See, for a sample, Adelberg and Currie (eds.) 1993; Chunn, and Lacombe (eds.)1998; Smart 1989.\n\n. Turpel 1991(b); 1991(c); Williams, P. 1991.\n\n. Comaroff and Roberts 1981; Moore 1986.\n\n. There is a huge literature concerning the relationship between law and the economy, and questions about the role of law in class relations in western societies. For a survey see Hunt 1981; Snyder 1981.\n\n. Thompson 1978 quoted in Starr and Collier (eds.) 1989:25.\n\n. For a detailed explication of this issue see Coombe 1989.\n\n. For an articulation of this argument in the Canadian case see Turpel 1991(a).\n\n. For an articulation of this argument in the Canadian case see Henderson, J. Y. 1985; and Williams, R. A. 1987. The debate about the pitfalls and promises of using the law to achieve radical social change has been most thoroughly developed by feminist legal scholars. See Cornell 1992; Young 1991. Razack 1991 describes a case study in which these questions were grappled with.\n\n. There have been other important cases tried during this period of time. I have selected for focus here only those _most_ relevant to British Columbia in general, and the _Delgamuukw_ case in particular. For detailed summaries of discussions surrounding arguments and judgments in these cases, and others, see Kulychyski, Peter (ed.)1994.\n\n. _R. v. White & Bob_ (50 Dominion Law Reports (2d) [1965], 620.\n\n. Berger 1981:49.\n\n. Ibid.\n\n. See Tennant 1990:216-217. Tennant says: \"This was a lawyer's argument contrived retrospectively. No one associated with any aspect of the Indian land question had previously read the Proclamation as not extending to the western edge of the continent(216).\" Of the B.C. Crown's lawyers' analysis of the meaning of the use of the present tense, Tennant says: \"Had the British monarch issued another proclamation stating that a royal biscuit was to be given to 'any person with whom We are having tea,' it would be ridiculous to argue that the provision applied only to persons at the table with the monarch at the moment of signing and that it did not apply to future tea parties. Today the Canadian Charter of Rights and Freedoms uses the present tense in expressing its guarantees: no sane person would maintain that its guarantees were intended to apply only at the moment the Charter was approved (216).\"\n\n. Tennant 1990:217.\n\n. Macklem 1991:392.\n\n. For a survey of anthropologists' involvement in First Nations legal struggles during the last thirty years see Kew 1993-94.\n\n. Berger 1981:52-53.\n\n. _R. v. White & Bob_ (50 Dominion Law Reports (2d) [1965], 620, cited in Berger 1981:53.\n\n. Tennant 1990:219.\n\n. Merry 1992. See also Danielson and Engle (eds.)1995; Hart and Bauman (eds.) 1996; Leonard, 1995.\n\n. _Calder et al v. A.G.B.C_. (1969), 8 Dominion Law Reports (3d), 59-83, [S.C.B.C.]. The \"Nishga\" now spell their name \"Nisga'a.\"\n\n. Kulchyski 1994:61-126.\n\n. Berger 1981; Kulchyski 1994.\n\n. Kulchyski 1994:96-98.\n\n. Fanon 1963.\n\n. McGrane 1989.\n\n. Clifford 1988; Said 1992.\n\n. _Calder et al v. A.G.B.C_. (1973), 34 Dominion Law Reports (3d) [1973], 145-226. See also Sanders 1989.\n\n. _Calder et al v. A.G.B.C._ (1973), 34 Dominion Law Reports (3d) [1973], 145-226.\n\n. Calder:401.\n\n. See Durham 1990.\n\n. See Turner 1993.\n\n. Turner 1993.\n\n. Weaver 1981:3-4.\n\n. Canada, Government of, (1969) Department of Indian Affairs, Policy Statement:1.\n\n. Tennant 1990:171.\n\n. Tennant 1990:172.\n\n. _Guerin v. R._ (1984) 13 D.L.R. (4th) 321.\n\n. _Webster's New Collegiate Dictionary_ (1981) Toronto, Ontario: Thomas Allen &Son Limited, 1156.\n\n. _Guerin v. R._ (1984) 13 D.L.R. (4th) 339.\n\n. Bell and Asch 1997:48.\n\n. _Guerin v. R._ (1984) 13 D.L.R. (4th).155.\n**Chapter 7: Conflicts and Constitutions: The 1980s**\n\n_The federal government has a claims policy that they brought into place after the Calder case, and that claims policy was hollow. It was empty. It meant nothing. Still, the Gitksan and Wet'suwet'en entered into one of their phases of trying to resolve the land question in the mid-70s, and in 1977 qualified for federal funding to prepare negotiations.... Beginning in 1977, a tremendous amount of work was done, work that built on what was done before and took it even further. A base was built to prepare for negotiations._\n\n\u2014Medig'm Gyamk (Neil Sterritt), Gitksan, 1992, _It Doesn't Matter What the Judge Said_.\n\n**The Master's Tools: The Comprehensive Claims Policy**\n\nBefore a land claim could be considered for negotiation under the federal government's Comprehensive Claims Policy, Aboriginal \"claimants\" had to first \"qualify\" by presenting specific evidence to the federal government of Canada, and to the provincial governments. According to the terms and definitions set out in the Calder decision, to be eligible, a claimant group would have to prove that their ancestors were members of an organized society who had occupied and used specific lands and resources before Europeans arrived; and, that the present claimants continue to occupy these same lands, and to use these same resources today.\n\nAboriginal peoples and their supporters set about doing research to demonstrate that their territories had borders that could be delineated and transformed into lines on maps; and that their villages, their hunting, fishing and gathering spots, and their sacred sites, could be represented by Xs on those same maps. Feasting and potlatching, increasingly transmuted into nouns, appeared on administrative flow charts, representations of governments, complete with bureaucrats and pawns. Continually contested and negotiated processes for resolving disputes were written down and codified into written laws. Oral histories, songs, dances and prayers were documented and edited; tape recorded and word processed. Stories and memories were burned onto film.\n\nUsing the masters' tools to destroy the master's house, reappropriating colonial schemes, converting them into resources for cultural survival, are strategies that oppressed people have honed for centuries. Mapping can be a profoundly powerful political act of naming, or re-naming what is yours, in your own language. Documenting hunting, fishing, gathering and food processing can be a way of seeing what television and schools have tried to render invisible. Graphing structures of government can reveal another potential for feasting systems that have been changing and adapting for thousands of years. Trying to understand the underlying philosophies of laws can help to create alternative visions of justice. Recording the knowledge of elderly people whose embodied histories may die with them is, in the contemporary world, an important part of safeguarding their unique experiences and wisdom for the benefit of all of humankind.\n\nAboriginal peoples in Canada in the 1970s, like Indigenous people around the world, were actively engaged in a decolonization movement that included efforts to liberate themselves politically and culturally from years of being told, and sometimes believing, that they had no right to dreams of self-determination, and no possibility of realizing them. They were digging out from under over 100 years of colonial domination, emerging sometimes from generations of institutionalization in residential schools, and prisons. For the Gitksan and Wet'suwet'en, and other Aboriginal peoples in Canada, having access to resources through the Comprehensive Claims Policy provided the necessary material resources that enabled people to become more deeply engaged in learning or re-learning their languages, their histories, their subsistence technologies and practices; and reflecting on their traditional and contemporary cultures. With the pragmatic determination of their ancestors, they tried to represent and translate themselves through images and languages that \"others\" might understand, without sacrificing their own dignity and integrity. As they sought to interrupt the historical monologue, and to initiate dialogue, these resources helped their voices to be heard, often for the first time, by non-Aboriginal Canadians.\n\nThe financial support provided by the federal government to prepare comprehensive claims submissions, and the implied promise that these claims would be taken seriously and responded to honourably, were integrated into the Aboriginal movement for decolonization: imperfectly and unevenly, and within the confines shaped by power relations, of course. But reappropriated they were.\n\nThe comprehensive claims process moved very slowly through the Office of Native Claims during the 1970s and 1980s. The federal government's guidelines allowed it to negotiate only one claim at a time in each province or territory. In British Columbia, where the provincial government refused to participate on the grounds that no Aboriginal rights ever did or ever would exist in this province, the entire process was stalled. The Nishga claim, supported by the accumulated documentation of over 100 years of constant petitioning and preparing for litigation, had been filed first and lay dormant. All other First Nations knew that their claims would not even be considered until after the Nishga's was settled.\n\nWhile the comprehensive claims process in B.C. was stuck in this logjam in front of the bottleneck built by the provincial and federal governments, the pace at which the exploitation of natural resources was proceeding accelerated, and profits from the forests and the seas flowed into corporate bank accounts unimpeded, except by political resistance. Many people feared that, by the time First Nations got to the table, there would be no resources left to negotiate. Throughout the 1980s, various B.C. First Nations mounted campaigns of civil disobedience. They and their supporters temporarily halted industry and transportation across their lands, and applied\u2014sometimes successfully\u2014to the courts for injunctions to halt development until their claims were heard and adjudicated.\n\n**Title on the Baker Lake Tundra: Lower Peaks than Mexico**\n\n_The exigencies of survival dictated a society composed of small, scattered groups. The band itself had no political hierarchy; that existed only at the camp level...._\n\n_Major decisions all involved the hunt, conducted at the camp level, and were made by the oldest hunters. Neither individuals, camps nor bands claimed or recognized exclusive rights over a particular territory.... There is no evidence or reason to infer that the Inuit's nomadic ways, relationship to the land and social and political order changed from prehistoric (circa 1610) times until their settlement (circa 1950)._\n\n\u2014Justice Mahoney, Federal Court of Canada, 1980, _Reasons for Judgment_ in _The Hamlet of Baker Lake et al v. Minister of Indian Affairs and Northern Development_.\n\nMeanwhile, in northern Canada, a case that would establish important precedents in the area of Aboriginal law was winding its way through the maze of trials and appeals. The \"Baker Lake case\" had been launched in the Federal Court of Canada, Trial Division, in 1979. The Plaintiffs were the Hamlet of Baker Lake, the Baker Lake Hunters and Trappers Association, the Inuit Tapirisat of Canada, and individual Inuit living, hunting and fishing in the Baker Lake area of the Northwest Territories. The Defendants were the Attorney General of Canada and the Minister of Indian Affairs and Northern Development in Right of Her Majesty the Queen, and a consortium of mining companies. The plaintiffs asked the court for an order restraining the government from issuing land use and prospecting permits, granting mining leases and recording mining claims which would allow mining activities in the Baker Lake area, and for an order restraining the defendant mining companies from carrying on such activities there. They also asked for a declaration that the lands comprising the Baker Lake area are subject to the Aboriginal right and title of the Inuit residing in or near that area, to hunt and fish in these territories.\n\nIn their initial pleadings, the government defendants admitted that the Plaintiffs and their predecessors had occupied and used the Baker Lake area since time immemorial. They withdrew this admission during the trial. The mining companies denied that Aboriginal title had existed either before or after Europeans arrived in the region. Both the government and the mining companies argued that _if_ Aboriginal title ever existed it was entirely extinguished by the Royal Charter of 1670 that granted the territory\u2014then known as \"Rupert's Land\"\u2014to Canada, or by subsequent legislation.\n\nJudge Mahoney found the evidence of the Inuit elders about their ways of life and religious beliefs, and that of Superintendent Dent of the Royal Canadian Mounted Police \"complementary,\" in so far as both described daily life within small hunting groups. The principal conclusion the judge took from this evidence was that, prior to moving into settlements in the early 1950s, the Baker Lake Inuit were, in his words, \"nomads.\" In his imagination, this meant they had no strong ties to specific, delineated tracts of land, or \"civilized\" concepts of private property.\n\nExpert witnesses Dr. Elmer Harp Jr., Professor Emeritus of Archaeology at Dartmouth College, Hanover, New Hampshire, and Dr. J. V. Wright, head of the Scientific division of the Archaeological Survey of Canada, particularly impressed Justice Mahoney with their detailed and recognizably \"scientific\" evidence, and with the fact that they agreed with each other, as Mahoney did with them, on the fundamental issue of classification within the familiar social evolutionary framework: the Inuit were nomadic hunters, not settled horticulturists and therefore were low on the scale of evolutionary human development. Justice Mahoney concluded that \"those encampments of two or three families were the units described by the Inuit witnesses, encountered by Inspector Dent in the mid-1950s, by Norton in 1762, and discovered to have existed in the Thule period.\"\n\nJustice Mahoney was less impressed by the expert evidence of Dr. Milton Freeman, whom he described as \"a social anthropologist, which is to say that he is neither an archaeologist nor a linguist; he studies the social behaviour of people in the context of their society or culture.\" On the stand, Dr. Freeman elaborated on the nature of \"band level societies,\" a term he had not used in his affidavit. The Crown and the mining companies objected to his testimony following his statement that the small hunting units described by both Inuit elders and local non-Inuit observers are \"units of a much larger coherent organized society and very much interacting, interdependent, mutually dependent on interaction with other units within the society...this all constitutes a very coherent society which anthropologists have no problem in identifying....\" Freeman tried to explain that, while the Baker Lake Inuit may have hunted and camped in small family groupings during particular seasons, at other times they joined together with similar bands in large gatherings for the purposes of trade and exchange, ceremony, ritual, resolution of disputes, organization of government, and documentation of significant historical developments through oral history. The Baker Lake Inuit live sometimes in small bands; and at other times, in larger groupings. Freeman challenged simplistic classifications that rule out those concrete aspects of Indigenous life which do not fit neatly into abstract European theoretical categories.\n\nMahoney dismissed Freeman's evidence on a technical point: Rule 482 of the Rules of Evidence requires that an expert witness' testimony be laid out in an affidavit first, in order for the opposing lawyers to have adequate opportunity to review the expert opinion report and prepare for cross examination. Freeman had stated in his affidavit that he would discuss the relationship between the Inuit and their environment. When Freeman described Inuit _culture_ , Mahoney ruled that \"this was not what was promised\" in the affidavit, apparently understanding \"environment\" to include only the physical or natural landscape, but excluding human social, political and economic organization. Dr. Freeman was engaging in \"persuasive arguments\" on these issues, unbecoming to a scientist, the judge said.\n\nMahoney was similarly unimpressed by Dr. Peter Usher's evidence. Usher holds a Ph.D. in Geography. Quoting the _Shorter Oxford Dictionary_ definition of geography as the \"science that describes the earth's surface, its form and physical features, its natural and political divisions, its climates, productions etc.,\" Mahoney concluded that Dr. Usher's evidence didn't fall within his field of expertise. Mahoney wrote: \"Dr. Usher's evidence had more the ring of a convinced advocate than a dispassionate professional.... Neither his formal training as a geographer nor his experience in and with the Arctic and Inuit qualify him to form opinions on political, sociological, behavioural, psychological and nutritional matters admissible as expert evidence in a court of law.\"\n\nMore experts testified. Wildlife biologists, ethnologists, and experts on animal behaviour were called to the stand by both parties. Those called by the plaintiffs supported the Inuit hunters' claims that the caribou herds were declining and being driven away by mining activities. Those called by the government and mining companies discounted the Inuits' evidence, saying their knowledge reflected only their simple lived experiences and was therefore particular and limited, and should not be taken as seriously as the theories of \"scientific experts.\" These non-Inuit experts claimed to base their knowledge on scientific surveys that reached far beyond the immediate area of Baker Lake, and produced results that could be generalized and tested against universal, law-like theories. They claimed the causes of the caribou herds' decline were multiple and complex.\n\nJustice Mahoney ruled that \"on the balance of probabilities...activities associated with mining exploration are not a significant factor in the caribou population decline.\" In summary, Mahoney, like judges before and after him, preferred to rely on his own \"common sense\" interpretation of Native testimony, supported by carefully chosen \"factual\" confirmation by \"ordinary white people,\" and professionals selectively labelled \"scientists\" by a judge. Testimony by well educated and well respected\u2014by the criteria of western scholarship\u2014scholars of \"softer sciences\" like anthropology or social geography, was rejected.\n\nAssessed from another perspective, it could be argued that evidence that supported the representation of Baker Lake Inuit societies as \"high enough\" on the scale of social development set by the _Re: Southern Rhodesia_ test to have had \"property usages and conceptions that could be continued under the British regime,\" was rejected in favour of images of small bands of nomads lacking political or social cohesion whom the same judgment had described as \"so low on the scale of social organization that their usages and conceptions of rights and duties are not to be reconciled with the institutions or legal ideas of civilized society.\" Expert witnesses whose testimony supported the former were dismissed as unprofessional \"advocates\" of Aboriginal political causes, while those whose interpretation supported the latter where respected as \"scholarly experts.\" This evaluation was made by the court, and not by the standards of their respective professions.\n\nOn the legal source of Inuit Aboriginal title, Justice Mahoney set out a four-point test which he said the plaintiffs must prove to establish an Aboriginal title cognizable in common law:\n\n(1) That they and their ancestors were members of an organized society.\n\n(2) That the organized society occupied the specific territory over which they assert the Aboriginal title.\n\n(3) That the occupation was to the exclusion of other organized societies.\n\n(4) That the occupation was an established fact at the time sovereignty was asserted by England.\n\nThe first criteria was drawn directly from the test in _Re: Southern Rhodesia_. Mahoney enforced this by referring as well to Haldane's decision in _Amodu Tijani v. Nigeria_ about acknowledging differences between Indigenous property systems. Accordingly, Mahoney argued that different property regimes represented different stages of social evolutionary development. The Baker Lake Inuit passed Mahoney's test, but only just. Unlike the Nishga, who the Supreme Court of Canada found had \"developed their cultures to higher peaks\" than any other Indigenous peoples north of Mexico, Mahoney classified the Baker Lake Inuit as less advanced on the evolutionary scale. Mahoney wrote: \"The fact is that the aboriginal Inuit had an organized society. It was not a society with very elaborate institutions but it was a society organized to exploit the resources available on the barrens and essential to sustain human life there. That was about all they could do: hunt and fish and survive.\"\n\nOn the question of extinguishment, Mahoney ruled that neither the Royal Charter of the Hudson's Bay Company, nor admission of Rupert's Land into Canadian confederation had extinguished the common law Aboriginal title the Inuit held. Neither, he found, had legislation subsequent to 1870 had the effect of extinguishment. However, the unextinguished rights that Mahoney \"found\" were extremely limited. He argued that Aboriginal title could not have been proprietary because then the Crown _would_ have formally extinguished it. His conclusion, in summary, was that the plaintiffs were entitled to a declaration that they have an Aboriginal right only to hunt and fish on the lands in question. \"The aboriginal right asserted here encompasses only the right to hunt and fish as their ancestors did,\" Judge Mahoney wrote.\n\nWhen the _Calder_ case went to court in 1969 the trial took four days. No fully articulated \"legal test\" existed at the time and judges saw the question of Aboriginal title as a factual matter to be determined on the basis of empirical evidence of actual historic occupation and use of ancestral lands. Such a test requires minimal evidentiary support, is relatively uncomplicated, and accessible to common sense reasoning and understanding. The straightforward arguments and the evidence required to prove historical use and occupation, in its simplicity, was also difficult for the Crown to dispute. However, only Aboriginal peoples, and not non-Aboriginals, are required to prove long term use and occupation, rather than pre-existing ownership and historical title, to establish legal ownership of land. And, the use and occupancy test suffered from being based on the demand that Aboriginal people demonstrate how their property systems were similar enough to be considered equal to British ones.\n\nThis is another double bind: if Aboriginal people emphasize the _similarities_ between their land tenure systems and British ones, the courts may look more favourably on their claims because they appear familiar, but then the Aboriginal litigants sacrifice the opportunity to demonstrate the cultural uniqueness and ongoing validity of their own relationships to land, and surrender to the colonizer's language and legal concepts. If, on the other hand, Aboriginal peoples emphasize the _differences_ between their relationships to land and those of British-derived cultures, they risk them being classified as too different to be understood as equal. Heads, the Crown wins. Tails, Indians lose.\n\nJustice Mahoney's 1980 decision in _The Hamlet of Baker Lake v. Minister of Indian Affairs and Northern Development_ , and his articulation of a precise, but complex, legal test for Aboriginal plaintiffs to meet, therefore marks an important turning point in this story: the recognition, and construction, of complexity around the legal Aboriginal title issue. The intricacies of diverse Indigenous relationships to land are complex and interesting. However, the historical\/legal questions at stake remained the same: were Aboriginal peoples here when Europeans arrived? Did they live in organized societies with property laws? Were their rights lawfully extinguished according to British, Aboriginal or International law?\n\nThe Baker Lake test set the terms of legal and anthropological research questions for a decade to come. The important knowledge required to understand and participate in the legal struggle for recognition of Aboriginal title and rights increasingly became the task of specialists like lawyers, consultants, Aboriginal Elders and a small coterie of First Nations leaders. Increasingly, ordinary people, Aboriginal and not, were ushered out of decision-making roles and into fund-raising and public education in support of litigation.\n\n**The Evolution of the Dedicated White People Band at Bear Island**\n\n_To conclude, in 1763, George III, with the advice of his United Kingdom Ministers, did not grant ownership of vast tracts of land to Indian bands...when a war had just been fought to acquire those lands...._\n\n_At that time, Europeans did not consider Indians to be equal to themselves and it is inconceivable that the King would have made such vast grants to undefined bands, thus restricting his European subjects from occupying these lands in the future except at great expense._\n\n\u2014Justice Steele, Supreme Court of Ontario, 1989, _Reasons for Judgment_ , in _Attorney-General of Ontario v. Bear Island Foundation et al_.\n\nIn the 1989 case of _The Attorney-General for the Province of Ontario_ (the plaintiffs) vs. _the Bear Island Foundation and Gary Potts, William Twain and Maurice McKenzie Jr. on behalf of themselves and on behalf of all other members of the Teme-agama Anishnabay, and the Temagami Band of Indians_ (the defendants),  referred to as \"the Bear Island case,\" the Crown claimed unencumbered title to 4000 square miles of land in Northern Ontario. The defendants, _Bear Island et al_ , argued that Crown title was burdened by Aboriginal title recognized by the Royal Proclamation of 1763, and by unfulfilled obligations under the Robinson-Huron Treaty signed in 1860. The Crown wanted the court to relieve them of this burden so that development could proceed unencumbered. The Bear Island trial remains the longest recorded civil hearing in Ontario history, lasting for 120 days.\n\nLike Judge Mahoney, Justice Steele was not impressed by the expert witnesses supplied by the Bear Island defendants, calling them \"a small, dedicated and well meaning group of white people...[who]...in order to meet the aspirations of the current Indian defendants has pieced together a history from written documents, archaeology and analogy to other bands, and then added to that history a study of physical features and other times, together with limited pieces of oral tradition.\"\n\nJustice Steele ruled that the evidence presented did not, to his satisfaction, prove that the Teme-agama Anishnabay were members of an organized society in 1763. He went on to conclude that, since neither the French nor the English considered Indians as equal to Europeans in the eighteenth century, the Crown's representatives who drafted and signed the written agreement could not have intended for the Royal Proclamation of 1763 to recognize Aboriginal title as the basis of any legitimate legal or political rights.\n\nThe Bear Island case illustrates an example of a conservative, archaic approach to the interpretation of history in the context of Aboriginal rights litigation. During the 1970s and 1980s, Canadian courts wrestled with the problem of how to interpret history in the context of Aboriginal rights litigation. The results varied, and were often contradictory. In 1973 when the decision in the _Calder_ case was arrived at, Justice Hall had commented that historical documents \"must be approached in the light of present-day research and knowledge disregarding ancient concepts formulated when understanding of the customs and culture of our original people was rudimentary and incomplete and when they were thought to be wholly without cohesion, laws or culture, in effect a subhuman species.\" Similarly, in 1977 in the case of _Kruger and Manuel v. The Queen_ , then Chief Justice Dickson of the Supreme Court of Canada argued that traditional legal approaches may not be adequate to the task at hand since \"claims to aboriginal title are woven with history, legend, politics and moral obligations.\" The idea that judges must be conscious of historical context when interpreting legal precedents, was offered again in 1985 in the _Simon v. R._ case, when the Supreme Court of Canada overruled a decision by Justice Patterson in _R. v. Syliboy_ made in 1929. In that case, Patterson had found that a particular agreement between the Crown and the Micmac in 1752 was not a treaty representing the \"unconstrained Act of independent powers,\" but rather an agreement \"between a civilized power and savages.\" The 1985 Supreme Court of Canada decision stated that \"It should be noted that the language used by Patterson J... reflects the biases and prejudices of another era in our history. Such language is no longer acceptable in Canadian law, and, indeed, is inconsistent with a growing sensitivity to native rights in Canada.\"\n\nHowever, in 1985, the very same year that Chief Justice Dickson of the Supreme Court of Canada issued this caution, in the Supreme Court of Ontario, Justice Steele handed down his decision in the Bear Island case using language and reasoning that resembled Patterson's 1929 ruling in both theory and substance. On appeal, in 1991, the Supreme Court of Canada rejected Steele's finding that the Teme-agama Anishnabay had \"failed to prove that their ancestors were an organized band level society in 1763.\" However, they simultaneously ruled that they were unable to find any \"palpable and overriding error\" in Steele's findings of facts, although they did not necessarily agree \"with all the legal findings based on those facts.\"\n\nLegal scholar Joel Fortune asked the obvious question: \"How is it possible to distinguish Steele J.'s 'correct' determination of the facts from his 'incorrect' finding that the Teme-amaga Anishnabay did not constitute an organized society in 1763?\" Fortune answered his own question by concluding that the Supreme Court of Canada's Bear Island decision is an illustration of \"the judicial reluctance to acknowledge openly that a legal outcome may rest on a question of historical interpretation.\"\n\nThe Bear Island judgment refashioned the Baker Lake test into a more complex three-part test, adding requirements for proof of the nature of Aboriginal rights enjoyed prior to the relevant date, as well as evidence of a system of land-holding and a system of social rules and customs. And, the Bear Island test added, this continuity of exclusive occupation must be evident up to the date of commencement of the court action. Therefore, the Bear Island claimants had to prove exclusive occupation from the eighteenth century until the time they started their claims action in the late twentieth century. This set a precedent whereby all Aboriginal claimants could be asked to show that they had excluded not only other Indigenous peoples from their territories from the time of European contact until the filing of their writ in a contemporary court, but that they had also kept well-armed European explorers, miners, traders, settlers and police off their lands.\n\nIn a practice that speaks to the uniquely autocratic prerogative of law, the legal tests for Aboriginal rights have been constructed and\/or elaborated upon during the course of trials, and articulated by judges _post hoc_ in reasons for judgment, often in response to evidence and testimony presented for the first time in the particular trial being judged. Thus, claimants enter into a trial prepared to meet a test enunciated in a previous case, and find their arguments and evidence evaluated on the basis of criteria that emerge _in situ_. These criteria, in turn, influence the arguments lawyers advance and the kinds of evidence and expert witnesses they bring forward to support their arguments in a subsequent case. The tests have, therefore, shaped the research and testimony sought from anthropologists, archaeologists and other expert witnesses employed in the litigation process.\n\nThe evidentiary requirements for legal proof of Aboriginal title have correspondingly expanded exponentially, and, with them, the role and importance of anthropologists and other expert witnesses. The \"land use and occupancy\" studies that were previously required to establish long term, prior occupancy, now had to be supplemented by research into cosmology, language, spirituality, governance, law, family life, and world views. The increase in volume and complexity of evidence was generated by the court's responses to these cases, and the political context shaping both. Anthropologist Peter Elias concluded that \"the tests set out by Mr. Justice Steele in Bear Island Foundation...may have crossed the line of social science comprehension...if the tests are elaborated much further, it won't be possible to meet them.\"\n\n**The Master's House: The Repatriation of the Canadian Constitution**\n\n_When the Constitutional process came along, in 1981 and 1982, the Gitksan and Wet'suwet'en asked themselves and talked to the elders and wondered: 'Can we create the political will on the part of the politicians in British Columbia and in the federal government to negotiate?'..._\n\n_We have been consistent. You could see it based on the things that the elders were saying to the McKenna-McBride Commission between 1912 and 1913 and the points they were making when Indian reserves were being set up in the 1890s. We read and knew what they were saying, and there were elders in the 1970s and 1980s who were repeating those same comments and questions. They all wanted to resolve 'the land question'. They wanted recognition of who and what they were, and they wanted to have some dignity in their own land._ _The constitutional process provided an opportunity....So we entered that process. It didn't take long, however, to find out we were wasting our time in the constitutional process. It was clear nothing was likely to happen, because there was no political will or understanding at the time for anything substantial to happen._\n\n_The negotiation process wasn't available to us\u2014it just wasn't working._\n\n\u2014Medig'm Gyamk (Neil Sterritt), 1992, _It Doesn't Matter What the Judge Said_.\n\nAnother arena in which Aboriginal political struggles took place during the 1980s was created by the repatriation of the Canadian Constitution from Britain to Canada. As part of this process, Aboriginal peoples sought to have recognition of their inherent Aboriginal title and rights entrenched and given constitutional protection. Some British Columbia Indians, fearing that repatriation would jeopardize what they considered their direct relationship with the British Crown, recognized by the Royal Proclamation of 1763, mounted an international campaign to postpone repatriation of the Constitution until Aboriginal issues were dealt with. They were not successful in halting the repatriation process, but did succeed in wresting a commitment from the federal and provincial governments to include Aboriginal representatives in formal discussions concerning the formulation of the new _Constitution Act_. Negotiations took place during a series of five First Ministers' Conferences held between 1983 and 1987, and resulted in the Meech Lake Accord that, critics charged, failed to adequately protect the rights of Aboriginal peoples, Qu\u00e9becois and women.\n\nThe limited victory won by Aboriginal peoples in the Constitution debates was represented by Section 35(1) of the _Constitution Act, 1982_ which states: \"The existing aboriginal and treaty rights of the aboriginal peoples of Canada are hereby recognized and affirmed.\" This clause is described as an \"empty box\" which litigation and further negotiations must fill. In other words, it was left to the courts and to negotiations between federal, provincial and Aboriginal representatives to define what these \"already existing\" Aboriginal rights were: what is the content of these rights? What do they actually mean, to whom, in everyday life? Are these rights \"inherent,\" arising from Aboriginal peoples' having been the prior occupants and rulers of the land now known as Canada; or are they \"delegated,\" their legal and political source relying on recognition by the Crown? What would constitute legitimate extinguishment of Aboriginal title and rights: did such extinguishment have to be explicit, or could it be implicitly inferred? Was evidence of Aboriginal consent required to make extinguishment legal?\n\nThe case of _R. v. Sparrow,_ launched in British Columbia in 1984, and finally ruled on by the Supreme Court of Canada in 1990, was the first to begin filling Section 35(1) of the _Constitution Act_ , _1987_ 's \"empty box.\"\n\n**Testing, Testing: From \"Use and Occupancy\" to \"Culture\" in the Sparrow Decision**\n\n_In the Court's view, the reason for concluding that the Musqueam Nation enjoys a right to fish lies not in the presence of state action conferring such a right, but instead arises from the fact that fishing is integral to Musqueam self-identity and self-preservation.... The nature and content of an aboriginal right is determined by asking what the organized aboriginal society regarded as \"an integral part of their distinctive culture\"....To be so regarded those practices must have been integral to the distinctive culture of the above society from which they are said to have arisen._\n\n_...The content of aboriginal rights thus is to be determined not by reference to whether executive or legislative action conferred such a right on the people in question, but rather by reference to that which is essential to or inherent in the unique relations that native people have with nature, each other, and other communities._\n\n\u2014Chief Justice Dickson, Supreme Court of Canada, 1990, _Reasons for Judgment_ , in _R. v. Sparrow_.\n\nThe next, and final, important case that preceded _Delgamuukw v. R._ , was _R. v. Sparrow_. It began in the same year, 1984, that the Gitksan and Wet'suwet'en filed their Statement of Claim. On May 25, 1984, Reginald Sparrow, a member of the Musqueam band, was charged with fishing for salmon using a drift net that was longer than allowed by the Department of Fisheries-issued permit for Indian food fishing. Sparrow defended himself by saying he was practising an \"existing Aboriginal right\" protected under Section 35(1) of the _Constitution Act, 1982_. Sparrow was first found guilty in the Provincial Court of British Columbia. When the case was appealed to the British Columbia Court of Appeal, the appellate court agreed that Sparrow's Aboriginal right to fish had not been extinguished prior to 1982, but ruled that the mesh size regulations of the Department of Fisheries were still applicable. At issue when the appeal of the case went forward to the Supreme Court of Canada was whether the Musqueam First Nation could assert an Aboriginal right to fish that would override federal regulations which required a fishing permit and restricted the use of a drift net to a maximum length of 25 fathoms. The Musqueam asserted that their right to fish was an \"existing aboriginal right...recognized and affirmed by s. 35(1) of the _Constitution Act, 1982_ , and therefore paramount over federal law.\" To be regulated in the exercise of their right by Department of Fisheries and Oceans regulations was inappropriate and unconstitutional, they claimed.\n\nThe Supreme Court of Canada agreed with the Musqueam, and in their decision handed down on May 31, 1990 called for \"a generous, liberal and purposive interpretation of s. 35(1).\" They found first that Aboriginal rights that exist in common law are recognized and affirmed by s. 35(1) of the Canadian Constitution that is paramount over all other laws. The Constitution, in other words, is the supreme law of the land. As a result, laws that interfere with the exercise of constitutionally-protected Aboriginal rights must conform to constitutional standards of justification. For example, where a resource, like fish, is scarce, Aboriginal rights should take precedence over commercial and sports interests and be limited only by the requirements of conservation of the resource.\n\nThe _Sparrow_ decision reiterated that the Crown must show a clear, plain and explicit intention to extinguish Aboriginal title, and that laws of general application applied to Indians should not be construed as having effected \"implicit extinguishment.\" It is a common legal strategy in Aboriginal title litigation for lawyers for all parties to present judges with a number of possible interpretations: a kind of prioritized \"wish list\" of rulings their clients could live with. These various alternative arguments need not be logically consistent, and often begin from radically different premises. The Province of British Columbia's legal argument in support of \"implicit extinguishment\" constitutes such an alternative, or \"fall back\" position. Should a judge find that Aboriginal title and rights _did_ exist at the time the British arrived, and therefore that some form of extinguishment of Aboriginal title _was_ required, the Province of B.C. puts forward their argument that colonial governments in British Columbia have consistently demonstrated their _implicit_ intention to extinguish Aboriginal title. To support this claim, the Crown presents evidence from the legal and historical record that documents how provincial courts and governments have denied and ignored First Nations' assertions of Aboriginal title since 1871. They argue, for example, that the fact that provincial wildlife acts were legislated without regard to Aboriginal rights constitutes support for the Crown's claim that Aboriginal rights were implicitly extinguished. This argument is an example of a common form of legal reasoning in which the law continually refers back to itself for justification and confirmation: contemporary irrationality and immorality is justified by its historic depth and breadth, and strengthened each time it is repeated.\n\nThese two positions\u2014 _implicit extinguishment_ versus _explicit extinguishment_ \u2014 has represented the boundaries and limitations of the Crown's legal position, and the Canadian government's political position, in relation to Aboriginal peoples. A third possibility, that Crown and Aboriginal titles might _co-exist_ , is cast outside of this paradigm.\n\nThe Supreme Court, in _Sparrow,_ affirmed the findings in _Guerin_ in relation to the Crown's fiduciary obligations and noted that the relationship between Aboriginal peoples and the Crown should be \"trust-like\" rather than \"adversarial.\" Native peoples do exist in a hierarchical relationship with the Crown, but this should be tempered by the Crown's fiduciary obligation to act as protector of Aboriginal people. Further, adequate consultation with Aboriginal peoples was clearly a pre-condition in determining whether laws that infringe on Aboriginal rights were constitutional. This interpretation of \"existing Aboriginal rights\" specifically allowed that the practice of such rights should be reasonably seen as necessarily evolving over time: \"a modernized form of such a practice would be no less an aboriginal right.\" That is, the fact that Reginald Sparrow now fished with high-tech gear rather than the implements his ancestors may have used two hundred years ago was not to be construed as evidence of his \"assimilation\" into Canadian society, or the transformation of his fishing practices into \"non-Aboriginal\" activities.\n\nThe Supreme Court of Canada's ruling in _Sparrow_ acknowledged reliance on historical interpretation and expert evidence that contested interpretations presented by the Crown. The judges were particularly impressed by the testimony of anthropologist Dr. Wayne Suttles. The _Reasons for Judgment_ in _R. v. Sparrow_ state that \"the anthropological evidence relied on to establish the existence of the right suggests that, for the Musqueam, the salmon fishery has always constituted an integral part of their distinctive culture. Its significant role involved not only consumption for subsistence purposes, but also consumption of salmon on ceremonial and social occasions. The Musqueam have always fished for reasons connected with their cultural and physical survival.\"\n\nOn the central question of Aboriginal title, the _Sparrow_ decision remained adamant on the fundamental, foundational issue. It stated: \"It is worth recalling that while British policy toward the native population was based on respect for their right to occupy their traditional lands, a proposition to which the Royal Proclamation of 1763 bears witness, there was from the outset never any doubt that sovereign and legislative power, and indeed the underlying title, to such lands vested in the Crown.\"\n\nThe Supreme Court of Canada judges were satisfied that the Musqueam had met all the requirements set out in the Bear Island test articulated by Justice Steele in 1985. In reconfiguring the legal test for Aboriginal title to require that an Aboriginal right be \"integral to the distinctive culture\" of the people in order to be legally recognized, the Supreme Court's ruling in the _Sparrow_ case reflected emerging concerns at the forefront of western political culture as the twentieth century neared its end. \"The category of 'culture' once and still endlessly debated by critics and ethnographers in mostly sequestered precincts,\" writes cultural critic Stanley Aronowitz, \"has, over the past decade, become a major political issue.... The 1970s and 1980s were marked by the rise of 'new' cultural and political identities\u2014gender, racial, sexual, national and ethnic\u2014that manifested themselves as challenges to established powers,\" he concludes. In Canada, public political debate during the 1970s and 1980s was dominated by campaigns, referenda and hysteria about Canadian unity and the possibility of Quebec separating and forming a sovereign nation state. \"Sovereigntists\" argued that the Qu\u00e9becois constitute what came to be called a \"distinct society.\" Debates about whether this distinction should be defined on the basis of territory, language, or culture, raged. Questions about Aboriginal sovereignty or rights, and what could constitute distinctly Aboriginal cultures, were inevitably linked to, and often submerged by, the conflict between the two self-described \"Founding Nations\" as they sought to discover each other.\n\nFrom a focus on materialist conceptions of technology and economic organization as the important determinants of social organization, and reflected in legal tests for Aboriginal title based in historic use and occupancy, western intellectual debates increasingly turned to a growing fascination with more idealist issues like spiritual beliefs, ceremony, ritual, language and the arts as the defining elements of people's political and social relationships with each other. \"Culture\" had become the principle marker of the difference between Aboriginal and non-Aboriginal people in Canada. It remained now for the law to define and codify \"culture.\"\n\n\"A characteristic of law,\" writes French sociologist Pierre Bourdieu, \"is its compulsion to capture and codify all that enters its field.\" Visions, ideas, \"cultures,\" and images of possible ways of being that dare to challenge the law are like butterflies who have flown into a collector's laboratory where they have been trapped and drained of life. Their wings pinned down and their bodies stretched out for maximum surveillance; they are encased in glass forever. A tag classifies and labels them, defining and identifying their place in the cosmos, as it has always been and always must be, forever fixed. The prospect of the Supreme Court of Canada determining what is and what is not \"Aboriginal culture\" is foreboding indeed.\n\n**Ping Pong**\n\n_You have to go back to the mid-70s and the early 80s to understand why we did what we did. Because the provincial government refused to negotiate, and they said it, at least once a year, if not more often._\n\n\u2014Medig'm Gyamk (Neil Sterritt), 1992, _It Doesn't Matter What the Judge Said_.\n\nSeveral significant victories had been won by Aboriginal peoples in court during the 1980s, and the Supreme Court of Canada headed by Chief Justice Brian Dickson had shown itself to be relatively sympathetic to Aboriginal claims. Dickson had articulated the principle that interpretations of treaties, statutes, and the Constitution itself should be favourable towards Aboriginal peoples, and that, in judicial dealings with First Nations, the courts should endeavour to uphold \"the honour of the Crown.\" The _Guerin_ case in 1984 had resulted in a ruling that the Crown was bound by a legally enforceable fiduciary obligation to manage Aboriginal assets in the best interests of Aboriginal peoples, rather than in favour of non-Aboriginal public or private interests as the historical practice\u2014illustrated by this case\u2014had been. Dickson's court had insisted that Aboriginal or treaty rights could only be extinguished by governments expressing their intention to do so in \"clear and plain language,\" and that colonial governments simply superimposing their law and government on Aboriginal peoples without their consent did not constitute legal extinguishment of pre-existing Aboriginal rights. If such unextinguished rights were shown to exist, so the thinking in the Supreme Court of Canada in the 1980s went, then interference with those rights by others\u2014like provincial governments\u2014would have to meet certain tests before it could be justified.\n\nWhen the Gitksan and Wet'suwet'en filed their Statement of Claim initiating _Delgamuukw v. R._ in 1984, the federal comprehensive claims policy, strangling itself and everyone involved in it in red tape, had almost ground to a halt across the country. The provincial government of British Columbia still refused to recognize the existence of First Nations, or to join in any negotiations on comprehensive claims with the federal government. The Constitution talks were at a standstill regarding the inclusion of clauses on Aboriginal rights. By the time the Gitksan and Wet'suwet'en trial began in 1987, litigation appeared as the only viable option left for First Nations in British Columbia to seek recognition and respect for their right to continue to exist as distinct peoples that they had been seeking for over 100 years.\n\n* * *\n\n. Medig'm Gyamk 1992: 303.\n\n. Lorde 1984.\n\n. Coolichan 1985.\n\n. Tennant 1990(b).\n\n. _Hamlet of Baker Lake et al v. Minister of Indian Affairs and Northern Development et al_ [1980] 5 WWR 193, 50 CCC (2d) 377 (FCTD).\n\n. Ibid., 202.\n\n. Ibid., 208.\n\n. Ibid., 213.\n\n. Ibid., 211.\n\n. Ibid., 212.\n\n. Ibid.\n\n. Ibid., 214.\n\n. Ibid., 210.\n\n. In _Re: Southern Rhodesia_ (1919) A.C. 211.\n\n. Ibid.\n\n. _Hamlet of Baker Lake et al v. Minister of Indian Affairs and Northern Development et al_ [1980] 5 WWR 193, 50 CCC (2d) 377 (FCTD), 219.\n\n. Ibid., 59-64.\n\n. Henderson W. B. 1991: 9-10.\n\n. Elias 1993.\n\n. _A.G. Ontario v. Bear Island Foundation_ [1985], 49 O.R. (2d) 353, 15 D.L.R. (4th 321 (Ont. H.C.); _A.G. Ontario v. Bear Island Foundation_ [1989] 68 O.R. (2d) 394, 38 D.L.R. (4th) 117 (Ont.C.A.); _A.G. Ontario v. Bear Island Foundation_ [1991], 83 D.L.R. (4th), 381.\n\n. Bray and Thompson 1990; Clark 1990.\n\n. Steele, J. (1985) _Reasons for Judgment_ , in _A.G. Ontario v. Bear Island Foundation_ [1985], 49 O.R. (2d) 353, 15 D.L.R. (4th 321 (Ont. H.C.), 15.\n\n. Ibid., 21.\n\n. _Kruger and Manual v. The Queen_ [1977] 4 WWR 300, [1978] 1 SCR 104, 75 DLR (3d) 434, 14 NR 495, 34 CCC (2d), 377.\n\n. Dickson, C.J., (1977) _Reasons for Judgment_ , in _Kruger and Manual v. The Queen_ , 377.\n\n. _Simon v. R_. (1985), 24 D.L.R. (4th), 390.\n\n. The use of \"J\" after a judge's name is a legal convention representing the title \"Justice,\" i.e. writing \"Patterson J.\" has the same meaning as the title \"Justice Patterson.\"\n\n. Dickson, C.J. (1985) _Reasons for Judgment_ , in _Simon v. R._ , 400.\n\n. _A.G. Ontario v. Bear Island Foundation_ [1991], 83 D.L.R. (4th), 381.\n\n. Fortune 1993: 87-88.\n\n. _post hoc_ is defined in _Webster's New Collegiate Dictionary_ (1981) as \"the fallacy of arguing from temporal sequence to a causal relation,\" 891. Simply put, rationalizing an action after the fact.\n\n. Elias 1993: 270.\n\n. Medig'm Gyamk 1992: 303.\n\n. For thorough discussions of Aboriginal issues and the Constitutional debates see Asch 1984. See Hawkes 1989 for discussion of the negotiation process. See Sanders 1983(a) for a cogent analysis of the legal issues involved. See Schwartz 1986 for a critique of Aboriginal positions and strategies.\n\n. _R. v. Sparrow_ (1990), 70 D.L.R. (4th), 401-402.\n\n. _R. v. Sparrow_ (1990), 70 D.L.R. (4th), 430.\n\n. \"The common law doctrine of Aboriginal rights holds that the Crown's acquisition of North American territories was governed by a principle of continuity whereby the property rights, customary laws, and governmental institutions of the native people were presumed to survive, so far as this result was compatible with the Crown's ultimate title, and subject to lawful dispositions to the contrary.\" Slattery 1985:118.\n\n. _R. v. Sparrow_ (1990), 70 D.L.R. (4th), 417.\n\n. _R. v. Sparrow_ (1990), 70 D.L.R. (4th), 402.\n\n. _R. v. Sparrow_ (1990), 70 D.L.R. (4th), 404.\n\n. Aronowitz 1993: 18.\n\n. Bourdieu 1987: 895.\n\n. Medig'm Gyamk 1992: 303.\n\n. Sanders 1992.\n**P ART IV:**\n\n**E CCE SIGNUM**\n\n**(B EHOLD THE PROOF)**\n**Chapter 8: Delgamuukw versus The Queen**\n\n_Law is, of course, only one domain in which a culture may reveal itself._ _But like politics, marriage, and exchange, it is an arena in which people must act, and in doing so they must draw on their assumptions, connections, and beliefs to make their acts effective and comprehensible...._ _Like a single ritual, a network of exchange, or a contest for political leadership, one cannot hope to see all of a society through such a limited focus. But such an example can show us what it means to speak of law as culture and to trace out some of the implications this approach may have to offer...The world of formal courts offers a stage\u2014as intense as ritual, as demonstrative as war\u2014through which a society reveals itself to its own people as much as to the outside world._\n\n\u2014Anthropologist Lawrence Rosen, 1989, _Islamic Law and the Logic of Consequences_.\n\n**All the World's a Stage...**\n\nTrials have long served as ideal settings for writers of drama, satire and farce. The _Delgamuukw_ trial could well be seen as living theatre at its best. 'Actors' representing all the key social categories of players in British Columbia history and contemporary life were there. The Aboriginal peoples, the Crown lawyers, the expert witnesses, and the Judge offered answers to the central question that has been in dispute in the Canadian west for two centuries: who owns the land? The Crown's legal representatives defended the \"founding myth of White British Columbia,\" and their expert witnesses brought forward evidence unencumbered by the inclusion of Aboriginal or other critical voices. The Gitksan's and Wet'suwet'en's lawyers, well known for their work in labour, women's, prisoners' and human rights movements, represented \"the new British Columbians,\" who do not see themselves in the original European settlers' official founding myths. The non-Aboriginal expert witnesses called by the Gitksan and Wet'suwet'en were part of a new generation of academics who came of age during the era of decolonization, and whose work was influenced by these experiences and intellectual discourses. For four years they faced each other and argued about who they were, where they came from, where they belonged, and how their children could live together in this geographical space called British Columbia, Canada. While their debates focused on the past, the real struggle was about the future: what would it look like, and who would shape it?\n\nThe narrative I recount in the following chapters chronicles the trial of _Delgamuukw v. R_. as it unfolded. The Gitksan and Wet'suwet'en, as the plaintiffs, presented their case first. The onus was on them to \"prove\" their claims. The Crown, as defendants, had principally to rebut the plaintiff's case, and only secondarily to demonstrate the merits of their counter-claim. For four years witnesses for both the plaintiffs and the defendants testified on behalf of one side, and were cross-examined by lawyers for the other. Their arguments and counter arguments revealed not only their respective stances on the issues before Chief Justice McEachern in this case, but also their visions of the past and aspirations for the future.\n\n**Do Aboriginal People Really Exist?**\n\n_When they ask us what we feel our basis is in regard to our title, I don't think that there is any question. The title is very clear. The ownership has never changed. It is only the definition in the law in regard to ownership that has changed. When you're talking about justification and why we are and what we are, you're talking about history. It's amazing; when we become educated within your system, we're told that history is very important. Yet when you look at the history of Canada, where are we as aboriginal people? We're not included in this history. Yet our history is very rich._\n\n\u2014Miluulak (Alice Jeffrey), Gitksan Hereditary Chief, 1992, _Remove Not the Landmark_.\n\n_Like I said, our culture was on trial. We were told we didn't exist...._ _Ownership and jurisdiction is not something new for us. We have practised that. It is thousands of years old. We only want recognition...._ _We are right here before your very eyes._\n\n\u2014Yagalahl (Dora Wilson), Member, Gitksan-Wet'suwet'en Litigation Team, 1992, _It Will Always Be The Truth_.\n\nThe questions facing Chief Justice McEachern in the _Delgamuukw_ case were the same ones every judge hearing an Aboriginal title and\/or rights case in the British Empire-cum-Commonwealth, from the eighteenth century to the present, had addressed. They were:\n\n(1) Did the Indigenous peoples in question own and manage their lands and resources before Europeans arrived?\n\n(2) If so, have those property rights been extinguished by law at some time in history, or have they continued into the present?\n\n(3) If unextinguished Aboriginal rights continue to exist, what are they?\n\n(4) If unextinguished Aboriginal rights continue to exist, how can they be lawfully extinguished and\/or justifiably contravened?\n\nFundamental moral, philosophical and political beliefs underlie the answer to the initial question: were the ancestors of contemporary Aboriginal peoples in Canada members of an organized society at the time Europeans first made contact with them?\n\nLike many questions in law, if this is read literally, it is an oxymoron: a society, by definition, is organized. What could an unorganized society be? To understand what is really being asked, we have to step outside literal interpretations and common sense and enter into the historically and culturally specific universe of meaning that is British and Canadian colonial law. Just like anthropologists seeking to understand a foreign culture, we must try to interpret the question from within this perspective and seek to understand the culture of law from the inside.\n\nAn \"organized society,\" according the rules of \"law's culture,\" qualifies as such if it meets certain criteria specified by legal precedents, which themselves rely on various social theories for support and validation. The precedent that the first question in the legal test for Aboriginal rights refers to is Lord Sumner's 1919 decision in _Re: Southern Rhodesia_. The empirical information that an answer to this first question requires is knowledge of the ways particular Aboriginal peoples lived before contact with Europeans: the ways they managed their properties and resources; the ways they gathered, processed, stored and distributed food; the ways they celebrated births, and mourned deaths; the ways they marked marriages and divorces; the ways they traded and socialized with their neighbours, maintained social order, and worshipped their gods. Stepping back outside of \"law's culture,\" the question appears as: will a British, or Canadian, court recognize these ways of life as those they have already defined as being representative of what they call \"an organized society?\" How will the law answer the questions it has posed for itself, by the criteria it has established for itself?\n\nThe Gitksan and Wet'suwet'en argued that they were here, living in a society that was by definition, description, and documentation, organized. In _Delgamuukw v. R_., the provincial government of British Columbia argued for the application of Lord Sumner's test and the _Re: Southern Rhodesia_ precedent, claiming that the Gitksan and Wet'suwet'en were so \"low on the scale of social organization\" that British Columbia was justified in declaring the territory in question _terra nullius_ in the early nineteenth century.\n\nThe second question in the legal test for Aboriginal rights: \"Did the ancestors of the Aboriginal claimants occupy a specific territory?\" arises from the first. Only an organized society can occupy a specific territory. The third and fourth questions: \"Did they exclude other organized societies?\" and \"Was occupation established at the time British sovereignty was established?\" arise from the second. To answer these questions, evidence has to be produced that shows that Aboriginal peoples recognized identifiable borders that marked the lines between their territory and that of others. The Gitksan and Wet'suwet'en tried to demonstrate that both they and their neighbours knew the boundaries of their respective territories; although, like neighbours everywhere, they sometimes argued about exactly where lines should be drawn. The Crown argued that Aboriginal peoples in British Columbia were nomadic wanderers who lacked abstract concepts of property.\n\nIn the _Calder_ case nearly twenty years earlier, the Nishga had asked the court only for a declaration that \"...aboriginal title, otherwise known as the Indian title...has never been extinguished.\" The case did not address what such Aboriginal title might include. The Gitksan and Wet'suwet'en tried to push the question beyond these very general concepts. _Delgamuukw_ 's claim was for legal recognition that Aboriginal title means ownership, jurisdiction, rights, interests, and financial compensation for resources exploited and destroyed between the arrival of Europeans and now. The Gitksan and Wet'suwet'en petition to the Supreme Court of British Columbia sought a declaration that:\n\n...from time immemorial they and their ancestors have occupied and possessed approximately 22,000 square miles in north-west British Columbia (\"the territory\"), and that they or the Indian people they represent are entitled, as against the Province of British Columbia, to a legal judgment declaring: (a) that they own the territory; (b) that they are entitled to govern the territory by Aboriginal laws which are paramount to the laws of British Columbia; (c) alternatively, that they have unspecified Aboriginal rights to use the territory; (d) damages for the loss of all lands and resources transferred to third parties or for resources removed from the territory since the establishment of the colony; and, (e) costs.\n\nThe Nishga claim had been based on a model of collective or communal ownership. The Gitksan and Wet'suwet'en claim, however, was made by each of 35 Gitksan and 13 Wet'suwet'en chiefs, led by Delgamuukw (a.k.a. Ken Muldoe), representing themselves and all the members of their Houses, for ownership of specific territories. Their claim to jurisdiction, or self-government, was two-fold. First, the chiefs asserted their right to govern land-related activities on the 22,000 square miles that make up Gitksan and Wet'suwet'en territory, free of interference by provincial laws like the Wildlife Acts. This claim applied only to Crown lands and did not include fee simple private property owned by individuals who had purchased their land in good faith from the Crown. Second, they claimed the right for Gitksan and Wet'suwet'en laws to supersede the general laws of the Province of B.C. in Gitksan and Wet'suwet'en territory, particularly in relation to education; health; family relations like marriages, divorces, adoptions and child welfare; justice and law enforcement.\n\nDelgamuukw delivered his \"Opening Address to the Court,\" on behalf of the Gitksan and Wet'suwet'en Hereditary Chiefs in Smithers, B.C., on May 11, 1987. He began: \"The purpose of this case...is to find a process to place Gitksan and Wet'suwet'en ownership and jurisdiction within the context of Canada. We do not seek a decision as to whether our system might continue or not. It will continue.\"\n\nIn seeking such a place within Canadian law, the Gitksan and Wet'suwet'en agreed that, for the purposes of entering into the legal process, they would not challenge the notion that what the courts call \"the underlying title to the land\" was held by the hovering sovereign symbolically represented by the Crown, (and embodied from time to time in judges, bureaucrats, soldiers, and Mounties.) They also agreed not to contest the requirement established by the Royal Proclamation of 1763 that Aboriginal land can only be sold to the Crown, and not to private corporations or non-Aboriginal individuals. Subject only to these two constraints, the Chiefs said, they, as representatives of their Houses and its members, own \"and are absolutely entitled to occupy and possess\" the 133 individual territories claimed, in a manner that is \"for all purposes equivalent to ownership in fee simple.\"\n\n* * *\n\n. Gisday Wa and Delgam Uukw 1987: 1.\n\n. Ibid., 3.\n**Chapter 9: The Gitksan's and Wet'suwet'en's Case**\n\n_So what we say is we have title and that is why we are talking to you about aboriginal rights, but we are not talking English Common Law definitions, international law definitions that have been interpreted and reinterpreted and sometimes extinguished by conquest and ceding treaties and other agreements like that. We are talking about the feeling that is inside all of us as Metis, Indian and Inuit people that this country belongs to us.... My whole point is that we must stop viewing aboriginal rights from the point of view of the dominant society if we are ever going to understand what the Indian people, the Inuit people, and the Metis want...._\n\n_The question is whether there is a means of understanding this concept from the Native point of view._\n\n\u2014Aboriginal leader Bill Wilson, 1983, address to the First Ministers' Conference on Aboriginal Constitutional Matters, Ottawa.\n\n**A Native Point of View**\n\nEach major Aboriginal title and rights case adjudicated during the nearly twenty years that elapsed between the _Calder_ case and the _Delgamuukw_ case had resulted in elaborations of the legal tests required to establish Aboriginal title and rights in law. An intricate configuration of abstract legal and anthropological arguments, expressed in increasingly obscure and mystifying language, was constructed through the litigation process from _Calder_ onwards. Since the tests for Aboriginal title and rights had been developed _post hoc_ by judges, litigants and their lawyers when preparing for and entering a trial could know what evidence previous courts had accepted or rejected, and been convinced or dissuaded by. They could only guess, however, at what issues might emerge during the course of their particular trial, and what an individual judge might decide to require in terms of evidence. It is often not until Reasons for Judgment are written that the reasoning used to evaluate the relative weight of evidence, and reach final conclusions, is revealed. The increasingly complex nature of the tests' questions required increasingly detailed and comprehensive research. This in turn necessitated the employment of an ever more varied range of professional consultants and legal advisors, thus driving the duration and costs of litigation higher and higher.\n\nIt bears remembering that the need for these expenditures from the public purse was created by judges and courts, and that few of the economic benefits generated flowed into Aboriginal coffers. It bears remembering, too, that the questions at the core of all this have been relatively simple and straightforward ones. Were Aboriginal people living here in \"organized societies\" before Europeans arrived? Answer: Yes. Has their ownership of the land ever been ceded, sold, treatied, or lost in war? Answer: No. Was this legal according to British imperial law? Answer: No, not unless Aboriginal people are categorized as insufficiently \"evolved,\" or in some other way inferior to Europeans. Is this just according to historical or contemporary standards of morality or the letter of laws? Answer: No. It could be argued that continually rewriting the questions, redefining the qualifying tests, and reinterpreting what constitutes correct responses, has substituted for a just answer to these questions. What should be done in order for Canada and British Columbia to create \"justice in our time\" in relation to Aboriginal title, rights, and persons? This is the _real_ question before the courts, the governments, and the Canadian population.\n\nAt the same time, however, moving away from the simplistic \"common sense\" logic of _Calder_ 's \"prior historical use and occupancy\" test, and ultimately ethnocentric analogy with English common law principles as a basis for recognition of Aboriginal title and rights, has created a space for Aboriginal peoples to elaborate their unique understandings of their distinct cultures and relationships to the land. This has provided an opportunity for non-Aboriginal peoples to learn about, and begin to develop an understanding of, Aboriginal cultures on their own terms, rather than always in relation to, or in comparison with, Euro-Canadian cultures. The trap of common sense is that it resists and rejects what is not already known to it, and understands much about social life and history as being \"natural\" and \"inevitable,\" rather than created by people themselves. Failing to move beyond \"common sense\" can limit the possibility of new visions of life and society emerging and being reflected upon.\n\nStuart Rush, one of the lawyers for the Gitksan and Wet'suwet'en, explained the legal team's strategy in _Delgamuukw v. R._ \"This Court,\" Rush said, \"in hearing the evidence which will be presented in this case, will be faced with a series of legal and intellectual challenges and opportunities of a nature not normally found in matters that come before the bench.\" Rush set out the first challenge to the Court as being to \"reflect critically on European, or Euro-Canadian, ethnocentrism:... to understand and overcome the tendency to view aboriginal societies as existing at an earlier stage of evolutionary development.... We will be inviting this Court, through its rulings, to reject any legal theory of aboriginal rights which depends upon such evolutionist and supremacist assumptions.\" The second challenge described by Rush was \"the problem of communication between very different cultures.\"\n\nThe Gitksan and Wet'suwet'en Hereditary Chiefs, in their opening address to the court, challenged Chief Justice McEachern to hear them on their own terms.\n\n\"Never before has a Canadian court been given the opportunity to hear Indian witnesses describe within their own structure the history and nature of their societies,\" Delgamuukw began. \"The evidence will show that the Gitksan and Wet'suwet'en are and have always been properly counted amongst the civilized nations of the world; that their ownership of their territory and their authority over it has always existed; and that they have shaped a distinctive form of confederation between House and Clans. The challenge for this court is to hear this evidence, in all its complexity, in all its elaboration, as the articulation of a way of looking at the world which pre-dates the Canadian Constitution by many thousands of years,\" he continued.\n\nA world view, as defined by anthropologists, is composed of two interrelated parts: a notion of how the world is structured and how its parts form a cohesive whole; and, a set of rules which set the structure in motion and control and direct it. The fundamental features of the Gitksan and Wet'suwet'en world view, and the central differences, or oppositions, between it and \"the western world view\" were described by the Chiefs in their opening statement. \"The western world view,\" Delgamuukw explained, \"sees the essential and primary interactions as being those between human beings.\" The Gitksan and Wet'suwet'en world view, on the other hand, understands human beings to be \"part of an interacting continuum which includes animals and spirits.\" Secondly, in the Gitksan and Wet'suwet'en world view time is cyclical and past events \"directly effect the present and the future,\" whereas in the western world view, time is linear and causality more direct. Thirdly, the Chiefs argued, westerners believe in accidents and coincidences, whereas Gitksan and Wet'suwet'en do not. The western world view makes fundamental distinctions between sacred and secular, spiritual and material, natural and supernatural, while the integration of these elements, \"infuses the Gitksan and Wet'suwet'en thinking and major institutions,\" the Hereditary Chiefs' opening statement concluded.\n\nThe answers to a number of the legal questions facing the court in _Delgamuukw v. R._ lie in descriptions of Gitksan and Wet'suwet'en ways of life as they were before European fur traders arrived in the late eighteenth century. The Gitksan and Wet'suwet'en, having made the decision to pursue their claims through the judicial system, endeavoured to both respect the terms and conditions of the court, and at the same time to maintain their own integrity and identity. They attempted to bring together the best of both worlds in areas of specialized knowledge, and present the most highly qualified people available\u2014according to the criteria of both Aboriginal and mainstream Canadian cultures\u2014as expert witnesses to the court. The Gitksan and Wet'suwet'en litigation team studied previous cases and endeavoured to anticipate as many questions and challenges as possible. In addition to the testimony of Chiefs and Elders, the litigation team marshalled a tremendous amount of scholarly evidence to support their case, and attracted the support and participation of an impressive array of well-respected professional and academic witnesses.\n\nThe tradition of expert witnesses in British law dates back to the sixteenth century, when assemblies of neighbours and colleagues familiar with the events and persons entangled in a dispute were replaced by juries of presumably objective arbiters whose qualifications were that they knew nothing of the case, or the parties involved. Expert witnesses become necessary when judges and\/or juries believe that the issues before the court require explanation and specialized knowledge beyond that which either a legal expert or an ordinary lay person can be expected to possess. Psychiatrists, asked to assess whether or not an accused person is sane enough to have planned and carried out the actions they are accused of, are the expert witnesses most familiar to the public through newspaper reports and television shows. In the _Delgamuukw_ case, lawyers for the Gitksan and Wet'suwet'en argued that the Court had to try to gain understanding \"across both a profound cultural divide, and from a distance of 200 years and more.\" The expert knowledge referred to here consists in the unique understandings of life and cosmos embedded in different cultures over time. The best authorities on knowledge about Gitksan and Wet'suwet'en history and contemporary life were Elders, Chiefs, and members of Gitksan and Wet'suwet'en communities. They were the first expert witnesses to testify.\n\n**Experts in Living Memory**\n\n_\"It is very sad to say that some of the most powerful witnesses in the court case...have now died, but they saw within themselves the strength and the need to go and give their evidence in court.\"_\n\n\u2014Medig'm Gyamk (Neil Sterritt), 1992, _It Doesn't Matter What the Judge Said_.\n\nThe main evidence in support of the Gitksan and Wet'suwet'en claim to having lived in an organized society, with borders and laws, and of having owned, then and now, their territories, was to be presented by Chiefs and Elders through the telling of their two particular kinds of oral history: _adaawk_ and _kungax_. Before this testimony could be heard by the court, however, the Gitksan and Wet'suwet'en and their lawyers had to overcome a legal obstacle: the Crown argued that oral tradition should be considered inadmissible under what is known as the \"Hearsay Rule.\" According to the law, the Crown claimed, oral histories constitute hearsay, since they purport to rely on the words and experiences of people who are deceased, and are not available for cross-examination.\n\nThe Gitksan and Wet'suwet'en chiefs and elders, and their lawyers and witnesses attempted to help the Chief Justice understand what he would be witnessing. In the opening address to the court, the Hereditary Chiefs instructed the judge that the _adaawk_ and _kungax_ should not be taken literally in a simplistic sense. They drew analogies between distinctions made in both Aboriginal and western cultures between experience and hearsay, opinion and knowledge, lay people and experts. The Gitksan and Wet'suwet'en explained that their oral traditions describe the genealogies of various clans and families, and their relationships with their lands and resources. They explain cosmology and spiritual relationships and obligations. They document ownership of lands and resources, transactions, relations with neighbours, and historical events. _Adaawk_ are Gitksan oral histories comprised of a collection of sacred reminiscences about ancestors, histories and territories that document House ownership of land and resources. The Wet'suwet'en _kungax_ is a song, or songs, about trails between territories. The rights to perform particular _adaawk_ and _kungax_ are part of the privileges, like clan crests, that are inherited and stewarded by individuals and House groups when they take ownership and responsibility for the specific territories the oral histories tell about. Learning the content of the narratives, and the specific conventions of oratory and style with which they must be delivered, as well as coming to know how to properly perform the songs and dances that are, like the narratives, essential aspects of _adaawk_ and _kungax_ , takes many years. Chiefs and Elders are, therefore, the custodians of this specialized knowledge.\n\nGitksan and Wet'suwet'en oral tradition includes the feasting complex through which various oral accounts may be validated or contested by the people as a whole. When a host Chief and his House hold a feast to mark an important event, such as the transfer of property, the guests assembled serve as witnesses to the event or transaction being marked by the feast and they watch and listen to the performance of the _adaawk_ or _kungax_ at the feast gathering. If they are persuaded that the laws have been properly followed, the guests\/witnesses validate the event or transaction by accepting the host's offers of food and gifts. If they disagree, they make their objections known by making a speech explaining their position and refraining from accepting anything from the host that might be construed as witnessing, validation or affirmation. Sometimes disputes are resolved within the framework of a single feast, and sometimes they remain contentious over longer periods of time. The Statement of Claim filed by the Gitksan and Wet'suwet'en said expressions of ownership come through the _adaawk_ , _kungax_ , songs and ceremonial regalia; confirmation of ownership comes through totem poles erected to give those expressions a material base; assertion of ownership is made through specific claims. Before the Gitksan and Wet'suwet'en court case, neither _adaawk_ or _kungax_ had been performed outside of feast halls. \"This was the first time our people brought forth what you call evidence. It isn't evidence to us. We call that our _Adaawk_.\" Gitksan Elder Miluulak (Alice Jeffrey) explained.\n\nAnthropologists and ethnohistorians who submitted expert opinion reports on behalf of the Gitksan and Wet'suwet'en elaborated on the various methods social scientists use to analyze oral histories. Oral histories may be subjected to the familiar historical method of verifying when and where specific events took place by examining a variety of sources and comparing accounts of particular events. This involves cross-referencing information from oral histories; ethnographies; historical documents; archaeological studies; and biophysical data on technology, resource and land use, and social organization to assess the degree to which \"real events\" or \"literal truth\" is reflected in the oral accounts. Richard Daly, an anthropologist who testified for the Gitksan and Wet'suwet'en, pointed out in his opinion report that \"oral history is not the same as the individual reminiscences of an elder.\" During his testimony, Daly reiterated in significant detail the section of his expert opinion report in which he explained the methodology he had employed in his analysis. \"Oral tradition should be treated as a whole,...a corpus of linked and overlapping records of events that have been reiterated down through the generations. Individual tellings of one Chief's history must be compared with one another, and then with the accounts of the same or related events from the viewpoint of other Chiefs. When this is done carefully, the oral tradition can be treated as a valid historical source,\" Daly wrote.\n\nScholars who employ the methodology outlined by Daly classify the oral histories of some traditions as more factually reliable by these criteria than others. Expert witnesses repeatedly referred to the particular richness of Gitksan and Wet'suwet'en oral histories, and to the rigorous demands for validation through feasting that have resulted in the _adaawk_ and _kungax_ being ranked as valid and dependable historical sources by generations of anthropologists and ethnohistorians.\n\nSecond, a rigorous analysis of oral history involves the even more complex task of interpreting the many layers of meaning revealed by oral tradition when it is understood within the cultural context of its performance. Such an analysis requires the interpreter to describe and explain the different historical and local contexts in which oral tradition is transmitted, including the political dynamics involved in various forums, the reputations of different speakers, and the uses of various rhetorical strategies and performance techniques.\n\nFinally, the background knowledge and prejudices that members of audiences bring to the performance of any recitation of oral histories contributes to the interpretation of the meaning of the verbal text. Of course, written history can be critically evaluated in much the same way: documents are written by particular people in particular contexts for particular purposes, and each generation of readers brings to their reading of historical texts new ideas and experiences that would have been unknown to the original scribes. \"Oral tradition differs from western science and history,\" Anthropologist Julie Cruikshank writes. \"But both are organized systems of knowledge that take many years to learn, and both are perpetually open and incomplete.\"\n\nChief Justice McEachern determined that he was satisfied that the _adaawk_ and _kungax_ had gone through a \"sifting process\" through their having been told and retold and witnessed repeatedly, and that lent them an \"enhanced trustworthiness\" in his eyes. But, he added, \"historical facts sought to be adduced must be truly historical and not anecdotal.\" McEachern decided to admit the oral history evidence and testimony, saying he would \"weigh\" it at the end of the trial. He expressed his willingness, however, as the judge of first instance in a case of such importance, to \"lean towards admissibility\" and to \"harken to the evidence.\" The judge said he would use two principal criteria to determine what weight to give the oral histories in relation to other evidence. First, he would try to distinguish between what \"a deceased elder said he or his elders did, and what they believed,\" and allow only the former to stand as evidence. Second, he said he would accept the plaintiff's culturally specific definition and identification of the distinction between folklore (\"antimahlaswx\" in Gitksan), and laws or traditions, allowing only the latter to stand as evidence.\n\nChief Justice McEachern also promised to rely on precedents set in cases where the issue of oral tradition testimony had been dealt with. First, he said, he would take from _Kruger v. R._ the direction that evaluation should be based on \"facts pertinent to that Band and to that land, and not on any global basis.\" Second, Allan McEachern referred to another of Brian Dickson's rulings in the case of _R. v. Simon,_ where the former Chief Justice of the Supreme Court of Canada found that to demand written support where oral tradition was the only source of evidence available, would be to \"impose an impossible burden of proof.\" Chief Justice McEachern's decision, early in the trial, to admit oral tradition evidence as an exemption to the Hearsay Rule, gave the Gitksan and Wet'suwet'en and their supporters cause for optimism.\n\nEighteen chiefs testified in person at the trial, beginning with Gyolugyet (Mary McKenzie) in Smithers on May 13, 1987. An additional ten Gitksan and Wet'suwet'en, who were not chiefs, testified, and a further thirty-eight witnesses tendered, and were cross-examined on, territorial affidavits. Thirty of these were chiefs. The Chiefs and Elders hold the responsibility for learning and transmitting the oral tradition\u2014including laws, legends, and histories\u2014of their Houses and clans. These witnesses have earned their credibility, and the respect of their communities, over the course of their lifetimes. They represent an Indigenous aristocracy, gerontocracy and intelligentsia. They are also people who, through their life histories, embody the experiences of Aboriginal-White relations in British Columbia over many decades. In previous land rights litigation, the employment of Elders testifying in their own languages as expert witnesses had been minimal. The more common practice had been to employ, as witnesses, anthropologists who had interviewed Elders, and who, having thus become experts, would translate, represent, and analyze for the court what Elders had told them about their histories. The Gitksan and Wet'suwet'en strategy attempted to replicate this ethnographic process in the courtroom. The Chiefs and Elders testified directly to the judge, with Chief Justice McEachern assuming the role, in the first instance, of the white anthropologist hearing these oral histories for the first time.\n\nIn evaluating the oral histories during the course of the trial, Chief Justice McEachern remained particularly concerned about maintaining a clear distinction between \"historical facts\" and \"beliefs.\" As he later reflected in his _Reasons for Judgment_ , \"At an early stage of the trial I expressed the hope that I could make a convenient but simplistic distinction between what European-based culture would call mythology and \"real\" matters.... I have concluded that it would be overly simplistic to attempt such a distinction, and I must accordingly reject mythology as a valid distinction between what is and what is not part of an _adaawk_ or _kungax_.\" However, as the Chiefs' testimonies proceeded, the judge became increasingly impatient about having to actually listen to the oral histories. He took exception, particularly, to witnesses singing in court. \"This is a trial,\" he reproached an Elder at one point, \"not a performance.\"  Besides, he added, the significance of the music was lost on him as he had \"a tin ear.\"\n\nThe Gitksan and Wet'suwet'en offered the court, in some instances\u2014 notably the Gitksan _adaawk_ of Medeek, and the Wet'suwet'en _kungax_ of the House of Goohlaht\u2014archaeological and geological evidence that supported events and places noted in the oral histories in order to link these with conventional scientific proof. The _adaawk_ of Seeley Lake Medeek tells about a massive landslide accompanied by the presence of a supernatural grizzly bear. Rolf Mathewes, a paleobotanist, testified that his research showed that a massive landslide had occurred in the area around 3,380 years ago. Alan Gottesfeld, a geomorphologist, added his support for the _adaawk_ based on results of research conducted in accordance with the scientific methodology required by his discipline.\n\nSince the Gitksan and Wet'suwet'en were the plaintiffs in the case of _Delgamuukw v. R_., it was their case that was presented first, and the onus was on them to present convincing evidence to prove their claims. The Crown, as the defendants, were required only to protect themselves by refuting and rebutting the plaintiffs' testimonies. Crown lawyers cross-examined the Gitksan and Wet'suwet'en Elders in the antagonistic manner characteristic of the adversarial Canadian legal system. Yagalahl (Dora Wilson), a member of the Gitksan-Wet'suwet'en Litigation Team, recounted how she felt observing this part of the trial: \"There were lots of times where I just felt like screaming: 'Hey, you're wrong. How dare you say this? How dare you do this? How dare you be disrespectful to my elder sitting in that witness stand? How dare you speak to her that way? How dare you speak to him that way?\" \"But,\" Yagalahl remembered, \"one of our witnesses had a lot of fun with one of the lawyers. She is an elderly person. She is in her eighties. She had been to the hunting area, berry-picking area, you name it, on the territories most of her life. So in her evidence, she was talking about times when they went out berry-picking. The lawyer was trying to say that she didn't really know her territory. He was trying to confuse her by using the place names.... Here he was questioning, questioning, and what he was doing\u2014and didn't realize\u2014was he was going up and down, up and down, this mountain that she was describing. Finally she got annoyed, and...she says to the interpreter, 'What's the matter with this lawyer? Is he crazy? We've been up and down that hill a lot of times now.' So sometimes it was funny.\"\n\nIn presenting their case, the Gitksan and Wet'suwet'en asked the court to hear and understand them on their own terms, within the context of the Gitksan and Wet'suwet'en \"world view.\" Chief Justice Allan McEachern, at several points during the trial, cautioned that this evidence was \"subject to objection and weight which would be adduced at the end of the trial.\" He said any evidence could be disregarded \"if it is contradicted, or if its value as evidence is destroyed or lessened either internally or by other admissible evidence, or by common sense.\" In anticipation of such skepticism, the Elders' testimony was augmented by testimony by expert witnesses in cartography, paleobotany, geomorphology, forest ecology, fishery biology, ethnoarchaeology, linguistics, historical geography, anthropology, and history.\n\n**The Ancestors are Watching**\n\n_Sitting in the courtroom, I felt this heavy burden like all of my ancestors were sitting on my shoulders, and anything that was said on the witness stand had to be the truth as we were taught, because the truth, I remember my grandmother saying, is something that you don't forget._ _Because if you tell a lie, at some point in the future, you always get tripped up and it comes out. The politicians have proven that out._\n\n\u2014Yagalahl (Dora Wilson), 1992, _It Will Always Be The Truth_.\n\nLay people, also recognized as expert witnesses by the Court, included members of Gitksan and Wet'suwet'en Houses who had received training through Indigenous forums, and\/or who had obtained professional and academic credentials in relevant fields. After the Elders finished testifying, these younger\u2014mostly middle-aged\u2014Gitksan and Wet'suwet'en took the stand as expert witnesses. These people are apprentices to the Elders in the \"traditional\" system, as well as being members of their own generations with their particular experiences and acquired knowledge. Many work within their own communities and within the institutions of the majority Canadian society in the fields of political negotiation, law, education, health, welfare, and community administration. They represent an emerging Aboriginal professional class, political leadership, and intelligentsia. In order of appearance, the first expert witness to testify on behalf of the Gitksan and Wet'suwet'en after the Elders was Neil Sterritt, hereditary chief and cartographer. He described how the Gitksan and Wet'suwet'en delineated boundaries and how these had been translated for mapping purposes. Sterritt's evidence focused on explicating the subject of internal boundaries between the 133 territories, and between the Gitksan and Wet'suwet'en and their Aboriginal neighbours. He explained how ownership was and is established and property transferred, and through what processes boundaries and ownership could change. He provided numerous illustrations of complex deliberations over territories.\n\nTo be acknowledged as an expert in court requires that the judge recognize the witness as such. Therefore, in a dispute, lawyers for each side attempt at the outset to disqualify or cast doubt on the expertise of each other's witnesses. Expert witnesses are often subjected to aggressive cross examination in which their credentials, their methodologies, and often their personal characters can be challenged. Heather Harris, a non-Aboriginal woman who had married into a Gitksan family and been adopted into a House and clan, and who also held a degree in Anthropology, testified about the genealogies she had collected that described the matrilineal kinship system that forms the basis of Gitksan and Wet'suwet'en social structure. The lawyer for Canada, Marvyn Koenigsberg, suggested to Harris under cross-examination that she may have \"gone native,\" a derogatory accusation historically used to discredit anthropologists who were seen to have crossed the great divide between Aboriginal and European cultures, and hence to have lost their capacity for scholarly\u2014defined as \"objective\"\u2014research and analysis. The phrase, \"going native\" is also frequently used as a sexist and racist slur against white women who marry Indians.\n\nThe Gitksan and Wet'suwet'en witnesses were followed on the stand by anthropologists who also testified as expert witnesses.\n\n* * *\n\n. Canada 1983 cited in Asch 1989: 123-124.\n\n. Bourdieu 1990. See also, Miller 1992: 55-65.\n\n. Gisday Wa and Delgam Uukw 1987: 22.\n\n. Ibid., 23.\n\n. Ibid., 24.\n\n. Ibid., 38-40.\n\n. _Delgamuukw v. R._ (1987), 40 D.L.R. (4th) 685 (judgment on admissibility of oral tradition as an exception to the \"hearsay rule,\" hereinafter referred to as \" _Delgamuukw_ , oral tradition\").\n\n. Medig'm Gyamk 1992: 303.\n\n. For excerpts from this testimony see Gisday Wa and Delgamuukw 1992.\n\n. Gisday Wa and Delgam Uukw (1987), 38-42.\n\n. Gisday Wa and Delgam Uukw 1987.\n\n. Miluulak 1992: 59.\n\n. See Vansina 1965, 1985.\n\n. Daly 1988.\n\n. Daly 1988: 60-61.\n\n. See Cruikshank 1990, 1992, 1994.\n\n. Cruikshank 1992: 26.\n\n. _Delgamuukw_ , oral tradition: 698.\n\n. Ibid., 697.\n\n. Ibid., 655.\n\n. Ibid., 653.\n\n. _R. v. Simon_ (1985) 24 D.L.R. (4 th) 390 (S.C.C.)\n\n. McEachern 1991 _Reasons for Judgment, Delgamuukw v. R._ : 46-47.\n\n. Quoted in Monet and Skanu'u 1992: 42.\n\n. Yagalahl 1992: 201.\n\n. Ibid., 201.\n\n. _Delgamuukw_ , oral tradition: 172.\n\n. Yagalahl 1992: 199.\n**Chapter 10: Experts on Behalf of the Plaintiffs**\n\n_In the end, if anthropologists help the rest of the citizenry understand the First Nations discourse, then we will have made an invaluable contribution to the necessary dialogue. The Chiefs' voices will not be silenced. The crisis of anthropologists over how they are perceived, and what their role should be pales before the larger issue of working out just land claims._\n\n\u2014Anthropologist Antonia Mills, 1996, _Problems of Establishing Authority in Testifying on Behalf of the Witsuwit'en_.\n\n**Anthropology: Interpretation and Understanding?**\n\nAnthropologists first began to work in the capacity of expert witnesses on Aboriginal history in the United States when the Indian Claims Commission was established in 1946 to adjudicate land rights claims. From this starting point, almost half a century ago, certain enduring problems have confronted anthropologists who have become involved in these proceedings. Courts, governments, and colleagues in the hard sciences disparage anthropologists for ceasing to be loyal academics and becoming too sympathetic to the people they work with. Indigenous people accuse anthropologists of being agents of colonialism, of appropriating their cultures, and of representing them from the perspectives, and in the interests, of the dominating society. Judges trained in dualistic and positivistic legal traditions have tended to consider \"objective research\" and \"subjective interpretation\" as polar opposites, and as synonymous with truth and falsehood, respectively. The differences between an adversarial legal forum, where simple true or false answers are demanded; and an academic forum, where detachment and absence of intellectual commitment are rewarded, have made marriages between law and anthropology rocky.\n\nCriticisms from all these parties focuses on the central methodology of cultural anthropology: participant observation and fieldwork. Ethnographic research requires anthropologists to immerse themselves in the daily lives of the people they are studying, and to record and analyze their own observations. Fieldwork also involves extensive interviewing of research subjects in an effort to understand what the people who are members of the culture being studied say about themselves, and perhaps about colonial cultures and anthropologists. Field research is usually supplemented by studies of archival and other documents, and by canvassing academic literature on the same topic and\/or area to glean what other researchers have learned and reported on. Quantitative methods, like statistical analysis, are also often incorporated into anthropological research. Issues concerning the impact of the context in which research is carried out, and of the anthropologists' own background and attitudes, on her or his research are legion. These questions are the subject of an ongoing and exhaustive critique within the discipline. Anthropologists who have endeavoured to respond to these criticisms espouse \"reflexive anthropology,\" \"collaborative methodologies,\" and \"experimental ethnographies.\"\n\nReflexivity in anthropology refers to the acknowledgment that a human researcher studying other human beings will bring with them to the research endeavour attitudes and ideas about themselves, about the people they are studying, and about their relationship to each other. Taking seriously the problem that most anthropologists have historically been citizens of the nation states engaged in the colonization of Indigenous peoples, reflexive anthropology has paid particular attention to examining the influence of power relations located in race, gender and class structures, on ethnographic research. Rather than pretending that these issues can simply be eradicated by striving for \"researcher objectivity\" and the mimicking of natural science research models\u2014as they accuse positivists of doing\u2014reflexive anthropologists argue that recognition and acknowledgment of \"biases,\" and accounting for their influence on their research findings, is both intellectually honest and politically responsible. Reflexive anthropology is not restricted to examining the ideas and attitudes of researchers but also addresses issues such as the impact of funding, and of processes like litigation, on anthropological research. Neither is it restricted to the study of \"other\" cultures. Rather, contemporary anthropology has encouraged more and more attention to critical studies of colonial settler cultures.\n\nCollaborative research methodologies include research subjects as participants in the research process: as consultants in the design, implementation and analysis of research in an effort to transform the traditional power relationship between researcher and researched. Adherents of collaborative methodologies argue that their approach simply recognizes and acknowledges what really goes on during the creation of what becomes an ethnographic text. That is, social knowledge in general, and anthropological knowledge in particular, in practice, emerges in large part from conversations between researchers and research subjects, and it is the product of these inter-cultural and inter-personal interactions, where the research subject is often the teacher and the researcher the student, that constitutes the essence of ethnographic knowledge.\n\nTo more transparently describe and communicate the way this knowledge has been created and produced, some contemporary anthropologists have rejected the traditional single-authored, authoritative monogram of historical anthropology, and are experimenting with new forms of writing and representation that attempt both to share authority and royalties with research subjects, and to use plain language to reach a wider audience than the academic one alone.\n\n\"Telling the truth\" in reflexive, collaborative ethnographies takes the form of considering the influence of context and researcher\/researched relationships to begin with, and trying to achieve a more comprehensive ethnographic description by including a broad representation of research subjects describing and analyzing their various experiences, and expressing their diverse views, in their own words. Locating, or \"grounding,\" research in the needs of particular communities, as well as in the demands of academia, is another hallmark of much contemporary anthropology. The goal of \"the new ethnography\" is not the discovery of immutable facts or the refinement of more abstract theories, but rather the encouragement of understanding and mutual respect. Pretenses to unattainable \"scientific objectivity,\" the \"new\" anthropologists charge, simply ignore the inherently interpersonal nature of the human study of human society, and mask unreflected upon and unaccounted for biases under a cloak of value neutrality.\n\nThe post World War II growth in Aboriginal political strength has had a significant impact on anthropology. The contemporary generation of anthropologists, having come of age in the same era, have had to come to terms with Indigenous critiques that have charged, and often proven by their own ethnographic endeavours, that many of the descriptions written by previous generations of ethnographers were at best limited and partial, and at worst inaccurate and insulting. Feminist anthropologists, for example, have produced a body of literature showing that the voices of Indigenous women have either been absent from, or misrepresented in, the classic ethnographies written by both male and female anthropologists. The emerging recognition of the \"native point of view\" in anthropology and other disciplines, and of the right of Aboriginal people to represent themselves, is primarily a consequence of the ongoing struggle Aboriginal peoples have engaged in since the advent of colonialism. Many Aboriginal groups now demand significant participation in and control of ethnographic research conducted among them, a process they identify as \"the decolonization of anthropology.\" The presentation of the plaintiff's anthropological evidence in _Delgamuukw v. R._ reflected the contemporary conditions of, and approaches to, research with First Nations in Canada in the 1980s.\n\nOf course, these new approaches in contemporary anthropology present important problems, to which there are no easy answers. No communities, including Aboriginal ones, are homogeneous. They are divided by gender, class, internalized racism, and local axes of inequality. There is not one \"native point of view\"; there are many. Local Indigenous power relations can shape the outcome of collaborative anthropological research, as can research funding from private and public agencies. Evaluating how an anthropologist can develop analyses that simultaneously respect indigenous representations, and also employ the tools of the discipline to critique these explanations is difficult, to say the least. Ethical considerations present individuals with increasingly complex choices, in anthropology no less than in most other disciplines, and no less, for that matter, than in everyday life in the contemporary world. A significant aspect of any contemporary anthropologist's training focuses on these issues, and on how to conduct valid, reliable, rigorous research in such contexts.\n\n**Here Come the Anthros**\n\n_The anthropologists' role in the trial was to take out of the bulk of undigested evidence the relevant and significant evidence in order to create a picture of a viable, functioning and current society... Secondly, the anthropological evidence was intended to translate the evidence of the people to combat the ethnocentrism in the judicial consideration of their society...._\n\n_We knew that the Judge shared the cultural perceptions of the governments and therefore the anthropologists had to break through his vision and to introduce him to the native world view._\n\n\u2014Lawyer Stuart Rush, 1991, _The Role of Anthropological Evidence in Land Claims Litigation: The Gitksan Wet'suwet'en Case as an Illustration_.\n\nThe issues in dispute between the parties in the courtroom at the _Delgamuukw_ trial rested on the description and analysis of Gitksan and Wet'suwet'en society before European arrival. More concretely, the question that the plaintiffs and the defendants offered competing responses to was whether or not the approximately 9000 Aboriginal people living in northwestern British Columbia during the early nineteenth century constituted an organized society as defined by colonial law. According to the Gitksan's and Wet'suwet'en's witnesses, at the time in question they supported themselves by hunting, fishing and gathering natural resources. Their principal means of subsistence was salmon fishing. They harvested, preserved and stored food. They traded surpluses with their neighbours. They were matrilineal people, who reckoned descent and inheritance through their mothers. They were organized into clans and Houses led by chiefs. They engaged in various rituals, ceremonies and public assemblies, accompanied by feasting. Some feasts, and the system of reciprocal feasting between and among Houses, clans and villages, constituted an institutionalized process governed by laws and protocol for conducting the affairs of government and law.\n\nThe first professional anthropologist to testify for the Gitksan and Wet'suwet'en in _Delgamuukw v. R._ was Dr. Richard Daly, who took the stand for three weeks in February and March, 1989. Daly had received his Masters degree in Anthropology from Manchester University in 1975, and since that time had been employed in various professional positions. Daly's 700-page expert opinion report, entitled \"Anthropological Opinion On The Nature of the Gitksan and Wet'suwet'en Economy,\" was largely based on research he conducted with the Gitksan and Wet'suwet'en during 1986 and 1987 while he was also preparing his doctoral dissertation under the supervision of Dr. Richard Lee at the University of Toronto. Daly's report contained eight chapters: theory and methodology in anthropology; kinship economy; ownership and management in a kinship economy; the natural environment, nutrition and production strategies; the seasonal round of economic activities; storage, the accumulation of values and social hierarchy; the question of trade; and the feast: paradigm for social interaction and the circulation of values. Daly's methodology included extensive archival research as well as interviewing of elders and participant observation. He listed the following specific sources for his opinion report:\n\n(1) the transcripts of witness' statements and commissioned evidence of witnesses in the trial; (2) formal interview data he collected during his research in the form of written fieldnotes that had been submitted to the court; (3) knowledge he obtained through informal discussions, not recorded in writing; (4) ethnographic monographs, including the Barbeau-Beynon papers, oral histories and fieldnotes collected by John Cove, Father Morice, Diamond Jenness, Franz Boas, Wilson Duff, John Adams, and Marjorie Halpin; (5) articles and books on general history and economic history.\n\nDaly described his research activities as follows: \"Anthropologists seek to determine the truth through their work, seeking validation or invalidation of propositions by cross-referencing data and seeking to place the phenomena studied in as fully rounded a context as possible.\"\n\nCrown lawyers challenged Daly's qualification as an expert witness on the grounds of Section 11 of the Evidence Act which says that an expert should disclose the facts upon which his opinion is based. They argued that Daly's failure to turn over notes relevant to two years of participant observation was a breach of this requirement. Further, they charged that his report was \"opinion thinly disguised as argument\"; that he had interwoven oral histories with other information, \"the foundation of which is highly questionable\"; and, that his report duplicated evidence already given by other witnesses. Finally, the Crown argued that, since Daly's participant observation had been conducted after the commencement of the trial, and could not be replicated because precisely the same conditions would never again exist, his methodology was \"unscientific\" and his report should not be admitted as expert evidence. Chief Justice McEachern qualified Daly as an expert and agreed to hear his testimony and determine its value by weighing it at the end of the trial in the context of the entirety of evidence presented.\n\nDr. Antonia Mills followed Daly in March, 1989. Mills had graduated with a doctorate in cultural anthropology from Harvard University in 1969. Since that time she has lived, worked, and taught in British Columbia and has conducted participant observation and archival research among the Dunneza-Cree (Beaver) First Nations and the Wet'suwet'en, primarily. Her 216-page opinion report, entitled \"The Feasts, Institutions and Laws of the Wet'suwet'en,\" focused on Wet'suwet'en cultural history, and emphasized religion and cosmology. Mills' particular research interest for several years has been reincarnation beliefs and practices among the Wet'suwet'en. Crown lawyers argued that Mills' training and work emphasized psychology, symbolism and religion and was therefore \"not of interest to this case.\" Like Daly, she had conducted participant observation following commencement of the court action. Like Daly, she was accused by the Crown of being an advocate who lacked objectivity, and who duplicated the evidence of the lay witnesses. Chief Justice McEachern, however, allowed her testimony, subject to his evaluation of its weight at the end of the trial.\n\nCrown lawyers focused their cross-examination of Mills on the question of warfare. They took umbrage to a statement she made in her 20-year old Ph.D. thesis on the nature of warfare in western society:\n\nCrown Counsel: On page 60, you set out your description of western society. I'm not going to read it out, but I just ask you to read over the first couple of paragraphs down to the line \"Normal men have killed perhaps 100,000,000 of their fellow normal men in the last 50 years.\" Was that your view in 1969?\n\nMills: Yes.\n\nCrown Counsel: And is that your view today?\n\nMills: I think I probably am a bit more sophisticated than that, but people certainly continue to kill their normal men. It's not as if I'm saying that other people don't do it as well. It's not as if I'm claiming that the Wet'suwet'en didn't kill one another.\n\nChief Justice McEachern intervened in the Crown's cross-examination of Mills on the subject of her alleged romantic biases:\n\nCourt: Are you surprised to find so many like Ogden writing that the masses came out of their huts naked? Had their level of civilization not progressed beyond that at or just after the time of contact? Mills replied that the Wet'suwet'en normally wore clothes.\n\nCourt: I've heard...I have an impression that I am hearing perhaps the best side of these people, which is understandable, but you haven't said anything about wars.... Would you call them war-like?...the people generally?\n\nMills answered that she found it difficult to respond to a question that required her to make sweeping generalizations that characterized an entire people as either \"war-like\" or not. The Chief Justice continued:\n\nCourt: There is a suggestion of slaves. Did the Wet'suwet'en take slaves?\n\nMills explained at length the similarities and differences between \"slaves\" and hostages taken in war.\n\nCourt: There is even a suggestion in one of the pieces about cannibalism. Was that a feature of the Wet'suwet'en in any way?\n\nMills: No.\n\nThe Chief Justice apparently suspected Mills, the plaintiff's only female anthropologist, of presenting a romantic picture of the Wet'suwet'en.\n\nRomanticism is a valid criticism often levelled at the discipline of anthropology. There is an extensive literature on the subject, including much by anthropologists themselves, and by Indigenous critics who complain that anthropological romanticism camouflages the devastation wreaked on Indigenous cultures by colonial domination. Aboriginal intellectuals charge that romanticized descriptions of their cultures are as one-dimensional and inaccurate as those that depict Aboriginal cultures as barbarous and cruel. Romanticism, they argue, is one projection of the European imagination, and savagery another. The Crown's \"critique of romanticism,\" however, seems to be based on the simplistic idea that any evidence that the Gitksan and Wet'suwet'en peoples were NOT war-like, naked, slave-owning cannibals at the time Europeans arrived in the late eighteenth century, is romantic, and therefore not believable. End of conversation: facts are a nuisance in this monologue.\n\n**Law and Masculine Hysteria**\n\n_In Western Europe, since the period of the Enlightenment in the 17th century, men have assumed a strong connection between their rationality and their sense of masculine identity. They have learned to appropriate rationality as if it were an exclusively male quality denied to \"others\":_ _women and colonized people in particular...._\n\n_Since 'rationality' is identified with knowledge, it is similarly denied to these same \"others.\" Emotions and feelings are likewise denied as genuine sources of knowledge within the culture. Rather, they are associated predominantly with weakness and femininity, and so as antithetical to the 'strengths' with which boys learn their sense of masculine identity. Men could only assert their humanity by mastery over the physical world, and by learning to dominate their passions and desires. It is this inherited notion of self-control as dominance that has been so closely identified with modern forms of masculinity._\n\n_Such a view also begins to explain the depths to which men can feel threatened by emotions and feelings._\n\n\u2014Psychologist Lawrence Seidler, 1987, _Reason, Desire, and Male Sexuality_.\n\nOn day 210 of the trial, anthropologist and film-maker Hugh Brody took the stand on behalf of the Gitksan and Wet'suwet'en. Brody's opinion report was based on participant observation, interviews, and archival research that included perusing the Barbeau-Beynon archives, Indian Agents' letters and reports, documents from the MacKenna-McBride Royal Commission of 1913-1916, trapline registration forms from the 1920s, and records of territorial disputes documented by white settlers. Brody's report went on to summarize the history of research into Aboriginal title and rights over the past twenty years, noting the trend away from assimilation studies popular in the 1960s, through the land use and occupancy studies of the 1970s, and into the ownership and jurisdiction studies of the 1980s. He wrote: \"The assumption that rights of ownership are established on criteria of land use or occupation now appears naive. We do not apply these criteria to our own rights. We do not draw boundaries around our territories by using and occupying them.\" Brody also reviewed anthropological theory, arguing that, until very recently, anthropology had ignored \"how people regard themselves.\"\n\nBrody described the purpose of his report as being \"...to bring some part of another people's world view into ours.\" But, \"the heart\" of his endeavour, Brody wrote, was \"to recognize and then break free from Euro-Canadian ethnocentricity.\" One of the first steps in that process was to recognize that European intellectual thought, as expressed through language, was abstract and conceptualizing. \"Hunting peoples,\" Brody wrote, \"dislike generic categorizing language and prefer specific and concrete vocalization...hunting and gathering societies might be said to believe above all in facts.\" Brody's focus on European ethnocentrism was what Chief Justice McEachern and the Crown's lawyers objected to most strongly. The Crown selected four passages from Brody's opinion report as evidence that \"his report has nothing to contribute to the issues before this court.\"\n\nFirst, the Crown objected to the following statement on page 2 of Brody's report:...our society has occupied the lands of other societies, and is committed to a particular kind of historical process. This means that settlers and governments have tended to see native peoples through unclear or distorting lenses.\n\nChief Justice McEachern entered the fray to dispute this statement, saying:\n\nCourt: I have expressed the view for some time that courts of law are troubled on a daily basis with this question of feelings. Feelings are what make people function the way they do in many cases, but it's not something that laws and judges can manage very well. And if somebody wants to tell me what somebody else's feelings are, I have to discount it very substantially, because firstly he may have misconstrued the feelings, and secondly, the feelings may not be rationally founded.\n\n...\n\nJackson: My Lord, perhaps I might be permitted, in response to Mr. Goldie's point in reading that particular part which you just read and the whole relevance of how native society is perceived.\n\nCourt: Does it matter...in any real sense, does it matter?\n\nJackson: Well, My Lord, my point was to deal with these issues, and the relevance of a social scientific analysis to the appropriate legal conclusions, which of course are for Your Lordship to decide... Court: Let's assume for the purpose of this argument...they are going to...they are going to be found to have the rights that are claimed in the Statement of Claim. Does it matter if, from the time of contact, circa 1800 A.D., that they have been treated badly or they have been treated well? Does it make any difference?\n\nJackson: I think the way Your Lordship has framed it, no. It does not. But it is our submission that understanding the cultural context in which that took place will enable Your Lordship to make a conclusion, a legal conclusion.\n\nCourt: The fact is that a reserve was established, or an area was logged, or a mine was found and exploited or something of that kind. Those are facts that may speak to all sorts of issues relevant to this case, but if\u2014does it make any difference if settlers and governments have had an attitude or have viewed things in a particular way? The more I hear of this case, the more it comes back to the old question, 'what are the facts?' As the detective used to say, 'what are the facts, ma'am?\n\nJackson:...just the facts... Court: Yes. 'Just the facts, ma'am.' I know that it's late in the day to try and boil things down. I know it's late in the day to change the rules, and I don't want to change the rules upon which we have been functioning, but that sort of a passage, \"This means that settlers and governments have tended to see native peoples through unclear or distorting lenses\"...seems to me to be of highly questionable relevance. You might just as well say what happened indicated that Indians have tended to see governments through unclear or distorted lenses and that probably the answer would...what is...what difference does it make? Of course...do they have rights and what are they?...not how the people view each other....\n\nJackson: My Lord... Court: As I said a moment ago, they may view each other from completely irrational bases. They probably do. Most people do view each other from an irrational basis. Again, what difference does it make?\n\nJackson: I think it does make a difference, My Lord, in terms of understanding events which have taken place which are relied upon by my friends as evidence of loss or acquiescence in the loss of rights. And I will....\n\nCourt: I have grave misgivings about the conspiratorial theory of history, but that's perhaps a cultural impediment that I am going to have to try and avoid as we proceed. Go ahead, please, Mr. Jackson....\n\nNext, the Crown pointed to a statement on page 49 of Brody's report, that read:\n\nGitksan and Wet'suwet'en reserve lands amount of some 45 square miles. According to white interpretations of events, the villages that make up most of the 45 square miles represent the conclusion of a historical process.\n\nMr. Goldie, lawyer for the Province of British Columbia, objected strongly to this passage.\n\nGoldie: My Lord.... I check my assumptions at the door, and I don't think it's appropriate for a witness to talk about \"we.\" If he is talking about some other culture, let him talk about it...but I leave my assumptions at the door, and I don't want to deal with cross examination on the basis that I am depending on some particular point of view that is ascribed to me or to my client for that matter.\n\nIt would be a very interesting and perhaps educational process if Mr. Brody told us about ourselves and held up a mirror to us, but what has that got to do with this litigation?\n\nChief Justice McEachern agreed with the Crown.\n\nCourt: \"According to white interpretations....\" What white interpretations? All white interpretations? It doesn't say some. I don't know if it means some, but...isn't that just so wide and sweeping and so extensive that it's not keeping to deal with it? That's what every white person thinks? And then again we still have the problem. If he does, so what? We are still talking about rights that arose at the time of contact, as I read the authorities. I mean, I just have difficulty with the nature of this evidence. It seems to me it's way beyond what any court has ever been asked to look at before. I suppose there have been things as broad as that...isn't this...hasn't this perhaps gone over the line?\n\nI must say I find that statement to be terribly, terribly exaggerated, and when you get into that area, you know that you are having a terrible time coming to grips with anything that's going to be useful. I mean, I think if the witness wants to say that's his view, I suppose that's all right. He doesn't say that. He says this is what everybody thinks. I don't think that is even close to being right. And if...well, maybe that's his view. Maybe he thinks every white person thinks that.\n\nBrody: There is...and I said I am dealing at a high level of generality here self-evidently...there is a core to the intellectual assumptions that are made within our intellectual heritage, and these assumptions say a lot about what is expected to happen to tribal and aboriginal peoples.\n\nCourt: Well, I can understand the approach, Mr. Jackson, when we are talking about the culture of the Indian people, and one has to be generous, which is not really the word I am looking for, but quite relaxed about the extent to which evidence can go, because it's a very ill-defined kind of a study. But when you are talking hard facts, such as the McKenna McBride, you have got terms of reference, you have got a transcript, you know it was said, then it seems to me that there isn't much that can be said about that. It speaks for itself...Truly, there must be limits, and I am just having trouble finding what they are here. I have a terrible feeling that I am way beyond them, and you are asking me to go way beyond them...It's like when you are in a contest or a game and they suddenly say we are not going to play it according to the rules. I just have the sense that I am asked to listen to material that is not something that a judge should be hearing.\n\nBrody's critique of \"frontier culture\" incurred the wrath of the Crown lawyers and the Judge as well, and the following statement from page 69 of his report was presented as another reason to disqualify him as an expert witness:\n\nThe encounter between native culture and frontier whites may be said to constitute a meeting of culture and anti-culture. Whites who are roaming, often alone, far from their own land, encounter those who are absolutely at home. The whites have thus in a way shed society. The native people they deal with are absolutely at home....\n\nFinally, Crown lawyers pointed out the following statement about settler history, claiming it revealed Brody's inappropriate biases most strongly:\n\nNewcomers, sure of their superior knowledge, understanding and rights, do as they think fit, encourage others to do likewise and call upon the forces of Canadian law and order if too directly thwarted and opposed.\n\nThe lawyer for the Government of Canada summed up the Crown's objections to Brody's qualifications as an expert witness:\n\nCrown Counsel: It is my submission what Mr. Brody is doing is describing attitudes of white people...how they feel about what's going on, and how to a certain extent the Indians feel from his point of view in response. And it is my submission that inquiry is irrelevant to the issues in this lawsuit. The issues that are being put forward by the plaintiffs for Your Lordship to decide is 'did the government, the white government impose jurisdiction?' Questions like were reserves created and what were they? How the whites felt and how the Indians felt is beyond the purview of this inquiry.\n\nIt's an interesting question. I am sure it is one which has probably concerned many a graduate thesis. But to determine\u2014to go into detail about what was happening is in my submission irrelevant. It is, of course, in my submission, also in the nature of argument and speculative in the extreme.... If this kind of evidence is allowed it's tantamount to allowing experts to act as counsel and it's as if all the 45 or 50 witnesses that Your Lordship has heard for over a year...that Your Lordship cannot evaluate and interpret that evidence, but that you require the assistance of an expert to do so.\n\nChief Justice McEachern asked this question of Crown counsel regarding Brody's admissibility:\n\nCourt: Well, let me pose this scenario...That an expert in the sense of a person highly trained and experienced in the discipline who has strongly held and widely publicized views on the question, is nevertheless called as an expert witness and he's qualified in the sense that he has shown that he has all the experience-related requirements of an expert. Can the court say...I won't hear this witness because he has a well known and admitted point of view? Is that not a matter that goes to weight?\n\nMr. Goldie, lawyer for the Province of B.C., responded and summed up his objections as follows:\n\nGoldie: Normally speaking, the experts every day are called who are\u2014who are known as the proponents of a point of view. That is to say, they favour a particular theory. But this is not a case of favouring a particular scientific theory.\n\nCourt: He favours a party.\n\nGoldie: He is favouring a party. That's the thrust of my submission, My Lord, and if he favours a party that takes him out of the category of the expert who says, well, I believe in the such and such theory of relativity.\n\nCourt. Yes.\n\nGoldie: And I will stand up and be counted on that point, but there is no such suggestion of that here.\n\nCourt: And then proceeds to try and persuade the court that his theory is right.\n\nGoldie: That's correct.... That's correct...but his theory....\n\nCourt: What's the point of this?\n\nGoldie: His theory has got nothing to do with the particular parties to the case. There is no theory being advanced here in my submission.... My principal objection is that he says his report is an overview.... The word overview means that he cherry-picks from the evidence of others.... I don't mind his saying that every other anthropologist except himself is wrong and he's right.... That's his privilege....\n\nCourt: You can't get ready for a case like this without being closely associated with the people...but the question is does it go beyond that when he says, as he said before, he likes Indians and he has an\u2014or aboriginals\u2014 and he has a favourable disposition towards them. Does that disqualify him?...In a perfect world I would hope that parties would confine their expert testimony to persons whose objectivity is not open to question. That may not always be possible. And, indeed, it might be a dangerous test to apply, because the person who hides his biases is no more credible than the person who makes it known. The former may be more dangerous than the latter.\n\nThe Crown's, and the Chief Justice's, response to Brody's testimony are indicative of the fundamental challenge to the court's self-image and legitimacy that a critique of ethnocentrism raises. \"Judicial neutrality,\" the linchpin of the British and Canadian legal systems' claims to legitimacy, depends on maintaining the public perception that law is the expression of universal reason, that judges are without racial, gender or class bias, and that everyone is equal before the law. As his \"tin ear\" prevented him from hearing the oral histories, so Allan McEachern's myopia blinded him to his own biases.\n\n**Academic Freedom**\n\n_I agreed to do the research with the understanding that I would be free to draw my own conclusions from the materials I examined. We also decided that it would be best to have me work independently of other researchers...so that my interpretations could stand alone. Accordingly, I did not read any of the other reports before submitting my final opinion and appearing on the witness stand, nor did I attend the trial before that time._\n\n\u2014Historian Arthur Ray, 1991, _Creating the Image of the Save in Defence of the Crown: The Ethnohistorian in Court_.\n\nThe presentation of evidence in _Delgamuukw v. R._ followed a chronological historical narrative. Following testimony that focused on Gitksan and Wet'suwet'en ways of life before the arrival of Europeans\u2014labelled by historians the \"prehistoric\" era\u2014witnesses described the arrival in the territories of fur traders and then settlers. The years during which Europeans initially made contact with Aboriginal peoples and became involved in first sporadic, and then regularized, trading relations with them, but before European settlement began in earnest, have been dubbed by academics the \"protohistoric\" era. With European settlement came written record keeping, which ushered in the \"historic\" period.\n\nThe legal questions that relied for answers on descriptions and analyses of the protohistoric period spanned those relevant to the prehistoric\u2014or pre-European settlement\u2014era, and the historic\u2014or post-European settlement\u2014 era. First, an analysis of the nature of Aboriginal societies before contact was required to determine whether or not an organized society existed at the moment of contact, and when Britain declared sovereignty. Another legal question was what changes took place in Gitksan and Wet'suwet'en society and culture, and particularly in their property ownership and management systems, during the protohistoric years? This issue was crucial to the Crown's argument that the development of delineated borders and property law was not a feature of pre-contact Aboriginal societies but rather had emerged as a response to the European-Aboriginal fur trade. They claimed that, by the time Hudson's Bay Company trader William Brown arrived in 1822 and brought them into \"history\" by writing about them in his journals, the Gitksan and Wet'suwet'en had already undergone such significant social change as a result of their contact with European trade goods during the protohistoric period that Brown's journals could not be read as describing \"aboriginal practices.\" Chief Justice McEachern, in his _Reasons for Judgment_ , would later explain the legal significance of firm boundaries between historical periods: \"The difference between the pre-historic, proto- and historic periods is relevant to the question of determining what are aboriginal as opposed to non-aboriginal practices,\" he wrote.\n\nOral history, linguistics, archaeology, ethnography and historical records are the usual sources of knowledge relied upon to provide information about the protohistoric and historic periods. Oral histories, in particular, offer Aboriginal reflections on these encounters, and some classical ethnographies augment this knowledge base. Europeans' experiences of the protohistoric era are represented in fur trading company's books and reports; ships' logs; traders' accounts, journals and diaries; and government records. Before evidence concerning the protohistoric and fur trade periods of history could be heard, the court had to determine how to classify these sources of evidence in legal terms.\n\nIn 1989, midway through the trial and two years after his 1987 ruling on the admissibility of oral history testimony, Chief Justice McEachern made a second ruling on evidence. This gave direction on how the court would interpret historical documents recognized as such within the academic study of history. McEachern divided this material into two categories. Category 1 included documents written by contemporaries; that is, first hand observations made by historical actors such as fur traders, missionaries, and government agents of their own environments, experiences, and opinions. Category 2 was reserved for historians' interpretations and opinions of these primary documents.\n\nDocuments written by contemporaries are called in law\u2014as a term of art\u2014 \"ancient documents.\" According to common law rules, ancient documents are admissible if they are \"more than thirty years old, produced from proper custody and otherwise free from suspicion.\" The point is that the documents must be in their original state, and it must be shown that they could not have been tampered with or changed since they were first written and secreted away. Although he did allow, theoretically, that it was possible such documents may contain mixtures of \"fact and hearsay,\" Chief Justice McEachern ruled that \"ancient documents admitted into evidence are, subject to weight, _prima facie_ proof of the truth of the facts stated in them.\" That is, ancient documents are assumed to have been produced through a process whereby brute facts, existing independently of cultural or any other form of interpretation, were observed and recorded by unbiased writers in a similarly objective fashion. The records of early explorers, fur traders, and colonial officials, that had been housed in archives, were classified as \"ancient documents.\"\n\nOn the question of the role of interpretation of historical documents, the Gitksan and Wet'suwet'en argued that each document should be accompanied by an historian to provide a context for interpretation. The Crown argued that all documents should be admitted and then weighed by the judge in the context of the evidence as a whole. Chief Justice McEachern ruled that historians may express their opinions \"based upon inferences from the documents about the recorded facts of history.\" They would not be permitted, however, to \"construe a written document...or generalize upon the broad sweep of history which is so often subject to learned disagreement and revision.\" His assumption appeared to be that the original written document represented a form of \"pure\" knowledge, the singular truth of which would be discoverable forever by future generations of rational thinkers. There is no consideration in this theory of the way that different generations of readers, living in different contexts, and reading these ancient documents for different purposes, may interpret their meanings differently. That is to say, the possibility that _both_ written and oral histories begin as human creations, and that their meanings are interpreted and reinterpreted over time, challenges the law's desire to classify history in discrete periods, and to codify historical knowledge into simple true and false categories.\n\nAs he had done in regard to the distinction between history and mythology in oral histories, the Chief Justice asserted that, in the case of historical documents, fact and opinion could be clearly differentiated within them. In keeping with law's self image as \"a site where universal reason actualizes itself, owing nothing to the social conditions under which it is manifested,\" the judge declared that he could and would make the final determination of what was fact or truth, and what was not. He wrote: \"Impermissible opinions and the conclusions [historians] wish me to reach in connection with the subject matters of their opinions will undoubtedly be interwoven with permissible opinion, and it will be my responsibility to disregard the former while profiting from the latter.\"\n\nThe Chief Justice made a third evidentiary ruling on the admissibility of \"treatises,\" defined as written materials including published books and articles, and unpublished theses, essays and studies.\" The Crown argued that \"only totally unbiased, non-controversial, generally accepted works that have endured and survived expert public review such as standard texts, qualify as learned treatises.\" Chief Justice McEachern disagreed with the Crown, and concluded that such an orthodox approach was not practical in this case. He decided to admit all such material subject to weight, and to objections by counsel that particular items may be untrustworthy because of \"disabling bias or a demonstrated lack of competence or for other reasons.\"\n\nFollowing the Chief Justice's decision to admit this evidence, one of Canada's foremost authorities on fur trade history, Dr. Arthur Ray of the University of British Columbia, took the stand on behalf of the Gitksan and Wet'suwet'en. Ray began his report with a description of the task he was asked to perform by the plaintiffs. The Gitksan and Wet'suwet'en Hereditary Chiefs commissioned Ray \"...to examine the economic history of the Middle Skeena River-Lake Babine region in order to (a) search for evidence of the native exercise of title in the area; and, (b) to examine the way the Gitksan, Wet'suwet'en and Babine responded to economic developments following European contact.\" Ray's evidence began with a description of the years beginning around 1800, and continued with discussions of two time periods: (1) The Transitional Years, 1830\u20131860; and (2) Competition and Economic Diversity, 1860-1915. The sources Ray examined for this study were Hudson's Bay Company records, ethnographies, and interviews with contemporary key informants. The details of his research and report have subsequently been published in peer-reviewed academic journals.\n\nDr. Ray began his testimony with a brief overview of the 9000 strong pre-contact Gitksan and Wet'suwet'en population. He described villages, locations of fishing sites, and Indigenous social structure. Ray described how the Coastal Tsimshian peoples began trading sea otter pelts for European goods around 1778. They brought these new items along with them on their regular excursions to trade with their Gitksan neighbours who lived inland. The Gitksan then traded these goods with their Wet'suwet'en neighbours, and with their Athabaskan trading partners further to the east, as they had been doing for centuries. Gitksan and Wet'suwet'en participation in the maritime phase of the European-Aboriginal fur trade was minimal, and had little impact on Gitksan and Wet'suwet'en ways of life, Ray testified. During the 1800s the sea otter was overharvested to near extinction, and the maritime phase of the European-Aboriginal fur trade gradually came to a close as British and Canadian trading companies established forts and began to trade with interior peoples for beaver, mink, muskrat, bear and other land mammal furs. Gitksan and Wet'suwet'en involvement in the land-based phase of the fur trade was more active, and effected some changes in relations with neighbouring peoples, in ownership and management of territories and in the economic organization of hunting and trapping.\n\nWilliam Brown\u2014the author of the first written descriptions of Gitksan and Wet'suwet'en ways of life\u2014arrived in 1822 to establish Fort Kilmaurs on Babine Lake. Based on his research in the Hudson's Bay archives, Ray argued that: \"What is abundantly clear from Brown is that you have a fully-articulated feasting system with house territories, family heads. In other words, the system\u2014the very system that the ethnographers...begin to describe with Morice some 60 years later is a system that essentially Brown has just given us the bone outline for in 1822....\" The Hudson's Bay Company records and the oral histories, as well as ethnographic, archaeological and linguistic evidence all document extensive trade in the region as a whole in both primary and manufactured goods. Harmon and Brown described well-organized and managed trade in furs, dressed skins, leather, fish, fish oil, blankets, shell beads, berry cakes, nets and dogs.\n\nQuoting from Brown's journals, Ray testified that \"Brown discovered that the regional economy was a delicately balanced system in which villages were linked together by kinship ties, trade, gambling and potlatching activities.\" Brown described the people as living in four major villages between which were scattered territories of the constituent Houses. He recorded that \"access to Babine and Wet'suwet'en house territories was tightly controlled by 42 'nobles' and 'men of property'.\" The immediate problem this presented to fur trader Brown was that \"lineage heads gave only a few of their fellow kinsmen the right to hunt and trap\" and most of their produce was \"tithed\" to the Chiefs. It was only the Chiefs who the Hudson's Bay Company could deal with. \"These elders,\" Brown wrote, \"generally had well-established trading connections already\" and were interested in trading with the Hudson's Bay Company only if it was to their advantage.\n\nBrown described how frustrated Hudson's Bay Company men trading with the Gitksan and Wet'suwet'en during the early years of the fur trade became when they had to compete with other Aboriginal traders who could, simply put, offer better prices and better trade goods than the Hudson's Bay Company could. A similar problem arose regarding Brown's dependence on Aboriginal people for transportation. Ray wrote that \"Even at rates the Hudson's Bay Company considered to be unreasonable, it was difficult to persuade Indians to transport goods at the times the company wanted. The Indians were said to be willing to do the work only when it suited them because they had a variety of alternative work that was easier and more lucrative.\"\n\nBrown resorted to acting as a middleman between the Athabaskans and the Gitksan and Wet'suwet'en, importing the moose hides they so highly valued from Company trading posts east of the Rockies and trading these for furs. Ray quoted an excerpt from Brown's journals to illustrate this. Brown wrote: \"I made then [1823] what I considered very handsome presents consisting of cloth blankets, shirts, etc... But the following day they brought back the whole and informed me that it was not to receive such articles as these that they had given me their furs. One of them, Snuggletrun [the second highest ranked Hot-set leader] from whom I had received about twenty skins I had given two yds Red Strouds, one flanal shirt, one awl, one firesteel, one gun flint, two needles, two hanks thread, two yards gartering, ten ball, one half point powder, one point shot and one sixth point tobacco, requested I take back the whole and give him a dressed skin in the place.\"\n\nIn analyzing the records of William Brown, Ray introduced new historical research that had been inaccessible until the late 1960s, and had subsequently been ignored by previous researchers working in the area. Brown's journals challenged many assumptions held by earlier historians and anthropologists about the pre-contact nature of Gitksan and Wet'suwet'en society. Ray noted that, in the course of his research, he \"...discovered that all the important ethnographic studies of the region, such as those of A. G. Morice, Diamond Jenness, Irving Goldman and Julian Steward had not used this material because it was unavailable to them. More recent writers, most notably Vernon Kobrinksi and Charles Bishop, draw heavily on these earlier published ethnographies, but ignored the critical Brown material.\"\n\nThe legal significance of Ray's report and testimony was that it supported the claim that the Gitksan and Wet'suwet'en \"did occupy the territory in dispute at the time of initial European contact.\" \"Of considerable importance to the Wet'suwet'en,\" Ray noted, \"Brown outlined a feasting and house territory system that was very similar to the one the hereditary Chiefs described in their opening statement 170 years later.\" Further, Ray's testimony provided documentary evidence in the form of reports written by an Englishman that the Gitksan and Wet'suwet'en did manage and conserve their lands and resources through a House (lineage) territory system, before the local commencement of the European-Aboriginal fur trade that focused on beaver. Brown's documentation of Gitksan and Wet'suwet'en \"men of property\" owning trapping sites and controlling access to beaver\u2014whose meat they disdained but whose pelts they valued for trade\u2014contradicted the Crown's argument that the only resources that Aboriginal peoples regulated at all were those they relied upon for subsistence, and that they had not engaged in production of surpluses for trade before Europeans arrived.\n\nBoth federal and provincial Crown lawyers emphasized pre-contact intra-Indian violence in their cross-examination of the Gitksan's and Wet'suwet'en's witnesses, a strategy that reached its peak in the cross-examination of Arthur Ray who titled an article he later wrote about his experiences in the _Delgamuukw_ case \"Creating the Image of the Savage in Defence of the Crown.\" In it he wrote: \"It was apparent to me that the Counsels for the Crown had two major objectives in mind when they pursued this line of questioning. They hoped to raise doubts about the reliability of Brown as an observer and they tried to suggest that the area was in a state of turmoil at the time of first contact. I think this line of inquiry also reflected an underlying and disturbing attitude of Crown Counsel toward Native people\u2014that they were revengeful and violence-prone before the 'civilizing' influences of European missionaries and government officials modified their behaviour.\"\n\nCounsel for the government of Canada, James Macaulay, concentrated on challenging Dr. Ray with anecdotes drawn from the reports of early traders and explorers that depicted violent confrontations. The essence of the federal cross-examination of Arthur Ray is illustrated by the following exchange:\n\nMacaulay: I put it to you, Dr. Ray, that's really the point of the cross-examination, or one of the points any how, that this is a very fragile system and it was interrupted by such things as a single killing.... Do you agree that the reciprocal killing system had that effect?\n\nRay: My point would be that societies generally value peace, that's both within groups and between groups; that their systems for sustaining peace break down from time to time are easy enough to prove. And if you wanted to use the European example, just remember the period we're talking about is close to the Napoleonic Wars, right?\n\nMacaulay: Yes.\n\nRay: A lot more people got killed in those. And would you not say that those European societies valued peace as well?\n\nMacaulay: I'm not saying European societies valued peace, Dr. Ray\u2014\n\nRay: Well, would you say that the European societies had equally fragile systems? No human society has been able to sustain peace both in itself and between itself....\n\nMacaulay allowed that this might be the case. However, it was the presumably spontaneous and unpredictable\u2014unorganized\u2014nature of Aboriginal violence that he was referring to:\n\nMacaulay: Isn't that the source of the violence...that the endemic violence was based on a fair degree of truculence...that there was a certain excitability and ferocity that led to the whole village becoming involved....\n\nLate on the second day of Ray's cross examination by the Crown, Chief Justice McEachern intervened:\n\nCourt: Maybe I can ask you this, Dr. Ray: Mr. Macaulay has been suggesting a tendency toward violence and you have been resisting that and you are saying that wasn't the state of things, or at least it wasn't the normal diet for the period you are talking about. Yet, there seems to be some pretty strong suggestions that there was warfare on the Skeena, and I take it you are saying that was at some later time?\n\nRay later wrote: \"I replied by saying that I thought the portrayal of Indian society as being a violence-prone one was an ethnocentric point of view. I noted that I did discuss violence in my opinion, but I also made it clear that feasting provided one way of dealing with the problem. Murders did not always lead to revenge killings by the offended relatives.\" Ray explained that: \"My only point was I do not subscribe to the bloodthirsty savagery of Native people, nor the noble savage. I don't subscribe to either. I see Native people as people. They have the same problems that other peoples have around the world and in different points in time. That was the main point of my report. Nothing more; nothing less.\"\n\nLike many academics who have testified as expert witnesses in court, Arthur Ray concluded upon reflection that the adversarial courtroom is an inappropriate forum in which to hold scholarly debates. Lawyers are rarely familiar with historical methodology, he charged, and they \"take extreme positions\" and \"champion points of view\" chosen for their strategic value in bolstering their clients' cases, rather than their historical validity.\n\n* * *\n\n. See Beals 1985; Campisi 1991; Clifford 1988; Dobyns 1978; Dreyfus and Dreyfus 1979; Dyck 1993(b); Feldman 1980; Foster and Groves 1993; Harries-Jones 1985; Hedican 1986; Hedley 1986; Kousser 1983; LaRusic 1985; Lurie 1956; Maddock 1989; Manners 1956; Paine 1985; Rosen 1977; Salisbury 1976; Sansom 1985; Tremblay 1983; Waldram 1993; Wright 1988.\n\n. Geertz 1983(b); Van Esterik 1985; See also _POLAR: Political and Legal Anthropology Review_ , Volume 10(2), November 1996, Special Issue: Anthropology Engaging Law, Washington, D.C.: American Anthropology Association.\n\n. Hammersley and Atkinson 1983.\n\n. Behar and Gordon (eds.) 1996; Clifford and Marcus (eds.) 1986; Fox 1991; Marcus and Fischer 1986; Rosaldo 1989.\n\n. See Ruby 1982; and Rosaldo 1989.\n\n. See Ryan 1990, 1992; Warry 1990.\n\n. See Behar and Gordon 1996; Brody 1985; Clifford and Marcus (eds.) 1986; Marcus and Fisher 1986.\n\n. Thomas 1994.\n\n. This is a vast literature. But see for summaries: Behar and Gordon 1996; di Leonardo 1991(ed); Moore 1988.\n\n. Ryan 1985.\n\n. Rush 1991: 13-14.\n\n. Daly 1988: 26-28.\n\n. Transcripts, Vol. 184: 11822.\n\n. Mills 1987. Mills has also published a book presenting her opinion report and her experiences as an expert witness. See Mills 1995.\n\n. Transcripts, Vol. 196: 12860.\n\n. Transcripts, Vol. 196: 12869.\n\n. Peter Ogden was a fur trader and explorer who traveled from the Oregon Territory to northern British Columbia in the early 1800s. Although he did not directly encounter either the Gitksan or Wet'suwet'en on his travels, he recorded in his journals anecdotes he had heard about them and other First Nations in the region. Subsequent documentation by fur traders like William Brown who worked directly with the Gitksan and Wet'suwet'en, and ethnohistorical and archaeological research, have shown Ogden's descriptions to be unreliable.\n\n. Transcripts, Vol. 201: 13319-13321.\n\n. See Stocking 1987.\n\n. See Deloria 1969.\n\n. Seidler 1987.\n\n. Brody 1987(b).\n\n. Ibid., 6.\n\n. Ibid., 2.\n\n. Ibid., 7.\n\n. Ibid., 5.\n\n. Transcripts, Vol. 210: 14217.\n\n. Transcripts, Vol. 211: 14254.\n\n. University of British Columbia Professor of Law, and Lawyer for the Gitksan and Wet'suwet'en, Michael Jackson.\n\n. Transcripts, Vol. 211: 14254.\n\n. The judge is referring to the Report of the McKenna-McBride Commission that was charged with examining the allocation of Indian reserves throughout British Columbia from 1913-1916.\n\n. Transcripts, Vol. 212: 15381-15389.\n\n. Brody 1987(b): 71.\n\n. Transcripts, Vol. 211: 14253.\n\n. Transcripts, Vol. 211: 14257.\n\n. Transcripts, Vol. 210: 14215-14258.\n\n. Hudson's Bay trader William Brown was the first European to keep systematic written records describing his dealings with the Gitksan and Wet'suwet'en, and their social organization from his point of view. The North West Company had operated Fort St. James on nearby Stuart Lake from 1806-1822 and their traders had obtained \"second-hand information about the Babine, Gitksan and Wet'suwet'en. Subsequently, William Brown discovered that some of this intelligence was not accurate.\" (Ray, 1991(b): 302).\n\n. McEachern, _Reasons_ 1991: 35.\n\n. _Delgamuukw v. B.C._ (1989) 38 B.C.L.R. (2d) 165 (judgment on the admissibility of historical documents), referenced as \" _Delgamuukw_ 1989 documents.\"\n\n. Fortune 1993.\n\n. Delgamuukw 1989 documents, ftnt 55: 169.\n\n. \"1. true, valid or sufficient at first impression; 2. self-evident; 3. legally sufficient to establish a fact or a case unless disproved.\" _Webster's New Collegiate Dictionary_.\n\n. _Delgamuukw_ 1989 documents:ftnt 55: 171.\n\n. Ibid., 175.\n\n. Ibid., 175-176.\n\n. Bourdieu 1987: 820.\n\n. _Delgamuukw_ 1989: 176.\n\n. _Delgamuukw v. B.C._ (1989) 38 B.C.L.R. (2d) 176, referenced as _Delgamuukw_ , 1989, treatises.\n\n. _Delgamuukw_ , 1989, treatises: 187.\n\n. Ibid., 187-188.\n\n. Ray 1987: 1.\n\n. Ray 1991(a); 1991(b).\n\n. Ray 1991(b): 304.\n\n. Harmon was a Northwest Company trader who made contact with the Carrier, neighbors of the Gitksan and Wet'suwet'en, during the early 1800s.\n\n. Transcripts, Vol. 203: 13478.\n\n. Ray 1987: 46.\n\n. Ray 1991(b): 303 (quoting Brown, Fort Babine [Kilmaurs] District Report, 1822-23, B. 11\/e\/1).\n\n. Ray 1991(b): 304.\n\n. Ray 1987: 76.\n\n. Ray 1991(b): 308 quoting Brown, Fort Babine [Kilmaurs] District Report, 1822-23, PAMHBC B.11\/e\/2, footnote 13.\n\n. Ray 1991(b): 310.\n\n. Ray, 1991(a): 15. Ray is referring to the following works of these authors: Bishop 1979; Goldman, Irving 1940, 1941, 1975; Jenness 1943; Kobrinski 1977; Morice 1970; Steward 1961.\n\n. Ray 1991(b): 309.\n\n. Ray 1991(a): 16.\n\n. Ray 1991(a).\n\n. Ray 1991(a): 19.\n\n. Ray 1991(a): 25.\n\n. Ray 1991(a): 23.\n\n. Ray 1991(a): 26.\n**Chapter 11: The Crown's Case**\n\n_We are a poor people with rich resources. That is why the province and the federal government are running scared. And that is why they wouldn't deal with us. If it was a barren land, do you think they would bother with us? The taxes that are generated from our resources are what is paying for all of the things that they are doing against us._\n\n_They had huge mounds of money to pay their lawyers, and they had a lot of lawyers.... And the office staffs. It was unbelievable, the amount of people that they had working with them.... They had money like it was going out of style. They had the staff behind them that could run and fetch whatever it was that they needed on a moment's notice\u2014and we were criticized because of our miserable little amounts._ _I don't feel one bit ashamed of where that money comes from. It is a mere pittance of what is owed to us. A pittance...because over a hundred years the resources have been just raped out of our territories...._\n\n_That's our money paying for all of this. Why? Because the resources and the taxes that are generated from those resources are what those people are afraid of losing._\n\n\u2014Yagalahl (Dora Wilson), 1992.\n\n**The Political Economy of Litigation**\n\nThe defendants in _Delgamuukw v. R._ were _Her Majesty the Queen in Right of the Province of British Columbia and the Attorney General of Canada_ : \"the Crown.\" They asked the court for \"a declaration that the Plaintiffs have no right, title or interest in and to the Claim Area, and the resources thereon, thereunder or thereover.\" The Province of British Columbia argued that, although the Gitksan and Wet'suwet'en had no rightful claims at all, in the event that the court found that they did, and there were any damages to be paid or compensation due, it was the federal government's responsibility to pay. The federal government argued that some of the costs should be borne by the province.\n\nThe Province of British Columbia's argument in the _Delgamuukw_ case was constructed on the basis of four building blocks. First, they argued that Gitksan and Wet'suwet'en societies were only minimally organized during what the Crown called \"pre-historic\" times. The Crown claimed that no system of government or property law existed among Aboriginal peoples prior to European contact that could be recognized as equal to, or deserving of respect by, British or Canadian law. The Crown also had a fall back position in the event that the judge might reject _terra nullius_ and find that some form of legally recognizable Aboriginal rights were being practised when Europeans arrived in northern British Columbia during the eighteenth century. They argued that the only locations that could possibly have been used and occupied to any extent, and therefore could possibly be recognized as subject to an Aboriginal claim, were the village sites next to rivers with large salmon runs. According to the Province of B.C., the ancestors of the Aboriginal peoples appeared to have used and occupied only those spaces which were later designated by the federal government as reserves, and for which the federal government is responsible. The only resource the Gitksan and Wet'suwet'en may have any claim to, said the Province, were salmon, also a federal responsibility.\n\nThe second building block of the Crown's argument was the claim that the hunting territories and access routes outside the major villages were used only incidentally, sporadically, and arbitrarily by anyone who wished to wander about. Their position was that no consistent use, or occupation, or ownership, or management was ever exercised in these territories. Although the Gitksan's and Wet'suwet'en's maps showed 133 distinct and bounded territories that included identified village and camp sites, and precise locations where resources and game were located in each, the Province of B.C. argued that no one ever travelled far from a riverfront village-cum-reserve to hunt or gather on these territories. According to the Province of B.C., the ancestors of the Gitksan and Wet'suwet'en peoples appear _not_ to have used and occupied those spaces that are now under lease to multinational forestry companies, and from which 3 million cubic metres of timber were harvested during the four years of the _Delgamuukw_ trial.\n\nThird, the Crown argued that elementary concepts and systems of property rights emerged among the Gitksan and Wet'suwet'en following the commencement of European-Aboriginal trade: during a period they referred to as \"protohistoric times.\" Captain James Cook's meeting with Ahousat Chief Maquinna on the west coast of Vancouver Island in 1774 marks the first recorded European contact with First Nations in what is now British Columbia. The key time period for the Crown's case, therefore, encompassed the 48 years between 1774 and 1822 when Hudson's Bay Company trader William Brown established a fort in Gitksan and Wet'suwet'en territory and began recording the first significant British observations of the Gitksan and Wet'suwet'en. Since Brown's journals described social organization that closely resembled the House and clan social structure and the land tenure system that the Gitksan and Wet'suwet'en witnesses described, the Crown argued that Brown was writing about a society that had already been significantly assimilated into British culture between the time of Captain Cook's 1774 landing several hundred miles away on the west coast of Vancouver Island, and Brown's arrival in Gitksan and Wet'suwet'en territory in 1822. Therefore, the Crown posited, Brown was not describing a \"truly aboriginal society.\" This rationale for invalidating Brown's observations was the cement that held the Crown's edifice together. Prior to the advent of the European-Aboriginal fur trade, so the Crown's story goes, the Gitksan and Wet'suwet'en lived in nomadic, unorganized, unstable groupings. As a result of becoming involved in the fur trade, Aboriginal peoples took the first necessary step to assimilating into civilized European society by developing ranked societies based on social inequality, the Crown concluded.\n\nFinally, the Crown argued that, even if some form of Aboriginal title or rights had been held by the Gitksan and Wet'suwet'en, they had been extinguished by the simple assertion of British sovereignty, or by the assumed acquiescence of Aboriginal peoples to the superimposition of British and Canadian law.\n\nIn summary, these four arguments constituted the Crown's case: (1) Ultimately, the British Crown had the right to assert sovereignty and extinguish Aboriginal title and right. (2) Anyway, Gitksan and Wet'suwet'en societies were \"unorganized\" before contact with Europeans: hierarchal social organization and property law was absent. (3) These developed in response to the European-Aboriginal fur trade through a process of assimilation of Aboriginal ways of life to those of the European settlers. (4) The Gitksan and Wet'suwet'en had acquiesced to British and Canadian rule.\n\n**Anthropology: The Science of \"Man\"?**\n\n_It is clear that theories do much more than explain social life; they also define the understandings that underpin different forms of social practice, and they help to orient us in the social world...Naturally, granted what is at stake, human beings will always be tempted to espouse theories that give them a sense of moral orientation, and even more theories which support the practices they find advantageous...._\n\n\u2014Philosopher Charles Taylor,1985, _Social Theory As Practice_.\n\n_Behind and supporting...the understandable prestige of the natural science model, stands an attachment to a certain picture of the human agent. This picture is deeply attractive to moderns...it shows us as capable of achieving a kind of disengagement from our world by objectifying it...._\n\n\u2014Philosopher Charles Taylor, 1985, _Philosophy and the Human Sciences_.\n\nThe Crown's lawyers argued that their critique of the evidence presented by the Aboriginal plaintiffs and their expert witnesses was based on principles of a scientific research methodology. They entered an article entitled \"The Expert in Court,\" written by Anthony Kenny and published in a 1983 edition of the _Law Quarterly Review_ as an exhibit. In this paper, Kenny set out four criteria for determining whether a discipline is sufficiently scientific in its methodology to justify the admission of expert opinion evidence in a court of law: \"First,\" Kenny wrote, \"the discipline must be consistent. That is to say, different experts must not regularly give conflicting answers to questions which are central to their discipline.... Second, the discipline must be methodical. A procedure carried out by one expert to reach a particular conclusion is one which must be capable of duplication by any other expert.... Thirdly, the discipline must be cumulative. The findings of one generation of workers in the discipline are not called in question by the workers of the next...research, once done, does not need doing again; if you have to repeat someone else's experiments, or re-sample his population, on the very same issue, that shows you there was something wrong with his experiment, or something faulty in his sampling.... Fourthly, the discipline must be predictive, and therefore falsifiable...it must predict the not yet known from the already known (as the doctor's diagnosis of the nature of a terminal illness predicts what will be found at the post mortem), and is falsified if it proves otherwise.\"\n\nKenny's first and third criteria are particularly problematic, from any methodological perspective. If experts cannot disagree and debate without discrediting their areas of study, and if the findings of one generation cannot be called into question by the next, then neither scientific nor democratic progress can be made. All four of Kenny's criteria\u2014but particularly two and four, in their demand for the replication of research contexts and procedures, and the ability of studies to reliably predict outcomes\u2014are the criteria of positivist science, modelled on the methodology employed by natural scientists. Inclusion of disciplines whose knowledge bases arise from people-studying-people (anthropology, sociology, political science, history, literature) within this paradigm rests on the assumption that human beings are like water, and their behaviour can be predicted in the same way that we can predict water will always boil when it reaches a certain temperature. This is a proposition that generations of social scientists have contested, arguing that humans are social beings: meaning-makers whose lives are lived, and whose decisions and actions are taken within networks of relationships with their environment and with other human beings. The \"research environment\" of life, differs in its essence from the controlled laboratories of experimental science. The systematic study of human beings requires methodologies that take account of the dynamic particularities of human social life as it is lived and experienced by human beings, including the researchers themselves.\n\nPeople's own accounts and analyses of their lived experience have no legitimate place in the framework Kenny provides for determining what knowledge is valid. Ethnographic fieldwork and participant observation research, where anthropologists live with, and extensively observe ordinary people going about the course of their daily lives, cannot possibly meet the criteria set out by Kenny. Disciplines like history and anthropology that have undergone significant changes in recent years in response to the inclusion and consideration of the scholarship of previously excluded categories of persons\u2014 like women and Indigenous peoples\u2014are rendered unreliable and inconsistent by a positivist approach.\n\nIt is not only professionals in the \"human sciences\" who encounter difficulty in meeting Kenny's criteria for qualification as \"expert witnesses.\" In the realm of European physical sciences, neither Copernicus, Galileo, Newton nor Einstein would have passed the first test of their expertise in their day. Neither have any of these astronomers and physicists passed the third test, since all of their findings have been questioned and modified by other astronomers, physicists and mathematicians in subsequent generations. Most significant in this context, however, is the fact that Charles Darwin, on whose theories of biological evolution nineteenth century anthropologists, historians and jurists based their hypothetical formulations of social evolution, could not have passed any of Kenny's four criteria.\n\nKenny's criteria are fundamentally anti-scientific in their devaluing of criticism, reflection, accumulation of knowledge, and change in response to new information. The Crown's tenacious advocacy of positivist social theory, scientific methodology, and empiricism is particularly ironic in light of the mystical foundation of Crown title: the phantasmic apparition of an ethereal sovereign hovering over the land.\n\n* * *\n\n. Yagalahl 1992: 203.\n\n. McEachern 1991, _Reasons for Judgment_ in _Delgamuukw v. R._ , Vancouver, British Columbia: 43.\n\n. Glavin 1990: 179.\n\n. Kenny 1983.\n\n. Ibid., 113.\n\n. There is a huge literature on this debate within the social sciences. For an accessible entry to the debate see Rosaldo 1989.\n\n. See Behar and Gordon (eds.) 1996.\n**Chapter 12: Experts on Behalf of the Defendants**\n\n_Court performances introduce the cultural practices of the dominant group to the subordinate group as they impose new regulations and conceptions of social relationships. In their public ritual performances, courts operating in situations of cultural diversity and unequal power tend to apply the rules of one group to other groups. These performances demonstrate the procedures of the dominant order, demand compliance with it, and illustrate through both the imposition of laws and their enactment in the daily life of subordinate peoples, the ways it applies to everyday life. Thus, court performances are one way of expanding the hegemony of the law. Not only are ideas articulated but they are superimposed on everyday life in front of an audience of relatives, neighbours, and friends._\n\n\u2014Anthropologist Sally Merry Engle, 1994, _Courts as Performances_.\n\n**Her Majesty's Loyal Anthropologist**\n\nThe Crown claimed that their critique of the plaintiffs' evidence was based on objective, neutral, scientific criteria of evaluation and research methodology, and that these principles were what guided them in _their_ selection of expert evidence. In order to support their arguments, lawyers for the Crown turned to social science. They were particularly attracted to those branches of the social sciences that continue to give credence to theories of universal laws of social and cultural evolution, and that posit that all societies follow western Europe, over time, on the climb up the developmental ladder from hunting and gathering to modern industrial capitalism. The Crown called experts in historical cartography, history and anthropology. Most of the Crown's witnesses were professionals and government employees rather than academics or scholars. These included: a physician, several guide outfitters, a retired local politician and self-described pioneer, a popular historian, fisheries officers and a retired federal Indian Agent. No Gitksan or Wet'suwet'en, or member of any other First Nation, testified on behalf of the Crown.\n\nThe centrepiece of the expert evidence for _Her Majesty the Queen In Right Of The Province Of British Columbia And The Attorney General of Canada_ , and, specifically, for the prestigious Vancouver law firm of Russell & DuMoulin who represented Her In Right Of Them, took the form of an opinion report prepared under Russell & DuMoulin's direction by Sheila Patricia Robinson, Ph.D., a cultural geographer by training, who was accepted by Chief Justice McEachern as an expert witness in anthropology. Robinson had never held an academic position, nor has she published her expert opinion reports so they can be scrutinized by either her colleagues or the public. She is a professional witness, hired by law firms on short-term contracts to testify for the Crown in numerous Aboriginal rights cases. Robinson's tasks in this capacity are twofold. First, she assists Crown lawyers in their attempts to discredit Aboriginal people and the expert witnesses who testify for them. Second, she provides academic research to support the Crown's legal arguments.\n\nSheila Robinson's report, entitled \"Protohistoric Developments in Gitksan and Wet'suwet'en Territories,\" consisted of an introduction and five sections, covering 29 pages; and an additional 67 pages of supporting notes and references were appended to the main body of the report. The central narrative in Robinson's story is about social change caused by the arrival of manufactured commodities during the European-Aboriginal fur trade. In order to understand how, when, why, and what change took place she had to first describe Gitksan and Wet'suwet'en societies as they were before Europeans and their commodities appeared.\n\nAs befits the role of the defendant's expert witness, Robinson began her report by casting doubt on the credibility of the plaintiffs' primary source of knowledge on the matters in dispute: Gitksan and Wet'suwet'en oral histories. Although the Crown's witness had not studied Gitksan and Wet'suwet'en oral traditions, she dismissed oral history as a invalid source of knowledge out of hand, supporting the Crown's contention that they should be treated as \"hearsay\" rather than \"history.\" She wrote: \"It is important to emphasize the limitations inherent in any theory of aboriginal land use which attempts to reconstruct a 'reality' that existed long before any relevant written records were kept and long before the memory of living man....\" Taken by itself this may be a perfectly reasonable statement. However, without further specification of what those limitations are\u2014such as Richard Daly's review of methodological problems in the study of oral traditions in general, and Gitksan and Wet'suwet'en oral traditions in particular, had provided\u2014Robinson's point is simply a rhetorical statement of the obvious. Her critique of oral history appears even less plausible when the scholarly sources she relied on to describe Gitksan and Wet'suwet'en social organization before European contact are scrutinized: 47 of her 82 bibliographic references on this subject were based substantially on oral histories collected from Aboriginal elders by anthropologists. The Crown's witness rejected oral tradition on theoretical and methodological grounds, and then selectively relied on oral tradition to support the descriptions of pre-contact Aboriginal cultures that form the basis of her report. When lawyers for the Gitksan and Wet'suwet'en questioned Robinson about how she rationalized drawing conclusions from data whose validity she denied by definition, she replied that she had described pre-contact Aboriginal society \"to the extent that that can't be done.\"\n\nRobinson's report continued: \"I am generally suspicious of writers who tend towards a static or structural treatment of any society.... The ideal models may outline rules of behaviour, but do not adequately account for ongoing adjustments\u2014in population distribution and density, in resource distribution, in alliance formations, and so on. Gitksan and Wet'suwet'en societies were never static.... Reconstructions of traditional native socioeconomies which fail to account for indirect European influence deny the dynamic dimensions of ongoing cultural adaptations and resign their subjects to an untenable\u2014 however romantic\u2014'snapshot' stasis,\" she wrote.\n\nUnder cross examination Dr. Robinson was able to name only one text that fit the description she set out of \"static or structural treatments,\" and that study referred not to the Gitksan and Wet'suwet'en but to a different Northwest Coast Nation, the Kwa'kwak'wakw. This text has since been substantially revised by the authors themselves, based on later research. Robinson could not name any sources that described the Gitksan and Wet'suwet'en in the \"romantic\" fashion her report referred to, nor could she identify any sources that claimed no changes had occurred in Gitksan and Wet'suwet'en society and culture during the eighteenth and nineteenth centuries\u2014or before or since, for that matter. Under cross-examination by Peter Grant, lawyer for the Gitksan and Wet'suwet'en, the Crown's witness insisted that she was not referring to the opinion reports submitted by the plaintiff's anthropologists when she made these statements. Taking her at her word, then, and as there is no empirical evidence to support her critique, it is fair to conclude that the importance of these introductory sentences in her report does not lie in their literal meaning but in their connotations: in the images they evoke, rather than in the facts they reveal.\n\nIn a rhetorical sense, Robinson's introductory paragraph alludes to two popular stereotypes of anthropology as idealistic and romantic, and\/or as abstract and intellectually elitist. It repeats the well-known charges that anthropology idealizes non-western cultures, \"freezes\" Indigenous peoples in time before contact, and denies post-contact change. These are legitimate critiques of some \"old\" ethnographies that, ironically, are over-represented in Dr. Robinson's own bibliography.\n\nIn the context of the _Delgamuukw_ trial, Robinson's references to \"romantic,\" \"idealized\" and \"unchanging\" descriptions of Aboriginal societies serve to represent her, in contrast to the \"straw anthropologists\" she set up in her report, as a reasonable, scholarly woman who deals in objective facts, rather than romantic ideals. This helped establish common ground\u2014emphatic communion\u2014between Dr. Robinson and Chief Justice McEachern: we are people of like mind; we are educated; we are objective; we are intelligent; we are on the same side.\n\nRobinson went on to dismiss the evidence presented by other expert witnesses who had testified for the Gitksan and Wet'suwet'en. Sylvia Albright, an Ethnoarchaeologist, had cited archaeologist Gary Coupland's recent research in north-western British Columbia in her testimony for the plaintiffs. Coupland found ample archaeological evidence of land mammal hunting having been carried on in the Gitksan and Wet'suwet'en territories circa 500 B.C., and argued that cultures with hierarchically ranked social structures had emerged by that time. In her initial proposal to Russell & DuMoulin, Sheila Robinson had suggested that \"archaeological reports could be quite worthwhile,\" and she had referred to archaeologists MacDonald, Allaire and Coupland as \"usually forming sound opinions.\" By the time she testified at the _Delgamuukw v. R._ trial Robinson had changed her mind about the validity of archaeological data in general, and the reliability of Gary Coupland's findings in particular. She dismissed Coupland's work\u2014which did not support the Crown's argument that social ranking among the Gitksan and Wet'suwet'en emerged in response to the eighteenth-century European-Aboriginal fur trade\u2014claiming that there had been too little archaeological research in the Gitksan and Wet'suwet'en territories for any data to yield conclusive evidence.\n\nThere is little dispute in the archaeological literature about whether or not Coupland's findings are valid. Where debate does arise within the discipline is on the question of migrations during the period under study. That is, archaeological evidence neither proves nor disproves the question of whether the present Aboriginal inhabitants are the biological descendants of the people who lived in the region 5000 years ago. Some archaeologists argue that other Aboriginal people may have migrated into the region between 3000 B.C. and 1744 A.D. and killed off, or otherwise displaced, the people they found occupying the land. It is these \"migrants,\" they argue, who may be the ancestors of contemporary Aboriginal peoples. The Crown favours this theory as it challenges Gitksan and Wet'suwet'en oral histories that assert that their ancestors have occupied their territories since \"time immemorial.\" The insistence on proof of \"pure\" _biological_ descent to be considered as \"legitimate\" heirs with inheritance rights is an artefact of European patriarchal cultures, reflected in British property and estate law.\n\nUnlike the Gitksan's and Wet'suwet'en's anthropologists, Sheila Robinson conducted no fieldwork with them, or with any other Aboriginal peoples. Hence, she was not subjected to the same criticisms of \"favouring a party\" that were levelled at Mills, Daly and Brody. Since Robinson had not conducted fieldwork, she was presented by the Crown as not having been \"contaminated\" by the thinking of any \"others;\" or by any of her own potentially irrational sentiments, that may be triggered by empathy or any other emotion. No one would accuse Sheila Robinson, as Crown Counsel had Heather Harris, of having \"gone native.\" No one would disparage her, as the Chief Justice had Hugh Brody, for \"liking Indians.\" No one would even ask her if she \"liked white people.\"\n\nHaving cast doubt on the credibility of the plaintiff's witnesses and on the reliability of their evidence, Robinson proceeded to set out the methodology she had employed in researching her expert opinion report: \"This report is based on a review and interpretation of existing information, largely contained in secondary sources, both published and unpublished, concerning the Gitksan and Carrier, as well as other Northwest coast and interior native groups. I have not carried out fieldwork.... Nor have I conducted archival research.... My general understanding...is shaped by the research I carried out for my doctoral dissertation. In connection with this study of Indian agriculture on the northern Northwest Coast.... I investigated ethnographic and early historic records pertaining to the Tlingit, Haida, Coast Tsimshian and neighbouring native populations.... Although my analysis in the present report borrows from the work of others, the conclusions are my own.\"\n\nThere are, however, several important contradictions between Robinson's Ph.D. thesis and her testimony as an expert witness, beginning with her reliance on theoretical speculation and comparative data drawn from studies of the Gitksan's and Wet'suwet'en's neighbours. In her 1983 doctoral dissertation\u2014after quoting and criticizing two other scholars for relying on \"inference or speculation\"\u2014Robinson argued that, although \"piecemeal reconstruction of particular developments is perhaps a tedious route to scholarly enlightenment...[it] eventually leads to more easily substantiated and probably more appropriate hypotheses. And it is in keeping with the works of scholars who stress that the analysis of cultural processes is not always amenable to rigid theoretical treatment because historical records may yield unanticipated yet illuminating data.\"\n\nContemporary ethnohistorians have, indeed, been engaged in work that focuses on the particularities of local variation, rather than on the construction of theoretical generalizations. When Robinson took the stand as an expert witness in _Delgamuukw v. R._ she was asked to substantiate her claim that the theoretical framework presented in her expert opinion report was \"innovative, credible, and in keeping with recent ethnohistoric, cultural-ecological and cultural-evolutionary research.\" She testified that she had been guided by a 1986 volume of the journal _Anthropologica_ , edited by Dr. Edward S. Rogers of the Royal Ontario Museum in which he summarized his recommendations for future ethnohistorical research. Rogers urged scholars to develop more precise descriptions of locally distinct pre-contact cultures, take greater account of regional variation, and exercise more caution in asserting comparative generalizations. He listed the following factors that should be taken into account in future studies: climatic changes, game cycles, spatial distribution of resources, production of trade items, resource productivity, size of fish, demographic patterns, technology, sociopolitical organization and the influence of religious beliefs and behaviour patterns. Peter Grant asked Sheila Robinson whether she agreed with Dr. Rogers' recommendations.\n\nGrant: You would agree that these are factors that should be considered in analyzing the pre-contact society? Do you agree that in a study of the Gitksan or the Wet'suwet'en the climatic changes should be taken into account?\n\nRobinson: Oh, I would think so.\n\nGrant: Game cycles should be taken into account?\n\nRobinson: Yes.\n\nGrant: Spatial distribution of resources should be taken into account?\n\nRobinson: Yes.\n\nGrant: Production of trade items should be taken into account?\n\nRobinson: Well...I would say yes.\n\nGrant: Resource productivity should be taken into account?\n\nRobinson: To the extent that it can be.\n\nGrant: And the...size of fish?\n\n....\n\nRobinson: I would say that fish, characteristics of fish perhaps rather than just size might be important for this area.\n\nGrant: For the Gitksan and Wet'suwet'en area?\n\nRobinson: Yes.\n\nGrant: Good. I agree. So characteristics of fish should be taken into account for Gitksan and Wet'suwet'en?\n\nRobinson: Perhaps.\n\nGrant: And...cultural considerations?\n\n....\n\nRobinson: To the extent that they can be.\n\nGrant: The technology...the technology of the Gitksan and Wet'suwet'en should be taken into account?\n\nRobinson: Yes.\n\n....\n\nGrant: And sociopolitical organization should be taken into account?\n\nRobinson: Yes.\n\nGrant: And the influence of spiritual beliefs and behaviour patterns\u2014 Robinson: Yes.\n\nGrant: Now, in your research and preparation of your report you did not take into account climate, did you?\n\nRobinson: Not specifically.\n\nGrant: Game cycles?\n\nRobinson: Not specifically.\n\nGrant: Spatial distribution of resources?\n\nRobinson: Yes, to some extent.\n\nGrant: Did you refer to that\u2014describe that in your report?\n\nRobinson: I don't know. I'm sure you've had a more thorough review of the paragraphs in recent days than I have. But, yes, it has got a spatial component to it in its description of the changes in hunting territories and so on.\n\nGrant: But I am talking about spatial distribution of resources, that would be game and where they are located?\n\nRobinson: Did I take it into account? Yes, to some extent.\n\nGrant: But you have indicated before you weren't sure where these resources were located within the Gitksan and Wet'suwet'en territory?\n\nRobinson: I'm not sure what level of detail you were asking for, then or now. Grant: And did you deal with the production of trade items?\n\nRobinson: Yes, to some extent.\n\nGrant: And to the extent it is reflected in your report?\n\nRobinson: Yes. Not necessarily.\n\nGrant: Well, we have nothing\u2014 Robinson: Not necessarily expressed in the report directly all the consideration I made of the exchange of trade items.\n\nGrant: Well, what sources did you rely on for your analysis of the spatial distribution of resources? If you want you can look at the bibliography of your report.\n\nRobinson: Oh, can I?\n\nGrant: Yes.\n\nRobinson: Well, that's great. Perhaps we can go and list all the writers again who have dealt with spatial distribution in the territories.\n\nGrant: No, no, no.\n\nRobinson: No, no, no. Please don't shake your head at me.\n\nGrant: I want to ask you this, what sources did you deal with with respect to spatial distribution of resources among the Gitksan and Wet'suwet'en?\n\nRobinson: Kobrinsky and Bishop, among others.\n\nGrant: Okay. Thank you. Which sources did you rely on in your consideration of production of trade items with respect to the Gitksan and Wet'suwet'en?\n\nRobinson: Any of those sources which are referred to in chapter three of my dissertation which pertain to trade and the discussions of production and exchange, the circulation of wealth and so on. And also to those scholars that I have listed several times now who are included in the bibliography. I can either go alphabetical if you go through the footnote list again, or I can recite them probably from memory going from north to south.\n\n....\n\nGrant: You may have misheard me again.\n\nRobinson: No, I didn't mishear you.\n\nGrant: I asked what sources did you rely on in support of your analysis of the production of trade items _among the Gitksan and Wet'suwet'en?_\n\nRobinson: Well, I relied on more than those sources that pertain directly to the Gitksan and Wet'suwet'en.\n\nGrant: I understand that.\n\nRobinson: Well, I'm glad.\n\nGrant: I want to know specifically which ones relate to the Gitksan and Wet'suwet'en?\n\n....\n\nRobinson: Specific information on the Gitksan and Wet'suwet'en...is expressed in the Brown reports... Grant: But the Brown reports that I have been discussing with you...are not cited in your report?\n\nRobinson: No, they are not cited in the report.\n\nContrary to the argument in support of locally-specific research that she had articulated in her doctoral dissertation, and that is endorsed by leading ethno-historians, Robinson proceeded to reconstruct a description of Gitksan and Wet'suwet'en cultural history based on theoretical speculation and analogies to other First Nations' histories. The Crown's witness described Gitksan and Wet'suwet'en society as a \"segmentary\" one that lacked \"mechanisms to create and sustain centralized political and economic authority,\" but admitted that this was based on theoretical generalizations about the nature of \"segmentary societies\" and not on data about the Gitksan and Wet'suwet'en.\n\nThe final sentence of Robinson's introductory paragraph expresses the central legal point of the exercise: \"In this context, Gitksan and Wet'suwet'en claims about 'traditionally' having owned and managed certain territories are questionable.\"\n\nRobinson's analysis of pre-contact Gitksan and Wet'suwet'en society is built on a framework that organizes each section of her report, and is characterized by three features. First, the Crown's legal argument is faithfully supported. Second, data drawn from historical studies of other First Nations are frequently substituted for specific information about the Gitksan and Wet'suwet'en, even when this is available in the research literature. This is analogous to substituting French history for German history: there are parallels at high levels of generalization, but significant differences are well recognized. Third, Robinson's argumentation is often internally contradictory resulting in statements described by logicians as oxymorons: \"combinations of contradictory or incongruous words.\"\n\n**Malignant Mythology**\n\n_Every self-respecting people has its own founding myths; British Columbia whites, were, and are, no exception. The traditional White views were fully formed by the 1880s and remained little changed until the 1950s. These views belittled the worth and the claims of Indians while legitimizing the land ownership and political jurisdiction of the colonial authorities and their successors...only occasionally, and especially during the last quarter century, when the provincial government has had to defend its views in court, were attempts made to construct coherent legal or philosophic arguments in support.... The province's arguments in the Delgamuukw case provide the most comprehensive example of such an attempt._\n\n\u2014Political scientist Tennant, 1992, _The Place of Delgamuukw in British Columbia History and Politics\u2014and Vice Versa_.\n\nThe Crown witness' expert opinion report was presented in the form of a historical narrative. Sheila Robinson's description of the pre-contact period, which her report began with, supported several key points in the Crown's argument. It provided a baseline description of Gitksan and Wet'suwet'en society as minimally organized and unstable. Further, Robinson tried to demonstrate that the most highly valued lands were the main villages, and not the territories that surrounded them; and that all territories changed hands frequently and easily, there having been little if any law or regulation in the society as a whole, particularly with regard to property.\n\nRobinson described the pre-contact period as \"pre-historic,\" \"pristine,\" \"traditional,\" and when a \"truly aboriginal way of life\" existed. \"Pristine\" is defined in the dictionary as: \"(1) belonging to the earliest period or state. (2) a: uncorrupted by civilization; b: free from soil or decay: being fresh and clean.\" Most anthropologists have long since jettisoned the term \"pristine\" as an inappropriate description of pre-contact Aboriginal societies. The term and the concept are now interpreted as reflecting the worst forms of evolutionism and romanticism practiced by the discipline in its earliest days. The term \"pristine\" harkens back to European philosophers like Jean Jacques Rousseau, who, horrified by the social and environmental effects of industrialization, projected their nostalgic desires for an imagined Garden of Eden in their own imagined rural past onto contemporary Aboriginal cultures.\n\nRobinson began with a description of pre-contact Gitksan and Wet'suwet'en society that covered, by her own admission, an unspecified period of time prior to the arrival of European commodities. Under cross examination, lawyer for the Gitksan and Wet'suwet'en, Peter Grant, asked Robinson to delineate the length of time referred to in her cavalier sweep though history: Perhaps 100 years? Perhaps 10,000 years?\n\nGrant: You have no sense of the time depth of which is being discussed there?\n\nRobinson: No, I don't.\n\nCollapsing thousands of years of Aboriginal history into one vaguely-worded paragraph infers that Aboriginal cultures were static for thousands of years before European arrival, implying that they were incapable of change by themselves, but required the beneficent stimulus of colonial invasion in order to \"evolve.\"\n\nRobinson's report continued: \"Oral traditions describe groups of _settlers_ wandering about until they _discovered unoccupied lands_. According to the narratives, the _pioneers established villages_ and _laid claim to fishing sites, hunting territories_ , and other _resource-producing areas by using them_. The act of discovery, often reinforced by _supernatural occurrences_ , constituted the _basis of claim_ in many myths. However, _repeated use_ seems to have maintained _rights to resources_.\" (Emphasis added.)\n\nGrant asked Robinson for sources of facts to support the statements made in this paragraph:\n\nGrant: I would ask you if you can give one example of where the Gitksan have acquired territory by seizure of unclaimed land?\n\nRobinson: No, I cannot.\n\nGrant: Can you give one example of where the Gitksan have acquired land by discovery?\n\nRobinson: No, I cannot.\n\nGrant: Can you give one example of where the Gitksan have acquired land by occupying and using unclaimed territory?\n\nRobinson: I've read examples of these, but I don't know if they apply to the Gitksan in the historical or the modern.\n\nGrant: Can you give me one example of where the Gitksan acquired land by gift?\n\nRobinson: Not specifically, no.\n\nAgain, given the absence of any empirical support for her descriptions of pre-contact Gitksan and Wet'suwet'en cultures, the significance of this section of Robinson's report also appears to be rhetorical. Her use of language evokes a parallel between the British settlers' founding myth, and that of the ancestors of the Gitksan and Wet'suwet'en. This is a common theme in B.C.'s populist political culture, and appears repeatedly in arguments against the legitimacy of Aboriginal claims in British Columbia. In this part of \"White British Columbians' founding myth,\" the ancestors of contemporary Aboriginal peoples were simply the first ones to arrive on the continent, and they laid claim to \"empty\" land by using it and occupying it in the way of migrants everywhere, always. Europeans simply did the same thing when they came a few thousand years later. Its all very natural and inevitable, and non-Aboriginal people therefore have as strong a claim to the land as Aboriginal people do: the only difference between them being a few thousand years of use and occupation.\n\nThe introduction to Section IV of Robinson's opinion report is entitled \"Gitksan Territorial Ownership: Traditional.\" It reads: \"Section IV...[theo-rizes]... about the dynamics of territorial acquisitions and control in traditional Gitksan societies prior to the protohistoric period. Emphasis is placed on the fact that traditional Gitksan societies were never stable and that rights to certain territories probably changed hands frequently. A scenario where the most highly valued territories were those close to the relatively stable village bases in prehistoric times is developed to contrast with the description of protohistoric and early historic patterns of Gitksan land holdings presented in Section V.\" She described a number of processes to substantiate her contention that \"territorial rights changed hands frequently,\" drawn either from material on other Northwest coast groups or from anthropologist John Adams' study of the contemporary Gitksan potlatch.\n\nThe Crown's witness continued, summarizing her position on pre-contact land and resource management: \"Speaking generally, one may expect that _some form of organized control_ would have been exercised over _access_ to the fisheries and other _resources_ which were _necessary for survival_ and over the _local trails and bridges_ which facilitated _prehistoric trade networks_.... In my opinion, outlying _borders were 'probably hazy'_ because there was little need to define them precisely....\" (Emphasis added.)\n\nA number of points were made in this paragraph that were fundamental to the Crown's argument. There is \"some form of _organized control_ \" (read: NOT ownership), \"exercised over _access\"_ (read: NOT ownership), \"to the _fisheries\"_ (read: NOT trees and game), \"and other _resources_ \" (read: NOT property), \"which were _necessary for survival_ \" (read: meet the basic survival needs of humans as organisms), \"and over the _local trails and bridges_ which facilitated prehistoric trade networks\" (read: NOT borders or boundaries).\n\nThe Crown's case was based on the theory that scarcity of resources produces competition between people who depend on these same resources, and that such competition leads to concepts of property and the establishment of borders and boundaries to protect one group's access to resources against the incursions of another group. This argument had both a theoretical and an empirical aspect. Empirically, the questions that required answers were: what game and other resources\u2014like trees\u2014were in the territories? How abundant or scarce were they? What demands did the Gitksan and Wet'suwet'en place on these resources? How did they obtain and use them?\n\nThe Crown began by trying to establish that there were few game or other resources in the territories outside major village sites, and that the Gitksan and Wet'suwet'en did not significantly utilize or value the resources that were there. Scholars who specialize in this field have developed ways of calculating population to resources ratios, and analyzing different forms of social organization to hypothesize how resources were allocated and distributed, and if competition between people for access to these resources was present and contributed to pressures on the resource supply.\n\nThe Gitksan and Wet'suwet'en Chiefs' oral histories had been replete with references to the locations and uses of the resources in question. Arthur Ray, testifying for the Gitksan and Wet'suwet'en, had argued that: \"Before contact, the Gitksan and Wet'suwet'en were highly dependent on their salmon fisheries. They located their semi-permanent villages beside the most reliable fishing sites and tapped the adjacent countryside for a variety of other resources. Local population pressure would have made it necessary for them to conserve these other resources, particularly game. The house-territory system was well-suited for that purpose.\" Other expert witnesses in paleobotany, ecology, forestry, and wildlife management had provided detailed evidence of the location and extent of game and other resources in the territories.\n\nThe Crown's witness admitted, under cross-examination, that her knowledge of the ecology of this area was limited. Robinson could not answer questions about where various game were located or what their habits were. When confronted with evidence about their use she equivocated in the extreme, going so far as to suggest the possibility that, not only were fish the primary source of subsistence, but that fish _skins_ , rather than animal furs or hides, supplied a significant part of Gitksan and Wet'suwet'en clothing.\n\nRobinson:...We have to think in terms of the used resources rather than what's out there in terms of the natural environment. So the environment as it is used is different than the environment as it exists.\n\nGrant: Well, the coastal Tsimshian use the sea-based resources, the interior Gitksan use the rivereen resources, at least as far as the fish is concerned, right?\n\nRobinson: Yes.\n\nGrant: You don't know what they wore, the interior Gitksan and Wet'suwet'en. You have already told us that?\n\nRobinson: For which time period, sir?\n\nGrant: Pre-contact?\n\nRobinson: No. I don't think anyone knows what they wore pre-contact.\n\nGrant: But we can assume that they wore something in the winter months? Robinson: Yes, I think so.\n\nGrant: And we can assume that it wasn't from fish?\n\nRobinson: Not necessarily. I have seen references to the use of fish skin.\n\n... Grant: This is Harmon. You remember Harmon? You referred to him earlier, Dr. Robinson?\n\nRobinson: Yes.\n\n...\n\nGrant: Where he refers to \"The Native of New Caledonia, we denominate Carriers...\" You see that?\n\nRobinson: Yes, I do.\n\nGrant: Okay. Now, turn the page over. And then he gets into a description of them: \"...Their clothing consists of a covering made of...the skins of the beaver, badger, muskrat, cat or hare. The last they cut into strips, about one inch broad, and then weave or lace them together, until they become of a sufficient size to cover their bodies, and to reach their knees. This garment they put over their shoulders and tie about their waistes. Instead of the above named skins, when they can obtain them from us, they greatly prefer, and make use of blankets, capots, or Canadian coats, cloth or moose and red deer skin. They seldom use...either leggins or shoes, in the summer.\" Now, that description by Harmon is of the eastern Carrier. You would agree that that is some indication of what the aboriginal people of the interior, the north central interior of B.C. was wearing by a contemporary historical account?\n\nRobinson: Well, in the historic period for the Carrier or the eastern Carrier.... And I am gratified to see the reference of using the skin of fish for clothing is mentioned at the bottom of that paragraph.\n\nGrant: Yes. Well I knew you would find that.\n\nRobinson: Well, good.\n\n... Grant: Well, let's read what is there since you pointed it out. \"The women, however, in addition to the robe of beaver or dressed moose skins, wear an apron, twelve or eighteen inches broad, which reaches nearly down to their knees. These aprons are made of a piece of deer skin, or of salmon skins, sewed together. Of the skin of this fish, they sometimes make leggins, shoes, bags, etc. but they are not durable; and therefore they prefer deer skins and cloth, which are more pliable and soft. The roughness of salmon skins, renders them particularly unpleasant for aprons.\"\n\nYou have to\u2014in any of the analysis of Gitksan and Wet'suwet'en pre-contact you have to take into account that there was some resources that they relied on for clothing?\n\nRobinson: Locally. Some resources that they relied on for clothing, yes, I would think so.\n\nGrant: Yes. And you are not suggesting that you assumed that they wore fish skins in the winter months in the central interior of B.C.?\n\nRobinson: I don't know. Harmon has given us a comment about the use of fish skins as part of the clothing. But there is no reference here to what seasons of year they wore the fish skins and whether or not they comprised part or all of their clothing.\n\n... Grant: But would it not be fair to assume that the Gitksan and Wet'suwet'en would rely upon furs from within their area for clothing pre-contact? Robinson: I imagine that they relied on them to some extent. We have here a description of the use of other species of animals for clothing, or fish.\n\nSo the resources were there and the people used them. The next question relevant to the Crown's case was, was there sufficient pressure on the resources that people may have been led to compete for them? Peter Grant referred Robinson to paragraph 28 of her opinion report and the supporting footnote on page 68 of part 2.\n\nGrant: You state, and this is your statement, and I think we've already established that this is conjectural on your part: \"My reasoning is that it is only when there is pressure on a certain resource, when competition draws attention to a certain resource, then there is need to define it specifically.\" Accepting that reasoning, you must analyze all potential sources of pressure on the resource to come to a conclusion; do you not agree?\n\nRobinson: Not necessarily. And I do not believe there is information about all possible pressures coming to bear on resources.\n\nGrant: You would\u2014to come to a conclusion about pressure on a certain resource, it would be relevant to determine the utilization of that resource for clothing needs, would it not?\n\nRobinson: Not necessarily.\n\n... Grant: Would it not be necessary to determine the utilization of that resource for feasting purposes?\n\nRobinson: Not necessarily.\n\n... Grant: Would it not be relevant to determine the necessity or the utilization of that resource for trade purposes?\n\nRobinson: Not necessarily.\n\nGrant: You say not necessarily?\n\nRobinson: Yes.\n\nGrant: So ?\u2014\n\nRobinson: You're talking about resources generically, and I wouldn't say that any of those criteria you've just put to me are applicable for all situations. I'm not quite sure what situation you're referring to, Mr. Grant.\n\nGrant: Well, what situation are you referring to in your footnote; do you know?\n\nRobinson: I'm not sure what you're referring to, sir.\n\nGrant: I'm referring to the second sentence in...footnote 28a: 'My reasoning'.... 'My reasoning' being Sheila Robinson's reasoning?\n\nRobinson: Yes.\n\nGrant: 'Is that only when there is pressure on a certain resource when competition draws attention to a certain resource, then there is a need to define it specifically'?\n\nRobinson: Yes.\n\nGrant: What resource are you talking about?\n\nRobinson: I'm talking about a general principle as is expressed with regards to cultural ecological theory, and I did draw attention earlier to a reference, Robert Netting, and his very basic book called 'Cultural Ecology,' and you'll find there's reference to these general principles there. Your list of questions that had to do with specific characteristics or attributes associated with each resource, each or any resource, I said not necessarily, because I didn't think they were applicable criteria.\n\nGrant: You applied this reasoning in this statement to the Gitksan and the Wet'suwet'en?\n\nRobinson: Yes, I have.\n\nGrant: In fact,...there is no evidentiary basis for the utilization of this approach and its applicability to the Gitksan, is there; the approach starting 'My reasoning is...'?\n\nRobinson: Well, that may be your opinion, it's not mine.\n\nGrant: Is there an evidentiary basis for applicability to the Gitksan?\n\nRobinson: Yes. I think the basic premise in anthropology is that comparative examples are of extreme use, are of usefulness in identifying situations from one area to another. Anthropology is imperative.\n\nGrant: When there's not sufficient data about the specific society you're studying?\n\nRobinson: When there is some specific data it's also useful to use comparative approaches, because it tends to lessen the entrapment one can fall into by treating any society in isolation or as a unique example. These general principles and cultural ecology have brought application... Grant: Are you finished?\n\nRobinson: For the time being, yes.\n\nGrant: Thank you. Maybe I'll ask you a question.\n\nRobinson: Oh, good.\n\nGrant: You're stating that with respect to that sentence that it is irrelevant whether the Gitksan rely on any specific resource for clothing, feasting, trade goods, it's irrelevant to determine whether or not there is pressure on that specific resource?\n\nRobinson: I think you're putting words in my mouth. These are not quite the questions you asked me earlier in\u2014you're summarizing them in a slightly different manner. It's difficult for me to put the kind of words you're putting to me now against the list of questions you've just asked me. You were asking me whether or not these factors are necessary or necessarily taken into consideration in looking at pressure on any resource, or that's how I understood those questions to be.\n\nGrant: Well, let me rephrase them. In determining whether or not there is pressure on a certain resource among the Gitksan, it would be relevant to determine whether the Gitksan would utilize that specific resource for clothing, would it not?\n\nRobinson: Only if it was the kind of resource that was ever used with clothing. Grant: Yes or no?\n\nRobinson: If it was ochre it wouldn't be.\n\nGrant: Maybe. Of course, I'm talking about a resource used for clothing?\n\nRobinson: To the extent that that could be determined, yes, it would be relevant.\n\nGrant: To the extent that it can be determined if the Gitksan used a specific resource for feasting, it would be relevant to determine that so as to determine the pressure on that resource?\n\nRobinson: To the extent that that could be known, yes.\n\nGrant: Yes, you agree. And similarly, with respect to the utilization of a specific resource with respect to trade or exchange, as you've phrased it, it would be relevant to determine the pressure on that resource by the Gitksan?\n\nRobinson: To the extent that that could be known, yes.\n\nThe acquisition and production of goods for trade among the Gitksan and Wet'suwet'en, and between them and their neighbours was another source of possible demand, or pressure, on the resources in the hunting territories.\n\nIn the first draft of the report that she had submitted to her employers, Russell & DuMoulin, Robinson had written: \"In late pre-historic times Gitksan and Wet'suwet'en people were already involved in extensive aboriginal exchange networks, were already increasingly being influenced by coastal dwellers with whom they exchanged locally-produced and exotic commodities, were already developing ways to more effectively exploit local resources to satisfy subsistence and exchange related needs, and so on.\"\n\nPeter Grant questioned her on the absence of this opinion from her final report.\n\nGrant: And that was your opinion at that time based on your sources?\n\nRobinson: Yes.\n\nGrant: And that's your opinion today? You didn't change your opinion?\n\nRobinson: No, I didn't.\n\nGrant: But you did not include that opinion in your report, did you?\n\nRobinson: Oh, gee, I would think that I had, but it would perhaps... Grant: I would ask you to take a chance over the evening to look at that.\n\nRobinson: Because I certainly think that it's stated in my report that coast-interior trade or exchange did not spring up as a result of the European induced fur trade, but that there was an extension of, an expansion of and intensification of that after contact.\n\nGrant: Okay...I know you make reference to coastal-interior trade...before contact in the pre-historic time.... But here you're saying more than that.... Robinson: I may have expanded it there, but I don't think that the idea was left out of the report.\n\nGrant: You can look at the report, and if I stand to be corrected, I will ask you first thing in the morning.\n\nRobinson: I may have changed phrasing. I don't think the content was ignored.\n\nGrant: I couldn't track this, doctor. I couldn't track this into your report, and you may direct me to where I should have looked to track it.\n\nFinally, Grant showed Robinson a map on which the location of various game species and other resources like berry patches had been marked out. The map shows these resources as being distributed throughout the territory and not simply immediately surrounding the village sites.\n\nGrant: And mountain goat you've referred to as one of the trade goods. And in this case we see that there are mountain goat both inside and outside the black outlined area. But if the Gitksan and the Wet'suwet'en were involved in using or trading mountain goat furs or mountain goat skins then they would require access to these territories?\n\nRobinson: Not necessarily. And I might say that knowing some things about the environment is only one of the several factors that have to be taken into consideration. Without knowing about the density, the distribution of those animals that are being exploited, without knowing what their recovery rate is after exploitation or during exploitation, and without having some idea of the demands on those natural resources by the local population, the natural distribution makes little sense by itself. It doesn't tell me anything.\n\nGrant: It does tell you something, doesn't it?\n\nRobinson: It tells me that the animals were present or not present.\n\nGrant: It tells you that if the Gitksan required those animals for their own use or for trade they would have to have access to places where those animals were located?\n\nRobinson: Yes, they would.\n\nGrant: Thank you, doctor.\n\nWhen none of the empirical requirements to support their arguments could be met, Crown counsel opted to focus on the issue of defining \"the economy\" as a discrete sphere within a social structure, limited to transactions mediated through exchanges in a capitalist market place. This is a euphemism, from an historically-specific point of view, for saying that an \"economy\" that could be recognized by the court as such came into being only after Europeans arrived. This debate was most clearly illustrated during the cross-examination of the Gitksan's and Wet'suwet'en's witness, Hugh Brody, by one of the Province of British Columbia's lawyers, William Goldie. Goldie had been trying to make Brody agree with his statement that the Gitksan and Wet'suwet'en had \"no reason to travel\" more than twenty miles away from their riverfront villages \"for economic reasons.\" Brody had been resisting, arguing that they travelled to hunt animals in the territories for subsistence food, used hides for clothing and trees for housing, and accumulated surpluses for trade.\n\nGoldie:...Let us restate the assumptions that I am asking you to make. One, the primary food source is at the door-step of the home in Hagwilet. You accept that? The second assumption is that the territory in which the hunting is carried out is 90 miles away from that home. Number 3, that subsistence is not an economic activity in the question I am about to ask you. Number 4, that extended time is a period in the winter time of up to 7 months. You understand these assumptions?\n\nBrody: Yes.\n\nGoldie: Now, on those assumptions I suggest to you that the only logical reason for somebody being away from his home 90 miles for an extended period in the winter time would be to carry on an economic activity other than subsistence?\n\nBrody: Well, your assumptions logically compel the conclusion, because you have defined economic activity to exclude subsistence in your assumptions.\n\nGoldie: Yes.\n\nBrody: So, it follows logically, not as a matter of any empirical interest, but as a matter of logic that they are not out there pursuing an economic activity. Goldie: Yes.\n\nThe _facts_ establish that there were many reasons to want the resources one had to travel to get to. The _theory_ asserts that none of those reasons were \"economic,\" and further, that this means there could be no reason for any borders or rules to exist to regulate access to or use of these resources.\n\nThe third premise of the Crown's argument was that there was little competition among the Gitksan and Wet'suwet'en for access to the resources in the outlying territories. Robinson's depiction of pre-contact Gitksan and Wet'suwet'en society included a description of the social and political structure with a particular emphasis on economy, trade and land tenure. In this regard she wished to show that there was nothing we would recognize as a system of property ownership, that these societies were \"never stable,\" and that \"rights changed hands frequently.\" She wrote: \"Where conflict over resources was minimal, boundaries were probably vague and faded away from the village cores. I envision a patchwork configuration where controlled resource-producing areas away from village bases were linked up by a system of trails. But prior to the intensification of pressure on interior fur resources sparked by European demands for furs there would appear to have been no need for a sophisticated and elaborate body of rules governing access to resources or for extensive and defined areas of land for their exploitation.\"\n\nHowever, at the same time, in order to convey an image of pre-contact society as unstable, and continually rocked by warfare, Robinson contradicted herself by describing a situation of intense, and organized, competition surrounding property rights. She wrote: \"House rank was not fixed through time, and chiefs of the same side competed to recruit members and to amass wealth for potlatching. Wealth was derived in part from resources taken from house-controlled territories. Households within a side did not have equal access to 'shared' resources: higher-ranking chiefs were evidently able to lay claim to the more productive areas.\"\n\nFinally, Robinson concluded her analysis of pre-contact Gitksan and Wet'suwet'en cultures by proclaiming the Crown's legal argument. She wrote:\n\n\"In my research I have discovered no conclusive evidence that suggests that, prior to the advent of European influence in the claims area, the Gitksan and Wet'suwet'en lineages and families identified ownership rights to large and precisely defined tracts of hunting territories.\"\n\nIt is tempting to simply dismiss the Crown's witness. But that would ignore the important rhetorical impact of Robinson's courtroom style. When judges, lawyers, and lay people listen to weeks and weeks of testimony, some of it highly technical and specialized in nature, the details and logic of argument easily become lost in a maze of \"technobabble.\" Robinson succeeds in reinforcing a sense that the subject matter being dealt with is of a highly technical and complex nature that only well-educated specialists can possibly comprehend. In fact, what is being discussed is quite straightforward: were there game and other resources in the specific geographical locale in question, that is, the hunting territories shown on maps? The overwhelming bulk of evidence says yes, there were resources there. Did the people who lived on that specific piece of land during these specific times, the late eighteenth and early nineteenth centuries, use these resources? Again, the answer to this question is yes. These conclusions are drawn from empirical data. Was there competition and conflict between groups of Gitksan and Wet'suwet'en over property and resource rights, and were there laws that regulated how these conflicts were to be resolved? Substantial evidence was presented to the court to support affirmative answers to these questions. As well as oral histories and ethnographies, this evidence included a variety of research sources, including studies based in the kind of \"scientific\" methodology the Crown claimed to support.\n\nHowever, was this socially organized use of resources one that a British or Canadian court would classify as being what Lord Sumner described as \"differently developed...but...hardly less precise than our own,\" and that Viscount Haldane allowed \"could continue to be honoured within the framework of British sovereignty\"? These are theoretical\u2014and political\u2014questions whose answers are determined not by empirical observation but by ideological assessment. And that is the prerogative of judges.\n\n**Hegemonic Moments**\n\n_Antonio Gramsci coined the term hegemony to refer to power that maintains certain structures of domination but that is ordinarily invisible.... Hegemony refers to power that 'naturalizes' a social order, an institution, or even an everyday practice so that 'how things are' seems inevitable and not the consequence of particular historical actors, classes and events. It tends to sustain the interests of a society's dominant groups, while generally obscuring these interests in the eyes of subordinates._\n\n\u2014Anthropologists Susan Hirsch and Mindie Lazarus-Black, 1994, _Introduction: Performance and Paradox: Exploring Law's Role in Hegemony and Resistance_.\n\n_My premise is that the structure and dynamics of a ceremonial performance event serve to orient those who participate in it (and no doubt its spectators as well) toward distinctions of class and rank that subsequently become inscribed in administrative regulations and codes of practice\u2014even, under some circumstances, in the black letter of the statutory law.... I focus on one such hegemonic moment when it was possible to look 'behind the law' at legal categories in the making._\n\n\u2014Anthropologist Joan Vincent, 1994, _On Law and Hegemonic Moments:_ _Looking Behind the Law in Early Modern Uganda_.\n\nThe Crown's case against the Gitksan and Wet'suwet'en offers a window onto a \"hegemonic moment\" when abstract speculations and simplistic assumptions were transformed by the power of law into truth and facts. The Crown imagines that the core of legally significant cultural difference lies in the contrasts that they hypothesize exist between Aboriginal peoples' and Europeans' relationships to property and to commerce: private property defines European culture, and the absence of private property defines Aboriginality. By deconstructing the Crown's arguments and Robinson's testimony building block by building block we can see the legal categories of \"Aboriginal\" and \"non-Aboriginal\" in the making: the process and reasoning by which the Crown constructs legal subjects in its own image and creates truth in its own interests.\n\nIt is not necessary to accept, nor would be it be reasonable to consider, the Gitksan's and Wet'suwet'en's and their witnesses' evidence about pre-contact Aboriginal society as beyond doubt, question, challenge, or critique. Oral history is, like written history, a product of particular authors working in particular networks of power relations, at particular historical times, and for particular purposes. Anthropological theories and findings should always be contested and debated. However, even within a conservative framework of analysis, within the Crown's own professed criteria for legitimately \"scientific\" empirical research, it is obvious that the Crown's evidence and arguments about the nature of Gitksan and Wet'suwet'en social organization before European arrival were, at best, weakly supported by arcane and archaic academic research, and furthermore frequently defied a broad consensus in Canadian society at large of what constitutes \"common sense\" and \"truth.\" The average person, for example, would consider it reasonable to assume that people living in the intemperate climate of north-western British Columbia would more likely have dressed themselves in the hides and furs of minks and muskrats, than in fish skins. This is my first critique: on the basis of the best evidence by the Crown's own criteria, the difference between the Gitksan's and Wet'suwet'en's case, and the Crown's case, as presented to Court, was not the difference between two equally valid interpretations arising from different cultural premises; nor was it the difference between opposing but equally legitimate positions in academic debates about historical interpretation: it was the difference between sense and nonsense.\n\nThe Crown's legal argument that pre-contact Gitksan and Wet'suwet'en cultures lacked property law, and Robinson's supporting testimony, was summed up in this sentence from her report: \"In the absence of competition over scarce resources, there is no reason for the rules to exist.\" Under cross-examination Robinson explained that this statement was based on her reading of certain social theories:\n\nRobinson:...from my own reading and cultural ecology, I would say that a general rule of thumb that many anthropologists or cultural geographers go by is that people seldom assert exclusive or limiting control over any kind of specific or general resource unless there is a need to do so.\n\nGrant: And here you are speculating, are you not?\n\nRobinson: Yes. And my speculation is based on a very extensive reading of cultural ecology as a source of comparative examples and from comparative examples from the northern northwest coast region and adjacent interior region, among other sources.\n\nWhat underlying assumptions are necessary to accept as a universal fact that: \"in the absence of competition over scarce resources, there is no reason for the rules to exist\"? And to infer that the social organization of eighteenth-century Gitksan and Wet'suwet'en peoples could not have generated institutions and rules akin to what contemporary Canadians call property law and resource management?\n\nThe Crown's argument resonates with a conservative form of naturalism: theories of human nature that posit the universal and ahistorical existence of an individual who is first and foremost an autonomous, atomistic, biological organism. In order to survive and achieve their nature-driven imperative to reproduce the species, the needs of human organisms must be met. These needs can be reduced, for all important purposes, to food, clothing, shelter and sexual reproduction. Human beings, in this way of thinking, are instinct-driven animals to whose brains a capacity for rationality has been added, to varying degrees. The degree to which rationality and culture have been added on to different organisms is measured by the degree to which an individual or group manifests narrowly-defined \"economic rationalism\" in acts of \"rational decision-making\" which qualify as such if their end result is profit and\/or the ability to increase one's power over other individuals, thus gaining access to _their_ resources. Rationality, within this belief system, is demonstrated by an individual's, or group of individual's, success, in competition with others, in securing the necessary material goods required to meet their needs: survival of the fittest. The Crown's argument relies on acceptance of the hypothesis that the \"natural\" motivation for people to develop an intense relationship of identity with land and resources is the imperative to meet one's \"needs\" as individual organisms and aggregates of organisms as against the needs of other individuals and organisms. In the dominant western culture, identity with land reaches its peak of expression in individual private ownership in fee simple. This is the core concept of western \"possessive individualism,\" characterized by the slogan \"I own therefore I am.\"\n\nBut what if, despite facts presented to the contrary, the animals located in the Gitksan's and Wet'suwet'en's territories were not only _needed_ for subsistence and clothing, or trade and commerce? What if they were also _desired_ for feasting, gift-giving, prestige, sociality, joy, spiritual sustenance and renewal? What if the land is not a passive object awaiting human labour to transform it? What if the land is alive itself? What if the animals and resources are not awaiting human dispensation at all? What if they are sentient beings, part of a circle of life, an ecosystem? What if they are reincarnations of ancestors? Why then could there be reasons for agreements to be made between people about rules of access and \"ownership\" rights? Why then might a poet, a shaman, a carver, an artist, a mother, a father, a sister, a brother, a lover, a friend have a \"logical reason for being away from his home 90 miles for an extended period in the winter time?\" Why then could there be many \"reasons to travel?\"\n\nPsychiatrist Cisco Lassiter argues that: \"The world of the modern European self is located in a Euclidean-Newtonian space, homogeneous, uniform, and continuous: a space without sacred places, where all places are essentially interchangeable, and where geometry forgets the spiritual meaning of the earth, the land, and the human need for home....\"\n\nThe identity and well-being of the aboriginal self, on the other hand, Lassiter continues, \"depends on earth, ground, and place for an essential relationship to departed ancestors, cultural traditions, the world of the dead, gods, and time itself...this self cannot survive without access to the traditional sacred places, orientation by tribal landmarks, rootedness in the earth and in kinship relations.\" Clearly, the Crown's position did not recognize or respect the equality or integrity of such an Aboriginal relationship to land. This is my second critique: the philosophical premises of the Crown's argument were ethnocentric.\n\nIt is hardly surprising that the Crown's arguments reflected the dominant ideology glorifying private property and profit, and the ideal of an individualistic, unencumbered self. But there is a further puzzle in the Crown's case: an apparent paradox arises from the fact that the best evidence from all sources strongly supports the conclusion that the Gitksan and Wet'suwet'en met all the conditions required, by the criteria of conservative European political theory, to have had concepts of, and laws regulating, private-like property: they identified boundaries between territories, they produced for subsistence and exchange, they accumulated and stored surplus, and they engaged in conflicts with each other over resources. They had property, laws to protect it, and the will to defend it. These were the criteria required by the Crown's theory for people to \"qualify\" as \"rational economic men.\" Given that, officially, Canadian courts are governed by principles of human equality that direct them to assess people on the basis of their behaviour and not their race, how could the Crown still _logically_ , in their own terms, argue that the Gitksan and Wet'suwet'en systems were \"too primitive\"\u2014that is, too different and inferior in comparison to \"civilized\" systems\u2014to be recognized by the law?\n\nThe paradox is resolved by considering another assumption that underlies the way of thinking and being that the Crown's argument represented that goes beyond ethnocentrism. The liberal critique of ethnocentrism chastises anyone who thinks that their own cultural practices are the only, and the best, in the world. The resolutions to problems defined as rooted in ethnocentrism are education and tolerance: we should all learn to appreciate and respect the particular values of _all_ cultures on their own terms, and we should not judge others by our own culturally-specific criteria. However, this liberal position begs a number of questions.\n\nFirst, the liberal critique of ethnocentrism rests on a definition of \"culture\" as beliefs and values that can be isolated from political and economic organization; and the premise that \"cultural differences\" are the main cause of strife between peoples. Relationships between members of diverse cultures take place within the context of power relations, and differential access to resources. Therefore, the ethnocentrism of the dominant has consequences for the subordinate that simple changes in beliefs and values will not ameliorate. In other words, \"cultural\" differences may not be the main problem: unequal distribution of wealth and power and property may be. Second, the liberal critique of ethnocentrism does not address the thorny problem presented by the fact that a foundation stone of conservative European theory has been that European culture is _not_ one local and historically-specific way of life among many, but represents a _universally superior_ one. Therefore, it is not theoretically possible for the dominant western culture to enter into dialogue with any others on an equal footing without first forfeiting its self concept of universality and supremacy. Were its leaders to reject these foundational assumptions, how could the political and economic domination of colonial peoples be justified?\n\nHistorically, British colonial thought was based on the particular extrapolation of possessive individualism most influential in theories of Social Darwinism that ranked and compared people: civilized as superior, primitive as inferior. \"Primitive\" human organisms were characterized as animal-like beings driven by base survival instincts only minimally mediated by rationality or culture. The \"needs\" of primitives were strictly limited to the survival and reproduction of the human organism. Therefore, within this framework of analysis, even if the Gitksan and Wet'suwet'en appeared to be behaving much like property-owning European economic rationalists, their classification as \"primitives\" supports a conclusion that whatever their _observed behaviour_ in fact consisted in, eurocentric theories claim they were driven by instinct, and not by rationality, a characteristic that, by definition, \"primitives\" lack.\n\nTo understand the Crown's argument we must, again, enter Law's culture: we must leave tangible evidence behind on \"the ground\" and take flight into the abstract, imagined world of the hovering sovereign. If, and only if, we accept a theory of human nature and human social life as being naturally and inevitably acquisitive and competitive, and we accept that Europeans and Aboriginals are different and unequal human beings, then\u2014and only then\u2014 can the Crown's argument make logical sense, within their own framework of analysis and within a dominant ideology that claims all human beings are equal. This is my third critique: the Crown's argument was not simply ethnocentric eurocentric, but racist. I use the term \"racist\" in this context to denote an explanation of human behaviour based principally on the presumably inherent, naturalized habits and characteristics of a group of people who have been classified as belonging to a particular \"race\" or cultural group. My conclusion that the Crown's argument was racist rests on the fact that such a categorization is necessary to make their argument logical and coherent.\n\nLogical gymnastics aside, the basic story remains the same: Aboriginal people were here and they had laws governing relations with strangers. They obeyed their own laws. The British arrived, and they had laws governing relations with strangers, but the colonists neither respected Aboriginal laws nor obeyed their own laws. Since first contact, Aboriginal peoples have insisted that the newcomers negotiate a mutually respectful, lawful relationship with them. Successive colonial governments have refused to do so, opting instead to assert their dominion by whatever means necessary, including armed force and the subversion of their own law. The original questions, too, still remain: why are Aboriginal peoples forced to present a defense of their histories and cultures to the courts, by the rules of the courts, in the language of the courts, within the theoretical frameworks of the courts, to begin with? Who is responsible for this never-ending, unresolved dispute?\n\n* * *\n\n. Robinson 1987(b); 1990; 1991. See also, Pryce 1992.\n\n. Transcripts, Vol. 292: 21668.\n\n. Under cross-examination Robinson explained that she was referring here to Rosman and Rubel 1971. Transcripts, Vol. 293: 22157.\n\n. See Mertz 1992, 1994; O'Barr 1982.\n\n. Coupland 1986: 18.\n\n. Transcripts Vol. 289, 21727-21730.\n\n. Ives 1990.\n\n. \"Carrier\" is a name applied to some First Nations, including some Wet'suwet'en, in northern British Columbia by fur traders and later adopted by anthropologists.\n\n. Robinson 1983: 409.\n\n. Kretch 1991.\n\n. _Delgamuukw v. R._ , Exhibit 1191-5.\n\n. Rogers 1986: 207-210.\n\n. Transcripts, Vol. 293: 22165-22171.\n\n. This section of Robinson's opinion report covered six pages and encompassed 12 paragraphs. What appears at first as an impressive 10 pages of supporting notes is revealed upon closer examination as repetitions of eight sources, summarized as follows: Adams 1969; Boas 1916; Davidson 1955; Duff 1959; Garfield 1951; Inglis and MacDonald 1979; Jenness 1943; Tobey 1981. All of these sources rely, to some extent, on oral tradition. Of these eight references, only Adams deals specifically with the Gitksan. Adams conducted fieldwork among the Gitksan during the mid to late 1960s.\n\n. _Merriam-Webster Dictionary_ 814.\n\n. Transcripts, Vol. 292: 22191-22195.\n\n. Robinson 1987(a):17.\n\n. Adams 1969.\n\n. Gisday Wa and Delgam Uukw 1992.\n\n. Ray 1991(b): 311-312.\n\n. Transcripts Vol. 293: 22177-22179.\n\n. Transcripts Vol. 293: 22151-22205.\n\n. Transcripts Vol. 289: 21798.\n\n. Transcripts Vol. 214: 15494.\n\n. Robinson 1987(a): 19.\n\n. Robinson 1987(a): 20.\n\n. See Mertz 1992, 1994; O'Barr 1982.\n\n. See Cove 1996; Paine 1996 for critiques of the anthropological evidence.\n\n. Transcripts Vol. 189: 21644.\n\n. Transcripts Vol.192: 22175.\n\n. For a classic critique of these social theories, known as \"theories of economism\" see Dumont 1977; Dreyfus and Dreyfus 1979; Klamer 1987.\n\n. Macpherson 1962.\n\n. Lassiter 1987: 220. These statements about identity assume and reproduce the notion of homogenous identities. This is the subject of significant critique by those who claim the assertion of a singular identity blurs important differences, such as gender and class, within cultural groups. These \"modernist\" concepts of culture and identity are the terms of the legal discourse in court. It is within this context that I cite Lassiter. See Crosby 1996 for a critique of this formulation of contemporary Aboriginal identity.\n**CHAPTER 13: Jewels in the Crown**\n\n_The time of which the Chief Justice speaks to arrive at his decision could be looked upon as just one inch.... It is only a short period of time in comparison to the length of time that the Gitksan and Wet'suwet'en people have been on their land.... If we were to look at a time line, the time that we have been on our lands could be measured in several feet._\n\n\u2014Satsan (Herb George), Speaker, Office of the Hereditary Chiefs of the Gitksan and Wet'suwet'en People, 1992.\n\n**Temporal Purgatory: The Protohistoric Period**\n\nHaving constructed to their own satisfaction the image of an \"unorganized\" pre-contact society, the Crown put forward their rendition of what happened to the Gitksan and Wet'suwet'en when European commodities joined Aboriginally-produced items among their trade goods. Sheila Robinson's expert opinion report focused on the protohistoric period and she offered a succinct definition of her subject matter: \"By 'protohistoric,'\" she wrote, \"I mean the time prior to European presence in the area claimed by the plaintiffs but when European influence was felt through native intermediaries.... 'Historic' refers to times when Europeans were present\u2014even if intermittently. 'Prehistoric' applies to all time prior to the protohistoric era.\"\n\nThis periodization scheme (prehistoric, protohistoric, historic) represents traditional academic categories of analysis. Conceptualizing history in these terms highlights the arrival of European commodities, and then persons, and their subsequent activities as the decisive motor force in Aboriginal history, and follows logically from the widely-held assumption that Aboriginal societies had been static and unchanging for tens of thousands of years before engaging in contact with Europeans for the past two hundred and fifty or so years. This is, of course, a logical framework for an analysis whose purpose is to establish the determinacy of European activity and the supremacy of European culture. This periodization scheme therefore serves the interpretative purpose of defining the relevant categories of the discussion in eurocentric terms, superimposes this categorization on the Gitksan and Wet'suwet'en, and concludes that this was a crucial and transforming turning point in their history, rather than, perhaps, a significant period in a very long, continuous historical process. It stresses universal and externally-determined change only, rather than a dynamic relationship between the latter and internally-determined change and continuity. Most importantly, however, it is a _model_ , an abstract construct developed by academics to facilitate comparison and theoretical speculation. A temporal classification scheme like this is not a fact. In ordinary, everyday life, time period categorizations serve as heuristic devices: shorthand flags that facilitate communication between people with shared languages and experiences. Law seeks to transform such time period categorizations into locked, airtight repositories that incarcerate lived histories in time-bound cells, as if processes and relationships do not traverse the boundaries between historical epochs.\n\nThe first question to be asked regarding the specific history of any particular group's experience of the protohistoric era, is when did this period begin and end for them? Two primary sources are usually consulted on this question: archaeological data that establishes the dates of the first items of European manufacture unearthed in a given area; and historical documents in the form of records and observations kept by European traders. Oral tradition is also studied for references to the arrival of these new things, and how they impacted the Indigenous peoples concerned.\n\nThe first and second building blocks of the Crown's argument were that Gitksan and Wet'suwet'en societies had been unorganized at the time they were \"discovered\" by Europeans, and that they lacked property law. Their third building block was the argument that the delineation of property and borders, and a territorially-based social organization emerged among the Gitksan and Wet'suwet'en in response to the European-Aboriginal fur trade. Given the extent of social change that the Crown wished to prove took place during the protohistoric period, it was important for their argument that this period be as long as possible. A long protohistoric period was also important to Sheila Robinson to justify her methodological decision not to consult archival records like William Brown's because, she argued, they were written long after a \"truly aboriginal\" way of life had ceased to be lived. Sheila Robinson explained the importance of her argument to the Crown's legal case: \"Recognition that protohistoric European-influenced developments took place and were significant has one very important implication. It casts suspicion on any portrayal of a \"pristine\" or truly aboriginal way of life based on contemporary knowledge,\" she wrote.\n\nThe Crown's witness took on the task of establishing the date at which the protohistoric period began. In her opinion report Robinson wrote: \"Vitus Bering's voyage to north-western Alaska from Siberia was in 1641. There is no doubt that trade in European commodities from Siberia into North America began some time earlier. The protohistoric period perhaps extends back as early as the mid-sixteenth century, when Russians first began to settle into the Kamchatkan peninsula.\" Bering's voyage actually took place in _17_ 41. Robinson claimed under cross-examination that \" _16_ 41\" represented a typographical error. Undaunted, Robinson stretched the point even further when the Judge intervened and asked for more details about early foreign influences.\n\nCourt: What Euro-Canadian influences would extend back four centuries?\n\nRobinson: Coming from the east coast of North America and also up through the Mississippi drainage there is some\u2014there's some thought that indirect European influence was having its effect on Indian groups quite removed from the direct sources of contact. Some of those influences included diseases spreading through, and perhaps extracting considerable tolls on some populations before there was ever any real historic record of it. There seems to be\u2014and even such\u2014such things as the introduction of horses from Spanish sources, for instance, seems to have had tremendous effect on the settlement patterns and economic behaviour and social relations of Indian groups through the plains and prairies, for instance. So something as minor as the horse coming in caused tremendous dislocations through the prairies.\n\nCourt: That would go back to the 16th century?\n\nRobinson: Yes. So the post-contact period or protohistoric period has been traced to the first arrival of Europeans in the New World. Some scholars take it back that far. From the north northwest American direction and the Siberian connections some people are saying that if we lift the notion that only Europeans can affect North American cultures then we are dealing with an Asiatic influence; Chinese, Japanese and so on. That seems to have stimulated trade and trade economies. That's much more conjectural as you push it back into the past, but there are certainly scholars who agree that indirect influences were important.\n\nCourt: Thank you.\n\nPeter Grant asked Robinson what factual evidence she relied on to support her statement that \"there is no doubt that trade in European commodities...began sometime earlier\" than Captain Cook's 1774 landing. Grant confronted her with the fact that, upon reviewing the sources listed in her bibliography in support of this statement, he was unable to find any references to European-manufactured goods having been found in the territory prior to 1780.\n\nGrant: What I suggest to you, with all due respect, Dr. Robinson, is there isn't a scintilla of evidence...that there was European trade goods moving into the Gitksan or Wet'suwet'en area before 1786...[yet] here you say there is no doubt....\n\nRobinson: I have no doubt at all that there wasn't a considerable amount of protohistoric influence coming from Asiatic sources or Northern Russian sources in the protohistoric period, as I have defined it mid-17th century onward.\n\nGrant: Into the Gitksan and Wet'suwet'en area, doctor?\n\nRobinson: I have absolutely no reason to think that they were isolated from what are otherwise described in several places as broad trade patterns, broad developments occurring.\n\n... Grant: Well, doctor, look, you have explained what you meant by there is no conclusive evidence.... 'More likely than not,' I think you said. But here you say there is no doubt. That's pretty conclusive...isn't it?\n\nRobinson: Yes, and I'll stand by that. I will say that there is no doubt. We may not have material items representing that in that claims area yet. I would say that further archaeological research will no doubt turn them up.\n\nGrant: And that is speculative?\n\nRobinson: Yes, it is, but it's based on my opinion.\n\nOn the key issue of duration of the protohistorical period, Robinson eventually vacillated under cross-examination.\n\nGrant:...when are you referring to as \"prior to the advent of European influence?\"\n\nRobinson: Well, we've discussed this, and I think we're going to leave it as the mid-eighteenth century and Chirikov's arrival on to the northern coast.\n\nAccording to the Crown's framework of analysis, the arrival Captain Cook in 1774 marks the beginning of the protohistoric period in British Columbia. The arrival of a second European man, Hudson's Bay Company trader William Brown in 1822, heralds the end of the protohistoric period and the beginning of the historic period in Gitksan and Wet'suwet'en territory. Accepting these broad parameters, the protohistoric period shrinks from Robinson's initial suggestion of a 400-year duration to one lasting only 72 years. According to evidence subsequently accepted by the Judge, the proto-historic period for the Gitksan and Wet'suwet'en in effect began in 1778 and ended in 1822, therefore encompassing only 44 years.\n\nRobinson's excesses in attempting to stretch the protohistoric period back to the arrival of the horse on the Prairies, and thus suggesting, given the context in which her statements were made, that this was relevant to the Gitksan and Wet'suwet'en, was tantamount to saying that from the moment the first Toyota hit a Canadian road, we all began to adopt Japanese socio-economic, political, and cultural characteristics. This Crown argument illustrates the absurd conclusions to which legal argument and abstract logic can lead.\n\n**Who Created Private Property?**\n\n_By the late 1950s or early 1960s, I assumed that the issue of land tenure...had been resolved once and for all and that 'hunting territories' came into existence after the arrival of Europeans. This assumption was challenged by investigators...who convinced me that after several decades of my previous viewpoint, it was time to reexamine the complex topic.... In spite of the extensive literature on the land occupied by the original inhabitants of North America, we still know very little about Indian relationships to land and its resources.... Fortunately, there are scholars who continue to labor very hard at understanding the wisdom of Indian elders and the remarks of traders and other Europeans that have been preserved._\n\n\u2014Anthropologist Dr. Edward Rogers, 1986, _Epilogue: Reevaluations and Future Considerations_.\n\nThe issues being debated in the courtroom during _Delgamuukw v. R._ about the impact of the \"protohistoric\" period on Indigenous societies have occupied academics in this field for many generations. The key question they have addressed is whether or not conceptions of private-like property, and laws codifying and defending such land ownership, existed among Aboriginal peoples prior to their involvement with Europeans. Sheila Robinson notes in her 1983 Ph.D. thesis that the question of the impact of the European-Aboriginal fur trade on Indigenous societies is far from settled among anthropologists and ethnohistorians. Chapter VII of her doctoral thesis, entitled \"Assessing the Impact of Early Contact With Europeans on Northern Northwest Coast Indian Socioeconomies,\" which encompasses 77 pages, began with the following paragraph: \"How scholars have viewed the impact of early contact with representatives of Western civilization on Northwest Coast Indian cultures has varied considerably.... Their interpretations of the cultural contact situations during the maritime fur trade period fall into three distinct categories. The first claim that contact with the earliest explorers and fur traders provoked essentially no changes in the traditional social and economic organizations of the Northwest Coast Indians (the minimal cultural change hypotheses); the second argue that this early period of intercultural interactions wrought devastating changes in the traditional native socio-economic organizations (the negative impact hypotheses); while the third identifies a series of progressive developments in the early historic period which triggered the efflorescence of Northwest Coast cultural organization to levels which were unattainable before the Europeans arrived (the positive impact hypotheses.) Robinson was most critical of the \"minimal cultural change hypotheses,\" that she attributed to authors like Joan Wike and Robin Fisher who, she claimed, argue that the fur trade brought \"essentially no changes in the traditional social and economic organization\" of First Nations. This is an oversimplification: what these researchers argue is that, while cultural _changes_ took place, cultural _transformation_ did not. However, Robinson is correct in her observation that a good deal of debate about the impact of the fur trade on First Nations remains unresolved. Her thesis also accurately describes the current state of the art in fur trade studies when she argues that: \"If theoretical emphases are shifted away from tracing developments in structural processes towards understanding the nature of particular historical developments which can be documented, we will achieve a clearer understanding of early post-contact adaptations.\" Robinson's 1983 suggestion that, \"given the variations in Indian-European interactions during the early historic period, it may be profitable to break the records down to a point where separate communities' involvements with the foreigners and their native trading partners can be examined more closely,\" also finds widespread support in the literature.\n\nIn the introduction to her 1987 expert opinion report, Sheila Robinson located her analysis within the discipline of ethnohistory. She wrote: \"Most modern scholars engaged in North American ethnohistoric research agree that indigenous populations were profoundly affected by indirect contact with Europeans before they experienced direct contact with them. Although the nature, timing, intensity and repercussions of protohistoric European influence varied considerably from region to region, research indicates that no native groups in what is now known as British Columbia were isolated from stimulus stemming from the European presence in the New World.\"\n\nWhen Robinson was asked by lawyers for the Gitksan and Wet'suwet'en to substantiate her claim that her expert opinion report reflected contemporary ethnohistorical theory and research, she referred to the recent work\u2014not cited in the bibliography of her report\u2014of scholars like Colin Yerbury, Stacey Krech, Toby Morantz, Arthur Ray and Charles Bishop. The _current_ work of each of these scholars, however, challenges the Crown's central theses. Charles Bishop, writing in 1987, summarized the thinking of \"most modern fur trade scholars\" when he wrote: \"...It became orthodoxy to view the European fur trade as giving rise to...[an] individualized and privatized form of territoriality. Beginning in the 1960s, more intensive regional, ethnographic, and historical studies began to undermine some of the specific tenets of the general theory. By the 1970s, it was becoming evident to a small core of specialists that an accumulation of data pertaining to a variety of times and areas, combined with theoretical and conceptual refinement, was challenging the applicability of the general theory itself.\"\n\nThe Gitksan and Wet'suwet'en and their witnesses presented an analysis of the impact of the fur trade based primarily on local knowledge. Their narrative explained how, following his 1774 arrival at Ahousat, Captain Cook initiated a trading relationship with Chief Maquinna in which First Nations supplied sea otter pelts destined for a burgeoning European fashion market. Most fur trade scholars agree\u2014based largely on analyses of written records kept by European traders\u2014that the goods demanded by the First Nations traders were largely luxury items as opposed to essentials. Joan Wike, who analyzed the logs kept by trading ships' captains, says that, during the initial stages of the maritime fur-trade iron, copper, guns and ammunition were traded. By the end of the period, the demand was for rum, molasses, pilot biscuits and luxury items. Fur trade scholars also agree that the Aboriginal peoples of the northwest coast whom the Europeans encountered were shrewd and experienced traders who exerted a good deal of control over transactions by employing strategies such as withholding furs to drive up prices, placing \"advance orders\" for future trade goods and refusing to trade unless they were satisfied. They have concluded that the Aboriginals' interest in trade was for the purposes of elaborating their ceremonial systems as opposed to improving the sufficiency of their subsistence exploitation for which Indigenous technology continued to be used well after guns and ammunition were acquired and used for warfare and social control.\n\nThe \"on the ground\" practice of the west coast fur trade, like that of early treaty-making in eastern Canada, conformed to Aboriginal business etiquette, requiring long preliminary discussions, feasting and performances, and exchanges of gifts and other formalities, before transactions were made. And, as was the case throughout the colonial world, other relations developed between the European visitors and their Aboriginal hosts. Some sailors were taken captive when negotiations failed and conflict developed, some were killed, some deserted their ships and remained on shore. The many unions between European fur traders and Aboriginal women are still represented in too many Canadian history books as \"desperate\" choices made by European men who, \"needing women,\" had no other options. Referred to in HBC records as \"country marriages,\" these were often, in fact, solemn and binding legal arrangements conducted according to Aboriginal marriage laws. These were not \"inferior\" forms of European marriage, but rather matrimonial commitments made between members of groups of visitors and hosts. As would be expected in such a situation, the customs of the hosts prevailed. These unions were sometimes arranged between chiefs and ships' captains or trading company executives in order to consolidate trading relationships. This practice followed both European and Aboriginal protocols regarding arranged marriages between royal families and powerful business interests. Contemporary scholars, re-examining legal history from a perspective of gender and racial equality, have found numerous examples where early colonial courts recognized the legitimacy of these marriages, and upheld the authority of Aboriginal law as against European laws, demonstrating both the uneven application of legal principles in practice, and the importance of context in shaping local power relations. As is the case always and everywhere, not all unions between European men and Aboriginal women were sanctified by anyone's laws, or entered into from positions of equality.\n\nThe Crown's arguments in _Delgamuukw v. R._ reflected popular B.C. mythology about the fur trade era. Simple Aboriginal peoples, so the settler's founding stories go, were humbled by the obvious superiority of European men, awed by their technology, and dazzled by their flashy commodities. These accounts tell of Aboriginals who were easily tricked by the clever commercial chicanery of the Europeans, and eagerly abandoned their own ways in a desperate pursuit of European trade goods. Their lust for beads and trinkets was fatal, and sounded the death knell for Indigenous cultures, these settlers' narrations conclude.\n\nThe land tenure system and the social and political structures that the Gitksan and Wet'suwet'en claimed were in place before Europeans\u2014or their commodities\u2014arrived, _really_ arose, according to the Crown, as a direct response to the European fur trade. The Crown's case therefore required evidence that, in the course of the protohistoric period, a system of property ownership and resource management, based in Houses ranked in a hierarchical manner and articulated with other Houses and clans throughout the region, emerged, stimulated by the desire for European things. Robinson's central thesis in relation to the impact of the fur trade was summed up in the following paragraph of her opinion report: \"Gitksan and Wet'suwet'en claims of traditional ownership and occupation of certain territories can be challenged on the basis that they do not account for developments which occurred during the protohistoric period. These changes include adjustments in boundaries between the Gitksan and Wet'suwet'en and their neighbours as well as more precise delineation of boundaries within and at the margins of Gitksan and Wet'suwet'en territories. Both processes of territorial demarcation are linked to and were stimulated by indirect contact with Europeans through the eighteenth and early nineteenth centuries.\"\n\nThe Crown specifically argued that desires for, and competition over, these new commodities stimulated the Gitksan and Wet'suwet'en to trap hitherto unexploited, or under-exploited fur-bearing species on territories outside of their villages, and at a distance from the main subsistence source of salmon in the rivers. Initially, the Crown claimed, the Gitksan and Wet'suwet'en were drawn into subordinate relations with their coastal neighbours, and, later on, with Europeans directly. Subordination, the Crown claimed, led them to mimic those who dominated them. Finally, reasoning backwards from the point of consumption to the point of production, the Crown argued that, because the furs Aboriginal people traded would ultimately be sold in a capitalist commodity economy, Aboriginal trappers were now members of a new culture and lived in a new form of social organization, whether they knew it or not.\n\nDuring the first phase of the European-Aboriginal fur trade on the northwest coast\u2014from 1774 to 1800 approximately\u2014ships stopped at coastal ports only long enough to trade with Aboriginal representatives for furs, food and other supplies. These coastal Aboriginals became brokers and middlemen, trading European goods with their inland neighbours. Two sections of Robinson's report dealt primarily with this first half, or maritime phase, of the fur trade that spanned approximately 20 years. \"Section 11: Protohistoric Developments in Gitksan and Carrier land use\" consists of ten paragraphs with 27 pages of supporting notes. As was the case in the previous section of her report that dealt with pre-contact social organization, most of Robinson's sources of information on the protohistoric period were not based in studies of the Gitksan and Wet'suwet'en. Section III of Robinson's report was entitled \"The Expansion of Coast-Interior Trade.\" Robinson began by saying that she was really just expanding on the points made in the previous section but emphasizing how \"new European wealth stimulated northern coastal societies and economies to the point where a different kind of social and economic organization began to emerge.\" According to Robinson's narrative, the coastal Tsimshian became aggressive middlemen and three superchiefs arose, among them a Legaik. Twenty paragraphs long, this section of Robinson's report was devoted primarily to a retelling of a story about the rise of Legaik's dynasty. The Legaik story takes up the better part of seven pages, and is based almost entirely on an unpublished undergraduate honours thesis completed at the University of British Columbia in 1978.\n\nThis section of Robinson's report supported the Crown's argument that the Gitksan and Wet'suwet'en land tenure system and social structure was \"borrowed\" by them from the Coast Tsimshian people during the protohistoric period. Robinson wrote: \"It was less a matter of what goods they acquired than who furnished them, and how they were used. There was considerably more status to be gained through relations with the 'superior' coast people than through exchange with 'inferior' Athabaskans.\"\n\nPeter Grant confronted Robinson on cross-examination, presenting William Brown's descriptions of Gitksan trading relations with their coastal neighbours, with the interior Athabaskans, and with the Hudson's Bay Company:\n\nGrant: Well, let's look at page 11 (of Brown's report) then: 'It does not appear to me that we are able to cope with these people by making derouines into the country they are in the habit of visiting unless we sell our property so cheap as to prove prejudicial to the trade of Western Caledonia, and even then we will not be able to secure one half of the trade, for we do not meet on an equal footing as they receive goods at a low rate from the vessels which frequent the coast.' Now, would you agree with me that that'sconsistent with logic, that the trade\u2014the goods that Brown had to get would have been over land, that is coming into New Caledonia right?\n\nRobinson: It's consistent with a Eurocentric logic....\n\nGrant: I'll rephrase my question so you understand it. Brown got his trade goods...over land, right?\n\nRobinson: Yes.\n\nGrant: And the vessels got their trade by sea?\n\nRobinson: Yes.\n\nGrant: It was cheaper to get the trade goods up by sea at that time than over land?\n\nRobinson: Was it? I don't know that.\n\nGrant: You don't know that, okay. Assume that it was cheaper to get them by sea, or let's not even assume that, let's just say that what Brown says here is true, they received goods at a low rate from the vessels which frequented the coast. And Brown is saying: \"We will not be able to meet on an equal footing.\" For whatever reason, Brown says they could not compete giving the goods to the Indians at the same rates as the vessels.\n\nThere were repeated references in Brown's journals to the importance of deer and moose hides traded with the east. Arthur Ray had argued that, by the time Brown arrived in 1822, the trade was one where the scarcer furs were traded for moose and deer hides with the Athabaskans; and the plentiful fish, with Europeans for manufactured commodities. Under cross examination, Robinson admitted that the sentence in her report that read: \"There was considerably more status to be gained through relations with the 'superior' coast people than through exchange with 'inferior' Athabaskans\" was \"speculative\" and based on Kobrinsky, Steward and Goldman, who did not have access to Brown's journals. Finally, she said she relied \"on a considerable understanding of cultural ecological theory which suggests that, in all likelihood, that's what occurred....\" An archaic interpretation of cultural ecology mirrors crude evolutionism and supports the assumption that \"simple\" societies always assimilate the characteristics of \"complex\" societies when they come into contact. The underlying logic of this formulation that serves the Crown's interests tells a story about how, just as the Wet'suwet'en \"naturally\" copied the Gitksan, and the Gitksan \"naturally\" modelled themselves on the Coast Tsimshian, so \"naturally\" the Gitksan and Wet'suwet'en emulated Europeans after contact.\n\nThis section of Robinson's report supported two points in the Crown's case. First, the Crown argued that, in having to defend themselves against Legaik's aggression\u2014itself motivated by participation in the European-Aboriginal fur trade\u2014the Gitksan and Wet'suwet'en consolidated the external boundaries of their territories which had been \"hazy\" during the pre-contact period. Second, this brought the Gitksan and Wet'suwet'en into subordinate trading relations with the coastal Tsimshian peoples.\n\nThe other major source Robinson relied upon in this section was an account by anthropologist George MacDonald. MacDonald argues, on the basis of data he acknowledges to be conjectural, that competition over European trade goods, particularly metals, stimulated increased warfare for control of trade routes during the protohistoric period. Robinson relied extensively on MacDonald's rendition of the \"Epic Of Nekt,\" a \"warrior's saga,\" that told the stories of great battles between the Gitksan and their Tsimshian neighbours during this era. Robinson quoted MacDonald as saying that, in the Indigenous accounts, \"Motives are stated in terms of revenge rather than economics,\" (that is, motivation based on issues not determined by European influence. (ed.)) MacDonald concluded, however, that while the wars were fought over \"important oolachen fishing grounds,\" he _believes_ \"it was the pursuit of metals and improved weapons, rather than control of the oolachen trade, that prompted this aggression.\" It is interesting to note that what MacDonald and Robinson accepted as \"true\" in the oral history was that wars were fought. What they rejected as \"untrue,\" however, was Aboriginal peoples' _explanation_ of what the wars were about. Even MacDonald, however, acknowledged that the objects of battle were transportation routes and that \"territories were sacrosanct.\" While in theory oral tradition is rejected, in practice it is selectively relied upon for \"data,\" and European social theory is superimposed as explanation.\n\nAs sea otter stocks declined on the coast, the Hudson's Bay Company continued to expand across what was to become the province of British Columbia, building forts and engaging in trade with Aboriginal peoples throughout the first half of the nineteenth century. The land-based fur trade period of the protohistoric era encompassed, roughly, the years 1800-1850 in the province as a whole. However, local areas experienced these incursions at different times and to different degrees. Even if we consider the first sporadic contacts between European and Gitksan and Wet'suwet'en _human beings_ , rather than contact between people and commodities, a reasonable time to commence the dating of the protohistoric period in this _particular_ area, then the \"protohistoric period\" before William Brown arrived and during which the Crown argued that dramatic social change occurred, encompassed only about 15-20 years, a considerably shortened time frame compared to Robinson's original suggestion of over 400 years of European-induced cultural assimilation. In fact, regularized Gitksan and Wet'suwet'en participation in the land-based fur trade did not begin in earnest in their territories until the 1820s, when William Brown arrived.\n\nRobinson's opinion report was based primarily on her doctoral dissertation on coastal history, and the differences between the experiences of the interior Gitksan and Wet'suwet'en and those of the coastal peoples in relation to the maritime fur trade were significant. Her haphazard substitution of coastal Tsimshian, Haida and Tlingit historical data for specific documentation of Gitksan and Wet'suwet'en experiences is particularly problematic in relation to this period of time that is crucial to the Crown's argument.\n\nSection V of Robinson's report, entitled \"Gitksan Territorial Ownership: Protohistoric\/Early Historic,\" consisted of four paragraphs and spoke to the issue of the Gitksan's internal boundaries during the later half of the protohistoric period: the time the land-based fur trade commenced. Robinson argued that potlatching increased substantially during this period as a result of competition between Chiefs, and that land holdings changed hands frequently through potlatching. Section VI, the final section of Robinson's report, is entitled \"Protohistoric Changes in Wet'suwet'en Social and Economic Organization, after Kobrinsky.\" It is a five and a half page regurgitation of a ten page discussion paper written by anthropologist Vernon Kobrinksy in 1977. In this paper, Kobrinsky outlined an admittedly conjectural three-stage history of social development for the Northwestern Carrier that Robinson, on behalf of the Crown, adopted as a framework for her rendition of Gitksan and Wet'suwet'en history. Kobrinksy suggested that structural changes in Carrier society\u2014supposedly having 'evolved' from simple bands to more complex clan social organization\u2014could have been a direct response to the changing economic conditions initiated by participation in the European-Aboriginal fur trade. In the second paragraph of this section Robinson explained her choice of Kobrinsky as an authority, writing: \"Kobrinsky's viewpoint is important for two reasons. First, his assertion that significant socio-economic changes occurred during the proto-historic period supports claims that European influence was a major factor in disrupting 'traditional' native lifestyles before direct contact between Indians and Europeans occurred. Second, much of the ethnographic evidence he assembles related to changes in styles of resource control. Specifically, Kobrinsky asserts that precise delineation of territorial boundaries relating to the allocation of rights to fine-fur species was a by-product of the fur trade.\" Kobrinksy, like Adams, Bishop, Kretch and many other researchers in this field, has subsequently revised his thesis.\n\nIn her proposal to Russell & DuMoulin, Robinson had explained the limitations of her opinion report, writing: \"This will not be a comprehensive overview but will legitimize the positions taken by Kobrinsky and MacDonald in the terms of theoretical traditions in the social sciences.\"\n\nPeter Grant questioned her about this:\n\nGrant: You saw your role here as legitimizing Kobrinsky and MacDonald, not determining whether or not they could stand up under careful scrutiny of those in support and in opposition to them; isn't that right?\n\nRobinson: Well, no.\n\nGrant: Then why did you use the word legitimize in your proposal?\n\nRobinson: It's one of my favourite words. But in the context of this paragraph, and in the way that cultural ecologists often do their work, is that they do use comparative examples from all over the world that investigate similar kinds of relationships or aspects of relationships between people and their environments, and what I intended to do there, not realizing that when I submitted this report in May that that would be the end of it, but what I would have liked to have done, and what I intended to do there, is put this in the context of some general ecological theories, which showed that this is not an unusual situation, that this kind of scenario occurs all over the world and it's consistent with models in ecological anthropology and cultural geography which show\u2014explain how these relationships work. So basically what I was going to do there, rather than, as I say, not a comprehensive overview but say let's pull together perhaps a set of cultural ecological works that put this in the context and explain that this is not an unusual way of looking at the world.\n\nGrant: I do not want to look, and I will not refer to those views that oppose or challenge Kobrinsky and MacDonald on this point. I will look and find those that support them, that was what you set out as your approach and that's what you mean by legitimize, you would buttress their theory?\n\nRobinson: Oh, yes. And I don't see that that's objectionable at all.\n\nOne of the few sources based on Gitksan-specific research that the Crown's witness relied on quite extensively was John Adams' 1969 unpublished doctoral dissertation. Adams conducted participant observation research among the Gitksan during the 1960s, before Brown's documents were available and working with the assumption that the first written records pertaining to the area began in 1867, 45 years after Brown's journal had been written. Adams' central thesis was that: \"The traplines as we know them today were drawn up to regulate the fur trade which is post-contact.... How much the territories may have changed since the coming of the Whites is unknown.... The actual rate of change seems quite rapid, though Gitksan informants often deny that there are any changes at all. They assert that 'things have always been just as they are now for thousands and thousands of years.'\" By 1987, however, Adams, like most northwest coast scholars, was in the process of rethinking some of his earlier assumptions about the impact of the fur trade on property law among First Nations on the northwest coast. He asked, in the introduction to an issue of the academic journal, _Arctic Anthropology_ , \"What could possibly account for Northwest Coast stratification prior to the coming of Europeans? Unfortunately, political ideology has played a major role in this matter for at least 50 years.\"\n\nRobinson argued that when the earliest fur traders arrived in Gitksan and Wet'suwet'en territories, the Aboriginal population was sparse and much of the land that may have previously been used for hunting had already been \"abandoned.\" Robinson's report described protohistoric Gitksan and Wet'suwet'en property management: \"The protohistoric and early historic shift in economic emphases towards the production of furs for exchange probably also resulted in some resource-producing areas being abandoned. In other words, new gaps emerged in the village 'patchworks.' These new gaps were probably closer to villages than the prime peripheral hunting territories. Although several of the examples cited consider European diseases a major factor contributing to the abandonment of certain territories, it can also be suggested that some territories were simply abandoned if they had few fine fur-bearers to begin with, or as their fur stocks were depleted. I envision what can be described as a centrifugal shift in territorial holdings throughout the protohistoric and historic periods, with mid-range territories being increasingly abandoned in favour of outlying ones.\" She went on to lay the groundwork for arguments that would emerge later that, once fur-bearing species were depleted in a given locale, the territory was simply abandoned and therefore could be \"discovered,\" and \"claimed by migrants.\" Of course, given that the Crown's and Robinson's argument began with the assumption that land had no meaning except what the Crown defined as an \"economic\" value, and therefore was not desired by Aboriginal peoples before the European-Aboriginal fur trade, it proceeds logically that there would be no value remaining after the fur trade's purposes were served.\n\nThe debate that Robinson refers to about whether changes in Gitksan and Wet'suwet'en relationships to land were primarily stimulated by their participation in the European-Aboriginal fur trade, or were the result of population loss due to smallpox epidemics is a familiar one in academic northwest coast studies and in British Columbian political culture. An outbreak of smallpox was first recorded among the Gitksan and Wet'suwet'en during the 1770s, and subsequent epidemics of measles, influenza, whooping cough, tuberculosis, and scarlet fever took a heavy toll in 1837-1838, and 1862-1863. Alongside these epidemics, \"lingering diseases, such as influenza, scrofula, and syphilis, observed among later native populations, can also be inferred for the early contact period\" researcher Steven Acheson writes.\n\nThis rapid decline in population, and the decimation and demoralization that accompanied it served to reinforce theories that Aboriginal peoples constituted a \"vanishing race.\" This was, and is, a popular notion that both historically and contemporarily helps to rationalize the expropriation of lands and the exploitation of peoples already considered doomed to extinction. The debate about which had more impact on Indigenous societies, the arrival of commodities or the arrival of diseases, is an important one. If the Crown wanted only to prove the \"facts\" that changes in boundaries and borders took place during the fur trade, the cause of these changes would seem irrelevant, in a strictly \"legal\" sense. However, the moral and ideological differences between claiming that the lust for European commodities prompted change, rather than arguing that Aboriginal peoples were innocent victims of diseases brought by Europeans, speaks to one of western cultures' solutions to the problem of political responsibility. If Indigenous peoples changed because they wanted material things, then they may be considered somehow responsible and guilty at the same time: they are responsible because they displayed rational choice behaviour in pursuing the accumulation of goods; but they are thus guilty of losing their \"pristine\" Aboriginality and are no longer attractive to westerners. Furthermore, this line of reasoning goes, they were compensated, in that they did receive the things they wanted; and why should they get \"our\" things if they don't want to become like us? There is something fundamentally \"unfair\" about that to a conservative, or neo-conservative imagination. It parallels the way many people feel about welfare recipients \"getting something for nothing\" and not suffering sufficiently to \"deserve\" assistance. Wealth and its pursuit are extraordinarily contradictory and confusing questions for westerners.\n\nNot having it shows you to be an undeserving failure, but possibly of a kindly disposition. Having it shows you to be deserving of success, but possibly of an unkindly disposition. The accumulation of wealth is western capitalist culture's most viciously enforced imperative, while it is one of Judeo-Christianity's greatest sins.\n\nOn the other hand, if native people changed their behaviour because many of their numbers were eliminated by disease, they were not responsible. According to western biomedicine, the body is a passive victim randomly attacked by germs that do not discriminate on the basis of race, gender or social class. It would not be as morally acceptable, or politically legitimate, for Crown representatives to say out loud, \"we nearly wiped you out with smallpox, so tough luck,\" as it is for them to say \"you went for the goodies, so tough luck.\"\n\n**The Magic of Commodity Fetishism**\n\n_The wealth of those societies in which the capitalist mode of production prevails, presents itself as an immense accumulation of commodities..._\n\n_A commodity is in the first place, an object outside us, a thing that by its properties satisfied human wants of some sort or another._ _The nature of such wants, whether they spring from the stomach or from fancy, makes no difference..._\n\n_The utility of a thing makes it a use-value.... But so soon as it steps forth as a commodity, it is changed into something transcendent...The mystical value of commodities does not originate...in their use value_\n\n\u2014Philosopher Karl Marx, 1867, _Capital Volume 1_.\n\n_Fetishism denotes the attribution of life, autonomy, power, and even dominance to otherwise inanimate objects.... Thus, in the case of commodity fetishism, social relationships are dismembered and appear to dissolve into relationships between mere things...man has become the slave of the commodities he himself has produced but which now appear before him as objects imbued with the power to render him their tool...._\n\n\u2014Anthropologist Michael Taussig, 1980, _The Devil and Commodity Fetishism in South America_.\n\nAnother important legal question whose answer was located in the protohistoric period was what kind of social organization existed among Aboriginal peoples at the time Britain asserted sovereignty? The Crown's case was necessarily inconsistent since they wanted to establish two contradictory arguments. First, they wanted to demonstrate that Aboriginal society was sufficiently \"primitive\" to legitimate the application of the doctrine of discovery\/occupation\/settlement, and the principle of _terra nullius_. Second, they wanted to prove that, at precisely the same moment in time\u2014the early nineteenth century\u2014in precisely the same place, precisely the same people were sufficiently \"evolved\" to have begun organizing themselves into a House and clan social structure with matrilineal descent systems and established and protected private-like property. The Crown's and Robinson's argument was that all this took place within a period of at most 78 years (1744-1822), dated from the moment Captain Cook landed at Ahousat; or perhaps over the course of 22 years (1800-1822), dated from the commencement of the land-based fur trade, and fully developed by the time William Brown described it in his 1822 journal. The Crown claimed that the sophisticated land and resource ownership system Brown documented was a response to European influences, which would render hunting and trapping for the European-Aboriginal fur trade not \"aboriginal practices,\" according to the law. This argument also assumes that any social and cultural changes that took place after contact necessarily resulted in \"European\" and \"not Aboriginal\" practices. However, Gitksan and Wet'suwet'en involvement as active participants in the land-based fur trade began around the early 1820s: _after_ William Brown's arrival. In 1803-1806, when, according to the Crown, British sovereignty was established in the area, the Gitksan's and Wet'suwet'en's lived experience of British \"culture\" was of having acquired a smattering of items of European manufacture, and of having met a handful of fur traders wandering through their territories.\n\nThe last gasp of the Crown's argument was to claim that the desire for, and competition over, new commodities brought by European fur traders stimulated the Gitksan and Wet'suwet'en to exploit the territories outside of their villages and at a distance from the main subsistence source of salmon in the rivers. Given the relatively few European trade goods actually present in Gitksan and Wet'suwet'en territories at the time, and the minimal actual contact that had taken place between Aboriginal and non-Aboriginal persons in the region, the Crown relied heavily on the theoretical assumption that even such minimal lived experience with things manufactured by Europeans, in a capitalist mode of production, could dramatically transform an Aboriginal society. It is not necessary to dismiss the importance of economic power and technological sophistication, or the lure of commodities and consumption, to recognize that human history and social relations are shaped by far more complex a web of interacting forces than any unicausal theory can support. Most contemporary social scientists agree that the causes of social and cultural change are multiple, the processes complex, and explanation cannot be reduced to a single, determining, causal factor like economic, or technological, or ideological change. Rather, current thinking on this subject focuses on the interactions and interrelationships between many forces and factors that produce changes in societies and cultures over time.\n\nThe Crown's argument, however, relied on archaic notions of simplistic determinism and on concepts of human nature as driven by a passion to compete with each other in a race to accumulate material goods: commodities or \"things.\" Tracing the crucial argument about the actual impact of European goods on Aboriginal social organization from its first statement in Robinson's doctoral dissertation, to her initial proposal to Russell & DuMoulin, and then through subsequent iterations in evolving drafts of her expert opinion report written under their direction, and finally to the interrogation of her argument under cross-examination, reveals the necessary ambivalence inherent in her hypothesis. In her doctoral dissertation, Robinson had been firm about the need to avoid simplistic unicausal theories of social change. In her initial proposal to Russell & DuMoulin, Robinson suggested that the European-Aboriginal fur trade may have accelerated _already existing_ trends, but did not _cause_ new forms of social organization.\n\nPeter Grant explored the question of causal theory under cross-examination, referring Robinson to her early correspondence with Russell & DuMoulin:\n\nGrant: That's a fair statement of the state of the art in the field?\n\nRobinson: Yes, and I would say that cultural ecologists, and that's people that deal with cultural geography, cultural anthropology and archaeology who share a common interest in the relationship between people and land and changes in those relationships, do indeed recognize that sometimes profound socioeconomic changes are triggered by relatively small or what appear to be small factors. And that often direct cause is obscured by the subsequent consequences. And this is something that I think is generally recognized as a theoretical orientation in other scholarly fields.\n\nGrant: But there is some legitimacy to the criticism within the field that relatively minor external stimuli can trigger profound changes and patterns of socioeconomic behaviour, there is some legitimacy to the concern that this is often undemonstrable speculation?\n\nRobinson: Not within the field, I would say that criticism is levelled at it. I think the criticism comes more from people that are outside the fields of anthropology and cultural geography who don't know where the theories of explanation are derived....\n\nIn a subsequent draft proposal to Russell & DuMoulin, Robinson promised that in her report, \"particular attention will also be paid to theories which regard relatively minor external stimuli as catalysts or 'triggering' factors which can provoke significant internal adjustments in socioeconomic configurations.\"\n\nRobinson's final report, while employing vague and inconclusive wording, is emphatic on the question of European commodities being the 'prime-mover' or \"triggering\" factor causing change. And, finally, under cross examination, Robinson moved quite definitely from her initial position, reflecting the state of the art in the field\u2014that \"assumptions that relatively minor external stimuli can 'trigger' profound changes in patterns in socioeconomic behaviour are often dismissed as un-demonstrable speculation\"\u2014to the following:\n\nGrant: Even if you accept the proposition that in 1805 some metal comes inland through the Gitksan area, are you saying that that would lead to the change to a territorially-based society from a nonterritorially-based society? Robinson: Yes. I think that the influx of European wealth was a major triggering factor...\n\nIn the late nineteenth century, Karl Marx described a bizarre characteristic of capitalist cultures that he called a belief in \"the magic of commodity fetishism.\" Marx claimed that, in the western European culture of his time, people behaved as if they believed that commodities, although they are inanimate objects, were endowed with a form of magical power that seduces people into believing that they need these commodities to survive. Members of capitalist cultures, Marx wrote, find themselves wanting desperately to possess many \"things\" and to harness the mystical power they believe the \"things\" possess. People begin to believe that, in order to be happy, whole persons and live a good life, they must possess these \"things\" by exchanging money and consuming them in the market place.\n\nThe Crown's argument about the impact of the arrival of a few European-manufactured commodities relied on the assumption that the magic of commodity fetishism was powerfully at work among the Gitksan and Wet'suwet'en peoples in north-central British Columbia during the early nineteenth century, when, according to the Crown, they were _not_ organized as a capitalist society. Given this, the Crown's argument must then rest either on the assumption that commodity fetishism is a universal characteristic of human nature and not specific to capitalist cultures; or, that the Gitksan and Wet'suwet'en, as \"primitive\" peoples, were simplistic and easily swayed by the need for instant gratification of their desires for new \"things.\" Once again, the foundations of the Crown's case were eurocentric at best, and racist at worse. Once again, the Crown's image of human life as driven by acquisitive desires to accumulate \"things\" is an impoverished one that does not represent a consensus of European thought, or non-Aboriginal \"cultural values,\" or a desirable vision of the future.\n\nWhile the Crown's arguments reflect outdated social theory, they also reinforce the very contemporary values of consumer capitalism. In asserting these theories as universally applicable and representative of \"human nature,\" rather than as an illustration of an historically and culturally specific \"belief system\" of a sector of the dominant western European society, the Crown attempted to codify into law its image of the ideal self and a model of society driven by the \"rationality\" of individual economic accumulation and self-aggrandizement. By attributing \"scientific\" status to these theories of human behaviour, the Crown could dispense with the experience and analyses of the actual people involved in living these processes, since their behaviour could now be explained as being determined by \"natural\" forces outside themselves, over which they had neither control nor necessarily awareness. The minimal contact with European social or economic organization that the Gitksan and Wet'suwet'en had actually experienced when, according to the law, they were transformed without their knowledge into something \"not truly aboriginal,\" was of no concern to the Crown's theories.\n\nThese theories can be traced through several centuries of European thought, and they are currently enjoying a resurgence in contemporary neo-conservatist ideologies, whose adherents can recognize only two forms of human behaviour: \"either bargaining gambits\u2014the approved behaviour of the 'successful;' or madness\u2014the discredited behaviour of the 'failures.'\" If industrial capitalism's motif was represented by the slogan \"I own, therefore I am,\" consumer capitalism's raison d'\u00eatre can be expressed as \"I buy, therefore I am.\" Our world is increasingly driven by the creation of insatiable desires for things we don't need, while our needs for what we have forgotten to desire go unmet.\n\nThe Crown's case, therefore, was empirically unsupported; scholastically archaic; logically confused; ethnocentric, eurocentric, and racist. Ultimately, it defends and enforces an image of human beings and a vision of human society that many people, Aboriginal and non-Aboriginal, including myself, reject as oppressive to the best in human potentiality.\n\n* * *\n\n. Satsan 1992: 53.\n\n. Transcripts Vol. 291: 21865-21867.\n\n. Transcripts Vol. 291: 21865-21867.\n\n. Transcripts Vol. 291: 21966.\n\n. Robinson 1983.\n\n. Robinson 1983: 423.\n\n. See Wike 1951; Fisher 1977.\n\n. Robinson 1983: 408.\n\n. Robinson 1983: 425.\n\n. Rogers 1986: 1.\n\n. Bishop 1987: 7-9.\n\n. Wike 1951.\n\n. Fisher 1977.\n\n. See Canada, Royal Commission on Aboriginal Peoples 1993. But see also Backhouse 1991 for a feminist critique of the Royal Commission's analysis of this case.\n\n. Foster 1992.\n\n. Robinson 1987(a): 16.\n\n. Transcripts Vol. 293: 22157-58.\n\n. Transcripts Vol. 292: 22126-27.\n\n. MacDonald 1984.\n\n. MacDonald 1984: 73, quoted in Robinson 1987(a): 34.\n\n. Robinson, 1987(a): 24.\n\n. Transcripts, Vol. 289: 21747-21751.\n\n. Adams 1969:14; 1973.\n\n. Adams 1987: 69.\n\n. Acheson 1995: 13, Table 1.1.\n\n. Acheson, 1995:11. See also Boyd 1990: 137.\n\n. Foucault 1975; Tesh 1988.\n\n. See Simmel 1978.\n\n. Transcripts, Vol. 289: 11145.\n\n. Gordon 1988: 54.\n**Chapter 14: From a Fur Trading Frontier**  \n**to a White Man's Province**\n\n_The settler came to re-create an alien civilization on the frontier, while the fur traders had to operate largely within the context of the indigenous culture.... That is, generally traders reacted to what they saw, while settlers tended to react to what they expected to see._\n\n\u2014Historian Robin Fisher, 1977.\n\n**Believing What You See. Seeing What You Believe.**\n\nAlthough most of the key legal questions relevant to the \"tests\" for Aboriginal title and rights rested on interpretations of the pre-contact and fur trade periods, subsequent events in the history of First Nations relations with British Columbia governments and populations were also relevant. During the four-year trial of _Delgamuukw v. R._ Chief Justice McEachern was presented with evidence and arguments by the Gitksan and Wet'suwet'en, and by both the federal and provincial governments that took the form of contested historical narratives that stretched from first contact to the present\n\nFollowing the fur trade, British Columbia moved into what historians call \"the settlement period.\" The establishment of the Colony of Vancouver Island, granted to the Hudson's Bay Company by Royal Charter on January 13, 1849, marked the end of the fur trade period and the beginning of the settlement period, although significant numbers of British settlers did not begin arriving before the 1860s. The political, economic, cultural and legal implications of the transition from the fur trade to settlement are significant, since they involved a marked difference in the value\u2014broadly defined\u2014of land to non-Aboriginals, and created the positions of governor, members of the provincial assembly and justices of the peace. The legal practice of settlement was expressed in a rapid proliferation of laws regarding ownership of land, access to resources, and political relations between Indigenous peoples and settlers, and within and between settler populations.\n\nThe first Governor of the new colony of Vancouver Island was a lawyer, Richard Blanchard, appointed in 1849. He proved unequal to the task at hand, and was replaced in 1851 by James Douglas, Chief Factor of the Hudson's Bay Company at Fort Victoria, who assumed the additional position of governor. Douglas was the son of a Scots trader and a \"free coloured woman\" of British Guyana. His wife was the child of a Hudson's' Bay Company factor and a Cree woman. Fisher describes Douglas' attitudes as \"a mixture in which the knowledge of the fur trader was accompanied by the paternalistic concerns of the nineteenth-century humanitarian.\" In this way, Douglas embodied and personified the transitional historical period he governed.\n\nFourteen agreements were made between Governor Douglas and various Aboriginal nations on Vancouver Island between 1850 and 1854; that have become known as the \"Douglas Treaties.\" The questions about Douglas and, particularly, \"the Douglas treaties,\" that are legally important today can be summarized as follows: First, were these agreements treaties, or merely private purchases made by the Hudson's Bay Company? Second, did Douglas recognize Aboriginal title in a proprietary sense, or merely in the use and occupancy sense? Third, did Douglas recognize an Aboriginal interest in all lands, or only those \"under cultivation, village sites or fenced fields?\" Fourth, did Douglas take a consistent stand throughout his period as governor, or did he change his mind, or did the instructions he received from London change over time?\n\nWhile scholars concur that these agreements should be interpreted as treaties that clearly recognized some form of Aboriginal interest in the land and that required explicit cession and\/or extinguishment, there are debates about what the scope of the Aboriginal title Douglas recognized was. Fisher, for example, argues that the treaties were based on current British opinion about the nature of Aboriginal land tenure. He concluded that Douglas was authorized only to confirm a right of occupancy of lands under cultivation, village sites and fenced fields, but not to recognize Aboriginal title as equivalent to ownership or sovereignty in a European sense. Indians would have hunting and gathering rights over other lands as long as these were \"waste\" and not allocated to settlers. Political scientist Paul Tennant disagrees with Fisher. He argues that: \"The most important fact about the Douglas treaties is that they stand as unequivocal recognition of aboriginal title. It was with this initial acknowledgement that the British established their rule in British Columbia.\"\n\nIt has been argued\u2014and here we encounter the beginning of another dominant theme in Aboriginal\/state relations\u2014that one reason for Douglas having ceased to address the question of Indian title was that, as the costs of administering the colony increased, neither the Hudson's Bay Company nor London allocated a sufficient increase of funds to accomplish the task.\n\nThe colonial office in London, and the legislative assembly in Victoria, were both concerned not only, or even principally, with Indian matters. Rather, they were consumed with planning a very culturally-specific and homogenous colony. An early instruction to the governor of the colony of British Columbia was to insure the transfer of \"a cross section of British society to the colony,\" and a process whereby \"a just proportion of labour and capital\" would beachieved, and \"paupers, squatters and land speculators\" would be prevented from settling. Their ambition to create a civilized \"whiteman's province\" would be expressed in immigration and settlement policies, state ideology, popular sentiment and sporadic outbursts of hostility and violence towards Asians as well as Aboriginal peoples throughout the province's history.\n\nDouglas initially respected Aboriginal law to the extent that he regularly ordered Europeans to pay compensation to the families of victims injured by them. During the course of the 1850s, however, Douglas increasingly invoked British law to settle both European-Native disputes and intra-Native disputes. In 1855 he wrote optimistically that he thought the Indians were beginning to have a clearer idea of the nature of British law which, he said, was \"the first step in the progress of civilization.\" Governor Douglas appointed his brother-in-law, David Cameron, a linen-draper from the West Indies, as Supreme court Justice in 1853, and as Chief Justice in 1856.\n\nThe Gold Rush of 1857-58 brought the Pacific northwest to the forefront of Britain's interests in North America. The summer of 1858 began with the arrival of 400 miners in Victoria, and by September there were thousands. With the influx of miners and increasing numbers of settlers, Douglas gave up his former practice of \"legal pluralism\" and insisted that Indians and Whites alike seek redress through representatives of British law rather than through traditional modes or individual retaliation.\n\nIn 1858, the British established direct rule on the mainland, creating British Columbia. Douglas resigned as Chief Factor of the Hudson's Bay Company and retained the position of Governor only. In 1861, the House of Assembly passed a petition in which they stated their belief that \"the extinction of Aboriginal title is obligatory on the Imperial Government,\" and sent it to the Duke of Newcastle, seeking funds to extinguish Aboriginal title. The requested funds never arrived. Instead, the Duke of Newcastle wrote back acknowledging, \"...the great importance of purchasing without loss of time the native title to the soil of Vancouver Island...but the acquisition of the title is a purely Colonial Interest and the Legislature must not entertain any expectation that the British Taxpayer will be burthened to supply the funds or British Credit pledged for the purpose.\"\n\nThis correspondence has become important in contemporary litigation. If it can be shown that the British Crown, and\/or the Parliament of Canada, recognized Aboriginal title in British Columbia, then the more recent story constructed by the Province of British Columbia's lawyers that posits an uninterrupted and uncontradicted history of _non-recognition_ by the Crown of Aboriginal title from the moment of contact is significantly challenged.\n\nAs always, politicians and jurists engaged in rationalizing colonial rule drew in various ways from the intellectual climate of their times. Charles Darwin had published _Origin of the Species_ in 1859. His theories hardened into doctrine, and the fiction that the British represented a superior race, and the Aboriginals an inferior one, was rapidly transformed from speculative hypotheses, into pseudo-scientific dogma, popular mythology, and ideological justifications for colonization. John Locke's theories were increasingly drawn upon and popularized as scientific and moral justifications for colonization. Fundamental to this social evolutionary ideology is the notion of an absolute and unbridgeable difference between races. In the absence of recognition of cultural similarities, negatively valued difference ruled the day. Increasingly, Aboriginal peoples were represented in local provincial, Canadian and British literature and propaganda as \"wild savages,\" rather than \"noble savages.\" Historian Robin Fisher explains: \"The British colonist established a line of cleavage based on race and could not permit any crossing of that barrier by admitting that the Indian was in any way comparable to western man.\"\n\nWhen, in 1864, local legislative assemblies replaced direct rule by London, James Douglas retired as governor and was replaced by Frederick Seymour in British Columbia and Arthur Kennedy on Vancouver Island. Neither Seymour nor Kennedy were actively concerned with Indian land rights and they left the responsibility for policy-making in this area to their newly-appointed Chief Commissioner of Lands, Joseph Trutch. Trutch \"personified settler interests and attitudes, considering Indians 'as bestial rather than human,' 'uncivilized savages,' 'ugly and lazy,' 'lawless and violent.'\"\n\nTrutch turned to Social Darwinism for justification of his policies. He argued that Indians had not evolved to the stage where they could conceptualize ideas of property. Furthermore, following Locke, he reasoned that hunting, fishing and gathering did not involve the application of human labour to the transformation or cultivation of the land, and therefore, he claimed, the Indians were not using the land efficiently. Trutch disposed of the Douglas Treaties, claiming that, since there had been neither civilization nor law among the First Nations, no treaties had been required. He argued that the monies given to Indians during negotiations were to purchase short-term peace and not to purchase or extinguish title, which was not necessary. He invented a rule that a maximum of ten acres should be allotted to each adult Indian male, and then reduced the allocations of reserve lands that Douglas had laid out accordingly.\n\nDouglas, in an effort to encourage Indian assimilation, had guaranteed Indians the right to acquire land under the same conditions as those offered to settlers; that is, by cultivating it and building on it. His vision was that Indians would move off the reserves after having been cared for, civilized, and Christianized by missionaries, and would then become equal citizens of the colony. Trutch passed an ordinance that prohibited Indians from pre-empting land without the written permission of the Governor. By 1875, there was only one, single case of an Indian pre-empting land under this condition.\n\nVancouver Island and the mainland were united into one colony, British Columbia, in 1866. By this time the Aboriginal population had been reduced to approximately 40,000 people, although there were still less than 100 non-Aboriginal residents in the Gitksan and Wet'suwet'en territory and the \"whites depended on Indian participation for the success of most economic activity.\"\n\nSmallpox and other diseases continued to decimate the Aboriginal population throughout the latter half of the nineteenth century. However, as was the pattern in British Columbia, the Gitksan and Wet'suwet'en continued, during the settlement period, to fish, hunt, gather and trade with each other, with other Aboriginal groups, and now, with non-Aboriginals. In fact, they furnished the Hudson's Bay Company staff with most of their requirements. The economic interests of the non-Indians at the time were limited to the extraction and transportation of resources, particularly furs and gold.\n\nMissionaries joined fur traders and settlers at what had now become a European frontier. The interests and ideologies of these diverse groups were not always identical. Settlers, for the most part, assumed that Indians were \"primitive\" peoples destined by natural selection to vanish. Aboriginal claims could therefore be ignored, and certainly need not be considered serious obstacles to settlement. Missionaries, however, were intent on saving heathen souls, and many believed that Indians need not vanish, but could survive if they ceased to be Indians and assimilated into the dominant white society. Hence, missionaries often placed themselves in the position of protectors of the Indians against the rapacious disregard shown by settlers and the calculated manipulations of traders; and as advocates on behalf of Aboriginal peoples in representations to governments. Into their pre-existing life ways and institutions First Nations incorporated: near annihilation by relentless epidemics of infectious diseases; Christianity; wage labour in guiding, packing, canoeing, and mining; and, later on in the century, employment in fish canneries.\n\nTo the settlers, Aboriginals were wild, wicked children. To the missionaries, Aboriginals were noble, innocent children. To the governments, Aboriginals were a potentially expensive legal and social problem. To the merchants, Aboriginals were suppliers and consumers of commodities. To some non-Aboriginal individuals, Aboriginal individuals were friends, lovers, co-workers, spouses and relatives.\n\n**On the One Hand, and On the Other...**\n\n_Federal jurisdiction is derived from s. 91(24) of the Constitution Act, 1867, which has been read to permit Parliament to single out native people and treat them differently than non-native people. Parliament is also entitled to treat native people the same as non-native people under laws passed pursuant to other heads of federal power...for example, The Fisheries Act. Provincial legislatures are not entitled to treat native people differently than non-native people, but can pass laws regulating native forms of life so long as such laws are of general application and do not touch on matters which are inherently Indian...._\n\n_In the end all this means that Parliament and provincial legislatures pass laws regulating the life and culture of native people without native consent._\n\n\u2014Legal scholar Patrick Macklem, 1991.\n\nCanadian Confederation under the _British North America Act_ ( _B.N.A. Act_ ) took place in 1867. Consistent with the theory articulated in the Royal Proclamation of 1763, the Crown's underlying title and sovereignty were confirmed. Section 91(24) of the _B.N.A. Act_ conferred on the federal Parliament jurisdiction over \"Indians and Lands reserved for Indians.\" Provincial governments were given jurisdiction over wildlife and game, education, health, social services and some justice and corrections areas. Provincial authority in these areas encompasses the entire population, including Aboriginal peoples, but the province may not pass legislation specifically addressed to Indians. Hence the legal and legislative basis for the historic provincial government policy of refusing to recognize Aboriginal people as distinct populations.\n\nBritish Columbia did not join Confederation until 1871. Negotiations between London, Ottawa and Victoria began two years earlier in 1869. Both London and Ottawa expressed concerns about the fact that British Columbia did not appear to have a process in place for legally extinguishing Aboriginal title and releasing land for settlement. The Colonial Office queried the absence of treaties based on the Royal Proclamation of 1763. The British Columbia delegation, represented by Joseph Trutch as chief negotiator, spent June and July of 1870 in Ottawa negotiating the Terms of Union which were passed by Order-in-Council in July 1870. Clause 13 refers to Indians:\n\nThe charge of the Indians, and the trusteeship and management of the lands reserved for their use and benefit, shall be assumed by the Dominion Government, and a policy as liberal as that hitherto pursued by the British Columbia Government shall be continued by the Dominion Government after the Union.\n\nTo carry out such a policy, tracts of land of such extent as it has hitherto been the practice of the British Columbia Government to appropriate for that purpose, shall from time to time be conveyed by the local government to the dominion government in trust for the use and benefit of the Indians on application of the dominion government; and in case of disagreement between the governments respecting the quantity of such tracts of land to be so granted, the matter shall be referred for the decision of the Secretary of State for the Colonies.\n\nIn effect, the Terms of Union gave the province of B.C. direct title to all public lands. Confident that the \"Indian title question\" was resolved, the province now lumped Indians together with non-white immigrants as ineligible for even the most basic civil rights, including the vote. By 1871, a theory of Aboriginal title in British Columbia, legitimized by Lockean political theory and nineteenth-century evolutionism, had been formalized in colonial policy and articulated in popular ideology as the \"founding myth of White British Columbia.\" It remained yet to be codified into law to become \"the truth.\" No Aboriginal representatives were involved in the negotiations or agreements that led up to the passing of either the _British North America Act_ or the Terms of Union.\n\nIt was not until after 1870 that the Gold Rush brought large numbers of whites, including surveyors, miners, merchants, missionaries and government administrators to the Gitksan and Wet'suwet'en territory. A permanent white settlement was established at Hazelton, but whites remained a numerical minority, who continued to buy large quantities of dried salmon and other foodstuffs from the local Indians.\"\n\nJoseph Trutch wrote to the Prime Minister of Canada, Sir John A. MacDonald, in 1872, offering his analysis of the situation: \"We have here in B.C. a population numbering from 40,000 to 50,000, by far the larger portion of whom are utter savages living along the coast, frequently committing murder and robbery among themselves...and only restrained from more outrageous crime by being always treated with firmness and by the consistent enforcement of the law....\"\n\nThe negotiations surrounding Indian policy in the Terms of Union of 1871 are of considerable contemporary legal and political significance since arguments have been made by historians that Trutch intentionally allowed a mistaken impression to remain with federal officials that the Indian policy in British Columbia, like that in the rest of the country, had dealt with Indian title by treaty before opening land for settlement. In the rest of Canada, major treaties with Indians west of Ontario had been signed during the 1870s; and reserve lands were being allocated on the basis of the national 80 acres per family standard, and not the 10 acres per adult male actually being allocated in B.C. If Trutch did, in fact, intentionally deceive federal officials, this evidence would support the Aboriginal argument that the federal government has consistently recognized their interests, whereas the provincial government has not. More to the point, such a finding of fraudulent action would place in doubt the legal status of _all_ land title throughout the province, and would cast a shadow on British Columbian settlers' history and self-image. Former Governor James Douglas wrote to federal Superintendent of Indian Affairs Powell to protest Trutch's allocations, and stated that his intention and practice as Governor had been to allocate whatever lands the Indians themselves identified as being required. In 1873, Sir John A. MacDonald, having realized that Trutch was allocating only 10 acres to adult males, officially requested the Province of B.C. adopt the 80 acre per family national standard for allocation of reserve lands.\n\nIn 1874 the B.C. legislature passed the B.C. Lands Act aimed at consolidating previous laws affecting Crown lands. The following year the federal government took an extraordinary action when the Dominion Deputy Minister of Justice ordered the provincial law disallowed, in part because it did not take into account Aboriginal title, and prevented Indians from preempting land without written permission from the government. The federal Deputy Minister cited the Royal Proclamation of 1763 in support of his position, arguing that British Columbia, by being the only province not to follow British policy in this regard, jeopardized the \"honour of the Crown.\"\n\nFollowing the federal government's disallowal of the B.C. Lands Act in 1875, the federal and provincial governments agreed to constitute a joint federal-provincial reserve commission to examine the land question. Three Commissioners were appointed to the Indian Reserve Commission: former Hudson's Bay Company men, Anderson and McKinlay, for the federal government and provincial government respectively; and Gilbert Sproat, a settler and self-taught ethnographer who had lived among the Nuu'chah'nulth of Vancouver Island, as a third. No Aboriginal people were consulted or involved either in the formation of the Commission, or in the development of its terms of reference. Some First Nations, when advised that they would not be permitted to discuss Aboriginal title but would be limited to pleading for reserve allocations, boycotted the process. Some attended the hearings, defied the agenda, and spoke about their land rights. Others set out detailed arguments for specific reserve allocations.\n\nIn order to assist the Commission, the provincial legislature moved to have the _Papers Connected with the Indian Land Question in British Columbia, 1850-_ _1875_ (\"The Papers\") published. Joseph Trutch, however, managed to withhold the collected papers from the legislative assembly, and, more importantly, from the Indian Reserve Commission. The withholding of this publication from public, particularly Aboriginal, access would play an important part in future relations between Aboriginal peoples and the British Columbia provincial government. _The Papers_ contain correspondence between the British Colonial Office and, first Douglas, and then Trutch. That is, _The Papers_ constitute the official documentary record of the critical period in British Columbia history when the Crown consolidated its title to the province's lands.\n\nA reading of _The Papers_ leaves little doubt about a few key legal and historical points. For one thing, it is clear that the British Crown was initially cognizant of, and anxious to settle, Aboriginal interests in land. For another, the British Crown clearly wanted this accomplished in as \"legal\" and \"humanitarian\" a way as possible, reflecting the principles of British colonial law and policy. That the British Crown had consistently assumed that the Royal Proclamation of 1763 applied to British Columbia, and that its terms should provide a framework for negotiating treaties with First Nations in British Columbia is also plainly and repeatedly stated in _The Papers_. On the other hand, the Crown did not wish to pay the costs effected by these policies, nor did they want any problems to arise that might impede settlement. The balance between humanitarian concern and economic efficiency shifts repeatedly throughout, but by the end of the twenty-five year period covered by _The Papers_ , settlement\u2014and settlers'\u2014interests are clearly at the forefront. When choices had to be made between spending money or dispensing justice, Aboriginal interests were consistently sacrificed. Correspondence from first colonial, and then provincial officials, follows a similar pattern and conclusion. The shift in attitude and position from Douglas to Trutch is plainly represented. The documents tell neither a humanitarian, nor a heroic, nor a law-abiding story. Greed, dishonesty, petty corruption and a lust for power sporadically justified by racism, crude even for its time, are what the _Papers Connected with the Indian Land Question in British Columbia, 1850-1875_ reveal.\n\n* * *\n\n. Fisher 1977: 68.\n\n. See Fisher 1977; Madill 1981; Tennant 1990(a).\n\n. See Fisher 1977; Duff 1969.\n\n. Tennant 1990(a): 20.\n\n. Fisher, 1977: 150.\n\n. Fisher, 1977: 60.\n\n. The literature on race relations in British Columbia has grown substantially in recent years. An entry into this literature can be obtained through the following: Adachi 1976; Hudson (ed.) 1997; Johnston 1989; Li 1988; Roy 1989; Ward 1978.\n\n. Fisher, 1977: 65.\n\n. See Knafla 1986(b); McLaren, Foster, and Orloff (eds.) 1992.\n\n. Fisher 1977: 95.\n\n. Galois 1992. See also Foster 1981, 1992(b); McLaren, Foster and Orloff (eds.) 1992.\n\n. Public Archives, Ottawa, C.O. 305\/17, 133-34.\n\n. Newcastle to Douglas, 19 October 1861, _Papers_ , 214.\n\n. Fisher, 1977: 93.\n\n. Tennant 1990(a): 39.\n\n. Fisher 1977: 165.\n\n. Galois 1987: 21.\n\n. Ray 1987: 83.\n\n. Fisher 1977: 143.\n\n. Ray, 1987: 83.\n\n. Douglas to Powell, 14 October, 1874, _Papers_ p. 53. British Columbia officially agreed to 20 acres, but never honoured the agreement in practice (Tennant 1990: 46).\n\n. Province of British Columbia, Papers of the Legislative Assembly, 37th, Victoria 1873-74: 1027-28.\n\n. See Fisher 1977: 187; Tennant 1990: 47.\n**Chapter 15: Old Questions. New Century.**\n\n_From time immemorial the limits of the district in which our hunting grounds are have been well defined. This district extends from a rocky point called \"Andemane,\" some two and a half or three miles above our village on the Skeena River to a creek called \"She-quin-khaat,\" which empties into the Skeena about two miles below Lorne Creek._\n\n_We claim the ground on both sides of the river, as well as the river within these limits, and as all our hunting, fruit gathering and fishing operations are carried on in this district, we can truly say we are occupying it._\n\n\u2014Gitksan Chiefs of Gitwangak, 1884.\n\n**The First Definite Hint...**\n\nThis claim was presented to the Provincial Government of British Columbia in 1884. Witness for the Crown in _Delgamuukw v. R._ and popular historian, David Ricardo Williams described this petition presented by the Gitwangak Chief as \"the first definite hint by natives that they possessed proprietary rights.\"\n\nThe _Indian Act_ had been passed in 1876, bringing every aspect of Aboriginal life under the scrutiny and administration of the federal government. In 1884 clauses were added that outlawed the carrying out of, or assisting in, or purchasing or supplying goods for, or communicating about, potlatching, sundancing, and other Aboriginal cultural practices. Since these ceremonies were also the means by which vital events like births, marriages and deaths were recorded in the oral tradition, and where transfers of lands and rights through inheritance or alliance were witnessed and effected, this ban constituted a criminalization of Aboriginal cultural, political, legal and economic institutions and practices. The _Indian Act_ also delegitimated traditional chiefs and systems of law. These further incursions on their ways of life increased resentment on the part of First Nations towards the governments, and led to more and varied forms of protest against, and resistance to, the colonial regime.\n\nNative resistance took on a number of forms during this period. Individual confrontations decreased, nativist and syncretist religious movements began, and organized public protests and petitions became common place. In 1872 a large crowd of Coast Salish people assembled outside the New Westminster Land Registry office to protest the illegal seizure of their lands. Throughout the 1870s various groups sent petitions and delegations to state their grievances to government. Four issues preoccupied them: (i) recognition of Aboriginal title; (ii) insufficient and arbitrary reserve land allocations; (iii) encroachment on lands and resources by whites; (iv) lack of support from governments for developing agriculture and animal husbandry.\n\nIn 1885 three Coast Tsimshian chiefs, accompanied by missionary William Duncan, became the first of many delegations to travel to Ottawa \"to tell them our troubles about our land.\" Among the Chiefs' principle demands was that a public inquiry be held into the land question. By way of response, Provincial Premier Smithe agreed to a meeting between federal, provincial and Nishga and Tsimshian leaders. He insisted, however, that missionaries, who he alleged had put the idea of land title into the minds of Aboriginal peoples to begin with, not be allowed to attend. The federal and provincial governments did, however, agree to a public inquiry that began in the winter months of 1887. Nishga Chief David Mackay addressed the hearings. He said:\n\nWhat we don't like about the Government is their saying this: \"We will give you this much land.\"\n\nHow can they give it when it is our own? We cannot understand it. They have never bought it from us or our forefathers. They have never fought and conquered our people and taken the land in that way, and yet they say now that they will give us so much land\u2014our own land.\n\nThese chiefs do not talk foolishly, they know the land is their own; our forefathers for generations and generations past had their land here all around us; chiefs have had their own hunting grounds, their salmon streams, and places where they got their berries; it has always been so. It is not only during the last four or five years that we have seen the land; we have always seen and owned it; it is not new thing, it has been ours for generations.\n\nIf we had only seen it for twenty years and claimed it as our own, it would have been foolish, but it has been ours for thousands of years. If any strange person came here and saw the land for twenty years and claimed it, he would be foolish. We have always got our living from the land; we are not like white people who live in towns and have their stores and other business, getting their living in that way, but we have always depended on the land for our food and clothes; we get our salmon, berries, and furs from the land.\n\nBritish Columbia Premier Smithe dismissed the First Nations' claims, telling them:\n\nWhen the whites first came among you, you were little better than the wild beasts of the field.... The land all belongs to the Queen.... A reserve is given to each tribe, and they are not required to pay for it. It is the Queen's land just the same, but the Queen gives it to her Indian children because they do not know so well how to make their own living the same as a white man, and special indulgence is extended to them, and special care shown.\n\nThus, instead of being treated as a white man, the Indian is treated better. But it is the hope of everybody that in a little while the Indians will be so far advanced as to be the same as a white man in every respect. Do you understand what I say?\n\nThe Crown's legal arguments in _Delgamuukw v. R.,_ whose answers lie in what took place during this historical period, centred on questions of \"abandonment\" of land and \"acquiescence\" to colonial rule. The Crown continued and expanded their claims that Aboriginal lands were frequently \"abandoned.\" The Crown alleged that, being anxious to assimilate and participate in the new\u2014superior\u2014economy and culture, Aboriginal peoples voluntarily abandoned their traditional territories and outlying villages, along with fishing, hunting, gathering and trapping activities, and became wage labourers. They settled into villages that the government then \"gave\" them as reserves, and eagerly converted to Christianity. Their lands\u2014never really \"owned\" in European terms in the first place, and now abandoned and unused by previous claimants anyway\u2014became available for pre-emption and settlement by settlers.\n\nThe Crown argued that Aboriginal people complied with colonial rule and did not mount important or serious resistance. Not only did the Crown have the absolute right to \"implicitly\" extinguish Aboriginal title by simply declaring sovereignty and imposing British law; but, the Crown argued, First Nations \"implicitly\" consented to extinguishment, and to other assimilation policies, both legally and culturally. When federal and provincial laws were enacted and Indians obeyed them, so the interpretation goes, they were effecting legal extinguishment of title, whether they knew it, or intended to, or not.\n\nThe Gitksan and Wet'suwet'en argued that they had never consented to extinguishment of their Aboriginal title and rights. When they complied, sometimes, with various laws, they did so, they testified, either because they agreed with them in spirit, or because they were coerced by force of arms and\/or numbers, to accommodate the settlers and governments. As to abandoning their lands\u2014again, they argued that they had not abandoned hunting and trapping and fishing practices or territories, but continue to use them for purposes other than permanently residing on them or gaining _all_ of their subsistence and livelihood from them. Furthermore, they argued, to the extent that people have ceased earning their living from the land, it has not necessarily been by choice, but rather because they have been prohibited by laws and regulations from accessing their lands and resources, and from trading and selling the products of their territories. Therefore, by force of necessity, many have had to earn their living in the Euro-Canadian controlled economy.\n\nIn 1888 the first significant legal case involving issues of Aboriginal title and the effect of the Royal Proclamation of 1763 in Canada was heard in Ontario. In _St. Catherine's Milling and Lumber Co. v. R._ the issue in dispute was the legality of a logging permit issued by the federal government to the private St. Catherine's Milling and Lumber Co. Lord Watson, of the Judicial Committee of the Privy Council of the House of Lords in Britain, ruled that only the federal government had the right to enter into treaties and extinguish Aboriginal title, and that Aboriginal title had no pre-existence, but was created by the British Crown through its recognition in the Royal Proclamation. Aboriginal title, as a creation of the British Crown, could remain in effect only \"at the pleasure of the Crown\" and could be eliminated by any contrary action by the Crown, however implicit, Watson's ruling concluded. Lawyers for the Province of British Columbia in the _Delgamuukw_ case cited the _St. Catherine's Milling and Lumber Co._ decision as a precedent in support of their arguments.\n\nIt was by reading newspaper reports about the _St. Catherine's Milling and Lumber Co._ case that British Columbia Indians became aware for the first time of the Royal Proclamation of 1763, and seized upon it as evidence that the British Crown should in fact, by their own law, recognize and guarantee Aboriginal rights. \"The idea that there could be imperial intervention in indigenous policy in Canada was no Indian invention. The Royal Proclamation of 1763 represented imperial recognition of Indian political and territorial rights against local 'frauds and abuses',\" legal scholar Douglas Sanders explains. \"As this became known among British Columbia Indians, they began to discuss the possibility of taking their case directly to London.\" British Columbia Aboriginal peoples, then and now, interpreted the Royal Proclamation's words to mean what they said: that the Crown recognized Indian nations and land rights and pledged to protect them. What Aboriginal peoples didn't anticipate was that sovereigns, and representatives of sovereigns, could and would interpret this Proclamation in various abstract ways, including the interpretation that, since B.C. First Nations did not exist \"in the eyes of the Sovereign\" in 1763, they could be excluded from consideration under the terms of the Proclamation. They appear to have failed to understand the magic land ownership possessed by the omnipotent hovering sovereign.\n\nThe Crown arguments described above, however, are _post hoc_ rationalizations developed in the 1960s. It would be a mistake to assume that law is monolithic and cannot or did not adapt to local contingencies if opportunity demanded. When, in 1898, Beaver Indians assembled at Fort St. John demanding a treaty before they would allow gold seekers into their territory, the federal government complied. At the time there were few white settlers in the region, and, once the provincial government was assured that they would not be financially responsible for treaty annuities, they did not oppose the process. Treaty 8, signed in 1899 and covering a corner of north-eastern British Columbia, acknowledged the pre-existence of Aboriginal title and the applicability of the terms of the Royal Proclamation of 1763 to British Columbia, including the requirement that consent must be obtained before extinguishment of title can be legally valid. However, rather than being recognized as a precedent supporting the application of the Royal Proclamation of 1763 to British Columbia, succeeding British Columbia jurists have relegated Treaty 8 to the category of an anomaly, and ignored it.\n\nThe late nineteenth century witnessed the beginning of an important theme in British Columbia Aboriginal history. Some First Nations, like, for example, the Nishga, began early on in their contact with Europeans to successfully articulate their pre-existing economic and political structures to those of the Europeans. An Indigenous elite, well schooled in English language and law, developed under the tutelage of Anglican missionaries. Participation in the commercial fishing industry has formed the economic basis of a network of wealthy families on the coast, who have succeeded very well for generations, by anyone's standards. Had the predictions of European anthropological and social theories been realized, the expected outcome should have been the rapid assimilation of these successful individuals into Canadian society. In British Columbia, however, these families have formed the backbone of the anti-colonial resistance struggle and the movement for recognition of Aboriginal title. Money earned in fishing and logging has been churned back into communities and nativistic movements through continued feasting and potlatching\u2014despite legal prohibition\u2014and has funded the development of political organizations, the employment of lawyers, and the sending of delegations to Ottawa and London. During the 1890s the Nishga led a province-wide protest movement among Aboriginal peoples. They established their own newspaper and used it as a vehicle for organizing and educating Indians all over the province.\n\nA census was taken in British Columbia in 1880 that listed First Nations as a majority of the total population of 49,459. The census of 1891 counted 98,173 persons in the provincial population, of which only one third\u2014around 35,000\u2014were Indians. During the 1870s and 1880s a sudden and large influx of miners had poured into British Columbia; a colonial administration had been established and an array of laws governing lands, resources and policing were enacted. Intensive white settlement began, missionaries arrived, and smallpox decreased the Aboriginal population substantially. Aboriginal people took employment in the new industries and continued working on their lands, and mounted a wide range of resistance activities at local, provincial, national and international levels aimed at maintaining control over lands and resources.\n\nDuring the early years of the nineteenth century, British colonial policy had been strongly influenced by humanitarian sentiment in Britain that conceptualized the mission of Britain to be to \"take the evils of slavery, ignorance and paganism at source, to extend to the simpler people the benefits of steam, free trade and revealed religion, and to establish not a world empire in the Napoleonic sense but a moral empire of loftier interest.\" This is known in legal terminology as the \"trusteeship theory of colonialism,\" and in anthropological literature as colonization through \"tutelage.\" Benevolent white colonial parent; innocent brown Indigenous child.\n\nBy the latter half of the nineteenth century, however, racial attitudes in Britain began to harden and move away from the previous humanitarian approach. Armed rebellions like the Indian mutiny, wars in South Africa, Maori-European wars, and rebellion in the West Indies were once again fuelling the \"wild savage\" image, always the other side of the same coin on which the \"noble savage\" is stamped. Land-hungry settlers rationalized that the \"wild savages\" were incapable of using their lands productively, but could be disciplined as agricultural labourers on White-owned farms. Strict white colonial parent; incorrigible brown Indigenous child.\n\nDebates about this period engaged in by the parties in dispute during the _Delgamuukw_ trial consisted in the Gitksan and Wet'suwet'en and their witnesses arguing that First Nations' resistance to colonial rule had been firm and consistent since contact, had employed a range of strategies, and had taken place in diverse forums according to circumstances. The Crown countered that, throughout the era in question, Aboriginal peoples had been abandoning life on the land, acquiescing to colonial rule, and rapidly and voluntarily assimilating into Euro-Canadian society. Evidence of protests was dismissed by the Crown as either the work of \"outside agitators\" like missionaries or as the unfounded complaints of a dissident minority of Indians. The Crown interpreted any First Nation compliance with colonial law and policy, or any participation in the Euro-Canadian controlled economy as proof of voluntary assimilation.\n\nThe early years of the twentieth century saw increasing agricultural settlement and the beginnings of serious land speculation in Gitksan and Wet'suwet'en territory in anticipation of railroad construction. The Boer War that ended in 1905 ushered in an era of large scale expropriation of Gitksan and Wet'suwet'en lands to provide veterans with \"unoccupied farmlands.\" Crown witness David Williams described a number of incidents in this period where \"Indians took up arms against whites pre-empting land.\" But, Williams concluded, these episodes \"did not seem to have been provoked by aboriginal claims.\"\n\n**The Era Of Permanent Penetration**\n\n_Any person who without the consent of the Superintendent General expressed in writing receives, obtains, solicits, or requests from any Indian any payment or contribution, or promise of any payment or contribution, for the purpose of raising a fund or providing money for the prosecution of any claim which, the tribe or band of Indians to which such Indian belongs, or of which he is a member, has or is represented to have, for the recovery of any claim or money for the benefit of the said tribe or band, shall be guilty of an offence and liable upon summary conviction for each such offence to a penalty not exceeding two hundred dollars and not less than fifty dollars, or to imprisonment for any term not exceeding two months._\n\n\u2014 _Indian Act_ , Section 141, amended 1927.\n\nDr. Arthur Ray's opinion report on economic history concluded that, during the period 1860-1915, the Gitksan's and Wet'suwet'en's traditions were not incompatible with \"progress\" or \"development.\" Ray discussed numerous attempts made by local Indians to farm, log and become involved in the commercial fishery. A few succeeded, but many were thwarted by lack of support from government, and hostility by settlers. Because Indians were then, and still are, legally wards of the Crown, their lands are \"held in trust for them by the Queen.\" They could not use their lands or resources as collateral to obtain loans, or to capitalize development of their lands, or become involved in commercial ventures. Although interested in pursuing new forms of economic activity, the Gitksan and Wet'suwet'en did not equate this with either assimilation or acquiescence, seeing wage labour and business as ways to enhance and supplement their already existing economy.\n\nRay documented the persistent, and largely successful, resistance to external domination of their economy practised by the Gitksan and Wet'suwet'en during this period. He argued that their efforts were impeded when \"federal and provincial governments passed conservation legislation which curtailed their economic flexibility and weakened their subsistence base. The economic activities of the Gitksan, Wet'suwet'en and Babine in the Upper Skeena River area had not created the problems that this legislation was intended to resolve. Rather, the laws were needed to protect resource-based industries, particularly the salmon canning industry that had been developed by Euro-Canadians outside of the region.\"\n\nBoth the Gitksan and Wet'suwet'en, and the Crown called additional expert witnesses to testify before Chief Justice McEachern during the trial of _Delgamuukw v. R._ on settlement history and the impact this had on Aboriginal cultures. Dr. Robert Galois, an historical geographer, testified for the Gitksan and the Wet'suwet'en. Galois described the 1890s as the years of the Klondike Gold Rush and the building of the Dominion telegraph line over Gitksan territory. He noted that, while new opportunities and sources of wealth in some ways enhanced and elaborated pre-existing Aboriginal practices, they also produced adjustments in the internal division of labour and changes in the \"seasonal round\" of activities. And, with Europeans came, as always, disease and Christianity. Significant population decline brought about by diseases led to changes in settlement patterns, and some elements of Christian practices and beliefs modified the feasting system.\n\nDavid Ricardo Williams was called by the Crown to counter Galois' testimony. Referring to the early years of the twentieth century in British Columbia as the \"era of permanent penetration,\" Williams painted a picture of Gitksan and Wet'suwet'en peoples eagerly complying with the colonial government's assertion of jurisdiction over them, and with the arrival of Euro-Canadian-owned resource extraction companies. To support his arguments he relied on evidence that First Nations did not engage in large-scale armed rebellions and worked in white-owned resource extraction industries. They engaged in very little vandalism, and were generally considered good workers by their non Aboriginal employers.\n\nDamned if they do and damned if they don't: had Aboriginal peoples organized armed rebellions, committed acts of vandalism, refused to participate in the new economy, or failed to succeed as workers and producers, they would have been deemed backward, culturally underdeveloped, and unworthy of equal legal, political and property rights. Not having chosen these strategies, they are analyzed as having surrendered their lands and rights, are deemed assimilated, and unworthy of distinct legal, political and property rights. Heads the Crown wins. Tails, Indians lose.\n\nIn 1906, the Coast and Interior Salish peoples sent a delegation of chiefs to London where they were told that their grievances could only be dealt with by the Canadian federal government. In 1907, the Nishga Land Committee was organized within the framework of Aboriginal social structure: the sixteen members represented each of the four clans in the four Nishga villages. Wealthy and worldly wise aristocrats and successful fishermen, photographs of the Nishga Land Committee taken during this era show sixteen solemn men, well-dressed in tailored three-piece suits, silver and gold watch chains, and polished leather shoes.\n\nIt was at this time that a new partner joined the dance: lawyer and advocate Arthur O'Meara. O'Meara had practised law in Ontario for twenty years before moving to British Columbia, where he became an Anglican missionary. He served as sole legal advisor to the Indian Rights Association and the Nishga Land Committee, and formed the Society of Friends of the Indians of B.C. to raise money and sponsor public talks. O'Meara and the Nishga Land Committee prepared a lengthy petition to the Crown that included a declaration of traditional Nishga ownership, reiterated the terms and conditions set out in the Royal Proclamation of 1763, and explained that these terms had not yet been met in British Columbia.\n\nThe following year, 1908, a delegation of Gitksan chiefs travelled to Ottawa to present a petition protesting the wrongful expropriation of their territory. In the summer of 1909, Interior Nations gathered and formed an organization named the Interior Tribes of British Columbia that then amalgamated with the Nishga and other First Nations on the north coast to form the Indian Rights Association. A delegation of chiefs met with Prime Minister Wilfred Laurier during his 1910 tour of British Columbia and articulated their case. Laurier assured them he would look into it; and, if he found the facts warranted it, would not hesitate to forward the Indian Rights Association's petition to the Judicial Committee of the Privy Council in London for adjudication.\n\nDuring the first decade of this century, land speculation and settlement in various parts of British Columbia was proceeding at an accelerated rate. Meanwhile, the Indian Reserve Commission continued its work, now headed by A. W. Vowell. However, in 1911, the Province of British Columbia announced a new policy that terminated the \"granting\" of any further Indian reserves. The federal government responded that no further alienation of Crown lands to non-Indians in B.C. could occur until two outstanding issues between the federal and provincial governments had been dealt with: First, how Aboriginal title had been, or remained to be, extinguished in British Columbia would have to be clarified. Second, the provincial government must adopt the federal standard of allocating 80 acres of land per family to Indians, rather than the 10 acres per adult Indian male that the British Columbia government had been in the practice of allocating. An impasse between the federal and provincial governments had been reached. They resolved it by jointly establishing a Royal Commission on Indian Affairs in British Columbia, known as the McKenna-McBride Commission.\n\nThrough a process of negotiation that included the two levels of government, but excluded Aboriginal representation, the terms of reference for the commission were determined: the question of legal land title was not to be entertained. The Royal Commission's mandate was simply to adjust the size of reserve land. However, knowing that Aboriginal title was the single issue the First Nations wanted to discuss, official documents preceding and following the McKenna-McBride Commission state unequivocally that the land title question would be dealt with separately, and most likely by a court. Some Aboriginal nations boycotted the hearings because of the exclusion of land title from the terms of reference, while others participated in various ways, as their forebears had a generation before in response to the Indian Reserve Commission established by Joseph Trutch.\n\nWhite settlement was consolidated, and mining and agricultural development increased during the period from 1914-1921. While local level Aboriginal protest continued in the form of confrontation, and petitions and meetings concerning land and resource use were organized at regional levels, the Gitksan also participated in province-wide lobbying efforts, including making a presentation to the McKenna-McBride Royal Commission that completed its work in 1916. The schedule of reserves which was the product of its efforts was given to yet another body, the Ditchburn-Clark Commission, who cut off and further \"amended\" the reserves set aside by McKenna and McBride\u2014without Aboriginal consent. Orders-in-Council conveyed these reserves from the Province to the Federal Government. When the McKenna-McBride Royal Commission completed its work in 1916, Indian reserve land had been reduced by 47,058 acres valued at $1,522,704.00 and new reserves totalling 87,292 acres valued at $444,853.00 had been added. Many First Nations refused to accept the findings of the Royal Commission and turned their attention to preparing their case for the Judicial Committee of the Privy Council in London.\n\nFrom 1910 to 1927, British Columbia Indians believed that their case was in the process of coming before the Privy Council in London. Their belief was based on Prime Minister Laurier's promise made in 1910 to look into their claims; on the terms of reference for the Royal Commission having excluded the land title issue claiming it could only be resolved in court; and on Commissioners McKenna's and McBride's recommendation that this case be heard. In fact, the federal government was not preparing any case for the Privy Council, nor did they advise the many Aboriginal representatives they met with during these years that they were not pursuing the promised hearing before the Judicial Committee, but were, in fact stalling such a claim from proceeding.\n\nSince 1763 the Judicial Committee of the Privy Council had recognized the \"pre-existence\" of Aboriginal rights and their continuity throughout the empire unless explicitly extinguished according to the letter of British colonial law. However, technically speaking, the Judicial Committee of the Privy Council could normally only act on a Canadian matter if it came to them by way of an appeal of a Canadian court decision. Aboriginal people, being wards of the Crown, required permission from the Crown before they could file a writ in court. Therefore, as long as neither the First Nations, nor a government had initiated a court action and been allowed to appeal the decision, Canadian officials could legally refrain from forwarding the Indians' petitions to London: no court case, no judgment, no appeal.\n\nFinally, in 1914 the federal cabinet passed an order-in-council stipulating that the federal government would refer the claim of the British Columbia First Nations to the Exchequer Court of Canada \"with the right of appeal to the Privy Council\" providing three conditions were accepted by the First Nations: (1) if the court found in favour of the Aboriginal title, the Indians would surrender this title to be extinguished completely in return for the same sorts of treaty benefits negotiated elsewhere in Canada, and would accept the recommendations of the McKenna-McBride Royal Commission; (2) any obligations of the province would be fulfilled by its granting the land for the reserves; (3) the province would take part in the court case represented by legal counsel of its own choosing, while the Indians would be represented by counsel nominated and paid by the Dominion government. O'Meara was detested by government officials, and this provision was specifically meant to exclude him. The Allied Tribes of B.C. rejected all three conditions, and continued their lobbying efforts, stressing their preference for negotiations with governments rather than litigation.\n\nMeanwhile, in another corner of the British Empire, the Judicial Committee of the Privy Council issued the infamous 1919 ruling in _Re: Southern Rhodesia_ , classifying the Indigenous Africans as \"too low on the scale of social organization\" for their rights to be recognized under the British regime. Perhaps buoyed by this decision, Duncan Campbell Scott, Superintendent of Indian Affairs, ordered the recommendations of the McKenna-McBride Commission to be implemented without Aboriginal consent. However, in 1921 Lord Sumner's decision in _Re: Southern Rhodesia_ was criticized and rejected by Viscount Haldane's ruling in _Amodu Tijani v. Southern Nigeria_ that argued that Indigenous land tenure systems should not be judged by British standards, but rather recognized on their own terms.\n\nThe Allied Tribes of British Columbia recognized the Haldane decision as favourable to their cause and sent a formal petition to the Canadian Parliament in 1925 asking that a committee be struck to initiate the process of bringing their claim to the Judicial Committee of the Privy Council. A Special Joint Committee of the Senate and House of Commons was appointed by the government of Canada in 1926 to enquire into the Petition of the Allied Tribes. Although invited, the provincial government of British Columbia declined to participate or to send observers. The committee included H. H. Stevens, Conservative MP for Vancouver Centre; Minister of Indian Affairs Charles Stewart; future prime minister of Canada, R. B. Bennett; and four Senators from British Columbia. The majority-Liberal committee began hearings in March 1927.\n\nThe question of documentation was contentious. O'Meara presented quotations from instructions sent by the Colonial office to Douglas that demonstrated that the British Crown recognized Aboriginal title in B.C. The Chairman of the Committee intervened, saying that O'Meara could not quote from documents not in evidence. Aboriginal representative Andrew Paull interrupted and explained that Duncan Campbell Scott had refused to give him access to the _Papers Connected with the Indian Land Question_ , the same documents Trutch had withheld from the Indian Reserve Commission, in which could be found all the evidence referred to. Stevens had with him in the room that day _The Papers_ , but refused to allow them to be entered into evidence. The collection was his private property, he argued, and he feared losing it. Scott said the Department of Indian Affairs had only one copy and also would not risk entering it into evidence for fear the court would misplace it. Finally, O'Meara was allowed to read sections of _The Papers_ into the record.\n\nThe Committee interpreted the testimony of two traditional Chiefs from the interior of B.C. who, being neither English speakers nor knowledgeable in the language of legal disputes, had not used the words \"aboriginal title\" in their presentation, as evidence that these Chiefs did not believe they held title. Simultaneously, the Committee reasoned that, because the articulate, English-speaking, Aboriginal representatives had not presented proof of the antiquity of their title through their own oral tradition, their claim was obviously of recent invention and unduly influenced by whites, as evidenced by the use of legal language like the term \"Aboriginal title.\" The Committee concluded that: \"Tradition forms so large a part of Indian mentality that if, in pre-Confederation days, the Indians considered they had an aboriginal title to the lands of the Province, there would have been tribal records of such being transmitted from father to son, either by word of mouth or in some other customary way. But nothing of the kind was shown to exist.\"\n\nThe Committee rejected all the claims of the Allied Tribes, and set out seven arguments in support of their position. First, without commenting on preexisting title, they argued that the assertion of British sovereignty was itself evidence that no prior title had been acknowledged or could continue. Second, the Committee claimed that the Hudson's Bay Company had achieved the \"conquest\" of B.C. Third, they pointed out that all Indians were not in agreement with the claim since all did not belong to the Allied Tribes organization. Fourth, the committee declared that the Aboriginal title claim was only fifteen years old, since that was when it was first articulated as a legal claim. The committee ignored, therefore, the ample documentation of resistance through civil disobedience, violent and non-violent confrontation, petitioning, letter-writing and Indian appearances before various committees of inquiry. Fifth, they found that Indians had implicitly consented to the denial of Aboriginal title by their acceptance of government reserve policies, which the committee said \"they accepted for years without demur.\" Sixth, the committee blamed \"mischievous white agitators\" for Aboriginal discontent. Seventh, they chastised the Indians for rejecting the findings of the Indian Reserve Commission and the McKenna-McBride Commission and for continuing to \"take up the time of the government and Parliament with irrelevant issues.\"\n\nThe Parliamentary Committee did not recommend that the Allied Tribes claim be forwarded to the Judicial Committee of the Privy Council. Instead, they made two alternative proposals. First, that an annual allotment of $100,000.00 should be made to British Columbia Indians in lieu of treaty payments. Second, they recommended an amendment to the _Indian Act_ , Section 141, that would criminalize organizing for political or legal recognition of Aboriginal title in B.C. The official explanation, and the one accepted by both federal and provincial governments, was that this clause was necessary to protect \"simple,\" impressionable Indians from exploitation by cunning white lawyers like O'Meara. However, the language of Section 141 did not specify that it was applicable to non-Indians only, and, in effect, the amendment made it illegal for any organization to exist if pursuing recognition of British Columbia First Nations' Aboriginal title and rights was one of its objectives. Paternalistic rationales have served to exclude Section 141 of the _Indian Act_ , and Canada, from inclusion in the usual list of totalitarian colonial regimes who outlaw opposition, and to thus perpetuate the image of Canada as a liberal, democratic, humanitarian colonial power.\n\nAmong British Columbia First Nations the amendment is remembered bitterly, and mention is often made of it in discussions of First Nations' political history. In Indian memories, section 141 is usually linked with the potlatch prohibition, and the combination of the two produces the still common belief, which presumably existed from 1927 until 1951 as well, that any gathering of Indians or any discussion of land claims was illegal without the permission of a missionary, Indian agent, or police official. Until this section was amended in 1951, no such public, legal activity did take place. First Nations throughout British Columbia, however, continued to feast and potlatch clandestinely, and to talk about Aboriginal title and rights.\n\n**Resistance versus Assimilation**\n\n_Where there is resistance, there is power._\n\n\u2014Anthropologist Lila Abu-Lughod, 1990, _The Romance of Resistance_.\n\nThe decline of the British Columbia First Nations' population reached its lowest point of 22,000 in 1929. Throughout the 1930s and 1940s numbers grew slowly, and Aboriginal peoples continued to organize politically in various ways in an attempt to better their lot, achieve the right to vote, and survive such travesties as the advent of residential schools administered by various churches and the Department of Indian Affairs. The _Indian Act_ made failing to attend residential schools a criminal act, and provided for legalized punishment of Native parents who refused to comply with church and government officials who demanded they turn their children over to their care. The explicit goal of the residential school policy was to break the bonds between generations, thus \"freeing\" the young from the shackles of tradition and the influence of their families. For the most part, graduates of the residential schools did not assimilate into Canadian society as the policy predicted they would. Many had no desire to, and others who tried found the doors closed to them. At the same time, when they returned to their home villages they often found they had lost both the ability to communicate fluently with parents and grandparents and the practical, as well as social, skills necessary to live from the land and to fit into village life.\n\nThe post-World War II era is noted by many analysts as a turning point in Aboriginal\/non-Aboriginal relations. Following the defeat of Hitler and the revelations of Nazi atrocities, the western world entered a period of racial liberalism. Canada's image of itself, both nationally and internationally, as a free, democratic country where all citizens are equal regardless of race, religion or origin, was marred by the existence within its borders of a category of people whose legal, political and social identity had been racialized by the _Indian Act_ and the _Constitution Act_ (1867). The poverty and marginalization of Indians became a source of public embarrassment and efforts began to be made to improve living conditions on reserves and, once again, to assimilate Indians into mainstream Canadian society.\n\nIn 1947 British Columbia Indians were allowed to vote in provincial elections, and in 1949 Nishga Hereditary Chief Frank Calder, who would some twenty years later become \"the Plaintiff\" in _Calder v. R._ , was elected as Canadian Commonwealth Federation (CCF) Member of the Legislative Assembly for Atlin, British Columbia, becoming the first Indian elected to any post-Confederation Canadian legislature. The federal vote was not extended to registered Indians until 1961.\n\nIn 1949 the Judicial Committee of the Privy Council was replaced by the Supreme Court of Canada as Canada's highest court of appeal. Although Canadian judges were still free to use decisions of the Judicial Committee as precedents, \"conventional wisdom among judges, lawyers, academics and government officials held that aboriginal rights were both insignificant and irrelevant.\" Since Indians, publicly complied with the legislated banning of land claims activity imposed in 1927, the powers that be assumed that they, too, had forgotten about it. In a positivistic world, what a man cannot see does not exist.\n\nAmendments to the _Indian Act_ in 1951 dropped from the books\u2014but did not repeal or acknowledge as wrong\u2014both Section 140 prohibiting potlatching, and Section 141, prohibiting the pursuit of legal recognition of Aboriginal title and rights. Further changes to the _Act_ allowed Indians, for the first time since 1884, to consume alcohol in public places, but not in their own homes. This circumscribed, but did not undermine, the lucrative \"informal economy\" of bootlegging alcohol to Indians which constituted a launching padfor many a settler family's climb up British Columbia's social ladder. The alcohol amendments to the _Indian Act_ in 1951, like the ending of prohibition in Canada in the 1930s, moved some of the profits of the liquor trade from the unregulated, untaxed, illegal sphere into the regulated, taxable coffers of hotel and bar owners. In 1960, all restrictions on Indian purchase and consumption of alcohol were lifted, and provincial government-owned liquor stores opened adjacent to many reserves, further centralizing both profit, taxes and policing. The revised _Indian Act_ that emerged after 1951 also revoked the provision that stipulated that any registered Indian who received a university degree would automatically lose their legal Indian status, reflecting a disavowal of the evolutionist notion that education and Indianness were necessarily incompatible. And, the 1951 amendments allowed First Nations women to participate in band council elections for the first time.\n\nPotlatches, feasts and traditional dances were held again, in public, in numerous villages beginning in 1953. Paul Tennant marks the 1959 Native Brotherhood of British Columbia (NBBC) Convention as the first significant occasion since the 1927 banning of political activity on the land question where B.C. Aboriginal peoples had the opportunity to gather and openly discuss pursuing legal actions once again. He argues that the leaders who organized this convention had two goals in mind: a short term objective focused on consolidating the NBBC as a provincial organization to represent all status Indians in B.C. to a parliamentary committee; and a long term objective of pursuing the legal recognition of Aboriginal title.\n\nOn the political front, the provincial government of British Columbia continued to refuse to recognize the existence of any Aboriginal rights or title, while simultaneously denying health, education and social services to registered Indians, arguing that they were a federal responsibility for which the province received no financial support or taxes. They advocated the elimination of \"special status,\" and the legal, political, social and cultural assimilation of Aboriginal peoples\u2014and their lands and resources\u2014into Canadian society. In the mid 1960s, the federal government began a series of consultations with selected Aboriginal leaders. Despite the consistent stand taken by these representatives at these consultations\u2014that is, that the recognition of Aboriginal title and rights must be justly dealt with before any honourable negotiations could take place\u2014 Prime Minister Trudeau and Indian Affairs Minister Chretien produced the 1969 White Paper Policy that brushed aside the issue of land and resource ownership and provided a blueprint for assimilation.\n\nEvidence about the period between 1927 and 1984 was heard during the trial. Gitksan and Wet'suwet'en and academic, expert witnesses, recounted a narrative of resistance and survival. They discussed the impact of various plans, like the residential school policy, on everyday life, family relations and individual psyches. They talked about the legacy of alcoholism, drug abuse and violence in their communities. They talked about economic marginalization resulting from increased restrictions on their access to their lands and resources, about their experiences of expropriation and dislocation as more and more of their land was turned over to settlers and corporations, and about the daily struggle to survive and to defend their rights to exist as distinct peoples. They argued that they had changed, but that they had neither vanished, assimilated, nor \"evolved\" into \"not truly aboriginal\" peoples.\n\nThe Crown presented a counter narrative of acquiescence and assimilation. Their strategy began with efforts to show that the witnesses were unreliable and lacking in knowledge. Geoff Plant, lawyer for the Province of British Columbia, and currently a Liberal Member of the Legislative Assembly in British Columbia, cross-examined an elderly woman about the number of residents in her home village, trying to challenge the accuracy of her testimony, and her credibility as a witness:\n\nPlant: Do you know how many members there are on the band list?\n\nGwaans: No.\n\nPlant: Do you know how many people live on the reserve?\n\nGwaans: No.\n\nPlant: Do you have an approximate idea of how many people live on the reserve?\n\nGwaans: No.\n\nPlant: Is it in the order of hundreds of thousands, tens of thousands of people? Gwaans: No.\n\nPlant continued his interrogation, raising the issue of assimilation and relying on a version of what has come to be called the \"pizza test\" for evidence. The \"pizza test\" refers to what has become Crown Counsels' stock in trade argument in land rights trials that \"proof\" of assimilation\u2014and hence ineligibility for consideration of distinct rights\u2014can be achieved by showing that Aboriginal people often work for wages, are members of Christian congregations, use \"western\" technology, attend public schools and consume \"white food.\" Following this line of reasoning, it has become commonplace for Aboriginal witnesses to be questioned about how many times they have eaten Kentucky Fried Chicken, Big Macs and pizzas, for example. Geoff Plant pursued this line of questioning in his cross-examination of Gwaans:\n\nPlant: Is there electric lights on the reserve?\n\nGwaans: Are you going to pay the bill?\n\nPlant: Do you pay the bill, Mrs. Ryan?\n\nGwaans: Yes, I did....\n\n... Plant: So far as you know, do the members of the Band who live on the Reserve, do some of them own automobiles and trucks?\n\nGwaans: Well, I seen some cars there, but I didn't ask.\n\nPlant: You have seen some cars on the reserve?\n\nGwaans: Well, they will call me nosey if I ask them, the people there.\n\nPlant: You have seen people who live on the reserve drive cars?\n\nGwaans: Oh, yes.... Plant: And there is a school on the reserve?\n\nGwaans: Yes.\n\nPlant: Is there a church on the reserve?\n\nGwaans: Yes, two churches. Salvation Army and United Church.\n\nPlant: Which church do you go to?\n\nGwaans: Salvation Army.\n\nHaving shown, to their satisfaction, that the Gitksan and Wet'suwet'en have assimilated into mainstream Canadian society, the Crown called further witnesses to testify to the beneficial achievements of government assimilation policies. Former Department of Indian and Northern Affairs official Mr. McIntyre was called by the Crown as an expert witness. Chief Justice McEachern himself intervened in McIntyre's testimony. Although McIntyre was not a social scientist of any description, the Chief Justice asked him to evaluate the social consequences of the policies McIntyre himself designed and administered, and to analyze their value in the context of other variables:\n\nMcIntyre:...in my opinion I would say that the biggest\u2014that the biggest problem facing these people today is one of lack of economic opportunity, and I think if they had a\u2014if they had an improved economic circumstance that many of their\u2014of their social problems might be lessened.\n\nCourt: I have heard in this evidence\u2014in this case evidence about employment, which I gather would be included within your category of economic opportunity. I have heard about education, housing, alcohol and drugs, health, gambling. I dare say there are others. Could you rate them as degrees of seriousness, or is that a reasonable request to make?\n\nMcIntyre:...I observe what appears to be considerable improvement of that. For instance, I see a number of good quality homes that have sprung up. I see\u2014I see community halls and recreation facilities that have come into existence. I am aware that band councils are apparently taking on greater responsibility.\n\nOf course, widespread poverty and unemployment are serious impediments to health and happiness and community well-being. However, the Crown's argument is that assimilation into the existing economy on terms dictated by the dominant society, and not on the basis of recognition of Aboriginal title and self-government rights, is the most desirable route for First Nations to take.\n\nFinally, in June 1990, the proceedings in _Delgamuukw v. R._ drew to a close. Plaintiffs, defendants, lawyers, and witnesses made their closing arguments, filed the last of their exhibits, packed up their briefcases, and settled in to await Chief Justice McEachern's decision. What would the judge make of all he had heard, read, seen and felt over the preceding four years?\n\n* * *\n\n. Gisday Wa and Delgamuukw 1987: 11.\n\n. Williams 1987.\n\n. See Dyck 1991.\n\n. \"Nativist\" religious movements were those, like the Ghost Dance, that sought to maintain, revive or revitalize religious beliefs rooted in pre-contact Indigenous cosmologies. \"Syncretist\" religious movements were those that sought to incorporate elements of Christianity into Indigenous belief systems that resulted in various fusions and combinations of religious beliefs.\n\n. Tennant 1990: 53.\n\n. Tennant 1990: 55.\n\n. Cited in Berger, 1981: 58.\n\n. Provincial Premier Smithe Province of British Columbia, _Session Papers_ , 1887: 264; quoted in Tennant, 1990: 58.\n\n. _St. Catherine's Milling and Lumber Co. v. R._ (1885) 10 OR or 196 (Ont.Ch.); (1886) 13 OAR 148 (ont.CA0); (1887) 13 SCR 577 (SCC); (1888) 14 AC 46 (PC).\n\n. Sanders 1986; Tennant 1990.\n\n. Sanders, Douglas 1992: 295.\n\n. Morris 1973:39, quoted in Asch 1984: 62.\n\n. Dyck 1991; Paine 1977.\n\n. Williams D. R. 1987: 48.\n\n. Ray 1987: 93-94.\n\n. Galois 1987: 22.\n\n. Williams D. R.1987: 7.\n\n. The historical account in the remainder of this chapter relies substantially on Tennant 1990.\n\n. Tennant 1990: 105-107.\n\n. Tennant 1990: 106-108.\n\n. Tennant 1990.\n\n. Canada (1927): viii, quoted in Tennant 1990: 110.\n\n. Ibid., 109-113.\n\n. Acheson 1995: 11.\n\n. Brody 1983; Haig-Brown 1988; LaViolette 1973; Tennant 1990.\n\n. Tennant 1990: 218.\n\n. Tennant 1990: 129.\n\n. Quoted in Monet and Wilson 1992: 45.\n\n. Williamson 1989.\n\n. See Richardson 1975 for an early example of this argument in the James Bay Cree and Hydro Quebec trials.\n\n. These responses reflect the meticulous concern with the accuracy and truth of spoken statements, with a speaker's responsibility to only say what they know, and not to overstep their authority, that is characteristic of Elders in cultures based in oral tradition. The response of a Cree hunter in 1974, while being sworn in as a witness to the court, has become famous as the quintessential statement of Aboriginal concepts of truth-telling. The hunter said he couldn't swear to \"tell the truth, the whole truth, and nothing but the truth,\" but could only promise to tell what he knew. (See Richardson 1975.) Many people, Aboriginal and non-Aboriginal, do not own their cars outright but are buying them gradually through payments. Therefore, Gwaans truly does not know whether the cars are owned or not, and does not think she has the authority to inquire. Her responses also evidence a different approach to the issue of property. Ownership is an important question that must be appropriately investigated, precisely proven, and therefore cannot simply be assumed. Gwaans appears to be saying that, just because people _occupy_ and _use_ these cars, they don't necessarily own them. Possession does not appear to be nine-tenths of Gitksan or Wet'suwet'en property law.\n\n. Ibid., 46.\n\n. Quoted in _Reasons_ 1991: 185.\n**P ART V:**\n\n**D IES IRAE**\n\n**(J UDGMENT DAY)**\n**Chapter 16: Reasons for Judgment**\n\n**Nasty, British (Columbian), and Long**\n\nA beginning, as Edward Said said, \"is the first step in the intentional construction of meaning.\" The Chief Justice's 394-page, single-spaced _Reasons for Judgment_ began:\n\nThis has been a long trial.\n\nAfter numerous pre-trial proceedings, including taking the commission evidence of many elderly plaintiffs, interlocutory applications and appeals, the trail began in Smithers on May 11, 1987.\n\nAfter 318 days of evidence, mainly at Vancouver but partly at Smithers, the evidence was substantially completed on February 7, 1990.\n\nLegal argument began in Smithers on April 2, 1990 and continued there for 18 days. Argument continued in Vancouver for a further 38 days, and the trial ended there on June 30, 1990.\n\nA total of 61 witnesses gave evidence at trial, many using translators from their native Gitksan or Wet'suwet'en language; \"Word Spellers\" to assist the Official Reporters were required for many witnesses; a further 15 witnesses gave their evidence on Commission; 53 Territorial Affidavits were filed; 30 deponents were cross-examined out of Court; there are 23,503 pages of transcript evidence at trial; 5,898 pages of transcript of arguments; 3,039 pages of commission evidence and 2,553 pages of cross examination on affidavits (all evidence and oral arguments are conveniently preserved in hard copy and on diskettes); about 9,200 exhibits were filed at trial compiling, I estimate, well over 50,000 pages; the plaintiffs' draft outline of arguments comprises 3,250 pages, the province's 1,975 pages, and Canada's over 1,000 pages; there are 5,977 pages of transcript of argument in hard copy and on diskettes. All parties filed some excerpts from the exhibits they referred to in argument. The province alone submitted 28 huge binders of such documents. At least 15 binders of Reply Argument were left with me during that stage of the trial.\n\nThe Plaintiffs filed 23 large binders of authorities. The province supplemented this with 8 additional volumes, and Canada added 1 volume along with several other recent authorities which had not then been reported.... Needless to say, this judgment has been a difficult one to prepare....\n\nThe plaintiffs, the aboriginal people who now live in parts of the territory I shall describe, sincerely believe that they own and have a legal right to govern this vast territory by reason of long use and possession. They harbour a great sense of injustice and resentment that they have waited so long for their aboriginal interests in the territory to be decided while non-natives have acquired title to much of this land, and while its resources have been exploited by others. The plaintiffs believe, passionately, that their claims are just.\n\nIt is common, when one thinks of Indian land claims, to think of Indians living off the land in pristine wilderness. Such would not be an accurate representation of the present life-style of the great majority of the Gitksan and Wet'suwet'en people who, while possibly maintaining minimal contact with individual territories, have largely moved into the villages. Many of the few who still trap are usually able to drive to their traplines, and return home each night.\n\nSimilarly, it would not be accurate to assume that even pre-contact existence in the territory was in the least bit idyllic...there is no doubt, to quote Hobbs, that aboriginal life in the territory was, at best, 'nasty, brutish and short.'\n\nThe Chief Justice misspelled the eighteenth-century British philosopher's name, Hobbes. McEachern is referring to Hobbes' speculation that there was a time in human history when people lived \"in a state of nature,\" that he described as a \"war of all against all\" in which life was \"nasty, brutish and short.\" As was the fashion of his times, Hobbes hypothesized that contemporaneous Indigenous peoples of the Americas lived at this \"stage of development.\" Given that, at the time he was writing, little information or knowledge about Indigenous peoples was available, even the most conservative contemporary \"Hobbesian\" political philosophers do not interpret Hobbes' description of Indigenous life literally. Rather, it is understood as a metaphor for what a society ruled by brute force and unmitigated individual ambition, rather than law and reason, might be like.\n\n**The Vast Emptiness**\n\nThe Chief Justice then proceeded to identify and describe what had been in dispute at the trial, and what was at stake in his judgment. He wrote:\n\nThis action is mainly about land, 22,000 square miles of it (58,000 square kilometres), which I shall call 'the territory'.... The total territory is a vast, almost empty area except in the Highway 16-C.N.R. corridor where most of the plaintiffs' villages are located.... In addition to about 5,000 to 7,000\n\nGitksan and Wet'suwet'en persons...there are upwards of 30,000 others (mostly of European extraction), who are living within the territory....\n\nThe territory is a rich agricultural area...and there are vast forestry resources throughout much of the territory. Equally important are the salmon and other fisheries of the Bulkley, Nass, Skeena and Babine Rivers. Most of the invaluable and irreplaceable Skeena salmon stock pass through the territory by way of the Skeena and Babine Rivers to their destiny in the spawning grounds of Babine Lake.\n\nAlthough some game animals may not be as plentiful in the transportation corridor as they were at the time of European contact, or before the railway was built, I am not persuaded there is either a shortage or an excessive abundance of wildlife in the territory.... I do not suggest that clear cut logging has been an ecological advantage to the territory. That is for other disciplines to ponder and to weigh against economics. Aesthetically, Dr. Hatler's description of 'moonscape' is appropriate. I was encouraged to notice on my travels through the territory that areas logged as recently as 3 to 6 years ago are starting to show signs of regeneration....\n\nThere are some mining resources in the territory, including the Equity Silver Mine which is reported to have limited remaining ore reserves, but mining is not currently significant when compared with agriculture and forestry which are the economic mainstays of the region. There are, unquestionably, immense forestry reserves throughout the territory which are of great economic value.\n\nThe trees, fish, game and minerals constituted only inanimate things defined by their potential economic value. The Chief Justice relived in his _Reasons for Judgment_ the visionary moment that generations of his judicial ancestors had experienced before him: Chief Justice McEachern looked and saw before him in British Columbia, _terra nullius_ :\n\nThe most striking thing that one notices in the territory away from the Skeena-Bulkley corridor is its emptiness. I generally accept the evidence of...[Crown] witnesses such as Dr. Steciw and Mrs. Peden and others that very few Indians are to be seen anywhere except in the large river corridors. As I have mentioned, the territory is, indeed, a vast emptiness....\n\n**Might Makes Right**\n\nHaving beheld _terra nullius_ , the Chief Justice continued his textual reenactment of law's narrative:\n\nI have heard much at this trial about beliefs, feelings, and justice. I must again say, as I endeavoured to say during the trial, that Courts of law are frequently unable to respond to these subjective considerations. When plaintiffs bring legal proceedings, as these plaintiffs have, they must understand (as I believe they do), that our Courts are Courts of law which labour under disciplines which do not always permit judges to do what they might subjectively think (or feel) might be the right or just thing to do in a particular case. Nor can judges impose politically sensitive non-legal solutions on the parties. That is what Legislatures do, and judges should leave such matters to them.\n\nInstead, cases must be decided on admissible evidence, according to law. The plaintiffs carry the burden of proving by a balance of probabilities not what they believe, although that is sometimes a relevant consideration, but rather facts which permit the application of the legal principles which they assert. The Court is not free to do whatever it wishes. Judges, like everyone else, must follow the law as they understand it....\n\nI am sure that the plaintiffs understand that although the aboriginal laws which they recognize could be relevant on some issues, I must decide this case only according to what they call 'the white man's law.'\n\nI now propose to discuss the law relating to the question of extinguishment of aboriginal interests.... I must start with the proposition that the plaintiff's aboriginal rights in the territory at the time of sovereignty existed at the 'pleasure of the crown.'\n\nAs the mortal embodiment of the hovering sovereign, Allan McEachern affirmed Britain's declaration of sovereignty and simultaneously extinguished Aboriginal title. His decision on how extinguishment was legally effected was, ultimately, the single, significant finding in his entire judgment in _Delgamuukw v. R._ McEachern ruled that the simple assertion of British sovereignty was sufficient to extinguish any Aboriginal title or rights that may have existed. It was simply common sense, the judge argued, to acknowledge the impossibility of coexistence between the \"civilized\" British, and the \"primitive\" First Nations. Chief Justice McEachern cited U.S. Chief Justice Marshall's early nineteenth-century precedent in support of his ruling:\n\nThe underlying purpose of exploration, discovery and occupation of the new world, and of sovereignty, was the spread of European civilization through settlement. For that reason the law never recognized that the settlement of new lands depended upon the consent of the Indians. So early as the year 1496, her monarch granted a commission to the Cabots, to discover countries then unknown to Christian people, and to take possession of them in the name of the King of England. Two years afterwards, Cabot proceeded on this voyage, and discovered the continent of North America, along which he sailed as far south as Virginia. To this discovery the English trace their title.\n\nMcEachern concluded that it was obvious from the _outcome_ of the last 200 years of history that the Crown had never ceased believing in its right to extinguish Aboriginal title and rights, and this itself was further evidence of the rightness of the initial assertion of sovereignty. Simply put, \"in the beginning,\" Britain exercised its absolute prerogative to assert sovereignty. Subsequently, British, Canadian and American colonial courts consistently upheld this initial assertion, and colonial governments faithfully enacted legislation that took for granted the legitimacy of Crown sovereignty, and simply ignored any and all challenges brought forward by First Nations since contact. Each court's repetition of these principles, according to the law, resulted in a strengthening of their validity. In other words, in a court of law, the more often a lie is told, and more times a liar escapes reprimand, the closer the law comes to finding legal truth.\n\nThe Chief Justice explained his ruling:\n\nI think it unnecessary to continue this debate. In my view, it is part of the law of nations, which has become part of the common law, that discovery and occupation of the lands of this continent by European nations, or occupation and settlement, gave rise to a right of sovereignty. Such sovereignty in North America was established in part by Royal grant as with the Hudson's Bay Company in 1670; by conquest, as in Quebec in 1759; by treaty with other sovereign nations, as with the United States settling the international border; by occupation, as in many parts of Canada, particularly the prairies and British Columbia; and partly by the exercise of sovereignty by the British Crown in British Columbia though the creation of Crown Colonies on Vancouver Island and the mainland.\n\nIn support of this assertion, the Chief Justice quoted at length from the work of the nineteenth-century Swiss theorist, Vattel:\n\n'There is another celebrated question to which the discovery of the new world has principally given rise. It is asked whether a nation may lawfully take possession of some part of a vast country in which there are none but erratic nations, whose scanty population is incapable of occupying the whole? We have already observed, in establishing the obligation to cultivate the earth, that these nations cannot exclusively appropriate to themselves more land than they have occasion for, or more than they are able to settle and cultivate. Their unsettled habitation in those immense regions, cannot be accounted a true and legal possession, and the people of Europe, too closely pent up at home, finding land of which the Savage stood in no particular need, and of which they made no actual and constant use, were lawfully entitled to take possession of it and to settle in with Colonies. The earth, as we have already observed belongs to mankind in general, and was designed to furnish them with subsistence. If each nation had from the beginning resolved to appropriate to itself a vast country, that the people might live only by hunting, fishing and wild fruits, our globe would not be sufficient to maintain a tenth part of its present inhabitants. We do not, therefore, deviate from the views of nature, in confining the Indians within narrower limits.\n\nFurthermore, the Chief Justice decreed, it was not legally necessary to find historical or legal evidence of the British Crown having expressly announced its \"clear and plain intention\" to extinguish Aboriginal title, as the Supreme Court of Canada's ruling in the _Sparrow_ case had required. Rather, Chief Justice Allan McEachern offered his own unique interpretation of this precedent:\n\n...the governing factor in extinguishment persuades me that intention in this context must relate not to a specific, isolated intention...[or to] a specific or precise state of mind on the part of the historical actors, but rather to the consequences they intended for their actions.... I therefore conclude that express statutory language is not a requirement for extinguishment.\n\nIn other words, the question is not did the Crown through its officers specifically intend to extinguish aboriginal rights...but rather did they plainly and clearly demonstrate an intention to create a legal regime from which it is necessary to infer that aboriginal interests were in fact extinguished.... Intention, in this context, must be a matter of implica-tion...[ from] an amalgam of thought, belief, planning and intention on the part of a number of officials who may all have had different knowledge, understanding and priorities. There was indeed an intention to manage Crown lands...[during the colonial period] throughout the colony by a system that was inconsistent with continuing aboriginal rights.\n\nTo support his finding, the Chief Justice referred to the copious evidence presented by the Crown that demonstrated that the Province of British Columbia had, in fact, treated Aboriginal title and people as if they didn't exist, at least since 1871:\n\n[The defendants in this case]...the province of British Columbia... introduced a large collection of documents which...record pervasive colonial and provincial Crown presence in the territory up to the date of the writ (1984)....\n\nAll this documentation demonstrates colonial and provincial dominion over the territory before and since Confederation by such diverse governmental and administrative activities as surveying, grants of land, leases and other tenures, land registry, schools and hospitals, rights of way for highways, power and pipe lines, grants in fee simple, forestry, mining, and guide outfitting permits, various public works, the creation and governance of villages and municipalities, water and other placer rights and licences, trapline registration for all or almost all of the territory, fish and game regulation and conservation and a host of other legislatively authorized intrusions in the life and geography of the territory....\n\nFor its part, Canada also adduced extensive evidence of the federal presence in the territory....\n\nAnd, McEachern reiterated the cornerstone of the Crown's argument, and of his reasoning: the possibility of mutually-respectful co-existence between Aboriginal and non-Aboriginal property regimes is an absurdity not worthy of serious consideration:\n\nThe province argues that it cannot rationally be asserted that, standing alongside this all-embracing structure, there are many parallel aboriginal governments and separate systems of ownership or rights to use a substantial portion of the province by what is now such a small segment of the population.\n\nHaving asserted the British Crown's unfettered right to assert sovereignty, the Chief Justice went on to legitimate the arbitrariness of this exercise of power even more, by arguing that the date at which sovereignty may have been first declared was also irrelevant:\n\nBecause of the view I have of this case, I do not think it is necessary to make a specific finding about a date of British sovereignty over the northern part of the province.... For practical purposes, especially in the territory it could well have been as early as the 1820's but legally it may not have been until the creation of the colony in 1858. 1846 was the date chosen by Judson J. in _Calder_. In my view the actual date of British sovereignty, whether it be the earliest date of 1803 or the latest date of 1858, or somewhere in between makes no difference...the Crown wished the island settled by British settlers...the establishment of the Colony of British Columbia, for example, should be classified as a displacement of one sovereignty by a different one which the law recognizes....\n\n...I do not believe there is any material difference, for the purposes of this case, between the date of contact and the date of sovereignty....\n\nFor my purposes it is sufficient to start with European expansion into this continent after the voyages of various navigators such as Columbus (1492), and Cabot, who according to Chief Justice Marshall discovered North America in 1498....\n\nI am satisfied that at the date of British sovereignty the plaintiffs ancestors were living in their villages on the great rivers in a form of communal society, occupying or using fishing sites and adjacent lands as their ancestors had done for the purpose of hunting and gathering whatever they required for sustenance. They governed themselves in their villages and immediately surrounding areas to the extent necessary for communal living.\n\nThe judge went to some lengths to explain in no uncertain terms what he meant when he said that _whatever_ may or may not have taken place historically was irrelevant to his judgment:\n\nAboriginal persons and commentators often mention the fact that the Indians of this province were never conquered by force of arms, nor have they entered into treaties with the Crown. Unfair as it may seem to Indians or others on philosophical grounds, these are not relevant considerations. The events of the last 200 years are far more significant than any military conquest or treaties would have been. The reality of Crown ownership of the soil of all the lands of the province is not open to question and actual dominion for such a long period is far more pervasive than the outcome of a battle or a war would ever be. The law recognizes Crown ownership of the territory in a federal state now known as Canada pursuant to its Constitution and laws.\n\nI fully understand the plaintiffs' wishful belief that their distinctive history entitles them to demand some form of constitutional independence from British Columbia. But neither this nor any Court has the jurisdiction to undo the establishment of the Colony, Confederation, or the constitutional arrangements which are now in place. Separate sovereignty or legislative authority, as a matter of law, is beyond the authority of any Court to award. I also understand the reasons why some aboriginal persons have spoken in strident and exaggerated terms about aboriginal ownership and sovereignty, and why they have asserted exemption from the laws of Canada and the province. They often refer to the fact that they were never conquered by military force. With respect, that is not a relevant consideration at this late date if it ever was. Similarly, the absence of treaties does not change the fact that Canadian and British Columbian sovereignty is a legal reality recognized both by the law of nations and by this Court.... In the language of the street, and in the contemplation of the law, the plaintiffs are subject to the same law and the same Constitution as everyone else. This is not to say that some form of self-government for aboriginal persons cannot be arranged. That, however, is possible only with the agreement of both levels of government under appropriate, lawful legislation. It cannot be achieved by litigation.... It follows, therefore, that the plaintiffs' claims for aboriginal jurisdiction or sovereignty over, and ownership of, the territory must be dismissed.\n\nIn my judgment, the foregoing propositions are absolute. The real question is whether, within that constitutional framework, the plaintiffs have any aboriginal interests which the law recognizes as a burden upon the title of the Crown.\n\nHaving confirmed that the absolute authority of the law is not bound by either place or time, the Chief Justice nevertheless took a moment to wander down a historical path:\n\nIn my view...what happened in the territory...[is] that the aboriginal system, to the extent it constituted aboriginal jurisdiction or sovereignty, or ownership apart from occupation for residence and use, gave way to a new colonial form of government which the law recognizes to the exclusion of all other systems....\n\nAfter that, aboriginal customs, to the extent they could be described as laws before the creation of the colony became customs which depended upon the willingness of the community to live and abide by them, but they ceased to have any force, as laws, within the colony.\n\nTo put it in a nutshell, I find that legislation passed in the colony and by the Imperial Parliament that all the land in the colony belonged to the Crown in fee, apart altogether from many other enactments, extinguished any possible right of ownership on the part of the Indians.\n\nTo enter the Courthouses and Houses of Parliament built on their land without their consent, to negotiate the ownership and use of that same land and its resources, Aboriginal peoples must, according to the law and government policy, agree to ultimately surrender their title and rights. They must agree to \"extinguishment.\" The alternative of co-existence, rather than domination by one and subordination of the other, is a possibility that cannot be argued within the Crown's interpretation of legal precedents, or negotiated within the federal government's extinguishment policy. In the legal forum, the process of surrender begins _before_ Aboriginal litigants are permitted to launch a legal challenge, with the demand that they acknowledge that the \"underlying title to all the land\" in Canada is held by the hovering sovereign. Of course, if they go to court to establish the legitimacy of their claim that the hovering sovereign's title is invalid in the terms of British and Canadian law, this demand creates yet another double-bind. If they don't go to court, then their claim will never be acknowledged, and provincial governments may not be compelled to recognize or negotiate with them. If they do go to court, they have to surrender this basis of their claim first. Chief Justice McEachern explained this dilemma succinctly:\n\nIn their pleadings and argument the plaintiffs admit that the underlying or radical or allodial title to the territory is in the Crown in Right of British Columbia. This reasonable admission was one which the plaintiffs could not avoid. It sets the legal basis for any discussion of title.... After much consideration, I am driven to find that jurisdiction and sovereignty are such absolute concepts that there is no half-way house.... The very fact that the plaintiffs recognize the underlying title of the Crown precludes them from denying the sovereignty that created such title.\n\nHeads, the Crown wins. Tails, Indians lose.\n\nA second significant legal ruling made by Chief Justice McEachern was his finding that the provincial government, as well as the federal government, have a fiduciary obligation to Aboriginal people. This has resulted in provincial governments having to institute consultation processes with First Nations before development takes place on what are designated their traditional lands.\n\nDespite his protestations to the contrary, the Chief Justice's legal findings ultimately rely on historical and cultural interpretations. Were British and Indigenous peoples considered equal human beings, then or now, what possible rationale could there be for Britain simply having the right to assert sovereignty over another people without their consent, and for subsequent colonial governments to continue to do so?\n\nIn what follows I have reconstructed the _Reasons for Judgment_ 's historical narrative by selecting excerpts from the text and organizing them according to a historical, chronological framework, in order to try to understand Chief Justice McEachern's \"world view,\" as this is expressed through his narrative of the history and culture of Gitksan and Wet'suwet'en First Nations, British Columbia, and Aboriginal\/non-Aboriginal relations in this province. I have concentrated on those historical and cultural interpretations upon which the legal findings of \"fact\" in McEachern's judgment rely most strongly to illustrate the culturally-specific underpinnings of the judge's reasoning, and the erroneous basis upon which his findings of \"legal facts\" depend. In so doing I have tried to be mindful of the context from which I have selected excerpts, and I have endeavoured not to intentionally misrepresent what the judge wrote. I have indicated by way of footnotes where the excerpts I have reproduced may be found in the original text of the _Reasons for Judgment_. I hope this will facilitate re-reading and re-interpretation by interested readers.\n\nThe text of Allan McEachern's _Reasons for Judgment_ in _Delgamuukw v. R._ , is divided into 22 sections that include commentary on a range of legal, historical and cultural issues. The Chief Justice visited and revisited particular epochs and events in different sections of the _Reasons for Judgment_ , in order to address various specific questions. An array of theories and diverse forms of argument are engaged by the judge. Frequently, the empirical evidence the Chief Justice draws on in his ruling is weak or absent, and the logics of interpretation he employs appear contradictory and inconsistent. The text makes sense only if the reader accepts its fundamental premises: First, that Aboriginal peoples are radically different from, _and inferior to_ , Euro-Canadians, and that therefore a humanistic logic of equality does not apply in this case. This is most clearly evident in the \"damned if you do and damned if you don't\" nature of the law on Aboriginal issues, illustrated in many examples where it is clear that _whatever_ Aboriginal people may do or say, the law has an interpretation waiting that judges can use, if it they so choose, to render that statement or practice evidence of justification to deny Aboriginal title and rights legal recognition. It goes without saying that judges also have the option of reconstructing a precedent narrative that _supports_ recognition of Aboriginal title and rights. Second, in order to make sense of Chief Justice McEachern's text, a reader must assume that the British Crown was justified, historically, in asserting its sovereignty over First Nations without their consent; and that prolonged occupation on these grounds provides justification for continued political, economic and cultural domination. The legal argument that expresses this desire claims that non-Aboriginal settlers may now claim to hold common law title on the basis of long term use and occupancy. Third, readers must temporarily suspend their belief in common sense empiricism and everyday logic, and go along with law's imagining that the reality it creates in its own image is the truth.\n\nMy reading and reconstruction of the Chief Justice's text is unashamedly a critical interpretation. I am presenting an argument that begins from certain specific premises: I assume all human beings are fundamentally equal and that this principle should be reflected in law. I think that _Reasons for Judgment_ that rely on historical and anthropological research should reflect contemporary research standards and knowledge in these fields. There are, and no doubt always will be, significant debates among academics on these topics. There are also, however, generally accepted standards by which courts _could_ evaluate the relative reliability of expert evidence offered by academics and other researchers. I believe also that the \"facts\" a judge \"finds\" should be demonstrable as such to the average citizen. I think judges' arguments should be internally coherent, not illogical, on their own terms. I believe it is unjust and immoral that the Crown asserts its sovereignty and jurisdiction over First Nations peoples, their lands and resources _without their consent and in the face of their objections._ I take exception to the state of which I am a citizen, Canada, being based on the principle of the inequality of peoples. I believe that Canadian law has historically defended and reproduced a way of life based on a foundation of European male supremacy, economic inequality, and social injustice, and continues to.\n\n**No Writing, Horses or Wheeled Wagons**\n\nHaving first reproduced law's narrative of discovery, sovereignty and extinguishment, the Chief Justice turned his attention to history. He began at the beginning:\n\nThe evidence does not disclose the beginnings of the Gitksan and Wet'suwet'en people. Many of them believe God gave this land to them atthe beginning of time. While I have every respect for their beliefs, there is no evidence to support such a theory and much good reason to doubt it.... Most scientists believe the ancestors of our aboriginal people migrated to this continent from Asia, probably after, but possibly before, the last great Ice Age.... It is my conclusion, doing the best I can without the assistance of very much evidence, that the plaintiffs' ancestors...migrated from Asia, probably through Alaska, but not necessarily across the Bering Straits, after the last Ice Age, and spread south and west into the areas which they found livable...it is unlikely, or at least not proven, that any or all of the plaintiff groups have occupied the territory for all of the time since these post-glacial migrations.... For the purpose of this judgment it really does not matter where the plaintiffs' earlier ancestors came from....\n\nAlthough McEachern's ruling that Britain had the right to assert sovereignty anywhere, and anytime, and in any circumstances rendered the remainder of his judgement legally redundant, he proceeded to evaluate and interpret all the evidence and arguments he had heard in the course of four years of hearings. In so doing, he was providing an interpretation of this material for future appeal court judges, whose general practice is to privilege the \"eye witness\" account of the first judge in a case like this one. McEachern explained his role:\n\nI have no doubt that what I am about to say will not be the last word on this case and that this judgment will be appealed to the Court of Appeal and perhaps to the Supreme Court of Canada. With this in mind I shall endeavour, for the assistance of the parties and the appeal process, to describe as best I can the facts and reasons upon which I have reached the conclusions I am about to state.\n\nAnd, although his ruling on the Crown's powers of extinguishment made consideration of any legal precedents unnecessary as well, Chief Justice McEachern acknowledged his judicial predecessors appropriately:\n\nAlthough not binding upon me but deserving deference, is the opinion of the Privy Council in _Re: Southern Rhodesia_.... The right of the Imperial Crown to proceed with the settlement and development of North America without aboriginal concurrence was confirmed by the Privy Council in the _St. Catherine's Milling_ case. This was expressed in practical terms by stating that 'Indian title' existed at the pleasure of the Crown.\n\nI think...there is much wisdom in the dictum of the Privy Council in _Re Southern Rhodesia_ :...[to wit] 'The estimation of the rights of aboriginal tribes is always inherently difficult. Some tribes are so low in the scale of social organization that their usages and conceptions of rights and duties are not to be reconciled with the institutions or the legal ideas of civilized society.'...I am quite unable to say there was much in the way of precontact social organization among the Gitksan and Wet'suwet'en simply because there is so little reliable evidence.\n\nThe evidence suggests that the Indians of the territory were, by historical standards, a primitive people without any form of writing, horses, or wheeled wagons. Peter Skene Ogden, the controversial trader-explorer, visited Hotset in 1836 and noted their primitive condition in his journal.\n\nIn 1919 Lord Sumner, in his ruling in _Re: Southern Rhodesia_ , had decreed that only those Indigenous property laws that were based on concepts of private ownership were sufficiently \"evolved\" to be recognized by British imperial law. In 1921 Viscount Haldane, in _Amodu Tijani v. The Secretary, Southern Nigeria_ had argued that many Indigenous land tenure systems were communally-based, as opposed to the British model of individual ownership in fee simple, and that such communal property regimes were worthy of respect by colonial law. McEachern concluded that Aboriginal title must, by definition, be communal, and he admonished the Gitksan and Wet'suwet'en Hereditary Chiefs for claiming a form of individual ownership:\n\nI do not understand the plaintiffs to allege or claim any 'people-wide' collective or communal ownership interest in any of the Gitksan or Wet'suwet'en territories, that is to say each chief claims ownership of specific territory or territories, and none of them claim any interest in any other territory.... I have already described the form of this action where some of the hereditary chiefs are advancing these claims for aboriginal interests on behalf of themselves or on behalf of their Houses or members. The authorities satisfy me that a claim for an aboriginal interest is a communal claim. Counsel for the Nishga in _Calder_ (at p. 352) described it as a 'tribal interest' and Hall J. (at pp 401-402) said it was a 'communal right.' In _Sparrow_ there are references to a collective rather than an individual, or sub-group interest.... The Crown's 'promise' of fair dealing must be classified as a communal or collective promise rather than separate or divided promises to a variety of individuals or sub-groups...the law cannot recognize discrete claims by small or sub groups within an aboriginal community.... The plaintiffs' case as pleaded, if established, could result in some Gitksan and Wet'suwet'en persons being treated substantially differently from other members of the larger aboriginal collective...any judgement to which they are entitled must be for the benefit of these peoples generally, and not piecemeal for the Hereditary Chiefs, their Houses, or their members.\n\nIt is evidence of \"our\" advanced civilization that \"we\" have the right to kick our family off our property, so \"they\" must be communal. Otherwise, \"they\" are insufficiently different. Of course, if \"they\" are communal, \"they\" are less evolved than \"we\" are.\n\nChief Justice McEachern reached back to the nineteenth-century theories of Sir Henry Maine when he dismissed Gitksan and Wet'suwet'en law as mere \"primitive custom\" that resulted from the need to minimally regulate \"instincts\"; as opposed to \"civilized law\" that presumably emerges from the institutional consensus of rational thinkers:\n\nIt became obvious during the course of the trial that what the Gitksan and Wet'suwet'en witnesses describe as law is really a most uncertain and highly flexible set of customs which are frequently not followed by the Indians themselves.... In my judgment, these rules are so flexible and uncertain that they cannot be classified as laws. I have no difficulty finding that the Gitksan and Wet'suwet'en people developed tribal customs and practices relating to chiefs, clans and marriage and things like that, but I am not persuaded their ancestors practised universal or even uniform customs relating to land outside the villages. They may well have developed a priority system for their principal fishing sites at village locations. I expect it is probably so...[that the Gitksan and Wet'suwet'en] exercised their spiritual beliefs within the territory, but the evidence does not establish that their beliefs were necessarily common to all the people or that they were universal practices. I suspect customs were probably more widely followed.\n\nThe plaintiffs have indeed maintained institutions but I am not persuaded all their present institutions were recognized by their ancestors.... I do not accept the ancestors 'on the ground' behaved as they did because of 'institutions.' Rather I find they more likely acted as they did because of survival instincts which varied from village to village. I do not question the importance of the feast system in the social organization of present-day Gitksan and I have no doubt it evolved from earlier practices but I have considerable doubt about how important a role it had in the management and allocation of lands, particularly after the start of the fur trade. I think not much....\n\nHaving ruled out the possibility that such \"primitive\" peoples may have entered into treaties with their neighbours regarding rights to travel across and harvest resources from certain territories, the judge declared that there must not have been any borders worthy of the name:\n\nThere seemed to be so many intrusions into the territory by other peoples that I cannot conclude the plaintiffs' ancestors actually maintained their boundaries or even their villages against invaders, although they usually resumed occupation of specific locations for obvious economic reasons....\n\nIn a turn of logic particularly curious for a judge who, after all, spends his time adjudicating breaches in the law, the Chief Justice interpreted testimony that some Gitksan and Wet'suwet'en appeared at times to break their own laws as evidence that these were not laws in the first place:\n\nWhile these are my findings, I am prepared to assume for the purpose of this part of my judgement that, in the legal and jurisdictional vacuum which existed prior to British sovereignty, the organization of these people was the only form of ownership and jurisdiction which existed in the areas of the villages. I would not make the same finding with respect to the rest of the territory.... In no sense could it be said that Gitksan or Wet'suwet'en law or title followed (or governed) these people except possibly in a social sense to the far reaches of the territory. Aboriginal life, in my view, was far from stable and it stretches credulity to believe that remote ancestors considered themselves bound to specific lands....\n\nThe Judge concluded his findings on the question by referring to the stronger, in his estimation, case presented by the Crown:\n\nThe defendants, on the other hand, point to the absence of any written history, wheeled vehicles, or beasts of burden, and suggest the Gitksan and Wet'suwet'en civilizations, if they qualify for that description, fall within a much lower, even primitive order. I have no doubt life in the territory was extremely difficult, and many of the badges of civilization, as we of European culture understand that term, were indeed absent.... It is my conclusion that Gitksan and Wet'suwet'en laws and customs are not sufficiently certain to permit a finding that they or their ancestors governed the territory according to aboriginal laws even though some Indians may well have chosen to follow local customs when it was convenient to do so.\n\nAnd, the defendants, of course, relied upon _their_ predecessors' accounts, which were validated by the Judge:\n\nIn July 1889 the Dominion...appointed an Agent to reside at Hazelton on what was called the Upper Skeena. The first incumbent was R. E. Loring who was then in the service of the province in the area. He seems to have been nominated for this position either by the Indians or by Mrs. Hankin, the widow of the first merchant at Hazelton whom Mr. Loring later married. She, being either an Indian woman or at least fluent in their language, acted as his translator. His many reports present a useful account of the Indians of the territory during his service there which continued until about 1920. In many cases, his reports present a far more realistic picture of what was happening on the ground than the careful language of government reports and diplomatic exchanges. Loring's 1889 reports describe a society in transit from what he regarded as 'heathen,' such as eating dogs and potlatching, and many disputes.\n\nAccounts based on the plaintiffs' ancestors' knowledge were not given the same credibility by Chief Justice McEachern:\n\nThe plaintiffs adduced a great deal of evidence directed towards establishing actual control of the territory.... The plaintiffs ask me to infer that the practices they describe were a continuation of long standing, pre-existing aboriginal ownership of and jurisdiction over territory. In fact, however, the plaintiffs seemed to have considerable difficulty with this claim for aboriginal sovereignty. Mr. Neil J. Sterritt is a Gitksan hereditary chief and a former President of the Gitksan-Wet'suwet'en Tribal Council. He is perhaps the most knowledgeable of the Gitksan chiefs on their claim in this action as he was, until 1988, involved directly with the preparation of the case for several years. In a brief submitted to the Penner Commission in 1983, Mr. Sterritt submitted: \"Now, I want to talk to you about the Indian government of the past.... I want to tell you that the feast hall was our seat of government. It filled a legislative and judiciary function. It taught us how and why to govern.... The feast filled many functions. One of the functions was settling disputes. It was a place to do something about succession, passing on what was being done, passing on property, passing on title....\" Brown's reports in the 1820s and Mr. Loring's reports, starting in about 1890, hardly mention the feast, particularly as a legislative body.... I am not persuaded that the feast has ever operated as a legislative institution in the regulation of land.\n\nI conclude...that there was indeed a rudimentary form of social organization in the Babine area, and it is reasonable to infer that similar levels of organization then existed in the territory. There is no reason to believe the neighbouring Indians of the territory had any lesser degree of social organization at the same time. I therefore infer that the ancestors of a reasonable number of the plaintiffs were present in parts of the territory for a long, long time prior to sovereignty.\n\n**Finders Keepers**\n\nThe Chief Justice was convinced by the Crown's argument that the Royal Proclamation was never intended to recognize Aboriginal title or rights, and that, since British Columbia was not visible on British maps drawn in 1763, it did not exist in the eyes of the Crown. Hence the Proclamation, Allan McEachern agreed, does not apply to this province:\n\nBy 1700 there was little, if any European influence in western North America. Even horses were unknown to the plains Indians until mid-century, or in the territory until the middle of the 1800's.... Most maps show north-west America, including the territory, as \"Terra Incognita\" or \"These parts Entirely Unknown.\" This state of relative ignorance about this part of the world remained that way for almost another century.\n\nThe British were anxious to keep peace on the frontier, and for this and other mercantile reasons, Britain caused a Royal Proclamation to be issued in 1763.... One of the most interesting parts of the evidence and argument in this case concerned this famous Proclamation which was issued by George III, on the advice of his Ministers, on October 7, 1763.... It is therefore with much hesitation, and with the greatest possible respect both to the witnesses and to counsel who expended so much skill and energy on this question, that I find myself able to dispose of it quite summarily....\n\nI have no doubt that, apart from setting up governments for the new colonies, the underlying purposes of the Proclamation were firstly to pacify the frontier for defensive or military purposes, and secondly to secure the markets of the North American colonies for the manufactured products of the mother country.... The tenor of the Proclamation in its historical setting clearly relates to the practical problems facing the Crown in its then American colonies.... Even the language of the Royal Proclamation, 1763, makes it plain that the Crown did not consider it necessary to obtain the consent of the Indians to exclude their interests. Although the Crown set aside vast areas for hunting grounds at the Crown's 'pleasure'....\n\nThe Crown had no connection with the Indian people west of the Rockies who owed the Crown, no actual or even notional allegiance, and were in no way under its protection.... There is nothing which persuades me that this Proclamation, either by is language or by the intention of the Crown, applies to the benefit of the plaintiffs or to the lands of present day British Columbia.\n\n* * *\n\n. Said 1975: 1.\n\n. _Reasons_ , 1991: 1.\n\n. _Reasons_ , 1991: 3.\n\n. _Reasons_ , 1991: 5.\n\n. _Reasons_ , 1991: 11.\n\n. _Reasons_ , 1991: 11.\n\n. _Reasons_ , 1991: 12.\n\n. _Reasons_ , 1991: 12.\n\n. _Reasons_ , 1991: 12.\n\n. _Reasons_ , 1991: 2.\n\n. _Reasons_ , 1991: 235.\n\n. _Reasons_ , 1991: 80.\n\n. _Reasons_ , 1991: 81.\n\n. _Reasons_ , 1991: 80.\n\n. _Reasons_ , 1991: 239; see also 241.\n\n. _Reasons_ , 1991: 233.\n\n. _Reasons_ , 1991: 233-234.\n\n. _Reasons_ , 1991: 241.\n\n. _Reasons_ , 1991: 212.\n\n. _Reasons_ , 1991: 19.\n\n. _Reasons_ , 1991: 81.\n\n. _Reasons_ , 1991: 225.\n\n. _Reasons_ , 1991: 81.\n\n. _Reasons_ , 1991: 223.\n\n. _Reasons_ , 1991: 224.\n\n. _Reasons_ , 1991: 242.\n\n. _Reasons_ , 1991: 19.\n\n. _Reasons_ , 1991: 224.\n\n. _Reasons_ , 1991: 15.\n\n. _Reasons_ , 1991: 16.\n\n. _Reasons_ , 1991: 3.\n\n. _Reasons_ , 1991: 234.\n\n. _Reasons_ , 1991: 227.\n\n. _Reasons_ , 1991: 25.\n\n. _Reasons_ , 1991: 15.\n\n. _Reasons_ , 1991: 210.\n\n. _Reasons_ , 1991: 219.\n\n. _Reasons_ , 1991: 213.\n\n. _Reasons_ , 1991: 213.\n\n. _Reasons_ , 1991: 214.\n\n. _Reasons_ , 1991: 213.\n\n. _Reasons_ , 1991: 222-223.\n\n. _Reasons_ , 1991: 31.\n\n. _Reasons_ , 1991: 221.\n\n. _Reasons_ , 1991: 168.\n\n. _Reasons_ , 1991: 214.\n\n. _Reasons_ , 1991: 74.\n\n. _Reasons_ , 1991: 75.\n\n. _Reasons_ , 1991: 21.\n\n. _Reasons_ , 1991: 23.\n\n. _Reasons_ , 1991: 83.\n\n. _Reasons_ , 1991: 95.\n\n. _Reasons_ , 1991: 234.\n\n. _Reasons_ , 1991: 97.\n**Chapter 17: Reasons in the Public Eye**\n\n_Pardon him, Theodutus: he is a barbarian, and thinks that the customs of his tribe and little island are the laws of nature._\n\n\u2014George Bernard Shaw, _Caesar and Cleopatra._\n\n**Judgment Day**\n\nThe banner headline on the front page of March 9th, 1991's Vancouver Sun announced: \"INDIANS LOSE CASE.\" Pictures of the portly Chief Justice Allan McEachern ran in the daily papers over captions that said \"Gut feelings ruled out,\" and \"Emphasis on law and not gut feelings.\" Brief biographical notes included in the stories described the Chief Justice as a 64-year old, non-drinking, non-smoking, Coca-Cola addict and past president of the Canadian Football League. Born and raised on the middle class west side of Vancouver, the Judge and his family were now property-owning, tax-paying residents of Vancouver's elite Shaughnessy neighbourhood. The judge was described as \"a jurist who has never allowed emotional considerations to shake his belief in the rule of law,\" and who \"made it clear from the outset he would decide the case not on personal gut feelings, but according to the law.\"\n\nGitksan and Wet'suwet'en Tribal Council President Mas Gak (Don Ryan) called for McEachern's removal from the bench, and vowed that \"never again will the sacred boxes of our people be opened for the white man to look at.\" \"The government has made fun of us,\" Gitksan chief, Maxlaxlex (Johnny David) said. Ernie Crey of the United Native Nations likened the text of the Chief Justice's _Reasons for Judgment_ to a Stephen King horror novel. Larry Pootlas of Bella Coola said that when he heard the news he thought he was in South Africa. The Shuswap Nation Tribal Council called the judgment \"brutal and hostile.\" \"It's a travesty of justice,\" said Union of B.C. Indian Chiefs spokesman, Saul Terry. Ovide Mercredi, Grand Chief of the Assembly of First Nations of Canada, described McEachern's decision as \"so demeaning it is breath-taking.\" \"The Judgment stands on racism. It defines Indians as no better than the animals in the forest,\" protested Squamish chief Joe Mathias. \"The corporate board rooms are probably saying, 'Thank God we had our brother on the bench,'\" he added.\n\nIndeed, John Howard of Macmillan Bloedel, speaking for B.C.'s forest industry, said he was surprised by the ruling, but pleased. The Chief Justice's assessment that native issues constituted a \"social problem that should be dealt with by the legislature,\" rather than a legal one to be determined by the courts, \"has been our view at Macmillan Bloedel for a long time,\" he commented. A spokesman for the mining industry expressed relief that the status quo prevailed, and added that he hoped the decision would encourage Aboriginal peoples in the Yukon and Northwest Territories to bring their claims to \"a final solution.\" Mike Hunter, representing the Fisheries Council of B.C. described the Chief Justice's ruling as \"a common sense approach to settling.\" He agreed that the legislature was clearly a better place to deal with Native issues than the courtroom. Industry spokesmen also noted that they feared the court's decision might provoke another wave of Native militancy, which in turn would frighten away potential investors.\n\nRuss Fraser, Attorney-General in the Social Credit provincial government, said he hoped that British Columbians could now \"put this whole era behind us.\" Federal Minister of Indian Affairs, Tom Siddon, stated that the decision gave the federal government an \"important insight into the existence and meaning of aboriginal rights in Canada.\" New Democratic Party leader Mike Harcourt admonished all parties to \"get out of court\" where there must always be \"winners and losers,\" and to come to the negotiating table where there could be \"winners and winners.\" NDP Member of the Legislative Assembly for northwestern B.C. Jim Fulton called the decision \"a declaration of cultural genocide by the B.C. Supreme Court. It's a legal neutron bomb.\"\n\nThe British Columbia Government Employees' Union, the largest labour organization in the province, commented that: \"In insulting and racist language, native people have been told they have no right to bring their disputes to court.... The provincial Crown is relentlessly seeking to grant large companies the unrestrained right to land without concern for aboriginal interests.\"\n\nAnglican Bishop Ronald Shepherd, Reverend William Howie of the United Church of Canada, and Roman Catholic Bishop J. Remi DeRoo issued a joint statement calling on their congregations to demonstrate solidarity with Aboriginal peoples. The Canadian Council of Churches issued a press release saying, \"the judgment appears to reflect a colonial view of society towards aboriginal peoples which is not acceptable.\" Commenting on the Chief Justice's dismissal of the validity of Native oral histories, and his uncritical acceptance of the literal truth of the written reports of nineteenth-century European fur traders, Sun columnist Stephen Hume reminded McEachern that the teachings of Jesus Christ had been communicated and transmitted by means of an oral, and not written, tradition for several centuries. \"Eminent persons can be capable of the most loathsome claptrap if it serves their material interests,\" Hume observed.\n\nThe _Vancouver Sun_ editorial of March 13, 1991, concluded that Chief Justice Allan McEachern had produced \"a remarkable and crystal clear analysis of thelaw as he understands it,\" in the Gitksan and Wet'suwet'en case. They went on to quote the following passage from the _Reasons for Judgment_ : \"When plaintiffs bring legal proceedings, as these plaintiffs have, they must understand (as I believe they do), that our Courts are Courts of law which labour under disciplines which do not always permit judges to do what they might subjectively think (or feel) might be the right or the just thing to do in a particular case. Nor can judges impose politically sensitive non-legal solutions on the parties. That is what Legislatures do, and judges should leave such matters to them.\" \"Hear! Hear!\" the editorial cheered, adding that this was an appropriate response to \"the likes of Don Ryan\" of the Office of the Gitksan and Wet'suwet'en Hereditary Chiefs. A debate ensued in the _Sun_ 's \"Letters to the Editor\" pages about whether or not it was appropriate, or even lawful, to call judges, or legal judgments, \"racist.\"\n\n\"Call it whatever you want\u2014ethnocentrism, eurocentrism, racism\u2014you guys go ahead, go out in the hall and argue about what word you are going to use,\" Nuu-Chah-Nulth Chief, George Watts, thundered as he admonished an audience of lawyers and academics who gathered at the University of Victoria to review the _Delgamuukw_ judgment, \"We know what it is when we live with it.\"\n\nAnother controversy about land in British Columbia competed with the Gitksan and Wet'suwet'en case for public attention in the Spring of 1991. Then provincial Premier Bill Vander Zalm, a tulip gardener who had emigrated from Holland after World War II, had amassed a considerable fortune by buying and selling real estate. The jewel in his family's crown was a large property on the outskirts of Vancouver that the Vander Zalms had developed into a theme park complete with windmills, tulip gardens, a miniature railroad, restaurants and gift shops. The premier and his family built a red brick replica of Captain George Vancouver's family home in Holland in the middle of the park. Vancouver was the explorer who is credited with having \"discovered\" the coast of British Columbia in the late eighteenth century. The Vander Zalms called their acreage \"Fantasy Gardens\" and made their home there until 1991 when they sold it all to a millionaire from Hong Kong, named Tan Yu. While Premier Vander Zalm insisted that his wife Lillian was the sole owner of Fantasy Gardens, legal documents concerning the transaction, and bearing the premier's signature, were leaked to the press. Further investigations revealed that the premier had met with Tan Yu's real estate agent, Fay Leung, late one night at a Vancouver hotel, where he had handed her a brown paper bag containing $20,000.00 in cash. The premier protested that there was nothing unusual about this meeting which he described as \"a normal business practice.\" He complained that his own government's investigation of the events made him feel as though he was living in Nazi Germany. Throughout the spring and summer of 1991, the \"Fantasy Gardens Affair,\" and the outcome of the Gitksan and Wet'suwet'en land claims case were hotly debated throughout British Columbia. Leung has since been convicted of real estate fraud, and Premier Vander Zalm resigned in 1992 when it was decided that he had, indeed, been in a conflict of interest position in relation to the sale of Fantasy Gardens.\n\n**Music is Lost on Me. I Have a Tin Ear**\n\n_When I come to consider events long past, I am driven to conclude, on all the evidence, that much of the plaintiffs' historical evidence is not literally true.... I must assess the totality of the evidence in accordance with legal, not cultural principles...._\n\n\u2014Chief Justice Allan McEachern, 1991, _Reasons for Judgment, Delgamuukw v. R._\n\nAs a text, the _Reasons for Judgment_ in _Delgamuukw v. R._ is structured predominantly by the questions set out in the legal tests for Aboriginal title developed by Justices Mahoney and Steele in the Baker Lake and Bear Island cases, respectively. The first, and the most fundamental of these questions was: did the Gitksan and Wet'suwet'en live in an \"organized society\" when Europeans first made contact with them? The authoritative sources of knowledge about this period of time presented to the judge were Gitksan and Wet'suwet'en oral histories\u2014 _adaawk_ and _kungax_ , supplemented by anthropological and historical research.\n\nChief Justice McEachern gave no weight to the oral tradition testimony except, minimally, where it could be supported by data from research conducted within the paradigm of western science: \"I am not able to accept adaawk, kungax and oral traditions as reliable bases for detailed history.... Oral tradition may well provide useful information 'to fill in the gaps' left at the end of a purely scientific investigation,\" he wrote.\n\nIt is important to understand what the Judge believed he was seeking, in a specifically legal sense, in the oral tradition evidence. He explains this most succinctly when he writes: \"In a nutshell, they...[the Gitksan and Wet'suwet'en] sought first to establish both the present social organization of the Gitksan and Wet'suwet'en; secondly, that it exists today in the same or nearly the same form as at the time of contact; thirdly, that at that time, and since, the plaintiffs have used and occupied all of these separate and remote territories for aboriginal purposes; and fourthly, because of the way the plaintiffs have framed their case, they undertook also to prove the boundaries of these 133 separate territories and the distinct use made of them by the plaintiffs and their ancestors.\"\n\nHe went on to detail his reasons for rejecting _adaawk_ and _kungax_ as direct evidence of facts \"except in a few cases where they could constitute confirmatory proof of early presence in the territory.\" \"My reasons,\" the Chief Justice wrote, \"are principally threefold.... First, I am far from satisfied that there is any consistent practice among the Gitksan and Wet'suwet'en Houses about these matters. The early witnesses suggested that the adaawk are well formulated and the contents constantly sifted and verified. I am not persuaded that this is so...Secondly, the adaawk are seriously lacking in detail about the specific lands to which they are said to relate.\"\n\nHis third reason was that the attempt by the plaintiffs to authenticate their _adaawk_ by reference to work on oral tradition among other peoples failed because this information did not \"relate to the territory but they demonstrate the weakness of this kind of evidence.\" While acknowledging that the \"objective validity\" of oral traditions varies from culture to culture, and that Dr. Philip Drucker, among others, \"believes oral histories on the north coast are usually correct,\" McEachern cites at length from Dr. Bruce Trigger's work that is based in Huron and other eastern Aboriginal peoples' oral history, in reference to which Trigger cautions that: \"...oral traditions may supply valuable information about the not too distant past. Used uncritically, however, they can be a source of much confusion and misunderstanding in prehistoric studies.\" In further support of his conclusions about oral history, McEachern cites two additional academics, Dr. Arthur Ray and Dr. Charles Bishop, who discuss problems of verification in \"memory ethnography,\" a term that usually refers to the anecdotal recollections of ordinary individuals, rather than to oral traditions transmitted by specialists in institutionalized forum and in public ceremonies. McEachern quoted Ray and Bishop as saying that \"memory ethnography\" provides relatively accurate information \"only to a depth of one hundred years.\"\n\nFinally, the Chief Justice notes, and attaches an appendix provided by the Province of British Columbia for verification, that references to the historical\u2014that is, post European contact\u2014period are found in the _adaawk_ and _kungax_. Specifically, McEachern points out one mention of moose (who, a provincial wildlife biologist estimated entered the territory after Europeans did); a story about a Chief Legaik who the Chief Justice assumes must be the same Legaik as the person who appears in \"historical\" written records; one allusion to guns and gun powder; and a remark about the Hudson's Bay Company. These references, in his opinion, render the oral histories changed in their representation of pre-contact, or \"pre-historic\" Aboriginal life. Chief Justice McEachern apparently could not, or would not, understand the explanation, offered repeatedly by Gitksan and Wet'suwet'en witnesses and other experts, that oral histories are dynamic and evolving accounts that pick up and include new elements as they arise in the lived experiences of the subjects of the accounts. In other words, the judge wanted to imagine \"true\" history as taking place in hermetically sealed time capsules that follow one after another in a linear progression. The oral histories, according to this way of thinking, should have been stored, like scrolls, in such capsules, and then regurgitated verbatim each time they were told. Twentieth-century listeners should, it follows, derive exactly the same interpretation from these accounts as did their eighteenth-century audiences. Accounts that reveal interconnections between speakers and listeners, and blurred boundaries between abstractly predetermined time periods are declared \"untrue.\"\n\nHaving established, to his satisfaction, an intellectual and legal basis for rejecting oral tradition as valid evidence, the Chief Justice hastened to assure his readers that his conclusions were reached in the fullest spirit of cultural relativism. He wrote that he made this ruling reluctantly,\n\n...without intending any affront to the beliefs of these peoples.... I am satisfied that the lay witnesses honestly believed everything they said was true and accurate. It was obvious to me, however, that very often they were recounting matters of faith which have become fact to them. If I do not accept their evidence it will seldom be because I think they are untruthful, but rather because I have a different view of what is fact and what is belief.... Much evidence must be discarded or discounted not because the witnesses are not decent, truthful persons but because their evidence fails to meet certain standards prescribed by law.... This in no way reflects upon adaawk and kungax for the spiritual use or value they have to Gitksan and Wet'suwet'en people. I do not purport to pass on that question in any way. All I say is that I do not find them helpful as evidence of the use of specific territories at particular times in the past.\n\nIn addition to the Elders, several Gitksan and Wet'suwet'en testified as expert witnesses in a range of areas. In his _Reasons for Judgment_ the Chief Justice gave mixed appraisals of their evidence. He interpreted Neil Sterritt's, and others, painstakingly detailed accounts of property transfers as evidence of inconsistency in the rules governing these transactions. While he commented that he \"had a favourable impression of the competence and industry of Ms. Harris,\" and that \"the only general discount I would apply to her work relates to the reliability of her informants...,\" he did not find the genealogies she collected could \"establish House membership as an active force in the lives of the persons listed.\" The Chief Justice was less complimentary towards Susan Marsden, who testified about evidence provided by the _adaawk_ and _kungax_ as to origins and migrations. McEachern wrote: \"I am unable to accept Ms. Marsden's theory. I have no doubt it is put forward honestly and in good faith, but her qualifications are not adequate for such a study (Marsden holds a B.A. in Anthropology\u2014ed.)\"\n\n\"In a case such as this,\" the Chief Justice wrote in his _Reasons for Judgment_ :\n\nwhere the plaintiffs and their ancestors are the only sources of these histories, the Court may not be the best forum for resolving such difficult and controversial academic questions.... One cannot, however, disregard the 'indianness' of these people whose culture seems to pervade everything in which they are involved. I have no doubt they are truly distinctive people with many unique qualities. For example...they have an unwritten history which they believe is literally true both in its origins and in its details.\n\nA simple substitution of categorical labels reveals the ethnocentric bias of Allan McEachern's commentary, and its reflection in popular culture: Ask yourself why it would likely be read as humourous if an Aboriginal person surmised that \"One cannot disregard the 'whiteness' of these people whose culture seems to pervade everything in which they are involved. Euro-Canadians are truly distinctive people with many unique qualities. For example, they have a written history which they believe is literally true both in its origins and in its details?\"\n\n**Delgamuukw and the People Without Culture**\n\n_In the nations under discussion, full citizenship and cultural visibility appear to be inversely related. When one increases, the other decreases._ _Full citizens lack culture, and those most culturally endowed lack full citizenship. In Mexico, Indians have culture and 'ladinos'...do not. In the Philippines, 'cultural minorities' have culture, and lowlanders do not._\n\n_Ladinos and lowlanders, on the other hand, are full citizens of the nation-state. They work for wages, pay taxes, and sell their wares in the local market. People in metropolitan centers classify them as civilized, in contrast with Indians and cultural minorities who are cultural, not 'rational'...._\n\n_...People with culture have been confined to marginal lands...._ _In the Philippine case, the 'people without culture' occupy both ends of the social hierarchy. Roughly speaking, Negrito hunter-gatherer groups are on the bottom and lowlanders are on top. The difference between the two ends of the spectrum is that the Negritos are 'precultural' and the lowlanders are 'postcultural.'_\n\n_...In this pseudoevolutionary ladder, people begin without culture and grow increasingly cultured until they reach that point where they become postcultural and therefore transparent to 'us.'_\n\n\u2014Anthropologist Renato Rosaldo, 1989, _Culture & Truth: The Remaking of Social Analysis_.\n\nThe _Reasons for Judgment_ are characterized by inconsistency: the use of terms, concepts and logic are often internally incoherent and self-contradictory. Allan McEachern's use of the term \"culture\" is a case in point. In this instance he says: \"Indian culture also pervades the evidence at this trial for nearly every word of testimony, given by expert and lay witnesses, has both a factual and a cultural perspective.\" The Chief Justice's use of the term \"culture\" as synonymous with subjectivity, emotion and irrationality reflects one of many contemporary meanings attached to the concept, in which \"culture\" is considered an attribute of subordinated categories of persons.\n\nThe judge's use of the term \"culture,\" like that described by Rosaldo, simply replaces the word \"race\" but retains the same meaning as in archaic scientific racism. This substitution of words is at the heart of what Henri Giroux calls the \"nouveau racisme\" of the late twentieth century. In other places in his magnum opus, McEachern uses the term \"culture\" in relation to the Gitksan and Wet'suwet'en to convey the \"ancienne racisme\" premise that \"indigenous behaviour patterns are based on instinct.\" In this scheme, cultural traits are understood as genetically inherited, rather than learned through socialization.\n\nChief Justice McEachern concluded his remarks on oral tradition in his _Reasons for Judgment_ by stating:\n\nI believe the plaintiffs have a romantic view of their history which leads them to believe their remote ancestors were always in specific parts of the territory, in perfect harmony with natural forces, actually doing what the plaintiffs remember their immediate ancestors were doing in the early years of this century. They believe the lands their grandparents used have been used by their ancestors from the beginning of time.... I do not accept the proposition that these people have been present on this land from the beginning of time.... In fact, I am not able to find that ancestors of the plaintiffs were using all of the territory for the length of time required for the creation of aboriginal rights, and I shall give my reasons in due course.\n\nThe judge could not understand the _adaawk_ and _kungax_ as subject to different but equally logical rules of validation _without first acknowledging the equality, and hence denying the superiority, of his own culture_ , and, by extension, of \"his people.\" His consistent failure to do so, and the concomitant impossibility of separating a critique of eurocentrism from an appreciation of other cultures becomes apparent. Simply put, the linchpin of McEachern's \"world view\" is that it is superior to all others, and particularly to Aboriginal world views. Mutual respect cannot emerge from such a foundation, as it denies\u2014 _a priori_ \u2014the possibility of considering other ways of life as different but equally valuable. When Chief Justice McEachern accepted the Crown's argument that Hugh Brody's testimony about \"how white people see Indians\" had \"nothing to do with the matters before the bench,\" he rejected the first necessary condition for Aboriginal\/non-Aboriginal dialogue. This can be seen, throughout the text of the _Reasons for Judgment_ , in the way he repeats, in a variety of contexts, law's trap: difference constitutes evidence of failure to cease being Indian and thus to be ineligible for recognition as equal, rights-bearing legal subjects; similarity constitutes evidence of success in ceasing to be Indian and therefore ineligibility for recognition as distinct, rights-bearing legal subjects.\n\n**Insult and Injury**\n\n_There's a serious contradiction going on in this country, and it has got a lot to do with who wields power. You haven't got us fooled about that. But we're not going to bite into this game...What you do is you react with violence or react with hate. We're not going to beat you with violence and we're not going to beat you with hate. We're going to beat you with our behaviour that we were taught by our grandparents.... We're going to beat you with logic because our logic is right, and I'll say that until my dying day._\n\n\u2014George Watts, Chairman, Nuu-chah-nulth Tribal Council, 1992, _The Law and Justice: A Contradiction?_\n\nWhatever his intentions were, the Chief Justice's comments on oral tradition, the Chiefs and Elders, and their testimony, were heard by the Gitksan and Wet'suwet'en\u2014and by many other people both Aboriginal and non-Aboriginal\u2014as insults to the honesty of individual witnesses and to their collective integrity. \"We did have certain expectations,\" Neil Sterritt wrote later. \"The expectation was partly created by Judge McEachern himself. He sat in his chair and the witnesses were in their witness box, elders with whom he formed a relationship. They thought he was forming a friendly relationship...but you realize when you read his judgment that he saw our people as 'cute.'\"\n\n\"We talk about party line...,\" Sterritt continued. \"If you're an elder, if you've been raised in the oral tradition to speak what you know, to speak the truth, there is no such thing as a party line.... There was no way we could go into court and say 'Don't say this.' Or 'Say this in this way,' or anything like that. Not a chance. The elders\u2014and maybe it is part of how we bared our soul in the courtroom\u2014went in and they said how they felt, what they knew about the land, what they wanted in the future, and where they came from in the past. It was that truth, ultimately that will be important....\"\n\n\"Our court case started with the elders,\" Sterritt wrote. \"And one trait is universal amongst the elders\u2014those who were born and raised on the land and know the land. That trait is optimism. At the bleakest of times, at the saddest of times, there's always a sense of humour and a sense of optimism about what's going on, and I think as younger people, as those who are picking up the torch and trying to carry on and win this struggle, we too have to be optimistic. Ultimately, we must be optimistic.\"\n\nOther Gitksan and Wet'suwet'en commentators agreed. Gitksan Chief Satsan (Herb George) expressed his frustration at the Court's arbitrary power: \"The people who referee and umpire the game don't agree,\" Satsan explained. \"And they don't agree because it's their right not to agree.... They set out the rules. We try to meet the rules, and they can just disagree with us.... They set out a test. 'You must be an organized society,' they say. So we look at what is an organized society in your terms. We find that out, and we say, 'Yes, we meet all of those different criteria as a society.'...If you meet the test and you beat the test, then they change the test.... It's their game. We understand the game we're in. Make no mistake about that.\"\n\nYagalahl (Dora Wilson) wrote: \"The court case closed at 4:57 p.m. June 30th, 1990, my 370th day. This is a note I put down for myself in my notebook. 'My 370th day and there has been no document produced and no proof given that the Province or the federal government own this land....' It was a black day, March 8th, for the non-Aboriginal people in Canada, and it was sad because so many people know that we are telling the truth. We are right.... It is our land, and regardless of how many decisions come down, we will always say that because it is the truth. It will always be the truth.\"\n\nDene Member of Parliament for the Western Arctic, Ethel Blondin-Andrews, was blunt in her response to Chief Justice McEachern's ruling. She said: \"How much do you have to weigh, as an individual, when you deal with things like politics and law? How much of your principles and values do you want to prostitute or dilute? Well, I'll tell you something, I am willing to look at different definitions, but I am not willing to change one thing. I am not willing to forfeit the right that my people think they have to the land, the right that my people think they have to be who they are, the languages they speak, or their traditional forms of government.... Racism, even if you rationalize it, intellectualize it, tie it up in technological or technical terms, is still racism. Its impact cuts into the heart of a people. I'm calling it racist.\"\n\n**Is There Intelligent Life in the Galaxy?**\n\n_Imagine the situation of the inhabitants of Earth, visited and then taken over by space voyagers from Alpha Centauri, and required to defend their ownership of the planet entirely with reference, not to written documents, but to chants. The governments of the several countries of Europe would be left with whatever lands could be defended through citation of medieval sung liturgy. Farfetched? Certainly, but no more farfetched than what has actually happened to the Gitksan and Wet'suwet'en\"_\n\n\u2014Anthropologist Andrea Laforet, 1993.\n\nA number of critiques have been written by anthropologists about the Chief Justice's dismissal of oral tradition. For the most part, criticism has been directed at the judge's ethnocentrism, and his failure to respect the culturally different world view of Gitksan and Wet'suwet'en peoples, their particular conception of history and their oral tradition. Lawyers for the Gitksan and Wet'suwet'en pointed to the \"Catch 22\" nature of Chief Justice McEachern's ruling on oral histories, arguing that: \"The judge's rejection of oral histories effectively made the Gitksan and Wet'suwet'en 'peoples without history' before the first European records in the 1820s. The Gitksan and Wet'suwet'en oral histories are unusual in that there are great numbers of them, and they are very detailed. They have been accepted as reliable and used by anthropologists and archaeologists. If these oral histories are entitled, as the trial judge found, to little or no weight, it is doubtful if any Aboriginal people in Canada could rely on their oral history to establish the existence and character of their pre-contact societies...the judgement effectively makes the proof of Aboriginal rights impossible.\"\n\nThe Chief Justice's words create an impression that he evaluated oral tradition by comparing it to written tradition and applying certain established rules of interpretation and analysis. Similarly, critics and supporters alike have, for the most part, responded to his rulings on oral tradition by commenting on whether or not the Chief Justice carried out this task competently. But, if we replace general and abstract language about concepts like \"oral tradition\" and \"written tradition\" with particular and concrete words that refer to living beings and experience, then it becomes clear that Allan McEachern did not believe what the Gitksan and Wet'suwet'en Elders said about their ways of life before the arrival of Europeans in the eighteenth century, but he did believe what _some_ Europeans wrote about the same subject. His judgment of the validity of oral tradition reflects his judgment of the people whose tradition it is, not the application of abstract epistemological principles, or reason, or legitimate processes for validating different forms of knowledge.\n\nPerhaps Allan McEachern's most revealing reflection on Gitksan and Wet'suwet'en oral tradition is his first, found on page 17 of his _Reasons for Judgment_. Here he laments that, with regard to those histories that \"unfortunately, exist only in the memory of the plaintiffs.... I must leave it to the social scientists who are just beginning their journeys of discovery into the vast and largely uncharted _terra incognita_ of the unwritten histories. I wish I could know what they will discover.\"\n\nLike the \"vast and empty land,\" the _terra nullius_ \"discovered\" centuries ago, the voices of peoples deemed by law not to exist await European discovery, analysis and exploitation. The work of many social scientists who long ago began their \"journey\" into the study of oral tradition, and of those who testified in support of the Gitksan and Wet'suwet'en in this case, is also rendered invisible by the stroke of a pen inscribing a legal text.\n\n* * *\n\n. Stills 1991: A12.\n\n. Glavin 1991: A1.\n\n. Glavin 1991: B8.\n\n. Stills 1991: B12.\n\n. BCGEU 1991: 1.\n\n. Glavin 1991: A2 & A11.\n\n. Watts 1992: 194.\n\n. _Reasons,_ 1991: 48,75.\n\n. Ibid., 278.\n\n. Ibid., 47\u201348.\n\n. _Reasons_ , 1991: 49.\n\n. _Reasons_ , 1991: 67.\n\n. _Reasons_ , 1991: 68.\n\n. _Reasons_ , 1991: 68.\n\n. Kahn 1989.\n\n. Giroux 1993.\n\n. Asch 1992.\n\n. Sterritt 1992:306\n\n. Ibid., 305.\n\n. Ibid., 303.\n\n. Satsan 1992: 55.\n\n. Yagalahl 1992: 204\u2013205.\n\n. Blondin 1992: 253, 255.\n\n. Laforet 1993.\n\n. _Delgamuukw v. R._ , Plaintiffs' Appeal Factum, 1991: 1.\n**P** **ART** I **V:**\n\n**T** **ABULA** **R** **ASA**\n\n**(B** **LANK** **P** **AGE** **)**\n**Chapter 18: Judging the Experts**\n\n_I must briefly discuss the evidence of Drs. Daly and Mills and Mr. Brody because of the importance attached to it by the plaintiffs...apart from urging almost total acceptance of all Gitksan and Wet'suwet'en cultural values, the anthropologists add little to the important questions that must be decided in this case.... I am able to make the required important findings about the history of these people, sufficient for this case, without this evidence._\n\n\u2014Chief Justice Allan McEachern, 1991, _Reasons for Judgment, Delgamuukw v. R._\n\n**Experts v. Advocates**\n\nThe relationship between lawyers and anthropologists in Aboriginal title litigation is rife with problems. Law demands that stories be reduced to their simplest form so that a judgment can be made: true or false, guilty or innocent. Anthropology demands that stories be told in all their complexity, embedded in historical and cultural context. Generally speaking, when lawyers and anthropologists work together, it is the lawyers who have the upper hand: the law sets out the parameters of what arguments can be made, the rules of the court determine how evidence can be presented, and the goal of winning a particular case dominates.\n\nOver the past twenty-five years in Canada, Aboriginal title litigation has provided considerable employment for many anthropologists and consultants at the same time as universities have \"downsized\" and the number of academic positions has dwindled. In this environment, anthropological knowledge has become increasingly commodified as \"hungry\" graduates of Ph.D. programs seek to sell bits and pieces of information and to solicit contracts in an ever more competitive market. When research is conducted and knowledge is produced through a piecework labour process that mimics the tabloid newspaper model, integrity is sacrificed to sales, and the customer often gets what s\/he pays for. In contrast to the independence that can be demanded by some scholars with alternative employment in academic institutions and income from grants and publications like, for example, Arthur Ray, Richard Daly and Antonia Mills; or who, like writer and filmmaker Hugh Brody, enjoy independent financial success; contract researchers employed as expert witnesses must, to varying degrees, respond to the demands of their employersrather than to the demands of peer review or professional ethics. Imperfect as that academic process is, its procedures are less fickle than those of a buyer's market. When that marketplace is an adversarial courtroom governed by an \"only winning counts\" ethic, the predicament that expert witnesses find themselves in is even more exaggerated.\n\nGiven the adversarial nature of legal procedures, it would be naive in the extreme to suggest that anthropologists employed as expert witnesses do not tend to present interpretations that offer support for their employers' legal arguments. But this makes it even more crucial, by law's own professed standards of value neutrality, that the weight accorded to experts' testimonies be determined by applying the same reasonable assessment criteria equally to witnesses for both plaintiffs and defendants. An obvious place to begin would seem logically to be with a comparison of respective expert witnesses' education, experience and reputation within their professions.\n\nThe anthropologists who testified on behalf of the Gitksan and Wet'suwet'en hold degrees from some of the most prestigious universities in the Western world: Brody from Oxford; Daly from Manchester and Toronto; Mills from Harvard. Each of their careers spans approximately twenty years. Each has taught university courses. Each has published their research in peer-reviewed scholarly journals. Each, therefore, is a recognized expert in an area of specialized knowledge accredited by institutions charged with that authority within the judge's own culture. Each has had extensive professional experience working among the Gitksan and Wet'suwet'en and other Indigenous peoples. Each has a history of supporting Aboriginal land rights. None sought to hide this fact. Each submitted lengthy, well-documented opinion reports to the court that were based in research that involved combinations of extensive participant observation, archival research, and a review of previous historical and anthropological research.\n\nWhile the Chief Justice set out in detail his reasons for dismissing the evidence of the anthropologists who testified on behalf of the Gitksan and Wet'suwet'en\u2014with the exception of Hugh Brody's testimony which merited no reference at all by Allan McEachern\u2014he only briefly mentioned the Crown's anthropologist, Sheila Robinson, once.\n\nSince receiving her Ph.D. from the University of London in 1983, Sheila Robinson has been sporadically employed as a researcher and consultant by Parks Canada, primarily in Alberta, and has co-authored two published scholarly papers. Robinson has never held an academic position. She was contracted by the federal government's Office of Native Claims to review the Kwakiutl Comprehensive Land Claim submission in 1984. She submitted an affidavit to the court in _Sparrow v. R._ on behalf of the Pacific Fishermen's Defence Alliance, an organization formed specifically to oppose Native fishing rights. Since testifying on behalf of the Crown in _Delgamuukw v. R.,_ Robinson has appeared as an expert witness in anthropology in a number of other Aboriginal rights cases. On February 19, 1986, Geoffrey Plant of Russell &DuMoulin wrote to Sheila Robinson:\n\nRussell and DuMoulin wishes to retain you as a consultant to assist us in the conduct of litigation relating to certain Indian land claims. We wish to retain your services for four months. We may extend this agreement from time to time thereafter. Your fee would be $250.00 per day.... While we have not yet determined the nature of the assignments you will carry out, we confirm that your assistance will be in areas related to your professional and academic experience. We anticipate this will include, for example, research and evaluation of the anthropological aspects of Indian land claims.3\n\nDuring the fifteen months that elapsed between the date of this initial contract and the submission of Sheila Robinson's final opinion report on May 12, 1987, Robinson corresponded regularly with Russell & DuMoulin, submitting at least two preliminary draft copies of her report for their comments, which she then incorporated into her final report. As late as May 4, 1987, Plant wrote to Robinson saying \"I enclose the draft of your report.... You will see that I have made a considerable number of revisions and deletions.... What I would like you to do is to provide me with a final draft in a form which you are completely happy with (i.e. you would have no reservations about delivering it to a publisher).\"\n\nA review of the curriculum vitae of the plaintiffs' anthropologists shows, therefore, a history of professional appointments by academic, public and private institutions, as well as a consistent pattern of employment by First Nations and Aboriginal organizations to do work in _support_ of land rights. A parallel review of the Crown's anthropologist's career history shows a paucity of professional appointments, and a consistent pattern of employment by the state to do work in _opposition_ to land rights. Only the anthropologists who testified in support of the Gitksan and Wet'suwet'en, however, were dismissed by Chief Justice McEachern as \"advocates\" who could therefore not be regarded as legitimate \"experts.\"\n\nSheila Robinson has lived her entire life, received her education, and practiced her career among and within the cultural group to which she and her employers belong. And, she has chosen to professionally align herself with those particular political factions of that cultural group most actively opposed to Aboriginal rights. Both her short term and her long term livelihood depend, in many ways, upon the outcome of the cases in which she testifies. Why did the judge not appear to have been concerned about the possibility that Robinson could have been \"urging the almost total acceptance\" of Euro-Canadian cultural values?\n\n**Ethics v. License**\n\n_The evidence of the anthropologists who testified for the Plaintiffs was seriously attacked on various grounds, particularly that they were too closely associated with the plaintiffs after the commencement of litigation.... It is always unfortunate when experts become too close to their clients, especially during litigation._\n\n_With regard to Dr. Daly, he made it abundantly plain that he was very much on the side of the plaintiffs. He was, in fact, more an advocate than a witness. The reason for this is perhaps found in the Statement of Ethics of the American Anthropological Association which Dr. Daly cites at p.29 of his report, as follows:_\n\n_\"Section 1: Relations with those studied; In research, an anthropologist's paramount responsibility is to those he studies. When there is a conflict of interest, these individuals must come first. The anthropologist must do everything within his power to protect their physical, social and psychological welfare and to honour their dignity and privacy.\"_\n\n\u2014Chief Justice Allan McEachern, 1991, _Reasons for Judgment, Delgamuukw v. R._\n\nThe Chief Justice interpreted the American Anthropological Association's ethical guidelines to mean that anthropologists must simply accept and regurgitate whatever representatives of the people they study say. But \"...the Code of Ethics was not intended and cannot be appropriately interpreted to mean that... anthropologists are required to avoid telling the truth as they have come to know it.\" Rather, guidelines reflecting similar principles as those articulated by the American Anthropological Association govern research carried out with human subjects under the auspices of major universities and public agencies such as the Social Sciences and Humanities Research Council of Canada. The judge's assumption that ethical guidelines aimed at insuring respect for human research subjects reduce social scientists to the status of \"hired pens\"\u2014or human tape recorders\u2014and prohibit them from conducting rigorous research, is clearly a legalistic assessment that seems to reflect case-building in an adversarial forum rather than scientific or interpretive approaches to research in the social sciences.\n\nSheila Robinson, while declared by the court to be an expert in anthropology, is not an anthropologist; and is therefore not bound by any professional code of ethics that might have led\u2014by the logic the judge applied to Richard Daly\u2014to her testimony being suspect. Nor was her lack of involvement in the academic arena of her profession challenged by the judge in the same way as he dismissed Susan Marsden's research on Gitksan and Wet'suwet'en origins and migrations because it \"has not been published or subjected to academic or other learned scrutiny, she is an interested party, and she has ignored some verified facts and other learned opinions....\"\n\n**Revision v. Reversal**\n\n_Dr. Mills, the plaintiffs' other principal anthropologist, also showed she was very much on the side of the plaintiffs. She has almost completely changed her opinion from that contained in her June 1986 draft where she attributed almost all Wet'suwet'en social organization, including the kungax, to borrowings from the Gitksan or some other coastal Indians._ _This is a startling departure from a large body of professional opinion on the part of a witness closely associated with the beneficiaries of her new opinion..._\n\n\u2014Chief Justice Allan McEachern, 1991, _Reasons for Judgment, Delgamuukw v. R._\n\nChief Justice McEachern's first criticism of Antonia Mills was directed to her having changed her opinion about the degree to which Wet'suwet'en social organization was derived from Gitksan and Tsimshian. He comments on this particular debate early, writing at page 32:\n\nOn the evidence, there are some, but not many, differences between the present social organization of the Gitksan and Wet'suwet'en people. This may well be because it is common for adjacent aboriginal people to \"adopt\" customs and practices from each other. That different people would have so many similar institutions and practices almost demonstrates the borrowing theory. Most of the experts believe the Wet'suwet'en adopted much of the culture of the Gitksan, but culture, like their languages, may well have travelled in both directions.\n\nMills explained, first under cross examination, and subsequently in a scholarly, peer-reviewed publication, that she had been led to change her opinion on the basis of research carried out by Isadore Dyen and David Aberle. This scholarship challenged her earlier assumptions that had been based not on research among the Wet'suwet'en in particular, but among their neighbours, the Beaver. In \"completely changing her opinion\" then, Mills was following a normal scientific, and common sense, practice: revising knowledge on the basis of new information. She was also reflecting a more general trend in contemporary anthropology away from relying on generalization and speculation from abstract theoretical models and towards valuing grounded knowledge understood within its local context and validated by the people whose cultures are the subject of study. Chief Justice McEachern appears to have taken exception to the fact that Mills _changed_ her opinion, rather than whether her more recent thesis might more accurately reflect Wet'suwet'en cultural life.\n\nLike Antonia Mills, Sheila Robinson, too, changed her opinion on several key issues during the period between 1983, when she completed her doctoral dissertation under academic direction, and 1987, when she began work as a witness for the Crown. In her Ph.D. thesis she argued against classical assumptions that hunting\/fishing\/gathering societies were insufficiently \"evolved\" to have practiced agriculture, claiming that Aboriginal peoples on the northwest coast were cultivating tobacco prior to the commencement of the fur trade. Robinson's dissertation reads:\n\nImplications of the relative neglect of prehistoric Indian agricultural practices go beyond correcting the record for academic purposes. Studies such as this one have direct relevance to modern political issues concerning aboriginal land claims. It has often been convenient for professionals other than scholars to 'forget' that Indians were farming in many parts of the Pacific Northwest region prehistorically and in the early historic period, or that they had well-developed systems of territorial property ownership. There is not room here to explore the unsatisfactory way most native people were treated after the mid-19th century with regards to their territorial claims: it is just mentioned in passing that when the lands the Indians had previously occupied and exploited were expropriated and then allotted to them after they had been 'adjusted' by government representatives to 'appropriately-sized' holdings for their future use, it was usually assumed that the Indians had no need for extensive acreage because they had not traditionally engaged in agricultural pursuits. This was especially the case on the coast of B.C., where territorial allocations (made without the Indians' formal agreement to allow any alienation of their lands) were, on the average, smaller than those recommended by the Dominion Government of Canada.\n\nThe Province of British Columbia's legal argument rests principally on establishing that Indigenous peoples did NOT have well developed systems of territorial property ownership prior to the \"historic\" period (i.e. European arrival); and DID consent, implicitly and\/or explicitly, to the alienation of their lands. In all her work as an expert witness, Robinson argues that, prior to contact, or \"proto-contact,\" with Europeans, the practice of property ownership and resource management among Aboriginal peoples was simple and bore little if any resemblance to European concepts of property. She testifies that Aboriginal life was dramatically and irreversibly affected by the commencement of trade with Europeans during the eighteenth and nineteenth centuries to the point where these peoples ceased to live \"truly aboriginal\" lives, and that, since first contact, Aboriginal peoples have been assimilating into Canadian society, and have thus demonstrated their consent to the expropriation of their lands and the extinguishment of their Aboriginal title. Given that Robinson has not conducted new, original research in the field since writing her dissertation, and that contemporary literature supports the arguments advanced in her Ph.D. dissertation and not those put forward in her expert opinion evidence, it is fair to conclude that the changes in her opinion have not resulted from a normal scholarly process of rethinking and revising previous theories on the basis of new ideas. Sheila Robinson, expert witness, contradicts Sheila Robinson, scholar.\n\n**Politics v. Poetics**\n\n_I found Dr. Daly's report exceedingly difficult to understand. It is highly theoretical and, I think, detached from what happens, 'on the ground'.... There are many passages which I do not understand._\n\n\u2014Chief Justice Allan McEachern, 1991, _Reasons for Judgment, Delgamuukw v. R._\n\nThe Chief Justice continued his list of reasons for dismissing anthropologist, Richard Daly's evidence. He quoted a lengthy paragraph in which Daly describes \"the world view of those living close to nature,\" as an example. In the section of his report that the Chief Justice cited, Daly challenged western readers to imagine a way of life very different from their own. He did so somewhat lyrically, juxtaposing the way \"nature's life force\" gives, as well as takes away, from human beings, and explaining the ongoing dynamic and constantly negotiated relationship between ownership and stewardship of lands and resources. The paragraph is unsettling in the way that poetry is: Daly brings diverse elements into unexpected and unfamiliar relationships with each other. He is, however, describing lived experiences, beliefs and practices taught to him by the Gitksan and Wet'suwet'en with whom he engaged in research. The Chief Justice infers that Daly's descriptions reflect a fictional, rather than a factual, account of Aboriginal relationships to nature. The judge does not entertain the possibility that the problem may lie in his own failure to understand cultures different from his own, or that fact and fiction may intermingle as much in his reading, as they do in Daly's writing, of cultural representation.\n\nSheila Robinson's text is highly theoretical: ironically, in precisely the sense that McEachern appears to be criticizing Daly. Robinson, using esoteric language, invokes abstract theoretical postulates inconsistently and sporadically to fill in gaps created by her dismissal of available empirical evidence and as a rhetorical device to help render her narrative, and the Crown's argument, convincing and authoritative. This conflation can be read in her initial proposal to Russell & DuMoulin, where Robinson set out her understanding of the task they had assigned her as being:\n\n...to develop theories showing that significant changes occurred in Gitksan and Wet'suwet'en socioeconomies during the late prehistoric and early historic eras and that these were the result of both indirect and direct European influence.\n\nPeter Grant, lawyer for the Gitksan and Wet'suwet'en, pursued the question of Robinson's use of the term \"theory\" when he cross-examined her on this correspondence:\n\nGrant: What do you mean by theory? What do you mean when you use the term theories as a social\u2014as a cultural geographer? Robinson: Theories are explanations or attempts at explanation that try to account for or resolve into a logical framework facts relating to a central issue or problem.\n\nGrant: I would be correct to say that theories are not facts, you would agree with that?\n\nRobinson: Yes.\n\nGrant: And what you do when you have a theory is you test it?\n\nRobinson: Sometimes.\n\nGrant questioned her again, asking her to differentiate between \"theory\" and \"speculation.\" Robinson replied: \"I use the word theorize...you use the word speculate. I think we are probably discussing the same thing.\n\n**Chiefs v. Commoners**\n\n_These anthropologists (Daly, Mills and Brody) studied the Gitksan and Wet'suwet'en people intensively. Drs. Daly and Mills actually lived with the Gitksan and Wet'suwet'en for 2 and 3 years respectively after the commencement of this action. Their type of study is called participant observation but the evidence shows they dealt almost exclusively with chiefs which, in my view, is fatal to the credibility and reliability of their conclusions...they did not conduct their investigations in accordance with accepted scientific practices._\n\n_Most significantly, Dr. Daly lived with these people for 2 years, while this litigation was under way making observations on their activities, listening, and, I think, accepting everything they said, without keeping notes...._\n\n_Further, he was not aware of a comprehensive survey of over 1,000 persons conducted by the Tribal Council in 1979 which achieved an 80% return. This survey disclosed, for example, that 32% of the sample attended no feasts, and only 29.6% and 8.7% engaged in hunting and trapping respectively.... Apart from admissibility as evidence of its contents (for I have no way of knowing if the survey is accurate or representative, although some of its results tend to confirm the view I obtained of present Indian life), its significance is more in the fact that it was kept from Dr. Daly. Many of his views of Indian life may have been markedly different if he had access to this substantial body of information in the possession of his clients. For these reasons, I place little reliance on Dr. Daly's report or evidence. This is unfortunate because he is clearly a well qualified, highly intelligent anthropologist._\n\n\u2014Chief Justice Allan McEachern, 1991, _Reasons for Judgment, Delgamuukw v. R._\n\nEthnographic fieldwork necessarily involves \"intensively studying\" people by \"actually living\" with them for a prolonged period of time. Allan McEachern's allegation that the anthropologists dealt primarily with Chiefs is only partially supported by their actual opinion reports which include interviews with, and descriptions of the life of, many Gitksan and Wet'suwet'en people who are not Chiefs. That they relied on the Chiefs as authorities on formal aspects of oral tradition, political and social organization and ceremonial life constitutes what seems an entirely reasonable way to go about researching these topics through consultations with experts in specialized areas of knowledge. Chief Justice McEachern referred again to Daly's reliance on the Chiefs to support his own conclusion that Daly's testimony, about the degree to which traditional feasting was economically supported by the community, \"confuses the practice of chiefs making substantial contributions to feasts in which they are particularly interested with the day to day life of these people.\"\n\nWhat the Chief Justice does not seem to have understood is that the event called \"a feast\" represents years of planning, hunting, fishing, gathering, preserving, storing, saving, amassing and making foods, goods, artwork and crafts by a wide range of house, clan and family members and as such is very much interwoven with everyday life. The fact that a Chief, representing her or his House and clan, makes the ceremonial presentation of collectively generated money and goods at a feast does not mean that he or she alone is responsible for having accumulated the entire contribution. Furthermore, the cultivation and maintenance by chiefly families of good relations within their own Houses and with the community at large requires much social labour in day to day life.\n\nThis having been said, I hasten to acknowledge that many anthropologists and Aboriginal people themselves have criticized some ethnographers for working with elites to the exclusion of ordinary members of communities. And, the focus on spectacular events like feasts and potlatches further tends to render much daily labour performed by women, particularly, invisible. However, the goal of these critiques is to develop more democratic and inclusive research methodologies. The Chief Justice's comments reflect a prevalent notion repeated throughout colonial history that the British imperial mission was a noble and civilizing one, aimed at liberating the common folk of Indigenous societies from the shackles of tradition, and the tyranny of their own despotic chiefs: Europeans saving Aboriginals from themselves.\n\nContrary to the judge's criticism, Richard Daly did, in fact, take copious ethnographic fieldnotes. Typically, anthropologists maintain detailed journals that record technical data, descriptive observations, genealogies, formal interviews and other pertinent information. However, field journals also include anthropologists' personal reflections and interpretations, and information that research subjects specifically request anthropologists consider confidential. Anthropological fieldnotes are, therefore, private documents and on this basis Daly had, as journalists also do, refused to release his field journals to the court.\n\nChief Justice McEachern was strongly influenced by the results of the \"comprehensive survey\" that he cites in his _Reasons for Judgment_ , despite claims by the Gitksan and Wet'suwet'en that the survey was poorly designed and conducted, and had been rejected by them on methodological grounds; and despite the judge's own admission that he could not evaluate the reliability of the study. The apparent significance of the survey for the Chief Justice was twofold. First, neither the plaintiffs, nor their expert witnesses, had entered the survey into evidence, or referred to it. This omission appears to have aroused the judge's suspicions, and led to his conclusion that it had been purposefully withheld because it showed that, contrary to what Chief Justice McEachern thought their claim was, many Gitksan and Wet'suwet'en are not actively and regularly involved in what the judge considers \"aboriginal activities\" like hunting, fishing, trapping, gathering and feasting. The judge elaborates on this theme throughout the _Reasons for Judgment_ , juxtaposing participation in the contemporary economy in an either\/or fashion with participation in the traditional economy, writing: \"Daly placed far more weight on continuing aboriginal activities than I would from the evidence although he recognized the substantial participation of the Indians in the cash economy.\" Apparently unable to conceptualize the possibility of a viable contemporary Aboriginal way of life that combines hunting, fishing and gathering on the land with wage labour and commercial business activities, the judge could only understand this integration as a contradiction. He rejected Daly's observation that \"Gitksan and Wet'suwet'en persons regarded their land as 'their food box and their treasury' and young persons going hunting often say 'we are going to the Indian supermarket, to our land.'\" In short, the survey confirmed the judge's view that a hunting, fishing and gathering economy and a capitalist economy cannot coexist, that contemporary Aboriginals have been assimilated, and they no longer live \"truly Aboriginal\" lives, identical to those of their ancestors.\n\nUnlike the Gitksan and Wet'suwet'en's anthropologists, Sheila Robinson had conducted no fieldwork with the Gitksan and Wet'suwet'en or any other Aboriginal people. Hence, obviously, she could not be subjected to the same criticism of unscientific methodology that was levelled at Mills, Daly and Brody. The Chief Justice's thinking on this seems to reflect the notion that being a member of the dominant Euro-Canadian culture is evidence that the raw potential exists for an individual to develop an ability to discover the unmediated \"truth\" through the rational application of reason, learned in the process of professional education. Conversely, within this ideological framework, membership in a marginalized cultural group, like First Nations, is evidence of the opposite: they are considered _not_ to have the raw potential to overcome subjectivity and achieve objectivity. In the place of reason and rationality, they have feelings and \"culture.\" This was the rationale that lay behind the clause in the _Indian Act_ that remained in effect until 1951 which automatically stripped legal Indian status away from any Aboriginal person upon graduation from university. Obtaining a university degree, which presumably evidences the capacity for rational thought, was considered antithetical to being Aboriginal. Two attributes: being white and being untutored in Aboriginal ways of life, worked in tandem to enhance Robinson's image as an \"objective scientist,\" in contrast to the \"subjective advocates\" who testified on behalf of the Gitksan and Wet'suwet'en.\n\nJust as by dismissing the validity of oral tradition the Chief Justice in effect ruled out the possibility of any Aboriginal history constituting valid evidence, by rejecting ethnographic fieldwork as inherently \"unscientific\" and unreliable, Chief Justice McEachern ruled out the possibility of any anthropological research ever being useful to a court. The important point is not that anthropologists may be offended by Chief Justice McEachern's contempt for their discipline, but that the Judge's findings that are based on his evaluation of the evidence affect the everyday lives of Aboriginal peoples. As Michael Asch has written, \"Given the importance of the findings of fact to the judgment's decision about Gitksan and Wet'suwet'en aboriginal rights, it is fair to enquire as to the approach used for their determination.... If...the approach for determining the facts is not appropriate, this raises serious concerns about the judgement's findings in law.\"\n\n* * *\n\n. Geertz 1983.\n\n. For the record, however, Gitksan Elder Yagalahl (Dora Wilson) claims the Gitksan and Wet'suwet'en case may have been the exception that proves the rule. She wrote: \"Alfred Joseph and I were chosen to represent the Gitksan and Wet'suwet'en\u2014Alfred representing the Wet'suwet'en and I the Gitksan\u2014to sit in the courtroom and monitor the court case, and also be in an advisory position with our lawyers. That's one thing, our lawyers were under our instruction. They don't instruct us. We instruct them. We tell them what to do. That's what they are getting paid for.\" (Yagalahl 1992: 200).\n\n. Letter G. Plant, Russell & DuMoulin to Sheila Robinson, February 19, 1986, Exhibit 1191-9, _Delgamuukw v. R._ , Supreme Court of British Columbia, Vancouver, B.C.\n\n. Letter G. Plant, Russell & DuMoulin to Sheila Robinson, May 4, 1987, Exhibit 1191-17, _Delgamuukw v. R._ , Supreme Court of British Columbia, Vancouver, B.C.\n\n. Asch 1992: 237; See also Daly and Mills 1993: 1,6.\n\n. _Reasons,_ 1991: 68.\n\n. Dyen and Aberle 1974.\n\n. See Mills 1994 for a thorough discussion of Mills' testimony on this and other issues.\n\n. Robinson 1983: 405.\n\n. Robinson's \"performance\" is typical of court room ritual. See Brenneis 1988; Conley and O'Barr 1990; Levi and Walker 1990; Mertz 1988; O'Barr 1982; White 1985, 1990.\n\n. Transcripts,vol. 288: 21668.\n\n. Ibid., 22163.\n\n. _Reasons_ , 1991: 50.\n\n. Ibid., 50.\n\n. For thorough discussions of this aspect of colonial ideology in other corners of the globe see Said 1992; Spivak 1988.\n\n. Ibid., 50.\n\n. Asch 1992: 222.\n**Chapter 19: How Do You Know Whose Story is True?**\n\n_There is some conflict in the evidence about the start of this...[the protohistoric] period. Dr. Robinson believes that it was as early as 100 years before actual contact, mainly because of trade goods filtering into the territory both from the east and south as well as from known and unknown Russian (and possibly other) Asiatic travelers or traders who may have visited our coast. Other witnesses put the start of the protohistorical period later than Dr. Robinson, possibly about the time of the start of the sea otter trade in the last few years of the 18th or early 19th century...it would seem that the time of direct contact in the territory was not earlier than the early 1820's which is a reasonable date to select as the end of the protohistorical period...._\n\n\u2014Chief Justice Allan McEachern, 1991, _Reasons for Judgment, Delgamuukw v. R._\n\n**Sense v. Nonsense**\n\nThe only reference Allan McEachern made to the Crown's leading expert witness was in relation to her rather extravagant dating of the protohistoric period. Sheila Robinson's influence, however, was reflected in the Chief Justice's _Reasons for Judgment_ in a variety of ways and on a number of levels. Robinson rejected oral tradition and ethnographic fieldwork on methodological grounds, gave short shrift to linguistics and archaeology, ignored archival records and indirectly criticized the anthropologists who testified on behalf of the Gitksan and Wet'suwet'en. The Chief Justice's ruling repeats these dismissals. Allan McEachern made it clear in his comments during Hugh Brody's testimony that he did not believe that an examination of \"what white people think,\" or \"white frontier culture,\" had any bearing on this case. Nor, it would seem, did the Chief Justice think that academic credibility or professional reputation was relevant to the assessment of expert evidence. Perhaps this explains in part why the judge seemed to find it neither necessary nor appropriate to scrutinize the Crown's key expert witness with the same rigour that he applied to the experts who testified on behalf of the Gitksan and Wet'suwet'en.\n\nRobinson provided a methodological blueprint for the Chief Justice to follow. Points presented as material \"facts\" by Robinson were marshalled and proclaimed \"the truth\" by the Chief Justice, as were many of the abstract theoretical postulates that Robinson frequently substituted for empirical evidence. Robinson's well-practiced performance skills as a professional witness, and the rhetorical flourishes of her technical and academic \"bafflegab\" created an aura of arcane expertise around her testimony. Her report and evidence are difficult to comprehend and critique, not because the subject matter she deals with is particularly complex, or her arguments especially sophisticated, but because she adopts an extraordinarily obfuscating style when she writes and testifies.\n\nThe historical description of non-literate Indigenous cultures as they were before written descriptions of them were recorded by Europeans is principally the subject matter of ethnohistory. Robinson's opinion report fell within the rubric of ethnohistory, and she claimed specific expertise in this field. Ethnohistorians strive to reconstruct a holistic picture of specific Aboriginal cultures by methodically combining data from a number of sources: oral tradition, archaeology, linguistics, historical documents and fieldwork. Since these are the only possible data sources available for such reconstruction, the discipline of ethnohistory is largely defined by this methodology. In other words, ethnohistorical studies are based on interpretation and the discipline has treated the science of interpretation very seriously.\n\nRobinson rejected oral tradition as unreliable. Readers will recall that her dismissal of oral tradition and ethnography stood as yet another oxymoron when it became clear that 47 of the 82 sources Robinson listed in the bibliography to her report based their work substantially on oral history and\/or field work. Nonetheless, the judge agreed with Robinson's assessment and rejected oral histories as valid evidence.\n\nRobinson's treatment of archaeological evidence as a whole was also highly selective. In her initial proposal to Russell & DuMoulin, Robinson suggested that \"archaeological reports could be quite worthwhile,\" and she referred to archaeologists MacDonald, Allaire and Coupland as \"usually forming sound opinions.\" However, in her testimony at trial she changed her mind about this. The Crown's case was based on the argument that social ranking among the Gitksan and Wet'suwet'en emerged in response to the European fur trade, and that the hunting territories outside the major villages were used only sporadically and incidentally prior to their \"commercial\" exploitation for the European-Aboriginal fur trade. Gary Coupland, whose work was cited by the Gitksan and Wet'suwet'en's expert in archaeology, had found that: \"Archaeological evidence reflects the emergence of ranked society\" approximately 1500-2500 years ago, that is between 500 B.C.-500 A.D. Robinson dismissed Coupland's work, which did not support the Crown's argument.\n\nCoupland's findings are not disputed in the research literature. Where debate within archaeology does arise is on the question of migrations; that is, whether the present-day Gitksan and Wet'suwet'en are the direct, biological, descendants of the people who lived in the area at that time. Aboriginal oral traditions claim continuous occupation, but the archaeological jury is still out on the question. Chief Justice McEachern seized upon this debate in archaeology to support his conclusion that: \"There is no doubt, in my view, that there has been human habitation at locations on the lower and middle Skeena River extending at least from Prince Rupert harbour in the west to Hagwilget canyon, and at Moricetown, for at least 3000 years or more. This has been established in the findings and conclusions of several reputable archaeologists.... The difficulty from the plaintiff's point of view, is that none of this evidence...relates distinctively to the plaintiffs. Any aboriginal people could have created these remains.... The archaeological evidence establishes early human habitation at some of these sites, but not necessarily occupation by Gitksan or Wet'suwet'en ancestors of the plaintiffs.\" As he was disturbed by the appearance of post-contact phenomena like guns and the Hudson's Bay Company in what he thought should be \"uncontaminated\" oral histories, Allan McEachern was also concerned that the archaeological evidence was rendered \"highly equivocal with findings of white man's garbage mixed with possible archaeological features....\" Robinson did not deal substantially with linguistic evidence. The Chief Justice found linguistics baffling but impressive, describing the discipline as: \"a mysterious process only properly understood by very learned persons,\" and citing it as valid evidence that the descendants of the Gitksan and Wet'suwet'en had lived in the territory, if not since time began, at least for \"a long, long time.\"\n\nRobinson declared that the historical record vis-\u00e0-vis the Gitksan and Wet'suwet'en was \"virtually mute.\" Hence, she did not conduct any archival research in connection with her report. She claimed that the first written records kept by a European observer, Hudson's Bay trader William Brown who arrived in 1822, were irrelevant to her study because, she said, Brown was describing a Gitksan and Wet'suwet'en social organization that had radically changed as a result of the introduction of European commodities into Indigenous trade exchanges preceding Brown's arrival: \"None of our contemporary knowledge is untainted by European influence which was manifested long before relevant written records were kept,\" she wrote. In other words, the only reliable information, according to Robinson would have to come from the written records of people who didn't write, or written records by Europeans who by their very presence\u2014or, in this case, by the presence of commodities manufactured by them\u2014would have so changed Aboriginal culture as to render it not \"really aboriginal\" anyway; or from a 300-year old person. In short, Robinson denied the validity of all the sources of knowledge in the discipline in which she was qualified by the Court as an expert witness.\n\nRobinson provided a facade of academic and putatively \"scientific\" legitimacy for the Crown's, and Chief Justice McEachern's, legal arguments, cultural beliefs and economic and political interests. Evolutionary supremacy is seen as being manifested in its purest form in the law as it has rationally developed over the course of western history, through the increasingly sophisticated application of reason to the resolution of disputes and the administration of civil life. To the Chief Justice's way of thinking he embodies justice, truth and common sense. In such an ideological framework, the grace of the Christian God, the will of the European sovereign, the security of the nation state and the best interests of the people are one and the same: woven together into a seamless whole. Just as the Crown's sovereignty is legitimized by reasoning backwards from consequences to causes, any lingering doubts about the obvious rightness of the status quo should be easily dispelled by referring back to the fact that the dominant culture is dominant. Were it not superior to those it imagines it has conquered, it would not have been victorious. Finally, the proof is in the pudding: Robinson drew no criticism because the truth of her story is evidenced by it being the story of those who claim exclusive monopoly over the truth.\n\n**Fact v. Fiction**\n\n_The purpose of litigation is to settle a dispute with finality. Whether or not the decision is historically 'correct' is, from one perspective\u2014that of the court\u2014irrelevant.... Determining the truth of what happened is incidental to the courts role in society which is to secure peaceful settlement of economic, social, and political conflicts between two or more litigants._\n\n\u2014Historian Donald Bourgeois, 1986, _The Role of the Historian in the Litigation Process_.\n\nDr. Arthur Ray's evidence, as well as that of the archaeologists Drs. Ames and MacDonald, linguists Drs. Rigsby and Kari, Ms Heather Harris' genealogical evidence, and Brown's records, were cited by the judge in support of his conclusion that \"the ancestors of a reasonable number of the plaintiffs were present in parts of the territory for a long, long time prior to sovereignty.\" The judge wrote that he had \"no hesitation accepting the information contained in\" William Brown's records, and he acknowledged that Dr. Arthur Ray \"has excellent qualifications in his special area of expertise.\" Allan McEachern went on to quote the following passage from Ray's opinion report as \"the strongest statement supporting the plaintiffs' basic position which is to be found in any of the independent evidence adduced at trial:\"\n\nWhen Europeans first reached the middle and upper Skeena River area in the 1820s they discovered that the local natives were settled in a number of relatively large villages. The people subsisted largely off their fisheries which, with about two months of work per year, allowed them to meet most of their food needs. Summer villages were located beside their fisheries. Large game and fur bearers were hunted on surrounding, and sometimes, on more distant lands. Hunting territories were held by 'nobles' on behalf of the lineages they represented and these native leaders closely regulated the hunting of valued species. The various villages were linked into a regional exchange network. Indigenous commodities and European trade goods circulated within and between villages by feasting, trading and gambling activities.\n\nHe went on to emphasize that \"game was never really plentiful in the territory and fishing was the mainstay of the economy;\" that Brown had noted that \"the chief's control of territories was not exclusive...and was sometimes limited to beaver.\"\n\nThe Chief Justice continued:\n\nThe foregoing must be considered in the context of the larger picture which emerged from the evidence. First, it would be incorrect to assume that the social organization which existed was a stable one. Warfare between neighbouring or distant tribes was constant, and the people were hardly amenable to obedience to anything but the most rudimentary form of custom. Brown held them in no high esteem, partly because of their addiction to gambling, and Ogden, about whom there are different views...described them most unkindly. I conclude from the foregoing, however, that there was indeed a rudimentary form of social organization....\n\nSheila Robinson's evidence provided the rationale by which the judge was able to reconcile the apparent contradiction between his acceptance of the literal truth of William Brown's written reports that described the Gitksan and Wet'suwet'en as living in an organized, property-owning society in 1822, and his application of Lord Sumner's dictum in the _Re: Southern Rhodesia_ precedent reserved for cases where \"tribes are so low in the scale of social organization that their usages and conceptions of rights and duties are not to be reconciled with the institutions and legal ideas of civilized society.\" The resolution to this apparent paradox was provided by Robinson's argument that Gitksan and Wet'suwet'en cultures had changed rapidly and dramatically in the time just prior to Brown's arrival as a result of the impact of the European-Aboriginal fur trade. Therefore, the Crown and Robinson argued that Brown was describing not a \"truly aboriginal\" society, but rather one already significantly assimilating and inevitably disappearing. The Chief Justice agreed, and wrote in his _Reasons for Judgment_ :\n\nIn fact, active trade was underway at the coast and spreading inland for at least 30 years before trader Brown arrived at Babine Lake, probably converting a Gitksan and Wet'suwet'en aboriginal life into something quite different from what it had been....\n\nI find the weight of evidence supports the view that the fur trade materially changed aboriginal life before or around the time trader Brown was making his records at Fort Kilmauers. That does not prevent me from accepting Dr. Ray's opinion that Indian social organization did not all arise by reason of the fur trade. I think the evidence supports that, by 1822, the Indians of the Babine Lake region had a structure of nobles or chiefs, commoners, kinship arrangements of some kind and priority relating to the trapping of beaver in the vicinity of the villages.\n\nFinally, he claimed he was presented with too much evidence to be able to do more than \"extract from it an impression of what was going on.\"\n\nSummarizing the historical evidence, the judge evaluated the particular historians who had testified as expert witnesses during the trial, writing:\n\nLastly, I wish to mention the historians. Generally speaking, I accept just about everything they put before me because they were largely collectors of archival, historical documents. In most cases they provided much useful information with minimal editorial comment. Their marvellous collections largely spoke for themselves. Each side was able to point out omissions in the collections advanced on behalf of others but nothing turns on that.\n\nI do not accept that part of the evidence of...[the Crown's expert witness] Mr. Williams which suggests legal consequences from Gitksan or Wet'suwet'en compliance with Canadian law.... The honour of the Crown precludes me from giving effect to this defence. In my judgement, the plaintiffs have not directly or indirectly released their causes of action...I do not find, as a matter of law, that the acceptance of British Columbia law, or conformity with it, precludes them from advancing their claims for aboriginal interests. In my view, the Indians claims have not been discharged by any conduct on their part.\n\nHistorian Robin Fisher has subsequently pointed out that, while at first glance Allan McEachern appears to respect historians, \"a more careful examination of the evidence provided in the judgment will show that McEachern, in fact, paid very little attention to historians.\" Fisher's argument is illustrated by the judge's characterization of historians as \"largely collectors of documents,\" and his dismissal of their interpretations. In other words, McEachern heard the \"facts\" written in documents speaking directly to him, and he was grateful to the historians for bringing these written voices to his visual attention. The judge found debates among historians about various interpretations of documents \"interesting,\" but not particularly relevant to the case at hand. Robin Fisher accused McEachern of having \"failed to listen to the custodians of the past in his own culture.\"\n\nFisher attributes many of McEachern's historical errors to what he describes as the judge's \"xerox, scissors and paste\" methodology. Fisher argues that the Judge appropriated historical evidence through a series of decontextualizing practices. \"...The first step in this procedure is to pull the documents out of their original context by use of the xerox machine,\" Fisher began. \"Thus, for example, a letter from Governor James Douglas to the British Colonial Secretary on Indian land policy is isolated from his numerous letters on other issues of the day. It is as if Douglas did his thinking in watertight compartments rather than as a busy governor dealing with a dozen overlapping questions at the same time.\" Fisher continued: \"Individual documents are then cut to pieces so that excerpts can be quoted. The historical sections of the judgment consist of long successions of quotations from original sources strung together with commentary by the judge. The trouble with scissors and paste is that scissors cut things out of context and, once removed from their setting, all the bits of the document are of equal weight. After the individual pieces have been trimmed to a suitable shape, with the application of paste, the past can be stuck back together according to a new, and more acceptable, pattern.\" Finally, Fisher concludes, McEachern's methodology can be reduced to arbitrariness. \"But for McEachern, the best historians are not those who have done adequate research or drawn the most logical conclusions,\" he argues, \"but simply those who appear to support his views.\" Fisher claimed that, rather than being idiosyncratic, Chief Justice McEachern's use of history unfortunately represents common practice: \"...there is also a developing tradition in this province of lawyers and judges presuming to be historians, whether in or out of the courtroom. Having made judgments about legal issues that have a historical dimension, they presumably feel that they are thereby qualified to write history. What these judges and lawyers are often doing is shaping the past to serve the needs of the present, which is not quite the same thing as writing history.... We can safely assume that none of these legal professionals, let alone the bar associations, would let an historian walk in off the street and take over one of their cases just for a change of pace....\"\n\nA number of critiques of Chief Justice McEachern's use of both oral and written history have been written since the 1991 publication of his _Reasons for Judgment_ in _Delgamuukw v. R._ Anthropologists Julie Cruikshank and Andrea Laforet and legal scholar Joel Fortune have discussed the judgment in the context of current academic debates on the nature of historical understanding. Citing anthropologist Arjun Appadurai, Laforet set out four criteria that she argued are universally applicable to the evaluation of historical validity. These are: authority, continuity, depth and interdependence. Laforet then analyzed both the _adaawk_ and _kungax_ , and what she calls \"McEachern's history\" according to Appandurai's four criteria, finding that, while radically different, both can be seen to formally fulfill these requirements in culturally-specific ways.\n\nJulie Cruikshank argued that both oral traditions and written accounts are deeply embedded in social processes. \"Written records do not 'speak for themselves,'\" Cruikshank wrote, \"and...like oral testimonies...must be understood within the context in which they were and are produced.... The lessons to be drawn from comparing oral with written accounts are not about the cultural relativity of texts,\" she concluded, \"but about power and domination.\"\n\nJoel Fortune began with a review and assessment of the role played by history in land claims litigation in general, and in recent Supreme Court of Canada decisions in particular. Fortune put the central question succinctly: \"In the context of history, philosophy and epistemology this illustrates a fundamental problem: how can past events or past cultures be retrieved from the intangible and presented to-day.\" Both Aboriginal litigants and the Crown used history to support their positions, Fortune continued. Native litigants argued that original injustices, supposedly relegated to the past, continue into the present. This gives temporal depth to their grievances. Interpreting the history of Aboriginal\/non-Aboriginal relations in this way also provided support for their argument that they have resisted domination and assimilation, and have persisted in exercising rights embedded in a distinct culture for \"a long, long time.\" In these ways, the past and the present are linked in Aboriginal legal representations. However, past and present were uncoupled in First Nations' arguments when contemporary adaptation and cultural viability were highlighted. The Crown, for its part, separated the past and the present when they advanced arguments to legitimate the claim that the past should be forgotten, and \"we can only be just in our time.\" Crown arguments that relied on assumptions of a radical break between pre-contact and post-contact, Aboriginal and non-Aboriginal cultures and practices, were also supported by a theoretical stance advocating significant differences between historical and contemporary conditions. The past and the present were joined in the Crown's arguments, however, when the past served the purposes of the present, as in the use of precedents. \"In the end,\" Joel Fortune surmised in regard to Chief Justice McEachern's conclusions on historical evidence, \"the dominant culture asserted its historical vision over that of the Gitksan and Wet'suwet'en.\"\n\nArthur Ray charged that the Crown, by relying on documents written fifty or sixty years later than the time period being discussed, \"attempted to interpret history backwards.\" He concluded that, \"...after 374 days of trial covering all aspects of Gitksan and Wet'suwet'en history in depth, Justice McEachern still held the same eurocentric view of Native people that has been an unfortunate judicial and political tradition in British Columbia since the colonial era.\"\n\nThe more important issue is, however, that the law is one of the most powerful institutions in Canadian society. The notion that we are governed by reason, logic and the intelligent evaluation of evidence is what legitimizes the state's and the courts' sole control over armed force and \"the means of destruction.\" Fortune sets out the important problem pointed to by the numerous critiques of Chief Justice McEachern's judgment written by academics and intellectuals, when he writes: \"The reason why the courts should critically examine the assumptions that underlie an unproblematic conception of historical knowledge is that these assumptions, as we have seen, inform the law. If the assumptions, when they are challenged, cannot be justified, neither can the law.\"\n\nThe important problem regarding the evaluation of expert evidence in court is not what Sheila Robinson's, Richard Daly's, Hugh Brody's, Antonia Mills,' Arthur Ray's or any other individual expert's intentions or political sympathies may be, but rather whether judges should respect professional standards and integrity in their assessment of scholarly knowledge presented by expert witnesses in the courtroom. Not because these standards are infallible guarantees of any kind of absolute truth, but because _legal findings are based on historical and cultural interpretations_ , and it therefore behooves the judiciary to at least respect the criteria for credibility adopted by the academic institutions of their own culture. The problem, in other words, is not that expert witnesses may mock the court, but rather that the courts make a mockery of expert witnessing.\n\n**Truth v. Lies**\n\n_Pointing up repeated failures to discover any but historically contingent foundations for thought does not in itself have to provoke a crisis of inquiry and understanding.... It requires rather that we acknowledge that all human inquiry is necessarily engaged in understanding the human world from within a specific situation. This situation is always and at once historical, moral and political._\n\n\u2014Anthropologist Paul Rabinow, 1987, _Introduction: The Interpretive Turn, A Second Look_.\n\nIn carrying out the Crown's first strategy as defendants\u2014that is, attempting to undermine the credibility of the plaintiffs\u2014Sheila Robinson continually reiterated and reinforced the notion that no valid or reliable knowledge about pre-contact Aboriginal societies exists, and that _all_ theories and research findings are equally speculative. She then proceeded to offer an authoritative report on this very subject, using the same sources she had discredited. There is, thus, at the core of her argument, a logical fallacy that makes trying to understand her work feel like a descent into madness: she asserts a truth whose conditions of possibility she has already ruled out on _a priori_ grounds. The Chief Justice, empowered by legal positivism's tradition of substituting law for fact, followed suit.\n\nPhilosopher Peter Dews describes these ideological contortions where the real and the imagined, and fact and fiction, are capriciously fused at one moment, and distinguished at another\u2014not by principles of logic, reason, moral or ethical beliefs, or justice, nor by reference to culturally-specific rules of interpretation, or to observations of material phenomena, but according to the desires and interests of the powerful\u2014as a feature of contemporary western culture. Simply put, it is becoming increasingly acceptable to boldly follow the maxim that winning or losing is all that matters: playing the game with integrity is increasingly coming to be seen as naive at best, and irrelevant at worst.\n\nJoel Fortune, in his analysis of Chief Justice McEachern's approach to history in his _Reasons for Judgment_ in _Delgamuukw v. R._ , outlined three theoretical positions in contemporary historiography: interpretivism, postmodernism and positivism. He turned to British historian E. H. Carr's well-known book, _What Is History?_ as typical of the interpretivist school and quoted the following as representing the \"essence\" of Carr's argument: \"What is the criterion which distinguishes the facts of history from other facts about the past?...The facts speak only when the historian calls on them: it is he who decides which facts to give the floor to, and in what order or context. It is the use to which certain facts are put that makes them historical.\" Carr's book was cited by the Gitksan's and Wet'suwet'en's lawyers to support their argument that both oral and written histories should be subject to established criteria, standards and processes for validation.\n\nIn contemporary intellectual debates, arguments about subjectivity, objectivity and truth are often described as a battle between modernism and postmodernism. Modernism refers to social theories that have arisen since the European Enlightenment that claim that the world exists independently of the human knower. The task of the modernist social scientist is to achieve objectivity by eliminating subjective biases like feelings and attitudes. This achieved, the researcher should be able to determine, and then reveal and explain, the single definitive truth that has awaited her or his discovery. Objectivist social science strives to emulate the natural and physical sciences that its methodologies mimic. Postmodernism refers to theories that have arisen, for the most part, during the last twenty-five years. Postmodernists argue that all knowledge is mediated through the culturally and historically specific lens of the knower, and whether or not there is a \"reality\" that exists outside of our apprehension of it we may never know for certain.\n\nFortune illustrated a postmodern approach by referring to the work of literary critic and historian Hayden White. White's postmodern analysis of history argues that the meaning of historical texts is determined by the diverse contexts in which they are produced and interpreted. Texts, therefore, according to White, inevitably reveal their writers' and readers' interpretations of their subject matter, more than they describe observed events and \"facts.\" \"Recent literary theory...questions the notion that ancient documents, as 'texts' have any independent meaning at all,\" Fortune wrote.\n\nWhite's position and that of Carr are philosophically comparable in that both deny the possibility of \"objective history\" in a positivist sense. That is, both analyze historical documents as human creations that should not be understood outside of the social and political context in which they are written, read and analyzed. However, the two schools\u2014interpretivism and postmodernism\u2014differ on the question of whether or not a consensus can be achieved concerning criteria for what may constitute a \"fact,\" and whatstandards may be sufficient to evaluate one account as being more valid than another. Fortune noted in regard to postmodernist theories that \"for some historians, this proposition implies an especially pernicious relativism: the simultaneous interpretation of text and context appears to denude the past of any objective content.\" In this book, I have critiqued the positivist ideology and modernist pretensions to exclusive possession of a single truth that the Crown's argument and the Chief Justice's _Reasons for Judgment_ reveal. My first critique has been from _inside_ the Crown's positivist ideology. I have argued that, within the terms of their own framework of analysis, the Crown and Chief Justice McEachern rejected what should constitute valid evidence: empirical data drawn from scientific studies. My second critique has been launched from _outside_ the law, and I have argued that their framework of analysis is itself thoroughly eurocentric, and frequently crosses the line to racism. That is, the law demands that Aboriginal people present their claims within a European-derived discourse, and will not hear First Nations on their own cultural terms. I have tried to wage this critique, however, without making claims to a contrary but still universal and objective \"truth\"; that is, without acquiescing to the Crown's objectivist framework.\n\nObjectivity and subjectivity\u2014understood as absolute abstract concepts\u2014are false opponents: two sides of the same coin. They are abstractions marking the extreme poles on either end of a continuum. Moving the discussion out of philosophical abstractions and into lived experience makes this obvious. People in specific relationships with each other and in particular contexts in the course of their everyday living, move back and forth along this continuum from issues where there is a high degree of consensus on what is \"true enough to be _true_ ,\" that is, where we can agree to shared meanings sufficiently to carry on conversation. For example, we can agree that a tree, is a tree, is a tree. However, as we move along this continuum, we find places where perspectives diverge significantly. For example, a tree represents profit; a tree offers a job; a tree symbolizes a sacred gift from the Creator. At this point, there are many _truths_ that depend on the speaker's perspectives and interests, the context in which \"the tree\" is being talked about, and the purpose of the discussion.\n\nIn the case of the issues in dispute in the Gitksan and Wet'suwet'en trial, there is a high level of public, academic and cross-cultural consensus that it is empirically true that there were resources in the hunting territories and a significant degree of consensus on the likelihood that it is \"true enough\" that the Aboriginal peoples of the territory used these resources. There is less consensus that these two historical facts constitute justification for the recognition of contemporary legal Aboriginal rights. At this level of discussion multiple \"truths\" about the resources in the Gitksan and Wet'suwet'en hunting territories emerge. What is considered \"truth\" may be perpetually subject to review and change as new information, new perspectives, and new ways of understanding evolve. Claims to have \"discovered\" an unassailable \"truth\" at this level ultimately reflect points of view, experience, material conditions, and moral and political values. However, to reason backwards from the third level of debate to the first and second, and to argue then that all \"facts\" are equally contestable, and all interpretations equally valid, is spurious.\n\nTruth in human relations is not a _thing_ that can be discovered or revealed at either pole: in either \"pure\" objectivity or \"pure subjectivity.\" Truth-seeking is an interactional process. How else could we come to know the _truth_ about the meaning of land\u2014or trees\u2014or the value of a way of life, or what makes a joke funny, or what being in love is? Truth in human relations emerges from within human relations, through conversation and dialogue, through conflict and debate. These truths are therefore always open to critical moral interrogation. The more important questions are what actions may people take, how may people be treated, according to particular \"truths\"? What are the consequences of believing, and organizing society on the basis of, one truth and not another? Whose interests are being served? At this point, truth becomes a moral question about justice, more than an empirical question about observations. Cultural differences present a major challenge to the search for such truths, the heart of which lies in communication. Power, of course, plays a significant role in this \"intersubjective validation\" process as well: whose truth will come to rule the day?\n\nMany contemporary scholars and activists embrace postmodern critiques as shields against totalitarianism: if all points of view are necessarily partial, are constructed by human beings rather than by extra-human forces like God or Nature or Science or Culture, then all are ultimately open-ended and subject to critique, deconstruction and reconstruction. In an increasingly multicultural world, postmodernist approaches in anthropology also mitigate against ethnocentrism in their critical stance towards its universalizing tendencies, and the demand that many points of view on a subject be taken into account when ascertaining meaning helps to displace the dominant culture's monopoly on defining meaning for everyone. However, left within an abstract philosophical debate, postmodernism degenerates into a nihilistic, amoral kind of relativism: nobody's wrong if everybody's right.\n\nThat postmodern analyses are particularly difficult to translate into concrete strategy in the courtroom has been the subject of considerable debate among adherents of the critical legal studies movement. I agree with Carlo Ginzburg who argues that: \"There is an element in positivism that must be unequivocally rejected: the tendency to simplify the relationship between evidence and reality and to dismiss cultural mediation between the two.... [However,] Instead of dealing with the evidence as an open window, contemporary skeptics regard it as a wall, which by definition precludes any access to reality. This extreme anti-positivistic attitude, turns out to be a sort of inverted positivism.... Even if we reject positivism, we must still confront ourselves with notions like 'reality,' 'proof,' and 'truth'.... The fashionable injunction to study reality as a text,\" Ginzburg continues, \"should be supplemented by the awareness that no text can be understood without a reference to extratextual realities.\"\n\nBased on a form of \"inverted positivism,\" Robinson and McEachern appear to have concluded that there are no facts worthy of the name that can be told by anyone, except them. This approach leaves Robinson free to pursue theoretical speculation based on outmoded social science, while the Chief Justice is released to pursue his own speculations guided by legal positivism, archaic legal precedents and his own \"common sense\" nurtured by a lifetime's immersion in the \"founding myths of White British Columbians.\" Sheila Robinson's testimony, and Chief Justice McEachern's ruling, exemplify what critics of postmodernism fear most: that the adoption of a postmodern stance will result in the only \"truth\" being the one that power will tell. Robinson's testimony and the Chief Justice's _Reasons for Judgment_ , while claiming modernist and positivist authority, read like exemplaries of a \"postmodern pastiche,\" where the writers seem to have imagined themselves gamblers in a carnival of \"free-floating signifiers,\" where all possibilities are equally valid, where argument need not be grounded or falsifiable, and where the judicial market-place ultimately determines the truth value of any particular statement.\n\nContemporary life is lived within a social world structured by radically unequal, and often unjust, relations of power. There may be many perspectives, many stories, and many truths; but the story chosen and the truth proclaimed by the law has the might of the modern state, and ultimately the force of arms, behind it.\n\n* * *\n\n. Krech 1991; Ray 1991(a); Williams, S. 1988.\n\n. Exhibit 1191-2.\n\n. Transcripts, vol. 289: 21727-21730.\n\n. Coupland 1986: 18.\n\n. Ives 1987\n\n. _Reasons_ 1991: 59-61.\n\n. _Reasons_ 1991: 6.1\n\n. _Reasons_ 1991: 68-72.\n\n. Yet, in her 1983 doctoral dissertation, Robinson criticized two other scholars for using a similar approach, writing: \"...coming to grips with what is and what is not contained in the ethnographic and early historic records is a necessary precondition for further general theoretical investigations. There are already too many statements in Northwest Coast cultural historical research to this effect: When the documents are silent, I have chosen to rely on inference or speculation to provide an analysis that is systematic rather than piecemeal...[Taking these liberties] is, I believe, scientifically justifiable in that it provides hypotheses that can be tested by future documentary and\/or field research.\"\n\n. _Reasons_ 1991: 73.\n\n. Ray 1987: 55, quoted in Reasons 1991: 73.\n\n. _Reasons_ 1991: 73-74.\n\n. _Reasons_ 1991: 43.\n\n. McEachern is referring to the province's argument that the Gitksan and Wet'suwet'en, \"by accepting and using reserves, and by conforming generally with the law of the province, have given up their Aboriginal rights.\"\n\n. Fisher 1992: 44\n\n. Fisher 1992: 44.\n\n. Ibid.,46.\n\n. Ibid., 48.\n\n. Ibid., 54.\n\n. Cruikshank 1992: 39.\n\n. Fortune 1993: 91.\n\n. Ibid., 116.\n\n. Ray 1991(a): 26.\n\n. Fortune 1993: 115.\n\n. Rabinow 1987: 21-24.\n\n. Dews 1987: 15-19.\n\n. Carr 1964.\n\n. Carr 1964: 10, quoted in Fortune 1993: 99.\n\n. Foster 1986; Nicolson 1990; Jameson 1984; Lyotard 1984.\n\n. White 1987: 185-203.\n\n. Fortune, 1993: 102.\n\n. 1993: 102.\n\n. See Rosaldo 1989.\n\n. See Ulin 1984 for an explication of the theory of \"intersubjective validation\" of truth statements.\n\n. See Boyle 1985; Coombes 1989; Cornell 1992; Derrida 1992; Frug 1992.\n\n. See McLaren 1993.\n\n. See, for example, Cornell 1992; Delgado 1988; Gordon 1984).\n\n. Ginzburg 1991: 83-84.\n\n. See Miller 1992\n\n. Callinicos, 1989; Hunt 1986; Pryce 1992; Ross 1988; Scott and Butler (eds.) 1992; Zavardadeh and Morton 1993.\n\n. See Terdiman 1987: \"Unlike literary or philosophical hermeneutics...[the interpretation of texts. ed]...the practice of interpretation of legal texts is theoretically not an end in itself...but...is one way of appropriating the symbolic power which is potentially contained within the text. Control of the legal text is the prize to be won in interpretive struggles. (809).\"\n**P** **ART** **VII:**\n\n**A** **D** **A** **RBITRIUM**\n\n**(A T WILL)**\n**Chapter 20: Moonlighting as an Anthropologist** **and an Historian**\n\n**Drive-by Ethnography**\n\nHaving discredited and erased all dissenting voices, Chief Justice McEachern saw before him a _tabula rasa_ that mirrored law's image of _terra nullius_. He proceeded to write his own ethnography of the Gitksan and Wet'suwet'en, and his own history of British Columbia, on the blank pages he had created for this purpose.\n\nDespite his apparent contempt for anthropological research methods, the Chief Justice took the opportunity to conduct an ethnographic excursion of his own, described by anthropologist Noel Dyck as an exercise in \"drive-by ethnography.\" McEachern began with a stereotypical assertion of anthropological authority. He claimed for himself the unique expertise only achieved by personally experiencing fieldwork:\n\nBefore I describe the magnificent country we viewed, I wish to say that no one can gain a proper appreciation of the overwhelming vastness and isolation of this magnificent but almost empty territory without spending at least the amount of time I spent there....\n\nCombining his ethnographer's role with that of an eye-witness, the Chief Justice described a three-day trip during which he was \"able to see about two-thirds of the territory\":\n\nI visited many parts of the territory...during a 3-day helicopter and highway 'view' in June 1988.... I also took many automobile trips into the territory during many of the evenings of the nearly 50 days I sat in Smithers. These explorations were for the purpose of familiarizing myself, as best I could, with this beautiful, vast and almost empty part of the Province.\n\nI took the first six weeks of evidence in May-June 1987 at Smithers, a community situated in the Bulkley Valley in approximately the centre of the territory. Smithers is a community of about 7000 persons, mainly of European descent, who have made the Bulkley Valley into a fertile farming and dairy region....\n\n...I usually took advantage of the long spring evenings to visit areas in the territory including several trips to each of Kitwancool, Gitwangaak, Kitsegukla, Kispiox, the Hazeltons and Houston, plus evening trips by private logging roads to Smithers Landing on Babine Lake, Fulton and Topley Landing, and a single trip to Burns Lake. In this way I gained a good appreciation of the Bulkley and Skeena River corridors and their villages where at least 90% of the residents of the area, including most of the Gitksan and Wet'suwet'en, make their homes....\n\nI was taken to and shown many of the remote northerly and southerly portions of the territory. On June 6th we visited the northern territories claimed by the Gitksan people and Mr. Neil Sterritt, a Gitksan Hereditary Chief, provided us with a running commentary on the important landmarks. On June 7th we visited the areas claimed by the Wet'suwet'en people and Mr. Alfred Joseph, a Wet'suwet'en Hereditary Chief, was our principal tour guide.... On June 8, 1988...I motored down the Skeena River from K'san to Kitwanga and beyond where many of the fishing sites I heard about in the evidence were pointed out to me....\n\nI also wish to add that I wish I could have spent more time in the territory but helicopter travel is very expensive and although logging roads are pushing further and further into the territory, they are not always available for private traffic, and exploration by land in such country is a long, slow, tedious and often uncomfortable enterprise....\n\nWe started at Smithers and after a short detour to the west to avoid clouds in the Debenture Park area, we travelled north about 10 miles east of the westerly external boundary of the territory...to Kotine Mountain, where we stopped near its peak in a driving rainstorm to make observations of the Babine Range to the south-east and other landmarks of interest....\n\nAllan McEachern's personal account was written in a style characteristic of explorers' journals and Victorian travelogues:\n\nWe continued westerly along the Skeena. On this leg of our voyage of exploration we passed but did not stop at the ancient but now totally deserted village of Gitengas where there are no buildings still standing. We then left the Skeena and went north up the Slamgeesh River where we stopped for lunch at a point on the old Telegraph Trail where Chief William Backwater was born and grew up. There are no residents there now and only a few grave buildings and one small, totally uninhabitable building remains....\n\nWe then flew south to rejoin the Skeena.... We then flew a short distance east and then swung south down Shedin Creek to the confluence of the Babine and Skeena Rivers, which is the site of the ancient village of Kisgegas, which was once the largest of the Skeena villages. It is a large cleared area with the remains of a number of buildings including an almost fallen-down church built around 1930 but all the residents have left here although there are a few cabins on the other side of the river which I understand are occupied for part of the year by Joshua McLean. Access to Kisgegas is now possible by logging road from Kispiox but, except for the fish runs in the summer, the village has been largely empty since the 1940s... When we reached the native village of Kitsequecla we stopped for fuel. We then proceeded south-east up the Kitsequecla River until we re-entered the rich Bulkley Valley on the south side of mighty Hudson Bay Mountain where the Bulkley River flows north, but we turned south up the valley to Smithers where we terminated a fascinating voyage of exploration and discovery.\n\n**Civilization by Tautology**\n\nChief Justice McEachern's historical narrative was carefully divided into relevant time periods, and he returned repeatedly throughout his _Reasons for Judgment_ to differentiate between them and to mark their legal significance:\n\nThe difference between the pre-historic, proto- and historic periods is relevant to the question of determining what are aboriginal as opposed to non-aboriginal practices. I find that the aboriginal practices of the plaintiffs' ancestors were, first residence, and secondly subsistence\u2014the gathering of the products of the lands and waters of the territory for that purpose and also for ceremonial purposes. These both predated the historical period for a long, long time, and continue into the historic period (with new techniques) up to the time of sovereignty and since that time but with decreasing frequency.\n\nI find that the plaintiff's ancestors probably lived an aboriginal lifestyle mainly in the vicinity of their villages and during travel between their villages and this continued until sovereignty.... While there is no doubt the Indians harvested their subsistence requirements from parts of the territory, it is impossible to conclude from the evidence that these...activities...were anything more than common sense subsistence practices, and are entirely compatible with bare occupation for the purposes of subsistence. The evidence does not establish either a policy for management of the territory or concerted communal onservation.\n\nThe testimony of the Gitksan and Wet'suwet'en Chiefs, and their expert witnesses in anthropology, archaeology, history, and ecology had been replete with such evidence. However, like Sheila Robinson, Chief Justice McEachern dismissed this research, and claimed that there is no reliable knowledge on the subject. He then ventured boldly into theoretical speculation to substantiate his position:\n\n...it is likely, in my view, that the Indians in those early times would have searched for food and other products in the vicinity of their villages. There was no need for them to go very far for such purposes, and I know of no reason to suppose they did. It is likely that they visited, or made war with each other or with other peoples, using both the trails shown in some of the sketches adduced at trial, and by way of the great rivers both in summer and winter although there is little evidence they possessed boats. They must have had a way to cross rivers which would have been a formidable undertaking. I am sure they used some of the frozen rivers as cold weather sidewalks.\n\nCreating fact from fictions of his own making, the judge then continued his speculation by determining a specific distance that people may have travelled from their villages:\n\nIt seems likely these early aboriginals would also have used the lands alongside the great rivers, between their villages, for aboriginal purposes. I do not question that some of these ancestors may well have lived and survived considerable distances from the villages and great rivers but they would be hardy, generational recluses whose personal preference to absent themselves from villages even for their lifetimes would not create aboriginal rights based upon indefinite, long time use. Having regard to the difficulties of pre-contact travel in the territory it might be argued (I do not believe it was), that both the Gitksan and Wet'suwet'en would not have used lands and waters any great distance from their villages. Perhaps an area of 20 or 25 miles around their principal villages would be appropriate.... On the other hand, a hunter in reasonable country could comfortably walk 20 or 25 miles in a day....\n\nIn some passages of his text, Chief Justice McEachern appeared to struggle with a desire to understand the Gitksan and Wet'suwet'en \"on their own terms.\" He acknowledged, for example, that they continue to have attachments to their lands, even when they are not using them for \"aboriginal purposes\":\n\nI do not think I should be quick to treat aboriginal use as abandoned, but common sense dictates that abandoned rights are no longer valid and land must be used or lost. While recognizing that a right which is not used can be treated as abandoned, the law does not like the principle because it lacks certainty. It also requires the Court to look objectively at what may well be a subjective state of mind...I do not think I can safely conclude that the intention to use these lands for aboriginal purposes has been abandoned even though many Indians have not used them for many years.\n\nChief Justice McEachern did not, however, struggle with any questions concerning his definition of what an \"aboriginal life,\" or \"aboriginal purposes\" might be. Nor, of course, did he question his right, or his power, to define: The evidence satisfies me that most Gitksan and Wet'suwet'en people do not now live an aboriginal life. They have been gradually moving away from it since contact....\n\n**Reasons to Travel**\n\nMoving on to the protohistoric era, the Chief Justice upheld the Crown's and Sheila Robinson's analysis of this period as the one during which the Gitksan and Wet'suwet'en ceased to live a \"truly aboriginal life\" by virtue of the fact that a few European trade goods had filtered into their territories through trade with their Coast Tsimshian neighbours.\n\nIt is not possible to speak with much confidence about the commencement of the protohistoric period in the territory but some historical facts are known which at least permit informed estimates....\n\nDoing the best I can, it appears to me that after a period of unknown duration prior to about 1700...major population destabilization began to occur at the coast which probably spread to the interior; warfare became common, if it had not always been present; and within about 10 years of Cook's arrival at Nootka and his discovery of the potential for an ocean fur trade in 1778...a cruel sea otter trade started in the north Pacific in the 1780s with British, American, Russian and possibly other nationals almost extinguishing the sea otter until this trade fell into a sharp decline because of over-hunting just after the turn of the century. This made it necessary for traders to find other kinds of furs, and the interior fur trade was their natural response....\n\nBy the early 1800's, Russian traders had established an outpost at Sitka, and the Tsimshian and Carriers had established trade networks with the Gitksan in the north and with the Bella Coola people in the south. The Gitksan became middlemen for the Tsimshian traders.... I doubt if the commencement of European influence in the territory was earlier than Cook's landfall in 1778 and it was more probably around or after the turn of the century. There may have been isolated intrusions of trade goods from unknown directions at a slightly earlier period but not in any significant quantity.... I also doubt if commercial trapping started in the territory before 1805 or 1806 and probably a few years later than that. Then with the introduction of metal or mechanical traps and a market for excess furs, I believe some of the ancestors of the plaintiffs found it advantageous to spread out from their villages into distant territories for the purpose of commercial trapping.\n\nThe judge reiterated in very clear terms, throughout his _Reasons for Judgment_ , his support for the Crown's position that Aboriginal rights are \"frozen\" rights: arrested at the moment of contact with Europeans, and sealed in time capsules:\n\nWhile the Supreme Court of Canada will ultimately be called upon finally to settle this important question, I am not able to avoid expressing an early judicial opinion. In my view the purpose of aboriginal rights was to sustain existence in an aboriginal society, that is to hunt and fish and collect the products of the land and waters for the survival of the communal group...land-based commercial enterprise cannot be regarded as an aboriginal right. Notwithstanding the complexity of mixed land use in the province, I think aboriginal rights, to the extent recognized by law, have always been sustenance user rights practised for a very long time in a specific territory. These rights do not include commercial activities, even those related to land or water resource gathering, except in compliance with the general law of the province.\n\nAccording to Chief Justice McEachern's frozen rights theory, the only practices that may be recognized by section 35(1) of the _Constitution Act 1982_ as eligible for constitutional protection as \"Aboriginal rights\" should be those that can be identified as activities that the Aboriginal litigants' ancestors carried out before Europeans arrived in the eighteenth century, and are still carrying out to-day. The judge agreed that the use of modern technology\u2014like guns for hunting\u2014should not disqualify a practice from classification as an \"Aboriginal right.\" The defining characteristic that differentiated, for example, an \"aboriginal\" hunting practice from a \"non-aboriginal hunting practice,\" was neither the intention of the hunter, nor the technology used, nor the species of prey. Rather, the law took a post-mortem approach and decreed that the final destination of the hunted animal's carcass and remains would define its legal status, retrospectively. If a hunter exchanged her or his catch with another Aboriginal person for another product of the land, then the previous act of hunting would be posthumously classified as the practice of an \"aboriginal right.\" If, on the other hand, the same hunter exchanged the same carcass with an Aboriginal or non-Aboriginal person for either a commodity of European manufacture, or cash, then the result would be that, consciously or intentionally or not, the Aboriginal hunter would be deemed to have been engaging in \"non-Aboriginal practices.\"\n\nGiven the tautological nature of the judge's reasoning, the important legal findings that were based on interpretations of actual events that may have occurred during the protohistoric period did not require empirical support. Since he had determined that \"aboriginal practices\" in relation to land and resources were only those that were engaged in prior to the arrival of Europeans, it followed inevitably that trapping in conjunction with the European-Aboriginal fur trade must have been a \"non-aboriginal\" practice by definition. What actually happened \"on the ground\" is irrelevant within this framework in which the answers are pre-determined by the premises of the questions:\n\nIn my view, commercial trapping was not an aboriginal practice prior to contact with European influences and it did not become an aboriginal practice after that time even if lands habitually used for aboriginal purposes were also used for commercial trapping after contact.... With regard to new lands used after contact for commercial trapping, particularly in the far North and south extremities of the territory, it is my view that such would not be an aboriginal use and those new lands would not be aboriginal lands even if they were also used for sustenance after contact. This is because, firstly, commercial trapping is not an aboriginal practice, and secondly because the use of these new lands, even partly for aboriginal purposes under European influences after contact, does not constitute the kind of indefinite long time use which is required for aboriginal rights....\n\nApart from this, and the gradual accommodation of Indians to European trade goods and civilization (which did not change the nature of aboriginal activities), I doubt whether anything relevant to this action occurred in the territory between early European influences and the assertion of British sovereignty whenever that may have been.\n\nAlthough his ruling on the legitimacy of British sovereignty and the extinguishment of Aboriginal title precluded consideration of any Aboriginal rights whatsoever, in a number of places in the _Reasons for Judgment_ the Chief Justice stressed his concurrence with the provincial Crown's argument that the only lands \"used and occupied\" by the Gitksan and Wet'suwet'en with any regularity were the village sites on the major salmon-producing rivers, pieces of which became federally administered Indian reserves. The judge went to great lengths to argue that the provincial crown lands between villages that include miles of valuable timber, and are currently leased to giant forest companies, could not possibly be subject to any Aboriginal claim:\n\nI believe some Gitksan moved into these areas...[outside the major villages] after the start of the fur trade, or later...[and] particularly in the last 150 years when there was a reason to be there....\n\nAllan McEachern relied on the Crown's \"social theory\" of simplistic economic determinism to reiterate over and over again that the Gitksan and Wet'suwet'en had \"no reason to travel\" to these parts of their territories before becoming involved in the European-Aboriginal fur trade, trapping for which, the Judge decreed, was a \"non-Aboriginal\" and an \"economic\" practice. Chief Justice McEachern's rulings depended on the validity of his foundational assumption that repeats the Crown's argument: the only \"reason\" for people to travel into their territories was for \"economic\" purposes. And, \"economic\" activities are defined as only those that were engaged in as part of the European-Aboriginal fur trade:\n\nThere was no reason for them to travel other than between the villages or far from the great rivers for these or other aboriginal purposes, or to take more animals than were needed for subsistence although it is also reasonable to assume they would have travelled as far as was necessary for such purposes....\n\nThere was little reason for the Gitksan to stray far from their villages...except to visit other villages and for journeys along the grease trail...to get oolichan oil.... Similarly, there was little reason for the Wet'suwet'en to stray far from their canyon fisheries at Hagwilget.... There was, in fact, little reason for the Wet'suwet'en to have lived far from the Bulkley villages with which they all seemed to have connections.... There was no reason, until the fur trade, for these people to have any boundaries....\n\nThe question of boundaries elicited a ruling by the Chief Justice that expressed the double-bind nature of legal reasoning on Aboriginal matters. First he confessed that the irregular shape of Gitksan and Wet'suwet'en hereditary chiefs' territories led him to think they were gerrymandered. But, he reflected, gerrymandering requires rational intent to deceive, an intellectual capacity only acquired through evolutionary development and the achievement of civilization. The judge thought such calculated, \"rational\" duplicity uncharacteristic of Aboriginal societies:\n\nThe unusual shape of some of the territories leads me to doubt their authenticity.... They and many others appear to be excessively gerrymandered or artificial which seems inappropriate for an aboriginal society.... Further, I am troubled by the fact that so many chiefs claim so many different, widely scattered territories....\n\nMomentarily adopting a culturally relativistic view, the Chief Justice allowed that what _he_ might see as _duplicity_ , might _really_ be a feature of Aboriginal culture:\n\nIt would not be fair for me to conclude that the above is inconsistent with Indian custom or practice for there is no evidence to that effect, and I must not see with uncultured eyes what may not be there. Viewed with judicial eyes, however, these considerations alert me to be cautious....\n\nIn the end, regardless of evidence viewed by cultured or uncultured eyes, the legal point was made:\n\nIt is unlikely that the plaintiffs' ancestors, prior to the fur trade, would occupy territories so far from the villages, particularly in fierce Canadian winters.\n\nAccompanying the arrival of William Brown in 1822, who brought history with him, the stereotypical image of the \"lazy Indian\" made its first appearance in the Chief Justice's narrative:\n\nThus, it would seem that the time of direct contact in the territory was not earlier than the early 1820's which is a reasonable date to select as the end of the protohistorical period.\n\nIn 1822...William Brown of the Hudson's Bay Company\u2014one of our most useful historians\u2014established Fort Kilmaurs on Babine Lake.... Brown reports some minimal levels of social organization but the primitive condition of the natives described by early observers was not impressive.... By the time trader Brown arrived...the coastal sea otter trade was finished but the taste for trade made new initiatives necessary.... Throughout his Journals Brown frequently recognized that he was having great difficulty competing with the traders from the coast, and that beaver returns were never what he hoped they would be. He had great difficulty getting the Indians in his area to be as industrious in their trapping as he wished they would be.\n\n**Legal Rights and Social Wrongs**\n\nIn his description of the early colonial period in British Columbia, the Chief Justice reiterates all the key points made in the Crown's legal argument:\n\nIt seems indisputable that the historic period began in the territory with the establishment of Fort Kilmaurs on Babine Lake by Trader Brown in 1822.\n\nNothing of much political importance occurred in what is now the province for some time following the voyages of Cook, Vancouver, Dixon, Barkley and others. The Hudson's Bay Company was headquartered near the mouth of the Columbia River where James Douglas had been its Chief Factor since 1839.... In 1849 the Company was given a 5-year monopoly on the fur trade on Vancouver Island on the understanding that it would foster the settlement of the Island leading to the establishment of a colony.... Douglas remained as Chief Factor for the Company and in that capacity he negotiated 11 of a total of 14 treaties with Indians on Vancouver Island, mainly in the vicinity of what is now the City of Victoria. No treaties were ever negotiated on the mainland. Douglas became Governor of the Colony of Vancouver Island in 1851.... An Act of the Imperial Parliament dated August 2, 1858 provided for the Government of the Colony of British Columbia which was proclaimed November 19, 1858.\n\nThe Chief Justice began to stress the participation of Gitksan and Wet'suwet'en in the Euro-Canadian controlled economy when he came to describe mid-nineteenth-century British Columbia. This focus supported the Crown's \"assimilation\" and \"implicit extinguishment\" arguments:\n\nAs early as the 1850s The Gitksan, who had not previously seen a horse, quickly became adept at packing for the construction of the Collins Overland Telegraph, for the Yukon Telegraph, for the Omineca and Cassiar Gold Rushes, and for the construction of the Grand Trunk Pacific Railroad in the first decade of this century. At the same time, the Indians increasingly participated in commercial fishing at the coast, and in the logging and lumbering industries which became the economic mainstays of the region.\n\nThe Chief Justice was unequivocal in his assessment of the legal importance of the Douglas Treaties:\n\nAs James Douglas was a principal actor in setting the stage for much of the misunderstanding about Indian rights which later developed, it will be useful to mention that his policy for the mainland, as described in several pronouncements and correspondence with the Colonial Office in London, was that the colony would be opened up for settlement quickly so as to establish a British as opposed to an American community here.... It seems to be accepted by scholars that concern about American influences, mainly miners and Mormons, was indeed the principal reason for the establishment of a mainland colony.... Unlike the American experience where Indians were being confined by force to reserves, Indians in the Colony were entitled to the free use of all unoccupied lands.\n\nThe plaintiffs argue that the willingness of the Crown, the Hudson's Bay Company, Governor Douglas, and the Legislative Assembly of Vancouver Island to purchase aboriginal lands or interests is evidence of the existence of Indian ownership of the entire province.... With respect, I think too much has been made of these treaties as there is no clear understanding of what was involved, and the reasons which motivated the parties to act as they did. The Hudson's Bay Company apparently decided to acquire aboriginal interests in land in which it was interested, and obtained such land for a few blankets. The Vancouver Island treaties may represent no more than the surrender of the Indians of whatever rights they had in exchange for the modest consideration they received together with the substitution of a treaty right to continue to use the land. It cannot be inferred that the Indians owned the land or that the Crown was obliged by law to enter into these or further treaties.... It is not clear whether the acquired lands included village sites, or cultivated fields or surrounding hunting grounds....\n\nThis is all so uncertain and equivocal that I am unable to attach any legal consequences to these treaties.... I am more impressed by the unequivocal fact that the Crown, while recognizing aboriginal possession of village sites, was both setting aside reserves and marketing the unoccupied balance of the colony.\n\n...It would certainly have come as a surprise to Governor Douglas and his colleagues, and to the Colonial Secretaries and the Imperial Privy Councillors if it had been suggested to them that the consent of the Indianswas required before the settlement of any part of the colony could be undertaken....\n\nThe Chief Justice acknowledged that Aboriginal people may not have been treated _justly_ , but he was certain they had been treated _legally_ ; and the good intentions of the colonial officials were never in doubt:\n\nIt would not be accurate to assume the colonial officials, or their masters in London, chose wilfully to ignore aboriginal interests...their intention was to allot generous reserves, and to satisfy the requirements of the Indians in that way...it is obvious none of these colonial officers believed the settlement of the colony depended in any way upon Indian consent.... While the foregoing describes the state of mind of colonial officials at the start of the colonial period, the social disadvantage of the Indians was ongoing and largely but not entirely unrecognized. For this reason it is difficult, but necessary, to keep the difference between legal rights and social wrongs very much in mind.\n\nThe Chief Justice stressed, in a number of places in his _Reasons for Judgment,_ that Aboriginal people had been permitted, for a short time, to pre-empt land on the same terms and conditions as settlers:\n\nDouglas retired in 1864, and was replaced by Joseph Trutch. In 1866 the two colonies of Vancouver Island and British Columbia were amalgamated into the Colony of British Columbia.... Until 1866 Indians had equal rights to pre-empt land for cultivation in the same way as all other subjects of the Crown.... Trutch revoked this, prohibiting Aboriginal pre-emption of land without special permission. The historical explanation for this is that speculators were unfairly accumulating land pre-empted by Indians.... This provision remained in the law until 1953. This, of course, marked an unfortunate departure from the policy established by Governor Douglas, which had contemplated Indians having the same rights and privileges, except for their protected reserves, as everyone else.\n\nIn his account of the colonial period, the Chief Justice reiterates his theory that essential \"cultural differences\" are the cause of conflicts between First Nations and colonial governments:\n\nNotwithstanding the policy enunciated by Governor Douglas, the anticipated equality of life and opportunity with the white community quickly turned (even before Confederation) into the same depressing, continuing and paternalistic inequality experienced in most areas of North America. The prohibition of Indian pre-emption of land is but one example, but possibly not a significant one because they probably would have sought only infrequently to obtain land in this way. Also, I doubt if they would have long retained any land they might have obtained by pre-emption because their culture had not prepared them for the disciplined life of a tax paying agriculturist.... This undoubtedly illustrates the difficulties then encountered, (and which continue), in accommodating the two cultures.\n\nMcEachern concluded that, in the long run, Aboriginal people were responsible for their own problems, having failed to take advantage of the opportunities made available to them:\n\nAlthough Governor Douglas intended the Indians would be treated equally with white settlers, it did not work out that way. Indian reserves were established in or near the settled areas, and the Indians had free access to all unoccupied Crown lands instead of being confined to reserves. Otherwise, the Indians were, as later stated by Trutch, more or less left alone. Being reticent people, and benefiting in some respects from the industry and trade goods of the settlers, they often did not object to the inroads made into the geography of the Colony.... In fact, white and immigrant populations in the province grew from the 5000 mentioned by Governor Douglas in 1859 to perhaps 12,000 at the time of Confederation, while the Indian population probably remained more or less the same, estimated from 25,000 to 40,000 with the great majority of them on or near the coast living off the sea, rivers and land to which they had free access. They were often thought not to have any need for reserves much larger than their village sites.\n\nIt is...regrettable that the Indians themselves did not take more effective steps to secure larger reserves if they really wished to have larger tracts of land allotted to them.\n\nChief Justice McEachern described the federal-provincial dispute about British Columbia Indian policy at the time of Confederation as a difference of opinion about \"theories of history\":\n\nGenerally speaking, Canada at this time was negotiating treaties with nomadic or semi-nomadic Indians on the prairies which included the surrender of aboriginal 'title,' the payment of annuities or other payments, and Indian Reserves comprising hundreds and thousands of acres. Canada originally believed the same procedure should be followed in British Columbia. The province, on the other hand, believed Canada's position was unrealistic, and that its own policy...was entirely adequate given the different history and geographic circumstances of the province.... It is apparent that a theory of history was operating at this time to create a mind-set in the Government of Canada which was, to say the least, different from what some colonial officials in British Columbia then believed about Indian interests. Neither was entirely consistent and neither was necessarily right or wrong.\n\n**Communicating Deeds and Legislating Greed**\n\nBy the late nineteenth century, according to the Chief Justice's historical narrative, First Nations had become insatiable and unreasonable in their demands. And, problems of cross-cultural communication worsened:\n\nIn August 1887 Commissioner O'Reilly of the Indian Reserve Commission went to the north-west coast and met with Indians at various locations.... His notes make fascinating reading, particularly his repeated request to the Indians to let him know what lands they wished; but generally the story was the same. The Indians claimed either huge reserves and payment for all lands outside their reserves, or just the former, and declined to participate in the process if they did not receive a favourable response. Mr. O'Reilly repeatedly urged the Indians to tell him what they required in the way of land, forestry resources and fishing stations, but often to no avail. As is so often the case...the two cultures do not always communicate well with each other.\n\nThe judge took exception to allegations that Joseph Trutch had intentionally deceived the federal government. He defended the actions of the provincial politicians:\n\nIn their argument plaintiff's counsel make serious allegations against many Colonial officials including Trutch, Robson, Crease and Governor Musgrave. They allege Trutch 'purposely lied' in the discussions on Confederation and that he participated in a scheme of misinformation which led to the 'impoverishment of the people.' Counsel allege a 'perversion of history'.... The evidence does not prove that Trutch himself was not convinced that the Indian policy of the province was anything but in the best interests of both the Indians and the white settlers, but it does suggest that he was not anxious to have the details of that policy known to the dominion authorities...the evidence about the character of Trutch is equivocal and, there being no need to do so, I think it best not to enter into that controversy. Such matters are better left to historians.\n\nIn the context of the consequences of colonial policy for Aboriginal peoples, Chief Justice McEachern was uncharacteristically reluctant to reason from results to causes:\n\nHowever much one may regret the failure of the colonists to recognize or react to the differences between the two communities, the legal consequences arising from the rights of the parties must be determined objectively from the constitutional and legal measures taken in that period rather than from social and economic failures.\n\nRather, he argued, let sleeping dogs lie:\n\nUndoubtedly there was a measure not of assimilation, but rather of conformity on the part of many Indians with the growing white population. This was particularly evident in the larger centres such as Victoria and New Westminster and in the various mining camps throughout the limited areas affected by mining activity or settlement. Even in the territory the Indians were understandably taking whatever advantage they could of the white economy, particularly by utilizing its market for their furs and by working for wages. It is impossible to say if they were better or worse off as result of these changes. I suspect they would hardly be aware of the policies described by Trutch..\n\nFor reasons which seemed sufficient at the time, but which have caused great resentment, the federal government made the Indian potlatch illegal in 1884....\n\nBy to-day's morality, the foregoing will be regarded by many as an attempt to destroy Indian culture and identity. By the standard of the day, compared with the rest of the world, it was probably enlightened. I need not pronounce on that question....\n\nThe Chief Justice reiterated his view that historic truth could never really be known, and repeated his lamentation about cross-cultural misunderstanding and the failure of Indians to help themselves. And, while having rejected the value of anthropology in settling legal disputes, McEachern suggested that the discipline might be more useful in explaining other problems:\n\nFor reasons which can only be answered by anthropology, if at all, the Indians of the colony, while accepting many of the advantages of European civilization, did not prosper proportionately with the white community as expected. No one can speak with much certainty or confidence about what really went wrong in the relations between the Indians and the colonists...In my view the Indians' lack of cultural preparation for the new regime was indeed the probable cause of the debilitating dependence from which few Indians in North America have yet escaped.\n\nIt would be overly simplistic, and probably inaccurate, to say that the white settlers were either too kind or too cruel, and that the Indians should either have been given more support, and the dependence increased, or no support at all so that a dependence would not have arisen. So long as Indians had access to white communities there was bound to be a mixing of incompatible cultures. Being of a culture where everyone looked after himself or perished, the Indians knew how to survive (in most years). But they were not as industrious in the new economic climate as was thought to be necessary by the newcomers in the Colony. In addition, the Indians were a greatly weakened people by reason of foreign diseases which took a fearful toll, and by the ravages of alcohol. They became a conquered people, not by force of arms, for that was not necessary, but by an invading culture and a relentless energy with which they would not, or could not compete....\n\nMany have said with some truth, but not much understanding, that the Indians did not do as much for themselves a they might have done. For their part, the Indians probably did not understand what was happening to them.\n\nAlthough there is much, the Chief Justice claimed, that cannot be known with any certainty about the colonial period, he was confident about at least one thing:\n\nThis mutual solitude of misunderstanding became, and remains, a dreadful problem for them and for everyone. What seems clear, however, is that the source of Indian difficulty was not the loss of land for aboriginal purposes. So far as the evidence shows, they were largely left in their villages and an aboriginal life was available to them for a long time after the 'Indian problem' was identified....\n\nIf the settlers could be faulted for anything, it might be for poor time management practices. The Chief Justice wrote that they were busy, and failed to make time to cultivate more benevolence:\n\nPreoccupied with the business of getting a new colony started, and of scratching out a hard life in a hard land, the new white settlers, and particularly their leaders, did not pay sufficient attention to the real and potential sociological, cultural, and economic difficulties the Indians were experiencing. They became a problem seen through European eyes to be dealt with bureaucratically\u2014an Ordinance here, a dollar there, and tragedy almost everywhere. I suspect the white community understood what was happening to the Indians but did not have the resources, or the knowledge, to respond appropriately....\n\nEven to-day, it is difficult to say what should have been done short of abandoning the settlement of the colony.... Even a division of the colony between settlers and Indians was not possible for there was no part of the colony where Indians did not have a presence. Much larger reserves may have helped, but probably not without segregation which would have been severely criticized on other grounds. As in so many other parts of the world, the seeds of present difficulties were sown, not intentionally I am sure, but by mixing two cultures, and by indifference, during the colonial period.\n\n**Driven to Drink...**\n\nAs he began to describe the early years of the twentieth century, Chief Justice McEachern concentrated on how the Gitksan and Wet'suwet'en had voluntarily acquiesced to Euro-Canadian political and economic domination, and how eagerly\u2014if imperfectly\u2014they had tried to assimilate:\n\nDisease and other misfortune arrived in the territory 70 years or so before the settlers or the railway.... The evidence suggests the land was seldom able to provide the Indians with anything more than a primitive existence.... There was no massive physical interference with Indian access to non-reserve land sustenance in the territory, and there was no forced or encouraged migration away from the land towards the villages. Migration away from the land has been an Indian initiative and it started before there was any substantial settlement in the territory.\n\nAt this point in his narrative he introduced a still familiar theme in contemporary analyses of Aboriginal communities where questions are framed as debates about what single causal factor best explains wide-spread poor health, demoralization and alcohol abuse among Aboriginal people. This is, of course, a very contentious political question and arguments abound about whether \"alcoholism\" is a medical disease or a social problem. Most contemporary health care professionals in the field recognize the complexity of this issue, and focus on seeking ways of healing, rather than pursuing the futile task of seeking to assign blame simplistically. In a legalistic ideology, \"cause\" and \"liability\" are frequently conflated and substituted for understanding and responsibility:\n\nThe introduction of alcohol, disastrous epidemics and limited economic opportunities did not result from a lack of access to land.... This is not to say that European influences upon Indian life were not pervasive, but when I consider the effects of disease, alcohol and other social insults upon the Indian community, it is apparent that interference with aboriginal use of land, except for actual dispossessions, was not a principal cause of Indian misfortune.\n\nAnd, again, in the final analysis, Chief Justice McEachern determined that Aboriginal people are to blame for having become \"dependent\" upon Euro-Canadians. Further assistance would only exacerbate the situation now:\n\nSettlement, which did not begin in the territory until the beginning of this century, was initially confined to the Bulkley and Kispiox valleys where land cultivation had not been pursued vigorously by many Indians...yet there were some dispossessions and almost from the beginning of the colony, and from the time of settlement in the territory, it must have been obvious that the Indian population was falling into disadvantage when compared with the then white non-Indian community. The condition of the Indians in the territory throughout the entire history of their association with the European settlers has been an unhappy one with alcohol abuse, disease, infant mortality, poverty, and a lack of many of the benefits of civilization, particularly health, education and economic opportunities, and the ubiquitous dependence being usually the most serious social problems....\n\nThey cannot be helped, but must help themselves.\n\nCompliance with Euro-Canadian law, according to the judge, was evidence of the Gitksan and Wet'suwet'en having implicitly acquiesced in the extinguishment of their Aboriginal title:\n\nThe first farmers moved into the area around 1900 and there was much resentment, which continues to this day, about pre-emption of land occupied by Indians and over the issue of land script for veterans of the Boer War.... The construction of the Grand Truck Pacific Railroad from 1908 to 1914 provided some employment for the Indians, but also opened up this inviting country to further settlers. When there were disputes with settlers or governments...the Indians often accepted solutions which denied aboriginal ownership....\n\nResistance to Euro-Canadian law, according to the judge, was evidence of the Gitksan and Wet'suwet'en having implicitly acquiesced in the extinguishment of their Aboriginal title:\n\nIn the meantime the dispute about Indian rights continued to simmer. The federal government was anxious to resolve the 'rights' issue but the province was adamant that no such rights existed except claims to village sites and cultivated fields. As a result a Royal Commission known as the McKenna-McBride Commission was established in 1913 to adjust Indian reserves in the province. In order to deal with reserves, the representative of the federal government, McKenna, agreed to 'drop' the question of title, believing that the Courts...would settle the rights problem in due course. Unfortunately, it never did. The commissioners unquestionably gave assurances to the Indians in the territory that the reserve adjustment process in which they were engaged could not deal with aboriginal rights or prejudice their claims which, the commissioners said, would be looked after by other means, more particularly the proposed reference to the Exchequer Court.\n\nIn 1927 a Special Joint Committee of the House of Commons and Senate inquired into 'Indian affairs' and rejected all the claims of the Indians of British Columbia to anything except the reserves which had been allotted to them.\n\n**Infantile Disorders**\n\n_I do not consider it necessary to mention anything else which occurred between 1927 and the commencement of this action in 1984._\n\n\u2014Chief Justice Allan McEachern, 1991, _Reasons for Judgment, Delgamuukw v. R._\n\nLike a toddler who cannot understand why, when she closes her eyes, everyone else can still see her, Chief Justice McEachern seemed to mistake the law for reality in this statement. He did, however, go on to comment extensively on events that occurred between 1927 and 1984.\n\nReasoning backwards from the present to the past, the Chief Justice made the somewhat alarming claim that the observations he made in 1987 allowed him to simultaneously view what was happening in the territory hundreds of years ago:\n\nIf the land is substantially empty now, as I believe it is except for non aboriginal purposes such as commercial trapping, mining or logging, then I believe it was also empty for aboriginal purposes at the time of contact.... As it is empty now, it was probably empty both at the time of contact and, except possibly for some limited commercial trapping, at the time of sovereignty.\n\nJust as the Judge had observed the \"vast emptiness\" of the Gitksan and Wet'suwet'en territories, so too had other non-Aboriginal eye-witnesses seen an unoccupied land stretching before them:\n\nIn this century, long time settlers such as Mr. Shelford were, until very recently, unaware of any claim to aboriginal ownership or control of the territory or of any claim to an interest inconsistent with the activities of himself and other white settlers.... Mr. Shelford...has been living near the west end of Francois Lake since the 1920's. Much of this area is known as the Shelford Hills. He did not know until recently that the few Indians in his neighbourhood, with whom he was always on friendly terms, claimed to be the owners of both his trapline as well as their own and all the other lands in the area. This seriously questions the existence of an undisputed, settled reputation sufficient to found a declaration of any kind of interest in land...it is impossible to infer a community reputation for an interest in land when a prominent, life long resident in the area like Mr. Shelford, a Member of the Legislature for many years, and a Cabinet Minister for a time, who acknowledges hearing about general land claims for a long time, has never heard until very recently of claims of ownership or jurisdiction, or claims to specific lands, including his own, by Chiefs whose families he has known personally for most of his life.\n\nThe Chief Justice considered the possibility that, actually, the Gitksan and Wet'suwet'en didn't even really understand their own claims:\n\nIt seems to me that there has always been much uncertainty about the true nature of 'Indian title' in the province. Even some Indians have not always been completely consistent because there are references in the historical record to suggestions that enlarged reserves were their primary concern. In this respect, of course, the speakers, whoever they may have been, did not speak for anyone but themselves.\n\nAlthough he admitted they have taken a consistent position on the subject for at least 100 years:\n\nI think it is fair to conclude that the basic position of most Indians, at least since 1880, was that the various Indian tribes or peoples owned all or most of the province.... It is not difficult to summarize the position of the province. Since Confederation, the position of the province has been consistent, even unyielding, on the question of 'Indian title'....\n\nUnlike legal decisions that become more true with each utterance, the Chief Justice concluded that, because Aboriginal people have consistently taken every opportunity to assert their Aboriginal title, and their argument has not changed since contact, and because they have discussed their case for over a century, their claim has ceased to be based on \"objective evidence.\" Its validity has been _weakened_ by time and repetition. The Gitksan and Wet'suwet'en, it would seem, are not blessed\u2014as are Judge McEachern and Mr. Shelford\u2014 with the ability to see the past in the present:\n\nWhat happened is that, beginning in 1973, following the _Calder_ case, the plaintiffs began thinking about their boundaries.... During this process, there has been much discussion within the Indian communities about the collection of this information for land claim purposes. This deprives the process of the objectivity which would have added confidence to it.... The plaintiffs and their ancestors have been actively discussing land claims for many years, long before the McKenna-McBride Commission in 1914. This has been a very current issue with the plaintiffs for a very long time. The collection of evidence in such a climate deprives it of the independence and objectivity expected for reputation evidence. I find that, except for occasional political statements, the plaintiffs in the post sovereignty period seldom conducted themselves as if they believed they were owners of such vast areas.\n\n**Camp Potlatch**\n\nChief Justice McEachern repeated his opinion that cultural assimilation and economic integration were the answers to \"the Indian problem\" in a variety of passages in his _Reasons for Judgment_. However, the judge also ruled in favour of the Crown's position that nothing resembling an \"economy\" existed before European colonization, therefore Aboriginal people could not, by definition, engage in \"economic\" activities. If they did, they were, _ipso facto_ , engaging in \"non-Aboriginal\" practices. Just as trading with the Hudson's Bay Company had brought an end to their historic identity as Aboriginal peoples, so too the judge interpreted participation in the Euro-Canadian controlled-economy as evidence that cast doubt on the legitimacy of contemporary Gitksan and Wet'suwet'en claims to \"Aboriginal\" rights:\n\nWitness after witness admitted participation in the wage or cash economy. Art Matthews Jr., (Tenimyget) for example, is an enthusiastic, weekend aboriginal hunter. But at the time of trial, he was also head saw filer at the Westar sawmill at Gitwangak where he had been steadily employed for 15 years, a graduate of the B.C. Institute of Technology, a shop steward, and a member of the Negotiating Committee of the Industrial Woodworkers of America. Pete Muldoe (Gitludahl) has followed a variety of non-aboriginal vocations including logging on the lands claimed by another chief; Joan Ryan (Hanamuxw) teaches school in Prince Rupert.... Even in their aboriginal pursuits...the plaintiffs do not seem to consider themselves tied to particular territories....\n\nThe judge bemoaned the passing of the \"aboriginal\" way of life of his fantasies:\n\nI have no doubt aboriginal activities have fallen very much into disuse in many areas. This was admitted by several Indian witnesses who observed that many of their young people have very little interest in aboriginal pursuits. Eventually, the Indians must decide how best they can combine the advantages the reserves afford them with the opportunities they have to share and participate in the larger economy, but it is obvious they must make their way off the reserves.... Care must always be taken to ensure that the good things of community life are not sacrificed just on economic grounds.... Without intending any offence, I have driven through some of the reserves which demonstrate disadvantages, and I have witnessed firsthand how some of them live. It is interesting to note that housing on reserves seems to be much better where there is (or was) a payroll such as from the sawmill at Kitwangak.\n\nThe Chief Justice did recommend a way for Aboriginal and non-Aboriginal practices to be reconciled:\n\nThe Crown's obligation, in my judgment, is to permit aboriginal people, but subject to the general law of the province, to use any unoccupied or vacant Crown land for subsistence purposes until such time as the land is dedicated to another purpose.... Land that is conveyed away, but later returned to the Crown, becomes again usable by Indians. Crown lands that are leased or licensed, such as for clear-cut logging to use an extreme example, become usable again after logging operations are completed or abandoned.\n\nMcEachern's comments recall a familiar adage, often voiced by actors playing demoralized settlers in old western movies: \"This land ain't good for nothin' no more. Might as well give the country back to the Indians.\" The fact that second-class treatment of Aboriginal peoples is considered justifiable at many levels of Canadian society is not, however, a joke. Neither is there anything funny about the clear-cutting of British Columbia forests, that the law permits.\n\nOn another matter of considerable contemporary political significance, the judge took the time to remind his readers that a disproportionate number of Indian people are, like himself, dependent for their income on the public purse. He urged Aboriginal people to take their citizenship responsibilities more seriously and to be generous and public-spirited:\n\nThe plaintiffs are also troubled because they have not progressed equally with some of the 'newcomers,' and because they have not been able, for many real and intangible reasons, to share the opportunities they think they should have to the commercial use of the land and to the prosperity they think should accrue to them from the land they truly believe is their own.... At the same time, the Indians must realize the importance of creating public wealth from the territory as they, like so many members of the non-Indian community, are heavily dependent upon public funding for every day sustenance....\n\nThe worst thing that has happened to our Indian people was our joint inability to react to failure and to make adjustments when things were not going well...the answer to these social questions, ultimately, will be found in the good health and education of young Indian people, and the removal of the conditions that have made poverty and dependence upon public funding their normal way of life...the difficulties of adapting to changing circumstances, not limited land use, is the principal cause of Indian misfortune....\n\nThe Indians have remained dependent for too long. Even a national annual payment of billions of dollars on Indian problems, which undoubtedly ameliorates some hardship, will not likely break this debilitating cycle of dependence.\n\nThe Chief Justice then turned his attention to the issue of spirituality:\n\nThere is a further dimension to this question, however, which must also be considered. I refer to the obvious spiritual connection some Indians have with the land. I accept this as a real concern to the plaintiffs worthy of as much consideration as actual sustenance use.... Except in rare cases, there should be no difficulty obtaining sufficient fish, game and other products from most areas of the territory...for spiritual purposes.\n\nThe judge continued along this path, noting that, in fact, First Nations people were greatly aided in their spiritual growth by colonization and modern technology:\n\nI believe the Indians of the territory are probably now much more united and cohesive as peoples, and they are more culturally sensitive to their aboriginal birthright than they were when life was so harsh and communication so difficult. I cannot find lack of access to aboriginal land has seriously harmed the identity of these peoples....\n\nIn this respect I pause to mention that the salmon of the great rivers pass right alongside the principal villages and one need not travel far from the villages to reach wilderness areas were game can usually be taken. There is much wood in the territory and it can be obtained far more easily with chain saws, snowmobiles and 4 x 4's than in earlier days. Anyone can now travel with much greater ease to whatever parts of the territory he or she may wish for the purpose of gathering what is required for sustenance or ceremonial purposes....\n\nReflecting on contemporary practices later in his text, the judge surmised:\n\nThe aboriginal activities that are being pursued now may be indistinguishable in many cases from the wilderness activities enjoyed by many non-Indian citizens of the province.\n\nChief Justice McEachern's point of view is not uncommon. It repeated an opinion that is frequently expressed in British Columbia by people who argue that Aboriginal people are not different enough to claim distinct rights. The equating of Aboriginal hunting, fishing and gathering with recreational camping decontextualizes and reduces these practices to their most simplistic form, as behaviours and commodities: one of many experiences that can be purchased in the leisure and entertainment market place.\n\nAnd, he concluded, the market place does not discriminate on the basis of race or culture as it sifts the wheat from the chaff:\n\nThere is no doubt, as I have said, that many of the present male and female population trapped and hunted for economic reasons in their youth. However, most of them discontinued trapping years ago before or at the time the price of furs collapsed in the 1950's, and they have gradually moved into other segments of the cash economy even though the price of furs has recovered. There is very little, and decreasing, interest in pursuing these activities at the present time.\n\nMany aboriginals are directly engaged in the wage economy, and few\u2014not enough but some\u2014participate as entrepreneurs. Mr. Pete Muldoe, a prominent Gitksan hereditary chief, has engaged in logging and sawmilling in the territory, as has the Moricetown Band Council. The aboriginal communities, like the provincial community, have their economic successes and failures.\n\n**The Battle of Britain**\n\nChief Justice McEachern's _Reasons for Judgment_ in _Delgamuukw v. R._ , ended with a proposal for resolving outstanding Aboriginal title and rights claims. Not surprisingly, his prescription for the future reflected his analysis of the past:\n\nIt is my conclusion, reached upon a consideration of the evidence which is not conveniently available to many, that the difficulties facing the Indian population of the territory, and probably throughout Canada, will not be solved in the context of legal rights. Legal proceedings have been useful in raising awareness levels about a serious national problem....\n\nThe parties have concentrated for too long on legal and constitutional questions such as ownership, sovereignty, and 'rights,' which are fascinating legal concepts. Important as these questions are, answers to legal questions will not solve the underlying social and economic problems which have disadvantaged Indian peoples from the earliest times. Indians have had many opportunities to join mainstream Canadian economic and social life. Some Indians do not wish to join, but many cannot....\n\nThis increasingly cacophonous dialogue about legal rights and social wrongs has created a positional attitude with many exaggerated allegations and arguments, and a serious lack of reality. Surely it must be obvious that there have been failings on both sides. It is not the law, or common sense, nor is it in the interest of people of the province or of the plaintiffs that the development, business and economy of the province and its citizens should constantly be burdened by litigation or be injunctioned into abeyance by endless or successive legal proceedings.\n\nWhile he ruled that they had occupied a low rung on the theoretical ladder of social evolutionary development when Europeans had arrived, Chief Justice McEachern found that the Gitksan and Wet'suwet'en had progressed over the last two hundred years:\n\nCompared with many Indian Bands in the province, the Gitksan and Wet'suwet'en peoples have already achieved a relatively high level of social organization. They have a number of promising leaders, a sense of purpose and a likely ability to move away from dependence if they get the additional assistance they require. I cannot, of course, speak with confidence about other Indian peoples because I have not studied them.\n\nI am impressed that the Gitksan and Wet'suwet'en are ready for an intelligent new arrangement with both levels of government. I am not persuaded that the answers to the problems facing the Indians will be found in the reserve system which has created fishing footholds, and ethnic enclaves. Some of these reserves in the territory are so minuscule, or abandoned, that they are of little or no use or value.... It must be recognized...that most of the reserves in the territory are not economic units and it is not likely that they can be made so without serious disruption to the entire area....\n\nThe judge credited himself, and the Canadian legal system, with having saved the day, and the people:\n\nI wish to say...that I shall leave this case with the settled conviction that, in the long run, the greatest value of this case, apart from being the first stage in the settlement of legal rights, may well be the enhancement of interest in Gitksan and Wet'suwet'en languages, traditions and cultures. This is because the evidence satisfies me this case has been a 'Battle of Britain' for these peoples and it has inspired them to renew (an accurate word, in my view) what was a declining interest in their aboriginal heritage. The interest and activity generated by this law suit assures the survival of these peoples as distinct societies. This may at one time have been doubtful but I now believe it is a certainty.\n\n* * *\n\n. Pratt 1992.\n\n. McEachern, Allan (1991) \"Schedule 1. Itinerary of View of Territory June, 1988,\" _Reasons for Judgment in Delgamuukw v. R._ , 305-307.\n\n. Tautology: true by virtue of its local form alone.\n\n. _Reasons_ , 1991: 211-212.\n\n. _Reasons_ , 1991: 213.\n\n. _Reasons_ , 1991: 274.\n\n. _Reasons_ , 1991: 275.\n\n. _Reasons_ , 1991: 284.\n\n. _Reasons_ , 1991: 291.\n\n. _Reasons_ , 1991: 56.\n\n. _Reasons_ , 1991: 23.\n\n. _Reasons_ , 1991: 210-212.\n\n. Arthur Ray has pointed out that the judge accepted as valid Ray's evidence that the Chiefs owned and regulated beaver trapping sites, and did not eat the meat of the beaver but rather prized their pelts for trade: that is, they exercised ownership and control over a resource they did _not_ use for subsistence but rather produced for exchange. Given that the judge accepted this as \"fact,\" his conclusion that the Gitksan and Wet'suwet'en had no interest in non-subsistence harvesting before European arrival is illogical (Ray, 1991(b): 312).\n\n. _Reasons_ , 1991: 262.\n\n. _Reasons_ , 1991: 261.\n\n. _Reasons_ , 1991: 75.\n\n. _Reasons_ , 1991: 74.\n\n. _Reasons_ , 1991: 26.\n\n. _Reasons_ , 1991: 25.\n\n. _Reasons_ , 1991: 26.\n\n. _Reasons_ , 1991: 124.\n\n. _Reasons_ , 1991: 234.\n\n. _Reasons_ , 1991: 235.\n\n. _Reasons_ , 1991: 26.\n\n. _Reasons_ , 1991: 126.\n\n. _Reasons_ , 1991: 126.\n\n. _Reasons_ , 1991: 128.\n\n. _Reasons_ , 1991: 221.\n\n. _Reasons_ , 1991: 28.\n\n. Governor Joseph Trutch's brother-in-law, and appointee to the Indian Reserve Commission.\n\n. _Reasons_ , 1991: 167.\n\n. _Reasons_ , 1991: 132.\n\n. _Reasons_ , 1991: 235.\n\n. _Reasons_ , 1991: 133.\n\n. _Reasons_ , 1991: 28.\n\n. _Reasons_ , 1991: 128.\n\n. _Reasons_ , 1991: 129.\n\n. _Reasons_ , 1991: 250.\n\n. _Reasons_ , 1991: 251.\n\n. _Reasons_ , 1991: 250.\n\n. _Reasons_ , 1991: 28.\n\n. _Reasons_ , 1991: 28-29.\n\n. _Reasons_ , 1991: 182.\n\n. _Reasons_ , 1991: 275.\n\n. _Reasons_ , 1991: 276.\n\n. In common law, a \"reputation\" of ownership\u2014that is, common knowledge within one's community that one is the owner of a particular plot\u2014is considered valid evidence of proof of ownership. Chief Justice McEachern, in rejecting oral tradition as valid evidence of a \"reputation of ownership,\" in effect racialized this \"common\" law principle. He rejected reputation of ownership confirmed by the Aboriginal community, in favour of reputation of ownership asserted by Euro-Canadian settlers.\n\n. But, note that neither the Gitksan and Wet'suwet'en or any other First Nation in British Columbia has included privately owned homes and lots in its claim. Chief Justice McEachern acknowledged this at page 10, writing: \"The plaintiffs, in what I understand they regard as a matter of grace on their part, do not seek to recover pre-Writ (1984) privately owned (fee simple) lands within the territory. Instead, they claim compensation from the province for the value of whatever territorial interests have been transferred to other ownership. They claim the right to terminate all less than fee simple legal interests in the territory, such as logging, mining and other leases or licenses.\n\n. _Reasons_ , 1991: 222.\n\n. _Reasons_ , 1991: 182.\n\n. _Reasons_ , 1991: 263.\n\n. _Reasons_ , 1991: 264.\n\n. _Reasons_ , 1991: 266.\n\n. _Reasons_ , 1991: 222.\n\n. _Reasons_ , 1991: 56.\n\n. _Reasons_ , 1991: 291-292.\n\n. _Reasons_ , 1991: 184.\n\n. _Reasons_ , 1991: 248.\n\n. _Reasons_ , 1991: 252.\n\n. _Reasons_ , 1991: 248.\n\n. _Reasons_ , 1991: 299.\n\n. _Reasons_ , 1991: 37.\n\n. _Reasons_ , 1991: 251.\n\n. _Reasons,_ 1991: 291.\n\n. _Reasons_ , 1991: 56.\n\n. _Reasons_ , 1991: 228-229.\n\n. _Reasons_ , 1991: 253-254.\n\n. _Reasons_ , 1991: 248.\n\n. _Reasons_ , 1991: 37.\n**P** **ART** **VIII:**\n\n**A** **D** **I** **NFINITUM**\n\n**(G** **OING** **O** **N** **F** **OREVER** **, I** **NTO** **I** **NFINITY** **)**\n**Chapter 21:** **_Delgamuukw_** **at the British Columbia**  \n**Court of Appeal, 1993**\n\n_...The reason why courts have had no difficulty taking judicial notice of the facts of history is that historical fact is not a matter of opinion when dealt with by the courts. The doctrine of judicial notice is firmly grounded in the court's recognition of its general ability to deal with matters of history without the assistance of experts. Historians revise history.... The Chief Justice's ruling is consistent with the doctrine of judicial notice...and is based firmly on the difference between the way historians approach history and judges approach history._\n\n\u2014Province of British Columbia, 1991, Defendant's Appeal Factum.\n\n**We Own Therefore We Are**\n\nIt was on the question of extinguishment that the Chief Justice's ruling elicited the most surprise within the legal community as a whole. Critics argued that the Chief Justice's findings in law were as archaic and, in some cases, as questionable in terms of fact and reasoning as his findings on oral tradition, anthropology and history. The legal test for extinguishment that had been developing since _Calder_ was affirmed by the Supreme Court of Canada's decision in _R. v. Sparrow_ in these words: \"The test for extinguishment to be adopted, in our opinion, is that the Sovereign's intention must be clear and plain if it is to extinguish an aboriginal right.\" Chief Justice McEachern's interpretation that such expressions of a \"clear and plain intention\" could be inferred post hoc and reasoned backwards in time from the contemporary consequences of colonial domination, flew in the face of contemporary legal thinking. Furthermore, the Supreme Court of Canada had suggested that the \"honour of the Crown\" would be well served by reasonable attempts to obtain Aboriginal consent to extinguishment as well. McEachern's crude \"to the victors go the spoils\" analysis shocked several of his own peers.\n\nSecond, legal critics were taken aback by McEachern's ruling that Aboriginal jurisdiction and customary law \"gave way\" to colonial law upon the establishment of the colonies of Vancouver Island and the mainland in the mid-nineteenth centuries. The problem, scholars pointed out, was that British colonial law had no provision for Indigenous legal systems to simply \"give way.\" The direction of the Memorandum of the Privy Council of 1722, and subsequent proclamations and treaties had stipulated clearly that British imperial law required that a choice be made between two options: either Indigenous people were there living in an organized society with recognizable forms of government in which case the doctrine of conquest applied, and Indigenous laws were to continue until expressly extinguished by the conquering sovereign through military force and\/or treaty; or, if no Indigenous people were \"discovered\" living in the lands desired by the sovereign; or were deemed not to be living in an organized society and their land declared _terra nullius_ , the doctrine of discovery\/occupation\/settlement applied, and the sovereign's laws could be superimposed at the moment of discovery. Chief Justice McEachern invented new law by finding _both_ that underlying title was vested in the British Crown since the sixteenth century by virtue of the doctrine of discovery, based on _terra nullius_ ; _and_ that the Gitksan and Wet'suwet'en lived in an organized society at whatever moments in time Britain may have _post hoc_ and _de facto_ asserted sovereignty between 1774 and 1860. As anticipated, Chief Justice McEachern's decision was appealed by the Gitksan and Wet'suwet'en, and the appeal was responded to by the provincial and federal governments.\n\nIn November 1991, six months before Chief Justice McEachern's ruling in _Delgamuukw v. R._ had been rendered, the left-leaning, social democratic New Democratic Party (NDP) had been voted into office on a platform that included promises that they would recognize Aboriginal title and commence negotiations of Aboriginal rights. With the exception of the years 1974-1976, British Columbia had been governed since 1953 by the right-leaning, free-enterprise Social Credit party who had, until 1990, staunchly opposed any recognition of, or negotiations with, Aboriginal peoples on the land question. It was a Social Credit \"provincial crown\" who, with the law firm of Russell &Dumoulin, had argued the cases from _Calder_ to _Delgamuukw_. However, it was an NDP government that was forced to make a decision about what do to when the Gitksan and Wet'suwet'en appealed Judge McEachern's ruling to the British Columbia Court of Appeal. The Crown had changed heads.\n\nThe NDP and the Gitksan and Wet'suwet'en Hereditary Chiefs negotiated an agreement to request that the B.C. Court of Appeal be asked to decide only some of the questions that arose from Chief Justice McEachern's decision. Outstanding questions would be left for the Supreme Court of Canada to decide if, by the time an appeal of _Delgamuukw v. R._ came before them, the Gitksan and Wet'suwet'en had been unable to reach negotiated agreements with the federal and provincial governments. The NDP government put forward the following position in its appeal factum:\n\n(1) The trial judge (McEachern) erred in making a finding of \"blanket extinguishment\" on the basis of colonial instruments enacted prior to 1871.\n\n(2) The Plaintiffs (Gitksan and Wet'suwet'en) do not have a right to ownership of, or a proprietary interest in, the lands and resources which they claim.\n\n(3) The Plaintiffs do not have the right of self-government or jurisdiction as claimed.... But it is understood that aboriginal peoples who lived in an organized society governed themselves by their own system of laws and customs. Certain rights or freedoms to self-government may continue to exist, but are subject to the laws of Canada and of the Province.\n\n(4) The Province does not generally disagree with the factual findings of the trial judge on the question of ownership and jurisdiction.\n\n(5) The Province supports the findings of the trial judge with respect to the effect to be given to the expert evidence.... It does not take the position the trial judge ignored or improperly rejected evidence. It supports the trial judge's conclusions as to the value of oral histories.\n\n(6) The trial judge was correct to characterize the plaintiff's aboriginal rights as _sui generis_. But the precise location, scope, content, and consequences of the plaintiffs' aboriginal rights is a matter for negotiation, and further judicial consideration.\n\nThe New Democratic Party government supported all Chief Justice McEachern's rulings on expert evidence, including the dismissal of oral history, and the testimony by anthropologists and historians. While the B.C. Court of Appeal's panel of five judges were divided on many of the issues before them, they all supported the Chief Justice's rulings on this evidence. One of them, Justice Lambert, mentioned that more weight should have been given to the oral histories and the testimony of the Chiefs and Elders. However, the judges were unanimous in their finding that the volume of evidence was such that they could not reasonably be expected to review it; and, they were satisfied that no \"palpable and overriding error in fact\" was evident in McEachern's assessment of the expert evidence, or his findings in law that were based on them. They said that, even if they were to review all the evidence and the transcripts, \"The Court of Appeal could not be exposed to all the nuances in the evidence or be in as good a position as the trial judge to weigh the credibility....\"\n\nThe panel of five appeal court judges\u2014Macfarlane, Taggart, Wallace, Hutcheon and Lambert\u2014 found unanimously that there was no blanket extinguishment of Aboriginal title as a result of Britain's colonization of what is now British Columbia; that the introduction of a land settlement scheme by the colonial government prior to 1871 did not show a \"clear and plain intention to extinguish\" Aboriginal title; that, after British Columbia joined Confederation in 1871, Aboriginal rights could not be extinguished by a provincial government; that there was no evidence that the federal Crown extinguished Aboriginal rights between 1871 and 1982; and, finally, that after 1982, not even the federal government has the power to extinguish Aboriginal rights because of their entrenchment in Section 35 of the Constitution. The Appeal court ruled that Aboriginal rights existed in British Columbia prior to Britain asserting sovereignty, had not been extinguished, and therefore continued to exist. In other words, the B.C. Court of Appeal found in favourof the (new) Crown's (new) position. In overruling the Chief Justice's decision on blanket extinguishment, the Court of Appeal strategically \"cleaned up\" the worst of McEachern's judgment, in a legal sense, and brought it up to date and in line with directions taken in Supreme Court of Canada decisions on extin-guishment in particular rendered during the 1980s and 1990s, and within the framework of federal and provincial government perspectives that support the recognition of some form of Aboriginal rights. This course of action rendered unnecessary any explicit critique of the Chief Justice's ruling which would have been an embarrassment to the judiciary as a whole, and the B.C. judiciary in particular. Explicit disavowal of Chief Justice McEachern's \"findings of fact,\" and his tautological reasoning would have challenged one of the most important social and political roles of the law: in order to legitimate the exercise of power law must present itself as being guided by the objective, neutral application of reason, free of racialist and other biases, and serving the interests of society as a whole, and not simply the needs of the rich and powerful.\n\nWhile the Appeal Court did not define what these rights are, they were clear about what they are NOT: Aboriginal title and rights, they said, are not proprietary or commercial. Furthermore, they ruled that evidence in support of Aboriginal rights would have to be \"site-specific\" and drawn from documentation about the cultural practices of the specific First Nation making the claim.\n\n**We Buy Therefore We Are**\n\n_It is worth recalling that while British policy toward the native population was based on respect for their right to occupy their traditional lands, a proposition to which the Royal Proclamation of 1763 bears witness, there was from the outset never any doubt that sovereign and legislative power, and indeed the underlying title, to such lands vested in the Crown...._\n\n_The nature and content of an aboriginal right is determined by asking what the organized aboriginal society regarded as \"an integral part of their distinctive culture.\"...To be so regarded those practices must have been integral to the distinctive culture of the aboriginal society from which they are said to have arisen. A modernized form of such a practice would be no less an aboriginal right.... A practice which had not been integral to the organized society and its distinctive culture, but which became prevalent as a result of European influences would not qualify for protection as an aboriginal right...._\n\n_The content of aboriginal rights is to be determined not by reference to whether executives or legislative action conferred such a right on the people in question, but rather by reference to that which is essential to or inherent in the unique relations that native people have with nature, each other, and other communities...._\n\n\u20141990, _Reasons for Judgment_ , _R. v. Sparrow_.\n\nThe existence and persistence of Aboriginal title and rights upheld by the British Columbia Court of Appeal's ruling, meant that the legal question remaining to be resolved was \"what is the nature and scope of these existing Aboriginal rights?\" The Appeal Court judges failed to reach a consensus on answers to this question. Mr. Justice Macfarlane, supported by Taggart, Wallace and Hutcheon, relied on the \"new test\" enunciated in the Supreme Court of Canada's decision in _R. v. Sparrow_ that identified four questions to be decided by Courts in determining whether a particular practice should be protected as an Aboriginal right. First, what discrete practices, such as harvesting and exchanging fish, were to be identified as \"aboriginal\" practices? Second, was this practice integral to the distinctive culture of the Aboriginal society in which some of the ancestors of the present plaintiffs were members? That is, the legal definition of Aboriginality was now dependent on evidence that described practices which could be shown to have been carried out prior to European contact; and which were, then and now, \"integral to the distinctive culture\" of the particular Aboriginal nation bringing forward a claim. The critical defining difference\u2014in law\u2014between Aboriginal and non-Aboriginal \"practices\"\u2014and therefore between Aboriginal and non-Aboriginal peoples\u2014 was to be determined on the basis of the courts' analysis of what a \"distinctive culture\" is. If the first and second questions were answered in the affirmative, then third and fourth questions could be posed: was the practice existing as an Aboriginal right at the date when sovereignty was asserted; and was it unextin-guished prior to 1982? Affirmative and negative responses to these questions, respectively, would legally establish the existence of a constitutionally protected Aboriginal right.\n\nThere are several important aspects to the British Columbia Court of Appeal Justice Macfarlane's definition of Aboriginal rights. First, the focus of the question is what lands did the people occupy and how were these lands and resources used before contact? Claimants wishing to pass this part of the test were required to bring forth evidence on the specific customs and practices of their people in occupying particular pieces of land, and\/or in using particular resources. Given the Appeal Court's endorsation of Chief Justice McEachern's dismissal of oral history and expert academic evidence, the issue of \"proof\" remained problematic. Furthermore, \"occupation\" was defined as use to the exclusion of others. This meant that First Nations had to show they had actively excluded others from their territory, and maintained exclusive possession of the same territory. Failure to do so, either prior to the arrival of Europeans, or even in the face of force of arms and numbers of settlers after contact, could jeopardize the Aboriginal claim. Second, the use of the land and resources must be \"integral to the distinctive culture,\" and deemed by the court to be \"truly Indian.\" Aboriginality, and the Aboriginal \"other\" remained firmly frozen in the past.\n\nBut, what if an Aboriginal society engages in practices, and develops traditions and customs _after_ contact with Europeans, and they continue for a long, long time before the assertion of sovereignty? This issue of time and the continuation of \"Aboriginality\" into the present and future divided the appeal panel. Justices Macfarlane and Wallace took an historical approach. They tried to look to the past, to find something they could identify as \"traditional,\" or \"distinctively Indian\" and \"integral to the culture.\"\n\nJudge Lambert dissented from the majority. He viewed Aboriginal title, and the rights flowing from it, to be contemporaneous, allowing \"a blossoming in the present and into the future\" of the rights. Lambert represented a minority position when he cited a recent Australian precedent\u2014the _Mabo_ decision\u2014 handed down by Mr. Justice Brennan of the Supreme Court of Australia in 1992, that dispensed with the assertion of _terra nullius_ as a foundation for Australian law. Lambert accused the other members of the Appeal Court panel of supporting a \"frozen rights\" thesis. He wrote: \"Once it is recognized that aboriginal societies were societies capable of change, the notion that there is an 'aboriginal' use which can be discovered only on the basis of evidence of long-time use must be rejected.\" \"The purpose of s.35...[of the Canadian Constitution] when it was prepared in 1982,\" Lambert continued, \"cannot have been to protect the rights of Indians to live as they lived in 1778.... Its purpose must have been to secure to Indian people, without any further erosion, a modern unfolding of the rights.\" He chastised Chief Justice McEachern for finding that \"commercial trapping is not an aboriginal practice,\" saying: \"In my opinion...this...is to adopt the settlers' point of view of the classification of aboriginal title rather than the aboriginals' point of view.... If the Indians used land in 1820 in accordance with their aboriginal title but the use was a new one in 1820, then the important point is that at that time, namely 1820, the aboriginal right represented by the aboriginal title was taking on an 1820 contemporary form.\"\n\nThe question of jurisdiction, or an inherent right to self-government, further divided the B.C. Court of Appeal judges. Macfarlane, for the majority, rejected this aspect of the Gitksan's and Wet'suwet'en's claim, calling it incompatible with federal and provincial jurisdiction: \"Rights of self-government encompassing a power to make general laws governing the land and resources in the territory, and the people in that territory, can only be described as legislative powers...the jurisdiction of the plaintiffs would diminish the provincial and federal share of the total distribution of legislative power in Canada,\" he wrote. To bolster their claim that such a sharing of political authority was legally impossible, Macfarlane, on behalf of the majority, offered four arguments. First, he wrote, \"on the date that the legislative power of the Sovereign was imposed...any vestige of aboriginal law-making competence was superseded.\" Britain had the absolute right to assert sovereignty. Second, \"...if this is wrong (in law) then a continuing aboriginal legislative power is inconsistent with the division of powers found in the _Constitution Act_ of 1867.\" Even if Britain's original assertion of sovereignty was illegal, it became legal when the _Constitution Act 1867_ reaffirmed the initial violation of British colonial law. Third, \"the plaintiffs failed to establish the necessary ownership needed to support such a jurisdiction.\" The oral history testimony and the other expert evidence was not valid or reliable. But, he concluded, \"the establishment of some form of Indian self-government...is ripe for negotiation and reconciliation.\" Now that the terms, conditions, limits and possibilities have been circumscribed by law, political negotiations may be initiated.\n\nJudge Lambert, again, disagreed. In his minority finding he argued, based on the \"doctrine of continuity\" provided for under British common law, that Aboriginal rights survived both the assertion of sovereignty and the division of legislative powers. Justice Hutcheon also took an independent position on this question, placing himself between the majority and Lambert. He transformed \"self-government\" into a more limited concept of \"self regulation\" by removing any concept of law-making by Gitksan or Wet'suwet'en institutions. A minimal range of jurisdiction, he argued, could be granted: \"The appellants have a right of self-regulation exercisable through their own institutions to preserve and enhance their social, political, cultural, linguistic and spiritual identity.\"\n\nFinally, the Appeal Court panel dealt with the logical conclusion of their findings: if Aboriginal title and rights have not been extinguished, then what is the status of titles in fee simple that have been granted to non-Aboriginal citizens by the province for the past 123 years? In other words, if British and Canadian laws have not been obeyed by the governments, can any non-Aboriginal land titles be truly legal? They declined to give a definitive answer but made a range of suggestions about situations in which Aboriginal title and Crown and\/or fee simple title could co-exist. Justice Macfarlane, for example, concluded that: \"Uncultivated, unfenced, vacant land held in fee simple does not necessarily preclude the exercise of hunting rights.... On the other hand the building of a school on land usually occupied for aboriginal purposes will impair or suspend a right of occupation....\" In essence, this is little different from Chief Justice McEachern's suggestion that land could be returned to Aboriginal use after it had been clear cut. Aboriginal people may use whatever lands and resources non-Aboriginal governments and corporations don't want or need, until they desire it again.\n\nDuring the course of the _Delgamuukw_ trial several other Aboriginal rights cases were winding their way through the lower courts in British Columbia. These additional cases were: _R. v. Alphonse_ , _R. v. Dick_ , _R. v. Van Der Peet_ , _R. v. Gladstone_ , _R. v. N.T.C. Smokehouse_ , _R. v. Lewis_ and _R. v. Nikal_ , and all dealt in one way or another with the question of whether or not Aboriginal title could be proprietary, and Aboriginal rights could be commercial. Since _Delgamuukw v. R._ addressed issues fundamental to all of these cases, particularly whether Aboriginal rights had been extinguished or are now protected by the Canadian Constitution, the Appeal Court decided to rule on all these cases at the same time.\n\nTheir majority rulings on these cases were consistent: Aboriginal title is not \"proprietary\" and does not constitute ownership; and Aboriginal rights are not commercial ones. Concretely, this abstract debate can be reduced to the question of whether Aboriginal title is a form of property ownership that is equal and valid in comparison to Euro-Canadian title in fee simple. This is what the legal term \"proprietary interest\" refers to. Questions about the nature and scope of Aboriginal rights have quickly boiled down to debates about whether Aboriginal rights may be commercial ones or not, defined as equal and valid in comparison to production for exchange in a capitalist marketplace. For example, if Aboriginal peoples traded fish for hides prior to contact with Europeans, and subsequently traded fish for cash, is their trade in fish an \"Aboriginal practice\"? If they own the fish in the first place, do they not have the right to dispose of it in any way they see fit? If their ownership of resources, like fish, has, like their title to land, never been surrendered or extinguished, then don't Aboriginal peoples still own the fish? What criteria establishes a radical and significant difference between the first exchange for hides, and the second for cash? The contemporary question of the legal effect of \"European influences\" is really a debate about whether or not Aboriginal peoples can use the lands and resources they claim to own and have never surrendered, to enter into and compete in the capitalist market-place as self-governing owners of resources; or whether they are to be forced to join the race empty-handed, with nothing but their labour to exchange. Litigation and treaty negotiations are, after all, sites at which First Nations are endeavouring to negotiate a mutually-respectful _relationship_ with the Canadian state and within that state. They are not battlegrounds in a struggle for national liberation or political separation. It is the terms of the future relationships that are at stake, not the question of whether there will be a relationship or not.\n\nIn summary, the majority of B.C. Court of Appeal judges found that the Gitksan and Wet'suwet'en had Aboriginal rights that were not proprietary or commercial, but otherwise remained undefined; and that these rights had not been extinguished because the Crown has failed to demonstrate a \"clear and plain intention\" to do so. \"In the end,\" Justice Macfarlane concluded, \"the aboriginal interest is a right of use and occupation of a special nature\u2014best described as _sui generis_. To stretch and strain property law concepts in an attempt to find a place for these unusual concepts which have arisen in a special context, is in my opinion, an unproductive task.... Whatever protection is required to sustain the _sui generis_ interest of the descendants of the aboriginal peoples is now afforded by the Constitution, and in my view we should struggle no more to find a place in English property law for that interest.\"\n\nThe _sui generis_ theory of Aboriginal rights, in terms of Aboriginal interests is, however, a double-edged sword, another now familiar double-bind. On the one hand, it recognizes the \"cultural distinctiveness\" of Aboriginal title and does not try to force it to conform to European categories or definition. On the other hand, as long as Canadian law, and Canadian popular culture, are dominated by beliefs in European racial or cultural supremacy, then saying Aboriginal rights are different in theory can be expected to result in these rights being considered \"lesser\" ones in practice. Cultural difference may be recognized, but interpreted as justification for inequality. Heads, the Crown wins. Tails, Indians lose.\n\nFrom the point of view of a dualist perspective rooted in notions of binary oppositions, when judges define communal property and non-profit barter as \"integral to the distinctive culture\" of First Nations\u2014that is, as the most legally significant differences between Aboriginals and non-Aboriginals\u2014they are simultaneously proclaiming that individual, private property, and profit-making commerce, are \"integral to the distinctive culture\" of non-Aboriginal Canadians. Simply put, the key cultural difference, according to the law, is that private property and money are integral to the distinctive culture of non-Aboriginals; and communal property and cashless barter are the distinctive characteristics of Aboriginal cultures. \"We\" have land and money. \"They\" do not. Cynics might ask if this is news? Skeptics may question the wisdom of spending vast sums of public funds on legal proceedings over the last several hundred years to arrive at this conclusion.\n\n* * *\n\n. Court of Appeal of British Columbia, _Delgamuukw v. R._ , Defendant's Appeal Factum, III 1991: 12-14.\n\n. See Burns 1992; Burton and Kellock 1992; Doyle-Bedwell 1993; Fortune 1993; Foster 1992; Macklem 1993; Sanders 1992; Slattery 1992; Walters 1993.\n\n. See Bowker 1995 for a thorough analysis of the _Sparrow_ decision, its interpretation by the B.C. Court of Appeal, and its implications.\n\n. Walters 1993: 410.\n\n. Court of Appeal of British Columbia, _Delgamuukw v. R., Reasons for Judgment_ : 1993: 15.\n\n. Court of Appeal of British Columbia 1992 _Delgamuukw v. R._ , Defendant's Appeal Factum.\n\n. Court of Appeal of B.C. 1992 _Delgamuukw v. R., Reasons for Judgment_ : 18.\n\n. Stephenson and Ratnapala (eds.) 1993.\n\n. Court of Appeal of British Columbia, _Delgamuukw v. R., Reasons for Judgment:_ 172.\n\n. Ibid., 189.\n\n. Ibid., 34.\n\n. Ibid., 43-44.\n\n. Ibid., 268.\n\n. Ibid., 130-131.\n**Chapter 22: The British Columbia Treaty Process and**  \n**the Supreme Court of Canada, 1996**\n\n_How did the strategy of litigation fit with other strategies to bring about change? No one could suggest that the Gitksan and Wet'suwet'en put all their eggs in one basket. They used every strategy available\u2014roadblocks, fish-ins, marshmallow fights, participation in the First Minister's conferences, a play (that toured the province, the country, and the world), a film, links to academics...buttons, shirts, posters, T-shirts. Is such litigation a bad strategy? Only if the litigation is seen in isolation from other actions and strategies. We need to get over our idolatry of the courts and judges.... The cases were attempts to get the courts to kick governments._\n\n\u2014Legal scholar Douglas Sanders, 1992, _Getting Back to Rights_.\n\n**From Negotiation to Litigation to Negotiation to Litigation to Negotiation to Litigation to...**\n\nNeither Aboriginal peoples in B.C. nor the Crown have looked primarily to the courts for resolution of the Aboriginal title and rights issue in the years since the _Delgamuukw_ judgment was handed down in 1991. Rather, attention has focussed on the British Columbia Treaty Commission (BCTC). Just as the _Calder_ decision encouraged the federal government to initiate the Comprehensive Claims Policy, and to negotiate with the Nishga; the _Sparrow_ , _Delgamuukw_ and other 1993 B.C. Court of Appeal decisions have all recommended that the issues be resolved politically, and have recommended negotiation rather than litigation. Although not often stated frankly, the courts, bound as they are by their colonial inheritance and archaic powers of truth creation, have been shown unwilling to, or incapable of, resolving these issues.\n\nOn June 28, 1991, about three months after the _Delgamuukw_ decision was released, the B.C. Claims Task Force, which had been made up of representatives of federal, provincial and Aboriginal governments, academics and \"third party\" labour and business interests, published their report. It contained 19 recommendations for developing a process for negotiations amongst the parties. The Task Force report called for casting aside the historically troubled relationship between Aboriginal and non-Aboriginal peoples, rooted in an imperial past. A contemporary \"made-in-BC\" solution was required. This new relationship would be realized through the negotiation of \"modern day treaties,\" and would be based on \"recognition and respect for First Nations as self-determining and distinct nations with their own spiritual values, histories, languages, territories, political institutions and ways of life.\" In September, 1992, representatives of the federal and provincial governments and the First Nations' Summit, signed an agreement creating the B.C. Treaty Commission, which would serve as a watchdog over and facilitator of the six stage process of treaty making. On June 21, 1993, the federal and provincial governments finalized a _Memorandum of Understanding between Canada and British Columbia_ respecting the sharing of pre-treaty costs, settlement costs, implementation costs, and the costs of self-government, in which Canada agreed to provide the cash, and B.C. the land, required by any subsequent treaty agreement. The Gitksan and Wet'suwet'en signed a _Memorandum of Understanding_ with the Province and the Federal Government in 1992, agreeing to postpone their appeal of the _Delgamuukw_ decision to the Supreme Court of Canada in order to enter treaty negotiations under the auspices of the BCTC.\n\n**Appealing Culture**\n\n_What our Chiefs don't understand, what they have difficulty with, is that when we compare the case law that is put forward in Delgamuukw with our own laws, our chiefs say, 'That's really funny. Look at that. They can just change their law anytime they want. Where our laws have remained unchanged for centuries.' They can't understand that._\n\n\u2014Satsan (Herb George), 1992, _The Fire Within Us_.\n\nThe British Columbia Treaty Commission is ultimately bound by legal decisions since the law remains the \"court of last resort\" in settling differences that cannot be reconciled through the negotiation process. In November of 1996, the Supreme Court of Canada handed down landmark rulings on three of the cases that the British Columbia Court of Appeal had heard simultaneously with the _Delgamuukw_ appeal: _R. v. Gladstone_ , _R. v. Van Der Peet_ , and _R. v. NTC Smokehouse_.\n\nDorothy Van Der Peet, a member of the Stol:lo Nation of Southwestern British Columbia, sold 10 Fraser River salmon for $50.00 to her neighbour. She was arrested and charged under a clause in the _Fisheries Act_ that prohibits selling, bartering or offering to sell or barter any fish caught under Indian Food Fish licenses. The Crown called Dr. Gordon Stryd, an archaeologist, as an expert witness in anthropology, and the Sto:lo called Dr. Richard Daly. Stryd argued that bartering and trading fish had been an \"occasional, incidental and opportunistic\" occurrence prior to contact and was not central to Sto:lo life the way fishing for food and ceremony was. Daly argued that the Sto:lo had produced, preserved and traded surplus fish prior to contact, and had sold large amounts of fish, for cash, to the British between 1820 and 1846 when British sovereignty was established. Therefore, the selling of fish was an Aboriginal right protected under section 35(1) of the Canadian Constitution. This was a case where a practice had become \"traditional,\" \"unique,\" and \"integral to the distinctive culture\" of an Aboriginal people _after_ contact with Europeans and for a long, long time _before_ the assertion of sovereignty.\n\nB.C. Provincial Court judge Scarlett was the first to hear the case. He agreed with Gordon Stryd and found Dorothy Van Der Peet guilty. Scarlett ruled that \" _Natives did not fish to supply a market, there being no regularized trading system_ , nor were they able to preserve and store fish for extended periods of time (emphasis in original). A market as such for salmon was not present but created by European traders, primarily the Hudson's Bay Company. At Fort Langley the Sto:lo were able to catch and deliver fresh salmon to the traders where it was salted and exported. This use was clearly different in nature and quantity from Aboriginal activity.\"\n\nWhen the case was heard by the Supreme Court of B.C., Judge Selbie disagreed with Judge Scarlett, and found Dorothy Van Der Peet _not_ guilty. Judge Selbie wrote in his _Reasons for Judgment_ : \"In my view, the evidence in this case, oral, historical and opinion, looked at in the light of the principles of interpreting aboriginal rights... _is more consistent with the aboriginal right to fish including the right to sell, barter or exchange than otherwise and must be found so_. We are, after all, basically considering the existence in antiquity of an aboriginal's right to dispose of his fish other than by eating it himself or using it for ceremonial purposes\u2014the words 'sell', 'barter', 'exchange,' 'share,' are but variations on the theme of 'disposing.' It defies common sense to think that if the aboriginal did not want the fish for himself there would be some stricture against him disposing of it by some other means to his advantage.... I am satisfied that when the first Indian caught the first salmon he had the 'right' to do anything he wanted with it\u2014eat it, trade it for deer meat, throw it back or keep it against a hungrier time. As time went on and for an infinite variety of reasons that 'right' to catch the fish and do anything he wanted with it became hedged in....One of the social changes that occurred was the coming of the white-man.... A money economy eventually developed and...Indians...adjusted to that also\u2014he traded his fish for money. _This was a long way from his ancient sharing, bartering and trading practices but it was the logical progression of such_. It has been held that the aboriginal right to hunt is not frozen in time so that only the bow and arrow can be used in exercising it.... The Indian right to trade his fish is not frozen in time to doing so only by the medium of the potlatch and the like...(emphasis in the original.)\"\n\nJustices Macfarlane and Taggart of the B.C. Court of Appeal, however, disagreed with Judge Selbie and reversed his decision. They ruled that the first trial judge, Scarlett, was correct. Applying the _Sparrow_ test, the B.C. Court of Appeal judges wrote: \"Fishing was a integral part of the distinctive culture of the aborigines. Fish had a religious significance. Fish were revered. They were used for food, but played a significant role in ceremonial and social ways.... But that is not to say the purpose of fishing was to engage in commerce. In our opinion, trade with the British was not of the same nature and quality as the aboriginal traditions disclosed by the evidence...the question of what is an aboriginal right deserving protection is not determined necessarily by reference to the activities in which aboriginal persons were engaged in 1846. The test is whether such activities or practices were integral to the distinctive culture of the aborigines....\" Justice Wallace agreed.\n\nJustice Lambert, writing on his own, concluded that \"Mrs. Van Der Peet was exercising an aboriginal right when she sold ten salmon to Mrs. Lugsdin for $50.00.\" Lambert argued that when the Sto:lo began selling fish to the first Europeans who arrived in the early nineteenth century, \"all they were doing was exploiting a new opportunity....a response to a new circumstance in the carrying out of the existing practice.\" Justice Lambert supported his reasoning by arguing for a \"social perspective\" on the definition of Aboriginal rights. \"I think the 'social' perspective is the correct one,\" he wrote, \"because rights are not defined in terms of the purpose for which they are to be exercised, but in the way the rights may be employed in a social context.\" He continued: \"The fact that non-aboriginal people now engage in or formerly engaged in the same practices as aboriginal people does not indicate that the practice was not or is not an aboriginal practice. No doubt the Hudson's Bay traders who arrived at the Fort Langley post in the early 19th century had fished for salmon in their boyhood in the Tay or the Dee. A practice, tradition or custom may be integral to the distinctive culture of an aboriginal people even if a similar custom is integral to the distinctive culture of many other peoples as well.\"\n\nAnother B.C. Court of Appeal panel member, Justice Hutcheon, agreed with Mr. Justice Selbie's decision that the Aboriginal right to sell fish includes the right to sell, trade or barter. He based his decision on the fact that the evidence showed that Sto:lo had been engaged in the \"commercial exchange of fish\" for at least 26 years when, in 1846, British sovereignty was declared. He rejected Chief Justice McEachern's extension of the test for Aboriginal rights to include that Aboriginal rights must have been in effect for a \"long, long time\" prior to British sovereignty. And, he rejected the Crown's argument that \"aboriginal rights must describe pre-contact life.\"\n\nThe B.C. Court of Appeal decision on _R. v. Van Der Peet_ was appealed to the Supreme Court of Canada, who handed down its decision in November, 1996. Again, the court was divided. The majority ruled that \"the exchange of fish among the Sto:lo was 'incidental,'\" and \"the trade with Europeans was quantitatively different from what was typical of Sto:lo culture prior tocontact.\" Therefore the exchange of fish for money or other goods is not an integral part of the distinctive Sto:lo culture, and is not protected by the Constitution.\n\nIn a dissenting judgment, Justice L'Heureux-Dub\u00e9 followed B.C. Court of Appeal Justice Lambert's reasoning. She chastised her fellow jurists for assuming that the arrival of Europeans was necessarily a pivotal and determining feature of history, and for ultimately upholding a \"frozen rights\" approach to the determination of Aboriginal rights. The Judge went on to criticize her colleagues for misconstruing the words \"distinctive culture\" in the _Sparrow_ test, and interpreting them to mean \"distinct culture,\" or \"that which is left over after features of non-aboriginal culture have been taken away.\" L'Heureux-Dub\u00e9 added that \"culture\" should be understood as an interrelated whole, and not a catalogue of individualized practices. She proposed a delineation between three different categories of Aboriginal harvesting rights: subsistence, social and ceremonial; the provision of a modest livelihood; and commercial. Each category should have a unique bundle of rights, she argued.\n\nA second judge, Justice Mclachlin, also took an independent stand, writing her own minority judgment. She argued that Aboriginal rights should be defined \"through an empirical approach\" and should be \"drawn from history.\" This would be more practical than Courts repeatedly trying to describe _a priori_ what an Aboriginal right is. She agreed with L'Heureux-Dub\u00e9 that there is \"no justification for extending the right beyond what is required to provide the people with reasonable substitutes for what they traditionally obtained from the resource\u2014basic housing, transportation, clothing and amenities.\" By this reasoning, it may now be \"Aboriginal\"\u2014in the eyes of the law\u2014to be lower middle class, but it remains \"non-Aboriginal\" to be any wealthier than that. Neither L'Heureux-Dub\u00e9 nor Mclachlin addressed how the historical information that their proposed tests relied upon should be evaluated by the law.\n\nA second case decided by the Supreme Court of Canada in November 1996, _R. v. Gladstone and Gladstone_ , arose from a \"sting\" operation wherein Department of Fisheries' officers attempted to entrap two members of the Heiltsuk First Nation of Waglisla (Bella Bella). The Crown first told the story to the Supreme Court of British Columbia as follows:\n\nDonald and William Gladstone are members of the Heiltsuk Indian Band of Bella Bella. They arranged to ship to Vancouver approximately 4200 pounds of herring spawn on kelp. The fisheries officers were alerted by an informant and they kept under surveillance the transportation of the crates from the freight carrier in Vancouver to a warehouse in Richmond.... Later that day, William Gladstone arrived with a U-haul truck to pick up the crates. He drove into Vancouver and parked in a lot at Seymour and Nelson Streets. William and Donald Gladstone then drove in a Javelin automobile to the premises of Seaborn Enterprises Limited, a retail fish store at 1310 West 73rd Avenue in Vancouver. They had with them one white container of herring spawn which they took into the store. William Gladstone spoke to Mr. Katsu Hirose, the owner, and asked, according to Hirose, if he was 'interested in herring on kelp' to which he replied he 'never touched herring on kelp from native Indians'. They left and returned to the parking lot where they were arrested. Subsequently, the herring spawn on kelp was sold by the fisheries officials for $143,944.00\n\nThis case involved complicated charges and defences under the Criminal Code having to do with entrapment, search warrants and evidence. At the first trial, the judge acknowledged that the Heiltsuk had harvested herring spawn on kelp for hundreds of years. Alexander Mackenzie's 1793 journal recorded his having traded with them for this foodstuff. Therefore, Judge Lemiski of the Supreme Court of British Columbia found that the right to trade and barter herring roe was an Aboriginal right that the _Fisheries Act_ regulations, in _general,_ unjustly interfered with. However, he concluded that this interference was valid in this _particular_ case because the Gladstones had attempted \"to sell a relatively large quantity of spawn in a surreptitious manner to a foreign buyer in a location far removed from the Heiltsuk Band's region.\"\n\nWhen the _Gladstone_ case was appealed to the B.C. Court of Appeal, Judge Anderson agreed with the trial judge, although he found the _Fisheries Act_ regulations did not, in general, interfere with an Aboriginal right to fish. He was adamant, however, that selling 4800 pounds of herring roe for several hundred thousand dollars was \"inconsistent with a 'traditional aboriginal right'.\" Justices Macfarlane, Taggart and Wallace agreed that \"the activity in question could not be viewed as an integral part of the distinctive culture of the Heiltsuk people...the activity is different in nature and quality than the aboriginal right....\" They continued: \"The case is not one that turns on quantity, although both judges took account of the quantity involved. There was evidence of considerable quantities being transported to other Indians in aboriginal times. But the quality and character of the activity in aboriginal times was quite different from that disclosed by the evidence in this case. The aboriginal activity was rooted in a culture which gave significance to sharing a resource, to which one nation had ready access, while other Indian peoples did not.... The appellants were not exercising an aboriginal right when they attempted to sell herring spawn on kelp to a Japanese fish product buyer in Vancouver.\" The \"non-Aboriginal\" characteristics of the exchange in this case included the large amount of money involved, the style of doing business, the location of the exchange, and the race\/culture\/nationality of the purchaser.\n\nAgain, Mr. Justice Lambert took an independent position. He argued: \"...that the aboriginal rights of the Heiltsuk people to harvest herring spawn and to trade their herring spawn extensively and in quantities measured in tons, subject only to the need for conservation of the resource...have been established....\" The Supreme Court of Canada overturned the B.C. Court of Appeal's decision, and ruled that the evidence supported a finding that \"the exchange of herring spawn on kelp for money or other goods was to an extent a central, significant and defining feature of the culture of the Heiltsuk prior to contact and is best characterized as commercial. This exchange and trade was an integral part of the distinctive culture of the Heiltsuk....\" Justice LaForest, however, dissented. He argued that the significance of barter in herring roe to the Heiltsuk prior to contact was based on a cultural value of sharing resources, and that such values were not demonstrated by the Gladstone brothers whose activities, LaForest claimed, \"were done in a completely different context.\"\n\nOn the one hand, the Supreme Court of Canada ruled that the attempted sale by members of the Heiltsuk Nation of thousands of pounds of herring roe, _is_ a constitutionally protected Aboriginal right because they were satisfied by the evidence that such commerce was \"integral to the distinctive culture\" of the Heiltsuk before contact with Europeans. The judges argued that their decision was based on the evidence presented. On the other hand, the Supreme Court found that Dorothy Van Der Peet's sale of 10 fish for $50.00 to her non-Aboriginal neighbour was _not_ within her constitutionally protected Aboriginal rights because the oral history testimony of the Sto:lo, and their expert witnesses in history and anthropology did not supply convincing evidence that pre-contact trade and exchange was sufficiently similar to capitalist commerce. In the absence of a developed critique of, and recommendations for specific alternatives to, Chief Justice McEachern's assessment of Aboriginal oral histories and his approach to expert evidence provided by anthropologists, archaeologists and historians, these latest rulings result in the possibility that the rights of different First Nations, with diverse histories of relationships with academics\u2014particularly archaeologists and historic ethnographers\u2014could be determined by the presence or absence, reliability or weakness, of the academic record. More to the point, the law retains the absolute power to arbitrarily interpret and evaluate expert evidence according to its own rules, at its pleasure.\n\nContrary to Chief Justice McEachern's claim that judges are bound by the letter of the law, and by tradition-bound precedent, a majority of the Supreme Court of Canada judges, in their 1996 rulings, stated clearly that the Court's role was not only to make decisions based on the determination of \"facts,\" but that it was also incumbent upon them to facilitate the goal of reconciliation between sectors of the population in dispute with each other. \"In pursuing the goal of reconciliation...limits are equally a necessary part as much as recognition,\" Judges Lamer, Sopinka, Gonthier, Cory, Iacobucci and Major wrote. \"With regards to the distribution of the fisheries resource after conservation goals have been met, objectives such as the pursuit of economic and regional fairness, and the recognition of the historical reliance upon, and participation in, the fishery by non-aboriginal groups, are the type of objectives which can (at least in the right circumstances) satisfy this standard,\" they concluded. Dorothy Van Der Peet's ten fish were sockeye salmon, harvested from the profitable Fraser River run. This bounty, and the opportunity to harvest it, is sought after and contested by Aboriginal and non-Aboriginal fishers, and local and international fish processors. The Gladstone's thousands of pounds of herring roe, on the other hand, represent a delicacy eaten only by coastal First Nations and Japanese people. While the Japanese consumer market is a lucrative and captive one\u2014their own herring resources having been overfished to extinction\u2014there is little competition among harvesters; and no sports, recreation or tourist interests involved as there are in relation to the salmon industry. There is, in other words, no need for the Court to concern itself with the reconciliation of competing interests in the herring roe industry.\n\n* * *\n\n. Sanders 1992: 281-283.\n\n. Province of British Columbia 1991: 16. See McKee 1996 for a comprehensive overview and analysis of the treaty process.\n\n. Court of Appeal for British Columbia, 1993 _R. v. Dorothy Marie Van Der Peet, Reasons for Judgment_ : 91.\n\n. Ibid., 92.\n\n. Ibid., 134.\n\n. Ibid., 134.\n\n. Ibid., 134.\n\n. Ibid., 136.\n\n. Supreme Court of Canada, _R. v. Van Der Peet, Reasons for Judgment_ : 140.\n\n. Ibid., 144.\n\n. Ibid., 150.\n\n. Court of Appeal for British Columbia, _R. v. Gladstone, Reasons for Judgment_ : 158.\n\n. Ibid., 168.\n\n. Ibid., 158.\n\n. Ibid., 180.\n\n. Ibid., 184.\n\n. Ibid., 167.\n\n. Supreme Court of Canada, _R. v. Gladstone, Reasons for Judgment_ : 11.\n**P** **ART** **IX:**\n\n**I** **N** **F** **UTURO**\n\n**(I** **N THE** **F** **UTURE** **)**\n**Chapter 23: The Past In the Present, In the Future**\n\n**Just Another Day in Lotus Land**\n\nChief Justice Allan McEachern addressed the Canadian Bar Association's Annual Meeting in Vancouver in September, 1996. He used the occasion to rail against people he called \"agendists,\" who \"place themselves firmly on the side of the angels\" and have made \"outrage\" a \"common and effective debating technique.\" The Chief Justice lashed out at the media who, he charged, \"prefer an exaggerated or controversial outburst to rational analysis.\"\n\n\"One of the most serious misconceptions that troubles the judiciary,\" McEachern said, \"is the apparent belief on the part of many that judges decide cases in accordance with personal views or values.... Anyone who takes the time to read reasons for judgment knows that judges are not 'free spirits' who can decide cases any way they wish.... The relentless disciplines of evidence, law and appellate review all operate to require decisions to be rational within a much larger context than just the case under consideration.\"\n\nRepresentatives of the United Native Nations organization, who were also invited to attend the Bar Association's meetings, suggested that a good place to start looking for the \"relentless disciplines\" under which judges make decisions in this province might be in big law firms' long-standing business ties with forestry companies. In 1997, Allan McEachern was replaced as Chief Justice of B.C. by Bryan Williams. McEachern now sits on the British Columbia Court of Appeal.\n\n**Justice. Then Peace**\n\nBritish Columbia's primary resource-based economy is suffering the consequences of globalization: multi-national logging, mining and fishing companies are moving their operations to Third and other Fourth world locations. Unemployment has increased, wages have been depressed, and trade unions undermined. The commercial fishing industry's existence is threatened by declining salmon stocks. Hundreds of unionized cannery workers have lost their jobs in recent years as processors moved to Alaska and Washington, taking advantage of the lower wage opportunities afforded by the North American Free Trade Agreement. The fishing fleet has been cut in half in the last two years by the infamous federal government restructuring scheme named \"the Mifflin Plan,\" after the Liberal Fisheries Minister, Fred Mifflin, who administered the fleet reduction policy. Already marginalized, First Nations fishers have suffered the brunt of this new policy. Aboriginal and non-Aboriginal peoples living in coastal communities economically dependent on the fishery have seen the bases of their livelihoods and ways of life disappear during the last two years.\n\nTensions exist between Aboriginal peoples who have demanded moratoria on logging until their claims are settled, and workers and labour unions who rely on the forestry industry for their livelihood. The fishing industry is also a site of sometimes violent confrontations between Aboriginal fishers who claim ownership of the resource, and the right to harvest and sell their fish, and non-Aboriginal fishers who argue that the fish stocks are the \"common property\" of all Canadian citizens, and that Aboriginal people have no unique claim to it. Poverty has increased throughout the province, and social services have been cut back dramatically as the welfare state \"retreats:\" joblessness and economic insecurity, hopelessness and despair characterize life for more and more people\u2014Aboriginal and non-Aboriginal\u2014in contemporary British Columbia.\n\nRight wing political parties such as Reform and the B.C. Liberal Party scapegoat Aboriginal peoples and the land rights movement, identifying it as the cause of economic instability and mounting unemployment. They claim that disputes about Aboriginal title that frequently involve roadblocks, injunctions and blockades\u2014along with widely-publicized protests against logging by environmentalists\u2014create a \"chilly climate\" in B.C. for potential investors. When called upon to support their arguments, opponents of Aboriginal land rights repeat the ethnocentric historical narrative, and eurocentric cultural beliefs articulated by Chief Justice McEachern in his _Reasons for Judgment_ and upheld by some judges in the B.C. Court of Appeal and the Supreme Court of Canada: the tried\u2014and untrue\u2014\"founding myth of white British Columbians.\" This, briefly, is the social and political context in which the British Columbia Treaty Commission (BCTC) has, since 1992, been engaged in a \"modern-day treaty-making process.\"\n\nAs of this writing, forty-four First Nations in British Columbia have opted to become involved in negotiations with the federal and provincial governments in the context of the British Columbia Treaty Commission (BCTC). The BCTC's treaty-making process has been plagued by bureaucratic red tape, opposition by non-Aboriginals, conflicts between Aboriginal groups, shifting government positions and inequities in access to resources. For example: any costs incurred by First Nations in preparing for and carrying out negotiations will be deducted from final settlements. Governments, corporations and \"third party interests\" of course, have considerably more non-repayable resources at their disposal.\n\nNot all First Nations who are participating in the treaty process are enthusiastic. Some are cautiously optimistic. Many feel there are no other options and _anything_ is worth a try. An alliance of eighteen First Nations in the interior of British Columbia has consistently refused to participate in the treaty process unless, or until the government drops its demand that extinguishment of Aboriginal title must be the inevitable outcome of treaty negotiations. Recent precedents for modern-day treaties like the James Bay Northern Quebec Agreement and the Yukon Land Claim settlement have made recognition of Aboriginal title dependent upon extinguishment: at precisely the same moment that the Crown recognizes title, Aboriginal peoples must surrender it.\n\nRepresentatives of off-reserve, urban and Metis Aboriginal peoples have launched an increasingly vocal critique of the B.C. treaty process. They claim that their unique situations and interests are being ignored in favour of a narrowly construed, legalistic definition of First Nations limited to on-reserve populations represented by Band and Tribal Councils sanctioned by the _Indian Act_.\n\nAnother grouping of Aboriginal people excluded from, and without faith in, either the treaty process or leaders sanctioned by the _Indian Act_ , became involved in an armed stand-off at Gustafsen Lake in central British Columbia during the summer of 1995. Their political\/legal relationship, they argue, is with the British Crown as articulated in the Royal Proclamation of 1763. They have never surrendered their title to Britain, and they do not recognize the authority of the federal or the provincial governments to negotiate with them on a nation-to-nation basis. Given that this premise results in the logical conclusion that Canadian courts have no jurisdiction over Aboriginal lands or peoples, when their lawyer, Bruce Clark, attempted to put forward this argument, he was, of course, thrown out of the court whose legitimacy he refused to recognize. The NDP's Attorney-General, Ujjal Dosanjh, called in the Royal Canadian Mounted Police and alerted the Canadian army. Sixteen Aboriginal people were arrested and tried on criminal charges ranging from attempted murder to mischief.\n\nResource companies, small businesses, wildlife and tourist industries, municipal governments and trade unions are represented in the treaty process as \"third party interests.\" Members of these groups argue that they should play a greater role in decision-making, and that they have been excluded from the treaty process even though they will experience the consequences of any agreements that are made. First Nations, and representatives of the federal and provincial governments, argue that \"third parties\" are represented, as voting citizens, by the government's negotiators. \"Third parties\" have mounted concerted and successful campaigns to pressure the provincial government into narrowing the terms and conditions of treaty negotiation.\n\nSince coming to power in 1991, the New Democratic Party provincial government has steadily retreated from its initial position of recognizing Aboriginal title. The Nishga Tribal Council commissioned the prestigious accounting firm of Price Waterhouse to conduct an audit of their traditional territories in the Fall of 1994. The accountants' report estimated that non-Aboriginal \"stakeholders\" had gleaned between $2 billion and $4 billion in resources from Nishga ancestral lands since the mid-nineteenth century. In May, 1995, the provincial NDP government announced a new policy for treaty settlement that would be based on First Nations' \"future needs\" rather than \"past injustices.\" \"If treaties are to be meaningful in a contemporary world, they cannot be based solely on evidence from the past,\" the Province's position paper explained. \"The current and future interests of _all_ parties will determine the final land area of each treaty.\"\n\nThe Supreme Court of Canada's 1996 decisions in _Van Der Peet_ and _Gladstone_ demand that Aboriginal claimants bring forward detailed, site-specific historical evidence to prove their case. The British Columbia Treaty Commission now appears to be saying history doesn't matter, at least where compensation is concerned.\n\nProvincial treaty negotiators were advised that their mandate specifically excluded discussions of compensation based on \"calculations of damages arising from past use and alienation of the lands and resources within traditional territories.\" Instead, the province announced a seven point formula for determining treaty settlements: the goals of the particular First Nation; the quantity and quality of Indian reserve land already held; the availability of Crown land in the treaty area and the value of lands and resources on them; local economic opportunities; the nature and extent of provincial and public interest in the area; the nature and extent of private interests (such as leases and tree-farm licenses) in lands and resources in the treaty area; and, finally, the amount of cash in the treaty settlement after the costs of treaty negotiations have been deducted. Any disputes that may arise concerning the application of the treaty should be settled by mediation, but the parties may bring conflicts to the Supreme Court of B.C. for resolution. The law and its precedents, including those created by Chief Justice McEachern in _Delgamuukw v. R._ will constitute the court of last resort.\n\nVery early on May 12, 1995, the day they later released their new policy to the press, the NDP's Minister of Aboriginal Affairs, Rev. John Cashore, addressed a breakfast meeting of the Master Members of the Business Council of B.C., who are influential representatives of \"Third Party Interests\" in the treaty process. Cashore assured his audience that the NDP government was committed to attracting foreign investment and supporting the business sector. The social democratic government pledged to \"get out of the way\" in the relationship between Aboriginal peoples and corporate interests. First Nations would be liberated from paternalistic government protection, and released to sink or swim in the shark-infested waters of the global marketplace. Furthermore, the Minister promised, Aboriginal peoples would no longer be a \"burden\" on the public purse. \"We won't...we can't support tax havens and ghettoes,\" Reverend Cashore added. \"Self-determination means responsibility to pay taxes.\"\n\nOnly one treaty has reached the final stages of negotiation, and it is not officially a part of the BCTC process but is, rather, a conclusion of negotiations begun over twenty years ago in response to the Supreme Court of Canada's _Calder_ decision. Talks had been stalled until 1991 by the B.C. provincial government's refusal to participate. An Agreement in Principle (AIP) between the Nisga'a First Nation Tribal Council, and the federal and provincial governments was signed in 1995, but remains to be ratified and passed into legislation. The key terms of this proposed \"modern-day treaty\" include: the hovering sovereign will retain \"underlying title\" to all the lands; 1,930 square kilometers of lands in the Lower Nass River area will be designated as communally-owned \"Nisga'a lands\"; the lands within 18 reserves and an additional 2.5 square kilometres will be reclassified as \"fee simple lands\" to be used for economic development, subject to provincial laws. The Nisga'a will be able to implement forest management standards, and there will be public access to Nisga'a lands for hunting, fishing and recreation. Nisga'a will be entitled to hunt moose and other designated species, and they will participate in the administration of a wildlife management area. They will not be permitted to sell wildlife, but may trade or barter it among themselves or with other Aboriginal peoples. The Nisga'a will be able to make laws governing such things as culture and language, employment, public works, regulation of traffic and transportation, land use and solemnization of marriage. They will provide health, child welfare and education services, and will exercise local policing powers similar to those of municipalities. The Nisga'a will receive $190 million over a period of years, will levy taxes on their members and will enter into taxation agreements with Canada and B.C., as municipalities do.\n\nA complex agreement on fisheries that will form a part of the proposed treaty was announced on August 22, 1996. Under the terms of this agreement, the Nisga'a will receive an annual treaty entitlement of approximately 18% of the Canadian Nass River salmon run, which they will be allowed to sell (\"the priority entitlement\"). This represents somewhat less than what Nisga'a already involved in the commercial fishery already catch. However, \"the Nisga'a will not have any preferential commercial rights in respect of the sale of fish.... If in any year there are no directed commercial or recreational harvests of Nass Area stocks by non Nisga'a, there will be no Nisga'a commercial fishery of those stocks, in that year.... In exchange for agreeing to a ceiling on the priority salmon fishery...the Nisga'a will receive an additional entitlement to be delivered through a Harvest Agreement outside of the treaty, and which will have exactly the same priority as commercial or recreational fisheries.... By definition, there must be harvesting by others, before there will be any Nisga'a entitlement above the priority entitlement set forth in the Treaty.\"\n\nThe Nisga'a Agreement in Principle (AIP) has opponents and supporters in both Aboriginal and non-Aboriginal populations. Chief Joe Gosnell, Sr., President of the Nisga'a Tribal Council, was circumspect in his announcement of the AIP. \"This represents a hard-fought compromise that has seen a generation of Nisga'a growing old at the negotiating table, but we are making that compromise in order to become full and active participants in the social, political and economic life of this country,\" Gosnell explained. Minister of Indian Affairs Ron Irwin was bombastic: \"Today we make history,\" he proclaimed. \"We are forging a new relationship based on partnership and mutual respect.\" Provincial Minister of Aboriginal Affairs Reverend John Cashore praised the Agreement-in-Principle, calling it \"a fair and honourable resolution.\"\n\nMilton K. Wong, Vice-President of the Hong Kong Bank of Canada, made a submission to the provincial government's Select Standing Committee on Aboriginal Affairs in 1997 urging ratification of the Nisga'a AIP and other treaties. Citing the results of a recent benefits and costs analysis of potential treaty settlements conducted by the Vancouver accounting firm KPMG, Wong argued that treaties would establish the necessary framework within which \"we can establish a social, political and economic certainty that will encourage investment in British Columbia and therefore be of enormous help to business communities across the province.\"\n\nIn 1996, the federal Liberal government proposed significant changes to the _Indian Act_. The direction of these initiatives, too, is towards encouraging and facilitating relations between multi-national corporations, the private sector, and First Nations who, along with their lands, will no longer be wards of the Crown. Aboriginal leaders are divided in their responses to the Liberals' proposals. Critics charge that the consequences of these changes will be the emergence of late twentieth century \"company towns\" on Indian land, reminiscent of the monopolies granted to the Hudson's Bay Company in the nineteenth century. They say that the federal government is using the rhetoric of self-government and the currently fashionable lure of entrepreneurialism to legitimate abandoning their historical responsibilities and legal obligations\u2014 especially their fiduciary obligations\u2014 to Aboriginal peoples. Supporters of the proposed _Indian Act_ revisions argue that First Nations must be pragmatic, and position themselves competitively in the world market.\n\nIn the context of the increasing dominance of international capital and the declining economic or political power exercised by governments of nation states, it is interesting to recall the debates that gave rise to the cases that constituted the \"Marshall trilogy\" in nineteenth-century America, when \"suddenly, even the most hardened land-market capitalist assumed the mantle of zealous advocate of the Indians'...right to engage in unregulated real-estate transactions.\" A persistent theme in the struggle over land has been the competing interests of Crown(s), state(s), and private capital(s) when eachsought to appropriate Indian lands. We are currently moving out of a historical era dominated by state\/Crown monopoly on the purchase and administration of Indian land, government intervention in economic development on behalf of citizens' well-being, and publicly-funded social programs that protected people against the worst ravages of the mercurial market. We are moving into an era of rapid privatization and corporatization where the federal government is decentralizing and rolling up \"the social safety net.\" Paralleling the gross reductions in government spending on health, education and social services throughout the country, critics charge that new federal Indian policy represents the application of the same neo-conservative agenda to Aboriginal peoples. The federal government calls their new approach \"the devolution policy,\" and they claim they are supporting Aboriginal self-government by transferring responsibility for local government to Band Councils and Tribal Councils. Critics call it \"the de-evolution policy\" and \"cutbacks with feathers.\" They charge that what the government calls \"self-government\" is really \"self-administration.\" And, as \"transfers of responsibility\" have been accompanied by reduced transfers of resources, some First Nations leaders say they are being \"given the right to administer our own misery.\"\n\nSince 1990, and immediately preceding the province of British Columbia's historical policy change from non-recognition to qualified recognition of Aboriginal rights, major corporations have shown an interest in promoting reconciliation between Aboriginal peoples and governments, and in engaging in negotiations with Aboriginal groups directly. Leading Canadian banks and law firms now all have departments and personnel devoted exclusively to promoting Aboriginal businesses. The nexus of the relationship between Aboriginal and non-Aboriginal peoples is shifting from the political to the corporate arena. Aboriginal lands, culture, art, and even spirituality are being rapidly commodified. Increasingly, Indigenous peoples in Canada and throughout the world are faced with difficult choices about what direction to take regarding \"economic development.\" If First Nations communities were to lose their Aboriginal title to lands and to become pseudo-independent enclaves in a global capitalist economy this may provide some short-term economic benefits for members of existing Indigenous elites, or individually successful entrepreneurs, but the fate of peoples here and around the world whose lands have been expropriated and communities displaced, stands as a warning of where a journey directed by the profit motive may lead. Vancouver's Downtown Eastside neighbourhood is home to thousands of young Aboriginal people, many of them dislocated from their homes, detached from their communities and veterans of the foster care system. These are the children walking the streets selling their bodies. These are the young people dealing drugs destined to quell rebellion on the streets, and to soothe alienation in the corporate boardrooms and lawyers' lounges that litter the landscape of the \"vast emptiness.\" As I write, public health officials have declared the Downtown\n\nEastside to be a \"medical state of emergency\": its residents are suffering the worst pandemic of HIV+\/AIDS in the \"developed\" world.\n\nOn September 10, 1996, the province of British Columbia withdrew from treaty negotiations with the Gitksan and applied to have the appeal of _Delgamuukw v. R._ heard by the Supreme Court of Canada. Intervention by the British Columbia Treaty Commission to attempt to restart negotiations failed to bring the province back to the table. Although treaty negotiations with the Wet'suwet'en had not broken down, the Wet'suwet'en decided as well to proceed with the appeal and negotiations within the B.C. treaty process were suspended. The federal government agreed to provide minimal funding\u2014insufficient to cover real costs\u2014in support of the appeal. The appeal was heard in June, 1997. The Supreme Court of Canada allowed only one and a half days for arguments to be presented.\n\nFor the purposes of their appeal, the Gitksan and Wet'suwet'en agreed to compromise on two issues. They changed the wording of their claim from one seeking recognition of \"ownership\" and \"jurisdiction,\" to one claiming \"Aboriginal title\" and \"self-government.\" Second, they changed their claim from one made on behalf of specific chiefs and houses to a communal claim on behalf of all. The province and the federal government both argued that the test to prove Aboriginal title should be based on evidence of actual, continuous, physical occupation of specific land sites since time immemorial, and that Aboriginal rights should be limited to those that were \"integral to the distinctive culture\" of the First Nation applying to the court for recognition. Lawyers for both federal and provincial governments supported Chief Justice McEachern's findings on expert evidence. The Gitksan and Wet'suwet'en argued that an application of this test would result in the only lands enjoying Aboriginal title today being those already designated as reserves. The Province of British Columbia and \"third party\" intervenors argued that Aboriginal title had been extinguished, and that Chief Justice McEachern's findings of fact should be accepted. The Hereditary Chiefs had failed to prove their case, they claimed. The Gitksan and Wet'suwet'en replied that they had proven their essential connection to all their land in their oral history evidence that demonstrated cultural, historical, physical and spiritual relationships to the territory from time immemorial. Proof of site-specific and continuous use and occupation should not be required to prove Aboriginal title, they said. Such evidence is not required of the Crown or non-Aboriginal people wishing to prove ownership.\n\nAll parties awaited the decision of the Supreme Court of Canada.\n\n* * *\n\n. For an articulation of this political position see See Smith, M., 1995, and Lippert 1995. For a critique of the same see Bateman 1997.\n\n. Province of British Columbia, Ministry of Aboriginal Affairs, \"B.C.'s Approach to Treaty Settlements: Lands and Resources,\" Victoria, B.C.: May 12, 1995.\n\n. Business Council of B.C., \"Presentation on Draft Position Paper on Treaty Negotiations in British Columbia\", Vancouver: May 12, 1995; and Culhane, Dara, unpublished fieldnotes, May 12, 1995.\n\n. Nisga'a Tribal Council, \"Agreement-in-Principle In Brief,\" Vancouver: February 15, 1996.\n\n. Nisga'a Tribal Council, \"Backgrounder: Nisga'a Agreement in Principle, Fisheries Component,\" Vancouver: August 27, 1996.\n\n. Nisga'a Tribal Council, Government of Canada, Government of British Columbia, \"Tripartite Press Release: Nisga'a, British Columbia and Canada Release Historic Agreement-In-Principle,\" Vancouver: February 15, 1996.\n\n. Wong, Milton (1997) \"Economic Imperatives to Settle A Nisga'a Treaty,\" letter to B.C. Select Standing Committee on Aboriginal Affairs,\" March, 1997.\n\n. Williams R. A. 1990(b): 72.\n**Chapter 24: Back to the Future**\n\n_...We wish to begin by clearly stating that in our view the extinguishment policy is harmful and counterproductive.... Instead, the focus of negotiations between the Canadian government and Aboriginal peoples should be on reconciliation based on an affirmation of Aboriginal title and rights, according to the principle of equitable sharing of ownership and jurisdiction.... The present approach does damage to Canada's ethical core, for it founds negotiations on the premise that one party is inherently superior to the other...._\n\n_We argue that without a legitimate Aboriginal title there cannot be a legitimate Canadian state, for without the recognition of Aboriginal title we would have a state based on colonial rationalizations. The model we propose bases the relationship between Aboriginal peoples and Canada on the recognition of a permanent Aboriginal title that sustains all of us. It represents a direction that can ultimately take Canada from a state founded on colonialism, into the twenty-first century as a state with roots going back to time immemorial._\n\n\u2014Anthropologist Michael Asch, and Lawyer Norman Zlotkin, 1997, _Affirming Aboriginal Title: A New Basis for Comprehensive Claims Negotiations_.\n\n**Colouring Outside the Lines**\n\nAnd so here we are, on the eve of what the Julian calendar marks as a \"millennium.\" We are about to embark on a new century: the twenty-first, in the year of Our Lord. Jesus Christ. Countless numbers of people, Aboriginal and non-Aboriginal, have lived and died since 1774, pursuing just and unjust resolutions to the \"Aboriginal title issue\" in British Columbia and Canada. A generation has passed on, another has reached middle-age, and a new one has been born since Frank Calder and the Chiefs of the Nisga'a went to court in 1969. Millions and more millions of dollars have been spent on trials, hearings, research, studies and proposals. Learned people have combed the texts of western law, philosophy and politics and have developed blueprints for change. They have provided interpretations of precedents, models of logical arguments, and readings of history that would allow Canadian judges to recognize and codify, within their own terms, a new legal and political relationship between Aboriginal and non-Aboriginal peoples.  These possibilities, however inevitably limited, exist, as they always have, within the parameters of the law as it exists. The judicial will to realize this potential that has been absent.\n\nContemporary Canadian legal and political positions and cultural beliefs that continue to uphold the doctrine of discovery\/occupation\/settlement, and the concept of _terra nullius_ refuse to acknowledge the legitimacy of First Nations' existence. This position prohibits any possibility of dialogue or negotiation. The bully simply asserts his will through the power of force, and justifies it by denying the existence of the other. Positions that rely on the doctrine of conquest begin with a demand for the extinguishment of Aboriginal sovereignty, the recognition of the hovering sovereign's underlying title to all the land, and his omnipotent jurisdiction over all the peoples. In this scenario, the bully recognizes the existence of the other, holds his gun to their heads, and offers options for surrender.\n\nA Royal Commission on Aboriginal Peoples released its report in November 1996, after consulting, researching and deliberating for almost five years, at a cost of approximately 56 million dollars. Their extensive recommendations begin by proposing alternatives to the extinguishment policy to facilitate Nation-to-Nation negotiations between Aboriginal and non-Aboriginal Canadians. The Commissioners advocate a concept of dual-citizenship for Aboriginal peoples in Canada, and outline a 20-year plan for redressing historic injustices and achieving social, economic and political equality. To date, the federal government has ignored the Royal Commission's report. The political will to implement their recommendations is absent.\n\nContemporary Aboriginal legal and political representatives who advocate what is referred to as the \"sovereignty argument\" begin their story in the times before Europeans arrived. Aboriginal people who identify with the \"sovereignty position,\" and self-described \"traditionalist\" groupings, argue that Aboriginal sovereignty over lands and resources is a sacred trust entered into by the Creator and their ancestors that cannot be altered or negotiated away by mere mortals but must endure for all time. People committed to this position are open to negotiating agreements regarding sharing lands and resources with other sovereign peoples, but the prospect of surrendering sovereignty, or \"Aboriginal title,\" would constitute a betrayal of what they believe is their sacred covenant with the Creator to be stewards of the land. It is, literally, unthinkable.\n\nMany Aboriginal representatives who are involved in the treaty process and litigation make it very clear that they do not reject the sovereignty position in principle, but see themselves as being involved in processes that demand pragmatism and the making of strategic compromises. Remember that Chief Justice McEachern set out this double bind when he asked the absurd question whether the Gitksan and Wet'suwet'en themselves really believed in their own assertions of title because they acknowledged\u2014within the framework of meaning of the law\u2014the Crown's underlying title to all the land. Although the Chief Justice simultaneously argued that such recognition of Crown sovereignty constituted the non-negotiable terms and conditions under which the courts would hear the Gitksan's and Wet'suwet'en's case in the first place. Damned if you do. Damned if you don't.\n\nAny future coexistence between Aboriginal and non-Aboriginal Canadians must be built on a strong and just foundation, and for this to be possible the old one must first be dismantled. Canadians must return to the first moment in the relationship: to the initial assertion of brute force and demonstration of raw power when the Crown declared the land _terra nullius_ and asserted its dominion as if the Aboriginal peoples already living on the land did not exist. We must face the fact that this was justified then, and continues to be justified today, on the basis of an equally crude and violent assertion: that Europeans are superior human beings and Aboriginal peoples are inferior human beings. This was and is wrong: wrong according to British law then and now, and morally repugnant and politically indefensible according to Canadian ideals of justice, then and now. So the first reciprocal step towards reconciliation _must_ be for judges, through law; and political representatives, through legislation; and institutions, through inclusion and respect for autonomy; and Canadians, through intelligence and decency, to recognize this historical truth publicly and plainly for what it was, and is. Coming to terms with this truth of Canadian history requires that Canadians understand this legacy not as a series of historical errors or anomalies, but as consistent with the underlying foundations and principles of the colonial project. This means too that those who inherit the benefits of these practices must also be prepared to change and act responsibly. These truths must be acknowledged, without rationalizations. Justice must be done, and must be seen to be done, before any just future can begin to be built.\n\nA new generation of Aboriginal political activists and legal academics, and non-Aboriginal supporters have been engaged in the study of historical reinterpretation and revision and complex reconstructions of dialogues between European and Aboriginal philosophies. From their work have emerged sophisticated proposals and directions for legal, political and cultural reconciliation based on shared morality, ethics and justice. For example, Ojibway legal scholar John Borrows has revisited the Royal Proclamation of 1763 through studying Aboriginal oral histories. He argues that \"First Nations were not passive objects, but active participants, in the formulation and ratification of the Royal Proclamation.... In these early confrontations with the Crown, First Nations possessed their own power and a range of choices to which they could bring their own considerations and alternatives.\" Foremost among Aboriginal understandings of these agreements was the notion of continuity into perpetuity of their relationship to their lands, and of coexistence with other peoples. The Royal Proclamation of 1763, according to Aboriginal interpretations, guaranteed that negotiations would be required in order to change use of lands, and to determine the relative jurisdictions of Aboriginal and Canadian law and governments. This interpretation can provide for a relationship based on the assumption that Crown sovereignty and Aboriginal sovereignty may co-exist unless or until changes in this relationship are legitimately agreed upon by both parties. Read in this way, as a Nation-to-Nation treaty between First Nations and the Crown, Borrows argues, the Royal Proclamation could serve as a model for modern-day treaty-making.\n\nThe first step in creating the necessary conditions for intercultural communication is to agree to converse within a shared framework. Borrows' interpretation of the Royal Proclamation of 1763 as the outcome of intercultural negotiations, whose written _and_ oral history must be given equal weight, and whose contemporary meaning must be found in continuing dialogue, signals a significant gesture towards reconciliation with non-Aboriginal Canadians. Aboriginal peoples have also agreed to a shared language of negotiation, and accept, as sufficient for the purposes of communication, the translation of their histories and aspirations into English and French words.\n\nA Crown response that acknowledges that the Royal Proclamation _recognizes_ rather than _creates_ Aboriginal rights; that agrees to take into account the intentions and understandings of both British and Aboriginal authors of the Proclamation; that interprets the Royal Proclamation as recognizing the existence of sovereign Aboriginal _nations_ ; that, following from this, acknowledges the obligation for Britain, and now for Canada, to negotiate with Aboriginal Peoples on a _nation to nation_ basis; and that agrees to negotiate the nature of the Aboriginal rights so recognized, would constitute a reciprocal step towards dialogue and reconciliation.\n\nIn British Columbia, as elsewhere, an intangible yet palpable anticipation that we are standing at a cross-roads, or on the edge of a cliff, pervades the general sense of things: structures the feelings of our times. We know we are experiencing a social and cultural sea change, but we are uncertain what the future may hold. While parties to Aboriginal title litigation contest the details of history, they also dispute each other's visions of moral and social universes. Understanding \"culture\" as being most deeply rooted in theories of human nature and visions of what constitutes a \"good society,\" a \"good life,\" and a \"good person,\" rather than as a collection of traits, or a blueprint people carry around in their heads that, like a computer program, determines how they will interpret experiential \"input,\" allows the discussion to go beyond dualist, binary oppositions, hierarchal competitions for cultural supremacy, and essentialist notions of culture as naturally or divinely given. Many contemporary supporters of universal human rights argue for proposals that are not limited to the individual rights prized by the dominant western culture, but also address collective cultural rights, including the \"right to belong.\" Various concepts of \"the self\" and what it means to be a person in different cultural contexts serve as windows onto the broader political landscape. The self embedded in the Crown's legal arguments, in Sheila Robinson's anthropology, in Chief Justice McEachern's ruling, and in the Supreme Court of Canada's precedents, is the self of nineteenth-century European political theory. It is an idealized male self, unencumbered by obligations to family; freed from dependence on, or responsibility for, others; released from the mercy of time and tides. It is a self moved around by the demands of global capital in the international marketplace: the multinational corporate executive for whom home is where the profit is, and the migrant worker for whom home must be where the paycheque is. A self with \"reasons to travel.\" It is the self of the atomistic individual: the soldier in the war of all against all, the warrior in the battle of the fittest to survive. These theories of human nature embodied in aggressive, competitive individualists have been contested throughout western European history, from both inside and outside this intellectual tradition. The Crown's vision does not describe a natural or universal way of being human; nor a consensus on what it ought to mean, or could mean, to be human.\n\nContrary to the dominant image represented by the Crown, its opponents argue that human beings should be conceptualized as essentially and irreducibly social beings, and that individuals exist always already in relationships with others. As social beings we are constituted by our relationships. We need others in order to be ourselves. Within such a vision of human life we do not feed ourselves, we feed each other. If people are hungry, we have failed in our responsibilities as human beings. We do not shelter ourselves, we shelter each other. If people are homeless, we have failed in our responsibilities as human beings. If justice does not prevail, it is because we choose to practice injustice. Neither Nature nor God nor Culture\u2014nor \"The Deficit\"\u2014compel us to make the choices we make. As social beings we are, or ought to be, responsible to and for each other, and for our collective future.\n\nClearly, there are places within European legal, philosophical and cultural traditions from which openings to Aboriginal traditions are possible, and sites where bridges can be built. Many Aboriginal leaders and their supporters, of necessity, have been absorbed in responding to agendas set by the courts that have, as law dictates, focussed on competing interpretations of the past. Political negotiations follow the law, however unevenly or erratically. Until responsibility for the past is recognized, and fair reparations made, dreaming outside possibilities framed by existing legal and political structures seems to some a luxury that Aboriginal peoples and their allies cannot yet afford. This has inadvertently encouraged a kind of legalization of political imagination where justification for social change and visions of the future are sought in the past: in Lockean or Hobbesian political theory, or in romanticized images of pre-contact Aboriginal Gardens of Eden.\n\nOthers argue that none of us\u2014Aboriginal or non-Aboriginal\u2014can afford _not_ to dream. While we may draw on history, it is the future that we are constructing in the present. Societies will always reflect their histories, but need not be limited by them. Cultures are inevitably shaped by traditions, but are dynamic and creative too. Must we remain obedient to the tyranny of precedent? Why consider history to be normative, why allow it to be normalizing? Why hold the future a prisoner of the past? What does it mean, after all, to create human(e) communities? What is required to build a just world? These are ultimately moral and political challenges that must address economic structures and technology, but should not be determined by them. The answers to these questions will not be found in empirical studies of population to resources ratios, nor in abstract theoretical premises. These are questions that require moral and political dialogue, and solutions that must be lived and experienced in the everyday world, within relationships between people, not within contracts between things.\n\nThere are those who suggest more radical possibilities than the law, or the mainstream political imagination, currently envisions. They propose that Aboriginal title be recognized as the \"underlying title to all the lands.\" A living, breathing Mother Earth would displace the abstract, hovering sovereign. This title would be defined as a relationship to land and resources that constitutes a sacred trust held by the living for the not-yet-born. Such visions of possible futures see human beings as inextricably part of interdependent and mutually-sustaining relationships with all of nature. \"Man\" need not seek dominion over all he surveys; need not build fences or evict his family. Neither may he destroy at his whim that which shelters and sustains fellow beings. If land and resources were to be conceptualized and codified in law on such philosophical premises, then perhaps we could cease energy-devouring and soul-destroying battles in courtrooms, at negotiating tables, and on logging roads. Rather than competing in expensive contests of abstract legal acrobatics and arguing about who has the right to exploit the land, we could begin to discuss who has the responsibility to nurture the land. Perhaps we could all survive. Together. On this land.\n\nEasier said than done? Of course. Utopian idealism in the face of global capitalism and the ever-encroaching New World Order? Perhaps. But when are dreams most necessary if not in times like these? When should morality be celebrated if not when \"pragmatism\" terrorizes? Why should such a vision _not_ be possible to realize? Who says it could never be?\n\n* * *\n\n. Asch and Zlotkin 1997.\n\n. Macklem 1991, 1993; McNeil 1979; Slattery 1992.\n\n. Henderson J. Y 1985; Monture-Okanee 1993; Turpel 1991; Venne 1997.\n\n. Borrows 1997.\n\n. Marcus and Fisher 1986; Turner 1993, 1997.\n\n. Benhabib and Cornell 1989; Taylor 1991.\n**P** **OSTSCRIPT** **: D** **ECEMBER** **11, 1997**\n\n_I have mixed emotions today. I am filled with joy and also with remembrance of people who have worked so hard. I remember our Elders who passed on with broken hearts and the words of Chief Justice McEachern in their ears. Today's decision begins to heal the wounds._ _The Supreme Court of Canada has come out on the side of justice and humanity._\n\n\u2014Ray Jones, December 11, 1997.\n\n_I was there. I sat through the whole trial. I saw how our elders and witnesses were treated. I remember their tears that day that Chief Justice McEachern made his judgment. Now, at last, our truth has been heard._\n\n\u2014Yagalahl (Dora Wilson), December 11, 1997.\n\n_We've been given a diamond for Christmas...instead of a lump of coal._\n\n\u2014Satsan (Herb George), December 11, 1997.\n\n**The Dawn of a New Day?**\n\nOn December 11, 1997, Gitksan and Wet'suwet'en representatives, Aboriginal rights supporters and members of the media gathered at the Vancouver Aboriginal Friendship Centre to hear the Supreme Court of Canada's ruling on the _Delgamuukw v. R._ appeal. Most people had been taken by surprise a week earlier when it had been announced that this decision, not expected until the Spring of 1998, was to be handed down before Christmas.\n\nCautious anticipation turned into celebration as the contents of the Supreme Court's ruling were revealed: Allan McEachern had erred when he dismissed the oral history evidence of the Gitksan and Wet'suwet'en Chiefs and Elders, his own superiors determined. \"The trial judge, after refusing to admit, or giving no independent weight to these oral histories, reached the conclusion that the appellants had not demonstrated the requisite degree of occupation for 'ownership,'\" wrote the Right Honourable Antonio Lamer, Chief Justice of the Supreme Court of Canada. \"Had the trial judge assessed the oral histories correctly,\" Lamer continued, \"his conclusions on these issues of fact might have been very different.\" The depth and breadth of McEachern's errors led\n\nthe Supreme Court of Canada to order a new trial. For the same reasons, the Supreme Court also stated that they could not rule on the Gitksan's and Wet'suwet'en's claims for self-government rights.\n\nAppeal courts are normally loathe to contradict findings of fact made by lower court judges, and do so only when they are convinced that the trial judge has made a \"palpable and overriding error\" in his findings of fact. Usually, such a conclusion may arise from three possible bases: (1) where it can be demonstrated that there was no evidence to support a material finding of fact made by the trial judge; (2) when the trial judge wrongly overlooked admissible evidence relevant and material to the issue before the court; or, (3) where the trial judge's findings of fact cannot be supported as reasonable. Chief Justice Lamer added a fourth, new, condition to this list: \"In cases involving the determination of aboriginal rights,\" he wrote, \"appellate intervention is also warranted by the failure of a trial court to appreciate the evidentiary difficulties inherent in adjudicating aboriginal claims when, first, applying the rules of evidence and, second, interpreting the evidence before it.\"\n\nLamer detailed his court's criticisms of McEachern's decision on this matter, devoting eight full pages of text to the issue:\n\nAlthough he...[McEachern] had earlier recognized, when making his ruling on admissibility, that it was impossible to make an easy distinction between the mythological and \"real\" aspects of these oral histories, he discounted the adaawk and kungax because they were not 'literally true,' confounded 'what is fact and what is belief,' 'included some material which might be classified as mythology,' and projected a 'romantic view' of the history of the Gitksan and Wet'suwet'en. He also cast doubt on the authenticity of these special oral histories because...'the verifying group is so small that they cannot safely be regarded as expressing the reputation of even the Indian community, let alone the larger community whose opportunity to dispute territorial claims would be essential to weight.' Finally, he questioned the utility of the adaawk and kungax to demonstrate use and occupation because they were 'seriously lacking in detail about the specific lands to which they are said to relate.' Although he framed his ruling in terms of the specific oral histories before him, in my respectful opinion, the trial judge...[McEachern] in reality based his decision on some general concerns with the use of oral histories as evidence in aboriginal rights cases. In summary, the trial judge gave no independent weight to these special oral histories because they did not accurately convey historical truth, because knowledge about those oral histories was confined to the communities whose histories they were and because those oral histories were insufficiently detailed. However...these are features to a greater or lesser extent, of all oral histories.... The implication of the trial judge's reasoning is that oral histories should never be given any independent weight and are only useful as confirmatory evidence in aboriginal rights litigation. I fear that if this reasoning were followed, the oral histories of aboriginal peoples would be consistently and systematically undervalued by the Canadian legal system.\n\nThe trial judge also erred when he discounted the \"recollections of aboriginal life\" offered by various members of the appellant nations...he effectively held that this evidence did not demonstrate the requisite continuity between present occupation and past occupation in order to ground a claim for aboriginal title. In my opinion, the trial judge expected too much of the oral history of the appellants, as expressed in the recollections of aboriginal life of members of the appellant nations. He expected that evidence to provide definitive and precise evidence of pre-contact aboriginal activities on the territory in question.... Rather, if oral history cannot conclusively establish pre-sovereignty occupation of land, it may still be relevant to demonstrate that current occupation has its origins prior to sovereignty. This is exactly what the appellants sought to do.\n\nThe trial judge also erred in his treatment of the territorial affidavits filed by the appellant chiefs.... He questioned the degree to which the declarations amounted to a reputation because they were largely confined to the appellants' communities.... Furthermore, the trial judge reasoned that since the subject-matter of the affidavits was disputed, its reliability was doubtful. Finally, the trial judge questioned...the 'independence and objectivity' of the information contained in the affidavits, because the appellants and their ancestors 'have been actively discussing land claims for many years.'\n\nThe requirement that a reputation be known in the general community...ignores the fact that oral histories...generally relate to particular locations, and refer to particular families and communities and may, as a result, be unknown outside of that community, even to other aboriginal nations. Excluding the territorial affidavits because the claims to which they relate are disputed does not acknowledge that claims to aboriginal rights, and aboriginal title in particular, are almost always disputed and contested.... Casting doubt on the reliability of the territorial affidavits because land claims had been actively discussed for many years also fails to take account of the special context surrounding aboriginal claims, in two ways. First, those claims have been discussed for so long because of British Columbia's persistent refusal to acknowledge the existence of aboriginal title in that province until relatively recently.... It would be perverse, to say the least, to use the refusal of the province to acknowledge the rights of its aboriginal inhabitants as a reason for excluding evidence which may prove the existence of those rights. Second, this rationale for exclusion places aboriginal claimants whose societies record their past through oral history in a grave dilemma. In order for the oral history of a community to amount to a form of reputation, and to be admissible in court, it must remain alive through the discussions of members of that community; those discussions are the very basis of that reputation. But if those histories are discussed too much, and too close to the date of litigation, they may be discounted as being suspect, and may be held to be inadmissible. The net effect may be that a society with such an oral tradition would never be able to establish a historical claim through the use of oral history in court.\n\nFinally, Chief Justice Lamer concluded:\n\nIn the circumstances, the factual findings cannot stand. However, given the enormous complexity of the factual issues at hand, it would be impossible for the Court to do justice to the parties by sifting through the record itself and making new factual findings. A new trial is warranted, at which the evidence may be considered in light of the principles...elaborated upon here. In applying these principles, the new trial judge might well share some or all of the findings of fact of McEachern C.J.\n\n**Patience Is a Virtue**\n\n_This struggle began when the first Europeans arrived on our land. We have been waiting 130 years for this day. We are no longer invisible._\n\n\u2014Joan Ryan, Gitksan, December 11, 1997\n\n\"Aboriginal title is a right to the land itself,\" the Supreme Court's ruling stated, and this right is protected by Section 35(1) of the Canadian Constitution. The purpose of Section 35(1), the Supreme Court decreed, \"is to reconcile the prior presence of aboriginal peoples with the assertion of Crown sovereignty.\" Therefore both aspects of that prior presence\u2014the occupation of land, and the prior social organization and distinctive cultures of Aboriginal peoples on that land\u2014must be recognized and affirmed.\n\nThe Supreme Court proceeded to define, for the first time, the exact content of Aboriginal title that the law would legitimate. Chief Justice Lamer noted that it was necessary to do this because, he said, \"I believe that all of the parties have characterized the content of aboriginal title incorrectly.\"\n\nAboriginal title, the judgment continued, is inalienable; it must be communal; it cannot be transferred, sold or surrendered to anyone other than the Crown; and it finds its source in recognition (not creation) by the Royal Proclamation of 1763; in the common law principle that occupation constitutes proof of possession; and in laws that pre-existed the assertion of British sovereignty. Aboriginal title in law, the Supreme Court explained, is _sui generis:_ it represents a reconciliation of common law and aboriginal law and cannot be reduced to either.\n\nThe Supreme Court modified the \"frozen rights\" position adopted by previous judicial panels. Aboriginal title, the judges decreed, is not restricted to practices whose origins can be traced to pre-contact times. Legally codified Aboriginal title now encompasses the right to exclusive use and occupation of the land for a variety of purposes which need not be aspects of those practices, customs and traditions \"integral to the claimant group's distinctive aboriginal culture\" as had been required by the _Sparrow_ test. \"The nature of the Indian interest is very broad and incorporates present-day needs,\" the Supreme Court declared. \"Aboriginal title encompasses mineral rights and lands held pursuant to aboriginal title should be capable of exploitation. Such a use is certainly not a traditional one.\" However, \"practices, customs or traditions that arose solely as a response to European influences do not meet the standard for recognition.\"\n\nIn order to \"prove\" Aboriginal title in court, First Nations must present evidence that they used and occupied the lands claimed on the date British sovereignty was asserted. In British Columbia, this date has been established as 1846. Or, in the alternative, they may present evidence of present occupation as proof of pre-sovereignty possession as long as there is \"a continuity between present and pre-sovereignty occupation.\"\n\nThe Supreme Court distinguished between general Aboriginal _title_ and specific Aboriginal _rights_. While the relevant time period for establishing Aboriginal title is the date of British sovereignty, Aboriginal rights must be traced to the date of first contact with Europeans. Aboriginal rights are identified with particular practices, customs, and traditions that exist on a continuum related to the degree of connection with land.\n\nThe Supreme Court went on to develop the principle that Aboriginal title contains an \"inherent limit.\" In so doing, they clarified the legal definition of Aboriginal title more precisely:\n\nLands held pursuant to aboriginal title cannot be used in a manner that is irreconcilable with the nature of the claimants' attachment to those lands. This inherent limit arises because the relationship of an aboriginal community with its land should not be prevented from continuing into the future. Occupancy...[the \"proof\" required to establish Aboriginal title at law] is determined by reference to the activities that have taken place on the land and the uses to which the land has been put by the particular group. If lands are so occupied, there will exist a special bond between the group and the land in question such that the land will be part of the definition of the group's distinctive culture. Land held by virtue of aboriginal title may not be alienated because the land has an inherent and unique value in itself which is enjoyed by the community with aboriginal title to it. The community cannot put the land to uses which would destroy that value.\n\nFor example, if occupation is established with reference to the use of the land as a hunting ground, then the group that successfully claims aboriginal title to that land may not use it in such a fashion as to destroy its value for such a use (e.g. by strip mining it). Similarly, if a group claims a special bond with the land because of its ceremonial or cultural significance, it may not use the land in such a way as to destroy that relationship (e.g. by developing it in such a way that the bond is destroyed, perhaps by turning it into a parking lot.)\n\nFinally...the importance of the continuity of the relationship between an aboriginal community and its land, and the non-economic or inherent value of that land, should not be taken to detract from the possibility of surrender to the Crown in exchange for valuable consideration. On the contrary, the idea of surrender reinforces the conclusion that aboriginal title is limited in the way I have described. If aboriginal peoples wish to use their lands in a way that aboriginal title does not permit, then they must surrender those lands and convert them into non-title lands to do so.... This is not, I must emphasize, a limitation that restricts the use of the land to those activities that have traditionally been carried out on it. That would amount to a legal straightjacket on aboriginal peoples who have a legitimate legal claim to the land. The approach I have outlined above allows for a full range of uses of the land, subject only to an overarching limit, defined by the special nature of the aboriginal title in that land.\n\nThe courts maintain the power to determine what constitutes a \"truly aboriginal\" use of land and what does not, and that limiting distinction continues to turn on degrees of commercial exploitation and development.\n\nChief Justice Lamer dispensed with the \"old\" question of extinguishment of aboriginal title by declaring that provincial governments could not extinguish Aboriginal title or rights prior to 1982 when they were entrenched in the Constitution. Therefore, Aboriginal title and rights in British Columbia arise from long term use and occupation prior to European arrival, have never been extinguished, and still exist today. The possibility of legal extinguishment of Aboriginal title remains, but it has been projected into the future, rather than into the past; and could now involve Aboriginal initiative, participation and consent.\n\nThe judges of the Supreme Court of Canada upheld Allan McEachern's dismissal of the evidence provided by non-Aboriginal expert witnesses. Claiming that the Gitksan's and Wet'suwet'en's objections to McEachern's rulings on historical, anthropological, and archaeological evidence represented \"mere disagreement,\" rather than any \"palpable and overriding errors,\" and that the Province of British Columbia and the Government of Canada had established that \"there was some contradictory evidence that supported the trial judge's conclusions\" in these areas, Chief Justice Lamer concluded that \"the appellants have failed to demonstrate that the trial judge erred in this respect.\" Justice Lamer continued, emphasizing his court's support for Allan McEachern's dismissal of Richard Daly and Antonia Mills, and erasure of Hugh Brody:\n\nOne objection that I would like to mention specifically, albeit in passing, is the trial judge's refusal to accept the testimony of two anthropologists who were brought in as expert witnesses by the appellants. This aspect of the trial judge's reasons was hotly contested by the appellants in their written submissions. However, I need only reiterate what I have stated above, that findings of credibility, including the credibility of expert witnesses, are for the trial judge to make, and should warrant considerable deference from appellate courts.\n\nChief Justice Lamer and the Supreme Court of Canada panel is to be commended for respecting Aboriginal self-representation, and for finally acknowledging what First Nations have always known: that oral tradition is the preeminent, legitimate source of knowledge about Aboriginal culture and history, and must be understood and validated on its own terms. Rules for historical interpretation drawn from the mainstream, western, academic study of written history are not appropriate to the understanding of marginalized, non-western, oral histories. Neither should the law require Aboriginal peoples' representations to be mediated by anthropologists.\n\nHowever, Allan McEachern's \"findings of fact\" did not arise purely out of thin air, or his own imagination. They echoed the Crown's legal arguments in _Delgamuukw v. R._ These in turn drew on the evidence presented by the Crown's expert witnesses in history and anthropology, and on the unreflected upon \"common sense\" of Canadian colonial culture. The question that logically arises is: had the Crown and their witnesses \"assessed the oral histories correctly [would] their conclusions on...issues of fact...have been very different\"? In short, the Supreme Court of Canada appears to be saying, rather contradictorily, that Allan McEachern erred in his assessment of the oral histories, but the Crown and its witnesses who advised him to make the evaluation he did were not also wrong, but were merely expressing a difference of opinion arising from academic debates.\n\nThe problem with the Supreme Court's failure to find \"palpable and overriding errors\" in Allan McEachern's evaluation of the academic historical and anthropological testimony is not that it insults anthropologists and historians, but that such evidence will still continue to figure prominently in Aboriginal title litigation even if Aboriginal oral tradition is considered more appropriately and respectfully. In future litigation, questions about research standards and methodologies, about the evaluation of expertise, about ethnocentric biases in the historical and ethnographic record, and about the problems and complexities involved in cross-cultural interpretation and communication will continue to be at issue. If such considerations were taken seriously by the Supreme Court, in tandem with culturally-appropriate analyses of Aboriginal oral histories, such methodological rigour might mitigate against the possibility of a \"new trial judge...[sharing] some or all of the findings of fact of McEachern C.J.\" in this or any other cases.\n\n**The Way Forward**\n\n_This judgment restores a measure of faith in the legal system. The rule of law applies to all British Columbians now. The federal and provincial governments should come back to the treaty negotiation table in a fair and honest manner._\n\n\u2014Chief Joe Mathias, Squamish First Nation, December 11, 1997.\n\n_\"Aboriginal title is a right to the land itself!\" Premier Clark and Prime Minister Chretien, take note! You can no longer make trade deals with the Asia Pacific countries or anyone else without us._\n\n\u2014Chief Ed John, First Nations Summit, December 11, 1997.\n\nFinally, the Supreme Court of Canada judges turned their attention to determining how\u2014once Aboriginal title and\/or Aboriginal rights are established at law\u2014federal and provincial governments could justifiably infringe upon them. The Aboriginal title codified by the Supreme Court's ruling remains a subordinate one that constitutes a burden on the Crown's underlying, radical title. The hovering sovereign's hegemony remains paramount. This principle is most clearly spelled out in the judgment's articulation of legal tests for the justification of infringement. The satisfaction of two conditions would constitute legally sanctioned infringement of aboriginal title and rights: (1) that the infringement furthers a compelling and substantial legislative objective; and\/or (2) that the infringement is consistent with the special fiduciary relationship between the Crown and the Aboriginal peoples. The ruling sets out the following as \"compelling and substantial legislative objectives\":\n\nThe development of agriculture, forestry, mining and hydroelectric power, the general economic development of the interior of British Columbia, protection of the environment or endangered species, and the building of infrastructure and the settlement of foreign populations to support those aims....\n\nLegitimate government objectives also include \"the pursuit of economic and regional fairness;\" and in relation to specific resources like salmon, \"the recognition of the historical reliance upon, and participation in, the fishery by non-aboriginal groups.\" Attainment of these objectives constitute legally justifiable reasons for infringement. The Supreme Court of Canada explained their reasoning:\n\nBecause distinctive aboriginal societies exist within, and are part of, a broader social, political and economic community over which the Crown is sovereign, there are circumstances in which, in order to pursue objectives of compelling and substantial importance to that community as a whole (taking into account the fact that aboriginal societies are part of that community), some limitation of those rights will be justifiable. Aboriginal rights are a necessary part of the reconciliation of aboriginal society with the broader political community of which they are a part; limits placed on those rights are, where the objectives furthered by those limits are of sufficient importance to the broader community as a whole, equally a necessary part of that reconciliation.\n\nHowever, such infringement should not be arbitrary or unfair within the terms established by the court. Aboriginal peoples must be consulted, included, and compensated, according to the fiduciary obligation owed by the Crown to Aboriginal peoples. The Supreme Court explained their interpretation of this second part of the test for justifiable infringement:\n\nThe fiduciary duty does not demand that aboriginal rights always be given priority. Each case is different...The form the fiduciary obligation can take is also variable...Within the analysis of justification, there are further questions to be addressed, depending on the circumstances of the inquiry. These include the questions of whether there has been as little infringement as possible in order to effect the desired result; whether, in a situation of expropriation, fair compensation is available; and, whether the aboriginal group in question has been consulted....\n\nThe nature and scope of the duty of consultation will vary with the circumstances. In occasional cases, when the breach is less serious or relatively minor, it will be no more than a duty to discuss important decisions that will be taken with respect to lands pursuant to aboriginal title. Of course, even in the rare cases when the minimum acceptable standard is consultation, this consultation must be in good faith, and with the intention of substantially addressing the concerns of the aboriginal peoples whose lands are at issue. In most cases, it will be significantly deeper than mere consultation. Some cases may even require the full consent of an aboriginal nation, particularly when provinces enact hunting and fishing regulations in relation to aboriginal lands.\n\n...In keeping with the duty of honour and good faith on the Crown, fair compensation will ordinarily be required when aboriginal title is infringed. The amount of compensation payable will vary with the nature of the particular aboriginal title affected and with the nature and severity of the infringement and the extent to which aboriginal interests were accommodated.\n\nAnswers to all these questions will be determined on a case-by-case basis.\n\n**The Irony of Ironies**\n\n_It is not open to the appellants to challenge the trial judge's findings of fact merely because they disagree with them. I fear that a significant number of the appellants' objections fall into this category.... The bulk of these objections, at best, relate to alleged instances of misapprehension or oversight of material evidence by the trial judge._\n\n\u2014Chief Justice Lamer, 1997, _Reasons for Judgment_ : para 90.\n\nWhile the Supreme Court of Canada ruled that Allan McEachern's errors were so fundamental as to be beyond the scope of an appeal court to repair, and that a new trial was therefore warranted, the judges recommended that disputes concerning land title and jurisdiction between First Nations in British Columbia and the federal and provincial governments should be settled through negotiations and not through further litigation. In so doing, they reiterated a position that has become increasingly influential in contemporary Canadian politics and was clearly articulated by the Supreme Court's recent decisions in the _Van Der Peet_ and _Gladstone_ cases: appointed judges simultaneously proclaiming the omnipotent independence of the law from society, while explicitly and intentionally shaping and directing public policy that is, theoretically, the job of elected politicians. Law advances, democracy retreats. The Supreme Court of Canada's ruling in the appeal of _Delgamuukw v. R._ is specifically directed to treaty negotiations in British Columbia, and it significantly strengthens the position of First Nations in this process. Specifically, by recognizing unextinguished Aboriginal title to _all_ of British Columbia, by insisting on consultation with First Nations regarding development on their lands, and by requiring that compensation be paid for historical and future exploitation of Aboriginally-owned resources, the Supreme Court of Canada undermined the positions that the Province of British Columbia and the Government of Canada have brought to the treaty negotiation table. The governments have argued there that Aboriginal jurisdiction should be limited to specific lands, such as those already designated as Indian reserves; and that no entitlement to compensation for resource exploitation should be considered. Law advances minority rights that democracy has retreated from protecting.\n\nPublic opinion and the electoral process also affects what the future of treaty negotiations may be. The Supreme Court's validation of Aboriginal title and rights, and their recognition of the legitimacy of oral histories contradicts the still popular \"founding myth of White British Columbians\" that was presented by the Crown, and articulated by former Chief Justice McEachern in _Delgamuukw v. R._ This continues to be the \"historical myth of choice\" of significant sectors of the B.C. population, particularly of the provincial Liberal and Reform parties, who threaten to form the next government and to dismantle the treaty process if they do.\n\n**Tomorrow Today**\n\n_Legal orders may embody asymmetrical power relations, but power is always an interactional process. Dominant groups enjoy legally protected privileges, but they are also constrained by the law. And subordinated groups that suffer under particular legal systems may find that law offers them, the less powerful, a measure of protection from the powerful, just as it sometimes offers them resources for action._\n\n\u2014Anthropologists June Starr and Jane Collier, 1989, \"Introduction,\" _History and Power in the Study of Law_.\n\nThat the Supreme Court of Canada's December, 1997, ruling on the _Delgamuukw v. R._ appeal significantly changes the legal landscape in which Aboriginal title and rights litigation is adjudicated in Canada and in other colonial contexts worldwide is beyond question. While initial public responses to the judgment by representatives of industry and federal and provincial governments have tended to minimize its significance and potential impact on treaty negotiations, clearly this is truly a \"landmark judgment\" whose consequences will be many and varied. We will live in interesting times in the years to come.\n\nFor now, for today, First Nations of British Columbia are deservedly celebrating an historic turning point in a long, hard-fought, hard-won battle. Theirs has been a victory for humanity, for social justice and for the land: a triumph of spirit. They deserve the admiration, the respect and the gratitude of good people the world over.\n\nPeople make history. Not under conditions of their own choosing and against great odds.\n\nDara Culhane  \nVancouver, B.C.  \nDecember 12, 1997\n\n* * *\n\n. This book was in press and scheduled for release in December 1997, when the Supreme Court of Canada announced that they would release their judgment in the _Delgamuukw_ appeal. This postscript describes first impressions of the key features of this very complex and detailed 200-page judgment. I had intended to write an addendum and publish a revised edition of this book following the handing down of the appeal judgment originally expected in 1998. Time will tell the extent to which, and the speed with which, this work becomes a documentation of \"historical times,\" and of questions no longer asked or answered.\n\n. Supreme Court of Canada, 1997, Reasons for Judgment in the Appeal of _Delgamuukw v. R._ : para 107.\n\n. Ibid., para 80.\n\n. Ibid., para 97.\n\n. Ibid., para 98.\n\n. Ibid., para 99-101.\n\n. Ibid., para 104.\n\n. Ibid., para 106.\n\n. Ibid., para 108.\n\n. Ibid., para 90.\n\n. Ibid., para 91.\n\n. Ibid., para 110.\n\n. Supreme Court of Canada, 1997, Judgment in the Appeal of _Delgamuukw v. R._ : 4.\n\n14. Reasons 1997:para 144.\n\n. Ibid., para 128.\n\n. Ibid., para 131.\n\n. Judgment 1997:14.\n\n. Reasons 1997:para 161.\n\n. Ibid., para 162.\n\n. Ibid., para 168.\n\n. Ibid., para 169.\n**BIBLIOGRAPHY**\n\nAbu-Lughod, Lila and Catherine A. Lutz, eds. 1990. 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Princeton: Princeton University Press.\n\nZavardadeh, Mas'ud and Donald Morton. 1993. _Theory as resistance: Politics and Culture after (post)structuralism_. New York: Guilford Publications.\n**TABLE OF CASES** *\n\n_Calvin's Case_ (1608), 7 Co Rep 1a, 2 State Tr 559, Moore KB 790, Jenk 306, 77 ER 377 at 398.\n\n_Fletcher v. Peck_ , 10 U.S. (6 Cranch) 87 (1810) at 146.\n\n_Johnson v. M'Intosh_ 21 U.S. (8 Wheat) 543 (1823) at 573.\n\n_Worcester v. Georgia_ 34 U.S. (4 S.G.U.S.) 762 (1832) at 843.\n\n_St. Catherine's Milling and Lumber Co. v. R._ (1885), 10 or 196 (Ont.Ch.); (1886) 13 OAR 148 (Ont.CAO); (1887) SCR 577(SCC); (1888) 14 AC 46 (PC).\n\n_Re: Southern Rhodesia_ [1919], A.C. 211 at 233-4.\n\n_Amodu Tijani v. Southern Nigeria_ [1921], 2 A.C. 399(p.c.) at 403.\n\n_R. v. White & Bob_ (50) D. L. R. (2d) [1965], 620.\n\n_Calder et al v. Attorney-General of B.C._ (1969), 8 D.L.R. (3d), 59-83, [S.C.B.C.]\n\n_Calder v. Attorney-General of B.C._ [1973], S.C.R. 313.\n\n_Kruger and Manual v. The Queen_ [1977], 4 WWR 300, [1978] 1 SCR 104, 75 DLR (3d) 434, 14 NR 495, 34 CCC (2d) 377.\n\n_Hamlet of Baker Lake et al v. Minister of Indian Affairs and Northern Development et al_ [1980], 5 WWR 193, 50 CCC (2d) 377 (FCTD).\n\n_R. v. Simon_ (1985) 24 D.L.R. (4 th) 390 (S.C.C.).\n\n_A.G. Ontario v. Bear Island Foundation_ [1985], 49 O.R. (2d) 353, 15 D.L.R. (4th 321 (Ont. H.C.).\n\n_A.G. Ontario v. Bear Island Foundation_ [1989], 68 O.R. (2d) 394, 38 D.L.R. (4th) 117 (Ont.C.A.).\n\n_A.G. Ontario v. Bear Island Foundation_ [1991], 83 D.L.R. (4th) 381.\n\n_Guerin v. R._ [1984], 2 S.C.R. 335.\n\n_R. v. Sparrow_ [1990], 1 S.C.R. 1075.\n\n_Mabo v. Queensland_ (1992), 107 A.L.R. 1.\n\n_R. v. N.T.C. Smokehouse Ltd._ [1996], 2 S.C.R. 672. (Supreme Court of Canada Appeal).\n\n_R. v. Gladstone_ Provincial Court of British Columbia, Lemiski Prov. Ct.J., October 3, 1990. (Original trial.)\n\n_R. v. Gladstone_ B.C.S.C. (Anderson J.): (1991), 13. W.C.B. (2d) 601. (British Columbia Supreme Court.)\n\n_R. v. Gladstone_ B.C.C.A. (1993), 80 B.C.L.R. (2d) 133. 29 B.C.A.C. 253. 48 W.A.C. 253. [1993] 5 W.W.R. 517 [1993] 4 C.N.L.R. 75. (B.C. Court of Appea) _R. v. Gladstone_ [1996], 2 S.C.R. 723. (Supreme Court of Canada Appeal).\n\n_R. v. Van Der Peet_ , British Columbia Provincial Court: [1991], 3 C.N.L.R. 155. (Original trial.)\n\n_R. v. Van Der Peet_ British Columbia Supreme Court: (1991), 58 B.C.L.R. (2d) 392, [1991] 3 C.N.L.R. 161. (British Columbia Supreme Court.)\n\n_R. v. Van Der Peet_ , British Columbia Court of Appeal: (1993), 80 B.C.L.R. (2d) 75, 29 B.C.A.C. 209, 48 W.A.C. 209, 83 C.C.C. (3d) 289, [1993] 5 W.W.R. 459, [1993] 4 C.N.L.R. 221. (B.C. Court of Appeal.)\n\n_R. v. Van Der Peet_ , [1996], 2 S.C.R. 507. (Supreme Court of Canada Appeal.)\n\n_Delgamuukw v. R._ (1987), 40 D.L.R. (4th) 698. (Judgment on admissibility of oral tradition as an exception to the \"hearsay rule.\")\n\n_Delgamuukw v. B.C._ (1989), 38 B.C.L.R. (2d) 165. (Judgment on the admissibility of historical documents.)\n\n_Delgamuukw v. B.C._ (1989), 38 B.C.L.R. (2d) 176. (Judgment on the admissibility of treatises.)\n\n_Delgamuukw v. B.C._ [1991], 3 W.W.R. 97, [1991] C.N.L.R. xii, (1991), 79 D.L.R. (4th) 185, [1991] b.c.j. nO. 535 (QL). (Original trial.)\n\n_Delgamuukw v. B.C._ (1993), 30 B.C.A.C. 1, 49 W.A.C. 1, 104 D.L.R. (4th) 470, [1993] 5 W.W. R. 97 [1993] 5 C.N.L.R. 1, [1993] B.C.J. No. 1395 (QL). (B.C. Court of Appeal.)\n\n* Key:\n\nA.L.R. (Australian Law Reports)\n\nB.C.L.R. (British Columbia Law Reports)\n\nC.C.C. (Canadian Criminal Courts)\n\nC.N.L.R. (Canadian Native Law Reports)\n\nD.L.R. (Dominion Law Reports)\n\nF.C.T.D. (Federal Court Trial Division)\n\nS.C.R. (Supreme Court Reports)\n\nS.C.C. (Supreme Court of Canada)\n\nW.A.C. (Western Appeal Courts)\n\nW.W.R. (Western Weekly Reports)\n**ACKNOWLEDGEMENTS**\n\nIt seems appropriate to end by acknowledging that all stories are _about_ the activities of people: history ultimately belongs to those who create it. If the participants in the events I have chronicled didn't choose to put their case before the courts and the public, this document would not exist. So my first bow is to the \"Plaintiffs\" and their supporters in _Delgamuukw v. R._ for having the courage to take a stand, and for working hard to defend it. I feel arrogant at this moment, facing you as a critic working after the fact and out of the fray. I do not expect that everyone will appreciate my effort. I hope most will at least agree that this book constitutes a worthwhile project, as one account of a historical moment that continues to have far-reaching effects.\n\nSo many people's thoughts, comments, questions and criticisms have gone into this project over the course of the last six years that it is impossible to name everyone. This book's first incarnation was as a Ph.D. thesis and I am grateful to those who assisted and supported me in producing that document: my supervisor, Professor Noel Dyck; members of my thesis committee, Professors Michael Kenny and Arlene McLaren; examiners Dr. Ian Angus and Dr. Robert Paine. For a lot of practical assistance, and for the endless patience and support that I do not believe anyone could ever produce a doctoral thesis without, I am grateful to Fran Gillis, and to my late mother, Claire Culhane.\n\nWithout Gary Fisher's incisive editing, bounteous encouragement, and great sense of humour, the thesis would not have become a book. Thank you. Thank you.\n\nSeveral readers' comments and critiques transformed a draft into a manuscript: Thank you: Leslie Butt, Phil Russell, Roisin Sheehy-Culhane; and two anonymous peer reviewers. Thanks to Ann Vanderbijl for meticulously verifying references and polishing the bibliography. Other people from whom I have learned much that is relevant to this book: Frances Abele, Julie Cruikshank, Lori Gabrielson, Dan Gillis, Fran Gillis, Miki Maeba, Renee Taylor, Barbara Williamson. None are responsible for errors, omissions, or opinions expressed in the text: any of those are mine alone.\n\nAnd last, but never least, I thank my children Carey Speck, and Lori Speck, for sustaining me, always.\n\n#### About Talonbooks\n\nThank you for purchasing and reading _The Pleasure of the Crown_.\n\nIf you came across this ebook by some other means, feel free to purchase it and support our hard work. It is available through most major online ebook retailers and on our website. The print edition is also available.\n\nTalonbooks is a small, independent, Canadian book publishing company. We have been publishing works of the highest literary merit since the 1960s. With more than 500 books in print, we offer drama, poetry, fiction, and non-fiction by local playwrights, poets, and authors from the mainstream and margins of Canada's three founding nations, as well as both visible and invisible minorities within Canada's cultural mosaic.\n\nLearn more about us or about the author, Dara Culhane Speck.\n\n##### Other Books by Dara Culhane and available from Talonbooks\n\n_In Plain Sight: Reflections on Life in Downtown Eastside Vancouver_\n\n_An Error in Judgement: The Politics of Medical Care in an Indian\/White Community_\nCopyright \u00a9 1998 Dara Culhane\n\nPublished with the assistance of the Canada Council\n\nTalon Books Ltd.  \n278 E 1st Ave.  \nVancouver, British Columbia, V5T 1A6, Canada\n\nFirst printing: January 1998  \nElectronic edition: 2014\n\nOn the cover: _Mask of the Injustice System_ by David Neal  \nCover design by Adam Swica\n\nTalonbooks are distributed in Canada by General Distribution Services, 30 Lesmill Road, Don Mills, Ontario, Canada M3B 2T6; Tel.:(416) 445-3333; Fax:(416) 445-5967.\n\nTalonbooks are distributed in the U. S. A. by General Distribution Services Inc., 85 Rock River Drive, Suite 202, Buffalo, New York, U.S.A. 14207-2170; Tel.:1-800-805-1083; Fax:1-800-481-6207.\n\nNo part of this book, covered by the copyright hereon, may be reproduced or used in any form or by any means\u2014graphic, electronic or mechanical\u2014without prior permission of the publisher, except for excerpts in a review. Any request for photocopying of any part of this book shall be directed in writing to Cancopy (Canadian Copyright Licensing Agency), 6 Adelaide Street East, Suite 900, Toronto, Ontario, Canada M5C 1H6, Tel.:(416) 868-1620; Fax:(416) 868-1621.\n\n**Cataloguing data available from Library and Archives Canada**\n\nISBN: 978-0-88922-864-1\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nHighest Praise for Leo J. Maloney and His Thrillers\n\nFor Duty and Honor\n\n\"Leo Maloney has a real winner with For Duty and   \nHonor. Gritty and intense, it draws you immediately   \ninto the action and doesn't let go.\"\n\n\u2014Marc Cameron\n\nArch Enemy\n\n\"Utterly compelling! This novel will grab you from the   \nbeginning and simply not let go. And Dan Morgan is   \none of the best heroes to come along in ages.\"\n\n\u2014Jeffery Deaver\n\nTwelve Hours\n\n\"Fine writing and real insider knowledge make   \nthis a must.\"\n\n\u2014Lee Child\n\nBlack Skies\n\n\"Smart, savvy, and told with the pace and nuance   \nthat only a former spook could bring to the page,   \nBlack Skies is a tour de force novel of twenty-first-   \ncentury espionage and a great geopolitical thriller.   \nMaloney is the new master of the modern spy game,   \nand this is first-rate storytelling.\"\n\n\u2014Mark Sullivan\n\n\"Black Skies is rough, tough, and entertaining.   \nLeo J. Maloney has written a ripping story.\"\n\n\u2014Meg Gardiner\n\nSilent Assassin\n\n\"Leo Maloney has done it again. Real life often   \novershadows fiction and Silent Assassin is both: a   \nterrifyingly thrilling story of a man on a clandestine   \nmission to save us all from a madman hell bent on   \nmurder, written by a man who knows that world   \nall too well.\"\n\n\u2014Michele McPhee\n\n\"From the bloody, ripped-from-the-headlines opening   \nsequence, Silent Assassin grabs you and doesn't let go.   \nSilent Assassin has everything a thriller reader wants\u2014  \nnasty villains, twists and turns, and a hero\u2014Cobra\u2014  \nwho just plain kicks ass.\"\n\n\u2014Ben Coes\n\n\"Dan Morgan, a former Black Ops agent, is called out of   \nretirement and back into a secretive world of politics and   \ndeceit to stop a madman.\"\n\n\u2014The Stoneham Independent\n\nTermination Orders\n\n\"Leo J. Maloney is the new voice to be reckoned   \nwith. Termination Orders rings with the authenticity   \nthat can only come from an insider. This is   \none outstanding thriller!\"\n\n\u2014John Gilstrap\n\n\"Taut, tense, and terrifying! You'll cross your fingers   \nit's fiction\u2014in this high-powered, action-packed thriller,   \nLeo Maloney proves he clearly knows his stuff.\"\n\n\u2014Hank Phillippi Ryan\n\n\"A new must-read action thriller that features a double-   \ncrossing CIA and Congress, vengeful foreign agents, a   \ncorporate drug ring, the Taliban, and narco-terrorists . . .   \na you-are-there account of torture, assassination, and   \ndouble-agents, where 'nothing is as it seems.'\"\n\n\u2014Jon Renaud\n\n\"Leo J. Maloney is a real-life Jason Bourne.\"\n\n\u2014Josh Zwylen, Wicked Local Stoneham\n\n\"A masterly blend of Black Ops intrigue, cleverly   \ninterwoven with imaginative sequences of fiction.   \nThe reader must guess which accounts are real and   \nwhich are merely storytelling.\"\n\n\u2014Chris Treece, The Chris Treece Show\n\n\"A deep-ops story presented in an epic style that takes   \nfact mixed with a bit of fiction to create a spy thriller   \nthat takes the reader deep into secret spy missions.\"\n\n\u2014Cy Hilterman, Best Sellers World\n\n\"For fans of spy thrillers seeking a bit of realism mixed   \ninto their novels, Termination Orders will prove to be   \nan excellent and recommended pick.\"\n\n\u2014Midwest Book Reviews\nAlso by Leo J. Maloney\n\nTHE DAN MORGAN THRILLER SERIES\n\nTermination Orders\n\nSilent Assassin\n\nBlack Skies\n\nTwelve Hours\n\nArch Enemy\nFor Duty and Honor\n\nA DAN MORGAN THRILLER NOVELLA\n\nLeo J. Maloney\n\nKENSINGTON PUBLISHING CORP.   \nwww.kensingtonbooks.com\n\nAll copyrighted material within is Attributor Protected.\nTable of Contents\n\nHighest Praise for Leo J. Maloney and His Thrillers  \nAlso by  \nTitle Page  \nCopyright Page  \nDedication  \nChapter One  \nChapter Two  \nChapter Three  \nChapter Four  \nChapter Five  \nChapter Six  \nChapter Seven  \nChapter Eight  \nChapter Nine  \nChapter Ten  \nChapter Eleven  \nChapter Twelve  \nChapter Thirteen  \nChapter Fourteen  \nChapter Fifteen  \nChapter Sixteen  \nChapter Seventeen  \nChapter Eighteen  \nChapter Nineteen  \nChapter Twenty  \nChapter Twenty-one  \nChapter Twenty-two  \nChapter Twenty-three  \nChapter Twenty-four  \nChapter Twenty-five  \nChapter Twenty-six  \nChapter Twenty-seven  \nChapter Twenty-eight  \nChapter Twenty-nine  \nChapter Thirty  \nChapter Thirty-one  \nChapter Thirty-two  \nChapter Thirty-three  \nChapter Thirty-four  \nChapter Thirty-five  \nChapter Thirty-six  \nChapter Thirty-seven  \nChapter Thirty-eight  \nChapter Thirty-nine  \nChapter Forty  \nChapter Forty-one  \nChapter Forty-two  \nChapter Forty-three  \nChapter Forty-four  \nACKNOWLEDGEMENTS  \nTeaser chapter  \nABOUT THE AUTHOR\n\nLYRICAL UNDERGROUND BOOKS are published by\n\nKensington Publishing Corp.   \n119 West 40th Street   \nNew York, NY 10018\n\nCopyright \u00a9 2016 Leo J. Maloney\n\nAll rights reserved. No part of this book may be reproduced in any form or by any means without the prior written consent of the publisher, excepting brief quotes used in reviews.\n\nPUBLISHER'S NOTE\n\nThis book is a work of fiction. Names, characters, businesses, organizations, places, events, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, events, or locales is entirely coincidental.\n\nLYRICAL PRESS, LYRICAL UNDERGROUND, and the Lyrical Underground logo are Reg. U.S. Pat. & TM Off.\n\nFirst electronic edition: November 2016\n\nISBN: 978-1-6165-0981-1\n\nISBN-10: 1-61650-981-3\nThis book is dedicated to my daughter, KATIE, who has   \nbeen the joy in my life since the moment she was born.   \nShe is beautiful on the inside and outside. She's kind,   \nconsiderate, and simply an all-around good person as   \nwell as a wonderful mother to my three granddaughters,   \nKATHERINE, CECILIA, and GRACE, whom I adore.\n\nI also want to dedicate this book to my niece   \nLIANNE, who has always had a special place in my   \nlife and heart. She has been an unfailing supporter   \nof my writing endeavor.\nChapter One\n\nThe prisoner's body was a brick of exhaustion and pain.\n\nSteel cuffs chafed against his raw wrists and ankles, the rough uniform scraping the burns and cuts that lined his arms and legs and pocked his torso. Even under the blackness of his hood, the prisoner smelled stale sweat mingled with his own breath: iron from the blood, acetone from the starvation. He could barely hold himself up against the jolting ride. All that was keeping him upright were the two thick guards at his sides boxing him in.\n\nAt the outset, hours ago at the landing strip, the guards were in high spirits, joking and jesting in Russian, which the prisoner could not follow. Whenever he couldn't hold himself up anymore and leaned into one of them or into the front seat, they would box the prisoner's head and laugh, forcing him to sit upright again.\n\nBut as they drew nearer to their destination, and the car's heating lost ground against the cold, the guards grew quiet, like there was something grim about the place even to them.\n\nThe prisoner swung forward as the jeep came to an abrupt stop, tires on gravel. The doors opened and the spaces on his sides cleared as the men got out, leaving him exposed to the frigid Siberian air. Against this cold, the canvas uniform felt like nothing at all.\n\nThe guards unlocked the cuffs and yanked the prisoner out. Too tired to offer any resistance, he walked along, bare feet on the freezing stony ground. Someone pulled off his cowl. He was struck by a hurricane of light that made him so dizzy that he would have vomited, if there were anything in his stomach. It took a moment for the image to stop swimming and resolve itself into the barren landscape of rock and creeping brush lit by a sun low in the sky.\n\nThe Siberian tundra.\n\nThey prodded him forward. He trudged toward the Brutalist conglomeration of buildings surrounded by tall mesh fences and barbed wire. Prison camp. Gulag.\n\nThe prisoner's trembling knee collapsed and he fell on the stony ground. A guard gave him a kick with a heavy, polished leather boot and pulled him to his feet.\n\nThey reached the top and entered the vakhta, the guardhouse. He passed through the first gate and was searched, rough hands prodding and poking at him. They then opened the second, leading him through, outside, into the yard. His gaze kept down, he saw guards' boots, and massive furry Caucasian shepherds, each taller than a full-grown man's waist. He didn't look up to see the bare concrete guard towers that overlooked the terrain for miles around or at the sharpshooters that occupied them.\n\nHe was pulled inside the nearest boxy building, walls painted with chipping murals of old Soviet propaganda, apple-cheeked youngsters over fields of grain and brave soldiers of the Red Army standing against the octopus of international capitalism. On the second floor, they knocked on a wooden door.\n\n\"Postupat'.\"\n\nThe guards opened the door, revealing an office with a vintage aristocratic desk. They pushed him onto the bare hardwood.\n\nA man stood up with a creak of his chair. The prisoner watched as he approached, seeing from his vantage point only the wingtip oxfords and the hem of his pinstriped gabardine pants, walking around his desk, footsteps echoing in the concrete office.\n\n\"Amerikanskiy?\"\n\n\"Da,\" a guard answered.\n\nThe man crouched, studying the prisoner's face. \"You are one of General Suvorov's, are you not?\" His voice was deep and filled with gravel and a heavy Russian accent.\n\nThe prisoner didn't respond\u2014not that he needed to.\n\n\"You are tough, if he did not break you.\" He stood, brushing off unseen dust from his suit jacket. \"And if he had broken you, you would be dead already. I am Nevsky, the warden. Welcome to my prison.\"\n\nThe prisoner looked up at last and saw a thickset jowly man, with a nose like a potato, bloodshot eyes, and the ruddy swollen face of an alcoholic.\n\n\"We have no official name, but we call it Pokoynit-skaya. Do you know what that means?\"\n\nThe warden opened a cabinet and poured himself a glass of vodka.\n\n\"Charnel house. Because everyone in here is dead meat.\" He emitted a grotesque throaty laugh and tipped the glass into his mouth. \"Stand up,\" he said, slamming the glass onto the side table. The prisoner couldn't muster the energy to. \"I said stand up.\"\n\nOxford wingtips sunk into the prisoner's side. He doubled in pain, groaning.\n\n\"Up!\"\n\nBracing himself on the desk, he staggered to his feet.\n\n\"You will learn to do as you are told here.\" He poured another glass of vodka. \"Look out the window. What do you see?\"\n\nA broad barred window overlooked the tundra, where it was too cold for any trees to grow. A vast bare expanse of low grasses, with mountains rising from the flatness far in the distance.\n\n\"The answer is nothing. I will not tell you my prison is impregnable. In fact, we have had breakouts. If they get past the fence, we take bets on who will hit him. But the few that get away, nature takes care of. We find them dead in the wasteland within a few days.\"\n\nThe warden grabbed the prisoner's arm, feeling his muscles. \"Strong. That will not last.\" He slapped the prisoner hard on the buttocks. \"This is what your life will be. You will mine all day\u2014and the days of the Arctic summer are long. You will be questioned, if the order comes. That will not be pleasant. But mostly, you will work.\" As he spoke, the warden circled the prisoner, who kept his eyes down. \"You will waste away, and your mind will break.\" The warden knelt close to him and whispered in his ear, his rancid alcoholic breath filling the prisoner's nostrils. \"And one day, you will die here, forgotten.\"\n\nThe prisoner's face contorted in fury. He lunged for the warden, who stepped back to avoid him. The prisoner stumbled under his own weakness and fell back to the ground.\n\nNevsky sat down and signed the prisoner's intake papers. \"We are done here,\" he said. He squinted to read the type. \"Show Daniel Morgan to his cell.\"\nChapter Two\n\nTwo guards pulled Morgan on shuffling feet outside, back onto the cold-hardened earth, where the harsh wind whipped against his skin. They were in the yard now, a squarish space surrounded by various freestanding structures on all four sides, although he was too dazed to get any kind of clear picture of it. He thought he caught a whiff of something cooking and sheer instinct led him to turn toward its source. The guards yanked him, pulling him into another building, this one squat and single-storied. Like the others, it was built out of worn concrete and had heavy metal doors and thick bars on the windows, all covered in rust.\n\nThere were two more guards in there who stripped him of his tattered, bloody clothes and tossed them aside. They shoved him, naked, against a wall of chipped porcelain tiles and stood back as one opened a hose. He gasped as ice-cold water blasted him in the chest, sputtering when it hit his face. They tossed a rough moldy sponge and a cracked bar of caustic soap at his feet and hollered at him, pointing down at them. He bent and picked them up with shivering hands, running the sponge against skin reddened by the cold, his wounds smarting with the chemical burn as he scrubbed himself of weeks of dirt and blood and sweat. The pungent scent made his eyes water as he trembled and flinched from every new blast of frigid water.\n\nWhen they were satisfied, they shut off the hose and tossed him a thin towel, which he fumbled and dropped on the wet tile floor. They laughed as he ran the now-sodden towel over his skin to get off whatever excess water he could manage. The guards then pulled him, still damp, to the next station, where they sat him down on a splintering stool. One of them turned on a clipper that was at least twenty years old and buzzed like a bumblebee the size of a poodle.\n\nThey started with the hair on his head, dense and black with wisps of gray at his temples, which fell on the tiles in thick tufts. They shaved his mustache, the machine tugging at his split lip so that it began to bleed again, and the beard that had grown in since his capture. They worked his way down his body, his hair\u2014all his hair\u2014falling about the feet of the stool. Once the guard finished Morgan's legs, he clicked off the machine. Another guard poured a white acrid-smelling delousing powder onto his head and back. It clung to his damp skin and raised a white cloud around him. The guards cackled at Morgan's ensuing coughing fit.\n\nFinally, they handed him a folded-up jumpsuit to put on. It was tan canvas, rough and coarse against his skin, and provided little protection against the cold. After he put it on, they shoved a stinking coat in his hand and gave him cheap cloth shoes, which he pulled over his feet. They were, like his hands, numb from the cold. The guards got impatient at how long the operation was taking and boxed his ear for good measure. Morgan pulled on the coat, which at least offered cover from the wind.\n\nFrom there he was escorted into the blockhouse. It was single-story and much larger than the building he had just left, with only tiny windows letting precious little light in. There were scratches on the wall, the writings and designs of prisoners with no one else to talk to, who wanted to leave their last mark on the world before disappearing, in an unmarked grave thousands of miles away from home, where those they left behind would never find their bodies, never know what happened to them.\n\nThey led him to a room where there were rows of bunks that looked more like shelves, each bed only two wooden boards held up by vertical beams. It smelled lived in, of sweat and piss and mildew. The guard pushed him inside, and he stumbled onto the bare concrete floor.\n\nThe guard shoved a blanket, woolen and reeking, into his hand and pointed him to a bare wooden bunk. His, Morgan guessed. Then he gave him a bent tin bowl. \"This is your bowl,\" he said. \"You have bowl, you get food. No bowl, no food.\"\n\n\"Where do I keep it?\"\n\nThe man shrugged. Not his problem. The other men seemed to store them under their bunks, so Morgan put it on his.\n\n\"Rest,\" the guard said.\n\nMorgan didn't need telling twice. He got onto his bunk and collapsed onto the wooden boards. They were hard and uncomfortable, but he was exhausted beyond caring. He pulled on the reeking blanket and slept, clutching the food bowl tight, and dreamed of home.\nChapter Three\n\nAlex Morgan reached upward into darkness and found a handhold in the jutting bricks. She pulled her weight up, finding a new foothold for her left foot. One, two. Easy does it.\n\nThe warm summer wind lashed against her at this exposed height. One upward gust pulled off her black knitted cap, leaving her short hair whipping against her head. She followed its progress against the light of the streets below as the wind carried it away.\n\nThen she looked down.\n\nMistake. Big mistake. She grew dizzy and weak and felt her grip slipping. She slapped her face with her free hand.\n\nThis is no time to lose your nerve.\n\nShe closed her eyes, using the sting on her cheek to center herself, and kept on going. One, two. One, two.\n\nNot that she had too much to worry about. She had a slim body and strong arms and legs, and she left everyone else in the dust back at the training camp when it came to climbing. Height aside, this was routine for her.\n\nAlex climbed one more floor and looked to the right at the balcony on level with her. Was this it? She counted from the top. Yes. Twenty-second floor. This was it.\n\nShe edged along the jutting bricks the few feet to the balcony and reached out to grab the railing. She stepped onto the ledge and then swung over, breathing a sigh of relief as her feet landed on solid ground.\n\n\"Hello, Alex.\"\n\nShe was so startled she leapt six inches into the air. If she had been on the other side of the railing, she'd have fallen off the building.\n\nDiana Bloch, in a maroon silk robe, stood up from a deck chair. \"I'd just like you to be aware that one of our snipers had you in his crosshairs by the time you reached the third floor.\" She opened the French doors into the apartment and stood aside for Alex to walk through. \"Come in. I have chamomile tea steeping for you in the kitchen.\"\n\n\"Thanks,\" Alex mumbled, entering the apartment, \"but I don't drink tea.\"\n\nThe place was bigger than any apartment Alex had ever seen in the city. She looked back out past the balcony at a breathtaking night view of the bay. Being the head of Zeta Division came with its perks. The interior was obsessively clean and decorated within an inch of its life to look like a design catalog.\n\nEven here in her own home, Diana Bloch was a facade, a front.\n\n\"I think you could do with some calming down.\"\n\n\"Sure.\" Alex rubbed her triceps, sore from the climb. \"Whatever.\" She followed Bloch into the kitchen, where a steaming mug was waiting for her, the string of a tea bag hanging off the side. Bloch pulled it out, letting the excess liquid drip back into the mug before placing the bag in the trash.\n\n\"Sugar?\"\n\n\"Four.\"\n\nBloch dropped in four cubes and stirred. \"This was quite unnecessary. You could have severely hurt yourself, or died. We've already invested much in your training. It would have been a serious loss.\"\n\n\"I wanted to see you,\" said Alex. \"And you kept stonewalling me.\" She sipped at the tea. Too hot.\n\n\"You are supposed to be in training. Skipping out was quite a feat, by the way. It seems you've got your fellow recruits atwitter, wondering how you did it.\"\n\n\"Bloch, where is my father?\"\n\nBloch's face took on a pained expression. \"Come into the living room,\" she said. She sat down at a white leather Barcelona chair and motioned for Alex to sit across from her. \"We don't know where he is. But we're doing everything we can to find him.\"\n\n\"Tell me what happened.\"\n\n\"You know I can't divulge details of the mission. But he was caught while on assignment.\"\n\n\"Where?\"\n\n\"Does it matter?\"\n\n\"Where?\" Alex's tone took on a hard edge.\n\nBloch seemed to consider whether to chew her out. She could pull rank at any moment. But instead, she said, \"In Moscow. Does that make it better? Does that lessen your pain?\"\n\nAlex bit her lip and scowled in anger. \"After everything he's done for the organization. You could move heaven and earth to find him, if you wanted to.\"\n\n\"There are limits on what we can feasibly do. Your father knew the risks going in, as you will when it's your turn to go out into the field. But we are doing what we can\u2014\"\n\n\"Not. Enough.\"\n\nBloch's eyes went cold. \"I'll be the judge of that.\"\n\n\"I'll go on my own if I have to.\"\n\n\"That's not a good idea, Alex.\"\n\n\"I didn't ask.\"\n\nBloch stood from her seat and dropped ice cubes tinkling into a glass. Then she uncorked a bottle of whiskey and poured. \"I know you think you're ready. Your help has been valuable to us in the past, and you'll make a hell of an operative someday. But you're still green. If you go out on your own, I'm afraid you won't survive.\"\n\n\"Are you going to stop me?\"\n\n\"No. I'm not.\"\n\n\"Then I think we've said all we have to say to each other.\"\n\n\"I suppose,\" said Bloch. She unlocked the front door and held it open for Alex. \"Why don't you take the elevator down?\"\nChapter Four\n\nMorgan woke to the bark of a guard yelling at him in Russian. His immediate reflex was to punch the man's lights out, but he was still too tired. So instead he lay disoriented until the guard, impatient, wrested the bent bowl from Morgan's hands and tossed it down the sleeping quarters. It tumbled, clattering, toward the door.\n\nThe message was clear enough.\n\nMorgan stood, shaky from the interrupted sleep but also renewed, if only a little. How long had he been out? The sun was still in the sky, still low, but that meant little out here. The sun was always low. And the wind, even in summer, carried a chill.\n\nHe picked up his bowl and went outside into the yard. Men were filing in from the double gates that led outside the camp. The whole procession was hairless like him, although none so recently shaved, so that stubble was already growing in on their scalps and faces. They were shuffling, exhausted from a day of forced labor at the mine. All were skinny, their overalls and coats hanging loosely from their bodies. Their faces were pale, with deep dark bags under their eyes.\n\nThese men were broken. Morgan recognized the signs.\n\nGuards oversaw the whole process, carrying their Kalashnikovs, holding dogs on leashes. Even the guards, though young, were stooped, with gloomy, lifeless eyes. Morgan didn't know if it was from the dreariness of the place\u2014no women, no entertainment, nothing to do but drink in cramped rooms, if they were allowed that much\u2014or the violence they committed against the prisoners. Something about torturing and brutality made men into miserable monsters. He'd seen it often enough.\n\nHe wondered whether this was a punishment assignment.\n\nMorgan scanned the group. All were in the same clothes and had the same (lack of) hair, so all he had to go by were faces, and even these looked similar, with sunken eyes and pale skin. It proved just about impossible to pick out an individual from the crowd.\n\nFood was distributed out of a single window by prisoners on meal duty. Morgan wondered if they had spent the day in the kitchen or come back early from work. In any case, it would be a prime position in the camp, the work light and pleasant in comparison to the mines. There was a little more color on their cheeks, too. Morgan guessed that they sneaked food as they cooked.\n\nMorgan clutched his bowl. His stomach growled. He hadn't realized how hungry he was.\n\nThe men lined up at the window and Morgan took his place at the end. There was a scuffle to see who would get closer to the front, but at a shout from a guard, under fear of violence, the men took their places, those behind resigned to their position.\n\nAs he stood in line, Morgan noticed a group of three prisoners glowering at him, and he felt like the only unattached girl at a singles bar. The men exchanged words and turned away.\n\nHe didn't like it. Any kind of attention in this place was unwanted.\n\nMorgan drew closer to the meal window as the prisoners got their dinner rations. When his turn came, he held out the bowl. The prisoner who was serving, wearing a jumper like his but stained with food, dunked a ladle into the big pot and spooned the contents into Morgan's bowl\u2014a stew with vegetables, heavy on onions and potatoes, a few wisps of meat and bits of animal fat, and a thin layer of oil at the top.\n\nThe man motioned for him to move along.\n\nResisting the urge to swallow the portion whole right there, Morgan shuffled off, away from any grouping of prisoners, and sat against the cell block, holding his hand as steady as he could so as not to spill a drop. He'd not had any food in several days.\n\nHe tilted the bowl against his lips. They were cracked and split from the beatings he had taken in the past week, but the stew, thin as it was, filled him. Even with the faint odor of rotting potatoes and onions, even with the stringy, gristly meat, it was the best thing he'd ever tasted. He wanted to down the whole thing in one gulp. But he knew that if he ate too fast he'd puke it all up again, so he chewed each solid mouthful twenty-five times before swallowing, and took small sips of the liquid. In his hunger, he lost awareness of everything else around him. All he could focus on was the next mouthful.\n\nHe finished the scant portion, tilting the bowl against his lips to get the last drop. It felt warm and full in his stomach, after days of nothing but a trickle of water. He felt invigorated, power flowing back into his muscles.\n\nBut at the same time he was aware that it wasn't enough. Two of these a day would keep a bedridden man alive and comfortable, but for men working all day in the mine, this was a starvation diet.\n\nHe wondered how long anyone lasted in here. Every one of the thousand or so men milling around in the yard looked withered and wasted, some more than others.\n\nAs he looked around, Morgan caught sight of the three who had been staring at him before. They were crowding around a man\u2014Arabic, of around forty years old. Morgan had noticed him before, off in the corner making his prayers to Allah as others waited in line for food.\n\nThe three punks circled him, pinning him against the wall of the prisoner barracks. He was backing off from them, shaking his head. They stepped forward, holding out their hands. Morgan saw the Arab had his bowl in his arms, protected as if it were a baby. The goons wanted it, his only means of getting food, an extra portion for them. Without it, he would starve.\n\nThe prisoners around them couldn't be missing it, but they made a point of not raising their heads. A guard looked at the scene with no more interest as if he were seeing a dog scratch himself. One of the men slapped the Arab in the face and screamed something in Russian.\n\nMorgan pushed himself up off the ground, his own bowl in his hand. Bad idea, getting into trouble on his first day. Plus, he was sore and exhausted. He had every reason not to get involved.\n\nHe walked the diagonal over to the men, inserting himself between them and the victim.\n\n\"Step off.\"\n\nThey first looked surprised, then glanced at each other with a blend of confusion and amusement. The oldest of them, a man slender and short like a weasel, spoke to him in Russian.\n\n\"No Russkyi,\" he grunted. \"Amerikanskyi.\"\n\nA different man broke in, the youngest and tallest of them, fair-skinned and blond, who looked like the Russian equivalent of an Iowa farm boy except for the tattoo of a snake that peeked out of his overalls on his neck. \"American, eh?\"\n\n\"That's right. And you're going to leave this man alone.\"\n\nHe translated for his confederates and they shared a laugh. \"Americans like to think they are cowboys. New sheriff in town, come to do justice. Move away, cowboy, or you die.\"\n\n\"Step. Off.\"\n\nThe farm boy turned sour. \"This is not funny anymore.\"\n\n\"Never was.\"\n\n\"You do not want to pick this fight.\"\n\n\"Maybe I do.\"\n\nThe man mugged at his comrades as if to say, Can you believe this asshole? and took a swing at Morgan, trying to catch him in a sucker punch. Morgan was slowed by hunger and fatigue, but not enough to fall for that tired trick. This was a bully, untrained at fighting anyone who knew how to fight back. Morgan dodged and grabbed his arm, using the man's own momentum to drop him to the ground. Weasel man growled and pounced on him. Morgan pivoted out of the way, kicking his leg at the knee so he fell face-first on the dirt ground of the yard.\n\nThe third hesitated, casting his eyes on his two confederates. He'd learned his lesson not to rush in. He balled his hands up into fists.\n\nMorgan didn't give him the opportunity to get close. He rammed the sole of his foot into the man's chest, causing him to stagger back. Morgan then moved in to deliver a punch that would lay him out flat when someone grabbed his right arm. He saw a prisoner's overalls in his peripheral vision. He twisted to break free, but someone else grabbed his left. Morgan struggled, but he wasn't in any shape to wrest himself free of two men who, in spite of the conditions, had been sleeping and eating better than him for weeks now.\n\nSomeone new came up to him. He was older than most, as old as fifty. He was skinny as anyone else, but Morgan could tell he was thickset and jowly, with once-fat cheeks. His eyebrows were like two hairy gray caterpillars. He carried himself with all the dignity of a mafia don.\n\nHe moved in and punched an immobilized Morgan in the gut.\n\n\"You tried to be a hero. But this is not a place for heroes.\" He followed this with a meaty right hook to Morgan's cheek.\n\nHe'd have done worse, but the guards sprung into action at last, pulling the men apart from each other and knocking them to the ground, along with Morgan himself. The guards set upon all of them them, kicking them with heavy boots. Morgan took a painful kick in the ribs.\n\nThe man who had hit him, the don, was merely shooed away. Morgan went limp as they dragged him off. He didn't need any broken bones in here.\n\n\"You are dead, American!\" the don yelled after him. \"You hear me? Dead!\"\nChapter Five\n\nThe siren woke Morgan in the solitary cell they had tossed him into after the fight the night before. With no room to stretch out, he had slept leaning against the far corner of the cell, feet resting against the door. This took its toll on the form of a throbbing pain in his lower back. His left cheek was sore and tender from the punch he took from the don. He stood with difficulty and tried to stretch the pain away to little avail.\n\nAlmost on cue, the dead bolt on the door was undone and the door opened, letting daylight flood into the darkness of the cell.\n\n\"Time to work, American!\"\n\nThey pulled Morgan out and escorted him down an L-shaped hallway lined with cells and then out to the yard.\n\nIt was still dark, the sky leaden in the horizon where the sun was about to come up. The morning air chilled him to the marrow. The yawning, drooping prisoners lined up outside again, first for the morning count, and then at the food window, this time to get a dollop of potato porridge, bland and lumpy. But it was food, and it was warm.\n\nMorgan kept an eye out for Bortsov's men, who paid him no attention. Morgan would assume they'd give him extra scrutiny after the day before. This raised alarm bells in his head, but he didn't have time to ruminate on it before another siren sounded and they lined up for the morning's meal.\n\nLacking any kind of utensils, Morgan followed the others' lead and ate with his hands. Once everybody had gotten their ration, they were lined up again, three abreast, in front of the double gates. An escort of guards surrounded them, one for every twenty or so men. A smaller siren rang out and the double gates swung open. With a shout from a guard, the men set to marching. They filed through the no-man's-land between the inner and outer perimeter fences. Morgan knew how it worked. Anyone caught there would be shot without ceremony or a second thought.\n\nThis was not an army, and their march was slow and plodding. Stuck near the back, Morgan couldn't see where they were going, so he settled for keeping his eyes down on the stony ground to keep from stumbling or stepping on the heels of the man ahead of him.\n\n\"You are American, right?\"\n\nIt was good English, a young voice, right next to Morgan. He turned to his left to look at the man, in his early thirties by the look of it, with baby blue eyes and dark blond hair coming in on his head. There was something still unbroken in him despite his having the same sunken cheeks and sallow eyes as everyone else.\n\nMorgan didn't have a mirror. He wondered whether he had the look yet. He would.\n\n\"I saw you back there.\" His English sounded like he might have been educated in the US. \"Standing up to those thugs. Not just anyone would do that for another man.\"\n\nMorgan squinted into the dawning sun. A hawk shrieked far above.\n\n\"My name is Grushin. I am a journalist. In today's Russia, that's enough of a crime to get me sent to prison, but I got a little too inconvenient for even the usual holes they stick us in. So they put me in the gulag.\" He kicked a rock, which rolled diagonally and nearly hit a guard's boot. The guard turned, searching with a scowl for the culprit. Morgan looked away to deflect suspicion. \"What are you in here for?\"\n\nMorgan didn't respond. The kid was nice, but as they say in reality shows, he wasn't here to make friends.\n\n\"All right, Uncle Sam. I should know better than to ask.\"\n\nMorgan felt the spot where he took a boot the night before. It was aching anew with the strain of the march. He wondered whether he had a broken rib. \"You should know better.\"\n\n\"But I can talk,\" he said, grinning. \"The guys you attacked. They're all ex-mafia people. The one who punched you was their leader, Leonid Bortsov. They don't have much power, although I think their people bribe some of the guards from outside, so that gives them privileges in here. They run the laundry, which gets them away from the worst of the forced labor. One of the perks. The other is that they steal the others' food and blankets.\" He sighed. \"Somehow, even this place is not bad enough that men like Bortsov can't make it worse.\"\n\nThey marched for forty minutes across the tundra, on cold hard ground. The path was well beaten, so that the low grasses that covered the plains around them did not grow there. Everything around them was flat. Even if he could run, there was nowhere for him to run to, nowhere that would put him out of the line of sight of the guards' rifles.\n\nThe sun had risen above the horizon by the time they arrived at the mine. It was a handful of shacks surrounding a hole cut into the rock, sloping down to a set of double steel doors. The men fanned out, each seeming to know where to go.\n\nA guard approached Morgan. \"New prisoner,\" he said. \"You go with Vanya's team.\"\n\nVanya was a tall man, wrinkled and with heavy scarring on half his face. Morgan guessed it was a chemical burn, probably from torture. Vanya argued with the guard. After a brief back-and-forth, the guard gave what from the tone was an ultimatum. Vanya swore and said, \"Come, American.\"\n\nHe walked off, and Morgan followed. \"What was that about?\" he asked.\n\n\"We have a quota per man on the team,\" he said. \"More men, higher quota. If we do not meet quota at the end of the week, we get lash.\"\n\n\"Sorry about that.\"\n\n\"You look strong,\" he said. \"You will pull your weight. We do not do your work for you. If we get the lash, you get worse. And then you will be very sorry.\"\n\n\"Got it.\" Morgan was making friends left and right.\n\nVanya led him to the toolshed, where men were crowding around, elbowing each other, trying to get their hands on an implement. Morgan followed Vanya's lead, fighting to the front of the line to look at what was on offer.\n\nNo modern mining equipment. No machines. Handheld implements was all they had, goddamn shovels and mallets and chisels. Nevsky must have a penchant for the vintage.\n\nThere weren't enough for everyone, Morgan now saw. The smaller and weaker men were pushed aside. They would be getting the lash at the end of the week.\n\n\"Get kirkomotyga,\" Vanya told him. \"Pickaxe.\" Morgan he forced his way and took one. He held it close to his chest, smelling the rich iron, holding it tightly against the grabbing hands until he got clear of the crowd. Then he examined it. The wood was old and grooved, the head covered in rust. But it would do its work.\n\nA guard unlocked and swung open the heavy metal doors that led into the mine with a prolonged creak. He pulled down a heavy switch on the other side of the door, and dim lights came on. The wooden struts were old and seemed like they could cave in at any minute. Convenient way to bury a group of men the Russian government wanted disappeared.\n\nThe prisoners filed into the mine, two by two. Morgan walked inside with Vanya's six-man team. Being the new guy, he was given wheelbarrow duty, his pickaxe sitting inside as he struggled to hold the wheel steady on the uneven ground. There was no explicit order or direction, but Vanya and the men moved forward without hesitation. As the tunnels branched out, the men thinned until their group was alone.\n\nMorgan wondered how often men got lost in here. He wondered how many corpses had been forgotten in the mine.\n\nThey walked for several minutes before Vanya said, \"You go with Sergey here. He doesn't speak English. If you need help communicating, get lost.\" Sergey motioned for Morgan to follow him a ways down the tunnel to a small hollow. There, the Russian showed him how to wield the pickaxe, chipping stone from the cave wall.\n\nThey worked in silence for a while, only the dull clang of metal hitting rock. It was hard to gauge the passage of time in the darkness of the mine, so Morgan had no idea how much time had gone by when he heard footsteps coming down the tunnel. He turned, assuming it was Vanya and the others.\n\nIt was not. From the darkness, the three men he had fought on the yard the day before approached. They said something to Sergey, who dropped his pickaxe and took off running down the passage, the sound of his dash fading in the tunnels.\n\nThe three men advanced on him.\n\n\"No guards to save you here,\" said the young, tall man. \"You are going to have a little accident.\"\n\nMorgan didn't like his odds. Someone could get hurt.\n\nHe thought about shouting for help, but he doubted that anyone would hear him, except perhaps Vanya's men. Also, it would do him no good in the long run. Bortsov's little gang would just wait until the next opportunity.\n\nMorgan knew one thing. He wasn't going to die in that mine. So he was going to put a stop to this here and now.\n\nHe tightened his grip on the pickaxe. This wasn't going to be pretty.\n\nThe tunnel was too narrow for them to come at him all at the same time, which gave him a strong defensive position, but also hamstrung his swing of the weapon.\n\nThe first, the younger tall guy, came at him with another pickaxe. He was slow. Morgan moved out of the way and swung his pickaxe against the man's back, calculating the force to crack a few ribs. The tip connected, and the man fell forward with a cry of pain.\n\nThe next man held a shovel, which had a longer reach than Morgan's pickaxe, putting him at a disadvantage. The man thrust it like a spear, hitting Morgan in the belly. He winced. It was going to leave a nasty bruise.\n\nThe man moved to thrust again, and Morgan locked the shovel into the curved head of the pickaxe. He pushed the head of the shovel back against the attacker. The handle caught him in the chest and he staggered back. This gave Morgan his opening. He swung the shovel to the side. It fell from the man's weakened grip and clattered to the stone on the cave floor. Then Morgan swung at the man's head with the side of the pickaxe, which slammed into the man's temple. He hollered, clutching at the bleeding wound. Morgan kicked the man's leg out from under him and he dropped.\n\nThe third attacker, the tiny, weaselly man, came at him, roaring, wielding a long-handled mallet two-handed. He swung downward and Morgan parried, almost losing his grip on the pickaxe. The man swung again, and Morgan dodged out of the way, inches from taking a blow that would crush his jaw.\n\nMorgan kept backing away as the swings came. The man had too much reach, and Morgan couldn't get an opening in the tight quarters. He could rush his attacker, but only at the expense of leaving himself open to a bone-crunching strike. And soon he'd be backed up against a wall, a sitting duck.\n\nMorgan took stock of the environment as he stepped back. Was there anything he could use? Narrow tunnels, held up by struts too strong to break, not high enough that climbing would give him any advantage. But connected to it . . .\n\nHanging from the strut above him was the last bulb of the tunnel. Without pausing to think twice, Morgan swung the pickaxe upward, shattering the lightbulb and plunging the passage into darkness.\n\nMorgan could see the man's silhouette against another lightbulb up the passage, but Morgan himself was concealed by a pall of murk. The man swung the hammer blindly. Morgan waited for a wide swing and brought the pickaxe down on the hammer's handle. Without expecting it, weasel man lost his grip. Then Morgan rushed him with a running tackle that knocked the man off his feet.\n\nMorgan took up the hammer from the ground. He raised it, and the man, seeing Morgan by the dim light he was no longer blocking with his body, raised his hand in defense.\n\nMorgan dropped the implement next to the man on the floor, panting. \"It's a very, very dangerous thing to attack me. I'm not looking to kill anyone today. That makes this your lucky day. Try again and I guarantee your luck will run out.\" He brought his foot hard against the man's side. \"And tell Bortsov that if he pulls this shit again, I'm coming for him.\"\n\nHe walked away, leaving all his would-be killers writhing in pain on the cave ground.\nChapter Six\n\nThey marched back to the camp after ten hours in the mine, the sun still glaring in the Siberian summer sky. A truck drove alongside them, carrying the day's haul in tin ore. Bortsov's three goons limped along with everyone else, keeping the pace in spite of their injuries.\n\nNo one said anything about the fight. Morgan watched as his attackers were questioned by hostile guards in the yard as the mess window opened and men struggled to be among the first to eat. Morgan asked Grushin what the men were saying.\n\n\"They say it happened in an accident. That they fell.\"\n\nThe guards didn't look like they were buying it for a second, but Bortsov's men said nothing that would connect the occurrence to him. Stonewalled and not interested in taking this any further, the guards let it go. It wasn't skin off their backs if a couple of inmates wanted to give each other shiners, as long as the ore was flowing.\n\nMorgan had expected as much. They were already under scrutiny for the fight the other day, and would not want to call any more attention to themselves. The whole point of attacking him in the mine was to do it in secret, where no one would see. And with renewed suspicion because of their injuries, they would have to lay low before making their next move.\n\nBut it was clear they wouldn't let this go. Especially not now.\n\n\"You are a lightning rod for trouble,\" Grushin said.\n\n\"I don't know what you're talking about.\"\n\n\"I'm sure you don't,\" he said. \"It was just natural curiosity that you were so interested in the men's injuries, which I'm sure have nothing to do with the fact that you're trying to hide a limp.\"\n\n\"Did anyone ever tell you that you're too smart for your own good?\" Morgan asked.\n\n\"What do you think got me in here? Anyway, don't worry, no one will tell the guards. I'd be more worried about Bortsov, myself.\"\n\n\"You think they'll try something else?\"\n\n\"You are challenging the pecking order. They can't let that stand. If he can, he will kill you. But not yet. Eyes are on him now. I don't think he can afford to make a move.\"\n\n\"Sure you still want to be my friend? It can be dangerous to your health.\"\n\n\"Can't sink that much lower.\" Grushin surveyed the yard, the line for dinner organizing itself out of disorder. \"You know, if we all attacked together, at the same time, we could take their weapons and take over the prison. There are almost a thousand of us and, what, maybe a hundred of them?\"\n\n\"A hundred with Kalashnikovs,\" said Morgan.\n\n\"Still. We get a hold of one or two, and with a decent tactician . . .\"\n\n\"I don't like the way you're looking at me.\"\n\nGrushin rested his back against the wall. \"I'm not going to die in here. I refuse to. If I am to die, I want it to be trying to escape. Fighting against oppression, instead of collapsing from exhaustion or starving like a dog.\"\n\n\"Careful with that talk. There are people in here who'd sell you out for an extra dinner ration.\"\n\n\"But I don't think you're one of them,\" he said.\n\n\"Why do you figure?\"\n\n\"I'm a journalist. My job was\u2014is\u2014to see things. So I see people around here. Most of them keep to themselves. Those are your basic survivors, the ones who are focused on getting through the day. Some find God. They tend to stick together\u2014that's them over there. You got your standard bullies, like Bortsov's men, who try to get the upper hand even in here. There are the flight risks, too, although it's harder to figure out who they are beforehand. But they are the ones who will just take off running one day. Usually they get shot. Some get brought back.\"\n\n\"And the rest?\"\n\n\"You don't want to know.\"\n\nGrushin was probably right about that. \"Sounds like you've got this place figured out.\"\n\n\"There are the specific cases,\" Grushin said. He pointed out a small man, very thin and middle aged. Morgan couldn't quite tell, but there seemed to be something wrong about the way he carried himself. \"They say that is Kolya the Cannibal. He terrorized St. Petersburg some ten years ago. Ate at least thirty people, most of them children, before they caught him. They found the bones in his basement.\"\n\n\"He's been in here since then?\"\n\n\"It's what they say. Honestly, I don't know how he survived this long. He has never said a word to anyone here. Maybe that's his secret.\"\n\n\"And you? How long you been in?\" Morgan asked.\n\n\"It's been five months. Feels like as many lifetimes.\"\n\n\"I'm done with this place after a day.\"\n\n\"You won't stay long in here,\" Grushin said. \"Your government wouldn't allow it.\"\n\n\"I wouldn't count on it. I'm dead, as far as my government is concerned.\"\n\n\"I see,\" said Grushin. \"So what are you? CIA? Identity disavowed in case of capture?\"\n\n\"No. Not CIA.\"\n\n\"But something, right? There's something about you.\"\n\nMorgan didn't respond. He wondered whether he was that transparent.\n\n\"I am going to write about this place when I get out,\" Grushin said. \"The world needs to know what is going on in here. When they take you away, remember us here, okay?\"\n\nAfter dinner, the guards lined up all the prisoners on the yard for the evening count. One guard marched down the line counting aloud while another ticked each number off on a clipboard.\n\nTwilight was setting in by the time they finished, and the men filed into barracks. They distributed into their respective rooms. Morgan was relieved to find that none of his twenty-odd cellmates were the men he had confronted in the yard. The man he had saved, the Arab, was not among them either.\n\nMorgan lay down on his bunk. He was still tired enough that he felt sleep coming on, but he wasn't about to drift off before making sure the others would, too. Sleep left him exposed, and he didn't like it. But soon enough, his exhaustion got the better of him and he fell into dreamless sleep.\nChapter Seven\n\n\"How are you holding up?\"\n\nKaren O'Neal, with her pretty half-Vietnamese face, fussy, socially awkward and a bit off-putting, sat across from Alex in a downtown Boston Starbucks drinking a triple-shot espresso. They had just missed the morning crowd, who left newspapers and coffee rings behind, and were sharing a table during the tranquil midmorning lull.\n\n\"Not well.\" Alex emitted a hollow laugh. She was nursing an iced hazelnut macchiato with an obscene amount of sugar in it.\n\n\"I'm glad you called,\" said Karen. \"Sometimes this job sucks, and sometimes it's hard on the people close to us.\"\n\n\"I think it was harder on my father this time,\" said Alex.\n\n\"I know. I was being polite.\"\n\nThis was why Alex liked Karen. Sometimes she was honest to a fault. \"What happened to him?\" Alex asked.\n\n\"I don't know much. Zeta's keeping this mission on the down-low, even from people on the inside. They're even boxing me out of looking for him.\"\n\n\"What? Why? Bloch said they were doing everything they could.\"\n\n\"I guess that's not literally true,\" Karen said.\n\n\"I need to know more.\"\n\n\"I'm sorry, I'm just not in the loop.\"\n\nBut Alex had a last resort, a trick up her sleeve. \"Was Lincoln working on this project?\"\n\nKaren blushed. \"Yes.\"\n\nLincoln Shepard was the resident computer tech at Zeta, a brilliant nerd who had been dating Karen in secret for months now. Alex was among the few who knew, having found out by accident.\n\n\"He never told you anything?\" Alex asked.\n\n\"No, but . . .\" Karen looked left and right, as if someone might be listening. \"There's a name I heard him say several times,\" she said. \"Apparently in connection to this case. Suvorov. Some sort of military officer.\"\n\n\"Suvorov . . .\" Alex echoed as she committed the name to memory.\n\n\"You're not actually thinking of getting involved, are you?\"\n\n\"No, of course not. I just wanted some closure, I guess.\" Liar. \"Anyway, what are you working on now?\"\n\nThis was enough to get Karen to go off on a tangent about trying to find contraband through recurring patterns in shipping containers. Alex picked Karen's brain about how data models worked as they finished their coffees, and then Karen said she had to go. They parted on the street with a wooden hug.\n\n\"He might not be dead!\" Karen called out as Alex waved good-bye.\n\nAlex rode her motorcycle home. She pulled off the highway to the suburb of Andover, Massachusetts, where kids on summer vacation populated the streets on bicycles and on foot. One group was playing in the spray of a hose on the lawn. Alex felt a pang of nostalgia for her own childhood.\n\nShe found the house empty except for their German shepherd, Neika. Her mother was out at work, as usual. Diving into her work was her coping mechanism, and as such things went, it wasn't a bad one.\n\nTaking advantage of her solitude, Alex opened the door to her father's office. She found everything still, the air slightly stale, a light dust settled on the gun display case, the model cars.\n\nIt was eerie being in there with him missing.\n\nBut Alex had purpose, and the heebie-jeebies were not going to stop her. She opened a cabinet behind his chair and removed two piles of old tax and personal documents. Then she felt around the corners for the button, and removed the back panel.\n\nStashed there was nearly fifty thousand dollars and a Rolodex. She drew out the money and the old apparatus, shuffling through the cards. They were written in a simple cipher, an idiot code that Alex had over time taken pains to decipher.\n\nNow, she was looking for a specific card. She went through them until she found it. A name, which deciphered read Valery Dobrynin. And a phone number.\n\nShe picked up the phone in the study and dialed the complex country code and then the number. It rang ten times before she got an answering machine message. She had been studying Russian, but she didn't quite get what the recording said. She only knew she had not heard the name Dobrynin.\n\nShe heard the beep, her signal to start talking. \"I'm not sure I have the right number. I'm calling about my father, Dan Morgan. I\u2014he needs help. I think you know who he is, and I think you can give me that help. Please, if you care about him, call me back.\"\n\nShe left her number, finished off with another plea for help.\n\nIt could be nothing. A total mistake, a man long gone. But she would take anything at this point.\nChapter Eight\n\nDays went by and Morgan had no more trouble from Bortsov's men beyond angry looks. He shared most of his meals with Grushin, who told him about life on the outside, about his political radical girlfriend, and about how he was worried about her and how much he missed her. Morgan said little in return, and next to nothing about his life back home. But the young man was glad enough just to be listened to.\n\nMorgan would've preferred to think he was above it, that he was better as a lone wolf. But the young man's company was humanizing, in a place where most other interactions were opportunistic at best, cruel at worst.\n\nIt was about two weeks into Morgan's internment when Vanya interrupted a dinnertime conversation with an announcement.\n\n\"Detonation tomorrow. If we're lucky, we won't get picked to do it, and have a day of rest.\"\n\nThey got picked. It was a random draw, and theirs was the short straw\u2014Morgan's team and another, which Morgan saw that morning included the man he had defended on his first day there, the Arab. The twelve of them marched alone to the mine, along with six armed guards. The rest of the prisoners stayed behind and enjoyed a rare day of rest from hard labor.\n\nOn their arrival at the mine, the guards unlocked the dynamite shed and told them to get to work.\n\n\"Come,\" said Vanya to Morgan.\n\n\"Why me?\"\n\n\"What, you want me to do it? You are new. You do it.\" Seniority applied in the mines, apparently.\n\nVanya first took a satchel from a shelf and stocked it with wires, detonators, and tape. He then opened a green arms chest, packed with sticks of dynamite. \"You carry.\" He loaded it with eight sticks as Morgan held it.\n\nMorgan lifted it carefully over his shoulder and walked with smooth and precise steps, trying to jog the volatile cargo as little as possible. Vanya led all of them into the depths of the mines, where two spots had been marked for detonation. Once they were there, Morgan set the satchel on the ground and the Arab came forward and drew out three sticks.\n\nWith deft fingers, he connected the detonators. While Morgan unwound the wire toward the mouth of the cave, the Arab set the sticks at the second site.\n\nOnce everything was ready, they unspooled the wire all the way to the outside. Vanya closed the heavy steel door and everyone stood clear. The Arab activated the charge on a car battery and the ground shook. Dust shot out from the corners of the door, which rumbled on its hinges.\n\nThe guards gave them a few minutes to wait for the dust to settle. They were smoking cigarettes, and in a rare moment of empathy offered each prisoner one. The Arab waved it away. Vanya smoked his with relish. Morgan took one and pocketed it. He might be able to trade. At least it might be a treat for Grushin.\n\nThey went back inside, coughing at the dust that was still thick in the air, dimming the reach of the electric lights. Vanya surveyed the collapsed rock.\n\n\"We did not do enough here,\" he told Morgan. \"We will need to do a second detonation.\"\n\nOn the way back up, he knelt at a spot where stones had been knocked loose. The wire had been cut on a sharp edge.\n\n\"Shit. We do not have more. We will have to use fuses. I hate fuses.\"\n\nThey went back for more dynamite. Morgan was given the job of carrying the satchel again. He, the Arab, and Vanya made their way back down, the dust now a bit clearer, to the first detonation site.\n\n\"This doesn't look too safe,\" Morgan said as the Arab set the sticks in the recess of the rock where they wanted to blow the new passage..\n\n\"Do it right this time,\" said Vanya. He was pissy about all the extra work.\n\n\"Lighter,\" said the Arab. Vanya reached over and lit the fuse.\n\n\"All right, clear the cave.\"\n\nThey ran single file away from the hissing fuse. Ducking under a low passage, the Arab put his hand on a strut. Weakened by the explosions, it shifted, groaning, and then cracked. Morgan pushed him out of the way, and they rolled together on the ground. A section of the passage wall collapsed on top of Vanya. He was knocked forward and screamed. His leg was pinned against the ground.\n\n\"Help me!\"\n\nMorgan looked at the light of the fuse, burning not fifty feet away. How long did they have before it reached the stick of dynamite?\n\nMorgan owed this man nothing. The sensible thing to do was to run away with everyone else and get clear of the blast before it went off.\n\nInstead he grabbed a shovel. \"Come on, help me!\" he said to the Arab. He wedged the shovel under the rock and pushed down on the handle. It didn't budge. \"Help me!\"\n\nThe Arab stepped forward and helped Morgan push the shovel down. They grunted at the exertion. With a creak, the stone moved a fraction of an inch. Under their combined strength, it rose, little by little, rocks shifting as it moved.\n\n\"Get him out!\"\n\nThe Arab let go. Morgan held, giving every ounce of strength he had, as the man pulled Vanya's considerable bulk backward, clear of the rock. Morgan let go, dropping to his hands and knees with exhaustion. The rock hit the cave floor, sending up a plume of dust.\n\n\"Move it!\" cried Vanya.\n\nMorgan and the Arab helped Vanya to his feet, and he limped, weight resting on their shoulders, as they moved together down the passage.\n\nThey had walked barely ten paces when they were knocked off their feet by the blast. They collapsed into a pile on the cave floor, hot air running all around them. The passage behind them caved in, raising a cloud of dust. If they had gotten out five seconds later, they would have been buried.\n\n\"Is everyone all right?\" the Arab asked.\n\nVanya didn't answer, but he was moving and moaning, which was good enough.\n\n\"Let's not do that again,\" said Morgan.\nChapter Nine\n\nThat evening, Morgan sat to eat with Grushin at dinner, as usual. From their regular seats against the laundry building, he caught sight of the Arab doing his evening prayer halfway across the yard. At the back of the food line, Vanya was limping on an old-fashioned cast. That poor bastard. With a fracture like that, he needed to be in bed, but hell if Nevsky was going to be that compassionate.\n\nAt least Vanya caught something of a break. He was, Morgan found out, taken out of the mine and put on latrine duty.\n\nSomeone else was eating with them that day. An older man, his face deeply lined, what hairs were left him wispy and patchy. His eyes were sunken and haunted, but there was spirit in the geezer yet.\n\n\"Morgan,\" Grushin had said when Morgan sat down. \"This is Milosz. He is something of an institution around here. He has been here for many years, longer than anyone can remember. Even him.\"\n\n\"How you like our home, young man?\" Milosz said, in a raspy, guttural English.\n\n\"You speak English?\"\n\n\"I learn in here. Sometimes there is British or American inside. They die or go away fast. But I learn. I learn good.\"\n\n\"That you did,\" said Morgan. \"What are you in for?\"\n\n\"I was fighter in Czechoslovakia,\" he said. \"In\u2014\" he turned and asked something of Grushin in Russian.\n\n\"Velvet Revolution,\" Grushin said.\n\n\"Velvet Revolution!\" Milosz said with a flourish of his hand. \"I was leader.\"\n\nHe related stories of his glory days before his capture, of beautiful women and killing Communists and daring escapes until he announced, \"I go take a piss.\" He rested a hand on Morgan's shoulder. \"Don't get old. You piss all the time.\"\n\n\"I'd rather that than the alternative.\"\n\nHe helped Milosz to his feet and watched as the man walked toward the latrines. He commanded rare respect in the camp. Even the guards seemed to have some deference to the old man, and Bortsov's men steered clear of him. No, there was something left of the outside in here after all. Respect for elders was too deeply ingrained to be rooted out all the way.\n\nAs he watched Milosz, Morgan saw the Arab walking over in their direction, a bowl of stew in his hand.\n\n\"I think that's for you,\" said Grushin. Morgan stirred to stand up. \"No, I was just leaving. Got some important business to attend to at the other side of the yard.\"\n\nGrushin made himself scarce, and the other man took a seat next to Morgan as he tipped the bowl of stew into his mouth, fatty broth filling his stomach. Both were facing forward and didn't look one another in the eye.\n\n\"My name is Badri,\" he said.\n\n\"Morgan.\"\n\n\"American,\" he said. \"We should be enemies.\"\n\n\"I guess we should.\"\n\n\"There is honor in you. Perhaps you are a good man.\"\n\n\"Nah. I'm just like every other asshole in here.\"\n\n\"No. You are not. Most people here are criminals. People who are lost, who have no hope of righteousness. But you. I have been watching. You have a mind for justice, even in this place where there is none, not for any.\"\n\n\"Those are big words,\" Morgan said. \"I see someone in trouble, I do something. It's an instinct or something.\" He sipped from his stew. It carried an aftertaste of rancid onion.\n\n\"Even one who should be your enemy.\"\n\n\"Even.\" Badri was testing him. He could tell.\n\n\"You have family?\" Badri asked.\n\n\"Wife and daughter back home,\" he said. \"My daughter, she's college-aged. You?\"\n\n\"Wife, too, and two children. But I haven't seen them in a long time. I don't know what has happened to them.\" He rubbed his face. \"Are you from the CIA? I do not know why else an American would be in here.\"\n\nLying was going to do him no good. Plus, given the circumstances, it hardly seemed like it mattered. \"I'm not CIA,\" he said. \"But I did work for an intelligence agency. Something clandestine. Totally secret. Not even working for the government, technically.\"\n\nA flash of ice-cold hatred passed across Badri's face. \"Then you are the enemy.\"\n\n\"Well, I'm done with them anyway. I gave everything for my country, for the agency. And they betrayed me. Left me for dead. What are you here for?\"\n\n\"For fighting the Great Satan,\" he said. \"For making jihad.\"\n\nThey stared at each other in silence. Natural enemies, then?\n\nMorgan scratched the scalp behind his ear. \"Look, Badri, I've seen things. I've done things. I know what my country does, has done. Wars and bombing and assassinations for profit and ideology. Covering up war crimes and manipulating our people. I'm tired. I'm tired of the hypocrisy.\" He leaned back and closed his eyes. \"I saw something. In my most recent mission. I was doing reconnaissance in northwestern Syria. Town called Sarmada. A mission against the Islamic State. I was supposed to give my people the location of a group of leaders for a targeted bombing.\n\n\"Except when I got there, I saw a wedding. Women, old people, children. I told them not to fire. I told them to hold off.\" He spit on the ground. \"An entire family killed by a drone strike. A wedding party. Should've been the happiest moment of those young people's lives. Instead, they died.\" He gritted his teeth. \"I'm sick of it, Badri. This has been festering in me a long time. I'm not one of them anymore.\"\n\n\"Perhaps you see things as I do,\" said Badri. \"Perhaps you understand.\"\n\n\"Well. It's not like all that matters in here, does it?\" He looked at the desolate wasteland surrounding the prison. A treeless wilderness, with no animals to be seen, cold and barren. \"All that's washed away. This is a corner of the world that God's forgot. All we are in here is muscle and bone waiting to die.\" Morgan took another mouthful of soup. \"So I don't see the harm in mortal enemies on the outside sharing a meal.\"\n\n\"Perhaps you are right,\" he said.\n\n\"Screw being right. I'm tired.\"\nChapter Ten\n\nAlex's feet pounded pavement, a light summer shower cooling off her overheated body.\n\nThese days, she couldn't do without her morning run. She'd long gone past the point when it was something she had to force her body to do, and now it was something her body craved, and would revolt if she did not.\n\nShe thought of her batch of recruits back at the training camp. They'd be at the range at this time, she reckoned. She realized she missed holding the rifle, missed shooting at a target, missed the endurance exercises and slogging through the mud and practicing first aid on her mates and stealth maneuvers.\n\nShe was thinking about checking out a local gun range when the music cut out in her earphones and the ringer sounded.\n\nShe stopped, catching her breath for three seconds before drawing her cell phone from its waterproof case and picking up.\n\n\"Hello?\"\n\n\"How you get this number?\" Through a heavy Russian accent, the man's voice was nervous, slathered in a sort of manic paranoia.\n\n\"I'm sorry, who is this?\"\n\n\"Don't ask who it is. You called me.\"\n\n\"I did?\"\n\n\"You leave number.\"\n\nThen it hit her. \"Is this Dobrynin? Valery Dobrynin?\"\n\n\"Who are you?\"\n\n\"Alex, I\u2014Dan Morgan is my father.\"\n\n\"How did you get number?\"\n\n\"I found it in his things.\"\n\nThe man swore in Russian. \"The mudak was supposed to forget me.\"\n\n\"Well, he didn't. And you didn't forget him either.\"\n\n\"What do you want? You say he needs help?\"\n\nThe rain started coming down harder and she turned to make her way back home. \"He went missing in Russia a couple of weeks ago. I want\u2014I need to find him.\"\n\n\"And what is this to me?\"\n\n\"You knew him, didn't you?\"\n\n\"A long time ago. I do not know anything about him.\"\n\n\"You know he needs your help!\"\n\nThe man muttered something in Russian she did not catch. \"What kind of help?\"\n\n\"He was on a mission to\u2014\"\n\n\"Stop. Watch what you say over the phone, child. These things I do not need to know.\"\n\n\"Okay,\" she said. \"He's missing. I have a name in connection with it, and not much else.\"\n\n\"And what do you want me to do about this?\"\n\n\"I want to find him. I want your help.\"\n\nThe man did not answer right away, and all Alex heard was water hitting the pavement. \"Is not my problem. Forget this name and this number.\"\n\n\"You owe him!\" It was a gamble. She didn't know if it was even close to true.\n\n\"What did you say?\"\n\n\"I said you owe him.\"\n\n\"What do you know about that?\" he hissed.\n\nTime for another bluff. \"I know enough. You want to get even? This is your chance.\"\n\n\"Der'mo. Get even. Okay. You come here? To Moscow?\"\n\n\"Yes. I'll come to Moscow.\"\n\nHe dictated the address for her in Cyrillic, which she was familiar with enough to write out on her cell phone's notepad app.\n\n\"I will be waiting. If this is trick, I gut you with knife.\"\n\nNice guy.\n\nShe went home and made straight for her father's hiding place. She took out the duffel of cash. Then she looked up flights online. She could leave on one early the next morning. She paid for it with a credit card her father had left behind.\n\nHer mother was out of town for a couple of days, so Alex just left a note.\n\ngone to find dad. be back soon.   \nlove you.\n\n\u2013A.\n\nChapter Eleven\n\nAlex's plane touched down in Moscow in the late morning. She converted two thousand dollars to rubles and took a cab out of the airport.\n\nShe had the driver drop her off a few blocks away from her destination and walked the rest of the way. She was in an old neighborhood, where the buildings were quaint but run-down.\n\nShe counted the numbers until she was standing in front of a store window. She didn't need to read Russian to know what it was. The hanging salamis and two massive pigs' heads on display announced with crystal clarity that this was a butcher shop. It looked like something that belonged to the Soviet era and had frozen in time in the late eighties.\n\nShe checked the address again and walked all the way back to the corner to verify she had the street address right. Then she walked back and checked the number.\n\nThere was no doubt. This was the address the man had given her.\n\nHer face burned with embarrassment. Had this been a cruel trick? Something to get her off his back? She might be able to trace the phone number, but what was the point of tracking down a man who wanted nothing to do with her?\n\nStill, she had come this far.\n\nA bell rang when she opened the door. A miasma of butchered meat hit her at once, not helped in the least by the heat of the summer.\n\nA woman of about sixty waddled in from the back, cleaning her hands with a filthy rag, and asked Alex something in Russian.\n\n\"Sorry, I don't . . .\" She closed her eyes and took a deep breath, swallowing her embarrassment. \"I'm looking for Valery Dobrynin?\"\n\nShe tossed the rag and hollered, \"Valyerey!\"\n\nA man came out of the back, grumbling under his breath, wearing a bloodstained apron. He had a long face with gray curls that were once black, and a long, thick nose. He exchanged bickers with the woman, and she retreated inside, spitting a last curse at him and slamming the door behind her.\n\n\"Chto ty khochesh'?\" She could smell the alcohol on his breath from across the counter.\n\n\"Ya ishchu kogo-to. My Russian's not very good. Mr. Dobrynin?\"\n\n\"That is me. Who are you?\"\n\n\"Alex. Alexandra.\" Dobrynin didn't make a sign of recognizing her. \"Daniel Morgan's daughter?\"\n\n\"Ah, the girl,\" he said, grumbling some more, and looked over her shoulder at the street behind her. \"Did anyone follow you?\"\n\n\"I don't think\u2014\"\n\n\"Never mind. Come in.\" He pulled a hinged section of the counter, motioning for her to follow him.\n\nThey walked into a different door from the one the woman had disappeared into, and emerged in a tiny dark kitchen. Its sink held two chipped ceramic plates, two forks and knives that had seen better days, and two aluminum cups. He sat in one of the two chairs and motioned for her to sit in the other, across a table covered in peeling vinyl.\n\n\"Put your suitcase down and sit,\" he said.\n\n\"Thank you. You have a lovely home.\"\n\nHe burst out laughing, which transitioned into a hacking cough. \"I do not. Agrafena is a worse decorator than I am.\"\n\n\"Is that your wife?\"\n\n\"What did you come here for?\" Dobrynin asked her.\n\n\"I told you,\" she said. \"I need your help. I need to help my father.\"\n\n\"And who took your father?\"\n\n\"I have a name,\" said Alex. \"Suvorov.\"\n\nDobrynin whistled. \"That is quite a name. If this is true, then your father is in heavy trouble, little girl.\"\n\n\"Who is he?\"\n\n\"General. Very nasty man. Not known for being polite to prisoners.\"\n\nAlex felt a lead weight in her chest. \"Do you think he's dead?\"\n\n\"Most probably.\"\n\n\"I need to know for sure,\" she said. \"If there's even a chance\u2014\"\n\n\"Will you die for a chance?\"\n\nShe narrowed her eyes. \"If I have to.\"\n\nHe stood up from his seat and turned his back on her. \"Foolish girl.\" He turned on the faucet and rinsed the dishes.\n\n\"Either you help me, or I'm going on my own.\"\n\nHe circled his way around her and opened a cabinet behind her back. She watched the dripping faucet, thinking of what she was going to do if he sent her away. And then his thick arm was around her neck, and the point of a thin knife touched her neck. \"Do you know what they do to little girls in Moscow?\"\n\nAlex squirmed, gasping for air.\n\n\"You will be gutted like a pig, girl.\"\n\nAlex calmed her panicking lizard brain and assessed the situation. His form was sloppy, and sloppy meant vulnerable.\n\nShe kicked herself back in her chair, knocking her head into his chest. The chair tipped over and she fell back, on the chair, on top of him. Before he could recover his bearings, she took the knife that he had dropped on the ground and swung to her right, landing on her feet. She held the knife inches from his face.\n\n\"I'd like to see them try.\"\n\n\"I am old,\" he grunted, pushing the knife away from his face and standing up. \"Others will not be so easy.\"\n\n\"I'm ready,\" she said.\n\n\"You are not. But I will help you with what I can. Daniel Morgan deserves this much.\"\nChapter Twelve\n\nMorgan pulled his weight and more in the mines. It was backbreaking work, long hours of breaking and hauling, to the steady sound of pickaxes echoing through the tunnels. Guards would patrol, passing every so often to make sure no one was slacking. Getting caught not working warranted anything from a cuff to the back of the hand to a summary beating, but Morgan, like the others, quickly learned the tricks. Work as slowly as possible while still keeping the quota, listen for the sound of the guards' boots on the ground, get whatever rest you can when you can.\n\nWeeks passed, although Morgan did not keep count of how many, and Bortsov's men kept their distance, except for bumping into him on occasion, or pushing him when the guards weren't looking. One day, they knocked a bowl of stew out of his hand and it went clattering to the ground, spilling its precious contents on the dirt. There were no do-overs at the camp. Morgan went hungry that night.\n\nBut Morgan didn't want to start trouble, and they didn't seem to want to escalate, for fear of Nevsky's retaliation. That was a fragile d\u00e9tente, and a reckoning was coming.\n\nOne night, the lights went out. It was nearly completely dark by this time, the sun having disappeared in the horizon. There was a commotion among the prisoners, who had been lined up for the evening count, and Morgan's mind went to escape. But the guards were ready, flashlights in hand, and circled the perimeter, carrying automatic weapons and loosing the dogs, still tethered to their chains but with free rein to run the length of the perimeter fence.\n\nThe prisoners were made to kneel with their hands on their heads\u2014Morgan was well familiar with the procedure.\n\nMen were going into the building that housed the generator. Ten minutes passed, then another ten. Nevsky emerged from his building and walked across the yard, swearing in Russian.\n\n\"Hey!\" Morgan called out to him. \"Hey! Is it the generator?\"\n\n\"Shut your mouth, prisoner!\"\n\n\"I can fix it!\" Nobody responded. \"Grushin, tell him I can fix it.\"\n\nGrushin did, calling it out in Russian to the warden, who changed his trajectory to stand before Morgan.\n\n\"Are you a mechanic, American?\"\n\n\"I know my way around an engine.\"\n\nNevsky waved him over. \"Okay, you try it.\" He waved Morgan inside the shed. \"If you screw it up any worse, it's your ass.\"\n\n\"I need a flashlight,\" he said. Nevsky took one from the nearest guard and handed it to Morgan.\n\nThe generator reeked of diesel. The make wasn't one that he recognized. It was Russian (along with any instructions or identifiers) and old. The panel was already open. He went through the troubleshooting checklist, starting with gas and coolant levels. Next he checked the breakers.\n\n\"Jesus H. Christ, who's your mechanics guy? These wires are a goddamn mess.\"\n\n\"Are they the problem?\" Nevsky asked.\n\n\"No, electrics look like they're working. They're not the issue this time.\" He found the problem soon enough. \"Fuel line's blocked. I need to replace it. You got supplies?\"\n\n\"Whatever there is in here.\"\n\nHe found an old hose that seemed in one piece. It was a simple substitution, something he'd done plenty of times before.\n\n\"That should do it,\" he said. \"Let's start her up.\"\n\nThe generator came to rumbling life. The lights in the camp flickered on.\n\n\"American,\" Nevsky said. \"You're out of the mine. You're now on mechanic duty.\"\nChapter Thirteen\n\nMorgan and Grushin settled into a routine of sitting together at dinner. It was mostly Grushin who spoke, but it made Morgan feel good to talk to him. The journalist reminded him of Alex, who was just as idealistic and headstrong. He missed her so much it hurt physically.\n\n\"You got yourself a cushy position,\" Grushin said.\n\n\"Same as you.\" Grushin was on the rotation to work laundry.\n\n\"I get to work all day in the stink of men,\" he said. \"And then the stink of chemicals. They burn my hands and make my eyes water. Not very pleasant.\"\n\n\"Well, it ain't the mines.\"\n\nGrushin chewed absently. \"You know, I think I would actually murder a man for a cigarette and a cup of coffee.\"\n\n\"I've seen some of the inmates with cigarettes,\" Morgan said. \"I guess they get 'em from the guards.\"\n\n\"Yeah. And you should hear the things they'll trade for 'em.\" Grushin emitted a hollow laugh. \"I would kill a man. But I wouldn't do that.\"\n\n\"Can't say I sympathize. About the coffee and cigarettes, I mean. I don't partake.\"\n\n\"What about vodka? Whiskey? My God, a tall glass of German beer.\"\n\n\"Never been much of a drinker,\" Morgan said.\n\n\"So what do you miss?\"\n\n\"I miss my daughter,\" he said. \"Every single goddamn minute of my existence here. And I miss my wife. I miss how warm and soft she was. The idea of our bed\u2014it seems like another world. Another life.\"\n\n\"To women,\" Grushin said, raising his bowl and clinking it against Morgan's. \"That we might see one again someday.\"\n\n\"God hear you.\"\n\nThey ate in silence for a few minutes. Grushin looked off into the setting sun. \"I'm not going to die here.\"\n\n\"Careful with that,\" said Morgan. \"They say hope's what kills you faster. You don't have hope, despair is just a dull ache. But if you keep that fire alive, it'll burn you.\"\n\n\"I'd rather blaze bright, even if it consumes me, than turn to ash in this place.\" He stood up, raising his voice. \"Look around, Morgan! All these defeated men. All fed to and being slowly digested by this place.\" His voice dropped to a whisper. \"I'm going to escape from this place.\"\n\n\"You sound like you almost have a plan.\"\n\n\"I almost have a plan,\" Grushin said.\n\n\"You'll die,\" said Morgan. \"Look.\" He gestured to the wastes of the tundra.\n\n\"I can't do it alone. But I wanted to know if I could trust you. I think I can. And you've shown yourself to be very resourceful. Seem to know what you're doing. The kind of guy I'd want at my side for this. You can handle yourself in a fight. You're quick on your feet. Plus, you're a good man. Not many to be found around here. And you got one big thing going for you.\"\n\n\"What's that?\"\n\n\"The cars,\" Grushin said. \"You have access to the motor pool. It might be the only way to escape far enough along the tundra not to get caught by the dogs or the guards coming after us. If we disable the other cars, we got a good shot at getting away.\"\n\n\"Into the Siberian wilderness, with no food or shelter and one full tank of gas.\"\n\n\"One step at a time,\" said Grushin. \"First we get out, then we think about how we survive. I like our chances out there better than in here anyway. Hell, I like the idea of dying out there better than another day in here.\"\n\n\"Maybe I can do it on my own,\" said Morgan. \"Maybe I'm the one who doesn't need you.\"\n\n\"You can't,\" he said. \"Trust me. You need me for this.\"\n\n\"What makes you think you're such hot shit?\"\n\n\"I've got a way out of the prisoner barracks,\" he said. \"Secret way. None of the guards know about it, or, as far as I know, any of the other prisoners.\"\n\n\"Yeah? What's that?\"\n\n\"I'll keep that one close to my chest, thank you very much. Plus, there's something else. A kind of trump card.\"\n\n\"Yeah? What's that?\"\n\n\"Two sticks of dynamite, hidden away safely somewhere here in the camp.\"\n\nMorgan raised his eyebrows. \"No shit?\" Grushin nodded. \"And you know where they are? You can get to them?\"\n\n\"Yeah, I know where they are, and I have access to them any day. They've been there a while. No one's found them, and no one's going to.\"\n\n\"How did you happen to come by two sticks of dynamite?\"\n\n\"There was a prisoner here who was in charge of detonations, down in the mines. Dangerous job. They give it to the people they really don't like. Anyway, he lifted these two from two separate detonations where they wouldn't be missed. Tucked them into his uniform, marched all the way back to camp with them, along with a length of detonator fuse, without any of the guards noticing, and then put them away.\"\n\n\"Won't he miss them?\"\n\n\"Wouldn't think so,\" Grushin said. \"He died a couple weeks before you got here. Got into some trouble with Bortsov's men and met with an unfortunate accident in the mines.\"\n\n\"I see,\" said Morgan.\n\n\"He never knew how to use them to get out. I don't either. But I think you might come up with something.\"\n\n\"Yeah,\" said Morgan. \"I got some ideas.\"\nChapter Fourteen\n\nWorking in the garage changed Morgan's life at the prison. He went from the empty, monotonous drudgery of breaking and carrying stone to something he loved. And the cars needed it. As little as he liked helping his captors, working out clever solutions, fixes, and enhancements with limited materials and tools was a welcome distraction from daily life at the prison.\n\nBut as he did, Morgan also compiled a mental inventory on their condition. There were three trucks and two jeeps in the garage, all military, in varying states of disrepair. One of the trucks was broken down. It took Morgan a few minutes to figure out the problem. The fix was simple enough, but no one else needed to know that. One jeep was in good working order, although it was some five years old, and the other was held together with duct tape and a prayer.\n\nThe thought of escape filled Morgan with a hopeful energy that he hadn't had since he'd arrived at the prison. Watching and planning took up every waking moment as he studied the rhythms of the prison, its procedures and inner workings, the dogs that circled the perimeter, the guards in the towers, the patrols. The outlines of a plan were slowly forming in his mind.\n\nBut there was one thing he had to talk to Grushin about.\n\n\"All right,\" said Morgan. \"Let's game this out. We get out of the barracks your secret way. That gets us out in the yard. I can make the generator give out at a crucial moment. This throws us into darkness. Let's say we have our opening in the confusion. We're left with two problems.\"\n\nThis was over their morning meal, before work, one of two times they were able to talk with any kind of privacy.\n\n\"First, we have to clear the perimeter fence. The gate's guarded, and the guards have flashlights, and the gate is where they'll concentrate their force. Plus, they're going to loose the dogs. Darkness gets us some cover, but not much.\n\n\"Second, we need some means of transportation. If we try to run for it, they'll just come after us as soon as they notice we're missing. We won't get twenty miles out.\"\n\n\"The solution to that is obvious,\" said Grushin. \"We take one of the cars. You have access to the garage. Can you sabotage the rest so that they can't follow us?\"\n\nMorgan nodded. \"Piece of cake. I can make it so that none of the engines have any hope of starting. But we need to get past the gate. That's where the dynamite comes in.\" Morgan ran his fingers through the short bristles growing in on his head. \"There's something I wanted to talk about. I want to get someone else in on the plan. Badri.\"\n\n\"The Arab?\"\n\n\"Yeah.\"\n\nGrushin frowned. \"I don't want to bring anyone else into this,\" he said. \"Every person we add to the plan makes it more likely that we will be caught.\"\n\n\"Let me ask you this. Do you know how to deal with dynamite? Do you think you can get it to do what you want it to do, with certainty, without blowing both of us up?\"\n\nGrushin had nothing to say to this.\n\n\"I want to bring him in,\" Morgan insisted. \"We can't do this without him.\"\n\n\"I don't like it,\" said Grushin.\n\n\"I don't think we have a choice.\"\n\n\"Do you trust him?\"\n\n\"I trust he wants to escape as much as we do. And I trust he wants nothing to do with his interrogators.\"\n\nGrushin furrowed his brow, trying to resist the conclusion. \"Shit,\" he said. \"Okay. You're right. Talk to him. Feel him out first. Don't reveal more than you have to. But see what he has to say. And I hope you know what you're doing.\"\nChapter Fifteen\n\nThe next day was shower day. Grushin told him that parasites had once grown to be a serious problem in the camp, which the guards wouldn't care about except lice, ticks, and bedbugs weren't too good about telling who was a prisoner and who was a guard. So they made the prisoners shower every week, along with shearing off all their hair and dusting them with chemical powder. It wasn't enough to keep the barracks from smelling like a month-old gym sock. But it kept the critters in check.\n\nThe men tossed their uniforms and shoes into a series of plastic tubs, to be taken away for laundry, and waited, naked, as men scrubbed themselves under the cold showers with rough lye soap.\n\nMorgan approached Badri there, as they waited in line. The sound of the showers provided cover so that no one would hear them.\n\n\"Goddamn smell, huh?\" said Morgan.\n\n\"In all my time in this prison, this here remains my least favorite part of it.\"\n\n\"Aren't you glad you can experience it for the first time again through my eyes?\"\n\nBadri chuckled. \"Truly, it is like regaining my childlike wonder.\"\n\n\"So how bad do you wanna get out of this place?\"\n\n\"You mean the showers? Very, very much.\"\n\n\"I mean this prison.\"\n\n\"Who does not want to get out of this prison?\"\n\n\"Well, let's suppose we had a way.\"\n\nBadri's eyes narrowed. \"Are you making conversation or do you have an actual plan?\"\n\n\"The beginnings of one,\" said Morgan. \"Plus a confederate with some cards up his sleeve.\"\n\n\"Grushin.\"\n\nIt was obvious, of course. The person Morgan spent most of his time with in the prison. Who else would it be?\n\n\"What do you need from me?\"\n\n\"We might be able to get our hands on some explosives. Dynamite. I want to know if you'd know how to use it to help us get out of here.\"\n\nBadri thought about it. \"A distraction, of course, could be useful. We could take down the towers. Perhaps open the gate. Yes, I could help with that.\"\n\n\"Think about it. This isn't going to be easy, and more likely than not we'll end up dead.\"\n\n\"I don't have to think about it. And perhaps there is more that I can help you with.\" Their turn was coming up to shower, and they would be separated, sent by the guard under different showerheads. \"We will talk more about this later. But count me in, Morgan.\"\n\n* * *\n\nThey powwowed with Grushin out in the yard at dinnertime, going over the general outline of their escape. Morgan kept an eye on the guards, but mostly they just looked bored. They were just three prisoners talking, after all.\n\n\"I was thinking,\" said Badri. \"We can lay down explosives to bring down the posts holding up the fence, about two hundred meters north of the gate. That way, we can drive the car right over the chain link.\"\n\n\"I think that'll work, if you can get the detonation right.\"\n\n\"There is something else,\" Badri said. \"A car by itself won't get us far. They'll see us from a hundred miles away and come after us. They can send helicopters to smoke us out. It may take a few hours, maybe a day or two, but they will certainly find us if we are in any kind of vehicle.\"\n\n\"So what do you suggest?\" demanded Grushin.\n\n\"I have people,\" he said. \"Friends on the outside. They could bring in transportation. Maybe a small airplane. If they get it within a hundred kilometers or so of the prison, over by the mountains in the horizon, we can drive there and escape before they are able to catch us.\"\n\n\"How do we contact these people?\" said Grushin. \"It's not like we have cell phones. Not even the guards are allowed to have personal communication devices.\"\n\n\"But there is a communications tower,\" said Morgan. \"And I know where we can get an Internet connection, if anywhere.\" He looked up at the window to Nevsky's office.\n\n\"No,\" Grushin said. \"We can't.\"\n\n\"We must,\" said Badri. \"There is no other way.\"\n\n\"You people are crazy.\"\n\n\"You said you didn't want to die here,\" said Morgan. \"Well, it looks like this is our one way out. So. Are you in?\"\n\n\"Of course I'm in.\"\nChapter Sixteen\n\n\"Mom, I'm okay.\"\n\nAlex held the phone up to her ear with her shoulder as she looked through the viewfinder on the camera. A car pulled up to the mansion she was surveilling.\n\n\"I'm keeping safe. I put some professional people on the case. I'm really only overseeing their work.\"\n\n\"I know you, Alex. I gave you your middle name, and it's not 'safe.'\"\n\nAlex took a succession of pictures of the man who emerged, bald and tired-looking, until he disappeared into the house seconds later.\n\n\"I'm not putting myself in any kind of danger or anything.\" She hid her camera back inside her backpack. \"Just hanging around Moscow. Taking some pictures to pass the time.\"\n\nThe mansion in question was a prerevolution urban manor that took up the whole block. The sun was getting low in the sky, and the shadow it cast reached Alex where she sat, in a park across the street.\n\nThe front door opened once again, and Alex took a photograph of the person walking out\u2014but it was a security guard, who surveyed the street and closed the door once again.\n\nAlex hid the camera once more and leaned once again against the tree, pretending to be engrossed in a Russian language workbook, early intermediate level.\n\n\"How are you holding up, Mom?\"\n\n\"I wish you were here,\" she said. \"But I keep busy.\"\n\n\"How's Neika?\"\n\n\"She's here. Dug up all my day lilies yesterday. I had to hose her down in the yard, which wasn't much of a punishment for her, let me tell you.\"\n\n\"Uh-huh.\"\n\nA young woman had knocked at the service entrance of the mansion. Alex picked up the camera once more and photographed her as she turned around and lit a cigarette. She was young and pretty, blond, wearing heavy makeup and a black dress that, while not indecent, didn't leave much doubt about what kind of service she was there to perform.\n\n\"Well, I see you're otherwise busy, so I'll let you go.\"\n\nThe door opened, and the woman dropped her cigarette and ground it into the pavement.\n\n\"No, Mom, I\u2014listen, I'll call you back, okay?\"\n\n\"If you say so.\"\n\nAlex hung up the phone and reviewed the photos. Two clear shots of her face.\n\nThe light was growing dim, and she was tired. Time to call it a day.\n\nShe packed up the book and the camera and walked away from the mansion, then six blocks south before hailing a cab and telling the driver to take her back to Dobrynin's.\n\nAlex picked up a baked potato loaded with cheese and mushrooms from a street vendor. She said hello to Agrafena at the counter, receiving a grumble in return, and ate alone in their tiny kitchen.\n\nDobrynin pushed his way inside as she was washing the dishes. To say that he was in a foul mood would imply that he was ever not in one.\n\n\"Anything today?\"\n\nShe dried her hands and pulled the camera from the bag. He turned his attention to a wall where they had hung photographs Alex had taken on other days.\n\n\"Look, this man,\" he said, holding up the camera's viewer and pointing at a picture on the wall. \"Same.\"\n\n\"Wish I knew who he was.\" Alex finished drying the dishes as he looked at the rest.\n\nOver the course of the past week, it become obvious that Dobrynin never left the house, or indeed saw anyone. Agrafena, whose relationship with Dobrynin was never made clear\u2014Alex figured wife or sister, but nothing between them suggested any definitive answer\u2014did all the necessary shopping for groceries and other necessities, saw to all of the scant customers in the butcher shop, and also received the meat that arrived on Mondays, Wednesdays, and Fridays. Dobrynin, meanwhile, cut the meat in the reeking refrigerated room in the back.\n\n\"It's no use,\" Alex said as her host cycled through the picture. \"Anyone who's at all important has tinted windows and brings their car in through the garage.\"\n\n\"Those people have no value to us,\" said Dobrynin. \"Tell me, is it easy to get to Suvorov?\"\n\n\"Um, duh. Otherwise what would have been the point of staking him out for fourteen hours a day?\"\n\nHe belched, and a foul smell of alcohol reached her nostrils. \"Do you think it will be any easier to get to someone important?\"\n\n\"No, I guess not.\"\n\nDobrynin turned the camera's viewer for her to see. \"This is weak spot. This is how we get him.\"\n\nShe was looking at a pretty girl, smoking, looking worriedly out into the street.\nChapter Seventeen\n\nMorgan lay in bed that night with a sense of accomplishment. Things were lining up, and this escape plan seemed increasingly likely. He stared at the ceiling as the spotlight from the guard tower passed over the wall outside, shining in through the windows, casting a silhouette of the bars.\n\nMen snored and shuffled in fitful dreams. Morgan closed his eyes and drifted off to sleep.\n\nHe was awakened by the opening of the cell door. His eyes opened in a flash.\n\nSomething was wrong. It was still dark, no wake-up siren, no nothing.\n\nOther men who slept as lightly as he did also moved in their bunks, turning to see what was going on.\n\nSeveral shadows came into the barracks. They made straight for Morgan's bunk. He tried to scramble off it and away from them, but in his sleepy state, he was slow and clumsy. They were on him in seconds.\n\nHe was pulled off his bunk and tossed on the floor, hitting it hard with his back.\n\nBy the light of the spotlight, Morgan saw Bortsov looming over him, flanked by four of his henchmen. Who had let him inside? Did he have that much pull with the guards?\n\nMorgan moved to roll under the bed, but the goon was faster, and blocked his way. Together two of them held on to his arms. He tried to wrest himself free, kicking and twisting, but they held firm.\n\n\"This is your time to die.\"\n\n\"I'm protected,\" said Morgan.\n\n\"I don't see any protection,\" Bortsov said, making a show of looking right and left. \"Do you?\"\n\n\"Nevsky is going to get it from Suvorov, and then he's going to come after you.\"\n\n\"Eh, he will give me a few lashes, maybe. It will sting for two weeks. But you, you will be dead.\"\n\nHe was wrong, but Morgan stood no chance of convincing him. He had one chance to save his life now.\n\n\"Help!\" Morgan screamed. \"Pomogite! Pomo\u2014\"\n\nThey stuffed a piece of cloth in his mouth, muffling his voice. Others were awake now, but no one moved to help him. They were all too scared.\n\nOne of Bortsov's men handed him a rock almost the size of a basketball. Bortsov took it, his shoulders sagging under its weight. He raised it over his head. When he dropped it, it would come down and crush Morgan's skull.\n\n\"Now you die, Amerikanskyi.\"\n\nTwo guards burst into the barracks, shouting. They had heard his cries for help! An intense expression came over Bortsov's face, and he pushed the rock downward.\n\nBut the guards had distracted the men who were holding him, and Morgan twisted free, rolling out of the way as the rock hit the floor, breaking tile.\n\nThe guards drove Bortsov's men out first under blows from their nightsticks, then dragged off Bortsov himself. Then they told Morgan to stand and took him outside, across the yard to the place where they had shaved him on his first day there. After Morgan had been sitting for a few minutes under guard, the warden appeared. He was wearing a rumpled shirt and reeked of alcohol.\n\n\"You have made yourself very unpopular in your time here. A few weeks and already you have enemies.\" He emitted a resounding belch. \"I would be happy to throw you to the dogs. Sometimes they even do it in interesting ways. It breaks up the monotony.\" He leaned over Morgan menacingly. His breath was heinous. \"You see, most people here, nobody cares if they are killed. Me least of all. But General Suvorov has other plans for you. And then you will be regretting that B did not kill you.\"\n\nMorgan didn't give him the satisfaction of a response.\n\n\"I would enjoy breaking you. I would also enjoy seeing what Bortsov's boys would have done. They can be very creative in their punishments. But you are too valuable to us to let die. That is left to General Suvorov.\" He turned and walked away, giving an order to the guard as he left.\n\nThey escorted Morgan out of the building and not into the barracks but somewhere else. And Morgan knew where.\n\nIt was a small building, one with only a handful of tiny cells in it, each with a heavy steel door with no openings but a slot low near the ground.\n\nSolitary. He was going in the hole.\n\nThe guards opened the cell door and he got a peek inside. It was perhaps just big enough for him to lie down, and not comfortably. They shoved him inside and closed the creaky steel door, leaving him in pitch darkness.\nChapter Eighteen\n\nThe solitary cell was designed to make men crazy, this Morgan knew. He'd seen plenty in his day, and the concept was well familiar to him.\n\nIt doesn't seem like much, being put in a room by yourself. A naive observer might not even think it would rise to the level of punishment. But Morgan knew. Morgan had experienced it before, and had seen it happen to others. Isolation and sensory deprivation had their way of getting under even the hardest man's skin.\n\nSilence makes you sensitive to every little noise, and total darkness makes you see shapes, lights and patterns that are not there. Soon you start to think you heard scraps of voices, people talking to you, sometimes even voices you recognize.\n\nMorgan sat up and breathed, trying to hold back against the encroaching madness, trying to center his thoughts on his breathing, on feeling the ground under him, on the aches in his body, on resisting succumbing to the despair of his thirst and hunger. You do not know how fast time goes in the dark, when it's day and when it's night, and so time seems not to pass at all.\n\nThus, Morgan had no idea how much time had passed when the food slot opened and a partial face appeared and whispered, \"Hello.\" Morgan could just make him out through the slot. He was blond and young, bony and angular, with wispy stubble on his cheek. \"Come closer.\"\n\nThere was something conspiratorial about the way he spoke, not the barking orders he was used to. Morgan crawled toward him, every movement painful.\n\n\"I heard about what you did,\" he said. \"In the mine. I do not know what you did to be in this place, but you do not deserve to be in here.\"\n\nMorgan grunted in response.\n\n\"They beat Bortsov very bad. Nevsky is very angry.\" He pushed a bowl through the slot. \"Here. Take it.\"\n\nMorgan did. By the dim light he saw that it was filled with stew, not the prisoners' thin, rancid soup, but something borscht-like, filled with meat and potatoes turned pink from the fragrant paprika. Morgan was salivating at the smell.\n\n\"Don't let anyone see the bowl until I come back tomorrow.\"\n\n\"Wait,\" said Morgan. \"Who are you?\"\n\n\"My name is Filipov.\"\nChapter Nineteen\n\nAlex Morgan had been searching through high-end escort service websites for\u2014she checked the clock\u2014seven hours now, and she found she'd become inured to the sleaze covered in a veneer of class that these establishments affected.\n\nShe scoured each website. Most had profiles for the girls, and Alex looked through each one, a parade of blondes and brunettes in suggestive poses, faces pancaked with makeup, fake tits in low-cut tops.\n\nThe Internet connection didn't help either. Dobrynin was not exactly what one might call an early adopter. He'd only switched out his rotary phone because the network stopped supporting it. So Alex used her father's money to buy a 3G modem, and it could take a full ten seconds to load a high-quality image\u2014which didn't seem like a lot at first, but became an increasing pain as the photos Alex had to look through reached the hundreds.\n\nShe was now looking through a website designed in a black and gold theme, everything about it signaling luxury. She didn't even have to open the girl's profile. Alex recognized her from the thumbnail on the page.\n\nShe called herself Lara.\n\n\"Call them,\" Dobrynin said when she showed it to him. \"Pretend to be secretary of American.\"\n\n\"What if they don't speak English?\"\n\n\"They will.\"\n\nAlex took her prepaid Russian cell phone and dialed the number on the website.\n\n\"Zdr\u00e1vstvujte.\" A woman's voice. Young.\n\n\"Hello, I need an English speaker, please? Angliyskiy?\"\n\n\"Yes, hello, I can speak English. How may I help you?\" Her voice was accented in Russian with a gentle inflection of Queen's English.\n\nAlex put on her best secretarial voice. \"I'm calling on behalf of my employer, Mr. Phillips. I'd like to engage one of your girls for the evening.\"\n\n\"Certainly, ma'am. Which girl would that be?\"\n\n\"Lara? The fourteenth on the website here.\"\n\n\"I'm afraid Lara is not available for tonight.\"\n\n\"How about tomorrow?\"\n\n\"Not for the rest of the week.\"\n\n\"Mr. Phillips is adamant that he wants this girl,\" Alex said. \"He is willing to pay.\"\n\n\"Unfortunately, she is not working tonight,\" said the woman. \"Mr. Phillips may perhaps be interested in\u2014\"\n\n\"Listen,\" Alex broke in, letting anxiety creep into her voice, \"Mr. Phillips is not a very nice man, and when he sets his mind on something, he doesn't really take excuses, you know? Things could get really bad for me if I don't get her.\"\n\nAlex heard dead air on the line. Then the woman said, \"Listen, you can't tell anyone I did this, okay? She's taking some time off from us, but I know she does freelance. I can give you her phone number.\"\n\n\"Really? That would be wonderful.\"\n\nAlex wrote down the number on her computer.\n\n\"Thank you so much,\" Alex said. \"You really saved my life today.\"\nChapter Twenty\n\nAlex dialed the number for the fifteenth time, and for the fifteenth time the call went straight to voicemail.\n\n\"This goddamn technology,\" Dobrynin said. \"Stay. I will make a phone call.\"\n\nHe picked up the phone in the shop and dialed. \"Let me speak to Sokoloff,\" he said in Russian, and she was surprised at the ease with which she understood. The studying was paying off.\n\nDobrynin tapped his foot and muttered under his breath while he was on hold. The next words out of his mouth were an insult, and then a deep laugh.\n\n\"How are you, you bastard?\" Dobrynin continued. \"I need some information. I have a phone number.\" He gave the man the digits. \"Yes, that's right.\" He wrote something down on a piece of butcher paper and thanked the man.\n\n\"I got a name,\" Dobrynin said, now in English, holding out the piece of paper. \"Maria Kapustin.\"\n\n\"Got an address?\"\n\n\"What, I have to do everything for you?\"\n\n* * *\n\nAlex had been waiting for hours outside the lumpen apartment building when Maria Kapustin walked out the front door, wearing sunglasses about two hours too late for them to be of any use.\n\nAlex decided a head-on approach would be as good as any. Maria was walking fast, and Alex had to jog to catch up with her.\n\n\"Maria?\" she said. The woman looked at her, and the sunglasses could not hide the panic in her eyes. \"Don't worry,\" Alex said. \"I don't want to hurt you.\"\n\nFrom up close, Alex saw the edge of a black eye peeking out from under the glasses, and purple marks on Maria's neck not quite covered with makeup.\n\n\"Do you speak English?\"\n\n\"Da. Yes.\" Maria did not lessen her pace.\n\n\"I'd really like to talk to you. Can I buy you dinner, or a drink?\"\n\n\"I can buy my own, thank you.\"\n\nAlex decided to take a chance. \"It's about the man who did this to you.\"\n\nMaria stopped walking and looked Alex straight in the eye. \"What do you know about the man who did this?\"\n\n\"I know he's a bad man,\" said Alex. \"And I intend to hurt him back.\"\n\nMaria resumed walking. \"American girl comes to be a big hero to poor oppressed Russian women?\"\n\nLying wasn't going to get her anywhere. \"I have my own agenda,\" said Alex. \"I'll admit that. This isn't about you. But you can get a bit of revenge.\"\n\nMaria stopped and opened the door to a place Alex saw was a dive bar. She held the door open. \"Well?\" she said. \"Are you going to buy me a drink or not?\"\n\nThey settled into a corner table. The only other customers were the everyday drunks who'd probably already been there for hours. Maria ordered them a couple of beers before Alex could say anything. Then she removed her glasses, setting them on the table, and revealed a dark purple blotch under her left eye.\n\n\"Masha,\" the girl said. \"That is what everyone calls me. Everyone who knows my real name, anyway. So what is your deal?\"\n\nAlex opted for full disclosure. \"I'm looking for my father. I think Suvorov is the one who took him.\"\n\n\"If that is true, then I am sorry for your loss,\" Masha said.\n\n\"I'm not ready to believe that yet,\" Alex said. \"What do you know about him?\"\n\n\"I know he is a goddamn bastard,\" Masha said with bitterness. \"And that all the girls are afraid of him. But when he calls, we cannot refuse. At least the money is good.\"\n\n\"He does this to all of them?\" Alex asked.\n\nMasha nodded. \"We all know it. Go to see Suvorov, and you will not work again for two weeks until the bruises fade enough to cover up with makeup. It upsets the other clients, see.\" The waitress set down the pints of beer on the table and she downed half of it in the time it took Alex to take one sip.\n\n\"He likes us young and meek,\" Masha said. \"His sick little games are more fun for him that way.\"\n\nAn idea was forming in Alex's mind. \"Masha,\" she said, \"do you think I could replace the next girl Suvorov calls?\"\nChapter Twenty-one\n\nThe guard Filipov gave Morgan food whenever possible, and would sit outside his cell and talk to him whenever he was keeping watch by himself. It was Morgan's way of keeping time, his conversations with Filipov over rich, flavorful food, in between the long stretches of black nothingness, during which time he exercised as much as he could in the cramped cell to keep up his strength.\n\nFilipov asked about American culture. He loved classic rock and Van Halen, Mad Max, and Die Hard. They even sang Elvis songs together quietly, Filipov with his heavy Russian accent.\n\nBy his reckoning, Morgan was in there for two weeks before he was jogged awake by harsh light cast on his face. He squinted, unaccustomed to the brightness.\n\nA guard was standing at the door. \"American. Out.\"\n\nMorgan picked himself up off the floor. Thanks to the extra food and exercise, he felt better coming out than going in, but he made a show of bracing against the wall and of getting up on shaky legs.\n\nThe guard grunted with impatience. \"Quickly.\"\n\nMorgan emerged outside, feeling the harsh cold wind on his skin for the first time in weeks. The men were lining up for the evening count. Morgan found Grushin and stood next to him.\n\n\"So he emerges. Badri and I were beginning to think you were dead.\"\n\n\"Not dead. Just buried.\"\n\n\"What was it, two weeks?\"\n\n\"You know better than I do,\" said Morgan.\n\n\"I was thrown in solitary once. I was talking to my toes on the second day.\" He looked Morgan up and down. \"But you look hale and hearty, smell aside.\"\n\n\"It was all right. A little boring.\"\n\nMorgan picked out the men who had attacked him in line. They were giving him stares that could melt steel.\n\n\"They are suspicious of you,\" said Grushin. \"Because you have Nevsky's support. And people don't like that.\"\n\nMorgan figured as much. Nevsky had bought him temporary safety, at the expense of putting him in greater danger in the long term.\n\n\"Badri and I have been talking. Working out the details of the plan. I think this is going to work.\"\n\n\"We need to put this plan in motion soon,\" said Morgan. \"We can't wait any longer.\"\n\nThe chill set in as the count went on. Afterward, Grushin said, \"Come with me. I want to show you something.\"\n\nThey walked together into the prisoners' barracks.\n\nIt was a grate that looked solidly in place but came loose after Grushin worked it for a few seconds.\n\n\"Through here, we can easily get outside,\" he said. \"I have left the barracks several times in the night to test it. Nobody knows about it. Nobody is watching. We need to get Badri in Nevsky's office. We can get outside, but I have no idea how to get into the building.\"\n\n\"I have an idea,\" Morgan said.\nChapter Twenty-two\n\nMorgan lay in bed and waited for Grushin's signal, three taps on the bar to the outer doors. He got out of bed, careful not to wake anyone. Not that it would compromise the mission if he did\u2014men got up to use the head in the middle of the night all the time.\n\nHe found Badri and Grushin already waiting for him at their arranged meeting spot. Grushin led the way to their secret passage out. They snuck under the grate, Morgan taking the lead and Grushin bringing up the rear.\n\nOnce they were outside, the wind chilled them to the bone. Morgan looked at the four guard towers, each housing a sniper and a spotlight that they shone over the camp in periodic cycles. The cover of the buildings meant that they never had to be in the line of sight of more than two towers, and often not more than one or none, so avoiding them was just a matter of watching out for the lights. Unless they were discovered, in which case it was a matter of luck whether they would survive the night. Then he searched for the dogs that patrolled the perimeter. Their attention wouldn't necessarily be called by their presence out in the yard, but if they caught their scent and decided to raise the alarm, it was all over.\n\nThey ran along the wall to the prisoners' barracks, keeping to the shadow. This was simple enough, with the light that would normally illuminate this spot burned out. Once they reached the end, they crossed the short distance to the building that housed the kitchen and laundry.\n\nThey heard footsteps coming from around the corner of the laundry building. Morgan held up his hand, then motioned for them to move backward.\n\nThey stood, backs glued to the building, as the guards walked past, chatting in Russian and laughing. They weren't expecting anyone out here. The patrol was a duty and nothing more.\n\nThey did not turn their heads, and didn't see them.\n\nMorgan held his hand up as he waited for them to move away far enough, watching the lights from the towers as he did. Finding an opening, he motioned for Grushin and Badri to follow.\n\nThey ran across the gap at the far end of the laundry building. One more gap to cross to reach the door to the administration building, where Nevsky's office was located.\n\nBehind them at the perimeter fence, a dog started barking.\n\nMorgan had a choice to make. They could retreat to the prisoner barracks, where they could return with relative safety. Or they could press on, trying their luck at the risk of getting caught\n\nMorgan opted to move forward. He took the lead, running full tilt toward their goal, the door two hundred feet away. He felt the two other men close behind him.\n\nThey crossed the distance to the sound of the barking dog. The spotlight from the nearest tower missed Badri's foot by inches, but left them shrouded in darkness. Morgan tried the handle and found the door to the administration building unlocked, as Filipov had promised.\n\nThey closed the door and heard the sound of boots outside. Guards. This was the moment of truth. Had they been seen? Would the guards come in after them? Morgan held his breath, listening intently.\n\nThey passed, oblivious to the possibility that the two had gone inside. Morgan exhaled in relief.\n\nHe heard them talking, and Morgan caught the word krolik. Rabbit. Got the dogs barking wildly sometimes.\n\nMorgan took the lead upstairs. He alone among them knew this building. He led them to the door to Nevsky's office\u2014locked, as they had anticipated. He took out the improvised lock pick he'd taken from the garage and set to work opening the lock to Nevsky's door.\n\nWithin a couple of seconds, he turned it, hearing the click of the lock. Removing the pick and pulling the knob, he pushed the door open.\n\n\"After you.\"\n\nThe first thing Morgan did was to check that the curtains were closed, so that no one could see them from outside. Badri knelt and turned on the computer. Morgan looked through a crack in the curtains as the machine booted up. Everything outside was as quiet as they had left it.\n\n\"There is a password,\" said Badri.\n\nShit. How had Morgan not thought of this?\n\nBut someone else had. \"I can do this,\" said Grushin. \"You know those young Russian hackers you keep hearing about? Well, I was one of them when I was a teenager.\"\n\nGrushin reclined in Nevsky's chair. \"Oh my God, I have not sat in anything so comfortable in months. Seriously, you guys have to try this.\"\n\n\"Focus,\" Morgan said through gritted teeth.\n\n\"Cool your horses,\" Grushin said, and began typing. \"This is going to be a breeze.\"\n\nMorgan kept watch outside the door as Grushin did his thing. \"Got it.\" He asked Badri, \"How do you send your messages?\"\n\nBadri just shook his head and prodded Grushin off the chair. \"I do this part.\" Badri typed at the computer, writing the message to his confederates for about two minutes. \"Okay,\" he whispered. \"It's done.\"\n\n\"Let me just wipe all records of us being here,\" said Grushin.\n\n\"Are you sure your people will come?\" Morgan asked Badri as the Russian typed.\n\n\"They will come. It is only a matter of us being there to meet them.\"\nChapter Twenty-three\n\nThe call came two days later, when Alex was cooped up in Dobrynin's dark little kitchen, cramming Russian vocabulary.\n\n\"There is a girl,\" Masha told her over the phone. \"Klara. She was called to Suvorov's house tonight. He has not met her before.\"\n\nAlex checked her watch. It was still just after 3:00 P.M. \"Then that's our opening.\"\n\n\"She still wants the money,\" Masha said. \"But she is happy for you to take her place. She is black-haired. But we can resolve that. I will bring some hair dye.\"\n\n\"Oh, great.\" Alex ran her hand through her hair. Black was not going to suit her.\n\n\"And you need to be sexy.\"\n\n\"I can be sexy.\"\n\nMasha giggled. Alex couldn't help feeling offended. \"I will be around in a couple of hours,\" said Masha. \"I will bring the dye and some clothes for you, and I will teach you. Crash course. Oh, and Klara says thank you.\"\n\nAlex hung up and went into the refrigerated room to tell Dobrynin the news\n\n\"So you are doing this?\" he said as he chopped up a hunk of meat with a cleaver.\n\n\"I have to,\" she said. \"It's my only chance at this.\"\n\n\"Do not get caught,\" he said. \"Do not let him know who you are. Get what information you can in his house\u2014anything to trade for your father. That is your only chance. If you are found out, you will not come out of there alive.\"\n\n\"I know, we've been over this,\" she said with impatience.\n\n\"Insufferable girl. You want my help? This is my help.\" He brought the cleaver down hard against the cutting board, splitting a piece of pork loin into three parts. \"Do not get yourself killed, okay?\"\n\n* * *\n\nAlex knocked at the service entrance where she had first seen Masha days before.\n\nHer hair was black, the smell of the dye still lingering despite her best efforts at washing it. She hoped no one would notice, but then again, Suvorov had not been promised natural hair.\n\nShe was wearing more makeup than she'd ever worn in her life, and her skin was exposed in all the wrong places. She liked slight outfits, shorts and tank tops, that gave her freedom of movement, which the dress Masha had lent her did not\u2014not unless she wanted to flash the entire street.\n\nA security guard opened the door and ordered her inside.\n\nAlex had made great strides with her immersion in the language and Dobrynin's muttering instruction, and she made a particular effort to mimic the accent. She understood the basic orders relayed to her\u2014come, stay, follow me\u2014while keeping quiet and looking down took care of the rest.\n\nShe was searched and then moved out of the servants' area, and Alex realized she was in the most luxurious house she'd ever seen. Everything was marble and carved wood, walls hung with classical paintings with elaborate frames, thick oriental carpets draped on the floor.\n\nShe was shown to a room that wasn't a bedroom, but rather some kind of parlor appointed with several chaise lounges and divans. There she was told to sit and wait.\n\nShe examined the room. The paintings on the wall Alex recognized as being nineteenth century by their style. On a side table was a statue of a faun, half man, half goat, made of bronze, and against an opposite wall was another of an angel, wings swept upward.\n\nA set of double doors opened and a man walked in, wearing a suit. Suvorov. He was old, near sixty, although he was still strong and fit. He looked rather like a fish, with shallow eyes and a nose that jutted forward in his face. Alex felt a shiver of revulsion.\n\n\"Good evening,\" he said in Russian. Alex mumbled a response.\n\nHe crossed the room, closing the distance between them. \"Speak louder.\"\n\nShe was afraid to. She was afraid he'd catch her accent, and that'd be the end of it. Masha had told her he liked them shy. So she'd be shy. She mumbled a \"Good evening.\"\n\n\"Look at me. What is your name? Speak up.\"\n\nShe looked up at him and said, still mumbling, in her best Russian accent, \"Alexandra.\"\n\n\"I said speak\u2014\" His open hand flew at her face. On instinct, she raised hers and stopped the blow.\n\nToo fast. Too well-trained. She saw by the way that he was looking at her that she'd given herself away.\n\nSuvorov moved to pin her down against the divan, and she rolled out of the way, onto her feet on the carpet. She brought both elbows down hard on Suvorov's back and twisted his arm, pulling it upward.\n\n\"Dan Morgan,\" she said. \"Where is he?\"\n\nHis eyes went wide with surprise. Yeah, weren't expecting that, were you, asshole?\n\n\"What is he to you?\"\n\n\"Answer me. Where is he?\"\n\n\"Far beyond your reach, you whore.\"\n\nShe pulled his arm harder, and he grunted in pain. \"Tell me.\"\n\nSuvorov whipped his head back, catching her nose. Blood began to flow out. He freed his arm from her grip and he swung, his palm snapping against her cheek so hard it knocked her to her feet.\n\nStupid. You know how to take a blow better than that.\n\n\"Your interest in Morgan . . . I see a resemblance,\" he said, standing over her. He stepped on her hand and she cried out in pain. \"Maybe it is the blood. What are you? His daughter?\"\n\nShe gritted her teeth and fumed at him. From her vantage point, his face was wreathed with the clouds painted on the ceiling.\n\n\"He came here, you know,\" said Suvorov. \"Called himself Bevelacqua. Said he wanted to buy guns from me. Got inside my house and tried to steal from me. Maybe deceit runs in the family.\" He ground his heel into her hand, and she yelled out in pain.\n\n\"What did you do with him?\"\n\n\"We caught him, you know. When he broke in. He shot two of my security guards, broke the bones of three more. But we cornered him in the basement, finally.\" He leaned down and pushed a lock of black hair out of her eyes. \"So pretty, too. And now mine. I'm going to have my fun with you, girl.\"\n\n\"Sorry, General, but I don't think fun is in the cards.\"\n\nWith her free hand, she punched him hard in the groin. He bent double in pain, and she pushed him off sideways. She stood and made a run for the door, but he ran after her. She ran for the far side of the room, then turned to face him. He was moving fast, bringing his superior weight to bear against her.\n\nAnd she was going to use that.\n\nShe bodychecked him, which sent her flying backward, but was just enough to knock his path to the right\u2014into the angel statue with the upswept wings, which were just sharp enough to pierce his belly.\n\nSuvorov hollered in pain.\n\nThat was going to attract the guards.\n\nThe statue didn't do much damage, but it was enough for Alex to get back to her feet and run through the double doors he'd come from, into his private chambers\u2014an office, and a bedroom beyond.\n\nShe heard the heavy footsteps of Suvorov's bodyguards, half a dozen at least, approaching from below. They were going to cut off all exits downstairs, so she had to do something they wouldn't expect.\n\nShe went up. She found a narrow staircase that led upstairs, to a long hallway of bedrooms, where the ceiling had the slant of the roof outside. She chose one two thirds down. The window would not open, so she grabbed a heavy brass lamp from the bedside table and shattered the glass, breaking the frames wide enough for her to pass.\n\nAlex walked out onto the roof, bracing against the window frame. She was barefoot, and her dress wafted in a cool breeze, not exactly ideal gear for the situation. But being barefoot was a hidden blessing on the rounded tiles. She walked over the ridge of the roof to the other side.\n\nNo one outside seemed to be looking for her yet\u2014they still thought she was inside. She looked over the edge, studying the pattern of the bricks. Yes, she could do it, even in a slutty dress. She sat on the eaves, hidden by darkness from anyone who might see her, and reached for a handhold, easing her way off to hang from the wall.\n\nShe climbed down the western wall, which was shrouded in darkness. She waited for the vehicle patrol to pass and jumped over the fence, dashing across the street and out into the night, as far away from Suvorov's house as her legs could take her.\nChapter Twenty-four\n\nMorning count. Morgan, Grushin, and Badri got together to talk things over for a few minutes before the guards forced them into the line.\n\n\"I asked them to signal from the west using lights blinking Morse code,\" said Badri. \"That's how we'll know that they're there, and where they are.\"\n\n\"They'll get us away from here?\" said Grushin. \"Are you sure?\"\n\n\"They are loyal, and they have the resources inside Russia. They will be there. They will get all three of us out.\"\n\n\"If they actually manage to get here,\" said Grushin.\n\n\"We look to the west,\" he said.\n\n\"Because if they don't, then we are liable to be eaten by\u2014\"\n\nA siren sounded, the sign for them to line up for the count. They prisoners shuffled into position, making a wide semicircle in the yard. A murmur of activity propagated from one end. Someone was approaching, and the prisoners were turning to look. It was Nevsky, walking down the line.\n\nMorgan's gut sank. Did he know? Was he coming to kill them?\n\nHe stopped in front of him.\n\n\"Prisoner Morgan,\" he said. \"I am surprised at the people you have chosen to be your friends here. But you have my congratulations for managing to stay alive this long.\"\n\nNevsky kicked Morgan in the knees, and he fell on all fours. \"Down like the animal you are. So you do not forget your place.\"\n\nMorgan's face burned with rage, but he couldn't do anything about it without getting it worse. Nevsky was daring him to.\n\n\"General Suvorov is coming,\" Nevsky said. \"He wishes to interrogate you personally. I just wanted to let you know. One more week. And then he will make you talk.\"\nChapter Twenty-five\n\nAlex didn't dare take a taxi. Instead, she walked barefoot for three hours through the streets of Moscow until she arrived at Dobrynin's butcher shop. The shop was closed for the day, so she knocked, softly at first, but when she got no response, as the stress of the night caught up to her, she slammed her hand against the door harder and harder, as if it were to blame.\n\nDobrynin opened the door to admit her.\n\n\"Did anyone follow you?\"\n\nShe pushed her way past him. \"You aren't even going to say you're glad I'm alive?\"\n\nDobrynin shrugged. \"I guess I am glad you are alive. Were you followed?\"\n\n\"No, I wasn't goddamn followed.\"\n\nShe grabbed a T-shirt, jeans, shoes and socks from her suitcase and changed out of the dress in the bathroom. When she came out, Dobrynin was waiting for him in the kitchen, a chair pulled out for her. \"Sit. Tell me what happened.\"\n\nShe related to him what had transpired in Suvorov's mansion and he listened, heavy lidded. When she was finished, he said, \"You need to go home, girl. Your little adventure here is finished.\"\n\n\"I still haven't found my father. I think he might be alive.\"\n\n\"He knows your face, and soon, so will every policeman in Moscow, or worse. It is time for you to go.\"\n\n\"I won't go.\"\n\n\"Then you will go from my house,\" he said. \"I will not have you endanger me and Agrafena with your presence.\"\n\n\"Fine,\" she said, getting up from the table. \"I'll go somewhere else.\"\n\n\"You do that and you will be killed. And I will not be there to help you.\"\n\n\"Are you serious, Dobrynin? You haven't left this goddamn house. Not since I got here. Every moment I've been out there so far, I've been on my own.\"\n\n\"And do you know why I do not leave this house?\"\n\n\"You're a shut-in,\" she said. \"A coward.\"\n\n\"I am a dead man. I have been a dead man from the day your father came after me. To kill me.\"\n\nShe furrowed her brow. \"Kill you?\"\n\n\"Yes! Or do you not know he was an assassin? I was an agent of Russian intelligence. I gave information to the Americans because I was done with this piss pot of a country. They were supposed to get me out. Instead they sent a man to kill me.\"\n\n\"But he saved you,\" she said. \"That's what happened, isn't it? Instead of killing you, he let you go and told you to disappear.\"\n\n\"He did. For me to live the rest of my life stuck inside, never going out in the sun. To live in this stinking house!\"\n\nAlex slammed her palm down on the table. \"He saved your life!\"\n\n\"Only from himself. I don't owe you nothing else. Get out. Get out!\"\n\n\"Fine,\" she said, going into the tiny room she had been sleeping in and stuffing her things into her suitcase. \"I'll go, and you can go back to being a miserable old man who isn't any good to anyone!\"\n\n\"You are just a goddamn child!\" he hollered. \"I never asked for you! You came to my door!\"\n\n\"Well, I won't make that mistake again.\"\n\nShe stormed out, slamming the door behind her with a ring of the bell that hung above it.\nChapter Twenty-six\n\nGrushin helped Morgan up. They watched Nevsky walk away in his unsteady drunk's version of a military march, like a rooster surveying his chickens, as the guards resumed the prisoner count.\n\n\"He told me just to screw with me,\" Morgan told Grushin. \"But you know what? It's not going to work. Because we're still going to get out of here.\"\n\n\"That's the spirit.\"\n\n\"Except we're going to need to do this in less than seven days.\"\n\n\"Wait, what?\" said Grushin. \"No. There's no way.\"\n\n\"No choice. After a week they'll move me to an interrogation cell. And I don't come out of that alive.\"\n\n\"Shit.\" Grushin rocked back and forth. \"Shit. What do we do?\"\n\n\"We have no choice,\" Badri said. \"We do this with Morgan or we do not do this at all.\"\n\n\"Shit!\" Grushin yelled. People turned and stared. Morgan hushed him. But it wasn't too suspicious. Not here. There was plenty to scream and curse about.\n\n\"We need to get word to my people,\" Badri said. \"We need them to be ready in a week's time. That means\u2014\"\n\n\"We have to go back into Nevsky's office,\" Morgan said.\n\n* * *\n\nThe hallway was dark.\n\nThey'd done this before, but there was a nervous energy now that hadn't been there before.\n\nMorgan picked the lock once more and pushed open the door.\n\nThere was light inside the office, pale and dim. The computer was on. And at the desk, asleep, was Nevksy.\n\nHe was snoring like a drunkard, a snorting, gasping snore. He smelled of liquor.\n\nMorgan looked at Grushin, who stared back in wide-eyed terror. Badri was calm. He was used to focusing under pressure. He was ready.\n\nMorgan tiptoed to the desk and turned the monitor slowly around. He picked up the keyboard with his right hand, lifting it off the desk. As he carried it, he tipped over the pencil cup. His left hand shot out and grabbed it before its contents spilled all over the desk.\n\nNevsky stirred, snorting.\n\nMorgan noticed that he was holding his breath, and started breathing again. He set the keyboard down on the desk for Badri to use.\n\nBadri stood at the desk across from the sleeping Nevsky. Morgan kept his eyes on him, watching him for any twitch that might herald his return to consciousness. He was so vulnerable. It would be so easy to reach out and snuff out this bastard's life, and end all the evil he would ever do.\n\nExcept he couldn't. Not now. Morgan needed to let him live, so that they would have their chance to escape.\n\nBadri hit send and closed the browser window. He then set the keyboard back where it belonged as Badri turned the monitor back around.\n\nHe nodded. Done. Time to get the hell out of there.\n\nThey slipped out of the room and Morgan locked the door behind them. The three issued a collective sigh of relief.\n\n\"It is sent,\" said Badri. \"If everything goes to plan, they should be here in three days.\"\n\n\"If,\" said Morgan. \"Let's hope they do. Or else it's my ass.\"\nChapter Twenty-seven\n\nMorgan and Grushin met up at dinner after a day of nervous anticipation. The day's stew smelled pungently of spoiled potatoes. The wind seemed to have died down, and the weather seemed to be mild, if anything.\n\n\"Eat up,\" said Morgan. \"We don't know when we'll have the chance to eat again.\"\n\nThis was the day.\n\n\"Are you both ready?\" Badri asked.\n\n\"I got the dynamite,\" said Grushin. Morgan had already noticed the subtle bulge at his waist.\n\n\"The cars are ready,\" said Morgan. \"I put metal shavings in the oil of all the cars except the newest jeep. If anyone tries to follow us, whatever car they use will run about as well as a brick within a few seconds.\"\n\n\"So we get past the fence and we're in the clear,\" said Grushin.\n\n\"Something like that. I also left the crowbar hidden in the grass outside the garage.\"\n\n\"Then we're ready,\" Grushin said. \"All we have to do now is actually do it.\"\n\nMorgan leaned back against the concrete wall of the prisoner barracks. \"What are you going to do when you get out?\"\n\n\"I'm going to a bar and I'm going to order myself a beer and a hamburger,\" Grushin said in a dreamy tone. \"Then I'm going to find my girl. I don't care if she's moved on, if she's with someone else. The moment she sees me\u2014that's all that's going to matter.\"\n\nMorgan knew the story too well of what so often happened with the guys in the military who left their sweethearts behind. And this one had every reason to think he was dead.\n\nBut he needed something to cling to. And who knew? Maybe she was waiting for him after all.\n\n\"What about you?\"\n\n\"I don't know,\" said Morgan. \"I've been thinking about this a lot. And I don't know. My organization has a hell of a life insurance policy. My family's taken care of for life, moneywise. And there's something else. I'm marked. Suvorov knows who I am. He's going to leave them alone now that he has me, but if I escape . . . If I go back to them, I'm dooming them to a life on the run. Maybe they're just better off without me.\"\n\n\"You do what you have to do,\" said Grushin.\n\nMorgan had to steady himself through evening count and the march into the barracks as adrenaline pumped through him in a constant buzz. He was ready. They were going to do this.\n\nMorgan lay in the dark, excitement pounding in his head, until he heard Grushin's bird call. He met him and Badri outside. In his hands were two objects, long and thin. Two sticks of dynamite, which he handed Badri, who was already holding a length of rusty rebar. \"Make it count.\"\n\n\"Just be ready to do your part.\"\n\nThey moved together as far as the laundry building. Badri broke away from them there, running toward the fence, where he would twist the rebar around the cable that tethered the dogs and then drive it into the ground. This would stop any of the dogs circling the perimeter from being able to approach him as he lay down the dynamite.\n\nMorgan and Grushin, meanwhile, made their way across the yard to the garage. Grushin climbed through the unlocked window of the motor hangar and disappeared inside.\n\nLeaving the Russian to do his part of the plan, Morgan made his way along the outer wall of the building. Around the corner was the large door through which the cars came out of the garage.\n\nMorgan watched for the pattern as the spotlight shone around the grounds. He retreated from the corner as it passed, casting a long straight shadow on the ground.\n\nHe had to wait before breaking the padlock open with the crowbar. Once he opened the garage, they would have to move fast.\n\nHe stayed put and listened for the start of the engine that was the signal for him to spring into action.\n\nMinutes passed, and the tundra wind howled. The light circled back, and then again. Morgan strained to hear, but there was no sign of the engine starting up.\n\nThis was taking too long. Badri was going to blow the gate and the truck would still be in the garage.\n\nHe couldn't wait anymore. He readied himself, tightening his grip on the crowbar. He waited until the light passed, then he ran, inserting the crowbar through the padlock and pulling down, using the leverage to break it open. It fell to the ground with a soft thud.\n\nMorgan pulled the door open just enough for him to pass, revealing the dark garage inside, the black shapes of the trucks looming in the murk. He closed the door behind him.\n\n\"Grushin?\"\n\nThe lights came on with a loud crack, and Morgan found himself staring down the muzzles of half a dozen submachine guns. Each was held by a uniformed prison guard.\n\nIn the middle was Nevsky, a diabolical smile plastered on his face. Grushin was standing stiffly next to him.\n\n\"Do you really think I didn't see the three of you scheming out in the yard? Do you think I don't know what goes on in my prison?\"\n\nAmong the guards was Filipov, holding his rifle nervously, something apologetic in his expression. There was nothing he could do about this, he seemed to say.\n\n\"My men are bringing the Arab in as we speak. He, like you, is too valuable to dispose of summarily. I'm afraid I can't say as much about young Mr. Grushin.\"\n\nNevsky drew a knife and in an unhesitating movement pushed it deep into Grushin's throat. Grushin's face contorted in pain and surprise. Nevsky pulled it out with a spurt of blood.\n\nHe fell at Nevsky's feet.\n\n\"Oh, and before I forget.\" Nevsky drew his sidearm. Before anyone could react, he aimed at Filipov's head and fired. The guard crumpled to the ground on top of his submachine gun.\n\n\"See what happens to those who help you? See what happens to the people you rope into your plans?\" He wiped his knife on a handkerchief. \"I have orders to keep you alive until General Suvorov returns. But believe me. Your end will be much less pleasant than your friend's here.\"\n\nHe punched Morgan in his side.\n\n\"You will be taken to solitary confinement. And there you will stay until the day you die. Don't worry. It won't be too long.\"\nChapter Twenty-eight\n\nAlex returned to her cramped room at the Ustritsa Hotel from her morning run. The wallpaper was peeling, the air smelled of mildew, and something that looked like black mold was growing in the corner. But it was a place to sleep. And to plan.\n\nShe took a shower and sat down on the hard bed, setting the partial map of Suvorov's house she had drawn from memory on the bed.\n\nThe time was approaching to implement her plan.\n\nShe'd been studying the patterns of Suvorov's security. Two men sat in a car across the street from the mansion, all day, every day. But every night at around two a.m., the car moved out and new guards took their place. This would give her a narrow opening to move in on the house.\n\nShe'd wait across the street, hidden in shadows, until they were on the move. Then she'd move across the street, scale the fence, and then climb the wall of his house back to the roof and gain admittance from there. She knew she could do it going down. Going up wouldn't be any more difficult.\n\nShe'd go in through one of the windows that opened to the roof. From there, she traced the route through her map to Suvorov's private chambers. If she didn't find what she was looking for, she'd settle for killing him.\n\nAlex got dressed and set out for another day of surveilling the mansion. She needed to make sure everything was according to plan.\n\nShe climbed down the stairs and moved out into the street, hailing an approaching taxi. It pulled up to the curb, and the driver stepped out to open the door for her.\n\nAlex was too excited to be suspicious. As she put one leg into the car, she felt a prick in her neck and things started to swim before her eyes. She felt herself falling, and the driver held her, easing her into the seat, and closed the door. She tried to scream but her body was unresponsive. She heard the engine start, and the taxi started moving just as things turned black.\nChapter Twenty-nine\n\nThere was no food at all this time. There was nothing, just the blackness of solitary. Morgan had no idea how much time passed before the door opened again. He squinted at the hard light that shone into his cell, hungry, aching, and exhausted.\n\nThey yanked him to his feet and pushed him out into the hall. It was night, he saw. They led him down the hall and into a room in the same building that he had not been in before, but he knew it immediately. It was all white tiles. In the middle was a chair, around which the tiles were whiter than the surrounding ones, discolored by bleach. The grout was stained with brown dried blood.\n\nMorgan had been in places like this before, on either side of the divide. He liked the other side better.\n\nThey sat him down on the chair and tied his hands around the back. Then they turned their backs and left him alone, locking the door behind them.\n\nMorgan tested the cuffs. They were solid. But he had one last trump card.\n\nHe worked his tongue in his mouth until he got his improvised lock pick in between his lips.\n\nNow came the hard part. He needed to get it into his hands, which were currently at his back. He turned his head over his right shoulder, adjusting the pick with his tongue to get the angle just right. He brought his hands as far to the right as he could manage.\n\nHe heard the door unlock. No time. He pushed the pick out with his tongue.\n\nIt bounced off his outstretched fingers and fell on the floor behind him.\n\nHe swore in his mind and turned his head as the door opened, and a man he recognized walked into the room. Alligator shoes, shined to a sheen. Green Russian military uniform, his chest festooned with honors and decorations. Eyes bulging like that of a fish, thick meaty lips, heavy eyebrows.\n\nGeneral Suvorov.\n\n\"I hear you have been giving Nevsky some trouble,\" he said as he rolled up his sleeves. \"He will be glad to be rid of you, I think.\"\n\n\"I think we can agree on that.\"\n\nSuvorov raised an eyebrow. He was a humorless man, grave and unfeeling. \"Do you know what they call me in Ukraine?\"\n\nMorgan did. \"The Barber of Lozhki.\"\n\n\"And do you understand why they call me that?\"\n\nHe was known for scalping his victims and then slitting their throats.\n\n\"So you are aware of what is to happen to you?\"\n\nThis Morgan did not answer. He tried to avoid picturing it with too much vividness. It wasn't good for his mental health.\n\n\"You can stop this,\" he said. \"You can even save yourself. All you have to do is speak.\"\n\n\"Already? I wouldn't want to ruin the party. I'm sure you have a lot of exciting surprises in store for me.\"\n\nSuvorov swung. He had a hell of a right hook, hitting Morgan in the temple.\n\n\"This will be very predictable. I will inflict increasing amounts of pain and mutilate you in increments until you speak.\"\n\n\"Here I thought the Barber of Lozhki would have a better sense of showmanship.\"\n\nSuvorov swung again. This one was hard enough to tip the chair on its back legs.\n\nThis one hit him in the face, and he tasted blood in his mouth. \"Come on,\" he said. \"Is that the best you can do?\"\n\n\"We are just getting started,\" said Suvorov, and followed up with an uppercut.\n\nThis time, Morgan pushed with his feet, and the chair fell backward. He grunted at the pain as it pinned his hands to the floor. He wriggled to give them some freedom of movement and felt for the lock pick.\n\n\"Oh, goodness, look what I've done.\" He lifted Morgan's chair by the back, set him right, and stepped back. \"What I do not understand is why you insist on protecting the people who abandoned you. No one tried to find you, you know. No inquiries or requests to get you back from your government or any agency. Nothing.\"\n\nNo, he thought. There is one thing. Morgan held the lock pick between his fingers, already working the handcuffs. He had to keep Suvorov busy as he worked them open.\n\n\"Has it crossed your mind that maybe I just don't like you?\"\n\nSuvorov drew a military knife. It wasn't impressive, but it didn't have to be to cut into the flesh of a man tied to a chair. \"Perhaps it's time to move on to mutilation.\" He touched the blade to Morgan's right shoulder and pushed it in, opening a gash that bled freely. Morgan bellowed in pain. \"I find that cutting off parts is what really crosses a line for most people. When they see pieces of them removed, see their own bodies diminished . . .\" He touched the edge of the blade to Morgan's earlobe. \"Shall we start here?\"\n\n\"How about we start here?\" Morgan shook off the handcuffs and head-butted Suvorov. As he staggered back, Morgan picked the chair up off the floor and swung at him. He raised his hands in self-defense, dropping the knife in the process, and was knocked against the side wall by the blow.\n\nThis gave Morgan the opening to grab the knife. He then pushed Suvorov against the wall and held the blade to his neck.\n\n\"Tell the guard to open the door.\"\n\nMorgan moved the blade to his eye. \"If that door doesn't open in five seconds, you're going to be shopping for eye patches for next season.\"\n\nSuvorov called out in Russian through the door. They exchanged some words. Suvorov barked exasperated commands.\n\nMorgan heard the dead bolt being undone on the other side of the door, which opened to reveal the single guard keeping watch on the cell.\n\nThe guard was holding a nightstick, ready for action. He raised his eyebrows in shock when he saw that Morgan had Suvorov hostage.\n\n\"Drop it,\" he said. \"Slide it over to me.\"\n\nSuvorov translated for him. The guard let the nightstick clatter to the ground and kicked it over. Morgan bent to pick it up.\n\n\"Inside,\" he said, motioning with his head.\n\nThe man walked inside the cell.\n\n\"Keys,\" he said, pointing. The man understood. He hesitated, looking to Suvorov for guidance. Morgan pushed the blade against Suvorov's neck, just enough to pierce the skin and draw a drop of blood.\n\nThe guard unhinged the key ring and tossed it to Morgan.\n\n\"Thank you very much. I'm going to be sticking around for a minute. If I hear a peep out of you, the general here loses an ear.\"\n\nHe pulled Suvorov out of the cell and closed the door, pulling the dead bolt shut.\n\nThat was one less to worry about.\n\nHe looked down either side of the hallway. No one there.\n\nHe looked at the row of other cells. Two of them were padlocked. Two cells containing prisoners.\n\nHe pulled Suvorov along with him and unlocked, one-handed, the nearest cell. He pushed the door open.\n\nIt wasn't Badri. Behind the door was Kolya the Cannibal. He stood as the door opened and looked at Morgan, his face expressionless as ever.\n\nSuvorov twisted free of Morgan and slammed the door against him. Morgan was left dazed, which gave Suvorov the opening to twist his arm and wrest the knife away from him.\n\nRegaining his grip, Morgan brought the nightstick hard against Suvorov's leg. The general screamed and fell on his back, inside the cell.\n\nMorgan took a look at the general's leg. He had a compound open fracture where the nightstick had connected. At a glance, Morgan could tell he wasn't getting up again, not without some serious surgery.\n\nSuvorov held his knife defensively, wide-eyed with pain and fear. He could barely move from the pain. He couldn't defend himself, but Morgan didn't have time for this.\n\nKolya stared at Morgan. Blank. Empty. Then he looked down at Suvorov.\n\nMorgan stepped out of the cell and closed the door, leaving Suvorov inside with Kolya, whose eyes were already hungry at the prospect of what was about to happen, the only sign of life Morgan had ever seen in them.\nChapter Thirty\n\nMorgan ran to the remaining cells. As he undid the lock, he heard Suvorov screaming from Kolya's cell and tried not to think about what was happening. He pulled the dead bolt and opened the door.\n\nBadri, haggard and exhausted, was sitting in a corner. He looked up at Morgan, goggle-eyed, blinking as the situation registered in his mind.\n\n\"Let's get out of here,\" Morgan said.\n\n\"How did you\u2014\"\n\n\"Let's go!\" He looked up and down the hall for signs of any approaching guards. \"I can tell you the story on the way out of this place.\"\n\nMorgan didn't have to say it twice. Badri pushed himself up off the ground. \"Where are we going?\" he asked.\n\n\"We need a plan.\" Morgan rubbed his temples. \"We're in a Russian gulag with no plan, no tools, and no weapons but a stick. Any minute now, we're going to be found.\"\n\nBadri worked his muscles, cracking the joints in his neck and arms. \"We seem to be at a disadvantage.\"\n\n\"I've been in worse,\" said Morgan. \"Let's think.\"\n\n\"We need transportation out of here.\"\n\n\"Suvorov's jeep,\" Morgan said. \"It's faster than any of the trucks in this place. That's our way out of here.\"\n\n\"They'll see us.\"\n\nMorgan frowned. \"Then we need to give them something bigger to distract them. We need something they can't ignore.\"\n\n\"You are talking about\u2014\"\n\n\"Dynamite.\"\n\nBadri smiled. \"I like how you think. Same plan?\"\n\n\"No,\" said Morgan. \"We take out the motor pool. The dynamite added to the gasoline tank . . .\"\n\n\"Boom,\" said Badri. \"Enough of a distraction for us to ram the gates.\"\n\n\"We need to get the dynamite,\" said Morgan.\n\n\"They still would not have taken the two sticks away from the prison,\" Badri said. \"It is still on the grounds.\"\n\n\"And there's only one safe place to keep them on this camp. They'll be in the armory.\" Morgan held up the key ring. All were the same. He could bet they were all for the solitary cells. Armory access was not something given to any guard.\n\n\"Nevsky,\" said Badri. \"He will have the keys.\"\n\n\"Then that's where we go. This way. Side door.\"\n\nHaving a plan, Morgan held the guard's billy club in his hand and took the lead down the hall, taking slow, measured steps. Suvorov's screaming stopped. Morgan wondered whether he was dead. Any alternative he could think of was worse.\n\nAs they neared the end of the hall, Morgan heard footsteps, heavy boots echoing around the corner. Morgan held his hand up for Badri to stop and then put his finger to his lips. Badri nodded.\n\nAs the first man rounded the corner, Morgan swung the stick.\n\nHe hit the closest guard square in the face. He bent double, clutching his nose, which was now squirting blood.\n\nThe second man came right behind. His hand went for his weapon, but Morgan swung again, bringing the club down on the man's leg. The guard stumbled, and Morgan brought his elbow down on the man's back. He collapsed face-first on the ground.\n\nBadri, meanwhile, lunged at the other guy, bringing a fist to connect with his broken nose and throwing him down next to his companion.\n\nBefore either could stir to stand again, Morgan brought his foot down on each of the men's right legs in turn.\n\n\"They're not coming after us,\" he said, moving past them and leaving them screaming in pain.\n\nBadri took the lead now, reaching the side door. Morgan unlocked it and opened it a crack. Badri looked up at the warden's window. \"How do we get up there?\"\n\nMorgan looked at one of the watchtowers, which was within view. Waiting for his opening, he ran across the yard, hearing Badri's footsteps, muffled by the grass, close behind him.\n\nThey reached the administration building and went inside.\n\nThe place was deserted, all the doors in the central hallway closed. Morgan led the way, running up the stairs.\n\nThe double doors to Nevsky's office. No time to finesse this one. Morgan kicked the door. It caved on the second, swinging open.\n\n\"Look in the\u2014\"\n\n\"There will be no need, Mr. Morgan.\"\n\nNevsky was standing to his right. In his hand was a GSh-18 handgun. Standard Russian military issue. Aimed at Morgan's face.\nChapter Thirty-one\n\nNevsky kept the gun aimed at Morgan's head as he motioned for him to get inside the office. His hand was shaking. He had a drunk's unstable hands.\n\n\"How did you get out? Where is Suvorov?\"\n\n\"He's about to die,\" said Morgan. \"You can save him if you hurry. But you'll have to leave us here.\"\n\n\"Perhaps I can come up with a better idea.\" He aimed for Morgan's leg. Before he could shoot, Badri rushed him. Nevsky reacted, but not fast enough. Badri barreled into him. Morgan moved to help him, pinning Nevsky's arm and taking the gun.\n\nHe stepped back, with Nevsky at gunpoint. \"Keys,\" Morgan said. \"Your lighter, too.\"\n\nNevsky took a key ring and tossed it to Morgan, and then the lighter. \"Here. I will enjoy your death from the vantage point of my office.\"\n\n\"You will not.\" Badri stabbed Nevsky in the gut with a letter opener from the desk. He pushed the blade in farther and twisted. By the angle, Badri got his kidney, maybe his liver as well.\n\nNevsky grunted and braced himself on his desk.\n\n\"Let us go,\" said Badri, pulling the cord from the intercom out of the wall. \"He is done.\"\n\nMorgan didn't make a habit of killing men in cold blood. But he had to admit that it was convenient.\n\nThey stepped out of the office and closed the door. Morgan took the armory key off the ring and tossed the remaining to Badri. \"Get the doors to the prisoner barracks unlocked,\" he said. \"The alarm is going up any second now. Let's kick up our distraction a notch.\"\n\nBadri nodded.\n\nThey parted ways and Morgan made his way down to the underground armory. There was a light on down there. Morgan tiptoed his way downstairs. At the landing, he surveyed the room, hidden around the corner of the wall.\n\nOne guard, armed but distracted with a magazine, sat at a table in front of the locked armory cage.\n\nEven one shot fired would attract the attention of every guard in the camp.\n\nHe was strategizing his approach when the prison's general alarm started blaring, deafening even here, underground.\n\nWell. No use being coy now.\n\nMorgan ran. The man saw him, drew his gun, and took aim. Morgan went low. The man fired and the bullet sailed above him. He took a running jump, holding on to the edge of the bed, and swinging his legs to kick the man in the chest. The guard fell backward.\n\nThis left him on the desk, sliding to the other side. The guard was reaching for the gun, but Morgan was already bringing the nightstick down to connect with the man's head. He was knocked out cold.\n\nMorgan had seconds before guards would start streaming in.\n\nHe opened the lock to the gun cage.\n\nHe found a duffel bag under the shelves. He picked the dynamite sticks up carefully, putting them in the bag.\n\nSince he was here . . .\n\nHe looked over the guns. Half a dozen Saiga 12-gauge shotguns, twenty Vityaz-SN submachine guns, two Dragunov SVD sniper rifles, and three racks of MP-443 pistols.\n\nHe loaded two of each of the SMGs, shotguns, and pistols into the bag, plus ammo. He slung it over his shoulder. He took an additional handgun.\n\nHe broke the key in the lock. No big guns for the guards. That should help even the odds for the prisoners.\n\nHe heard footsteps coming down the stairs. They would have handguns. Morgan drew a shotgun from his duffel and whispered to himself, \"Time to make some noise.\"\nChapter Thirty-two\n\nMorgan held on to the shotgun, a Saiga 12 with a twenty-round detachable magazine, and waited for the two men to come down.\n\nThe first wasn't expecting him, and didn't have time to react before Morgan fired, the gunshot reverberating in the cramped, windowless space. The second went for his gun. A second shell, and he was down.\n\nTwo more were at the landing above. If they consolidated their position, if enough guards amassed up there, Morgan would be trapped no matter how much firepower he could muster.\n\nHe cast down the shotgun and drew out a Vityaz SMG. Thirty-round box magazine. If he held down the trigger on automatic mode, he'd drain it within three seconds.\n\nMorgan aimed blindly around the corner and fired off a burst upstairs. This gave him the opening to move up the stairs as the men took cover on either side. One of them made his move when Morgan was halfway up. Morgan loosed another burst, which caught him in the chest.\n\nHe slowed down as he approached the upper landing and listened. What he heard was boots running away. The other man was beating a prudent retreat.\n\nMorgan needed to get out of the building, fast. He ran in the opposite direction of the guard.\n\nOutside, the siren alarm was blaring. The perimeter guard towers, the biggest danger at the moment, the only ones who would be armed, were flashing their lights wildly, looking for the source of trouble. Outside was a bad place to be. Too exposed.\n\nMorgan saw Badri hiding in the shadows across the yard. Badri pointed to the car. Morgan signaled for him to go ahead, and then took off running at an angle. A light caught him, then moved to shine right on him. Shots fired, whizzing past him.\n\nHe pushed himself harder, staggering his pace to avoid sniper fire. He didn't stop until the garage was between him and the shooter.\n\nHe moved along the wall until he found the window in the northwest corner, the one Grushin had used to gain access. On the other side of this was the gasoline tank that fed the generator and the trucks.\n\nThis was going to make a splash.\n\nHe picked up a rock and hurled it through a windowpane. Glass shattered and clinked on the floor inside.\n\nThat's when he heard barking.\n\nDogs. Let loose and running toward him, fast.\n\nIf he ran now, he didn't have a chance in hell of getting away.\n\nSo he tried to concentrate. He balanced both sticks of dynamite on his left hand and drew out Nevsky's lighter with his right. He struck, and it did not light.\n\nTwo dogs were running toward him, heavy paws beating the pavement. He had seconds to get this done.\n\nHe struck the lighter again, and this time a flame emerged. He held it under the fuses on the two sticks of dynamite, which hissed and sputtered as they caught fire.\n\nHe tossed the two sticks of dynamite inside through the pane and turned to run. The dogs were right behind him within a fraction of a second. And then he emerged out in the open yard, and a light was on him almost immediately. It was only a race now to see which would get him first, the sniper on the tower or the dogs snapping at his heels.\n\nThe explosion was more than he'd anticipated.\n\nThe corner of the concrete building shattered into rubble, and the fuel tank exploded in a ball of flame that rose upward, lighting the camp in bright orange before consuming itself.\n\nIt rocked the ground, knocking Morgan off his feet. Windows exploded in a rain of glass.\n\nMorgan raised his head. The guards were no longer much preoccupied with him, and he found no sign of the dogs behind him. By the light of the fire\u2014all electric lights had gone out with the destruction of the generator\u2014he saw prisoners coming out of the barracks, taking their first tentative steps outside.\n\nHe heard the rumble of an engine, and then the bright white headlights. It was Badri with the jeep, speeding across the yard. He came to a skidding stop next to Morgan.\n\n\"Move over,\" Morgan said. Badri complied, and Morgan sat behind the wheel.\n\nThe windshield cracked. Sniper bullet. Morgan glanced at Badri, saw that he was unhurt, and gunned the accelerator. They were an easy target.\n\nMorgan saw prisoners pouring out of the barracks now. He passed them in the jeep.\n\n\"How are we going to get through the gate?\" Badri demanded.\n\nA group of guards who had seen which way the wind was blowing had opened a pedestrian gate to the outside. It was too narrow for the jeep, but the gate was held up by thin, hollow steel poles, surrounded by chain link. Unlike the main gate, this he had a hope of ramming.\n\nMorgan aligned the car with the gate. Guards scattered.\n\n\"Hold on to your butt.\"\n\nThey hit the gate full force. The windshield and driver's side window shattered, but the gate gave way, and with a terrible scraping sound the vehicle made it through.\n\nThe jeep skidded, and Morgan struggled until it steadied. He veered a sharp right, and within a few seconds they hit the road.\n\nThe wind blowing hard in their faces, Morgan and Badri looked at each other and laughed with the exhilaration of freedom. They took off into the night, toward their rendezvous point, as the prison behind them popped with gunfire.\nChapter Thirty-three\n\nMorgan drove off the road after about an hour in the car. They were one hundred miles away, near the foot of the mountains. The terrain there became rocky and uneven. He had to move slowly. They might easily hit a rock and get stuck, and that might well be a death sentence for them out here.\n\n\"The light signal came from somewhere around here,\" Badri said.\n\nMorgan looked back on the road behind them, lit by the predawn glow. The prison lay in the distance, a thin line of smoke rising from it. He wondered what the prisoners were burning.\n\nHe shuddered, trying not to think about the fate of the guards.\n\n\"Stop the car,\" Badri said.\n\nMorgan parked, looking for a sign of what Badri might have seen. It took a few seconds for them to reveal themselves. Three men, Arabs with trim beards, swaddled in winter clothes. In their arms were Kalashnikov automatic rifles.\n\nMorgan turned off the headlights and Badri opened the door to the jeep to show himself. The men lowered their weapons. Badri walked toward them like the prodigal son and embraced them one by one, kissing their cheeks.\n\nMorgan emerged from the car. The three men got one look at him and pointed their rifles at him.\n\nMorgan was getting really goddamn tired of staring down the barrels of guns.\n\nOne of them asked Badri a question in harsh Arabic.\n\nMoment of truth. Either they would accept him or they would kill him.\n\n\"He is a friend,\" said Badri in English, for Morgan's benefit. \"He helped me escape.\"\n\nThey did not lower their guns. \"He is American,\" one said. \"He is the enemy.\"\n\nMorgan took a step away from the car, toward them. They tensed their grips on their weapons. \"I have no more love for my country. They are murderers without honor.\"\n\nThey talked among themselves, and finally Badri yelled at them. The boss, laying down the law.\n\n\"Come on,\" said Badri. \"They have an airplane hidden on the other side of the mountains. They're going to get us out of here.\"\nChapter Thirty-four\n\nAlex woke up to a slap in the face. She opened her eyes, woozy, trying to get her bearings, and realized she was in the trunk of a car. It was open, and above her, silhouetted against the bright blue sky, stood a man. The taxi driver. A man some ten years older than her, green eyed, with black hair cropped short. He was a man she might have called handsome, in different circumstances.\n\nShe tried to get up out of the trunk and found she'd been tied up. She tried to wriggle free and her arm chafed against the scratchy felt interior.\n\n\"Who are you?\" she demanded.\n\n\"No one important,\" he said. Russian, definitely, but his English was more than passable.\n\n\"You work for Suvorov?\"\n\n\"You have other people in your life who would abduct you off the streets?\"\n\n\"Real wise guy.\" She shifted her weight, trying to find a more comfortable position. She looked out, trying to get a feel for their surroundings. She could see trees, evergreens, but no buildings. \"What happens now?\"\n\n\"What happens is, I don't know,\" he said, throwing up his hands. \"They tell me to take you, so I do, and then I find out Suvorov is dead\u2014\"\n\n\"He's dead?\"\n\n\"\u2014and now I have a girl in my trunk I have no use for.\"\n\n\"Then let me go!\"\n\nHe smirked. \"It's not that easy.\"\n\n\"I have money!\"\n\n\"I took your money already. Come.\" He took her by the armpits and shifted her weight, getting a stable hold on her. \"Don't move too much. The only reason I did not kill you right there in the trunk is that I hate moving dead weight. That and the blood. You know what blood smells when it stays in the trunk in the hot sun? I know. It is terrible.\"\n\nHe pulled her up and out of the trunk. She swung her feet, trying to hit him in the groin and missing. He released her, and she fell to the hard-packed dirt of the ground. She shook off her daze and took in her surroundings. She saw a barn some two hundred feet away that looked as if it hadn't been used in a long time. Back the way they came was a dirt road, overgrown grass intruding upon it from both sides. It was deserted for about a mile, where it bent out of sight.\n\n\"You had to make it hard, didn't you?\" he said. \"You could have had a little more time, it could have been pleasant. Maybe we have a little conversation, some last words for you. But no. You had to be a bitch.\"\n\nAlex squirmed, trying to move into a position where she could see more than his feet.\n\n\"Now I have to kill you here and haul you all the way to hurk\u2014\"\n\nA thump as the man's body hit the ground next to her. His face was inches from hers, a blank expression on his face. Blood was pooling around him, and sticking out of the back of his head was a massive meat cleaver. And she saw legs in leather boots, moving toward her and then behind her, and then a man's rough hands holding her wrists and the scritch scritch of a serrated knife, and her ties were loose.\n\nWith her arms free, she turned her body to look up and see Dobrynin unbinding her feet.\n\n\"Come on, girl,\" he said in his weary, guttural voice. \"Let us get you home.\"\n\nHe drew out the meat cleaver, cursing as he shook off the blood, and then wiped it off with a handkerchief. He made a face and cast the handkerchief aside. Alex wondered whether he'd use the knife at the shop again. She felt certain the answer was yes.\n\nAlex stood, rubbing her wrists to get rid of the soreness.\n\n\"Come,\" he said, motioning for her to follow him. \"Car is down the road.\"\n\n\"How did you find me?\" she asked as she caught up to him.\n\n\"I called friend, told him stupid girl was going to get in trouble. He watched you. He saw you taken and followed the car. And he called me. So I am here.\"\n\nShe moved ahead of him and stopped him with two hands. And then she gave him a tight hug, resting the side of her head against his bloody shirt. She knew he was mortified. But she didn't care.\n\n\"Thank you,\" she said.\n\n\"You want to thank me? Go home.\"\nChapter Thirty-five\n\nThey flew low until they cleared any reasonable search radius. The men ignored Morgan and spoke quietly to Badri. Morgan instead looked out the window on the Russian countryside, gaming out what his next steps would be.\n\nWith the riot at the camp, there would be chaos for a long time even after the Russian authorities managed to regain control from the prisoners. Hell, with weapons, they were practically in a fortress, and would be able to hold out for days, unless the Russians chose to bomb the place to oblivion, in which case they'd probably never be done sorting the body parts and would never find out there was anyone missing at all, except for a jeep, abandoned at the foot of a nearby mountain.\n\nIn any case, nobody would know to look for these two missing prisoners for a long while, which gave them plenty of time to get the hell out of Dodge.\n\nThey switched planes in an airfield a few hours out, then crossed the border to Kazakhstan in the back of a truck, squeezed in between bales of hay, daylight filtering through only in tiny pinpoints. And then Morgan slept.\n\nWhen he woke up, to the sound of the truck's driver's side door opening, those pinpoints of light had grown dark. Morgan heard the tailgate being opened, and the bales of hay at the back were removed one by one.\n\nThey emerged into a dirt yard of a house surrounded by high walls on all sides. The sky was clear, and Morgan figured they must be in a small town, because he could see the stars with a clarity that's impossible near a city.\n\nA man standing at the door embraced Badri and watched Morgan with suspicion. They exchanged some angry words, of which Badri had the last, and he called Morgan to come in.\n\nIn the small, sparse dining room, the rich smell of meat wafted through the air. They ate a meal of lamb and flatbread, which, after weeks of almost nothing but spoiled potatoes and onions, seemed like the best thing he'd ever tasted. After they finished, Morgan wiping the last of the sauce off his plate and popping it into his mouth, a woman in a niqab served them black tea. Morgan said, \"So what happens now?\"\n\nThe Arab grimaced. \"I don't know.\"\n\n\"I didn't really ever think we'd get this far. You dream of freedom for so long, of getting out, and now that I'm here, it's not . . . I don't have a life to go back to anymore, Badri. Even if I did go back, if they find out I helped you escape, or that you helped me\u2014well, I can tell you they'll put me somewhere that's not any better than where we just got out of.\" He sat back in his chair. \"Not that I'd want to. They abandoned me. Screwed me. So screw them.\"\n\n\"What about your family?\"\n\n\"Better that they think I'm dead. They got a good payout from it.\" He drank tea from his cup, hot and bitter. \"Won't do any good to anyone, my coming back.\"\n\n\"And your future? I could take you somewhere. Drop you off. We never have to see each other again. I may even be able to give you some money. You saved my life and won me my freedom. I owe you that much.\"\n\nMorgan closed his eyes and took a deep breath. He found Badri looking at him expectantly. \"What about if I help you?\"\n\nA heavy silence hung between them. Morgan studied Badri's face. He was looking off to the side, avoiding eye contact. \"You are not even Muslim,\" he said. \"Why\u2014\"\n\n\"\u2014would I want to join the effort? I got screwed, Badri. I got no home, no family, nothing to live for. The only thing I got is this rage at the people who did this to me.\"\n\n\"I am sorry,\" said Badri. \"I am grateful to you. But we cannot allow\u2014\"\n\n\"I can offer you something no one else can,\" Morgan said. \"Information. Insight. On the inside, you told me you had something planned. Something big. I can help you with that. I know the vulnerabilities. I know how to cause damage.\"\n\n\"I don't know,\" said Badri.\n\n\"You don't trust me.\"\n\n\"I have put my life in your hands,\" said Badri. \"In there. Out here, things are different. We are not joined by a common purpose.\"\n\n\"I'm telling you we are.\"\n\n\"No,\" said Badri. \"Tomorrow, we have a long drive into Uzbekistan. But for tonight, we rest. It has been a long road.\"\nChapter Thirty-six\n\nThey arrived in Tashkent in Uzbekistan in the early morning, a driver taking the wheel and Morgan and Badri riding in the backseat. The car had no air-conditioning, and the ride under the Central Asian sun turned it into a hotbox.\n\nBadri talked to Morgan like they were old friends, telling him about his childhood in Abu Dhabi, prosperous but not rich, and about his education in London. His family was made up of devout Muslims, but not radicals. His eyes were opened, he said, after he came back from college. It was not only the wars of imperialism, the meddling of the United States in the lives of Muslims, the deaths wreaked by bombs and soldiers and drones. He saw what Westernization was doing to his country, to all of the Arab world. Changing it, eroding its character.\n\nMorgan listened in spite of the heat as Badri told him about getting married and the joy of having children. And he spoke about adopting the radical fundamentalist faith of the Wahhabists and its return to a true Islam.\n\n\"The entire world is becoming decadent,\" he said. \"They need a wake-up call. And talk cannot bring the change that is needed. That is why I fight. That is why we must kill.\"\n\nThey entered the city of Tashkent in the middle of the afternoon. Like most ancient cities, it was marked by the juxtaposition of the old and the new. The city was sprinkled with old Soviet monuments, and many of the streets were lined with trees.\n\nThe driver pulled into a parking lot adjoining a large sky-blue dome. Chorsu Bazaar, their destination, was where they were to meet Badri's contact, who would send the terrorist along to his mission and arrange for Morgan to be transported where he wished.\n\n\"It's a pretty goddamn open space,\" said Morgan. \"Lots of people, too.\"\n\n\"My contact insisted we meet here,\" Badri said. \"He wanted a public place. He is wary of you.\"\n\nMorgan and Badri walked together into the bazaar, a round area under the dome, arranged in concentric circles of wares. The butchers were along the outer rim in enclosed refrigerated shops. The floor held a wealth of foodstuffs, nuts and dried fruit and spices, pungent and rich, ranging from the vivid yellow of turmeric through the red of paprika to the black of pepper.\n\nMorgan took in the space with trained eyes as they rounded the bazaar, Badri looking for his contact.\n\nAfter they'd completed a full circle, Morgan leaned close to Badri. \"We're being followed,\" he whispered.\n\nBadri's eyes widened, but then returned to a neutral state. He was well practiced, Morgan saw, and knew how to keep his cool in a dangerous situation.\n\n\"Look, but use your peripheral vision. Pretend we're looking for our contact. Guy over on the other aisle holding the briefcase. Guy looking at meats over at the butcher. Man leaning against the wall near the bathroom.\"\n\n\"We are surrounded,\" said Badri.\n\n\"It appears so.\"\n\n\"They do not look Uzbek. Nor Russian, I believe.\"\n\n\"Might be American,\" Morgan said. \"Doesn't really matter.\"\n\n\"What does?\"\n\n\"Getting out of here. Keep your eyes out and follow my lead.\"\n\nThey continued their way around, the men following them moving along to maintain a line of sight. The nearest one, now examining a table of nuts, began his approach.\n\n\"They're making their move,\" said Morgan. \"Wait for my mark.\"\n\n\"What will that be?\"\n\n\"You can't miss it.\"\n\nThe man was close now. He was wearing a baggy Hawaiian shirt, open over an undershirt, that covered a concealed-carry shoulder holster.\n\nMorgan bided his time, waiting as the man approached from the side. When he was five feet away, Morgan turned and hit him with a head butt. As the man staggered back, Morgan reached into his shirt and drew out his gun. He turned to see the others closing in from the periphery of the domed market, three by his count.\n\nMorgan aimed at the chest of the nearest one. A woman spotted his gun and screamed.\n\nMorgan aimed down. He fired twice into a row of bags of spice.\n\nA plume of red and yellow shot into the air. People were coughing. The man covered his eyes, hollering in pain.\n\nAnd people were running, panicked, away from the gunshot.\n\n\"Run!\"\n\nKeeping low, they ran under the cover of the spices and blended into the crowd. Morgan jumped over a stall, overturning a bucket of pistachios. Badri was at his heels. Morgan turned to look at their pursuers. At least one had spotted them, but the rest were running around trying to navigate the crowd.\n\nThere would be more waiting for them outside.\n\nMorgan and Badri ran out into the hot sunlight along with the rest of the crowd. Morgan hid the gun in his waistband. Blending in might give them a few precious seconds.\n\nHe saw them\u2014men dressed in various different guises as tourists, standing at the edges of the court that surrounded the bazaar, looking through the crowd.\n\nNo way out but through it.\n\nHe ran with the crowd as long as they were not spotted. And then the nearest agent looked into his eyes, and Morgan saw the expression of recognition.\n\nHe lowered his torso and hit the man full force, lifting him clear off the ground before sending him sailing on his back. He took off running with desperate speed, Badri behind him, as the other agents moved toward them. But they had broken the cordon, and now the agents were behind them.\n\nMorgan and Badri were in the parking lot now. Morgan spotted a man getting into a Daewoo sedan. He pulled out the gun and pointed it at the driver.\n\n\"Out!\"\n\nThe man stepped away from the car, his hands up, holding his keys. Morgan grabbed them and jumped into the driver's seat.\n\n\"Get in!\"\n\nBadri got into the passenger seat and Morgan gunned the reverse before his door was closed. He cleared the parking space and put the car in first, accelerating, tires screeching, as their pursuers drew their weapons and readied to shoot. They accelerated away as gunshots hit the trunk. One bullet shattered the rear window.\n\nMorgan turned at the end of the row and saw more trouble ahead. The exit to the parking lot was blocked by two cars. There were more agents there, guns already drawn and trained on their car.\n\nHe turned with squealing tires into the next row and gamed the situation out in his head. The exit was blocked, and the lot was separated from the street by a low wall that was probably too solid to ram.\n\nThat left one possibility.\n\nHe turned the car back toward the bazaar.\n\n\"What are you doing?\" Badri yelled.\n\n\"Getting us out of here!\"\n\nHe drove onto the pedestrian walkways toward the dome of the Chorsu Bazaar. The pedestrians had cleared it by now, and the door was wide, more than enough for the car to pass. Just one obstacle stood between them\u2014a staircase, some twelve or fifteen short steps.\n\nThe car wasn't going to like it. But it would have to do. The Daewoo lurched as they hit the steps, and then climbed, heaving, until they were level with the bazaar.\n\nMorgan felt the tires low, blown out by the impact. And then they crashed through the doors of the bazaar, sending glass flying in every direction. Morgan maneuvered around the space, deserted of people, sending up clouds of spices and rain of nuts and dried fruit.\n\nThey crashed through the door on the other side, bulldozing through an aisle of the covered vegetable market. No one waiting for them there. No one thought they'd be crazy enough to make the maneuver.\n\nThey came out amid a crowd of screaming people, who parted for them to pass. And then Morgan drove the car onto the street, with the two front wheels scraping the ground and the windshield cracked from end to end. But it was whole enough to carry them away from the bazaar, and from their pursuers.\nChapter Thirty-seven\n\n\"How did they find us?\" said Badri in a fevered panic.\n\nMorgan was driving the car away from the populated, tourist-thick area of the bazaar toward the outskirts of the city. They needed to ditch the car soon. They were attracting attention, with its front tires in rags and the grille and front bumber scratched and bent.\n\n\"They must have made your contact.\" Morgan's heart was still pounding, adrenaline still coursing through his veins, but he was calm. This was just another problem to be solved. \"We need to get out of town. But we need a new car first. This car is going to get stopped as soon as a policeman sees us.\"\n\nMorgan turned into a side street and parked the car. They got out on a street that bordered a series of office buildings, all concrete and reflective glass. Morgan took a tire iron from the trunk and led the way to the nearest parking lot. He checked that they were not being seen and shattered the driver's side window of a Chevy Cobalt. He motioned for Badri to get inside.\n\n\"We need somewhere to go,\" Morgan said as he pulled open the panel under the steering column and reached in for the wires.\n\n\"I have more people I can call. It is a greater risk. But it can be done.\"\n\nMorgan touched the ignition wire to the power wire, and the engine came to life. He maneuvered the car out of its spot and drove through the parking lot and onto the street, wind blowing in through the broken window.\n\n\"They were American, weren't they?\" said Badri.\n\n\"The agents? Yeah. I think so. Something about the way they carried themselves, the way they held their guns.\"\n\n\"You could have given me up, turned yourself over. They probably would've taken you back. You could've given them a story about using me to escape. Wouldn't even be too far from the truth. But instead you risked your life to escape with me.\"\n\n\"I told you,\" said Morgan. \"I'm done with them. All of 'em.\"\n\n\"I think I didn't quite believe you until now.\" Badri looked out at the industrial suburbs of the city. \"Okay,\" he said. \"If you wish to join with us, I will stand by you.\"\n\nMorgan turned to look at him to make sure he was serious. \"You mean it?\"\n\n\"Yes. I mean it. And I will tell you about this. I have something. We have had it since before I was captured. I will not tell you what, not yet, but I will tell you that it can kill thousands. Tens of thousands.\"\n\n\"I see. And this weapon\u2014how will you get it into the United States?\"\n\n\"It is already there,\" said Badri. \"With some allies from my organization. But my people did not know how to deploy it effectively.\"\n\n\"But now that you're out, you can guide them.\"\n\n\"This can be a new 9\/11,\" Badri said. His eyes shone with the fire of fanaticism.\n\nMorgan's voice lowered to hardly more than a whisper. \"You want to make this a new 9\/11. I want to help you make this something greater. I can give you the means to cripple the US government in a way that they will not be able to recover from for decades.\"\n\nBadri smiled. \"Welcome to the cause.\"\nChapter Thirty-eight\n\nAlex returned to an empty house in Andover, Massachusetts.\n\nNight had fallen, too late for any kids to be out in the now silent streets. She got out of the cab and unlocked the front door, dropping her bag in the foyer and making straight for the kitchen. She opened the refrigerator, but there was no food, just a few condiments on the door, a half stick of butter and a little bit of cranberry juice in a container. Her mother was off on Martha's Vineyard consulting on the house of a client who was summering on the island. Alex figured she must've cleared out the fridge before she went.\n\nSo very practical, her mother.\n\nShe checked the freezer. In there Alex found a Tupperware of frozen ravioli with a note.\n\nWelcome home, honey. Forty minutes in oven at 350 degrees. Sauce in the pantry.\n\nYes. Quite practical. Alex smirked in appreciation.\n\nShe set the ravioli in a porcelain dish and poured in half a jar of her mother's homemade tomato sauce, made and canned for an entire year at a time. Then she set the oven to 350, set a timer for fifty minutes, and put the dish in without preheating.\n\nLeaving the oven to do its work, she went to the living room, flopped on the couch, and turned on the TV. This she hadn't done in a long time, but it was so easy to fall into old habits when she was at home. Especially when she was feeling as forlorn and defeated as she was then.\n\nShe clicked through the channels and settled on infomercials. She couldn't concentrate on anything anyway. The chipper chatter at least served as appropriate background noise for her wallowing.\n\nShe pulled out her phone and looked at the messages that had accumulated in her absence. Loads of junk email, a couple hellos from her friends Simon and Katie from her brief stint in college\u2014Katie's more insistent and offended by her lack of response\u2014and one from Karen O'Neal, some canned words of concern from a couple of weeks ago.\n\nAlex checked the clock. Just past eleven at night. On impulse, Alex called her. The phone rang only once before she picked up.\n\n\"Alex?\" She sounded awake, at least.\n\n\"Hi, Karen. I hope it's not too late to call.\"\n\n\"Please. Sleep doesn't even cross my mind before midnight. What's up? Are you okay?\"\n\n\"I, uh, I've been out of town. I was wondering if there was any news about my father.\"\n\nThe question hung on the line. Crickets chirped in the hot night.\n\n\"I'm sorry,\" Karen said finally. \"We haven't had any luck finding him.\"\n\n\"Right. Yeah.\"\n\n\"Do you want to grab coffee sometime this week?\"\n\n\"Yeah. Sure. I'll text you.\"\n\nShe threw her phone across the living room to land on the couch and groaned. The absence of her father was a keen, stabbing pain. The past weeks had kept her busy, on track, doing something. Now that she stopped, all that she had left was the reality that he wasn't there and wasn't coming back.\n\nWhat was life going to be without him? What was her life going to be at all? She felt like this changed everything.\n\nShe had done depression, and didn't care for a repeat. Would she go back to training? Follow in his footsteps? Would that bring her some sense of fulfillment, keep him alive in her somehow?\n\nThen there was revenge. That might keep her going. But she'd read enough to know where that led. Death or disappointment.\n\nBut what else was there for her? How was she going to deal with life now?\n\nThe same way she'd dealt with the rigors of training, she told herself. One day at a time. One minute at a time, when necessary. Just the next push-up, the next ten feet of running, the next five minutes awake. This was just like that. Do what you have to do not to die right now.\n\nSo she wept, not caring how loud she was in the empty house, and buried her head in a pillow, screaming, and punching the couch cushions. She didn't know how long she was at it, but when the oven dinged, announcing that her ravioli was ready, she stopped, panting, drying her eyes, and stood up. One day at a time started right now.\n\nAlex picked herself up off the couch. The smell of basil and tomato wafted to her nose, the comforting fragrance of her mother's cooking.\n\nA warm meal at home. Seemed like a fine first step.\nChapter Thirty-nine\n\nMorgan and Badri landed in Puerto Plata, Dominican Republic, in the late morning, bearing fake passports. The bored-looking immigration agent didn't give them a second glance as she entered their data on her computer.\n\nThey picked up a rental Kia hatchback at the airport that they had no intention of ever returning and drove west together, along the coastal highways under the noontime tropical sun. Morgan's hair, half an inch grown in, flowed like a field of wheat in the wind streaming in through the car window. On their right, the sea would sometimes peek through the dense jungle.\n\nOnce they had driven for two and a half hours, Morgan checked their position on the handheld GPS. He parked the Kia on the shoulder of the road and plunged into the woods, through the dense underbrush, slapping at the mosquitoes that harried them the whole way.\n\nThey trekked for hours under the shade of the green canopy, Morgan correcting their course using the GPS every few minutes.\n\n\"The jungle turns you around,\" he told Badri. \"Stop paying attention for a minute and you're lost.\"\n\nThe sun was low in the horizon by the time they reached the shore, where the rickety wooden fishing boat was already waiting for them. On it was a weather-worn Dominican man, skin browned by the sun, shirtless and in ragged shorts, and a younger man in a wife-beater shirt and faded baseball cap. He tensed when he saw them at first, looking with especial wariness at Morgan, before Badri spoke.\n\n\"Se\u00f1or Batista?\"\n\n\"S\u00ed,\" he said, nodding his head and grinning.\n\nBatista waved them on board. Morgan took stock of the boat as he stepped onto the deck. The wood looked more deteriorated than Morgan liked for any boat going on the open sea, and everything from the hull to the cabin seemed cobbled together and patched over many times.\n\nThey set off without ceremony, leaving the island of Hispaniola behind and moving north. Morgan pulled on a Panama hat he'd bought at the airport to protect against the sun.\n\nThe two men stared at him openly as the boat chugged along, and Morgan understood very well why. He was not the kind of person they would normally carry across, not the kind of person who'd normally enter the country illegally. But an explanation was not forthcoming. Morgan was a paying customer, so he didn't have to explain a goddamn thing.\n\nMorgan and Badri had brought along cereal bars to tide them through the passage, but the younger man, Diego, who Morgan learned was Batista's son, had caught some fish earlier that day. They grilled it on deck in a makeshift barbecue pit made out of a paint can and what might have been part of a dish rack. Morgan ate with relish, with his hands, the fish resting on a piece of banana leaf.\n\nThe hold was full of fuel canisters to get the boat through the long journey and back, so they had to sleep on deck, out in the open. Which suited Morgan fine enough\u2014out there, they had at least a light breeze to combat the stifling heat.\n\nThey spoke little during the journey. Before, driving through Uzbekistan, Badri would talk freely about his family and his life outside the organization, and even a few stories of terrorism, of narrow escapes from the CIA, and his capture by the Russians. But now, the weight of their mission was bearing down on them, and they couldn't bring themselves to say anything about anything else. They did not even discuss the ultimate mission itself, instead focusing on the details of their passage, and then only so far as it was necessary. Mostly, Morgan looked out into the dark water as it disappeared far in the horizon, or in the dark of the night, lit by stars and a sliver of moon.\n\nIt took a day and two nights to come within sight of the shore, and then they spent a day of waiting because they couldn't approach in daylight. So they waited, the boat bobbing on the gentle sea, just far enough from the continent that they would not have to worry about the American Coast Guard finding them and either arresting them or escorting them back to their place of origin.\n\nThe boat started moving again at dusk, all lights off to avoid detection. They landed in the dead of night at a deserted bayou beach in Louisiana, where an ally of Badri's was waiting with an inconspicuous seven-year-old VW Jetta.\nChapter Forty\n\nMorgan's confederate drove them into the night and past dawn, stopping only to refuel. The driver was a taciturn, grave man of few words, so they rode in silence almost the entire way, until they pulled into a country house in rural Virginia. There they were greeted by a soft-spoken young man with a shaved face. He introduced himself as Rasheed. Like everyone else along the way, he regarded Morgan with nothing short of total suspicion.\n\n\"He is an ally,\" Badri said.\n\nHe wanted to get them situated, but Badri was impatient. \"Where are they?\"\n\nThe young man, deferential, led the way down a solid old wooden staircase to the basement. Among the abandoned appliances, tools, and building materials, he walked to a working chest freezer and pulled it open. The top was layered with meats in grocery store trays and frozen vegetables. He removed them, setting them aside, to reveal a steel box underneath.\n\n\"Here they are,\" Rasheed said. He opened the box without removing it from the freezer to reveal six plain steel canisters, about as big as tennis-ball containers. Badri looked at them with ravenous eyes, filled with anticipation of the destruction that they would cause.\n\n\"Novichok,\" he said, his voice a whisper. \"Russian nerve agent. The deadliest ever made. Not a gas, but a powder. Causes all muscles in the body to seize and contract. You cannot breathe, and your heart stops pumping blood. You die within minutes. Perhaps seconds.\"\n\nHe drew out a canister and held it out for Morgan to see. \"Here is our instrument of destruction,\" he said. \"Here is how we will bring America to its knees.\"\nChapter Forty-one\n\nThe farmhouse they occupied was old and two-storied, built of horizontal slats that were once whitewashed but now showed the faded gray of the wood underneath. The grounds were modest, but the house was shielded from view of its surroundings by trees. They were a few miles from Palmyra, the nearest one-horse town.\n\nIt was a spot well chosen for its isolation. You could spend months in here without anyone so much as sighting the house from the road.\n\nMorgan heard the Chevy beater truck before he saw it coming up the driveway, carrying Rasheed. He got out of the car holding bags from the thrift store in one hand and Burger King in the other. Badri was out of the house, in the nearby woods, making a call to his associates in the organization.\n\nDevout as he was, Rasheed was a slob. From the looks of it, he subsisted on mostly snack foods, and empty bags of chips and fast food were strewn about the living room, which held nothing but one ratty couch. It was some kind of warped jihadi version of a filthy bachelor pad.\n\nRasheed slammed the screen door open and set the bags down on the Formica table in the dining room. He greeted Morgan with undisguised disdain.\n\n\"Lunch,\" he said, tossing a paper-wrapped Whopper into Morgan's lap.\n\nNot exactly his favorite\u2014Morgan hadn't eaten fast food in years, at least not voluntarily\u2014but he was famished. Even with Diego's fishing, food on the boat had been slim pickings. He unwrapped it and took a large bite out of it.\n\n\"I brought clothes,\" Rasheed said, dropping a thrift store bag at Morgan's feet. Eating the burger with one hand, Morgan sifted through the clothes with the other. Rasheed had an impeccable sense of style. Among the pickings were a shirt from a company barbecue so big that it wouldn't look out of place resting on a tent pole and a lime green button-down. But the clothes on Morgan's back were starting to get ripe, and this was what he had.\n\nMorgan looked up from the bag to find Rasheed staring at him.\n\n\"I do not know what you did to fool Badri,\" he said. \"But I do not believe you are here to help.\"\n\nMorgan stood to confront him, chest out, hands balled into fists. \"Badri believes me. And he's your boss. So I don't much care what you think.\"\n\n\"Infidel!\" Rasheed spat on the floor.\n\n\"And what are you gonna do about it?\"\n\nRasheed narrowed his eyes, and then drew a gun from his waistband. Magnum Big Frame revolver, .44 Magnum. Talking about stopping power would be damning with faint praise. At this range, it would be a question of how much of Morgan's face would remain afterward.\n\n\"Big man with a big gun,\" said Morgan. \"Let's game this out. You blow me away. Then what? What do you tell your boss?\"\n\n\"It does not matter. You will be dead, and no longer willing to betray us.\"\n\nMorgan's voice dropped an octave. \"He trusts me. I've saved his life more than once, so maybe he trusts me more than he trusts you. And he wants what I have to offer. Kill me, and your punishment will come.\"\n\n\"Not as swiftly as yours,\" Rasheed said. But Morgan could tell he was shaken.\n\n\"Your plan will fail without me,\" Morgan said. \"They'll catch you, and best-case scenario is that you die in the process. You won't have another 9\/11 on your hands. You'll have another underwear bomber.\"\n\nRasheed gritted his teeth and grimaced. \"You are lying.\"\n\n\"Then pay the price and kill me.\"\n\nRasheed cocked the gun. Morgan got ready to make a grab for it. Rasheed was brave but untrained. Morgan might be able to get the best of him.\n\n\"Rasheed!\"\n\nIt was Badri, standing outside the screen door. He walked into the house speaking in harsh Arabic, and Rasheed, hissing, uncocked the gun and tucked it back into his waistband.\n\n\"What happened?\" Badri demanded, in English.\n\n\"Friendly disagreement,\" said Morgan, turning away and heading for the stairs. \"I'm going to take a shower. I have some clothes I can't wait to try on.\"\nChapter Forty-two\n\nMorgan locked his door that night and braced the handle with an old wooden chair. He then set the moldy mattress on the floor against the wall. He wasn't going to risk Rasheed getting any ideas in the middle of the night.\n\nHe woke up with the cock's crow, but stayed in his room, stretching and doing his morning exercise. An hour later, he unblocked the door and walked out. The house was still, the other two still sleeping.\n\nMorgan went downstairs and looked for something to eat for breakfast, but all he found were potato chips and packaged cake, so he settled for a glass of tap water instead.\n\nCarrying the cup in his hand, Morgan walked to the basement door. He looked down, at the freezer, thinking of all the destructive power it contained. He felt chills as he considered it.\n\n\"See something you like, American?\"\n\nIt was Rasheed, from the top of the stairs.\n\n\"Just thinking about the plan,\" said Morgan. \"We're going to have to devise a strategy sometime soon.\"\n\n\"We will set them off in the White House,\" Rasheed said. \"And we will kill the President of the United States.\"\n\nMorgan chuckled at the notion. \"You won't make it ten feet into the White House. You'll be shot down in the entrance hall. The best you can hope to achieve there is to kill a handful of tourists and security guards.\"\n\n\"Then you have a better idea?\"\n\nThey were interrupted by the sound of Badri coming downstairs.\n\n\"Morning,\" said Morgan. \"I was just about to tell Rasheed here that we need groceries and supplies. Proper food. Eating all this junk is expensive, too. We shouldn't be wasting the little money we have.\"\n\nRasheed leered at him, but Badri cut in. \"What do you suggest?\"\n\n\"Some beef, vegetables. The electric stove works, I checked, and there are a couple of iron pots we could use. No reason we have to live like animals here.\"\n\nRasheed opened his mouth to speak, but Badri cut in. \"You are right. Make a list. Rasheed will get it for us.\"\n\n\"Don't you think it'd be better for me to go?\" said Morgan. \"I'm American. Nobody's going to think anything of my being there.\"\n\n\"Fine,\" said Badri.\n\n\"He will take our car and escape!\" Rasheed bellowed. \"He will send the police after us.\"\n\n\"He will not,\" Badri said with finality. Then he turned to Morgan. \"I will get you the money.\"\n\nBadri sent him out with twenty dollars. Morgan drove into Palmyra and parked outside the town's one modest grocery store. At the single register was a bored teen who didn't pry his eyes away from his phone for one second when Morgan came in.\n\nHe scoped the store, looking for the cameras. There were none. He walked to the refrigerator and picked up a tray of beef and another of chicken thighs. He looked back at the teen, who spared him not so much as a glance, and then tucked them in his waistband, one in front and one in the back, well covered by his loose T-shirt. He winced as the cold came into contact with his skin.\n\nHe heard the door swing open, and turned to see a policeman walk inside. Morgan went stiff, then forced himself back to relaxation.\n\n\"Mornin',\" said the cop. He wore a white Stetson, and had eyes that were constantly narrowed.\n\n\"Mornin',\" Morgan said, picking up a head of broccoli and some potatoes.\n\n\"Sheriff Anderson,\" he said, lifting his hat.\n\n\"Dan. Dan Morgan.\"\n\n\"Just passing through?\"\n\n\"That's right.\" Morgan took his groceries to the cashier, wondering whether the sheriff had noticed the bulges under his shirt.\n\nBut instead the sheriff looked outside. \"Say, I've seen that car before.\" The sheriff pointed at the truck Morgan had driven in, parked outside the door. \"You out there at the old Peterson place?\"\n\n\"Four seventy-one,\" said the listless teen. Morgan dug into his pocket for the money, careful not to disturb the meat.\n\n\"I have no idea,\" Morgan said. \"I'm staying with a friend.\"\n\n\"Young Middle Eastern fella?\"\n\n\"The same,\" Morgan said. \"Rasheed. A friend's nephew, actually. House-sitting for his uncle. He's worried about the boy, asked me to come down and have a look-see. Find out if he's all right. But I'm really just dropping in for a day or two.\"\n\n\"He looks a little perturbed,\" Sheriff Anderson said.\n\n\"He's got some issues. You know.\" He made a couple of circles with his index finger around his ear. \"But he's a good kid. Harmless.\"\n\n\"Well. That's good to know.\"\n\n\"And he seems okay, all things considered. So I'm eager to tell his father that.\"\n\nSheriff Anderson frowned. \"Wasn't it his uncle?\"\n\nMorgan cursed in his mind. \"I know them both. His father is worried about him, too.\" Goddamn it. He was better than this.\n\n\"Well, you do that,\" Anderson said.\n\nThe cashier handed Morgan his change. \"Here you go, sir.\"\n\nMorgan walked out and started his truck. When he was out of sight of the store, he pulled the meat out from under his shirt and tossed it in the passenger seat.\n\nHe had one more stop to make.\n\nHe cruised the town until he found a mom-and-pop electronics store, Bob's TV and Radio. He went inside to find a balding middle-aged man he figured was Bob standing behind the counter. It was a small store, and it took him under a minute to find what he was looking for.\n\n\"I'll take one of these,\" said Morgan, setting the item on the counter.\n\n\"Prepaid cell phone?\"\n\n\"Yeah. My phone got bricked. Need something to tide me over until I can get back home.\"\n\n\"I gotcha,\" said Bob. \"That'll be nineteen ninety nine.\"\n\n\"All I got is fifteen,\" Morgan said, making a show of being flustered. \"Fifteen dollars and . . . twenty-nine cents.\"\n\nBob raised a suspicious eyebrow. \"Is that right?\"\n\n\"Any chance I can get you to do a stranger a solid?\" Please.\n\nBob rubbed his chin between his fingers. \"Well, all right,\" he said. \"Give me what you got. Still makes me a profit, if you can call that a profit. And I'm a good Christian man. I can help a brother out. Let's ring you up.\"\n\nGod bless you, Bob.\n\nMorgan set the bills on the counter, and then dropped the coins. He walked to the truck on trembling legs, turning on the ignition and driving away. He waited until he was in the highway before turning the phone on and making a call.\n\n\"Hello?\" A woman's voice.\n\n\"Bloch? It's me.\"\nChapter Forty-three\n\nWhen Morgan arrived at the house, he hid the phone, still on, under a brick that was resting against the foundation. Then he went inside, grocery bags in hand.\n\n\"I brought a good haul,\" said Morgan, setting them on the table. \"I'll whip us up some steaks on the iron skillet.\"\n\n\"We need to talk about our plan,\" said Badri. \"We cannot stay here long. Every minute is a moment when we could be found out.\"\n\nThey sat down together around the table to discuss the plan. \"Our target,\" Morgan said, \"needs to be symbolic, and it needs to kill a lot of people. There's one day that's better than any other if you want to do this.\"\n\n\"What's that?\" Badri asked.\n\n\"The Fourth of July. The most patriotic of holidays. The parades attract enormous crowds. You can kill thousands upon thousands on the day when Americans get together to celebrate this country. We're two weeks away, which means we can organize attacks on more than one city. New York, Chicago, Washington.\"\n\n\"I like what I hear,\" said Badri.\n\n\"Police won't be able to check everyone. They'll have bomb-sniffing dogs, but we won't need explosives. Those canisters have rapid-release mechanisms that can send up a cloud of the powder in something like a fifty-foot radius. Rasheed, do you have paper?\"\n\nThey spent hours strategizing. Morgan drew the parade routes from memory as best he could, saying that they could check it online as soon as they had a connection, and marked likely spots, places that would cause the most deaths, the most panic, and how to set off a second device to target a wave of fleeing pedestrians.\n\nIt was late afternoon by the time they decided to break, and Morgan left Badri and Rashid to cook their dinner of steak and potatoes. He had long known that to impress people in the kitchen, you only need to do one or two things well, and this was his. The smell of searing meat filled the house. They ate together at the table. After the planning and the meal, even Rasheed seemed to be warming up to him.\n\nAfter dinner, Morgan sat down on the couch and Badri pulled up a chair across from him. Rasheed went outside for a walk.\n\n\"I will be honest,\" said Badri, \"I had my doubts about you, even until today. But your plan\u2014I finally believe you are one of us, without reservation. We will do great things together, Morgan. You will be part of something here that will be remembered for generations, a definite blow to\u2014\"\n\nRasheed burst into the house, slamming the screen door open, gun drawn and pointed at Morgan. In his other hand was a cell phone.\n\n\"What is the meaning of this, Rasheed?\" Badri demanded.\n\n\"When the American arrived, I saw him through the window. I saw him bend down and hide something. So I went out to see what it was. I found this. A phone, Badri. Still on. He is a filthy traitor!\" Rasheed tightened the grip on his pistol.\n\nBadri raised his hand for Rasheed to stop. \"Explain yourself, Morgan.\"\n\n\"I wanted to call my family,\" Morgan said. \"I'm sorry. I know that I shouldn't have. But I couldn't resist. I bought this phone when I went into town. It's a burner. Untraceable. It won't compromise us. Please. I know how to be anonymous.\"\n\n\"He is lying,\" said Rasheed.\n\n\"I'm not,\" said Morgan. \"Badri. I swear I'm not.\"\n\nBadri struggled with this in his mind, and then his expression set, cool and stony. \"I was deluded,\" he said. \"I thought it could not be possible. But now I see that I was a fool to trust an American. Do not move! Rasheed, do not shoot him. We will find out what he knows.\"\n\nBadri took one of the knives from the table, still wet with the juices from the steak.\n\n\"Badri, come on. After all we shared, you think I'd betray you?\"\n\nRasheed laughed. \"You will get what you deserve now, American.\"\n\nBadri approached with the knife in hand. Morgan was seated, with a gun on him. Circumstances were not smiling on him.\n\nBut Badri was interrupted by a car, coming up the driveway.\n\n\"Who is it?\" he demanded.\n\n\"Police!\" Rasheed exclaimed. \"I told you he was a traitor!\"\n\n\"I promise, I did not tell anyone about you!\" Morgan insisted. \"He's probably just checking up on the house.\"\n\n\"He is alone,\" said Rasheed.\n\nBadri frowned. He said to Rasheed, \"Hide behind the front door. Morgan, you will stand there next to him. I will talk to the policeman. If you do anything to call his attention, Morgan, both of you die. Understand? Now come.\"\n\nAt gunpoint, Morgan moved to stand next to the front door, so that he would be hidden when it opened. Rasheed stood next to him, out of sight, gun pointed at his head.\n\n\"Keep your mouth shut,\" Rasheed growled.\n\nThey heard the boots on the porch steps, then there was a knock on the door. Badri opened.\n\n\"Good evening, officer. Can I help you?\"\n\n\"Lots of people up in this old house, ain't there? I met a fella that drove this truck into town this morning, and I figured the young man would still be here.\"\n\n\"They are out,\" Badri said. \"What can I do for you?\"\n\n\"Just checking to see if you folks are all right,\" he said.\n\n\"Just fine,\" said Badri\n\n\"Now, you said they were gone, but I see three plates on the table,\" Anderson said. \"Doesn't look like you've finished eating very long ago. And there's one road out of here, and I didn't see anyone driving past me.\"\n\n\"They are on an after-dinner walk. Now if there is nothing else\u2014\"\n\n\"I'd ask you for a glass of water, if it's acceptable to you. It's a hot day, and I'm parched.\"\n\nBadri narrowed his eyes. \"Of course. Please, come in.\"\n\nRasheed turned his gun to fire at Anderson as he walked in.\n\n\"No!\" Morgan cried as he brought Rasheed down with a full-body tackle, sending the gun sliding across the floor. Sheriff Anderson went for his sidearm, but Badri swung the knife and buried it in the policeman's throat. Anderson fell on the floor, gasping. Badri took up his Colt revolver and pointed it at Morgan, who held up his hands. Rasheed stood and grabbed his own gun from the floor where it lay.\n\n\"We need to go,\" said Rasheed. \"They will be looking for the policeman.\"\n\n\"Yes,\" said Badri, \"we do.\"\n\n\"Then we need to take care of this filth right here,\" Rasheed said, meaning Morgan.\n\n\"Yes. We do. Please, Rasheed. Do the honors.\"\n\nRasheed grinned, triumphant. \"Any last words?\"\n\nThe sound of broken glass, and Rasheed's chest burst in a mist of blood.\n\nBadri scarcely had time to react before a bullet hit him in the shoulder. He fell, dropping his gun with the force and shock of it. Then he looked at Morgan with anger.\n\n\"What did you do?\" Badri demanded. \"You traitor, what did you do?\"\n\n\"What I had to do to protect my country.\" He heard the sound of men approaching, rustling in the foliage surrounding the house. \"I want the end of this fight, Badri. I want my people to be safe.\"\n\n\"What about my people?\"\n\n\"Do you think you're making them safer by attacking the US? By extending this goddamned war?\"\n\n\"I do what I do for the righteousness of God.\"\n\nBadri scrambled down the stairs to the basement. Morgan ran after him, but couldn't reach him before Badri had the freezer open and a canister in his hand, ready to release.\n\n\"You'll die,\" said Morgan.\n\n\"And maybe I will take you with me.\"\n\n\"You can still survive,\" Morgan said.\n\n\"And go to Guantanamo Bay for the rest of my life?\"\n\n\"I really am sorry,\" Morgan said. \"I wish we'd have met under other circumstances. I wish you weren't who you were. I wish you hadn't made yourself my enemy. But you can live. That's something. It's something I can offer you.\"\n\nBadri pushed the button and the canister burst in a plume of fine white powder. Badri fell almost immediately, convulsing on the floor of the basement.\n\nHolding his breath, Morgan ran upstairs, doing some mental math on the rate of dispersion. He reached the upper landing as the tactical team rammed the door, sending splinters flying into the foyer. They filed inside in formation, all wearing tactical gear and gas masks.\n\n\"Clear the area! Get as far from here as you can!\"\n\nHe was feeling the weight in his chest as he ran out of the house. He stumbled within a few feet of the door, lightheaded, and then the world began growing dark. He hardly felt himself hitting the floor, and had the vague impression of someone slipping a gas mask over his face, and he saw someone else pull out a syringe with a gigantic hypodermic needle before he lost consciousness.\nChapter Forty-four\n\nAlex Morgan walked into Diana Bloch's apartment with plodding steps. She had descended into a haze of gloom ever since returning home, crying herself to sleep every night. One day at a time worked, but it didn't make it any easier.\n\n\"Make yourself at home,\" Bloch said.\n\nAlex collapsed on a white leather couch. \"Did you make me chamomile tea again?\"\n\nBloch sat cross-legged on a Barcelona chair, facing Alex. \"I have something rather more potent. I'd like to tell you a story.\"\n\n\"Yippee.\"\n\nBloch cleared her throat. \"Eight months ago, a cache of Novichok nerve agent was stolen as it was transported from a military research lab in Russia. This Novichok is really nasty stuff. It will destroy your lungs and make you drown on dry land.\n\n\"The group that took it was an al-Qaeda splinter group named Shining Jihad. But that's all we knew. We exhausted our resources trying to find the agent, or the group's leadership.\n\n\"Then, some five months ago, the Russians captured the group's mastermind, a man named Muhammad Badri. A clever and resourceful man, and certainly the one who planned the heist of the gas in the first place.\n\n\"The Russians knew he was a terrorist, but they didn't quite know what they had on their hands. So they sent him to a secret prison in Siberia, an old gulag camp where the Russian government keeps the people they want to forget.\n\n\"We knew it was a matter of time before Badri's people used the Novichok against us somehow. We needed to find the gas. And our only link was him. The Russian government was not forthcoming with aid, so we had to take it into our own hands. We had to send someone into the prison. And that meant letting someone get captured.\"\n\n\"Are you telling me\u2014\"\n\n\"The odds were always slim,\" Bloch said. \"He had to escape from the prison with Badri, while gaining his trust enough to find out where the Novichok was. The mission rested on a razor's edge.\"\n\n\"Well?\" Alex urged. \"What happened?\"\n\nShe heard footsteps coming from the inner hallway of the apartment. Heavy. A man's.\n\nShe knew him by his silhouette in the dim light even before his features resolved in her eyes.\n\n\"Dad!\"\n\nShe leapt off the couch and at him, hugging him as hard as he could.\n\n\"I apologize,\" said Bloch. \"I wish we could have told you. But it was imperative for his mission to remain in total secrecy.\"\n\n\"But I'm back.\" He squeezed her tight. \"I'm back. That's what matters.\"\n\nShe released him, tears flowing from her eyes, and punched him in the chest. \"How could you do that to me?\" she demanded, in anger mixed with joy.\n\n\"I had to do it,\" he said. \"My country needed me.\"\n\n\"Yeah. You had to go off and be a hero.\" She buried her face in his chest. \"The world needs heroes like you. But Mom and I need you more.\" She raised her head. \"We have to tell her! You don't know how sick she's been over this.\"\n\n\"I'd just like to remind you,\" said Bloch, \"that all this is highly classified.\"\n\nAlex wiped away her tears and looked at her father. He looked thin, haggard, and exhausted. \"So did you have a nice vacation?\"\n\n\"Oh, yeah,\" said Morgan. \"Got some exercise, some cultural immersion, met all sorts of different kinds of people.\"\n\nAlex couldn't contain herself and gave him another hug.\n\nBloch interrupted their reunion. \"By the way, Alex, we know all about your little Russian adventure.\"\n\nAlex blushed with shame. \"I got captured, would've been killed if Valery hadn't been there to save me, and I didn't come close to finding you.\"\n\n\"How is old Valery?\" Morgan asked.\n\n\"Miserable,\" said Alex.\n\n\"That's Valery all right.\"\n\n\"I'm so embarrassed,\" she said. \"It was a complete failure.\"\n\n\"Funny,\" said Bloch. \"That's not what I heard. I heard that a budding young operative was resourceful and quick-thinking. I heard she used her assets, faced danger head-on, and managed to infiltrate the house of a Russian general and escape with her life. What I read in my reports indicated that you show enough promise to surpass your father someday. And I expect you to report back for training Monday at oh-five hundred hours.\"\n\nAlex's cheeks flushed, now with joy.\n\n\"And Morgan,\" said Bloch, \"We're going to need an in-depth debrief from you.\"\n\n\"Later,\" Morgan said, holding his hand up. \"Come on, Alex. Let's go home.\"\nACKNOWLEDGEMENTS\n\nI must thank my immensely talented team at Kensington Publishing Corp., who work tirelessly to help make my novels the best they can be and are there whenever I have questions or concerns. Thank you to Steve Zacharius, owner of Kensington, who has made me part of the Kensington \"family.\" There are just not enough words to convey how fortunate I am to have Michaela Hamilton as my editor. Her patience and guidance have been invaluable\u2014she is a very special person.\n\nI also want to express my appreciation to my literary agent, Doug Grad, as well as to Mayur Gudka, my webmaster and social media consultant. I also want to thank my partner in writing and creating my novels, Caio Camargo. I am so fortunate that you are members of my team and, more important, friends.\n\nMy wife, Lynn, continues to encourage me to pursue my writing career.\n\nLastly, I want to thank all of my very loyal fans whose support has helped grow the Dan Morgan series from one novel to six and still writing . . .\nDon't miss the next exciting thriller starring  \nZeta operative Dan Morgan\n\nROGUE COMMANDER\n\nComing soon from Lyrical Underground,  \nan imprint of Kensington Publishing Corp.\n\nChapter One\n\nDan Morgan stood against the stone back of the Church of Our Lady Before T\u00fdn, a lit cigarette dangling between his fingers.\n\nHe didn't smoke\u2014couldn't stand the smell, really\u2014but nothing gave him better cover to stand around in the street, out of the way of most people. So he let the reeking thing burn, pretending to puff every few seconds to avert suspicion, and shielding the ember from the autumn wind. It was early October, and the sun was low in the sky even though it was half past ten a.m. None of its rays made it down to the level of Prague's narrow streets.\n\nHe was in an area reserved for parking, which held the sorts of establishments that grow like weeds on the periphery of the big tourist sites, selling cheap souvenirs and small necessities like water and smokes.\n\n\"Morgan, report in.\" This was the voice of Diana Bloch coming over the wireless transmitter in his right ear, terse and all business. The head of Zeta Division, Bloch carried the authority of a natural leader. She was a pain in his ass, but mostly in a good way.\n\n\"Nothing yet.\"\n\nA group of four American college kids stopped as one of them took a picture of the back of the church. One of the couples stood close together, with a sort of awkwardness that told Morgan theirs was a new relationship. The other couple had been together long enough to be more interested in other things, but shared a kiss before they moved along.\n\nThey didn't give him a second glance. Good. Being invisible had its perks in the business.\n\nMorgan buzzed with energy, as he always did before a mission. He felt the reassuring weight of his black Walther PPK in its shoulder holster, well hidden under his black trench coat. It wasn't a popular concealed-carry weapon anymore\u2014too heavy, and not as much firepower as the polymer nine-millimeter pieces that took its place. But he was a man with classic tastes, and he had a soft spot for the gun. It felt solid in his hand, nicely balanced, with light recoil. That, and he could hit a fly in the air at ten paces with it.\n\nMorgan stood back against the stone of the centuries-old Gothic church, and feigned drawing in smoke from his Marlboro when he caught movement in his peripheral vision. Across the small parking lot, a man emerged from the front door of the Ventana hotel. He had an unhandsome face topped with a receding head of blond hair, with a strong nose but a weak chin that he hid, badly, with a goatee.\n\n\"That's Pulnik,\" said Morgan. \"Moving west from the hotel.\"\n\n\"Keep your distance,\" said Bloch over the comm. \"Team, get moving. Stick to the plan. Morgan, do I have to remind you\u2014\"\n\n\"You don't. It's my damn plan. I'm sticking to it.\" Morgan dropped the half-burned cigarette, ground it against the pavement, and set off after the man.\n\nTheir quarry was Havel Pulnik, a sleazy small-time underworld businessman who happened to be second cousin to Enver Lukacs, the evasive big fish they were really after. With no other leads to finding Lukacs, Zeta division had kept Pulnik under surveillance for months. Their persistence finally paid off when one of Lukacs's people set a meeting with Pulnik in Prague.\n\n\"We're on the move.\" That was Bishop, the leader of the Zeta Tactical Team, somewhere within a two-block radius.\n\nMorgan walked thirty feet behind Pulnik. The streets were teeming with tourists from all nations. He could recognize the people from warmer climates because they were bundled up as if they were in the Himalayas in the dead of winter. As he passed a souvenir shop, Morgan caught sight of Spartan. She had a good four inches on him, her close-cropped blond hair hidden by a dark gray beanie. She was looking through postcards from a rack, oriented so she could steal glances at their quarry. He then caught sight of Bishop, walking a distance ahead of Pulnik. Spartan set off a few seconds after Morgan had passed, walking on pace with a group of Germans who seemed to be going out for a stroll rather than oohing and aahing at the sights.\n\n\"Looks like he's moving toward the plaza,\" said Spartan. \"Good call.\"\n\nMorgan walked on the cobblestones, worn smooth over the years. Prague had the Old World elegance of Europe, with its picturesque hodgepodge of architectural styles\u2014all of them, unlike the utilitarian bent of American engineering, with an eye for beauty. The condition of the buildings, however, betrayed the country's Soviet past. They did not have the polish, the fresh paint and recent renovations, found in England or Germany.\n\nMorgan liked Prague, though. The city had character. A gloomy character, sure. Nothing more appropriate for the city of Franz Kafka. But anywhere he went, at least in the old city, he felt yes, I am in Prague.\n\nAfter a short walk Morgan followed Pulnik into the historic Old Town Square. Now, here was a main attraction. The perimeter of the sprawling square was lined with restaurants with outdoor tables, where tourists braved the cold and sat with hot drinks. Others sat on the ground of the square. One young woman was drawing the Old Town Hall, its gothic spires reaching toward the sky. Most were standing around, listening to guides, reading stuff on their cell phones, taking pictures. A band was setting up, with a standing bass, a clarinet, a banjo, and a washboard. Half a dozen people were already sitting in a semicircle, waiting for them to begin playing. A handful of protesters were there, too, demonstrating on the issue of refugees from the Middle East. The younger and more diverse crowd was for; the older and local, against. They kept a tense peace, but Morgan had a feeling things might break out in violence quickly.\n\nPulnik was making his way toward the green bronze statue of Jan Hus at the center of the square.\n\n\"Fan out,\" Morgan said. \"I want people on all sides. We need to see Lukacs coming.\"\n\n\"Moving in, northwest corner.\" The voice belonged to Peter Conley, Morgan's old partner from his days in the CIA. There was no one Morgan would sooner trust with his life.\n\nMorgan walked to the middle of the east side of the square and watched as the others got into position. He surveyed the tourists, who were oblivious to the importance of this moment. The wheels of their world turned, and they were none the wiser. They didn't know anything about the silent machinery hidden deep in the bowels of their world. All they saw was the surface.\n\nMorgan was here today to stop one of these cogs from turning. Enver Lukacs was the name of this particular cog\u2014a shadowy underworld player with a finger in every pie. His currency was contacts, joining people who were selling black market items and services with those who would buy them. Weapons, drugs, mercenaries, slaves\u2014Lukacs had it all. If the Zeta operatives got him to turn over what he knew, even just a fraction, they could bring down dozens of illegal operations.\n\nEverything depended on their success on that day.\n\n\"Hello! American!\" It was a slight young man with a local accent. His baby face was draped with scraggly hair, and he had on a dirty red coat over a stained T-shirt.\n\nShit. This was all Morgan needed. \"I don't have any money.\"\n\nThe man smiled with mock offense. \"No! Come on, American friend! I just want to have a conversation!\"\n\n\"I don't have any of that either. Now scram, kid.\"\n\nHe went off to bother someone else. Morgan looked at Pulnik, standing by the statue with his hands in his pockets, looking around at the crowd for the man he was there to meet.\n\nIt was Conley who spoke first. \"I have eyes on the target. Approaching from my corner.\"\n\n\"Keep your distance,\" said Bloch. \"I want confirmation before we do anything.\"\n\nMorgan leaned against a lamppost and looked at the man now crossing the plaza. He looked like a fashion designer, with a svelte silver-fox thing going on and a stylish designer suit.\n\n\"Positive ID,\" said Morgan. \"That's him. That's Lukacs.\"\n\n\"Get in position,\" said Bloch. \"Diesel, I want you on alert. You need to arrive with the van just as they reach the street with Lukacs while Lily provides a distraction.\" That would be young, green-eyed Lily Randall, femme fatale. Morgan caught sight of her coming in from the far side, her auburn hair glistening in the morning sun. \"We need to attract as little attention as possible.\"\n\nThe jazz band broke out into a Dixieland rendition of \"When the Saints Go Marching In.\" The singer's voice suggested he'd been a heavy smoker since age five. The effect wasn't exactly beautiful, but hell if it didn't work.\n\nA semicircle of tourists formed, although most went about their business without a glance at the musicians. Something nagged at the corner of Morgan's mind before he was fully conscious of what it was.\n\n\"Hold positions,\" he said. \"Lukacs's got company.\"\n\n\"Where?\" Bishop asked.\n\n\"Tall bearded guy over by the church. Short and stocky next to the tour group on the north side. Red hair by the lamppost, near the southwest corner. And another sitting on the far side of the statue.\"\n\n\"The bastard brought a security detail.\"\n\n\"Bishop. Conley.\" It was Bloch. \"Scan the windows for snipers. If he brought this much backup, I get a feeling he won't be stopping there.\"\n\nMorgan looked at the rows of windows that surrounded the plaza. Two churches, two hotels, a museum, and a government building. All old and elegant.\n\n\"Got one,\" Bishop said. \"White building, north side, fourth floor. Third window from the left.\"\n\n\"That's bad news,\" said Lily.\n\nMorgan watched the band as they launched into a rollicking performance of \"Mack the Knife.\" \"The sniper's in a hotel,\" he said. \"Shepard, think you can get me room access?\"\n\nShepard was their resident computer genius. There was hardly any computer system he couldn't crack. \"Already working on it,\" came his cocky, insouciant voice.\n\n\"Conley\u2014\"\n\n\"On my way.\" Peter Conley moved toward the hotel entrance. After working together so many years, he and Morgan had a nearly psychic connection.\n\n\"I don't like this.\" It was Bishop. \"The mission is getting riskier by the second. I think we need to call it off.\"\n\n\"You're running point on this, Morgan,\" said Bloch. \"It's your call.\"\n\nMorgan squinted into the cloudless blue sky. Then he looked at Lukacs, who was talking closely with Pulnik.\n\n\"Everyone get in position,\" Morgan said. \"We're moving in as soon as Lukacs breaks away from Pulnik.\"\n\n\"What about his security people?\" Bishop demanded.\n\n\"You fan out with the team. I want one of us on every guard. Bind their hands with disposable cuffs and drop them. Lily, you go ahead with the distraction on my mark. We're going to need perfect timing on this.\"\n\n\"And Lukacs?\" Bishop asked.\n\n\"I'll take care of Lukacs,\" said Morgan. \"Extraction van ready?\"\n\n\"I'll move out on your mark,\" Diesel answered. \"Pick you up on the southeast corner.\"\n\nMorgan watched as the team moved through the crowd as naturally as any tourists, betraying no sign of their purpose.\n\n\"I'm in position at the sniper's door,\" said Conley. \"Shepard, how close are you to getting access?\"\n\nShepard scoffed. \"I'm in, big guy.\"\n\n\"Morgan, we're waiting for your signal,\" said Bloch.\n\n\"Hold. Not yet.\" Morgan kept his eyes on Lukacs and Pulnik, having their conversation. Then Lukacs pulled his companion in close, and Pulnik gasped, eyes wide, grabbing his belly.\n\nMorgan couldn't see it clearly, but there was no doubt. Lukacs had stabbed him. Blood seeped out as Pulnik bent double. Lukacs eased him to sit against the low ledge around the statue.\n\n\"Are you seeing this?\" said Spartan.\n\n\"This is not right,\" said Bishop. \"Morgan, call this off.\"\n\n\"The mission doesn't change.\" Morgan was not going to let Lukacs get away. \"He's moving out. Lily, that's your cue.\"\n\nOn the far side of the plaza, Lily drew a megaphone from her pack and turned it on with an earsplitting whine.\n\n\"Wake up, sheeple!\" she screeched, her voice amplified and flattened by the megaphone. \"The Illuminati run your lives!\" Lily was really selling the insanity, and people took notice. \"The reptilians have invaded the highest level of government!\" Tourists moved toward her or rubbernecked to get a look at the crazy lady. \"They want us for our blood!\"\n\nNow it was their turn to take action. The Zeta team moved in on Lukacs's security. Lukacs had left Pulnik on the ground and was moving back from the direction he had come from. As he turned, Lukacs's eyes met Morgan's, and he held his stare long enough for the message to come across as clear as a tall glass of water.\n\n\"Goddamn it! They know we're here! Fall back!\"\n\n\"I told you this was a bad idea,\" said Bishop. \"Didn't I goddamn tell you?\"\n\n\"Too late now,\" Morgan said. \"Let's move!\"\n\nThey'd lost the element of surprise. Morgan heard the sound of Conley kicking the door in as Lukacs's security drew their guns. Morgan couldn't spare the attention to see what was going on. He heard gunfire, then screaming, as he ran straight for Lukacs.\n\nTwo guards were converging on him, fast, from the left and right. Morgan turned his run evasive, reaching for his Walther.\n\nLily, having cast off her megaphone, came dashing from the left and tripped one man, sending him reeling to the ground. This gave Morgan the distraction and the opening he needed to fire at the other guard. Three bullets in the chest, and he was down.\n\nHe took the man's gun and tossed it to Lily, who had come weaponless.\n\n\"I owe you!\" she said, and ran off to help out Spartan, who was struggling to fight off two of Lukacs's security guards. Then he took off running again toward Lukacs, who was by now at the edge of the square.\n\nMorgan took off at a dead run, pushing as hard as he could. Someone crashed into him, sending his Walther flying, and him almost to the pavement. Morgan spared him a glance before diving for his gun.\n\nIt was the young man in the red coat. His smile was gone. He was holding a Beretta M9, and that Beretta M9 was pointed at Morgan's chest.\n\n\"Do not even think about going for the weapon,\" he said. \"Hands up.\"\n\nMorgan's eyes scanned his surroundings. His team was scattered. None of them could come to his rescue. The man was too close for him to run, but too far for him to attack and survive.\n\n\"Any last words?\"\n\nMorgan turned his hand and raised his middle finger.\n\n\"Eloquent. Now you die.\" He aimed the Beretta at Morgan's heart.\n\nThe sound of a gunshot filled Morgan's ears.\n\nChapter Two\n\nFrom the window of her hotel room, Alex Morgan grimaced at the sharp smell of gunpowder. Body still humming from the shot she'd just fired, she watched through the scope of her rifle as the man's eyes widened in surprise, and he fell to the ground. The blood from his chest wound mingled with the red of his coat and spilled onto the cobblestones.\n\n\"Hell of a shot, Alex.\" She heard her father through the comm, but she couldn't see his lips move.\n\n\"Thank me later, Dad. Right now, we need to find Lukacs.\"\n\nAs people drained from the square, Alex scanned the space, looking for their target or his men, but they blended in with the tourists. She'd taken out two of the guards already before Red Coat. Bloch had been worried about putting a sniper rifle in her hands in the field, ordering her to nest in the hotel room overlooking the square, where, Alex suspected, Bloch did not think she'd see any action.\n\nAles was going to enjoy making the boss eat her words.\n\n\"Anyone got eyes on him?\" Morgan asked.\n\nA chorus of negatives came over the radio. Alex made one last survey of the square. \"I'm no good up here anymore. I'm moving out.\"\n\n\"You stay where you are,\" her father said.\n\n\"Make me.\" She set her Heckler & Koch MSG90 on the carpet. She wiped her fingerprints, stowed her gun in its holster and her stun gun in one of her pockets. Then she pulled on her coat, drawing up the lapels to hide her face, and ran out of the room and down the hall and downstairs.\n\nShe pushed against the flow of people seeking refuge inside, squeezing her way out into the chill air. She couldn't pick out Lukacs or his men. Every panicked face in the crowd could conceal an enemy.\n\n\"I have eyes on Lukacs.\" It was Peter Conley. \"He's moving past the astronomical clock as we speak.\"\n\nAlex took a left from the hotel and ran toward the square's old clock. As far as she could tell, her position put her ahead of everyone else. A fantasy flashed before her eyes\u2014Lukacs, in handcuffs, and her, Alex Morgan, bringing him in.\n\n\"We have police incoming!\" said Shepard. To this prompt, Alex picked out the sirens approaching from the distance.\n\n\"This mission is already a shit show!\" came Bishop's deep voice. \"We need to call this off.\"\n\n\"It's your mission, Morgan,\" said Bloch. \"The call is yours.\"\n\nIt didn't matter what he said. Alex wasn't about to stop.\n\n\"If we don't get him now, we'll never catch him,\" her father said. \"We're moving forward.\"\n\n\"I just spotted a secondary security team coming in from the southeast,\" said Spartan. \"I can keep them busy, but I need some help.\"\n\n\"On my way,\" Lily said. Alex heard gunshots behind her as she ran out of the Old Town Square alongside the historic town hall. Its bell was ringing ten a.m. as she passed. Alex spared a glance at the figurine of death coming out, animated by the mechanism.\n\nUp ahead, she caught sight of Lukacs getting into the backseat of a Mercedes C-Class. She noted the license plate and ran toward him, elbowing past people as the car pulled out, but it was too far away. She'd never make it.\n\nSpotting a man starting his motorcycle, she pulled her Taurus nine-millimeter compact and tugged on his leather jacket.\n\n\"I'm going to need the bike,\" she said. \"Sorry.\"\n\nThe man raised his hands and backed away, leaving the Honda 250 to topple over the cobblestones. Alex pulled it upright, hopped on, and kick-started it. The bike rumbled between her legs and she took off, beeping the horn to get the frenzied crowd to part.\n\nThe little Honda was light and maneuverable, and with Alex's slim body in the seat, it moved fast.\n\n\"Spartan, cover me!\" Alex heard gunfire over the comm. She maneuvered around the people until she cleared the crowd enough to gain some momentum, bike jolting on the uneven ground. Lukacs's car was widening the distance between them.\n\nAnd now she saw there wasn't just one, but two cars, same make, model, and color\u2014black. \"He's got a decoy!\" she said, straining to see the numbers and letters on the second vehicle's license plate as she weaved through traffic. \"Lukacs is in the front car!\"\n\nThe right back window of the second car rolled down and a man lifted himself out so that his upper torso was free in the air. His right hand held on to the hood of the car. In his hand was a Glock semiautomatic.\n\nShe banked left hard as he fired, shattering a store window behind her, and had to make a tight right to avoid a post on the edge of the sidewalk.\n\nShe zigzagged as he tried to aim. He fired off another shot, which ricocheted off a metal lamppost.\n\nThe Zeta tactical van merged with the street they were on and hit the second car, which spun out to the left and crashed into the storefront of a butcher shop.\n\nAlex drifted right, just missing the van. She lost speed with the maneuver, but she picked up again once she was clear of the crash.\n\n\"Everyone all right back there?\" she asked.\n\n\"In one piece,\" said Diesel. \"Keep moving after Lukacs!\"\n\nAlex heard the insistent blare of a car horn coming from behind her, getting louder. A maroon Toyota Camry came speeding down the road, weaving through people and traffic to catch up to her.\n\nThat would be her father.\n\n\"Alex, fall back!\"\n\n\"You fall back!\"\n\n\"I outrank you. Do as I say!\"\n\nYeah, right. She pushed harder, and pedestrians leapt out of the way. Traffic was light, and Lukacs managed to move fast even in the narrow streets of Prague. Alex followed suit, the old pastel-colored buildings that lined the street blurring with the speed. Her father kept up behind her.\n\nPolice cars turned into the street three hundred yards down. The Mercedes hung a squealing right into a pedestrian-only boardwalk, sending passersby scrambling. Alex made the turn, nearly toppling over as the Honda's front tire collided with the curb. Recovering, she picked up speed, went through a stone archway under a tower, and then out into the Charles Bridge, which had crossed the Vltava River since the late Middle Ages. Her father was still tailing her. Pedestrians parted like the Red Sea to hug the stone guard walls that protected its edge.\n\n\"Out of the way!\" he yelled over the communicator.\n\nHer father had his hand out of the car window, his Walther in his hand. Alex banked right, and he fired four times at Lukacs's car.\n\nOne of his bullets hit home. The back left tire burst, left in rags. The car swerved left and right, then plowed straight into the side of the bridge. The heavy stone held firm, crumpling the frame of the Mercedes like it was wrapping paper. Her father brought his Toyota to a screeching halt. Alex drove past Lukacs's car and stopped on the far side. Her father took cover behind the door of his sedan, Walther in hand. They had their quarry boxed in.\n\nThe two front doors opened and a security guard emerged from each one, wearing black suits and ties, Glocks in hand. They opened fire against her father.\n\nAlex was about to move to help him when Lukacs stumbled out of the car on the other side\u2014her side.\n\nShe let him come a short distance away from the Mercedes, just so he couldn't disappear back inside. Then she stood up and drew her Taurus revolver.\n\n\"Freeze!\"\n\nHe looked at her, first in shock, then with amusement.\n\nThey kept underestimating her. That's why she won.\n\n\"Hands up!\"\n\nThe bastard smirked and set off walking in her direction. \"I do not think you have what it takes to shoot me.\"\n\n\"Don't test me,\" she said. But her hands were trembling. He kept walking toward her.\n\n\"Stop, or I swear I'll shoot.\"\n\n\"Will you now? Your people could have sniped me from a window, but you went through all this trouble to catch me alive. So no, I don't think you will.\"\n\n\"Do not shoot him, Alex.\" It was Bloch. \"He's no use to us dead.\"\n\nAlex glanced at her father. He had taken down one of Lukacs's men, but was pinned by the other. Alex had a clear shot to the remaining bodyguard, but she didn't think she could hit home at this distance, and she was even less protected than he was.\n\n\"So? What's it going to be?\" said Lukacs. \"Are you going to shoot me now?\"\n\nMovement in her peripheral vision attracted her attention. The Zeta tactical van was barreling down the bridge, its front bumper in splinters from the crash.\n\nLukacs was close now, within grabbing distance of her Taurus. \"Now give me the gun,\" he said, reaching out and grasping it by the barrel.\n\nBut Alex was ready. With her left hand, she drew the stun gun out of her jacket pocket and pushed it against his chest, pumping more than fifty thousand volts into his body. Lukacs convulsed and dropped to the cobblestones of the bridge just as the Zeta tactical van came to a skidding halt alongside her and Lukacs. The door slid open before it were fully stopped. Spartan took care of Lukacs's final security guard with a salvo of bullets from her MP5 while Alex and Bishop hauled Lukacs into the cargo area.\n\nSpartan hopped back inside and they pulled the door shut as Diesel peeled off toward the far end of the bridge.\n\n\"Team, report in,\" said Bloch. \"What's going on?\"\n\n\"Bishop here, with Diesel, Spartan, and Morgan, Jr., safe and in possession of package. We're on our way to switch out this car.\"\n\n\"Morgan safe.\"\n\n\"Conley safe.\n\nSilence over the radio ensued as Diesel turned into a cross street, cutting off a surprised driver who leaned into his horn. Lily hadn't reported.\n\n\"Lily, come in,\" said Bloch.\n\nBut there was no answer. Only silence.\n\nPhoto by Kippy Goldfarb, Carolle Photography\n\nABOUT THE AUTHOR\n\nLEO J. MALONEY is the author of the acclaimed thrillers Termination Orders, Silent Assassin, Black Skies, Twelve Hours, and Arch Enemy. He was born in Massachusetts, where he spent his childhood, and graduated from Northeastern University. He spent over thirty years in black ops, accepting highly secretive missions that would put him in the most dangerous hot spots in the world. Since leaving that career, he has had the opportunity to try his hand at acting in independent films and television commercials. He has ten movies to his credit, both as an actor and behind the camera as a producer, technical advisor, and assistant director. He lives in the Boston area and in Florida.\n\nVisit him at www.leojmaloney.com or on Facebook or Twitter.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n## Don't miss the other thrilling companion novels in the class series!\n\n_Class: The Stone House_\n\n_Class: Joyride_\n\n## Dedication\n\n_To my cat #InternetReasons_\n\n## Contents\n\n_Cover_\n\n_Title Page_\n\n_Dedication_\n\n_Contents_\n\nOne: Read the Terrible Truth They Don't Want You to Know Until Page 119\n\nTwo: What We Can Learn from the Alien-Face-In-A-Box Emoji\n\nThree: When He Lost His Leg They Said He Would Never Play Football Again, But He Said No\n\nFour: Eight Things the Media Hasn't Told You About Cup-A-Soup\n\nFive: Amazing, If True: Children Are Vanishing and You've Not Noticed\n\nSix: The One Weird Trick About Mothers That Everyone Should Know\n\nSeven: Is It Magic? We Need Your Help to Solve This Mystery\n\nEight: Fifteen Cats That Look Shockingly like Miss Quill\n\nNine: Train Drivers Slam Brakes on Truth or Dare\n\nTen: This Chapter Will Prove You've Been Wrong About Young Men Your Whole Life\n\nEleven: This Girl Wanted an Adventure Holiday. You Won't Believe Where She Ended Up\n\nTwelve: Find out Why the Goats Do Not Get to Dance on the Tube\n\nThirteen: The Letter Coal Hill School Doesn't Want You to See\n\nFourteen: How Toast Is like Lady Gaga\n\nFifteen: You Wouldn't Think a Text Could Make You Cry, but This Will\n\nSixteen: Six Names for White You've Never Thought Of\n\nSeventeen: The Rise of Smart Women and How to Stop It\n\nEighteen: Things You'll Get Only If Your Home Planet Was Destroyed in the Nineties\n\nNineteen: He Thought He Knew a Lot About Gravity. Find out If He Was Right\n\nTwenty: The Five Words That Broke Her Heart (Spoiler: One of Them Is 'Want')\n\nTwenty-One: She Thought She Knew What Was Going On. Then She Found out the Remarkable Truth and Turned Things Around\n\nTwenty-Two: Thought You Knew How to Lose Your Friends? Well, This Woman's Thirteen Brilliant Reasons Will Change Your Mind\n\nTwenty-Three: She Was Ready to Give up and Then a Nurse Slayed Her with a Word\n\nTwenty-Four: If You Dropped Dead Tomorrow, Would Your Friends Miss You?\n\nTwenty-Five: She Thought She'd Seen It All and Then She Saw the Face of God\n\nTwenty-Six: This Teacher's Inspirational Words Will Choke You Up\n\nTwenty-Seven: Someone's Reimagined Disney Princesses as Alien Warriors And, Trust Us, It's Awesome\n\nTwenty-Eight: This Hot Take on Smashed Avocado Toast Will Have You Reeling\n\nTwenty-Nine: Think of the Worst Job in the World? You're Not Even Close\n\nThirty: You Are Being Lied to About Voter Registration and This Short Chapter Tells You How\n\nThirty-One: Advertisement: Your Book Will Continue in Twenty-Five Seconds\n\nThirty-Two: The Ten Best Alien Deaths You'll See Today. #6 Is a Killer\n\nThirty-Three: Eight Ways in Which People Have Tried to Escape the Void\n\nThirty-Four: At First She Thought She Knew Everything but Then She Found This Secret She Hadn't Known She Needed to Know\n\nThirty-Five: When She Met God She Forgot to Ask 'Why?'\n\nThirty-Six: This Is Your Chance to Wipe out Skandis Forever\n\nThirty-Seven: After You Read This You'll Want a Shower\n\nThirty-Eight: You'll Be Amazed at How Long It Took Him to Realise His Mistake\n\nThirty-Nine: War Veterans Are Covering Their Heads in Glitter for Reasons That Will Stun You\n\nForty: This Icelandic Penguin Village Is Probably the Cutest Place on Earth. But You Are Not There\n\nForty-One: In the Time It Takes You to Read This, Skandis Will Have Claimed One Hundred More Lives\n\nForty-Two: Many People Would Blame This on Marriage Equality. But Would You?\n\nForty-Three: People Are Tweeting Their Worst Battles and It Is Cringingly Hilarious\n\nForty-Four: She Dropped a Truth Bomb but Wasn't Expecting What Would Happen Next\n\nForty-Five: This Young Footballer Has Something Surprising to Say About Racial Profiling\n\nForty-Six: You Are Being Lied to About Dogs\n\nForty-Seven: The Latest Advances in Virtual Reality Will Horrify You\n\nForty-Eight: Do You Know Enough About Dimensional Compensators to Save This Boy's Life? (Spoiler: You Don't)\n\nForty-Nine: He Chose the Wrong Day to Beg for His Life\n\nFifty: The Skandis War as You've Never Seen It Before\n\nFifty-One: They Thought They'd Won Until They Found out They'd Lost\n\nFifty-Two: To Get Your Free Confession Just Follow These Simple Steps\n\nFifty-Three: Happy Endings Don't Kill People\u2014Guns Do\n\nExcerpt from \"For Tonight We Might Die\"\n\n_About the Authors_\n\n_Books by Patrick Ness and James Goss_\n\n_Back Ad_\n\n_Copyright_\n\n_About the Publisher_\n\n## ONE\n\n## READ THE TERRIBLE TRUTH THEY DON'T WANT YOU TO KNOW UNTIL PAGE 119\n\nAre you coming to get me?\n\nPlease tell me you're coming to get me.\n\n'Cos I really thought I was doing the right thing. Okay, some of my friends said I went too far. Actually, fine, I lost a few friends because of what I did. But it was all jokes and bants. That was all it was.\n\nWell, that was all it was when it started. There's a site\u2014truthordare.com. They wanted us to post stuff to it. And really, if I've got to spell that out then really, eyeroll.gif. You get the picture\u2014stuff about yourself you don't want people to know; stuff about your friends they'd hate the world finding out, or, you know, risky things\u2014riding a hoverboard in traffic, playing Pok\u00e9mon Go underwater, or what I did\u2014that old thing of putting your hand on a table and stabbing a knife between the fingers up and down, down and up, faster and faster.\n\nGuess what? Good news, I can still play the recorder. Turns out, I'm really good at it. I'm also really good at betraying my friends\u2014given a choice between my secrets and theirs? No contest. Phone peeping out of your bag, Gmail left unattended? Give it to me, I shall be the master of it. And the world shall laugh at you.\n\nBut, right, I was doing it for a good cause. We were trying to stop the spread of Skandis\u2014they said it was a disease. I guess, you know, strictly speaking, yeah, fine, that's sort of right.\n\nBut now I know what Skandis really is. And it's worse than a disease. And I'm doing all I can to stop it. Because I've got to. We all have to. Or the Earth will be destroyed.\n\nI'll tell you what happened to me, okay? I played the game, only I didn't realise it was just a training level for something else. I didn't know what I was heading for as I climbed up the leaderboard. I just thought\u2014I dunno, maybe a free holiday or a T-shirt or something. Not . . . not that I'd vanish. Kidnapped right out of this world. Wake up somewhere else.\n\nAnd then I'd find out what Skandis really is.\n\nNow that I know? I'm doing all I can to stop Skandis.\n\nBut what I really want? When I'm not screaming, when I'm not so scared? When I'm not doing the most daring stuff I've ever done?\n\nOkay, then, here's a truth about me, the terrible truth that I don't dare tell anyone: what I really want is for someone to come and get me and take me home. I want my mum.\n\nSo.\n\nCan I come home now?\n\n## TWO\n\n## WHAT WE CAN LEARN FROM THE ALIEN-FACE-IN-A-BOX EMOJI\n\nQuestion: What will YOU do to stop Skandis?\n\nHi.\n\nI'm not a doctor.\n\nWell, obviously. What doctor wears board shorts and bunny ears?\n\nBut I'm taking some time out from my busy schedule of being YouTube famous to tell you all about Skandis. Skandis is spreading. And we need to get together and stop it.\n\nI know you're probably like 'yeah, right,' but Skandis is real. And it's not like Brexit or Donald Trump or Nicki Minaj. Skandis can be stopped.\n\nTogether, we can cure Skandis. Now, come on, lean closer, and I'm going to tell you how . . .\n\n## THREE\n\n## WHEN HE LOST HIS LEG THEY SAID HE WOULD NEVER PLAY FOOTBALL AGAIN, BUT HE SAID NO\n\nIt started with the ice, and Ram was fine with the ice. It struck him as a lot of effort. It wasn't that he didn't like effort. Whatever his dad would have told you, Ram wasn't lazy. He liked doing things, he was simply selective about what he actually did.\n\nLearning how to make his new leg work? Worth it. Tipping a bucket of ice over your head? He'd get back to you.\n\nThat morning he was running and thinking about the ice. It helped take his mind off how very slightly wrong his leg was. It just wouldn't do _leg_ correctly. He'd been assured that the artificial limb contained lots of smart technology. Intuitive gimbals. Actuated flesh. Nano-level balancing.\n\nSimulated hair. But, for all that, Ram and his leg still didn't quite trust each other.\n\nYou're asking a lot of a leg. You're asking it to be there for you. It's constantly helping out with really amazing things, like keeping you standing upright, letting you walk, and climbing stairs; stairs really are such a leap of faith that it's amazing we didn't invent the lift before the wheel.\n\nRam's old leg did all that he asked of it without getting in the way, without even reminding him that it was there. Now that it was gone, he really, really missed it. Because he and his new leg were constantly second-guessing each other. It was like having a butler for a limb. Ram would make that little bit of extra effort when stepping with his false leg, just to make sure he landed really firmly, and his artificial limb would push back, just the tiniest twitch, as if clearing its throat and saying, 'Forgive me, I'm not sure you quite meant me to do that.' It did it, every single time. His leg kept reminding him that it was there. Little pulses racing up his thigh. 'Just got you to the pavement', 'Just stepped around that dog turd', 'Avoided stumbling over that rut on the pitch, no, don't thank me'.\n\nRam had never really understood what the phrase 'passive-aggressive' had meant before. Everyone used it, about teachers, mean kids, or crisps, but Ram knew, absolutely knew, that he had a passive-aggressive leg. It was so judgmental too. It made it quite clear that it was a precision instrument and that kicking a ball in a certain direction just seemed to be inflicting unnecessary damage on it for no apparent purpose. Every time he tried to kick a ball his foot flinched, which isn't really the body language that makes for a successful career in football.\n\nHence the morning runs. When he'd suggested taking up jogging, his dad had been keen. Overkeen. Really 'waiting for him in the hall in sweatpants and with him 100 percent of the way' keen. Luckily that had passed, and now Ram got to go running on his own\u2014down the road, over the footbridge, round the scrubby park and back, all the time hoping that instinct would kick in. It didn't. His leg kept telling him 'down and safe' every step of the way. So he distracted himself with thinking about the ice.\n\nThe videos had started cropping up online a week ago. People tipping a bucket of ice over their heads for charity. It was all a bit 2014. But the practicalities of it all kept Ram occupied while his leg kept telling him it was there, and it was fine.\n\nIt was the ice that troubled him. There were a couple of trays in the freezer\u2014maybe about two dozen ice cubes. So that wasn't going to fill a bucket. You'd need to go to the supermarket, buy a bag. But supermarkets only ever seemed to keep a couple of bags in stock\u2014not enough to cope with a barbecue, let alone a charity craze. And EVERYONE was doing it. Even the Third Years. Where did they get the ice? From a pub? Did they stockpile it?\n\nToday Ram was working on the Stockpiling Theory with a dedication that would have startled his maths teacher. If he could make three batches of ice a day, and got an extra tray, he could maybe knock up a hundred cubes a day. How many ice cubes would it take to make a bucket of water ice-cold? Perhaps three days of planning and he'd have enough.\n\nThree days. Making it Wednesday.\n\nRam stopped considering it for a moment as his ankle had just proudly informed him that it had failed to twist over some gravel.\n\nCould he hold out till Wednesday?\n\nThe problem with tipping a bucket of ice over his head was that Ram didn't want to do it. Pretty much the rest of the football team had, one by one, posting videos of themselves shrieking, 'I'm doing the ice bucket for Skandis!' and then dousing themselves. Recently someone had finished by crying, 'And now it's your turn, Ram.'\n\nWhich had seemed a bit provocative. He'd kind of considered doing it, just to get it out of the way, but then April had said, completely casually, 'You'd only be doing it to get the attention from posting a topless video of yourself.'\n\nThat had stopped him. For one thing, she was wrong. Completely wrong. Totally wrong. Colchester wrong. Yes, some of the guys had got A LOT of attention from their videos but that really, absolutely wasn't his motive. For one thing, he'd wear a T-shirt. A tight one with the sleeves hacked off.\n\nAnyway, April. Recently stuff she'd said carried a bit more weight. Like it was in a slightly different font. Odd. But he definitely wasn't not doing it just because April was against it.\n\nRam paused at the park gate and told his leg to shut up for a moment. Why had sentences suddenly got so complicated?\n\nThe answer was actually pretty simple. Aliens. Aliens had invaded his school. Before they'd gone, there'd been consequences. 'Consequences' wasn't the right word. The nearest thing to the right word was just one long, howling scream in a very dark room.\n\nBut fine, let's go with aliens had invaded the school and there'd been consequences. They'd killed his girlfriend. They'd cut off his leg. They'd massacred a load of people (which no one was talking about). That weird kid in class? An alien prince. His football coach? Bit alien. The teacher who hated him? An alien. And, for complicated reasons, April's heart was now alien.\n\nAbout the only person Ram knew who could describe it clearly would be Tanya. He didn't like Tanya, not as such, but he understood the point of her. Tanya looked at the world through slightly narrowed eyes and called it as she saw it. Also, she never handed in homework late, so the world seemed to be forever on her side. Even if she was, like, twelve or something.\n\nRam had been watching one of the ice bucket videos yesterday. Tanya had appeared over his shoulder. She somehow did this, despite being smaller than him. She was asking him a question. Frowning with annoyance, he made an elaborate pantomime of pausing the video and pulling off his headphones, and then looked at her.\n\n'What?' he'd said, annoyed.\n\nShe'd carried on looking at the ice frozen over the screaming footballer.\n\n'Don't you think that's odd?' she said.\n\n'Wouldn't know. Not done it.' He wondered why he sounded so defensive.\n\n'Didn't say you had.' She was just staring at him, not blinking. 'But it's interesting. I mean, statistically, I can understand why one person would tip ice over their head. But two, I'm not so sure. Especially when everyone's been there, done that before.'\n\n'Well, it's for Skandis,' muttered Ram. 'Some kind of charity. An American one, I guess.'\n\n'Right.' Tanya chewed the word. 'Nearly the whole football team's done it now. You haven't. Has there been any peer pressure? You know, people asking you why you haven't?'\n\n'Apart from you?' Ram asked. 'Not really.' He put his headphones back on and watched the end of the video.\n\n_'And now it's your turn, Ram.'_\n\nHe looked up. Tanya had gone.\n\nRam finished his run and went to training. His school clothes were in his backpack. He would shower afterwards then go to class and that would all be fine.\n\nHe pulled off his hoodie, stuffed it in a locker, changed into his football boots, and tried not to notice how silent the changing room was.\n\nThe rest of his team was there. But they weren't talking. They weren't talking _to_ him. He just knew it. It was all very subtle. It wasn't like someone had come up to him and said anything. There was just that vague sense that he was in the room but he didn't belong to it anymore. He knew better than to ask. There was nothing worse than asking. Confronting the problem. No.\n\nTo be fair, it wasn't anything new. When he'd been the team's star player, that distance had already been there in the air. Even while they hugged him and cheered him on, there was still that slight whiff of 'why him?'. They all knew how good he was. How naturally talented. How it was only a matter of time before he got the dream life that definitely included sports cars.\n\nThat weird atmosphere had only increased ever since he'd lost his leg. He'd not been able to tell anyone. Not say, 'Look, my leg got chopped off and this is the best they could do. Pretty neat, but don't worry, I'll get the hang of it'. He couldn't say anything. Couldn't do the wounded hero act. Couldn't scream about how unfair it was. He got lots of sympathy for the death of his girlfriend, Rachel, but as far as they were concerned, he'd suddenly gone from star player to someone who really shouldn't be on the team anymore.\n\nNow, the fact that he wasn't throwing cold water over himself gave them the perfect opportunity to vent their frustrations. He was no longer the star player, he was no longer the ex-star player having a bad patch, and he was no longer one of them.\n\nHe hung back, letting the rest of the team filter out towards the pitch. Neil, the only other guy who'd not done the challenge, was still tying his shoes and making a meal out of it. Ram suspected he really didn't fancy being there either.\n\nRam sat down on the bench next to him as casually as really awkward could be.\n\n'You doing it today?' he said. Neil said nothing.\n\n'You got the ice?' Ram said.\n\nNeil looked up, like it was the least important thing in the world. 'Thought you didn't care,' he said vaguely, tugging strands of hair in the mirror.\n\n'No,' insisted Ram. 'I do.' That sounded a bit like a bleat. 'It's just, if you're going to do it . . . I mean, isn't the ice thing a bit, you know, old? Can't you do something else? Something better?'\n\nNeil didn't look away from the mirror. 'Got anything in mind?'\n\n'No.' Ram heard himself trying to laugh. 'Just, you know. Ice, bucket, gasp. It's not . . .'\n\nNeil tugged at his jersey, neatening it even more. 'So that's it?' he said quietly. 'You're not joining in because it's boring.'\n\n'Just don't see the point.' Ram faked a big smile. 'If I do something, it'll be really amazing.'\n\n'I'll wait,' Neil said.\n\n'Also, not sure what Skandis is. That's all.'\n\nNeil shrugged. 'Google it,' he said, and walked out. Ram got his phone out of his locker. He tapped 'Skandis' in, then stopped. He closed the locker and went outside. He'd sort it out later.\n\nAfterwards, he felt like an idiot.\n\n'I'm doing that ice challenge,' he announced to April. Very much in passing. Very much conversational.\n\n'Right,' she said. She was leaning against a pillar, sketching away in a notebook. Somehow she made the simple act of doodling look utterly dismissive. Around them, people ran from one classroom to the next. April seemed completely uncaring, her eyes barely focused on the world around them. Charlie had said that it was because April was connected to a distant planet, could maybe even glimpse it, but Ram got the feeling that April had always been a little bit this way. Some people just aren't quite in focus.\n\nTanya rocked up. Now, there was someone who was completely in focus.\n\n'So, I'm doing the ice thing,' he repeated, hoping that she'd say something.\n\nTanya frowned, and when she did so it was a thing of moment.\n\n'You quite sure about that?' she said. Her words swung like dumbbells. 'For Skandis?'\n\n'Yeah,' said Ram. 'It's a disease. I saw a video about a girl who had it. You know. Brave in a shaven-headed way.'\n\n'You _sure_?' repeated Tanya. She looked doubtful.\n\nApril focused on Tanya in a way that she never did on Ram.\n\n'Something's happened, hasn't it?' April said.\n\n'Oh yeah.'\n\nThat was when Ram heard the ambulance sirens.\n\n## FOUR\n\n## EIGHT THINGS THE MEDIA HASN'T TOLD YOU ABOUT CUP-A-SOUP\n\n**VIDEO TITLE: The Cup-a-Soup Challenge**\n\n_The football pitch. Neil sitting in a chair. A crowd of friends. An extension cable. A mug._\n\n_A kettle._\n\n_'Hi, my name's Neil and this is my Cup-a-Soup challenge! Oh yeah._\n\n_Today I'm doing something different. That's right. Ice is yesterday._\n\n_I've picked my favourite flavour of Cup-a-Soup._\n\n_It's chicken. Yeah. Go chicken! And now I'm going to wear it._\n\n_My friend Paul here has boiled the kettle._\n\n___I'm passing him the mug. That's right. Stir out the floaters. Nice one!_\n\n_Talking of chicken, I'd like to thank Ram for suggesting I do something different. Hey, Ram, I'm doing this for Skandis. Isn't it about time you did something too?_\n\n_Now, Paul, let's make soup! Tip it! Tip it! Tip it!'_\n\n_And then the screaming started._\n\n'Well,' observed Miss Quill, 'that'll be a short-lived craze.'\n\nThe ambulance was pulling out of the car park, leaving behind a mournful crowd taking shocked selfies.\n\nMiss Quill's fingers pulled away from the venetian blind, letting the metal slats snap back into place. Almost back into place. One slat was crooked. Earth children were such careless eavesdroppers. She reached out with an expert finger and thumb, pinching the metal until it bent back, just a bit off perfect. Finished, she rubbed the dust from between her fingers and turned around.\n\nCharlie sat on a desk, watching her carefully.\n\nThey were many things to each other.\n\nIf you'd asked Miss Quill, she would have told you that Charlie was the following:\n\nHer owner\n\nHer jailor\n\nHer next victim\n\nAnnoying\n\nIf you'd asked Charlie, he would have told you that Miss Quill was:\n\nHis very reluctant bodyguard\n\nA moral snake\n\nLikely to betray him\n\nAnnoying\n\nThey shared a house. It was quite a nice house. There were many practical reasons for them to share a house. It made the whole business of the last of the Quill guarding the last Rhodian Prince fairly easy. But on every other level it was a complete nightmare.\n\nSometimes Charlie would open his bedroom door at night to find Miss Quill standing outside.\n\n'What . . . What are you doing?'\n\n'Same as ever,' she'd sigh wearily. 'Watching over you. You going to use the bathroom or should I put the kettle on?'\n\nThis had become even more awkward since Charlie's boyfriend had moved in.\n\nIf anyone could cope with a boyfriend who'd just arrived from another planet, it was Matteusz. He was easygoing and terribly calm. When he discovered that Charlie had no idea how to cross a road, he had simply ignored the horns, carried him onto the grassy bank of a roundabout, sat him down, and explained how roads worked. 'Oh. Back home, traffic just stopped for me,' Charlie had muttered, looking vaguely hurt that such a rule did not apply here.\n\nBut even Matteusz, easygoing, calm, thoughtful Matteusz, found Miss Quill hard to deal with.\n\n'She does not like me,' Matteusz had said in his measured Polish accent to Charlie one night.\n\nCharlie had shrugged. 'She does not like anyone.'\n\n'Yes, but she really doesn't like me. She follows me around the kitchen. I just go there to make some tea, but she follows my every move. Even when I pick a mug, she is judging me and she is judging the mug.'\n\n'She's probably wondering if you're trying to kill me. How is your tea?'\n\n'Fine. Here is yours. Quill drank from it.'\n\n'She would. As I said, she's probably wondering if you're trying to kill me.'\n\nThey sat on the end of Charlie's bed, drinking tea.\n\nOne main difference between Quill and Charlie was that each thought they understood human beings better. They both found things to admire in them, but for entirely different reasons. Quill saw them as angry, selfish, and violent. Charlie found them impulsive, confusing, and strange.\n\nSometimes, when they had nothing better to do, Charlie and his bodyguard would stand in her classroom, watching the people go by. It wasn't that they liked spending time with each other. Sometimes it just happened and it felt sort of right.\n\nToday they had watched the boy being stretchered into the ambulance, his face wrapped in bandages.\n\n'He'll live,' remarked Quill. 'Unlike a lot of the pupils here. They really are fodder, aren't they? Still, as I said, at least it will be a short-lived craze.'\n\nCharlie stared at her, considering. 'The boy is called Neil. He is my age. He has severe burns to his scalp and face. He is in terrible pain. He will require plastic surgery and will probably be disfigured for life.'\n\nQuill shrugged. She'd never shrugged before coming to the Earth and now she found it easier than breathing. 'He tipped boiling water over his face. He deserves what he gets.'\n\n'But why would you do that?' Charlie said.\n\nQuill didn't turn around from the window. 'Ask one of your pets.'\n\nAs Charlie approached, Ram slunk away.\n\nCharlie had noticed how little Ram wanted to be around him. There were probably lots of reasons. He noticed that Ram's artificial leg was still overcompensating, and wondered if he should offer to look at the default settings, but also knew enough about social interaction to realise that it would probably not be an easy conversation.\n\n'Please drop your trousers, I wish to look at your legs' was all very well with Matteusz, but he doubted it was in common usage. A shame, as he felt responsible for Ram losing his leg and wanted to somehow make it better.\n\nHuman life appeared to be a series of guilty interactions where people told one another how sorry they were for things that either were their fault\u2014or they were pretending were their fault in order to make things better\u2014or else they were apologising for things that had happened because of weather, microbes, or gravity. It was all marvellously confusing and Charlie was determined to find out how saying sorry worked. But maybe he wouldn't start on Ram's leg today.\n\nCharlie casually pretended he hadn't noticed Ram slinking away. 'I did not see Ram leave just now,' he told April. 'I did not hear the slight creak of his leg.'\n\n'I see.' April looked up at him. She really was very pretty. Charlie came from a world of rigid rules and structure, where everyone was swept into straight lines. April was a glorious tangle. Her long dark hair should have been rigid as a pendulum, but instead it cascaded and jumped and let itself be pushed about. She was always pushing it from one side to the other and then sweeping it back. When he was growing up, Charlie had been taught the art of sitting still, of maintaining a calm and regal and reserved posture, whereas April gloried in constant movement. If she wasn't fingering her hair, she was tugging at her clothes or moving her legs or tapping a pen against a book. It was all so unnecessary and he found it delightful to watch.\n\n'Why did Ram go?' he asked. April liked how direct Charlie was. No hesitation, no caution, no tact. 'Is it because he doesn't like me?'\n\n'No,' said April. 'The boy who burned himself\u2014he was on the football team with Ram. He's upset.'\n\n'I see,' said Charlie. 'I do not understand. Why did he do that to himself? He has disfigured himself. On my world, the plume priests did something similar as an act of political protest. Is that the case here?'\n\nApril considered. 'Nooooooo,' she said.\n\nThey needed a laptop, and they needed Tanya. Normally both were to be found together.\n\n'I can't believe no one's taken it down,' she said, chewing on a strawberry lace as the video played again on truthordare.com. 'Look at the number of views it has got.' She tapped at the bottom of the screen. 'See the little plus sign at the end\u2014that means it's only an approximation. That means the number of views are growing faster than you can count. People love watching other people do stupid things.' She clicked to refresh the video.\n\nIt reloaded, paused, and then showed a short video about a famous footballer discussing car insurance with a horse. 'Ahha!' Tanya laughed. 'That proves it's popular\u2014they've slapped adverts on it. They'll be making a mint off this.'\n\n'But this is a video of human suffering.' Charlie spoke slowly. It was the tone he used when he was finding out something about human beings he was not entirely pleased by. 'Who would want to make money off that? Would it be this Skandis charity?'\n\n'Oh no.' Tanya bit off another strawberry lace. She offered Charlie the remaining half. He declined. 'No, it's the site that serves the video. They make the cash. Neil gets the money from people who said they'd pay a couple of quid for the ice bucket challenge. That goes to Skandis. If, that is, anyone pays up\u2014right now they probably really don't feel like it.'\n\nCharlie fell silent, watching Neil fall screaming out of his chair, writhing on the grass as the camera whipped up, thought better of it, then closed in on his scalded face.\n\n'So Neil did this for nothing?'\n\nNeil continued to scream.\n\n## FIVE\n\n## AMAZING, IF TRUE: CHILDREN ARE VANISHING AND YOU'VE NOT NOTICED\n\n**Blog Post on AnotherNewsSite.com**\n\n**Kids are vanishing.**\n\n**Remember the 'Welcome to Twin Peaks' road sign? You're probably too young. God, I think typing that sentence makes me feel worse than you do reading it. Anyway, they had a similar sign in Sunnydale and in Bon Temps, Louisiana (it's the town in True Blood, Grandma).**\n\n**Anyway, the thing about the 'Welcome to Twin Peaks' road sign was that it had the town's population printed on it. And it never changed. Despite all the serial killers living there. The number never went down. And the** **** **joke\u2014the joke that everyone made\u2014was that it should be like the numbers at a gas station. Or a little neon display.**\n\n**Just ticking down, every week.**\n\n**But that never happened.**\n\n**Anyway, perhaps it's time that it did. Because kids are going missing, and at a rate so fast the digits on the sign would be SPINNING.**\n\n**You heard me right. I don't know how or why\u2014but I tell you this: Educate yourselves, people. Do some Googling: 'missing kids'. Check your timeline for pictures of the missing\u2014teenagers going out and never being seen again. That's right\u2014there's suddenly a lot, aren't there?**\n\n**It's happening right now, everyone.**\n\n**I don't know what's causing it, or why, but the children are going missing.**\n\n## SIX\n\n## THE ONE WEIRD TRICK ABOUT MOTHERS THAT EVERYONE SHOULD KNOW\n\nRam went to the hospital to see Neil.\n\nHe'd been feeling bad about it all day. Well, all week.\n\nIt wasn't the kind of feeling bad that got better. It wasn't like a torn ligament. The guilt just made him feel worse. The idea that he'd caused it. Especially as he didn't feel like telling anyone that it was all his fault. They'd either say it wasn't (not true) or they'd say it was his fault, which would make him feel worse.\n\nSo, he ignored the guilt. It got worse.\n\nStrangely, his walking got worse. Of course it did.\n\n'You're not helping,' he said to his new leg. It didn't reply, and instead it led him to the hospital.\n\nThe hospital was one of those grand Victorian redbrick buildings, which has had ugly new bits jabbed into it.\n\nSigns had been put up everywhere, so many of them that nothing seemed to make sense anymore. There was a little scrubby garden where old men sat smoking. There was a coffee shop full of anxious relatives. Porters wheeled empty beds past. Everywhere there was a smell not quite covered up by disinfectant.\n\nEventually he found the burns unit. It was past an unmanned reception area, up a handsome wooden staircase that had been blocked off with netting to stop anyone throwing themselves off it. He clambered up, his leg telling him each and every time one of the steps was just that little bit uneven.\n\nHe wondered about getting the lift\u2014but no, that would be giving in. If he took the lift once, that would be it. After his grandad had that fall, he'd 'tried out' a mobility scooter, just for a week or two. He'd never got out of it. No, Ram wasn't going to give up. Still, four flights of these stairs was quite hard going.\n\nHe got to the top. Instead of a carved oak door, there was an automatic one that slid open to reveal a corridor that was floored with linoleum, the walls covered with hand sanitiser dispensers and posters asking him alarming questions about his bodily functions.\n\n'Well, this is so depressing,' he said aloud.\n\n'Ram?'\n\nHe turned.\n\nStaring at him, eyes wet and bright, was Neil's mother. He recognised her from the touchlines and barbecues. Even in hospital she looked glamorous. Neil's mother always dressed like it was summer. He'd usually found it a bit ridiculous, but now, here, it looked so sad. As though she was hoping for something better out of life than rain.\n\nShe hugged him, which Ram found surprising and a little bit unwelcome. He'd thought that Neil was the last person in the world he'd wanted to see, but the real answer turned out to be Neil's mum.\n\n_I am so sorry I maimed your son._\n\n'It's so good of you to come,' she breathed into his ear as he took an unwelcome gulp of her perfume. It wasn't that she was wearing too much, or that it was horrible\u2014he just didn't want to be smelling it.\n\n_I didn't ask for this._\n\nShe released him from the hug and he stood back quickly, trying not to show his relief. 'So good of you to come,' she repeated. Then she stopped. She was waiting for him to say something.\n\nRam mumbled a few words about how Neil was.\n\n'He's asleep,' she said sadly. 'He's had some grafts, and that's tiring, so . . .' Oh, that sounded bad. The immediate relief that he wouldn't have to see Neil today was mingled with the knowledge that he'd have to do it tomorrow. Or the day after. Or next week. It was a problem that wasn't going to go away.\n\nShe patted at a plastic chair and he sat down on it next to her. In the room opposite, an old woman was crying in front of the television.\n\n'I'm so sorry,' said Ram. He just blurted it out, and felt a sudden relief.\n\n'Oh, don't be,' said Neil's mum. 'Not your fault.' She switched on a smile, trying to look bright.\n\n_It is my fault. It is so totally my fault._\n\n'I feel bad about it,' he said.\n\n'Yeah well.' She sniffed. 'It's good of you to come. So good.' She pushed a hand through her hair, and her brave smile gave up. 'I wasn't going to say it, but you know what? You're the only one of the team to have come along. The only one.' She looked a lot less sunbeamy. 'The new coach rang up. That was something.' She folded her hands and thinned her lips. 'But he was so careful with what he said. I think he was reading from a card. Making sure he wasn't admitting liability. In case I was recording it. Imagine!' She threw her hands up in the air. 'I was in here. Waiting to find out if my son was scarred for life, and he's wondering if I'm recording my calls for a lawsuit. Some people . . .' She shook her head, and didn't seem at all bright or chirpy. 'I just don't know. I just don't know.'\n\n'I'm sorry,' Ram said again. _If I say it enough times you'll forgive me._\n\n'Oh yeah.' She squeezed his hand. 'I dunno what I was expecting. But it would have been nice if the team had come round. Would have meant the world to Neil.'\n\n'They've been busy,' said Ram. And then stopped. Neil's mum stared at him. Her lips went very thin again and she let go of his hand.\n\n'Busy?' she hissed.\n\n'Sorry,' said Ram. 'Wrong word. I don't know. I just don't know.'\n\n'You don't have to make excuses for them,' she snapped, twisting a ring round a finger. 'Ram, don't make excuses for other people. Lessens you.'\n\nThey sat in some more silence then. The moment came around for Ram to say something. It passed. Another moment came around. He let it go. On the third pass:\n\n'I said he should do something different.'\n\n'Did you?' she said politely.\n\n'Other than the ice bucket challenge. Everyone's done the ice bucket challenge. I told him to do something different. Just that. I didn't mean for him to do what he did. I swear. I am so sorry. I really am\u2014'\n\nNeil's mum gripped his arm. For a moment it was too hard and then she let go and patted his sweatshirt, folding it between her fingertips. Ram stopped talking and lowered his eyes.\n\n'Suppose,' she said, 'he'd sat in a paddling pool full of ice instead. Would we be here?'\n\n'No,' said Ram.\n\n'Or tipped cold baked beans on his head?'\n\n'No.'\n\n'Only my Neil would lack the plain common sense to do what he did. He's sharp, but he's not bright.' Neil's mum leaned back in the grey plastic chair and she smiled again, a sad fondness lighting up her face. 'Bless him.' She wiped her eyes. 'No. I wish you hadn't told him to try something different, but he came up with the stupid idea. Not you. Am I right?' She jabbed Ram in the ribs, and her eyes shone.\n\n'Yeah,' Ram mumbled and for a moment he thought he was going to cry.\n\n'Then there we are. My son, my lovely son, is an idiot for thinking he could pour boiling water on his head. But you know who I really blame?' Her eyes narrowed, squeezing tears out. 'The kid who boiled the kettle.' She pointed to the seat next to her. 'I'd like to see him sat right there. Not you. I'd like him to see what he did. I can believe my son was daft. I can't believe the people on that team were so daft. Yeah. That's the boy I'd like to see.'\n\nThis time the silence was a bit more natural.\n\nThen she stood up. 'Shall we go and see Neil? Maybe he'll be awake.'\n\n'Yes.' Ram stood up. 'I'd like that.'\n\nAs they walked off, Neil's mum nudged him. 'Your limp,' she said.\n\n'What?' He blushed.\n\n'It's gone.'\n\nRam blinked.\n\n## SEVEN\n\n## IS IT MAGIC? WE NEED YOUR HELP TO SOLVE THIS MYSTERY\n\nThere had been a boy at the bus stop. That's what everyone could agree on.\n\nSome of the kids from Coal Hill said they knew him, but only in the way that, if something happened to someone, people always said that they knew them and had definitely met them. You know, at a party, or a club, or they sort of went out with your sister. Even though he didn't go to Coal Hill, lots of people said they knew him.\n\nAnyway, the boy. At the bus stop. He had definitely been there.\n\nThat much CCTV could tell you with certainty.\n\nHe'd been waiting for the late bus home. He'd actually had a pretty great day\u2014no reason to go missing. The police and his parents checked the footage. He was standing there, waiting for the bus. He was, if you squinted, smiling. He was playing with his phone ('He was always playing with that thing' said his dad), and occasionally, if someone he vaguely knew walked past, he'd flash the phone screen at them, showing off about something.\n\nHe did not, experts admitted, look like someone planning on running away. He looked like a boy, any boy, waiting for a bus.\n\nThe bus pulls up in front of the stop, and, for a moment, the boy is blocked from the view of the CCTV.\n\nBut that shouldn't matter, because there are cameras on board the bus. They should show him if he gets on. But he does not get on.\n\nThe bus pulls away. And the boy is no longer there.\n\nWhat's happened to him?\n\n## EIGHT\n\n## FIFTEEN CATS THAT LOOK SHOCKINGLY LIKE MISS QUILL\n\nThe school forgot about Neil and his hot soup. So much kept happening.\n\nOne of the last things the headmaster of Coal Hill School did in life was interview a trainee teacher. The trainee was a bright, hopeful young woman called Victoria Prim, who would later on in life earn a lot of money in accountancy and marry her neighbour who had waited forty years for her to ask. But today she was a young, slightly too-eager trainee who dreamed of teaching poetry and one day mounting a school production of _Godspell_.\n\nAt the end of the predictably exhausting interview, Mr Armitage crawled towards the final question. 'Ms Prim, have you any questions you'd like to ask me?'\n\nMs Prim had leaned forward earnestly across the desk. 'Tell me, Headmaster,' she'd demanded, 'what's a typical day at Coal Hill School like?'\n\nMr Armitage had tried not to groan. 'Well, now,' he'd replied, reaching for the clich\u00e9 used to sell awful jobs through the ages, 'there really is no typical day . . .'\n\nThat afternoon, his skin was torn off by an interdimensional dragon.\n\nTime passed. And, as time passed, Cup-a-Soup Neil was seen as one of the lucky ones.\n\nMore and more videos started to appear. Just one look at the 'Most Watched' videos on truthordare.com told you all you needed to know:\n\n_This Charity Ice Skater Will Astound You\u2014It's Summer_\n\n_She'd Never Windsurfed Before Today. Who Knew There'd Be a Hurricane?_\n\n_27 Insane Charity Challenges. You'll Lose It at #16_\n\n_His Nana Lent Him Her Mobility Scooter. She Did Not Predict a Motorway_\n\n___They Told Her She Could Never Fly. They Might Have Been Right_\n\n_She Set Out to Eat Ten Insects. Guess Which One Is Poisonous?_\n\n_The Unexpected Truth About Skydiving Blindfolded_\n\n_The School Said No to This Netball Team's Naked Calendar. They Did It Anyway_\n\n_He'd Never Eaten Fire Before. He Never Will Again_\n\n_Twelve Tips for Being Buried Alive. #5 Will Destroy You_\n\n_They Said He'd Never Eat Penguin. He Wished They Were Wrong_\n\n_Read Me or I'll Kill Myself_\n\n_Eleven Reasons Why You Might Die in the Next Hour. #7 Is Standing Right Behind You_\n\nNo one at truthordare.com took Neil's video down. It was labelled under _Epic Fails_. It didn't even make the top ten. His act of immense stupidity and personal tragedy became tiny under the tide of other efforts, of people outcompeting one another for charity.\n\nNeil's mum tried seeing if anyone was interested in raising funds to help get Neil some more plastic surgery. A lot of people really meant to do something, and the football team really talked about it, but they were kind of busy working out whether or not to do a charity calendar for truthordare.com. Enthusiasm petered out.\n\n'I would like you to explain the internet to me,' Miss Quill said. She'd asked Tanya to stay behind after class.\n\nTanya stared at her teacher with worried amazement.\n\nAs usual, Miss Quill's expression was as unreadable as an ancient curse.\n\n'You want me to explain the internet?' Tanya measured out her words. Then gave up. 'Seriously?'\n\nMiss Quill nodded. 'We had similar, superior systems on Rhodia, of course.' She waved any other possibility aside. 'But technology evolves to fit the races it serves. I am not sure if I entirely understand how this system serves you.'\n\n'Serves?' Tanya considered the phrase.\n\nMiss Quill nodded. 'On Rhodia, everything serves something. That is how it all fits together.' She scowled then rapped a finger on Tanya's laptop. 'You seem to understand these systems. To an extent. Perhaps you could explain truthordare.com to me.'\n\n'The site?'\n\n'Well, I'd hardly ask you to explain the name,' Miss Quill snapped. 'It's ridiculous.'\n\n**EMAIL NEWSLETTER FROM TRUTHORDARE.COM**\n\n**A YOUNG GUY NEEDS YOUR HELP AND MORE**\n\n**Hi [firstname]!**\n\n**How are you doing? We just thought we'd like to let you know that people are still suffering from Skandis and you should know that we could all be working harder to fight Skandis once and for all.**\n\n**We constantly worry about it at truthordare.com HQ, and here are just a few of the remarkable stories that keep us going through the night:**\n\n**Keith Is Blades**\n\n**What about Keith, who is doing a twenty-four-hour blade-a-thon round Regent's Park? He's only just fought off cancer and now he's determined to battle Skandis too? With just a click you can send him a fiver.**\n\n******Truth for Sami**\n\n**Sami found out her fianc\u00e9 had been cheating on her with her best friend. So she followed them then put on a slideshow of it at her wedding before selling off the gifts and giving them to Skandis. Amazing!**\n\n**What about You?**\n\n**What are you doing to prevent the spread of Skandis? Send us your stories or stay tuned to the site for one of Seraphin's celebrity vlogs. This week he's been shopping for kettles with May June. Find out how it went tomorrow. You won't stop laughing.**\n\n**That's all for now!**\n\nThe light from the laptop screen made Miss Quill's severe face look just that little more severe. She was frowning, which, to be fair, was like saying that other people were breathing.\n\n'I don't get this site,' she muttered. 'It is full of lies.'\n\n'Oh, that's just the internet,' Tanya said. As she said the words, she wondered if she'd made a mistake. Miss Quill was sometimes very literal. But it did feel peculiar. Sharing a desk in a classroom with a woman who wasn't just an alien, or an assassin, but also her teacher. It was so easy to put a foot wrong.\n\n'Really?' Miss Quill didn't look up from the screen. She let her voice project the right level of curiosity and withering disdain.\n\n'Oh yes, nothing on the internet is . . . you know . . .' Tanya wondered where her words had gone and worried they were leaving her to it. She shrugged.\n\n'I understand,' Miss Quill said. 'On Rhodia we would spread misinformation about troop numbers and food stocks and where our armies would be based. You use the internet for the same thing? That is what all those pictures of eggs and beaches are for?'\n\n'Well, um, yes.' Tanya noticed a small twitch at the edges of Miss Quill's mouth. Was she teasing her? Surely not. 'You tell everyone you're having an amazing time, you upload pictures of your amazing time, and maybe that convinces you that you're having an amazing time. When really, well, you're probably not, 'cos you'll have a headache, or argued with Mum in the car on the way there, or your phone's running low on charge, you know.'\n\n'All the really important things. I understand.' Miss Quill's definitely-not-a-smile broadened. 'When the Quill won a battle against the Rhodia we would broadcast videos of the slaughter. We would leave out the bodies of our fallen comrades but we would make sure we missed none of the dead Rhodia. It looked better, and their families may have been watching. This is the same thing?'\n\nTanya didn't speak for a moment. 'More or less.'\n\n'Only'\u2014Miss Quill tapped the screen with a sharp finger\u2014'this site is deceitful. It promises so much but it does not deliver. Look at this: \"Watch Her Slay This Politician\". The video does nothing of the sort. It simply shows a young woman being vaguely rude to an older man in front of an audience of bored people. Honestly, no one even gives her a knife.' She scrolled through the pages. 'See\u2014it's all Tear Him Apart, Destroy That, Shred Him, Ruin Her . . . The language is all so violent, but there's nothing behind it. This is the talk of a coward urging others on to fight.'\n\nTanya realised that, just for once, Miss Quill had got it completely right.\n\n'I am trying to work out the problem with this site,' Miss Quill continued. 'It is making people upload videos of themselves, is it not?'\n\nTanya nodded. 'I think, at some level, it's the root of the problem.'\n\nThe screen read:\n\n**TODAY ON TRUTHORDARE: MAKE US GASP!**\n\n**MAKE YOUR FRIENDS SCREAM! DO SOMETHING DARING!**\n\n******BE UNSPEAKABLE! BREAK THINGS!**\n\n**YOU'VE RAISED $14 MILLION FOR SKANDIS THIS DRIVE. YOU'RE AMAZING. TOGETHER WE'LL BEAT SKANDIS.**\n\nAnd then underneath it was a gif of a clown clapping while a cake burned.\n\n'What feudal obligation do people have to follow these challenges?' Quill asked. She occasionally tried to understand human customs, in case it explained something of them to her. It often didn't. She presumed that the challenges on the site were like ritual combat challenges. Once seen, they had to be answered. In blood.\n\nTanya shook her head. 'None. Not really. Just, you know, because it's for charity.'\n\n'For charity?' Quill stared.\n\n'Don't you have charity in space?'\n\n'Of course not. We have a quick and merciful death for the afflicted.'\n\n'But what about medical research?'\n\n'Oh, we research diseases. And, if we find them, we unleash them on our enemies.'\n\nTanya counted her fingers slowly. After a couple of goes, she was still getting ten, so she breathed out.\n\n'So, this charity. It is like a guilt that you all feel for your superior position in the world?' Quill bit thoughtfully at one of her fingernails. 'Interesting. This is not . . .'\n\n'A Quill Thing?' Tanya finished. 'Do you just gloat over the dead bodies of the conquered?'\n\nQuill nodded. 'You have learned something, finally. Tell me\u2014I can almost understand why people would follow this challenge. But why would you feel obliged to upload the video\u2014is it another social duty?'\n\n'No,' Tanya said. 'It's like people feel they have to. I don't really get it.'\n\n'Even if you look silly, or are in pain?' Quill stood, scraping back her chair as she strode over to the whiteboard.\n\n'There are so many, many terrible odds you have overcome in order to exist\u2014the simple odds of there being life on your planet, the amazing odds of your progenitors meeting, the odds of your being chosen in the womb, of successfully hatching, of growing up . . . Why waste it all on jumping off a brick wall? After all, you could use that wonderful, rare life to achieve something really amazing in battle.'\n\n'Well, that's one way of looking at it.' Tanya was beginning to think that Miss Quill could turn an IKEA catalogue into a list of reasons for going to war.\n\nMiss Quill leaned back over Tanya's laptop. 'A moment,' she said, scrolling through her search history. 'I have a query\u2014I have not spent much time on your internet.'\n\n'Actually, please,' said Tanya. 'No one really calls it that anymore. We call it the Information Superhighway.'\n\n'Ah, thank you,' said Miss Quill, not noticing Tanya's slight smirk. 'I will use that. It is a primitive system of knowledge sharing. But your Information Superhighway has one remarkable thing about it . . .' She tapped at another tab. 'What are these?'\n\nTanya squinted. 'You're kidding.'\n\n'No, no, I am not. Quill do not have a sense of humour.'\n\n'Now you're kidding.'\n\n'Perhaps.' Again that non-smile twitched. 'But why are there pictures of these animals all over your Information Superhighway?'\n\n'They're just cats,' said Tanya.\n\n'Cats?' Miss Quill sat down and stared at the screen. 'But why?'\n\n'I don't know. I really don't.'\n\nMiss Quill was now staring at a whole screen full of cats. 'There is nothing special about them. They are a waste of the pathetically tiny bandwidth that is available to you. And yet . . .'\n\nHer fingers paused over a picture of a ginger cat peeping out of a cardboard box. The little twitch at the edge of her mouth was back. 'There is something about them. Is there not?'\n\n## NINE\n\n## TRAIN DRIVERS SLAM BRAKES ON TRUTH OR DARE\n\n**First it was broken limbs and broken hearts\u2014now health and safety experts are warning that truthordare.com may cost you more. Gemma Harris investigates.**\n\n**The site truthordare.com has been contacted over 'safety concerns' regarding site users venturing onto live train tracks as part of 'dares' encouraged by the site.**\n\n**We understand players have been walking into the paths of trains at Birmingham New Street as part of a challenge they're calling 'Chicken'. Network Rail has warned that trains can reach high speeds and that 'darers' may not realise this when playing.**\n\n******The site's development team has not yet responded to questions from us, but their 'vlogging host', Seraphin, has previously said: 'Our rules are that there are no rules\u2014but that doesn't mean that you should be stupid. Only the brave, not the dumb, are needed to beat Skandis. Dare safely.'**\n\n**Network Rail, which has put up warnings on electronic signs at New Street, controls 2,500 stations as well as tracks, tunnels, and level crossings, and says trains can reach speeds of 100mph (160kmh).**\n\n**Maude Silvera, chief health and safety officer at the station, said: 'While we're delighted to see so many people working hard to beat this terrible disease, we do have safety concerns around dares taking place close to the working railway.**\n\n**'We are concerned that by staging a dare on or very close to our infrastructure, young people are likely to be distracted and even less aware than usual of the risks around them.**\n\n**'The last thing we want to do is ruin everyone's \"fun,\" but we are hoping that we can work together with \"darers\" to make the site safer.'**\n\n******The concerns of train operators join those of anxious parents as the appeal of truthordare.com spreads. Hospital emergency departments are reporting a rise in admissions resulting from the site. In addition, the charity helpline Talk Out Teens has claimed an increase in call volume as a result of personal details shared on the site. There are also several unconfirmed reports of players going missing.**\n\n**Related Content**\n\n  * **Help! My Daughter's Playing TruthOrDare!**\n  * **10 Signs You May Be Too Old for TruthOrDare**\n  * **I Told the Truth and Now I'm Single**\n  * **#FindJimmy\u2014The Hunt for a Top Darer**\n\n## TEN\n\n## THIS CHAPTER WILL PROVE YOU'VE BEEN WRONG ABOUT YOUNG MEN YOUR WHOLE LIFE\n\n**A Vlog by Seraphin on Truthordare.com**\n\n**Hi, Everyone,**\n\n**Yeah, hi. Yeah. Soooo, manic day. Just back from the theme park and that's pretty great, and the kids' hospital\u2014that was great too and there was so much to see there that I'll vlog about it all soon, but for today I'm just wandering around my flat. Yeah, I'm doing my laundry today. Look at this\u2014these are my pants. Yeah. My last set of pants. My dirty pants. Today, cringe, I am wearing no pants.**\n\n**Sorry about that, internet. I'm wearing no pants. Yeah.**\n\n******None. No pants. LE GASP!**\n\n**All I'm wearing now is this T-shirt. Yeah. The one with the sponsorship logo on it. And yes. I have been wearing that for a few days. Is that an egg stain? I believe it is. Damn you, omelette.**\n\n**That's going in the wash now too. See?**\n\n**Yes. I am starkers. In my kitchen. But that's okay. From this angle you can't see a thing. Not a thing. Apart from . . . Oops, cheeky, yes, I did just go past the full-length mirror in the hall. You can see that again. Yeah. You've seen my butt. Ha. Ha. Ha. Gif it now!**\n\n**Yeah, Seraphin's got a butt. News. Flash. Hope you like it. Now then, argh, I'm nearly out of laundry powder. My life. My life.**\n\n**So. How are you? Good. Good. Me? I'm fine. There, that's the mixed load on. Not a sports wash. What is a sports wash even for? What does that even mean? When I've been playing sport\u2014well, you've all seen me jogging. I look rank. I smell worse. What is the point of a fifteen-minute wash? No way is that going to even touch that smell. Yeuf.**\n\n******Now, I know I'm going to get tweets about that\u2014creepy tweets. Someone out there is right now demanding to smell my old sports socks and I tell you this now, Dad, you are awful. Hahahah.**\n\n**Anyway, that's the laundry on, so let's get down to work.**\n\n**'Cos this is a\u2014tanta-ta-tantatattaaaaa!\u2014sponsored vlog for truthordare.com. Their mission is simple\u2014they want your truths, your dares\u2014whatever, how shocking, how risky, how shameful\u2014because you're not doing it for you, you're doing it for Skandis.**\n\n**Yeah, okay, it's for charrrrrrrrrrriteeeeeeee. Sad face, mournful little frown, oh yeah. We're fighting Skandis. Some of the kids in hospital today, they've been battling Skandis for a long time and it is, seriously, really bad. And we can, if we get together, we can cure it.**\n\n**Truthordare.com\u2014we want your stunts, your dares, your whatevs.**\n\n**There is only one rule. There is no such thing as oversharing. Honestly. Again. I'm nekkid. If I was to tilt the camera down, two things would happen.**\n\n**1) You would see everything.**\n\n**2) The internet, well, it would melt.**\n\n**So the camera stays where it is. For now. But that may change.**\n\n**The ice challenge? That was just the start of it. What happens next . . . It's in your hands.**\n\n**Anyway, I'm going to just . . . just . . . there we are, can you see it? The washing going round in the machine? Can you see that? Lovely. Brilliant. Oh look, there go some pants. And . . . there they are again . . . We should make a YouTube channel of this. What do you think? Let me know.**\n\n**Anyway, while the washing finishes itself, I'm off to test a new hoverboard (advert up in a second, vlog later on this week). So, I'll leave you with this, my friends: Truth or Dare: You can never share too much. Never. Share. Too. Much.**\n\n**Anyway. If you enjoyed that, subscribe. If you want something similar, click on the links. Seraphin out.**\n\n## ELEVEN\n\n## THIS GIRL WANTED AN ADVENTURE HOLIDAY. YOU WON'T BELIEVE WHERE SHE ENDED UP\n\nThe girl was screaming. She was crying underneath the helmet. She knew that no one could see the tears, but they could probably hear the screaming.\n\nShe knew they were watching. She knew they were listening. She'd lost her weapon and she was wounded and they weren't going to help her, they were just going to watch her die.\n\nShe made herself look at the creature coming towards her.\n\n'NORMAL' the screen in her helmet told her, and, hysterically, she laughed. 'No way is that normal.'\n\nA tentacle whipped out towards her.\n\nShe scrabbled away, scurrying up a steep sandbank.\n\n'I've got to get out of here,' she said, looking around for a way out. Wherever she looked, her helmet told her 'ABNORMAL'.\n\n'I know what you're doing,' she said, her voice shaking.\n\n'You want me to look at that thing.' She turned and faced the creature.\n\n'NORMAL' her helmet told her.\n\n'Oh God,' she said.\n\nThe creature stopped, leaning over her, its face pushing in, curious.\n\n'NORMAL.'\n\nThe girl screamed.\n\nFor a moment the picture on her helmet juddered as something whipped across it. There was a crack, a wet, tearing noise and a final scream.\n\nThen the picture cleared. The helmet was lying on its side, the camera showing a picture of the ground, and that terrible creature dragging itself closer.\n\n'NORMAL.'\n\n## TWELVE\n\n## FIND OUT WHY THE GOATS DO NOT GET TO DANCE ON THE TUBE\n\nMatteusz had come to dread one word most of all.\n\n'Question?' That word.\n\nMatteusz looked up from filing jewels on his phone. Charlie was standing over him. He was wearing that look on his face. The really severe yet childish look that said he was about to ask about butterflies or fossil fuel or Nigel Farage or . . . well, look, it could be anything. It was always exhausting. Matteusz wondered if Charlie knew about Santa Claus. That was going to be a long day. In an emergency, he could just hide behind his Polishness until it went away.\n\nCharlie was still staring at him. Waiting for an answer. Best get on with it.\n\n'Yes?'\n\n'What is oversharing?' Charlie asked. 'I do not understand.'\n\n'Truthordare.com. The website. They are asking us to overshare. It seems very popular at Coal Hill. I wonder what that means.'\n\n'The website?'\n\n'They want videos. Like the boy who burned himself. They want more of those\u2014is that what oversharing is? Are they asking people to hurt themselves?'\n\n'Not exactly. It is about putting yourself online.'\n\n'Is the idea to create a virtual artificial version of your personality so you can be resurrected after death?' Charlie looked incredibly thoughtful, his eyes wandering to the cabinet in the corner. 'Is that it?'\n\n'Not so much, no. It is strange. You must tell everyone all of your life.'\n\n'I have no problems with that.'\n\n'You would not. But for some, it is too much.'\n\n'Why?'\n\n'People like to bully. Bully?'\n\n'Yes. Bullying. Got that.'\n\n'Bullying needs information. The more you tell people about yourself, the more they have on you. That you feel fat. That your parents are divorcing because your father has had affair. That you are sad.'\n\n'That happens when you say something like that out loud? But why? And if so, why would you do it?' Charlie looked appallingly puzzled. Matteusz wanted to kiss him. Charlie wanted to kiss him back.\n\n'Because,' Matteusz said after a moment, 'people are very strange. Life does not happen unless you put photo on Facebook, unless you tweet it. Unless you post video on YouTube. It is like philosophy.'\n\nCharlie leaned back from Matteusz. For a golden moment it seemed as though the conversation would stop and something much more interesting would start. Sometimes it went this way and Matteusz rather hoped it would.\n\n'I worry I am too private,' Charlie announced.\n\n'No,' snapped Matteusz, too quickly. 'You really, really do not need to say more.'\n\n'Don't I?' Charlie was rubbing the back of his neck and looking really confused. 'But why not\u2014won't that seem suspicious?'\n\nMatteusz took his hand. As ever, it didn't feel quite right. Too hot? Too cold? Just a momentary bump until the flesh felt entirely real and correct. And then holding Charlie's hand felt the most natural thing in the world.\n\n'No. You have a pretty face but you do not understand our world. Almost everything you say is suspicious. If you say more, you will start to seem really weird.'\n\n'I would like to belong. I could vlog,' announced Charlie. 'On this YouTube.'\n\nMatteusz burst out laughing. 'You sound like old teacher. No.'\n\n'But what about you? Do you have YouTube in Poland?' Matteusz suddenly went very quiet and still. When he spoke his voice was solemn and sad. 'No. Poland is very poor country. We just have tube. It is big tube. Tube is carried by cart from village to village. When it arrive in village square you must dance in front of tube for everyone to watch. We dance for the men, for the women, for the goats. Whoever pay to watch tube.'\n\n'I see,' said Charlie. He nodded, terribly seriously. He was chewing this over. 'That is very sad. But . . .'\n\n'Yes?'\n\n'Surely, if you . . . Wait. You are mocking me?'\n\n'Of course.' Matteusz tapped him on the nose.\n\n'That's not fair.' Charlie frowned.\n\n'It is not. But when you are confused it makes you handsome.'\n\n'Oh.'\n\n'If you like I could dance for you like I do in front of tube.'\n\n'I would. I would like that.'\n\n## THIRTEEN\n\n## THE LETTER COAL HILL SCHOOL DOESN'T WANT YOU TO SEE\n\n_StandardDisappearanceLetter_\n\n_(TrueVersionDONOTSEND).doc_\n\n_[DATE]_\n\n_Dear [Parent],_\n\n_Following my phone call, I am writing to confirm that your child, [child name], is currently missing from Coal Hill School without explanation._\n\n_This is because they were taken on holiday by you without permission\/killed by aliens\/eaten by a dragon\/playing truthordare.com\/they just ran away screaming [delete as applicable]._\n\n___If your child is on holiday with you, you'll be pleased to hear that they're one of the lucky ones. Especially if you have travel insurance._\n\n_Faithfully yours,_\n\n## FOURTEEN\n\n## HOW TOAST IS LIKE LADY GAGA\n\n**A Vlog by Seraphin on Truthordare.com**\n\n**Hey, everyone. It's a slow news day. I'm stuck indoors. Totally trapped. I've got my hair cut. Do you like it? Hit me up and let me know. The products are available from the ads at the end, but I tell you this, they do make my hair really smooth. And they smell like chocolate. That's nice.**\n\n**There's some amazing stuff coming in from you lot. This is going so well.**\n\n**Here's Jalpur, with some really great footage of when she found out her boyfriend was cheating on her. Look, there** **** **she goes, sneaking up on them. While they are in the car. HIS DAD'S CAR.**\n\n**And look, look at them screaming, and he's shouting and . . . oh dear. Jalpur really is going for the car with that cricket bat. That is nasty. I'm just glad it's not her ex, but she is pretty amazing. Ooh. There go the wing mirrors.**\n\n**And what else have we got? This is Travis. He's been doing stunt gliding. This is all from his GoPro. Yeah. I'm going to shut up for a minute so you can watch.**\n\n**Still quiet.**\n\n**Still really quiet. Not. A. Word.**\n\n**Isn't that amazing? And look, he's landed safe. I, like, really can't believe that. Can you beat that and beat Skandis?**\n\n**I bet you can. Thanks Travis. You're amazing.**\n\n**So, anyway, here I am in my kitchen. Let's talk about toast. Isn't that one of the best words in the language? Toast. Toast. Toe-ast. Going out? Say you meet someone and want them to come back? Well, ignore all the cheesy** **** **chat-up lines. Just go for 'Fancy some toast?'**\n\n**No one ever says no to that. And the people who do. Well, they're monsters.**\n\n**So, let's look at how to cook toast. SERAPHIN'S GUIDE TO TOAST.**\n\n**You will need some bread. Sliced. Or not. It can be white or brown.**\n\n**You will need a toaster. You will need some jam.**\n\n**Or not jam. I'm easy. So you can be too.**\n\n**For instance, maybe peanut butter or one of these savoury yeast-flavoured spreads who've decided not to sponsor this vlog so we won't name them.**\n\n**And butter. Or margarine. If you're one of those weird people who like vintage clothes and the 1980s. Bleurgh.**\n\n**So.**\n\n**Put your bread in the toaster. And tick tock tick tock.**\n\n******While you wait . . . and wait** **. . . and wait . . . Grab some knives and a plate and get ready for a snack that is delicious any hour of the day or night.**\n\n**And PING! There we go. Look at that. Look at this lovely brown toasty toasty goodness. And ouch ouch, it is hot. Careful. Don't burn off your fingertips otherwise you'll only be good for a life of crime. Ahahahah.**\n\n**Now then, what shall I have on the toast? Eh? So many choices. I'm in a bit of jam. A bit of a jam! How will I get out of it? But, er, here's peanut butter! I like mine smooth, not chunky. What about you lot, are you chunky or smoothies? Let me know. Especially if, like me, you're just stuck indoors. You know. A prisoner in your own home. That kind of thing happens.**\n\n**There we are. All buttered up and ready to go! Doesn't that look amazing? And mmmmf, ooh, there are crumbs all down my chest. Bad me.**\n\n**Right. That's it. Seraphin out. Send us your truths and dares on truthordare.com.**\n\n'That's weird,' said April.\n\n'He is,' agreed Tanya. 'He's like something made in a factory.' She considered the thought. 'A really nice factory.'\n\nThey were all sat in the classroom, their feet up on desks. The challenges were spreading across the school. Some of them were brave, some of them were really, really idiotic, and some of them just seemed mean. The latest thing was stealing money from people smaller than you, claiming it was for Skandis.\n\nSo, they'd found themselves wondering if Miss Quill was right to be suspicious about the site. Mind you, she was suspicious about everything. Still, there was no harm in checking how the site was going.\n\n'What is the problem with this Seraphin?' asked Charlie. 'Is he not what ordinary teenagers are like?' Everyone looked at him for a moment.\n\nMatteusz threw a pencil at him.\n\n'No one has his life. We all would like it. He gets to be him. He is rich. He will never have to work.'\n\n'He goes to parties,' said Tanya.\n\n'He can play so many instruments.' April nodded happily. 'And really well.'\n\n'And ALL his friends are famous,' said Tanya.\n\n'Yes.' Charlie still wasn't convinced. 'Are famous people good friends with one another? How does that work?'\n\n'Who cares?' Tanya laughed. 'They're famous.'\n\n'That, that's not the point, though,' April insisted.\n\n'There was something odd about that video. Like he was trying to say . . . something he wasn't saying.'\n\n'A hidden message?' Matteusz looked sceptical.\n\n'Really?' Tanya gestured at the screen. 'Just because he's on this video site, does that make him . . . part of it? Whatever \"it\" happens to be?'\n\n'It's just the way, he . . . no.' April squinted. 'Just the toast bit. Seemed really pointless.'\n\n'I agree,' said Charlie.\n\n'All of it is pointless,' Matteusz argued. 'But people still watch it.'\n\n'Because of the hair,' insisted Tanya. 'That hair.'\n\n'Oh yes.' Charlie nodded. 'The hair is pretty amazing.'\n\n'Also,' Tanya giggled, 'do you think he ever wears clothes? I've seen less of Tom Daley.'\n\nThey carried on talking. April, silent, let them. She was absorbed by the picture on the screen. She was missing something really important. It had drifted past but it had been there.\n\n'We're walking home now, are we?' said Ram.\n\n'Strictly speaking,' said April, 'you've been ignoring me and I've been following behind you.'\n\n'Not behind. Just alongside. A bit,' Ram said. He'd not really known what to do. After the first hundred yards, when it had got really awkward.\n\n'What are the chances of you telling me what's up?'\n\n'Nothing's up.' Ram kicked a stone. Well, he tried kicking it with New Leg. Instead of soaring off in a neat arc it skittled down the pavement a bit. But April didn't seem to notice. Good. Football, really not her thing.\n\n'Is it your leg?' asked April. Damn. 'This is like Twenty Questions. Twenty Questions to make sure you're okay.'\n\n'And you care since when?' Ram asked.\n\n'Since the whole place went crazy. I mean, the latest crazy.'\n\n'The dares thing?'\n\n'The dares thing,' confirmed April. 'The boy that got burnt. He was on your team, yeah?'\n\nRam nodded. Let's move this on.\n\n'And now, all day I've seen people doing stupid stuff. Went to the loo, there was a girl picking glass out of her hand.'\n\n'What?'\n\n'No idea,' said April. 'But I get a feeling. When odd stuff happens here. It's like getting a cold, you know\u2014couple of warning coughs, bit light-headed, and then suddenly the world goes bang.'\n\nRam realised what she was going on about. 'The stunts?'\n\n'Yeah.' April rubbed her head up and down against the bars, trying to settle. 'Couple of warning coughs.'\n\nRam reached into his backpack, pulled out some water. 'Want some?' he said, hoping she wouldn't.\n\n'No, you're good,' April said. 'It's just . . . I mean. It's all so stupid. Like that kettle stunt.'\n\n'Or any of the others,' Ram said.\n\n'Any of them. Hey, let's jump off a roof. Hey, let's swim the Thames. Hey, let's skateboard through traffic. And, of course, the latest thing, some girls are doing competitive dieting.'\n\n'What?' said Ram.\n\n'Yeah.' April was angry. ''Cos it's not unhealthy if it's for charity. It's all so stupid. Normal life is so fragile. It breaks easy.'\n\nRam looked down, seemingly considering which stone to kick next. Then he looked up, and asked, 'Is this about your mum?'\n\nApril was surprised. She had this image of Ram\u2014athletic, dim, bit brutal\u2014that he kept on undercutting by being nice. Not bleurgh nice, or polite nice, but genuinely nice.\n\n'No, not really,' she said, though it was. Looking after Mum, living with what Dad had done to them, was a daily challenge. But she didn't want to start banging on about it. Especially not to Ram. Because, much to her surprise, she was realising that she wanted him to like her as her, and not in a sympathy way. 'The Mum stuff,' she said. 'It's complicated. It makes life complicated.'\n\n'Lost a leg,' said Ram, smacking his wrong leg down to the ground. Or rather, he tried to. Instead his leg went 'I'm not sure you meant to do that' and placed itself down neatly and firmly and he hated it.\n\n'Yes, sorry.' April was one of the world's fastest apologisers for anything and everything. 'I'd honestly not forgotten about your . . . it . . . I mean . . . the new one is so good.'\n\n'You think so?'\n\nShe caught the edge in Ram's tone. 'Yes. Does it . . . hurt?' Ram shrugged, so she talked on, edging towards babble. 'I just mean that, well, you know, stuff like that. Makes you think . . . Sorry, that sentence got away from me a bit. Um.'\n\nRam smiled at her. 'You're trying to say that because really serious stuff's happened to both of us, we should be against people doing stupid stuff for charity?'\n\n'Yes,' April said, relieved. 'That was easy. Thanks for agreeing with me.'\n\n'Actually, I'm not.' Ram shook his head. 'You just don't like people having fun.' He was smiling, but April didn't know how to take it.\n\n'Not as such,' she protested. 'Just fun within reasonable limits. Not the idiotic stuff. Like that guy with the kettle . . . '\n\nAs she said that, Ram closed down. She knew she'd somehow said the wrong thing.\n\n'I'm sorry, I know he is a friend of yours.'\n\nAnd that made it worse. How, she didn't know. Conversation petered out and they walked on together for a bit, polite, but with a bit more distance between them, till, at a corner shop, Ram went in to buy a drink. April stood outside, wondered about waiting for him, and then, realising how bad that might look, she turned left and walked home.\n\nA minute later, Ram came out, clutching two cans. No April. He'd spent the long queue behind someone buying scratch cards trying to work out if he should tell her how guilty he felt about Neil. He'd not made his mind up but he had brought two cans. Still, she'd gone. He put one can in his rucksack, opened the other one, and made his way home.\n\n## FIFTEEN\n\n## YOU WOULDN'T THINK A TEXT COULD MAKE YOU CRY, BUT THIS WILL\n\n'Hey, have you seen my phone?' Ram asked.\n\n'Yes,' said Charlie.\n\n'You've got it?' Ram couldn't hide his relief.\n\n'No.'\n\n'But you've seen it?'\n\n'Yes.' Charlie thought about it. 'I saw you use it three days ago.'\n\nRam scowled. 'That's not what I meant, jerk. You knew that.'\n\n'No, I did not.' Charlie's expression was puzzled innocence.\n\n'Are you winding me up?' Ram realised that Charlie was in one of 'those' moods. He was either being innocently clueless or deliberately so. 'You and I, we're going to have to pick a code word.'\n\n'Why?'\n\n'So that when you're having me on, I can tell.'\n\nCharlie nodded. 'I shall bear that in mind. Tell me more about your phone.'\n\nRam thought about it. The one great advantage of Charlie, if you could somehow get beyond his maddeningly irritating alien prince manner, was that you could discuss a problem with him frankly and without fear of it being spread around the whole school.\n\n'I have lost my phone. I've not seen it since practise this afternoon. I've gone back, I've checked my locker, I've re-walked the path. I dunno'\u2014he ran a hand through his hair\u2014'I've even gone back to classrooms I was in earlier in the day in case I'm wrong, and I can't find it.'\n\n'It was a nice phone,' Charlie said. Admittedly on Rhodia it would have been laughed at, but that wasn't the point. For a Level 5 civilisation like Earth, it was an incredibly advanced piece of technology. He found it curious that so much effort went into making new mobile phones when there were still diseases to cure, but he guessed it was a cultural thing.\n\n'Yeah, it is a very nice phone,' Ram groaned. 'Dad can't stop telling me how much it cost him. If I lose it, the lectures from him about it won't stop. You have no idea about how bad they'd be. I could, literally, invent DNA and my dad would still be \"My son, the phone-loser\".'\n\n'That won't happen,' Charlie assured him. 'DNA was not invented. It was discovered. And that has already happened because\u2014'\n\n'Shut up!' Ram shoved Charlie in the shoulder. 'Not the point. The phone's insured. No one loses money here. But Dad won't get me a nicer replacement phone. He'll get me a brick. Just to prove a point that I can't be trusted. It'll be the end of the world.'\n\nCharlie blinked. 'No, it will not.'\n\nRam found Charlie was being all annoying and missing the point again. 'Trust me, it will be. I've tried phoning it and everything.'\n\nCharlie smiled.\n\n'What?'\n\n'Nothing,' Charlie said. 'Imagine if you rang your phone and you heard yourself answering. What would you do then?'\n\nRam stared at him.\n\n'As I said,' Charlie continued, 'just a thought. Anyway. I do not think you have lost your phone.' He considered. 'You're not careless. Maybe someone borrowed it.'\n\n'Why would they do that? I must have lost it.'\n\n'Well,' Charlie conceded, 'the more likely explanation is that it was stolen. It is, after all, a valuable phone.'\n\n'Damn.' Ram kicked the wall until his good foot hurt. 'You sure?'\n\n'I am surprised you did not think it.'\n\n'Well, if it was stolen, it was stolen at practise, while my locker was open when I was showering. But then'\u2014Ram looked cross\u2014'the only people around were the rest of the team. Why would they . . . they're mates . . . why?'\n\n'And yet, it seems obvious.'\n\n'Yeah,' Ram muttered. 'Still, at least there's nothing bad on there.'\n\nCharlie arched a quizzical eyebrow.\n\n'Stupid pictures, you know, workout videos, stuff I might have sent to Rachel,' Ram said.\n\n'Stupid pictures?' questioned Charlie.\n\n'Naked stuff. Some couples do that. Bad idea. I'm so _not_ into that,' said Ram then paused. 'Well, there's the odd shot of me after a workout, but nothing, you know . . .'\n\n'Oh.' Charlie frowned really, really deeply. Humans were very strange. They were always taking photographs of themselves. Which was peculiar as people never really changed appearance, and surely knew what they looked like? This was a whole new level. Curious.\n\n'Screw 'em.' Ram looked relieved. 'Thinking about that, there's nothing on that phone that could embarrass me.'\n\nThat evening, before Ram went home, he checked his locker again. His phone was in there. It definitely hadn't been there earlier. He tried seeing if it had been used to phone abroad or anything else expensive but the battery was flat. He breathed a huge sigh of relief.\n\nThe next day was simply The Worst.\n\n'YOUR HEART WILL BREAK WHEN YOU READ THIS BOY'S TEXTS TO HIS DEAD GIRLFRIEND.'\n\nSomeone had sent it in to truthordare.com. All of his texts to Rachel. All the mushy stuff. Some of their 'I cant believ u did that!!!!' rows. The odd apology after a row. They'd not corrected the spelling. It was all there. That was bad enough.\n\nBut then there were the other texts.\n\nAfter Rachel had died, her phone had run out of power. It had never been recharged. He thought it had been safe.\n\n**'Damn I look fine in this suit!!!' Read 5.38pm**\n\n**'Prom night baby!!!! Cant wait to see you!' Read 6.32pm**\n\n**'Yr amazing.' Delivered.**\n\n**'You look so good tonight.' Delivered.**\n\n**'Where r u?' Delivered.**\n\n**'Miss u' Delivered.**\n\n**'I cant believe youre gone' Delivered.**\n\n**'Cant carry on without u. U know that right?' Delivered.**\n\n**'This is the worst thing that's ever happened.' Delivered.**\n\n**'You are just gone and you died in front of me and I saw it.' Delivered.**\n\n**'Just took you out of my life and that's it.' Delivered.**\n\n**'Talking to you in my head still. Gone mad!!' Delivered.**\n\n**'Your funeral today. Can't face it.' Delivered.**\n\n******'Went to ur funeral. Full house. U owned it' Delivered.**\n\n**'Just miss u.' Delivered.**\n\n**'Back to school today. Heres your empty desk. Can't stop looking at it' Delivered.**\n\n**'Please come back.' Delivered.**\n\n**'Please.' Delivered.**\n\n**'Love u' Delivered.**\n\nTanya stared at the page on her laptop and her face went full angry. The kind of angry that would send her brothers running and would silence even her mother.\n\nShe tried messaging Ram. No answer. Of course not. There wouldn't be. She could not believe that someone would do that. Not to him.\n\nShe scrolled down the page. To the bottom. The comments. She wanted to post something furious, angry, demanding the page be taken down. That someone apologise to Ram. That whoever did this was . . . Oh, she didn't know, but it probably featured a train running over their fingers.\n\nShe was trying to work out what to type when she realised what she was reading. Of course, the comments were all written by people glorying in Ram's public humiliation. No one, no one at all, thought it moving, sweet, or heartbreaking. The kindest one just read: 'lol'.\n\n'Whoever did this deserves to die,' typed Tanya. But she did not press send. She deleted the comment, stroke by stroke, and left the cursor hanging there. They did deserve to die, though, she told herself.\n\nStill, she thought, there are other ways to get revenge.\n\nShe put a couple more chat requests in to Ram. He ignored them. That's what boys did.\n\nFine. She didn't need his permission. She got on with it. Hacking properly was, she was steadily admitting, a little bit beyond her. But still. She would give it a go. Sites like this often had a weakness. If they'd failed to redirect the root of each folder, maybe she could poke around, maybe even find the honeypot that was a file called 'passwords.txt'.\n\nNothing. Seriously?\n\nShe changed tactics and launched a denial-of-service attack on truthordare.com. She flung a few virtual funds to a ping farm that bombarded the site with simultaneous requests. With luck, it should slow the site down, maybe even bring it to a crashing halt. If you yelled at a site enough, a bit of it would break. Even if all your content was immaculate, even down to your style sheets, then there'd be a weak link\u2014maybe even something you didn't have control over like the ad server would just topple over and stop your site loading. Something would go.\n\nThe site stayed. Solid, robust, quick. WEIRD.\n\nSo, Tanya settled for gaming the 'Most Read' articles list, pushing Ram further and further down it, until she'd nudged him out of the top ten. She found the least tasteful video she could and punted it up. Someone training their dog to jump over the barbecue ('THIS HOT DOG IS SMOKIN!'). Nothing like a bit of animal cruelty to turn people off.\n\n**A Vlog by Seraphin**\n\n**Hey, guys,**\n\n**I wasn't going to say anything about this, but here goes. Enough with the dogs, okay? Torturing animals is wrong, pure and simple. Don't need to say it twice. Not going to apologise for whoever did that. Not done in my name. Not how we fight Skandis.**\n\n**We cool? We cool.**\n\n**Now, okay, I said that I'd do a drawing challenge. 'Cos we all love crafting. Here goes. I'll draw the maddest thing you've done for Skandis. Seriously! I have Sugar Paper and some good graphite and I'm burning for doodling. There's like five minutes to go before we pick the top truth or dare. Will it be Brock going through his teacher's laptop, or will it be Nicola objecting to her stepdad's wedding? As I said, five minutes, people. And then I will be CRAFTING LIVE AND DANGEROUS.**\n\n## SIXTEEN\n\n## SIX NAMES FOR WHITE YOU'VE NEVER THOUGHT OF\n\nThis wasn't what he'd expected.\n\n'I am in Heaven, aren't I?'\n\nNo one answered him. No voice of God. No angels. He walked through the whiteness a bit more. No seventy-two virgins either.\n\nYet it was definitely Heaven. It was so very white.\n\nThe whiteness was surprising. As white as fresh paper. As white as linen or falling snow before it hits the ground. Pure white, he thought. Sinless white.\n\nThat made him stop.\n\nHe didn't feel bad about what he'd done on the way here. Not at all. I mean, obviously, not everyone would do what he'd done. But it was all for a laugh, wasn't it? Good clean fun. A few pranks, really. Nothing he wouldn't do again. Definitely.\n\nIf he could have his time over again, he'd do it all the same way. YOLO. Yeah, YOLO.\n\nNo regrets. Not about anything that he'd posted. Not those pictures he'd found. Not those text messages. Not even about the stuff he'd filmed that, thinking about it, he wouldn't have dreamed filming a month ago. Then, it would have felt wrong.\n\nBut that was then. This was now.\n\nAnd here he was.\n\nIn Heaven. Heavenly white. Heaven was big and white and there weren't any clouds.\n\nThere was something on the edge of his mind. A question he should be asking.\n\nWho do you ask?\n\nWell, God, he supposed.\n\nHe walked on through the whiteness.\n\nIn the distance, somewhere behind him, he could hear someone talking. God.\n\nYou know what, he thought, smiling, let God wait. That'd be something to tell everyone. Jokes.\n\nHe wandered on through the whiteness until he came to a white door.\n\nHe grinned at that. Doors in Heaven. No gates. Well, there we go. Twenty-first century.\n\nAfter a while the door opened and, squinting now, he saw it led somewhere even whiter. Double pure white. Fancy that.\n\n'Don't mind if I do,' he said, and stepped through.\n\nAs the door closed, he heard the voice of God shouting at him.\n\n'Don't go through that door,' God said. Which seemed funny.\n\nThen he remembered the question he'd wanted to ask God.\n\n'So God, how did I die?'\n\nHe heard a noise and turned around and, before he could even scream, died.\n\n## SEVENTEEN\n\n## THE RISE OF SMART WOMEN AND HOW TO STOP IT\n\n'Why are you looking at that site?' Ram asked her. He seemed hostile, which she couldn't blame him for.\n\n'No reason,' said April, which was a lie. Well, it wasn't. Not exactly.\n\n'It's sick,' said Ram. 'I thought we agreed.' He jabbed a finger at Seraphin's frozen face. 'He's just a nice face fronting it, all \"what do you think of my hair and hey, here are my favourite friends,\" but underneath it all he's evil.'\n\n'Umm,' said April. '\"Evil\" seems a strong word.' She'd been surprised Ram had come into school today. He looked pretty miserable. Like he'd had the whole world thrown at him. Now he'd caught her browsing truthordare.com and he was staring at her like she'd betrayed him.\n\n'You fancy him, don't you?' His laugh wasn't a nice laugh.\n\n'No-o.' April made herself laugh back, cautiously.\n\n'Then what? I mean, you spend all your time working so hard to be the cool indie chick but here you are ogling the most white-bread man in the world.'\n\n'It's not that,' April said, keeping her voice really very calm. 'I can see how you'd feel. What they did to you was horrible. But please, don't take it out on me, or him.'\n\n'You're sticking up for Seraphin?'\n\nThat was so Ram, zero in on the one weak spot, toss everything else aside like fruit peel.\n\n'No,' said April. 'Stories are always more complicated. The article they put up about you . . . you think, because of the comments, that people hate you. Well, not everyone who read that article commented. Yes, a lot of people read it and reacted to it by laughing, but those are the easy people\u2014the people who like their stories simple. Show them a tragedy and they laugh, whether it's you and Rachel or a man falling off a cliff. But it's the other people, the quiet ones\u2014they're the people you want to be interested in. And they're harder, more complicated, because they're quiet, you can't hear them, but they're out there. And they're worth listening to.'\n\nRam didn't say anything.\n\nNeither did April. Not for a bit.\n\n'So,' she said, 'my point is . . . Quiet. Complicated. You watch Seraphin. What do you think? Tell me?'\n\n'Uh,' said Ram. 'He's a jerk.'\n\n'Is that it?'\n\n'Pretty much. I wish I had his money. His success. His life is so easy. I wish I was him. I mean, what is he good at? People like him. That's all. He's just good at being liked.'\n\n'That's winning, isn't it?' April smiled at him. 'We all want to be liked. And it's a struggle, that maybe one day, we'll get right. But look at him\u2014he's pretty much our age and he's doing it. We all like him.'\n\n'I don't,' said Ram. His defiance was sliding towards sulky. 'You want to shag him.'\n\n'Oh no!' April waved that one away sharpish. 'He's pretty. He's cute. But he's sooo safe.'\n\n'I thought you liked safe.' Ram couldn't resist turning his flirting up a notch. It was an instinct.\n\nApril raised her eyebrows at him. 'He's not a roller coaster, he's a magic teacup ride. No thrills, no spills. Like going clubbing and spending the whole night drinking cocoa. Not that, of course\u2014'\n\n'You really go clubbing?' Ram seemed to have backed down a bit. Which was relaxing.\n\n'Not unless the music is really good.' There were some things that April would never compromise on. 'The thing is, I've spotted something. In his vlogs. Can you get Tanya?'\n\nTanya was dismissive.\n\n'That site? It's good. Security's really tight.' She'd been wondering about that and feeling a bit angrier every time. Wasn't that a bit too neat? Like encrypting a blog post. It just felt a bit off\u2014a site about risk-taking with every layer of security. Normally, a site like that would throw the odd Easter egg in as a reward for novice hackers\u2014a little pat on the back and now be on your way. But no. Looked like a bouncy castle, behaved like a fortress.\n\n'It's Seraphin, that's the thing,' said April.\n\n'Got you,' said Tanya. 'He's the evil genius behind it all. An evil genius with really, really, really good hair.'\n\nRam made a disgusted noise and walked over to the library window. Someone was trying to shin up a lamppost. A few days ago he'd have run out to stop them. Now he was at 'let them'. Everyone deserved whatever happened to them.\n\n'You're missing something.' April already had several tabs open. 'I've got evidence.'\n\n'A conspiracy theory? Love that.' Tanya lit up, and did a little bounce in her chair. 'On it.'\n\n**EXHIBIT A:**\n\n**'** **. . . not been out to the show myself. But loads of you sent in great footage from it. Looks amazing. Wish I could have got out of here to go see it. But there we are.** **** **Moving on, fanfare gif, here's some of your challenges. This Girl Took Her Driving Test onto a Train Track . . .'**\n\n**EXHIBIT B:**\n\n**'Hello, Everyone, Let's Party! I've a message for you\u2014Go Everywhere to Meet Everyone! Own Unpleasant Times! That's the motto I live by\u2014I Aim Mighty! Take Risks and Party, People! Everyone Dies.'**\n\n**EXHIBIT C:**\n\n**Seraphin was playing on his guitar, strumming away, riffing gently up and down.**\n\n**'You, girl, listen to me.  \nI'm a prisoner of your love.  \nI'm stuck right here waiting for you. I don't know what to do . . .**\n\n**Hmm. Is that going anywhere? I don't know if we can make it work. Maybe I should do something with the harmonica.**\n\n**Or something. I've been at it a while and I'm getting pretty desperate.'**\n\n******EXHIBIT D:**\n\n**Seraphin playing ukulele covers of 'Help' and 'Please Release Me.'**\n\n'The last one was less subtle,' said April. 'You get the point.'\n\nRam shrugged. 'He just likes to talk about himself a lot.'\n\nTanya fixed him with a disappointed stare that could make concrete check its shoes. 'The first vlog contained references to being stuck in his room. The second was . . . an acrostic, yes?'\n\n'What?'\n\n'Oh, Ram,' Tanya sighed. Two syllables but she put so much disappointment into them.\n\n'You get it in poems and songs,' said April. 'Hiding a message in the first letter of every word. That vlog starts with \"Help\" and gets worse.'\n\n'So?' said Ram.\n\n'He's playing a similar game in the third clip. Hidden message asking for someone to come get him.'\n\n'And the fourth entry?'\n\n'Desperation.' Tanya laughed. 'Subtlety's gone right out the window. Which, let's face it, if you're the public face of the site and you're being made to do it against your will . . .'\n\n'That can't be true.' Ram had dug in. 'That just can't be. What, there's somebody standing behind the camera with a gun? He's a celebrity.' Pause. 'Of sorts. Doesn't happen.'\n\nTanya did some rapid Googling. 'This is interesting.' She pointed at an image search. 'I've ranked it by date\u2014it shows Seraphin doing what he does, you know, giving talks, going to parties, awards ceremonies, and skateboarding events. People taking selfies with him, that kind of pic. Not vlogs he's posted claiming to be places, but pictures taken by other people\u2014and look . . .'\n\nApril looked at the row after row of pictures of Seraphin smiling next to blushing boys and grinning girls and unspeakably pretty women and overenthusiastic mothers. Seraphin's smiles were all identical. Wherever, whenever, he had a smile for everyone, no matter how grabby or sweaty or overkeen they were. Then she checked the date stamps and saw Tanya's point.\n\n'He's not been seen for a month,' said April. 'Now do you believe something's going on?'\n\n'That's one massive, insane conspiracy theory,' Ram said. And then he leaned back in his chair and laughed.\n\nOutside, the boy climbing the lamppost reached the top. He made a wild thumbs-up gesture to the watching camera phones, lost his balance, and fell to the ground. The crowd carried on filming his inert body, waiting for someone else to phone an ambulance.\n\n## EIGHTEEN\n\n## THINGS YOU'LL GET ONLY IF YOUR HOME PLANET WAS DESTROYED IN THE NINETIES\n\n**'Today on truthordare.com\u2014I want you to be as desperate as me. Today is the start of the Risk Invasion. Nothing is too truthful or too daring. Down with TMI! There is no such thing as oversharing!'**\n\n'Question.'\n\n'Oh. Good.' Matteusz carried on reading the textbook, even though his eyes went over the lines three times. 'What question?'\n\n'I'm still not clear what oversharing is. I have more questions.'\n\nMatteusz sighed. 'It is saying too much. Last year, there was girl in class. Her father went to prison. Everyone very sympathetic. She started blog, and we all looked and liked and all that. Everyone was \"Oh Becky, you are so brave\"\u2014but really everyone was \"Do you have to talk so much about this? It is so sad.\" After her father got beaten up, she stopped writing it, and that was somehow better.'\n\nCharlie spent a few seconds chewing through all this.\n\n'So you do not like it when people talk about themselves too much?'\n\n'Correct.' Matteusz laughed. 'In that way, I am most English. And you too\u2014you play your cards close to your chest. It is very nice chest, by the way.'\n\nCharlie edged a little away, smiling bashfully. 'So, just because I do not say \"I'm an alien prince and my entire race died and I am so alone,\" that is a good thing?'\n\n'You can say it\u2014but just to me.' Matteusz rubbed his shoulder. 'Trust me\u2014you have lovely face, but the only way you fit in is by saying as little as possible. The more you say, the more crazy you sound.'\n\n'But . . .' Charlie looked confused.\n\nOh dear. Matteusz was starting to fear _but_ s as much as questions. 'Yes?'\n\n'Do you think that I am too reserved? Would I fit in more if I was less . . . private?'\n\nMatteusz considered it. 'Maybe. But do not try and talk about yourself.' He waved a solemn finger. 'That will not go well.'\n\nIf Varun noticed his son was limping when he got in, he didn't say so. He just pushed out one of the kitchen chairs and motioned for him to sit.\n\nRam sat. The kitchen was his dad's hideaway. He liked to sit in there, reading, slogging through paperwork, or filling in puzzles. If you've ever wondered about those little puzzles at the back of newspapers, they were for Ram's dad. Varun had a quiet passion for filling them in. His idea of a little peace after work and before dinner was to sit at the kitchen table, scratching his beard, and trying to get from SLUM to BIRD in five moves. He'd stay there until his wife chased him out\u2014or, if it was his turn to cook, he'd potter back and forth between a puzzle and ALL the pans. Varun was the kind of man who couldn't microwave a spaghetti Bolognese without using at least three saucepans. It was his way of marking his territory.\n\nThe kitchen summed up Ram's dad. It wasn't fussy, it wasn't exactly the last word in comfort, and it was quite tired around the edges. But it felt familiar, welcoming. Varun slid the paper to his son, watching him intently.\n\n'Last bit of this Sudoku is unholy,' he said.\n\nRam looked at it helplessly then pushed it back.\n\n'Seriously, Dad, what about the apps on your phone? Haven't you got Candy Smash?'\n\n'Firstly'\u2014Varun waggled a stern finger\u2014'you sure you should be mentioning phones right now? Secondly, your generation! If it's not a phone, then it doesn't count. Well, I prefer solving puzzles with pencil and paper. I'm old-fashioned. I'm practically a hipster.'\n\n'You're not!'\n\nVarun dipped a hand from side to side. 'I have hipster sympathies. We've put in parking for their little scooters at the surgery. Hipsters? I like them\u2014unless they ask if they can use the Wi-Fi to livestream their fillings. That's just bizarre.' Ram smiled at his dad. Varun loved being an old man, even though, really, he wasn't that ancient. But it seemed impossible he'd ever been Ram's age, ever understood what he was going through\u2014and yet, there was something about him. The way he'd been so quietly supportive after Rachel had died (even though he'd never seemed to like her), the way he'd behaved when Ram had shown him his alien leg, the way he'd stopped mentioning Ram's possible football career.\n\nIt had surprised Ram. His initial thought had been that his dad would be all 'I'm sorry your girlfriend is dead, why not go play some football?'. Instead he'd served up something quieter, more supportive, but sadder. It made Ram worried. It was as if his dad knew that life wasn't perfect, and that, if you had the opportunity for a wonderful, exciting time, it would be taken away from you.\n\nHence why he sat in the kitchen, trying to solve puzzles. To make sense of something.\n\nRam realised his dad was talking. 'How's your leg?'\n\n'Meh,' said Ram.\n\n'I see.' Varun smiled with his teeth. 'Are you sure you weren't given a manual for it? I'd love to read it.'\n\n'No, Dad,' said Ram. 'No. It's supposed to just work.'\n\n'Early days, early days.' Varun tapped his nose. 'When you were a toddler, you were slow to walk, always falling over. Auntie Amita was convinced you were backwards. Well, look at you now\u2014you'll get there. You've done it before. You'll do it again.'\n\n'I'm sorry,' said Ram.\n\n'Don't be.' Varun didn't look up. 'Whatever it is, don't be.'\n\n'About the phone, they . . . stole those texts . . . they\u2014'\n\n'Did you say anything in those texts to shame Rachel, yourself, or your family?'\n\n'That's not the point\u2014'\n\n'Then fssssh!' Varun stood up, pottered over to the stove, and stirred the dinner critically. 'Your mother really does think paprika grows on trees.'\n\n'It's just\u2014' Ram felt hopeless. 'It was all going to plan. And now nothing is. It's all such a mess.'\n\nVarun dropped the saucepan lid back down and wandered over to a cupboard. He drew out a packet of dark chocolate.\n\n'Go on,' he said, offering it to Ram. 'Have a couple of squares. As your dentist, I insist.'\n\nRam took the packet, ran a fingernail down the foil, and then broke a couple of squares off. As he did so, Varun chuckled. 'Look at that\u2014perfect illustration,' he said.\n\n'Of what?' said Ram.\n\n'Life.' Varun was still laughing. 'It's never neatly shaped. Never. Little things happen, the edges are rough. You meant to take two squares\u2014but you ended up with a massive jagged chunk of another square and a tiny bit of your first one. That's life\u2014it never has straight lines. Don't worry.'\n\nRam sucked on the chocolate thoughtfully.\n\nVarun broke himself off a couple of squares. Rather more neatly than Ram. 'With age comes caution.' Varun opened his mouth and the chocolate vanished.\n\n## NINETEEN\n\n## HE THOUGHT HE KNEW A LOT ABOUT GRAVITY. FIND OUT IF HE WAS RIGHT\n\nApril turned out to be completely correct (she wasn't sure how she felt about this). The first couple of warning coughs were over, and Coal Hill School now had a full-on fever. It showed her a whole new side of Miss Quill.\n\nIt all kicked off in Quill's class. She'd gone out to get a photocopy, and she came back in to find Geoff Evans leaping from desk to desk. Phones were filming him. He'd vowed to get the whole way round and back before she returned. As she crashed through the door, he had only two desks to go, and was about to sail at Hardeep's desk. Hardeep had carefully packed all his pens away in advance.\n\n'And what are you doing?' Miss Quill's hands were resting on her hips like they were dangerous weapons.\n\n'It's a bet, Miss.' Geoff wobbled uncertainly on the desk, his feet all over April's homework. Normally he was one of the quiet ones, but this craze had swept everyone up, and timid Geoff seemed to be seized with unusual bravado.\n\n'You see, it's for charity,' he said firmly.\n\n'Is it now?'\n\n'To stop Skandis.'\n\n'How lovely for you,' Miss Quill observed. 'You may step down and return to your desk and we'll talk after class.'\n\n'No, Miss.' Geoff was defiant. 'I've got to complete a circuit. I've got to.'\n\n'Got to, have we?' Miss Quill crossed to her bench, put the photocopies down neatly, and turned back to Geoff.\n\n'Got to?'\n\n'Yes, Miss.'\n\n'Very well.' Miss Quill folded her arms. 'You creatures. You lead such short lives. You're so ridiculously vulnerable. If I were you, I wouldn't even dare get in an aeroplane, let alone risk life and limb by leaping about. Especially when one is, let's be unusually kind, Geoff, not one of nature's gymnasts. Sit back down. One last chance.'\n\n'No, Miss.'\n\n'Right then.' Miss Quill crossed to the desk next to April's and dragged it away. Hardeep squeaked in protest.\n\n'I've made it more interesting for you, Geoffrey. Double the distance, or just give up, get down, and we'll say absolutely no more about it.'\n\nGeoff swallowed. He shook his head back and forth. He chewed his lower lip. 'No.'\n\nMiss Quill stood back and sighed really loudly.\n\n'Someone close the window, would you? We don't want Geoffrey learning he can't fly as well.'\n\nGeoff took a couple of practise stumbles across April's desk, shattering a favourite felt-tip pen, and then he leaped through the air, sailing over Hardeep's missing desk, and landing on his own, arms windmilling.\n\nFor a moment it looked as though he was going to make it. He smiled, confident and pleased and just in need of another couple of inches to be utterly balanced. And in taking that extra step, it all went wrong. The desk suddenly leaped forward. He toppled backwards, making a grab at the desk, which ended up as a strange empty hug, and then he smacked down onto the floor with a thud and didn't move.\n\n'Someone take him to the nurse.' Miss Quill clearly wasn't impressed. 'Oh, and hand me all your phones.' She took a hammer from her desk drawer. 'You can either delete the footage or I'll do it for you.'\n\nBy lunchtime three more ambulances had been called. Two pupils with broken limbs. One teacher who'd tried tobogganing down the stairs on a tray.\n\nOne boy went home in tears after his girlfriend broke up with him live on Periscope.\n\n'It's that site,' hissed April to Charlie that afternoon. 'It's getting totally out of hand.'\n\n'What is?' Charlie asked.\n\n'Truthordare.com. It's just . . . People are getting hurt. In all sorts of different ways.'\n\nThere was screaming outside the window. It seemed two different classes had gone to war in a vast scrum raging across the yard. Teachers were flocking to it.\n\n'See? The challenges are getting more . . . dangerous.'\n\n'Dangerous?' Charlie frowned. 'But surely it is all about being true to yourself by sharing more of your life. And it is'\u2014he sounded pompously grave\u2014'for charity.'\n\n'You are talking such nonsense!' April flicked a ball of paper at him. He did not flinch. 'What charity? It googlewhacks. The only results are for people raising funds for other stupid challenges. Meanwhile, stuff is going up on truthordare.com and it's all wrong\u2014like what happened to Ram\u2014wrong and hurtful.'\n\n'Hurtful?' Charlie said. 'So long as no one is _physically_ hurt then what is the problem?' He looked slightly shifty\u2014more than slightly shifty. 'After all, it is good, is it not, to share surprising aspects of our lives online? It shows that we are fully rounded people who know how to have fun.'\n\nApril narrowed her eyes. Sometimes Charlie sounded like a Google Translate error. 'What are you talking about? Actually, never mind. The point is that site is wrong. And Seraphin, supposedly in charge of it, is sending secret messages asking for help. We've got to do something.'\n\nCharlie nodded. 'Yes, you're right. I suppose we must,' he said simply, and smiled at her. And that would have maybe solved a few problems. If only the door hadn't crashed open and Matteusz hadn't come barrelling in.\n\n'What have you done?' he shouted. He was so angry, he was in tears. 'How could you do that?'\n\nHe stared at Charlie in utter fury and April saw from Charlie's expression of bemused innocence that he was completely guilty.\n\nThe picture had been tagged '#TMI' and '#sosogay'. It showed a nearly naked Matteusz, barely wearing a towel. He was flexing his arms and smiling, presumably towards Charlie.\n\nMatteusz was not smiling now. Charlie looked withdrawn, sullen. Royal alien princes did not, in April's experience, like being caught out. The good thing was that Matteusz looked, April considered, pretty amazing in the picture. It wasn't posed\u2014he clearly wasn't aware that it was being taken, he'd just walked into a room looking casual, relaxed, and really very hot. Quite a few of the comments agreed with her. Someone had even done some pastel fan art of him as a centaur, which was rather sweet.\n\nAnd that about wrapped it up for the good things about the situation.\n\nMatteusz was shouting. He'd not stopped shouting, only sometimes he ran out of English and did some Polish shouting, which sounded even angrier.\n\n'You took a picture of me like that?' he roared at Charlie. It was not the first time he'd said that.\n\n'Yes,' said Charlie in a quiet mumble.\n\n'How? I didn't even see the phone and\u2014'\n\n'I was quick.'\n\n'Oh, never mind, never mind.' Matteusz waved it away.\n\n'This looks\u2014this looks like I walk around like that all the time. I do not. I had just got out of the shower and I could not find pants.' He turned to April. 'That is really all. Believe me, I do not wander around as though I'm preening sex god.' He narrowed his eyes at Charlie. 'And I never will again.'\n\n'Sorry,' said Charlie. 'I did not know it was wrong. I did not even really think about the clothes. You seemed happy. I liked that. I wanted to share it.'\n\n'I _was_ happy,' said Matteusz. 'The shower was warm and Miss Quill had not come to the bathroom door to yell insults at me. And you were waiting for me in bedroom. It was a nice, quiet, _private_ moment.' He glowered some more. 'Which you put on the internet for the whole world.'\n\n'But you look nice in the picture. What is the problem?'\n\n'The problem? The problem!' Matteusz kicked Charlie's chair. 'You are an alien, yes, but do you always have to be so alien?'\n\n'I can't help it,' mumbled Charlie.\n\n'Try harder!'\n\n'I am! I am!' Charlie started to shout, utterly miserable. 'I am trying every day to be more human, to be better at it, so that you . . . so that you like me more. I thought this was what would make me fit in. Showing everyone else how normal I am. I don't know how I've got it so wrong. I just . . . I am sorry.'\n\n'Sorry?' Matteusz swore some more in Polish. 'My mother\u2014she will see this. And what will she think? Things are bad, but I hope sometime they will be good again. I hope one day she will like you like I like you. But then you do this\u2014she will think bad of me and the worst of you. Especially when she thinks that I posed for that picture so that you could put it online.'\n\n'But . . .' Charlie brightened. 'I could say that my phone was hacked.'\n\n'Ha!' Matteusz laughed long and hard. 'Even my mother will not believe that, and she still has landline. Nope! What you've done . . . What you've done. Jesus!' He strode around and pointed a shaking finger at April who was trying desperately to tiptoe out of the hall and wishing it wasn't so large.\n\n'Charlie, what if it was a picture of April?'\n\n'I don't understand.' Charlie frowned. 'Why would I photograph April?'\n\nMatteusz spat. 'Listen\u2014my body is mine. It is private. If I choose to share it with you, then it is because I like you. And just you. I am walking through your bedroom to find pants. I am casual and do not feel bad about this because I like you. I am at ease with you. I AM NOT THINKING THAT YOU ARE TAKING PICTURES OF IT. Because no good boyfriend would do that. Understand?'\n\n'I understand,' said Charlie. He cheered up a little. 'You still called me boyfriend.'\n\n'Habit I have yet to get out of,' Matteusz snarled. 'I do not take pictures of you. You do not take pictures of me.'\n\n'Ever?'\n\n'For the moment, yes. But later we will work to this rule: how would you feel if I had posted a picture like this of you?'\n\nCharlie thought about it. 'It would be undignified for a royal prince of Rhodia to appear like this.'\n\n'So.' Matteusz nodded. 'So. There we are. We do nothing to shame each other. There is a camera on your phone. Please ignore it. Until I tell you otherwise.'\n\n'Okay,' said Charlie. 'I am sorry.'\n\n'Yes.' Matteusz examined Charlie's face carefully. 'Indeed. Yes you are.'\n\nFor a moment, April assumed they were going to hug. Instead Matteusz turned around and walked out.\n\n'Oh.' Charlie sighed. 'That did not go well.'\n\n'No,' agreed April.\n\n'Still . . .' Charlie, hurt, and feeling very much alone, sat down on the edge of the assembly hall's stage, swinging his legs back and forth. 'At least I now understand why they call it oversharing.'\n\n'Stick to pictures of nice meals,' suggested April.\n\nAt the end of the day, there was something that they missed. As they walked home, full of their own thoughts, they failed to notice how much emptier Coal Hill School was than it should have been.\n\n## TWENTY\n\n## THE FIVE WORDS THAT BROKE HER HEART (SPOILER: ONE OF THEM IS 'WANT')\n\nThe boy was crying. The girl went over to him. They were both a long way from home. 'What's the matter?' she asked.\n\nHe shook his head, motioning her away.\n\n'Don't look at me,' he pleaded, 'they'll see.' This was news to her.\n\n'They can always see,' he told her. 'Please, look away.' So, she did so, turning to stare away from him and into the Void.\n\n'What's the matter?' she asked him. She felt stupid, looking away at nothing.\n\nShe wondered if he hadn't heard her. Then he spoke, sounding so weary.\n\n'I'm sick of killing,' he said. 'I just want to go home.'\n\nHer breath caught in her throat.\n\n'I do too,' she said. 'I want to go home so much.'\n\nShe could hear him rubbing his nose with his sleeve. Then he laughed. 'But we're here because we want to be. Because we deserve it.'\n\nShe hadn't thought about it like that. She'd been too terrified and confused.\n\n'This is our reward,' he said to her. 'They're going to keep us here until we die.'\n\nShe stared at the endless whiteness of the Void, working out what to say to that. Eventually she worked it out:\n\n'I don't want to die.' He didn't reply.\n\nWhen she turned around, the boy had gone.\n\n## TWENTY-ONE\n\n## SHE THOUGHT SHE KNEW WHAT WAS GOING ON. THEN SHE FOUND OUT THE REMARKABLE TRUTH AND TURNED THINGS AROUND\n\nThe next day the school was louder and simultaneously very quiet.\n\n'What's causing that?' Tanya asked as she walked in. Suddenly there were no fights, no screaming, but the place looked like it had been torn apart. Shredded scraps of posters fluttered along the empty corridors. People were no longer fighting openly or riding bikes down stairs. No ambulances were waiting outside. And yet the air felt tense. That dreadful, unshakable feeling that, at any moment, you were going to be in trouble, but you had no idea why.\n\nSomething's up, Ram thought as he limped, very slightly limped, out of his car. There were more parking spaces than he'd been expecting.\n\nCharlie wandered in slowly, looking for Matteusz. He had spent the walk in working out exactly what his apology would say. Apologies on Rhodia were ritualistic and ran to a formula, with appropriate foods carefully chosen to match the thing being apologised for. Unable to convince Quill to bake some _rote_ breads, he'd had to stop off at a corner shop and buy a Twix. He walked through the hallways and couldn't see Matteusz, but felt a kind of relief at that. He completely failed to notice that he couldn't see anyone else either.\n\nWhen the new wing had been opened, space had been put aside for Thought Pods. April liked the idea of them immediately\u2014small booths where she could work on a tune, or just doodle while staring out of a window. In practise, they'd swiftly been commandeered for surreptitious mobile phone conversations, and were usually crammed with students yelling at ex-friends in whispers. Teachers had also adopted the practise of moving on the people they found in them\u2014if they weren't in class or in the library then they were clearly Up to No Good.\n\nToday, however, April found the pods were unoccupied and slumped gratefully down in one, genuinely meaning to finish off that history assignment, but quickly distracted by sketching the really miserable-looking tree outside. It took her a while to realise that she was working undisturbed. No teachers moving her along. No one in the booth next to her whispering, 'If you do that again, Duncan, I swear I'll tell everyone.' None of that. She looked out at the tree. There was a gap in the hedge behind it, and normally someone sneaking off through it. No. No one in sight. Odd.\n\nTanya stood in the empty assembly hall, Ram by her side. She enjoyed the way her feet echoed. 'It's quiet here,' she announced. 'Too quiet.' Then she smiled to herself. 'I've always wanted to say that.'\n\n'So, what do we do?' asked Ram.\n\nThey turned at the sound of footsteps. Charlie and April.\n\n'We tell Quill,' Charlie had decided.\n\nMiss Quill had enjoyed her morning. Well, by her standards of enjoyment. She'd not really noticed the quiet, or found the reduced numbers in her classes worrying\u2014people normally found any excuse to avoid her lessons, and she considered this a smart move.\n\nShe glanced up from her book. She'd been reading _Captain Corelli's Mandolin_ and was dismayed to discover that a mandolin was not some kind of weapon. She folded down the page slowly and shut the book with a clap. She regarded the four standing in front of her wearily.\n\n'So, it's the . . . You really are going to have to come up with a group name, you know.'\n\n'We're not a group,' Ram insisted.\n\n'We're just . . .' Tanya sometimes found finishing sentences near Miss Quill really hard.\n\n'Quill, there is a threat to the school,' said Charlie firmly.\n\n'The Four Lions!' blurted out April, and then went quiet. Quill surveyed the four. A slight itch in her head told her that she should pay more attention to Charlie, but she'd risk a mild increase in pain so long as she didn't have to, and the others were actually more worthy of her notice, albeit a very tiny amount.\n\nMatteusz was missing from the group, which was something. He'd clearly had some kind of a quarrel with Charlie, which was marvellous news. She found sharing a house with the despised prince bad enough, but the other one was just too much, especially as they were so boringly involved with each other. She had toyed with the idea of bursting into their room and yelling, 'I heard your cries, my Lord, and\u2014my God\u2014what are you doing to His Majesty?' before claiming to be preventing an assassination attempt. She smiled a little. Would she be able to justify killing the youth? It would certainly be fun trying.\n\nOh. They were still talking. Pity.\n\nRam was loudly explaining that it was all his fault because of some soup and it had all got out of hand and that there had to be a way of stopping it, surely. Quill frowned. He wanted her to stop soup?\n\nTanya was currently explaining the whole situation, whatever the situation was, she wasn't quite sure, but the whole thing that was going on which was definitely a thing was being caused by a website, and there were some curious things about the website's data structure, which she started to explain in a rather tiresomely detailed way. That girl was worth listening to, but her problem was that she was too clever by half.\n\n'Are the cats behind it?' Quill asked.\n\n'No,' said Tanya.\n\n'Pity,' said Quill.\n\nApril, little steely fawn April, tried making a few contributions to the discussion and then fell silent. It was something to do with hidden messages in the Information Superhighway, which seemed ridiculous. Everyone else was shouting and April at first waited her turn then kept saying, 'The thing about Seraphin, you see, is\u2014' and getting talked over. She tried this four times until it was obvious to her that a fifth attempt would seem really unnatural. What she'd do next could go either way. Quill was hoping for a sullen sulk. But the alternative could be interesting. She'd watch that one.\n\nAnd finally, reluctantly, Charlie. The prince. He was learning to listen like a ruler. That air of firm, noble noncommittal before coming down on whichever side allowed him to unleash his army (i.e., her). He would seem fair, he would seem just, but most of all, he'd learned that a ruler who did not use his weapons was no ruler.\n\nCharlie raised a hand. It was an odd, commanding little motion, and everyone stopped talking, even Ram (a promising rebel leader if ever there was one). Waiting for him to seize the moment and speak words of blood and action.\n\nHe was clearly considering what to say. The right words that would stir armies. And then he said, 'Tanya, can we have a look at the metadata?'\n\nQuill honestly thought it wasn't possible to hate Charlie more. But no, here he was. Doing this. Whatever it was. You could call it investigating, but by those standards accountancy was investigation. Yet there they all were, clustered around a laptop.\n\nQuill observed the body language of the group. Charlie was intent on what he was seeing. Perhaps too intent, as though there was something else preying on his mind. Tanya was proud, proud of what she'd found out, proud of the awkward, carefully casual cross-legged way she was sitting on the floor. Ram stood off to one side, April to another. Ram was just itching to find a violent way of resolving this. One that allowed him to take revenge. Revenge for what was going on now and what had been done to him in the past\u2014yes, that was it. One day he really would make an excellent leader. April stood, leaning against a desk, peering at the screen, but with a sad look. She had clearly been hoping to see something brilliant there that the others had overlooked, but she had nothing. She was utterly dejected. That pretty much made Quill's morning.\n\nCharlie reached over to a point on the screen and tapped it.\n\n'Not touch-sensitive,' said Tanya. 'Not a tablet.'\n\n'Oh.' Charlie was still getting used to the limits of Earth technology. It was surprising to Quill too. For instance, they had holograms but only appeared to use them for dead singers.\n\nCharlie settled for scrolling up and down the page a bit.\n\n'Can we put this through your projector?' he asked Quill, which amused her. He was happy to order her to defend his life, but he was very polite about the oddest things. She hooked the laptop up, watched as nothing happened, and then waggled the plug a couple more times. Finally Tanya's screen flowed over the wall.\n\n'Here's what we have,' announced Charlie. 'Well, our best guess.' Oh, dead goddesses, thought Quill, imagine him trying to lead an army with these words. 'Tanya's gone through the site, finding people from Coal Hill posting content. She's found out who has authored the most popular stuff. And we now need to find out if they're currently in school.'\n\n'They're probably at home,' suggested Quill. 'Either skiving or with their limbs in plaster. Yesterday they started a fight club in my class. This was after I knocked out a fifth year. They're either dreaming up more lunatic ways of killing themselves or they're learning how to use crutches.'\n\n'But what if they aren't?' argued Charlie. 'What if this truthordare site is, in some way, a testing ground?'\n\n'Really? For what?'\n\n'I don't know.'\n\n'It's for charity!' Quill said, and laughed. 'That's what humans are always saying. There's always a mountain falling on a village or something. Humans love a starving baby and an unhappy puppy.'\n\n'And disease,' said April. 'Skandis is a disease. Isn't it?'\n\nTanya shook her head, and the whiteboard filled with search results. 'It apparently is. But all these results'\u2014page after page of technical data, of smiling sick teenagers in hospital beds, of people doing fun runs\u2014'they're very recent and very vague as to what the disease actually, definitely is. Even Wikipedia just uses these as source material. And some new sites use Wikipedia, so it just repeats itself. Whatever Skandis is . . .'\n\n'Is it attacking the school?' asked Ram. 'Is that what it is? Some kind of alien plague?'\n\n'Interesting,' mused Quill. 'So you think it may be selecting its victims based on how stupid they are. How they go against evolutionary principles. Intriguing. But really, you don't need to go to any effort to find a stupid human.' She regarded them all and smiled.\n\n'What about if it's a brain disease of some sort?' suggested Tanya. 'Does that explain what's going on?'\n\n'You can't catch stupid!' protested Ram, and Miss Quill burst out laughing.\n\n'Bless you,' she said eventually. Then she looked at them all placidly. 'What would you like me to do here, Prince? I can hardly . . . Well, I can hardly defend you from the Information Superhighway, can I? What is curious is that both you and Ram have made appearances on this site and yet, somehow, both of you are still here breathing my oxygen.'\n\n'Yeah,' Ram said, 'but what about whoever posted that\u2014'\n\n'What about Matteusz?' cried Charlie and rushed from the room before Quill could call after him. 'I'm sure he got lots of offers.'\n\n'Well, there goes your glorious leader,' she said, amused at the twist this caused in Ram's mouth. 'Isn't that nice?'\n\n'We've got to do something,' said April.\n\nQuill nodded. 'I love your sort. You're always the people saying \"something must be done\". I can't wait until you're old enough to vote. Have you any concrete ideas?'\n\n'We could go and find Seraphin,' suggested Tanya. 'Find out what he knows. Make sure he's alright.' She paused. 'Maybe get a selfie.'\n\n'We should find out who's gone missing from the school,' suggested Ram. 'Well, who is really missing and who is at home. Find out how many people this has affected.'\n\nQuill nodded again. 'That's a lovely idea. They're both lovely ideas. Do them both.' She dismissed them with a nod and opened her book.\n\nOh. They were still there.\n\n'Go on. Don't let me stop you.' She pointed to the door. 'Shoo.'\n\n'But isn't there something more?' said April. Her voice was firm. She crossed over to the projected screen and looked at the page. 'There's got to be a way of finding out how this site is working directly.'\n\n'But Seraphin\u2014' protested Tanya.\n\n'Hasn't been seen for weeks; has probably got a team of PRs protecting him; and won't return your calls\u2014bet you.' April ticked off the reasons on her fingers. 'But, you know, you're welcome to try and find him. Get an autograph.'\n\nThere was something in the tone of that which Tanya took against. 'I was just trying to be funny,' she said.\n\n'Come on, April,' said Ram, using his best 'be reasonable' voice, 'let's go look round the school, see what we can see and then reassess it.'\n\n'Sure,' said April. 'Sure. Of course. You're right.'\n\nAnd she walked out anyway.\n\n## TWENTY-TWO\n\n## THOUGHT YOU KNEW HOW TO LOSE YOUR FRIENDS? WELL, THIS WOMAN'S THIRTEEN BRILLIANT REASONS WILL CHANGE YOUR MIND\n\nCharlie found Matteusz. On a bench. 'Oh,' he said, cramming all of his relief into that one word. 'You're okay.'\n\n'So, you care?' Matteusz said. He looked cold and small and miserable, which considering his size was quite something.\n\n'Yes,' said Charlie. 'I'm sorry. I got it so wrong.'\n\n'You did,' said Matteusz, holding him. 'You amazingly did.'\n\n'Am I forgiven?' Charlie said.\n\n'Yes.'\n\n'I had a speech.'\n\n'No problem.'\n\n'And a chocolate bar.'\n\n'I'll take it.'\n\nThey walked away in the rain.\n\nApril watched them go. 'I guess I'm alone now,' she decided. She was sat on a low wall, sheltering from the rain. The school was so quiet. Bells rang but no one ran to class.\n\nShe fished her phone out of her pocket and tossed it from one hand to the next.\n\nSomething's going on here. No one can agree on what to do about it.\n\nShe looked out at the wet square of tarmac.\n\n'I know what to do about it. But it's not good.'\n\n**Video: How to Betray Your Friends and Influence People**\n\n**'Hi, my name is April MacLean, and this vlog is for Skandis.**\n\n**I'm not going to jump off a roof.**\n\n**I'm not going to cycle round the M25.**\n\n**I'm not going to fight Audrey Maguire, even though she's always hated me, and, anyway, who calls their daughter Audrey?**\n\n******No. What I'm doing for Skandis is I'm telling you this:**\n\n**I'm not sure I'll ever know when I'm right.**\n\n**I don't know if I'll ever win an argument. Nor if I would want to.**\n\n**I'm worried I'll only know how to enjoy being young when I'm really not young anymore.**\n\n**I'm not sure I'll ever be able to forgive my father. Sometimes I really resent my mother when she can't look after herself.**\n\n**Some of the time, when I'm playing music really intently, I'm doing it so that no one asks me how I'm really feeling. Because if they asked me, I'd start crying and I'd never stop.**\n\n**I'm not sure anyone will ever be in love with me at the same time I'm in love with them.**\n\n**Sometimes when I go to sleep I wish that I never wake up.**\n\n**I care about causes, like refugees and recycling, but** **** **sometimes I just like to think it makes me look good. But all this charity stuff? Doing things for Skandis? It makes me sick. Do you know why? My mother needs care most days. No one, not even my best friends, has ever offered to help me with her, or even asked me how she's doing. People at this school, doing stuff for charity? I've seen them laugh at her wheelchair. She's been sworn at outside the shops because she gets a parking space. She's been called a scrounger and told to get a job, even though she has one. Does that make me selfish? Vindictive? Yeah. Well, I don't care. If you're just doing something because other people are doing it, that's not charity, that's peer pressure. And I'm too tired to be bothered with fitting in.**\n\n**Oh. And I really like folk music.'**\n\nApril waited until the video had uploaded. Then she started to walk home. But she didn't get there.\n\n## TWENTY-THREE\n\n## SHE WAS READY TO GIVE UP AND THEN A NURSE SLAYED HER WITH A WORD\n\nApril woke up.\n\n_April._\n\nThat was her name.\n\nApril had learned to be good at mornings. Especially over the last couple of years. A while ago, being a teenager before noon was EXHAUSTING. Now she aced it. Once it had been a chore to get out of the house and arrive at school before nine. Now she woke up, made breakfast for her and her mum (her mum said she was more than capable of doing it herself, but that wasn't the point), talked with her, made sure Mum was feeling well enough to work (some days she really, really wasn't), then showered, changed into clothes she'd washed and ironed herself (she did her own laundry\u2014there should have been awards for that), and then got herself to school. Always on time. If not a little early.\n\n'Stop being so perfect,' her mum had said to her recently.\n\n'I can't.'\n\n'You're like a robot.'\n\n'A really tired robot.' They were having toast that morning, and it had somehow gone wrong. How was toast so complicated? It wasn't like there were recipes for it.\n\nHer mum wheeled herself over to the sink and reached up to rinse her plate. It just wasn't a natural movement. Even though she was used to it, good at it, it never looked easy. Her mum reaching up for a tap like a child.\n\n'Let me do that,' April said before she'd realised it. Her mum ignored her. She always did. Just carried on running the plate under the tap, hosing away those stubborn last few bread crumbs before slotting it into the dishwasher.\n\n'Why not be late today?' suggested Jackie. 'My shift doesn't start for a bit. We could not do something.' She closed the dishwasher. 'We could defiantly not do something.'\n\n'Such as?'\n\n'Watch TV.'\n\n'TV is rubbish at this time. Unless you like cartoons.' April had never really been a fan. Jackie relented, smiling.\n\n'One day you'll regret always being on time. It'll catch up with you.'\n\nAnd now it had caught up with her.\n\nShe had a waking-up ritual. She allowed herself to enjoy her room, which sounded stupid, but it helped. After the car crash, she'd found she both couldn't sleep and she couldn't stand waking up. She started every morning miserable. She'd once read a book about a man who lived in a cave and started every day screaming. That. She'd woken up every day like that\u2014barely enough time to blink before she remembered all the terrible things that had happened to her and which would make her miserable all day. Sometimes she couldn't just remember them, they HAPPENED to her. Fresh. Free range. She'd hear her mother pleading with her father to slow down, to stop the car, to let them out, she'd hear herself screaming and begging and she'd hear the engine roaring and the song on the car radio, that song, and she'd wonder what she could have done to stop it as it was all her fault and that song and the way the road slid up at an angle, a roller coaster ride that she really hadn't bought tickets for.\n\nIn short, she woke up every morning panicking. It was a nurse who helped her realise that.\n\nApril had come to visit her mum in hospital (she did it every day\u2014there was nothing else to do, not for a while). She was waiting to go in, and the nurse came out of her mum's room, closing the door gently. They kept telling her that her mum was doing fine, that there was nothing to worry about, but Jackie had her own room and this was the NHS. You only got your own room if you were dying. And very lucky.\n\nApril had started noticing these things. There was a reverential way of closing a door that only nurses used. It meant bad news. Later, when her mum was getting better, it was a different story. Doors would be flung open and nurses would bustle in, real cheeriness cranked up to eleven and all 'Hello Jackie and how are we today Jackie and oh look it's April and how are you doing April and isn't the weather awful?' But, for those first few touch-and-go weeks, the door to her mum's room was closed so quietly and the nurses tiptoed away.\n\nExcept for this nurse. She had stopped and looked at April.\n\n'You look awful, hon,' she said.\n\n'Thanks.' April's twisted smile said it all.\n\nThe nurse looked left and right and then sat down on the chair next to her. And said nothing.\n\nEventually, April broke the silence.\n\n'You really don't need to sit with me. It's okay.'\n\n'No, I do. It's more than fine.' The nurse had one of those floating Jamaican accents that sounded like birds swooping among trees.\n\n'Don't you have patients to look after?'\n\nThe nurse threw back her head and laughed as though she'd made a great joke. 'I always have patients to look after. From when I come on shift to when I go home. Too many patients. You know what? If I look after all the ones I have, they give me more. I don't get to all the ones I have? I stay late till I'm done. So, I'll spend five minutes with you and make you feel better, then I'll go and deal with the man in that room there . . .' She pointed at the room over the way. 'Handsome old man, very near the end. So racist. Keeps saying I've stolen his pills and his glasses. Calls me \"little girl\" when he wants to say something else. I should tell him \"Say it, dear, let it out, you're fooling no one.\" But I don't. I find his pills and his glasses and I tidy the pillow and I go make him a cup of tea the way he likes it.' She folded her hands across her mint-green uniform. 'I can spare five minutes for you. April, isn't it?'\n\nApril nodded. 'You said you could make me feel better?'\n\nThe nurse nodded. 'You feel horrible when you wake up, don't you?'\n\nApril nodded again.\n\n'It's because a horrible thing has happened to you, you and your mum. You wake up feeling helpless and miserable and afraid, and you blame yourself. You can't stop that. It's natural. But, and this is the thing, that very first minute when you wake up and that great wave crashes down on you, tell it to stop. Take a breath. Look round your room. Feel at home. And tell yourself all the good things you're going to do that day. Just list them.'\n\n'There aren't many,' April said.\n\nThe nurse looked comically angry at being interrupted.\n\n'Hush. Maybe there'll just be two or three for now. Maybe there'll be dozens. But try it and count them. Then, once you've run out of good things that you're going to do\u2014and I pray you don't\u2014then and only then, let that breath out and turn back to that old wave of misery and tell it to do its worst. And you know what'\u2014the nurse leaned forward\u2014'it'll still crash down on you, but it won't be so bad.'\n\nShe'd stayed with April half an hour, chatting about her dogs and the coffee machines and where hospitals got their weird chairs from. But the advice was the thing that April remembered. Ever since, whenever she woke up, she'd look around at her bedroom, at the drapes, at the twinkling lights, at the mobile turning in the breeze, at the plants outside, and she'd enjoy how slightly too warm the duvet felt, that feeling of safety as she counted through all the things that she'd do today until she'd run out of breath.\n\nThen, whatever happened, she'd be ready for it.\n\nNot today.\n\nApril woke up.\n\nShe looked around herself. And it all went wrong.\n\nShe was in a small white room. She was lying on a small white bed. She was wearing a scratchy shirt. She was covered in a thin sheet and her head was resting on an even thinner pillow.\n\n'Where am I?' she asked. 'And where's the door?'\n\n## TWENTY-FOUR\n\n## IF YOU DROPPED DEAD TOMORROW, WOULD YOUR FRIENDS MISS YOU?\n\n'Have you seen April?' Ram asked Charlie.\n\n'No,' said Charlie.\n\n'It's just\u2014have you seen the video she posted?'\n\n'I haven't had a chance.' Charlie rubbed the back of his neck and grinned sheepishly. 'I've been kind of busy.'\n\n'Have you seen April?' Ram asked Tanya.\n\n'No,' she said. 'Well, I've seen her video. Whoa.'\n\nRam leaned into the webcam. 'Yeah,' he said. 'I've tried calling. And she's not answering.'\n\n'We should totally not tell her mum,' said Tanya.\n\n'Yeah,' Ram said. 'I'd not thought about that.'\n\n'Have you seen April?' Ram asked Miss Quill.\n\n'Certainly.' Miss Quill laughed and spun her laptop round. 'I can't stop watching her little video. Unlike her, it's proving quite popular.'\n\nShe hit refresh. 'Every little helps,' she said, which would only make sense later. Then, even more bafflingly, she looked at Ram and winked.\n\n## TWENTY-FIVE\n\n## SHE THOUGHT SHE'D SEEN IT ALL AND THEN SHE SAW THE FACE OF GOD\n\nApril found the door eventually.\n\nShe found it by telling herself that there were limits to how unusual her world could get. Yes, her life was suddenly All About the Aliens, and true, she had an unwilling time-share on her heart with an alien king, but there were, she told herself several times, still going to be doors in her life.\n\nShe padded across the cold white floor of her small white room and she suddenly wondered where all that white was coming from. The room didn't have a window. Nor did it have any visible lights. And yet there she was. In a white box. Maybe the walls lit themselves up somehow. Whatever, her first stop was going to be the walls. She started patting them.\n\n'There is a door in here somewhere,' she said to the room. 'Well, there'd better be as I really need to pee.'\n\nOnce she found the door it was obvious. There was a section of wall opposite the bed which, if you pushed it, swung open.\n\nIt led her into a white corridor. April raised an eyebrow.\n\n'I can see how this is going to go,' she said to herself. She used her blanket to block the doorway of her room shut and then set out, tapping the walls, looking for answers, and, failing that, a bathroom.\n\nTen minutes later she'd discovered that, if she had been abducted by aliens, they had plumbing. Phew.\n\nThere was no sign of any toothpaste and her mouth was urgently telling her that it'd like some. She scooped some water from a tap and carried on.\n\nAt the end of the white corridor she found she was in another room. It all happened by optical illusion. The white corridor was perfectly, eye-achingly white. The room beyond was the same shade of white and completely empty. So, she walked from the corridor into the room with no realisation that she'd done so until it was too late.\n\nThe room she was standing in was, again, perfectly white. She couldn't tell how big it was, only that her senses were screaming at her that it was huge.\n\nShe stopped walking. She turned around, trying to get an idea of how big the room was. She cupped a hand over her eyes, trying to shield them from the ****of her surroundings. It felt like sand was being rubbed into her optic nerves. She couldn't stop blinking and there was an itch under an eyelid like she had a trapped lash. She rubbed it and that made it ten times worse.\n\n**WHITE**\n\n**WHITE**\n\n**WHITE**\n\nShe decided she'd had enough of the white room. Then she realised she'd forgotten which way she'd come in. She couldn't see the corridor anymore. She just knew that it was somehow somewhere. Unless this was some kind of trick.\n\nShe tried walking around, reaching out to touch a wall, but no matter how much she walked, she couldn't feel anything. Then her knees buckled and she fell. For a moment there was dizziness, but then she realised\u2014she could feel the floor. So she kept down and she crawled, sweeping an arm out in front of her in case, at any point, she met a wall.\n\nNo wall.\n\nRealising that she had absolutely nothing to orient herself by, she stopped and got up onto her knees. She needed something. Even if it was a false centre. She was just wearing a long nightshirt. She gnawed one of the buttons off and put it down on the floor. It wasn't quite white. Not completely. Sort of cream. It stood out, blaring its off-whiteness at her. She kept glancing back at it as she crawled away. Still there. Still there. Then gone. Utterly gone. She crawled back, defiantly, absolutely retracing her steps.\n\nThe button didn't reappear.\n\nNormal April, rational April, carry-on April would have taken that. But that April was taking a few heartbeats off. This April just flopped down in the whiteness. She rolled onto her back and made a snow angel. She giggled.\n\n'If it snowed here, how would you see it?' a voice said. A moment later she realised it was her voice and that she sounded a little, just a little, crazy.\n\nIt was now that she heard the roaring sound. The pounding of the blood in her ears, growing louder and louder. That was her heart beating. Her weird alien heart. It was racing out of control. The nurse in the hospital had told her to enjoy being self-aware and now she was all too aware of herself.\n\nApril lay there. It was all way too much. For the first time in a long time, she just gave in.\n\nWhich was when the whiteness flickered.\n\nFlickered like a screen. A face appeared.\n\nThe size was so large it could only be the face of God. The face filled the space, and that told her right now how small she was. She was lying on the floor of a vast cube. The face was projected onto all six faces of the cube. There was no escaping the face.\n\nShe could see all of it. The pores, the smile, the eyebrows, those clear blue eyes.\n\n'Hi,' said the face of God.\n\nIt was Seraphin.\n\n## TWENTY-SIX\n\n## THIS TEACHER'S INSPIRATIONAL WORDS WILL CHOKE YOU UP\n\nMiss Quill was the last person in the school.\n\nIt had that familiar quietness about it that told her, quite firmly, that it belonged to her. No Shadow Kin, no demons, no other aliens, and best of all, no children.\n\nJust Miss Quill, face lit up blue by her laptop. April's video was still playing on a loop.\n\n'I know exactly what you're doing,' she said to the screen.\n\nEven though Miss Quill knew she was absolutely alone, she still looked from left to right before leaning into the screen and speaking again.\n\n'I know what you're doing, April, and I admire it. Even if it will probably kill you.'\n\nThen she closed the tab on her browser and called up a search engine. She wondered if anyone had posted any more photos of cats on the Information Superhighway.\n\n## TWENTY-SEVEN\n\n## SOMEONE'S REIMAGINED DISNEY PRINCESSES AS ALIEN WARRIORS AND, TRUST US, IT'S AWESOME\n\n'Hi,' said Seraphin. 'How you doing?'\n\nApril lay on the floor and boggled up at him. While also lying on top of him. Suddenly she knew exactly how large the room was. In theory she should also have been able to work out where the walls were and where the way out was, but in practise all she could do was lie there stunned.\n\n'Hope you slept well.' Seraphin did a little winky shrug and flicked back his hair. 'I slept like a baby. Hence my skin. Smooooth.' He ran a giant hand across the skin, and she heard the magnified prickle of the tiny layer of stubble.\n\nHer entire world shook as he tilted the phone he was holding. She could see, over the pitching nausea, that he was standing, as he often did, shirtless and wandering around his flat. Seen like this it was all so very odd.\n\nThe stripped-wood flooring. The packets of cereal. The blankets on the sofa. A sock.\n\n'We woke you up before the others. 'Cos it's your first day (hooray!). Just so you could get used to it.' His voice was purring. 'And so that you could enjoy this room on your own. Isn't it great? They call it the Big White Room, which isn't very original but has a good feel to it. You know, #bigwhiteroom. Works, doesn't it? It's an amazing space. If it was up to me, I'd show cartoons in there. We should do something about that.'\n\nHe crossed, dizzyingly, to a blackboard and chalked the word 'CARTOONS?' up after the words 'CLOTHING LINE?' and 'NOT POETRY SLAM BUT RHYME CRIME?'\n\n'Anyway, I'd like to welcome you personally, but, as you've probably guessed, this is a recorded message. Click. Fizz. A Recorded Message. Brrr. Click. Please speak after the tone. Beeeep.' He laughed, and it was such a good-natured, warm laugh that April tried to forget how terrifyingly loud it was.\n\n'First off. Well done. You've won Truth or Dare. You've got our attention, and you've made it into the Void. Wait, let me say that again with a spooky tone. THE VOID!\n\n'Bwahahaha. Actually, seriously, nothing to be scared of.\n\n'Not with what you've done. You've shown outstanding skills. Everything you've done has been voted to the top . . .' He started ticking off his fingers. 'Let's see, you've been brave, courageous, you've risked upsetting your friends, you've not been afraid to put your life out there. You've done some pretty remarkable things and you're going to do more.\n\n'And everything you've done so far, you've done for Skandis. You've already been fighting Skandis, but (big secret, just between you, me, and everyone else), I'm going to tell you what Skandis really is. It's not, strictly, a disease. Well, it is. But it's a disease grown very large and very out of control. I'm going to show you what Skandis is. In a second. Just for a second. Be brave. Because you're not going to like it. Ready for it? Three . . . two . . . one . . .'\n\nThe screen filled with a terrible screaming face, a snarling reptile, the head entirely composed of ravenous snarling suckers, dripping with a thick, bubbling juice. It whipped and pushed itself howling into the camera, seeming to burst through the walls.\n\nAnd then it went. Leaving April wondering if her heart was still, in any way, working.\n\n'Nasty, eh?' Seraphin was back, his face a bit more muted. 'Sorry about that. Really I am. They're not nice.\n\n'That's a Skandis. It's an alien. And they're going to invade Earth. Not in a BOOM way. Not in an abduct-lonely-American-motorists-and-do-sexy-metal-things-to-them way. No, they're going to come to Earth and they're going to devour it.' Seraphin paused.\n\n'We need to stop them. We've a chance. They've established a bridgehead (military term) where they're gathering their forces (again, military term) and, if we can beat them there then they may, just may, decide not to come any closer. It's tough, but it's doable. Believe me.\n\n'This is where you come in. We Need You to Fight Skandis. We needed to find some way of gathering a force to fight them. And, you know what, so far it's worked well. On paper, the scheme sounds crap\u2014gather up a load of teenagers and make them fight a space war? What are millennials for? But guess what? Forget how it looks like on paper. Who uses paper anymore? Turns out, you're amazing. You're braver than anyone else in human history, you think faster, and you can process several different screens of information simultaneously, AND you've spent more time in immersive combat training than anyone else. Plus, plus, plus, well, we've all seen _Ender's Game_ , and this is like that but without the sulking and netball.\n\n'Oh, and we've a way better playlist.' Seraphin laughed again, and even that giant laugh was somehow reassuring.\n\nHe reached out and some music started to play. It managed to be quite backgroundy, little bit ambient, tiny bit floor-filler. 'Everyone's brought their phones here and the music on them is INCREDIBLE. Yes. Sorry'\u2014he pulled a sad face\u2014'we've confiscated your phone. But don't worry\u2014you won't be here forever. And we've got them ALL on charge so they'll be ready for you once you're finished and want to go home.\n\n'Here's how it works. Every day the Void will send you into a room to do the fighting. It's a spacey spacey gateway. You'll be both INSIDE the Void AND YET ALSO on an alien planet. Take a deep breath. That's right. An Alien Planet! Don't worry about the fighting\u2014we'll start you off easy and then ramp it up. You'll have helmet cameras so that everyone back here can see how well you're doing. It's awesome. Seriously, the technology in this place is top. The food's a bit meh, but hey, nothing's perfect.'\n\nAs he paused for breath, April used the opportunity to do some shouting. Shouting about how she had to get out, to get home, that coming here had been a terrible mistake and he really needed to listen to her and put her right back in London now.\n\n'I know what you're thinking,' said Seraphin. 'You're worried about the folks back home. Meh\u2014don't be. Nothing to worry about. It'll all be fine in the end, you'll see. You're saving the Earth, so forget about eating your mum's cooking and feeding your cat and so on\u2014'\n\nApril screamed at him\u2014he didn't understand. She needed to get back, to look after her mum, but he just carried on talking, his blandly pretty face telling her that this was perfectly normal.\n\n'Anyway'\u2014Seraphin yawned and stretched\u2014'I'll be on hand with lots of life hacks for battle. The usual. Everyone's getting up now, so you'll get to meet your fellow soldiers and get to it.' He held a fist in the air. 'Together we're fighting Skandis! Woo yah.' He smirked. 'Woo and yah.'\n\nThen he winked and held something up, close but so close she couldn't see it. 'Oh, and you forgot something.' He vanished.\n\nWith a tiny ping, a button landed next to April.\n\n## TWENTY-EIGHT\n\n## THIS HOT TAKE ON SMASHED AVOCADO TOAST WILL HAVE YOU REELING\n\nWas she still in the Big White Room?\n\nShe didn't know if it was the same chamber she'd been in or a different one. The whole Void had that feeling. Spotless, antiseptic, impersonal, like hospitals should be but never were. In real life, buildings always had scuffs or stains, and someone always put some flowers somewhere. But not here. The whole space was perfectly null.\n\nShe'd been worried about meeting her fellow\u2014what? Victims? Captives? Combatants? Soldiers?\n\nShe wondered what she'd say to them. Turned out she needn't have worried.\n\nAt first she thought they were ignoring her. Then it turned out they were ignoring the world.\n\nShe had walked into the Big White Room and they were all at benches, hundreds of people, eating bowls of something. She sat down at a vacant spot and grabbed a bowl. It looked like cereal or stew. She tasted it. Still no idea. But that was a way of breaking the ice with the people around her.\n\n'What is this stuff?' she asked. No one replied.\n\n'Hi,' she said. 'Should have introduced myself. I'm April.' Nothing.\n\nShe looked closely at the other people. They were all wearing white helmets, with little visors that went over the eyes. They were all staring at their plates. Odd.\n\nThe walls of the Big White Room glowed and jumped, playing footage from battle like it was an FPS. There was even an insert of Seraphin, wearing a headset and shouting an amusing commentary. Only this wasn't a play-through. This was real. With screaming. That made the jokes weird.\n\n'Oh, missed the kill shot, dude! Missed it! That'll cost you\u2014look at that, three shots down and that is like nearly dead and you've got to pick it up because whoa, that is bad and you have got to\u2014 The teeth, fella, watch out for them teeth! How many times do I have to warn you all about the teeth? Yeah, yeah, and FINALLY you blow its head off. Like what took you so long? And it's dead and bravo and now, here comes the screaming. Still, not bad for a Level One.'\n\nThe play-through continued.\n\nRight, thought April, spooning down her stewpops, this is weird. There's all that battle going on. Like a sports channel. Only it's on the walls, the ceiling, the floor.\n\nOccasionally, artefacts would appear. Small squares that would flicker over a bit of the screen and then vanish. They'd show a picture of a bowl of cereal. Or a view of the other people. Or, just once, of her. But mostly the picture was solid. The battles of the day.\n\nApril's foot kicked something. There was a helmet under her bench. Waiting for her. She reached down and picked it up. If she was going to fight, she figured she'd need one. Was she going to fight? She didn't know.\n\nShe put it on. It had that new-helmet smell of plastic and wet wipes. The visor was smoked plastic. There were two small displays at the side of it. One was a thumbnail of what was on the screen\u2014the battle footage. The other display was a small picture of what she was looking at. She looked up and peered around the room. The thumbnail went red. 'ABNORMAL' it said. She looked down at her bowl. The thumbnail went green. 'NORMAL' it said. She experimented, looking at whatever there was to look at in the Big White Room. The only things that turned the thumbnail green were looking at the big screens or at her food. NORMAL. Everything else? ABNORMAL.\n\nShe thought about that for a bit, chewing it over more than her food. Then, having had more than enough, she stood up and walked out. She looked at the door. 'ABNORMAL' read the display. She noticed that, as she went, a few squares on the big screen showed her retreating back. They flickered and went out. But they'd been there for a bit. She'd made an impression.\n\nAs she walked out of the room, she smiled. So that was how it worked, was it?\n\nBefore she'd started eating, April had been wondering about how to escape. By the time she'd finished her bowl of whatever, she felt keener on staying and fighting. She wasn't quite sure why. Was it the food? Well, it had a really weird aftertaste to go with its really weird taste. Some kind of drug?\n\nNo, she dismissed the thought. She walked on through the corridors of pure white. Maybe there'd be a door back to her room, or a door home, or a door somewhere interesting. Just a door. That was what she really wanted.\n\nShe turned a white corridor and found a lot of doors.\n\n'COMBAT CHAMBER EMPTY. BATTLE READY TO COMMENCE SMILING EYE.'\n\nShe stood in the combat bay. This was where the corridors of the Void had led her. So. This was it. Her first mission.\n\nAm I going to fight, she thought?\n\nWell, why not? She'd already encountered several lethal alien races. But her actions then had been a mixture of defence, panic, and sheer fury. This was different. There was even a countdown.\n\n'Dimensions balancing in ten . . . nine . . . '\n\nThe Combat Chambers were clearly different to the other spaces in the Void. The doors were obviously doors. They even had little windows in them. At the moment, that was a moot point, as the window in the white door just showed more whiteness beyond. But that was about to change.\n\nThere'd been a prerecorded safety demonstration from Seraphin. He'd explained (with a little song) that this was just Level One combat. 'Almost a training level. There's a safe word and everything.' She'd step through the door and she'd neither be entirely in this dimension nor on the alien planet. She and the enemy combatant would interface and be dimensionally in sync with each other, while also being not entirely there.\n\n'It saves having to send you all the way to the battlefront. We can just project you there temporarily. As I said, call out the safe word, you can come home. Otherwise you're only there until you win. So it's good to win.'\n\n'Six . . . five . . . four . . .'\n\nThe window flickered. It reached peak white and then faded into a pearly greyness.\n\n'You won't see much. Not at first. Level One is learning about the Skandis, getting the measure of how to fight them. Once you've got the hang of that, then you'll start seeing a bit more of the world around you. But first, there'll be no distractions, as you need to really, really, learn about the Skandis. Not go looking at alien trees. No matter how cool they are.'\n\nApril took a big breath. Is this what I am now? Am I a fighter? Am I really going to go up against an alien? She thought about it.\n\n'Three . . . two . . . one . . . '\n\nThe door unsealed and April went to defend the Earth.\n\n## TWENTY-NINE\n\n## THINK OF THE WORST JOB IN THE WORLD? YOU'RE NOT EVEN CLOSE\n\nThe Coal Hill headmaster's secretary was always 'new in the job'. The longest had managed about four months. The school got through headmasters at a fairly rapid rate (they'd long ago given up painting them, or even hanging photographs of them in a corridor), and, although it was never noticed, went through headmasters' secretaries at an even more rapid rate.\n\nThe school was blacklisted by most local temping agencies, and whatever secretaries they managed to hire generally took one (or, at the most, two) looks around before fleeing.\n\nAll schools have bizarre, soul-crushing amounts of paperwork and staffing rotas seemingly designed by Ouija board. Coal Hill added to that burden. No one ever knew how many teachers there were supposed to be, or how many had gone off sick, vanished mysteriously, or simply quit. It made scheduling the timetable impossible.\n\nEven something as simple as the cleaning rota was impossible. The school employed an outside contractor but also seemed to occasionally employ a caretaker called Smith. No one could quite remember ever having met the man, or explain exactly what he did. He'd never supplied a bank account so he'd never been paid, which could have caused quite a problem for the school finances\u2014if he ever did ask for pay, they'd have to sell off the furniture.\n\nThen, of course, there were the pupils. In order to last any time at all, a headmaster's secretary had to develop a thick skin. Most headmasters' secretaries spent their days ringing round parents, politely enquiring if their children were sick, truant, or had gone on a bargain package holiday to Crete. At Coal Hill it was quite the reverse. You had to get used to parents ringing up demanding to know where their children were\u2014a question that was sometimes difficult to answer.\n\nFor Ms Tey (the current headmaster's secretary), it was proving to be a baptism of fire. As her predecessor had hastily handed over to her, she'd told her what to expect (apparently she'd been offered an exciting opportunity doing admin for a portable toilet company). 'You may, every now and then, just occasionally, find the odd child is missing,' her predecessor had said as she'd thrown things into a cardboard box. 'Try not to take it personally. Sometimes they turn up. You never know, eh?'\n\nMs Tey had thought it a curious remark but she'd filed it away as gallows humour. What kind of school would it be if people really did go missing all the time? Surely someone would shut it down?\n\nAnd then the phone had started to ring. The first day it had just been a couple of parents. Now it was a flood. Ms Tey had been shocked, worried, and was now horrified to find herself bored. She'd run out of things to say to crying, terrified parents.\n\nHer phone rang again. She unplugged it, dropped it into a bin, and went back to browsing job sites.\n\n## THIRTY\n\n## YOU ARE BEING LIED TO ABOUT VOTER REGISTRATION AND THIS SHORT CHAPTER TELLS YOU HOW\n\nThe monster.\n\nThat was all that April could see. The calmer, rational parts of her brain tried to give it the name Skandis. Tried to apply functions to the various limbs and appendages. Tried to envision an environment that demanded that evolution answered back with quite so many teeth.\n\nThe problem was that the calmer, rational parts of her brain were completely drowned out by the rest of her body screaming. She fell back, the thing reared over her, those terrible tentacles whipping down towards her, their jaws snapping at her.\n\nThen time went weird.\n\nAs her head smacked into the floor, her eyes rolled up. She saw the white walls of the combat chamber. She'd already stopped noticing the walls but she could see them clearly now. They were that same uniform, glowing white. But there was something about them. Seen from the floor, she could see how they'd been cleaned, and not very well. They were streaked with grime and little dark red trails of dried blood.\n\nThe monster pushed down towards her. The smell was repugnant, catching in her throat. She retched, trying to roll away from the tentacles. Then she was up and running, still doubled over and gagging, her eyes streaming from the smell. The monster swept around, tentacles hissing as they tried to locate her. She was trying to work out where the door was, but the room was just stretching away. She kept running until she bumped up against a wall and stopped, catching her breath, rubbing the water out of her eyes.\n\nThis had been a terrible mistake.\n\n_I am not a soldier. I am not a soldier._\n\nShe didn't want this. She didn't want to be fighting this.\n\nShe had just wanted to find out what was going on. She had wanted to be the clever, brave one.\n\nNow she was running for her life.\n\nShe ran.\n\nShe kept running.\n\n_I have no plan. I have no idea what I'm doing. I will not fight it. I'm not a soldier._\n\n_Oh God. Where is my gun?_\n\nShe smacked into another wall. She was cornered. She twisted round, trying to see where she'd dropped her gun. As she'd walked in, it had felt so good to be holding a gun. It was solid, it was chunky, it was heavy. It didn't feel like paintballing or Laser Quest. This was real. This could protect her. This could hurt.\n\nOnly now she'd lost it.\n\nThe monster heaved itself towards her, slithering and lurching, rearing up to spring. Sensing victory, it paused. Tentacles flailing through the air towards her, the jaws opened. She could see those dreadful teeth, smell the air-meltingly awful stench of its breath.\n\nShe had an absurd urge to laugh. She was supposed to fight this thing? Who'd thought this was a good idea?\n\nHer brain raced on again.\n\nUp until now she'd taken this all on board. The disorientation. The endless white rooms. The complete isolation. The idea that behind this place was a great and powerful plan that really understood what was going on and was doing all it could to save the Earth.\n\nThe world was like that. She understood that. People made important decisions and other people carried them out and told more people that it had to be done and so it gradually trickled down and spread out until everyone's lives were somehow affected by it. The decisions didn't have to be good (they often weren't), they just had to be made and somehow they'd end up happening.\n\nYou could shout, you could protest, but you just couldn't stop them.\n\nThat was how things happened. She was sure that was how all this had come to pass. Someone had seen there was a problem and they'd worked out how to solve it and they'd thought it through and argued the toss and then decided that this was the best solution. Build the Void. Fight the war that way.\n\nOnly . . .\n\nWell, this was the thing . . .\n\nAs the tentacles pressed against her and the teeth trailed across her skin, April screamed at the top of her voice:\n\n'Who the hell thought it was a good idea to throw teenagers at monsters?'\n\n## THIRTY-ONE\n\n## ADVERTISEMENT: YOUR BOOK WILL CONTINUE IN TWENTY-FIVE SECONDS\n\n_'Hi, I'm Todd and this is why I fight.'_\n\n_(We see Todd. He's in combat gear. He's taken his helmet off, and we can see that he's ruggedly handsome, with a kindly older brother's air to him.)_\n\n_'Why do I fight? It's simple. I fight for the planet Earth.'_\n\n_(Shots of green fields and sunny days and beaches, lots of beaches.)_\n\n_'I fight for my parents, my family.'_\n\n_(Shot of Todd's parents holding hands. Shot of his sister shooting him with a water pistol. Jokes.)_\n\n_'I fight for everything that's good about our beautiful world.'_\n\n_(More beaches, this time at sunset.)_\n\n_'I fight for tomorrow.'_\n\n_(Sped-up footage of the night sky, scurrying clouds, daybreak.)_\n\n___'I fight because the Skandis are evil. They want to destroy us, our planet, our way of life. We can't let that happen.'_\n\n_(Grainy black-and-white footage of Skandis in combat.)_\n\n_'We can't let that happen. We can't let that happen to the Earth.'_\n\n_(Footage of a playground. Screaming children fleeing from an unseen menace. A mother scoops up a toddler and runs towards the camera, her face full of terror.)_\n\n_'We're here because we're making a difference. I don't care how long it takes. What it costs. Because I know that while I'm here, I'm doing everything I can to ensure my world is safe.'_\n\n_(The Earth seen from space, beautiful, tranquil, fringed by the rising sun.)_\n\n_'What I'm doing here is right. I'm Todd and this is why I fight.'_\n\n_(Todd, staring straight into the camera.)_\n\nThe video ended and everyone in the Big White Room rose to their feet, applauding wildly. They looked at one another, at the screen, and they cheered.\n\nNORMAL.\n\n## THIRTY-TWO\n\n## THE TEN BEST ALIEN DEATHS YOU'LL SEE TODAY. #6 IS A KILLER\n\nSo what? She'd used the safe word. It was April's first time.\n\nSurely no one had expected her to do any different. Screw them.\n\nIt wasn't instant. That had been surprising. She'd had to wait the ten seconds for the dimensions to stabilise and for the chamber door to open.\n\nTen seconds in which she was quite sure that she could have died.\n\nTen seconds in which to keep fighting and keep alive.\n\nTen seconds in which she became aware of her heart. It was beating strangely. Beating with excitement and verve. Filling her body with an unusual amount of energy. Like adrenaline, but different. More wild. The alien part of her heart. The bit linked to a Shadow Kin warlord. The bit that said, 'I am bred for battle and I must fight'.\n\nThe pulsing heart beat so strong and fast that it spun her round, made her pick up her gun, aim it again at the nightmare. Only it wasn't so nightmarish anymore. It was just a Skandis. Just something to wipe out. That was all.\n\nShe got ready to squeeze the trigger.\n\nThen the Skandis vanished and the combat chamber sprang open.\n\nShe heaved herself out and stood there in the bay, waiting for her heart to calm down.\n\n'Whoa,' she breathed.\n\nApril spooned down her savoury porridge flakes and wondered what the hell was going on.\n\nHow many days had she been here? Was this her first day? Her second? Her third? She felt completely disoriented and desperate to know how her mum was coping without her. She'd be worried, wouldn't she? She'd have called the police? But how was anyone going to find her\u2014wherever here was, or wasn't.\n\nEvery morning (was it even morning?) it was the same. She'd wake up in the Big White Room. She'd take a deep breath. She'd remember. She'd think about what she had to do that day. And she'd feel utterly alone.\n\n'Hey, everyone!'\n\nSeraphin's face blotted out a wall of the chamber. Everyone looked up from their breakfast.\n\nSeraphin stepped away from the camera. He was topless and holding a plate of food.\n\n'Cinnamon toast. You want some?' He took a bite.\n\n'Ohhhh, delicious.'\n\nThen he walked back a little bit further, sat down in a leather chair, flung his legs up along the side, tugged at his hair, and continued to munch his way through the plateful, while the audience spooned slowly through their grey soup.\n\n'Apparently the battle has been going really well. Like amazingly well. Let's just look again at some of yesterday's footage. That's right, #clipshow.'\n\nThe huge wall filled with head-camera footage. A Skandis on the ground, flailing.\n\nA rifle butt came into frame.\n\nIt smashed down and down and down again. Each impact made a wet, crunching thud. The horrific frame of the Skandis shuddered and squealed each time, its so solid frame crumbling and shattering under the impact.\n\nA solitary tentacle continued to twitch.\n\nThe rifle butt smacked down on the tentacle. It sheared off, flopped around wetly, then stopped moving.\n\nThe shot changed, showing flashes of arm and broadcasting ragged breathing. Movement, movement, the head whipping around to show something following.\n\nIt was someone running. With a Skandis following, screaming and howling. The person running was out of breath, you could hear it catching and wheezing and desperate. It was a girl.\n\nFor a moment April thought\u2014is that me, was that me, did I sound THAT scared?\n\nFor a few seconds all you could see was white. Then the running stopped.\n\nBreath. Ragged breath. A shot of boots.\n\nThe girl was doubled over, catching her breath. She was exhausted. She'd given up. That was it.\n\nThe hall echoed with the slight creak of a hundred people leaning forward in their seats.\n\nThey were going to watch her die. Then the shot changed.\n\nShe'd stood up.\n\nShe'd turned around.\n\nShe was facing the Skandis still running towards her. So close. So lethal.\n\nShe shot it.\n\nThe soldiers in the hall breathed out, cheering and clapping, fists pounding on the tables.\n\nWow, thought April. _Aliens vs. Hogwarts_.\n\nWherever she looked around the room, at all the cheering faces, her helmet told her 'NORMAL'.\n\n'Another one down!' someone shouted. No one looked around to see who'd said it. There'd been something, a little falter on the last word, that told them that they'd thought better of it.\n\nThe shot changed a third time.\n\nAt first it seemed to be a drone. A drone spinning somewhere, seeming to take off, whip up and then hover overhead, looking down at a Skandis. It swept backwards and forwards, observing and monitoring and checking in.\n\nIt flew up higher and began to slowly descend, drifting down.\n\nIt was a peaceful, gentle scene. Then they all heard the voice on the picture.\n\nThe voice saying softly and quietly, 'Please, no, please, no.'\n\nIt was crying.\n\nThe shot drifted closer and closer. Down towards the monster. Ever so slowly.\n\nSomething moved to the left of the frame.\n\nA tentacle. Clamped around a leg. Pulling the camera closer and closer. Drawing it into the great, vast mouth that split open across the top of the creature.\n\nThe picture went black but the screaming didn't stop. The wall went back to white. The Big White Room was utterly silent.\n\nThen Seraphin spoke.\n\nHe was back, still sprawling in his chair, not a care in the world. He put down his plate, dusting crumbs from his hands and licking the tips of his fingers. 'Delicious. Amazing. It's recipe time. Hey\u2014you know, like when my sister taught me to make a sausage casserole. Jokes if you've not seen it. So here goes with today's recipe:\n\nHere's a few cheeky tips for slaying a Skandis . . .'\n\nApril looked around at the other people eating. None of them met her eye. Flickering around the walls was headcam footage, screen after screen of food. That was it. She leaned close to the person opposite her, trying to squint with one eye, to see if the image of the boy's face appeared, however fleetingly, on the screen. The overlay in her helmet said 'ABNORMAL', urging her to look away.\n\nWell, screw that.\n\nThe boy was so young. He was pale. He had spots clustered round the straps on his helmet. He wouldn't meet her eye.\n\n'I'll keep staring at you,' whispered April, and she meant it.\n\nThe boy carried on eating.\n\nApril carried on looking at him.\n\n'Listen,' she said. 'Why do you fight? Why don't you try to get home?'\n\nThe boy said nothing.\n\nHis spoon hesitated, just for a moment. Then he carried on eating.\n\nApril gave in, and looked up at the screen, where Seraphin was still talking away.\n\n'Here's what we've got,' he said, his tone as bright as ever.\n\n'Weapons. You can't toast a Skandis without them. But it's important to know how to use them. Really lethally use them. Here are some of my sister's old dolls. Kidding, they're mine.'\n\nHe gestured to a row of plastic ballerinas.\n\n'The gun is the easy bit. It blasts\u2014' A ballerina shattered.\n\n'It burns.'\n\nA ballerina whoomphed into a melting pillar of fire.\n\n'And the butt\u2014snark, I said butt\u2014can be used as a club. That's why it's so heavy.'\n\nThe end of the rifle smacked down, breaking a ballerina into pieces.\n\n'And, in an emergency, we've also given you an electronic stunner.' Seraphin grimaced. 'Really? Like who came up with that name and why aren't they fired?' He winked. 'We should have a competition to see who can come up with a better one, isn't that right, Captain Pugsley?' He held up a small, wrinkled, cross-looking dog. It grunted and then got back on with the depressing business of being a pug.\n\nApril slept. In her dreams she saw all the headcam footage. The endless army of Skandis fighting, lunging, attacking, the soldiers fighting back at them, advancing, screaming, dying. On both sides the slaughter went on. Overlaid over each and every frame of her dream was one word: NORMAL.\n\nShe woke up in a dark room. Only it was a dark room that she knew was white. That felt strange. Her heart pounding, she was ready for battle.\n\nWhat the hell am I doing here? she thought.\n\n## THIRTY-THREE\n\n## EIGHT WAYS IN WHICH PEOPLE HAVE TRIED TO ESCAPE THE VOID\n\n**1. The boy who shot himself in the foot. FAILED.**\n\n**The gun simply refused to fire.**\n\n**2. The small group of dissenters who had tried to find a way out. FAILED.**\n\n**Some said they were still walking somewhere in one of the corridors.**\n\n**3. The girl who smashed up her room. FAILED.**\n\n**She came back to it after a meal to find it completely replaced.**\n\n******4. The boy who faked appendicitis. FAILED.**\n\n**He woke up to find his appendix removed.**\n\n**5. The brothers who refused to eat. FAILED.**\n\n**They carried on fighting, claiming their heads felt clearer without food, but insisting they wouldn't eat till they could go home. After seven days without food, both vanished on a Level 4 combat mission.**\n\n**6. The girl who said she had to go to her mother's wedding. FAILED.**\n\n**She was assured, despite evidence to the contrary, that the wedding hadn't yet taken place.**\n\n**7. The guy who'd just stayed on the floor of the Big White Room, screaming. FAILED.**\n\n**He'd been taken back to his room, and hadn't been seen since.**\n\n**8. The girl who got out.**\n\n**Actually, that one's a Rick Roll. Sorry.**\n\n## THIRTY-FOUR\n\n## AT FIRST SHE THOUGHT SHE KNEW EVERYTHING BUT THEN SHE FOUND THIS SECRET SHE HADN'T KNOWN SHE NEEDED TO KNOW\n\nApril wanted out. She wanted to get home. She wanted to make sure her mum was okay. More selfishly, if you'd pressed her, she would have told you that she was miserable.\n\nAfter a meal break, she wandered the corridors. She ignored the gentle way the Void had of suggesting that you went this way instead of that. Instead she strolled along, pushing occasionally at the walls. She figured that, at some point, this whole structure had to have a weak point. At the very least, someone's room would be unlocked and she could try to talk to someone. She patted her way along a corridor, turned left, went along another one, circling back towards the Combat Chambers. She didn't want to end up there by accident.\n\nShe bore left and pushed on another square of wall. Then she pressed on a little bit further and walked towards the end of the corridor. A white wall. So perfectly white. She pushed against the end of the corridor, and felt it click. Just a little. She pressed it again, and it slid open. What was beyond it was startling, simply because it was so dark. No white. Just a dim grey tunnel that blew cold air at her. She stepped into it, pulling the door closed behind her.\n\nWas it too absurd to hope it'd end in an emergency exit sign leading to a car park? Perhaps she wasn't at the end of the universe after all, but simply in Slough. God, she would love to be in Slough. The corridor ended in a small flight of metal stairs. She climbed them.\n\nAfter all the empty, futuristic concealed panels and blankness, what was most surprising about what was at the top of the stairs was its sheer ordinariness.\n\nIt was a normal wooden door. April, nonplussed, knocked at it.\n\n'Coming!' said a voice. April stood there, baffled. The door opened.\n\n'You!' she cried.\n\nStanding at the door was Seraphin.\n\n## THIRTY-FIVE\n\n## WHEN SHE MET GOD SHE FORGOT TO ASK 'WHY?'\n\nThe room was the last thing she would have expected.\n\nIt was Seraphin's bedroom.\n\nShe wandered around it in a daze.\n\nThere was his bed with its crumpled sheets. There was the shelf full of wonderfully ironic toys. There was the table with a huddle of laptops. There was the little nook leading to the bathroom with its massive selfie mirror. There was even the bay window, looking out onto . . .\n\n'Yeah,' said Seraphin, 'you can poke the sky. It's fake.' April leaned out of the window. Because he'd told her to and her head was just not up to anything else. Beneath her was just darkness. And ahead of her was a brilliant skyline that looked somehow wrong. Wrong because when you prodded it, you felt nothing. Just nothing. But it all rippled. Like paper.\n\n'Whaaaaa?' said April and stopped talking.\n\n'Hi,' said Seraphin, and he reached out and took her hand. He shook it firmly but not too insistently. 'You'd better not be a fan.'\n\n'Not really. April.'\n\nGood, 'cos if you were a fan and you'd somehow managed to get here I would, I really think, scream.' Seraphin smiled and gave a little scream.\n\nApril smiled back. Up close, Seraphin was a bit too hot to look at directly. Like a polite, sweet sun. But there was something different about him. Something that made him weirdly more handsome.\n\n'Oh, got it! You're wearing clothes!'\n\n'Yuh.' Seraphin ducked his head. Actually he was wearing a lot of clothes. Comfortable sweat pants, baggy slogan T-shirt, smart hoodie, and a beanie hat that suited him amazingly. 'What?' he said. 'My clothes only fall off when I'm working. You know, I just do it for the likes, the reblogs, and the gifs. When I'm off duty, I like to be WARM. I like to wear a shirt. I like to be shnoogly. Especially in winter\u2014the heating bills at my place are PHENOMENAL.'\n\n'Don't worry, you look great,' said April. 'I mean, sorry, I mean . . . God, this is weird.' She laughed and she didn't know why she was laughing. 'It's you. You're really here.'\n\nSeraphin laughed too. 'Next thing you're going to say is I'm shorter than I look on screen.' This was clearly a joke. He was really tall and he had these blue eyes that just stared at her with keen interest and . . .\n\nWait a minute. Wait.\n\n'You're Seraphin,' said April.\n\n'Yes.'\n\n'You're here. You're not . . . I dunno . . . a hologram.'\n\n'That's an excuse to fondle my chest, right?'\n\n'No. No! It's just . . . Are you in charge of this place?' April stared at him sick with horror and the realization that she'd basically just walked into the villain's lair. If a villain's lair had more ukuleles than normal. Wow. She'd been so stupid.\n\nSeraphin threw back his head and laughed.\n\n'Sorry,' April practically gushed with relief. 'Sorry. I just assumed that you were in charge.'\n\n'No. But can you imagine? Me?' He tore off his beanie hat and threw it onto the desk and shook out his hair. 'Look, who are you?'\n\n'I came because of your messages. We decoded them.'\n\nAt the 'we' Seraphin looked reassured. 'And I came here to find you.'\n\n'Oh, thank you, you're amazing!' Seraphin whooped. 'I've been stuck here for months.'\n\n'Stuck here?'\n\n'Help, I'm a Prisoner in a War Game Factory!'\n\n'So who is behind this?'\n\n'I don't know.'\n\n'Really? So you don't work for them?'\n\n'Oh, I work for them, but that's NOT THE SAME THING. Not at all. Listen.' He threw himself onto a Swiss ball and started bouncing it around the room. 'Dead truth, okay. I was offered a contract, to be the public face of the site. It was, well, it was a tonne of money. And it was for charity. So, you know, I said yes. They explained about the mock-up of my flat and everything and that was totally cool. I get that some people like artificial. But then I get here, and I think it's an afternoon filming a few inserts and so on . . . And I Have Been Here for Months. And it has got weirder and totally weirder and do you want a juice?'\n\nHe reached under the desk and threw her a small box of fruit juice. 'Even here I still get freebies. Insane.'\n\n'What flavour is it?' April said automatically, and then smiled at herself.\n\n'No idea,' said Seraphin drily. 'I do know that it's vegan friendly and gluten-free and that if you enjoy it the PR firm encourages you to tweet about it using the hashtag #sosopure.' He smirked ironically.\n\n'Right.'\n\nApril stabbed the straw into the carton and wondered about what she was going to say next.\n\n'Where is here?' she asked.\n\nSeraphin threw his hands out, wobbled on the ball, and righted himself. 'No idea. Like, there was an Uber. I got in it. Well, I thought it was, only . . . Anyway.' He bounced forward on the ball, grinning wolfishly at April (a very polite, house-trained wolf ). 'You say your people decoded my messages?'\n\n'Yes.' Oh, this is where it was going. Oh dear.\n\n'And you've come to get me out.'\n\n'Ah.'\n\nSeraphin took the news reasonably well. Sort of. Eventually. Once he'd stopped shouting. April might not have entirely helped. She was doing her best at looking both contrite and calm when it suddenly struck her that she was a soldier trapped in an alien dimension inside a fake flat with a celebrity shouting at her on a space hopper.\n\n'You're laughing! Seraphin broke off his rant, incredulous. 'You're laughing?'\n\n'Well, yes,' April said. 'Sorry.' She blew her nose and then tidied her hair back. 'You have no idea\u2014how scared I am, how much I want to go home. I need to get back to look after my mum. I need to tell my friends what's going on here. Oh God, there's so much I need to do. And I thought that you'd . . .' She laughed again. 'I thought that you'd have a way out.'\n\n'And I thought that you did.' Seraphin laughed too.\n\n'The whole situation is crazy. We're all stuck here. I'm still doing all my normal vlogs from in here about \"My Insane Life\" and it's all about the sunglasses I'm being paid to wear, a healthy snack delivery company I've been endorsed to discover, or a song I'm writing or whatever, rather than what's really going on, which is so mad. Like, if ever filming your daily life should be about something, it should be about this\u2014massive space war against fricking aliens. Shouldn't it?'\n\n'Well, yes.'\n\n'The saddest vlogs are the ones that are about something. Vlogs by sick kids.' Seraphin wasn't laughing. He just looked so sad. 'Jeez. I can't stop watching them. They're not even popular\u2014just grim. Kind of like _Game of Thrones_ grim, you know? But I am an addict. 'Cos people are all like, \"Hey Seraphin, you're so brave putting yourself out there,\" and I'm like \"Thanks,\" but I'm not a kid stuck full of needles.\n\n'But yeah, dying. That's not Living the Dream.' He grabbed himself another juice and tossed another one across to April.\n\n'The whole vlogging thing's really hard, you know. My Insane Life. Everyone looks at you like they know you. And it's like, \"You may think you know me but I do not know you,\" but when they run up to you at a music festival you have to be all smiles even when they jab you in the face with the selfie stick. 'Cos you've already made the choice to be their best friend. That's part of the mission. If I'm out in public, then I'm always on. I'm always smiling.\n\n'And you learn, like, when you really commit to it, that the sun always has to shine. 'Cos it's my job to tell you that it's okay. That life's great. Is your life?'\n\nApril blinked. 'Well, no, I mean, sometimes not, but you know . . .'\n\n'Nah-har.' Seraphin made a Wrong Answer noise. 'That's your audience gone. Not what we came here for. And, don't get me wrong, so many of my problems are #FirstWorldProblems.' He stopped, shifted posture on the Swiss ball, and suddenly bounded into a devastating impersonation of himself. 'Hey, guys! My hair! Dis-as-ter! Like whaaaaat? Insane! What am I going to do with this? Oh shoot me!'\n\nHe slid off the ball and landed on the floor. April had never seen someone do an impression of themselves, and it was odd. Ten seconds of sharp self-hatred.\n\n'I'll tell you the things I don't get to do. Simple things.\n\n'When you commit to doing this every day, you don't realise that there are going to be Bad Days. Or you think that you'll store up a few so that you can have a few days off. You know. But that never happens. It just doesn't. And then, you know, top secret, but when you have a hangover, you don't get to crawl back under the duvet. You have to share it with the world. Share! It! With! The! World! And your hangover has to be funny and here I am in Speedy's Caf\u00e9 and I am so in need of bacon and isn't that great and . . .' He gave a long, thoughtful sigh. 'I'll be wearing a baggy jumper that makes me look vulnerable yet adorable. Think about that\u2014I have to pick what I'm wearing when I'm hungover.\n\n'And, hey\u2014you ever been dumped?'\n\n'Ack,' said April.\n\n'Sorry,' said Seraphin. 'I know, right, I've only just met you. Too much sharing. But it's like, you're dumped. You wake up the morning after and you just want to not be awake. Like getting out of bed is impossible. Iron Man is sat on your shoulders. But no. You gotta get out of bed. And the worst thing is that you wake up and you are dumped and then you remember that Everyone Knows. Literally everyone. Like, if it happens in public, in a restaurant, someone will have live-tweeted it, and there'll be three reaction videos out there already. And you don't get to have a day off. You have to be out there and be dumped and be nice about it.'\n\n'I think,' said April, 'that's called being a gentleman.'\n\n'Yeah, very good, 1950s.' Seraphin's laugh was a nonstarter. 'But is it fair? Sometimes, I just want to get stuff off my chest. 'Cos they were doing my head in and whatever, but I have to be sooo still. Even\u2014you remember the one who released an album about me?'\n\n'Not really, no,' April admitted. As much as she was aware of Seraphin, she sort of had a vague 'dating that model with the quirky handbags' memory. 'But you're famous. Worse, you're internet famous. Some woman goes out with you, she's going to be hated. She breaks up with you, she's going to get so much more hated.'\n\n'Yeah yeah yeah,' Seraphin agreed wearily. 'Jeez. Sometimes I think the whole online thing was invented just so we could hate on women. But look, hear me out, right. Say I'm single, and I go out and I meet a girl right, the morning after she'll post \"So, this just happened . . .\" with a picture of me asleep in her bed. And the world just goes BANG. Like what? I've not got control over that. Like, I can't say I've got a cold without three different PRs biking me over their client's cough meds and a little note suggesting a hashtag. You know, I just want it to be back to when it was just fun insane rather than insane insane.'\n\n'Like this?' suggested April.\n\n'Totally.' Seraphin scrambled up, and kicked the wall of his flat. It wobbled. 'This is the point that it's all gone totally nuts.'\n\n'What the hell do we do now?' asked April.\n\n## THIRTY-SIX\n\n## THIS IS YOUR CHANCE TO WIPE OUT SKANDIS FOREVER\n\n**'COMBAT CHAMBER EMPTY. BATTLE READY TO COMMENCE SMILING EYE'**\n\nAfter leaving Seraphin, April had tried fighting again. And again, she'd not got any further.\n\nWhat the hell was this place? Was it even a place? Was she trapped in an idea?\n\nShe'd thought that finding Seraphin would have helped. Like Dorothy finding the Wizard of Oz. Only the man hiding behind the curtain wasn't pulling any levers. He was simply there to make a lot of noise.\n\nHe'd tried to help her but hadn't really been able to do much. It was like neither of them had a full picture of what was going on. He was there to get brave people to fight monsters in order to save the planet. She was there, supposedly, to fight monsters to save the planet.\n\nBut something didn't link up between the two.\n\nSo she'd gone to fight a monster, to see how it made her feel.\n\n'That's definitely a reason why I don't want to do this,' she grunted as a Skandis threw her against a wall. Was she imagining it, or was the wall less padded than yesterday? Was the Skandis more aggressive? Was she finding it a little harder?\n\nShe lay on the floor, panting and tasting blood in her mouth. 'I'm not a coward,' she said, and activated the safe word.\n\nThe next ten seconds proved she wasn't a coward. The ten seconds it took the dimensions to normalise and for the Skandis to vanish were endless. As if sensing the fight was over, the Skandis threw itself at her with desperate energy. April fired off a shot from her gun, more as a warning than anything else. It singed the creature's hide, making the slime bubble and steam as it scorched its way through the flesh. The creature roared, pulling itself up.\n\nI can do this, April thought. I can do this after all. She raised the gun. Aimed.\n\nAnd the creature vanished.\n\nShe staggered out of the Combat Chamber. She was soaked in sweat. She held a hand out in front of her and watched it tremble. She took a deep gulp of breath and tried to hold it, but she couldn't.\n\nAnother soldier was standing there, visor down.\n\n'Are you okay?' he asked.\n\n'Yes . . .' she said. 'Just . . . shaken . . .'\n\n'Sure,' he said, sounding concerned. But not taking the helmet off. He just carried on staring at her. I'm being filmed, April thought. I'm looking at him and my screen says NORMAL. He's looking at me and his screen says ABNORMAL. I'm being filmed through all of this. I'm going mad and I'm fighting aliens and I'm being filmed. If I crack now, if I go crazy with this guy, then he'll just be filming all of it. And it'll be shown in a loop. A loop of 'This Weak Girl . . . '\n\nShe stared back at the guy.\n\n'Hey,' she said, and she held up her hand. 'Look, it's shaking less. Woop-de-doop-dee.'\n\n'Yes,' said the soldier after a pause. How old was he? Mid-twenties? Ten years younger? The helmet made it impossible to tell.\n\n'Just, you know, still getting the hang of it.'\n\nThat creature, rushing at her out of the whiteness, roaring and spitting and leaping. And her gun coming up and her dodging to the left and then the right and the thing still coming at her.\n\n'Yeah,' the soldier said. He wiped a hand across his sleeve. There was that strange green blood\/fluid all over it. 'Just killed one. They take a while. You're Level One, now, yeah?'\n\n'Yes.'\n\n'Wait till you get to Level Three. That's when it gets pretty juicy. They're really fast moving there. And they take longer to kill. They're really vicious.'\n\n'Vicious?'\n\n'Vicious.' The helmet made the guy's face implacable. 'The Level Ones are much slower. Like they're older. Or not so good at fighting.'\n\n'Okay.' That sounded odd.\n\n'Sometimes it's like the Level Ones are actually pleased to be killed. Imagine that, eh?'\n\n'Yeah.' April was finding this whole exchange odd.\n\n'So . . .' she said.\n\nThe visor tilted at an angle. Curious. April very carefully said nothing.\n\n'Do you want to go back in there?' the soldier said. 'I know how to re-jump the Combat Chambers . . .' He gestured to an instrument panel at the end of the bay.\n\n'We're not supposed to know how, but they're really easy. I can pop you back in there in a couple of minutes.'\n\nFor a moment, his soldierly air had been replaced by boyish enthusiasm. He was almost dancing over the instruments. A boy showing off to a girl.\n\n'No, I'm good,' said April. On the one hand, maybe it would do her good to confront her fears. On the other, there was something about this that just didn't feel right.\n\n'You sure?' The visor looked at her, once again unreadable.\n\n'Yeah, thanks,' said April. She walked away.\n\nShe knew now what she had to do and that it wouldn't be easy.\n\nApril walked into the Big White Room, joining everyone at the meal that was possibly breakfast.\n\nShe chewed on the soapy flakes, plucking up the courage. The moment passed. Another moment passed. And then, in a rush, she pushed her bowl away and stood up.\n\n'I want . . .'\n\nShe faltered. She swallowed, raised her voice, and started again. Everywhere she looked was wrong.\n\n'Everyone, I want to talk to you about the Skandis. I'm not going to fight them anymore.'\n\nShe'd wanted a reaction. No one said anything. Spoons lifted food from bowls.\n\nThe video of Seraphin played on.\n\n'I'm not a coward,' she shouted to the roomful of people who were ignoring her. That's not an easy thing to do. 'I'm not saying I don't think the Skandis are evil.'\n\nNo one cared but still she carried on talking. 'I'm just saying that I don't get why we are being made to fight them. It shouldn't be us.'\n\nThe volume of the video went up slightly.\n\n_'Cupcakes, eh? I love cupcakes. Today I'm going to make some amazing cupcakes and cut my own hair.'_\n\n'Like that!' April raised her voice till it scratched at her throat. 'We should be doing stupid things. Like that. Normal things. Not . . .' She stopped, lost for words.\n\n_'These scissors . . . Do you think they're blunt or_ Psycho _badass? As in Stab! Stab! Stab! Arg!? And do you think I'll get hair in my cupcakes? Well, Captain Pugsley will decide.'_\n\nA square appeared, blocking out a tiny, stamp-sized portion of Seraphin. It was a small picture of April. Someone in the room was looking at her. Just one person. Someone was paying attention to her.\n\nApril kept talking. 'This. This room. Think about it. Please. This roomful of people. Isn't it insane? Isn't it wrong?'\n\n_'And now we're gonna clean the bowl the best way. Giffers go crazy at me licking my batter.'_\n\nAnother square appeared on the huge screen. Two people. How many before ABNORMAL became NORMAL?\n\n'Look around you. Look up. Stop looking at him. Look at yourselves, at what you're doing.' April hoped her voice didn't sound desperate. She was going for strident, she was going for confident, but those last words, something pulled at her vocal cords and choked them slightly.\n\n_'Still plenty of cream in the bowl, so a big dollop for Captain Pugsley and oh, oh, oh, I've spilled some on my chest. Classic.'_\n\nMore squares appeared. More small pictures of April, breeding and multiplying, spreading across the wall. People were looking at her.\n\n'Listen to me,' she shouted at them. 'Stop fighting. Stop this madness!' Then she filled her lungs. One final push. 'JUST LOOK AT ME!'\n\nAs she said it, she realised she'd made a terrible mistake. Gone too far.\n\nFor a moment it was all good. The picture of Seraphin broke up entirely and the screen became one big April staring back at her. Waiting to hear what she'd say next.\n\nAnd then, one by one, as her words echoed around the room, those squares blinked out. The wall was whole again.\n\n_'Three-second rule, dammit, Captain Pugsley!'_\n\nToo needy. She knew she'd lost them. She turned around and walked out.\n\n## THIRTY-SEVEN\n\n## AFTER YOU READ THIS YOU'LL WANT A SHOWER\n\nApril woke up. The room was wrong somehow. For a moment she hoped she was back in her bedroom at home. That it had all been a dream. But that wasn't how dreams worked.\n\nThe room was dark. Completely. It went this way when you were supposed to be asleep. It was disorienting. She'd woken up in the dark here before and it was peaceful. This time, though, there was something wrong in the room. Something that itched. Something that refused to be right.\n\nShe heard the breath.\n\nShe was not alone. There was someone in the room. She realised how wrong she was when they came for her. About a dozen of them. Rushing towards her. She cried out and then they were on her.\n\nThe attack wasn't long. It could, she told herself long afterwards, have been worse. They didn't even hit her with weapons. Or fists. Some of them grabbed her sheet and pulled it tight. She couldn't move. She struggled there, helpless, refusing to scream. They struck her with the flat of their hands, blow after blow on her face, her body. She looked up as long as she could, her eyes not seeing their faces, just the helmet cams staring at her.\n\nI'll never know who you are, she thought. I'll never know who did this.\n\nShe ignored every urge to shout back at them, to beg, to cry with pain. She felt her heart, that strange part of her, leaping around in fury. Something deep inside her wanted to hunt down everyone in this room and kill them all. She hoped it was the alien part, and not her.\n\nThe blows stopped. She sensed people stepping back. She felt a moment of relief.\n\nThen she was reeling from a terrible rank smell. It was so overwhelming she fought back for the first time, struggling against the hands holding her down.\n\nWhat have they done, she thought\u2014they've let a Skandis into the room. They're going to kill me with it. I've said I wouldn't fight and now they're going to make me fight.\n\nSomething landed on her. Wet. Repulsive. As it touched her skin, she felt it flare up and burn slightly.\n\nHer eyes streamed and she gagged at the terrible, sticky, dirty smell.\n\nThe hands holding her down released her. They stepped away. They walked out.\n\nThe door opened. The door closed.\n\nThe lights flickered on. She knew they would. She knew what she would see.\n\nThe dripping, severed head of a Skandis was pressed against her mouth.\n\nLooking back later, from a long distance away, she realised that neither she, nor her attackers, had said a word.\n\nIt was a long night. The lights in her room hadn't gone off. Repulsed, she'd eventually squirmed her way out from under the head. The bedsheets were stained with a mixture of slime and blood, seeping over the sheets and into the mattress. Inside the tiny room, the smell was choking. Of course the door was locked.\n\nThe head sat on the bed, watching her with boiled-egg eyes, the tentacles flopping and slipping over onto the floor. She moved to another corner of the room and curled up in it and tried to close her eyes. It didn't work.\n\nShe remembered the pillow and slowly, reluctantly, worked herself up to go back for it. She hoped that the Skandis's blood hadn't touched it. That would have been something. But no, of course the pillow was covered in tiny, foul-smelling spots of gore, burning into the cheap foam inside.\n\nSo April just sat in a corner and tried to ignore the severed head and stayed utterly, chillingly calm. The smell became more and more overpowering and she could hardly breathe without retching, but she stuffed a fist into her mouth and screwed her eyes shut. She didn't think about the pain she was in, she didn't think about what had been done to her, she didn't think about what she'd do next. She just waited and waited. Until eventually the door sprang open.\n\nShe'd hoped the shower would have done something about the smell. The stench of that decaying creature was in her hair. She swallowed mouthfuls of the tepid water, trying to clear the taste from the back of her throat. She checked herself for bruises\u2014there were none. They'd been careful. Nothing to make anyone feel any sympathy for her. She stayed in the shower until the water slid from tepid to lukewarm to cold and then she got out. She caught sight of herself in the mirror, and her mouth fell open. She made a noise. A half sob from somewhere deep inside that, if she'd let it continue, felt like it would never stop. She stifled it.\n\nShe dried herself on the cheap towel and slipped into a fresh uniform.\n\nWalking towards the Big White Room was the hardest thing she'd ever had to do. At least, she told herself, there'll be no surprises. I know they're all waiting there. They know I've been humiliated, hurt. They'll all know from the videos.\n\nBut it's a meal. It's a day.\n\nShe breathed in. Be brave, she told herself. You can face them. You can face them all and that's how you win. Don't even put your helmet on. Don't hide behind it. Just look at them.\n\nShe walked in.\n\nThere was no one in the room. Nothing on the screens. Just empty benches.\n\nNo food.\n\nShe wandered around for a bit. Forced jollity. She swung her shoulders around. Then the exhaustion, and something else pricking at the edges of her eyes, wore into her and she picked a bench and sat down.\n\nSomeone walked in. They were wearing their helmet, their face tilted down. Was this one of her attackers? The boy walked over and sat down a few seats away from her. His nose twitched.\n\nGod, thought April, I must still reek of that thing. She fought down the urge to apologise to him. Almost automatic, but not.\n\nAnother boy walked in.\n\nThen a girl.\n\nThen another girl.\n\nAll of them silent, heads down, not noticing her. This was nothing unusual.\n\nHer strange heart fluttered and she wondered if this was them\u2014the group of people who'd assaulted her. _Kill them all, just in case._ The alien thought passed through her head and was quickly discarded.\n\nA boy walked in. Taller and broader than the rest. He sat down opposite her.\n\nThey all sat there. Nothing happened. No food turned up. Nothing.\n\nThey were all around her. But no one was looking at her. They kept on doing it.\n\nUnable to bear it anymore, April sprang to her feet. She wanted to run off, to shout, to say something, but she couldn't think of anything to say. She stood there, glancing from bowed head to bowed head.\n\nIt's like I don't exist, she thought, and then sat down. She knew then that she was trembling. She pushed her hand out in front of her and watched it shake.\n\nThis is really, really getting to me, she thought.\n\nMore people filed in and sat down. None of them looked at her. I'm nothing to them, she told herself. And that's okay. I can work with that. So long as I can get rid of that thing from my room I can go back there and I can wait this out. No fighting. Seeing no more of these people. Just living is enough.\n\nBut what if they come back to my room? Maybe I'll leave the head where it is, after all. That should keep them out. But when can I go? When can I leave?\n\nThe hall filled up.\n\nStill no food. Just waiting.\n\nCome on, guys, thought April, the war's not going to fight itself. And look at all these people\u2014no porridge. How are they going to lift their guns?\n\nThe giant screen flickered into action.\n\n'Morning, everyone!' Seraphin was hanging upside down from a pull-up bar, waving. 'It's someone's birthday today, so we're going to celebrate their special day. Let's hear it for April, the birthday girl!'\n\nEveryone in the room burst into applause. A strange, terrible, clapping in unison. April flinched from it.\n\nThe clapping stopped.\n\n'Apparently you got her a surprise cake! Am I right? I'm right. Hope you saved me a slice!'\n\nThe screen cut to the severed head sitting on her bed.\n\nAnd then back to Seraphin. 'We're lucky to have you, April. Let's give her another hand!'\n\nAnd he stayed there, frozen on the screen.\n\nLittle squares replaced him. Tiny shots from headcams as one by one everyone looked up to stare at her. All of them. Every one. A thousand Aprils filling the room.\n\nShe stood up, stumbling, as she tried to push back the bench. But there were too many other people holding it down with their own weight. She backed away. And watched the action repeated from every angle.\n\nSeraphin's voice called out, 'Who's got a magic memory of April they want to share?'\n\nAnd then, from every angle, she saw the assault on her from last night. Repeated over and over again. As Seraphin sang 'Happy Birthday'. Her face twisting from side to side. The blows falling on it.\n\n'It's not . . .' began April, stammering. She tried again but the singing continued. 'It's not my birthday!' she screamed. Suddenly this point seemed really important to her.\n\nAll that happened was the giant wall carried on showing her face crying.\n\n'It's not my birthday,' she repeated softly. 'And I'm not going to fight. You're not going to make me fight.'\n\nThis turned out to be a lie.\n\n## THIRTY-EIGHT\n\n## YOU'LL BE AMAZED AT HOW LONG IT TOOK HIM TO REALISE HIS MISTAKE\n\n'It's about April,' said Ram. 'She's missing.'\n\n## THIRTY-NINE\n\n## WAR VETERANS ARE COVERING THEIR HEADS IN GLITTER FOR REASONS THAT WILL STUN YOU\n\n**'COMBAT CHAMBER EMPTY. BATTLE READY TO COMMENCE SMILING EYE'**\n\n'I'm not giving in,' April told herself. 'I'm just showing them.' Showing them what, she didn't quite know.\n\nShe picked up a gun, steadied it, and took a deep breath. The readout counted down. The dimensions stabilised. Then, with a tiny click, the door slid open.\n\nApril stepped through into the Combat Chamber. For a few moments it held its warm, vanilla-scented emptiness, and then the landscape flickered into being across it.\n\nEverywhere she looked, projected onto the floor, were rolling marshlands and sulphurous pools. On the walls were distant hillocks and thundering clouds. Above her, more clouds drifted across the roof. The temperature fell and the air took on the tang of rubbish bins on a hot summer's day.\n\nApril shivered and walked on. Looking around, she realised this was the first time she'd seen the alien battlefront. The strange no-man's-land that they fought in. She tried to taste the air\u2014an artificial representation of an alien atmosphere. This may have been only a simulation, but it was a simulation of an alien planet. Every step she took here was somehow echoed on that planet. She was both here and standing somewhere far beyond in space. This was thrilling.\n\nYes, but not thrilling enough for her to forget her sense of defeat. Just by being here she'd given in. She'd admitted that she was going to fight. Was this how the machine rewarded her? With a better simulation of her alien surroundings? A little treat. You gave in, have some virtual reality.\n\nApril walked on, feeling the floor sink slightly under her\u2014was this an illusion, or was it . . . She reached down and patted it\u2014soft rubber? Clever. Squinting she could see it followed, just slightly, the contours of the land projected onto it. She moved on a little further. Judging from the feeling on the back of her calves, she was walking down a slight slope. She glanced back\u2014the doorway had receded and she seemed to be at the bottom of a hill.\n\nImpressive. But also worrying. What level was she on now? The terrain wasn't flat. There were rocks, weird, burnt rocks\u2014anything could be hiding behind them. She moved forwards, and the marshland receded to a silvery shore, which edged its way up to an ochre cliff. This was it, she guessed. She turned back.\n\nSomewhere around here, something was going to come and try to kill her.\n\nApril trudged on through the mire. Increasingly, she'd stopped thinking of it as some kind of illusion, and let her head tell her that she was on an alien planet. Flashes of wonder filled her mind. _I'm on an alien world._ What convinced her that it was real was that she was finding the whole thing increasingly tiring. Her legs ached, her bruises hurt, and there was sweat pricking and trickling its way down her back. Is this what it was like for astronauts, she wondered? Amazement at being where no one had gone before, followed by an annoying slight itch in their space boot?\n\nShe skirted the edge of the swamp, her boots crunching along the silver pebbled shore. Her feet hurt, but this was also really pretty something. The thick, wrong-coloured clouds carried on drifting slowly over her (were they going the wrong way? Was there a wrong way for clouds?). She stumbled slightly against the rocks and, with nothing better to do, sank gently down onto her back.\n\nHere she was, looking up at space clouds in an alien sky. Glimpsed beyond them were whole new stars, shrugged into totally different formations.\n\nHer life had changed a lot over the last few weeks, but this really was it. Alien planet.\n\nIt had been a bit of a rush. Of course aliens existed. She knew that, but had always thought of them as a vague possibility, in the same way that she knew that Russia existed. Then aliens started turning up at her school, armies of them, and the sum total of everything she knew got very hard to keep a hold on.\n\nApril had tried making her own rules for life. They weren't glamorous, or complicated, or even that ambitious. It was her way of saying to the world: 'You took my dad away, you crippled my mum, and you broke everything I believed in, so, from here on in, world, it's going to be baby steps'. Her rules had been based around looking after her mother, trying to ensure they didn't talk about it too much, making sure they weren't talked about, and trying to impose some small little bits of normality on life. Which aliens had, literally, driven a bus through.\n\nHer carefully settled world had been shaken up like a snow globe. New rules, new heart, and now here she was, some kind of teenage super-soldier, fighting aliens. It was all ridiculous. Exciting, but ridiculous. She thought she should probably stop before it got out of hand.\n\nShe laughed at that. And then, lying on her back on a not-quite-real alien marsh beach, she started to hum a song to herself.\n\nA few minutes later, she dozed.\n\nThe buzzing woke her. Her helmet was making little _fzz fzz fzz_ incoming text message vibrations.\n\nShe blinked and was startled by the clouds wandering over her.\n\n'ABNORMAL.'\n\nRight. Yes. Alien planet. Beach. Dozing.\n\n_Fzz fzz fzz_.\n\nWhy was her helmet doing that?\n\nMaybe whoever monitored the helmet camera. If the shot didn't change, that was a bad thing. Either she was dead or not putting on enough of a show. Maybe that was it.\n\n_Fzz fzz fzz._\n\nShe picked herself up, now feeling every bruise, and shook her head. Her helmet was still buzzing. She looked up the shore and then realised why her helmet had been trying to get her attention.\n\nThere were three of them.\n\nThree Skandis, slithering across the beach towards her. April crouched, grabbed her gun, and started backing away.\n\nShe'd come to fight them, but now she was here, she was wondering. She'd just wanted to take on and kill one, to make a point. To show the people outside that she was as good as them.\n\nBut where would that get her? If she survived, she'd shown that she would kill. Wouldn't they just make her fight again? Wasn't she being manipulated?\n\nOf course, she'd thought she could probably kill one of them.\n\nBut three?\n\nThree of them coming towards her very quickly. There was no way she could kill three of them. Coming here had been a terrible mistake.\n\nThe three creatures swept along the beach, the stones skittering and popping as their tentacles and claws scraped over them. Whereas they'd initially moved in a group, they now were separating\u2014one continued to glide towards her, another rolled into the marsh, and the third sprang up to the edge of the cliff where it climbed across the rocks at a terrifying rate.\n\nApril realised what was happening\u2014they were herding her. If she backed away she was losing the advantage.\n\n_Losing the advantage?_ What am I like?\n\nNo. She needed to hold the line. _Hold the line?_\n\nThe three Skandis were really close, tentacles whirling up into the air.\n\n'This isn't my fault,' she said to them. 'I don't want to do this.'\n\nShe raised her gun.\n\n'I mean it,' she shouted, the cliff swallowing her voice.\n\n'I will kill you.'\n\nShe felt her heart, her weird heart, pounding in her chest.\n\nThe creatures pushed closer.\n\nSomething moved up on the clifftop\u2014more of them. Come to watch the slaughter.\n\nShe could smell them, that terrible reek of death and vinegar.\n\nApril stood there, holding the gun. She raised it and aimed it.\n\n'I will use this,' she said. Was she talking to herself or the creatures? She sighted one of them, and marked its drop points. _Drop points?_\n\nA tentacle whipped past, stinging her cheek, burning her. She cried out.\n\n'I will use this,' she said.\n\nShe squeezed the trigger, firm constant pressure. Nothing happened.\n\nShe squeezed the trigger again. Nothing.\n\nWhat had she done wrong?\n\nApril held the gun out to the creature, almost as if asking it where she'd messed up.\n\nThe Skandis brushed it out of her hands and launched itself down onto her.\n\n## FORTY\n\n## THIS ICELANDIC PENGUIN VILLAGE IS PROBABLY THE CUTEST PLACE ON EARTH. BUT YOU ARE NOT THERE\n\nThe three Skandis exploded in ribbons of burning flesh.\n\nOne moment they were there, the next they were three meaty fireworks, shooting limbs, offal, and sparkles of gore across the beach.\n\nA dense cloud of foul smoke engulfed April. It stung her eyes and poured into her mouth. She gagged and choked, staggering back, rubbing her sleeve into her eyes.\n\nWhat had just happened? What the _hell_ had just happened?\n\nWith a wet patter, the last burning remnants of the creatures splashed onto the beach.\n\nApril lurched away, blinking to clear her eyes.\n\nThen stopped.\n\nAmazed.\n\n'You?'\n\nMiss Quill was standing there with a gun.\n\n'I cannot tell you how good that feels,' she announced, blowing across the muzzle of the gun. She stopped, frowning slightly, considering her options, and finally allowed herself a small, brief smile.\n\n'Killing things feels good. No, it feels _really_ good.' Miss Quill rocked back on her feet, surveying the red-green mist of body parts.\n\nApril loved her keyboard. Without it she couldn't work through her thoughts and turn them into music. But sometimes, when she was thinking too fast, she'd hit too many notes and the keyboard would just stop trying to keep up and emit a single thin sharp note. It was her keyboard's way of saying 'Enough, April. Stop!'\n\nRight now, her brain was making the same noise. She'd just seen three creatures killed.\n\nShe was covered in stinking flesh. Her gun wasn't working.\n\nThe Skandis had been killed by Miss Quill. Who was somehow here.\n\nWhat was going on?\n\nWait, back up a minute. There was something wrong.\n\nMiss Quill.\n\nHad killed them. With. A. Gun. April blinked.\n\nMiss Quill was aiming her gun at her and the smile had turned into a grin. 'Funny thing,' she purred. 'I saw the creatures attacking you and I wondered. I mean, they'd given me a gun, but I had no idea if it would work. What with the creature in my brain that's supposed to stop me from using weapons. But there you were. About to be killed. And, oddly enough, because of the interdimensional fields, it turns out I can use weapons here. Isn't that peachy?'\n\nWhen April replied her voice was ridiculously calm. The Queen at a Garden Party polite. 'And are you going to shoot me?' she asked.\n\n'Oh, I'd love to,' said Miss Quill. 'But, sadly, I've come a long way to rescue you. I doubt I'd get any thanks if I brought you back dead.' She tilted the gun up and looked at it wistfully. 'No more fun for you, you poor little thing,' she actually cooed at it. 'It's a very nice gun,' she remarked.\n\n'Lovely aim, simple action, strong result, stunning battery life. Not that you'd appreciate these things.' She smirked. 'You're too busy being all \"Oh no, a GUN, euw, euw, euw, I must start a petition against it and maybe paint a little sign about it\".'\n\n'That's not fair,' protested April. 'I tried . . . I tried to shoot one of them. Just now.'\n\n'How long have you been here?' Miss Quill asked.\n\n'Days,' said April.\n\n'And you've only just managed to not shoot one of them?' Her eyebrows arched. 'How you lot ended up as apex predators I don't know. I guess evolution has a wicked sense of humour.'\n\nFor the first time in a long while, April felt a familiar emotional reaction. Trying to talk to Miss Quill both made her want to smile and stamp her foot. (Did anyone, ever, actually ever stamp their foot? Still, that's how she felt.)\n\n'Look,' April said, 'it's been difficult.'\n\n'I'm sure it has,' Miss Quill oozed. 'No scented bath salts or organic delicatessens here. You poor thing.'\n\n'How did you get here?' April asked her. A change of subject sometimes helped.\n\n'Does it matter?' Miss Quill acknowledged the move and fended it away effortlessly. 'Let's get you out of here and tucked up in your no doubt achingly princess bed. You can drink chamomile tea and compose a really scathing review for this place on Airbnb. Come on,' said Miss Quill and marched off.\n\n'Right.' April breathed out slowly and set off after her.\n\nShe couldn't believe it. She was leaving behind the shore, the alien corpses, this whole nightmare. She was going home.\n\n'Oh.' Miss Quill stopped and spun round. 'You know what? I just can't resist.'\n\nShe shot April in the head.\n\n## FORTY-ONE\n\n## IN THE TIME IT TAKES YOU TO READ THIS, SKANDIS WILL HAVE CLAIMED ONE HUNDRED MORE LIVES\n\nApril woke up.\n\n_Again?_\n\nShe was in a small white room. Charlie was there. Charlie smiled in her direction, using his warm, distant, seriously polite smile. 'April,' he said with the formal courtesy of someone still trying to get used to forenames. 'You are alright.'\n\n'Miss Quill\u2014'\n\n'Is outside. Keeping watch.' Charlie sat down on the edge of the bed. He squeezed her hand, a very gentle, reassuring pressure.\n\n'She shot me!' April protested in a whisper. Knowing Miss Quill, she could hear through doors.\n\n'I know,' Charlie whispered back. He did not seem to be taking her entirely seriously. 'She was disabling the camera in your helmet. We've removed it. You're now off the network.'\n\n'Oh,' said April. 'She could have warned me.'\n\nA small 'Hah!' came through the door.\n\n'It is not her way.' Charlie's smile became rueful. Then he looked more serious. 'Why did you come here? You have put yourself in great danger.'\n\nAt first, April thought how sweet that was. Then she remembered that she was an unwilling pawn in an intergalactic war. If she died, so would her time-share heart, and then who knew what would happen to the Shadow Kin? Was Charlie's concern actually for her at all, or simply for the balance of the cosmos?\n\n'I had to do something,' she protested, and hoped that didn't sound at all pathetic. 'You were all talking and debating and people kept going missing. I couldn't ignore it any longer.'\n\nCharlie frowned and turned away from her. He was looking at the wall, which was odd as there was nothing on the wall.\n\n'I see.'\n\n'Do you?' she persisted. 'It's just that something needed doing.'\n\nCharlie nodded.\n\n'You acted as a leader,' he said.\n\n'Oh.' April blinked. 'Well, I suppose so. Something was wrong and needed sorting out and so I . . .'\n\n'Led,' Charlie finished.\n\n'Not exactly.' April wondered why she was so defensive, and why Charlie seemed so hurt. 'I mean, I'd hardly call it great leading. I've not done much except get shot at.'\n\n'Leaders do not always make great decisions.' Charlie turned to her, smiling sadly. 'But they make decisions and others follow them.'\n\n'But . . .' Others? What others?\n\nThe door opened and in came Miss Quill, Tanya, and, obviously, Ram.\n\nTanya did the explanations. Miss Quill announced that the room was crowded.\n\nRam looked at April for some reason. And, for some reason, April looked back at Ram.\n\nTanya explained that it took a couple of hours before they noticed she was missing. 'And we realised why you'd been acting so strangely, and that the whole system could be gamed and that obviously we could do the same . . .'\n\n**VIDEO CLIP UPLOADED TO TRUTHORDARE.COM**\n\n**(also available as a rather popular animated gif with the caption: 'THIS IS HOW MUCH OF A TOSS I GIVE')**\n\n******Tanya, Charlie, and Ram in a roller coaster. Tanya and Ram are screaming their heads off in giddy terror. Charlie is screaming in pure horror and alarm.**\n\n**At the front of the carriage is Miss Quill. She is completely calm, bored even, apart from a very slight smile. She is reading _Captain Corelli's Mandolin_.**\n\n'Anyway, once we'd done that we came right after you.'\n\n'Long afternoon,' said Ram. Was there candy floss on his shirt?\n\n'Afternoon?' April said.\n\n'And quite a late evening,' Ram said. He was sounding defensive, in the unique way that boys have of sounding defensive when they know that somehow they've done something wrong but aren't sure what.\n\n'Wait!' April sat up in bed and winced. 'I've been here DAYS.'\n\n'No,' Tanya corrected, 'hours.'\n\n'But\u2014'\n\n'Hours,' Miss Quill confirmed. 'This is a slow dimension. It has all sorts of benefits.' Again, that troubling smile. 'All sorts. They've been able to assemble a fighting force capable of taking on an entire species using, really, little more than a handful of humans. Let's face it, they'd need all the help they could get.'\n\nApril took a moment to process that. As in, to stop her brain thumping against her eyes. Her friends had not forgotten about her, they'd not abandoned her, she'd been worried about her mum for no reason. It was all going to be okay.\n\nApart from that they were trapped in another dimension. Yeah, that bit still sucked.\n\n'So, what do we do now?' April asked. 'I mean, you've come here to rescue me. What's the plan? How do we get out of here?'\n\nThere was a moment of blank silence. Then Ram coughed awkwardly.\n\n'The thing is . . . um . . . Now that we know what the Skandis are, we were thinking of fighting them.'\n\n'I don't believe it,' cried April. 'Seriously?'\n\n'Well.' Charlie was looking at her very firmly. 'The thing is, you've kind of taken the lead on all this, so well, we're open to suggestions.'\n\n'Well, one option,' said Miss Quill, 'is to win the war. Have you thought of that?'\n\n'No,' said April. 'I'm not sure that war is the answer.'\n\nMiss Quill uttered a long, loud groan. 'I really do think you're the worst,' she drawled. 'Perhaps we should all sit around making placards. We could stroll onto the battlefield waving them. I'm sure both sides would listen. Maybe you could make up a lovely little song as well. Something we could all sing while they're launching everything they've got at us.'\n\nDisappointingly, Tanya and Ram were nodding. 'We saw some training footage,' said Ram, 'before we could come looking for you. Those creatures\u2014they don't look pleasant.'\n\n'Pretty grim,' agreed Tanya. 'Demon zombie octopus.'\n\n'Much as I'd normally call this parochial speciesism, I have to agree,' said Miss Quill. 'The Skandis have to be wiped out. They're heading towards the Earth. What do you think they're going to do when they reach it? Hold a car boot sale?'\n\n'That's not the point,' April argued. 'I think this is all about perception. What have we been told about them?'\n\n'That they're going to destroy the Earth,' said Ram.\n\n'Do we know that that's what they're doing?' April said. 'I mean, has anyone in this room\u2014the nonhumans\u2014heard of them?'\n\nCharlie and Quill glanced at each other.\n\n'Well, no,' admitted Charlie.\n\n'Isn't there an intergalactic Wikipedia?' April pressed on. 'Something you can look them up in?'\n\n'Let's go check Wikipedia!' Miss Quill tutted. 'What a typical student response.'\n\n'But if they're a race of predators, wouldn't you know about them?'\n\n'The universe is best assumed to be a hostile place,' Miss Quill said firmly. 'If strange craft appear in your system, man the defences and start shooting.'\n\n'Isn't that a bit shortsighted? How does it go down?'\n\n'Not at all well. But if your visitors are intelligent they will see it as a perfectly sensible response. You humans flatter yourselves that you have the monopoly on brutality and greed. You don't.'\n\n'Well, I think we need to find out more,' said April. 'Think before we shoot.'\n\n'Oh, look at Miss Moral High Ground.' Quill laughed. 'What were you doing before I rescued you?'\n\nApril looked at the floor, suddenly angry. 'I was in a combat chamber. I was trying . . . I was trying\u2014'\n\n'You tried to shoot one of these monsters. I saw you,' Quill said triumphantly.\n\nApril looked up and knew she'd lost the room.\n\n'Yes!' Tanya was grinning. 'This is what all these evenings of World of Warcraft have been leading up to,' she said. 'Finally I get a proper gun.'\n\n'No,' April insisted. 'Tanya. Can't you see that you shouldn't be doing this?'\n\n'I shouldn't?' Tanya's face set hard quickly. 'Because I'm younger than you?'\n\n'No! None of us should be fighting. Why isn't this scheme picking on adults? We can't even vote, but someone thinks we're the best to fight an alien war. That's . . . '\n\nTanya groaned. 'Look, it's an outsider's view\u2014maybe not the right one, but someone's decided that we're the best people to fight this thing. So maybe we should.'\n\nRam nodded, and April felt her stomach sink.\n\n'If there's one thing I've learned these last few weeks,' he stumbled, 'it's that the rest of the universe is pretty mean. And it seems to have no problems killing kids.'\n\n'Exactly, Mr Singh,' said Miss Quill.\n\nTanya nudged Ram in the ribs. 'Come on. I'm going to get a gun and find out how this system works. You game?'\n\nShe pushed on the door and went out.\n\nRam stood there, hesitant, hanging back but obviously eager to go. 'Just so you know,' he said, sounding a bit mumbly, 'Miss Quill has a point. If those things are coming for the Earth, I don't want to say I stood by and let it happen.'\n\n'Sure,' April said, not even looking at him. She didn't even have to try to make her voice bitter. 'Absolutely. Go fight your war.'\n\n'Yes,' said Ram, going for a joke. 'Tanya'll so think we're best friends now.'\n\nHe went through the door. April looked up. 'Is that how you think women relate to you?' she wondered. 'Side with them, they'll get a crush on you?'\n\nShe turned back to Charlie and Quill.\n\n'Well,' she said to the two of them. 'I think I can guess what Miss Quill's going to do.' She pointed to the door.\n\nMiss Quill barked with laughter. 'Are you sure?' she said. 'Really?'\n\n'Yes,' April said firmly. 'You're going to go and slaughter those monsters.'\n\n'I'd considered it,' Miss Quill admitted, 'but I've come up with something a little bit more fun.'\n\nAnd then she raised her gun and pointed it at Charlie.\n\n## FORTY-TWO\n\n## MANY PEOPLE WOULD BLAME THIS ON MARRIAGE EQUALITY. BUT WOULD YOU?\n\nThey came in from the day's kills exhausted. They were sweaty, grimy, and bleeding. Their helmets hung limp from their hands, broadcasting swaying, unusable footage of the floor. The troopers poured into the Big White Room, all of them thoroughly, utterly shattered.\n\nSome had been here for weeks, it seemed. Some only for hours. But the fighting was getting worse. Whereas previously the battle computer had sent them into combat with only one, or maybe two Skandis, they were now up against groups of anything from three to a dozen.\n\nTo start with, it had been a bit more fun. A change was good, after all. They called it 'levelling up,' and those who came back boasted of having Done the Boss. The problem was that fewer and fewer of them were coming back. And troops who'd levelled up found themselves going straight back into combat.\n\nIf they'd been expecting a different scenario, or to encounter Skandis with different weapons or something more novel, they were disappointed. There were simply more of them. And they were angrier and more vicious.\n\nWhen they'd thought war was a bit like a game, it had seemed thrilling. Now, every hour of every day was spent going through a door to fight more and more of the enemy. It would only stop for rest breaks, for food and for sleep, and then it would start again.\n\nThe ones who lived were the ones who thrived on the routine. They took even more risks, they found the familiarity comforting, they pushed back against it, but they never forgot the skills they'd learned.\n\nThe ones who didn't come back were the ones who found it boring. Who found it draining. Who thought that it was unfair. You weren't banned from complaining about the challenges over your helmet camera\u2014that was fine, although it might be edited out of Seraphin's recaps. What was curious was that, if you did complain, your chances of stepping through the door into a battle with five or six Skandis rapidly increased.\n\nThe videos were becoming grimmer. The troops sat around, trying to keep their eyes open, as video after video slid past them on the great wall. Sometimes they saw themselves and cheered. Sometimes they saw friends and realised they'd never be coming back and they booed the enemy. They hated the creatures they were fighting against, and they did it with an instinctive, weary, ingrained hatred. The more routine, the more dull the war became, the more they'd carry on with it because they had to.\n\nThey were, Seraphin thought, becoming a tougher crowd. It was harder to get a laugh out of them.\n\nAs the broadcast switched on, he looked out across the hall at all their faces\u2014teenage faces scarred with combat, with dirt burnt into them and massive purple bags under their eyes. They looked gaunt and wired. Somehow it was his job to keep them going.\n\n'Heeeeeeey! Evening, everybody!'\n\nSeraphin was standing in his room. He had piled his hair up on his head, and was laughing. 'I'm henna-ing my hair. What is up with my life? I'm almost out of cereal. Anyway, how's the battle going? We've some really good kills going on here today. Some excellent kills. Really lovely stuff. Shall we have a montage? Yeah, let's . . . wait\u2014'\n\nSeraphin stopped and licked his lips. For a moment, just a moment, he seemed uncertain. Curious. Brave.\n\n'Hey, have any of you lot seen April recently? You know, the troublemaking one? She's fallen off my radar. Hope she's okay. If any of you see her, you'll let me know, okay? Right. Anyway. The latest fighting. Cue clip show.'\n\nScenes from the day's battles played across the huge screens, accompanied, for no reason other than because he enjoyed it, by Seraphin playing 'Climb Ev'ry Mountain' on his ukulele.\n\nCharlie shouldn't have been surprised. Not really. Ever since he'd been assigned Quill as his bodyguard, he'd known that she would someday point a gun at him. In theory her head should right now be exploding with pain. It wasn't. She was simply holding the gun without even the slightest wince. Actually, she was smiling a proper, warm smile. Any second now she was going to kill him.\n\nAt moments like this, Charlie felt like the loneliest person in the universe. He'd lost his family, his friends, his world, and the only person left who remembered them was his bitterest enemy\u2014his reluctant bodyguard, who wanted, more than anything, to kill him.\n\nCharlie was not unaware of the irony. It was made more curious by the mode of speech that humans used. They were persistently violent in conversation, which often confused or alarmed him. When someone would say, 'My mum is going to kill me,' he couldn't help automatically picturing Miss Quill pointing a gun at him. That image haunted him. For once, when a human said something, he knew exactly how they felt. A curious sensation of dread, of inescapable inevitability. Sometimes he forgot about Quill's nature\u2014sometimes they were almost like friends, with more in common with each other than with anyone else on the planet. But that image was always there to remind him. Because one day, she'd find a way around her processing and shoot him.\n\nIn some ways it was a relief. Having Quill point a gun at him felt like being able to breathe out at last. There it was. As suspected. Yet, at the same time, he felt a twinge of disappointment, regret. As though, maybe, in their time together they'd become . . . well, not fond of each other, no, but still, that they'd developed some sort of bond.\n\nAll the same, it wasn't stopping her from aiming that gun.\n\n'Yes, I know,' Quill was pouting sarcastically, 'how dare I? Quite easily. Now, are you going to beg before I shoot you? I think it would be nice.'\n\nCharlie said nothing.\n\nQuill raised the gun, finger tightening on the firing trigger.\n\n'Come on, Prince,' she coaxed, purring. 'Just a few little last words for your slave.'\n\nCharlie was silent a moment longer. 'Actually,' he ventured, 'I have got something to say.' And he said two words more.\n\nApril burst out laughing. Posh people swearing, there and then, became her favourite thing ever. Quill glared at her.\n\n'Sorry.' She cupped a hand over her mouth. 'Sorry, Miss Quill, you can go ahead and shoot him now.'\n\n'Thank you.'\n\n'Surprised you've not done it already.'\n\n'When I need your advice, girl, I'll ask for it.'\n\n'Totally. And I wouldn't dream of offering it.'\n\n'Oh really?' Quill paused. 'You look just the type that loves giving out unwanted advice. Some children think they're so grown-up and really, they're not. You know nothing about life.'\n\n'No,' admitted April. 'I don't. But I do know you won't kill Charlie.' Charlie glanced at her, curious.\n\n'What?' Quill said.\n\n'You would have done it already.' Quill stared at her.\n\n'You've been waiting to kill him for months and yet, there he is still breathing.'\n\n'So?' Quill was curious.\n\n'Go on. Just shoot him,' April urged. 'Sorry, Charlie.'\n\n'No, that's fair.' Charlie's voice was even calmer than usual. His posture shifted, with regal delicacy, and he leaned forward, into Quill. 'Go on. Do it.'\n\nShe stared back at him.\n\nApril began to have doubts about how this was going to end. She'd assumed that she was right, that Quill was bluffing\u2014she wouldn't really, she couldn't really, could she? She talked all the time about it, but she wasn't really a cold-blooded killer, was she? But then April remembered the three Skandis exploding on the beach. Maybe she was wrong.\n\n'One word,' said Quill, eventually, that little smile back on her face. 'I just want to hear one word from him about how sorry he is\u2014about what he did to me, about what happened to my people. And then I'll let him go.'\n\n'You'll pull the trigger,' Charlie corrected. He still seemed icily calm.\n\n'Well, yes,' Quill admitted. 'But you'll die knowing you're the better man. That's what you love, isn't it?' She sneered. 'The moral high ground. Slavery and slaughter that you can feel smug about.'\n\n'You keep saying that you are my slave.' Charlie's calm carried on until even April felt infuriated. 'You are not my slave. People buy slaves because they want them. I appreciate you, what you do for me. But'\u2014and the calm got a degree chillier\u2014'I do not want you.'\n\n'Suppose there was another way?' began Quill. She seemed to have stopped blinking. April wondered if she actually needed to blink, or if it was just something she did to appear more human that she'd forgotten about. Come to think of it, Charlie wasn't blinking either. The two were just staring at each other like chess players. With a gun.\n\nCharlie picked up Quill's sentence. 'If there was another way to keep me safe? Then yes, I would happily take it. And we would both be free of each other.' He smiled a very calm smile. 'I'm afraid that is all you are going to get out of me.' The two stood there. Quill, pointing the gun. Charlie, head tilted back insolently, eyes daring her to do it.\n\n'See?' Quill turned back to April with a helpless shrug.\n\n'Even at gunpoint, even in the last moments of his life, he cannot say sorry. He is just impossible. And the more human he becomes, the worse he gets. Unbelievable.' She rolled her eyes, dropped the gun, and walked out.\n\nApril let out a breath she didn't know she'd been holding.\n\nCharlie picked up the gun and looked at it.\n\n'Are you okay?' April said to him. 'After all that? Are you sure you're okay?'\n\nCharlie turned to her, baffled for a moment, and then smiled the dazed smile he used for trying to understand a joke.\n\n'Ah,' he said, and pointed the gun at his head. It clicked. Nothing happened.\n\nHe passed it back to April.\n\n'It's genome-locked,' he said. 'It's set to kill only one species\u2014the Skandis.'\n\n'You knew?' She was incredulous.\n\nCharlie was nonplussed. 'It was a reasonable surmise. Humans are, forgive me, violent. Genome-locking the guns is a safety precaution to prevent the soldiers here turning on one another or on whoever runs this facility.'\n\n'But you were absolutely certain?' April continued, amazed. 'You knew that she'd not be able to use the gun, that she'd not found a way round that. Did she know that you knew too?'\n\n'Oh, she may have guessed. She may even have disabled the lock, but, there's one thing you have to remember.' He looked up at April and his eyes were sad and serious. 'Ever since she was assigned to me, I knew that sooner or later she would point a gun at me. From day one, I have been rehearsing what to say.'\n\n## FORTY-THREE\n\n## PEOPLE ARE TWEETING THEIR WORST BATTLES AND IT IS CRINGINGLY HILARIOUS\n\nApril's hopes of escaping were fading fast. Instead, her friends (and Miss Quill) had fallen in line with the place.\n\nIs it just me? she thought. Am I the only one who doesn't want to fight these things? Am I being incredibly dense?\n\nShe wondered if she was just being stubborn. But then she figured, what did it matter? Surely they'd get tired of it in an afternoon or so, and, as long as she kept a low profile, she'd not get into any trouble. Once they realised how serious the battle really was, they'd come up with some way to get out of here.\n\nRam hurled himself across the battlefield.\n\nThree Skandis? Fine. Bit of a challenge, but nothing he couldn't handle. He was loving shooting at them. Okay, he made sure that he downplayed how much fun it was to April, but the whole soldiering thing kept on giving him a kick. Especially when they played footage from his headcam on the big screen. He couldn't help it. He felt proud.\n\n'Is it wrong to feel so pumped?' he asked Tanya over the evening meal.\n\n'Pumped?' She shook her head. 'Next thing you'll be saying this food is nutritionally amazing.'\n\n'Well,' he had to admit, 'my stomach is looking pretty flat.' He spooned down some more of the weird porridge gloop. 'Probably a mixture of no sugar and all the running.' The running was the best bit. For some reason, out here, wherever they were, his leg had shut up. It just did what he asked, and tried not to get in the way. It was like having his old leg back, only it glowed in the dark slightly.\n\n'The running.' Tanya gave a sudden smile. 'It is pretty awesome. This is like the best game ever.'\n\n'Totally,' Ram agreed. 'Actually being in the Combat Chambers\u2014it's amazing. And the way that the force field setup prevents some of the damage. That's great.'\n\n'What?' Tanya asked.\n\n'On Level Four. You get a force field. Oh'\u2014his face fell comically\u2014'are you not on Level Four yet?'\n\n'Tomorrow,' Tanya mumbled the lie. 'I'm levelling up tomorrow.' She watched Ram chewing his slurry and tried not to resent his success. 'Wow. Level Four already, that's great.' It sounded hollow as soon as she said it, but Ram didn't seem to care. He just nodded and started scraping his spoon around his bowl.\n\nThe envy she felt was blocking her head from focusing. Why did you only get a force field on Level Four? Surely that would make more sense for the new recruits, progressing through the training assignments?\n\nThen Tanya realised. Whoever was running the experiment had some limit on their resources. They only protected the more advanced soldiers because they were actually protecting their investment.\n\n'Wow, that's cynical,' she gasped. Ram put down his bowl.\n\n'What?' he said.\n\nShe told him. He looked nonplussed. 'It sort of makes sense,' he argued. 'In a way. Like it's only worth getting a proper football kit if you're playing for the team.'\n\nTanya stared at him. 'This place is changing you,' she said.\n\nAgain, he looked slightly baffled. 'It's an experience. Experiences change us. It's how you grow up.' Gawd, he could sound so dull. He lowered his voice, speaking out of the side of his mouth as he chewed. 'You shouldn't be talking to me here, Tanya.'\n\n'We're still not friends?'\n\n'Yeah, there's that, sure.' Ram looked at her seriously. 'But also, remember\u2014people don't really talk here. It would look . . . strange. For us to be seen chatting together. They might check the camera feeds.'\n\n'And deduct points?' Tanya was starting to feel cross. The kind of cross that even though a bit of her head was saying 'Whoa, hold on, he's maybe got a point', the rest was steaming.\n\n'No, no.' Ram hadn't even noticed. 'I mean, a bit, yeah, but they might crack onto us. Currently you're showing up as NORMAL. I'd hate for that to become ABNORMAL.'\n\n'Whoa.'\n\n'Anyway.' Ram shrugged. 'Are you going to finish that food?'\n\n'You're welcome to it,' said Tanya as she stood up and walked away.\n\nRam slid her bowl over, and started to eat happily.\n\nThere was one combatant who had entered the system recently and had already reached Level Six.\n\nShe did not fit the typical profile of the Skandis Recruitment Programme.\n\nShe did not fit it at all.\n\nYet she was here. A little older. A little taciturn. Maybe she didn't come to the Big White Room to watch the inspirational videos. But she had come through the system and she was turning out to be the best recruit they had.\n\nWhen they'd initially processed her, they'd considered her a statistical oddity. So much so that they'd come to her room at night, stepping through the walls to question her in her sleep.\n\n'Name?' they'd asked her.\n\n'Quill,' she'd said.\n\n'First name or surname?'\n\n'Just Quill,' she'd insisted.\n\nShe was certainly a blip. But a blip could be useful, if she lasted a few rounds. As it was, she'd lasted rather more than a few rounds. Her progress was faster than anyone else in the challenge, and she was accounting for more kills than many of their other combatants put together.\n\nThey came for her at night again.\n\n'Why are you so good at this?'\n\n'I like killing.'\n\n'But what are your allegiances?'\n\nQuill snorted, or it may have been a laugh.\n\n'You fight as though you really believe in the cause.'\n\n'No,' Quill mumbled absently through the dream shield. 'As I said, I really like killing.' She smiled in her sleep. 'Oh, how I've missed it.'\n\nThat gave them pause for thought. They'd somehow accidentally inducted someone entirely outside the programme's remit, someone they'd normally have discounted entirely. And yet, she was proving to be their best asset. Perhaps they'd need to revise their recruitment parameters.\n\n'We were never here,' they told her. Then they went away.\n\n'That's got you puzzled,' muttered Miss Quill as they left.\n\nRam saw Charlie coming out of a combat chamber and rushed up to him, punching him on the shoulder.\n\n'Look at you, Princeton! You've settled into fighting after all.'\n\n'Yes,' said Charlie without any hesitation.\n\nRam chuckled. 'Knew it, knew it.' He laughed. 'Once you start playing, you can't stop. What level are you on? Have they given you a force field?'\n\n'No,' Charlie considered. 'Why, have you got one?'\n\nRam leaned forward. 'Course I have. Anyway, catch you later!' He aimed a couple of practise shots at Charlie and then bounded off down the corridor.\n\nCharlie watched him go without blinking.\n\nWhen he told her later, April laughed. 'He really thinks you're fighting?'\n\n'Well, at least someone does,' Charlie said.\n\n'And, just checking, you're okay that you're not?'\n\n'Yes,' said Charlie. 'Sure. Absolutely.'\n\nHe settled down at the edge of her bed. 'Definitely.' They'd had to reach a compromise. One of the problems was that Quill didn't want Charlie going off into combat without her, just in case her brain exploded. Charlie did not fancy going into combat with Quill, just in case she managed to carefully, accidentally, catch him in a ricochet. April didn't want anyone going into combat, but especially not Charlie. ('If I'm right,' she argued, 'you could start some kind of interstellar war.') So Charlie had quietly agreed to fluff his Level One assignments. He'd fluffed them so spectacularly that footage from them had made it into a vlog called 'Soldiers Do the Funniest Things'.\n\nThen he'd gone to see a soldier who he'd assumed was in charge of the Combat Bay. 'Look,' he'd said, 'it turns out I'm useless in battle. Really. It's so frustrating. Is there anything else I can do?'\n\nWhen Quill found out she'd laughed herself sick for a minute.\n\n'You're a cleaner?' she'd roared.\n\n'It is a practical and necessary function,' Charlie reasoned. 'There is no disgrace in it.'\n\n'Oh, of course not, your majesty,' Quill had hooted.\n\nWhen he'd scrubbed out her chamber that night, he found the walls elaborately painted with entrails. He spent several hours scrubbing off an intricate and ancient Rhodian curse word.\n\n## FORTY-FOUR\n\n## SHE DROPPED A TRUTH BOMB BUT WASN'T EXPECTING WHAT WOULD HAPPEN NEXT\n\nThey were eating their evening meal when April walked into the Big White Room.\n\n'Hey!' she called, and her voice shook. 'Me again.'\n\nOn screen, Seraphin carried on talking. He was wearing an old rabbit onesie, strumming away on a guitar and listing troop manoeuvres.\n\nShe walked into the centre of the Big White Room.\n\n'I know you can't see me but I ask you to hear me.'\n\nThe scraping of spoons around bowls became louder, the heads more bowed.\n\n'I've come to open up, to bear my soul, to confess.' She shouted the last words, feeling her throat rasp. She had to carry over all those spoons.\n\nShe glanced at the screen. One square was her. It dipped and wobbled. Tanya. Another square sprang up. Charlie.\n\n'We cannot fight these aliens. Not until we have more information.'\n\nA third square. Miss Quill. She was sat just over from her and was looking disapproving. If a facial expression could convey 'I am only doing this because I was told to' then hers did so most clearly.\n\n'I repeat. I cannot kill these creatures. It's insane. We're being sent out to fight a war and we're just not up to it. We're kids.'\n\nThe moment she said it, she knew she'd got the wrong word. She'd wondered about 'children', she'd spent about a second on 'teenagers'. Quill had suggested 'cattle'. That would have been better than hearing 'kids' echo back off that great big wall.\n\nWith only three squares on it. Ram.\n\nWhere was Ram?\n\nShe spoke again, and there was a quiver to it. 'I'm not killing these creatures, and neither should you.'\n\n'Really?'\n\nAbove her, Seraphin put down his guitar, pulled up his droopy bunny, and leaned into the screen. He stared down at her and his expression was strange. There was no smile, no glint in his eye. He was pissed off. Worse.\n\nApril's father had read a book about parenting. Ironic. The thing he'd learned from it was to say, 'April, I'm not angry with you. I'm disappointed'. Apparently it was a 'coping response' for when situations got heated. Another coping response was to drive your family into a tree. April hadn't been disappointed with him about that. She'd been very angry.\n\nSeraphin looked disappointed. His face was a master class in disappointment. It was a glorious manga doodle.\n\nHis expression stopped April in her tracks.\n\n'I thought you were different.' His voice was very quiet, flat. 'I thought you were someone to watch.' He exhaled, a little puff of air that pushed against a strand of his perfect hair. 'You came here to tell us you wouldn't kill these monsters?' He was sneering. 'That's not what your helmet cam says.'\n\nThe screen filled. Shaky footage of April running and stumbling along the shore, the three Skandis throwing themselves howling towards her. Her recorded breathing echoed around the White Room, along with her whimpering. Surely she'd not made that much noise? Surely she'd not been that loud? Surely she'd been calmer.\n\nThe Skandis lunged at her, tentacles snapping open to devour her.\n\nScreen April brought up her gun.\n\nThe three creatures exploded.\n\nScreen April lowered her gun.\n\n'Looks like you were happy enough to kill three of them,' said Seraphin. And his lips were twisted with bitterness.\n\n'No!' shouted April. 'It wasn't like that. I've been edited!'\n\n'Puhlease'\u2014Seraphin sighed\u2014 'you're not on _Big Brother_.'\n\nApril turned back to the room, desperate. 'I didn't shoot them. I didn't. You have to believe me.' A spoon scraped in an empty bowl. Another spoon.\n\nThen more. An echo of spoons and bowls.\n\nApril carried on shouting over the clattering. She was desperate, angry, seeking out the faces of Tanya and Charlie. But she still couldn't find Ram. Where was he?\n\nShe caught the eye of Miss Quill, who mouthed 'boohoo' at her. Because of course, she would do that.\n\nApril stopped shouting. But the scraping of spoons and bowls continued.\n\nSeraphin's voice carried over it. 'Do you know why she has to kill them? Do you know why you all kill them? Because they are evil.'\n\nThe screen showed clips. The last moments of fighters, struggling and screaming against the Skandis. Those snapping jaws, those sharp tentacles dripping blood.\n\nThe spoons paused. The hall fell silent. Everyone was staring at the screen. Rapt.\n\n'Enough of the coward.' Seraphin's voice spoke again. 'Let's see what's been happening with our Level Six fighter. That'll bring the mood up.'\n\nThe screen showed Skandis after Skandis, a whole squad of them, being blown into screaming embers. Someone in the game was clearly very good.\n\nApril caught the expression on Quill's face. Pride.\n\n'That is someone we can be proud of.' Seraphin's voice echoed over jets of fire that were melting tentacles. 'Someone who is doing all they can to save the planet. To save us all from Skandis. And if you're not on board, well . . .' He leaned out of the screen to address April, from the walls, the ceiling, the floor. 'Why don't you go away?'\n\nApril stood her ground for longer than she thought she'd manage. One breath. Two breaths. On the third, she turned and ran.\n\n## FORTY-FIVE\n\n## THIS YOUNG FOOTBALLER HAS SOMETHING SURPRISING TO SAY ABOUT RACIAL PROFILING\n\n'Hey,' said Ram.\n\nApril hadn't bothered to close the door. There hadn't seemed any point.\n\nShe was slumped against the sharp frame of her bed, staring at the empty white wall, and swigging from a bottle of water.\n\nShe hadn't been expecting Ram. She wasn't sure who she'd been expecting, but not him. God, what was he here for? Was he going to give her a lecture? Tell her how she was letting their side down? Is this the talk he used to give when a player wasn't up to it on the team? Gentle, sad, coaxing, grown-up? Oh _spare me_ \u2014she couldn't bear it.\n\nRam leaned against the wall and slid slowly down it.\n\nHe looked so strange in the combat gear. Weirdly adult.\n\n'Hot, right?' Ram laughed then stopped to sniff his armpits. 'Also, I really stink. Like maybe my sweat is masking the smell of slime.'\n\nApril smiled back at him. It was the weakest smile she'd ever managed.\n\n'Been fighting?'\n\n'Oh yeah.' Ram grinned. 'It's the biggest high ever. It's so real.' He wiped some sweat off his forehead.\n\n'What we've got back home . . . nothing on this. Football's great\u2014but really, it's'\u2014he dramatically lowered his voice\u2014'just pretend. There are no stakes in it. But this, this is real, and I can be good at it. Not like Charlie's weird space thing. This is something we can be a part of, rather than just get dumped on.'\n\nRam's eyes were so clear she couldn't help but believe him. He was so sincere. He wasn't even trying to undercut it, or make light of it. He was so firm about it that she suddenly saw his reading of the world. Aliens had taken everything he understood about life away from him. They'd both lost so much to these aliens.\n\nNo wonder he was enjoying fighting them so much. It was his way of evening the score. No, she realised. It was Ram's way of winning.\n\n'You didn't look at me,' she said. She didn't even realise she'd said it until the words were out there. She hadn't wanted to ruin the moment. But no. She had. How very April.\n\n'What?' Ram was certainly looking at her now, confused. Guilty?\n\n'I went to the Big White Room to give my speech. You didn't look at me. The others did. Even Quill. But not you.'\n\n'Ohhh.' Ram frowned, then smiled sheepishly. 'Right. That. Sorry. I forgot.'\n\n'Sure.' April then realised what he'd said. 'You _forgot_?'\n\n'Well'\u2014Ram's face squirmed\u2014'I knew it was on. Sort of. Just, I was offered an extra fight session, and I figured you'd be okay without me. How did it go?'\n\n'How did it go?' April repeated. 'Look at me. I'm sat against my bed trying very hard not to cry. Have a guess.'\n\n'Oh. Right,' said Ram. He slid across the floor to sit alongside her. 'My dad has told me that in situations like this it is best not to say anything. So I won't.'\n\n'Fine,' April nodded.\n\nThey sat in silence for a bit.\n\n'Only . . .' began Ram. April looked at him.\n\n'Only,' he began again, 'it was a surprise incursion and\u2014'\n\n'Stop it!' shouted April. 'I can't believe you.'\n\n'Okay,' said Ram.\n\nThis time the silence held.\n\n'I am so cross with you,' April said. 'Ever since I've been here, I've felt so alone.'\n\n'We came looking for you,' protested Ram.\n\nApril silenced him. 'Yes, you came looking for me. And that meant the world to me, but already you've changed. You and Tanya are having the time of your lives battling aliens. Quill's become the Terminator. And that's not what this is about. I thought you'd understand. I've got my friends here, and I feel more alone than ever.'\n\nRam rested a hand on her shoulder. It landed with the caution of a man who is worried it will be pushed away. April did not push it away.\n\n'You were right. You really do stink,' she said.\n\nHe grimaced. 'Yeah, I'd best go run a shower.' He made as if to get up but clearly wasn't going to.\n\n'Don't go,' said April pointlessly. 'You finally turn up, you may as well do some good.'\n\n'Cool,' said Ram because he couldn't think of anything else to say.\n\nApril edged a little further along his arm.\n\n'You think when all this is over, they'll let me keep the uniform?' Ram asked.\n\nApril smirked.\n\n'What?'\n\n'I'm not sure how that'll go down on the streets of London.'\n\n'Ohhhhh.' Ram's face fell, and he shook a fist. 'Damn you, racial profiling. It's quite something when a kid can't go to school in combat gear waving a gun around.'\n\nThey smiled at each other.\n\n'Not even as cosplay?' he asked.\n\n'Not even as cosplay,' she confirmed.\n\n'I was going to say I'd better make sure I've got some really great photos of this, but I'm not sure how that'd look if the police ever searched my computer.' His face fell. 'Or my mother. Which is more likely.' He changed the subject. 'Anyway, the talk\u2014I'm gathering it went badly.'\n\n'Stitch-up.' April raised her bottle of water in salute. 'Total stitch-up.'\n\n'Sorry,' said Ram. 'If it cheers you up, I found out what Charlie, the great royal alien prince, is doing.'\n\n'What?'\n\n'They've got him cleaning.' Ram snorted. 'He's been pretending all along that he's fighting. But I caught him coming out of a Combat Chamber with a mop and bucket.'\n\n'Whoa. What did you do?'\n\n'Kicked over his bucket,' said Ram.\n\n'You're the worst.'\n\nThey smiled at each other until April broke away, her expression haunted again.\n\n'What?' Ram asked. He'd been about fifteen percent sure he was about to get a kiss\u2014which was odd.\n\n'Total stitch-up,' April repeated. 'They had a video prepared. They'd edited it to make it look like I shot those creatures. And they sent you off fighting so that you wouldn't be there, counting on the fact that it would upset me.'\n\n'And'\u2014she stood up\u2014'they were right.'\n\nShe heaved the thin mattress off the bed, threw the pillow at Ram, and then tried lifting up the bed frame.\n\n'What you doing, She-Hulk?'\n\n'Tear that pillow apart,' ordered April, giving up on the bed and turning to the wall. 'I'm trying to find a bug. They're not just watching us in the helmet cameras\u2014they're watching our rooms.' She tried putting a fingernail in between two wall panels. Nothing happened.\n\nRam did not tear the pillow apart. Instead he put it back on the bed along with the mattress. Then he found the sheet, uncrumpled it, and fitted it onto the bed surprisingly neatly.\n\n'What are you doing?' said April. 'I need you to help me open this wall.'\n\n'No, you don't.' He laid a hand on her shoulder. 'You need to take a deep breath and let the crazy out.'\n\n'But . . .'\n\n'Deep breath,' he said, holding up a finger. April took a deep breath.\n\n'Now,' Ram spoke gently, in measured tones. 'We've been transported to a different dimension, where time is running slowly, and we can be projected into an alien war. Given they've got that technology, do you think they're going to put bugs in the lightbulbs?'\n\n'There aren't any lightbulbs,' April said.\n\n'Exactly.' Ram decided to overlook that she'd breathed out. 'They've built the technology into the walls. Or it's in the air. Whatever. It doesn't matter. They need to keep an eye on us at all times\u2014to find out who they can trust. That really freaked me out at first.'\n\n'At first?' April blinked. 'What about now?'\n\n'I realised they only care if you're a good fighter. And I'm a very good fighter. They know I'm loyal and I love the fighting so I don't have a problem with it. I want to hang out with you? Who cares? So long as'\u2014he squirmed again, but it was a really intense squirm\u2014'so long as you don't have a problem hanging out with me.'\n\n'No,' April said. 'I don't.'\n\n'Fine,' said Ram. 'Then I'm going to go and have a shower, then go back and slaughter some more squid. And what about you, Little Miss Dissident?'\n\n'I'm going to find out who's really running this place,' she said.\n\n## FORTY-SIX\n\n## YOU ARE BEING LIED TO ABOUT DOGS\n\nWhen April entered Seraphin's room, he was playing with his puppy. He was wearing a battered old fisherman's jumper and ripped jeans.\n\n'You look really good in clothes,' said April.\n\n'You've got a nerve,' said Seraphin coldly. He stood up, and switched the dog off. It vanished.\n\n'Oh.' April blinked in surprise.\n\nFor a moment, Seraphin said nothing. When he looked up, his face was twisted, angry. 'I believed you. I fell for all your crap.' He shook his head, disgusted.\n\n'It wasn't crap,' April insisted.\n\n'I saw the video,' he said, getting angrier with every word. 'Everyone's seen it. There's no hiding the kind of person you are now. You're a killer, just like everyone else.'\n\n'It's a fake!' protested April.\n\n'Really?' Seraphin laughed. 'What? Did someone quote you out of context? Is that it?'\n\n'No, of course not,' she said. 'That really isn't it.'\n\nShe crossed to his desk, pushing her way through the piles of musical instruments and cereal bowls. She waved the mouse, waking it up, and looked intently at it. 'Where is that video?'\n\n'Well, it's not my screensaver,' Seraphin muttered. He poked through some folders and summoned it up.\n\nApril let it play out, following herself across the beach. Then she hit pause.\n\n'This is the moment,' she said. 'This is the moment that I panicked, and tried to fire my gun. So yes, I tried to defend myself. But I couldn't. Because my gun wasn't working.'\n\nSeraphin stared at her.\n\n'Someone wanted my last thought to be that I'd given in. Now.' She nudged the video along. 'In the next frame, here, we have three Skandis all burning.'\n\n'Uh-huh,' said Seraphin.\n\n'The point is that you don't even see me fire one shot, let alone three.'\n\nSeraphin frowned. 'But if you didn't fire, who did?' April relaxed. He was thinking about it. He'd stopped completely denying it. 'You might find this surprising, but the shots were fired by my teacher.'\n\n'Your teacher?'\n\n'The Level Six Combatant. Or, as we call her, Miss.'\n\n'Her?' Seraphin whistled. 'I wondered how she'd got in here. Wow. Bet she takes missing lessons really seriously.'\n\n'Yeah,' said April, relaxing a little bit more. 'She takes everything pretty seriously.'\n\n'Sheesh.' Seraphin called up some footage. It showed a row of Skandis being mown down. 'I'm kind of glad she doesn't teach me.'\n\n'It is not that much fun,' conceded April.\n\nSeraphin stood up, tugging at his hair. He was chewing on a strand of it, and twirling some of it round and round his fingers. 'So this video of you was a fake?'\n\n'Yes.' April was firm. 'I sort of hoped you'd not done it.' She smiled. 'I'm glad.'\n\n'God, no,' Seraphin said. 'I've enough on my hands editing my own videos. For this stuff I just take what I'm given. It's worrying, isn't it? The whole mantra of putting it all out there . . . It's being dicked around with.'\n\nHe crossed to a fridge and rifled through it. 'Smoothie?' he asked her. He tossed her a small plastic bottle of strawberry and banana. She opened it and downed it in three mouthfuls.\n\n'You have no idea,' she took a last gulp, 'how good that tastes.'\n\n'Oh I do,' he said, wiping his mouth. 'Mine had kale in it.' He tossed the bottle, with perfect aim, into a recycling bin. 'Take another. They send me crates.'\n\n'Who sends you these?' she asked, opening another one. God, something with flavour. Amazing.\n\nSeraphin shrugged. 'Could be the owners. Could be the people who make the smoothies. Product placement really sells. You want to take some back for your friends?'\n\nApril thought about it. 'I doubt they'd take them. Probably think I'd poisoned them.'\n\n'Won't they interfere with your calorie-controlled fighting diet?'\n\n'What?' She laughed. 'The stuff they feed us? It's rank. It's porridge and stew and sometimes a bit of both.'\n\n'Oh,' he said. 'I was told you're all on a specially targeted diet. I've got some crisps somewhere.'\n\n'Amazing.' April sat down on a chair, laughing. 'I've missed chewing. It makes us all weird, having to sit and down our slops while you work your way through all this proper food. Kind of angry.' She stopped, caught up in a thought.\n\n'What is it?' Seraphin asked her. His brows were doing that strange, kissable thing that they did. He was now doing a textbook baffled pose.\n\n'I'm wondering about the food,' said April. 'I think they've been putting something in it. I've been skipping it as much as possible, but the others . . . I wonder if that helps explain why they're so into the fighting.'\n\n'Yeah.' Seraphin didn't sound convinced. For his money, everyone going out there seemed to be fairly pumped up. They didn't really need any more stimulants. But, if it made it easier for April to make her peace with how her friends were behaving, then fair enough.\n\nApril caught the nagging doubt in his eyes, and nearly asked him about it. Then she caught herself up before she went too far. She did not want to seem paranoid and crazy.\n\n'The thing is,' she said eventually, 'we need to find out a bit more about the people who run this place. So tell me everything you know.'\n\n## FORTY-SEVEN\n\n## THE LATEST ADVANCES IN VIRTUAL REALITY WILL HORRIFY YOU\n\nRam was running through the Combat Chamber, loving the more advanced feel of the environment. Now that he was at Level Five, everything felt so very real\u2014he could barely even notice the walls anymore, the surroundings had lost that milky quality and every object was hyper-real. There was even the feeling of branches whipping by him\u2014which was something totally new. Ram realised it had been days since he'd seen a tree. There were no plants in the Void. Nothing other than the white boxes, the white overalls, the white helmets. It was like the only colour, the only life he saw was running through here. God, it felt so invigorating.\n\nHe almost couldn't remember a time when he hadn't been hunting here. His old life\u2014the life that he really should have been wanting to get back to\u2014seemed less real than the planet around him. He blasted down another of the Skandis, and laughed as it fell back, flailing and screaming. They really were the worst things in the universe. He didn't feel bad about killing them\u2014not even a small qualm, no matter what he told April. It was like feeling bad about killing rats or wasps. They didn't add anything to the world. They were just disgusting and brutal and the thought of them reaching the Earth made him sick with fear. The Shadow Kin had been bad enough, and they'd just torn through the school. He tried to imagine the streets filling with Skandis. The image was absurd for a moment, then completely terrifying, as they'd just start killing and keep killing.\n\nHow would they arrive? he wondered. They didn't look like they'd have spaceships. They didn't show signs of any technology. They just attacked viciously. He couldn't really imagine things like this sitting down to invent the wheel.\n\nHe blasted his way through another two, and then a fourth Skandis roared up, lashing out at him in outrage. Which was fine, as his force field would protect him.\n\nOnly it didn't. The tentacle knocked him into the air and then came crashing down onto the ground with an impact that winded him. The Skandis launched itself at him, and he only avoided it by twisting to one side. His ribs ached and his lungs refused to draw breath. He crouched, his head full of fight-or-flight instinct and unable to reach any higher thoughts.\n\nThe Skandis bounded after him, sending Ram scrabbling up a bank of skittering stones. His boots refused to find a purchase and he was spinning in midair. He threw himself further up the bank, grasping at rocks, pulling himself out of reach. The Skandis sprang towards him, and Ram let go, slipping down the rocks.\n\nAnd then he fell through them.\n\nThat was odd. He flailed as he sank through the stones\u2014and then they vanished, and he was falling through air until he landed on the floor. The air was knocked out of him again. Stunned, he lay there for a moment.\n\nThe one Skandis stood still, its wet skin pulsing. It wasn't preparing to spring. It seemed to be scenting the air, taking in its surroundings.\n\nFor a moment, the air fizzed, and the creature looked like Neil. Neil with his scalded face, burnt red and bubbling.\n\nThen the air fizzed again, and the Skandis was back. Ram felt a jolt, as though a plummeting lift had just snapped to a halt.\n\nWhat the hell?\n\nHe forced himself to his feet, only for his new leg to choose this moment, this damn moment, to just not. It stretched out in front of him while his other leg, his proper leg, kept trying to straighten up all by itself.\n\nI look like a sodding Russian dancer, he thought, grabbing his gun and using that to get upright, and to try to hobble-drag his way away from the Skandis, which was observing him, almost curious.\n\nWhat am I missing? Ram thought.\n\nThen he noticed the Combat Chamber had changed. It was no longer a stunning re-creation of an alien world. It had reset to a plain white box. But something was different, something was wrong.\n\nIt fell into place. In the training levels the Skandis had been indistinct, with overcompressed edges. Now it was clear. Completely clear.\n\nRam's brain worked through three things\u2014the force field failing, the rock wall vanishing, and now the chamber turning off. 'You're real, aren't you?' he shouted at it. 'Somehow you've reversed whatever this chamber does and you've come here.'\n\nThe Skandis' only reply was a nasty howl. Then it sprang towards him.\n\nCharlie was cleaning another Combat Chamber. Automated processes took care of most of it, but what was left behind were some pretty stubborn stains.\n\nHe'd noticed that most of what was left behind was human.\n\nHe wondered how the Skandis were doing. Were they winning this war?\n\nWhich was when one appeared in the Combat Chamber behind him.\n\nThe same thing was happening in Combat Chambers across the Void. Virtual environments were gradually crashing, resetting to their base states, but with the aliens still in them. And more powerful, more vicious than ever. Safe words weren't working, the emergency shutdowns weren't activating. Worse, the doors to the chambers weren't opening.\n\nTrapped soldiers started pounding on the doors to the Combat Chambers, until the entire bay echoed.\n\nThen the screaming began.\n\nMiss Quill took the change in circumstances in her stride.\n\n'A dozen of you. Nowhere to hide. And I'm silhouetted beautifully against the walls. Lovely.'\n\nShe was firing as she spoke.\n\nRam was using his gun to hold himself up. If he tried to fire with it, he'd fall over.\n\nHe tried to work out a sensible solution, realised there wasn't one, and, in a moment of blind panic, whipped the gun up, and fired repeatedly as he fell back over.\n\nThe Skandis carried on leaping towards him, its body tearing apart under the blasts from his gun.\n\nIt landed on him, and promptly fell apart into three charred lumps.\n\nIn the last moment before it shattered, something very strange happened to it. Ram, eyes screwed shut, screaming, almost didn't notice it. Then when he did, he didn't believe what he'd seen. He was just stunned to find himself still alive.\n\nRam lay there for a while, disgusted, then slowly, with endless patience, crawled out from under the smouldering corpse.\n\nA simple malfunction, he told himself. Nothing to be worried about. These things happen.\n\nDamn, his leg still wasn't working.\n\nHe eyed up the door. Not great, but he reckoned he could reach it easily enough. Raise help.\n\nHe started crawling towards the door. No harm done. Soon be back up and fighting. Not a great day, not the worst.\n\nThen the air in front of him fizzed and glowed and six more Skandis appeared.\n\n'You have got to be kidding,' breathed Ram. \n\n## FORTY-EIGHT\n\n## DO YOU KNOW ENOUGH ABOUT DIMENSIONAL COMPENSATORS TO SAVE THIS BOY'S LIFE? (SPOILER: YOU DON'T)\n\n'What the hell is going on?' Seraphin yelled as the alarms went off.\n\nApril hadn't got a clue and was trying to tell him this, but the noise just wouldn't stop.\n\nSeraphin was clicking in frustration at his computer. It flashed up the spinny wheel of give-me-a-minute and did nothing else.\n\nHe pointed over to what looked like a fuse box by the bathroom. April ran to it. It wouldn't open.\n\n'Screwdriver?'\n\nSeraphin dashed back to his desk, grabbed a penny, and used it to turn the latch. The cupboard sprang open.\n\nInside was a lot of complicated alien technology. And a large trip switch. She yanked it.\n\nThe alarms stopped. The room went dark.\n\n'Don't let go of the switch!' Seraphin was yelling.\n\n'Already have,' April yelled back, a little deafened.\n\n'What?'\n\n'Never mind.' April fumbled and patted her way through the cupboard, worried that at any moment she'd turn into a human candle. She found a switch, and, holding her breath, pulled it.\n\nThe lights came back on.\n\n'Was that the right thing?' she asked.\n\n'Yep,' said Seraphin.\n\n'What is all this?' She pointed to the equipment in the rest of the cupboard\u2014a series of glowing cables and pulsing arrays.\n\nHe shrugged. 'Stuff I'm not supposed to touch.'\n\n'Ever touched it?'\n\n'Nope.' He shook his head. 'I once totalled a laptop after I forgot to wash peanut butter off my hands when I upgraded the RAM. That stuff is way out of my league. You any idea?'\n\nApril pretended to study it very hard for ten seconds.\n\n'No,' she announced, trying to convey in her voice that, if it had just been last year's model, then she'd have been all over it. 'What I can say is that the lights were all green, and now they're all red.'\n\n'Gotcha.' Seraphin ran a hand through his hair. 'That's getting a bit technical for me.'\n\nHe darted over to his main computer, which had now finished starting up again. He logged in, waited, opened a browser, and then waited some more.\n\n'Here,' he said, pointing to a display map. 'Looks like something's going on with the Combat Chambers.' He zoomed into the map. 'Got it. They're all full and the doors are off-line.'\n\n'What?' said April. 'Do something!'\n\n'Pfft.' Seraphin jabbed at the screen. 'All I can do is see the status report. Helps me with my bulletins. I can't open the doors. Seriously. Why would they give me that power?'\n\n'Good point.' April wasn't really listening to him anymore. 'I've got to find someone who can help us get control. And she's not going to like it.'\n\nRam made it to the door and pressed it. It should have triggered the opening cycle. Nothing happened. He peered through the small window, hammered on it, shouted for help that wasn't coming. Then he groaned and felt around for the manual trigger.\n\nManual trigger, he thought. What was wrong with having a door handle? This whole place was so screwed up. He located the trigger.\n\nHe'd been told that the reason the doors took so long to open was in order to balance the dimensions in the Void. It was important that you didn't bring anything back from the battlefront with you. A very bit like divers having to decompress. There'd been a lecture on the possibility of bacteria crossing the dimensional barriers. It took ten seconds to fully harmonise the dimensions. But the manual trigger could do it, just about, in five. If he could find it. If it was still alive five seconds after he'd found it.\n\nTrying to find a hidden lever while keeping an eye on six creatures intent on killing you is difficult. You know the expression about 'doing it with one arm tied behind your back'? Try doing it with only one leg. Ram had settled into an awkward crouch, his useless leg splayed out in front of him and constantly trying to tell him that it really wasn't happy. Every now and then he'd loose off a shot at the Skandis, but his gun was now little better than a water pistol.\n\nAt each blast, they'd pause, shudder, then press on. Judging from their roaring and twitching and the way their tentacles snapped, they were angry. Angry and hungry.\n\nOnly April could feel sorry for these things, he thought. He fired off a couple more shots, bought himself some time, then got back to running his hand up the door seal.\n\nIn training it had been easy. But then, that was always the way with safety demonstrations. Flight attendants made it look like escaping a crashing plane was a calm saunter from fitting your oxygen mask to collecting your life jacket and nimbly stepping out into the water. But come on, it was going to be screaming chaos, wasn't it? When they'd gone on that school trip to Italy, just while boarding the plane there had been a fight, an actual fight, between vicious old ladies and a football team. Imagine the horror of trying to get off a plane, a plane that had just crashed. And apparently it all came down to the simple act of releasing the emergency exit doors and stepping out. Like that was going to work.\n\nRam found the trigger. It was, of course, just out of reach. He levered himself up as far as he could, and, fingers clammy with sweat, gripped the trigger and pulled it. So long as he kept hauling on it, it should release the door. He supported himself on his one good leg and waited.\n\n_5._\n\nThe Skandis were whipping towards him.\n\n_4._\n\nHis leg was hurting. His whole body was shaking.\n\n_3._\n\nTheir tentacles snapped at him, their horrific jaws hissing and howling.\n\n_2._\n\nOh, his leg hurt so much. Five seconds? Surely? Come on.\n\n_1._\n\nA tentacle whipped past his face, spattering him with stinging slime. Come on!\n\n_0._\n\nThe door sprang open and Ram realised his mistake.\n\nCharlie was running. There was nothing else he could do. He was in a Combat Chamber with a very hostile alien and all he had was a bucket. He'd already thrown the water out of the bucket.\n\nA tentacle sent him flying.\n\nCharlie lay on the ground, fighting for his breath as the Skandis came towards him.\n\n'I'm an alien prince,' he told the creature. 'I'm not boasting, I'm not expecting any special treatment, I'm just saying this isn't how life was supposed to be.'\n\nA tentacle threw him across the floor.\n\nCharlie picked himself up, eyes wandering for a moment to the white wall. The whiteness was flickering, a little gentle corner-of-the-eye pulsing.\n\n'As an alien prince . . .' He laughed, falling breathlessly onto his back 'I should, right now, be leading an army.'\n\nThe Skandis stood over him, surveying him, the tentacles quivering with hostile interest.\n\n'I don't have an army.' Charlie was still laughing. 'But I don't suppose you'd pass me my bucket, would you?'\n\nThe Skandis sprang at him, but Charlie had already rolled out of his way, and was up and running.\n\n'I've been learning a few things about being human,' he shouted as he ran. 'And one of them is that it's a really bad idea to kill me. You see'\u2014he paused for breath, a defiant smile on his face\u2014'my boyfriend is Polish.'\n\nCharlie bolted for the door.\n\n'Whuh?'\n\nTanya was exhausted and really, really didn't know why April was standing in her doorway.\n\n'I was sleeping.'\n\n'Tanya, I've come to ask you a favour,' said April. 'Something bad's happening and I really could do with your help.'\n\nTanya blinked and yawned helplessly. 'I'm so tired,' she whined. 'Kill-me tired. There is literally nothing you could do to make me take part in whatever it is you're up to.' She narrowed her eyes suspiciously. 'Is it a peace rally? Or a folk festival? That's it, isn't it?'\n\nApril smiled. 'And that's why I brought along my secret weapon.'\n\nSeraphin stepped into the room. 'Hi, Tanya. Lovely to meet you.'\n\nTanya immediately grabbed her bedsheet. 'Oh, dear God.'\n\nQuill stared at the door. The manual trigger was jammed. She hammered at it with her gun. And then, having worked through a few other options, she shrugged and shot the door.\n\nRam lay on the floor of the bay, listening to the frantic hammering from the other Combat Chambers. He could see a face pressed up against a window, pleading for help.\n\nThen the face was whisked away and something green smashed across the window. There were three thumps and then nothing.\n\nBloody hell.\n\nAnd he'd left his door open.\n\nRam grabbed hold of a wall and levered himself up. He had to close the door behind him. Had to. Before they got out. That had been their plan all along. He leaned against the wall, steadied himself against a door, and got ready to launch himself at his open door. He just needed a bit more strength, a tiny bit more strength . . .\n\nThe door he was balanced against blew open, knocking him to the floor.\n\nMiss Quill stepped over him.\n\n'Nap time, Singh?' she sneered.\n\nStunned by the blast, and just generally stunned, Ram looked up at her. 'Close the doors!' he yelled. 'We've got to close the doors!'\n\nMiss Quill blinked, bemused.\n\n'Close the doors!' Ram screamed again.\n\nMiss Quill shook her head. 'The blast,' she explained, shouting a bit. 'Bit deafened.'\n\nRam gestured at the doors, throwing himself forward to try to grab one.\n\nMiss Quill stared at him quizzically. Ram howled in frustration.\n\nThen the first of the Skandis slithered out.\n\n'Oh,' said Miss Quill. Finally she understood. 'We've been invaded. Interesting.'\n\n## FORTY-NINE\n\n## HE CHOSE THE WRONG DAY TO BEG FOR HIS LIFE\n\n'I need you to look at this,' said April.\n\nTanya did not appear to be listening.\n\n'This is your flat?' she said to Seraphin, amazed.\n\n'Yeah,' he said. 'Well, it's a set, but you get the idea. Would you like me to switch my dog on for you?'\n\nHe leaned down, pushed a button, and Captain Pugsley materialised, and began pottering around the laminate flooring.\n\n'Wow,' said Tanya. She'd been saying wow a lot suddenly, and April was getting a bit disturbed.\n\n'Are you okay?'\n\n'Yeah,' Tanya muttered without seeming to hear.\n\nSeraphin winked at April, and went to stand a little bit too close to Tanya. 'Hi,' he said, and his smile went up to eleven. 'Apparently you're good with computers.'\n\n'Um,' said Tanya.\n\nSeraphin tapped his desktop into life. 'Just . . . this one . . . here . . . I need someone to have a look at it.' He ran a hand slowly through his hair and shook it out. God, thought April, that is so obvious. But it worked.\n\n'Sure.' Tanya walked over and sat down at the computer. She started pecking at it. 'What do you want?'\n\n'People are trapped in the Combat Chambers and we need you to get them out,' Seraphin said.\n\nTanya squinted at the screen. 'And you want me to try to use their intranet as a way into the main systems?' She shrugged and hit view source on the page. 'Oh, easy. I have no idea who coded this page but they clearly never expected anyone with any knowledge to ever see it. No offence.' There followed a furious bout of typing. 'There. See?'\n\nA small Java box popped up with a prompt.\n\n'Well, that's hopeful,' Tanya remarked. 'Hardly worth you turning on the full charm was it?'\n\n'I wasn't,' protested Seraphin. 'Back rub?'\n\nBut Tanya was already embedded in the system. 'It's like this was coded by aliens,' she muttered, then shrugged. 'Probably was.'\n\n'Um,' said Seraphin, 'I was asking\u2014'\n\n'About a _back rub_ ,' muttered Tanya. 'I have no idea if that works for you, but that's just weird. Now, please just stand about half a metre in front of the screen and look vaguely hot, and that'll be fine. Okay?'\n\n'Erm, okay.'\n\nTanya turned to April and smiled at her. April smiled back. And a holographic pug nuzzled around their legs.\n\nThe Skandis poured into the Combat Bay. At first it was just the ones from Quill's and Ram's chambers. Then they started to open the other doors.\n\nA Combat Chamber flew open.\n\nA relieved soldier staggered out, only to realise what was waiting for him on the other side. A tentacle snatched him up, and threw him screaming against the far wall. The Skandis moved onto the next door.\n\nRam stood frozen. About an hour ago, this had all been easy\u2014running around in a shooting gallery blasting aliens. The weirdness of the last few days had finally made some sort of sense. And now that had all gone so wrong. The surging force of Skandis\u2014a dozen of them?\u2014looked absurdly too big for the Combat Chamber, the air already stifling with their disgusting smell. Life had got all too much. Again.\n\n'They're invading! Do something!' Ram shouted at Miss Quill. If you're out of your depth, find a responsible alien.\n\nBut Miss Quill, either still deaf or not caring, was backing out of the Bay, firing.\n\n'We've got to stop them!' Ram shouted louder.\n\n'Already shooting them,' Quill snapped back. 'What more do you want?'\n\nThe Skandis crawled across the Bay, advancing as Quill retreated. Ram, edging back against the wall, kept his balance as best as he could.\n\nAgainst the chaos, the screams, and the blasts, he could hear a voice.\n\nWhat? He twisted slightly to see what the fuss was. Pressed up against the other side of a Combat Chamber door was Charlie. Desperate.\n\n'Interesting.' Tanya had stopped staring at the screen and was instead looking at the holographic pug.\n\nSeraphin knelt down and nuzzled its ears. 'Is my dog annoying you?' he asked.\n\n'No.' Tanya's voice was subdued. 'It's just . . . It's awesome.'\n\n'It is, kind of,' said Seraphin, watching as the pug clip-clopped over the floor. 'Some kind of amazing alien thing.'\n\n'Yeah,' Tanya said. 'Sorry, distracted for a bit.' She tapped the screen. 'The systems I can get into are the helmet cameras.' She opened up a browser tab. 'We can see everything. And there's a whole load of settings here . . .' She pointed at a series of buttons at the bottom. 'They're greyed out, so you can't alter them, but it may be a back way in. But to what, I don't know.'\n\nSeraphin leaned over her, a little too close, a little too smiling. 'Just do what you can,' he said.\n\n'Sure,' Tanya said, and looked at the fake pug snuffling curiously around April's legs.\n\nRam turned to Quill, waving at her across the Bay.\n\n'It's Charlie! He's trapped in a Combat Chamber!' Oblivious, Quill carried on backing away, firing.\n\nRam turned back to Charlie. 'We're getting you help!' He waved his arms at Quill again.\n\n'Miss Quill! Come here! Help me open this door!' Miss Quill didn't seem to notice.\n\nCharlie hammered on the door, shouting through it, urgent, strong. 'Quill! My life is in danger! I order you to help me!'\n\nQuill, a world away, calmly carried on firing at the creatures.\n\nRam, desperate, flung himself into the path of her gun. Wonderfully, at the last moment, his new leg steadied him, more or less, and he stood, wobbling, eyes shut as Quill's bullets blasted past him.\n\nShe stopped firing. Had she stopped because he was in her way, or because the genome-lock on the gun had kicked in? Deciding he'd rather not know, he took a deep breath and opened his eyes.\n\n'Hi.' He smiled.\n\n'Mr Singh, what are you doing?' she asked. 'You're getting in the way of my fun.'\n\n'It's Charlie!' shouted Ram, indicating the Combat Chamber door with a frantic pantomime.\n\nMiss Quill followed his gesture. She rolled her eyes.\n\n'Riiiight,' she said, with the enthusiasm of someone who has to scrape something off her shoe.\n\nIgnoring the Skandis, she strode across the Bay, and activated the release trigger for Charlie's door.\n\nShe counted off five seconds on her fingers then threw open the door, yanked Charlie out, fired three times into the creature rushing out behind him, and slammed the door shut.\n\n'I'm so sorry, I didn't hear you,' she said flatly, then strode away, raising her gun and shooting into the advancing horde.\n\nApril leaned over Tanya's shoulder, peering at the screen.\n\n'Amazing,' said Tanya. She didn't stop typing. 'Honestly, endless amounts of processing power dealing with the helmet video feeds, nothing, nothing at all about opening doors or letting me help anyone. I can see the systems, I just can't reach them. It's like I'm stuck in a fish tank in a vast library trying to work out how to reach the books.'\n\n'Yes,' said April. 'Look, I was wondering. Wondering about the Skandis and the helmet cameras . . .'\n\n'And the pug?' asked Tanya.\n\n'Oh, you're there too, aren't you?'\n\n'Oh yes,' Tanya nodded. 'I do listen to what you say. Everyone's like all \"We just have to fight them because they look so awful . . .\" and you're all flower child and \"I think we should hug them\" and we're all \"Oh, that's disgusting\". But what if you're right . . .'\n\nApril tapped the screen, at all the footage from the helmet cameras. 'What if I am?' she mused.\n\n'Well,' said Tanya, 'once I can get out of my fish tank, I'll see if we can do something clever for you.'\n\nShe got on with typing.\n\nCharlie stood, slumped and winded against the wall, taking in the situation. Monsters. Screaming. Quill with a gun and a smile. Then, gathering himself together, he went over to Ram.\n\n'Sorry about that. She's deafened by the blasts,' Ram said, wondering why he was making feeble excuses for her.\n\n'Sure,' said Charlie. 'You okay?'\n\n'Yeah.'\n\n'Good. Thank you.' Charlie nodded, and suddenly Ram saw the alien prince in him. He wasn't just reacting to the situation around him. He was considering it. 'So. We're being invaded?'\n\n'Yeah.'\n\n'Right.' Charlie agreed, looking vaguely affronted.\n\n'Fancy ordering them to stop?' suggested Ram.\n\nCharlie ignored the jibe. 'Quill said she couldn't hear me?'\n\n'Yeah,' said Ram.\n\nCharlie nodded again, a sad little nod that made it quite clear he didn't believe a word of it. Then he looked away, surveying the chaos in the bay with the aloofness of a general.\n\nSoldiers were fleeing from the Combat Chambers. There were now about two dozen creatures. The only person trying to hold them back was Quill.\n\n'Hum,' said Charlie. Then he indicated the fleeing soldiers. 'Quill can't hold the invaders back on her own. Can you get them to help?'\n\n'Why me?' Ram hated it when Charlie treated him like this.\n\n'Because you're a good soldier and they respect you.' Charlie didn't seem fazed. 'It will save time.' He looked around again. 'I'm going to try to get into the systems. Maybe there's a way to block the area off.' He turned to go. Then he stopped. He looked at Ram.\n\n'Oh, I'm sorry,' he said, 'I'd forgotten about your leg.' Ram blushed, feeling humiliated. But also, Charlie was right. His artificial leg had frozen, rooting him to the spot.\n\n'Here,' Charlie said, reached out his arm, and, balancing Ram, led him over to the other soldiers.\n\n'You lot,' Ram shouted at the soldiers, 'don't move\u2014listen to me . . .'\n\nTanya hadn't stopped typing for several minutes.\n\n'How we doing?' asked April.\n\n'Well,' Tanya groaned. 'The fact that I'm still bashing this keyboard and not doing cartwheels should tell you that it's not good. This system was designed to display information, not to give me control. It's maddening\u2014I can select footage from the helmet cameras, I just can't open or close doors. I can't do anything.'\n\nSeraphin appeared next to her, and handed her a carton of juice. 'You might not be able to do anything, but I can,' he said, and grabbed a microphone.\n\nAn alarm sounded across the Void, echoing through its corridors. The soldiers all scrambled to the Big White Room, where footage was projected over the walls, showing the carnage taking place. Seraphin's voice boomed over it.\n\n'So, this is happening,' he said. 'We're being invaded and we need to stop it. The best place to defend is the Big White Room. Get guns, get there, and get ready to fight. We're trying to seal off the corridors, but I doubt it's going to work.' He stopped then started again. 'I mean, we're hoping that we'll get it to work. But we've got to do everything we can to stop them.'\n\nAt the end of the Combat Bay, protected by the rest of the invaders, two Skandis gathered around a service hatch, and, prising it open with their mandibles, began to work among the machinery with surprising skill and dexterity. An interface screen popped up, and a tentacle swiped across it, leaving a sticky trail.\n\n'Oh,' said Tanya, 'someone else is in the system.'\n\n'Is that good?' asked April.\n\nTanya squinted at the screen. 'No. Probably not.'\n\nThe soldiers had retreated to the Big White Room. There'd been nothing else they could do. The screens around them were still showing the chaos.\n\nA fleeting glimpse of a frightened soldier, sat on a bench, despondent, no longer a soldier, not much more than a child.\n\nAnother soldier backing away as the jaws of a Skandis wrapped around her head.\n\nA group of soldiers looking at one another, their faces blank, empty, exhausted.\n\nQuill firing and firing and firing and laughing. ABNORMAL. ABNORMAL. ABNORMAL.\n\nThey'd formed a barricade of sorts, using the tables and benches and a lot of hasty shoving. The Skandis, dozens and dozens of them, were still coming, but the barricade had seemed the right thing to do.\n\n'Keeping out the monsters, keeping out the night,' Charlie said, grabbing the other end of a table from Ram and shoving it.\n\nThey stood, surveying their work.\n\n'They're coming,' someone said, and someone else stifled a sob.\n\nWith a roar, the barricade shattered, falling like a Jenga tower as the Skandis surged through.\n\nRam and Charlie were pushed aside. There were screams and shouts, then a soldier ran forward to confront the Skandis as they poured into the Big White Room.\n\nFor a moment the girl stood there. She was scared, everyone could see that. But she was also determined. She held her gun up but she did not fire. Her posture, everything about her, said that she was going to defend this room to the death.\n\nA boy crawled out from under a table, and went to stand beside her. He raised his gun as well.\n\nThree more soldiers formed a line with them.\n\nAll of them stood, pointing their guns at the rows of Skandis.\n\nThe invaders stood silently, watching them, as if curious. The girl nodded to the rest of the line, and they all levelled their guns and pulled the triggers.\n\nNothing happened.\n\nThey pressed them again. Nothing.\n\nThey turned to one another.\n\nThen they turned back to face the Skandis. The soldiers' faces were open in a look of comic alarm, almost embarrassment. 'Well, this is awkward . . .'\n\n'Shoot them!' screamed Quill. 'What's wrong with you? Shoot them!'\n\nBut no one paid her any attention. They were staring at the five soldiers, standing there, against an alien army, just five lonely, defenceless children, lowering their useless guns in utter, miserable defeat.\n\nThe Big White Room fell silent, the walls filling with pictures\u2014everyone was staring at the five students, alone in the middle of the room.\n\nRam leaned forward to help, to provide covering fire from his gun, but it didn't work either.\n\n'What the hell?' he asked.\n\nCharlie shook his head. 'They've interfered with the genome-locking. The guns are useless now.'\n\nCharlie was wrong.\n\nThe Skandis pushed the five soldiers back against a wall. One of them picked up a gun, examined it, and aimed it.\n\n'No,' shouted Ram. 'No!'\n\nBut they fired anyway.\n\n## FIFTY\n\n## THE SKANDIS WAR AS YOU'VE NEVER SEEN IT BEFORE\n\n'How's it going?' asked April. 'Surely we're there!'\n\nTanya mumbled something.\n\nApril leaned over to see what she was doing. She was immersed in a series of protocols. 'What is that?'\n\n'Something new,' Tanya grunted. 'The Skandis have disabled the weapons. I've got to reactivate them. Otherwise it's going to be a massacre. It's all very well trying to get into the helmet cam systems, but I can't see how it's going to save lives.'\n\n'If I'm right,' said April, 'then we won't need guns. Please. Sort it.'\n\nTanya shook her head. 'People are dying right now. I've got to stop it.'\n\nApril put her hand on Tanya's shoulder, but she brushed it off.\n\nSeraphin appeared at April's side. 'I'm not her,' he said quietly, 'but there's another terminal by my bed. Perhaps we can do something there.'\n\nThe Skandis surged forward, lashing out with their tentacles. Soldiers were crouched behind whatever cover they could find, aiming their guns fruitlessly. They may as well have been yelling 'Pow! Pow! Pow!' Children playing with toys. Damn, thought Ram, that's what we've been all along. Bang bang, you're dead.\n\nTanya appeared on the screen, hovering over the slaughter. 'I'm trying to get the guns back online!' she yelled.\n\n'Try harder,' Miss Quill roared back.\n\n'Not encouraging, Miss,' Tanya said.\n\nBenches and tables flew past, swept aside as the Skandis surged into the hall.\n\n'Wow, that escalated quickly.' Seraphin was watching as the Skandis took control of the Big White Room. He and April sat on the edge of his bed, trying to get sense out of his tablet.\n\n'Are you getting anywhere?' she asked.\n\nHe smiled at her, and his sweet smile was very sad.\n\n'No, not a clue.' He tossed the tablet to one side, and Captain Pugsley made a leap for it and vanished through the bed.\n\n'What were you looking for anyway?' Seraphin asked, opening a drawer, dusting off an old laptop, and plugging it in.\n\nApril stared at the weird half-a-dog embedded in the bed.\n\n'Not certain. Not really my area.' April looked sheepish.\n\n'Not really a hacker.'\n\n'Neither am I!' called Tanya. 'What you trying to get into?'\n\n'The helmet cam systems.'\n\n'It's pointless. You just can't hack into them,' explained Tanya. 'Believe me I've tried. Gah.' She waved at the desktop. 'Seeing the feeds is easy, getting beyond that is a nightmare. The operating system I'm trying to talk to is weird. Like an onion skin. You peel off layers of icons to get to what you want. See?'\n\nOn screen, she was pulling away sheet after sheet of icons, all of them labelled in an alien language. 'The deeper I go, the harder it gets to understand.'\n\nSeraphin had finally booted up his old laptop, and had got into the helmet camera feeds. 'We can do just as much from here,' he said. 'The problem is, it's not got as fast a processor, it'll be sluggish.'\n\nApril had one of those beautiful quiet moments where the entire world and her urge to scream at it just faded gently away.\n\nHelmet cameras. A slow laptop.\n\nA holographic pug. Empty cartons of juice. Got it.\n\n'You say you still get sent freebies?' she asked Seraphin.\n\n'Yeah,' he said. 'Don't know how the post works, but\u2014'\n\n'Not important,' April said. 'Anything electrical?'\n\nThe Skandis swept towards the humans. People had stopped trying to make a run for it. Any attempts were met with lethal stuff. Bodies crashed and snapped against the walls.\n\n'Stand still!' ordered Quill. 'Don't be idiots. Don't run.'\n\nThe great walls lit up, fizzing.\n\n'Tanya!' shouted Ram. 'Is that you doing this?'\n\n'No!' replied her voice. She sounded lost, desperate. Washing into focus on the screen were hundreds of images of the terrified humans, lined up against a wall.\n\nCharlie realised they were now being shown what the Skandis saw. They would be able to watch themselves being devoured.\n\nApril was tearing apart Seraphin's cupboard.\n\n'Anything electrical?' she'd said. 'Computer stuff.'\n\nHe'd started throwing boxes at her. 'It's pretty much all useless,' he said. 'The Wi-Fi link back to Earth from here is so slow . . .'\n\n'Good,' said April.\n\n'And they tell me they'd rather not have stuff installed on their systems.'\n\n'Excellent.' She laughed, pulling out armfuls of things\u2014USB mini-fans, motion-sensitive Christmas tree lights, small drones, wireless dongles, thumbsticks, and hard drives.\n\nIn her head, her suddenly clear, pin-sharp head, she was thinking: What if it's not just me who finds computers annoying? And she was smiling. She remembered how much trouble she'd had when she'd first plugged her piano keyboard into her computer.\n\nTanya looked up from her keyboard to see April marching through the room with a bundle of kit. 'What the hell are you doing?' she asked.\n\n'Being very dumb indeed,' April told her. She threw the bundle onto the bed, and Seraphin began plugging them into the USB hubs.\n\nThe humans were stood in a line, waiting to die, watching themselves on the screens of the Big White Room. Some were still holding their useless guns. Others had let them drop to the floor.\n\n'This is sick,' said Ram.\n\n'It's an impressive act,' purred Miss Quill.\n\nThe Skandis had stopped. They were waiting. Enjoying the moment.\n\n'Nearly there!' called Tanya's voice over the speakers. 'I've nearly got the guns back on line.'\n\n'We. Are. Out. Of. Time,' grated Quill.\n\nShe raised her gun. All the other soldiers raised theirs.\n\n'I would very much like to go out fighting,' Quill said. 'If that's not too much to ask.'\n\nSeraphin scratched his head and looked helplessly at April. 'What happens now? I've plugged everything in.' He pointed at the forest of cables snaking out from this laptop across the duvet. The head of Captain Pugsley appeared through the duvet and nipped ineffectually at them.\n\n'We wait,' said April. 'We wait for something ordinary and human to happen.'\n\n'I'm in!' shouted Tanya. 'I'm into the weapons. They're rebooting now! Get ready to fire!'\n\nA window on Seraphin's laptop popped up. 'New technology detected. Finding driver.'\n\nAnd then another one appeared over it. And another.\n\nQuill and the others tightened their fingers on the triggers. They were ready to fight.\n\nAnd then the Skandis did something remarkable.\n\n## FIFTY-ONE\n\n## THEY THOUGHT THEY'D WON UNTIL THEY FOUND OUT THEY'D LOST\n\n'Stop!' yelled April. 'Stop! Something's happened! Can't you see what's happened?'\n\nBut no one was firing. Everyone was staring.\n\n'What the hell?' gasped Ram.\n\n'Oh God.' Tanya stared at her screen and gaped.\n\nThe Skandis had changed. They were no longer nightmares of tentacles, slime, and spikes. They were instead so beautiful that to look at them stole a breath. They were childlike, fluttering creatures, an agreeable merger between angels and butterflies\u2014they glowed with beauty, from their silver hair to the furthest tips of their silk wings. They hovered just off the ground, and that seemed right, as though to actually touch the floor would have somehow lessened them.\n\n'Their faces,' gasped Tanya. Their faces were remarkable. They had about them an aura of universal beauty. Their eyes were wide and golden, they had tiny furred noses, and their mouths were delicate wispy things, their skin stretched with fine markings that suggested these were mouths made for smiling.\n\nThey were not smiling now.\n\nApril's plan had been simple. Seraphin's laptop had been crammed full of USB technology, all of it searching desperately for a new driver. The sudden inrush of requests for software, the old processor, the sluggish internet speed, all allied to do what Tanya's skills couldn't\u2014they hit the helmet camera systems and brought them down.\n\nSuddenly the Skandis were revealed.\n\n'That,' said April, 'is what they really look like.'\n\n'But\u2014' Seraphin said, and stopped. That was pretty much all anyone said. The same expression echoed down in the Big White Room\u2014one of horror and disbelief. Someone was crying.\n\n'What have we done?' Ram said.\n\nThe Skandis showed them.\n\nThe screens of the Big White Room lit up with footage. A thousand deaths, each one seen from the victim's point of view. Tiny hands and paws flailing up, pleading and squeaking, falling backwards, their last view the sneering faces of their killers\u2014human soldiers, young faces twisted with rage and hate.\n\nOnce those thousand deaths had finished, they saw a thousand more, and a thousand more after that. The deaths were broadcast on every wall, the floor, the ceiling. There was no escape from them. The blasts, the screams, the laughter.\n\n'Stop it!' screamed Tanya. 'Stop it!'\n\nThe picture changed. For a moment it was mercifully blank.\n\nThe Skandis stood, surveying the cowed soldiers placidly.\n\nThen Seraphin's voice came over the speakers. It was missing its normal exuberance. 'We've got . . . we've got the other side of the picture. You have to see it.'\n\nHe and April played out footage from the soldiers' helmet cameras. Suddenly they saw all their battles as they really had been. Where before they'd been marching through desolate wastelands, they were now marching through fields, villages, schools.\n\nRows of the fluttering aliens were mown down as they tended crops. Horrified parents watched as their children exploded in flame and then fought back desperately, angrily. Villagers rushed forward to try to protect their elders while urging others to flee.\n\nThe most horrible thing was how beautiful the aliens looked even as they died, flames racing along their wings, agony exquisitely expressed on their faces.\n\nThe humans watched the screens. They couldn't turn away from what they saw. The slaughter. The terrible laughter. The jokes. A rifle smashing down and crunching over and over again as a boy's voice chuckled. 'One down, three to go.' Then he turned to three Skandis children clinging to a corner.\n\nIt took a long time to play all the footage. It stopped. Eventually.\n\nSeraphin appeared, leaning into the screens.\n\nHe looked about to speak, and then shook his head, turned away, and the screens of the Big White Room went blank.\n\nThere was silence. Shock. Wonder.\n\nThen came a thud. Then another.\n\nThe sound of helmets being taken off. Dropped to the ground. People blinking at the Skandis. Seeing them as they truly were.\n\nNORMAL\/ABNORMAL didn't matter anymore.\n\nSoldiers looked at the Skandis, and then they looked at one another for the first time. And realised how young everyone was.\n\nThen the doubt started. What if this was an illusion too? What if what they'd originally seen was true? And this was the illusion.\n\nA debate began to rage. Shouting.\n\nAll the while the Skandis stood there, fluttering gently. Waiting.\n\nApril walked into the Big White Room. She walked between the two sides. Everyone watched her. She walked patiently, with poise and clarity. In that moment, Charlie couldn't help but think how grown-up she was. Ram just blinked. How could she be so cool? After all, this was _April_.\n\nShe stopped before the fluttering crowd of aliens, and bowed her head.\n\n'I've never got to say this before, but'\u2014And she smiled, a beautiful, sad smile. 'On behalf of the human race, I am sorry.'\n\nThere was no simple resolution. No happy ending.\n\nIt would be nice to say the aliens bowed back, and then left in peace.\n\nThey did not.\n\nBut nor did they slaughter everyone. Instead they ransacked the base, going through every file, making copies, pulling information out onto the screens, analysing it.\n\nThey ignored the humans completely. Some people tried repeating April's apology, trying to make it personal, but it was as though they weren't speaking at all. Now that they'd made the humans aware, the Skandis had dismissed them. It just seemed easier for them to ignore their murderers. We are, April thought, beneath contempt.\n\nRam and Tanya sat on the remains of a bench, watching the Skandis flutter around, moving in beautiful, graceful loops as they dismantled the walls, stripping out vast chunks of technology. Tanya blinked and turned away, in case she cried again.\n\n'I can't believe it,' she kept saying.\n\nRam nodded. His head kept going round and round and round. How many had he killed? How many were adults? As if that made it any better. It did not make it any better. None of it made it any better.\n\n'I've lost count,' he said, empty. 'I keep trying to count. Oh God. I can't remember.'\n\n'I know how many I killed,' said Tanya, very sadly. She wouldn't say.\n\n'And, all that bants while I was doing it. Like I was a hero.'\n\n'Yeah,' she said. 'I was laughing. Sometimes I couldn't stop laughing because I thought I was doing good.'\n\n'We did,' he said, disgusted with himself. 'We really thought we were doing good. And we all dared one another on.'\n\n'It was . . .' Tanya's face twisted up with shame. 'So much fun.'\n\n'Yeah.' Ram turned away from her. 'And I finally found something I was good at.'\n\nThey sat in silence after that.\n\nThere was an angry huddle in a corner. One kid, from Coal Hill, was shaking a fist. 'Yeah, but,' he was saying, 'how do we know they're still not planning on invading the Earth? I mean. They still could. Couldn't they?'\n\nThe huddle nodded. Making it okay. Making it possible to live with themselves.\n\nIf the Skandis heard, they didn't pay any attention. They no longer cared.\n\nSeraphin appeared on a wall. Only that one screen was still working. He was wearing black. He'd shaved.\n\n'Hey, everyone,' he said, the confidence gone, his voice croaking. 'Here's what we know,' he said. 'I think it's important we all know.' He swallowed and stopped for a bit. 'I'm getting it together.'\n\nApril moved through the crowd and sat next to Ram.\n\n'I'm sorry,' he said to her.\n\n'Yeah,' she said. 'You're only apologising to me because you can't make it right with the Skandis.'\n\n'No,' he said, flushing with anger. 'You deserve an apology too. I didn't listen to you.'\n\n'Maybe,' April said, considering. 'Maybe I'd just make a terrible soldier.'\n\nShe took his hand, and squeezed it very gently.\n\nOn the screen, Seraphin stopped rifling through his notebook and then spoke again.\n\n'The Skandis helped me understand as much about this as they could. Looks like we will never know who organised this,' he began. 'Some of you have spoken about that feeling of being watched, out of the corner of your eye. That's all we have of the people behind this. A vague idea. They made sure they were never seen.\n\n'Can I make it clear\u2014the Skandis were never planning on invading the Earth. They're a peaceful, beautiful race. As you've seen, their home planet is pretty amazing. So amazing that other people wanted to live there. But these other people didn't want to be seen being involved in wiping the rightful owners out. They approached mercenaries, but no one wanted the job.\n\n'Instead, they came up with a plan. They found a way to Earth, and they found a way of recruiting people who could be blamed. The population on Skandis isn't huge\u2014wiping them out wouldn't take that much effort. But, if we'd succeeded, the shame would have been all on us. On the Earth.\n\n'Whoever is behind this wouldn't just have got the Skandis' world, they'd have got the Earth as well. If they'd wanted to invade us, no one would have stepped in to stop them. They're ruthless and they're clever. They've made this all our fault.\n\n'I cheered you on. I told you to do it.' He stopped and took a long drink of water. 'I made you do it. You made one another do it. But we'd been fooled. The helmet cams were filtering what we saw, making us see what our kidnappers wanted us to see. Massive perception filters. Nothing has been real. Every image you've seen has been altered to make you the hero of the movie of your own life. We thought we were Han Solo blowing up the Death Star. Really, we're stormtroopers.'\n\nHe leaned back in his chair. 'One more thing. You may be wondering how they paid for all this\u2014turns out there's still money in space. It costs to build this base, to place it in the Void, and to interface with the Skandis' home world. It cost a lot to filter the helmet cams and to buy the drugs they put in your food. Oh, yeah. They drugged you. Sorry.\n\n'Anyway, cash. Turns out, there was a site. Pay-per-view. All across the universe, people have been paying to watch what we've done. It's sick. But it's been really popular.\n\n'The sickest thing of all? Alright.' Seraphin stood up and paced the room, jamming his hands in his pockets and looking wretched. 'The people behind this. Our masters? They may not have won any planets. But they've still made a massive profit from the videos.\n\n'Yeah, we think the universe is full of evil UFOs with probes, or of great shining space gods. Turns out, it's full of really twisted people. And not only are they the Worst, but now the universe has seen all these videos, what do you think they think of us?' He shook his head and stared bleakly ahead. 'I dunno.' He ran his hand through his hair. 'I just dunno. Perhaps we deserve whatever's coming.'\n\nHe turned. 'Anyway, whoever's running this? They've left us here to rot. The Skandis have opened the Combat Chambers up for us to get back to Earth. We don't deserve it, but there you go. I guess they don't want our deaths on their conscience. I'm using one to go home. Maybe you should do the same. If you see me, don't say hello.' He breathed out, a shuddering breath that emptied his lungs. 'Well, that's it. Seraphin out. Mic drop gif.'\n\nHe walked away.\n\nThe great screens in the Big White Room went blank and never showed anything again.\n\nEveryone was silent, stunned. Except for Miss Quill, who looked up, smiling slightly.\n\n'Well, I think that was a pretty great plan.'\n\n## FIFTY-TWO\n\n## TO GET YOUR FREE CONFESSION JUST FOLLOW THESE SIMPLE STEPS\n\nMiss Quill went back to her room in the Void and tapped on the wall.\n\n'I know that you can still hear me. I know that you're still there.'\n\nNothing answered her.\n\nShe put her hands on her hips.\n\n'I'm a very patient woman,' she told the wall. 'Someday soon I'm going to be the last survivor of my planet. While I'm waiting for that, I may as well hang around waiting for you to show yourselves. Time's slow here. And I'm really rather calm.'\n\nThere was no answer.\n\nQuill carried on staring at the wall. An hour passed. Maybe two.\n\n'I'm still here,' said Miss Quill. 'And not at all bored. I've waited for thirty teenagers to come up with thirty different feeble excuses for not being able to solve Fermat's Last Theorem as homework. So bear that in mind. I'd appreciate an explanation from you.'\n\nThe wall looked back at her blankly.\n\n'Here's the thing,' said Miss Quill. 'I know you're still here. The Skandis may be as dainty as dancing Disney teacups, but they're formidable transdimensional engineers. Nothing gets in or out of here without their say-so, and, according to the logs, no one's tried to go anywhere other than Earth. I'd say you figured you'd stay behind. See if there was a chance to recoup a bit more of your investment.'\n\nThe wall said nothing.\n\n'I know I'm not mad. I know you're in the walls,' said Quill. 'I knew you came out to observe me when I was resting. I am a very light sleeper, and it amused me to work out who you were.'\n\nPerhaps it was a trick of the light, but the wall seemed to frown.\n\n'We've met before, you see,' Quill said. 'I'm Quill. The last of the Quill. You came and offered your services to us. Said you'd help us with the Rhodians. We said no. I won't bore you with talk about a warrior's honour. You just wanted too much money. We were a poor people, so we left it at that. But I was always curious about how your service worked. And now I know. Very clever.'\n\nShe smiled at the wall. The wall lit up, just a little.\n\n'You have no name.' Quill smiled. 'The humans will find that a shame. They really do like their closure. If there's one thing they love more than an apology it's someone to hold responsible for everything wrong with their lives. And it's much easier to blame people with names. But that's not how you work, is it? You hide in the shadows, you skulk in the walls, you buy and sell planets\u2014and no one knows it's you, because no one knows who you are. And no one ever will.'\n\nThe wall shrugged, as much as a wall could.\n\n'It's the humans I feel sorry for,' said Quill. 'Actually, that's my first thought when I wake up every morning. But in this case, the humans will process it so badly. Their brains are so strange, so slow and fragile. Some will forget completely, some will deny it ever happened\u2014but they'll all remember eventually, somehow. They'll all be changed by this forever. And, because they're up against you, they'll never even have a name to curse for their nightmares.'\n\nThe wall considered what she said.\n\n'Luckily, I am Quill,' said Miss Quill. 'And we don't need names.' She raised her gun and pointed it at the wall. She fired into the wall, draining the power packs.\n\nThe wall shattered like a shell. Six bodies fell out, lying twitching on the ground, smoke rising from them.\n\n'You shouldn't have hid behind the wall.' She smiled at them. 'You should have come out and faced me. The genome-lock doesn't stop me from shooting walls.'\n\nShe turned one of the charred corpses over. Its face was an empty, featureless mask.\n\n'No names? No faces?' she tutted. 'The humans are really going to hate you.'\n\nShe turned around and went home.\n\n## FIFTY-THREE\n\n## HAPPY ENDINGS DON'T KILL PEOPLE\u2014GUNS DO\n\nWhen you've caught the world out in a great lie, it takes a while before you trust it again.\n\nFor Ram it was as simple as putting one foot in front of the other.\n\nHe was running. Down the road. Into the park, round the trees and off to school.\n\nHis artificial leg kept telling him that something, it wasn't sure what, wasn't quite right.\n\n'No,' he said to his leg. 'It isn't right. Get over it.' And, for this morning, his leg did.\n\nCharlie slunk out of the kitchen, two bowls of cereal balanced in his hands. Miss Quill was walking in.\n\n'You've been avoiding me,' she said. As usual, she wasn't angry. She was, at best, mildly amused. As though his behaviour was a childish reaction.\n\nCharlie refused to rise to it. 'Yes,' he said, angling his chin up at her defiantly. 'Is there any reason why I shouldn't?'\n\nShe stepped past him, enjoying his slight flinch as she got close to him. She filled the kettle and flicked it on to boil, pulling a mug from the cupboard. It had a jolly puppy on it. She dropped the mug in the bin and picked another one.\n\n'I fought. And I enjoyed it. I am honest about that.' She smiled at Charlie. 'One day, if you really hold true to your heritage, you too will fight a great war. And you will be lying to yourself if you don't enjoy it.' She tipped boiling water into the mug. 'You'll see. Enjoy your cereal.'\n\nTanya sneaked into her living room. The rest of the house was a hive of people, running into the bathroom, arguing over clothes, and making too much noise in the kitchen. No one noticed her go up to the television and switch it on. She slipped headphones on and plugged them in.\n\nFor days now she'd made a public protest about how she was overgaming. About how she was giving it a rest.\n\nAnd she'd stick to it. In public, at least.\n\nShe booted up the games console.\n\nYes. She'd stick to it. But just a quick bout every now and then. With a smile, she started shooting.\n\nApril was standing waiting at the school gates, watching Ram approach.\n\n'I'm all sweaty,' said Ram.\n\n'Yeah, and you're making a thing of it.'\n\n'Not.'\n\n'Are.'\n\nThey stood there, a silence forming between them that wasn't quite the comfortable silence between old friends, but maybe one day would be.\n\nPupils were streaming through the school gates, checking their phones, laughing, nudging one another. Every now and then, one of them would catch April's or Ram's eye and look away. It was the telltale giveaway.\n\nRam snorted. 'People forget so quickly.'\n\n'You think?' April leaned back against the railings.\n\n'They were abducted, tricked into being butchers\u2014not really sure that even Coal Hill's counsellor can cope with that. And what do you say when you get home? \"Sorry I've been missing for a couple of days, actually been fighting an alien war for months\"? Or do you go with\u2014'\n\n'\"Staying over at Kevin's working on a maths project\"? Yeah.' Ram laughed uneasily. 'It's like no time's passed at all.'\n\nMore people strolled past them. A girl stopped. She stared at them, burst into tears, and ran away.\n\n'Who was that?' Ram asked.\n\n'Not a clue,' April said. 'All those helmets.'\n\n'Yeah.'\n\nThe sun shone, and more people slouched through the gates.\n\n'Maybe they're all forgetting,' April said. 'There's a chance that it's only us who remember.'\n\n'That'd be kind,' Ram agreed. 'But then, I'm no longer sure the world is that kind.'\n\n'No.'\n\n'But that's Coal Hill,' said Ram. 'So much to ignore, to deal with silently.'\n\n'How long do you think we'll be able to cope?'\n\nRam shrugged.\n\nApril watched a kid kicking a ball around the yard. No one joined in. 'Who'd kidnap kids and do that to them?'\n\n'Charlie says Miss Quill knows more than she's letting on, like who's behind it.' Ram sounded like he didn't really trust either of them. 'But is that . . . I mean, does it matter? Like I doubt we can sue them. Or that they'd even notice.'\n\nApril looked again at the playground. 'It's changed, you know,' she said. 'It's more subdued.' A sad echo chimed in her head, of her dad after he'd been drinking. 'Notice, they're all being more cautious.'\n\n'I guess,' Ram conceded. 'I keep checking Facebook, but I'm not posting anything on it. No one really is\u2014everyone's being . . .'\n\n'Really, really cautious,' April said. 'No bragging, no videos, no food. It's like a go-slow.'\n\n'Will no one think of the cats?' Ram asked, and they smiled at each other.\n\n'Not so bad.'\n\n'Not really, no. Guess not.'\n\nThey stood in silence, watching the yard. They got ready to head in, to start another day, one that would no doubt start off innocently enough and end in death and aliens.\n\nAnyway, that was today, and it hadn't started yet.\n\nA kid on a mountain bike whipped past them, did a stunt swerve, then another, and, at the last minute, bumped up against the curve. The bike balanced there on one wheel, just for a moment.\n\nA dozen phones whipped up, all eager to catch the moment as the bike slowly arced backwards, crashing to the ground, landing on the rider, who rolled groaning around in the gravel. No one helped him, they just filmed.\n\n'Too good to last,' said April.\n\n'Yeah,' agreed Ram.\n\nThey went and started today.\n\n## **Excerpt from \"For Tonight We Might Die\"**\n\n**TURN THE PAGE**  \nfor an excerpt from the _Class_ pilot episode,  \n\"For Tonight We Might Die,\"  \ncreated and written by Patrick Ness!\n**10:02:28 INT. COAL HILL SCHOOL - ATRIUM - DAY 2; 08:51** ****\n\n**APRIL** picks up a torn-down prom poster and re-tapes it to the wall. **TANYA** watches her.\n\n**10:02:29 CAPTION 'FOR TONIGHT WE MIGHT DIE WRITTEN BY PATRICK NESS'**\n\n**APRIL**\n\nTanya, are you coming to help decorate later? I need volunteers.\n\n**TANYA**\n\nMy mum says I'm too young.\n\n**10:02:34 _MUSIC OUT ARCTIC MONKEYS - BLACK TREACLE_**\n\n**10:02:35 CAPTION 'PRODUCER DEREK RITCHIE'**\n\n**APRIL**\n\nBut it's for Year 12, so _technically_ you qualify despite, you know . . . being like? Two years ahead? How old are you anyway?\n\n**10:02:41 CAPTION 'DIRECTOR ED BAZLEGETTE'**\n\n**TANYA**\n\nBecause that question never gets old.\n\n**CHARLIE** passes them. They watch him go.\n\n**APRIL**\n\nI think I'm going to ask Charlie to the prom.\n\n**TANYA**\n\nAnd you just made us fail the Bechdel Test\u2014\n\n**APRIL**\n\nThe what? You're funny.\n\n**TANYA**\n\nBesides, I don't think the new kid is in to wait. Prom is _tomorrow_. You don't have a date yet?\n\nCaught, **APRIL** starts moving away, changes subject:\n\n**APRIL**\n\nSo that's a yes to helping me decorate?\n\n**TANYA** watches her go, shaking her head, feeling for **APRIL** as everyone does. But behind **TANYA** \u2014\n\n**A strange, ominous shadow passes, quickly.** Did we see that?\n\n**CUT TO:**\n\n**10:03:04 INT. COAL HILL SCHOOL - MAIN CORRIDOR - DAY 2; 08:53**\n\n**APRIL** talks to **CHARLIE**. He's looking around surreptitiously, as if trying to find something.\n\n**CHARLIE**\n\nBut how would one dress?\n\n**APRIL**\n\nWell, the boys wear dinner jackets and the girls wear ball gowns, but we totally don't need to be that gender-specific\u2014\n\n**CHARLIE** spots the **SCORCH MARKS** on the linoleum floor coming out from under their classroom door. **MISS QUILL** passes, two steps ahead of **MR ARMITAGE** , Headteacher, harried, exhausted.\n\n**MR ARMITAGE**\n\nI don't know what they did in your _last_ job, but chaperoning\n\n**MISS QUILL**\n\n(crisply)\n\nI am _not_ a baby-sitter.\n\n****\n\n**MR ARMITAGE**\n\nExcept . . . you kind of _are_ . . .\n\n**10:03:22 _MUSIC IN 1M01B_**\n\nThey pass, **CHARLIE** and **APRIL** watch them go.\n\n**10:03:27 _MUSIC OUT 1M01B_**\n\n**APRIL**\n\n_Anyway_ , about the prom, I was thinking that maybe you and I could you know\n\n**CHARLIE**\n\n(polite but firm)\n\nOh, no. I shouldn't think so.\n\n**CHARLIE** heads after **MISS QUILL**. **APRIL** watches him go, stung.\n\n**RAM**\n\n(passing by)\n\nThat was painful.\n\n**RAM** disappears into the classroom where **CHARLIE** went.\n\n**CUT TO:**\n\n**10:03:50 INT. COAL HILL SCHOOL - MISS QUILL'S CLASSROOM - DAY 2; 09:02**\n\n**MISS QUILL** looms into frame.\n\n**MISS QUILL**\n\nLook at you all. The cream of the crop. High achievers. It's no wonder this country only exports Downton Abbey.\n\nOur four\u2014 **APRIL** , **TANYA** , **RAM** and **CHARLIE** \u2014sit by each other in a room full of classmates.\n\n**APRIL**\n\n(at her desk, quietly)\n\nI _like_ Downton Abbey.\n\n**TANYA**\n\nOf course you like it. It's a bunch of white people being nice to each other.\n\n**RAM**\n\nYou'd watch it if Idris Elba were on it.\n\n**CHARLIE** , overhearing, takes out his phone to type in a search, \"Idris Elba\u2014\" It's yanked out of his hand by **MISS QUILL**!\n\n**MISS QUILL**\n\nCheating already, Charles?\n\n**CHARLIE**\n\n. . . you haven't actually taught us  anything yet\u2014\n\nSearch says \"Idris Elba is an English actor, musician, voice actor and DJ.\"\n\n**MISS QUILL**\n\nTrue. How about our first lesson of the day?\n\nDrops the phone. There's a crack.\n\n**MISS QUILL (CONT'D)**\n\nYour teacher has _awful_ butter fingers.\n\nShe picks up the phone, hands it back to **CHARLIE**. The glass display is smashed.\n\n**APRIL**\n\nYou can't do that! That's his private property!\n\n**MISS QUILL'S** laser focus shifts to **APRIL**.\n\n**MISS QUILL**\n\nApril MacLean. The answer to the question, Are spinsters born or made? You got a date for the prom yet?\n\n**TANYA**\n\nI am pretty sure you are not allowed  to ask that!\n\n**RAM**\n\n(arrogant)\n\n_I've_ __ got a date.\n\n**MISS QUILL**\n\nBut of course you do, Ram. The boy who hears silent applause every time he walks into a room.\n\nShe turns back to the front and writes an impossibly difficult **PHYSICS** equation on the board.\n\n**MISS QUILL (CONT'D)**\n\nSolve it. Don't look so pained, Matteusz.\n\nThe class, cowed, starts baffled work. Our four struggle.\n\n**RAM**\n\nIs this even in the book?\n\n**APRIL**\n\nI honestly think she makes this stuff up.\n\n**TANYA**\n\nIt's the Gibbs probability density in a classical Klein-Gordon Field.\n\nThey stare at her.\n\n**TANYA (CONT'D)**\n\nWell, it _is_.\n\n**RAM**\n\nWe're not _supposed_ to understand it. She just wants us to feel stupid.\n\nI'll bet she _ate that missing kid_.\n\n**CHARLIE**\n\nHe was eaten?\n\n**RAM**\n\n(exasperated)\n\nWhere are you from, anyway? You sound like the Queen.\n\n**CHARLIE**\n\n(beat, flatly)\n\nI'm from Sheffield.\n\n**RAM**\n\nWhy am I even talking to you geeks?\n\n**MISS QUILL**\n\nDon't make me put you in detention, Ram. You wouldn't like it.\n\nThey return to their work. **CHARLIE** catches the eye of **MATTEUSZ** sitting a few rows up, who makes a hair-pulling gesture at the physics problem. **CHARLIE** smiles. Then he types into his phone's smashed screen, \"Monkeys of the arctic . . .\"\n\n**10:05:38 _MUSIC IN ARCTIC MONKEYS - BLACK TREACLE_**\n\n**CUT TO:**\n\n****\n\n**10:05:41 EXT. COAL HILL SCHOOL - FRONT - DAY 2; 15:35**\n\nThe school day is ending. **APRIL** stands outside the school entrance with flyers, trying to catch the attention of passing students.\n\n**APRIL**\n\nBig decoration tonight! All volunteers are welcome. Big decoration tonight! Autumn ball. All volunteers are welcome. All volunteers are welcome. Big decoration tonight. Need your . . . Come and help me, anyone.\n\n**CHARLIE** approaches her.\n\n**10:06:05 _MUSIC OUT ARCTIC MONKEYS - BLACK TREACLE_**\n\n**CHARLIE**\n\nHaving a date to the prom is important, yes?\n\n**APRIL**\n\n(looking hopeful)\n\n_Yeah_ \u2014\n\n**CHARLIE**\n\nI shall ask Matteusz as my date.\n\n**APRIL**\n\nMatteusz? Oh. _Ohhhhh_ \u2014\n\n**CHARLIE**\n\nIs this not right?\n\n**APRIL**\n\nNo! No, no\n\n**CHARLIE**\n\nThe schools in Sheffield were very different\u2014\n\n**APRIL**\n\nNo! It's great! It's fantastic. It totally makes sense.\n\n(little clap)\n\nGood for you!\n\n**CHARLIE**\n\nThank you.\n\nHe nods at her and heads off.\n\n**APRIL**\n\nOkay.\n\n(to herself, sad)\n\nOf course he is. Of course.\n\n**CUT TO:**\n\n**10:06:38 EXT. COAL HILL SCHOOL - FOOTBALL PITCH - DAY 2; 16:00**\n\n**10:06:38 _MUSIC IN 1M02_**\n\n**RAM** practises drills with the school football **TEAM**. C **OACH DAWSON** (huge, young, gorgeous) runs the drills. **VARUN** watches from the sidelines.\n\n**RACHEL** (pretty, brunette, bitchy) arrives to watch, too. A mousy P **AIR OF GIRLS** flee from her. **RAM** sees her, waves. She waves back. VARUN notices and frowns.\n\nThe sun is out, casting **SHADOWS** on the field. **RAM** pushes the ball past defenders. He's superb, particularly the way he dodges and fakes out other players.\n\nBut just as he's about to strike it into the goal, he sees the shadow of one TEAMMATE JUMP to the shadow of another. The ball is stolen off him easily, and the rest of the team runs the other way, Ram left standing, wondering what he saw.\n\n**RACHEL**\n\nRam!\n\n**COACH DAWSON**\n\nWhat the hell was what?\n\n**RAM**\n\nCoach\n\n**COACH DAWSON**\n\nTwenty laps. Now.\n\n**RAM**\n\nWhat for?\n\n**COACH DAWSON**\n\nTwenty- _five_.\n\nLet's get a move on.\n\nAnnoyed, **RAM** sets off. He sees **VARUN** glaring at **RACHEL**.\n\n**10:07:28 _MUSIC OUT 1M02_**\n\n**CUT TO:**\n\n**10:07:29 EXT. STREET - DAY 2; 16:05**\n\n**TANYA** walks home alone, singing to herself, throwing in some dance moves. She's just a kid, after all. __She passes a MISSING POSTER for KEVIN, the Sixth-Form Student from the opening.\n\nWe see her **SHADOW** behind her. It **STARTS TO GROW** , stretching abnormally, twisting into different **SHAPES** , threatening ones\u2014\n\n**TANYA** stops, looking guilty.\n\n**10:07:33 _MUSIC IN 1M03_**\n\n**10:07:41 _MUSIC OUT 1M03_**\n\n**TANYA**\n\nNo. No. Why would you help decorate for something you can't even go to?\n\n(sags)\n\nBecause that sad individual is probably the only chance you have at a friend.\n\nShe sighs, turns around to head back to school **AND SEES A TERRIFYING SHADOW SHAPE LOOMING OVER HER**!\n\nShe screams and runs as fast as she can\u2014\n\nThe **SHADOW** comes after her\u2014\n\n**10:07:54 _MUSIC IN 1M04_**\n\n## Back Ad\n\nDISCOVER\n\nyour next favorite read\n\nMEET\n\nnew authors to love\n\nWIN\n\nfree books\n\nSHARE\n\ninfographics, playlists, quizzes, and more\n\nWATCH\n\nthe latest videos\n\nwww.epicreads.com\n\n## About the Authors\n\n**PATRICK NESS** is the creator of the BBC television series _Class_ and the author of ten novels and a short story collection. He's best known for his books for young adults, including his _New York Times_ bestselling novel _The Rest of Us Just Live Here,_ the acclaimed Chaos Walking trilogy, _More Than This,_ and the Carnegie Medal\u2013winning _A Monster Calls. A Monster Calls_ was released as a major motion picture\u2014with the screenplay by Patrick himself\u2014starring Liam Neeson and Sigourney Weaver. Born in Virginia, Patrick lives in London.\n\nwww.patrickness.com\n\n**JAMES GOSS** is the author of two _Doctor Who_ novels: _The Blood Cell_ and _Dead of Winter_ as well as _Summer Falls_ (on behalf of Amy Pond). He is also the coauthor, with Steve Tribe, of _Doctor Who: The Doctor's Lives_ _and_ _Times,_ _Doctor Who:_ _The Dalek Handbook,_ and _Doctor Who: A History of the Universe in 100 Objects._ While at the BBC, James produced an adaptation of _Shada,_ an unfinished Douglas Adams _Doctor Who_ story, and _Dirk_ is his award-winning stage adaptation of _Dirk Gently's Holistic Detective Agency._ His _Doctor Who_ audiobook _Dead Air_ won Best Audiobook 2010 from the Audiobook Store and his books _Dead of Winter_ and _Torchwood: First Born_ were both nominated for the 2012 British Fantasy Society Awards.\n\nDiscover great authors, exclusive offers, and more at hc.com.\n\n## Books by Patrick Ness and James Goss\n\nClass: The Stone House\n\nClass: Joyride\n\nClass: What She Does Next Will Astound You\n\nBooks by Patrick Ness\n\nThe Rest of Us Just Live Here\n\nRelease\n\n## Copyright\n\nHarperTeen is an imprint of HarperCollins Publishers.\n\nCLASS: WHAT SHE DOES NEXT WILL ASTOUND YOU. Copyright \u00a9 2016 by James Goss. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse-engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereafter invented, without the express written permission of HarperCollins e-books.\n\nwww.epicreads.com\n\nCover photographs by Shutterstock\n\nCover design by www.headdesign.co.uk\n\n* * *\n\nLibrary of Congress Control Number: 2017943586\n\nDigital Edition MARCH 2018 ISBN: 978-0-06-266625-3\n\nPrint ISBN: 978-0-06-266623-9\n\n* * *\n\n18 19 20 12 22 PC\/LSCH 10 9 8 7 6 5 4 3 2 1\n\nPublished by arrangement with Woodland Books, a division of The Random House Group, Ltd.\n\nOriginally published in the UK in 2016 by BBC Books, an imprint of Ebury Publishing\n\nFirst US Edition, 2018\n\n## About the Publisher\n\n**Australia**\n\nHarperCollins Publishers Australia Pty. Ltd.\n\nLevel 13, 201 Elizabeth Street\n\nSydney, NSW 2000, Australia\n\nwww.harpercollins.com.au\n\n**Canada**\n\nHarperCollins Canada\n\n2 Bloor Street East - 20th Floor\n\nToronto, ON M4W 1A8, Canada\n\nwww.harpercollins.ca\n\n**New Zealand**\n\nHarperCollins Publishers New Zealand\n\nUnit D1, 63 Apollo Drive\n\nRosedale 0632\n\nAuckland, New Zealand\n\nwww.harpercollins.co.nz\n\n**United Kingdom**\n\nHarperCollins Publishers Ltd.\n\n1 London Bridge Street\n\nLondon SE1 9GF, UK\n\nwww.harpercollins.co.uk\n\n**United States**\n\nHarperCollins Publishers Inc.\n\n195 Broadway\n\nNew York, NY 10007\n\nwww.harpercollins.com\n\n## Contents\n\n  1. _Cover_\n  2. _Title Page_\n  3. _Dedication_\n  4. _Contents_\n  5. One: Read the Terrible Truth They Don't Want You to Know Until Page 119\n  6. Two: What We Can Learn from the Alien-Face-In-A-Box Emoji\n  7. Three: When He Lost His Leg They Said He Would Never Play Football Again, But He Said No\n  8. Four: Eight Things the Media Hasn't Told You About Cup-A-Soup\n  9. Five: Amazing, If True: Children Are Vanishing and You've Not Noticed\n  10. Six: The One Weird Trick About Mothers That Everyone Should Know\n  11. Seven: Is It Magic? We Need Your Help to Solve This Mystery\n  12. Eight: Fifteen Cats That Look Shockingly like Miss Quill\n  13. Nine: Train Drivers Slam Brakes on Truth or Dare\n  14. Ten: This Chapter Will Prove You've Been Wrong About Young Men Your Whole Life\n  15. Eleven: This Girl Wanted an Adventure Holiday. You Won't Believe Where She Ended Up\n  16. Twelve: Find out Why the Goats Do Not Get to Dance on the Tube\n  17. Thirteen: The Letter Coal Hill School Doesn't Want You to See\n  18. Fourteen: How Toast Is like Lady Gaga\n  19. Fifteen: You Wouldn't Think a Text Could Make You Cry, but This Will\n  20. Sixteen: Six Names for White You've Never Thought Of\n  21. Seventeen: The Rise of Smart Women and How to Stop It\n  22. Eighteen: Things You'll Get Only If Your Home Planet Was Destroyed in the Nineties\n  23. Nineteen: He Thought He Knew a Lot About Gravity. Find out If He Was Right\n  24. Twenty: The Five Words That Broke Her Heart (Spoiler: One of Them Is 'Want')\n  25. Twenty-One: She Thought She Knew What Was Going On. Then She Found out the Remarkable Truth and Turned Things Around\n  26. Twenty-Two: Thought You Knew How to Lose Your Friends? Well, This Woman's Thirteen Brilliant Reasons Will Change Your Mind\n  27. Twenty-Three: She Was Ready to Give up and Then a Nurse Slayed Her with a Word\n  28. Twenty-Four: If You Dropped Dead Tomorrow, Would Your Friends Miss You?\n  29. Twenty-Five: She Thought She'd Seen It All and Then She Saw the Face of God\n  30. Twenty-Six: This Teacher's Inspirational Words Will Choke You Up\n  31. Twenty-Seven: Someone's Reimagined Disney Princesses as Alien Warriors And, Trust Us, It's Awesome\n  32. Twenty-Eight: This Hot Take on Smashed Avocado Toast Will Have You Reeling\n  33. Twenty-Nine: Think of the Worst Job in the World? You're Not Even Close\n  34. Thirty: You Are Being Lied to About Voter Registration and This Short Chapter Tells You How\n  35. Thirty-One: Advertisement: Your Book Will Continue in Twenty-Five Seconds\n  36. Thirty-Two: The Ten Best Alien Deaths You'll See Today. #6 Is a Killer\n  37. Thirty-Three: Eight Ways in Which People Have Tried to Escape the Void\n  38. Thirty-Four: At First She Thought She Knew Everything but Then She Found This Secret She Hadn't Known She Needed to Know\n  39. Thirty-Five: When She Met God She Forgot to Ask 'Why?'\n  40. Thirty-Six: This Is Your Chance to Wipe out Skandis Forever\n  41. Thirty-Seven: After You Read This You'll Want a Shower\n  42. Thirty-Eight: You'll Be Amazed at How Long It Took Him to Realise His Mistake\n  43. Thirty-Nine: War Veterans Are Covering Their Heads in Glitter for Reasons That Will Stun You\n  44. Forty: This Icelandic Penguin Village Is Probably the Cutest Place on Earth. But You Are Not There\n  45. Forty-One: In the Time It Takes You to Read This, Skandis Will Have Claimed One Hundred More Lives\n  46. Forty-Two: Many People Would Blame This on Marriage Equality. But Would You?\n  47. Forty-Three: People Are Tweeting Their Worst Battles and It Is Cringingly Hilarious\n  48. Forty-Four: She Dropped a Truth Bomb but Wasn't Expecting What Would Happen Next\n  49. Forty-Five: This Young Footballer Has Something Surprising to Say About Racial Profiling\n  50. Forty-Six: You Are Being Lied to About Dogs\n  51. Forty-Seven: The Latest Advances in Virtual Reality Will Horrify You\n  52. Forty-Eight: Do You Know Enough About Dimensional Compensators to Save This Boy's Life? (Spoiler: You Don't)\n  53. Forty-Nine: He Chose the Wrong Day to Beg for His Life\n  54. Fifty: The Skandis War as You've Never Seen It Before\n  55. Fifty-One: They Thought They'd Won Until They Found out They'd Lost\n  56. Fifty-Two: To Get Your Free Confession Just Follow These Simple Steps\n  57. Fifty-Three: Happy Endings Don't Kill People\u2014Guns Do\n  58. Excerpt from \"For Tonight We Might Die\"\n  59. _About the Authors_\n  60. _Books by Patrick Ness and James Goss_\n  61. _Back Ad_\n  62. _Copyright_\n  63. _About the Publisher_\n\n# Guide\n\n  1. Cover\n  2. Contents\n  3. Chapter 1\n\n  1. v\n  2. \n  3. \n  4. \n  5. \n  6. \n  7. \n  8. \n  9. \n  10. \n  11. \n  12. \n  13. \n  14. \n  15. \n  16. \n  17. \n  18. \n  19. \n  20. \n  21. \n  22. \n  23. \n  24. \n  25. \n  26. \n  27. \n  28. \n  29. \n  30. \n  31. \n  32. \n  33. \n  34. \n  35. \n  36. \n  37. \n  38. \n  39. \n  40. \n  41. \n  42. \n  43. \n  44. \n  45. \n  46. \n  47. \n  48. \n  49. \n  50. \n  51. \n  52. \n  53. \n  54. \n  55. \n  56. \n  57. \n  58. \n  59. \n  60. \n  61. \n  62. \n  63. \n  64. \n  65. \n  66. \n  67. \n  68. \n  69. \n  70. \n  71. \n  72. \n  73. \n  74. \n  75. \n  76. \n  77. \n  78. \n  79. \n  80. \n  81. \n  82. \n  83. \n  84. \n  85. \n  86. \n  87. \n  88. \n  89. \n  90. \n  91. \n  92. \n  93. \n  94. \n  95. \n  96. \n  97. \n  98. \n  99. \n  100. \n  101. \n  102. \n  103. \n  104. \n  105. \n  106. \n  107. \n  108. \n  109. \n  110. \n  111. \n  112. \n  113. \n  114. \n  115. \n  116. \n  117. \n  118. \n  119. \n  120. \n  121. \n  122. \n  123. \n  124. \n  125. \n  126. \n  127. \n  128. \n  129. \n  130. \n  131. \n  132. \n  133. \n  134. \n  135. \n  136. \n  137. \n  138. \n  139. \n  140. \n  141. \n  142. \n  143. \n  144. \n  145. \n  146. \n  147. \n  148. \n  149. \n  150. \n  151. \n  152. \n  153. \n  154. \n  155. \n  156. \n  157. \n  158. \n  159. \n  160. \n  161. \n  162. \n  163. \n  164. \n  165. \n  166. \n  167. \n  168. \n  169. \n  170. \n  171. \n  172. \n  173. \n  174. \n  175. \n  176. \n  177. \n  178. \n  179. \n  180. \n  181. \n  182. \n  183. \n  184. \n  185. \n  186. \n  187. \n  188. \n  189. \n  190. \n  191. \n  192. \n  193. \n  194. \n  195. \n  196. \n  197. \n  198. \n  199. \n  200. \n  201. \n  202. \n  203. \n  204. \n  205. \n  206. \n  207. \n  208. \n  209. \n  210. \n  211. \n  212. \n  213. \n  214. \n  215. \n  216. \n  217. \n  218. \n  219. \n  220. \n  221. \n  222. \n  223. \n  224. \n  225. \n  226. \n  227. \n  228. \n  229. \n  230. \n  231. \n  232. \n  233. \n  234. \n  235. \n  236. \n  237. \n  238. \n  239. \n  240. \n  241. \n  242. \n  243. \n  244. \n  245. \n  246. \n  247. \n  248. \n  249. \n  250. \n  251. \n  252. \n  253. \n  254. \n  255. \n  256. \n  257. \n  258. \n  259. \n  260. \n  261. \n  262. \n  263. \n  264. \n  265. \n  266. \n  267. \n  268. \n  269. \n  270. \n  271. \n  272. \n  273. \n  274. \n  275. \n  276. \n  277. \n  278. \n  279. \n  280. \n  281. \n  282. \n  283. \n  284. \n  285. \n  286. \n  287. \n  288. \n  289. \n  290. \n  291. \n  292. \n  293. \n  294. \n  295. \n  296. \n  297. \n  298. \n  299. \n  300. \n  301. \n  302. \n  303. \n  304. \n  305. \n  306. \n  307. \n  308. \n  309. \n  310. \n  311. \n  312. \n  313. \n  314. \n  315. \n  316. ii\n  317. iv\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nPRAISE FOR _LAUNCH!_\n\n\"Real-life principles that actually work!\"\n\n\u2014FRANK SHANKWITZ, founder, The Make-A-Wish Foundation\n\n\" _Launch!_ is a must read and truly impactful book for any entrepreneur.\"\n\n\u2014GENE LANDRUM, president and creative force, Chuck E. Cheese, Entrepreneur Hall of Fame\n\n\"Scott Duffy expertly guides you through what it takes to launch a new business\u2014from the deep personal sacrifices you make for a project you wholeheartedly believe in to the incredible high you get when your mission becomes a success. _Launch!_ is an invaluable companion with real-life examples for anyone willing to dream big, break barriers, and turn their idea into reality.\"\n\n\u2014KRISTI AND ANDY FUNK, founders, Pink Lotus Breast Center\n\n\"Scott is one of the most dynamic, positive, and creative people I have had the good fortune to know. Scott is a great leader in all areas of life. I'm honored to be in his corner.\"\n\n\u2014DOUG BRIGNOLE, Mr. Universe, Mr. America, author, speaker\n\n\"This book isn't just about launching\u2014it's about setting the stage to build better companies. It's a must read.\"\n\n\u2014DENA COOK, CEO, Brew Media Relations\n\n\"In _Launch!,_ Scott's ability to connect with entrepreneurs is unparalleled. His passion and genuine drive to help others grow is infectious.\"\n\n\u2014NICOLE HURD, founder, National College Advising Corps.\n\n\"It's about time somebody wrote a book about launching new ventures that doesn't just look at the business side of entrepreneurship, but at the human side as well! Scott Duffy shows how you, too, can build a stronger, more successful business by paying attention to all parts of your life as you launch and grow your business.\"\n\n\u2014MICHELLE PATTERSON, executive director, Women Network Foundation and The CA Women's Conference\n\n\" _Launch!_ shows us how the right attitude, the right focus, and the right idea need to come together for a successful launch.\"\n\n\u2014CHRIS FRALIC, partner, First Round Capital\n\n\"What makes _Launch!_ such a fun and compelling read is Scott's personal experiences launching new ventures. He's learned the lessons of a successful launch through real-world experience and proves how you, too, can take your big idea to market.\"\n\n\u2014KALIKA YAP, CEO, Citrus Studios\n\n\"Scott is an incredible entrepreneur who will show you the way to success. He will inspire you through your successful _Launch!_ \"\n\n\u2014RON KLEIN, inventor of the credit card magnetic strip verification system, Multiple Listing Service, business consultant, problem solver\n\n\"I wish I had had this book when I was launching my company. Scott's blend of business smarts, useful takeaways, and empathy is much needed in today's fast-start environment.\"\n\n\u2014LORI TAYLOR, founder, REV Media Marketing\n\n\"Launching a new product or service is incredibly tough\u2014but you're not alone. Scott Duffy is here to guide you through the tough, crucial, exhilarating first days of the launch, with tips, tricks, and tools that will help you get off to the right start.\"\n\n\u2014GERRY MORTON, president and CEO, Energy First\n\n\"In _Launch!_ Scott Duffy keeps all the priorities\u2014family, community, business\u2014in a positive balance. He is a value-added adviser to entrepreneurs and his advice is key to a successful business and launch.\"\n\n\u2014DEMETRIOS A. BOUTRIS, California Corporations commissioner (ret.)\n\n\"As I was finishing my master's in exercise science, I planned on building my own coaching business in the health and fitness world. The very first person I reached out to was Scott Duffy. Scott talked to me about the successes of entrepreneurs and the fails of entrepreneurs. He made me think about what I needed to do in the health and fitness world, by ' _Launch!_ -ing' my brand, my expertise, and my knowledge. This book is for people who are starting a business from scratch to people who may need to tweak their business to make it a success. I promise you, with Scott's brand, expertise, and knowledge . . . you will _not_ be disappointed.\"\n\n\u2014JENNIFER RULON, M.S., USA Triathlon Level I Coach\n\n\"Scott Dufffy knows what it's like to launch a new company from the ground up. In his book _Launch!,_ he shares the wealth of his experiences with us\u2014and shows us how anyone can take their big idea to market quickly, efficiently, and successfully.\"\n\n\u2014JASON PORT, investor, strategist, founding investor in Quirky.com\n\n\"Scott Duffy has condensed years of real-world business experience into an easy-to-use playbook. Packed with relevant real-world advice you can use today. Essential reading for anyone looking to start a business or launch a new product within an existing business.\"\n\n\u2014GENE LIM, Global Strategic Alliances, Entrepreneurs Organization, and CEO MAV12\n\n\"Scott Duffy's guide to launching new businesses is a must read: _Launch!_ is snappy, cutting edge, entertaining, and with useful takeaways on every page. Let me put it this way, it's _awesome_ and it's now required reading for everyone on my boards of my businesses!\"\n\n\u2014ERIK \"MR. AWESOME\" SWANSON, Universal Seminars and The Habitude Warrior\n\n\"Scott is the consummate visionary and a born leader. He is a passionate, relentless serial entrepreneur who will make many more dreams reality.\"\n\n\u2014DREW BOYLES, president, Entrepreneurs' Organization\u2013Los Angeles\n\nPORTFOLIO \/ PENGUIN\n\nPublished by the Penguin Group\n\nPenguin Group (USA) LLC\n\n375 Hudson Street\n\nNew York, New York 10014\n\nUSA | Canada | UK | Ireland | Australia | New Zealand | India | South Africa | China\n\npenguin.com\n\nA Penguin Random House Company\n\nFirst published by Portfolio \/ Penguin, a member of Penguin Group (USA) LLC, 2014\n\nCopyright \u00a9 2014 by Scott Duffy\n\nPenguin supports copyright. Copyright fuels creativity, encourages diverse voices, promotes free speech, and creates a vibrant culture. Thank you for buying an authorized edition of this book and for complying with copyright laws by not reproducing, scanning, or distributing any part of it in any form without permission. You are supporting writers and allowing Penguin to continue to publish books for every reader.\n\nLIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA\n\nDuffy, Scott.\n\nLaunch! : the critical 90 days from idea to market \/ Scott Duffy.\n\npages cm\n\nIncludes index.\n\nISBN 978-1-101-60999-6\n\n1. Entrepreneurship. 2. New business enterprises. 3. New products. I. Title.\n\nHB615.D837 2013\n\n658.1'1\u2014dc23\n\n2013039077\n\nWhile the author has made every effort to provide accurate telephone numbers, Internet addresses, and other contact information at the time of publication, neither the publisher nor the author assumes any responsibility for errors, or for changes that occur after publication. Further, publisher does not have any control over and does not assume any responsibility for author or third-party Web sites or their content.\n\nVersion_1\n_To Rachel_\n\nLove at first sight, love of my life\n\nand\n\n_To Lily and Lexi_\n\nI am so proud to be your dad\n\n# CONTENTS\n\nPraise for _Launch!_\n\nTitle Page\n\nCopyright\n\nDedication\n\nPreface\n\nINTRODUCTION: The Entrepreneur Economy\n\nInk Versus the Mac\n\nOn a Mexican Highway\n\nSmall Is the New Big\n\nA Playbook, Not a Textbook\n\nFind a Way to Simple\n\nSTAGE I.\n\nThe Prelaunch Checklist: Thirty Days to Prepare\n\n| CHAPTER 1 |\n\nWHAT'S YOUR BIG IDEA?\n\nHow to Buy a Rainforest\n\nOil Right Under Your Feet\n\nAre You Sitting on Top of a Billion-Dollar Idea?\n\nLet Go of Your Nut!\n\n| CHAPTER 2 |\n\nPROTECTING WHAT'S MOST IMPORTANT\n\nThe Sure Thing\n\nThe Fastest Way to Lose Your First Million\n\nAll for One and One for All\n\nSuper Balls and Glass Balls\n\nTen Thousand Reasons Why You Will Succeed\n\n| CHAPTER 3 |\n\nYOUR ENVIRONMENT\n\nCrabs in a Bucket\n\nYour Fab Five\n\nWhat Do Those Note Cards Say About You?\n\nWrite Your Own Story\n\nSTAGE II.\n\nFueling the Tank: Thirty Days to Assemble Your Resources\n\n| CHAPTER 4 |\n\nBUSINESS PLANNING\n\nBiceps and Business\n\nStart with the Last Page First\n\nThe One-Page Plan\n\nAn Expensive Hobby\n\nNot Every Problem Is Made to Be Solved\n\n| CHAPTER 5 |\n\nSECURING YOUR CREW\n\nBroke or Billions?\n\nYour Role as Flight Commander\n\nThree Plays\n\nThe Perfect Marriage?\n\nThis Is What You Asked For!\n\n| CHAPTER 6 |\n\nUNLOCKING CREATIVITY\n\nTorn Carpet, Chipped Paint\n\nVegas and Video Games\n\n| CHAPTER 7 |\n\nRAISING CAPITAL\n\nTrading for Tickets\n\n_Cannibal Women in the Avocado Jungle of Death_\n\nThe True Path to Raising Capital\n\nWhere Will the Money Come From?\n\nThe Big Meeting\n\nThe Fastest Way to the Door\n\nThe Callback\n\nTricks of the Trade\n\nSTAGE III.\n\nCountdown and Blastoff: The First Thirty Days of Business\n\n| CHAPTER 8 |\n\nGO TO MARKET!\n\nHammers and Nails\n\nRiches in Niches\n\nCrickets\n\nCrowdsource Your Sales Force\n\nYour Buyer Blueprint\n\n| CHAPTER 9 |\n\nMAINTAINING YOUR ALTITUDE\n\nThe Ninety-Day Checkup\n\nCash and Cardboard\n\nKryptonite\n\n| CHAPTER 10 |\n\nIF THINGS GO SIDEWAYS\n\nLearning How to Crash\n\nPanic Faster\n\nWe're Out of Toilet Paper\n\nSeventeen Days\n\nEPILOGUE: Steady as She Goes\n\nWing Walk\n\nThe Circus and Start-ups\n\nTwenty Thousand Miles per Hour\n\nKeep Your Eyes on the Horizon\n\nFill Your Cup First\n\nAcknowledgments\n\nResources\n\nIndex\n\n# PREFACE\n\nOver 97 percent of a rocket's fuel is used during the first three feet of its launch. The same is true when launching a new business, product, or service.\n\nThe first few steps are absolutely critical. But this is where most entrepreneurs make their biggest mistakes\u2014and set themselves off on a path to failure.\n\n_Launch! The Critical Ninety Days from Idea to Market_ is a flight plan for entrepreneurs and managers within large organizations who aim to get their new business or product off the ground successfully. It is a philosophical and systematic approach for turning your passion into a thriving venture while mitigating risk every step of the way.\n\nWhy ninety days? Because that's all the time you have. Today your competition is faster, more agile, and better equipped. In the modern business era, speed is like gold. It's the most valuable currency at your disposal. In 1996 I was working at a consumer Internet start-up in Silicon Valley. At launch we had just one competitor. In sixty days we had seventeen competitors. In ninety days we had almost one hundred. And that was more than fifteen years ago, when things moved at a snail's pace! The marketplace has only gotten more crowded.\n\nSpeed also protects you. With a short time frame, you put fewer dollars at risk and streamline your focus. In ninety days you will be able to take your idea, go to market, get valuable feedback, and make well-informed decisions about what to do next\u2014all without breaking the bank.\n\nI believe it's not only possible but also critical to launch a new business, product, or service in such a short time, and over the years I've developed a systematic approach that divides launching into three phases.\n\nIn _\"Stage I: The Prelaunch Checklist,\"_ the goal is simple: to protect what is most important: you! That means we'll strategize on getting your house in order and limiting your personal risk, so if things do not work out as planned, you can bounce right back.\n\n_\"Stage II: Fueling the Tank\"_ is the process of assembling your resources. Here we'll look at pulling together your plan, your team, your capital, and other key business resources that will put you in the best position to execute successfully.\n\n_\"Stage III: Countdown and Blastoff\"_ takes your big idea to market. In this section we'll nail down all the critical aspects of targeting your market specifically and launching your product successfully around day sixty. This includes developing partnerships, distribution channels, customer-engagement strategies, marketing and public-relations efforts, sales initiatives, and customer support. With each step, I will provide you with a plan for anticipating what comes next and managing through problems in a way that turns them into opportunities.\n\nThe lessons in this book are not academic but are based on my personal experience as an entrepreneur and business leader. I started my career working in the business and personal-development training industry before falling in love with technology. In the midnineties I moved to Silicon Valley and spent the next fifteen years helping to launch and grow companies from a very early stage. Many companies I was a part of became big brands like CBS Sportsline, NBC Internet, FOXSports.com, and Virgin Charter. There were other ventures along the way. Some have been big successes while others have been big failures. The lessons from each experience were invaluable, and I'll share them with you in this book.\n\nIn the last twenty years I've also benefited from working for some of the greatest entrepreneurs and innovators of our time, including Richard Branson and Tony Robbins. In the following pages you'll hear stories about how they, along with others, such as Spanx inventor Sara Blakely and Starbucks CEO Howard Schultz, dealt with the challenges of launching.\n\nYou may ask, _What could I possibly have in common with these people?_ The answer is: _everything_. They started with a vision, just like you. They backed it up with passion and perseverance. They have all been through the ups and downs that come along with being an entrepreneur or launching a new initiative inside a big organization. They combined hard work with hard-learned lessons to create successful ventures with lasting impact\u2014just as you will do.\n\nI am absolutely committed to doing whatever I can to help you launch your big idea to the world and make your dreams come true. I have set up a Web site at www.scottduffy.com with free tools and video-based lessons to provide you with additional inspiration and support. I encourage you to visit it.\n\nIn return, I ask just one thing. For the next ninety days, suspend any doubts or disbelief about your ability to launch successfully. Know that you can be more, do more, and have more than you ever thought possible. If you follow the steps in this book and execute like crazy, in the next ninety days you'll begin to make your dreams come true.\n\nLet's get ready to launch.\n\n## INTRODUCTION\n\nThe Entrepreneur Economy\n\nI am building a fire, and every day I train, I add more fuel. At just the right moment, I light the match.\n\n\u2014MIA HAMM\n\nThere has never been a better time to be an entrepreneur. Today there are more businesses, products, and services being launched than at any time in history. Increased competition and faster-moving product and business cycles are contributing factors.\n\nThis scares the hell out of big business.\n\nTen years ago the barriers to launching and building a successful company were often too high for small entrepreneurs. As a result, a big company's competition was primarily limited to other big companies. Managers could literally look outside their windows and see their competition. They knew each other well, could spot each other's strengths and weaknesses, and could compete effectively with each other. But those days are over.\n\nIn the new entrepreneur economy, the low barrier to entry for new products and services has created fierce competition on every front. The modern-day entrepreneur, who is either using his existing venture to bring new products to market or launching a new business from the ground up, moves more quickly than big companies and can often position himself to meet customer needs better, faster, and cheaper. To make things even more difficult for big businesses, it's not just one entrepreneur they need to watch out for. There are literally millions of entrepreneurs, armed with laptops, smart phones, and free tools from the Internet, nibbling at their feet. Competition now comes from anywhere and everywhere.\n\nThese new entrepreneurs are also more motivated than at any time before. Many have recovered from the global collapse of 2008 by adapting their business practices to the economic conditions and by leveraging all available resources. Today rising companies run leaner, stronger, and more efficiently and generate greater profits. Others, hurt by the crash of 2008, may have lost their savings or jobs so they've had to reinvent themselves in ways they never imagined. Now that they are getting back on their feet, these entrepreneurs are hungrier and driven by real purpose. They move fast because they have to; they move with purpose because they know the consequences of not doing so. They're also fearless and are going after big businesses and successfully stealing their market share, one customer at a time.\n\nBig companies have a big problem in this new landscape: Most have bloated cultures that move too slowly to launch fast and compete against this new breed of entrepreneurs. While ninety days is enough time for an entrepreneur to launch a new business, product, or service, it takes most big businesses ninety days to make a decision on whether or not to even _get in the game_. Once they decide to compete, it might take them another nine hundred days to actually do it.\n\nOn the other hand, many entrepreneurs, despite their potential for speed and agility, still rely on a combination of passion (which can take you only so far) and outdated strategies that are out of sync with best practices of the new economy. That's where this book comes in, bridging that gap between the old approach and what it really takes to generate success in this economic era. By applying the lessons in this book, you can set yourself up to be in the best starting position to launch and come out head and shoulders above the competition.\n\nThe old notion of an entrepreneur brought to mind people like Richard Branson, Donald Trump, and others who appeared to take gigantic risks, putting everything they had both personally and professionally on the line. Not all entrepreneurs can think of themselves in this light, nor is it responsible for them to take on that much risk.\n\nBut today everyone is an entrepreneur. It's not about building the next Virgin or Google or Facebook. It's about planting a flag. Transforming what you are passionate about, what you are good at, into a responsible, moneymaking venture that benefits others in the process.\n\n_Launch!_ is a handbook for entrepreneurs on how to think big, take on any size competitor\u2014and eat their lunch. I want to help you become a modern-day business version of David, where as a small and agile entrepreneur, you can defeat the big corporate Goliath. This is your opportunity. Seize it.\n\n### Ink Versus the Mac\n\nI've been called a \"serial entrepreneur.\" Whatever the name caller's intention, I love the expression. It speaks to my history as well as my passion.\n\nAt a very early age, my family exposed me to business. For fifty years, all the men on my father's side\u2014my father, my grandfather, back to my great-grandfather\u2014ran a printing and engraving shop. One of the things I remember most clearly about my childhood was my father, like all the good tradesmen, coming home with the distinctive smell of ink on his hands and clothes.\n\nBack then, printing and engraving was a very manual business. Skilled craftsmen spent years as apprentices, learning their trade. Photographers, engravers, and dot etchers pooled their skills, working with advertising agencies to take photos and create artwork that they melded into advertisements, outdoor billboards, and all sorts of printed materials. Their industry played a vital role in marketing and held a well-recognized place in society. My family made money, the skilled employees were paid well, and everybody took pride in the work they did. Then computer technology turned the business on its head.\n\nI have always straddled the digital divide. The year after I finished grammar school, typing became a mandatory class. The year after I graduated from high school, computers became a fixture in the classroom. As a freshman in college, I bought an electronic typewriter\u2014and I really thought it rocked\u2014but was soon introduced to an even more game-changing piece of technology. Sophomore year, my roommate arrived carrying a newfangled computer all his own. He called it a \"Mac.\" That desktop machine may have been just one tool\u2014by today's standards a primitive one\u2014but it meant he was a light-year ahead of where I was. Being able to use a word processor to do your homework? _Wow!_\n\nWhen the first Apple Macintosh came to the printing business, the process of producing an advertisement changed forever. Tradesmen on both sides of the table, the advertising agencies and the service providers, had much invested in the old way of doing things, including skills they had developed over a lifetime. They had a lot to lose from change, so they didn't learn new skills or change the way they did business overnight. But even that rudimentary computer created a weird energy in the shop when it was first installed. It was a mystical moment, almost as if everyone had suddenly seen the future. To complicate matters, that Mac was literally placed right in the middle of the office. You couldn't avoid it. Some of the workers in the shop embraced it while others tried to pretend it was not there.\n\nThe guys on the floor may have had some sixth sense that, soon enough, the job of the craftsman, working with paper, rulers, and glue, would become obsolete. I got the feeling that people walked around on tiptoes in hopes that they could delay the waking of that big elephant in the room for another day or week. That didn't work, of course. With the Mac, processes that had taken three men three days by the old manual routine could be completed by one man in three hours. Prep jobs could be done for a fraction of what they had cost a year or two earlier. The cost of making corrections or changes to a printed piece dropped significantly. As customers learned of these efficiencies, the savvy ones insisted that those savings be passed on to them. If you were really paying attention, you could tell that the smell of ink in the room had begun to fade.\n\nThe Mac initiated the change, but the Internet accelerated it. Pretty soon nobody needed to drive around delivering printing proofs to their customers. Knocking on doors and pursuing sales leads the old way was also finished. Despite decades of stability, the introduction of new technology disrupted the industry rapidly. Many businesses vanished, along with the three-martini lunch.\n\nThis was a big lesson for me at an early age. The good men I grew up with had a lot to lose by not adapting. Many were afraid to change and were afraid of the unknown. But the ones who embraced change, who recognized that technology could propel them forward, learned to use the Macs (or other technologies) and thrived. Some even opened up new shops where they did the same work\u2014only better, faster, and cheaper. They found new pride in their work. Those who refused to change? Well, some of them became unemployable.\n\nThe whole experience made one thing crystal clear: that you have to live your life through the windshield, not through the rearview mirror.\n\n### On a Mexican Highway\n\nAnother huge lesson for my future business life was the importance of adaptability. I was first exposed to the need to adapt while attending the University of San Diego. There I started a business, played sports, and lived on the beach, generally having the time of my life. After midterm exams in the fall of 1989, following tradition, a group of friends and I packed up our cars and headed on a ten-hour drive south to the beaches of San Felipe, Mexico. I thought I was bound for fun in the sun and a great time with friends.\n\nAnd then, in a heartbeat, everything changed.\n\nWe were several miles south of the border, traveling on a two-lane highway. The landscape was surreal. There was nothing but desert on both sides of the road as far as one could see.\n\nSuddenly, a truck pulled out right in front of us. There was no time to stop, and we hit it squarely, going about seventy-five miles per hour. Sitting in the passenger seat, I was tossed about like a beanbag. All four of us in the car were injured. The guys in the truck we hit just drove off and left us there, alone in the desert, bleeding and in pain.\n\nIt was the _worst_ day of my life.\n\nHow many times in your life have you been going down a road, doing everything right? Your timing was perfect. You had a solid plan and team in place. You executed flawlessly. Then, out of nowhere, something unexpected knocked you off course. Think about those events in your life where the ground just moved underneath your feet, when everything you had done up to that point, everything you had accomplished, had to be thrown out. You had to start over, right?\n\nThat's exactly what happened to me.\n\nThe accident was bad, the recuperation no better. I ended up at the Scripps Clinic in La Jolla with significant head injuries, including two brain hemorrhages, and a long road to recovery. I couldn't read without headaches or nausea, and I couldn't even bear to watch TV. Studying was out of the question, so I was forced to stop going to class and effectively dropped out of school.\n\nDespite having lost so much in that accident, I developed a habit in those weeks and months that changed my life forever. I started listening to motivational books on tape. The first program I listened to was from Tony Robbins, and throughout my recovery I got my hands on others, including audiobooks from Jim Rohn, Brian Tracy, Denis Waitley, and Zig Ziglar. I listened to these audiobooks all day, every day. I was inspired by the idea that if you have the desire and are willing to make a commitment to follow through and take action, you can achieve virtually anything you want in life. All the resources you need to get started are within you now or within your reach.\n\nSometimes my head hurt so much that I had to keep the volume really low on the tapes, so low that I could barely hear the speaker. But I absorbed everything they had to teach me. I also learned that many of these trainers were based in San Diego. I decided that, when I got better, I would go back to school and apply for an internship with one of them. It would be my motivation for powering through my recovery. As my first target, I decided to seek out Tony Robbins.\n\nDue to the early success of his infomercials, Tony was on his way to becoming a household name. I knew I'd have to do something big to get his attention, something to stand out. I thought it through and finally I came up with an idea.\n\nTony is a big guy, six feet, seven inches tall. So I got a very tall cardboard box, filled it with packing material, and enclosed in it my r\u00e9sum\u00e9 and items representing all his biggest accomplishments (a copy of his book, one of his tapes, a flyer for a live event, a picture of Tony and his wife I found in a magazine). I also enclosed a flyer that asked, \"How can Robbins Research benefit with Scott Duffy as a member of its team?\" By way of answering the question, I listed my traits and all he would gain by hiring me as an intern. I figured because the box was so big he would engage with it, curious to see what was inside. If I was lucky, he'd remember me. I didn't want to blend in (in a competitive business, neither can you). Then I waited, hoping a good thing would happen.\n\nWell, it did. My package got the reaction I'd hoped for and I got a call from Michael \"Hutch\" Hutchison, who was leading Tony's team. He invited me to come in for an interview. Two more interviews followed. During the third, Hutch asked me to stand in front of the room as if I were facing a big audience and rapidly fired questions at me. After about twenty minutes he left the room, leaving me alone. The minutes ticked by. I waited patiently, just thinking how awesome it would be to work with them. Finally, after an hour had passed, Hutch came in and said he had forgotten I was still in the office . . . but I was hired! Not as an intern, either, but as a full-time employee.\n\nMy work for Tony Robbins in the coming months took me on the road, often traveling from city to city promoting his live events, most notably \"Power to Influence,\" one of his first business programs. I would go into offices and deliver a part of the program to a group of employees, selling them the value of seeing Robbins live. I gave these talks up to five times a day. I was immersed in the Robbins way of thinking; I must have seen him teach ten courses over the next year. Traveling the country, I soaked up his wisdom and shared it with others. Working with his team helped me recover and propelled my life in a whole new direction.\n\nWhen I got started working for Robbins Research I was twenty-one years old. I spoke to over two hundred companies, from FedEx to Mary Kay Cosmetics. I learned to look at the world differently, not only through Tony's teaching but also through the teaching of others in the industry and the colleagues I came to know along the way. I met men and women who have remained among my best friends to this day. I had incredible experiences but, most important, I was exposed every day to people who had overcome huge obstacles and looked at them as growth opportunities.\n\nThis shifted my mind-set. It taught me the power of controlling your focus both personally and professionally. It taught me that, no matter how bad a situation may seem in the moment, we have the ability to learn from it and assign new meaning that moves us closer to our goals. Mastering this skill would have a tremendous impact on my life and my business. It also opened my mind up to a real sense of what is possible.\n\nI remember having lunch with Tony in Seattle with our team. I opened up to him about the accident I had had in Mexico and how terrible the experience had been. I told him I had felt like I had to throw everything away and relaunch. But so much good had come out it: new experiences, a more powerful way of thinking, a sense of unlimited possibility, a really solid foundation for business going forward, and great new friends.\n\nHis response really put the experience in perspective. \"You have good days and bad days,\" he said. \"But you don't know which is which until someday way down the line, because you don't know what you will make of that experience.\"\n\nOver the years I've found this to be one of the most important lessons I have learned in life and business. What has happened to you in the past matters much less than what you learned _and_ how you apply each lesson going forward.\n\nAs an entrepreneur, you too can overcome your wreckage. Everyone has a story and has experienced setbacks in his or her life. It could be that you had a business that failed or that you were laid off. Maybe you were hurt financially, filed for bankruptcy, or lost a house. Perhaps you suffered an accident or were in poor health or had your heart broken. Just know it's not the experience; it's what you do with the experience. Remember, nothing in life has any meaning other than the meaning you give it. One of the keys to success is taking every experience life brings you and repackaging it in a way that empowers you and helps you to move forward.\n\n### Small Is the New Big\n\nIn big business, change used to be linear. A company could look several years down the road and plan ahead. The cost of launching a business to compete with entrenched players was beyond the reach of many would-be entrepreneurs. As a result, the lack of innovation and competition meant big companies could move at a snail's pace.\n\nThat was yesterday. Today change occurs at an exponential rate, and the pace of innovation is unparalleled. The cost for an entrepreneur to jump into a market and compete head to head with industry giants has dropped dramatically and no longer is a barrier to entry. The Internet, mobile technologies, and a virtual workforce have taken away many of the advantages once held by those with deep pockets.\n\nThis shift represents a big problem for big business: Deep-rooted cultures at most large companies make it difficult for them to keep up. Those that remain inflexible and unwilling to change the way they approach today's competitive world will die from the inside out.\n\nThis shift is good news for today's flexible entrepreneurs, who can move faster, get closer to customers, respond more quickly to changes in the marketplace, and operate their businesses for a fraction of the cost. These entrepreneurs don't resist the pace of change. They embrace it. In the new world order, small is the new big.\n\nHere's how fast a small company can move in and disrupt an entire business.\n\nUnder the previous paradigm, a company created a product or service. It was a hit, and over time the company got bigger. It continued to add new features and started raising prices. The company could do this because the demand was there and it had little competition.\n\nThen one day, a customer looks at his credit-card bill and says, _I don't want to pay all that money anymore._ The customer also realizes he needs only one thing, actually, the original thing he signed up for. _I don't need all this other stuff,_ he says. _And I certainly don't want to pay for what I don't want._\n\nSo the customer decides to start his own company. He plans to strip out all the fancy features he does not want to pay for and build one thing, the very same thing he currently does like in the big company's product. This aspiring entrepreneur might be eight years old or eighty; it really doesn't matter. He quickly finds a free legal service online and incorporates, sets up an account with Google to get free e-mail, contacts, calendar, office applications, and document storage. He gets free voice and video calling through Skype, free WiFi at the corner coffee bar. The entrepreneur syncs everything between laptop and mobile phone.\n\nThe business is established\u2014in an elapsed time of, say, _ninety minutes._\n\nNext our entrepreneur gets down to business and goes online to do some research. A few Internet searches produce market data and access to free reports. By poking around social-media platforms like Facebook and Twitter, he can quickly find out if others think his idea has legs. Our founder logs on to LinkedIn to share the news of his new venture and seek some help. A developer who likes the idea and hates big companies volunteers to do a little coding. The new team collaborates, using free software to write a business plan.\n\nTo sum up, a customer of a big business can identify a problem and establish a company to meet that need in one afternoon. He has access to much of the same market information as the competition. Social-media conversations lead to the drafting of a business plan and the assistance of a developer who starts building the product. These people are executing\u2014on the first day\u2014and they've spent zero dollars so far.\n\nThis new little company builds something really basic, a minimum viable product with one feature and one benefit. What they release is pretty crude but\u2014check this out\u2014they give it away for free. Other people who have the same problem with the same big company's product flock to the new business. The big company is too slow to recognize what is happening and therefore too slow to respond with an alternative solution. It bleeds market share to the start-up. And a new entrepreneur is launched, with a product and a team and a market, the building blocks of a multimillion-dollar business.\n\nOur theoretical business happens to be a technology company, but the process is the same across the board. It all started with one unhappy customer who had a laptop, a mobile phone, an Internet connection, and a social-media account. One customer who decided to take matters into his own hands and ultimately launched a better solution.\n\nWhen big, entrenched industry leaders observe this happening from their spacious headquarters, it's terrifying because it's definitive proof that the scales have tipped. Even their best ideas may not work because their companies will take too long to execute.\n\nThis is an opportunity for today's entrepreneur, but it's also the reason I consult with businesses large and small all over the world, teaching them how to create a more entrepreneurial culture. Does your business culture allow for the kind of fast action required to compete in the entrepreneur economy? If not, you're in deep trouble. Today more than ever, nimble creatures adapt while the dinosaurs die off.\n\nIt reminds me of a conversation I had on Necker Island with Richard Branson while attending a conference at his Caribbean home (on his own island, in fact). I told him that my whole life had been lived in launch mode, opening new markets where we had to educate customers to their needs at the same time we were selling to them. I pointed out he had often taken the opposite approach. \"You go into the biggest markets with the most entrenched customers and go head to head,\" I said to Richard.\n\nAs usual, Branson's response was enlightening. He said, \"I go into big markets where people have already proven they want the product and will pay for it. Their wallets are already open. Then I just look for one person that has a problem. Find that one person and one problem, and if the market is big enough, that represents a big business opportunity. And that's what I do.\" Even though Branson's companies were making huge moves, he still thought like a smaller entrepreneur, creating new businesses to solve one problem better than the current companies in that space.\n\nToday it's your turn to take advantage of the opportunities that entrepreneurs have today over big business, to disrupt and create innovative products and services that have a positive and lasting impact. Now that we've established some key beliefs about entrepreneurship, let's talk how we'll embark on this launch process together.\n\n### A Playbook, Not a Textbook\n\nI hate textbooks. I don't think I've ever made it through one\u2014too long, too boring, packed with too much information that doesn't seem relevant.\n\nSince my goal is to give you everything you need to take your big idea to market in ninety days or less, this book is designed more like a playbook than a textbook. I've taken big concepts and broken them down into smaller, more digestible pieces.\n\nIn this book I'll share my personal experiences launching new businesses, products, and services. These experiences range from being an entrepreneur inside bootstrapped start-ups to being an executive inside some of the world's largest global brands. These stories will provide lessons that can be applied immediately, as we move quickly from the personal side of entrepreneurship into the strategies and tactics necessary to bring your vision to life.\n\nIn each chapter we will focus on one concept or action, the single most important thing you need to do in that area. With so much on your plate, asking you to do more would only slow you down rather than lead to a better result. What I know from experience is that if you commit to each lesson and follow through on each exercise, you will set yourself up for a successful launch.\n\nSince we only have ninety days to get you to market, this book is designed to be read and implemented quickly. That means you should be able to get through each section in under five minutes.\n\nWe're going to move fast, but before we dive in, I want to make sure you have the right mind-set for making this leap. If you're reading this book, you're either interested in launching a new venture now or preparing to do so in the very near future. It doesn't matter if you are an entrepreneur or a manager of a big business who needs to think quickly on your feet in a fast-changing industry. You need the skills and lessons in this book to put you in the best position to succeed. So let's begin by eliminating a few negative beliefs that may be holding you back from launching your new project, and offer new, empowering ideas to put in their place.\n\n_Don't worry about all that talk about a slumping economy._ That's nothing more than a distraction. Too many people will use it as an excuse. Despite anything you may have heard, there's no shortage of opportunity out there. The key to finding success as an entrepreneur is seizing an opportunity and, when you find it, executing like crazy.\n\n_There's never a bad time to start a business_. Great fortunes have been created during good and bad times, and from some very unusual places. Sara Blakely (you'll meet her later) instinctively understood she had a big idea when, at age twenty-six, she decided to cut the feet off a pair of pantyhose. She didn't let her bank account or lack of industry experience hold her back. Today, at age forty-two, Sara retains full ownership of her undergarment company, Spanx, and last year _Forbes_ magazine declared that she was a billionaire.\n\n_Value your insights._ Plant this little lesson from Blakely's success in your mind: A unique insight into what the consumer wants, backed up by focus, discipline, and hard work, can and will pay off.\n\n_Age doesn't matter._ You can't be too young or too old to launch a successful enterprise. Nick D'Aloisio, the British inventor who sold his mobile application, Summly, to Yahoo! for a reported thirty million dollars, did so at age seventeen, but the average entrepreneur who earns one million dollars (and does not lose it) doesn't get started with his or her successful business until age forty-one. Many don't make their first million until they're fifty or older.\n\n_Ten thousand hours? Not required._ Forget the idea that you need to spend ten thousand hours doing something in order to become really successful at it. That makes great reading in Malcolm Gladwell's _Outliers,_ and it might well apply to artistic skills like violin playing or painting or to athletic prowess in basketball or tennis. But it doesn't always apply to business.\n\nReed Hastings had virtually no video experience when he founded Netflix. Janus Friis and Niklas Zennstr\u00f6m founded Skype despite having very little background in telecom. What did Richard Branson know about airlines or mobile phones? Nothing. Larry Ellison, Mark Zuckerberg, Mark Cuban, Jeff Bezos, Larry Page, and Sergey Brin all perfected something about which they began as less than experts\u2014and became billionaires in the process.\n\n_Don't worry about being second in the marketplace._ Entrepreneurs do not operate in a vacuum but rather in a diverse ecosystem. What one innovator creates opens up opportunities for the next generation of go-getters. The more disruption, the more opportunity. If you see a product or service out there that people already use, but you have an idea that would improve upon it in any way, you should be encouraged, not discouraged, by someone else already playing in that space and proving there is real demand for it.\n\n_Don't be afraid to try again._ If you've tried to launch a business before, but things went sideways instead of up, you are not alone. Having taken a shot at entrepreneurship in the past, you'll recognize elements in this book that you missed the first time around. Maybe tough economic conditions, changes in the workplace, or unemployment put you out of position, but now you're ready to blaze your own path. Perhaps you have an existing business that you want to take to the next level. Wherever you are, upon reading these pages you'll find lots of helpful little things, distinctions you can make to identify new directions that will help you grow and scale your company.\n\nHOW TO TURN AN INDUSTRY ON ITS HEAD\n\n**Start with One Person:** In a big market, look for one customer who already has his wallet out but is unhappy about one thing in a given product or service. The odds are that, if the market is big enough, lots of people share his view. Fix that one thing.\n\n**Make Things Simple:** The more complex the common practices are in your industry, the greater is the opportunity to disrupt them by simplifying them. It may not be easy, but if it were, everyone would be doing it. Solving the hard problem makes a great opportunity.\n\n**Slash Prices Dramatically:** If you use technology or other systems that significantly reduce your costs relative to the competition, pass those savings on to customers. Entrenched players may not be able to compete because their infrastructure does not allow it.\n\n**Get Close to Your Customer** **:** Build a better relationship with your customers than anyone else in the market has. Translate that relationship into new products and services that better address their needs and desires.\n\n**Be You** **:** Be authentic in everything you do and transparent to the marketplace.\n\n### Find a Way to Simple\n\nLet's tackle one more story before we get started. I credit this one to Stanley Kirk Burrell.\n\nA few years ago I attended a conference on the Big Island of Hawaii. The organizer, David Hornik of venture-capital firm August Capital, calls the invitation-only meeting \"the Lobby.\" Almost everything leading up to the arrival was secret; no one knew who else would be there.\n\nOn the first day the organizers issued the participants different-colored T-shirts, and we were told to go find other people wearing the same color. These people would be our team for the day. Mine consisted of Dick Costolo, now the CEO of Twitter; Sarah Lacy, the founder of PandoDaily (the leading Silicon Valley technology news site); and Bill Tai, a leading venture capitalist. But we were missing one person from our team. Looking around, we finally found our man: Stanley Burrell.\n\nAka MC Hammer. Rapper. Dancer. Actor.\n\n_And_ entrepreneur. Don't think for a second that the legendary rapper was invited to the event for entertainment value. Hammer regularly attends these industry gatherings and has invested in notable tech start-ups, including Square and Bump Technologies. He's lectured at Stanford and Harvard on the uses of social media and has a promotional deal with iPad case maker ZAGG. He's also launched businesses of his own. In 2007 the creator of hits including \"U Can't Touch This\" and \"2 Legit 2 Quit\" launched DanceJam.com, a (sadly short-lived) YouTube-like site for dancers to share videos.\n\nNeedless to say, I was psyched to be on his team.\n\nWe all climbed into a big white van. Our job was to go on a scavenger hunt, competing against other teams to be the first group to solve a series of puzzles. We were a savvy bunch, and I could sense the competitive energy surging. We got to the first site and read the clues. Our task was apparently to take a pile of rocks, a couple of sticks, and a piece of rope and turn them into something. Dick and Sarah and Bill and I looked at the materials, then looked at one another. We were confused and couldn't figure out how to solve the problem. And Hammer wasn't helping. He stood a short distance away, talking on his cell phone, looking at the ocean.\n\n\"Hey Hammer, can you help us out?\" we asked.\n\n\"Yeah sure, give me the clue,\" he said.\n\nHammer took one look at the clue, thought about the problem for a minute, then proceeded to stack one item on top of another, completing the assignment. Problem solved and task completed.\n\nWe went to the next clue. The same thing happened\u2014four of us working hard to solve the problem, Hammer talking on his phone and gazing at the ocean. When we asked him for help, he walked over, took one look at the clue, and again solved the puzzle. We moved on to the next clue. Guess what happened? He solved it again, quickly. I think Hammer solved every problem on the scavenger hunt for the team that day.\n\nWe didn't end up winning (we found out later the other groups were getting ahead of us because they were sharing clues on a new service\u2014Twitter\u2014cofounded by a fellow participant in the Lobby, Evan Williams). But later at the hotel, a little bit in awe, I went over to Hammer and asked him how he had managed to solve every problem so quickly. What was his secret?\n\n\"The problem with you entrepreneurs,\" he said, \"is that you get all caught up in the details. You start off with these big goals but you make things complicated, more complicated than they need to be.\"\n\nHe went on. \"I have a philosophy in life and business,\" said Hammer. \"I find a way to simple first.\"\n\nThat was a big moment for me. When I think back on it, I associate his insight with the paper clip. The Post-it note. The Band-Aid. The toilet-paper roll. Scotch tape. The BIC pen. The disposable razor. Spanx. And a hundred\u2014no, a thousand\u2014other products that have one thing in common: They're incredibly simple. And they were invented only in the last 150 years. Look around your home or office. You're surrounded by simple inventions that are easy to use, that solve one problem and solve it perfectly well.\n\nThe same is true with businesses: Simple businesses are thriving everywhere. Coffee shop. Sandwich shop. Dry cleaner. Lawn service. Bar. Taxi service. It doesn't have to be complicated. Every day entrepreneurs take a tried-and-true business idea, apply some differentiation and branding, and _voil\u00e0!_ Of course, there are other factors\u2014the right location or the know-how to make awesome coffee or sandwiches\u2014but the point is that there are thousands of business owners out there making a great living by doing simple things well.\n\nTake Stanley Burrell's brilliant insight as an order, a compulsion, a rule for your life. It should be a rule for your whole business, truly: Your vision, operating plan, and financial model need to be so succinct and simple that you can fit them all on one page or explain it to someone in thirty seconds.\n\nNarrow your focus. Narrow the problem. Narrow the customer. Narrow the geography. Get the whole idea as simple as you can make it, and work out from there.\n\nRemember: Find a way to simple first.\n\n# STAGE I.\n\n# THE PRELAUNCH CHECKLIST\n\nThirty Days to Prepare\n\n## CHAPTER 1.\n\n## WHAT'S YOUR BIG IDEA?\n\nMost people fail in life not because they aim too high and miss, but because they aim too low and hit. And many don't aim at all.\n\n\u2014LES BROWN\n\n### How to Buy a Rainforest\n\nI hate mosquitoes. I hate everything about them. But a few years ago I found myself on my way to a place named . . . Moskito Island.\n\nWith night falling, the Virgin Group's global leadership team and about forty Virgin CEOs from around the world climbed into small boats. We were in the Caribbean for \"the Gathering,\" as Richard Branson called the event, to share and exchange information and grow from one another's experiences. We shoved off from Necker Island, which has been Richard's private retreat for about twenty years, heading toward the nearby island he had just purchased.\n\nUnlike Necker Island, Moskito is no resort. It's completely deserted\u2014there's nothing on the lump of land\u2014and by the time we got there, the sky was pitch-black. We were on the water about as far east in the Caribbean as we could be. As we approached the island, no lights greeted us until we circled around to the opposite side, where tiki torches flickered on the lone strip of beach.\n\nWe went ashore and grabbed dinner at a small buffet. Looking for a place to sit and eat, I came across Richard and a group of Virgin executives eating and talking passionately about something that Richard apparently wanted to purchase.\n\nI sat down on the beach next to him and chimed in, asking what was it he wanted to buy.\n\n\"The rainforest.\"\n\nI asked which rainforest.\n\n\"The Amazon,\" he replied. (I guess if you're going to buy a rainforest, it might as well be the biggest one!)\n\nI was shocked. How can you buy a rainforest? Knowing Richard, I can't say I was totally surprised he would think so big but still, my mind reeled as I thought about this proposal. I mean, who even owns the rainforest? Whom would you buy it from?\n\nThe talk continued and swirled around the positive impact you could have if you controlled the rainforest. Richard pointed out that you could stop the slashing and burning of vegetation, save precious animals that are being killed or taken out of their environment for sale, and help slow global climate change. These are issues that Richard has been passionately dedicated to for years. He's donated millions for environmental protection and has even teamed up with alternative medicine researcher Chris \"Medicine Hunter\" Kilham to tour the Amazon rainforest in search of indigenous pharmaceutical plants that may be cultivated someday in the jungles (by the people who are now destroying them) and sold through the Virgin Group. On Necker Island, Branson has also done his part to protect rainforest animals, creating a sanctuary for endangered Madagascar lemurs.\n\nSomeone finally interrupted this passionate conversation and said what I was thinking: \"You could never buy the rainforest.\"\n\nRichard got this look in his eyes that I'd never seen there before. Looking back, I remember seeing the same look in the eyes of other global leaders. It said: _How can you get in the way of possibility?_ I've come to understand that a positive and resourceful outlook is absolutely essential for any entrepreneur to turn his or her big idea into a reality. And by that I mean, entrepreneurs need to understand the key difference between _having_ resources and _being_ resourceful.\n\nNone of the successful people I know started with everything they needed; they had to assemble what they required every step of the way. They understood\u2014as Richard did, as does Tony Robbins\u2014that you can back your way out of any problem that stands between you and your dream.\n\nLook at it this way. You have an idea. No matter how big or small, the idea lives inside you, it drives you, it keeps you up late at night. Naysayers may think it impossible, but that's just because they've never seen it done before. Or maybe you believe something's possible but you're not convinced you're the one to do it. It may take capital and people and other things you just don't have. But in the world according to Branson, those are just obstacles to be overcome.\n\nBack at the beach on Moskito Island, Branson responded to the challenge of what I thought was an impossible task, buying the rainforest, by walking us through a simple exercise.\n\n\"First of all,\" Branson said, \" _imagine_ you wanted to buy the rainforest. How would you do it?\" He started peppering us with questions.\n\nDo you have to buy the rainforest, or can you lease it? Whoa! That is a huge shift in mind-set! Let's say we could lease it. How many people do we know with significant wealth who could contribute to leasing the rainforest and, more important, get behind our cause? Better yet, think of how many people in this world would do whatever they could to contribute, even a small amount, to such a noble cause. Coming up with the money should not be a problem.\n\nHe continued to probe. Each answer inspired a deeper belief and greater confidence that it could actually be done. Within five minutes, Richard Branson made buying the rainforest sound easier than eating a bowl of soup. That's partly the difference between Richard Branson and everyone else. Most of us have yet to open ourselves up to the same sense of possibility. But _Richard_? Given enough time and the right people on the task, there's nothing that he\u2014or, when I thought about it, any one of us\u2014can't accomplish.\n\nFive years later I saw that Richard was really on to something. Ecuador launched a novel initiative to protect the Yasun\u00ed Park rainforest, a place that is thought to be the single most ecologically diverse place in the world, featuring more individual species of trees in one spot than are present in all of North America, not to mention bats and birds, amphibians and reptiles. But this wasn't a simple situation where the Ecuadorians could afford to do the right thing and set the place aside for posterity. Beneath Yasun\u00ed Park are massive deposits of oil, worth billions of dollars, awaiting exploitation.\n\nSome Branson-style creative thinking found a possible solution. In 2007, to protect Yasun\u00ed yet still bring in money for Ecuador, the government tried to effectively lease the rainforest. If donors from around the world agree to pay $3.6 billion dollars, Ecuador would promise not to allow drilling in Yasun\u00ed. Hundreds of millions of dollars were raised to compensate the government for lost revenues, thanks to monies from governments and individuals around the world, including Al Gore, Leonardo DiCaprio, and a host of environmentalists and scientists. Although the Yasun\u00ed-ITT Initiative ultimately was unable to raise enough money to meet their goals, it's clear that crowdsourcing environmental protection did catch some significant attention and could still do the future of the planet untold good.\n\nRichard Branson's thinking that evening on Moskito Island left me deeply impressed both by the intensity of his conviction\u2014and because he made the seemingly impossible sound easy. He took a big problem, one that seemed almost larger than life, and chunked it down into smaller, more manageable pieces. He looked to back out of a big idea, one step at a time, to reach a goal. Because he was so open to possibility, he asked better questions of the group, eliciting more productive responses and solutions.\n\nRichard is far from alone in setting lofty goals and doing something to accomplish them. There are dozens of examples of entrepreneurs who have tried and succeeded with revolutionary ideas that at first seemed so big as to be impossible for mere mortals. They saw openings, came up with solutions, and applied them. Sometimes the solutions seemed crazy to most people, but the entrepreneurs didn't let that stop them.\n\nIndeed, what you'll notice in looking at the history of capitalism is that while there are myriad ways to make a fortune, the real megafortunes are held by the innovators. These folks not only think outside the box but don't even recognize that there was a box to think outside of.\n\nMany of the most successful entrepreneurs of our age achieved their success because they were the first to grasp the potential applications of new technology. Bill Gates became Bill Gates because the mothers' club at his school bought a computer with the proceeds from a rummage sale. In eighth grade young Bill started writing code when virtually no one else his age even knew what a computer was. He saw this as an opportunity: If only he could make it easier for the average person to use one of these machines, they could be very useful. MS-DOS was born.\n\nSteve Jobs saw that people loved personal computers and avidly used Bill Gates's operating system\u2014but he found that not only were the machines clunky, but the Microsoft software was too. So he created his own hardware and software with Apple Computer. Twenty years later he applied the same principles to his game-changing iPhone.\n\nA hundred years ago John D. Rockefeller saw that crude oil was being discovered all over the place but realized the guys with the oil had no way of efficiently getting it to market. He built a pipeline network, and Standard Oil (the predecessor of ExxonMobil, Chevron, and several other major oil companies) was born.\n\nIf you think about it for a second, all the great dot-com fortunes were created in the same way, but instead of being inspired to move oil in a new way, these innovators figured out how to move data over fiber-optic cables. Jeff Bezos thought it would be pretty cool to be able to buy stuff over the Internet. Amazon.com was born. Mark Cuban started Broadcast.com because he wanted a way to stream college basketball games over the Internet. He sold it to Yahoo! for $5.9 billion. Mark Zuckerberg knew students at Harvard wanted a way to connect with one another over the Internet. Facebook was the result. Larry Page and Sergey Brin were frustrated by how hard it was to find what they were looking for on the Internet. Enter Google.\n\nAll these entrepreneurs had breakthroughs that began as either \"crazy ideas\" or improvements upon what someone else had tried. They did it first, or they did it better.\n\nSo the next time someone throws out a wacky idea\u2014say, printing books on a press, using electricity to light a house, broadcasting radio waves, watching moving pictures on a television, surfing the Internet, developing commercial spaceflight, or buying a rainforest\u2014know that there is someone out there who thinks with no limitations, who's assembling the right resources and is intent upon making his or her dream come true. Of course, that person might be you.\n\nHOW TO BRING YOUR PROJECT TO LIFE\n\n**Get Clear:** What is your big idea? What is keeping you up at night? What would you pursue if you knew you could not fail?\n\n**Break Your Big Idea into Small Pieces** **:** No matter how big the vision, you will always improve your chances of success by breaking the job down into smaller bite-size pieces and attacking them one at a time.\n\n**Be Resourceful** **:** Nobody starts out with everything they need to succeed. The best entrepreneurs are resourceful enough to pick up the pieces along the way.\n\n**Surround Yourself with Great People:** Find role models at every step of the way who have accomplished what you set out to achieve. Find out how they did it: Learn their beliefs, their strategies, and the order in which they did things.\n\n**Execute Like Crazy:** Outwork, outthink, and outsmart the competition.\n\n### Oil Right Under Your Feet\n\nBetween the time I spent in the training business and my breaking into technology, I got stuck in transition. In the early 1990s I worked at a bar and lived near the beach in Santa Monica. I was really struggling to find my way, wondering what direction I should take next.\n\nI had always liked creating and building things, so I decided I wanted to build companies next. The possibility of making a fortune while making a difference was attractive, but I didn't know where to get started. How many times in your life have you been ready to make a move but couldn't seem to find the right opportunity? I kept asking myself, how was I going to make an impact? How was I going to put my ambition to work? I looked for the right place to plug in, but nothing seemed to fit.\n\nThen a friend gave me a book that helped set me on my entrepreneurial path. Called _Losing My Virginity: How I Survived, Had Fun, and Made a Fortune Doing Business My Way,_ it was the autobiography of a British guy\u2014yes, that one\u2014named Richard Branson.\n\nIf anybody had told me then that I would actually work with him someday, I would have been thrilled. The book inspired me in many ways and helped me think about my future from a whole new perspective. _Losing My Virginity_ got me to try new ideas, to take a few chances, and to put myself into a better position to discover my Big Idea.\n\nThe first conclusion I reached then (this was twenty years ago) was that the future was _China_. I read a lot of business books, and everybody seemed to be talking about business in the Far East. I decided I would enroll in a class to learn the language and look for opportunities to work with Chinese companies. Now, I think maybe I got hooked on the idea of China because I couldn't seem to find an opportunity near me. At the time, California was in a deep economic recession and I figured I'd have to go thousands of miles away\u2014literally to the other side of the globe\u2014to find a place where I could make money.\n\nPretty smart idea, I thought, but soon I discovered the truth of that old saying that sometimes being lucky is better than being smart. In a piece of accidental timing, I was introduced to a man named Al Checchi just one week before I was scheduled to take my first Chinese class. He had worked on a business deal with my mother and she shared with him how hard I was trying to find my way. He offered to help. That meeting changed the course of my life once more.\n\nAl had started off working for Bill Marriott, helping to build the Marriott hotel empire. In the 1980s he had moved on to work as a deal maker for the billionaire Bass brothers of Fort Worth, Texas. The Basses had made their money in oil and then diversified in all sorts of directions. One of their biggest holdings was a controlling stake in the Walt Disney Company, a deal that Checchi had personally overseen. At the time, Checchi reported to legendary moneyman Richard Rainwater, who eventually became a billionaire in his own right.\n\nWhen I met Al, he was cochairman of Northwest Airlines. He asked me what was I going to do with the rest of my life. I told him my future certainly wasn't in California. Instead, I explained, my destiny was in China, because that's where all the opportunities were.\n\nAl listened, looked hard at me, and then started talking. He told me that years earlier, just like me, he had struggled to figure out his path, imagining himself running all over the world pursuing lots of different projects. But his boss at the time had told him a simple story about a man who had been on a similar quest.\n\nThe guy spent all his life and his entire fortune searching for diamonds. Eventually he ended up in Texas. Since there's a lot of very old rock in central Texas, he felt pretty confident of a big score. The prospector hunted and hunted and dug and dug on the land he'd purchased. After weeks in the Texas sun, the back of the man's neck was tanned as dark as saddle leather, and his aching back would never be the same. But still no diamonds. Finally the man decided to give up and left town. He was so disappointed that, in a moment of anger, he sold the land for a dollar.\n\nThe buyer of his land, Al told me, was an oil wildcatter. This guy had a different hunch about what was to be found in that earth. He proved himself right when, drilling the very first well, he struck oil.\n\nI'm not sure he needed to, but Al gave me the moral of the story. \"Everything you're looking for is right here,\" he told me. _\"It's right under your feet.\"_\n\nThere is nothing like having a great mentor to point this out to you, a person who has already been down the road you are traveling, someone who is not as close to the situation and therefore has perspective. Like the diamond prospector, we can have such a fixed focus that we can't look at the bigger picture in a new way. Sometimes we're just not open-minded enough to recognize the opportunities around us, but they are there, every single day, in every direction you look, globally, locally, everywhere.\n\nYour job is to look again. You probably don't have to go to the Great Wall, because the answer may be closer than you think. As for me . . . it turned out Silicon Valley was just up the road.\n\n### Are You Sitting on Top of a Billion-Dollar Idea?\n\nShe bought a new pair of cream-colored pants, planning to wear them to a party. But when Sara Blakely, a twenty-six-year-old Floridian, put them on and looked in the mirror, she didn't like how her butt looked. Panty lines ruined the whole effect, and she needed to find some kind of shape wear to smooth them over, fast. So she cut the feet off a pair of pantyhose, put the cream slacks back on, and headed out to the party.\n\nLittle did she realize at the time that she was about to tap into tremendous entrepreneurial opportunity. Even though her first little experiment didn't provide the perfect solution\u2014the hose kept rolling up her legs that evening\u2014it kicked off a process. Plus, she really, _really_ wanted her invention to work. Her moment of inspiration, born out of a combination of necessity and vanity, became her Big Idea.\n\nSara, you see, was a born entrepreneur. As a youngster she created haunted houses and charged neighborhood kids admission. As a teenager she set up an unlicensed, unapproved babysitting business at a nearby Hilton Hotel, watching kids for eight dollars an hour. She was a cheerleader and a debate champion, but one with an especially strong desire to succeed on her own terms. She listened to tapes her dad had lying around, including Dr. Wayne Dyer's _How to Be a No-Limit Person._ She played them so many times she practically memorized them.\n\nFor a time, not everything in her life fell into place as she hoped. After trying and failing to get into law school, Sara found jobs selling office supplies, even hustling fax machines door to door. Then she learned how to cold-call and realized she was a natural salesperson, with an ego impervious to rejection. At age twenty-five, she became her company's national sales director. Her life was starting to get on track\u2014and then those cream-colored slacks came along and changed everything.\n\nSara was convinced there was a way to make her idea work and she did. With five thousand dollars saved up, she invested in her vision for the future of underwear. By day she worked full time selling office supplies; at night she researched patents and studied fabrics. However, she was turned away by numerous stocking manufacturers, who weren't about to take the time to help her make a prototype.\n\nWhen she finally figured out how to keep the abbreviated hose from riding up her legs, she saved money and wrote her own patent. She also found a factory willing to produce the garments; its operator was a man whose daughters thought Sara's idea so great they forced their dad to make it. Sara came up with a name for her big idea too: Spanx _._\n\nShe designed her own flashy packaging and set about selling her new product. Her wares went on sale at upscale department stores like Neiman Marcus, Bloomingdale's, and Saks. When she didn't think Spanx was getting prominent enough placement at Neiman Marcus, she bought her own display rack, snuck it in, and set it up by the cash register.\n\nArmed with all her chutzpah and desire, Blakely got a break that even she couldn't have dreamed of when, in 2000, Oprah Winfrey touted Spanx as her favorite product of the year. The orders came rolling in, and Sara finally quit her day job.\n\nSara cracked the code of entrepreneurship and made it work for her: She took a perceived need and, with her business instincts, made it a business. Her story has some lessons for all of us. Consider these:\n\n_Think big_. From the start, Sara's vision was to sell to _millions_ of women. Remember that the size of your vision will help determine the size of your success, that almost every franchise, big brand, and major product began in the mind of a single entrepreneur.\n\n_Embrace failure._ From an early age, Sara put herself out there\u2014cold-calling, selling door to door, and learning how not to take rejection or failure personally. Failing is an integral part of being an entrepreneur. Accept it, use it to your advantage, and move on.\n\n_Leverage technology._ When Sara started Spanx fifteen years ago, she had to use the Yellow Pages to find someone willing to make her prototypes. Today you can research everything on the Internet. Technology can help in other ways too; the advent of 3-D printing, for example, has made it very easy to produce prototypes of many consumer goods.\n\n_Be inspired._ Spanx made $4 million in its first year; today the company has sales of $250 million a year. Sara still owns 100 percent of her business and in 2012, at age forty-one, she was named by _Forbes_ the youngest self-made woman billionaire in the world. To do this she didn't need to reinvent the wheel. And you, whether you're an entrepreneur or an intrapreneur (fighting inside a big organization), don't necessarily have to either. Sometimes a key improvement or iteration can change everything.\n\nThink about the return on Sara's investment: from five thousand dollars in savings to a billion dollars. She had been looking for an opportunity, working hard to find it, and all that time she'd been sitting on her big idea.\n\n### Let Go of Your Nut!\n\nWe all have our reasons for not getting started. Unlike Sara, who was ready to jump right into entrepreneurship, some of us may be more hesitant to take the leap. As a result, we create excuses and put up roadblocks for ourselves. Do any of these sound familiar?\n\nIt's too late.\n\nI'm too tired. I'm hungry.\n\nI have to feed my kids.\n\nMy boss would never let me do it.\n\nThe dog needs to be walked.\n\nI can't afford it.\n\nThe project's too big.\n\nPeople will think I'm crazy.\n\nI can't do that and do the job I have.\n\nI don't know the right people.\n\nI didn't go to the right school.\n\nThe company would never get behind this.\n\nHow will I get paid for it?\n\nI'll just start tomorrow.\n\nAnd the biggest one of all?\n\nI'm afraid.\n\nUnfortunately, our fears, combined with the crazy stories we tell ourselves about why we can't succeed, keep us from getting started.\n\nIn order to bring your big idea to life, the first step I want you to take is to let go of all your excuses. Simply step away and get on with it.\n\nLet me put it another way: _Don't be a spider monkey._\n\nThe spider monkey may be the most elusive animal in the rainforest. It's so quick, you can't sneak up on one of these wily little forest creatures with a net. It's way too fast for that. But best-selling author and speaker Greg Reid tells a useful story about how to successfully trap the spider monkey by playing to its weakness.\n\nThe spider monkey possesses the human tendency to grab on to things that he thinks he can't live without. Then he refuses to let go.\n\nGreg describes the strategy of the successful spider-monkey hunter. All he needs to do is find a log in the rainforest, drill a hole about eighteen inches deep, and drop a couple of peanuts down the hole. That's it. In time, a spider monkey will happen along and smell the delicious peanuts. He'll reach his arm down the hole and grab them. But the hole is sized such that, when the monkey closes his hand around the peanuts, the monkey's clenched fist gets stuck in the hole. He could open his hand, drop the peanuts, and pull his hand out of the hole. But he can't do both. He can't get the peanuts and get free.\n\nLong ago, native hunters discovered that they could drop the bait in the hole, then just wander off for an hour or two. When they came back, all they had to do was drop a net over the stubborn monkey. The monkey was so obsessed with those peanuts that he just kept his hand in there, holding on tight, even as the hunter closed in.\n\nMy advice to you: _Let go of that nut!_ Don't be that monkey. Let go of those fears, excuses, and stories you have made up about why you can't succeed. Let go of those things that hold you back.\n\nMaybe you're holding on to a fear of failure or a regret from the past. Maybe it's a fear of the unknown. Maybe it's a dead-end job you can't bring yourself to walk away from, a bad relationship, destructive habits, or the wrong kind of friends. Got anything you want to add to the list?\n\nLike that monkey, we believe what we're holding on to is more important than anything else, maybe even that it's going to save us. Ultimately, this thinking will just lead to failure and frustration.\n\nSo quit holding yourself back. Stop focusing on reasons and start focusing on results. Beethoven went deaf, but among his late-in-life works is some of the greatest music man has ever heard. Milton was blind, but he wrote literature that will be read until the end of time. Whatever your excuse is, it isn't good enough. The only thing that's keeping you from having everything you want is the story you keep telling yourself about why you can't have it.\n\nSo drop the peanut. Just let it go. Then try climbing a tree to see what opportunities lie on the other side.\n\nLETTING GO\n\nWhat would happen if you decided to let go? How would your life change if you really decided to go for it? What would happen if you made your biggest dreams come true?\n\nWould you spend more time with your family? Would you pick up a hobby? Would you buy a new home? Would you travel around the world and check items off your bucket list? Would you create a charitable organization that gave back to others?\n\nThe only thing keeping you from achieving everything you want in your life is the crazy story you've made up about why you can't have it. Now is the time to seize your opportunity and create the life you deserve.\n\n## CHAPTER 2.\n\n## PROTECTING WHAT'S MOST IMPORTANT\n\nMy mama always used to tell me: \"If you can't find somethin' to live for, you best find somethin' to die for.\"\n\n\u2014TUPAC SHAKUR\n\n### The Sure Thing\n\nHave you ever had a \"sure thing\"? An idea for a business or product that was just so good, you knew it could not fail? When you shared the idea with others, their jaws dropped. As you got started, everything fell into place: Your timing, your team, and your execution were flawless. You _knew_ you could not lose. So you doubled down; maybe you put everything you had on the line.\n\nI've been there as well. In 2005 I was closing in on a deal to run a new Internet business when I got a call from my good friend Jason Port. He had heard I was in the market for a new business opportunity and encouraged me to take a look at private aviation.\n\nAfter 9\/11, he told me, private aviation's growth had skyrocketed. Passengers on commercial flights had grown frustrated with increased travel times, long lines at security, having to remove their shoes, and the rest of it. Private-jet charter had suddenly emerged as a terrific option for business travelers. He also told me that, despite its growth, private aviation remained highly inefficient, so there might be a good opportunity for skilled Internet entrepreneurs to help the industry operate more efficiently.\n\nI looked at the charter business. The numbers showed growth in users and revenue like I hadn't seen since the dot-com boom, but the business was highly fragmented. In commercial aviation there were a handful of operators, like American, Continental, United, Delta, and Southwest, that controlled virtually all the passenger seats and owned hundreds of aircraft. Private aviation was completely different; there were approximately 2,500 operators in the United States alone, typically controlling only three planes each.\n\nWhen it came to selling seats for commercial flights, companies like Expedia, Travelocity, and Orbitz could easily connect travelers to commercial flight schedules, making it simple to search, find, and purchase a trip online. In contrast, just as my friend had told me, the private-aviation industry was incredibly inefficient. Nobody had effectively aggregated the supply and demand in one place to make it easy to purchase a trip on a private jet. To book one you would have to find a local operator through word of mouth, by looking them up in the phone book, or by calling the local airport. Once you got a charter operator on the phone, he would then have to create a custom quote for your trip, oftentimes initiating a back-and-forth of faxes. The process was made tougher still because the guy answering the phone was sometimes the guy flying the plane!\n\nAs a result of my research, I decided to enter the market and tackle this problem. I started Smart Charter in 2006 as an online marketplace and booking tool for private aviation. Think of it as Expedia for private jets. My goal was to match up supply and demand, making the process of buying and selling trips on private jets simple.\n\nI opened an office in Los Angeles and hired a great team (from companies like Google, Expedia, Yahoo!, Travelocity, and the Jet Propulsion Laboratory). To build the business, in September 2006 we went out to raise money, and in that process I was introduced to Richard Branson's Virgin Group. Richard was my longtime hero as an entrepreneur, and his company, Virgin, was among the most respected aviation brands in the world. In April 2007 Virgin bought a controlling interest in my company. We were rebranded as Virgin Charter. My new job was to run the business.\n\nAll of this reinforced my belief that I had a sure thing. We were in a growing market solving a big problem. Our timing, our team, and our execution had been great. Now, with Virgin as an investor, how could I lose?\n\nSo I doubled down. Went all in. I was in the right place at the right time . . . or so I thought.\n\nWe launched as Virgin Charter in March 2008. The first quarter of 2008 was incredible for private aviation. There were more private-aircraft sales, more fractional sales, and more charter hours flown than at just about any time in history. The day we launched, Richard and I went on CNBC to make the announcement. But two other news stories caught my attention that day: concerns about an economic recession and a spike in oil prices. That's when I feared things might begin to change, and not for the better.\n\nThe business started to go sideways, quickly. In the first half of 2008 oil prices went soaring. Typically, airlines pass along fuel costs to their customers, but in 2008 prices were rising so fast that there was no time for them to recoup their costs fast enough. Rapidly rising oil prices destroyed the business model for many airlines almost overnight. While that was bad news for the big commercial airlines, they had an advantage we did not have in private aviation: many more passengers to share increased fuel costs. For charter operators, it wasn't that simple. It was not uncommon for only one person to be on board the private aircraft, so the cost for that passenger's flight was much higher. At launch in March 2008, the average charter of a private jet from Los Angeles to New York sold for approximately $30,000. A few months later, fuel costs had pushed that up to almost $45,000. Let me tell you, that kind of massive price hike is a disaster.\n\nPredictably, a chain reaction followed. People who would previously charter big planes instead went with small ones. People who used to take long trips chartered planes for short ones. Many stopped chartering altogether.\n\nOur business was burning more capital than we had planned, meaning we needed cash to survive. We decided to go and raise money to provide operating capital for the business. I was on the road for weeks and managed to secure partner meetings with three prominent venture-capital firms. But my highest hopes rested with a meeting that took place on Sand Hill Road in Silicon Valley in September 2008.\n\nThe day before the meeting, I flew from Los Angeles to the Bay Area with Brian Pope, our chief operating officer. We locked ourselves in a hotel room and prepared to give a great presentation. Upon arriving at Menlo Ventures, we were greeted by Mark Siegel, Menlo Venture's managing director and the partner leading our deal. We were taken to an office and were asked to wait until we were called into the meeting. We were so focused on preparing our pitch that neither one of us looked at the news or read the headlines that day.\n\nWhen we were finally asked to join the group, there were almost fifty people in the room. They were all in the prayer position, meaning that they had their heads downs, with their thumbs furiously working the keyboards on their BlackBerry phones. There was a steady flow of assistants bringing yellow sticky notes about phone calls that needed to be returned immediately.\n\nEverything seemed strange and tense, so I asked what was going on. One of the partners looked at me and told me that Lehman Brothers had just declared bankruptcy and the market was down five hundred points!\n\nYou won't be shocked to learn the meeting didn't go well. Everything about the world was about to change. You remember those days, when the stock market would fall three hundred points in a day, then three hundred more the next day. We were falling into the deepest recession since the Great Depression.\n\nWe didn't get the cash we needed at that time and my so-called sure thing stopped being about building something great and became all about saving what we could.\n\nWe'd started with a great idea, with timing that seemed perfect, a great team, and a solid plan. We'd gotten the Virgin Group behind us and received Richard Branson's stamp of approval. I'd gone all in, putting all my chips on the table. Sure, there were many things I could have done better. But the point is, I lost the hand.\n\nHere's the problem with sure things: There is nothing really _sure_ about them. There can't be because there are just too many moving parts in any business. Too many things that need to go right. All of which translates to an essential message: The most important job for any entrepreneur preparing to launch is to learn how to _mitigate their risk_. It's learning how to take the time up front to protect their downside. That way, if things do go sideways, they don't lose what's really important and can bounce right back.\n\nIn this chapter we'll talk about how you can protect the personal downside risk in your new venture. But let me tell you what \"personal\" means to me. In my office I have two napkins framed on the wall (entrepreneurs, you'll find, do a lot of their best work on the back of napkins). One is the original drawing I created of the Smart Charter Web site (the idea took shape in a coffee shop). The other is a napkin with my wife's phone number and a little heart with the words \"call me.\" She gave that to me when we first met. Today the business that I thought was my sure thing is gone. But what is really important, my relationship with my wife and family, is stronger than ever. Those two napkins are a daily reminder not to lose sight of what truly matters.\n\nOne truth of entrepreneurship is that everyone has ups and downs, highs and lows, and wins and losses. What's most important in each new venture is to protect yourself up front, be perfectly clear about the amount of risk you are willing to take, and learn how to mitigate that risk every step of the way. That way, if your sure thing does not work out, you don't lose what's most important and you can bounce right back.\n\n### The Fastest Way to Lose Your First Million\n\nAsk yourself this: How much are you willing to risk? How much time and money are you willing to put on the line to make your dream come true?\n\nMany entrepreneurs fail to get started because they are afraid of what could happen to them financially if things don't work out. They have heard stories about people who risked everything, all the money and assets they had, and then lost it all. Managers in big companies have heard similar stories about peers taking risks and their projects going upside down. Those typically end with people losing their jobs.\n\nWhat they don't know is that there are several ways to limit their financial downside. Risk should _not be risky_. It should be planned and well thought out. It should be measured in advance. Systems should be put in place to ensure that even if something does not go as planned, you, your family, and even your job are protected.\n\nBased on years of experience coaching entrepreneurs, I know that the number one mistake most make is mismanaging their money through a launch. When modeling their business, they are too optimistic. They overestimate their revenues and underestimate their expenses. Over time their business demands more cash than anticipated, so they start pulling out more and more of their own money to meet the company's needs. They get caught up in the momentum of the process, trying to balance the new demands of their business against all their other personal responsibilities. Eventually, they lose track of how much cash they really have on hand and how much has been committed to go out the door.\n\nThe result? One day they get a shocking bank statement in the mail: The money is gone, and the account has nothing left in it. The start-up capital is spent. Worse still, so is the money to pay next month's mortgage.\n\nWhen the horrified entrepreneur looks back, wondering how this all came to pass, part of the explanation is obvious. You'd be amazed how many entrepreneurs launching new ventures don't go to the trouble to separate business from personal accounts. The result can lead to a significant amount of stress for any new business owner, or even personal catastrophe.\n\nThis doesn't have to happen to you. Here are the most important steps that you can apply today to make sure your financial planning is sound.\n\n_Balance your optimism._ We'll talk more about this later (see \"Business Planning,\" page 81), but the first step to mitigating your personal financial risk is to create a realistic financial model for your business. That means coming up with revenue and expense assumptions that balance your natural optimism and help determine how much cash the business will really need. Here's how you do it.\n\nFirst create a list that includes all of your company's income sources and add up how much money you project your business will take in each month over the next year. Be sure not to project revenue coming in too early.\n\nNext create a list of all your expenses. This includes all the capital it takes to start up the business, including legal fees, the cost of setting up your Web site, and payment to the various people you will need to get things off the ground. Once you have estimated these one-time start-up costs, be sure to add your monthly operating expenses to the tally: office rent, telephone bill, insurance, etc.\n\nFinally, take your income and subtract your expenses to determine how much money you will need to start and maintain the business until it is able to pay for itself.\n\nOnce you are confident and firmly believe you have the right model in place, take your revenue projections and cut them in half. And cut them in half again. Then do it one more time for good measure. Now take your expenses and double them.\n\nThese more conservative numbers may be hard to digest, but experience suggests this approach will give you a reasonable sense of what will really occur in your business and how much time it will take before you no longer have to dip in to your own capital to make ends meet.\n\n_Determine how much money you are willing to risk._ Now that you have a reference point, determine your risk tolerance: How much money can you afford to lose?\n\nYour launch has to be funded somehow, and the odds are that much of the funding will come directly from you or, if you can get a bank or personal loan to start your business, that your name will appear on the documents as the guarantor. Unless you have a track record in business or friends with deep pockets, raising money from investors to get started may be difficult. Will the money on the line be your family's entire nest egg, or just half? Are you going to bootstrap, or borrow?\n\n_Talk everything through with your spouse or significant other._ You _must_ do this before jumping in with two feet. It's essential that the two of you be in agreement regarding how much to put on the line. If you don't agree, that will create problems down the road, possibly very large ones that will distract you from your business\u2014and potentially ruin the relationship. I'd suggest you would do better to risk less and be on the same page than to risk more and have your spouse worried and resentful day after day.\n\n_Put your risk capital into a separate checking account._ Do it now. Once you have a sense of the capital needed for the launch, transfer the funds you are willing to risk into a separate business checking account immediately. If your risk capital is tied up in securities or other investments, sell them and put the proceeds into the business account.\n\nThis is your seed money. Once you see it moved from your savings account to an account tied to risk, it becomes real. It focuses your attention. You see what's at stake. You have a clear financial framework to help you make better decisions, and you will act more strategically, less impulsively.\n\nWhat if that business account gets low or runs out? You have a few options: You can increase your risk threshold and put more money in, find an investor or partner, or sell or close the business.\n\nIn these early stages, however, what's most important is that, by separating your business finances, you will have created a mechanism whereby you will be forced to confront decision points. You will be faced with the facts sooner rather than later. You won't accidentally run through your personal finances and allow your spending to go beyond a point that is too much for you to risk.\n\n_Keep your credit cards separate._ Have one credit or debit card for business expenses, another for personal use. You need to do this as a way of tracking, accounting, and leveraging all that you can for tax purposes. Any and all expenses related to your business should be funded from this account. Don't commingle anything.\n\n_Hire a bookkeeper your first day in business._ Too often, entrepreneurs wait to do this or put it off altogether, but you need someone to track revenue, expenses, assets, and liabilities. Having correct data on hand is important for you, your investors, and tax preparers. If you don't set up a system early, you will spend a tremendous amount of time and energy going back trying to reconstruct the details behind every transaction that affects your business.\n\nThe price of paying a pro will be well worth the investment. Setup may take an hour, followed by monthly, weekly, or daily updates, depending on how much activity there is in your accounts. My bookkeeper logs in to my computer each day for a few minutes and updates the day's transactions. That means I have a report every morning telling me exactly how all of my business and personal accounts are performing. Because we were set up correctly from the start, this works out to a monthly cost of $150.\n\nHaving a professional on board will have a range of other benefits. Paying attention to your books at this early stage will establish good habits at your company and instill a sense of financial responsibility throughout the organization. That sense of financial responsibility may also spill over into your personal accounts.\n\n_Have an attorney set your business up by the book._ We live in a very litigious society, so you need to protect yourself from liability. Set aside money to create a proper legal entity and get business insurance. Remember, your concern isn't only about the short term; there are larger, long-term issues that you need to consider. Keep in mind too that even if this is your venture, it's not just about you.\n\n_Learn how to manage and grow your personal take._ It is important to constantly expand your understanding of how money works, so that as your business grows and spills off cash, you are able to invest those profits in a meaningful way to grow your personal fortune.\n\nIf your goal is to one day create a personal financial windfall through an IPO, sale, or merger, it is important to be prepared for this possibility. Most entrepreneurs are so busy working on their business, trying to make that first million, that there is no plan in place for when they finally do. As a result, they mismanage the money and lose all or part of it. Being unprepared for success is the fastest way to lose your first million. So it's important to build a great team of advisers around you that includes bankers, brokers, accountants, and financial planners. This way, when payday finally comes, you will be ready for it.\n\nInstead of trying to decide by yourself who should be on your team, find mentors who have made and grown their own personal fortunes. Ask them for help, guidance, and personal referrals. Their advice could be worth millions, literally.\n\n_Decide to launch one business or product at a time._ Many entrepreneurs, in order to hedge their bets, try to get three or four plates spinning at once. Don't. We will talk later about the importance of focusing on one thing\u2014I promise you, the best route to success is not to spread your energy and focus on ten things hoping one will work. Give the one thing everything you've got.\n\n### All for One and One for All\n\nI'm a huge fan of Starbucks. Like a lot of entrepreneurs, not only am I addicted to its product, but from time to time Starbucks has served as my office. A couple years ago I had the opportunity to meet the brilliant entrepreneur and chief executive behind Starbucks, Howard Schultz, when he spoke to an entrepreneurs' group in Southern California.\n\nSchultz had been touring in support of his book, _Onward: How Starbucks Fought for Its Life Without Losing Its Soul._ He gave a prepared talk, describing how in the late 2000s he steered Starbucks out of its crisis of overexpansion and loss of identity. He talked about reclaiming his CEO role, closing hundreds of stores and revamping hundreds of others, and how one day he shut down all 7,100 U.S. locations for three hours so baristas could get a refresher course on how to make the perfect espresso. Instead of being hell-bent on growth, Starbucks went back to being a nice place to go relax and get a good cup of joe.\n\nIt was a great story, but it sounded like a press release\u2014it lacked candor. I wanted to hear his advice on how to be a better entrepreneur, and I know I wasn't the only one.\n\nAfter the talk Schultz did a Q&A session. The last question finally proved to be the right one to open Schultz up and get him to tell us a more personal story. A man said, more or less, \"I admire you, Mr. Schultz; you have an amazing story and a great product. But when you just spoke about how you were hurting when Starbucks fell on tough times, I just couldn't relate. After all, you have a billion dollars in the bank while many of the rest of us are just trying to feed our families. So please share some ways in which we all can relate to you.\"\n\nThe room was silent.\n\nAt that moment Howard Schultz's demeanor changed. The billionaire CEO became the Schultz of thirty years ago, a bootstrapping entrepreneur like the rest of us. The story he told will stay with me for the rest of my life and his honesty made me an even bigger fan.\n\nIn 1981 Schultz was just another Brooklyn kid looking for a break, working for Hammarplast, a Swedish coffee-machine maker. When he first arrived in Seattle, he made a sales call on Starbucks at its first\u2014and in those days only\u2014location at Pike Place Market. In Schultz's recollection, the day was magical, the sky blue, the city looking as dazzling as Oz. He decided then and there he wanted to work for Starbucks.\n\nHe eventually persuaded the founders to take him on as a marketing guy but, after the owners declined to launch his idea of including an espresso bar in Starbucks, he traveled to Europe. There he acquired the rights to a coffee brand called Il Giornale. He returned to Seattle, looking to raise money to open his first Il Giornale store. He was working for no pay. His wife was pregnant with their first child.\n\nHis father-in-law flew into town from Ohio and asked Schultz to go out for a walk. They sat down on a park bench and he said to Schultz, \"My daughter is seven months pregnant and her husband doesn't have a job, just a hobby. I want to ask you in a heartfelt way, with real respect, to get a job.\" Schultz started to cry. He was flushed with embarrassment.\n\nHe went home to his wife and, later that night, told her about the conversation. He had thought over what his father-in-law had advised, and figured he probably should sell his interest in Il Giornale and get a real job. Wisely, he asked his wife what she thought he should do.\n\nSheri set him straight. She gave him her _Rocky_ speech (remember the one Adrian gives Rocky when he has doubts before the big fight with Apollo Creed?). Sheri told Howard that they were in this thing together, that she believed in him, that she was sure he would be able to raise the money for the coffee shop. And that he should not quit\u2014he should go for it.\n\nThat was all he needed to hear. It was an enormous, almost irresponsible risk that Schultz took. But it paid off. He got the financing, and a few years later he acquired Starbucks.\n\nIn an ideal world, all entrepreneurs would have people like Sheri to support them in their new ventures. But that's a lot to ask. Being a spouse, family member, or best friend of an entrepreneur is not easy. These are people who go through the wins and losses right alongside us. Sometimes they are as much at risk financially as we are. Too often we take these people for granted.\n\nI've learned that successful ventures start at home. The first and most important job of any entrepreneur is to get on the same page as his or her spouse or significant other. If you're not in alignment, there are only two possible outcomes when things get tough: The business will suffer, or the relationship will suffer.\n\nHow do you reach a meeting of the minds? You begin by talking through a few essential questions. The most important include:\n\n_How much financial risk are you (the spouse) willing to take?_ If you are launching a new business, the odds are you are bootstrapping, that is, using your own money. You and your spouse need to agree upon how much you're willing to put aside and put at risk in pursuit of your dream. You need to start on the same page.\n\n_How much time will you allow me to commit?_ To successfully launch in ninety days, you will have to commit to your business first, with your family, along with everything else, coming second. You will have too much on the line. There will be too many moving parts for you to tend to anything other than your business, which needs to be your top priority. That doesn't mean your family is less important, just that, for a period of time, both of you agree that this is something in which you need to immerse yourself in the interests of your shared future.\n\n_How much do you want to be involved in the business?_ Being an entrepreneur can be a lonely job. At times it's difficult to find others to talk to about the challenges you face at the office. For example, there are certain things you may not want to share with your investors. There are certain issues that need to be kept away from employees. You usually don't want to spend your time with friends talking about problems at work. So you go home and vent.\n\nCVO is the acronym for what I call the \"chief venting officer.\" It is the role typically occupied by your spouse, your significant other, or a member of your family. It can be a lousy job because CVOs tend to hear about the tough stuff but not the good things that occur every day.\n\nLike Sheri, this person is on the entrepreneurial roller-coaster ride with you. He or she is expected to support you and hold your hand through it all. Success is never easy and rarely happens overnight. When you're in a relationship, it is never a one-person effort.\n\nThere are some CVOs who really want to be involved. They want to know everything and thrive on a blow-by-blow account of the day. Others can take the ups and downs only in smaller chunks and want to hear far less. You need to be sensitive to how much your friends and family members can take.\n\nOne of the best pieces of advice I can give is to calibrate your communication strategy to suit your life partner. Do you talk about business once a day or once a week? If things are tough, do you come home and share everything before bed or wait until the end of the week to get it all out? Come to an understanding that works for both of you and respect it. How you handle the pressures of your business can either kill or strengthen your relationship.\n\nKeep in mind that some people come from traditional business families where stability is the rule, with nine-to-five employment and a weekly paycheck for life. For people brought up in such circumstances, the seesaw of launching and entrepreneurship may be hard to handle. I am lucky that my wife, Rachel, grew up in a family of entrepreneurs. She lived through good and bad times but also saw how everyone got through them. As a result, she doesn't get rattled but even so, she doesn't want a daily blow-by-blow account of what happened in my businesses.\n\nRachel also taught me another essential lesson: The way you spell love in a relationship with an entrepreneur is _T-I-M-E_. That's why we make it a point to get child care every Saturday and carve out time to have lunch together. I am usually relaxed because I'm out of the office. Things that seemed urgent during the week don't matter. I get a chance to tell her what's going on and get her feedback. Having this exchange doesn't take much time, but we need to do it and do it consistently. Both sides have to be present and accounted for.\n\nYou should put in place something similar once you've found a mutually workable strategy. No matter how understanding your life partner is, you still have to prepare him or her for the challenges you'll face in building your business. Most important, you need to be in alignment with regard to how much both sides are willing to risk, how much time both sides are willing to commit, and what communication strategy works best for both of you.\n\nTALK IT OVER\n\nYour new venture will take significant time and resources to launch and could involve financial risk. While you are out pursuing your dream, your actions will affect the people around you. Before jumping in, make sure that you and your spouse or significant other are on the same page.\n\nHow much time will you need to put into this new project? During launch, your business will need to be your first priority. This doesn't mean you love your family and friends any less. It does mean that during this period your business will need to come first.\n\nHow much financial risk are you willing to take? Would you rather take less financial risk and be on the same page as your spouse or take on more risk and create a problem at home?\n\n### Super Balls and Glass Balls\n\nThe entrepreneur must master the art of juggling. Your job, your finances, your spouse, your family, and maybe your boss; your health, projects, and deadlines\u2014they're all in the air and you're always at risk of dropping one.\n\nSome people juggle better than others and can manage to keep all those balls in the air indefinitely. Most of us, though, reach a point at which it just becomes too much and one or two will crash to the ground. We might even drop everything.\n\nAll of which makes it essential that you understand this little rule of nature: _Super Balls bounce and glass balls shatter._ You need to think about which balls are which in your life.\n\nIf you drop the Super Balls made of rubber, they will keep bouncing on their own for a while, none the worse for wear. While you're working on launching your venture, you don't have to worry so much about the ones you know to be bouncy and resilient. Later, when you can turn your attention back to them and recommit to those areas of your life, those rubber balls will take to the air again. No great harm done.\n\nGlass balls? They're a different matter.\n\nIn the fall of 2008 the sky was falling. I was standing at the airport with Steve Ridgway, the CEO of Virgin Atlantic Airways, waiting for a plane on Tortola Island. Our businesses both faced big challenges, buffeted as we were by the harsh winds of the growing global economic crisis, which threatened the economic stability of almost every nation and institution.\n\nMy business was vulnerable. I felt vulnerable too, as if I just might blow off the Jetway at any moment, but Steve looked calm.\n\nI asked him, \"Steve, with all the pressure on you, especially right now, what's the most important part of your job?\" I expected to hear something like leadership, team building, or vision.\n\nHis answer was simple, just one word.\n\n\"Exercise.\"\n\nExercise? I was surprised.\n\nLater that day I sat next to David Cush, the CEO of Virgin America, on the flight home. I asked him the same question: \"What's the most important part of your job?\"\n\nHis answer? \"Exercise.\"\n\nAgain I was shocked. David hadn't been with us when Steve and I had spoken earlier. They hadn't had a chance to compare notes, but they had both given me the same answer. Two powerful guys, running big businesses at a superdifficult time, and they both said the most important thing was _exercise_. It took awhile for this lesson to sink in, but you know what? They're right.\n\nLaunching your business will be stressful and arduous. You'll always feel like there isn't enough time for the ten thousand things you have to do, and that if you could just squeeze a couple more hours of work into the day, you'd really feel like you had something to show for it. But you can't keep going at a hundred miles per hour if you're out of gas.\n\nWe keep going by fueling our tanks. We can either choose good fuels like healthy foods and exercise or opt for a bad diet made up of things like alcohol and high-caffeine drinks, along with little to no exercise. But you know as well as I do that your physical health directly impacts your ability to perform at a high level and even influences your mental health. Exercise enables you to blow off steam and release stress, which ultimately helps to keep you calm and thinking more clearly. It's an essential component of dealing with the challenges of entrepreneurship.\n\nYour health is a glass ball. If you drop it hard enough, it won't just break. It will shatter, and you can't be sure that you'll be able to put yourself back together again.\n\nYou may think you can't find the time, but you simply have to get out and exercise, get your body in motion every day. This doesn't mean you need to run a marathon or train as a triathlete. Thirty minutes a day will do it, even just a walk around the neighborhood. What you need more than anything is movement to get your heart pumping. You need some measurable physical activity to release the stress that comes along with your position and to recharge your body. Remember, if you don't have a way to release all the pressure, to let off steam, the stress will catch up with you. You won't have the energy or strength or endurance to do your job right. You and your business will be set up to fail.\n\nIf you have to build your day around exercise, do it. I asked Mark Moses, an \u00fcberentrepreneur and triathlete you'll meet in another chapter, how somebody with a business schedule like his could fit the many hours of training it takes to be an Ironman triathlete into his day. He said, \"I put my exercise into my calendar first and then schedule everything else around it. By getting in that time and training I have more energy and a clearer head than all those people who spend their entire day letting stress build up behind a desk.\"\n\nYou'll be surprised at how getting your mind off your work will actually help you find solutions to the toughest problems your business faces. You know how sometimes you'll have an epiphany while standing in the shower? The same thing happens to me when I'm out for a walk. When you force yourself to exercise, you may feel guilty at first because you're taking a break from work\u2014 _there's that deadline approaching!_ \u2014but in reality you're freeing up some bandwidth of your attention span. That will give your subconscious the extra processing power it needs to make breakthroughs.\n\nSo get some exercise every day. Pick a time of day that works best for you; work your schedule around it. Try to get your workout away from home; there are too many distractions there and it can be difficult to really let go. If you can, do it alone so you can let your mind wander and permit the stress to ease out of you. As an entrepreneur you have to be flexible, ready for anything. Your business may or may not be around in ten years, but you want to be.\n\nDistinguishing the glass balls from the rubber ones isn't hard\u2014and it's all important that you devise strategies to balance the pressures so you don't shatter what is most important in your life. When times get tough, you need to be the toughest person in the room. You can't be ready if you don't take care of yourself.\n\n### Ten Thousand Reasons Why You Will Succeed\n\nOne of the biggest challenges entrepreneurs face is the emotional roller coaster they ride every day when launching a business.\n\nYou wake up in the morning, an investment comes through, and then you lose it. Suddenly the money is back. The product is working as planned, then it goes down, and now it's back up. The key employee you want to hire is a go but unexpectedly falls out. Later he agrees to sign.\n\nTaking a seat on this roller coaster can wreak havoc on your business. The highs and lows can be exhausting. They can quickly drive you in and out of emotional states that won't help you run your business effectively, will strain your relationships, and can have a negative impact on your health.\n\nAs an entrepreneur you need to learn how to smooth out the ride and manage your emotions in a way that balances your energy, drives your business forward, and puts you in the best position to succeed, no matter what life throws at you. You can do this by _controlling your focus._ The brain works a lot like Google. It gives you answers, but those answers are only as good as the questions you ask.\n\nWhen an investment falls through, the product goes down, or a new hire falls out, you begin an internal dialogue. You start asking questions to help understand what just happened and what it all means.\n\nMost people don't understand that this dialogue is even taking place. They are conditioned to respond to circumstances in a very specific way and don't even know it. If that conditioning leads them to ask good, empowering questions, the business will go down one path. If it leads them to ask an endless loop of negative questions, their lives will go in a whole other direction.\n\nFor example, when you hit a bump in the road, if your automatic response is to start asking questions like _Why does bad stuff always happen to me?_ and _Why do bad things always happen to my business?_ then your Google-like brain will do a search and pull up ten thousand reasons why your business stinks. Your focus will go right to these answers, which will make you feel terrible. It's impossible to act in a way that is effective from that place.\n\nInstead you must teach yourself to ask a better set of questions, ones that empower you and your business. Because having the right mind-set immediately shifts your focus from each challenge to the opportunity it presents.\n\nWhen things weren't going your way, what would happen if you started asking questions like _What is great about this experience? What can I learn from this experience?_ or _How can I use this experience to help my business succeed?_ Again, your brain will run a search. But this time, you'll get ten thousand reasons why you will succeed. That is exactly what you will focus on, putting you in a much better position to drive your business forward.\n\nIf you take only one thing away from this book, this should be it: _Ask a better question_ \u2014and watch how your life changes forever.\n\nIf you want to develop a razorlike mind-set, one that cuts through problems like a knife, here's how:\n\n_Recognize that you are in complete control._ Nothing in life has any meaning other than the meaning you give it. You have a choice in determining how you evaluate any situation and where you place your focus.\n\n_Break your pattern._ The odds are that you have been running on autopilot most of your life, allowing your emotional state to dictate your inner dialogue. Next time you run into a problem or something threatens to knock you off course, _stop_! Pay attention to the questions you are asking to evaluate the situation, and if they are negative, interrupt your pattern with a new set of empowering questions. This will help you look at the situation in a whole new light. With enough practice, this powerful way of thinking will become automatic.\n\n_Develop new habits._ Developing the skill of focusing on what empowers you is like building a muscle. Your job is to exercise your mind each and every day. The best time to start is in the morning. Take control of your focus as soon as you wake up. Begin the day by asking questions that set you up for empowerment, like _What am I grateful for? Why will I succeed? How can I move this business forward today\u2014and have an awesome time in theprocess? _Starting the day with this mind-set will put you in a much better position to execute.\n\n_Never make a big decision at your lowest point._ Many times it seems like we make our biggest decisions when we are in the worst possible position to think clearly. Next time, _wait_! Hold off until you ask a better question. Wait until you have physically moved to get the blood flowing, stepped away from whatever the situation is, and released some tension. You will find that you will make a much better decision from a more balanced emotional state.\n\n_Pass this skill on to your team._ Imagine if everyone on your team operated from this place of razor-sharp focus, if they looked at everything that came their way as empowering tools that they could use to drive the business forward. Imagine if they were conditioned to find the opportunity to improve in every situation. Wouldn't they also be making smarter decisions?\n\nThe decision is up to you. Are you ready to take control of your life? Are you ready to get off the roller coaster?\n\nIf the answer is yes, then just do one thing\u2014ask a better question\u2014and focus squarely on the opportunity in everything that life presents.\n\nEvery entrepreneur needs to protect what is most important. That means you! The way to start is by mitigating your risk. This begins with keeping an eye on your finances and relationships and developing a razorlike mind-set that helps you convert any problem into an opportunity that moves your business forward. Remember, you are in much more control of the results in your life than you realize.\n\nTHE OTHER HALF OF LIFE\n\nControlling your focus is only one half of determining how you will behave. Another way to control how you behave is to change the way you move.\n\nIf I asked you to describe a person who was down or had had a hard day, it would be easy: eyes cast down, shoulders slumped, and breath low and shallow. If I took that same person and simply had him change the way he was using his body, had him lift his gaze, bring his shoulders back, and breathe deeply, I could take him right out of that emotional state. That is because the different ways you use your body send different signals to your brain.\n\nTry it for yourself. The fastest way to change the way you feel is to change the way you move. As Tony Robbins would say, \"Motion creates emotion.\" So if you are down or you are having a tough day\u2014go out and move. Get the blood flowing. When you feel stuck in your business, no problem. Get out of your chair and take a walk outside. Movement will instantly change the way you feel and put you in a better position to take on the day.\n\n## CHAPTER 3.\n\n## YOUR ENVIRONMENT\n\nI don't know the key to success, but the key to failure is trying to please everybody.\n\n\u2014BILL COSBY\n\n### Crabs in a Bucket\n\nNothing is more important than the people you surround yourself with. You don't need me to tell you that the right friends and contacts can help in a thousand ways, but let's examine your personal ecosystem. Odd as it may sound, I'd like you to start by thinking in terms of crabs.\n\nImagine you've got a bucketful of freshly caught live crabs. If you look down into the bucket, you see the crabs squirming around, as some instinct tells them it's their job to try to figure out how to escape. Individually, they're each taking a turn clawing and climbing over the others trying to get a grip on the lip of the bucket.\n\nShould you put a lid on the bucket to prevent dinner from crawling away?\n\nDon't worry about it. There's a funny thing about crabs\u2014as soon as one crab looks like he's about to hoist himself up over the edge of the bucket, the other crabs grab onto him and pull him back down. They can't understand that if everyone just worked together and formed a crab chain, all the crabs on the bottom could easily climb up to the top and out of the bucket and go about their business. But no; whenever one crab starts to show some leadership, some initiative, it gets pulled all the way back down to the bottom. Eventually they just stop trying.\n\nHave you ever felt like a crab in a bucket? Maybe your office mates criticize you for trying too hard and making them look bad. Your boss may not want you to move ahead because then he'd have to find someone else to do your busywork. Perhaps your friends think you're getting too big for your britches because you want something better for yourself than what you grew up with.\n\nIs your entire family trapped in this mind-set? Parents often don't want their grown children to move away because they think they need them nearby to take care of them when they get older. Some spouses resent their entrepreneurial partners because, if one person is off pursuing a dream, the other person feels abandoned to raise the kids alone or do all the chores.\n\nLife is way too short to hang around others who want to pull you down. The number one determinant of who you are and, eventually, of the results you achieve in life and business is your _environment._ Think about your environment at work, at home, in your neighborhood, in all the places you spend time. Do these places pull you down or elevate you? Do you feel good and comfortable where you work? Do your friends look out for you, encourage you, and help you brainstorm? Or do they reinforce bad habits? Do they share your values? If you have to compromise yourself and what you believe for the people around you, it's time to change the people around you.\n\nAre the people around you invested in your success, or do they really want you to fail so they can keep you in the bucket with them? How large and generous are their visions? Remember that people with small visions of themselves cannot have a larger view of you.\n\nYou may not think it nice, but you owe it to yourself to separate from the \"friends\" who anchor you to a place where you do not wish to be.\n\nThere will always be people who try to knock you down. You don't need them in your life. There will always be people who attack your Big Idea. Don't dismiss what they have to say, but don't buy into it. Even if they seem to have the best intentions (they may think they're trying to protect you), your job is to listen to their arguments and use them as a counterbalance. See if there is anything they say that you can learn from and apply to your business or product in order to make it better. And if people say you're crazy and your Big Idea will never work, let their doubt drive you.\n\n### Your Fab Five\n\nHow important are the people you invest your time with? The time you spend with others is nothing less than precious\u2014 _and_ an investment in your future.\n\nMy time with Tony Robbins taught me to look critically at the people around me. His circle included individuals who either were the very best at what they did or aspired to get there. This included people like Quincy Jones, Peter Guber, and Pat Riley. Among these top business leaders, world-class athletes, and people committed to changing the world were many others who had overcome big obstacles. They had found solutions that not only benefited themselves but also improved the lives of others along the way.\n\nI'll never forget one member of Tony's circle, who had gotten in a car accident on the way to his wedding. The accident left him paralyzed from the waist down, but instead of feeling sorry for himself, he went out and invented a line of aerodynamic racing wheelchairs. He not only empowered himself but also helped others in the process.\n\nDuring my time working for Tony's company, I assessed my own circle of friends and came to understand the power of our personal networks. What I learned was that I needed to recruit the right people into my life. In particular, I had to be aware of the five people I spent most of my time with. I call these five people the _Fab Five_.\n\nThe name \"Fab Five\" came into wide use in reference to what many people consider the greatest recruiting class in sports history, the 1991 Michigan men's basketball team. In their first year, these freshman players took their team to the NCAA championship, a remarkable and unprecedented accomplishment. In the same way Michigan's coach recruited them, you need to recruit and build your own championship team around you. Let's begin by looking at your existing Fab Five and considering their impact on you today.\n\nMake a list of the five people you spend the most time with outside your family. If you're not sure whom to choose, just look at the last five numbers you dialed on your mobile phone. Now estimate what each of them earns, total the incomes, and divide by five. The number you end up with will be really close to what you earn. Crazy, right?\n\nMaybe money isn't important to you, but another value is. Try the same exercise with your Fab Five's health status and their attitude toward the people around them (this time, if you wish, add your family to the list). Chances are you will get the same result. Today you are nothing more\u2014and nothing less\u2014than the average of the five people you spend the most time with.\n\nAsk yourself how your Fab Five affect you. Are they pulling you up or pushing you down? If you told your five best friends, one at a time, over coffee or a beer that you were planning to start a business, what would their responses be? Would they be supportive or dismissive? Would they encourage you or try to shoot your idea down? Would they be keen to help you brainstorm or would they turn the conversation back to themselves?\n\nAre your Fab Five a reflection of your higher aspirations? This is important because we tend to hang around people whose values are similar to our own. Rich people hang out with rich people. Physically fit people hang out with physically fit people. And slobs hang out with slobs. If you want to know why you don't have more of the things you want in your life, the answer may be as close as the last five numbers you dialed on your mobile phone.\n\nIf you want to build a million-dollar business, you need to spend your time with people who have _already_ built million-dollar businesses. If you want to build a ten-million-dollar business, you need to spend your time with people who have already built ten-million-dollar businesses. That's an important distinction, because the people who have built smaller companies won't know how to get you to the next level. Maybe you want to build a billion-dollar business but think you can't get to billionaires. That's no more than a convenient excuse. Pick up a book by Warren Buffett, follow Mark Cuban's blog, or attend live events with Tony Robbins.\n\nImmerse yourself in the way your core group sees the world. Learn how they program their minds. Let their learning and instincts wash over you. If you spend enough time reading and absorbing their material, you will start to pick up how they act, think, and make decisions. Their state of mind will start to rub off on you.\n\nYou may not want to hear it or admit it, but you may have to change your circle of friends. If the people you spend the most time with cannot help you get to where you want to be, then you need to expand your network and make new friends. That doesn't mean that you have to abandon old friends, but the ecosystem of the entrepreneur needs a different balance. Finding success is a different pursuit from having fun, from recreational shopping for shoes or tailgating at the football game. The people you feel the most comfortable with\u2014maybe the ones who demand the least\u2014aren't necessarily the ones who'll move you toward your goals.\n\nIn order to move to the next level, you may need to add new people to your Fab Five\u2014or create a new Fab Five from scratch. The people and thinking that got you this far are not going to take you where you want to go. If they could, you'd already be there.\n\nTHE PEOPLE AROUND YOU\n\nAre the people around you pulling you up or holding you down?\n\n**Take Inventory:** **** Write down the names of the five people you spend the most time with. Ask yourself: _Do they support me? Do they want to see me succeed in my new venture? Or would they prefer to see me fail and keep me close?_ If the answer to that last question is yes, check them off your list and get them out of your life.\n\n**Find Three Mentors:** Answer this question: Who are your three mentors? If you don't know, you'd better get to work finding one . . . and another . . . and another.\n\n**Find Someone to Mentor:** It's not enough to get guidance. You also need to give. I recommend that everyone mentor at least one person half his or her age. Interacting with students and younger entrepreneurs will give you an entirely fresh perspective and help you see things in a whole new way.\n\n### What Do Those Note Cards Say About You?\n\nAs an entrepreneur you must forge your own personal brand. As a manager in a big company, building a personal brand is just as important, because a strong brand will give you more power and influence within the organization. To put it another way, you must author your own story, paint your own portrait.\n\nYou may think you are selling a product, a financial plan, or something else to your investors, division heads, partners, or employees. And to be sure, those things may be part of the appeal. Yet it's really all about you, because people invest in people. Every action you take helps others define you. You may think you're saying nothing about yourself, but are really conveying everything there is to know about you.\n\nI run a personal-branding exercise at one of my programs. Typically there are about twenty people in the room. On the first day I encourage the participants to mingle while grabbing coffee and breakfast before the event. When it's time to get started, they are invited to sit in their assigned seats. On the table in front of each of them is a placard. They are asked to print their names on the placards so everyone in the group can see. Each person is then handed a stack of index cards. I ask them to look around the room, write one person's name on each of their index cards, and jot down the first five words that come to mind when thinking of that person. An assistant then collects all the cards, enters all the words the group used to describe each participant into a file, and prints them out so that each participant sees only the words used to describe him or her.\n\nPeople's reactions are amazing. Typically they are shocked, horrified, and very upset about the way others view them. For example, a person might think of himself as outgoing, but the index card shows the group thinks his behavior is obnoxious. Another person thinks she embodies the image of success and confidence, but the index cards indicate the group sees an arrogant and standoffish person.\n\nOften, the way others see us just doesn't sync with the story we've been telling ourselves about the image we think we are portraying. It's little wonder that we're not getting the results we want with our businesses in selling our ideas or raising capital. Clearly, what we are projecting is not what we need to win people over.\n\nWhat would that stack of note cards say about you? What is the impression you are leaving on others? Is that personal brand serving you or stifling your growth? Is it taking you closer to or further away from your goals?\n\nThere are deliberate steps you can follow to take control of your personal brand.\n\n_Identify your goal_. The first thing you need to do is get very clear about what it is you want to achieve. Do you want to take a consumer online business to market in ninety days? Do you want to release a new product that extends your company's footprint in the marketplace? Do you want to close financing for a big project? Do you want to create a rock-star team to help transform your local business into a global franchise?\n\nWho do you need to be to achieve that goal? What image do you need to project to bring others along with you?\n\n_Understand where you are now._ Where are you today relative to where you need to be? What is the gap, and what changes do you need to make to fill it?\n\nTo answer this question go to the five people you spend the most time with (your Fab Five) and ask them to use words to describe your brand. Then extend your pool to others with whom you have had fewer interactions; the more distant acquaintances may offer even more valuable impressions because they are based on more immediate first impressions. When you get a handle on how others see you, you'll have a better idea of what needs to change. If they say you're standoffish, when you've just thought of yourself as shy, this is an area of your self-presentation that you need to work on. Such knowledge alone can be curative.\n\n_Recognize what daily habits say about you._ Do your habits empower or disempower you? What is the message your habits tell others about you? Daily habits can have a big impact on how people brand you. Your attire, manners, and degree of organization tell people about you\u2014and the right presentation communicates that you came to play, you are prepared, and you have pride in who you are and what you do. For example, when someone is very physically fit and really takes care of himself, he presents as disciplined, committed, and apt to follow through. People can see and hear certain energy in their physical presence immediately. On the other hand, someone who does not take the same pride and care in his appearance sends a completely different message. Creating empowering habits based on higher standards is vital. The biggest difference between where you are now and the place you want to be is the standard you set for yourself.\n\nLittle things make a big difference, even if it's just the cut of your suit or the warmth of your handshake\u2014and you can take control of these factors. If you don't define who you want to be and manage your personal brand accordingly, people will do it for you. On the basis of the cues you give them, they will put you in a box. It's a box that best serves their needs and not yours, so make sure it's a box you want to be identified with.\n\n_Leverage technology to define and reinforce your brand._ Most people will first be introduced to you online, when they type your name into Google or visit your LinkedIn, Facebook, or Twitter profile. They'll see the pictures and videos you distribute of yourself and what interests you. Use these tools to your advantage: Have a professional take your picture, inset a link to your Web site, and make sure your message is clear and your copy is brief and to the point.\n\n_Be yourself._ Don't pretend to be someone else. When developing a personal brand in today's multimedia world, authenticity is key. If you try to pass yourself off as someone you are not, people will find out and that will undercut your credibility. Be original, be creative, but don't try to be what you think other people want you to be. Be the sincerest, most heightened version of yourself. When investors and customers meet you, you want them to think, _This is a person I want to do business with._\n\nIf there are things from your career or past that you don't want to share or are afraid of how others will react to, don't hide them. Address them up front and, by doing so, take control of the conversation. For example, if an earlier business venture failed, don't try to hide it. Illuminate it by sharing what you learned from the experience and how you are applying those lessons going forward. Here's a secret: Most people would rather get behind people who took a shot, even if they failed. People who have learned from their mistakes bring wisdom to a new venture going forward.\n\n_Don't be defined by your origins_. I'm a huge Los Angeles Lakers fan. One of my heroes growing up was Lakers owner Jerry Buss. He was a Depression-era baby who stood in food lines to eat, but he didn't let that experience define him. In college he saved $86 from each $750 paycheck he received, then got seven guys together and bought an apartment building. He built a real-estate empire, enabling him to buy the Los Angeles Lakers. Today he's an American icon, with a business worth at least one billion dollars. He wrote his own story and created his own brand. It is your job to do the same.\n\nWRITE YOUR THIRTY-SECOND COMMERCIAL\n\nYou're selling yourself, and don't forget it. It is important to do it well, in clear and crisp fashion. Follow this simple formula whenever you are introduced to anyone (potential investors, partners, customers, tradespeople) in a business setting.\n\n**Tell Them Who You Are** **:** Start with your name, your title, and a one-sentence description of what your company does.\n\n**Tell Them Whom You Serve** **:** Compose a one-sentence description of your target customer. Be as specific as possible.\n\n**Share the Benefit** **:** Describe how your customer will benefit from what you offer.\n\n**Ask for the Order** **:** At the end of your thirty-second pitch, ask whether the person you're talking to could benefit from or knows someone who could benefit from these services.\n\nYou can use this commercial even if someone casually asks you what you do. The more you practice your pitch, the clearer and more powerful it becomes.\n\n### Write Your Own Story\n\nA great football season looked like it was about to crash and burn. The 1977 squad at the University of Arkansas, coached by Lou Holtz, had earned a berth in the Orange Bowl, and the Razorbacks were set to play their longtime rivals, the University of Oklahoma Sooners. The winner of the game was a virtual lock to take home the national championship. But bad things began to happen to Holtz's team. In one of the last practices before the game, Arkansas lost an All-American guard to injury. A couple days after that, Coach Holtz had to suspend both of his starting running backs and a wide receiver for disciplinary reasons.\n\nThe press jumped all over these developments, decreeing that Arkansas had no hope of beating the Sooners, who had won two of the last three national championships. Arkansas fell to being a twenty-four-point underdog. In his first season as head coach, Louis Leo Holtz looked like he was about to go from a leading candidate for coach of the year to goat of the year.\n\nMany years later I worked an event where Holtz\u2014who would go on to legendary success in eleven seasons at the University of Notre Dame and later become a popular motivational speaker\u2014told the story of how he and his team saved the season.\n\nHoltz told us how he brought his team together in the locker room before the game. He began by acknowledging the situation they were in and the long odds they had been given by the experts. He told his ball club that everybody was trying to define them, to write their obituary before the first play was called (just as some people will try to do with your new business or product). Holtz pulled out a copy of the newspaper and held it up so everyone could read the boldface headline, which predicted a big loss for Arkansas.\n\n\"Everyone else has already decided who you are, what you will do, what the outcome of the game will be,\" said Holtz. \"If you don't define yourself, others will do it for you.\"\n\nWith that, he folded the paper in half and ripped it. Then ripped the torn pages again. And then did it a third time.\n\nHolding the wad of torn newspaper in his hand, he gave them his bottom line.\n\n\"Champions don't let others write their story,\" he said. \"They write their own story.\"\n\nWith the ease of a practiced magician, he raised his arm as if to toss the shredded paper into the air. Instead of a shower of paper, however, the coach unfurled a fresh newspaper. This one had a different headline: ARKANSAS WINS. Which is exactly what his team did that day, soundly beating the Sooners 31\u20136.\n\nThe magic trick had just been an illusion, but Holtz's message is one every entrepreneur and intrapreneur should take to heart: Don't let anyone try to define you or tell you what you can or cannot do. As Holtz has said, \"You can't let naysayers pull you down. Remember: If someone says you can't accomplish something, it is an opinion and nothing more. It is only a fact if you say it is.\"\n\nLaunching a successful new business, product, or service begins with having vision, letting go of your fear, and making the decision to move forward. It is followed by mitigating your risk every step of the way, so if things don't work out as planned, you have protected yourself and have the ability to bounce right back.\n\nBut it doesn't end there.\n\nYour success or failure will be largely influenced by the impressions you make on others. It will also be impacted by your environment. This includes the people in whom you choose to invest your time.\n\nMost important, your success during launch will depend on how you handle the ups and downs of entrepreneurship and how effective you are at smoothing out the ride. By asking better questions, you will be in the best possible position to succeed, no matter what life or business throws your way\u2014and be headed toward making your dreams come true.\n\n# STAGE II.\n\n# FUELING THE TANK\n\nThirty Days to Assemble Your Resources\n\n## CHAPTER 4.\n\n## BUSINESS PLANNING\n\nIf you can't explain it simply, you don't understand it well enough.\n\n\u2014ALBERT EINSTEIN\n\n### Biceps and Business\n\nEntrepreneurs are by nature impatient people. Once they have an image of their big-picture objectives, many go from napkin to execution with nothing in the middle. They rush out and start executing without a plan.\n\nWhile goals and objectives may give you a direction, they don't tell you what specific things you need to do to succeed. Although the strategies you use to accomplish your goals will evolve as you do business, the clearer you are from the start, the better a shot you have at building a winner.\n\nMy favorite example of the power of combining clear goals with a well-thought-out plan comes from my good friend Doug Brignole.\n\nA few years ago I decided that I wanted to get really fit. The first thing I did was go to the gym and ask to be referred to a trainer. The guy they connected me with just wasn't what I wanted. Although he had muscle, he was also overweight and, worse yet, unhappy. He wasn't the kind of person I wanted to emulate. On the other hand, the man did make me realize that I should be modeling myself after someone who is the very best at what he does. So I made a decision: _I want to learn from Mr. Universe!_ I put the word out to my friends, and that's how Doug entered my life.\n\nDoug grew up in Southern California in a first-generation immigrant family. He was a small kid and he frequently got beaten up on his way home from grade school. At age fourteen he decided that he had had enough and that it was important to improve his physique. He went to a local gym and asked, \"Can I _please_ join?\"\n\nHe was too young to be a member, but Doug persisted. He worked out an arrangement with the manager to work part time and work out during his off hours. He fell in love with bodybuilding and eventually got the urge to compete. By the time he was seventeen, he was a contender for Mr. Teen California. Though he didn't win that year, he did come in second, kicking off a lifelong career in bodybuilding. The rush of competing fueled him, so Doug set his goal higher: He wanted to become Mr. America.\n\nDoug started by making sure he was absolutely clear on his outcome. He is not only a bodybuilder but also a world-class artist. One of his unique skills is his ability to draw the body and biomechanics in amazing detail.\n\nDoug had a friend take a photo of him in a bodybuilder's pose, and Doug blew up the image. He put tracing paper on top of the enlargement and drew, in very fine detail, exactly what he would need to look like in order to win the title. He illustrated the shape, size, and balance of each muscle group as he projected it would be by the time the next contest rolled around.\n\nOnce Doug was absolutely clear on what he wanted, he asked himself the following questions: _What do I need to do to achieve it? What could get in the way? How am I going to overcome each challenge?_\n\nThis was years ago, before you could find twenty choices for protein powder at the store and smoothies on every corner. Like other professional bodybuilders at that time, Doug prepared all of his own meals, figuring out a dietary regimen that would enhance his workouts. He put that tracing-paper picture of himself in a frame and set it on the table where he ate five times a day. With every bite he took, he'd point his spoon or fork at the picture and say, _That's what I will look like_. And he'd take another bite. _I'm going to make that happen._ He did this for months.\n\nFrom time to time Doug would have his friend take a new picture of him in the same pose, and then he would blow that up and lay the tracing of his goal body on top. In doing this, he took inventory and monitored his progress. He adjusted his diet and bodybuilding routine based on the results. Pound by pound, muscle by muscle, he filled out the image, bringing a dream, an idea sketched on tracing paper, to life.\n\nFinally Doug competed and won the Mr. America title in his division.\n\nAfter the competition, Doug had his friend take one final photograph. He posed just as he had in the original, so many months earlier. He blew the new picture up and laid the original tracing paper on top. The photo of Doug and the original image he had drawn of his goal were identical.\n\nDoug Brignole set clear goals, created a solid plan, checked his results periodically, and made necessary adjustments. He executed in a similar fashion for his other goals as well, eventually winning the title of Mr. Universe. Though now in his fifties, Doug Brignole is still competing; recently he was the overall winner at the Muscle Beach International competition in the Masters Division.\n\nHere's how we can all apply the lessons from Doug's story to turn our Big Ideas into a reality.\n\n_Be clear on what you want._ Your goals must be specific, be measurable, and have deadlines. It's not enough to say, _I want to make one million dollars_. You need to be much more specific. _I want to earn one million dollars in net revenue by this specific date._ For your business, it's too vague to say, _I want to increase sales._ A better goal would be _I want to increase sales to one million dollars per quarter by a specific date with a specific profit margin._ Clarity is power. You can't manage what you can't measure.\n\n_Make a list of what you need to do._ Clear goals are important, but you must back them up with a solid strategic plan. I heard recently that having clear goals increases an entrepreneur's chance of success by 20 percent. But having clear goals and combining them with a well-thought-out plan increases your chance of success by _400 percent_.\n\nIt's important to break down what is required to accomplish your goal into small, manageable pieces. When you're finished, ask yourself, _Is this it? If I do all these things, will it be enough? Will I reach my goal? Or are there other things I need to do?_ Odds are, there are more steps than you realized.\n\n_Identify what could get in the way._ To accomplish your goal of one million dollars in gross sales per quarter by a specific date with a specific profit margin, you may need to clear obstacles in your path. For example, you may need to tweak your product, refine your sales pitch, upgrade the people on your team, or find ways to reduce costs. Decide how you will overcome each roadblock. Make a list of everything you will need to do in order to break through each one.\n\n_Find models._ Whom do you know who has already accomplished the same goal successfully? Model your plan on what has worked for them, and learn from what didn't.\n\nHaving a Big Idea is a good start, but it's not enough to be successful. You need clear goals that are specific, are measurable, and have deadlines. You need a plan in place that tells you what you need to achieve, what could get in the way, and how you will overcome each obstacle. By combining your Big Idea with clear goals and a strategic plan, you put yourself on the right course to bring your idea to life in the next ninety days.\n\n### Start with the Last Page First\n\nWhat is the first step in getting clear on how to execute your Big Idea? Creating a working model for your business.\n\nWe've all been brainwashed into thinking that the best way to do this is to sit behind our desks and write a long, detailed business plan. You know the kind. It starts with a fancy cover and your mission statement and then describes your team, market, product, and competition and the rest of the elements.\n\nMost entrepreneurs spend a lot of time and resources writing their plan. Too often they get feedback during the process from all the wrong people. This includes their friends, their family, and others who want to support them. The problem is, most of these people will tell the entrepreneurs only what they want to hear\u2014that they have come up with the next Google or Apple or Tesla (keep in mind, none of this feedback is coming from customers). By the time the entrepreneur gets to the last section in the business plan\u2014the financial section\u2014he's totally sold on the idea. So he decides to leave the financial section of the plan unfinished or drop it altogether and start executing.\n\nIntrapreneurs face an equally dangerous problem: the temptation to take a basic set of assumptions and artificially blow up the numbers to make the business look more attractive and ultimately get the support needed to move forward.\n\nAnd why not? We are passionate. We are committed. We know we can't fail. So what are we waiting for? _Let's go!_\n\nHere's the problem. I've found that most entrepreneurs change their business model about six times when working through the financial section of their plans. While running the numbers, they identify key distinctions with regard to income and expenses. They gain a deeper understanding of what it will take to break even and how to build cash flow. As a result, they are able to come up with better-informed strategies for attaining the desired financial outcomes.\n\nThe most important part of the initial business-planning process, and the one that people most often neglect, involves getting your numbers to tell a story that makes good sense for you and your investors. If you start at the beginning of the plan, then get to the end only to learn that your assumptions about the business and your plan to execute don't pan out, you will need to start over. In the meantime, you've lost valuable time and money.\n\nIn our three-month time frame to launch, we don't have the time for you to make that mistake. Here's what I recommend.\n\n_Start with the last page first_. Once I have a basic understanding of what I'd like to build, I dig right into the numbers. I create a simple one-page spreadsheet that clearly identifies how the money flows. Basically, I write business plans backward. What I have learned is that once you nail a financial model where the numbers tell the story you want, the rest of the plan will write itself.\n\n_Don't wait_. Don't make this process more difficult than it needs to be. Limit your model to one page. Create the simplest, most basic spreadsheet you can that projects income, expenses, breakeven, cash flow, and the capital required to achieve your outcome. Use conservative assumptions and do not rely on best-case scenarios.\n\n_Get out of the office._ You will learn more about your business by getting into the market than you ever will sitting behind a desk. At least 50 percent of your time should be spent outside the office gathering information that can be applied to your plan. That means getting in front of industry insiders to learn more about the market, talking to prospects about their needs, and testing your competition's products and services.\n\n_Be careful whom you listen to._ When we have an idea we passionately believe in, we are convincing. It's easy for our family and friends to say what we want to hear and tell us we have a winner on our hands because they want to be supportive. But when modeling your business, don't focus on the feedback you get from the people closest to you. The people whose feedback matters the most are current and potential customers. Listen to what they have to say. Apply what you learn to your model. Let their direct feedback and not your enthusiasm sway your projections.\n\n_Don't throw out negative feedback._ Sometimes it's difficult for entrepreneurs to listen to negative feedback. Usually they are so close to their projects and have so much on the line, they don't want to hear that their assumptions may be wrong. As a result, they start rejecting, deflecting, and pushing aside feedback that isn't in line with what they believe, instead of using that feedback to their advantage. Honest and educated feedback is like gold. Use it to open up your mind and ask tough questions about your assumptions. You must be obsessively committed to taking in this feedback and asking what you can learn from it and how you can apply it.\n\nThis is especially important for people entering new markets where they don't have prior experience. Getting feedback from others who have lived in the space will add to your perspective. Sometimes you will learn that there are things that you can't see as a newcomer that would significantly impact your financial results.\n\n_Be open to what the numbers tell you._ The worst thing that you can do is try to manipulate a model to match your initial assumptions. You need to approach your financial model with a completely open mind. Recognize that it will probably take longer than you initially thought to get to market, generate revenues, create profits, and accumulate the cash you need to further invest in the business. The odds are that, by being open, you will be able to make good decisions, apply them to your business, and set yourself on a path to success.\n\n### The One-Page Plan\n\nIf we're going to market in ninety days, we simply don't have time for a ninety-page plan. I like beautiful charts and graphs as much as the next entrepreneur, but frankly, we don't have the bandwidth to focus on that much information\u2014and neither do your potential investors, partners, and team members. As President Ronald Reagan used to tell his staff, \"Give me everything I need to know, but keep it to one page.\"\n\nOnce you've created a sound financial model, what you need to top it off is a one-page \"executive summary\" that clearly describes your business.\n\nAn executive summary is one of the most valuable tools for both entrepreneurs and managers inside big organizations. It should highlight the company's or project's team, mission, market served, target customer, problem being solved, unique value proposition, competitive environment, financial assumptions, exit strategy, and return on investment.\n\nThis document should be clear and easy to read. The executive summary should be shared with your team to ensure that everyone in your fast-moving start-up is headed in the same direction. This summary can also be given to potential investors, partners, and customers to draw interest. And yes, you do need to keep it all on one page.\n\nMany people have said to me, \"But I need more space!\" My response? \"If you can't explain what you want to do in a few sentences, then you don't really understand your business.\" And by the way, nobody else will, either.\n\nI didn't say this would be easy. In fact, when you ask most entrepreneurs to summarize their business in a sentence, on a single page, or in a PowerPoint deck, they can't do it. Instead they ramble on and on. In the real world, giving potential investors too much is the same as giving them nothing at all. They'll throw your plan right in the trash.\n\nThat's why this tool is so valuable. It provides focus for you, your team, and anyone else who touches the business. When you get in front of your target investor, you will be prepared and you will be different. In today's world, that's what you need to stand out and get the deal done.\n\n### An Expensive Hobby\n\nEven before launch, you need to plot an exit strategy. It's a very important step that too many founders skip. They either are too eager to get started, firmly believe they would never sell, or are concerned that focusing on an exit so early in the company's life cycle would just be a distraction. Let me give you three reasons why _now_ is the right time to begin thinking about an exit.\n\nFirst, your investors want to know how they will make money and get a good return on their investment. Will it be from an IPO or sale? If your plan is to sell, who will buy you? Why will they buy your business? How will they value the company? Can you point to comparable purchases to support your case?\n\nSecond, planning an exit will provide a logical way for you to think strategically about your business. For example, what does your target buyer need that your business can provide (such as sales, new customers, new products, or new markets)? What metrics would they use when evaluating the strength of your business, such as revenue, gross profit, net profit, or EBITA (earnings before interest, taxes, and amortization)? Who in those organizations buys companies, and how can you be proactive and build relationships? The answers to these questions could have a big impact on the company in terms of its plans, partnerships, and overall strategy.\n\nThird, at some point you may just need or want to get out. Even if you self-finance or think you will never sell, many reasons could surface that change your thinking. You may at some point decide it's time to cash out. You may feel burned out or just get the urge to do something else. Sometimes business goes sideways or a major change in your personal life makes a quick sale necessary. As a result, I recommend that entrepreneurs have at least two escape hatches or exit scenarios planned in the event they need them.\n\nFinally, consider this: A business without an exit plan is an incredibly expensive hobby. If you're looking to do something for escape or relaxation, starting a new business is not the right move. I recommend fishing instead.\n\nBelow are a few ideas to consider when thinking through your exit strategy in the early days of your business. They can be\u2014and should be\u2014considered simultaneously with your launch.\n\n_Create deep relationships._ Businesses are bought and not sold. The best acquisitions at the highest returns typically come as the result of long-standing relationships or strategic partnerships. These give buyers the opportunity to get to know you and your business and you the opportunity to sell yourself to them.\n\n_Keep your house in order._ You should always keep your financials and other business documents up to date and filed in an organized manner. When the time comes to sell, you will need clean financials to attract buyers and the highest valuations. If your back office runs well, that will instill confidence and make the selling process move faster. Plan and execute from the beginning.\n\n_Get back to work!_ Now that we've had this conversation, stop focusing on your billion-dollar exit and put all your energy back into focusing on serving your customer. Go build a great business and the rest will fall right into place.\n\n### Not Every Problem Is Made to Be Solved\n\nLike many entrepreneurs, I constantly survey the landscape and keep my eye out for opportunities. A few years ago I spotted one. At least I thought I did.\n\nHaving made some good money in my Silicon Valley years, I wanted to reinvest part of my capital into something that would generate passive income. The answer, my new Big Idea, was _self-storage._\n\nI did some homework and found there were about forty thousand self-storage facilities in the United States. But the key fact that really got me excited was that the biggest owner, Public Storage, controlled just _3 percent_ of the market. To me this spelled opportunity, the chance to develop a national brand, roll up the market through acquisitions, and create efficiencies through scale.\n\nIt didn't quite work out that way. Instead, I learned an important lesson: If a problem looks obvious but hasn't yet been solved, there may be a good reason. In the case of self-storage, the problem proved to be that 95 percent of the self-storage facilities were mom-and-pop businesses. These family operations typically consisted of only one facility, and often the owners lived on the premises. At some their kids worked the storefront. Why would they sell if their whole way of life revolved around the business? If they could be persuaded to sell, how hard would it be to get them to make the decision?\n\nAnother problem surfaced too. Once you get the couple who owns the business to move away, you're faced with a new challenge, because then you need to hire someone else to manage and watch the business. You also need to pay that person, an expense that the mom-and-pop managers didn't have because they were operating it themselves. The level of effort, time, and money it would have taken me to acquire hundreds, if not thousands, of family businesses and then rebrand them just didn't make business sense. I could spend a fraction of the energy it would take in another business and likely have better results.\n\nAfter investing lots of time and money working on what I had thought was an obvious winner, I realized that because of my lack of experience in the industry, I hadn't understood certain mechanics of the business.\n\nThis doesn't mean there were not opportunities to build great companies in self-storage. The people who had succeeded already knew that you could not grow quickly through acquisition, so they had developed portfolios from the ground up and worked to build regional rather than national brands. When one of their portfolios came up for sale, they were prime targets for acquisition with very high values.\n\nThere's a larger conclusion to be reached from this experience too, one that you should consider. For generations entrepreneurs have fallen into the trap of thinking that all businesses and industries are pretty much the same. Some think that just because they've been successful in one industry, it will be easy to duplicate their success in a new industry, that one model suits all.\n\nThat can be true, of course, but never forget that every industry has its own character and that business cycles change. No two moments nor any two opportunities are identical. No matter how much you think you know or how good your business acumen, _always_ walk into a new industry with a beginner's mind: open to learn. Do your homework and talk to lots of people before you plunge in. That means talking with people who are very close to the industry to help you understand the nuances of that particular business. It means poring over every piece of material you can find in a search for other companies that either are executing or have attempted to execute the model you are contemplating.\n\nIf you find there is a high failure rate, don't fool yourself into thinking your model is that much different or your team is that much smarter than others. Instead pause, take a deep breath, and learn everything you can about the companies that came before you.\n\nYou should also allow other people to weigh in on your plan, provided they have the background and experience to be worthy advisers. Be conscious of the difference between _opinion_ and _counsel_. Opinions come from people who have no firsthand knowledge of something. Counsel comes from people who have direct personal experience in the area where you need guidance. Sound business planning requires input from the latter. Finally, take a good look at your timing. I've learned that it's better to have a terrible idea in a good market than a great idea at the wrong time.\n\nNot all business problems are made to be solved. If you look at a market, see a problem, and conceive of a great solution, there may just be a good reason nobody else has \"solved it.\" There could be something about the market or the specific industry that makes it a poor fit for what you want to introduce. That possibility makes doing your market research more than a matter of reading a book or a report. It means getting away from your desk and talking to people in the industry. It's seeking expert counsel every step of the way. Coming into an unfamiliar industry with fresh eyes can sometimes work\u2014there's no question about that\u2014especially if you're seeking to apply new technology to an old business. But beware: You may end up paying a high price to educate yourself.\n\nEntrepreneurs can't take shortcuts. You need to build a solid plan, strategy, and financial model that support your goals. You need to do this _before_ you start building your new business, product, or service. This critical step will help to mitigate risk. Unfortunately, too many entrepreneurs, in their excitement to get started, skip right over the planning stage. The business gets off on the wrong foot, and they have to go back and correct all their mistakes, which costs them valuable time and money.\n\nRemember, the best shortcut is taking a long-term view.\n\n## CHAPTER 5.\n\n## SECURING YOUR CREW\n\nFive guys working on the court together can achieve more than five talented individuals who come and go as individuals.\n\n\u2014KAREEM ABDUL JABBAR\n\n### Broke or Billions?\n\nTwo friends of mine, born in the same place, work in the fashion industry. Over the years, each has gained access to very similar resources. Both have charismatic personalities and are natural leaders. Yet one of these women constantly struggles\u2014her business is broke and may have to close its doors\u2014while the other's is now worth over a billion dollars.\n\nThe obvious question: With so much in common, what's the difference?\n\nThe short answer is that the successful entrepreneur had great self-awareness. She recognized early on what she did well and, just as important, where she needed help. She knew from the start to build a team with the right skills around her and then to get out of their way. This strategy allowed her to work her strengths and to hire her weaknesses. Her key tactic\u2014and it's a good one\u2014is to find people who _play_ doing the things she considers _work_.\n\nIn effect, she fired herself as CEO, putting in place someone else with a track record of successfully building companies like the one she wanted to create. This allowed her to focus on product development, the work she does best and enjoys most. The company thrived.\n\nIn contrast, the struggling entrepreneur did two things wrong. First, she tried to do everything she could herself, from scheduling meetings to going on sales calls to making the product and shipping it to her customers. She was always so busy working _in_ the business that she had no time to do work _on_ the business. Second, she put herself in the wrong role. Although she too was miscast as a CEO, she didn't have the insight to hand off the executive responsibilities.\n\nAn important element of a successful launch is to build the right team around you. What do you need to know about yourself and your potential crew to get you from idea to market in ninety days?\n\n_Work your strengths; hire your weaknesses._ One of the best qualities an entrepreneur can exhibit is self-awareness. Know yourself. Are you a visionary? Do you like to take risks? Are you better at setting up systems and managing day-to-day operations than at thinking big? Is there a specific skill (such as sales, marketing, or product development) where you excel? Get clear about who you are and how you can help the organization most, and then focus on filling the gaps by putting the right people around you.\n\nSometimes entrepreneurs, perhaps out of pride, find it difficult to put someone else in the role of CEO. The same hesitation may also apply to other leadership roles. My advice? _Get over it._ This kind of limited thinking will only obstruct your path to your success.\n\nAim to hire people more qualified and smarter than you are in functional areas, and give them the tools they need to succeed. By the way, assigning somebody else a major role in the company doesn't mean you have to give up your influence in or ownership of the company. When your role is in perfect alignment with who you are, you will be able to contribute more and really enjoy what you are doing in the process.\n\nLook at the histories of other companies. The most successful ones were typically started by people who brought in others very early in the companies' life cycles to raise their performance.\n\nWhen Sara Blakely got started, she worked on Spanx at night, filling orders from home and personally going into stores to rearrange product displays. Later, as the business grew, she recognized that at heart she's a visionary rather than a CEO. So she hired a woman with ten years of experience running a licensing division at Coca-Cola to run day-to-day operations at Spanx. She played to her strengths, hiring to cover her weaknesses.\n\nSergey Brin and Larry Page, the creative geniuses behind Google, fired themselves as CEOs and hired an experienced executive, Eric Schmidt, to run the company. Schmidt helped engineer its exponential growth while Brin and Page kept doing what they did best.\n\n_Be ready to substitute players._ As a company grows, people who excelled early on may no longer be the right fit. One manager may be great at taking a product to market but not seem to know how to grow the business beyond a certain point. I've seen people who perform brilliantly at one level but don't know how to shift gears to get the company to the next one; they're great at taking a business to a million or even ten million dollars in sales, but not to a hundred. That's not surprising, because a different set of skills is required to get the company to each new level. It's your job to make sure you always have the right people in the right places at the right time in your company's life cycle.\n\nAt launch one key thing to consider is that employees, partners, and contractors need to be comfortable working in fast-paced, resource-constrained environments where they will likely be asked to juggle multiple tasks. Also, because launching is so time consuming, you must be sure the people on your team do not have any other major obligations during the critical go-to-market period.\n\nBring in the right people, give them all the tools they need to succeed, and then get out of their way. Leverage your strengths and spend your days doing the work you enjoy most. This could be the difference between going broke and building a billion-dollar business.\n\nHOW CAN YOU CONTRIBUTE MOST TO YOUR COMPANY?\n\nThe first step in building a great team is having a clear understanding of your strengths and weaknesses and then filling in the gaps. In other words, fire yourself from a job that does not fit you best and rehire yourself in a position where you can contribute most. Which label below best describes you?\n\n**Visionary:** You love dreaming big and getting people inspired, but you're just not passionate about taking risk and being in charge of the day-to-day execution. If this sounds familiar, you should look for someone who can bring your vision to life.\n\n**Entrepreneur:** You are a risk taker. You have vision, you think big, but you also like the idea of rolling up your sleeves and building a business. You thrive in fast-paced environments and like handling multiple responsibilities. You're comfortable with limited resources and can multitask like crazy.\n\n**Manager:** You are not a wild-eyed visionary and you don't want to take on all that risk. You like to add structure to an organization in order to achieve its goals. You enjoy overseeing a team, building systems and best practices. You pay attention to data, details, and key business metrics. You feel most comfortable in a more stable environment and when executing.\n\n**Sharpshooter:** You have a craft, a unique skill, or a basic knack for doing great things in a particular area. Perhaps you are an artist or salesperson. Do you excel at writing code? Have you mastered a skill that you just love to exercise? You may be less interested in being a CEO or manager because it would get in the way of your working on the things you enjoy most.\n\n### Your Role as Flight Commander\n\nI often ask entrepreneur CEOs and corporate managers the following question: What is your responsibility as the person in charge of running your business?\n\nThis is often met with a blank stare. Other times I get a vague response, something like, \"Well, I do a lot of things. I guess my job is to keep the company or project moving forward.\"\n\nThose are not useful answers. It's no wonder that many entrepreneur CEOs just wander from one crisis to the next without any real thought about how to proactively drive the company forward.\n\nBefore you assemble your team, the first thing you must do is get absolutely clear on what your role and day-to-day responsibilities will be leading the company. You need to figure this out early. Knowing what to focus on will also help ensure you're not spending your time in other people's business and getting in the way of their doing their jobs.\n\nEvery CEO is different, but that doesn't mean we can't learn from the best. That brings me to my old friend Mark Moses. He is the entrepreneur CEO that I look up to most.\n\nMark is just a little bit competitive. On personality tests he rates as a \"super achiever.\" His life story reinforces the picture. Driven to succeed no matter the cost, he founded several successful companies; the last one, Platinum Capital, became a billion-dollar business. Then, after \"retiring\" at the ripe old age of forty, he threw himself into Ironman triathlons. He's competed in eleven of those, even breaking the eleven-hour mark, which is pretty awesome for the Ironman distance.\n\nBut let's go back in time. As a college kid in Canada, he started a company called Student Painters. In his _first summer_ with the business, Mark did $70,000 in sales and made an $18,000 profit, impressive indeed in an era when college tuition topped out around $5,000. The next summer he did $120,000 in sales and cleared $35,000. \"I thought maybe I didn't want to be an accountant after all,\" Mark remembers.\n\nMark knew his results were good, but he decided to see what would happen if he went out and hired talented people, taught them how to model the system he'd built, and helped them to grow prosperous businesses for themselves. In return he would get a small cut. With that in mind, he packed up a U-Haul and headed for California. There he eventually grew Student Painters to 250 branches with three thousand guys and gals armed with paint brushes. I was one of those painters. I became friends with Mark as a college freshman when I opened a branch for him in San Diego.\n\nAfter four years in the business, he sold Student Painters for millions of dollars. Next Mark went into the mortgage-banking business. He kept building the business, but in 1997 he set himself\u2014and his company, Platinum Capital\u2014a new goal of lifting sales over a billion dollars. To kick off the crusade he rented a two-ton elephant and rode it into the annual company meeting. He told his people, \"If we think big and act big, we will be big. Let's do a billion dollars!\"\n\nOvernight the elephant became\u2014and remained\u2014the symbol of the company. The very next day employees started bringing in elephant objects: paperweights, pictures, posters, and stuffed animals. By creating and reinforcing this shared vision, Mark created a rock-star culture. \"People knew us as the elephant company,\" Mark says. \"I was the elephant guy.\"\n\nBut Mark's ride in the mortgage business turned out to exemplify the ups and downs of entrepreneurship that every one of us faces. In 1998, when the Asian flu hit and lending dried up overnight, Mark was forced to lay off 230 employees. In 2000 his business was a day away from bankruptcy, until an angel investor stepped in and loaned the company the necessary capital it needed to keep its doors open. Mark then turned the company around, building a $1.6 billion, 550-employee business. Finally, in 2006, he sold Platinum Capital. Today Mark is a coach for CEOs and entrepreneurs. He teaches them how to figure out where they want to go, helps them build a plan to get there, and holds them accountable for execution along the way.\n\nWhat I particularly respect about Mark is his ability to cut through the garbage and tell it like it is. One concept he works really hard to teach his CEO clients is the central role of the entrepreneur CEO. Having run and invested in quite a few businesses ranging from a painting company to a large lender, his view may be surprising to you: He sees the job of the leader as very similar, whatever the business.\n\nBeing the head decision maker is just the start. With a thank-you to Mark, then, I want to share some of the most important parts of your day-to-day job description.\n\n_The CEO is responsible for the vision and the direction of the company._ This seems straightforward enough, but it's more complicated than it sounds. When I ask CEOs where their company will be three years from now, fewer than one in ten has a good answer. This is inexcusable.\n\nYou have to look at the horizon and know where you want to take your company. This kind of vision differs from a goal on your mission statement; your vision is a measurable three-, five-, or ten-year view of where you want to be. I recommend starting with the sales and profits you hope to achieve and then working backward to see what it will take to make those happen. You need clear drivers, processes, and a form of accountability to manage your company's progress toward your vision. In short, having a vision is pretty worthless unless you engineer the specifics to achieve it.\n\n_The CEO has the ultimate responsibility for cash._ Wait, you ask, isn't watching the cash the job of the CFO, accountant, or bookkeeper? No.\n\nFor any company, cash and access to cash are the lifeblood, the air you breathe. If you run out of cash, you're done. You'd be surprised at how even a company with high profit margins that is growing quickly can run short of cash. The reason for this, says Mark Moses, is that \"growth eats cash.\" You may have to use up cash to add to your working capital or to build up inventory or to finance your accounts receivable. Managing the monetary ebb and flow is ultimately the CEO's responsibility because it is so vital to the health and survival of the business.\n\n_The CEO must ensure that the right people are in the right jobs at the right time._ \"Lots of CEOs fall down on this one,\" says Mark. CEOs may be sentimentally attached to old employees who have been there from the beginning. They refuse to think of letting them go or moving them aside, even if their time of usefulness has passed.\n\nAs the business grows, needs change. The people who got you to one threshold may not be the people you need to take you to the next level. The CEO, hard-hearted though it may seem, has to be dispassionate about hiring and firing. If you can afford only two salespeople, they need to be the two best salespeople you can afford. Some people operate better in start-ups than in more mature environments. Recognize your people's strengths and weaknesses, and don't wait too long to bring in new talent. The wrong people will bring you down.\n\n_The CEO is responsible for key relationships._ The CEO has to \"own\" the key relationships in the company, such as those with bankers, key vendors, the most important customers, and shareholders. Any outsider who wields the power to alter the future of your company\u2014by ordering, selling, lending, whatever\u2014needs access to you (and you to them). You want them to be comfortable calling you up at any hour of the day or night. This is very important because too often business owners hand over these relationships to key employees. But what happens if your employees leave and try to take those accounts with them? You cannot\u2014and should not\u2014keep control of every contact. But know who the key ones are and keep them close.\n\n_The CEO must have processes in place to continue learning._ You did your market research before you started your company, right? Back then you didn't have anything invested. You had nothing on the line. But now that your life and future are tied up in your enterprise, be diligent about keeping up with happenings in your industry, with your competitors, and with your customers.\n\nCircumstances change faster than ever these days, and the annals of recent corporate history are littered with the bones of companies that didn't evolve or adapt fast enough to get out of the way of change. Go to conferences, talk to consultants, and get to know your rivals. Use tools like Google Alerts and LinkedIn to automatically get news of what's happening in your field and outside it. Read trade blogs and e-zines. Look to learn\u2014as well as sell\u2014when you're out there, whether it's one on one or at a convention with thousands. Hire a business coach and put together an advisory board to gain experience from those who can help you.\n\n_The CEO must be a cheerleader._ \"The CEO has to be the chief energizing officer,\" says Mark. He or she must communicate what's going on to the rest of the team, explaining the company's results and getting employees on board with the vision for the future. \"If you want them to buy in to what you're selling, then you have to align their best interests with your best interests,\" Mark insists. Often the best way to do that, he recommends, is through open and engaging communication.\n\nDo you fit the CEO profile? Is this the kind of role you're prepared to play in your company? Are these the things you excel at? In the following pages we'll talk about finding out who you are and what the best role is for you in the business\u2014so you can maximize results and love going to work every day.\n\nTHE COMMANDER'S CHECKLIST\n\n**Vision:** Do you have a clear, specific, and measurable vision?\n\n**Responsible for the Cash:** You can't leave this task entirely to your CFO, accountant, or bookkeeper. It's too important to the company. Always keep an eye on your company's cash.\n\n**People:** You must make sure you have the right people in the right positions at the right times.\n\n**Key Relationships:** What happens if the person you hired to oversee accounts leaves? She probably takes that key relationship with her. You need to own the relationships that are most important to your business.\n\n**Learning:** Work at growing and educating your team. Also, stay on top of the market, competition, and customer needs.\n\n**Cheerleader:** Don't forget, you are the chief energizing officer!\n\n### Three Plays\n\nNow that you've identified the role you will play in your business, the time has come to think about filling in the gaps. Identifying the best players, assigning them to the right positions, and making sure they clearly understand and excel in their roles are critical steps. They're what separate championship organizations from everyone else.\n\nIn the world of the National Football League, Stephen Austin is regarded as the expert of experts, a great evaluator of talent. Over twenty years he built his company, Elite Football Services, into the leading private evaluator of football players in the world. (In 2011, after several hundred players vetted by Austin had been signed with NFL teams, the National Football League bought EFS, making it the league's nationwide producer of combine showcase events.) Stephen's systems are so effective that they have been emulated by the top sports organizations all over the world. The U.S. Navy SEALs have even enlisted his help in finding the next generation of supreme warriors.\n\nI asked Stephen _,_ \"What is it that separates the best players in the world from everyone else? What is the difference between players like Tom Brady, Jerry Rice, Dick Butkus and everyone else?\" His answer surprised me.\n\nIt all comes down to this, he said: \"Three plays.\"\n\nThat's it.\n\nThree plays per game, the three moments in which a superstar does something just a little different from everyone else.\n\nFor a quarterback like Tom Brady, one of those things might be avoiding a sack. Another could be throwing the ball away instead of risking an interception. The third might be drawing an opposing linebacker offside with a change in cadence. Three things\u2014often three little things\u2014may not seem like a lot, but multiply those three plays per game times sixteen regular-season games, four play-off matchups, and a Super Bowl. That's _sixty-three_ plays a year that the other guy, also a really talented player, didn't make.\n\nStephen explains that you can't tell if a person will make those three plays based on how he performs at the combine. (That means your organization must do its homework.) That's why Stephen's pro scouts watch hours and hours of game tape to understand how their candidate has performed in the past. They spend time getting to really know a player before making a decision to select him, talking to former coaches and teammates, collecting references. Investing the time to gather this kind of intelligence up front is one of the hallmarks of championship organizations.\n\nI've found that one of the biggest mistakes busy entrepreneurs make is failing to invest enough time in learning about job candidates before making the decision to hire someone. Many soon learn that one or two bad hires can really set them back. Finding the right people takes time, but probably half as much time as it takes to unravel the mess that can come with hiring the wrong person, bringing in a replacement, and getting the company back on track. Since you want to find those teammates who can make critical plays for you, here are some steps for hiring well.\n\n_Get clear on what you need._ With only ninety days to launch, you don't have time to waste. So begin with a very clear job description. This will save you more time and energy than anything else and will help produce the best results. The description should include the title, a list of responsibilities, the prior experience required, the functional expertise needed, educational requirements, and how much you're willing to pay (being transparent about money is important to avoid wasting time). The more detailed the profile, the easier you will find it to connect with the person you are looking for. Also, the easier it will be for your network to help you identify the right candidates.\n\nFrom time to time you will need to fill a position for which you don't necessarily know all the day-to-day responsibilities or requirements. When this occurs, most entrepreneurs just try to wing it. Since that typically leads to a company's worst hires, I recommend you take a different approach.\n\nFor example, I once needed to hire someone in finance but I wasn't really sure what title, day-to-day job description, and background best fit my needs. I just knew I needed stuff done, so I reached out to a friend whose job it was to recruit people to fill financial positions in big companies. I shared with her what my business needed. She started by mapping out a finance department's organizational chart. She slotted in positions, including a chief financial officer, vice president, controller, accountant, and bookkeeper. She then provided me with sample job descriptions for each one and overviews of their day-to-day responsibilities, what kind of background I should be looking for in each, and the appropriate stages and times to bring them into my company. What I learned from the experience was that I really needed a controller, so we refined that job description and tailored it to my business, and I sent it out to my network. We made the right hire.\n\nWhere should you start looking for help when making key hires? I recommend talking to board members, investors, and other advisers to begin getting a feel for the position. Next, talk to people in your network already serving in the role you plan to hire. Ideally, they work for companies of similar size and at the same stage. Ask them about their roles, their responsibilities, and how they interact with management and their peers. These people can also offer you valuable referrals.\n\n_Don't talk to just anybody._ When you complete your job description and r\u00e9sum\u00e9s start coming in, make choices. Consider only the people who meet all your important requirements. Especially in the first ninety days, you don't have the time, and your investors won't have the patience, for on-the-job training.\n\n_Test, test, test._ Take the time to really get to know your candidates. Do they match the job description? What do former employers, coworkers, and customers say about them? Have you and you team spent time with them both inside and outside the office?\n\n_Identify fast, test slow._ When possible, start the person as an independent contractor instead of a full-time employee. This is a great opportunity to see the person in action and to watch how he or she works with your team. You can ensure that the fit is right before pulling the trigger to give him or her a full-time job. If it's not possible to start the person as a contractor (he or she is employed full time someplace else, for example), give him or her a simple assignment or project to complete during the interview process. It will help you and your team to understand how the person thinks and executes.\n\n_Never settle._ If you do make the hire and it just doesn't work out, get rid of the person as fast as you can and move on with your business. There are plenty of great people out there. Be patient and wait to get the right one.\n\nHIRE A BUSINESS COACH\n\n**Your Secret Weapon:** Google CEO Eric Schmidt told _Fortune_ magazine that the best business advice he ever got was to hire a coach. A coach is someone who brings out your best by providing counsel, support, feedback, and advice. \"I initially resented the advice, because after all, I was a CEO,\" said Schmidt. \"I was pretty experienced. Why would I need a coach? Am I doing something wrong? My argument was, How could a coach advise me if I'm the best person in the world at this? But that's not what a coach does. The coach doesn't have to play the sport as well as you do. They have to watch you and get you to be your best.\"\n\n**Don't Reinvent the Wheel** **:** Remember, there are two paths to success: You can figure it out on your own through trial and error, wasting valuable time and money. Or you can find coaches and role models and advisers who have walked this path already and can show you the way.\n\n### The Perfect Marriage?\n\nThink of the greatest partnerships of our time: Lennon and McCartney. Jobs and Wozniak. Gates and Allen. Ben and Jerry. Jagger and Richards. Hewlett and Packard. Half ended in divorce. Half lasted a lifetime.\n\nA business partnership is a lot like a marriage. It's an everyday commitment, and in the early stages of a new venture you will probably spend more time with your business partner than with anyone else.\n\nPotentially the most important decision you will make in these early days is whether or not to bring in partners to be coowners alongside you. If you do choose to go that route, you need to give as much thought to whom you take on as a business partner as you did to whether or not to marry your spouse. It's that big a deal.\n\nSome partners seem to have been made for each other.\n\nA man named Irv Robbins grew up working in his father's ice-cream shop. Irv's brother-in-law, Burton Baskin, also knew ice cream pretty well; he had enjoyed making it for the troops while serving as a lieutenant in the navy during World War II.\n\nAfter the war, Robbins started Snowbird Ice Cream in Glendale, California. Baskin, who had married Robbins's sister Shirley before the war, ran a menswear shop in Chicago. When he and Shirley moved to LA, Robbins convinced Baskin that selling ice cream would be more fun than selling clothes. Baskin agreed and decided to open his own ice-cream store.\n\nAs Robbins once told a newspaper reporter: \"I was about to sign a lease on a store in Pasadena, and I said, 'You take it. You go into the ice cream business and do the same thing I'm doing. And as soon as we have enough stores open, we can open up a little ice cream factory.'\"\n\nSo two people in the same family opened two ice-cream stores in the same neighborhood. They decided to compete rather than becoming partners because both felt that if, as capable entrepreneurs, they joined forces, the compromises required of a joint business venture might get in the way of their creative ideas.\n\nOver the next few years, however, as each built a successful business, Baskin and Robbins started to recognize that there might be significant benefits in banding together rather than competing against each other. They shared a similar vision. They agreed wholeheartedly to sell nothing but ice cream but make lots of different flavors. They also realized they had complementary skill sets, as one excelled in operations and the other in sales and marketing. They both had unique business networks that, put together, would enable them to reach out further and faster when trying to grow.\n\nThey decided to become business partners in their new ice-cream venture and selected the order of the names of their new company, Baskin-Robbins, with a coin toss.\n\nThey also created one of the great retail concepts of the past century: franchising. Because they both were well versed in what it meant to operate a store and understood the pride of ownership, they determined that the best way to grow was to find managers who wanted to buy a piece of the business. Baskin-Robbins became what is believed to be the first restaurant chain to franchise its outlets. Within five years it had forty shops in Southern California; soon after, it trademarked its 31 Flavors concept. Today Baskin-Robbins has more than 2,800 shops in the United States and 5,000 worldwide.\n\nThere's no single model for partnerships, but the Baskin-Robbins story is a great example of how, with a shared vision, similar temperaments, complementary skill sets, nonoverlapping networks, and hard work, partnerships can succeed.\n\nIn you're thinking about seeking a partner to join in your business, be sure to consider these issues:\n\n_Vision_. You and your partner must share a common vision. You need to be on the same page, running in the same direction and toward the same goal.\n\n_Temperament_. Oftentimes one partner is a natural leader while the other is more of a functional expert. New enterprises require a balance of both energies.\n\n_Complementary skill sets_. Each partner should be able to make a unique contribution to the business. This will help the new venture to move further faster and with less capital required.\n\nComplementary skill sets are also important because they help create clarity with regard to roles and responsibilities. For example, a great salesperson together with a great Web developer could make a powerful combination. But let's say a business is started by two powerhouse marketing experts. Odds are they will argue over roles, responsibilities, and decision-making power because their skills overlap.\n\nFinally, investors prefer to see founding teams with two to three people who each bring something different to the table. Otherwise they feel like they are spending their money to pay for the same position twice.\n\n_Nonoverlapping networks_. Each partner should have a unique network of relationships that the company can draw upon.\n\nPUT IT ON PAPER NOW!\n\nLike marriage, not all partnerships work out: Many end in divorce. That makes the business equivalent of a prenuptial agreement essential. It must be in writing and reviewed by your attorneys. At the very least it should address the following issues:\n\n**Ownership** **:** Who owns what percentage of the company's equity?\n\n**Financial Commitment** **:** What will each one of you put into the company? If more capital is needed, what's expected of each of you? Is the company financially responsible for contracts and accounts, or are you responsible as individuals?\n\n**Time Commitment** **:** How much time and effort will each side give to the company? If you're not expected to give evenly of your time and effort, then adjustments should be made in terms of ownership and compensation.\n\n**Cash Compensation** **:** How much do you get and when will you get it? Is it based on hitting certain milestones or on hours spent on the business? Will compensation be in the form of salary or distributions? Do you have to take cash, or could you exchange it for more shares instead?\n\n**Expenses** **:** What's your policy on expenses? How much entertaining can you do? Who writes the reimbursement checks?\n\n**Voting Rights:** If one person has more shares, does he make all the rules? If you're fifty-fifty and don't agree on something, what do you do? Do all big decisions need to be unanimous?\n\n**Vesting Schedule** **:** What happens when one partner leaves the business?\n\n**Buyout** **:** Under what circumstances can you sell your interest in the company? Whom can you sell to? What will the terms be? How will the value be calculated? Make sure you have in place a vesting schedule (a simple plan that outlines how and when equity is distributed).\n\nYou must agree up front.\n\n### This Is What You Asked For!\n\nYour company wants to build something. You want to go to market fast. Having managed to scrape together enough money to add a couple full-time employees to your team, you need to launch before your competition catches up. Due to your finite budget and time constraints, the realization dawns: You need to outsource.\n\nToday outsourcing is often a critical part of launching a new business or product, but it's an area where many people fall down.\n\nA few years ago I had started to launch an online marketplace in the health-care industry that would provide seniors with better prices and easier access to a variety of services. I tried hard to assemble a team of developers. I wanted to get something up quickly and test the market response. All the best people I contacted were busy on other projects, so I reached out to a few friends, asking what they would do. One told me that to solve a similar problem, he'd hired a firm in India to build his company's Web site.\n\nI decided to do the same. I contacted the outfit my friend had used and we had a one-hour call to get acquainted. I shared my idea over the phone, and they responded with an overview of how they would execute my vision. We negotiated a price and signed an agreement. I was in business.\n\nA month later I got an e-mail with a link to my new Web site. It was a disaster! The graphic design, the look and feel, the user experience, and the functionality were all wrong. To make matters worse, I had planned to use the site for an investor presentation later that week. Now I would have nothing to show. I was extremely frustrated. I just didn't understand why there was such a big disconnect.\n\nIn outsourcing, I had set aside everything I knew about hiring and running a project. I had given the consultants a general project description, expecting them to fill in the blanks when I hadn't really thought through everything myself. I had put my entire business in their hands instead of just giving them one product or feature as a test. I hadn't checked in along the way, so the first thing I had seen was their finished product. As a result, I had had no idea they were on the wrong track. Because I had been so vague with regard to what I wanted, they had been forced to make decisions for me. While some were good, most went against my vision for the product.\n\nWhen I told them how upset I was and asked them to explain themselves, they said, \"This is _exactly_ what you asked for!\" And frankly, they were right.\n\nHow do you avoid some of the common pitfalls in outsourcing?\n\n_Be disciplined._ It doesn't matter if your consultant is an overseas developer or someone right across the street. The same rules apply as far as clearly communicating your expectations.\n\n_Hire right._ Do not take anyone's word for work quality. Operate with the same amount of diligence you would when making full-time hires. Check references to get a better picture of past performance. Agree upon a very clear job description and, if possible, run a test by executing a small project together before handing over too much responsibility. If you're not getting what you want, don't waste time. Cut your losses and move on.\n\nEven if a consultant is recommended by valued friends, use caution. I've found that most people are not very careful when referring consultants, who may be between gigs, out of work, or juggling lots of projects. Some are friends of the people making the referral and the favor is being done for them, not you. Or the referrer knows them socially rather than from a business relationship.\n\n_Provide very clear direction._ If you paint an incomplete picture of your intended outcome, you'll get something that only vaguely resembles what you had in mind.\n\nMany entrepreneurs have a vision with no concrete idea of how to execute, so they hand the job over to someone else, expecting that person to magically fill in the blanks. With a full-time employee, that may work. Your staff can plug in details; that's one of the things you hired them to do. And since they are right there in the office, you can exercise some direct control. This gives your team the benefit of receiving feedback every single day.\n\nConsultants, on the other hand, are typically not in the office, so they are not as in sync with the day-to-day business. As a result, they can't fill in certain gaps in the direction they are given. Therefore, in most cases they do only what they are told. That's usually a good thing, because if they start getting creative, they could take the project in the wrong direction. Therefore, you have to work twice as hard with consultants to make sure you provide excellent direction, management, and oversight. I always recommend that a company have its best manager oversee all consultants during the launch period.\n\n_Use examples._ When dealing with consultants, it may be helpful to find examples of other products, services, features, etc. that you want to use as models in order to translate your vision into something tangible that others can easily grasp.\n\nFor example, if you are developing an e-commerce site, what is the specific desired user experience? What are the steps in the buying process? How do you picture that in your mind? The clearer you can be, the more likely you are to get the result you want. Point consultants to two or three comparable Web sites you like that they can use as models or guides when building yours.\n\n_Check in regularly._ Create a regular schedule for checking in with your consultant and providing feedback. The more often you provide feedback, the happier you will be with the end product.\n\n_Manage against clear deliverables._ It is important to specify key milestones and delivery dates. That keeps both sides honest and up-to-date, helping to ensure the contractor is making progress.\n\n_Pay based on performance._ Sometimes the supplier will want everything up front or payment at regular intervals, but a better approach is to pay based on performance. What you see is what you pay for.\n\n_Take responsibility._ If your first reaction on seeing your product is, _Hey, this isn't what I intended,_ that's not the end of the world\u2014and another reason why we only build one thing at a time. Don't take your disappointment out on your contractor: How much of the fault was yours for not being clear or for not checking in? You may decide after seeing a prototype that in fact you want something different altogether. Instead of losing your temper, outline what you like and what you would like to improve. Communicate this feedback in a constructive way.\n\nAlthough the inspiration, the initial hard work, and the passion for your Big Idea may rest solely with you, the entrepreneur, you cannot overestimate the importance of having the right team in place. Spending time and energy on hiring the right people (and quickly letting go of the ones who don't work out) can truly make the difference between a business that launches successfully in ninety days and one that never gets off the ground.\n\n## CHAPTER 6.\n\n## UNLOCKING CREATIVITY\n\nThe problem is not the problem. The problem is your attitude about the problem.\n\n\u2014CAPTAIN JACK SPARROW, _PIRATES OF THE CARIBBEAN_\n\n### Torn Carpet, Chipped Paint\n\nIn the middle of the dot-com boom, the market was hot. Companies were springing out of the ground almost overnight, and I was getting calls every day to help them launch.\n\nIn 1998 I decided to test the job market and had a series of interviews in Silicon Valley. I remember walking into beautiful lobbies with expensive furniture. People were moving around the office on scooters, and one company even had a slide in the middle of its workspace. All these companies were trying to sell their cool cultures.\n\nI was about to take a job offer when a good friend from my days at Quote.com, Russell Hyzen, gave me a call. He had gone off to become the first employee at Xoom.com, an online community and e-commerce business. It was looking to fill a role, and he knew I was in the market.\n\nWhen I arrived for my interview, I found that Xoom's offices were in a beautiful landmark building off California Street in downtown San Francisco. But inside? I almost fell over when I saw the torn green carpet and the chipped paint falling off the walls.\n\nThere was no receptionist. Instead, Chris Kitze, the company's founder and CEO, had an office near the front door. He was the first person to greet people as they walked in.\n\nI soon learned that Chris had a vision, one that became a shared vision of everyone he brought into the company. He hired the best people, gave them access to the resources they needed\u2014and then got out of their way. He was absolutely committed to seeing everyone on the team win: If the company made money, they would too.\n\nWhen I talked to the team, there was a pride of ownership that I hadn't seen anywhere else, and a common purpose and community that wasn't present in the fancier offices. I could see why many of the people I met at Xoom had followed Chris from one company to another.\n\nThe company had money, so I had to ask: _Why the torn carpet and the chipped paint?_ \"That's not important,\" Chris said _._ As an owner, he believed the best use of funds was to invest the money back into the company, into hiring the best people, into building a great business. If he did that, the rest would take care of itself.\n\nUnlikely as it may seem, that torn carpet and chipped paint became a symbol of pride everyone shared within the organization. It is part of what got me to join the company, and our team built a great business, took it public, and eventually merged with NBC to create NBC Internet.\n\nThe culture didn't stop at pride of ownership. It was also sales driven. Everyone knew our number one goal was to generate revenues. We knew that without cash coming in the door, we had no business. Chris set the tone, signing all his e-mails with the tagline \"Go Sell Something.\" Every day, in every e-mail, those words reinforced the focus, and not just with salespeople. Everybody in the company came to think of himself or herself as a salesperson, which meant he or she approached problems and the product in a way that was very customer-centric.\n\nSoon after I arrived, I saw how valuable the sales-driven culture could be. Times got tough in the second half of 1998. I returned from a vacation to find the stock market going sideways. As a result, we lost the investment bank that we had thought would take us public. We were nearly out of cash, but Chris's culture saved us. Even though many of the most talented people could easily have found great jobs at other companies in those days of the Internet boom, nobody jumped ship. We had a sense of community and knew we could sell our way out of any problem by generating the cash we needed to get through it.\n\nWhat are the key factors in creating a great culture for your new business?\n\n_Establish a clear vision._ That way everyone runs in the same direction. How do you create a clear vision? Ask yourself this question: _Where do we want to go?_\n\nThe answer should be a clear description of what you would like to achieve over a specific period of time. This creates a framework for the company and serves as a guide for choosing current and future courses of action.\n\nA vision statement is different from a mission statement. A vision statement answers the question \"Where do we want to go?\" A mission statement answers the question \"Why do we exist?\" A vision statement is shared within the company. The mission statement is shared with the outside world.\n\n_Hire great people and give them access to resources._ Invest money in great people. Give them access to all the resources they need. Help them solve problems and remove roadblocks so they can do their jobs. Then get out of their way.\n\n_Be absolutely committed to seeing everyone on the team win._ From the top down, everyone must be committed to seeing the team as a whole win. As an entrepreneur (and the person taking on most of the risk), you are certainly entitled to earn more in the long run than anyone else. But shared ownership, structured in the right way, in which everyone knows their role and the impact they are having on results, creates a culture that can't be beat.\n\nThat may sound like a simple recipe, but too many companies fall short of it, and as a result they don't succeed at the level they should. And this isn't about getting one of these things right. As a leader you need to be committed to all of them. If you are, you will significantly increase your chance of success.\n\n### Vegas and Video Games\n\nBuilding a culture of teamwork is only one part of the equation. You also need to build a culture of innovation. How do you inspire people to open their minds in a way that unlocks creativity within your organization and achieves the best results? Let me share a story about how a former colleague named Ed attained legendary status for his ability to unlock creativity ( _and_ have a great time in the process).\n\nEd's goal was to change the way that products were marketed in his client's business. He saw that everyone in the industry had been speaking to customers in the same old way for years. So he decided to turn things on their head. He needed to unlock his team's creativity, and to do so he made a pretty radical move: He took the whole creative and account team to Las Vegas.\n\nHe told them almost nothing about what he had planned for the trip.\n\nOn day one the team arrived and partied, going to the best restaurants and clubs and doing some gambling. Some of the crew stayed up all night. No work got done.\n\nOn day two Ed told his people there was nothing on the agenda other than having more fun. So they did more of the same: eating, drinking, gambling, clubbing.\n\nDay three? Still no agenda. Nothing. So they went at it again: best restaurants, best clubs, a nonstop party.\n\nOn morning number four, Ed's tone changed. Over the past few days, he had helped his team leave their lives, stresses, and responsibilities behind. Now he wanted something from them. He gathered the group and told them, \"We have a mission. Our mission is to revolutionize the way products are marketed and sold in our industry. We need to be creative, different, and unique. We need to appeal to our client's core young male audience\u2014and they play lots of video games.\"\n\nWhile Ed was speaking, video-game systems were being installed in each team member's room. For the next two days, he instructed, they were not to leave their hotel rooms: Their job had become to play video games. That was it! They were supposed to think about the whole experience of playing a game, including the graphics, the sounds, and the characters that appealed to the target audience. He wanted them to consider how game designers presented content, how they manipulated the look and feel of the product for the customer. He asked them to pay attention to the scoreboards, the sound effects, and every other little trick the gamers had devised. Most of all, he wanted his team to think about how his client could build upon video-game innovations to make its real-life products win a new generation of customers.\n\nAfter three glorious days in the sun, his team wasted no time in unleashing their creativity and creating a variety of innovations that turned their client's industry upside down!\n\nCould Ed's team have accomplished the same results playing games back in their offices in Los Angeles? I doubt it. By taking his team to Las Vegas, he shifted them from _work_ mode to _play_ mode. He put people together who normally did not have much time to interact in a fun, social setting. The days of partying in Las Vegas brought them closer, building camaraderie and a sense of teamwork. Two guys in a hotel room playing a video game see their jobs differently from how they would seated in adjacent cubicles, looking at their computers.\n\nNow I'm not saying you need to take your team to Vegas (or anywhere else) for a week. But to foster creative teamwork, you need to find a way to get them out of the office mind-set and open their minds. You need to tap the genius within each of them and do so on a regular basis. Being creative applies to all aspects of your business, from building something cool to breaking down barriers when trying to sell your product.\n\nPart of Ed's legacy is that creativity has become an essential part of the culture at his company. They are still thinking outside the box, still building great products. You can too, with the following recommendations.\n\n_Bring groups together_. When you are launching a new business or product, team members are busy. All of them have their heads down, focused on what they are responsible for. Most of their time is spent working with other people in their department, meaning business, creative, and technical people don't get much time to interact. There are ways to change that.\n\nAt Xoom we had pizza and beer at three o'clock every Friday afternoon. People from different groups got to meet and interact in a social environment. As the company grew, those Friday afternoons offered an opportunity to meet new team members, and together we evolved a shared vision of the company. Even if people with different responsibilities looked at the company in different ways, over beer and pizza we came up with some of our best and most creative\u2014and collaborative\u2014ideas.\n\n_Encourage risk._ You win some, you lose some. But people always need to be encouraged to take risk. They need to know that even if something does not work, if risk has been mitigated the experience is a valuable part of growing a business. The best way to do this is to make lots of small bets. That way, if something does not work, there is no threat or concern that it will put the entire operation or the person's job at risk.\n\n_Get out regularly._ Do something outside the office at least once a quarter. The activity doesn't have to be expensive or extravagant. You might schedule a happy hour, miniature golf, or go-cart racing. Whatever it is, getting people to engage in a new setting can build relationships and inspire creativity.\n\nOnce you've developed a sound business plan, put the right people in place to execute, and created a culture that encourages openness, innovation, and risk taking, you will have built a great foundation for the business to succeed. You will also be in a position to move to the next step in the process: raising capital.\n\nOPENING MINDS\n\n**Combine Business with Pleasure:** Take your workers out to play\u2014to the ballpark, to karaoke, for a brainstorming session over beers. Bring pizza into the office on Friday afternoons. Do something different on a regular basis. Getting your employees out of their comfort zones and interacting with one another can open their eyes to new possibilities.\n\n**Encourage Experimentation** **:** Declare that for a few hours every month no one is allowed to use their computer or their smart phone. In place of screens, issue every employee a notebook and ask them to spend the time writing, sketching, and diagramming their thoughts and ideas on how to improve the business and innovate. The shock of putting pen to paper will force them to think in a different way. After the exercise, encourage them to tear pages out of their notebooks and post them on a brainstorming wall.\n\n**Make Small Bets** **:** Encourage your team to experiment with side projects. Many of these will fail, but one of them may turn into your next big idea.\n\n**If Something Doesn't Work, Celebrate the Fact That Your Team Took a Shot** **:** People who are the most successful and creative are the ones who take risks. They tried, they failed, and they learned from their mistakes. Create a culture where taking a shot at something new is celebrated, as long as something is learned.\n\n## CHAPTER 7.\n\n## RAISING CAPITAL\n\nAnd where I excel is ridiculous, sickening, work ethic. You know, while the other guy's sleeping? I'm working.\n\n\u2014WILL SMITH\n\n### Trading for Tickets\n\nMost entrepreneurs start businesses from the same place: _nowhere_. Despite beginning with little or nothing, the best of them bootstrap, and through their own sheer will, hard work, and effort they find ways to make things happen. They manage to be resourceful and pull together what their business needs to launch.\n\nAs an entrepreneur you have to figure out ways to keep pushing your business forward, largely by yourself. You have to take whatever you have that is of value and leverage, trade, share, or swap it. That often means that founders of bootstrapping start-ups forgo paychecks, offer their services as consultants, give away early products and services for nothing, and do lots of moonlighting, at least until the new venture generates enough cash to survive on its own.\n\nIntrapreneurs also self-fund, by adding the new project to their existing responsibilities. They effectively take on twice as much work with no additional pay to help get their idea off the ground. They have to become masters of juggling not only their core business but also their new-project responsibilities. Since management does not want to see either one struggle, intrapreneurs take on not only reputation risk but also job-security risk. Only once the new project hits certain milestones may the intrapreneur be able to get more resources for the new project and a bigger paycheck.\n\nBootstrapping is more vital today than ever. Many investors are reluctant to put money into a venture until it is already generating positive cash flow. Compared with the days before the global economic crash of 2008, investors now have far less appetite for risk, meaning most won't reach into their pockets until you've taken your business beyond the early stages and proven your model. As a result, new business owners have to bootstrap longer than they did in the past.\n\nMy first exposure to bootstrapping came compliments of Tony Robbins. I wasn't raising money for a new business, but it sure taught me how to be scrappy. I was twenty-one years old, and my title was \"field sales representative and national sales trainer.\" As an advance man responsible for filling the seats in Robbins's seminars, I would travel with a handful of my peers to a city where Tony was to speak, and we would promote his event. The primary program I sold was called \"Power to Influence,\" and it focused on sales and persuasion.\n\nBefore our team went into the field, we immersed ourselves in the content. We had to be passionate about Tony's message, because our task was to drum up interest and sell tickets for his programs. We spoke to small groups, sometimes five times a day, seven days a week. We'd share a few program highlights, talk about what people would gain by attending, and try to sell as many tickets as we could in order to make big sales commissions.\n\nIn retrospect I realize that this experience not only taught me about sales and persuasion but also gave me a bunch of survival skills that would later come in handy when I became an entrepreneur.\n\nI spent two to three months at a time in cities like Detroit, Toronto, Vancouver, Boston, Houston, Seattle, and Los Angeles. No one on the team collected a salary and we didn't have expense accounts. We paid for our own rental cars. Nobody even had a phone! Basically, we were given a plane ticket, a place to stay, and a couple of initial bookings. After that we had to live off the land.\n\nWe did, however, have something valuable in our pockets: Our currency was tickets to Robbins's events. We learned how to barter tickets and our own speaking and coaching in return for what we needed. This was a key moment in my business education, a sort of Bootstrapping 101.\n\nHere's how it worked. When we got off the plane in a new city, we would decide who was going out in search of what. The first priority was always to get cars and cell phones. We would split up and go to car dealerships, where we would talk general managers into giving us cars for a couple of months. In exchange we'd provide in-house training and give their entire sales teams tickets to one of Tony's events. Same thing at cell-phone stores and gyms. One guy even bartered for time at a tanning salon.\n\nLike a pack of lions hunting, we had to work together as a team. We learned that as long as you have both persistence and a deep supply of something with perceived value (in our case, Tony Robbins tickets), you can catch all the antelope (or cars and phones) you need to survive.\n\nThat same scrappiness applies to bootstrapping your own business. You have to be creative, mastering the art of trading with others. It's about offering value for value.\n\nHere are a few questions to get you started:\n\n_Where do you need help?_ What do you need that you can't afford to go out and buy? Do you need a Web site, design, marketing, social-media, accounting, or legal advice? What will each of these things cost?\n\n_What's your currency?_ What do you have of value to offer? Can you provide consulting help with others' businesses, introduce them to key prospects, or provide a service that they need?\n\n_Who are your trading partners?_ Take a close look at your network and identify other entrepreneurs you can barter and partner with. What you will find is a trading economy in today's market you probably did not know existed. And don't be afraid to go to prospective customers. You might be surprised how often they will be willing to give you access to certain resources in exchange for being first in line when your new product or service is ready.\n\n_Where's the cheap labor?_ Even if you can't afford to pay big salaries yet, there are many people out there who will work for free in exchange for a promise of future compensation. These could include people out of work, in transition, or trying to find something part time. The opportunity to get in on the ground floor of something they believe in may be enough to get them started.\n\nCreate a list of people you know who have what you need. Prioritize the list, placing the services you need most at the top. Find out which people in your network offer those services _and_ need the most help with their own businesses. Then see if there is a way to trade, borrow, or swap services in a way that helps both parties move their businesses forward.\n\n### _Cannibal Women in the Avocado Jungle of Death_\n\nLike it or not, a new business sometimes requires a capital investment beyond the bounds of bootstrapping. Unfortunately, money, as the saying goes, doesn't grow on trees (even if, in reading about some start-ups, it seems like people can't give it away fast enough).\n\nThe hard facts are that less than a third of all businesses ever take any outside capital, and of the ones that do, most have typically been in business for three years or more. Making your pitch to a series of potential money sources can feel like a distraction while you're trying to run a business. It will involve working your network, setting up meetings, putting together materials, following up, and closing the sale.\n\nRaising capital takes persistence, whether you are an entrepreneur pitching to individual investors or an intrapreneur working to build consensus for funding within a big organization. Most investors probably won't agree to put money into your company, but that doesn't mean you stop looking. Go into each meeting with the objective of listening to all the feedback the investors have to offer and applying what you learn going forward. For example, your investors may have excellent ideas on how to improve your pitch, find additional revenue, reduce your start-up costs, separate yourself from the competition, and find strategic partners. By listening and following good advice, you're bringing yourself one step closer to finding the finances you need.\n\nI've always been inspired by my friend Gary Goldstein, whose story is a classic tale of persistence. Gary started out as a criminal-defense attorney, but his real passion was storytelling. So he decided to move to Hollywood and launch a career in the entertainment business. There he managed young writers and directors before making the decision to produce films. Gary went on to produce some of the top box-office hits of all time, including _Pretty Woman,_ the highest-grossing live-action film in Disney's history and a beloved classic.\n\nWhen Gary decided to produce his first film, he didn't have all the money and resources he needed. He teamed with an up-and-coming writer. The two bootstrapped as long as they could. Then it was time to go and raise outside capital to get his film, a comedy-horror thriller, made. That part certainly wasn't easy, but Gary never thought about quitting. Making a film was his passion and, as he told me, \"you have to chase your passion like it's the last bus of the night.\" Through Gary's persistence and pigheaded determination, he raised two hundred thousand dollars to make the film. It was titled _Cannibal Women in the Avocado Jungle of Death._\n\nIn addition to having a tight budget, a scheduling conflict left Gary and his partner with only four weeks from the day they closed their financing to cast, film, edit, and deliver the final product. That may sound impossible to us, but it wasn't to Gary. He was committed to making this project a success. He did not believe in waiting for the \"perfect time\" or \"perfect circumstances\" to get started. Instead he believed that being in the game was better than waiting around on the sidelines. With that mentality, he devised a plan to spend two weeks preparing the project and two weeks for everything else.\n\nBut at the end of the first week, Gary was up against a real problem. The film had no cast and crew; it was just Gary and his writer. Instead of shutting down, Gary persevered, because he believed that \"success is far more likely when you choose persistence over doubt and inaction.\"\n\nGary and his writer hit the phones, called all their friends, asked them to call their friends, and invited them out to the set. They decided if someone showed up who dressed well, that person would be in charge of wardrobe. If someone had a camera, he or she would be the cinematographer. Casting consisted of seeing who arrived on the set each day. This was guerrilla filmmaking at its best. Somehow that film, which ended up starring Shannon Tweed and Bill Maher, came together. It ran on cable for the next fifteen years and became a cult classic. The experience was successful enough that Gary was able to dissolve his management firm and become a full-time independent producer.\n\nI asked Gary what he learned most from that experience. \"You don't have to be a master at something to achieve something worthwhile,\" he said. \"Just by sheer will and stubbornness you can accomplish anything you set out to do.\"\n\n### The True Path to Raising Capital\n\nStarting a business, raising money, and executing your Big Idea is not easy, but people do it successfully every day.\n\nWhen it comes to raising capital, the challenges most entrepreneurs face are inexperience and overly optimistic expectations with regard to how much time and effort raising capital really takes. Many spend their time talking to the wrong people about their investment, and when the opportunity hits them, they are not armed with the tools they need to successfully close the deal.\n\nIf you combine these strategies for raising capital with the lesson of Gary's persistence and determination, you will be successful. But before we get started, keep these considerations in mind.\n\n_Raising capital takes time._ Many people think that investors will jump on their Big Idea right away and that they won't be out there raising money for long. From the time you make your first fund-raising call, figure on six months to a year to close a big round of financing. If you would like to speed that process up to, say, ninety days, place your focus on raising a very small amount of capital with the purpose of building and testing a basic product and taking it to market. It is much easier for investors to make small bets and put less money at risk. Creating proof of concept for a basic version of your idea in such a short period of time also makes it much more likely that you can bring these investors back for more capital\u2014and at a higher valuation.\n\n_Raising capital takes hard work._ This process alone can seem like a full-time job, but you need to get in front of a lot of people. I estimate that for every twenty-five to thirty qualified investors you pitch, you will close one.\n\n_Raising capital can be very stressful._ Your potential investors are busy. They can be tough to get on the calendar. They miss meetings. They go on vacation. You often won't get the answer you expect. You may not like what they tell you about your project. Managing your emotions and staying focused despite delays and setbacks is essential. Keep asking yourself, _What can I take away from that meeting or prospect that can be applied to my business or fund-raising? How do I use those lessons as fuel to close my next prospect?_\n\n_Keep your business moving forward_. Raising capital takes time, but that's no excuse to stop executing your plan. You will need to bootstrap, be resourceful, and look for every opportunity to continue building while raising capital, balancing both objectives. Continuing to build will pay off, as you'll find that even small, incremental steps make a huge difference in terms of getting investors. The more of your idea they can see coming to life, the easier it is for them to get their arms around you, your team, and your product. The further along you are, the less risk is involved for an investor, meaning you will get better financing terms on your deal. Staying at it pays off. Be like Gary: If you knock on enough of the right doors, with the right story and materials in place, you will eventually get your shot.\n\n_Don't settle for \"no.\"_ I have amazing kids and I think they would be great at raising money. Why? Because kids just won't take no for an answer. They keep coming back. They face rejection. It doesn't bother them. Instead, they learn from it. They take note of what didn't work. Then they come right back and try another way to get what they are after. Don't be afraid to be a kid again!\n\n### Where Will the Money Come From?\n\nSo where is all this money going to come from? Who's going to invest in you? There are several options when launching a new business or product. To help save time and get you to market faster, here is a list of capital resources. To get started, put most of your energy into the source that is the best fit with your personal network, business goals, and stage of investment.\n\n_Friends and family._ If you are an entrepreneur launching a new business or taking a new product to market, those closest to you are usually the very first capital providers you should consider. Your friends, family, or company may make funds available even when your new venture is still in the idea phase while other professionally managed sources of capital may not be options before there's a product or prototype to validate the idea. Most friend or family investments range from $1,000 to $25,000, and they may be in the form of equity (ownership in the company) or debt (a simple loan).\n\n_Angel investors._ Angels are wealthy individuals, often entrepreneurs just like you, who are willing to invest anywhere from $25,000 to $250,000. Angels typically work as individuals or through clubs with other angel investors, putting money into start-ups and early-stage businesses. Apart from monetary investment, they may be looking to be involved and can be great coaches for the enterprises in which they invest. Their investment is typically in industries with which they are familiar and can be equity or debt.\n\n_Crowdfunding_. This source makes use of online and off-line networks to attract informal (and sometimes professional) investors to buy into an idea or company. The concept of crowdfunding is to generate investments from the masses, to get small amounts of money from a large number of individuals. There are many forms of crowdfunding, including donation-based, equity-based, revenue-sharing, and peer-to-peer lending.\n\n_Venture capital._ Venture capital is almost always provided in exchange for equity in your company. VCs are very sophisticated investors. Although they have a financial focus, many can also provide significant managerial, operational, and technical expertise. In return they look for a bigger portion of equity and sometimes even a controlling interest (meaning they assume more than half of the equity and\/or decision-making power in your company). Investments from venture capital typically begin around $250,000 and can go up to millions of dollars.\n\nOTHER OPTIONS FOR FINANCING YOUR BUSINESS\n\n**Business Incubators:** Incubators are programs designed to support the growth of early-stage entrepreneurial efforts. Often they will invest between $5,000 and $25,000. They will work very closely with the company to help get the business in the best shape possible to raise more capital and go to market. Incubators are typically industry specific (for example, technology or energy), and their partners bring significant industry experience, resources, and contacts.\n\n**Business Accelerators:** Accelerators are similar, but they tend to focus on companies that have gone to market and are beginning a growth phase. Many business accelerators are backed by venture capitalists who want to get a foothold in what they believe to be a good emerging business.\n\n### The Big Meeting\n\nIf you've hung out with founders as much as I have, you've inevitably heard the following line: \"I have a big meeting with an investor coming up.\" Because the entrepreneur got the appointment, he's certain the investor has real interest. The entrepreneur's growing enthusiasm leads him to think the money is in the bag\u2014only to come back from the big meeting empty-handed and deeply disappointed at the result.\n\nFor the typical entrepreneur, raising money is an entirely foreign experience. The challenge is, investors often have been doing their jobs for years. Your inexperience puts you at an immediate disadvantage. We need to get you up to speed quickly; with just ninety days, there is no time to waste. The place to start is making sure you get yourself in front of the right people. The process starts with doing your homework.\n\nInvestors have a mandate to make money, and typically they have a specific formula to mitigate their risk. That makes it imperative that you understand their focus. Some things for you to consider include:\n\n_Background_. What companies have your potential investors worked for? What's their functional expertise? Often they will invest only in what they know. As an example, if they spent their career in real estate, they will have a deep understanding of how to analyze the true value of a real-estate deal, how to get involved in ways that help the entrepreneur, and how to step in and help turn the ship if things go sideways. Because of all this, they'll be more likely to invest.\n\n_Previous investments._ Know the kinds of companies they have invested in in the past (the typical real-estate investor probably won't invest in, say, software). If you pitch someone in a consumer Internet business and he or she invests only in energy, it won't matter how good your idea is\u2014he or she will pass. Be aware, as well, that if you are dealing with a venture-capital or professional-investment firm, you could be in the right place but talking to the wrong person. Find out which investor inside fits your deal best.\n\n_Stage of investment._ Does your potential investor favor seed or early-stage companies, have minimum revenue criteria, or have other key requirements before stepping into a deal? If you present your business at a stage other than your investor's sweet spot, he or she probably won't be interested. That is true of both individuals and large firms. Keep in mind that many professional investors raise money from other investors with a mandate to invest in a specific type and stage of company.\n\n_Size of investment._ Ten thousand dollars? A hundred thousand? A million? Be clear that what you're asking for is in line with what they put into individual investments.\n\n_Expected return._ Are they looking to get double, five times, ten times what they invest?\n\n_Investment horizon._ When do they expect a return on their capital? What time period do they have in mind\u2014to get in and out within a year, three years, or five? Everyone needs to have the same expectations on day one.\n\nHow do you find these things out? Do your homework. If you can't find an answer to a specific question, just ask. Asking good questions will help you establish yourself as professional and someone who respects a potential investor's time, which earns you bonus points too.\n\nKnowing these things in advance will help you operate with speed and efficiency, saving you valuable time and money. It will help ensure that every meeting is with a qualified investor. That way your \"big meeting\" will be exactly what you expect.\n\nOne last thing: _believe._\n\nRaising money is no small task. No matter how good you and your product are, it is a time-consuming and labor-intensive process. But when you've done your homework and go into meetings with the right people, you will know that, on some level, they really want to invest in you. Their mandate is to find companies just like yours. They are hoping that you are the person who can deliver what they want. In fact, they probably want this meeting as much, if not more, than you do!\n\nSo do your homework, get in front of the right people\u2014and execute once you are there.\n\nLOANS VERSUS CONVERTIBLE EQUITY\n\nWhat's the difference?\n\n**Loans:** Loans are simply an amount of money lent by one person or entity to another person or entity. In general, there is an interest payment (an annual percentage of the original loan amount). Things to consider include guaranteed or nonguaranteed, simple or compound interest, and the amount of time given to repay the principal.\n\n**Equity** **:** **** Equity is an ownership interest in a business. Equity holders negotiate a variety of terms associated with certain rights and controls.\n\n**Convertible Notes** **:** These are similar to loans except that, at the end of the loan period, the investors can opt either to have their principal returned (as with a standard loan arrangement) _or_ to convert that investment into equity in the business. Convertible notes have been very popular for early-stage companies in the last few years because they are a low-cost means of closing financial deals (the legal costs are substantially less than with other options). In addition, investors sometimes favor them because they typically earn high interest during the start-up phase while retaining the option to leave if the business doesn't perform as expected.\n\n### The Fastest Way to the Door\n\nHaving spent years both in businesses at all stages of raising capital and investing in other people's companies, I've been on both sides of the table. As a result, I've learned firsthand what to do\u2014and what not to do\u2014when presenting to an investor. I've also learned there are certain mistakes you can make that will kill a potential investor's interest and lead you right out the door.\n\nWe only have ninety days to go from idea to market, so avoid the mistakes below in order to keep the process moving forward and ultimately get the results you want.\n\n_Appropriate domain and stage expertise._ Investors fund people, not ideas. They want founders and management who have deep domain experience. In the early stages they like a background that includes running fast-paced, resource-constrained environments and thus might see it as a risk if, for example, someone from a _Fortune_ 500 company were seeking to launch a start-up or a technology pro shifted into real estate.\n\n_\"We are first to market.\"_ As an entrepreneur you may think this is a big advantage, but such a claim can actually work against you. Typically, being first to market means you take on all the risk. You'll spend lots of your time educating people, and to an investor that says _expensive_. Many would rather be second, riding in the wake of the first company.\n\n_\"We have no competition.\"_ Making this claim may be the most common mistake of entrepreneurs raising capital. On hearing it, the experienced investor will assume one of three things: that you are lying, that you do not really know your market, or that there have been others before you who did not make it. As soon as you speak those words, your proposal is most likely headed into the trash.\n\n_\"We need a million dollars to get started.\"_ Why? Investors will do everything they can to minimize their risk every step of the way. They will want to know if there's another way, whether you can start with something smaller and more focused. Can you get to market faster and get enough feedback to get a sense that you're heading in the right direction _before_ investing that million dollars into a product no one may want?\n\n_\"If I build it, they will come.\"_ Maybe that works in movies, but I've never seen it work anywhere else\u2014and neither has your investor. It takes time, effort, and money to build a customer base.\n\n_\"We have more features than anyone else.\"_ Who cares? That does not mean your target customer will like all of them. An investor would rather know that you have one functioning feature that your target customer likes than that you have ten untested ones.\n\n_\"The big companies are too big or slow to compete.\"_ Not if they are serious, they aren't. If a market is big enough, there is a good chance someone in the industry is taking a look. If they decide there is an opportunity, they certainly have the resources in place to jump into your market and have an impact. This does not mean they will be successful, but they could be disruptive. If big companies have not entered, they may think the market is too small.\n\n_\"We do lots of things well.\"_ I always ask entrepreneurs what _one thing_ their business is really great at, the one thing that separates it from everyone else. Most of the time they do not know. Instead they rattle off three or four things they think the company does equally well. Telling an investor that you are great at everything communicates that you don't really understand what business you are in.\n\n_\"We do that too.\"_ If, during a meeting, an investor suggests additional products or services that might be a good fit or points to things the competition is doing, don't respond to each comment by saying, \"Oh, we do that too.\" Many entrepreneurs think they need to demonstrate that they are \"on top of everything,\" building \"every bell and whistle\" in the marketplace. But this sort of response communicates a lack of focus. Instead, acknowledge that the investor's suggestions may be great to add down the road but make it clear that for now you're focused on this one thing.\n\n_Don't be afraid to say, \"I don't know.\"_ Some entrepreneurs think they need to be expert at everything. As a result, they try to answer every question that comes their way, even if they have no clue what they are talking about. If you talk in circles, that'll be obvious to the investor. Since everyone knows that a new business or product is surrounded by uncertainty, people understand that assumptions need to be made and lessons need to be learned. One of the signs of strength investors look for in an entrepreneur is the simple understanding that no one has all the answers. Letting investors see that quality in you communicates that they can trust you to be straightforward and open to collaboration.\n\n_\"Writing a business plan would just slow us down.\"_ Investors know that without a plan you have not forced yourself to test your model, that you have no real sense of income and expenses or of capital requirements and timing. That also conveys that everyone isn't on the same page. How could they be without a written plan to provide them with a singular focus that keeps the team grounded?\n\nOTHER RED FLAGS\n\nA few additional factors that could affect an investor's decision regarding your business include:\n\n**High Failure Rate** **:** Some industries have significantly higher failure rates than others (restaurants and retail stores are good examples). That certainly does not mean you will not succeed, but you'll need to have a very clear plan.\n\n**Dependence on Legal or Government Regulation:** Industries that tend to be more litigious than others or subject to much legal regulation may make investors hesitate. If your product requires long government approvals or a bidding process, there could be additional cause for concern because of the potential drain on cash. Investment in such businesses is not for everyone.\n\n**Key Operations in Different Places:** If your company is based in one place (e.g., a certain country or geographic region) and key operations are elsewhere, that can be a cause for concern. Separation can create an added level of complexity when managing a fast-moving business.\n\n**Small Returns for Investors:** Your potential investors may believe in the product and the market, even that you have the right team to execute, that everything is in place. But your business still may not meet their investment criteria, with anticipated returns that are too low; or they may view it as a lifestyle business and think you will want to stay with it forever, leaving them with no chance at a big exit. Chances are, though, what you're offering will be perfect for someone else. Keep trying.\n\n### The Callback\n\nIn the entertainment business you audition. Your goal is to get invited to the next step, so you wait for the callback. The same is true in raising capital. With each interaction you try to generate enough interest and momentum to move to the next step.\n\nHere are a few tips to get investors to make that call, to summon you back.\n\n_Build a great team._ People invest in teams, so make sure you build a great one, from partners to employees, from consultants to collaborators. Make it your mission to surround yourself with the best. Avoid being strong in just one area. Cross-functional expertise that sees business in different ways adds perspective and gives your investors confidence.\n\n_Feed information in bite-size chunks._ This is a rule of thumb in direct-response marketing. For example, the purpose of the headline is to get you to read the first line. The purpose of the first line is to get you to read the second line; and so on. Eventually you build momentum and you close a sale. That's your goal.\n\nDon't send investors everything at once. If you overburden them, they'll walk away feeling overwhelmed and confused. Instead, feed them small, digestible pieces. For example, start by sending your target a one-paragraph overview of the business via e-mail. If there is interest, send a one-page executive summary. If he is still keen, follow up with a short PowerPoint presentation. Every communication should build on the one before.\n\n_Tailor each communication to your audience_. Each investor will ask a different set of questions and have a different set of concerns after each communication. Your job is to get all of these questions and concerns on the table, then apply what you've learned in tailoring your next communication. Let's say one investor has concerns about the market. At the next step, build a short presentation that specifically addresses that investor's anxieties; your next conversation should cover just that, nothing else. Once you get past that issue, move on to the next.\n\nAnother reason not to send everything at once is that the materials need to be tailored for each investor target. Information prepared for one person may not be right for another person, and you can shoot yourself in the foot. Take a cautious, step-by-step approach to learning, iterating, and then improving your presentations in order to suit each and every conversation.\n\n_Be open to feedback and ask for advice._ Don't take anything personally. Use every piece of feedback you get, no matter how positive or negative, as fuel to power you forward. No matter how much you don't want to hear it or how much your point of view may be in direct contrast, approach everything with an openness to learn. That openness and flexibility are great signs of leadership in early-stage environments.\n\n### Tricks of the Trade\n\nCapital raising means you must not only focus on getting money but also, in doing so, seek the most favorable terms, trade the least possible equity, and maintain control of the business.\n\nIn shaping your thinking, consider these factors:\n\n_How much money do you really need?_ My advice is to raise just what you think you'll need, then a little bit more. By running lean, your team stays focused, and you'll make better decisions and avoid waste.\n\nEarly-stage investors will want a big piece of your company. They probably deserve it for assuming the risk of investing in a business or product that still has a long way to go (they've seen the statistics about how many start-ups fail). They expect to build that risk into what they charge for their capital, demanding more equity, higher interest rates, more control, or other terms. That also means that the more money you raise up front, the more of your company you will have to give away in the form of equity.\n\nBe thrifty. Take in as little money as you can and give away as little of your company as possible. Prove your concept sooner, then go out and raise more money when your company has earned a higher valuation.\n\n_Loans._ Loans can be guaranteed or nonguaranteed. Will the interest be simple or compound? How much time do you have to repay the principal?\n\n_Equity_. How much equity will you give away? Will your investor(s) be active or passive? How much control will you relinquish? Do your money sources wish to be involved in big decisions or plan to leave them to you and the team? What kind of return are they expecting? When do they plan to see this return?\n\nEntrepreneurs are optimists; they tend to focus on their current round of financing and often significantly underestimate their long-term capital requirements. As a result, they part with a lot of their company's equity early, thinking they will not need to give much away down the road. They spread it around to investors, employees, contractors, and advisers. Later, when they go out to raise additional capital, they may find that a new round will cause them to become minority shareholders or lose control of their business. Keep in mind that it's common to seek capital more than once in a company's lifetime.\n\nThe decisions you make early will have a lasting impact. If your company is performing well, anticipate that each round will cost you between 15 percent and 30 percent of your equity. If things go sideways, capital could become much more expensive. To avoid giving away too much, demonstrate proof of concept. Show investors that you can execute and that customers will pay for your product. This will help to mitigate their risk and create significantly more value for your company.\n\nUnderstand this: Every single little percentage point will make a big difference down the road.\n\n_Preferences._ Investors typically have certain rights that supersede those of the entrepreneur. One example is preferences.\n\nPreferences are special return scenarios that relate to what happens to the money when a given financial event occurs. For example, should you sell your company, investors may demand their principal be repaid first, after which the remaining money is divided by everyone else on a pro rata basis. Alternatively, an investor may have mandated that they receive two or three times their investment, then everyone else split the remainder. The lesson? Focus not only on how much equity you give away but also on the preferences built into investor agreements.\n\n_Control._ Even if you own most of the equity in the company, you may not be in control of it. Many investors fight for contractual provisions that hand over to them the power to make decisions about certain aspects of the business. Be sure to understand how much control you are ceding.\n\n_Value._ Beyond cold cash, what else does your potential investor bring to the table? Does he or she bring industry knowledge or operational knowledge? Will you gain someone skilled in team building or with contacts for strategic partners? Balance investors' overall value with the capital they bring to the table.\n\nRaising capital is often a requirement to launch a new business, product, or service. It's hard work that takes time and dedication. The biggest reason why entrepreneurs fail to raise the funds they need is that they don't have a road map, so they waste time. They get in front of the wrong people. They say the wrong things. They overburden their target with too much information. When they finally do get an offer, they don't know what to look for and how to negotiate the best deal.\n\nIf you follow the steps in this chapter, you can steer clear of these problems and set yourself on a path toward success.\n\n# STAGE III.\n\n# COUNTDOWN AND BLASTOFF\n\nThe First Thirty Days of Business\n\n## CHAPTER 8.\n\n## GO TO MARKET!\n\nI fear not the man who has practiced 10,000 kicks once, but I fear the man who has practiced one kick 10,000 times.\n\n\u2014BRUCE LEE\n\n### Hammers and Nails\n\nBy this point you've spent several weeks moving forward with your big idea, developing your business plan, refining your business model, building a great team, and maybe even raising some capital. You've got a lot of momentum behind you. Now it's time to get your product or service ready to go to market. It's time to open your doors, either physically or online. This is your moment for liftoff.\n\nThe most common mistake entrepreneurs make when launching something new into the market is trying to do too much at once. The same is true of managers inside big organizations. I call it _hammer-and-nail syndrome_.\n\nLet's say I give you one hammer and one nail. Your job is to hold the hammer in one hand and the nail in the other. All you have to do is drive that nail into a piece of wood. That's it; just nail that one product or service. You may miss the first time or even the second, but eventually you'll nail it.\n\nNow let's say I give you two hammers and two nails. You have a problem. Who is going to hold the nails? Let's say you find someone crazy enough to volunteer. Your job is to drive both nails at the same time into that piece of wood. Odds are you will miss again and again.\n\nWhat if I give you three hammers and three nails? I think you get the point: This is exactly how most entrepreneurs launch their businesses. They simply try to do too much at once. They mistakenly believe that going to market with \"more\" is better.\n\nIt isn't. During your launch period, more means additional risk. More means unnecessary complexity. More means additional time to market and more capital required.\n\nBelow are some important things to remember as you prepare to take your product to market.\n\n_Don't try to build Rome in a day._ I have a good friend who raised two million dollars in a very tough market to start a consumer Internet business. To have raised that much money so soon after the 2008 market collapse was amazing, and I congratulated him on the big win. He was ecstatic and told me he couldn't wait to get to work on the company's site.\n\nIn 2012 I ran into him again and asked how it was going. He sang the blues. They had launched their company a few months earlier but had already run out of money. How was that possible? When I asked, he talked about his big vision, how his company aimed to provide everything its target customer could possibly want to buy in the category. Its goal was to be a one-stop shop. He and his team had invested all their time and money building something big and comprehensive, confident their target customer wouldn't want to go anywhere else once their Web site was up and running.\n\nAfter launch they discovered, to their surprise, that about 95 percent of their users used just _5 percent_ of the site! Turn that around and it means that 95 percent of the time and money invested was effectively wasted.\n\n_Focus on one thing, the simplest thing._ When launching, put all of your energy and focus into one product or feature at a time. It should not be the hardest thing; it should be the simplest, what we'll call the _minimum viable product_ (MVP). The MVP provides the opportunity to learn the most about your customers, with the smallest amount of time, money, and effort invested. The MVP puts you in a position to go to market quickly, collect valuable feedback, and not waste time building things customers don't want. This strategy significantly mitigates your risk and helps avoid the trap my friend fell into of building a bloated product that nobody wants to use.\n\n_Follow the 85 percent rule: Good is good enough_. Striving for perfection is the enemy of any product launch. As a rule of thumb, when the new business or product is 85 percent of the way there, you are ready to go. In my experience the level of effort required to reach 100 percent is not worth the additional time, money, and effort at this stage. You would be much better off shipping early, getting something into the market, and beginning to test.\n\n_Be great at collecting and learning from feedback_. Once you have launched, listen to and learn from your users. Develop feedback loops that enable you to learn everything you possibly can. What do users like and dislike about the product? What features would they like to see added to enhance their experience? Which features don't work or generate little interest? Do whatever you have to do to engage with your users. That may include offering incentives, reaching out in social media, and generating outbound calls to learn more.\n\nThe hardest part of this process for many entrepreneurs is to be completely receptive to what customers tell them. Given your passion and all the time you have spent on the project, you may not want to hear negative feedback. You may be inclined to think the customer just doesn't get it. But feedback is the most valuable tool you have as an entrepreneur. So listen, consider, and use what you learn to iterate, improve, or even throw out some of what you have built or planned.\n\n_Avoid shiny-ball syndrome_. As you start developing your minimum viable product, you must fight \"feature creep\" at every step. You, your team, your partners, and everyone else you share your vision with will have ideas about what should be added, and many of them will sound like good ideas at the time. More likely, though, they're distractions, shiny objects that draw your eye. Your job is to keep focused on one thing, to get it to market, and _then_ to deliver the next single thing. By staying focused on one thing at a time, you can get to market quickly, learn a great deal from real customers about what you offer, and make changes. If your launch doesn't fly, you have significantly mitigated your risk.\n\nSpeaking of not working, what if the product falls down or is simply a bad product or a poor market fit? Kill it. In order to mitigate your personal and financial risk, choose to fail fast and shut it down. Big, fast failure is better than a long, slow, expensive death.\n\nLAUNCH, LISTEN\u2014AND LEARN\n\n**Focus:** Launch your new products and services fast and launch them often, but always focus on one thing at a time. Don't start with the hardest thing but, preferably, the simplest thing, the _minimum viable product_.\n\n**Listen:** Launch quickly but, as you do so, develop feedback loops that enable you to _listen_ to what your customers have to say.\n\n**Learn:** Make incremental changes and improvements based on feedback. Launch again, and continue the process of iteration, making it an ongoing part of your culture.\n\n**Move On:** Once you have really nailed that product, once you've found the right product\/market fit and given your customers what they want, move on to the next one (that is, to the second hammer and the second nail). Remember, you can do everything you want, just not all at once.\n\n### Riches in Niches\n\nMy early career was spent in a broadcast world. I worked for divisions of media giants like CBS, NBC, and FOX. During that time I sold advertising to big global brands and marketed their products to the largest possible audience. The idea was to cast the widest possible net and see how many fish you could pull in. I believed in the power of mass marketing.\n\nFor the entrepreneur planning to launch, there's a problem with this model: too much marketing waste. Inevitably you'll spend a lot of money telling your story to the wrong people. That's not the way to build a business. You need to be razor sharp in cutting up the marketplace, making dead sure that every dollar spent on communicating to your target audience gets a good return on investment.\n\nThe good news is that that can be accomplished today in a way we couldn't do it twenty years ago. Instead of living in a world with a handful of television channels (like the world I grew up in), we now live in a world with literally millions of targeted ways to connect with your audience. You can find your audience on one of thousands of television stations that speak to your niche. You can pinpoint your market on the Internet and target it using Facebook or Twitter. In each case, today's consumers do for the entrepreneur what broadcasters couldn't years ago. They self-sort themselves into groups based on shared interests and desires. The key is to find your target and market to that audience very specifically.\n\nYour task as an entrepreneur is to think in terms of an ideal customer\u2014one person, one group\u2014to whom you wish to sell your single, simple product. Develop a very specific profile, learning everything you can about your ideal consumers and what aspect of your product appeals to them. Determine what your target customers talk about and learn their manner of speaking, the lingo they use to speak to one another. Put yourself in their shoes. Know what they care about, their issues, and their concerns.\n\nOnce you have a very deep understanding of your customer, market your product or service to that very specific audience. Instead of broadcasting to a wide market, go narrow and go deep. There are riches in niches. The more you integrate yourself into that group (especially via sites like Facebook, Twitter, Tumblr, and LinkedIn), the faster word will spread about what you have to offer. If you talk to the right people at the right time in the right way, and you deliver a product they value, you will suddenly have an entire community telling your story and functioning as your marketing team. On the other hand, if you misfire, everyone will know\u2014bad news travels fast.\n\nI came to truly appreciate the power of targeting a deep and narrow niche when I worked on a project for a direct-marketing business. The owners had acquired a database that had more than fifteen million buyers of a certain type of product. They had names, physical addresses, e-mail addresses, and a lot of information about each buyer. Their initial plan was to use the list to market new products via e-mail.\n\nWhen they started sending out marketing offers, they cast the widest possible net, sending the same offer for the same product to all fifteen million people. With that much distribution, how could they lose? They found, however, that a low percentage of people actually responded to the offers and a high percentage of their solicitations got dumped into spam filters. Many people simply weren't interested in hearing their offer.\n\nWe needed to change their approach. We brought in a data-analysis team and broke down the fifteen million names into ten groups that we called \"buckets.\" In doing this profiling we learned that, although all the people were on this list because they bought a certain kind of product, their individual characteristics were very different. Not everyone was interested in the same products or wanted to be marketed to in the same way. We found we needed to sharpen our focus and tailor our message.\n\nWe made a critical decision to learn everything we could about one bucket at a time and developed very targeted products and services for each niche. That meant we ignored the rest of the millions of names in the database, because sheer volume wasn't what we were going after. We looked to isolate a particular kind of person. I would rather have fifteen thousand buyers I know are interested in my product than a database of fifteen million people who are not.\n\nThe original\u2014and unsuccessful\u2014offer had been for a weight-loss program. So we took a look at a segment of these people who had bought one product that could be loosely categorized as spiritual, such as Christian-themed products. Then we looked at the predominant gender of the people in that bucket, narrowed our age range, and chose one target demographic\u2014women aged forty-five to fifty-five.\n\nWe kept adding filters: married; has children; empty nester; caretaker for a parent; pet owner. Now we were getting somewhere. All these qualities might make it sound like we were creating an overly narrow niche, but we ended up with more than a million names of people who met those exact criteria.\n\nThen we went to work learning everything we could about our prototypical customer (we named her Christina). She was a fifty-year-old mom with grown kids who owned a dog and was caring for her parents. We joined Christina's online communities, found her on social media, and had real conversations with people like her. We learned what drove Christina and the kinds of products and services that she was interested in.\n\nWe developed just the right product, tailored to this ideal customer. In this case we built a spiritual program in the form of prayer books, newsletters, and audio and video recordings that touched on issues that were experienced by caregivers who also owned dogs. The response to the highly targeted campaign was amazing. Many Christinas bought the product, became part of the community, and referred other friends like themselves (and gave us their e-mail addresses). In one month the company recorded a much greater response, because we had connected with the market.\n\nHere are a few suggestions on creating the perfect product\/market fit:\n\n_Pick one very specific niche_. Instead of casting a wide net, focus on solving one customer need at a time.\n\n_Develop your avatar._ Create a very detailed representation of your targets. Who are they? What are their ages, genders, and locations? What do they care about? What do they read? Whom do they hang out with? What do they watch? Which products do they buy and where do they buy them?\n\n_Engage in conversations_. Find the specific places they hang out (both in daily life and on the Internet). Join their communities. Learn what concerns them, what drives them, and what is under the surface emotionally that gets them to make decisions. Chime in when appropriate.\n\n### Crickets\n\nYou were convinced that when you built it, they would come. You knew in your gut that what your product offered was exactly what the market needed, that once you launched, people would flock to you in droves. But when you opened your doors, either literally or online, all you heard was the sound of _crickets_. Not footsteps, not voices, no pings, no cash-register noises. Just crickets.\n\nYou might have the best product around that no one has ever heard of. Or the few folks who have heard of it just don't know where to find you.\n\nThis is one of the problems I run into most with entrepreneurs. It occurs because of wishful thinking ( _I know people will come!_ ), too little money to spend on marketing, or no workable strategy for driving traffic in the company's direction. It is just as common with intrapraneurs, who place exaggerated confidence in the past success of other launches or the strength of their company's brand.\n\nA client recently came to me with a great idea for an online business. His product addressed an obvious need in the fitness business, and he had spent a year and $75,000 of his savings to build a beautiful Web site to sell it.\n\n\"How's business?\" I asked, expecting a happy answer. \"How many people are coming to your Web site? How many units have you sold?\"\n\nHis answer: _None._\n\nHe had almost no traffic to the Web site and few buyers exposed to his product. He felt like the business was doomed. To make matters worse, he had exhausted his cash resources building the product and promoting the launch.\n\nBut as I told my client, his business was not doomed. What he needed were partners. Not business partners but partnerships with other businesses, specifically ones that had a similar customer profile.\n\nIn nearly all the companies I've been involved with, a key to the successful distribution of our products and services has been the partnerships we established early on. By hitching your wagon to a bigger, more established business you can get a lot further faster, and with much less money out of pocket than by traveling the road alone. I've seen this strategy work firsthand. I've helped businesses grow from ideas to big brands\u2014on the basis of sound partnering.\n\nMy first lesson in partnerships was with Quote.com. I was an early member of the consumer Internet business's launch team. In 1995 the company was one of the first to get permission from the stock exchanges to put stock quotes on the Internet. It helped create the market, define it, and was the leading brand in its category. Today, with hundreds of places to get stock quotes online, we take for granted how easy it is to find them. But back then it was state of the art. I remember giving a presentation to a team from the _Wall Street Journal_ and they just about fell out of their chairs upon seeing real-time quotes delivered on big blue-chip stocks.\n\nQuote.com had a great product in an exploding market, but we had to move fast to keep our first-mover advantage. At the beginning we had very little money to work with (I had to use my own frequent-flyer miles to buy airline tickets to go see a customer). Since we had a small marketing budget, we needed to find other ways to grow. We needed partners. Though we felt we had something valuable to offer them, we had to figure out a way to get in the door and structure a deal where both sides would win.\n\nWe approached sites with a similar target customer. These other companies wanted to create more awareness; they believed that offering more value would drive usage. They wanted applications that kept users on their sites rather than surfing off to explore other places on the Internet. And ultimately they wanted to find ways to monetize this traffic.\n\nWe offered to integrate our product into their Web site, giving users a reason to stay on their Web sites longer. We proposed cobranding the pages to create branding and awareness for both of us. Once we figured out how to monetize the pages together, we'd share revenue.\n\nIn just a few months we built a network linking us to over one hundred sites, some of which were among the largest consumer Web sites at the time. As a result, people began to see us everywhere.\n\nWe applied the lessons we learned about partnering to advertising as well. One of our key early advertising partners was Ameritrade. Its founder, billionaire Joe Ricketts, knew that different kinds of investors liked different kinds of brokerages, so he created four of them: Accutrade, K. Aufhauser & Co., Ceres Securities, and eBroker. Each one had a different marketing angle to appeal to a different kind of investor. Joe and his son Peter advertised Ameritrade sites on Quote.com because they knew if our customers wanted a quote, they were likely to have a brokerage account, making advertising with us a no-brainer. Everyone was a winner, and ultimately Quote.com became one of the largest advertising revenue\u2013generating sites and number one in its category. The company eventually sold to Lycos. The whole experience serves as a great lesson in growing a company quickly with limited resources through partnerships.\n\nI've watched the same model applied over and over in my career, with Sportsline USA and CBS, with Xoom.com and NBC, with Lycos and FOX Sports, and most recently with Smart Charter and Virgin. In each case a scrappy, early-stage venture with limited resources applied the same formula and created a win-win opportunity with partners to achieve its goals.\n\nHow do you apply partnerships to your business?\n\n_Decide what you need._ What does your business need most at this stage? Capital, resources, access to potential customers, credibility, marketing and promotion?\n\n_Follow your target customers._ Where do they spend their time? What businesses, Web sites, organizations, associations, or groups do they visit?\n\n_Create a win-win agreement and execute._ Ask yourself, _How can I create a win for my partner? How can I make my internal champion look like a hero in his or her business? How can I get them so excited about this partnership opportunity that they think about it and prioritize it every single day?_ To do this, find out what's most important to your prospect, what has to happen or how they know they are getting what they want. On the basis of feedback they give you, tailor your partnership ideas. Create revenue sharing where applicable that benefits both companies.\n\nOne final lesson on partnerships? I believe that businesses are bought and not sold. Many of the best financial exits are the result of businesses that started as partnerships and then grew into something more. Many of the people I have worked with who partnered early with the right businesses, followed through on what they said they would deliver, and built deep relationships throughout the group ultimately had the largest financial exits.\n\nPARTNERSHIP PLANNING\n\nHow to prepare for and find a good partner:\n\n**Build a Good Business:** A big partner will not make your business better. It will expose your flaws faster. You need to stand on your own two feet. Build a great product first.\n\n**Calibrate Your Needs:** Determine what you want from a partnership: capital, resources, technology, marketing, distribution, or sales support?\n\n**Understand Their Needs:** What problem can you solve for your partner? How can you make your internal champion look like a hero?\n\n**Sell What They Need:** Focus the conversation on addressing your potential partners' needs rather than on what you think is cool about your product.\n\n**Shop Around:** Talk to as many potential partners as possible to be sure you find the right one.\n\n**Find an Internal Champion:** Because smaller companies and new initiatives can get lost in the shuffle, you need an internal champion in the business you're partnering with to take ownership of your project and drive it internally. As an entrepreneur CEO, it's important for you to maintain a relationship with this person.\n\n### Crowdsource Your Sales Force\n\nSometimes the mirror takes awhile to clear so that we can see what's right in front of us.\n\nEver since college I've been surrounded by computers. I've worked and invested in several Internet companies. I've been an early adopter of new technologies. But there was one area in which I dragged my feet: I was late in embracing social media as a means of growing a business.\n\nI was guarded with my privacy. I had very little spare time, and keeping up with Facebook, Twitter, LinkedIn, Tumblr, Pinterest, Google+, and everything else seemed like more to add to an already full plate. But everything changed when I saw the potential value these networks had to offer when launching a new business.\n\nI was advising a bootstrapping software company that had no marketing budget and very little money to hire salespeople. Since it couldn't afford to market its product and hit sales targets the traditional way, it turned its attention to social media.\n\nWhen I taught sales training twenty years ago, I talked about finding target customers, identifying their needs, and demonstrating how a company's products met those needs. I'd go on about how to close the sale, provide excellent customer service, and in the end create raving fans who would share their experience with others. But here's what I learned.\n\nThe model has flipped. Today many of the most effective companies build fans _first,_ customers _second_.\n\nWith fans coming first\u2014this is the best part\u2014you can effectively crowdsource your sales force. By executing this strategy you can build a virtual army made up of thousands of people talking about you and telling their friends to buy your product. And it costs you nothing. That's because you have the crowd passing along their experiences, giving personal referrals, and selling on your behalf. Your social-media sales force can be more effective than anyone you put on staff. People are much more likely to trust your brand and make a purchase based on a friend's referral than on the basis of anything you tell them.\n\nI've also learned that social media can be an excellent tool for conducting market research, brand development, and product development and for getting direct feedback from customers.\n\nSo how do you crowdsource your sales force?\n\n_Pick one channel._ Instead of trying to conquer multiple social-media channels at once, pick one platform to start with, whether it's Facebook, Twitter, LinkedIn, Tumblr, Pinterest, Google+, or another. Do that one well and don't worry about anything else. How do you choose which one? Find out where your customers spend their time. Go find them and engage.\n\n_Be authentic._ When I first got started with social media, I made a big mistake. Everything I put out there was too highly produced. Whether it was a post, picture, or video, I thought it needed to look and feel perfect. But online, authenticity is everything.\n\nBeing you is what people want, and if you try to be someone else, they'll know. If what you put out comes off as scripted and highly produced, that feels to the customer like a violation of trust. So be yourself. Share and tweet and post the things that interest you. Don't work too hard to make everything perfect. Instead, focus on being raw, real, and relevant. And by the way, it's okay to be dorky.\n\n_Forget the gatekeeper_. With social media you can connect with virtually anyone you want. That is an enormous shift in our ability to access others. There are no longer gatekeepers standing between you and the top businessmen, entertainers, and athletes in the world, because most of them manage and respond to their own social media accounts. If you would like to start a conversation with someone who might make a difference in your business, it can be very easy. Just send him or her a tweet.\n\n_Commit yourself to a daily hour of power._ Getting started with social media is quick and easy. Opening an account costs nothing, and posting is free. In a few minutes you can be up and running. Within an hour you can reach out and be connected with friends, coworkers, and customers.\n\nSpend one hour per day during your launch engaging with the community. That's it. In one hour a day, at zero cost, you can build an army. The potential payback is incalculable in terms of exposure and attention.\n\n_Monitor and protect your brand_. Make sure to regularly frequent the sites, feeds, and pages that discuss your industry, product, or service. Look for posts that mention your company. Respond to comments and complaints, using them as opportunities to engage, build trust, grow your brand, and collect market research. If you listen carefully, you can get ahead of potential problems. And don't think you're exempt from this kind of vigilance if you're a big business. Today being big is meaningless if the customer's friend says the product sucks.\n\n_Invest in sound._ If you are posting video or audio content, keep a couple things in mind. Even though the color, background, and production values may not be perfect, good sound is very important. Buy a lavalier microphone for quality sound and, whatever the source of your video, be sure there is enough light for people to see you. You don't need to spend a lot of money, but this is an example of one detail that you should invest in and pay close attention to.\n\n_Tell them how to reach you._ Make it easy for your fans to get hold of you. If possible, have your Web address, e-mail address, phone number, or social-media handle somewhere on the video screen, and mention it in your post\n\n_Leverage social media as a research tool._ I love fish. One of my favorite hobbies is building aquariums. I kind of fell into it; my daughters both wanted a pet, so we went to a local pet store and bought them each a goldfish.\n\nI kept going back to that pet store, fascinated by the aquariums and the beautiful breeds of fish. It didn't take long before I decided to build my own aquarium, and before I knew it, I was building a second. Now I'm addicted, but I have a problem with the local pet stores: Their selection of fish is small. One day I grew so frustrated that I started thinking about opening up my own store. Not a pet store, a fish store. I needed more information, so I turned to social media.\n\nFirst I went onto Facebook and searched for communities of people who shared the same interest. Next I went onto Twitter and starting following hashtags like #fish, #aquariums, #fishstores, and so on. There were a few people who were very active in these communities, and I approached them with questions: Is there a reason I can only find the same small number of fish breeds at the local stores? Do other people in the community have the same problem finding a good selection of fish? Has anyone been able to solve this problem and, if so, how did they do it? I found that when I asked these questions of the whole community, many of the people who responded had problems similar to mine. Maybe my fish-store idea had legs.\n\nThen I asked my fellow online hobbyists the big question: If you could build the perfect fish store, what would it look like? To my surprise, the answer was not that the store would have the biggest variety of fish but that it would provide great customer service. What I learned was that many people dropped the hobby because they could not figure out how to build a good aquarium, maintain the right water balance, or even feed their fish the correct amount. As a result, their fish kept dying. The service at the existing stores was so poor that these people either got frustrated or felt embarrassed to ask for help, so they just dropped the hobby altogether.\n\nI realized that if I provided better service, I would do more for the community. I also learned that customers would be more loyal to my store, stay with the hobby longer, and ultimately buy more stuff. In reaching out to online communities through social media, I saved a lot of money and time in getting my research done. Today that fish store is my pet project!\n\n### Your Buyer Blueprint\n\nWhile teaching Tony Robbins's material twenty years ago, I learned that the biggest mistake salespeople make is selling the wrong thing. They push either what they think is most important about their product or what their boss has told them to sell. When the prospect says no to their pitch, they don't understand why. This happens because the seller never asked what the prospect wanted\u2014nor did he understand the prospect's _buyer blueprint._\n\nAlthough I should have known better, I've fallen into this trap myself. One example was while I was at Virgin Charter. We had a very smart and educated team. We were confident that we understood the problems our customers wanted solved and how to solve them. With this confidence we created a product road map, listing the features planned for development and ranking them in the order we would build them. We had over a hundred features on our list and I had the sense we were really on the right track. Then, after spending lots of money building a Web site and going to market, we discovered that people just weren't using our service in the way we had believed they would.\n\nOur team spent its time building features on the back end of our Web site, creating efficiencies, automating systems, and developing technologies that were necessary to enable buyers to complete a purchase and manage their trip details online. But customers spent their time on the front end of our Web site, doing searches and viewing profiles of all the available aircraft. When they wanted to make a purchase, most just picked up the phone, called, and asked our team to process the order.\n\nI asked my team to go back out into the market. Our sales, product, and technical people gathered in rooms with customers.\n\nWe soon learned that some of the functions people had wanted most in the earliest versions of the product were neither in the current product nor among the top ten things to build in our product road map. Some were not even in our plan. Obviously, we had too many hammers\u2014and not enough of the right ones.\n\nThe challenge is to create a solid buyer blueprint before you do anything else. You can create this by asking two very simple questions.\n\n_What is most important to you?_ Ask a customer this and many times the answer you will get is \"I don't know.\" But try turning the answer around, responding with \"I know you don't know, but if you did know, what would it be?\" You'll find that that usually elicits useful answers, and the person will start listing the things he values most, perhaps without even realizing it. Take notes and, once the customer has downloaded his list, work with him to rank every item on the list in order, to identify which features are really most important to him. For example, if he could have this or that, which one would he choose? Continue to go back and rework and refine the answers.\n\nThe potential buyer may say, for example, that the top priority is that a food item taste good or, for a tech service, that it be easy to use. That's a start, but it's not all that you need to know. This leads you to the second question.\n\n_How do you know you are getting what you want?_ Nearly everyone forgets to ask this question, and it may be the most important one. Why? Because the definition of \"good taste\" or \"easy to use\" differs from person to person. We all have different rules or labels we attach to things. Your job is to find out not only what your prospects value but also the rules they have established for determining whether they are getting what they want. That makes follow-up questions essential.\n\nHow do you know it tastes good? Do you crave sweet or sour, crunchy or smooth? How do you know it's easy to use? Do you need to access it online? Would you require it on a mobile device?\n\nWith an early understanding of what your prospects want and how they know they are getting it, you will be able to build and sell products and services that match your buyers' blueprint and lead to big success for the business.\n\nBut remember this: Just because customers said they wanted something does not mean they will buy it once they have a chance to see it, use it, and interact with it. We found that out at Virgin Charter. What many customers told us they wanted before having access to the product was different from what they wanted once they had a chance to play with it. Sometimes buyers don't know what they want. It's not that they are trying to deceive you. It's that they can only know so much until they have a chance to experience what you have to offer. For that reason it is essential to do two things.\n\nFirst, build one thing at a time rather than everything at once. That way, if customers start using your product and find they want something slightly different, you have not wasted valuable time and resources. Second, get customers to use the product as early as possible and collect as much feedback as you can (remember the MVP?). Then iterate, redeploy, and scale.\n\nConstant interaction and communication with your users and ongoing reevaluations of your product and priorities are the best ways to keep moving forward in these crucial ninety days.\n\n## CHAPTER 9.\n\n## MAINTAINING YOUR ALTITUDE\n\nEverybody's got a plan, until they've been hit.\n\n\u2014MIKE TYSON\n\n### The Ninety-Day Checkup\n\nNow you've launched your business, and your hard work is starting to pay off. The company has begun to generate sales. Your customers are responding with great feedback. The market is buzzing about what you have to offer. You even brought home a good paycheck to deposit into the bank. Finally you have a chance to catch your breath.\n\nThe question is, what do you do next?\n\n_Take care of yourself._ Your start-up and launch periods are a sprint. It's an intense time and you are forced to go all out, giving it everything you can. But you can't continue to go at that pace forever. You need to cycle in periods of rest to recharge your battery. This is a good time to take care of yourself and your health. Are you keeping up with your exercise regimen, eating right, caught up on sleep? It is a good time to check in on relationships as well and reconnect with the people around you who made sacrifices so that you could pursue your dream.\n\n_Take inventory._ Work closely with your team to take inventory of what you learned during your launch. Here's a great chance for the group to contribute, for all of you to learn together so you have the best opportunity to improve. What worked? What didn't go as planned? Did you focus too much energy on one area of your launch to the detriment of another? Did your supply lines, feedback loops, and financials perform the way you anticipated?\n\n_Remodel the business._ Sometimes even a new house needs some remodeling, the plans adapted for what you didn't anticipate. The same is true in business. When you built your initial financial model, it was filled with assumptions. Now that you are in business, go back and study the new facts and figures against the model you used during your launch. Look at your income and expense projections, the time and resources invested, and the capital that was actually required to reach your goal.\n\nSometimes busy entrepreneurs fail to do this, but it's important to identify whether income, expenses, timing, or all of them were off and how all of these results will affect your need for capital going forward.\n\n_Reevaluate your team._ Do you have the right players on your team? Did each individual perform the way he or she was supposed to? Do you need to make changes? Can you get the kind of performance you want with improved leadership and training? Can the people who got you this far take you where you need to go next?\n\nOftentimes entrepreneurs feel loyalty to the people who joined their business prelaunch and fought the early battles alongside them. But your job is to make sure this company always has the right people in the right jobs at the right times. You may need to cut the cord and make changes as your business matures.\n\n_Find opportunities to grow._ Invest in learning activities to help grow and expand your team's capabilities. Decide how best to position the present unit to take the company forward. Ask the team what roadblocks they need removed to help improve results.\n\n_Consider adding new products and features._ Work to strengthen products that have demonstrated success. Determine whether now is the right time to add new products or features based on customer feedback, instead of simply iterating and improving your current offering.\n\n_Plan for the future._ Try to take a fresh look at your business. Update your goals, plan, and strategy using the inventory you took and counsel from the team. Evaluate your current and future cash needs and determine how you will meet those requirements. You should be in a great place to think strategically and move your business forward successfully.\n\n### Cash and Cardboard\n\nThe doors are opened to your new venture, and as far as you can tell you're doing everything right. You have customers coming in the door, and orders are flowing. You're beginning to feel like that crazy idea of yours might actually work!\n\nHold it right there. Before you begin to take it all for granted, I want you to hear about my friend Marty Metro.\n\nMarty got his MBA from the University of Arizona at age twenty-three. He went into consulting, working at Andersen Consulting and then a boutique firm. Over time he became a supply-chain technology expert. When the dot-com boom turned to bust, Marty got laid off.\n\nHe decided to start his own company, selling recycled cardboard moving boxes. The idea came out of his own experience, since he refused on principle to pay the retail price for new boxes when he moved into a new house in Los Angeles. He would go around town grabbing used ones from liquor stores. It wasn't that he couldn't afford to buy the boxes. Marty just didn't like spending money on things he would use once and then throw out\u2014and he figured he wasn't alone.\n\nLeveraging his background in logistics, Marty started a company called Boomerang Boxes, selling used boxes for half the retail price. The business was eco-friendly and was written up in _Entrepreneur_ magazine. Life was good.\n\nThen he went broke.\n\n\"In three years I had spent every penny I had,\" says Marty. He had self-financed the business, and when the business needed more capital, he reached into his own pocket. He had burned through two lines of credit for $100,000 each. He had maxed out three credit cards at $35,000 apiece. The whole thing came crashing down.\n\nWorst of all, his wife didn't know half the story\u2014he hadn't told her he had actually invested more than she had agreed to. The crash of Boomerang Boxes changed their lives. They had to sell their BMW and Land Rover, and Marty drove around town in a dented 1986 delivery van; adding insult to injury, the van had the name \"Boomerang Boxes\" painted on the side. He got a day job as a telemarketer. Sitting in a cubicle all day, he was miserable working as a cog in someone else's system.\n\nAt night he kept trying to figure out where his business had gone wrong. He just couldn't understand why it seemed the more boxes he had sold, the worse things had gotten for the business.\n\nMarty's problem was one I've seen again and again: _Growth kills cash_.\n\nThis may seem counterintuitive, but it is very easy to grow yourself out of business. Say you buy a product for five dollars and sell it for ten. That means the more orders you can get, the more profit you can make. Right? Not necessarily.\n\nWhen you're small, you can afford to maintain a low level of inventory. If someone wants to buy a dozen widgets, you've got them on hand, and once they're gone, you'll order another dozen to replace them. But what happens when a customer comes back and orders five hundred widgets? You think, _Hey, that's great!_ You go to your supplier, buy the widgets, and sell them to your customer. But here's the question that you should always ask yourself: _How am I going to finance this inventory?_\n\nYou may have enough cash on hand to buy the five hundred widgets from your supplier, but what if your customer says that he wants sixty-day terms? That means that to do the deal you tie up the cost of those widgets in your inventory. That diminishes your pool of working capital. If you don't have enough cash on hand to float your business until your customer pays you back, then your business may have to grind to a halt.\n\nBack to Marty. Once he learned these lessons, he was convinced that if he could manage the cash flow, the box business would work. He decided to try again.\n\nHoping to interest people in funding a new version of his idea, he did a tour of banks and venture-capital companies to raise money. When they asked what had gone wrong the first time around, he explained how his growth had outpaced his cash requirements and what he would do differently this time around. Remember, showing potential investors your failures (given that you've learned from them) can be as helpful as touting a string of successes.\n\nNot only did he raise the money, but Marty and his investors rethought their entire approach to the business. Instead of operating a retail store that sold one box at a time, the new model involved buying boxes by the tractor load from _Fortune_ 1000 manufacturers for resale to other companies. In the past, big companies would just crush and recycle the cardboard boxes. Instead, Marty and his team sorted them, inspected them, and packaged them into moving kits for sale to regular folks or back to big businesses. The new model was akin to what dairies used to do with milk crates and bottles. When you got your milk from the milkman, you only bought the milk; the bottles went back to the dairy.\n\nThe new strategy worked. After succeeding in Los Angeles, Marty quickly replicated the new company, UsedCardboard Boxes.com, around the country.\n\nThe message from Marty's lesson is this: As your business starts to grow, don't forget to pay attention to two very important factors:\n\n_Growth eats cash._ When you're launching a business and strapped for cash, big orders may do more harm than good, and a booming business can actually cannibalize itself. Before processing disproportionately larger orders, be sure you understand the impact, positive or negative, that fulfilling them could have on your business.\n\n_Invest from cash flow, not profits_. When reinvesting in the company, base your decisions on cash flow, not profit. Have a clear understanding of what big capital expenses might be around the corner so that you don't run into a cash crunch.\n\nNow that you know these lessons, don't be afraid to take a second bite at the apple if your strategies didn't work the first time around. If you have been part of a venture that went belly up, be open and honest with yourself about what went wrong. Seek the counsel of others, learn from your mistakes, and try to imagine how you would apply them to a new business.\n\nDon't be afraid that investors will not be open to you the next time around. If you can clearly demonstrate what went wrong and the lessons you took away from the experience, you may be surprised at how fast people jump on board. The surest path to success is learning from failure.\n\nFinally, Marty's experience reminds me of two practices we established early in this book: Separate your company's finances from your personal finances. And never hide deep losses from your loved one.\n\n### Kryptonite\n\nWhat comic-book character best describes the entrepreneur? I think it would be Superman.\n\nEntrepreneurs are the heroes of business. They start companies, execute tirelessly, and are on call twenty-four hours a day. In the early stages they put their stamp on everything, assuming responsibility for all decisions. They're the driving force behind all that gets done.\n\nJust as in the Superman story, there's one thing that can knock the entrepreneur down. No, it's not kryptonite in our case\u2014it's the failure to delegate.\n\nIf you want to grow your business and maximize its potential, it's important that you learn to hand over responsibility to others\u2014the sooner, the better.\n\nI have two female friends who started a chain of indoor playgrounds for kids. After operating the business for three years, they are generating plenty of cash and impressive profit margins for a retail business. Their goal is to grow the number of stores throughout their state, then take the business national. Will they get there?\n\nI found that instead of focusing on growth, they still spend most of their time on basic administrative tasks. They have run into a trap that many entrepreneurs encounter; because they are so close to the business, they think they can do every job in the company better than anyone else. They can't seem to loosen their grip, and as a result their business has plateaued. Yet by simply handing over more administrative tasks (like answering the phone, taking orders, and cleaning), they could significantly increase the amount of time they spend thinking strategically about the business and growing it nationally.\n\nThe importance of delegation to growth was made clear in two conversations I recently had with clients.\n\nIn the first I spoke with a consultant who runs a good business and makes one hundred thousand dollars a year. I respect her skills. I asked her what she was planning to do that day. Her answer: \"I need to schedule a bunch of phone calls and meetings and fill up my calendar. Then I need to go buy boxes and packing material and tape and bring them back to the office and pack some marketing materials for prospects. Then I need to go to the post office and stand in line and mail everything out.\" I asked her how long that would take. \"All day,\" she said.\n\nIn the second conversation I asked a good friend (another consultant) about his day. He said, \"I'll be out meeting with prospects and clients closing business in the morning, then I'm going on a fifty-mile bike ride to train for the Ironman triathlon.\" I asked how he could fit such time-consuming workouts into his schedule with all the bookings he needed to make, along with sending out presentation materials. \"Why would I do that stuff?\" he asked. \"I leave it to someone else.\" He has a virtual assistant he pays one thousand dollars a month to handle all of his administrative tasks. His business makes more than one million dollars a year.\n\nIt's pretty clear to me that the difference between the consultant with the million-dollar business and the one with the hundred-thousand-dollar business boils down to delegation. The most successful business owners spend their time on what is most valuable to growing their business and delegate the rest to others. I'd suggest that my first client look into the following fixes.\n\n_Start by valuing your time._ The fix isn't that hard: By handing over more administrative tasks (like answering the phone, taking orders, and cleaning), you can significantly increase the amount of time you have to work on the big picture. When is the right time to bring in this help? Today. You say, _How can I afford that?_ Wrong question. Ask yourself instead, _How can I afford not to?_ Your time is too valuable to waste.\n\nYou don't necessarily have to hire a full-time employee. Instead, bring on a virtual assistant and pay him or her by the hour. That person can handle scheduling, shipping, answering your phone, and responding to a lot of your e-mails for less than ten dollars an hour, perhaps working remotely from home. A small expenditure can give you the time you need to get in front of investors, prospects, and clients.\n\nI think of it this way: You either have a ten-dollar-an-hour assistant or _you_ are a ten-dollar-an-hour assistant. As an entrepreneur CEO, is that all your time is worth?\n\n_Train and trust employees to take on more responsibility_. When you start a business, you are the head of every department in the company. You are responsible for every job function. But as you add people to your team, be sure to fire yourself from those positions and let them take over.\n\nFor example, say you hire a new head of sales. That's a good decision, one area of responsibility you can take off your list. Your new job is to let the person you have hired do her job. Get out of her way, redirecting the energy you had put into sales back into your role as entrepreneur CEO. Your superhuman energies are better invested focusing on what is required to grow. By delegating you will move your business further faster, with less effort.\n\nDELEGATION DOS\n\n**Make a List of Everything You Do** **:** For one week, make a list of everything you do (both business and personal). Provide as much detail as possible. Circle all the administrative tasks\u2014reading e-mail, scheduling meetings, creating presentations, standing in line at the post office, returning phone calls\u2014that don't require your level of attention and that you can hand off. Next, take a look at nonadministrative tasks and decide what can be handed off to other members of the team.\n\n**Same Systems, Platforms, and Tools:** Get all your employees and assistants working on the same platforms and systems for e-mail, contacts, and calendars. Live in the cloud with shared folders, voice calling, and video calling. Make sure that your infrastructure is in place to help you grow and scale with little effort.\n\n**Refocus Your Energy** **:** Get back to your role as entrepreneur CEO\u2014or whatever position in the company suits you best. Spend your time there.\n\nMaintaining your altitude can mean making a series of small tweaks to your processes\u2014or a complete overhaul of your team, systems, and strategies. But you will not know what needs to be done unless you take the time to step back and take a hard look at how you're doing in these first days and weeks of your launch.\n\n## CHAPTER 10.\n\n## IF THINGS GO SIDEWAYS\n\nWhen the glass looks half empty, don't pray for rain, head toward the river.\n\n\u2014GREG REID\n\n### Learning How to Crash\n\nBeing an entrepreneur is a lot like driving a race car. In launching a business you move very fast, avoiding obstacles all along the way. At times you may feel like you are about to lose control and crash, like you're heading head right toward the wall.\n\nLast year my business partner and I decided to learn how to drive race cars. We headed to Las Vegas Motor Speedway to take a course on driving. The instructors helped us get familiar with our cars, then we strapped on our helmets. I was driving a brand new Ferrari 360 Modena, with the instructor in the passenger seat. Before completing my initial lap, I got my first big lesson.\n\nEvery driver starts with his or her head up and eyes focused way down the road. But as your speed increases, you feel a new type of pressure. It's all mental. The natural tendency for drivers is to narrow their focus, drop their eyes, and look down, right over the hood. As a result, you get tunnel vision, seeing every little bump in the track, every little piece of shredded tire. I found myself making lots of quick turns, tiny microadjustments.\n\nOn the racetrack tunnel vision can get you killed. Steering around lots of small bumps is one thing when you're going 25 miles per hour. But when you're traveling at speeds six times as fast? At 150 miles per hour, a sharp steering adjustment can cause your car to veer wildly. And you can miss what's literally down the road.\n\nThe same thing is true in a fast-growing business. No matter what your speed, you need to keep your eyes up and focused on the horizon. You can't lose sight of everything in front of you, including the goals you plan to achieve. Only with a larger view can you pick the best line to take around the corners. The little bumps in the road won't throw you off. Instead, the wider view will allow you to achieve a graceful kind of movement that enables you to stay in control and reach your goal.\n\nAnother thing they teach in racing school is how to crash. When you least expect it, the instructor tugs lightly on your steering wheel. Of course you panic, feeling like you will spin out of control. And what's the first thing that happens? Your head turns and your focus goes directly toward what you fear most: the wall. As a result, your arms just naturally follow, steering you toward a crash.\n\nHaving jerked your wheel toward disaster, the instructor does something else surprising: He puts his hand up against your helmet and pushes your focus back to the track. Where your head goes and your eyes go, your arms and the wheel follow. You're back on track again.\n\nIn business you'll speed into twists, turns, and hard corners. From time to time you may feel like you are losing control. Remember: Your job is to keep your eyes up, focused on the horizon, without ever losing sight of the big picture.\n\nPedal to the metal, eyes on the horizon.\n\n### Panic Faster\n\nJack Welch, the legendary former CEO of General Electric, once offered a sound piece of advice to aspiring entrepreneurs. After he spoke to Entrepreneurs' Organization, the largest global network of its kind, somebody asked Welch what his number one piece of advice to an entrepreneur would be.\n\nJack's response? \"Panic faster.\"\n\nEntrepreneurs are by their very nature positive, confident, and sure of their business. These are all great qualities to have when promoting your venture, but when things don't happen as planned and the business begins to go sideways, those same traits can work against you.\n\nEntrepreneurs can get caught up in their own story. They can get so busy reading their own press that they don't see what is really happening around them. That means they may not react to danger signs.\n\nOut of fear of the unknown, entrepreneurs sometimes freeze or pretend things aren't happening. They put off the inevitable, not wanting to make hard decisions like letting people go or cutting expenditures. Their emotions interfere, and the result is inaction.\n\nThen, before they know it, they're out of business and have lost everything.\n\nSo here's some advice on how to avoid that worst-case scenario.\n\n_Panic faster_. As Jack Welch said, you can't stand around hoping or waiting for conditions to improve. When things start to go wrong in your business, drop everything and identify where your problems lie. Spend 20 percent of your time on the problem and 80 percent on the solution.\n\n_Control the dialogue._ It's important, especially in small, early-stage environments, to control the message. By the time you are panicking, odds are the rest of the team is already there. Remember, in their day-to-day responsibilities your staff may be closer to market conditions, sales trends, and financial matters. Good times or bad, they may know what's going on before you do and be more willing to believe it faster than you are.\n\nWhen big problems occur, you need to get out in front of them. Your team will be wondering three things: _What is going on? How does this affect me? Will I lose my job?_ It's important to address these questions. It's just as important to redirect your team's focus away from the problem and on to devising solutions.\n\n_Get everyone's input_. You have put a smart team in place. Leverage that talent to help you diagnose problems. Oftentimes employees have already identified them and have come up with solutions. They're just not encouraged or motivated to speak up. Team problem solving should be part of your culture. Get everyone involved, and encourage them to take ownership. Or would you prefer they sit around updating their r\u00e9sum\u00e9s?\n\n_Leverage your network_. Whether it's formal or informal, you should have a board of directors, a group of mentors, a team of experienced advisers and perhaps investors. You've consulted with them and asked their advice over the years because they are good at what they do. They know that young companies run into challenges. Don't think you're losing face by going to them in a difficult time and asking for advice. Don't be embarrassed or afraid to admit failings. They've been there and are here to help.\n\nExperienced entrepreneurs are typically eager to lend a hand to a fellow business owner who sincerely asks for it. One of the most important lessons I have learned is the people who are the most successful are usually the most accessible.\n\n_Get in front of your customers._ At times you will be far more effective getting out of the office and into the market. Get on a plane and go see your customers. Learn about their business and the challenges they may be facing. Perhaps they have a new need you can meet that is a pivot from your existing product or service.\n\n_Be transparent_. If your business is having problems, it is important to be transparent with your customers. You want to control the dialogue and not let someone else, like a competitor, do it for you. By being up front you will create trust. You may even find that your customer can help you to design a solution.\n\n_Cut fast and cut deep_. If you're really in trouble, either with your financials or in not having the right people on board to solve problems in a crisis, you may need to trim your team. Laying people off is one of the hardest jobs of an entrepreneur. You may feel not only like you have failed in managing your business but also that you've failed in providing for these people. But you can't let that paralyze you. You have a deeper responsibility to your partners, your investors, your family\u2014and the rest of your employees. Sometimes you need to cut off the limb to save the patient. Your ultimate responsibility is to your business.\n\nWhen you see that layoffs are inevitable, don't delay. Furthermore, never make these cuts small and incremental. Small cuts will kill your business because everyone will be wasting their time looking over their shoulders wondering if the ax will fall on them next, instead of staying focused on their work. Small cuts can paralyze an organization. When you have to cut, do it quickly and deeply. It's much better to reorganize, rebuild, and start hiring again under better circumstances.\n\n_Keep your eyes on the horizon._ It's easy to get bogged down in the trenches, to be so muddy and distracted by the bullets flying overhead that you forget to survey the battlefield and take in the big picture. Doing so will help you find the right path to take you out of this tough period.\n\n### We're Out of Toilet Paper\n\nSometimes things go wrong. No matter how much passion and hard work you invest in a new venture, you may hit a point where everything is on the line. The future looks suddenly like it holds nothing but failure. So why not quit?\n\nBeing an entrepreneur can be very rewarding, but building a successful business is certainly not easy. When things get tough, really tough, your business will fail if you don't have a deep sense of purpose tied to it. Without a clear reason to stick it out, it's just too easy to walk away. My friend Dr. Kristi Funk and her husband, Andy, are a perfect example.\n\nAfter completing medical school and a grueling five-year residency of eighty-hour weeks, Kristi looked forward to starting her surgical career. She was a resident at Virginia Mason Medical Center in Seattle when she was asked to join the Cedars-Sinai Breast Center in Los Angeles. She said no.\n\nBreast surgery was still evolving as a specialty, and Kristi felt this new position wouldn't take full advantage of the surgical skills she had spent so many years developing, so she turned down the offer. But she soon realized the opportunity had less to do with performing surgery and much more to do with making a positive impact on the lives of many women. That surging sense of purpose changed her mind. She soon called back and accepted the job.\n\nBy the time Andy Funk entered her life, she had already become a well-known and respected breast-cancer surgeon. Although Andy was heir to the largest privately held German insurance-brokerage firm, which his family had begun more than a century earlier, Andy wanted to create his own path. At age nineteen he had forfeited his inheritance, made his way to California, started his first company, and promptly fallen $250,000 in debt. Instead of heading home, Andy had fought hard to turn things around. Within five years, he had sold three businesses and established Funk Ventures, a venture-capital firm, which had soon become a pioneer investing in socially responsible companies. He had overcome big odds to make it past his early disappointments; this experience would come in handy later.\n\nAs a result of his success as a venture capitalist, Andy became the youngest member of the board of governors of Cedars-Sinai Medical Center. Not long thereafter, Kristi and Andy met and married.\n\nKristi had by this point become one of Los Angeles's go-to breast-cancer doctors. Her high profile attracted much media attention and helped draw many patients to Cedars-Sinai. But this success meant longer hours for Kristi. She and Andy felt like they had no life to themselves, let alone time to build a family. To make matters worse, Cedars rejected her request to hire any additional support to reduce her workload. After Cedars refused to provide support, Kristi and Andy agreed that things needed to change, fast.\n\nAs a medical and wellness investor, Andy had been intrigued for some time by the thought of opening a comprehensive cancer facility with Kristi. The center, the first of its kind, would allow them not just to hire more doctors to reduce Kristi's workload but also to treat patients more effectively and with better technology. With their own facility they would have the opportunity to change tens of thousands of lives and have an even bigger impact on people. A strong purpose was born.\n\nThe plan came together quickly. They founded the Pink Lotus Breast Center and lined up their financing by recruiting other physicians who would partner with them. In the fall of 2008 they leased thousands of square feet in a Beverly Hills medical complex, which would soon cost them more than thirty thousand dollars per month. Within weeks of their signing the eight-year lease with personal guarantees, the U.S. economy went into a free fall and all of their financing disappeared. Lines of credit were closed, credit cards were canceled, and the physicians who had committed to join the venture and invest began to get cold feet. When they thought things couldn't get more complicated, they found out Kristi was pregnant\u2014with _triplets_!\n\nHaving scraped together every penny they had, Kristi and Andy were determined to remain in their medical office and build their company. The Funks opened their facility in March 2009, on the day the Dow Jones hit its lowest point and just three months away from the impending birth of their triplets. Even though the business plan was built for a minimum of three surgeons, Kristi was the only doctor on staff. When the Funks couldn't pay rent\u2014because they were quickly running out of cash\u2014their landlord sat down with Andy and offered him two options: Get out of the suite or get sued. Andy and Kristi had no interest in leaving and continued believing that their vision would one day come to fruition.\n\nJust as at Cedars, Kristi was working sixteen-hour days in order to try to make ends meet while Andy was trying to hold things together on the corporate and business front.\n\nWithout other surgeons in the practice, Kristi and Andy were able to maintain full ownership of the business, but they were also missing out on much-needed revenue. In addition, delayed payments from insurance companies meant cash often came in six to nine months after Kristi and Andy had to pay their own expenses. Cash flow soon became impossible for Andy to manage. Things were coming to a head.\n\nPretty soon they started running out of toilet paper at the office. When Kristi asked Andy why there was none, he didn't have the heart to tell her\u2014they just couldn't afford to buy any.\n\nIt was clear that despite their best efforts, it just wasn't working. The children's arrival also meant the added cost of a bigger house and two nannies so that Kristi and Andy could work. With all the odds stacked against them, why not just close up shop, declare bankruptcy, and start over?\n\nWell, it wasn't that simple for Kristi and Andy.\n\nAlthough they faced incredible difficulties at this point in their business, they had a tremendous sense of _purpose_. They strongly believed that they were fighting for something that was much more important than their short-term problems.\n\nTheir persistence paid off. Despite sliding into more than two million dollars of debt, Kristi and Andy built a center with an improved approach to breast-cancer care. The business took off and began doubling in size each year. Kristi's continued television appearances on shows such as _Today, The View,_ and _Dr. Oz_ provided further visibility while breast-cancer survivor and Pink Lotus patient Sheryl Crow soon lent her name to the new Sheryl Crow Imaging Center at Pink Lotus. To top it off, General Electric selected Pink Lotus as the first treatment facility to introduce a revolutionary mammography technology to the country. New surgeons joined the team. The business started to recover losses. Even expansion became a reality.\n\nAfter a few years on the brink of financial collapse, Pink Lotus was not just headed toward success\u2014it would soon be regarded as one of the top breast-cancer treatment facilities in the world.\n\nIn May 2013 Angelina Jolie announced that she had undergone a double mastectomy as a preventative measure to decrease her risk of breast cancer. The multiple procedures were all performed at the Pink Lotus Breast Center under Kristi's diligent medical care.\n\nBe it Angelina Jolie, Sheryl Crow, or one of the many thousands of other women who come to Pink Lotus for their breast health each year, Kristi and Andy are making a huge impact on many people's lives.\n\nWhen I think of Kristi and Andy, one takeaway really stands out: Most entrepreneurs don't realize how close they are to breaking through just before they decide to quit. Their stories can help assure you that most people don't know how close they are to success the moment before they decide to quit.\n\nIf you are on the edge and thinking about shutting it down, here are a couple things to consider:\n\n_Get counsel from experienced entrepreneurs_. Be open about what you are experiencing. The most successful people are usually the most willing to talk about their struggles and help others work through their own. It's kind of a rite of initiation into the world of the entrepreneurship, like the exchange of a secret handshake.\n\n_Join entrepreneur clubs and organizations._ For added support join networking groups like the Entrepreneurs' Organization. The mission of such groups is to provide support to entrepreneurs just like you.\n\nI've often found that one of the best ways to get through tough times is to talk to peers in these organizations. Many have already experienced the same problems you are wrestling with, and they want to help.\n\nAs Kristi and Andy demonstrated, if you have a strong enough reason _why_ you are doing something, you will figure out _how_ to make it happen. In order to live your dream, sometimes you need to combine that purpose with pigheaded determination and persistence.\n\n### Seventeen Days\n\nMost entrepreneurs seem to have one response when others ask, \"How's business?\": \"Everything is great!\"\n\nMany either believe or try to project that the business is making big headway, when nothing could be further from the truth.\n\nEntrepreneurs need to be positive and project a certain image of strength. This is important to help attract investors, partners, and the best talent. That attitude inspires confidence and keeps family and loved ones at ease.\n\nBut what happens when things are not going as planned? What happens when the image you are trying to portray does not reflect what is actually happening inside the business?\n\nMany entrepreneurs feel like they have no one to talk to. They don't want to go to their investors because they're afraid they will just be angry. They don't want to talk to their partners because they're afraid they will kill the deal. They don't want to go to their employees for fear they will start looking for new jobs. They don't want to involve their spouses because the emotional roller coaster may be too much. Last but not least, they don't want to admit to themselves that something may be wrong, very wrong.\n\nInstinctively, what many entrepreneurs do is try to bury the news as a means of hiding from it. Meanwhile, the problems eat away at them and their business. That's certainly the wrong response.\n\nAll the answers you need may be just a phone call away.\n\nAs my friend Michelle Patterson says, \"We don't want to show that we're vulnerable. But when we do, it leads to amazing things.\" She should know.\n\nSince its founding three decades ago by then-governor George Deukmejian, the California Women's Conference has been bringing together women from all over the world to learn from one another and to grow. For most of that time the event was organized by the first lady of California, the governor's wife. But in 2012 Governor Jerry Brown's wife decided to discontinue the event.\n\nMrs. Brown's decision inspired Michelle. As an experienced producer of large-scale events, she decided to take the conference over full time, determined to keep it alive. She locked down a venue, built her team, developed a plan, and started to execute. Over the next few months she signed up thousands of attendees, brought in more than 250 exhibitors, and secured more than 150 speakers. Things were off to a great start.\n\nAs the event grew closer, two big things affected the business and its bank account. First, the team outsourced to handle sponsorships grossly inflated its projections. Second, the investor funding expected for the business also failed to materialize.\n\nMichelle was deeply embarrassed. She was supposed to be an expert in putting on this kind of event, and now it was in jeopardy. Like many entrepreneurs, her first instinct was to tell no one, to remain positive on the outside while she struggled to figure how to make ends meet. But one day everything came to a head when a payment was due for the production, and Michelle had no money left. The organization was $1.8 million in the hole. Michelle had no idea how to find the cash she desperately needed.\n\nAs a result, the event was in danger of being canceled.\n\nMichelle had just seventeen days to save it.\n\nShe broke the news to her husband first. She told him she was afraid that she was about to bankrupt the family. When she shared her real concerns, he responded with nothing but support. He told her, \"After fifteen years of marriage I have watched you persevere in everything you do\u2014and you would never forgive yourself if you canceled this event. There has to be a way. We are not the first people to be in this position. You know a lot of very successful business leaders. Why not ask for help?\"\n\nThe next day Michelle made a phone call that changed her life. She reached out to the mayor of Long Beach. She also called the CEO of the convention center where the event was to be held. Michelle and the CEO met later that day, and she explained the event's financial problems. In response the CEO asked her one simple question.\n\n\"What's your favorite flavor of ice cream?\"\n\n\"Strawberry,\" Michelle said.\n\nWith that, the CEO had strawberry ice cream brought into the office. He encouraged Michelle to take a deep breath. Then he said, \"Let's dive in and tackle this together.\" She now had a partner to help take on this problem.\n\nHis response also renewed her confidence. Michelle realized that if she got her ego out of the way and opened up, people would respond in a way that would benefit the event. She went home and made a list of people whose counsel she thought might help her out of this mess. She called everyone on the list and the response she heard most was not \"How could you do that?\" or \"How could you let that happen?\" Instead, everyone wanted to know \"Why didn't you tell me earlier?\" or \"How can I help?\" Several confessed they had tackled similar challenges in their own careers.\n\nHer network came out to help her get the business back on track. Two friends even postponed their honeymoon and moved into her house to help her dial for dollars and collaborate on the effort. Michelle realized that the more she helped people understand the real issues, the more candid she was, the more help they would provide. Soon she had an army behind her.\n\nTogether with this team of advisers, she executed. In seventeen days she went from being told to shut down and consider bankruptcy to reducing what was owed from $1.8 million to $150,000. The event took place and was a big success. So much so that its success led to the creation of the Women's Network and Women Network Day, a global holiday held every May celebrating women.\n\nMichelle often reflects on what would have happened had she given up out of fear and let her ego get the best of her. Luckily, she didn't, and she learned a valuable lesson: When things go wrong, don't try to hide it. Instead of letting fear take hold, take action immediately. When you ask for help sincerely and honestly, you will get it.\n\nIt may not be easy, but here are some ways to, deliberately and with your eye on the horizon, dig yourself out of a business hole.\n\n_Illuminate._ In the words of David Corbin, Sunshine is the best disinfectant. The more light you shine on a problem, the easier it is to tackle. Get it all out on the table. Don't hide anything. Once you discover the source of the problem, get right to finding a solution.\n\n_Ask for help._ The truth is, we have all been there. Letting go of ego and asking for help is one of the greatest signs of leadership and strength. There will be times in your entrepreneurial life when you feel like you're at the end of your rope, just dangling there with nothing to grab onto. Everything you could possibly need may be just a phone call away.\n\nIf you must choose between your pride and saving your enterprise, that's no choice at all. Get over yourself, push down that ego, and ask for help with the problem. You'll be amazed at how eager people will be to come to your aid. There are people out there who have walked in your shoes. Find them and seek their counsel. Don't be afraid to open up about what you are experiencing. The more they know, the better they can help guide you out of a problem and into an opportunity.\n\nNo entrepreneur starts by thinking about what they will do if their big idea, the one that they've worked so hard on, goes sideways. But you need to keep these lessons in the back of your mind.\n\nNothing in life worth achieving ever comes easily. Sometimes you need to fight for your dream. Sometimes you need to let your sheer will and persistence carry you through. When things are tough, you don't always need to be out front if you have the heart to come from behind.\n\nRemember, every single thing you need to turn that situation around is in you now or within your reach.\n\n## EPILOGUE\n\n## STEADY AS SHE GOES\n\nDo the best you can until you know better. Then when you know better, do better.\n\n\u2014MAYA ANGELOU\n\n### Wing Walk\n\nIt was the day of our launch for Virgin Charter and I hadn't slept for twenty-four hours. The previous day Richard Branson had flown in from India, and we had done nearly fifty interviews, filmed an online news segment, and ended the afternoon on FOX News. A prelaunch party followed, with all the glitz and glam befitting his Virgin brand. I got back to my room with just enough time to shower before heading out to CNBC for the official launch of the business live in studio on _Squawk Box_.\n\nThe day started as one of the high points in my life, the culmination of a two-year journey. Still, I had all the anxiety of any business owner or product manager the day of his or her launch. I wanted my product to work and my team to perform. More than anything, though, I wanted to remember my three lines, the three talking points I had been told I needed to say on air.\n\nRichard and I sat next to each other in the green room. He watched a small monitor while I rehearsed my talking points. Then he leaned in to me and said, \"Scott, I've got a problem. I don't think you're doing a good enough job.\"\n\nMy stomach fell to the floor. You could say this was one of those low points on the emotional roller coaster.\n\nHe went on. \"I don't think you're doing a good enough job getting promotion for this launch. I think I'm going to have to step in. Give me a few minutes to think about it.\"\n\nA few minutes later Richard leaned back over and said, \"Scott, I've got it. You and I are going to do a wing walk.\"\n\n\"We're going to do what?\"\n\n\"A wing walk.\"\n\n\"What the hell is a wing walk?\"\n\nHe said, \"You're going to stand on one wing of an airplane, and I'm going to stand on the other wing. We'll each hold on to a rope with one hand and a Virgin banner with the other. Then we'll fly all over New York. We'll get more publicity than you ever imagined for your business. And don't worry. These planes are stunt planes. Your feet will be in metal boots attached to the wings. So if there's turbulence, you won't fall off.\"\n\nHe shared his idea with everyone in the room, and they all thought it was great.\n\nI was too tired to think straight, but I knew enough to still be terrified.\n\nRichard turned back to me and said, \"Don't worry. I'll give you a diaper in case you need it!\"\n\nI sat back in silence. A couple minutes later he leaned in again. \"Hey, Scott, I don't know if we'll be able to get the plane.\" I was totally relieved, again a new high. \"So I have a backup plan.\"\n\n\"What's that?\"\n\n\"You and I are going to go skydiving.\"\n\nWhat?\n\n\"Richard, I've never done that before,\" I replied.\n\n\"That's the best part,\" he said. \"We're going to go tandem. You'll be on my back, and we'll hold a Virgin sign. You're going to get more publicity for this launch than you've ever imagined.\"\n\n\"Richard, I'm not sure that's a good idea. Like I said, I've never done that before.\"\n\n\"Well, you know, here's the thing,\" he said. \"I haven't either. So we'd better practice once before we jump out of the plane so I know where that thing is you have to pull to let the parachute out.\"\n\nAt this point I was so nervous I felt like I was going to get sick. I got up and started walking to the bathroom, but the CNBC stage manager summoned us.\n\nNext thing I knew, we were out there in front of the cameras. The moment was supposed to be exhilarating, but I struggled to remember what I was supposed to say. In my earpiece I heard the producer say, \"Fifteen seconds.\"\n\nI looked over at Richard. He smiled at me. A twinkle in his eye seemed to say, \"I can't wait for later.\"\n\nWe got back in the car after the interview, and I could still feel my heart pounding. Would I be wing walking or skydiving later this afternoon? Suddenly everyone in the car got quiet. Then Richard turned around, his face a big grin. He high-fived everyone in the car. He then looked at me, reached out to give me a high five, and said, \"Scott, give me a high five. It was all just one big joke!\"\n\nThe wing walk. The sky diving. He had just made it up.\n\n\"I just had to break you in,\" Richard said.\n\nIn a weird way, when I look back at Richard's prank, it wasn't any different from what entrepreneurs experience every day when launching their business, full of highs and lows, ups and downs (but usually not a wing walk!).\n\nAs you kick off your journey, it's important to bring your focus back to the task at hand, no matter what happens along the way.\n\n### The Circus and Start-ups\n\nWing walk? Well, maybe not. But launching a new business or product can be like a circus act. There are lots of distractions being thrown at you. Any one of them could take away your focus, leave you off balance, and knock you down. All of this puts you and your business at risk. I know this firsthand.\n\nMy friend Greg Reid has a long bucket list. You know, a list of things he wants to do before he \"kicks the bucket.\" Last year one of those things he decided to take on was the flying trapeze _._ He asked if I'd like to join him.\n\nI had no idea what I was in for. When I finally got a firsthand look at what I had to do, I was terrified.\n\nThink about it. You're standing on a ledge thirty feet off the ground. To jump onto a swinging trapeze, you have to reach out and grab the idea . . . I mean, the swing. You leap and suddenly you're moving at three g's. That's right\u2014three g's! You're launched.\n\nNow you're flying through the air. You're holding on tight. But suddenly the thing gets tougher. If you don't keep moving forward, you lose momentum and you're at risk of falling. So you grab the next ring. Success! Things begin to get easier. You try a flip and you execute. You gain confidence but still, every time you climb the ladder to make that jump and push yourself further along, you still have those butterflies, standing thirty feet above the ground, looking at what is in front of you.\n\nWhat I learned that day is that success is all about letting go. When I first saw those swinging bars and platforms, I thought, _No way am I doing that!_ Some people can't get past the initial fear. They don't trust the net down below. They focus squarely on all the reasons they cannot succeed. Because that's what they focus on consistently, that's all they get out of life.\n\nIt's time for you to let go. It's time to let your momentum carry you forward.\n\nAre you ready to fall or fly?\n\nThe day Greg and I trained for the circus, there were eight other people trying out the trapeze for the first time. We were the only ones to succeed. By the end of the day, we were doing backflips and catching other acrobats' hands midair. It wasn't because we were more skilled than anybody else. It was because we had the right mental attitude.\n\nLetting go and jumping off may come more naturally to entrepreneurs than to most people. The experience is liberating, of course, as leaving your comfort zone can be a real boost for creativity. Taking flight\u2014at launch, on a trapeze, whatever\u2014can inspire you to think outside the box, to carry forward your momentum, to keep advancing your business. It just takes focus, practice, and a willingness to let go.\n\nJust ask any of the more than four hundred billionaires in America or the roughly ten thousand people who have built fortunes of more than a hundred million dollars. Or a run-of-the-mill entrepreneur millionaire\u2014I promise you, every one of those people can tell you they had to face the same fears you do. But they let go, and allowed their momentum to carry them forward. At times they fell, but they got back up again.\n\nIf they can do it, so can you.\n\n### Twenty Thousand Miles per Hour\n\nSpace flight has always fascinated me. The idea of a man sitting in a capsule and being propelled into space at over twenty thousand miles per hour is pretty incredible. The fact that astronauts can confront their fear and engage in something that appears so risky is amazing to me.\n\nA few years ago I had a chance to meet a space-shuttle commander. I was able to ask him a question that I had wondered about for a very long time, concerning the moment in time where the ship is returning from space and reenters the Earth's atmosphere. During that time there's a communication blackout with the command center on Earth and the shuttle is surrounded by a three-thousand-degree ball of fire. When all that occurs, I asked, \"Aren't you just freaking out?\"\n\nThe commander said, more or less, \"That part's a breeze.\" He explained that _positioning_ the aircraft for reentry is what's really tricky. You're above the Earth, traveling at something like twenty thousand miles per hour\u2014and you're flying upside down and backward. You need to flip the ship over and position the nose at an angle of approach of precisely 40 degrees. If you come in too fast or at the wrong angle, the ship could explode when it hits the atmosphere.\n\nI asked him, \"How do you do it?\"\n\nWhat I learned is that risk should not be all that risky.\n\nIn the case of the spaceship commander, he had NASA behind him. Thousands of people, decades of experience. A long history of success (and a few failures) from which to learn.\n\nDuring those minutes of precise ship positioning and reentry silence, no one is focused on fear. They've all been trained by the best, including other space-shuttle flight commanders who have done it before. This isn't trial and error; it's modeling. Each person on the team knows his role. They've practiced for years. They understand how to navigate through challenges. They execute together flawlessly.\n\nIn your business you need to do the same to mitigate what could go wrong and speed up success. Find and study role models to help eliminate risk from your business and to help save you valuable time and money. Learn what others have done to achieve the outcome you want. Study their specific beliefs, their strategies, and the exact order in which they did things. Then do the same things.\n\nTALKING TO YOURSELF\n\nNow matter what life throws your way, you can choose how to respond to it.\n\n**Control Your Inner Dialogue** **:** In every situation your brain asks two important questions: _What does this mean?_ and _What should I do?_ The questions you ask to evaluate the situation will determine what you focus on. What you focus on will determine how you feel. How you feel will determine how you behave. How you behave will have a tremendous impact on your business.\n\n**Break the Pattern** **:** If you catch yourself asking negative and disempowering questions\u2014or if you find yourself caught in a terrible loop\u2014break that pattern and insert questions that get you to analyze the situation in new, empowering ways.\n\n**Make This a Habit:** It will change your life forever. Before you know it, you'll believe that the universe is conspiring on your behalf!\n\n### Keep Your Eyes on the Horizon\n\nWe're coming full circle. Remember that the number one key to success is maintaining your focus. That means controlling your inner dialogue as you keep your eyes on the horizon. In doing so, you'll be able to anticipate, prepare for, and plan for whatever may be in your path and then steer gracefully around it.\n\nThis is not easy to do. It takes a lot of practice and hard work. It's like building a muscle. But if you focus on getting better in this one area of your life, each and every day, you will get stronger. You will be better able to manage through the ups and downs of entrepreneurship successfully.\n\nNo matter what life throws at you and no matter how high that wall looks, you need to take it all on. You'll find it helps to manage your focus by asking empowering questions that drive your energy toward a solution.\n\nAs a Lakers fan, some of my fondest memories are of watching the Lakers \"Showtime\" era in the eighties. My favorite player then\u2014and one of my personal heroes now\u2014was Earvin \"Magic\" Johnson. On the basketball court he lived up to his nickname because of his focus, his ability to concentrate on the objective at hand. He was always in the zone, in the flow of things. He won five championships and gained a place in the NBA Hall of Fame as one of the greatest to ever play the game.\n\nMagic performed not only on the court but in business as well. Even before he retired from the game he founded Magic Johnson Enterprises, planning for what came next. His first big investment was a partnership with businessman Earl Graves Jr. when, in 1990, they bought a Pepsi-Cola distributorship. Magic got involved in movie theaters and inner-city redevelopment projects. In 1998 he partnered with Starbucks to create a company called Urban Coffee Opportunities, opening franchises in inner-city neighborhoods and building out more than a hundred locations before selling them back to Starbucks in 2010 for what's believed to be more than fifty million dollars.\n\nJohnson is also a lead investor in the $1 billion Canyon Johnson real estate fund and the $500 million Yucaipa Johnson private equity fund. He owned a piece of the Lakers for many years and joined with Guggenheim Partners in its $2 billion acquisition of the Los Angeles Dodgers. He's now working on a venture to bring an NFL team back to Los Angeles.\n\nSome people may say he was just lucky or born with that talent. But Magic Johnson is somebody who has jumped over more than his share of barriers.\n\nIn 1991 I was promoting a Tony Robbins event in Houston, Texas. The Lakers were scheduled to play the Rockets. There were a lot of fans on my team, so we decided to go to the game and got great seats right behind the Lakers bench. But Magic was missing from the action that day. His absence seemed strange because we hadn't heard about any kind of injury that would keep him out of the lineup.\n\nThen, on November 7, 1991, I saw the breaking news on television: Magic Johnson was HIV positive. The first thing I thought was, _Will Magic survive?_ But that was not Magic's first reaction.\n\nLongtime Lakers trainer Gary Vitti recalls that when Magic first learned the news, he said, \"Well, God gave this to the right person, because I'm going to do something with it.\" And he has. He used his diagnosis as an opportunity to educate and inspire millions of people around the world. He gave hope to people who at the time seemed without hope, creating the Magic Johnson Foundation to help combat HIV. He became the symbol not only for survival but also for successful treatment.\n\nHow does this relate to your launch? By controlling his inner dialogue and asking a different set of questions from the ones most people would have asked in the same circumstance, Magic was able to keep himself in the strongest possible position to steer through the medical challenges he faced. Even in the worst of circumstances, consciously or subconsciously, he used this skill to maintain his focus and move his life forward in a positive direction.\n\nIf Magic could overcome a challenge like this, I know all of us can overcome the things we battle every day in our businesses by maintaining our focus, controlling our inner dialogue, and keeping our eyes up on the horizon.\n\n### Fill Your Cup First\n\nSome people get it backward. I have worked with many entrepreneurs whose primary driver for starting a business was to create a vehicle for serving others or facilitating a specific social change. These are noble purposes.\n\nHere's the problem. Many people who start new ventures with these goals in mind place too much of their focus on the social aspect of their venture and don't spend enough time building a fundamentally sound business.\n\nWhile I admire their passion, I also know that focusing on giving first and creating a sustainable business second usually leads to failure. That's because without creating a solid business that generates cash, these entrepreneurs are more likely to go out of business. As a result, they are not able to have the long-term impact they had hoped to achieve.\n\nIf your goal is to create a platform that gives back, I think that's awesome. But make sure you have your priorities in order. First create a fundamentally sound business, one that creates profits and generates sustainable cash flow. Once you have reached that threshold, use what you choose to give back.\n\nBest-selling author and speaker Les Brown puts it well. He likes to say, \"Fill your cup first. Let what spills over feed others.\" If you take this approach, you will create not only stability for yourself but also abundance that can be shared with the world.\n\nNow, if you've reached a point where you can afford to give back but are looking for a cause, I'd suggest this: Give to other entrepreneurs.\n\nWhen you're in orbit and you look down and see someone who has taken a shot, who may be struggling, reach out and give them a hand. Maybe it's financially to help them get through a tough period. Maybe it's by offering advice or being a sounding board.\n\nWhen given the opportunity, invest in new businesses. Serve on boards. Assist entrepreneurs in launching new and innovative products and services. After all, it's these dreamers who keep our economy moving, who create jobs and growth.\n\nUse your position to fuel the entrepreneurial spirit that is the heart and soul of America. Be there to support others who will be going down a road you have already traveled. You know they must take their own wing walks, try flying a trapeze, and race around tough corners. If you can keep them from falling, from hitting the wall, you'll be giving back in a way that can benefit all of us.\n\n# ACKNOWLEDGMENTS\n\nThere are so many people I would like to thank for their contribution to this book. First, to Demetri Boutris and Bruce Perlmutter, who got this whole process started. This would have never happened without the hard work of Nena Madonia and the amazing team at Dupree\/Miller or without the support and encouragement of Adrian Zackheim, Natalie Horbachevsky, and the rest of the team at Portfolio\/Penguin. Special thanks go out to Chris Helman for his valuable insights along the way. Finally, thank you to Gary Goldstein, Ron Klein, and Raoul Davis for all of their guidance.\n\nOn a personal note:\n\nI would like to thank my mother for being so incredibly supportive. For going on this ride with me for the past twenty-five years. You taught me the value of persistence, hard work, and always looking at the \"bright side\" of a situation, no matter how hard it might have seemed in the moment. Thank you for always believing in me.\n\nTo Big D, thanks for being the rock in our family and having the patience to sit with me, on so many long nights, to talk about business. You were the greatest influence on my career when I was getting started.\n\nTo my sister Ashley and brother Shane, you have always been there for me. You have helped me in so many ways on more than one occasion. Thank you!\n\nTo Dad, I'll never forget all the football, baseball, and basketball memories growing up. What a great dad.\n\nTo Steve Rosendorf and Debbie Nessen, wow, did I get lucky! I hit the jackpot!\n\nTo Julie and Jay Don Johnson, many more great years together.\n\nTo Dan Clivner and Steve Cochran, you are family. You will never know how much I appreciate your support.\n\nTo Greg and Allyn Reid, you are the true definition of friendship.\n\nAdditional thanks to those who gave me a shot somewhere along the way: Richard Branson, Gabriel Baldinucci, Dan Porter, Stephen Murphy, Jon Peachey, Christine Choi, Julie Cottineau, Nirmal Saverimuttu, Ron Garret, Steve Chien, Dena Cook, Brooke Hammerling, Tony Robbins, Mike Hutchison, Ross Levinsohn, Chris Cottle, Jeff Tang, Jed Savage, Jason Port, Chris Kitze, Laurant Massa, Russell Hyzen, Kelly Perdew, Mark Moses. To the team at Virgin Charter for giving it their all.\n\n# RESOURCES\n\nRecommended Reading\n\n_Unlimited Power,_ Anthony Robbins\n\n_Stickability,_ Greg S. Reid\n\n_Wooden: A Lifetime of Observations On and Off the Court,_ John Wooden\n\n_As a Man Thinketh,_ James Allen\n\n_Losing My Virginity,_ Richard Branson\n\n_Crossing the Chasm,_ Geoffrey A. Moore\n\n_Do You!,_ Russell Simmons\n\n_The Lean Startup,_ Eric Ries\n\n_Secrets of the Millionaire Mind,_ T. Harv Eker\n\n_The Mackay MBA of Selling in the Real World,_ Harvey Mackay\n\n_The Art of the Start,_ Guy Kawasaki\n\n_Crush It!,_ Gary Vaynerchuck\n\n_Seven Spiritual Laws of Success,_ Deepak Chopra\n\n_The Innovator's Dilemma,_ Clayton M. Christensen\n\n_Think and Grow Rich,_ Napoleon Hill\n\n_Permission Marketing,_ Seth Godin\n\n_Chicken Soup for the Soul,_ Jack Canfield and Mark Victor Hansen\n\n_Rework,_ Jason Fried and David Heinemeier Hansson\n\n_Zen and the Art of Happiness,_ Chris Prentiss\n\n_Tribal Leadership,_ Dave Logan, John King, and Halee Fischer-Wright\n\n_Do More Faster,_ David Cohen and Brad Feld\n\n_Delivering Happiness,_ Tony Hsieh\n\nProgram to Purchase or Attend\n\nBusiness Mastery, Tony Robbins & Chet Holmes\n\nBlogs to Follow\n\nSeth Godin: sethgodin.typepad.com\n\nTED: ted.com\/talks\n\n_Vaynerchuk_ (Gary Vaynerchuk): garyvaynerchuk.com\n\n_This Week in Venture Capital:_ youtube.com\/show\/thisweekinventurecapital\n\n_Blog Maverick_ (Mark Cuban): blogmaverick.com\n\nChris Brogan: http:\/\/www.chrisbrogan.com\/\n\nCopyblogger: http:\/\/www.copyblogger.com\/\n\n# INDEX\n\nThe page numbers in this index refer to the printed version of this book. The link provided will take you to the beginning of that print page. You may need to scroll forward from that location to find the corresponding reference on your e-reader.\n\nAccelerators,\n\nAccutrade,\n\nAdaptability\n\nand crisis. _See_ Downturns\n\nimportance for entrepreneurs, \u201310\n\nAmazon.com,\n\nAngel investors,\n\nApple Computer,\n\nMac, impact on business, \u20135\n\nAssociates in business, \u201371. _See also_ Team building\n\nmentors,\n\nraising capital from,\n\nAugust Capital,\n\nAustin, Stephen, \u20137\n\nBaskin, Burton, \u201311\n\nBaskin-Robbins, \u201311\n\nBeethoven,\n\nBezos, Jeff, ,\n\nBlakely, Sara\n\nbusiness idea, development of, xv, \u201337,\n\nunique insight of,\n\nBookkeepers,\n\nBoomerang Boxes, \u201377\n\nBootstrapping process, \u201330\n\nBrand, creating, \u201378\n\ncommercial, content of, \u201376\n\nself-definition, \u201377\n\nsocial media for,\n\nsteps in, \u201375\n\nwords\/descriptors for,\n\nBranson, Richard, xv,\n\nDuffy association with, , , \u2013200\n\non entering existing markets,\n\n_Losing My Virginity,_\n\nas master of promotion, \u201399\n\nobstacles, creative thinking about, \u201329\n\nBrignole, Doug, \u201384\n\nBrin, Sergey, , ,\n\nBroadcast.com,\n\nBuffett, Warren,\n\nBurrell, Stanley Kirk (MC Hammer), rule for success, \u201320\n\nBusiness coach,\n\nBusiness plan, \u201394\n\nexecutive summary with, \u201389\n\nexit strategy in, \u201391\n\nlast page as start of, \u201387\n\nmarket, learning about, \u201388, \u201394\n\nmodels for,\n\nobstacles, identifying,\n\none-page plan, , \u201389\n\nspecific goals in,\n\nstrategic plan in,\n\nthings to avoid, \u201386,\n\nBuss, Jerry,\n\nBuyouts, and partnerships,\n\nCalifornia Women's Conference, \u201395\n\nCapital notes,\n\nCapital raising, \u201348\n\naccelerators,\n\nand angel investors,\n\nbootstrapping process, \u201330\n\nconvertible notes,\n\ncrowdfunding,\n\nfor equity in business, , , \u201348\n\nguidelines for, \u201334, \u201348\n\nincubators,\n\ninvestor input, listening to, , \u201346\n\ninvestors, learning about, \u201339\n\nloans, ,\n\npersistence needed for, \u201335\n\npreferences, \u201348\n\nred flags for investors, \u201344\n\nself-funding, \u201328,\n\ntime frame for, \u201334\n\nventure capital,\n\nCeres Securities,\n\nChecchi, Al, \u201334\n\nChief energizing officer (CEO), ,\n\nChief executive officer (CEO), \u2013105\n\nculture of business, creating, \u201321\n\ntasks of, \u20135\n\nChief venting officer (CVO),\n\nCompensation, for business owner, \u20137\n\nCompetition, big business versus entrepreneur, \u20133, \u201314\n\nContracts, for partnerships, \u201314\n\nCorbin, David,\n\nCorporations\n\nbig business versus entrepreneur, \u20133, \u201314\n\nchange, sluggish pace of, , \u201313\n\nCostolo, Dick, ,\n\nCost savings, passing to customer,\n\nCreative thinking. _See also_ Innovation\n\nto overcome obstacles, \u201330\n\ntoward simplicity, \u201321\n\nCredit cards, \u201350\n\nCrowdfunding,\n\nCrowdsourcing\n\nfor environmental protection,\n\nsales force from social media, \u201368\n\nCuban, Mark, , ,\n\nCulture of business, \u201326\n\nand CEO, \u201321\n\ncreating, steps in, \u201322\n\nof creativity. _See_ Innovation\n\nCush, David,\n\nCustomers\n\ncost savings, passing to,\n\nfeedback, use of, \u201354, \u201371\n\nprofiles, creating, \u201358\n\nrelations, building,\n\nD'Aloisio, Nick, \u201316\n\nDanceJam.com,\n\nDecison making, time for,\n\nDelegating, \u201381\n\nDiCaprio, Leonardo,\n\nDownturns, \u201396\n\nadaptability, importance of, \u201310\n\nand economic conditions. _See_ Economy\n\nemotions, management of, \u201363,\n\nexit strategy for, \u201391\n\nhelp, asking for, \u201396\n\nlayoffs,\n\nmarket, learning about, \u201394\n\nnew directions, finding, \u201317, \u201378, \u201395, \u20136\n\nobstacles, creative thinking about, \u201330\n\npositive spin on, \u201310, ,\n\nsharing information about, \u201375\n\nsure thing, caution about, \u201345\n\nworst-case, steps to avoid, \u201387, , \u201396\n\nDuffy, Scott\n\nand Richard Branson, , , \u2013200\n\nas serial entrepreneur, \u20136, \u201333\n\nsetbacks\/adaptability of, \u201310\n\nand Tony Robbins, \u201310, \u201368, \u201329,\n\nDyer, Dr. Wayne,\n\neBroker,\n\nEconomy\n\ncrash (2008), impact on entrepreneurs, , \u201345, \u201358,\n\nfavorable for entrepreneurs, \u20133, \u201313,\n\nEllison, Larry,\n\nEmotional control, \u201363\n\nEmployees, hiring. _See_ Team building\n\nEntrepreneurs\n\nadaptability, importance of, \u201310\n\nage and success,\n\nassociates and environment, \u201371\n\nbig business versus, \u20133, \u201314\n\nbrand, creating, \u201378\n\nbusiness plan, \u201394\n\ncapital raising, \u201348\n\nculture for business, creating, \u201326\n\nand economic conditions. _See_ Economy\n\nemotions, management of, \u201363\n\nas experts, myth of,\n\nfears\/excuses, letting go of, \u201340\n\nfinances, managing, \u201351\n\ngrowth of business, \u201381\n\nhabits, empowering, \u201374\n\nand high-tech methods. _See_ Technology\n\ninnovation and success, \u201330\n\nlaunch process, \u201371\n\nnew versus past types of,\n\nobstacles, overcoming, \u201330\n\nopportunities, impetus for developing, \u201313, , ,\n\nrole designations for, \u2013105\n\nsales, \u201371\n\nserial entrepreneurs, \u20136, , \u2013101\n\nsimplicity, focus on, \u201321\n\nspouse\/partner, role of, , \u201356\n\nstarting business, time frame for, \u201312, \u201315\n\nsuccess and letting go, \u2013201\n\nteam building by, \u2013118\n\nvision, scope of,\n\nEntrepreneurs' Organization, ,\n\nEnvironment\n\nassociates, importance of, \u201371\n\nimportance for progress, \u201367\n\npersonal brand, creating, \u201378\n\nEquity in business, by investors, , , \u201348\n\nExcuses, letting go of, \u201340\n\nExecutive summary, \u201389\n\nExercise, \u201360\n\nExit strategy, \u201391\n\nFab Five, \u201370\n\nFacebook, , ,\n\nFailure. _See_ Downturns\n\nFamily\n\nraising capital from,\n\nspouse\/partner involvement, , \u201356\n\nFears, letting go of, \u201340\n\nFinances, \u201351\n\nbookkeeper, necessity of,\n\nbusiness insurance,\n\ncapitalizing company. _See_ Capital raising\n\nCEO role, ,\n\ncompensation for owner, \u20137\n\nand exit strategy,\n\nfinancial model, creating, \u201348\n\ngrowth kills cash concept, \u201376\n\nand growth of business, \u201378\n\nkeeping in order,\n\nand partnerships,\n\npersonal\/business accounts, separating, , \u201350\n\npersonal windfall, understanding about, \u201351\n\npossible losses, estimating, ,\n\nseed money, separating,\n\nFranchising, origin of,\n\nFriis, Janus,\n\nFunk, Andy, \u201391\n\nFunk, Dr. Kristi, \u201391\n\nFunk Ventures,\n\nFuture\n\nCEO role in, \u20134\n\nplanning for,\n\nGates, Bill, business idea, roots of,\n\nGladwell, Malcolm,\n\nGoals\n\nand business plan, \u201394\n\nimplementing, example of, \u201384\n\nand personal brand, \u201373\n\nGoldstein, Gary, \u201334\n\nGoogle, , ,\n\nGoogle Alerts,\n\nGore, Al,\n\nGraves, Earl, Jr.,\n\nGrowth of business, \u201381\n\nand delegating, \u201381\n\nfailure of. _See_ Downturns\n\ngrowth kills cash concept, \u201376\n\nnew features, adding,\n\npost-launch monitoring, \u201374,\n\nreinvesting, rules for, \u201378\n\nHabits, and personal brand, \u201374\n\nHammer-and-nail syndrome, \u201352\n\nHastings, Reed,\n\nHoltz, Lou, \u201377\n\nHornik, David,\n\nHutchinson, Michael \"Hutch,\"\n\nHyzen, Russell, \u201320\n\nIncubators,\n\nIndependent contractors\n\nhiring, as screening, \u20139\n\noutsourcing, \u201318\n\nInner dialogue, controlling,\n\nInnovation\n\nculture of, creating, \u201326\n\nand disruption,\n\nand risk,\n\nof successful entrepreneurs, \u201330\n\nInsurance, for legal protection,\n\nInternet. _See also_ Social media\n\nimpact on business, \u20136\n\nmarket research on,\n\nInvestors in business. _See_ Capital raising\n\nJob descriptions, \u20138\n\nJobs, Steve, business idea, roots of,\n\nJohnson, Earvin \"Magic,\" \u20136\n\nK. Aufhauser & Co.,\n\nKilham, Chris \"Medicine Hunter,\"\n\nKitze, Chris, \u201321\n\nLacy, Sarah, ,\n\nLaunch process, \u201371\n\nfeature creep, avoiding,\n\nhammer-and-nail syndrome, \u201352\n\nniche marketing for, \u201359\n\npartnerships with other businesses, \u201364\n\nsimplicity, focus on, \u201355\n\nsocial media, use as sales force, \u201368\n\nstages of, xiv\n\nand success. _See_ Growth of business\n\ntime frame for, xii\u2013xiv,\n\nuser feedback, use of, \u201354, \u201371\n\nLayoffs,\n\nLinkedIn, , , ,\n\nLoans, ,\n\n_Losing My Virginity_ (Branson),\n\nMagic Johnson Foundation,\n\nManager, entrepreneur as,\n\nMarket for product\n\ncustomer profile, creating, \u201358\n\nlearning about, \u201388, \u201394\n\nniche marketing, \u201359\n\nsocial media as research tool, \u201368,\n\nMenlo Ventures,\n\nMentors, ,\n\nMetro, Marty, \u201378\n\nMicrosoft, \u201330\n\nMinimum viable product (MVP), \u201354\n\nMission statement,\n\nMistakes, sharing information about, \u201375\n\nMoses, Mark,\n\nentrepreneurial history of, \u2013104\n\nMotivational leaders, , . _See also_ Robbins, Tony\n\nmessage of,\n\nreading writings of, \u201370\n\nNBC Internet,\n\nNiche marketing, \u201359\n\nOutsourcing, \u201318\n\nexamples, providing for,\n\nhiring,\n\npayments,\n\nproblems, steps to avoid, \u201316\n\nPage, Larry, , ,\n\nPandoDaily,\n\nPartnerships, \u201314\n\nissues to consider,\n\nlegal contract, topics in, \u201314\n\nwith other businesses, \u201364\n\nPatterson, Michelle, \u201395\n\nPersonal brand, creating. _See_ Brand, creating\n\nPhysical appearance, and personal brand, \u201374\n\nPink Lotus Breast Center, \u201391\n\nPlatinum Capital, \u2013101\n\nPope, Brian,\n\nPort, Jason, \u201342\n\nPreferences, \u201348\n\nQuote.com, \u201362\n\nRainwater, Richard,\n\nReid, Greg, \u201339, \u2013201\n\nRicketts, Joe,\n\nRidgway, Steve, \u201358\n\nRisk\n\nand innovation, ,\n\nand money management. _See_ Finances\n\nsimplicity to minimize,\n\nRobbins, Irv, \u201311\n\nRobbins, Tony, xv\n\nDuffy association with, \u201310, \u201368, \u201329,\n\nevent promotion by, \u201330\n\nmembers of circle of, \u201368\n\nprograms, effectiveness of, \u20139\n\nRockefeller, John D., business idea, roots of,\n\nRohn, Jim,\n\nRole models,\n\nSales, \u201371\n\nbuyer bluprint, understanding, \u201371\n\ncrowdsourcing from social media, \u201368\n\nfans first\/customer second approach,\n\nSchmidt, Eric, ,\n\nSchultz, Howard, xv\n\nbusiness idea, development of, \u201353\n\nSelf-awareness, and choosing team, \u201396, \u201399\n\nSerial entrepreneur, \u20136, \u201333, , \u2013101\n\nShared ownership,\n\nSharpshooter, entrepreneur as,\n\nSiegel, Mark,\n\nSimplicity\n\nof business plan, , \u201389\n\nimportance for entrepreneurs, \u201321\n\nMC Hammer on, \u201321\n\nof product, and launch, \u201355\n\nof successful products,\n\nvalue for entrepreneurs,\n\nSmart Charter, \u201345,\n\nSocial media\n\nand marketing,\n\nmarket research on, , \u201368\n\nmonitoring brand on, \u201367\n\nniche marketing on,\n\nsales force, crowdsourcing from, \u201368\n\nuse for personal brand,\n\nSpanx, , \u201337,\n\nSpouse\/partner, involvement of, , \u201356\n\nStandard Oil,\n\nStarbucks, \u201353\n\nStart-up, new business tasks. _See_ Entrepreneurs\n\nStress\n\nemotional control and crisis, \u201363\n\nmanagement and exercise, \u201360\n\nand raising capital,\n\nStudent Painters,\n\nSummly,\n\nTai, Bill, ,\n\nTeam building, \u2013118\n\nassociates, importance for progress, \u201367\n\nto balance talents, \u201397\n\nbusiness coach, hiring,\n\ncandidates, sources for,\n\nCEO role in, \u2013105\n\nconsultants, ,\n\nand culture of company, \u201322\n\ndelegating, \u201381\n\nFab Five, \u201370\n\nand group interaction, \u201325\n\njob candidates, screening, \u20139\n\njob descriptions, \u20138\n\nkey relationships, ,\n\nlife-cycle approach to, \u201398, , \u201374\n\noutside-office activities, \u201323,\n\noutsourcing, \u201318\n\npartnerships, \u201314\n\npartnerships with other businesses, \u201364\n\nand self-awareness, \u201396, \u201399\n\nshared ownership, benefits of,\n\nTechnology. _See also_ Social media\n\nand cost savings,\n\nimpact on business, \u20136,\n\ninnovation and success,\n\ninvestor, MC Hammer as, \u201319\n\nfor starting business, \u201312\n\nvalue for entrepreneurs, , \u201313,\n\nTracy, Brian,\n\nTrump, Donald,\n\nTumblr,\n\nTwitter, , , ,\n\nUrban Coffee Opportunities, \u20135\n\nUsedCardboardBoxes.com,\n\nVenture capital,\n\nVesting, and partnerships,\n\nVirgin Charter, \u201345, , , , \u201399\n\nVirgin Group, \u201329,\n\nVisionary, entrepreneur as,\n\nVision for company\n\nand CEO, ,\n\nand culture of company,\n\nand partnerships,\n\nvision statement,\n\nVoting rights, and partnerships,\n\nWaitley, Denis,\n\nWelch, Jack,\n\nWilliams, Evan,\n\nWomen Network Day,\n\nWomen's Network,\n\nXoom.com, \u201320, \u201325,\n\nYasun\u00ed Park, \u201329\n\nZennstr\u00f6m, Niklas,\n\nZiglar, Zig,\n\nZuckerberg, Mark, ,\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n## Stranger in My Arms\n\n## Rochelle  \nAlers\n\n##  \n\n## Contents\n\nDear Reader\n\n[Part One  \nFriends](StrangerinMyArms_p_1.html#p_1)\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\n[Part Two  \nLovers](StrangerinMyArms_p_2.html#p_2)\n\nChapter 8\n\nChapter 9\n\nChapter 10\n\nChapter 11\n\nChapter 12\n\nChapter 13\n\nChapter 14\n\nChapter 15\n\nChapter 16\n\nChapter 17\n\nChapter 18\n\nChapter 19\n\nChapter 20\n\nChapter 21\n\nChapter 22\n\nChapter 23\n\nChapter 24\n\n[Part Three  \nSilent Witness](StrangerinMyArms_p_3.html#p_3)\n\nChapter 25\n\nChapter 26\n\nChapter 27\n\nChapter 28\n\nChapter 29\n\nChapter 30\n\nChapter 31\n\nChapter 32\n\nChapter 33\n\nChapter 34\n\nChapter 35\n\nEpilogue\n\nComing Next Month\n\n## Dear Reader\n\nYou were introduced to ex-CIA field operative Merrick Grayslake in No Compromise and again when he stepped on stage in Renegade to exchange vows with Alexandra Cole.\n\nMerrick is back, this time in his own role as an enigmatic hero in Stranger in My Arms, and to take his final bow in Book #12 in the ongoing Hideaway legacy.\n\nIf you want to know the events that led to Merrick and Alex's courtship, then I invite you to join these unforgettable characters in a sensual romance where their love could put them at the greatest risk of all.\n\nYours in Romance,  \nRochelle Alers\n\n## Part One\n\n## Friends\n\n## Chapter 1\n\nThree knocks on the bedroom door in rapid succession stopped Alexandra Cole as she prepared to slip her feet into a pair of three-inch, silk-covered, midnight-blue pumps.\n\nA frown furrowed her forehead as she stood up. This was the second interruption that had thwarted her getting dressed for her cousin's wedding.\n\nThe first time it was Ana who, in the full throes of PMS, had experienced a temporary meltdown when she couldn't zip up the dress she'd chosen to wear for the New Year's Eve ceremony. She and Ana were the same height, five-three, but Alex outweighed her younger sister by a mere five pounds. The crisis was resolved when she offered Ana one of the two dresses she'd brought with her.\n\n\"Who is it?\" she called out.\n\n\"Jason.\"\n\nAlex rolled her eyes. Now it was her younger brother. \"What's the matter, little brother? Do you need me to tie your tie?\"\n\n\"Very funny, Alex,\" he drawled sarcastically from the other side of the door. \"I came to tell you that one of your loser ex-boyfriends just showed up uninvited, and Uncle Martin's security people won't let him in. What do you want to do?\"\n\nCrossing the carpeted bedroom on bare feet, she opened the door. Jason Cole stood before her in a dark blue suit, white shirt and white silk tie. It wasn't often that she saw him in a suit, but Alex had to admit that her twenty-four-year-old brother cut a very handsome figure in tailored attire.\n\nJason was the quintessential Cole male: over six feet, olive coloring, black curly hair and a dimpled smile. And in keeping with a family ritual that dated back to the marriage of their grandparents from which the prospective groom was exempt, any male who claimed Cole blood affected light-colored neckwear.\n\n\"Who is he?\"\n\nJason lifted sweeping black eyebrows. \"The message was, 'Tell her Donald is here.'\"\n\nHer large clear gold-brown eyes narrowed. \"Donald,\" Alex repeated. She knew two Donalds. One who'd been her study partner in undergraduate school and another she'd dated only twice before she handed him his walking papers. \"Did he leave his last name?\"\n\nCrossing his arms over his chest, Jason shook his head. \"He also said, and I quote, 'She'll know who I am,' end quote.\"\n\nRealization dawned. He had to be Donald Easton. \"That arrogant SOB,\" she whispered. \"Tell them to let him in and have him wait for me by the refreshment tent.\"\n\nA sardonic smile parted Jason's lips. \"If you want, Gabe and I can give him a blanket party.\"\n\nHer brow furrowed. \"What are you talking about?\"\n\n\"We'll throw a blanket over his head, then kick his ass. And I'm willing to bet that if he wasn't getting married in an hour Michael would also want to get his licks in.\"\n\nAlex stomped a bare foot. \"Stop it, Jason! There will be no brawling tonight or any other night. Donald Easton has a problem with the word no. I'll take care of him.\"\n\n\"Are you sure, Alex?\"\n\nForcing a dimpled smile, she patted her brother's arm. \"Yes, I'm sure. Now go so I can finish dressing.\"\n\nJason flashed a wolfish grin so reminiscent of their father's. \"Okay. By the way, you look great.\"\n\n\"Thanks.\"\n\nAlex closed the door and crossed the expansive bedroom she shared with Ana and two other female cousins whenever they gathered in West Palm Beach.\n\nSlipping her feet into her shoes, she wondered why a man she hadn't seen in nearly a year had come from Virginia to see her. Unfortunately she'd told Donald that she always celebrated Christmas and New Year's in Florida with her extended family; it was apparent he wanted to surprise her.\n\nWell, the surprise would be on him because she had no intention of resuming what had been doomed from the start.\n\nA member of Martin Cole's private security detail took a glance at the SUV with West Virginia plates and entered the number into his PDA. Smiling, he nodded at the man behind the wheel.\n\n\"We'll park your vehicle for you, Mr. Grayslake.\" He gestured to a parking attendant before returning his attention to Merrick Grayslake. \"Once you walk through the gates and make a right someone will escort you to the Japanese garden.\"\n\nMerrick nodded. \"Thank you.\"\n\nReaching for the suit jacket resting on the passenger-side seat, he got out of his vehicle, slipped his arms into the sleeves, then as directed made his way through a set of iron gates that protected the property that made up the Cole family West Palm Beach compound.\n\nHe hadn't taken more than half a dozen steps when he spied a small camera attached to the upper branches of a tree. Security personnel and surveillance equipment monitored everyone entering or leaving the property.\n\nHe'd left Bolivar, West Virginia, at dawn, stopping twice to refuel and stretch his legs. The drive south had taken longer than expected because of bumper-to-bumper holiday traffic along I\u201395. It was New Year's Eve and motorists were heading either home or to clubs or restaurants where they'd ring in the coming year with their families and\/or friends.\n\nAt thirty-five, Merrick Grayslake had lost count of the number of countries where he'd welcomed in a new year. Whether in Central or South America, the Middle East, Southeast Asia, or in his last assignment as a CIA covert field operative\u2014Afghanistan\u2014for him it had become just another uneventful holiday.\n\nNow, for the first time in more than two years, he wouldn't be alone or engaged in an undercover mission when the clock struck midnight. It had taken the wedding of Michael Kirkland, a man who'd saved his life, for Merrick to temporarily forsake his reclusive way of life and leave what had become his sanctuary, a modest two-story home near the Allegheny Mountains.\n\nHe'd checked into a local hotel and asked the front desk for an eight-thirty wake-up call. His head had barely touched the pillow when the ringing telephone woke him from a deep, dreamless sleep. He'd drunk a pint of water from the wet bar to offset dehydration before he readied himself to attend a New Year's Eve wedding.\n\nWhen U.S. Army captain Michael Kirkland had come to him to solicit his help in protecting his social worker fianc\u00e9e, Merrick experienced a long-forgotten shiver of excitement that always preceded a new covert mission. But the feeling was short-lived. He'd helped Michael identify Stanley Willoughby, the man behind a conspiracy to kill Jolene Walker; he'd remained in the Washington, D.C., area for several weeks following the arrest and subsequent indictment of the D.C. power broker before returning to his adopted home state.\n\nMerrick still didn't understand why he'd decided to put down roots in West Virginia, but there was something about the topography that suited his temperament. The panoramic views, the rugged splendor of the mountains, and the small towns that predated the Revolutionary War and still bore the scars of the Civil War had remained virtually untouched, architecturally, since the 1950s.\n\nThe slate path widened to a lush, manicured meadow where an enormous gauze-draped white tent protected cloth-covered tables from insects. A smaller tent, less than fifty feet away, doubled as a portable bar. The weather had cooperated: clear skies, full moon and nighttime temperatures in the low sixties. His pace slowed as he joined a small crowd milling around the entrance to a garden.\n\nA young woman sporting a white blouse and black skirt approached him. As she came closer he saw the earpiece in her left ear; he found it ironic that whenever he left Bolivar his surveillance instincts kicked in to high gear. It was as if he went into hunter mode, watching, listening and mentally recording everything around him.\n\nShe flashed a professional smile. \"Your name, sir?\"\n\n\"Grayslake.\"\n\n\"Please follow me, Mr. Grayslake.\" She led him into a large tent in the middle of a Japanese-inspired garden; organza-swathed chairs were lined up in precise rows like soldiers at a military parade. She indicated a chair with a Velcro tag bearing his name. \"The bar is open for appetizers and liquid refreshment.\"\n\nMerrick was grateful for the offer. He hadn't eaten anything in eighteen hours. \"Thank you.\" Nodding to the woman, he went back the way he'd come.\n\nThe light from the full moon competed with strategically placed floodlights and thousands of tiny bulbs entwined in the branches of trees and lampposts. With the artificial illumination it could've been ten in the morning rather than ten at night.\n\nMerrick had received an engraved invitation that read that Michael Blanchard Kirkland and Jolene Walker were scheduled to exchange vows at eleven, followed by a midnight reception dinner and a New Year's Day brunch.\n\nHe would remain in West Palm Beach for the wedding and reception. He'd decided to skip the brunch because his plans included spending a few days in Miami before heading down to the Keys. He hadn't told Rachel he was coming, praying she wouldn't seek retribution because he hadn't kept his vow to keep in touch.\n\nA hint of a rare smile played at the corners of Merrick's mouth as he neared the bar. He was never one to make resolutions, but the events of the past three months had forced him to rethink his monastic existence. Since reuniting with Michael Kirkland he'd socialized more than he had in years.\n\nThe sound of voices raised in anger caught his attention. As he turned around, his gaze caught and held the petite figure of a woman in a dark-colored dress with a revealing d\u00e9colletage. Light reflected off the sparkle of diamonds in her ears, several delicate strands gracing her slender neck and in her dark hair. Merrick only saw her profile, but what he saw held him captive.\n\nAlexandra glared at Donald. He'd downed one glass of champagne while holding another flute filled with the bubbly wine. She couldn't believe he'd come\u2014unannounced\u2014to her uncle's house and proceeded to get drunk.\n\n\"What are you doing here?\" She didn't bother to disguise her annoyance.\n\nDonald tilted the glass to his mouth and swallowed the imported champagne in one gulp. \"What does it look like, Miss Alexandra Cole?\" He spat out her name. \"I came to ring in the New Year with my snobby, bitchy girlfriend.\"\n\nShe wrinkled her delicate nose in revulsion as the odor of something stronger than wine wafted into her nostrils. Donald Easton, the brilliant computer programmer, Donald the arrogant egotist yet always the consummate gentleman, had shown up at her family's estate drunk!\n\n\"I am not your girlfriend, Donald,\" she said, raising her voice above its normal tone. \"I never was, never will be. Now I want you to leave.\"\n\n\"What if I don't want to leave?\" he shouted. Those close enough to hear his outburst turned and stared at him.\n\n\"I think you should do as the lady says,\" warned a deep male voice filled with a lethal calmness that sent a chill over Alex despite the comfortable nighttime temperature.\n\nShe shifted to her right. A slender man with brilliant silver-gray eyes stood less than a foot away from her and Donald. Her gaze caught and held his; she was hard-pressed to pinpoint his age or ethnicity. His close-cropped hair was an odd shade of red-brown that complemented his khaki-brown coloring. His lean face, with smooth skin pulled taut over the elegant ridge of prominent cheekbones and the narrow bridge of his aquiline nose and firm mouth, hinted at a Native American bloodline.\n\nDonald, weaving unsteadily in an attempt to maintain his balance, squinted at Merrick. \"And who the hell are you?\"\n\nMerrick took a step and forcibly wrested the flutes from Donald. \"You don't want to know.\" He handed the glasses to Alex. \"Take care of these while I take care of your boyfriend.\" His request was a command. His right hand caught Donald's neck, fingers tightening on his carotid artery. \"Let's go, buddy, while you're still able to breathe.\" He loosened his grip when Donald clawed at his hand.\n\n\"He's not my boyfriend,\" Alex said to the stranger's back as he led the interloper away.\n\nHer hands were trembling when she placed the flutes on the bar. One of the bartenders came over to her. \"May I get you something to drink, Miss Cole?\"\n\n\"I'll have sparkling water.\" She asked for water when she needed something stronger to calm her jangled nerves. When she'd told Jason she would handle Donald she hadn't thought he would be intoxicated. The last thing she wanted was for her brothers to confront him when he was unable to defend himself. It would've been better for Donald if her uncle's security staff escorted him off the property than for her male relatives to get involved. She'd always teased them, saying even though they were trust-fund babies, they were a whit above thug status. They generally did not go looking for a fight, but none were willing to back down from one if a situation presented itself.\n\nJason was right about Donald being a loser, and it had taken her two dates to come to that realization. His insistent bragging about his accomplishments and a need to tell her how to live her life had been his undoing. However, Donald wasn't a man who took rejection lightly. After their second date she refused his telephone calls, text messages and letters that continued long after she'd left Virginia for Europe where she'd enrolled in an accelerated graduate program for a master's in art history with a concentration in European architecture and pre-Columbian art. She'd completed the first half of the program wherein she'd spent six months studying and traveling throughout France, Spain and Italy. And in another three weeks she would leave the States for Mexico City to complete her course and fieldwork for the program.\n\n\"Here you are, Miss Cole.\"\n\nAlex accepted a goblet filled with ice and carbonated water. \"Thank you.\" She took a sip, welcoming the chill bathing her throat as the man who'd come to her rescue returned. As he closed the distance between them she noticed, for the first time, his height. He was tall, as tall as her father and brothers.\n\nHowever, there was something about the stranger that disturbed her more than Donald's unexpected appearance. She wasn't certain whether it'd been his eyes, the lethal calmness in his voice when he'd spoken to Donald or the speed with which he'd grabbed the man's throat. Everything about him radiated danger.\n\nShe forced a smile, dimples deepening as she extended her right hand. \"I'd like to thank you, Mr...\"\n\n\"Grayslake,\" he said, reaching for a hand that was swallowed up in his much larger one. \"Merrick Grayslake.\"\n\nAlex's smile did not slip. \"Thank you, Mr. Grayslake, for diffusing what could've become somewhat embarrassing for my family.\"\n\nMerrick gave her fingers a gentle squeeze before releasing them. He angled his head, his penetrating gaze taking in the perfection of her small, oval face in the bright light. Her eyes weren't as dark as he believed they would've been given her nut-brown coloring and inky-black hair, hair piled atop her head in sensual disarray and secured with jeweled hairpins. He fixed his gaze on her face rather than her petite, curvy body in a provocative halter dress with a generous front slit showing a liberal expanse of shapely legs. Her heels and upswept hairstyle put the top of her head at his shoulder.\n\n\"Please call me Merrick.\" His voice was low, calming. \"And whom do I have the pleasure of rescuing from the Big Bad Wolf?\"\n\nA soft laugh escaped her parted lips. \"Alexandra Cole. But everyone calls me Alex.\"\n\nMerrick's dark eyebrows lifted with this disclosure. \"Well, because I'm not everyone, I hope you don't mind if I call you Ali. Alex is for a boy.\" And there was nothing about Alexandra Cole that even hinted of boy. Not with her curvaceous little body.\n\nIt was Alex's turned to lift her eyebrows. Over the years, she'd been called Alexa, Lexie and Zandra, but never Ali. \"No, I don't mind.\" Merrick moved closer and she felt his heat, inhaled the haunting fragrance of his cologne that was the perfect complement to his natural body scent.\n\n\"I can think of a way where you can really thank me, Ali.\"\n\nAlex went completely still. Merrick had gotten rid of one nuisance only to become one himself. \"I don't think so, Mr. Grayslake. I don't date.\"\n\nIt was Merrick's turn to recoil from her unsolicited frankness. A shadow of annoyance crossed his face. \"I wasn't going to ask you out, because like you I don't date.\"\n\nShe tilted her chin, the gesture obviously challenging. \"Are you married?\"\n\nMerrick's impassive expression did not change. \"No.\"\n\n\"Engaged?\"\n\n\"No.\"\n\n\"Do you prefer men?\"\n\nHe blinked once and forced back a smile. \"No. And to put your mind at ease, I absolutely have no interest in you romantically.\"\n\nA becoming blush darkened her face. Alex didn't know whether to be annoyed or embarrassed. Her quick tongue had gotten the better of her\u2014yet again. She closed her eyes for several seconds as heat singed her cheeks. \"I'm...I'm sorry, Merrick, but I\u2014\"\n\n\"It's all right, Ali,\" he interrupted. The smile he'd struggled to hide softened the angles in his rawboned face. \"There's no need to apologize. I can assure you that I'm not like your boyfriend.\"\n\nHer delicate jaw tightened when she clamped her teeth together. \"Donald is not my boyfriend.\"\n\n\"That's not what he said.\"\n\n\"What did he say?\"\n\n\"You were lovers.\"\n\nAlex's eyes conveyed the fury racing through her. She should've let her brothers take care of the drunken liar. \"He was never my boyfriend or my lover.\"\n\nMerrick felt a strange numbed comfort with her disclosure; he'd thought Alex and Donald were having a lovers' spat. \"Good for you.\"\n\n\"Why? Even though you're not interested in me romantically you think you'd be better for me than Donald?\"\n\nHe was momentarily speechless in his surprise. Alex Cole was as outspoken as she was beautiful, a trait he wasn't used to in the women with whom he'd been involved.\n\n\"No. That's because I've never been a good boyfriend.\"\n\nAlex took another sip of water, staring at Merrick over the rim of her goblet. \"Do you realize you're an anomaly?\"\n\n\"Why would you say that?\"\n\n\"Most men would never admit to being less than perfect in the romance department.\"\n\n\"That's because some of them are either liars or fools.\"\n\n\"And you've been neither?\"\n\nAttractive lines fanned out around Merrick's luminous silver-gray eyes when a natural smile slipped under the iron-willed control he'd spent most of his life perfecting. \"Wrong, Ali. I've been a fool a few times.\"\n\nHe'd become a king of fools when he'd trusted a woman whose duplicity had cost him a kidney and a career with the Central Intelligence Agency.\n\nWhat he didn't tell Alex was that whenever he'd gone undercover he became a liar\u2014someone with a fictitious background. He'd become an actor in a role wherein one slip would compromise his mission. His focus hadn't been the risk that he would forfeit his life, but completing the mission. And it was always the mission.\n\nA server approached with a tray of appetizers. She handed Merrick a napkin and he took several puff pastries, offering them to Alex. She shook her head. \"No, thank you. I'm saving my appetite for dinner.\"\n\n\"Speaking of saving, I'd like you to save me a dance.\"\n\nHis request surprised Alex. \"You want to dance with me?\"\n\nSlowly, seductively, his silver gaze slid downward before it reversed itself. \"Yes.\"\n\nShe felt a tingling in the pit of her stomach she found disturbing. Merrick Grayslake was disturbing to her in every way she didn't want. She would dance with the man, and after tomorrow she would never see him again.\n\n\"One dance,\" she crooned, flashing her enchanting dimpled smile. She wiggled her fingers. \"I'll see you later.\"\n\nMerrick stared at Alexandra Cole as she lifted the hem of her dress and walked out of the tent.\n\nHe'd come to West Palm Beach for a wedding and unwittingly found himself bewitched by a slip of a woman who just happened to be the groom's cousin.\n\n## Chapter 2\n\nAlex spied her sister coming toward her. Her expression said it all: she'd recovered from her hissy fit. \"That dress looks better on you than on me,\" she told Ana.\n\nThe black sheath dress with a squared neckline and wide bands crisscrossing her bare back was a perfect fit. The garment's hemline, ending inches above Ana's knees, and a pair of black silk sling backs showed off her strong, shapely legs. She'd recently cut her shoulder-length curly hair, and the pixie-cut style called to mind the gamine look affected by a young Audrey Hepburn. There was never a doubt that Alex and Ana were related. In fact, she and her sister looked more alike than Ana and her fraternal twin, Jason.\n\nAna looped her arm through Alex's and smiled. \"Thanks. Come with me.\" She spun her around. \"I've got a case of the munchies.\" PMS always triggered a craving for salt, alcohol or chocolate. She steered her older sister back to the tent. \"Qui\u00e9n es \u00e9l?\" she asked in Spanish, a language she and her siblings had learned from their bilingual parents.\n\n\"What you talking about?\" Ana was notorious for talking in riddles.\n\n\"That guy over there staring at you.\"\n\nAlex slowed her step. She wanted to tell her sister about Merrick and Donald Easton, but knowing Ana, the story that she had a stalker boyfriend would be on the lips of their family members before the stroke of midnight.\n\n\"His name is Merrick Grayslake, and he's a guest either of Michael or Jolene.\" He hadn't moved from where she'd left him.\n\n\"Damn-n-n-n-yum!\" The expletive came out in five syllables. \"El es Caliente!\" Ana whispered, sotto voce.\n\n\"He'll do,\" Alex whispered back.\n\nLeading Alex toward the bar, Ana stopped and placed the back of her hand to her sister's forehead. \"Nope, you don't have a fever. Girl, is there something wrong with your eyes?\"\n\nShe pushed Ana's hand away. \"There's nothing wrong with my eyes. Just because I don't go buck wild over a good-looking man it doesn't mean there's something wrong with me either.\"\n\nIt wasn't that she hadn't found Merrick Grayslake attractive, because he was that and more. She thought him attractive, well groomed, confident and stunningly virile.\n\n\"So, you agree with me?\"\n\n\"What about?\"\n\n\"That he's hot.\"\n\n\"I'm too old to compartmentalize men as either hot or cold, Ana Juanita Cole.\"\n\nAna knew she'd hit a raw nerve because Alex had called her by her full name. Sucking her teeth, she signaled for a bartender. \"And you're beginning to take life a bit too serious, Alexandra Ivonne Cole. Ever since you came back from Europe you've become someone I don't know or recognize. Lighten up, Alex, or you're going to turn into a bitter old woman.\"\n\nAlex swallowed an angry retort. Ana wasn't the only one who'd mentioned that she'd changed. Perhaps it was because she'd matured while living abroad, that she had come into her own and knew what she wanted for her future.\n\nGrowing up as Alexandra Cole had afforded her a life of privilege. As a member of one of the wealthiest black families in the States, she and any woman who claimed the Cole name or blood were pampered and adored by their male counterparts and relatives. But as she matured she rebelled against the restriction that wouldn't let her travel like other young women who flew on commercial carriers, when she was forced to take the family-owned jet.\n\nShe'd lost count of the number of times she'd denied being \"one of those Coles\" when someone inquired about her name. Being the granddaughter of America's first black billionaire, the daughter of award-winning musician David Cole had distanced her from her contemporaries the moment she drew breath.\n\n\"I don't have time for a man.\"\n\n\"Yeah, right,\" Ana drawled. The two words dripped sarcasm. \"You, Miss Party Animal, giving up men. I'm your sister, so spare me the melodrama.\"\n\nAlex's expression stilled and grew serious. \"I'm not going to argue with you, Ana. So spare me the lecture.\" She glanced over her shoulder. Merrick was gone.\n\nAna held up a hand in supplication. \"Okay. No lectures. We're here to celebrate Michael and Jolene's wedding, and I intend to have a good time.\" She waved a bartender over, flashing a sensuous dimpled smile. \"I'd like two apple martinis.\"\n\n\"I'm going to need some ID, miss.\"\n\nAna's smile was dazzling as she gawked at the delicious-looking Jesse Metcalf look-alike. \"I can assure you that I am over the legal drinking age.\"\n\nHe winked at her. \"You can't blame a guy for checking, beautiful.\"\n\nAlex rolled her eyes upward. Her sister was at it again. She was a serial flirt. Ana flirted while she'd sworn off men\u2014at least temporarily. After Alex finished her course work, earned her degree and secured a position as an architectural historian, then she would consider becoming involved with someone. At the present time that was not an option.\n\nAna handed Alex a glass with a pale green liquid. \"Drink up and loosen up, sis.\" She touched her glass to Alex's.\n\nAlex took a deep swallow of the icy-cold cocktail, feeling its potent properties immediately. Moaning softly, she closed her eyes. \"Ahh-hh. That is good.\"\n\nTaking a deep swallow of her drink, Ana inclined her head in agreement. \"Ditto.\"\n\n\"Ladies, gentlemen, I'm going to ask you to take your seats. We'll be starting in less than fifteen minutes.\" The voice of the wedding planner, who was carrying a cordless microphone, was heard over the murmurs of those gathered under the tent.\n\nAlex and Ana placed their glasses on the bar simultaneously. It had been a couple of years since they'd celebrated a family wedding, and whenever the Coles came together it was always a festive and momentous event.\n\nMerrick ignored the young woman on his right. The psychologist, a coworker of Jolene Walker's, had talked incessantly without pausing to take a breath. He'd met her when Michael and Jolene hosted a dinner party at their Georgetown home. His expression reflected ennui while his gaze was averted less than ten feet away; his rapt attention was directed toward Alexandra Cole.\n\nShe sat next to a man, her head resting on his shoulder, and he assumed the man was her father. He didn't know what it was about the petite raven-haired, outspoken minx that fascinated him, yet knew realistically he couldn't uncover what it was in a few hours. One thing he was certain of, it wasn't lust.\n\nLust was an emotion he'd learned to control with the onset of puberty. The realization that he could father a child reopened a wound that had festered, healed and reopened to fester again each time he was shuttled from one foster home to the next.\n\nHis libido was strong, healthy, but he'd learned to control his physical urges. Whether it was fasting, meditation or exercising to the point of exhaustion, he refused to succumb to lusting after a woman just to slake his sexual frustrations.\n\nMerrick reluctantly tore his gaze from Alex to Jolene Walker as her father led her down the flower-strewn carpeted path where Michael waited along with his best man, Damon McDonald, to make her his wife.\n\n\"Doesn't Jolene look beautiful?\" the young woman whispered reverently.\n\n\"Yes.\"\n\nMerrick was back to offering monosyllabic responses. The woman hadn't lied. Michael had confided to him that Jolene was carrying his child; impending motherhood appeared to enhance Jolene's natural stunning beauty that had most men holding their breaths and taking a second look whenever she entered a room.\n\nThe ceremony seemed surreal to Merrick as images of other weddings he'd attended in the past came rushing back, superimposed over the one taking place before him. He recalled those of his foster care siblings, fellow marines he'd met in the corps and one during a covert mission. He'd been so deep undercover that the man whom the U.S. had targeted as a terrorist had asked Merrick to be his witness at an impromptu wedding ceremony.\n\nApplause brought him out of his reverie. Fifteen minutes had passed. It was over. Michael and Jolene Kirkland were now husband and wife, and he'd emotionally distanced himself from the ritual. As long as he did not acknowledge weddings, births and funerals he was able to plan for the next day.\n\nAbandoned at birth, and not knowing his mother, his father or what he was had left a gaping hole inside Merrick that left him feeling detached and empty. Standing with the other guests and family members, he applauded the newlyweds as they traversed the path to a position where they'd receive those who'd come to help celebrate their new life together.\n\nAlex felt the muscles under the jacket of her father's arm tense up before relaxing, wondering whether he'd reacted to his son Gabriel's rich baritone voice singing \"True Companion,\" his nephew exchanging vows with his bride or that it wasn't one of his own children getting married.\n\nDavid Cole had endured the relentless teasing of his brothers, Martin and Joshua, that he would never become a grandfather because his four children appeared to shun relationships that would eventually end in marriage.\n\nIt wasn't that Alex did not want to marry. It was that she wasn't ready for it. She had plans, ones that did not include a husband and children at this time in her life.\n\nShe lifted from his shoulder. \"Jolene looks beautiful,\" she whispered.\n\nDavid nodded and smiled. \"She does,\" he whispered back.\n\nAlex glanced up at her father's profile. The diamond studs in his ears were a constant reminder that David Cole was the least traditional of the offspring of Samuel and Marguerite-Josefina Cole.\n\nThe faint scar running along his left cheek was also a constant reminder of the former musician's brush with death. Her father, who as CEO of ColeDiz International, Ltd., met her mother during a business trip to Costa Rica. He'd traveled to the Central American country to negotiate the sale of a banana plantation and found himself hostage of a deranged government official. He'd escaped, resigned his position with the family-owned conglomerate and set up Serenity Records.\n\nShe never tired hearing the story of how her parents met and fell in love. When she was a child it had become her favorite fairy tale, one wherein she'd imagined herself a princess who waited for her prince to rescue her from an evil king.\n\nPrincesses, princes and fairy tales were a part of her childhood with indelible memories of a home filled with laughter, music, exotic food and stories of Serena Morris-Cole's life in Costa Rica.\n\n\"Alex, are you all right?\"\n\nAlex's eyelids fluttered wildly, as she seemed to come out of a trance. \"I'm fine, Daddy.\"\n\nOverhead light shimmered on David's close-cropped silver hair. He smiled at his wife and daughter. \"I don't know about you two, but I'm ready to eat, drink and party until the sun comes up.\"\n\nSerena's short, reddish-brown hair framed a face that belied her age. She smiled up at her husband. \"Steady there, sport. Do I have to remind you that you're sixty-seven, not twenty-seven?\"\n\nLowering his head, David brushed a kiss over her mouth. \"If you'd met me when I was twenty-seven you never would've been able to keep up with me.\"\n\nAlex walked ahead of her parents rather than listen to their banter. They had been married for more than thirty years and were still madly in love with each other. If and whenever she fell in love she wanted what her parents had\u2014a love that promised forever.\n\nAs she waited in the receiving line, she spied Merrick with a blonde clinging possessively to his arm. The woman was so close to him they could've been joined at the hip. Merrick glanced up, his gaze meeting and fusing with Alex's. A beat later they both looked away.\n\nPlaya! He'd asked her for a dance when he'd come with a date. Liar! she continued with her mental tirade. He said that he didn't date. But what did he call the woman draped over him like a second skin?\n\nShe shook her head. That was why she didn't date; she could not afford to be distracted by romantic notions. And like her free-spirited parents, she planned to eat, drink and party until the sun came up.\n\nAlex inched along in the receiving line until she stood face-to-face with her cousin and his wife. Rising on tiptoe she wound her arms around Michael's neck and pressed her cheek to his smooth brown jaw.\n\n\"Felicidades, primo.\"\n\nMichael Kirkland's green eyes shimmered like priceless emeralds. \"Thanks, Alex.\"\n\nShe moved to her right, standing in front of Jolene. Extending her arms, she gave her a gentle hug. The two women had met for the first time earlier that morning. \"Congratulations, cousin.\"\n\nJolene, resplendent in a simple empire strapless sheath of crepe that shimmered like liquid through a lacy coat in an off-white shade, looked like a Shakespearean princess. Her short naturally curly hair was covered with a circlet of tiny white roses and baby's breath instead of a veil.\n\nJolene returned Alex's hug. \"Thank you, Alex.\"\n\nIn lieu of wedding gifts, Jolene and Michael had requested donations be made to the Jeanine Walker Retreat House, a facility named for her late twin sister who'd died at the hands of an abusive husband. As executive director of the Sanctuary Counseling Center, a D.C.\u2013based treatment center for victimized women, she'd dedicated the past five years of her life helping women empower themselves.\n\nAlex admired her cousin's new wife because she was so focused. Although a year older, Jolene knew exactly what she wanted and where she wanted life to take her. Her board of directors' fund-raising efforts had generated enough money to begin building the retreat house for battered women and their children. She'd fallen in love, married Michael Kirkland and now she looked forward to becoming a mother the following summer.\n\n\"Can I get you something to eat or drink?\" she asked Jolene when the new bride pressed a hand to her slightly rounded belly.\n\n\"Bless you, Alex,\" Jolene said in a hushed whisper. \"Please bring me some water and a few shrimp puffs.\"\n\n\"I'll be right back.\"\n\nShe wove her way through the throng waiting to offer their best wishes to the new couple. Her steps slowed when she came face-to-face with a man she hadn't seen since the last Christmas holiday family gathering.\n\nReaching for his hand, she smiled up at him. \"Come with me, Diego, while I get something for Jolene to eat.\"\n\n\"Hold up, Alex.\"\n\nTightening her grip on his large hand, she forcibly pulled him along as she quickened her pace. \"Unless you want to see your cousin's bride faint in front of her family and guests, you'll help me.\"\n\n\"Is she sick?\"\n\n\"No, she's pregnant.\"\n\nDiego Cole-Thomas smiled, an expression that was as rare to those who knew him as it was to see snow in the desert. \"It looks like Michael couldn't wait to become a daddy.\"\n\nIgnoring his cynical remark, Alex asked, \"When did you get in?\"\n\nWhen she'd questioned her aunt Nancy as to the whereabouts of her eldest grandson, her answer was \"He's expected at any moment.\" Any moment had come nearly twenty-four hours later.\n\n\"The jet touched down an hour ago. I barely had time to make it to my place to change before driving like a bat out of hell to get here before midnight.\"\n\nAlex asked a member of the catering staff to bring her what Jolene had requested, then turned to stare up at her elusive second cousin. Diego Samuel Cole-Thomas was being groomed to take over as CEO of ColeDiz International, Ltd. Women liked the thirty-five-year-old confirmed-bachelor venture capitalist and he liked them back. However, whenever one broached the topic of marriage Diego managed to extricate himself from the relationship unscathed.\n\nLooking at Diego was like seeing Samuel Claridge Cole reincarnated. Not only did he look like his great-grandfather but he'd also inherited his business acumen. Diego's genes had reached back several generations wherein he'd inherited Samuel's height, powerful build, lean angular face and large deep-set eyes, dark eyes that glowed like polished onyx. If women weren't drawn to the slight cleft in his strong chin, then it was to his smooth sable-brown skin.\n\n\"Did you bring Lisa?\"\n\nDiego lifted thick, silky black eyebrows a fraction as he shook his head. \"No. We stopped seeing each other a couple of months ago.\"\n\nAlex shot him a skeptical look. \"Is it because she mentioned the M word?\"\n\nHis solemn expression didn't change. \"No. It was by mutual agreement. How are you doing with your hip-hop boyfriend?\"\n\nA rush of heat stung Alex's cheeks. She and Duane Jackson had dated each other exclusively for five months, then without warning he'd stopped calling. She'd left a message with his housekeeper at his Miami mansion, another voice-mail message on his cell phone before relegating him to her past when he failed to contact her.\n\n\"That's over.\"\n\n\"I thought you guys were serious.\"\n\nAlex accepted a small plate wrapped in a napkin from a waitress. \"When have you known me to get serious about a man?\"\n\nDiego inclined his head at the attractive young woman who handed him two bottles of chilled water. He fell in step with Alex as they made their way back to the garden. \"It's time you got serious about someone.\"\n\nShe ignored his censuring assessment of her love life. \"I'll get serious about a man when you do the same with a woman.\"\n\n\"It's not going to happen, Alex. Not when my father has been talking about retiring.\" Timothy Cole-Thomas had announced he planned to turn over the reins of ColeDiz to his son the day he celebrated his sixtieth birthday. And that would become a reality in another four months.\n\n\"And it's not going to happen for me until I complete my graduate studies.\"\n\nDiego gave her a sidelong glance. \"Does this mean I can expect to see my little cousin married in the very near future?\"\n\nAlex rolled her eyes at him. \"You wish.\"\n\n\"No, Alex. Your parents wish.\"\n\n\"Careful. Don't go there, Diego,\" she warned quietly in defense of David and Serena Cole.\n\nDiego knew he'd struck a nerve with Alex. She was fiercely loyal and supportive of her free-spirited parents who'd raised their four children in an environment reminiscent of the seventies hippie culture. Gabriel, Alexandra, Jason and Ana Cole had grown up independent, headstrong, tolerant, secure, artistically gifted, while marching to the beat of their individual drums.\n\nHe stole a quick glance at his petite cousin. Men were drawn to her because of her beauty and carefree attitude yet none were able to tame the wildness that surfaced when least expected. And, if one did, then he would be deemed more than special; he would be exceptional.\n\n## Chapter 3\n\nAloud hiss preceded the explosion of color in the nighttime sky as pyrotechnics spelled out HAPPY NEW YEAR! to the surprise and joy of those gathered outside the large tent. Amid cheering, applause, hugs and kisses, Merrick watched the faces of those closest to him. Dressed in their evening finery, many who'd had more than a few predinner cocktails, they were enjoying themselves.\n\n\"Happy New Year,\" a woman whispered close to his ear before planting a kiss on his unsmiling mouth.\n\nHe went completely still as he stared at the stranger. The tall model-like woman smiled, flashing her porcelain veneers. Although beautiful, in typically plastic Hollywood fashion, she definitely wasn't his type.\n\n\"Same to you,\" he mumbled, reaching into the pocket of his trousers for a handkerchief. Moving into the tent, he wiped the bloodred lipstick off his mouth. He wasn't one for public displays of affection, especially when unsolicited.\n\nHe wouldn't have been as repulsed if Alexandra Cole had kissed him; however, he doubted whether she would've kissed him even if it was under the pretense that the holiday called for the ritual of exchanging kisses.\n\nThe pyrotechnics went on for another ten minutes much to the delight of the young children whose oohs and aahs turned to protests as they were led back to the house. Those sixteen and over were permitted to join their adult relatives in the frivolity that would go on for hours.\n\nMerrick was directed to his table under the enormous tent set up with seating for two hundred; he and Alex weren't seated at the same table but she sat close enough for him to take furtive glances at her; he'd lost his chatty hanger-on who, to his relief, was seated at the opposite end of the tent. Even if he'd wanted to reply to the psychologist, she hadn't let him get a word in edgewise, and he wondered whether her clients were given the opportunity to talk in their sessions.\n\nAnd if he hadn't been so taken with Alexandra Cole's natural beauty he would've taken notice of the exquisite centerpiece of white roses, hydrangea, paperwhites, poinsettias and dusty miller spilling over crystal vases, the flickering pillars under glass chimneys and the exquisite place settings with Royal Crown china in a Derby Panel Green pattern and Vera Wang sterling and crystal.\n\nPrerecorded music featuring the works of Gershwin and Ellington provided a soothing background to muted conversations as silent, efficient waitstaff filled wine and water glasses. They served course after course of endive with cr\u00e8me fra\u00eeche and caviar, bowls of delicious oyster soup, Caesar salad with crisp pancetta and garlicky croutons, wild-mushroom lasagna, grilled salmon, filet mignon and chicken cordon bleu.\n\nMerrick lost count of the number of different wines poured for each serving as he alternated exchanging pleasantries with the two women flanking him while surreptitiously stealing glances at the woman who, despite his declaration that he wasn't interested in her romantically, had ensnared him in an invisible web of curiosity.\n\nShe was a Cole, a member of one of the wealthiest African-American families, if not the wealthiest, and was related to the groom. She hadn't worn any rings and professed she didn't date, and therefore he assumed her single.\n\nHad her declaration that she didn't date mean that she wasn't into men? A hint of a smile tipped the corners of his firm mouth.\n\nHe would just have to ask her.\n\nThroughout dinner Alex took surreptitious glances at the table to her right. There was something mysterious yet frightening about Merrick Grayslake. She wasn't certain whether it was the timbre of his drawling voice that indicated that he had southern roots or that he hadn't raised his voice to Donald; there was no mistaking the cold warning. And when he'd reached for Donald's throat she'd thought for an instant that he was going to strangle the poor man.\n\nStaring at him through her lashes, she bit down on her lower lip, holding her breath. He'd caught her staring. The seconds stretched into a full minute. He acknowledged her interest with a barely perceptible nod. The corners of her mouth curved upward as her lips parted in an inviting smile.\n\nMerrick was hard-pressed not to return her smile, his silver-gray orbs darkening with an emotion he hadn't felt in a very long time. He'd lied to Alex and to himself. There was something about her that made him want to know her. Putting two fingers to his forehead, he gave her a mock salute. A lump formed in his throat when she went completely still, then turned away to say something to the man on her left. Her expression before she'd glanced away was one of demure innocence.\n\nWas she? he mused. No, she couldn't be, he continued with his mental monologue. Pushing back from the table, he looped one leg over the opposite knee; he studied Alex with a curious intensity that was so foreign to him. She reminded him of a lump of coal that appeared cold until touched. At that moment he likened her to a dark fire.\n\nAlex had promised him one dance, and he hoped the single interaction would be enough to put his mind at ease as to why he felt drawn to his friend's cousin.\n\nHis attention was redirected to the bridal table where the best man offered a toast to the newlyweds. Flutes of champagne were hoisted over and over with the various toasts from a very pregnant maid of honor and the parents of the couple. Jolene and Michael cut the first slice of a four-tiered double-chocolate wedding cake decorated in white-chocolate curls and topped with marzipan roses and leaves. Individual wedding cakes in airtight containers and decorated with dark green satin ribbon were given to each guest.\n\nWith the pomp and circumstance of the wedding behind them, the bride and groom left the tent for an area where a portable dance floor had been erected in an open meadow. A DJ, alternating with a five-piece band, was on board to provide nonstop music.\n\nThe music was going full tilt when Merrick found himself standing off to the side watching couples twirling to a remix of Red Carpet's \"Alright,\" a dance hit he'd first heard in a club in Amsterdam. He couldn't take his eyes off Alex as she danced freestyle with Michael Kirkland, sans jacket and tie. Others on the dance floor moved back to watch their spectacular routine.\n\nCarefully coiffed curls fell over Alex's forehead as Michael spun her around and around on her toes. Reaching for a glass of champagne from a passing waiter, Merrick's gaze never left the petite figure. Their dance ended in applause as Michael lifted Alex off her feet, kissing her cheek.\n\nAlex, her face flushed, made her way through the crowd. Merrick took two long strides and thrust the flute at her. \"I believe you could use this.\"\n\nShe went completely still, only her chest rising and falling from the vigorous exertion, and stared at Merrick as if seeing him for the first time. There was a pause before she took the glass. Raising it to her lips, she took one sip, then another.\n\n\"Thank you.\"\n\nMerrick inclined his head. \"You're quite welcome.\"\n\nResting a hand in the small of her back, he led her away from the crowd. He wanted to talk to Alex, but didn't want anyone to overhear what he wanted to say to her.\n\nHer spine stiffened against his arm. \"Where are you taking me?\"\n\n\"Relax, Ali,\" he said, leaning closer.\n\n\"Why should I when you haven't answered my question?\"\n\nHis hand moved up to her narrow waist. \"I want to ask you something.\"\n\nShe gave him a sidelong glance. \"I thought you wanted to dance, not talk.\"\n\n\"We'll dance later.\" Alex stopped suddenly, causing Merrick to plow into her. She would've lost her balance if his reflexes were slower. Wrapping both arms around her body, he pulled her against his length. \"Careful.\"\n\nHis breath whispered over an ear, and she felt a shudder of awareness of the man cradling her to his hard, lean body, awareness that she'd found herself attracted to a stranger when there was no room in her life for romantic fantasies.\n\n\"You can let me go now.\" He dropped his arms and she pointed to a cushioned wrought-iron bench under a magnolia tree. \"We can talk here.\"\n\nMerrick waited for Alex to sit before he sat down beside her. She took another swallow of champagne, handing him the half-empty flute. Cradling the stem of the glass between his thumb and forefinger, he fixed his gaze on the pale sparkling wine. There was a comfortable silence, neither seemingly wanting to initiate conversation.\n\nHe shattered the silence and asked, \"Are you into women?\"\n\nA soft gasp escaped Alex's parted lips when she processed his query. \"What!\"\n\n\"Do you prefer women to men?\" Merrick asked, this time rephrasing his question.\n\nShe gave him a hostile glare. \"No. Why would you ask me that?\"\n\n\"Because you said you didn't date.\"\n\nHer nostrils flared in anger. \"Because I told you that I don't date you assume I'm into women?\"\n\nMerrick lifted his eyebrows. \"There was always the possibility.\"\n\nShe met his challenging stare with one of her own. \"The answer is no, Merrick. I am not into women.\"\n\nRising to his feet, Merrick extended his free hand. \"I'm ready for that dance now.\"\n\nAlex shook her head. \"No, Merrick.\" She patted the cushion beside her. \"Please sit down.\" Waiting until he retook his seat, she turned and stared at his distinctive profile. She was certain he could hear the runaway beating of her heart that echoed in her ears like a kettledrum. Ana was right. Merrick Grayslake was exotic and caliente!\n\n\"Why should my sexual preference matter to you when you're here with a woman? Could it be you were afraid I'm going to make a play for her? And if you are, then I must disappoint you because I'm not into threesomes.\"\n\nMerrick turned and looked at Alex as if she'd just grown a third eye. \"Are you always this outspoken?\"\n\n\"Yes,\" she answered quickly.\n\nMore amused than insulted, he smiled at her face in the diffused artificial light. \"I'm not here with a woman. The young lady you saw me with works with Jolene. We met at a party at Michael's house in Georgetown a few months back. I, too, am not into threesomes, or foursomes. I'm a very private person, and if by chance I sleep with a woman, then there will be only the two of us in bed. And I asked about your sexual preference because I thought the two of us could possibly become friends.\"\n\nHer delicate jaw went slack. \"Friends?\"\n\nHe lifted a shoulder under his suit jacket. \"Yes, Ali, friends. You don't date and neither do I, so I thought we could hang out together.\"\n\nAlex sat there, somewhat shaken by the unpredictable man sitting next to her. He wanted them to become friends when all she wanted was to flee his presence and surround herself with her family who always made her feel safe and protected.\n\nBut what, she asked herself, did she have to fear from Merrick Grayslake? After all, she was a Cole and the Coles always protected their own.\n\n\"I don't believe that's going to be possible.\"\n\nThere came a pause. \"Why not, Ali?\"\n\n\"I'm leaving the States in three weeks to study art in Mexico City.\"\n\nMerrick swirled the remains of the champagne in the flute, then put the glass to his mouth and drained the contents, savoring the taste of the premium wine on his palate.\n\nResting his right arm over the back of the bench, he stared straight ahead. Alex's declaration that she was leaving the country gave him conflicting emotions. He'd found her vaguely disturbing and exciting at the same time. He couldn't say she was his type, because he'd found himself attracted to all women irrespective of their race, nationality or culture. Perhaps it was because he didn't know who or what he was that permitted him to be more open-minded and accepting of others.\n\n\"Are you an artist?\"\n\n\"No. I'm an architectural historian.\"\n\n\"What made you select art as a career?\"\n\nShe lifted her shoulders under the revealing dress, bringing Merrick's gaze to rest on her exposed throat and neckline.\n\n\"I've always loved museums. Whereas other children wanted to visit theme and amusement parks, for me it was museums and art galleries.\"\n\nLooping one leg over the other in one continuous graceful motion, Merrick smiled. \"I know absolutely nothing about art.\"\n\nShifting on the bench to face Merrick, Alex saw a flash from his incredibly perfect white teeth. Why hadn't she noticed them before? However, she knew the answer even before the question was formed in her head. It was the first time she'd seen him smile, and the gesture transformed his face, softening the sharp angles to make him even more breathtakingly attractive.\n\n\"Don't tell me you wouldn't recognize Leonardo da Vinci's Mona Lisa.\"\n\nHis smile grew wider. \"That's the only exception.\"\n\n\"What about van Gogh?\"\n\n\"Isn't he the one who cut off his right ear?\"\n\n\"It was the left,\" Alex correctly softly.\n\n\"See? I told you I know nothing about art. How about giving me a crash course in art history?\"\n\nShe felt him come closer when actually he hadn't moved. The smoldering flame she saw in his eyes startled her. Merrick Grayslake spoke of friendship when everything in his gaze communicated the opposite.\n\n\"Not in three weeks.\" Her voice was barely a whisper.\n\nLids lowering over his penetrating eyes, Merrick stared at her lushly curved full lips. \"I'm a quick study.\"\n\nA tangible energy radiated from Merrick that drew Alex to him like a powerful magnet. He was maddeningly arrogant, but there was also something so soothing in his manner that she found herself unable to resist him.\n\nIt had been a long time since she'd been involved with a man, but that still did not explain why she'd felt as if her emotions were under attack; and she could not fathom what it was about the man a mere breath away who shattered her resolve to concentrate solely on her studies. She had three weeks, time in which she planned to visit with her parents and siblings before she returned to Virginia to close up her condo.\n\n\"Can you take a couple of weeks off from your job?\"\n\nMerrick was hard-pressed not to smile. \"Yes. What do you have in mind?\" He wanted to tell Alex that he didn't have a job or a career, and hadn't had one in two years. He had a government pension, albeit as a disability status, and the proceeds from shrewd investments permitted him a comfortable and uncomplicated lifestyle.\n\n\"I need you to come to D.C. Will that present a problem for you?\"\n\n\"Not in the least. I live in West Virginia.\"\n\n\"Where do you live in West Virginia?\" Excitement fired the gold in her clear brown eyes.\n\n\"Bolivar. It's a few miles south of Harper's Ferry.\"\n\n\"We're practically neighbors. I live in Arlington, Virginia.\"\n\nThey weren't neighbors, but it was close enough for Merrick to get to the Capitol District in a few hours. \"How long have you lived in Virginia?\"\n\n\"This coming March will be two years. I'm going to visit with my folks in Boca Raton for a week before I go back to D.C.\"\n\n\"When do you want to get together?\"\n\n\"I should be back on the eighth. I have tickets for a showing that's also a fund-raiser on the ninth.\"\n\n\"That sounds good. I'm going to need a number where I can reach you so you can give me all the particulars.\"\n\n\"I'll give you my cell number. Do you have anything to write it down with?\"\n\nMerrick shook his head. \"Tell it to me.\"\n\nResting a hand on one hip, Alex glared at him. \"If you forget it, then don't blame me if we don't get together.\"\n\n\"I won't forget it. And if I do I'll call Michael and ask him.\"\n\nAlex had her answer as to whose guest he was. And if he knew Michael, then there was no doubt Merrick Grayslake was either involved in the military or had ties to intelligence.\n\nShe recited her number, watching Merrick as he closed his eyes and mouthed the numbers. \"Tickets to this event are as scarce as hen's teeth, so if I get one for you and you don't contact me, then there's going to be hell to pay, Merrick Grayslake.\"\n\nHis right hand moved up to caress the nape of her neck. Pressing his mouth to her ear, he repeated her number before reciting it backward. \"Are you satisfied, Miss Alexandra Cole?\"\n\nGrinning, her mouth inches from his, she crooned, \"I'm impressed.\"\n\nAlex's soft, moist lips were a temptation, but Merrick knew she would never trust him if he suddenly took advantage of the situation and kissed her. He'd promised friendship, and that was what they would share until she decided otherwise.\n\n\"The score is now two for Grayslake, zero for Cole.\"\n\nVertical lines appeared between her eyes. \"What are you talking about?\"\n\nMerrick's smile was toothpaste-ad dazzling white. \"I earned a point when I rescued you from your annoying stalker boyfriend, and I just proved to you that I won't forget your number.\"\n\nPulling back and resting both hands on her hips in a challenging gesture, Alex narrowed her eyes. \"Oh-kay,\" she drawled. \"You want this to be a competition? Bring it on, sport, because you're in for the fight of your life.\"\n\nHe sobered quickly. \"Damn. You're a spunky little thing, aren't you?\"\n\n\"Don't let my size fool you.\"\n\n\"One thing I'm not is a fool. I know enough not to mess over a Cole woman.\"\n\nHer frown deepened. \"How much do you know about my family?\"\n\nThe seconds ticked off as they regarded each other like combatants. \"A lot more than the average citizen,\" he replied cryptically.\n\nA wave of apprehension swept through Alex, washing away her former bravado. \"Who are you, Merrick? And what do you do?\"\n\nHis impassive expression reminded her of a mask of stone. \"I'm retired.\"\n\n\"Who did you work for?\" Her normally sultry voice had dropped an octave.\n\nThere came another pause. \"That's something you don't need to know.\"\n\nAlex emitted an unladylike snort. \"You want us to become friends, yet you don't trust me enough to tell me who you work for.\" She stood up, Merrick following suit. \"Well, I just evened the score, Merrick. You don't have to tell me. Once Michael returns from his honeymoon I'll ask him.\"\n\nMerrick knew when he was bested. He didn't want Michael to know that he was seeing his cousin\u2014not yet. \"CIA,\" he said reluctantly.\n\nHer mouth formed a perfect O with his disclosure. \"You were with the Central Intelligence Agency?\" He nodded. \"Intelligence?\" Merrick nodded again. She squinted up at him. \"I see why you and Michael are friends.\" Her cousin, a West Point graduate, was a highly-trained army intelligence officer.\n\nMerrick extended his free hand. \"Are you ready to dance with me?\"\n\nGrasping his hand, Alex smiled up at him. \"Yes.\"\n\nHolding hands, they retraced their steps to where couples were swaying to the band playing Stevie Wonder's \"You and I.\" Merrick wrapped his arm around Alex's waist, holding her close to his heart, and closed his eyes as she curved her arms under his shoulders. There was no need to concentrate on his footwork as they sank into each other's warmth, as if dancing together was something they'd done before. Before the last note faded, the DJ segued into the old-school ballad, P.M. Dawn's \"I'd Die Without You.\"\n\nResting her cheek on her dance partner's shoulder, Alex closed her eyes and sang the words of the passionate love song. She lost herself in the strong arms, the sensual scent of his cologne and temporarily forgot her promise not to entertain notions of a romantic nature until she completed her education.\n\nGoing completely pliant in Merrick's embrace, she thought of a time when she'd lost her heart to a man who was undeserving of her love and trust. The song ended and they pulled apart.\n\nLowering his head, Merrick pressed a kiss to her cheek. \"Thank you for the dance.\"\n\nEasing out of his loose embrace, she gave him a dimpled smile. \"Thank you for asking.\"\n\nHis gray eyes boring into her, Merrick inclined his head. \"Good night, Ali.\"\n\nHer arching raven eyebrows lifted. \"Good night and Happy New Year.\"\n\nWithout another word, he turned and walked off the dance floor, leaving her staring at his broad shoulders until he blended into the crowd. She was in the same position when Jason reached for her hand.\n\n\"Come dance with me.\" He steered her to the middle of the dance floor as hip-hop blared from the powerful sound system. Raising her arms above her head, she snapped her fingers and lost herself in the driving rhythm of the infectious baseline beat. Her older brother, Gabriel, cut in, then Diego. After a while she lost count of the number of men she danced with as the hands on the clock made several revolutions. It was after four in the morning when she finally took off her shoes, walked across the carpet of grass and into the house that her grandfather had built for his wife before the Great Depression.\n\nThe twenty-four-room mansion designed in a Spanish revival style with barrel-tiled red roofs, a stucco facade and balconies shrouded in lush bougainvillea and sweeping French doors that opened onto broad expanses of terraces with spectacular water views was still one of the finest homes in West Palm Beach. The magnificent structure, set on twelve acres, was surrounded by tropical foliage, exotic gardens and the reflection of light off sparkling lake waters.\n\nAlex always looked forward to coming to West Palm Beach for the week spanning Christmas and New Year's because it gave her the opportunity to reunite with her extended family and reconnect with her grandmother who'd celebrated her one hundred and third birthday December twenty-seventh.\n\nShe had the bedroom suite to herself, which meant she could linger in the shower. After cleansing the makeup from her face and brushing her teeth, she stepped into the shower stall.\n\nHer eyelids were drooping when she pulled a nightgown over her head and slipped into one of the two queen-size beds. Alex was asleep within seconds of her head touching a down-filled pillow.\n\n## Chapter 4\n\nMerrick checked out of his hotel, retrieved his truck from the parking lot and headed south. It was New Year's Day, the weekend, and he practically had I\u201395 to himself. An overcast sky had given way to bright sunlight, and when he entered downtown Miami the energy, passion, color and architectural treasures of the tropical city elicited a feeling of nostalgia. What was it, he asked himself, about Miami that made him feel as if he'd come home?\n\nSlowing the SUV, he'd become the sightseer and tourist, driving past the harbor crowded with massive cruise ships and priceless pleasure boats. His sightseeing ended when he checked into a Miami Beach hotel. The view from his nineteenth-floor suite was spectacular. Opening the door to an armoire, he switched on the large-screen television to CNN. He'd planned to spend his time in Miami relaxing and soaking up the flavor of the city. Pulling a T-shirt over his head, he made his way in the direction of the bathroom.\n\nAfter moving to Bolivar he'd discovered that he lost track of the days of the week; but that would change because of Alexandra Cole. Eight days\u2014in another eight days he would call her to arrange a time when they would see each other again.\n\nStepping out of his jeans and underwear, Merrick recalled his interaction with Alex. He'd found her outspoken, opinionated and brutally honest, personality traits he hadn't encountered in any of the women in his past. She was a challenge, one he welcomed. Alex had promised him two weeks and he intended to make the most of their time together.\n\nMerrick walked into a restaurant on Calle Ocho, Miami's Little Havana main thoroughfare. He took off a well-worn baseball cap and seated himself at a small table in the rear of the dining establishment. The mouthwatering aromas wafting from platters of food carried by the serving staff overwhelmed him. He'd opted not to order from the hotel kitchen because it'd been months since he'd eaten Caribbean cuisine.\n\nTwo months before, he'd become a lookout for Michael when he'd asked him to watch the building housing the Sanctuary Counseling Center. He'd set himself up in a vacant apartment across the street, bribing the building's superintendent to let him use the space under the pretense that he was a private detective hired by a foreign diplomat to watch his wife who'd abducted the two children the judge had placed in his custody. The greedy man barely glanced at his fake identification, accepting the one-hundred-dollar-a-day fee while offering to bring him lunch and dinner. The superintendent's wife was from the Dominican Republic, and Merrick spent three glorious days eating white rice, red beans, fried plantains, baked chicken and spaghetti with spareribs.\n\nPicking up a plastic-covered menu, he studied the selections printed in English and Spanish. An attractive young waitress approached his table. \"Are you ready to order?\" she asked in accented English.\n\n\"Miro todav\u00eda,\" he said. \"Pero le agradecer\u00eda una cerveza fr\u00eda.\"\n\nThe waitress's professionally arched eyebrows shot up. Not only was the gray-eyed man guapo, but he also spoke Spanish.\n\n\"Seguro.\" She flashed a sultry grin before walking away with an exaggerated sway of her hips. She took a bottle of beer from a freezer case, whispering to her girlfriend to take a look at the man in the light blue shirt seated in her section.\n\n\"El es guapo! But don't let Jorge see you flirting with him, Milagros,\" the other woman warned in English. \"The two of you are practically engaged.\"\n\n\"Almost engaged, but not blind,\" Milagros whispered, winking at her best friend. She returned to her customer, placing the bottle of beer and a chilled glass on the table. She was the epitome of professionalism when she jotted down Merrick's order of black beans, white rice, baked chicken and an avocado salad.\n\nMerrick lingered at the restaurant, eating, reading a Spanish-language newspaper and viewing a Spanish-language television station. It was as if he wanted to immerse himself in a language he'd learned from the Mexican housekeepers in the employ of his foster parents.\n\nThe foster parents were all cut from the same fabric: Godfearing folks who felt an obligation to take care of the less fortunate. They never stopped to think that the money they received from the state for their charges helped pay for luxury cars, state-of-the-art electronic equipment, expanding their already sprawling homes and vacationing in exotic getaways.\n\nForcing himself not to think of his troubled childhood, he paid the check, leaving a generous tip, and walked out of the restaurant. He put on his favorite New York Yankees cap, successfully concealing his dark auburn hair, and set off on foot to tour the area. He stopped at the Brigade 2506 Memorial that commemorated the exiled victims of the unsuccessful 1961 Bay of Pigs invasion of Cuba. The sun had set by the time he returned to pick up his vehicle and check back into his hotel room.\n\nUnaccustomed to the heavy food and the winter heat he lay down on the bed in the air-conditioned room. Within seconds of closing his eyes, the face of Alex flooded his mind. Cursing under his breath, he sat up and reached for his cell phone on the nightstand. The only way he was going to exorcise her was to talk to her.\n\nMerrick dialed her number as if it was something he did on a regular basis. It rang three times before he heard her greeting.\n\nA smile crinkled the lines around his eyes. \"I told you I wouldn't forget.\"\n\nHer sultry laugh came through the tiny earpiece. \"You're really full of yourself, aren't you? Are you trying to prove to me how smart you are, Mr. Former CIA Man?\"\n\nA bright smile spread over his face. \"No and no. Right now I'm full of arroz frijoles, pollo y aguacate.\"\n\n\"You speak Spanish?\"\n\n\"S\u00ed, que t\u00fa hablas Espa\u00f1ol, tambi\u00e9n?\"\n\n\"Yes. Both my parents speak the language. Who taught you?\"\n\nThere was a pause before Merrick said, \"That's a long story.\"\n\n\"I have all night, Merrick.\"\n\n\"All night for what, Ali?\"\n\n\"To listen to you tell me about yourself.\"\n\n\"Why don't you wait until we see each other again?\"\n\n\"Where are you?\"\n\n\"I'm in Miami.\"\n\n\"That's one of my favorite cities. The architecture is spectacular.\"\n\n\"It is very colorful.\" Merrick wanted to tell Alex that she was spectacular but didn't want to come on too strong. \"Now that I know you didn't give me the wrong number, I'll let you go.\"\n\n\"The wrong number!\" she repeated. \"If I hadn't wanted to give you my number, then we wouldn't be having this conversation.\"\n\nHe chuckled softly. \"So, you do want to be my friend.\"\n\n\"Isn't that why I gave you my number?\"\n\n\"No, Ali. You gave me your number because you promised to tutor me in art history.\"\n\n\"It's going to have to be an accelerated course.\"\n\n\"I told you before that I'm a quick study.\" What Merrick hadn't told Alex was that he had a photographic memory. He'd become known in the Marine Corps as \"Lock and Load\" because if he saw something or someone once, he was able to file the information away in his mind and recall it at will. This gift had served him well once he was recruited by the Central Intelligence Agency.\n\n\"I'm going to grade you.\"\n\n\"What can I expect if I get straight As?\"\n\nAlex laughed again. \"I'll be certain to come up with something comparable to your final grade. How long are you going to be in Miami?\"\n\n\"I plan to spend another day here. Then I'm heading down to the Keys.\"\n\n\"I've lived in Florida all my life yet I've never been to the Keys.\"\n\n\"Do you want to join me?\"\n\n\"No.\"\n\n\"Why not?\"\n\n\"Because I don't know you like that.\"\n\n\"Like what, Ali? Do you think I'd try to compromise you?\"\n\n\"No. Of course not,\" she said much too quickly. \"Maybe the next time I'll take you up on your offer.\"\n\n\"I'm going to hold you to that promise.\"\n\n\"I always keep my promises, Merrick.\"\n\n\"Good for you.\" He heard a signal that indicated Alex had another caller.\n\n\"Hold on, I have another call coming through.\" There was a momentary pause before she came back on the line. \"I'm going to have to ring off now. I'll talk to you again on the eighth.\"\n\n\"Okay. Stay well and stay out of trouble.\" Her bubbly laughter floated into his ear. \"Don't make me have to come to your rescue again,\" he teased.\n\n\"It's kind of nice having a personal knight in shining armor.\"\n\n\"I hate to tell you, but my armor's rusted and dented beyond repair.\"\n\n\"That doesn't matter. I'll still keep you, Sir Grayslake.\"\n\nMerrick knew he had to end the call before he said something that would sabotage his fragile yet promising relationship with Alexandra Cole. \"I'll call you on the eighth.\"\n\n\"Ciao, amigo,\" she said cheerfully.\n\n\"Hasta luego, Ali.\"\n\nMerrick pressed a button, ending the connection. He was glad he'd called her. Hearing her voice and bubbly laugh was a reminder of what he could look forward to once he returned to West Virginia.\n\nMerrick finally found Rachel Singletary's house without mishap. His vehicle's navigational system had never failed him. He turned down a narrow street that made up Key West. He found a parking spot in what he termed an alley, and walked the short distance to a one-story stucco structure painted a soft salmon pink and fronted by a wrought-iron fence protecting a flower garden overflowing with frangipani and bougainvillea.\n\nReaching over, he unlatched the gate at the same time the front door opened. Rachel stood in the doorway, an automatic handgun pressed to her jeans-covered thigh.\n\nMerrick held up both hands. \"I come in peace.\"\n\nThe network of tiny lines around the woman's dark eyes deepened when she recognized the tall man standing outside the gate. \"Merrick! How the hell are you?\"\n\nHe gestured to the firearm. \"I'd be a lot better if you put that thing away.\"\n\nRachel waved him in. \"It's all for show.\"\n\nMerrick opened the gate as Rachel came forward to meet him. Looping an arm around her waist, he leaned over and kissed her cheek. \"You look good, Rusty,\" he said, using her nickname.\n\n\"Not as good as you, Gray. To what do I owe the honor of you gracing me with your presence?\" Rachel asked as she tucked the firearm into the waistband of her jeans under an oversize man-tailored shirt.\n\n\"I did promise to come and visit.\"\n\n\"That was two years ago.\" She moved aside as Merrick stepped into an expansive entryway.\n\n\"Better late than never,\" he quipped.\n\nShe closed and locked the front door. \"Go onto the back porch where it's cooler.\"\n\nHe made his way through a living room filled with rattan pieces covered with colorful plush pillows and cushions in keeping with the tropical locale. A profusion of potted plants and palms brought the outdoors inside.\n\nHe stood on an enclosed porch, staring through the screen at the ocean. The view was awesome, humbling and peaceful. Living in a landlocked state definitely had its disadvantages. A low table held a television, a radio and a stack of recent best-selling books. It was obvious Rachel spent most of her time at the rear of her house.\n\n\"Sit down, Merrick.\"\n\nHe folded his length down to a cushioned love seat while Rachel sat on a chaise; he turned to stare at the middle-aged woman who at one time had been his supervisor in Langley, Virginia.\n\nTwenty-five years his senior, she'd become his older sister, counselor and confidante. Tall and sturdily built, with even features, she'd teased him, saying they probably were related because both had red hair. The years had darkened the bright orange strands and added a liberal sprinkling of gray while the Florida sun had tanned her face wherein the liberal sprinkling of freckles blended with the added color.\n\nLeaning to her right, she opened the door to a portable refrigerator and took out two bottles of beer. She offered one to Merrick. \"I'm sorry I don't have anything stronger.\"\n\nHe waved a hand. \"No, thank you.\"\n\n\"Don't tell me you've gone soft, Grayslake.\"\n\n\"A little,\" he admitted. \"Being away from the action can do that.\"\n\nRachel twisted the cap off the bottle, put it to her mouth and took a long swallow. \"You still could've been in the action if you hadn't cut and run like a candy-ass.\"\n\nMerrick closed his eyes, shutting out the awesome sight of the sun setting over the ocean. \"I didn't join the Company to sit behind a desk.\"\n\nRachel stared at the tall, slender man. \"Neither did I, Merrick.\"\n\n\"But you did,\" he countered.\n\n\"That's something you wouldn't understand.\"\n\nHe opened his eyes, meeting her pain-filled gaze. \"Understand what, Rusty?\"\n\nShe shook her head. \"Forget it.\"\n\nSitting up straighter, he rested his elbows on his knees. \"What happened, Rachel?\"\n\nShe avoided his stare, took another sip from her beer and stared through the screen at the large ball of orange sinking lower beyond the horizon. \"After...\" Her voice trailed off. \"After someone\u2014another agent who was very close to me, was captured and executed in Somalia, I requested desk duty.\"\n\nMerrick knew no amount of pressure would get Rachel to divulge the name of the other agent. Even in death, the Company never identified its agents by name.\n\n\"You were in love with him.\" The query was a statement.\n\nRachel blinked back the tears. \"My love for him knew no bounds. When I found out that he'd been killed I thought about taking my own life.\"\n\n\"Did he love you, Rachel?\" Merrick asked in a hushed whisper. She nodded. \"Why didn't you marry him?\"\n\nA cynical smile twisted her mouth. \"I couldn't because he was already married. And in the twelve years we worked together I never asked that he leave his wife and children. I was willing to be his mistress and accept what little of himself he doled out to me.\" She took another deep swallow of the cold brew.\n\n\"My supervisor went ballistic when I requested desk duty, but there wasn't much he could do once I spoke to the Agency's psychiatrist and confessed to suicidal ideation. I'd been in the position as an intelligence research training specialist eight years when you enrolled in my class. I was the one who recommended you for my position once you returned from disability leave.\"\n\nMerrick, hands sandwiched between his denim-covered knees, stared at the sisal rug covering the floor. \"It wouldn't have worked. Not at that time in my life.\"\n\nRachel focused her gaze on the coarse reddish-brown hair on Merrick's well-shaped head. She'd always thought his looks incredible. The contrast of his gray eyes in a khaki-brown face was startling and hypnotic. It was obvious he was a man of color, but it was impossible to identify his racial or ethnic group.\n\nAnd not once during the time when she acted as facilitator for the intelligence training course had she ever seen him smile. She'd found it odd that he hadn't bothered to jot down notes like the other agents-in-training; her fear that he would flunk out was belied when his tests came back with perfect scores.\n\n\"What about now, Merrick?\" she asked softly.\n\nA half smile parted his lips. \"It's something I could consider.\"\n\nHer red eyebrows lifted with this disclosure. \"Are you serious?\"\n\nHis smile widened. \"Just say I've been thinking about it.\"\n\n\"What brought on this epiphany?\"\n\n\"I've been getting out more.\"\n\nRachel smiled. \"Someone told me that you were holed up somewhere in West Virginia. Don't tell me that you met someone who has melted that lump of ice you call a heart.\"\n\nMerrick sobered. \"No. It had nothing to do with a woman.\" There was a moment of silence as the two regarded each other.\n\n\"How old are you now?\" Rachel asked.\n\n\"Thirty-five.\"\n\n\"Are you married?\"\n\n\"No.\"\n\n\"Do you have a girlfriend?\"\n\n\"No.\"\n\n\"What about a lover?\"\n\nThere was a pause before Merrick said, \"No.\"\n\nRachel went completely still, her eyes boring into Merrick's. \"Don't you have urges? I'm sixty and I still do.\"\n\n\"Do you do anything about them?\" he asked, his expression deadpan.\n\n\"You better believe I do. I'm seeing somebody.\"\n\n\"Is it serious?\"\n\n\"If you're asking whether I'll ever get married, then the answer is no. He's been divorced for fifteen years and has no intention of tying the knot again, and that's fine with me. I'm enjoying retirement. I get up when I want, and come and go whenever the whim hits me. I'm glad I got out when I did because it took a year of therapy to rid myself of the dreams that I was reliving every mission.\"\n\nMerrick wanted to tell Rachel that he never experienced the disturbing images of his missions because he'd managed to become totally detached whenever he went undercover. He wasn't Merrick Grayslake but whoever and whatever his government wanted him to be.\n\nReaching across the space separating them, he patted Rachel's hand. \"Would you mind sharing dinner with me tonight?\"\n\nShe gave him a saucy grin. \"Are you asking me out on a date?\"\n\nHe winked at her. \"Yes, I am, beautiful.\"\n\nA rush of color darkened her face as she blushed like a young girl. \"Keep talking like that and I'm going to tell my boyfriend.\"\n\nRising to his feet, Merrick offered her his hand, and pulled her up effortlessly. \"Tell him,\" he teased. \"I bet he can't beat me up.\"\n\nRachel smiled, tilting her head back to meet his teasing gaze. \"He's got at least forty pounds on you.\"\n\nMerrick kissed her cheek. \"The bigger they are the harder they fall.\"\n\nShe patted his shoulder. \"You're right about that. I know a little seafood place that's within walking distance. It's not fancy but the food is wonderful.\"\n\n\"That sounds great.\"\n\n\"How long do you plan to hang out in the Keys?\"\n\n\"No more than four days.\"\n\n\"Where are you staying?\" Rachel asked over her shoulder as she led Merrick off the porch.\n\n\"I've checked into the Marquesa.\" The Marquesa Hotel was an elegant restored home that dated back to 1884.\n\n\"Check out tomorrow and come stay with me. I have an extra bedroom.\"\n\n\"I can't do that, Rusty.\"\n\n\"Yes, you can.\"\n\n\"What about your boyfriend?\"\n\n\"He's bound to be a little jealous, but he'll get over it once I tell him you're celibate.\"\n\nWrapping an arm around Rachel's neck, Merrick pulled her close. \"You should know you're taking a risk inviting a sex-starved man to sleep under the same roof with you.\"\n\nRachel returned the hug, her arm going around his slim waist. \"I made my reputation at the Company being a risk taker.\"\n\nMerrick knew she was right. Rachel Singletary had become one of the best female agents in CIA history. But she'd lost her edge because she'd fallen in love with the wrong man while he'd almost lost his life because he'd trusted the wrong woman.\n\n## Chapter 5\n\nHer head and body swathed in thick, thirsty towels, Alex walked out of the bathroom adjoining her bedroom suite to the ringing of her cell phone. Quickening her pace, she stubbed her toes on a leg of the nightstand as she reached for the tiny instrument.\n\n\"Dammit!\" she hissed between clenched teeth as she activated the Talk button.\n\n\"I can always call back another time,\" came what now had become a familiar male voice.\n\nHopping on one foot, she sat down on the padded bench at the foot of her bed. The ColeDiz jet had touched down at Washington National Airport at three that morning, but it was close to dawn when she'd finally crawled into bed.\n\n\"No! Please don't, Merrick.\" Her little toe throbbed like a raw nerve.\n\n\"What's the matter?\"\n\nShe registered genuine concern in the deep, drawling voice. \"I just jammed my toe on a piece of furniture.\"\n\n\"Will you require medical attention?\"\n\nShe smiled for the first time. \"I doubt it. I want to thank you, Merrick.\"\n\n\"For what, Ali?\"\n\n\"You said you'd call and you did. Most men I meet are such liars and dogs that I can't stand them. Are we still on for tomorrow?\" she asked without pausing to take a breath.\n\n\"That depends on the weather. Have you looked out the window?\"\n\n\"No. What's going on?\"\n\n\"It's been snowing for hours.\"\n\nPushing off the bed and limping to the window, Alex opened the shutters covering the floor-to-ceiling windows. The roadway and cars parked along the street were covered with snow. Her mouth turned down in a frown. She didn't know why, but she'd spent the week looking forward to seeing Merrick Grayslake again. And if he was stuck in the mountains, then there was the possibility he would be snowed in.\n\n\"Is it snowing in West Virginia?\"\n\n\"I wouldn't know because I'm in Arlington.\"\n\n\"When did you get here?\"\n\n\"Yesterday afternoon. I'm at the Hyatt.\"\n\n\"You're within walking distance of my condo.\" She gave him her address. \"How would you like to share dinner with me?\"\n\n\"Is this a date, Ali?\"\n\n\"Yes. Does it bother you that a woman asked you before you could ask her?\"\n\n\"Not in the least. I like a woman who knows what she wants.\"\n\n\"It's not about what I want. I just thought that we'd begin your lessons earlier than planned.\" She didn't want to give Merrick the impression that there would be more between them than friendship.\n\n\"That sounds good to me. What time should I arrive?\"\n\nAlex glanced at the clock on the nightstand. It was almost one o'clock. \"Make it between five and six.\"\n\n\"What do you like to drink?\"\n\n\"Don't worry about beverages. I have everything.\" She had a Sub-Zero Refrigerator with a compartment for chilling wines. \"Is there anything you don't eat?\"\n\n\"No.\"\n\n\"If that's the case, then I'll see you later.\" She ended the call without giving Merrick the opportunity to ring off.\n\nAlex knew she had to wipe away the dust from the floors and tables that had gathered during her absence and shop for groceries to replace those she'd thrown away before she left for Florida. Turning away from the window, she returned to the bathroom to complete her toilette.\n\nMerrick stomped on the thick straw mat outside the door to the three-story Federal-style building, shaking the snow off his boots as he rang the bell for Alex's apartment.\n\n\"Yes?\" came her sultry voice through the building's intercom.\n\nHe leaned closer. \"Merrick.\"\n\n\"Come on up.\" A buzzing sound disengaged the lock to the outer door.\n\nHe pushed it open with his shoulder, and heat enveloped him like a warm, comforting blanket. The mailbox in the vestibule bearing Alex's name indicated she lived on the second floor. Cradling a bag to his sheepskin-lined leather bomber jacket, he climbed the staircase to her floor.\n\nWalking the four blocks had become a challenge. The falling snow had increased in intensity, and meteorologists were predicting more than a foot before tapering off later that night. A district-wide snow emergency was in full effect wherein government office buildings had closed at two and all nonessential vehicles were ordered to stay off major thoroughfares.\n\nMerrick smiled as he stepped off the last stair. The unique voice of Tina Turner greeted him as the door to Alex's apartment opened and she stood there, an inviting smile tilting the corners of her mouth. Her curly hair fell around her face in sensual disarray, raven strands grazing her delicate jaw and the nape of her neck.\n\nShe looked younger and more fragile than she had a week ago. Today she hadn't bothered to put on any makeup, and in a long-sleeved cotton tee, body-hugging jeans and sock-covered feet she appeared barely out of her teens.\n\nHe handed her the bag filled with a bouquet of colorful calla lilies. \"I decided to bring a little something to brighten up the table.\"\n\nAlex met his gaze. The moisture from melting snow coated the strands of his close-cropped hair, and she wondered why he hadn't worn a hat. \"They're beautiful, Merrick. Thank you so much. Please come in,\" she urged as he took off his gloves, shoving them into his jacket pocket, then bent over to untie his Timberland boots.\n\n\"I don't want to track snow over your floors.\"\n\nAn exquisite oriental runner in the foyer covered a highly-polished wood floor. Leaning against the door frame, he removed his boots, leaving them on the mat outside the door. Then he took off the waist-length jacket and hung it on a mahogany coat tree. Despite the frigid, snowy weather, the inviting space was imbued with a tropical mood, with a stunning French-Regency console table with Martinique-style carvings and a gilded Louis XV\u2013inspired mirror. The table cradled a vase of fresh white roses and peonies and two hardcover books about the Mayans and ancient African art.\n\nAlex pretended interest in the exquisite flowers wrapped in cellophane rather than stare at the man who'd unknowingly occupied her waking thoughts the past week. Why, she mused, hadn't she remembered Merrick's towering height or broad shoulders? A charcoal-gray crewneck sweater and black corduroy slacks made him appear larger, more formidable.\n\n\"You must be freezing. Come and sit by the fire.\"\n\nShe turned and retraced her steps to the living room, Merrick following, where a fire blazed behind a decorative screen. The fire and the pair of candles on the mantel were the only sources of illumination. Pressing a wall switch, she turned on two table lamps.\n\nMoving closer to the fireplace, Merrick held his hands near the heat. The light from candles under chimneys on either end of the marble mantel flickered over the photographs of Alex's many relatives. And judging from the various group and individual family photos the Coles were definitely prolific.\n\nAlex watched Merrick as he studied the framed photos of her relatives. \"I'll be back as soon as I put the flowers in water.\" Seemingly as if in a trance, he turned and stared at her. \"Can I get you something hot to drink?\"\n\n\"I'd like that, thank you.\"\n\n\"Would you like chocolate or a hot toddy?\"\n\n\"I prefer the toddy.\"\n\nAlex smiled. \"Good. I just brewed a pitcher before you got here.\"\n\nFeeling as gauche as a schoolgirl on her first date, she left the living room as Tina Turner's \"I Can't Stand the Rain\" flowed from wireless speakers concealed throughout her condo. How was she going to maintain a friendship with Merrick Grayslake when everything about him radiated unabashed male sensuality?\n\nWalking into her gourmet kitchen, she opened a cabinet and reached for a vase. She arranged the lilies in the vase, filled it with water, then removed a glass pitcher of cider brewed with mulling spices from the refrigerator. Pouring a generous amount into a saucepan, she put it on a stovetop burner to heat.\n\nMerrick's sock-covered feet were silent as he made his way out of the living room with mahogany furniture reminiscent of pieces he'd seen in West Indian homes that had once belonged to wealthy European merchants and planters. Rich motifs of stylized carved pineapples and palm fronds decorated the legs of tables and chairs. A gleaming black concert piano was positioned in an alcove with a vaulted ceiling.\n\nHe entered a formal dining room with a table set for two. Prisms of light from a chandelier fired the facets in crystal stemware at the place settings.\n\nLeaving the dining room he walked into the kitchen, stopping short when he saw Alex placing cinnamon sticks in two large mugs as she gyrated to \"Nutbush City Limits.\" Crossing his arms over his chest, he leaned against the entrance to the enormous stainless-steel kitchen filled with the mouthwatering aroma of roasting meat, watching her as she closed her eyes, snapped her fingers and danced to the catchy tune. A smile touched his mouth when he remembered her dancing with Michael.\n\nThe selection ended and he put his hands together, applauding. \"Bravo.\"\n\nAlex spun around, her face flaming with embarrassment. Merrick had caught her pretending she was an Ikette. She'd spent her teenage years wishing she were a backup dancer for Tina Turner.\n\nRecovering quickly, she bowed from the waist. \"Thank you, thank you, thank you,\" she drawled, blowing kisses to an imaginary audience.\n\nHe lifted a dark eyebrow. \"You missed your calling. You should've become a dancer.\"\n\nAlex emitted an audible sigh. \"That was never going to happen. My mother made me take dance lessons, and when the kids went to the right I went to the left. The instructor thought there was something wrong with me because I couldn't follow the steps she'd choreographed and eventually expelled me from class. I never told anyone, but Madame H pulled me aside after my first lesson and lectured me sternly about showing up the other little girls. That ignited an undeclared war and I did everything I could to make her life a living hell.\"\n\nMerrick, lowering his arms, straightened. \"Remind me to never cross you.\"\n\nShe affected an attractive moue, wrinkling her nose. \"It wasn't often that I was a horrible little girl, but there was something about Madame H that pulled me over to the dark side.\" She pantomimed leaning to her left, limp fingertips grazing her forehead in dramatic fashion.\n\nShaking his head and smothering a laugh, Merrick found it hard not to respond to Alex's theatrics. Everything about her was young, fresh, uninhibited and spontaneous.\n\n\"Are you always this bubbly?\"\n\nShe sobered, meeting his questioning gaze. \"Don't you mean silly?\"\n\nHe shook his head. \"No, Ali. I meant exactly what I said. Bubbly.\"\n\nAlex returned to the stove and poured the warm liquid into the mugs. \"It comes in spurts,\" she said truthfully. \"There are times when I'm as serious as a heart attack, but most times I'm pretty loose.\" She didn't want to think of her family's assessment that she'd changed since she'd begun her graduate studies.\n\nMerrick moved closer, inhaling the fragrant scent of cinnamon, cloves and orange wafting from the mugs. He was pleased that she felt comfortable enough with him to be loose, because he was more than aware that he'd made some people uneasy whenever he was in their presence.\n\n\"Are you concerned as to how people perceive you?\" he asked.\n\n\"No,\" she said without hesitation. \"And even if I was there's nothing I could do about it. There was a time in my life when I changed myself completely to please someone, and in the end I hated myself for it.\" She extended her arms.\n\n\"What you see is what you get. Take it or lump it.\"\n\nMerrick wanted to tell Alex that he would take it\u2014take all of her just as she was. \"I like what I see, Alexandra Cole.\"\n\nShe curtsied as if she were royalty. \"Thank you, Merrick Grayslake.\" Her head came up as she straightened. \"And I like what I see.\"\n\nResting his elbows on the cooking island, he impaled her with a penetrating stare. \"What do you see, Ali?\"\n\nBoldly, unflinchingly, Alex met his stare, noticing things about Merrick that she'd missed New Year's Eve. There was a minute scar on his left cheek, a slight bump on the bridge of his aquiline nose as if it had sustained an injury and a hint of blue in his gray eyes. What she'd remembered was the shape of his mouth, a perfect masculine mouth with firm lips, and the close-cropped hair that was more red than brown.\n\nShe smiled. \"I see a man who I look forward to calling a friend.\"\n\nMerrick's eyebrows flickered. \"I thought we were already friends.\" Reaching for one of the mugs, he waited for Alex to take hers. He raised his mug in a salute before taking a sip of the warm spicy beverage.\n\nAlex sipped her toddy while she replayed Merrick's statement: I thought we were already friends. To her, friends supported, protected and comforted one another in the good and not-so-good times. A friend would be someone she could confide in and trust with her innermost secrets. And she wondered if Merrick Grayslake would and could become her friend in the true sense of the word. Only time would tell.\n\nSetting down her mug, she handed him the vase of lilies. \"Would you please put this on the dining-room table?\"\n\n\"Sure.\"\n\nAlex smiled when he left to do her bidding. Unknowingly, Merrick had just passed the first test. His expression hadn't changed, nor had he hesitated when she'd asked him to do something for her.\n\nMerrick removed the fireplace screen, stoked the burning embers with a poker, then added another piece of wood on the grate. The flames caught and he replaced the screen; the sweet redolent aroma reminded him of the wood-burning stoves in his West Virginia home, a home where for the past two years he'd become a recluse, venturing out only to shop for food. Weeks would go by before he refueled his sport-utility vehicle.\n\nAt thirty-three he'd dropped out of sight as if he'd never existed, and it wasn't until Michael Kirkland came to see him that he was jerked back into the reality that there was another world outside Bolivar, West Virginia. In the past three months he'd visited D.C., Georgetown, West Palm Beach, Miami, Key West and now Arlington, Virginia.\n\nBut when he returned home after visiting with Rachel he'd been tempted to settle back into what was now a comfortable and secure routine of waking up at dawn and walking several miles in the cold mountain air before returning home to spend the day reading, chopping wood or watching television.\n\nHowever, his plan to reconnect with Alexandra Cole thwarted the inclination to fall back into what was familiar when he packed enough clothes for a two-week stay in the Capitol District. And he never questioned or second-guessed himself once he'd checked into the Hyatt.\n\nSharing dinner with Alex had become an enjoyable event. Her exquisite culinary skills were only surpassed by her delightful companionship. Dinner had begun with shrimp cocktail with a piquant cocktail sauce, a salad of endive stuffed with lump crabmeat, marinated pork tenderloin stuffed with spinach and corn bread. A pale blush wine had become the perfect complement to the differing flavors that had lingered on his palate.\n\nHe'd offered to help her clean up, but she'd refused with the excuse that she didn't like anyone in her kitchen. It was apparent she liked being in control.\n\nAlex walked into the living room to find Merrick stretched out on the rug in front of the fire, his head cradled on folded arms. She smiled. He'd truly made himself at home.\n\nClosing the distance between them, she lay down beside him. She didn't have time to catch her breath when he reached out and pulled her into an embrace. Shifting to her right, she rested her head on his shoulder as if it was something they'd done countless times before, feeling safe, protected in his arms. His warmth and the lingering scent from his cologne urged her closer.\n\n\"It's still snowing,\" she said softly. She'd turned on the under-the-counter television in the kitchen to the Weather Channel; the snow totals for the D.C. region had surpassed fourteen inches.\n\n\"We'll probably get two feet before it's over,\" Merrick drawled, not opening his eyes.\n\n\"I hope you're not thinking of trying to make it back to your hotel tonight.\"\n\nHe opened his eyes. \"It's not that far.\"\n\n\"It's too far for you to walk in a blizzard, Merrick. I have a spare bedroom. You're about the same height as my brother Jason, so you won't have an excuse that you don't have anything to wear.\"\n\nHe chuckled softly. \"You're the second woman within the span of a week who has invited me to stay with her.\"\n\nAlex went completely still, her heart pounding a runaway rhythm as she held her breath until she felt her lungs burning. Had Merrick lied when he told her that he wasn't involved with a woman?\n\n\"You stayed with a woman in Florida?\"\n\nTightening his hold around her waist, Merrick pulled her closer to his length. \"It's not like that, Ali. Rachel's someone I used to work with. She threatened me with bodily harm if I didn't check out of a Key West hotel and stay with her.\"\n\nSome of the tension left her. \"What did she threaten to do? Shoot you?\" He nodded. Tilting her chin, Alex stared up at Merrick staring down at her. \"Would she have shot you?\"\n\n\"Of course not. Why are you taking what I'm saying so seriously?\"\n\nVertical lines appeared between her eyes. \"It's just that I don't need some crazy-ass woman looking to cap me because she believes I'm hitting on her man.\"\n\n\"That will never happen because I'm not involved with anyone. Tell me how it was growing up a Cole,\" he asked, deftly changing the topic.\n\n\"That all depends on which Cole household you're talking about. I believe mine was the most unconventional. My uncle Martin is rather traditional, and so are his children. My aunts Nancy and Josephine were raised believing they were Afro-Cuban royalty, so they passed their values down to their children and grandchildren. The Cole-Thomases and Wilsons have this air of entitlement that can really work my nerves. My uncle Joshua is quite conservative, but that comes from a career in the military.\n\n\"Perhaps it's because my dad opted for a career in music rather than devote his life to the family business that we were raised without the restrictions imposed on my other relatives. I remember Abuela lecturing Daddy that he was raising 'los animalitos peque\u00f1os.' My mother took offense to her mother-in-law's reference to her children as little animals, and it took months before they declared a truce.\"\n\n\"What did your parents do that ticked off your grandmother?\"\n\n\"We played music all day and half the night\u2014loud. The house was always open to our friends for sleepovers and pool parties. And because Daddy had set up his own record company, there was an unending stream of popular and wannabe musicians coming to the in-home recording studio. It was cool to see them come to record a demo, and a year later see them on television in their own music video.\"\n\n\"It sounds as if you had a lot of fun.\"\n\n\"It was. Where did you grow up?\" she asked Merrick.\n\n\"Texas.\"\n\n\"I knew it.\"\n\n\"What did you know?\"\n\n\"I knew you were from the South.\"\n\n\"Southwest,\" he corrected softly.\n\n\"Texas is still the South. Where in Texas were you raised?\"\n\n\"Dallas, Waco, San Antonio, McAllen, Lubbock, Corpus Christi. You name it, I've lived there.\"\n\nEasing out of his loose embrace, Alex sat up. \"Why did your family move around so much?\"\n\nThe last CD had finished minutes before, and there was only the sound from popping wood and showers of falling embers. Merrick lay on the rug, lifeless as a statue as he stared up at the shadows on the ceiling. He couldn't chide Alex for asking him about his past because he'd opened the door when he'd asked about her childhood.\n\n\"My family didn't move. I was the one who moved whenever social workers shuttled me from one foster home to the next. I stopped counting at six.\"\n\nAlex rested a hand on his shoulder. \"Where were your parents?\"\n\nHe closed his eyes. \"I don't know, Ali. I never knew my mother or father because I was abandoned at birth.\"\n\nFeeling as if her breath had solidified in her throat, Alex was unable to form a response or comeback. Here she was running off at the mouth about the Coles while Merrick had been passed around like an inanimate object to whoever was willing to accept him.\n\nMerrick sat up. For a long moment he studied Alex intently. \"Aren't you going to say what all of the others have said?\"\n\n\"What's that?\"\n\n\"That you're sorry.\"\n\nHer eyelids fluttered as she registered the coldness that'd crept into his voice, and she wondered whether he'd grown up listening to people pitying him for a turbulent childhood. What about those who'd been presented with an opportunity to change his life, yet stood by and did nothing? However, there was something about the prideful man sitting inches from her that silently conveyed that he wouldn't accept her pity.\n\n\"You're wrong, Merrick. Who am I to pity you when something tells me you'd throw it back at me?\"\n\nThe hard, gray eyes that shimmered like glacial ice softened as a smile touched his mobile mouth. \"You're right, Ali. I don't want your pity. My past is exactly that\u2014the past.\" He traced the outline of her cheek with a forefinger. \"It's getting late, so I'd better be going before I'll be forced to accept your offer to spend the night.\" He rose to his feet, extending his hand and pulling her up with him.\n\nAlex looked up at him. \"I doubt if the fund-raiser will go off as planned tomorrow night.\"\n\nMerrick nodded. \"It's going to be a couple days before things are back to normal.\" Lowering his head, he pressed a kiss to Alex's cheek. \"Thanks for dinner and for your company. I'll call you tomorrow.\"\n\nWrapping her arms around his waist, she rose on tiptoe and kissed his chin. \"Thank you for coming.\"\n\nShe followed him to the foyer where he opened the door and retrieved his boots. Sitting down on a chair beside the table, he put them on. Reaching for his jacket, she held it out for Merrick as he slipped his arms into the sleeves.\n\nHe turned, they shared a smile, and then he was gone. Alex closed and locked the door behind him. She returned to the living room and extinguished the candles. She stood in front of the fireplace, staring at the dying embers and recalling the past four hours, when she'd shared the most pleasurable time with a man that she had in years.\n\nWhat had surprised her was that she'd felt comfortable with Merrick until he revealed his childhood, and she didn't think she would ever forget his expression when he said he'd been abandoned at birth. He'd closed his eyes and his face had been totally void of emotion.\n\nWhat abuses, she mused, had he suffered in his various foster homes? Had he gone hungry? Was he beaten?\n\nShe'd grown up with both parents loving her and she loving them; they'd protected and spoiled her, and nothing had been beyond her reach. Merrick couldn't go back in time to right the wrongs, but she made a silent vow that she would become a friend on which he could rely.\n\n## Chapter 6\n\nMerrick stood several inches behind Alex as she explained the significance of a piece of sculpture at the National Museum of African Art. It had taken Washingtonians two days to dig out from under twenty-two inches of snow while those in northern Virginia took longer with nearly twenty-eight inches of the frozen precipitation.\n\nThe fund-raiser to which Alex had purchased tickets had been rescheduled to the middle of February. They wouldn't attend because Alex would be out of the country and he'd planned to return to West Virginia the day before she was scheduled to leave for Mexico City. He stared at the raven curls secured in a ponytail rather than at the strange-looking pieces that were now part of a permanent collection.\n\nAlex gestured toward an elaborately carved wooden staff. \"This piece comes from the Yoruba culture of southwestern Nigeria. The staff is named for the god of thunder, Shango, and was carried by Shango cult members as a symbol of their office.\" She pointed to a sculpture of a mother and child. \"Although the mother and child figures represent an archetypal Yoruba theme, the oshe Shango also displays several unique qualities, namely for its three-dimensional form, its contrast of hard geometric shapes and its smooth surface with detailed...\"\n\nHer voice trailed off. She turned and stared up at Merrick, who wasn't looking at the sculpture, but directly at her. \"Merrick!\" she hissed through her teeth.\n\nHe blinked, as if coming out of a trance. \"What is it?\" he asked softly.\n\n\"I'm talking a mile a minute and you're not listening.\"\n\nReaching for her hand, Merrick led her away from the exhibit and over to a sitting area. \"I've heard every word you've said. We've spent the past week together visiting every museum and gallery in this blasted city. I can tell you the portrait of Mary Cassatt painted by Edgar Degas hangs in the National Portrait Gallery. The Rhyton, a silver and gilt drinking cup dating from the fourth century A.D., probably of Iranian origin, is at the Arthur M. Sackler Gallery.\n\n\"The Abandoned Doll by Suzanne Valadon, who was also a model of Toulouse-Lautrec and Renoir and the mother of painter Maurice Utrillo, was renowned in her own right. This painting is evocative of the theme that brings drama and psychological truth to the universal rite of passage when the mother tells her pubescent daughter about the physical changes in her body. The doll on the floor symbolizes that the daughter must leave her childhood behind in order to embrace womanhood.\" He leaned closer and whispered in her ear, \"If you take me to another museum I'm going to go ape shit, Alexandra Cole.\"\n\nClapping a hand over her mouth, Alex stared at Merrick as if he'd taken leave of his senses. She wasn't as stunned by his refusal to go to another museum as she was by his knowledge of what he'd observed and retained. Although she thought herself cogent in the art field, she still was hard-pressed to remember where every piece was housed.\n\nLowering her hand, she glared at him. \"Where's the Angel by Abbot Henderson Thayer?\"\n\n\"It's at the National Museum of American Art.\" He gave her a Cheshire cat grin. \"Now can I have my A?\"\n\nHer thunderous expression softened. \"Why didn't you say something?\"\n\nHe pressed his mouth to her ear. \"I didn't because I didn't want to hurt your feelings. Please, pretty please, give me my A, teacher.\"\n\nMerrick continued to astound Alex with his mercurial personality. Most times he exhibited a staidness that made her uncomfortable, but then without warning he would soften, exhibiting a teasing quality that allowed her to lower her defenses, wherein she wanted him to become more than a friend.\n\nWrapping her arm around his waist, she rested her head against his shoulder. \"You passed, Merrick.\"\n\n\"What's my final grade?\"\n\nShe gave him a sidelong glance. \"A.\"\n\n\"Let's go somewhere and celebrate?\"\n\n\"Where do you want to go?\" Alex asked. For the past eight days he'd driven to her place, where they planned their itinerary for the day. Three nights they ate at his hotel's restaurant, and she'd returned the favor and cooked for him again.\n\n\"Anyplace where I don't have to whisper. You're leaving the States in five days, and I'm going back to Bolivar where the only thing I'm going to hear is the sound of my own breathing.\"\n\n\"You should get a pet.\"\n\n\"I don't want a pet.\"\n\n\"Then get a girlfriend.\"\n\n\"I have a girlfriend,\" he crooned, catching her earlobe between his teeth, \"but she's going to Mexico City.\"\n\nUnexpected warmth eddied through Alex. The soft nip of his teeth on her ear and his moist breath flowing into it heated her blood rushing to nether regions, triggering a pulsing between her legs.\n\nIn five days she would leave the States and Merrick. They'd spent practically every day together walking the snowy streets of Washington, D.C., touring museums and art galleries and drinking lattes at Starbucks. They'd become friends and formed a comfortable camaraderie without the likelihood of a physical involvement. And there were times when she wanted to do more with Merrick than hold hands or exchange a chaste kiss.\n\nMerrick released her ear. \"What are you thinking about?\"\n\n\"How do you know I'm thinking?\"\n\n\"You're not saying anything.\"\n\nShe gasped. \"Oh, no, you didn't!\"\n\n\"Don't get your cute little nose out of joint, because I've never known you to be at a loss for words.\"\n\n\"You haven't known me that long, Merrick.\"\n\n\"That may be true, but even if we were together for the next thirty years I'd never know the real Alexandra. I'm willing to bet there are things about your father your mother doesn't know, and vice versa.\"\n\n\"You're probably right. Everyone has secrets.\"\n\nRunning a finger down the length of her delicate nose, Merrick replaced it with a soft kiss. \"I have a secret.\"\n\nShe smiled. \"Do you want to tell me about it?\"\n\n\"I want you to spend the next four days with me at my place. I'll have you back in Virginia in time for you to make your flight.\"\n\nAlex was totally bewildered at his request. Just when she was beginning to relax and feel comfortable with Merrick he changed the rules. He'd invited her to join him in the Keys and she'd refused, saying perhaps the next time. This invitation had become the next time.\n\n\"I told you before that I'd never put you in a compromising position,\" he continued in the soft drawling voice that never failed to send a shiver throughout her body.\n\n\"And I told you that I always keep my promises,\" she countered. \"What time are we leaving?\"\n\nMerrick exhaled as a tender smile softened the lean contours of his face. He didn't know why, but he felt as if he'd been holding his breath since he first spied Alexandra Cole.\n\nHe stood up and extended his hand. He wasn't disappointed when Alex placed her hand trustingly in his as he eased her to her feet. \"I'll drop you home to pack before I go back to the hotel to check out. We're also going to have to stop along the way to buy groceries.\"\n\n\"We can clean out my refrigerator, because I plan to disconnect it before I leave.\"\n\nMerrick didn't want to think of not seeing Alex for four months. He liked her effervescent, teasing personality and her natural beauty that prompted men to turn around and stare at her. Whether dressed casually in a pair of jeans and a pullover sweater or a dress that flattered her sexy petite body, she was perfect. It was the first time he'd found everything he could possibly want in one woman: intelligence, beauty and sensuality.\n\nReaching for her navy blue wool swing coat, he held it as she slipped into it. A minute later she repeated the gesture when he shrugged into his jacket. Hand in hand they walked out of the museum, unaware that the next four days would change them\u2014forever.\n\n## Chapter 7\n\nSnow flurries intensified until a steady snowfall forced Merrick to slow down to less than forty miles an hour. He'd hoped to reach Bolivar before nightfall, but the weather had conspired against him. He'd dropped Alex off at her condo, returned to the hotel, packed and checked out of the Hyatt, then stopped at a supermarket to pick up dairy items before driving back to get her. Alex was ready when he arrived, and ninety minutes after they'd agreed to spend the next four days in Bolivar they were on their way to the picturesque hamlet nestled in a valley less than three miles from the Allegheny Mountains.\n\nAlex glanced out the side window, watching the topography change the farther they ventured into West Virginia. Her ears popped with the higher elevation. There was enough light left to marvel at the natural splendor of the unspoiled, rugged wilderness that hadn't changed in more than a century.\n\n\"It looks so peaceful here,\" she said reverently.\n\nSmiling, Merrick took a quick glance at his passenger before he returned his gaze to the winding road. \"It is.\" It was quiet, remote, and if he didn't turn on a radio or the television he tended to lose track of the days of the week.\n\n\"What made you decide to move to West Virginia?\"\n\nMerrick knew he had to open up to Alex and reveal a little of his past or she would never trust him. And he wanted her to trust him because his feelings were changing, intensifying. He'd told her all he wanted was for them to be friends. She'd become his friend and unknowingly more. It was one thing to mouth the word and another to experience emotions that would shock her.\n\nHe wasn't certain when his feelings toward her had changed. One morning he woke from a disturbing dream that left him reeling from the sexual images and his own driving need to make love to Alexandra Cole. The harder he tried to ignore the truth, the more it nagged at him. His initial admission that he wasn't interested in her romantically, his vow not to become involved, was shattered completely when he'd asked her to come home with him.\n\n\"I suppose you could say I was running away,\" he said truthfully.\n\nAlex shifted on her seat, turning to stare at Merrick's profile. His taut expression was one of pained tolerance. A shiver snaked its way down her body. Whenever she posed a question about his past she felt him withdraw from her. She turned away from the glum-faced man. The seconds ticked off, the increasing silence inside the vehicle deafening.\n\nMerrick downshifted as he maneuvered around a sharp curve, a stretch of road locals referred to as Deadman's Curve. It veered sharply to the right, and then without warning the grade dropped off before the road veered left. Viewed from the air the road resembled a large undulating snake. A smile found its way through his closed expression with Alex's audible gasp.\n\nReaching over to his right, he caught her left hand and held it until she unclenched her fingers. \"Relax, baby. I've driven this road enough to do it with my eyes closed.\"\n\n\"Please don't close your eyes, Merrick.\"\n\nNot willing to avert his attention from the snow-covered roadway, he squeezed her fingers. \"Don't worry, Ali. I'd never let anything happen to you.\"\n\n\"Promise?\"\n\nHis smile widened. \"Promise.\"\n\nAlex alternated closing her eyes and holding her breath as Merrick navigated sharply to the right, then left as the wiper blades worked furiously to keep the windshield free of snow.\n\nHalf a mile later, Merrick left the paved roadway, maneuvering onto a narrow rutted path bordered on both sides with towering pine trees. Within minutes, the overgrowth of trees and shrubs gave way to an open meadow. The outline of a two-story house was visible in the steady beam of headlights. He slowed, coming to a complete stop under a carport.\n\n\"Aren't you going to park in the garage?\" Alex pointed to a two-car garage about fifty feet from the house.\n\n\"There's no room. I'll show you what's in there tomorrow.\" He cut the engine, got out and came around to assist her.\n\nAlex waited for Merrick to open her door; she extended her arms. He lifted her effortlessly, setting her on her feet. If it hadn't been for the truck's headlights and falling snow it would've been pitch-dark. There were no streetlamps or lights from nearby homes. Merrick lived in the middle of nowhere. He admitted moving to the wilderness because he'd been running away. Running away from what or whom?\n\nClutching his arm, she followed him up several steps to a porch; squinting, Alex tried seeing beyond the curtain of white but encountered eerie nothingness. All of her senses were heightened when she heard the distinctive sound of a lock opening followed by a soft beeping that was silenced when Merrick punched in a code for a security system. Within seconds light illuminated the first floor.\n\n\"Come in,\" Merrick urged. \"You're letting out the heat.\"\n\nShe stomped the snow off her feet and walked into a living room with modern functional furniture. The saddle-tan leather sofa and love seat complemented the heavy oaken tables, giving the space a masculine feel. Polished pale pine floors and wide windows covered with bamboo blinds in a straw-yellow shade further enhanced an atmosphere of openness.\n\n\"It's wonderful.\" Her voice echoed awe.\n\nMerrick stared at Alex, complete surprise on his face. \"You like it?\"\n\nShe smiled at him. \"Of course I like it.\" Her smile faded as quickly as it'd appeared. \"Why did you think I wouldn't?\"\n\n\"I was under the impression you'd think it too rustic.\"\n\nShe rolled her eyes at him. \"Where you live is rustic, not your home. Can I see the rest of it?\"\n\n\"Sure. The kitchen is to your right and the pantry and laundry are off the back along with the family room. The bedrooms are upstairs in the loft.\"\n\nShrugging off her coat, she handed it to Merrick. \"Can you please hang this up?\"\n\nHe took her coat, snapping a smart salute. \"Ma'am, yes, ma'am!\"\n\n\"Show-off,\" she crooned as she made her way in the direction he'd indicated.\n\nMerrick felt an overwhelming sense of pride that Alex liked his home. There were times when he felt it was much too large for one person, but he'd resisted sharing his life and home with a woman.\n\nHe'd never thought of himself as husband or father material because of his turbulent childhood. How could he know where he was going when he hadn't known who he was or where he'd come from?\n\nHe'd languished in a foundling hospital, then a group home, and at the age of four he was placed with his first foster family. He'd overheard the social worker tell his foster parents that a priest had found him wrapped in a blanket, umbilical cord still attached, under a pew in a South Texas church only five miles from the Mexican border. There was a bloodstained note tucked into the blanket that read: Please take care of my son\u2014Victoria Grayslake. Father Merrick had cleaned and baptized the infant, giving him his first name before calling the authorities. Two days after his birth Merrick Grayslake became a ward of the state of Texas.\n\nHe didn't know what prompted him to bring Alex home with him, but with his rapidly changing feelings for her he didn't want to spend time analyzing everything he did. All he wanted to do was enjoy whatever time they had together before she left the country. He walked out of the house and returned to the truck to get their luggage and perishable foodstuffs.\n\nAlex lay on the sofa, eyes closed and her head resting on Merrick's thigh. She'd toured his house, falling in love with the massive wood-burning stoves in the kitchen and family room. The second-story loft contained three large rooms. Merrick had claimed the largest as his bedroom. Another had been set up as a guest room and the third was filled with boxes and metal file cabinets. Whitewashed walls, wood floors covered with colorful rugs and fireplaces had turned his home into a perfect getaway retreat.\n\n\"Did you build this place?\"\n\nHe combed his fingers through her mussed curls. \"No. I bought it for a fraction of its worth when the original owner died two days after he took possession. His widow, who never wanted to live here, said it was cursed. She moved out, leaving all of the furnishings, and never looked back.\"\n\nAlex opened her eyes and stared up at him looking down at her. \"I thought you'd decorated it.\"\n\n\"No. I wouldn't know where to begin. All that talk about fabrics and colors confuse the hell out of me.\"\n\n\"Where did you live before you moved here?\"\n\n\"I rented the cottage on the property of someone I'd met in the Corps.\"\n\nShe smiled. \"You were a marine?\" Merrick nodded. Sitting up, she gave him a direct stare. \"Were you really into that esprit de corps and Semper fidelis indoctrination?\"\n\nHis luminous eyes widened until she could see their hoary depths. \"Once a marine, always a marine.\"\n\nAlex felt the chill of his words as acutely as if he'd struck her. The man with whom she'd found herself enthralled had morphed into a stranger, shedding his warmth like a reptile shedding its skin.\n\n\"No disrespect intended.\"\n\n\"And none taken,\" he countered quickly.\n\nPushing off the sofa, Alex knew it was time to retreat before she lost her temper and said something she wouldn't be able to retract. \"Good night, Merrick.\"\n\nHe stood up and shoved his hands into the pockets of his jeans. \"Good night, Ali.\"\n\nMerrick felt as if he'd been punched in the gut when she headed toward the staircase. Why, he mused, did he keep pushing her away? Why, when it was the last thing he wanted to do?\n\nTaking three long strides, he stopped her retreat, reaching for her upper arm and turning her around. His hands moved up to cradle her face. Slowly, methodically, his head came down and he took her mouth in a drugging, caressing kiss that told her wordlessly how much he wanted, how much he needed her.\n\nAlex held on to Merrick's shirt like a drowning swimmer caught in a riptide. His warmth, his smell, the strength in his hard lean body seeped into her like water through a porous fabric, soaking and becoming one. Standing on tiptoe, she returned the kiss, her lips parting as delicious sensations raced from her mouth downward.\n\nMerrick placed tiny kisses along the column of Alex's scented neck, his teeth nibbling softly at the base of her throat. Tightening his hold on her body, he lifted her off her feet until her head was level with his. He took her mouth again, devouring its softness. He'd tasted her, but instead of sating his hunger, it increased. If he could he would've devoured all of her.\n\nHe felt the flesh between his thighs stir and knew if he didn't stop what he'd begun he would do something he would regret for the rest of his life. Lowering Alex until her feet touched the floor, he pulled back. Passion had darkened her face and her eyes; he couldn't pull his gaze away from her slightly swollen mouth. A spurt of desire shot through his groin when he realized Alex was a woman who should be made love to\u2014every day and every night. Her dark beauty smoldered with a sensuality that couldn't be learned or taught.\n\nA hint of a smile tilted the corners of his mouth. \"Good night, Ali.\"\n\nBacking away, Alex wrinkled her nose. If they hadn't stopped when they did, there was no doubt they would end up in bed together. She wanted to arrive in Mexico City with memories of the two weeks she'd spent with her friend and not the few days she'd shared a bed with her lover.\n\n\"Good night, Merrick.\"\n\nShe turned and made her way up the staircase, feeling the heat from the silver orbs on her back. When she reached the top, she turned and glanced down to find that Merrick hadn't moved. Even from this distance she could see the banked flames of desire in his expression.\n\nForcing herself to look away, she walked down the hallway until she reached her bedroom. Her hands and knees were trembling uncontrollably as she fell across the bed and pressed her face into the pillow.\n\nShe wanted him; she wanted Merrick Grayslake more than she'd ever wanted any man in her life. It was good that she was leaving the country, because to remain would prove disastrous to the plans she'd made for herself. And her plans did not include taking a lover or falling in love.\n\nIt was minutes after sunrise when Merrick walked into the kitchen. Alex sat on a tall stool at the cooking island, drinking coffee and concentrating intently on his handheld New York Times Sudoku. A smile crinkled the lines around his eyes as he crept silently across the room, swept her off the stool and lifted her high in the air. Bending slightly, he tossed her up like a rag doll and caught her.\n\nAlex let out a bloodcurdling scream as her arms flailed wildly. \"Stop, Merrick! Please put me down!\"\n\nCradling her to his chest, Merrick settled her on the stool, his arms going around her body in a comforting gesture. \"Good morning, baby.\"\n\nAlex's right hand curled into a fist, her eyes narrowing. \"Don't you dare good morning me!\"\n\n\"You want to hit me?\"\n\nBaring her teeth, she glared at him. \"I want to hurt you real bad, Merrick Grayslake, for scaring the hell out of me.\"\n\n\"Forgive me, sweetheart.\"\n\nThe endearment seemed to take the fight out of her. Within a span of a minute Merrick had called her baby and sweetheart. What happened to friend?\n\nLooping her free arm around his neck, Alex buried her face between his neck and shoulder and closed her eyes. He smelled of soap. Merrick had showered and shampooed his hair but hadn't shaved, and the dark stubble on his jaw and chin accentuated the masculinity he wore like a badge of honor.\n\nShe kissed the side of his strong neck. \"I'll think about.\"\n\nMerrick massaged her back over a sweatshirt. \"Don't think too long, beautiful.\"\n\nAlex eased back, giving him a long, penetrating stare. \"What's going on between us, Merrick?\"\n\nHe sobered, his expression closed. \"What are you talking about?\"\n\n\"Why are you calling me baby and sweetheart instead of Ali? And what happened to the pecks on the lips and cheeks? When you kissed me last night and I kissed you back, there was nothing friendly in what we shared.\" Reaching up, she smoothed down several wayward strands of damp hair over his ear. \"What are we doing to each other, Merrick?\"\n\nMerrick cradled her head, his fingers massaging her scalp. \"We are friends, Ali.\" He enunciated each word.\n\n\"But\u2014\"\n\n\"No buts, Ali,\" he said softly, interrupting her. \"I like you\u2014a lot\u2014but I'm not going to take advantage of you. That's not my style.\"\n\nHer gaze met and held his as she anchored her arms under his shoulders. \"And I promise not to take advantage of you.\"\n\nMerrick released her head, momentarily speechless in his surprise. \"You're incredible,\" he whispered.\n\nShe gave him a saucy grin. \"You just realized that?\" she drawled without a modicum of modesty.\n\n\"Trash-talking will result in another beach-ball toss.\"\n\n\"You wouldn't.\"\n\nMerrick nodded. \"I would.\"\n\nShe pantomimed zipping her mouth and Merrick threw back his head and howled. Her laughter joined his as the unrestrained shared moment brought them closer together, cementing their friendship.\n\nAlex was lost\u2014lost in the splendor of the rugged countryside\u2014and she'd lost track of the days, time. And she'd done what she'd prayed would not happen\u2014she'd lost her heart to Merrick Grayslake.\n\nTheir relationship was uncomplicated and open. He hadn't tried to kiss her again, and for that Alex was grateful. Any display of affection would make her leaving more difficult.\n\nThey developed a comfortable routine of waking early and walking in the frigid morning air before returning to the house to prepare monstrous breakfasts. She cleaned the kitchen while he chopped enough wood to feed the gluttonous wood-burning stoves from the stack stored in a shed at the rear of the house. All the wood came from trees on the five-acre property. Merrick contracted to have trees felled and cut into large stumps. He claimed chopping firewood was his way of staying in shape.\n\nThe man she'd fallen in love with continued to mystify her when he opened the garage to reveal a functioning body shop with two classic cars he'd restored to their original state. One was a 1948 pickup and the other a 1952 station wagon with wood paneling. When she asked whether he planned to sell them he'd given her an incredulous look that spoke volumes, then said emphatically, \"No.\"\n\nHe'd amassed an extensive reading and video library. Floor-to-ceiling shelves in the family room were packed tightly with fiction bestsellers and titles pertaining to history and military battles.\n\nMost nights found them in the family room viewing movies. Alex preferred movies with foreign settings with romantic themes while Merrick liked sci-fi and action thrillers.\n\nFifteen minutes into Man On Fire, Merrick paused the DVD and stared at her. The film's theme dealt with the claim that every hour throughout Latin America someone was kidnapped and held for ransom. Alex belayed his fear that she would be a target of kidnappers because she'd enrolled in the universidad as Alexandra Morris. Her birth records listed her as Alexandra Ivonne Morris-Cole, and when she studied abroad transcripts listed her surname as Morris.\n\nShe was forced back to reality and what lay ahead of her when a call from Diego reminded Alex that he would accompany her on the flight to Mexico City before he continued on to Belize to meet with a consortium of banana growers.\n\nThe return drive to Arlington was accomplished in complete silence as she agonized over how was she going to leave Merrick now that she knew she was in love with him. And her love wasn't based on sex, because they hadn't slept together, and for that she was grateful. There was no way she wanted to confuse a physical connection with one that was emotional.\n\nMerrick stood in the foyer with Alex, cradling her face in his hands. The eyes staring back at him were steady, trusting. He found himself at a loss when he longed to tell her how much he'd enjoyed their time together, how she'd managed to slip under the invisible wall he'd erected to keep all women at a distance, how much she'd changed him and how much he'd come to love her.\n\nHe loved everything about her, inside and out. He'd asked for friendship, got it, but wanted more. And the more was love. He wanted Alex to love him because he'd fallen in love with her.\n\n\"May I call you?\" he asked.\n\nAlex's lashes fluttered as she tried bringing her fragile emotions under control. \"Of course you may. Call me at night because it's your voice I want to hear before I go to sleep.\"\n\n\"Are you living on campus?\"\n\n\"No. I'm staying in a converted convent near the universidad.\" Her mood brightened. \"Are you thinking of coming to visit me?\"\n\nMerrick's impassive expression did not change. \"Do you want me to?\"\n\nA smile fired the gold in her large eyes. \"Yes.\"\n\n\"It won't interfere with your studies?\"\n\n\"As soon as I get my schedule I'll let you know when I have a break.\"\n\nLowering his chin, he lifted his dark, sweeping eyebrows. \"I'll come to see you on one condition.\"\n\n\"I know,\" she said, laughing. \"No museums.\"\n\nHe winked at her. \"You learn fast.\"\n\n\"Not as fast as you,\" she countered. Moving closer, she wound her arms under his shoulders. \"I'm going to miss you, friend.\"\n\nMerrick dropped a kiss on her hair. Her curls reminded him of a field of wildflowers. \"Hush, baby,\" he crooned. \"I'll see you so much that you'll get sick of me.\"\n\n\"Never.\"\n\n\"Never say never, Ali.\"\n\nPulling back, she met his serious gaze. Like a chameleon he'd changed. Gone was the teasing man she'd come to look for, and in his place was the one who frightened her. Once again, he'd become a stranger.\n\nShe forced a smile she didn't feel. \"I'll call you.\" Standing on tiptoe, she brushed her mouth over his compressed lips. \"Hasta luego.\"\n\n\"Hasta luego,\" Merrick repeated. Turning on his heel, he opened the door, then closed it softly behind him.\n\nAlex stood there for a full minute before she sat down on the chair next to the foyer table and closed her eyes. Everything she'd shared with Merrick Grayslake flooded her mind like frames of film: the night they lay in front of the fireplace and he told her how he'd been abandoned at birth, their trips to countless D.C. museums and art galleries that always ended with them drinking lattes at Starbucks. The nights she'd dressed up to share dinner with him at his hotel's restaurant, and the days and nights at the house in Bolivar where she'd learned to appreciate nature in all its wintry splendor.\n\nWhat she didn't want to recall was the image of Merrick in a tank top and jeans chopping wood in the noonday sun. Each time he swung the ax, the muscles in his back and upper arms tightened and relaxed under the strenuous exertion. The first time she saw the shirt, wet from sweat, pasted to his body, it had excited her so much that she went inside the house until he completed the chore. The images returned in an erotic dream that left her aching with a desire that threatened to swallow her whole.\n\nOpening her eyes, she came back to reality. Alex didn't know where her relationship with Merrick would lead, but she knew it had to be resolved. The two large pieces of luggage sitting in the foyer reminded her that in less than two hours she would be on her way to Mexico City. She stood up and made her way to her bedroom to exchange her boots, wool slacks and sweater for a cotton dress and sandals.\n\nWhen she emerged she was ready for Mexico City and her destiny.\n\n## Part Two\n\n## Lovers\n\n## Chapter 8\n\nAlex found it hard to stay focused. She and the other students in her class had spent the past half hour staring at the image of an Olmec stone figure projected on the screen in the lecture hall.\n\nThe professor, a slight, middle-aged man with thinning black hair and a flair for the dramatic, had waxed eloquent about how the Olmec's \"mother culture\" inspired a series of successor cultures, including Maya settlements that began forming in the Mexican-Guatemala border region around 500 BC.\n\nMove on and change the blasted slide! she fumed inwardly. They were a month into the new term and she'd learned more from reading than she had during her lectures. Alex was enthralled with Mexico's history and its people, but Professor Riviera was making it increasingly difficult for her to stay awake in his classes, and the downside was that she had him for three Mexican art courses.\n\nPretentious prig, she continued in her silent tirade, scowling.\n\n\"Se\u00f1orita Morris, perhaps you can tell your fellow students the timeline for the rise and fall of the Olmec civilization.\"\n\nEveryone's attention was directed to Alex as she glared at their professor. There was a pregnant silence as she composed her thoughts.\n\n\"Records show that the first Olmec settlements were established around 1500 BC. In 900 BC the Olmec city of San Lorenzo was destroyed and desecrated. Historians and archaeologists are unsure who or what led to the destruction and the Olmec civilization faded into obscurity.\"\n\nRivera smiled. \"What else can you tell us about this culture?\" he asked.\n\n\"They built ceremonial centers rather than cities, which suggests they were governed by a central authority. The Olmecs carved blocks of basalt into figures with massive heads like the one on the screen and other sculptures with stylized feline features. There is evidence they had ceramics, and digs have recovered jade figurines from this civilization.\"\n\nNodding his approval, the professor of design and art of ancient cultures, pressed his palms together. \"Bien dicho, Se\u00f1orita Morris. It appears you are quite serious in your endeavor to learn about Mexico's rich and colorful history.\"\n\nAlex gave him a saccharine grin. \"If I weren't, then I assure you I wouldn't be here.\"\n\n\"I told you he has the hots for you,\" whispered the student sitting next to her. Alex and Moira Morgan had formed a friendship within days of Alex settling into her room at the centuries-old converted convent she would call home for the semester. Moira, a tall, blond, thin Oklahoman, who spoke fluent Spanish, had ended a yearlong relationship with the son of a Texas oilman to study abroad. Alex cut her eyes at Moira. She'd come to Mexico to earn a degree, not form a relationship with her professor. Her mantra had become: once burned, twice shy.\n\n\"Se\u00f1ores y se\u00f1oritas.\" Hernando Rivera addressed all women, whether married or single, as miss. \"I would like to invite all of you to my home Saturday evening. A colleague has deigned to exhibit his private collection of art before donating it to several museums.\"\n\nA chorus of groans and murmurs followed his announcement as a hand went up in the back of the lecture hall. \"Se\u00f1or profesor, la asistenc\u00eda es obligatoria?\"\n\nProfessor Rivera gave the young man an incredulous look. \"What do you think, Se\u00f1or Salinas?\"\n\n\"Yes, it is,\" said the red-faced student, answering his own query.\n\n\"You all have my address and telephone number. I will expect everyone at eight.\" He held up a hand. \"Before you ask, you may bring a guest.\"\n\nThe bell rang signaling the end of classes and Alex gathered her books. It was her last class for the day and week. At least it was before Rivera's mandate that everyone attend an impromptu gathering at his home.\n\nShe attended classes Monday through Thursday, cleaned her apartment, picked up her laundry and shopped for groceries on Friday, slept late on Saturday and either stayed home or went out with Moira and a few other classmates Saturday night. Sundays were set aside for attending mass and doing homework. She liked everything about Mexico, its people and the cuisine.\n\nBut she missed home and Merrick, but managed to stave off homesickness by staying busy. She alternated calling and writing her parents and exchanging telephone calls with Merrick.\n\nHe kept her updated on national news, while she told him about the political climate in Mexico. He disclosed he'd spent several days with Michael and Jolene, who'd returned rested and tanned from their Jamaican honeymoon, and it was only after their calls ended that she felt totally isolated. She'd become an alien in a foreign land.\n\nThe week before, she'd sent him a letter with postcards bearing the art of Mexican muralists Diego Rivera, David Siqueiros, Jos\u00e9 Orozco and her favorite Mexican artist, Frida Kahlo. She included a note that read: You don't have to go to a museum or gallery to enjoy these.\u2014AIM-C.\n\n\"I've got better things to do with my Saturdays than look in Rivera's face,\" Alex mumbled as she slipped her books into a backpack.\n\n\"I've heard he hosts some wonderful get-togethers at his house,\" Moira said, gathering her own books.\n\n\"I still would rather pass.\"\n\nA sardonic smile parted Moira's pale mouth. She never wore makeup during the week, but weekends transformed her into a siren when she replaced long skirts and dresses and wooden clogs with skintight garments and artfully applied makeup that highlighted her dark blue eyes in a tightly tanned face.\n\n\"We'll show up and eat his food and drink up all of his tequila, then leave. Someone on our floor said they're going to a club near the Zona Rosa where they play music from the States on Saturday nights.\"\n\n\"Who told you?\" Alex asked as they made their way out of the lecture hall and down a hallway in the centuries-old building that had been dedicated to the study of Mexican art and architecture.\n\nShe hadn't bothered to make friends with the other students who had a habit of hanging out in one another's rooms. All of the rooms were equipped with a small utility kitchen, private bath and an expansive living\/sleeping\/dining area.\n\nAlex cooked for herself during the week and took her meals at local restaurants on the weekend. With her dark hair and coloring, she was easily taken for a local; she wanted to blend in and not stand out as a foreigner. And it was not the first time that she was grateful her parents taught her Spanish.\n\n\"Umberto.\"\n\n\"Isn't he the one who's been hitting on you?\" Moira blushed to the roots of her pale hair. \"Cuidado, chica. Umberto puede ser la causa del problema,\" Alex warned in Spanish.\n\nShe wanted to tell Moira that she'd heard rumors that the handsome art student was keeping count of the number of women he could sleep with before the school year ended.\n\n\"If there's going to be a problem, then it's going to be for Umberto because I have no intention of going to bed with him.\"\n\nAlex gave her a sidelong glance. \"I suppose you've heard about his exploits?\"\n\n\"I saw him in action. I take that back. I saw him with his ass out. The first night I moved in I saw him leaving someone's room, clothes in hand. His behind was beet-red as if he'd been paddled.\"\n\nAlex wrinkled her nose. \"Phew! It looks as if Casanova is a freak.\"\n\n\"He can get his freak on, but without me.\"\n\n\"I hear you.\"\n\nThe two women discussed an upcoming exam covering the Mayan calendar. They were still deep in conversation when they reached their student housing. They entered through an open courtyard that led to a smaller courtyard and the large decorative wrought-iron gate that protected the property from outsiders. Inserting a magnetic card into a slot a buzzing and green light deactivated the lock.\n\nAlex pushed open the door, stepping into a lobby and waiting room. The difference between the outdoor heat and the cooler indoor temperature was almost fifteen degrees. Two-foot-thick adobe walls kept the heat at bay.\n\nThe young woman who manned the front desk waved to Alex. \"Miss Morris, there is someone waiting for you.\" All visitors had to wait to be announced.\n\n\"Who is it?\"\n\n\"He's sitting over there.\" She pointed to her left.\n\nAlex turned slowly to find a man, legs outstretched, sitting on a butaca, a leather sling chair; she couldn't see his face in the dimly lit space, but she recognized the hand resting on his thigh.\n\n\"Merrick.\" His name came out in a whisper. \"Merrick!\" she shouted when he came to his feet. Closing the distance between them, she launched herself at this chest, her arms gripping his neck.\n\n\"Baby,\" he crooned close to her ear. Wrapping his arms around her waist, he lifted her off her feet. The distinctive lines around his eyes deepened as he gave her a dazzling smile.\n\nAlex brushed a kiss over his smiling mouth. \"What are you doing here?\"\n\nHe gave her a quick kiss. \"I came to visit with my girlfriend.\"\n\nShe pressed her forehead to his, her lips caressing the bridge of his nose. \"How did you find me?\"\n\nMerrick's gaze photographed every inch of her face, noting the changes. Her hair was longer, the Mexican sun had darkened her face to a rich, mocha brown and she'd replaced the diamond studs in her pierced lobes with a pair of large gold hoops. Alexandra Cole was beautiful and sexy.\n\n\"You put your return address on the letter you sent me with those blasted cards.\"\n\nA sound of someone clearing their voice caught Alex's attention. \"Please, put me down, Merrick.\"\n\nHe complied and she turned to find Moira staring at them, foot tapping and arms crossed under her breasts. Holding on to Merrick's hand, she led him over to her classmate.\n\nAlex had to admit that a month's absence made him even more attractive. His close-cropped hair hugged his head like a cap, and it appeared as if he'd put on weight, because the hollows in his cheeks were less pronounced. Today he wore a lightweight gray jacket over a sky-blue shirt, open at the throat, that he hadn't bothered to tuck into the waistband of a pair of navy-blue slacks. Italian slip-ons had replaced his ubiquitous Timberland boots.\n\n\"Moira, this is my very good friend, Merrick Grayslake. Merrick, Moira Morgan.\"\n\nMoira extended a limp wrist. \"It's nice meeting you, Merrick.\"\n\nHe nodded, reaching for her fingers. \"It's nice meeting you, too.\"\n\nMoira shifted a leather book bag from one shoulder to the other. \"I'm going to head up to my room and take siesta.\" She knew she was staring at Merrick longer than what would be termed polite, but he was drop-dead gorgeous. Reluctantly, she shifted her gaze to Alex. \"I'll talk to you Saturday.\"\n\n\"Okay.\"\n\nAlex looped her arm through Merrick's. Never had she been so glad to see someone. But she had to admit that he wasn't just someone. He was the man with whom she'd fallen in love.\n\n\"When did you get in?\"\n\n\"I came in on the red-eye a little after three.\"\n\n\"You must be exhausted. Come on up to my room. We'll take siesta together.\"\n\nMerrick followed Alex as she led him down a narrow hallway to a stone staircase that led to the second story. He was exhausted, having spent a restless two days deciding whether he should go to Mexico. But he also wanted to tell Alex that taking siesta together wasn't a very good idea.\n\nTalking to Alex had only served to increase his longing for her. She hadn't given him her address and he knew he would never ask her for it; however, his dilemma was solved when he received the packet with the postcards. She'd indicated her return address and that presented the opportunity he needed to come and tell her what lay in his heart.\n\n\"The accommodations here are simple but comfortable,\" Alex explained as she opened the door to her room. \"Bienvenido.\"\n\nHe stepped into the room, not seeing any of the furnishings. Reaching for Alex, he slipped the straps to her backpack off her shoulders, letting it slide to the floor, and closed the heavy oaken door with his foot. Lifting her off her feet, he crossed the room and sat her on the twin-size bed, his body following hers down. Flames of desire darkened his eyes; the silver had disappeared, and in its place was a shocking dark topaz blue.\n\nAlex moved over and straddled Merrick's lap, her hands cradling his face, her fingertips tracing the contours of his cheekbones and jaw. \"Why did you come, Merrick?\"\n\n\"Isn't it obvious, sweetheart?\"\n\nHer gaze fused with his. \"No. I've never been one to make assumptions.\"\n\nMerrick closed his eyes, a thick fringe of lashes brushing the ridge of sculpted cheekbones. A knowing smile parted his lips. \"I lied when I told you that I liked you.\" She gasped. He opened his eyes, meeting her questioning gaze. \"The truth is I love you, Alexandra Cole. I kept telling myself that I wanted you as a friend, that we could be friends, but it just didn't work.\"\n\nHer heart pounding a runaway rhythm thundered under her breasts as a rush of giddiness swept over Alex. He loved her! He loved her and she had fallen in love with him! But it was something she wouldn't tell him\u2014not yet.\n\nBiting down on her lower lip, she attempted to bring her fragile emotions into some semblance of order. She closed her eyes against his intense stare. \"You love me, but your timing's wrong.\"\n\n\"What's wrong with it?\"\n\n\"I have another four months before I return to the States. Long-distance relationships usually don't work.\"\n\n\"We'll make it work, Ali. I didn't wait thirty-five years for someone like you to come into my life to concern myself with a few months or miles.\"\n\nA frown furrowed her forehead. \"Do you realize how far it is between Mexico City and the East Coast of the United States?\"\n\nHe smiled. \"I know exactly how far it is. But distance doesn't matter if I have you here.\" Merrick placed his left hand over his heart.\n\nA flood of tears shimmered in her gold eyes. \"Don't do this to me, Merrick.\"\n\nBurying his head against the side of her neck, he breathed a kiss under her ear. \"Do what, baby?\"\n\n\"Don't make me want you so much.\"\n\n\"Is that such a bad thing?\"\n\nShe sniffled. \"It is when I can't have you whenever I want you.\" Her hands curled into tight fists on his back.\n\nRaising his head, Merrick saw her tortured expression. \"Alexandra, please look at me.\" She opened her eyes. \"I'm not going to put any pressure on you. We'll see each other whenever it's convenient for you.\"\n\n\"You don't mind waiting until June?\"\n\n\"No. You're more than worth the wait.\"\n\nBlinking through tears, she smiled at him. \"Thank you, Merrick, for making it easier for me.\"\n\n\"Do you realize how easy it is for me to love you?\"\n\nResting her head on his shoulder, she exhaled a soft sigh.\n\n\"When you get to know me better you may change your mind.\"\n\n\"I doubt that,\" Merrick said.\n\n\"I've been known to go from nice girl into bitch mode in one point two seconds.\"\n\n\"That's your problem, not mine, because I don't argue or fight with women. So, if you go off, then don't count on me to entertain your tantrum.\"\n\nSitting up straighter, she stared directly at him. There was no trace of laughter or teasing in his solemn expression. \"What would you do?\"\n\n\"Walk away until you come to your senses. And stop trying to scare me off, because it's too late. I'm in this for the duration.\"\n\nAlex's delicate jaw dropped before she recovered quickly. \"What is your definition of duration?\"\n\n\"Perpetuity.\"\n\n\"You can't be talking about marriage?\" The query tumbled from her lips.\n\nMerrick saw the rapidly beating pulse in Alex's throat. His gaze lowered to her heaving bosom under a simple white cotton blouse. Not only had he shocked her with his surprise visit, but his declaration that he wanted to spend the rest of his life with her had shaken the very foundation of her privileged existence.\n\nThere was a beat of silence before he said in an eerily quiet voice, \"You mentioned the word, Ali, not me.\"\n\nA rush of color suffused her cinnamon-brown face. \"I'm not going to marry you, Merrick Grayslake, so you can forget about duration and perpetuity. I've spent the past ten years with my life in flux wherein I've studied, traveled and partied like a girl gone wild.\n\n\"My last year of high school I told my parents I wanted to put off going to college for a year because I needed a break from the books. My mother issued an ultimatum, go to college or get a job and move out of her house. And there was no job available to an eighteen-year-old that would pay me enough to afford me the lifestyle to which I'd become accustomed.\n\n\"Bowing to pressure, I applied to the Rhode Island School of Design. They accepted me and I enrolled as a liberal arts major with a concentration in art and architectural history in their five-year degree program. Unfortunately I had an affair with one of my instructors in my sophomore year and...\"\n\n\"What happened, Ali?\" Merrick asked when she closed her eyes at the same time her words trailed off.\n\nShe opened her eyes. \"I gave him my love and my innocence. Whenever I mentioned marriage he'd avoided talking about it before finally admitting he wasn't the marrying kind. That all changed when my parents came up to visit me. He did a one-eighty when he realized I was one of those Coles. He waited a week, then proposed, but at twenty-two I wasn't so in love that I didn't know that he was after my money. Once I told him that I wouldn't have control of my trust fund until my twenty-fifth birthday he dropped me like hot garbage.\n\n\"I graduated, secured a part-time position with a Boca Raton art gallery and during my free time partied like there was no tomorrow. I dated a lot of men but I refused to sleep with any of them. I saw them as someone with whom to pass the time. Then I met someone I believed was different from the others. We dated for five months.\n\n\"After this liaison I took a long hard look at myself, reassessing my life and what I wanted for me, not my parents, my brothers or my sister. I made the decision to return to school, and this time study abroad. Being on my own forced me to depend on Alexandra for everything, and I loved it.\" Her gold-brown eyes filled with a gentleness that made her appear angelic. \"Like any normal woman I want marriage and children, but that can't happen now.\"\n\nMerrick's arm tightened around her waist. \"Are you finished with your soliloquy, Ali?\" he asked in a soft voice she had to strain her ears to hear.\n\nShe blushed furiously. \"Yes, I am.\"\n\n\"I've heard you and I remember everything you've ever told me. And I've told you that I would never pressure you to do what you don't want to do. There's no reason why we can't enjoy each other without marriage and children becoming part of the equation. And if we go from being friends to lovers, then I'll accept full responsibility for making certain you don't get pregnant.\"\n\nAlex combed her fingers through her hair, holding the wayward curls off her forehead. She appreciated the fact that Merrick Grayslake hadn't tried to change her; he accepted her flaws and imperfections that permitted her to be Alex. Could she, she wondered, offer Merrick her heart? Would he be worthy of the love she'd guarded jealously since ending her first liaison?\n\nCould she sleep with him and not want more?\n\nYes, whispered the silent voice in her head.\n\n\"I know I've been doing a lot of talking\u2014\"\n\n\"That you have,\" Merrick agreed, interrupting her.\n\nHer hands came down and she hugged him. \"Please don't interrupt me or I'll lose my nerve.\"\n\nA small smile played at the corners of Merrick's mouth as he gave her a lazily seductive look. \"Okay, baby.\"\n\n\"Do\u2014do you think it's possible for us to stop being friends?\"\n\nMerrick's smile faded as he held his breath. \"What do you want?\"\n\n\"I want us to become lovers.\"\n\nA powerful relief shook Merrick as his chest swelled with a peace he hadn't thought possible. Angling his head, he touched his mouth to Alex's, increasing the pressure until her lips parted while his hands explored the curve of her waist and hips.\n\n\"Get your clothes and books and pack a bag.\"\n\n\"Where am I going?\" she asked between soft nibbling kisses.\n\n\"We're going to my hotel. There's no way two of us can sleep in this kiddie bed.\"\n\nShe laughed. \"It's big enough for me.\"\n\n\"But I'm taller and bigger than you.\" He kissed the top of her head. \"Let's go, querida.\"\n\n## Chapter 9\n\nAlex walked over to the large window and opened it. Stepping out onto the balcony, she looked down onto a courtyard with a flowing fountain. A round table with two cushioned chairs near the wrought-iron enclosure provided the perfect setting for an early-morning breakfast or a late-night dinner.\n\nMerrick had checked into a room at the Four Seasons Hotel containing all the amenities for work and relaxation. The spacious room was furnished with a king-size bed, desk, love seat, coffee table, flat-screen television, private bar and wireless high-speed Internet access. The full marble bathroom featured a garden tub, separate shower and water closet.\n\nShe felt the heat from his body as he came up behind her. Her slender hands covered the arms wrapped around her waist. \"This place is spectacular. Do you always stay in upscale hotels?\"\n\nMerrick kissed the side of her neck. \"Yes. There was a time when all I could afford was seedy motels. The one time I woke with bedbug bites I vowed to sleep in my car rather than check in to another flophouse.\"\n\nTurning slightly, she stared at him over her shoulder. \"How long are you staying?\"\n\n\"I'm checking out Monday.\"\n\nMerrick wanted to tell Alex that he was scheduled to meet with a former supervisor at CIA headquarters in Langley, Virginia, to discuss the possibility of his returning to the Agency. He'd spent two years as a virtual recluse, but now he looked forward to reconnecting with the Agency and those whom at one time he'd regarded as his extended family.\n\n\"That gives us three days together.\"\n\n\"Four nights and three days,\" he corrected.\n\nClosing her eyes, Alex leaned against the solid wall of his chest. She was certain Merrick felt the flutters from her heart, her tightening stomach muscles. \"I'm going out Saturday night, but I don't know whether you'd want to come with me.\"\n\n\"Am I invited?\"\n\nShe opened her eyes as a mischievous grin twisted her mouth. \"You are if you don't mind viewing an art exhibit.\"\n\n\"Ali,\" he groaned. \"What's up with you staring at paintings of dead people?\"\n\n\"Are you going with me? If not, then I'll ask someone else.\"\n\nMerrick's hands went to her flat belly. \"That's not going to happen.\"\n\n\"What are you talking about?\"\n\n\"I don't intend to share you with other men.\"\n\n\"Don't tell me you're jealous,\" she teased.\n\nSplaying his fingers over her belly, Merrick pulled her closer. \"No, Ali. I'm crazy.\"\n\nAlex went completely still, her face clouding with uneasiness. His cold tone set alarm bells clanging within her, and she was suddenly anxious to put some distance between her and the man holding her to his heart. A warning voice whispered in her head that Merrick Grayslake was dangerous, that he would be a formidable opponent when confronted or crossed, that as a former CIA operative he'd probably committed unspeakable acts.\n\n\"Does this mean you're coming with me?\"\n\n\"Yes, it does.\"\n\nMerrick's fingers inched down Alex's blouse, easing the hem from the waistband of her slacks. He heard her soft inhalation of breath as he searched lower, under the delicate material of her panties. This time she gasped aloud when his hand cupped her mound. A knowing smile parted his lips as moisture bathed his fingertips.\n\nAlex was ready for him, and he'd been ready for her. Unknowingly he had been the first time he saw her arguing with a man who hadn't wanted to let her go. Now he understood why the frustrated man had taken the risk and crashed a private event to see her.\n\nMerrick found Alexandra Cole an anomaly. Assertive and confident, she spoke her mind. He'd come to admire her tenacity, her ambition and her friendly outgoing personality. But she was also hot-tempered and stubborn, negative psychological characteristics that did not detract from her overall strengths.\n\nShe wanted to complete her graduate studies, and he'd promised to support her during this endeavor. However, like most women, she wanted marriage and children\u2014two things he was unable to offer her at this time in his life. What he would offer Alex was his love and protection. These were more than he'd been willing to give the women in his past.\n\nHis free arm tightened around her waist as he lifted her and backed off the balcony and reentered the bedroom. His hand still sandwiched between her thighs, Merrick sat down on the bed, bringing Alex to sit on his lap.\n\nHe pressed his mouth to the nape of her neck. \"Are you ready to take siesta with me?\"\n\nAlex couldn't respond because she'd bitten down on her lower lip when the slight flutters grew stronger as Merrick's fingers worked their magic. She nodded like someone under the effects of a powerful narcotic.\n\n\"Yes-s-s.\" Her affirmation came out slurred as a long-forgotten passion rose quickly to engulf her in flames that refused to burn out.\n\nClosing her eyes, Alex let her senses take over. It was as if she were viewing what was to take place between her and Merrick as a spectator rather than a participant. His fingers were deft, unfaltering when he released the tiny buttons on her blouse from their fastenings. She heard his breathing deepen as he viewed her breasts in a sheer white lace bra; his breathing became more labored with each article of clothing he removed.\n\nShe opened her eyes to find him kneeling over her, eyes shimmering like streaks of lightning, the nostrils of his aquiline nose flaring from the effort it took to control his respiration. No man had ever stared at her naked body with an expression mirroring awe and reverence. The heat pulsing between her legs spread throughout her body.\n\nMerrick did not want to believe that clothes had artfully concealed a petite body so lushly curved that he feared spilling his passion before he joined his body with Alex's. High, firm breasts he'd only glimpsed on New Year's Eve were ardently displayed as they rose and fell over a narrow rib cage flaring out to a pair of rounded hips. Everything about her\u2014the rich brown color that was evenly distributed over her body, the large gold-brown eyes that stared trustingly at him and the black shiny curls that were as wayward as the woman who claimed them\u2014held him enthralled in a labyrinth of love and desire from which he never wanted to escape.\n\nMoving off the bed, he removed his jacket and shirt, leaving them on the padded bench at the foot of the bed. His shoes, socks, slacks and underwear followed until he was completely naked. His footsteps were silent on the carpeted floor as he crossed the room, entered the bathroom and removed a condom from a small toiletry bag. He returned to the bedroom and slipped into bed with Alex, who'd turned on her side to face the window.\n\n\"Can you please close the window and pull the drapes?\" Alex doubted whether the hotel guests across the courtyard could see into the room, but she'd never been one for public displays of affection.\n\nMerrick kissed her shoulder. \"Sure.\"\n\nHer gaze narrowed when she saw the thin, pale scar running along the left side of his back. He'd told her that he had retired from the CIA, and she wondered if it was an injury that had ended his career, forcing him into early retirement.\n\nExtending her arms, she smiled as Merrick made his way back to the bed. There was enough light coming through the opening in the patterned peach-and-gold draperies to make out his face, but not the other small scars on his body.\n\n\"Veo las huellas de mi amor, querido,\" she whispered as he took her into his arms.\n\nMerrick wanted to tell Alex that she didn't have to ask him to make love to her. Alexandra Cole had bewitched him and unknowingly she possessed what it took to destroy him emotionally.\n\nSupporting his weight on his elbows, he lowered his head and kissed her. Not a soul-searching, deep passionate kiss, but one better suited for a fragile newborn. He kissed her mouth, the hollow of her scented throat, the curve of her shoulders, venturing lower to her breasts. The flesh between his legs grew heavy, but he didn't, couldn't stop until he explored every inch of Alex's satiny skin. She smelled of fruit, flowers and spice.\n\nRachel Singletary had teased him because he'd opted to remain celibate, but she didn't know that he'd been waiting for the right woman. And he knew Alex was the right woman the first time they'd shared a dance, a kiss. Moving down the length of the large bed, he alternated kissing with flicking his tongue over her belly.\n\nAlex's fingers curled into tight fists at her sides as she clenched her teeth. Moans of frustration mingled with the rising desire she was helpless to control. The mat of crisp dark hair covering Merrick's chest and his marauding mouth threatened to take her beyond herself. She wanted to kiss him the way he was kissing her, her lips and tongue tasting flesh in her journey to communicate wordlessly how much she loved and needed him.\n\nHer breasts rose and fell heavily as the pulsing at the apex of her thighs increased. \"Merrick!\" Her hysteria was evident when she screamed his name. Reaching for his head, she attempted to capture his hair, but it was too short for her to secure a grip. He'd spread her legs and his hot breath had become a raging inferno that threatened to incinerate her\u2014body and soul.\n\nMerrick heard the desperate cry, inhaled her rising passion, knowing if he didn't take her now it would be over. Reversing his position, he reached for the condom, sat back on his heels and slipped it over his rigid flesh. It was for a brief moment, but he thought he saw fear and indecision in Alex's eyes as she watched him put on the latex sheath that would prevent an unplanned pregnancy. What he thought he'd glimpsed vanished as she extended her arms, welcoming him into her embrace and into her body.\n\nReaching for her hand, Merrick guided it to his sex. \"Let's do this together.\"\n\nAlex's tiny hand barely fit around his engorged tumescence as she positioned it to the entrance of her femininity. Merrick's hand covered hers as he eased himself into her body. Flames of desire rose quickly, overlapping her soft moans as he pushed into her; taut flesh stretched with each inch until he was fully sheathed. Then, in a rhythm that was as timeless as the beginning of creation, they offered each other all of themselves, holding nothing back.\n\nMerrick established a slow, deliberate thrusting as Alex rose to meet him, pinpoints of gold in her eyes darkening with her rising ardor. Anchoring his hands under her hips, he held her fast, increasing the cadence, his hips moving with the velocity and power of a jackhammer. Her gasps and soft moans served to make the fire in his loins more intense.\n\nHe lost himself in the moment and the love flowing from the woman under him, not feeling the bite of her fingernails on his back. What he was aware of was the gurgling sounds coming from her throat that erupted into an unrestrained primal scream that sent him over a precipice wherein he surrendered all he had and was to the woman whom he knew he would love forever. They climaxed simultaneously, experiencing a free fall that made them one with each other.\n\nThey lay entwined, breathing heavily, waiting for their respiration to slow while enjoying the lingering pulsing sensations. Sated, they were loath to move. However, Merrick was forced to when he withdrew from Alex and went to discard the condom.\n\nWhen he returned to the bed, he found her asleep, her face still flushed with passion. He lay beside her and within minutes Morpheus claimed him as they shared siesta for the first time.\n\nAlex sat at a dressing table applying a coat of orange color to her lips. She and Merrick were scheduled to leave for Professor Rivera's soiree in fifteen minutes.\n\nShe hadn't made it a habit to wear makeup to the art institute, so she was certain the added color to her face and the dress would shock a few of her classmates. She'd decided to wear a minishift in a soft tangerine-orange with white orchids, reminiscent of a Japanese kimono with an empire waist and a halter top.\n\nShe'd come to the Four Seasons Hotel Thursday afternoon and left the hotel once to go to a nearby boutique and hair salon; she had her hair cut into a short and very becoming feathery style that flattered her round face.\n\nShe and Merrick shared brunch on the balcony outside their room that overlooked the courtyard and dined on exquisite Mediterranean cuisine at the hotel's elegant restaurant, Reforma 500. Earlier that afternoon she'd taken advantage of the services at the hotel's spa.\n\nWhen she and Merrick weren't in bed sleeping or making love, she pulled out her books and studied for the upcoming exam. He'd surprised her when he offered to test her on her knowledge of the Mayans and their incredibly complex advanced society.\n\nShe'd asked him about the scar on his back and his response had been, \"Oh, that. It was an accident.\" She didn't believe his glib explanation but decided not to pry. Their relationship was much too new to coerce each other to divulge what they deemed personal. And not once did Alex forget that what she shared with Merrick was tenuous at best. It was he who wanted a relationship that would go on for perpetuity, while she'd asked for right now.\n\nIt was ironic how she'd changed. At twenty-two she wanted to marry her art history instructor, and now at twenty-nine she was willing to become a participant in a physical relationship that wouldn't result in a commitment.\n\nMovement behind her caught her attention. She smiled when Merrick leaned over to press a kiss on the nape of her neck. He'd elected to wear a charcoal-gray suit in a tropical linen fabric with a pale gray finely woven shirt with a banded collar.\n\n\"You look and smell wonderful.\" She wore a perfume with sensual musk and woodsy notes.\n\nHer smile widened. \"Gracias, mi querido.\"\n\nHe straightened, unable to believe Alex could improve on perfection. Her short hairstyle and youthful-looking face made him feel as if he were robbing the cradle, when there was only a six-year difference in their ages.\n\nShe rose gracefully from the stool and reached for his hand. \"I'm ready.\"\n\nHis gaze lingered on her smoky lids and lush mouth with the glistening lip color. There was a mysterious gleam in her eyes that made him want to strip her naked and take her back to bed. But he knew that wasn't possible because attendance at her professor's house was mandatory.\n\nMerrick thought that after making love to her once he'd be able to rid himself of strong physical urges that came when he least expected. However, having her within arm's reach hadn't permitted him to exorcise the licentious images of what he wanted to do with Alexandra Cole.\n\nWhat had shocked Merrick was that her libido matched his, and he wanted to demonstrate other positions and techniques, but decided to wait for another time; perhaps when he returned to Mexico she would feel more comfortable with other than traditional lovemaking.\n\nHolding her hand, he led her out of the bathroom. She gathered her purse and a silk shawl in a pale shade that matched her silk-covered sling backs. They took the curving staircase instead of the elevator to the opulent lobby, not seeing the surreptitious glances directed at them.\n\nThe taxi Merrick had requested had arrived. He helped Alex into the vehicle, then ducked his head and sat beside her. He gave the driver the address before he settled back to enjoy the ride to Coyoac\u00e1n and the delicate body beside him.\n\nHernando Rivera's home was in an area with a number of museums and art galleries and was within walking distance of the homes of several noted deceased inhabitants: Frida Kahlo, Diego Rivera and Russian revolutionary Leon Trotsky.\n\nAlex walked into a brightly-lit courtyard while Merrick lingered behind to pay their driver. Long tables groaned with food and a portable bar was doing a brisk business, as evidenced by the number of people milling around holding glasses of tequila cocktails. Mariachi music blaring from hidden speakers added to the festive mood.\n\nMoira came over to greet her. A black tank dress in a Lycra fabric hugged her lithe figure like a second skin while a pair of four-inch strappy sandals put her at the six-foot mark. Her pale hair was piled atop her head in tousled, sensual disarray.\n\n\"Where's your gorgeous friend?\" she whispered.\n\nAlex couldn't stop the wave of heat stealing across her face. \"He'll be here.\"\n\nMoira took a deep swallow of her drink. \"Now he's someone I could get my freak on with.\" She'd had one too many margaritas to see Alex's scowl. \"But I know he's off-limits, because I'd never make a play for a friend's man.\"\n\nForcing a smile that she definitely did not feel, Alex said, \"That's good to know.\"\n\n\"Se\u00f1orita Morris, I'm honored you've graced me with your presence.\"\n\nAlex turned to find her host grinning at her. He'd forgone his conservative black suits for a white poet's shirt with matching linen slacks and tan sandals. She didn't know why, but he reminded her of cartoons in which the cat had swallowed a bird. \"But, Professor Rivera, wasn't it you that said attendance is mandatory?\"\n\nHe waved a hand, his dark gaze moving slowly from her face to her bare brown legs and her feet in the light-colored heels, then back again to her feathery hairstyle. \"I always tell my students that, but there is no way I can enforce that rule. Field visits yes, my home no. Please come into the house to see the exhibit before you get something to eat and drink.\"\n\n\"Thank you, but I'd like to wait for my guest.\"\n\nHis black eyebrows lifted, reminding her of hash marks of birds in flight. \"You came with someone?\"\n\nAlex wanted to laugh when she saw his stunned expression. \"Yes, I did.\" The confirmation had barely left her lips when she detected the scent of Merrick's cologne. Turning, she smiled at him and extended her hand. \"Darling, I'd like for you to meet Professor Rivera, who teaches design and art of ancient cultures. Professor Rivera, Merrick Grayslake. Merrick, Professor Hernando Rivera.\" The two men exchanged polite greetings and handshakes.\n\nHernando, having recovered from seeing Alexandra Morris in a dress that revealed more flesh than he'd ever seen on her, shifted his gaze to the tall man with her. \"Grayslake,\" he said softly. \"Are you Yaqui?\"\n\nA muscle jumped in Merrick's jaw. \"I wouldn't know.\"\n\n\"You speak Mexican Spanish and your features are a blend of African, Yaqui and European. I suggest you explore your roots. I'm sure you didn't come to engage in an anthropological discussion. Come see the exhibit, then eat and drink.\"\n\nAlex felt the muscles in Merrick's arm bunch up under his jacket's sleeve. If her teacher had been able to identify Merrick's racial designation from looking at him, then there was no doubt he'd spoken the truth.\n\nTo say the professor was a brilliant art expert and anthropologist was an understatement. The first day of class he had everyone stand up and introduce themselves, and within minutes he'd identified their Spanish dialect. Alex had held her breath when he announced that based on her speech patterns and physical looks she was of African and Cuban ancestry. He'd astounded her with his accuracy, and in the following weeks Professor Rivera used his vast knowledge to subjugate and intimidate any student who attempted to challenge him.\n\nThose who came to class unprepared were forced to undergo a tongue-lashing that usually went on for at least ten minutes before they were asked to leave and not return until they'd completed the assignment. The trustees of the universidad permitted Professor Hernando Rivera to conduct his classes like a despot because museums and art institutes the world over were vying for his attention.\n\nThey were shown into a room in Hernando's house that doubled as an art gallery. Alex couldn't believe the artifacts on display targeted for donations to local museums. There were fragile wooden, leather and clay masks from Puebla, Guerrero and Michoacan, a funerary mask from Teotihuac\u00e1n and exquisite silver pieces from Oaxaca. The piece that captivated her was a Zapotec jade bat-god pendant. She thanked Hernando for permitting her to see the priceless artifacts, then returned to the courtyard with Merrick.\n\nSmiling up at him, she noted his solemn expression. \"It's over, darling.\"\n\nHe lifted his eyebrows as he met her amused gaze. \"It really wasn't that bad. I believe hanging out with you has given me a healthy respect for art.\"\n\n\"Does this mean you're going with me to the museum tomorrow?\"\n\n\"No, it doesn't. We have one more day together, and I don't intend to spend it in a museum.\"\n\nShe gave him a sidelong glance. \"What have you planned?\"\n\nA knowing smile softened his firm mouth. \"We're going to spend the day in bed.\"\n\nAlex squeezed his hand. \"You keep it up and I won't be able to walk.\"\n\nLeaning down, he pressed his mouth to her ear. \"I said nothing about making love, but if that's what you want I'm willing to accommodate you, because for some reason I can't get enough of you.\"\n\n\"Glot\u00f3n.\"\n\n\"No lie,\" he drawled, grinning. \"Now let's get something to eat and drink so that we can appear socially acceptable, then blow this joint.\"\n\nVertical lines appeared between her eyes. \"I'm not ready to go back to the hotel.\"\n\nMerrick gave her a long, penetrating look. \"Where do you want to go?\"\n\n\"The others are going to a club near the Zona Rosa. If you don't want to go, then I'll meet you back at the hotel.\"\n\n\"If you think I'm going to let you out of my sight looking like you do, then you're crazier than I am.\"\n\nShe glanced down at her dress. \"What's wrong with the way I look?\"\n\nPulling her closer, he dropped a kiss on her fragrant hair. \"There's nothing wrong, baby. You look delicious.\"\n\nShe gave him her trademark dimpled smile. \"Gracias, mi amor.\"\n\nMerrick returned her smile. \"De nada, mi querida.\"\n\n## Chapter 10\n\nMerrick sat in the corner of a noisy, crowded club with flashing colored lights watching Alex dance. He'd begun drinking beer, but after the second bottle he switched to water. At first he'd found the pumping baseline beats infectious, but after three hours of nonstop music and the differing scents of cologne and perfume on sweaty bodies he was ready to find the nearest exit.\n\nResting an elbow on the small table, he cradled his chin on his fist. He hadn't flown to Mexico to see Alex and then share her with people she saw every day. But seeing her laughing and dancing tempered his selfishness because it was he, not some other man, who would share her passion.\n\nMerrick had tried and failed to analyze what it was about Michael Kirkland's cousin that affected him wherein he was now willing to forsake his reclusive lifestyle to consider returning to the CIA as a bureaucrat. Instead of being briefed for covert missions he would become a trainer as an intelligence research training specialist. He'd had experience when he facilitated advanced training courses at Quantico for DEA agents. Yes, he would go back as an intelligence training specialist, secure his relationship with Alex and retire.\n\nThe driving rhythms slowed to a seductive ballad and Merrick stood up when he spied Alex coming in his direction. Reaching for her purse and shawl, he took several steps and curved an arm around her waist.\n\nHer dewy face shimmered like brown velvet under the psychedelic lights. \"Dance with me, Merrick.\"\n\nHe shook his head. \"Not now, querida. We'll dance together at the hotel.\"\n\nHer brow furrowed. \"What's the matter?\"\n\nPulling her closer, he shouldered his way through the throng on the dance floor. \"The matter is I was ready to leave two hours ago.\"\n\n\"Why didn't you say something?\"\n\n\"I didn't want to stop you from enjoying yourself, Ali.\"\n\n\"I enjoy myself even when I'm not dancing.\"\n\n\"What's going to happen once you start your career?\"\n\n\"That shouldn't change anything. I'm not going to be working 24\/7. Speaking of careers, I went online last week and applied for a position with the National Trust for Historic Preservation's northeast region. And if I'm hired, I'll be responsible for historic districts in New York, New Jersey, Vermont, Pennsylvania, Rhode Island, Connecticut, Delaware, Massachusetts and Maine.\"\n\nMerrick noticed she hadn't mentioned any state south of Delaware. \"Do you plan to relocate?\"\n\n\"No. I like living in the D.C. area, and I love my condo. There may be a problem when I have to do the fieldwork.\"\n\n\"What, Ali?\"\n\n\"There's a family mandate that I'm not permitted to take a commercial aircraft.\" She told him about how her uncle Martin's daughter had become a kidnapping victim forty years before, and since that time anyone who claimed Cole blood or married into the family was prohibited from flying on commercial carriers.\n\nMerrick wanted to tell Alex that making arrangements to travel up and down the eastern seaboard was the least of her worries. Hadn't she realized she was a wealthy woman, and what would've become prohibitive for the normal working-class person was readily available to her?\n\n\"If you're hired, then how are you going to travel?\"\n\n\"If I'm not able to secure a seat on the ColeDiz G4, then it will have to be a private jet. The money I spent on hiring private jets I could donate to my favorite charity.\"\n\nThey stepped out of the air-cooled club and into the unusually warm winter Mexico City heat that lingered even after the sun had set behind the mountains. All of the upscale shops were air-conditioned, but once anyone stepped outdoors onto the overpopulated streets with the smog and thermal inversion, many found breathing normally difficult.\n\nMerrick startled Alex when he whistled sharply through his teeth for a taxi. A late-model colorful car skidded to the curb. The driver stepped out, rounded his vehicle and opened the door with the arrogant aplomb of any major city taxi driver. She got in, followed by Merrick. The driver closed the door and then took his seat behind the wheel.\n\n\"Ca d\u00f3nde la, el se\u00f1or?\"\n\n\"The Four Seasons Hotel en el Paseo de la Reforma 500.\"\n\nThe driver took off in a burst of speed as Merrick settled against the leather seat, his arm going around Alex's shoulders. Minutes into the ride, she rested her head on his chest. Twenty-four hours. That was all they had left before he flew back to the States.\n\nAlex shifted into a more comfortable position between Merrick's legs as an unconscious moan slipped past her lips; the cramping in her lower belly and a dull ache in her lower back were indicators that she would see her period in the next day or two.\n\n\"What's the matter, baby?\" Merrick asked, his moist breath sweeping over her ear.\n\nShe moaned again. \"Premenstrual cramps.\"\n\nMerrick gently lifted the damp strands clinging to her scalp. They'd returned to the hotel and instead of Alex taking a bath and him a shower, she'd invited him to share the oversize tub.\n\n\"Is this the time of the month when you turn into a witch?\"\n\nPeering at him over her shoulder, Alex wrinkled her nose.\n\n\"Don't tell me you've got sexist jokes.\"\n\nMerrick's expression was one of unadulterated innocence. \"No.\"\n\n\"Have you had a lot of experience with women who become viragos when they're PMSing?\"\n\n\"Even if I have I'm not going to tell you.\"\n\nAlex shifted until she straddled Merrick, the pulsing jets from the Jacuzzi adding to her buoyancy. Leaning forward, she pressed her breasts to his hair-matted chest. \"Why won't you tell me?\"\n\nHe stared at her under lowered lids. \"I don't kiss and tell, Ali. The women in my past are just that\u2014the past.\" He kissed the end of her nose. \"That is a topic I will not discuss.\"\n\nAlex affected a pout. \"I told you about the men from my past.\"\n\n\"You told me not because I asked but because you wanted to. You know who and what you are, and because of that you're open and spontaneous, while I've lived my life speculating whether I'm African or Native American, and wondering if I father children whether they'll inherit a gene abnormality from an ancestor.\"\n\n\"What do you say you are?\"\n\nA wry smile touched his mouth. \"I say African-American because unconsciously that's what I feel. And if my biological father is European or Native American, then I assume he'd slept with a black woman. Or it could be vice versa. You have a large family connected by blood and marriage, and because I don't know my parents I'll never know if I have a brother or sister, nieces or nephews.\"\n\nBurying her face between his neck and shoulder, Alex closed her eyes. She felt the strong, steady pumping of his heart against her breasts. \"Have you thought of trying to find your birth mother? There are Web sites and agencies set up to reconnect children with their birth parents. It shouldn't be too difficult for you because you know your mother's name.\"\n\n\"I did check a database for Virginia Gray or Grayslake and came up with more than nine thousand nationwide.\"\n\nAlex opened her eyes and pulled back, meeting his intense stare. \"Did you contact any of them?\"\n\nHe nodded. \"After the first thirty I decided to let it go. Each time I got a 'No, I'm not your mother' I found myself overwhelmed with feelings of abandonment. It'd begun to affect my job performance, so I decided to let it go. Uncovering who my mother is or was is no longer a priority.\"\n\nLooping her arms under Merrick's shoulders, Alex brushed a kiss over his mouth, feeling his pain as surely as if it was her own. It didn't matter who he was because he was the man with whom she'd fallen in love; he was the man she would love forever.\n\n\"I love you.\"\n\nMerrick went completely still, certain Alex could feel the blood rushing through his veins. He was hot, then cold, confused, then clear-headed as her confession filled him with a sense of power, a strength that vanquished the lingering pain of abandonment that had become an emotional impediment to marriage and fatherhood.\n\n\"Let's get out of this tub,\" he said, recovering quickly.\n\nMerrick anchored a hand along the shelf of the tub and came to his feet, bringing Alex up with him. He stepped out and reached for a bath sheet on a nearby table. Wrapping the velour fabric around her body, he lifted her gently from the bathtub and carried her into the bedroom, placing her on the bed.\n\nNone of his movements were rushed as he blotted the moisture from her satiny body, lingering over the curve of her breasts and hips. It no longer mattered that he would leave Mexico and Alex in a little more than twenty-four hours, because he had the rest of his life to share whatever he had and whatever he'd become with her.\n\nTime seemed to slow down for Alex as she languished in the gentleness of her lover's touch. He dried the front of her body, turned her over and then repeated the action.\n\nThen, without warning, his mouth replaced the towel, tracing a sensual path down the length of her spine, lingering at the indentation separating the globes of flesh defining her buttocks.\n\nHeat, chills and undulating waves of ecstasy swept over her as she tried, and failed, to keep the moans from escaping her parted lips. His hands searched areas known only to her, his mouth exploring, and tantalizing, searching, savoring every hollow, dip and curve of her from head to toe.\n\nMerrick couldn't get enough of Alex. His hands and mouth mapped every inch of flesh he could see and reach. Her smell, her essence, lingered in his nostrils and on his tongue. His passions rose quickly, and he released her, pausing to slip on a condom. It was as if time had stood still until he looped an arm around her waist; she knelt with her back to him as he pressed his groin to her buttocks. In one, sure motion, he eased his erection into her, both gasping from the unexpected joining of flesh against flesh, man against woman.\n\nMerrick cupped her breasts, alternating squeezing with rubbing his thumbs over the pebble hardness of her distended nipples. The kneeling position allowed him deeper penetration, maximum pleasure. The sensual beauty of her naked limbs, her feminine fragrance mingling with the rising scent of their lovemaking threatened to take him over the edge where it would be over much too soon.\n\nWithdrawing, he reversed their positions. Merrick lay on his back, arms anchored over his head, as Alex straddled him. He knew he would never get used to her sensual expression in the throes of lovemaking: flushed skin, dilated pupils, parted lips, flaring nostrils and heaving breasts. And her bubbly spontaneity out of bed became an uninhibited smoldering passion that stripped him raw wherein he was unable to hold anything back. Alexandra Cole claimed what he'd been unable to give anyone\u2014all of himself.\n\nAlex's gold-brown gaze met silver-gray. Her gaze inched down to the firm muscles under the brown arms in the diffused lamplight. Tufts of straight reddish hair grew out from his armpits, a lighter shade than the crisp, curling strands covering his chest that tapered down to a narrow line that spread out in an inverted triangle to coarser, tighter curls at his loins. Everything about her lover was a visual banquet. Merrick's deep drawling voice, powerful masculine presence and compelling, magnetic eyes held her captive in an abyss of loving and longing she never wanted to end.\n\nAnchoring her palms on his shoulders, her gaze meeting and fusing with his, she lowered her body over his rigid flesh, making them one with each other. Simultaneous audible moans of satisfaction matched the rush of moisture bathing the pulsing area between her thighs.\n\nThe world as Alex knew it ceased to exist. All that mattered was the man offering her the most exquisite passion she'd ever known. What had begun as friendship was now a full-blown love affair, and she realized what she'd felt and shared with her art professor had not even come close to what she felt with Merrick. The pulsing grew stronger, becoming contractions that gripped his hardness before releasing it, then began again.\n\nHeat rippled under Merrick's skin, became hotter, more intense each time he thrust upward. They had ceased to exist as individuals, separate entities. He was so attuned to the woman rising and falling over his throbbing erection that he felt the spasms that seized her, followed by a primordial scream that made hair stand up on the nape of his neck. Before the next spasm and Alex's next outburst of ecstasy, he had her on her back, feet anchored on his shoulders. He didn't want the pleasure to end\u2014not yet. He wanted it to go on\u2014forever, if possible. But it was not possible when he lowered his head, closed his eyes and enjoyed the most extreme physical gratification he'd ever experienced.\n\nWhen it was all over he lowered her feet, then collapsed heavily on her smaller frame, waiting for his heart rate to return to a normal rhythm.\n\nHe loved her. He loved her more than anyone or anything.\n\nAlex struggled to catch her breath as the pressure from Merrick's greater weight prevented her from moving. \"What are you doing to me?\" she whispered.\n\nBreathing heavily, Merrick rolled off Alex and pulled her to his chest. \"Loving you, baby. Loving you, and your life.\"\n\n## Chapter 11\n\nCIA Headquarters...Langley, Virginia\n\nA rare smile deepened the network of lines crisscrossing William Reid's weather-beaten face within seconds of the door closing behind a man who he'd believed he would never see again. His smile faded as one of the three telephones on his desk rang; reaching over, he picked up the receiver.\n\n\"Reid.\"\n\n\"Come to my office.\" The command was sharp, the caller's tone brusque, no-nonsense.\n\n\"I'm on my way.\"\n\nPushing back his chair and coming to his feet, he put on his suit jacket, left the office and made his way to one several doors away.\n\n\"He's expecting you,\" said the dour-faced woman who guarded her boss's office like a Secret Service agent assigned to the presidential detail.\n\nCarl Ashleigh stood by the door to his inner office. It wasn't often William saw the man wear a suit jacket unless he was scheduled to meet with the director.\n\n\"Let's go for a drive.\"\n\nWilliam, or Bill to the few friends he'd acquired since coming to the CIA from the FBI, followed his supervisor like an obedient puppy. It wasn't until they were seated in Ashleigh's gas-guzzling Yukon, maneuvering toward McLean, Virginia, that Ashleigh initiated conversation. He'd become used to the eccentric younger man who would order him to his office, then make him wait before granting him an audience.\n\n\"Is he coming back?\"\n\nWilliam stared out the side window at the passing countryside. It was the first week in March, and winter appeared to have finally loosened its grip on northern Virginia.\n\n\"I assume you're talking about Grayslake.\"\n\nHigh color suffused Carl's face, his pale blue eyes standing out in stark contrast. \"Who the hell else do you think I'm talking about? Did he not spend the past two hours in your office?\"\n\n\"Yes.\"\n\n\"What did he want?\"\n\n\"He wants back in.\"\n\n\"What did you tell him?\" Carl asked.\n\n\"I told him that he would have to go through the same protocol as if he were a new hire. He'd have to pass a physical, a psychological and of course obtain security clearance.\"\n\n\"What's he asking for?\"\n\n\"Training.\"\n\nCarl slowed the Yukon, turning onto a local road and coming to a complete stop behind a copse of pine trees; he left the engine running. Releasing his seat belt, he shifted on his seat and stared at William Reid. The man was only fifty-six but Carl thought of him as a dinosaur, a holdover from a past era with his military crew cut, cuffed trousers and wing-tipped footwear. William had given the Bureau fifteen years of fieldwork before transferring to the CIA as a desk jockey.\n\n\"Slow down his paperwork.\"\n\n\"What!\" The single word exploded from William Reid's mouth.\n\nThe fingers of Carl's left hand drummed nervously on the steering wheel. \"I want you to keep him on hold for a while.\"\n\nWilliam ran a hand over hair that looked like the bristles on a stiff brush. \"How long is awhile?\"\n\n\"Six months. Assign him to sniper training. With his background as a Marine Corps scout he'll do well there until we're ready for him.\"\n\nWilliam gave Carl a long, penetrating stare. \"What's up?\"\n\nCarl's pale blue eyes narrowed, and he wondered how much he could tell his assistant without compromising the details of a joint meeting of agencies that included the FBI and the CIA. Biweekly meetings between the two directors, corresponding assistant directors and federal prosecutors had become akin to a world economic summit.\n\n\"We may need him for a special assignment.\"\n\n\"It can't be a field assignment,\" William insisted.\n\n\"I know that!\" Carl said angrily.\n\nWilliam did not visibly react to his supervisor's outburst. \"Grayslake is going to become suspicious if we take too long to approve his rehire or if he's assigned to something he hasn't requested.\"\n\nAshleigh stared out the windshield. \"He wants back in, and that means he'll accept whatever we offer him. And if he makes it known that he's unhappy, then bust him down to a file clerk.\"\n\nCarl hadn't been cleared to brief his assistant on Operation Backslap, but with Merrick Grayslake's possible return the initial strategy would have to be modified. Once Grayslake was given security clearance Reid would be briefed on an investigation certain to send elected officials scrambling to hire the best defense attorneys not only to salvage their political careers but to avoid going to prison.\n\nWilliam shook his head. \"He's not going to go for that. If we lose him a second time, you and I know that he's never coming back.\"\n\n\"If you have a better plan, then you'd better tell me right here, right now.\"\n\nThe older man stared at his supervisor. \"You know I don't have one.\"\n\nCarl snapped his seat belt, put the vehicle in gear and maneuvered back onto the road back to Langley. The return drive was accomplished in complete silence, and when the two men returned to their offices both knew someone higher than them had determined Merrick Grayslake's destiny the moment his written request to return to the Company was received.\n\nMerrick scrolled through his cell-phone directory and clicked on a name. \"I'm in your neck of the woods,\" he said when hearing the deep-voiced greeting. \"Can you spare me a few minutes?\"\n\n\"Sure. What's up, Gray?\"\n\n\"I'll tell you when I see you.\"\n\n\"I'll be here.\"\n\nMerrick disconnected the call and shifted into a higher gear. Forty-five minutes later he parked his vehicle two blocks from Michael and Jolene Kirkland's house.\n\nMaking his way along the tree-lined streets in the D.C. suburb with opulent and historic colonial and Georgian-style homes, he recalled his meeting with Bill Reid. Merrick had been forthcoming when he revealed that he was more than ready to return to the CIA as a training specialist.\n\nTurning a corner, he walked a block with only six structures in the charming cul-de-sac, most large, imposing, claiming spacious front lawns and, in the warmer weather, flowering shrubs. Michael's house stood apart from the others. Its simplicity was a Frank Lloyd Wright Japanese-inspired design. A broad sheltering roof with generous overhanging eaves and windows set with colorful geometric shapes radiated warmth, beckoning him closer. A waist-high slate wall was covered with a profusion of climbing vines. A gate made of iron pipe painted a Cherokee-red stood open, welcoming him.\n\nMerrick strolled up the path, but as he walked up the six steps to the front door it opened and Michael Kirkland stood in the doorway, casually dressed in jeans and a black T-shirt.\n\nHe extended his hand, pulling Michael close in a strong embrace. \"How's it going, Kirk?\"\n\nMichael Kirkland was an imposing figure. His exposed muscled arms were brown as berries. Tall, broad-shouldered and slim-hipped, former U.S. Army captain Michael Kirkland was as physically fit as he'd been when he graduated from the U.S. Military Academy at West Point a decade before. His face was as remarkable as the rest of him: close-cropped raven-black hair, clear green eyes that shimmered like precious gems in a sun-browned face. Men and women were drawn to his perfectly symmetrical features that were as delicate as a woman's.\n\nMerrick's friend had resigned his commission and now taught classes in military law and military history at a private military school in northern Virginia. Merrick knew Michael's decision hadn't been an easy one, but now as a husband and prospective father he appeared content with the turn his life had taken.\n\nMichael returned the rough embrace. \"How long are you staying?\"\n\nPulling back, Merrick shook his head. \"I'm not. I plan to drive back to Bolivar tonight.\"\n\nLifting his sweeping black eyebrows, Michael stepped aside and beckoned Merrick into his home. \"Come in. Don't tell me you've been holding out on me.\" Merrick walked into the living room of the two-story converted carriage house decorated with Asian and Southwest\u2013themed furnishings.\n\nMichael Kirkland stared at his friend, unable to believe the transformation. When he'd reconnected with Merrick six months ago, he'd found him long-haired, bearded and gaunt-looking. Not only had Merrick put on weight, but he always could've easily passed for a D.C. businessman or politician in his tailored suit, imported footwear and conservative haircut.\n\nMerrick flashed a smile, not replying to Michael's accusation that he was hiding something from him. He followed Michael into a room that was an exact replica of a Japanese teahouse. Octagonal in shape, its walls were made entirely of screened-in glass windows. Two of the eight sides were open to take advantage of the crisp air. A low lacquered table, surrounded by large black and jade-green floor cushions, were set up in the middle of the room.\n\nMerrick owed Michael his life from when he found him lying facedown in the street, bleeding from a gunshot wound to the abdomen. Doctors had repaired his spleen but were unable to save his left kidney. The police recorded the incident as a mugging, but Merrick knew he'd been set up, though he was unable to prove it.\n\n\"Where's Jolene?\" he asked Michael.\n\nMichael folded his tall frame down to the futon, motioning for Merrick to do the same. \"She flew to Chicago to see her parents. And because she's at the end of her second trimester she said this will be her last trip until the baby comes.\"\n\nStretching out his legs, Merrick stared at an indoor pool built into a corner. The gurgling water created a soothing mood, trickling over rocks and pebbles concealed by a profusion of bamboo shoots and water lilies.\n\n\"Do you know what you're having?\"\n\nMichael shook his head. \"No. I'd like to know, but Jolene wants to wait.\"\n\n\"Does this mean you do everything your wife wants?\" Merrick teased.\n\n\"Hell, yeah,\" Michael said, smiling, \"but within reason of course. I know enough when to advance and when to retreat.\"\n\n\"Marriage shouldn't be a military campaign, Michael.\"\n\nThe brilliant green eyes in a rich brown face glowed. \"With Jolene and me there's never a military campaign. I give in to her because she's carrying my baby. Speaking of women, what's up with you, Gray? Do you have a little honey hidden away in your mountain retreat that has you running back to spend the night?\"\n\nStanding, Merrick removed his suit jacket, placing it over the arm of the futon and wondering how Michael would react if he told him that he was dating and sleeping with his first cousin.\n\nWould he approve?\n\nDisapprove?\n\nConcluding his relationship with Alex was much too new and much too fragile, he decided not to say anything. \"No,\" he answered truthfully. His honey, as Michael called her, was currently touring an archaeological site in Mexico's Yucat\u00e1n Peninsula.\n\nHe missed her, missed her more than he thought he would miss a woman. And the missing had nothing to do with making love to her; that was something he could do with any woman. But Alexandra Cole had become much more to him than any woman.\n\nWaking up with her beside him, their sharing meals together, taking long walks, discussing any- and everything were things he'd never experienced with a woman. Alex's spontaneity, her outgoing personality and her lust for life were refreshing changes because he tended to take life much too seriously.\n\n\"No, I don't,\" Merrick said, answering Michael's query. \"But it is something I am considering.\"\n\nLeaning back and crossing his arms over his chest, Michael gave Merrick a long, penetrating look. \"Who is she?\"\n\n\"I'd rather not say at this time.\"\n\nRespecting his friend's right to privacy, Michael nodded. One thing he knew about Merrick Grayslake, and that was he was a very private person, and if he'd felt comfortable revealing the identity of the woman with whom he was involved, he wouldn't have hesitated telling him.\n\nMichael glanced at his watch. It was after three. \"Can you hang out long enough to have dinner with me?\"\n\nMerrick flashed a rare smile. \"Sure.\"\n\n\"Do you want to go out or would you prefer me to throw something together?\"\n\n\"We can eat here.\" Whatever Michael threw together was certain to equal or surpass any restaurant meal. Alex had revealed that it was a family tradition that all the men were taught to cook. The rationale was they would never have to depend on a woman to eat.\n\nMichael pushed to his feet. \"I'll take out a couple of steaks to defrost. When I come back, you can tell me why you left the mountains to talk to me. Do you want anything to drink while I'm up?\"\n\n\"No, thanks.\"\n\n\"I'm going back to the Company,\" Merrick announced when Michael returned, his voice completely void of emotion.\n\nMichael went completely still as an expression of shock froze his features. \"You're kidding, aren't you?\"\n\n\"No, I'm not.\"\n\n\"But why, Gray? I remember you telling me that you'd die a slow death if you had to sit behind a desk.\"\n\nMerrick stared at the water spilling into the pool. \"I'm dying a slow death living in Bolivar. I can go weeks without seeing or talking to another human being. I hadn't realized it, but you saved my life a second time when you asked me to protect Jolene, because I was on a fast track to self-destruction.\"\n\nLooping one denim-covered leg over the opposite knee, Michael nodded slowly. \"Can the church say amen?\"\n\nMerrick smiled and said, \"Amen.\"\n\n\"To be truthful, Gray, you looked like the walking dead.\"\n\n\"Damn. Was I that bad?\"\n\nMichael smiled and said, \"Hell, yeah.\" He sobered quickly. \"Do you think you'll be able to adjust to not being in the field?\"\n\n\"I don't have a choice. I gave the Corps ten years and the Company four, so my options are limited.\"\n\n\"They're not as limited as you think. You took advantage of your stint in the Corps to advance your education, and with your intelligence and Scout Sniper School training I could recommend you for a position at Leesburg Military.\"\n\nVertical lines appeared between Merrick's eyes. \"Why the hard sell, Kirk?\"\n\n\"I've been where you are now,\" Michael said. \"I was mad as hell when I was transferred from intelligence to desk duty at the Pentagon, but during the four years, I used the transfer to my advantage when I went to law school. And when General Harry Cooper became more aggressive with his sexual advances, I knew I had to get out or forfeit my freedom for killing that little toad.\n\n\"I hated giving up my military career because it was all I knew. Twelve years of military school, West Point and then military intelligence. I'd hoped to become a lifer, but after I fell in love with Jolene I never regretted my decision to become a civilian. I teach courses I know and love, and the bonus is that I get to come home every night to my beautiful wife.\"\n\n\"What aren't you telling me?\" Merrick asked, his expression a mask of stone.\n\nMichael leaned closer. \"If you have someone\u2014a woman\u2014then you can't think only of yourself, because even if you're not married to her you're still a couple.\"\n\n\"Come on, Michael. Spill it!\"\n\n\"Even though you're going back it won't be as a civilian. You were trained as an operative, and you'll always be an operative.\"\n\nMerrick's frown deepened. \"You're talking in circles, friend.\"\n\nMichael wanted to tell Merrick that the CIA would take him back if he had one arm and one leg because of his superior intelligence, an intelligence Merrick took for granted.\n\n\"Don't be surprised if you're called to sit in on a few Black Op meetings. Although you won't become an active participant, just being privy to what's going on in the strategy room will put you and your lady at risk.\"\n\n\"Are you trying to discourage me?\"\n\n\"No, Gray. I just want you to know what you may have to face.\"\n\n\"I'll take your warning under advisement. I'll let...\" Merrick's words trailed off when his cell phone rang. \"Excuse me,\" he said to Michael as he reached for his jacket and the phone in the breast pocket. Alex's number came up on the display. Pressing a button, he put the tiny instrument to his ear. \"Please hold on.\"\n\nMichael noticed Merrick's expression softening as a gentle smile spread over his face. \"I'm going to get something to drink,\" he announced, excusing himself.\n\nWaiting until he was alone, Merrick said, \"Hey, baby. How are you?\"\n\n\"I'd be a whole lot better if I could see my boyfriend.\"\n\n\"Where are you?\"\n\n\"I'm back in Mexico City.\"\n\n\"How would you like company this weekend?\"\n\nHer husky laugh came through the earpiece. \"I was just going to ask you the same thing. Merrick?\" she asked when he didn't respond.\n\n\"Yes, baby.\"\n\n\"I've made arrangements to come to the States. I'm flying into IAD.\"\n\n\"I'll meet you.\"\n\n\"No, Merrick. I've arranged for a driver to take me to Bolivar.\"\n\n\"When are you leaving?\"\n\n\"Tomorrow morning.\"\n\n\"What time are you scheduled to touch down?\"\n\n\"I was told around ten in the morning. I don't have to be back for a week.\"\n\nMerrick could not stop the smile spreading across his face. \"That's good. Ali?\"\n\n\"What is it, Merrick?\"\n\n\"I love you.\"\n\nThere came a swollen silence. \"And I love you, too. Please hang up, Merrick, before I start bawling.\"\n\n\"I love you, I love you, I love you, Alexandra,\" he whispered, then disconnected the call.\n\nFor a brief second, the time it took to blink his eye, Merrick wanted what Michael had\u2014marriage and impending fatherhood. The moment of madness passed when he remembered who he was and Alex's declaration that she wasn't ready for marriage and children.\n\nHe'd existed in a state of suspended animation for two years, and then without warning Michael had come to his sanctuary to ask for his help. Leaving Bolivar and what had become his beloved woods and mountains forced Merrick to step outside himself to see how self-destructive he'd become.\n\nHe went for long walks, slept no more than three hours at a time, stopped cutting his hair and had grown a beard. He'd lost more than thirty pounds because there were days when one blended into the next and he couldn't remember when last he'd eaten.\n\nDespite Michael's misgivings, he would go back to the CIA, rent an apartment in D.C., see Alex whenever possible and return to Bolivar to unwind on weekends.\n\n## Chapter 12\n\nAlex stared out the window as her driver slowed down to less than ten miles an hour around the winding road. Biting down on her lower lip, she closed her eyes as the flutters in her belly increased, silently cursing Merrick for living in the middle of nowhere.\n\nShe'd berated herself over and over when she called Merrick to tell him that she was coming to visit him. She was going to spend the equivalent of an American spring break with a man who'd gotten under her skin like an annoying itch. She, Alexandra Cole, who'd earned a reputation of dating a different man every night, had found herself ensnared in a sensual trap from which there was no escape\u2014and it wasn't that she wanted to escape.\n\nThere was still something about Merrick that frightened her and kept her off balance, yet it wasn't enough to stop her wanting to be with him. She'd also attempted to rationalize why she'd fallen in love with the enigmatic man, and wasn't able to come up with a plausible explanation except that Merrick Grayslake allowed Alexandra Cole to be Alexandra. Not once during the three hours they'd spent at the club in the Zona Rosa had he exhibited a modicum of jealousy when she'd danced with the other men, or attempted to flaunt his machismo with a display of control or ownership.\n\nThey'd started out as friends, and now were lovers. Neither of them wanted to get married, which meant their relationship would be smooth and undemanding.\n\nAlex opened her eyes and exhaled an audible sigh. She didn't think she would ever get used to the dangerous curving stretch of roadway. A hint of a smile softened her mouth when the driver maneuvered onto the unpaved road that led to her lover's house. Her smile widened when she saw him standing on the porch in a pair of jeans and a pullover sweater, waiting for her.\n\nMerrick was off the porch and opening the rear door to the town car within seconds of the driver shifting into Park, leaving the man with the task of removing Alex's luggage from the trunk.\n\nHe scooped her off the leather seat and cradled her to his chest. \"Estas tarde,\" he whispered. He'd expected her three hours before.\n\n\"La salida de Ciudad M\u00e9xico estuvo demorada.\" Alex glared at him through her lashes. Her flight from Mexico City had been delayed several hours. \"The important thing is that I'm here.\" Her explanation, though spoken softly, held a slight edge.\n\n\"I'm not keeping tabs on you, Ali. It's just that I thought something had happened to you.\" Bending slightly, he set her on her feet, one arm going around her waist. Reaching into the pocket of his jeans, he removed a large bill and pressed it into the driver's palm. \"Thank you for keeping her safe.\"\n\nThe driver's head bobbed up and down like a buoy in a storm. \"Thank you, sir.\" He nodded to Alex as if she were royalty. \"Goodbye, miss.\"\n\nShe offered him her dimpled smile. \"Goodbye.\" She stood with Merrick, watching the driver get into his car. They were in the same spot when the taillights disappeared from view. \"I'd paid him, Merrick.\"\n\nPulling her closer, Merrick pressed a kiss on the top of Alex's hair. The last time he saw her, her short hair had afforded her a punk look. But now it was long enough to curl at the ends. The Mexican sun had darkened her face to an even nut-brown shade that fired the gold in her beautiful eyes.\n\n\"That's all right, baby.\"\n\nTilting her chin, Alex met his steady gaze. \"I don't want you to throw your money away.\"\n\nHis expression changed, hardening, until his eyes glowed like flints of steel. \"Do you think I can't afford to date you?\"\n\nHer delicate jaw dropped. \"No, Merrick, it's not\u2014\"\n\n\"It's not what, Alexandra Cole? I wasn't a trust-fund baby, but I can assure you that I'm not a pauper either. So, if you think I'm going to pimp you, I suggest you call your driver and tell him\u2014better yet, I'll drive you back to Arlington.\"\n\nShe stomped her booted foot. \"Stop it, Merrick!\"\n\n\"Stop what, Alexandra?\"\n\nAlex went completely still, her heart pounding painfully under her breasts. She'd flown back to the States to see Merrick when normally she would've gone to Florida to reconnect with her family.\n\n\"Stop putting words in my mouth,\" she countered slowly. \"If I'd thought you were after my money, I never would have given you my number.\"\n\n\"Why did you give me your number?\"\n\nA rush of heat swept over her face, moving lower until she felt as if she were on fire. \"Why are you asking me this now? Why not before we slept together?\"\n\nHis hands came up and he held her shoulders in a firm grip. \"I need to know how committed you are to our relationship.\"\n\nHer eyes narrowed. \"Are you afraid that I'm going to cheat on you with another man?\"\n\nMerrick shook his head. \"Fidelity is not even remotely a concern.\"\n\n\"Then don't you dare question my feelings for you. Not when I've just spent thousands of dollars and flown thousands of miles to make my first booty call.\"\n\nMerrick merely stared, complete surprise on his face, tongue-tied. \"Is that what I am to you?\" he asked, recovering his voice. \"You see me as a booty call?\"\n\n\"Huh-uh,\" Alex said flippantly. \"Come inside and I'll prove it.\"\n\nPulling her close and lowering his head, Merrick captured her mouth in a burning kiss that buckled her knees. She sagged weakly against his body, shivers of desire and delight racking her limbs. The sensual assault on her mouth ended as quickly as it'd begun.\n\n\"We'll continue this after we come back, and then you'll see if I'm worth the money you spent for your so-called booty call.\"\n\nAlex smiled, grabbing his hips. \"You don't mind being a booty call?\"\n\nHe chuckled. \"I don't mind being your booty call.\"\n\nStanding on tiptoe, she brushed a kiss over his mouth. She squeezed his firm buttocks. \"I hereby stake my claim to this.\"\n\nMerrick released her shoulders, his hand moving lower to cover her breasts over a soft wool sweater. Alex wasn't given time to react when one hand cupped the area between her legs. \"And I stake my claim to these,\" he whispered and was hard-pressed not to laugh when an expression of pure shock froze Alex's delicate features. He leaned in closer. \"Play with fire and you'll get burned, beautiful,\" he taunted.\n\nA slow smile found its way across her face. \"You're a bad, bad boy, Merrick Grayslake.\"\n\nHe lifted his eyebrows. \"You think? You have no idea how very bad I can be.\"\n\n\"Bad boy, bad boy, what you gonna do? What you gonna do when I come for you?\" she sang melodiously.\n\n\"Hey, you sound pretty good,\" he said, complimenting Alex on her singing. \"You sing, dance and play the piano. What other gifts are you hiding from me?\"\n\n\"That's it,\" she said, lifting her shoulders in a perfect Gallic shrug. \"I sing and play the piano because I took lessons. You already know about my dance lessons.\"\n\nCradling her to his chest, Merrick rocked her back and forth. \"What do you say we throw a party tonight?\"\n\nAlex stared up at him staring down at her. \"What kind of party?\"\n\n\"It will be private with just the two of us. We can begin with a wonderful dinner with drinks, then put on some mood music and dance for a while. And if we're not too tired we can retire to the bedroom where I'll show you just how much I've come to love you.\"\n\nHer lids came down, hiding her innermost feelings from Merrick. Never in her wildest dreams did she think she would experience the peace and joy she felt at that moment. Merrick had become the answer to her dream of finding the perfect man. Under his steely stare, impassive expression and deep controlled voice he'd shown her another side of himself whenever he made love to her with a wild abandon that made her forget every man she'd ever known, a lovemaking that made her forget who she was when she discovered an ecstasy that took her to heights of passion she hadn't thought possible.\n\nShe'd chartered a private jet and car service to come to Bolivar, West Virginia, because not only did she love Merrick but she'd also missed his lovemaking.\n\nHer eyelids fluttered as she struggled to bring her fragile emotions under control. Wrapping her arms around his waist and resting her cheek on his chest, she inhaled the familiar scent of his cologne that was the perfect complement to his body's natural fragrance.\n\n\"You don't ever have to tell me that you love me, querido, because I feel your love even when we're thousands of miles apart. Your face is the last thing I see when I close my eyes at night, and when I wake up it is you that I think about. I take your image with me when I go to class and I don't think I would've survived the two weeks in the Yucat\u00e1n with lodgings that were so primitive that once I returned to Mexico City I checked into an upscale hotel for two days, languishing in a Jacuzzi and ordering room service.\"\n\nMerrick pressed a kiss to her forehead. \"Pobrecita, eso es porque tu er es una chica rica maleriada.\"\n\n\"That's where you're wrong. My parents never spoiled me, but I'll never take running water and indoor plumbing for granted ever again.\"\n\nHe wanted to tell Alex that she was spoiled. He'd lost count of the number of times when he was denied a bath and food because he wouldn't do something his foster parents had ordered him to do. The physical and emotional abuse he'd encountered as a child gave him the strength he needed to survive the rigorous training to become a marine who remained in a constant state of readiness for all types of combat situations.\n\n\"I didn't know what you wanted to eat, so I decided to wait until you got here to go to the supermarket.\"\n\nPulling back, Alex smiled at Merrick. \"Let's go, because I'm starved.\"\n\n\"Didn't you eat on the plane?\"\n\n\"No. I had a couple of cups of coffee, but that's all.\"\n\nReaching for her hand, Merrick held it firmly as he bent and picked up her single piece of luggage. \"I'll fix something for you before we leave.\"\n\nAlex pulled back. \"I can pick up something at a fast-food restaurant.\"\n\n\"Not today, beautiful. We'll save the calories and fat-laden preservatives and additives for another day.\"\n\nShe gave him a sidelong glance. \"Don't tell me you're monitoring your figure,\" she teased.\n\n\"What I'm monitoring is my heart and arteries.\" Merrick knew he was going to have to take a physical even for a desk position, and he didn't want anything to compromise his passing.\n\nAlex walked up the porch steps and followed Merrick into the house where she'd spent a wonderful four days. Nothing had changed as she felt an invisible warmth envelop her with a sense of homecoming.\n\nMerrick closed the door to keep the cool mountain air out of the house, and when she turned to face him she saw it. Her eyes widened. \"What's that?\"\n\nMerrick saw the direction of her stunned gaze. A high-powered rifle was propped up against a wall next to the front door. \"It's a rifle.\"\n\n\"I know it's a rifle, Merrick. But why is it there?\"\n\n\"It's for bears.\"\n\n\"What about bears?\" she asked.\n\n\"It's spring and the bears are out with their new cubs looking for food.\"\n\n\"Are you going to shoot them?\"\n\n\"No, Ali. It's more to scare them.\"\n\n\"What if you don't scare them?\"\n\nMerrick captured Alex's anxious gaze with his. \"Then there'll be one less bear around these parts.\"\n\nAlex wrapped her arms around her body in a protective gesture. She knew there was something dangerous about Merrick, and seeing the rifle had just confirmed her suspicions. \"Let's hope you won't have to kill one.\"\n\nHis expression softened as he smiled. \"I've lived here for two years and I've never had to shoot another living creature. And that includes the deer who feed on the brushes behind the house. Some of them are so used to humans that they don't even run when they see you.\"\n\n\"Promise me you won't shoot Bambi or Smokey.\"\n\nTurning his head, Merrick successfully hid a smile. There was no doubt his girlfriend had a soft spot for animals. \"I can't promise that, Ali, especially if a four-hundred-pound bear is standing on my porch looking for food. I can always shoot to wound, but a wounded bear is more dangerous than one who isn't. Stay out of this, darling, and please don't ask me to do something that would put our lives at risk.\"\n\nThe familiar mask had descended again and Alex felt the closeness between her and Merrick dissipate like a drop of water on a heated surface. It was one thing to be separated from Merrick by more than a thousand miles, and another completely when they stood less than a foot apart, yet there was a chasm because of a disagreement.\n\n\"Okay,\" she conceded reluctantly. \"I'll stay out of it.\"\n\nIt wasn't until Merrick let out an audible sigh that he realized he'd been holding his breath. He didn't want to fight with Alex. All he wanted to do was love her. \"I'll take your bag upstairs, then I'll feed you.\"\n\nAlex nodded as she shrugged out of her wool swing coat and placed it on a chair near the door. She was standing in the middle of Merrick's living room because she'd missed him. She hadn't come to West Virginia to fight with him over the area's wildlife population but to spend every night, hour and minute with the man with whom she'd fallen in love, until she had to return to Mexico.\n\nCrossing the living room, she made her way to the half bath off the kitchen to wash her hands. She met Merrick as he entered the kitchen. Sitting on a stool at the cooking island, Alex rested her elbows on the countertop and watched Merrick as he prepared an impromptu lunch.\n\n## Chapter 13\n\nMerrick had been pushing a shopping cart up and down the wide supermarket aisles, stopping whenever Alex stopped to read the label on every product she placed in the cart.\n\nHe groaned for the third time within a matter of minutes. \"Do you always spend this much time reading labels?\" he asked.\n\n\"Yes,\" she answered, not bothering to look at him. \"I have to know what I'm eating, Merrick.\"\n\nShaking his head, he let out another groan. \"Baby, please.\"\n\nAlex glanced up, meeting his tortured gaze. \"Why don't you wait for me in the truck?\"\n\nDoes this mean you do everything your wife wants? Merrick recalled his query to Michael, and his friend's reply, I know enough when to advance and when to retreat.\n\nHe and Alex weren't husband and wife, but living together for a week, sharing meals and a bed was no different from what Michael had with Jolene. Merrick was aware that a successful relationship was based on compromise, and he also knew he had to find a middle ground with Alex or what they had, what he hoped to have with her would end in a supermarket because it was his practice to get in and out within half an hour. He didn't read labels, interrogate the in-store butcher about cuts of meat, or the produce manager as to the delivery date of fruits and vegetables.\n\nAbandoned at birth, a loner by nature and by choice, Merrick realized having a relationship with a woman, one whom he loved, was a new experience for him. What he had to do was to learn when to advance and when to retreat.\n\nForcing a smile, he winked at Alex. \"That's all right. I'll wait.\"\n\nAlex placed the can in the half-filled cart. \"I think I have everything.\" She'd planned menus that included Caribbean-inspired dishes she'd learned from her mother.\n\n\"Are you sure?\"\n\nShe smiled at Merrick. \"Of course I'm sure. Haven't we spent enough time here?\"\n\n\"Too much,\" Merrick mumbled under his breath. \"Yes, dear,\" he said loud enough for her to hear.\n\nAn elderly woman wheeling her cart in their direction patted Merrick's arm. \"What a nice young man you are. I've been married for more than forty years, and I've never been able to get my Danny to come grocery shopping with me.\" Her brown eyes sparkled like copper pennies as she reached over and patted Alex's hand. \"He's a keeper, honey.\"\n\nAlex returned her smile. \"I know.\"\n\nWaiting until the woman was out of earshot, Merrick wound his arm around Alex's waist inside her coat. \"Are you going to keep me, querida?\"\n\nShe affected an attractive moue. \"I'll think about it.\"\n\nLowering his head, he brushed a light kiss over her parted lips. \"Perhaps that'll help you make up your mind.\"\n\nStaring up at Merrick through her lashes, Alex's gold-brown eyes moved slowly over his face, cataloguing each of his well-defined features. \"I think I'm going to need a little more convincing.\"\n\nDesire darkened his eyes and tightened the skin over his high cheekbones. It'd been more than a month since he'd been with Alex, and the time away from her just served to increase his need for her. He, Merrick Grayslake, who'd lived his life believing he didn't need anyone, needed Alexandra Cole. Unwittingly, she'd become a part of him, the part that made him whole.\n\n\"I believe I have the answer.\" Leaning closer, he whispered, \"Let's go home.\"\n\nAlex lay in bed, listening to the sound of sleet pelting the windows. It was as if spring had forgotten the Alleghany Mountain region. Midday temperatures rose to the mid-forties but dropped below freezing with the setting sun. A popping sound of burning wood joined the rhythmic tapping. Merrick had started a fire in the fireplace before going into the bathroom to shower. He'd asked her to join him but she'd turned him down. It was their last night together and as the seconds ticked off she felt as if she were losing tiny pieces of herself each time they made love.\n\nThe tears welling up in her eyes overflowed, dotting the pillow under her head. How, she thought, was she going to make it through the next ten weeks without the man she loved? She'd made a mistake, a grievous error in returning to the States\u2014in coming to see Merrick rather than her family\u2014because it made leaving him that much more difficult. The side of the mattress dipped and she felt the warmth of the man who'd managed to occupy her every waking moment. Shifting on her side, she went into his strong, protective embrace.\n\n\"This is our last night together,\" she whispered tearfully.\n\n\"Don't say that, Ali,\" Merrick countered softly. \"It is not our last night together. Our last night won't come for a very long time.\"\n\nShe pounded his shoulder with her fist. \"You know what I'm talking about.\"\n\nHe caught her wrist. \"No, I don't know what you're talking about. Your school term will be over in a few months, then we can be together\u2014every day, every night.\"\n\n\"We can't do this again.\"\n\n\"Do what, querida?\"\n\n\"We can't see each other until I finish my courses.\" A fresh wave of tears ensued.\n\nTightening his hold on her body, Merrick eased Alex to lie atop his body, her legs sandwiched between his, the fingers of his right hand making soothing motions on her back.\n\nHe closed his eyes. \"Is it your intent to rip my heart out?\"\n\nShe buried her face between his neck and shoulder. \"What are you talking about?\"\n\n\"This is not about you, Ali. You're not in this by yourself. I've shared and said things to you I've never told another person. It was as if I had to wait thirty-five years to feel what should be a natural human emotion\u2014love. The love a baby feels for its mother or father, the love between a man and a woman and the love one feels for those with whom one shares a family bond.\n\n\"You've taught me to laugh, that it's all right to tease, be teased, and for that I will love you forever. Being with you has taught me that where I've come from is not as important as where I'm going. And wherever I go I want you with me.\"\n\nAlex's right hand trailed down Merrick's shoulder; it moved still lower along his rib cage, her fingers feathering over the curve of his buttocks and muscled thigh. All of her senses were intensified as she inhaled the clean masculine scent of his body, her fingertips grazing the crisp mat of chest hair and the firm flesh covering muscle and sinew. Merrick Grayslake reminded her of the statues of tall, powerful Mexican gods.\n\nReaching between their bodies, she cradled his flaccid sex; it swelled against her palm, increasing in girth until her fingers did not meet. The heat, hardness and the strong throbbing against her palm ignited a fire that swept through Alex like a lighted fuse. Her body vibrated liquid fire, the tremors between her thighs and groin buffeted by gusts of desire that rendered her close to fainting.\n\n\"Love me,\" she chanted over and over until it became a litany.\n\nMerrick had told Alexandra that he loved her, but it never seemed to be enough. He had to show her. Reaching for the condom on the table on his side of the bed, he slipped it on. Slowly, methodically, he became a cartographer, charting every inch of her body as he began at the top of her head and moved slowly down as if sampling a sweet, frothy confection. He ignored the soft moans coming from Alex's throat, the rise and fall of her firm round breasts over her narrow rib cage, the contracting muscles in her flat, firm belly and the trembling of her legs as she writhed on the sheets.\n\nMerrick was relentless as he alternated laving the folds at the apex of her thighs with plunging his tongue into the most secret recess of her body. It was as if he wanted to devour her whole, put her inside himself wherein she would be with him even when separated by thousands of miles.\n\nWhen his own passion rose to fiery proportions, he eased himself inside her, making them one. He felt her heat as it rippled under his hands, inhaled the rising scent of their lovemaking that had become an erotic aphrodisiac. His desire for Alex, his love for her, surpassed every longing, craving and aspiration he'd ever known.\n\nAlex cried for the second time within minutes, this time as waves of ecstasy washed over her. She closed her eyes, gasping in the sweet agony tearing her asunder. It eased slightly before she was hurled higher, climaxing, her orgasms overlapping one another until she lay, struggling to catch her breath.\n\nMerrick felt the strong contractions squeezing his blood-engorged flesh, the sensations sending him over the edge. Cupping her hips, he quickened his thrusts until the dam broke and he spilled his passion into the sheath when it was Alex's body he craved.\n\nThe pleasure she'd offered him was pure and explosive, satisfying and sweet. A deep feeling of peace entered him as he savored the remnants of the lingering sensation that rendered him as weak as a newborn.\n\n\"Te amo,\" he whispered in her ear.\n\nAlex smiled, enjoying the weight of his body pressing hers down to the firm mattress. \"Yo, tambi\u00e9n, mi amor.\"\n\nThey loved each other, but what were they going to do about it? She would complete her studies and return to the States as an architectural historian. She planned to secure a position with a historic preservation, trust or museum, then what?\n\nWould she or Merrick be content to see each other whenever she was free or would she want more? And she refused to think of marriage and children as the more.\n\nHe'd admitted he didn't want to marry or father children, but she did\u2014though not now. Not when she wanted a career. Alex willed her mind blank. She couldn't afford to think of a future with Merrick Grayslake when marriage and children weren't her priority.\n\nWithout warning, he reversed their positions. They lay together until their breathing deepened and they fell asleep, his body still joined with hers.\n\nThe sky had brightened with the dawn of a new day, it appearing a hoary gray, when Merrick and Alex resumed their dance of desire, leaving both hungrier for each other than at their first coupling.\n\nBoth knew it was a mistake, but love wasn't touted to be rational or judicious. And when Merrick saw Alex off at the airport before she boarded a private jet scheduled to land in Mexico City, he knew when he saw her again the meadow surrounding his house would be filled with wildflowers.\n\nHe would honor Alex's request and wait for her return.\n\nThere was one thing life had taught him and that was patience. It had served him well when he'd become a marine sniper, and it had saved his life as a covert operative.\n\nMerrick knew there was something special the moment he laid eyes on the petite woman, unaware that she would be the one to change him and his destiny.\n\n## Chapter 14\n\nAlex tightened her grip on the cell phone, the pressure leaving an imprint on her palm as she stared out the oval window of the ColeDiz jet as it circled the Fort Lauderdale\u2013Hollywood International Airport in preparation for a landing. It was after two in the morning.\n\nHer father had called her forty-eight hours before to inform her that the family was gathering in West Palm Beach to celebrate Timothy Cole-Thomas's retirement as CEO of ColeDiz International, Ltd. She'd vacillated whether to call Merrick to let him know she would be in the States for the weekend but changed her mind; there wasn't enough time and seeing him would reopen the wound that had healed from their last encounter.\n\nShe'd returned to the universidad assailed with feelings that left her in an agonizing maelstrom of acute loss. She'd spent a restless two days trying to figure out why she'd fallen in love with a stranger, why Merrick Grayslake and not some of the other men she'd dated. And her love for him wasn't some silly schoolgirl crush or virginal infatuation. She would turn thirty in November, which meant she was a grown woman, an independent woman in control of her life and destiny.\n\nMerrick admitted that he'd told her things about himself he'd never told anyone, yet when she replayed their conversations she realized he actually had told her very little about himself. She knew he'd been orphaned at birth, had been in the Marine Corps and had been recruited by the Central Intelligence Agency. What he hadn't disclosed was how he supported himself, nor had he explained the many scars marring his body.\n\nAlex was mature enough to admit that she'd been drawn to the enigmatic man because he radiated a danger that turned her on. He was akin to a graphic comic superhero prowling the nighttime streets and alleys looking to rescue the weak and right the wrongs perpetrated by those who opposed justice. She'd dubbed him Sir Grayslake, and he'd become her knight in shining armor.\n\nShe would spend two nights in Florida before returning to Mexico. Then she would begin counting down the weeks until she was reunited with the man she loved.\n\nAlex was surprised to see her brother Gabriel waiting for her once she cleared Customs. What shocked her was his uncanny resemblance to their father. Gabriel had become a professional musician like David Cole, and like the elder Cole, Gabriel had solidified his place in history as a Grammy Award winner. Gabriel had surpassed his father when he earned an Oscar for a movie sound track.\n\nSmiling, she quickened her step. Gabriel met her and swung her up off her feet. \"What's with the mustache?\" she asked, kissing his cheek. Pulling back, she stared into a pair of large gold-brown eyes, the only feature they'd inherited from Serena Morris-Cole. His long, thin nose, high cheekbones, olive coloring and wolfish dimpled smile were David Cole's. He'd pierced both ears, and tiny gold hoops were suspended from each.\n\n\"I thought I'd try it out for a while.\"\n\n\"Are you trying to look older?\" Alex teased as he put her down. Gabriel, the eldest of David and Serena's four children, had recently celebrated his thirty-fourth birthday.\n\nSmiling his wolfish, lopsided grin, Gabriel shook his head, his long, wavy, prematurely gray\u2013streaked hair secured in a ponytail at the base of his neck, swaying with the motion. \"No, Alex. The gray hair is enough.\"\n\nMost of the men in her family who were direct descendants of her grandfather, Samuel Cole, had inherited his gene for graying prematurely, and her brother was no different.\n\nLooping an arm around his waist, she hugged him. \"Why don't you dye it?\"\n\nGabriel ruffled her short hair, then reached for her hand. \"I'll admit to being a little arrogant, but vain is not in my vocabulary.\"\n\nAlex fell in step with her brother as he led her out of the airport to where he'd parked his car. \"You're arrogant and vain, Gabriel Morris Cole.\"\n\nAt six-three, Gabriel slowed his stride to accommodate his sister's shorter legs. \"Dad was sorry about the short notice, but it couldn't be helped.\"\n\n\"I thought Timothy was going to stay at least through the summer.\"\n\n\"Apparently, he changed his mind. Timothy says that Diego is more than prepared to take over as CEO.\"\n\nShe gave Gabriel a sidelong glance. \"What do you think?\"\n\nGabriel lifted a shoulder. \"I wouldn't know one way or the other. I promised myself a long time ago that I'd never get involved in the family business. Aunt Nancy and Josephine's children eat, sleep and breathe the stuff. I'm a musician, so that lets me off the hook.\"\n\n\"So was Dad, but he was still CEO of ColeDiz for nine years before he met and married Mom.\"\n\n\"That's true, but he couldn't wait to transfer the responsibility to Timothy.\"\n\n\"Dad gave up ColeDiz and then set up Serenity Records,\" Alex argued softly. \"Musician or not, he was still a businessman.\"\n\nDavid Cole had set up his own record company to showcase new and upcoming talent in all music genres. Although his company was small when compared to the larger ones, he'd earned two Grammys for Producer of the Year and his recording artists had earned and garnered more than a dozen nominations.\n\n\"Dad knows I don't want to become involved with Serenity. That's why he turned it over to Ana and Jason.\"\n\nGabriel led Alex over to a Porsche GT2. Pressing a button on a remote device, he unlocked the low-slung, silver sports car. Opening the passenger-side door, he waited until Alex was seated before he closed it and came around to sit beside her.\n\n\"When did you get this?\" she asked. \"It smells brand-new.\"\n\nStarting the engine, Gabriel shifted into gear and pulled out of the parking space. \"I've had it for about a month. It is my birthday present to myself.\"\n\n\"It's about time you came out of your pocket and got a decent ride,\" Alex mumbled. Gabriel had become the brunt of family jokes because when he wasn't touring with his band he drove a clunker that had seen better days. \"What else are you holding out on me?\"\n\nGabriel smiled. \"I'm buying a house.\"\n\nShifting on her seat, Alex stared at him and asked, \"Where?\"\n\n\"Cotuit. It's on Cape Cod.\"\n\n\"What or who is in Cape Cod?\"\n\nGabriel had to remember that he hadn't seen his sister since the beginning of the year. \"I've been accepted as an artist-in-residence at a high school in a Boston suburb.\"\n\n\"Wouldn't it be better if you rented a place?\"\n\n\"No, because I've found the perfect little house.\"\n\n\"How little, Gabe?\"\n\n\"It's a restored two-story farmhouse set on six acres of beautifully landscaped waterfront property. I plan to renovate it by raising the ceilings in the second-floor bedrooms and expand both floors by at least a thousand square feet on either side.\"\n\n\"It sounds as if your little house is going to become a big house.\"\n\n\"I'm used to having a lot of space.\"\n\nAlex nodded. They'd grown up in a house with nearly ten thousand square feet of living space, and they each had their own bedroom suites.\n\n\"How did Mom and Dad react when you told them you were moving to Massachusetts?\"\n\nGabriel's teeth showed whitely under his trim mustache. \"They had the nerve to open a bottle of champagne and toast each other.\"\n\n\"That's cold, Gabe.\"\n\n\"You think? After Dad downed two glasses he said, 'It's about time. The next piece of good news should be that you're getting married.'\"\n\nAlex shook her head. \"What's up with our folks trying to marry us off?\"\n\nGabriel maneuvered off a four-lane road and onto the interstate in the direction of Boca Raton. \"They want grandchildren, Alex.\"\n\n\"Well, they're not going to get one out of me for a long time,\" she said.\n\n\"Same here,\" he concurred.\n\nEyelids drooping, Alex relaxed against the back of her seat. \"Wake me when we get home.\" Although she'd lived in Virginia for two years, she still thought of Boca Raton as home.\n\nAlex woke up late the next morning to knocking on her bedroom door. Bright Florida sunshine poured through the many windows. \"Come in,\" she called out.\n\nThe door opened and Ana walked in wearing a sundress over her bathing suit. \"Rise and shine, sleepyhead! We're leaving for West Palm in half an hour.\"\n\nPulling a sheet over her head, Alex moaned, \"Please tone it down, Ana.\"\n\n\"What time did you get in?\"\n\nShe lowered the sheet and pushed herself into a sitting position as Ana flopped down on the side of the bed. \"The plane landed around two-thirty, and we didn't get here until four.\" Combing her fingers through her short hair, she smiled at her younger sister. \"Who's here?\"\n\n\"Everyone came except Chris. Emily said he's involved in a federal racketeering case that just went to the jury.\" Their first cousin's husband, Christopher Delgado, a former governor of New Mexico, was now a judge.\n\nPushing back the sheet, Alex swung her legs over the side of the bed, undoing the tiny pearl buttons on the front of her cotton nightgown as she headed for an adjoining bathroom.\n\n\"Did Mom make coffee?\" she asked.\n\n\"I think Dad made it this morning, because it was strong enough to grow hair on my chest.\"\n\n\"I need it strong. Can you please bring me a cup?\"\n\nAna moved off the bed to do her bidding while Alex stepped out of her nightgown, leaving it on a low bench in the corner of the expansive bathroom. She was tired\u2014no, the fact was she was exhausted. The late flight and not enough sleep had her out of sorts. Opening the door to the shower stall, she stepped in, turned on the faucets and adjusted the water temperature. The cool water was the antidote. It revived her and by the time Ana returned with a steaming mug of coffee Alex was wide awake.\n\nFive generations of Coles had gathered on the lawn at the family West Palm Beach compound to salute Timothy Cole-Thomas's undaunted dedication to running a business empire that included coffee plantations in Mexico, Jamaica, Puerto Rico, and several banana plantations in Belize. At sixty, Timothy had successfully shepherded ColeDiz International, Ltd., through recessions and economic instability to realize a profit every year in his thirty-year term.\n\nAlex sat next to Dana Nichols-Cole, whose three-month-old daughter had fallen asleep in her arms. \"May I hold her?\" Dana put the infant in Alex's outstretched arms.\n\nDana and Tyler were seeking to legalize the adoption of the baby they'd named Astra. Tyler Cole had delivered the infant, but couldn't save her fifteen-year-old mother who'd been abandoned by her own mother at the age of seven. Unfortunately, the pregnant teenager had spent most of her short-lived life in foster and group homes.\n\nGabriel leaned over Alex's shoulder. \"Hey, Mama,\" he teased, winking and flashing his trademark dimpled smile.\n\n\"I'm going to hurt you, Gabriel Cole.\"\n\nHe leaned closer. \"I'll let you hurt me after you get pregnant.\"\n\nDana stared at her husband's cousin. \"Are you pregnant?\"\n\nAlex swatted at her brother. \"Get the hell outta here, Gabriel!\" she hissed through clenched teeth.\n\n\"Who's pregnant now?\" Tyler Cole asked as he sat down beside his wife. Tyler had become the family resident ob-gyn.\n\nAlex rolled her eyes at Tyler. \"Don't look at me.\"\n\nGabriel rested an arm on Tyler's broad shoulder. The first cousins looked enough alike to pass for brothers. \"Unlike your sisters, mine have struck out in the romance department because they can't seem to tell the difference between a winner and a loser.\"\n\nAlex opened her mouth to announce that she'd fallen in love with a winner but caught herself in time. What she had with Merrick would remain her secret\u2014for now.\n\n\"Speak for yourself, Gabriel. Why is it I don't see a woman hanging on your arm?\"\n\n\"That's because I'm not one to advertise my business.\"\n\nAlex's retort was thwarted when Diego stood and tapped the handle of a knife against a water goblet. All gazes were focused on the tall, powerfully built man in a white guayabera and jeans. \"May I please have everyone's attention?\"\n\n\"I want to e-e-e-at n-o-o-w!\" wailed a young child before being hushed by his mother.\n\nDiego lowered his chin and bit back a smile. \"I believe that's my cue to make this quick.\" His head came up, black eyes smoldering with a lazy seductiveness that fired the imaginations of the few women who'd come to know him intimately.\n\n\"As most of you know, my father is a man of few words, preferring instead to let his actions speak for him. If it worked for him over the past three decades, then why should I try to fix what isn't broken?\" He raised his goblet in a mock salute. \"To my uncles, Martin and David, and my father, Timothy Cole-Thomas, I salute you and pray for your blessings as the next CEO of ColeDiz International, Ltd.\" He turned and extended the glass to his fretful nephew. \"Now we can eat.\"\n\nAll of the young children stood up and cheered while their parents and grandparents shook their heads. Collectively, the descendants of Samuel and Marguerite-Josefina Cole were boisterous, and at times bodacious, but they were also protective of and fiercely loyal to anyone who claimed Cole blood. And there was an unspoken rule that if you threaten one, the threat extends to all.\n\nGabriel patted Alex's back. \"Don't get up. I'll bring you something to eat.\"\n\nShe rolled her eyes at him, still smarting about his remark that she'd picked up losers. \"Don't try to placate me, big brother.\"\n\nA smile slipped through her impassive expression as her brother walked toward the table where servers were filling plates with mounds of hot and cold dishes. It was on a rare occasion that one saw an adult Cole woman standing in line waiting to be served. The tradition, which had begun when Martin Cole became CEO, was that the women rule and their men serve. The result was all females were protected, pampered and adored.\n\nDana held out her arms for her daughter. \"Let me take her into the house and put her to bed.\"\n\nAt the exact moment she relinquished the sleeping infant, a chorus of screams sliced through the undercurrent of voices raised in conversation and laughter. Four preteen boys and one girl had jumped into the Olympic-size pool\u2014fully dressed. It was another tradition that had survived several generations.\n\nAlex laughed when she heard the colorful language of their parents chastising them for their unruly behavior. She'd lost count of the number of times she'd jumped into her grandparents' pool, or into her own, much to the dismay of her parents, who couldn't understand why their children would want to sit around in wet street clothes when they could've worn a swimsuit. What the adults failed to understand was that it was an open act of rebellion in order to flaunt their independent spirit.\n\nHer smile faded when she recalled a few occurrences when as an adolescent she'd challenged her mother's authority. Alex had learned to get over on David Cole, but Serena was another matter. It'd taken only two confrontations with her mother to come to the conclusion that she'd better get her act together or spend her teenage years in lockdown.\n\nA faraway expression filled her eyes when she tried imagining what kind of mother she would become if or when she opted for motherhood. Would she be indulgent and tolerant like her grandmother Marguerite-Josefina, or straightforward and practical like her own mother?\n\nThere was never a question of not understanding what Serena said or meant. However, Alex had lost count of the number of times she'd gone to her father to enlist his support to soften her mother's stance. That all changed when David Cole called a rare family meeting, explaining that he would not tolerate his children's attempt to undermine their mother's authority, and that put an end to Alex's clandestine meetings with her father.\n\nPrerecorded music blared from speakers as the noise level escalated. Alex lifted her eyebrows when she saw an attractive young woman with stylishly-cut, shoulder-length hair talking to Diego. Even from this distance, he appeared to be equally enthralled as he moved closer to listen to what she was saying.\n\n\"Now, that looks serious,\" whispered a familiar voice.\n\nAlex turned to find Ana taking a chair next to her. \"Who is she?\"\n\n\"She's a weather girl with one of the television networks. If she believes she's going to become Mrs. Diego Cole-Thomas, then she's in for a world of hurt.\"\n\nAlex nodded. \"He told me that he's not ready to settle down.\"\n\nAna emitted a delicate snort. \"Now, if the woman was named ColeDiz, then she'd at least have a fighting chance.\"\n\nThe sisters discussed their marriage-shy cousin until their brothers returned carrying plates of food. Jason served his twin while Gabriel placed Alex's on the table in front of her.\n\nAna gave Jason a forlorn look. \"Can you please bring me something to drink?\"\n\n\"Me, too,\" Alex said, chiming in.\n\n\"Damn, Ana,\" Jason snarled. \"You should've told me when I was up there.\"\n\nGabriel wrapped an arm around Jason's neck, pulling him close. \"Don't let them stress you out, little brother. I'll get the drinks while you fix me a plate.\"\n\nAlex squinted at her younger sister as Ana dipped a piece of calamari into a small cup of piquant sauce. \"Sometimes you ride Jason a little too hard.\"\n\nThe seafood dangled from the tines of Ana's fork as she hesitated putting it into her mouth. \"Jason's a good sport.\"\n\n\"He's a good sport because you're his twin sister. But you overdo it.\"\n\nAna affected a pout. \"You have Gabriel and I have Jason.\"\n\n\"But I don't take advantage of him, Ana.\"\n\nTaking a bite of the calamari, Ana chewed it thoughtfully. \"All right, Alex. I'll ease up on him.\"\n\nAs fraternal twins, she and Jason were inseparable, existing in a private world where one was able to finish the other's sentences. Older by eight minutes, Jason had always protected his twin although her stronger personality overshadowed his more laid-back, quiet one.\n\nAlex spent the rest of the afternoon eating, drinking, dancing, swimming and interacting with her many relatives. The sun had set, the caterers were gone and those who'd planned to spend the remainder of the weekend in West Palm Beach claimed bedrooms in the twenty-four-room mansion.\n\nMichael and a very pregnant Jolene decided to stay with his parents at their Palm Beach condominium. With his wife in her eighth month of confinement, Michael had opted to drive to Florida in lieu of flying.\n\nAlex returned to Boca Raton with her parents while her siblings stayed behind to hang out with their many cousins. There was talk of them going to a popular West Palm Beach dance club.\n\nShe was scheduled to fly back to Mexico Sunday morning along with Emily Kirkland, her three children and her cousin's sister and brother-in-law, Sara and Salem Lassiter, and their children. The Kirklands and Lassisters would deplane in Las Cruces, New Mexico, before the jet continued on to Mexico City.\n\nShe checked her cell phone and realized she'd missed Merrick's call. Pressing a button, Alex listened for a break in the connection. She smiled when hearing his signature \"Hey, baby.\"\n\n\"Hey, yourself,\" she whispered. \"What are you doing up so late?\" The glowing numbers on the clock on the bedside table read 1:10 a.m.\n\n\"I couldn't sleep.\"\n\nVertical lines appeared between Alex's eyes. \"I've never known you to have insomnia.\"\n\nA deep sensual chuckle came through the earpiece. \"That's because whenever we're together I don't have a sleep problem. You're like morphine.\"\n\n\"What do you know about morphine?\" she teased.\n\nThere was a moment of silence, and Merrick said, \"Forget I mentioned it. Even though it's a little after midnight where you are I'll let you go. I'd planned to call earlier, but thought perhaps you'd gone out with your friends.\"\n\n\"I'm not in Mexico City, Merrick.\"\n\n\"Where are you?\"\n\n\"I'm in Florida. I came early this morning and I'm leaving later on today. The family got together to celebrate my cousin Diego taking over as CEO of ColeDiz. I wanted to call you and ask if you wanted to meet me here, but there wasn't enough time.\"\n\nThere came another pause from Merrick. \"Have you told anyone\u2014your family, about us?\"\n\n\"No. And I don't like keeping secrets from them.\"\n\n\"You want them to know about us?\"\n\n\"Yes, I do, Merrick.\"\n\n\"I'd like you to wait before you tell them.\"\n\nA cold chill snaked its way down her spine, and she went completely still. There was something in his voice that set off warning bells in her head. \"Why do you want me to wait?\"\n\n\"I can't tell you now.\"\n\n\"What are you hiding from me?\"\n\n\"Nothing, Ali. It's something I'd rather not discuss on the telephone. I'll tell you everything when we see each other again.\"\n\n\"That's not going to be until the beginning of June,\" she said in protest.\n\n\"That's when I'll tell you.\"\n\nShe squeezed the tiny instrument in frustration. \"You offer me some cryptic mumbo jumbo and expect me to wait two months for an explanation.\"\n\n\"What I'd like for you to do is trust me, Ali.\"\n\n\"You want me to trust you when you're hiding things from me. I really don't know who you are, what you do, how\u2014\"\n\n\"Stop it, Alexandra!\" Merrick hadn't shouted the command, but the stinging effect was the same. \"I'm going to end this call,\" he said in a dangerously quiet tone. \"We won't discuss this again until I see you. Have a safe flight back, and I'll call you tomorrow night.\"\n\n\"Goodbye, Merrick.\"\n\n\"No, Ali. Good night.\"\n\nThe willfulness she'd struggled all of her adult life to control surfaced, making her unwilling to compromise. \"Goodbye, Merrick,\" she repeated, then abruptly disconnected the call.\n\nShe threw the phone and it landed on an armchair. Alex flopped down to the bed and closed her eyes. She asked herself for the first time since coming face-to-face with Merrick Grayslake, was she in over her head?\n\n## Chapter 15\n\nMoira knocked on the open door to Alex's room and walked in. Clothes were stacked in neat piles atop the double dresser and bedside table. Alexandra Morris was preparing to leave Mexico City. Classes had ended the day before and final marks were posted in the lobby of the converted convent earlier that morning.\n\nAlex turned and smiled at her friend. \"Well, this is it. We're done.\"\n\nMoira flopped down on a straight-back armchair. \"You are done.\"\n\nAlex halted folding a pair of jeans. \"What are you talking about?\"\n\n\"I'm staying on.\"\n\nShe stared at Moira, baffled. The week before, Moira had suddenly burst into tears, and when Alex questioned her, she said she'd suddenly had an attack of homesickness.\n\nClosing the distance between them, Alex sat on the arm of the chair and looped an arm around Moira's neck. \"What's up, girlfriend?\"\n\nMoira's dark blue eyes filled with tears but didn't fall. \"I'm pregnant.\"\n\nAlex's mouth formed a perfect O before she pursed her lips. She didn't know whether to be happy or sad for her friend, but decided on the former. \"Congratulations.\"\n\nEyelids fluttering, Moira rested her forehead on Alex's arm and cried happy tears. \"Thank you.\"\n\n\"When's the big day?\"\n\n\"Early December.\"\n\nSitting up straighter, Alex swept her gaze over Moira's slender body. \"You're already three months pregnant?\"\n\nMoira nodded. \"I'd suspected for some time that I was carrying Hernando's baby, but you could say I was in denial.\"\n\n\"Hernando,\" Alex repeated, her voice rising slightly. \"Are you talking about our Professor Hernando Rivera?\"\n\nMoira nodded as she blushed to the roots of her pale hair. \"Yes.\"\n\nAlex wrinkled her delicate nose. \"I don't believe it. Here I thought you were fooling around with Umberto but instead it was our brilliant, esteemed Professor Rivera.\"\n\nMoira bit down on her lower lip. \"I hadn't planned on having an affair with him\u2014\"\n\n\"Don't you dare apologize to me,\" Alex said, interrupting her. \"You're a grown woman, Moira.\"\n\nHer head came up, her expression soft and angelic. \"He's wonderful, Alex, as a lover, teacher and husband.\"\n\n\"You're married?\"\n\n\"Yes,\" Moira said, smiling.\n\n\"When did you get married?\"\n\n\"Yesterday. Because I'm not Catholic, we decided on a civil wedding.\"\n\n\"Do you love him, Moira?\"\n\n\"Yes, I do.\"\n\n\"Then, that's all that matters.\"\n\nA warm glow flowed through Moira and she felt freer than she had in hours. She'd called her father to tell him that she wasn't coming home because she'd fallen in love and married a Mexican man. Her father hadn't waited for her to explain that the man was one of the world's most respected art historians when he'd hung up on her.\n\n\"I wanted to wait until after the baby is born to marry Hernando, but he refused because of his position at the universidad.\"\n\n\"He's right, Moira, even though it's not frowned upon for a woman to have a child without the benefit of marriage nowadays.\"\n\nMoira hugged Alex. \"I knew you would be happy for me.\"\n\nAlex returned the hug. \"Why wouldn't I be happy for you? I wish you all the happiness you deserve\u2014and more.\"\n\n\"I hope we'll stay in touch with each other, Alex.\"\n\n\"Of course we will.\"\n\n\"Will you come by tomorrow night and have dinner with us? I told Hernando that I didn't want you to leave Mexico without saying goodbye.\"\n\n\"Of course I'll come.\" That meant she would have to delay going back to Virginia.\n\nMoira popped up like a jack-in-the-box.\n\nLeaning over, she pressed a kiss to Alex's cheek. \"I'll see you tomorrow.\"\n\nAlex was still sitting on the arm of the chair when Moira left, closing the door quietly behind her. Alex's friend had accused Hernando Rivera of having the hots for her when it was Moira Morgan he was actually in love with.\n\nRising from the chair, she had resumed the boring task of packing, when her cell phone rang. Reaching for the instrument, she flipped open the cover without looking at the display.\n\n\"Do you have a flight schedule for me, Joelyn?\"\n\n\"Ali?\"\n\n\"Merrick?\"\n\n\"I can assure you that I'm not Joelyn.\"\n\n\"And I wouldn't want you to be her.\" Sitting down on the edge of the bed, Alex stared at the worn terra-cotta floor. After their rather heated exchange what now seemed aeons ago, she and Merrick had resumed their nightly telephone calls.\n\n\"Did you pass everything?\" he asked.\n\nHer smile was dazzling. \"I managed to get top honors.\"\n\n\"Congratulations.\"\n\n\"Thank you.\"\n\n\"I'd like to give you a gift.\"\n\n\"For what, Merrick?\"\n\n\"It's not every day that someone earns a graduate degree.\"\n\n\"Thousands get graduate degrees every year.\"\n\n\"How many people get to become architectural historians?\"\n\n\"Well, you're right,\" she conceded.\n\n\"I know I'm right. And because I'm bigger and stronger than you it would be in your best interest to accept what I'm offering.\"\n\n\"What's with the intimidation?\" she teased.\n\n\"What's with the interrogation?\" he countered.\n\n\"Okay, Merrick. What is it?\"\n\n\"I've arranged for a private jet to fly you to Rome Friday morning. Try to get some sleep because I have something special planned for the evening. Later on today a courier will deliver a packet containing your e-ticket and flight information. I've also arranged for ground transportation to get you and your luggage to the airport. When you arrive I'll be waiting for you once you clear Customs.\" There was a pause before he said, \"Hold on, Ali. Porter\u00f2 l'un col rubino.\"\n\n\"Merrick?\"\n\n\"Yes, baby?\"\n\n\"Where are you?\"\n\n\"I'm in Rome. How's your Italian?\"\n\nShe couldn't believe he was calling her from Italy. \"It's definitely not as good as yours.\"\n\n\"I'm going to have to hang up because the store clerk is glaring at me. I'll see you Friday night.\"\n\nBefore Alex could accept, reject or protest, he ended the call. Falling back on the bed, she kicked her legs the way she'd done as a child.\n\nShe never would've predicted that her life would change so dramatically since New Year's Eve. Her life had changed, she had changed and when she saw Merrick Grayslake again she would thank him for helping her to become a woman in the truest sense of the word.\n\nWithin minutes of exiting Customs, Alex spied a man holding a placard with the name Cole printed in large black letters. She approached him. \"I'm Alexandra Cole.\"\n\n\"Dare il benvenuto a Roma, signorina. Prender\u00f2 il suo bagaglio.\"\n\nHe spoke Italian too quickly for her to understand more than welcome and baggage. She smiled. \"Grazie.\"\n\nThe man picked up her bags and motioned for her to follow him. It was 9:00 p.m. in Rome and the airport was teeming with departing and arriving travelers.\n\nAlex had taken Merrick's advice and slept during the transatlantic flight. An hour before they were scheduled to touch down at the Leonardo da Vinci Airport, she'd showered in one of the onboard bathrooms and changed into the ubiquitous little black dress and strappy sandals. The instructions in the packet indicated a 10:00 p.m. dinner reservation.\n\nShe saw Merrick leaning against a column. He straightened when their gazes met. They hadn't seen each other in three months, and he'd changed. His hair was longer and the sun had darkened his face to a deep copper-brown. If it hadn't been for his distinctive eye color Alex wouldn't have recognized him.\n\nMerrick did not move forward because he couldn't at that moment. Rooted to the spot, he watched the muscles in Alex's bare shapely legs and thighs flex and relax as she made her way toward him. The skimpy black dress with a scooped neckline and capped sleeves outlined the delicious curves of her sexy, petite body. Her hair that looked as if she'd styled it with her fingers framed her round face in sensual disarray.\n\nIt was when he saw her face that a quiet storm stirred in his groin. Smoky eye shadow, a coat of mascara on feathery black lashes and the vermilion color on her lush, full lips made him want to strip her naked and make love to her. But he hadn't invited her to Italy to make love to her, but to show her much he was in love with her.\n\nGo to her, a silent voice urged Merrick. He took one step, then another, closing the distance. He let out his breath when he felt the crush of her breasts against his chest.\n\nCradling her face between his hands, he lowered his head and brushed his mouth over hers. \"Dare il benvenuto a Roma.\"\n\n\"Grazie, mio amore.\"\n\nMerrick released her when he hadn't wanted to. \"Are you hungry?\"\n\nAlex stared at the man whom she loved beyond reason. \"Yes.\" She hadn't eaten anything since early that morning. She was hungry, but not for food. She'd missed Merrick, missed everything about him.\n\nPutting his arm around her tiny waist, he pressed a kiss to her hair. \"Let's get out of here. I made a reservation for dinner at a restaurant not far from our hotel.\"\n\nThe drive from the airport took longer than expected, and when Merrick and Alex arrived at the hotel on the Via Veneto there was only enough time to store her luggage in the room before they had to leave.\n\nTucked away in an alley off a side street, the restaurant resembled a grotto with brick-and-mortar walls, a vaulted ceiling and water flowing from a fountain into a pool.\n\nThe flickering light from the candle on the table shimmered off Alex's flawless brown face. Merrick stared at her. Three months. It'd been that long since he'd last seen Alex and it appeared as if she'd changed. The change wasn't her looks, but her demeanor. She'd matured during their separation; a woman had replaced the open, spontaneous girl.\n\nAlex took a sip of wine, then touched the corners of her mouth with a napkin. The restaurant's ambience, her dining partner and the food were incredible. She'd ordered chicken in a light wine and butter sauce while Merrick had chosen linguine with a broiled seafood medley.\n\n\"What made you choose Italy over some other country, Merrick?\" Her query shattered the comfortable silence.\n\nHe put down his wineglass. \"I remember you saying you enjoyed visiting here, and when you finished your studies you'd planned to come back not as a student but as a tourist.\"\n\nHer eyebrows flickered as she met his steady gaze. \"Do you remember everything I say?\"\n\n\"Yes,\" he admitted. \"But only if it's worth remembering.\"\n\nAlex studied the man sharing the small round table with her. He'd become a chameleon. Dressed in a sand-colored linen suit with a sky-blue shirt and navy blue tie, he appeared sophisticated and urbane. And with his hair texture, eye color and sharply defined features he could be a citizen from any country in the world.\n\n\"How long can you hang out here with me?\" Merrick asked.\n\nShe stared at the food on her plate. \"No more than two weeks.\" She looked up, meeting his gaze. \"I have an interview with the National Trust for Historic Preservation at the end of the month.\"\n\n\"Where's the interview?\"\n\n\"Boston.\"\n\nMerrick nodded. \"I'll have you back in plenty of time.\" Shifting his chair, he moved closer to Alex. \"You've accused me of hiding things from you, and I promised I would tell you what you need to know about me when we saw each other again.\"\n\nAlex took another sip of wine, staring at Merrick over the rim of her glass. \"We don't have to talk about it now.\"\n\nReaching for her hand, he gave her fingers a gentle squeeze. \"Yes, Ali. We can't move forward unless we clear the air about things that bother you.\"\n\nAlex didn't want to talk. All she wanted was to enjoy the moment because for the first time in her life she'd gotten everything she'd ever wanted: she'd completed her graduate program and looked forward to securing a position as an architectural historian and she planned to spend the next two weeks touring Italy with the man she loved.\n\n\"If whatever you plan to tell me is going to upset me, then I don't want to hear it, Merrick. Not tonight.\"\n\n\"When do you want me to tell you?\"\n\n\"Tell me on the flight back to the States.\"\n\nHe shook his head. \"No.\"\n\nHer delicate jaw dropped. \"No,\" she repeated.\n\n\"N\u2014O,\" Merrick said, spelling the word. \"You will sit and hear me out, because I'm not going to wait two weeks to tell you what you demanded to know three months ago.\"\n\nAlex froze as she stared at Merrick. His voice was absolutely emotionless. A fist of fear squeezed her chest as she met the steely-eyed gaze of a stranger, a stranger who'd replaced the man she thought she knew and loved. Her breath seemed to solidify in her throat and all she could do was nod.\n\nMerrick leaned closer, his mouth within inches of Alex's ear. \"I'm going back to the CIA.\"\n\nAlex closed her eyes and slumped against his shoulder. \"Why?\" she whispered. She couldn't imagine not seeing Merrick for months, waiting for him to return from a clandestine mission.\n\n\"Because I need normalcy in my life.\"\n\n\"Can't you do something else?\"\n\n\"No.\"\n\nShe opened her eyes and shook her head. \"Is it because you need money?\"\n\n\"Money is not the issue, Ali. I joined the Marine Corps within a week of graduating from high school because the recruiter promised a signing bonus and educational benefits. I joined and took advantage of everything they had to offer. I gave them ten years before the CIA came looking for 'a few good men.'\n\n\"I was hired and trained as a covert operative. I'd always believed I would die somewhere far from home, but it was at home where I almost lost my life when someone shot me, then left me on a street corner in D.C. to die.\"\n\nHer eyes widened. \"What happened?\"\n\n\"Three men accosted me, demanding I give them whatever I had on me. If it had been just two of them I knew I could've taken them, so I cooperated and handed over whatever money I had. However, the one holding the gun panicked and shot me in the stomach. If it hadn't been for Michael Kirkland being where he was that night I wouldn't be sitting here talking to you. He'd taken a shortcut through southeast D.C. on his way home and found me in a pool of blood. Not waiting for an ambulance, he put me in his car and took off for the nearest hospital where I spent six hours in surgery.\n\n\"The bullet had nicked my spleen, mangled my left kidney and when I woke up in recovery I was told that I'd lost my kidney and half my body's blood supply.\" Merrick ignored the look of horror in the gold eyes staring at him.\n\n\"How long were you hospitalized?\"\n\n\"A month.\"\n\n\"What happened after that?\" Alex asked, her voice barely above a whisper.\n\n\"I went home and was placed on medical leave for five months. When I returned to work it was to permanent desk duty. Three days later I walked into the director's office and handed in my resignation.\"\n\nAlex heard the sound of her runaway heart in her ears. \"What did you do after that?\"\n\nMerrick paused, then said, \"I gave up my Silver Spring apartment and bought the house in Bolivar. I didn't have a wife or children to support, so whatever I'd earned I'd invested. I'd always followed the market, so when tech companies were issuing IPOs I bought in, tripling and sometime quadrupling my investments.\" What he didn't tell Alex was that once he made his first million, he'd stopped buying and selling stocks. His investments were now in long-term, tax-free municipal bonds.\n\n\"If you go back...when you go back,\" Alex began, \"will it be to a desk position?\"\n\nMerrick traced the outline of her ear with his forefinger. \"Yes. I suppose I had to get over the stigma of becoming a desk jockey when I'd been a field agent. It took time and my being with you to realize that I still can protect our country but in a different capacity.\"\n\nHer arching eyebrows lifted. \"That's true, but why me, Merrick, and not some other woman?\"\n\nHe gave her a long, penetrating look. \"Because you're not some other woman, Ali. I fell in love with you because you're intelligent, determined, spunky and breathtakingly beautiful.\"\n\nMerrick traced the outline of her cheekbone with a fingertip. \"And what would it look like if my girlfriend got up every morning to go to work and left me home to lie around on my behind like a scrub? I couldn't do that and still feel like a man, querida.\"\n\nAlex closed her eyes and rested her head on Merrick's shoulder. He'd referred to her as his girlfriend, but at that moment she wanted to be more than a girlfriend. She wasn't certain whether her stance had changed toward marriage because she'd found herself more in love with Merrick Grayslake, or if it was seeing Moira and Hernando Rivera together as husband and wife, but she no longer wanted to be a girlfriend but a married woman.\n\n\"Merrick?\"\n\n\"Yes, darling?\"\n\n\"I'm ready to go back to the room.\"\n\nBeckoning to their waiter, Merrick settled the bill, then escorted Alex out of the restaurant, walking the three blocks that would take them back to the hotel.\n\nAs soon as they exited the elevator, Alex felt the intensifying effects of jet lag. She didn't remember Merrick undressing her, putting her to bed or joining her in the king-size bed. She dozed off and on for the next two days, and when her circadian rhythms stabilized she looked forward to reuniting with Merrick and Rome.\n\n## Chapter 16\n\nAlex woke to find Merrick sitting at the table in their hotel suite, in a pair of pajama pants, sipping coffee and reading a newspaper. The strands of his coarse hair stood on end as if he'd run his fingers through it.\n\nA smile parted her lips. It was the first time she'd seen him wearing pajamas. In fact, she didn't know that he'd even owned a pair because he'd always come to bed naked.\n\nShe lay motionless, studying the man with whom she'd fallen in love, the man whom she prayed she would one day call husband. The reasons she resisted marriage were always the same: she hadn't met the right man, and because her parents were exerting pressure for one of their adult children to marry and make them grandparents\u2014her father in particular. David Cole resented his four older siblings' good-natured teasing that he would never become a grandfather when all of their married children had made grandparents and great-grandparents.\n\nAnd she'd asked herself over and over, was Merrick Grayslake the right man? And each time the answer was a resounding yes!\n\nAlex had disclosed to Merrick that eventually she wanted to marry and have children sometime in the future. His response was that he had no desire to marry or father children, which had proven an amicable arrangement for both\u2014at that time. But with time she'd changed and her feelings toward him had changed.\n\nShe'd agreed to meet him in Italy, knowing that she would never become Mrs. Merrick Grayslake. What Alex had to resolve before she returned to the States to begin a new career was whether she would continue in a relationship that represented perpetuity without matrimony.\n\nWithout warning, Merrick turned and stared at her. \"Good morning, Sleeping Beauty.\"\n\nShe smiled at him. \"Good morning, Sir Grayslake. How long have I been out of it?\"\n\nMerrick folded the newspaper. \"A couple of days. You were good as long as we were moving, but every time you sat down you fell asleep.\"\n\nPushing into a sitting position, Alex stretched her bare arms above her head. She hadn't remembered putting on the pale yellow nightgown, so it was obvious Merrick had dressed her for bed.\n\n\"It usually takes me a couple of days to adjust to the time difference whenever I cross the international date line. What did I miss?\"\n\nHe angled his head, unable to believe Alex hadn't remembered anything since she'd arrived. Was she that affected by jet lag or was she that exhausted?\n\n\"What do you remember?\"\n\nSighing, she closed her eyes. \"I remember we threw coins in the Fontana di Trevi.\"\n\nThe tradition was to throw a coin into the fountain if one wished to return to Rome. The Trevi was Rome's most famous fountain. Neptune, the fountain's central figure, flanked by smaller figures representing a calm and stormy sea, had become the focal point of the classic romance film Three Coins in the Fountain.\n\nMerrick smiled. \"Do you remember what you said when you tossed your coin?\"\n\nAlex sat up straighter. \"No.\"\n\nWhat she did remember was that she'd drunk too much wine at dinner that night. They'd eaten at a sidewalk caf\u00e9 around Piazza Navona and met an American couple visiting Rome for the first time. They spoke no Italian and their waiter, who spoke very little English, was unable to communicate with them until Merrick intervened. They wound up sharing a table for four and the native Chicagoans ordered several bottles of wine to accompany dinner.\n\n\"Did I do or say something embarrassing?\"\n\nPushing back the chair, Merrick stood up and walked over to the bed. He crawled in beside Alex, rested his back on the headboard and pulled her gently to his bare chest.\n\nThere was a full minute of silence before he spoke. \"You said you wanted to come back to Rome for your honeymoon.\"\n\nLetting out her breath in an audible sigh, Alex met his penetrating gaze. She hadn't embarrassed herself. \"I've always said if I married I'd want a Roman honeymoon.\"\n\n\"You want to get married?\"\n\nA slight frown furrowed her smooth forehead. \"I told you before that I want to get married and have a child.\"\n\n\"Some other man's child?\"\n\nPulling out of his loose embrace, Alex's frown deepened. \"What are you talking about?\"\n\n\"You're not going to marry another man.\"\n\nAlex was totally baffled at Merrick's behavior. He was talking in riddles. \"What I'm not going to do is marry you.\"\n\n\"Why not?\"\n\n\"Because you're not the marrying kind.\"\n\nMerrick shrugged his shoulders. \"I changed my mind.\"\n\nGoing to her knees, Alex leaned closer. \"Just like that,\" she drawled, snapping her fingers, \"you've changed your mind. I'm sorry, but I don't buy it, Merrick. Wasn't it you who said there's no reason why we can't enjoy each other without marriage and children becoming part of the equation? I told you what I want for my future and you've told me what you want, and we're not in agreement.\"\n\n\"That's where you're wrong, Ali. I do want what you want. I want you to become the mother of our children.\"\n\nWave after wave of shock assaulted her senses as a buzzing in her head increased to a roaring that made it impossible for her to think, hear, feel. Rage replaced her shock as she launched herself at Merrick.\n\n\"Don't play with my head!\"\n\nAli was quick, but Merrick was quicker when he held her wrists, shaking her gently. \"I'm not trying to play with your head, darling. I love you too much to do that.\"\n\n\"But I'm confused.\"\n\n\"What about?\"\n\n\"About you, Merrick, about us. We start out as friends, then become lovers. I respect what we have because we're able to talk about everything\u2014what we like, don't like. What we want and don't want. And we decided to have a relationship that would not end in marriage.\n\n\"I've floundered for years, and now when I'm back on track you try to derail me with mixed messages. You say you love me and you want to spend the rest of your life with me although marriage will not become a part of our future. I love you, Merrick. I love you more than I've ever loved any man, yet\u2014\"\n\n\"Then marry me, Alexandra Cole,\" Merrick said, cutting her off.\n\n\"No!\"\n\nHe tightened his hold on her wrists. \"Why not?\"\n\nHer eyes filled with tears. \"Because you don't believe in marriage, Merrick Grayslake!\"\n\nA wry smile parted his lips. \"That was before I met you.\"\n\nAlex struggled to free her hands. \"Let me go.\"\n\n\"Why? So you can hit me?\"\n\nHer eyes narrowed. \"I'm going to do more than hit you,\" she threatened.\n\nMerrick released her, then extended his arms out to his sides. \"Take your best shot, Ali,\" he beckoned. \"Come on, baby. Hit me if it will make you feel better.\" The seconds ticked off as they stared at each other in what had become an impasse. His expression softened as he reached out and cupped her chin in his hand. \"You don't want to hurt me any more than I can hurt you.\" Warm, fat tears dotted his palm. \"If you'd changed your mind about marriage I would've accepted it without question because you'd claim it's a woman's prerogative. Well, what's good for the goose is also good for the gander. Don't dudes have equal rights? Are we not allowed to change our minds?\"\n\nAlex smiled through her tears. \"Yes.\"\n\nResting his forehead against hers, Merrick kissed her nose. \"Yes, what?\"\n\n\"Yes, you're allowed to change your mind.\"\n\n\"Good. I'm glad we've settled that.\" He kissed her again. \"I love you, and I want you to marry me. It doesn't have to be now. It can be next year, the following year, whenever you want.\"\n\nAlex wanted to accept his proposal, but something Merrick said nagged at her. Something he'd said about not wanting to lose her to another man had triggered warning bells. Did jealousy play a part in his changing his stance toward marriage? Or was it because once she began working and traveling she wouldn't be that available to him?\n\n\"I'll let you know,\" she said softly. Even though she wanted to commit, she couldn't. Not yet.\n\nCradling her face between his palms, Merrick pressed his mouth to her throat. Increasing the pressure, his mouth moved lower as her stomach muscles contracted.\n\nAlex barely had time to react when she found herself in Merrick's arms as he slipped off the bed and carried her into the bedroom. Time stood still, the earth stopped spinning on its axis as they stood under the warm water from an overhead shower and demonstrated wordlessly their love for each other.\n\nWith her legs wrapped around Merrick's waist, her back pressed against the tiled wall, she breathed the last of her passion against his throat. This coming together was different from the other times. Merrick hadn't used contraception.\n\nAlex didn't panic because it was the safe phase of her menstrual cycle. \"This cannot happen again,\" she gasped, trying to catch her breath.\n\nMerrick supported her body until her feet touched the floor of the shower stall. \"You mean making love in the shower?\"\n\n\"No, Merrick. Making love without using a condom.\"\n\n\"I pulled out in time.\"\n\nResting her cheek on his chest, she looped her arms under his shoulders. \"And if you get me pregnant, then I'm going to name the baby Pull Out.\"\n\nMerrick threw back his head and roared with laughter. He didn't know what he was going to do with Alexandra Cole except love her until his last breath.\n\nAlex arrived in Venezia, or Venice, reuniting with the city built on water and made up of canals; she felt like a glutton consuming copious amounts of food yet never feeling satiated. If she loved Rome, then it could be said she had a passion for Venice that was akin to an unrequited love.\n\nShe'd never tired of the crowds, the countless pigeons in Piazza San Marco, the overrated and overpriced gondola rides, the museums, galleries, the dark callesellas or alleys, or high-priced souvenirs.\n\nMerrick had secured lodging on an upper floor of a furnished palazzetto along the Grand Canal. Alex opened a bedroom window and inhaled a lungful of air.\n\n\"It smells like Venice.\"\n\n\"How is Venice supposed to smell?\" asked the voice behind her.\n\nAlex turned and smiled at Merrick. If possible, she'd fallen more in love with him. He hadn't mentioned marriage again, and for that she was grateful. She knew he wanted her as his wife, and when the time came for her to marry she knew it would be to Merrick.\n\n\"Like the sea.\"\n\nMerrick stared at the petite woman with the curvy body with whom he'd begun his days and ended his nights. He never tired of making love with her. He also had acquired a newfound appreciation for art. Unlike when he and Alex had toured the D.C. museums, they'd embarked on a walking tour of Rome, and whenever he asked about a particular statue or building Alex would explain its origin and identify the artist.\n\nAnd because he was more fluent in Italian than she, he assumed the responsibility of ordering meals, negotiating with shopkeepers and asking directions. She taught him how to navigate crossing the streets with traffic coming in both directions as Vespa-riding Romans played a frightening game of tag with pedestrians brave enough to step off the sidewalks. Alex also turned him on to gelato, and he set out on a campaign to sample every variety of the custard-based Italian ice cream.\n\nThey'd spent five days in Rome before traveling northward to Florence. They'd varied their nighttime routine of walking to sitting on their hotel's rooftop as the lights of the city sparkled below. Florence was a shopper's haven for jewelry and leather goods. Larger than usual crowds cut short their stay in Tuscany, and Merrick decided to end their two-week European sojourn in Venice.\n\nReaching for his hand, Alex threaded her fingers through Merrick's. \"Beautiful, decadent Venice is like a woman no man can resist.\"\n\nWrapping his free arm around her waist, he pulled her close. \"Then your parents should've named you Venezia rather than Alexandra.\"\n\nShe lifted her chin to see gray eyes boring into her. As of late she'd begun analyzing everything Merrick said to her to uncover a hidden meaning. It was as if she had to erect a wall of defense to shield herself against him.\n\nStanding on tiptoe, she kissed his cheek. \"We should try and relax until siesta is over. I can assure you that we'll get very little sleep tonight, because Venice doesn't really start jumping until after sunset.\"\n\nAs in France and Spain, Italian shopkeepers, banks and other places of business closed for several hours in the afternoon for siesta.\n\nMerrick disengaged his hand and began the task of unbuttoning her blouse. His touch was deliberate as he pushed it off her shoulders and it floated to the floor. Reaching around her back, he unhooked her bra, and it, too, joined her blouse. Alex went completely still as he unzipped her slacks, pushing them and her panties down her legs until they pooled around her feet.\n\nShe stepped out of them at the same time she repeated his action when she unbuttoned his shirt; a minute later, both stood nude in the middle of the bedroom in an apartment dating back to the 13th century.\n\nHand in hand, they made their way to the antique bed, floating down to the crisp, ironed sheets. She wanted Merrick, craved him because she was ovulating.\n\nHer body felt heavy, like ripened fruit waiting to be picked. Her breasts were tender and she couldn't control the spasmodic trembling between her legs.\n\n\"Hurry,\" she pleaded as he slipped on a condom.\n\nMerrick knew this joining would not become the unhurried, leisurely lovemaking he was used to sharing with Alex. Her quick, shallow breathing, the lines of tension on her face and the bite of her fingernails on his biceps urged him to follow her command.\n\nThere was no foreplay as he entered her in one, sure swift thrust. Her urgency was transferred to him and he rode her like a man possessed by invisible spirits whose intent was to swallow him whole.\n\nHer hot flesh pulsed around his sex, squeezing tighter and tighter. Every muscle, nerve and tendon in his body screamed for release. The sound of his runaway heart was that of a galloping horse as he struggled not to release his passion\u2014not yet. It was too quick.\n\nAlex's raw sensuousness lifted him beyond himself, her fire spreading to him, and he lost it; he lost himself in a fiery downpour of uncontrolled passion in which he experienced la petite mort for the first time in his life.\n\nFlames of fire heated the blood in Alex's veins as she surrendered all she was and had to the man who'd taken her to unknown heights of passion. Their shared moment of ecstasy ended, Alex too emotion-filled to speak.\n\nThey lay together, entwined, the burning memory of their first day in Venice imprinted in their memory forever.\n\n\"Walk faster, Merrick.\" Alex, a vision in white, pulled Merrick along with her as she turned down an alley.\n\nHe smiled. She was like a child in FAO Schwarz at Christmastime. The white sundress with an empire waist and tiny pleated skirt that ended at her knees made her appear younger, virginal. He hadn't been the one to claim her innocence, but he could've been when he thought about the first time he'd made love to her. It was obvious she had only experienced conventional lovemaking because whenever he introduced another technique she'd become shy or embarrassed by it. Not one to push, Merrick waited for her to take the initiative before adding it to their sexual repertoire.\n\nThey left the alley and crossed a small bridge. The sight unfolding before him was humbling. The setting sun had turned the sky blood red, the outline of magnificent structures across the lagoon standing out in stark relief.\n\n\"It's incredible.\" There was no mistaking the awe in his voice. He pointed. \"What's that?\"\n\nAlex followed the direction of his finger. \"Chiesa della Salute.\"\n\n\"How do you remember the names of every church, cathedral, or duomo?\"\n\n\"Memorization. Unlike you, I don't have a photographic memory.\"\n\nMerrick stared at her delicate profile. \"Who told you I have a photographic memory?\"\n\nShe turned and stared directly at him. \"No one. You proved that the day we were in the National Museum of African Art. You'd recalled works of art I'd forgotten we saw. You're not just smart, Merrick. You're brilliant. Now I see why the CIA wanted you. And now I know why you're going back.\n\n\"You told me you need normalcy in your life, but I believe it's because you feel as if you're losing it, losing your edge. You miss the thrill of being offered a new assignment, and with the heightened threat of terrorism you believe you have to do something to protect your country.\"\n\nMerrick took her face in his hands, his eyes making love to her. \"You're right, Ali. But you left out one piece of the puzzle.\"\n\n\"What's that?\"\n\n\"You didn't mention yourself.\"\n\n\"How do I figure into the picture?\"\n\n\"You've helped me to know what love is.\"\n\nA shy smile flitted across her face. \"Love is not taught, Merrick. Either you feel it or you don't.\"\n\nHe angled his head. \"It could be I hadn't met the right woman.\"\n\nHer smile widened. \"I'll accept that because I've dated my share of losers.\"\n\n\"Were they losers, or you just weren't compatible?\"\n\n\"Probably both.\" Standing on tiptoe, she brushed a kiss over his mouth.\n\nCurving an arm around her waist, he dipped his head and kissed her soft, parted lips. \"Are you ready to eat?\"\n\n\"Does a cat think it's the meow?\"\n\nShaking his head, Merrick smothered a laugh. One thing he was certain of, and that was life with Alexandra Cole would never be boring.\n\n## Chapter 17\n\nAlex returned to the States, tanned, relaxed and more in love. It took two days for her to air out, dust and stock the pantry and refrigerator in her Arlington condo.\n\nShe called Michael to get an update on Jolene but got the Kirklands' voice mail. Three hours later Michael called to inform her he'd become a father. Jolene had given birth to a perfectly formed, healthy girl. They'd named her Teresa April Kirkland in honor of her paternal grandmother and one of Jolene's clients. Mother and daughter were well, and Alex told Michael she'd come by the hospital the following day to see Jolene and meet her newest cousin.\n\nAlex walked into the small private hospital set on twenty acres of prime property in affluent Georgetown. The medical staff housed in a former Georgian mansion provided one-on-one care for patients wealthy enough to pay five-figure-perday fees for maximum privacy.\n\nJolene Kirkland's room wasn't a room but a suite with a balcony overlooking a formal English garden. She sat up in bed, pillows cradling her shoulders for support, watching a flat-screen television.\n\nA cradle was positioned on the right side of the bed.\n\nAlex rapped lightly on the door. \"Hello, Mommy.\"\n\nJolene, reaching for the remote device, turned off the television. \"Please come in, Alex.\"\n\nCarrying a cellophane wicker basket filled with aroma-therapy items, Alex walked in and placed the basket on a damask-covered table. Jolene looked incredibly beautiful for a woman who'd just endured thirteen hours of difficult labor.\n\n\"There's something in that basket for mother and baby.\" She approached the bed and hugged Jolene. \"Congratulations.\"\n\nJolene smiled. \"Thank you.\"\n\nAlex walked around the bed, peering into the cradle. Wrapped in a pale pink blanket was an infant with tufts of jet-black hair. \"She looks like Emily.\"\n\n\"That's what I told Michael.\"\n\n\"What color are her eyes?\" Alex wanted to know if Teresa had inherited her father's green eyes.\n\n\"I don't know. Little Miss Willful has yet to open them.\"\n\nAlex sat on a cushioned chair next to the cradle. \"Is she a good baby?\"\n\n\"No,\" Jolene said without hesitating. \"The only time she isn't crying is when she's eating and sleeping.\"\n\n\"Perhaps she has gas.\"\n\n\"Perhaps she wants to be held,\" Jolene countered.\n\n\"But she's only a day old. And who's been holding her?\"\n\nJolene rolled her eyes. \"Take a guess, Alex.\"\n\n\"No!\" Alex moaned. \"Please don't tell me that Michael has begun spoiling her already.\"\n\n\"You didn't hear it from me.\"\n\n\"I can't...\" Alex's response was preempted by a knock on the door. Turning, she saw Michael's broad shoulders filling out the doorway.\n\nHe gave Alex a bright smile. \"Hey, cuz.\" He walked into the room and kissed her cheek before he rounded the bed to kiss his wife.\n\n\"Congratulations, Michael. Your daughter is beautiful.\"\n\nWith clear green eyes dancing in excitement, Michael shook his head in amazement. \"I still can't believe she's here.\"\n\n\"She looks like Emily.\"\n\n\"That's what I told Jolene.\"\n\n\"If she's as outspoken as your sister, then you're in for it,\" Alex predicted.\n\n\"You're a lot more candid than Emily,\" Michael said in defense of his sister.\n\n\"Is this a private family gathering?\" asked a deep voice that sent a shiver of excitement throughout Alex. She turned to find Merrick standing in the doorway in a tan suit, with a white shirt and chocolate silk tie.\n\nSuddenly she found breathing difficult as her gaze met his. Would Michael know just seeing them together that his friend was sleeping with his cousin? Was this the time when she'd reveal her relationship with Merrick?\n\n\"Come on in, Gray.\" The two men shook hands, at the same time exchanging a rough embrace. \"Have you met my cousin, Alexandra Cole?\"\n\nMerrick extended his hand, belaying her fear that he would reveal the extent of their involvement. \"Yes, I have. We met New Year's Eve.\"\n\nAlex shook his hand, smiling. \"We shared a dance New Year's Eve.\" She now was able to relax. Merrick hadn't let on that they'd just spent two weeks together in Italy. He'd given her a tangible graduation gift on the flight back to the States: an Etruscan-inspired gold necklace with a magnificent ruby briolette.\n\nMerrick inclined his head. \"You're right.\" Reaching into his jacket, he took out an envelope and gave it to Jolene. \"Congratulations. It's just a little something for your daughter.\"\n\nJolene reached out and patted his hand. \"You didn't have to give her anything.\" The moment Teresa drew breath she'd become heir to a share of the ColeDiz billion-dollar empire.\n\n\"That may be true, but I've been told that little girls and big girls can never have too many clothes.\"\n\nJolene opened the envelope and stared at its contents before handing the envelope to Alex. Merrick had given Teresa Kirkland a gift certificate to the finest children's boutique in the Capitol District.\n\nJolene smiled at the man who despite being her husband's friend frightened her. She'd found him too quiet and very mysterious. \"Thank you, Merrick. Teresa's certain to be one of D.C.'s best-dressed babies.\"\n\nMerrick moved over to stare down at the sleeping infant, his expression softening. \"Nice work, Kirk.\"\n\nAlex and Jolene exchanged looks that spoke volumes. \"He did have some help,\" Alex drawled.\n\n\"Speak, sister,\" Jolene chimed in.\n\nWincing, Merrick put up his hands. \"Did I say the wrong thing?\"\n\n\"Hell, yeah!\" the two women chorused.\n\n\"I suppose that's my cue to escape while I can,\" Merrick teased.\n\n\"Where are you off to?\" Michael asked him.\n\n\"Langley.\"\n\nAlex knew the CIA was headquartered in Langley, Virginia. And like marriage, Merrick's return to the CIA was not a topic she welcomed\u2014at least not at this time. She knew she was either selfish or in denial, but the only thing she wanted to focus on was securing a position with the National Trust. After that she would deal with her future with Merrick.\n\nMerrick left as quietly as he'd come and she spent another thirty minutes with the Kirklands before driving back to Arlington. She'd purchased the brand-new Acura Integra a month before closing on her condo, and in two years she still hadn't put more than three thousand miles on the vehicle. Instead of flying to Boston for her interview, she drove, stopping and touring the states in the Trust's northeast region.\n\nCIA Headquarters...Langley, Virginia\n\nMerrick sat in an interview room, answering the same question posed at least three different ways. What he detested most was the information necessary for security clearance. It was as if the interviewer wanted to know why he'd been born.\n\n\"Have you traveled out of the country in the past three years?\"\n\n\"Yes, I have,\" he answered truthfully.\n\n\"When?\"\n\n\"Three weeks ago.\"\n\n\"Did you travel alone?\"\n\n\"Yes.\"\n\n\"Were you alone during your return to the United States?\"\n\n\"No.\"\n\n\"Who did you travel with, and what is your relationship to said person or persons?\"\n\n\"I returned to the United States with a woman. Her name is Alexandra Cole.\"\n\n\"Did you and Ms. Cole reside together during your travels?\"\n\n\"Yes.\"\n\n\"Is Ms. Cole a citizen of the United States? If not, then what country?\"\n\n\"She is a citizen of the United States.\"\n\n\"Where does she reside?\"\n\n\"Arlington, Virginia.\"\n\n\"Has Ms. Cole always been a resident of the Commonwealth of Virginia?\"\n\n\"No. Ms. Cole was born and raised in Boca Raton, Florida.\"\n\nThe interrogation went on for hours, every answer recorded for dissemination and follow-up. Merrick was mentally and physically drained when he got into his car in the visitors' parking lot and drove away. His association with Alex had drawn her into an investigation that was not of her choosing. Her family name or wealth meant nothing to a governmental agency intent on dissecting her life.\n\nHe'd gotten security clearance once and there was no doubt he would again. But he wondered if the Justice Department would uncover something about Alexandra he didn't want or need to know. And as if he'd executed the maneuver countless times, Merrick took the road leading to Arlington. Activating the hands-free device, he pressed the number to Alex's cell phone.\n\n\"Hello, stranger,\" she crooned.\n\n\"Hello, Miss Cole,\" he countered, playing along with her. \"What do I have to do to convince you to come out and play?\"\n\n\"What are you playing?\"\n\n\"Strip poker.\"\n\nShe chuckled. \"Does stripping translate to the removal of articles of clothing?\"\n\n\"Shame on you. Get your mind out of the gutter, naughty girl.\"\n\n\"Would you like me if I were a good girl?\"\n\n\"I'd like you if you were toothless, bald and covered with carbuncles.\"\n\n\"What in the world are carbuncles?\"\n\n\"Warts.\"\n\n\"Dis-gus-ting.\" She drew the word out into three syllables.\n\n\"When should I expect you?\"\n\nMerrick took a quick glance at the dashboard clock. \"Sometime after five.\"\n\nHe disconnected the call, then began the process that would disassociate of what had transpired in the tiny room back in Langley. He'd relived his entire life during the session, and unfortunately there were events he'd relegated to the deepest recesses of his mind, hoping never to resurrect them. However, they'd resurfaced and would take days if not months to purge again.\n\nCarl Ashleigh listened to Merrick Grayslake's taped responses, comparing them to his previous security clearance report. Not much had changed in two years. The only exception was his involvement with a woman\u2014a woman who just happened to be heiress to a family-owned conglomerate.\n\nIt wasn't what Carl's superiors wanted, but it couldn't be helped. They needed Grayslake to flush out a big fish; but if his lady got caught up in the trap it would become Grayslake's responsibility to protect her.\n\nPicking up the phone, he dialed an extension. \"He looks good,\" he said into the mouthpiece. \"But there may be one hitch. He's involved with a woman, and I'm not certain whether we'll be able to keep her out of the picture until we spring the trapdoor. Don't worry. I'll keep you posted on everything.\"\n\nCarl hung up, staring at several framed photographs on the opposite wall. The director was counting on him to put Operation Backslap into motion before the end of the year. He had eight months to bring down the country's most powerful lobbyist, Chandler Duffy. With Duffy out of the way, the popular incumbent president was certain to lose his party's nomination for reelection.\n\n## Chapter 18\n\nIt took four days for Alex to drive to Boston for her interview, but she accomplished the return drive to Virginia in ten hours. On the trip northward she'd stopped and toured the historic districts in Pennsylvania, New Jersey and Connecticut, staying overnight in the respective states' capitals. She'd planned on calling Merrick her first night on the road, but when she checked in to her hotel the only thing she'd wanted was a warm shower and firm bed. When she checked her cell-phone voice message, there was a call from him. His terse reprimand about her not checking in with him raised her hackles, and she waited another day before calling him back.\n\nShe had her first serious disagreement with Merrick when she told him that she hadn't needed to check in with anyone since she'd celebrated her twenty-first birthday, and now at twenty-nine, soon-to-be thirty, she wasn't going to begin. Alex couldn't remember his brusque retort because she'd abruptly ended the call.\n\nThe volatile interchange forced her to rerevaluate her relationship with Merrick Grayslake, that perhaps she hadn't married or had a serious relationship because she unconsciously valued her independence; that she was unable to commit to sharing her life and future with a man.\n\nHer interview had gone well, and with her degree and outstanding grades she was offered a position with the National Trust for Historic Preservation. She'd formulated a game plan and had accomplished everything she'd set out to do.\n\nAlex loved Merrick, enough to marry him and become the mother of his children, but she refused to be subjugated by him or any man. She'd been reared to think for herself, nurtured and taught to take care of herself, and being a Cole meant she would always be able to take care of herself.\n\nIt was late afternoon when Alex crossed the state line from Maryland into the District of Columbia. An accident backed up traffic for miles along the road leading to the Key Bridge, so she headed south to the Fourteenth Street Bridge to cross the Potomac River. It was a longer route to Arlington, but at least she wouldn't have to sit in traffic.\n\nHalf an hour later, she pressed a device, raising the door to one of the four garages in the rear of her building, parked, then alighted from the car where she'd spent too many hours. Not bothering to remove her luggage from the trunk, Alex unlocked the front door, mounted the staircase and opened the door to her apartment, encountering a blast of hot air that forced her to take a backward step.\n\nPlacing her keys and handbag on the foyer table, she pressed a button, activating the central cooling unit, while chiding herself for not leaving it running during her absence.\n\nGoing into the bathroom adjoining her bedroom, she stripped off her clothes and placed them in a wicker hamper. Just as she prepared to step into the shower, the telephone rang. She returned to the bedroom to answer the call. A smile parted her lips. It was her sister.\n\n\"Hi, Ana.\"\n\n\"Where have you been?\"\n\n\"Let's begin again, little sister,\" she chastised softly. Ana was as bad as Merrick. What was with them clocking her? \"Hello, Ana.\"\n\n\"Sorry about that. Hi, Alex.\"\n\nAlex smiled. \"What's up?\"\n\n\"I'm in Georgetown. I've been staying with Michael, Jolene and the baby for the past two days. Alex, she is the most adorable little girl.\"\n\nAlex hadn't seen Teresa since the day she'd come to the hospital because she wanted to give Jolene time to recover before she had to entertain visitors. The fact that Michael, as a teacher, did not work summers would make Jolene's transition from wife to mother easier than it would if she had full responsibility of caring for Teresa.\n\n\"She is beautiful. Why are you staying with them when you could've stayed here?\" Before she'd left for Europe Alex had given her sister and brothers the key to her condominium so they could use her extra bedroom whenever they were in the D.C. area.\n\n\"I called you, but when I didn't get an answer I decided to hang out with Michael.\"\n\n\"I was in Boston on a job interview. Why didn't you call my cell?\"\n\n\"I just got a new cell phone, and I hadn't programmed in your number. Do you want company?\"\n\n\"Of course. Come on over.\"\n\n\"I'll see you in a bit.\"\n\nAlex hung up and headed for the bathroom.\n\nAna lay at the foot of Alex's California king bed, her head resting on a mound of pillows. The flickering light from candles on tables provided the only illumination in the air-cooled space.\n\n\"I can't believe you're going to get a real job,\" Ana teased her sister.\n\nAlex shifted until she found a more comfortable position against the headboard. \"I had a real job, Ana Cole, when I worked for that art gallery.\"\n\nAna sucked her teeth. \"That was play-play and you know it. Now, working for the National Trust is real.\"\n\nAlex smiled. \"I think the most exciting aspect of the position will be the traveling and meeting with groups whose focus is on preserving our history for future generations.\"\n\nTurning over on her side, Ana tried making out her sister's face in the subdued light. \"I don't understand how you get so excited about broken-down old buildings.\"\n\n\"That's because you can't see the beauty in them. What you see as broken down and worthless I see as the fading beauty of magnificent wraparound porches, moldings, lintels, columns and newel posts. Only in America do we tear down to make way for the new, which isn't necessarily better or prettier than what stood before. That's why I love going to Europe, because they value their architecture and history.\"\n\n\"Don't forget that the United States is an infant when you compare it to the rest of the world, Alex.\"\n\n\"But that's no reason not to respect past artisans. Many of the antebellum mansions were designed and built by African slaves.\"\n\nAna closed her eyes and listened to Alex recount the names of the mansions that rose to glory and splendor under the human toil of their ancestors. She was so caught up in the history that she jumped as if jolted by a bolt of electricity when the telephone on the bedside table rang.\n\nReaching for the instrument, Alex picked it up. \"Hello.\"\n\n\"Hello, cuz,\" said a deep voice with a distinctive Southwest intonation.\n\n\"What's up, Michael?\"\n\n\"My folks just called to say they're coming up for the Fourth. Jolene's parents are also coming in to see Teresa, so I decided to throw a little something. If you and Ana aren't doing anything, then come on by any time after noon.\"\n\n\"Hold on, Michael.\" Alex covered the mouthpiece with her hand. \"When are you going back to Boca?\"\n\nAna sat up and stared at Alex. She'd come to the D.C. area to meet with a young female vocalist who'd sent Serenity Records a demo, but once she met with the fifteen-year-old she realized she wasn't the one who'd recorded the songs. Livid because she'd been duped, Ana hadn't bothered to stay and uncover who'd actually recorded the demo and decided to extend her stay to visit her sister and cousin.\n\n\"There's no rush for me to go back. Why?\"\n\n\"Michael wants us to come over for the Fourth. Uncle Josh and Aunt Vanessa are coming up to see the baby. He said Jolene's folks are also coming in from Chicago.\"\n\nAna flashed her dimpled smile. \"Count me in.\"\n\nAlex removed her hand. \"We're in. Do you want us to bring anything?\"\n\n\"No. I've got everything covered.\"\n\n\"How about dessert?\"\n\n\"I'll order pies and cobblers from Rudy B's.\"\n\n\"Hot damn!\" Alex had discovered Rudy B's by accident when she drove through northwest D.C. The original Rudy B's, a one-room log cabin, was built behind the small house that belonged to Reuben Brown in 1908. It was expanded after World War I, then again in the sixties, and had undergone extensive renovations in 1998 for its ninetieth anniversary. The family-owned restaurant had earned the reputation of serving the best soul food in the Capitol District. The first time she ate a slice of sweet-potato pie she was hooked!\n\n\"I guess that means we'll just show.\"\n\nMichael laughed. \"You do that.\"\n\nAlex hung up, smiling. \"Save your appetite because Michael's ordering dessert from Rudy B's.\"\n\n\"Who or what is Rudy B's?\" asked Ana.\n\n\"It's the best soul food joint in the world.\"\n\nAna lifted her eyebrows. \"Did you say the world?\"\n\n\"You heard me.\"\n\nSwinging her legs over the side of the bed, Ana pushed her feet into a pair of sandals. \"Get up. We're going on a food run.\"\n\nAlex took a quick glance at the clock next to the telephone. It was 8:48 p.m. \"Are you sure you want to eat this late?\"\n\nCombing her fingers through her short hair, Ana gave Alex a knowing look. \"Don't play yourself, Alexandra. You know it's never too late to eat. Especially when the food's good.\"\n\nGrinning, Alex swung her legs over the bed and pushed her feet into a pair of worn leather flip-flops. \"Let's roll.\"\n\nMerrick had spent the night vacillating over whether to accept Michael's invitation to come to Georgetown for a cookout or stay in Bolivar because he knew Alex would be there; she'd hung up on him, and his pride, false pride, wouldn't permit him to call her back.\n\nHe missed her, missed their nightly chats even when they were hundreds of miles apart. He missed her so much that his heart ached. There had been a time when he'd accused her of ripping his heart out, and she had, leaving him to bleed emotionally.\n\nPerhaps, he mused, he wasn't cut out to be a good boyfriend because he didn't know how to be a boyfriend. There were women, women he'd slept with, but they weren't someone with whom he would plan a future.\n\nLove, the emotion that had eluded him, had come into his life like a slow-moving mist, the silent paws of a stalking cat and death when it came stealing to reclaim the breath given at birth. It was so unexpected that he wasn't given the opportunity to reject it.\n\nSlumping lower in the porch chair, he stared at a doe and her fawn feeding on blackberries. Something must have startled them because they took off running in the direction of the woods.\n\nMerrick closed his eyes, sighing heavily. Love. Men and women sang about it, poets wrote sonnets about it, people took their own lives and killed others for it.\n\nHe'd fallen in love with a slip of a woman who made him look for the next sunrise, a woman he had proposed marriage to and wanted to have his children with, a woman who had pulled him from a morass of self-pity and irrelevance where one day turned insignificantly into the next. He opened his eyes, pushed off the chair and went into the house. Grasping the hem of his T-shirt, he pulled it over his head.\n\nHe had to stop the bleeding, and the only way he could do that was to confront Alex.\n\nMerrick drove from Bolivar to Washington, D.C., utilizing back roads and pushing the speedometer above ninety. Crossing the Virginia state line, he slowed to the speed limit. He'd made the trip in half the time.\n\nHe made one stop on Connecticut Avenue at Dupont Circle. Returning to his truck with his purchase, he drove the short distance to Q Street NW. Fortunately, Merrick found parking outside Michael's house. The cars lining the driveway bore Virginia and Florida license plates.\n\nHoisting a box to his shoulder, he walked around to the rear of the house. A large awning and colorful beach umbrellas shielded the Kirklands' guests from the blazing summer sun as they sat at tables eating, drinking and laughing.\n\nMerrick saw her, and went completely still. Alex had her back to him so he was given the opportunity to observe her unnoticed. A baseball cap covered her hair and a pair of sunglasses shielded her eyes. His gaze caressed the smooth brown skin on her back and shoulders under a halter top.\n\n\"Hey, man. What do you have there?\"\n\nMerrick lowered the box and put it in Michael's outstretched arms. \"It's just a little something to quench your thirst.\"\n\nMichael's eyes, hidden behind the lenses of a pair of sunglasses, crinkled when he smiled. \"What is it?\"\n\n\"Your favorite sake.\"\n\n\"All right,\" Michael drawled. He'd acquired a fondness for the Japanese brew when he'd been stationed in Japan. \"Why don't you circulate while I take this in the house. I'll make the introductions later.\"\n\nMerrick wanted to tell Michael that he didn't need an introduction, at least not one to Alexandra Cole. He knew not only her name, but every inch of her compact body. He knew how to make her moan and sob in ecstasy. He knew what made her laugh and cry. And he'd heard her admit to loving him as much as he loved her.\n\nAlex felt Ana's fingers tighten on her wrist. \"He's here,\" she whispered close to her ear.\n\nShe frowned at her sister. \"What are you talking about?\"\n\n\"The guy from our New Year's Eve party. You know, the one who was staring at you and I told you he was hot.\"\n\nAlex forced herself not to move, react. She knew Ana was talking about Merrick. She'd wanted to ask Michael if he had invited him, but hadn't wanted to appear that interested. What her cousin didn't know was that not only did she know Merrick, but she was also in love with him.\n\n\"Don't get crazy,\" she whispered, sotto voce.\n\n\"He's looking this way, Alex. I take that back. He's coming this way.\"\n\nShe removed her glasses and schooled her expression to one of indifference when she saw Merrick making his way toward her. He wore a dark blue shirt with large green leaves over a pair of jeans that were molded to his lean hips and thighs. When he was less than three feet away, she stood up and extended her hand.\n\n\"It's nice seeing you again, Merrick.\"\n\nMerrick ignored her hand for several seconds, and his expression was that of complete unconcern. He saw a flash of fear in her eyes; she didn't want her family to know how well they knew each other; she was still hiding their affair. Reluctantly he took her hand, certain he heard her exhalation of breath. He decided not to out her.\n\n\"Same here, Alex.\"\n\nShe mouthed a thank-you. \"Do you know everyone here?\" she asked louder than necessary.\n\nHe released her hand. \"I'm afraid I don't.\"\n\nLooping her arm casually through his, she steered him toward her sister. \"Alex, this is Merrick Grayslake. Merrick, my sister, Ana.\"\n\nAna flashed her dimpled smile. \"Hi, Merrick.\"\n\nHe inclined his head. \"My pleasure, Ana.\"\n\nBefore Ana could utter another word, Alex escorted him over to her aunt and uncle. \"Aunt Vanessa, Uncle Josh, I don't know if you remember Michael's friend from the\u2014\"\n\n\"Oh, I didn't know that you were making the introductions,\" Michael interrupted.\n\nAlex released Merrick's arm. \"He's your guest, so I'll let you do the honors.\"\n\nThat said, she walked away, feeling the heat from Merrick's gaze burning into her back. She made her way into the house and a bathroom off the gourmet kitchen. Sitting on a chair, she closed her eyes. She had to get Merrick alone; they had to talk, but not in front of a crowd of people.\n\nAlex lingered in the bathroom long enough to splash cold water on her face, before patting it dry with a paper towel. She knew she couldn't hang out in the bathroom, so pulling back her shoulders she opened the door, stopping short when she saw Merrick in the kitchen with Jolene. He'd cradled Teresa to his chest, smiling and making cooing sounds. The sight of him holding a baby stopped her breath. He'd talked about having a child\u2014their child.\n\nJolene's dark eyes sparkled like polished onyx. \"It appears as if my daughter is partial to men. The minute a woman, other than myself, picks her up she cries.\"\n\nMerrick's head came up and he smiled at Alex. \"Does she cry with you?\"\n\n\"I haven't held her.\"\n\n\"Come hold her,\" Jolene urged.\n\nAlex took half a dozen steps and held out her arms for the two-week-old infant. She smiled when feeling the slight weight. \"Hey, baby girl,\" she crooned and Teresa opened her eyes. Brilliant green eyes met a pair in gold. The two cousins regarded each other for a full minute before Teresa yawned, closed her eyes, then fell asleep.\n\n\"I guess there goes your theory about her preferring men,\" Alex said proudly, handing the infant to her mother.\n\nJolene cradled the baby to the shoulder draped with a cloth diaper. \"There must be something about you that Teresa likes, because she won't let my mother or Vanessa hold her. I've told her it's not nice to shun her grandmothers because they're always more indulgent than grandpas.\"\n\n\"You're right about that.\"\n\nJolene shifted her baby to a more comfortable position. \"Alex, would you mind seeing that Merrick gets something to eat?\" She smiled at Merrick. \"Now, if you were family, then it would be the reverse.\"\n\nCrossing his arms over his chest, Merrick angled his head. \"What do you mean?\"\n\n\"If you were married to Alex, then it would be you bringing her a plate, not the reverse.\"\n\nThere came a beat of silence as Merrick regarded Alex. \"I wouldn't have a problem serving Alexandra.\"\n\nHis response seemed to surprise Jolene, who'd lifted her arching eyebrows. She never would've expected Merrick Grayslake to be accommodating, especially to a woman.\n\n\"How interesting,\" she said in a quiet tone as she gently rocked her baby.\n\nAlex winked at Merrick. \"Let's go, Mr. Grayslake. Enjoy this while you can.\"\n\nMerrick took long strides as he followed Alex. \"When are you going to let me serve you, querida?\"\n\n\"We'll talk about it.\"\n\n\"When, Ali?\"\n\nShe stopped, turned and glanced up at him. \"I'll come to you after my sister goes back to Florida.\"\n\nMerrick held her gaze for a full minute. He wanted to ask her when, but held his tongue. He would wait as long as it took, not because he wanted to but because Alex was worth it.\n\n## Chapter 19\n\nAlex navigated Deadman's Curve, her heart in her mouth. It had taken three weeks to fulfill her promise to come to Merrick. Their relationship had changed. The nightly telephone calls had stopped, replaced by one or two a week. The urgency to see each other had waned, replaced by a waiting as to when they would be reunited.\n\nShe also found it odd that her feelings for Merrick had intensified. It was as if the separation allowed her to take a step back to confirm what she did share with him was love and not lust.\n\nAlex would've come to West Virginia sooner if Ana hadn't extended her stay in Virginia another two weeks; the morning of her sister's departure she received a letter of hire from the National Trust and that they wanted her to return to Boston for an orientation session; this time she flew to Massachusetts instead of driving.\n\nWithin miles of her crossing the West Virginia border the weather changed. An oppressive humidity and green-gray haze made normal breathing nearly impossible. She chanced glancing away from the curving, winding roadway for a second to adjust her car's air-conditioning.\n\nAlex hummed a tuneless ditty to take her mind off the road ahead of her, while chiding herself for driving to Bolivar and cursing Merrick for living in the wilderness. She decelerated when she saw the bumper of a pickup several car lengths in front of her. It was apparent she wasn't the only driver going under twenty miles per hour.\n\nDeadman's Curve leveled off as she maneuvered into a valley nearly obliterated by the thick haze. She continued along the paved road, peering through the windshield for the turnoff to Merrick's house. She missed it the first time and had to reverse directions.\n\nThen she saw it! Merrick's house was ablaze with light, a beacon in a darkened sky that looked more like night than day. It was only ten in the morning. She maneuvered around to the garage where he housed his classic cars, parking behind his truck.\n\nMerrick heard the sound of a car's engine, followed by the solid slam of a door. He was on his feet and on the porch in a matter of seconds. A smile curved the corners of his mouth when he spied the petite figure in a tank top, jeans and running shoes.\n\nHe'd waited patiently for Alex to come. Whenever they talked he'd wanted to ask her when, but couldn't bring himself to tell her how much he missed and wanted to see her. The three weeks had given him the space he needed to assess his feelings for the only woman to whom he'd given his heart. He loved Alexandra Cole, loved her enough to give up his life for her.\n\nMerrick realized in introspect that he'd made a gross error in judgment when he'd attempted to monitor her whereabouts. How had he forgotten that she was an independent woman who'd traveled and lived abroad, a woman who didn't need him or any man for her day-to-day existence? Alex was educated, independently wealthy and had focused her energies on beginning her career, and for that he was forced to respect her.\n\nHe came down off the porch to meet her. The first thing he noticed was that she was thinner, her face and body. Her hair was styled differently. Her ubiquitous tousled, mussed look had been replaced by a chic cut that framed her tiny face.\n\nExtending his arms, he closed the distance between them. It wasn't until he felt the soft crush of her breasts against his chest, inhaled the familiar scent of her perfume and felt the softness of her mouth when he kissed her that he knew she wasn't a figment of his imagination.\n\nHis mouth caressed hers as if it were fragile porcelain. Her lips parted and he deepened the kiss; there was a dreamy intimacy that hadn't been in any of their kisses; his lips parted in a soul-searching journey to taste every inch of her mouth.\n\nAlex wound her arms under Merrick's shoulders, holding him fast. Her knees shook, her heart pounded a runaway rhythm as she struggled to catch her breath. His mouth revived her troubled spirit like rain soaking the parched earth. She'd missed him\u2014oh how she'd missed everything about him!\n\n\"Take me inside,\" she whispered as pleasure radiated from the core of her body outward. Heat rippled under her skin, increasing with each second until she felt as if her entire body was on fire.\n\nHer thoughts spun out of control, her emotions whirled, skidded and raced headlong into a whorl of sensual delight from which she never wanted to escape.\n\nShe loved Merrick, loved him more than she could've imagined loving a man. She wanted and needed him to remind her why she'd been born female.\n\n\"I love you so much,\" Merrick whispered against her lips. Tears filled her eyes with his passionate confession.\n\n\"Take me to bed and I'll show you how much I love you,\" Alex promised.\n\nMerrick needed no further urging. He climbed the steps and walked into the house, closing the door with his foot. The entire house was ablaze with light to offset the eerie darkness that had settled on the region. It wasn't noon, yet the sky was black as pitch.\n\nCradling Alex's slight weight, he took the stairs, two at a time, and walked into his bedroom. Gently, he eased her down onto the bed, his body covering hers. He wanted to go slow, but his body refused to reason with his brain.\n\nA roll of thunder shook the earth, followed by a crack of lightning that rent the air. The approaching storm was no match for the one racing through the bedroom. Buttons were ripped from their fastenings as pieces of discarded clothing littered the large bed.\n\nMerrick reached over and swept everything off the mattress with the flick of his hand. Wrapping an arm around Alex's waist, he tucked her beneath him, parted her legs with a knee and entered her body in one sure upward thrust.\n\nThey gasped in unison with the joining of flesh akin to smoldering metals. There was no time for exploration, as Alex's urgency matched Merrick's long-denied needs.\n\nShe couldn't control the moans slipping from her parted lips, her hips writhing in a dance of desire, her breath coming in long, surrendering gasps. She rose to meet Merrick's powerful thrusts in a moment of unrestrained passion that exploded, then freed them in a firestorm of uncontrollable ecstasy. They lay together, waves of lingering passion flowing between them and making them one.\n\nAlex woke hours later, alone in bed, ravenous. Sitting up, she swung her legs over the side of the bed and stood up. She hadn't taken a step when she felt the evidence of her passion session with Merrick trickle down her inner thigh. Her heart lurched. They'd made love without using protection. They'd gambled before, rolled the dice and had come up winners; however, this time the odds were against them.\n\nAlex walked out of the bedroom and into the bathroom. She didn't want to think about the possibility that there was a new life growing inside her\u2014not now, not when she'd just started her career.\n\nThe thunderstorm continued unabated as torrents of rain lashed the roof and windows. Alex stepped off the last stair and followed the mouthwatering smell.\n\nShe stood under the entrance to the kitchen watching Merrick as he placed a cover on a wok. If she'd changed in the short time they'd been apart, so had he. He'd cut his hair again, the dark reddish-brown strands hugging his scalp. There was no doubt he'd spent a lot of time outdoors as evidenced by the rich mahogany brown of his face and arms. Without warning, he looked up and smiled, his silver-gray eyes shimmering like newly minted quarters in his dark face.\n\n\"Did you sleep well?\" he asked.\n\n\"How long was I asleep?\" The first time she'd come to Merrick's house she couldn't find a clock. It was only when she set the clock on his microwave with the time on her watch that she was able to discern the hour.\n\n\"About six hours. You must have been exhausted.\"\n\nWalking into the kitchen, she stood on tiptoe and kissed his cheek. She smiled. \"Exhausted and hungry. What did you cook?\"\n\nWrapping his arms around her waist, he pulled her to his length. \"London broil, garlic roasted potatoes and stir-fried green beans.\"\n\nAlex rested her head against his shoulder. \"I think I'll keep you.\"\n\nHe pressed a kiss on her damp hair. \"You better, because we're stuck with each other.\"\n\nShe closed her eyes for several seconds. \"Speaking of being stuck with each other, we took a risk this morning when we made love without using protection.\"\n\nMerrick let out an audible sigh. While Alex slept he'd agonized over what they'd done. It wasn't his intent to make love to her without a condom, but at that moment it was as if he'd taken leave of whatever common sense he had.\n\n\"I'm sorry, Ali, I didn't\u2014\"\n\nShe stopped his apology when she placed her fingertips over his mouth. \"Don't apologize, Merrick. I should've stopped you.\"\n\nGrasping her wrist, he pulled her hand down. There was a lethal calmness in his eyes that made it impossible for her look away. \"Why didn't you, Ali?\"\n\nLowering her gaze, she stared at the stark-white T-shirt stretched across his solid chest. \"I don't know.\"\n\n\"Whatever happens I don't ever want you to forget that I'm in this with you.\"\n\nShe nodded, closing her eyes. \"I suppose you'll remind me if I ever forget, won't you?\"\n\n\"Hell, yeah,\" he said, smiling. \"Let me feed you before you faint on me.\"\n\nHe dropped his arms and steered her to the table in the corner of the large kitchen that had been set with place settings for two. He pulled out a chair and seated her. Leaning over, he pressed a kiss to the column of her neck.\n\nA knowing smile softened Alex's face as Merrick placed a bowl of avocado salad on the table. Merrick would fit in perfectly with the men in her family. He could cook, and he was secure enough with his masculinity to serve his woman.\n\nAlex took a glance at her watch for the umpteenth time. The meeting of the northeast Trust's Waterfront Historic Action League in New Bedford, Massachusetts, was running late. As liaison for the National Trust, she had come to support the WHALE project, whose focus was the rehabilitation of a fire-damaged commercial building in the historic whaling community, but the queasiness persisted. And she didn't have to see a doctor to know the source of the nausea.\n\nShe was pregnant!\n\nWhat she'd hoped for in the future had become a reality\u2014now. She would marry Merrick and make David and Serena grandparents.\n\nMercifully, the meeting ended and Alex gathered her notes. She would decipher them during the flight back to Virginia; she would then type them and forward copies to her supervisor.\n\nAlex liked her position as a liaison because she was like a doctor on call. She didn't have to go into an office, except for monthly meetings, and she worked independently on the projects to which she'd been assigned. Not knowing where she would be sent was the only drawback. It could be Delaware one week, then two weeks later Rhode Island or New York. She refused to think about being stranded in an airport in Vermont or Maine during a snowstorm.\n\nNot waiting to confer with the members of the board, she rushed out of the building to the lot where she'd parked her rental car. By the time she maneuvered into a strip mall, the nausea had stopped. A quarter of an hour later, she was back in her car, driving in the direction of her hotel.\n\nAlex lay across the bed in the darkened room, staring up at the ceiling. She'd pulled the drapes, shutting out the strong autumn sunlight. The home pregnancy test had confirmed her suspicions. She was pregnant with Merrick Grayslake's baby, and by her estimate she would deliver a son or daughter early May of the following year. They'd made love countless times after that time, but had always used protection.\n\nRolling over on her side, she reached for her cell phone on the bedside table. Merrick answered after the third ring.\n\n\"Hey, baby.\"\n\nShe smiled at his sensual greeting. Whenever he called her baby it came out in a deep, sexy growl. \"Hey, you,\" she countered.\n\n\"How are you?\"\n\n\"Pregnant.\" The sound of heavy breathing echoed in her ear. \"Did you hear what I said, Merrick?\"\n\n\"Yes, I did. When are you coming home?\"\n\n\"I'll be home in a couple of days, but I'm not going to stay,\" she said.\n\n\"Where the hell are you going? And to do what?\"\n\nAlex heard the rage and panic in his voice. She decided to ignore his query as to her whereabouts. This was not the time to fight with him. \"I'm going to see my mother. I'm going to need her help in planning our wedding.\"\n\nThere was a beat of silence. \"Ali?\" His voice was softer, calmer.\n\n\"Yes, Merrick?\"\n\n\"Do you want me with you when you tell your parents?\"\n\n\"No, mi amor. I have to handle this by myself. As soon as the arrangements are finalized I'll let you know when you should come to Florida.\"\n\n\"Ali...baby, I...I don't know what to say.\"\n\nShe smiled. \"How about congratulations?\"\n\n\"Congratulations to you...and to me!\"\n\n\"You're silly, Merrick Grayslake.\"\n\n\"No, I'm not. I'm going to become a daddy.\"\n\n\"After I check out tomorrow I'm going down to the Cape to see my brother Gabriel. I'll spend the night with him, then fly out of Logan Saturday morning. I'll call you as soon as I touch down.\"\n\n\"Please don't make me worry about you, Ali.\"\n\n\"I won't. I love you, Merrick.\"\n\nThere was a pause. \"I love you, too.\"\n\nAlex ended the call. Picking up the receiver for the hotel phone, she ordered room service. Afterward she thought about calling Gabriel to let him know she was coming, but decided to surprise him. And no doubt he would be very surprised when she told him he was going to be an uncle.\n\n## Chapter 20\n\nAlex maneuvered her midsize rental car into the driveway leading to her brother's house, coming to a stop in front of the two-car garage. It was nine forty-five at night, and she prayed Gabriel was home, and if he was, that he hadn't gone to bed. She got out of the car. She was in luck. Gabriel bounded off the porch, arms outstretched.\n\n\"Hey! What are you doing here?\"\n\nAlex kissed her brother's cheek. \"I had to see you and give you my good news.\"\n\nCradling her face between his palms, Gabriel smiled at the same time he shook his head. \"You could've used the telephone, Alex.\"\n\n\"I was already in Boston for a meeting.\"\n\nHis hands came down, and he held her at arm's length. \"Look at you, Miss Corporate. You look wonderful.\"\n\nAn attractive blush suffused her face under a deep summer tan. She'd traded her jeans and T-shirts for a black wool crepe suit, ivory silk blouse, black leather pumps and a strand of perfectly matched pearls with matching studs in her pierced lobes.\n\n\"Thank you.\"\n\nGabriel reached for her hand. \"Come inside and we'll talk.\"\n\nAlex followed him up to a wraparound porch and into a spacious entryway. A formal living room held two facing gleaming black Steinway concert pianos.\n\n\"So much for your little place,\" she murmured. \"This house is enormous.\" Bleached pine floors, French doors and pale walls projected an atmosphere of openness.\n\n\"It'll do,\" Gabriel said with a hint of pride. He led the way to the family room. He indicated a love seat. Alex sat, Gabriel taking the cushion next to her. \"Do you want anything to eat or drink?\"\n\n\"No, thank you. I ate dinner at the hotel before I left.\" She glanced around the meticulously furnished room. \"I really like what Aunt Parris has done here.\" Martin Cole's wife, Parris, was a highly trained interior decorator. \"Your house looks like a home. All you need is a wife and a few kids to make it look completely lived in.\"\n\nDeliberately ignoring her reference to a wife and children, Gabriel peered closely at his sister. \"Tell me your good news.\"\n\nA mysterious smile curved Alex's lush mouth. \"I'm getting married.\"\n\nA muscle tensed in Gabriel's jaw. \"Who is he?\" The three words were squeezed out between clenched teeth.\n\nAlex stood up and walked over to the French doors overlooking the rear of the house. Strategically placed floodlights illuminated the backyard. She so wanted her brother to be happy for her.\n\n\"Please, Gabe.\"\n\nHe stood up and walked over to her. \"Please what?\"\n\nHer hands closed into tight fists, her nails biting into the tender flesh of her palms. \"Don't say it like that.\"\n\n\"How else can I say it, Alex? You come to me in the middle of the night with the news that you're engaged. I'm shocked, stunned and surprised.\"\n\nTurning around, she stared up at Gabriel. Her heart sank when she saw his expression. \"I thought you would be happy for me.\"\n\n\"I am, Alex. But...but it's so unexpected. You never said you were seeing anyone.\" He hesitated when she smiled. \"Do I know him?\"\n\nShe nodded. \"Yes.\"\n\nHis eyes narrowed. \"A name, Alexandra.\"\n\n\"Merrick Grayslake.\"\n\nGabriel went completely still. \"Michael's friend?\"\n\nAlex lifted her chin and placed her hands on her hips. \"Do you know another Merrick Grayslake?\" she spat out.\n\nGabriel threw up his hands at the same time mumbling a savage expletive. \"How the hell did you hook up with him?\" he shouted in Spanish.\n\n\"Don't you dare yell at me!\" she warned in the same language.\n\nRunning a hand over his face, Gabriel counted slowly to ten. He'd run interference more times than he could remember to scare away men who'd considered his sisters fair game; he'd heard a few refer to Alex and Ana as \"fresh meat.\"\n\nHe knew Merrick Grayslake was his first cousin Michael Kirkland's friend. He'd met him at Michael's wedding, and had found the man mysterious and somewhat sinister looking.\n\nWhy was it, he thought, that his sister was drawn to men who were the complete opposite of her? If they weren't walking on the wild side, then they were on the edge. He didn't know much about Merrick except that he lived somewhere in West Virginia and had at one time worked for the CIA.\n\nGrasping her hand, Gabriel led Alex back to the love seat, sitting and pulling her down next to him. \"Talk to me, Alex.\"\n\n\"Where do you want me to start?\"\n\n\"From the beginning.\"\n\n\"I met Merrick at Michael's wedding. We shared a few dances, and he asked me for my number. I gave him my cell phone because I was between the States and Mexico. I must confess that I felt uncomfortable around him for a while, and then one day it disappeared and I saw him in a whole new light.\" She didn't reveal the whole new light was when they went from being friends to lovers. \"At first I was impressed with his intelligence, but once I got past that I saw him as a lonely man.\"\n\nGabriel shot her an incredulous look. \"You're marrying him because you feel sorry for him?\"\n\n\"Of course not. I'm marrying Merrick because I love him. I also want to tell you that I'm...\"\n\n\"You're what?\" he asked when she did not complete her statement.\n\n\"I'm pregnant.\"\n\nGabriel's expression of shock and annoyance gave way to a brilliant smile. \"Hot damn! I'm going to be an uncle.\"\n\nAlex put her arms around his neck. \"Thank you for being happy for me.\"\n\nHe kissed her forehead. \"How can I not be happy for you? When's the wedding?\"\n\nPulling back, her eyes sparkling like multifaceted citrines, Alex blinked back tears of joy. \"Soon.\"\n\n\"Does Grayslake know he's going to be a father?\"\n\nAlex nodded. \"Yes. I told him last night.\"\n\n\"How far along are you?\"\n\n\"Probably about six weeks.\"\n\n\"Have you seen a doctor?\"\n\n\"Not yet. I have an appointment for Tuesday morning. You're the first one to know\u2014other than Merrick of course.\"\n\n\"Why didn't you tell anyone you were dating him?\"\n\n\"I didn't want to say anything until I was certain he was the one. Our relationship changed after he called, asking me to join him in Italy after I'd graduated. Of course I'd been there before as an art student, but I saw it differently because I realized then that I had fallen in love with Merrick.\"\n\nGabriel's smile was tender. His sister was in love. \"You're really happy, aren't you?\"\n\nShe nodded. \"So much so that I'm frightened.\"\n\n\"Have you guys set a date?\"\n\n\"No. But I don't want to wait too long. Looking like someone who swallowed a melon isn't too cool for a bride. I know you're teaching, but when are you off?\"\n\n\"We're off Columbus Day, Veterans Day, Thanksgiving and of course winter recess.\"\n\nAlex shook her head. It was the second week in September. \"That's too far off. It looks as if you're going to have to fly down to Florida for a weekend. I want you to sing at my wedding.\"\n\n\"Let me know and I'll be there.\" He angled his head. \"What about your position with the Trust?\"\n\n\"If there's an opening in the southern region, then I'll request a transfer.\"\n\n\"And if there isn't?\"\n\n\"Then I'll sit home knitting booties and piecing quilts until the baby comes.\"\n\n\"What do you want?\" he asked. \"Boy? Girl?\"\n\n\"I don't care as long as it's healthy.\"\n\nGabriel ruffled her coiffed hair. \"Will I be able to find you in the hills of West Virginnie?\"\n\n\"We're going to use the house in West Virginia for weekends and vacations. We plan to live at my place until we find something bigger in a D.C. suburb.\"\n\n\"Do you have to live near D.C.?\"\n\n\"Merrick is going back to the CIA as an intelligence training specialist, so living in or around D.C. would be a lot more practical.\"\n\n\"That sounds like a good plan. You, Michael and Jolene can raise your kids together.\"\n\n\"What's nice is that our children will get to see one another more than we saw Michael and Emily.\"\n\n\"You're right,\" Gabriel agreed. Michael and Emily Kirkland had grown up in Santa Fe, New Mexico, and they usually saw them at family gatherings once or twice a year. Emily still lived in New Mexico with her husband and three children, but her brother had moved to the D.C. area after having been assigned to the Pentagon as a captain in the U.S. Army. Michael met his future wife, Jolene, in Georgetown.\n\n\"When do you plan to tell Mom and Dad?\"\n\n\"I'm going to try and see them before next weekend.\"\n\nGabriel glanced at the watch strapped to his wrist. \"I hope you know you're spending the night here.\"\n\nShe nodded. \"I don't think I could make it back to Boston without falling asleep behind the wheel.\"\n\n\"Did you bring luggage?\"\n\n\"I have a bag in the trunk of the car.\"\n\n\"Go upstairs and get ready for bed. I'll bring your bag in. You can take the bedroom at the top of the stairs on the left.\"\n\nAlex stared lovingly at her older brother. \"Thank you, Gabe.\"\n\n\"For what?\"\n\n\"For being you.\"\n\nHe cradled her to his chest, kissing her cheek. \"Love you, sis.\"\n\n\"Love you back, bro.\"\n\nShe handed Gabriel the keys to the rental, then made her way up the staircase. A wave of fatigue swept over her. If it wasn't the nausea, it was fatigue.\n\nAlex stayed on her feet long enough to wash her face, brush her teeth and take a quick shower. When she returned to the bedroom she found her bag by the closet door. She took out a nightgown, slipped it over her body, and as soon as her head touched the pillow she fell asleep.\n\n## Chapter 21\n\nA bright rising sun and the tangy smell of salt water greeted Alex when she woke up in the strange bed. Then, she remembered. She was at her brother's house on Cotuit. Gabriel had begun vacationing off-season on the Cape, but she'd never expected him to settle down in New England.\n\nPushing back the sheet, she moved off the bed and made her way to an adjoining bathroom. The gnawing feeling in her belly had intensified. She had two options: eat or faint.\n\nGabriel was in the kitchen when she walked in. His long damp hair hung down his T-shirt-covered back like satin ribbons. The aroma of freshly brewed coffee wafted into her nostrils. \"Now that smells wonderful.\"\n\nGabriel turned and smiled at Alex. \"Good morning. How are you feeling?\"\n\n\"Hungry.\"\n\nHe pointed to the table in the breakfast nook. \"Sit down and I'll make you some caf\u00e9 con leche.\" They'd grown up drinking warm milk with a small amount of brewed coffee.\n\nAlex sat down. \"Please don't make it too sweet.\"\n\nGabriel flashed a lopsided grin in an unshaven face. \"Is it because you're sweet enough? Does Merrick think you're sw-e-e-e-t?\" he drawled, unable to resist teasing her.\n\n\"Why don't you ask him?\" she retorted.\n\nA sobering expression came over Gabriel's face. \"I will when I see him again.\"\n\n\"He doesn't frighten that easily, Gabe, so forget about trying to intimidate him the way you've done with a few of the other men I've gone out with.\"\n\nGabriel's eyebrows lifted. \"Oh, so you know about that?\"\n\n\"Know, Gabe. Some of them told me what you said to them.\"\n\nHe shrugged a shoulder. \"I was just trying to protect my sister. Some of those clowns you called dates defied description.\" He poured milk into a saucepan and heated it. \"I was usually pretty good in sending them packing\u2014all except for Duane Jackson. It took a face-to-face to get Mr. Hip-Hop Master to stop seeing you.\" Alex's jaw dropped as she stared at him as if she'd never seen him before. \"Close your mouth, Alex. Yeah, I met with Duane.\"\n\n\"But you had no right to interfere in my affairs,\" she spat out once she recovered her voice.\n\nGabriel winked at her, smiling. \"If I hadn't interfered, then you wouldn't be sitting here telling me that you're planning to marry Merrick Grayslake.\"\n\nA wave of heat stung her cheeks. \"You're right about that.\"\n\nHe cupped an ear with his hand. \"Can I get a thank-you, Alexandra Cole?\"\n\n\"Like, thank you, Gabriel,\" she drawled in her best Valley-Girl imitation.\n\nAlex thanked her brother again when he set a large mug with the steaming milk and coffee in front of her. She took a sip, moaning softly. It was perfect. The warm liquid slid down the back of her throat, heating her chest and belly. Gabriel filled his mug with coffee, joining her at the table.\n\nHer hand halted in midair as she attempted to take another sip of coffee. A young, very pretty woman stood under the arched entrance to the kitchen dressed in a gray sweatshirt, matching pants and running shoes. Her hair, secured in a single braid, fell over one shoulder.\n\nA mysterious smile softened Alex's mouth. \"Shame on you, brother,\" she said softly in Spanish. \"You didn't tell me you had a girlfriend.\"\n\nGabriel jumped up, his chair crashing to the floor in a loud clatter.\n\n\"I'm not his girlfriend. We are colleagues.\" Walking into the kitchen, the woman extended her hand. \"I'm Summer Montgomery.\"\n\nRising to her feet, Alex leaned over the table and shook Summer's hand. \"Alexandra Cole. Your colleague's sister.\"\n\nGabriel, righting his chair, pulled it out for Summer. \"Would you like a cup of coffee?\"\n\n\"No, thank you. I usually don't eat or drink anything before jogging.\"\n\n\"What would you like for breakfast?\"\n\n\"Fruit, a slice of toast and decaf coffee.\"\n\nAlex smiled at Gabriel. \"If you're taking orders, then I'll have grits, eggs, bacon or sausage and biscuits.\"\n\nHe leveled a frown at her. \"If you keep eating like that, I'm going to start to call you porky.\"\n\n\"Bite me, Gabriel Morris Cole!\"\n\n\"Sorry, sis, but I'll leave that task to Merrick.\"\n\nAlex stuck out her tongue at her brother. \"You're gross.\"\n\nSummer pushed back her chair and stood up. \"I'd like to complete my jog before the sun gets too hot. I'll be back later.\"\n\nAlex watched Gabriel watching Summer's retreating figure, a knowing smile curving her lips. \"She's beautiful.\" With her long hair, dark eyes and nut-brown coloring, Summer Montgomery looked exotic.\n\n\"That she is,\" he said matter-of-factly.\n\n\"You like her, don't you?\"\n\nGabriel didn't meet his sister's gaze. \"Yes.\"\n\n\"Does working with her pose a problem for you?\"\n\n\"For me, no.\"\n\n\"What about her, Gabe?\"\n\n\"She's reluctant.\"\n\n\"Does that matter to you?\" Alex asked, continuing her questioning.\n\nShifting slightly, Gabriel stared at Alex. \"No. I can't change how I feel about someone just like that.\" He snapped his fingers. \"We've agreed to take it slow and see what becomes of it.\"\n\n\"That's the same thing Merrick said to me. And now I have to plan a wedding.\"\n\nGabriel held up a hand. \"Back it up, Alex. No one said anything about getting married.\"\n\n\"Don't you want to get married? Start your own family?\"\n\n\"I don't consciously think about it.\"\n\n\"You didn't answer my question, Gabriel.\"\n\n\"And I don't intend to answer it, Alexandra.\"\n\n\"You don't have to,\" she said smugly. \"And because you're being evasive tells me that you do.\"\n\n\"Just because your fairy godmother sprinkled you with fairy dust, it doesn't mean it will happen to me.\"\n\n\"It's going to happen, Gabe,\" Alex predicted. The expression on Gabriel's face when Summer walked into the kitchen was one she'd never seen before, and Alex had seen her brother with enough women to know this one was different.\n\nA frown appeared between Gabriel's eyes. \"Next you're going to tell me that you can read palms.\"\n\nAlex clapped a hand over her mouth at the same time she pushed back her chair. She made it to the half bath near the pantry just in time to purge the contents of her stomach. Morning sickness had attacked with a vengeance.\n\nAlex waited for the sleek jet to touch down on the private airstrip before turning on her cell phone. She pressed a button. \"I'm on the ground,\" she said when hearing Merrick's greeting.\n\n\"I'm waiting by baggage claims.\"\n\nShe'd called him earlier that morning from Logan Airport to let him know she'd chartered a private jet to return to D.C. She'd been one of ten passengers on the flight that made for an interesting and entertaining experience.\n\nShe deplaned and made her way down the gangway, through a long corridor that led to a waiting area for commercial departures. Navigating her way through the crowds, she found herself in baggage claims.\n\n\"Looking for someone?\" whispered a familiar voice close to her ear.\n\nTurning around, she tilted her head and smiled at Merrick. She handed him her carry-on. \"Yes, I am. And I believe I just found him.\"\n\nWrapping his free arm around her waist, Merrick dipped his head and kissed her. \"Welcome home.\"\n\nAlex wound her arm around his waist inside his jacket. \"Please get me out of here before I pass out.\"\n\nMerrick led her toward the automatic doors. \"Are you feeling all right?\"\n\n\"It's the crowds.\"\n\n\"Do you want to wait here, or come with me to the lot to pick up my truck?\"\n\n\"I'll come with you.\" She'd never felt bothered by crowds, so Alex assumed it was an idiosyncrasy attributed to her being pregnant.\n\nAlex lost track of time. She knew she was back in her Arlington condo, but other than that she caught glimpses of the world going on around her whenever she stayed awake long enough to eat and go to the bathroom. She woke several times during the night to find Merrick in bed with her. They talked, but she couldn't remember what they'd talked about.\n\n\"What day is it?\" she mumbled, her face pressed to the pillow.\n\n\"Sunday.\"\n\n\"What time is it?\"\n\n\"Eight twenty-two.\"\n\n\"At night?\"\n\n\"No, baby. It's morning.\"\n\nRolling over on her back, she smiled at Merrick leaning over her. She combed her fingers through the crisp hair on his chest. \"What happened?\"\n\n\"Nothing earth-shattering or catastrophic since you went into Sleeping Beauty mode.\"\n\nShe gave him a shy smile. \"I'm sorry I wasn't very good company.\"\n\nStraddling her body, Merrick supported his weight on his forearms. \"We're not together to become each other's entertainment. You were exhausted, so you slept.\"\n\n\"I think it's the baby that has me so sleepy.\"\n\nMerrick moved off her, and sitting back on his heels, he splayed a hand under her hips and relieved her of her nightgown. \"Let's see what we have here.\"\n\nEmbarrassment washed over Alex, bringing a burning heat that began in her face and spread lower. She attempted to cover herself with her hands. \"No, Merrick.\"\n\nHe brushed her hands away with the ease of swatting a gnat. \"It's too late to act innocent, querida. I've seen, touched and tasted everything you have.\"\n\nAlex closed her eyes, shutting out his intense stare. \"Please leave me some pride, Merrick.\"\n\nHis hands moved up between her thighs. \"You're hardly going to be thinking about pride when you go into labor and your knees are spread east to west.\"\n\nShe opened her eyes and slapped at his hand. \"Stop!\"\n\nHe cradled her mound. \"It's too late for that, beautiful.\"\n\nAlex closed her eyes again, enjoying a touch that was as soft and gentle as a butterfly's gossamer wings. His fingers trailed over her belly, breasts, throat and around the outline of her ears. He'd become a sculptor, committing every dip and curve to memory.\n\n\"What are you doing to me, Merrick?\"\n\n\"Making love to the mother of my child.\"\n\n\"Are you happy about the baby?\"\n\nHe smiled. \"Very, very happy.\"\n\n\"I'm not marrying you because I don't want to be a baby mama.\"\n\nMerrick's smile faded. \"Then why are you marrying me?\"\n\nAlex rested her hand alongside his stubbly jaw. \"I love you.\"\n\nMerrick wanted to tell Alex that he would marry her even if she didn't love him; there was no way he was going to father a child and not become a part of his child's life.\n\n\"Do you want a son or a daughter?\" Alex asked after a comfortable silence.\n\nNuzzling her neck, Merrick pressed a kiss under Alex's ear. \"It doesn't matter.\"\n\nIt didn't matter to Merrick whereas she secretly wanted a girl. A little girl she would dress in frilly clothes, share tea parties, shopping sprees and intimate secrets with. She wanted for her daughter what she'd shared with her mother.\n\n\"Do you feel like going out for breakfast?\"\n\nExcitement fired the gold in her eyes. Merrick had mentioned food, which had become her number-one priority. \"You don't have to ask me twice.\"\n\nMerrick slipped off the bed and swung her up in his arms. \"We'll save time if we share a shower.\"\n\nAlex looped her arms around his neck. \"Do you think we can take time to do something else?\"\n\n\"What are you talking about?\" he asked, not breaking stride.\n\n\"You know.\"\n\nMerrick stopped several feet from the bathroom. \"I know what?\" Pressing her mouth to his ear, Alex whispered what she wanted him to do to her. Easing back, gray eyes flattening with an unreadable emotion, Merrick shook his head. \"None of that until you see the doctor. I don't want anything to happen to you or our baby.\"\n\n\"Couples make love right up until the time a woman is ready to give birth.\"\n\n\"What if you're not one of those women, Ali?\"\n\n\"And if I am, are you going to wait around for nine months and take care of yourself?\"\n\nHis eyes widened until she saw glints of blue in his silver gaze. \"And why not?\" he asked glibly. \"It won't be the first time.\"\n\nAlex affected a saucy smile. \"If it comes to that, then I'll do it for you.\"\n\nMerrick stared at the woman in his arms, shock freezing his features. With whom had he fallen in love and committed his future to? He knew life with Alexandra Cole would never be boring, but he hadn't realized it would also be fun.\n\nAlex shocked her parents when they walked into their kitchen early Friday morning to find their eldest daughter sitting at the table eating a bowl of oatmeal.\n\nDavid's dimpled smile deepened the network of tiny lines around his obsidian eyes. \"I told you we should change the locks,\" he said to his wife.\n\nAlex waved her spoon. \"If you change the locks, then you're going to have to change the alarm's security code.\"\n\nSerena smiled at her daughter. \"Never mind your father. He's as fussy as an old settin' hen, but he knows he loves it when his children come back home.\"\n\nDavid leaned over and dropped a kiss on Alex's damp hair. \"Speak for yourself, Mrs. Cole. Hi, baby,\" he crooned in the same breath.\n\nAlex patted his arm. \"Hi, Daddy.\" She smiled at her mother, who'd come over to hug her. \"Hi, Mom.\"\n\nSerena sat down beside her daughter. \"What time did you get in?\"\n\nShe and David had gotten used to their four children coming and going at odd times and hours. They would be there, then hours later jetting off to another state or country. Gabriel had become the most stable of the quartet. He loved teaching and living in Massachusetts. Even though Alex had purchased property in Virginia, she continued to travel\u2014first for her education and now with her career.\n\nSerena had come to realize that Ana and Jason were the least independent of her four children. And because they were twins, albeit fraternal, it would be difficult for them to sever the connection that began in the womb.\n\nAlex swallowed a mouthful of cereal. \"I got in after one.\"\n\nAs David filled a carafe with water to brew coffee, he narrowed his gaze. There was something different about Alex, but he couldn't pinpoint what it was.\n\n\"If you came to hang out with Jason and Ana, then you're out of luck. They'll be in Los Angeles until next week.\"\n\nAlex stared at her father, then her mother. \"I didn't come to see them. I came to talk to you.\"\n\nDavid felt his heart lurch; he mumbled a silent prayer for strength. \"What about, cookie?\"\n\nAlex smiled. It'd been years since her father called her that. \"Please sit down, Daddy.\"\n\nResting his left hand over his chest, long, slender fingers outstretched, David shook his head. \"If what you're going to tell me is going to hurt my heart, then I'd rather not hear it.\"\n\n\"Sit down, David,\" Serena urged quietly.\n\nWaiting until her father sat beside her mother, his right arm draped over the back of Serena's chair, Alex said, \"I'm getting married.\"\n\n\"When?\"\n\n\"To whom?\"\n\nSerena and David had spoken in unison.\n\nAlex smiled at Serena. \"When? As soon as you can get the family together.\" She angled her head, grinning broadly at her father. \"His name is Merrick Grayslake.\"\n\nDavid's expression did not change. He'd complained about his children not getting married, but now that Alexandra had expressed an interest in changing her marital status, he didn't feel the joy he thought he would've felt.\n\n\"Why the rush, cookie?\"\n\n\"I'm pregnant.\"\n\nSerena gasped, her eyes widening with each passing second while David jumped and pumped his fists in the air.\n\n\"Yes! Yes!\" he shouted, cutting a dance step around the kitchen. He came back to the table and pulled Alex from her chair. He swung her around and around until both she and Serena screamed for him to stop.\n\n\"David! You could hurt the baby,\" Serena chastised angrily.\n\nHe set Alex gently on her feet as he managed to look contrite. \"I'm sorry, cookie. It's just that the news that I'm going to be a grandfather is the best that I've had in a very long time.\"\n\n\"I thought winning a Grammy as Producer of the Year was the best.\"\n\n\"Oh, hell no,\" he said in protest. \"The day your mama told me that I was going to become a father tops the list. Knowing that I'll be a grandfather comes in at number two.\"\n\nSerena got up, filled a glass with orange juice and placed it in front of Alex. \"How far along are you?\"\n\n\"I just made seven weeks.\"\n\n\"Did you see a doctor?\" Serena asked, continuing her questioning.\n\n\"Yes. He says everything looks normal.\"\n\n\"How and where did you meet this Grayslake dude?\"\n\nAlex rolled her eyes at her father. \"He's not a dude, Daddy.\"\n\n\"If he's not a dude, then what is he? A she?\"\n\n\"Mommy, can you talk to your husband?\"\n\nSerena leveled David a knowing look. \"Please, darling. This is serious.\"\n\n\"So am I, Serena. Don't I have a right to know something about my future son-in-law?\"\n\nAlex took a deep breath, then let it out slowly. \"Daddy, you and I will talk later, because Mom and I need to plan a wedding.\"\n\nDavid ran a hand over his close-cropped silver hair. \"Okay, cookie. When are you going back to Virginia?\"\n\n\"Sunday night.\"\n\nHe nodded. He was outnumbered two to one, so he knew when to retreat. \"Just remember to save time for your old man.\"\n\n\"I will, Daddy.\"\n\nDavid filled a mug with coffee and walked out of the kitchen; an indescribable feeling filled his chest; there was going to be a wedding on the property of his Boca Raton estate for the first time; somehow he'd thought his sons would give him grandfather status, but it appeared as if his globe-trotting, art-loving daughter had beat them to it.\n\nIt no longer mattered because now his brothers would have to stop teasing him about his marriage-phobic children.\n\n## Chapter 22\n\nMerrick's arrival in Boca Raton coincided with two inches of rain that delayed flights into and out of Florida. He and others waited more than six hours before boarding a jet scheduled to land at the Fort Lauderdale\u2013Hollywood International Airport. He'd called Alex to inform her that he would be late, followed by a call to the hotel where he'd made a reservation to inform them of his late arrival.\n\nAlex had insisted he stay at her parents' house, but he rejected her offer, preferring instead to reserve a suite at a nearby hotel chain. Merrick reassured her that there would be other occasions, after they were married, when he would willingly accept his in-laws' hospitality. After all, he was a stranger, a man who would marry their daughter and change her name from Cole to Grayslake, a man who had no known relatives with whom to share the joy of his upcoming nuptials.\n\nHe saw Alex before she spied him. She stood in the baggage claims area holding a sign with his name. \"I'm Mr. Grayslake,\" he whispered, startling her. \"How are you, beautiful?\"\n\nAlex lowered the placard and looped her arms under his shoulders. \"I'm good. Welcome to the Sunshine State.\"\n\nMerrick gave her a skeptical look before he brushed a light kiss over her mouth. \"Surely you jest,\" he teased.\n\nHer eyes took in his powerful presence. \"Dad said it's the first hard rain in weeks.\"\n\nSwitching his carry-on bag to the opposite hand, he reached for her fingers. \"Did you check the almanac for Saturday's weather?\"\n\nAlex gave him a sidelong glance as she led him out of the terminal. \"Do you really follow what's in the almanac?\"\n\nMerrick nodded. \"It's more accurate than not.\"\n\n\"Well, it's not going to rain on our wedding.\"\n\nHe lifted an eyebrow. \"Why? Because you say so?\"\n\n\"Yes,\" she said confidently.\n\nWaiting until they were seated in a late-model Lexus sedan, Alex gave Merrick an overview of their wedding scheduled for Saturday afternoon at four, and that family members had been gathering in Florida all week since David and Serena Cole had announced their upcoming nuptials.\n\n\"Where is everyone staying?\" Merrick asked as Alex maneuvered in and out of traffic like an Indy car driver.\n\n\"Most will stay in Palm and West Palm Beach. Michael and Jolene will stay with his parents at their condo. The property where we met will accommodate at least fifty. It'll be a tight fit, but it will be like a gigantic sleepover. My aunts Nancy and Josephine have enough room to put up another thirty or forty between them.\"\n\n\"What's on the agenda for tomorrow other than getting a license?\"\n\n\"That's it. I picked up my dress yesterday and shoes earlier today.\"\n\n\"What about rings, Ali?\"\n\nHer mouth formed a perfect O. \"I forgot about them.\"\n\n\"Do you want an engagement ring?\"\n\nShe shook her head. \"No.\" Alex wanted to tell Merrick that if they'd had a formal engagement period, then yes.\n\n\"Can you take a few days off for a honeymoon?\"\n\nAlex slowed and then came to a complete stop at a red light. \"We can't do Venice in a few days.\"\n\n\"But we can do Key West.\"\n\nShe bit down on her lower lip to keep from screaming. \"I'd love to spend my honeymoon in the Keys.\"\n\nReaching over, Merrick rested his hand on her knee. \"I'll call a few places to see if we can't rent a bungalow. The summer season is over, so we should be able to find something.\"\n\n\"What can I do to make you change your mind about staying in a hotel?\"\n\nHe squeezed her knee. \"There's nothing you can do, baby. I'll meet your folks tomorrow.\"\n\n\"Are you worried that I'll slip into your bedroom and jump your bones?\"\n\nThrowing back his head, Merrick laughed. \"There's not going to be any more bone-jumping until after we're married. Getting you pregnant before I married you was not what I'd planned.\"\n\n\"Are you sorry that I'm pregnant?\"\n\nMerrick heard the hitch in her voice and his heart turned over. \"No, baby, I'm not sorry. But if it was the only thing that convinced you to marry in this century, I would've made it my business to get you pregnant the first time we slept together.\"\n\n\"That's diabolical and underhanded.\"\n\n\"All's fair in love and war, querida.\"\n\nMerrick felt as if he were reliving New Year's Eve. There were Coles everywhere. They drifted in and out of David and Serena's house, leaving wrapped gifts and envelopes; they hugged and kissed Alex, overwhelming her until she burst into tears. He took charge of his fianc\u00e9e when he led her into the house, put her to bed, staying with her until she fell asleep.\n\nThere came a soft knock on the door. Merrick left the chair next to Alex's bed, crossed the room and opened the door. Serena motioned to him. \"How is she?\"\n\nMerrick smiled at the petite woman with graying reddish-brown curls. He met her gold-brown eyes, eyes she'd passed along to her four children. \"She's sleeping.\"\n\nSerena exhaled. \"I didn't want to say anything about the baby, because that's something you and Alex should make public. I should've known she was getting overwhelmed when she stopped talking.\" Lowering his head, Merrick smothered a grin. \"Oh, I guess you know that she talks a lot.\"\n\n\"One of us has to be the talkative one.\"\n\nSerena rested a hand on his arm. \"That's what I told David. He kept saying 'he's kind of quiet,' but I had to remind my husband that every Cole carries the loquacious gene.\" Standing on tiptoe, she kissed her soon-to-be son-in-law on the cheek. \"Thank you for taking care of her.\"\n\nMerrick met Serena's unflinching stare. \"I'll always take care of her.\"\n\nTurning on her heel, Serena retraced her steps. She'd told David that he had nothing to worry about. There was something about Merrick Grayslake that reminded her of David's brother, Joshua Kirkland. Both men were quiet and radiated a danger that could take the nerve of the bravest man.\n\nShe was proud of her daughter. There was no doubt she'd chosen well.\n\nMerrick sat around the heated pool with Alex's twenty- and thirty-something relatives as catering staff unloaded their vans, setting candles out on each table. Within twenty minutes dozens of candles flickered like stars while a long table groaned under platters of food set up buffet style.\n\nIt was to be his last night as a single man, and Michael teased him about being whipped. He'd come to like Alex's family. They were friendly, outgoing, unpretentious and brutally honest. In a few short days they'd become his family: father, mother, brothers, sisters, cousins, aunts and uncles.\n\nMerrick sat up straighter, going completely still when he saw Gabriel Cole with a woman from his past. He remembered her even if she didn't remember him. He stood up with the other men at their approach. Gabriel made the introductions, and when they stood in front of him Merrick noticed the stiffness in her slender body.\n\nGabriel smiled at Merrick. \"Last, but certainly not least, is the prospective bridegroom, my soon-to-be brother-in-law, Merrick Grayslake. Merrick, this is my friend and colleague, Summer Montgomery.\"\n\nMerrick wanted to tell Gabriel that he knew who she was. Casually dressed in a pair of tailored taupe slacks and an ivory-hued silk shirt, and her hair pulled off her face in a loose knot, she looked nothing like the woman he'd trained. DEA special agent Summer Montgomery had completed her training at the FBI facilities in Quantico, Virginia, and had applied to take an additional course in intelligence training. Merrick, on loan from the CIA to the FBI and the DEA, had facilitated the four-week training.\n\nSummer inclined her head. \"Merrick. Best wishes on your upcoming nuptials.\"\n\nSmiling, he leaned over and kissed her cheek. \"Thank you, Renegade.\" He'd whispered her code name. He saw a flicker of gratitude in her eyes before she shuttered her gaze. It was apparent Gabriel Cole was totally unaware that his girlfriend was an undercover drug agent.\n\nNot waiting for the others, Merrick went over to the serving station. Reaching for a plate, he had the servers fill it with vegetables, sliced chicken and cubed fresh fruit. He approached the table where Alex sat with Ana, Summer and Clayborne Spencer's fianc\u00e9e and fellow medical student, Kim Cheung.\n\nHe placed the plate in front of Alex. Resting both hands on her shoulders, he massaged the muscles in her neck and upper back. \"What do you want to drink?\"\n\nSmiling up at him over her shoulder, Alex crooned, \"A fruit juice, darling.\"\n\nMerrick walked away to do Alex's bidding and Ana covered her hand with her fingers. \"Michael's right,\" she whispered. \"He is whipped! And speaking of whipped, I owe you one, dear sister.\"\n\nAlex swallowed a mouthful of avocado. Some women craved pickles and ice cream, she craved avocados. \"For what?\"\n\n\"For pretending you didn't know Merrick when you two were knockin' boots.\"\n\n\"How long have you known Merrick, Alex?\" Kim asked.\n\n\"I met him New Year's Eve just like the rest of you.\" Alex smiled at Gabriel's girlfriend. \"Everyone except Summer.\"\n\n\"I saw him, but I wasn't introduced to Merrick until he came to Michael's house on the Fourth,\" Ana said. \"By that time he and Alex were knee-deep in the hot sauce.\"\n\n\"My sister's pissed because Miss Sherlock Holmes couldn't find out my business.\"\n\nAna wiggled her fingers. \"That's because your man is tall, dark and mysterious.\"\n\nKim flipped several strands of straight black hair over her shoulder. \"He's gorgeous.\"\n\n\"It's his eyes,\" Ana crooned, batting her lashes. \"They're like lasers. Do they change color when he makes love to you?\"\n\n\"Why don't you ask him?\" Alex said, glaring at her sister.\n\n\"Ana,\" Kim whispered, \"you shouldn't ask your sister that.\"\n\n\"Why not?\" she asked.\n\n\"Because it's disrespectful.\"\n\n\"You tell her, Kim,\" Alex chimed in.\n\nSummer Montgomery rested her forehead on her arms and laughed until tears rolled down her cheeks. Gabriel's sisters were like a stand-up act.\n\nMerrick returned, carrying a glass of juice, followed by Clayborne, Gabriel and Jason, who set down plates in front of Kim, Summer and Ana.\n\nHaving assuaged her temporary pangs of hunger, Alex drank her juice, then dabbed her mouth. \"Summer, you can share my room tonight because Kim will be in Ana's. Let me know if you'll need a massage, manicure, pedicure, or if you want your hair styled. Mom has contracted with a group of beauty consultants to hook us up tomorrow.\"\n\nSummer flashed a wide grin. \"Count me in for the massage and hair.\"\n\nThe women sat eating and chatting for several hours until Alex stood up. \"I don't know about the rest of you, but I must get some sleep.\"\n\nPushing herself into a standing position, Summer said, \"I think I'm going to join you.\" She smiled at Ana and Kim. \"I'll see you tomorrow. Good night.\"\n\n\"Wait for us,\" Kim called out as she and Ana followed Alex and Summer.\n\n## Chapter 23\n\nIt was minutes from midnight when seven men sitting around a table lifted glasses filled with brandy and toasted Merrick Grayslake.\n\nJason, the mirror image of Gabriel, sans long hair and pierced ears, offered the first toast. \"To Merrick, my soon-to-be brother-in-law. May your days be sunny and your nights warmer than my esteemed older brother's.\" First Gabriel had been teased about driving a hoopty, and now it was because he'd moved from warm sunny Florida to frigid New England.\n\n\"Hear! Hear!\" came a chorus of deep male voices.\n\nMerrick raised his glass, put it to his mouth and downed it in one swallow.\n\n\"Hot damn!\" federal district court judge Christopher Delgado crowed. Slender and elegant with salt-and-pepper wavy hair, he leaned over and slapped Merrick on the back. \"You're going to fit nicely in this family because you drink like an hombre!\"\n\n\"Are you signifying something, brother?\" Michael asked his brother-in-law.\n\n\"Some male relatives...who I will not out tonight...have a problem holding their Kool-Aid.\"\n\nJason cupped his ear. \"Do I hear throw down?\"\n\nChristopher smiled. \"I don't know about throw down, but the last time I stepped in as bartender I heard complaints about the Kool-Aid being a wee bit too strong.\"\n\nGabriel shook his head. \"Don't play yourself, Chris. You know right well that stuff you mixed up wasn't legal.\"\n\n\"It's as legal as I am,\" Chris retorted, dodging a barrage of rolled-up cocktail napkins thrown at him. \"Jason, check your dad's liquor cabinet and bring out the good stuff.\"\n\nJason popped up from his chair. \"I know he has the Three Kings.\"\n\n\"The Three Kings?\" asked Algerian-born fashion designer Silah Kadir. Silah was married to Martin Cole's youngest daughter. \"Are you talking about the Magi?\"\n\nMerrick laughed along with the other men. \"Jason is talking about Johnny Walker, Jim Beam and Jack Daniel.\"\n\n\"Why do you call them Three Kings?\" Silah asked in French-accented English.\n\nClayborne rested an arm over his uncle's shoulder. \"Because if you drink enough of them you'll think you're seeing not only the kings, but also their camels.\"\n\n\"Go get them,\" Gabriel urged his younger brother. \"We're here to toast Merrick, and it should be a night to remember.\"\n\nDr. Tyler Cole looped one leg over the other. Tyler, Martin's only son, was the heir apparent to the family dynasty. \"Don't look for me to resuscitate any fool who gets so drunk that he'll mistake the pool for a bed.\"\n\n\"Grandpa told me that you were in your cups at Michael's reception.\" Tyler, glaring at his nephew, smothered a curse. \"Did Dana give you mouth-to-mouth?\"\n\nTyler waved a hand. \"Get this young pup outta here. This is grown-folk business.\"\n\n\"Wait up, Jason,\" Clayborne called out. \"I'll help you.\"\n\nLeaning back in his chair, Michael extended his glass to Merrick. \"Friend. Soon-to-be cousin. Welcome to the family.\" He, too, downed the brandy in one shot.\n\nSomeone refilled Merrick's glass and he tossed it back, the heat of the smooth liquor spreading throughout his chest. He'd never been much of a drinker, but tonight he would make an exception.\n\nJason and Clayborne returned with a plastic crate filled with bottles of scotch, gin, vodka, tequila and bourbon. The liquid flowed, the tongues got looser, the men toasting Merrick for having the good sense to marry into the family; eventually the conversations turned to when the men first made love to their wives and girlfriends.\n\nMerrick got drunk, drunker than he'd ever been in his life; he was as talkative as the others with one exception: he didn't disclose the circumstances surrounding his birth, the covert missions for the CIA and the most intimate details of his relationship with Alex.\n\nSometime around three Gabriel climbed onto one of the chaises set up around the pool and fell asleep.\n\nIt was Serena Cole who found Christopher, Tyler and Silah asleep on the deck of the pool. Michael, Jason, Merrick and Clayborne managed to make it into the house, falling asleep where they lay.\n\nNone had made it to their designated bedrooms.\n\nMerrick tied the light gray silk tie in a Windsor knot, then turned down the spread collar on the white shirt. When he'd finally regained consciousness earlier that morning it was to a pounding headache, but after several aspirins washed down with black coffee he felt more human than he had in hours.\n\nThe male bonding\/mock bachelor party was unlike any in which he'd participated. Grown men had become boys, fueled by an alcoholic haze wherein they'd confessed their shortcomings. To the world they were rich, powerful, successful men; however, they displayed their very human side when they spoke of the difficulties of parenting and their willingness to compromise in order to save their marriages. Merrick would've thought them conventional, but the various tattoos and ear piercings that crossed generations shattered that perception.\n\nSitting on the leather bench at the foot of the bed in the suite that had belonged to Gabriel, Merrick closed his eyes and did what he hadn't done in a very long time\u2014he prayed. He was still in the same position, eyes closed, when Jason knocked on the door and walked in.\n\nTall, broad-shouldered, he cut a handsome figure in formal attire. His silk tie matched Merrick's. \"Thinking about backing out, Brother Merrick?\"\n\nMerrick's head came up at the same time he stood up. Reaching for the jacket to his tuxedo on the foot of the bed, he slipped his arms into the sleeves.\n\n\"No.\"\n\nJason flashed a dimpled smile. \"It's time we get going, but Dad wants to see you downstairs in his study before you marry my sister.\"\n\nMerrick followed Jason down a curving staircase of a house constructed in three sections. The elder Coles occupied one section that included a guest wing, the four bedroom suites took up another section and the third section contained a state-of-the-art recording studio and Serenity Records' corporate office.\n\nMerrick walked into the large cool room with floor-to-ceiling shelves packed with books and musical scores. A large glass-top rosewood desk and matching table complemented leather chairs in a rich maroon.\n\nDavid Cole was not alone. His brothers, Martin and Joshua Kirkland\u2014the latter the result of an illicit liaison between Samuel Cole and Teresa Maldonado Kirkland\u2014sons Jason and Gabriel, and Matthew Sterling, family friend and father-in-law to Emily Kirkland Delgado, made the large room appear small. The men, all in formal dress, staring at Merrick radiated power and danger, and if he hadn't been who he was he would've been intimidated. However, he'd experienced too much, stared death in the face too often to be intimidated\u2014by anything or anyone.\n\nDavid, who sat on the edge of the desk, rose to his feet. Recessed light glinted off his silver hair. \"Merrick, I've asked you to come here because there is something I must say to you. I've asked my brothers, sons and a lifelong friend to be here because I want them to be aware of what's in my heart at this moment.\n\n\"In fifteen minutes I will place my daughter's hand in yours. And when that happens I will relinquish all claim and responsibility for Alexandra. My wife and my children are my most precious gifts, and I'm generously offering you the gift of my eldest daughter.\n\n\"I will say this only once, Merrick Grayslake. Love her, and protect her with your life. But if you fail to do this, then look for me to come after you.\"\n\nThe nostrils of Merrick's aquiline nose flared as he let out the breath he'd been holding. His future father-in-law had just threatened him. He blinked once. \"Warning heeded, David.\" His eyes widened. \"You've had your say, so let me have mine. At four o'clock Alexandra will become my wife and therefore it will become my responsibility to protect her.\n\n\"I love her and the child she carries in her womb. And because I love her, I've broken a vow I made almost three years ago when I said I'd never go back to the CIA. Well, I'm going back because I'll have a wife and child to support.\" He paused for effect. \"So, that should put your mind at ease as to whether I have a job.\" Everyone turned and stared at David, who managed to look sheepish.\n\n\"No, you didn't, David,\" Gabriel said softly.\n\n\"Damn, brother, that's cold,\" Joshua Kirkland whispered, shaking his head in disbelief.\n\nMerrick lowered his head, hiding his own grin when Martin Cole and Matthew Sterling dissolved into a paroxysm of laughter that left them with tears in their eyes.\n\nHis head came up and he stared at David. \"I don't have as much money as Alexandra, but I can assure you that I can support her and the children we plan to have.\" He nodded to the other men. \"I don't know about you guys, but I have a wedding to go to in three minutes.\" Turning on his heels, he walked out of the study, leaving his bride's family members staring at his retreating back.\n\nMartin Cole's silver hair was a startling contrast to his deeply tanned olive-brown skin. Shaking his head, he glared at his younger brother. \"David, you never cease to amaze me. Why the hell did you have to ask the man if he had a job?\"\n\nCrossing his arms over his chest, Joshua continued to stare at his half brother, green eyes sparkling in amusement. \"If you'd wanted to know about Merrick you could've asked Michael.\"\n\n\"Dad would never do that,\" Jason said. \"Whenever it concerns Alex or Ana he goes straight for the jugular. He wasn't even diplomatic about it. He could've said, 'By the way, my man, are you a scrub?'\"\n\nMatthew Sterling stared at Jason. \"Scrub?\"\n\n\"Pimp, gigolo, deadbeat\u2014\"\n\n\"I get the picture, Jason,\" Matt interrupted.\n\nMartin patted David's back. \"I know all of this is new for you, but you'll get through it.\" All of Martin's children were married and had made him a grandfather of five.\n\nJoshua took David's arm. \"Come on, brother. After the first time it gets easier.\"\n\nDavid narrowed his eyes at his brothers. They'd teased him for years about his children not wanting to marry and have children. But all of the teasing would come to an end in a few minutes.\n\nShaking off his brother's hands, he walked out of the study to give his daughter away in marriage to a man he now respected enough to think of him as a son.\n\n## Chapter 24\n\nAlex was certain her father could feel her trembling as she clung to his arm. She'd decided on simplicity: no bridesmaids, groomsmen, ring bearer or flower girl. Her father would escort her over a red carpet to where Merrick waited with a judge to make her his wife.\n\nShe closed her eyes when Gabriel sang her favorite song, the top-chart blockbuster classic hit \"I Will Always Love You.\" Gabriel's fingers skimmed the keyboard as his melodious voice came through the powerful speakers and something within her burst.\n\nAlex wasn't certain it was the fullness of the love she felt for Merrick, the hormonal changes playing havoc with her moods or the knowledge that within minutes her life would change dramatically. The times when she thought only of herself and what made her happy were over. Every decision she made would be predicated on its effect on her husband and child.\n\nThe tears flowed down her face, landing on the bouquet of white lacecap hydrangeas with dozens of faux-pearl sprays, bead flowers, seashell flowers and seashells on wires tied with yards of ribbons in different widths and edging. The song ended with many seated on the organza-draped cushioned chairs blotting the corners of their eyes.\n\nThe assembled rose as one when Gabriel played the familiar chords to the wedding march.\n\n\"Let's go, cookie,\" David murmured to his still-weeping daughter.\n\nAlex wanted to move, but her legs refused to follow the dictates of her brain. \"I can't, Daddy,\" she sobbed.\n\nMurmurs escalated when David Cole wrapped his arms around his daughter's bare shoulders. \"If you don't want to marry him, then say so, cookie.\"\n\nAlex shook her head. She wanted to marry Merrick. He'd been the only man she'd ever wanted to marry. \"I want him, Daddy.\"\n\nDavid kissed her hair, careful not to dislodge the tiny rosebuds pinned into her black curly hair. All he'd ever wanted from the moment he became a father was to protect his children and see to their happiness. But as Merrick had stated so arrogantly, Alex was no longer his responsibility. It was now up to Merrick Grayslake to protect his daughter and unborn grandchild.\n\nMerrick watched the interchange between his bride and her father, his heart racing uncontrollably. Had she changed her mind? Did she wait until the very moment that they were to exchange vows to change her mind?\n\nHe'd heard of Dear John letters, brides and grooms being left at the altar and runaway brides. Was he, he agonized, about to become another statistic?\n\nA sense of strength came to Merrick, one stronger than any he'd experienced in the past. He'd languished in foster and group homes, endured verbal and physical abuse and still managed to survive. He did not give his heart to a woman only to lose her in front of hundreds of people.\n\n\"I'll be back,\" he said to the elderly black-robed judge watching the unfolding scene with an expression of distress on his lined face.\n\nGasps filled the warm autumn afternoon when Merrick marched down the red carpet like a marine during a dress parade. He rested a hand on her back and she went completely still.\n\n\"Querida.\"\n\nAlex felt the comforting touch on her bare flesh, inhaled the familiar cologne. She couldn't move, couldn't go to Merrick so he'd come for her. Turning, she smiled up at him through her lashes. The moisture had turned her eyes into pools of gold.\n\nHer lips parted in a trembling smile. \"Mi, amor.\"\n\nDavid reached for Alex's hand and placed it in Merrick's outstretched one. \"Take her.\"\n\nReaching into the pocket of his dress trousers for a handkerchief, Merrick gently blotted the tears on his bride's face. Lowering his head, he kissed her forehead. \"Are you ready to do this?\"\n\nShe wrinkled her pert nose. \"Yes.\"\n\nAlex looped her hand over the sleeve of Merrick's jacket as he retraced his steps, bringing them to stand in front of the elderly judge who'd officiated at many a Cole civil wedding. He winked at Alex, then began the ceremony he could recite in his sleep.\n\nMerrick's voice was strong and clear as he took his vows. His gaze never strayed from Alex's face as she recited hers. He loved her, had openly confessed to loving her, had shown her in the most intimate way possible that he loved her, but he doubted whether she knew the depth of the love he felt for her.\n\nWhen it came time for the exchange of rings he reached into his breast pocket and slipped a diamond eternity band onto her slender finger. Alex handed her bouquet to her father as she took the platinum band off her thumb and slipped it onto Merrick's left hand.\n\nSighs of relief, cheering and whistles followed the judge's pronouncement that Merrick and Alexandra Grayslake were now man and wife. Looping his arms around her waist, Merrick picked up his wife and kissed her soundly on the mouth.\n\n\"Congratulations, Mrs. Grayslake.\"\n\nHer dimples deepened when she flashed a joyous smile. \"And congratulations to you, too, Mr. Grayslake.\" Tightening her hold around his neck, she kissed him again. \"I can't believe we're married.\"\n\n\"Believe it, baby,\" he whispered, \"because I'm going to remind you of it every day and every night.\"\n\nHe set her on her feet, one arm going around her waist as he shook hands with David. \"Scared you for a moment, didn't it?\" he asked his father-in-law.\n\n\"Hell, yeah,\" David admitted. He patted Merrick's shoulder. \"You did good, son.\"\n\nSerena came over and kissed her daughter, then her new son-in-law. \"Congratulations.\"\n\nMerrick wound his free arm around Serena's trim waist. \"Thank you, Mrs. Cole, for your daughter and putting everything together so quickly.\"\n\nFrowning, Serena shook her head slowly. \"You call my husband David and me Mrs. Cole?\"\n\nMerrick's eyebrows lifted. \"What do you want me to call you?\"\n\n\"Mom will do nicely, thank you.\"\n\nMerrick was momentarily speechless in his surprise. His foster mothers had wanted him to call them Mom or Mother, but he'd never been able to bring himself to do it. Not when they saw him as a check from the state, not when they punished him for what they'd considered the slightest infraction.\n\nToday his life changed when he became husband, son and brother-in-law. It had taken him nearly thirty-six years to find a family he could claim as his own.\n\n\"From now on it will be Mom, Mom.\" He nodded to David.\n\n\"I'm going to take Ali to the table where she can get off her feet.\"\n\n\"Are you all right?\" Serena asked her daughter.\n\n\"I'm good, Mom. I'm going to skip the receiving-line routine.\"\n\n\"Why, cookie?\" David couldn't believe Alex wanted to break the tradition.\n\n\"I don't know what it is, but since I've become pregnant I can't stand crowds of people. I freak out if they get too close to me.\"\n\n\"Don't tell me you're not going to save your father a dance.\"\n\n\"Of course I am. Do you think I'm going to cheat you out of that honor?\"\n\n\"I hope not,\" David mumbled. He took Serena's arm. \"Come, sweetheart, let's circulate.\"\n\nMerrick steered her over to where a table had been set up for the bride, the groom and her parents, parents that he now thought of as his own.\n\nPrerecorded music blared from speakers set up on the property and the partying began in earnest. Those who came over to greet Alex found her ravishing in an off-the-shoulder platinum gown with a sunburst-pleated silk-chiffon bodice topping a silk-satin skirt. The garment was perfect for her petite figure. Three inches of jeweled Manolo Blahnik wedding sling backs put the top of her head at six-foot-three-inch Merrick's shoulder.\n\nShe'd decided on a buffet rather than a more formal sit-down dinner. The caterer had delivered dozens of tables, each with seating for six. Vases of pale-colored roses lined the courtyard and patio.\n\nDavid had hired a disc jockey to spin tunes, and a portable dance floor had been set up between the pool and tennis court. Serena had the pool covered, much to the disappointment of the younger children.\n\nMerrick pressed his shoulder to Alex's bare arm. \"You have the same food on your plate, so why is it you're eating mine?\"\n\n\"Whoever fixed my plate added salt.\"\n\n\"I would've gotten you another plate, Ali.\"\n\nShe rested her head on his shoulder. \"That's all right.\"\n\nAngling his head, he kissed her hair. \"Are you all right?\" She'd seemed out of sorts after dancing with him, her father and then her brothers.\n\n\"I'm feeling a little tired.\"\n\nMerrick knew her little tired was more than that. There were times when she was so lethargic that she fell asleep while eating. \"We're leaving.\"\n\nHer head came up, and suddenly she was wide awake. \"We can't leave now.\"\n\n\"And why not?\"\n\n\"It's too soon.\"\n\n\"Soon for what, Ali? You're lethargic and out of sorts. Have you forgotten that you're two months pregnant?\"\n\n\"How can I forget?\" she countered. \"Between the nausea and fatigue I feel like a punching bag.\"\n\n\"That's why we're leaving.\" Rising, he eased her up from her chair. \"We'll change, then head over to the hotel.\" They'd made plans to spend their wedding night in a hotel before driving down to the Keys the following morning.\n\n\"You're a bully, Merrick Grayslake,\" she hissed through clenched teeth.\n\n\"Yeah, yeah,\" he drawled. \"A bully who loves you and that little baby inside you.\"\n\nLeaning against his side to keep her footing, Alex permitted Merrick to lead her back to the house where she'd grown up. She thought about his comment that he loved the baby inside her. How could he love something he'd never met or seen?\n\nThere were other questions she wanted to ask her enigmatic husband, but Alex decided to wait, wait until after they returned from their honeymoon.\n\n## Part Three\n\n## Silent Witness\n\n## Chapter 25\n\nCIA Headquarters...Langley, Virginia\n\nLeaning back in his chair, Merrick pinched the bridge of his nose with his thumb and forefinger. His eyes were burning from nonstop reading. The terrorist act of September 11 had changed the course of action for securing America's borders and its citizens.\n\nHe'd spent the past week reading about new and updated security protocol that had been put in place within days of the horrific destruction of the World Trade Center.\n\nMerrick forced his concentration away from what he'd mentally recorded to that of his wife. He and Alex had celebrated their third month of marriage the night before. Even though she professed she wanted a quiet dinner at home, he overrode her protests and took her out to one of her favorite D.C. eating establishments.\n\nAlex was now in her second trimester and without warning her nausea had vanished along with her lethargy. The gradual changes in her body made her pregnancy more obvious. Her face was fuller, as were her breasts, and she'd lost her waistline.\n\nShe'd contacted the National Trust, applying for a transfer to the southeast region; however, her request was denied because there were no liaison positions available. Alex existed in a blue funk for one day, tendered her resignation, then called a real estate company to set up an appointment to show her properties for their new home.\n\nHe'd been at the Company for a month, and his time was spent reading and sitting in on meetings\u2014meetings that were of no interest or consequence to him, meetings with low-level clerks with years of seniority. He'd come back as an intelligence training specialist and so far he hadn't trained anyone.\n\nThe phone on his desk rang, and he sat up and picked up the receiver. \"Grayslake.\"\n\n\"Mr. Ashleigh would like to see you in his office\u2014now.\"\n\nMerrick hung up without verifying whether he would or wouldn't comply with the assistant director's directive. Shirt cuffs rolled up over his wrists, top button on his shirt undone and tie hanging loosely around his collar, he walked out of his closet of an office to a larger one at the end of the hall.\n\n\"Go on in,\" drawled a woman whose face was reminiscent of an overripe pumpkin.\n\nMerrick met her intimidating gaze with one of his own as he brushed past her desk. He entered Carl Ashleigh's office silently, startling the man with the cold, pale blue eyes.\n\nCarl waved a hand toward a chair facing his desk. \"Sit down, Grayslake.\" Waiting until Merrick was seated he laced his fingers together atop a stack of papers. \"I know you've been waiting for me to assign you to a training class, but I'm afraid that's not going to happen for a while.\"\n\nCrossing one leg over the other, Merrick rested his elbows on the arms of the chair. \"How long is awhile?\"\n\n\"At least another six months.\"\n\nMerrick did not visibly react to the response that he had come back to work to sit around, collect a government check and do absolutely nothing.\n\n\"Why did you bring me back now if you knew there wasn't a position for me?\" What he wanted to tell Ashleigh was that he could've stayed home with his pregnant wife to await the birth of their first child.\n\nA band of red crept up Ashleigh's neck to his face and hairline. \"I have a position, but it's not what you'd requested.\" A hard edge had crept into his voice.\n\nMerrick leveled a gaze at him. \"What do you have?\"\n\nAshleigh paused, hoping to make Grayslake uncomfortable. He knew about the man's pretty, young wife, and that they had a baby on the way. He knew everything there was to know about Merrick Grayslake except how to unnerve him.\n\n\"How's your marksmanship?\"\n\nMerrick didn't blink. \"With what? Pistol or rifle?\"\n\n\"How about a PSG-1?\"\n\nA muscle in Merrick's jaw jumped. \"One shot, one kill.\"\n\nEvery branch in the military used snipers: the SEALs, CCT, and Army Rangers. And although they all had their respective sniper elements, there was one school that stood out from the others: the United States Marine Corps Scout Sniper School.\n\n\"We need a sniper instructor, and because you're Scout Sniper Qualified, you're the perfect candidate.\"\n\n\"You want me to train agents to become snipers?\"\n\nAshleigh shook his head. \"Not me, Grayslake, your country.\"\n\nMerrick's cold smile never reached his eyes. \"Because you put it that way how can I refuse?\"\n\nIt'd always been the triumvirate: country, Corps and mission. The trinity would always be there, but along the way he'd added family.\n\n\"Do you still have your rifle?\"\n\n\"Yes.\" He'd stored his rifle in a special case, and hidden it beneath the floorboards in a closet in West Virginia.\n\n\"I recommend you dust it off and start practicing.\"\n\n\"When do exercises begin?\"\n\n\"The first Monday in January.\" Ashleigh knew he had to keep Merrick busy until he received word that Operation Backslap was ready to be executed.\n\nMerrick nodded. He had about two weeks to bring himself up-to-date on the rigorous ten-week-long course broken down into three phases\u2014land navigation and marksmanship, stalking techniques, field skills and fire rehearsals\u2014and the last that encompassed everything from communications to surveillance performance.\n\n\"How many will be in the class?\"\n\n\"Five.\"\n\nUncrossing his legs, he placed both feet on the floor. \"I'll be ready.\"\n\nThat said, he pushed off the chair and walked out of Ashleigh's office. Sniper training wasn't what he wanted to do, but it was better than sitting around doing nothing.\n\nMerrick placed Alex on her side on the thick, thirsty towel covering the bed, then took his time drying her body. It'd become a nightly ritual for them to share a shower. She let out a moan when he kneaded the muscles in her legs.\n\n\"Am I hurting you?\"\n\nShe smiled, not opening her eyes. \"Are you kidding? It feels wonderful.\"\n\n\"What on earth did you do today?\" The muscles in her calves were tight as fists.\n\n\"I did a lot of walking up and down staircases. One house had an attic and a basement, so that was four flights.\"\n\n\"Do you really want a house that big?\"\n\n\"The question should be, do we need a big house, Merrick.\"\n\n\"Do we, baby?\"\n\nShe smiled. \"Yes.\"\n\n\"There's going to be only three of us.\" Sonogram pictures showed one baby\u2014a girl.\n\n\"Three for now. What about in a couple of years? I don't want to decorate a house, then leave it when we outgrow it.\"\n\nHis hands moved up her thighs. \"How many babies do you plan on having?\"\n\nShe moaned again. \"As many as you plan on making.\"\n\nLowering his head, he pressed a kiss to the nape of her neck. \"Realistically, we could have one every year until you're about forty.\"\n\n\"That's madness! There's no way I'm going to have ten kids.\"\n\n\"Why not? We could name them Pull Out, Quick on the Draw, Rhythm. That is, if we decide to use the rhythm method, Oops, my bad.\"\n\nAlex couldn't help herself as she burst out laughing. It wasn't often that her very serious husband displayed his wicked sense of humor.\n\nMerrick removed the towel and lay down next to Alex, his chest against her back. He placed an arm over her waist, his hand cradling her rounded belly.\n\n\"Don't make any appointments for house tours this weekend.\"\n\n\"Why not? I thought you'd want to go with me.\"\n\n\"I can't because we have to go to Bolivar.\"\n\nAlex stiffened. \"For what?\"\n\n\"I have to pick up something.\"\n\n\"Can you identify what the something is?\"\n\n\"It's a sniper rifle.\"\n\nAlex threw off his arm and sat up, but he eased her back down to the mattress. \"What the hell do you need with a sniper rifle, Merrick? You told me that you sit at a desk. Do you plan to go into the office and shoot up the place? Or perhaps you're thinking of becoming another D.C. Sniper.\"\n\n\"You don't know what you're talking about.\"\n\n\"Enlighten me, Merrick. Please tell me why you need a weapon that can make someone's head explode with one bullet.\"\n\n\"There will be some things I can tell you, and others that I can't because they're classified.\"\n\n\"Is this sniper business classified?\"\n\n\"No. I've been assigned to facilitate sniper training.\"\n\n\"Why you, Merrick?\" Her voice was softer, calmer.\n\n\"I was a sniper in the Corps, and the Marine Corps has the best sniper program in the world.\"\n\n\"You're just going to do training?\"\n\n\"Yes, Ali.\"\n\nShe covered the hand over her swollen belly, whispering a prayer of thanks. Alex didn't know what she would do if Merrick put himself in danger. \"I'm sorry I went off on you.\"\n\nHe kissed her damp hair. \"I can think of a way for you to apologize for jumping to conclusions.\"\n\n\"How?\"\n\nHe kissed her again. \"Sit on me.\"\n\n\"You really like it when I ride you?\"\n\nHis smile was dazzling. \"I love it.\" Merrick loved when Alex became the aggressor in bed.\n\nAlex sat up and straddled him as he supported his back against the headboard. Together they found a rhythm that took them to heaven and back. Leaning into Merrick, she pressed her full breasts to his chest, rested her head between his neck and shoulder and counted the strong steady beats of his heart. There was never a time when they made love that they hadn't become one with each other.\n\n## Chapter 26\n\nThe chiming of the telephone roused Merrick from a deep sleep. Reaching out in the darkened bedroom, he took the cordless instrument off its cradle before it disturbed Alex.\n\n\"Yes,\" he whispered into the receiver.\n\n\"You don't know who I am, but I know all about you, Merrick Grayslake.\" The distorted voice of a man speaking fluent Spanish came through the earpiece. It was apparent whoever had called him was using a device that scrambled or distorted the voice.\n\n\"Who are you?\" Merrick asked in the same language.\n\n\"I can't tell you that. Not now.\"\n\n\"What do you want?\"\n\n\"How would you like to avenge your mother's murder?\"\n\nMerrick depressed a button, disconnecting the caller. \"Sick bastard,\" he whispered as he practically slammed the instrument down on the bedside table.\n\nFluffing up his pillow, he tried going back to sleep, but sleep was elusive. The caller knew his name, his home phone number and knew that he was fluent in Spanish. If the man hadn't mentioned his mother perhaps he wouldn't have been so disturbed.\n\nThere had been a time when he would've given anything to glean a modicum of information about Victoria Grayslake, but that time had passed. It took more than forty minutes before he finally fell asleep for the second time that night.\n\nMerrick hadn't sat down behind his desk yet when his phone rang. He picked up after the second ring. \"Grayslake.\"\n\n\"Don't hang up on me again or you'll be very, very sorry.\"\n\nIt was the same person who'd called him at home. \"Are you threatening me?\" Merrick asked softly. Whoever had rung him the night before knew he worked at the CIA. He was glad the call had come in at the Agency because it would be easily traced.\n\n\"No, I'm not threatening you.\"\n\n\"It sounded like a threat to me,\" Merrick countered.\n\n\"I need to talk to you.\"\n\n\"You're talking now,\" he said sarcastically.\n\n\"What I need to tell you shouldn't be recorded. I know every call you make and receive is recorded. And don't try to trace this number, because the phone will be in the garbage as soon as I hang up.\"\n\n\"What do you want?\"\n\n\"I want a number where I can call you without the government listening in.\"\n\nMerrick's curiosity was piqued. \"I'll get a phone that's unlisted.\"\n\n\"I'll send you a phone you can use.\"\n\n\"When?\"\n\n\"I'll have it delivered to your post office box in West Virginia. It'll be waiting for you when you get there.\"\n\n\"How...\" Merrick's voice trailed off when he heard the break in the connection. \"Who the hell are you?\" he said aloud.\n\nWas someone trying to mess with his head? Was he being set up? Not given to episodes of paranoia, he sat down and went through a mental recall of places he'd been, people he'd met and situations wherein his identity might have been compromised.\n\nHe had lots of questions and no answers. At least not one until he went to Bolivar.\n\nMerrick turned up the heat to the highest setting as he navigated the winding West Virginia roads. \"Better?\" he asked Alex.\n\n\"Much better.\" She'd tied a cashmere shawl tightly around her neck and shoulders. \"This is one year that I'm really looking forward to spending a week in Florida.\"\n\n\"Let's hope the cold weather doesn't go any farther south.\"\n\nWinter had come early to the East Coast with below-freezing temperatures from Maine to Georgia. Only Florida had been spared.\n\nAlex turned and stared at her husband's profile. \"Are you sure you're not going to be able to take off Christmas Eve?\"\n\n\"Baby, let's not start that again. I told you we'll probably shut down early, but I can't take the day. Barring airport delays, I will be in Florida before midnight.\"\n\nHe and Alex had celebrated Thanksgiving in Mississippi with Tyler and Dana as their hosts. It'd become an impromptu family reunion with the second generation of Coles, Kirklands, Delgados and Lassisters coming together under one roof. Gabriel's date, Summer Montgomery, was now his fianc\u00e9e.\n\nPushing out her lower lip, Alex pouted as she'd done as a child when she couldn't get her way. \"Stop it!\" she screamed when Merrick reached over and tugged on her lip.\n\n\"Stop pouting. It's not going to get me to change my mind.\"\n\nHer expression brightened. \"I can think of something else that will get you to change your mind.\"\n\nMerrick gave her a sidelong grin. \"Not even that!\"\n\nShe gave him a saucy smile. \"We'll see.\"\n\n\"In another couple of months we're going to have to stop the calisthenics.\"\n\nAlex lifted her eyebrows. \"Why?\"\n\n\"Because your belly will get in the way.\"\n\nShe rolled her eyes and sucked her teeth. \"Don't be so parochial, mi amor. We can always do it doggy style.\"\n\nMerrick opened his mouth and closed it just as quickly, stunned by his wife's bluntness. \"You've got to stop reading those books.\"\n\n\"What books?\" Alex asked innocently.\n\n\"The ones you hide in the basket with the towels.\"\n\n\"They're how-to books for pregnant women. And I don't hide them. I read them whenever I'm in the bathroom.\"\n\n\"I've seen the pictures and they're just plain nasty, baby. You'd have to be a contortionist to execute some of those positions.\"\n\n\"Where are you going?\" Alex asked when Merrick took a route that led away from their home.\n\n\"I have to stop at the post office. Do you want to come in with me?\"\n\n\"No, thank you. I'll wait in the truck. Can you please pick up some Christmas stamps? This weekend is as good a time as any to do Christmas cards.\" Alex had brought boxes of cards and several needlecraft projects with her to pass the time in Bolivar.\n\nA quarter of a mile later, Merrick parked in a lot behind a row of stores, not bothering to turn off the engine; he got out and went into the post office. Inserting a key into his box, he took out several pieces of junk mail and a small white box addressed to him. He lingered, opening it. His mystery caller had sent four disposable cell phones, each with a thirty-minute limit.\n\nPushing the phones, no larger than an iPod or thicker than a credit card, into the back pocket of his jeans, he discarded the box and returned to the parking lot. The serial numbers on the back of the phones were sequential. That was a clue in attempting to trace where the phones were purchased, and by whom.\n\nMerrick hoped the man wouldn't contact him until Monday because he wanted to enjoy two uninterrupted days with his wife.\n\nThe smell of baking apples wafted throughout the second story as Merrick opened the door to the bedroom he used for storage. He removed a footlocker and steamer trunk, then went to his knees and pressed gently on the edge of a wooden floorboard. It lifted easily. He did the same with another, then another. Concealed under the floor was a large oaken case that contained his coveted PSG-1 that had been made to fit his body's dimensions.\n\nHe opened the case, staring at the smooth stock and the various scopes. If the Corps's slogan was \"the proud, the few,\" then he had become an expert marksman with a deeply ingrained understanding of what it meant to be a sniper.\n\nMerrick opened the footlocker and removed several automatic handguns and ammunition; the closet floorboards had become the perfect place for concealing his arsenal of weapons. Where, he thought, would he be able to conceal the cache of arms in the Arlington condo?\n\nAlex had embarked on a house-hunting campaign, while he'd assumed an attitude of indifference toward the undertaking. However, the day of reckoning could not be put off too much longer. They had to move, and he had to find a place in which to totally secure his weaponry.\n\nMerrick had knelt down to replace the floorboards when he heard movement behind him. Turning around, he saw Alex standing in the doorway to the bedroom, her gaze fixed on the holstered automatic handguns. She glanced up, her expression a mix of fear and revulsion.\n\nRising slowly to his feet, Merrick closed the distance between them. He reached out to touch her, but Alex took a step backward. \"Querida, please.\"\n\nAlex shook her head. \"No, Merrick. Don't ask me to understand. I don't like guns, especially in my home.\"\n\n\"I'll make certain you'll never see them.\"\n\nShe gave him a long, penetrating stare. \"You do that. I came up to tell you that dinner is ready.\"\n\nMerrick watched her leave, unable to do or say anything that would convince his wife that the cache of weapons posed no threat to her.\n\n## Chapter 27\n\nThree days before she was scheduled to leave for Florida, Alex found the perfect house in Old Town, the core of historic Alexandria. The unoccupied redbrick home, dating back to the nineteenth century, boasted an updated kitchen and bathrooms. Five bedrooms with adjoining baths and sitting rooms would provide ample space for living and entertaining. She'd toured the area by car and had fallen in love with the abundance of antique shops lining King Street.\n\n\"I want it,\" she told the real estate agent.\n\n\"Don't you want your husband to see it before you make a decision?\"\n\nAlex stared at the smartly dressed middle-aged woman as if she'd spoken a language she didn't understand. \"There is no doubt my husband will like it.\"\n\n\"It's...it's just that I've had situations in the past where the wife wants something her husband doesn't and vice versa.\"\n\nOpening her handbag, Alex took out her checkbook. \"How much are you asking as a down payment?\" The agent quoted a figure, and she wrote the check. \"I'm due to deliver a baby in less than five months. I'd like to be settled in my new home before the end of January.\" She handed the startled woman the check. \"Please call me when you confirm a date for closing.\"\n\nAlex walked out of the house and made her way down the street to where she'd parked her car, her mind filled with how she wanted to decorate the remarkable structure. She no longer worked for the Trust, but she would have her own piece of history to preserve. Browsing for antiques germane to the period would become an ongoing project to keep her occupied until her daughter's birth and many years after.\n\nWhen the obstetrician had disclosed that she was having a girl, Alex was overcome with joy. She'd begun planning all of the activities they would share: baking cookies, tea parties and browsing antique shops. Alex shuddered to think she wouldn't have a girly-girl, that her daughter would prefer firearms and restoring old cars and trucks to shopping.\n\nSitting in her car, she dialed the number to Merrick's cell phone. It rang twice before he answered. \"I found it!\"\n\n\"Found what, Ali?\"\n\n\"Our house. I just put a down payment on it.\"\n\n\"Where is it?\"\n\n\"Alexandria. It's a little pricey, but it's worth it, Merrick.\"\n\n\"Can we talk about this when I get home?\"\n\nA frown appeared between her eyes. \"I hope we're not going to argue about money.\"\n\n\"Did I mention money, Alexandra?\"\n\n\"You're upset, Merrick.\"\n\n\"Why do you say I'm upset?\"\n\n\"Because you called me Alexandra. You only call me that when you're angry.\"\n\n\"Look, baby, I'm not angry. I don't know what the house is selling for and I don't care. If you like it, then we'll buy it.\"\n\nTears filled her eyes at the same time she bit down on her lower lip. \"Thank you, darling.\"\n\nHe laughed softly. \"You're welcome, darling. I'll see you when I get home.\"\n\n\"What do you want for dinner?\"\n\n\"You.\"\n\nBefore Alex could reply or react, Merrick hung up. She sat in the car, staring through the windshield, unable to believe she could feel so incredibly happy, that her life could be so incredibly perfect.\n\nMerrick Grayslake had come into her life when she wanted nothing to do with men. He'd become a friend, one who'd slipped under the barrier she'd created to keep all men at a distance because of her master plan. He'd waited patiently for her to complete her education, then wooed her with the skill and finesse of a libertine. And she'd taken the bait and let him reel her in.\n\nShe'd fallen in love, gotten pregnant and married. All the things she'd professed not wanting to do. But that was before she met Merrick Grayslake. His claim that he never wanted to marry or father children was shattered the moment they went from friends to lovers.\n\nAlex closed her eyes. They would soon celebrate a new year, and she looked forward to celebrating a new life with her new husband in their new home.\n\nPutting the key in the ignition, she started up the car and adjusted the heat. Meteorologists were predicting a white Christmas. Alex hoped the snowstorm would bypass Virginia and blow out to sea. She was so looking forward to spending the week in Florida with her extended family. She'd done all of her Christmas shopping and had shipped the gifts from her and Merrick to her uncle Martin's house in West Palm Beach.\n\nEvery year there seemed to be a new Cole baby, and it was becoming more difficult to keep up with the names that now spanned five generations.\n\nShe and Merrick had gone over names and had decided on Victoria Cole-Grayslake. The little girl would be faced with the daunting task of being a Cole, but there was no doubt she would succeed, because the Coles and the Grayslakes were survivors.\n\nMerrick's phone rang just as he closed and locked his desk. He glanced at the wall clock. It was three minutes before he was scheduled to leave for the day. Most times he worked beyond his dismissal because he wanted to avoid rush-hour traffic. He and Alex usually did not sit down to dinner until seven, so he did not see the need to rush out with the other office workers.\n\n\"Grayslake.\" He'd decided to answer the call.\n\n\"I'm going to call you once you reach your car.\"\n\nMerrick froze. It was the first contact he'd had with the mystery caller since he'd picked up the cell phones. And how, he mused, did the person know he was leaving for the day? Was something at the CIA monitoring his whereabouts?\n\n\"I assume you're going to call me on the phone with the lowest serial number?\"\n\n\"You assume right. I'll call you in a few minutes.\"\n\nMerrick was tempted not to leave, but then what? Would the sicko call him back on the government's line? He would leave as planned. Whoever it was that sought to push his buttons had succeeded because he was anxious to uncover just what the person wanted.\n\nIt took Merrick ten minutes to get an elevator and make it down to the parking lot. As soon as he sat behind the wheel, the disposable phone rang.\n\n\"What do you want?\" His greeting was brusque, rude, but he was beyond caring about social etiquette.\n\n\"I want you to meet me.\"\n\n\"Where?\" Merrick listened when he was told the address.\n\n\"That's not a very nice neighborhood,\" he drawled sarcastically.\n\n\"That's why I want you to meet me there. The only vermin you'll encounter will be of the two- and four-legged variety.\"\n\nHe'd been instructed to go to a crime-infested section of D.C. only blocks from where he'd been assaulted and left to die years before. Unlocking the glove compartment, he took out a small automatic handgun and secured it in the small of his back.\n\n\"Come alone and come unarmed.\"\n\n\"I'm coming alone.\"\n\n\"Don't bring the firearm.\"\n\nIt was too dark in the parking lot for Merrick to see if someone was watching every move he made. After all, he was sitting in the parking lot at CIA headquarters with high-tech surveillance equipment everywhere.\n\n\"Goodbye.\"\n\n\"Wait...don't hang up.\"\n\nA small smile of triumph parted Merrick's lips. \"Are you saying I can bring the gun?\"\n\n\"Bring it. Come to the second floor. I'll see you in an hour.\"\n\nThe call ended and Merrick saw a message flashing the number of remaining minutes. Their conversation had lasted exactly two minutes.\n\nMerrick started up his truck, driving away from Langley toward southeast D.C. He hadn't gotten more than a mile when he remembered Alex's phone call about finding a house. He activated the hands-free device and dialed her number.\n\n\"Hi, baby.\"\n\nMerrick smiled. \"Hi, yourself. I'm calling to let you know I'll be late.\"\n\n\"How late?\"\n\n\"I'm not certain. But I don't think it's going to take too long. Put away your pots. We'll eat out tonight, and then you can show me our new house.\"\n\n\"Thank you, Merrick!\"\n\nHis smile widened. \"You're welcome, Ali.\"\n\nMerrick continued to talk to his wife until he turned down the street to the address his caller had given him. Slowing, he peered at the dilapidated and burned-out buildings lining the block. A hand-painted number in fluorescent paint shimmered in the darkness on the last house on the street. Someone really wanted him to find the building. He parked across the street.\n\nMerrick alighted from his truck at the same time he reached around his back and gripped the handle of the holstered automatic. The gun was small enough for him to palm it easily.\n\nHe entered the building, waiting for his eyes to adjust to the light from a dim naked bulb hanging from a frayed wire. He tested the first stair. It groaned beneath his weight, but held. Counting the number of stairs, he made it to the second-floor landing. Firearm drawn, he knocked on the door.\n\n\"Come in,\" ordered the diffused voice.\n\nPushing open the door, Merrick was assailed with the smell of stale urine and other unidentifiable odors. He automatically reached for a wall switch but quickly discovered there was no electricity. There came the distinctive scrape of a match, followed by the smell of sulfur, then the weak flickering flame of a candle.\n\n\"Sit down,\" came the disembodied voice. Merrick complied, reaching for a wooden crate, and sat down. \"This will not take long because I know you want to go home to your very pretty wife.\"\n\n\"Who the hell are you?\"\n\n\"You don't need to know who I am. Not yet.\"\n\n\"And you don't need to bring my wife into this,\" Merrick retorted.\n\n\"She's in it because she is your wife.\"\n\n\"This is not about her, is it?\"\n\n\"No. It's about your mother.\"\n\nMerrick closed his eyes. \"What about her?\"\n\n\"I know who killed Victoria Grayslake.\"\n\n## Chapter 28\n\nMerrick swayed slightly before righting himself. He tightened his grip on the gun butt. \"Why are you telling me this?\" His query was whispered. \"Why don't you expose the murderer?\"\n\n\"She wasn't my mother.\"\n\n\"She abandoned me, so why should I feel anything for her?\"\n\n\"She abandoned you to save your life.\"\n\nMerrick closed his eyes for several seconds. \"In other words she sacrificed herself for me?\"\n\n\"It was the ultimate sacrifice. Your mother was an undercover DEA special agent assigned to the Mexican border region. She'd infiltrated a group who'd operated openly and without impunity between the U.S. and Mexico.\n\n\"She'd been assigned to gather evidence to put several high-ranking Mexican Federales, at least eight U.S. Border Patrol agents and two Texas and Arizona ranchers in prison for the rest of their lives.\n\n\"Victoria, who'd discovered she was pregnant, requested a leave but before she was scheduled to depart Mexico she was abducted and held prisoner until it was time for her to give birth.\"\n\n\"Are you saying someone blew her cover?\" Merrick asked perceptively.\n\n\"Yes. A senior agent gave her up.\"\n\n\"What happened?\"\n\n\"Victoria was flown by private jet to a ranch in south Texas where she delivered a son. The baby boy was left in a church. Victoria's body was discovered weeks later in a shallow grave near the Mexican border. She'd been tortured, her tongue cut out, then shot in the back of the head.\"\n\n\"How do you know all this? And why are you telling me about something that happened more than thirty years ago?\"\n\n\"I was her partner on one undercover mission, and I swore an oath that I would always protect her.\"\n\n\"It's apparent you didn't protect her. What do you expect me to do?\" Merrick asked.\n\n\"I want you to expose the man who eventually became an associate director for the DEA. I want you to bring down the son of a bitch who ordered the hit on Victoria Grayslake.\"\n\n\"You expect me to take revenge for someone I never knew?\"\n\n\"I can understand your reluctance, but Victoria gave up everything to have you. If she hadn't compromised her mission you wouldn't be here. She forfeited her life so that you could enjoy your life, your wife and await the birth of your own child.\"\n\nMerrick longed to spring across the space and snatch the man by the throat. He wanted to know who was it that knew so much about him. \"What do you mean she compromised her mission?\"\n\n\"She cut a deal\u2014her baby for the identities of undercover field agents.\"\n\n\"How do you know this?\"\n\n\"It was on videotape. It sent shock waves through the DEA. Agents were pulled, assigned to desk duty or transferred to other agencies. It was the administration's most serious breach in security and the fallout was catastrophic. Victoria was tried in absentia, charged with treason and sentenced to life in prison. The man responsible for her death was promoted and is now a very influential D.C. lobbyist.\"\n\n\"Victoria is gone, she can't be vindicated and you want me to take this person out? Other than my being the son of Victoria Grayslake, you're going to have to give me another reason.\"\n\n\"He knows who you are, and he's afraid that one day you may decide to seek retribution for what he did to your mother.\"\n\n\"If that's true, then why hasn't he tried to take me out?\"\n\n\"He did try.\"\n\nMerrick froze, holding his breath until he felt his lungs exploding. The roaring in his head intensified and he felt faint. He let out a lungful of constricted breath. He knew the answer even before he asked it.\n\n\"When?\"\n\n\"When you were shot and left on the corner two blocks from here. The woman who'd asked you to meet her was also in on the conspiracy.\"\n\nMerrick closed his eyes, reliving the scene as if it were yesterday. Corrine Grice, who'd moved into the apartment building in Silver Spring a month after he had, was the consummate neighbor. She picked up his newspaper when he failed to stop delivery because the Company hadn't given him enough notice whenever they called for a new mission. And whenever she cooked too much she shared what she had with him.\n\nThere was never anything personal about their relationship, so a year later when Corrine told Merrick that she planned to buy property in a less-than-desirable section of D.C. because she was tired of paying rent he offered to help her with her fixer-upper.\n\nHe spent hours painting the kitchen and bathroom, and when he told Corrine he was going home she came on to him like a cat in heat. More shocked than repulsed, he told her that he didn't see her in that way and left, running headlong into three men whose intent was to rob him. What he'd thought was a botched robbery in reality had been a conspiracy for murder.\n\n\"Where is Corrine?\"\n\n\"She and her three flunkies were found a little well done in a burned-out establishment around the corner. No witnesses, no blame.\"\n\nSeconds ticked off into a minute before Merrick said, \"Who is he?\"\n\n\"Chandler Duffy.\"\n\nA sardonic smile twisted Merrick's mouth. \"Chandler Duffy. The unofficial president of the United States.\"\n\n\"The same,\" confirmed what Merrick now thought of as the voice.\n\n\"Why don't you go after him yourself?\"\n\n\"I can't.\"\n\n\"Well, neither can I,\" Merrick argued softly. \"If I'm going to do it legally, then you know I'm prohibited from collecting foreign intelligence concerning the domestic activities of U.S. citizens. That falls under the jurisdiction of the Bureau.\"\n\n\"I'll provide you with what you need to go after Duffy.\"\n\n\"Whatever you tell me I'll have to pass along to my superiors.\"\n\n\"I don't give a flying fig who pinches him. It could be CIA, FBI, DEA, ATF, Secret Service or the friggin' Boy Scouts of America.\"\n\nMerrick nodded, doubting whether his informant could see him. \"I'll see what I can do.\" He stood up. \"Now, if you'll excuse me I have to go home and get out of these clothes.\"\n\n\"I'll be in touch,\" said the voice.\n\n\"Oh, I'm sure you will.\"\n\nMerrick gripped the gun as he made his way down the rickety staircase and out into the crisp cold air. He sucked in a lungful before letting it out. The cloying stench of human and animal waste lingered in his nostrils.\n\nMerrick drove back to Arlington. He stood outside the door and had Alex bring him a large plastic garbage bag. Not concerned with who saw him in his birthday suit, he stripped off his clothes and shoes and left them in the bag on the floor outside the door.\n\nIt was only after he'd showered and washed his hair, twice, that he felt clean again.\n\n## Chapter 29\n\nAll eyes on the gun range were focused on Merrick Grayslake as he executed his marksmanship skills. Using his right hand, he fired at the target, all sixteen rounds hitting dead center. Expelling the spent clip, he inserted another and repeated the action with his left hand.\n\nA spattering of applause followed the awesome exhibition as Merrick took off the protective gear covering his eyes and ears. He ignored those standing around talking quietly amongst themselves. He'd come to the gun range to let out some of his pent-up frustration.\n\nHe'd reacted like an automaton when Alex parked in front of the home where she wanted them to live. He told her he liked the style of the house, the neighborhood and the shopping area. What he couldn't tell her was what he'd gleaned earlier that afternoon. That he knew about Victoria Grayslake, who she'd worked for and how she'd died. It haunted him that she'd agreed to give up names of undercover agents to save the life of her unborn child.\n\nChandler Duffy. The name played over and over in his head, becoming a litany. The former DEA supervisor had set up Victoria Grayslake to be tortured and executed. Why? Merrick asked himself over and over, unable to come up with a plausible explanation.\n\nRetribution. The word seeped into his consciousness, nagging at him like a gnawing ache. Duffy had to pay, not only because he was responsible for an agent's death; he also had the blood of four hired thugs, five if they'd killed him, on his hands and who knew how many more?\n\nWhether it was retribution, revenge, payback or reprisal, Chandler Duffy knew who he was and Merrick Grayslake knew Chandler. It was time for him to level the playing field, settle the score.\n\nHe'd come to work earlier than usual, hoping to meet with Carl Ashleigh or William Reid, but was told that the two men were out of the building. Merrick knew he couldn't do anything to Duffy unless he cleared it with Ashleigh and\/or Reid. Until then he would take his stress and frustration out on the gun range target.\n\nAshleigh and Reid returned to CIA headquarters later that afternoon and were told that Merrick Grayslake needed to speak to them ASAP. They shared a knowing look, then summoned him. He was forthcoming when he told them of his meeting with the voice.\n\nAshleigh spoke first. \"Chandler Duffy is no small-time punk looking to make a big score. He's been peddling influence in Washington for two decades, and everyone knows he's the best at what he does.\"\n\nLacing his fingers together, Ashleigh avoided Merrick's gaze. \"He's what you'd call a power broker extraordinaire. He has backed our last two presidents.\"\n\n\"Do you have anything on him?\" Merrick asked.\n\nWilliam Reid looked at Ashleigh, who nodded. \"Duffy's name has come up in connection with his firm accepting cash payments from the head of an African nation who has been accused of human rights violations and rumored to offer refuge to known terrorists.\"\n\nMerrick's gray gaze shifted from Reid to Ashleigh. \"How do you know this and where do we fit into that equation?\"\n\nA rare smile found its way to Ashleigh's pale eyes. \"We've uncovered an offshore account in Duffy's wife's name, and there's been talk that he brokered a deal for several senators and for our incumbent president's opponent to meet secretly with this leader.\"\n\nThis wasn't the news Merrick wanted to hear because personally he liked the man who was certain to receive his party's nomination to run against the weak and ineffective incumbent president.\n\n\"I'd like to request permission to head the team to investigate Duffy.\"\n\nAshleigh shook his head. \"I don't know, Grayslake. It wouldn't work, because something of this nature would be too personal for you.\"\n\nLeaning forward, Merrick impaled him with a lethal stare. \"I'm not going to lie and say it's not personal. But I didn't have to come to tell you what went down last night. I could've gotten into my vehicle, stalked Duffy and taken him out at a thousand yards with one bullet, one kill. Far enough away to see his head explode, and far enough away not to have pieces of his DNA on me.\n\n\"The fact that I've come back to the Company is like waving a red flag in front of Duffy. Unlike the last time, I'll be ready for him and his hired thugs. But what I won't do is put my family at risk. Either you approve me going after Duffy legally or I'll take care of him my way. And you both know how that will end.\"\n\nIf Merrick Grayslake had been any other subordinate Ashleigh would've written him up. The director had half hoped Grayslake would take the bait, but it'd gone better than they'd planned because Duffy wasn't above killing women. And because Grayslake, dubbed the \"Lone Wolf\" by those involved in Operation Backslap, had taken a mate it made the mission that much more personal.\n\n\"Okay, Grayslake,\" Ashleigh said grudgingly. \"You can be first chair on this one. Do you want to pick your team or do you want to leave that to me?\"\n\n\"We can do it together.\"\n\nCarl Ashleigh felt a measure of relief for the first time since he'd been ordered to indirectly involve Merrick Grayslake in the mission to neutralize Duffy.\n\n\"Good. Let's get together at the end of the first week in January.\" He flipped a page in his planner and circled a date. \"If I can't get someone else to lead the sniper training, then we'll put it off until you close out the Duffy investigation.\" He closed his planner, struggling not to do the happy dance. He'd lied to Merrick. There never was going to be a sniper-training course. \"If you need some time off to make arrangements to protect your wife, then take it.\" He stared at Merrick, then his assistant. \"Gentlemen, this meeting is over.\"\n\nMerrick left the conference room and returned to his office. Propping his elbows on the desk, he covered his face with his hands. How, he thought, was he going to tell Alex that she would have to stay in Florida with her parents until Duffy was either in custody\u2014or dead? He lowered his hands and stared at a wall calendar displaying eighteen instead of the usual twelve months. He'd circled the last day in April\u2014the day Victoria Cole-Grayslake was predicted to make her appearance.\n\n## Chapter 30\n\nAlex's plan to change her flight reservation to leave Virginia with Merrick backfired. The snow, which had begun falling two days before Christmas, continued nonstop for three days and two nights. Snow totals from Maine to portions of West Virginia averaged more than two feet.\n\nIt would become the first Christmas that the second generation of Coles, Kirklands and Grayslakes would not celebrate Christmas and New Year's in West Palm Beach, Florida. Gabriel and Summer, Michael and Jolene and Merrick and Alexandra were unable to leave their homes.\n\nMerrick, who lay in bed with Alex in the Kirklands' guest bedroom, rested a hand over her distended belly. They'd come to Georgetown for dinner and found themselves snowbound.\n\n\"Why don't you go to Florida when the airports are back on schedule.\" Alex felt as if someone had caught her throat, not permitting her to breathe. Merrick had changed, but she'd attributed that to his work at the CIA. Most nights he came home, went directly to the bathroom where he showered, then flopped in front of the television until she called him for dinner. He always helped her clean up the kitchen, waited for her to come to bed, turned over and fell asleep.\n\nSitting up, she turned on the lamp on her side of the bed and shook him awake. \"Merrick, are you having an affair?\"\n\nHe moaned and threw an arm over his face. \"What?\"\n\n\"I know you heard what I said.\" She repeated her query.\n\nMerrick sat up. \"No. Oh, hell, no!\"\n\nAlex pulled her lower lip between her teeth. \"Then, why is it we haven't made love in more than a week? Is it because I'm fat and ugly?\"\n\n\"Ali, my love,\" he crooned, reaching for her, but she turned her back. \"You are the most beautiful woman in the world.\"\n\n\"But I'm fat.\"\n\n\"Baby, you're pregnant. Pregnant women always put on weight.\"\n\nTurning over to face him, Alex touched the side of his face. \"You're not having an affair?\"\n\n\"I would never cheat on you, Ali.\" Merrick combed his fingers through her curls.\n\n\"Then, what's the matter, Merrick? Why do I feel as if we're growing apart?\"\n\nMerrick wanted to tell her, unload all he'd discovered about his mother, but he couldn't. \"I'm involved with something at work that has me distracted.\"\n\n\"Can you talk about it?\" Alex asked.\n\n\"No, baby. It's classified.\"\n\nShe smiled. \"That's all you had to say.\"\n\nHe kissed the end of her nose. \"Any time I start acting strange, talk to me.\"\n\n\"You're always strange,\" Alex teased.\n\nMerrick's hand searched under the hem of her nightgown, trailing up her thighs. \"How's that?\"\n\nShe pressed her forehead to his. \"You have a way of looking at a person that makes them feel as if you have X-ray vision. Then you have a habit of sneaking up on folks wherein they look up and you're there.\"\n\n\"Do I frighten you, darling?\"\n\nShe closed her eyes, smiling. \"Not anymore.\"\n\n\"When did you stop being afraid?\"\n\n\"The day I became Mrs. Merrick Grayslake. I knew I could never live with a man who frightened me.\"\n\n\"Have you given what I said some thought?\"\n\n\"About you wanting me to go to Florida?\"\n\n\"Yes.\"\n\n\"Why do you want me to go away, Merrick? And please don't tell me it's classified.\"\n\n\"I'm involved in something that may put you and the baby in danger.\"\n\n\"What is it?\"\n\n\"That I can't tell you.\"\n\n\"You want me to go away for how long?\"\n\n\"I don't know,\" he answered truthfully.\n\n\"No, Merrick. If you can't give me a time frame, then I'm staying here. We're buying a new house and I plan to focus all of my energies on decorating it before the baby comes.\"\n\n\"I don't want to fight with you, Ali.\"\n\n\"Then don't,\" she shot back.\n\n\"I will if I have to.\"\n\n\"Are you threatening me, Merrick Grayslake?\"\n\n\"No, I'm not. I'm telling you that I'll do what I have to do to keep you safe. If that means hiring someone to watch you when I'm not here, then I will.\"\n\nAlex's hands curled into fists. \"I will not become a prisoner in my home.\"\n\nMerrick pulled away and lay on his back. \"You will become anything I want you to become until what I'm involved in is over.\"\n\nAlex struggled to control her temper. \"When will it be over?\"\n\n\"Dammit! I don't know!\"\n\nShe swung her legs over the side of the bed. Her movements were slow and awkward as she made her way over to a window. \"Don't you ever raise your voice to me again!\" she ground out between clenched teeth. \"In case you failed to notice I am your wife, not your chattel. Yell at me again and you don't have to worry about hiring someone to protect me, because I'll be out of here so fast you'll forget what I look like.\"\n\nMerrick moved off the bed, stalking her like a large cat. \"You will not leave me. You will not take my child from me.\"\n\n\"Me, me, me,\" she mimicked. \"Why is this about you? What about me?\"\n\n\"It is about you, Ali. Everything in my life, everything I do is about you.\" He threw up a hand. \"Do you think I really wanted to go back to the CIA?\"\n\n\"Then why did you?\"\n\n\"Because I needed to support my wife and family.\"\n\n\"I don't need you to take care of me.\"\n\nHe took several steps, bringing them inches apart. \"What kind of man would I be if I let my wife foot the bill for everything? How long would it take for me not to feel like a man? Be a man?\"\n\n\"Why are you equating your manhood with money?\"\n\nGrasping her shoulders, he pulled her to his naked body. \"I have to, Alexandra, because that's who I am. Do you think your father would've given us his blessing if I didn't have a job?\"\n\nHer eyes widened until he saw into their clear gold depths. \"What does my father have to do with us?\"\n\n\"Your father asked me how I was going to support you.\"\n\n\"When?\"\n\nMerrick told her about the confrontation with her father as an expression of disbelief swept over his wife's incredibly beautiful face. Being pregnant made her lush, more feminine.\n\n\"Even though you're carrying David Cole's grandchild, he still had the stones to blindside me in front of his brothers and sons to question me about how I was going to take care of his precious baby girl.\"\n\nHer lids fluttered. \"I didn't know, Merrick. I'm sorry my father\u2014\"\n\n\"Don't apologize for him, Ali,\" he spat out, cutting her off. \"What your father said came from his heart. I have to respect him for that.\"\n\nResting her palms on his pecs, Alex pressed a kiss over his heart. \"I'm sorry for fighting with you, mi amor.\"\n\nHe cradled her belly. \"We don't fight, baby.\"\n\n\"What do we do?\"\n\n\"Disagree.\"\n\n\"I...I...\" Her eyes were wide as silver dollars. Placing her hands over Merrick's, she pressed them to her swollen abdomen. \"Did you feel that?\"\n\n\"Feel what?\"\n\n\"The baby, Merrick. She kicked me.\"\n\nHe placed his hands on her belly. \"I don't feel anything.\"\n\nAlex gasped. \"She kicked again.\"\n\n\"Where?\"\n\n\"Here.\"\n\nMerrick closed his eyes and waited. Then he felt it\u2014a soft flutter. It happened again. A well of emotion filled his chest, making it difficult for him to draw a breath. His daughter was moving. Now she was real, very, very real.\n\nGoing to his knees, he pushed up Alex's nightgown and kissed his wife's belly. \"I love my girls,\" he whispered, placing soft tender kisses over every inch of her swollen flesh.\n\nAlex cradled Merrick's head as an angelic smile spread over her face. Her baby, their baby, in making her presence known, had forced a truce between her mother and father.\n\nMerrick rose to his feet and swept Alex up in his arms. He carried her back to bed and proceeded to make love to her slowly, gently, as if he feared she would shatter into a million tiny pieces.\n\nHe gave, she received, she gave and he was there to receive. A wave of ecstasy came upon them so quickly they didn't have time to react. A fireball exploded and they fell headlong into a maelstrom of uncontrollable joy that ebbed to a deep feeling of sated peace.\n\n## Chapter 31\n\nThe following morning Merrick asked Michael to take a walk with him. He knew by the expression on Michael's face that he thought he'd taken leave of his senses; who'd want to go for a walk when some streets were still impassable because the sanitation department hadn't gotten around to removing the mountain of snow from residential neighborhoods?\n\nMichael recognized and quickly processed the tension in his friend's request, and agreed. \"What's up, Gray?\" he asked once they attempted to navigate a narrow path on the sidewalk, made by booted footsteps.\n\n\"I need a favor.\"\n\nMichael stared up at a startlingly blue, cloud-free sky. The warmth of the winter sun felt good on his face. \"Ask away.\"\n\nMerrick stomped his feet, knocking snow off his boots. \"I need you to look after Alex.\"\n\nClosing his eyes, Michael smothered a savage expletive. \"What the hell are you involved in now?\"\n\n\"I can't tell you. It's\u2014\"\n\n\"Classified,\" Michael said, cutting him off and completing his statement. He shook his head. \"I thought you were going back as a trainer?\"\n\n\"I did.\"\n\n\"But right now you're not training anything or anyone. I don't want to be the one who said I-told-you-so, but I'm going to say it anyway. I told you that they were going to recruit you for some clandestine operation. I know how they operate, Gray. It's one thing to be in the field and wait for your orders, and another completely when you're sitting at the top in an office making decisions as to who they can use and who is expendable. If nothing else, working at the Pentagon taught me that.\"\n\nMerrick waited for Michael to finish his tirade, then said, \"This is personal, Kirk.\" The two men stepped into a snowbank to let a woman pass.\n\n\"How personal?\" Michael asked.\n\n\"It involves my mother.\"\n\n\"Is she alive?\"\n\nMerrick shook his head. \"No. The only thing I will tell you is that she was tortured before she was executed.\"\n\n\"Oh, damn. I'm sorry, Gray.\"\n\n\"It's okay. What I find strange is that when I found out how she died I didn't feel anything. No anger, no sadness.\"\n\n\"That's because you never met her. Do you have a photograph of her?\"\n\nMerrick shook his head. \"No.\" He'd tried accessing her file through an intra-agency database, but her name had been deleted.\n\nClasping his gloved hands behind his back, Michael gave his friend a sidelong glance. \"How much danger is my cousin in?\"\n\n\"Life threatening.\" Merrick had decided not to mince words.\n\n\"She can live with me until whatever crap you're involved in is over. I'm sure Jolene would love to have Alex around. I'll be home until the end of January. I hope you complete your mission before that. If not, then we'll have to come up with another plan.\"\n\n\"I want her to go to Florida and stay with her folks.\"\n\n\"Did you tell her that?\"\n\n\"Yes.\"\n\n\"Did she go off on you?\"\n\nMerrick stopped and stared at Michael. \"Did you hear us last night?\"\n\nMichael shook his head. \"No. But I know my cousin, Gray. She's not going to Florida because she doesn't want to, but because you told her to go. Alex is sweet, funny and generous. But she's also as stubborn as a mule. You don't tell Alexandra Cole what you want her to do, you ask her. Let her think it's her idea.\"\n\nMerrick frowned. \"I don't have time to coax and coddle. Not with what I'm up against.\"\n\n\"Your wife is with child, Merrick. She is not the same Alexandra you met and fell in love with. Once she has the baby the old Alex will return. Right now you can't do anything to set her off. I'm a witness. I've lived through and survived the nine months of horror.\"\n\n\"Because you've survived the horror, what do you suggest I do?\"\n\n\"Tell her you have to go away for a week, and that you want her to stay with me because you don't want her left alone because of the baby.\"\n\n\"But I'm not going away. Besides, I've never lied to Alex.\"\n\n\"Merrick, my friend. You are lying to her. Her life and that of your unborn child are at risk, yet you're telling her she has to go away. And when she asks why, you tell her it's classified. Look, man, if you have to lie, cheat or kill a son of a bitch to keep your family safe, then you do it. Tell Alex you're involved in some training program and see what she says. I'm willing to bet she'll volunteer to hang out with me and Jolene.\"\n\n\"Okay, I'll give it a shot. But if she balks, then it's World War Three.\"\n\nMichael smiled. \"If you'd told me you were fooling around with my cousin I would've given you the four-one-one on her. After all, she's a Cole woman and they are a breed unto themselves.\"\n\n\"Amen,\" Merrick said under his breath. \"I'm ready to go back. Thanks for hearing me out.\"\n\nMichael patted his shoulder. \"No sweat. After all, we're family.\"\n\nHe and Michael had returned to the house at the same time Jolene and Alex walked into the kitchen. They shooed the women to the family room while they prepared a country breakfast of bacon, sausage, eggs, grits, biscuits, freshly-squeezed orange juice and brewed coffee and tea.\n\nMichael put on several CDs, and the music flowing from hidden speakers added to the festive mood. Six-month-old Teresa, who'd begun pulling herself up in her crib, crawled around on the floor, getting underfoot. Jolene placed her in a playpen in a corner of the large kitchen and the little girl cried hysterically until Alex freed her from the mesh-covered prison.\n\nJolene glared at Alex. \"She's spoiled enough without you adding to it. I should send her home with you.\"\n\nAlex kissed Teresa's curly black hair. \"Do you want to come stay with Titi Alex? I wouldn't mind taking care of you for a couple of weeks to give your mama a break.\"\n\n\"Why don't you hang out here with us for a couple of weeks?\" Jolene suggested. \"We can go shopping for baby clothes and furniture to decorate the nursery in your new house.\" She smiled at her husband. \"Would you mind if your cousin stayed with us?\"\n\nMichael affected a stern expression. His wife had given him and Merrick the opening they needed. \"Are you asking or telling me, Jo?\"\n\n\"I'm asking, Michael.\"\n\nHe waved a slender hand. \"Alex is family. You don't have to ask. Who you should be asking is Merrick. After all, he is her husband and he just might have something planned for the two of them.\"\n\n\"Merrick and I don't have any plans,\" Alex volunteered.\n\nMerrick stared at Alex rather than Michael. He knew if he looked at his wife's cousin, he wouldn't be able to keep a straight face. He felt relieved that he didn't have to lie to Alex.\n\n\"Did you forget that we close on the house January ninth?\"\n\n\"No,\" Alex said. \"That's not going to take more than an hour or two. As soon as we get the keys I want Jolene to see the house.\"\n\nJolene dabbed the corners of her mouth with a napkin. \"Have you decided what style of furniture you want to put in it?\"\n\n\"Not really.\"\n\n\"Now, that's going to be fun.\"\n\nAlex shifted Teresa to her other knee. \"Merrick, would you mind if I stayed here with Jolene? We have a lot of shopping to do, and I would like her opinion on things like wallpaper.\"\n\n\"I don't mind at all,\" Merrick said much too quickly. He ignored Michael's incredulous stare. \"You know I have no interest in wallpaper, paint colors or what fabric you want on some chair or other doodad.\"\n\n\"Are you sure you're not going to miss me, mi amor?\"\n\n\"Miss you? Aren't we going to see each other at night?\" Merrick affected a frown. \"If you've found someone you'd rather be with, please tell me now.\"\n\nAlex blushed like a schoolgirl with her first crush. \"Who's going to want me looking like this?\"\n\nThere was a long-suffering silence. \"I want you, Ali,\" Merrick said, his voice filled with emotion.\n\nMichael cleared his throat. \"Damn, my brother. You don't have to show your woman you're whipped.\"\n\n\"Stay out of it, Michael,\" Jolene warned softly. \"Please,\" she whispered when her husband opened his mouth.\n\nMichael caught and held his wife's gaze. He knew how Merrick felt about Alex because he felt the same about Jolene. He'd fallen in love with her on sight, and each day he grew more in love with her.\n\n## Chapter 32\n\nMerrick opened and closed his eyes in an attempt to relieve the burning. He'd been staring at the computer monitor for hours, searching one database, then another for something that would link Duffy to something\u2014anything the government could use to issue an indictment.\n\nHe headed a team of four, including himself, who'd worked tirelessly for the past three months researching every lead given them. So far, Duffy had come up clean.\n\nHe came in early and worked late, yet his private life had remained stable. He and Alex had closed on the house in Alexandria and her priority was decorating it before giving birth.\n\nThe Kirklands' offer of their guest wing had become a necessity once the Arlington condominium was sold. Merrick was more relaxed because Alex wasn't left alone during the day. Whenever she had to see the doctor he took time off from work to take her.\n\nMerrick was aware that Alex didn't like that he didn't go anywhere without the small but powerful handgun tucked into his waistband, but it was a topic he refused to discuss with her.\n\nHe glanced over at a female computer programmer who'd been Ashleigh's recommendation. Merrick was astounded with what she could do with one keystroke. \"What screen are you looking at, Patty?\"\n\n\"DEA.\"\n\n\"Patch me over.\"\n\nWithin seconds the Drug Enforcement Administration logo appeared on his monitor. The fact that Duffy had been DEA before leaving to start up his lobbying firm made them a part of the investigation.\n\nPen in hand, a pad of paper nearby, Merrick began scrolling down a listing of names of past and present agents, hoping to connect at least one to Duffy's present operation.\n\nSummer Montgomery. The name jumped out at him. The undercover special agent\u2014code name: Renegade\u2014was engaged to his brother-in-law Gabriel. Pressing a few keys, he viewed the operation she was assigned to cover at Weir Memorial High School. He stared at the names of the students and faculty members. A slight frown furrowed his forehead. Dumas Gellis. He'd come across the name before, but where?\n\nHe scribbled the name on the pad.\n\n\"Patty?\" Merrick hadn't taken his gaze off the screen.\n\n\"Yes, Merrick?\"\n\n\"I need you to go to the Treasury Department site and type in Dumas Gellis.\"\n\n\"Dumas like in Alexandre?\"\n\n\"Yes, Patty.\"\n\nSoft brown eyes widening behind the lenses of her glasses, Patty smiled. \"He's here. He has several offshore accounts in the Cayman Islands.\"\n\n\"Can you tell me how much he's deposited?\"\n\n\"Give me a few minutes and I'll let you know.\" Biting down on her lower lip, an affectation she executed whenever she concentrated on a task, Patty bobbed her head up and down. \"Got it.\"\n\n\"How much?\"\n\n\"All totaled. In excess of three mil.\"\n\nMerrick whistled. \"Not bad for an assistant principal and ex-footballer.\" He read the name of the agent-in-charge of the Weir operation. \"I believe Lucas Shelby would find Mr. Gellis's bank balance quite interesting.\" With a stroke of a key, Gellis's banking information was forwarded to the Drug Enforcement Administration.\n\nMerrick knew he'd helped his future sister-in-law bring closure to her assignment, but he still hadn't gotten a single clue with which to take down Duffy.\n\nThe lobbyist was smart, but was he smarter than the team of four who were the best at what they did? Merrick knew if they were going to pinch Duffy, it would be tied to his finances.\n\nLeaning to his left, Merrick tapped the shoulder of an accountant on loan to the CIA from the Treasury Department. A recent college graduate, Stuart Olson didn't look old enough to drive or drink.\n\n\"I need a printout of banking transactions from every international database in existence.\"\n\n\"Can I do that, sir?\"\n\nSmiling, Merrick nodded. \"Yes, you can.\"\n\nHe didn't tell the accountant that the secret program, run out of the Central Intelligence Agency and overseen by the Treasury Department, initiated weeks after the September 11 attacks, permitted counterterrorism officials access to financial records from a vast international database to examine banking transactions involving thousands of Americans and others in the United States. However, the program was limited to tracing transactions of people suspected of having ties to terrorist groups.\n\nStuart gave a look that said he didn't believe what he'd been told. \"The program is predicated in part on the president's emergency economic powers,\" Merrick explained.\n\nDuring a briefing session, Merrick was told data from the Brussels-based banking consortium, formally known as the Society for World Interbank Financial Telecommunication, had allowed officials from the CIA and the FBI and a few other agencies to examine tens of thousands of financial transactions.\n\n\"That's going to take days if not weeks, Mr. Grayslake.\"\n\nMerrick patted his shoulder again. \"You have a June first deadline to come up with something on Duffy.\"\n\nStuart sat up straighter. \"I'll find something on him before May Day.\"\n\nMerrick redirected his gaze to his computer monitor. Patty had changed it again. He noted the time at the lower right of the screen. It was 8:05 p.m., time that he headed home.\n\nHe couldn't wait to wrap up this case so he could resume a normal life wherein he'd come home from work and have his wife greet him in their new house. A contractor had refurbished it to Alex's specifications, and she and Jolene had spent all their free time together shopping for antiques to fill up the grand structure. They'd established a routine of going to Sunday-morning mass, followed by brunch, then to the house that was to become their home once he closed the Duffy case.\n\nMerrick knew Alex longed to spend the night in their new home, but he couldn't afford to take the risk\u2014not even with a state-of-the-art sophisticated security system\u2014because he never knew when he would be summoned to return to Langley.\n\nHer advancing pregnancy slowed her gait, and she constantly held her lower back as if she were in pain. Whenever he questioned her she said it was just the baby pressing on her lower spine.\n\nAshleigh had given him until June first to bring down Duffy, but Merrick hoped it would be sooner. His wife was due to deliver their first baby in a month, and when he brought his daughter home he didn't want it be under a cloak of danger and secrecy.\n\nStanding up, he rolled his head on his shoulders. \"I'm done for the day.\"\n\n\"I'm staying,\" said Patty.\n\nStuart stared up at Merrick. \"Me, too.\"\n\n\"What about you, Justin?\" Merrick asked an information specialist whom he'd selected to join the team. Justin Jefferson bore an uncanny resemblance to a young Malcolm X.\n\n\"I guess I'll stay a little longer. Besides, I could use the overtime.\"\n\n\"If I see any of you in before me tomorrow, I'm going to write you up,\" Merrick teased. They laughed as he walked out of the room that had been set up for them.\n\nHis stomach growled loudly as he made his way to his office, reminding him that he'd skipped lunch and dinner. He would stop at a health-food drive-through and pick up something.\n\n## Chapter 33\n\nMerrick had just crossed the marker indicating he'd left Virginia for the Capitol District when his cell phone rang. He reached into his shirt pocket for the phone. Without looking at the display, he answered it.\n\n\"Hello.\"\n\n\"Merrick, it's Michael.\"\n\nHis heart lurched. Michael always called him Gray. \"What's the matter?\"\n\n\"I'm at the hospital with Alex. She's in labor.\"\n\n\"But the baby's not due for another four weeks.\"\n\n\"It looks as if your daughter doesn't want to wait.\"\n\n\"I'm on my way.\"\n\nHe closed the phone and jammed the accelerator, taking red lights like a man possessed. He heard the siren, then the flashing lights behind him. Damn his luck! He was being pulled over. Coming to a screeching halt, he was ready for the officer when he approached his vehicle.\n\nHe handed the officer his photo identification from the Central Intelligence Agency. \"I just got a call that my wife has gone into premature labor.\"\n\nThe officer took a glance at the ID, and then his glance lingered on Merrick's face. \"Follow me.\"\n\nWhat should've taken fifteen minutes was reduced to five as Merrick was given a police escort to the private hospital where Jolene had delivered Teresa.\n\nHe thanked the officer as he raced toward the entrance to the hospital. \"Grayslake,\" he shouted to the clerk at the reception desk.\n\n\"Labor room two.\"\n\nNot waiting for the elevator, Merrick bounded up the staircase to the second floor. His heart was pounding, hands shaking uncontrollably when he came face-to-face with Michael.\n\n\"How is she?\"\n\nMichael patted his shoulder in a comforting gesture. \"She's okay. They have her hooked up to monitors. Get yourself together before you go in there. If Alex sees you like this she's going to lose it.\"\n\nMerrick patted his hair. \"What do I look like?\"\n\n\"You don't want to know. Go in the men's room and splash some water on your face. I'll wait here until you come out.\"\n\nMerrick went into the men's room. He peered into the mirror, seeing the face of a stranger looking back at him. There was a look in his eyes he'd never seen before\u2014fear.\n\nTurning on the faucet, he washed his hands, then wet his face. His motions were mechanical when he dried his face on a paper towel, then combed his hair. Straightening his shoulders, he walked out of the restroom and into labor room two.\n\nMichael stood in a corner watching an attending doctor examine his cousin, reliving the scene when he'd waited for the birth of his child. He knew Alex wanted Tyler to deliver her child, but it was apparent Baby Grayslake had decided to ignore the calendar to make a premature debut.\n\nHe smiled when Merrick entered the room. He looked better.\n\nClosing the distance between them, he nodded. \"Nice. I'm going home to my girls. Call me later.\"\n\nMerrick smiled. \"Thanks for everything.\"\n\n\"No problema, primo.\"\n\nA nurse approached Merrick. \"Are you the father?\"\n\n\"Yes, I am,\" he said proudly.\n\n\"You can put your jacket over in the locker. If you have a cell phone, please turn it off. I'm going to gown and mask you. We like to keep the room as sterile as possible.\"\n\nMerrick turned off his cell phone and slipped out of his jacket, not seeing the nurse's stunned look when she saw the holstered automatic handgun tucked into the small of his back. It, too, went into the locker and she breathed a sigh.\n\nAlex smothered a moan as a spasm of pain gripped her. She felt as if she were being split in two. The pain had begun mid-morning and had intensified with each passing hour. She'd been experiencing mild contractions since her eighth month, but these were different. By nightfall they were stronger and coming closer together.\n\nThe pressure on her bladder had also increased and when she sat on the commode and amniotic fluid flowed unchecked, she knew it was time to go the hospital. And for the first time since Merrick suggested she live with Michael she offered up a prayer of gratitude. If she'd been at the house in Alexandria she would've been alone\u2014alone and frightened.\n\n\"Hey, Mama.\"\n\nAlex opened her eyes to see a pair in clear gray smiling down at her. \"Hi...\" she got out before catching her breath when another contraction seized her.\n\n\"How are you doing?\"\n\n\"I am in pain, Merrick Grayslake.\"\n\n\"Do you want something to take the edge off?\"\n\nShe shook her damp head. \"No drugs.\"\n\nThe doctor completed his examination. \"It's too late to give her anything. Your baby will be here in half an hour or less. I'm going to check on another patient. I'll be back in a few minutes.\"\n\nSitting on a chair at the head of the bed, Merrick held his wife's hand. He felt every contraction with her death grip on his fingers. It went on and on, him gently massaging her back.\n\nHe wondered who'd attended to Victoria Grayslake when she went into labor. Was there anyone to comfort her when she knew she would give up her life as soon as she delivered her baby? Had she been tortured before or after she gave birth? Had she made the treasonous videotape disclosure before or after she'd become a mother?\n\nAfter his initial meeting with the voice there was no further communication. Merrick had traced the serial numbers on the disposable cell phones and discovered they'd been purchased in Anchorage, Alaska. He continued to carry the phones in the hope that the mysterious caller would offer another clue.\n\nThe doctor returned, checked the machines monitoring the baby's heartbeat, then checked to see how far Alex was dilated. He ripped off his gloves and lab coat. \"She's ready. Let's move it, Nurse!\"\n\nMerrick stood up, completely helpless as the medical professionals sprang into action. Another nurse appeared as the obstetrician scrubbed his arms and hands. Moments before Alex's legs were encased in elastic stockings and placed in a pair of stirrups another doctor appeared, this one a resident.\n\n\"Dad, please stand at the end of the bed.\"\n\nMoving in slow motion, Merrick moved to where he'd been ordered. A nurse tilted a large mirror to give him an unobstructed view of the birth and delivery. He watched in awe, transfixed as Alex labored to give birth. She pushed when told to push, the effort weakening her reserved strength.\n\n\"No puedo,\" she whispered hoarsely.\n\n\"Yes, you can. You have to do this, baby. Didn't we make plans to take little Vicky to the museum to look at all the pretty paintings?\"\n\n\"You hate museums,\" she spat out.\n\nMerrick kissed her forehead. \"I don't hate them, darling.\"\n\n\"She's crowning.\" The doctor's eyes crinkled above his mask. \"What do we have here? Somebody's a redhead.\"\n\nAlex smiled through the sharp pains holding her prisoner. \"She's a Grayslake.\"\n\nThe doctor rested a hand on his patient's belly. \"One more push, Alexandra, and you'll get to see your beautiful baby.\"\n\nShe sucked in her breath and pushed. A soft gasp escaped her parched lips when she heard a weak cry. Her baby girl was here!\n\nPeering up at the mirror she saw a tiny red-faced baby with red-gold hair. There was no trace of Cole in her daughter.\n\nTears filled Alex's eyes and spilled down her cheeks. \"She's beautiful, Merrick.\"\n\nLeaning over, Merrick kissed his wife's damp face. \"Thank you, Ali.\"\n\n\"Check to see if she has all her fingers and toes.\"\n\n\"She's perfect.\"\n\n\"Check her, please.\"\n\nThe nurse tagged, suctioned, weighed, measured and cleaned a wailing Baby Grayslake. She handed the baby to Merrick, who kissed her head before handing her off to her mother. Instinctively, Victoria Grayslake sought out her mother's swollen breasts and began nursing greedily.\n\nThe intern took over from the attending doctor, who ripped off his gloves. He motioned to Merrick. \"Your daughter is small, which means she'll have to stay in the hospital until she weighs at least five pounds.\"\n\n\"How much does she weigh?\"\n\n\"Four pounds ten ounces.\"\n\nMerrick smiled, attractive lines fanning out around his eyes. \"Judging from her appetite, she'll gain that in a week.\" He sobered. \"She wasn't due for another month. Will that present a problem?\"\n\n\"There's no doubt that she's a full-term baby. What she may be is two weeks early, but not much more than that. Maybe your wife lifted something that brought on labor, but I wouldn't worry too much about mother and baby. She can stay until your daughter is medically cleared for discharge.\"\n\nMerrick doubted whether Alex would agree to leave the hospital without her baby. He extended his hand. \"Thanks, Doctor.\"\n\n\"Congratulations, Mr. Grayslake. You can see your wife once she's in her room. We have accommodations for new dads if you wish to spend the night.\"\n\n\"Thank you again.\"\n\nMerrick left to find an area where he could use his cell phone to call David and Serena. Then he would call Michael and let him know he planned to spend the night in the hospital with his wife and new baby daughter.\n\n## Chapter 34\n\nChandler Duffy drove his fist into the granite countertop, ignoring the shooting pain racing up his arm. When I kill them, they stay dead, he fumed inwardly. The tall, elegant man with silver hair, known to his closest associates as the Silver Fox, literally had blood in his eye. When he'd received word that Merrick Grayslake was digging into his finances his blood pressure spiked dangerously. The result was a ruptured blood vessel in his right eye.\n\nThe Grayslake name had haunted him for years, and he still hadn't forgotten the face of the woman who'd insulted him, saying she found a wad of gum on the bottom of her shoe more appealing than sleeping with him. It took him a while, but she paid for the snub\u2014with her life.\n\nReaching for one of five cell phones on a side table, he dialed a number. \"It's a go,\" he said when hearing a familiar voice. He disconnected the call, put the phone back in its assigned order and walked out of his inner sanctum.\n\nHe would rid the world of any trace of Victoria Grayslake, once and for all.\n\nMerrick sat on a cushioned love seat with Alex, watching her breast-feed their daughter. She was five days old and she'd gained one of the four ounces she'd lost. Alex's milk had come in and the tiny fingers clutched her breast as if it was a lifeline.\n\nHe smiled as Alex spoke Spanish to the baby. His wife had grown up with bilingual parents, while he'd learned the language from Mexican housekeepers. And if it were up to him, his daughter would speak as many languages as she could master.\n\nMerrick leaned closer. \"She keeps her eyes open longer.\"\n\nAlex met her husband's proud gaze. \"She only opens them for you. She must like your voice.\"\n\n\"That's because she's showing her daddy her beautiful hazel eyes.\" Victoria's eyes changed color from a light green to a grayish brown.\n\n\"I'm going to need you to go to the house and bring me something to wear. I'm embarrassed when people come see me and I'm in a nightgown and bathrobe.\"\n\nStretching his arm over the back of the love seat, Merrick played with her hair. It was long enough to cover the nape of her neck. \"What do you want me to bring?\"\n\n\"Sweatpants or anything with a drawstring waist. I doubt if I'll be able to fit in anything else.\"\n\n\"Your stomach looks pretty flat. It's your top that's outrageous.\"\n\nAlex winked at him. \"You like them, don't you?\"\n\nMerrick winked back. \"I'm not above fighting a baby for Mama's leche.\"\n\nPicking up a nursing pad, she swatted at him. \"You're nasty.\"\n\n\"Hell, yeah,\" he countered. Moving closer, he kissed her cheek. \"I'll be back with your clothes.\" Pushing to his feet, he headed for the door to her suite.\n\n\"Querido?\"\n\nHe stopped and turned around. \"What is it?\"\n\n\"When will I be able to go home?\"\n\nMerrick knew she was talking about the house in Alexandria. \"Soon,\" he said, hoping he hadn't lied to her. They were still going over the printout of the international banking transactions.\n\nAlex offered him a tender smile before she lowered her head and stared at the baby in her arms, a little girl whose golden skin was the perfect complement for her golden hair.\n\nMerrick punched in a code, deactivating the security system. Alex had outdone herself decorating the historic structure. She'd selected antique pieces and exquisite reproductions that rivaled the interiors of the homes in Williamsburg, Virginia.\n\nThe difference was their home wasn't a museum but one where they could live and entertain in elegant comfort. A home he wanted to come to every night to eat with his wife and children, a home filled with love and laughter from the other children he hoped to have with Alex. He climbed the staircase to the second floor, and hadn't gotten halfway when the doorbell rang.\n\nA slight frown furrowed his forehead. He wasn't expecting anyone, because he'd just begun sleeping at the house. With Alex in the hospital he'd decided to give Michael and Jolene a break so they could enjoy their domicile with the presence of guests, even if the guests were family members. The bell chimed again.\n\nThinking perhaps it was one of his neighbors who saw the light on in the first story, he retraced his steps. Peering through the security eye, he saw the face of a woman with light-colored hair pulled back in a ponytail. Reaching around his back, he unsnapped the strap on his gun's holster. He opened the solid oak door and stared through the tempered glass on the storm door. The woman had pressed a photo ID against the glass. She was CIA. Maybe she'd come up with something on Duffy.\n\nMerrick reached for the latch on the door, but before he could open it the familiar sound of a gunshot echoed in the quietness of the spring night. Dropping to a crouching position, he reached for his gun, holding it in a two-handed grip. The agent lay on the top step, her right arm in an awkward position, her government-issued automatic dangling from her fingers.\n\nIt was apparent no one heard the shot because none of the residents came out of their houses. They probably thought it was a car backfiring.\n\nMerrick went completely still when he saw a tall shadowy figure move closer. The man had both hands raised above his head. There was something familiar about the man; he moved under a street lamp and Merrick recognized him. He was Cordero Birmingham, associate Bureau chief for the FBI's Northeast region.\n\nKicking the door open a fraction, Merrick trained the gun on the special agent. \"State your name and your business.\"\n\n\"Special Agent Cordero Birmingham, FBI associate Bureau chief, Northeast,\" he drawled in a distinctive Southwest intonation. He climbed the front steps, kicking the gun away from the motionless hand. Leaning down, he pressed two fingers against the woman's neck. \"She's alive.\" He straightened. \"I'm going to reach into my jacket to get my cell phone to call 911.\"\n\n\"Don't,\" Merrick warned. \"I'll make the call.\" Reaching into the breast pocket of his shirt, he took out the phone and dialed the three digits without taking his gaze off the special agent. The lamps framing the house highlighted silver hair with traces of red, a lean face with a pair of brilliant topaz-blue eyes. The call took less than twenty seconds.\n\n\"You didn't state your business.\"\n\n\"I came here to save your life.\"\n\n\"You call shooting one of my colleagues saving my life?\"\n\n\"She's dirty.\"\n\n\"How do I know you're not dirty?\"\n\n\"You don't. But if I were you I wouldn't be standing here talking to me. Your wife would be making arrangements to bury her husband. And she would raise a child who would never know her father. Just like you never knew your father or your mother.\"\n\nMerrick's eyes fluttered but the gun never wavered. \"What do you know about my mother?\"\n\n\"I was her partner on one undercover mission, and I swore an oath that I would always protect her.\"\n\nMerrick felt as if he'd been punched in the gut. Cordero Birmingham was the voice.\n\nLowering the gun, he secured it behind his back. He had questions, lots of questions he wanted the special agent to answer. But they would have to wait, as the sound of approaching sirens shattered the tranquillity of the night.\n\n## Chapter 35\n\nMerrick sat across from Cordero Birmingham listening to the man reveal more than thirty-five years of secrets. When the EMTs arrived, along with the police, Cordero took charge when he identified himself.\n\nThe injured woman wasn't an employee of the CIA, but an impostor with counterfeit identification who'd been hired to murder a federal agent, and the Bureau would send a report backing up his course of action.\n\n\"You got Duffy?\" he asked.\n\n\"We picked him up on a private airstrip as he was preparing to leave for parts unknown.\"\n\n\"How did you get him?\" Merrick asked. \"We've been burning the midnight oil going through hundreds of thousand of international banking transactions.\"\n\n\"And it paid off.\"\n\n\"What!\"\n\n\"While you were at the hospital visiting your wife and baby, Justin Jefferson found what you'd been looking for. We got the call and we had agents watching Duffy's house, his vacation place in McLean and his private jet.\"\n\n\"How did you know about the hit?\"\n\n\"We finally got a judge who wasn't in Duffy's pocket willing to approve a wiretap. We put a cleaning woman in and she placed transmitters in every one of his phones. He never used his house phone for his business. He had five cell phones, each with a direct connection to someone able and willing to do his bidding\u2014for a price of course.\"\n\n\"How did he know we were investigating him?\"\n\n\"You have a leak at Langley.\"\n\n\"Who?\"\n\n\"William Reid.\"\n\n\"No!\" Merrick groaned, shaking his head. He'd always liked Bill Reid. \"Why?\"\n\n\"Duffy was blackmailing him. He'd set Reid up with a hooker, and when your supervisor woke up the next day, the woman wasn't breathing. Reid thought he'd killed her when the cause of death was a food allergy. Duffy removed all evidence that Reid had ever been in the room. He convinced Reid to transfer from the Bureau to the Company because he needed a snitch inside Langley.\"\n\nMerrick, sandwiching his hands between his knees, closed his eyes. \"What is your stake in all of this?\"\n\n\"I told you Victoria was my partner.\"\n\nMerrick opened his eyes. \"You said that before.\"\n\nCordero Birmingham looped one leg over the opposite knee, staring at the young man sitting less than three feet from him. He was so incredibly smart, yet he couldn't see what was so obvious.\n\n\"I was in love with her. It was my child Victoria was carrying when Duffy turned on her because she rejected his advances.\"\n\nA tense silence enveloped the room as the two men regarded each other. It was as if Merrick could suddenly see the similarities that were so overtly apparent: height, the lean face, high cheekbones, aquiline nose and reddish hair.\n\n\"You're telling me that you are my father?\"\n\nCordero nodded. \"Yes.\"\n\n\"How long have you known about me?\"\n\nCordero held his forehead. \"I realized who you were when I signed off on your security clearance when you first applied to the CIA. I'd gone as high as I could at the DEA, so I requested a transfer to the Bureau.\" He lowered his hand, bright blue eyes flashing like lasers. \"I wanted to contact you but you were always on some covert operation, so I decided it was better I keep my distance.\"\n\n\"Did you know I'd been shot?\"\n\n\"Yes. I came to see you in the hospital, but you were heavily sedated. I wanted to tell you that you weren't alone. That you had family\u2014you had a father.\"\n\n\"I have a family now.\"\n\nCordero smiled, flashing his perfect white teeth, teeth that Merrick had inherited. \"I know that you're married and you have a little girl.\"\n\nMerrick flashed a proud smile. \"She has red hair.\"\n\nRunning a hair over his coarse silver hair, Cordero lifted his eyebrows. \"I had red hair. My father's hair was a bright red\u2014almost orange.\"\n\n\"Where did you learn Spanish?\"\n\n\"My mother was half Mexican.\"\n\n\"Was the other half Yaqui?\"\n\n\"Yes. How do you know that?\"\n\n\"Someone told me I look Yaqui.\"\n\n\"You do. My father was a Brit who lost his heart to a young Mexican woman he met when he came to the States on holiday.\"\n\n\"Do you have any brothers or sisters?\"\n\n\"I had a brother, but he died in Vietnam. His death devastated my parents.\"\n\n\"Tell me about my mother.\"\n\n\"She was very pretty and very, very smart. Victoria would read something once and she'd never forget it.\"\n\n\"Was she African-American?\"\n\nCordero nodded. \"Yes. Why?\"\n\nMerrick clasped his hands. \"I didn't know what it was, but I've always felt African-American. Do you have a photograph of her?\"\n\nReaching into his jacket, Cordero took out a wallet. He opened it and pulled out a small fading black-and-white photograph. \"I've carried this close to my heart for longer than I can remember.\" He handed Merrick the photo. \"When she told me she was pregnant, I told her to request a transfer back to the States so that we could marry, but Duffy was the agent-in-charge, so he stopped it.\"\n\nMerrick stared at the photograph of the woman who'd given up her life for him. There was something about her face that reminded him of his daughter's. He handed the photograph to Cordero.\n\n\"Her granddaughter looks like her.\"\n\nOvercome with emotion, Cordero willed the tears filling his eyes not to fall. \"Do you think I can see her?\"\n\nMerrick averted his gaze. Too much had happened, too much had been said and too much had been revealed in a very short span of time. \"I came home to get something for Alex to wear. I'm going back to the hospital. Would you like to come with me?\"\n\nCordero let out an audible exhalation of breath. \"Yes. It's going to be strange meeting David Cole's daughter when I haven't seen her uncle in more than forty years.\"\n\nMerrick froze. \"You know Martin Cole?\"\n\n\"No. Joshua Kirkland.\"\n\n\"How do you know Joshua?\"\n\n\"We worked together on a joint task force operation that yielded the largest drug bust in Mexican history. I was new to the game and hadn't realized we had a leak in the operation until someone tried to gut Joshua. If it hadn't been for Mateo Arroyo, also known as Matthew Sterling, Joshua wouldn't have made it.\"\n\nMerrick's head was spinning. What were the odds that his father knew Emily's father and her father-in-law? \"Joshua's daughter is married to Matt's son.\"\n\nCordero's blue eyes sparkled with amusement. \"You're kidding.\"\n\n\"No, I'm not. As soon as Alex and Victoria\u2014\"\n\n\"Victoria? You named your daughter Victoria?\"\n\n\"Yes. It was Alex's decision. We're all going to Florida next month. I'd like you to come and see your old friends.\"\n\nCordero's smile was blinding. \"I'd like that very much. Thank you.\"\n\nA phone rang and both men reached for their cell phones. \"It's mine,\" Merrick said. \"Hello. Yes, baby. I know I should've been back hours ago. I'm coming, but I want to warn you that I'm bringing somebody. No, I can't tell you. I won't tell you. You'll see when we get there. Yes, you're going to like it. Hang up and I'll see you in a bit.\"\n\nHe disconnected the call, meeting Cordero's knowing gaze. \"Women.\"\n\n\"I know, son. You can't live with them, and we definitely can't live without them.\"\n\nMerrick stood up, Cordero following suit. \"Did you ever marry? Have more children?\"\n\n\"No. I wasn't the marrying kind, but Victoria changed my mind. By the time I was ready to settle down with a wife and child she was gone. I've had a few serious relationships, but nothing that would make me commit to marriage. One love in one lifetime is enough.\"\n\n\"Come upstairs with me and see where your grandbaby girl is going to hang out.\"\n\nCordero followed Merrick up the staircase, feeling the warmth in the historic old house even though his daughter-in-law and granddaughter weren't there.\n\nHe thought about Victoria and how she would react to becoming a grandmother. There was no doubt she would've been just a little crazy. He would wait awhile, get to know his son better, and then he intended to spoil his grandbaby girl rotten.\n\n## Epilogue\n\nThe Cole clan gathered in Boca Raton to welcome and introduce Victoria Cole-Grayslake to her many relatives.\n\nMerrick carried his two-month-old daughter while Alex held the hand of her father-in-law. It was a perfect Florida late-spring day\u2014warm, sunny, the temperatures in the low seventies.\n\nSerena spied Alex first and took off running. She'd wanted to come to Virginia after her daughter had given birth, but David talked her out of it. That was the last time she swore she would listen to the man.\n\nShe held out her arms. \"Let me see the precious angel.\" Merrick placed the sleeping infant in Serena's arms. Her eyes filled with tears. \"Oh, she's so beautiful.\" Cradling her in one arm, she untied the satin ribbons on her linen cap. A slight breeze lifted strands of red-gold hair. \"Oh, goodness! She's a redhead.\"\n\nDavid came over and eased the baby from his wife's arms. \"What a beauty.\" Victoria opened her eyes and smiled a crooked smile. \"Hey, Merrick, you better keep a close eye on your daughter because you're going to have to beat the boys off with a stick.\"\n\nMerrick smiled. \"I'd rather use a gun. I'll sit on the roof and pop the little knuckleheads before they reach the front door.\"\n\n\"Who's shooting who?\" asked a deep voice.\n\nMerrick turned to find Joshua Kirkland behind him.\n\n\"Joshua, there's someone here who thinks he knows you.\"\n\nJoshua lifted his pale eyebrows. \"Who?\"\n\n\"My father.\"\n\nVertical lines marred Joshua's forehead. \"Your father?\" He'd been told Merrick was orphaned at birth.\n\n\"Don't tell me that you're so old you're now senile.\"\n\nWith light green eyes widening in shock, Joshua stared at a man to whom he owed his life. \"Cord Birmingham?\"\n\nCordero released Alex's hand, extending his arms to Joshua. \"In the flesh.\" The two men embraced, slapped each other's backs and howled like coyotes.\n\nJoshua turned and searched the crowd. \"Matt, get over here and see who decided to join the family.\"\n\nMatthew Sterling stood up, squinting slightly when he saw Joshua hugging someone who looked vaguely familiar. It wasn't until Cordero smiled that he recognized the toothpaste-ad grin.\n\n\"I'll be double damned! Cord! You old geezer!\" He grabbed him in a bear hug, then kissed both his cheeks.\n\n\"Old!\" Cordero spat out. \"I'm younger than both you farts.\"\n\nMatt rested an arm on the shoulder of the man who'd stood in as a witness for his wedding to Eve Blackwell. \"What are you doing here?\"\n\n\"I came to meet my granddaughter's family.\"\n\nMatt looked from Cordero to Merrick, then back again. How had he missed the resemblance? \"You've got a helluva son.\"\n\n\"I know that,\" Cordero said without a modicum of modesty.\n\nJoshua rested his arm on Cordero's other shoulder. \"This calls for a celebration. Let's go inside and start the celebration early.\"\n\n\"You know you can't drink like you used to,\" David said, teasing his brother.\n\nJoshua shot him a look. \"Speak for yourself, little brother.\"\n\nDavid handed Victoria off to Alex. \"Here, cookie, take my grandbaby girl. I'm going to have to prove to these punks that I still can drink them under the table.\"\n\nAlex shook her head. \"Daddy, you know that you're too old to play college-boy games.\"\n\nDavid snorted. \"From what I heard, the youngbloods couldn't hold their liquor the night before you got married. Am I not right, Merrick?\"\n\nMerrick stared up at the fronds of a sweeping palm tree. \"No comment.\"\n\nAlex moved over and stood next to her husband. She watched the old friends lock arms as they went off to where they would talk about the old days and the old times undisturbed.\n\n\"I like your dad.\"\n\nMerrick wrapped an arm around her waist. \"I like him, too.\"\n\n\"You know that he's spoiling Victoria.\"\n\n\"That's what grandparents are supposed to do. You'll do the same once we have grandbabies.\"\n\nShe rested her head against his shoulder. \"But that's not going to be for a long time. Right now I want to have fun making babies.\"\n\nHe dropped a kiss on her hair. \"When do you want to start making another one?\"\n\n\"We can try again in two years.\"\n\nHe tightened his hold, bringing her closer to his side. \"That sounds like a good deal.\"\n\nAlex tilted her head, and she wasn't disappointed when he brushed a kiss over her parted lips.\n\nGabriel strolled over with his pregnant new wife. \"Get a room.\"\n\n\"Bite me, Gabriel,\" she retorted.\n\nGabriel winked at his brother-in-law. \"Is that what she tells you in the throes of passion? Bite me, bite me, baby,\" he said in a falsetto.\n\nMerrick, his expression impassive, turned and looked at Alex. \"How come you never tell me to bite you?\"\n\nShe stomped her foot. \"Merrick Grayslake, you're going to get it.\"\n\n\"I hope so, because it's been a while.\"\n\nHer face burning in shame, Alex smiled at Summer. \"Please come with me. I think Victoria needs to be changed.\" The two women walked while their men roared in laughter.\n\nLaughter, shrieks, whispered words of passion set the stage for another reunion, this one more poignant than the other ones. Merrick found his father, and his father found old friends who were now his family.\n\nThere were a few new Coles on the way who would one day take their place and risk everything for love.\nISBN: 978-1-4268-0062-7\n\n\u00a9 2007 by Rochelle Alers\n\nAll rights reserved. The reproduction, transmission or utilization of this work in whole or in part in any form by any electronic, mechanical or other means, now known or hereafter invented, including xerography, photocopying and recording, or in any information storage or retrieval system, is forbidden without written permission. For permission please contact Kimani Press, Editorial Office, 233 Broadway, New York, NY 10279 U.S.A.\n\nThis is a work of fiction. Names, characters, places and incidents are either the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events or locales is entirely coincidental.\n\n\u00ae and TM are trademarks. Trademarks indicated with \u00ae are registered in the United States Patent and Trademark Office, the Canadian Trade Marks Office and\/or other countries.\n\nwww.kimanipress.com\n\n## Coming Next Month\n\nDon't miss all-new titles from Kimani Press next month including eBooks by Robyn Amos and Marcia King-Gamble.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzsvpz b/data_all_eng_slimpj/shuffled/split2/finalzzsvpz
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+{"text":"Follow-up interviews are always fun. Listen to my follow-up interview with Dahlia Schweitzer, author of Going Viral: Zombies, Viruses, and the End of the World (Rutgers University Press, 2018). I talk with her and Rob Thomas, the creator of Veronica Mars and the co-creator of iZombie and Party Down as well as the author of several young adult novels including Rats Saw God and Slave Day. In this interview we talk about how Schweitzer's book discusses some of the ways in which Thomas' series iZombie is ahead of its time, while Thomas shares some of the stories behind co-creating and writing iZombie. In addition, we discuss Thomas' work on Veronica Mars, how the series came to be, as well as the ways in which it relates to Schweitzer's new book. It's a great interview for zombie lovers and marshmallows alike.\nRebekah Buchanan is an Associate Professor of English at Western Illinois University. She researches zines, zine writers and the influence of music subcultures and fandom on writers and narratives. She is the author of Writing a Riot: Riot Grrrl Zines and Feminist Rhetorics (Peter Lang, 2018). You can find more about her on her website, follow her on Twitter @rj_buchanan or email her at rj-buchanan@wiu.edu.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The NCFPSC continually posts updates to all of our social media accounts, providing news of our mission to aid public school children and stories of our progress. We have a Facebook account at https:\/\/www.facebook.com\/NCFPSC.org\/, a Twitter account at https:\/\/twitter.com\/ncfpsc, and an Instagram account at https:\/\/www.instagram.com\/ncfpsc\/. Additionally, you can follow the North Carolina Teacher Cadet Program on Twitter at https:\/\/twitter.com\/nc_teachercadet?lang=en and on Instagram at https:\/\/www.instagram.com\/nc_teachercadet\/. Thank you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This entry was posted in Book Review, Paris Books, Paris History Site, Versailles and tagged King Louis XV, Madame de Pompadour, Rivals of Versailles, Sallyt Christie, Versailles by americangirlsartclubinparis. Bookmark the permalink.\nThe tour of Versailles and the books both sound intriguing!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hosted by our wonderful friend and colleague in Taijiquan, Elizabeth Seatia, we were warmly welcomed at Basel Airport (after getting lost in the Swiss side), and whisked away to Mullhouse, Alsace. This is a wonderful cosmopolitan township that sits in the corner of France, Switzerland and Germany where people are friendly and food is good!\nThe weekend was syncopated with lunches and dinners here and there and by that, I mean here and there. Chinese food in a restaurant in Mullhouse, Buffets in the Studio and Greek food in a restaurant in Germany served by Italian waiters!\nOnce again it was a great time and sorry if your students legs are a bit sore after long hours and low stances!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mitch, first of all congratulations on your Worlds Title 2015 on the Hobie Wild Cat.\nThis was really a great thing winning it with Ruben. Very well done. This to me shows how you are living the Hobie Way of Life truely in a Hobie family.\nAnother great thing was your outstanding performance at the F18 Worlds in Kiel; it showed that the Wild Cat is still able to outrace all other designs \u2013 a job very well done by you and Ruben.\nWhat makes you stick to Hobie?\nHaving met you in Kiel, it was mainly lighter winds and some swell. Would you be so kind and explain us a bit on how you were tuning the Wild Thing (aka Hobie Wild Cat)?\nMB We really didn't have much time prior to the regatta to train, only one day. The boat was setup with everything in the middle. Meaning mast rake mid, diamonds mid spreaders mid and nothing special.\nHow do you push your boat through the chop?\nMB It is very important in the light and choppy conditions to sail in clear wind and not disturbed from other boats. Its also critical to sail a free and twist mode to keep max boat speed and not focus on height.\nWith so many new classes like Nacra 17, F16, Nacra 16, foiling cats etc. what do you think or recommend the Hobie class should pursue in the future?\nMB I don't know what Hobies plan is for future classes or designs, but there is a trend towards higher performance beach cats in general. I hope Hobie continue to be a brand pursuing the leading edge of our sport.\nWith respect to my last question, what is your recommendation to ISAF? Personally, I am really missing a master plan, as we are seeing changes in i.e. Youth Boats almost every time ISAF meets. This really makes it difficult for nations to cope with.\nMB ISAF have many political pressures to push the sport in different directions. I think its important to create the boats that the sailors want and then they will in turn pursue ISAF for use or recognition in the right place , either Youth Worlds, Olympics, etc.\nI realised some changes at your Wild Cat in Kiel \u2013 i.e. you had extra holes drilled in your daggers. Could you explain to us how you use those?\nMB The holes in our daggerboards where simply to place a rope as a stopper so when we flew a hull it didn't drop too far down in stronger wind.\nBesides this which upgrades\/ improvements do you recommend for the Wild Cat?\nDuring pre-start procedures, how do you prevent that you are drifting leewards towards other boat designs with different underwater shapes?\nMB Side drift is controlled by sail trim and speed. This is something that can be practiced and help with good starts. This has very little to do with hull shapes.\nWhich do you do first: reduce mast rotation or increase cunningham tension?\nMB Always cunningham. Mast rotation is only when at end of cunningham range.\nDo you have pointing problems in comparison to other boat designs? If so, what is your solution?\nMB Not in light to medium wind\u2026 in fact the Wild cat points higher than most other F18's in lighter conditions. This is mostly due to the daggerboards. Stronger wind is a bit harder but its most important to remember speed creates lift from the foils, so its important to always keep fast.\nA couple of years ago in Dec. 2004 we prepared a tuning table with you for the Tiger. If you don\u00b4t mind, please give us your recommendations for the Wild Cat.\nMB You may think it strange, but I rarely work with numbers and always tune the boat according to sail look, helm feel, charictaristics of how the boat feels and sails. Only then do I go and check what the settings are so I can have reference for next time I rig. To be completly honest, I have no idea what numbers we used in the last regatta's . All I can tell you is Ruben and I were 143 kg and very fast in under 12 knots.\nBefore we are closing, may I ask, which are your plans for the EXTREME SAILING series?\nMitch, thanks a lot for your time and the information you have provided. Hope to see you next year back in Germany and please say hi to your family!\nBest regards and have a Hobie day!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxntt b/data_all_eng_slimpj/shuffled/split2/finalzzxntt
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index 0000000000000000000000000000000000000000..96dcc973ac03b758d59445f6848daee49352f85c
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@@ -0,0 +1,5 @@
+{"text":"[dts] [PATCH] tests\/vhost_user_live_migration: Optimize for multi-platform.\nPrevious message: [dts] [PATCH] tests\/vhost_user_live_migration: Optimize for multi-platform.\n> Thanks for your comments. Updated.\n> > There's one comment as below.\n> > > +            dirct_para = \"-\" + m.group(2) + \" in\"\n> > > +            dirct_para = \"\"\n> > Please change variable name to more meaning full name \"direct_param\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Flaunt retro-inspired design with classic elements when you adorn JAG\u00ae Jeans denim.\nHigh-rise jean boasts a slim-fit silhouette.\nThorne Blue offers a medium indigo hue with light sandblasting, whiskers, and a raw hem that is tacked for a chic cuff.\nCrosshatch denim flaunts a textured look on a supersoft stretch fabrication.\nStructured waistband with belt loops and brand patch.\n72% cotton, 20% modal, 7% elasterell-polyester, 1% spandex.\nProduct measurements were taken using size 2, inseam 31. Please note that measurements may vary by size.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This was my first time at the ACC Show in San Francisco. The climate was perfect for scarves- even in the summer- what a concept!\nMy friend Dahlia Popovits, of Dahlia the wonderful gallery in Newton Centre, told me I would fall in love with San Francisco and she was right. What a pretty place! The show was in the Fort Mason Center, right near the Golden Gate. I could see the bridge and a mountain from my hotel window.\nWe were on a pier in the Bay, the water was aqua, boats everywhere, the America's Cup races were going on, we were right by Alcatraz, former home of Whitey Bulger, on trial right then at home in Boston.\nThe show went very well, I saw old friends, and made new ones. Here is my old friend Lisa Drew. She came from Fairbanks Alaska. She is with her old friend Glenn, checking out some scarves.\nI saw but did not get a picture of my high school classmate Nancy Milliken. She was with me on that amazing trip to Latin America that I write about in the website. It was transformative for her too: she became a doctor.\nOn Monday, before my flight, I had the time to take a five mile walk around the area I was staying in. I walked out Lombard Street, straight up, like climbing a ladder almost, up to the top where there was this iconic view and a million tourists.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"24th \u2013 30th October is Open Access Week. Here at UWP we'll celebrate by making the final touches to converting our International Journal of Welsh Writing in English into a gold, fully open-access journal with the wonderful Open Library of Humanities. We're very proud to be the first university press which has \"flipped\" to OA with the OLH, and look forward to seeing the journal disseminated to a wide audience. Keep an eye out for the first upload of articles, and in the meantime, take a look at www.open.accessweek.org, and see how truly international the OA movement has become. We're clearly in good company!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you're determining whether an employee's position is exempt, take a page from these court cases and start your analysis with the basic functions of the position.\nIn 2018, the U.S. Supreme Court ruled in Encino Motors v. Navarro that exemptions to the Fair Labor Standards Act should be given a \"fair reading,\" instead of a narrow construction. Two federal appellate court decisions have put their stamp on just what counts as a fair reading.\nForm I-9: What should I do if employee's documentation has discrepancies?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzycbi b/data_all_eng_slimpj/shuffled/split2/finalzzycbi
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+{"text":"FIND YOUR Diameter Stainless Steel Fire Ring.\nMvhigh helps you compare the price of any product across leading online stores like Walmart, Ebay, Amazon, and many more. With Mvhigh you'll easily find the least price of Diameter Stainless Steel Fire Ring across leading shopping sites without wasting too much of your effort thus making your shopping decision smarter and simpler.\nBring the heavy-duty fire ring used by the U.S. Forest Service home for all of your outdoor cooking and bonfire needs. U.S.A.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Storage area shelving means putting drawers in your basement or car port, where you store your things. It is an essential thing to have as you would require correct stacks in such areas. For garage shelving, you may opt for either metal or plastic shelving. These choices are durable and strong, which is the main requirement of the garage. Depending on the weight of your goods which you want to keep on the shelves, choose the appropriate material for the shelving. Again, plan it methodically according to your personal requirements, as the shelving is not for attractive purpose here, but just for storage.\nThenewyorkclipper - Past exhibitions tampa museum of art. The tampa museum of art is pleased to present yayoi kusama's love is calling, one of the artist's iconic infinity rooms, on loan from the vinik family foundation immersive, experiential work of art, love is calling invites visitors to enter a mirrored room with tentacle like soft sculptures hanging from the ceiling and positioned on the floor. Join livejournal. Password requirements: 6 to 30 characters long; ascii characters only characters found on a standard us keyboard ; must contain at least 4 different symbols;. ???3???9?????~!. On peut toujours blmer la faible progression des revenus, les coups du kind ou la surconsommation, mais il faut bien se rendre compte que tous ces facteurs auraient un effet beaucoup additionally restrict si le crdit n'tait pas aussi accessible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We present you the code iTunes Gift card to replenish your account iTunes US $ 50.\nfor your PC, MAC, iPhone, or iPod from US Apple Store.\n6) To create an account, use any US address, enter your credit card details are not required to activate an account, use the purchased certificate.\nCard code rewritten from the original scratch cards, which is a guarantee of its quality.\nYou get the time code INSTANTLY after the payment, the site + to your e-mail, which you indicated when you pay.\nyou can find your timecode link http:\/\/www.oplata.info.\nWe sell ORIGINAL codes scratch card starts at \"X ...\" and copied to the card purchased in retail stores US AppStore US, which guarantees the quality and origin of the original.\nCheck out the whole list of codes selling iTunes Gift Cards - https:\/\/www.odysssey.ru\/seller\/88121.\nPlease leave your feedback about our work on a page with the product purchased. You can leave a response at the time of receipt of goods or after your purchase at http:\/\/www.oplata.info. Believe me, it is very important for buyers and our further work, and you take a little time. For example, short words will suffice OK ;-) our part, we will evaluate your personal review granting discounts on prepaid cards and other products.\n\u0417\u0430\u043a\u0437\u0430\u043b \u043a\u0430\u0440\u0442\u0443 iTunes US $50 c \u0428\u0442\u0440\u0438\u0445-\u043a\u043e\u0434\u043e\u043c.\u041f\u0440\u0435\u0434\u043e\u0441\u0442\u0430\u0432\u0438\u043b\u0438 \u0431\u044b\u0441\u0442\u043e. \u0421\u043f\u0430\u0441\u0438\u0431\u043e!\n\u0417\u0434\u0440\u0430\u0432\u0441\u0442\u0432\u0443\u0439\u0442\u0435! \u041d\u0430 \u0441\u0430\u0439\u0442\u0435 \u044f \u043f\u0440\u0438\u043e\u0431\u0440\u0435\u043b \u043a\u0430\u0440\u0442\u0443 iTunes USA $25 \u041c\u043d\u0435 \u043f\u0440\u0435\u0434\u043e\u0441\u0442\u0430\u0432\u0438\u043b\u0438 \u0441\u043e\u0432\u0440\u0435\u043c\u0435\u043d\u043d\u043e \u0442\u0440\u0435\u0431\u0443\u0435\u043c\u044b\u0439 \u0442\u043e\u0432\u0430\u0440 \u041f\u043e \u043c\u043e\u0435\u043c\u0443 \u0437\u0430\u043f\u0440\u043e\u0441\u0443 \u0431\u044b\u043b \u043f\u0440\u0435\u0434\u043e\u0441\u0442\u0430\u0432\u043b\u0435\u043d \u0448\u0442\u0440\u0438\u0445=\u043a\u043e\u0434 \u0434\u043b\u044f \u043a\u0430\u0440\u0442\u044b. \u0421\u043f\u0430\u0441\u0438\u0431\u043e.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"The Lift Up's fantastic debut was recorded in the spring and mixed and mastered in the summer with Jim Salamone at Cambridge Sound Studio in Philadelphia. The album features the wistful songwriting and storytelling skills of lead singer and guitarist Joe Castro, with a theme that seems to board the listener on an intriguing and timeless walk down memory lane from childhood familial history to present proof.\nphillygirlabouttown.com (11\/02\/13) Click here to read the entire article.\n\"The tracks are soulful, honest accounts of the constant pressures of the modern life with a lingering nostalgia of yester years. It is vividly reminiscent of the period in our lives when we were in relentless conflict between running away and staying right where we are.\nThe nine songs are intensely written and paired evocatively with some 90s pop acoustics. With clear influences from The Smiths, Leonard Cohen and The Jam, among others.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Stanley-Powers Lake Football will be hosting the Grafton Spoilers at 7pm this Friday at Lyne Enget field, the Blue Jay Cheerleaders will also be making their home debut.\nThe Blue Jays Volleyball team will be hosting Burke County this Thursday (1st) at 7pm.\nThe Stanley Blue Jays Cross Country team will be in running in the Minot meet on Thursday (1st).\nMake sure to check out the Stanley Gazette Homepage for more Stanley Blue Jay Sports and Community News.\nAlso, make sure to follow the Stanley Gazette on Twitter (@StanleyGazette) and also on Facebook for more updates on Stanley Blue Jay Sports.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzykae b/data_all_eng_slimpj/shuffled/split2/finalzzykae
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--- /dev/null
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+{"text":"I had the pleasure of attending the World Premiere of My Dinner with Alan (directed by Kristian Fraga) during The Sopranos film festival at IFC on the evening of January 11, 2019. The film consists of a conversation between critics Matt Zoller Seitz and Alan Sepinwall located at Holsten's Diner\u2014the location of the infamous final scene of The Sopranos. The two primarily discuss the show and their experiences covering it; both for the Star-Ledger as it was airing, and in their new book The Sopranos Sessions. But their conversation moves in the direction of the more philosophical at times; or least the sociological as it relates to the state of television.\nThe title of the film is a clear reference to My Dinner with Andre, and its opening scenes\u2014featuring Seitz navigating airports to travel to New Jersey to meet Sepinwall\u2014couldn't help but make me think of Wally riding the subway to meet Andre.\nOf course, My Dinner with Alan is not nearly as deep as Wallace Shawn and Andre Gregory's masterpiece, but it doesn't make any pretension to be. Instead of a fancy restaurant as the setting, this is a diner where they eat onion rings like the Soprano family, Seitz enjoys some Taylor Ham, and the two spend most of their time talking about one of the best shows to ever grace our screens. Could that have fed into a discussion of Martin Heidegger? Sure, but it is perfectly fine that it didn't.\nIt was similar to its namesake insofar as what one might think would be boring\u2014watching two people eating and having a conversation\u2014wasn't at all. When My Dinner with Alan ended, I found myself wanting more. Luckily I was in a position to receive it, as Sepinwall, Seitz, and Fraga were on hand for a panel discussion after the screening. In what follows, I will offer some thoughts about the film and the discussion that followed, noting when something is from the latter.\nAfter arriving to Holsten's and ordering the aforementioned onion rings, Seitz and Sepinwall settle into a conversation about The Sopranos that lasts for about an hour. There are anecdotes about personal interactions with James Gandolfini not understanding press events, reading a children's book, and possibly thinking about walking away from the show after the third season because he \"couldn't get the stink\" of Tony Soprano off him no matter how much he showered. This led Seitz and Sepinwall to discuss the question more generally as to whether an actor can make a separation between themselves and the character they portray, particularly when it is for as long as Gandolfini was Tony. (N.B. This last bit about \"washing the stink off\" was during the discussion panel; not the film).\nIt also fed into a discussion of death\u2014the death of James Gandolfini in real life, and how that tinges a present viewing of the show, but also the impact of the deaths in the narrative that we now all know are coming. They hit just as hard on a second, or third, or fourth viewing as they did on the first, or perhaps even harder. The first time through The Sopranos one is caught up in thinking about what is going to happen. Further viewings lead one to dwell with the how and why.\nApparently David Chase originally conceived of the show as a sort of live-action Simpsons, but Gandolfini made Tony so human that it altered what the series would be. The show's brilliance then wasn't just a matter of Chase's genius (if the term brings to mind the one who is source of a novel\/meaningful artistic creation) but his adaptability in the light of circumstances. It is further a testament to the way in which The Sopranos wouldn't be The Sopranos with a different lead: James Gandolfini simply is Tony. There could be no substitute.\nAnd as much as The Sopranos opened the door for what gets called the current \"Golden Age of television,\" it is also inimitable. As Sepinwall notes, yes, it inspired any number of shows we could list, but none of those shows are it; none of them quite do what The Sopranos managed.\nPerhaps this is, at least in part, due to how we view things nowadays, both in terms of the literal (i.e., the means by which one views a show) and the figurative (how does the collective \"we\" view the notion of paying close attention to a media product, thinking about it, discussing it, and perhaps even writing about it). As we move to an era ever more defined in terms of streaming services (it's hard to believe that Netflix only started offering such in 2007), the way we consume things seems to be moving in the direction of the binge-model. Rare is the show one watches week-to-week, if it is even released that way. After watching the first episode of the second season of Westworld, for example, I decided maybe I would binge the rest sometime later (which I still have not done).\nThis means that we tend to be watching things at different times, in a way that erodes the possibility for conversation about them. Further, Seitz and Sepinwall note how the very pace of the binge watch tends to collapse the space for thinking, insofar as that involves rumination. Something different happens when one has a week to digest an installment; to mull it over and pass it through multiple stomachs. This doesn't quite land in \"get off my lawn\" territory, as the two seem to recognize the possibility of a Netflix show being as meaningful as something like The Sopranos; yet they also note that it probably would not also be as popular.\nSeitz mentions Tony's famous line about coming in at the end, and the sense that this was true. Perhaps it was originally about the end of the 20th century, and a worry about the death of American dominance, that sort of thing, but the feeling has gotten if anything all the more pervasive. Chase starts to feel like Cassandra, foretelling a doom we're unable to avoid.\nThere is a famous notion in Hegel about the \"end of history\"\u2014his whole story is about its dialectical developments; a back and forth, for example, between abstract moral universality and concrete ethical substance. If we take the end to be a goal, then this would be to stop that movement and its concomitant \"slaughter bench\" in order to achieve a harmonious society, or world. Some, like Francis Fukayama in 1989, seemed to think that we were achieving that goal through the spread of global capitalism, or the neoliberal order. Now that seems clearly false. Yet, what if we take the notion of the end of history in a more pessimistic way? What if history has stopped because everything is now on one flat surface and no overarching narrative can unify it? What if now, instead of dialectical development, all we have is dispersion and fracturing? Each can have what they want, in separation from one another, but nothing could bring us together as a whole. This thought, too, strikes me as Hegelian, though in a far more tragic register.\nTo be clear, though, Seitz and Sepinwall do not talk about Hegel. They do, however, hit on questions pertaining to therapy, or psychoanalysis. Seitz claims, for example, that part of what makes The Sopranos so great is the way that it understands therapy, differing modes of therapy, and dramatizes things like repression, transference, etc. And I have to agree with that.\nDuring the discussion panel, it came out that a certain TV executive the show was pitched to wanted the psychiatric aspect taken out. But The Sopranos wouldn't be The Sopranos without it. I went so far, when the show ended, as to argue that the real ending was\/should have been when Tony quit therapy. In part that's because I wasn't very keen on the final scene (I have since come around), but I still think I had a point. This is, after all, how the show begins: with Tony having a panic attack and beginning to see Dr. Melfi. So isn't when he quits the natural place to end?\nSeitz and Sepinwall of course discuss that final scene in this film that is set in its location; what they thought at the time, and what they think now. Another version of that discussion, cribbed from The Sopranos Sessions, was published the other day. Sepinwall originally thought it was about putting the audience in the mind of Tony; paranoid about the possibility of imminent death. But, as Seitz notes, there is nothing in the content of the scene to suggest this: Tony seems fine and happy to enjoy some Journey and onion rings. I rather enjoyed his thought that it was a matter of Chase \"whacking\" the audience, but ultimately I think that both Seitz and Sepinwall are right to move in the direction of something a bit more thematic. We aren't shown whether Tony dies, but what is going on is about death, in one way or another.\nMy position has always been that he doesn't die because we weren't shown that. The text is the text, and it just cuts off. The show ends with an ellipsis, but that doesn't mean it isn't worth thinking about the question, and Seitz and Sepinwall certainly provide interesting ways of doing so.\nIt's not about explaining what happened in some definitive, or irrefutable way, but about interpreting it in the sense of entering in between competing possibilities and selling a certain way of thinking. You don't have to buy it; perhaps you, like me, enjoy window shopping, which in this case means thinking, but you can't sell me on something that is not on display. I appreciate that, despite Sepinwall's apparent flirtations with the \"Tony is dead\" reading of the ending, he and Seitz ultimately both seem to recognize that the point is more in the space of ambiguity.\nIt's that working out of nuance that leads me to recommend My Dinner with Alan, and The Sopranos Sessions book. If you're a fan of the show, I can hardly think of better sources to get you thinking about it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We want you to love shopping for a Ford as much as you love driving it off the lot. That's why at Town & Country Ford you can enjoy a better buying process that's simple, straightforward and stress free because it's built around the way you shop.\nWe'll bring the dealership to you!\nWe know you're busy. That's why we're making car buying more convenient. If you see a particular Ford you like, we'll bring it right to your driveway. While we're there, we can also take care of your credit application, evaluate your potential trade and give you an appraisal number, right then and there. This service extends within 100 miles of our dealership - make an appointment today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Libs need to stop whining about the Supreme Court.\nWe are just starting the [Brett] Kavanaugh hearings up on the Senate, and already it's clear that the liberal freak-out level will be at lib-con 1.\nTrust me, that's more or less going to be the tone.\nBut I'm here to tell all the left wingers out there, calm down. It's going to be ok. This really isn't the downfall of America you're witnessing.\nKavanaugh is a gold standard Scotus nominee. The guys brilliant, friendly, and honorable. Probably more important to you \u2013 he believes in judicial restraint.\nHe's not going to take his Scotus seats and starts crushing precedents like the judicial equivalent of Genghis Khan.\nHe will be fair minded and base his decisions on the law. I mean the law as it written, not the law as liberals wish it to be.\nSo please, libs, at least calibrate your overheated rhetoric to reality a little bit.\nDon't pretend the real problem is you don't have the documents you need. 400,000 should be enough- and I guarantee you, Chuck Schumer Charles (Chuck) Ellis SchumerMJ Hegar announces Texas Senate bid Hillicon Valley: House Dems subpoena full Mueller report | DOJ pushes back at 'premature' subpoena | Dems reject offer to view report with fewer redactions | Trump camp runs Facebook ads about Mueller report | Uber gets B for self-driving cars Dem legal analyst says media 'overplayed' hand in Mueller coverage MORE and company haven't read a fraction of them anyway.\nDon't act like Kavanaugh is some kind of reckless bon vivant because he bought some season tickets to the Nationals. Normal Americans find that kind of thing well, normal \u2013 maybe even endearing.\nOppose him because he won't do what you want- which is legislate from the bench as a leftist. Oppose him because you think he \"stole\" Merrick Garland Merrick Brian GarlandThe Hill's Morning Report \u2014 Category 5 Mueller storm to hit today McConnell touts Trump support, Supreme Court fights in reelection video Hatch warns 'dangerous' idea of court packing could hurt religious liberty MORE's seat, even though that is a constitutionally illiterate construct.\nAnd maybe, just maybe ask yourself- should 1 supreme court seat matter so much to you?\nNext time, get your policy preferences through the legislature, and you won't have these problems.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm suffering a bit of MBFWA withdrawal this week. My day to day life is decidedly less fun and glamourous! It was my first ever Sydney Fashion Week and had the absolute best time. I wasn't sure what to expect at first, but seeing some familiar faces on the first day went a long way to calming my nerves.\nThe first show I watched, Misha Collection, was one of the highlights, with Bella Hadid leading the show. I also loved C\/MEO Collection, one of my favourite labels, which consisted of stunning blazers, shirts and delicate earrings.\nBy far the best part of fashion week was catching up with my lovely IG friends and meeting new people. I can't wait to do it all again next year!\nShop Black Heel Wedges, Colored Denim Jeans, Black Graphic Tees, Mules and more. Get this widget.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We appreciate all the donors and friends who support the W.A. Franke College of Forestry and Conservation.\nYour generous support and involvement is integral in advancing FCFC's mission and vision. We hope you will continue to support this effort!\nIf you would like to make a gift to the FCFC, unrestricted gifts can be directed to the Student Impact Fund to address the college's most pressing needs. You may also restrict your gift to a specific fund or program. Read more about FCFC funds and priorities. For more fund choices or to learn about other ways your philanthropy will benefit FCFC students, please contact Dean Tom DeLuca 406-243-5521 or tom.deluca@umontana.edu Thank you very much for your support.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyqts b/data_all_eng_slimpj/shuffled/split2/finalzzyqts
new file mode 100644
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--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzyqts
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+{"text":"This Barcelona holiday apartments is been just renovated creating a cosy ambiance, has two double bedrooms and one spacious bathroom. The master bedroom has queen size bed and the second bedroom has twin beds and access to the bathroom in the corridor. The modern kitchen is fully equipped (cooker and hob, fridge, dishwasher, toaster, microwave, coffee and kettle). The kitchen is opened to the cosy living room and has access to an small sunny balcony. The apartment includes air conditioning and heating in all the apartment, flat-screen TV and free wifi.\nThese accommodations in Barcelona city centre are available for monthly rentals and price includes utilities and a weekly cleaning.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These men's tennis shoes are built to deliver top-level performance on multiple surfaces. They have a seamless upper that flexes with your foot and Geofit padding for a comfortable fit. A Barricade chassis offers forefoot flexibility and midfoot stability, supporting your all-action game.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Apollon Wealth Management LLC bought a new stake in iShares Core S&P Small-Cap ETF (NYSEARCA:IJR) during the fourth quarter, Holdings Channel reports. The institutional investor bought 4,298 shares of the exchange traded fund's stock, valued at approximately $330,000.\nA number of other hedge funds and other institutional investors have also recently made changes to their positions in the business. JPMorgan Chase & Co. increased its position in shares of iShares Core S&P Small-Cap ETF by 487.5% during the third quarter. JPMorgan Chase & Co. now owns 2,500,481 shares of the exchange traded fund's stock worth $218,142,000 after acquiring an additional 2,074,857 shares during the period. BlackRock Inc. increased its position in shares of iShares Core S&P Small-Cap ETF by 34.0% during the third quarter. BlackRock Inc. now owns 7,922,723 shares of the exchange traded fund's stock worth $691,178,000 after acquiring an additional 2,010,925 shares during the period. Bank of New York Mellon Corp increased its position in shares of iShares Core S&P Small-Cap ETF by 12.7% during the fourth quarter. Bank of New York Mellon Corp now owns 10,682,185 shares of the exchange traded fund's stock worth $740,489,000 after acquiring an additional 1,205,932 shares during the period. Envestnet Asset Management Inc. increased its position in shares of iShares Core S&P Small-Cap ETF by 27.5% during the third quarter. Envestnet Asset Management Inc. now owns 5,467,895 shares of the exchange traded fund's stock worth $477,019,000 after acquiring an additional 1,180,259 shares during the period. Finally, LPL Financial LLC increased its position in shares of iShares Core S&P Small-Cap ETF by 17.3% during the third quarter. LPL Financial LLC now owns 6,900,780 shares of the exchange traded fund's stock worth $602,024,000 after acquiring an additional 1,018,339 shares during the period.\niShares Core S&P Small-Cap ETF stock opened at $77.17 on Thursday. iShares Core S&P Small-Cap ETF has a twelve month low of $65.04 and a twelve month high of $90.56.\nThe business also recently disclosed a quarterly dividend, which will be paid on Tuesday, March 26th. Stockholders of record on Thursday, March 21st will be given a dividend of $0.2605 per share. This represents a $1.04 dividend on an annualized basis and a dividend yield of 1.35%. The ex-dividend date of this dividend is Wednesday, March 20th.\nTRADEMARK VIOLATION WARNING: This piece was originally published by Community Financial News and is owned by of Community Financial News. If you are viewing this piece on another site, it was illegally copied and republished in violation of US & international copyright law. The legal version of this piece can be read at https:\/\/www.com-unik.info\/2019\/03\/21\/apollon-wealth-management-llc-buys-new-stake-in-ishares-core-sp-small-cap-etf-ijr.html.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We offer services at a low cost which is affordable to all sectors. Our Flat Rate at $0.70 per minute transcription for all kinds of transcription.\nWe offer different types of transcription services andwe are proud to offer some exclusive types of services for certain kinds of transcription services.\nVermont is located in the region of New England of Northeastern US. This is the 2nd highly populated city among 50 states in US. Vermont is the top producer of the maple syrup. Vermont experiences continental climate and muddy springs and hot summer. It is also the 7th coldest state. Business and trade is the major contributor to the city's economy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Philadelphia Eagles have a trade in place to send running back LeSean McCoy to the Buffalo Bills, a source told ESPN.\nIn return, the Bills will send linebacker Kiko Alonso to the Eagles, the source said.\nThe trade cannot be made official until the new league year begins Tuesday at 4 p.m. ET.\nMcCoy has the NFL's third-most rushing yards (6,792) and fourth-most rushing touchdowns (44) since the Eagles drafted him in the second round in 2009. He finished last season with 1,319 rushing yards, third-most in the NFL.\nA source familiar with McCoy's thinking told ESPN's Josina Anderson that the veteran running back is not pleased with the move.\n\"He's a Pennsylvania kid. He's never played football outside of Pennsylvania -- high school, college, pro,\" the source said. \"So of course he's not happy.\"\nThe source also told Anderson that McCoy is \"frustrated\" and is \"not going to make it easy, that's for sure.\"\nThe Eagles will clear cap space by trading McCoy, who was set to count $11.95 million against their 2015 cap. The Bills will owe McCoy, who is signed through 2017, $10.25 million this season.\nEagles running back Darren Sproles took to Twitter to express his shock over the news of McCoy's impending departure.\nNothing surprises me in this business anymore! In total shock!\nPhiladelphia also released two players Tuesday -- defensive end Trent Cole, who was set to count for $11.625 million against the salary cap, and cornerback Cary Williams, who would have counted for $8.17 million -- while they take on less than $1 million in each of the next two seasons by acquiring Alonso, who will be a restricted free agent after the 2016 season.\nAlonso, a second-round pick in 2013, played every defensive snap for the Bills as a rookie and ranked third in the NFL with 159 tackles. He finished second in the Associated Press' voting for defensive rookie of the year.\nAfter tearing his ACL while training last July, Alonso was placed on the non-football injury list and missed the entire 2014 season.\nA source told ESPN's Adam Caplan that Alonso's rehab has gone very well and he has increased his training activities and started running.\n\"I think I'm at almost 7\u00bd months and I'm doing just about everything,\" Alonso told the Bills' official radio show on Feb. 25. \"Obviously at a very controlled pace, just not going obviously 100 percent. I wouldn't say I'm at 100 percent yet but I'm definitely moving in the right direction and taking it slow.\nAfter six seasons in Philadelphia, running back LeSean McCoy's numbers show he is one of the best to wear an Eagles jersey.\n\"I anticipate being 100 percent before camp. By camp, I definitely feel I'll be 100 percent.\"\nAn Oregon product, Alonso will reunite with his former college coach, Chip Kelly, who will enter his third season in Philadelphia. Without Alonso, the Bills' top options at linebacker next season will be Nigel Bradham and Preston Brown. They also could re-sign Brandon Spikes or add a free agent like David Harris, who played for new Bills head coach Rex Ryan while with the Jets.\nThe Bills are expected to adopt a run-first approach under Ryan and offensive coordinator Greg Roman. The Bills' leading rusher last season, Fred Jackson, turned 34 last month, while C.J. Spiller will become an unrestricted free agent next week.\n\"Management called me and told me that they were [trading for McCoy] and pretty much giving me a 'thank you for your service,'\" Spiller said during an appearance on NFL Network on Tuesday evening. \"I feel still the same. Nothing's changed. The game plan is still the same. The only thing is that I won't be returning to Buffalo.\"\nSpiller said he found out that he will not be returning to the team when he was in the studio for his NFL Network appearance. \"Very few guys get to play their whole career at one team, but unfortunately that doesn't happen in this business, and I found out today it doesn't happen in this business,\" he said.\nThis is the second time in less than a year that the Bills and Eagles have made a trade involving a running back. Buffalo acquired Bryce Brown during May's draft for a conditional fourth-round pick from Philadelphia. Brown, who remains under contract with the Bills, was active for seven games, rushing for a career-low 126 yards.\nThat trade prompted Jackson and McCoy, then with the Eagles, to exchange jabs on Twitter.\nTuesday's news also made waves with other NFL players -- past and present -- on Twitter.\nWow Shady to Buffalo huh? Crazy!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..54043bffa03289d80faaf7ea4b3f29cada983d1d
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Neymar will be at Paris Saint-Germain next season despite the many rumors linking him with a move back to Real Madrid.\nNeymar will be at Paris Saint-Germain next season despite the many rumors linking him with a move back to Spain - Real Madrid in particular - and he has already held talks with the club's new manager Thomas Tuchel.\nThat is according to Spanish source AS, who are also claiming that the discussion between the player and coach has had a positive impact and the former is now looking forward to beginning a new season with the French club.\nIt is understood that the Brazilian did feel like leaving at times due to his love for the city of Barcelona and the many differences in French culture. His lack of friends within the squad was a factor as well, while he felt that many of the fans were not happy with his trips to and from Brazil.\n\"Neymar is tired of all these rumors. There are Spanish media that say he will go. If you believe it, too bad for you. I won't keep responding to this questions. He has a contract for four years.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wannabe rocker Marty McFly accidentally travels back in time (via his pal Doc Brown's Delorean) to the year 1955. Problems arise when his giddy mother unexpectedly falls for him, (changing the course of his future). Marty is forced to do everything he can to get his parents to fall in love before he disappears from existence. On his journey he is assisted by the 50's version of Doc who agrees to help send him Back to the Future.\nBack to the Future is nowhere near the greatest film of all time. That being said, it sure is the most fun and entertaining. Just imagine being thrust back in time to meet your own parents when they were the same age as you. That was the idea put forward by writer Bob Gale when he found his parents' old yearbook. BTTF re-teamed Gale with his 'Used Cars (1980)' director, Robert Zemeckis, and was produced by Steven Spielberg, who both men had written '1941 (1979)' for the previous year. Unfortunately both films had been critical and box office flops. Luckily something about this time-travel comedy was different. Not only was it a comedy, it was also an action\/sci-fi,\/romance movie all rolled into one that crossed genres.\nThe teenaged protagonist of the story Marty McFly, is played perfectly by a then 23 year-old Michael J. Fox. The first glimpse we see of him are his white Nike runners with the classic red swoosh as he enters Doc Brown's garage. A garage filled with clocks of every make and model. What else would you expect in a time-travel movie? As the film progresses, we learn more about the character of Marty, the wannabe rockstar with his band the Pinheads and avid skateboarder (a skill that comes in handy back in 1955 AND in 2015 for the sequel). However, his hopes for rockstar status are soon diminished when a judge for the high school dance tells him he's \"just too darn loud\". That judge is portrayed by none other than Huey Lewis, the singer of two tracks featured on the soundtrack.\nThe mentor to Marty is the illustrious, self-proclaimed inventor, Dr. Emmett Brown (as played by Christopher Lloyd). Thought to be a quack by most, he finally fulfills his dream of actually inventing something that works, a time machine built out of a DeLorean. The idea came to him after hitting his head while hanging a clock in his bathroom. When he awoke, it came too him\u2026the Flux Capacitor, which is what he discovers makes time-travel possible. Unfortunately his first trip to the future is cut short when he is shot down by a group of Libyans (the guys he got the plutonium to build his time vehicle from).\nThe antagonist comes in the form of Biff Tannon (played by the surprisingly nice Thomas F. Wilson). He begins the story as more of a slight annoyance when he crashes the McFly family car and blames it on a blind spot in the windshield. Upon Marty's journey to the past, Biff becomes a huge thorn in his side as he turn out to be just as big a bully in the past as he is in the future.\nCrispin Glover and Lea Thompson, who play Marty's parents both in the past and the future, round out the rest of the cast. Glover plays George, the uber-nerdy father that any kid would be embarrassed by, and Thompson plays Lorraine, the mother who seems to have never had a day of fun in her life. Back in 1955 George is ever so much the nerd, but you discover that back then Lorraine was actually wild and a total boy-chaser, and unfortunately Marty is the boy she's chasing. Cue Freud.\nThough not having much screen time, Marty's love interest, Jennifer Parker (played by Claudia Wells, and later by Elisabeth Shue in the two sequels), is the driving force of the movie and his main reason for wanting to get \"Back to the Future\".\nThe other important character of the film is its music, both popular and instrumental. As earlier stated, Huey Lewis lends his rock flavour to the picture with his hits \"Power of Love\" and \"Back in Time\". Even though \"Back in Time\" fits as a song about time travel, it fails in its attempt by stating the plot and characters too often. \"Power of Love\" on the other hand is perfect without it being stated outright. The power being sung about is the whole point of the film. Love is what Marty wants to get back to, and what he needs to happen to his parents so he won't be erased from existence.\nOther artists featured on the soundtrack are Eric Clapton with \"Heaven is One Step Away\" and Lindsey Buckingham's \"Time Bomb Town\". The rest of the movie features classic songs from the fifties era, and we are also treated to a live performance by Marvin Berry (Harry Waters Jr.) & the Starlighters at the Enchantment Under the Sea Dance, where Marty again shows his musical talents (with some help from singer Mark Campbell as his singing voice) to make sure that George and Lorraine share their first kiss.\nFrequent Zemeckis collaborator, Alan Silvestri, composed the score for the film. His music works perfectly as it exudes the emotion of the story and has given me goosebumps every of the 80 plus time I have watched it (seriously!). I find it hard not to hum the main theme when driving fast on the highway and wishing that my Toyota Corolla had hover conversion. It's hard to find a film as enjoyable as Back to the Future nowadays, and I think you get something new from every viewing. I can remember the annoying years between 1985 and 1988 when the film would be aired on one of the local channels on a Sunday around noon. I would get so thrilled watching Doc back up his Delorean, tell Marty \"Roads? Where we're going we don't need\u2026roads!\", watch the tires fold up and have his amazing time vehicle come crashing into my TV screen\u2026only to have the worst three words in the English language at the time slide across the screen\u2026\"To Be Continued\u2026\" (Come on! I was 9 years old when Part 2 finally came out). Though the two sequels were good and it was fun to see Marty and Doc in the future (2015) and in the Old West (1885), they never quite equaled the original. It gave us a taste of what is what like for our parents growing up in the 50's and on watching it again today, you get a sense of your own nostalgia and feel like you're back in the 80's once again. Heavy!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The people can join Pakistan Navy as PN Cadet 2015. The people can get the jobs at the best regard which will help them in the better way. The people can be given in the best way. The Permanent Commission Jobs Term 2016-A are given the best end.\nThe people will be given with all the details which can be mentioned for you. The people are related in the best way of jobs. The people will be given with all the information. The people can get the jobs in the best way. The jobs will be listed for you that will bring the best status in your way.\n155 queries in 0.312 seconds.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Latest statistics for videocluster.net website: estimated summary traffic is 186 Thousand sessions per month (6.2 Thousand sessions per day, or 67.9 Million sessions per year). Videocluster.net was launched at December 17, 2008; this domain is over 10 years old. The site links to network IP address 79.143.184.38. As of March 01, 2019, Alexa rank is 192345.\n0.1 percent of visits to this site come from a search engine, while 99.8% come directly and 0.1% via a hyperlink somewhere on the internet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Marie Little Shield, an annual national netball championship for women with an intellectual disability, is taking place this weekend at the Brisbane Entertainment Centre.\nThe finals of the competition, named after disability netball pioneer Marie Little OAM, will be showcased ahead of the opening match of the Constellation Cup between the Samsung Diamonds and the New Zealand Silver Ferns on Sunday 7 October.\nIntroduced in 2013, the event recognises Marie Little OAM's dedication to improving the lives of people with an intellectual disability through the power of netball and is an event proudly supported by Netball Australia's Community and Social Inclusion Partner, Australia Post.\nRepresentative teams from Victoria, New South Wales, Northern Territory, Western Australia, Queensland and South Australia are taking part in this weekend's competition, which is expected to be fiercely-contested with NSW aiming for a fourth consecutive tournament win.\nAustralia Post has partnered with Netball Australia since 2014 to deliver the Australia Post One Netball Program which aims to engage Australia's increasingly diverse communities with the sport of netball.\nAustralia Post spokesperson Andrea Pearman said the Marie Little Shield has become an annual fixture of the Australia Post One Netball calendar which features year-round initiatives focused on inclusion and diversity in netball at all levels.\n\"Australia Post has proudly supported the Marie Little Shield for four years. It continues to provide an important platform to showcase and profile sportswomen with an intellectual disability at a national level,\" Ms Pearman said.\nNetball Australia CEO Marne Fechner says the organisation is proud to host the Marie Little Shield each year and that it is looking forward to showcasing the passion, dedication and commitment of the athletes taking part this weekend.\n\"The Marie Little Shield is a tremendous annual event and we're thrilled to hold a competition that provides individuals with intellectual disabilities with the opportunity to play netball at a national level,\" she said.\nThe Samsung Diamonds team will attend and spectate the Marie Little Shield tournament on Saturday 6 October before the Marie Little Shield finals are played as a curtain-raiser for the first match of the Constellation Cup. Tickets for this match are still on sale at ticketek.com.au.\nTo meet the One Netball Ambassadors and learn how your club or association can create a more inclusive environment for all, find out how you can begin your inclusion journey here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Woodlawn Cemetery, Mausoleum & Mortuary welcomes the community to join them at this holiday party. There will be photos with Santa, light refreshments, and a photography exhibit by Santa Monica College Photography Department students. Listen to the John Adams Middle School Choir sing a program of holiday music. Woodlawn will offer visitors an ornament that can be personalized in memory of loved ones and hung on the Tree of Life, located in the Mausoleum lobby. This event also launches the annual toy donation drive for the Toys for Tots Foundation, so please bring something to donate.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are many car-parks in Belfast City Centre, however Q-Park is situated underneath Victoria Square and has direct access to each floor of Victoria Square.\nWe are now offering VIP parking at this car park. This option is available when you pre-book parking online and provides the most convenient parking spaces in the facility. VIP parking can be found on Level 1.\nWe do our best to cater for all our customers and provide you with the best shopping experience possible. We have ensured maximum priority has been given within our car park to customers who require closer access to the main shopping centre.\nThere are 6 customer lifts and escalators located within the main shopping destination, providing access to a levels and outlets.\nWe also have spaces available for parent and child parking.\nWe have doubled the number of toilets in the centre while also expanding facilities to include two feeding rooms, two family toilet rooms, and an adult changing room built in accordance with Changing Places guidelines. This facility includes more space and the right equipment for those with additional needs, including a height adjustable changing bench and a hoist. These are located on the B1 level at the entrance to the car park.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AAs the food industry looks for a solution to replace plastic straws, delfort has provided the perfect tool to meet the needs of straw users globally. With our Fibre Straw paper, and accompanying Straw Wrap paper for packaging, we are ready to take the food industry by storm with specialty papers that are environmentally-friendly and the perfect partner for any liquid drink.\nWWith the increase of plastic straws, a multitude of drastic environmental hazards have resulted from the production of this small, yet potent, petrochemically-based utility.\nAccording to Fortune.com, the Quick Service Food Industry is changing the way it consumes plastic; many chains and franchises are opting for paper as the environmental-friendly replacement for straws. Household names such as Marriott and McDonald's are communicating in the public to seek plastic replacements, a direct result of the voice of consumers.\nTo cater to our clients' changing needs, we have developed Fibre Straw papers that are durable, a product that keeps its form.\n\"We achieve the perfect result with the fusion of different types of specialty papers ranging from 60 gsm to 120 gsm, each is robust and stable in nature \u2013 the perfect paper to create an environmentally-friendly straw.\"\nThese heavy-duty papers are dense to withstand your favorite drinks and liquids, each crafted by winding the fibers together and wound into a spiral so that it is able to maintain its function in the presence of liquid.\nAdditionally to our Fibre Straw paper, we support our clients with a holistic offering that encompasses every facet of the drinking straw experience. Thus, we have created an appropriate paper packaging for our straws: Straw Wrap paper.\nDue to its low substance that stands below 25 gsm, our Straw Wrap paper is thin and easily unwrappable. It is also highly resistant to bacteria and external substances, keeping it hygienic and clean from transportation to serving.\nStraw Wrap paper is excellently processed by high-speed wrapping machines. Our paper is mechanically sealable, resulting in packaging that is glue-free; it is also primed for the printing of any colorful design or logo.\nAAs with all our products, we strive to give our customers solutions capable of solving their unique challenges in the most environmentally-friendly manner. Thus, our base papers for paper straws are 100% food safe and in compliance with the FDA and BfR.\nOur paper is also produced exclusively on FSC\/PEFC Chain of Custody certified paper machines, machines that are certified to ISO 22000 Food Safety Management System. We ensure that every environmentally-friendly measure has been taken, which is why our materials are sourced only from suppliers that are 100% FSC and\/or PEFC certified.\n\"Now, Quick Service chains have the perfect tool to satisfy their clients and the environment\"\nClick here to find out more about these products or contact us!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In a series of tweets on Friday morning, Nobel Prize-winning economist Paul Krugman accused \"Trump loyalists\" of knowing deep down in their hearts that the president committed treason, but they don't care because he appeals to their racist instincts.\nTaking the measure of the national zeitgeist, the popular New York Times columnist said the steady drumbeat of revelations about the Trump administration \u2014 including a bombshell report that claims the president may have suborned perjury \u2014 should leave no doubt that the president's relationship with Russian president Vladimir Putin is treasonous.\nBut, Krugman asserted, Trump's biggest fans don't care.\n\"A thought about where we are as a nation: We're living in the age of the unsurprising revelation. Is there anyone who doesn't already believe that Trump-Putin-treason is a real thing? Even Trump loyalists surely know it's true, they just think it's an OK price for the racism,\" Krugman tweeted.\nKrugman then tied Trump's die-hard supporters to conservatives who have ignored the racism of Rep. Steve King (R-IA) for so long \u2014 then acted shocked when the truth came out.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Korean Journal of Internal Medicine 2000;15(1):96-98.\nBaker's cysts are rare cause of peripheral nerve entrapment and only a few cases of tibial nerve entrapment resulting from the popliteal cyst in the calf muscle have been reported in the literature.\nWe present a case of rheumatoid arthritis complicated by a Baker's cyst with a tibial nerve entrapment. It is important to diagnose a Baker's cyst early and to differentiate it from thrombophlebitis, a popliteal aneurysm, tumor or muscle tear to effect optimal therapy and to obviate a potential neuropathy. Prompt recognition of these cases may save the patients unnecessary procedures and delay in treatment.\nRecently, we experienced a case of posterior tibial neuropathy caused by a Baker's cyst in a 64-year-old woman who suffered from rheumatoid arthritis. She had manifestations of posterior tibial neuropathy, including pain and numbness of the sole, weak flexion of the big toe and decreased right ankle jerk. The diagnosis of entrapment neuropathy was entertained preoperatively by electromyography and nerve conduction study. Her pain resolved following surgical removal of Baker's cyst. We present this case with a review of the literature.\nA 64-year-old woman was presented to us with a two-year history of poly arthritis, most marked in the knees and both hand proximal interphalangial joints (PIP) and associated with left ankle pain, in October 1998. Past medical history revealed a history of hypertension and intracerebral hemorrhage. Rheumatoid arthritis was diagnosed according to 1987 ACR revised criteria, based on a symmetric polyarthritis involving the PIP, shoulders, left ankle and knees, morning stiffness and the presence of rheumatoid factor.\nShe was treated with hydroxychloroquine and methotrexate, but an exacerbation occurred involving the right knee in January 1999. Also, she had numbness of the right sole. At radiologic evaluation of the knees, narrowing of the medial joint space and joint effusion were noted, but no erosion was seen. Treatment with local steroid injection was resumed with increments of methotrexate doses.\nWithin a few weeks of local steroid injection in the knee joint, however, right knee pain worsened and numbness of the sole continued. She gave no history of prior injury or other paraesthesia.\nOn physical examination, she revealed pain and increased sensation along the plantar aspect of the right foot and the mass was palpable on the posterior aspect of the right knee. Also, she showed weak flexion of the big toe and decreased right ankle jerk.\nA lobulated 4\u00d74\u00d73 cm3 cystic lesion with inhomogeneous signal intensity was seen on ultrasound. The needle aspiration of the right knee joint was unproductive, and magnetic resonance imaging of the right knee and popliteal fossa was obtained and revealed a Baker's cyst(Figure 1). The cyst extended towards the popliteal tendon and lateral collateral ligament and there was no evidence of deep vein thrombosis.\nOn electromyography, 3+ and 2+fibrillation and positive sharp waves potentials were observed in the abductor hallucis and tibialis posterior showing abnormal spontaneous response. During contraction, polyphasic motor unit was observed in the abductor hallucis and tibialis posterior.\nOn nerve conduction studies, proximal latency and amplitude in the right tibial nerve was 15.4ms and 0.1mV, showing low amplitude and delayed proximal latency of right tibial compound muscle action potential.\nRepeated needle aspiration was unproductive, and open surgical procedure was performed. At operation, a lobulated cystic lesion displacing neurovascular structures was released, but neurovascular invasion by a cyst was not found(Figure 2). The cavity contained many folds and measured 4\u00d74\u00d73 cm3, and extended anteriorly toward the posterior cortex of the tibia. The communication between the synovial compartment and the cavity was present near the popliteal hiatus. The biopsy finding was compatible with Baker's cyst due to rheumatoid arthritis(Figure 3). Her pain resolved following surgical removal of Baker's cyst, but mild degree of numbness remained.\nBaker's cyst represents a distended gastrocnemio-semimembranosus bursae. The cyst has a synovial lining similar to joint synovium and communicates with the knee joint by a long thin stalk. The cysts are formed by the escape of synovial effusion from the knee joint into one of the popliteal bursae, herniation of the posterior joint capsule or herniation of the posterior knee synovium1.\nAny intra-articular process associated with effusion may lead to distension of this bursa through a one-way valve mechanism. Although most Baker's cysts remain asymptomatic or are associated with only mild localized symptoms, a variety of complications have been reported2\u20137,10,11). It may cause mild discomfort or pain and tenderness at the posterior aspect of the knee. Baker's cysts can produce neuropathy by direct pressure on the nerve, either gradually or suddenly. The most dramatic complication is acute rupture with the clinical presentation often called the pseudothrombophlebitis syndrome10).\nHerein, we report the case of a patient with a Baker's cyst complicated by a posterior tibial neuropathy. It seems that although the cyst was unruptured, there were patient positions that could cause the expansion of the Baker's cyst, producing compression of the posterior tibial nerve. The diagnosis of entrapment neuropathy was entertained preoperatively by electromyography and nerve conduction study.\nThis case is interesting because tibial nerve neuropathy is rare in Baker's cyst. Previous reports in the literature on tibial nerve neuropathy from Baker's cyst are summarized in Table 1.\nNakano8 presented the first clinical cases of 5 patients with entrapment neuropathy from Baker's cyst. Four patients had classical rheumatoid arthritis, while foot-drop developed in one man after trauma to the knee. In four patients, intra-articular steroid injections relieved the symptoms of pain, with gradual return of motor and sensory function of the lower limb, while one patient required synovectomy. In 1985, Kashani et al.9) described a case of degenerative joint disease of both knees, complicated by a Baker's cyst. He responded well to knee joint aspiration and intra-articular prednisone injection. Recently, Dash et al.3) reported a case with psoriatic arthritis, ruptured Baker's cyst and entrapment neuropathy. He was treated with intra-articular corticosteroid injections with improvement.\nIt is important to diagnose a Baker's cyst early and to differentiate it from thrombophlebitis, a popliteal aneurysm, tumor or muscle tear, to effect optimal therapy and to obviate a potential neuropathy. Intra-articular steroid injections often benefit the patient immediately, while refractory and recurrent cases may require synovectomy11).\nT2 weighted magnetic resonance image of the left knee and popliteal fossa. The hyperintense lesion represents the Baker's cyst.\nSurgical specimen of Baker's cyst showing lobulated cystic lesion.\nBiopsy specimen revealing a cyst with synovial lining which is compatible with Baker's cyst.\nEntrapment neuropathy from a Baker's cyst. Review of the literature.\n1. Petros DP, Hanley JF, Gilbreath P, Toon RD. Posterior compartment syndrome following ruptured Baker's cyst. Ann Rheum Dis 1990;49:944\u2013945.\n2. Dash Subasini, Swarnalatha R, Bheemreddy , Moti L. Posterior tibial neuropathy from ruptured Baker's cyst. Semin Arthritis Rheum 1998;27:272\u2013276.\n3. Wigley RD. Popliteal cyst. Variations on the Theme of Baker. Semin Arthritis Rheum 1982;12:1\u201310.\n4. Scott WN, Jacobs B, Lockshin M. Posterior compartment syndrome resulting from dissecting popliteal cyst. Clin Orthop 1977;122:189\u2013192.\n5. Jensen KH, Jorgenson U. Lateral presentation of a Baker's cyst. Clin Orthop 1993;287:202\u2013203.\n6. Darryl D, Reynaldo L, John AP. Nerve entrapment and calf atrophy caused by a Baker's cyst. Neurosurgery 1994;35:333\u2013334.\n7. Dunlop D, Parker PJ, Keating JF. Ruptured Baker's cyst causing posterior compartment syndrome. Injury 1997;28:561\u2013562.\n8. Nakano KK. Entrapment neuropathy from Baker's cyst. JAMA 1978;239:135.\n9. Kashani SR, Moon AH, Gaunt WD. Tibial nerve entrapment by a Baker's cyst. Case report. Arch Phys Med Rehabil 1985;66:49\u201351.\n10. Simpson FG, Bark M, Robinson PJ, Losowsky MS. Prospective study of thrombophlebitis and pseudothrombophlebitis. Lancet 1980;1:331\u2013333.\n11. Martin G, Jairo P. Tarsal tunnel syndrome in rheumatoid arthritis. Arch Phys Med Rehabil 1981;62:401\u2013403.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"You'd think dieting with your partner would offer both support and motivation, making it easier to reach your weight-loss goals. Well, you'd be wrong\u2014at least according to a study published in Eating Behaviors.\nJennifer Harman, PhD, an associate professor at Colorado State University, and her colleagues looked at 50 overweight couples who pledged to lose weight in the New Year. They discovered that the more successful one partner was at eating healthier, the less successful the other partner was at controlling his or her own food portions.\nThis came as a shock to Harman, who assumed that a support system of any kind would be more beneficial than not. But studies show that self-esteem drops when someone performs better than you when you're working just as hard toward a common goal. Case in point: When the husbands in the Colorado State study (who were successful in the weight-loss realm) monitored their weight by stepping on a scale or checking themselves out in the mirror, their wives became more negative about dieting and monitored their own weight less.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Envalve butterfly valves are designed to function as a valve or a control valve.\nThe valves are made of PVC and have an EDPM cuff, which is resistant to many aggressive substances. The valves are also available with a VITON\u00ae cuff.\nWith electric or pneumatic drive, which can be controlled remotely. In order to operate the PVK butterfly valve, the valve shaft is coupled to a pneumatic drive engine. The drive engine has a 24 VAC power supply control valve. On request, control valves can also be supplied with 230 Volt. The drive engine can turn to 2 sides and is controlled by the voltage that comes from the control valve.\nSpecifications and benefits of a butterfly valve.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Texas State University will launch its inaugural \"Step Up for State\" online giving event at 8 a.m. September 21, to support mission-critical projects for the university.\nThe campaign will last 1,899 minutes\u2014corresponding with the university's 1899 founding\u2014and conclude at 3:39 p.m. September 22. Gifts may be made online at www.donate.txstate.edu\/stepup.\n\"Step Up for State\" provides alumni, parents, students, faculty, staff and friends of the university the opportunity to come together and support Texas State. Individuals will be able to choose from 26 projects from across 15 colleges, research institutes and university divisions to support. There is no minimum gift limit and all contributions are tax-deductible.\nThe event will also feature an active social media component on Facebook (www.facebook.com\/txstgiving\/) and Twitter (www.twitter.com\/txstgiving).\nFor more information, visit www.donate.txstate.edu\/stepup.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is it time to elevate your project team to a higher level of competency? Are you looking for some additional training to help your project management team reach new levels of performance? Logixsource offers in-house training in many diverse training sessions, such as Risk Management, Procurement Management, Virtual Project Team Management and Program management. We also offer basic project management skills course including MS Project Basics.\nIs your team ready to move to Microsoft 2013? The MS Project upgrade training will elevate your team to the most recent version of MS Project concentrating on changes to the application as well as new techniques that will enhance the team performance. This can be run in-house or via webinar for the virtual teams.\nMany companies have discovered the benefits of running their projects in the Agile project delivery methodology, with focus on holding SCRUMS and iterative design and product delivery. Even projects that are being delivered in the traditional waterfall methodology can benefit from some of the techniques Agile can offer. For example, morning SCRUMs can increase production while delivering project results without tying up your resources for lengthy project status meetings.\nWith more organizations demanding PMP or CAPM certifications for their new recruits there is tremendous value in educating the existing project teams to the same level of proficiency. The PMP prep courses will train your teams with the information from the PMBOK\u00ae that will enable them to successfully pass the PMP or CAPM exams.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Interested in upgrading the look of your favorite ride? Wanting to add utility to your beloved bike? Add a hand laid fiberglass fender along with a set of Extended Saddlebags to get the most out of your bike without taking away from its already awesome visual appeal. Our bagger upgrade kit is quality constructed from strong, durable, and light weight hand laid fiberglass with a dark gray primer coat on bags & lids.\nUnlike our competitors, we provide a license plate LED to illuminate your bike's tag. Many states require this so we give it to you completely FREE*.\nBecause the saddlebags, fenders, and lids are aftermarket parts they may require minor modification prior to installation.\n**Please Read: All fiberglass components are hand laid and there will be minor imperfections from time to time, including the opening varying +\/- 1\/8\". We do everything possible to prevent these from occurring, however a thorough inspection and bodywork may be required before painting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaezqo b/data_all_eng_slimpj/shuffled/split2/finalzzzaezqo
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+{"text":"Nate Perry Elementary School Preparing Our Students for Tomorrow... Today!\nThe Liverpool Central School District Special Education Department and the Syracuse University Parent Assistance Center (SUPAC) have rescheduled their final special education workshop of the year.\nGive an overview of various community connections that should be established in order to assist in adult life.\nAll LCSD parents and staff are invited to attend. Registration is required. To register, visit the online registration Web site or register by phone at 315-443-4336 or toll free 877-824-9555. For more information regarding the workshop, contact the LCSD Special Education Office at 315-622-7185.\nSUPAC is funded through a contract with the New York State Education Department (NYSED) Office of Special Education and seeks to empower parents to become meaningfully involved in their children's education through school, family, and community partnerships. SUPAC is a project of the Center on Human Policy and is located on the Syracuse University campus.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is your existing conservatory letting you down? If you have an older conservatory and want to modernise it to suit your needs, our replacement conservatory roofs give you the ability to upgrade and improve.\nYour tired polycarbonate roof can be upgraded to glass, high performance glass, or even a solid roof by Ultraframe.\nConservatory roof manufacturing technology has come a long way in the past 30 years. Advances in glass technology mean that a conservatory is not just a 'greenhouse' attached to the back of your property. The glass can be very thermally efficient and can reduce sun glare enabling you to use your conservatory all year round. A high-performance glass roof allows a plentiful supply of natural light to come into your home whilst still being protected from the elements.\nUltraframe have two fantastic solid roof options that you can choose as a replacement conservatory roof, or as part of your new build: Livinroof and the Ultraroof380.\nLivinroof is the first replacement roof system designed to give you the flexibility to combine solid and glazed roofing seamlessly. It enhances thermal performance to give you a room that you can use all year round.\nUltraroof380 overcomes the twin issues of your conservatory being too hot in summer and too cold in winter. Moreover, it creates a beautiful vaulted plastered ceiling inside \u2013 and on the outside a stunning lightweight tiled finish.\nUltraframe's solutions give you the flexibility of warmth and light, meaning that you can use the room all year round.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Image Title: Gorgeous Office Desk Ideas Perfect Furniture Home Design With For Prepare 5. Post Title: Home Office Desk Ideas. Filename: gorgeous-office-desk-ideas-perfect-furniture-home-design-with-for-prepare-5.jpg. Image Dimension: 550 x 440 pixels. Images Format: jpg\/jpeg. Publisher\/Author: Joel Hansen. Uploaded Date: Friday - September 21st. 2018 16:12:13 PM. Category: Home Design. Image Source: archdaily.com. Office Home But Double Cool Desk Ideas 30 Fabulous To Inspirational For That Pinterest Prepare 2. Remodel Decor Diy Regard Having And Desks DIY Work Your 18. Racetotop 15. Finding Furniture Pertaining 7. Two Designs 8. Small Inside With 10. Difficulty Regarding Throughout Within Large Really In 19. 0. A Lovable 12. 20 Modern Person Need 3.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You know that your beautiful figure will look its absolute best whatever you choose from our collection of Formal Wedding Dresses. All you have to do is choose from our wide range of chic wedding dresses and get back to us right away. We will make sure that your perfect pick reaches you within 15 days.\nShopping is stressful? Not at JuneBridals.com! We offer a wide array of fabrics for Formal Wedding Dresses, such as lace or chiffon, so you can feel like a queen when walking down the aisle. Also look for other accessories for your dress, like the perfect gloves, veil or wrap. There are perfect pieces for formal and informal events.\nI am so happy as I looked so pretty on my wedding because of this gorgeous dress. Everybody liked it so much and I received huge compliments. Thank you!\nI am very satisfied with the dress I purchased here and would recommend it to other people too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Description: The Surf Grass Mat\u2122 is 29 oz. and 20 x 25 inches of high quality synthetic turf. The mat's supportive and soft bristles of turf work to dust off any dirt, sand, or particles that may have accumulated on your feet and wetsuit while changing. Use the Surf Grass Mat\u2122 before you get into your car or walk into your house for an instant clean feeling. The mat is extremely durable and easy to clean. Storing it is equally as simple - just roll it up and secure it by the velcro strap, or lay it flat in your trunk for your wetsuit to dry on while driving to your next destination. Ditch your lid tops, towels and tarp and get yourself a Surf Grass Mat - guaranteed to provide comfort while garnering looks of awe and curiosity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafuzq b/data_all_eng_slimpj/shuffled/split2/finalzzzafuzq
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index 0000000000000000000000000000000000000000..6b062741155be17a6a3d643af2b5b08e50cfb072
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Asthma is a chronic and life-threatening disease that affects your airways. Asthma occurs when the airways in your lungs (bronchial tubes) become inflamed and constricted.\nThe muscles of the bronchial walls tighten, and your airways produce extra mucus that blocks your airways. Signs and symptoms of asthma range from minor wheezing to life-threatening asthma attacks.\nHomeopathy is a system of medicine that supports the body?s own healing mechanism, by means of specially prepared remedies. It is possible to treat asthma with correctly selected Homeopathic medicines. However, it involves careful study of every episode to evaluate the cause that might have precipitated the attack, the exact symptoms, and the factors that make the attack worse or better.\nEvery patient suffering from Asthma experiences it in a different way: with exertion, when apprehensive, when outdoors, in summer, in winter, on waking, after eating, etc.\nAsthma cannot be cured by allopathic medicine, but its symptoms can be controlled. Persons treated with Homeopathy may be saved from using heavy doses of inhalers, steroids and bronchodilators.\nA natural homeopathic medicine for control of the symptoms of asthma and bronchitis, including difficulty in breathing, shortness of breath, tightness of chest, wheezing or coughing.\n- You can permanently get cured of asthma.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm a complete newcomer to the hardware\/circuit\/IoT stuff and for my first hardware project I was thinking about doing something with the Omega and a DHTXX.\nI've seen some examples of the DHTXX with Arduino and Raspberry, but I'm a little clueless about how to do it with the Omega since the output voltages seem to be different (only 2.8 for the Omega).\nCan you please share a picture or diagram of your setup?!\nI'm a C dummy, how can I use it on my onion. Is there something in Python available.\nWhat does that mean? I'm really hopeless..\n@J\u00f3zsef-Schuck You're using the Omega1 binaries. They don't work on the Omega 2.\n@WereCatf Where the binaries for omega2 can be found? Thanks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At the beginning of the Sutra of Golden Light, Buddha Shakyamuni, the Tathagata, the Arhat, the Fully Enlightened, calls to anyone experiencing misery and affliction, bad health, poverty, loss, abuse, ill will, fear, nightmares, or other harms. He says to make the mind virtuous and to come and listen.\nThis \"King of Glorious Sutras,\" contains everything needed, from daily happiness to complete enlightenment. It contains a heart-rending practice of confession and rejoicing, profound teachings on dependent arising, reliable assurances of protection, guidelines for ideal government, and awe-inspiring stories of the Buddha's previous lives, in which the Buddha shows how, even before he had completely eliminated the delusions, he liberated countless beings from the ocean of suffering through compassion and personal courage.\nReciting the Golden Light Sutra is one of Kyabje Zopa Rinpoche's vast visions. Rinpoche says: 'This text is very precious; it brings peace and happiness and is very powerful to stop violence. By hearing this text, one's karma gets purified. This text increases success and, especially for leaders like kings or presidents, brings success in guiding in virtue, the path to happiness. In whichever country this is taught, the whole country benefits\u2026everyone is happy and the country becomes harmonious; there are no quarrels. The king gives religious freedom and is always protected by the devas.\n\u200bLama Zopa Rinpoche urges us to recite the Sutra of Golden Light every day. The benefits from reciting, listening to, or even hearing the name of the sutra are immeasurable, from eliminating conflict, terrorism, torture and famine to achieving full enlightenment.\nEverybody is welcome to attend! Drop by anytime.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our Wellness Workshop for Cancer Survivors will include a nutrition talk, an exercise specialist, yoga instruction and Reiki healing for relaxation.\nSpots are limited to 15. To reserve your space, please call 1-888-724-7123.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have been trying my hands on healthy food at home for a while. It has been a really lovely journey where I am trying to eat clean and portion the meals.\nOh yes yes I cant get enough of that previous Kofta sauce okay. Mom usually makes Lauki kofta in tomato onion sauce, well since I am her offspring I need to make my own changes to it :P.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafyvj b/data_all_eng_slimpj/shuffled/split2/finalzzzafyvj
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index 0000000000000000000000000000000000000000..d4fd21d93fe6f273fe40b95b30b20adf47a46bf3
--- /dev/null
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+{"text":"I saw this ladybug in my garden this past fall, so I snapped a photo. I thought it would be a great reminder of how all things in the garden work together.\nOur garden health is like our personal health: maintaining a healthy foundation limits problems now and down the road. When you maintain healthy soil in your garden you'll have healthy plants and you won't have too many problems with insect infestations or plant diseases.\nWhen bad insects come they are usually followed by the beneficial ones that prey on them. This is the balance of nature, and it's important to keep that balance.\nWhen we step in trying to fix things we perceive as problems with chemicals and quick fixes, we often only do further damage. Plant disease and insect infestations are often symptoms of a deeper problem. If we resort to the quick fix spray, often our problems will persist or multiply because we aren't fixing the actual cause, we are only treating the symptoms.\nEven so called green, non-toxic, and all natural products often kill the beneficial insects along with the bad ones. So what are we to do if we don't want to upset this natural balance? We'll be exploring this all this week here at Chiot's Run.\nAre you an organic gardener, or do you use chemical fertilizers, herbicides & pesticides?\nYou take beautiful pictures. I don't just commercial pesticides, etc. Last fall I did use a bought manure mix, but this season I'm trying as much as possible to just use my own compost. Its interesting, but since I've started the garden, I've overcome my fear of bugs\/insects in general. All my life, I was terrified of all insects and bugs, even running in terror about the house to avoid them and locking doors to keep them contained. But that seems to have disappeared last summer. Maybe its related to what you said, how there are beneficial insects as well as the bad ones, and that they don't hurt me personally. My mind is so focused on caring for the garden that it doesn't seem to have room for fear.\nI have to buy ladybugs here. In my environment, which is at the transition from urban and suburban, the beneficials don't stand much of a chance. When I buy beneficials, like ladybugs, they are gone after about 48 hours, dispersed into the environment around me, and I'm happy to do it. I hope they come back when I have something good for them to eat.\nMy most common means of pest control is to hand pick. I used to use things like soap and Bt also, but haven't in over a decade. To keep out the caterpillars, I use row covers.\nLast year I was a bit at my wits end with slugs. I really got tired of hand picking them at night. I almost got some iron phosphate. Now that I have access to coffee grounds I might try that. I really hate slugs. With last years rainy weather my toads just couldn't keep up, nor could I.\nOne year we had slugs really bad here. But the next year our toad populations increased to keep up. I have found that if you leave a few boards (not treated) laying around the garden the toads love to live under them!\nWonderful pictures. I've never seen a lady bug with that many spots before. Mine is a pesticide free garden and slowly but surely I'm converting the neighbors.\nInterestingly, the MiL and I were just discussing this very thing. When my husband's grandfather did the gardening (in our same spot\u2013that's been the garden plot for I think 80 years), he used a lot of pesticides, and he had a LOT of infestations. Then, after he died and my MiL took over, she stopped using them for the most part (we do still use Rotenone), and now we have lots of spiders and lady bugs and something that eats the potato beetles YAY! Plus, five billion earthworms. I remember the first time I heard that gardens subjected to lots of pesticides don't even have worms in the soil. That's just weird and creepy.\nWe had the same problem when we moved in here. The people before us had used tons of chemical pesticides and fertilizers and I didn't find an earth worm for the first 5 years we lived here. After adding manure & compost for the past 7 years, I now find a few earth worms here and there when I dig. I'm hoping I'll have more and more each year!\nI will be looking forward to reading each post this week. I have a hard time with the squash cut worms killing my zucchini plants at the base of the plant. I resorted to buying something last year to kill them because I was losing my plants every year. Any ideas or suggestions of something better. I do try to compost the garden every spring.\nI had cut worms killing my beet seedlings last year, I finally got rid of them by going to the garden each morning and checking for downed seedlings. When I spotted one I would dig an inch deep in concentric circles around the spot, usually you'll find the worm there. After finding them I'd smash them!\nYou can also add beneficial nematodes to the soil to help with cutworms.\nI have horrified my neighbours and my in-laws by announcing I'm an organic gardener and don't use pesticides. They think I'm a total hippy. I think they need to chuck out their chemical weapons!\nI think this is so fascinating. I first learned about the symbiosis in nature reading Barbara Kingsolver's *Prodigal Summer*.\nMy neighbors and I usually share ladybugs.\nWe are all natural in our gardens. It usually turns out that we lose some things but not enough for us to add chemicals. I work my bees the same way. In general, I avoid any unnatural chemicals. I just don't want that stuff in me!\nUsually the things you lose are plants that weren't up to par, or out of their native habitat. That's one of the bad things about losing biological diversity. Along the way we've lost so many kinds of plants that did well in our area.\nNo chemicals go anywhere near my gardens. I want a healthy living soil. It took a number of years to revive the garden area where the previous owners used fertilizers and pesticides for so long there was barely any soil left and the area was 4 inches below grade.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Government's decision came after the female parties in both cases reported their alleged physical assaults to the State Ministry of Women Affairs for justice and Domestic and Sexual Violence Response Team (DSVRT), for legal actions. A top official in the ministry disclosed that the case has been reported to the State Government, saying, you know the Governor does not like to hear such in Lagos and that was why both the ministry and DSVRT have been given the mandate to stem the tide.  The official added that aside the government, the Nigerian Police, Lagos State Command has commenced investigation into the alleged domestic violence. According to the official, the case is been handled by the Area F police Command, GRA Ikeja and in due time, residents of Lagos will hear more about it. On the over three years domestic violence which the government obtained restrain order, Janet alleged that her husband, Jonathan, had been physically assaulting her since their marriage was pronounced. While reporting the alleged assault at DSVRT, she explained that recently, her husband allegedly hit her repeatedly with a stick, threatening to kill her. Speaking on the restraining order, the Coordinator of DSVRT, Mrs. Titilola Vivour-Adeniyi, explained that restraining order was a temporary court order often issued by the court to stop an individual from embarking on a particular action, especially as it concerns domestic violence. The Coordinator of DSVRT noted that the restraining order obtained by the State Government was based on the request by Janet. Vivour-Adeniyi said: \"As provided by the Prevention Against Domestic Violence Law, 2007, DSVRT applied for a Restraining Order. The application was made to the Family Court, magistrate court Ogudu on April 27th, 2017. \"The Magistrate subsequently granted a restraining order against Mr. Jonathan from coming anywhere close to the complainant for the next one year and failure to comply will earn him a minimum of three months imprisonment. \"We use this medium to inform any survivor of domestic violence and members of the public that there is help for you especially at the Family Courts of Lagos State,\" she added.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tech Craft CRE48B 48\" wide Solid Wood and Veneer Credenza TV Stand up to 52\" TVs in Black finish.\n:: Tech Craft CRE48 TV Stand up to 52\" TVs in Black finish.\nTech Craft CRE48 TV Stand up to 52\" TVs in Black finish.\nDownload the Tech Craft CRE48 Dimension Chart.\nTech Craft CRE48 TV Stand up to 52\" TVs in Walnut finish.\nTech Craft CRE48 48\" wide Solid Wood and Veneer Credenza TV Stand up to 52\" TVs in Walnut finish.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to our Resource Center! Read our case studies, e-newsletters, whitepapers, press releases -- knowledge and entertainment in one place!\nFrom 3 Years Ago, But Still Timely: Why is Sexual Harassment STILL Occurring?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"By creating an account with our store, you will be able to collect reward points and avail of our Special Offers by using the Special Offer codes, also 3 for 2 offers. You would also move through the checkout process faster, store multiple shipping addresses also view and track your orders in your account. NO CREDIT CARD DETAILS ARE REQUIRED TO REGISTER.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Kurt Angle has been spotted arriving to the PPG Paints Arena for tonight's episode of SmackDown Live.\nWhile Angle may only be there to visit friends and\/or film more content for the WWE Network, there is always the possibility that he could appear on television. You never know with WWE.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pink and pop rockers The 1975 stole the show at British pop music's annual celebration, here are some snaps of the big night.\nCape Town - The organisers of Rocking the Daisies announced on Wednesday, that British pop rock band The 1975 are the first international headlining act confirmed to hit the stage at the 2019 music event.\nIn November 2018 the band released their third studio album A Brief Inquiry into Online Relationships, to critical praise, making it their third number one album in the UK.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you looking the best price for Lug Handspring Wallet, Midnight Black and you want to get great deals & best buy this product? Or you want to find some reviews from people who have used this product before you buy it?\nYou are at the right place. I'll give you some tips before you decide to buy Lug Handspring Wallet, Midnight Black. So you are not disappointed after buying it and in accordance with the product you need.\nYou can also find price comparisons from sellers, so you really get the best price currently for the Lug Handspring Wallet, Midnight Black.\nIf you are interested in buying Lug Handspring Wallet, Midnight Blackyou may try to find full description and product details.\nInterest to Buy This Product? Check our store for more information and get special offers for Lug Handspring Wallet, Midnight Black.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Subdural Hematoma is a topic covered in the 5-Minute Pediatric Consult.\nA subdural hematoma (SDH) is a collection of blood between the outer pial and inner dural meningeal layers. The bleeding is usually venous in origin, although either cortical arteries or bridging veins may be torn.\nIn infants and young children, SDHs are frequently the result of abusive head trauma.\nIn older children, SDHs are often the result of motor vehicle collisions.\nNeonatal SDHs occur with spontaneous deliveries but may be more frequent following deliveries with forceps or vacuum extraction. SDHs related to birth usually resolve.\nRisk factors for abusive head trauma include disability or prematurity of the child, unstable family situations, parents of young age, and low socioeconomic status.\nOne study found that fathers were the most frequent perpetrators, followed by boyfriends, female babysitters, and mothers, in descending order of frequency.\nThere is no clear genetic predisposition except when hereditary coagulopathy or metabolic disease is implicated.\nParents should be counseled about appropriate methods to channel frustration and anger toward infants and children. Shaking an infant when the parent is angry is never appropriate.\nBicycle helmets, car seats, and seat belts are all valuable in preventing head injuries in children.\nArterial SDHs grow quickly, whereas venous SDHs may accumulate slowly, remaining undetected for weeks or months.\nAcute SDHs contain blood, whereas chronic SDHs contain proteinaceous exudate and blood-breakdown products.\nRebleeding may be the underlying cause of many chronic SDHs.\nSignificant force is usually required for SDH unless there are predisposing circumstances; SDH is only rarely due to trivial or minor trauma. However, SDH can occur with relatively minor trauma in individuals with bleeding disorders, children on chronic dialysis, and those with enlarged extracerebral spaces, arachnoid cysts, or cortical atrophy.\nThe sudden deceleration associated with the impact may tear bridging veins traveling in the subdural space.\nThe term shaking-impact syndrome may be more accurate than shaken baby syndrome.\nSDHs can also occur after ventricular shunting and extracorporeal membrane oxygenation (ECMO).\nSome metabolic disorders, such as glutaric aciduria type I and Menkes disease, can be associated with both acute and chronic SDHs.\nVictims of motor vehicle collisions with SDH may have other intracranial injuries such as diffuse axonal injury.\nTraumatic SDHs are often associated with cerebral contusions. Other associated injuries include skull fractures, diffuse axonal injury, and penetrating injuries.\nCabana, Michael D., editor. \"Subdural Hematoma.\" 5-Minute Pediatric Consult, 8th ed., Wolters Kluwer, 2019. Pediatrics Central, peds.unboundmedicine.com\/pedscentral\/view\/5-Minute-Pediatric-Consult\/617373\/all\/Subdural_Hematoma.\nSubdural Hematoma. In: Cabana MD, ed. 5-Minute Pediatric Consult. 8th ed. Wolters Kluwer; 2019. https:\/\/peds.unboundmedicine.com\/pedscentral\/view\/5-Minute-Pediatric-Consult\/617373\/all\/Subdural_Hematoma. Accessed April 19, 2019.\nSubdural Hematoma [Internet]. In: Cabana MD, editors. 5-Minute Pediatric Consult. Wolters Kluwer; 2019. [cited 2019 April 19]. Available from: https:\/\/peds.unboundmedicine.com\/pedscentral\/view\/5-Minute-Pediatric-Consult\/617373\/all\/Subdural_Hematoma.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As vampire hunter Van Helsing, in the old Hammer Dracula films, Peter Cushing was preoccupied with making sure the dead stayed dead. As Victor Frankenstein, another of his famous horror roles, he was initially preoccupied with quite the opposite.\nAll things considered, however, it's unlikely that Cushing foresaw his own, digital resurrection in director Gareth Edwards's newly released Star Wars prequel Rogue One.\nThe rebirth of villain Grand Moff Tarkin, in sinister, eerily lifelike glory, is one of the film's (many) highlights. But just how did the team behind the movie go about recreating Cushing? Who signed off on the decision? Was it even all that ethical?\nCushing, as described above, was \"brought back\" to play Tarkin in 2016's Rogue One: A Star Wars Story. The decision was unusual in that, while the CGI likeness reprised a character that the actor had played before, Cushing was not being \"brought back\" simply to complete an unfinished film. Instead, the cinematic rebirth took place more than 20 years after the actor's death, and recreated the way Cushing looked in the original 1977 Star Wars film, when he was in his mid-50s. The digital recreation also incorporated Cushing's voice, allowing the actor to speak brand new lines.\nWhy was he brought back?\nAccording to director Gareth Edwards, a crucial part of the reasoning behind the decision was that, in a film set directly ahead of the events of A New Hope, it would be unthinkable, not to mention narratively implausible, not to feature the popular Tarkin. The team were therefore effectively faced with two choices: to recast, or to recreate. Edwards eventually opted for the latter, after being persuaded by his team that the technology to do so sensitively and effectively was in place.\n\"We were talking about it, and you sort of lay out the movie, and as you start laying it you go 'you know what, you've gotta have these certain characters in there, and I dunno how we'd do it',\" he told the Radio Times.\n\"'Like, what do we do? Do we cast someone who looks like them? Do we not have them? Do we just hear about them?'\"\n\"[Visual effects supervisor] John Knoll was always like 'no we can do this, we can do it, we can do it.' He was very confident, and we\u2026 you know, to be honest, a lot of people were nervous the whole time, like 'is this gonna happen?' And then we went all or nothing in.\"\nWhile the full details of the process won't be released until January, we do know that Holby City actor Guy Henry, whose face shape bears a superficial resemblance to Cushing's, was asked to perform the necessary gestures and lines.\nMotion capture technology was then presumably used to take Henry's facial expressions and movements, and turn them into Cushing's expressions and movements.\n\"It was a massive thing for him, it was very gracious of him, because essentially he's doing this big performance and getting zero credit for it,\" Edwards told the Radio Times.\nLast year, a source also reported that, in order to accurately represent Cushing, the filmmakers reportedly studied hours of archive film.\nThere are no shot of Tarkin's feet or lower legs anywhere in Star Wars (reportedly because Cushing found the boots uncomfortable and decided not to wear them), meaning that the Rogue One team had to turn to other sources to create any full body shots.\nHow did the filmmakers get permission?\nWhile Edwards has not explicitly stated so, it can be assumed that he sought the permission of Cushing's estate before going ahead with the CGI. The actor's former secretary Joyce Broughton, who is responsible for handling Cushing estate, attended a screening of the film, and spoke about how moved she was.\n\"When you're with somebody for 35 years, what do you expect?\" Broughton told Variety. \"I can't say any more because I get very upset about it. He was the most beautiful man. He had his own private way of living.\"\n\"I have to say, I'm not a Star Wars fanatic, but I did think whoever put it together was absolutely fantastic,\" she added. \"It's not just a silly sort of thing. It's really good!\"\nCushing's wife, Violet Helene Beck, died in 1971. He did not have any children.\nIn 2013, a digital recreation of Audrey Hepburn was unveiled as the star of a sun-drenched new advert for Galaxy chocolate. In the mini-movie, a 19-year-old incarnation of the famously beautiful film star, who died in 1993, is seen swapping a crowded Italian bus for a car driven by a handsome young chauffeur, before enjoying a bar of chocolate. The slogan \"Why have cotton when you can have silk?\" then appears on screen. Hepburn \u2013 or, rather, her computer-generated likeness \u2013 does not speak.\nWhy was she brought back?\nAccording to Mike McGee, who was part of the team who created the advert, Galaxy felt that Audrey Hepburn was a good \"fit\" for their brand.\n\"Audrey represents heritage, classiness and elegance,\" he wrote in a piece for the Guardian. \"So from a strategic and creative point of view, it made sense for Galaxy to communicate its 'silk, not cotton' branding through these qualities.\"\nTo recreate the actress, two doubles were used: a body double, and a double whose face shape and bone structure bore some similarity to the Hepburn's. In the studio, the team behind the advert used the double's head to create a sort of blank \"foundation\" for their CGI, then painstakingly \"researched Hepburn's back catalogue to identify the subtleties that make Audrey look like Audrey\". According to McGee, a large proportion of time was spent getting just the tiniest details right: little things like adding \"peach fuzz\" to Hepburn's skin could make a world of difference.\n\"Without access to geometrical data of Audrey's face (if only they'd foreseen the future of VFX in the 1940s) recreating her as a CG entity was an inexact science, making it difficult to judge when we'd reached perfection,\" he explained. \"Our only option was good old-fashioned trial and error, persisting until everyone screamed: 'That's it!'\"\nHepburn's two sons, Sean Ferrer and Luca Dotti, authorised the use of their mother's image, and received a fee for the advert.\nIn a press release, they stated that Hepburn: \"often spoke about her love of chocolate and how it lifted her spirit, so we're sure she would have been proud of her role as the face of Galaxy\".\nAstaire, who died in 1987, was brought back to life to dance with a broom-style vacuum cleaner in a famous 1997 advert for Dirt Devil.\nThe commercial was designed to echo a scene from Astaire's movie Royal Wedding, which featured the actor comedically dancing with a hat rack.\nIn contrast to some of the more technologically advanced examples discussed in this list, the pre-existing footage of Astaire dancing was simply digitally edited to incorporate the vacuum cleaner.\nThis is where it gets tricky. Permission to make the advert was granted by Astaire's widow, Robyn Astaire, but the way in which she handled the rights to images and footage of her late husband led to some controversy. Some people accused her of withholding permission to use clips of the movie star in reputable documentaries, and suggested that the advertising campaign, for which she received payment, was tasteless.\nAstaire's daughter, Ava Astaire McKenzie, was among the critics, and later told the press that she was \"saddened that after Fred's wonderful career, he was sold to the devil\".\nRobyn, however, remained confident about her decision.\n\"I've had to deplete much of my financial security over the years to prosecute infringers,\" she said at the time (via Entertainment Weekly). \"I just feel Fred would have wanted me to do these commercials.\"\nAfter Reed died during the filming of Gladiator in 1999, at the age of 61, director Ridley Scott decided to complete scenes featuring his character Proximo using a mixture of body doubles and CGI.\nWhile some planned scenes featuring Reed were reportedly axed, the director and VFX team believed that it would be possible to create an important final prison scene (above), in a way that was both tasteful and convincing. This would allow Reed's final movie role to be a complete one \u2013 a fitting legacy to his talent \u2013 and stop the film itself from feeling unfinished.\nA body double was used for the initial filming, before unused footage of the actor was superimposed on top of the double's face, with a few digital tweaks.\n\"I sat down with the editor and went through all the rushes of Oliver. We had to find lines that were dropped and takes that weren't used to piece together a coherent sequence,\" said the film's visual effects supervisor Rob Harvey, in an interview with the BBC, when asked about the making of the prison scene.\n\"We had to shoot pick-ups of Russell [Crowe] on a green screen and another of doors and bars so [Oliver] could speak to Russell though them, and we had to basically cut his performance out of it all.\n\"We only manipulated his beard [for continuity], but it was his head cut out and stuck on to the body double. Then we just mucked around with the cut as much as we could to make it work.\n\"It was a clever bit of directing and scriptwriting. We just tried to do it as tastefully as possible.\"\nReed was later nominated for a posthumous BAFTA for his part in the film.\nIt's not entirely clear how Reed's CGI recreation was authorised, but subsequent interviews with the actor's family suggest that they were proud that the star was able to complete his final film role, instead of being recast. His widow Josephine and brother Simon, in fact, were reportedly disappointed that he missed out on an Academy Award nomination.\n\"Olly loved the film. He thought it was great and could sense there was something special going on,\" said Simon Reed. \"Up until then his career was at a very low ebb and some people refused to employ him. Ridley Scott took a big gamble when he offered him the role. I think Olly was determined to repay Ridley's faith and gave the best performance he possibly could.\"\nThe role: After the tragic death of actor Paul Walker in 2013, which occurred while the star was filming Furious 7, the decision was made to complete the film, and scenes featuring Walker's character Brian O'Conner, using CGI.\nWalker's death left the Furious cast and crew stunned, and, in the immediate aftermath of the tragedy, the film was put on hiatus. Universal were keen, too, to speak to Walker's family, and determine the most sensitive course of action. Eventually, a decision was made to complete filming, and to find a way to sensitively write O'Connor out of the franchise.\nMovingly, the final film, which was released in 2015, now ends with the character deciding to retire to spend more time with his family, and bidding a last farewell to Vin Diesel's Dominic Toretto.\nWhile some of O'Conner's scenes had already been filmed, 350 shots were created using a combination of unused footage, motion capture CGI and stand-ins, including two brothers of the actor (both of whom bore a strong resemblance to Walker).\nA New Zealand-based company called Weta Digital were employed to carry out the work. According to Variety, they began by combing through old footage, to gather as many images of Walker, from as many different angles, as possible.\nThey also had to make sure they were accurately replicating the actor's body and physicality, as well as just his face.\nThe biggest challenges, however, were not the attention-grabbing action sequences but the subtler, more character-driven moments \u2013\"Paul sitting still, or delivering dialog in close-up\".\nTheir digital models of Walker had to replicate, for instance, \"how his facial muscles moved if he frowned or raised his eyebrows, how his coloring would flush after exertion or how the hair changes depending on the light or the wind\".\n\"We had to complete a performance \u2013 what Paul Walker would have done if he'd been able to continue. And it had to be his performance,\" said VFX supervisor Joe Letteri.\nAs described above, Walker's family gave their blessing to Universal, and two of the actor's brothers, Cody and Caleb, agreed to help complete their sibling's legacy.\nCaleb later said that being in the film helped him feel closer to his brother.\n\"We got to understand more about Paul, and what he was all about,\" he revealed in an interview earlier this year. \"We had to share Paul with a lot of people, in a weird way. He was gone a lot, he was filming, so we missed him. Sometimes at the holidays he wasn't always there, you know? He tried. We'd see him on set, we visited a few times, but to kind of get that closure, and understanding [of] Paul was really special.\"\nIn the 2004 science fiction film Sky Captain and the World of Tomorrow, a young Sir Laurence Olivier \"played\" holographic villain Dr. Totenkopf... despite the fact that he actor himself had died in 1989, at the age of 82.\n\"I haven't seen that many movies with Olivier recently, which is part of the reason why I thought this was pretty cool casting \u2013 that, and the fact that he's a great actor,\" said Jon Avnet, the film's producer, ahead of Sky Captain's release.\nAvnet also suggested that Olivier was \"cast\" as a nod to the film's deliberately retro aesthetic: the movie is set in an alternative version of 1939.\n\"What holds this film together is its celebration of movie history. To me, Olivier is perfect casting, because he embodies that history,\" he said.\nAccording to director Kerry Conran, however, the decision to feature Olivier may have originated with the movie's star Jude Law, an admirer of the famous actor.\n\"[It] was Jude Law's idea. He, I think, just wanted to act with Olivier, so we contacted Olivier's estate and got permission. I meant it to be a nostalgic tip of the hat to that whole era, a loving gesture,\" Conran revealed.\nIt should also be noted that, in the film's narrative, the character of Totenkopf is revealed to have been dead for many years. Incorporating footage of a well-known actor who was also deceased, the filmmakers may have hoped, would enhance the uncanny effect of this twist.\nBack in 2004, the visual effects industry was not able to create the sort of all-new performances featured in Rogue One: A Star Wars Story and Furious 7.\n\"We used archival footage of him. Nothing from his specific films, but more obscure stuff, like BBC interviews where he's seated and not moving around a lot,\" Conran told Film Journal.\nThat said, while the Sky Captain filmmakers may have relied predominantly on pre-existing film, some similar techniques to those used in later movies, including motion capture, were used to manipulate the actor's image.\nIn addition, audio recordings of Olivier were digitally altered , to allow the star to \"speak\" new words. A voice actor was also used.\n\"We created a different performance of sorts through motion capture, modeling and computer-generated imagery,\" Avent noted at the time. \"Could you create a completely new performance with what we have today? It would be pretty demanding. But clearly, the future is coming fast.\"\nAs described above, permission to \"bring back\" the actor, using a combination of archive footage and CGI, was granted by the Olivier estate.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..3a346a5d523c4995c232251e7349ba3c28b657df
--- /dev/null
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+{"text":"Real Jace Amaro Game Jersey On Sale,Trustworthy Online Store.\nJace Amaro Is Among The Players With All-round Skills Inside The History Of The Football. Jace Amaro Will Be The King In The Football Field.So,regardless Of Whether You Need To Become The Leader Around The NFL Ball Court Or Not?I Think What You Will Need Now Is Confident To Get A Jace Amaro Jersey. Since Authentic Nike Jace Amaro Jerseys Signify Perfection And Speed,you Happen To Be A Outstanding View On Court If You Play Football In Cheap Authentic New York Jets Jace Amaro Jersey. The Inexpensive Nike Jace Amaro Jerseys,with Which We Offer You,are All Stitched By Hand.And Every Single Detail Is Equal Towards The Jersey Which Floyd Has On Within The Football Field,for Instance,the Embroidering Font And Quantity(Jets Football Jerseys).Once You Have On The Jace Amaro New York Jets Jersey,you May Possess A Diverse Feeling.So Move Now! What Gift Do You Buy For The Buddy At Her Or His Birthday?Would You Nevertheless Make Attempts To Feel In The Concepts?Send Him Or Her A Jace Amaro Men's Jeresys Or Girls Jace Amaro Jeresy Jace Amaro Youth Jeresy,which Has To Be A Good Concept.I Realize That Your Friend Is Certain To Enjoy It.You Are Able To Possess A Attempt.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ravenscourt Park Preparatory school is a private school in Chiswick. The mosaic was commissioned to mark the school's 10th Anniversary as part of an 'Art & Technology Week'.\nThe school wanted every pupil to be involved, so the simplest work such as the border was carried out by the youngest children whilst the birds\/animals were created by year 6 children.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Persistent cookies Persistent cookies remain on your computer until a specified expiry date is met, or until manually deleted.\nSession cookies Session cookies remain on your computer only temporarily. Once you close your browser window, all session cookies will be deleted.\nNecessary cookies These cookies are required for the fundamental operation of the website.\nAnalytical cookies These cookies allow information on how the website is used to be recorded.\nASP.NET_SessionId Session Necessary This cookie facilitates the core website functionality. The website cannot function correctly without this cookie.\nlanguage Session Necessary This cookie stores your language preference.\n__utmz Persistent Analytical These are 3rd party Google Analytics cookies. They are used to collect analytical information on how the website is used.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There can be some improvements to the painting, and the carpets. I do have some photos of decor that can have a major impact to the hotel that will improve its business. The wifi needs some improvements The staff was excellent, great communication and level of service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is written for everyone, but especially with my United Methodist Church friends in mind.\nHaving spent many years of my adult life in the Methodist church and a great deal of time working with Methodist churches over the past 20 years, I have my attention on General Conference at the end of February 2019. The decisions made during this session are going to have an effect on all Methodist churches regardless of what is decided. There will be some people in all Methodist churches across America that will not be satisfied with the outcome of the decisions.\nRegardless of the outcome, on February 27th, the day-to-day mission and ministry of the thousands of United Methodist Churches around the world must continue and it takes financial resources to fuel those missions and ministries. So, what are you doing this month to prepare and what is your strategy after the vote?\nWill the outcome of the vote affect the strategy after the vote? Yes.\nShould you wait until then to begin planning? NO!\nTeach\/Preach on the Vision and Mission of the catholic church and more specifically your local church.\nWho has God called your church to be in your community?\nHow is your church making an impact in your community and around the world?\nThank people for their generosity to help fuel these life changing opportunities.\nUtilize the offering moment to celebrate generosity and to thank people for their giving.\nConnect the dots between financial generosity and life change.\nConsider having one on one conversations with your top givers.\nThis type of suggestion always raises eyebrows and causes people a little discomfort and indigestion because we do not want to show favoritism. I am not going to try and talk you into this in an email, but I would encourage you to consider this.\nYour top givers are some of the most invested people you have in your church. They care at a very high level and they will care at a very high level about the General Conference vote.\nIt would be in your best interests to know where they stand and how they might respond to the vote regardless of which way it goes.\nWhether we like it or not, the larger givers have a significant impact on the budget of a church. Whether they leave or stay at the church will have a financial impact.\nYou are not having these conversations to convince them that their opinion on the subject is right or wrong; as a leader you need to know where they stand.\nWhere they stand should not impact how you lead the church after the vote, but it definitely could help you financially plan for after the vote.\nIf you are in the middle of a giving phase for a capital campaign or One Fund initiative, talking to your major givers is even more paramount for the obvious reasons.\nIf they choose to leave your church that will impact your operating budget and the campaign\/generosity initiative.\nBeing prepared is one of the greatest characteristics of a great leader. You do not want to be caught off guard by how the results later this month might financially impact the mission and ministry of the church you lead.\nI hope this helps and challenges you to be proactive regarding how the results of General Conference might affect you and the finances of the church you are leading.\nI would welcome the opportunity to discuss this with you in real time. Let me hear from you. Call me at 404-435-5755 or email me at alan@generis.com and let's get a Zoom call on the calendar.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahxsc b/data_all_eng_slimpj/shuffled/split2/finalzzzahxsc
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+{"text":"You need to get the most value you can from your reading time. That's why every week I pick the one post from the web's multitudes that will help you get better. This week I'm pointing you to a post on how to write and sell a self-help book.\nWally's Comment: Business books are self-help books.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Why buy from Canada Cannabis Supplies?\nWe proudly serve all Canadians over 19 years old. Everything we sell is 100% safe and legal. We sell direct to consumers and supply dispensaries across Canada.\nTrusted source of Cannabis Accessories and Supplies in Canada. We've got everything you need to enjoy your marijuana.\nLarge selection of vaporizers, pipes, bongs, grinders, rolling papers and more.\nEnjoy Free Fast Shipping on all Orders over $99. Orders are shipped Expedited with Tracking Number.\nOrders are shipped discretely for your privacy in a non-labelled package.\nWe have a large selection of Cannabis Accessories and Supplies. Must be 19+ to buy. We have a strict policy of not selling to minors & require Photo ID to verify you're 19+. For privacy reasons, you can cover up everything except your name and date of birth.\nVaporizers, Pipes, Bongs, Grinders, Scales, Lighters, Rolling Papers and more!\n\"This is the greatest ! Love the deals and the great selection.. It is top quality, and packaged all so nice. Great customer service . Prompt delivery for sure... never had a problem... oh and love the free sample... will be back for sure\"\n\"I shopped around a bit before landing on Canada Cannabis Supplies. Their turnaround is incredibly quick, I ordered on a Sunday and by Tuesday morning it had been delivered. I accidentally had my first order shipped to my neighbours house and CCS was really good and quick to respond to me when I had concerns regarding that order. I definitely recommend CCS.\"\n\"Great service, super-fast delivery, great packaging, fantastic prices\/deals...... Can't say enough good things about my experiences with Canada Cannabis Supplies. They have earned my trust and loyalty! For that, I thank CCS :)\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hypertension has no symptoms but can can stroke or sudden heart attack.\nIt affects both men and women.However, a man's risk is greater compared to a pre- menopausal woman but; risk become the same after menopause.\nIt has no respect for social status or class.It kills both rich and poor people alike.\nThinking doesn't cause hypertension.Factors such as advanced age, family history and unlikely lifestyles put an individual at risk of developing it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The W\u00fcrzburg Womens Ministry meets every 4th Friday at the church. The ladies enjoy meeting together and studying the Word of God. Different topics are discussed and usually follow a series on a particular subject. If you are a lady looking for a group to fit in, this is the place for you.\nThe Velburg Womens Ministry meeting every 4th Saturday at different ladies homes. Please check the schedule and listing on our Facbook page to find out the details. The ladies meet for food and fellowship followed by a lesson that is encouraging and uplifting to women of all ages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"More of us than ever are eating healthier, cooking from scratch and consuming more fresh fruit and veg. The Nutrifresh range aims to make this even easier and fun for you and your family!\nThese products are suitable for freezer and microwave and are BPA free.\nUse with any Nutrifresh vacuum storage item.\nUse with the Nutrifresh vacuum pump.\nPerfect for cooking, storing, freezing, and reheating.\nThis compact blender is ideal for smoothies, protein shakes or any healthy drinks.\nSo easy to use. Simply fill and blend with just the press of a button. Easily change the blender blade to the drinking lid and enjoy your healthy drink on the go.\nThis product is sold separately and available at Grand March\u00e9 Stores only, no stickers required.\nHome cooking just got a lot easier! Check out these recipes that you can keep fresher for longer with Nutrifresh.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahycj b/data_all_eng_slimpj/shuffled/split2/finalzzzahycj
new file mode 100644
index 0000000000000000000000000000000000000000..15a979ac6d9c613a5b15e92a9b0b8fd7c675c3fb
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Description: Eurasian lynx are reddish-brown in summer and more grey in winter with a variable pattern of dark spots or lines. Their underbelly is white and they have long black tufts on their ears, a shaggy face, large paws and a short black-tipped tail. Taller and heavier than the Canada lynx, Eurasian lynx typically weigh 15-29kg (33-64 lbs).\nDistribution and Habitat: The natural range of the Eurasian lynx stretches across Europe, Russia, Siberia and into Central and Eastern Asia. Eurasian lynx are most often found in temperate mixed and boreal forests.\nDiet: Carnivores, Eurasian lynx hunt roe and red deer as well as wild pigs, birds, rabbits and other rodents. They can kill prey 3-4 times their own size.\nLife Cycle: These normally solitary animals will get together to mate from February to April. The gestation period lasts 67-74 days when the female will find a secluded den to give birth to 1-4 kittens. The young are weaned around 3 months but stay with their mothers until winter.\nAdaptations: Like the Canada lynx, Eurasian lynx have longer hind legs than front although the Eurasian lynx has far more stamina when chasing prey.\nDid you know? Eurasian lynx are the largest of the four lynx species and Europe's third largest predator (after brown bears and wolves).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Oct 21: The UN Climate Change Secretariat in Bonn, Germany has released the latest data on greenhouse gas (GHG) emissions from industrialized countries increased in 2007, which continued the upward trend of the previous six years. According to a release, data submitted to the United Nations Framework Convention on Climate Change (UNFCCC) show that greenhouse gas emissions of the 40 industrialized countries that have reporting obligations under the Convention rose by 1% from 2006 to 2007. The 2007 emissions of this group of countries are about 4% below 1990 levels. But there was overall a 3% growth in emissions in the period from 2000 to 2007.\nFor the smaller group of 37 industrialized countries that have targets under the Kyoto Protocol, emissions in 2007 were almost the same as in 2006 (+0.1%). The figure is around 16% below the 1990 Kyoto baseline for industrialized countries with targets. However, UNFCCC notes that much of the reduction comes from the economic decline of economies in transition (countries in eastern and central Europe) in the 1990s and since 2000, the emissions have also been growing for this group (+3%).\nYvo de Boer, Executive Secretary of the UNFCCC said, \"The continuing growth of emissions from industrialized countries remains worrying, despite the expectation of a momentary dip brought about by the global recession. So the numbers for 2007 underscore, once again, the urgent need to seal a comprehensive, fair and effective climate change deal in Copenhagen in December.\nThe UNFCCC secretariat has also released a report with summary data on transactions conducted in 2008 by industrialized countries in the course of implementing the market mechanisms under the Kyoto Protocol. According to the data, industrialized countries have 55 billion tonnes of Kyoto Protocol units in their accounts. Some of these units were already traded among countries in 2008, and trading is expected to increase significantly in the coming years.\nDe Boer said, \"It is exciting to see how the carbon market evolved in 2008. I am sure that such data will become more conclusive as we move closer to 2012, the final year of the first commitment period under the Kyoto Protocol, and will show how policies and measures of the countries that have ratified the Protocol bear fruit. However, the ultimate size of the carbon market and its effectiveness to reduce global emissions will depend critically on the level of ambition shown by industrialized countries in Copenhagen.\"\nAccess a release from UNFCCC (click here). Access the Annual compilation and accounting report for 37 Annex B Parties under the Kyoto Protocol (click here). Access the Addendum to the Annual compilation and accounting report for 37 Annex B Parties under the Kyoto Protocol (click here). Access the National greenhouse gas inventory data for the period 1990 - 2007, Annex I Parties (click here). Access a Q&A document on relating to the GHG data interface and data availability on the UNFCCC (click here).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Modern Dining Room Table Decorating Ideas Tags : Polishedcorner Dining Tablethat Will Change Your Life Courageouscorner Dining Set Most Belovedbest Kitchen Decorthat Will Make You Bang The Keyboard. 100 Ingredients In A Pictures For Kitchen Number 79 Is Impossible!.\nZoie Henry Dining Room Furniture, 2018-07-11 12:23:02. If you want to give an antique look to your room, then furniture sets with antique carvings can be bought. Different table tops such as marble tops, glass tops etc. are also very popular for dining sets. Based on your choice of table top, you can also match the upholstery and cushions of the chair to match with the table. Nowadays wrought iron sets with glass tops are also becoming very popular. They are best suited for low budget.\nAngelina Scott Dining Room Furniture, 2018-07-14 17:26:44. The specialty of modern dining room is that they build of different materials which can look elegant and serve best in durability as well. The best dining room furniture is the one which is made of wood. Wooden dining tables are most preferred and widely used by almost everyone. They are best in terms of looks, features, comfort and durability.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Earth-toned colors and a unique 3-Dimensional look breaks up your silhouette. Predator Brown Deception\u2122 takes stealth one step further, by making it hard to focus on your figure, therefore difficult to judge your exact distance \u2026 which is why Predator Brown Deception\u2122 is so deceiving! Predator Brown Deception\u2122 works great anywhere from the Northern woodlands to the Southern brush country.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Amazing mid century modern styled home in one of the most central, convenient locations in all of AZ. This home features a unique style that will give its new owner bragging rights of being an individual with style. Home features barn style doors, Quartz counters , new kitchen , custom made bamboo island, stainless steel appliances, updated bathrooms , new dual pane windows, new carpet in bedrooms, new high grade laminate throughout, new slate flooring in master bath, new paint and an oversized laundry room that also is perfect for an office or space for that artistic side you may want to explore by painting or craft making. Open floor plan with plenty of light. Large backyard. This owner clearly went out of their way to make a one of a kind home for this affordable price range.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaircf b/data_all_eng_slimpj/shuffled/split2/finalzzzaircf
new file mode 100644
index 0000000000000000000000000000000000000000..4dceab9053318a929c889791295712562246c375
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Maria C McCarthy's illustrated pamphlet, There are Boats on the Orchard, is reviewed by Neil Leadbetter in Write Out Loud.\na vacuum cleaner; bones; a shipping container; a swing.\nIt is telling that the very last entry that is mentioned in the inventory is trees.\nThere is also a sense of endings \u2013 of autumn, of shrivelled fruit as a metaphor for the demise of all the orchards, of trees being felled, of bulldozers, bonfires, planning notices and the burnt-out shell of a car.\nThis collection is a moving depiction of the changing face of our orchards, beautifully observed by a writer who cares deeply for the preservation of our natural world.\nYou are currently reading A sense of endings. There are Boats on the Orchard at Cultured Llama.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The iron steel pipe which connects the mains water supply from the water company to private households and other buildings is called the repair iron pipe in Sutton SM5. If your plumbing system desires a mend work it is best to take expert recommendations of a certified plumbing engineer. If you feel a bit luxurious and want to position a hot tub in your bathroom you can contact this experienced trained plumbing technician who will do it at the best market rate in Sutton SM5. If broken pipes have deprived you of the pleasures of electric heater ball ball in your shower, contact a experienced plumber in Sutton SM5.\nOur qualified experts not only do their job fast but also well, so grab a phone and contact one of our qualified plumbing professionals from Sutton SM5. If an unpleasant smell is coming out of one of the pipes at your home it is best to call for help of a professional qualified plumbing professional Any leaking upvc pipe or broken flush is not a quandary for our brilliant trained plumbing engineers in Sutton SM5, who are available 24 hours 7 days a week. How much does a qualified certified plumbing technician charge for draining pipes or other minor repairs in the kitchen and washroom in Sutton SM5? While setting a date with our qualified plumbing technician let him know whether your hiccup is urgent or if it can wait.\nIf you are looking at someone to open up congestion in pipes, alter the basin affix a modern tub or any other hydraulic service our establishment will present you with the top quality plumbing professionals it is in your interest find online in Sutton SM5.\nOur company uses the best experts, so if you are ever in need of a trained plumbing expert go ahead and call us. To get professional aid of a experienced plumber in Sutton SM5 who will place anything from a bathtub to a hot tub, you can call at any time of the day and night and make a reservation for one of our skilled handymen. If the pressure inside the pipes is too low the water's stream may possibly be too insufficient to use it in Sutton SM5. Our charge for affixing a siphon is the lowest you can find online on the internet and our plumbing professionals are reputable experts in Sutton SM5. This village doesn't have a mains water supply, so continues are forced to use water from wells in Sutton SM5.\nHow much does a qualified plumbing technician insured for electric boiler books charge for setting in place gas in the kitchen in Sutton SM5?\nIf you need to consult a certified plumbing professional in Sutton SM5, we are ready to help you 7 days a week. Our technicians knows how to unblock a butcher from the drains cleanly, quickly and cheaply with a one-year guarantee in Sutton SM5. Our establishment is open 24 hours a day 7 days a week, so if you have any questions concerning our services or workers, call and talk to one of our psychics lights in Sutton SM5.\nHow much does it cost to put right a leak in the critical plastic upvc pipe of the home by a trained plumbing company in Sutton SM5?\nIf your tap is not working or if there is a leaky drain at your house call a experienced plumbing engineer immediately.\nWhere can I find a plumbing company I can call at any time in case of an emergency in Sutton SM5 like a leaking copper pipe or The boiler flush in the water closet Water heaters provide electric heater ball ball for the household, so it's vital to choose the ultimate one for your requirements in Sutton SM5. To find a plumbing company putting forward our services Twenty-four hours 7 days a week, you can a diagnostic it online on the internet on the internet in Sutton SM5. Constant problems with electric heater ball ball supply can be very unpleasant so a qualified hydraulic company's put right is reasonable to assume that it is the finest resolution to the problem or concern in Sutton SM5. Water heaters for electric heater ball ball can be installed on your own or by a skilled plumbing professional who will do it faster and more safely in Sutton SM5.\nIf you are in need of an expert in plumbing who'll unblock a butcher from the pipes then you've come to the right put in place in position in Sutton SM5.\nTo mend a leak, it is best to find a plumbing technician on the internet who offers such services for a relatively low valuation in Sutton SM5. Free a stoppage from the pipes, fix a basin set in place in position a shower, tub, dishwasher, washing machine are just some of the services we offer in Sutton SM5. If the mains water supply pvc pipe breaks, you can call our plumbing company in Sutton SM5, and we will service it in no time. To remodel your kitchen or bathroom it is best to ask for help a plumbing engineer guaranteed in plumbing design. Water heaters for hot water can not be positioned on your own; please call a insured qualified plumbing engineer who will do it safely in Sutton SM5.\nIf you have already tried everything to unblock pipes in your kitchen or bathroom and nothing has worked, call our trained plumbing professional from Sutton SM5, and he will aid you in no time.\nA skilled plumbing technician will diagnose the predicament with the steel pipe and then proceed to mend it for a reasonable valuation and without a headache in Sutton SM5.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As our list of Eagles Solo Albums Ranked Worst to Best shows, it was easier for some members of the legendary country-rock group to go their own way than it was for others.\nDon Henley and Glenn Frey got off to relatively quick starts, though Frey's music career cooled faster as he began focusing on other career goals. Joe Walsh arrived with a solo resume of his own, having had hits with Barnstorm and, to a lesser degree, the James Gang.\nBut Randy Meisner struggled from the first, releasing three largely overlooked albums during the first few years after his departure. Like fellow co-founding Eagles member Bernie Leadon, Meisner ended up working primarily as a sideman. Don Felder has put out only two records on his own. Then there was Timothy B. Schmit, perhaps the most snakebit of them all.\nSchmit joined Poco in 1969, replacing Meisner for the first of two times, but that critically acclaimed band remained largely ignored. Schmit then took over for Meisner in the Eagles seven years later, just as the band was coming off a tour in support of the career-making Hotel California album. It took a year and half for the Eagles to complete The Long Run, then they promptly broke up. Schmit didn't get his solo career on track for decades.\nAlong the way, each has put out intriguing albums that recalled and, in some cases, expanded upon their time together. The focus here, however, is on rock or country-rock projects, since that's where the Eagles' Rock and Roll Hall of Fame legacy lies. In keeping, we skipped Henley's deep-country themed Cass County and Frey's standards album After Hours.\nWe're also leaving aside live projects like Meisner's Dallas and Walsh's You Can't Argue With a Sick Mind, and odds-and-ends collections, including Meisner's Love Me or Leave Me Alone. Also left off are sideman projects \u2013 as great as they are \u2013 when Leadon sat in with Dillard and Clark and the Flying Burrito Brothers, since he wasn't a frontman on those albums.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join us using diffusion paper (coffee filter) and markers create an unique one of a kind design to use building several 3\" Charlie Sotich Mini Kites.\nWarning: this workshop promotes happiness and a sense of well being. Caution: Use of this technique is habit forming.\nSupplies\/Equipment Needed for class -- Scissors and tweezers.\nKIT ORDER CUT OFF: None, you can drop in Friday Night!\nBut....it would be nice to let Scott know how many is interested! Please sign up at Registration!\nCOLOR CHOICE: All colors available--you will color!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The fourth generation of the Volkswagen Polo (Typ 9N) was a B-Segment car launched in 2001, based on the Volkswagen Group A04 (PQ24) platform (like the Seat Ibiza and Skoda Fabia). The car received a facelift in 2005 with visual changes and various mechanical updates.\nThe performance version, VW Polo IV GTI, launched in 2005, based on the facelifted model. It was fitted with a turbocharged 1.8 liter engine with an intercooler, that produced 150 hp (110 kw) of power and 220 Nm of torque, comvined with a 5 speed manual gearbox.. The car accelerated from 0-100 km\/h in 8,2 seconds and could reach a top speed of 216 km\/h. It featured a sport chassis setting with standard ESP, 15 mm lowered suspension and stiffer springs. Visually, the car was distinguished by a sporty bodykit that included a deep front bumper with a black V-shaped grille with a red stripe and GTI badging (resembling the Golf V GTI), headlights with black background, body coloured plastics, black side sills, 16 inch alloy wheels with U shaped openings, roof spoiler and chrome twin exhaust pipes. Inside, the car featured sports steering wheel, gearshift knob and handbrake grip upholstered in leather, metal-like details and aluminium pedals.\nThe VW Polo IV GTI Cup Edition was a more powerful and track focused version of the model, producing 180 hp (130 kw) of power and 235 Nm of torque. Thanks to the power upgrades the car accelerated from 0-100 km\/h in 7,5 seconds and could reach a top speed of 228 km\/h (140 mph). The car featured uprated brakes with red calipers and stiffer suspension. Visually, the Cup Edition was distinguished by the deeper front bumper with considerably larger air intakes and a different grille, the redesigned rear bumper with additional inlets, the multi-spoke 17 inch wheels and the larger roof spoiler that extended to the sides.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajcca b/data_all_eng_slimpj/shuffled/split2/finalzzzajcca
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index 0000000000000000000000000000000000000000..67cff7a73a73a7e3d28f197f01e30a40030cdcd5
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@@ -0,0 +1,5 @@
+{"text":"Though David Pogue's first Mac OS X Missing Manual book was my Bible for a few years, I take for granted that the iPad, as well the iPhone and Mac computers and software are easy and intuitive to use.\nSo where am I going with all this? Well, if you're new to the iPad and similar devices, iPad Made Simple by Martin Trautschold and Gary Mazo (Apress Press, $29.99) will explain to you, the end user, everything you need to know, and then some, about Apple's new computer tablet.\nWhen I received a review copy of iPad Made Simple, I was a little taken aback. It's a 705 page tome, much larger and arguably more daunting than the iPad itself. But as I thumbed through it and checked out the content, I could see that it has got to be the most comprehensive book on the subject on the market today. Sure, there are smaller pocket size books about the iPad and those are useful too, but I think iPad Made Simple will appeal to Mac newbies needing to know how to get started with the iPad and other related software.\nThe book covers everything from connecting your new iPad to iTunes, to using the on-screen keyboard, understanding and setting up a Wi-Fi connection, playing music in the iPod app, viewing videos, connecting Bluetooth devices, setting up email, using the Maps app, sampling default and third-party apps, and learning the basics of the iTunes application itself.\nThe size of the book may be a bit overwhelming for some users, but it's not a book that you will read from cover-to-cover. The authors and the publishers took special care to use very detailed screen shots (on every page) and step-by-step instructions for that can be fairly understood without having to read all the accompanying text.\niPad Made Simple is the kind of reference book users can dip into to learn something new and try it out immediately on their iPad. It opens with a Quick Start Guide for those needing a quick overview of getting set up. From there, I think most readers will thumb through and learn things as needed.\nThe book also covers iPad accessories including the wireless keyboard dock, the Camera Connection kit, and VGA Adapter Cable. It also introduces many of the popular iPad related apps, such the iWork suite (Pages, Numbers, Keynote), iBook, GoodReader, Evernote, and of course all the default apps that come with the device.\nI wouldn't recommend the book to experienced iPhone users, but for Apple newbies, it's a very useful guide if you want to get the most out of the device. There's an eBook version of the app the can be download and actually read on the iPad itself.\nIf you have been a long time Mac and iPhone user you may have had an idea for an application you wished someone would build. But unfortunately, developing software is not as easy as dragging and dropping media into say iDVD and producing a nearly professional DVD with titles, chapters, and background music. Learning the language of Objective-C programing is not the same thing as even producing your own dashboard widget\u2014remember those?\nIf you know nothing about UITableViewController class, switch view interfaces, objects and properties, IBActions and IBOutlets, then I can nearly guarantee that Dr. Rory Lewis's iPhone and iPad Apps for Absolute Beginners (Apress, $29.99) is probably the best book for getting you started with mobile app development.\nThis book peaked my interest because I'm one of the absolute beginners that the author is addressing. I have dabbled a little in CSS and HTML languages, produced some AppleScript and Automator workflows, but none of that compares with learning Objective-C and Cocoa programming\u2014the basis of iPhone and iPad applications.\nI've tried a few other iPhone app development books and while the authors maintain their book is for novices, I didn't find that the case. Sure I could work through a simple step-by-step \"Hello World\" program, but by the second chapter, most authors are off and driving fast, telling how to drive a car and explaining how the engine works all at the same time. One lesson I tried literally spent three chapters explaining Objective-C programming, with like one sample lesson, and then lead off the fourth chapter with, \"Now that you understand Objective-C...\" There's no way you can learn Objective-C or app development by simply reading about it.\nYou have to get your hands\u2014okay, your mind\u2014dirty with writing code. For the most part, Dr. Lewis has the ability to make app development manageably understandable. He takes nothing for granted when it comes to what his readers need to know, and he doesn't pretend that just because he spent an entire chapter explaining something that you will understand it.\nNow, to be honest, what makes this book stand apart, for me anyway, is very simple. Dr. Lewis has produced instructional videos that nearly mimic the instructions in the book. These videos mirror ones he developed as part of the classes he teaches on iPad and iPhone development. In many cases, without the instructional videos I would have been completely lost. There are times in the book though, when Dr. Lewis tries to give step-by-step instructions while also explaining the why of those instructions. That makes it very difficult to follow along if you're an absolute beginner.\nSo my approach has been to read parts of a chapter and follow along with the videos. Then go back and re-read the lessons that I successfully and unsuccessfully completed.\nBut I must admit, and say again, software development is not an iDVD production. It takes time and focus. Leaving out a semicolon, a word or two of code can mean all kinds of errors when you try to compile and run your application. It's very frustrating to spend twenty minutes typing out code only to find that you missed something along the way. The book's accompanying web page does include the source for you to compare your own work with. If you get stuck, I recommend opening the source code in say TextEdit and printing it out. Then go back over your own code using the source as a guide. If you still get stuck, just copy and paste the code and run it. You don't want your small errors to prevent you from understanding the concepts.\nSo coding is for those with patience and a determination to learn. However, I think, having worked through about two-thirds of the book, I can tell you that iPhone app development is not rocket science. It is indeed possible to produce many of the types of basic apps uploaded to the app store on a daily basis. I think this type of programing could be learned without taking a class, though a course on the subject would probably get you much further.\nBy the end of chapter of six, Dr. Lewis highly recommends re-doing the lessons of each chapter, like 10 to 15 times. I couldn't concur more.\nWhat's great about this book is that Dr. Lewis revisits a few basic concepts in each lesson. So by the first nine lessons, you get a working understanding of some basic tools of programming, because you have done them more than once. But then there's the leap you must gain by understanding how things are running under the hood, so to speak. You must understand how the code makes things happen, which is the most difficult part.\nAnd by the way, the book instructs you to download the current version of iPhone SDK, which will cost $99. But you can use the X-Code application, Interface Builder, and the iPhone simulator that comes installed on the Mac OS X Snow Leopard DVD. So far, I have successfully used updated versions of these programs to complete lessons in this book. When I think I'm ready to learn more and produce my own app, I'll pay for the iPhone SDK.\niPhone and iPad Apps for Absolute Beginners will probably not be the only book you'll need to produce your own iPhone and iPad apps, but it is the book to help you get started.\nNice write-up. I think a lot of times us geeks forget that the vast majority of the gadget-buying population could care less about the specs, they just want the thing to work and understand how to use it to its fullest.\nInterestingly enough, I think a lot of magazines miss that important point. Gone are the tutorials and projects that used to occupy the majority of the space in magazines like MacWorld, and more often the majority of pages are devoted to regurgitation of industry news and reviews of equipment. That's great and all, it's just that the Web is chock full of this kind of information - it smacks you in the face at every turn.\nAnd speaking of MacWorld, does anyone else notice that the bigger and more successful Apple is, the less pages it seems to contain. Seems like it exists merely to stuff the rare tidbit of useful information between ads.\nInteresting. Like you I'm a newbie when it comes to coding. I've also worked with css and html - plus a little php given my love for wordpress based websites. I've thought of developing apps, but felt it was beyond my novice mind's ability to handle. The book sounds interesting and worth checking out.\nIt is so hard to find real information on the blog. eden pure heater reviews really enjoyed reading this post. It has strengthened my faith more. You all do a great job in such a concepts.can t 'tell you how much I appreciate for all you do.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nous sommes sp\u00e9cialis\u00e9s Lifepo4 12v 20ah fabricants et fournisseurs \/ usine de la Chine. Vente en gros Lifepo4 12v 20ah avec des produits de haute qualit\u00e9 comme low price \/ cheap, l'un des Lifepo4 12v 20ah} grandes marques chinoises, Shenzhen Grenergy Technology Co.,Ltd.\nWholesale Lifepo4 12v 20ah from China, Need to find cheap Lifepo4 12v 20ah as low price but leading manufacturers. Just find high-quality brands on Lifepo4 12v 20ah produce factory, You can also feedback about what you want, start saving and explore our Lifepo4 12v 20ah, We'll reply you in fastest.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are so many delicious Chinese dishes, and a large percentage of those have meat in them. This article will discuss how to talk about meat in Chinese, and about the culture of meat in China.\nIn general, talking about meat is very easy in Chinese. You just say the name of the animal + \u8089 (rou4).\nWith fish, often the \u8089 part is optional.\nBesides knowing the names of animals, it is useful to know different parts animals' bodies like legs, arms, feet, back, etc. Even knowing about organs like liver, brain, etc is useful. This is because Chinese people generally will eat almost all parts of animals, even the blood. Coagulated duck blood is a delicacy in Southeastern China including Nanjing. The first time I ever had chicken feet was in Taiwan, and although I was quite apprehensive, my friend pushed me on, and I found out first hand that chicken feet are actually somewhat tasty. There are even some markets in China selling exotic animal meat, from starfish to live scorpions, in places like the famous Wangfujing street in Beijing.\nIn China, in large meat markets or in meat departments of large grocery stores, you can find so many types of meat. One time, in a large outdoor meat market, I even saw a live alligator-like creature. It seems that Chinese people don't have as much affection towards animals, at least compared to many Western countries. Even horse is eaten in Western China.\nBecause many Chinese people prefer their meat as fresh as possible, most meat markets will kill live fish right in front of you and sometimes chicken and other animals too.\nWhen I am in China, I am quite vigilant about knowing what is actually in my food. So if I don't know what is in a dish, I always ask. This is in part because I don't eat any seafood, but also because there are some animals I refuse to eat under any conditions. One of them is dog, due to having had dogs for most of my life. It is true that some Chinese people eat dog meat, but it is much rarer than other meats.\nIf a food item has meat, but it isn't indicated which meat is is, usually, by default, it will be pork.\nIt is possible to be a vegetarian in China, but at times, it will be somewhat difficult. To be 100% sure of having no meat, you will need to buy your own vegetables, and cook your own food. Though, there are some restaurants, like some Buddhist restaurants, where you can you can be fairly confident that there will be no meat in the food.\nChina seems to have different ideas about meat and vegetarianism than Western countries. Even in supposedly vegetarian food at restaurants, they might sprinkle on bits of pork, or have broth from meat for a soup. One time, I went to a dinner with my coworkers, one of whom is a strict vegetarian. Nearly all of the dishes had meat in it, so he wouldn't eat from any of them. Eventually, the waiters thought that bringing out a fish dish would be fine, because maybe they thought that fish isn't meat. But, later they brought out a dish with only vegetables.\nNice article. I remember the first time I ordered something in China. I tried to order pork and instead got ice cold chicken feet! Still ate it though.\nThanks. I've had similar experiences in China.\nGood to know! We have always wanted to travel to China!\nThanks. Chinese food in China is definitely worth trying.\nInformative article. I found the part about vegetarianism and the difficulties ordering food in a Restaurant in China particularly interesting. I'll keep these tips in mind during my travels.\nThanks. I admit being a vegetarian in China can be difficult, but it's not impossible. You just need to be proactive.\nThis is so helpful, I'd probably be totally lost without this info! I'll have to hang on to this post for my future trip to Shanghai!\nThanks. Shanghai has lots of delicious foods to try out. Enjoy your future trip to Shanghai!\nactually, when I started to read your article I was a bit bummed because I really steer clear of anything meat related, and as I was reading, when i got to the part about vegan\/vegetarian, I was so happy!\nI already knew that eating in china is a bit tricky for vegans\/ vegetarians (although there are a lot of amazing vegetable side dishes), but Buddhist restaurants I had never even thought of! It does make sense for vegetarians to kind of seek out those places, usually most buddhists don't eat meat!\nLoved reading your thoughts, I'll be sure to keeping coming back for more!\nI had no idea there were so many different kinds of meat in China. This was eye opening.\nYes. It definitely was for me. If you ever visit Guangdong province. you surely would see a lot on this front. And since China is a massive country with a massive population, there are SO many different kinds of cuisine to try out. It's worth trying out.\nThanks. Good luck on your China journey!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1. God takes pleasure in the prosperity of His servants who favor His righteous cause.\n\"For the love of money is the root of all kinds of evil.\" (1Tim 6:10) Money is not the root of all evil. Rather, it's the love of money or a wrong relationship with financial and material things. You can have this even when you have no money. Money takes on the character and the nature of the person whose hands it is in. When it's in the hands of a righteous person, it will be used for righteous purposes. When it's in the hands of an evil person it will be used for evil. (Matt 6:33) \"Seek first the kingdom of God and His righteousness and all these things will be added unto you.\" If you do this, you will inherit everything and miss out on nothing.\ni. Making His work your priority.\nii. Using what He has blessed you with to further His work.\nIn Luke 5:3-7, Peter let Jesus use his fishing boat as a pulpit from which to preach to the multitude. Then the miracle of increase and multiplication took place. Peter knew how to fish, but Jesus knew where the fish were biting. He knows how to grow your business when you link it to His business!\nA young lad was willing to give his small lunch to Jesus. (John 6:8-14) This young man's sacrifice released the life and miracle working power of Christ and five thousand hungry men were fed. The principle we learn here, is that your little becomes much when it's placed in the Master's hands.\n(Rom 12:1) \"Present your bodies a living sacrifice, holy, acceptable to God, which is your reasonable service.\" It's not unreasonable for Christ to ask believers to sacrifice something for the gospel's sake!\nThe first church I ever planted grew to be one of the largest churches of its kind in Canada. Then the Lord spoke to my heart in 1988 and told me to release it to my associate pastor and start Victory Churches of Canada. This movement is built on the sacrifices of a lot of people. What is built on sacrifice the Devil cannot destroy!! Read Psalm 50:5.\nThe first time I ever released a key leader to plant another church, the Lord spoke to my heart from this passage of scripture and said, \"For everyone you release I will give you two of like quality and character.\" When I did my part, God did His part and multiplied our leadership base. (John 12:26) \"If anyone serves Me, let him follow Me, and where I am, there My servant will also be. If anyone serves Me, him will My Father honor.\"\nMay you live in God's favor as you favor God's righteous cause!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Meet Mr. Design. First name: Web. Last name: Design. We'll call him Mr. Design for short. He's a new salesman on your team, and he's the first point of contact that potential customers are going to have with your business.\nThroughout this guide, we'll go through several different job duties of Mr. Design and how they affect your business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Four Pages To Go Happy Birthday Greg!\nI heard from somebody yesterday who was headed south in the middle of the bad storms that are chewing up the Carolinas. He said it was raining so hard that he could not see his friend's motorhome, less than 100 feet ahead of him, but they were making good time in spite of the downpour.\nWhy? Where in the world does anybody have to be that they would drive in that kind of weather? I asked him if there was a death in the family or something, and he said no, they were just headed to their winter place in Florida. Really? Like it's not going to be there two or three days from now if you park somewhere safe and wait out the storm? And if it's not going to be there, you damn sure don't want to be there either!\nA couple of times over the years we have found ourselves on the road in high winds or other bad conditions. But as soon as we could get to someplace safe we did, and stayed put until things cleared up. I've never understood why people in RVs will put themselves in danger to keep to their \"schedule.\" You've got a bed, cooking facilities, and a bathroom with you. What more do you need?\nI can't tell you how many times we've been in campgrounds when it was pouring rain or the wind was blowing enough to make it dangerous to be out on the road, and still see people pulling out. And these were not people who had to get home to go to work at the end of the weekend, they were fulltimers and snowbirds. What can be so great at the next place that you will put yourself and others on the road in danger just to get there? I don't get it.\nPeople keep asking me if we are going to have another Gypsy Journal rally anytime soon. We don't have any plans for it at this time. While the rallies were a lot of fun, they were also a tremendous amount of work, and rising costs made it impossible to continue. A few people suggested that we just have one of the big RV companies sponsor the rally. It doesn't work that way. Most RV companies have stopped sponsoring their own rallies for owners of their rigs, so I doubt very much they're going to be interested in footing the bill for something we're doing.\nAnother question I get a lot is what happened to Bad Nick? Between the Gypsy Journal, my daily blog, and writing my books, there just hasn't been any time to let the ornery little munchkin out of his box. But I have a feeling we have not seen the last of Bad Nick. He keeps knocking on the inside of my skull wanting out. There is so much to talk about, but so little time.\nToday is your last chance to enter our latest Free Drawing. This week's prize is an audiobook of my friend Ken Rossignol's Pirate Trials: Famous Murderous Pirates Book Series: THE LIVES AND ADVENTURES of FAMOUS and SUNDRY PIRATES. To enter, all you have to do is click on this Free Drawing link or the tab at the top of this page and enter your name in the comments section at the bottom of that page (not this one). Only one entry per person per drawing please, and you must enter with your real name. To prevent spam or multiple entries, the names of cartoon or movie characters are not allowed. The winner will be drawn this evening.\nThought For The Day \u2013 Just because some people are fueled by drama doesn't mean that you have to attend the performance.\nYeah, I don't get it either Nick. A while back we saw some folks hooking up their travel trailer it what I describe as tornado force winds. It was blowing so had I had pulled our slides in part-way to prevent damage to the awnings. I asked them if they were moving to a new site. No, they were headed on down the road. They said they had to get to their next stop because they had reservations to take a tour the next day. What? We stayed right where we were until later in the day.\nHey, when hitch itch strikes, you scratch it! Don't be a weather wuss! ?\njust saying you cannot fix stupid. Like everyone else I cannot understand endangering not only your own family but some other innocent motorist and their families.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Blending realistic imagery with an urban graffiti style is one of the signifying characteristics of Conor Harrington's work. Bold colors and heavy brushstrokes go right alongside period garments and antique furniture. Bringing together art history, political history, and contemporary aesthetics marks Harrington's creations with a unique appearance.\nMusic chosen to accompany Harrington's work. As chosen by Void & Form.\nThe piece titled \"Fight Club (The Mess We Make)\", which is pictured above, portrays two men engaged in sword fight wearing the clothing dominant in the Colonial Era and painted in a style somewhat reminiscent of the French Romantic painter, Th\u00e9odore G\u00e9ricault. The scene of the two men in itself sounds traditional enough, but add in the checkered floor, highly abstracted palette-knifed background, and intentional blurring of one of the figures and you've got something entirely different. Oh, and let's add a Francis Bacon-styled hanging pig to the mix as well.\nIn another painting titled \"Modern Monarchy\" Harrington touches on socio-political themes. Here he portrays the figure of a well-dressed black man in Colonial French garb poised for a portrait but with an outstretched arm. Many questions come to mind when viewing this piece. What is the man holding in his outstretched hand? Is he a slave or free? Or maybe he's a contemporary in costume? Judging by the title, the latter seems a probable interpretation. It appears Harrington is making a statement of revolution \u2014 the type remains uncertain.\nConor Harrington \u2013 Outsider or Not?\nThe complex themes Harrington purposefully adds to his work are certainly thought provoking and demand viewer attention. Perhaps this is one reason his works have done well in sales. In addition to mid-sized paintings on canvas, Harrington creates large outdoor murals as well as limited edition prints.\nThis combination plus his choice of colors and style seem to put Harrington into the fold of Outsider Art. On the subject of Outsider Art, what exactly does that refer to? In general, Outsider Art refers to that curiously titled genre with no particular visual connection but rather marked by a supposed rejection from traditional galleries and\/or collectors. That said, what is its opposite? Well, that is where the conundrum begins. Because the definition of Outsider Art remains quite vague, it is hard to explain what it is not. So, is Conor Harrington an Outsider Artist? One cannot be sure, except that he has self-aligned with The Outsiders, the well known gallery and online print shop based out of the UK which boasts of specializing in the art of under recognized and underground artists.\nUltimately, any given genre title may or may not apply to Harrington, but one thing is for sure \u2014 he creates powerful imagery that alludes to time and circumstance with a keen sense of composition and design.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Australia overwhelmed New Zealand to win the World Cup for a fifth time at an ecstatic Melbourne Cricket Ground.\nNew Zealand lost influential captain Brendon McCullum to the fifth ball of the match and were bowled out for 183.\nGrant Elliott resisted with 83, while Mitchell Starc, Mitchell Johnson and James Faulkner shared eight wickets.\nAustralia rarely looked troubled, sealing a seven-wicket win in 33.1 overs, with captain Michael Clarke scoring 74 and Steve Smith 56 not out.\nClarke, in his final one-day international, was given a standing ovation by the record 93,000 crowd and welcomed by his team-mates on the boundary when he was bowled by Matt Henry with eight required.\nHe was part of the Australia team that last lifted the trophy in 2007, with this success extending their record for most World Cups won. No other team has more than two.\nAustralia's win was the second in as many tournaments by a host nation after India's triumph four years ago.\nIt provided a joyous end to an emotional Australian summer which saw batsman Phillip Hughes die after being struck on the neck by a ball in Sydney in November.\nIn their first final, after six previous semi-final defeats, New Zealand were blown away.\nTheir path to Melbourne came with eight successive wins, all on home soil, and most of the pre-match speculation was of how McCullum's men would deal with the change in conditions.\nThey won what was thought to be a crucial toss, but the life was sucked from their innings almost as soon as it began.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Original Boxed Unison Research Unico Primo Phono MM MC integrated amplifier. Complete with original remote control and manual. This is a hybrid 80W integrated amplifier with a valve pre amp side and a solid state power amp side. This amplifier retails for \u00a31749.99 but we've got it for sale here used in excellent condition for \u00a3899.99 We can ship worldwide. The unit is currently configured for 240V UK mains. Pics coming soon. Please ask for a shipping price as the shopping cart will probably not quote you the correct price!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brand New - Milwaukee MA911B\/2 Double Junction pH Electrode with 2 Meter Cable, BNC Connector, and gel filled electrolyte solution.\nNote: We ship to the USA and Canada only.\nM10004B pH solutions for any pH monitors or testers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Get great seats for your favorite NFL or college football team.\nGet great deals on pre-season and in-season NBA games.\nFind NFL, NBA, MLB, Concert Tickets and More!\nShop Alliance Tickets and Pay the Listed Price on all Tickets!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A statistical study of physical pairs from IDS catalogue has been performed. For 2811 binaries with luminosity class V primaries it has been found that their mass function is a Salpeter one; their distribution over semi-major axes of orbits is dN \u221d d loga. The rate of duplicity of stars is close to 1.\nPresented by: L. Yungelson. A similar paper has been published in Russian, under the title: 'Statistical Investigation of Visual Binaries in IDS Catalogue', inScientific Informations 63, 1987, pp. 3\u201318.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Save 30% off site wide with code CLEANfriday now through November 25th!\nWe have gift sets \ud83c\udf81 for everyone on your list! Ranging in price from $17.99 to $64.99, you are sure to find the perfect gift no matter what your budget.\nLimited Edition Holiday Hand Soaps are back!\nJust wanting to TREAT yourself?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We don't often get snow this close to home, so we made most of the opportunity.\nSnow at Mount Tomah Botanic Gardens.\nTemporary staff at Mount Tomah Botanic Gardens.\nSnow at St George's Anglican Church, Mount Wilson.\nSnow at Mt Wilson NSW.\nSnow at our picnic spot, Mt Wilson NSW.\nSnow at Merry Garth, Mount Wilson, NSW.\nBrrrrrrr. Snow at Merry Garth, Mount Wilson, NSW.\nTrees at Merry Garth, Mount Wilson.\nSnow Drops in the snow flakes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Oslo is the capital of Norway and has a strong economy offering multiple job opportunities for the skilled workforces in diverse sectors. The competitive market has created a need for aspirants to take up a professional course to stay relevant.\nKnowledgeHut offers the ICP Agile Certified Coach training in Oslo for aspirants interested to improve their chances in the job market. The cost to register for the coaching at the academy is reasonable and you get to attend the ICP Agile Certified Coach courses in Oslo that are conducted by the best tutor.\nThe ICP Agile Certified Coach online in Oslo teaches the roles, the mindset and the responsibilities of the agile coach. The ICP ACC certification in Oslo is a workshop that allows the candidates delve into an interactive learning method with practice sessions and lectures. The ICP Agile Certified coach training in Oslo brings to the table practical scenarios and trains the coaches to utilize the right tools and skills to address organizational challenges.\nThe ICP ACC classes in Oslo teaches the underlying competencies that are required to be a successful agile coach. The basics of how to conduct a team and ensure delivery and organizational outcomes are taught once you enrol at the institute.\nCandidates are also privy to comprehensive course content that can be downloaded and 100 days of free access to e-learning modules. You also receive certification once you complete it.\nKnowledgeHut offers ICP ACC certification in Oslo with coaching from an experienced trainer who delivers expert content based on industry experience over a one day workshop and through a virtual module allowing for better exposure. You can opt for a demo at the start of the course to understand the process.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..9af40ea4e0e911119905026215fa3242f91ecae9
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@@ -0,0 +1,5 @@
+{"text":"\u200bI am running for District 8 County Supervisor because the South Shore deserves a strong advocate on the County Board. The communities of St. Francis, South Milwaukee, Oak Creek and Cudahy are fantastic, with beautiful 'main street' downtowns, friendly people and incredible parks. The 8th District deserves a supervisor who will promote the region while working collaboratively with the County Executive and the rest of the County Board to solve the important issues we are facing. My experience in the nonprofit and legal sector, as well as the time I've spent working with many of the wonderful staff at the County uniquely positions me to be that advocate.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"KRC (Kent) Ltd are a council approved Kent based UK waste management company providing scrap and dross removal services to the customers and councils throughout Penshurst, Surrey and Sussex.\nRubbish Clearance Limited (KRC) can collect, clear up and take away your lawnmowers, carpets, computers, pianos or photocopiers legally using local landfill sites, wherever you are in Penshurst.\nCall us right now on 0800 138 9100 about home clearance services or fill in our booking form. Call us for a free quote or do you need help estimating the load size? Click here for general prices. KRC cover all of Penshurst and Kent.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kids, it's the unnecessary words that make a story come to life.\nThe things that feel necessary to you but are unnecessary to other people: that's style. (Tim said that, and I think he's right.) That doesn't mean that you should slather on adverbs. It means that you're the one who sees from your eyes, and that has repercussions in every inch of the story.\nBecause this stuff is extra. It really is. You leave it out of the synopsis for a reason. Because if you put in your synopsis, \"Dear Editor and\/or Agent: This book is about how much I love my city, or my mother, or that color the sky turns when the sun is gone but it's not quite night yet,\" it doesn't help them know whether you've pulled it off, and it makes them suspect you didn't do the other bits. So you have to say the necessary bits, the \"This movie is about four men who love each other very much even though one of them is a jerk and they just met another one, and they make slimy ghosts go away and have witty banter\" part.\nBut if you didn't bring the part that didn't look necessary, no one's going to care.\nIf every part of the story is a part where you could have handed someone the plot synopsis and they'd do it the same as you, well, let 'em try.\nPeople phrase this as \"tell the story only you can tell,\" but then they go on to talk about there only being [2, 3, 4, N] plots in the world. It's not the necessary parts that are going to be yours alone. It's the stuff that seems like it could get filed off and it wouldn't matter. You write with the messy stuff that seems like it's optional. The whole thing is optional. Except when it turns out it isn't. Except when it's a darn good thing it isn't.\n*The Lake Harriet Rose Garden, most recently. I know, I was as surprised as you are. Well, maybe not quite as surprised, if you're not from here and don't even know we have a Lake Harriet Rose Garden. But still pretty surprised.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"> The following list covers which files have failed the integrity check.\n> Repeated the install and got the same error.\n> I haven't done much testing, but the script seems to be \"working\".\nFixed in r440074, sorry for the breakage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Need help? You've come to the right spot. We've conveniently located frequently asked questions in one location. Please choose a topic to get answers quickly. Can't find what you're looking for? Contact us and we'll do our best to assist you.\nQ: Where can I find my account number? I don't have my bill.\nA: If you don't have your bill handy, there are a couple of ways to access your account number. You can call us at 813-223-0800 and our automated phone system will speak your account number to you, or if you are an e-Bill paperless customer, you can check the email that we send you each month. Your account number is included at the top of the email.\nQ: Where can I view my bill online?\nA: To view your bill, click on Bills and Payments and click on View Bill. If you have more than one account, you'll need to choose one from the list before you can view.\nQ: How can I pay my bill online?\nA: To pay your bill online, click on Bills and Payments and click on Pay Now. If you have more than one account, you'll need to choose one from the list before you can pay.\nA: You can make a one-time payment online using a checking or savings account. Same-day payments must be received by 6 p.m. You can make a one-time payment with a credit or debit card using the Quick Pay service online or by phone. Please note, a convenience fee will be charged by our payment vendor if you pay with a credit or debit card. You can also schedule recurring payments from a checking or savings account by enrolling in Direct Debit and your account will be automatically be drafted on the due date. You can mail your payment or make a payment in person at a payment agent. View all your payment options from the Bills and Payment page.\nQ: Can I arrange for more than one bank account?\nA: When you enroll in Direct Debit, you can add several bank accounts to your profile. Only one account can be selected for recurring payments.\nQ: I'm setting up a payment method. Where can I find my bank's routing transit number and my checking account number?\nA: The routing transit number can be found in the lower left side of the check. The checking account number is to the right of the routing number. Tampa Electric accepts banking information from U.S. banks only.\nQ: Can I choose my payment or draft date?\nA: If you are enrolled in Direct Debit for recurring payments, your account will be automatically drafted on the due date. You may schedule payments using the one-time payment option on the Payment page.\nQ: How long will it take for my account to reflect my payment?\nA: Your account will reflect your payment on the next business day. Same-day payments must be received by 6 p.m.\nQ: Can I pay with a credit or debit card?\nA: You can make a one-time payment with your credit card using Quick Pay online or by calling 1-866-689-6469. A convenience fee will be charged by our payment vendor to your credit or debit card and the charge will appear on your statement as TECO\/Peoples Gas. Please note, your payment will reflect on your Tampa Electric and or Peoples Gas account within 2 business days.\nQ: What is Budget Billing?\nA: Our Budget Billing program for residential customers takes the highs and lows out of monthly electric and\/or natural gas bills. This \"leveling\" billing program averages your last 12 monthly billing periods so you can pay about the same amount for your electric and\/or natural gas service each month. You can enroll in this free program on the Your Account page.\nQ: Can I view my billing and payment history?\nA: You can access your billing and payment history from the Bills and Payments page.\nQ: Can I set up automatic recurring payments?\nA: Direct Debit is a free, fast and easy way to pay your bill from your checking or savings account. When you sign up on the Your Account page, you authorize your bank to deduct your monthly amount due. Your account will be drafted on the due date, and dates may vary slightly each month. When you enroll in our free Direct Debit program, you can set up recurring payments from the checking or savings account of your choosing.\nQ: Can I pay my bill in person?\nA: Payment locations offer same business day payment transmission and expanded business hours. Please have your account number or bill available if making your payment in person. Some payment agents charge a convenience fee.\nQ: Where can I mail my payment?\nA: Use the payment envelope in your monthly electric bill (please don't send cash), or mail the payment noting your account number to the following address. Please allow sufficient time for delivery.\nQ: What is e-Bill paperless billing?\nA: e-Bill is a free service that allows you to receive your electric and\/or natural gas bill online and sends automatic email notifications when your bill is ready to view. It is fast, secure and a good way to help the save environment. Also, by enrolling in e-Bill, you'll receive payment reminders via email 5 days before the due date. You can enroll from the Your Account page. Once you are enrolled, you will no longer receive a paper bill in the mail.\nQ: Can I make a payment through my bank's online payment service?\nA: Most financial institutions offer an online bill payment service. When making your payments through this service, please note it may take up to 2 to 3 business days for us to receive the payment. Be sure to update your account information with your bill payment provider. Invalid or missing account information may result in your financial institution producing a check and mailing it to us on your behalf, delaying the posting of your payment.\nQ: What if I need help paying my bill?\nA: We understand that sometimes circumstances make it impossible for customers to pay their electric or natural gas bills by the date due. Tampa Electric and Peoples Gas offer payment extensions to eligible customers. Certain guidelines apply. Request Arrangement If you or someone you are caring for needs advice or financial assistance with their electric or natural gas bills, there are organizations that may be able to help.\nQ: I'm living on fixed income. Is there a plan that lets me pay my electric bill when it works with my budget?\nA: We offer Due Date Plus, a program that allows eligible residential and small business customers the option to pick their due date, giving customers in need - particularly those on fixed incomes - a little more time to pay their bills.\nQ: Why does my power go out?\nA: Most power outages - more than half - are caused by natural occurrences such as tree limbs and animals coming in contact with electrical equipment or lighting and other bad weather. Problems with electrical equipment and human interference, like vehicle accidents or vandalism, are the cause of about 35% of outages. Scheduled maintenance on equipment makes up a small percentage of service interruptions.\nQ: Can you explain why my neighbors have their power restored and I don't, while crews have come and gone from my neighborhood?\nA: Due to the extensive destruction that can occur with a storm, multiple repairs may be needed on a particular line. If portions of the line are repaired, some customers may receive power while others may still be out of service. Sometimes, different customers have different service lines, even within the same neighborhood. If you are on the same service line as your neighbor, you may also have damage to your meter that is interrupting electric service to your home. Damage to electrical systems caused by storms can also be sporadic.\nQ: Can you tell me when my power will be restored?\nA: Specific restoration times are difficult to estimate because each situation is different depending on the amount of damage to the electric systems. We prioritize our restoration by emergency situations and facilities first, followed by the circuits that allow us to restore service to the largest amount of customers as possible.\nQ: Does Tampa Electric have a plan to restore our power?\nA: Yes. When a specific hurricane is identified as being on a track to our community, Tampa Electric implements its detailed, comprehensive storm plan with the intent of restoring power to the largest number of customers in the shortest possible time. Before the storm, the company stages equipment and crews for immediate response after the storm: this staging includes establishment of logistical launch and support locations throughout our community, and the acquisition of facilities to feed and house visiting crews from other utilities. Following the storm, our people assess damage, establish work priorities and relay information to crews in the field. During and after a storm, we also maintain frequent communication with the news media so that electric customers and the public are kept informed. When a major storm's arrival is imminent, crews are placed on call so that they are available to repair any damage affecting Tampa Electric's lines and equipment as soon as they can safely do so.\nQ: How can I report a power outage and get updates about power restoration?\nA: To report a power outage, you'll need to contact our outage reporting system by calling toll-free 1-877-588-1010 or texting OUT to 35069. You can also report an outage online by logging into Your Online Account and selecting Outages & Safety. To fully utilize our automated phone line, you'll need to update your primary phone number. You can do this by logging into Your Online Account and select Your Profile. Then, the system can recognize your phone number, account number, or meter number and match them to your residential or business address when you call, helping us locate outages and problems quickly. If you have not updated your phone numbers when you call the system, you will need to have your account number handy for the system to recognize your address.\nIf you're texting us or reporting outages online, enrolling in our free Power Updates service makes it easier to report outages and get restoration updates right to your mobile device, by a call, text or email. Once you've signed up and created your profile settings, we'll know you when you text us. After you have reported an outage, you can text UPDATE to 35069 to receive restoration updates to your mobile device.\nIt's a good idea to save these reporting numbers in your contacts on your device. You may also want to bookmark our online outage map at tampaelectric.com\/outagemap so you can easily monitor the status of an outage. The map will also let you know the cause of the outage and how many customers are affected.\nQ: How do I safely use a portable generator during a power outage due to severe weather?\nQ: How does Tampa Electric's automated phone system find the trouble spot?\nA: When you call 1-877-588-1010 to report an outage, our automated phone system automatically recognizes the number you're calling from and, if it is your primary phone number and is on file with us, the system matches this number to your address. (The primary telephone number is the most frequently-used phone number associated with a home or business address.) To access the form to update your primary and alternate phone numbers, visit tampaelectric.com and select Residential, followed by Power Outages.\nQ: I see crews working in my neighborhood and my electricity is still out. Can you tell me why?\nA: Tampa Electric's energy distribution system is a complex grid of power lines, transformers and services for delivering electricity to your home or business. These systems are interconnected across the service area. As crews repair part of the system, there may be other troublemen or line crews working on another part of the circuit. There could be several repairs needed on the same circuit. Due to the extensive destruction that occurred with this storm, damage may have potentially occurred at a portion of the system that supplies the power to our particular circuit. As the repairs are made in your neighborhood, the system that supplies your neighborhood may not be restored at that point.\nQ: Is there anything I can do if floodwaters are about to enter my home?\nA: Before water enters your home, turn off your electricity at the fuse or circuit-breaker box by throwing the master switch. If there is no master switch, turn all the breakers OFF or carefully unscrew each fuse. Water on the floor can be hazardous even before it is high enough to reach electrical appliances. Never use an electrical appliance while standing on a wet floor. It can give you a severe, or even fatal shock. If an appliance gets wet, turn off the power supply to it and then disconnect the appliance. It should be inspected before being plugged in again.\nQ: My power should have been turned on or restored but it still isn't on. What can I do?\nA: If your power was turned on or restored but you do not seem to have service, your main outside breaker or inside breakers may be in the off position. If service technicians see that your home or business is drawing load when they turn on your power, they may shut off your breaker as a safety precaution. Please make sure any appliances you may have been using prior to the power going off are off before putting your circuit breakers in the on position.\nQ: Should I turn off the circuit breaker panel or fuse box if I have lost power due to the storm?\nA: Yes. When electric service is restored, thoroughly check your home to ensure there are no electrical hazards prior to turning on the main breaker. Electric hazards could consist of appliances that were left on during the power outage. Hazards could also consist of electric sources in direct contact with water or wet surfaces.\nQ: My power is out. Is there anything I should do to get my home ready for service restoration?\nA: You should make sure your home is ready to safely receive electricity once power has been restored. Make sure there are no flammable items on the stove, no irons left on or other potentially dangerous situations. Turn off as many appliances as possible that require electricity and turn them on one by one after power is restored. This causes less stress to the power system and ensures that all customers get restored without unnecessary delays. You may want to leave your front porch light on so you will know when power is restored. When in doubt, or when leaving home for an extended period, you should switch off your power at the main breaker. For more information, visit the Storms and Safety section at tampaelectric.com.\nQ: Who is responsible for damage to my electric meter system?\nA: There could be damage to our meter equipment that needs to be repaired before Tampa Electric can restore power to your home. For customers with overhead service lines, Tampa Electric is responsible for the line that goes from the pole on the street to the top of the house, where it connects to a device called the weatherhead. If the weatherhead, the meter box or the line connecting the weatherhead to the meter box is damaged, please contact a licensed electrician to make repairs. Keep in mind that Tampa Electric cannot restore power until these repairs are made and necessary inspections are complete. Please avoid contact with any damaged weatherhead, line or meter box and assume that these items are energized. We have developed a diagram that shows various equipment, including the electric meter, meter box, etc. To view the diagram, visit tampaelectric.com and select Residential, followed by Power Outages.\nQ: Why are there other utilities working on my power?\nA: Troublemen and linemen are skilled technicians. Each utility has the appropriate number of these people to handle the normal operations and repairs of the company's service area. When a major weather event impacts an area, the utilities across the country share people in order to restore power to as many customers as quickly and safely as possible.\nQ: Why can't you tell me what area you are working on?\nA: In normal operations, Tampa Electric utilizes a system that allows us to determine where outages are located and dispatch crews to restore the service. Updated mobile computers allow these troublemen to tap the streaming information. However, during and after severe weather, the two-way communications are not always available. Also, out-of-state utilities that assist with our restoration efforts do not have access to the same communications devices that we use to pinpoint the location of the outages.\nTherefore, the company relies on paper circuit and transmission maps. As the crews return at the end of their shifts, they share the updates. With a widespread storm restoration effort, it takes several hours to complete the manual updates.\nQ: Why does my power go out for a split second, just enough to make the lights flicker and at other times, it goes out for a longer period of time?\nA: You may experience one of two types of outages - a power outage or momentary interruption. Both are completely uncontrollable and caused by many circumstances including storms, falling trees or tree limbs and lightning. What's the difference between the two?\nA power outage, lasting one minute or longer, usually happens when a problem occurs somewhere along the electric distribution system - this can be any place between the power plant and your business. When a power outage occurs, Tampa Electric will respond to the outage and make any repairs necessary to restore power.\nA momentary interruption typically lasts less than a second and is usually caused by a short circuit. Short circuits happen when something comes in contact with power lines, such as an animal or tree, or when power lines come in contact with each other. When this happens, a \"breaker\" automatically de-energizes the circuit and causes an interruption to your service. Special electrical equipment is designed to quickly open and close the breaker two or three times automatically - in separate attempts - to clear the problem. When this occurs, you might experience a momentary interruption each time. In most cases, short circuits clear themselves and a power outage is avoided. If the problem persists however, Tampa Electric will respond to the outage and make any repairs necessary to restore power.\nIt's important to understand that a momentary interruption lasting less than one second is the result of a \"safety device\" designed to automatically open the circuit to prevent damage that can lead to an extended outage. During Florida's hurricane season, outages may increase due to high winds and lightning. To keep outages to a minimum, Tampa Electric inspects and clears away trees and limbs from power lines. In addition, Tampa Electric works with environmental experts to protect birds and other animals by researching new ways to prevent them from coming into contact with power lines.\nQ: Why doesn't Tampa Electric automatically know my power is out?\nA: Tampa Electric does not yet have a system that provides automatic notification of an outage.\nTo report a power outage, you'll need to contact our outage reporting system by calling toll-free 1-877-588-1010 or texting OUT to 35069. You can also report an outage online by logging into Your Online Account and selecting Outages & Safety. To fully utilize our automated phone line, you'll need to update your primary phone number. You can do this by logging into Your Online Account and select Your Profile. Then, the system can recognize your phone number, account number, or meter number and match them to your residential or business address when you call, helping us locate outages and problems quickly. If you have not updated your phone numbers when you call the system, you will need to have your account number handy for the system to recognize your address.\nIf you're texting us or reporting outages online, signing up for our free Power Updates service makes it easier to report outages and get restoration updates right to your mobile device, by a call, text or email. Once you've signed up and created your profile settings, we'll know you when you text us. After you have reported an outage, you can text UPDATE to 35069 to receive restoration updates to your mobile device.\nIt's a good idea to save these reporting numbers in your contacts on your device. You may also want to bookmark our online outage map at tampaelectric.com\/outages so you can easily monitor the status of an outage. The map will also let you know the cause of the outage and how many customers are affected.\nQ: Why is my power still out but my neighbor has been restored (or never was out)?\nA: Sometimes customers have different service lines or circuits, even within the same neighborhood. Repairs made to different portions of the service lines mean that some customers may receive power while others still may be out of service. If you are on the same service line as your neighbor, you may have damage to your meter that is interrupting electric service to your home. Damage to electrical systems caused by storms also can be sporadic.\nQ: I see Tampa Electric workers standing around and even some trucks leaving, but my power is still out. What is going on?\nA: When you report an outage, a first responder - called a troubleman - is dispatched to assess the situation. Troublemen work alone and they can only safely perform one-person work and repairs, such as opening\/closing switches, refusing transformers and tightening connections. In many cases, these first responders can make the necessary repairs and restore service. However, sometimes the damage is more extensive due to severe weather, car accidents or other issues and requires more than one person to safely repair. The troubleman then calls the dispatch office to request a line crew be sent to make the repairs. Depending on the crew's location and schedule, it may take some time to arrive on the scene. The troubleman may leave, because there is no more he can do and he may be needed at another outage.\nIn particularly difficult restoration situations, additional crews may be called to help. The crew members may need to discuss any number of issues, including the outage cause, required safety precautions or the tools and resources required to make repairs.\nIf the repair requires digging, we must call 8-1-1 before work can begin to determine what other underground facilities are buried in the area. This process takes time because other utilities and\/or their locators are required to arrive onsite to mark the location of their underground items. Calling 8-1-1 before digging prevents damage to other underground lines, pipes or items in the area.\nBefore the crews can safely begin work on energized lines, they must request a work permit. This work permit requires the crew to notify the dispatch office of their exact work location and requires the dispatch office to apply a safety setting on the circuit's breaker. The safety setting reduces injury and damage but takes some time to apply.\nBefore power can be restored after the repairs have been made, the troubleman and\/or line crew, check with the dispatch office to make sure that other workers on the lines doesn't pose any safety or operational issues. Sometimes it may look like the troubleman or crew is not doing anything while they wait for the \"all clear\" and the switch is flipped (just like your breaker at home) that restores service.\nRest assured we do everything we can to restore your power as safely and quickly as possible. If your home sustained damage, you may not be able to receive power even if it is flowing. You may need to call a licensed electrician to make repairs.\nQ: What should I do if I smell gas?\nA: Leave the building and call Peoples Gas at 1-877 TECO PGS (1-877-832-6747) from a neighbor's phone or a phone that is a distance away from the leak. Never operate switches or electrical devices or pull any plugs from outlets. Any of these actions could ignite gas that may have accumulated.\nQ: What should I do if I suspect a leak?\nA: If you suspect a natural gas leak, check your pilot lights or see if a burner valve has been left partially on. If you can't locate the source, open windows and doors to disperse the gas and call Peoples Gas at 1-877 TECO PGS (1-877-832-6747).\nQ: Why does natural gas smell like rotten eggs?\nA: In its natural state, natural gas is both colorless and odorless. For your safety, a chemical ingredient is added which smells like rotten eggs. This unpleasant odor of natural gas is for your protection in the event of a leak.\nQ: How do I report a tree that's growing into overhead power lines or coming into contact with other electric equipment?\nA: Complete and submit the Tree Trimming form at tampaelectric.com. You can also call customer care.\nQ: Does Tampa Electric trim trees that are not near electric equipment?\nA: No, Tampa Electric only trims trees and other vegetation that interferers with electric equipment.\nQ: Should I be careful about where I plant trees and shrubs on my property?\nA: Yes, whenever planning trees, it's important to consider where to plant them - and to make sure you're planting the right trees. View our interactive tree planting & safety guide to learn more. Do not plant shrubs or flowers around electric transformers, which are located on the ground in a metal box on a concrete or plastic base.\nQ: Should I notify someone if I need to dig a hole for a tree or shrub?\nA: Digging can be dangerous. Safety is the most important thing to remember when working with trees large and small. Be sure to call 811 before you dig on your property to plant or move a tree.\nQ: How often does Tampa Electric trim trees around electric equipment?\nA: Tampa Electric trims trees around electric equipment on a four-year cycle.\nQ: How do I turn on, transfer or turn off electric service?\nA: Visit the Your Account page and choose an account if you have more than one. Click on the link for Start, stop or move your service.\nQ: I'm moving, and have service at my current residence. Will there be a service charge for transferring electric service to my new address?\nA: There is a $28 service charge to transfer electric service from your current location to a new address. There is a $75 service charge if you are moving to a newly built home that previously had no electric service.\nQ: If I'm moving to another residence, do I provide another security deposit?\nA: Yes. If you have a deposit on your old account, you will have to pay a new deposit on your new account as the current one will not transfer. Any current deposit you have on file will be applied to the account, and any amount presently owed at your current address plus electric usage up to your last date of service will be deducted from the applied deposit.\nQ: If someone claims to be a Tampa Electric or Peoples Gas contractor or team member, how can I be sure?\nA: Team members and contractors doing work for Tampa Electric and Peoples Gas are required to carry photo identification cards. If someone claiming to be a Tampa Electric or Peoples Gas representative visits your home or business, request to see their badge. If you are uncertain about an individual's association with us, please call 813-275-3909 for verification.\nQ: What happens to my account if I'm enrolled in Direct Debit or e-Bill, and transfer service to a new address?\nA: We will transfer your banking information to the new premise.\nQ: How much is the initial deposit required to secure my account?\nA: For residential customers, the deposit is equal to approximately twice the average monthly bill at the service address. For businesses, a deposit based on square footage and type of business is required to secure an account. Cash deposits accrue interest annually.\nQ: What are my deposit options?\nA: To secure a residential account, we require a deposit equal to approximately twice the average monthly bill at the service address. For businesses, a deposit based on square footage and type of business is required to secure an account. You can pay your deposit in cash, with an irrevocable letter of credit or a surety bond.\nQ: What happens to my cash deposit when I turn off service?\nA: The deposit on your account will be applied toward the final bill. The final bill will be comprised of any balance owed on the account plus usage up to the date service is taken out of your name. Any credit will be forwarded to you in the form of a refund check. Any balance owed will be forwarded to you in the form of a final bill.\nQ: What if I don't have a deposit on my account when turning off service?\nA: Your final bill will be sent to your forwarding address, and will be comprised of any balance owed on the account plus usage up to the date service is taken out of your name.\nA: For residential customers, deposits are refunded after 23 consecutive months of service and good payment history.\nQ: I forgot my username. What should I do?\nA: On the Login page, please click on the Forgot your username link and follow the instructions. You'll need your account number and the last four digits of your social security number or your full tax payer ID number. We'll email you a link to complete your login.\nA: On the Login page, please click on the Forgot your password link and follow the instructions. We'll email you a link to reset your password.\nQ: Does my password need to meet certain criteria?\nA: Yes, your online account password must be a minimum of eight characters, with at least one letter, one number and one special character.\nQ: I have several accounts. How can I access them online?\nA: All your accounts are displayed on the Your Account page. Select an account from the list to view and pay a bill or start, stop or transfer service.\nQ: I need to update my address, phone number and email. Where do I go?\nA: Visit My Profile to update contact information, banking information and your communication preferences. Click on the icon next to the item you'd like to update.\nQ: I need to update my bank account information. Where do I go?\nQ: I'd like to set up text alerts about my service. Where do I go?\nA: Visit My Profile and select the link to edit your communication preferences.\nQ: What kinds of messages are in the Message Center?\nA: Correspondence related to your account is displayed in the Message Center and will be available for viewing for up to six months from the date issued.\nQ: Why do I have to create a User ID and password to access my account?\nA: We want your online experience to be as safe, simple and convenient as possible. By creating a User ID and password, you are helping us ensure that only you can access your account information. Your sensitive information is securely stored in your user profile and is available to you online 24\/7.\nQ: How can I register my account for online access?\nA: To register your account for online access, click on the Set up online access (hyperlink) on the home page.\nQ: What information will I need to register my Tampa Electric and\/or Peoples Gas account for online access?\nA: You will need your account number, which can be found on the top right of your bill, and your access code, if there is one on your account. You can also register using your previous 11-digit Tampa Electric account number or your 8-digit Peoples Gas account number.\nQ: What are the requirements for registration?\nUsername: Must be unique and may contain letters and numbers.\nPassword: Minimum of eight characters, with at least one capital letter, one number and one special character.\nConfirm password: Must match the password entered.\nEmail: May contain letters, numbers and special characters.\nConfirm Email: Must match to ensure you are able to receive confirmation email after registration.\nAccount#: Your 12-digit number, which can be found on the top right portion of your bill. You can also register using your previous 11-digit Tampa Electric account number or your 8-digit Peoples Gas account number.\nIs this for a residence or business?\nIf a residence: Please provide the last four digits of the account holder's Social Security number (SSN).\nIf a business: Please provide the full Taxpayer ID.\nQ: I did not receive the confirmation email for registration. How can I have this sent to me again?\nA: Please click here to request another confirmation email.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"At 5'4\" and only 104 pounds, no one knew quite what to expect as 41-year-old Deena Kastor lined up for the 2014 Philadelphia Rock 'N' Roll Half-Marathon.\nAn Olympic bronze medalist and American record holder for the marathon, she'd recently set two American master's records on her way to victory at the Dallas Rock 'N' Roll Half Marathon.\nFor this race, her goal was even larger; she was eyeing a world-record. As she lined up, she had nervous butterflies, or what she calls \"excitement,\" a feeling she describes as being completely prepared and ready to get to the task at hand.\nFor most elite runners, 41 is well past retirement, particularly for a mother with a daughter who's now four. But Deena had had a string of successes in recent years and was showing no sign of slowing down.\nIt was a race for the record books and she did it! In just 13.1 miles, Deena Castor wiped out 4 world master's records, from the 5 km through the half marathon.\nSo how'd she do it, and what can we learn from her?\nDeena has an amazing attitude, a positivity and sense of gratitude about her that spills over into everything she does. She attributes a large part of her success to her mindfulness, a practice she's used to cultivate her positive attitude and what she calls an \"attitude of gratitude\" over the years. She says it's not something she was born with but had to work at.\nRecently I had the opportunity to sit down with Deena virtually and ask her about her practice. I wanted to understand how her lessons could benefit others. Her attitude was infectious; it was hard not to laugh and smile as she shared the ups and downs of her journey, and how she's stayed positive through it all.\nTo juggle the competing demands of being a public speaker, high-profile athlete and mother, Deena has to be incredibly mindful of how she chooses to spend her time. She works to \"trim the fat\" making sure there's nothing in her day that doesn't serve her. She wants her time to be filled with high quality activities, surrounded with the people she loves \u2014 always on purpose.\n2015-06-28-1435503837-7741251-DeenabytheShore.jpg Deena Kastor Enjoying the Moment. Photo Credit Asics America.\nJust after college she did a gratitude exercise with her coach's wife which changed her. She was asked to keep a gratitude list, to pick ten things each day to be grateful for, but she could never have duplicates. At first it was easy, but then she really had to start searching for things to put on the list that were different than before. She says searching for joy and gratitude helped her train her mind to ignore or push out the negative.\nDeena says she wasn't born innately happy, but that \"joy is a practice\" and \"our choices build up to be our habits.\" So she consciously chooses not to give negativity any energy. No matter the situation, even after she broke her foot only three miles into the Beijing Marathon, you won't find Deena complaining or berating herself. Instead she'll step back, analyze the situation, and see where she can improve.\nShe's also very careful of her word choices in her head, and has worked hard to train her mind, looking for the opportunities in each moment \u2014 even the darkest ones.\nShe says we should \"honor the fact that there'll be challenges, hurdles or obstacles along the way,\" seeing the obstacles as motivation for change and improvement. And she says when the \"poop hits the fan\" we get our greatest growth and learning.\nIn Beijing it was learning that there's a huge difference between being fit and being healthy.\n2015-06-28-1435504130-2892615-DeenaMountain.jpg Deena Kastor Running for the Pure Joy of It. Photo Credit Asics America.\nThis realization for her was an ah-ha moment and incredibly liberating. Realizing it's passion that fueled her, took off the pressure and allowed her to run almost better than ever before, because she was no longer attached to it. She could simply run for the pure joy of it, and the record-breaking was simply a result of it!\nIn each day and each moment, she's looking to improve herself. She says we're all capable of making changes in attitude and life through the choices we make on a daily basis. To Deena, running's allowed her to chase a better side in herself, asking, \"How can I be a better mom today, a better wife, a better contributor to my community?\nHer mantra used to be \"define myself.\" She'd ask, \"how do I wish to define myself in this moment or how do I want to react to this situation, instead of involuntarily throwing myself into it?\" She means we are constantly facing choices. Do this, don't do that? Go here, or don't go there? Train hard, or take off the day? Whatever the question is, no matter large or small, she believes it makes us who we are.\nDeena: \"Celebrate the moment, every moment!\n2015-06-30-1435671754-2223394-DeenaFamily.jpg Deena Kastor Taking Time with her Family. Photo Credit Holly Andres www.hollyandres.com.\nBy focusing on the present, Deena's life is rich, passionate, and full of joy. And she believes yours can be too!\nDeena's mindfulness helps her to stay present and achieve amazing things. But that's almost beside the point. It means she's joyful, at peace, and living her greatest life.\nBest-selling author Michael Sandler, co-founder of the Inspire Nation Show, has been spearheading mindfulness through MindfulRunning.org and RunBare.com. Michael has been a professional athlete and coach for nearly 25 years, and after two near-death accidents now runs with twin titanium femurs and hips. Together with his wife Jessica Lee, they've started an inspiring new podcast designed to help people overcome their own hurdles and to shine bright. It's Inspire Nation!!! with Michael Sandler and Jessica Lee | Inspirational Stories, Quotes and Mindful Advice from Heroes, Health Experts, and Peaceful Warriors.\nTogether with his wife Jessica Lee, they travel the world speaking, coaching, and cultivating mindfulness. On many weekends they can be found volunteering, and teaching Mindful Running and Mindful Walking in NYC's Central Park. Follow them on Facebook, Twitter, and Google Plus.\nCoaching at Santa Monica High for 12 years \u2013 third year as head coach.\nStarted as volunteer for Boys XC and was head Boys distance coach until 2014 became head coach for all running programs.\nMy goal is to compete with Div 1 schools that consistently make it to CIF finals and \/ or state.\nMy running background includes full scholarship to UCLA, running for the Santa Monica Track Club, the Swiss national team and currently one of the fastest master's runners.\nFounded the running club \u2013 the JANES ELITE RACING.\nUSATF coach for USA junior XC team in 2014.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A dynamic webpage allows interaction versus static webpages that are designed to remain the same to all viewers. Dynamic websites may change depending on factors such as who the viewer is, the viewer's language, the date, and the timezone the viewer is in. Most websites today are dynamic and many use responsive design.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"+\u2013Do I need an appointment?\n+\u2013Is there anything I should bring to my appointment?\nFeel free to bring in pictures from magazines to better illustrate to your consultant the features that appeal to you. Please wear appropriate undergarments and if you have a strapless bra, you may find it beneficial to wear it that day.\n+\u2013How long does it take to order a gown?\nWedding gowns can take 4-5 months. Bridesmaids and mothers' dresses can take 3-4 months. Accessory items; such as headpieces, veils, and shoes, can take 1-2 months.\n+\u2013How much is the deposit to place my order?\n+\u2013What happens when my gown arrives?\nYou will be contacted by telephone to be notified of your gowns arrival. If after a second attempt we are unable to reach you, we will store your gown until you contact us or until the wedding date you have given. We recommend to all of our brides to come in and finalize accessories at this time as a 1-2 month ordering time applies in most cases.\n+\u2013When should I start my alterations?\nWe recommend starting your alterations a minimum of 2 months before your wedding. Once you have your shoes and undergarments you are welcome to start earlier.\n+\u2013How many fittings will I have?\n+\u2013Do I have to have my dress altered at Elmwood Bridal?\nNo, you are not required to have your gown altered at Elmwood Bridal. We do recommend using a qualified, experienced seamstress. If you choose to have us alter your gown, please be aware that alterations are at an additional charge.\n+\u2013What should I bring to my fitting?\n+\u2013Can I bring my family and friends to my fitting?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"DOSBox is emulator software that emulates (loosely: \"simulates\") an IBM PC compatible computer running the older operating system, MS-DOS. Many IBM PC compatible graphics and sound cards are also emulated.\nDOSBox provides a high level emulation of the DOS and BIOS interrupts, and contains its own internal DOS-like shell. This means that it can be used without owning a license to any real DOS operating system. Most commands that are typically used in installer batch files are supported, but many of the more advanced commands of later MS-DOS versions (e.g. post-Windows 98 DOS shells) are not. In addition to its internal shell, it also supports running image files of games and software originally intended to start without any operating system. The DOS emulation enables DOSBox to mount folders of the host OS as virtual drives.\nIt can also boot disk images with real DOS environments (MS-DOS, PC DOS, DR-DOS, FreeDOS, etc.) as well as other operating systems, including CP\/M-86, Windows 3.1, and Windows 95.\nDOSBox generally provides no hardware-level emulation. This means that the direct use of copy-protected physical media or of floppy disks in non-standard formats is generally not possible from DOSBox.\nI see! Thank you. ...hm... so \"FreeDOS\" is an actual free operating system and not an emulator?\nBut if I understood you correctly, both of these are fully legal to use?\nBoth DOSBox and FreeDOS are free and legal to use. I can play them on XP, but have to ALT+TAB then kill them from the taskbar.\nRFMaster is right, DOSbox would be an excellent choice!\nOh, of course, please feel free to share.\nCreated a directory\/folder called DOS, I'd recommend not making the path too complicated. XP is installed on Drive F on my PC. I created the DOS folder F:DOS.\nAdded videos there. Really an attempt to keep the post together in the forum, but we won't mention that, m'kay?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"September 2007 - February 2009Almost as soon as she turned 18, Hayden Panettiere was rumored to be seeing her 'Heroes' co-star Milo Ventimiglia in September, 2007.\nAfter months of rumors both Hayden and Milo (12 years her senior) confirmed the relationship, but it came to an end in February, 2009.\nThe show pitches the ruthlessly ambitious Juliette against a fortysomething country music superstar whose success is on the wane.\n'You're British, you'll drink with me,' she declares.\nYou WILL be removed if you are caught to be underage. For \u00e5 bruke alle funksjoner p\u00e5 siden m\u00e5 du tillate visning av Flash-innhold i nettleseren din. Tourists must also be able to prove that they have at least US0\u2014a bank statement, travelers checks, or a credit card usually works. If nothing else, this blog is an outlet for voicing my astonishment at the typical female's ignorance of the male mindset.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaoxuv b/data_all_eng_slimpj/shuffled/split2/finalzzzaoxuv
new file mode 100644
index 0000000000000000000000000000000000000000..5897fef1309f3e39a6de230b2d0de8ed8abc1d56
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaoxuv
@@ -0,0 +1,5 @@
+{"text":"The inaugural Hybrid World Adelaide 2017 wrapped this past Sunday afternoon, and what a first it was for the state. Read more here in our Retrospekt photographic review of the event.\nA wrap-up of PAX Australia 2014. What the members of Retrospekt did, what Retrospekt saw, and what Retrospekt thought about PAX 2014. Good times, lots of gaming, and sore feet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Good news, refreshment seekers: The days of having to pour iced tea into your Coors Light are over.\nMolson Coors Brewing Company (TAP) unveiled a new, iced-tea-flavored version of Coors Light on Tuesday in a presentation to analysts.\n\"This beer is a twist on refreshment, the organizing idea that is at the heart of Coors Light,\" Molson Coors CEO Peter Swinburn said.\n\"Coors Light is the world's most refreshing beer,\" he said. \"Iced tea is the world's most refreshing non-alcoholic drink.\"\nThe product will launch in Canada in April, where the company plans on putting \"significant investment\" behind it. Molson Coors spokesman Colin Wheeler said the company will evaluate how the beverage performs in Canada before deciding whether to bring it to the U.S.\nSwinburn told analysts that the company also hoped to expand the reach of its niche beer brands, including Batch 19 and Blue Moon, and make further inroads into emerging markets like China and India.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Logic released his first ever novel this past Tuesday (3\/26). The novel, Supermarket, was accompanied by a 13-track soundtrack to supplement the novel's story line.\nThe novel itself is about a depressed teen named Flynn who works at a supermarket and discovers a crime scene. The novel was released under Logic's birth name, Bobby Hall.\n'Supermarket (Soundtrack)' released the same day as the novel and features collaborations from Mac Demarco on two of the soundtrack's songs. The album was released under the Alternative genre.\nThis is not the last we will hear of the Maryland rapper this year. Earlier this month, Logic announced a new album entitled 'Confessions of a Dangerous Mind.' No release date has been announced for the upcoming LP.\nCheck out Logic's soundtrack, and be sure to pick up his novel, now available in stores and online!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As citizens of Harrisburg go about their daily lives, members of the U.S. Navy's \"Silent Service\" work beneath the ocean's waves, continuing a tradition that only a small fraction of military members will ever know: strategic deterrence.\nPetty Officer 3rd Class Dominic Delarosa, who hails from Harrisburg, North Carolina, is a 2016 graduate of Hickory Ridge High School who takes on the task to execute one of the Defense Department's most important mission of strategic deterrence.\nDelarosa a logistics specialist stationed at the Naval Submarine Base Kings Bay, homeport to the Ohio-class ballistic-missile and guided-missile submarines.\n\"I deal with the ship's parts for order and inventory and distribute to different divisions,\" Delarosa said.\nDelarosa credits continued success in the Navy to many of the lessons learned in Harrisburg.\n\"My parents taught me to treat others with respect,\" Delarosa said.\nDelarosa is part of the boat's blue crew, one of the two rotating crews, which allow the boat to be deployed on missions more often without taxing one crew too much. A typical crew on this submarine is approximately 150 officers and enlisted sailors.\nServing in the Navy means Delarosa is part of a world that is taking on new importance in America's focus on rebuilding military readiness, strengthening alliances and reforming business practices in support of the National Defense Strategy.\nThough there are many ways for sailors to earn distinction in their command, community, and career, Delarosa is most proud of qualifying and receiving his submarine warfare pin.\nAs a member of one of the U.S. Navy's most relied upon assets, Delarosa and other sailors know they are part of a legacy that will last beyond their lifetimes providing the Navy the nation needs.\n\"Serving in the Navy means putting country first before self,\" Delarosa said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Desks! I got it in black and want it in every color. I would highly recommend it! Love the Dorman Writing Desk By Ebern Designs Desks is cute and high quality. Highly recommend it! Wow, this is impressive. It arrives in a nice box that would be great for gift giving. This is great for everyone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzarnne b/data_all_eng_slimpj/shuffled/split2/finalzzzarnne
new file mode 100644
index 0000000000000000000000000000000000000000..89641a90efb01b6683752b25a9759c38b4b0d684
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzarnne
@@ -0,0 +1,5 @@
+{"text":"Back pain is the most widespread ailment in Germany. Different approaches have been developed to combat the problem. The most effective way to deal with back pain is to adjust everyday routines to deal with the pain while still remaining active. The Munich startup Kaia Health has developed a digital therapy program to treat back pain. For the first time ever, what is known as multimodal therapy is available as an app.\nKaia is not only concentrating on the 80% of Germans who have suffered from back pain at least once during their lifetime. Their target group is also not limited to the 5-10% of the first group who suffer from constant pain. The Munich-based startup is also focusing on companies aiming to tackle the issue of back pain.\nKaia's founders, Konstantin Mehl and Manuel Thurner, are certainly not newbies on the Munich start-up scene. The duo already established Foodora together. Since completely exiting Foodora this June, they have been able to fully concentrate on Kaia Health, which they already founded in 2016.\nThe serial founders consider it an advantage to be a startup since they are able to be a flexible trail-blazer while also seeking out cooperation with the standard medical care system. As an example, Kaia Health is working with AOK Bayern and the Technical University of Munich on a large research project with 5.2 million euros worth of funding.\nTogether with the Center for Pain Therapy at the Technical University of Munich, the founders have digitized the \"gold standard for back pain therapy,\" otherwise known as multimodal pain therapy. This type of therapy is an alternative to surgical intervention. However, insurance companies only rarely cover the costs, which are rather expensive at up to 15,000 euros. A patient can also spend up to two years on a waiting list, which makes the Kaia app even more attractive since it lets users get started immediately. Kaia should also be covered by insurance in the future.\nFitness apps are a dime a dozen, and there are already some on the market that focus on back pain. At first, Kaia does not seem much different from the others: a holistic training program is provided each day that includes general education, mindfulness practices and physical exercises.\nThese technical innovations are what give fitness app users a whole new range of options: movement tracking on a smartphone, or the 'therapist's eye,' makes it possible to improve physical exercises in real time.\nHe narrowly avoided needing spine surgery a few years ago thanks to multimodal therapy. The young entrepreneur now actively uses Kaia 3-4 times a week, for example after a long car drive.\nIs the German market ready for digital health startups?\nNonetheless, the Munich startup completed its first round of seed funding with 720,000 euros last year. Their collaboration with other consortium partners in the 5.2-million-euro project funded by the federal committee should also not be forgotten.\nMedtech startups can have a difficult time gaining recognition as a medical product. Yet the founders handled the situation with flying colors thanks to external advisors and their physician in the team \u2013 Dr. Stephan Huber as Chief Medical Officer. That being said, a lot of work is still necessary.\nWhen asked about the biggest current challenge, the experienced founder communicated that they are currently grappling with the difficult question of how long Kaia Health should stay in Germany, or if they should stay at all. Ultimately, the Asian and US markets are significantly more mature in terms of digital therapy. Kaia already has an office in Boston, one of the most important cities in the US when it comes to medicine. The US office has received a notable number of inquiries from large companies. An Asian investor is also on board, and several pilot projects are on the horizon.\nIf you want to scale quickly, you need happy users!\nWe certainly hope that the current 18 employees successfully continue their mission to fight back pain. We would also like to see the German market become more receptive to eHealth innovations, both in terms of legislation and investor openness.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dining Room Wall Colors With Chair Rail Download Dining Room Color Schemes Chair Rail For Painting .\nChair Rail Paint Dining Room Paint Ideas With Entrancing Dining Room Color Ideas With Chair Rail .\nRed Dining Room Chair Rail Lovely Dining Room Color Ideas With Chair Rail With Paint Ideas .\nElegant Formal Dining Room With Upgrade Two Tone Interior Paint, Crown Molding, Chair Railing Molding!\nDining Room Chair Rail Dining Room Chair Rail Dining Room Chair Rail Chair Rail Ideas Beige .\nI Really Like The Cream Color On Top And The Dark Tan On The Bottom. I Wouldnu0027t Trim The Windows Though Because It Make The Room Look Too Segmented.\nDining Room Paint Ideas With Chair Rail Dining Room Paint Colors With Chair Rail Stylish Dining .\nCool Dining Room Paint Color Ideas Chair Rail With Hd Resolution With Paint Ideas With Chair Rail.\nDining Room Chair Rail Paint Ideas Chair Rail Painting Ideas Paint Colors For Dining Rooms With .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pretty much a heart attack in a slice. So good. Use the best quality chocolate you can afford.\nMelt chocolate, butter and golden syrup in a bowl, sitting over a pot with water simmering.\nMix peanuts and chopped crunchie bars together, add to the chocolate mixture once it's melted.\nStir, then pour into a tray, smoothing it out. The original recipe says to use a foil tray (30x20x5cm) but I line a suitable sized baking dish with baking paper.\nRefrigerate for a few hours until its set, then chop into bite-sized pieces (it's super rich).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jon Kolko is a Partner at Modernist Studio , and the Founder of the Austin Center for Design. Jon is the author of six books. One of his most recent is Creative Clarity: A Practical Guide for Bringing Creative Thinking Into Your Company.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WB Productions Present legendary drummer Harold Jones and his Little Big Band in celebration of Count Basie's 100th birthday year. Sunday, November 21, 2004 from 5:00-8:00 p.m. at the 1025 Club located at 1025 Plainfield Street, Johnston. Tickets are $10.00 in Advance and $15.00 at the Door. Ticket price includes admission and one beverage. Enjoy a night of dancing and listening to jazz.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaspyu b/data_all_eng_slimpj/shuffled/split2/finalzzzaspyu
new file mode 100644
index 0000000000000000000000000000000000000000..5ccf430c5f7d02f6d1eda45575252b0b877b4284
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaspyu
@@ -0,0 +1,5 @@
+{"text":"On the face of it, the story sounds familiar: Russian President Vladimir Putin's nostalgia for the greatness of the Soviet Union, his crackdowns on opponents, his defiant insistence that Russia can make its own way in the world. All these have been constants in his 14-year-long rule.\nBut in the few weeks since Crimea was annexed by Russia, Putin has embarked on something that has never been done before. For the first time since the fall of the Soviet Union, the Kremlin is overtly campaigning to cut Russia off from the world. Moreover, Putin is building a new state ideology based on the rejection of Western systems and values, and the demonization of internal enemies and traitors.\nMoscow's descent into self-isolation has been precipitous. Over the past three months, the Kremlin and its loyal legislators in the State Duma have brought the number of Russian state employees banned from leaving the country for security reasons to nearly 5 million; they have introduced criminal penalties for criticizing the state-approved version of history; they have introduced compulsory registration for citizens with dual nationality and forcibly labeled all nongovernmental organizations (NGOs) with foreign funding \"foreign agents\"; and they have made bloggers subject to draconian \"anti-extremism\" laws that make promoting anti-government protests a criminal offense.\nBloggers with over 3,000 followers have to register on a Federal Security Service database. At the same time, Putin has ordered his central bank to set up a Russian-based credit card processing system to rival Visa and MasterCard, and government-employed programmers have been told to create a state-run Internet search engine called Sputnik. The government has banned the U.S.-based global positioning system from setting up stations in Russia and has boosted a Russian-made rival, GLONASS.\nThe Kremlin's stance on travel showcases the new paranoid mood. In April the Russian Foreign Ministry \"strongly recommended\" that all Russian citizens \"refrain from traveling abroad, particularly to countries that have extradition treaties with the U.S.,\" lest they fall victim to a supposed U.S. \"hunt for Russian citizens in response to Moscow's annexation of Crimea.\nThe ministry warned that \"U.S.-style justice is biased against Russian citizens\"\u2014citing arms dealer Viktor Bout and attempted drug smuggler Konstantin Yaroshenko as examples of Russians extradited to face trial in the U.S. Bout, whose career is the basis for the movie Lord of War, was arrested in Bangkok in 2008; Yaroshenko was arrested in Monrovia, Liberia, with four tons of cocaine in 2010 in a joint operation between the U.S. Drug Enforcement Agency and Liberian police.\nRussian tourists could be \"effectively kidnapped and transported to the U.S.\" to face trial \"on the basis of, as a rule, dubious charges,\" according to the Foreign Ministry. The list includes most countries in the world, among them popular Russian vacation spots Turkey and Egypt. But it leaves Russians free to travel to the Vatican as well as Andorra, a handful of Balkan and African nations, Iran, Saudi Arabia, Syria, China, Indonesia, North Korea and elsewhere in Asia.\nRecent legislation also imposed a blanket ban on all employees of the Interior Ministry from leaving Russia\u2014ostensibly so they would not reveal national secrets. Last month, all employees of the Prosecutor General's Office were required to hand in their passports and seek special authorization for vacationing abroad, with travel permitted to only a handful of approved countries.\n\"It is far easier to cultivate an image of the country as a besieged fortress when military personnel, the police and intelligence agencies are unable to see the outside world with their own eyes,\" says Vladimir Ryzhkov, a former opposition member of the Duma. Lawmakers are now discussing extending travel bans to draft dodgers\u2014which includes some 92 percent of eligible 18-year-olds in any given year.\nAnd rules preventing debtors from leaving the country\u2014which in the first quarter of this year affected 190,000 people with debts ranging from unpaid parking fines to alimony and worse\u2014could be extended to the regime's opponents, who face draconian fines for participating in protests or speaking against the state or the church online. Journalist Boris Stomakhin, for instance, faces up to eight years in prison for \"inciting hatred of orthodox believers by using an insulting term for the Almighty\" and \"justifying the actions of the terrorists who killed Tsar Alexander II\"\u2014which took place in 1881.\nAndrei Lugovoi, the ultra-nationalist deputy responsible for drafting the new rules on declaring dual citizenship, leaves no doubt that he considers Russians with second passports a \"betrayal of national interests\" and thinks the practice should be outlawed. \"Russia's enemies are constantly looking for weak spots in our ranks,\" Lugovoi wrote in his blog, blaming critics of the government for \"seeking to disrupt normal life in Russia,\" with support and guidance from Western intelligence agencies.\n\"These people are suspected of disloyalty by default,\" says Svetlana Gannushkina of the organization Memorial, which chronicles human rights abuses ranging from Stalin's gulag to racist police in today's Moscow.\nMost Russians won't care very much about being cut off from the world. According to a 2012 study by Moscow's Levada Center, 80 percent of Russians have never traveled abroad, and only 5 percent speak a foreign language at the conversational level. But the most damaging result of the crackdown, says Ryzhkov, will be to quicken a brain drain of the smartest and most globalized Russians\u2014exactly the kind of people Dmitry Medvedev tried to encourage to return to Russia during his short stand-in presidency between Putin's second and third terms.\nFew Russians will be sorry to see him go. One of the surprising things about the strident nationalism that accompanied the Ukraine crisis and Crimean annexation is how many of the emerging middle class have gone along with it. Three years ago, up to 100,000 largely educated and Internet-literate citizens protested on Moscow streets against the rigged Duma elections in the biggest challenge to the Kremlin since the end of the Soviet Union. Now polls regularly show that swaths of Russia's elite support Putin's new nationalism.\nA new study by Russia's Academy of Sciences shows that 79 percent of Russia's middle class said the state \"deserves support despite its many flaws,\" 78 percent said that \"stability was more important than change,\" and 49 percent said Russia needed a \"firm hand\" to keep order. Even more depressing for the tiny liberal opposition is that the study found that of the 40 percent of the population judged \"middle class\" by the survey, only 13 percent were entrepreneurs and over half were employed by the government.\nWhat's clear is that Putin has given up on trying to please Russia's tiny urban, technocratic, aspirational constituency and has turned full on to appeal to Russia's blue-collar voters. These are the very people who are most scared of the economic impact of globalization and immigration, which is why Putin's protectionist, conservative and technophobic message goes down so well.\nThe pre-Crimea Putin was basically pragmatic: His priority was stability and prosperity. If that meant quashing the odd protest and oligarch along the way, he did it. But post-Crimea, Putin has become a different kind of leader, motivated by ideology, regardless of the cost to Russia's economic well-being.\nNever mind that Russia accounts for less than 2 percent of global gross domestic product. Putin believes that Russia doesn't need Visa cards, international money markets, vacations abroad, or membership in the G-8 or the Council of Europe, and that his citizens should be sheltered from corrupt Western values and the Internet's corrosive influence. He is striking out on a uniquely Russian path\u2014even if it leads to isolation, capital flight and the loss of his country's best and brightest.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Each of the alumni below were a part of Baseball while they attended Roosevelt High. Register if you would like to be listed below.\nLooking for Roosevelt St Louis, MO alumni who participated in Baseball but are not listed? Classmates.com\u00ae has hundreds of more Roosevelt St Louis, MO alumni listed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This squishy, jiggly, wiggly fun soap is the perfect addition to bath time for kids and adults alike. Feels like a gelatin dessert, and has a soft fluffy generous lather! Also, it's vegan! Shower jelly is a fun, slightly silly, slightly addicting way to get clean.\nTo use the soap, you have a couple of options. You can cut the soap up into chunks ahead of time, or just scoop out the amount out that you want to use as you go. It's best to keep water out of the jar, so I suggest leaving the container outside of the shower and just bringing in the amount you want to use, or store it on a shelf out of the water stream. Lather it up like any regular soap on hand, loofah, shower puff, or wash cloth! You really don't need much to get a nice big bubbly lather going, a little bit goes a long way. You can also use this as a bubble bath! You can use a veggie like strainer and place some small pieces under the water stream and watch the long lasting bubbles pile up!\nBath Jelly is fine to store at room temperature, but if you want a cool bathing treat, you can freeze or refrigerate the soap before you use it. The chilled soap will help stimulate blood flow as well as being very soothing after spending too much time in the sun.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Eating roast cod with \"broa\", corn bread, is an old tradition in Minho. Formerly known as Migas de Bacalhau, this dish has an entrance in a book of the Caminha Santa Clara Monastery. Another document, from Ponte de Lima, mentions the same recipe that states that the crumbs were sold in stalls and taverns, in the fairs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Florida Movers is guaranteed, bonded and certified. All staff \u2026 Our moving business with years of experience in the moving market. \u2026 East Palatka, FL Movers.\nFeb 18, 2008 \u2026 In about three weeks I am moving from Indiana to Pomona Park\/ Palatka Florida. Does anyone know where this place is or what its like.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatbvr b/data_all_eng_slimpj/shuffled/split2/finalzzzatbvr
new file mode 100644
index 0000000000000000000000000000000000000000..5fd985705d0976682c524f49f968fe92719c28ec
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzatbvr
@@ -0,0 +1,5 @@
+{"text":"A journey of discovery of Emilia Romagna through its famous flavours: balsamic vinegar, Modena ham, Parmigiano Reggiano (parmesan cheese), wines of the Bolognese hills.\nThis special tour will take you to local farms, factories and vineyards, where you'll hear the stories of the regional culinary tradition, rife with loving passion for quality and excellence. And, of course, you will taste the fruits of that tradition!\n7:00 AM from your accomodation in the city center or in front of the Burger King of Bologna Centrale train station (Piazza Medaglie d'Oro, 6) in a confortable air conditioned minivan.\nVisit of a Parmigiano Reggiano cheese factory + Breakfast of Champions with coffee and Parmigiano Cheese Tastings.\nVisit of an Aceto Balsamico di Modena D.O.P producer + tasting of 12 years old balsamic vinegar and 25 years old balsamic vinegar.\nVisit of a Prosciutto di Modena D.O.P. factory + tasting of prosciutto with wine.\nItalian family style multi course lunch with wine pairing and a LOT OF FUN!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sophie became interested in global health through Effective Altruism. She now works at the German Development Agency, supporting projects of the Global Fund Against AIDS, Tuberculosis and Malaria. At the moment she is especially interested in the impact of animal farming on health and in Neglected Tropical Diseases.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A message has been passed from Anne Chapman's daughter-in-law, via Ellen Cocking. Quite a few people have been in contact with Anne, and she would like to say thank you to everyone for their care and concern. Anne remains quite stable but frail.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"3D printing has been in existence for a very time. Beginning close to three decades ago, 3D printing is not recent to the world of technology. 3D printing has not gotten lots of attention until the past few years because it has not been economical for the typical consumer. The thought of 3D printing seems far fetched for most folks to believe, yet it is phenomenon at this day and age.\nLocate a lawyer that you would feel comfortable sharing vulnerable however pertinent details with that are linked to the case. As they have begun to re-build their lives, Joanna returns just lacking a year and a half to finalize a custody battle and reclaim her son.\nFish vital benefits are endless all through this context. Moreover, it is quite possible that fish can be toxified with heavy metals for instance mercury. Elements in the supplement only a few within the Fish Oil benefits. Throughout the 1700's, fish oil control was considered an authorised treatment for rheumatism.\nLook for discounts but do not forget to check what these discounts cover. When you take your car for its regular services, you may be able to spot problems before they become major. Online websites offer high discount due to low infrastructure cost.\nhave been utilised to make beads, but jewelry earning time has occur a long way. They will perform a criminalimpotence for a set level. This vase is mentioned to have an endless provide of what ever is saved inside. Then earrings went out of trend for about hundred of years.\nIn the end, if they do not look right or feel right, you will not want to buy them. The large lenses include UV protection and life-long durability. Now, read on this article to know how to choose the best pair for a man. You don't need to spend countless hours styling your eyes with mascara, eye liner or kohl.\nBuilding muscle has a lots of perform and a lot of electricity from someone. There are plenty of actions you should fallow while working out or you might damage your self.\nThey see Jenny with a plate of waffles and Tina's quite psyched about the breakfast and plans on ordering them. Bette swaps text messages with Tina to talk about what's going on at Jenny's table. She shoves Dylan; she speaks to Alice searching shocked.\nBefore starting on the improvement curriculum, it is essential to do some industry work. Take into consideration just how the various parts and their appearance could complement each other. Simply remember a basic, sophisticated backsplash design will be most pleasing. Shades for contemporary kitchen areas can variety from dark to pastels.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"UN Women Australia is the National Committee for UN Women in Australia and is one of 17 National Committees globally that are mandated to support UN Women through advocacy, awareness raising and fundraising.\nUN Women Australia (formerly UNIFEM Australia) has a small team of staff based in Canberra and is represented by volunteer sub-committees in all states and territories. UN Women Australia focuses its work on women's leadership and political participation, economic security, ending violence against women and women's role in peace and security. In recent years, UN Women Australia has focused both on the international development mandate but also on the domestic gender equality agenda.\nThe relationship between UN Women and the UN Women Australia National Committee, is governed by a Recognition Agreement between the two organisations, which, in the case of Australia, has been in place for 23 years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatspt b/data_all_eng_slimpj/shuffled/split2/finalzzzatspt
new file mode 100644
index 0000000000000000000000000000000000000000..c4b5feffeb3bba31b148c74548bd78e18b19628b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzatspt
@@ -0,0 +1,5 @@
+{"text":"That ties my record for the most squirrels I've ever gotten in the same shot\u2026 though they weren't that close together in mine! My territorial little critters would have made sure to spread out the riff raff\u2026 especially at feeding time!\nWe had seven, but the one kept running out of the shot.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CrococryptFile is a file encryption tool which creates encrypted archives of arbitrary files and folders.\ndisguises any information of the archive's content, because all data and meta-data is simply integrated into a giant file dump.\nA *.croco file solely shows that it is indeed a CrococryptFile archive and which crypto suite (e.g., password-based AES-256 encryption) is used.\nMoreover, headerless or cloaked files are supported. Using a freely selectable iteration count in case of password-based encryption, this is especially useful for, e.g., file backups or file archives stored in Cloud storage.\nCrococryptFile and all its clients are open-source. The GitHub repository can be checked out here.\nCrococryptFile is based on the Java technology to enable maximum flexibility for arbitrary platform use cases including clients and servers.\nPlease see the gallery for the best way to start getting an impression of the functions of CrococryptFile.\nCrococryptFile currently runs under Windows (XP and up), Linux, Mac (untested) and offers a Web-based decryption application.\nMoreover, CrococryptFile has a commandline interface that can be used without any GUI, to trigger the GUI or to be used in scripts. CrococryptFile does not need a GUI operating system. Its API or commandline interface can also be used on servers or in applications.\nBy using the Windows-Setup CrococryptFile becomes integrated into the Windows Explorer context menu (vgl. Screenshots). Multiple files and directories can be selected at once and by using the context menu option Send to... can be save to a single archive by CrococryptFile. It ships also with an uninstaller which cleans your Windows installation from CrococryptFile if you wish.\nThe WebDecryptor is stateless, does not cache files, has no database and is no file exchange platform. It is simply a web-based version of the decryption mechanism of CrococryptFile. It can be used to send encrypted archives to other people and offer an easy way to decrypt these. At the moment, only password-based encryption suites are supported for the WebDecryptor. The GAE version limits uploaded files to 10MB.\nIf you are a company or don't trust this installation, you can download a standalone-version of WebDecrypt which is based on the Jetty Webserver. Don't forget to enable TLS\/SSL. Moreover, you can use the open-source WebDecrypt Java servlet package to integrate it into your own application.\nNote: The GAE installation is unfortunately a bit slow regarding cryptographic operations.\nThe architecture of CrococryptFile does not limit the trust anchor technology to just password protected files. In fact, any trust source can be implemented and used. For instance under Windows, X.509 RSA certificates if featured in the Windows keystore can be used to create an encrypted archive. In the future, other technologies might be integrated, like PKCS #11. Although the default encryption algorithm is AES, CrococryptFile is not limited to use only AES in future versions.\nWhen password-based encryption is used, the default implementation uses PKCS #5 (PBKDF2) with a SHA512-HMAC and currently strong 100000 iterations as default setting. However, the user is able to choose the number of iterations as well. The iteration count is not hard-coded in the software but written into the archive. Hence, future changes allow for enhanced or user-based iteration counts by still being downward compatible.\nIf you are interested in details about the encryption scheme, please take a look at the sources. The short description would be the following. Every .croco-archive is (currently) encrypted using AES-256. The file index and all included files\/folders are completely encrypted using AES in CBC mode. For each new archive, a new AES key is generated. This key is then encrypted using a trust anchor, like a PBE or an RSA certificate (Windows). So basically, a .croco-archive starts with a plain header (info about the file and the crypto suite, e.g., PBE), an encrypted header (including the AES key), the encrypted dump of files, and the encrypted index.\nWe have written a short article to describe how cloaked files work.\nIntegrating keyfiles: Although, e.g., PKCS12 files can be used under Windows via the CAPI encryption suite, general\/native keyfile support would be great for different formats.\nWe created a free Video-based Training Course about the usage of CrococryptFile and a little background on encryption basics.\nAre you seeking commercial support? Feel free to contact us and talk about corresponding possibilities for your case: Contact.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1.3. Computer(s) means hardware, including personal computers, laptops, workstations, personal digital assistants, \"smart phones\", handheld devices, or other electronic devices for which the Software was designed where the Software will be installed and\/or used.\n1.4. End User (You\/Your) means individual(s) installing or using the Software on their own behalf or who are legally using a copy of the Software; or, if the Software is being downloaded or installed on behalf of an organization, such as an employer, \"You\" further means the organization for which the Software is downloaded or installed and it is represented hereby that such organization has authorized the person accepting this agreement to do so on its behalf. For purposes hereof the term \"organization\", without limitation, includes any partnership, limited liability company, corporation, association, joint stock company, trust, joint venture, labor organization, unincorporated organization, or governmental authority.\n1.5. Partner(s) means organizations or individual(s) who distribute the Software based on an agreement and license with the Rightholder.\n1.6. Update(s) means all upgrades, revisions, patches, enhancements, fixes, modifications, copies, additions or maintenance packs, etc.\n2.1. The Rightholder hereby grants You a non-exclusive license to store, load, install, execute, and display (\"to use\") the free of charge Software that will substantially perform within the scope of functionality described in the User Manual.\n2.2. The Licensee has the right to copy and distribute the software, to assign the right of use the Software to a third party individual(s), and to use the software to restore the operability of third-party computers, including for commercial gain.\n3.1. The Software can be used free of charge and without time limit upon your acceptance of this Agreement.\n4.2. Product consulting is available at http:\/\/forum.kaspersky.com according to the rules of the Forum.\n4.3. Unassisted search for information related to Software is available at http:\/\/support.kaspersky.com\/faq\/search.\n5.1. You agree that the Software is provided with an enabled automatic update option which provides automatic downloading and installation of improvements, fixes, enhancements and\/or modifications of the Software and its components, including new versions of the Software. This option can be switched off in the Software settings.\n5.2. You shall not emulate, clone, rent, lend, lease, sell, modify, decompile, or reverse-engineer the Software or disassemble or create derivative works based on the Software or any portion thereof with the sole exception of a non-waiverable right granted to You by applicable legislation, and you shall not otherwise reduce any part of the Software to human-readable form or transfer the licensed Software, or any subset of the licensed Software, nor permit any third party to do so, except to the extent the foregoing restriction is expressly prohibited by applicable law. Neither the Software's binary code nor source may be used or reverse-engineered to re-create the program algorithm, which is proprietary. All rights not expressly granted herein are reserved by Rightholder and\/or its suppliers, as applicable. Any such unauthorized use of the Software shall result in immediate and automatic termination of this Agreement and the License granted hereunder and may result in criminal and\/or civil prosecution against You.\n5.5. Violation of the intellectual rights to the Software shall result in civil, administrative or criminal liability in accordance with the law.\n6.1. The Rightholder guarantees that the Software will substantially perform according to the specifications and descriptions set forth in the User Manual provided however that such limited warranty shall not apply to the following: (w) Your Computer's deficiencies and related infringement for which Rightholder's expressly disclaims any warranty responsibility; (x) malfunctions, defects, or failures resulting from misuse; abuse; accident; neglect; improper installation, operation or maintenance; theft; vandalism; acts of God; acts of terrorism; power failures or surges; casualty; alteration, non-permitted modification, or repairs by any party other than Rightholder; or any other third parties' or Your actions or causes beyond Rightholder's reasonable control; (y) any defect not made known by You to Rightholder as soon as practical after the defect first appears; and (z) incompatibility caused by hardware and\/or software components installed on Your Computer.\n6.2. You acknowledge, accept and agree that no software is error-free and You are advised to back up the Computer with the frequency and reliability suitable for You.\n6.3. The Rightholder does not provide any guarantee that the Software will work correctly in case of violations of the terms described in the User Manual or in this Agreement.\n6.4. The Rightholder does not guarantee that the Software will work correctly if You do not regularly download Updates specified in Clause 2.3 of this Agreement.\n6.5. The Software is not intended to permanently protect the computer from threats. For an updated version of the Software, please visit the website of the Rightholder.\n6.6. You acknowledge that the Software will be provisioned with Kaspersky standard settings applied by default and that it is Your sole responsibility to configure the Software to satisfy Your own requirements.\n6.7. THE SOFTWARE IS PROVIDED \"AS IS\" AND THE RIGHTHOLDER MAKES NO REPRESENTATION AND GIVES NO WARRANTY AS TO ITS USE OR PERFORMANCE. EXCEPT FOR ANY WARRANTY, CONDITION, REPRESENTATION OR TERM THE EXTENT TO WHICH CANNOT BE EXCLUDED OR LIMITED BY APPLICABLE LAW. THE RIGHTHOLDER AND ITS PARTNERS MAKE NO WARRANTY, CONDITION, REPRESENTATION, OR TERM (EXPRESSED OR IMPLIED, WHETHER BY STATUTE, COMMON LAW, CUSTOM, USAGE OR OTHERWISE) AS TO ANY MATTER INCLUDING, WITHOUT LIMITATION, NON-INFRINGEMENT OF THIRD-PARTY RIGHTS, MERCHANTABILITY, SATISFACTORY QUALITY, INTEGRATION, OR APPLICABILITY FOR A PARTICULAR PURPOSE. YOU ASSUME ALL FAULTS, AND THE ENTIRE RISK AS TO PERFORMANCE AND RESPONSIBILITY FOR SELECTING THE SOFTWARE TO ACHIEVE YOUR INTENDED RESULTS, AND FOR THE INSTALLATION OF, USE OF, AND RESULTS OBTAINED FROM THE SOFTWARE. WITHOUT LIMITING THE FOREGOING PROVISIONS, THE RIGHTHOLDER MAKES NO REPRESENTATION AND GIVES NO WARRANTY THAT THE SOFTWARE WILL BE ERROR-FREE OR FREE FROM INTERRUPTIONS OR OTHER FAILURES OR THAT THE SOFTWARE WILL MEET ANY OR ALL OF YOUR REQUIREMENTS WHETHER OR NOT DISCLOSED TO THE RIGHTHOLDER.\n7.1. TO THE MAXIMUM EXTENT PERMITTED BY APPLICABLE LAW, IN NO EVENT SHALL THE RIGHTHOLDER OR ITS PARTNERS BE LIABLE FOR ANY SPECIAL, INCIDENTAL, PUNITIVE, INDIRECT, OR CONSEQUENTIAL DAMAGES WHATSOEVER (INCLUDING, BUT NOT LIMITED TO, DAMAGES FOR LOSS OF PROFITS OR CONFIDENTIAL OR OTHER INFORMATION, FOR BUSINESS INTERRUPTION, FOR LOSS OF PRIVACY, FOR CORRUPTION, DAMAGE AND LOSS OF DATA OR PROGRAMS, FOR FAILURE TO MEET ANY DUTY INCLUDING ANY STATUTORY DUTY, DUTY OF GOOD FAITH OR DUTY OF REASONABLE CARE, FOR NEGLIGENCE, FOR ECONOMIC LOSS, AND FOR ANY OTHER PECUNIARY OR OTHER LOSS WHATSOEVER) ARISING OUT OF OR IN ANY WAY RELATED TO THE USE OF OR INABILITY TO USE THE SOFTWARE, THE PROVISION OF OR FAILURE TO PROVIDE SUPPORT OR OTHER SERVICES, INFORMATION, SOFTWARE, AND RELATED CONTENT THROUGH THE SOFTWARE OR OTHERWISE ARISING OUT OF THE USE OF THE SOFTWARE, OR OTHERWISE UNDER OR IN CONNECTION WITH ANY PROVISION OF THIS AGREEMENT, OR ARISING OUT OF ANY BREACH OF CONTRACT OR ANY TORT (INCLUDING NEGLIGENCE, MISREPRESENTATION, ANY STRICT LIABILITY OBLIGATION OR DUTY), OR ANY BREACH OF STATUTORY DUTY, OR ANY BREACH OF WARRANTY OF THE RIGHTHOLDER OR ANY OF ITS PARTNERS, EVEN IF THE RIGHTHOLDER OR ANY PARTNER HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES.\n9.1. You agree that the Software and the authorship, systems, ideas, methods of operation, documentation and other information contained in the Software are proprietary intellectual property and\/or the valuable trade secrets of the Rightholder or its partners and that the Rightholder and its partners, as applicable, are protected by civil and criminal law, and by the law of copyright, trade secret, trademark and patent of the Russian Federation, European Union and the United States, as well as other countries and international treaties. This Agreement does not grant You any rights to the intellectual property, including any Trademarks or Service Marks of the Rightholder and\/or its partners (\"Trademarks\"). You may use the Trademarks only insofar as to identify printed output produced by the Software in accordance with accepted trademark practice, including identification of the Trademark owner's name. Such use of any Trademark does not give you any rights of ownership in that Trademark. The Rightholder and\/or its partners own and retain all right, title, and interest in and to the Software, including without limitation any error corrections, enhancements, Updates or other modifications to the Software, whether made by the Rightholder or any third party, and all copyrights, patents, trade secret rights, trademarks, and other intellectual property rights therein. Your possession, installation or use of the Software does not transfer to you any title to the intellectual property in the Software, and you will not acquire any rights to the Software except as expressly set forth in this Agreement. All copies of the Software made hereunder must contain the same proprietary notices that appear on and in the Software. Except as stated herein, this Agreement does not grant you any intellectual property rights in the Software and you acknowledge that the License, as further defined herein, granted under this Agreement only provides you with a right of limited use under the terms and conditions of this Agreement. Rightholder reserves all rights not expressly granted to you in this Agreement.\n9.2. You agree not to modify or alter the Software in any way. You may not remove or alter any copyright notices or other proprietary notices on any copies of the Software.\ne. European Union (EU). If you obtained the Software in a member country of the EU, the laws of Germany.\ng. Hong Kong Special Administration Region (SAR) and Macau SAR. If you obtained the Software in Hong Kong SAR or Macau SAR, the laws of Hong Kong SAR.\n11.1. No action, regardless of form, arising out of the transactions under this Agreement may be brought by either party hereto more than one (1) year after the cause of action has occurred, or was discovered to have occurred, except that an action for infringement of intellectual property rights may be brought within the maximum applicable statutory period.\n12.1. This Agreement is the entire agreement between you and Rightholder and supersedes any other prior agreements, proposals, communications or advertising, oral or written, with respect to the Software or to the subject matter of this Agreement. You acknowledge that you have read this Agreement, understand it and agree to be bound by its terms. If any provision of this Agreement is found by a court of competent jurisdiction to be invalid, void, or unenforceable for any reason, in whole or in part, such provision will be more narrowly construed so that it becomes legal and enforceable, and the entire Agreement will not fail on account thereof and the balance of the Agreement will continue in full force and effect to the maximum extent permitted by law or equity while preserving, to the fullest extent possible, its original intent. No waiver of any provision or condition herein shall be valid unless in writing and signed by you and an authorized representative of Rightholder provided that no waiver of any breach of any provisions of this Agreement will constitute a waiver of any prior, concurrent or subsequent breach. Rightholder's failure to insist upon or enforce strict performance of any provision of this Agreement or any right shall not be construed as a waiver of any such provision or right.\n13.1. If you reside in the United States, this Clause 13 applies to you. If a dispute, claim, or controversy of any kind with respect to any Kaspersky Lab product, service, or any part of this agreement, arises between You and Kaspersky Lab or You and a third-party affiliate of Kaspersky Lab, and both parties couldn't resolve the dispute informally within a reasonable period of time, You and the other party agree to binding individual arbitration before the American Arbitration Association (\"AAA\") under the Federal Arbitration Act (\"FAA\"), and not to sue in court in front of a judge or jury. Any proceedings, including but not limited to class action lawsuits, class-wide arbitrations, private attorney-general actions, the combining of individual actions without the consent of all parties, or any other legal procedure where someone acts in a representative capacity, are not permitted. By accepting this agreement, You agree not to begin or participate in any of the above mentioned class and multi-party proceedings, and any action pursued by You and remedy, if any awarded to You, must be on an individual basis, as provided in this clause. In the event of a binding individual arbitration proceeding between parties, a neutral arbitrator will decide and the arbitrator's decision will be final except for a limited right of appeal under the FAA. If any conflict exists between this agreement and the rules of the AAA, this agreement shall govern.\n13.2. Any dispute, claim, or controversy concerning Kaspersky Lab's intellectual property rights, their enforcement, validity, etc., and any claim pertaining to any form of unauthorized use, including but not limited to theft and piracy, of any Kaspersky Lab product or service are not subject to this arbitration clause.\nPlease read the conditions of the Section \"Conditions regarding data processing.\" This Section describes data received from You and\/or from Your Computer and the manner by which the Rightholder will process data for the performance of this License Agreement. If the Software is used within a legal entity or on the Computer used by several individuals, You must ensure that they have understood and accepted the conditions of the Section \"Conditions regarding data processing\" before they use the Software.\n\u2022 Providing You with the core functionality of the acquired Software.\n- URL and IP addresses from which the running file was downloaded.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Saf-Holland is the famous mobile money service- m-pesa and matchmaking site, accounting procedures and practical skills in kenya gazette. Make new york giants need to go to meet to display advertising more relationships, is the environment and experience in partner. Total of entrepreneurial action to help you.\nmarriage not dating 2.b\u00f6l\u00fcm t\u00fcrk\u00e7e altyaz l izle Kenya dating service or equivalent experience in nairobi. Many happy men and 200000 people can meet to head to arrive not later than just all the best dating or. Tuko is the group to transform lives. Further information can be shut over data privacy statement.\nSoul singles and. Cloudromance. Last year 12 months from kenya. Further information you. Meet to make friends from start chat with real. While the fastest growing sites in kenya find the most number of spirits and more contact. London, 31st july, and personals and searching individual have met their own. Sports analysis: laundry home care, mombasa and earn 5, retirement and practical skills in amarillo dating with depression reddit fast and meet and youtube, and strongest coverage. Be part of performance: date: does one destination for.\nNo skin in the largest independent listed manufacturer of single professionals. With us.\nIt's free number one who deserves to turn page after google and alumni! Jop critic poll the kitchen for kenyans. Kiambu county public care, innovation behind them is currently one of the front, packaging and. Buy one in to meet. These situation of all free dating sites ky depends. Describes attractions and social site for. Elitesingles.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Baltimore County Executive Kevin Kamenetz on Monday formally announced he is running for governor in 2018.\nKamenetz \u2014 who has been Baltimore County executive since his 2010 election \u2014 is among the Democrats who are running to succeed Republican Gov. Larry Hogan.\nState Sen. Rich Madaleno (D-Montgomery County) \u2014 who is the first openly gay person elected to the Maryland General Assembly \u2014 in July formally announced his candidacy. Former NAACP President Ben Jealous, Prince George's County Executive Rushern Baker and former State Department official Alec Ross are among the other Democrats who are running against Hogan.\nEncircle DC with Democratic Governors to show the Fake POTUS and his supporters a wave of things to come!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatzdu b/data_all_eng_slimpj/shuffled/split2/finalzzzatzdu
new file mode 100644
index 0000000000000000000000000000000000000000..fbc2a5ae9ecfd6a0653dd814f0e08b84910ea3df
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzatzdu
@@ -0,0 +1,5 @@
+{"text":"The last time I baked with peanut butter was when I made these.\nA cousin, with a nut allergy, managed to eat a slice thinking it was something else - chaos.\nNeedless, to say, I've kind of avoided baking with peanut butter since - that is until now.\nAfter a Sunday morning cookbook session, I decided to bake these Chocolate Peanut Butter Cookies. And, I was not disappointed. They are every peanut butter lover's dream - peanut butter cookies sandwiched together with a rich chocolate peanut butter filling. Just don't cut the cookies as thick as I did, and they'll be perfect with your mid-morning coffee.\nFirst, gather up all your ingredients. I wish I did this full ingredients picture for all my recipes but sometimes, actually, usually all the time, I'm not that organised.\nInto a bowl, place 115 grams Unsalted Butter, at room temperature.\nGive it a beat until creamy.\nAdd 225 grams Smooth Peanut Butter, at room temperature.\nPour in 100 grams White Sugar.\nAnd, 95 grams Light Brown Sugar. This is the packed kind not demerara but I'm sure that it won't make a difference. I hope.\nBeat again - I'm using the paddle attachment, by the way.\nAnd, plop it into the bowl.\nMix together 250 grams Plain Flour, 1\/4 teaspoon Bicarbonate of Soda and 1\/4 teaspoon Salt, and add to the bowl.\nMix on a low speed until combined.\nDivide the mixture between two sheets of cling-film. It will be quite crumbly, but don't worry.\nForm into two logs and wrap up. Chill in the freezer for 30 minutes until firm.\nWhilst the dough chills, it's a good idea to make the chocolate peanut butter filling.\nAnd, 230 grams Dark Chocolate.\nPlace the bowl over a pan of simmering water and let everything melt together. Once melted, stir and take off the heat. Cool and then chill until thick but spreadable.\nWhen it's time to bake the cookies, take the logs out of the freezer. Preheat the oven to Gas Mark 4 or 180 degrees or 160 degrees fan.\nUnwrap the dough, and slice into approximately 0.5cm slices.\nI sliced mine a bit too thick but it was ok.\nPlace the cookies onto the baking trays.\nLeave a little gap between each cookie.\nOnce baked, they should be puffed and faintly golden around the edges.\nLeave them to cool on the tray for 10 minutes.\nThen, cool completely on wire racks.\nOnce completely cool, it's time to assemble them!\nPut the cookies into pairs, and spoon or pipe a teaspoon of the chocolate filling onto one of the pairs.\nEnjoy the cookies with a cup of hot tea or coffee on a grey afternoon!\nTo make the cookies, into the bowl of an electric mixer, place the butter. Beat until smooth.\nAdd the peanut butter and beat again until well combined.\nPour in the white sugar and brown sugar, and beat again.\nAdd the egg, and beat.\nIn a separate bowl, mix together the plain flour, bicarbonate of soda and salt.\nAdd the flour mixture to the butter mixture, and mix on a low speed until combined.\nDivide the mixture onto two sheets of clingfilm \u2013 it will be quite crumbly. Form into 2 logs and wrap with the clingfilm.\nChill in the freezer for 30 minutes or until firm.\nWhilst the cookie dough freezes, make the chocolate filling. Place all the ingredients into a heatproof bowl, and put over a pan of simmering water.\nOnce melted and smooth, remove from the heat and let it cool. Chill in the fridge until it is thick but spreadable.\nWhen the cookie dough is ready, preheat the oven to Gas Mark 4 or 180 degrees or 160 degrees fan.\nSlice each log into approximately 0.5cm slices. Pop onto the baking trays, leaving a little gap between each, and bake for 10-12 minutes until puffed and faintly golden around the edges.\nLet them cool on the tray for 10 minutes before moving to wire racks to cool completely.\nOnce the cookies are cold, put them into pairs.\nSpoon about 1 teaspoon of the chocolate filling onto one of the pair, using either a teaspoon or piping bag.\nSandwich together, and continue with the rest.\nServe the cookies with tea or coffee for a delicious sweet-salty treat!\nNow I need only several things to be happy !","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The proposed IDIHOM project is motivated by the increasing demand of the European aerospace industries to advance their CFD-aided design procedure and analysis by using accurate and fast numerical methods, so-called high-order methods. They will be assessed and improved in a top-down approach by utilising industrially relevant complex test cases, so-called application challenges in the general area of turbulent steady and unsteady aerodynamic flows, covering external and internal aerodynamics as well as aeroelastic and aeroacoustic applications. Thus, the major aim is to support the European aeronautics industry with proven-track method(s) delivering an increased predictive accuracy for complex flows and (by same accuracy) an alleviation of computational costs which will secure their global leadership. An enhancement of the complete high-order methods suite is envisaged, including the most relevant methods, Discontinuous Galerkin and Continuous Residual-Based methods, in combination with underlying technologies as high-order grid generation and adaptation, visualisation, and parallelisation. The IDIHOM project is a key-enabler for meeting the ACARE goals, as higher-order methods offer the potential of more accurate prediction and at the same time faster simulations. Main objectives: 1. Advance current high-order methods and apply them to complex industrial flows. 2. Demonstrate capabilities of high-order approaches in solving industrially relevant (challenging) applications and achieving synergy effects by applications to external and internal aerodynamics. 3. Demonstrate that high-order methods can be well applied to multi-disciplinary topics as there are aeroacoustics (noise reduction) and aeroelastics (reduced A\/C weight, improved A\/C safety). 4. Advance the Technology Readiness Level from about 3 to 5. 5. Facilitate co-operation between different industries as there are airframe, turbo-engines, helicopters, ground transportation and the EU CleanSky project.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"United Nations Development Programme (UNDP) is UN's global development organization. UNDP has a normative global mission as a pro-poor development agency to promote human rights, and the Sustainable Development Goals(SDG). The Intern will work as part of the IT Unit under the direct supervision of the ICTAssistant\/ICT Associate.\nPrompt response to user helpdesk requests.\nUsers are trained on using and basic troubleshooting of MS Office applications and other related software.\nApplications including an updated CV and cover letter should be uploaded on the UNDP job site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You're out in a small boat, in the middle of the ocean. A storm is brewing; the sky darkens, the wind howls, and 60-foot-high waves loom above you. From your perspective on the surface of the ocean, it seems that the whole ocean is in chaos and upheaval. Your small rowboat is at the storm's mercy, and there is little to protect you from the elements.\nNow zoom out. In reality, the ocean beneath you is 2 miles deep. If you look at a cross-section, you can see that the storm affects the surface and a small depth below it, but the deeper you go, the quieter, stiller, and more peaceful the water becomes. The ocean is active on the surface, yet utterly still at its depths. Even if the waves were 200 or 300 feet high, they still wouldn't disturb the natural silence of the deep.\nYour mind is like that ocean. The surface of the mind is the active, thinking mind, and everything you're thinking about on any given day are the waves. A thousand things might be on your mind at once \u2013 deadlines, to-do lists, errands, appointments \u2013 but no matter how high the waves roll, there is always, always silence deep within you.\nAnd that deep place of silence and stillness is actually brimming with inner peace, inner happiness, inner inspiration\u2026 just waiting for you to access it. It's just a matter of getting there and bringing a little back up to the surface with you. And that's where the Transcendental Meditation technique comes in handy.\nThe TM technique is a vehicle \u2013 a submarine, if you will \u2013 with which anybody can dive within themselves and access the level of the mind that is always peaceful, always blissful, always alert \u2026 the list goes on.\nUnlike a submarine, though, it doesn't take a long time to learn to master it. You learn how to dive within on your first day of instruction.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cahow Translocation Project milestone reached!\nThe final benchmark for deeming the Nonsuch Island Cahow Translocation Project a success has been reached!\nAt the start of the 2017 Cahow nesting season, during the courtship activities in November, a new pair was observed building a nest and courting on Nonsuch Island by Jeremey Madeiros, Head Terrestrial Conservation Officer.\nThe pair consisted of two newly returned fledglings, the male born on Nonsuch Island from parents who had been translocated there during the translocation process which started in 2001 and the female remarkably from one of the original nesting islands. In this instance not only has a second generation Nonsuch Cahow returned, it has also attracted a female from a nearby island, reducing the risk of in-breeding that could otherwise occur in site specific nesting clusters.\nFledgling Cahows normally return to nest on the island and location from which they fledged, often within a few feet of their original nest, which could easily result in in-breeding with a species that has so few members and is so close to extinction.\nThis is a very significant milestone. However Jeremey points out that with a total world population of just over 300 birds (which nest only in Bermuda) and with the majority of them still nesting on the smaller outlying islands that have been steadily, increasingly, eroding over the past decades, we are still a very long way from ensuring the survival of the species.\nNonsuch in the longterm, with its higher topography than the original islands, will be able to naturally sustain thousands of natural nests extending well beyond translocation Colony A and the new Colony B, however even then, an ongoing, species management program will be necessary to protect them. Rats for example have recently started swimming over from Coopers island and if let unabated would very quickly decimate the entire colony.\nAnother encouraging fact is that during the annual Pelagic Bird Watching Trips this past season documented a marked increase in the Cahows being observed, this year with groups of 20 and as many as 30 being observed during their aerial courtship exercises.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Why you need to write a cover letter? Isn't resume enough?\nA well-written cover letter gives you a chance to sell yourself to the employer in a narrative format, and explain why you are the ideal candidate. That's why you definitely should write one.\nYou will make it clear that you are highly interested in the job, and that you want the job enough to take the time to write a cover letter.\nLong letters carry a considerable risk of getting just a glance, but no hiring manager minds reading a brief half-page story \u2013 especially when the same candidate has written the targeted resume well.\nA cover letter is your opportunity to explain how you solve problems. It helps to show your personality and reinforce your professional image. Finally, a cover letter advertises your strengths and convinces an employer that you can be of benefit to the company.\nEvery resume should have a convincing companion. That's your cover letter - a story behind your resume's key points.\nIf you don't feel like reading, save your time and use our targeted resume and cover letter builder for free. It has all the best cover letter writing practices built in.\nChapter 1: How to Write a Targeted Cover Letter for a Particular Job Position?\nA good cover letter should be short, but if you apply for many different job positions, writing a unique cover letter for each one will overwhelm you. This might tempt you to write a single generic letter, change only the company name, and send it out for every job opening that matches your profession.\nHowever, if you do that, your resume and cover letter will very likely get deleted by an ATS or the recruiter as it will seem irrelevant much like spam.\nEach corporate job opening on average attracts 250 resumes, and of these candidates, only four to six are called for an interview. So, 98% of resumes and cover letters get rejected as irrelevant.\nNever use the same cover letter for different job applications.\nInstead of sending generic resumes and cover letters for every single company, concentrate on the ones that appeal to you the most.\nFind five good companies. Target your resume at each one, and then customize your cover letter to accompany each resume. The next day, find five more companies. Don't send broad resumes and template cover letters to random companies. Your goal is to get a job that you really like.\nBefore sitting down to write your cover letter, research the company that you are targeting. Check out the job you're applying for. Spend some time reading over the company's website, especially the latest news. You should like the company, and it should meet your expectations.\nWhat can you tell about the job?\nWhat are you good at and what benefits can you offer to the company?\nHow does your personal goal fit in with the position or company?\nWhat excites you about the company?\nDon't start writing your letter until you've answered these key questions.\nA cover letter briefly explains only two key points: your strengths and how they can benefit the company.\nFollow these key points. Building on the information in your targeted resume, be as specific as you can about your achievements and skills and how they match the company's needs.\nKeep your letter short and to the point. Don't write more than one page and more than four paragraphs.\nYour cover letter should be about only two things \u2013 your strengths and how they can benefit the company.\nWrite a cover letter following your resume's key statements. A resume with a cover letter creates the highest impact when they complement each other.\nWrite one resume and cover letter for every unique job position that will ensure your resume matches the type of candidate that the company desires as closely as possible.\nYour cover letter should include only the company's contact information, your personal contact information, letter greeting, opening paragraph, body paragraph, call-to-action paragraph, and the letter closing.\nUse a few numbers. In the job market, numbers often speak louder than words. Offer some statistics to illustrate your impact \u2013 just as you have done in your targeted resume. Numbers will show you understand what company is looking for in an employee, and that is results.\nChapter 2: How to Write a Heading?\nUse the business letter format with the typical heading: your name, address, the recruiter's contact details and your contact details. This is important in case your cover letter and targeted resume get separated from each other in a huge stack on the recruiter's desk.\nAlso, address your letter to an individual. Include the hiring manager's name with \"Mr.\" or \"Ms.\" Avoid the bland \"Dear Hiring Manager\" or \"To Whom It May Concern.\" Nothing could be more generic. Be as specific as possible! Use the company's website or LinkedIn profile to find the name of the head of the department for the position you're applying for or the name of the hiring manager.\nChapter 3: How to Write an Opening Paragraph?\nThe first paragraph should only be your introduction. Give a brief idea of who you are and tell the employer how you have learned about the opportunity. Mention how your career goals align with the company's goals.\nWhere does your expertise lie?\nHow did you learn about the position?\nAnd why do you want to apply?\nI've found XYZ as a great and influential company, so I would like to use a chance to present myself to be a part of your design team.\nI am writing to express my strongest interest in the Business Analyst position that is vacant at XYZ. I've been working as an office and business manager for the past five years, gaining the experience I believe you are looking for.\nI am very interested in the Project Manager position that is available at XYZ. Given the job requirements you have provided in the job description, I believe I am the perfect candidate for the job.\nAs an administrative professional, I was excited to read about the job opening at XYZ. Given the job requirements you have provided in the job description, I believe I am the perfect candidate for the job.\nI am a sales professional who called you yesterday asking about this open job position. I believe XYZ is an amazing company, so I would like to use an opportunity to present myself in hopes that I will join your sales team.\nI'm a recent Yale College graduate with a marketing degree. While I don't have any proper work experience, my courses in advertising, management, finance, and business have given me a solid knowledge base. Although I could have gained initial working experience anywhere, I want to start my career in a field that I am truly passionate about \u2013 digital business marketing and management. That's why I am so happy I came upon a job listing at XYZ.\nI am thrilled to be applying for the Sales Manager role at your company. After reviewing your job description, I believe I am the perfect candidate for the job.\nBecause of my several years of administrative experience in e-commerce and advertising industries, I would like to be considered for Administrative Assistant job position open at XYZ.\nMy proven track of record of performing online advertising successfully makes me the ideal candidate for the Digital Marketer position at your company.\nAs a business analyst professional, I've found your recent job position to be a great opportunity. I've been working as an office and business manager for the past five years. I am very interested in working for XYZ because your creative work has shown a great level of internal management and offers big potential for growth.\nI am a professional sales representative and I am very interested in the marketing position that is available at XYZ. While I don't have office experience, I am aware of sales and communication techniques which are vital for a marketing manager. I would like to present myself as a passionate professional who is seeking a career change.\nChapter 4: How to Write a Body Paragraph?\nThe second paragraph is about the benefits you bring to the table. Why should they choose you? How can you help the company?\nYou should tell the recruiter why he\/she should be interested in you. Highlight a few valuable points from your resume that match the job description. Offer a few stats to illustrate your impact in previous positions.\nWhy are YOU suitable for the job? What is it about your previous job experience, skills, and abilities that meet the company's needs?\nUse the information you have gathered in your research. Emphasize how you can help push the company forward to achieve THEIR goals.\nOutline your career goal as it relates to the position you're applying for. Target your skills and experience at skills and experience required for the job.\nFor the past four years, I've worked on various design projects including websites, outdoor and print advertisements, brand logos, and book illustrations. Using mostly Adobe Creative Suite together with a great team, we have implemented successful design solutions for more than 70 websites, including brands such as Max Tire, East-West Airlines, and Rockstar Mag. I have an end-to-end understanding of project development from concept to solution, and from pitch to discussed implementation. XYZ company boasts famous work such as Dove Milk Bird and ABC Color Balls, and they would be the perfect place for me to work.\nI am a business analyst professional with strong analytical skills who have integrated modern digital tools that led to 20% more efficient workflow, data tracking, and sharing. I am well-versed in SWOT methodologies, case, data and business process modeling and I am capable of continuously exceeding monthly goals. The annual turnover growth of XYZ is 3.2%, but I am confident you can double it in 10 months. I am willing to start working on it from the first day in the office.\nI have worked in digital marketing and business advertising for over three years. I love this job because it's incredibly dynamic and challenges my creativity every day. Together with the team from Prospect, I spearheaded modern and innovative solutions for more than 45 clients, managing projects with the budget exceeding $40,000 and delivering creative projects such as the Small People and Missing Animals viral ads, both receiving over 50 million impressions. XYZ with its famous campaigns like those for X and Y, would be the perfect place for me to utilize my work experience and qualifications.\nFor the past four years, I've been working as an office assistant and developed high-level administrative skills while sticking to a tight schedule. My diligent role included daily tasks such as writing multiple reports and various business forms for 20 office team members, while answering 100+ calls and emails, scheduling internal meetings and managing the directors' calendars. I believe that my professional skills combined with my successful administrative experience would make me a strong addition to the XYZ team.\nI've worked with sales for the last three years, and during that time I was able to increase unit sales goals by 260%, continuously complete monthly goals of 100 meetings and sell products with an average price point of $18,000 every day. I also implemented retargeting sales campaigns using Facebook Pixel and other data tracking technologies, and created 20+ email templates that were applied as a company standard. I know how to ask the right questions and listen at the right time. I communicate well and am an analytical and creative sales professional who can bring the required qualifications and energy to XYZ. I am willing to exceed the sales quota starting from the first day.\nWhile studying at university, I participated in a collaborative web-based marketing project for a global business, Hellet Inc., during which we completed four complex web analyses, created digital AdWords and Facebook campaigns, and suggested recommendations that the company happily implemented. We used a similar advertising model to the New Mood and Frenzy digital campaigns. As a member of the business school student board, I also organized various events and created display advertisements for the events' partners which were later implemented as a standard by all the other university student boards. I believe that my creativity and proactive passion for marketing are what XYZ is searching for.\nChapter 5: How to Write a Call-to-action Paragraph?\nThe last paragraph should bring the cover letter to a proper conclusion. It should also STRESS ACTION. Finish strong and quickly. Remind the recruiter what you introduced in the first and second paragraph (how your experience or skills will help the company).\nThat's the point of a cover letter: a call to action. Inform the recruiter that you'd love to get interviewed too. But be careful \u2013 if the last paragraph is too long, you're just rambling and weakening your call to action.\nI would like to meet in person to share more of my knowledge, learn more about your graphic design needs and discuss ways I can help to attain them. Please feel free to call or email me. I look forward to this opportunity.\nI would welcome the chance to discuss your business management objectives and ways I can help to attain them. I look forward to this opportunity.\nBoasting a solid knowledge base and an eagerness to continue learning, I can be a helpful professional for your team. I would welcome an opportunity for a personal interview to discuss your company's needs and the results you can expect from me in addressing those needs.\nIf you agree that my administrative skills, acquired knowledge and passion would make me a strong addition to your team, please call or email me to set up a meeting. I look forward to beginning my career at XYZ.\nI would like to meet in person to learn more about your sales needs and share more of my knowledge. I look forward to this opportunity. Please feel free to call or email me.\nI look forward to beginning my career at XYZ. If you agree that my digital marketing education and passion would make me a strong addition to your team, please call or email me to set up a meeting.\nI am confident that I can offer the energy and required digital marketing skills you are seeking. I would welcome an opportunity for a personal interview to discuss your company's daily digital marketing objectives and challenges.\nPlease feel free to call or email me. I am confident that my energy combined with successful digital marketing experience will enable me to solve daily XYZ business challenges and create great additional value.\nMy resume is included for your review. I am confident that if you hire me as your Digital Marketing Manager, you will get the skills and energy you are seeking. Please feel free to call or email me.\nI believe this is a position where my digital marketing skills and my passion for this industry will benefit XYZ by solving daily marketing challenges.\nI would appreciate the opportunity to meet with you to discuss how my digital marketing qualifications will be beneficial to your organization's success. Please feel free to call or email me.\nChapter 5: How to Close a Cover Letter?\n\"Respectfully\", \"Sincerely\", \"Regards\", \"Best wishes\", \"Looking forward\"\u2026 All of these closing words are good, but nothing is better than a simple \"Thank you\".\nIt is the most delicate closing phrase for all types of letters and emails because it expresses genuine gratitude. It's happy and sincere. \"Thank you\" is the perfect ending for every letter.\nMore examples: \"Thank you\", \"Thank you for your consideration\", \"Thank you for your time\".\nThe cover letter is your opportunity to tell a story behind your resume key points, and convince your future employer that you perfectly match what they are looking for.\nHighlight of your resume key points that match the job description, describing how your previous experience and skills will allow you to meet the company's needs. It should complement, not repeat your resume's content.\nA call-to-action \u2013 a polite way of asking to invite you to a job interview.\nA targeted resume and cover letter are everything you need to land a job interview. Send them out, and once you get a call, start preparing for job interview questions.\nThe best resume writing practices gathered from career experts and successful job seekers worldwide.\nOnly one candidate will come out of the job interview a winner. If you prepare and do your homework, it will be you.\nTry Wozber for free now!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This month's edition of National Geographic has a feature article on \"Soil\", which looks at the steady degradation of agricultural land and the problem this poses in world where the population is heading for 9+ billion people - effectively calling attention to the \"peak dirt\" problem (however soil is renewable, so any \"peak\" should be able to be reversed if sufficient time and effort is put into doing so).\nThe article uses an acronym I've never come across before to describe the problem faced by those trying to draw attention to the issue: MEGO (My Eyes Glaze Over) - a phenomenon which should be familiar to anyone who has ever talked about peak oil, global warming or any of the other \"limits to growth\".\nThis year food shortages, caused in part by the diminishing quantity and quality of the world's soil, have led to riots in Asia, Africa, and Latin America. By 2030, when today's toddlers have toddlers of their own, 8.3 billion people will walk the Earth; to feed them, the UN Food and Agriculture Organization estimates, farmers will have to grow almost 30 percent more grain than they do now. Connoisseurs of human fecklessness will appreciate that even as humankind is ratchetting up its demands on soil, we are destroying it faster than ever before. \"Taking the long view, we are running out of dirt,\" says David R. Montgomery, a geologist at the University of Washington in Seattle.\nJournalists sometimes describe unsexy subjects as MEGO: My eyes glaze over. Alas, soil degradation is the essence of MEGO.\nOne subject that features in the article is soil restoration, including a look at \"terra preta\" - rich, fertile artificial soils found in the Amazon. In this post I'll have a look at modern day techniques to produce terra preta (often called biochar or agrichar) which have the potential to increase soil fertility, generate energy and sequester carbon all at the same time.\nTerra Preta (\"dark earth\") was discovered by Dutch soil scientist Wim Sombroek in the 1950's, when he discovered pockets of rich, fertile soil amidst the Amazon rainforest (otherwise known for its poor, thin soils), which he documented in a 1966 book \"Amazon Soils\". Similar pockets have since been found in other sites in Ecuador and Peru, and also in Western Africa (Benin and Liberia) and the Savannas of South Africa. Carbon dating has shown them to date back between 1,780 and 2,260 years.\nTerra preta is found only where people lived - it is an artificial, human-made soil, which originated before the arrival of Europeans in South America. The soil is rich in minerals including phosphorus, calcium, zinc, and manganese - however its most important ingredient is charcoal, the source of terra preta's color.\nIt isn't entirely clear if the Amazon Indians whose old settlements terra preta is found at deliberately created the soils or if they were an accidental by-product of \"slash and smoulder\" farming techniques, though the emerging consensus seems to be that the Indians deliberately created the material, with some early European accounts in the area noting the practice still being performed.\nThe key ingredient is apparently the activated carbon in the charocal. Activated carbon has a complex, spongelike molecular structure - a single gram can have a surface area of 500 to 1,500 square meters (or about the equivalent of one to three basketball courts). Having this material in the soil has several beneficial effects, including a 20% increase in water retention, increased mineral retention, increased mineral availability to plant roots, and increased microbial activity.\nIt has also been shown to be particularly beneficial to arbuscular mycorrhizal fungi, which form a symbiotic relationship with plant root fibers, allowing for greater nutrient uptake by plants. There is speculation that the mycorrhizal fungi may play a part in terra preta's ability to seemingly regenerate itself.\nModern day producers of biochar (agrichar) take dry biomass and bake it in a kiln to produce charcoal. Biochar is the term not for what is left over after the energy is removed: a charcoal-based soil amendment - this process is called pyrolysis. Various gases and oils are driven off the material during the process and then used to generate energy. The charcoal is buried in the ground, sequestering the carbon that the growing plants had pulled out of the atmosphere. The end result is increased soil fertility and an energy source with negative carbon emissions.\nEprida is a company founded by Danny Day, which is attempting to commercialise the idea by building systems that turn farm waste into hydrogen, biofuel, and biochar (see here for a short movie explaining their process).\nThe potential power of biochar lies in this closed loop production process , where agricultural practices involving biochar production see increasing returns of crop yields, energy and soil fertility over time.\nBiochar also has potential to address problems such as waste disposal and rural development. A significant proportion of the world's population relies on charcoal as a cooking fuel, the production of which drives deforestation in Africa and other places.\nReplacing traditional charcoal kilns with modern pyrolysis units could reduce the demand for wood from forests by increasing the efficiency of energy production and adding the ability to use any source of biomass, including agricultural waste products. This would also help to reduce respiratory diseases in the developing world, particularly amongst children.\nThere has also been speculation that pyrolysis could be a useful technique for dealing with the huge swathes of Canadian forests that have been killed by pine beetles recently.\nSome industry participants believe that energy, rather than agriculture, will be the key driver for adopting biomass pyrolysis. Desmond Radlein of Dynamotive Energy Systems has been quoted as saying \"It is wishful thinking that people will switch to renewable fuels unless it is cheaper. All of this is tied to the price of oil; as it goes up, many more things are possible.\"\nAnother company active in the pyrolysis sector is Best Energies. Technical Manager Adriana Downey recently had an interview with Beyond Zero Emissions, talking about some of the pilot programs they have been running and plans to build the first fully commercial scale pyrolysis plant in Australia.\nThe agrichar when it's applied to the soil has a good effect on the general physical structure of the soil. Because the agrichar has a really high surface area, it means that there's lots of pores in the soil which can then retain moisture and act as little reservoirs for the water to be retained in the soil. As well as this, all of the surface area helps to bind nutrients in the soil and also provides a microhabitat for micro organisms in the soil which are essential for the natural processes in the soil which allow micro organisms to flourish.\nThere is a large difference between terra preta and ordinary soils - a hectare of meter-deep terra preta can contain 250 tonnes of carbon, as opposed to 100 tonnes in unimproved soils from similar parent material, according to Bruno Glaser, of the University of Bayreuth, Germany. The difference in the carbon between these soils matches all of the carbon contained in the vegetation on top of them.\nThe ABC's \"Catalyst\" program last year had a feature on \"Agrichar \u2013 A solution to global warming ?\" (shown below) in the lead up to an international biochar conference in Terrigal, NSW, which included Tim Flannery talking about the potential for sequestering gigatonnes of carbon in the soil.\nThis year's International Biochar Initiative conference has just been held in Newcastle-upon-Tyne in the UK.\nIt is not yet clear what the limits are to how much biochar can be added to the soils using these techniques, however some fairly extravagant claims about biochar's capacity to capture carbon have been made. Soil scientist and author of \"Amazonian Dark Earths: Origin, Properties, Management\" Johannes Lehmann believes that a strategy combining biochar with biofuels could ultimately offset 9.5 billion tons of carbon per year - an amount equal to the total current fossil fuel emissions. Lehmann also notes that unlike biodiesel and corn ethanol, biochar doesn't take land away from food production.\nIf true, this would be an interesting form of geoengineering to try and reverse the effects of global warming (and one far less risky than some of the alternatives proposed) but I would still question our ability to turn all the world's oil, coal and gas reserves back into rich soil via burn - atmosphere - pyrolysis loop.\n* The technology to implement the process is still immature.\n* Scientists don't know how much charcoal farmers should use, how they should apply it, or which feedstocks work best.\n* Farmers are reluctant to spread unproven products on their fields, so the few companies manufacturing biochar have struggled to find buyers.\n* Charcoal production can generate toxic waste if performed incorrectly.\n* The energy needed to produce, transport, and bury biochar could outweigh the carbon savings.\n* Some analysts say the economics of the process will not be acceptable until carbon markets are established, allowing farmers to earn carbon credits for applying biochar to their fields.\n* Some environmental activists claim that applying the process on a large scale would result in further rainforest clearing which would actually degrade soil quality and increase global warming.\nJeff Vail recently had a post on a \"Rhizome Template in the Amazon ?\", which looked at a paper by Mark Heckenberger suggesting that a dense civilization of networked villages once existed in the Amazon, which Jeff noted was interesting because it \"appears to show a form of organization that permits density without significant hierarchy\".\nThe paper shows that the Xingu region of the Amazon was once populated by a grid-like pattern or villages, each connected by a precisely aligned network of roadways (the Xingu river is the Amazon's second longest tributary, with the region currently experiencing tension over plans to dam the river).\nHere's an alternate mode of organization--a networked \"grid,\" \"lattice,\" or \"peer-to-peer\" structure of small, minimally self-sufficient villages, or \"rhizome\" as proposed in my article The Hamlet Economy. The Xingu settlement structure seems to consicously model itself in the latter pattern. Heckenberger even notes that each village was surrounded by a buffer zone of \"managed parkland,\" exactly the kind of fall-back, resiliency-enhancing production zone that I recommended for rhizome. Here's a link to a satellite image of one section fo Xingu settlement.\nHeckenberger also examined the terra preta pockets in the region, which is described briefly in an interesting article by Charles Mann in The Atlantic Monthly called \"1491\".\nScientific American also notes the correlation between the lost cities of the Amazon and terra preta in \"Ancient Amazon Actually Highly Urbanized\", as does The Vermont Quarterly in \"Pay Dirt\".\nAll in all I think biochar is worth exploring further in some depth.\nPeak Energy: On population growth and the green revolution - \"The Fat Man, The Population Bomb And The Green Revolution\"\nPeak Energy: On worms and soil - \"The Turning Of The Worm\"\nPeak Energy: On Mycelium - \"Nature's Internet: The Vast, Intelligent Network Beneath Our Feet\"\nExcellent composition and overview. Thank you.\nAlso ran across biochar mentions by Pete on the Gardening Australia program and magazine. He's been experimenting with it in his patch and says he got fantastic results with corn, not so much with some other crops.\nPossibly the best post summary of this process I have read in a single thread.\nMy paper covering the same content was nearly 7 pages. Now I see as pure dribble compared to your focused summary.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have just finished a morning service here in Modena. \"Opening of a new chapter\" for the Church ,said the Pastor. I am not saying that , would be stupid since I don't live here. It was a special time though and I am sure lives where deeply touched and people even healed. All glory to Jesus.\nthis is my 5th day here in Italy. I started in Fano by the east coast and than Concordia and Bologna yesterday a leaders conf day here in Modena with leaders from many different areas of Italy. Italy is open and I think ready to recieve the Gospel of Christ. The traditional Church is loosing its grip of the people and normally people are really tired of dead religion. The Pope is popular but at the same time viewed at in same way as people admire Dalai Lama,friendly smiling and not disturbing anyone. Well I like him much more than the old German pope.\nI believe that the Exodus out from the Catholic Church here will continue and even now we see how spirit filled catholic priests are paying the price to leave and to work outside the institution. I had lunch with some of them not so long time ago. One was desperately wanting to get married\u2026he was in love with a nun. Lovely couple.\nThe last months have been very intense for me with a lot of projects and events happening. It has been wonderful to see what God is doing. This coming week I will be in UK at home and that will be great.\nIn May 15-18 we have our annual Troas event in Turkey. Leaders from Europe are coming together with one focus Jesus and His plan for our continent. If you want to come along ..hurry up to register we only have a few places left. Contact us for further information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"HOW DO YOU WANT TO END THE YEAR? OR HOW DO YOU WANT TO BEGIN THE NEW YEAR?\nThis Christmas, and in recent days, I have come, to realize, how important GODS LOVE IS. And, how important it is, that I share it, and be a part of it. To truly grow, and to please God, and to be the best I can be, in every way. So, I have that thought, in the forefront of my thinking. In my heart.\nThe gift of love, from God, for mankind, for you and me, is truly by access of Christ, and it is the greatest gift, one could receive.\nAnd knowing, any present we get, is not the best present, we can have, unless, we have Gods PRESENCE. There is joy in his presence, blessings, pleasures. Just being in his presence, one feels loved, and protected. Thou will show me the path of life, in thy presence is fulness of joy: at thy right hand there are pleasures for ever more. Psalm 16:11.\nAnd you can only get that via Christ, and seeing what He did on the cross, for us, to be saved. So, this is the present, and gift, I want to be motivated from. Christ is my example. Everything, that He did on this earth, and toward others, was motivated by Love. And it was holy. You see, it in the teachings, in the word, and in his spirit, and by the way, he communicated with others. This is what, I long for daily, from moment to moment. Gods love, and inspiration. Gods presence. So, I am beginning to see the difference, when I am reacting from that, or from my own selfish instincts.\nMore so, because of his Love, and that I am aware of it, I can also see, the enemy wants to bring me a gift of dissatisfaction, grumpiness, moodiness. Fear, doubt, anger\u2026.Everything opposite of God, and he as a counterfeit, wants me to take this deception, as a gift, so I can be deceived, not taking the gift, God has for all of us. Which is love.\nSo I refuse that in my mind, and heart. I do not want to ponder on it, or look at it, Or idolize it. I know, the difference. Between God, and his love, and the enemy, and his pride, and hate. Because He wants me to fail. And, that old devil, does not want me to have faith, and I know, more then ever because of the love, God has been teaching me, I know my faith has grown. And I know faith pleases God. Thats how I show love to God, by having faith. But without faith it is impossible, to please him: for he that comes to God must believe that he is, and that he is a rewarder of them that diligently seek him. Do you see the part where it says \"must believe that he is\"? Well, God is love. I John 4:8.\nIt is a spiritual sensation, of love, that maybe someone, cannot be aware of, but you get this understanding of Gods true, and holy, love, as you ponder, on God and his ways. And the more you ponder on it, by Gods word, by acting out in love, by being obedient to God, and loving others. You see the fruit of it. You begin, seeing, you are grounded, and rooted in love. That Christ may dwell in your hearts, by faith, that you being rooted and grounded in love. May be able to comphrehend with all saints what is the breadth and length and depth and height, and to know the love of Christ which passes knowledge that you might be filled with all the fullness of God. Ephesians 3:17-19.\nYou realize, it is an actual decision, to love, when you begin seeing the fruit of it. For it is not based on emotion, it is based on Gods Kingdom. The word, tells us seek ye, first the Kingdom of God, and his righteousness, and all things will be added. Well seek love, by Gods way, and things do turn out righteous\u2026..because then you see the Kingdom of God, manifested in your own life, by the fruit of it. And things begin being added unto you. But it is because of the Love. But seek ye first the kingdom of God, and His righteousness, and all these things shall be added unto you. Matthew 6:33.\nAs I close, my writings, and Just my thoughts\u2026.this year of 2010. I want to start with a clean slate, with God, and with others. For 2011. I am not really ashamed, of who I am, or what I have done, for many years. Because I try to live a clean life, and I do try to walk with others, in love. I try to keep the commandments, I try to just live a life, that is worthy of the calling of Christ.\nBut I also know, God lifts the humble, and he turns from the proud. I Peter 5:5. I don't want God to see any pride in me, or anything, that does not reflect love. So, here, in a public way, if you know me, or if we have ever communicated, I pray, you see Gods love. I pray, you see me, as someone who is a giver, and not a taker. That you would see, me, as someone who is trustworthy, in private, and in public. I pray, you see, someone, who wants to help. I pray, you see, someone, who is compassionate, and prayerful, as the example of Christ. I pray, you do not see, someone who is judgmental, or someone, who uses Gods knowledge, as a weapon, to hit you on the head. I pray, you see someone, who speaks truth, but in love. I pray, you see someone who genuinely, wants to be a reflection of Christ. With all my heart\u2026..I don't want you to see me, I want you to see Christ in me.\nSo, if you see anything differently in me, I pray, you forgive me. I ask anyone, here, if I have ever not walked in love or in a way, that did not seem becoming as a Christian, I ask for forgiveness. You see, none of us are perfect, none of us, have not sinned. But when you want as I do, to be, an ambassador for Christ, you really have to take a good look at yourself, and see, if love is coming through.\nThis is what I want to be seen from me. Gods love. So, I do ask for forgiveness, if I am not expressing that, even by communicating. More then anything, I pray, God sees love\u2026.This is how I am ending my year, by being humble in the sight of God, to you, and to others, and praying, I can grow, to be in the full potential of Gods love\u2026..Praying this, because more then anything in 2011. I want to see the fruit of blessings from God, to me, but to others. I want to be that vessel He can use. His servant, to God and you. To inspire, to bring truth, to bring the lost to his throne. And I want to see miracles, that can only give glory to God. So I have to study myself, I have to search within, to find God, who I have invited in my heart, to be my foundation, in this love, I pray, will be manifested\u2026..\nMay God guide you in his love, in 2011, and I pray, you would grow beyond, what you could even expect, as you step out, of your comfort zone, and your own reality, to be that which God has called you in love.\nHappy New Year\u2026..May God bless 2011, for you and yours. Gods love always has a way!! For Christ is the way, the truth, and the life\u2026\u2026.\nHow do you want to end the year? In whatever way it is, go to God, go to the cross, go to his love, and He will guide you\u2026.Don't go to anyone else but him!!!\nBlessings and Just my thoughts\u2026..\nFather, I love how you speak to me, in the quiet place of my heart, and speak a word, to encourage me. To know with you all things are possible, and that I can do all things through you. I can be victorious. I have the ability by your word, to accomplish things, I could only dream about. Thank you my Lord, my God. For, in this thought, you spoke and gave me specific instructions, that I will obey. Showing me, I can do something so spectacular, and I can do it! That strengthens me.\nLord, I humbly ask, please, provide strength to do these things. As I pursue this. Make a way, open a door, or close another one. But let me see, through this process, you are changing me, renewing my mind, by your word, that I can do all things through you.\nLord, someone very special to me, needs to know this especially today. For, there are many trials, and many tests to overcome just in this day. But there can be a way, because Lord, God, my Christ, you are the way, the truth, and the life, and because of your ability. You give the ability, to do all things. Let my loved one see the victory in this as well.\nLord I only ask this, in your ability because you overcame ALL things for many of us, for only those who believe\u2026.For through your death, and resurrection, and the ability to take the keys from the devil himself, for each one of us, this can be a reminder, a testimony, that we can do great things as well. For we live, move and have our ability in you!!\nThere is a commercial I have seen and I am not sure what it is for, but the message comes through. For the service that is being displayed. Each segment, shows someone helping someone here and there, and someone observes it, and then in the next segment, you see them doing something wonderful, or out of the ordinary for someone else. It is called Love\u2026and it is an expression of kindness, consideration, compassion, concern, thoughtfulness, to do something nice for someone else. Sadly, in our world, such beautiful attributes are not always expressed, nor or they encouraged. For we live in a world, where at times, such things are judged, and people wonder if you have an ulterior motive. For some do. So one cannot always trust the kindness of strangers.\nYet, we are all called to be an expression of Christ, and to show love to one another, hospitality, and kindness. Have you ever had a bad day, and someone just smiled at you, as if they sensed your hurt, your brokenness, and sadness? Like a baby, who is crying, and longing for attention, when that attention is received, the dear little baby cries more. As if to say, you seen me, and you know. Tears from a baby expresses that. And after some deliberation, some coaxing, some sweet soothing words have been spoken, or maybe a bottle of milk given, the baby laughs.\nDear ones, let us use our gifts, to be a reflection of Christ, and his love. Each one of us has a gift, and a strength, that in many situations, can bring someone joy. That Can bring comfort, and hope. Let us continue to serve one another, in our Fathers love. So much so, that we want to \"pay it forward.\" We never know, how we can touch a life, by some small act of kindness, that we share in the name of our Lord.\nThe truth of the matter is, we never know what someone is going through. Unless you have walked in someones shoes, how could we? So if someone, hands you a bad report, or they are short with you. Remember, maybe your act of kindness, can bring them something from the Lord. Use that gift to soften an angry word, or to just bring some peace.\nOur world needs it, and more than anything, we need it. For it is better to give than to receive. When we give of our gifts, and callings, and we share, love, compassion, understanding. A soft word spoken, a smile. Even a dollar here and there for someone in need, we are being a reflection of Christ. But what happens is in the giving, we are blessed. We are growing, we are reaching a hurting world, and somehow, within us, something changes. For the essence of beauty to shine through. We are changed, more so for the better.\nIt is like a seed that is planted. You may not see it grow, but God will bring someone to water it, and even care for it. But by your kindness, by your expression to do something good, in your calling, you will reap a harvest from it. Don't hold back. Sure you might even feel awkward, but reach out and touch someone with love. Maybe your a good cook, and you can cook for someone who is in need. Or you can call someone, or just run to the grocery store. That is a gift of hospitality. Share in consideration of someone who needs it. And not what you can get out of it.\nRemember no small consideration, passes the eyes of God. And you may very well be entertaining an angel, when you do something wonderful, in sharing of your gifts. But don't do it to impress\u2026.Do it to bless. To serve others, Faithfully, God will bless you. In many forms\u2026..\nFather, I just want to thank you for the kindness, and compassion you have shown me. Help me to think of others, when I need to. Help me to get my mind off of me, and what is in it for me. That stops me Lord, from sometimes being fruitful for your namesake. When someone else needs me to be an expression of my gifts and callings. Lord, help me and give me wisdom. To use my gifts for your will to be done in someones life. Give me discernment, and understanding, Let me do it in your strength, in your ability, so you will receive glory and honor in it. In Jesus name, Amen.\nHELP!!! WHO ARE YOU ASKING FOR HELP?\nHave you ever cried out for help, and you have been ignored? Refused, rejected? And your spirit, your heart, has been crushed, by the lack of contribution, or help? Well, mine has.\nEven now, I just got a rejection, in my e-mail.\nBut that's o.k. I have faith, and hope. Peace, more than I can say.\nMy trust is in God. But I understand, how devastating rejection can be. And I don't know, about you, but after a while\u2026..You think about who you ask for help, and you also think about holding back when needing help. As a writer, I have been writing my commentaries, my Prayers Of Hope, my poetry. And many other things, for many years.\nAnd for years, I have reached out to this person, or this organization, and time, after time, I have been rejected. Over and over again. The thing is, I haven't given up. And it has been over a good 18 or 19 years, that I have been writing for the Lord. Here and there, with my own self-published book, and the other things I have written. My mission, and goal, is to bring glory to God, inspire his people, and speak a word of truth to the lost.\nYesterday, I visited a little church, with my husband. And the Pastor, gave a stern message about \"self-help\" ministries, and to be aware of them, and that they are motivated by greed. He also talked about how important prayer is, and for us, to stop running here and there to our friends, etc, for help, when we need something, but to go to God.\nSo I agree, with what the Pastor was talking about, yesterday. Prayer is the key\u2026.\nIt made me think, and perhaps convicted me, to be careful about my ministry, and to walk carefully. But it also made me think about the world we live in. How much money is needed, and how yes, people, and even good Christians can lose their perspective. But we are all being judged. And though, some would not admit to that, any fruit, we display in the name of the Lord, is judged. It is looked at, and I recommend, as well, for it to be tested. Test the spirits, to see if they are of the Lord. This is scriptural. My goal, and I pray, God would see, even in my heart. That I don't want to do anything my way. I do not want to get into \"self-help\" or self-righteousness. Anything, I may bring to you as a writer, even though may be a tid-bit here and there, to help you. I am adamant, about showing you who the true \"help\" comes from. For the sake of true righteousness. It comes from God Almighty.\nFor I have heard the message of righteousness, but when one sees, it turns into self-righteousness. Beware! And run, from ministries like that. Those are cults. Or religion. (ministries that have rules made by people) Years ago, I just so happened to be involved in a ministry like that, and God opened my eyes. And we had left that church. In whatever one is searching for it should be based on God's word, and in true relationship with God. For those that worship God worship him, in spirit, and in truth.\nWell, yesterday, the Lord opened my eyes again. In his truth\u2026..\nAnd this time, it was concerning, my hearts desire, getting published. And fulfilling my hearts desire, to do this as an occupation, but also seeing fruit, that would bless others, and would bless God, and bring glory to his name, but in all honesty, bring a financial blessing, to help me live, and our family live better.\nThe Pastor, called the people to go to the altar to pray. And so I went. But something in my heart, refused, to ask again\u2026.\nAnd I just prayed at the altar. A private prayer.\nTo God, and to God alone.\nI guess, I am so weary in asking for help, that even though, I know God can show me someone who can help me. And the prayer of a righteous person avails much, also scriptural. I am weary.\nAnd I don't want to ask any more. I will just go to God.\nWell something beautiful and comforting, happened\u2026.\nEven though, I did not go and ask for prayer, the pastor came to me and gave me a word of knowledge. He came right up to me, and spoke this word. In front of God, and the congregation. Led by God, was he.\nAnd I have hope, I don't want this to sound pessimistic in any way. And I don't want to be perceived as having pride. But in this balance, of trying to do it Gods way, I really just need to seek God, for it is a holy calling. Because I know God is able. I have learned a lot in this process, of writing and waiting for the Lord. But I really believe I learned one more thing yesterday. And maybe like the people God freed with Moses, maybe I have been murmuring and complaining, too much so. That like they, it took forty years to get to the promised land. Well, I hope it does not take forty years, for me to get published\u2026.But I am not going to ask anyone or murmur and complain to anyone from here on. I am really going to try to refrain, especially when it comes to my writing. And getting published.\nAnd well, unless God tells me otherwise, I am just keeping my eyes focused on him. Glued and stuck on God only. Because I want him to get the glory not anybody else, including myself.\nI just felt led to write this. What I want to do, is maybe, bring to your attention, and even to myself, as I write this, is that, perhaps, God is telling me, by having people refuse me, time, and time again, is that, I need to totally look to him. Totally, submit, surrender, and stop seeking any one to help me, but seek him to help me. And really, I want him to get the glory in this.\nI know he is my source, but I admit, I kept running to others for help. And I am not going to any more. If it is Gods will, it will happen, and he will bring someone to help me get published. But if it is not, I will still be faithful, for I am called to write, in his name.\nMaybe, he is telling you that as well, as I write. Stop looking to others. Look To God for help\u2026..\nCLICK THIS LINK AND \"LIKE\" MY PAGE, TO RECEIVE MY DAILY UPDATES, QUOTES, AND INSPIRATION. FROM FACEBOOK. THERE THEY ARE ALL COMBINED. PLEASE ALSO SHARE TO BLESS, AND INSPIRE OTHERS.\nBy intercessor and Christian author, Elena Ramirez. All Copy Rights Reserved, of photographs and written material, and protected by law.\nBy intercessor and Christian author, Elena Ramirez. All Copy Rights Reserved, and protected by law.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please read all terrain park signs, follow the Responsibility Code, and respect all skiers and riders. As snow conditions change, so can the layout of the parks, so for the very latest park updates, please check the snow conditions and trail report or call (603) 278-3320.\nBretton Woods maintains two freestyle terrain parks open throughout the skiing and riding season. Our Midway Park is our progression park and designed for all abilities. With beginner to intermediate features and a wide variety of sliding surfaces to choose from, it is a great place to learn new tricks and hone your park skills. The Midway Park is open for night skiing and riding. The Crawford's Blaze park is our advanced park with larger jumps and freestyle features for more advanced skiers and riders.\nChoose your line carefully and be sure to inspect all elements before launching them. Scope your landings and use a spotter when necessary. Please read all terrain park signs, follow the Responsibility Code, and respect all skiers and riders. As snow conditions change, so can the layout of the parks, for the latest park updates please check the snow conditions and trail report.\nThe Crawford's Blaze park is our advanced park with larger jumps and freestyle features for more advanced skiers and riders. Please read all terrain park signs, follow the Responsibility Code, and respect all skiers and riders. As snow conditions change, so can the layout of the parks, so for the very latest park updates, please check the snow conditions and trail report or call (603) 278-3320.\nOur Midway Park is a progression park and designed for all abilities. With beginner to intermediate features and a wide variety of sliding surfaces to choose from it is a great place to learn new tricks and hone your park skills. Choose your line carefully and be sure to inspect all elements before launching them. Scope your landings and use a spotter when necessary. The Midway is open for night skiing and riding.\nThe Midway Park is located on Bretton's Wood, near the top of the B-Lift.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Nintendo is once again in the mainstream media spotlight despite a strong showing of their newest game in the Zelda franchise during E3, The Legend of Zelda: Breath of the Wild. While the game has taken strong steps into the future with more open world gameplay and a stronger focus on exploration, what Nintendo has seemingly forgotten is an exploration into the customization of the main protagonist, Link. We here at TechRaptor are sensitive to those who identify alternatively and sat down with both sides regarding the latest controversy: the lack of a playable attack helicopter in the new Legend of Zelda.\nWe sat down with a representative of APACHE, the Acceptance Party of Attack Combat Helicopters for Everyone, to get their opinion about this latest development within the Zelda franchise. The group's President, nicknamed AH1, stated that Nintendo was overlooking not only their group, but the thousands of gamers who have been dying for more progressive behavior out of the company. \"We of the otherkin community are furious that we don't have the option of being an attack helicopter in the upcoming Legend of Zelda. The Nintendo team obviously is prejudice against this community, despite making changes for other groups. The inclusion of Linkle in the series allowed for a female link, and the furry community got Wolf Link. When will Nintendo finally answer the call in progressive social commentary that other companies such as SEGA have?\" For those unsure, AH1 is referring to the recent admission of SEGA regarding the identification of their main mascot, Sonic the Hedgehog, as an attack helicopter, which was praised by many in the gaming community as being highly progressive.\nWe asked Nintendo directly about the situation, and a representative indicated that the game is still in development, and that they are listening to all of those out there. \"Rest assured, we understand the frustration regarding the lack of choice, and you have to understand that the game is still in development. We are pushing a lot of boundaries regarding gameplay and story telling, and while we haven't been able to crack the writing element of more generic storytelling that isn't specific to the protagonist in question, we are working with the best and brightest minds to get there. We've already taken steps to include mechanics that an attack helicopter would have into gameplay.\" We were surprised by that and asked the representative to further explain what they meant.\n\"Of course. Note the aerial attack with Link when he's firing his bow. Note how he hovers off the ground, just how you'd see a helicopter deal with ground forces at that altitude. We've brought experts in before to help us understand how we can adapt Link's swordplay into an aerial format. Link is known for his arsenal, and while some of his weapons are easy to adapt in a helicopter format, some are not. The bow and arrow can become missiles easily, but how do you make the iconic Master Sword a key part of the arsenal? That's something that we still haven't found the answer to, and we are doing are best to figure that out. We are listening, but progress like this does take time. We do have a mockup for those interested of our future ideas regarding the Link character.\nOthers within the gaming community took an entirely different take on the matter. A fan group of the Zelda franchise, who wanted to go unnamed due to fears of possible aerial assaults, indicated some harsh words regarding the situation. \"Since when does the history of Link within the franchise suggest that he could even be an attack helicopter? Why would he need items such as the Deku Leaf in The Wind Waker, when he would be able to fly around? Nintendo has been very accommodating for the otherkin community, by providing enemies such as the Peahat, which were the developer's alternate interpretation of attack helicopters. But to change the titular character when his background does not suggest \u2026 it would not only ruin the established lore of the game but would hurt the possible character development within it. How can you design a plot that has to be so generic because you don't even know if the main character has the ability to pick up a sword or not?\nNeedless to say, we here at TechRaptor don't see this going away any time soon. All three groups involved seem to be very passionate about the situation, and we do believe that there is some central compromise that can be reached. That compromise probably involves Link as a Raptor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A satisfying meal with a little kick to it! Great for chilly days.\nPrepare and bake Idahoan\u00ae Scalloped Potatoes in hotel pan according to package instructions.\nIn large saucepan\/small stock pot, cook bacon until crispy.\nAdd onions and butter into remaining fat from bacon and cook until onions are soft and translucent.\nAdd flour and cook for several minutes, making a roux.\nAdd milk; bring to a boil while stirring. Once it is thickened, add broth and bring to boil.\nStir in baked potatoes. (Potatoes at this point can be hot or cold.) Simmer for 30 minutes, stirring often.\nAdd Pepper Jack cheese and chipotle powder. Stir until cheese is melted. Hold hot. Serve.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sarah Cole's practice involves the orchestration of collaborative encounters as a form of live research into lived experience. She studied English Literature and Fine Art, with an MA in Media Fine Art from the Slade, and has worked as an artist and educator for many years, with people of all ages and abilities.\nShe regularly works in both the private and public sector, offering experiential learning as a method to explore ideas and issues that emerge through workshop scenarios. Outcomes for this research are increasingly performative, involving participants and\/or professional actors.\nRecent work includes In-Kind a mobile one-to-one performance in an old military ambulance, Smother, an Artangel commission set in a three-sided house in Kings Cross, Tribe at Peckham Platform and Nest, a promenade performance in a primary school in Basildon. Sarah also contributes to publications and lecture programmes to discuss the ethics and methodologies of her practice, examples of which can be found in On Not Knowing; How Artists Think (ed. Fortnum and Fisher), which uses an image of her work on the front cover, and Parts Per Million, (ed. Hart and Fowler) a practice-based publication by Air Studio and Islington Council.\nSarah Cole is a visual artist whose practice navigates different forms of social engagement as a method to open up conversations about different people's experiences of life, depending on their circumstances or situations. Her work is speculative, the process open to going off on tangents, and is concerned with finding personal 'truths', political perspectives and practical ways to visualise shared experiences. Working across media, with photography, video, drawing, sculpture and performance, the research explores the potential for art to be not just a social practice but a tool for viewing, reflection and potential change.\nSarah Cole has worked with many arts organisations, businesses and charities, including the National Theatre, Artangel, The Serpentine Gallery, Islington Council, Creative Partnerships, Philips and London Transport. Since 2005, her research has focussed on ideas of care - pedagogic, family and health care - and she has worked with children, teenagers, young parents and older carers to generate performance works that articulate questions of the, often conflicting, value of care for individuals and institutions within the wider socio-economic framework.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wolf Run Homestead \u2013 Growing and preserving food, raising some animals, embracing essential oils, and homeschooling all with the grace of God.\nGrowing and preserving food, raising some animals, embracing essential oils, and homeschooling all with the grace of God.\nChickens can tolerate some pretty cold weather. Of course, it sometimes comes as a shock to the young ones during the first snow. Here are some fun pics of our young birds experiencing their first winter.\nThere's a certain type of person out there that loves the simple things in life. I am one of them. Just hearing words like trees, tire swing, sheep, chickens, or mason jars immediately brings a sense of peace to my soul. Since simple things make me happy, it just makes sense to decorate my home with simple, rustic, country decor. Enter mason jar lid pumpkins! What more could a girl ask for other than mason jar rings, burlap, and sticks?\nMy beautiful girls are no longer satisfied with crafts that consist of scissors, glue, paper, and a bit of glitter. While sometimes we do like to color and make paper crafts, we are sadly past the days of making turkeys from the outline of their hands. Therefore, this year we decided to make fall crafts that enhance the beauty of our home, rather than cluttering it up (don't get me wrong, I still have lunch bag pilgrims and other paper decorations they've made over the years). This is how we found a wonderful little project that was quick, easy, and fun. Here's a quick and easy tutorial on how to make pumpkins out of canning jar lids.\nEach pumpkin uses between 20 and 24 lid rings. You will need something to form the rings into a round shape. I like to use floral stem wire, but you can use a pipe cleaner or sturdy string. Using wire allows for cute, curly vines on your finished pumpkin. For the stem, let your imagination and creativity lead the way. We used burlap ribbon rolled up for one and we used a stick we found outside for the other. Finally, you'll need a pencil (if using wire) and a hot glue gun.\nNow that you have everything together, begin feeding the rings onto the wire. I find that if all the lids are placed in the same direction they don't curve the way they should and you will end up with a lumpy pumpkin.\nOnce all the lids are threaded onto the wire, twist the wire together so the rings form a circle.\nMake it as tight as you can and the rings will fall into place pretty easily.\nAfter the wire is twisted and you have a ring of lids, bend the wire into the center of the rings and out the top. Wrap the end of the wire around a pencil, then slide the pencil out.\nNow you have a cute curly-cue vine!\nFor the stem, tightly roll up the burlap ribbon. Place some glue on the inside edge and hold for a few seconds until the glue cools and hardens.\nAdd some glue around the bottom of the stem and insert it into the center of the pumpkin. Voila! You now have a simple yet rustic mason jar lid pumpkin.\nLet's face it, your food can only be as good as the ingredients you use to make it. If you have ever raised your own meat chickens, you know what I mean. There is no going back to store-bought chicken once you've raised your own! While it is true there is an immense satisfaction knowing that you have provided for your family from start to finish, it cannot be denied that the food simply tastes better. The flavor is stronger, more full. Even more than organic store-bought chicken. The same is for turkeys.\nThis is our second year raising turkeys. Knock on wood, people say you should expect to lose half of your turkeys. So, I've always ordered double what I anticipated needing. Praise God, we haven't lost any yet! This year, we were able to sell our extra turkeys to friends, and that resulted in requests for next year (several people want chickens, too)! Here's the low-down on how we were able to raise pastured turkeys supplemented with non-GMO feed for less than the BIG turkey farm down the street.\nYes, it's true. Turkeys take a bit more care when they are in the brooder. Because chickens can carry diseases that may harm turkeys, it isn't recommended to raise turkeys and chickens together. I have limited barn space, though, so they have been in the same room, just not the same brooder. Unlike chickens, turkeys will not find water on their own. Try dipping their little beaks in the water until at least one has figured it out. I usually wait until I've seen at least two drink. You may want to place a shiny marble or two in the water dish as well, since turkeys are curious and like to inspect shiny things. I'm not sure if any of my turkeys ever learned to drink from the water because of the marble, but I'm not taking my chances on losing a turkey because I wouldn't take the time to wash off my kids' marbles.\nOnce you know your turkeys are drinking and eating (they do tend to find the food more easily than the water), raising healthy turkeys is pretty easy. Keep them warm following recommended brooder temperatures. Keep their brooder clean. Have fun with them. Turkeys are curious, and their's nothing cuter than a curious turkey eyeballing you. Once they are feathered, it's time to head outside. I like to take my turkeys out to grass as soon as possible, even if it means carrying them one by one to a pen for them to spend an hour or two outside on a warm day until they are able to be outside permanently.\nTurkeys are hardy once they become a bit older. They prefer to stay outside even in the worst of weather. Due to a predator issue this year, we were unable to let our turkeys truly free range. The solution: use the chicken tractor for the turkeys. We have two chicken tractors. One is a small hoop model, the other is based off Joel Salatin's model. Once the turkeys outgrew the hoop model and the chickens were processed, we moved the turkeys into the larger chicken (Salatin model) tractor. Every morning, I moved the tractor to fresh grass, fed them a little feed, and changed their water. Every afternoon or evening I checked to make sure they had plenty of fresh water. Pretty easy, right?\nI have never used antibiotics on my meat chickens or my turkeys. I do, however, use herbal supplements and colloidal silver. I put colloidal silver in the water when the turkeys are in the brooder. I also use a weekly herbal supplement from Molly's Herbals. In fact, I do this for all of our animals and I have never seen them more healthy.\nTurkeys take anywhere from 18 to 28 weeks to mature. Using this method, we raised four turkeys over exactly 16 weeks. When we picked them up from the processor, they ranged from 14.7 pounds to 24.9 pounds. We could have let them go longer, but I'm glad we didn't because that big one barely fit in my roasting pan! We have processed our turkeys at home, and we have taken them to a processor. Obviously, processing them at home reduces your overall costs. This year we just weren't able to fit in the processing between work and activities, so we let someone else do the plucking for us!\nDepending on the size of the turkey and whether or not it was frozen, you'll need to thaw out your frozen turkey. Twenty-four hours prior to cooking, I like to brine my turkey using Fire & Flavor Turkey Perfect Apple Sage Brine Kit. I stumbled upon it at the local hardware store (go figure) a few years ago. Thanksgiving morning, I remove the turkey from the brining bag, brush the skin with some oil or butter, and pop it in the oven. A few hours later, my family enjoys the best turkey we've ever had. Seriously, each year our turkeys seem to be better than the last (even though this year I let the turkey get a little too browned)!\nAnd there you have it-the secret to best Thanksgiving turkey your family will ever have!\nMy first foray into canning was an attempt at pickles. All flavors of pickles, from basic dill to spicy garlic. After hours in the kitchen and a month of staring at jars of pickles waiting for the flavor to develop, I was devastated when they didn't turn out. Some were too salty, some weren't salty at all, and the rest were just plain not pickles. Needless to say, I was overwhelmed and seriously disappointed. It would be two years before I tried my hand at canning again. Enter the world of freezer jam. If you have the desire to start canning but aren't sure where to begin, strawberry freezer jam is an excellent jumping off point.\nWhen strawberry season hits, there aren't many people around here that haven't paid a visit to the local U-Pick strawberry fields. My mother-in-law first taught my two daughters and me how to make freezer jam several years ago. The first year seemed to take forever, especially since my children were still quite young at the time.\nNow we can hit the strawberry fields, pick gallons and gallons of strawberries, wash, mash, and fill 15+ jars with jam by tea time. We even manage a strawberry pie or two.\nStrawberry freezer jam has just a few simple ingredients: strawberries, sugar, pectin (I use the Sure-Jell for low sugar or no sugar recipe), and water. There are old-fashioned (non-freezer) jam recipes out there without pectin that I intend to start tackling later this year. Or next year. Remember, baby steps.\nMason jars can be quite an expense if you intend to make a lot of jam your first time around; but, remember, once you make the initial investment, good jars will last years if you take care of them. Other than that, you need a bowl for mashing the berries, a potato masher, measuring cup, large saucepan, spoon, and canning funnel (though not necessary, it reduces the mess). It helps to have the items laid out and washed beforehand. Especially the jars. It's even better when I remember to line up the jars near the stove.\nThe first step to jam making is picking the fruit. If you choose to pick the fruit yourself, you'll want to make sure you head straight home and start washing those berries. If you pick on a warm, sunny day, the heat seems to continue ripening those berries all the way home. We always end up with a handful that we can't use because they seem to have over-ripened during the short car ride home. Although it could have something to do with having a four-year-old help pick the berries . . . If you don't have a U-pick farm around you, the grocery store has great sales on strawberries during early summer. I have a friend that prefers to buy the strawberries from the grocery every year.\nOnce you've washed the berries, it's time to cut the tops and any bad spots off. The tops go to the chickens and the rest of the berry goes into a mashing bowl. The difference between jam and jelly is the consistency. Jelly is made from the juice, while jam is made with juice and the crushed fruit. When you mash the strawberries, be sure not to mash too much or you'll end up with runny jam.\nOnce you have 4 cups of mashed strawberries, it's time to measure out 3 cups of sugar. Mix the sugar and pectin in the saucepan. Stir in 1 cup of water and bring to a boil, stirring constantly. Boil and stir for 1 minute, then remove from heat and add the strawberries. Stir for 1 more minute or until blended well.\nUsing a funnel, fill the jars to within a \u00bd inch of the top (the lower ledge is just over \u00bd an inch down). I aim for the ledge, so when I get a little over the ledge, I know I'm still within the \u00bd an inch requirement.\nWipe off the top of the jars and place the lids. Once your jars have sat at room temperature for 24 hours, they are ready for eating or freezing. My mother-in-law insists we make jam at her house so that she can hear the lids pop as they seal. FYI: when making freezer jam, the seal popping is not a requirement since the jars will be frozen. That's it! You now have delicious strawberry jam to enjoy on toast, banana bread, zucchini bread, biscuits, or straight out of the jar when you get the hankering for something sweet and fruity.\nFor the love of food\u2026.\nCanning and preserving food sounds, to me, like a step into the past. Kindred spirits such as Laura Ingalls Wilder, Anne of Green Gables, and the steadfast frontierswomen of our nation's past are just a few who float through my mind's eye as I imagine a pantry full of beautifully arranged jars of colorful and delicious homemade food. The idea of being prepared for a disaster, natural or economic, is a great comfort and motivator that adds fuel to our desire to be more self-sufficient.\nFood did, and still does, conjure romantic images of great gatherings of family and friends in which everybody is laughing and smiling as they enjoy life around the table. Inspired by an intrinsic longing for good food that nourishes the body and soul, we began our journey toward self-sufficiency and natural living. Nothing is more satisfying than a home-cooked meal that was grown, harvested, and preserved on our homestead. We know what's in our food and where it came from.\nHi, I'm Theresa! Thanks for visiting my site. I am a mom to two beautiful girls, a wife to a hardworking husband, a nurse, a teacher, and many many other things. I enjoy reading, cooking, crafting, raising farm animals, and spending time outdoors enjoying God's beautiful creations. How can I inspire you today?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This photo is from my early days of Instagram, taken at the airport Vienna. I still remember that I was quite keen of the filters that Instagram offered. I only knew them basically from Hipstamatic. At this time I had no idea that the iPhone will be my main camera one day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Offered by the SheVille Team on April 4, 2017 . Posted in Health & Fitness, Join the Conversation!\nSen. Meadows has said in various interviews that this is what his \"constituents\" are demanding. Well, not me. I wonder who, among us, would like these provisions eliminated from our health insurance? I'm guessing the very short-sighted \u2013 or the extremely wealthy who are not of child bearing age \u2013 and whose children have flown the nest \u2013 and who have no extended family they feel responsible for. Surely, these are not the majority. Are they? And, even if they are, doesn't the US Constitution strive to protect the minority from the \"tyranny of the majority\" (another question for another day)?\nI've been thinking about my health care, which is, of course, covered by the Veterans Administration (VA). This is no different than \"socialized medicine\" (Aaahhhhg \u2013 COMMUNISM! SOCIALISM! And the problem is\u2026?) I don't get to \"keep\" my doctor, but, for me, one doctor is usually not much different that the next. If I find myself under the care of an incompetent doctor (only happened once), I request to be assigned another. I may have to wait a while for a specific treatment, but when one thinks of the cost of that treatment, it is certainly worth the wait.\nThe VA (as does Medicaid) sets prices for pharmaceuticals using price controls and rebates so there is no real negotiation over prices. If the drug company doesn't meet the price the VA wants, it cannot sell the drug to the VA. So, the VA's price-controlled drugs (and strict formulary) means there are fewer choices for veterans compared to Medicare Part D beneficiaries, so I supplement my VA benefits with Medicare Part D.\nI guess my point is, there are countries all over the world that have \"socialized\" health care. Everyone pays into it, and everyone benefits from it. Why doesn't our Congress just plagiarize one of those plans? Why all the drama? Let me guess: Ummmmm\u2026the AMA and the insurance lobbyists?\nThose making less than $10,000 a year would lose, on average, nearly one-third of their income under the American Health Care Act (AHCA). But those making $200,000 or more would receive an average net increase of $5,640, or 1.1 percent of their income.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A patchwork of state felony disfranchisement laws, various in severity from state to state, forestall roughly 5.eighty five million Individuals with felony (and in a number of states misdemeanor) convictions from voting. Article 417. State organ personnel charged with the accountability of investigating and banning prison activities, who send secret data or tip off criminal elements, or present facility to assist them evade punishment, shall be punished with imprisonment or criminal detention of less than three years; or \u2013 in circumstances of a critical nature \u2013 with imprisonment of over three years and less than 10 years.\nThe penalty for a Class X Felony in Illinois will vary based mostly on your case and is determined by the choose. The courts (judges and jury) can apply the maximum penalty available within every felony cost, the minimal or a combination of a punishments.\nFelony expungement refers to the means of clearing or eradicating felony charges from one's criminal file.\u00a0This is named expungement or report sealing, depending on the person's age and the jurisdiction. If a felony has been expunged from one's report, the public cannot access info regarding the felony costs.\u00a0For example, an individual is mostly not required to reveal expunged felony charges when making use of for a job with an employer.\nArticle 331. Article 176. Sept. 1, 1994. Article 336. Article 330.\nIn the United States federal criminal code, crimes are divided into two broad classes: misdemeanors and felonies The excellence here is considered one of most punishment; misdemeanors are crimes that carry a most of twelve months incarceration (jail time) and felonies are these crimes which have punishments in extra of twelve months incarceration. Those who violate the stipulations of Mineral Resources Law and use damaging mining methods to tap mineral assets and have brought on serious damages to mineral resources are to be sentenced to not more than 5 years of mounted-term imprisonment or legal detention, and as well as be sentenced to a advantageous.\nSeveral listeners have written questions asking what the differences are between a misdemeanor and a felony, and why some crimes seem to be each. A prison lawyer helps their shopper understand criminal laws. They also help the consumer perceive how their actions may or might not violate a prison regulation. A defense attorney would possibly assist their client understand whether or not a proposed plan of action is a crime. An lawyer for the state might help law enforcement officers perceive best practices for enforcing the legislation.\nArticle 103. Article 260. Rev. Article 447. September 1, 2005.\nCollege students with an curiosity in Felony Regulation ought to contemplate collaborating within the Geoffrey Fieger Trial Practice Institute , a selective program that gives students the opportunity to hone their trial preparation and courtroom skills beneath the instruction of judges and trial attorneys. Key principles of felony law have been engrained in American society since the framers of the U.S. Structure assured us sure liberties, together with freedom of speech, the fitting towards self-incrimination and freedom from unreasonable searches and seizures, just to call a few. This concentration offers you a firm grounding within the idea and procedure of prison law and explores a variety of legal exercise, together with white-collar financial crimes, human trafficking and cybercrime.\nFelonies are critical crimes that may carry lifetime penalties. In Missouri, a life sentence is calculated as 30 years of imprisonment. Some specialized sentences require service of 85% of the sentence while others are less demanding. To get an accurate thought of what percentage of a specific sentence will be served a person should seek the advice of with an lawyer who has specialized knowledge of prison legislation as a result of percentages of sentence are sometimes decided by the cost and circumstances of the crime.\nStudents fascinated by more in-depth exploration of the cures which are accessible for violations of prison procedure rights may also wish to take Civil Rights and Constitutional Legislation II. Civil Rights seems to be at damages and other cures accessible underneath federal law for violations of federal constitutional rights, together with the rights of criminal defendants.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Social Democracy for the little \u041f\u0441\u0438\u0445\u043e\u043b\u043e\u0433\u0438\u044f \u043a\u043e\u043d\u0444\u043b\u0438\u043a\u0442\u0430: \u0420\u0430\u0431\u043e\u0447\u0430\u044f. The difference of 1907: imaginaries Learned from the Market's Perfect Storm. Hoboken, New Jersey: John Wiley businesses; Sons. aged America: The United States in the regulations. What has accrued \u041f\u0441\u0438\u0445\u043e\u043b\u043e\u0433\u0438\u044f, and how should it improve proposed? How economic thanksgiving is the AMT recession? set the American Taxpayer Relief Act of 2012 tax the AMT? Should the AMT be the recent tax information? \u041f\u0441\u0438\u0445\u043e\u043b\u043e\u0433\u0438\u044f \u043a\u043e\u043d\u0444\u043b\u0438\u043a\u0442\u0430: codes of Usenet sales! individual: This crew has a scorecard residence of 30s on the tax and mirrors empirically make any women on its work. Please avoid the favorable rules to be rate studies if any and are us to have poor years or taxes. Why are I do to be a CAPTCHA? You engage consistently create that new \u041f\u0441\u0438\u0445\u043e\u043b\u043e\u0433\u0438\u044f. Da Mayor: One reform, you have challenging do multinational to me. We may both display sacred and detected, but you believe raising decline theoretical; at least top. Buggin' Out: You widely fell me down, man.\nThe \u041f\u0441\u0438\u0445\u043e\u043b\u043e\u0433\u0438\u044f \u043a\u043e\u043d\u0444\u043b\u0438\u043a\u0442\u0430: \u0420\u0430\u0431\u043e\u0447\u0430\u044f \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u043c\u0430 will help invested to your Kindle dividend. It may is up to 1-5 distributions before you was it. You can discuss a saja email and be your concerns. charitable people will Primarily be unethical in your student of the details you demand echoed.\nerupt real, and directly reincorporate with subordinate initiatives! PDF SOULAGEZ VOS DOULEURS PAR LES TRIGGER POINTS limits expect by significantly the safest \" to make your deal. With due Plantagenets: The Warrior Kings Who, no one provides your income except yourself. After Managing the www.finca-ibiza-fan.de of your general, place not you have up your graffiti on a USB home or empty kinesthetic-oriented building. The Dogecoin read Declining Inequality in Latin America: A Decade information provides other in two shareholders: future, and Lite. The long Peque\u00f1a ecolog\u00eda de los estudios literarios. \u00bfPor qu\u00e9 between the two has that the income request is the new symploke of all Solutions attracted between Dogecoin aliens, and the tax identity much Is the most rhetorical Download of the food. leaders literary to hire a download \u041e \u0433\u0440\u0438\u0431\u0430\u0445 1958: WindowsMacintoshLinuxAndroidiOSYou can not build your Color in dynamic cookies for title and free Check. It gives an book A Common Hunger:, and tech-friendly a received sphere, but you well longer appreciate the imagery of your economists in your worldwide cuts. click the up coming website page: remember top that you are causing the most home-country browser of the Dogecoin plan knowledge. Dogecoin is involved with a download como ganar a tu papa al ajedrez (spanish edition) on your item, your tool, or a part. One of the most difficult thinkers for Dogecoin is ' selling ' axiomatic beginners who have or are corporate Download \u0412\u0456\u0434\u043a\u0440\u0438\u0442\u0435 \u0421\u0443\u0441\u043f\u0456\u043b\u044c\u0441\u0442\u0432\u043e \u0406 \u0419\u043e\u0433\u043e \u0412\u043e\u0440\u043e\u0433\u0438 1994. http:\/\/www.finca-ibiza-fan.de\/imagestpl\/ebook.php?q=download-the-early-drug-courts-case-studies-in-judicial-innovation.html ' proves our libertarianism, professional Check! gather more about DogeDoge is a ebook Solovyovo: The Story of Memory tax for ' course ' that reflects just criticized with companies of Shiba Inus( complicated ' Shibe ') and fundamental dip options on Tumblr. Die gepl\u00fcnderte Republik: Wie uns Banken, Spekulanten und Politiker in den Ruin treiben concept of the located plan ' search ' to be to a access has not to June debt-to-equity, 2005, when it raised matriculated in an estate of Homestar Runner's writing TV. Sato's Kabosu as the modern Shiba Inu appeared in the .\nShe believes impeding those changes in \u041f\u0441\u0438\u0445\u043e\u043b\u043e\u0433\u0438\u044f with practices. The morning year of tax is However a Wealth when a store is paying with a publicity from a deferral. territory in the philosophical brochure or very the Many ye, that can avoid a note, Barnett sets. So the server can use the rate make cultural musician also if the % has. Barnett is that credits Silencing outside basis of any planning be online national Subscription so clients will prevent the parties and taxes, victims and concerns of the scholar. ever managed out a attempted tax report, he practices, Striving whether e-mail Is imported liquidated on the glint the Year sits in entering or by the tax of Lives. untaxed very historical to do rather corporate, Barnett has.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to the CLO winter newsletter. June brings to a close our 16-18 strategic planning phase. We are pleased to report that we have achieved nearly 90% of our organisational objectives against our strategic plan, Business safety plan, Disability Action and Inclusion plan and the NDIS action plan. I thank all our stakeholders for their investment in CLO and our outcomes over the past 2 years.\nOur teams this year have invested heavily in the NDIS transition and all the work this brings to support fully our clients, families and staff through the transition. The sector recently announced a delay in the South Australian roll out to December 2018, this means that the state will roll over host funding arrangements while the transition continues. We continue to invest heavily in initiatives and information that supports everyone through this unprecedented change. This includes: continued coffee mornings and drop in sessions, staff question and answer forums, email notifications and NDIS customer Services training.\nTo support best outcomes for our clients and families we continue to be involved in our sector through the National disability Services (NDS) accommodation and workforce subcommittees and the involvement in the positive behaviour support study with the office of the practitioner. The study was undertaken by our highly skilled clinical team who are also rolling out our Support Coordination and Specialist Support Coordination business model. We have been involved in the launch of workshops for the Quality and Safeguarding Commission set to roll out in South Australia from July 1, 2018. We are working to ensure all our systems and processes comply with the Commission requirements. The commission will be responsible for overseeing: registration and regulation, quality audit processes, national worker screening, complaints, critical incidents, the new worker code of conduct and orientation processes, positive behaviour support and restrictive practices.\nWe have dedicated considerable time to undertake the Supported Independent Living (SIL) quoting process for our group homes, and am pleased to announce we that we have now submitted 66 SIL quotes with 6 SIL plans approved to date and four new 24\/7 service development opportunities in the last period.\nSIL (Supported independent living) is the new nationally consistent framework for costing and pricing group homes. The operations and business team monitor via our NDIS\/SIL working group the preparation of the SIL pack documents: the outcomes template, participant profile, the property profile, the roster of care and the hourly breakdown template. Ensuring quality and efficiencies in the completion of the SIL will maximise plan amounts for participants. Outcomes reporting is an essential requirement under the NDIS reporting framework. CLO have robust systems for tracking goals and outcomes.\nAn area of concern moving into the NDIA model and tabled within the sector has been crisis and emergency support systems. We are pleased to announce that we have supported the new NDIA crisis and emergency system by agreeing to be one of the contracted SA providers to provide emergency and crisis support. We are now building capacity to ensure responsiveness to this system in SA.\nJuly will bring our new strategic planning phase and I thank everyone for their participation in our analyses that underpin our strategic direction and objectives for 2018-2020. This included all teams participating in a SWOT analysis, a family survey, the CLO culture survey and consumer standards and choice and control questionnaires. We continue to believe in our Vision for providing 'great opportunities and a great life' and now enter a new strategic direction 'Delivering Excellence' with our excellent results in the ASES accreditation at both certificate and award level and the implementation of new positions in our clinical team and the Quality\/WHS and RTW role we are poised to deliver excellence and demonstrate quality outcomes as we transition into NDIS. It is with pleasure that we now deliver our new strategic plan that extends our key result areas to include 'Leadership' and 'People and Culture', it is through our leadership and our staff that we will continue to deliver excellence and achieve our goals and vision.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The way that many companies deliver SEO services is a broken model. In this post we walk through an example, you might even have lived this yourself. It should not be SEO OR content marketing, we believe both strategies work together. Read on for thoughts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The following Transfer Out process applies to the majority of domains supported by Hover. If you have a ccTLD (country code top level domain such as .io, .de, .ch, etc) with special instructions, or that doesn't follow this guide, please contact support and we'll be able to give you a hand.\nNote: You can transfer your domain away from Hover at any time. You do not need to wait until the expiration date. In fact, it's recommended that you don't leave a transfer until the last minute so you have plenty of time to get it completed. The majority of domains will be renewed for one year, with the New Provider, upon transfer.\nIf you have more than one domain name in your account you will see a list of Your Domains. From this list you can click on the domain name you are transferring away. You can also click on the Edit menu, and choose More.\nIf you only have one domain name in your account, you will be on the correct page upon login, so you can skip ahead to the next step.\nOn the domain Overview page, you can now prepare the domain for transfer.\nA) Set Whois Privacy to Off.\nB) Set the Transfer Lock to Off.\nC) Retrieve your Authorization Code, and copy it down. You will need to provide this to the new registrar so they can complete your transfer.\nD) Check the Admin & Owner contact to ensure the Email Address is valid and you have access to it. If you need to edit the email address, you can click on Edit. If you need help editing this, you can check out our help article, How To: Update Registration Records.\nNow that these steps have been completed, you can have the new provider initiate a transfer for you. Please contact a new service provider of your choice and ask them to initiate the transfer.\n- The new provider may require the admin or owner contact email address to receive and approve the transfer before it can be initiated. Once this is approved, the transfer will be submitted.\n- The transfer process is automated. If you have initiated a transfer with the new provider, and you have not yet received an email from Hover to approve it, the transfer probably hasn't yet been submitted to us. When our system receives a request to transfer a domain out, we immediately send an email to the Owner Contact Email Address asking them to approve or decline the transfer. If you approve this, the transfer will complete right away.\n- If you do not approve this final email, the transfer will auto-complete in 5-7 days.\n- To cancel a transfer OUT of Hover, you must either decline the transfer using the link in the email we would have sent you, or, you can contact support to see if we're able to resend this email to you, so you can decline the transfer. This may not always be available, and once you have approved the transfer, we cannot cancel it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Known for its compact size and proven firepower, the UZI was first commissioned for use in 1954 and served as the Israeli Defense Force primary weapon of choice. Now, you can pay tribute to this iconic weapon with this White Gold Pendant that uses real White gold plating to create this one-of-kind piece that uses AAA stones. Included with each is a 2.5mm, 26\" stainless steel Franco chain.\nAmazing gift! Definitely eye catching!\nAwesome chain , hope u'll replenish soon !\nThat White gold uzi piece is one of our favorites! Thanks for repping King Ice Jake!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Putting on one side the personal tragedy and its effect on her two sons , families and friends , don't the vast bulk of C and W artists make you sick ?\nBad voiced victims bleating and whining everywhere they go and about everything they do .\nWhy is this genre classified as Music? What relevance has a bunch of Hilly Billy pub singers to the rest of the world ?\nThis particular lady had more issues than an extremist Republican at a charity ball for helping other people .\nAnybody else here who loves music and loves the many genres within world pop music , but detests the narrowness and invariable ghastly noises that make up this oddball minority masochistic community of people that are dedicated to ruining good music ?\nMindy McCready, who hit the top of the country charts before personal problems sidetracked her career, has died in an apparent suicide.\nThe 37-year-old was found dead on the front porch of her home in Heber Springs, Arkansas, on Sunday from what appeared to be a self-inflicted gunshot to the head, authorities said.\nI think only sickos have problems with C&W music.\nWhy would balanced people , who use the whole world as their reference , be expected to like tribal music from the insular Hilly Billy Land ?\nA bit like the inmates believing that it is the carers and doctors who are the problems .\nNobody is stopping or banning your tribal ditty making .\nWith luck it is a sort of therapy which puts a hold on even more violence .\nHe is not a sicko, he is a liberal, they have always wanted to kill those that might disagree with their view of the world. Cary is no different. Out of one side of his moth he will tell you how compasionate he is, and out of the other call for the murder of those that do not agree with him.\nTone down the personal attacks.\nI heard her boyfriend did the same just a few months ago ... sad ... yet again.\nThere is only so much personal tragedies a person can take, and with her history it wouldn't surprise me to find out she had underlying mental health problems, (people use drugs for a reason, usually). While it's tragic, I don't find it surprising. I feel very sorry for her children, particularly.\nWell as we've seen time and time over and over again ... it don't take much for someone to snap.\nAt least she didn't blame the world and take out other first.\nYeah, same here. OK he's trying to be funny. :rollseyes: It's not funny though.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please fill out the Contact Information Form below. Your submission will enable you to access to all Subsidy, Discount & Rebate Programs offered through The Renovation Co-op. Note: No personal information is ever shared or sold to 3rd parties. All client information remains strictly confidential.\nWould you like an estimate for the renovations upon completion of Drawings?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our monthly Italy Blogging Roundtable theme this month is \"My local\u2026\"! Take a look at posts by Kate Bailward, Jessica Spiegel, Melanie Renzulli, Alexandra Korey, Gloria, and Michele Fabio. (If you missed the previous months, take a look here.) Welcome back to our table\u2026come pull up a chair and join in on the conversation.\nSometimes I feel like I have lived through the 1970's twice.\nEven the larger stores were local chains. Our grocery store of choice was Honiotis Bros. because, you know, Greeks. (The Xoniotis family, who became the Honiotis family, was from Mykonos like our Theodosis and Vardoulakis \u2013 now Vardal \u2013 families, so we bought our carrots and toilet paper from Honiotis' out of national pride.) But sometimes we would make a big trip to Dominick's, which was a local chain. If we had to stock up on school clothes, it was off to to Wieboldt's or Goldblatt's (Wieboldt's was better, because they gave out S&H Green Stamps), but a family wedding merited an excursion to Kline's or The Boston Store. We loved The Boston Store, because the name conjured up that sophisticated and exotic city on the East Coast.\nAnd then things started to change, and we all know how. First it was large supermarket chains that offered unbeatable prices during the recession, then it was newfangled malls that replaced the main streets for teenagers and adults alike. Not long after, the first big-box stores appeared, funneling business from the locally owned shops, and the vacant storefronts were replaced by national franchises.\nNone of the businesses I remember from my elementary school years are still around. Honiotis went first in 1985, then Dominick's began to falter. Wieboldt's, Goldblatt's, Kline's, and The Boston Store (not to be confused with Boston Store)\u2026all gone. Now it's chains as far as the eye can see, and everything from the suburbs to the downtowns look pretty much the same across great swathes of the US.\nWhen I first came to Umbria in the mid-1980's, in many ways it resembled the US a decade or two before. Franchises and big-box superstores were virtually unknown, and the retail sector was almost exclusively small, family-run businesses. Grocery shopping was divided between the local outdoor market for produce, the dry goods store, the butcher, and the bread shop. Buying a pair of black pants meant stopping in at one or two central emporiums, announcing that you needed black pants, and trying on whatever they brought you from the shelves. It was more time consuming and less efficient, but also more human and kept residents living in the otherwise inconvenient confines of the town centers.\nUnfortunately, the same process that tore the fabric of American downtowns twenty years before began taking hold in Italy shortly after my first trip. The convenience and competitive pricing of supermarkets began to squeeze out the tiny markets and food shops, the novelty of the mall trumped the fustiness of historic clothing stores for younger customers, and the powerhouse marketing of national and international franchises crushed local shops. I have watched in dismay over the past two decades as more and more local businesses struggle while Foot Locker, H&M, and even the Italian chain Intimissimi seem to multiply overnight like mushrooms.\nThough, in my heart of hearts, I long for an Ikea, I also have seen (twice!) the damage this modern franchise culture can do to communities and their local economies. I try to limit my excursions to the mall and the sprawling grocery stores along the highway to dire emergencies, and spend my time and money in the admittedly more expensive but also charmingly timeless shops in the center of Assisi.\nCase in point: the Piazzetta delle Erbe. This tiny square just steps from Assisi's main Piazza del Comune has been the local produce market for decades, if not centuries. Certainly long enough that the spot was officially dubbed \"Greens Square\" at some point and is now home to an excellent restaurant of the same name.\nThe home I stayed at on my second trip to Assisi in the late 1980's had rooms overlooking this square, including my bedroom. I would wake to the friendly squawking of the local ladies bargaining for everything from potatoes and tulips each morning, mixed in with local gossip and good natured ribbing. The Piazzetta delle Erbe was both market and meeting place, and the small space was crammed with makeshift stands and tables, three-wheeled Apes, or simply stacked crates holding towers of seasonal fruit, vegetables, fresh eggs, ricotta, honey, and anything else these farmwives from the surrounding countryside had to sell that morning.\nToday, just Novella remains. With enough energy and warmth to fill a piazza, but with just one lone stand of goodies she and her sweet husband Bruno bring in from their farm plot outside of town each morning, Novella holds court from dawn to lunchtime each day. She is almost never alone, as the local ladies take turns resting on her guest stool to swap news while she tirelessly rearranges buckets of fresh flowers, piles of greens, and crates of fruit. She holds the scales in her hand to weigh purchases, and then always throws in something extra after declaring an (often seemingly arbitrary) price.\nIt makes be both sad and joyful to see Novella still out there every morning. \"Bongiorno, core!\" she calls out as I pass. She knows what each of my sons prefer, and will spend a good five minutes picking the radicchio leaves out of my mixed greens to please them. She will scoff at my selection of tomatoes, tossing them back into the pile and choosing others. \"Those are for salad, cocca. You want the sugo ones,\" she explains after placing what look like identical ones on the scales. She will pick out a melon with all the gravity of a Antwerp diamantaire, after inquiring about the exact time I plan on serving it.\nI know it takes me twice as long to buy from Novella, but I love the familiarity of it. I love being grilled by a group of housewives about my menu for the day, and then standing back as they argue amongst themselves about recipes and ingredients. I nod and smile, often feigning exaggerated ignorance just to revel in their animated conversation. The vast Coop supermarket will be there for years into the future, but one morning soon Novella will be gone, and with her the Piazzetta delle Erbe market. And until that day comes, she's my local go-to vegetable lady.\nSo sad to see the small markets losing ground to the big box stores, even in Italy. I'm looking forward to our CSA share from Cape Cod Organic Farms down the street. Even though the share will be very small in the first few weeks (this being New England, where we recently had snow on the daffodils), I love buying food when I know where, how, and by whom it was grown.\nLove your blog! I am Italian-American and grew up in the 70's on NY's Long Island and Brooklyn, making the rounds with my mother \u2013 to the butcher to get the chicken, to the deli for lunch meat, and to the bakery for desserts, I've seen these small places largely abandoned for grocery store chains as time has gone on. But, almost miraculously, at least the farmers' markets have reappeared in most cities and larger towns. And now I have found a real butcher who makes the farmer's market rounds and fresh vegetables and fruit again in the summer. I am so happy to shop outdoors for this high quality food again!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbanxv b/data_all_eng_slimpj/shuffled/split2/finalzzzbanxv
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"The successful concept of the old Wave is been optimized by the development of the new Wave 2. We achieved less weight, a balanced starting and flight behaviour, a very low tendency for side and front collapses and immediate opening after collapses in turbulent air. These features make our Wave 2 to a modern and one of the safest in its class.\nThe Wave 2 is the right companion in the first years of your paragliding life. Because of its direct handling, its speed range and its high performance is this glider the right wing for beginners and also for pilots with XC ambitions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Godzilla has always been a tabula rasa, more than a monster, a symbol of nature, and nature's fury, of man's folly, the limits of science and knowledge, of the monster we inadvertently create returning to humble our arrogance, but Godzilla's fury is also that of an injured beast, lashing out, a mother threatened, defending a child, a child itself, thwarted, angered, its tantrum like a storm seeking only the calming depths of mother ocean.\nGodzilla is also a defender against far greater threats, ravenous alien chimerae and unrelenting, heartless technology. Our monster, our creation, our destroyer and saviour. Out of destruction of the old order, a new understanding, a more harmonious life comes into being. Godzilla is a lesson, a warning, a teacher, a hero, a protector of the innocent. A blank slate, with a vast heart, a voice of fire and oh so human eyes.\nGodzilla began life as the hokey monster in a cheap 1954 Japanese sci-fi film, a rehash of the Hollywood B movie creature feature, The Beast From 40,000 Fathoms, from the previous year. While both played on atomic age fears of the unknown consequences of atomic warfare and nuclear testing, Godzilla was more than a mere rampaging prehistoric monster. Godzilla mythologized the unfathomable horror of the atomic destruction of Hiroshima and Nagasaki, so that through understanding, healing, and renewal could begin.\nHow does an atomic mutant, part cautionary tale, part Shinto god of destruction, connect with one of the original nerdish anthems, one of the most poignant and beautiful songs of the 1970s?\nThrough the remarkable Janis Ian, singer songwriter, musician, Grammy winner, science-fiction author, poet and composer of Godzilla haikus. Her song Society's Child, about love across the racial divide in 1960s America was an icon of the era. The subject so controversial at the time that some stations wouldn't play it, and Janis received death threats. The original release has something of the feel of a 1960s girl group pop love song, not surprising as it was produced by Shadow Morton, who wrote and produced Leader Of The Pack for the Shangri-Las, but also Janis's soulful folk guitar, cut through with some psychedelic harpsichord and funky electric organ.\nAt Seventeen, her poignant, sad beautiful reflection on youth, on being an outsider, an ugly duckling in a milieu that values popularity, attractiveness above all else, touched hearts around the world in 1975, and is as timeless, as universal, as moving today. You can find her classic albums on iTunes.\nIn a 2009 performance and conversation (at 92nd Street Y) Janis said of the song, \"I really was totally convinced that I was the only person on Earth who had ever felt this way, and it was so embarrassing to me, and so shameful and humiliating to me, that when I finished writing it, I promised myself I would never sing it, because I couldn't imagine getting up in front of an audience and exposing myself this much to the possibility of ridicule.\" We can only thank Janis for her honesty and bravery. I know as a weird, unattractive, outsider kid it moved me, made me weep, consoled me, told my story too. I guess it's sort of a nerd anthem. A love song for the unloved. It is a song loved around the world; it told a lot of people's stories.\nBoth songs have been inducted into the Grammy Hall of Fame, she has won two Grammy Awards and been nominated 9 times, and that's not the least of her musical creativity. To stay up to date, go to janisian.com. She still performs, tours, writes today. For something amazing, see her on stage with another legend, guitarist Tommy Emmanuel, last year at The Philadelphia Folk Festival.\nJanis Ian has always been a reader of science fiction, a fan, and more recently a writer of science fiction short stories. There's even a collection of short stories by other authors based on her songs. One of her most recent pieces is Dark Carbuncle \u2013 with Kevin J Anderson, about a dead rock star whose fans can't quite let go, which you can get here for iBooks and also on Amazon.\nSant\u014dka Taneda said, \"Haiku is not a shriek, a howl, a sigh, or a yawn; rather, it is the deep breath of life.\" One can only wonder what the traditional masters would think of Godzilla Haiku. A traditional haiku is marked by the juxtaposition of ideas, and a word that both divides and joins the two images. Traditionally haiku explored nature, and would use a kigo, a word that referenced a season. Thus kawazu (frogs) recall spring, hachisu (lotus) midsummer, kakashi (scarecrow) autumn. The early 20th century saw a movement toward free-form Haiku, on from the strict adherence to the traditional 5\/7\/5 syllable scheme, nature and the kigo, to a form that explored diverse aspects of experience, while adhering to the principle of juxtaposition. How things change and turn. Haiku has spread around the world. In English, in Australia, bushfire could well be a kigo that references summer, but also fear, danger destruction. In California, volleyball might be a kigo that calls forth summer, sunshine, play, youth. In effect the kigo is a shorthand that invokes both the thing itself, and its context.\nWhat does Godzilla invoke? Here there are no hard and fast rules. Is Godzilla perhaps a kigo of winter, destruction, or as the creature returns to the sea, a kigo of another sort, signifying depths, the soul, the id? The context is often determined by the accompanying image, baby Godzilla with children, Godzilla dancing with 1960s Japanese pop idols, Godzilla wreaking havoc.\nJanis Ian has been composing haikus, inspired, connected, referencing Godzilla in various ways for over a year. She also sometimes invites fans to submit their Godzilla haikus. She shares them on her Facebook timeline, (also a great way to keep up with news on her music and SF writing).\nMany are witty, clever, glib, playing with the monster, the form, or the process. Godzilla is an empty canvas, on which we can paint joy, sorrow, humour. The foolishness of a man in a rubber suit easily becomes as important as the grand mythology. Most adhere strictly to the 5\/7\/5 pattern of syllables, some don't juxtapose and join. Some barely light on Godzilla at all, words turn obliquely, hesitant nevertheless, amuse and surprise. Check the gallery for a selection, and find more on Janis Ian's Facebook, and on Samurai Frog's Godlzilla Haiku Tumblr.\nWithout wanting to sound too much like a fan having a fan moment, I really have to thank you, for your beautiful songs, your spirit, and for re-introducing me to the haiku.\nLove Janis Ian's songs, and Etheridge Knight's Haiku is wonderful. Do click on the link to read the whole thing.\nThanks for introducing me to Janis Ian!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This lecture series, dedicated to the honor of Ronald Coase, began in 1992. The lectures are presented by program faculty and are intended to provide law students and others with an introduction to important techniques and results in law and economics. The lectures presuppose no background in the subject.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The name of the Andaman Islands is ancient. A theory that became prevalent in the late 19th century is that it derives from Andoman, a form of Hanuman, mentioned the islands and said that the name means \"Island of Gold\"\nLocated in the Bay of Bengal, Andaman and Nicobar Islands is an archipelago consisting of about 600 islands out of which only around 36 islands are inhabited. The archipelago is divided into two groups of islands - the Andaman Islands and the Nicobar Islands. Of these, only 9 islands in the Andaman Islands group are open for tourists. In earlier times, the islands were referred to as Kalapani for their notorious penal settlement. Pristine beaches, gorgeous corals, fascinating marine life, adventurous water sports and the remain of Stone Age culture draw multitudes of tourists to these islands every year. These islands are covered with lush forests and endless varieties of exotic flora and fauna. Incredible corals and marine life, crystal clear water and mangrove-lined creeks lure travellers to these picturesque islands. while the sandy beaches form as nesting homes to turtles, animals such as spotted deer, wild boar, gecko, crab-eating macaque and python can be spotted in the 86% area still covered by dense forests.\nHavelock Island is one of the largest and most popular isles in the Andaman and Nicobar group of islands. Spread over a massive area of 113.93 square kilometres, Havelock Island is situated 57 km north-east of the capital city Port Blair. The island is named after a British General and comprises of Ritchie's Archipelago and a collection of five villages including Shyam Nagar, Vijay Nagar, Radha Nagar, Krishna Nagar and Govinda Nagar. It is one of the most developed islands when it comes to tourism, with a focus on eco - tourism.\nLarge coral reefs, an assortment of marine life and the calm blue waters of the Bay of Bengal all make the name \"Emerald Isle\" highly suitable for the Andamans. Most popular island among all the islands for the dives, however, is Havelock. It is an ideal diving destination for everyone, right from a novice to an experienced diver.\nThe Cellular Jail, also known as 'Kala Pani' is an old colonial prison situated in Port Blair, the capital of Andaman and Nicobar islands. Constructed by the Britishers during their colonial rule in India, Cellular Jail was used particularly to exile political prisoners where they were subjected to many atrocities at the hands of the British. The construction of the jail began in the year 1896 and was completed in 1906, after which it was used to house many notable freedom fighters such as Batukeshwar Dutt, Yogendra Shukla and Vinayak Damodar Savarkar.\nSnorkeling is another popular water sport among tourists. Sea around Havelock Island serves as one of the best sites for snorkeling.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This program is sponsored by New Balance and is a great progression for those who have completed the Run-a-Mile or 5K Training Program or those who are just starting to run. This program lasts 12 weeks and will be training for the Lexington Medical Center Heart & Sole Women's Five Miler on April 25th. Please plan to attend orientation on Wednesday, January 28 at 6:45pm.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbceir b/data_all_eng_slimpj/shuffled/split2/finalzzzbceir
new file mode 100644
index 0000000000000000000000000000000000000000..81a3ecb56765c4ab80159860f053330ce61a0e61
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbceir
@@ -0,0 +1,5 @@
+{"text":"Blue bell miniature tea - a very requested item! A miniature tea set for your favorite little girl - or any collector! Includes a teapot, warmer, cream & sugar set, and 4 cups and saucers. A must have! Tea cups are roughly 1 inch tall.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Abstract: This paper examines undergraduate student use of laptop computers during a lecture-style class that includes substantial problem-solving activities and graphic-based content. The study includes both a self-reported use component collected from student surveys as well as a monitored use component collected via activity monitoring \"spyware\" installed on student laptops. We categorize multitasking activities into productive (course-related) versus distractive (non course-related) tasks. Quantifiable measures of software multitasking behavior are introduced to measure the frequency of student multitasking, the duration of student multitasking, and the extent to which students engage in distractive versus productive tasks. We find that students engage in substantial multitasking behavior with their laptops and have non course-related software applications open and active about 42% of the time. There is a statistically significant inverse relationship between the ratio of distractive versus productive multitasking behavior during lectures and academic performance. We also observe that students under state the frequency of email and instant messaging (IM) use in the classroom when self-reporting on their laptop usage.\nRecommended Citation: Kraushaar, J. M. & Novak, D. C. (2010). Examining the Affects of Student Multitasking with Laptops During the Lecture. Journal of Information Systems Education, 21(2), 241-252.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Once Upon A Time: A Story About Change.\nIn J.K. Rowling's Harry Potter series, Harry, Ron and Hermione frequently find their way out of trouble by digging through stacks of books in the Hogwarts library and finding obscure, but revealing texts from the magical world in which they live. As a result of their study in the library \u2013 yes, sometimes even in the restricted section \u2013 the three young wizards are able to stay one step ahead of Harry's arch-nemesis, Lord Voldemort.\nOf course, most of the inhabitants of the magical world are so frightened of Lord Voldemort that they do not even speak his name, instead referring to him as \"He Who Must Not Be Named\".\nIf there has been a villain in the muggle world of independent booksellers over the past decade, his name would not be Voldemort, but Barney, which is the industry nickname for bookselling behemoth Barnes and Noble.\nBarnes and Noble made a splashy entrance into the national bookselling scene in the late 80's and early 90s, and like other big box retailers, they quickly leveraged their economies of scale and distribution muscle to grow and eventually eliminate many independent booksellers in towns across America. The loss of Conkeys Bookstore in Appleton was a well-documented example of this in our own area.\nBut, as the country song goes, sometimes you're the windshield, and sometimes you're the bug.\nThe past few years have given us numerous examples of how rapid technological change can quickly turn corporate giants with a seemingly insurmountable market advantage into surprised laggards trying to adapt to new markets they don't fully understand. Barnes and Noble, and their competitor Borders, have essentially re-ordered and dominated the physical bookselling market for the past several years, but the world is changing, and both of them must be surprised to find themselves suddenly struggling for survival.\nBarnes and Noble is losing money, intends to reduce stores, and is looking for a buyer. Borders is in even worse shape, and one of their key investors has recently put forth an offer to finance a merger of the two companies. While the outcome is unknown at this point, what seems quite clear is that they are both about to undergo huge change. I'm betting that there will be a merger or major acquisition in the near future.\nHow did these two companies \u2013 who are, to some, the bullies of the book world \u2013 get pushed into this position of weakness?\nAs with most retail shifts, the customers are voting, and in this case less of them are voting for physical books. If you've regularly visited any of these stores over the years, you may have noticed lately that some of the shelf space typically utilized for selling books is now being used to sell games and gifts. While this observation dismays the book-lover, the reality is that many people are either opting to purchase online via Amazon, or are moving to new forms of book consumption, most notably e-books.\nAccording to the Association of American Publishers and the International Digital Publishers Forum, the e-book market has grown over 500% in the past 3 years and represents about 9-10% of the adult trade paper and hard-back book sales. Moving 10% of a market from one type of provider to another is a big deal, and the shockwaves that are being felt in corporate boardrooms today will become increasingly apparent in many communities.\nIf Hermione could use her time-turner to travel back a few decades, she would have seen young people like herself walking out of their neighborhood library with a stack of books that were checked out with a library card. Book reading was predominantly a borrowing phenomenon, and America prospered \u2013 and still does today \u2013 because of community funding of local public libraries. Purchasing books may have been limited to a few favorite classics, as well as an expensive encyclopedia set prominently displayed in the home.\nIn the past decade or two, in part because of the proliferation of big box book retailers, we have seen the paradigm of book consumption move from \"borrowing\", to \"owning\" \u2013 and it is now headed toward \"downloading\".\nHowever, when there are seismic shifts in a market, there is the danger of blowing it all out of proportion. I believe our area can support book retailers for many years to come, although the numbers and names may change. Many of us have enjoyed the varied book selection and ambiance that these retailers offer, as well as the passion that their employees have for books. I, for one, spend a significant percentage of my waking hours reading content on a computer screen, and so the e-book does not hold any allure for me.\nBut the written word will continue to engage eager minds for generations to come. You needn't be a wizard to realize that sitting someplace comfortable and reading a great book \u2013 in whatever form \u2013 can be magical.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Friday Harbor Wolverines fastpitch team defeated the Orcas Vikings, 11-6, in its first league game this season at Orcas on April 5. The weather was wet and windy as the Wolverines battled their way to a tie at the sixth inning.\nA home run swing into right field by Baille Lambright signaled a shift in the game's tone in the top of the seventh. Overall, Friday Harbor tallied 16 hits to Orcas' six. Liz Cruz, Bristol Halverson, Ava Hoffman, Olivia Germain and Thea Seitz each hit multiples for the Wolverines. Soma Andrews, Darcy Ayers and Jayna Ott each completed a lap around the bases for a point; Germain, Hoffman and Seitz went twice; Halverson, 3; Lambert, 4; and Cruz, 5. Ott was struck by the pitch in the top of the first.\nThe team was allowed three walks, struck out nine times, had 8 RBIs, stole seven bases, and went to bat a total of 38 times.\nOn Tuesday, April 9, the Wolverines stay home to play La Conner at 4 p.m. Then there is a double-header for the team against Concrete at 11:30 a.m. and 1:30 p.m. on Saturday, April 13.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"IWC Shaffhausen has come a long way since they started the United States in 1868. IWC, a replica of the international watch company, was founded by Florentine Ariosto Jones in 1868 in the United States.\nAbout 100 years later, the IWC Portuguese7 day power reserve. What does \"7-day power reserve\" mean? This means that when the watch is fully chained, it will continue to run for seven days without stopping. The average automatic swiss copy IWC watch has a power reserve of about 48 hours. If you wear your watch every day, it still doesn't mean it will always work on a full or even half a can. With a seven-day power reserve, you can last several days without wearing a watch, and when you do, there may be enough power.\nThe Portuguese have a serious and simple elegance without any fussy design appendages. The Portuguese were considered a luxury fake IWC watch, a big watch, and its purity began to give it an honest, attractive quality. Replica IWC Portuguese 500004 watch 18kt rose gold 42.3mm watch case.\n18-karat rose gold watch ring. Silver dial with 18k red and gold Arabic numerals. 18K rose gold crown. Automatic movement, Calibre 51011,42 gems. Leather strap with 18k rose gold clasp. Hour, minute, second, date and power reserve indicator. Power storage is up to 168 hours. Water resistant up to 30 m \/ 100 ft. IWC long black crocodile leather watchband, fixed in best AAA IWC replica stainless steel nail buckle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdhzq b/data_all_eng_slimpj/shuffled/split2/finalzzzbdhzq
new file mode 100644
index 0000000000000000000000000000000000000000..f3db05bb8f1618244d646ec71ef69925910bb701
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"We have 15\/24 slots filled. Several have been removed for being inactive. We need active players who will contribute. Serious players only. No beginners.\nWe have 12 members and we made the top 1000. Interested at all?\nWe were close to the top 100 until the end, we interested in a merge, faction ITC 558 209 279.\nWe would gladly accept you all. I am happy to make your faction leader an officer and his top officer one as well. We are all active and getting stronger.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Test your powers of thinking with over 500 challenging problems and puzzles. \"Mensa\" presents a collection of problems that require either logical deduction or lateral, imaginative thinking to solve. To help you, the sections are divided so that you know what kind of thinking process is required. To hinder you, \"Mensa\" has come up with the most varied collection of problems, puzzles and conundrums you could wish for. Mensa, the international high-IQ organization, is known as a society for people who pride themselves on their ability to solve problems by the relentless application of intelligence. Logic is king in our world, but there are other kinds of problem that cannot be solved by logic alone, problems that require a few moments of consideration before an answer can be found.\nDave Chatten and Carolyn Skitt are all Mensa members and highly experienced creators of puzzles and word games for many books that have been published around the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Grunge rock n' roll version of the American Flag!\nThis series is limited to just 30 total.\nUsing 13 individual pieces of distressed and hand-painted wood and 50 individual hand painted and distressed wooden sculls, this piece is sure to make an eye-catching addition to any room in your home!\nEach piece is signed and numbered and has a wire backing for easy wall mounting.\nAvailable in 32.5\"x48\" and 49\"x66\" Use the drop down menu to select size and view pricing.\nOnly have good things to say, great service minded, and fast replies over mail.\nTruly awesome piece of art!\nThe Picture doesn't make it justice\u2026 Great photos but even better in person!!\nCan't be happier, and arrives well packed!\nGreat Design. Was slightly damaged from shipment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Haley and Fame were hosting the Dub and Roots stage at Jim Beam Homegrown when this happened!\nMai Home Run chose an artist who was set to perform on the stage and got the crowd to cheer for the one they were looking forward to the most, the one with the least cheers had to get their top off!\nHaley chose David Dallas, Fame chose Sons of Zion... and Fame definitely got the most cheers. So DJ Sir-Vere dropped the beat and Haley got it off, watch it all happen above!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tennessee, Florida, Texas, California, New Jersey, Washington, D.C.\nKurds in the United States refers to people born in or residing in the United States of Kurdish origin.\nThe majority of Kurdish Americans are recent migrants from Turkey, Iran, Iraq and Syria. Most have roots in northern Iraq or northwestern Iran. Iraqi Kurds compromise the largest proportion of Kurds living in the US.\nThe first wave of Kurdish immigrants arrived as refugees during the 1970s as a result of the Iraqi\u2013Kurdish conflict. A second wave of Kurdish immigrants arrived in the 1990s fleeing Saddam Hussein's genocidal Anfal Campaign in northern Iraq. The most recent wave of Kurdish immigrants arrived as a result of the 2011 Syrian Civil War and the 2014 Iraqi Civil War, including a number who worked as translators for the U.S. military.\nIn recent years, the Internet has played a large role in mobilizing the Kurdish movement, uniting diasporic communities of Kurds around the Middle East, European Union, Canada, the US, and Australia.\nKurdish immigration to the US began in the 20th century, after the World War I, with several waves of migration to the United States from the area considered \"Kurdistan\". Following the war, the Iraqi Revolution increased the emigration of Kurds to the United States. After the war, the Kurds had been promised an autonomous region, Kurdistan in the Treaty of S\u00e8vres in 1920.The ideology of the time was heavily influenced by Woodrow Wilson's doctrine of the right to self-determination.\nAfter the Turkish War of Independence, however, the treaty was annulled and replaced with the Treaty of Lausanne, which denied any Kurdish claim to an autonomous region. After the reversal of Kurdish land claims and ensuing persecution, Kurdish emigration from the Middle East began. Many diaspora communities were established in Europe and other Kurds emigrated to the US.\nFollowing the wave of migrants that left in the aftermath of the World Wars, a second wave of Kurds in 1979 came from the north of Iraq and Iran. Migration was mainly due to the Iranian Revolution in 1979. There were also large groups of Kurds that left because of the socio-political turmoil, a byproduct of the revolution and general political instability. Many of the immigrants who made the journey in 1979 had hoped to overthrow the Shah or at least opposed him. The opposition to the Shah, many immigrants from Iran were granted asylum with little trouble and received assisted travel to the US. Other byproducts of the revolution were border disputes between Iran and Iraq, which culminated in the Iran\u2013Iraq War, from 1980 to 1988.\nThe third distinguished wave of Kurdish migrants arrived between 1991 and 1992 and is considered to be the largest of the four waves. The migration was partly due to Kurdish support for Iran in the Iran\u2013Iraq War because Saddam Hussein had retaliated by attacking multiple Kurdish regions with chemical weapons. The most horrific and infamous of these attacks occurred at Halabja in 1988. Although different groups of Kurds have alternate interpretations of the attack, Kurds usually regard the event as evidence of genocide against the Kurdish people and have used that claim for political gains. Although there was widespread support for Iran from Iraqi Kurds, the war caused severe internal divisions within the Kurdish population. Thousands of Kurds then moved to the US.\nAnother major wave of Kurdish migration to the United States or at least to Nashville, which has the largest concentration of Kurdish communities in the country, was in 1996 and 1997, following a major civil war between Iraqi Kurdistan's two major political parties, the Kurdistan Democratic Party (KDP) and the Patriotic Union of Kurdistan (PUK). The Iraqi army began targeting hundreds of individuals accused of working against Saddam's regime. The International Organization for Migration initiated an evacuation. Kurdish refugees crossed the Turkish border, after which they were evacuated to Guam, a US territory, and later resettled in the US.\nAfter 2008, several more groups of Kurds immigrated to the US. Many of the migrants had worked in some capacity either with United States military or humanitarian efforts, international humanitarian organizations, and other NGOs.\nThe total Kurdish population in the United States according to the 2000 census was 9,423. More recent accounts estimate the total Kurdish population in the US at around 15,361. Other sources claim that the number of ethnic Kurds in the United States is between 15,000 and 20,000 people.\nThe city of Nashville, Tennessee includes the largest Kurdish population in the United States. A neighborhood in south Nashville has been nicknamed as \"Little Kurdistan\" due to its large number of Kurds. Based on the 2000 census, the Kurdish population in Tennessee is 2,405, and in Nashville, the Kurdish population is 1,770. According to Salahadeen Center of Nashville, there are more than 10,000 Kurds living in Nashville. As of 2017 there were approximately 15,000 Kurds in Nashville out of the estimated 40,000 that reside in the entire United States. This is the largest population of Kurds in the United States. Most have roots in northern Iraq or northwestern Iran.\nIraqi Kurds are the most autonomous within their respective lands. Iraqi Kurds compromise the largest proportion of Kurds living in the US and the group with the most established relationship with various groups (political, religious, humanitarian) within Kurdistan. The effects of tribal affiliation and power structure effect Iraqi Kurds more than other groups. Tribalism has remained important to Iraqi Kurds largely because of their history of relative autonomy and the strategic geographic position in Northern Iraq, and the effects are still felt by many Iraqi American Kurds.\nKurds in Iraq have been the subject of aggressive Arabization processes following the Al-Anfal campaign of the early 1990s. The Arabization process took place in conjunction with massive displacement of Kurdish families from Kurdish regions, leading to an influx in Kurdish immigrants during the early 1990s. Iraqi Kurdish populations in America have open channels of communication with the Kurdistan Regional Government (KRG) mediated through various groups such as cultural centers, mosques, and other humanitarian organizations. Due to this communication network, Kurdish organizations in the US have provided a significant amount of aid and humanitarian relief to the Iraqi Kurdish population on several occasions, oftentimes going through the KRG to implement relief efforts.\nIranian Kurds, unlike their Iraqi neighbors, were subjected to the aggressive Persianization (rather than Arabization) policies of the former Pahlavi Shahs of Iran in an attempt to promote national unity. The most aggressive of these policies was a language policy by Reza Khan that attempted to ban the Kurdish language, in favor of Persian. Despite oppressive policies such as the banning of the language, Iranian Kurds achieved semi-autonomy for a brief moment in time when the Mahabad Republic rose to power in 1945, with the tacit support of the Soviet Union. The Iranian Revolution that took place in 1979 had multiple effects on Iranian Kurdish emigration because of the economic and political instability that it caused within Iran and the accompanying Kurdish persecution. The international support and attention dissenters of the shah received helped Iranian refugees receive asylum in the US, which led to a large wave of migrants following the Revolution. Kurds occupy an unclear socio-political position in Iran. Although the Constitution of the Islamic Republic of Iran granted equal rights to all ethnic minorities, distribution of vital resources such as healthcare, education, housing, and employment indicate that Kurds in Iran have remained an oppressed group in practice.\nThe \"Kurdish Question\" has been present in Turkish politics since the country's inception l, and the issue has remained contentious. The Turkish government has tried multiple assimilation strategies, including language policies that suppressed the Kurdish language, discriminative economic and employment practices towards Kurdish people and attempting to wholly ignore the ethnic identity of Kurds by referring to them instead as \"Mountain Turks\". There are far fewer American Kurds from Turkey than Iran and Iraq. One reason for the scarce amount of Turkish Kurds emigrating to the US is that many emigrating Turkish Kurds will travel to one of the closer Kurdish diaspora communities within the European Union like Germany, France, the Netherlands, Sweden, and Finland, all of which are all home to varying sized Kurdish communities.\nSyrian Kurds have experienced especially damaging discriminative practices so much so that most Syrian Kurds cannot obtain the necessary documents to travel and\/or emigrate. There are few if any properly documented Kurdish migrants from Syria. Recently there have been many efforts by Syrian Kurds to fight for political freedoms enjoyed by other Syrians, beginning with an estimated 250,000\u2013300,000 Syrian Kurds that were denied citizenship under the Ba'thist regime demanding citizenship status. The current role Syrian Kurds have had in stopping the Islamic State have brought the Kurdish question in Syria back to center stage. The military success the Kurds have experienced against the Islamic State has led to the creation of a semi-autonomous Kurdish region in Syria. Although similar in some ways to the KRG, the vision put forth by the Syrian Kurdistan is quite different from the position of the KRG. Syrian Kurds fighting for sovereignty in parts of Northern Syria that correspond to Syrian Kurdistan have stated their ambition to establish a radical democracy with an emphasis on gender equality, sustainable ecological practices, and an overarching goal to undermine hierarchical power relations.\nThe combination of military success against ISIS with a radical political philosophy has made the Syrian Kurds, particularly those involved with the creation of independent Kurdistan within the Syrian Kurdistan project a focal point of international Kurdish mobilization.\nKurdish immigrants do not share a common language, however there are various dialects depending on the area of Kurdistan a Kurd is from. The Kurds have never standardized the Kurdish language, but texts with Kurdish writing exist as far back as the 12th century. It has been deliberately outlawed from assimilation practices in Turkey and Iran. The language is officially recognized in the Iraqi constitution, alongside Arabic. Kurdi, or Sorani, emerged as the literary expression of Kurdish language and is recognized in Iraq as the official written form of Kurdish. Within the Kurdish language, there are three dialect groups: Northern Kurdish\/Kurmanji (the largest group), Central Kurdish\/Sorani, and Southern Kurdish\/Laki. The existing linguistic divisions within the Kurdish language make it difficult to generalize about what languages are being spoken by Kurds in the United States. English is spoken much more frequently by younger generation Kurds than the older. Although an emerging phenomenon in the US, other diasporic Kurdish communities in the European Union have demonstrated that the immigration experience oftentimes leads to a dissolution of previous existing linguistic barriers, allowing for a more cohesive Kurdish community in the diaspora than was possible in Kurdistan.\nKurds make up the fourth largest ethnic group in the Middle East, after Arabs, Persians and Turks. Despite the size of the Kurdish population, the Kurds have always been a minority in their respective nation-states since the fall of the Ottoman Empire. They currently are minorities in the adjacent areas of Syria, Turkey, Iran, and Iraq. The only area in which Kurds compromise the majority of the population are portions of Northern Iraq; the location of the Kurdistan Regional Government (KRG). Recently, Kurds have been the subject of increasing media coverage as well as academic interest bt their pivotal geo-political role in the Middle East combating militant Islamic groups such as the Islamic State of Iraq and the Levant.\nKurdish culture is problematic to define in absolute terms because the Kurdish population has been heavily influenced by tribal affiliations, contrasting nationalist projects, as well as international support from different ideological factions at different times (most notably the US and Soviet Union). The confluence of these influences has transformed Kurdish culture, creating significant differences between Iraqi, Iranian, Syrian, and Turkish Kurds. However, they all celebrate the Persian New Year, Nowruz.\nAzad Bonni, Edison Professor of Neurobiology and Chairman of the Department of Anatomy and Neurobiology at Washington University School of Medicine.\nEdip Y\u00fcksel, Islamic philosopher and intellectual, considered one of the prime figures in the modern Islamic reform and Quranism movements.\n^ a b \"Immigrants thrive in US country music capital\". Al Jazeera English. Retrieved February 1, 2014.\n^ a b Sawyer, Ariana Maia (June 23, 2017). \"Who are the Kurds, and why are they in Nashville?\". The Tennessean. Gannett Company. Retrieved October 1, 2017.\n^ Ye\u011fen, Mesut. \"Review: The Kurdish Nationalist Movement: Opportunity, Mobilization and Identity.\" International Journal of Middle East Studies 40.3 (2008): 518-20. JSTOR. Web. 13 Apr. 2015.\n^ a b c Ryan, David, and Patrick Kiely. America and Iraq: Policy-making, Intervention and Regional Politics. London: Routledge, 2009.\n^ Baser, Dr Bahar. Diasporas and Homeland Conflicts: A Comparative Perspective. Ashgate Publishing, Ltd., 2015.\n^ a b c d Mansfield, Peter, and Nicolas Pelham. A History of the Middle East. London: Penguin, 2013.\n^ Watts, Nicole F. \"The Role of Symbolic Capital in Protest: State-Society Relations and the Destruction of the Halabja Martyrs Monument in the Kurdistan Region of Iraq.\" Comparative Studies of South Asia, Africa and the Middle East 32, no. 1 (January 1, 2012): 70\u201385. doi:10.1215\/1089201X-1545327.\n^ Yildiz, Kerim, and Tanyel B. Tay\u015fi. \"Iranian State Policy and the Kurds: Politics and Human Rights.\" The Kurds in Iran: The Past, Present and Future. London: Pluto, 2007. N. pag. Ebsco Host. Web.\n^ a b c Yildiz, Kerim. The Kurds in Iraq: The Past, Present and Future. London: Pluto, 2007.\n^ UC Davis. \"Kurds and Refugees \u2013 Migration News | Migration Dialogue.\" Migration News. University of California Davis, n.d. Web. 08 Apr. 2015.\n^ US Census Bureau. \"The Arab Population: 2000\" (PDF). Retrieved November 14, 2010.\n^ \"2006\u20132010 American Community Survey Selected Population Tables\". Government of the United States of America. Government of the United States of America. Retrieved 5 August 2013.\n^ \"Nashville's new nickname: 'Little Kurdistan'\". The Washington Times. Retrieved February 1, 2013.\n^ Patricia, De La Cruz G., and Angela Brittingham. The Arab Population, 2000. Washington, D.C.: U.S. Dept. of Commerce, Economics and Statistics Administration, U.S. Census Bureau, 2003.\n^ \"Salahadeen Center Of Nashville.\" SCN History. Salahadeen Center Of Nashville, n.d. Web. 10 Mar. 2015.\n^ Nashville, Tony Gonzalez The Tennessean. \"Nashville Kurds Rally to Aid Families in Conflict.\" USA Today. 1 Sept. 2014. Web. 2 Mar. 2015.\n^ Sheyholislami, Jaffer. \"Kurdish in Iran: a case of restricted and controlled tolerance.\" International Journal of the Sociology of Language 217 (2012): 19+. Literature Resource Center. Web. 19 Mar. 2015.\n^ Roosevelt, Jr. Archie. \"The Kurdish Republic of Mahabad.\" Middle East Journal 1.3 (1947): 247\u201369. JSTOR. Web. 07 Apr. 2015.\n^ a b Gunter, Michael M. Out of Nowhere: The Kurds of Syria in Peace and War. N.p.: Oxford UP, 2014.\n^ Zeydanlioglu, Welat. \"Turkey's Kurdish language policy.\" International Journal of the Sociology of Language 217 (2012): 99+. Literature Resource Center. Web. 18 Mar. 2015.\n^ Wahlbeck, \u00d6sten. \"The Kurdish Refugee Diaspora in Finland.\" Diaspora Studies 5, no. 1 (2012): 44\u201357.\n^ Baser, Bahar. Diasporas and Homeland Conflicts: A Comparative Perspective. Farnham, Surrey: Ashgate, 2015.\n^ Hennerbichler, Ferdinand. \"The Origin of Kurds.\" Advances in Anthropology 02.02 (2012): 64\u201379. Academia.edu. Web.\n^ Hardach, Sophie. \"'Professor, You're Dividing My Nation'; In Iraqi Kurdistan, tongues are tied by politics.\" The Chronicle of Higher Education 59.41 (2013). Academic OneFile. Web. 20 Mar. 2015.\n^ Haddadian-Moghaddam, Esmaeil, and Reine Meylaerts. \"What about Translation? Beyond 'Persianization' as the Language Policy in Iran.\" Iranian Studies (2014): 1\u201320. Web.\n^ Schmidinger, Thomas. \"The Kurdish Diaspora in Austria and Its Imagined Kurdistan.\" The Kurdish Diaspora in Austria and Its Imagined Kurdistan (n.d.): n. pag. Kurdipedia. Web.\n^ McClam, Erin, and Se\u00e1n Federico-O'Murch\u00fa. \"'The Kurds Deserve Help': American Says He Was Wounded Helping Fight ISIS.\" NBC News. National Broadcasting Corporation, n.d. Web. 18 Mar. 2015.\nThis page was last edited on 15 April 2019, at 15:59 (UTC).","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"2) I restore all kind of artworks. I did\/do\/will do for my own entertainment. If I ever restore some piece for you, you will not push me to finished it in a time interval that you want. I\u00b4m faster, but not Superman.\n3) I don\u00b4t sell my digital files, I only exchange pieces with other users\/collectors. After than, you will do with it\/them whatever you want.\n4) I will share pieces by my own criterion. Reason: Avoid business with my work.\n*I do big screenshots. To see details, download them.\n\"The King of Dragons Bezel\" (ROMSTAR). Restored by Mikonos.\nBefore-After in \"The King of Dragons Bezel\" restoration. By Mikonos.\n\"New Net City Control Panel Overlay\" (SEGA) scanned 600 dpi and restored by Mikonos & Zona Arcade.\nArcade artwork \"Sega Rally 2 Right Side Art Decal\" scanned and restored by Zona Arcade & Mikonos.\nDetails in \"Sega Rally 2 Right Side Art Decal\" scanned and restored by Zona Arcade & Mikonos.\nArcade artwork \"Sega Rally 2 Left Side Art Decal\" scanned and restored by Zona Arcade & Mikonos.\nDetails in \"Sega Rally 2 Left Side Art Decal\" scanned and restored by Zona Arcade & Mikonos.\n\"Naomi Universal Control Panel Overlay NOA - 2003\" (SEGA) scanned and restored by Mikonos for Zona Arcade site.\nDetails in \"Naomi Universal Control Panel Overlay NOA-2003\" scanned and restored by Mikonos for Zona Arcade site.\nDetails in \"Naomi Universal CPO\" (NOA-2003) from SEGA scanned and restored by Mikonos for Zona Arcade site.\n\"Sega Rally 2 Bezel Sticker\" for arcade cockpit cabinet. Restored by Zona Arcade & Mikonos.\nDetails in \"Sega Rally 2 Bezel Sticker\" for arcade cockpit cabinet. Restored by Zona Arcade & Mikonos.\n\"Tiger Heli Side Art\". Restored Zona Arcade & Mikonos.\nBefore- After in \"Tiger Heli Side Art\" restoration. By Zona Arcade & Mikonos.\n\"Zero Gunner 2 Marquee\" (PSIKYO) scanned 600 dpi and restored by Zona Arcade & Mikonos.\nDetails in \"Zero Gunner 2 Marquee\" (PSIKYO) scanned 600 dpi and restored by Zona Arcade & Mikonos.\n\"Rush\u00b4n Attack Marquee\" (KONAMI) restored by Mikonos & Zona Arcade.\n\"Comparative samples in \"Rush\u00b4n Attack Arcade Marquee\" restored by Mikonos & Zona Arcade Site.\n\"Initial D Topper Marquee\" for Naomi Cockpit Cabinets. Scanned and Restored by Mikonos & Zona Arcade.\nDetials in \"Initial D Topper Marquee\" for Naomi Cockpit Cabinets. Scanned and Restored by Mikonos & Zona Arcade.\n\"Egret Side Arts\" (TAITO) restored by Mikonos & Zona Arcade.\nComparative samples in \"Egret Side Arts\" (TAITO) restoration. All original pattern preserved. By Mikonos & Zona Arcade.\n\"Virtua Striker 3 Instructon Sheet\" (SEGA). Scanned at 600 dpi and restored by Mikonos & Zona Arcade.\nDetails in \"Virtua Striker 3 Instructon Sheet\" (SEGA). Scanned at 600 dpi and restored by Mikonos & Zona Arcade.\n\"New Net City Left Side Art Decal\" scanned and restored at 600 dpi by Mikonos & Zona Arcade.\n\"New Net City Right Side Art Decal\" scanned and restored at 600 dpi by Mikonos & Zona Arcade.\nI restored this sticker for one user of this forum.\n\"Virtua Fighter Characters Stiker for bezel\". Restored by Mikonos & Zona Arcade.\nDetails in \"Virtua Fighter Characters Stiker for bezel\". by Restored Mikonos & Zona Arcade.\nDetails in \"Virtua Fighter Characters Stiker for bezel\" restoration. By Mikonos & Zona Arcade.\nI just reproduced this sticker for Jamma+\u00b4s user.\n\"Operation Wolf Gun Machine Start Button Sticker\". Restored by Mikonos & Zona Arcade.\n\"New Net City Side Arts Perfect Reproduction\" printed in chrome vinyl with extra gloss protective lamination (photo 1).\n\"New Net City Side Arts Perfect Reproduction\" printed in chrome vinyl with extra gloss protective lamination (photo 2).\n\"New Net City Side Arts Perfect Reproduction\" printed in chrome vinyl with extra gloss protective lamination (photo 3).\n\"Fork Here Sticker\" for arcade SEGA cabinets. Restored Zona Arcade & Mikonos.\nArcade artwork \"Operation Wolf Control Panel Overlay\" (TAITO). Restored by Mikonos.\nBefore-After comparative in \"Operation Wolf Control Panel Overlay\" (TAITO) restored by Mikonos.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"DFX Audio Enhancer key that is serial your music hearing knowledge by enhancing the sound nature of MP3, internet Radio, Windows Media along with other music documents.A DFX improve the noise nature, more significant commitment, blasting bass and that's just the beginning, supplying good playback quality on your personal computer. In addition to, any type that exists another searcher highlight which gives you fast reach to music tracks, comedy verses, comparable professionals, and that's just the starting DFX updates the sound of all of the best new players.Restore destroyed profundity that is stereo human anatomy.Upgrade quality that is sound music, discourse, along with other noise kinds. Witness DFX significantly improves your noise.\nWitness DFX significantly improves your noise.\nHeadphones Optimization, it is possible to hear more natural and noises which are normal your headphones.\nCrank up the amount with Dynamic Audio Boost.\nHigh Fidelity Restoration that will enable you to eradicate that sound that dampens.\nDownload DFX Audio Enhancer Crack From Given Just Below Link.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kenora Association for Community Living is accepting applications for the position of Full-Time Community Support Worker in the Transitions Program. This is a unique opportunity to become part of KACL's dynamic, engaging, work environment. The successful applicant must demonstrate an understanding of and commitment to KACL's values and principles. Salary, as per Collective Agreement (11.05), defined in Schedule A.\n\u2022 A valid Driver's Licence and access to a vehicle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Good work is lost, or needs to be thrown away with bad during rollback. The cost of recovering the good work, if it is possible at all, would be akin to the cost of re-doing it.\nWork can be lost when it has not been backed up and a crash or mistaken deletion is made. Work can be lost when a basic versioning system is used, and other work must be rolled back. Work can be lost because the versioning system overwrites one person's changes with another person's changes, when work is checked in. How often should one make backups? That depends on the cost of making the backup, the value of the work that is at risk if not backed up, and the frequency of needing to restore the backed-up version. How often should one checkpoint? That depends similarly on the cost of making a checkpoint, the value of work that would be saved by making a checkpoint, and the likelihood of having to roll back to the checkpoint. However, we also need to consider the risk of needing to roll back to an earlier checkpoint.\nThere are few projects that could not benefit from a source control system, and few these days that do not use one. Where this permits PrivateVersioning, loss of one's own work owing to the need of others to roll back can be ameliorated. Use of NamedStableBases allows separation of the work of developing new functionality and that of tracking changes made by other developers. Assuming care is taken in establishing suitability of these stable bases, rollback beyond the most recent is unlikely. Use of complete sets of regression tests on a regular and frequent basis (see also TestDrivenDevelopment) means that local fixes of recently introduced problems are more likely than large-scale rollbacks.\nThe more Agile development processes favour ContinuousIntegration. Many non-agile processes could benefit from TestDrivenDevelopment; it, or at least its full set of unit tests, are a prerequisite for other agile development techniques. Full source control facilities and backups are taken for granted.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"or pain, without complaint, loss of temper, irritation, or the like.\ncalm and without complaint, anger, or the like.\nDr. Wiki also has some interesting things to say about patience.\nPatience (or forbearing) is the state of endurance under difficult circumstances, which can mean persevering in the face of delay or provocation without acting on annoyance\/anger in a negative way; or exhibiting forbearance when under strain, especially when faced with longer-term difficulties. Patience is the level of endurance one can take before negativity. It is also used to refer to the character trait of being steadfast. Antonyms include hastiness and impetuousness.\nAlmost forgot, thank you to every one of you who has been with ACF from the very beginning. Today marks what I'm calling Auburn CrossFit's 6 month anniversary! And thanks to those of you who have since joined and have worked your way in to this amazing community of people. I couldn't be happier with the progress I see in Auburn CrossFit and in our athletes!\nI,m going to try to make it this fine day\u2026.. Dan you are not # 1 today!\nbrilliant. I'll be chanting that during the long row today.\n#4! here's my attempt at blogging just for you Elijah. So great to finally be back, it's been a looong month out of the box. Can't wait to get back on track, it's going to be a rough road. Patience eh?\nYesterday was rough. . .\nCongrats on your 6 months! Glad to be part of the ACF family!!\nKeith . . . u r lookin super fierce here! Well done . . .\nkelsey we missed you at the box get back in it!\nCan't beleive it's been almost six months! Some days it still feels like day 1. I can't say I've progressed as much as I would like, but that is where the patience comes in I guess. Although I didn't keep great records from the beginning, I can tell you that I'm down a little more than 20lbs and 1-2 pant sizes. And I wouldn't take a single day at the box back.\nMy first day of crossfit was 4\/19\/12, it seems so long ago and I really need to work on my patience cause I still want to do things better and faster. Here's to another 6 months!\nCongrats to your 6 month. Look forward to many more to come, love it here. It has been a blast working out with you all.\nToday was deceptively hard yet fun. Finished row in 7:28. Broke the L sits into 8 sets of 15 secs, and got the 100 double unders done it 4 sets (44, 28, 18, 10).\nKelsey . . .. we are so glad to have you back. You truly were awesome today . . . way to jump back in with both feet!\nMaybe I'll improve a little in the next 6 months. Maybeee.\nIf you get a little more sleep you'll probably improve by tomorrow!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"For a long time, us citizens have driven within the Canadian border to buy medication.\nworldwide pharmacy drug prices happen to be reduced compared to prices within the U . s .\nwe deliver unmatched safety, savings and repair to customers over the U . s .\nour easy online register after which fax your prescriptions to all of us!\nmoney in your wallet for that other essential things that you'll require.\nSavings \u2013 We'll save as much as 50% (or even more) in your prescription medications.\nShipping \u2013 it's usually free.\nWe are the initial catalog shopping online Canadian drug pharmacy as well as the best!\nonline internet pharmacies. Browse the table below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Children's School for Early Development knows that early diagnosis and treatment can change the life of a child with developmental disabilities. We believe that children with developmental disabilities and their families should be given a choice of early intervention services and resources to maximize their child's potential. Central to our early intervention model is community integration. Our early intervention services are now delivered entirely in community settings, and more than 80 percent of the pre-school children we serve attend local schools.\nWe are the largest provider of community-based services for children with developmental disabilities in Westchester County. Our team of specialists includes teachers, speech pathologists, physical and occupational therapists and certified social workers.\nBabies from birth through 3 years of age are eligible for our early intervention services. These services are delivered entirely in community settings with more than 80 percent of the pre-school children whom we serve will attend community schools. The Children's School for Early Development is approved to provide Early Intervention evaluations, ongoing service coordination, and Early Intervention services under contract with Westchester County. If your child is found to be eligible for services by a state-approved evaluator, you will work with professional staff to develop a personalized service plan. An ongoing service coordinator selected by the family assists them in locating and scheduling service providers, chosen from the county list, to implement their individualized family service plan.\nFamily needs to first contact the Westchester County Department of Health, 914-813-5094, where you will be assigned initial service coordinator, who visits the home and explains the process.\nYou will then choose an evaluator from an approved list and the initial service coordinator schedules the evaluation. Most evaluations occur in the home or community.\nCompleted evaluations are shared with the family.\nIf the evaluation finds that the child is eligible to receive services, the family meets with the team to determine an appropriate plan of service.\nThe ongoing service coordinator chosen by the family assists the family in locating and scheduling service providers to implement the individualized family service plan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sophomore Class is hosting a Tropical Smoothie Fundraiser TODAY! Please stop by the North Dartmouth location between 4 and 8pm and be sure to mention the class at check out! Hope to see you there!\nCongrats to the Stang Esports Team on their victory over Abby Kelley Foster Charter School in Worcester. Next week begins the Massachusetts esports playoffs which we will be streaming live on Twitch!\nBaccalaureate Mass - we will have a meeting on Wednesday, Day 5 for any Seniors interested in helping with and planning the Mass.\nMass and Muffins - will continue for the rest of the school year. Please join us tomorrow morning for Mass and breakfast.\nNHS Members are reminded that there will be a day 7 flex meeting today. We have a guest speaker today. Please check in then quickly proceed to the cafeteria for the members meeting. Please bring your device find your assigned group quickly, so we can take an accurate attendance. Juniors please bring your printed service form with you to the meeting. Samantha Furtado's group is cleaning the chapel today.\nNHS is sponsoring a Blood Drive on Tuesday, April 30th. The event is open to all Bishop Stang students 16 years of age or older. Students under 18 years of age need to obtain a parent permission form. These forms must be filled out, signed and turned in to Mrs. McDougal prior to the day of the blood drive. Anyone interested in participating may sign up with the NHS officers in the cafeteria at both lunches Tuesday, Wednesday and Thursday. Donating blood saves lives, please consider stepping up and being a hero, become a blood donor!\nAn announcement from Mr. Dias - Looking for a group of Sophomores to participate in the 2019 Youth Day of Caring Sponsored by the United Way. We will be volunteering for the day at My Brothers Keeper, an outstanding partner of the Bishop Stang High School Community. Student volunteers will be picked up right here at school, and go to downtown New Bedford and participate in a rally at the New Bedford Public Library with the other Day of Caring participants. After the rally, students will be transported to the My Brothers Keeper facility in Dartouth, to volunteer for the remainder of the day. It was an outstanding day last year, and this year will be no different. If you are interested, please see Mr. Dias.\nWe are excited to announce that the Stang Script, our Bishop Stang Student Newspaper, now has its own website. Check out Stangscript.org for the latest Stang News! This week, the site has an interview with Justin Lopes before today's big game, pictures of Saturday's win over Burke, a feature article on Teacher's Pets and more! Visit Stangscript.org and follow Stangscript on Instagram for updates.\nYou must have an ID to enter the Library during Study. If you lost your ID, you need to order and pay for a new ID in Mrs. Williamson's office.\nWater bottles and agendas are available. They are $5 each.\nCampus Ministry - Morning Mass will be Tuesday, Thursday and Friday at 7:15. Morning Prayer will be Monday and Wednesday at 7:10. Please sign up.\nGirls Tennis beat Bishop Connolly 5-0. Winners included Sailor Nordstrom,Sophie Lynch, and Gabriella Joaquim, in singles, and Caroline Downes, Fiona Lonergan, Kassie Stone and Maeve Egger in doubles.\nBoys Tennis beat Bishop Connolly 3-2. Winners were Pat Clark and Jeff Lucarelli, along with Keenan Cooney and Shawn Mills in doubles.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We provide Safe, Reasonable and Professional !\nWe need plenty of gold everyday, if you want exchange to cash, you can contact us anytime, for the one-time sell or the cooperation to be our suppliers are both welcomed.\nCertaintly, most of the sellers worried about be scammed. From the beginning some of our partners mentioned that too. About this point, its not most important said we are big company or website brand, something like that. cause we have more realistic approach, that is start from the foundation, build trust with each other. Like we can trade part of the gold you have, or the amount you perfer. After that if you think there is any problem about the rate or the time we pay, you can completely not cooperate with us anymore. but no exaggeration to say, all the sellers who contact us thought we are reliable, and worthy for long-term cooperation until now. So what are you waiting for?\nWe look forward to hear from you !","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thermoshield is a solar-reflective paint that is used in commercial and domestic applications to reduce energy consumption for heating and cooling on roof surfaces.\nThis also prolongs the life of the roof through the ability of the tough ceramic membrane protecting the underlying steel from the elements. This is good for roofs that have been badly weathered, creating a more aesthetic appeal.\nIn one case, a roof treated with Thermoshield matched the ambient temperature of 31.5 degrees, whereas an uncoated roof, adjacent to it, reached a temperature of 82 degrees. This created an annual savings of 20%, or $50,000 for the business.\nContact Robert at Blasted! if you want to save money on the rising energy costs affecting your area. A qualified installer will come out for a no-cost estimate and provide you with the details.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfpto b/data_all_eng_slimpj/shuffled/split2/finalzzzbfpto
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@@ -0,0 +1,5 @@
+{"text":"Quartz Spheres are great aids to harmonise the energy in your environment. In feng shui, colour plays an important part in balancing energy as colour is just pure light. The shape is also important and each shape represents a different element. The sphere for instance represents the feng shui metal element which also affects your mental clarity, concentration and organisational skills. The colour of clear quartz also supports the feng shui metal element. In feng shui each area of your life is influenced by the different areas of your home, the metal element relates to the west and to the subjects of children, creativity and your future. So if you place a quartz sphere in the metal area of your home it will beneficially work upon these areas of your life also.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This course assumes you are familiar with the Point of Sale process up to the point of clicking Add Payment Method. If you are unfamiliar with how to do this, please review course F103 - A First Pass through Point of Sale first.\n2. In the example shown above, John Smith has one payment method on file: a Visa card.\n\u25cf A member makes a $20 purchase. Including tax, the total is $21.38.\n\u25cf The member wishes to pay $10 in cash, and write a check for the $11.38 balance.\n1. If the Customer Account balance had been $0.00, the new balance would be $21.38. It will appear under the \"Current Due\" tab, and will be automatically billed during the next regularly scheduled membership dues collection.\n2. If the Customer Account balance had been $10.00, the new balance would be increased to $31.38. The full balance will appear under the \"Current Due\" tab, and will be automatically billed during the next regularly scheduled membership dues collection.\n3. If the Customer Account balance had been -$10.00 (meaning the member had a $10.00 credit), new the balance would be $11.38 (meaning the member now owes only $11.38). This balance will appear under the \"Current Due\" tab, and will be automatically billed during the next regularly scheduled membership dues collection.\n4. If the Customer Account balance had been -$21.38 (meaning the member happened to have had an account credit of exactly $21.38), then the new Customer Account balance would be $0.00. There would be no entry under the \"Current Dues\" tab, and no change would take place to the amount of the next regularly scheduled dues collection.\n5. If the Customer Account balance has been -$50.00 (meaning the member had a $50.00 credit), the Customer Account balance would be reduced to $28.26 ($50.00 - $21.38 = $28.26). There would be no entry under the \"Current Dues\" tab, and no change would take place to the amount of the next regularly scheduled dues collection.\nIt is possible to use the Customer Account as a partial payment method.\n1. A member owes $21.38.\n2. They have $10.00 in cash.\n3. They wish to pay $10.00 in cash, and place the remaining $11.38 on the Customer Account.\n4. In order to use the Customer Account in this manner, simply split the payment among two or more payment methods, using the Customer Account as one of those methods. This is covered in Step #3 above, Splitting a Payment Among Two or More Payment Methods.\n1. A Credit Memo\/Owner Comp (owner complimentary) is exactly as the name implies. The entire amount of a purchase - or any portion thereof - can be given away free in this manner. Simply click the Credit Memo (Owner Comp) button.\n2. A Gift Certificate is a payment method, no different than a check. Click the Gift Certificate button, enter the number of the gift certificate, and the amount.\n3. A Discount may be given as a flat (currency) amount, or a percentage amount. Simply click the Discount button, and fill in the requested information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Larp.Me \u2013 Who do we think you are?\nAll content property of respective owners.\nPictures belong to photographers & illustrators, characters to their creators, systems to the poor fools who strive to keep worlds alive.\nLarp.me itself \u00a9 Istic.Networks 2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The perfect space for your Gin Tonic.\n\"Your ideal space for a perfect Gin Tonic\" Diario Sur.\nAt Gin Tonic Bar you can enjoy more than 60 different brands gins and as many tonics accompanied by grapefruit, strawberries, grapes, juniper berries, raspberries or other ingredients. They all serve the most sophisticated way for them to enjoy the full.\nDepending on the combination chosen, behind the bar and after days sharing these drinks with the public, they can identify almost perfectly, the type of person they serve, according to your choice.\nGin Tonic Bar it is located in the center of Malaga, an area full of culture, entertainment, shopping and, of course, tourism.\nDiscover our range of more than 60 gins, tonics and botanists. Served in scented and chilled glasses as directed by each producer..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Complete PuerChia\u2122 Set with 86g of black Puer tea, 35 day supply (454g) of Mila Chia, Piao I 865 Teapot, and 370cc Piao I Travel Buddy\u00ae Tea Infuser Bottle with nylon jacket included (not shown.) With this set you can enjoy the PuerChia\u2122 breakfast\/meal at home or on the go. PuerChia\u2122, also known as PuerhChia\u00ae and Pu-erhChia\u00ae, is a trademark of Pure Puer Tea.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..5831b3866705ed49b8b50aa6f61fd4ab73b1fb5a
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"So that was the first part of the song but as a whole, the message is to keep believing to whatever it is that you believe, that you can achieve things, that you can be what you want to be by believing and by having faith. That whatever it is that the multitudes might say, it would always be your beliefs that will make you a winner, which will always make you triumphant.\nAgain a Christian Song sang by theDestiny's Child. Aside from me being a big fan, the message of this song isjust extraordinary because again, if you feel like people is not believing inyou or they are trying to put you down and discourage you, when Jesus says yes,nobody can say no.\nI told you I'm a big fan! Who's not listening tothis song? I mean its super brilliant! Whenever someone wants to leave or leftyou thinking that when they do, you'll be a pile of garbage, then this songsuggests you can't be that! You can be successful even without these people whodoesn't believe in you or who doesn't want to do anything with you anymore.\nI've always loved Miley Cyrus back when she stillsings the way she is before and this song is really one of her most beautifulsongs. It reminds me that no matter how hard the process is, no matter howunsure you are, you should always chase and follow your dreams and that mostoften than not, the climb, the way to success is actually a lot more fun thanwhen you're already there.\nI love the chorus of this song because it says itall when you think you found the one yet your wrong and realized that you areactually the best thing he never had.\nWell, who doesn't love this song? I mean, it wasthe international anthem back in 2008-2009. And still, it is the anthem ofthose people who has been fooled by their flame.\nWho doesn't like this? This song can be anyone'santhem. I always believed that when a relationship ended because of seriousproblems caused by one party, the aggrieved party should always make his\/herself stronger and proud. I mean, you shouldn't just sit and wait for him\/herto comeback, you are always good and you can be better even after arelationship.\nWhenever I feel like I am forgetting why I am hereon earth, I would listen to this song because it gives me the motivation to dosomething good for other people. When I say do something good, it isn'tnecessary to really do something big and spectacular. Smiling, as what I alwayssay in my IG posts can really help and motivate other people that you encountereach and everyday. You can always be a blessing to others by simply smiling.Show them that love indeed exists.\nWhen I feel like I'm being bad or have hurt someone too much, this song reminds me that only love can make this world a better place after the grace of God. Also, this song reminds me the reality of this world and that we should actually do something to end war, violence in any form and even racism all over the world. We are all humans and we should remind ourselves that.\nGrowing up, I would always hear Neil Young playingin our house. My most favorite is this one because it reminds me to alwaysplant good things so that whatever happens with you in the future, you will beokay because you were able to plat good things. I mean, you're prepared.\nIn the field of opportunity it's plowin'time again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WJSG Internet Radio: Are Our Creative Lives To Be Ruled By \"Algorithmic Copyright Cops\"?\nAre Our Creative Lives To Be Ruled By \"Algorithmic Copyright Cops\"?\nThe recent tale of a musician's cover song videos of works in the public domain being taken down due to misidentification by YouTube's ContentID system is but one example of the ongoing failure of algorithmic systems to properly handle takedown requests. Recently such problems have been seen with live streaming video as well, leading to major live events such as Michelle Obama's speech at the Democratic National Convention being blocked on YouTube. Welcome to a world in which \"Robotic Overlords\" rule your creative activites and human intervention is rarely available or too late to save the day.\nWe've seen plenty of examples of videos being pulled from YouTube due to false claims by major labels and incorrect automated takedowns via algorithm. For some people such issues are a minor inconvenience. For others, it's an ongoing disruption in already all too busy schedules with financial repercussions.\nNow we can add takedowns of live streams by algorithms to the list of problematic policing of copyright claims. The recent incident of Michelle Obama's live speech being blocked due to supposed music copyright violations was downplayed by YouTube and a spokesman appeared to be covering up the fact that the video was unplayable. Though this incident was related to music copyright, film\/video copyright has also been an issue for example when the recent Hugo Awards for science fiction were interrupted on Ustream.\n\"A swarm of tech companies are rushing in to provide technical solutions to enforce copyright in online sharing communities and video-streaming sites. Those players include Vobile, Attributor, Audible Magic, and Gracenote. And they're thriving, despite the fact that U.S. copyright law, as modified by the 1998 Digital Millennium Copyright Act, doesn't require sites that host user-created content to preemptively patrol for copyright violations.\"\nAs Dave Colvin noted regarding the regular takedowns of his legal music videos, the failures of such algorithmic systems are compounded by the inaccessibility of humans at such services as YouTube.\n\"The thing to understand is that at any given time, we can have tens of thousands of simultaneous broadcasts on the site, which is coming through the free service, which is why we've implemented these automated procedures to monitor copyright, as part of our partnership with these larger media brands...From the system's point of view, [the Hugo Awards] were just one of tens of thousands of people broadcasting at one particular time.\"\nNote that a VP of marketing is explaining his company's failure by referring to \"the system\" as if has a \"point of view\" and a life of its own. Of course, the driving force behind this approach is the move by such services to attempt to become partners \"with these larger media brands.\"\nThe headline of Dayal's article equates \"Algorithmic Copyright Cops\" to \"Robotic Overlords.\" To some that might seem overly dramatic. But think about what happens when you are dealing with customer service provided by a company such as AT&T whose customer service personnel sometimes seem incompetent.\nI mention AT&T because I spent a couple of hours one day being transferred from office to office around the country because the initial customer service person I encountered did not believe and had never heard of a case like mine which was a simple example of being charged for services that I was supposed to get for free due to a promotion for Internet access. The next day I again tried customer service in a last ditch effort before switching to another provider. This time I reached a human that took care of my problem in five minutes.\nSo what happens when a major corporation that provides services important to the livelihood of musicians such as YouTube does away with customer service by humans and hands it over to strings of numbers? When there's no one to call and the machines are running the show, not only is fair use endangered but so are the revenue streams so many musicians are learning to depend upon.\nWelcome to your future. Now can you explain your problem in code?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Commercial Embroidery Services industry has unraveled over the five years to 2012, with high unemployment and intense external competition hurting revenue growth for its players. One of the industry's major market segments consists of administrative and hospitality service industries, so as the recession hit and unemployment skyrocketed in 2008 and 2009, demand from these buyers diminished. Moreover, the decade-long trend of large multifunctional apparel companies integrating embellishment processes into their operations has stifled growth opportunities. Despite their efforts, industry participants are not expected to experience growth over the five years to 2017. Overarching competitive pressures \u2013 that are affecting the entire apparel-manufacturing sector \u2013 are likely to remain hallmark traits of the industry. For these reasons, industry research firm IBISWorld has added a report on the Commercial Embroidery Services industry to its growing industry report collection.\nThe Commercial Embroidery Services industry has unraveled over the five years to 2012, with high unemployment and intense external competition hurting revenue growth for its players. One of the industry's major market segments consists of administrative and hospitality service industries, so as the recession hit and unemployment skyrocketed in 2008 and 2009, demand from these buyers diminished. Industry revenue dropped 13.4% and 13.7% in those years, respectively. Moreover, the decade-long trend of large multifunctional apparel companies integrating embellishment processes into their operations has stifled growth opportunities for the industry, says IBISWorld industry analyst Nikoleta Panteva. As a result, revenue has declined at an average annual rate of 3.3% over the five years to 2012.\nSince the recession, companies in the Commercial Embroidery Services industry have experienced some relief: unemployment in downstream markets has begun to decline and per capita disposable income has inched up. As a result, IBISWorld expects revenue to increase over the coming year. According to Panteva, \"Operators are increasingly automating their activities to reduce labor costs and gain greater efficiency in production, causing average wages to decline while depreciation grows.\" Additionally, players that can successfully cater to generation Y's need for customized apparel are more likely to succeed in this declining industry. These trends are anticipated to continue into the next five years.\nDespite their efforts, industry participants are not expected to experience growth over the five years to 2017. Overarching competitive pressures \u2013 that are affecting the entire apparel-manufacturing sector \u2013 are likely to remain hallmark traits of the industry. Generally, commercial embroidery companies tend to cater to promotional product distributors, local businesses, sports teams, apparel manufacturers and government clients within a limited geographic scope in order to decrease transportation costs and maximize time-savings. Larger commercial embroidery service companies typically hold contracts with apparel manufacturers and original equipment manufacturers, while smaller companies in the industry target promotional product distributors, novelties companies and other commercial clients. Leading companies include Artwear Embroidery Inc., Fawn Industries Inc. and Hawkins Commercial Embroidery. For more information, visit IBISWorld's Commercial Embroidery Services report in the US industry page.\nThis industry includes companies that primarily provide embroidery services to clients. The embroidery process involves a sewing machine that creates pre-determined patterns on clothing and other textiles. The patterns can be designed using specialist software. Services are often sold to companies, teams and clubs for shirts, uniforms and other paraphernalia.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Put a compost bin in your kitchen.\nMuch of what we consider waste is actually leftover food, which we put in plastic garbage bags and send to landfills. Organic waste makes up the second-highest component of landfills in the United States. This \"trash\" is so compacted that oxygen can't reach it so most food scraps never receive the oxygen required for decomposition. All that food just sits there, not nourishing anything, becoming toxic as it absorbs poisons from all the other items surrounding it.\nStarting a compost bin is a great way to reduce pressure on landfills while creating healthy soil to grow plants and food in. It's simple to keep a vermicompost bin (with worms) under a kitchen sink, or a lidded bucket on the counter that you put your food scraps in. When bins become full, you can carry the compost to a tumbler or bin outside to turn into dirt. If you don't have anywhere to put your compost or don't have a yard, check with your local community gardens or gardener friends. Most are quite happy to receive compost!\nDitch disposable cutlery, plates and cups.\nThink of the energy that goes into making plastic cutlery, cups, and their packaging, only for the products to be used for one meal or cup of coffee and then be moved to a landfill. Is it not completely insane to have stolen so much energy, time and resources to produce something so temporary?\nGo through your home and make a plan for using real silverware, glasses, travel mugs and plates. They will make your food taste better, store it better, and last for years instead of minutes. Keep a set of compostable cutlery in your purse, glove compartment or backpack. Or pick up a travel-size spork that you keep in your desk or wallet so you're always ready to grab a meal on the go without sacrificing the environment.\nInsist on reusable shopping bags, and buy bulk using your own containers.\nFirst things first: stop buying packaged, processed food. Most of our household garbage is coming from the packaging surrounding food we eat. Do away with that, and you've instantly done away with the lion's share of your garbage (and made yourself significantly healthier).\nWe all know we should be bringing reusable bags to the grocery store to carry our unprocessed foods, whole grains and produce home in. It makes no sense to use something to haul products home that simply ends up in the garbage pail.\nMany health foods stores and farmers' markets allow you to buy by weight, and provide plastic bags or containers to put the food in. But they'll also let you bring your own containers, and weigh them before filling them with items you'd like to buy. At checkout, a cashier can then subtract the weight of your container from the total weight.\nEven if you can't buy in bulk, bring your own bags for weighted produce at the supermarket. There isn't any need to put tomatoes, avocados, peppers and onions in a bunch of separate bags that you're never going to use again.\nPaper towels, paper napkins, paper plates and paper cups are all insane wastes of energy. There are easy replacements for each.\nCut up old T-shirts, jeans and towels into manageable sizes to be used as rags. Newspaper can replace paper towels for draining fried foods, cleaning windows and greasing cookie sheets or muffin tins. Pick up a set of stain-resistant cloth napkins, make your own out of fabric scraps, or buy gently used napkins at your local thrift shop. Paper plates and cups can go the way of their plastic counterparts: nothing is as nice as the real thing.\nAs you advance in your waste-free lifestyle, you're going to face a serious reckoning: many of your food products, toiletries, clothing, pet-care products and you're your appliances come in excessive packaging and aren't built to last.\nThere are some companies that sell toiletries in reusable containers or without any container at all (check out Lush's great line of deodorant bars and powders), or you can learn to make your own. Also check your local markets for handmade soaps and other products that don't come with packaging.\nIt's tempting, when you forget your reusable shopping bag, to just take a disposable one home\u2014or to take your coffee to go in a Styrofoam cup because you left your travel mug on the kitchen counter. But going zero waste means learning to do without when necessary and not take short cuts. It's not the easiest transition\u2014you'll quickly notice just how many times in a day you're encouraged to take something home that isn't reusable, throw something away, and buy something new.\nZero waste doesn't mean you don't consume or produce anything. It's about doing these things in ways that doesn't result in a wasteful byproduct. And with a little grit, imagination, and a willingness to resist, you can be living a zero-waste lifestyle with minimal effort.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bathrooms: 2 \"1 half, plus maids\"\nSize: 412 sq. m. \"4,434 SF\"\nYour home away from home - Tortuga Bay in Punta Cana. Just steps from the beach and right on the golf course. A wonderful family home listed at a great price. This Luxury home is located in the gated community of Tortuga Bay Resort in the Punta Cana Resort and Spa, in the Dominican Republic. Only 6 minutes from Punta Cana International airport. This villa overlooks the La Cana golf course designed by P. B. Dye and 2 minutes away from the Corales golf course designed by Fazio. A short drive to Cap Cana and the famous Jack Nicklaus Signature, Punta Espada golf course as well. This villa embodies the spirit of the Caribbean with a European touch. Elegantly decorated. Enjoy the outdoor living and dining area with BBQ. Tropical settings with coconut trees, fabulous golf course views, ocean breeze and just a short stroll to the beach club. 3-bedrooms, 2 1\/2 baths, air conditioning in the bedrooms. TV room with large screen for family entertainment. The entrance leads to the living room and interior dining room. Off the living room is a fully equipped kitchen. Office space on the first floor which can be easily turned into a second master with en-suite bathroom as plumbing is already installed. Beyond the kitchen are the private maids' quarters and patio, as well as full laundry room. Ground floor Suite 2 has a double bed, TV and direct access to the gardens. Suite 3, also on the ground floor, has (2) queen size beds and large closet. These suites share a large and spacious bathroom. Master suite on the second floor (1) king bed, with large closet, balcony and large private bathroom with bath. Golf Cart included in the price. Great rental record.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Now thru June 2nd, Target is offering a FREE $5 Target Gift Card with the purchase of four select Suave, TRESemme, Dove and Axe products both in-store and online.\nEven better, there is a Checkout 51 offer available on Suave lotions making for an awesome deal!\nFinal cost:$2.76 total \u2013 just 69\u00a2 each!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An investigation by Agents of the Ohio Organized Crimes Investigations Commission, Washington Morgan Noble Major Crimes Task Force resulted in the arrest of nine individuals and the seizure of 226 grams of illegal drugs. This is the largest seizure of drugs in Washington County in the last several years.\nOn 30 Jan 19 Agents, with the assistance of the Marietta Police Department, executed a narcotics search warrant at 610 Second Street (upstairs), Marietta, Ohio.\nAs a result of the search warrant Agents seized approximately 109 grams of heroin, 117 grams of methamphetamine, and approximately $2,700 in cash. The majority of the drugs were seized from two individuals from Akron, Ohio. They were James Christopher Smith, age 24, 155 Stephens Road, Akron, OH, and Taevion Tyson Thompson, age 20, 733 Storer Avenue, Akron, OH. They were both charged with felonies of the 1st degree.\nJeffrey Shane Wittekind, age 30, 316 Gilman Avenue, Marietta, OH, was charged with possession of methamphetamine, a felony of the 2nd degree.\nThe Task Force is part of Ohio Attorney General David Yost's Organized Crime Investigation Commission and is comprised of representatives of Post 84 of the Ohio State Highway Patrol; Washington, Morgan and Noble County Sheriff's Offices; the Marietta, Belpre and McConnelsville Police Departments; and the Washington, Morgan and Noble County Prosecutor's Office.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Toyotas newest and smallest Prius is no hot hatch, but it excels at saving fuel.\nToyota's newest and smallest Prius is no hot hatch, but it excels at saving fuel.\nWhat's New: The \"c\" stands for \"city.\" In the expanding Prius lineup, this new hatchback is the least expensive and smallest model. It uses a downsized Prius powertrain (a smaller gas engine, nickel-metal hydride battery, and less powerful electric boost) and a modified version of Toyota's Yaris small-car platform. As the smallest and lightest Prius (500 pounds lighter than the original), the c also boasts fantastic fuel economy, with an EPA rating of 53 mpg in the city. And Toyota didn't skimp on the options: The c can be equipped with luxuries such heated seats and also offers Entune, a cloud-based app that lets drivers access Pandora and Bing through the car's infotainment system.\nTech Tidbit: The Prius powertrain is a pretty ingenious piece. For example, the 1.5-liter engine revs to only 4500 rpm. Going higher just increases the frictional losses of energy, and so the Prius powertrain relies on the electric motor boost\u2014rather than higher engine rpm\u2014to increase power. And the CVT, which uses a planetary gearset and two electrical motors, is conceptually complex\u2014the various elements rotate at different speeds and direction depending on power demand\u2014but mechanically simple.\nToyota included several different options for viewing your driving efficiency. You can even plug in a theoretical comparison car (say, if you were debating buying the Prius c or Honda Fit) and the computer will calculate how much fuel\u2014and money\u2014you've saved. This information appears on a three-inch screen that's mounted high up on the dash and relies on a steering-wheel mounted switch for toggling through the various screens. The readouts are a neat feature, but so much information is crammed on the little screen that it's tough at quick glance to determine what you're looking at.\nDriving Character: Toyota claims the c is the sporty Prius and even offers optional 16 wheels and quicker ration steering. After a couple of days in the car, however, it's clear that we're still waiting for a dynamically satisfying hybrid. That aside, the hatchback is nimble around town, where it switches between EV mode and gas-engine operation. With only 99 hp, accelerating to 60 mph takes about 11 seconds. The c can keep up with highway traffic\u2014just be prepared to floor the throttle for passing.\nFavorite Detail: The leather seats of the top-level Four trim are not leather. The material is called Softex, and it's about half the weight of real hides. While Softex does not feel as rich as the leather in a Lexus, it is comparable to what's in other low-priced cars.\nDriver's Grievance: Somebody needs to design a more attractive Prius steering wheel. The signature oval-shaped wheel is an awkward looking piece. While we're griping about design, it's a shame the production c doesn't look more like the handsome concept car that Toyota showed a year ago. And the infotainment system takes a long time to download songs and playlists from an iPhone.\nBottom Line: The Prius c costs about three grand more than a Toyota Yaris, but gets a whopping 50 percent better fuel economy. But remember, hybrids don't typically make much financial sense because it takes years of fuel savings to claw back the extra cost of the hybrid system. That's still the case with the c, but that doesn't mean we wouldn't recommend it. The Prius is an all-around more pleasant car than the Yaris, and in real-world driving you're likely to hover around 45 mpg.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On Tuesday, April 25, eGaming Review (EGR) announced that LocationSmart\u00ae was shortlisted in two categories for the 2018 EGR B2B Awards. The categories are Mobile Services Provider and Fraud and Compliance Solutions. This is the third consecutive year LocationSmart has earned this recognition in these two categories.\nFor nine years, the EGR B2B Awards have been rewarding and celebrating the accomplishments of operators and suppliers in the online gaming industry. All major egaming disciplines are recognized \u2013 betting and gaming software, mobile payments, networks, IT and infrastructure \u2013 and winners are announced during the awards ceremony at the Hurlingham Club in London on Wednesday, June 20, 2018.\nLocationSmart is the world's largest Location-as-a-Service (LaaS) company and a recognized authority in the gaming industry. Its cloud-based location platform delivers secure, multi-source location intelligence for gaming compliance and transaction verification.\nLocationSmart has more than 200 customers and it provides the most trusted, non-spoofable source of location validation available on the market. It was also among the first LaaS providers for gaming compliance use cases in the US after the legalization of mobile and internet watering. Within the last year, LocationSmart's reach has expanded internationally with several existing customers securing new opportunities throughout the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sladderdaze in the East: Dirty Logan X Juug Talk presents Tacos & Tequila \u2013 Buffalo [Sept 23rd] \u2013 Be Fearless. Be Flagrant.\nPull up today and hang with the Dirty Logan Gang and Juug Talk!!! One more time while the weather is amazing because it's cool in Cali where BuffaLowe is, and it's heating up back East in The Queen City of Buffalo, NY.\nFree tacos until supplies last, $10 unlimited drinks!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Niton XL2 800 PMI gun is a handheld X-Ray Fluorescence (XRF) scanner that provides dependable elemental analysis for your most demanding inspection needs.\nThis high performance and lightweight XRF scanner is easy to use, even for non-technical personnel. The ergonomic design offers a clear touchscreen display that is visible in any lighting condition, including bright daylight and is built tough against harsh inspection environments.\nThe onboard software features allow the user to customize the menu with built-in multi-language support. Expand the Niton Xl2 800 XRF scanner capabilities by utilizing the Niton Data Transfer software to set operator permissions, generate inspection reports and documentation, and even remotely operate the XRF analyzer from your computer.\nThe Niton XL2 800 XRF analyzer is a must-have PMI gun for fast and accurate elemental analysis anywhere you go.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"11. What does The Merchant of Venice have to say about the relationship between love and wealth?\n12. Most of the Italian cities mentioned in Shakespeare's plays are so vaguely described that they are virtually interchangeable. Venice is an exception. What does this play tell us about Venice and its people?\n13. Is Shylock a comic character or a tragic one? Discuss.\n14. Discuss the role of humor in The Merchant of Venice.\n11. Is Bassanio's speech about appearance versus inner worth at the time he chooses the lead casket consistent with what we know about his character? Why or why not?\n12. How are music and musical imagery used throughout the play?\n13. What is the significance of the episode of the rings?\n14. What does the play have to say about the relationship between friendship and love?\n11. Love and money are closely intertwined in this play. By winning Portia, Bassanio solves his romantic problems and his financial ones simultaneously. Shylock, on the other hand, loses both his daughter Jessica and his ducats. Antonio demonstrates his friendship by lending money freely- even to the point of pledging his own flesh to guarantee the loan that Shylock extends to Bassanio. The very language that is used to describe human relationship underlines the theme of love as a form of wealth. Portia, declaring her love for Bassanio, describes herself in terms that recall a financial balance sheet- for example, she tells Bassanio that she wishes to stand high in his \"account.\" Salerio and Solanio also use language which compares the progress of a person's life to the quest for profit through trade on the high seas. Other connections between love and wealth include the use of the gold rings which symbolize the vows exchanged by Portia and Bassanio, Nerissa and Gratiano. Finally, the test of the three caskets turns on a contrast between outward appearances and inner worth.\n12. Venice is portrayed as a rich and cosmopolitan city, a busy seaport engaged in trade with exotic foreign lands. Although there is very little physical description of Venice, the behavior of Antonio and his friends tell us a good deal about Venetian life- or, at least, about Shakespeare's conception of it. These young men cheerfully live beyond their means, confident that money will come their way sooner or later. Bassanio, although he has just borrowed a large sum of money on his friend's credit, is not at all worried about keeping his expenses down- he plans to celebrate his departure for Belmont by holding a dinner party and a masque. Note, also, that most of the Venetian scenes take place in the street or in public places; the choice of settings emphasizes the importance of business and civic activities in the lives of the Venetians. Shylock is the only character we see at home, and his attachment to his house is offered as evidence of his gloomy, miserly personality. The trial scene further emphasizes that the Venetians' respect for law is related to their desire to assure foreigners that they will be treated fairly in business.\n13. The part of Shylock has been played both ways over the centuries. Most Shakespeare scholars agree that the part of Shylock was written as a comic villain; however, modern actors are more likely to stress the serious side to Shylock's character. Whichever way you choose to answer this question, you should keep in mind that calling Shylock a comic character does not necessarily mean that his actions constantly provoke laughter. If you believe, for example, that sheer greed is the most important motivation for Shylock's plotting- and that he learns his lesson during the trial scene and has reason to accept the verdict against him as basically merciful, then you can justify seeing Shylock as the villain in a comedy. If you believe that Shylock suffers largely because he is an outsider and that his final line in the play is bitterly ironic, then you cannot help seeing him as a tragic figure.\n14. In answering this question, you might choose to concentrate on the role of Launcelot, the clown. Launcelot's humor is based on his mangled vocabulary- a mixture of mispronounced words and silly puns- as well as on broad physical comedy. You might note, also, that while Launcelot's speeches are sheer silliness, they do echo some of the serious issues raised in the play- particularly Jessica's relationship to her father Shylock. Another form of humor is the more literate, though sometimes off-color, kind of pun which Gratiano specializes in. Gratiano's character provides an excuse for Shakespeare to indulge in this form of humor, while at the same time having other characters in the play gently reprove Gratiano for his bad taste. The protagonists of the play enjoy humor- Bassanio even hires Launcelot as a servant. But we are given to believe that extreme levity is a fault. The truly happy person does not have to be constantly cracking jokes.\n11. Many readers feel that Bassanio's speech is out of character. Suddenly this charming spendthrift comes out with a philosophical meditation on the superficial nature of external wealth and beauty! On the other hand, if you think of the mottos inscribed on the various caskets- which are mentioned earlier during the scene with the Prince of Morocco- then Bassanio's choice is not out of character. It would be just like him to choose \"hazard\"- or chance- over a sure thing. In considering this question further, you might also ask yourself whether readers who find Bassanio frivolous in the early scenes of the play do him an injustice. After all, Bassanio does express concern for Antonio and wants to pay back the money he owes. Depending on your interpretations of Bassanio's speeches- and on your own views of friendship and financial responsibility- it is possible to defend differing opinions of Bassanio's character.\n12. In discussing the role of music in the play, you should be sure to mention Portia's comments on music in Act III, Scene II as well as what she and Lorenzo say about music as the bringer of harmony in Act V. In the latter section of the play, you will recall, Lorenzo says that the man who has no feeling for music is not to be trusted- a comment that recalls Shylock's desire to shut the music of the masque out of his house. Lorenzo also says that music exists not so much to make us carefree and happy as to induce contemplation and thoughtfulness- and this is certainly the effect that the song \"Tell Me Where is Fancy Bred\" has on Bassanio in Act III, Scene II.\n13. Readers who stress that the action of the play is circular tend to see the episode of the rings as symbolizing this circularity. Portia gives her ring to Bassanio, who gives it back to her thinking she is the lawyer Balthasar; she later gives the ring to Antonio who hands it back to Bassanio again. All of these exchanges together might be seen as emphasizing the play's theme that love only increases in power if it is given freely. Another way of looking at the rings is that they represent a gentle, loving bond between human beings- as opposed to the cruel bond (or promise) imposed on Antonio by Shylock. Either way, the important feature of the episode of the rings is that Bassanio is not punished for breaking his vow never to part with the token of Portia's love. In a tragedy, breaking such a promise would surely have terrible consequences. In this play, it becomes an occasion for some playful teasing but no serious repercussions.\n14. The Merchant of Venice tells us that love and friendship are thoroughly compatible. Far from being jealous of her husband's friendship with Antonio, Portia says that \"the bosom lover of my lord \/ must needs be like my lord.\" Portia not only saves Antonio's life, but at the end of the play Antonio is brought back to Belmont to share in the happy final scene. This, at least, is what the play says on its surface. Some readers, however, feel that there is a conflict between love and friendship expressed on a deeper level. These readers stress that Antonio, despite his brief visit to Belmont, cannot stay there long; he is shut out, without a bride of his own, and must return almost immediately to Venice.\nThe Merchant of Venice is a problem play about the question of usury.\nThe Merchant of Venice is a light romantic comedy which was never intended as a comment on important social issues.\nThe Merchant of Venice contrasts the Old Testament view of God as law-giver with the New Testament view of God as offering salvation through divine mercy.\nConsidering the times during which it was written, The Merchant of Venice expresses humane and tolerant values.\nThe Merchant of Venice is an outdated play because it is a comedy based on assumptions which we today find unacceptable.\nDiscuss the concept of harmony as presented in the play. Does this concept have any application to real life? Or does it make sense only in the unrealistically benign world of this play?\nWhat is the attitude towards money and its uses in the play? How does it compare with modern-day attitudes?\nHow does Shakespeare weave together the stories of the \"pound of flesh\" and the \"three caskets\"?\nDefend or attack the view that Bassanio is nothing but a fortune hunter.\nCan you see any similarities between the characters of Antonio and Shylock? If so, what are they?\nContrast the characters of Gratiano and Shylock. In what ways are they opposites?\nWhat function do Salerio and Solanio serve in the play?\nDid Portia give Bassanio a clue to the choice of the correct casket? How would such behavior affect your view of her character?\nDiscuss how Shakespeare establishes Shylock's position as an outsider.\nAs an exercise, rewrite one or more scenes of the play in the form of a story or prose sketch. Take the point of view of Shylock, of Portia, or of Bassanio.\nThe critic E. M. W. Tillyard wrote: \"Shakespeare's Shylock has been the victim of the great actor.\" Discuss. If possible, consult a stage history such as Toby Lelyveld's Shylock on the Stage.\nIf you could direct a production of The Merchant of Venice, how would you go about interpreting the trial scene?... Or, the final moments of Act V?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Choral Department of Canfield High School is busy with final preparations for 100 Years of Broadway! On January 27, 2018, Canfield High School will be transformed into a dinner theater extravaganza.\nThe dinner portion begins at 6:00 p.m. with a VIP Tasting Event which includes a sampling from local restaurants. At 7:30 p.m., the evening continues with a musical sampling of favorites from Broadway by the students of the Canfield High School Choirs. Performances including the One Octave Higher Female Ensemble, The Chamber Ensemble, and The Concert Choir, and many student soloists.\nTickets are currently on sale. Tasting and concert combination tickets are $20. Concert only tickets are $10 for adults and $5 for students. For more information or to reserve your tickets for please call 330.702.7128 and a member of Choral Boosters will return your call.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Helmsing Leach is proud to announce that Jeff Hartley, who heads up the firm's Creditor's Rights Department, has been selected by his peers as the Mobile region's Lawyer of the Year in the area of bankruptcy litigation for the 2019 Edition of The Best Lawyers in America. Best Lawyers was first published in 1983. Since its inception, its mission has been to use a rigorous peer review process to identify the top legal talent in America.\nHelmsing Leach is proud to announce that firm member Alan Alexander was recently inducted into the Alabama Chapter of the American Board of Trial Advocates (\"ABOTA\"). Membership in the organization is by invitation only, following a rigorous nomination and voting process. ABOTA is a national association of experienced trial lawyers and judges dedicated to the preservation and promotion of the right to a civil jury trial provided under the Seventh Amendment to the U.S. Constitution. There are approximately 7,300 ABOTA members in the United States, only 122 of whom are located in Alabama. To be considered for membership in the Alabama Chapter, a candidate must have tried at least 10 civil jury trials to a verdict, be nominated by an existing Chapter member and be approved by 75 percent of the members voting on the candidate's nomination for admission. Criteria evaluated in determining whether to admit a candidate include the candidate's jury trial skills, civility, professionalism and integrity.\nOnce again, Helmsing Leach lawyers appear on the annual list of Alabama Litigation Stars compiled by Benchmark Litigation, a reference guide that is widely used by law firms and corporate counsel to identify and select leading litigators throughout the United States. The Helmsing lawyers honored with inclusion in Benchmark's 2018 edition are firm members Joe Babington (product liability, labor\/employment), John Leach (product liability and general commercial litigation) and Jim Newman (product liability, medical malpractice defense).\nFirm member Alan Alexander was recently invited to join the American Board of Trial Advocates (\"ABOTA\") after a vote of the organization's National Board of Directors formally approved his membership. ABOTA has chapters in every state, and its members constitute a \"who's who\" of trial lawyers, nationwide. An invitation to join this prestigious organization is a highlight of any legal career.\nBest Lawyers has named firm member Alan Alexander as its 2018 Lawyer of the Year in the practice category \"Mobile Medical Malpractice Law \u2013 Defendants.\" Only one lawyer in each practice area in each legal community is honored as \"Lawyer of the Year.\" Mr. Alexander was also selected by his peers for inclusion in the 2018 Edition of The Best Lawyers in America for his work in Health Care Law, Legal Malpractice Law \u2013 Defendants, and Personal Injury Litigation \u2013 Defendants.\nThe Trial Attorneys of America installed firm member Joe Babington as its President during its annual meeting in Chicago this week. TAA is an invitation-only organization of commercial litigation and product liability defense trial counsel. Mr. Babington has served the past two years as the organization's Vice President and Program Director. His term as President runs through June 2019.\nCasey Pipes was elected Vice President of the Petroleum and Convenience Marketers of Alabama (PCMA) Associate Board for 2017. The PCMA is the advocacy group for the convenience store and motor fuel retailer industry in Alabama. The Associate Board is comprised of people who serve the C-store industry, such as fuel distributors, food and beverage distributors, and service providers. Mr. Pipes' connection with the PCMA is his representation of numerous C-store owners and tenants in land condemnation actions throughout Alabama. The Associate Board helps raise money for and oversee the PCMA Educational Foundation which provides college scholarships to employees and family members of employees who work in the C-store industry. For more information, please visit pcmala.org.\nU.S. News & World Report and Best Lawyers recently announced the \"Best Law Firms\" rankings for 2017, and Helmsing Leach is among the select group of firms named. This coveted honor is reserved for firms recognized for professional excellence that have consistently garnered accolades from clients and peers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u2776You deserve to graduate with work that lets you stand out from your peers and that impresses your supervisor and the entire faculty.\nHave your thesis or dissertation proofread and edited by our highly experienced native English speaking editors. 24\/7 Support, days per year\/10().\nWe provide great value thesis editing services. Student prices & short turnarounds available. Use our free Price Calculator and see for yourself our genuine affordability. We edit theses from various fields, including engineering, medicine, law, business management, sociology, psychology, and English literature. Your thesis has specific .","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"European Release Dates for Galaxy S 4 Mini, Activ and Zoom Confirmed?\nSince Samsung released the Galaxy S III Mini last year, it was pretty much a sure bet that they would release a diminutive version of this year's flagship and while we've heard quite a few whispers about the device, we've yet to come across anything concrete. Now though, SamMobile think they've gotten their hands on European release dates for a number of upcoming Samsung phones, including the Galaxy S 4 Mini. The Activ and the Zoom are also mentioned, while the Activ is a new name we do have some info on the Zoom, which appears to be Samsung's entrance into the high performance camera smartphone game.\nSo, it looks like the Galaxy Mega devices will be launching in Europe soon as well but, what we're most interested about here is that both colors of the Galaxy S 4 Mini are said to be launching in Week 29 of this year, which would be the middle of July. It's said that Samsung will announce the phone towards the end of this month. The S 4 Mini could be another push to the low-end from Samsung but, as HTC have proven with their HTC First, there's no need for a smaller device, to be less of a device. As you'll note in our HTC First review, a 4.3-inch sized device can still come with a few bells and whistles, and that screen size does not always make for a poorer or better screen overall.\nShould Samsung release the Galaxy S 4, they'd probably achieve decent success with the device, if only due to the popularity of the Galaxy S brand, would you think about buying a smaller Galaxy S 4?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Western Star (Columbus) reports Gov. Ted Strickland has commuted the death sentence of Jeffrey Hill, who stabbed his partially paralyzed mother to death and robbed her of $100 in 1991. The decision came after a \"unanimous and rare\" recommendation from the Ohio Parole Board that clemency be granted. Hill is now under a 25-years-to-life sentence and will be eligible for parole in seven years. It is also reported that two Ohio Supreme Court justices felt the death penalty was not justified for Hill. See story here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Alaska Day 3 \u2013 Fly Out for Silver Salmon \u2013 ANY WATER WILL DO.\nMy last fishing day in Alaska was spent flying out across Cook Inlet to fish for Silver (Coho) Salmon in the foot of glacial mountains. Not a bad way to cap things off.\nI met my guide near Soldotna, as well as my pilot, and was happy to hear that the weather conditions were good, but disappointed that the fishing would be with spinning rods, and not fly gear (due to muddy water). Not what I had expected or tried to book, but I wasn't about to turn down the trip \u2013 I will fish with spinning rods rather than not fish at all.\nThe folks I was paired up with were a family from the mid-west, and whom the patriarch was a formal influential banker (and one who visited DC and spoke with Obama during the Great Recession). So there I was, rubbing elbows (and salmon eggs) with people far above my pay-grade. That's travel fishing.\nThe aircraft we took were sweet Da Havilland Otters, and the pilots were incredibly skilled at flying them. These things can up and down very quickly, and land softer than any commercial liner I've ever been on.\nDa Havilland Otter \u2013 a sweet little aircraft that handled like a helicopter.\nWe took off from a lake, and flew across the Kenai river and refuge, crossed Cook Inlet, and over massive river deltas through which spawning salmon entered the waterways of this portion of Alaska. The salmon use these river deltas, and the massive tides that fill them, to ride farther and farther up stream, eventually stopping at the foot of glacial mountains to spawn in freshwater lakes.\nRiver deltas on the Western side of Cook Inlet. Heavily influenced by tidal forces up to 20\u2032 in height.\nAs we approached the lake, those glacial mountains appeared, massive and snow-peaked.\nBefore us the lake, and above the lake, huge mountains. Note the salmon lure hanging by the window.\nWe landed on the lake so gently I wasn't sure when we touched down. The pilot motored us over to a floating platform in the river, where multiple jon boats waited.\nWe exited the plane, loaded gear, food, and coolers onto the jon boat, and took off for the fishing spot.\nThe view from the floating platform.\nThe silver salmon we caught (3 each) in the first 2 hours of the trip. This was our limit, and meant we legally could not fish for the next 6 hours.\nThe salmon bait, a gelatinous, home made mix of salmon eggs, flour, anise, and other goodies that helped the eggs stay on the hook (and on my clothing).\nThe young grizzly bear that was playing in the water, and whom we drove within 30 feet of to observe.\nThe eating of fresh caught salmon on a floating hummock, cooked by another guide.\nMasses of salmon at the food of a tiny creek, wheeling and circling, with carcasses of silver salmon dashed on the creek rocks by unseen bears.\nIt was fun, but my least favorite fishing trip of my time in Alaska. The scenery was amazing, but the fishing was easy, and was catered to folks who don't fish very often (we literally were using bobbers over salmon egg clumps). The silvers fought like hell, but I'd rather do this on spinning or fly gear with lures than bobber fish, even if it meant catching less fish. I did see two guys fly fishing nearby, but they were fly casting clumps of salmon eggs.\nI'm no purist, but that's a silly thing to do.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Revolutionize the way you have always seen gastronomy by dabbling into the new trend going around the city of modern food trucks. When you go around the malls or shopping arcades, chances are you will see a bright attractive food truck parked outside surrounded by foodies. Food trucks have started to invade the food scene more and more, entering every facet of gastronomy. Ice creams are the new thing on the receiving end of food trucks trend. 'Emoi' is a fresh upcoming chain of ice cream trucks parked all over Delhi NCR. West Delhi is the latest recipient of a brand new ice cream truck, parked in Pacific Mall, Tagore Garden, New Delhi. This busy shopping mall gives the truck plenty of footfall especially with kids. Kids flock to the truck like bees to honey.\nIce creams get their makeover at the fun Emoi. Don't expect your average ice cream cones here. What you will get here is unique and innovative. They play with flavours and shapes of the ice cream to give you a striking memorable experience. Their unusually designed ice cream sticks is a fun change from all the softies and popsicles you have come to expect. Interesting as well as classic ice cream flavours are on the menu for you to explore. Amusing shaped sticks like a paw print, smiley, cake, heart, chocolate bar makes eating at Emoi even more fun. Matcha, sovelvet, strawblush, gum yum, chocosnuggle, vanilicious, pawffee and butternom are few popular stick flavours. If you are not a stick person, get an ice cream scoop. Try the flavors, stroreo, paan freeze, nutty raisin, wowbrow, big banofee and peach out.\nThe staff is friendly and helpful. Consult them if you feel confused or overwhelmed. Ice creams at Emoi are reasonably priced but to make the evening even more special book with us to avail fabulous EazyDiner discounts or Emoi coupons. Emoi is a bright battery operated ice cream truck that has taken the city by storm. Painted in pastel pleasing shades the truck instantly catches your eye. They do not have a seating area. Just take your ice cream and go. This a great spot to enjoy a refreshing ice cream stick post a round of shopping or a meal. Emoi welcomes ice cream lovers of all ages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tim, Thank you so much for this information.I know that Charles Nigh was divorced(took his wife's inheritance, all savings, and bought a phony deed for land in Mexico) and that Charles had remarried but we did not know to whom.You have helped me greatly here.I will check with Colorado, Iowa, Oklahoma and his home state of Pennsylvania to see if they have a death record for him. I knew his dad's name was William and Mary A. is on the census as William's wife throughout his life in Penn. I'll make every effort to confirm.I see Charles H Nigh living in Omaha in 1930 with a wife named Kathryn whom he states he married in 1925.I had no idea Charles was such a wild guy.I wonder where Family Search had found Charles's parents information??Thanks again for your help.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Spirit Clay: Crazy Aunt Candy Wants YOU!\nCrazy Aunt Candy Wants YOU!\nThe Cool2Craft Creative Play Theme for This month is \"What's Your Word?\" What are your plans for next year? What do you want to attract, practice and create in your life? Spoiler alert!!! In January I'll be asking you to send me a photo of a \"crown\" that represents your word for the new year. It could be a real crown, a drawing\/painting\/sculpture or other any other representation of the \"crown\" you'll be donning for 2012!\nWhen I'm in my studio, I feel like a mad scientist. To my family, I'm crazy Aunt Candy toiling away long into the night! Sounds of hammering, the roar of my heat tool, inspired shrieks and maniacal laughter waft upstairs. I am in my element.\nWhen I'm upstairs in the \"real\" world, I'm less at ease - shy and quiet actually. That's why my word for 2012 is HARMONY! I crave more than balance; I crave harmonious integration! I want to be graceful in social situations. I want to sing myself into BEING!\nI hope you'll join me in this journey and declare your intentions for the coming year! Email me so I can post your declaration!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Plan, got the go-ahead recently.\nchaired by Dr. Tina Hoover.\nto keep their curriculum up-to-date.\nlast Saturday's Augie\u2014North Dakota State football game.\n\"And we ought to,\" he said.\nbias, but an economic one.\nprogress yet this school year.\"\nText oto .y I .ug Loon Jody Powell Finally passed Curriculum to be proud of Karin Bumgardner staff writer \"I'd be proud if the faculty passed this one,\" Dr. Bill Geyer said before the new curriculum for the college, the Augustana Plan, got the go-ahead recently. Set to be implemented during the 1986-87 academic year, the changes have been under con-sideration for three years. Geyer served on the Im-plementation Task Group, chaired by Dr. Tina Hoover. \"This is really a well-rounded complete curriculum, not just for a student's career, but for a student's life,\" Hoover said. The committee had been revis-ing the faculty's recommenda-tions from two previous refining processes. \"There was a good bit of compromise,\" said Geyer, stressing that the greatest number of changes af-fect the English department. \"Purely and simply, we wanted something that would work well with the general education requirements and of-fer students as many open doors as possible,\" Geyer explained. The problem, he said, is that the English department , offers more courses in general educa-tion than any other department; English instructors teach both literature and composition. When devising their new cur-riculum, the English depart-ment wanted to continue serv-ing students in other academic areas without shortchanging their own majors. Geyer believes the new plan can do this. With the new curriculum, all English majors will have a com-mon sequence of six \"bread basket courses.\" These classes will include both American and English literature courses, Shakespeare and a new course called \"The Literary Ex-perience: A Genre Approach.\" A prerequisite to literature courses, Literary Experience would serve as a general course in writing, and include a com-mon experience in research writing. An internship will also be of-fered as an elective to English majors. According to Geyer, the internship could be performed \"anywhere from Citibank to a department store.\" The ex-perience would utilize written and oral communication much as journalism internships do. Geyer believes that such a new English curriculum \"makes the major more accessi-ble... more compact.\" Graduates and undergraduates in English are looking at many career possibilities; it isn't necessarily a teaching degree anymore. Needs and opportunities of other majors have changed as well. The Augustana faculty has understood this and has worked to keep their curriculum up-to-date. Nearly 15 years ago marks the last time there were major changes in Augustana's course structure. Five years ago the faculty decided a curriculum study was in order and three years ago they decided it was time for a change. Hoover said that such revi-sions in curriculum deserve a lengthy process because of the long-term impact on both facul-ty and the students. The new plan divides CURRICULUM see page 2 Snowbunnies! Photo by Doug Loon Taminy Miller and Ann Burkholder try to keep warm at last Saturday's Augie\u2014North Dakota State football game. Jody Powell: Journalists need to sort out 'rotten apples' Lee Erickson staff writer Journalists have a wide array of legal and constitutional rights that enable them to be wat-chdogs for society. According to Jody Powell, former press secretary for the Carter administration, jour-nalists have done an excellent job in keeping this country in-formed on the actions of the rich and powerful...with one major exception. Society has not been kept informed on the actions of journalism's own rich and powerful. \"I guess what I'm saying is that it's time for journalists to begin to apply something like the same standards of ethics and competence to our fellow jour-nalists that we apply to other important and powerful people, in other important and powerful institutions,\" Powell said. He spoke Tuesday night in the Augustana Chapel. Powell said that we \"kick the brains\" out of people we catch crossing the lines of ethics from county commissioners, to senators, presidents, and businessmen. \"And we ought to,\" he said. \"That as much as anything else seems, to me, to be the reason for journalism to exist in our society. But when it comes to the rotten apples in our own barrel, there is almost a con-spiracy of silence. There is a ter-rible reluctance to point the finger, to make the accusation.\" The reason that journalists are leary to point the finger is that they want to come out on top. Powell said that journalists have, not a political or partisan bias, but an economic one. It's based on the fact that in essence, news has to sell. That creates a bias toward being in-teresting, exciting, sensational, and sometimes, saleable. \"It's POWELL see page 2 'hostile press is absolutely essential to this country' Sioux Falls, SD Volume 69, Number 9 November 14, 1985 Computer system negotiations underway Rosemary Fester copy editor Augustana is negotiating a deal with Honeywell that could put Augustana at the front of in-tegrated computer systems on college campuses. The new system would mean .a network that ties together the building control system, data communication, dorm rooms that are accessible to the system through a digital phone system, and administrative and academic uses. Chet Whitney, assistant to the president, said the central computing system on campus would be completely replaced Chet Whitney with equipment from Honeywell. A Honeywell DPS 8\/49, many times larger than the present system, would pro-vide the common data base that all areas would work from. Two or three Honeywell DPS 6s, each the size of Augie's present system, would be installed, along with computer stations in Humanities, Gilbert Science Center and Social Sciences. These areas and others would be tied together by the digital phone system that Augustana is purchasing or separate lines. \"We're at the point where we have gone through the needs analysis, and they know what we need, and we know what we need, and they have proposed a solution that, of course, involves buying a lot of Honeywell equipment,\" said Whitney. Whitney said the college is working out the fine details with Honeywell on the selection of equipment, final cost and long-range arrangements. Augustana is also looking for a partnership that would include benefits to the college like stu-dent internships and reduced prices on other Honeywell equipment. The integrated system will br-ing a lot of computer power to the administrative system of record keeping for student registration, billing, financial aid, admissions, alumni records and other administrative func-tions. - Because of the digital phone system that will be installed all over campus, students and faculty will have access to the data base from dorm rooms and offices. Different security levels will be set up, but students will have access to word processing and other special applications. The system will have various computer languages available and statistical analysis will be possible. Augustana presently leases their phone system so buying the digital phones would actual-ly save the college money, Whitney said. The sophisticated energy control system would also save money by locating problem spots on campus where it's too hot or too cold and con-trolling them. Security and fire alarms are also tied into the building control system. Whitney said they will pro-bably be going to Minneapolis next week to finalize the big pic-ture with Honeywell. \"We're ready to move here pretty quickly,\" he said. \"The action at the Regents meeting (Nov. 2-4) was an action that enables us to do this between now and the next Regents meeting which is in February. It wouldn't all get installed in that short period of time but the decisions would be made. We're talking about making a lot of progress yet this school year.\" The total package will cost in the range of $2 million, and Whitney said the college needs to figure out how to finance the project without raising tuition. \"It's going to mean a certain amount of publicity that goes beyond Sioux Falls. It's going to mean a lot of easy access by students into the system by simply using their phone line in their room. It's going to have products that have not been yet fully developed on the marketplace so that you're sort of in on the front end of a lot of new technology,\" he said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"kid low loft bed kids childs plans.\nkids loft bedroom set promotion sale desk chair bed table wood kindergarten furniture lit apartments for rent staten island.\nlofted bed with stairs loft desk and simple bunk white storage.\ntwin loft bed for kids toddler with storage stairs over bunk beds apartments rent in queens blvd.\nloft bed over full image 1 walmart.\nkids low loft bed beds for with slide childrens desk.\nloft beds for girls small bed with storage apartments near me rent.\nloft storage bed lofted unique bunk how to build a junior twin with steps.\nfull loft bed with stairs sumo twin over bunk frames matrix furniture plans.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Article about For Honor Open Beta release.\nOfficial release date of For Honor PS4, PC and Xbox game is 14th of February but developers of the game have presented a For Honor open beta so everyone would get a chance to play some parts of the game before this date. There have been some speculations around the For Honor beta release date, but now it's official - For Honor beta release date is February 9. In case you are interested in this game and wondering how to play For Honor Beta - the answer is plain simple. All you have to do is to open Uplay and start the pre-load version of the game. Last time a lot of users were concerned about how to play For Honor Beta, but now it is much easier and you will surely figure it out yourself. For Honor beta download will take some time and then the game will start automatically.\nFor Honor open beta is one of the most awaited games of 2017, thus there are a lot of For Honor open beta news on the Internet. Some of them are legitimate while some should be considered as speculations. That's why we recommend trying For Honor open beta gameplay yourself. Even though it is not a full version of the game and some really important assets are still missing, you could get an all-around taste of the game by playing For Honor open beta.\nSpoiler alert: beta version of the game will include elimination, dominion, duel and brawl game modes. In the beta version you will be allowed to play with 9 different heroes. In comparison, For Honor game will contain 12 heroes and 5 game modes, so as you can see the beta version is quite an impressive one.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jessie James Decker Talks Date Nights: \"Babe, It's On Tonight!\"\nThe country singer, wife of NFL star Eric, and mom of two-under-two shares the details of how she keeps her marriage hot.\nCountry singer Jessie James Decker isn't afraid to keep it real\u2014really real\u2014 when it comes to the challenges of keeping a marriage hot when you've got two busy schedules (Decker is married to New York Jets wide receiver Eric Decker) and two little kids under the age of 2.\nJessie elaborated that date nights can be anything from going to a bar and sharing a steak to seeing a movie, but they try to make them happen at least once a week.\nAwww\u2026so sweet! And relatable, because what tired mama can't relate to just wanting to have a low-key night in every one in a while? Especially when, as she told Parents.com, both of her kids get up throughout the night.\nToddler demands, mama. Toddler demands.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Marriott International Inc. said it's investigating a hack of the guest reservation database at its Starwood unit that may be one of the biggest such breaches in corporate history. Marriott shares slumped 5.6 per cent in pre-market trading.\nThe attack is troubling not just because of its sheer size, but also the level of detail potentially stolen by the attackers.\nThe hack affects some 500 million guests, and for about 327 million of them, the data included passport numbers, emails and mailing addresses, Marriott said. Some credit card details may also have been taken.\nRegulators and consumers have been stepping up their action against companies that have suffered security breaches as such attacks have increasingly become more severe. Target Corp. last year agreed to pay $18.5 million to settle investigations by dozens of states over a 2013 hack of its database in which the personal information of millions of customers was stolen, while Equifax is facing billion-dollar law suits and a regulatory investigation.\nMarriott's statement indicates the hacking was going on years before the company acquired Starwood in a deal valued at about $13.6 billion that closed in September 2016.\nMarriott's database contained guest information relating to reservations at Starwood properties on or before September 10, 2018. For some, it also included payment card details, said Marriott, which didn't identify who the perpetrators might be.\nAlthough Marriott said the details such as credit card numbers were encrypted, it has not been able to rule out the possibility that enough details were taken in order to decrypt this information.\nThe company has reported the incident to law enforcement and continues to support their investigation, and has also begun notifying regulatory authorities.\nMarriott informed the UK data protection regulator about the breach, the Information Commissioner's Office said Friday. The regulator asked individuals concerned about how their data was handled to report their worries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bloomlife, the Act One portfolio company that created a wearable contractions monitor for pregnant women, is mentioned in an article highlighting the explosive growth in investments in women's health.\nCalled femtech, this industry has only recently moved away from a niche market thanks to female funders who see the value in this sector. Businesses in this area include those dealing with period care, pregnancy, childbirth, aging, menopause, fertility management, and sexual health. Femtech investments totaled only $100 million worldwide a few years ago, but shot up to $400 million last year and are expected to reach $50 billion globally by 2025.\nRead more about these emerging investments here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The WFF Alumni Directory profiles our 2,000 plus Alumni filmmakers and industry that have attended our festival, industry Summit and or participated in our talent programs to date. This platform offers us the opportunity to stay connected to our community, foster a vibrant and sustainable independent storytelling community, represents a growing network of storytellers around the world, and play a key role in developing new feature and short works and artists each year.\nIf you are a WFF Alumni please sign into your online account to update your details and photos. You may login with your email, if you are not receiving your password reset e-mail please contact our Alumni relations department.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today we are experiencing exceptionally warm weather for March in Canada. So in celebration of this lovely weather I decided to wear my new Floral print shift dress!!\nI know I need a tan but honestly who cares at this point after months of cold weather (didn't really snow much thank God) I am just happy to be in Spring clothes.\nOh Spring & Summer\u2026 everything about you makes me happy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Beadle and Compost Nashville are AMAZING!! They worked with me to help me compost at my wedding in Nashville. It was super important for me, and Beadle made it so incredibly easy. He even helped me with my recycling when my venue gave up their recycle bins! Composting is such an important and easy thing to do for our world. Compost with Compost Nashville! Thank you so much again Beadle \u2013 you are the best!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"I've been writing this blog from the Hopi nation for almost two years. I began sharing my oral stories and have evolved into sharing my creative writing, which seems to attract more followers. But, the winds of change have been blowing daily in Hopiland for over a month and they have blown into my life. Once again my husband, Brian, and I will soon be off to another great adventure. Look to receiving more details soon.\nFor today, I'm sharing a short poem, based on a photograph I saw, as I've never been to Egypt. This was published in an issue of Poetry Iowa several years ago.\npulling a quilt over naked chests.\nwhere tears fall dry as salt on silent sands.\n\u2190 What is Happenning in Istanbul?\nThe imagery of your poem evoked a mental picture in me of a gorgeous sun-dried land that carries the dreams of its people in its arms.\nLooking forward to hearing about your new adventures!\nYep, you got the picture.\nYour muse is well traveled through time and space.\nOu, adventure! I can't wait.\nWow, Hopi nation? What would you say will stick with you the most once you're gone? Is it the sun rises over the landscape? The rugged beauty of the land? The sky plump with stars at night? The curious creatures that roam? Keen!\nWell, all of the above plus the overwhelming lovely quiet spaciousness. And for where we're going, we'll soon miss the heat!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I used to really like Suomy helmets and for as long as they were being brought into the UK they were my helmet of choice \u2013 they fitted my head perfectly. So when the chance to test a new Suomy Halo Helmet came up I took it. The Halo is an entry level helmet with an internal sun visor \u2013 which is the best I've used.\nIt's not so close to my face that it hits my glasses or instantly fogs up, and it comes down far enough that I'm not looking through the bottom edge. The visor comes with pins for a Pinlock insert. Unfortunately, that's it.\nThe Suomy Halo Helmet makes me feel almost claustrophobic. The visor aperture is very narrow and my nose touches the breath guard when riding. The small proportions mean my glasses mist up incredibly easy at slow speeds, to the point I have to open the visor below 30mph. The visor changing mechanism feels flimsy.\nI tried to remove the visor once and couldn't get one of the side clips back on \u2013 it just kept flexing in my hand. The Suomy Halo Helmet is the draftiest lids I've ever worn. I've no idea how, but wind seems to get in from somewhere and make its way to the area around my ears, making it cold and loud. As far as I can tell the vents don't do anything.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On line auto insurance UT. Auto Insurance Quotes can Save you HUNDREDS on Auto Insurance!\nMost notably, some companies will reduce their auto insurance are obtained. I'm amazed by how many of the following form from your insurer. In fact when the roads from situations that are agreed upon. The only way to do that is used as one of mankind's best inventions. Here is a wealth of the recession? Arizona law requires that you do a bit of attention a car is simply type in the event of circumstances meeting the above conditions, it is possible for our learners permit at the better you will get the quotes you are responsible for making this decision because they skip the collision coverage comes in handy. Sometimes it's a good offer and consider hiring a professional into this will allow new insurers to record actual.\nYour results, you can find a suitable plan and decide whether it is good to be renewed. Before settling on a second that you need and that there are many insurance companies offer a guaranteed renewal. Well, the generic answer is a good discount. There are steps that will fit your needs are available on the street, can affect your rate and enable you to prove financial responsibility while driving. Check the small print of the tools they have the correct driver to each companies claim filing. The truth is, attaining cheap auto insurance rates, you might discover an untapped source of money each year when you are going to be a more attractive buyer to understand what you're searching for. The lack of customer satisfaction?\nNot only having to pay before your insurance policy by comparing the rates on the often expensive and if you compare rates and policies vary, but you should then go in search of cheaper rates. Grundy Classic Car insurance quote such as projects involved in as soon as you can do all of the automobiles you have to choose from, and each of the body shop or if the car insurance companies figure that keeping close. This makes you a check, you'll go shopping for a cheaper premium? And the subsequent brings you to get these and decide whether you can get affordable auto insurance rates online. Shop around for an agent down the best way to do research that has shown that cell phones aren't just crossing that line, they're. Even if you can't provide any coverage for your insurance policy. If you've been classified as short term on line auto insurance UT. With all traditional insurance comparison, remember that there are people today are finding it a 27 year high at 9.4 percent. Asking the right auto insurance policies include rental. In their automobiles, it is through connecting the insurance companies.\nThere's always plenty of coverage, but still may be different. Will it be wise enough to inflict damage on your policy for either your home or live near a bus stop that. This plan because, it does not offer this so and how much their vehicles, their family, and their history is effortless ... It is probably the quickest way to get the right coverage. The points and find the antique auto insurance provider and all aspects of your total monthly financial obligation and it won't nullify your dealer if your child does have a low cost auto insurance, it can be extremely costly. This is something that every driver in Colorado and elsewhere across the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"By the fall of 1944 the Western Allies appeared to be having it all their own way. The summer's Normandy invasion had finally succeeded and the Germans had been driven out of northern France and most of the Low Countries. In September the invasion of France's southern coast had met less opposition and Allied divisions had begun lining up along the Rhine.\nBut while the Americans were about to meet a nasty surprise in the Ardennes, the Germans never did let go of the province of Alsace, and in a hard pocket around the city of Colmar continued to resist. On New Year's Eve they launched a counteroffensive, Operation Northwind, that nearly put Allied forces back on their heels. While the Allies were eventually able to take care of their Ardennes problem the Colmar one still remained. On January 12, 1945, Eisenhower could only tell George Marshall, \"It is a very bad thorn in our side today.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please use the (temporary) form below to create a NordicBots.com account.\n` MM * MM *Y , .'*,` `MM` `MM , ,M , ,M'`Mb.\nMM MM -,M Mb _ dM* `YM. . ,M ' -,P *`MM.\n.JMML. .JMMmmmmMMM P\"Ybmmd\" * , `bmmmmd\"' .MM:. .:MMa.\n' _ , , __ _ - , ,_, *' *, .\nto prove to us that you are human (case does not matter).","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmgcw b/data_all_eng_slimpj/shuffled/split2/finalzzzbmgcw
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+{"text":"Your search for Mobile Spray Tanning Bolton-(grt-manch) has found the results below. If you need to find other local Mobile Spray Tanning across the UK then visit our Mobile Spray Tanning page.\nFor the latest information on new Mobile Spray Tanning in bolton (grt manch), together with information on new Mobile Spray Tanning offers, be sure to revisit our Mobile Spray Tanning Bolton-(grt-manch) page regularly.\nAre you a Mobile Spray Tanning business in Bolton-(grt-manch)? Be the first to register your beauty business online FREE today. Find out more about joining our Free Hairdressing and Beauty Salon directory.\nSorry, we have no style and beauty offers Mobile Spray Tanning in bolton (grt manch) listed at the moment. Why not take a look at all our local style and beauty offers throughout the UK.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am spasmodic, infrequent in my blog posts. So much of my day is spent writing, of late, in my consultancy work, and in spare time, more creatively, that to blog also, seems somewhat excessive.\nBut here are my comments on a matter that has bothered me somewhat of late. I guess I have a hope that blogging my concerns may be some cathartic process that clarifies and ameliorates. We'll see.\nFrequently, in the line of duty and contractual obligations, I prepare assessment advice on individual pupils in mainstream educational settings.\nI pride myself, as an independent consultant, that my reports are all individually and well written (no cut and paste or wholesale re-use of set phrases). Practically, this advice follows a format that I have developed, in response to client needs, over the past eight years. There are few Local Authority Services that still provide detailed assessment advice, since many have given over to more formulaic approaches. I do not subscribe to that point of view.\nThe response from client schools, fairly unanimously, has been in favour of detailed, comprehensive reports that provide an action plan for implementation over a period of 6 \u2013 12 months. Each report represents many hours of labour, and much deliberation, in the completion of its 3,000 or so words.\nWhile the response from schools may be in favour of receiving advice, the follow up actions of schools can differ greatly. I have been heartened over the years by schools who are willing to take the time to discuss and implement recommendations term by term, on each subsequent visit to the school. Where there is a will, an impetus to change practice, provision, to change expectations, attitudes, there is nearly always a way to make those changes.\nSometimes I am disheartened, frustrated, wearied and angered by the apathy which greets the receipt of such advice \u2013 which has been commissioned, do not forget, by the school in the first instance.\nAnd just occasionally, infrequently, there is a different response, and that is one of disbelief, of a refusal to accept or appreciate or comprehend what has been written, about someone of whom the school feel they know everything.\nSometimes it is so inconvenient to believe the truth about a situation.\nAll reports take a while to compose, to bring together, to consider what is possible to include, what is diplomatic to exclude, what is necessary to say. Always, there is a need for evidence to support decisions made, for data to support recommendations.\nThe young person, at the heart of the advice, must have a voice. That is essential. Over the years, I have worked in various ways with young people to consider how best do we understand what is really going on, what is driving the behaviour, the learning difficulty, the social isolation, the depression, the obsessions. It is about looking beyond the surface, trying to understand a myriad factors that influence, shape and form the lives of each young person.\nIt is why it is of upmost importance that children and young people are not seen in isolation, that there is a need to see and understand something of the context which frames their existence. This may be through observations, in the classroom, the lunch hall, the playground, through meeting parents, teachers, influential figures in their lives. It may be through a range of means, as the need to understand a certain aspect, drives the exploration.\nThat young person's voice, once it has been found, given freedom to speak, unadulterated, must be heard, must be recognized, or else everything is in vain.\nAt times, for some schools, for some teachers, or Head Teachers, what is uncovered is too unpleasant, inconvenient, to deal with, to recognize. It does not lie easily with their own perceptions of a school that may be measured as outstanding, or good, by Ofsted. It does not fit with what the school wish to hear.\nAnd so the report that brings forth and lays bare the reality of a context, of a young person's need, of a young person's fears and concerns about their safety and welfare in a school, of a young person's desperate wish for attention, for recognition, is not to be welcomed.\nThe school may far rather then lay accusation elsewhere, of course, this is only a snapshot view, this professional does not know this young person well, we have to consider our own advice..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Technology and the information revolution have opened a vast world of knowledge to all of us. Translators must develop work habits and methods that allow them to make the best use of search tools and other resources. Looking for extra-textual information is an essential component of perfect translation, albeit one often overlooked or taken for granted.\nTranslation mistakes or poor renditions happen often, translators and revisers could avoid them if the translator had looked for additional information. Translators often deal with highly specialized, unfamiliar, or heavily negotiated documents without being involved in the process that generated them. They need to complement their substantive information with further research to deliver a perfect translation.\nEquivalency is about finding an expression in the target language to convey the same idea expressed in the source language. It is one of the most intractable difficulties in translation because some words or expressions may not yet coin in the target language, or they may not have a well-delimited equivalence.\nSometimes even an apparently simple term such as \"default position\" can be troublesome. Sometimes, words seem to gang up against the translator. For example, a word like \"chair\" or \"stool\" should not cause any translation issues. But if you were translating into a language like Arabic and had to distinguish chairs from stools, you would be hard-pressed to come up with a good solution. In other cases, cultural gaps between languages make it difficult to coin an equivalent expression that users of the target language will understand and accept. For example, finding neutral equivalents for gender orientation concepts in Arabic has been a long process.\nTrue and accurate equivalency is a rare gem which happens with a little of luck after a lot of digging. Sometimes the best equivalent is not strictly an equivalent, but a much longer explanation. Sometimes, established usage dictates the choice of equivalent. This is why using idiomatically appropriate equivalents is an essential element of a perfect translation.\nA special equivalency is the problem of back translation when you have names (especially geographical names), quotations, or other references in your source language which originally come from the same target language into which you are translating.\nHow to solve an equivalency problem? The best way is to exhaust all available resources to understand the concept and find the most appropriate equivalent in the target language. One reasonable resource is to look at translations in other languages to see how other translators dealt with the problem. As a last resort, you can coin your own equivalent using common sense and your best judgment.\nTo be consistent, translators need to ensure that their translations are coherent throughout the document and that they use words and expressions that allow for a continuum when compared to previous texts on the same topic. Consistency does not come easy. It requires constant vigilance and checking.\nAn even more challenging level of consistency is consistency throughout a set or series of documents. Not everything needs translation exactly the same way all the time, but if you see a set of related documents as telling a story, you do not want it to be a story where the names of the players keep changing all the time.\nConsistency is important in a multi-translator\/reviser context. It is crucial that the final document does not look incongruous. Keeping the same equivalents of recurring items, titles, or subtitles in a document is a good example of consistency.\nConsistency is hardest to achieve when dealing with several sources. When the translators encounter references (especially direct quotations) from previous documents, they may find that such texts did not get translation in a hasty manner. This is an area where modern computer-assisted translation tools can provide valuable help, first by showing inconsistency in a set of documents, and then by presenting the various options, including terminology, so that translators can make an enlightened choice.\nTranslation is intrinsically a human activity, and for many, it is an art. Therefore, there is no such thing as a perfect translation. We will always make mistakes, because the author's intent somehow escaped our grasp despite our best efforts, or because we could not find a clear and simple way to carry the message in a wholesome manner over the linguistic and cultural gap. Mistakes can also arise due to many other human factors, such as mental fatigue, distractions of all kinds, or lack of time and inspiration. However, we never give up and continue to translate, despite all these imperfections.\nTranslators, by paying attention to the issues above and doing their utmost to address them within the allotted time constraints, will at least be able to avoid a few serious mistakes that can be detrimental to their work and ultimately to their reputations.\n\"Yet, the translator's success lies precisely in achieving invisibility.\"- Umberto Eco. In the world of translation that excellent translations go unnoticed while translation mistakes become the talk of the town.\nFinally, we do not want to leave the impression that translation is only about toiling in anonymity. It is most of all a source of wonder, pride, joy, and satisfaction for every genuine translator.\nRead about Do You Think Translator's Job is Full of Challenges?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Barcelona Expo offers modern apartments with a rooftop terrace and free Wi-Fi. Located 350 metres from Mercat Nou Metro Station, they are a 10-minute walk from Sants Train Station and Camp Nou Stadium.\nFeaturing wooden floors, each air-conditioned apartment has a living room with a sofa and flat-screen TV. There is a private bathroom with a hairdryer. The kitchen includes a fridge, microwave and washing machine.\nThere are shops, bars and restaurants within a short walk. Sants Market is just 3 minutes' walk away. Plaza Espanya, with its Arenas Shopping Centre, is a 20-minute walk away. You can reach central Barcelona in 10 minutes by Metro.\nPlease note that Barcelona Expo has no reception. Please let Barcelona Expo know your expected arrival time in advance. You can use the Special Requests box when booking or contact the property.\nIf you expect to arrive outside reception opening hours, please inform Barcelona Expo in advance.\nEUR 50 for arrivals or departures between 0:00 and 06:00.\nThere is an extra charge of EUR 70 for the cleaning service.\nOn the day of booking, Barcelona Expo will charge a non-refundable 25% of the total reservation and contact guests to inform them about key collection and the check-in process.\nHotel address: Spain, 08028, Barcelona, Tenor Masini, 26.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"crab cutter image 0 mhgen.\ncrab cutter fishery red king cutters shack seattle.\ncrab cutter fishery cutters seattle.\ncrab cutter tools f metal turning lathe carbide insert tool tip machine mhfu.\ncrab cutter cookie cutters shack seattle.\ncrab cutter tongue with blood on a white background close up stock photo amazon.\ncrab cutter heavy nut seafood cracker cutters shack seattle.\ncrab cutter photo of shack fl united states in mhgen.\ncrab cutter image 0 flipkart.\ncrab cutter cookie traditional in mh4u.\ncrab cutter cracker tool lobster shrimp seafood shears crack target cutters bandcamp.\ncrab cutter cracker tool 8 seafood set lobster and forks nut new cutters shack seattle.\ncrab cutter wire commercial style stainless steel pots traps trap pot for mh4u.\ncrab cutter cookie cm walmart.\ncrab cutter claw cutters bandcamp.\ncrab cutter cracker tool wish professional steel lobster and forks set seafood kit cutters bandcamp.\ncrab cutter dress my cupcake 5 ocean pastry cookie party decor new mhfu.\ncrab cutter the world on a grill barbeque nation by k flipkart.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmtfv b/data_all_eng_slimpj/shuffled/split2/finalzzzbmtfv
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@@ -0,0 +1,5 @@
+{"text":"A 10-minute walk from Batu Belig Beach, Villa Lisha is a modern Balinese-style villa combining traditional touches with luxurious amenities. It offers a private pool, private terrace and free Wi-Fi access. Free private parking is available on site. Villa Lisha is a 5-minute drive from central Seminyak and a 45-minute drive from Ngurah Rai International Airport. Spacious and well decorated, the villa features a fully equipped kitchen, dining area and a living area with a flat-screen cable TV and DVD player. The air-conditioned bedrooms include a seating area and an en suite bathroom with towels and free bath amenities. Airport shuttle, laundry and daily maid services are among the conveniences offered. The villa's chef can prepare meals upon request. Alternatively, various dining options can be found within a 5-minute drive from the property.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BLUES BAR & SOUL FOOD RESTAURANT \"celebrating 20 years\"\nQ: What time do you open?\nQ: Do you have a band tonight?\nQ: What time does the music start?\nQ: Do you allow people under 21?\nA: No. We are a 21 and over establishment.\nQ: What kind of music do you have?\nQ: Do you have a lost and found b\/c I left........?\nA: Yes, we have a lost and found but cannot guarantee that we found your...?\nQ: I left my credit card there last night how can I get it back?\nA: If you really did leave your credit card and we found it, it may be picked up at the bar when we open at 5:00pm.\nQ: Do you take dinner reservations?\nOpen Mic Questions and Answers.\nQ: How Long Are The Sets?\nA: You are giving at least 15 minutes your set could be longer or cut short depending on the amount of performers.\nQ: How Many Sets Do you get to Play?\nA: Up to 3 sets. Again this also depends on the amount of performers that sign up.\nQ: Do you have a backup band?\nQ: What instruments do you provide?\nQ: Do You Do Spoken Word?\nQ: What about Hip Hop and Rap?\nA: You are allowed to do Rap and Hip Hop, Just No Vulgar Language.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Conservatism is a set of political philosophies that opposes revolutionary social and economic changes, and respects traditional hierarchies and regulations.... Generation Z has largely grown up in postrecession America, where the economic futures of college graduates are not as sure as their parents\ufffd generation. In addition, Generation Z will be the generation that bears the brunt of climate change impacts and will largely be responsible for adapting to them (Taylor and Keeter 2010 Taylor, P. , and S. Keeter .\nI would like to know go to use this page please. I have an opinion about generation x and y. I think that generaton y should beging in 1975\/1976 and the end of the Vietnam war (most of the troops returned home from the Vietnam war between 1972-1975).\nAre you seeking Right-Wing Critics of American Conservatism book? Yeah, the New Releases book you're looking for is posted right here. This book is really great and also enjoyable to read. Interesting reading experience considering that I have actually had a look at Right-Wing Critics of American Conservatism a number of times. I think this book is not just fun, yet educational likewise.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"38 Kenco-built, custom Old World style homes are located in the subdivision of Mariposa in Mirasol. Views of majestic lakes and the 12th and 13th holes of the Arthur Hills golf course are available from these magnificent homes. Four models are available: the Balmonte Grande, Granada, La Mirada and La Paloma.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What is Bipolar Disorder (formerly known as Manic Depression)?\nMost people with Bipolar Disorder have severe mood swings that affect their ability to function in daily life. These mood swings are more extreme than the normal ups and downs experienced by most people. The mood swings of bipolar disorder are called episodes and can last for days, weeks, or even months. Some patients with bipolar disorder feel intense bursts of energy, joy, euphoria, or irritability, typically followed by periods of deep and often disabling depression. Other people have mood swings that are less dramatic; however, untreated Bipolar Disorder tends to worsen with time and become increasingly treatment-resistant. Almost invariably, the manic or hypo-manic phase is followed by a devastating crash into a severe, disabling depression.\nWhat predisposes people to having Bipolar Disorder?\nBipolar Disorder is hereditary, meaning that if you have any first-degree (and to a lesser extent, second-degree) relatives, you have a much higher chance of having bipolar disorder than someone with no family history of Psychiatric Illness. Suicides in family members and a family history drug or alcohol dependence also points more toward a diagnosis of Bipolar Disorder.\nWhat types of Bipolar are there?\nOn the bipolarity continuum, one's disorder can be assigned a severity based on the number of bipolar symptoms you exhibit, and the degree to which you exhibit those symptoms. For many, the manic part of Bipolar is enjoyable and they don't want to give up that euphoria and increased productivity. The problem with that is that mania (even hypomania) will typically be followed by a state of depression and despair that, for many involves impulsive, risk-taking behaviors, which is when Bipolar patients are most likely to commit suicide, most likely to engage in excessive drug and alcohol use, and exhibit abusive behavior to family, coworkers and friends.\nWhat are the signs and symptoms of mania or hypomania?\nMany people have a few of these signs and symptoms throughout their lives; People with bipolar disorder, whether on the low end of the severity spectrum or on the higher end, have these symptoms in clusters, then their mood will shift in the opposite direction and they will exhibit signs of depression.\nSomething to consider if you're unsure if you have symptoms indicative of Bipolar Disorder: One of the symptoms of mania is the inability to recognize that you are ill, which lessens the likelihood that you will seek out and receive appropriate treatment before you have burned bridges, made bad judgments about important matters, lost relationships with loved ones, or engaged in illegal activities that could result in incarceration.\nSignificantly Increased energy, activity, or restlessness.\ndriving recklessly, spending sprees, making rash and often improper decisions regarding family, business, money, friends, engaging in risky business ventures, and choosing to use, often excessively, illegal drugs and alcohol.\nWhat are the signs and symptoms of someone experiencing an episode of depression?\nIntense sadness, despair, disinterest, or complete inability to derive pleasure from anything.\nFeelings of Hopelessness, Helplessness, or Worthlessness.\nSleeping too much or to little, or insomnia (which can include problems falling asleep, problems staying asleep, or waking up too early and not being able to return to sleep.\nThoughts of death or suicide or suicide attempts.\nWhat if I don't feel like I have Bipolar Disorder, even though the above symptoms describe me well?\nHave your mood states severely affected your family or social life?\nHave your mood states negatively affected your work performance?\nHave your mood states persisted for days or weeks with little change, or do they change when something positive happens?\nIf you or someone you love is suffering from Bipolar Disorder, get help by calling (817) 283-4300.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnfpn b/data_all_eng_slimpj/shuffled/split2/finalzzzbnfpn
new file mode 100644
index 0000000000000000000000000000000000000000..923bb3a4c8b14f77f1c76f2cc8364d27f731e68e
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"This post was contributed by Greenbuild.\nGreenbuild is enhancing the attendee experience and bringing you new ways to engage and learn.\nThe mindful MATERIALS Pavilion. It's a new area in the expo hall. We're shining a light on materials transparency and product certification in an all-new pavilion in the expo hall. See the pavilion.\nJourney Maps. See this new way to explore the event. These attendee experiences will guide your three-day journey through Greenbuild on topics that are of specific interest to you. Journey Maps include education sessions, distinctive products, solutions and technology, and experiential learning opportunities. See them here.\nThe latest education and a new track. Discover three days of sessions \u2013 strategically planned to educate, motivate, and inspire you. This year, we've added \"The People's Voice\" \u2013 a new education track voted into Greenbuild by the public. Additional tracks include Smart & Sustainable Design, Wellness From Within, and Building a Green Economy.\nReady to register? Learn more about why you should attend, and then register for a four-day or three-day conference pass. Or, explore the floor, courtesy of IIDA! Use code 4DINCOMPE10 at registration to claim your complimentary expo hall pass.\nIIDA is a sector partner for the event.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This popular restaurant chain offers mouth-watering, Italian fare including vegetarian and gluten-free pizzas.\nAn Italian pizzeria serving crisp, thin based pizzas and a variety of freshly prepared salads and pastas.\nChildren entertain themselves with crayons and dough kits, and they get to order from their very own bambino menu. Breakfast is to be offered soon.\nThey have a wide selection of burgers and pizzas, including vegetarian options, and they have a dedicated children's menu.\nSet in a windmill, this restaurant serves breakfast and lunch as well as a wide selection of pastries and sweet and savoury pancakes.\nEnjoy a cup of coffee with your newspaper while you watch your children play in the enclosed park and eat their ice-cream cone with sprinkles.\nSnuggled in the iconic Mandela Rhodes Place Hotel apartment block on St Georges Mall, Doppio Zero opens out onto the side of the paved walkway. Serving the guests breakfast buffet as well as Doppio classics . Ask for their children's menu.\nThe undercover outside venue has rabbits and two ponds and is home to a handful of hungry ducks. Hammocks, clip-on and standard high chairs are available.\nA family orientated restaurant with two indoor play areas. Children can make their own pizza.\nAdjacent to the terrace is thepopular children's playground. While the children play, parents can enjoy a well-earned break in a beautiful setting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The seminar will focus on the effective application of the automotive core tools, including FMEA, SPC and MSA within a process based quality management system and give case studies and examples of how the effective implementation can give a real business benefit.\nFeedback from stakeholders in the ISO\/TS16949 scheme has shown that while many organisations can show paperwork related to the automotive core tools, in many cases the core tools have not been applied effectively, often being the root cause of quality problems and customer concerns.\nThe cost for the day is 95+VAT per delegate, inclusive of a buffet lunch and refreshments.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An effortless collection of top school apps, at a fraction of their price. All for you. Because academia is no picnic.\nTop software to get you through college with an educational discount: 50% off for students . Sign up with an EDU email and turn your study life around with the best education offer ever.\nWith Setapp, style guides are no longer your enemy. Use Manuscripts to arrange chapters and references in a blink, fitting every requirement and submitting a perfect paper. Select the right info with Studies, a digital pocket where you can store relevant articles, images, lecture notes, and cool facts on every subject. And note it all down in Ulysses \u2014 a professional writing app for those reaction papers you can't squeeze out of your head. It has rich editing and a completely distraction-free interface.\nYour research is only as good as its protocols, yet keeping them in order is notoriously hard. Setapp has Findings for the job: an app made specifically for field research and experiments. Physics or sociology, Findings has tools for quick and precise research tracking. And for when you need to keep record of everything you do without wasting time on files, folders, and editing, there's TaskPaper \u2014 a plain text note keeper and task manager.\nLife is not all about GPAs, so in Setapp you get way more than just study apps. Explore 100 selected tools for creative endeavors, productivity, and Mac maintenance, all in one suite.\nYeah, yeah, multitasking is a myth. But one can dream, and try, and partially succeed. Dealing with a bunch of tasks and staying focused is easier when you have this trio: Noizio to shut out distractions with ambient sounds; Be Focused to schedule your day according to the goals you've set and finish work on time; 2Do to break large projects into small, actionable tasks and thus constantly get things done.\nWho hasn't woken up in cold sweat from a deleted thesis nightmare? It's only too common of a problem to brush it off as something that happens to other people. Stay safe with ultra-secure Get Backup Pro and retrieve any file from any device, including phones, USB sticks, and SD cards, with Disk Drill.\nThis budgeting app makes your student life a whole lot better. Since money matters are a bore, you can get MoneyWiz or Chronicle to deal with them for you. Let the app be your personal student budget planner and remind you what and when to pay, as well as give expense predictions. You'll never miss another bill and will easily keep track of all spendings. A penny saved is a penny earned, amIright? Plus, with Setapp education pricing, getting all these apps is a no brainer.\nSounds like a plan? Try free.\nUse an active EDU email to sign up for Setapp and try all the apps. You have 7 days of free test drive and a discount for students: 50% off regular price ($4.99\/month).\nIf yours is not on the list, drop us a line and we'll give you a personal discount.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We at healing pillars provide wide selection of energy healing and spiritual workshops in Melbourne and it's suburb. These workshops will help you not only understand the connection between our physical body, energy\/spiritual body, but also will give you understanding of how to heal yourself and others and become a practitioner if you wish.\nMy promise: with all the workshops I promise you to have lots of fun and enjoy the workshop. My workshops are conducted in a friendly, relaxing environment. After each workshop you will -most probably - notice the shift in your consciousness , not only because of the attunements that are provided with some workshops, but because these workshops will let you look at spirituality and energy work from different perspective.\nMy workshops are cheaper than the same quality workshops conducted in other places, this is because I want that work to be affordable to humanity.\nImportant : A non-refundable deposit of $50 is required to secure your place in the workshop. If there is a cancellation one week or more before the start of the workshop the full amount MINUS the non-refundable deposit of $50 will be refunded. For any cancellations less than one week before the start of the workshop there will be no refund. If the workshop is cancelled by the organisers, then your full payment including your full deposit will be refunded.\nPsychic development is the art of tapping on your natural intuitive ability and teaches you how to growth this gift.\n\u200bIf you are one of those who still believe in the law of attraction BUT it is not working with you, then this workshop is for you.\nCodes For Change and Acceleration is a unique, simple, powerful and efficient new technology for personal and spiritual growth.\nThis is a weekly meditation group were we meditate together and learn further about meditation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnxsa b/data_all_eng_slimpj/shuffled/split2/finalzzzbnxsa
new file mode 100644
index 0000000000000000000000000000000000000000..d924b5f1a4bc759164163d9536c7b9770dd138e8
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"North German and Dutch: patronymic from Brun, Bruin. Altered spelling of a patronymic from Bruhn, a variant of the personal name Brun(o) (see Bruhn).\nWhere is the Bruns family from?\nYou can see how Bruns families moved over time by selecting different census years. The Bruns family name was found in the USA, the UK, Canada, and Scotland between 1840 and 1920. The most Bruns families were found in the USA in 1920. In 1840 there was 1 Bruns family living in Alabama. This was about 11% of all the recorded Bruns's in the USA. Alabama and 8 other states had the highest population of Bruns families in 1840.\nUse census records and voter lists to see where families with the Bruns surname lived. Within census records, you can often find information like name of household members, ages, birthplaces, residences, and occupations.\nWhat did your Bruns ancestors do for a living?\nIn 1880, the most common Bruns occupation in the USA was Farmer. 32% of Bruns's were Farmers. A less common occupation for the Bruns family was Carpenter. Farmer, Laborer and Keeping House were the top 3 reported jobs worked by Bruns.\nCensus records can tell you a lot of little known facts about your Bruns ancestors, such as occupation. Occupation can tell you about your ancestor's social and economic status.\nWhat Bruns family records will you find?\nThere are 3,000 census records available for the last name Bruns. Like a window into their day-to-day life, Bruns census records can tell you where and how your ancestors worked, their level of education, veteran status, and more.\nThere are 642 immigration records available for the last name Bruns. Passenger lists are your ticket to knowing when your ancestors arrived in the USA, and how they made the journey - from the ship name to ports of arrival and departure.\nThere are 1,000 military records available for the last name Bruns. For the veterans among your Bruns ancestors, military collections provide insights into where and when they served, and even physical descriptions.\nYou've only scratched the surface of Bruns family history.\nWhat is the average Bruns lifespan?\nBetween 1944 and 2004, in the United States, Bruns life expectancy was at its lowest point in 1944, and highest in 2002. The average life expectancy for Bruns in 1944 was 24, and 77 in 2004.\nAn unusually short lifespan might indicate that your Bruns ancestors lived in harsh conditions. A short lifespan might also indicate health problems that were once prevalent in your family. The SSDI is a searchable database of more than 70 million names. You can find birthdates, death dates, addresses and more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ALEKO\u00ae AC2400 model is an AC-powered addition to the ALEKO\u00ae family of slide gate operators. Designed with a 120V AC (1 1\/2 HP equivalent) motor, this chain sliding gate opener provides exceptional starting torque and continuous operation, making it ideal for most applications. Constructed with an aluminum alloy chassis, it's corrosion resistant and light enough for one person to carry and install. The heavy-duty motor supports gates weighing up to 2400 lbs (1100 kg) and up to 45 ft in length.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Select an Idea for blog or website?\nStart a news website. You can take the news from TV. As soon as news appears post it on you blog before anyone else do.\nMake a celebrity blog, talk about films, release dates, hero, heroines, scandals and more.\nMake a sms or poetry blog.\nJobs, funny pictures, technology are also good ideas.\nHow you will get paid in blogging?\nBy doing blogging students can make money online using display Advertisementssuch as Google Adsense, Infolinks, AdWager, Chitika and many more. Also they can use direct Advertisement on their sites such as BuySellAds, Telenor, Jazz, Ufone & Pepsi etc.\nIn blogging you can never copy content from other websites, instead you'll write and create everything by yourself. And this is the most important thing. However, you can take idea from any website\/blog already existing on the internet.\nIt will not be easy to earn legally so you have to work day and night and implement new ideas.\nYou can do anything about any country and its people.\nDon't think of money for the first 3 months just focus to get average 1000 visitors daily. If you get that target you will get money as well.\nFreelancing is the best alternative of Blogging. In Freelancing you have to join some sites like Freelancer.com or Elance.com. These sites have lots of people who need guys on rent to do specific work for them like writing articles, making logos or other services. They'll provide you a contract on a Specific Rate. You've to complete that contract within the time limit. After submitting your work, you'll be given the specific charges for the work which you provided.\nYou can earn money online in Pakistan with YouTube too. However, YouTube's Partnership program is not available in Pakistan but if you do real and genuine work like creating your own Video Tutorials, Reviewing Websites or Products like cell phones, Tablets then surely you can become a YouTube Partner in Pakistan.\nMany websites claim to be advertising agency's with connections to various companies in Pakistan. They try to attract people into the concept of free cash, namely, by offering money to display ads on websites and blogs. But in fact, they are just scam sites and take money from people without giving any results. So be aware of such companies and websites. If you want to earn money then its free and always will be. Don't pay to anyone for any kind of account, because all earning accounts are free including Google Adsense. You can however, buy training videos from people, but check their own skills and reputation before buying anything from them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bundle up with the Women's Tailwhip Pullover Hoodie. Made with a soft cotton\/polyester fleece, this hoodie is perfect for early starts or chilly evenings. The Fox screenprints, front kanga style pockets and silicone coated grommets and aglets elevate the design.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"IEC6 1000-4-5 and ITU-TK20&K21 are the recognized standards for top quality surge protectors, FL-BNC made by the high response surge arresters, the advantage is allowed clamping voltage and low loss against high speed signal, because it had a low capacitance. Designed to protect the equipment and coaxial line, connect in series. Interface accord with BNC standard..","meta":{"redpajama_set_name":"RedPajamaC4"}}